Properties

Label 960.4.a.bk.1.2
Level $960$
Weight $4$
Character 960.1
Self dual yes
Analytic conductor $56.642$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [960,4,Mod(1,960)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(960, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("960.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 960 = 2^{6} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 960.a (trivial)

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: yes
Analytic conductor: \(56.6418336055\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{89}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 22 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 480)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(5.21699\) of defining polynomial
Character \(\chi\) \(=\) 960.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-3.00000 q^{3} -5.00000 q^{5} +24.8680 q^{7} +9.00000 q^{9} +O(q^{10})\) \(q-3.00000 q^{3} -5.00000 q^{5} +24.8680 q^{7} +9.00000 q^{9} +25.7359 q^{11} -60.6039 q^{13} +15.0000 q^{15} -28.6039 q^{17} -86.6039 q^{19} -74.6039 q^{21} -52.3398 q^{23} +25.0000 q^{25} -27.0000 q^{27} -6.00000 q^{29} +84.8680 q^{31} -77.2078 q^{33} -124.340 q^{35} +448.227 q^{37} +181.812 q^{39} +183.208 q^{41} -252.000 q^{43} -45.0000 q^{45} +41.9243 q^{47} +275.416 q^{49} +85.8117 q^{51} +228.792 q^{53} -128.680 q^{55} +259.812 q^{57} -179.849 q^{59} -480.039 q^{61} +223.812 q^{63} +303.019 q^{65} -855.472 q^{67} +157.019 q^{69} +675.775 q^{71} -621.623 q^{73} -75.0000 q^{75} +640.000 q^{77} -513.699 q^{79} +81.0000 q^{81} -1284.30 q^{83} +143.019 q^{85} +18.0000 q^{87} -1004.87 q^{89} -1507.10 q^{91} -254.604 q^{93} +433.019 q^{95} +300.416 q^{97} +231.623 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 6 q^{3} - 10 q^{5} + 12 q^{7} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 6 q^{3} - 10 q^{5} + 12 q^{7} + 18 q^{9} - 24 q^{11} - 8 q^{13} + 30 q^{15} + 56 q^{17} - 60 q^{19} - 36 q^{21} + 84 q^{23} + 50 q^{25} - 54 q^{27} - 12 q^{29} + 132 q^{31} + 72 q^{33} - 60 q^{35} + 104 q^{37} + 24 q^{39} + 140 q^{41} - 504 q^{43} - 90 q^{45} + 348 q^{47} + 98 q^{49} - 168 q^{51} + 684 q^{53} + 120 q^{55} + 180 q^{57} - 888 q^{59} + 172 q^{61} + 108 q^{63} + 40 q^{65} - 1560 q^{67} - 252 q^{69} + 144 q^{71} - 564 q^{73} - 150 q^{75} + 1280 q^{77} - 84 q^{79} + 162 q^{81} - 1512 q^{83} - 280 q^{85} + 36 q^{87} + 28 q^{89} - 2184 q^{91} - 396 q^{93} + 300 q^{95} + 148 q^{97} - 216 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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
\(3\) −3.00000 −0.577350
\(4\) 0 0
\(5\) −5.00000 −0.447214
\(6\) 0 0
\(7\) 24.8680 1.34274 0.671372 0.741121i \(-0.265705\pi\)
0.671372 + 0.741121i \(0.265705\pi\)
\(8\) 0 0
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) 25.7359 0.705425 0.352712 0.935732i \(-0.385259\pi\)
0.352712 + 0.935732i \(0.385259\pi\)
\(12\) 0 0
\(13\) −60.6039 −1.29296 −0.646481 0.762930i \(-0.723760\pi\)
−0.646481 + 0.762930i \(0.723760\pi\)
\(14\) 0 0
\(15\) 15.0000 0.258199
\(16\) 0 0
\(17\) −28.6039 −0.408086 −0.204043 0.978962i \(-0.565408\pi\)
−0.204043 + 0.978962i \(0.565408\pi\)
\(18\) 0 0
\(19\) −86.6039 −1.04570 −0.522850 0.852425i \(-0.675131\pi\)
−0.522850 + 0.852425i \(0.675131\pi\)
\(20\) 0 0
\(21\) −74.6039 −0.775233
\(22\) 0 0
\(23\) −52.3398 −0.474505 −0.237252 0.971448i \(-0.576247\pi\)
−0.237252 + 0.971448i \(0.576247\pi\)
\(24\) 0 0
\(25\) 25.0000 0.200000
\(26\) 0 0
\(27\) −27.0000 −0.192450
\(28\) 0 0
\(29\) −6.00000 −0.0384197 −0.0192099 0.999815i \(-0.506115\pi\)
−0.0192099 + 0.999815i \(0.506115\pi\)
\(30\) 0 0
\(31\) 84.8680 0.491701 0.245851 0.969308i \(-0.420933\pi\)
0.245851 + 0.969308i \(0.420933\pi\)
\(32\) 0 0
\(33\) −77.2078 −0.407277
\(34\) 0 0
\(35\) −124.340 −0.600493
\(36\) 0 0
\(37\) 448.227 1.99157 0.995785 0.0917169i \(-0.0292355\pi\)
0.995785 + 0.0917169i \(0.0292355\pi\)
\(38\) 0 0
\(39\) 181.812 0.746491
\(40\) 0 0
\(41\) 183.208 0.697860 0.348930 0.937149i \(-0.386545\pi\)
0.348930 + 0.937149i \(0.386545\pi\)
\(42\) 0 0
\(43\) −252.000 −0.893713 −0.446856 0.894606i \(-0.647456\pi\)
−0.446856 + 0.894606i \(0.647456\pi\)
\(44\) 0 0
\(45\) −45.0000 −0.149071
\(46\) 0 0
\(47\) 41.9243 0.130112 0.0650562 0.997882i \(-0.479277\pi\)
0.0650562 + 0.997882i \(0.479277\pi\)
