Properties

Label 1386.2.c.b.197.5
Level $1386$
Weight $2$
Character 1386.197
Analytic conductor $11.067$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1386,2,Mod(197,1386)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1386, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 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("1386.197");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1386 = 2 \cdot 3^{2} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1386.c (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: \(11.0672657201\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 4x^{10} - 8x^{9} + 17x^{8} - 16x^{7} + 88x^{6} - 48x^{5} + 153x^{4} - 216x^{3} + 324x^{2} + 729 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{5}\cdot 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 197.5
Root \(-0.833477 + 1.51833i\) of defining polynomial
Character \(\chi\) \(=\) 1386.197
Dual form 1386.2.c.b.197.8

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} -0.968524i q^{5} +1.00000i q^{7} +1.00000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{4} -0.968524i q^{5} +1.00000i q^{7} +1.00000 q^{8} -0.968524i q^{10} +(-3.27778 - 0.506135i) q^{11} +6.36261i q^{13} +1.00000i q^{14} +1.00000 q^{16} +7.67213 q^{17} +0.690475i q^{19} -0.968524i q^{20} +(-3.27778 - 0.506135i) q^{22} +3.04461i q^{23} +4.06196 q^{25} +6.36261i q^{26} +1.00000i q^{28} +9.36769 q^{29} +4.20608 q^{31} +1.00000 q^{32} +7.67213 q^{34} +0.968524 q^{35} -4.16234 q^{37} +0.690475i q^{38} -0.968524i q^{40} -0.349474 q^{41} -0.794780i q^{43} +(-3.27778 - 0.506135i) q^{44} +3.04461i q^{46} -8.19535i q^{47} -1.00000 q^{49} +4.06196 q^{50} +6.36261i q^{52} +7.24488i q^{53} +(-0.490204 + 3.17461i) q^{55} +1.00000i q^{56} +9.36769 q^{58} -12.0994i q^{59} -8.29965i q^{61} +4.20608 q^{62} +1.00000 q^{64} +6.16234 q^{65} -12.2823 q^{67} +7.67213 q^{68} +0.968524 q^{70} +4.98166i q^{71} +9.97054i q^{73} -4.16234 q^{74} +0.690475i q^{76} +(0.506135 - 3.27778i) q^{77} +2.37996i q^{79} -0.968524i q^{80} -0.349474 q^{82} +4.79593 q^{83} -7.43064i q^{85} -0.794780i q^{86} +(-3.27778 - 0.506135i) q^{88} -7.23148i q^{89} -6.36261 q^{91} +3.04461i q^{92} -8.19535i q^{94} +0.668741 q^{95} -2.78871 q^{97} -1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 12 q^{2} + 12 q^{4} + 12 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 12 q^{2} + 12 q^{4} + 12 q^{8} - 4 q^{11} + 12 q^{16} + 16 q^{17} - 4 q^{22} - 4 q^{25} + 16 q^{29} + 12 q^{32} + 16 q^{34} - 8 q^{35} + 24 q^{37} + 16 q^{41} - 4 q^{44} - 12 q^{49} - 4 q^{50} - 8 q^{55} + 16 q^{58} + 12 q^{64} - 48 q^{67} + 16 q^{68} - 8 q^{70} + 24 q^{74} + 8 q^{77} + 16 q^{82} + 16 q^{83} - 4 q^{88} - 48 q^{95} + 48 q^{97} - 12 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1386\mathbb{Z}\right)^\times\).

\(n\) \(155\) \(199\) \(1135\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) 0.968524i 0.433137i −0.976267 0.216568i \(-0.930514\pi\)
0.976267 0.216568i \(-0.0694865\pi\)
\(6\) 0 0
\(7\) 1.00000i 0.377964i
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) 0.968524i 0.306274i
\(11\) −3.27778 0.506135i −0.988287 0.152606i
\(12\) 0 0
\(13\) 6.36261i 1.76467i 0.470622 + 0.882335i \(0.344029\pi\)
−0.470622 + 0.882335i \(0.655971\pi\)
\(14\) 1.00000i 0.267261i
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 7.67213 1.86077 0.930383 0.366590i \(-0.119475\pi\)
0.930383 + 0.366590i \(0.119475\pi\)
\(18\) 0 0
\(19\) 0.690475i 0.158406i 0.996859 + 0.0792029i \(0.0252375\pi\)
−0.996859 + 0.0792029i \(0.974762\pi\)
\(20\) 0.968524i 0.216568i
\(21\) 0 0
\(22\) −3.27778 0.506135i −0.698825 0.107908i
\(23\) 3.04461i 0.634845i 0.948284 + 0.317422i \(0.102817\pi\)
−0.948284 + 0.317422i \(0.897183\pi\)
\(24\) 0 0
\(25\) 4.06196 0.812392
\(26\) 6.36261i 1.24781i
\(27\) 0 0
\(28\) 1.00000i 0.188982i
\(29\) 9.36769 1.73954 0.869768 0.493461i \(-0.164268\pi\)
0.869768 + 0.493461i \(0.164268\pi\)
\(30\) 0 0
\(31\) 4.20608 0.755435 0.377717 0.925921i \(-0.376709\pi\)
0.377717 + 0.925921i \(0.376709\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) 7.67213 1.31576
\(35\) 0.968524 0.163710
\(36\) 0 0
\(37\) −4.16234 −0.684284 −0.342142 0.939648i \(-0.611152\pi\)
−0.342142 + 0.939648i \(0.611152\pi\)
\(38\) 0.690475i 0.112010i
\(39\) 0 0
\(40\) 0.968524i 0.153137i
\(41\) −0.349474 −0.0545786 −0.0272893 0.999628i \(-0.508688\pi\)
−0.0272893 + 0.999628i \(0.508688\pi\)
\(42\) 0 0
\(43\) 0.794780i 0.121203i −0.998162 0.0606014i \(-0.980698\pi\)
0.998162 0.0606014i \(-0.0193019\pi\)
\(44\) −3.27778 0.506135i −0.494144 0.0763028i
\(45\) 0 0
\(46\) 3.04461i 0.448903i
\(47\) 8.19535i 1.19541i −0.801714 0.597707i \(-0.796079\pi\)
0.801714 0.597707i \(-0.203921\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 4.06196 0.574448
\(51\) 0 0
\(52\) 6.36261i 0.882335i
