Properties

Label 1560.2.l.c.1249.3
Level $1560$
Weight $2$
Character 1560.1249
Analytic conductor $12.457$
Analytic rank $0$
Dimension $6$
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] = [1560,2,Mod(1249,1560)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1560, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1560.1249");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1560 = 2^{3} \cdot 3 \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1560.l (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: \(12.4566627153\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.5161984.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 4x^{3} + 25x^{2} - 20x + 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1249.3
Root \(1.32001 + 1.32001i\) of defining polynomial
Character \(\chi\) \(=\) 1560.1249
Dual form 1560.2.l.c.1249.6

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000i q^{3} +(1.32001 + 1.80487i) q^{5} -1.00000 q^{9} +O(q^{10})\) \(q-1.00000i q^{3} +(1.32001 + 1.80487i) q^{5} -1.00000 q^{9} -4.64002 q^{11} -1.00000i q^{13} +(1.80487 - 1.32001i) q^{15} +4.24977i q^{17} +6.24977 q^{19} -2.24977i q^{23} +(-1.51514 + 4.76491i) q^{25} +1.00000i q^{27} +9.21949 q^{29} +9.28005 q^{31} +4.64002i q^{33} +7.28005i q^{37} -1.00000 q^{39} -5.67030 q^{41} +4.24977i q^{43} +(-1.32001 - 1.80487i) q^{45} +2.88979i q^{47} +7.00000 q^{49} +4.24977 q^{51} +9.21949i q^{53} +(-6.12489 - 8.37466i) q^{55} -6.24977i q^{57} -5.92007 q^{59} +0.969724 q^{61} +(1.80487 - 1.32001i) q^{65} -1.93945i q^{67} -2.24977 q^{69} +5.60975 q^{71} +12.5601i q^{73} +(4.76491 + 1.51514i) q^{75} -12.2498 q^{79} +1.00000 q^{81} +3.67030i q^{83} +(-7.67030 + 5.60975i) q^{85} -9.21949i q^{87} +9.67030 q^{89} -9.28005i q^{93} +(8.24977 + 11.2800i) q^{95} -6.00000i q^{97} +4.64002 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 6 q^{9} - 12 q^{11} + 2 q^{15} + 4 q^{19} - 10 q^{25} + 20 q^{29} + 24 q^{31} - 6 q^{39} - 20 q^{41} + 42 q^{49} - 8 q^{51} - 20 q^{55} + 12 q^{59} + 4 q^{61} + 2 q^{65} + 20 q^{69} + 16 q^{71} - 4 q^{75} - 40 q^{79} + 6 q^{81} - 32 q^{85} + 44 q^{89} + 16 q^{95} + 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1560\mathbb{Z}\right)^\times\).

\(n\) \(391\) \(521\) \(781\) \(937\) \(1081\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\) \(1\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.00000i 0.577350i
\(4\) 0 0
\(5\) 1.32001 + 1.80487i 0.590327 + 0.807164i
\(6\) 0 0
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) −4.64002 −1.39902 −0.699510 0.714623i \(-0.746598\pi\)
−0.699510 + 0.714623i \(0.746598\pi\)
\(12\) 0 0
\(13\) 1.00000i 0.277350i
\(14\) 0 0
\(15\) 1.80487 1.32001i 0.466016 0.340826i
\(16\) 0 0
\(17\) 4.24977i 1.03072i 0.856974 + 0.515360i \(0.172342\pi\)
−0.856974 + 0.515360i \(0.827658\pi\)
\(18\) 0 0
\(19\) 6.24977 1.43380 0.716898 0.697178i \(-0.245561\pi\)
0.716898 + 0.697178i \(0.245561\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 2.24977i 0.469110i −0.972103 0.234555i \(-0.924637\pi\)
0.972103 0.234555i \(-0.0753632\pi\)
\(24\) 0 0
\(25\) −1.51514 + 4.76491i −0.303028 + 0.952982i
\(26\) 0 0
\(27\) 1.00000i 0.192450i
\(28\) 0 0
\(29\) 9.21949 1.71202 0.856009 0.516962i \(-0.172937\pi\)
0.856009 + 0.516962i \(0.172937\pi\)
\(30\) 0 0
\(31\) 9.28005 1.66675 0.833373 0.552711i \(-0.186407\pi\)
0.833373 + 0.552711i \(0.186407\pi\)
\(32\) 0 0
\(33\) 4.64002i 0.807724i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 7.28005i 1.19683i 0.801185 + 0.598416i \(0.204203\pi\)
−0.801185 + 0.598416i \(0.795797\pi\)
\(38\) 0 0
\(39\) −1.00000 −0.160128
\(40\) 0 0
\(41\) −5.67030 −0.885552 −0.442776 0.896632i \(-0.646006\pi\)
−0.442776 + 0.896632i \(0.646006\pi\)
\(42\) 0 0
\(43\) 4.24977i 0.648084i 0.946043 + 0.324042i \(0.105042\pi\)
−0.946043 + 0.324042i \(0.894958\pi\)
\(44\) 0 0
\(45\) −1.32001 1.80487i −0.196776 0.269055i
\(46\) 0 0
\(47\) 2.88979i 0.421520i 0.977538 + 0.210760i \(0.0675938\pi\)
−0.977538 + 0.210760i \(0.932406\pi\)
\(48\) 0 0
\(49\) 7.00000 1.00000
\(50\) 0 0
\(51\) 4.24977 0.595087
\(52\) 0 0
\(53\) 9.21949i 1.26639i 0.773990 + 0.633197i \(0.218258\pi\)
−0.773990 + 0.633197i \(0.781742\pi\)
\(54\) 0 0
\(55\) −6.12489 8.37466i −0.825879 1.12924i
\(56\) 0 0
\(57\) 6.24977i 0.827802i
\(58\) 0 0
\(59\) −5.92007 −0.770728 −0.385364 0.922765i \(-0.625924\pi\)
−0.385364 + 0.922765i \(0.625924\pi\)
\(60\) 0 0
\(61\) 0.969724 0.124160 0.0620802 0.998071i \(-0.480227\pi\)
