Properties

Label 1280.3.e.j.639.3
Level $1280$
Weight $3$
Character 1280.639
Analytic conductor $34.877$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1280,3,Mod(639,1280)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1280, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1280.639");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1280 = 2^{8} \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1280.e (of order \(2\), degree \(1\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(34.8774738381\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\zeta_{24})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{18} \)
Twist minimal: no (minimal twist has level 80)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 639.3
Root \(-0.258819 - 0.965926i\) of defining polynomial
Character \(\chi\) \(=\) 1280.639
Dual form 1280.3.e.j.639.8

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.82843i q^{3} +(4.89898 - 1.00000i) q^{5} -8.48528 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-2.82843i q^{3} +(4.89898 - 1.00000i) q^{5} -8.48528 q^{7} +1.00000 q^{9} -13.8564 q^{11} -9.79796 q^{13} +(-2.82843 - 13.8564i) q^{15} +19.5959i q^{17} -13.8564 q^{19} +24.0000i q^{21} +25.4558 q^{23} +(23.0000 - 9.79796i) q^{25} -28.2843i q^{27} +22.0000i q^{29} +55.4256i q^{31} +39.1918i q^{33} +(-41.5692 + 8.48528i) q^{35} +48.9898 q^{37} +27.7128i q^{39} -22.0000 q^{41} +59.3970i q^{43} +(4.89898 - 1.00000i) q^{45} +8.48528 q^{47} +23.0000 q^{49} +55.4256 q^{51} +29.3939 q^{53} +(-67.8823 + 13.8564i) q^{55} +39.1918i q^{57} -13.8564 q^{59} -46.0000i q^{61} -8.48528 q^{63} +(-48.0000 + 9.79796i) q^{65} -59.3970i q^{67} -72.0000i q^{69} +27.7128i q^{71} +78.3837i q^{73} +(-27.7128 - 65.0538i) q^{75} +117.576 q^{77} -71.0000 q^{81} +76.3675i q^{83} +(19.5959 + 96.0000i) q^{85} +62.2254 q^{87} -146.000 q^{89} +83.1384 q^{91} +156.767 q^{93} +(-67.8823 + 13.8564i) q^{95} +58.7878i q^{97} -13.8564 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{9} + 184 q^{25} - 176 q^{41} + 184 q^{49} - 384 q^{65} - 568 q^{81} - 1168 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1280\mathbb{Z}\right)^\times\).

\(n\) \(257\) \(261\) \(511\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 2.82843i 0.942809i −0.881917 0.471405i \(-0.843747\pi\)
0.881917 0.471405i \(-0.156253\pi\)
\(4\) 0 0
\(5\) 4.89898 1.00000i 0.979796 0.200000i
\(6\) 0 0
\(7\) −8.48528 −1.21218 −0.606092 0.795395i \(-0.707263\pi\)
−0.606092 + 0.795395i \(0.707263\pi\)
\(8\) 0 0
\(9\) 1.00000 0.111111
\(10\) 0 0
\(11\) −13.8564 −1.25967 −0.629837 0.776728i \(-0.716878\pi\)
−0.629837 + 0.776728i \(0.716878\pi\)
\(12\) 0 0
\(13\) −9.79796 −0.753689 −0.376845 0.926277i \(-0.622991\pi\)
−0.376845 + 0.926277i \(0.622991\pi\)
\(14\) 0 0
\(15\) −2.82843 13.8564i −0.188562 0.923760i
\(16\) 0 0
\(17\) 19.5959i 1.15270i 0.817203 + 0.576351i \(0.195524\pi\)
−0.817203 + 0.576351i \(0.804476\pi\)
\(18\) 0 0
\(19\) −13.8564 −0.729285 −0.364642 0.931148i \(-0.618809\pi\)
−0.364642 + 0.931148i \(0.618809\pi\)
\(20\) 0 0
\(21\) 24.0000i 1.14286i
\(22\) 0 0
\(23\) 25.4558 1.10678 0.553388 0.832924i \(-0.313335\pi\)
0.553388 + 0.832924i \(0.313335\pi\)
\(24\) 0 0
\(25\) 23.0000 9.79796i 0.920000 0.391918i
\(26\) 0 0
\(27\) 28.2843i 1.04757i
\(28\) 0 0
\(29\) 22.0000i 0.758621i 0.925270 + 0.379310i \(0.123839\pi\)
−0.925270 + 0.379310i \(0.876161\pi\)
\(30\) 0 0
\(31\) 55.4256i 1.78792i 0.448143 + 0.893962i \(0.352085\pi\)
−0.448143 + 0.893962i \(0.647915\pi\)
\(32\) 0 0
\(33\) 39.1918i 1.18763i
\(34\) 0 0
\(35\) −41.5692 + 8.48528i −1.18769 + 0.242437i
\(36\) 0 0
\(37\) 48.9898 1.32405 0.662024 0.749482i \(-0.269698\pi\)
0.662024 + 0.749482i \(0.269698\pi\)
\(38\) 0 0
\(39\) 27.7128i 0.710585i
\(40\) 0 0
\(41\) −22.0000 −0.536585 −0.268293 0.963337i \(-0.586459\pi\)
−0.268293 + 0.963337i \(0.586459\pi\)
\(42\) 0 0
\(43\) 59.3970i 1.38132i 0.723177 + 0.690662i \(0.242681\pi\)
−0.723177 + 0.690662i \(0.757319\pi\)
\(44\) 0 0
\(45\) 4.89898 1.00000i 0.108866 0.0222222i
\(46\) 0 0
\(47\) 8.48528 0.180538 0.0902690 0.995917i \(-0.471227\pi\)
0.0902690 + 0.995917i \(0.471227\pi\)
\(48\) 0 0
\(49\) 23.0000 0.469388
\(50\) 0 0
\(51\) 55.4256 1.08678
\(52\) 0 0
\(53\) 29.3939 0.554601 0.277301 0.960783i \(-0.410560\pi\)
0.277301 + 0.960783i \(0.410560\pi\)
\(54\) 0 0
\(55\) −67.8823 + 13.8564i −1.23422 + 0.251935i
\(56\) 0 0
\(57\) 39.1918i 0.687576i
\(58\) 0 0
\(59\) −13.8564 −0.234854 −0.117427 0.993081i \(-0.537465\pi\)
−0.117427 + 0.993081i \(0.537465\pi\)
\(60\) 0 0
\(61\) 46.0000i 0.754098i −0.926193 0.377049i \(-0.876939\pi\)
0.926193 0.377049i \(-0.123061\pi\)
\(62\) 0 0
\(63\) −8.48528 −0.134687
\(64\) 0 0
\(65\) −48.0000 + 9.79796i −0.738462 + 0.150738i
\(66\) 0 0
