Properties

Label 20.4.c.a.9.1
Level $20$
Weight $4$
Character 20.9
Analytic conductor $1.180$
Analytic rank $0$
Dimension $2$
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] = [20,4,Mod(9,20)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(20, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("20.9");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 20 = 2^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 20.c (of order \(2\), degree \(1\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.18003820011\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-19}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 5 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 9.1
Root \(0.500000 + 2.17945i\) of defining polynomial
Character \(\chi\) \(=\) 20.9
Dual form 20.4.c.a.9.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-8.71780i q^{3} +(7.00000 + 8.71780i) q^{5} +8.71780i q^{7} -49.0000 q^{9} +O(q^{10})\) \(q-8.71780i q^{3} +(7.00000 + 8.71780i) q^{5} +8.71780i q^{7} -49.0000 q^{9} +20.0000 q^{11} +52.3068i q^{13} +(76.0000 - 61.0246i) q^{15} -69.7424i q^{17} -84.0000 q^{19} +76.0000 q^{21} -61.0246i q^{23} +(-27.0000 + 122.049i) q^{25} +191.792i q^{27} +6.00000 q^{29} -224.000 q^{31} -174.356i q^{33} +(-76.0000 + 61.0246i) q^{35} -122.049i q^{37} +456.000 q^{39} +266.000 q^{41} +305.123i q^{43} +(-343.000 - 427.172i) q^{45} -374.865i q^{47} +267.000 q^{49} -608.000 q^{51} +366.148i q^{53} +(140.000 + 174.356i) q^{55} +732.295i q^{57} -28.0000 q^{59} +182.000 q^{61} -427.172i q^{63} +(-456.000 + 366.148i) q^{65} -427.172i q^{67} -532.000 q^{69} +408.000 q^{71} -1081.01i q^{73} +(1064.00 + 235.381i) q^{75} +174.356i q^{77} +48.0000 q^{79} +349.000 q^{81} +200.509i q^{83} +(608.000 - 488.197i) q^{85} -52.3068i q^{87} -1526.00 q^{89} -456.000 q^{91} +1952.79i q^{93} +(-588.000 - 732.295i) q^{95} -557.939i q^{97} -980.000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 14 q^{5} - 98 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 14 q^{5} - 98 q^{9} + 40 q^{11} + 152 q^{15} - 168 q^{19} + 152 q^{21} - 54 q^{25} + 12 q^{29} - 448 q^{31} - 152 q^{35} + 912 q^{39} + 532 q^{41} - 686 q^{45} + 534 q^{49} - 1216 q^{51} + 280 q^{55} - 56 q^{59} + 364 q^{61} - 912 q^{65} - 1064 q^{69} + 816 q^{71} + 2128 q^{75} + 96 q^{79} + 698 q^{81} + 1216 q^{85} - 3052 q^{89} - 912 q^{91} - 1176 q^{95} - 1960 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/20\mathbb{Z}\right)^\times\).

\(n\) \(11\) \(17\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 8.71780i 1.67774i −0.544331 0.838870i \(-0.683216\pi\)
0.544331 0.838870i \(-0.316784\pi\)
\(4\) 0 0
\(5\) 7.00000 + 8.71780i 0.626099 + 0.779744i
\(6\) 0 0
\(7\) 8.71780i 0.470717i 0.971909 + 0.235358i \(0.0756264\pi\)
−0.971909 + 0.235358i \(0.924374\pi\)
\(8\) 0 0
\(9\) −49.0000 −1.81481
\(10\) 0 0
\(11\) 20.0000 0.548202 0.274101 0.961701i \(-0.411620\pi\)
0.274101 + 0.961701i \(0.411620\pi\)
\(12\) 0 0
\(13\) 52.3068i 1.11595i 0.829859 + 0.557973i \(0.188421\pi\)
−0.829859 + 0.557973i \(0.811579\pi\)
\(14\) 0 0
\(15\) 76.0000 61.0246i 1.30821 1.05043i
\(16\) 0 0
\(17\) 69.7424i 0.995001i −0.867464 0.497500i \(-0.834251\pi\)
0.867464 0.497500i \(-0.165749\pi\)
\(18\) 0 0
\(19\) −84.0000 −1.01426 −0.507130 0.861870i \(-0.669293\pi\)
−0.507130 + 0.861870i \(0.669293\pi\)
\(20\) 0 0
\(21\) 76.0000 0.789741
\(22\) 0 0
\(23\) 61.0246i 0.553239i −0.960979 0.276620i \(-0.910786\pi\)
0.960979 0.276620i \(-0.0892142\pi\)
\(24\) 0 0
\(25\) −27.0000 + 122.049i −0.216000 + 0.976393i
\(26\) 0 0
\(27\) 191.792i 1.36705i
\(28\) 0 0
\(29\) 6.00000 0.0384197 0.0192099 0.999815i \(-0.493885\pi\)
0.0192099 + 0.999815i \(0.493885\pi\)
\(30\) 0 0
\(31\) −224.000 −1.29779 −0.648897 0.760877i \(-0.724769\pi\)
−0.648897 + 0.760877i \(0.724769\pi\)
\(32\) 0 0
\(33\) 174.356i 0.919742i
\(34\) 0 0
\(35\) −76.0000 + 61.0246i −0.367038 + 0.294715i
\(36\) 0 0
\(37\) 122.049i 0.542291i −0.962538 0.271145i \(-0.912598\pi\)
0.962538 0.271145i \(-0.0874024\pi\)
\(38\) 0 0
\(39\) 456.000 1.87227
\(40\) 0 0
\(41\) 266.000 1.01322 0.506612 0.862174i \(-0.330898\pi\)
0.506612 + 0.862174i \(0.330898\pi\)
\(42\) 0 0
\(43\) 305.123i 1.08211i 0.840987 + 0.541056i \(0.181975\pi\)
−0.840987 + 0.541056i \(0.818025\pi\)
\(44\) 0 0
\(45\) −343.000 427.172i −1.13625 1.41509i
\(46\) 0 0
\(47\) 374.865i 1.16340i −0.813404 0.581699i \(-0.802388\pi\)
0.813404 0.581699i \(-0.197612\pi\)
\(48\) 0 0
\(49\) 267.000 0.778426
\(50\) 0 0
\(51\) −608.000 −1.66935
\(52\) 0 0
\(53\) 366.148i 0.948948i 0.880270 + 0.474474i \(0.157362\pi\)
−0.880270 + 0.474474i \(0.842638\pi\)
\(54\) 0 0
\(55\) 140.000 + 174.356i 0.343229 + 0.427457i
\(56\) 0 0
\(57\) 732.295i 1.70166i