\(48\) 0 0
\(49\) 275.416 0.802961
\(50\) 0 0
\(51\) 85.8117 0.235609
\(52\) 0 0
\(53\) 228.792 0.592963 0.296481 0.955039i \(-0.404187\pi\)
0.296481 + 0.955039i \(0.404187\pi\)
\(54\) 0 0
\(55\) −128.680 −0.315476
\(56\) 0 0
\(57\) 259.812 0.603735
\(58\) 0 0
\(59\) −179.849 −0.396852 −0.198426 0.980116i \(-0.563583\pi\)
−0.198426 + 0.980116i \(0.563583\pi\)
\(60\) 0 0
\(61\) −480.039 −1.00758 −0.503792 0.863825i \(-0.668062\pi\)
−0.503792 + 0.863825i \(0.668062\pi\)
\(62\) 0 0
\(63\) 223.812 0.447581
\(64\) 0 0
\(65\) 303.019 0.578230
\(66\) 0 0
\(67\) −855.472 −1.55989 −0.779944 0.625849i \(-0.784753\pi\)
−0.779944 + 0.625849i \(0.784753\pi\)
\(68\) 0 0
\(69\) 157.019 0.273955
\(70\) 0 0
\(71\) 675.775 1.12957 0.564787 0.825237i \(-0.308958\pi\)
0.564787 + 0.825237i \(0.308958\pi\)
\(72\) 0 0
\(73\) −621.623 −0.996651 −0.498325 0.866990i \(-0.666052\pi\)
−0.498325 + 0.866990i \(0.666052\pi\)
\(74\) 0 0
\(75\) −75.0000 −0.115470
\(76\) 0 0
\(77\) 640.000 0.947205
\(78\) 0 0
\(79\) −513.699 −0.731591 −0.365795 0.930695i \(-0.619203\pi\)
−0.365795 + 0.930695i \(0.619203\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) −1284.30 −1.69844 −0.849220 0.528039i \(-0.822928\pi\)
−0.849220 + 0.528039i \(0.822928\pi\)
\(84\) 0 0
\(85\) 143.019 0.182502
\(86\) 0 0
\(87\) 18.0000 0.0221816
\(88\) 0 0
\(89\) −1004.87 −1.19681 −0.598405 0.801194i \(-0.704198\pi\)
−0.598405 + 0.801194i \(0.704198\pi\)
\(90\) 0 0
\(91\) −1507.10 −1.73612
\(92\) 0 0
\(93\) −254.604 −0.283884
\(94\) 0 0
\(95\) 433.019 0.467651
\(96\) 0 0
\(97\) 300.416 0.314460 0.157230 0.987562i \(-0.449744\pi\)
0.157230 + 0.987562i \(0.449744\pi\)
\(98\) 0 0
\(99\) 231.623 0.235142
\(100\) 0 0
\(101\) 1649.25 1.62481 0.812407 0.583091i \(-0.198157\pi\)
0.812407 + 0.583091i \(0.198157\pi\)
\(102\) 0 0
\(103\) −1326.38 −1.26885 −0.634427 0.772983i \(-0.718764\pi\)
−0.634427 + 0.772983i \(0.718764\pi\)
\(104\) 0 0
\(105\) 373.019 0.346695
\(106\) 0 0
\(107\) −1189.43 −1.07464 −0.537322 0.843377i \(-0.680564\pi\)
−0.537322 + 0.843377i \(0.680564\pi\)
\(108\) 0 0
\(109\) −1864.42 −1.63834 −0.819168 0.573554i \(-0.805564\pi\)
−0.819168 + 0.573554i \(0.805564\pi\)
\(110\) 0 0
\(111\) −1344.68 −1.14983
\(112\) 0 0
\(113\) −1218.19 −1.01414 −0.507069 0.861906i \(-0.669271\pi\)
−0.507069 + 0.861906i \(0.669271\pi\)
\(114\) 0 0
\(115\) 261.699 0.212205
\(116\) 0 0
\(117\) −545.435 −0.430987
\(118\) 0 0
\(119\) −711.320 −0.547955
\(120\) 0 0
\(121\) −668.662 −0.502376
\(122\) 0 0
\(123\) −549.623 −0.402909
\(124\) 0 0
\(125\) −125.000 −0.0894427
\(126\) 0 0
\(127\) 1740.19 1.21588 0.607941 0.793982i \(-0.291996\pi\)
0.607941 + 0.793982i \(0.291996\pi\)
\(128\) 0 0
\(129\) 756.000 0.515985
\(130\) 0 0
\(131\) 456.303 0.304331 0.152166 0.988355i \(-0.451375\pi\)
0.152166 + 0.988355i \(0.451375\pi\)
\(132\) 0 0
\(133\) −2153.66 −1.40411
\(134\) 0 0
\(135\) 135.000 0.0860663
\(136\) 0 0
\(137\) 1329.51 0.829109 0.414555 0.910024i \(-0.363937\pi\)
0.414555 + 0.910024i \(0.363937\pi\)
\(138\) 0 0
\(139\) −1097.25 −0.669550 −0.334775 0.942298i \(-0.608660\pi\)
−0.334775 + 0.942298i \(0.608660\pi\)
\(140\) 0 0
\(141\) −125.773 −0.0751204
\(142\) 0 0
\(143\) −1559.70 −0.912087
\(144\) 0 0
\(145\) 30.0000 0.0171818
\(146\) 0 0
\(147\) −826.247 −0.463590
\(148\) 0 0
\(149\) −1590.16 −0.874299 −0.437150 0.899389i \(-0.644012\pi\)
−0.437150 + 0.899389i \(0.644012\pi\)
\(150\) 0 0
\(151\) −1166.60 −0.628721 −0.314361 0.949304i \(-0.601790\pi\)
−0.314361 + 0.949304i \(0.601790\pi\)
\(152\) 0 0
\(153\) −257.435 −0.136029
\(154\) 0 0
\(155\) −424.340 −0.219895
\(156\) 0 0
\(157\) −1770.57 −0.900041 −0.450021 0.893018i \(-0.648583\pi\)
−0.450021 + 0.893018i \(0.648583\pi\)
\(158\) 0 0
\(159\) −686.377 −0.342347
\(160\) 0 0
\(161\) −1301.58 −0.637138
\(162\) 0 0
\(163\) 4018.35 1.93093 0.965464 0.260538i \(-0.0838999\pi\)
0.965464 + 0.260538i \(0.0838999\pi\)
\(164\) 0 0
\(165\) 386.039 0.182140
\(166\) 0 0
\(167\) −2899.74 −1.34365 −0.671823 0.740712i \(-0.734488\pi\)
−0.671823 + 0.740712i \(0.734488\pi\)
\(168\) 0 0
\(169\) 1475.83 0.671748
\(170\) 0 0
\(171\) −779.435 −0.348567
\(172\) 0 0
\(173\) 3127.74 1.37455 0.687276 0.726396i \(-0.258806\pi\)
0.687276 + 0.726396i \(0.258806\pi\)
\(174\) 0 0
\(175\) 621.699 0.268549
\(176\) 0 0
\(177\) 539.546 0.229123