0.0620802 + 0.998071i \(0.480227\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 1.80487 1.32001i 0.223867 0.163727i
\(66\) 0 0
\(67\) 1.93945i 0.236941i −0.992958 0.118471i \(-0.962201\pi\)
0.992958 0.118471i \(-0.0377991\pi\)
\(68\) 0 0
\(69\) −2.24977 −0.270841
\(70\) 0 0
\(71\) 5.60975 0.665755 0.332877 0.942970i \(-0.391981\pi\)
0.332877 + 0.942970i \(0.391981\pi\)
\(72\) 0 0
\(73\) 12.5601i 1.47005i 0.678041 + 0.735024i \(0.262829\pi\)
−0.678041 + 0.735024i \(0.737171\pi\)
\(74\) 0 0
\(75\) 4.76491 + 1.51514i 0.550204 + 0.174953i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −12.2498 −1.37821 −0.689103 0.724663i \(-0.741995\pi\)
−0.689103 + 0.724663i \(0.741995\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 3.67030i 0.402868i 0.979502 + 0.201434i \(0.0645601\pi\)
−0.979502 + 0.201434i \(0.935440\pi\)
\(84\) 0 0
\(85\) −7.67030 + 5.60975i −0.831961 + 0.608463i
\(86\) 0 0
\(87\) 9.21949i 0.988434i
\(88\) 0 0
\(89\) 9.67030 1.02505 0.512525 0.858672i \(-0.328710\pi\)
0.512525 + 0.858672i \(0.328710\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 9.28005i 0.962296i
\(94\) 0 0
\(95\) 8.24977 + 11.2800i 0.846409 + 1.15731i
\(96\) 0 0
\(97\) 6.00000i 0.609208i −0.952479 0.304604i \(-0.901476\pi\)
0.952479 0.304604i \(-0.0985241\pi\)
\(98\) 0 0
\(99\) 4.64002 0.466340
\(100\) 0 0
\(101\) −15.2800 −1.52042 −0.760211 0.649677i \(-0.774904\pi\)
−0.760211 + 0.649677i \(0.774904\pi\)
\(102\) 0 0
\(103\) 1.21949i 0.120160i −0.998194 0.0600802i \(-0.980864\pi\)
0.998194 0.0600802i \(-0.0191356\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 5.28005i 0.510441i −0.966883 0.255221i \(-0.917852\pi\)
0.966883 0.255221i \(-0.0821481\pi\)
\(108\) 0 0
\(109\) 10.1892 0.975950 0.487975 0.872858i \(-0.337736\pi\)
0.487975 + 0.872858i \(0.337736\pi\)
\(110\) 0 0
\(111\) 7.28005 0.690991
\(112\) 0 0
\(113\) 1.03028i 0.0969202i −0.998825 0.0484601i \(-0.984569\pi\)
0.998825 0.0484601i \(-0.0154314\pi\)
\(114\) 0 0
\(115\) 4.06055 2.96972i 0.378648 0.276928i
\(116\) 0 0
\(117\) 1.00000i 0.0924500i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 10.5298 0.957256
\(122\) 0 0
\(123\) 5.67030i 0.511274i
\(124\) 0 0
\(125\) −10.6001 + 3.55510i −0.948098 + 0.317978i
\(126\) 0 0
\(127\) 11.0303i 0.978779i −0.872065 0.489389i \(-0.837220\pi\)
0.872065 0.489389i \(-0.162780\pi\)
\(128\) 0 0
\(129\) 4.24977 0.374171
\(130\) 0 0
\(131\) 7.21949 0.630770 0.315385 0.948964i \(-0.397866\pi\)
0.315385 + 0.948964i \(0.397866\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) −1.80487 + 1.32001i −0.155339 + 0.113609i
\(136\) 0 0
\(137\) 8.10929i 0.692823i 0.938083 + 0.346412i \(0.112600\pi\)
−0.938083 + 0.346412i \(0.887400\pi\)
\(138\) 0 0
\(139\) 19.0596 1.61662 0.808309 0.588759i \(-0.200383\pi\)
0.808309 + 0.588759i \(0.200383\pi\)
\(140\) 0 0
\(141\) 2.88979 0.243365
\(142\) 0 0
\(143\) 4.64002i 0.388018i
\(144\) 0 0
\(145\) 12.1698 + 16.6400i 1.01065 + 1.38188i
\(146\) 0 0
\(147\) 7.00000i 0.577350i
\(148\) 0 0
\(149\) −3.92007 −0.321145 −0.160572 0.987024i \(-0.551334\pi\)
−0.160572 + 0.987024i \(0.551334\pi\)
\(150\) 0 0
\(151\) 21.7796 1.77240 0.886199 0.463305i \(-0.153337\pi\)
0.886199 + 0.463305i \(0.153337\pi\)
\(152\) 0 0
\(153\) 4.24977i 0.343574i
\(154\) 0 0
\(155\) 12.2498 + 16.7493i 0.983925 + 1.34534i
\(156\) 0 0
\(157\) 21.5904i 1.72310i −0.507673 0.861550i \(-0.669494\pi\)
0.507673 0.861550i \(-0.330506\pi\)
\(158\) 0 0
\(159\) 9.21949 0.731153
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 1.28005i 0.100261i −0.998743 0.0501305i \(-0.984036\pi\)
0.998743 0.0501305i \(-0.0159637\pi\)
\(164\) 0 0
\(165\) −8.37466 + 6.12489i −0.651966 + 0.476822i
\(166\) 0 0
\(167\) 18.8898i 1.46174i −0.682519 0.730868i \(-0.739115\pi\)
0.682519 0.730868i \(-0.260885\pi\)
\(168\) 0 0
\(169\) −1.00000 −0.0769231
\(170\) 0 0
\(171\) −6.24977 −0.477932
\(172\) 0 0
\(173\) 22.9991i 1.74859i 0.485397 + 0.874294i \(0.338675\pi\)
−0.485397 + 0.874294i \(0.661325\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 5.92007i 0.444980i
\(178\) 0 0
\(179\) 4.49954 0.336312 0.168156 0.985760i \(-0.446219\pi\)
0.168156 + 0.985760i \(0.446219\pi\)
\(180\) 0 0
\(181\) −21.0596 −1.56535 −0.782675 0.622430i \(-0.786145\pi\)
−0.782675 + 0.622430i \(0.786145\pi\)
\(182\) 0 0
\(183\) 0.969724i 0.0716841i
\(184\) 0 0
\(185\) −13.1396 + 9.60975i −0.966040 + 0.706523i
\(186\) 0 0
\(187\) 19.7190i 1.44200i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −14.4390 −1.04477 −0.522384 0.852710i \(-0.674957\pi\)