\(67\) 59.3970i 0.886522i −0.896393 0.443261i \(-0.853822\pi\)
0.896393 0.443261i \(-0.146178\pi\)
\(68\) 0 0
\(69\) 72.0000i 1.04348i
\(70\) 0 0
\(71\) 27.7128i 0.390321i 0.980771 + 0.195161i \(0.0625228\pi\)
−0.980771 + 0.195161i \(0.937477\pi\)
\(72\) 0 0
\(73\) 78.3837i 1.07375i 0.843662 + 0.536874i \(0.180395\pi\)
−0.843662 + 0.536874i \(0.819605\pi\)
\(74\) 0 0
\(75\) −27.7128 65.0538i −0.369504 0.867384i
\(76\) 0 0
\(77\) 117.576 1.52695
\(78\) 0 0
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) 0 0
\(81\) −71.0000 −0.876543
\(82\) 0 0
\(83\) 76.3675i 0.920091i 0.887895 + 0.460045i \(0.152167\pi\)
−0.887895 + 0.460045i \(0.847833\pi\)
\(84\) 0 0
\(85\) 19.5959 + 96.0000i 0.230540 + 1.12941i
\(86\) 0 0
\(87\) 62.2254 0.715234
\(88\) 0 0
\(89\) −146.000 −1.64045 −0.820225 0.572041i \(-0.806152\pi\)
−0.820225 + 0.572041i \(0.806152\pi\)
\(90\) 0 0
\(91\) 83.1384 0.913609
\(92\) 0 0
\(93\) 156.767 1.68567
\(94\) 0 0
\(95\) −67.8823 + 13.8564i −0.714550 + 0.145857i
\(96\) 0 0
\(97\) 58.7878i 0.606059i 0.952981 + 0.303030i \(0.0979981\pi\)
−0.952981 + 0.303030i \(0.902002\pi\)
\(98\) 0 0
\(99\) −13.8564 −0.139964
\(100\) 0 0
\(101\) 70.0000i 0.693069i −0.938037 0.346535i \(-0.887358\pi\)
0.938037 0.346535i \(-0.112642\pi\)
\(102\) 0 0
\(103\) 25.4558 0.247144 0.123572 0.992336i \(-0.460565\pi\)
0.123572 + 0.992336i \(0.460565\pi\)
\(104\) 0 0
\(105\) 24.0000 + 117.576i 0.228571 + 1.11977i
\(106\) 0 0
\(107\) 42.4264i 0.396508i −0.980151 0.198254i \(-0.936473\pi\)
0.980151 0.198254i \(-0.0635272\pi\)
\(108\) 0 0
\(109\) 146.000i 1.33945i 0.742609 + 0.669725i \(0.233588\pi\)
−0.742609 + 0.669725i \(0.766412\pi\)
\(110\) 0 0
\(111\) 138.564i 1.24832i
\(112\) 0 0
\(113\) 39.1918i 0.346830i −0.984849 0.173415i \(-0.944520\pi\)
0.984849 0.173415i \(-0.0554803\pi\)
\(114\) 0 0
\(115\) 124.708 25.4558i 1.08441 0.221355i
\(116\) 0 0
\(117\) −9.79796 −0.0837432
\(118\) 0 0
\(119\) 166.277i 1.39728i
\(120\) 0 0
\(121\) 71.0000 0.586777
\(122\) 0 0
\(123\) 62.2254i 0.505898i
\(124\) 0 0
\(125\) 102.879 71.0000i 0.823029 0.568000i
\(126\) 0 0
\(127\) 110.309 0.868572 0.434286 0.900775i \(-0.357001\pi\)
0.434286 + 0.900775i \(0.357001\pi\)
\(128\) 0 0
\(129\) 168.000 1.30233
\(130\) 0 0
\(131\) −180.133 −1.37506 −0.687532 0.726154i \(-0.741306\pi\)
−0.687532 + 0.726154i \(0.741306\pi\)
\(132\) 0 0
\(133\) 117.576 0.884026
\(134\) 0 0
\(135\) −28.2843 138.564i −0.209513 1.02640i
\(136\) 0 0
\(137\) 215.555i 1.57339i 0.617339 + 0.786697i \(0.288211\pi\)
−0.617339 + 0.786697i \(0.711789\pi\)
\(138\) 0 0
\(139\) −13.8564 −0.0996864 −0.0498432 0.998757i \(-0.515872\pi\)
−0.0498432 + 0.998757i \(0.515872\pi\)
\(140\) 0 0
\(141\) 24.0000i 0.170213i
\(142\) 0 0
\(143\) 135.765 0.949402
\(144\) 0 0
\(145\) 22.0000 + 107.778i 0.151724 + 0.743293i
\(146\) 0 0
\(147\) 65.0538i 0.442543i
\(148\) 0 0
\(149\) 2.00000i 0.0134228i −0.999977 0.00671141i \(-0.997864\pi\)
0.999977 0.00671141i \(-0.00213632\pi\)
\(150\) 0 0
\(151\) 27.7128i 0.183529i 0.995781 + 0.0917643i \(0.0292506\pi\)
−0.995781 + 0.0917643i \(0.970749\pi\)
\(152\) 0 0
\(153\) 19.5959i 0.128078i
\(154\) 0 0
\(155\) 55.4256 + 271.529i 0.357585 + 1.75180i
\(156\) 0 0
\(157\) −68.5857 −0.436852 −0.218426 0.975854i \(-0.570092\pi\)
−0.218426 + 0.975854i \(0.570092\pi\)
\(158\) 0 0
\(159\) 83.1384i 0.522883i
\(160\) 0 0
\(161\) −216.000 −1.34161
\(162\) 0 0
\(163\) 110.309i 0.676740i 0.941013 + 0.338370i \(0.109876\pi\)
−0.941013 + 0.338370i \(0.890124\pi\)
\(164\) 0 0
\(165\) 39.1918 + 192.000i 0.237526 + 1.16364i
\(166\) 0 0
\(167\) 93.3381 0.558911 0.279455 0.960159i \(-0.409846\pi\)
0.279455 + 0.960159i \(0.409846\pi\)
\(168\) 0 0
\(169\) −73.0000 −0.431953
\(170\) 0 0
\(171\) −13.8564 −0.0810316
\(172\) 0 0
\(173\) 48.9898 0.283178 0.141589 0.989926i \(-0.454779\pi\)
0.141589 + 0.989926i \(0.454779\pi\)
\(174\) 0 0
\(175\) −195.161 + 83.1384i −1.11521 + 0.475077i
\(176\) 0 0
\(177\) 39.1918i 0.221423i
\(178\) 0 0
\(179\) 263.272 1.47079 0.735396 0.677638i \(-0.236996\pi\)
0.735396 + 0.677638i \(0.236996\pi\)
\(180\) 0 0
\(181\) 26.0000i 0.143646i 0.997417 + 0.0718232i \(0.0228817\pi\)
−0.997417 + 0.0718232i \(0.977118\pi\)
\(182\) 0 0
\(183\) −130.108 −0.710971
\(184\) 0 0
\(185\) 240.000 48.9898i 1.29730 0.264810i
\(186\) 0 0
\(187\) 271.529i 1.45203i
\(188\) 0 0
\(189\) 240.000i 1.26984i
\(190\) 0 0
\(191\) 110.851i 0.580373i −0.956970 0.290187i \(-0.906283\pi\)
0.956970 0.290187i \(-0.0937174\pi\)
\(192\) 0 0
\(193\) 333.131i 1.72607i 0.505148 + 0.863033i \(0.331438\pi\)
−0.505148 + 0.863033i \(0.668562\pi\)
\(194\) 0 0
\(195\) 27.7128 + 135.765i 0.142117 + 0.696228i
\(196\) 0 0
\(197\) 107.778 0.547094 0.273547 0.961859i \(-0.411803\pi\)
0.273547 + 0.961859i \(0.411803\pi\)
\(198\) 0 0