\(178\) 0 0
\(179\) 2852.53 1.19111 0.595553 0.803316i \(-0.296933\pi\)
0.595553 + 0.803316i \(0.296933\pi\)
\(180\) 0 0
\(181\) 1666.91 0.684532 0.342266 0.939603i \(-0.388806\pi\)
0.342266 + 0.939603i \(0.388806\pi\)
\(182\) 0 0
\(183\) 1440.12 0.581729
\(184\) 0 0
\(185\) −2241.14 −0.890657
\(186\) 0 0
\(187\) −736.147 −0.287874
\(188\) 0 0
\(189\) −671.435 −0.258411
\(190\) 0 0
\(191\) −3144.00 −1.19106 −0.595528 0.803334i \(-0.703057\pi\)
−0.595528 + 0.803334i \(0.703057\pi\)
\(192\) 0 0
\(193\) −1063.13 −0.396507 −0.198253 0.980151i \(-0.563527\pi\)
−0.198253 + 0.980151i \(0.563527\pi\)
\(194\) 0 0
\(195\) −909.058 −0.333841
\(196\) 0 0
\(197\) 4600.87 1.66395 0.831976 0.554812i \(-0.187210\pi\)
0.831976 + 0.554812i \(0.187210\pi\)
\(198\) 0 0
\(199\) −1834.83 −0.653606 −0.326803 0.945092i \(-0.605971\pi\)
−0.326803 + 0.945092i \(0.605971\pi\)
\(200\) 0 0
\(201\) 2566.42 0.900602
\(202\) 0 0
\(203\) −149.208 −0.0515878
\(204\) 0 0
\(205\) −916.039 −0.312092
\(206\) 0 0
\(207\) −471.058 −0.158168
\(208\) 0 0
\(209\) −2228.83 −0.737663
\(210\) 0 0
\(211\) −1896.80 −0.618867 −0.309434 0.950921i \(-0.600139\pi\)
−0.309434 + 0.950921i \(0.600139\pi\)
\(212\) 0 0
\(213\) −2027.32 −0.652160
\(214\) 0 0
\(215\) 1260.00 0.399680
\(216\) 0 0
\(217\) 2110.49 0.660229
\(218\) 0 0
\(219\) 1864.87 0.575417
\(220\) 0 0
\(221\) 1733.51 0.527639
\(222\) 0 0
\(223\) −5967.06 −1.79186 −0.895928 0.444200i \(-0.853488\pi\)
−0.895928 + 0.444200i \(0.853488\pi\)
\(224\) 0 0
\(225\) 225.000 0.0666667
\(226\) 0 0
\(227\) −6375.03 −1.86399 −0.931995 0.362472i \(-0.881933\pi\)
−0.931995 + 0.362472i \(0.881933\pi\)
\(228\) 0 0
\(229\) 1360.12 0.392485 0.196242 0.980555i \(-0.437126\pi\)
0.196242 + 0.980555i \(0.437126\pi\)
\(230\) 0 0
\(231\) −1920.00 −0.546869
\(232\) 0 0
\(233\) 96.7594 0.0272056 0.0136028 0.999907i \(-0.495670\pi\)
0.0136028 + 0.999907i \(0.495670\pi\)
\(234\) 0 0
\(235\) −209.621 −0.0581880
\(236\) 0 0
\(237\) 1541.10 0.422384
\(238\) 0 0
\(239\) −6068.00 −1.64229 −0.821143 0.570722i \(-0.806663\pi\)
−0.821143 + 0.570722i \(0.806663\pi\)
\(240\) 0 0
\(241\) −542.753 −0.145070 −0.0725349 0.997366i \(-0.523109\pi\)
−0.0725349 + 0.997366i \(0.523109\pi\)
\(242\) 0 0
\(243\) −243.000 −0.0641500
\(244\) 0 0
\(245\) −1377.08 −0.359095
\(246\) 0 0
\(247\) 5248.53 1.35205
\(248\) 0 0
\(249\) 3852.91 0.980595
\(250\) 0 0
\(251\) 4658.42 1.17146 0.585731 0.810505i \(-0.300807\pi\)
0.585731 + 0.810505i \(0.300807\pi\)
\(252\) 0 0
\(253\) −1347.01 −0.334727
\(254\) 0 0
\(255\) −429.058 −0.105367
\(256\) 0 0
\(257\) −6519.33 −1.58235 −0.791176 0.611588i \(-0.790531\pi\)
−0.791176 + 0.611588i \(0.790531\pi\)
\(258\) 0 0
\(259\) 11146.5 2.67417
\(260\) 0 0
\(261\) −54.0000 −0.0128066
\(262\) 0 0
\(263\) −1205.17 −0.282562 −0.141281 0.989970i \(-0.545122\pi\)
−0.141281 + 0.989970i \(0.545122\pi\)
\(264\) 0 0
\(265\) −1143.96 −0.265181
\(266\) 0 0
\(267\) 3014.61 0.690978
\(268\) 0 0
\(269\) 6083.66 1.37891 0.689456 0.724327i \(-0.257850\pi\)
0.689456 + 0.724327i \(0.257850\pi\)
\(270\) 0 0
\(271\) −3151.36 −0.706389 −0.353194 0.935550i \(-0.614905\pi\)
−0.353194 + 0.935550i \(0.614905\pi\)
\(272\) 0 0
\(273\) 4521.29 1.00235
\(274\) 0 0
\(275\) 643.398 0.141085
\(276\) 0 0
\(277\) −6780.54 −1.47077 −0.735385 0.677650i \(-0.762998\pi\)
−0.735385 + 0.677650i \(0.762998\pi\)
\(278\) 0 0
\(279\) 763.812 0.163900
\(280\) 0 0
\(281\) 5163.29 1.09614 0.548071 0.836432i \(-0.315362\pi\)
0.548071 + 0.836432i \(0.315362\pi\)
\(282\) 0 0
\(283\) −6543.64 −1.37448 −0.687242 0.726429i \(-0.741179\pi\)
−0.687242 + 0.726429i \(0.741179\pi\)
\(284\) 0 0
\(285\) −1299.06 −0.269999
\(286\) 0 0
\(287\) 4556.00 0.937047
\(288\) 0 0
\(289\) −4094.82 −0.833466
\(290\) 0 0
\(291\) −901.247 −0.181553
\(292\) 0 0
\(293\) −267.208 −0.0532780 −0.0266390 0.999645i \(-0.508480\pi\)
−0.0266390 + 0.999645i \(0.508480\pi\)
\(294\) 0 0
\(295\) 899.243 0.177478
\(296\) 0 0
\(297\) −694.870 −0.135759
\(298\) 0 0
\(299\) 3172.00 0.613516
\(300\) 0 0
\(301\) −6266.73 −1.20003
\(302\) 0 0
\(303\) −4947.74 −0.938087
\(304\) 0 0
\(305\) 2400.19 0.450606
\(306\) 0 0
\(307\) 646.800 0.120244 0.0601219 0.998191i \(-0.480851\pi\)
0.0601219 + 0.998191i \(0.480851\pi\)
\(308\) 0 0
\(309\) 3979.14 0.732573
\(310\) 0 0
\(311\) 5855.48 1.06763 0.533816 0.845601i \(-0.320757\pi\)
