Properties

Label 1183.2.c.d.337.3
Level $1183$
Weight $2$
Character 1183.337
Analytic conductor $9.446$
Analytic rank $0$
Dimension $4$
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] = [1183,2,Mod(337,1183)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1183, 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("1183.337");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1183 = 7 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1183.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(9.44630255912\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 91)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 337.3
Root \(0.707107 + 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 1183.337
Dual form 1183.2.c.d.337.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.41421i q^{2} -1.41421 q^{3} +4.41421i q^{5} -2.00000i q^{6} -1.00000i q^{7} +2.82843i q^{8} -1.00000 q^{9} +O(q^{10})\) \(q+1.41421i q^{2} -1.41421 q^{3} +4.41421i q^{5} -2.00000i q^{6} -1.00000i q^{7} +2.82843i q^{8} -1.00000 q^{9} -6.24264 q^{10} +4.24264i q^{11} +1.41421 q^{14} -6.24264i q^{15} -4.00000 q^{16} +1.41421 q^{17} -1.41421i q^{18} +1.24264i q^{19} +1.41421i q^{21} -6.00000 q^{22} +0.171573 q^{23} -4.00000i q^{24} -14.4853 q^{25} +5.65685 q^{27} +5.82843 q^{29} +8.82843 q^{30} -5.24264i q^{31} -6.00000i q^{33} +2.00000i q^{34} +4.41421 q^{35} +6.24264i q^{37} -1.75736 q^{38} -12.4853 q^{40} +3.17157i q^{41} -2.00000 q^{42} +5.00000 q^{43} -4.41421i q^{45} +0.242641i q^{46} -4.41421i q^{47} +5.65685 q^{48} -1.00000 q^{49} -20.4853i q^{50} -2.00000 q^{51} -5.82843 q^{53} +8.00000i q^{54} -18.7279 q^{55} +2.82843 q^{56} -1.75736i q^{57} +8.24264i q^{58} -11.6569i q^{59} +6.00000 q^{61} +7.41421 q^{62} +1.00000i q^{63} -8.00000 q^{64} +8.48528 q^{66} +2.48528i q^{67} -0.242641 q^{69} +6.24264i q^{70} +1.07107i q^{71} -2.82843i q^{72} +0.757359i q^{73} -8.82843 q^{74} +20.4853 q^{75} +4.24264 q^{77} -1.48528 q^{79} -17.6569i q^{80} -5.00000 q^{81} -4.48528 q^{82} +4.75736i q^{83} +6.24264i q^{85} +7.07107i q^{86} -8.24264 q^{87} -12.0000 q^{88} -4.41421i q^{89} +6.24264 q^{90} +7.41421i q^{93} +6.24264 q^{94} -5.48528 q^{95} -13.7279i q^{97} -1.41421i q^{98} -4.24264i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{9} - 8 q^{10} - 16 q^{16} - 24 q^{22} + 12 q^{23} - 24 q^{25} + 12 q^{29} + 24 q^{30} + 12 q^{35} - 24 q^{38} - 16 q^{40} - 8 q^{42} + 20 q^{43} - 4 q^{49} - 8 q^{51} - 12 q^{53} - 24 q^{55} + 24 q^{61} + 24 q^{62} - 32 q^{64} + 16 q^{69} - 24 q^{74} + 48 q^{75} + 28 q^{79} - 20 q^{81} + 16 q^{82} - 16 q^{87} - 48 q^{88} + 8 q^{90} + 8 q^{94} + 12 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1183\mathbb{Z}\right)^\times\).

\(n\) \(339\) \(1016\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.41421i 1.00000i 0.866025 + 0.500000i \(0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(3\) −1.41421 −0.816497 −0.408248 0.912871i \(-0.633860\pi\)
−0.408248 + 0.912871i \(0.633860\pi\)
\(4\) 0 0
\(5\) 4.41421i 1.97410i 0.160424 + 0.987048i \(0.448714\pi\)
−0.160424 + 0.987048i \(0.551286\pi\)
\(6\) − 2.00000i − 0.816497i
\(7\) − 1.00000i − 0.377964i
\(8\) 2.82843i 1.00000i
\(9\) −1.00000 −0.333333
\(10\) −6.24264 −1.97410
\(11\) 4.24264i 1.27920i 0.768706 + 0.639602i \(0.220901\pi\)
−0.768706 + 0.639602i \(0.779099\pi\)
\(12\) 0 0
\(13\) 0 0
\(14\) 1.41421 0.377964
\(15\) − 6.24264i − 1.61184i
\(16\) −4.00000 −1.00000
\(17\) 1.41421 0.342997 0.171499 0.985184i \(-0.445139\pi\)
0.171499 + 0.985184i \(0.445139\pi\)
\(18\) − 1.41421i − 0.333333i
\(19\) 1.24264i 0.285081i 0.989789 + 0.142541i \(0.0455272\pi\)
−0.989789 + 0.142541i \(0.954473\pi\)
\(20\) 0 0
\(21\) 1.41421i 0.308607i
\(22\) −6.00000 −1.27920
\(23\) 0.171573 0.0357754 0.0178877 0.999840i \(-0.494306\pi\)
0.0178877 + 0.999840i \(0.494306\pi\)
\(24\) − 4.00000i − 0.816497i
\(25\) −14.4853 −2.89706
\(26\) 0 0
\(27\) 5.65685 1.08866
\(28\) 0 0
\(29\) 5.82843 1.08231 0.541156 0.840922i \(-0.317987\pi\)
0.541156 + 0.840922i \(0.317987\pi\)
\(30\) 8.82843 1.61184
\(31\) − 5.24264i − 0.941606i −0.882238 0.470803i \(-0.843964\pi\)
0.882238 0.470803i \(-0.156036\pi\)
\(32\) 0 0
\(33\) − 6.00000i − 1.04447i
\(34\) 2.00000i 0.342997i
\(35\) 4.41421 0.746138
\(36\) 0 0
\(37\) 6.24264i 1.02628i 0.858304 + 0.513142i \(0.171519\pi\)
−0.858304 + 0.513142i \(0.828481\pi\)
\(38\) −1.75736 −0.285081
\(39\) 0 0
\(40\) −12.4853 −1.97410
\(41\) 3.17157i 0.495316i 0.968847 + 0.247658i \(0.0796610\pi\)
−0.968847 + 0.247658i \(0.920339\pi\)
\(42\) −2.00000 −0.308607
\(43\) 5.00000 0.762493 0.381246 0.924473i \(-0.375495\pi\)
0.381246 + 0.924473i \(0.375495\pi\)
\(44\) 0 0
\(45\) − 4.41421i − 0.658032i
\(46\) 0.242641i 0.0357754i
\(47\) − 4.41421i − 0.643879i −0.946760 0.321940i \(-0.895665\pi\)
0.946760 0.321940i \(-0.104335\pi\)
\(48\) 5.65685 0.816497
\(49\) −1.00000 −0.142857
\(50\) − 20.4853i − 2.89706i
\(51\) −2.00000 −0.280056
\(52\) 0 0
\(53\) −5.82843 −0.800596 −0.400298 0.916385i \(-0.631093\pi\)
−0.400298 + 0.916385i \(0.631093\pi\)
\(54\) 8.00000i 1.08866i
\(55\) −18.7279 −2.52527
