Properties

Label 306.2.b.a.271.1
Level $306$
Weight $2$
Character 306.271
Analytic conductor $2.443$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [306,2,Mod(271,306)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(306, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("306.271");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 306 = 2 \cdot 3^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 306.b (of order \(2\), degree \(1\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(2.44342230185\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 102)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 271.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 306.271
Dual form 306.2.b.a.271.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} -2.00000i q^{5} -2.00000i q^{7} -1.00000 q^{8} +2.00000i q^{10} -6.00000 q^{13} +2.00000i q^{14} +1.00000 q^{16} +(1.00000 - 4.00000i) q^{17} -2.00000i q^{20} -6.00000i q^{23} +1.00000 q^{25} +6.00000 q^{26} -2.00000i q^{28} +6.00000i q^{29} -10.0000i q^{31} -1.00000 q^{32} +(-1.00000 + 4.00000i) q^{34} -4.00000 q^{35} -2.00000i q^{37} +2.00000i q^{40} +4.00000 q^{43} +6.00000i q^{46} -8.00000 q^{47} +3.00000 q^{49} -1.00000 q^{50} -6.00000 q^{52} +6.00000 q^{53} +2.00000i q^{56} -6.00000i q^{58} +10.0000i q^{61} +10.0000i q^{62} +1.00000 q^{64} +12.0000i q^{65} +8.00000 q^{67} +(1.00000 - 4.00000i) q^{68} +4.00000 q^{70} +10.0000i q^{71} +16.0000i q^{73} +2.00000i q^{74} -6.00000i q^{79} -2.00000i q^{80} +16.0000 q^{83} +(-8.00000 - 2.00000i) q^{85} -4.00000 q^{86} -10.0000 q^{89} +12.0000i q^{91} -6.00000i q^{92} +8.00000 q^{94} -12.0000i q^{97} -3.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{8} - 12 q^{13} + 2 q^{16} + 2 q^{17} + 2 q^{25} + 12 q^{26} - 2 q^{32} - 2 q^{34} - 8 q^{35} + 8 q^{43} - 16 q^{47} + 6 q^{49} - 2 q^{50} - 12 q^{52} + 12 q^{53} + 2 q^{64}+ \cdots - 6 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/306\mathbb{Z}\right)^\times\).

\(n\) \(37\) \(137\)
\(\chi(n)\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) 2.00000i 0.894427i −0.894427 0.447214i \(-0.852416\pi\)
0.894427 0.447214i \(-0.147584\pi\)
\(6\) 0 0
\(7\) 2.00000i 0.755929i −0.925820 0.377964i \(-0.876624\pi\)
0.925820 0.377964i \(-0.123376\pi\)
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) 2.00000i 0.632456i
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 0 0
\(13\) −6.00000 −1.66410 −0.832050 0.554700i \(-0.812833\pi\)
−0.832050 + 0.554700i \(0.812833\pi\)
\(14\) 2.00000i 0.534522i
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 1.00000 4.00000i 0.242536 0.970143i
\(18\) 0 0
\(19\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 2.00000i 0.447214i
\(21\) 0 0
\(22\) 0 0
\(23\) 6.00000i 1.25109i −0.780189 0.625543i \(-0.784877\pi\)
0.780189 0.625543i \(-0.215123\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 6.00000 1.17670
\(27\) 0 0
\(28\) 2.00000i 0.377964i
\(29\) 6.00000i 1.11417i 0.830455 + 0.557086i \(0.188081\pi\)
−0.830455 + 0.557086i \(0.811919\pi\)
\(30\) 0 0
\(31\) 10.0000i 1.79605i −0.439941 0.898027i \(-0.645001\pi\)
0.439941 0.898027i \(-0.354999\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) −1.00000 + 4.00000i −0.171499 + 0.685994i
\(35\) −4.00000 −0.676123
\(36\) 0 0
\(37\) 2.00000i 0.328798i −0.986394 0.164399i \(-0.947432\pi\)
0.986394 0.164399i \(-0.0525685\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 2.00000i 0.316228i
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) 4.00000 0.609994 0.304997 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 6.00000i 0.884652i
\(47\) −8.00000 −1.16692 −0.583460 0.812142i \(-0.698301\pi\)
−0.583460 + 0.812142i \(0.698301\pi\)
\(48\) 0 0
\(49\) 3.00000 0.428571
\(50\) −1.00000 −0.141421
\(51\) 0 0
\(52\) −6.00000 −0.832050
\(53\) 6.00000 0.824163 0.412082 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 2.00000i 0.267261i
\(57\) 0 0
\(58\) 6.00000i 0.787839i
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) 10.0000i 1.28037i 0.768221 + 0.640184i \(0.221142\pi\)
−0.768221 + 0.640184i \(0.778858\pi\)
\(62\) 10.0000i 1.27000i
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 12.0000i 1.48842i
\(66\) 0 0
\(67\) 8.00000 0.977356 0.488678 0.872464i \(-0.337479\pi\)
0.488678 + 0.872464i \(0.337479\pi\)
\(68\) 1.00000 4.00000i 0.121268 0.485071i
\(69\) 0 0
\(70\) 4.00000 0.478091
\(71\) 10.0000i 1.18678i 0.804914 + 0.593391i \(0.202211\pi\)
−0.804914 + 0.593391i \(0.797789\pi\)
\(72\) 0 0
\(73\) 16.0000i 1.87266i 0.351123 + 0.936329i \(0.385800\pi\)
