Properties

Label 378.3.b.c.323.2
Level $378$
Weight $3$
Character 378.323
Analytic conductor $10.300$
Analytic rank $0$
Dimension $8$
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] = [378,3,Mod(323,378)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(378, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("378.323");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 378 = 2 \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 378.b (of order \(2\), degree \(1\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(10.2997539928\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.157351936.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + x^{4} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 323.2
Root \(-0.581861 + 1.28897i\) of defining polynomial
Character \(\chi\) \(=\) 378.323
Dual form 378.3.b.c.323.7

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.41421i q^{2} -2.00000 q^{4} +0.672556i q^{5} +2.64575 q^{7} +2.82843i q^{8} +O(q^{10})\) \(q-1.41421i q^{2} -2.00000 q^{4} +0.672556i q^{5} +2.64575 q^{7} +2.82843i q^{8} +0.951138 q^{10} +13.2647i q^{11} +4.62808 q^{13} -3.74166i q^{14} +4.00000 q^{16} +15.2020i q^{17} +4.04508 q^{19} -1.34511i q^{20} +18.7591 q^{22} -9.26566i q^{23} +24.5477 q^{25} -6.54510i q^{26} -5.29150 q^{28} +27.5450i q^{29} +44.0486 q^{31} -5.65685i q^{32} +21.4988 q^{34} +1.77942i q^{35} -2.90984 q^{37} -5.72060i q^{38} -1.90228 q^{40} -23.9470i q^{41} +62.8940 q^{43} -26.5294i q^{44} -13.1036 q^{46} +55.1568i q^{47} +7.00000 q^{49} -34.7156i q^{50} -9.25617 q^{52} +1.84323i q^{53} -8.92125 q^{55} +7.48331i q^{56} +38.9546 q^{58} -13.2450i q^{59} -20.5346 q^{61} -62.2941i q^{62} -8.00000 q^{64} +3.11265i q^{65} -75.2985 q^{67} -30.4039i q^{68} +2.51647 q^{70} -91.3391i q^{71} +40.4610 q^{73} +4.11514i q^{74} -8.09016 q^{76} +35.0951i q^{77} -84.4365 q^{79} +2.69022i q^{80} -33.8662 q^{82} +80.0194i q^{83} -10.2242 q^{85} -88.9456i q^{86} -37.5182 q^{88} +109.600i q^{89} +12.2448 q^{91} +18.5313i q^{92} +78.0035 q^{94} +2.72054i q^{95} +126.396 q^{97} -9.89949i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 16 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 16 q^{4} + 16 q^{10} - 40 q^{13} + 32 q^{16} + 40 q^{19} - 16 q^{22} - 40 q^{31} - 16 q^{34} - 8 q^{37} - 32 q^{40} + 40 q^{43} - 64 q^{46} + 56 q^{49} + 80 q^{52} + 56 q^{55} + 112 q^{58} - 88 q^{61} - 64 q^{64} + 240 q^{67} - 112 q^{70} - 288 q^{73} - 80 q^{76} - 424 q^{79} + 144 q^{82} + 200 q^{85} + 32 q^{88} + 56 q^{91} + 288 q^{94} + 600 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/378\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(325\)
\(\chi(n)\) \(-1\) \(1\)

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 − 0.707107i
\(3\) 0 0
\(4\) −2.00000 −0.500000
\(5\) 0.672556i 0.134511i 0.997736 + 0.0672556i \(0.0214243\pi\)
−0.997736 + 0.0672556i \(0.978576\pi\)
\(6\) 0 0
\(7\) 2.64575 0.377964
\(8\) 2.82843i 0.353553i
\(9\) 0 0
\(10\) 0.951138 0.0951138
\(11\) 13.2647i 1.20588i 0.797786 + 0.602941i \(0.206004\pi\)
−0.797786 + 0.602941i \(0.793996\pi\)
\(12\) 0 0
\(13\) 4.62808 0.356006 0.178003 0.984030i \(-0.443036\pi\)
0.178003 + 0.984030i \(0.443036\pi\)
\(14\) − 3.74166i − 0.267261i
\(15\) 0 0
\(16\) 4.00000 0.250000
\(17\) 15.2020i 0.894232i 0.894476 + 0.447116i \(0.147549\pi\)
−0.894476 + 0.447116i \(0.852451\pi\)
\(18\) 0 0
\(19\) 4.04508 0.212899 0.106449 0.994318i \(-0.466052\pi\)
0.106449 + 0.994318i \(0.466052\pi\)
\(20\) − 1.34511i − 0.0672556i
\(21\) 0 0
\(22\) 18.7591 0.852687
\(23\) − 9.26566i − 0.402855i −0.979503 0.201427i \(-0.935442\pi\)
0.979503 0.201427i \(-0.0645580\pi\)
\(24\) 0 0
\(25\) 24.5477 0.981907
\(26\) − 6.54510i − 0.251735i
\(27\) 0 0
\(28\) −5.29150 −0.188982
\(29\) 27.5450i 0.949829i 0.880032 + 0.474914i \(0.157521\pi\)
−0.880032 + 0.474914i \(0.842479\pi\)
\(30\) 0 0
\(31\) 44.0486 1.42092 0.710461 0.703736i \(-0.248486\pi\)
0.710461 + 0.703736i \(0.248486\pi\)
\(32\) − 5.65685i − 0.176777i
\(33\) 0 0
\(34\) 21.4988 0.632318
\(35\) 1.77942i 0.0508405i
\(36\) 0 0
\(37\) −2.90984 −0.0786445 −0.0393222 0.999227i \(-0.512520\pi\)
−0.0393222 + 0.999227i \(0.512520\pi\)
\(38\) − 5.72060i − 0.150542i
\(39\) 0 0
\(40\) −1.90228 −0.0475569
\(41\) − 23.9470i − 0.584073i −0.956407 0.292037i \(-0.905667\pi\)
0.956407 0.292037i \(-0.0943329\pi\)
\(42\) 0 0
\(43\) 62.8940 1.46265 0.731326 0.682028i \(-0.238902\pi\)
0.731326 + 0.682028i \(0.238902\pi\)
\(44\) − 26.5294i − 0.602941i
\(45\) 0 0
\(46\) −13.1036 −0.284861
\(47\) 55.1568i 1.17355i 0.809751 + 0.586774i \(0.199602\pi\)
−0.809751 + 0.586774i \(0.800398\pi\)
\(48\) 0 0
\(49\) 7.00000 0.142857
\(50\) − 34.7156i − 0.694313i
\(51\) 0 0
\(52\) −9.25617 −0.178003
\(53\) 1.84323i 0.0347779i 0.999849 + 0.0173889i \(0.00553535\pi\)
