Properties

Label 3040.2.a.l.1.3
Level $3040$
Weight $2$
Character 3040.1
Self dual yes
Analytic conductor $24.275$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3040,2,Mod(1,3040)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3040, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3040.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3040 = 2^{5} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3040.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(24.2745222145\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.148.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 3x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-1.48119\) of defining polynomial
Character \(\chi\) \(=\) 3040.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.67513 q^{3} +1.00000 q^{5} -4.15633 q^{7} -0.193937 q^{9} +O(q^{10})\) \(q+1.67513 q^{3} +1.00000 q^{5} -4.15633 q^{7} -0.193937 q^{9} +0.806063 q^{11} +0.481194 q^{13} +1.67513 q^{15} -3.61213 q^{17} +1.00000 q^{19} -6.96239 q^{21} +5.76845 q^{23} +1.00000 q^{25} -5.35026 q^{27} -8.57452 q^{29} -3.61213 q^{31} +1.35026 q^{33} -4.15633 q^{35} -9.18172 q^{37} +0.806063 q^{39} +0.312650 q^{41} +3.19394 q^{43} -0.193937 q^{45} +5.76845 q^{47} +10.2750 q^{49} -6.05079 q^{51} -8.21933 q^{53} +0.806063 q^{55} +1.67513 q^{57} +8.88717 q^{59} -5.58181 q^{61} +0.806063 q^{63} +0.481194 q^{65} -9.59991 q^{67} +9.66291 q^{69} -16.2374 q^{71} -2.12601 q^{73} +1.67513 q^{75} -3.35026 q^{77} -16.8872 q^{79} -8.38058 q^{81} +5.89446 q^{83} -3.61213 q^{85} -14.3634 q^{87} -11.2750 q^{89} -2.00000 q^{91} -6.05079 q^{93} +1.00000 q^{95} +17.4944 q^{97} -0.156325 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{5} - 2 q^{7} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 3 q^{5} - 2 q^{7} - q^{9} + 2 q^{11} - 4 q^{13} - 10 q^{17} + 3 q^{19} - 10 q^{21} + 6 q^{23} + 3 q^{25} - 6 q^{27} - 14 q^{29} - 10 q^{31} - 6 q^{33} - 2 q^{35} - 2 q^{37} + 2 q^{39} - 20 q^{41} + 10 q^{43} - q^{45} + 6 q^{47} - q^{49} + 12 q^{51} - 10 q^{53} + 2 q^{55} - 6 q^{59} - 18 q^{61} + 2 q^{63} - 4 q^{65} - 2 q^{67} - 2 q^{69} - 6 q^{71} + 2 q^{73} - 18 q^{79} - 13 q^{81} - 2 q^{83} - 10 q^{85} + 8 q^{87} - 2 q^{89} - 6 q^{91} + 12 q^{93} + 3 q^{95} + 6 q^{97} + 10 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.67513 0.967137 0.483569 0.875306i \(-0.339340\pi\)
0.483569 + 0.875306i \(0.339340\pi\)
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) −4.15633 −1.57094 −0.785472 0.618898i \(-0.787580\pi\)
−0.785472 + 0.618898i \(0.787580\pi\)
\(8\) 0 0
\(9\) −0.193937 −0.0646455
\(10\) 0 0
\(11\) 0.806063 0.243037 0.121519 0.992589i \(-0.461224\pi\)
0.121519 + 0.992589i \(0.461224\pi\)
\(12\) 0 0
\(13\) 0.481194 0.133459 0.0667296 0.997771i \(-0.478743\pi\)
0.0667296 + 0.997771i \(0.478743\pi\)
\(14\) 0 0
\(15\) 1.67513 0.432517
\(16\) 0 0
\(17\) −3.61213 −0.876069 −0.438035 0.898958i \(-0.644325\pi\)
−0.438035 + 0.898958i \(0.644325\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) −6.96239 −1.51932
\(22\) 0 0
\(23\) 5.76845 1.20281 0.601403 0.798946i \(-0.294609\pi\)
0.601403 + 0.798946i \(0.294609\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −5.35026 −1.02966
\(28\) 0 0
\(29\) −8.57452 −1.59225 −0.796124 0.605134i \(-0.793120\pi\)
−0.796124 + 0.605134i \(0.793120\pi\)
\(30\) 0 0
\(31\) −3.61213 −0.648757 −0.324379 0.945927i \(-0.605155\pi\)
−0.324379 + 0.945927i \(0.605155\pi\)
\(32\) 0 0
\(33\) 1.35026 0.235050
\(34\) 0 0
\(35\) −4.15633 −0.702547
\(36\) 0 0
\(37\) −9.18172 −1.50947 −0.754733 0.656033i \(-0.772233\pi\)
−0.754733 + 0.656033i \(0.772233\pi\)
\(38\) 0 0
\(39\) 0.806063 0.129073
\(40\) 0 0
\(41\) 0.312650 0.0488278 0.0244139 0.999702i \(-0.492228\pi\)
0.0244139 + 0.999702i \(0.492228\pi\)
\(42\) 0 0
\(43\) 3.19394 0.487071 0.243535 0.969892i \(-0.421693\pi\)
0.243535 + 0.969892i \(0.421693\pi\)
\(44\) 0 0
\(45\) −0.193937 −0.0289104
\(46\) 0 0
\(47\) 5.76845 0.841415 0.420708 0.907196i \(-0.361782\pi\)
0.420708 + 0.907196i \(0.361782\pi\)
\(48\) 0 0
\(49\) 10.2750 1.46786
\(50\) 0 0
\(51\) −6.05079 −0.847279
\(52\) 0 0
\(53\) −8.21933 −1.12901 −0.564506 0.825429i \(-0.690933\pi\)
−0.564506 + 0.825429i \(0.690933\pi\)
\(54\) 0 0
\(55\) 0.806063 0.108690
\(56\) 0 0