−0.351123 + 0.936329i \(0.614200\pi\)
\(74\) 2.00000i 0.232495i
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 6.00000i 0.675053i −0.941316 0.337526i \(-0.890410\pi\)
0.941316 0.337526i \(-0.109590\pi\)
\(80\) 2.00000i 0.223607i
\(81\) 0 0
\(82\) 0 0
\(83\) 16.0000 1.75623 0.878114 0.478451i \(-0.158802\pi\)
0.878114 + 0.478451i \(0.158802\pi\)
\(84\) 0 0
\(85\) −8.00000 2.00000i −0.867722 0.216930i
\(86\) −4.00000 −0.431331
\(87\) 0 0
\(88\) 0 0
\(89\) −10.0000 −1.06000 −0.529999 0.847998i \(-0.677808\pi\)
−0.529999 + 0.847998i \(0.677808\pi\)
\(90\) 0 0
\(91\) 12.0000i 1.25794i
\(92\) 6.00000i 0.625543i
\(93\) 0 0
\(94\) 8.00000 0.825137
\(95\) 0 0
\(96\) 0 0
\(97\) 12.0000i 1.21842i −0.793011 0.609208i \(-0.791488\pi\)
0.793011 0.609208i \(-0.208512\pi\)
\(98\) −3.00000 −0.303046
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) 18.0000 1.79107 0.895533 0.444994i \(-0.146794\pi\)
0.895533 + 0.444994i \(0.146794\pi\)
\(102\) 0 0
\(103\) −16.0000 −1.57653 −0.788263 0.615338i \(-0.789020\pi\)
−0.788263 + 0.615338i \(0.789020\pi\)
\(104\) 6.00000 0.588348
\(105\) 0 0
\(106\) −6.00000 −0.582772
\(107\) 8.00000i 0.773389i −0.922208 0.386695i \(-0.873617\pi\)
0.922208 0.386695i \(-0.126383\pi\)
\(108\) 0 0
\(109\) 14.0000i 1.34096i 0.741929 + 0.670478i \(0.233911\pi\)
−0.741929 + 0.670478i \(0.766089\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 2.00000i 0.188982i
\(113\) 4.00000i 0.376288i 0.982141 + 0.188144i \(0.0602472\pi\)
−0.982141 + 0.188144i \(0.939753\pi\)
\(114\) 0 0
\(115\) −12.0000 −1.11901
\(116\) 6.00000i 0.557086i
\(117\) 0 0
\(118\) 0 0
\(119\) −8.00000 2.00000i −0.733359 0.183340i
\(120\) 0 0
\(121\) 11.0000 1.00000
\(122\) 10.0000i 0.905357i
\(123\) 0 0
\(124\) 10.0000i 0.898027i
\(125\) 12.0000i 1.07331i
\(126\) 0 0
\(127\) 8.00000 0.709885 0.354943 0.934888i \(-0.384500\pi\)
0.354943 + 0.934888i \(0.384500\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0 0
\(130\) 12.0000i 1.05247i
\(131\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) −8.00000 −0.691095
\(135\) 0 0
\(136\) −1.00000 + 4.00000i −0.0857493 + 0.342997i
\(137\) 2.00000 0.170872 0.0854358 0.996344i \(-0.472772\pi\)
0.0854358 + 0.996344i \(0.472772\pi\)
\(138\) 0 0
\(139\) 16.0000i 1.35710i −0.734553 0.678551i \(-0.762608\pi\)
0.734553 0.678551i \(-0.237392\pi\)
\(140\) −4.00000 −0.338062
\(141\) 0 0
\(142\) 10.0000i 0.839181i
\(143\) 0 0
\(144\) 0 0
\(145\) 12.0000 0.996546
\(146\) 16.0000i 1.32417i
\(147\) 0 0
\(148\) 2.00000i 0.164399i
\(149\) −10.0000 −0.819232 −0.409616 0.912258i \(-0.634337\pi\)
−0.409616 + 0.912258i \(0.634337\pi\)
\(150\) 0 0
\(151\) −8.00000 −0.651031 −0.325515 0.945537i \(-0.605538\pi\)
−0.325515 + 0.945537i \(0.605538\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −20.0000 −1.60644
\(156\) 0 0
\(157\) −2.00000 −0.159617 −0.0798087 0.996810i \(-0.525431\pi\)
−0.0798087 + 0.996810i \(0.525431\pi\)
\(158\) 6.00000i 0.477334i
\(159\) 0 0
\(160\) 2.00000i 0.158114i
\(161\) −12.0000 −0.945732
\(162\) 0 0
\(163\) 4.00000i 0.313304i −0.987654 0.156652i \(-0.949930\pi\)
0.987654 0.156652i \(-0.0500701\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) −16.0000 −1.24184
\(167\) 18.0000i 1.39288i −0.717614 0.696441i \(-0.754766\pi\)
0.717614 0.696441i \(-0.245234\pi\)
\(168\) 0 0
\(169\) 23.0000 1.76923
\(170\) 8.00000 + 2.00000i 0.613572 + 0.153393i
\(171\) 0 0
\(172\) 4.00000 0.304997
\(173\) 14.0000i 1.06440i 0.846619 + 0.532200i \(0.178635\pi\)
−0.846619 + 0.532200i \(0.821365\pi\)
\(174\) 0 0
\(175\) 2.00000i 0.151186i
\(176\) 0 0
\(177\) 0 0
\(178\) 10.0000 0.749532
\(179\) 20.0000 1.49487 0.747435 0.664335i \(-0.231285\pi\)
0.747435 + 0.664335i \(0.231285\pi\)
\(180\) 0 0
\(181\) 10.0000i 0.743294i 0.928374 + 0.371647i \(0.121207\pi\)
−0.928374 + 0.371647i \(0.878793\pi\)
\(182\) 12.0000i 0.889499i
\(183\) 0 0
\(184\) 6.00000i 0.442326i
\(185\) −4.00000 −0.294086
\(186\) 0 0
\(187\) 0 0
\(188\) −8.00000 −0.583460
\(189\) 0 0
\(190\) 0 0
\(191\) 8.00000 0.578860 0.289430 0.957199i \(-0.406534\pi\)
0.289430 + 0.957199i \(0.406534\pi\)
\(192\) 0 0
\(193\) 4.00000i 0.287926i −0.989583 0.143963i \(-0.954015\pi\)
0.989583 0.143963i \(-0.0459847\pi\)
\(194\) 12.0000i 0.861550i
\(195\) 0 0
\(196\) 3.00000 0.214286
\(197\) 18.0000i 1.28245i −0.767354 0.641223i \(-0.778427\pi\)
0.767354 0.641223i \(-0.221573\pi\)
\(198\) 0 0
\(199\) 6.00000i 0.425329i −0.977125 0.212664i \(-0.931786\pi\)
0.977125 0.212664i \(-0.0682141\pi\)
\(200\) −1.00000 −0.0707107
\(201\) 0 0
\(202\) −18.0000 −1.26648
\(203\) 12.0000 0.842235
