Properties

Label 2240.2.e.e.2239.3
Level $2240$
Weight $2$
Character 2240.2239
Analytic conductor $17.886$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2240,2,Mod(2239,2240)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2240, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2240.2239");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2240 = 2^{6} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2240.e (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: \(17.8864900528\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.12745506816.5
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 8x^{6} + 55x^{4} - 72x^{2} + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 560)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 2239.3
Root \(1.00781 - 0.581861i\) of defining polynomial
Character \(\chi\) \(=\) 2240.2239
Dual form 2240.2.e.e.2239.8

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.41421i q^{3} +(1.22474 - 1.87083i) q^{5} +2.64575 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.41421i q^{3} +(1.22474 - 1.87083i) q^{5} +2.64575 q^{7} +1.00000 q^{9} -3.46410i q^{11} +2.44949 q^{13} +(-2.64575 - 1.73205i) q^{15} -4.89898 q^{17} -6.48074 q^{19} -3.74166i q^{21} +(-2.00000 - 4.58258i) q^{25} -5.65685i q^{27} +6.00000 q^{29} -4.89898 q^{33} +(3.24037 - 4.94975i) q^{35} -9.16515i q^{37} -3.46410i q^{39} +7.48331i q^{41} +5.29150 q^{43} +(1.22474 - 1.87083i) q^{45} -2.82843i q^{47} +7.00000 q^{49} +6.92820i q^{51} +9.16515i q^{53} +(-6.48074 - 4.24264i) q^{55} +9.16515i q^{57} -6.48074 q^{59} +11.2250i q^{61} +2.64575 q^{63} +(3.00000 - 4.58258i) q^{65} -5.29150 q^{67} +9.79796 q^{73} +(-6.48074 + 2.82843i) q^{75} -9.16515i q^{77} +6.92820i q^{79} -5.00000 q^{81} -9.89949i q^{83} +(-6.00000 + 9.16515i) q^{85} -8.48528i q^{87} -7.48331i q^{89} +6.48074 q^{91} +(-7.93725 + 12.1244i) q^{95} +14.6969 q^{97} -3.46410i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{9} - 16 q^{25} + 48 q^{29} + 56 q^{49} + 24 q^{65} - 40 q^{81} - 48 q^{85}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2240\mathbb{Z}\right)^\times\).

\(n\) \(897\) \(1471\) \(1541\) \(1921\)
\(\chi(n)\) \(-1\) \(-1\) \(1\) \(-1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.41421i 0.816497i −0.912871 0.408248i \(-0.866140\pi\)
0.912871 0.408248i \(-0.133860\pi\)
\(4\) 0 0
\(5\) 1.22474 1.87083i 0.547723 0.836660i
\(6\) 0 0
\(7\) 2.64575 1.00000
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 3.46410i 1.04447i −0.852803 0.522233i \(-0.825099\pi\)
0.852803 0.522233i \(-0.174901\pi\)
\(12\) 0 0
\(13\) 2.44949 0.679366 0.339683 0.940540i \(-0.389680\pi\)
0.339683 + 0.940540i \(0.389680\pi\)
\(14\) 0 0
\(15\) −2.64575 1.73205i −0.683130 0.447214i
\(16\) 0 0
\(17\) −4.89898 −1.18818 −0.594089 0.804400i \(-0.702487\pi\)
−0.594089 + 0.804400i \(0.702487\pi\)
\(18\) 0 0
\(19\) −6.48074 −1.48678 −0.743392 0.668856i \(-0.766784\pi\)
−0.743392 + 0.668856i \(0.766784\pi\)
\(20\) 0 0
\(21\) 3.74166i 0.816497i
\(22\) 0 0
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) −2.00000 4.58258i −0.400000 0.916515i
\(26\) 0 0
\(27\) 5.65685i 1.08866i
\(28\) 0 0
\(29\) 6.00000 1.11417 0.557086 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 0 0
\(33\) −4.89898 −0.852803
\(34\) 0 0
\(35\) 3.24037 4.94975i 0.547723 0.836660i
\(36\) 0 0
\(37\) 9.16515i 1.50674i −0.657596 0.753371i \(-0.728427\pi\)
0.657596 0.753371i \(-0.271573\pi\)
\(38\) 0 0
\(39\) 3.46410i 0.554700i
\(40\) 0 0
\(41\) 7.48331i 1.16870i 0.811503 + 0.584349i \(0.198650\pi\)
−0.811503 + 0.584349i \(0.801350\pi\)
\(42\) 0 0
\(43\) 5.29150 0.806947 0.403473 0.914991i \(-0.367803\pi\)
0.403473 + 0.914991i \(0.367803\pi\)
\(44\) 0 0
\(45\) 1.22474 1.87083i 0.182574 0.278887i
\(46\) 0 0
\(47\) 2.82843i 0.412568i −0.978492 0.206284i \(-0.933863\pi\)
0.978492 0.206284i \(-0.0661372\pi\)
\(48\) 0 0
\(49\) 7.00000 1.00000
\(50\) 0 0
\(51\) 6.92820i 0.970143i
\(52\) 0 0
\(53\) 9.16515i 1.25893i 0.777029 + 0.629465i \(0.216726\pi\)
−0.777029 + 0.629465i \(0.783274\pi\)
\(54\) 0 0
\(55\) −6.48074 4.24264i −0.873863 0.572078i
\(56\) 0 0
\(57\) 9.16515i 1.21395i
\(58\) 0 0
\(59\) −6.48074 −0.843721 −0.421860 0.906661i \(-0.638623\pi\)
−0.421860 + 0.906661i \(0.638623\pi\)
\(60\) 0 0
\(61\) 11.2250i 1.43721i 0.695418 + 0.718605i \(0.255219\pi\)
−0.695418 + 0.718605i \(0.744781\pi\)
\(62\) 0 0
\(63\) 2.64575 0.333333
\(64\) 0 0
\(65\) 3.00000 4.58258i 0.372104 0.568399i
\(66\) 0 0
\(67\) −5.29150 −0.646460 −0.323230 0.946320i \(-0.604769\pi\)