\(53\) 7.24488i 0.995161i 0.867418 + 0.497580i \(0.165778\pi\)
−0.867418 + 0.497580i \(0.834222\pi\)
\(54\) 0 0
\(55\) −0.490204 + 3.17461i −0.0660991 + 0.428064i
\(56\) 1.00000i 0.133631i
\(57\) 0 0
\(58\) 9.36769 1.23004
\(59\) 12.0994i 1.57521i −0.616183 0.787603i \(-0.711322\pi\)
0.616183 0.787603i \(-0.288678\pi\)
\(60\) 0 0
\(61\) 8.29965i 1.06266i −0.847164 0.531331i \(-0.821692\pi\)
0.847164 0.531331i \(-0.178308\pi\)
\(62\) 4.20608 0.534173
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 6.16234 0.764344
\(66\) 0 0
\(67\) −12.2823 −1.50052 −0.750261 0.661142i \(-0.770072\pi\)
−0.750261 + 0.661142i \(0.770072\pi\)
\(68\) 7.67213 0.930383
\(69\) 0 0
\(70\) 0.968524 0.115761
\(71\) 4.98166i 0.591214i 0.955310 + 0.295607i \(0.0955219\pi\)
−0.955310 + 0.295607i \(0.904478\pi\)
\(72\) 0 0
\(73\) 9.97054i 1.16696i 0.812126 + 0.583482i \(0.198310\pi\)
−0.812126 + 0.583482i \(0.801690\pi\)
\(74\) −4.16234 −0.483862
\(75\) 0 0
\(76\) 0.690475i 0.0792029i
\(77\) 0.506135 3.27778i 0.0576795 0.373537i
\(78\) 0 0
\(79\) 2.37996i 0.267766i 0.990997 + 0.133883i \(0.0427447\pi\)
−0.990997 + 0.133883i \(0.957255\pi\)
\(80\) 0.968524i 0.108284i
\(81\) 0 0
\(82\) −0.349474 −0.0385929
\(83\) 4.79593 0.526422 0.263211 0.964738i \(-0.415218\pi\)
0.263211 + 0.964738i \(0.415218\pi\)
\(84\) 0 0
\(85\) 7.43064i 0.805966i
\(86\) 0.794780i 0.0857034i
\(87\) 0 0
\(88\) −3.27778 0.506135i −0.349412 0.0539542i
\(89\) 7.23148i 0.766536i −0.923637 0.383268i \(-0.874799\pi\)
0.923637 0.383268i \(-0.125201\pi\)
\(90\) 0 0
\(91\) −6.36261 −0.666982
\(92\) 3.04461i 0.317422i
\(93\) 0 0
\(94\) 8.19535i 0.845286i
\(95\) 0.668741 0.0686114
\(96\) 0 0
\(97\) −2.78871 −0.283150 −0.141575 0.989928i \(-0.545217\pi\)
−0.141575 + 0.989928i \(0.545217\pi\)
\(98\) −1.00000 −0.101015
\(99\) 0 0
\(100\) 4.06196 0.406196
\(101\) −15.9272 −1.58482 −0.792408 0.609992i \(-0.791173\pi\)
−0.792408 + 0.609992i \(0.791173\pi\)
\(102\) 0 0
\(103\) −7.44969 −0.734039 −0.367020 0.930213i \(-0.619622\pi\)
−0.367020 + 0.930213i \(0.619622\pi\)
\(104\) 6.36261i 0.623905i
\(105\) 0 0
\(106\) 7.24488i 0.703685i
\(107\) 3.91251 0.378236 0.189118 0.981954i \(-0.439437\pi\)
0.189118 + 0.981954i \(0.439437\pi\)
\(108\) 0 0
\(109\) 15.5606i 1.49044i 0.666819 + 0.745219i \(0.267655\pi\)
−0.666819 + 0.745219i \(0.732345\pi\)
\(110\) −0.490204 + 3.17461i −0.0467391 + 0.302687i
\(111\) 0 0
\(112\) 1.00000i 0.0944911i
\(113\) 14.2751i 1.34289i 0.741055 + 0.671445i \(0.234326\pi\)
−0.741055 + 0.671445i \(0.765674\pi\)
\(114\) 0 0
\(115\) 2.94878 0.274975
\(116\) 9.36769 0.869768
\(117\) 0 0
\(118\) 12.0994i 1.11384i
\(119\) 7.67213i 0.703303i
\(120\) 0 0
\(121\) 10.4877 + 3.31800i 0.953423 + 0.301636i
\(122\) 8.29965i 0.751415i
\(123\) 0 0
\(124\) 4.20608 0.377717
\(125\) 8.77673i 0.785014i
\(126\) 0 0
\(127\) 19.2658i 1.70956i 0.518989 + 0.854781i \(0.326309\pi\)
−0.518989 + 0.854781i \(0.673691\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) 6.16234 0.540473
\(131\) 18.7832 1.64110 0.820550 0.571575i \(-0.193667\pi\)
0.820550 + 0.571575i \(0.193667\pi\)
\(132\) 0 0
\(133\) −0.690475 −0.0598718
\(134\) −12.2823 −1.06103
\(135\) 0 0
\(136\) 7.67213 0.657880
\(137\) 12.8552i 1.09829i −0.835726 0.549147i \(-0.814953\pi\)
0.835726 0.549147i \(-0.185047\pi\)
\(138\) 0 0
\(139\) 12.8642i 1.09113i −0.838068 0.545566i \(-0.816315\pi\)
0.838068 0.545566i \(-0.183685\pi\)
\(140\) 0.968524 0.0818552
\(141\) 0 0
\(142\) 4.98166i 0.418051i
\(143\) 3.22034 20.8552i 0.269298 1.74400i
\(144\) 0 0
\(145\) 9.07283i 0.753457i
\(146\) 9.97054i 0.825167i
\(147\) 0 0
\(148\) −4.16234 −0.342142
\(149\) −9.48765 −0.777259 −0.388629 0.921394i \(-0.627051\pi\)
−0.388629 + 0.921394i \(0.627051\pi\)
\(150\) 0 0
\(151\) 10.8618i 0.883923i 0.897034 + 0.441961i \(0.145717\pi\)
−0.897034 + 0.441961i \(0.854283\pi\)
\(152\) 0.690475i 0.0560049i
\(153\) 0 0
\(154\) 0.506135 3.27778i 0.0407855 0.264131i
\(155\) 4.07369i 0.327207i
\(156\) 0 0
\(157\) −3.89046 −0.310492 −0.155246 0.987876i \(-0.549617\pi\)
−0.155246 + 0.987876i \(0.549617\pi\)
\(158\) 2.37996i 0.189339i
\(159\) 0 0
\(160\) 0.968524i 0.0765685i
\(161\) −3.04461 −0.239949
\(162\) 0 0
\(163\) −15.8831 −1.24406 −0.622032 0.782992i \(-0.713693\pi\)
−0.622032 + 0.782992i \(0.713693\pi\)
\(164\) −0.349474 −0.0272893
\(165\) 0 0
\(166\) 4.79593 0.372236
\(167\) −3.23024 −0.249963 −0.124982 0.992159i \(-0.539887\pi\)
−0.124982 + 0.992159i \(0.539887\pi\)
\(168\) 0 0
\(169\) −27.4828 −2.11406
\(170\) 7.43064i 0.569904i
\(171\) 0 0
\(172\) 0.794780i 0.0606014i
\(173\) 6.30573 0.479416 0.239708 0.970845i \(-0.422948\pi\)
0.239708 + 0.970845i \(0.422948\pi\)
\(174\) 0 0
\(175\) 4.06196i 0.307055i
\(176\) −3.27778 0.506135i −0.247072 0.0381514i
\(177\) 0 0
\(178\) 7.23148i 0.542023i
\(179\) 14.7744i 1.10429i −0.833749 0.552144i \(-0.813810\pi\)