\(56\) 2.82843 0.377964
\(57\) − 1.75736i − 0.232768i
\(58\) 8.24264i 1.08231i
\(59\) − 11.6569i − 1.51759i −0.651328 0.758797i \(-0.725788\pi\)
0.651328 0.758797i \(-0.274212\pi\)
\(60\) 0 0
\(61\) 6.00000 0.768221 0.384111 0.923287i \(-0.374508\pi\)
0.384111 + 0.923287i \(0.374508\pi\)
\(62\) 7.41421 0.941606
\(63\) 1.00000i 0.125988i
\(64\) −8.00000 −1.00000
\(65\) 0 0
\(66\) 8.48528 1.04447
\(67\) 2.48528i 0.303625i 0.988409 + 0.151813i \(0.0485111\pi\)
−0.988409 + 0.151813i \(0.951489\pi\)
\(68\) 0 0
\(69\) −0.242641 −0.0292105
\(70\) 6.24264i 0.746138i
\(71\) 1.07107i 0.127112i 0.997978 + 0.0635562i \(0.0202442\pi\)
−0.997978 + 0.0635562i \(0.979756\pi\)
\(72\) − 2.82843i − 0.333333i
\(73\) 0.757359i 0.0886422i 0.999017 + 0.0443211i \(0.0141125\pi\)
−0.999017 + 0.0443211i \(0.985888\pi\)
\(74\) −8.82843 −1.02628
\(75\) 20.4853 2.36544
\(76\) 0 0
\(77\) 4.24264 0.483494
\(78\) 0 0
\(79\) −1.48528 −0.167107 −0.0835536 0.996503i \(-0.526627\pi\)
−0.0835536 + 0.996503i \(0.526627\pi\)
\(80\) − 17.6569i − 1.97410i
\(81\) −5.00000 −0.555556
\(82\) −4.48528 −0.495316
\(83\) 4.75736i 0.522188i 0.965313 + 0.261094i \(0.0840833\pi\)
−0.965313 + 0.261094i \(0.915917\pi\)
\(84\) 0 0
\(85\) 6.24264i 0.677109i
\(86\) 7.07107i 0.762493i
\(87\) −8.24264 −0.883704
\(88\) −12.0000 −1.27920
\(89\) − 4.41421i − 0.467906i −0.972248 0.233953i \(-0.924834\pi\)
0.972248 0.233953i \(-0.0751661\pi\)
\(90\) 6.24264 0.658032
\(91\) 0 0
\(92\) 0 0
\(93\) 7.41421i 0.768818i
\(94\) 6.24264 0.643879
\(95\) −5.48528 −0.562778
\(96\) 0 0
\(97\) − 13.7279i − 1.39386i −0.717139 0.696930i \(-0.754549\pi\)
0.717139 0.696930i \(-0.245451\pi\)
\(98\) − 1.41421i − 0.142857i
\(99\) − 4.24264i − 0.426401i
\(100\) 0 0
\(101\) −1.75736 −0.174864 −0.0874319 0.996170i \(-0.527866\pi\)
−0.0874319 + 0.996170i \(0.527866\pi\)
\(102\) − 2.82843i − 0.280056i
\(103\) 8.00000 0.788263 0.394132 0.919054i \(-0.371045\pi\)
0.394132 + 0.919054i \(0.371045\pi\)
\(104\) 0 0
\(105\) −6.24264 −0.609219
\(106\) − 8.24264i − 0.800596i
\(107\) 8.14214 0.787130 0.393565 0.919297i \(-0.371242\pi\)
0.393565 + 0.919297i \(0.371242\pi\)
\(108\) 0 0
\(109\) − 8.72792i − 0.835983i −0.908451 0.417992i \(-0.862734\pi\)
0.908451 0.417992i \(-0.137266\pi\)
\(110\) − 26.4853i − 2.52527i
\(111\) − 8.82843i − 0.837957i
\(112\) 4.00000i 0.377964i
\(113\) −20.3137 −1.91095 −0.955476 0.295067i \(-0.904658\pi\)
−0.955476 + 0.295067i \(0.904658\pi\)
\(114\) 2.48528 0.232768
\(115\) 0.757359i 0.0706241i
\(116\) 0 0
\(117\) 0 0
\(118\) 16.4853 1.51759
\(119\) − 1.41421i − 0.129641i
\(120\) 17.6569 1.61184
\(121\) −7.00000 −0.636364
\(122\) 8.48528i 0.768221i
\(123\) − 4.48528i − 0.404424i
\(124\) 0 0
\(125\) − 41.8701i − 3.74497i
\(126\) −1.41421 −0.125988
\(127\) −2.00000 −0.177471 −0.0887357 0.996055i \(-0.528283\pi\)
−0.0887357 + 0.996055i \(0.528283\pi\)
\(128\) − 11.3137i − 1.00000i
\(129\) −7.07107 −0.622573
\(130\) 0 0
\(131\) 2.82843 0.247121 0.123560 0.992337i \(-0.460569\pi\)
0.123560 + 0.992337i \(0.460569\pi\)
\(132\) 0 0
\(133\) 1.24264 0.107751
\(134\) −3.51472 −0.303625
\(135\) 24.9706i 2.14912i
\(136\) 4.00000i 0.342997i
\(137\) 7.41421i 0.633439i 0.948519 + 0.316720i \(0.102581\pi\)
−0.948519 + 0.316720i \(0.897419\pi\)
\(138\) − 0.343146i − 0.0292105i
\(139\) −2.24264 −0.190218 −0.0951092 0.995467i \(-0.530320\pi\)
−0.0951092 + 0.995467i \(0.530320\pi\)
\(140\) 0 0
\(141\) 6.24264i 0.525725i
\(142\) −1.51472 −0.127112
\(143\) 0 0
\(144\) 4.00000 0.333333
\(145\) 25.7279i 2.13659i
\(146\) −1.07107 −0.0886422
\(147\) 1.41421 0.116642
\(148\) 0 0
\(149\) 7.75736i 0.635508i 0.948173 + 0.317754i \(0.102929\pi\)
−0.948173 + 0.317754i \(0.897071\pi\)
\(150\) 28.9706i 2.36544i
\(151\) 18.2426i 1.48457i 0.670087 + 0.742283i \(0.266257\pi\)
−0.670087 + 0.742283i \(0.733743\pi\)
\(152\) −3.51472 −0.285081
\(153\) −1.41421 −0.114332
\(154\) 6.00000i 0.483494i
\(155\) 23.1421 1.85882
\(156\) 0 0
\(157\) −12.2426 −0.977069 −0.488535 0.872545i \(-0.662468\pi\)
−0.488535 + 0.872545i \(0.662468\pi\)
\(158\) − 2.10051i − 0.167107i
\(159\) 8.24264 0.653684
\(160\) 0 0
\(161\) − 0.171573i − 0.0135218i
\(162\) − 7.07107i − 0.555556i
\(163\) 8.48528i 0.664619i 0.943170 + 0.332309i \(0.107828\pi\)
−0.943170 + 0.332309i \(0.892172\pi\)
\(164\) 0 0
\(165\) 26.4853 2.06188
\(166\) −6.72792 −0.522188
\(167\) 21.3848i 1.65480i 0.561610 + 0.827402i \(0.310182\pi\)
−0.561610 + 0.827402i \(0.689818\pi\)
\(168\) −4.00000 −0.308607
\(169\) 0 0
\(170\) −8.82843 −0.677109
\(171\) − 1.24264i − 0.0950271i
\(172\) 0 0
\(173\) 0.727922 0.0553429 0.0276714 0.999617i \(-0.491191\pi\)
0.0276714 + 0.999617i \(0.491191\pi\)
\(174\) − 11.6569i − 0.883704i
\(175\) 14.4853i 1.09498i
\(176\) − 16.9706i − 1.27920i
\(177\) 16.4853i 1.23911i
\(178\) 6.24264 0.467906
\(179\) 9.00000 0.672692 0.336346 0.941739i \(-0.390809\pi\)
0.336346 + 0.941739i \(0.390809\pi\)
\(180\) 0 0
\(181\) −6.72792 −0.500083 −0.250041 0.968235i \(-0.580444\pi\)
−0.250041 + 0.968235i \(0.580444\pi\)
\(182\) 0 0
\(183\) −8.48528 −0.627250
\(184\) 0.485281i 0.0357754i