\(58\) 0 0
\(59\) −28.0000 −0.0617846 −0.0308923 0.999523i \(-0.509835\pi\)
−0.0308923 + 0.999523i \(0.509835\pi\)
\(60\) 0 0
\(61\) 182.000 0.382012 0.191006 0.981589i \(-0.438825\pi\)
0.191006 + 0.981589i \(0.438825\pi\)
\(62\) 0 0
\(63\) 427.172i 0.854264i
\(64\) 0 0
\(65\) −456.000 + 366.148i −0.870151 + 0.698692i
\(66\) 0 0
\(67\) 427.172i 0.778916i −0.921044 0.389458i \(-0.872662\pi\)
0.921044 0.389458i \(-0.127338\pi\)
\(68\) 0 0
\(69\) −532.000 −0.928192
\(70\) 0 0
\(71\) 408.000 0.681982 0.340991 0.940067i \(-0.389238\pi\)
0.340991 + 0.940067i \(0.389238\pi\)
\(72\) 0 0
\(73\) 1081.01i 1.73318i −0.499019 0.866591i \(-0.666306\pi\)
0.499019 0.866591i \(-0.333694\pi\)
\(74\) 0 0
\(75\) 1064.00 + 235.381i 1.63814 + 0.362392i
\(76\) 0 0
\(77\) 174.356i 0.258048i
\(78\) 0 0
\(79\) 48.0000 0.0683598 0.0341799 0.999416i \(-0.489118\pi\)
0.0341799 + 0.999416i \(0.489118\pi\)
\(80\) 0 0
\(81\) 349.000 0.478738
\(82\) 0 0
\(83\) 200.509i 0.265166i 0.991172 + 0.132583i \(0.0423271\pi\)
−0.991172 + 0.132583i \(0.957673\pi\)
\(84\) 0 0
\(85\) 608.000 488.197i 0.775845 0.622969i
\(86\) 0 0
\(87\) 52.3068i 0.0644583i
\(88\) 0 0
\(89\) −1526.00 −1.81748 −0.908740 0.417363i \(-0.862954\pi\)
−0.908740 + 0.417363i \(0.862954\pi\)
\(90\) 0 0
\(91\) −456.000 −0.525294
\(92\) 0 0
\(93\) 1952.79i 2.17736i
\(94\) 0 0
\(95\) −588.000 732.295i −0.635027 0.790862i
\(96\) 0 0
\(97\) 557.939i 0.584022i −0.956415 0.292011i \(-0.905676\pi\)
0.956415 0.292011i \(-0.0943244\pi\)
\(98\) 0 0
\(99\) −980.000 −0.994886
\(100\) 0 0
\(101\) 1246.00 1.22754 0.613770 0.789485i \(-0.289652\pi\)
0.613770 + 0.789485i \(0.289652\pi\)
\(102\) 0 0
\(103\) 845.626i 0.808952i 0.914549 + 0.404476i \(0.132546\pi\)
−0.914549 + 0.404476i \(0.867454\pi\)
\(104\) 0 0
\(105\) 532.000 + 662.553i 0.494456 + 0.615795i
\(106\) 0 0
\(107\) 1281.52i 1.15784i 0.815384 + 0.578920i \(0.196526\pi\)
−0.815384 + 0.578920i \(0.803474\pi\)
\(108\) 0 0
\(109\) 902.000 0.792623 0.396312 0.918116i \(-0.370290\pi\)
0.396312 + 0.918116i \(0.370290\pi\)
\(110\) 0 0
\(111\) −1064.00 −0.909824
\(112\) 0 0
\(113\) 1464.59i 1.21927i −0.792684 0.609633i \(-0.791317\pi\)
0.792684 0.609633i \(-0.208683\pi\)
\(114\) 0 0
\(115\) 532.000 427.172i 0.431385 0.346383i
\(116\) 0 0
\(117\) 2563.03i 2.02523i
\(118\) 0 0
\(119\) 608.000 0.468364
\(120\) 0 0
\(121\) −931.000 −0.699474
\(122\) 0 0
\(123\) 2318.93i 1.69993i
\(124\) 0 0
\(125\) −1253.00 + 618.964i −0.896574 + 0.442894i
\(126\) 0 0
\(127\) 2624.06i 1.83344i 0.399525 + 0.916722i \(0.369175\pi\)
−0.399525 + 0.916722i \(0.630825\pi\)
\(128\) 0 0
\(129\) 2660.00 1.81550
\(130\) 0 0
\(131\) 2940.00 1.96083 0.980416 0.196938i \(-0.0630997\pi\)
0.980416 + 0.196938i \(0.0630997\pi\)
\(132\) 0 0
\(133\) 732.295i 0.477429i
\(134\) 0 0
\(135\) −1672.00 + 1342.54i −1.06595 + 0.855908i
\(136\) 0 0
\(137\) 732.295i 0.456673i 0.973582 + 0.228336i \(0.0733286\pi\)
−0.973582 + 0.228336i \(0.926671\pi\)
\(138\) 0 0
\(139\) −364.000 −0.222116 −0.111058 0.993814i \(-0.535424\pi\)
−0.111058 + 0.993814i \(0.535424\pi\)
\(140\) 0 0
\(141\) −3268.00 −1.95188
\(142\) 0 0
\(143\) 1046.14i 0.611764i
\(144\) 0 0
\(145\) 42.0000 + 52.3068i 0.0240546 + 0.0299575i
\(146\) 0 0
\(147\) 2327.65i 1.30600i
\(148\) 0 0
\(149\) 254.000 0.139654 0.0698272 0.997559i \(-0.477755\pi\)
0.0698272 + 0.997559i \(0.477755\pi\)
\(150\) 0 0
\(151\) −2360.00 −1.27188 −0.635941 0.771738i \(-0.719388\pi\)
−0.635941 + 0.771738i \(0.719388\pi\)
\(152\) 0 0
\(153\) 3417.38i 1.80574i
\(154\) 0 0
\(155\) −1568.00 1952.79i −0.812547 1.01195i
\(156\) 0 0
\(157\) 2214.32i 1.12562i 0.826587 + 0.562809i \(0.190279\pi\)
−0.826587 + 0.562809i \(0.809721\pi\)
\(158\) 0 0
\(159\) 3192.00 1.59209
\(160\) 0 0
\(161\) 532.000 0.260419
\(162\) 0 0
\(163\) 1525.61i 0.733100i 0.930398 + 0.366550i \(0.119461\pi\)
−0.930398 + 0.366550i \(0.880539\pi\)
\(164\) 0 0
\(165\) 1520.00 1220.49i 0.717163 0.575849i
\(166\) 0 0
\(167\) 3059.95i 1.41788i −0.705269 0.708940i \(-0.749174\pi\)
0.705269 0.708940i \(-0.250826\pi\)
\(168\) 0 0
\(169\) −539.000 −0.245335
\(170\) 0 0
\(171\) 4116.00 1.84069
\(172\) 0 0
\(173\) 1900.48i 0.835207i −0.908629 0.417604i \(-0.862870\pi\)
0.908629 0.417604i \(-0.137130\pi\)
\(174\) 0 0
\(175\) −1064.00 235.381i −0.459605 0.101675i
\(176\) 0 0
\(177\) 244.098i 0.103659i
\(178\) 0 0
\(179\) −1972.00 −0.823431 −0.411716 0.911312i \(-0.635070\pi\)
−0.411716 + 0.911312i \(0.635070\pi\)
\(180\) 0 0
\(181\) −1330.00 −0.546177 −0.273089 0.961989i \(-0.588045\pi\)
−0.273089 + 0.961989i \(0.588045\pi\)
\(182\) 0 0
\(183\) 1586.64i 0.640917i
\(184\) 0 0
\(185\) 1064.00 854.344i 0.422848 0.339528i
\(186\) 0 0
\(187\) 1394.85i 0.545462i
\(188\) 0 0