\(204\) 0 0
\(205\) 0 0
\(206\) 16.0000 1.11477
\(207\) 0 0
\(208\) −6.00000 −0.416025
\(209\) 0 0
\(210\) 0 0
\(211\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(212\) 6.00000 0.412082
\(213\) 0 0
\(214\) 8.00000i 0.546869i
\(215\) 8.00000i 0.545595i
\(216\) 0 0
\(217\) −20.0000 −1.35769
\(218\) 14.0000i 0.948200i
\(219\) 0 0
\(220\) 0 0
\(221\) −6.00000 + 24.0000i −0.403604 + 1.61441i
\(222\) 0 0
\(223\) −16.0000 −1.07144 −0.535720 0.844396i \(-0.679960\pi\)
−0.535720 + 0.844396i \(0.679960\pi\)
\(224\) 2.00000i 0.133631i
\(225\) 0 0
\(226\) 4.00000i 0.266076i
\(227\) 12.0000i 0.796468i 0.917284 + 0.398234i \(0.130377\pi\)
−0.917284 + 0.398234i \(0.869623\pi\)
\(228\) 0 0
\(229\) 10.0000 0.660819 0.330409 0.943838i \(-0.392813\pi\)
0.330409 + 0.943838i \(0.392813\pi\)
\(230\) 12.0000 0.791257
\(231\) 0 0
\(232\) 6.00000i 0.393919i
\(233\) 4.00000i 0.262049i 0.991379 + 0.131024i \(0.0418266\pi\)
−0.991379 + 0.131024i \(0.958173\pi\)
\(234\) 0 0
\(235\) 16.0000i 1.04372i
\(236\) 0 0
\(237\) 0 0
\(238\) 8.00000 + 2.00000i 0.518563 + 0.129641i
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 0 0
\(241\) 20.0000i 1.28831i 0.764894 + 0.644157i \(0.222792\pi\)
−0.764894 + 0.644157i \(0.777208\pi\)
\(242\) −11.0000 −0.707107
\(243\) 0 0
\(244\) 10.0000i 0.640184i
\(245\) 6.00000i 0.383326i
\(246\) 0 0
\(247\) 0 0
\(248\) 10.0000i 0.635001i
\(249\) 0 0
\(250\) 12.0000i 0.758947i
\(251\) −12.0000 −0.757433 −0.378717 0.925513i \(-0.623635\pi\)
−0.378717 + 0.925513i \(0.623635\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) −8.00000 −0.501965
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 2.00000 0.124757 0.0623783 0.998053i \(-0.480131\pi\)
0.0623783 + 0.998053i \(0.480131\pi\)
\(258\) 0 0
\(259\) −4.00000 −0.248548
\(260\) 12.0000i 0.744208i
\(261\) 0 0
\(262\) 0 0
\(263\) 16.0000 0.986602 0.493301 0.869859i \(-0.335790\pi\)
0.493301 + 0.869859i \(0.335790\pi\)
\(264\) 0 0
\(265\) 12.0000i 0.737154i
\(266\) 0 0
\(267\) 0 0
\(268\) 8.00000 0.488678
\(269\) 6.00000i 0.365826i 0.983129 + 0.182913i \(0.0585527\pi\)
−0.983129 + 0.182913i \(0.941447\pi\)
\(270\) 0 0
\(271\) 32.0000 1.94386 0.971931 0.235267i \(-0.0755965\pi\)
0.971931 + 0.235267i \(0.0755965\pi\)
\(272\) 1.00000 4.00000i 0.0606339 0.242536i
\(273\) 0 0
\(274\) −2.00000 −0.120824
\(275\) 0 0
\(276\) 0 0
\(277\) 22.0000i 1.32185i −0.750451 0.660926i \(-0.770164\pi\)
0.750451 0.660926i \(-0.229836\pi\)
\(278\) 16.0000i 0.959616i
\(279\) 0 0
\(280\) 4.00000 0.239046
\(281\) −22.0000 −1.31241 −0.656205 0.754583i \(-0.727839\pi\)
−0.656205 + 0.754583i \(0.727839\pi\)
\(282\) 0 0
\(283\) 16.0000i 0.951101i 0.879688 + 0.475551i \(0.157751\pi\)
−0.879688 + 0.475551i \(0.842249\pi\)
\(284\) 10.0000i 0.593391i
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −15.0000 8.00000i −0.882353 0.470588i
\(290\) −12.0000 −0.704664
\(291\) 0 0
\(292\) 16.0000i 0.936329i
\(293\) 6.00000 0.350524 0.175262 0.984522i \(-0.443923\pi\)
0.175262 + 0.984522i \(0.443923\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 2.00000i 0.116248i
\(297\) 0 0
\(298\) 10.0000 0.579284
\(299\) 36.0000i 2.08193i
\(300\) 0 0
\(301\) 8.00000i 0.461112i
\(302\) 8.00000 0.460348
\(303\) 0 0
\(304\) 0 0
\(305\) 20.0000 1.14520
\(306\) 0 0
\(307\) −12.0000 −0.684876 −0.342438 0.939540i \(-0.611253\pi\)
−0.342438 + 0.939540i \(0.611253\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 20.0000 1.13592
\(311\) 10.0000i 0.567048i −0.958965 0.283524i \(-0.908496\pi\)
0.958965 0.283524i \(-0.0915036\pi\)
\(312\) 0 0
\(313\) 4.00000i 0.226093i −0.993590 0.113047i \(-0.963939\pi\)
0.993590 0.113047i \(-0.0360610\pi\)
\(314\) 2.00000 0.112867
\(315\) 0 0
\(316\) 6.00000i 0.337526i
\(317\) 18.0000i 1.01098i −0.862832 0.505490i \(-0.831312\pi\)
0.862832 0.505490i \(-0.168688\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 2.00000i 0.111803i
\(321\) 0 0
\(322\) 12.0000 0.668734
\(323\) 0 0
\(324\) 0 0
\(325\) −6.00000 −0.332820
\(326\) 4.00000i 0.221540i
\(327\) 0 0
\(328\) 0 0
\(329\) 16.0000i 0.882109i
\(330\) 0 0
\(331\) 12.0000 0.659580 0.329790 0.944054i \(-0.393022\pi\)
0.329790 + 0.944054i \(0.393022\pi\)
\(332\) 16.0000 0.878114
\(333\) 0 0
\(334\) 18.0000i 0.984916i
\(335\) 16.0000i 0.874173i
\(336\) 0 0
\(337\) 8.00000i 0.435788i 0.975972 + 0.217894i \(0.0699187\pi\)
−0.975972 + 0.217894i \(0.930081\pi\)
\(338\) −23.0000 −1.25104
\(339\) 0 0
\(340\) −8.00000 2.00000i −0.433861 0.108465i
\(341\) 0 0
\(342\) 0 0
\(343\) 20.0000i 1.07990i
\(344\) −4.00000 −0.215666
\(345\) 0 0