−0.522384 + 0.852710i \(0.674957\pi\)
\(192\) 0 0
\(193\) 21.7190i 1.56337i −0.623673 0.781685i \(-0.714360\pi\)
0.623673 0.781685i \(-0.285640\pi\)
\(194\) 0 0
\(195\) −1.32001 1.80487i −0.0945280 0.129250i
\(196\) 0 0
\(197\) 13.0109i 0.926988i −0.886100 0.463494i \(-0.846596\pi\)
0.886100 0.463494i \(-0.153404\pi\)
\(198\) 0 0
\(199\) −18.6888 −1.32481 −0.662406 0.749145i \(-0.730464\pi\)
−0.662406 + 0.749145i \(0.730464\pi\)
\(200\) 0 0
\(201\) −1.93945 −0.136798
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −7.48486 10.2342i −0.522765 0.714786i
\(206\) 0 0
\(207\) 2.24977i 0.156370i
\(208\) 0 0
\(209\) −28.9991 −2.00591
\(210\) 0 0
\(211\) −8.24977 −0.567938 −0.283969 0.958834i \(-0.591651\pi\)
−0.283969 + 0.958834i \(0.591651\pi\)
\(212\) 0 0
\(213\) 5.60975i 0.384374i
\(214\) 0 0
\(215\) −7.67030 + 5.60975i −0.523110 + 0.382582i
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 12.5601 0.848732
\(220\) 0 0
\(221\) 4.24977 0.285871
\(222\) 0 0
\(223\) 9.77959i 0.654890i −0.944870 0.327445i \(-0.893812\pi\)
0.944870 0.327445i \(-0.106188\pi\)
\(224\) 0 0
\(225\) 1.51514 4.76491i 0.101009 0.317661i
\(226\) 0 0
\(227\) 25.4499i 1.68917i 0.535423 + 0.844584i \(0.320152\pi\)
−0.535423 + 0.844584i \(0.679848\pi\)
\(228\) 0 0
\(229\) 8.24977 0.545160 0.272580 0.962133i \(-0.412123\pi\)
0.272580 + 0.962133i \(0.412123\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 11.5904i 0.759310i −0.925128 0.379655i \(-0.876043\pi\)
0.925128 0.379655i \(-0.123957\pi\)
\(234\) 0 0
\(235\) −5.21571 + 3.81456i −0.340236 + 0.248835i
\(236\) 0 0
\(237\) 12.2498i 0.795708i
\(238\) 0 0
\(239\) 6.88979 0.445664 0.222832 0.974857i \(-0.428470\pi\)
0.222832 + 0.974857i \(0.428470\pi\)
\(240\) 0 0
\(241\) 21.2195 1.36687 0.683434 0.730012i \(-0.260486\pi\)
0.683434 + 0.730012i \(0.260486\pi\)
\(242\) 0 0
\(243\) 1.00000i 0.0641500i
\(244\) 0 0
\(245\) 9.24008 + 12.6341i 0.590327 + 0.807164i
\(246\) 0 0
\(247\) 6.24977i 0.397663i
\(248\) 0 0
\(249\) 3.67030 0.232596
\(250\) 0 0
\(251\) −24.3397 −1.53631 −0.768154 0.640266i \(-0.778824\pi\)
−0.768154 + 0.640266i \(0.778824\pi\)
\(252\) 0 0
\(253\) 10.4390i 0.656294i
\(254\) 0 0
\(255\) 5.60975 + 7.67030i 0.351296 + 0.480333i
\(256\) 0 0
\(257\) 23.4693i 1.46397i −0.681319 0.731986i \(-0.738593\pi\)
0.681319 0.731986i \(-0.261407\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −9.21949 −0.570672
\(262\) 0 0
\(263\) 22.6282i 1.39532i 0.716431 + 0.697658i \(0.245774\pi\)
−0.716431 + 0.697658i \(0.754226\pi\)
\(264\) 0 0
\(265\) −16.6400 + 12.1698i −1.02219 + 0.747587i
\(266\) 0 0
\(267\) 9.67030i 0.591813i
\(268\) 0 0
\(269\) −12.2791 −0.748672 −0.374336 0.927293i \(-0.622129\pi\)
−0.374336 + 0.927293i \(0.622129\pi\)
\(270\) 0 0
\(271\) −22.9385 −1.39342 −0.696708 0.717355i \(-0.745353\pi\)
−0.696708 + 0.717355i \(0.745353\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 7.03028 22.1093i 0.423942 1.33324i
\(276\) 0 0
\(277\) 11.5298i 0.692760i 0.938094 + 0.346380i \(0.112589\pi\)
−0.938094 + 0.346380i \(0.887411\pi\)
\(278\) 0 0
\(279\) −9.28005 −0.555582
\(280\) 0 0
\(281\) −11.3288 −0.675819 −0.337909 0.941179i \(-0.609720\pi\)
−0.337909 + 0.941179i \(0.609720\pi\)
\(282\) 0 0
\(283\) 20.9991i 1.24827i −0.781318 0.624133i \(-0.785452\pi\)
0.781318 0.624133i \(-0.214548\pi\)
\(284\) 0 0
\(285\) 11.2800 8.24977i 0.668172 0.488674i
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −1.06055 −0.0623854
\(290\) 0 0
\(291\) −6.00000 −0.351726
\(292\) 0 0
\(293\) 4.26915i 0.249406i 0.992194 + 0.124703i \(0.0397979\pi\)
−0.992194 + 0.124703i \(0.960202\pi\)
\(294\) 0 0
\(295\) −7.81456 10.6850i −0.454981 0.622104i
\(296\) 0 0
\(297\) 4.64002i 0.269241i
\(298\) 0 0
\(299\) −2.24977 −0.130108
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 15.2800i 0.877816i
\(304\) 0 0
\(305\) 1.28005 + 1.75023i 0.0732953 + 0.100218i
\(306\) 0 0
\(307\) 20.0000i 1.14146i −0.821138 0.570730i \(-0.806660\pi\)
0.821138 0.570730i \(-0.193340\pi\)
\(308\) 0 0
\(309\) −1.21949 −0.0693746
\(310\) 0 0
\(311\) 3.71904 0.210887 0.105444 0.994425i \(-0.466374\pi\)
0.105444 + 0.994425i \(0.466374\pi\)
\(312\) 0 0
\(313\) 1.52982i 0.0864704i −0.999065 0.0432352i \(-0.986234\pi\)
0.999065 0.0432352i \(-0.0137665\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 16.1093i 0.904788i 0.891818 + 0.452394i \(0.149430\pi\)
−0.891818 + 0.452394i \(0.850570\pi\)
\(318\) 0 0
\(319\) −42.7787 −2.39515