\(185\) −27.5563 −2.02598
\(186\) −10.4853 −0.768818
\(187\) 6.00000i 0.438763i
\(188\) 0 0
\(189\) − 5.65685i − 0.411476i
\(190\) − 7.75736i − 0.562778i
\(191\) 20.8284 1.50709 0.753546 0.657395i \(-0.228342\pi\)
0.753546 + 0.657395i \(0.228342\pi\)
\(192\) 11.3137 0.816497
\(193\) − 2.48528i − 0.178894i −0.995992 0.0894472i \(-0.971490\pi\)
0.995992 0.0894472i \(-0.0285100\pi\)
\(194\) 19.4142 1.39386
\(195\) 0 0
\(196\) 0 0
\(197\) 23.6569i 1.68548i 0.538320 + 0.842741i \(0.319059\pi\)
−0.538320 + 0.842741i \(0.680941\pi\)
\(198\) 6.00000 0.426401
\(199\) 12.2426 0.867858 0.433929 0.900947i \(-0.357127\pi\)
0.433929 + 0.900947i \(0.357127\pi\)
\(200\) − 40.9706i − 2.89706i
\(201\) − 3.51472i − 0.247909i
\(202\) − 2.48528i − 0.174864i
\(203\) − 5.82843i − 0.409075i
\(204\) 0 0
\(205\) −14.0000 −0.977802
\(206\) 11.3137i 0.788263i
\(207\) −0.171573 −0.0119251
\(208\) 0 0
\(209\) −5.27208 −0.364677
\(210\) − 8.82843i − 0.609219i
\(211\) 17.9706 1.23714 0.618572 0.785728i \(-0.287711\pi\)
0.618572 + 0.785728i \(0.287711\pi\)
\(212\) 0 0
\(213\) − 1.51472i − 0.103787i
\(214\) 11.5147i 0.787130i
\(215\) 22.0711i 1.50523i
\(216\) 16.0000i 1.08866i
\(217\) −5.24264 −0.355894
\(218\) 12.3431 0.835983
\(219\) − 1.07107i − 0.0723761i
\(220\) 0 0
\(221\) 0 0
\(222\) 12.4853 0.837957
\(223\) 9.24264i 0.618933i 0.950910 + 0.309466i \(0.100150\pi\)
−0.950910 + 0.309466i \(0.899850\pi\)
\(224\) 0 0
\(225\) 14.4853 0.965685
\(226\) − 28.7279i − 1.91095i
\(227\) 21.1716i 1.40521i 0.711582 + 0.702603i \(0.247979\pi\)
−0.711582 + 0.702603i \(0.752021\pi\)
\(228\) 0 0
\(229\) 21.4558i 1.41784i 0.705288 + 0.708921i \(0.250818\pi\)
−0.705288 + 0.708921i \(0.749182\pi\)
\(230\) −1.07107 −0.0706241
\(231\) −6.00000 −0.394771
\(232\) 16.4853i 1.08231i
\(233\) 3.34315 0.219017 0.109508 0.993986i \(-0.465072\pi\)
0.109508 + 0.993986i \(0.465072\pi\)
\(234\) 0 0
\(235\) 19.4853 1.27108
\(236\) 0 0
\(237\) 2.10051 0.136442
\(238\) 2.00000 0.129641
\(239\) 20.4853i 1.32508i 0.749025 + 0.662541i \(0.230522\pi\)
−0.749025 + 0.662541i \(0.769478\pi\)
\(240\) 24.9706i 1.61184i
\(241\) − 22.2132i − 1.43088i −0.698675 0.715439i \(-0.746227\pi\)
0.698675 0.715439i \(-0.253773\pi\)
\(242\) − 9.89949i − 0.636364i
\(243\) −9.89949 −0.635053
\(244\) 0 0
\(245\) − 4.41421i − 0.282014i
\(246\) 6.34315 0.404424
\(247\) 0 0
\(248\) 14.8284 0.941606
\(249\) − 6.72792i − 0.426365i
\(250\) 59.2132 3.74497
\(251\) −19.4142 −1.22541 −0.612707 0.790310i \(-0.709919\pi\)
−0.612707 + 0.790310i \(0.709919\pi\)
\(252\) 0 0
\(253\) 0.727922i 0.0457641i
\(254\) − 2.82843i − 0.177471i
\(255\) − 8.82843i − 0.552858i
\(256\) 0 0
\(257\) 16.5858 1.03459 0.517296 0.855806i \(-0.326938\pi\)
0.517296 + 0.855806i \(0.326938\pi\)
\(258\) − 10.0000i − 0.622573i
\(259\) 6.24264 0.387899
\(260\) 0 0
\(261\) −5.82843 −0.360771
\(262\) 4.00000i 0.247121i
\(263\) −31.9706 −1.97139 −0.985695 0.168541i \(-0.946095\pi\)
−0.985695 + 0.168541i \(0.946095\pi\)
\(264\) 16.9706 1.04447
\(265\) − 25.7279i − 1.58045i
\(266\) 1.75736i 0.107751i
\(267\) 6.24264i 0.382043i
\(268\) 0 0
\(269\) 14.8284 0.904105 0.452053 0.891991i \(-0.350692\pi\)
0.452053 + 0.891991i \(0.350692\pi\)
\(270\) −35.3137 −2.14912
\(271\) − 20.0000i − 1.21491i −0.794353 0.607457i \(-0.792190\pi\)
0.794353 0.607457i \(-0.207810\pi\)
\(272\) −5.65685 −0.342997
\(273\) 0 0
\(274\) −10.4853 −0.633439
\(275\) − 61.4558i − 3.70593i
\(276\) 0 0
\(277\) −9.48528 −0.569915 −0.284958 0.958540i \(-0.591980\pi\)
−0.284958 + 0.958540i \(0.591980\pi\)
\(278\) − 3.17157i − 0.190218i
\(279\) 5.24264i 0.313869i
\(280\) 12.4853i 0.746138i
\(281\) 15.5563i 0.928014i 0.885832 + 0.464007i \(0.153589\pi\)
−0.885832 + 0.464007i \(0.846411\pi\)
\(282\) −8.82843 −0.525725
\(283\) −8.48528 −0.504398 −0.252199 0.967675i \(-0.581154\pi\)
−0.252199 + 0.967675i \(0.581154\pi\)
\(284\) 0 0
\(285\) 7.75736 0.459506
\(286\) 0 0
\(287\) 3.17157 0.187212
\(288\) 0 0
\(289\) −15.0000 −0.882353
\(290\) −36.3848 −2.13659
\(291\) 19.4142i 1.13808i
\(292\) 0 0
\(293\) 21.3848i 1.24931i 0.780900 + 0.624656i \(0.214761\pi\)
−0.780900 + 0.624656i \(0.785239\pi\)
\(294\) 2.00000i 0.116642i
\(295\) 51.4558 2.99588
\(296\) −17.6569 −1.02628
\(297\) 24.0000i 1.39262i
\(298\) −10.9706 −0.635508
\(299\) 0 0
\(300\) 0 0
\(301\) − 5.00000i − 0.288195i
\(302\) −25.7990 −1.48457
\(303\) 2.48528 0.142776
\(304\) − 4.97056i − 0.285081i
\(305\) 26.4853i 1.51654i
\(306\) − 2.00000i − 0.114332i
\(307\) − 4.75736i − 0.271517i −0.990742 0.135758i \(-0.956653\pi\)
0.990742 0.135758i \(-0.0433471\pi\)
\(308\) 0 0
\(309\) −11.3137 −0.643614
\(310\) 32.7279i 1.85882i
\(311\) 4.58579 0.260036 0.130018 0.991512i \(-0.458496\pi\)
0.130018 + 0.991512i \(0.458496\pi\)
\(312\) 0 0
\(313\) −19.2132 −1.08599 −0.542997 0.839734i \(-0.682711\pi\)
−0.542997 + 0.839734i \(0.682711\pi\)
\(314\) − 17.3137i − 0.977069i
\(315\) −4.41421 −0.248713
\(316\) 0 0
\(317\) 11.3137i 0.635441i 0.948184 + 0.317721i \(0.102917\pi\)
−0.948184 + 0.317721i \(0.897083\pi\)
\(318\) 11.6569i 0.653684i
\(319\) 24.7279i 1.38450i