\(346\) 14.0000i 0.752645i
\(347\) 12.0000i 0.644194i 0.946707 + 0.322097i \(0.104388\pi\)
−0.946707 + 0.322097i \(0.895612\pi\)
\(348\) 0 0
\(349\) −10.0000 −0.535288 −0.267644 0.963518i \(-0.586245\pi\)
−0.267644 + 0.963518i \(0.586245\pi\)
\(350\) 2.00000i 0.106904i
\(351\) 0 0
\(352\) 0 0
\(353\) −14.0000 −0.745145 −0.372572 0.928003i \(-0.621524\pi\)
−0.372572 + 0.928003i \(0.621524\pi\)
\(354\) 0 0
\(355\) 20.0000 1.06149
\(356\) −10.0000 −0.529999
\(357\) 0 0
\(358\) −20.0000 −1.05703
\(359\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) 0 0
\(361\) −19.0000 −1.00000
\(362\) 10.0000i 0.525588i
\(363\) 0 0
\(364\) 12.0000i 0.628971i
\(365\) 32.0000 1.67496
\(366\) 0 0
\(367\) 22.0000i 1.14839i −0.818718 0.574195i \(-0.805315\pi\)
0.818718 0.574195i \(-0.194685\pi\)
\(368\) 6.00000i 0.312772i
\(369\) 0 0
\(370\) 4.00000 0.207950
\(371\) 12.0000i 0.623009i
\(372\) 0 0
\(373\) −6.00000 −0.310668 −0.155334 0.987862i \(-0.549645\pi\)
−0.155334 + 0.987862i \(0.549645\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 8.00000 0.412568
\(377\) 36.0000i 1.85409i
\(378\) 0 0
\(379\) 4.00000i 0.205466i 0.994709 + 0.102733i \(0.0327588\pi\)
−0.994709 + 0.102733i \(0.967241\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −8.00000 −0.409316
\(383\) −24.0000 −1.22634 −0.613171 0.789950i \(-0.710106\pi\)
−0.613171 + 0.789950i \(0.710106\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 4.00000i 0.203595i
\(387\) 0 0
\(388\) 12.0000i 0.609208i
\(389\) 10.0000 0.507020 0.253510 0.967333i \(-0.418415\pi\)
0.253510 + 0.967333i \(0.418415\pi\)
\(390\) 0 0
\(391\) −24.0000 6.00000i −1.21373 0.303433i
\(392\) −3.00000 −0.151523
\(393\) 0 0
\(394\) 18.0000i 0.906827i
\(395\) −12.0000 −0.603786
\(396\) 0 0
\(397\) 22.0000i 1.10415i −0.833795 0.552074i \(-0.813837\pi\)
0.833795 0.552074i \(-0.186163\pi\)
\(398\) 6.00000i 0.300753i
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(402\) 0 0
\(403\) 60.0000i 2.98881i
\(404\) 18.0000 0.895533
\(405\) 0 0
\(406\) −12.0000 −0.595550
\(407\) 0 0
\(408\) 0 0
\(409\) −10.0000 −0.494468 −0.247234 0.968956i \(-0.579522\pi\)
−0.247234 + 0.968956i \(0.579522\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −16.0000 −0.788263
\(413\) 0 0
\(414\) 0 0
\(415\) 32.0000i 1.57082i
\(416\) 6.00000 0.294174
\(417\) 0 0
\(418\) 0 0
\(419\) 4.00000i 0.195413i −0.995215 0.0977064i \(-0.968849\pi\)
0.995215 0.0977064i \(-0.0311506\pi\)
\(420\) 0 0
\(421\) 2.00000 0.0974740 0.0487370 0.998812i \(-0.484480\pi\)
0.0487370 + 0.998812i \(0.484480\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) −6.00000 −0.291386
\(425\) 1.00000 4.00000i 0.0485071 0.194029i
\(426\) 0 0
\(427\) 20.0000 0.967868
\(428\) 8.00000i 0.386695i
\(429\) 0 0
\(430\) 8.00000i 0.385794i
\(431\) 10.0000i 0.481683i 0.970564 + 0.240842i \(0.0774234\pi\)
−0.970564 + 0.240842i \(0.922577\pi\)
\(432\) 0 0
\(433\) 34.0000 1.63394 0.816968 0.576683i \(-0.195653\pi\)
0.816968 + 0.576683i \(0.195653\pi\)
\(434\) 20.0000 0.960031
\(435\) 0 0
\(436\) 14.0000i 0.670478i
\(437\) 0 0
\(438\) 0 0
\(439\) 6.00000i 0.286364i −0.989696 0.143182i \(-0.954267\pi\)
0.989696 0.143182i \(-0.0457335\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 6.00000 24.0000i 0.285391 1.14156i
\(443\) −4.00000 −0.190046 −0.0950229 0.995475i \(-0.530292\pi\)
−0.0950229 + 0.995475i \(0.530292\pi\)
\(444\) 0 0
\(445\) 20.0000i 0.948091i
\(446\) 16.0000 0.757622
\(447\) 0 0
\(448\) 2.00000i 0.0944911i
\(449\) 24.0000i 1.13263i −0.824189 0.566315i \(-0.808369\pi\)
0.824189 0.566315i \(-0.191631\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 4.00000i 0.188144i
\(453\) 0 0
\(454\) 12.0000i 0.563188i
\(455\) 24.0000 1.12514
\(456\) 0 0
\(457\) −22.0000 −1.02912 −0.514558 0.857455i \(-0.672044\pi\)
−0.514558 + 0.857455i \(0.672044\pi\)
\(458\) −10.0000 −0.467269
\(459\) 0 0
\(460\) −12.0000 −0.559503
\(461\) −2.00000 −0.0931493 −0.0465746 0.998915i \(-0.514831\pi\)
−0.0465746 + 0.998915i \(0.514831\pi\)
\(462\) 0 0
\(463\) 24.0000 1.11537 0.557687 0.830051i \(-0.311689\pi\)
0.557687 + 0.830051i \(0.311689\pi\)
\(464\) 6.00000i 0.278543i
\(465\) 0 0
\(466\) 4.00000i 0.185296i
\(467\) −28.0000 −1.29569 −0.647843 0.761774i \(-0.724329\pi\)
−0.647843 + 0.761774i \(0.724329\pi\)
\(468\) 0 0
\(469\) 16.0000i 0.738811i
\(470\) 16.0000i 0.738025i
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) −8.00000 2.00000i −0.366679 0.0916698i
\(477\) 0 0
\(478\) 0 0
\(479\) 14.0000i 0.639676i −0.947472 0.319838i \(-0.896371\pi\)
0.947472 0.319838i \(-0.103629\pi\)
\(480\) 0 0