−0.323230 + 0.946320i \(0.604769\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) 9.79796 1.14676 0.573382 0.819288i \(-0.305631\pi\)
0.573382 + 0.819288i \(0.305631\pi\)
\(74\) 0 0
\(75\) −6.48074 + 2.82843i −0.748331 + 0.326599i
\(76\) 0 0
\(77\) 9.16515i 1.04447i
\(78\) 0 0
\(79\) 6.92820i 0.779484i 0.920924 + 0.389742i \(0.127436\pi\)
−0.920924 + 0.389742i \(0.872564\pi\)
\(80\) 0 0
\(81\) −5.00000 −0.555556
\(82\) 0 0
\(83\) 9.89949i 1.08661i −0.839535 0.543305i \(-0.817173\pi\)
0.839535 0.543305i \(-0.182827\pi\)
\(84\) 0 0
\(85\) −6.00000 + 9.16515i −0.650791 + 0.994100i
\(86\) 0 0
\(87\) 8.48528i 0.909718i
\(88\) 0 0
\(89\) 7.48331i 0.793230i −0.917985 0.396615i \(-0.870185\pi\)
0.917985 0.396615i \(-0.129815\pi\)
\(90\) 0 0
\(91\) 6.48074 0.679366
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −7.93725 + 12.1244i −0.814345 + 1.24393i
\(96\) 0 0
\(97\) 14.6969 1.49225 0.746124 0.665807i \(-0.231913\pi\)
0.746124 + 0.665807i \(0.231913\pi\)
\(98\) 0 0
\(99\) 3.46410i 0.348155i
\(100\) 0 0
\(101\) 3.74166i 0.372309i 0.982521 + 0.186154i \(0.0596025\pi\)
−0.982521 + 0.186154i \(0.940398\pi\)
\(102\) 0 0
\(103\) 16.9706i 1.67216i −0.548608 0.836080i \(-0.684842\pi\)
0.548608 0.836080i \(-0.315158\pi\)
\(104\) 0 0
\(105\) −7.00000 4.58258i −0.683130 0.447214i
\(106\) 0 0
\(107\) −15.8745 −1.53465 −0.767323 0.641260i \(-0.778412\pi\)
−0.767323 + 0.641260i \(0.778412\pi\)
\(108\) 0 0
\(109\) 2.00000 0.191565 0.0957826 0.995402i \(-0.469465\pi\)
0.0957826 + 0.995402i \(0.469465\pi\)
\(110\) 0 0
\(111\) −12.9615 −1.23025
\(112\) 0 0
\(113\) 9.16515i 0.862185i −0.902308 0.431092i \(-0.858128\pi\)
0.902308 0.431092i \(-0.141872\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 2.44949 0.226455
\(118\) 0 0
\(119\) −12.9615 −1.18818
\(120\) 0 0
\(121\) −1.00000 −0.0909091
\(122\) 0 0
\(123\) 10.5830 0.954237
\(124\) 0 0
\(125\) −11.0227 1.87083i −0.985901 0.167332i
\(126\) 0 0
\(127\) 10.5830 0.939090 0.469545 0.882909i \(-0.344418\pi\)
0.469545 + 0.882909i \(0.344418\pi\)
\(128\) 0 0
\(129\) 7.48331i 0.658869i
\(130\) 0 0
\(131\) −19.4422 −1.69867 −0.849337 0.527850i \(-0.822998\pi\)
−0.849337 + 0.527850i \(0.822998\pi\)
\(132\) 0 0
\(133\) −17.1464 −1.48678
\(134\) 0 0
\(135\) −10.5830 6.92820i −0.910840 0.596285i
\(136\) 0 0
\(137\) 18.3303i 1.56606i 0.621982 + 0.783032i \(0.286328\pi\)
−0.621982 + 0.783032i \(0.713672\pi\)
\(138\) 0 0
\(139\) 19.4422 1.64907 0.824534 0.565813i \(-0.191437\pi\)
0.824534 + 0.565813i \(0.191437\pi\)
\(140\) 0 0
\(141\) −4.00000 −0.336861
\(142\) 0 0
\(143\) 8.48528i 0.709575i
\(144\) 0 0
\(145\) 7.34847 11.2250i 0.610257 0.932183i
\(146\) 0 0
\(147\) 9.89949i 0.816497i
\(148\) 0 0
\(149\) −6.00000 −0.491539 −0.245770 0.969328i \(-0.579041\pi\)
−0.245770 + 0.969328i \(0.579041\pi\)
\(150\) 0 0
\(151\) 3.46410i 0.281905i −0.990016 0.140952i \(-0.954984\pi\)
0.990016 0.140952i \(-0.0450164\pi\)
\(152\) 0 0
\(153\) −4.89898 −0.396059
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −7.34847 −0.586472 −0.293236 0.956040i \(-0.594732\pi\)
−0.293236 + 0.956040i \(0.594732\pi\)
\(158\) 0 0
\(159\) 12.9615 1.02791
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 5.29150 0.414462 0.207231 0.978292i \(-0.433555\pi\)
0.207231 + 0.978292i \(0.433555\pi\)
\(164\) 0 0
\(165\) −6.00000 + 9.16515i −0.467099 + 0.713506i
\(166\) 0 0
\(167\) 5.65685i 0.437741i −0.975754 0.218870i \(-0.929763\pi\)
0.975754 0.218870i \(-0.0702371\pi\)
\(168\) 0 0
\(169\) −7.00000 −0.538462
\(170\) 0 0
\(171\) −6.48074 −0.495595
\(172\) 0 0
\(173\) 7.34847 0.558694 0.279347 0.960190i \(-0.409882\pi\)
0.279347 + 0.960190i \(0.409882\pi\)
\(174\) 0 0
\(175\) −5.29150 12.1244i −0.400000 0.916515i
\(176\) 0 0
\(177\) 9.16515i 0.688895i
\(178\) 0 0
\(179\) 24.2487i 1.81243i 0.422813 + 0.906217i \(0.361043\pi\)
−0.422813 + 0.906217i \(0.638957\pi\)
\(180\) 0 0
\(181\) 11.2250i 0.834346i 0.908827 + 0.417173i \(0.136979\pi\)
−0.908827 + 0.417173i \(0.863021\pi\)
\(182\) 0 0
\(183\) 15.8745 1.17348
\(184\) 0 0
\(185\) −17.1464 11.2250i −1.26063 0.825276i
\(186\) 0 0
\(187\) 16.9706i 1.24101i
\(188\) 0 0
\(189\) 14.9666i 1.08866i
\(190\) 0 0
\(191\) 6.92820i 0.501307i 0.968077 + 0.250654i \(0.0806455\pi\)
−0.968077 + 0.250654i \(0.919354\pi\)
\(192\) 0 0
\(193\) 9.16515i 0.659722i 0.944030 + 0.329861i \(0.107002\pi\)
−0.944030 + 0.329861i \(0.892998\pi\)
\(194\) 0 0
\(195\) −6.48074 4.24264i −0.464095 0.303822i
\(196\) 0 0
\(197\) 9.16515i 0.652990i −0.945199 0.326495i \(-0.894132\pi\)
0.945199 0.326495i \(-0.105868\pi\)
\(198\) 0 0
\(199\) −12.9615 −0.918815 −0.459408 0.888226i \(-0.651938\pi\)