0.833749 0.552144i \(-0.186190\pi\)
\(180\) 0 0
\(181\) −6.51181 −0.484019 −0.242009 0.970274i \(-0.577807\pi\)
−0.242009 + 0.970274i \(0.577807\pi\)
\(182\) −6.36261 −0.471628
\(183\) 0 0
\(184\) 3.04461i 0.224452i
\(185\) 4.03132i 0.296389i
\(186\) 0 0
\(187\) −25.1475 3.88314i −1.83897 0.283963i
\(188\) 8.19535i 0.597707i
\(189\) 0 0
\(190\) 0.668741 0.0485156
\(191\) 18.2020i 1.31705i −0.752559 0.658525i \(-0.771181\pi\)
0.752559 0.658525i \(-0.228819\pi\)
\(192\) 0 0
\(193\) 5.63864i 0.405878i 0.979191 + 0.202939i \(0.0650494\pi\)
−0.979191 + 0.202939i \(0.934951\pi\)
\(194\) −2.78871 −0.200217
\(195\) 0 0
\(196\) −1.00000 −0.0714286
\(197\) −10.6097 −0.755912 −0.377956 0.925824i \(-0.623373\pi\)
−0.377956 + 0.925824i \(0.623373\pi\)
\(198\) 0 0
\(199\) 16.4026 1.16275 0.581374 0.813636i \(-0.302515\pi\)
0.581374 + 0.813636i \(0.302515\pi\)
\(200\) 4.06196 0.287224
\(201\) 0 0
\(202\) −15.9272 −1.12063
\(203\) 9.36769i 0.657483i
\(204\) 0 0
\(205\) 0.338474i 0.0236400i
\(206\) −7.44969 −0.519044
\(207\) 0 0
\(208\) 6.36261i 0.441167i
\(209\) 0.349474 2.26322i 0.0241736 0.156550i
\(210\) 0 0
\(211\) 27.0878i 1.86480i −0.361425 0.932401i \(-0.617710\pi\)
0.361425 0.932401i \(-0.382290\pi\)
\(212\) 7.24488i 0.497580i
\(213\) 0 0
\(214\) 3.91251 0.267453
\(215\) −0.769764 −0.0524974
\(216\) 0 0
\(217\) 4.20608i 0.285527i
\(218\) 15.5606i 1.05390i
\(219\) 0 0
\(220\) −0.490204 + 3.17461i −0.0330495 + 0.214032i
\(221\) 48.8147i 3.28364i
\(222\) 0 0
\(223\) −0.796023 −0.0533057 −0.0266528 0.999645i \(-0.508485\pi\)
−0.0266528 + 0.999645i \(0.508485\pi\)
\(224\) 1.00000i 0.0668153i
\(225\) 0 0
\(226\) 14.2751i 0.949566i
\(227\) −22.1143 −1.46778 −0.733890 0.679268i \(-0.762297\pi\)
−0.733890 + 0.679268i \(0.762297\pi\)
\(228\) 0 0
\(229\) 26.8581 1.77483 0.887416 0.460969i \(-0.152498\pi\)
0.887416 + 0.460969i \(0.152498\pi\)
\(230\) 2.94878 0.194437
\(231\) 0 0
\(232\) 9.36769 0.615019
\(233\) −12.4954 −0.818604 −0.409302 0.912399i \(-0.634228\pi\)
−0.409302 + 0.912399i \(0.634228\pi\)
\(234\) 0 0
\(235\) −7.93739 −0.517778
\(236\) 12.0994i 0.787603i
\(237\) 0 0
\(238\) 7.67213i 0.497310i
\(239\) 11.4659 0.741669 0.370834 0.928699i \(-0.379072\pi\)
0.370834 + 0.928699i \(0.379072\pi\)
\(240\) 0 0
\(241\) 8.26479i 0.532382i −0.963920 0.266191i \(-0.914235\pi\)
0.963920 0.266191i \(-0.0857651\pi\)
\(242\) 10.4877 + 3.31800i 0.674172 + 0.213289i
\(243\) 0 0
\(244\) 8.29965i 0.531331i
\(245\) 0.968524i 0.0618767i
\(246\) 0 0
\(247\) −4.39322 −0.279534
\(248\) 4.20608 0.267086
\(249\) 0 0
\(250\) 8.77673i 0.555089i
\(251\) 1.83153i 0.115605i −0.998328 0.0578024i \(-0.981591\pi\)
0.998328 0.0578024i \(-0.0184093\pi\)
\(252\) 0 0
\(253\) 1.54098 9.97955i 0.0968808 0.627409i
\(254\) 19.2658i 1.20884i
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 8.02981i 0.500886i 0.968131 + 0.250443i \(0.0805763\pi\)
−0.968131 + 0.250443i \(0.919424\pi\)
\(258\) 0 0
\(259\) 4.16234i 0.258635i
\(260\) 6.16234 0.382172
\(261\) 0 0
\(262\) 18.7832 1.16043
\(263\) −17.5901 −1.08465 −0.542327 0.840167i \(-0.682457\pi\)
−0.542327 + 0.840167i \(0.682457\pi\)
\(264\) 0 0
\(265\) 7.01684 0.431041
\(266\) −0.690475 −0.0423357
\(267\) 0 0
\(268\) −12.2823 −0.750261
\(269\) 7.84431i 0.478276i −0.970986 0.239138i \(-0.923135\pi\)
0.970986 0.239138i \(-0.0768649\pi\)
\(270\) 0 0
\(271\) 18.4223i 1.11908i −0.828804 0.559538i \(-0.810978\pi\)
0.828804 0.559538i \(-0.189022\pi\)
\(272\) 7.67213 0.465191
\(273\) 0 0
\(274\) 12.8552i 0.776612i
\(275\) −13.3142 2.05590i −0.802877 0.123976i
\(276\) 0 0
\(277\) 28.2859i 1.69953i −0.527160 0.849766i \(-0.676743\pi\)
0.527160 0.849766i \(-0.323257\pi\)
\(278\) 12.8642i 0.771546i
\(279\) 0 0
\(280\) 0.968524 0.0578804
\(281\) 8.83955 0.527323 0.263662 0.964615i \(-0.415070\pi\)
0.263662 + 0.964615i \(0.415070\pi\)
\(282\) 0 0
\(283\) 12.9517i 0.769901i −0.922937 0.384951i \(-0.874218\pi\)
0.922937 0.384951i \(-0.125782\pi\)
\(284\) 4.98166i 0.295607i
\(285\) 0 0
\(286\) 3.22034 20.8552i 0.190423 1.23319i
\(287\) 0.349474i 0.0206288i
\(288\) 0 0
\(289\) 41.8616 2.46245
\(290\) 9.07283i 0.532775i
\(291\) 0 0
\(292\) 9.97054i 0.583482i
\(293\) −19.8507 −1.15969 −0.579846 0.814726i \(-0.696887\pi\)
−0.579846 + 0.814726i \(0.696887\pi\)
\(294\) 0 0
\(295\) −11.7185 −0.682280
\(296\) −4.16234 −0.241931
\(297\) 0 0
\(298\) −9.48765 −0.549605
\(299\) −19.3716 −1.12029
\(300\) 0 0
\(301\) 0.794780 0.0458104
\(302\) 10.8618i 0.625028i
\(303\) 0 0
\(304\) 0.690475i 0.0396015i
\(305\) −8.03841 −0.460278
\(306\) 0 0
\(307\) 26.9138i 1.53605i 0.640419 + 0.768025i \(0.278761\pi\)
−0.640419 + 0.768025i \(0.721239\pi\)
\(308\) 0.506135 3.27778i 0.0288397 0.186769i
\(309\) 0 0
\(310\) 4.07369i 0.231370i
\(311\) 13.4037i 0.760052i −0.924976 0.380026i \(-0.875915\pi\)
0.924976 0.380026i \(-0.124085\pi\)