0.533816 + 0.845601i \(0.320757\pi\)
\(312\) 0 0
\(313\) −3939.52 −0.711421 −0.355710 0.934596i \(-0.615761\pi\)
−0.355710 + 0.934596i \(0.615761\pi\)
\(314\) 0 0
\(315\) −1119.06 −0.200164
\(316\) 0 0
\(317\) −16.1943 −0.00286929 −0.00143464 0.999999i \(-0.500457\pi\)
−0.00143464 + 0.999999i \(0.500457\pi\)
\(318\) 0 0
\(319\) −154.416 −0.0271022
\(320\) 0 0
\(321\) 3568.30 0.620446
\(322\) 0 0
\(323\) 2477.21 0.426735
\(324\) 0 0
\(325\) −1515.10 −0.258592
\(326\) 0 0
\(327\) 5593.25 0.945894
\(328\) 0 0
\(329\) 1042.57 0.174708
\(330\) 0 0
\(331\) −1946.45 −0.323223 −0.161611 0.986854i \(-0.551669\pi\)
−0.161611 + 0.986854i \(0.551669\pi\)
\(332\) 0 0
\(333\) 4034.04 0.663857
\(334\) 0 0
\(335\) 4277.36 0.697603
\(336\) 0 0
\(337\) −3713.10 −0.600195 −0.300097 0.953909i \(-0.597019\pi\)
−0.300097 + 0.953909i \(0.597019\pi\)
\(338\) 0 0
\(339\) 3654.57 0.585512
\(340\) 0 0
\(341\) 2184.16 0.346858
\(342\) 0 0
\(343\) −1680.69 −0.264573
\(344\) 0 0
\(345\) −785.097 −0.122517
\(346\) 0 0
\(347\) −12200.6 −1.88750 −0.943751 0.330658i \(-0.892729\pi\)
−0.943751 + 0.330658i \(0.892729\pi\)
\(348\) 0 0
\(349\) 1455.52 0.223244 0.111622 0.993751i \(-0.464395\pi\)
0.111622 + 0.993751i \(0.464395\pi\)
\(350\) 0 0
\(351\) 1636.30 0.248830
\(352\) 0 0
\(353\) 364.526 0.0549625 0.0274813 0.999622i \(-0.491251\pi\)
0.0274813 + 0.999622i \(0.491251\pi\)
\(354\) 0 0
\(355\) −3378.87 −0.505161
\(356\) 0 0
\(357\) 2133.96 0.316362
\(358\) 0 0
\(359\) 11180.1 1.64363 0.821814 0.569756i \(-0.192962\pi\)
0.821814 + 0.569756i \(0.192962\pi\)
\(360\) 0 0
\(361\) 641.233 0.0934879
\(362\) 0 0
\(363\) 2005.99 0.290047
\(364\) 0 0
\(365\) 3108.12 0.445716
\(366\) 0 0
\(367\) 5558.37 0.790585 0.395292 0.918555i \(-0.370643\pi\)
0.395292 + 0.918555i \(0.370643\pi\)
\(368\) 0 0
\(369\) 1648.87 0.232620
\(370\) 0 0
\(371\) 5689.60 0.796197
\(372\) 0 0
\(373\) −9598.80 −1.33246 −0.666229 0.745747i \(-0.732093\pi\)
−0.666229 + 0.745747i \(0.732093\pi\)
\(374\) 0 0
\(375\) 375.000 0.0516398
\(376\) 0 0
\(377\) 363.623 0.0496752
\(378\) 0 0
\(379\) 5518.37 0.747915 0.373958 0.927446i \(-0.378001\pi\)
0.373958 + 0.927446i \(0.378001\pi\)
\(380\) 0 0
\(381\) −5220.58 −0.701990
\(382\) 0 0
\(383\) −7208.20 −0.961675 −0.480838 0.876810i \(-0.659667\pi\)
−0.480838 + 0.876810i \(0.659667\pi\)
\(384\) 0 0
\(385\) −3200.00 −0.423603
\(386\) 0 0
\(387\) −2268.00 −0.297904
\(388\) 0 0
\(389\) 8580.13 1.11833 0.559164 0.829057i \(-0.311122\pi\)
0.559164 + 0.829057i \(0.311122\pi\)
\(390\) 0 0
\(391\) 1497.12 0.193639
\(392\) 0 0
\(393\) −1368.91 −0.175706
\(394\) 0 0
\(395\) 2568.50 0.327177
\(396\) 0 0
\(397\) −10323.7 −1.30512 −0.652558 0.757738i \(-0.726304\pi\)
−0.652558 + 0.757738i \(0.726304\pi\)
\(398\) 0 0
\(399\) 6460.99 0.810661
\(400\) 0 0
\(401\) 6130.00 0.763386 0.381693 0.924289i \(-0.375341\pi\)
0.381693 + 0.924289i \(0.375341\pi\)
\(402\) 0 0
\(403\) −5143.33 −0.635750
\(404\) 0 0
\(405\) −405.000 −0.0496904
\(406\) 0 0
\(407\) 11535.5 1.40490
\(408\) 0 0
\(409\) 4015.45 0.485456 0.242728 0.970094i \(-0.421958\pi\)
0.242728 + 0.970094i \(0.421958\pi\)
\(410\) 0 0
\(411\) −3988.54 −0.478686
\(412\) 0 0
\(413\) −4472.47 −0.532871
\(414\) 0 0
\(415\) 6421.51 0.759566
\(416\) 0 0
\(417\) 3291.75 0.386565
\(418\) 0 0
\(419\) 11923.3 1.39020 0.695098 0.718915i \(-0.255361\pi\)
0.695098 + 0.718915i \(0.255361\pi\)
\(420\) 0 0
\(421\) 12939.9 1.49799 0.748993 0.662577i \(-0.230537\pi\)
0.748993 + 0.662577i \(0.230537\pi\)
\(422\) 0 0
\(423\) 377.318 0.0433708
\(424\) 0 0
\(425\) −715.097 −0.0816172
\(426\) 0 0
\(427\) −11937.6 −1.35293
\(428\) 0 0
\(429\) 4679.09 0.526594
\(430\) 0 0
\(431\) 5958.20 0.665885 0.332942 0.942947i \(-0.391959\pi\)
0.332942 + 0.942947i \(0.391959\pi\)
\(432\) 0 0
\(433\) 8063.83 0.894972 0.447486 0.894291i \(-0.352319\pi\)
0.447486 + 0.894291i \(0.352319\pi\)
\(434\) 0 0
\(435\) −90.0000 −0.00991993
\(436\) 0 0
\(437\) 4532.83 0.496189
\(438\) 0 0
\(439\) 6065.39 0.659420 0.329710 0.944082i \(-0.393049\pi\)
0.329710 + 0.944082i \(0.393049\pi\)
\(440\) 0 0
\(441\) 2478.74 0.267654
\(442\) 0 0
\(443\) 5019.57 0.538345 0.269172 0.963092i \(-0.413250\pi\)
0.269172 + 0.963092i \(0.413250\pi\)
\(444\) 0 0
\(445\) 5024.35 0.535229
\(446\) 0 0
\(447\) 4770.47 0.504777
\(448\) 0 0