\(57\) 1.67513 0.221877
\(58\) 0 0
\(59\) 8.88717 1.15701 0.578505 0.815679i \(-0.303636\pi\)
0.578505 + 0.815679i \(0.303636\pi\)
\(60\) 0 0
\(61\) −5.58181 −0.714677 −0.357339 0.933975i \(-0.616316\pi\)
−0.357339 + 0.933975i \(0.616316\pi\)
\(62\) 0 0
\(63\) 0.806063 0.101554
\(64\) 0 0
\(65\) 0.481194 0.0596848
\(66\) 0 0
\(67\) −9.59991 −1.17282 −0.586408 0.810016i \(-0.699458\pi\)
−0.586408 + 0.810016i \(0.699458\pi\)
\(68\) 0 0
\(69\) 9.66291 1.16328
\(70\) 0 0
\(71\) −16.2374 −1.92703 −0.963514 0.267658i \(-0.913750\pi\)
−0.963514 + 0.267658i \(0.913750\pi\)
\(72\) 0 0
\(73\) −2.12601 −0.248830 −0.124415 0.992230i \(-0.539705\pi\)
−0.124415 + 0.992230i \(0.539705\pi\)
\(74\) 0 0
\(75\) 1.67513 0.193427
\(76\) 0 0
\(77\) −3.35026 −0.381798
\(78\) 0 0
\(79\) −16.8872 −1.89995 −0.949977 0.312319i \(-0.898894\pi\)
−0.949977 + 0.312319i \(0.898894\pi\)
\(80\) 0 0
\(81\) −8.38058 −0.931175
\(82\) 0 0
\(83\) 5.89446 0.647001 0.323501 0.946228i \(-0.395140\pi\)
0.323501 + 0.946228i \(0.395140\pi\)
\(84\) 0 0
\(85\) −3.61213 −0.391790
\(86\) 0 0
\(87\) −14.3634 −1.53992
\(88\) 0 0
\(89\) −11.2750 −1.19515 −0.597576 0.801812i \(-0.703869\pi\)
−0.597576 + 0.801812i \(0.703869\pi\)
\(90\) 0 0
\(91\) −2.00000 −0.209657
\(92\) 0 0
\(93\) −6.05079 −0.627437
\(94\) 0 0
\(95\) 1.00000 0.102598
\(96\) 0 0
\(97\) 17.4944 1.77628 0.888142 0.459569i \(-0.151996\pi\)
0.888142 + 0.459569i \(0.151996\pi\)
\(98\) 0 0
\(99\) −0.156325 −0.0157113
\(100\) 0 0
\(101\) 7.69323 0.765505 0.382752 0.923851i \(-0.374976\pi\)
0.382752 + 0.923851i \(0.374976\pi\)
\(102\) 0 0
\(103\) 0.950170 0.0936230 0.0468115 0.998904i \(-0.485094\pi\)
0.0468115 + 0.998904i \(0.485094\pi\)
\(104\) 0 0
\(105\) −6.96239 −0.679460
\(106\) 0 0
\(107\) −13.7259 −1.32693 −0.663467 0.748205i \(-0.730916\pi\)
−0.663467 + 0.748205i \(0.730916\pi\)
\(108\) 0 0
\(109\) −19.9756 −1.91331 −0.956656 0.291221i \(-0.905939\pi\)
−0.956656 + 0.291221i \(0.905939\pi\)
\(110\) 0 0
\(111\) −15.3806 −1.45986
\(112\) 0 0
\(113\) 6.53198 0.614477 0.307238 0.951633i \(-0.400595\pi\)
0.307238 + 0.951633i \(0.400595\pi\)
\(114\) 0 0
\(115\) 5.76845 0.537911
\(116\) 0 0
\(117\) −0.0933212 −0.00862755
\(118\) 0 0
\(119\) 15.0132 1.37626
\(120\) 0 0
\(121\) −10.3503 −0.940933
\(122\) 0 0
\(123\) 0.523730 0.0472232
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 6.51151 0.577803 0.288902 0.957359i \(-0.406710\pi\)
0.288902 + 0.957359i \(0.406710\pi\)
\(128\) 0 0
\(129\) 5.35026 0.471064
\(130\) 0 0
\(131\) 16.4387 1.43625 0.718126 0.695913i \(-0.245000\pi\)
0.718126 + 0.695913i \(0.245000\pi\)
\(132\) 0 0
\(133\) −4.15633 −0.360399
\(134\) 0 0
\(135\) −5.35026 −0.460477
\(136\) 0 0
\(137\) −10.6253 −0.907781 −0.453890 0.891058i \(-0.649964\pi\)
−0.453890 + 0.891058i \(0.649964\pi\)
\(138\) 0 0
\(139\) 5.26916 0.446924 0.223462 0.974713i \(-0.428264\pi\)
0.223462 + 0.974713i \(0.428264\pi\)
\(140\) 0 0
\(141\) 9.66291 0.813764
\(142\) 0 0
\(143\) 0.387873 0.0324356
\(144\) 0 0
\(145\) −8.57452 −0.712075
\(146\) 0 0
\(147\) 17.2120 1.41962
\(148\) 0 0
\(149\) −10.7308 −0.879105 −0.439552 0.898217i \(-0.644863\pi\)
−0.439552 + 0.898217i \(0.644863\pi\)
\(150\) 0 0
\(151\) 7.66291 0.623599 0.311799 0.950148i \(-0.399068\pi\)
0.311799 + 0.950148i \(0.399068\pi\)
\(152\) 0 0
\(153\) 0.700523 0.0566340
\(154\) 0 0
\(155\) −3.61213 −0.290133
\(156\) 0 0
\(157\) 13.8496 1.10531 0.552657 0.833409i \(-0.313614\pi\)
0.552657 + 0.833409i \(0.313614\pi\)
\(158\) 0 0
\(159\) −13.7685 −1.09191
\(160\) 0 0
\(161\) −23.9756 −1.88954
\(162\) 0 0
\(163\) 9.05808 0.709484 0.354742 0.934964i \(-0.384569\pi\)
0.354742 + 0.934964i \(0.384569\pi\)
\(164\) 0 0
\(165\) 1.35026 0.105118
\(166\) 0 0
\(167\) −0.586734 −0.0454029 −0.0227014 0.999742i \(-0.507227\pi\)
−0.0227014 + 0.999742i \(0.507227\pi\)
\(168\) 0 0
\(169\) −12.7685 −0.982189
\(170\) 0 0
\(171\) −0.193937 −0.0148307
\(172\) 0 0
\(173\) 12.4568 0.947070 0.473535 0.880775i \(-0.342978\pi\)
0.473535 + 0.880775i \(0.342978\pi\)
\(174\) 0 0
\(175\) −4.15633 −0.314189
\(176\) 0 0
\(177\) 14.8872 1.11899
\(178\) 0 0
\(179\) 15.2144 1.13718 0.568589 0.822622i \(-0.307489\pi\)
0.568589 + 0.822622i \(0.307489\pi\)
\(180\) 0 0
\(181\) −25.8496 −1.92138 −0.960691 0.277619i \(-0.910455\pi\)