\(189\) −1672.00 −0.643493
\(190\) 0 0
\(191\) 1728.00 0.654627 0.327313 0.944916i \(-0.393857\pi\)
0.327313 + 0.944916i \(0.393857\pi\)
\(192\) 0 0
\(193\) 976.393i 0.364157i −0.983284 0.182079i \(-0.941717\pi\)
0.983284 0.182079i \(-0.0582825\pi\)
\(194\) 0 0
\(195\) 3192.00 + 3975.32i 1.17222 + 1.45989i
\(196\) 0 0
\(197\) 3783.52i 1.36835i 0.729318 + 0.684175i \(0.239838\pi\)
−0.729318 + 0.684175i \(0.760162\pi\)
\(198\) 0 0
\(199\) −1512.00 −0.538607 −0.269304 0.963055i \(-0.586794\pi\)
−0.269304 + 0.963055i \(0.586794\pi\)
\(200\) 0 0
\(201\) −3724.00 −1.30682
\(202\) 0 0
\(203\) 52.3068i 0.0180848i
\(204\) 0 0
\(205\) 1862.00 + 2318.93i 0.634379 + 0.790056i
\(206\) 0 0
\(207\) 2990.20i 1.00403i
\(208\) 0 0
\(209\) −1680.00 −0.556019
\(210\) 0 0
\(211\) 1644.00 0.536387 0.268193 0.963365i \(-0.413573\pi\)
0.268193 + 0.963365i \(0.413573\pi\)
\(212\) 0 0
\(213\) 3556.86i 1.14419i
\(214\) 0 0
\(215\) −2660.00 + 2135.86i −0.843770 + 0.677509i
\(216\) 0 0
\(217\) 1952.79i 0.610893i
\(218\) 0 0
\(219\) −9424.00 −2.90783
\(220\) 0 0
\(221\) 3648.00 1.11037
\(222\) 0 0
\(223\) 1089.72i 0.327235i 0.986524 + 0.163617i \(0.0523163\pi\)
−0.986524 + 0.163617i \(0.947684\pi\)
\(224\) 0 0
\(225\) 1323.00 5980.41i 0.392000 1.77197i
\(226\) 0 0
\(227\) 985.111i 0.288036i −0.989575 0.144018i \(-0.953998\pi\)
0.989575 0.144018i \(-0.0460023\pi\)
\(228\) 0 0
\(229\) 3934.00 1.13522 0.567611 0.823297i \(-0.307868\pi\)
0.567611 + 0.823297i \(0.307868\pi\)
\(230\) 0 0
\(231\) 1520.00 0.432938
\(232\) 0 0
\(233\) 4637.87i 1.30402i 0.758210 + 0.652010i \(0.226074\pi\)
−0.758210 + 0.652010i \(0.773926\pi\)
\(234\) 0 0
\(235\) 3268.00 2624.06i 0.907152 0.728403i
\(236\) 0 0
\(237\) 418.454i 0.114690i
\(238\) 0 0
\(239\) 3856.00 1.04361 0.521807 0.853063i \(-0.325258\pi\)
0.521807 + 0.853063i \(0.325258\pi\)
\(240\) 0 0
\(241\) 994.000 0.265681 0.132841 0.991137i \(-0.457590\pi\)
0.132841 + 0.991137i \(0.457590\pi\)
\(242\) 0 0
\(243\) 2135.86i 0.563850i
\(244\) 0 0
\(245\) 1869.00 + 2327.65i 0.487372 + 0.606972i
\(246\) 0 0
\(247\) 4393.77i 1.13186i
\(248\) 0 0
\(249\) 1748.00 0.444880
\(250\) 0 0
\(251\) −6300.00 −1.58427 −0.792136 0.610344i \(-0.791031\pi\)
−0.792136 + 0.610344i \(0.791031\pi\)
\(252\) 0 0
\(253\) 1220.49i 0.303287i
\(254\) 0 0
\(255\) −4256.00 5300.42i −1.04518 1.30167i
\(256\) 0 0
\(257\) 6695.27i 1.62506i −0.582922 0.812528i \(-0.698091\pi\)
0.582922 0.812528i \(-0.301909\pi\)
\(258\) 0 0
\(259\) 1064.00 0.255265
\(260\) 0 0
\(261\) −294.000 −0.0697247
\(262\) 0 0
\(263\) 1403.57i 0.329078i 0.986371 + 0.164539i \(0.0526137\pi\)
−0.986371 + 0.164539i \(0.947386\pi\)
\(264\) 0 0
\(265\) −3192.00 + 2563.03i −0.739936 + 0.594135i
\(266\) 0 0
\(267\) 13303.4i 3.04926i
\(268\) 0 0
\(269\) −5082.00 −1.15188 −0.575939 0.817493i \(-0.695363\pi\)
−0.575939 + 0.817493i \(0.695363\pi\)
\(270\) 0 0
\(271\) 2800.00 0.627631 0.313815 0.949484i \(-0.398393\pi\)
0.313815 + 0.949484i \(0.398393\pi\)
\(272\) 0 0
\(273\) 3975.32i 0.881308i
\(274\) 0 0
\(275\) −540.000 + 2440.98i −0.118412 + 0.535261i
\(276\) 0 0
\(277\) 1586.64i 0.344159i −0.985083 0.172079i \(-0.944951\pi\)
0.985083 0.172079i \(-0.0550485\pi\)
\(278\) 0 0
\(279\) 10976.0 2.35525
\(280\) 0 0
\(281\) −1254.00 −0.266218 −0.133109 0.991101i \(-0.542496\pi\)
−0.133109 + 0.991101i \(0.542496\pi\)
\(282\) 0 0
\(283\) 4646.59i 0.976010i −0.872841 0.488005i \(-0.837725\pi\)
0.872841 0.488005i \(-0.162275\pi\)
\(284\) 0 0
\(285\) −6384.00 + 5126.07i −1.32686 + 1.06541i
\(286\) 0 0
\(287\) 2318.93i 0.476942i
\(288\) 0 0
\(289\) 49.0000 0.00997354
\(290\) 0 0
\(291\) −4864.00 −0.979838
\(292\) 0 0
\(293\) 6538.35i 1.30367i −0.758362 0.651833i \(-0.774000\pi\)
0.758362 0.651833i \(-0.226000\pi\)
\(294\) 0 0
\(295\) −196.000 244.098i −0.0386833 0.0481761i
\(296\) 0 0
\(297\) 3835.83i 0.749419i
\(298\) 0 0
\(299\) 3192.00 0.617385
\(300\) 0 0
\(301\) −2660.00 −0.509368
\(302\) 0 0
\(303\) 10862.4i 2.05950i
\(304\) 0 0
\(305\) 1274.00 + 1586.64i 0.239177 + 0.297871i
\(306\) 0 0
\(307\) 6355.27i 1.18148i −0.806862 0.590741i \(-0.798836\pi\)
0.806862 0.590741i \(-0.201164\pi\)
\(308\) 0 0
\(309\) 7372.00 1.35721
\(310\) 0 0
\(311\) −5208.00 −0.949577 −0.474789 0.880100i \(-0.657476\pi\)
−0.474789 + 0.880100i \(0.657476\pi\)
\(312\) 0 0
\(313\) 4568.13i 0.824939i 0.910972 + 0.412469i \(0.135334\pi\)
−0.910972 + 0.412469i \(0.864666\pi\)
\(314\) 0 0
\(315\) 3724.00 2990.20i 0.666107 0.534854i
\(316\) 0 0
\(317\) 854.344i 0.151371i −0.997132 0.0756857i \(-0.975885\pi\)
0.997132 0.0756857i \(-0.0241146\pi\)
\(318\) 0 0
\(319\) 120.000 0.0210618
\(320\) 0 0
\(321\) 11172.0 1.94256
\(322\) 0 0
\(323\) 5858.36i 1.00919i
\(324\) 0 0
\(325\) −6384.00 1412.28i −1.08960 0.241044i