\(320\) 0 0
\(321\) −5.28005 −0.294703
\(322\) 0 0
\(323\) 26.5601i 1.47784i
\(324\) 0 0
\(325\) 4.76491 + 1.51514i 0.264310 + 0.0840447i
\(326\) 0 0
\(327\) 10.1892i 0.563465i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 15.5298 0.853596 0.426798 0.904347i \(-0.359642\pi\)
0.426798 + 0.904347i \(0.359642\pi\)
\(332\) 0 0
\(333\) 7.28005i 0.398944i
\(334\) 0 0
\(335\) 3.50046 2.56009i 0.191250 0.139873i
\(336\) 0 0
\(337\) 21.4087i 1.16621i −0.812398 0.583103i \(-0.801838\pi\)
0.812398 0.583103i \(-0.198162\pi\)
\(338\) 0 0
\(339\) −1.03028 −0.0559569
\(340\) 0 0
\(341\) −43.0596 −2.33181
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) −2.96972 4.06055i −0.159885 0.218613i
\(346\) 0 0
\(347\) 4.62065i 0.248049i −0.992279 0.124025i \(-0.960420\pi\)
0.992279 0.124025i \(-0.0395802\pi\)
\(348\) 0 0
\(349\) −23.9688 −1.28302 −0.641510 0.767114i \(-0.721692\pi\)
−0.641510 + 0.767114i \(0.721692\pi\)
\(350\) 0 0
\(351\) 1.00000 0.0533761
\(352\) 0 0
\(353\) 0.511357i 0.0272168i 0.999907 + 0.0136084i \(0.00433182\pi\)
−0.999907 + 0.0136084i \(0.995668\pi\)
\(354\) 0 0
\(355\) 7.40493 + 10.1249i 0.393013 + 0.537373i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 12.6694 0.668664 0.334332 0.942455i \(-0.391489\pi\)
0.334332 + 0.942455i \(0.391489\pi\)
\(360\) 0 0
\(361\) 20.0596 1.05577
\(362\) 0 0
\(363\) 10.5298i 0.552672i
\(364\) 0 0
\(365\) −22.6694 + 16.5795i −1.18657 + 0.867809i
\(366\) 0 0
\(367\) 0.841057i 0.0439028i −0.999759 0.0219514i \(-0.993012\pi\)
0.999759 0.0219514i \(-0.00698792\pi\)
\(368\) 0 0
\(369\) 5.67030 0.295184
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 29.0596i 1.50465i 0.658792 + 0.752325i \(0.271068\pi\)
−0.658792 + 0.752325i \(0.728932\pi\)
\(374\) 0 0
\(375\) 3.55510 + 10.6001i 0.183585 + 0.547385i
\(376\) 0 0
\(377\) 9.21949i 0.474828i
\(378\) 0 0
\(379\) −9.75023 −0.500836 −0.250418 0.968138i \(-0.580568\pi\)
−0.250418 + 0.968138i \(0.580568\pi\)
\(380\) 0 0
\(381\) −11.0303 −0.565098
\(382\) 0 0
\(383\) 7.38934i 0.377577i 0.982018 + 0.188789i \(0.0604561\pi\)
−0.982018 + 0.188789i \(0.939544\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 4.24977i 0.216028i
\(388\) 0 0
\(389\) 14.0000 0.709828 0.354914 0.934899i \(-0.384510\pi\)
0.354914 + 0.934899i \(0.384510\pi\)
\(390\) 0 0
\(391\) 9.56101 0.483521
\(392\) 0 0
\(393\) 7.21949i 0.364175i
\(394\) 0 0
\(395\) −16.1698 22.1093i −0.813593 1.11244i
\(396\) 0 0
\(397\) 26.2186i 1.31587i 0.753074 + 0.657936i \(0.228570\pi\)
−0.753074 + 0.657936i \(0.771430\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 10.2909 0.513905 0.256953 0.966424i \(-0.417282\pi\)
0.256953 + 0.966424i \(0.417282\pi\)
\(402\) 0 0
\(403\) 9.28005i 0.462272i
\(404\) 0 0
\(405\) 1.32001 + 1.80487i 0.0655919 + 0.0896849i
\(406\) 0 0
\(407\) 33.7796i 1.67439i
\(408\) 0 0
\(409\) 11.9007 0.588451 0.294226 0.955736i \(-0.404938\pi\)
0.294226 + 0.955736i \(0.404938\pi\)
\(410\) 0 0
\(411\) 8.10929 0.400002
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −6.62443 + 4.84484i −0.325180 + 0.237824i
\(416\) 0 0
\(417\) 19.0596i 0.933354i
\(418\) 0 0
\(419\) −26.2791 −1.28382 −0.641910 0.766780i \(-0.721858\pi\)
−0.641910 + 0.766780i \(0.721858\pi\)
\(420\) 0 0
\(421\) 22.3103 1.08734 0.543669 0.839300i \(-0.317035\pi\)
0.543669 + 0.839300i \(0.317035\pi\)
\(422\) 0 0
\(423\) 2.88979i 0.140507i
\(424\) 0 0
\(425\) −20.2498 6.43899i −0.982258 0.312337i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 4.64002 0.224022
\(430\) 0 0
\(431\) 12.0487 0.580367 0.290184 0.956971i \(-0.406284\pi\)
0.290184 + 0.956971i \(0.406284\pi\)
\(432\) 0 0
\(433\) 27.5904i 1.32591i −0.748660 0.662954i \(-0.769302\pi\)
0.748660 0.662954i \(-0.230698\pi\)
\(434\) 0 0
\(435\) 16.6400 12.1698i 0.797828 0.583499i
\(436\) 0 0
\(437\) 14.0606i 0.672607i
\(438\) 0 0
\(439\) −19.8789 −0.948768 −0.474384 0.880318i \(-0.657329\pi\)
−0.474384 + 0.880318i \(0.657329\pi\)
\(440\) 0 0
\(441\) −7.00000 −0.333333
\(442\) 0 0
\(443\) 17.9007i 0.850488i −0.905079 0.425244i \(-0.860188\pi\)
0.905079 0.425244i \(-0.139812\pi\)
\(444\) 0 0
\(445\) 12.7649 + 17.4537i 0.605115 + 0.827383i
\(446\) 0 0
\(447\) 3.92007i 0.185413i
\(448\) 0 0
\(449\) 17.2682 0.814938 0.407469 0.913219i \(-0.366411\pi\)
0.407469 + 0.913219i \(0.366411\pi\)
\(450\) 0 0
\(451\) 26.3103 1.23890
\(452\) 0 0
\(453\) 21.7796i 1.02329i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 30.4995i 1.42671i −0.700804 0.713354i \(-0.747175\pi\)