−0.960691 + 0.277619i \(0.910455\pi\)
\(182\) 0 0
\(183\) −9.35026 −0.691191
\(184\) 0 0
\(185\) −9.18172 −0.675053
\(186\) 0 0
\(187\) −2.91160 −0.212918
\(188\) 0 0
\(189\) 22.2374 1.61753
\(190\) 0 0
\(191\) 27.3865 1.98161 0.990807 0.135281i \(-0.0431937\pi\)
0.990807 + 0.135281i \(0.0431937\pi\)
\(192\) 0 0
\(193\) 1.64481 0.118396 0.0591981 0.998246i \(-0.481146\pi\)
0.0591981 + 0.998246i \(0.481146\pi\)
\(194\) 0 0
\(195\) 0.806063 0.0577234
\(196\) 0 0
\(197\) 0.261865 0.0186571 0.00932856 0.999956i \(-0.497031\pi\)
0.00932856 + 0.999956i \(0.497031\pi\)
\(198\) 0 0
\(199\) 1.92478 0.136444 0.0682219 0.997670i \(-0.478267\pi\)
0.0682219 + 0.997670i \(0.478267\pi\)
\(200\) 0 0
\(201\) −16.0811 −1.13427
\(202\) 0 0
\(203\) 35.6385 2.50133
\(204\) 0 0
\(205\) 0.312650 0.0218364
\(206\) 0 0
\(207\) −1.11871 −0.0777560
\(208\) 0 0
\(209\) 0.806063 0.0557566
\(210\) 0 0
\(211\) −9.58769 −0.660044 −0.330022 0.943973i \(-0.607056\pi\)
−0.330022 + 0.943973i \(0.607056\pi\)
\(212\) 0 0
\(213\) −27.1998 −1.86370
\(214\) 0 0
\(215\) 3.19394 0.217825
\(216\) 0 0
\(217\) 15.0132 1.01916
\(218\) 0 0
\(219\) −3.56134 −0.240653
\(220\) 0 0
\(221\) −1.73813 −0.116920
\(222\) 0 0
\(223\) −0.274084 −0.0183540 −0.00917701 0.999958i \(-0.502921\pi\)
−0.00917701 + 0.999958i \(0.502921\pi\)
\(224\) 0 0
\(225\) −0.193937 −0.0129291
\(226\) 0 0
\(227\) −12.1138 −0.804020 −0.402010 0.915635i \(-0.631688\pi\)
−0.402010 + 0.915635i \(0.631688\pi\)
\(228\) 0 0
\(229\) 17.5574 1.16022 0.580112 0.814537i \(-0.303009\pi\)
0.580112 + 0.814537i \(0.303009\pi\)
\(230\) 0 0
\(231\) −5.61213 −0.369251
\(232\) 0 0
\(233\) −10.1866 −0.667349 −0.333675 0.942688i \(-0.608289\pi\)
−0.333675 + 0.942688i \(0.608289\pi\)
\(234\) 0 0
\(235\) 5.76845 0.376292
\(236\) 0 0
\(237\) −28.2882 −1.83752
\(238\) 0 0
\(239\) 25.9003 1.67535 0.837677 0.546166i \(-0.183913\pi\)
0.837677 + 0.546166i \(0.183913\pi\)
\(240\) 0 0
\(241\) 3.84955 0.247972 0.123986 0.992284i \(-0.460432\pi\)
0.123986 + 0.992284i \(0.460432\pi\)
\(242\) 0 0
\(243\) 2.01222 0.129084
\(244\) 0 0
\(245\) 10.2750 0.656448
\(246\) 0 0
\(247\) 0.481194 0.0306177
\(248\) 0 0
\(249\) 9.87399 0.625739
\(250\) 0 0
\(251\) 0.312650 0.0197343 0.00986716 0.999951i \(-0.496859\pi\)
0.00986716 + 0.999951i \(0.496859\pi\)
\(252\) 0 0
\(253\) 4.64974 0.292327
\(254\) 0 0
\(255\) −6.05079 −0.378915
\(256\) 0 0
\(257\) 14.6678 0.914955 0.457477 0.889221i \(-0.348753\pi\)
0.457477 + 0.889221i \(0.348753\pi\)
\(258\) 0 0
\(259\) 38.1622 2.37128
\(260\) 0 0
\(261\) 1.66291 0.102932
\(262\) 0 0
\(263\) −7.06793 −0.435827 −0.217914 0.975968i \(-0.569925\pi\)
−0.217914 + 0.975968i \(0.569925\pi\)
\(264\) 0 0
\(265\) −8.21933 −0.504909
\(266\) 0 0
\(267\) −18.8872 −1.15588
\(268\) 0 0
\(269\) 24.2882 1.48088 0.740439 0.672123i \(-0.234618\pi\)
0.740439 + 0.672123i \(0.234618\pi\)
\(270\) 0 0
\(271\) −3.81924 −0.232002 −0.116001 0.993249i \(-0.537008\pi\)
−0.116001 + 0.993249i \(0.537008\pi\)
\(272\) 0 0
\(273\) −3.35026 −0.202767
\(274\) 0 0
\(275\) 0.806063 0.0486075
\(276\) 0 0
\(277\) 8.70052 0.522764 0.261382 0.965235i \(-0.415822\pi\)
0.261382 + 0.965235i \(0.415822\pi\)
\(278\) 0 0
\(279\) 0.700523 0.0419392
\(280\) 0 0
\(281\) −3.93937 −0.235003 −0.117501 0.993073i \(-0.537488\pi\)
−0.117501 + 0.993073i \(0.537488\pi\)
\(282\) 0 0
\(283\) 10.1055 0.600712 0.300356 0.953827i \(-0.402895\pi\)
0.300356 + 0.953827i \(0.402895\pi\)
\(284\) 0 0
\(285\) 1.67513 0.0992262
\(286\) 0 0
\(287\) −1.29948 −0.0767057
\(288\) 0 0
\(289\) −3.95254 −0.232502
\(290\) 0 0
\(291\) 29.3054 1.71791
\(292\) 0 0
\(293\) −9.57944 −0.559637 −0.279818 0.960053i \(-0.590274\pi\)
−0.279818 + 0.960053i \(0.590274\pi\)
\(294\) 0 0
\(295\) 8.88717 0.517431
\(296\) 0 0
\(297\) −4.31265 −0.250245
\(298\) 0 0
\(299\) 2.77575 0.160526
\(300\) 0 0
\(301\) −13.2750 −0.765161
\(302\) 0 0
\(303\) 12.8872 0.740348
\(304\) 0 0
\(305\) −5.58181 −0.319613
\(306\) 0 0
\(307\) 28.8242 1.64508 0.822541 0.568706i \(-0.192556\pi\)
0.822541 + 0.568706i \(0.192556\pi\)
\(308\) 0 0
\(309\) 1.59166 0.0905463
\(310\) 0 0
\(311\) −4.74543 −0.269089 −0.134544 0.990908i \(-0.542957\pi\)
−0.134544 + 0.990908i \(0.542957\pi\)
\(312\) 0 0
\(313\) −2.18664 −0.123596 −0.0617982 0.998089i \(-0.519684\pi\)