−0.999849 + 0.0173889i \(0.994465\pi\)
\(54\) 0 0
\(55\) −8.92125 −0.162205
\(56\) 7.48331i 0.133631i
\(57\) 0 0
\(58\) 38.9546 0.671630
\(59\) − 13.2450i − 0.224492i −0.993680 0.112246i \(-0.964196\pi\)
0.993680 0.112246i \(-0.0358044\pi\)
\(60\) 0 0
\(61\) −20.5346 −0.336633 −0.168316 0.985733i \(-0.553833\pi\)
−0.168316 + 0.985733i \(0.553833\pi\)
\(62\) − 62.2941i − 1.00474i
\(63\) 0 0
\(64\) −8.00000 −0.125000
\(65\) 3.11265i 0.0478869i
\(66\) 0 0
\(67\) −75.2985 −1.12386 −0.561929 0.827185i \(-0.689941\pi\)
−0.561929 + 0.827185i \(0.689941\pi\)
\(68\) − 30.4039i − 0.447116i
\(69\) 0 0
\(70\) 2.51647 0.0359496
\(71\) − 91.3391i − 1.28647i −0.765670 0.643233i \(-0.777593\pi\)
0.765670 0.643233i \(-0.222407\pi\)
\(72\) 0 0
\(73\) 40.4610 0.554261 0.277130 0.960832i \(-0.410617\pi\)
0.277130 + 0.960832i \(0.410617\pi\)
\(74\) 4.11514i 0.0556100i
\(75\) 0 0
\(76\) −8.09016 −0.106449
\(77\) 35.0951i 0.455780i
\(78\) 0 0
\(79\) −84.4365 −1.06882 −0.534408 0.845226i \(-0.679466\pi\)
−0.534408 + 0.845226i \(0.679466\pi\)
\(80\) 2.69022i 0.0336278i
\(81\) 0 0
\(82\) −33.8662 −0.413002
\(83\) 80.0194i 0.964089i 0.876147 + 0.482045i \(0.160106\pi\)
−0.876147 + 0.482045i \(0.839894\pi\)
\(84\) 0 0
\(85\) −10.2242 −0.120284
\(86\) − 88.9456i − 1.03425i
\(87\) 0 0
\(88\) −37.5182 −0.426344
\(89\) 109.600i 1.23146i 0.787957 + 0.615730i \(0.211139\pi\)
−0.787957 + 0.615730i \(0.788861\pi\)
\(90\) 0 0
\(91\) 12.2448 0.134558
\(92\) 18.5313i 0.201427i
\(93\) 0 0
\(94\) 78.0035 0.829824
\(95\) 2.72054i 0.0286373i
\(96\) 0 0
\(97\) 126.396 1.30306 0.651528 0.758624i \(-0.274128\pi\)
0.651528 + 0.758624i \(0.274128\pi\)
\(98\) − 9.89949i − 0.101015i
\(99\) 0 0
\(100\) −49.0953 −0.490953
\(101\) 119.202i 1.18022i 0.807324 + 0.590108i \(0.200915\pi\)
−0.807324 + 0.590108i \(0.799085\pi\)
\(102\) 0 0
\(103\) −132.242 −1.28390 −0.641952 0.766745i \(-0.721875\pi\)
−0.641952 + 0.766745i \(0.721875\pi\)
\(104\) 13.0902i 0.125867i
\(105\) 0 0
\(106\) 2.60672 0.0245917
\(107\) − 109.283i − 1.02134i −0.859777 0.510669i \(-0.829398\pi\)
0.859777 0.510669i \(-0.170602\pi\)
\(108\) 0 0
\(109\) −184.589 −1.69347 −0.846736 0.532013i \(-0.821436\pi\)
−0.846736 + 0.532013i \(0.821436\pi\)
\(110\) 12.6166i 0.114696i
\(111\) 0 0
\(112\) 10.5830 0.0944911
\(113\) − 35.8528i − 0.317281i −0.987336 0.158641i \(-0.949289\pi\)
0.987336 0.158641i \(-0.0507111\pi\)
\(114\) 0 0
\(115\) 6.23168 0.0541885
\(116\) − 55.0901i − 0.474914i
\(117\) 0 0
\(118\) −18.7313 −0.158739
\(119\) 40.2206i 0.337988i
\(120\) 0 0
\(121\) −54.9522 −0.454150
\(122\) 29.0403i 0.238035i
\(123\) 0 0
\(124\) −88.0972 −0.710461
\(125\) 33.3236i 0.266589i
\(126\) 0 0
\(127\) 96.3439 0.758613 0.379307 0.925271i \(-0.376163\pi\)
0.379307 + 0.925271i \(0.376163\pi\)
\(128\) 11.3137i 0.0883883i
\(129\) 0 0
\(130\) 4.40195 0.0338611
\(131\) − 76.2958i − 0.582411i −0.956661 0.291205i \(-0.905944\pi\)
0.956661 0.291205i \(-0.0940563\pi\)
\(132\) 0 0
\(133\) 10.7023 0.0804682
\(134\) 106.488i 0.794688i
\(135\) 0 0
\(136\) −42.9976 −0.316159
\(137\) 173.775i 1.26843i 0.773156 + 0.634216i \(0.218677\pi\)
−0.773156 + 0.634216i \(0.781323\pi\)
\(138\) 0 0
\(139\) −85.9389 −0.618265 −0.309133 0.951019i \(-0.600039\pi\)
−0.309133 + 0.951019i \(0.600039\pi\)
\(140\) − 3.55883i − 0.0254202i
\(141\) 0 0
\(142\) −129.173 −0.909669
\(143\) 61.3901i 0.429302i
\(144\) 0 0
\(145\) −18.5256 −0.127763
\(146\) − 57.2205i − 0.391921i
\(147\) 0 0
\(148\) 5.81969 0.0393222
\(149\) − 0.416605i − 0.00279600i −0.999999 0.00139800i \(-0.999555\pi\)
0.999999 0.00139800i \(-0.000444998\pi\)
\(150\) 0 0
\(151\) −170.051 −1.12617 −0.563084 0.826400i \(-0.690385\pi\)
−0.563084 + 0.826400i \(0.690385\pi\)
\(152\) 11.4412i 0.0752711i
\(153\) 0 0
\(154\) 49.6320 0.322285
\(155\) 29.6252i 0.191130i
\(156\) 0 0
\(157\) −146.513 −0.933202 −0.466601 0.884468i \(-0.654522\pi\)
−0.466601 + 0.884468i \(0.654522\pi\)
\(158\) 119.411i 0.755767i
\(159\) 0 0
\(160\) 3.80455 0.0237785
\(161\) − 24.5146i − 0.152265i
\(162\) 0 0
\(163\) 85.8374 0.526610 0.263305 0.964713i \(-0.415188\pi\)
0.263305 + 0.964713i \(0.415188\pi\)
\(164\) 47.8940i 0.292037i
\(165\) 0 0
\(166\) 113.165 0.681714
\(167\) − 244.178i − 1.46214i −0.682300 0.731072i \(-0.739020\pi\)
0.682300 0.731072i \(-0.260980\pi\)
\(168\) 0 0
\(169\) −147.581 −0.873259
\(170\) 14.4592i 0.0850539i
\(171\) 0 0
\(172\) −125.788 −0.731326
\(173\) 144.775i 0.836847i 0.908252 + 0.418424i \(0.137417\pi\)
−0.908252 + 0.418424i \(0.862583\pi\)
\(174\) 0 0
\(175\) 64.9470 0.371126
\(176\) 53.0588i 0.301470i
\(177\) 0 0
\(178\) 154.998 0.870774
\(179\) − 192.258i − 1.07407i −0.843561 0.537033i \(-0.819545\pi\)
0.843561 0.537033i \(-0.180455\pi\)
\(180\) 0 0
\(181\) 298.952 1.65167 0.825835 0.563912i \(-0.190704\pi\)