\(481\) 12.0000i 0.547153i
\(482\) 20.0000i 0.910975i
\(483\) 0 0
\(484\) 11.0000 0.500000
\(485\) −24.0000 −1.08978
\(486\) 0 0
\(487\) 18.0000i 0.815658i 0.913058 + 0.407829i \(0.133714\pi\)
−0.913058 + 0.407829i \(0.866286\pi\)
\(488\) 10.0000i 0.452679i
\(489\) 0 0
\(490\) 6.00000i 0.271052i
\(491\) −12.0000 −0.541552 −0.270776 0.962642i \(-0.587280\pi\)
−0.270776 + 0.962642i \(0.587280\pi\)
\(492\) 0 0
\(493\) 24.0000 + 6.00000i 1.08091 + 0.270226i
\(494\) 0 0
\(495\) 0 0
\(496\) 10.0000i 0.449013i
\(497\) 20.0000 0.897123
\(498\) 0 0
\(499\) 24.0000i 1.07439i 0.843459 + 0.537194i \(0.180516\pi\)
−0.843459 + 0.537194i \(0.819484\pi\)
\(500\) 12.0000i 0.536656i
\(501\) 0 0
\(502\) 12.0000 0.535586
\(503\) 34.0000i 1.51599i 0.652263 + 0.757993i \(0.273820\pi\)
−0.652263 + 0.757993i \(0.726180\pi\)
\(504\) 0 0
\(505\) 36.0000i 1.60198i
\(506\) 0 0
\(507\) 0 0
\(508\) 8.00000 0.354943
\(509\) −10.0000 −0.443242 −0.221621 0.975133i \(-0.571135\pi\)
−0.221621 + 0.975133i \(0.571135\pi\)
\(510\) 0 0
\(511\) 32.0000 1.41560
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) −2.00000 −0.0882162
\(515\) 32.0000i 1.41009i
\(516\) 0 0
\(517\) 0 0
\(518\) 4.00000 0.175750
\(519\) 0 0
\(520\) 12.0000i 0.526235i
\(521\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(522\) 0 0
\(523\) −16.0000 −0.699631 −0.349816 0.936819i \(-0.613756\pi\)
−0.349816 + 0.936819i \(0.613756\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) −16.0000 −0.697633
\(527\) −40.0000 10.0000i −1.74243 0.435607i
\(528\) 0 0
\(529\) −13.0000 −0.565217
\(530\) 12.0000i 0.521247i
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) −16.0000 −0.691740
\(536\) −8.00000 −0.345547
\(537\) 0 0
\(538\) 6.00000i 0.258678i
\(539\) 0 0
\(540\) 0 0
\(541\) 10.0000i 0.429934i −0.976621 0.214967i \(-0.931036\pi\)
0.976621 0.214967i \(-0.0689643\pi\)
\(542\) −32.0000 −1.37452
\(543\) 0 0
\(544\) −1.00000 + 4.00000i −0.0428746 + 0.171499i
\(545\) 28.0000 1.19939
\(546\) 0 0
\(547\) 8.00000i 0.342055i 0.985266 + 0.171028i \(0.0547087\pi\)
−0.985266 + 0.171028i \(0.945291\pi\)
\(548\) 2.00000 0.0854358
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) −12.0000 −0.510292
\(554\) 22.0000i 0.934690i
\(555\) 0 0
\(556\) 16.0000i 0.678551i
\(557\) −38.0000 −1.61011 −0.805056 0.593199i \(-0.797865\pi\)
−0.805056 + 0.593199i \(0.797865\pi\)
\(558\) 0 0
\(559\) −24.0000 −1.01509
\(560\) −4.00000 −0.169031
\(561\) 0 0
\(562\) 22.0000 0.928014
\(563\) −24.0000 −1.01148 −0.505740 0.862686i \(-0.668780\pi\)
−0.505740 + 0.862686i \(0.668780\pi\)
\(564\) 0 0
\(565\) 8.00000 0.336563
\(566\) 16.0000i 0.672530i
\(567\) 0 0
\(568\) 10.0000i 0.419591i
\(569\) 30.0000 1.25767 0.628833 0.777541i \(-0.283533\pi\)
0.628833 + 0.777541i \(0.283533\pi\)
\(570\) 0 0
\(571\) 20.0000i 0.836974i 0.908223 + 0.418487i \(0.137439\pi\)
−0.908223 + 0.418487i \(0.862561\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 6.00000i 0.250217i
\(576\) 0 0
\(577\) −2.00000 −0.0832611 −0.0416305 0.999133i \(-0.513255\pi\)
−0.0416305 + 0.999133i \(0.513255\pi\)
\(578\) 15.0000 + 8.00000i 0.623918 + 0.332756i
\(579\) 0 0
\(580\) 12.0000 0.498273
\(581\) 32.0000i 1.32758i
\(582\) 0 0
\(583\) 0 0
\(584\) 16.0000i 0.662085i
\(585\) 0 0
\(586\) −6.00000 −0.247858
\(587\) 12.0000 0.495293 0.247647 0.968850i \(-0.420343\pi\)
0.247647 + 0.968850i \(0.420343\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 2.00000i 0.0821995i
\(593\) 46.0000 1.88899 0.944497 0.328521i \(-0.106550\pi\)
0.944497 + 0.328521i \(0.106550\pi\)
\(594\) 0 0
\(595\) −4.00000 + 16.0000i −0.163984 + 0.655936i
\(596\) −10.0000 −0.409616
\(597\) 0 0
\(598\) 36.0000i 1.47215i
\(599\) 40.0000 1.63436 0.817178 0.576386i \(-0.195537\pi\)
0.817178 + 0.576386i \(0.195537\pi\)
\(600\) 0 0
\(601\) 40.0000i 1.63163i −0.578310 0.815817i \(-0.696288\pi\)
0.578310 0.815817i \(-0.303712\pi\)
\(602\) 8.00000i 0.326056i
\(603\) 0 0
\(604\) −8.00000 −0.325515
\(605\) 22.0000i 0.894427i
\(606\) 0 0
\(607\) 38.0000i 1.54237i 0.636610 + 0.771186i \(0.280336\pi\)
−0.636610 + 0.771186i \(0.719664\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) −20.0000 −0.809776
\(611\) 48.0000 1.94187
\(612\) 0 0
\(613\) 14.0000 0.565455 0.282727 0.959200i \(-0.408761\pi\)
0.282727 + 0.959200i \(0.408761\pi\)
\(614\) 12.0000 0.484281
\(615\) 0 0
\(616\) 0 0
\(617\) 12.0000i 0.483102i 0.970388 + 0.241551i \(0.0776561\pi\)
−0.970388 + 0.241551i \(0.922344\pi\)
\(618\) 0 0
\(619\) 4.00000i 0.160774i 0.996764 + 0.0803868i \(0.0256155\pi\)
−0.996764 + 0.0803868i \(0.974384\pi\)