−0.0617982 + 0.998089i \(0.519684\pi\)
\(314\) 0 0
\(315\) 0.806063 0.0454165
\(316\) 0 0
\(317\) −16.9805 −0.953719 −0.476860 0.878979i \(-0.658225\pi\)
−0.476860 + 0.878979i \(0.658225\pi\)
\(318\) 0 0
\(319\) −6.91160 −0.386975
\(320\) 0 0
\(321\) −22.9927 −1.28333
\(322\) 0 0
\(323\) −3.61213 −0.200984
\(324\) 0 0
\(325\) 0.481194 0.0266919
\(326\) 0 0
\(327\) −33.4617 −1.85044
\(328\) 0 0
\(329\) −23.9756 −1.32182
\(330\) 0 0
\(331\) −4.64974 −0.255573 −0.127786 0.991802i \(-0.540787\pi\)
−0.127786 + 0.991802i \(0.540787\pi\)
\(332\) 0 0
\(333\) 1.78067 0.0975802
\(334\) 0 0
\(335\) −9.59991 −0.524499
\(336\) 0 0
\(337\) −5.63023 −0.306698 −0.153349 0.988172i \(-0.549006\pi\)
−0.153349 + 0.988172i \(0.549006\pi\)
\(338\) 0 0
\(339\) 10.9419 0.594284
\(340\) 0 0
\(341\) −2.91160 −0.157672
\(342\) 0 0
\(343\) −13.6121 −0.734986
\(344\) 0 0
\(345\) 9.66291 0.520234
\(346\) 0 0
\(347\) −15.2301 −0.817596 −0.408798 0.912625i \(-0.634052\pi\)
−0.408798 + 0.912625i \(0.634052\pi\)
\(348\) 0 0
\(349\) −21.1998 −1.13480 −0.567400 0.823442i \(-0.692051\pi\)
−0.567400 + 0.823442i \(0.692051\pi\)
\(350\) 0 0
\(351\) −2.57452 −0.137417
\(352\) 0 0
\(353\) 22.6859 1.20745 0.603725 0.797192i \(-0.293682\pi\)
0.603725 + 0.797192i \(0.293682\pi\)
\(354\) 0 0
\(355\) −16.2374 −0.861793
\(356\) 0 0
\(357\) 25.1490 1.33103
\(358\) 0 0
\(359\) −16.6556 −0.879050 −0.439525 0.898230i \(-0.644853\pi\)
−0.439525 + 0.898230i \(0.644853\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) −17.3380 −0.910011
\(364\) 0 0
\(365\) −2.12601 −0.111280
\(366\) 0 0
\(367\) −17.4920 −0.913075 −0.456537 0.889704i \(-0.650911\pi\)
−0.456537 + 0.889704i \(0.650911\pi\)
\(368\) 0 0
\(369\) −0.0606343 −0.00315650
\(370\) 0 0
\(371\) 34.1622 1.77361
\(372\) 0 0
\(373\) −5.83146 −0.301941 −0.150971 0.988538i \(-0.548240\pi\)
−0.150971 + 0.988538i \(0.548240\pi\)
\(374\) 0 0
\(375\) 1.67513 0.0865034
\(376\) 0 0
\(377\) −4.12601 −0.212500
\(378\) 0 0
\(379\) −10.1768 −0.522747 −0.261373 0.965238i \(-0.584175\pi\)
−0.261373 + 0.965238i \(0.584175\pi\)
\(380\) 0 0
\(381\) 10.9076 0.558815
\(382\) 0 0
\(383\) −7.65069 −0.390932 −0.195466 0.980710i \(-0.562622\pi\)
−0.195466 + 0.980710i \(0.562622\pi\)
\(384\) 0 0
\(385\) −3.35026 −0.170745
\(386\) 0 0
\(387\) −0.619421 −0.0314869
\(388\) 0 0
\(389\) 28.6761 1.45394 0.726968 0.686672i \(-0.240929\pi\)
0.726968 + 0.686672i \(0.240929\pi\)
\(390\) 0 0
\(391\) −20.8364 −1.05374
\(392\) 0 0
\(393\) 27.5369 1.38905
\(394\) 0 0
\(395\) −16.8872 −0.849686
\(396\) 0 0
\(397\) 20.5139 1.02956 0.514781 0.857322i \(-0.327873\pi\)
0.514781 + 0.857322i \(0.327873\pi\)
\(398\) 0 0
\(399\) −6.96239 −0.348555
\(400\) 0 0
\(401\) 16.4894 0.823444 0.411722 0.911310i \(-0.364928\pi\)
0.411722 + 0.911310i \(0.364928\pi\)
\(402\) 0 0
\(403\) −1.73813 −0.0865827
\(404\) 0 0
\(405\) −8.38058 −0.416434
\(406\) 0 0
\(407\) −7.40105 −0.366856
\(408\) 0 0
\(409\) −18.3634 −0.908013 −0.454007 0.890998i \(-0.650006\pi\)
−0.454007 + 0.890998i \(0.650006\pi\)
\(410\) 0 0
\(411\) −17.7988 −0.877949
\(412\) 0 0
\(413\) −36.9380 −1.81760
\(414\) 0 0
\(415\) 5.89446 0.289348
\(416\) 0 0
\(417\) 8.82653 0.432237
\(418\) 0 0
\(419\) −29.1246 −1.42283 −0.711415 0.702772i \(-0.751945\pi\)
−0.711415 + 0.702772i \(0.751945\pi\)
\(420\) 0 0
\(421\) 23.3258 1.13683 0.568416 0.822742i \(-0.307557\pi\)
0.568416 + 0.822742i \(0.307557\pi\)
\(422\) 0 0
\(423\) −1.11871 −0.0543937
\(424\) 0 0
\(425\) −3.61213 −0.175214
\(426\) 0 0
\(427\) 23.1998 1.12272
\(428\) 0 0
\(429\) 0.649738 0.0313697
\(430\) 0 0
\(431\) 25.2750 1.21746 0.608728 0.793379i \(-0.291680\pi\)
0.608728 + 0.793379i \(0.291680\pi\)
\(432\) 0 0
\(433\) 7.98190 0.383586 0.191793 0.981435i \(-0.438570\pi\)
0.191793 + 0.981435i \(0.438570\pi\)
\(434\) 0 0
\(435\) −14.3634 −0.688674
\(436\) 0 0
\(437\) 5.76845 0.275942
\(438\) 0 0
\(439\) −21.3503 −1.01899 −0.509496 0.860473i \(-0.670168\pi\)
−0.509496 + 0.860473i \(0.670168\pi\)
\(440\) 0 0
\(441\) −1.99271 −0.0948908
\(442\) 0 0
\(443\) 1.76845 0.0840217 0.0420108 0.999117i \(-0.486624\pi\)
0.0420108 + 0.999117i \(0.486624\pi\)
\(444\) 0 0
\(445\) −11.2750 −0.534488
\(446\) 0 0
\(447\) −17.9756 −0.850215
\(448\) 0 0
\(449\) −4.64974 −0.219435 −0.109717 0.993963i \(-0.534995\pi\)