\(320\) − 35.3137i − 1.97410i
\(321\) −11.5147 −0.642689
\(322\) 0.242641 0.0135218
\(323\) 1.75736i 0.0977821i
\(324\) 0 0
\(325\) 0 0
\(326\) −12.0000 −0.664619
\(327\) 12.3431i 0.682578i
\(328\) −8.97056 −0.495316
\(329\) −4.41421 −0.243363
\(330\) 37.4558i 2.06188i
\(331\) − 18.0000i − 0.989369i −0.869072 0.494685i \(-0.835284\pi\)
0.869072 0.494685i \(-0.164716\pi\)
\(332\) 0 0
\(333\) − 6.24264i − 0.342095i
\(334\) −30.2426 −1.65480
\(335\) −10.9706 −0.599386
\(336\) − 5.65685i − 0.308607i
\(337\) 33.0000 1.79762 0.898812 0.438334i \(-0.144431\pi\)
0.898812 + 0.438334i \(0.144431\pi\)
\(338\) 0 0
\(339\) 28.7279 1.56029
\(340\) 0 0
\(341\) 22.2426 1.20451
\(342\) 1.75736 0.0950271
\(343\) 1.00000i 0.0539949i
\(344\) 14.1421i 0.762493i
\(345\) − 1.07107i − 0.0576644i
\(346\) 1.02944i 0.0553429i
\(347\) 5.65685 0.303676 0.151838 0.988405i \(-0.451481\pi\)
0.151838 + 0.988405i \(0.451481\pi\)
\(348\) 0 0
\(349\) 0.272078i 0.0145640i 0.999973 + 0.00728200i \(0.00231795\pi\)
−0.999973 + 0.00728200i \(0.997682\pi\)
\(350\) −20.4853 −1.09498
\(351\) 0 0
\(352\) 0 0
\(353\) 8.48528i 0.451626i 0.974171 + 0.225813i \(0.0725038\pi\)
−0.974171 + 0.225813i \(0.927496\pi\)
\(354\) −23.3137 −1.23911
\(355\) −4.72792 −0.250932
\(356\) 0 0
\(357\) 2.00000i 0.105851i
\(358\) 12.7279i 0.672692i
\(359\) 8.10051i 0.427528i 0.976885 + 0.213764i \(0.0685724\pi\)
−0.976885 + 0.213764i \(0.931428\pi\)
\(360\) 12.4853 0.658032
\(361\) 17.4558 0.918729
\(362\) − 9.51472i − 0.500083i
\(363\) 9.89949 0.519589
\(364\) 0 0
\(365\) −3.34315 −0.174988
\(366\) − 12.0000i − 0.627250i
\(367\) 1.75736 0.0917334 0.0458667 0.998948i \(-0.485395\pi\)
0.0458667 + 0.998948i \(0.485395\pi\)
\(368\) −0.686292 −0.0357754
\(369\) − 3.17157i − 0.165105i
\(370\) − 38.9706i − 2.02598i
\(371\) 5.82843i 0.302597i
\(372\) 0 0
\(373\) −8.48528 −0.439351 −0.219676 0.975573i \(-0.570500\pi\)
−0.219676 + 0.975573i \(0.570500\pi\)
\(374\) −8.48528 −0.438763
\(375\) 59.2132i 3.05776i
\(376\) 12.4853 0.643879
\(377\) 0 0
\(378\) 8.00000 0.411476
\(379\) 32.2426i 1.65619i 0.560585 + 0.828097i \(0.310576\pi\)
−0.560585 + 0.828097i \(0.689424\pi\)
\(380\) 0 0
\(381\) 2.82843 0.144905
\(382\) 29.4558i 1.50709i
\(383\) − 3.51472i − 0.179594i −0.995960 0.0897969i \(-0.971378\pi\)
0.995960 0.0897969i \(-0.0286218\pi\)
\(384\) 16.0000i 0.816497i
\(385\) 18.7279i 0.954463i
\(386\) 3.51472 0.178894
\(387\) −5.00000 −0.254164
\(388\) 0 0
\(389\) −18.3431 −0.930034 −0.465017 0.885302i \(-0.653952\pi\)
−0.465017 + 0.885302i \(0.653952\pi\)
\(390\) 0 0
\(391\) 0.242641 0.0122709
\(392\) − 2.82843i − 0.142857i
\(393\) −4.00000 −0.201773
\(394\) −33.4558 −1.68548
\(395\) − 6.55635i − 0.329886i
\(396\) 0 0
\(397\) − 24.2132i − 1.21523i −0.794233 0.607613i \(-0.792127\pi\)
0.794233 0.607613i \(-0.207873\pi\)
\(398\) 17.3137i 0.867858i
\(399\) −1.75736 −0.0879780
\(400\) 57.9411 2.89706
\(401\) − 17.6569i − 0.881741i −0.897571 0.440871i \(-0.854670\pi\)
0.897571 0.440871i \(-0.145330\pi\)
\(402\) 4.97056 0.247909
\(403\) 0 0
\(404\) 0 0
\(405\) − 22.0711i − 1.09672i
\(406\) 8.24264 0.409075
\(407\) −26.4853 −1.31283
\(408\) − 5.65685i − 0.280056i
\(409\) − 5.24264i − 0.259232i −0.991564 0.129616i \(-0.958626\pi\)
0.991564 0.129616i \(-0.0413744\pi\)
\(410\) − 19.7990i − 0.977802i
\(411\) − 10.4853i − 0.517201i
\(412\) 0 0
\(413\) −11.6569 −0.573596
\(414\) − 0.242641i − 0.0119251i
\(415\) −21.0000 −1.03085
\(416\) 0 0
\(417\) 3.17157 0.155313
\(418\) − 7.45584i − 0.364677i
\(419\) 32.8701 1.60581 0.802904 0.596109i \(-0.203287\pi\)
0.802904 + 0.596109i \(0.203287\pi\)
\(420\) 0 0
\(421\) − 18.7279i − 0.912743i −0.889789 0.456372i \(-0.849149\pi\)
0.889789 0.456372i \(-0.150851\pi\)
\(422\) 25.4142i 1.23714i
\(423\) 4.41421i 0.214626i
\(424\) − 16.4853i − 0.800596i
\(425\) −20.4853 −0.993682
\(426\) 2.14214 0.103787
\(427\) − 6.00000i − 0.290360i
\(428\) 0 0
\(429\) 0 0
\(430\) −31.2132 −1.50523
\(431\) − 23.6569i − 1.13951i −0.821814 0.569755i \(-0.807038\pi\)
0.821814 0.569755i \(-0.192962\pi\)
\(432\) −22.6274 −1.08866
\(433\) −8.97056 −0.431098 −0.215549 0.976493i \(-0.569154\pi\)
−0.215549 + 0.976493i \(0.569154\pi\)
\(434\) − 7.41421i − 0.355894i
\(435\) − 36.3848i − 1.74452i
\(436\) 0 0
\(437\) 0.213203i 0.0101989i
\(438\) 1.51472 0.0723761
\(439\) 17.5147 0.835932 0.417966 0.908463i \(-0.362743\pi\)
0.417966 + 0.908463i \(0.362743\pi\)
\(440\) − 52.9706i − 2.52527i
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) −26.3137 −1.25020 −0.625101 0.780544i \(-0.714942\pi\)
−0.625101 + 0.780544i \(0.714942\pi\)
\(444\) 0 0
\(445\) 19.4853 0.923691
\(446\) −13.0711 −0.618933
\(447\) − 10.9706i − 0.518890i
\(448\) 8.00000i 0.377964i
\(449\) − 26.8284i − 1.26611i −0.774106 0.633056i \(-0.781800\pi\)
0.774106 0.633056i \(-0.218200\pi\)
\(450\) 20.4853i 0.965685i
\(451\) −13.4558 −0.633611
\(452\) 0 0
\(453\) − 25.7990i − 1.21214i
\(454\) −29.9411 −1.40521
\(455\) 0 0
\(456\) 4.97056 0.232768
\(457\) 35.2132i 1.64720i 0.567168 + 0.823602i \(0.308039\pi\)
−0.567168 + 0.823602i \(0.691961\pi\)
\(458\) −30.3431 −1.41784