−0.459408 + 0.888226i \(0.651938\pi\)
\(200\) 0 0
\(201\) 7.48331i 0.527832i
\(202\) 0 0
\(203\) 15.8745 1.11417
\(204\) 0 0
\(205\) 14.0000 + 9.16515i 0.977802 + 0.640122i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 22.4499i 1.55290i
\(210\) 0 0
\(211\) 10.3923i 0.715436i −0.933830 0.357718i \(-0.883555\pi\)
0.933830 0.357718i \(-0.116445\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 6.48074 9.89949i 0.441983 0.675140i
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 13.8564i 0.936329i
\(220\) 0 0
\(221\) −12.0000 −0.807207
\(222\) 0 0
\(223\) 25.4558i 1.70465i 0.523013 + 0.852325i \(0.324808\pi\)
−0.523013 + 0.852325i \(0.675192\pi\)
\(224\) 0 0
\(225\) −2.00000 4.58258i −0.133333 0.305505i
\(226\) 0 0
\(227\) 18.3848i 1.22024i −0.792309 0.610120i \(-0.791121\pi\)
0.792309 0.610120i \(-0.208879\pi\)
\(228\) 0 0
\(229\) 11.2250i 0.741767i 0.928679 + 0.370884i \(0.120945\pi\)
−0.928679 + 0.370884i \(0.879055\pi\)
\(230\) 0 0
\(231\) −12.9615 −0.852803
\(232\) 0 0
\(233\) 18.3303i 1.20086i −0.799678 0.600429i \(-0.794996\pi\)
0.799678 0.600429i \(-0.205004\pi\)
\(234\) 0 0
\(235\) −5.29150 3.46410i −0.345180 0.225973i
\(236\) 0 0
\(237\) 9.79796 0.636446
\(238\) 0 0
\(239\) 10.3923i 0.672222i −0.941822 0.336111i \(-0.890888\pi\)
0.941822 0.336111i \(-0.109112\pi\)
\(240\) 0 0
\(241\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(242\) 0 0
\(243\) 9.89949i 0.635053i
\(244\) 0 0
\(245\) 8.57321 13.0958i 0.547723 0.836660i
\(246\) 0 0
\(247\) −15.8745 −1.01007
\(248\) 0 0
\(249\) −14.0000 −0.887214
\(250\) 0 0
\(251\) 6.48074 0.409061 0.204530 0.978860i \(-0.434433\pi\)
0.204530 + 0.978860i \(0.434433\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 12.9615 + 8.48528i 0.811679 + 0.531369i
\(256\) 0 0
\(257\) −14.6969 −0.916770 −0.458385 0.888754i \(-0.651572\pi\)
−0.458385 + 0.888754i \(0.651572\pi\)
\(258\) 0 0
\(259\) 24.2487i 1.50674i
\(260\) 0 0
\(261\) 6.00000 0.371391
\(262\) 0 0
\(263\) 15.8745 0.978864 0.489432 0.872041i \(-0.337204\pi\)
0.489432 + 0.872041i \(0.337204\pi\)
\(264\) 0 0
\(265\) 17.1464 + 11.2250i 1.05330 + 0.689545i
\(266\) 0 0
\(267\) −10.5830 −0.647669
\(268\) 0 0
\(269\) 18.7083i 1.14066i −0.821414 0.570332i \(-0.806814\pi\)
0.821414 0.570332i \(-0.193186\pi\)
\(270\) 0 0
\(271\) 25.9230 1.57471 0.787354 0.616501i \(-0.211450\pi\)
0.787354 + 0.616501i \(0.211450\pi\)
\(272\) 0 0
\(273\) 9.16515i 0.554700i
\(274\) 0 0
\(275\) −15.8745 + 6.92820i −0.957269 + 0.417786i
\(276\) 0 0
\(277\) 9.16515i 0.550681i −0.961347 0.275340i \(-0.911209\pi\)
0.961347 0.275340i \(-0.0887905\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 6.00000 0.357930 0.178965 0.983855i \(-0.442725\pi\)
0.178965 + 0.983855i \(0.442725\pi\)
\(282\) 0 0
\(283\) 21.2132i 1.26099i 0.776192 + 0.630497i \(0.217149\pi\)
−0.776192 + 0.630497i \(0.782851\pi\)
\(284\) 0 0
\(285\) 17.1464 + 11.2250i 1.01567 + 0.664910i
\(286\) 0 0
\(287\) 19.7990i 1.16870i
\(288\) 0 0
\(289\) 7.00000 0.411765
\(290\) 0 0
\(291\) 20.7846i 1.21842i
\(292\) 0 0
\(293\) 31.8434 1.86031 0.930155 0.367168i \(-0.119673\pi\)
0.930155 + 0.367168i \(0.119673\pi\)
\(294\) 0 0
\(295\) −7.93725 + 12.1244i −0.462125 + 0.705907i
\(296\) 0 0
\(297\) −19.5959 −1.13707
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 14.0000 0.806947
\(302\) 0 0
\(303\) 5.29150 0.303989
\(304\) 0 0
\(305\) 21.0000 + 13.7477i 1.20246 + 0.787193i
\(306\) 0 0
\(307\) 4.24264i 0.242140i 0.992644 + 0.121070i \(0.0386326\pi\)
−0.992644 + 0.121070i \(0.961367\pi\)
\(308\) 0 0
\(309\) −24.0000 −1.36531
\(310\) 0 0
\(311\) 12.9615 0.734978 0.367489 0.930028i \(-0.380218\pi\)
0.367489 + 0.930028i \(0.380218\pi\)
\(312\) 0 0
\(313\) −9.79796 −0.553813 −0.276907 0.960897i \(-0.589309\pi\)
−0.276907 + 0.960897i \(0.589309\pi\)
\(314\) 0 0
\(315\) 3.24037 4.94975i 0.182574 0.278887i
\(316\) 0 0
\(317\) 9.16515i 0.514766i 0.966309 + 0.257383i \(0.0828602\pi\)
−0.966309 + 0.257383i \(0.917140\pi\)
\(318\) 0 0
\(319\) 20.7846i 1.16371i
\(320\) 0 0
\(321\) 22.4499i 1.25303i
\(322\) 0 0
\(323\) 31.7490 1.76656
\(324\) 0 0
\(325\) −4.89898 11.2250i −0.271746 0.622649i
\(326\) 0 0
\(327\) 2.82843i 0.156412i
\(328\) 0 0
\(329\) 7.48331i 0.412568i
\(330\) 0 0
\(331\) 24.2487i 1.33283i 0.745581 + 0.666415i \(0.232172\pi\)
−0.745581 + 0.666415i \(0.767828\pi\)
\(332\) 0 0
\(333\) 9.16515i 0.502247i
\(334\) 0 0
\(335\) −6.48074 + 9.89949i −0.354081 + 0.540867i
\(336\) 0 0
\(337\) 9.16515i 0.499258i −0.968342 0.249629i \(-0.919691\pi\)
0.968342 0.249629i \(-0.0803086\pi\)
\(338\) 0 0
\(339\) −12.9615 −0.703971