\(326\) 0 0
\(327\) 7863.45i 1.32982i
\(328\) 0 0
\(329\) 3268.00 0.547631
\(330\) 0 0
\(331\) 7828.00 1.29990 0.649948 0.759978i \(-0.274791\pi\)
0.649948 + 0.759978i \(0.274791\pi\)
\(332\) 0 0
\(333\) 5980.41i 0.984157i
\(334\) 0 0
\(335\) 3724.00 2990.20i 0.607355 0.487679i
\(336\) 0 0
\(337\) 5370.16i 0.868046i 0.900902 + 0.434023i \(0.142906\pi\)
−0.900902 + 0.434023i \(0.857094\pi\)
\(338\) 0 0
\(339\) −12768.0 −2.04561
\(340\) 0 0
\(341\) −4480.00 −0.711453
\(342\) 0 0
\(343\) 5317.86i 0.837135i
\(344\) 0 0
\(345\) −3724.00 4637.87i −0.581140 0.723752i
\(346\) 0 0
\(347\) 2746.11i 0.424838i 0.977179 + 0.212419i \(0.0681341\pi\)
−0.977179 + 0.212419i \(0.931866\pi\)
\(348\) 0 0
\(349\) −8890.00 −1.36353 −0.681763 0.731573i \(-0.738787\pi\)
−0.681763 + 0.731573i \(0.738787\pi\)
\(350\) 0 0
\(351\) −10032.0 −1.52555
\(352\) 0 0
\(353\) 1534.33i 0.231344i −0.993287 0.115672i \(-0.963098\pi\)
0.993287 0.115672i \(-0.0369021\pi\)
\(354\) 0 0
\(355\) 2856.00 + 3556.86i 0.426988 + 0.531771i
\(356\) 0 0
\(357\) 5300.42i 0.785793i
\(358\) 0 0
\(359\) 9144.00 1.34429 0.672147 0.740417i \(-0.265372\pi\)
0.672147 + 0.740417i \(0.265372\pi\)
\(360\) 0 0
\(361\) 197.000 0.0287214
\(362\) 0 0
\(363\) 8116.27i 1.17354i
\(364\) 0 0
\(365\) 9424.00 7567.05i 1.35144 1.08514i
\(366\) 0 0
\(367\) 10365.5i 1.47431i 0.675722 + 0.737156i \(0.263832\pi\)
−0.675722 + 0.737156i \(0.736168\pi\)
\(368\) 0 0
\(369\) −13034.0 −1.83882
\(370\) 0 0
\(371\) −3192.00 −0.446686
\(372\) 0 0
\(373\) 3295.33i 0.457441i 0.973492 + 0.228721i \(0.0734542\pi\)
−0.973492 + 0.228721i \(0.926546\pi\)
\(374\) 0 0
\(375\) 5396.00 + 10923.4i 0.743062 + 1.50422i
\(376\) 0 0
\(377\) 313.841i 0.0428743i
\(378\) 0 0
\(379\) −2588.00 −0.350756 −0.175378 0.984501i \(-0.556115\pi\)
−0.175378 + 0.984501i \(0.556115\pi\)
\(380\) 0 0
\(381\) 22876.0 3.07604
\(382\) 0 0
\(383\) 4071.21i 0.543157i −0.962416 0.271579i \(-0.912454\pi\)
0.962416 0.271579i \(-0.0875457\pi\)
\(384\) 0 0
\(385\) −1520.00 + 1220.49i −0.201211 + 0.161564i
\(386\) 0 0
\(387\) 14951.0i 1.96383i
\(388\) 0 0
\(389\) −5426.00 −0.707221 −0.353611 0.935393i \(-0.615046\pi\)
−0.353611 + 0.935393i \(0.615046\pi\)
\(390\) 0 0
\(391\) −4256.00 −0.550474
\(392\) 0 0
\(393\) 25630.3i 3.28977i
\(394\) 0 0
\(395\) 336.000 + 418.454i 0.0428000 + 0.0533031i
\(396\) 0 0
\(397\) 9816.24i 1.24096i 0.784220 + 0.620482i \(0.213063\pi\)
−0.784220 + 0.620482i \(0.786937\pi\)
\(398\) 0 0
\(399\) −6384.00 −0.801002
\(400\) 0 0
\(401\) 370.000 0.0460771 0.0230386 0.999735i \(-0.492666\pi\)
0.0230386 + 0.999735i \(0.492666\pi\)
\(402\) 0 0
\(403\) 11716.7i 1.44827i
\(404\) 0 0
\(405\) 2443.00 + 3042.51i 0.299737 + 0.373293i
\(406\) 0 0
\(407\) 2440.98i 0.297285i
\(408\) 0 0
\(409\) 4186.00 0.506074 0.253037 0.967457i \(-0.418571\pi\)
0.253037 + 0.967457i \(0.418571\pi\)
\(410\) 0 0
\(411\) 6384.00 0.766179
\(412\) 0 0
\(413\) 244.098i 0.0290830i
\(414\) 0 0
\(415\) −1748.00 + 1403.57i −0.206761 + 0.166020i
\(416\) 0 0
\(417\) 3173.28i 0.372653i
\(418\) 0 0
\(419\) 2716.00 0.316671 0.158336 0.987385i \(-0.449387\pi\)
0.158336 + 0.987385i \(0.449387\pi\)
\(420\) 0 0
\(421\) 1966.00 0.227594 0.113797 0.993504i \(-0.463699\pi\)
0.113797 + 0.993504i \(0.463699\pi\)
\(422\) 0 0
\(423\) 18368.4i 2.11135i
\(424\) 0 0
\(425\) 8512.00 + 1883.04i 0.971512 + 0.214920i
\(426\) 0 0
\(427\) 1586.64i 0.179819i
\(428\) 0 0
\(429\) 9120.00 1.02638
\(430\) 0 0
\(431\) −11824.0 −1.32144 −0.660722 0.750631i \(-0.729750\pi\)
−0.660722 + 0.750631i \(0.729750\pi\)
\(432\) 0 0
\(433\) 2580.47i 0.286396i 0.989694 + 0.143198i \(0.0457385\pi\)
−0.989694 + 0.143198i \(0.954261\pi\)
\(434\) 0 0
\(435\) 456.000 366.148i 0.0502610 0.0403573i
\(436\) 0 0
\(437\) 5126.07i 0.561128i
\(438\) 0 0
\(439\) −13272.0 −1.44291 −0.721456 0.692461i \(-0.756527\pi\)
−0.721456 + 0.692461i \(0.756527\pi\)
\(440\) 0 0
\(441\) −13083.0 −1.41270
\(442\) 0 0
\(443\) 3112.25i 0.333787i −0.985975 0.166894i \(-0.946626\pi\)
0.985975 0.166894i \(-0.0533736\pi\)
\(444\) 0 0
\(445\) −10682.0 13303.4i −1.13792 1.41717i
\(446\) 0 0
\(447\) 2214.32i 0.234304i
\(448\) 0 0
\(449\) −2094.00 −0.220093 −0.110047 0.993926i \(-0.535100\pi\)
−0.110047 + 0.993926i \(0.535100\pi\)
\(450\) 0 0
\(451\) 5320.00 0.555452
\(452\) 0 0
\(453\) 20574.0i 2.13389i
\(454\) 0 0
\(455\) −3192.00 3975.32i −0.328886 0.409595i
\(456\) 0 0
\(457\) 7078.85i 0.724584i −0.932065 0.362292i \(-0.881994\pi\)
0.932065 0.362292i \(-0.118006\pi\)
\(458\) 0 0
\(459\) 13376.0 1.36021
\(460\) 0 0
\(461\) −9450.00 −0.954730 −0.477365 0.878705i \(-0.658408\pi\)
−0.477365 + 0.878705i \(0.658408\pi\)
\(462\) 0 0
\(463\) 3112.25i 0.312395i 0.987726 + 0.156197i \(0.0499236\pi\)