0.825835 + 0.563912i \(0.190704\pi\)
\(182\) − 17.3167i − 0.0951467i
\(183\) 0 0
\(184\) 26.2072 0.142431
\(185\) − 1.95703i − 0.0105786i
\(186\) 0 0
\(187\) −201.649 −1.07834
\(188\) − 110.314i − 0.586774i
\(189\) 0 0
\(190\) 3.84743 0.0202496
\(191\) − 225.355i − 1.17987i −0.807451 0.589935i \(-0.799154\pi\)
0.807451 0.589935i \(-0.200846\pi\)
\(192\) 0 0
\(193\) −343.910 −1.78192 −0.890958 0.454086i \(-0.849966\pi\)
−0.890958 + 0.454086i \(0.849966\pi\)
\(194\) − 178.752i − 0.921400i
\(195\) 0 0
\(196\) −14.0000 −0.0714286
\(197\) − 111.867i − 0.567851i −0.958846 0.283926i \(-0.908363\pi\)
0.958846 0.283926i \(-0.0916369\pi\)
\(198\) 0 0
\(199\) 240.040 1.20623 0.603116 0.797654i \(-0.293926\pi\)
0.603116 + 0.797654i \(0.293926\pi\)
\(200\) 69.4313i 0.347156i
\(201\) 0 0
\(202\) 168.577 0.834539
\(203\) 72.8773i 0.359002i
\(204\) 0 0
\(205\) 16.1057 0.0785644
\(206\) 187.018i 0.907857i
\(207\) 0 0
\(208\) 18.5123 0.0890016
\(209\) 53.6567i 0.256731i
\(210\) 0 0
\(211\) 216.294 1.02509 0.512544 0.858661i \(-0.328703\pi\)
0.512544 + 0.858661i \(0.328703\pi\)
\(212\) − 3.68645i − 0.0173889i
\(213\) 0 0
\(214\) −154.550 −0.722195
\(215\) 42.2998i 0.196743i
\(216\) 0 0
\(217\) 116.542 0.537058
\(218\) 261.048i 1.19747i
\(219\) 0 0
\(220\) 17.8425 0.0811023
\(221\) 70.3559i 0.318352i
\(222\) 0 0
\(223\) 410.400 1.84036 0.920180 0.391495i \(-0.128042\pi\)
0.920180 + 0.391495i \(0.128042\pi\)
\(224\) − 14.9666i − 0.0668153i
\(225\) 0 0
\(226\) −50.7035 −0.224352
\(227\) − 418.066i − 1.84170i −0.389916 0.920850i \(-0.627496\pi\)
0.389916 0.920850i \(-0.372504\pi\)
\(228\) 0 0
\(229\) −38.0368 −0.166099 −0.0830497 0.996545i \(-0.526466\pi\)
−0.0830497 + 0.996545i \(0.526466\pi\)
\(230\) − 8.81292i − 0.0383171i
\(231\) 0 0
\(232\) −77.9091 −0.335815
\(233\) 101.421i 0.435281i 0.976029 + 0.217641i \(0.0698361\pi\)
−0.976029 + 0.217641i \(0.930164\pi\)
\(234\) 0 0
\(235\) −37.0960 −0.157855
\(236\) 26.4900i 0.112246i
\(237\) 0 0
\(238\) 56.8805 0.238994
\(239\) − 403.709i − 1.68916i −0.535431 0.844579i \(-0.679851\pi\)
0.535431 0.844579i \(-0.320149\pi\)
\(240\) 0 0
\(241\) 452.244 1.87653 0.938266 0.345914i \(-0.112431\pi\)
0.938266 + 0.345914i \(0.112431\pi\)
\(242\) 77.7142i 0.321133i
\(243\) 0 0
\(244\) 41.0692 0.168316
\(245\) 4.70789i 0.0192159i
\(246\) 0 0
\(247\) 18.7210 0.0757933
\(248\) 124.588i 0.502372i
\(249\) 0 0
\(250\) 47.1267 0.188507
\(251\) 55.2568i 0.220146i 0.993923 + 0.110073i \(0.0351085\pi\)
−0.993923 + 0.110073i \(0.964891\pi\)
\(252\) 0 0
\(253\) 122.906 0.485795
\(254\) − 136.251i − 0.536421i
\(255\) 0 0
\(256\) 16.0000 0.0625000
\(257\) − 66.7940i − 0.259899i −0.991521 0.129949i \(-0.958519\pi\)
0.991521 0.129949i \(-0.0414815\pi\)
\(258\) 0 0
\(259\) −7.69873 −0.0297248
\(260\) − 6.22529i − 0.0239434i
\(261\) 0 0
\(262\) −107.899 −0.411827
\(263\) − 361.502i − 1.37453i −0.726406 0.687265i \(-0.758811\pi\)
0.726406 0.687265i \(-0.241189\pi\)
\(264\) 0 0
\(265\) −1.23967 −0.00467801
\(266\) − 15.1353i − 0.0568996i
\(267\) 0 0
\(268\) 150.597 0.561929
\(269\) − 172.459i − 0.641111i −0.947230 0.320555i \(-0.896130\pi\)
0.947230 0.320555i \(-0.103870\pi\)
\(270\) 0 0
\(271\) 91.2119 0.336575 0.168288 0.985738i \(-0.446176\pi\)
0.168288 + 0.985738i \(0.446176\pi\)
\(272\) 60.8078i 0.223558i
\(273\) 0 0
\(274\) 245.755 0.896917
\(275\) 325.617i 1.18406i
\(276\) 0 0
\(277\) −52.0409 −0.187873 −0.0939367 0.995578i \(-0.529945\pi\)
−0.0939367 + 0.995578i \(0.529945\pi\)
\(278\) 121.536i 0.437180i
\(279\) 0 0
\(280\) −5.03295 −0.0179748
\(281\) − 413.083i − 1.47005i −0.678042 0.735023i \(-0.737171\pi\)
0.678042 0.735023i \(-0.262829\pi\)
\(282\) 0 0
\(283\) 108.973 0.385063 0.192531 0.981291i \(-0.438330\pi\)
0.192531 + 0.981291i \(0.438330\pi\)
\(284\) 182.678i 0.643233i
\(285\) 0 0
\(286\) 86.8187 0.303562
\(287\) − 63.3578i − 0.220759i
\(288\) 0 0
\(289\) 57.9007 0.200348
\(290\) 26.1991i 0.0903418i
\(291\) 0 0
\(292\) −80.9221 −0.277130
\(293\) 403.517i 1.37719i 0.725145 + 0.688596i \(0.241773\pi\)
−0.725145 + 0.688596i \(0.758227\pi\)
\(294\) 0 0
\(295\) 8.90801 0.0301966
\(296\) − 8.23028i − 0.0278050i
\(297\) 0 0
\(298\) −0.589168 −0.00197707
\(299\) − 42.8822i − 0.143419i
\(300\) 0 0
\(301\) 166.402 0.552830
\(302\) 240.489i 0.796321i
\(303\) 0 0
\(304\) 16.1803 0.0532247
\(305\) − 13.8107i − 0.0452809i
\(306\) 0 0
\(307\) −91.2879 −0.297355 −0.148677 0.988886i \(-0.547502\pi\)
−0.148677 + 0.988886i \(0.547502\pi\)
\(308\) − 70.1902i − 0.227890i
\(309\) 0 0
\(310\) 41.8963 0.135149
\(311\) 12.1248i 0.0389864i 0.999810 + 0.0194932i \(0.00620527\pi\)
−0.999810 + 0.0194932i \(0.993795\pi\)
\(312\) 0 0
\(313\) 87.0221 0.278026 0.139013 0.990291i \(-0.455607\pi\)
0.139013 + 0.990291i \(0.455607\pi\)
\(314\) 207.200i 0.659874i