\(312\) 0 0
\(313\) 5.88512 0.332647 0.166323 0.986071i \(-0.446810\pi\)
0.166323 + 0.986071i \(0.446810\pi\)
\(314\) −3.89046 −0.219551
\(315\) 0 0
\(316\) 2.37996i 0.133883i
\(317\) 19.3875i 1.08891i −0.838790 0.544454i \(-0.816737\pi\)
0.838790 0.544454i \(-0.183263\pi\)
\(318\) 0 0
\(319\) −30.7052 4.74132i −1.71916 0.265463i
\(320\) 0.968524i 0.0541421i
\(321\) 0 0
\(322\) −3.04461 −0.169669
\(323\) 5.29741i 0.294756i
\(324\) 0 0
\(325\) 25.8447i 1.43360i
\(326\) −15.8831 −0.879686
\(327\) 0 0
\(328\) −0.349474 −0.0192965
\(329\) 8.19535 0.451824
\(330\) 0 0
\(331\) 23.2283 1.27674 0.638372 0.769728i \(-0.279608\pi\)
0.638372 + 0.769728i \(0.279608\pi\)
\(332\) 4.79593 0.263211
\(333\) 0 0
\(334\) −3.23024 −0.176751
\(335\) 11.8957i 0.649932i
\(336\) 0 0
\(337\) 22.5217i 1.22684i −0.789758 0.613418i \(-0.789794\pi\)
0.789758 0.613418i \(-0.210206\pi\)
\(338\) −27.4828 −1.49486
\(339\) 0 0
\(340\) 7.43064i 0.402983i
\(341\) −13.7866 2.12885i −0.746586 0.115283i
\(342\) 0 0
\(343\) 1.00000i 0.0539949i
\(344\) 0.794780i 0.0428517i
\(345\) 0 0
\(346\) 6.30573 0.338998
\(347\) −5.02864 −0.269951 −0.134976 0.990849i \(-0.543096\pi\)
−0.134976 + 0.990849i \(0.543096\pi\)
\(348\) 0 0
\(349\) 6.62260i 0.354500i −0.984166 0.177250i \(-0.943280\pi\)
0.984166 0.177250i \(-0.0567201\pi\)
\(350\) 4.06196i 0.217121i
\(351\) 0 0
\(352\) −3.27778 0.506135i −0.174706 0.0269771i
\(353\) 26.9128i 1.43243i −0.697882 0.716213i \(-0.745874\pi\)
0.697882 0.716213i \(-0.254126\pi\)
\(354\) 0 0
\(355\) 4.82485 0.256077
\(356\) 7.23148i 0.383268i
\(357\) 0 0
\(358\) 14.7744i 0.780849i
\(359\) 0.126745 0.00668936 0.00334468 0.999994i \(-0.498935\pi\)
0.00334468 + 0.999994i \(0.498935\pi\)
\(360\) 0 0
\(361\) 18.5232 0.974908
\(362\) −6.51181 −0.342253
\(363\) 0 0
\(364\) −6.36261 −0.333491
\(365\) 9.65670 0.505455
\(366\) 0 0
\(367\) 22.5235 1.17572 0.587859 0.808963i \(-0.299971\pi\)
0.587859 + 0.808963i \(0.299971\pi\)
\(368\) 3.04461i 0.158711i
\(369\) 0 0
\(370\) 4.03132i 0.209578i
\(371\) −7.24488 −0.376135
\(372\) 0 0
\(373\) 13.0593i 0.676185i −0.941113 0.338093i \(-0.890218\pi\)
0.941113 0.338093i \(-0.109782\pi\)
\(374\) −25.1475 3.88314i −1.30035 0.200792i
\(375\) 0 0
\(376\) 8.19535i 0.422643i
\(377\) 59.6029i 3.06971i
\(378\) 0 0
\(379\) −30.8782 −1.58611 −0.793054 0.609151i \(-0.791510\pi\)
−0.793054 + 0.609151i \(0.791510\pi\)
\(380\) 0.668741 0.0343057
\(381\) 0 0
\(382\) 18.2020i 0.931295i
\(383\) 14.8598i 0.759299i 0.925130 + 0.379649i \(0.123955\pi\)
−0.925130 + 0.379649i \(0.876045\pi\)
\(384\) 0 0
\(385\) −3.17461 0.490204i −0.161793 0.0249831i
\(386\) 5.63864i 0.286999i
\(387\) 0 0
\(388\) −2.78871 −0.141575
\(389\) 27.6308i 1.40094i 0.713684 + 0.700468i \(0.247025\pi\)
−0.713684 + 0.700468i \(0.752975\pi\)
\(390\) 0 0
\(391\) 23.3586i 1.18130i
\(392\) −1.00000 −0.0505076
\(393\) 0 0
\(394\) −10.6097 −0.534510
\(395\) 2.30505 0.115980
\(396\) 0 0
\(397\) 15.9319 0.799597 0.399798 0.916603i \(-0.369080\pi\)
0.399798 + 0.916603i \(0.369080\pi\)
\(398\) 16.4026 0.822187
\(399\) 0 0
\(400\) 4.06196 0.203098
\(401\) 17.3215i 0.864995i −0.901635 0.432498i \(-0.857632\pi\)
0.901635 0.432498i \(-0.142368\pi\)
\(402\) 0 0
\(403\) 26.7616i 1.33309i
\(404\) −15.9272 −0.792408
\(405\) 0 0
\(406\) 9.36769i 0.464911i
\(407\) 13.6432 + 2.10670i 0.676269 + 0.104425i
\(408\) 0 0
\(409\) 5.23472i 0.258840i 0.991590 + 0.129420i \(0.0413116\pi\)
−0.991590 + 0.129420i \(0.958688\pi\)
\(410\) 0.338474i 0.0167160i
\(411\) 0 0
\(412\) −7.44969 −0.367020
\(413\) 12.0994 0.595372
\(414\) 0 0
\(415\) 4.64497i 0.228013i
\(416\) 6.36261i 0.311952i
\(417\) 0 0
\(418\) 0.349474 2.26322i 0.0170933 0.110698i
\(419\) 22.0977i 1.07954i 0.841812 + 0.539771i \(0.181489\pi\)
−0.841812 + 0.539771i \(0.818511\pi\)
\(420\) 0 0
\(421\) −3.16019 −0.154018 −0.0770092 0.997030i \(-0.524537\pi\)
−0.0770092 + 0.997030i \(0.524537\pi\)
\(422\) 27.0878i 1.31861i
\(423\) 0 0
\(424\) 7.24488i 0.351842i
\(425\) 31.1639 1.51167
\(426\) 0 0
\(427\) 8.29965 0.401648
\(428\) 3.91251 0.189118
\(429\) 0 0
\(430\) −0.769764 −0.0371213
\(431\) −30.7860 −1.48291 −0.741454 0.671003i \(-0.765864\pi\)
−0.741454 + 0.671003i \(0.765864\pi\)
\(432\) 0 0
\(433\) −16.6539 −0.800333 −0.400167 0.916442i \(-0.631048\pi\)
−0.400167 + 0.916442i \(0.631048\pi\)
\(434\) 4.20608i 0.201898i
\(435\) 0 0
\(436\) 15.5606i 0.745219i
\(437\) −2.10223 −0.100563
\(438\) 0 0
\(439\) 31.6856i 1.51227i −0.654415 0.756136i \(-0.727085\pi\)
0.654415 0.756136i \(-0.272915\pi\)
\(440\) −0.490204 + 3.17461i −0.0233696 + 0.151343i
\(441\) 0 0
\(442\) 48.8147i 2.32188i
\(443\) 11.0473i 0.524874i −0.964949 0.262437i \(-0.915474\pi\)
0.964949 0.262437i \(-0.0845262\pi\)
\(444\) 0 0
\(445\) −7.00386 −0.332015
\(446\) −0.796023 −0.0376928
\(447\) 0 0
\(448\) 1.00000i 0.0472456i