−0.987726 + 0.156197i \(0.950076\pi\)
\(464\) 0 0
\(465\) −17024.0 + 13669.5i −1.69778 + 1.36324i
\(466\) 0 0
\(467\) 5361.45i 0.531259i 0.964075 + 0.265630i \(0.0855798\pi\)
−0.964075 + 0.265630i \(0.914420\pi\)
\(468\) 0 0
\(469\) 3724.00 0.366649
\(470\) 0 0
\(471\) 19304.0 1.88850
\(472\) 0 0
\(473\) 6102.46i 0.593216i
\(474\) 0 0
\(475\) 2268.00 10252.1i 0.219080 0.990316i
\(476\) 0 0
\(477\) 17941.2i 1.72216i
\(478\) 0 0
\(479\) 13216.0 1.26066 0.630328 0.776329i \(-0.282920\pi\)
0.630328 + 0.776329i \(0.282920\pi\)
\(480\) 0 0
\(481\) 6384.00 0.605167
\(482\) 0 0
\(483\) 4637.87i 0.436916i
\(484\) 0 0
\(485\) 4864.00 3905.57i 0.455387 0.365656i
\(486\) 0 0
\(487\) 2379.96i 0.221450i 0.993851 + 0.110725i \(0.0353173\pi\)
−0.993851 + 0.110725i \(0.964683\pi\)
\(488\) 0 0
\(489\) 13300.0 1.22995
\(490\) 0 0
\(491\) −12300.0 −1.13053 −0.565266 0.824909i \(-0.691226\pi\)
−0.565266 + 0.824909i \(0.691226\pi\)
\(492\) 0 0
\(493\) 418.454i 0.0382277i
\(494\) 0 0
\(495\) −6860.00 8543.44i −0.622897 0.775756i
\(496\) 0 0
\(497\) 3556.86i 0.321020i
\(498\) 0 0
\(499\) 15756.0 1.41350 0.706749 0.707464i \(-0.250161\pi\)
0.706749 + 0.707464i \(0.250161\pi\)
\(500\) 0 0
\(501\) −26676.0 −2.37883
\(502\) 0 0
\(503\) 19170.4i 1.69934i −0.527316 0.849670i \(-0.676801\pi\)
0.527316 0.849670i \(-0.323199\pi\)
\(504\) 0 0
\(505\) 8722.00 + 10862.4i 0.768562 + 0.957167i
\(506\) 0 0
\(507\) 4698.89i 0.411608i
\(508\) 0 0
\(509\) −2394.00 −0.208472 −0.104236 0.994553i \(-0.533240\pi\)
−0.104236 + 0.994553i \(0.533240\pi\)
\(510\) 0 0
\(511\) 9424.00 0.815838
\(512\) 0 0
\(513\) 16110.5i 1.38654i
\(514\) 0 0
\(515\) −7372.00 + 5919.38i −0.630775 + 0.506484i
\(516\) 0 0
\(517\) 7497.31i 0.637778i
\(518\) 0 0
\(519\) −16568.0 −1.40126
\(520\) 0 0
\(521\) 6874.00 0.578033 0.289017 0.957324i \(-0.406672\pi\)
0.289017 + 0.957324i \(0.406672\pi\)
\(522\) 0 0
\(523\) 16154.1i 1.35061i −0.737539 0.675305i \(-0.764012\pi\)
0.737539 0.675305i \(-0.235988\pi\)
\(524\) 0 0
\(525\) −2052.00 + 9275.74i −0.170584 + 0.771098i
\(526\) 0 0
\(527\) 15622.3i 1.29131i
\(528\) 0 0
\(529\) 8443.00 0.693926
\(530\) 0 0
\(531\) 1372.00 0.112128
\(532\) 0 0
\(533\) 13913.6i 1.13070i
\(534\) 0 0
\(535\) −11172.0 + 8970.61i −0.902818 + 0.724922i
\(536\) 0 0
\(537\) 17191.5i 1.38150i
\(538\) 0 0
\(539\) 5340.00 0.426735
\(540\) 0 0
\(541\) 7174.00 0.570119 0.285059 0.958510i \(-0.407987\pi\)
0.285059 + 0.958510i \(0.407987\pi\)
\(542\) 0 0
\(543\) 11594.7i 0.916344i
\(544\) 0 0
\(545\) 6314.00 + 7863.45i 0.496261 + 0.618043i
\(546\) 0 0
\(547\) 4332.75i 0.338674i −0.985558 0.169337i \(-0.945837\pi\)
0.985558 0.169337i \(-0.0541627\pi\)
\(548\) 0 0
\(549\) −8918.00 −0.693280
\(550\) 0 0
\(551\) −504.000 −0.0389676
\(552\) 0 0
\(553\) 418.454i 0.0321781i
\(554\) 0 0
\(555\) −7448.00 9275.74i −0.569640 0.709429i
\(556\) 0 0
\(557\) 9153.69i 0.696327i −0.937434 0.348164i \(-0.886805\pi\)
0.937434 0.348164i \(-0.113195\pi\)
\(558\) 0 0
\(559\) −15960.0 −1.20758
\(560\) 0 0
\(561\) −12160.0 −0.915144
\(562\) 0 0
\(563\) 13870.0i 1.03828i 0.854689 + 0.519140i \(0.173748\pi\)
−0.854689 + 0.519140i \(0.826252\pi\)
\(564\) 0 0
\(565\) 12768.0 10252.1i 0.950715 0.763381i
\(566\) 0 0
\(567\) 3042.51i 0.225350i
\(568\) 0 0
\(569\) −16070.0 −1.18399 −0.591994 0.805942i \(-0.701659\pi\)
−0.591994 + 0.805942i \(0.701659\pi\)
\(570\) 0 0
\(571\) −3932.00 −0.288177 −0.144089 0.989565i \(-0.546025\pi\)
−0.144089 + 0.989565i \(0.546025\pi\)
\(572\) 0 0
\(573\) 15064.4i 1.09829i
\(574\) 0 0
\(575\) 7448.00 + 1647.66i 0.540179 + 0.119500i
\(576\) 0 0
\(577\) 16180.2i 1.16740i −0.811968 0.583702i \(-0.801604\pi\)
0.811968 0.583702i \(-0.198396\pi\)
\(578\) 0 0
\(579\) −8512.00 −0.610961
\(580\) 0 0
\(581\) −1748.00 −0.124818
\(582\) 0 0
\(583\) 7322.95i 0.520215i
\(584\) 0 0
\(585\) 22344.0 17941.2i 1.57916 1.26800i
\(586\) 0 0
\(587\) 11272.1i 0.792589i −0.918123 0.396295i \(-0.870296\pi\)
0.918123 0.396295i \(-0.129704\pi\)
\(588\) 0 0
\(589\) 18816.0 1.31630
\(590\) 0 0
\(591\) 32984.0 2.29574
\(592\) 0 0
\(593\) 10670.6i 0.738935i 0.929244 + 0.369467i \(0.120460\pi\)
−0.929244 + 0.369467i \(0.879540\pi\)
\(594\) 0 0
\(595\) 4256.00 + 5300.42i 0.293242 + 0.365203i
\(596\) 0 0
\(597\) 13181.3i 0.903643i
\(598\) 0 0
\(599\) −6840.00 −0.466569 −0.233284 0.972409i \(-0.574947\pi\)
−0.233284 + 0.972409i \(0.574947\pi\)
\(600\) 0 0
\(601\) −10150.0 −0.688897 −0.344449 0.938805i \(-0.611934\pi\)
−0.344449 + 0.938805i \(0.611934\pi\)
\(602\) 0 0
\(603\) 20931.4i 1.41359i
\(604\) 0 0
\(605\) −6517.00 8116.27i −0.437940 0.545410i
\(606\) 0 0
\(607\) 1839.46i 0.123000i −0.998107 0.0615002i \(-0.980412\pi\)
0.998107 0.0615002i \(-0.0195885\pi\)
\(608\) 0 0