\(449\) 1830.39 0.192386 0.0961931 0.995363i \(-0.469333\pi\)
0.0961931 + 0.995363i \(0.469333\pi\)
\(450\) 0 0
\(451\) 4715.02 0.492288
\(452\) 0 0
\(453\) 3499.81 0.362992
\(454\) 0 0
\(455\) 7535.48 0.776414
\(456\) 0 0
\(457\) −7111.06 −0.727881 −0.363940 0.931422i \(-0.618569\pi\)
−0.363940 + 0.931422i \(0.618569\pi\)
\(458\) 0 0
\(459\) 772.305 0.0785362
\(460\) 0 0
\(461\) −2347.58 −0.237176 −0.118588 0.992944i \(-0.537837\pi\)
−0.118588 + 0.992944i \(0.537837\pi\)
\(462\) 0 0
\(463\) 8669.58 0.870215 0.435107 0.900379i \(-0.356710\pi\)
0.435107 + 0.900379i \(0.356710\pi\)
\(464\) 0 0
\(465\) 1273.02 0.126957
\(466\) 0 0
\(467\) 6448.02 0.638927 0.319463 0.947599i \(-0.396497\pi\)
0.319463 + 0.947599i \(0.396497\pi\)
\(468\) 0 0
\(469\) −21273.8 −2.09453
\(470\) 0 0
\(471\) 5311.70 0.519639
\(472\) 0 0
\(473\) −6485.45 −0.630447
\(474\) 0 0
\(475\) −2165.10 −0.209140
\(476\) 0 0
\(477\) 2059.13 0.197654
\(478\) 0 0
\(479\) −7603.95 −0.725330 −0.362665 0.931920i \(-0.618133\pi\)
−0.362665 + 0.931920i \(0.618133\pi\)
\(480\) 0 0
\(481\) −27164.3 −2.57502
\(482\) 0 0
\(483\) 3904.75 0.367852
\(484\) 0 0
\(485\) −1502.08 −0.140631
\(486\) 0 0
\(487\) 14109.5 1.31286 0.656429 0.754388i \(-0.272066\pi\)
0.656429 + 0.754388i \(0.272066\pi\)
\(488\) 0 0
\(489\) −12055.0 −1.11482
\(490\) 0 0
\(491\) −15808.3 −1.45299 −0.726493 0.687173i \(-0.758851\pi\)
−0.726493 + 0.687173i \(0.758851\pi\)
\(492\) 0 0
\(493\) 171.623 0.0156786
\(494\) 0 0
\(495\) −1158.12 −0.105159
\(496\) 0 0
\(497\) 16805.1 1.51673
\(498\) 0 0
\(499\) 14653.7 1.31461 0.657304 0.753625i \(-0.271697\pi\)
0.657304 + 0.753625i \(0.271697\pi\)
\(500\) 0 0
\(501\) 8699.23 0.775754
\(502\) 0 0
\(503\) 323.889 0.0287108 0.0143554 0.999897i \(-0.495430\pi\)
0.0143554 + 0.999897i \(0.495430\pi\)
\(504\) 0 0
\(505\) −8246.23 −0.726639
\(506\) 0 0
\(507\) −4427.49 −0.387834
\(508\) 0 0
\(509\) −2178.99 −0.189748 −0.0948741 0.995489i \(-0.530245\pi\)
−0.0948741 + 0.995489i \(0.530245\pi\)
\(510\) 0 0
\(511\) −15458.5 −1.33825
\(512\) 0 0
\(513\) 2338.30 0.201245
\(514\) 0 0
\(515\) 6631.89 0.567449
\(516\) 0 0
\(517\) 1078.96 0.0917845
\(518\) 0 0
\(519\) −9383.22 −0.793599
\(520\) 0 0
\(521\) −12371.2 −1.04029 −0.520145 0.854078i \(-0.674122\pi\)
−0.520145 + 0.854078i \(0.674122\pi\)
\(522\) 0 0
\(523\) −3963.16 −0.331351 −0.165676 0.986180i \(-0.552980\pi\)
−0.165676 + 0.986180i \(0.552980\pi\)
\(524\) 0 0
\(525\) −1865.10 −0.155047
\(526\) 0 0
\(527\) −2427.55 −0.200656
\(528\) 0 0
\(529\) −9427.54 −0.774845
\(530\) 0 0
\(531\) −1618.64 −0.132284
\(532\) 0 0
\(533\) −11103.1 −0.902305
\(534\) 0 0
\(535\) 5947.16 0.480595
\(536\) 0 0
\(537\) −8557.58 −0.687685
\(538\) 0 0
\(539\) 7088.07 0.566428
\(540\) 0 0
\(541\) −2426.23 −0.192813 −0.0964066 0.995342i \(-0.530735\pi\)
−0.0964066 + 0.995342i \(0.530735\pi\)
\(542\) 0 0
\(543\) −5000.73 −0.395215
\(544\) 0 0
\(545\) 9322.08 0.732686
\(546\) 0 0
\(547\) 1636.29 0.127902 0.0639512 0.997953i \(-0.479630\pi\)
0.0639512 + 0.997953i \(0.479630\pi\)
\(548\) 0 0
\(549\) −4320.35 −0.335862
\(550\) 0 0
\(551\) 519.623 0.0401755
\(552\) 0 0
\(553\) −12774.6 −0.982339
\(554\) 0 0
\(555\) 6723.41 0.514221
\(556\) 0 0
\(557\) 4160.34 0.316480 0.158240 0.987401i \(-0.449418\pi\)
0.158240 + 0.987401i \(0.449418\pi\)
\(558\) 0 0
\(559\) 15272.2 1.15554
\(560\) 0 0
\(561\) 2208.44 0.166204
\(562\) 0 0
\(563\) −16665.2 −1.24752 −0.623762 0.781614i \(-0.714397\pi\)
−0.623762 + 0.781614i \(0.714397\pi\)
\(564\) 0 0
\(565\) 6090.94 0.453536
\(566\) 0 0
\(567\) 2014.30 0.149194
\(568\) 0 0
\(569\) 12324.4 0.908025 0.454013 0.890995i \(-0.349992\pi\)
0.454013 + 0.890995i \(0.349992\pi\)
\(570\) 0 0
\(571\) 14208.0 1.04131 0.520653 0.853768i \(-0.325688\pi\)
0.520653 + 0.853768i \(0.325688\pi\)
\(572\) 0 0
\(573\) 9432.00 0.687657
\(574\) 0 0
\(575\) −1308.50 −0.0949009
\(576\) 0 0
\(577\) 13281.1 0.958232 0.479116 0.877752i \(-0.340957\pi\)
0.479116 + 0.877752i \(0.340957\pi\)
\(578\) 0 0
\(579\) 3189.39 0.228923
\(580\) 0 0
\(581\) −31938.0 −2.28057
\(582\) 0 0
\(583\) 5888.18 0.418291
\(584\) 0 0
\(585\) 2727.17 0.192743
\(586\) 0 0
\(587\) −24241.4 −1.70451 −0.852256 0.523124i \(-0.824766\pi\)
−0.852256 + 0.523124i \(0.824766\pi\)
\(588\) 0 0
\(589\) −7349.90 −0.514172
\(590\) 0 0
\(591\) −13802.6 −0.960683
\(592\) 0 0