\(449\) 22.1346i 1.04459i 0.852763 + 0.522297i \(0.174925\pi\)
−0.852763 + 0.522297i \(0.825075\pi\)
\(450\) 0 0
\(451\) 1.14550 + 0.176881i 0.0539394 + 0.00832900i
\(452\) 14.2751i 0.671445i
\(453\) 0 0
\(454\) −22.1143 −1.03788
\(455\) 6.16234i 0.288895i
\(456\) 0 0
\(457\) 26.2665i 1.22870i −0.789035 0.614348i \(-0.789419\pi\)
0.789035 0.614348i \(-0.210581\pi\)
\(458\) 26.8581 1.25500
\(459\) 0 0
\(460\) 2.94878 0.137487
\(461\) 17.7454 0.826486 0.413243 0.910621i \(-0.364396\pi\)
0.413243 + 0.910621i \(0.364396\pi\)
\(462\) 0 0
\(463\) 23.8222 1.10711 0.553556 0.832812i \(-0.313270\pi\)
0.553556 + 0.832812i \(0.313270\pi\)
\(464\) 9.36769 0.434884
\(465\) 0 0
\(466\) −12.4954 −0.578840
\(467\) 11.2649i 0.521279i −0.965436 0.260640i \(-0.916067\pi\)
0.965436 0.260640i \(-0.0839335\pi\)
\(468\) 0 0
\(469\) 12.2823i 0.567144i
\(470\) −7.93739 −0.366124
\(471\) 0 0
\(472\) 12.0994i 0.556919i
\(473\) −0.402266 + 2.60511i −0.0184962 + 0.119783i
\(474\) 0 0
\(475\) 2.80468i 0.128688i
\(476\) 7.67213i 0.351652i
\(477\) 0 0
\(478\) 11.4659 0.524439
\(479\) −9.58585 −0.437989 −0.218994 0.975726i \(-0.570278\pi\)
−0.218994 + 0.975726i \(0.570278\pi\)
\(480\) 0 0
\(481\) 26.4833i 1.20753i
\(482\) 8.26479i 0.376451i
\(483\) 0 0
\(484\) 10.4877 + 3.31800i 0.476712 + 0.150818i
\(485\) 2.70093i 0.122643i
\(486\) 0 0
\(487\) 6.21142 0.281466 0.140733 0.990048i \(-0.455054\pi\)
0.140733 + 0.990048i \(0.455054\pi\)
\(488\) 8.29965i 0.375708i
\(489\) 0 0
\(490\) 0.968524i 0.0437534i
\(491\) −9.06426 −0.409064 −0.204532 0.978860i \(-0.565567\pi\)
−0.204532 + 0.978860i \(0.565567\pi\)
\(492\) 0 0
\(493\) 71.8701 3.23687
\(494\) −4.39322 −0.197660
\(495\) 0 0
\(496\) 4.20608 0.188859
\(497\) −4.98166 −0.223458
\(498\) 0 0
\(499\) −40.7758 −1.82537 −0.912687 0.408659i \(-0.865996\pi\)
−0.912687 + 0.408659i \(0.865996\pi\)
\(500\) 8.77673i 0.392507i
\(501\) 0 0
\(502\) 1.83153i 0.0817450i
\(503\) −36.2530 −1.61644 −0.808220 0.588881i \(-0.799569\pi\)
−0.808220 + 0.588881i \(0.799569\pi\)
\(504\) 0 0
\(505\) 15.4259i 0.686442i
\(506\) 1.54098 9.97955i 0.0685051 0.443645i
\(507\) 0 0
\(508\) 19.2658i 0.854781i
\(509\) 11.3348i 0.502407i −0.967934 0.251203i \(-0.919174\pi\)
0.967934 0.251203i \(-0.0808263\pi\)
\(510\) 0 0
\(511\) −9.97054 −0.441071
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) 8.02981i 0.354180i
\(515\) 7.21520i 0.317940i
\(516\) 0 0
\(517\) −4.14795 + 26.8625i −0.182427 + 1.18141i
\(518\) 4.16234i 0.182883i
\(519\) 0 0
\(520\) 6.16234 0.270236
\(521\) 21.7715i 0.953826i −0.878951 0.476913i \(-0.841756\pi\)
0.878951 0.476913i \(-0.158244\pi\)
\(522\) 0 0
\(523\) 7.49377i 0.327680i 0.986487 + 0.163840i \(0.0523881\pi\)
−0.986487 + 0.163840i \(0.947612\pi\)
\(524\) 18.7832 0.820550
\(525\) 0 0
\(526\) −17.5901 −0.766966
\(527\) 32.2696 1.40569
\(528\) 0 0
\(529\) 13.7304 0.596972
\(530\) 7.01684 0.304792
\(531\) 0 0
\(532\) −0.690475 −0.0299359
\(533\) 2.22356i 0.0963132i
\(534\) 0 0
\(535\) 3.78936i 0.163828i
\(536\) −12.2823 −0.530515
\(537\) 0 0
\(538\) 7.84431i 0.338192i
\(539\) 3.27778 + 0.506135i 0.141184 + 0.0218008i
\(540\) 0 0
\(541\) 9.33253i 0.401237i −0.979669 0.200618i \(-0.935705\pi\)
0.979669 0.200618i \(-0.0642951\pi\)
\(542\) 18.4223i 0.791307i
\(543\) 0 0
\(544\) 7.67213 0.328940
\(545\) 15.0708 0.645564
\(546\) 0 0
\(547\) 22.7544i 0.972907i 0.873706 + 0.486454i \(0.161710\pi\)
−0.873706 + 0.486454i \(0.838290\pi\)
\(548\) 12.8552i 0.549147i
\(549\) 0 0
\(550\) −13.3142 2.05590i −0.567720 0.0876639i
\(551\) 6.46815i 0.275553i
\(552\) 0 0
\(553\) −2.37996 −0.101206
\(554\) 28.2859i 1.20175i
\(555\) 0 0
\(556\) 12.8642i 0.545566i
\(557\) 25.3558 1.07436 0.537180 0.843468i \(-0.319490\pi\)
0.537180 + 0.843468i \(0.319490\pi\)
\(558\) 0 0
\(559\) 5.05687 0.213883
\(560\) 0.968524 0.0409276
\(561\) 0 0
\(562\) 8.83955 0.372874
\(563\) 22.1493 0.933483 0.466742 0.884394i \(-0.345428\pi\)
0.466742 + 0.884394i \(0.345428\pi\)
\(564\) 0 0
\(565\) 13.8258 0.581655
\(566\) 12.9517i 0.544402i
\(567\) 0 0
\(568\) 4.98166i 0.209026i
\(569\) −14.6605 −0.614602 −0.307301 0.951612i \(-0.599426\pi\)
−0.307301 + 0.951612i \(0.599426\pi\)
\(570\) 0 0
\(571\) 14.4233i 0.603597i 0.953372 + 0.301798i \(0.0975869\pi\)
−0.953372 + 0.301798i \(0.902413\pi\)
\(572\) 3.22034 20.8552i 0.134649 0.872000i
\(573\) 0 0
\(574\) 0.349474i 0.0145867i
\(575\) 12.3671i 0.515743i
\(576\) 0 0
\(577\) −0.249866 −0.0104020 −0.00520102 0.999986i \(-0.501656\pi\)
−0.00520102 + 0.999986i \(0.501656\pi\)
\(578\) 41.8616 1.74121
\(579\) 0 0
\(580\) 9.07283i 0.376729i
\(581\) 4.79593i 0.198969i
\(582\) 0 0
\(583\) 3.66689 23.7471i 0.151867 0.983505i
\(584\) 9.97054i 0.412584i
\(585\) 0 0
\(586\) −19.8507 −0.820026
\(587\) 41.6743i 1.72008i −0.510224 0.860041i \(-0.670438\pi\)
0.510224 0.860041i \(-0.329562\pi\)
\(588\) 0 0