\(609\) 456.000 0.0303416
\(610\) 0 0
\(611\) 19608.0 1.29829
\(612\) 0 0
\(613\) 5004.02i 0.329707i −0.986318 0.164853i \(-0.947285\pi\)
0.986318 0.164853i \(-0.0527151\pi\)
\(614\) 0 0
\(615\) 20216.0 16232.5i 1.32551 1.06432i
\(616\) 0 0
\(617\) 24653.9i 1.60864i −0.594197 0.804319i \(-0.702530\pi\)
0.594197 0.804319i \(-0.297470\pi\)
\(618\) 0 0
\(619\) −27020.0 −1.75448 −0.877242 0.480049i \(-0.840619\pi\)
−0.877242 + 0.480049i \(0.840619\pi\)
\(620\) 0 0
\(621\) 11704.0 0.756305
\(622\) 0 0
\(623\) 13303.4i 0.855518i
\(624\) 0 0
\(625\) −14167.0 6590.66i −0.906688 0.421802i
\(626\) 0 0
\(627\) 14645.9i 0.932856i
\(628\) 0 0
\(629\) −8512.00 −0.539580
\(630\) 0 0
\(631\) −10648.0 −0.671775 −0.335888 0.941902i \(-0.609036\pi\)
−0.335888 + 0.941902i \(0.609036\pi\)
\(632\) 0 0
\(633\) 14332.1i 0.899918i
\(634\) 0 0
\(635\) −22876.0 + 18368.4i −1.42962 + 1.14792i
\(636\) 0 0
\(637\) 13965.9i 0.868681i
\(638\) 0 0
\(639\) −19992.0 −1.23767
\(640\) 0 0
\(641\) 1682.00 0.103643 0.0518214 0.998656i \(-0.483497\pi\)
0.0518214 + 0.998656i \(0.483497\pi\)
\(642\) 0 0
\(643\) 18542.8i 1.13725i 0.822595 + 0.568627i \(0.192525\pi\)
−0.822595 + 0.568627i \(0.807475\pi\)
\(644\) 0 0
\(645\) 18620.0 + 23189.3i 1.13668 + 1.41563i
\(646\) 0 0
\(647\) 30677.9i 1.86410i −0.362327 0.932051i \(-0.618018\pi\)
0.362327 0.932051i \(-0.381982\pi\)
\(648\) 0 0
\(649\) −560.000 −0.0338705
\(650\) 0 0
\(651\) −17024.0 −1.02492
\(652\) 0 0
\(653\) 1342.54i 0.0804559i −0.999191 0.0402279i \(-0.987192\pi\)
0.999191 0.0402279i \(-0.0128084\pi\)
\(654\) 0 0
\(655\) 20580.0 + 25630.3i 1.22768 + 1.52895i
\(656\) 0 0
\(657\) 52969.3i 3.14540i
\(658\) 0 0
\(659\) 21356.0 1.26238 0.631192 0.775626i \(-0.282566\pi\)
0.631192 + 0.775626i \(0.282566\pi\)
\(660\) 0 0
\(661\) −13762.0 −0.809803 −0.404901 0.914360i \(-0.632694\pi\)
−0.404901 + 0.914360i \(0.632694\pi\)
\(662\) 0 0
\(663\) 31802.5i 1.86291i
\(664\) 0 0
\(665\) 6384.00 5126.07i 0.372272 0.298918i
\(666\) 0 0
\(667\) 366.148i 0.0212553i
\(668\) 0 0
\(669\) 9500.00 0.549015
\(670\) 0 0
\(671\) 3640.00 0.209420
\(672\) 0 0
\(673\) 5858.36i 0.335547i 0.985826 + 0.167774i \(0.0536577\pi\)
−0.985826 + 0.167774i \(0.946342\pi\)
\(674\) 0 0
\(675\) −23408.0 5178.37i −1.33478 0.295282i
\(676\) 0 0
\(677\) 29588.2i 1.67972i 0.542807 + 0.839858i \(0.317362\pi\)
−0.542807 + 0.839858i \(0.682638\pi\)
\(678\) 0 0
\(679\) 4864.00 0.274909
\(680\) 0 0
\(681\) −8588.00 −0.483249
\(682\) 0 0
\(683\) 17270.0i 0.967521i −0.875201 0.483760i \(-0.839271\pi\)
0.875201 0.483760i \(-0.160729\pi\)
\(684\) 0 0
\(685\) −6384.00 + 5126.07i −0.356088 + 0.285922i
\(686\) 0 0
\(687\) 34295.8i 1.90461i
\(688\) 0 0
\(689\) −19152.0 −1.05897
\(690\) 0 0
\(691\) −1652.00 −0.0909480 −0.0454740 0.998966i \(-0.514480\pi\)
−0.0454740 + 0.998966i \(0.514480\pi\)
\(692\) 0 0
\(693\) 8543.44i 0.468310i
\(694\) 0 0
\(695\) −2548.00 3173.28i −0.139066 0.173193i
\(696\) 0 0
\(697\) 18551.5i 1.00816i
\(698\) 0 0
\(699\) 40432.0 2.18781
\(700\) 0 0
\(701\) 28790.0 1.55119 0.775594 0.631232i \(-0.217450\pi\)
0.775594 + 0.631232i \(0.217450\pi\)
\(702\) 0 0
\(703\) 10252.1i 0.550023i
\(704\) 0 0
\(705\) −22876.0 28489.8i −1.22207 1.52197i
\(706\) 0 0
\(707\) 10862.4i 0.577824i
\(708\) 0 0
\(709\) −7554.00 −0.400136 −0.200068 0.979782i \(-0.564116\pi\)
−0.200068 + 0.979782i \(0.564116\pi\)
\(710\) 0 0
\(711\) −2352.00 −0.124060
\(712\) 0 0
\(713\) 13669.5i 0.717990i
\(714\) 0 0
\(715\) −9120.00 + 7322.95i −0.477019 + 0.383025i
\(716\) 0 0
\(717\) 33615.8i 1.75092i
\(718\) 0 0
\(719\) 28336.0 1.46976 0.734878 0.678199i \(-0.237239\pi\)
0.734878 + 0.678199i \(0.237239\pi\)
\(720\) 0 0
\(721\) −7372.00 −0.380787
\(722\) 0 0
\(723\) 8665.49i 0.445744i
\(724\) 0 0
\(725\) −162.000 + 732.295i −0.00829866 + 0.0375128i
\(726\) 0 0
\(727\) 23511.9i 1.19946i 0.800202 + 0.599730i \(0.204726\pi\)
−0.800202 + 0.599730i \(0.795274\pi\)
\(728\) 0 0
\(729\) 28043.0 1.42473
\(730\) 0 0
\(731\) 21280.0 1.07670
\(732\) 0 0
\(733\) 3365.07i 0.169566i −0.996399 0.0847829i \(-0.972980\pi\)
0.996399 0.0847829i \(-0.0270197\pi\)
\(734\) 0 0
\(735\) 20292.0 16293.6i 1.01834 0.817683i
\(736\) 0 0
\(737\) 8543.44i 0.427004i
\(738\) 0 0
\(739\) 14300.0 0.711819 0.355909 0.934520i \(-0.384171\pi\)
0.355909 + 0.934520i \(0.384171\pi\)
\(740\) 0 0
\(741\) −38304.0 −1.89896
\(742\) 0 0
\(743\) 31671.8i 1.56383i 0.623386 + 0.781914i \(0.285756\pi\)
−0.623386 + 0.781914i \(0.714244\pi\)
\(744\) 0 0
\(745\) 1778.00 + 2214.32i 0.0874374 + 0.108895i
\(746\) 0 0
\(747\) 9824.96i 0.481227i
\(748\) 0 0
\(749\) −11172.0 −0.545015
\(750\) 0 0
\(751\) −11824.0 −0.574519 −0.287260 0.957853i \(-0.592744\pi\)
−0.287260 + 0.957853i \(0.592744\pi\)
\(752\) 0 0