\(199\) 249.415i 1.25334i 0.779283 + 0.626672i \(0.215583\pi\)
−0.779283 + 0.626672i \(0.784417\pi\)
\(200\) 0 0
\(201\) −168.000 −0.835821
\(202\) 0 0
\(203\) 186.676i 0.919587i
\(204\) 0 0
\(205\) −107.778 + 22.0000i −0.525744 + 0.107317i
\(206\) 0 0
\(207\) 25.4558 0.122975
\(208\) 0 0
\(209\) 192.000 0.918660
\(210\) 0 0
\(211\) 96.9948 0.459691 0.229846 0.973227i \(-0.426178\pi\)
0.229846 + 0.973227i \(0.426178\pi\)
\(212\) 0 0
\(213\) 78.3837 0.367998
\(214\) 0 0
\(215\) 59.3970 + 290.985i 0.276265 + 1.35342i
\(216\) 0 0
\(217\) 470.302i 2.16729i
\(218\) 0 0
\(219\) 221.703 1.01234
\(220\) 0 0
\(221\) 192.000i 0.868778i
\(222\) 0 0
\(223\) −229.103 −1.02737 −0.513683 0.857980i \(-0.671719\pi\)
−0.513683 + 0.857980i \(0.671719\pi\)
\(224\) 0 0
\(225\) 23.0000 9.79796i 0.102222 0.0435465i
\(226\) 0 0
\(227\) 296.985i 1.30830i −0.756363 0.654152i \(-0.773026\pi\)
0.756363 0.654152i \(-0.226974\pi\)
\(228\) 0 0
\(229\) 70.0000i 0.305677i −0.988251 0.152838i \(-0.951159\pi\)
0.988251 0.152838i \(-0.0488414\pi\)
\(230\) 0 0
\(231\) 332.554i 1.43963i
\(232\) 0 0
\(233\) 156.767i 0.672821i 0.941715 + 0.336411i \(0.109213\pi\)
−0.941715 + 0.336411i \(0.890787\pi\)
\(234\) 0 0
\(235\) 41.5692 8.48528i 0.176890 0.0361076i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 332.554i 1.39144i 0.718314 + 0.695719i \(0.244914\pi\)
−0.718314 + 0.695719i \(0.755086\pi\)
\(240\) 0 0
\(241\) −122.000 −0.506224 −0.253112 0.967437i \(-0.581454\pi\)
−0.253112 + 0.967437i \(0.581454\pi\)
\(242\) 0 0
\(243\) 53.7401i 0.221153i
\(244\) 0 0
\(245\) 112.677 23.0000i 0.459904 0.0938776i
\(246\) 0 0
\(247\) 135.765 0.549654
\(248\) 0 0
\(249\) 216.000 0.867470
\(250\) 0 0
\(251\) −290.985 −1.15930 −0.579650 0.814865i \(-0.696811\pi\)
−0.579650 + 0.814865i \(0.696811\pi\)
\(252\) 0 0
\(253\) −352.727 −1.39418
\(254\) 0 0
\(255\) 271.529 55.4256i 1.06482 0.217355i
\(256\) 0 0
\(257\) 78.3837i 0.304995i 0.988304 + 0.152497i \(0.0487316\pi\)
−0.988304 + 0.152497i \(0.951268\pi\)
\(258\) 0 0
\(259\) −415.692 −1.60499
\(260\) 0 0
\(261\) 22.0000i 0.0842912i
\(262\) 0 0
\(263\) −347.897 −1.32280 −0.661400 0.750033i \(-0.730038\pi\)
−0.661400 + 0.750033i \(0.730038\pi\)
\(264\) 0 0
\(265\) 144.000 29.3939i 0.543396 0.110920i
\(266\) 0 0
\(267\) 412.950i 1.54663i
\(268\) 0 0
\(269\) 142.000i 0.527881i −0.964539 0.263941i \(-0.914978\pi\)
0.964539 0.263941i \(-0.0850223\pi\)
\(270\) 0 0
\(271\) 332.554i 1.22714i 0.789642 + 0.613568i \(0.210266\pi\)
−0.789642 + 0.613568i \(0.789734\pi\)
\(272\) 0 0
\(273\) 235.151i 0.861359i
\(274\) 0 0
\(275\) −318.697 + 135.765i −1.15890 + 0.493689i
\(276\) 0 0
\(277\) 88.1816 0.318345 0.159173 0.987251i \(-0.449117\pi\)
0.159173 + 0.987251i \(0.449117\pi\)
\(278\) 0 0
\(279\) 55.4256i 0.198658i
\(280\) 0 0
\(281\) −406.000 −1.44484 −0.722420 0.691455i \(-0.756970\pi\)
−0.722420 + 0.691455i \(0.756970\pi\)
\(282\) 0 0
\(283\) 93.3381i 0.329817i 0.986309 + 0.164908i \(0.0527328\pi\)
−0.986309 + 0.164908i \(0.947267\pi\)
\(284\) 0 0
\(285\) 39.1918 + 192.000i 0.137515 + 0.673684i
\(286\) 0 0
\(287\) 186.676 0.650440
\(288\) 0 0
\(289\) −95.0000 −0.328720
\(290\) 0 0
\(291\) 166.277 0.571398
\(292\) 0 0
\(293\) −303.737 −1.03664 −0.518322 0.855186i \(-0.673443\pi\)
−0.518322 + 0.855186i \(0.673443\pi\)
\(294\) 0 0
\(295\) −67.8823 + 13.8564i −0.230109 + 0.0469709i
\(296\) 0 0
\(297\) 391.918i 1.31959i
\(298\) 0 0
\(299\) −249.415 −0.834165
\(300\) 0 0
\(301\) 504.000i 1.67442i
\(302\) 0 0
\(303\) −197.990 −0.653432
\(304\) 0 0
\(305\) −46.0000 225.353i −0.150820 0.738862i
\(306\) 0 0
\(307\) 280.014i 0.912099i 0.889955 + 0.456049i \(0.150736\pi\)
−0.889955 + 0.456049i \(0.849264\pi\)
\(308\) 0 0
\(309\) 72.0000i 0.233010i
\(310\) 0 0
\(311\) 138.564i 0.445544i 0.974871 + 0.222772i \(0.0715105\pi\)
−0.974871 + 0.222772i \(0.928490\pi\)
\(312\) 0 0
\(313\) 607.473i 1.94081i −0.241484 0.970405i \(-0.577634\pi\)
0.241484 0.970405i \(-0.422366\pi\)
\(314\) 0 0
\(315\) −41.5692 + 8.48528i −0.131966 + 0.0269374i
\(316\) 0 0
\(317\) 68.5857 0.216359 0.108179 0.994131i \(-0.465498\pi\)
0.108179 + 0.994131i \(0.465498\pi\)
\(318\) 0 0
\(319\) 304.841i 0.955614i
\(320\) 0 0
\(321\) −120.000 −0.373832
\(322\) 0 0
\(323\) 271.529i 0.840647i
\(324\) 0 0
\(325\) −225.353 + 96.0000i −0.693394 + 0.295385i
\(326\) 0 0
\(327\) 412.950 1.26285
\(328\) 0 0
\(329\) −72.0000 −0.218845
\(330\) 0 0
\(331\) −180.133 −0.544209 −0.272105 0.962268i \(-0.587720\pi\)
−0.272105 + 0.962268i \(0.587720\pi\)
\(332\) 0 0
\(333\) 48.9898 0.147117
\(334\) 0 0
\(335\) −59.3970 290.985i −0.177304 0.868611i
\(336\) 0 0
\(337\) 470.302i 1.39555i −0.716315 0.697777i \(-0.754172\pi\)
0.716315 0.697777i \(-0.245828\pi\)
\(338\) 0 0
\(339\) −110.851 −0.326995
\(340\) 0 0