\(315\) 0 0
\(316\) 168.873 0.534408
\(317\) 193.797i 0.611348i 0.952136 + 0.305674i \(0.0988817\pi\)
−0.952136 + 0.305674i \(0.901118\pi\)
\(318\) 0 0
\(319\) −365.377 −1.14538
\(320\) − 5.38045i − 0.0168139i
\(321\) 0 0
\(322\) −34.6689 −0.107667
\(323\) 61.4931i 0.190381i
\(324\) 0 0
\(325\) 113.609 0.349565
\(326\) − 121.392i − 0.372369i
\(327\) 0 0
\(328\) 67.7323 0.206501
\(329\) 145.931i 0.443560i
\(330\) 0 0
\(331\) −417.814 −1.26228 −0.631138 0.775670i \(-0.717412\pi\)
−0.631138 + 0.775670i \(0.717412\pi\)
\(332\) − 160.039i − 0.482045i
\(333\) 0 0
\(334\) −345.320 −1.03389
\(335\) − 50.6425i − 0.151172i
\(336\) 0 0
\(337\) 63.6695 0.188930 0.0944652 0.995528i \(-0.469886\pi\)
0.0944652 + 0.995528i \(0.469886\pi\)
\(338\) 208.711i 0.617488i
\(339\) 0 0
\(340\) 20.4483 0.0601422
\(341\) 584.291i 1.71346i
\(342\) 0 0
\(343\) 18.5203 0.0539949
\(344\) 177.891i 0.517125i
\(345\) 0 0
\(346\) 204.742 0.591740
\(347\) 310.877i 0.895901i 0.894058 + 0.447950i \(0.147846\pi\)
−0.894058 + 0.447950i \(0.852154\pi\)
\(348\) 0 0
\(349\) 415.607 1.19085 0.595426 0.803410i \(-0.296983\pi\)
0.595426 + 0.803410i \(0.296983\pi\)
\(350\) − 91.8490i − 0.262426i
\(351\) 0 0
\(352\) 75.0365 0.213172
\(353\) − 150.239i − 0.425605i −0.977095 0.212803i \(-0.931741\pi\)
0.977095 0.212803i \(-0.0682591\pi\)
\(354\) 0 0
\(355\) 61.4307 0.173044
\(356\) − 219.200i − 0.615730i
\(357\) 0 0
\(358\) −271.894 −0.759479
\(359\) − 14.0939i − 0.0392589i −0.999807 0.0196294i \(-0.993751\pi\)
0.999807 0.0196294i \(-0.00624865\pi\)
\(360\) 0 0
\(361\) −344.637 −0.954674
\(362\) − 422.782i − 1.16791i
\(363\) 0 0
\(364\) −24.4895 −0.0672789
\(365\) 27.2123i 0.0745543i
\(366\) 0 0
\(367\) −44.6289 −0.121605 −0.0608024 0.998150i \(-0.519366\pi\)
−0.0608024 + 0.998150i \(0.519366\pi\)
\(368\) − 37.0626i − 0.100714i
\(369\) 0 0
\(370\) −2.76766 −0.00748017
\(371\) 4.87672i 0.0131448i
\(372\) 0 0
\(373\) −259.881 −0.696732 −0.348366 0.937359i \(-0.613263\pi\)
−0.348366 + 0.937359i \(0.613263\pi\)
\(374\) 285.175i 0.762500i
\(375\) 0 0
\(376\) −156.007 −0.414912
\(377\) 127.481i 0.338145i
\(378\) 0 0
\(379\) −185.662 −0.489872 −0.244936 0.969539i \(-0.578767\pi\)
−0.244936 + 0.969539i \(0.578767\pi\)
\(380\) − 5.44108i − 0.0143186i
\(381\) 0 0
\(382\) −318.700 −0.834294
\(383\) − 604.452i − 1.57820i −0.614262 0.789102i \(-0.710546\pi\)
0.614262 0.789102i \(-0.289454\pi\)
\(384\) 0 0
\(385\) −23.6034 −0.0613076
\(386\) 486.362i 1.26000i
\(387\) 0 0
\(388\) −252.793 −0.651528
\(389\) − 506.617i − 1.30236i −0.758924 0.651179i \(-0.774275\pi\)
0.758924 0.651179i \(-0.225725\pi\)
\(390\) 0 0
\(391\) 140.856 0.360246
\(392\) 19.7990i 0.0505076i
\(393\) 0 0
\(394\) −158.203 −0.401531
\(395\) − 56.7883i − 0.143768i
\(396\) 0 0
\(397\) −232.366 −0.585305 −0.292652 0.956219i \(-0.594538\pi\)
−0.292652 + 0.956219i \(0.594538\pi\)
\(398\) − 339.468i − 0.852935i
\(399\) 0 0
\(400\) 98.1907 0.245477
\(401\) 647.469i 1.61464i 0.590116 + 0.807318i \(0.299082\pi\)
−0.590116 + 0.807318i \(0.700918\pi\)
\(402\) 0 0
\(403\) 203.861 0.505857
\(404\) − 238.404i − 0.590108i
\(405\) 0 0
\(406\) 103.064 0.253852
\(407\) − 38.5982i − 0.0948359i
\(408\) 0 0
\(409\) −148.863 −0.363967 −0.181984 0.983302i \(-0.558252\pi\)
−0.181984 + 0.983302i \(0.558252\pi\)
\(410\) − 22.7769i − 0.0555534i
\(411\) 0 0
\(412\) 264.484 0.641952
\(413\) − 35.0430i − 0.0848498i
\(414\) 0 0
\(415\) −53.8175 −0.129681
\(416\) − 26.1804i − 0.0629336i
\(417\) 0 0
\(418\) 75.8821 0.181536
\(419\) 418.288i 0.998301i 0.866515 + 0.499150i \(0.166355\pi\)
−0.866515 + 0.499150i \(0.833645\pi\)
\(420\) 0 0
\(421\) 494.465 1.17450 0.587251 0.809405i \(-0.300210\pi\)
0.587251 + 0.809405i \(0.300210\pi\)
\(422\) − 305.885i − 0.724847i
\(423\) 0 0
\(424\) −5.21343 −0.0122958
\(425\) 373.172i 0.878053i
\(426\) 0 0
\(427\) −54.3294 −0.127235
\(428\) 218.566i 0.510669i
\(429\) 0 0
\(430\) 59.8209 0.139118
\(431\) − 804.276i − 1.86607i −0.359786 0.933035i \(-0.617150\pi\)
0.359786 0.933035i \(-0.382850\pi\)
\(432\) 0 0
\(433\) −855.394 −1.97551 −0.987753 0.156023i \(-0.950133\pi\)
−0.987753 + 0.156023i \(0.950133\pi\)
\(434\) − 164.815i − 0.379757i
\(435\) 0 0
\(436\) 369.177 0.846736
\(437\) − 37.4803i − 0.0857673i
\(438\) 0 0
\(439\) −736.810 −1.67838 −0.839191 0.543837i \(-0.816971\pi\)
−0.839191 + 0.543837i \(0.816971\pi\)
\(440\) − 25.2331i − 0.0573480i
\(441\) 0 0
\(442\) 99.4983 0.225109
\(443\) − 138.024i − 0.311566i −0.987791 0.155783i \(-0.950210\pi\)
0.987791 0.155783i \(-0.0497900\pi\)
\(444\) 0 0
\(445\) −73.7121 −0.165645
\(446\) − 580.394i − 1.30133i
\(447\) 0 0
\(448\) −21.1660 −0.0472456
\(449\) 488.441i 1.08784i 0.839137 + 0.543921i \(0.183061\pi\)
−0.839137 + 0.543921i \(0.816939\pi\)
\(450\) 0 0
\(451\) 317.650 0.704323