\(589\) 2.90419i 0.119665i
\(590\) −11.7185 −0.482445
\(591\) 0 0
\(592\) −4.16234 −0.171071
\(593\) 25.5097 1.04756 0.523778 0.851855i \(-0.324522\pi\)
0.523778 + 0.851855i \(0.324522\pi\)
\(594\) 0 0
\(595\) 7.43064 0.304627
\(596\) −9.48765 −0.388629
\(597\) 0 0
\(598\) −19.3716 −0.792166
\(599\) 8.01287i 0.327397i 0.986510 + 0.163698i \(0.0523424\pi\)
−0.986510 + 0.163698i \(0.947658\pi\)
\(600\) 0 0
\(601\) 8.16202i 0.332936i 0.986047 + 0.166468i \(0.0532362\pi\)
−0.986047 + 0.166468i \(0.946764\pi\)
\(602\) 0.794780 0.0323928
\(603\) 0 0
\(604\) 10.8618i 0.441961i
\(605\) 3.21356 10.1575i 0.130650 0.412963i
\(606\) 0 0
\(607\) 24.6663i 1.00117i 0.865686 + 0.500587i \(0.166882\pi\)
−0.865686 + 0.500587i \(0.833118\pi\)
\(608\) 0.690475i 0.0280025i
\(609\) 0 0
\(610\) −8.03841 −0.325466
\(611\) 52.1438 2.10951
\(612\) 0 0
\(613\) 44.4293i 1.79448i 0.441541 + 0.897241i \(0.354432\pi\)
−0.441541 + 0.897241i \(0.645568\pi\)
\(614\) 26.9138i 1.08615i
\(615\) 0 0
\(616\) 0.506135 3.27778i 0.0203928 0.132065i
\(617\) 28.8338i 1.16080i 0.814330 + 0.580402i \(0.197104\pi\)
−0.814330 + 0.580402i \(0.802896\pi\)
\(618\) 0 0
\(619\) −30.6095 −1.23030 −0.615149 0.788411i \(-0.710904\pi\)
−0.615149 + 0.788411i \(0.710904\pi\)
\(620\) 4.07369i 0.163603i
\(621\) 0 0
\(622\) 13.4037i 0.537438i
\(623\) 7.23148 0.289723
\(624\) 0 0
\(625\) 11.8093 0.472374
\(626\) 5.88512 0.235217
\(627\) 0 0
\(628\) −3.89046 −0.155246
\(629\) −31.9340 −1.27329
\(630\) 0 0
\(631\) −16.7730 −0.667722 −0.333861 0.942622i \(-0.608352\pi\)
−0.333861 + 0.942622i \(0.608352\pi\)
\(632\) 2.37996i 0.0946697i
\(633\) 0 0
\(634\) 19.3875i 0.769975i
\(635\) 18.6594 0.740474
\(636\) 0 0
\(637\) 6.36261i 0.252096i
\(638\) −30.7052 4.74132i −1.21563 0.187711i
\(639\) 0 0
\(640\) 0.968524i 0.0382843i
\(641\) 8.74257i 0.345311i 0.984982 + 0.172655i \(0.0552347\pi\)
−0.984982 + 0.172655i \(0.944765\pi\)
\(642\) 0 0
\(643\) −18.1706 −0.716577 −0.358289 0.933611i \(-0.616640\pi\)
−0.358289 + 0.933611i \(0.616640\pi\)
\(644\) −3.04461 −0.119974
\(645\) 0 0
\(646\) 5.29741i 0.208424i
\(647\) 20.7346i 0.815163i −0.913169 0.407581i \(-0.866372\pi\)
0.913169 0.407581i \(-0.133628\pi\)
\(648\) 0 0
\(649\) −6.12392 + 39.6591i −0.240385 + 1.55676i
\(650\) 25.8447i 1.01371i
\(651\) 0 0
\(652\) −15.8831 −0.622032
\(653\) 3.99386i 0.156292i 0.996942 + 0.0781459i \(0.0249000\pi\)
−0.996942 + 0.0781459i \(0.975100\pi\)
\(654\) 0 0
\(655\) 18.1920i 0.710821i
\(656\) −0.349474 −0.0136447
\(657\) 0 0
\(658\) 8.19535 0.319488
\(659\) −5.75013 −0.223993 −0.111997 0.993709i \(-0.535725\pi\)
−0.111997 + 0.993709i \(0.535725\pi\)
\(660\) 0 0
\(661\) 42.5089 1.65341 0.826703 0.562639i \(-0.190214\pi\)
0.826703 + 0.562639i \(0.190214\pi\)
\(662\) 23.2283 0.902794
\(663\) 0 0
\(664\) 4.79593 0.186118
\(665\) 0.668741i 0.0259327i
\(666\) 0 0
\(667\) 28.5209i 1.10434i
\(668\) −3.23024 −0.124982
\(669\) 0 0
\(670\) 11.8957i 0.459571i
\(671\) −4.20075 + 27.2044i −0.162168 + 1.05022i
\(672\) 0 0
\(673\) 1.45886i 0.0562348i −0.999605 0.0281174i \(-0.991049\pi\)
0.999605 0.0281174i \(-0.00895123\pi\)
\(674\) 22.5217i 0.867504i
\(675\) 0 0
\(676\) −27.4828 −1.05703
\(677\) −3.94184 −0.151497 −0.0757485 0.997127i \(-0.524135\pi\)
−0.0757485 + 0.997127i \(0.524135\pi\)
\(678\) 0 0
\(679\) 2.78871i 0.107021i
\(680\) 7.43064i 0.284952i
\(681\) 0 0
\(682\) −13.7866 2.12885i −0.527916 0.0815177i
\(683\) 2.11146i 0.0807928i 0.999184 + 0.0403964i \(0.0128621\pi\)
−0.999184 + 0.0403964i \(0.987138\pi\)
\(684\) 0 0
\(685\) −12.4506 −0.475712
\(686\) 1.00000i 0.0381802i
\(687\) 0 0
\(688\) 0.794780i 0.0303007i
\(689\) −46.0963 −1.75613
\(690\) 0 0
\(691\) −15.1455 −0.576162 −0.288081 0.957606i \(-0.593017\pi\)
−0.288081 + 0.957606i \(0.593017\pi\)
\(692\) 6.30573 0.239708
\(693\) 0 0
\(694\) −5.02864 −0.190884
\(695\) −12.4593 −0.472609
\(696\) 0 0
\(697\) −2.68121 −0.101558
\(698\) 6.62260i 0.250669i
\(699\) 0 0
\(700\) 4.06196i 0.153528i
\(701\) 37.1475 1.40304 0.701521 0.712649i \(-0.252505\pi\)
0.701521 + 0.712649i \(0.252505\pi\)
\(702\) 0 0
\(703\) 2.87399i 0.108395i
\(704\) −3.27778 0.506135i −0.123536 0.0190757i
\(705\) 0 0
\(706\) 26.9128i 1.01288i
\(707\) 15.9272i 0.599004i
\(708\) 0 0
\(709\) −35.2497 −1.32383 −0.661916 0.749578i \(-0.730256\pi\)
−0.661916 + 0.749578i \(0.730256\pi\)
\(710\) 4.82485 0.181073
\(711\) 0 0
\(712\) 7.23148i 0.271011i
\(713\) 12.8059i 0.479584i
\(714\) 0 0
\(715\) −20.1988 3.11897i −0.755391 0.116643i
\(716\) 14.7744i 0.552144i
\(717\) 0 0
\(718\) 0.126745 0.00473009
\(719\) 43.0144i 1.60417i −0.597211 0.802084i \(-0.703725\pi\)
0.597211 0.802084i \(-0.296275\pi\)
\(720\) 0 0
\(721\) 7.44969i 0.277441i
\(722\) 18.5232 0.689364
\(723\) 0 0
\(724\) −6.51181 −0.242009
\(725\) 38.0512 1.41319
\(726\) 0 0
\(727\) 18.2866 0.678211 0.339106 0.940748i \(-0.389876\pi\)