\(459\) 8.00000 0.373408
\(460\) 0 0
\(461\) 3.17157i 0.147715i 0.997269 + 0.0738574i \(0.0235310\pi\)
−0.997269 + 0.0738574i \(0.976469\pi\)
\(462\) − 8.48528i − 0.394771i
\(463\) − 4.24264i − 0.197172i −0.995129 0.0985861i \(-0.968568\pi\)
0.995129 0.0985861i \(-0.0314320\pi\)
\(464\) −23.3137 −1.08231
\(465\) −32.7279 −1.51772
\(466\) 4.72792i 0.219017i
\(467\) −15.8995 −0.735741 −0.367870 0.929877i \(-0.619913\pi\)
−0.367870 + 0.929877i \(0.619913\pi\)
\(468\) 0 0
\(469\) 2.48528 0.114760
\(470\) 27.5563i 1.27108i
\(471\) 17.3137 0.797774
\(472\) 32.9706 1.51759
\(473\) 21.2132i 0.975384i
\(474\) 2.97056i 0.136442i
\(475\) − 18.0000i − 0.825897i
\(476\) 0 0
\(477\) 5.82843 0.266865
\(478\) −28.9706 −1.32508
\(479\) − 36.2132i − 1.65462i −0.561743 0.827312i \(-0.689869\pi\)
0.561743 0.827312i \(-0.310131\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 31.4142 1.43088
\(483\) 0.242641i 0.0110405i
\(484\) 0 0
\(485\) 60.5980 2.75161
\(486\) − 14.0000i − 0.635053i
\(487\) 39.4558i 1.78791i 0.448152 + 0.893957i \(0.352082\pi\)
−0.448152 + 0.893957i \(0.647918\pi\)
\(488\) 16.9706i 0.768221i
\(489\) − 12.0000i − 0.542659i
\(490\) 6.24264 0.282014
\(491\) 28.6274 1.29194 0.645969 0.763364i \(-0.276454\pi\)
0.645969 + 0.763364i \(0.276454\pi\)
\(492\) 0 0
\(493\) 8.24264 0.371230
\(494\) 0 0
\(495\) 18.7279 0.841757
\(496\) 20.9706i 0.941606i
\(497\) 1.07107 0.0480440
\(498\) 9.51472 0.426365
\(499\) − 38.7279i − 1.73370i −0.498569 0.866850i \(-0.666141\pi\)
0.498569 0.866850i \(-0.333859\pi\)
\(500\) 0 0
\(501\) − 30.2426i − 1.35114i
\(502\) − 27.4558i − 1.22541i
\(503\) 16.6274 0.741380 0.370690 0.928757i \(-0.379121\pi\)
0.370690 + 0.928757i \(0.379121\pi\)
\(504\) −2.82843 −0.125988
\(505\) − 7.75736i − 0.345198i
\(506\) −1.02944 −0.0457641
\(507\) 0 0
\(508\) 0 0
\(509\) − 24.8995i − 1.10365i −0.833960 0.551825i \(-0.813931\pi\)
0.833960 0.551825i \(-0.186069\pi\)
\(510\) 12.4853 0.552858
\(511\) 0.757359 0.0335036
\(512\) − 22.6274i − 1.00000i
\(513\) 7.02944i 0.310357i
\(514\) 23.4558i 1.03459i
\(515\) 35.3137i 1.55611i
\(516\) 0 0
\(517\) 18.7279 0.823653
\(518\) 8.82843i 0.387899i
\(519\) −1.02944 −0.0451873
\(520\) 0 0
\(521\) −17.6569 −0.773561 −0.386780 0.922172i \(-0.626413\pi\)
−0.386780 + 0.922172i \(0.626413\pi\)
\(522\) − 8.24264i − 0.360771i
\(523\) −2.97056 −0.129894 −0.0649468 0.997889i \(-0.520688\pi\)
−0.0649468 + 0.997889i \(0.520688\pi\)
\(524\) 0 0
\(525\) − 20.4853i − 0.894051i
\(526\) − 45.2132i − 1.97139i
\(527\) − 7.41421i − 0.322968i
\(528\) 24.0000i 1.04447i
\(529\) −22.9706 −0.998720
\(530\) 36.3848 1.58045
\(531\) 11.6569i 0.505864i
\(532\) 0 0
\(533\) 0 0
\(534\) −8.82843 −0.382043
\(535\) 35.9411i 1.55387i
\(536\) −7.02944 −0.303625
\(537\) −12.7279 −0.549250
\(538\) 20.9706i 0.904105i
\(539\) − 4.24264i − 0.182743i
\(540\) 0 0
\(541\) − 35.2132i − 1.51393i −0.653453 0.756967i \(-0.726680\pi\)
0.653453 0.756967i \(-0.273320\pi\)
\(542\) 28.2843 1.21491
\(543\) 9.51472 0.408316
\(544\) 0 0
\(545\) 38.5269 1.65031
\(546\) 0 0
\(547\) 18.5147 0.791632 0.395816 0.918330i \(-0.370462\pi\)
0.395816 + 0.918330i \(0.370462\pi\)
\(548\) 0 0
\(549\) −6.00000 −0.256074
\(550\) 86.9117 3.70593
\(551\) 7.24264i 0.308547i
\(552\) − 0.686292i − 0.0292105i
\(553\) 1.48528i 0.0631606i
\(554\) − 13.4142i − 0.569915i
\(555\) 38.9706 1.65421
\(556\) 0 0
\(557\) − 12.0000i − 0.508456i −0.967144 0.254228i \(-0.918179\pi\)
0.967144 0.254228i \(-0.0818214\pi\)
\(558\) −7.41421 −0.313869
\(559\) 0 0
\(560\) −17.6569 −0.746138
\(561\) − 8.48528i − 0.358249i
\(562\) −22.0000 −0.928014
\(563\) 17.6569 0.744148 0.372074 0.928203i \(-0.378647\pi\)
0.372074 + 0.928203i \(0.378647\pi\)
\(564\) 0 0
\(565\) − 89.6690i − 3.77241i
\(566\) − 12.0000i − 0.504398i
\(567\) 5.00000i 0.209980i
\(568\) −3.02944 −0.127112
\(569\) 23.1421 0.970169 0.485084 0.874467i \(-0.338789\pi\)
0.485084 + 0.874467i \(0.338789\pi\)
\(570\) 10.9706i 0.459506i
\(571\) 8.45584 0.353866 0.176933 0.984223i \(-0.443382\pi\)
0.176933 + 0.984223i \(0.443382\pi\)
\(572\) 0 0
\(573\) −29.4558 −1.23054
\(574\) 4.48528i 0.187212i
\(575\) −2.48528 −0.103643
\(576\) 8.00000 0.333333
\(577\) 26.9706i 1.12280i 0.827545 + 0.561400i \(0.189737\pi\)
−0.827545 + 0.561400i \(0.810263\pi\)
\(578\) − 21.2132i − 0.882353i
\(579\) 3.51472i 0.146067i
\(580\) 0 0
\(581\) 4.75736 0.197369
\(582\) −27.4558 −1.13808
\(583\) − 24.7279i − 1.02413i
\(584\) −2.14214 −0.0886422
\(585\) 0 0
\(586\) −30.2426 −1.24931
\(587\) 24.5563i 1.01355i 0.862079 + 0.506775i \(0.169162\pi\)
−0.862079 + 0.506775i \(0.830838\pi\)
\(588\) 0 0
\(589\) 6.51472 0.268434
\(590\) 72.7696i 2.99588i
\(591\) − 33.4558i − 1.37619i
\(592\) − 24.9706i − 1.02628i
\(593\) − 30.5563i − 1.25480i −0.778698 0.627399i \(-0.784119\pi\)
0.778698 0.627399i \(-0.215881\pi\)
\(594\) −33.9411 −1.39262
\(595\) 6.24264 0.255923
\(596\) 0 0
\(597\) −17.3137 −0.708603
\(598\) 0 0
\(599\) 10.7990 0.441235 0.220617 0.975360i \(-0.429193\pi\)
0.220617 + 0.975360i \(0.429193\pi\)
\(600\) 57.9411i 2.36544i
\(601\) −5.02944 −0.205155 −0.102578 0.994725i \(-0.532709\pi\)