\(452\) 71.7055i 0.158641i
\(453\) 0 0
\(454\) −591.235 −1.30228
\(455\) 8.23529i 0.0180995i
\(456\) 0 0
\(457\) −331.337 −0.725026 −0.362513 0.931979i \(-0.618081\pi\)
−0.362513 + 0.931979i \(0.618081\pi\)
\(458\) 53.7921i 0.117450i
\(459\) 0 0
\(460\) −12.4634 −0.0270943
\(461\) − 20.8885i − 0.0453112i −0.999743 0.0226556i \(-0.992788\pi\)
0.999743 0.0226556i \(-0.00721212\pi\)
\(462\) 0 0
\(463\) 106.427 0.229864 0.114932 0.993373i \(-0.463335\pi\)
0.114932 + 0.993373i \(0.463335\pi\)
\(464\) 110.180i 0.237457i
\(465\) 0 0
\(466\) 143.430 0.307790
\(467\) 472.833i 1.01249i 0.862390 + 0.506245i \(0.168967\pi\)
−0.862390 + 0.506245i \(0.831033\pi\)
\(468\) 0 0
\(469\) −199.221 −0.424779
\(470\) 52.4617i 0.111621i
\(471\) 0 0
\(472\) 37.4625 0.0793697
\(473\) 834.270i 1.76378i
\(474\) 0 0
\(475\) 99.2972 0.209047
\(476\) − 80.4412i − 0.168994i
\(477\) 0 0
\(478\) −570.930 −1.19442
\(479\) − 373.539i − 0.779832i −0.920850 0.389916i \(-0.872504\pi\)
0.920850 0.389916i \(-0.127496\pi\)
\(480\) 0 0
\(481\) −13.4670 −0.0279979
\(482\) − 639.570i − 1.32691i
\(483\) 0 0
\(484\) 109.904 0.227075
\(485\) 85.0087i 0.175276i
\(486\) 0 0
\(487\) −642.695 −1.31970 −0.659851 0.751396i \(-0.729381\pi\)
−0.659851 + 0.751396i \(0.729381\pi\)
\(488\) − 58.0806i − 0.119018i
\(489\) 0 0
\(490\) 6.65797 0.0135877
\(491\) − 805.881i − 1.64131i −0.571427 0.820653i \(-0.693610\pi\)
0.571427 0.820653i \(-0.306390\pi\)
\(492\) 0 0
\(493\) −418.738 −0.849368
\(494\) − 26.4754i − 0.0535940i
\(495\) 0 0
\(496\) 176.194 0.355231
\(497\) − 241.661i − 0.486239i
\(498\) 0 0
\(499\) 408.219 0.818075 0.409037 0.912518i \(-0.365865\pi\)
0.409037 + 0.912518i \(0.365865\pi\)
\(500\) − 66.6472i − 0.133294i
\(501\) 0 0
\(502\) 78.1449 0.155667
\(503\) 708.495i 1.40854i 0.709933 + 0.704269i \(0.248725\pi\)
−0.709933 + 0.704269i \(0.751275\pi\)
\(504\) 0 0
\(505\) −80.1699 −0.158752
\(506\) − 173.816i − 0.343509i
\(507\) 0 0
\(508\) −192.688 −0.379307
\(509\) − 842.863i − 1.65592i −0.560787 0.827960i \(-0.689501\pi\)
0.560787 0.827960i \(-0.310499\pi\)
\(510\) 0 0
\(511\) 107.050 0.209491
\(512\) − 22.6274i − 0.0441942i
\(513\) 0 0
\(514\) −94.4610 −0.183776
\(515\) − 88.9402i − 0.172699i
\(516\) 0 0
\(517\) −731.638 −1.41516
\(518\) 10.8876i 0.0210186i
\(519\) 0 0
\(520\) −8.80389 −0.0169306
\(521\) − 798.331i − 1.53231i −0.642659 0.766153i \(-0.722169\pi\)
0.642659 0.766153i \(-0.277831\pi\)
\(522\) 0 0
\(523\) 101.371 0.193825 0.0969127 0.995293i \(-0.469103\pi\)
0.0969127 + 0.995293i \(0.469103\pi\)
\(524\) 152.592i 0.291205i
\(525\) 0 0
\(526\) −511.240 −0.971940
\(527\) 669.625i 1.27063i
\(528\) 0 0
\(529\) 443.148 0.837708
\(530\) 1.75316i 0.00330786i
\(531\) 0 0
\(532\) −21.4045 −0.0402341
\(533\) − 110.829i − 0.207934i
\(534\) 0 0
\(535\) 73.4991 0.137381
\(536\) − 212.976i − 0.397344i
\(537\) 0 0
\(538\) −243.894 −0.453334
\(539\) 92.8529i 0.172269i
\(540\) 0 0
\(541\) 469.805 0.868402 0.434201 0.900816i \(-0.357031\pi\)
0.434201 + 0.900816i \(0.357031\pi\)
\(542\) − 128.993i − 0.237995i
\(543\) 0 0
\(544\) 85.9952 0.158079
\(545\) − 124.146i − 0.227791i
\(546\) 0 0
\(547\) −742.985 −1.35829 −0.679145 0.734004i \(-0.737649\pi\)
−0.679145 + 0.734004i \(0.737649\pi\)
\(548\) − 347.551i − 0.634216i
\(549\) 0 0
\(550\) 460.493 0.837259
\(551\) 111.422i 0.202217i
\(552\) 0 0
\(553\) −223.398 −0.403975
\(554\) 73.5970i 0.132847i
\(555\) 0 0
\(556\) 171.878 0.309133
\(557\) 506.745i 0.909775i 0.890549 + 0.454887i \(0.150321\pi\)
−0.890549 + 0.454887i \(0.849679\pi\)
\(558\) 0 0
\(559\) 291.079 0.520713
\(560\) 7.11767i 0.0127101i
\(561\) 0 0
\(562\) −584.187 −1.03948
\(563\) 900.954i 1.60027i 0.599818 + 0.800137i \(0.295240\pi\)
−0.599818 + 0.800137i \(0.704760\pi\)
\(564\) 0 0
\(565\) 24.1130 0.0426779
\(566\) − 154.111i − 0.272281i
\(567\) 0 0
\(568\) 258.346 0.454835
\(569\) 590.261i 1.03737i 0.854967 + 0.518683i \(0.173577\pi\)
−0.854967 + 0.518683i \(0.826423\pi\)
\(570\) 0 0
\(571\) 941.922 1.64960 0.824800 0.565425i \(-0.191288\pi\)
0.824800 + 0.565425i \(0.191288\pi\)
\(572\) − 122.780i − 0.214651i
\(573\) 0 0
\(574\) −89.6014 −0.156100
\(575\) − 227.450i − 0.395566i
\(576\) 0 0
\(577\) 236.677 0.410185 0.205092 0.978743i \(-0.434251\pi\)
0.205092 + 0.978743i \(0.434251\pi\)
\(578\) − 81.8839i − 0.141668i
\(579\) 0 0
\(580\) 37.0512 0.0638813
\(581\) 211.711i 0.364391i
\(582\) 0 0
\(583\) −24.4499 −0.0419380
\(584\) 114.441i 0.195961i
\(585\) 0 0
\(586\) 570.660 0.973822
\(587\) 920.582i 1.56828i 0.620582 + 0.784141i \(0.286896\pi\)
−0.620582 + 0.784141i \(0.713104\pi\)
\(588\) 0 0
\(589\) 178.180 0.302513
\(590\) − 12.5978i − 0.0213522i
\(591\) 0 0
\(592\) −11.6394 −0.0196611
\(593\) 760.444i 1.28237i 0.767388 + 0.641184i \(0.221556\pi\)
−0.767388 + 0.641184i \(0.778444\pi\)