\(341\) 768.000i 2.25220i
\(342\) 0 0
\(343\) 220.617 0.643199
\(344\) 0 0
\(345\) −72.0000 352.727i −0.208696 1.02240i
\(346\) 0 0
\(347\) 619.426i 1.78509i −0.450960 0.892544i \(-0.648918\pi\)
0.450960 0.892544i \(-0.351082\pi\)
\(348\) 0 0
\(349\) 214.000i 0.613181i 0.951842 + 0.306590i \(0.0991881\pi\)
−0.951842 + 0.306590i \(0.900812\pi\)
\(350\) 0 0
\(351\) 277.128i 0.789539i
\(352\) 0 0
\(353\) 352.727i 0.999225i −0.866249 0.499613i \(-0.833476\pi\)
0.866249 0.499613i \(-0.166524\pi\)
\(354\) 0 0
\(355\) 27.7128 + 135.765i 0.0780643 + 0.382435i
\(356\) 0 0
\(357\) −470.302 −1.31737
\(358\) 0 0
\(359\) 27.7128i 0.0771945i −0.999255 0.0385972i \(-0.987711\pi\)
0.999255 0.0385972i \(-0.0122889\pi\)
\(360\) 0 0
\(361\) −169.000 −0.468144
\(362\) 0 0
\(363\) 200.818i 0.553219i
\(364\) 0 0
\(365\) 78.3837 + 384.000i 0.214750 + 1.05205i
\(366\) 0 0
\(367\) 246.073 0.670499 0.335250 0.942129i \(-0.391179\pi\)
0.335250 + 0.942129i \(0.391179\pi\)
\(368\) 0 0
\(369\) −22.0000 −0.0596206
\(370\) 0 0
\(371\) −249.415 −0.672278
\(372\) 0 0
\(373\) −382.120 −1.02445 −0.512226 0.858851i \(-0.671179\pi\)
−0.512226 + 0.858851i \(0.671179\pi\)
\(374\) 0 0
\(375\) −200.818 290.985i −0.535516 0.775959i
\(376\) 0 0
\(377\) 215.555i 0.571764i
\(378\) 0 0
\(379\) −290.985 −0.767769 −0.383885 0.923381i \(-0.625414\pi\)
−0.383885 + 0.923381i \(0.625414\pi\)
\(380\) 0 0
\(381\) 312.000i 0.818898i
\(382\) 0 0
\(383\) −568.514 −1.48437 −0.742185 0.670195i \(-0.766211\pi\)
−0.742185 + 0.670195i \(0.766211\pi\)
\(384\) 0 0
\(385\) 576.000 117.576i 1.49610 0.305391i
\(386\) 0 0
\(387\) 59.3970i 0.153481i
\(388\) 0 0
\(389\) 670.000i 1.72237i 0.508296 + 0.861183i \(0.330276\pi\)
−0.508296 + 0.861183i \(0.669724\pi\)
\(390\) 0 0
\(391\) 498.831i 1.27578i
\(392\) 0 0
\(393\) 509.494i 1.29642i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −362.524 −0.913160 −0.456580 0.889682i \(-0.650926\pi\)
−0.456580 + 0.889682i \(0.650926\pi\)
\(398\) 0 0
\(399\) 332.554i 0.833468i
\(400\) 0 0
\(401\) 2.00000 0.00498753 0.00249377 0.999997i \(-0.499206\pi\)
0.00249377 + 0.999997i \(0.499206\pi\)
\(402\) 0 0
\(403\) 543.058i 1.34754i
\(404\) 0 0
\(405\) −347.828 + 71.0000i −0.858833 + 0.175309i
\(406\) 0 0
\(407\) −678.823 −1.66787
\(408\) 0 0
\(409\) 650.000 1.58924 0.794621 0.607106i \(-0.207670\pi\)
0.794621 + 0.607106i \(0.207670\pi\)
\(410\) 0 0
\(411\) 609.682 1.48341
\(412\) 0 0
\(413\) 117.576 0.284686
\(414\) 0 0
\(415\) 76.3675 + 374.123i 0.184018 + 0.901501i
\(416\) 0 0
\(417\) 39.1918i 0.0939852i
\(418\) 0 0
\(419\) 540.400 1.28974 0.644869 0.764293i \(-0.276912\pi\)
0.644869 + 0.764293i \(0.276912\pi\)
\(420\) 0 0
\(421\) 482.000i 1.14489i −0.819942 0.572447i \(-0.805994\pi\)
0.819942 0.572447i \(-0.194006\pi\)
\(422\) 0 0
\(423\) 8.48528 0.0200598
\(424\) 0 0
\(425\) 192.000 + 450.706i 0.451765 + 1.06048i
\(426\) 0 0
\(427\) 390.323i 0.914105i
\(428\) 0 0
\(429\) 384.000i 0.895105i
\(430\) 0 0
\(431\) 221.703i 0.514391i −0.966359 0.257195i \(-0.917202\pi\)
0.966359 0.257195i \(-0.0827984\pi\)
\(432\) 0 0
\(433\) 568.282i 1.31243i −0.754575 0.656214i \(-0.772157\pi\)
0.754575 0.656214i \(-0.227843\pi\)
\(434\) 0 0
\(435\) 304.841 62.2254i 0.700784 0.143047i
\(436\) 0 0
\(437\) −352.727 −0.807155
\(438\) 0 0
\(439\) 526.543i 1.19942i −0.800219 0.599708i \(-0.795283\pi\)
0.800219 0.599708i \(-0.204717\pi\)
\(440\) 0 0
\(441\) 23.0000 0.0521542
\(442\) 0 0
\(443\) 432.749i 0.976861i 0.872603 + 0.488430i \(0.162430\pi\)
−0.872603 + 0.488430i \(0.837570\pi\)
\(444\) 0 0
\(445\) −715.251 + 146.000i −1.60731 + 0.328090i
\(446\) 0 0
\(447\) −5.65685 −0.0126552
\(448\) 0 0
\(449\) −26.0000 −0.0579065 −0.0289532 0.999581i \(-0.509217\pi\)
−0.0289532 + 0.999581i \(0.509217\pi\)
\(450\) 0 0
\(451\) 304.841 0.675922
\(452\) 0 0
\(453\) 78.3837 0.173032
\(454\) 0 0
\(455\) 407.294 83.1384i 0.895151 0.182722i
\(456\) 0 0
\(457\) 137.171i 0.300156i −0.988674 0.150078i \(-0.952047\pi\)
0.988674 0.150078i \(-0.0479525\pi\)
\(458\) 0 0
\(459\) 554.256 1.20753
\(460\) 0 0
\(461\) 502.000i 1.08894i 0.838781 + 0.544469i \(0.183269\pi\)
−0.838781 + 0.544469i \(0.816731\pi\)
\(462\) 0 0
\(463\) 823.072 1.77769 0.888847 0.458204i \(-0.151507\pi\)
0.888847 + 0.458204i \(0.151507\pi\)
\(464\) 0 0
\(465\) 768.000 156.767i 1.65161 0.337134i
\(466\) 0 0
\(467\) 178.191i 0.381565i 0.981632 + 0.190783i \(0.0611025\pi\)
−0.981632 + 0.190783i \(0.938897\pi\)
\(468\) 0 0
\(469\) 504.000i 1.07463i
\(470\) 0 0
\(471\) 193.990i 0.411868i
\(472\) 0 0
\(473\) 823.029i 1.74002i
\(474\) 0 0
\(475\) −318.697 + 135.765i −0.670942 + 0.285820i
\(476\) 0 0
\(477\) 29.3939 0.0616224
\(478\) 0 0
\(479\) 775.959i 1.61996i 0.586460 + 0.809978i \(0.300521\pi\)
−0.586460 + 0.809978i \(0.699479\pi\)