−0.102578 + 0.994725i \(0.532709\pi\)
\(602\) 7.07107 0.288195
\(603\) − 2.48528i − 0.101208i
\(604\) 0 0
\(605\) − 30.8995i − 1.25624i
\(606\) 3.51472i 0.142776i
\(607\) −1.27208 −0.0516321 −0.0258160 0.999667i \(-0.508218\pi\)
−0.0258160 + 0.999667i \(0.508218\pi\)
\(608\) 0 0
\(609\) 8.24264i 0.334009i
\(610\) −37.4558 −1.51654
\(611\) 0 0
\(612\) 0 0
\(613\) − 16.0000i − 0.646234i −0.946359 0.323117i \(-0.895269\pi\)
0.946359 0.323117i \(-0.104731\pi\)
\(614\) 6.72792 0.271517
\(615\) 19.7990 0.798372
\(616\) 12.0000i 0.483494i
\(617\) 23.6569i 0.952389i 0.879340 + 0.476195i \(0.157984\pi\)
−0.879340 + 0.476195i \(0.842016\pi\)
\(618\) − 16.0000i − 0.643614i
\(619\) 32.9706i 1.32520i 0.748974 + 0.662599i \(0.230547\pi\)
−0.748974 + 0.662599i \(0.769453\pi\)
\(620\) 0 0
\(621\) 0.970563 0.0389473
\(622\) 6.48528i 0.260036i
\(623\) −4.41421 −0.176852
\(624\) 0 0
\(625\) 112.397 4.49588
\(626\) − 27.1716i − 1.08599i
\(627\) 7.45584 0.297758
\(628\) 0 0
\(629\) 8.82843i 0.352012i
\(630\) − 6.24264i − 0.248713i
\(631\) − 2.00000i − 0.0796187i −0.999207 0.0398094i \(-0.987325\pi\)
0.999207 0.0398094i \(-0.0126751\pi\)
\(632\) − 4.20101i − 0.167107i
\(633\) −25.4142 −1.01012
\(634\) −16.0000 −0.635441
\(635\) − 8.82843i − 0.350345i
\(636\) 0 0
\(637\) 0 0
\(638\) −34.9706 −1.38450
\(639\) − 1.07107i − 0.0423708i
\(640\) 49.9411 1.97410
\(641\) −26.6569 −1.05288 −0.526441 0.850212i \(-0.676474\pi\)
−0.526441 + 0.850212i \(0.676474\pi\)
\(642\) − 16.2843i − 0.642689i
\(643\) 4.48528i 0.176882i 0.996081 + 0.0884411i \(0.0281885\pi\)
−0.996081 + 0.0884411i \(0.971811\pi\)
\(644\) 0 0
\(645\) − 31.2132i − 1.22902i
\(646\) −2.48528 −0.0977821
\(647\) 50.5269 1.98642 0.993209 0.116344i \(-0.0371176\pi\)
0.993209 + 0.116344i \(0.0371176\pi\)
\(648\) − 14.1421i − 0.555556i
\(649\) 49.4558 1.94131
\(650\) 0 0
\(651\) 7.41421 0.290586
\(652\) 0 0
\(653\) −5.31371 −0.207941 −0.103971 0.994580i \(-0.533155\pi\)
−0.103971 + 0.994580i \(0.533155\pi\)
\(654\) −17.4558 −0.682578
\(655\) 12.4853i 0.487840i
\(656\) − 12.6863i − 0.495316i
\(657\) − 0.757359i − 0.0295474i
\(658\) − 6.24264i − 0.243363i
\(659\) −26.6569 −1.03840 −0.519202 0.854652i \(-0.673771\pi\)
−0.519202 + 0.854652i \(0.673771\pi\)
\(660\) 0 0
\(661\) 19.2426i 0.748452i 0.927338 + 0.374226i \(0.122092\pi\)
−0.927338 + 0.374226i \(0.877908\pi\)
\(662\) 25.4558 0.989369
\(663\) 0 0
\(664\) −13.4558 −0.522188
\(665\) 5.48528i 0.212710i
\(666\) 8.82843 0.342095
\(667\) 1.00000 0.0387202
\(668\) 0 0
\(669\) − 13.0711i − 0.505357i
\(670\) − 15.5147i − 0.599386i
\(671\) 25.4558i 0.982712i
\(672\) 0 0
\(673\) −40.9411 −1.57816 −0.789082 0.614288i \(-0.789443\pi\)
−0.789082 + 0.614288i \(0.789443\pi\)
\(674\) 46.6690i 1.79762i
\(675\) −81.9411 −3.15392
\(676\) 0 0
\(677\) 29.3553 1.12822 0.564109 0.825701i \(-0.309220\pi\)
0.564109 + 0.825701i \(0.309220\pi\)
\(678\) 40.6274i 1.56029i
\(679\) −13.7279 −0.526829
\(680\) −17.6569 −0.677109
\(681\) − 29.9411i − 1.14735i
\(682\) 31.4558i 1.20451i
\(683\) 26.8284i 1.02656i 0.858221 + 0.513281i \(0.171570\pi\)
−0.858221 + 0.513281i \(0.828430\pi\)
\(684\) 0 0
\(685\) −32.7279 −1.25047
\(686\) −1.41421 −0.0539949
\(687\) − 30.3431i − 1.15766i
\(688\) −20.0000 −0.762493
\(689\) 0 0
\(690\) 1.51472 0.0576644
\(691\) − 30.6985i − 1.16783i −0.811816 0.583913i \(-0.801521\pi\)
0.811816 0.583913i \(-0.198479\pi\)
\(692\) 0 0
\(693\) −4.24264 −0.161165
\(694\) 8.00000i 0.303676i
\(695\) − 9.89949i − 0.375509i
\(696\) − 23.3137i − 0.883704i
\(697\) 4.48528i 0.169892i
\(698\) −0.384776 −0.0145640
\(699\) −4.72792 −0.178826
\(700\) 0 0
\(701\) −28.7990 −1.08772 −0.543861 0.839175i \(-0.683038\pi\)
−0.543861 + 0.839175i \(0.683038\pi\)
\(702\) 0 0
\(703\) −7.75736 −0.292574
\(704\) − 33.9411i − 1.27920i
\(705\) −27.5563 −1.03783
\(706\) −12.0000 −0.451626
\(707\) 1.75736i 0.0660923i
\(708\) 0 0
\(709\) 24.7279i 0.928677i 0.885658 + 0.464338i \(0.153708\pi\)
−0.885658 + 0.464338i \(0.846292\pi\)
\(710\) − 6.68629i − 0.250932i
\(711\) 1.48528 0.0557024
\(712\) 12.4853 0.467906
\(713\) − 0.899495i − 0.0336864i
\(714\) −2.82843 −0.105851
\(715\) 0 0
\(716\) 0 0
\(717\) − 28.9706i − 1.08193i
\(718\) −11.4558 −0.427528
\(719\) 10.2426 0.381986 0.190993 0.981591i \(-0.438829\pi\)
0.190993 + 0.981591i \(0.438829\pi\)
\(720\) 17.6569i 0.658032i
\(721\) − 8.00000i − 0.297936i
\(722\) 24.6863i 0.918729i
\(723\) 31.4142i 1.16831i
\(724\) 0 0
\(725\) −84.4264 −3.13552
\(726\) 14.0000i 0.519589i
\(727\) −42.0000 −1.55769 −0.778847 0.627214i \(-0.784195\pi\)
−0.778847 + 0.627214i \(0.784195\pi\)
\(728\) 0 0
\(729\) 29.0000 1.07407
\(730\) − 4.72792i − 0.174988i
\(731\) 7.07107 0.261533
\(732\) 0 0
\(733\) 42.6985i 1.57710i 0.614968 + 0.788552i \(0.289169\pi\)
−0.614968 + 0.788552i \(0.710831\pi\)
\(734\) 2.48528i 0.0917334i
\(735\) 6.24264i 0.230263i
\(736\) 0 0
\(737\) −10.5442 −0.388399
\(738\) 4.48528 0.165105
\(739\) − 41.6985i − 1.53390i −0.641705 0.766952i \(-0.721773\pi\)
0.641705 0.766952i \(-0.278227\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −8.24264 −0.302597