0.700804 0.713354i \(-0.252825\pi\)
\(458\) 0 0
\(459\) −4.24977 −0.198362
\(460\) 0 0
\(461\) −2.64002 −0.122958 −0.0614791 0.998108i \(-0.519582\pi\)
−0.0614791 + 0.998108i \(0.519582\pi\)
\(462\) 0 0
\(463\) 16.0000i 0.743583i 0.928316 + 0.371792i \(0.121256\pi\)
−0.928316 + 0.371792i \(0.878744\pi\)
\(464\) 0 0
\(465\) 16.7493 12.2498i 0.776731 0.568070i
\(466\) 0 0
\(467\) 4.78051i 0.221215i −0.993864 0.110608i \(-0.964720\pi\)
0.993864 0.110608i \(-0.0352797\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) −21.5904 −0.994832
\(472\) 0 0
\(473\) 19.7190i 0.906682i
\(474\) 0 0
\(475\) −9.46927 + 29.7796i −0.434480 + 1.36638i
\(476\) 0 0
\(477\) 9.21949i 0.422132i
\(478\) 0 0
\(479\) −9.11021 −0.416256 −0.208128 0.978102i \(-0.566737\pi\)
−0.208128 + 0.978102i \(0.566737\pi\)
\(480\) 0 0
\(481\) 7.28005 0.331942
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 10.8292 7.92007i 0.491731 0.359632i
\(486\) 0 0
\(487\) 26.2791i 1.19082i −0.803422 0.595411i \(-0.796989\pi\)
0.803422 0.595411i \(-0.203011\pi\)
\(488\) 0 0
\(489\) −1.28005 −0.0578857
\(490\) 0 0
\(491\) 5.40115 0.243751 0.121875 0.992545i \(-0.461109\pi\)
0.121875 + 0.992545i \(0.461109\pi\)
\(492\) 0 0
\(493\) 39.1807i 1.76461i
\(494\) 0 0
\(495\) 6.12489 + 8.37466i 0.275293 + 0.376413i
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −22.9092 −1.02556 −0.512778 0.858521i \(-0.671383\pi\)
−0.512778 + 0.858521i \(0.671383\pi\)
\(500\) 0 0
\(501\) −18.8898 −0.843934
\(502\) 0 0
\(503\) 37.3094i 1.66354i −0.555117 0.831772i \(-0.687327\pi\)
0.555117 0.831772i \(-0.312673\pi\)
\(504\) 0 0
\(505\) −20.1698 27.5786i −0.897546 1.22723i
\(506\) 0 0
\(507\) 1.00000i 0.0444116i
\(508\) 0 0
\(509\) 0.700576 0.0310525 0.0155262 0.999879i \(-0.495058\pi\)
0.0155262 + 0.999879i \(0.495058\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 6.24977i 0.275934i
\(514\) 0 0
\(515\) 2.20103 1.60975i 0.0969891 0.0709339i
\(516\) 0 0
\(517\) 13.4087i 0.589715i
\(518\) 0 0
\(519\) 22.9991 1.00955
\(520\) 0 0
\(521\) 13.8401 0.606348 0.303174 0.952935i \(-0.401954\pi\)
0.303174 + 0.952935i \(0.401954\pi\)
\(522\) 0 0
\(523\) 43.6878i 1.91034i 0.296064 + 0.955168i \(0.404326\pi\)
−0.296064 + 0.955168i \(0.595674\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 39.4381i 1.71795i
\(528\) 0 0
\(529\) 17.9385 0.779936
\(530\) 0 0
\(531\) 5.92007 0.256909
\(532\) 0 0
\(533\) 5.67030i 0.245608i
\(534\) 0 0
\(535\) 9.52982 6.96972i 0.412010 0.301327i
\(536\) 0 0
\(537\) 4.49954i 0.194170i
\(538\) 0 0
\(539\) −32.4802 −1.39902
\(540\) 0 0
\(541\) 14.1892 0.610042 0.305021 0.952346i \(-0.401336\pi\)
0.305021 + 0.952346i \(0.401336\pi\)
\(542\) 0 0
\(543\) 21.0596i 0.903755i
\(544\) 0 0
\(545\) 13.4499 + 18.3903i 0.576130 + 0.787752i
\(546\) 0 0
\(547\) 44.3085i 1.89449i 0.320504 + 0.947247i \(0.396148\pi\)
−0.320504 + 0.947247i \(0.603852\pi\)
\(548\) 0 0
\(549\) −0.969724 −0.0413868
\(550\) 0 0
\(551\) 57.6197 2.45468
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 9.60975 + 13.1396i 0.407911 + 0.557743i
\(556\) 0 0
\(557\) 39.2077i 1.66128i −0.556808 0.830641i \(-0.687974\pi\)
0.556808 0.830641i \(-0.312026\pi\)
\(558\) 0 0
\(559\) 4.24977 0.179746
\(560\) 0 0
\(561\) −19.7190 −0.832538
\(562\) 0 0
\(563\) 18.0606i 0.761162i 0.924748 + 0.380581i \(0.124276\pi\)
−0.924748 + 0.380581i \(0.875724\pi\)
\(564\) 0 0
\(565\) 1.85952 1.35998i 0.0782305 0.0572146i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 42.3397 1.77497 0.887486 0.460835i \(-0.152450\pi\)
0.887486 + 0.460835i \(0.152450\pi\)
\(570\) 0 0
\(571\) −39.9301 −1.67102 −0.835510 0.549475i \(-0.814828\pi\)
−0.835510 + 0.549475i \(0.814828\pi\)
\(572\) 0 0
\(573\) 14.4390i 0.603197i
\(574\) 0 0
\(575\) 10.7200 + 3.40871i 0.447053 + 0.142153i
\(576\) 0 0
\(577\) 21.3406i 0.888421i 0.895923 + 0.444210i \(0.146516\pi\)
−0.895923 + 0.444210i \(0.853484\pi\)
\(578\) 0 0
\(579\) −21.7190 −0.902612
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 42.7787i 1.77171i
\(584\) 0 0
\(585\) −1.80487 + 1.32001i −0.0746223 + 0.0545758i
\(586\) 0 0
\(587\) 32.7905i 1.35341i 0.736255 + 0.676704i \(0.236592\pi\)
−0.736255 + 0.676704i \(0.763408\pi\)
\(588\) 0 0
\(589\) 57.9982 2.38977
\(590\) 0 0
\(591\) −13.0109 −0.535197
\(592\) 0 0
\(593\) 6.66938i 0.273879i −0.990579 0.136939i \(-0.956273\pi\)
0.990579 0.136939i \(-0.0437265\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 18.6888i 0.764880i
\(598\) 0 0
\(599\) −1.40115 −0.0572495 −0.0286247 0.999590i \(-0.509113\pi\)