\(620\) −20.0000 −0.803219
\(621\) 0 0
\(622\) 10.0000i 0.400963i
\(623\) 20.0000i 0.801283i
\(624\) 0 0
\(625\) −19.0000 −0.760000
\(626\) 4.00000i 0.159872i
\(627\) 0 0
\(628\) −2.00000 −0.0798087
\(629\) −8.00000 2.00000i −0.318981 0.0797452i
\(630\) 0 0
\(631\) −8.00000 −0.318475 −0.159237 0.987240i \(-0.550904\pi\)
−0.159237 + 0.987240i \(0.550904\pi\)
\(632\) 6.00000i 0.238667i
\(633\) 0 0
\(634\) 18.0000i 0.714871i
\(635\) 16.0000i 0.634941i
\(636\) 0 0
\(637\) −18.0000 −0.713186
\(638\) 0 0
\(639\) 0 0
\(640\) 2.00000i 0.0790569i
\(641\) 40.0000i 1.57991i 0.613168 + 0.789953i \(0.289895\pi\)
−0.613168 + 0.789953i \(0.710105\pi\)
\(642\) 0 0
\(643\) 36.0000i 1.41970i 0.704352 + 0.709851i \(0.251238\pi\)
−0.704352 + 0.709851i \(0.748762\pi\)
\(644\) −12.0000 −0.472866
\(645\) 0 0
\(646\) 0 0
\(647\) 32.0000 1.25805 0.629025 0.777385i \(-0.283454\pi\)
0.629025 + 0.777385i \(0.283454\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 6.00000 0.235339
\(651\) 0 0
\(652\) 4.00000i 0.156652i
\(653\) 34.0000i 1.33052i 0.746611 + 0.665261i \(0.231680\pi\)
−0.746611 + 0.665261i \(0.768320\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 16.0000i 0.623745i
\(659\) 40.0000 1.55818 0.779089 0.626913i \(-0.215682\pi\)
0.779089 + 0.626913i \(0.215682\pi\)
\(660\) 0 0
\(661\) 22.0000 0.855701 0.427850 0.903850i \(-0.359271\pi\)
0.427850 + 0.903850i \(0.359271\pi\)
\(662\) −12.0000 −0.466393
\(663\) 0 0
\(664\) −16.0000 −0.620920
\(665\) 0 0
\(666\) 0 0
\(667\) 36.0000 1.39393
\(668\) 18.0000i 0.696441i
\(669\) 0 0
\(670\) 16.0000i 0.618134i
\(671\) 0 0
\(672\) 0 0
\(673\) 36.0000i 1.38770i 0.720121 + 0.693849i \(0.244086\pi\)
−0.720121 + 0.693849i \(0.755914\pi\)
\(674\) 8.00000i 0.308148i
\(675\) 0 0
\(676\) 23.0000 0.884615
\(677\) 42.0000i 1.61419i 0.590421 + 0.807096i \(0.298962\pi\)
−0.590421 + 0.807096i \(0.701038\pi\)
\(678\) 0 0
\(679\) −24.0000 −0.921035
\(680\) 8.00000 + 2.00000i 0.306786 + 0.0766965i
\(681\) 0 0
\(682\) 0 0
\(683\) 44.0000i 1.68361i 0.539779 + 0.841807i \(0.318508\pi\)
−0.539779 + 0.841807i \(0.681492\pi\)
\(684\) 0 0
\(685\) 4.00000i 0.152832i
\(686\) 20.0000i 0.763604i
\(687\) 0 0
\(688\) 4.00000 0.152499
\(689\) −36.0000 −1.37149
\(690\) 0 0
\(691\) 40.0000i 1.52167i −0.648944 0.760836i \(-0.724789\pi\)
0.648944 0.760836i \(-0.275211\pi\)
\(692\) 14.0000i 0.532200i
\(693\) 0 0
\(694\) 12.0000i 0.455514i
\(695\) −32.0000 −1.21383
\(696\) 0 0
\(697\) 0 0
\(698\) 10.0000 0.378506
\(699\) 0 0
\(700\) 2.00000i 0.0755929i
\(701\) −22.0000 −0.830929 −0.415464 0.909610i \(-0.636381\pi\)
−0.415464 + 0.909610i \(0.636381\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 14.0000 0.526897
\(707\) 36.0000i 1.35392i
\(708\) 0 0
\(709\) 6.00000i 0.225335i −0.993633 0.112667i \(-0.964061\pi\)
0.993633 0.112667i \(-0.0359394\pi\)
\(710\) −20.0000 −0.750587
\(711\) 0 0
\(712\) 10.0000 0.374766
\(713\) −60.0000 −2.24702
\(714\) 0 0
\(715\) 0 0
\(716\) 20.0000 0.747435
\(717\) 0 0
\(718\) 0 0
\(719\) 26.0000i 0.969636i 0.874615 + 0.484818i \(0.161114\pi\)
−0.874615 + 0.484818i \(0.838886\pi\)
\(720\) 0 0
\(721\) 32.0000i 1.19174i
\(722\) 19.0000 0.707107
\(723\) 0 0
\(724\) 10.0000i 0.371647i
\(725\) 6.00000i 0.222834i
\(726\) 0 0
\(727\) 48.0000 1.78022 0.890111 0.455744i \(-0.150627\pi\)
0.890111 + 0.455744i \(0.150627\pi\)
\(728\) 12.0000i 0.444750i
\(729\) 0 0
\(730\) −32.0000 −1.18437
\(731\) 4.00000 16.0000i 0.147945 0.591781i
\(732\) 0 0
\(733\) 14.0000 0.517102 0.258551 0.965998i \(-0.416755\pi\)
0.258551 + 0.965998i \(0.416755\pi\)
\(734\) 22.0000i 0.812035i
\(735\) 0 0
\(736\) 6.00000i 0.221163i
\(737\) 0 0
\(738\) 0 0
\(739\) −40.0000 −1.47142 −0.735712 0.677295i \(-0.763152\pi\)
−0.735712 + 0.677295i \(0.763152\pi\)
\(740\) −4.00000 −0.147043
\(741\) 0 0
\(742\) 12.0000i 0.440534i
\(743\) 26.0000i 0.953847i −0.878945 0.476924i \(-0.841752\pi\)
0.878945 0.476924i \(-0.158248\pi\)
\(744\) 0 0
\(745\) 20.0000i 0.732743i
\(746\) 6.00000 0.219676
\(747\) 0 0
\(748\) 0 0
\(749\) −16.0000 −0.584627
\(750\) 0 0
\(751\) 10.0000i 0.364905i 0.983215 + 0.182453i \(0.0584036\pi\)
−0.983215 + 0.182453i \(0.941596\pi\)
\(752\) −8.00000 −0.291730
\(753\) 0 0
\(754\) 36.0000i 1.31104i
\(755\) 16.0000i 0.582300i
\(756\) 0 0
\(757\) −22.0000 −0.799604 −0.399802 0.916602i \(-0.630921\pi\)
−0.399802 + 0.916602i \(0.630921\pi\)
\(758\) 4.00000i 0.145287i
\(759\) 0 0
\(760\) 0 0
\(761\) 18.0000 0.652499 0.326250 0.945284i \(-0.394215\pi\)
0.326250 + 0.945284i \(0.394215\pi\)
\(762\) 0 0
\(763\) 28.0000 1.01367
\(764\) 8.00000 0.289430