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 18.5203 1.00000
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −15.8745 −0.852188 −0.426094 0.904679i \(-0.640111\pi\)
−0.426094 + 0.904679i \(0.640111\pi\)
\(348\) 0 0
\(349\) 11.2250i 0.600859i 0.953804 + 0.300429i \(0.0971300\pi\)
−0.953804 + 0.300429i \(0.902870\pi\)
\(350\) 0 0
\(351\) 13.8564i 0.739600i
\(352\) 0 0
\(353\) 4.89898 0.260746 0.130373 0.991465i \(-0.458382\pi\)
0.130373 + 0.991465i \(0.458382\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 18.3303i 0.970143i
\(358\) 0 0
\(359\) 24.2487i 1.27980i 0.768459 + 0.639899i \(0.221024\pi\)
−0.768459 + 0.639899i \(0.778976\pi\)
\(360\) 0 0
\(361\) 23.0000 1.21053
\(362\) 0 0
\(363\) 1.41421i 0.0742270i
\(364\) 0 0
\(365\) 12.0000 18.3303i 0.628109 0.959452i
\(366\) 0 0
\(367\) 8.48528i 0.442928i 0.975169 + 0.221464i \(0.0710835\pi\)
−0.975169 + 0.221464i \(0.928916\pi\)
\(368\) 0 0
\(369\) 7.48331i 0.389566i
\(370\) 0 0
\(371\) 24.2487i 1.25893i
\(372\) 0 0
\(373\) 9.16515i 0.474554i −0.971442 0.237277i \(-0.923745\pi\)
0.971442 0.237277i \(-0.0762548\pi\)
\(374\) 0 0
\(375\) −2.64575 + 15.5885i −0.136626 + 0.804984i
\(376\) 0 0
\(377\) 14.6969 0.756931
\(378\) 0 0
\(379\) 24.2487i 1.24557i 0.782392 + 0.622786i \(0.213999\pi\)
−0.782392 + 0.622786i \(0.786001\pi\)
\(380\) 0 0
\(381\) 14.9666i 0.766764i
\(382\) 0 0
\(383\) 19.7990i 1.01168i 0.862627 + 0.505841i \(0.168818\pi\)
−0.862627 + 0.505841i \(0.831182\pi\)
\(384\) 0 0
\(385\) −17.1464 11.2250i −0.873863 0.572078i
\(386\) 0 0
\(387\) 5.29150 0.268982
\(388\) 0 0
\(389\) 30.0000 1.52106 0.760530 0.649303i \(-0.224939\pi\)
0.760530 + 0.649303i \(0.224939\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 27.4955i 1.38696i
\(394\) 0 0
\(395\) 12.9615 + 8.48528i 0.652163 + 0.426941i
\(396\) 0 0
\(397\) −7.34847 −0.368809 −0.184405 0.982850i \(-0.559036\pi\)
−0.184405 + 0.982850i \(0.559036\pi\)
\(398\) 0 0
\(399\) 24.2487i 1.21395i
\(400\) 0 0
\(401\) −24.0000 −1.19850 −0.599251 0.800561i \(-0.704535\pi\)
−0.599251 + 0.800561i \(0.704535\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) −6.12372 + 9.35414i −0.304290 + 0.464811i
\(406\) 0 0
\(407\) −31.7490 −1.57374
\(408\) 0 0
\(409\) 22.4499i 1.11008i −0.831824 0.555039i \(-0.812703\pi\)
0.831824 0.555039i \(-0.187297\pi\)
\(410\) 0 0
\(411\) 25.9230 1.27869
\(412\) 0 0
\(413\) −17.1464 −0.843721
\(414\) 0 0
\(415\) −18.5203 12.1244i −0.909124 0.595161i
\(416\) 0 0
\(417\) 27.4955i 1.34646i
\(418\) 0 0
\(419\) −6.48074 −0.316605 −0.158302 0.987391i \(-0.550602\pi\)
−0.158302 + 0.987391i \(0.550602\pi\)
\(420\) 0 0
\(421\) 34.0000 1.65706 0.828529 0.559946i \(-0.189178\pi\)
0.828529 + 0.559946i \(0.189178\pi\)
\(422\) 0 0
\(423\) 2.82843i 0.137523i
\(424\) 0 0
\(425\) 9.79796 + 22.4499i 0.475271 + 1.08898i
\(426\) 0 0
\(427\) 29.6985i 1.43721i
\(428\) 0 0
\(429\) −12.0000 −0.579365
\(430\) 0 0
\(431\) 3.46410i 0.166860i 0.996514 + 0.0834300i \(0.0265875\pi\)
−0.996514 + 0.0834300i \(0.973413\pi\)
\(432\) 0 0
\(433\) 4.89898 0.235430 0.117715 0.993047i \(-0.462443\pi\)
0.117715 + 0.993047i \(0.462443\pi\)
\(434\) 0 0
\(435\) −15.8745 10.3923i −0.761124 0.498273i
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 12.9615 0.618618 0.309309 0.950962i \(-0.399902\pi\)
0.309309 + 0.950962i \(0.399902\pi\)
\(440\) 0 0
\(441\) 7.00000 0.333333
\(442\) 0 0
\(443\) 15.8745 0.754221 0.377110 0.926168i \(-0.376918\pi\)
0.377110 + 0.926168i \(0.376918\pi\)
\(444\) 0 0
\(445\) −14.0000 9.16515i −0.663664 0.434470i
\(446\) 0 0
\(447\) 8.48528i 0.401340i
\(448\) 0 0
\(449\) −6.00000 −0.283158 −0.141579 0.989927i \(-0.545218\pi\)
−0.141579 + 0.989927i \(0.545218\pi\)
\(450\) 0 0
\(451\) 25.9230 1.22066
\(452\) 0 0
\(453\) −4.89898 −0.230174
\(454\) 0 0
\(455\) 7.93725 12.1244i 0.372104 0.568399i
\(456\) 0 0
\(457\) 27.4955i 1.28618i −0.765789 0.643092i \(-0.777651\pi\)
0.765789 0.643092i \(-0.222349\pi\)
\(458\) 0 0
\(459\) 27.7128i 1.29352i
\(460\) 0 0
\(461\) 18.7083i 0.871332i 0.900108 + 0.435666i \(0.143487\pi\)
−0.900108 + 0.435666i \(0.856513\pi\)
\(462\) 0 0
\(463\) 26.4575 1.22958 0.614792 0.788689i \(-0.289240\pi\)
0.614792 + 0.788689i \(0.289240\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 7.07107i 0.327210i 0.986526 + 0.163605i \(0.0523123\pi\)
−0.986526 + 0.163605i \(0.947688\pi\)
\(468\) 0 0
\(469\) −14.0000 −0.646460
\(470\) 0 0
\(471\) 10.3923i 0.478852i
\(472\) 0 0
\(473\) 18.3303i 0.842828i
\(474\) 0 0
\(475\) 12.9615 + 29.6985i 0.594714 + 1.36266i
\(476\) 0 0
\(477\) 9.16515i 0.419643i
\(478\) 0 0