0.339106 + 0.940748i \(0.389876\pi\)
\(728\) −6.36261 −0.235814
\(729\) 0 0
\(730\) 9.65670 0.357411
\(731\) 6.09766i 0.225530i
\(732\) 0 0
\(733\) 7.08135i 0.261556i −0.991412 0.130778i \(-0.958253\pi\)
0.991412 0.130778i \(-0.0417474\pi\)
\(734\) 22.5235 0.831358
\(735\) 0 0
\(736\) 3.04461i 0.112226i
\(737\) 40.2587 + 6.21650i 1.48295 + 0.228988i
\(738\) 0 0
\(739\) 35.2950i 1.29835i 0.760640 + 0.649174i \(0.224885\pi\)
−0.760640 + 0.649174i \(0.775115\pi\)
\(740\) 4.03132i 0.148194i
\(741\) 0 0
\(742\) −7.24488 −0.265968
\(743\) 1.16019 0.0425633 0.0212817 0.999774i \(-0.493225\pi\)
0.0212817 + 0.999774i \(0.493225\pi\)
\(744\) 0 0
\(745\) 9.18902i 0.336660i
\(746\) 13.0593i 0.478135i
\(747\) 0 0
\(748\) −25.1475 3.88314i −0.919485 0.141982i
\(749\) 3.91251i 0.142960i
\(750\) 0 0
\(751\) 21.2096 0.773950 0.386975 0.922090i \(-0.373520\pi\)
0.386975 + 0.922090i \(0.373520\pi\)
\(752\) 8.19535i 0.298854i
\(753\) 0 0
\(754\) 59.6029i 2.17061i
\(755\) 10.5199 0.382860
\(756\) 0 0
\(757\) 22.5045 0.817938 0.408969 0.912548i \(-0.365888\pi\)
0.408969 + 0.912548i \(0.365888\pi\)
\(758\) −30.8782 −1.12155
\(759\) 0 0
\(760\) 0.668741 0.0242578
\(761\) −46.3962 −1.68186 −0.840930 0.541144i \(-0.817992\pi\)
−0.840930 + 0.541144i \(0.817992\pi\)
\(762\) 0 0
\(763\) −15.5606 −0.563333
\(764\) 18.2020i 0.658525i
\(765\) 0 0
\(766\) 14.8598i 0.536905i
\(767\) 76.9836 2.77972
\(768\) 0 0
\(769\) 7.34810i 0.264979i −0.991184 0.132490i \(-0.957703\pi\)
0.991184 0.132490i \(-0.0422971\pi\)
\(770\) −3.17461 0.490204i −0.114405 0.0176657i
\(771\) 0 0
\(772\) 5.63864i 0.202939i
\(773\) 4.73217i 0.170204i 0.996372 + 0.0851021i \(0.0271217\pi\)
−0.996372 + 0.0851021i \(0.972878\pi\)
\(774\) 0 0
\(775\) 17.0849 0.613709
\(776\) −2.78871 −0.100109
\(777\) 0 0
\(778\) 27.6308i 0.990612i
\(779\) 0.241303i 0.00864557i
\(780\) 0 0
\(781\) 2.52139 16.3288i 0.0902225 0.584289i
\(782\) 23.3586i 0.835303i
\(783\) 0 0
\(784\) −1.00000 −0.0357143
\(785\) 3.76800i 0.134486i
\(786\) 0 0
\(787\) 4.64274i 0.165496i 0.996571 + 0.0827480i \(0.0263696\pi\)
−0.996571 + 0.0827480i \(0.973630\pi\)
\(788\) −10.6097 −0.377956
\(789\) 0 0
\(790\) 2.30505 0.0820099
\(791\) −14.2751 −0.507565
\(792\) 0 0
\(793\) 52.8074 1.87525
\(794\) 15.9319 0.565400
\(795\) 0 0
\(796\) 16.4026 0.581374
\(797\) 10.2652i 0.363613i −0.983334 0.181807i \(-0.941805\pi\)
0.983334 0.181807i \(-0.0581945\pi\)
\(798\) 0 0
\(799\) 62.8758i 2.22439i
\(800\) 4.06196 0.143612
\(801\) 0 0
\(802\) 17.3215i 0.611644i
\(803\) 5.04644 32.6812i 0.178085 1.15329i
\(804\) 0 0
\(805\) 2.94878i 0.103931i
\(806\) 26.7616i 0.942639i
\(807\) 0 0
\(808\) −15.9272 −0.560317
\(809\) −18.8569 −0.662972 −0.331486 0.943460i \(-0.607550\pi\)
−0.331486 + 0.943460i \(0.607550\pi\)
\(810\) 0 0
\(811\) 47.0613i 1.65255i 0.563270 + 0.826273i \(0.309543\pi\)
−0.563270 + 0.826273i \(0.690457\pi\)
\(812\) 9.36769i 0.328741i
\(813\) 0 0
\(814\) 13.6432 + 2.10670i 0.478194 + 0.0738399i
\(815\) 15.3832i 0.538850i
\(816\) 0 0
\(817\) 0.548776 0.0191992
\(818\) 5.23472i 0.183028i
\(819\) 0 0
\(820\) 0.338474i 0.0118200i
\(821\) −22.6115 −0.789145 −0.394573 0.918865i \(-0.629107\pi\)
−0.394573 + 0.918865i \(0.629107\pi\)
\(822\) 0 0
\(823\) 17.1001 0.596073 0.298036 0.954555i \(-0.403668\pi\)
0.298036 + 0.954555i \(0.403668\pi\)
\(824\) −7.44969 −0.259522
\(825\) 0 0
\(826\) 12.0994 0.420991
\(827\) −0.366688 −0.0127510 −0.00637549 0.999980i \(-0.502029\pi\)
−0.00637549 + 0.999980i \(0.502029\pi\)
\(828\) 0 0
\(829\) −7.24917 −0.251774 −0.125887 0.992045i \(-0.540178\pi\)
−0.125887 + 0.992045i \(0.540178\pi\)
\(830\) 4.64497i 0.161229i
\(831\) 0 0
\(832\) 6.36261i 0.220584i
\(833\) −7.67213 −0.265824
\(834\) 0 0
\(835\) 3.12856i 0.108268i
\(836\) 0.349474 2.26322i 0.0120868 0.0782752i
\(837\) 0 0
\(838\) 22.0977i 0.763351i
\(839\) 16.1924i 0.559022i −0.960142 0.279511i \(-0.909828\pi\)
0.960142 0.279511i \(-0.0901724\pi\)
\(840\) 0 0
\(841\) 58.7536 2.02599
\(842\) −3.16019 −0.108907
\(843\) 0 0
\(844\) 27.0878i 0.932401i
\(845\) 26.6177i 0.915677i
\(846\) 0 0
\(847\) −3.31800 + 10.4877i −0.114008 + 0.360360i
\(848\) 7.24488i 0.248790i
\(849\) 0 0
\(850\) 31.1639 1.06891
\(851\) 12.6727i 0.434414i
\(852\) 0 0
\(853\) 19.4113i 0.664629i −0.943169 0.332315i \(-0.892170\pi\)
0.943169 0.332315i \(-0.107830\pi\)
\(854\) 8.29965 0.284008
\(855\) 0 0
\(856\) 3.91251 0.133727
\(857\) 15.2405 0.520607 0.260303 0.965527i \(-0.416177\pi\)
0.260303 + 0.965527i \(0.416177\pi\)
\(858\) 0 0
\(859\) 15.0502 0.513506 0.256753 0.966477i \(-0.417347\pi\)
0.256753 + 0.966477i \(0.417347\pi\)
\(860\) −0.769764 −0.0262487
\(861\) 0 0
\(862\) −30.7860 −1.04857
\(863\) 10.3687i 0.352956i −0.984305 0.176478i \(-0.943530\pi\)
0.984305 0.176478i \(-0.0564705\pi\)
\(864\) 0 0
\(865\) 6.10725i 0.207653i
\(866\) −16.6539 −0.565921