\(765\) 0 0
\(766\) 24.0000 0.867155
\(767\) 0 0
\(768\) 0 0
\(769\) 30.0000 1.08183 0.540914 0.841078i \(-0.318079\pi\)
0.540914 + 0.841078i \(0.318079\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 4.00000i 0.143963i
\(773\) −14.0000 −0.503545 −0.251773 0.967786i \(-0.581013\pi\)
−0.251773 + 0.967786i \(0.581013\pi\)
\(774\) 0 0
\(775\) 10.0000i 0.359211i
\(776\) 12.0000i 0.430775i
\(777\) 0 0
\(778\) −10.0000 −0.358517
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 24.0000 + 6.00000i 0.858238 + 0.214560i
\(783\) 0 0
\(784\) 3.00000 0.107143
\(785\) 4.00000i 0.142766i
\(786\) 0 0
\(787\) 32.0000i 1.14068i −0.821410 0.570338i \(-0.806812\pi\)
0.821410 0.570338i \(-0.193188\pi\)
\(788\) 18.0000i 0.641223i
\(789\) 0 0
\(790\) 12.0000 0.426941
\(791\) 8.00000 0.284447
\(792\) 0 0
\(793\) 60.0000i 2.13066i
\(794\) 22.0000i 0.780751i
\(795\) 0 0
\(796\) 6.00000i 0.212664i
\(797\) −18.0000 −0.637593 −0.318796 0.947823i \(-0.603279\pi\)
−0.318796 + 0.947823i \(0.603279\pi\)
\(798\) 0 0
\(799\) −8.00000 + 32.0000i −0.283020 + 1.13208i
\(800\) −1.00000 −0.0353553
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 24.0000i 0.845889i
\(806\) 60.0000i 2.11341i
\(807\) 0 0
\(808\) −18.0000 −0.633238
\(809\) 24.0000i 0.843795i −0.906644 0.421898i \(-0.861364\pi\)
0.906644 0.421898i \(-0.138636\pi\)
\(810\) 0 0
\(811\) 20.0000i 0.702295i −0.936320 0.351147i \(-0.885792\pi\)
0.936320 0.351147i \(-0.114208\pi\)
\(812\) 12.0000 0.421117
\(813\) 0 0
\(814\) 0 0
\(815\) −8.00000 −0.280228
\(816\) 0 0
\(817\) 0 0
\(818\) 10.0000 0.349642
\(819\) 0 0
\(820\) 0 0
\(821\) 10.0000i 0.349002i 0.984657 + 0.174501i \(0.0558313\pi\)
−0.984657 + 0.174501i \(0.944169\pi\)
\(822\) 0 0
\(823\) 14.0000i 0.488009i −0.969774 0.244005i \(-0.921539\pi\)
0.969774 0.244005i \(-0.0784612\pi\)
\(824\) 16.0000 0.557386
\(825\) 0 0
\(826\) 0 0
\(827\) 32.0000i 1.11275i 0.830932 + 0.556375i \(0.187808\pi\)
−0.830932 + 0.556375i \(0.812192\pi\)
\(828\) 0 0
\(829\) 30.0000 1.04194 0.520972 0.853574i \(-0.325570\pi\)
0.520972 + 0.853574i \(0.325570\pi\)
\(830\) 32.0000i 1.11074i
\(831\) 0 0
\(832\) −6.00000 −0.208013
\(833\) 3.00000 12.0000i 0.103944 0.415775i
\(834\) 0 0
\(835\) −36.0000 −1.24583
\(836\) 0 0
\(837\) 0 0
\(838\) 4.00000i 0.138178i
\(839\) 34.0000i 1.17381i −0.809656 0.586905i \(-0.800346\pi\)
0.809656 0.586905i \(-0.199654\pi\)
\(840\) 0 0
\(841\) −7.00000 −0.241379
\(842\) −2.00000 −0.0689246
\(843\) 0 0
\(844\) 0 0
\(845\) 46.0000i 1.58245i
\(846\) 0 0
\(847\) 22.0000i 0.755929i
\(848\) 6.00000 0.206041
\(849\) 0 0
\(850\) −1.00000 + 4.00000i −0.0342997 + 0.137199i
\(851\) −12.0000 −0.411355
\(852\) 0 0
\(853\) 14.0000i 0.479351i −0.970853 0.239675i \(-0.922959\pi\)
0.970853 0.239675i \(-0.0770410\pi\)
\(854\) −20.0000 −0.684386
\(855\) 0 0
\(856\) 8.00000i 0.273434i
\(857\) 48.0000i 1.63965i −0.572615 0.819824i \(-0.694071\pi\)
0.572615 0.819824i \(-0.305929\pi\)
\(858\) 0 0
\(859\) 20.0000 0.682391 0.341196 0.939992i \(-0.389168\pi\)
0.341196 + 0.939992i \(0.389168\pi\)
\(860\) 8.00000i 0.272798i
\(861\) 0 0
\(862\) 10.0000i 0.340601i
\(863\) 16.0000 0.544646 0.272323 0.962206i \(-0.412208\pi\)
0.272323 + 0.962206i \(0.412208\pi\)
\(864\) 0 0
\(865\) 28.0000 0.952029
\(866\) −34.0000 −1.15537
\(867\) 0 0
\(868\) −20.0000 −0.678844
\(869\) 0 0
\(870\) 0 0
\(871\) −48.0000 −1.62642
\(872\) 14.0000i 0.474100i
\(873\) 0 0
\(874\) 0 0
\(875\) −24.0000 −0.811348
\(876\) 0 0
\(877\) 38.0000i 1.28317i 0.767052 + 0.641584i \(0.221723\pi\)
−0.767052 + 0.641584i \(0.778277\pi\)
\(878\) 6.00000i 0.202490i
\(879\) 0 0
\(880\) 0 0
\(881\) 20.0000i 0.673817i −0.941537 0.336909i \(-0.890619\pi\)
0.941537 0.336909i \(-0.109381\pi\)
\(882\) 0 0
\(883\) −16.0000 −0.538443 −0.269221 0.963078i \(-0.586766\pi\)
−0.269221 + 0.963078i \(0.586766\pi\)
\(884\) −6.00000 + 24.0000i −0.201802 + 0.807207i
\(885\) 0 0
\(886\) 4.00000 0.134383
\(887\) 42.0000i 1.41022i 0.709097 + 0.705111i \(0.249103\pi\)
−0.709097 + 0.705111i \(0.750897\pi\)
\(888\) 0 0
\(889\) 16.0000i 0.536623i
\(890\) 20.0000i 0.670402i
\(891\) 0 0
\(892\) −16.0000 −0.535720
\(893\) 0 0
\(894\) 0 0
\(895\) 40.0000i 1.33705i
\(896\) 2.00000i 0.0668153i
\(897\) 0 0
\(898\) 24.0000i 0.800890i
\(899\) 60.0000 2.00111
\(900\) 0 0
\(901\) 6.00000 24.0000i 0.199889 0.799556i
\(902\) 0 0
\(903\) 0 0
\(904\) 4.00000i 0.133038i
\(905\) 20.0000 0.664822
\(906\) 0 0
\(907\) 8.00000i 0.265636i 0.991140 + 0.132818i \(0.0424025\pi\)
−0.991140 + 0.132818i \(0.957597\pi\)
\(908\) 12.0000i 0.398234i
\(909\) 0 0