\(479\) 25.9230 1.18445 0.592225 0.805772i \(-0.298250\pi\)
0.592225 + 0.805772i \(0.298250\pi\)
\(480\) 0 0
\(481\) 22.4499i 1.02363i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 18.0000 27.4955i 0.817338 1.24850i
\(486\) 0 0
\(487\) −21.1660 −0.959123 −0.479562 0.877508i \(-0.659204\pi\)
−0.479562 + 0.877508i \(0.659204\pi\)
\(488\) 0 0
\(489\) 7.48331i 0.338407i
\(490\) 0 0
\(491\) 10.3923i 0.468998i −0.972116 0.234499i \(-0.924655\pi\)
0.972116 0.234499i \(-0.0753450\pi\)
\(492\) 0 0
\(493\) −29.3939 −1.32383
\(494\) 0 0
\(495\) −6.48074 4.24264i −0.291288 0.190693i
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 17.3205i 0.775372i −0.921791 0.387686i \(-0.873274\pi\)
0.921791 0.387686i \(-0.126726\pi\)
\(500\) 0 0
\(501\) −8.00000 −0.357414
\(502\) 0 0
\(503\) 5.65685i 0.252227i −0.992016 0.126113i \(-0.959750\pi\)
0.992016 0.126113i \(-0.0402503\pi\)
\(504\) 0 0
\(505\) 7.00000 + 4.58258i 0.311496 + 0.203922i
\(506\) 0 0
\(507\) 9.89949i 0.439652i
\(508\) 0 0
\(509\) 3.74166i 0.165846i 0.996556 + 0.0829230i \(0.0264256\pi\)
−0.996556 + 0.0829230i \(0.973574\pi\)
\(510\) 0 0
\(511\) 25.9230 1.14676
\(512\) 0 0
\(513\) 36.6606i 1.61861i
\(514\) 0 0
\(515\) −31.7490 20.7846i −1.39903 0.915879i
\(516\) 0 0
\(517\) −9.79796 −0.430914
\(518\) 0 0
\(519\) 10.3923i 0.456172i
\(520\) 0 0
\(521\) 37.4166i 1.63925i −0.572900 0.819625i \(-0.694182\pi\)
0.572900 0.819625i \(-0.305818\pi\)
\(522\) 0 0
\(523\) 29.6985i 1.29862i 0.760522 + 0.649312i \(0.224943\pi\)
−0.760522 + 0.649312i \(0.775057\pi\)
\(524\) 0 0
\(525\) −17.1464 + 7.48331i −0.748331 + 0.326599i
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −23.0000 −1.00000
\(530\) 0 0
\(531\) −6.48074 −0.281240
\(532\) 0 0
\(533\) 18.3303i 0.793974i
\(534\) 0 0
\(535\) −19.4422 + 29.6985i −0.840561 + 1.28398i
\(536\) 0 0
\(537\) 34.2929 1.47985
\(538\) 0 0
\(539\) 24.2487i 1.04447i
\(540\) 0 0
\(541\) −38.0000 −1.63375 −0.816874 0.576816i \(-0.804295\pi\)
−0.816874 + 0.576816i \(0.804295\pi\)
\(542\) 0 0
\(543\) 15.8745 0.681240
\(544\) 0 0
\(545\) 2.44949 3.74166i 0.104925 0.160275i
\(546\) 0 0
\(547\) 26.4575 1.13124 0.565621 0.824665i \(-0.308637\pi\)
0.565621 + 0.824665i \(0.308637\pi\)
\(548\) 0 0
\(549\) 11.2250i 0.479070i
\(550\) 0 0
\(551\) −38.8844 −1.65653
\(552\) 0 0
\(553\) 18.3303i 0.779484i
\(554\) 0 0
\(555\) −15.8745 + 24.2487i −0.673835 + 1.02930i
\(556\) 0 0
\(557\) 27.4955i 1.16502i 0.812824 + 0.582510i \(0.197929\pi\)
−0.812824 + 0.582510i \(0.802071\pi\)
\(558\) 0 0
\(559\) 12.9615 0.548212
\(560\) 0 0
\(561\) 24.0000 1.01328
\(562\) 0 0
\(563\) 9.89949i 0.417214i 0.978000 + 0.208607i \(0.0668929\pi\)
−0.978000 + 0.208607i \(0.933107\pi\)
\(564\) 0 0
\(565\) −17.1464 11.2250i −0.721356 0.472238i
\(566\) 0 0
\(567\) −13.2288 −0.555556
\(568\) 0 0
\(569\) 36.0000 1.50920 0.754599 0.656186i \(-0.227831\pi\)
0.754599 + 0.656186i \(0.227831\pi\)
\(570\) 0 0
\(571\) 24.2487i 1.01478i −0.861717 0.507388i \(-0.830611\pi\)
0.861717 0.507388i \(-0.169389\pi\)
\(572\) 0 0
\(573\) 9.79796 0.409316
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 4.89898 0.203947 0.101974 0.994787i \(-0.467484\pi\)
0.101974 + 0.994787i \(0.467484\pi\)
\(578\) 0 0
\(579\) 12.9615 0.538661
\(580\) 0 0
\(581\) 26.1916i 1.08661i
\(582\) 0 0
\(583\) 31.7490 1.31491
\(584\) 0 0
\(585\) 3.00000 4.58258i 0.124035 0.189466i
\(586\) 0 0
\(587\) 9.89949i 0.408596i 0.978909 + 0.204298i \(0.0654911\pi\)
−0.978909 + 0.204298i \(0.934509\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) −12.9615 −0.533164
\(592\) 0 0
\(593\) 24.4949 1.00588 0.502942 0.864320i \(-0.332251\pi\)
0.502942 + 0.864320i \(0.332251\pi\)
\(594\) 0 0
\(595\) −15.8745 + 24.2487i −0.650791 + 0.994100i
\(596\) 0 0
\(597\) 18.3303i 0.750209i
\(598\) 0 0
\(599\) 27.7128i 1.13231i −0.824297 0.566157i \(-0.808429\pi\)
0.824297 0.566157i \(-0.191571\pi\)
\(600\) 0 0
\(601\) 22.4499i 0.915752i 0.889016 + 0.457876i \(0.151390\pi\)
−0.889016 + 0.457876i \(0.848610\pi\)
\(602\) 0 0
\(603\) −5.29150 −0.215487
\(604\) 0 0
\(605\) −1.22474 + 1.87083i −0.0497930 + 0.0760600i
\(606\) 0 0
\(607\) 42.4264i 1.72203i 0.508576 + 0.861017i \(0.330172\pi\)
−0.508576 + 0.861017i \(0.669828\pi\)
\(608\) 0 0
\(609\) 22.4499i 0.909718i
\(610\) 0 0
\(611\) 6.92820i 0.280285i
\(612\) 0 0
\(613\) 27.4955i 1.11053i −0.831673 0.555265i \(-0.812617\pi\)
0.831673 0.555265i \(-0.187383\pi\)
\(614\) 0 0
\(615\) 12.9615 19.7990i 0.522657 0.798372i
\(616\) 0 0
\(617\) 27.4955i 1.10693i −0.832874 0.553463i \(-0.813306\pi\)
0.832874 0.553463i \(-0.186694\pi\)
\(618\) 0 0
