Properties

Label 3040.2.a.m.1.1
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 1.1
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 \(-0.660660\pi\)
−0.483569 + 0.875306i \(0.660660\pi\)
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 4.15633 1.57094 0.785472 0.618898i \(-0.212420\pi\)
0.785472 + 0.618898i \(0.212420\pi\)
\(8\) 0 0
\(9\) −0.193937 −0.0646455
\(10\) 0 0
\(11\) −0.806063 −0.243037 −0.121519 0.992589i \(-0.538776\pi\)
−0.121519 + 0.992589i \(0.538776\pi\)
\(12\) 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 \(-0.705391\pi\)
−0.601403 + 0.798946i \(0.705391\pi\)
\(24\) 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 \(-0.394845\pi\)
0.324379 + 0.945927i \(0.394845\pi\)
\(32\) 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 \(-0.578307\pi\)
−0.243535 + 0.969892i \(0.578307\pi\)
\(44\) 0 0
\(45\) −0.193937 −0.0289104
\(46\) 0 0
\(47\) −5.76845 −0.841415 −0.420708 0.907196i \(-0.638218\pi\)
−0.420708 + 0.907196i \(0.638218\pi\)
\(48\) 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 \(-0.696364\pi\)
−0.578505 + 0.815679i \(0.696364\pi\)
\(60\) 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 \(-0.300542\pi\)
0.586408 + 0.810016i \(0.300542\pi\)
\(68\) 0 0
\(69\) 9.66291 1.16328
\(70\) 0 0
\(71\) 16.2374 1.92703 0.963514 0.267658i \(-0.0862496\pi\)
0.963514 + 0.267658i \(0.0862496\pi\)
\(72\) 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 \(-0.101106\pi\)
0.949977 + 0.312319i \(0.101106\pi\)
\(80\) 0 0
\(81\) −8.38058 −0.931175
\(82\) 0 0
\(83\) −5.89446 −0.647001 −0.323501 0.946228i \(-0.604860\pi\)
−0.323501 + 0.946228i \(0.604860\pi\)
\(84\) 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 \(-0.514906\pi\)
−0.0468115 + 0.998904i \(0.514906\pi\)
\(104\) 0 0
\(105\) −6.96239 −0.679460
\(106\) 0 0
\(107\) 13.7259 1.32693 0.663467 0.748205i \(-0.269084\pi\)
0.663467 + 0.748205i \(0.269084\pi\)
\(108\) 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 \(-0.593290\pi\)
−0.288902 + 0.957359i \(0.593290\pi\)
\(128\) 0 0
\(129\) 5.35026 0.471064
\(130\) 0 0
\(131\) −16.4387 −1.43625 −0.718126 0.695913i \(-0.755000\pi\)
−0.718126 + 0.695913i \(0.755000\pi\)
\(132\) 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 \(-0.571736\pi\)
−0.223462 + 0.974713i \(0.571736\pi\)
\(140\) 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 \(-0.600932\pi\)
−0.311799 + 0.950148i \(0.600932\pi\)
\(152\) 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 \(-0.615431\pi\)
−0.354742 + 0.934964i \(0.615431\pi\)
\(164\) 0 0
\(165\) 1.35026 0.105118
\(166\) 0 0
\(167\) 0.586734 0.0454029 0.0227014 0.999742i \(-0.492773\pi\)
0.0227014 + 0.999742i \(0.492773\pi\)
\(168\) 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 \(-0.692511\pi\)
−0.568589 + 0.822622i \(0.692511\pi\)
\(180\) 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 \(-0.956806\pi\)
−0.990807 + 0.135281i \(0.956806\pi\)
\(192\) 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 \(-0.521733\pi\)
−0.0682219 + 0.997670i \(0.521733\pi\)
\(200\) 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 \(-0.392944\pi\)
0.330022 + 0.943973i \(0.392944\pi\)
\(212\) 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 \(-0.497079\pi\)
0.00917701 + 0.999958i \(0.497079\pi\)
\(224\) 0 0
\(225\) −0.193937 −0.0129291
\(226\) 0 0
\(227\) 12.1138 0.804020 0.402010 0.915635i \(-0.368312\pi\)
0.402010 + 0.915635i \(0.368312\pi\)
\(228\) 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 \(-0.816087\pi\)
−0.837677 + 0.546166i \(0.816087\pi\)
\(240\) 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 \(-0.503141\pi\)
−0.00986716 + 0.999951i \(0.503141\pi\)
\(252\) 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 \(-0.430075\pi\)