−0.109717 + 0.993963i \(0.534995\pi\)
\(450\) 0 0
\(451\) 0.252016 0.0118670
\(452\) 0 0
\(453\) 12.8364 0.603106
\(454\) 0 0
\(455\) −2.00000 −0.0937614
\(456\) 0 0
\(457\) 0.0752228 0.00351877 0.00175939 0.999998i \(-0.499440\pi\)
0.00175939 + 0.999998i \(0.499440\pi\)
\(458\) 0 0
\(459\) 19.3258 0.902052
\(460\) 0 0
\(461\) −32.1768 −1.49862 −0.749311 0.662218i \(-0.769615\pi\)
−0.749311 + 0.662218i \(0.769615\pi\)
\(462\) 0 0
\(463\) −22.3331 −1.03791 −0.518954 0.854802i \(-0.673678\pi\)
−0.518954 + 0.854802i \(0.673678\pi\)
\(464\) 0 0
\(465\) −6.05079 −0.280598
\(466\) 0 0
\(467\) 10.9175 0.505201 0.252600 0.967571i \(-0.418714\pi\)
0.252600 + 0.967571i \(0.418714\pi\)
\(468\) 0 0
\(469\) 39.9003 1.84243
\(470\) 0 0
\(471\) 23.1998 1.06899
\(472\) 0 0
\(473\) 2.57452 0.118376
\(474\) 0 0
\(475\) 1.00000 0.0458831
\(476\) 0 0
\(477\) 1.59403 0.0729856
\(478\) 0 0
\(479\) −12.8061 −0.585124 −0.292562 0.956247i \(-0.594508\pi\)
−0.292562 + 0.956247i \(0.594508\pi\)
\(480\) 0 0
\(481\) −4.41819 −0.201452
\(482\) 0 0
\(483\) −40.1622 −1.82744
\(484\) 0 0
\(485\) 17.4944 0.794378
\(486\) 0 0
\(487\) 24.2351 1.09820 0.549098 0.835758i \(-0.314971\pi\)
0.549098 + 0.835758i \(0.314971\pi\)
\(488\) 0 0
\(489\) 15.1735 0.686168
\(490\) 0 0
\(491\) −12.5647 −0.567035 −0.283518 0.958967i \(-0.591501\pi\)
−0.283518 + 0.958967i \(0.591501\pi\)
\(492\) 0 0
\(493\) 30.9722 1.39492
\(494\) 0 0
\(495\) −0.156325 −0.00702629
\(496\) 0 0
\(497\) 67.4880 3.02725
\(498\) 0 0
\(499\) 41.4314 1.85472 0.927361 0.374167i \(-0.122071\pi\)
0.927361 + 0.374167i \(0.122071\pi\)
\(500\) 0 0
\(501\) −0.982857 −0.0439108
\(502\) 0 0
\(503\) −23.2301 −1.03578 −0.517890 0.855447i \(-0.673282\pi\)
−0.517890 + 0.855447i \(0.673282\pi\)
\(504\) 0 0
\(505\) 7.69323 0.342344
\(506\) 0 0
\(507\) −21.3888 −0.949911
\(508\) 0 0
\(509\) −22.0362 −0.976737 −0.488369 0.872637i \(-0.662408\pi\)
−0.488369 + 0.872637i \(0.662408\pi\)
\(510\) 0 0
\(511\) 8.83638 0.390898
\(512\) 0 0
\(513\) −5.35026 −0.236220
\(514\) 0 0
\(515\) 0.950170 0.0418695
\(516\) 0 0
\(517\) 4.64974 0.204495
\(518\) 0 0
\(519\) 20.8667 0.915946
\(520\) 0 0
\(521\) −11.6121 −0.508737 −0.254368 0.967107i \(-0.581868\pi\)
−0.254368 + 0.967107i \(0.581868\pi\)
\(522\) 0 0
\(523\) −12.7489 −0.557472 −0.278736 0.960368i \(-0.589915\pi\)
−0.278736 + 0.960368i \(0.589915\pi\)
\(524\) 0 0
\(525\) −6.96239 −0.303864
\(526\) 0 0
\(527\) 13.0475 0.568356
\(528\) 0 0
\(529\) 10.2750 0.446741
\(530\) 0 0
\(531\) −1.72355 −0.0747955
\(532\) 0 0
\(533\) 0.150446 0.00651652
\(534\) 0 0
\(535\) −13.7259 −0.593423
\(536\) 0 0
\(537\) 25.4861 1.09981
\(538\) 0 0
\(539\) 8.28233 0.356745
\(540\) 0 0
\(541\) −21.2447 −0.913382 −0.456691 0.889625i \(-0.650965\pi\)
−0.456691 + 0.889625i \(0.650965\pi\)
\(542\) 0 0
\(543\) −43.3014 −1.85824
\(544\) 0 0
\(545\) −19.9756 −0.855659
\(546\) 0 0
\(547\) −14.3249 −0.612487 −0.306244 0.951953i \(-0.599072\pi\)
−0.306244 + 0.951953i \(0.599072\pi\)
\(548\) 0 0
\(549\) 1.08252 0.0462007
\(550\) 0 0
\(551\) −8.57452 −0.365287
\(552\) 0 0
\(553\) 70.1886 2.98472
\(554\) 0 0
\(555\) −15.3806 −0.652869
\(556\) 0 0
\(557\) 38.3536 1.62509 0.812547 0.582896i \(-0.198080\pi\)
0.812547 + 0.582896i \(0.198080\pi\)
\(558\) 0 0
\(559\) 1.53690 0.0650041
\(560\) 0 0
\(561\) −4.87732 −0.205920
\(562\) 0 0
\(563\) 0.773377 0.0325939 0.0162970 0.999867i \(-0.494812\pi\)
0.0162970 + 0.999867i \(0.494812\pi\)
\(564\) 0 0
\(565\) 6.53198 0.274802
\(566\) 0 0
\(567\) 34.8324 1.46282
\(568\) 0 0
\(569\) −25.0640 −1.05074 −0.525368 0.850875i \(-0.676072\pi\)
−0.525368 + 0.850875i \(0.676072\pi\)
\(570\) 0 0
\(571\) 1.95509 0.0818182 0.0409091 0.999163i \(-0.486975\pi\)
0.0409091 + 0.999163i \(0.486975\pi\)
\(572\) 0 0
\(573\) 45.8759 1.91649
\(574\) 0 0
\(575\) 5.76845 0.240561
\(576\) 0 0
\(577\) −40.0625 −1.66783 −0.833913 0.551896i \(-0.813904\pi\)
−0.833913 + 0.551896i \(0.813904\pi\)
\(578\) 0 0
\(579\) 2.75528 0.114505
\(580\) 0 0
\(581\) −24.4993 −1.01640
\(582\) 0 0
\(583\) −6.62530 −0.274392
\(584\) 0 0
\(585\) −0.0933212 −0.00385836
\(586\) 0 0
\(587\) 22.8324 0.942394 0.471197 0.882028i \(-0.343822\pi\)
0.471197 + 0.882028i \(0.343822\pi\)
\(588\) 0 0
\(589\) −3.61213 −0.148835