−0.0286247 + 0.999590i \(0.509113\pi\)
\(600\) 0 0
\(601\) 0.349078 0.0142392 0.00711959 0.999975i \(-0.497734\pi\)
0.00711959 + 0.999975i \(0.497734\pi\)
\(602\) 0 0
\(603\) 1.93945i 0.0789804i
\(604\) 0 0
\(605\) 13.8995 + 19.0050i 0.565094 + 0.772663i
\(606\) 0 0
\(607\) 27.2800i 1.10726i −0.832762 0.553631i \(-0.813242\pi\)
0.832762 0.553631i \(-0.186758\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 2.88979 0.116909
\(612\) 0 0
\(613\) 9.34060i 0.377263i −0.982048 0.188632i \(-0.939595\pi\)
0.982048 0.188632i \(-0.0604052\pi\)
\(614\) 0 0
\(615\) −10.2342 + 7.48486i −0.412682 + 0.301819i
\(616\) 0 0
\(617\) 34.6694i 1.39574i −0.716226 0.697868i \(-0.754132\pi\)
0.716226 0.697868i \(-0.245868\pi\)
\(618\) 0 0
\(619\) −19.6509 −0.789837 −0.394919 0.918716i \(-0.629227\pi\)
−0.394919 + 0.918716i \(0.629227\pi\)
\(620\) 0 0
\(621\) 2.24977 0.0902802
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −20.4087 14.4390i −0.816349 0.577560i
\(626\) 0 0
\(627\) 28.9991i 1.15811i
\(628\) 0 0
\(629\) −30.9385 −1.23360
\(630\) 0 0
\(631\) −43.3993 −1.72770 −0.863850 0.503749i \(-0.831953\pi\)
−0.863850 + 0.503749i \(0.831953\pi\)
\(632\) 0 0
\(633\) 8.24977i 0.327899i
\(634\) 0 0
\(635\) 19.9083 14.5601i 0.790035 0.577800i
\(636\) 0 0
\(637\) 7.00000i 0.277350i
\(638\) 0 0
\(639\) −5.60975 −0.221918
\(640\) 0 0
\(641\) −4.56009 −0.180113 −0.0900564 0.995937i \(-0.528705\pi\)
−0.0900564 + 0.995937i \(0.528705\pi\)
\(642\) 0 0
\(643\) 8.62065i 0.339965i 0.985447 + 0.169983i \(0.0543711\pi\)
−0.985447 + 0.169983i \(0.945629\pi\)
\(644\) 0 0
\(645\) 5.60975 + 7.67030i 0.220884 + 0.302018i
\(646\) 0 0
\(647\) 7.40871i 0.291267i −0.989339 0.145633i \(-0.953478\pi\)
0.989339 0.145633i \(-0.0465220\pi\)
\(648\) 0 0
\(649\) 27.4693 1.07826
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 2.00000i 0.0782660i 0.999234 + 0.0391330i \(0.0124596\pi\)
−0.999234 + 0.0391330i \(0.987540\pi\)
\(654\) 0 0
\(655\) 9.52982 + 13.0303i 0.372361 + 0.509135i
\(656\) 0 0
\(657\) 12.5601i 0.490016i
\(658\) 0 0
\(659\) −40.3784 −1.57292 −0.786460 0.617641i \(-0.788089\pi\)
−0.786460 + 0.617641i \(0.788089\pi\)
\(660\) 0 0
\(661\) 39.0284 1.51803 0.759015 0.651073i \(-0.225681\pi\)
0.759015 + 0.651073i \(0.225681\pi\)
\(662\) 0 0
\(663\) 4.24977i 0.165047i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 20.7418i 0.803124i
\(668\) 0 0
\(669\) −9.77959 −0.378101
\(670\) 0 0
\(671\) −4.49954 −0.173703
\(672\) 0 0
\(673\) 11.1807i 0.430986i 0.976505 + 0.215493i \(0.0691358\pi\)
−0.976505 + 0.215493i \(0.930864\pi\)
\(674\) 0 0
\(675\) −4.76491 1.51514i −0.183401 0.0583177i
\(676\) 0 0
\(677\) 33.3406i 1.28138i −0.767798 0.640692i \(-0.778648\pi\)
0.767798 0.640692i \(-0.221352\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 25.4499 0.975242
\(682\) 0 0
\(683\) 31.8889i 1.22019i 0.792327 + 0.610097i \(0.208870\pi\)
−0.792327 + 0.610097i \(0.791130\pi\)
\(684\) 0 0
\(685\) −14.6362 + 10.7044i −0.559222 + 0.408992i
\(686\) 0 0
\(687\) 8.24977i 0.314748i
\(688\) 0 0
\(689\) 9.21949 0.351235
\(690\) 0 0
\(691\) 39.0303 1.48478 0.742391 0.669967i \(-0.233692\pi\)
0.742391 + 0.669967i \(0.233692\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 25.1589 + 34.4002i 0.954333 + 1.30488i
\(696\) 0 0
\(697\) 24.0975i 0.912757i
\(698\) 0 0
\(699\) −11.5904 −0.438388
\(700\) 0 0
\(701\) −44.5601 −1.68301 −0.841506 0.540248i \(-0.818330\pi\)
−0.841506 + 0.540248i \(0.818330\pi\)
\(702\) 0 0
\(703\) 45.4986i 1.71601i
\(704\) 0 0
\(705\) 3.81456 + 5.21571i 0.143665 + 0.196435i
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −23.4305 −0.879951 −0.439976 0.898010i \(-0.645013\pi\)
−0.439976 + 0.898010i \(0.645013\pi\)
\(710\) 0 0
\(711\) 12.2498 0.459402
\(712\) 0 0
\(713\) 20.8780i 0.781886i
\(714\) 0 0
\(715\) −8.37466 + 6.12489i −0.313194 + 0.229058i
\(716\) 0 0
\(717\) 6.88979i 0.257304i
\(718\) 0 0
\(719\) 15.0984 0.563075 0.281537 0.959550i \(-0.409156\pi\)
0.281537 + 0.959550i \(0.409156\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 21.2195i 0.789162i
\(724\) 0 0
\(725\) −13.9688 + 43.9301i −0.518788 + 1.63152i
\(726\) 0 0
\(727\) 3.40871i 0.126422i 0.998000 + 0.0632111i \(0.0201341\pi\)
−0.998000 + 0.0632111i \(0.979866\pi\)
\(728\) 0 0
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) −18.0606 −0.667994
\(732\) 0 0
\(733\) 12.4390i 0.459445i −0.973256 0.229722i \(-0.926218\pi\)
0.973256 0.229722i \(-0.0737818\pi\)
\(734\) 0 0