\(619\) 45.3652 1.82338 0.911690 0.410878i \(-0.134778\pi\)
0.911690 + 0.410878i \(0.134778\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 19.7990i 0.793230i
\(624\) 0 0
\(625\) −17.0000 + 18.3303i −0.680000 + 0.733212i
\(626\) 0 0
\(627\) 31.7490 1.26793
\(628\) 0 0
\(629\) 44.8999i 1.79028i
\(630\) 0 0
\(631\) 13.8564i 0.551615i 0.961213 + 0.275807i \(0.0889452\pi\)
−0.961213 + 0.275807i \(0.911055\pi\)
\(632\) 0 0
\(633\) −14.6969 −0.584151
\(634\) 0 0
\(635\) 12.9615 19.7990i 0.514361 0.785699i
\(636\) 0 0
\(637\) 17.1464 0.679366
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −24.0000 −0.947943 −0.473972 0.880540i \(-0.657180\pi\)
−0.473972 + 0.880540i \(0.657180\pi\)
\(642\) 0 0
\(643\) 4.24264i 0.167313i 0.996495 + 0.0836567i \(0.0266599\pi\)
−0.996495 + 0.0836567i \(0.973340\pi\)
\(644\) 0 0
\(645\) −14.0000 9.16515i −0.551249 0.360877i
\(646\) 0 0
\(647\) 39.5980i 1.55676i −0.627795 0.778379i \(-0.716042\pi\)
0.627795 0.778379i \(-0.283958\pi\)
\(648\) 0 0
\(649\) 22.4499i 0.881237i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 27.4955i 1.07598i 0.842951 + 0.537990i \(0.180816\pi\)
−0.842951 + 0.537990i \(0.819184\pi\)
\(654\) 0 0
\(655\) −23.8118 + 36.3731i −0.930403 + 1.42121i
\(656\) 0 0
\(657\) 9.79796 0.382255
\(658\) 0 0
\(659\) 10.3923i 0.404827i −0.979300 0.202413i \(-0.935122\pi\)
0.979300 0.202413i \(-0.0648785\pi\)
\(660\) 0 0
\(661\) 33.6749i 1.30980i −0.755714 0.654901i \(-0.772710\pi\)
0.755714 0.654901i \(-0.227290\pi\)
\(662\) 0 0
\(663\) 16.9706i 0.659082i
\(664\) 0 0
\(665\) −21.0000 + 32.0780i −0.814345 + 1.24393i
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 36.0000 1.39184
\(670\) 0 0
\(671\) 38.8844 1.50112
\(672\) 0 0
\(673\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(674\) 0 0
\(675\) −25.9230 + 11.3137i −0.997775 + 0.435465i
\(676\) 0 0
\(677\) 7.34847 0.282425 0.141212 0.989979i \(-0.454900\pi\)
0.141212 + 0.989979i \(0.454900\pi\)
\(678\) 0 0
\(679\) 38.8844 1.49225
\(680\) 0 0
\(681\) −26.0000 −0.996322
\(682\) 0 0
\(683\) 47.6235 1.82226 0.911132 0.412115i \(-0.135210\pi\)
0.911132 + 0.412115i \(0.135210\pi\)
\(684\) 0 0
\(685\) 34.2929 + 22.4499i 1.31026 + 0.857768i
\(686\) 0 0
\(687\) 15.8745 0.605650
\(688\) 0 0
\(689\) 22.4499i 0.855275i
\(690\) 0 0
\(691\) 6.48074 0.246539 0.123269 0.992373i \(-0.460662\pi\)
0.123269 + 0.992373i \(0.460662\pi\)
\(692\) 0 0
\(693\) 9.16515i 0.348155i
\(694\) 0 0
\(695\) 23.8118 36.3731i 0.903232 1.37971i
\(696\) 0 0
\(697\) 36.6606i 1.38862i
\(698\) 0 0
\(699\) −25.9230 −0.980497
\(700\) 0 0
\(701\) −6.00000 −0.226617 −0.113308 0.993560i \(-0.536145\pi\)
−0.113308 + 0.993560i \(0.536145\pi\)
\(702\) 0 0
\(703\) 59.3970i 2.24020i
\(704\) 0 0
\(705\) −4.89898 + 7.48331i −0.184506 + 0.281838i
\(706\) 0 0
\(707\) 9.89949i 0.372309i
\(708\) 0 0
\(709\) 10.0000 0.375558 0.187779 0.982211i \(-0.439871\pi\)
0.187779 + 0.982211i \(0.439871\pi\)
\(710\) 0 0
\(711\) 6.92820i 0.259828i
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) −15.8745 10.3923i −0.593673 0.388650i
\(716\) 0 0
\(717\) −14.6969 −0.548867
\(718\) 0 0
\(719\) −51.8459 −1.93353 −0.966763 0.255673i \(-0.917703\pi\)
−0.966763 + 0.255673i \(0.917703\pi\)
\(720\) 0 0
\(721\) 44.8999i 1.67216i
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −12.0000 27.4955i −0.445669 1.02116i
\(726\) 0 0
\(727\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(728\) 0 0
\(729\) −29.0000 −1.07407
\(730\) 0 0
\(731\) −25.9230 −0.958795
\(732\) 0 0
\(733\) 26.9444 0.995214 0.497607 0.867403i \(-0.334212\pi\)
0.497607 + 0.867403i \(0.334212\pi\)
\(734\) 0 0
\(735\) −18.5203 12.1244i −0.683130 0.447214i
\(736\) 0 0
\(737\) 18.3303i 0.675205i
\(738\) 0 0
\(739\) 51.9615i 1.91144i 0.294285 + 0.955718i \(0.404919\pi\)
−0.294285 + 0.955718i \(0.595081\pi\)
\(740\) 0 0
\(741\) 22.4499i 0.824719i
\(742\) 0 0
\(743\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(744\) 0 0
\(745\) −7.34847 + 11.2250i −0.269227 + 0.411251i
\(746\) 0 0
\(747\) 9.89949i 0.362204i
\(748\) 0 0
\(749\) −42.0000 −1.53465
\(750\) 0 0
\(751\) 31.1769i 1.13766i −0.822455 0.568831i \(-0.807396\pi\)
0.822455 0.568831i \(-0.192604\pi\)
\(752\) 0 0
\(753\) 9.16515i 0.333997i
\(754\) 0 0
\(755\) −6.48074 4.24264i −0.235858 0.154406i
\(756\) 0 0
\(757\) 45.8258i 1.66557i 0.553600 + 0.832783i \(0.313254\pi\)
−0.553600 + 0.832783i \(0.686746\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 37.4166i 1.35635i 0.734901 + 0.678175i \(0.237229\pi\)
−0.734901 + 0.678175i \(0.762771\pi\)
\(762\) 0 0
\(763\) 5.29150 0.191565
\(764\) 0 0