0.217914 + 0.975968i \(0.430075\pi\)
\(264\) 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 \(-0.462992\pi\)
0.116001 + 0.993249i \(0.462992\pi\)
\(272\) 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 \(-0.597105\pi\)
−0.300356 + 0.953827i \(0.597105\pi\)
\(284\) 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 \(-0.807444\pi\)
−0.822541 + 0.568706i \(0.807444\pi\)
\(308\) 0 0
\(309\) 1.59166 0.0905463
\(310\) 0 0
\(311\) 4.74543 0.269089 0.134544 0.990908i \(-0.457043\pi\)
0.134544 + 0.990908i \(0.457043\pi\)
\(312\) 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 \(-0.459213\pi\)
0.127786 + 0.991802i \(0.459213\pi\)
\(332\) 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 \(-0.365948\pi\)
0.408798 + 0.912625i \(0.365948\pi\)
\(348\) 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 \(-0.355147\pi\)
0.439525 + 0.898230i \(0.355147\pi\)
\(360\) 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 \(-0.349089\pi\)
0.456537 + 0.889704i \(0.349089\pi\)
\(368\) 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 \(-0.415825\pi\)
0.261373 + 0.965238i \(0.415825\pi\)
\(380\) 0 0
\(381\) 10.9076 0.558815
\(382\) 0 0
\(383\) 7.65069 0.390932 0.195466 0.980710i \(-0.437378\pi\)
0.195466 + 0.980710i \(0.437378\pi\)
\(384\) 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 \(-0.248055\pi\)
0.711415 + 0.702772i \(0.248055\pi\)
\(420\) 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 \(-0.708320\pi\)
−0.608728 + 0.793379i \(0.708320\pi\)
\(432\) 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 \(-0.329832\pi\)
0.509496 + 0.860473i \(0.329832\pi\)
\(440\) 0 0
\(441\) −1.99271 −0.0948908
\(442\) 0 0
\(443\) −1.76845 −0.0840217 −0.0420108 0.999117i \(-0.513376\pi\)
−0.0420108 + 0.999117i \(0.513376\pi\)
\(444\) 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 \(-0.326322\pi\)
0.518954 + 0.854802i \(0.326322\pi\)
\(464\) 0 0
\(465\) −6.05079 −0.280598
\(466\) 0 0
\(467\) −10.9175 −0.505201 −0.252600 0.967571i \(-0.581286\pi\)
−0.252600 + 0.967571i \(0.581286\pi\)
\(468\) 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 \(-0.405492\pi\)
0.292562 + 0.956247i \(0.405492\pi\)
\(480\) 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 \(-0.685029\pi\)
−0.549098 + 0.835758i \(0.685029\pi\)
\(488\) 0 0
\(489\) 15.1735 0.686168
\(490\) 0 0
\(491\) 12.5647 0.567035 0.283518 0.958967i \(-0.408499\pi\)
0.283518 + 0.958967i \(0.408499\pi\)
\(492\) 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 \(-0.877929\pi\)
−0.927361 + 0.374167i \(0.877929\pi\)
\(500\) 0 0
\(501\) −0.982857 −0.0439108
\(502\) 0 0
\(503\) 23.2301 1.03578 0.517890 0.855447i \(-0.326718\pi\)
0.517890 + 0.855447i \(0.326718\pi\)
\(504\) 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 \(-0.410085\pi\)
0.278736 + 0.960368i \(0.410085\pi\)
\(524\) 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 \(-0.400928\pi\)
0.306244 + 0.951953i \(0.400928\pi\)
\(548\) 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 \(-0.505188\pi\)
−0.0162970 + 0.999867i \(0.505188\pi\)
\(564\) 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 \(-0.513025\pi\)
−0.0409091 + 0.999163i \(0.513025\pi\)
\(572\) 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 \(-0.656178\pi\)
−0.471197 + 0.882028i \(0.656178\pi\)
\(588\) 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 \(-0.346873\pi\)
0.462722 + 0.886504i \(0.346873\pi\)
\(600\) 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 \(-0.813447\pi\)
−0.833119 + 0.553093i \(0.813447\pi\)
\(608\) 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 \(-0.452770\pi\)
0.147834 + 0.989012i \(0.452770\pi\)
\(620\) 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 \(-0.568517\pi\)
−0.213594 + 0.976923i \(0.568517\pi\)