\(593\) 18184.5 1.25927 0.629637 0.776889i \(-0.283204\pi\)
0.629637 + 0.776889i \(0.283204\pi\)
\(594\) 0 0
\(595\) 3556.60 0.245053
\(596\) 0 0
\(597\) 5504.49 0.377360
\(598\) 0 0
\(599\) 10505.1 0.716569 0.358285 0.933612i \(-0.383362\pi\)
0.358285 + 0.933612i \(0.383362\pi\)
\(600\) 0 0
\(601\) −4822.88 −0.327337 −0.163669 0.986515i \(-0.552333\pi\)
−0.163669 + 0.986515i \(0.552333\pi\)
\(602\) 0 0
\(603\) −7699.25 −0.519963
\(604\) 0 0
\(605\) 3343.31 0.224669
\(606\) 0 0
\(607\) −27083.2 −1.81099 −0.905496 0.424355i \(-0.860501\pi\)
−0.905496 + 0.424355i \(0.860501\pi\)
\(608\) 0 0
\(609\) 447.623 0.0297843
\(610\) 0 0
\(611\) −2540.77 −0.168230
\(612\) 0 0
\(613\) 10078.3 0.664042 0.332021 0.943272i \(-0.392269\pi\)
0.332021 + 0.943272i \(0.392269\pi\)
\(614\) 0 0
\(615\) 2748.12 0.180187
\(616\) 0 0
\(617\) 24141.2 1.57518 0.787591 0.616199i \(-0.211328\pi\)
0.787591 + 0.616199i \(0.211328\pi\)
\(618\) 0 0
\(619\) 22858.7 1.48428 0.742140 0.670245i \(-0.233811\pi\)
0.742140 + 0.670245i \(0.233811\pi\)
\(620\) 0 0
\(621\) 1413.17 0.0913184
\(622\) 0 0
\(623\) −24989.1 −1.60701
\(624\) 0 0
\(625\) 625.000 0.0400000
\(626\) 0 0
\(627\) 6686.49 0.425890
\(628\) 0 0
\(629\) −12821.0 −0.812732
\(630\) 0 0
\(631\) 569.330 0.0359187 0.0179593 0.999839i \(-0.494283\pi\)
0.0179593 + 0.999839i \(0.494283\pi\)
\(632\) 0 0
\(633\) 5690.39 0.357303
\(634\) 0 0
\(635\) −8700.96 −0.543759
\(636\) 0 0
\(637\) −16691.3 −1.03820
\(638\) 0 0
\(639\) 6081.97 0.376524
\(640\) 0 0
\(641\) 25927.2 1.59760 0.798802 0.601594i \(-0.205468\pi\)
0.798802 + 0.601594i \(0.205468\pi\)
\(642\) 0 0
\(643\) 19555.1 1.19935 0.599673 0.800245i \(-0.295297\pi\)
0.599673 + 0.800245i \(0.295297\pi\)
\(644\) 0 0
\(645\) −3780.00 −0.230756
\(646\) 0 0
\(647\) −8653.85 −0.525839 −0.262919 0.964818i \(-0.584685\pi\)
−0.262919 + 0.964818i \(0.584685\pi\)
\(648\) 0 0
\(649\) −4628.57 −0.279949
\(650\) 0 0
\(651\) −6331.48 −0.381183
\(652\) 0 0
\(653\) 27732.5 1.66195 0.830976 0.556308i \(-0.187782\pi\)
0.830976 + 0.556308i \(0.187782\pi\)
\(654\) 0 0
\(655\) −2281.51 −0.136101
\(656\) 0 0
\(657\) −5594.61 −0.332217
\(658\) 0 0
\(659\) 30254.7 1.78840 0.894201 0.447665i \(-0.147744\pi\)
0.894201 + 0.447665i \(0.147744\pi\)
\(660\) 0 0
\(661\) 20714.2 1.21890 0.609448 0.792826i \(-0.291391\pi\)
0.609448 + 0.792826i \(0.291391\pi\)
\(662\) 0 0
\(663\) −5200.52 −0.304633
\(664\) 0 0
\(665\) 10768.3 0.627936
\(666\) 0 0
\(667\) 314.039 0.0182303
\(668\) 0 0
\(669\) 17901.2 1.03453
\(670\) 0 0
\(671\) −12354.2 −0.710775
\(672\) 0 0
\(673\) −5092.56 −0.291685 −0.145842 0.989308i \(-0.546589\pi\)
−0.145842 + 0.989308i \(0.546589\pi\)
\(674\) 0 0
\(675\) −675.000 −0.0384900
\(676\) 0 0
\(677\) 27189.5 1.54354 0.771771 0.635901i \(-0.219371\pi\)
0.771771 + 0.635901i \(0.219371\pi\)
\(678\) 0 0
\(679\) 7470.72 0.422239
\(680\) 0 0
\(681\) 19125.1 1.07617
\(682\) 0 0
\(683\) −7705.95 −0.431713 −0.215856 0.976425i \(-0.569254\pi\)
−0.215856 + 0.976425i \(0.569254\pi\)
\(684\) 0 0
\(685\) −6647.56 −0.370789
\(686\) 0 0
\(687\) −4080.35 −0.226601
\(688\) 0 0
\(689\) −13865.7 −0.766678
\(690\) 0 0
\(691\) −15821.6 −0.871033 −0.435516 0.900181i \(-0.643434\pi\)
−0.435516 + 0.900181i \(0.643434\pi\)
\(692\) 0 0
\(693\) 5760.00 0.315735
\(694\) 0 0
\(695\) 5486.24 0.299432
\(696\) 0 0
\(697\) −5240.45 −0.284787
\(698\) 0 0
\(699\) −290.278 −0.0157072
\(700\) 0 0
\(701\) 2108.00 0.113578 0.0567888 0.998386i \(-0.481914\pi\)
0.0567888 + 0.998386i \(0.481914\pi\)
\(702\) 0 0
\(703\) −38818.2 −2.08258
\(704\) 0 0
\(705\) 628.864 0.0335949
\(706\) 0 0
\(707\) 41013.4 2.18171
\(708\) 0 0
\(709\) −301.312 −0.0159605 −0.00798027 0.999968i \(-0.502540\pi\)
−0.00798027 + 0.999968i \(0.502540\pi\)
\(710\) 0 0
\(711\) −4623.29 −0.243864
\(712\) 0 0
\(713\) −4441.97 −0.233314
\(714\) 0 0
\(715\) 7798.49 0.407898
\(716\) 0 0
\(717\) 18204.0 0.948175
\(718\) 0 0
\(719\) 12744.9 0.661061 0.330530 0.943795i \(-0.392772\pi\)
0.330530 + 0.943795i \(0.392772\pi\)
\(720\) 0 0
\(721\) −32984.3 −1.70375
\(722\) 0 0
\(723\) 1628.26 0.0837561
\(724\) 0 0
\(725\) −150.000 −0.00768395
\(726\) 0 0
\(727\) −3416.70 −0.174303 −0.0871515 0.996195i \(-0.527776\pi\)
−0.0871515 + 0.996195i \(0.527776\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) 7208.18 0.364712
\(732\) 0 0