\(910\) −24.0000 −0.795592
\(911\) 50.0000i 1.65657i −0.560304 0.828287i \(-0.689316\pi\)
0.560304 0.828287i \(-0.310684\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 22.0000 0.727695
\(915\) 0 0
\(916\) 10.0000 0.330409
\(917\) 0 0
\(918\) 0 0
\(919\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(920\) 12.0000 0.395628
\(921\) 0 0
\(922\) 2.00000 0.0658665
\(923\) 60.0000i 1.97492i
\(924\) 0 0
\(925\) 2.00000i 0.0657596i
\(926\) −24.0000 −0.788689
\(927\) 0 0
\(928\) 6.00000i 0.196960i
\(929\) 16.0000i 0.524943i 0.964940 + 0.262471i \(0.0845376\pi\)
−0.964940 + 0.262471i \(0.915462\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 4.00000i 0.131024i
\(933\) 0 0
\(934\) 28.0000 0.916188
\(935\) 0 0
\(936\) 0 0
\(937\) −2.00000 −0.0653372 −0.0326686 0.999466i \(-0.510401\pi\)
−0.0326686 + 0.999466i \(0.510401\pi\)
\(938\) 16.0000i 0.522419i
\(939\) 0 0
\(940\) 16.0000i 0.521862i
\(941\) 30.0000i 0.977972i 0.872292 + 0.488986i \(0.162633\pi\)
−0.872292 + 0.488986i \(0.837367\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 8.00000i 0.259965i −0.991516 0.129983i \(-0.958508\pi\)
0.991516 0.129983i \(-0.0414921\pi\)
\(948\) 0 0
\(949\) 96.0000i 3.11629i
\(950\) 0 0
\(951\) 0 0
\(952\) 8.00000 + 2.00000i 0.259281 + 0.0648204i
\(953\) 6.00000 0.194359 0.0971795 0.995267i \(-0.469018\pi\)
0.0971795 + 0.995267i \(0.469018\pi\)
\(954\) 0 0
\(955\) 16.0000i 0.517748i
\(956\) 0 0
\(957\) 0 0
\(958\) 14.0000i 0.452319i
\(959\) 4.00000i 0.129167i
\(960\) 0 0
\(961\) −69.0000 −2.22581
\(962\) 12.0000i 0.386896i
\(963\) 0 0
\(964\) 20.0000i 0.644157i
\(965\) −8.00000 −0.257529
\(966\) 0 0
\(967\) 8.00000 0.257263 0.128631 0.991692i \(-0.458942\pi\)
0.128631 + 0.991692i \(0.458942\pi\)
\(968\) −11.0000 −0.353553
\(969\) 0 0
\(970\) 24.0000 0.770594
\(971\) −32.0000 −1.02693 −0.513464 0.858111i \(-0.671638\pi\)
−0.513464 + 0.858111i \(0.671638\pi\)
\(972\) 0 0
\(973\) −32.0000 −1.02587
\(974\) 18.0000i 0.576757i
\(975\) 0 0
\(976\) 10.0000i 0.320092i
\(977\) 22.0000 0.703842 0.351921 0.936030i \(-0.385529\pi\)
0.351921 + 0.936030i \(0.385529\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 6.00000i 0.191663i
\(981\) 0 0
\(982\) 12.0000 0.382935
\(983\) 46.0000i 1.46717i −0.679597 0.733586i \(-0.737845\pi\)
0.679597 0.733586i \(-0.262155\pi\)
\(984\) 0 0
\(985\) −36.0000 −1.14706
\(986\) −24.0000 6.00000i −0.764316 0.191079i
\(987\) 0 0
\(988\) 0 0
\(989\) 24.0000i 0.763156i
\(990\) 0 0
\(991\) 10.0000i 0.317660i 0.987306 + 0.158830i \(0.0507723\pi\)
−0.987306 + 0.158830i \(0.949228\pi\)
\(992\) 10.0000i 0.317500i
\(993\) 0 0
\(994\) −20.0000 −0.634361
\(995\) −12.0000 −0.380426
\(996\) 0 0
\(997\) 18.0000i 0.570066i 0.958518 + 0.285033i \(0.0920045\pi\)
−0.958518 + 0.285033i \(0.907995\pi\)
\(998\) 24.0000i 0.759707i
\(999\) 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 306.2.b.a.271.1 2
3.2 odd 2 102.2.b.a.67.1 2
4.3 odd 2 2448.2.c.a.577.1 2
12.11 even 2 816.2.c.a.577.2 2
15.2 even 4 2550.2.f.e.1699.2 2
15.8 even 4 2550.2.f.j.1699.1 2
15.14 odd 2 2550.2.c.f.1801.2 2
17.4 even 4 5202.2.a.h.1.1 1
17.13 even 4 5202.2.a.n.1.1 1
17.16 even 2 inner 306.2.b.a.271.2 2
24.5 odd 2 3264.2.c.j.577.2 2
24.11 even 2 3264.2.c.i.577.1 2
51.2 odd 8 1734.2.f.h.829.1 4
51.8 odd 8 1734.2.f.h.1483.1 4
51.26 odd 8 1734.2.f.h.1483.2 4
51.32 odd 8 1734.2.f.h.829.2 4
51.38 odd 4 1734.2.a.d.1.1 1
51.47 odd 4 1734.2.a.e.1.1 1
51.50 odd 2 102.2.b.a.67.2 yes 2
68.67 odd 2 2448.2.c.a.577.2 2
204.203 even 2 816.2.c.a.577.1 2
255.152 even 4 2550.2.f.j.1699.2 2
255.203 even 4 2550.2.f.e.1699.1 2
255.254 odd 2 2550.2.c.f.1801.1 2
408.101 odd 2 3264.2.c.j.577.1 2
408.203 even 2 3264.2.c.i.577.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
102.2.b.a.67.1 2 3.2 odd 2
102.2.b.a.67.2 yes 2 51.50 odd 2
306.2.b.a.271.1 2 1.1 even 1 trivial
306.2.b.a.271.2 2 17.16 even 2 inner
816.2.c.a.577.1 2 204.203 even 2
816.2.c.a.577.2 2 12.11 even 2
1734.2.a.d.1.1 1 51.38 odd 4
1734.2.a.e.1.1 1 51.47 odd 4
1734.2.f.h.829.1 4 51.2 odd 8
1734.2.f.h.829.2 4 51.32 odd 8
1734.2.f.h.1483.1 4 51.8 odd 8
1734.2.f.h.1483.2 4 51.26 odd 8
2448.2.c.a.577.1 2 4.3 odd 2
2448.2.c.a.577.2 2 68.67 odd 2
2550.2.c.f.1801.1 2 255.254 odd 2
2550.2.c.f.1801.2 2 15.14 odd 2
2550.2.f.e.1699.1 2 255.203 even 4
2550.2.f.e.1699.2 2 15.2 even 4
2550.2.f.j.1699.1 2 15.8 even 4
2550.2.f.j.1699.2 2 255.152 even 4
3264.2.c.i.577.1 2 24.11 even 2
3264.2.c.i.577.2 2 408.203 even 2
3264.2.c.j.577.1 2 408.101 odd 2
3264.2.c.j.577.2 2 24.5 odd 2
5202.2.a.h.1.1 1 17.4 even 4
5202.2.a.n.1.1 1 17.13 even 4