\(480\) 0 0
\(481\) −480.000 −0.997921
\(482\) 0 0
\(483\) 610.940i 1.26489i
\(484\) 0 0
\(485\) 58.7878 + 288.000i 0.121212 + 0.593814i
\(486\) 0 0
\(487\) −347.897 −0.714367 −0.357183 0.934034i \(-0.616263\pi\)
−0.357183 + 0.934034i \(0.616263\pi\)
\(488\) 0 0
\(489\) 312.000 0.638037
\(490\) 0 0
\(491\) 928.379 1.89079 0.945396 0.325923i \(-0.105675\pi\)
0.945396 + 0.325923i \(0.105675\pi\)
\(492\) 0 0
\(493\) −431.110 −0.874463
\(494\) 0 0
\(495\) −67.8823 + 13.8564i −0.137136 + 0.0279927i
\(496\) 0 0
\(497\) 235.151i 0.473141i
\(498\) 0 0
\(499\) −512.687 −1.02743 −0.513714 0.857961i \(-0.671731\pi\)
−0.513714 + 0.857961i \(0.671731\pi\)
\(500\) 0 0
\(501\) 264.000i 0.526946i
\(502\) 0 0
\(503\) 704.278 1.40016 0.700078 0.714066i \(-0.253149\pi\)
0.700078 + 0.714066i \(0.253149\pi\)
\(504\) 0 0
\(505\) −70.0000 342.929i −0.138614 0.679066i
\(506\) 0 0
\(507\) 206.475i 0.407249i
\(508\) 0 0
\(509\) 170.000i 0.333988i −0.985958 0.166994i \(-0.946594\pi\)
0.985958 0.166994i \(-0.0534061\pi\)
\(510\) 0 0
\(511\) 665.108i 1.30158i
\(512\) 0 0
\(513\) 391.918i 0.763973i
\(514\) 0 0
\(515\) 124.708 25.4558i 0.242151 0.0494288i
\(516\) 0 0
\(517\) −117.576 −0.227419
\(518\) 0 0
\(519\) 138.564i 0.266983i
\(520\) 0 0
\(521\) −722.000 −1.38580 −0.692898 0.721035i \(-0.743667\pi\)
−0.692898 + 0.721035i \(0.743667\pi\)
\(522\) 0 0
\(523\) 246.073i 0.470503i −0.971935 0.235252i \(-0.924409\pi\)
0.971935 0.235252i \(-0.0755914\pi\)
\(524\) 0 0
\(525\) 235.151 + 552.000i 0.447907 + 1.05143i
\(526\) 0 0
\(527\) −1086.12 −2.06094
\(528\) 0 0
\(529\) 119.000 0.224953
\(530\) 0 0
\(531\) −13.8564 −0.0260949
\(532\) 0 0
\(533\) 215.555 0.404419
\(534\) 0 0
\(535\) −42.4264 207.846i −0.0793017 0.388497i
\(536\) 0 0
\(537\) 744.645i 1.38668i
\(538\) 0 0
\(539\) −318.697 −0.591275
\(540\) 0 0
\(541\) 362.000i 0.669131i −0.942372 0.334566i \(-0.891410\pi\)
0.942372 0.334566i \(-0.108590\pi\)
\(542\) 0 0
\(543\) 73.5391 0.135431
\(544\) 0 0
\(545\) 146.000 + 715.251i 0.267890 + 1.31239i
\(546\) 0 0
\(547\) 42.4264i 0.0775620i 0.999248 + 0.0387810i \(0.0123475\pi\)
−0.999248 + 0.0387810i \(0.987653\pi\)
\(548\) 0 0
\(549\) 46.0000i 0.0837887i
\(550\) 0 0
\(551\) 304.841i 0.553250i
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) −138.564 678.823i −0.249665 1.22310i
\(556\) 0 0
\(557\) −656.463 −1.17857 −0.589285 0.807925i \(-0.700590\pi\)
−0.589285 + 0.807925i \(0.700590\pi\)
\(558\) 0 0
\(559\) 581.969i 1.04109i
\(560\) 0 0
\(561\) −768.000 −1.36898
\(562\) 0 0
\(563\) 517.602i 0.919364i 0.888083 + 0.459682i \(0.152037\pi\)
−0.888083 + 0.459682i \(0.847963\pi\)
\(564\) 0 0
\(565\) −39.1918 192.000i −0.0693661 0.339823i
\(566\) 0 0
\(567\) 602.455 1.06253
\(568\) 0 0
\(569\) 842.000 1.47979 0.739895 0.672723i \(-0.234875\pi\)
0.739895 + 0.672723i \(0.234875\pi\)
\(570\) 0 0
\(571\) −401.836 −0.703740 −0.351870 0.936049i \(-0.614454\pi\)
−0.351870 + 0.936049i \(0.614454\pi\)
\(572\) 0 0
\(573\) −313.535 −0.547181
\(574\) 0 0
\(575\) 585.484 249.415i 1.01823 0.433766i
\(576\) 0 0
\(577\) 39.1918i 0.0679235i −0.999423 0.0339617i \(-0.989188\pi\)
0.999423 0.0339617i \(-0.0108124\pi\)
\(578\) 0 0
\(579\) 942.236 1.62735
\(580\) 0 0
\(581\) 648.000i 1.11532i
\(582\) 0 0
\(583\) −407.294 −0.698617
\(584\) 0 0
\(585\) −48.0000 + 9.79796i −0.0820513 + 0.0167486i
\(586\) 0 0
\(587\) 59.3970i 0.101187i 0.998719 + 0.0505937i \(0.0161113\pi\)
−0.998719 + 0.0505937i \(0.983889\pi\)
\(588\) 0 0
\(589\) 768.000i 1.30390i
\(590\) 0 0
\(591\) 304.841i 0.515805i
\(592\) 0 0
\(593\) 509.494i 0.859180i 0.903024 + 0.429590i \(0.141342\pi\)
−0.903024 + 0.429590i \(0.858658\pi\)
\(594\) 0 0
\(595\) −166.277 814.587i −0.279457 1.36905i
\(596\) 0 0
\(597\) 705.453 1.18166
\(598\) 0 0
\(599\) 859.097i 1.43422i 0.696961 + 0.717110i \(0.254535\pi\)
−0.696961 + 0.717110i \(0.745465\pi\)
\(600\) 0 0
\(601\) −598.000 −0.995008 −0.497504 0.867462i \(-0.665750\pi\)
−0.497504 + 0.867462i \(0.665750\pi\)
\(602\) 0 0
\(603\) 59.3970i 0.0985024i
\(604\) 0 0
\(605\) 347.828 71.0000i 0.574922 0.117355i
\(606\) 0 0
\(607\) −127.279 −0.209686 −0.104843 0.994489i \(-0.533434\pi\)
−0.104843 + 0.994489i \(0.533434\pi\)
\(608\) 0 0
\(609\) −528.000 −0.866995
\(610\) 0 0
\(611\) −83.1384 −0.136069
\(612\) 0 0
\(613\) −205.757 −0.335656 −0.167828 0.985816i \(-0.553675\pi\)
−0.167828 + 0.985816i \(0.553675\pi\)
\(614\) 0 0
\(615\) 62.2254 + 304.841i 0.101180 + 0.495676i
\(616\) 0 0
\(617\) 431.110i 0.698720i 0.936989 + 0.349360i \(0.113601\pi\)
−0.936989 + 0.349360i \(0.886399\pi\)
\(618\) 0 0
\(619\) 41.5692 0.0671554 0.0335777 0.999436i \(-0.489310\pi\)
0.0335777 + 0.999436i \(0.489310\pi\)
\(620\) 0 0
\(621\) 720.000i 1.15942i
\(622\) 0 0
\(623\) 1238.85 1.98853
\(624\) 0 0