\(594\) 0 0
\(595\) −27.0506 −0.0454632
\(596\) 0.833209i 0.00139800i
\(597\) 0 0
\(598\) −60.6447 −0.101412
\(599\) − 399.798i − 0.667442i −0.942672 0.333721i \(-0.891696\pi\)
0.942672 0.333721i \(-0.108304\pi\)
\(600\) 0 0
\(601\) 664.811 1.10617 0.553087 0.833124i \(-0.313450\pi\)
0.553087 + 0.833124i \(0.313450\pi\)
\(602\) − 235.328i − 0.390910i
\(603\) 0 0
\(604\) 340.103 0.563084
\(605\) − 36.9584i − 0.0610883i
\(606\) 0 0
\(607\) 372.670 0.613954 0.306977 0.951717i \(-0.400682\pi\)
0.306977 + 0.951717i \(0.400682\pi\)
\(608\) − 22.8824i − 0.0376355i
\(609\) 0 0
\(610\) −19.5312 −0.0320184
\(611\) 255.270i 0.417791i
\(612\) 0 0
\(613\) 512.686 0.836356 0.418178 0.908365i \(-0.362669\pi\)
0.418178 + 0.908365i \(0.362669\pi\)
\(614\) 129.101i 0.210262i
\(615\) 0 0
\(616\) −99.2639 −0.161143
\(617\) − 921.484i − 1.49349i −0.665110 0.746745i \(-0.731615\pi\)
0.665110 0.746745i \(-0.268385\pi\)
\(618\) 0 0
\(619\) −1104.63 −1.78454 −0.892270 0.451503i \(-0.850888\pi\)
−0.892270 + 0.451503i \(0.850888\pi\)
\(620\) − 59.2503i − 0.0955650i
\(621\) 0 0
\(622\) 17.1470 0.0275676
\(623\) 289.974i 0.465448i
\(624\) 0 0
\(625\) 591.280 0.946048
\(626\) − 123.068i − 0.196594i
\(627\) 0 0
\(628\) 293.026 0.466601
\(629\) − 44.2353i − 0.0703264i
\(630\) 0 0
\(631\) 119.817 0.189884 0.0949418 0.995483i \(-0.469733\pi\)
0.0949418 + 0.995483i \(0.469733\pi\)
\(632\) − 238.823i − 0.377884i
\(633\) 0 0
\(634\) 274.071 0.432288
\(635\) 64.7967i 0.102042i
\(636\) 0 0
\(637\) 32.3966 0.0508581
\(638\) 516.721i 0.809907i
\(639\) 0 0
\(640\) −7.60910 −0.0118892
\(641\) 82.6281i 0.128905i 0.997921 + 0.0644525i \(0.0205301\pi\)
−0.997921 + 0.0644525i \(0.979470\pi\)
\(642\) 0 0
\(643\) 126.741 0.197109 0.0985547 0.995132i \(-0.468578\pi\)
0.0985547 + 0.995132i \(0.468578\pi\)
\(644\) 49.0293i 0.0761324i
\(645\) 0 0
\(646\) 86.9643 0.134620
\(647\) − 738.224i − 1.14099i −0.821299 0.570497i \(-0.806750\pi\)
0.821299 0.570497i \(-0.193250\pi\)
\(648\) 0 0
\(649\) 175.691 0.270710
\(650\) − 160.667i − 0.247180i
\(651\) 0 0
\(652\) −171.675 −0.263305
\(653\) 69.1073i 0.105831i 0.998599 + 0.0529153i \(0.0168513\pi\)
−0.998599 + 0.0529153i \(0.983149\pi\)
\(654\) 0 0
\(655\) 51.3132 0.0783408
\(656\) − 95.7880i − 0.146018i
\(657\) 0 0
\(658\) 206.378 0.313644
\(659\) 427.081i 0.648075i 0.946044 + 0.324037i \(0.105040\pi\)
−0.946044 + 0.324037i \(0.894960\pi\)
\(660\) 0 0
\(661\) 93.3442 0.141217 0.0706083 0.997504i \(-0.477506\pi\)
0.0706083 + 0.997504i \(0.477506\pi\)
\(662\) 590.878i 0.892565i
\(663\) 0 0
\(664\) −226.329 −0.340857
\(665\) 7.19788i 0.0108239i
\(666\) 0 0
\(667\) 255.223 0.382643
\(668\) 488.356i 0.731072i
\(669\) 0 0
\(670\) −71.6193 −0.106894
\(671\) − 272.385i − 0.405939i
\(672\) 0 0
\(673\) 468.530 0.696181 0.348091 0.937461i \(-0.386830\pi\)
0.348091 + 0.937461i \(0.386830\pi\)
\(674\) − 90.0423i − 0.133594i
\(675\) 0 0
\(676\) 295.162 0.436630
\(677\) 1286.77i 1.90070i 0.311183 + 0.950350i \(0.399275\pi\)
−0.311183 + 0.950350i \(0.600725\pi\)
\(678\) 0 0
\(679\) 334.414 0.492509
\(680\) − 28.9183i − 0.0425269i
\(681\) 0 0
\(682\) 826.313 1.21160
\(683\) − 478.532i − 0.700632i −0.936632 0.350316i \(-0.886074\pi\)
0.936632 0.350316i \(-0.113926\pi\)
\(684\) 0 0
\(685\) −116.874 −0.170618
\(686\) − 26.1916i − 0.0381802i
\(687\) 0 0
\(688\) 251.576 0.365663
\(689\) 8.53061i 0.0123811i
\(690\) 0 0
\(691\) 886.514 1.28294 0.641471 0.767147i \(-0.278324\pi\)
0.641471 + 0.767147i \(0.278324\pi\)
\(692\) − 289.549i − 0.418424i
\(693\) 0 0
\(694\) 439.647 0.633497
\(695\) − 57.7987i − 0.0831636i
\(696\) 0 0
\(697\) 364.041 0.522297
\(698\) − 587.758i − 0.842060i
\(699\) 0 0
\(700\) −129.894 −0.185563
\(701\) − 503.322i − 0.718006i −0.933336 0.359003i \(-0.883117\pi\)
0.933336 0.359003i \(-0.116883\pi\)
\(702\) 0 0
\(703\) −11.7705 −0.0167433
\(704\) − 106.118i − 0.150735i
\(705\) 0 0
\(706\) −212.470 −0.300948
\(707\) 315.378i 0.446080i
\(708\) 0 0
\(709\) −102.128 −0.144046 −0.0720229 0.997403i \(-0.522945\pi\)
−0.0720229 + 0.997403i \(0.522945\pi\)
\(710\) − 86.8761i − 0.122361i
\(711\) 0 0
\(712\) −309.996 −0.435387
\(713\) − 408.139i − 0.572425i
\(714\) 0 0
\(715\) −41.2883 −0.0577459
\(716\) 384.516i 0.537033i
\(717\) 0 0
\(718\) −19.9318 −0.0277602
\(719\) 375.610i 0.522406i 0.965284 + 0.261203i \(0.0841192\pi\)
−0.965284 + 0.261203i \(0.915881\pi\)
\(720\) 0 0
\(721\) −349.880 −0.485270
\(722\) 487.391i 0.675057i
\(723\) 0 0
\(724\) −597.904 −0.825835
\(725\) 676.166i 0.932643i
\(726\) 0 0
\(727\) −967.189 −1.33038 −0.665192 0.746672i \(-0.731650\pi\)
−0.665192 + 0.746672i \(0.731650\pi\)
\(728\) 34.6334i 0.0475734i
\(729\) 0 0
\(730\) 38.4840 0.0527178
\(731\) 956.112i 1.30795i
\(732\) 0 0
\(733\) 44.8714 0.0612162 0.0306081 0.999531i \(-0.490256\pi\)
