Properties

Label 3040.2.a.e.1.2
Level $3040$
Weight $2$
Character 3040.1
Self dual yes
Analytic conductor $24.275$
Analytic rank $1$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 4 \) 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 1.2
Root \(-1.56155\) 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)
 
\(f(q)\) \(=\) \(q+1.56155 q^{3} -1.00000 q^{5} +0.438447 q^{7} -0.561553 q^{9} +O(q^{10})\) \(q+1.56155 q^{3} -1.00000 q^{5} +0.438447 q^{7} -0.561553 q^{9} -4.00000 q^{11} -1.56155 q^{13} -1.56155 q^{15} +3.56155 q^{17} -1.00000 q^{19} +0.684658 q^{21} +6.68466 q^{23} +1.00000 q^{25} -5.56155 q^{27} -7.56155 q^{29} +3.12311 q^{31} -6.24621 q^{33} -0.438447 q^{35} +7.12311 q^{37} -2.43845 q^{39} -8.24621 q^{41} -2.00000 q^{43} +0.561553 q^{45} -5.12311 q^{47} -6.80776 q^{49} +5.56155 q^{51} +2.43845 q^{53} +4.00000 q^{55} -1.56155 q^{57} -7.80776 q^{59} -13.1231 q^{61} -0.246211 q^{63} +1.56155 q^{65} -6.43845 q^{67} +10.4384 q^{69} -8.00000 q^{71} -5.80776 q^{73} +1.56155 q^{75} -1.75379 q^{77} -3.12311 q^{79} -7.00000 q^{81} -14.0000 q^{83} -3.56155 q^{85} -11.8078 q^{87} -10.0000 q^{89} -0.684658 q^{91} +4.87689 q^{93} +1.00000 q^{95} +7.12311 q^{97} +2.24621 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{3} - 2 q^{5} + 5 q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{3} - 2 q^{5} + 5 q^{7} + 3 q^{9} - 8 q^{11} + q^{13} + q^{15} + 3 q^{17} - 2 q^{19} - 11 q^{21} + q^{23} + 2 q^{25} - 7 q^{27} - 11 q^{29} - 2 q^{31} + 4 q^{33} - 5 q^{35} + 6 q^{37} - 9 q^{39} - 4 q^{43} - 3 q^{45} - 2 q^{47} + 7 q^{49} + 7 q^{51} + 9 q^{53} + 8 q^{55} + q^{57} + 5 q^{59} - 18 q^{61} + 16 q^{63} - q^{65} - 17 q^{67} + 25 q^{69} - 16 q^{71} + 9 q^{73} - q^{75} - 20 q^{77} + 2 q^{79} - 14 q^{81} - 28 q^{83} - 3 q^{85} - 3 q^{87} - 20 q^{89} + 11 q^{91} + 18 q^{93} + 2 q^{95} + 6 q^{97} - 12 q^{99}+O(q^{100}) \) 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
\(3\) 1.56155 0.901563 0.450781 0.892634i \(-0.351145\pi\)
0.450781 + 0.892634i \(0.351145\pi\)
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 0.438447 0.165717 0.0828587 0.996561i \(-0.473595\pi\)
0.0828587 + 0.996561i \(0.473595\pi\)
\(8\) 0 0
\(9\) −0.561553 −0.187184
\(10\) 0 0
\(11\) −4.00000 −1.20605 −0.603023 0.797724i \(-0.706037\pi\)
−0.603023 + 0.797724i \(0.706037\pi\)
\(12\) 0 0
\(13\) −1.56155 −0.433097 −0.216548 0.976272i \(-0.569480\pi\)
−0.216548 + 0.976272i \(0.569480\pi\)
\(14\) 0 0
\(15\) −1.56155 −0.403191
\(16\) 0 0
\(17\) 3.56155 0.863803 0.431902 0.901921i \(-0.357843\pi\)
0.431902 + 0.901921i \(0.357843\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) 0.684658 0.149405
\(22\) 0 0
\(23\) 6.68466 1.39385 0.696924 0.717145i \(-0.254552\pi\)
0.696924 + 0.717145i \(0.254552\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −5.56155 −1.07032
\(28\) 0 0
\(29\) −7.56155 −1.40415 −0.702073 0.712105i \(-0.747742\pi\)
−0.702073 + 0.712105i \(0.747742\pi\)
\(30\) 0 0
\(31\) 3.12311 0.560926 0.280463 0.959865i \(-0.409512\pi\)
0.280463 + 0.959865i \(0.409512\pi\)
\(32\) 0 0
\(33\) −6.24621 −1.08733
\(34\) 0 0
\(35\) −0.438447 −0.0741111
\(36\) 0 0
\(37\) 7.12311 1.17103 0.585516 0.810661i \(-0.300892\pi\)
0.585516 + 0.810661i \(0.300892\pi\)
\(38\) 0 0
\(39\) −2.43845 −0.390464
\(40\) 0 0
\(41\) −8.24621 −1.28784 −0.643921 0.765092i \(-0.722693\pi\)
−0.643921 + 0.765092i \(0.722693\pi\)
\(42\) 0 0
\(43\) −2.00000 −0.304997 −0.152499 0.988304i \(-0.548732\pi\)
−0.152499 + 0.988304i \(0.548732\pi\)