\(590\) 0 0
\(591\) 0.438658 0.0180440
\(592\) 0 0
\(593\) −9.22425 −0.378795 −0.189397 0.981901i \(-0.560653\pi\)
−0.189397 + 0.981901i \(0.560653\pi\)
\(594\) 0 0
\(595\) 15.0132 0.615480
\(596\) 0 0
\(597\) 3.22425 0.131960
\(598\) 0 0
\(599\) −22.6497 −0.925443 −0.462722 0.886504i \(-0.653127\pi\)
−0.462722 + 0.886504i \(0.653127\pi\)
\(600\) 0 0
\(601\) 35.1754 1.43483 0.717417 0.696644i \(-0.245324\pi\)
0.717417 + 0.696644i \(0.245324\pi\)
\(602\) 0 0
\(603\) 1.86177 0.0758173
\(604\) 0 0
\(605\) −10.3503 −0.420798
\(606\) 0 0
\(607\) 41.0517 1.66624 0.833119 0.553093i \(-0.186553\pi\)
0.833119 + 0.553093i \(0.186553\pi\)
\(608\) 0 0
\(609\) 59.6991 2.41913
\(610\) 0 0
\(611\) 2.77575 0.112295
\(612\) 0 0
\(613\) 26.7513 1.08048 0.540238 0.841513i \(-0.318334\pi\)
0.540238 + 0.841513i \(0.318334\pi\)
\(614\) 0 0
\(615\) 0.523730 0.0211188
\(616\) 0 0
\(617\) −20.6761 −0.832388 −0.416194 0.909276i \(-0.636636\pi\)
−0.416194 + 0.909276i \(0.636636\pi\)
\(618\) 0 0
\(619\) −7.35614 −0.295668 −0.147834 0.989012i \(-0.547230\pi\)
−0.147834 + 0.989012i \(0.547230\pi\)
\(620\) 0 0
\(621\) −30.8627 −1.23848
\(622\) 0 0
\(623\) 46.8627 1.87752
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 1.35026 0.0539243
\(628\) 0 0
\(629\) 33.1655 1.32240
\(630\) 0 0
\(631\) 10.7308 0.427188 0.213594 0.976923i \(-0.431483\pi\)
0.213594 + 0.976923i \(0.431483\pi\)
\(632\) 0 0
\(633\) −16.0606 −0.638353
\(634\) 0 0
\(635\) 6.51151 0.258401
\(636\) 0 0
\(637\) 4.94429 0.195900
\(638\) 0 0
\(639\) 3.14903 0.124574
\(640\) 0 0
\(641\) −37.2506 −1.47131 −0.735655 0.677356i \(-0.763126\pi\)
−0.735655 + 0.677356i \(0.763126\pi\)
\(642\) 0 0
\(643\) 7.38058 0.291062 0.145531 0.989354i \(-0.453511\pi\)
0.145531 + 0.989354i \(0.453511\pi\)
\(644\) 0 0
\(645\) 5.35026 0.210666
\(646\) 0 0
\(647\) −5.32979 −0.209536 −0.104768 0.994497i \(-0.533410\pi\)
−0.104768 + 0.994497i \(0.533410\pi\)
\(648\) 0 0
\(649\) 7.16362 0.281197
\(650\) 0 0
\(651\) 25.1490 0.985668
\(652\) 0 0
\(653\) 11.0230 0.431364 0.215682 0.976464i \(-0.430803\pi\)
0.215682 + 0.976464i \(0.430803\pi\)
\(654\) 0 0
\(655\) 16.4387 0.642312
\(656\) 0 0
\(657\) 0.412311 0.0160858
\(658\) 0 0
\(659\) −12.1114 −0.471794 −0.235897 0.971778i \(-0.575803\pi\)
−0.235897 + 0.971778i \(0.575803\pi\)
\(660\) 0 0
\(661\) 7.22425 0.280991 0.140495 0.990081i \(-0.455130\pi\)
0.140495 + 0.990081i \(0.455130\pi\)
\(662\) 0 0
\(663\) −2.91160 −0.113077
\(664\) 0 0
\(665\) −4.15633 −0.161175
\(666\) 0 0
\(667\) −49.4617 −1.91516
\(668\) 0 0
\(669\) −0.459126 −0.0177509
\(670\) 0 0
\(671\) −4.49929 −0.173693
\(672\) 0 0
\(673\) −6.06888 −0.233938 −0.116969 0.993136i \(-0.537318\pi\)
−0.116969 + 0.993136i \(0.537318\pi\)
\(674\) 0 0
\(675\) −5.35026 −0.205932
\(676\) 0 0
\(677\) 28.0933 1.07971 0.539857 0.841757i \(-0.318478\pi\)
0.539857 + 0.841757i \(0.318478\pi\)
\(678\) 0 0
\(679\) −72.7123 −2.79044
\(680\) 0 0
\(681\) −20.2922 −0.777598
\(682\) 0 0
\(683\) −29.5247 −1.12973 −0.564865 0.825183i \(-0.691072\pi\)
−0.564865 + 0.825183i \(0.691072\pi\)
\(684\) 0 0
\(685\) −10.6253 −0.405972
\(686\) 0 0
\(687\) 29.4109 1.12210
\(688\) 0 0
\(689\) −3.95509 −0.150677
\(690\) 0 0
\(691\) 8.03032 0.305488 0.152744 0.988266i \(-0.451189\pi\)
0.152744 + 0.988266i \(0.451189\pi\)
\(692\) 0 0
\(693\) 0.649738 0.0246815
\(694\) 0 0
\(695\) 5.26916 0.199871
\(696\) 0 0
\(697\) −1.12933 −0.0427765
\(698\) 0 0
\(699\) −17.0640 −0.645418
\(700\) 0 0
\(701\) −19.5672 −0.739044 −0.369522 0.929222i \(-0.620479\pi\)
−0.369522 + 0.929222i \(0.620479\pi\)
\(702\) 0 0
\(703\) −9.18172 −0.346295
\(704\) 0 0
\(705\) 9.66291 0.363926
\(706\) 0 0
\(707\) −31.9756 −1.20256
\(708\) 0 0
\(709\) 37.8251 1.42055 0.710276 0.703923i \(-0.248570\pi\)
0.710276 + 0.703923i \(0.248570\pi\)
\(710\) 0 0
\(711\) 3.27504 0.122824
\(712\) 0 0
\(713\) −20.8364 −0.780329
\(714\) 0 0
\(715\) 0.387873 0.0145056
\(716\) 0 0
\(717\) 43.3865 1.62030
\(718\) 0 0
\(719\) 39.8799 1.48727 0.743634 0.668587i \(-0.233100\pi\)
0.743634 + 0.668587i \(0.233100\pi\)
\(720\) 0 0
\(721\) −3.94921 −0.147076
\(722\) 0 0
\(723\) 6.44851 0.239823
\(724\) 0 0
\(725\) −8.57452 −0.318450
\(726\) 0 0
\(727\) 52.1319 1.93346 0.966732 0.255790i \(-0.0823356\pi\)