\(735\) 12.6341 9.24008i 0.466016 0.340826i
\(736\) 0 0
\(737\) 8.99908i 0.331485i
\(738\) 0 0
\(739\) 13.4693 0.495475 0.247737 0.968827i \(-0.420313\pi\)
0.247737 + 0.968827i \(0.420313\pi\)
\(740\) 0 0
\(741\) −6.24977 −0.229591
\(742\) 0 0
\(743\) 45.4111i 1.66597i 0.553293 + 0.832986i \(0.313371\pi\)
−0.553293 + 0.832986i \(0.686629\pi\)
\(744\) 0 0
\(745\) −5.17454 7.07523i −0.189580 0.259216i
\(746\) 0 0
\(747\) 3.67030i 0.134289i
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 1.56101 0.0569621 0.0284810 0.999594i \(-0.490933\pi\)
0.0284810 + 0.999594i \(0.490933\pi\)
\(752\) 0 0
\(753\) 24.3397i 0.886987i
\(754\) 0 0
\(755\) 28.7493 + 39.3094i 1.04629 + 1.43062i
\(756\) 0 0
\(757\) 38.8780i 1.41304i 0.707691 + 0.706522i \(0.249737\pi\)
−0.707691 + 0.706522i \(0.750263\pi\)
\(758\) 0 0
\(759\) 10.4390 0.378911
\(760\) 0 0
\(761\) 5.42809 0.196768 0.0983841 0.995149i \(-0.468633\pi\)
0.0983841 + 0.995149i \(0.468633\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 7.67030 5.60975i 0.277320 0.202821i
\(766\) 0 0
\(767\) 5.92007i 0.213761i
\(768\) 0 0
\(769\) −16.4390 −0.592805 −0.296403 0.955063i \(-0.595787\pi\)
−0.296403 + 0.955063i \(0.595787\pi\)
\(770\) 0 0
\(771\) −23.4693 −0.845225
\(772\) 0 0
\(773\) 42.3884i 1.52461i −0.647221 0.762303i \(-0.724069\pi\)
0.647221 0.762303i \(-0.275931\pi\)
\(774\) 0 0
\(775\) −14.0606 + 44.2186i −0.505070 + 1.58838i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −35.4381 −1.26970
\(780\) 0 0
\(781\) −26.0294 −0.931404
\(782\) 0 0
\(783\) 9.21949i 0.329478i
\(784\) 0 0
\(785\) 38.9679 28.4995i 1.39082 1.01719i
\(786\) 0 0
\(787\) 15.5592i 0.554625i −0.960780 0.277312i \(-0.910556\pi\)
0.960780 0.277312i \(-0.0894437\pi\)
\(788\) 0 0
\(789\) 22.6282 0.805586
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0.969724i 0.0344359i
\(794\) 0 0
\(795\) 12.1698 + 16.6400i 0.431620 + 0.590161i
\(796\) 0 0
\(797\) 36.4002i 1.28936i 0.764452 + 0.644681i \(0.223010\pi\)
−0.764452 + 0.644681i \(0.776990\pi\)
\(798\) 0 0
\(799\) −12.2810 −0.434469
\(800\) 0 0
\(801\) −9.67030 −0.341683
\(802\) 0 0
\(803\) 58.2791i 2.05663i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 12.2791i 0.432246i
\(808\) 0 0
\(809\) −5.59793 −0.196813 −0.0984064 0.995146i \(-0.531375\pi\)
−0.0984064 + 0.995146i \(0.531375\pi\)
\(810\) 0 0
\(811\) 28.1892 0.989857 0.494929 0.868934i \(-0.335194\pi\)
0.494929 + 0.868934i \(0.335194\pi\)
\(812\) 0 0
\(813\) 22.9385i 0.804489i
\(814\) 0 0
\(815\) 2.31032 1.68968i 0.0809271 0.0591868i
\(816\) 0 0
\(817\) 26.5601i 0.929220i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 33.8208 1.18035 0.590176 0.807274i \(-0.299058\pi\)
0.590176 + 0.807274i \(0.299058\pi\)
\(822\) 0 0
\(823\) 2.53830i 0.0884795i 0.999021 + 0.0442397i \(0.0140865\pi\)
−0.999021 + 0.0442397i \(0.985913\pi\)
\(824\) 0 0
\(825\) −22.1093 7.03028i −0.769747 0.244763i
\(826\) 0 0
\(827\) 12.6694i 0.440558i −0.975437 0.220279i \(-0.929303\pi\)
0.975437 0.220279i \(-0.0706967\pi\)
\(828\) 0 0
\(829\) 37.4693 1.30136 0.650681 0.759351i \(-0.274484\pi\)
0.650681 + 0.759351i \(0.274484\pi\)
\(830\) 0 0
\(831\) 11.5298 0.399965
\(832\) 0 0
\(833\) 29.7484i 1.03072i
\(834\) 0 0
\(835\) 34.0937 24.9348i 1.17986 0.862903i
\(836\) 0 0
\(837\) 9.28005i 0.320765i
\(838\) 0 0
\(839\) 19.3893 0.669394 0.334697 0.942326i \(-0.391366\pi\)
0.334697 + 0.942326i \(0.391366\pi\)
\(840\) 0 0
\(841\) 55.9991 1.93100
\(842\) 0 0
\(843\) 11.3288i 0.390184i
\(844\) 0 0
\(845\) −1.32001 1.80487i −0.0454098 0.0620895i
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −20.9991 −0.720687
\(850\) 0 0
\(851\) 16.3784 0.561446
\(852\) 0 0
\(853\) 7.77959i 0.266368i 0.991091 + 0.133184i \(0.0425201\pi\)
−0.991091 + 0.133184i \(0.957480\pi\)
\(854\) 0 0
\(855\) −8.24977 11.2800i −0.282136 0.385769i
\(856\) 0 0
\(857\) 26.9697i 0.921268i −0.887590 0.460634i \(-0.847622\pi\)
0.887590 0.460634i \(-0.152378\pi\)
\(858\) 0 0
\(859\) −5.93189 −0.202393 −0.101197 0.994866i \(-0.532267\pi\)
−0.101197 + 0.994866i \(0.532267\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 33.3288i 1.13452i −0.823537 0.567262i \(-0.808002\pi\)
0.823537 0.567262i \(-0.191998\pi\)
\(864\) 0 0
\(865\) −41.5104 + 30.3591i −1.41140 + 1.03224i
\(866\) 0 0
\(867\) 1.06055i 0.0360182i
\(868\) 0 0
\(869\) 56.8392 1.92814
\(870\) 0 0
\(871\) −1.93945 −0.0657157
\(872\) 0 0
\(873\) 6.00000i 0.203069i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 19.1589i 0.646952i 0.946236 + 0.323476i \(0.104851\pi\)