\(625\) 433.000 450.706i 0.692800 0.721130i
\(626\) 0 0
\(627\) 543.058i 0.866121i
\(628\) 0 0
\(629\) 960.000i 1.52623i
\(630\) 0 0
\(631\) 969.948i 1.53716i −0.639753 0.768580i \(-0.720963\pi\)
0.639753 0.768580i \(-0.279037\pi\)
\(632\) 0 0
\(633\) 274.343i 0.433401i
\(634\) 0 0
\(635\) 540.400 110.309i 0.851023 0.173714i
\(636\) 0 0
\(637\) −225.353 −0.353772
\(638\) 0 0
\(639\) 27.7128i 0.0433690i
\(640\) 0 0
\(641\) 166.000 0.258970 0.129485 0.991581i \(-0.458668\pi\)
0.129485 + 0.991581i \(0.458668\pi\)
\(642\) 0 0
\(643\) 144.250i 0.224339i 0.993689 + 0.112169i \(0.0357799\pi\)
−0.993689 + 0.112169i \(0.964220\pi\)
\(644\) 0 0
\(645\) 823.029 168.000i 1.27601 0.260465i
\(646\) 0 0
\(647\) −687.308 −1.06230 −0.531150 0.847278i \(-0.678240\pi\)
−0.531150 + 0.847278i \(0.678240\pi\)
\(648\) 0 0
\(649\) 192.000 0.295840
\(650\) 0 0
\(651\) −1330.22 −2.04334
\(652\) 0 0
\(653\) 29.3939 0.0450136 0.0225068 0.999747i \(-0.492835\pi\)
0.0225068 + 0.999747i \(0.492835\pi\)
\(654\) 0 0
\(655\) −882.469 + 180.133i −1.34728 + 0.275013i
\(656\) 0 0
\(657\) 78.3837i 0.119305i
\(658\) 0 0
\(659\) −623.538 −0.946189 −0.473094 0.881012i \(-0.656863\pi\)
−0.473094 + 0.881012i \(0.656863\pi\)
\(660\) 0 0
\(661\) 98.0000i 0.148260i −0.997249 0.0741301i \(-0.976382\pi\)
0.997249 0.0741301i \(-0.0236180\pi\)
\(662\) 0 0
\(663\) −543.058 −0.819092
\(664\) 0 0
\(665\) 576.000 117.576i 0.866165 0.176805i
\(666\) 0 0
\(667\) 560.029i 0.839623i
\(668\) 0 0
\(669\) 648.000i 0.968610i
\(670\) 0 0
\(671\) 637.395i 0.949918i
\(672\) 0 0
\(673\) 489.898i 0.727932i −0.931412 0.363966i \(-0.881423\pi\)
0.931412 0.363966i \(-0.118577\pi\)
\(674\) 0 0
\(675\) −277.128 650.538i −0.410560 0.963760i
\(676\) 0 0
\(677\) 146.969 0.217089 0.108545 0.994092i \(-0.465381\pi\)
0.108545 + 0.994092i \(0.465381\pi\)
\(678\) 0 0
\(679\) 498.831i 0.734655i
\(680\) 0 0
\(681\) −840.000 −1.23348
\(682\) 0 0
\(683\) 398.808i 0.583907i 0.956433 + 0.291953i \(0.0943052\pi\)
−0.956433 + 0.291953i \(0.905695\pi\)
\(684\) 0 0
\(685\) 215.555 + 1056.00i 0.314679 + 1.54161i
\(686\) 0 0
\(687\) −197.990 −0.288195
\(688\) 0 0
\(689\) −288.000 −0.417997
\(690\) 0 0
\(691\) −789.815 −1.14300 −0.571502 0.820601i \(-0.693639\pi\)
−0.571502 + 0.820601i \(0.693639\pi\)
\(692\) 0 0
\(693\) 117.576 0.169662
\(694\) 0 0
\(695\) −67.8823 + 13.8564i −0.0976723 + 0.0199373i
\(696\) 0 0
\(697\) 431.110i 0.618523i
\(698\) 0 0
\(699\) 443.405 0.634342
\(700\) 0 0
\(701\) 718.000i 1.02425i −0.858911 0.512126i \(-0.828858\pi\)
0.858911 0.512126i \(-0.171142\pi\)
\(702\) 0 0
\(703\) −678.823 −0.965608
\(704\) 0 0
\(705\) −24.0000 117.576i −0.0340426 0.166774i
\(706\) 0 0
\(707\) 593.970i 0.840127i
\(708\) 0 0
\(709\) 838.000i 1.18195i −0.806691 0.590973i \(-0.798744\pi\)
0.806691 0.590973i \(-0.201256\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 1410.91i 1.97883i
\(714\) 0 0
\(715\) 665.108 135.765i 0.930220 0.189880i
\(716\) 0 0
\(717\) 940.604 1.31186
\(718\) 0 0
\(719\) 221.703i 0.308348i 0.988044 + 0.154174i \(0.0492717\pi\)
−0.988044 + 0.154174i \(0.950728\pi\)
\(720\) 0 0
\(721\) −216.000 −0.299584
\(722\) 0 0
\(723\) 345.068i 0.477273i
\(724\) 0 0
\(725\) 215.555 + 506.000i 0.297317 + 0.697931i
\(726\) 0 0
\(727\) −347.897 −0.478537 −0.239269 0.970953i \(-0.576908\pi\)
−0.239269 + 0.970953i \(0.576908\pi\)
\(728\) 0 0
\(729\) −791.000 −1.08505
\(730\) 0 0
\(731\) −1163.94 −1.59225
\(732\) 0 0
\(733\) 676.059 0.922318 0.461159 0.887317i \(-0.347434\pi\)
0.461159 + 0.887317i \(0.347434\pi\)
\(734\) 0 0
\(735\) −65.0538 318.697i −0.0885086 0.433602i
\(736\) 0 0
\(737\) 823.029i 1.11673i
\(738\) 0 0
\(739\) −235.559 −0.318754 −0.159377 0.987218i \(-0.550948\pi\)
−0.159377 + 0.987218i \(0.550948\pi\)
\(740\) 0 0
\(741\) 384.000i 0.518219i
\(742\) 0 0
\(743\) 1349.16 1.81583 0.907914 0.419157i \(-0.137674\pi\)
0.907914 + 0.419157i \(0.137674\pi\)
\(744\) 0 0
\(745\) −2.00000 9.79796i −0.00268456 0.0131516i
\(746\) 0 0
\(747\) 76.3675i 0.102232i
\(748\) 0 0
\(749\) 360.000i 0.480641i
\(750\) 0 0
\(751\) 1163.94i 1.54985i 0.632053 + 0.774926i \(0.282213\pi\)
−0.632053 + 0.774926i \(0.717787\pi\)
\(752\) 0 0
\(753\) 823.029i 1.09300i
\(754\) 0 0
\(755\) 27.7128 + 135.765i 0.0367057 + 0.179821i
\(756\) 0 0
\(757\) −1205.15 −1.59201 −0.796003 0.605292i \(-0.793056\pi\)
−0.796003 + 0.605292i \(0.793056\pi\)
\(758\) 0 0
\(759\) 997.661i 1.31444i
\(760\) 0 0
\(761\) −530.000 −0.696452 −0.348226 0.937411i \(-0.613216\pi\)
−0.348226 + 0.937411i \(0.613216\pi\)
\(762\) 0 0
\(763\) 1238.85i 1.62366i
\(764\) 0 0
\(765\) 19.5959 + 96.0000i 0.0256156 + 0.125490i
\(766\) 0 0
\(767\) 135.765 0.177007
\(768\) 0 0
\(769\) 386.000 0.501951 0.250975 0.967993i \(-0.419249\pi\)
0.250975 + 0.967993i \(0.419249\pi\)