\(753\) 54922.1i 2.65800i
\(754\) 0 0
\(755\) −16520.0 20574.0i −0.796324 0.991741i
\(756\) 0 0
\(757\) 1342.54i 0.0644590i 0.999480 + 0.0322295i \(0.0102607\pi\)
−0.999480 + 0.0322295i \(0.989739\pi\)
\(758\) 0 0
\(759\) −10640.0 −0.508837
\(760\) 0 0
\(761\) 6762.00 0.322106 0.161053 0.986946i \(-0.448511\pi\)
0.161053 + 0.986946i \(0.448511\pi\)
\(762\) 0 0
\(763\) 7863.45i 0.373101i
\(764\) 0 0
\(765\) −29792.0 + 23921.6i −1.40802 + 1.13057i
\(766\) 0 0
\(767\) 1464.59i 0.0689482i
\(768\) 0 0
\(769\) 29442.0 1.38063 0.690316 0.723508i \(-0.257472\pi\)
0.690316 + 0.723508i \(0.257472\pi\)
\(770\) 0 0
\(771\) −58368.0 −2.72642
\(772\) 0 0
\(773\) 35830.1i 1.66717i −0.552393 0.833584i \(-0.686285\pi\)
0.552393 0.833584i \(-0.313715\pi\)
\(774\) 0 0
\(775\) 6048.00 27339.0i 0.280323 1.26716i
\(776\) 0 0
\(777\) 9275.74i 0.428269i
\(778\) 0 0
\(779\) −22344.0 −1.02767
\(780\) 0 0
\(781\) 8160.00 0.373864
\(782\) 0 0
\(783\) 1150.75i 0.0525216i
\(784\) 0 0
\(785\) −19304.0 + 15500.2i −0.877693 + 0.704748i
\(786\) 0 0
\(787\) 23912.9i 1.08310i 0.840667 + 0.541552i \(0.182163\pi\)
−0.840667 + 0.541552i \(0.817837\pi\)
\(788\) 0 0
\(789\) 12236.0 0.552108
\(790\) 0 0
\(791\) 12768.0 0.573929
\(792\) 0 0
\(793\) 9519.84i 0.426304i
\(794\) 0 0
\(795\) 22344.0 + 27827.2i 0.996805 + 1.24142i
\(796\) 0 0
\(797\) 17139.2i 0.761733i 0.924630 + 0.380867i \(0.124374\pi\)
−0.924630 + 0.380867i \(0.875626\pi\)
\(798\) 0 0
\(799\) −26144.0 −1.15758
\(800\) 0 0
\(801\) 74774.0 3.29839
\(802\) 0 0
\(803\) 21620.1i 0.950135i
\(804\) 0 0
\(805\) 3724.00 + 4637.87i 0.163048 + 0.203060i
\(806\) 0 0
\(807\) 44303.8i 1.93255i
\(808\) 0 0
\(809\) −8646.00 −0.375744 −0.187872 0.982193i \(-0.560159\pi\)
−0.187872 + 0.982193i \(0.560159\pi\)
\(810\) 0 0
\(811\) −18956.0 −0.820759 −0.410379 0.911915i \(-0.634604\pi\)
−0.410379 + 0.911915i \(0.634604\pi\)
\(812\) 0 0
\(813\) 24409.8i 1.05300i
\(814\) 0 0
\(815\) −13300.0 + 10679.3i −0.571630 + 0.458993i
\(816\) 0 0
\(817\) 25630.3i 1.09754i
\(818\) 0 0
\(819\) 22344.0 0.953312
\(820\) 0 0
\(821\) 13726.0 0.583484 0.291742 0.956497i \(-0.405765\pi\)
0.291742 + 0.956497i \(0.405765\pi\)
\(822\) 0 0
\(823\) 45951.5i 1.94626i −0.230264 0.973128i \(-0.573959\pi\)
0.230264 0.973128i \(-0.426041\pi\)
\(824\) 0 0
\(825\) 21280.0 + 4707.61i 0.898030 + 0.198664i
\(826\) 0 0
\(827\) 2135.86i 0.0898079i −0.998991 0.0449040i \(-0.985702\pi\)
0.998991 0.0449040i \(-0.0142982\pi\)
\(828\) 0 0
\(829\) −8778.00 −0.367759 −0.183880 0.982949i \(-0.558866\pi\)
−0.183880 + 0.982949i \(0.558866\pi\)
\(830\) 0 0
\(831\) −13832.0 −0.577409
\(832\) 0 0
\(833\) 18621.2i 0.774534i
\(834\) 0 0
\(835\) 26676.0 21419.6i 1.10558 0.887733i
\(836\) 0 0
\(837\) 42961.3i 1.77415i
\(838\) 0 0
\(839\) −18088.0 −0.744299 −0.372150 0.928173i \(-0.621379\pi\)
−0.372150 + 0.928173i \(0.621379\pi\)
\(840\) 0 0
\(841\) −24353.0 −0.998524
\(842\) 0 0
\(843\) 10932.1i 0.446646i
\(844\) 0 0
\(845\) −3773.00 4698.89i −0.153604 0.191298i
\(846\) 0 0
\(847\) 8116.27i 0.329254i
\(848\) 0 0
\(849\) −40508.0 −1.63749
\(850\) 0 0
\(851\) −7448.00 −0.300017
\(852\) 0 0
\(853\) 22927.8i 0.920320i −0.887836 0.460160i \(-0.847792\pi\)
0.887836 0.460160i \(-0.152208\pi\)
\(854\) 0 0
\(855\) 28812.0 + 35882.5i 1.15246 + 1.43527i
\(856\) 0 0
\(857\) 4498.38i 0.179302i −0.995973 0.0896510i \(-0.971425\pi\)
0.995973 0.0896510i \(-0.0285752\pi\)
\(858\) 0 0
\(859\) 40516.0 1.60930 0.804650 0.593750i \(-0.202353\pi\)
0.804650 + 0.593750i \(0.202353\pi\)
\(860\) 0 0
\(861\) 20216.0 0.800185
\(862\) 0 0
\(863\) 28010.3i 1.10484i 0.833564 + 0.552422i \(0.186296\pi\)
−0.833564 + 0.552422i \(0.813704\pi\)
\(864\) 0 0
\(865\) 16568.0 13303.4i 0.651247 0.522922i
\(866\) 0 0
\(867\) 427.172i 0.0167330i
\(868\) 0 0
\(869\) 960.000 0.0374750
\(870\) 0 0
\(871\) 22344.0 0.869228
\(872\) 0 0
\(873\) 27339.0i 1.05989i
\(874\) 0 0
\(875\) −5396.00 10923.4i −0.208478 0.422032i
\(876\) 0 0
\(877\) 33807.6i 1.30171i 0.759201 + 0.650856i \(0.225590\pi\)
−0.759201 + 0.650856i \(0.774410\pi\)
\(878\) 0 0
\(879\) −57000.0 −2.18722
\(880\) 0 0
\(881\) 19362.0 0.740434 0.370217 0.928945i \(-0.379283\pi\)
0.370217 + 0.928945i \(0.379283\pi\)
\(882\) 0 0
\(883\) 4942.99i 0.188386i 0.995554 + 0.0941930i \(0.0300271\pi\)
−0.995554 + 0.0941930i \(0.969973\pi\)
\(884\) 0 0
\(885\) −2128.00 + 1708.69i −0.0808270 + 0.0649005i
\(886\) 0 0
\(887\) 44016.2i 1.66620i 0.553124 + 0.833099i \(0.313436\pi\)
−0.553124 + 0.833099i \(0.686564\pi\)
\(888\) 0 0
\(889\) −22876.0 −0.863033
\(890\) 0 0
\(891\) 6980.00 0.262445
\(892\) 0 0
\(893\) 31488.7i 1.17999i
\(894\) 0 0
\(895\) −13804.0 17191.5i −0.515550 0.642065i
\(896\) 0 0
\(897\) 27827.2i 1.03581i
\(898\) 0 0