0.0306081 + 0.999531i \(0.490256\pi\)
\(734\) 63.1148i 0.0859875i
\(735\) 0 0
\(736\) −52.4145 −0.0712153
\(737\) − 998.812i − 1.35524i
\(738\) 0 0
\(739\) 790.331 1.06946 0.534730 0.845023i \(-0.320413\pi\)
0.534730 + 0.845023i \(0.320413\pi\)
\(740\) 3.91407i 0.00528928i
\(741\) 0 0
\(742\) 6.89673 0.00929478
\(743\) − 198.537i − 0.267210i −0.991035 0.133605i \(-0.957345\pi\)
0.991035 0.133605i \(-0.0426554\pi\)
\(744\) 0 0
\(745\) 0.280190 0.000376094 0
\(746\) 367.527i 0.492664i
\(747\) 0 0
\(748\) 403.299 0.539169
\(749\) − 289.136i − 0.386030i
\(750\) 0 0
\(751\) 527.315 0.702151 0.351076 0.936347i \(-0.385816\pi\)
0.351076 + 0.936347i \(0.385816\pi\)
\(752\) 220.627i 0.293387i
\(753\) 0 0
\(754\) 180.285 0.239105
\(755\) − 114.369i − 0.151482i
\(756\) 0 0
\(757\) −986.410 −1.30305 −0.651526 0.758626i \(-0.725871\pi\)
−0.651526 + 0.758626i \(0.725871\pi\)
\(758\) 262.565i 0.346392i
\(759\) 0 0
\(760\) −7.69485 −0.0101248
\(761\) − 1431.24i − 1.88073i −0.340166 0.940366i \(-0.610483\pi\)
0.340166 0.940366i \(-0.389517\pi\)
\(762\) 0 0
\(763\) −488.375 −0.640073
\(764\) 450.710i 0.589935i
\(765\) 0 0
\(766\) −854.825 −1.11596
\(767\) − 61.2990i − 0.0799204i
\(768\) 0 0
\(769\) 321.382 0.417922 0.208961 0.977924i \(-0.432992\pi\)
0.208961 + 0.977924i \(0.432992\pi\)
\(770\) 33.3803i 0.0433510i
\(771\) 0 0
\(772\) 687.819 0.890958
\(773\) − 731.933i − 0.946873i −0.880828 0.473436i \(-0.843013\pi\)
0.880828 0.473436i \(-0.156987\pi\)
\(774\) 0 0
\(775\) 1081.29 1.39521
\(776\) 357.503i 0.460700i
\(777\) 0 0
\(778\) −716.465 −0.920906
\(779\) − 96.8674i − 0.124348i
\(780\) 0 0
\(781\) 1211.59 1.55133
\(782\) − 199.201i − 0.254732i
\(783\) 0 0
\(784\) 28.0000 0.0357143
\(785\) − 98.5381i − 0.125526i
\(786\) 0 0
\(787\) −790.685 −1.00468 −0.502341 0.864669i \(-0.667528\pi\)
−0.502341 + 0.864669i \(0.667528\pi\)
\(788\) 223.733i 0.283926i
\(789\) 0 0
\(790\) −80.3108 −0.101659
\(791\) − 94.8575i − 0.119921i
\(792\) 0 0
\(793\) −95.0358 −0.119843
\(794\) 328.615i 0.413873i
\(795\) 0 0
\(796\) −480.080 −0.603116
\(797\) − 1157.18i − 1.45192i −0.687739 0.725958i \(-0.741397\pi\)
0.687739 0.725958i \(-0.258603\pi\)
\(798\) 0 0
\(799\) −838.491 −1.04943
\(800\) − 138.863i − 0.173578i
\(801\) 0 0
\(802\) 915.660 1.14172
\(803\) 536.703i 0.668373i
\(804\) 0 0
\(805\) 16.4875 0.0204813
\(806\) − 288.302i − 0.357695i
\(807\) 0 0
\(808\) −337.154 −0.417269
\(809\) − 951.837i − 1.17656i −0.808658 0.588280i \(-0.799805\pi\)
0.808658 0.588280i \(-0.200195\pi\)
\(810\) 0 0
\(811\) −737.263 −0.909079 −0.454540 0.890727i \(-0.650196\pi\)
−0.454540 + 0.890727i \(0.650196\pi\)
\(812\) − 145.755i − 0.179501i
\(813\) 0 0
\(814\) −54.5861 −0.0670591
\(815\) 57.7304i 0.0708349i
\(816\) 0 0
\(817\) 254.411 0.311397
\(818\) 210.523i 0.257364i
\(819\) 0 0
\(820\) −32.2114 −0.0392822
\(821\) − 1089.87i − 1.32750i −0.747957 0.663748i \(-0.768965\pi\)
0.747957 0.663748i \(-0.231035\pi\)
\(822\) 0 0
\(823\) 1121.28 1.36243 0.681216 0.732083i \(-0.261452\pi\)
0.681216 + 0.732083i \(0.261452\pi\)
\(824\) − 374.037i − 0.453928i
\(825\) 0 0
\(826\) −49.5583 −0.0599979
\(827\) 1322.75i 1.59946i 0.600362 + 0.799729i \(0.295023\pi\)
−0.600362 + 0.799729i \(0.704977\pi\)
\(828\) 0 0
\(829\) −303.153 −0.365686 −0.182843 0.983142i \(-0.558530\pi\)
−0.182843 + 0.983142i \(0.558530\pi\)
\(830\) 76.1095i 0.0916982i
\(831\) 0 0
\(832\) −37.0247 −0.0445008
\(833\) 106.414i 0.127747i
\(834\) 0 0
\(835\) 164.224 0.196675
\(836\) − 107.313i − 0.128365i
\(837\) 0 0
\(838\) 591.549 0.705905
\(839\) 48.7743i 0.0581339i 0.999577 + 0.0290669i \(0.00925360\pi\)
−0.999577 + 0.0290669i \(0.990746\pi\)
\(840\) 0 0
\(841\) 82.2710 0.0978252
\(842\) − 699.279i − 0.830498i
\(843\) 0 0
\(844\) −432.587 −0.512544
\(845\) − 99.2564i − 0.117463i
\(846\) 0 0
\(847\) −145.390 −0.171653
\(848\) 7.37291i 0.00869447i
\(849\) 0 0
\(850\) 527.746 0.620877
\(851\) 26.9616i 0.0316823i
\(852\) 0 0
\(853\) −795.011 −0.932017 −0.466009 0.884780i \(-0.654308\pi\)
−0.466009 + 0.884780i \(0.654308\pi\)
\(854\) 76.8334i 0.0899689i
\(855\) 0 0
\(856\) 309.100 0.361098
\(857\) 515.098i 0.601048i 0.953774 + 0.300524i \(0.0971616\pi\)
−0.953774 + 0.300524i \(0.902838\pi\)
\(858\) 0 0
\(859\) 731.616 0.851706 0.425853 0.904792i \(-0.359974\pi\)
0.425853 + 0.904792i \(0.359974\pi\)
\(860\) − 84.5995i − 0.0983715i
\(861\) 0 0
\(862\) −1137.42 −1.31951
\(863\) − 1675.40i − 1.94137i −0.240350 0.970686i \(-0.577262\pi\)
0.240350 0.970686i \(-0.422738\pi\)
\(864\) 0 0
\(865\) −97.3691 −0.112565
\(866\) 1209.71i 1.39689i
\(867\) 0 0
\(868\) −233.083 −0.268529
\(869\) − 1120.02i − 1.28887i
\(870\) 0 0
\(871\) −348.488 −0.400101
\(872\) − 522.095i − 0.598733i
\(873\) 0 0
\(874\) −53.0052 −0.0606467
\(875\) 88.1659i 0.100761i