\(770\) 0 0
\(771\) 221.703 0.287552
\(772\) 0 0
\(773\) 695.655 0.899942 0.449971 0.893043i \(-0.351434\pi\)
0.449971 + 0.893043i \(0.351434\pi\)
\(774\) 0 0
\(775\) 543.058 + 1274.79i 0.700720 + 1.64489i
\(776\) 0 0
\(777\) 1175.76i 1.51320i
\(778\) 0 0
\(779\) 304.841 0.391323
\(780\) 0 0
\(781\) 384.000i 0.491677i
\(782\) 0 0
\(783\) 622.254 0.794705
\(784\) 0 0
\(785\) −336.000 + 68.5857i −0.428025 + 0.0873703i
\(786\) 0 0
\(787\) 1298.25i 1.64962i 0.565413 + 0.824808i \(0.308717\pi\)
−0.565413 + 0.824808i \(0.691283\pi\)
\(788\) 0 0
\(789\) 984.000i 1.24715i
\(790\) 0 0
\(791\) 332.554i 0.420422i
\(792\) 0 0
\(793\) 450.706i 0.568356i
\(794\) 0 0
\(795\) −83.1384 407.294i −0.104577 0.512319i
\(796\) 0 0
\(797\) 1361.92 1.70880 0.854402 0.519613i \(-0.173924\pi\)
0.854402 + 0.519613i \(0.173924\pi\)
\(798\) 0 0
\(799\) 166.277i 0.208106i
\(800\) 0 0
\(801\) −146.000 −0.182272
\(802\) 0 0
\(803\) 1086.12i 1.35257i
\(804\) 0 0
\(805\) −1058.18 + 216.000i −1.31451 + 0.268323i
\(806\) 0 0
\(807\) −401.637 −0.497691
\(808\) 0 0
\(809\) 1006.00 1.24351 0.621755 0.783212i \(-0.286420\pi\)
0.621755 + 0.783212i \(0.286420\pi\)
\(810\) 0 0
\(811\) 651.251 0.803022 0.401511 0.915854i \(-0.368485\pi\)
0.401511 + 0.915854i \(0.368485\pi\)
\(812\) 0 0
\(813\) 940.604 1.15695
\(814\) 0 0
\(815\) 110.309 + 540.400i 0.135348 + 0.663067i
\(816\) 0 0
\(817\) 823.029i 1.00738i
\(818\) 0 0
\(819\) 83.1384 0.101512
\(820\) 0 0
\(821\) 482.000i 0.587089i −0.955945 0.293544i \(-0.905165\pi\)
0.955945 0.293544i \(-0.0948349\pi\)
\(822\) 0 0
\(823\) 636.396 0.773264 0.386632 0.922234i \(-0.373638\pi\)
0.386632 + 0.922234i \(0.373638\pi\)
\(824\) 0 0
\(825\) 384.000 + 901.412i 0.465455 + 1.09262i
\(826\) 0 0
\(827\) 398.808i 0.482235i 0.970496 + 0.241117i \(0.0775139\pi\)
−0.970496 + 0.241117i \(0.922486\pi\)
\(828\) 0 0
\(829\) 1106.00i 1.33414i 0.744996 + 0.667069i \(0.232451\pi\)
−0.744996 + 0.667069i \(0.767549\pi\)
\(830\) 0 0
\(831\) 249.415i 0.300139i
\(832\) 0 0
\(833\) 450.706i 0.541064i
\(834\) 0 0
\(835\) 457.261 93.3381i 0.547618 0.111782i
\(836\) 0 0
\(837\) 1567.67 1.87297
\(838\) 0 0
\(839\) 360.267i 0.429400i −0.976680 0.214700i \(-0.931123\pi\)
0.976680 0.214700i \(-0.0688774\pi\)
\(840\) 0 0
\(841\) 357.000 0.424495
\(842\) 0 0
\(843\) 1148.34i 1.36221i
\(844\) 0 0
\(845\) −357.626 + 73.0000i −0.423225 + 0.0863905i
\(846\) 0 0
\(847\) −602.455 −0.711281
\(848\) 0 0
\(849\) 264.000 0.310954
\(850\) 0 0
\(851\) 1247.08 1.46542
\(852\) 0 0
\(853\) −676.059 −0.792566 −0.396283 0.918128i \(-0.629700\pi\)
−0.396283 + 0.918128i \(0.629700\pi\)
\(854\) 0 0
\(855\) −67.8823 + 13.8564i −0.0793944 + 0.0162063i
\(856\) 0 0
\(857\) 1077.78i 1.25761i 0.777561 + 0.628807i \(0.216457\pi\)
−0.777561 + 0.628807i \(0.783543\pi\)
\(858\) 0 0
\(859\) −568.113 −0.661365 −0.330683 0.943742i \(-0.607279\pi\)
−0.330683 + 0.943742i \(0.607279\pi\)
\(860\) 0 0
\(861\) 528.000i 0.613240i
\(862\) 0 0
\(863\) −195.161 −0.226143 −0.113072 0.993587i \(-0.536069\pi\)
−0.113072 + 0.993587i \(0.536069\pi\)
\(864\) 0 0
\(865\) 240.000 48.9898i 0.277457 0.0566356i
\(866\) 0 0
\(867\) 268.701i 0.309920i
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 581.969i 0.668162i
\(872\) 0 0
\(873\) 58.7878i 0.0673399i
\(874\) 0 0
\(875\) −872.954 + 602.455i −0.997661 + 0.688520i
\(876\) 0 0
\(877\) 244.949 0.279303 0.139652 0.990201i \(-0.455402\pi\)
0.139652 + 0.990201i \(0.455402\pi\)
\(878\) 0 0
\(879\) 859.097i 0.977357i
\(880\) 0 0
\(881\) 550.000 0.624291 0.312145 0.950034i \(-0.398952\pi\)
0.312145 + 0.950034i \(0.398952\pi\)
\(882\) 0 0
\(883\) 636.396i 0.720720i −0.932813 0.360360i \(-0.882654\pi\)
0.932813 0.360360i \(-0.117346\pi\)
\(884\) 0 0
\(885\) 39.1918 + 192.000i 0.0442846 + 0.216949i
\(886\) 0 0
\(887\) −1230.37 −1.38711 −0.693555 0.720404i \(-0.743956\pi\)
−0.693555 + 0.720404i \(0.743956\pi\)
\(888\) 0 0
\(889\) −936.000 −1.05287
\(890\) 0 0
\(891\) 983.805 1.10416
\(892\) 0 0
\(893\) −117.576 −0.131664
\(894\) 0 0
\(895\) 1289.76 263.272i 1.44108 0.294158i
\(896\) 0 0
\(897\) 705.453i 0.786458i
\(898\) 0 0
\(899\) −1219.36 −1.35636
\(900\) 0 0
\(901\) 576.000i 0.639290i
\(902\) 0 0
\(903\) −1425.53 −1.57866
\(904\) 0 0
\(905\) 26.0000 + 127.373i 0.0287293 + 0.140744i
\(906\) 0 0
\(907\) 585.484i 0.645518i −0.946481 0.322759i \(-0.895390\pi\)
0.946481 0.322759i \(-0.104610\pi\)
\(908\) 0 0
\(909\) 70.0000i 0.0770077i
\(910\) 0 0
\(911\) 886.810i 0.973447i −0.873556 0.486723i \(-0.838192\pi\)
0.873556 0.486723i \(-0.161808\pi\)
\(912\) 0 0
\(913\) 1058.18i 1.15901i
\(914\) 0 0
\(915\) −637.395 + 130.108i −0.696606 + 0.142194i
\(916\) 0 0
\(917\) 1528.48 1.66683
\(918\) 0 0
\(919\) 193.990i 0.211088i 0.994415 + 0.105544i \(0.0336584\pi\)