\(743\) − 18.3431i − 0.672945i −0.941693 0.336472i \(-0.890766\pi\)
0.941693 0.336472i \(-0.109234\pi\)
\(744\) −20.9706 −0.768818
\(745\) −34.2426 −1.25455
\(746\) − 12.0000i − 0.439351i
\(747\) − 4.75736i − 0.174063i
\(748\) 0 0
\(749\) − 8.14214i − 0.297507i
\(750\) −83.7401 −3.05776
\(751\) 1.48528 0.0541987 0.0270993 0.999633i \(-0.491373\pi\)
0.0270993 + 0.999633i \(0.491373\pi\)
\(752\) 17.6569i 0.643879i
\(753\) 27.4558 1.00055
\(754\) 0 0
\(755\) −80.5269 −2.93067
\(756\) 0 0
\(757\) 4.51472 0.164090 0.0820451 0.996629i \(-0.473855\pi\)
0.0820451 + 0.996629i \(0.473855\pi\)
\(758\) −45.5980 −1.65619
\(759\) − 1.02944i − 0.0373662i
\(760\) − 15.5147i − 0.562778i
\(761\) − 43.2426i − 1.56754i −0.621048 0.783772i \(-0.713293\pi\)
0.621048 0.783772i \(-0.286707\pi\)
\(762\) 4.00000i 0.144905i
\(763\) −8.72792 −0.315972
\(764\) 0 0
\(765\) − 6.24264i − 0.225703i
\(766\) 4.97056 0.179594
\(767\) 0 0
\(768\) 0 0
\(769\) 9.78680i 0.352921i 0.984308 + 0.176460i \(0.0564648\pi\)
−0.984308 + 0.176460i \(0.943535\pi\)
\(770\) −26.4853 −0.954463
\(771\) −23.4558 −0.844742
\(772\) 0 0
\(773\) − 27.1716i − 0.977294i −0.872482 0.488647i \(-0.837491\pi\)
0.872482 0.488647i \(-0.162509\pi\)
\(774\) − 7.07107i − 0.254164i
\(775\) 75.9411i 2.72789i
\(776\) 38.8284 1.39386
\(777\) −8.82843 −0.316718
\(778\) − 25.9411i − 0.930034i
\(779\) −3.94113 −0.141205
\(780\) 0 0
\(781\) −4.54416 −0.162603
\(782\) 0.343146i 0.0122709i
\(783\) 32.9706 1.17827
\(784\) 4.00000 0.142857
\(785\) − 54.0416i − 1.92883i
\(786\) − 5.65685i − 0.201773i
\(787\) 33.2426i 1.18497i 0.805581 + 0.592486i \(0.201853\pi\)
−0.805581 + 0.592486i \(0.798147\pi\)
\(788\) 0 0
\(789\) 45.2132 1.60963
\(790\) 9.27208 0.329886
\(791\) 20.3137i 0.722272i
\(792\) 12.0000 0.426401
\(793\) 0 0
\(794\) 34.2426 1.21523
\(795\) 36.3848i 1.29044i
\(796\) 0 0
\(797\) 35.6569 1.26303 0.631515 0.775363i \(-0.282433\pi\)
0.631515 + 0.775363i \(0.282433\pi\)
\(798\) − 2.48528i − 0.0879780i
\(799\) − 6.24264i − 0.220849i
\(800\) 0 0
\(801\) 4.41421i 0.155969i
\(802\) 24.9706 0.881741
\(803\) −3.21320 −0.113391
\(804\) 0 0
\(805\) 0.757359 0.0266934
\(806\) 0 0
\(807\) −20.9706 −0.738199
\(808\) − 4.97056i − 0.174864i
\(809\) 0.514719 0.0180965 0.00904827 0.999959i \(-0.497120\pi\)
0.00904827 + 0.999959i \(0.497120\pi\)
\(810\) 31.2132 1.09672
\(811\) 45.9411i 1.61321i 0.591090 + 0.806606i \(0.298698\pi\)
−0.591090 + 0.806606i \(0.701302\pi\)
\(812\) 0 0
\(813\) 28.2843i 0.991973i
\(814\) − 37.4558i − 1.31283i
\(815\) −37.4558 −1.31202
\(816\) 8.00000 0.280056
\(817\) 6.21320i 0.217372i
\(818\) 7.41421 0.259232
\(819\) 0 0
\(820\) 0 0
\(821\) 8.14214i 0.284162i 0.989855 + 0.142081i \(0.0453794\pi\)
−0.989855 + 0.142081i \(0.954621\pi\)
\(822\) 14.8284 0.517201
\(823\) 6.48528 0.226063 0.113031 0.993591i \(-0.463944\pi\)
0.113031 + 0.993591i \(0.463944\pi\)
\(824\) 22.6274i 0.788263i
\(825\) 86.9117i 3.02588i
\(826\) − 16.4853i − 0.573596i
\(827\) − 1.45584i − 0.0506247i −0.999680 0.0253123i \(-0.991942\pi\)
0.999680 0.0253123i \(-0.00805803\pi\)
\(828\) 0 0
\(829\) −25.2721 −0.877736 −0.438868 0.898552i \(-0.644620\pi\)
−0.438868 + 0.898552i \(0.644620\pi\)
\(830\) − 29.6985i − 1.03085i
\(831\) 13.4142 0.465334
\(832\) 0 0
\(833\) −1.41421 −0.0489996
\(834\) 4.48528i 0.155313i
\(835\) −94.3970 −3.26674
\(836\) 0 0
\(837\) − 29.6569i − 1.02509i
\(838\) 46.4853i 1.60581i
\(839\) 38.8284i 1.34051i 0.742133 + 0.670253i \(0.233814\pi\)
−0.742133 + 0.670253i \(0.766186\pi\)
\(840\) − 17.6569i − 0.609219i
\(841\) 4.97056 0.171399
\(842\) 26.4853 0.912743
\(843\) − 22.0000i − 0.757720i
\(844\) 0 0
\(845\) 0 0
\(846\) −6.24264 −0.214626
\(847\) 7.00000i 0.240523i
\(848\) 23.3137 0.800596
\(849\) 12.0000 0.411839
\(850\) − 28.9706i − 0.993682i
\(851\) 1.07107i 0.0367157i
\(852\) 0 0
\(853\) − 14.2721i − 0.488667i −0.969691 0.244333i \(-0.921431\pi\)
0.969691 0.244333i \(-0.0785691\pi\)
\(854\) 8.48528 0.290360
\(855\) 5.48528 0.187593
\(856\) 23.0294i 0.787130i
\(857\) −18.7696 −0.641156 −0.320578 0.947222i \(-0.603877\pi\)
−0.320578 + 0.947222i \(0.603877\pi\)
\(858\) 0 0
\(859\) −2.97056 −0.101354 −0.0506771 0.998715i \(-0.516138\pi\)
−0.0506771 + 0.998715i \(0.516138\pi\)
\(860\) 0 0
\(861\) −4.48528 −0.152858
\(862\) 33.4558 1.13951
\(863\) − 28.2843i − 0.962808i −0.876499 0.481404i \(-0.840127\pi\)
0.876499 0.481404i \(-0.159873\pi\)
\(864\) 0 0
\(865\) 3.21320i 0.109252i
\(866\) − 12.6863i − 0.431098i
\(867\) 21.2132 0.720438
\(868\) 0 0
\(869\) − 6.30152i − 0.213764i
\(870\) 51.4558 1.74452
\(871\) 0 0
\(872\) 24.6863 0.835983
\(873\) 13.7279i 0.464620i
\(874\) −0.301515 −0.0101989
\(875\) −41.8701 −1.41547
\(876\) 0 0
\(877\) − 1.75736i − 0.0593418i −0.999560 0.0296709i \(-0.990554\pi\)
0.999560 0.0296709i \(-0.00944593\pi\)
\(878\) 24.7696i 0.835932i
\(879\) − 30.2426i − 1.02006i
\(880\) 74.9117 2.52527
\(881\) 49.1127 1.65465 0.827324 0.561724i \(-0.189862\pi\)
0.827324 + 0.561724i \(0.189862\pi\)
\(882\) 1.41421i 0.0476190i
\(883\) −2.00000 −0.0673054 −0.0336527 0.999434i \(-0.510714\pi\)