\(899\) −1344.00 −0.0498609
\(900\) 0 0
\(901\) 25536.0 0.944204
\(902\) 0 0
\(903\) 23189.3i 0.854588i
\(904\) 0 0
\(905\) −9310.00 11594.7i −0.341961 0.425878i
\(906\) 0 0
\(907\) 42656.2i 1.56160i −0.624778 0.780802i \(-0.714811\pi\)
0.624778 0.780802i \(-0.285189\pi\)
\(908\) 0 0
\(909\) −61054.0 −2.22776
\(910\) 0 0
\(911\) −15888.0 −0.577819 −0.288909 0.957356i \(-0.593293\pi\)
−0.288909 + 0.957356i \(0.593293\pi\)
\(912\) 0 0
\(913\) 4010.19i 0.145365i
\(914\) 0 0
\(915\) 13832.0 11106.5i 0.499751 0.401277i
\(916\) 0 0
\(917\) 25630.3i 0.922997i
\(918\) 0 0
\(919\) 7944.00 0.285145 0.142573 0.989784i \(-0.454463\pi\)
0.142573 + 0.989784i \(0.454463\pi\)
\(920\) 0 0
\(921\) −55404.0 −1.98222
\(922\) 0 0
\(923\) 21341.2i 0.761054i
\(924\) 0 0
\(925\) 14896.0 + 3295.33i 0.529489 + 0.117135i
\(926\) 0 0
\(927\) 41435.7i 1.46810i
\(928\) 0 0
\(929\) −20622.0 −0.728295 −0.364147 0.931341i \(-0.618640\pi\)
−0.364147 + 0.931341i \(0.618640\pi\)
\(930\) 0 0
\(931\) −22428.0 −0.789525
\(932\) 0 0
\(933\) 45402.3i 1.59315i
\(934\) 0 0
\(935\) 12160.0 9763.93i 0.425320 0.341513i
\(936\) 0 0
\(937\) 41182.9i 1.43584i 0.696123 + 0.717922i \(0.254907\pi\)
−0.696123 + 0.717922i \(0.745093\pi\)
\(938\) 0 0
\(939\) 39824.0 1.38403
\(940\) 0 0
\(941\) 40502.0 1.40311 0.701556 0.712615i \(-0.252489\pi\)
0.701556 + 0.712615i \(0.252489\pi\)
\(942\) 0 0
\(943\) 16232.5i 0.560556i
\(944\) 0 0
\(945\) −11704.0 14576.2i −0.402890 0.501759i
\(946\) 0 0
\(947\) 3844.55i 0.131923i −0.997822 0.0659615i \(-0.978989\pi\)
0.997822 0.0659615i \(-0.0210114\pi\)
\(948\) 0 0
\(949\) 56544.0 1.93414
\(950\) 0 0
\(951\) −7448.00 −0.253962
\(952\) 0 0
\(953\) 31976.9i 1.08692i −0.839436 0.543459i \(-0.817114\pi\)
0.839436 0.543459i \(-0.182886\pi\)
\(954\) 0 0
\(955\) 12096.0 + 15064.4i 0.409861 + 0.510441i
\(956\) 0 0
\(957\) 1046.14i 0.0353362i
\(958\) 0 0
\(959\) −6384.00 −0.214964
\(960\) 0 0
\(961\) 20385.0 0.684267
\(962\) 0 0
\(963\) 62794.3i 2.10126i
\(964\) 0 0
\(965\) 8512.00 6834.75i 0.283949 0.227998i
\(966\) 0 0
\(967\) 19100.7i 0.635198i −0.948225 0.317599i \(-0.897123\pi\)
0.948225 0.317599i \(-0.102877\pi\)
\(968\) 0 0
\(969\) 51072.0 1.69316
\(970\) 0 0
\(971\) 17556.0 0.580225 0.290113 0.956992i \(-0.406307\pi\)
0.290113 + 0.956992i \(0.406307\pi\)
\(972\) 0 0
\(973\) 3173.28i 0.104554i
\(974\) 0 0
\(975\) −12312.0 + 55654.4i −0.404410 + 1.82807i
\(976\) 0 0
\(977\) 4393.77i 0.143878i 0.997409 + 0.0719392i \(0.0229188\pi\)
−0.997409 + 0.0719392i \(0.977081\pi\)
\(978\) 0 0
\(979\) −30520.0 −0.996347
\(980\) 0 0
\(981\) −44198.0 −1.43846
\(982\) 0 0
\(983\) 9406.50i 0.305209i −0.988287 0.152605i \(-0.951234\pi\)
0.988287 0.152605i \(-0.0487661\pi\)
\(984\) 0 0
\(985\) −32984.0 + 26484.7i −1.06696 + 0.856723i
\(986\) 0 0
\(987\) 28489.8i 0.918783i
\(988\) 0 0
\(989\) 18620.0 0.598667
\(990\) 0 0
\(991\) 28576.0 0.915990 0.457995 0.888955i \(-0.348568\pi\)
0.457995 + 0.888955i \(0.348568\pi\)
\(992\) 0 0
\(993\) 68242.9i 2.18089i
\(994\) 0 0
\(995\) −10584.0 13181.3i −0.337221 0.419976i
\(996\) 0 0
\(997\) 31541.0i 1.00192i 0.865471 + 0.500960i \(0.167019\pi\)
−0.865471 + 0.500960i \(0.832981\pi\)
\(998\) 0 0
\(999\) 23408.0 0.741338
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 20.4.c.a.9.1 2
3.2 odd 2 180.4.d.a.109.1 2
4.3 odd 2 80.4.c.b.49.2 2
5.2 odd 4 100.4.a.d.1.1 2
5.3 odd 4 100.4.a.d.1.2 2
5.4 even 2 inner 20.4.c.a.9.2 yes 2
7.6 odd 2 980.4.e.a.589.2 2
8.3 odd 2 320.4.c.b.129.1 2
8.5 even 2 320.4.c.a.129.2 2
12.11 even 2 720.4.f.a.289.1 2
15.2 even 4 900.4.a.s.1.1 2
15.8 even 4 900.4.a.s.1.2 2
15.14 odd 2 180.4.d.a.109.2 2
20.3 even 4 400.4.a.w.1.1 2
20.7 even 4 400.4.a.w.1.2 2
20.19 odd 2 80.4.c.b.49.1 2
35.34 odd 2 980.4.e.a.589.1 2
40.3 even 4 1600.4.a.ck.1.2 2
40.13 odd 4 1600.4.a.cj.1.1 2
40.19 odd 2 320.4.c.b.129.2 2
40.27 even 4 1600.4.a.ck.1.1 2
40.29 even 2 320.4.c.a.129.1 2
40.37 odd 4 1600.4.a.cj.1.2 2
60.59 even 2 720.4.f.a.289.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
20.4.c.a.9.1 2 1.1 even 1 trivial
20.4.c.a.9.2 yes 2 5.4 even 2 inner
80.4.c.b.49.1 2 20.19 odd 2
80.4.c.b.49.2 2 4.3 odd 2
100.4.a.d.1.1 2 5.2 odd 4
100.4.a.d.1.2 2 5.3 odd 4
180.4.d.a.109.1 2 3.2 odd 2
180.4.d.a.109.2 2 15.14 odd 2
320.4.c.a.129.1 2 40.29 even 2
320.4.c.a.129.2 2 8.5 even 2
320.4.c.b.129.1 2 8.3 odd 2
320.4.c.b.129.2 2 40.19 odd 2
400.4.a.w.1.1 2 20.3 even 4
400.4.a.w.1.2 2 20.7 even 4
720.4.f.a.289.1 2 12.11 even 2
720.4.f.a.289.2 2 60.59 even 2
900.4.a.s.1.1 2 15.2 even 4
900.4.a.s.1.2 2 15.8 even 4
980.4.e.a.589.1 2 35.34 odd 2
980.4.e.a.589.2 2 7.6 odd 2
1600.4.a.cj.1.1 2 40.13 odd 4
1600.4.a.cj.1.2 2 40.37 odd 4
1600.4.a.ck.1.1 2 40.27 even 4
1600.4.a.ck.1.2 2 40.3 even 4