−0.994415 + 0.105544i \(0.966342\pi\)
\(920\) 0 0
\(921\) 792.000 0.859935
\(922\) 0 0
\(923\) 271.529i 0.294181i
\(924\) 0 0
\(925\) 1126.77 480.000i 1.21812 0.518919i
\(926\) 0 0
\(927\) 25.4558 0.0274605
\(928\) 0 0
\(929\) 1318.00 1.41873 0.709365 0.704841i \(-0.248982\pi\)
0.709365 + 0.704841i \(0.248982\pi\)
\(930\) 0 0
\(931\) −318.697 −0.342317
\(932\) 0 0
\(933\) 391.918 0.420063
\(934\) 0 0
\(935\) −271.529 1330.22i −0.290405 1.42269i
\(936\) 0 0
\(937\) 548.686i 0.585577i 0.956177 + 0.292789i \(0.0945832\pi\)
−0.956177 + 0.292789i \(0.905417\pi\)
\(938\) 0 0
\(939\) −1718.19 −1.82981
\(940\) 0 0
\(941\) 1654.00i 1.75770i 0.477094 + 0.878852i \(0.341690\pi\)
−0.477094 + 0.878852i \(0.658310\pi\)
\(942\) 0 0
\(943\) −560.029 −0.593880
\(944\) 0 0
\(945\) 240.000 + 1175.76i 0.253968 + 1.24419i
\(946\) 0 0
\(947\) 1417.04i 1.49635i −0.663502 0.748174i \(-0.730931\pi\)
0.663502 0.748174i \(-0.269069\pi\)
\(948\) 0 0
\(949\) 768.000i 0.809273i
\(950\) 0 0
\(951\) 193.990i 0.203985i
\(952\) 0 0
\(953\) 137.171i 0.143936i 0.997407 + 0.0719682i \(0.0229280\pi\)
−0.997407 + 0.0719682i \(0.977072\pi\)
\(954\) 0 0
\(955\) −110.851 543.058i −0.116075 0.568647i
\(956\) 0 0
\(957\) −862.220 −0.900962
\(958\) 0 0
\(959\) 1829.05i 1.90724i
\(960\) 0 0
\(961\) −2111.00 −2.19667
\(962\) 0 0
\(963\) 42.4264i 0.0440565i
\(964\) 0 0
\(965\) 333.131 + 1632.00i 0.345213 + 1.69119i
\(966\) 0 0
\(967\) 1450.98 1.50050 0.750250 0.661154i \(-0.229933\pi\)
0.750250 + 0.661154i \(0.229933\pi\)
\(968\) 0 0
\(969\) −768.000 −0.792570
\(970\) 0 0
\(971\) 817.528 0.841944 0.420972 0.907074i \(-0.361689\pi\)
0.420972 + 0.907074i \(0.361689\pi\)
\(972\) 0 0
\(973\) 117.576 0.120838
\(974\) 0 0
\(975\) 271.529 + 637.395i 0.278491 + 0.653738i
\(976\) 0 0
\(977\) 1704.84i 1.74498i 0.488633 + 0.872490i \(0.337496\pi\)
−0.488633 + 0.872490i \(0.662504\pi\)
\(978\) 0 0
\(979\) 2023.04 2.06643
\(980\) 0 0
\(981\) 146.000i 0.148828i
\(982\) 0 0
\(983\) 738.219 0.750986 0.375493 0.926825i \(-0.377473\pi\)
0.375493 + 0.926825i \(0.377473\pi\)
\(984\) 0 0
\(985\) 528.000 107.778i 0.536041 0.109419i
\(986\) 0 0
\(987\) 203.647i 0.206329i
\(988\) 0 0
\(989\) 1512.00i 1.52882i
\(990\) 0 0
\(991\) 775.959i 0.783006i −0.920177 0.391503i \(-0.871955\pi\)
0.920177 0.391503i \(-0.128045\pi\)
\(992\) 0 0
\(993\) 509.494i 0.513085i
\(994\) 0 0
\(995\) 249.415 + 1221.88i 0.250669 + 1.22802i
\(996\) 0 0
\(997\) −1420.70 −1.42498 −0.712489 0.701683i \(-0.752432\pi\)
−0.712489 + 0.701683i \(0.752432\pi\)
\(998\) 0 0
\(999\) 1385.64i 1.38703i
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 1280.3.e.j.639.3 8
4.3 odd 2 inner 1280.3.e.j.639.7 8
5.4 even 2 inner 1280.3.e.j.639.5 8
8.3 odd 2 inner 1280.3.e.j.639.2 8
8.5 even 2 inner 1280.3.e.j.639.6 8
16.3 odd 4 80.3.h.b.79.1 4
16.5 even 4 320.3.h.e.319.2 4
16.11 odd 4 320.3.h.e.319.4 4
16.13 even 4 80.3.h.b.79.3 yes 4
20.19 odd 2 inner 1280.3.e.j.639.1 8
40.19 odd 2 inner 1280.3.e.j.639.8 8
40.29 even 2 inner 1280.3.e.j.639.4 8
48.29 odd 4 720.3.j.e.559.4 4
48.35 even 4 720.3.j.e.559.3 4
80.3 even 4 400.3.b.h.351.1 4
80.13 odd 4 400.3.b.h.351.4 4
80.19 odd 4 80.3.h.b.79.4 yes 4
80.27 even 4 1600.3.b.t.1151.2 4
80.29 even 4 80.3.h.b.79.2 yes 4
80.37 odd 4 1600.3.b.t.1151.3 4
80.43 even 4 1600.3.b.t.1151.4 4
80.53 odd 4 1600.3.b.t.1151.1 4
80.59 odd 4 320.3.h.e.319.1 4
80.67 even 4 400.3.b.h.351.3 4
80.69 even 4 320.3.h.e.319.3 4
80.77 odd 4 400.3.b.h.351.2 4
240.29 odd 4 720.3.j.e.559.1 4
240.77 even 4 3600.3.e.bd.3151.3 4
240.83 odd 4 3600.3.e.bd.3151.4 4
240.173 even 4 3600.3.e.bd.3151.1 4
240.179 even 4 720.3.j.e.559.2 4
240.227 odd 4 3600.3.e.bd.3151.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
80.3.h.b.79.1 4 16.3 odd 4
80.3.h.b.79.2 yes 4 80.29 even 4
80.3.h.b.79.3 yes 4 16.13 even 4
80.3.h.b.79.4 yes 4 80.19 odd 4
320.3.h.e.319.1 4 80.59 odd 4
320.3.h.e.319.2 4 16.5 even 4
320.3.h.e.319.3 4 80.69 even 4
320.3.h.e.319.4 4 16.11 odd 4
400.3.b.h.351.1 4 80.3 even 4
400.3.b.h.351.2 4 80.77 odd 4
400.3.b.h.351.3 4 80.67 even 4
400.3.b.h.351.4 4 80.13 odd 4
720.3.j.e.559.1 4 240.29 odd 4
720.3.j.e.559.2 4 240.179 even 4
720.3.j.e.559.3 4 48.35 even 4
720.3.j.e.559.4 4 48.29 odd 4
1280.3.e.j.639.1 8 20.19 odd 2 inner
1280.3.e.j.639.2 8 8.3 odd 2 inner
1280.3.e.j.639.3 8 1.1 even 1 trivial
1280.3.e.j.639.4 8 40.29 even 2 inner
1280.3.e.j.639.5 8 5.4 even 2 inner
1280.3.e.j.639.6 8 8.5 even 2 inner
1280.3.e.j.639.7 8 4.3 odd 2 inner
1280.3.e.j.639.8 8 40.19 odd 2 inner
1600.3.b.t.1151.1 4 80.53 odd 4
1600.3.b.t.1151.2 4 80.27 even 4
1600.3.b.t.1151.3 4 80.37 odd 4
1600.3.b.t.1151.4 4 80.43 even 4
3600.3.e.bd.3151.1 4 240.173 even 4
3600.3.e.bd.3151.2 4 240.227 odd 4
3600.3.e.bd.3151.3 4 240.77 even 4
3600.3.e.bd.3151.4 4 240.83 odd 4