−0.946236 + 0.323476i \(0.895149\pi\)
\(878\) 0 0
\(879\) 4.26915 0.143995
\(880\) 0 0
\(881\) 53.8989 1.81590 0.907949 0.419080i \(-0.137647\pi\)
0.907949 + 0.419080i \(0.137647\pi\)
\(882\) 0 0
\(883\) 22.9385i 0.771943i 0.922511 + 0.385972i \(0.126134\pi\)
−0.922511 + 0.385972i \(0.873866\pi\)
\(884\) 0 0
\(885\) −10.6850 + 7.81456i −0.359172 + 0.262684i
\(886\) 0 0
\(887\) 47.5298i 1.59590i 0.602727 + 0.797948i \(0.294081\pi\)
−0.602727 + 0.797948i \(0.705919\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −4.64002 −0.155447
\(892\) 0 0
\(893\) 18.0606i 0.604373i
\(894\) 0 0
\(895\) 5.93945 + 8.12110i 0.198534 + 0.271459i
\(896\) 0 0
\(897\) 2.24977i 0.0751177i
\(898\) 0 0
\(899\) 85.5573 2.85350
\(900\) 0 0
\(901\) −39.1807 −1.30530
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −27.7990 38.0100i −0.924069 1.26349i
\(906\) 0 0
\(907\) 20.6206i 0.684697i 0.939573 + 0.342349i \(0.111222\pi\)
−0.939573 + 0.342349i \(0.888778\pi\)
\(908\) 0 0
\(909\) 15.2800 0.506807
\(910\) 0 0
\(911\) −41.2413 −1.36638 −0.683192 0.730238i \(-0.739409\pi\)
−0.683192 + 0.730238i \(0.739409\pi\)
\(912\) 0 0
\(913\) 17.0303i 0.563620i
\(914\) 0 0
\(915\) 1.75023 1.28005i 0.0578608 0.0423170i
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −37.2489 −1.22873 −0.614363 0.789023i \(-0.710587\pi\)
−0.614363 + 0.789023i \(0.710587\pi\)
\(920\) 0 0
\(921\) −20.0000 −0.659022
\(922\) 0 0
\(923\) 5.60975i 0.184647i
\(924\) 0 0
\(925\) −34.6888 11.0303i −1.14056 0.362673i
\(926\) 0 0
\(927\) 1.21949i 0.0400535i
\(928\) 0 0
\(929\) 59.0091 1.93602 0.968012 0.250903i \(-0.0807274\pi\)
0.968012 + 0.250903i \(0.0807274\pi\)
\(930\) 0 0
\(931\) 43.7484 1.43380
\(932\) 0 0
\(933\) 3.71904i 0.121756i
\(934\) 0 0
\(935\) 35.5904 26.0294i 1.16393 0.851251i
\(936\) 0 0
\(937\) 56.9679i 1.86106i 0.366216 + 0.930530i \(0.380653\pi\)
−0.366216 + 0.930530i \(0.619347\pi\)
\(938\) 0 0
\(939\) −1.52982 −0.0499237
\(940\) 0 0
\(941\) 2.35906 0.0769032 0.0384516 0.999260i \(-0.487757\pi\)
0.0384516 + 0.999260i \(0.487757\pi\)
\(942\) 0 0
\(943\) 12.7569i 0.415421i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 43.4481i 1.41187i −0.708276 0.705936i \(-0.750527\pi\)
0.708276 0.705936i \(-0.249473\pi\)
\(948\) 0 0
\(949\) 12.5601 0.407718
\(950\) 0 0
\(951\) 16.1093 0.522379
\(952\) 0 0
\(953\) 23.5904i 0.764167i 0.924128 + 0.382084i \(0.124793\pi\)
−0.924128 + 0.382084i \(0.875207\pi\)
\(954\) 0 0
\(955\) −19.0596 26.0606i −0.616755 0.843300i
\(956\) 0 0
\(957\) 42.7787i 1.38284i
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 55.1193 1.77804
\(962\) 0 0
\(963\) 5.28005i 0.170147i
\(964\) 0 0
\(965\) 39.2001 28.6694i 1.26190 0.922900i
\(966\) 0 0
\(967\) 31.5979i 1.01612i −0.861321 0.508060i \(-0.830363\pi\)
0.861321 0.508060i \(-0.169637\pi\)
\(968\) 0 0
\(969\) 26.5601 0.853233
\(970\) 0 0
\(971\) −0.499542 −0.0160311 −0.00801553 0.999968i \(-0.502551\pi\)
−0.00801553 + 0.999968i \(0.502551\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 1.51514 4.76491i 0.0485233 0.152599i
\(976\) 0 0
\(977\) 18.7081i 0.598526i −0.954171 0.299263i \(-0.903259\pi\)
0.954171 0.299263i \(-0.0967409\pi\)
\(978\) 0 0
\(979\) −44.8704 −1.43406
\(980\) 0 0
\(981\) −10.1892 −0.325317
\(982\) 0 0
\(983\) 3.83016i 0.122163i −0.998133 0.0610815i \(-0.980545\pi\)
0.998133 0.0610815i \(-0.0194550\pi\)
\(984\) 0 0
\(985\) 23.4830 17.1745i 0.748232 0.547226i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 9.56101 0.304022
\(990\) 0 0
\(991\) 6.43899 0.204541 0.102271 0.994757i \(-0.467389\pi\)
0.102271 + 0.994757i \(0.467389\pi\)
\(992\) 0 0
\(993\) 15.5298i 0.492824i
\(994\) 0 0
\(995\) −24.6694 33.7309i −0.782072 1.06934i
\(996\) 0 0
\(997\) 12.1505i 0.384809i 0.981316 + 0.192405i \(0.0616286\pi\)
−0.981316 + 0.192405i \(0.938371\pi\)
\(998\) 0 0
\(999\) −7.28005 −0.230330
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 1560.2.l.c.1249.3 6
3.2 odd 2 4680.2.l.e.2809.1 6
4.3 odd 2 3120.2.l.m.1249.6 6
5.2 odd 4 7800.2.a.bj.1.1 3
5.3 odd 4 7800.2.a.bp.1.1 3
5.4 even 2 inner 1560.2.l.c.1249.6 yes 6
15.14 odd 2 4680.2.l.e.2809.2 6
20.19 odd 2 3120.2.l.m.1249.3 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1560.2.l.c.1249.3 6 1.1 even 1 trivial
1560.2.l.c.1249.6 yes 6 5.4 even 2 inner
3120.2.l.m.1249.3 6 20.19 odd 2
3120.2.l.m.1249.6 6 4.3 odd 2
4680.2.l.e.2809.1 6 3.2 odd 2
4680.2.l.e.2809.2 6 15.14 odd 2
7800.2.a.bj.1.1 3 5.2 odd 4
7800.2.a.bp.1.1 3 5.3 odd 4