\(632\) 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 \(-0.546489\pi\)
−0.145531 + 0.989354i \(0.546489\pi\)
\(644\) 0 0
\(645\) 5.35026 0.210666
\(646\) 0 0
\(647\) 5.32979 0.209536 0.104768 0.994497i \(-0.466590\pi\)
0.104768 + 0.994497i \(0.466590\pi\)
\(648\) 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 \(-0.424197\pi\)
0.235897 + 0.971778i \(0.424197\pi\)
\(660\) 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 \(-0.308928\pi\)
0.564865 + 0.825183i \(0.308928\pi\)
\(684\) 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 \(-0.548811\pi\)
−0.152744 + 0.988266i \(0.548811\pi\)
\(692\) 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 \(-0.766900\pi\)
−0.743634 + 0.668587i \(0.766900\pi\)
\(720\) 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 \(-0.917664\pi\)
−0.966732 + 0.255790i \(0.917664\pi\)
\(728\) 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 \(-0.603701\pi\)
−0.320054 + 0.947399i \(0.603701\pi\)
\(740\) 0 0
\(741\) 0.806063 0.0296115
\(742\) 0 0
\(743\) 26.7880 0.982755 0.491378 0.870947i \(-0.336494\pi\)
0.491378 + 0.870947i \(0.336494\pi\)
\(744\) 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 \(-0.149771\pi\)
0.891333 + 0.453348i \(0.149771\pi\)
\(752\) 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 \(-0.472887\pi\)
0.0850762 + 0.996374i \(0.472887\pi\)
\(788\) 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 \(-0.659373\pi\)
−0.480026 + 0.877254i \(0.659373\pi\)
\(812\) 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 \(-0.441538\pi\)
0.182634 + 0.983181i \(0.441538\pi\)
\(824\) 0 0
\(825\) 1.35026 0.0470101
\(826\) 0 0
\(827\) −0.874947 −0.0304249 −0.0152124 0.999884i \(-0.504842\pi\)
−0.0152124 + 0.999884i \(0.504842\pi\)
\(828\) 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 \(-0.318474\pi\)
0.539867 + 0.841750i \(0.318474\pi\)
\(840\) 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 \(-0.600333\pi\)
−0.310011 + 0.950733i \(0.600333\pi\)
\(860\) 0 0
\(861\) −2.17679 −0.0741849
\(862\) 0 0
\(863\) −21.6751 −0.737830 −0.368915 0.929463i \(-0.620271\pi\)
−0.368915 + 0.929463i \(0.620271\pi\)
\(864\) 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 \(-0.514112\pi\)
−0.0443207 + 0.999017i \(0.514112\pi\)
\(884\) 0 0
\(885\) 14.8872 0.500427
\(886\) 0 0
\(887\) 11.1876 0.375643 0.187821 0.982203i \(-0.439857\pi\)
0.187821 + 0.982203i \(0.439857\pi\)
\(888\) 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 \(-0.504356\pi\)
−0.0136829 + 0.999906i \(0.504356\pi\)
\(908\) 0 0
\(909\) −1.49200 −0.0494865
\(910\) 0 0
\(911\) 27.3211 0.905188 0.452594 0.891717i \(-0.350499\pi\)
0.452594 + 0.891717i \(0.350499\pi\)
\(912\) 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 \(-0.295359\pi\)
0.599519 + 0.800361i \(0.295359\pi\)
\(920\) 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 \(-0.264679\pi\)
0.673757 + 0.738953i \(0.264679\pi\)
\(948\) 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 \(-0.454566\pi\)
0.142250 + 0.989831i \(0.454566\pi\)
\(968\) 0 0
\(969\) −6.05079 −0.194379
\(970\) 0 0
\(971\) 18.9262 0.607370 0.303685 0.952772i \(-0.401783\pi\)
0.303685 + 0.952772i \(0.401783\pi\)
\(972\) 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 \(-0.499174\pi\)
0.00259409 + 0.999997i \(0.499174\pi\)
\(984\) 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 \(-0.627555\pi\)
−0.390086 + 0.920778i \(0.627555\pi\)
\(992\) 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.m.1.1 yes 3
4.3 odd 2 3040.2.a.l.1.3 3
8.3 odd 2 6080.2.a.bt.1.1 3
8.5 even 2 6080.2.a.bu.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3040.2.a.l.1.3 3 4.3 odd 2
3040.2.a.m.1.1 yes 3 1.1 even 1 trivial
6080.2.a.bt.1.1 3 8.3 odd 2
6080.2.a.bu.1.3 3 8.5 even 2