0.966732 + 0.255790i \(0.0823356\pi\)
\(728\) 0 0
\(729\) 28.5125 1.05602
\(730\) 0 0
\(731\) −11.5369 −0.426708
\(732\) 0 0
\(733\) −46.1984 −1.70638 −0.853188 0.521603i \(-0.825334\pi\)
−0.853188 + 0.521603i \(0.825334\pi\)
\(734\) 0 0
\(735\) 17.2120 0.634875
\(736\) 0 0
\(737\) −7.73813 −0.285038
\(738\) 0 0
\(739\) 17.4010 0.640108 0.320054 0.947399i \(-0.396299\pi\)
0.320054 + 0.947399i \(0.396299\pi\)
\(740\) 0 0
\(741\) 0.806063 0.0296115
\(742\) 0 0
\(743\) −26.7880 −0.982755 −0.491378 0.870947i \(-0.663506\pi\)
−0.491378 + 0.870947i \(0.663506\pi\)
\(744\) 0 0
\(745\) −10.7308 −0.393148
\(746\) 0 0
\(747\) −1.14315 −0.0418257
\(748\) 0 0
\(749\) 57.0494 2.08454
\(750\) 0 0
\(751\) −48.8529 −1.78267 −0.891333 0.453348i \(-0.850229\pi\)
−0.891333 + 0.453348i \(0.850229\pi\)
\(752\) 0 0
\(753\) 0.523730 0.0190858
\(754\) 0 0
\(755\) 7.66291 0.278882
\(756\) 0 0
\(757\) 37.0376 1.34615 0.673077 0.739572i \(-0.264972\pi\)
0.673077 + 0.739572i \(0.264972\pi\)
\(758\) 0 0
\(759\) 7.78892 0.282720
\(760\) 0 0
\(761\) 28.5705 1.03568 0.517841 0.855477i \(-0.326736\pi\)
0.517841 + 0.855477i \(0.326736\pi\)
\(762\) 0 0
\(763\) 83.0249 3.00570
\(764\) 0 0
\(765\) 0.700523 0.0253275
\(766\) 0 0
\(767\) 4.27645 0.154414
\(768\) 0 0
\(769\) 27.0679 0.976094 0.488047 0.872817i \(-0.337709\pi\)
0.488047 + 0.872817i \(0.337709\pi\)
\(770\) 0 0
\(771\) 24.5705 0.884887
\(772\) 0 0
\(773\) −13.6350 −0.490416 −0.245208 0.969471i \(-0.578856\pi\)
−0.245208 + 0.969471i \(0.578856\pi\)
\(774\) 0 0
\(775\) −3.61213 −0.129751
\(776\) 0 0
\(777\) 63.9267 2.29336
\(778\) 0 0
\(779\) 0.312650 0.0112019
\(780\) 0 0
\(781\) −13.0884 −0.468340
\(782\) 0 0
\(783\) 45.8759 1.63947
\(784\) 0 0
\(785\) 13.8496 0.494312
\(786\) 0 0
\(787\) −4.77338 −0.170152 −0.0850762 0.996374i \(-0.527113\pi\)
−0.0850762 + 0.996374i \(0.527113\pi\)
\(788\) 0 0
\(789\) −11.8397 −0.421505
\(790\) 0 0
\(791\) −27.1490 −0.965308
\(792\) 0 0
\(793\) −2.68594 −0.0953803
\(794\) 0 0
\(795\) −13.7685 −0.488317
\(796\) 0 0
\(797\) 35.4798 1.25676 0.628379 0.777907i \(-0.283719\pi\)
0.628379 + 0.777907i \(0.283719\pi\)
\(798\) 0 0
\(799\) −20.8364 −0.737138
\(800\) 0 0
\(801\) 2.18664 0.0772612
\(802\) 0 0
\(803\) −1.71370 −0.0604751
\(804\) 0 0
\(805\) −23.9756 −0.845028
\(806\) 0 0
\(807\) 40.6859 1.43221
\(808\) 0 0
\(809\) −46.7024 −1.64197 −0.820985 0.570950i \(-0.806575\pi\)
−0.820985 + 0.570950i \(0.806575\pi\)
\(810\) 0 0
\(811\) 27.3404 0.960052 0.480026 0.877254i \(-0.340627\pi\)
0.480026 + 0.877254i \(0.340627\pi\)
\(812\) 0 0
\(813\) −6.39772 −0.224378
\(814\) 0 0
\(815\) 9.05808 0.317291
\(816\) 0 0
\(817\) 3.19394 0.111742
\(818\) 0 0
\(819\) 0.387873 0.0135534
\(820\) 0 0
\(821\) −52.6761 −1.83841 −0.919204 0.393782i \(-0.871167\pi\)
−0.919204 + 0.393782i \(0.871167\pi\)
\(822\) 0 0
\(823\) −10.4788 −0.365269 −0.182634 0.983181i \(-0.558462\pi\)
−0.182634 + 0.983181i \(0.558462\pi\)
\(824\) 0 0
\(825\) 1.35026 0.0470101
\(826\) 0 0
\(827\) 0.874947 0.0304249 0.0152124 0.999884i \(-0.495158\pi\)
0.0152124 + 0.999884i \(0.495158\pi\)
\(828\) 0 0
\(829\) 27.6023 0.958667 0.479333 0.877633i \(-0.340878\pi\)
0.479333 + 0.877633i \(0.340878\pi\)
\(830\) 0 0
\(831\) 14.5745 0.505584
\(832\) 0 0
\(833\) −37.1147 −1.28595
\(834\) 0 0
\(835\) −0.586734 −0.0203048
\(836\) 0 0
\(837\) 19.3258 0.667998
\(838\) 0 0
\(839\) −31.2750 −1.07973 −0.539867 0.841750i \(-0.681526\pi\)
−0.539867 + 0.841750i \(0.681526\pi\)
\(840\) 0 0
\(841\) 44.5223 1.53525
\(842\) 0 0
\(843\) −6.59895 −0.227280
\(844\) 0 0
\(845\) −12.7685 −0.439248
\(846\) 0 0
\(847\) 43.0191 1.47815
\(848\) 0 0
\(849\) 16.9281 0.580971
\(850\) 0 0
\(851\) −52.9643 −1.81559
\(852\) 0 0
\(853\) 39.3112 1.34599 0.672996 0.739647i \(-0.265007\pi\)
0.672996 + 0.739647i \(0.265007\pi\)
\(854\) 0 0
\(855\) −0.193937 −0.00663249
\(856\) 0 0
\(857\) −39.3293 −1.34346 −0.671732 0.740794i \(-0.734449\pi\)
−0.671732 + 0.740794i \(0.734449\pi\)
\(858\) 0 0
\(859\) 18.1721 0.620022 0.310011 0.950733i \(-0.399667\pi\)
0.310011 + 0.950733i \(0.399667\pi\)
\(860\) 0 0
\(861\) −2.17679 −0.0741849
\(862\) 0 0
\(863\) 21.6751 0.737830 0.368915 0.929463i \(-0.379729\pi\)
0.368915 + 0.929463i \(0.379729\pi\)
\(864\) 0 0
\(865\) 12.4568 0.423542
\(866\) 0 0