\(765\) −6.00000 + 9.16515i −0.216930 + 0.331367i
\(766\) 0 0
\(767\) −15.8745 −0.573195
\(768\) 0 0
\(769\) 44.8999i 1.61913i 0.587029 + 0.809566i \(0.300297\pi\)
−0.587029 + 0.809566i \(0.699703\pi\)
\(770\) 0 0
\(771\) 20.7846i 0.748539i
\(772\) 0 0
\(773\) −12.2474 −0.440510 −0.220255 0.975442i \(-0.570689\pi\)
−0.220255 + 0.975442i \(0.570689\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) −34.2929 −1.23025
\(778\) 0 0
\(779\) 48.4974i 1.73760i
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 33.9411i 1.21296i
\(784\) 0 0
\(785\) −9.00000 + 13.7477i −0.321224 + 0.490677i
\(786\) 0 0
\(787\) 38.1838i 1.36110i −0.732700 0.680552i \(-0.761740\pi\)
0.732700 0.680552i \(-0.238260\pi\)
\(788\) 0 0
\(789\) 22.4499i 0.799239i
\(790\) 0 0
\(791\) 24.2487i 0.862185i
\(792\) 0 0
\(793\) 27.4955i 0.976392i
\(794\) 0 0
\(795\) 15.8745 24.2487i 0.563011 0.860013i
\(796\) 0 0
\(797\) 36.7423 1.30148 0.650740 0.759300i \(-0.274459\pi\)
0.650740 + 0.759300i \(0.274459\pi\)
\(798\) 0 0
\(799\) 13.8564i 0.490204i
\(800\) 0 0
\(801\) 7.48331i 0.264410i
\(802\) 0 0
\(803\) 33.9411i 1.19776i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −26.4575 −0.931349
\(808\) 0 0
\(809\) 12.0000 0.421898 0.210949 0.977497i \(-0.432345\pi\)
0.210949 + 0.977497i \(0.432345\pi\)
\(810\) 0 0
\(811\) −6.48074 −0.227570 −0.113785 0.993505i \(-0.536297\pi\)
−0.113785 + 0.993505i \(0.536297\pi\)
\(812\) 0 0
\(813\) 36.6606i 1.28574i
\(814\) 0 0
\(815\) 6.48074 9.89949i 0.227010 0.346764i
\(816\) 0 0
\(817\) −34.2929 −1.19976
\(818\) 0 0
\(819\) 6.48074 0.226455
\(820\) 0 0
\(821\) −18.0000 −0.628204 −0.314102 0.949389i \(-0.601703\pi\)
−0.314102 + 0.949389i \(0.601703\pi\)
\(822\) 0 0
\(823\) 26.4575 0.922251 0.461125 0.887335i \(-0.347446\pi\)
0.461125 + 0.887335i \(0.347446\pi\)
\(824\) 0 0
\(825\) 9.79796 + 22.4499i 0.341121 + 0.781607i
\(826\) 0 0
\(827\) −47.6235 −1.65603 −0.828016 0.560704i \(-0.810530\pi\)
−0.828016 + 0.560704i \(0.810530\pi\)
\(828\) 0 0
\(829\) 56.1249i 1.94930i −0.223742 0.974648i \(-0.571827\pi\)
0.223742 0.974648i \(-0.428173\pi\)
\(830\) 0 0
\(831\) −12.9615 −0.449629
\(832\) 0 0
\(833\) −34.2929 −1.18818
\(834\) 0 0
\(835\) −10.5830 6.92820i −0.366240 0.239760i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −12.9615 −0.447480 −0.223740 0.974649i \(-0.571827\pi\)
−0.223740 + 0.974649i \(0.571827\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) 0 0
\(843\) 8.48528i 0.292249i
\(844\) 0 0
\(845\) −8.57321 + 13.0958i −0.294928 + 0.450509i
\(846\) 0 0
\(847\) −2.64575 −0.0909091
\(848\) 0 0
\(849\) 30.0000 1.02960
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 22.0454 0.754820 0.377410 0.926046i \(-0.376815\pi\)
0.377410 + 0.926046i \(0.376815\pi\)
\(854\) 0 0
\(855\) −7.93725 + 12.1244i −0.271448 + 0.414644i
\(856\) 0 0
\(857\) −9.79796 −0.334692 −0.167346 0.985898i \(-0.553520\pi\)
−0.167346 + 0.985898i \(0.553520\pi\)
\(858\) 0 0
\(859\) 6.48074 0.221120 0.110560 0.993869i \(-0.464736\pi\)
0.110560 + 0.993869i \(0.464736\pi\)
\(860\) 0 0
\(861\) 28.0000 0.954237
\(862\) 0 0
\(863\) 15.8745 0.540375 0.270187 0.962808i \(-0.412914\pi\)
0.270187 + 0.962808i \(0.412914\pi\)
\(864\) 0 0
\(865\) 9.00000 13.7477i 0.306009 0.467437i
\(866\) 0 0
\(867\) 9.89949i 0.336204i
\(868\) 0 0
\(869\) 24.0000 0.814144
\(870\) 0 0
\(871\) −12.9615 −0.439183
\(872\) 0 0
\(873\) 14.6969 0.497416
\(874\) 0 0
\(875\) −29.1633 4.94975i −0.985901 0.167332i
\(876\) 0 0
\(877\) 9.16515i 0.309485i 0.987955 + 0.154743i \(0.0494548\pi\)
−0.987955 + 0.154743i \(0.950545\pi\)
\(878\) 0 0
\(879\) 45.0333i 1.51894i
\(880\) 0 0
\(881\) 29.9333i 1.00848i −0.863564 0.504239i \(-0.831773\pi\)
0.863564 0.504239i \(-0.168227\pi\)
\(882\) 0 0
\(883\) −5.29150 −0.178073 −0.0890366 0.996028i \(-0.528379\pi\)
−0.0890366 + 0.996028i \(0.528379\pi\)
\(884\) 0 0
\(885\) 17.1464 + 11.2250i 0.576371 + 0.377323i
\(886\) 0 0
\(887\) 39.5980i 1.32957i 0.747035 + 0.664785i \(0.231477\pi\)
−0.747035 + 0.664785i \(0.768523\pi\)
\(888\) 0 0
\(889\) 28.0000 0.939090
\(890\) 0 0
\(891\) 17.3205i 0.580259i
\(892\) 0 0
\(893\) 18.3303i 0.613400i
\(894\) 0 0
\(895\) 45.3652 + 29.6985i 1.51639 + 0.992711i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) 44.8999i 1.49583i
\(902\) 0 0
\(903\) 19.7990i 0.658869i
\(904\) 0 0
\(905\) 21.0000 + 13.7477i 0.698064 + 0.456990i
\(906\) 0 0
\(907\) −5.29150 −0.175701 −0.0878507 0.996134i \(-0.528000\pi\)
−0.0878507 + 0.996134i \(0.528000\pi\)
\(908\) 0 0
\(909\) 3.74166i 0.124103i
\(910\) 0 0
\(911\) 24.2487i 0.803396i 0.915772 + 0.401698i \(0.131580\pi\)