−0.0336527 + 0.999434i \(0.510714\pi\)
\(884\) 0 0
\(885\) −72.7696 −2.44612
\(886\) − 37.2132i − 1.25020i
\(887\) −43.1127 −1.44758 −0.723791 0.690019i \(-0.757602\pi\)
−0.723791 + 0.690019i \(0.757602\pi\)
\(888\) 24.9706 0.837957
\(889\) 2.00000i 0.0670778i
\(890\) 27.5563i 0.923691i
\(891\) − 21.2132i − 0.710669i
\(892\) 0 0
\(893\) 5.48528 0.183558
\(894\) 15.5147 0.518890
\(895\) 39.7279i 1.32796i
\(896\) −11.3137 −0.377964
\(897\) 0 0
\(898\) 37.9411 1.26611
\(899\) − 30.5563i − 1.01911i
\(900\) 0 0
\(901\) −8.24264 −0.274602
\(902\) − 19.0294i − 0.633611i
\(903\) 7.07107i 0.235310i
\(904\) − 57.4558i − 1.91095i
\(905\) − 29.6985i − 0.987211i
\(906\) 36.4853 1.21214
\(907\) 31.9706 1.06157 0.530783 0.847508i \(-0.321898\pi\)
0.530783 + 0.847508i \(0.321898\pi\)
\(908\) 0 0
\(909\) 1.75736 0.0582879
\(910\) 0 0
\(911\) −10.0294 −0.332290 −0.166145 0.986101i \(-0.553132\pi\)
−0.166145 + 0.986101i \(0.553132\pi\)
\(912\) 7.02944i 0.232768i
\(913\) −20.1838 −0.667985
\(914\) −49.7990 −1.64720
\(915\) − 37.4558i − 1.23825i
\(916\) 0 0
\(917\) − 2.82843i − 0.0934029i
\(918\) 11.3137i 0.373408i
\(919\) 24.0000 0.791687 0.395843 0.918318i \(-0.370452\pi\)
0.395843 + 0.918318i \(0.370452\pi\)
\(920\) −2.14214 −0.0706241
\(921\) 6.72792i 0.221693i
\(922\) −4.48528 −0.147715
\(923\) 0 0
\(924\) 0 0
\(925\) − 90.4264i − 2.97320i
\(926\) 6.00000 0.197172
\(927\) −8.00000 −0.262754
\(928\) 0 0
\(929\) 17.1005i 0.561049i 0.959847 + 0.280525i \(0.0905085\pi\)
−0.959847 + 0.280525i \(0.909492\pi\)
\(930\) − 46.2843i − 1.51772i
\(931\) − 1.24264i − 0.0407259i
\(932\) 0 0
\(933\) −6.48528 −0.212319
\(934\) − 22.4853i − 0.735741i
\(935\) −26.4853 −0.866161
\(936\) 0 0
\(937\) 2.78680 0.0910407 0.0455203 0.998963i \(-0.485505\pi\)
0.0455203 + 0.998963i \(0.485505\pi\)
\(938\) 3.51472i 0.114760i
\(939\) 27.1716 0.886711
\(940\) 0 0
\(941\) 25.9289i 0.845259i 0.906303 + 0.422630i \(0.138893\pi\)
−0.906303 + 0.422630i \(0.861107\pi\)
\(942\) 24.4853i 0.797774i
\(943\) 0.544156i 0.0177202i
\(944\) 46.6274i 1.51759i
\(945\) 24.9706 0.812292
\(946\) −30.0000 −0.975384
\(947\) − 20.8284i − 0.676833i −0.940996 0.338416i \(-0.890109\pi\)
0.940996 0.338416i \(-0.109891\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 25.4558 0.825897
\(951\) − 16.0000i − 0.518836i
\(952\) 4.00000 0.129641
\(953\) 17.1421 0.555288 0.277644 0.960684i \(-0.410446\pi\)
0.277644 + 0.960684i \(0.410446\pi\)
\(954\) 8.24264i 0.266865i
\(955\) 91.9411i 2.97514i
\(956\) 0 0
\(957\) − 34.9706i − 1.13044i
\(958\) 51.2132 1.65462
\(959\) 7.41421 0.239417
\(960\) 49.9411i 1.61184i
\(961\) 3.51472 0.113378
\(962\) 0 0
\(963\) −8.14214 −0.262377
\(964\) 0 0
\(965\) 10.9706 0.353155
\(966\) −0.343146 −0.0110405
\(967\) − 41.6985i − 1.34093i −0.741940 0.670466i \(-0.766094\pi\)
0.741940 0.670466i \(-0.233906\pi\)
\(968\) − 19.7990i − 0.636364i
\(969\) − 2.48528i − 0.0798387i
\(970\) 85.6985i 2.75161i
\(971\) −7.45584 −0.239269 −0.119635 0.992818i \(-0.538172\pi\)
−0.119635 + 0.992818i \(0.538172\pi\)
\(972\) 0 0
\(973\) 2.24264i 0.0718958i
\(974\) −55.7990 −1.78791
\(975\) 0 0
\(976\) −24.0000 −0.768221
\(977\) − 24.0416i − 0.769160i −0.923092 0.384580i \(-0.874346\pi\)
0.923092 0.384580i \(-0.125654\pi\)
\(978\) 16.9706 0.542659
\(979\) 18.7279 0.598547
\(980\) 0 0
\(981\) 8.72792i 0.278661i
\(982\) 40.4853i 1.29194i
\(983\) − 33.0416i − 1.05386i −0.849907 0.526932i \(-0.823342\pi\)
0.849907 0.526932i \(-0.176658\pi\)
\(984\) 12.6863 0.404424
\(985\) −104.426 −3.32730
\(986\) 11.6569i 0.371230i
\(987\) 6.24264 0.198705
\(988\) 0 0
\(989\) 0.857864 0.0272785
\(990\) 26.4853i 0.841757i
\(991\) 34.9706 1.11088 0.555438 0.831558i \(-0.312551\pi\)
0.555438 + 0.831558i \(0.312551\pi\)
\(992\) 0 0
\(993\) 25.4558i 0.807817i
\(994\) 1.51472i 0.0480440i
\(995\) 54.0416i 1.71323i
\(996\) 0 0
\(997\) −28.4853 −0.902138 −0.451069 0.892489i \(-0.648957\pi\)
−0.451069 + 0.892489i \(0.648957\pi\)
\(998\) 54.7696 1.73370
\(999\) 35.3137i 1.11728i
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 1183.2.c.d.337.3 4
13.5 odd 4 91.2.a.c.1.2 2
13.8 odd 4 1183.2.a.d.1.1 2
13.12 even 2 inner 1183.2.c.d.337.1 4
39.5 even 4 819.2.a.h.1.1 2
52.31 even 4 1456.2.a.q.1.2 2
65.44 odd 4 2275.2.a.j.1.1 2
91.5 even 12 637.2.e.g.508.1 4
91.18 odd 12 637.2.e.f.79.1 4
91.31 even 12 637.2.e.g.79.1 4
91.34 even 4 8281.2.a.v.1.1 2
91.44 odd 12 637.2.e.f.508.1 4
91.83 even 4 637.2.a.g.1.2 2
104.5 odd 4 5824.2.a.bl.1.2 2
104.83 even 4 5824.2.a.bk.1.1 2
273.83 odd 4 5733.2.a.s.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
91.2.a.c.1.2 2 13.5 odd 4
637.2.a.g.1.2 2 91.83 even 4
637.2.e.f.79.1 4 91.18 odd 12
637.2.e.f.508.1 4 91.44 odd 12
637.2.e.g.79.1 4 91.31 even 12
637.2.e.g.508.1 4 91.5 even 12
819.2.a.h.1.1 2 39.5 even 4
1183.2.a.d.1.1 2 13.8 odd 4
1183.2.c.d.337.1 4 13.12 even 2 inner
1183.2.c.d.337.3 4 1.1 even 1 trivial
1456.2.a.q.1.2 2 52.31 even 4
2275.2.a.j.1.1 2 65.44 odd 4
5733.2.a.s.1.1 2 273.83 odd 4
5824.2.a.bk.1.1 2 104.83 even 4
5824.2.a.bl.1.2 2 104.5 odd 4
8281.2.a.v.1.1 2 91.34 even 4