\(867\) 0 0
\(868\) 4.20608i 0.142764i
\(869\) 1.20458 7.80098i 0.0408626 0.264630i
\(870\) 0 0
\(871\) 78.1474i 2.64793i
\(872\) 15.5606i 0.526950i
\(873\) 0 0
\(874\) −2.10223 −0.0711089
\(875\) 8.77673 0.296707
\(876\) 0 0
\(877\) 35.8867i 1.21181i 0.795538 + 0.605904i \(0.207189\pi\)
−0.795538 + 0.605904i \(0.792811\pi\)
\(878\) 31.6856i 1.06934i
\(879\) 0 0
\(880\) −0.490204 + 3.17461i −0.0165248 + 0.107016i
\(881\) 47.2646i 1.59238i 0.605044 + 0.796192i \(0.293156\pi\)
−0.605044 + 0.796192i \(0.706844\pi\)
\(882\) 0 0
\(883\) 24.7187 0.831849 0.415925 0.909399i \(-0.363458\pi\)
0.415925 + 0.909399i \(0.363458\pi\)
\(884\) 48.8147i 1.64182i
\(885\) 0 0
\(886\) 11.0473i 0.371142i
\(887\) 20.2723 0.680678 0.340339 0.940303i \(-0.389458\pi\)
0.340339 + 0.940303i \(0.389458\pi\)
\(888\) 0 0
\(889\) −19.2658 −0.646154
\(890\) −7.00386 −0.234770
\(891\) 0 0
\(892\) −0.796023 −0.0266528
\(893\) 5.65868 0.189361
\(894\) 0 0
\(895\) −14.3093 −0.478308
\(896\) 1.00000i 0.0334077i
\(897\) 0 0
\(898\) 22.1346i 0.738640i
\(899\) 39.4013 1.31411
\(900\) 0 0
\(901\) 55.5837i 1.85176i
\(902\) 1.14550 + 0.176881i 0.0381409 + 0.00588949i
\(903\) 0 0
\(904\) 14.2751i 0.474783i
\(905\) 6.30684i 0.209647i
\(906\) 0 0
\(907\) −20.0305 −0.665102 −0.332551 0.943085i \(-0.607909\pi\)
−0.332551 + 0.943085i \(0.607909\pi\)
\(908\) −22.1143 −0.733890
\(909\) 0 0
\(910\) 6.16234i 0.204279i
\(911\) 41.4786i 1.37425i 0.726541 + 0.687123i \(0.241127\pi\)
−0.726541 + 0.687123i \(0.758873\pi\)
\(912\) 0 0
\(913\) −15.7200 2.42739i −0.520256 0.0803349i
\(914\) 26.2665i 0.868819i
\(915\) 0 0
\(916\) 26.8581 0.887416
\(917\) 18.7832i 0.620277i
\(918\) 0 0
\(919\) 19.1193i 0.630688i −0.948977 0.315344i \(-0.897880\pi\)
0.948977 0.315344i \(-0.102120\pi\)
\(920\) 2.94878 0.0972183
\(921\) 0 0
\(922\) 17.7454 0.584414
\(923\) −31.6963 −1.04330
\(924\) 0 0
\(925\) −16.9072 −0.555907
\(926\) 23.8222 0.782847
\(927\) 0 0
\(928\) 9.36769 0.307509
\(929\) 33.3761i 1.09503i −0.836795 0.547516i \(-0.815573\pi\)
0.836795 0.547516i \(-0.184427\pi\)
\(930\) 0 0
\(931\) 0.690475i 0.0226294i
\(932\) −12.4954 −0.409302
\(933\) 0 0
\(934\) 11.2649i 0.368600i
\(935\) −3.76091 + 24.3560i −0.122995 + 0.796526i
\(936\) 0 0
\(937\) 33.6228i 1.09841i 0.835688 + 0.549204i \(0.185069\pi\)
−0.835688 + 0.549204i \(0.814931\pi\)
\(938\) 12.2823i 0.401031i
\(939\) 0 0
\(940\) −7.93739 −0.258889
\(941\) −2.54376 −0.0829241 −0.0414620 0.999140i \(-0.513202\pi\)
−0.0414620 + 0.999140i \(0.513202\pi\)
\(942\) 0 0
\(943\) 1.06401i 0.0346490i
\(944\) 12.0994i 0.393801i
\(945\) 0 0
\(946\) −0.402266 + 2.60511i −0.0130788 + 0.0846995i
\(947\) 14.4954i 0.471039i −0.971870 0.235519i \(-0.924321\pi\)
0.971870 0.235519i \(-0.0756791\pi\)
\(948\) 0 0
\(949\) −63.4386 −2.05930
\(950\) 2.80468i 0.0909959i
\(951\) 0 0
\(952\) 7.67213i 0.248655i
\(953\) −20.0850 −0.650616 −0.325308 0.945608i \(-0.605468\pi\)
−0.325308 + 0.945608i \(0.605468\pi\)
\(954\) 0 0
\(955\) −17.6291 −0.570463
\(956\) 11.4659 0.370834
\(957\) 0 0
\(958\) −9.58585 −0.309705
\(959\) 12.8552 0.415116
\(960\) 0 0
\(961\) −13.3089 −0.429319
\(962\) 26.4833i 0.853856i
\(963\) 0 0
\(964\) 8.26479i 0.266191i
\(965\) 5.46116 0.175801
\(966\) 0 0
\(967\) 34.8984i 1.12226i 0.827729 + 0.561128i \(0.189632\pi\)
−0.827729 + 0.561128i \(0.810368\pi\)
\(968\) 10.4877 + 3.31800i 0.337086 + 0.106644i
\(969\) 0 0
\(970\) 2.70093i 0.0867216i
\(971\) 15.1574i 0.486424i 0.969973 + 0.243212i \(0.0782010\pi\)
−0.969973 + 0.243212i \(0.921799\pi\)
\(972\) 0 0
\(973\) 12.8642 0.412409
\(974\) 6.21142 0.199027
\(975\) 0 0
\(976\) 8.29965i 0.265665i
\(977\) 37.3885i 1.19616i −0.801435 0.598081i \(-0.795930\pi\)
0.801435 0.598081i \(-0.204070\pi\)
\(978\) 0 0
\(979\) −3.66011 + 23.7032i −0.116978 + 0.757558i
\(980\) 0.968524i 0.0309384i
\(981\) 0 0
\(982\) −9.06426 −0.289252
\(983\) 1.12437i 0.0358619i −0.999839 0.0179310i \(-0.994292\pi\)
0.999839 0.0179310i \(-0.00570791\pi\)
\(984\) 0 0
\(985\) 10.2758i 0.327413i
\(986\) 71.8701 2.28881
\(987\) 0 0
\(988\) −4.39322 −0.139767
\(989\) 2.41979 0.0769450
\(990\) 0 0
\(991\) 21.3347 0.677719 0.338859 0.940837i \(-0.389959\pi\)
0.338859 + 0.940837i \(0.389959\pi\)
\(992\) 4.20608 0.133543
\(993\) 0 0
\(994\) −4.98166 −0.158009
\(995\) 15.8863i 0.503629i
\(996\) 0 0
\(997\) 28.9595i 0.917156i 0.888654 + 0.458578i \(0.151641\pi\)
−0.888654 + 0.458578i \(0.848359\pi\)
\(998\) −40.7758 −1.29073
\(999\) 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 1386.2.c.b.197.5 yes 12
3.2 odd 2 1386.2.c.a.197.8 yes 12
11.10 odd 2 1386.2.c.a.197.5 12
33.32 even 2 inner 1386.2.c.b.197.8 yes 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1386.2.c.a.197.5 12 11.10 odd 2
1386.2.c.a.197.8 yes 12 3.2 odd 2
1386.2.c.b.197.5 yes 12 1.1 even 1 trivial
1386.2.c.b.197.8 yes 12 33.32 even 2 inner