\(733\) 5965.41 0.300597 0.150298 0.988641i \(-0.451977\pi\)
0.150298 + 0.988641i \(0.451977\pi\)
\(734\) 0 0
\(735\) 4131.23 0.207324
\(736\) 0 0
\(737\) −22016.4 −1.10038
\(738\) 0 0
\(739\) −8546.50 −0.425424 −0.212712 0.977115i \(-0.568230\pi\)
−0.212712 + 0.977115i \(0.568230\pi\)
\(740\) 0 0
\(741\) −15745.6 −0.780606
\(742\) 0 0
\(743\) −33899.8 −1.67384 −0.836919 0.547326i \(-0.815646\pi\)
−0.836919 + 0.547326i \(0.815646\pi\)
\(744\) 0 0
\(745\) 7950.78 0.390999
\(746\) 0 0
\(747\) −11558.7 −0.566147
\(748\) 0 0
\(749\) −29578.8 −1.44297
\(750\) 0 0
\(751\) −18256.9 −0.887087 −0.443544 0.896253i \(-0.646279\pi\)
−0.443544 + 0.896253i \(0.646279\pi\)
\(752\) 0 0
\(753\) −13975.3 −0.676344
\(754\) 0 0
\(755\) 5833.02 0.281173
\(756\) 0 0
\(757\) −1117.87 −0.0536717 −0.0268359 0.999640i \(-0.508543\pi\)
−0.0268359 + 0.999640i \(0.508543\pi\)
\(758\) 0 0
\(759\) 4041.04 0.193255
\(760\) 0 0
\(761\) 28438.3 1.35465 0.677324 0.735685i \(-0.263140\pi\)
0.677324 + 0.735685i \(0.263140\pi\)
\(762\) 0 0
\(763\) −46364.2 −2.19987
\(764\) 0 0
\(765\) 1287.17 0.0608339
\(766\) 0 0
\(767\) 10899.5 0.513115
\(768\) 0 0
\(769\) 4566.26 0.214127 0.107063 0.994252i \(-0.465855\pi\)
0.107063 + 0.994252i \(0.465855\pi\)
\(770\) 0 0
\(771\) 19558.0 0.913572
\(772\) 0 0
\(773\) −29259.6 −1.36144 −0.680720 0.732543i \(-0.738333\pi\)
−0.680720 + 0.732543i \(0.738333\pi\)
\(774\) 0 0
\(775\) 2121.70 0.0983402
\(776\) 0 0
\(777\) −33439.5 −1.54393
\(778\) 0 0
\(779\) −15866.5 −0.729752
\(780\) 0 0
\(781\) 17391.7 0.796829
\(782\) 0 0
\(783\) 162.000 0.00739388
\(784\) 0 0
\(785\) 8852.83 0.402511
\(786\) 0 0
\(787\) −7865.23 −0.356246 −0.178123 0.984008i \(-0.557002\pi\)
−0.178123 + 0.984008i \(0.557002\pi\)
\(788\) 0 0
\(789\) 3615.50 0.163137
\(790\) 0 0
\(791\) −30293.9 −1.36173
\(792\) 0 0
\(793\) 29092.2 1.30277
\(794\) 0 0
\(795\) 3431.88 0.153102
\(796\) 0 0
\(797\) −11603.9 −0.515724 −0.257862 0.966182i \(-0.583018\pi\)
−0.257862 + 0.966182i \(0.583018\pi\)
\(798\) 0 0
\(799\) −1199.20 −0.0530970
\(800\) 0 0
\(801\) −9043.83 −0.398936
\(802\) 0 0
\(803\) −15998.1 −0.703062
\(804\) 0 0
\(805\) 6507.92 0.284937
\(806\) 0 0
\(807\) −18251.0 −0.796115
\(808\) 0 0
\(809\) −6069.60 −0.263777 −0.131889 0.991265i \(-0.542104\pi\)
−0.131889 + 0.991265i \(0.542104\pi\)
\(810\) 0 0
\(811\) −26995.1 −1.16883 −0.584417 0.811453i \(-0.698677\pi\)
−0.584417 + 0.811453i \(0.698677\pi\)
\(812\) 0 0
\(813\) 9454.07 0.407834
\(814\) 0 0
\(815\) −20091.7 −0.863537
\(816\) 0 0
\(817\) 21824.2 0.934555
\(818\) 0 0
\(819\) −13563.9 −0.578705
\(820\) 0 0
\(821\) −23158.9 −0.984472 −0.492236 0.870462i \(-0.663820\pi\)
−0.492236 + 0.870462i \(0.663820\pi\)
\(822\) 0 0
\(823\) −30150.7 −1.27702 −0.638511 0.769613i \(-0.720449\pi\)
−0.638511 + 0.769613i \(0.720449\pi\)
\(824\) 0 0
\(825\) −1930.19 −0.0814554
\(826\) 0 0
\(827\) 33988.7 1.42914 0.714572 0.699562i \(-0.246622\pi\)
0.714572 + 0.699562i \(0.246622\pi\)
\(828\) 0 0
\(829\) 11198.3 0.469158 0.234579 0.972097i \(-0.424629\pi\)
0.234579 + 0.972097i \(0.424629\pi\)
\(830\) 0 0
\(831\) 20341.6 0.849149
\(832\) 0 0
\(833\) −7877.96 −0.327677
\(834\) 0 0
\(835\) 14498.7 0.600896
\(836\) 0 0
\(837\) −2291.43 −0.0946279
\(838\) 0 0
\(839\) 20126.3 0.828174 0.414087 0.910237i \(-0.364101\pi\)
0.414087 + 0.910237i \(0.364101\pi\)
\(840\) 0 0
\(841\) −24353.0 −0.998524
\(842\) 0 0
\(843\) −15489.9 −0.632858
\(844\) 0 0
\(845\) −7379.16 −0.300415
\(846\) 0 0
\(847\) −16628.3 −0.674562
\(848\) 0 0
\(849\) 19630.9 0.793558
\(850\) 0 0
\(851\) −23460.1 −0.945009
\(852\) 0 0
\(853\) 24676.2 0.990500 0.495250 0.868751i \(-0.335077\pi\)
0.495250 + 0.868751i \(0.335077\pi\)
\(854\) 0 0
\(855\) 3897.17 0.155884
\(856\) 0 0
\(857\) −44115.5 −1.75841 −0.879204 0.476446i \(-0.841925\pi\)
−0.879204 + 0.476446i \(0.841925\pi\)
\(858\) 0 0
\(859\) 408.822 0.0162385 0.00811923 0.999967i \(-0.497416\pi\)
0.00811923 + 0.999967i \(0.497416\pi\)
\(860\) 0 0
\(861\) −13668.0 −0.541004
\(862\) 0 0
\(863\) 34369.3 1.35567 0.677836 0.735213i \(-0.262918\pi\)
0.677836 + 0.735213i \(0.262918\pi\)
\(864\) 0 0
\(865\) −15638.7 −0.614719
\(866\) 0 0
\(867\) 12284.5 0.481202
\(868\) 0 0
\(869\) −13220.5 −0.516082
\(870\) 0 0
\(871\) 51844.9 2.01687
\(872\) 0 0
\(873\) 2703.74 0.104820
\(874\) 0 0