\(876\) 0 0
\(877\) −1563.11 −1.78234 −0.891170 0.453670i \(-0.850115\pi\)
−0.891170 + 0.453670i \(0.850115\pi\)
\(878\) 1042.01i 1.18680i
\(879\) 0 0
\(880\) −35.6850 −0.0405512
\(881\) 751.835i 0.853389i 0.904396 + 0.426694i \(0.140322\pi\)
−0.904396 + 0.426694i \(0.859678\pi\)
\(882\) 0 0
\(883\) −1005.78 −1.13905 −0.569527 0.821973i \(-0.692873\pi\)
−0.569527 + 0.821973i \(0.692873\pi\)
\(884\) − 140.712i − 0.159176i
\(885\) 0 0
\(886\) −195.195 −0.220310
\(887\) − 1413.71i − 1.59381i −0.604106 0.796904i \(-0.706470\pi\)
0.604106 0.796904i \(-0.293530\pi\)
\(888\) 0 0
\(889\) 254.902 0.286729
\(890\) 104.245i 0.117129i
\(891\) 0 0
\(892\) −820.801 −0.920180
\(893\) 223.113i 0.249847i
\(894\) 0 0
\(895\) 129.304 0.144474
\(896\) 29.9333i 0.0334077i
\(897\) 0 0
\(898\) 690.760 0.769220
\(899\) 1213.32i 1.34963i
\(900\) 0 0
\(901\) −28.0207 −0.0310995
\(902\) − 449.224i − 0.498032i
\(903\) 0 0
\(904\) 101.407 0.112176
\(905\) 201.062i 0.222168i
\(906\) 0 0
\(907\) −158.562 −0.174821 −0.0874104 0.996172i \(-0.527859\pi\)
−0.0874104 + 0.996172i \(0.527859\pi\)
\(908\) 836.132i 0.920850i
\(909\) 0 0
\(910\) 11.6465 0.0127983
\(911\) − 85.1896i − 0.0935121i −0.998906 0.0467561i \(-0.985112\pi\)
0.998906 0.0467561i \(-0.0148883\pi\)
\(912\) 0 0
\(913\) −1061.43 −1.16258
\(914\) 468.581i 0.512671i
\(915\) 0 0
\(916\) 76.0735 0.0830497
\(917\) − 201.860i − 0.220131i
\(918\) 0 0
\(919\) −89.1494 −0.0970070 −0.0485035 0.998823i \(-0.515445\pi\)
−0.0485035 + 0.998823i \(0.515445\pi\)
\(920\) 17.6258i 0.0191585i
\(921\) 0 0
\(922\) −29.5408 −0.0320399
\(923\) − 422.725i − 0.457990i
\(924\) 0 0
\(925\) −71.4299 −0.0772215
\(926\) − 150.511i − 0.162538i
\(927\) 0 0
\(928\) 155.818 0.167908
\(929\) − 271.528i − 0.292280i −0.989264 0.146140i \(-0.953315\pi\)
0.989264 0.146140i \(-0.0466850\pi\)
\(930\) 0 0
\(931\) 28.3155 0.0304141
\(932\) − 202.841i − 0.217641i
\(933\) 0 0
\(934\) 668.686 0.715938
\(935\) − 135.620i − 0.145049i
\(936\) 0 0
\(937\) −1238.24 −1.32149 −0.660747 0.750609i \(-0.729760\pi\)
−0.660747 + 0.750609i \(0.729760\pi\)
\(938\) 281.741i 0.300364i
\(939\) 0 0
\(940\) 74.1921 0.0789277
\(941\) − 796.254i − 0.846179i −0.906088 0.423089i \(-0.860946\pi\)
0.906088 0.423089i \(-0.139054\pi\)
\(942\) 0 0
\(943\) −221.885 −0.235297
\(944\) − 52.9800i − 0.0561229i
\(945\) 0 0
\(946\) 1179.84 1.24718
\(947\) 716.676i 0.756786i 0.925645 + 0.378393i \(0.123523\pi\)
−0.925645 + 0.378393i \(0.876477\pi\)
\(948\) 0 0
\(949\) 187.257 0.197320
\(950\) − 140.427i − 0.147818i
\(951\) 0 0
\(952\) −113.761 −0.119497
\(953\) − 1105.45i − 1.15996i −0.814629 0.579982i \(-0.803060\pi\)
0.814629 0.579982i \(-0.196940\pi\)
\(954\) 0 0
\(955\) 151.564 0.158706
\(956\) 807.418i 0.844579i
\(957\) 0 0
\(958\) −528.265 −0.551424
\(959\) 459.766i 0.479422i
\(960\) 0 0
\(961\) 979.278 1.01902
\(962\) 19.0452i 0.0197975i
\(963\) 0 0
\(964\) −904.489 −0.938266
\(965\) − 231.299i − 0.239688i
\(966\) 0 0
\(967\) 176.547 0.182572 0.0912858 0.995825i \(-0.470902\pi\)
0.0912858 + 0.995825i \(0.470902\pi\)
\(968\) − 155.428i − 0.160566i
\(969\) 0 0
\(970\) 120.220 0.123939
\(971\) 1493.78i 1.53840i 0.639009 + 0.769199i \(0.279345\pi\)
−0.639009 + 0.769199i \(0.720655\pi\)
\(972\) 0 0
\(973\) −227.373 −0.233682
\(974\) 908.908i 0.933171i
\(975\) 0 0
\(976\) −82.1384 −0.0841582
\(977\) 720.120i 0.737073i 0.929613 + 0.368536i \(0.120141\pi\)
−0.929613 + 0.368536i \(0.879859\pi\)
\(978\) 0 0
\(979\) −1453.81 −1.48500
\(980\) − 9.41579i − 0.00960795i
\(981\) 0 0
\(982\) −1139.69 −1.16058
\(983\) − 727.718i − 0.740303i −0.928971 0.370151i \(-0.879306\pi\)
0.928971 0.370151i \(-0.120694\pi\)
\(984\) 0 0
\(985\) 75.2366 0.0763824
\(986\) 592.185i 0.600594i
\(987\) 0 0
\(988\) −37.4419 −0.0378967
\(989\) − 582.755i − 0.589236i
\(990\) 0 0
\(991\) −682.016 −0.688210 −0.344105 0.938931i \(-0.611818\pi\)
−0.344105 + 0.938931i \(0.611818\pi\)
\(992\) − 249.176i − 0.251186i
\(993\) 0 0
\(994\) −341.760 −0.343823
\(995\) 161.440i 0.162252i
\(996\) 0 0
\(997\) −210.841 −0.211476 −0.105738 0.994394i \(-0.533720\pi\)
−0.105738 + 0.994394i \(0.533720\pi\)
\(998\) − 577.309i − 0.578466i
\(999\) 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 378.3.b.c.323.2 8
3.2 odd 2 inner 378.3.b.c.323.7 yes 8
4.3 odd 2 3024.3.d.h.1457.5 8
9.2 odd 6 1134.3.q.f.701.6 16
9.4 even 3 1134.3.q.f.1079.6 16
9.5 odd 6 1134.3.q.f.1079.3 16
9.7 even 3 1134.3.q.f.701.3 16
12.11 even 2 3024.3.d.h.1457.4 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
378.3.b.c.323.2 8 1.1 even 1 trivial
378.3.b.c.323.7 yes 8 3.2 odd 2 inner
1134.3.q.f.701.3 16 9.7 even 3
1134.3.q.f.701.6 16 9.2 odd 6
1134.3.q.f.1079.3 16 9.5 odd 6
1134.3.q.f.1079.6 16 9.4 even 3
3024.3.d.h.1457.4 8 12.11 even 2
3024.3.d.h.1457.5 8 4.3 odd 2