−0.915772 + 0.401698i \(0.868420\pi\)
\(912\) 0 0
\(913\) −34.2929 −1.13493
\(914\) 0 0
\(915\) 19.4422 29.6985i 0.642740 0.981802i
\(916\) 0 0
\(917\) −51.4393 −1.69867
\(918\) 0 0
\(919\) 55.4256i 1.82832i 0.405351 + 0.914161i \(0.367149\pi\)
−0.405351 + 0.914161i \(0.632851\pi\)
\(920\) 0 0
\(921\) 6.00000 0.197707
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) −42.0000 + 18.3303i −1.38095 + 0.602697i
\(926\) 0 0
\(927\) 16.9706i 0.557386i
\(928\) 0 0
\(929\) 14.9666i 0.491039i 0.969392 + 0.245520i \(0.0789586\pi\)
−0.969392 + 0.245520i \(0.921041\pi\)
\(930\) 0 0
\(931\) −45.3652 −1.48678
\(932\) 0 0
\(933\) 18.3303i 0.600107i
\(934\) 0 0
\(935\) 31.7490 + 20.7846i 1.03830 + 0.679729i
\(936\) 0 0
\(937\) −19.5959 −0.640171 −0.320085 0.947389i \(-0.603712\pi\)
−0.320085 + 0.947389i \(0.603712\pi\)
\(938\) 0 0
\(939\) 13.8564i 0.452187i
\(940\) 0 0
\(941\) 26.1916i 0.853822i −0.904294 0.426911i \(-0.859602\pi\)
0.904294 0.426911i \(-0.140398\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) −28.0000 18.3303i −0.910840 0.596285i
\(946\) 0 0
\(947\) 15.8745 0.515852 0.257926 0.966165i \(-0.416961\pi\)
0.257926 + 0.966165i \(0.416961\pi\)
\(948\) 0 0
\(949\) 24.0000 0.779073
\(950\) 0 0
\(951\) 12.9615 0.420305
\(952\) 0 0
\(953\) 54.9909i 1.78133i −0.454660 0.890665i \(-0.650239\pi\)
0.454660 0.890665i \(-0.349761\pi\)
\(954\) 0 0
\(955\) 12.9615 + 8.48528i 0.419424 + 0.274577i
\(956\) 0 0
\(957\) −29.3939 −0.950169
\(958\) 0 0
\(959\) 48.4974i 1.56606i
\(960\) 0 0
\(961\) −31.0000 −1.00000
\(962\) 0 0
\(963\) −15.8745 −0.511549
\(964\) 0 0
\(965\) 17.1464 + 11.2250i 0.551963 + 0.361345i
\(966\) 0 0
\(967\) −42.3320 −1.36131 −0.680653 0.732606i \(-0.738304\pi\)
−0.680653 + 0.732606i \(0.738304\pi\)
\(968\) 0 0
\(969\) 44.8999i 1.44239i
\(970\) 0 0
\(971\) 6.48074 0.207977 0.103988 0.994579i \(-0.466840\pi\)
0.103988 + 0.994579i \(0.466840\pi\)
\(972\) 0 0
\(973\) 51.4393 1.64907
\(974\) 0 0
\(975\) −15.8745 + 6.92820i −0.508391 + 0.221880i
\(976\) 0 0
\(977\) 36.6606i 1.17288i 0.809994 + 0.586438i \(0.199470\pi\)
−0.809994 + 0.586438i \(0.800530\pi\)
\(978\) 0 0
\(979\) −25.9230 −0.828501
\(980\) 0 0
\(981\) 2.00000 0.0638551
\(982\) 0 0
\(983\) 11.3137i 0.360851i −0.983589 0.180426i \(-0.942252\pi\)
0.983589 0.180426i \(-0.0577475\pi\)
\(984\) 0 0
\(985\) −17.1464 11.2250i −0.546331 0.357657i
\(986\) 0 0
\(987\) −10.5830 −0.336861
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 48.4974i 1.54057i 0.637699 + 0.770286i \(0.279886\pi\)
−0.637699 + 0.770286i \(0.720114\pi\)
\(992\) 0 0
\(993\) 34.2929 1.08825
\(994\) 0 0
\(995\) −15.8745 + 24.2487i −0.503256 + 0.768736i
\(996\) 0 0
\(997\) 22.0454 0.698185 0.349093 0.937088i \(-0.386490\pi\)
0.349093 + 0.937088i \(0.386490\pi\)
\(998\) 0 0
\(999\) −51.8459 −1.64033
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 2240.2.e.e.2239.3 8
4.3 odd 2 inner 2240.2.e.e.2239.7 8
5.4 even 2 inner 2240.2.e.e.2239.5 8
7.6 odd 2 inner 2240.2.e.e.2239.6 8
8.3 odd 2 560.2.e.d.559.2 yes 8
8.5 even 2 560.2.e.d.559.6 yes 8
20.19 odd 2 inner 2240.2.e.e.2239.1 8
28.27 even 2 inner 2240.2.e.e.2239.2 8
35.34 odd 2 inner 2240.2.e.e.2239.4 8
40.3 even 4 2800.2.k.k.2351.7 8
40.13 odd 4 2800.2.k.k.2351.2 8
40.19 odd 2 560.2.e.d.559.8 yes 8
40.27 even 4 2800.2.k.k.2351.1 8
40.29 even 2 560.2.e.d.559.4 yes 8
40.37 odd 4 2800.2.k.k.2351.8 8
56.13 odd 2 560.2.e.d.559.3 yes 8
56.27 even 2 560.2.e.d.559.7 yes 8
140.139 even 2 inner 2240.2.e.e.2239.8 8
280.13 even 4 2800.2.k.k.2351.6 8
280.27 odd 4 2800.2.k.k.2351.5 8
280.69 odd 2 560.2.e.d.559.5 yes 8
280.83 odd 4 2800.2.k.k.2351.3 8
280.139 even 2 560.2.e.d.559.1 8
280.237 even 4 2800.2.k.k.2351.4 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
560.2.e.d.559.1 8 280.139 even 2
560.2.e.d.559.2 yes 8 8.3 odd 2
560.2.e.d.559.3 yes 8 56.13 odd 2
560.2.e.d.559.4 yes 8 40.29 even 2
560.2.e.d.559.5 yes 8 280.69 odd 2
560.2.e.d.559.6 yes 8 8.5 even 2
560.2.e.d.559.7 yes 8 56.27 even 2
560.2.e.d.559.8 yes 8 40.19 odd 2
2240.2.e.e.2239.1 8 20.19 odd 2 inner
2240.2.e.e.2239.2 8 28.27 even 2 inner
2240.2.e.e.2239.3 8 1.1 even 1 trivial
2240.2.e.e.2239.4 8 35.34 odd 2 inner
2240.2.e.e.2239.5 8 5.4 even 2 inner
2240.2.e.e.2239.6 8 7.6 odd 2 inner
2240.2.e.e.2239.7 8 4.3 odd 2 inner
2240.2.e.e.2239.8 8 140.139 even 2 inner
2800.2.k.k.2351.1 8 40.27 even 4
2800.2.k.k.2351.2 8 40.13 odd 4
2800.2.k.k.2351.3 8 280.83 odd 4
2800.2.k.k.2351.4 8 280.237 even 4
2800.2.k.k.2351.5 8 280.27 odd 4
2800.2.k.k.2351.6 8 280.13 even 4
2800.2.k.k.2351.7 8 40.3 even 4
2800.2.k.k.2351.8 8 40.37 odd 4