\(875\) −3108.50 −0.120099
\(876\) 0 0
\(877\) −41868.4 −1.61208 −0.806040 0.591861i \(-0.798394\pi\)
−0.806040 + 0.591861i \(0.798394\pi\)
\(878\) 0 0
\(879\) 801.623 0.0307600
\(880\) 0 0
\(881\) −2174.40 −0.0831526 −0.0415763 0.999135i \(-0.513238\pi\)
−0.0415763 + 0.999135i \(0.513238\pi\)
\(882\) 0 0
\(883\) −19480.7 −0.742442 −0.371221 0.928545i \(-0.621061\pi\)
−0.371221 + 0.928545i \(0.621061\pi\)
\(884\) 0 0
\(885\) −2697.73 −0.102467
\(886\) 0 0
\(887\) −14546.7 −0.550653 −0.275326 0.961351i \(-0.588786\pi\)
−0.275326 + 0.961351i \(0.588786\pi\)
\(888\) 0 0
\(889\) 43275.0 1.63262
\(890\) 0 0
\(891\) 2084.61 0.0783805
\(892\) 0 0
\(893\) −3630.80 −0.136058
\(894\) 0 0
\(895\) −14262.6 −0.532679
\(896\) 0 0
\(897\) −9515.99 −0.354214
\(898\) 0 0
\(899\) −509.208 −0.0188910
\(900\) 0 0
\(901\) −6544.35 −0.241980
\(902\) 0 0
\(903\) 18800.2 0.692836
\(904\) 0 0
\(905\) −8334.54 −0.306132
\(906\) 0 0
\(907\) −22357.6 −0.818492 −0.409246 0.912424i \(-0.634208\pi\)
−0.409246 + 0.912424i \(0.634208\pi\)
\(908\) 0 0
\(909\) 14843.2 0.541605
\(910\) 0 0
\(911\) −32284.2 −1.17412 −0.587060 0.809544i \(-0.699715\pi\)
−0.587060 + 0.809544i \(0.699715\pi\)
\(912\) 0 0
\(913\) −33052.7 −1.19812
\(914\) 0 0
\(915\) −7200.58 −0.260157
\(916\) 0 0
\(917\) 11347.3 0.408639
\(918\) 0 0
\(919\) 12767.9 0.458295 0.229147 0.973392i \(-0.426406\pi\)
0.229147 + 0.973392i \(0.426406\pi\)
\(920\) 0 0
\(921\) −1940.40 −0.0694228
\(922\) 0 0
\(923\) −40954.6 −1.46049
\(924\) 0 0
\(925\) 11205.7 0.398314
\(926\) 0 0
\(927\) −11937.4 −0.422951
\(928\) 0 0
\(929\) −17645.4 −0.623171 −0.311586 0.950218i \(-0.600860\pi\)
−0.311586 + 0.950218i \(0.600860\pi\)
\(930\) 0 0
\(931\) −23852.1 −0.839656
\(932\) 0 0
\(933\) −17566.4 −0.616398
\(934\) 0 0
\(935\) 3680.74 0.128741
\(936\) 0 0
\(937\) 45735.0 1.59455 0.797277 0.603614i \(-0.206273\pi\)
0.797277 + 0.603614i \(0.206273\pi\)
\(938\) 0 0
\(939\) 11818.6 0.410739
\(940\) 0 0
\(941\) 14982.6 0.519042 0.259521 0.965737i \(-0.416435\pi\)
0.259521 + 0.965737i \(0.416435\pi\)
\(942\) 0 0
\(943\) −9589.06 −0.331138
\(944\) 0 0
\(945\) 3357.17 0.115565
\(946\) 0 0
\(947\) −31821.4 −1.09193 −0.545964 0.837809i \(-0.683836\pi\)
−0.545964 + 0.837809i \(0.683836\pi\)
\(948\) 0 0
\(949\) 37672.8 1.28863
\(950\) 0 0
\(951\) 48.5830 0.00165658
\(952\) 0 0
\(953\) 19627.5 0.667154 0.333577 0.942723i \(-0.391744\pi\)
0.333577 + 0.942723i \(0.391744\pi\)
\(954\) 0 0
\(955\) 15720.0 0.532657
\(956\) 0 0
\(957\) 463.247 0.0156475
\(958\) 0 0
\(959\) 33062.3 1.11328
\(960\) 0 0
\(961\) −22588.4 −0.758230
\(962\) 0 0
\(963\) −10704.9 −0.358214
\(964\) 0 0
\(965\) 5315.65 0.177323
\(966\) 0 0
\(967\) −34374.9 −1.14315 −0.571573 0.820551i \(-0.693667\pi\)
−0.571573 + 0.820551i \(0.693667\pi\)
\(968\) 0 0
\(969\) −7431.62 −0.246376
\(970\) 0 0
\(971\) −33737.5 −1.11503 −0.557513 0.830169i \(-0.688244\pi\)
−0.557513 + 0.830169i \(0.688244\pi\)
\(972\) 0 0
\(973\) −27286.3 −0.899034
\(974\) 0 0
\(975\) 4545.29 0.149298
\(976\) 0 0
\(977\) 38438.8 1.25872 0.629358 0.777116i \(-0.283318\pi\)
0.629358 + 0.777116i \(0.283318\pi\)
\(978\) 0 0
\(979\) −25861.3 −0.844259
\(980\) 0 0
\(981\) −16779.7 −0.546112
\(982\) 0 0
\(983\) 18714.3 0.607217 0.303609 0.952797i \(-0.401808\pi\)
0.303609 + 0.952797i \(0.401808\pi\)
\(984\) 0 0
\(985\) −23004.3 −0.744142
\(986\) 0 0
\(987\) −3127.71 −0.100867
\(988\) 0 0
\(989\) 13189.6 0.424071
\(990\) 0 0
\(991\) −23659.4 −0.758392 −0.379196 0.925316i \(-0.623799\pi\)
−0.379196 + 0.925316i \(0.623799\pi\)
\(992\) 0 0
\(993\) 5839.36 0.186613
\(994\) 0 0
\(995\) 9174.15 0.292301
\(996\) 0 0
\(997\) −52118.0 −1.65556 −0.827780 0.561053i \(-0.810396\pi\)
−0.827780 + 0.561053i \(0.810396\pi\)
\(998\) 0 0
\(999\) −12102.1 −0.383278
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 960.4.a.bk.1.2 2
4.3 odd 2 960.4.a.bn.1.1 2
8.3 odd 2 480.4.a.o.1.1 2
8.5 even 2 480.4.a.r.1.2 yes 2
24.5 odd 2 1440.4.a.y.1.2 2
24.11 even 2 1440.4.a.s.1.1 2
40.19 odd 2 2400.4.a.bb.1.2 2
40.29 even 2 2400.4.a.y.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
480.4.a.o.1.1 2 8.3 odd 2
480.4.a.r.1.2 yes 2 8.5 even 2
960.4.a.bk.1.2 2 1.1 even 1 trivial
960.4.a.bn.1.1 2 4.3 odd 2
1440.4.a.s.1.1 2 24.11 even 2
1440.4.a.y.1.2 2 24.5 odd 2
2400.4.a.y.1.1 2 40.29 even 2
2400.4.a.bb.1.2 2 40.19 odd 2