\(867\) −6.62102 −0.224862
\(868\) 0 0
\(869\) −13.6121 −0.461760
\(870\) 0 0
\(871\) −4.61942 −0.156523
\(872\) 0 0
\(873\) −3.39280 −0.114829
\(874\) 0 0
\(875\) −4.15633 −0.140509
\(876\) 0 0
\(877\) 10.2701 0.346797 0.173399 0.984852i \(-0.444525\pi\)
0.173399 + 0.984852i \(0.444525\pi\)
\(878\) 0 0
\(879\) −16.0468 −0.541246
\(880\) 0 0
\(881\) −37.1187 −1.25056 −0.625281 0.780400i \(-0.715015\pi\)
−0.625281 + 0.780400i \(0.715015\pi\)
\(882\) 0 0
\(883\) 2.63401 0.0886415 0.0443207 0.999017i \(-0.485888\pi\)
0.0443207 + 0.999017i \(0.485888\pi\)
\(884\) 0 0
\(885\) 14.8872 0.500427
\(886\) 0 0
\(887\) −11.1876 −0.375643 −0.187821 0.982203i \(-0.560143\pi\)
−0.187821 + 0.982203i \(0.560143\pi\)
\(888\) 0 0
\(889\) −27.0640 −0.907696
\(890\) 0 0
\(891\) −6.75528 −0.226310
\(892\) 0 0
\(893\) 5.76845 0.193034
\(894\) 0 0
\(895\) 15.2144 0.508561
\(896\) 0 0
\(897\) 4.64974 0.155250
\(898\) 0 0
\(899\) 30.9722 1.03298
\(900\) 0 0
\(901\) 29.6893 0.989093
\(902\) 0 0
\(903\) −22.2374 −0.740015
\(904\) 0 0
\(905\) −25.8496 −0.859268
\(906\) 0 0
\(907\) 0.824162 0.0273658 0.0136829 0.999906i \(-0.495644\pi\)
0.0136829 + 0.999906i \(0.495644\pi\)
\(908\) 0 0
\(909\) −1.49200 −0.0494865
\(910\) 0 0
\(911\) −27.3211 −0.905188 −0.452594 0.891717i \(-0.649501\pi\)
−0.452594 + 0.891717i \(0.649501\pi\)
\(912\) 0 0
\(913\) 4.75131 0.157245
\(914\) 0 0
\(915\) −9.35026 −0.309110
\(916\) 0 0
\(917\) −68.3244 −2.25627
\(918\) 0 0
\(919\) −36.3488 −1.19904 −0.599519 0.800361i \(-0.704641\pi\)
−0.599519 + 0.800361i \(0.704641\pi\)
\(920\) 0 0
\(921\) 48.2842 1.59102
\(922\) 0 0
\(923\) −7.81336 −0.257180
\(924\) 0 0
\(925\) −9.18172 −0.301893
\(926\) 0 0
\(927\) −0.184273 −0.00605231
\(928\) 0 0
\(929\) −36.0263 −1.18199 −0.590993 0.806677i \(-0.701264\pi\)
−0.590993 + 0.806677i \(0.701264\pi\)
\(930\) 0 0
\(931\) 10.2750 0.336751
\(932\) 0 0
\(933\) −7.94921 −0.260246
\(934\) 0 0
\(935\) −2.91160 −0.0952196
\(936\) 0 0
\(937\) 44.4241 1.45127 0.725636 0.688079i \(-0.241546\pi\)
0.725636 + 0.688079i \(0.241546\pi\)
\(938\) 0 0
\(939\) −3.66291 −0.119535
\(940\) 0 0
\(941\) −9.31406 −0.303630 −0.151815 0.988409i \(-0.548512\pi\)
−0.151815 + 0.988409i \(0.548512\pi\)
\(942\) 0 0
\(943\) 1.80351 0.0587303
\(944\) 0 0
\(945\) 22.2374 0.723384
\(946\) 0 0
\(947\) −41.4676 −1.34751 −0.673757 0.738953i \(-0.735321\pi\)
−0.673757 + 0.738953i \(0.735321\pi\)
\(948\) 0 0
\(949\) −1.02302 −0.0332087
\(950\) 0 0
\(951\) −28.4445 −0.922377
\(952\) 0 0
\(953\) −17.2424 −0.558535 −0.279267 0.960213i \(-0.590092\pi\)
−0.279267 + 0.960213i \(0.590092\pi\)
\(954\) 0 0
\(955\) 27.3865 0.886205
\(956\) 0 0
\(957\) −11.5778 −0.374258
\(958\) 0 0
\(959\) 44.1622 1.42607
\(960\) 0 0
\(961\) −17.9525 −0.579114
\(962\) 0 0
\(963\) 2.66196 0.0857804
\(964\) 0 0
\(965\) 1.64481 0.0529484
\(966\) 0 0
\(967\) −8.84700 −0.284500 −0.142250 0.989831i \(-0.545434\pi\)
−0.142250 + 0.989831i \(0.545434\pi\)
\(968\) 0 0
\(969\) −6.05079 −0.194379
\(970\) 0 0
\(971\) −18.9262 −0.607370 −0.303685 0.952772i \(-0.598217\pi\)
−0.303685 + 0.952772i \(0.598217\pi\)
\(972\) 0 0
\(973\) −21.9003 −0.702093
\(974\) 0 0
\(975\) 0.806063 0.0258147
\(976\) 0 0
\(977\) 49.2081 1.57431 0.787153 0.616758i \(-0.211554\pi\)
0.787153 + 0.616758i \(0.211554\pi\)
\(978\) 0 0
\(979\) −9.08840 −0.290466
\(980\) 0 0
\(981\) 3.87399 0.123687
\(982\) 0 0
\(983\) −0.162664 −0.00518819 −0.00259409 0.999997i \(-0.500826\pi\)
−0.00259409 + 0.999997i \(0.500826\pi\)
\(984\) 0 0
\(985\) 0.261865 0.00834372
\(986\) 0 0
\(987\) −40.1622 −1.27838
\(988\) 0 0
\(989\) 18.4241 0.585851
\(990\) 0 0
\(991\) 24.5599 0.780172 0.390086 0.920778i \(-0.372445\pi\)
0.390086 + 0.920778i \(0.372445\pi\)
\(992\) 0 0
\(993\) −7.78892 −0.247174
\(994\) 0 0
\(995\) 1.92478 0.0610195
\(996\) 0 0
\(997\) −43.3112 −1.37168 −0.685840 0.727752i \(-0.740565\pi\)
−0.685840 + 0.727752i \(0.740565\pi\)
\(998\) 0 0
\(999\) 49.1246 1.55423
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 3040.2.a.l.1.3 3
4.3 odd 2 3040.2.a.m.1.1 yes 3
8.3 odd 2 6080.2.a.bu.1.3 3
8.5 even 2 6080.2.a.bt.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3040.2.a.l.1.3 3 1.1 even 1 trivial
3040.2.a.m.1.1 yes 3 4.3 odd 2
6080.2.a.bt.1.1 3 8.5 even 2
6080.2.a.bu.1.3 3 8.3 odd 2