\(44\) 0 0
\(45\) 0.561553 0.0837114
\(46\) 0 0
\(47\) −5.12311 −0.747282 −0.373641 0.927573i \(-0.621891\pi\)
−0.373641 + 0.927573i \(0.621891\pi\)
\(48\) 0 0
\(49\) −6.80776 −0.972538
\(50\) 0 0
\(51\) 5.56155 0.778773
\(52\) 0 0
\(53\) 2.43845 0.334946 0.167473 0.985877i \(-0.446439\pi\)
0.167473 + 0.985877i \(0.446439\pi\)
\(54\) 0 0
\(55\) 4.00000 0.539360
\(56\) 0 0
\(57\) −1.56155 −0.206833
\(58\) 0 0
\(59\) −7.80776 −1.01648 −0.508242 0.861214i \(-0.669705\pi\)
−0.508242 + 0.861214i \(0.669705\pi\)
\(60\) 0 0
\(61\) −13.1231 −1.68024 −0.840121 0.542399i \(-0.817516\pi\)
−0.840121 + 0.542399i \(0.817516\pi\)
\(62\) 0 0
\(63\) −0.246211 −0.0310197
\(64\) 0 0
\(65\) 1.56155 0.193687
\(66\) 0 0
\(67\) −6.43845 −0.786582 −0.393291 0.919414i \(-0.628663\pi\)
−0.393291 + 0.919414i \(0.628663\pi\)
\(68\) 0 0
\(69\) 10.4384 1.25664
\(70\) 0 0
\(71\) −8.00000 −0.949425 −0.474713 0.880141i \(-0.657448\pi\)
−0.474713 + 0.880141i \(0.657448\pi\)
\(72\) 0 0
\(73\) −5.80776 −0.679747 −0.339874 0.940471i \(-0.610384\pi\)
−0.339874 + 0.940471i \(0.610384\pi\)
\(74\) 0 0
\(75\) 1.56155 0.180313
\(76\) 0 0
\(77\) −1.75379 −0.199863
\(78\) 0 0
\(79\) −3.12311 −0.351377 −0.175688 0.984446i \(-0.556215\pi\)
−0.175688 + 0.984446i \(0.556215\pi\)
\(80\) 0 0
\(81\) −7.00000 −0.777778
\(82\) 0 0
\(83\) −14.0000 −1.53670 −0.768350 0.640030i \(-0.778922\pi\)
−0.768350 + 0.640030i \(0.778922\pi\)
\(84\) 0 0
\(85\) −3.56155 −0.386305
\(86\) 0 0
\(87\) −11.8078 −1.26593
\(88\) 0 0
\(89\) −10.0000 −1.06000 −0.529999 0.847998i \(-0.677808\pi\)
−0.529999 + 0.847998i \(0.677808\pi\)
\(90\) 0 0
\(91\) −0.684658 −0.0717717
\(92\) 0 0
\(93\) 4.87689 0.505710
\(94\) 0 0
\(95\) 1.00000 0.102598
\(96\) 0 0
\(97\) 7.12311 0.723242 0.361621 0.932325i \(-0.382223\pi\)
0.361621 + 0.932325i \(0.382223\pi\)
\(98\) 0 0
\(99\) 2.24621 0.225753
\(100\) 0 0
\(101\) 5.12311 0.509768 0.254884 0.966972i \(-0.417963\pi\)
0.254884 + 0.966972i \(0.417963\pi\)
\(102\) 0 0
\(103\) −16.0000 −1.57653 −0.788263 0.615338i \(-0.789020\pi\)
−0.788263 + 0.615338i \(0.789020\pi\)
\(104\) 0 0
\(105\) −0.684658 −0.0668158
\(106\) 0 0
\(107\) 0.684658 0.0661884 0.0330942 0.999452i \(-0.489464\pi\)
0.0330942 + 0.999452i \(0.489464\pi\)
\(108\) 0 0
\(109\) 15.5616 1.49053 0.745263 0.666770i \(-0.232324\pi\)
0.745263 + 0.666770i \(0.232324\pi\)
\(110\) 0 0
\(111\) 11.1231 1.05576
\(112\) 0 0
\(113\) 10.2462 0.963882 0.481941 0.876204i \(-0.339932\pi\)
0.481941 + 0.876204i \(0.339932\pi\)
\(114\) 0 0
\(115\) −6.68466 −0.623348
\(116\) 0 0
\(117\) 0.876894 0.0810689
\(118\) 0 0
\(119\) 1.56155 0.143147
\(120\) 0 0
\(121\) 5.00000 0.454545
\(122\) 0 0
\(123\) −12.8769 −1.16107
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 9.36932 0.831392 0.415696 0.909504i \(-0.363538\pi\)
0.415696 + 0.909504i \(0.363538\pi\)
\(128\) 0 0
\(129\) −3.12311 −0.274974
\(130\) 0 0
\(131\) −16.0000 −1.39793 −0.698963 0.715158i \(-0.746355\pi\)
−0.698963 + 0.715158i \(0.746355\pi\)
\(132\) 0 0
\(133\) −0.438447 −0.0380182
\(134\) 0 0
\(135\) 5.56155 0.478662
\(136\) 0 0
\(137\) 17.8078 1.52142 0.760710 0.649092i \(-0.224851\pi\)
0.760710 + 0.649092i \(0.224851\pi\)
\(138\) 0 0
\(139\) −14.2462 −1.20835 −0.604174 0.796852i \(-0.706497\pi\)
−0.604174 + 0.796852i \(0.706497\pi\)
\(140\) 0 0
\(141\) −8.00000 −0.673722
\(142\) 0 0
\(143\) 6.24621 0.522334
\(144\) 0 0
\(145\) 7.56155 0.627953
\(146\) 0 0
\(147\) −10.6307 −0.876804
\(148\) 0 0
\(149\) −5.12311 −0.419701 −0.209851 0.977733i \(-0.567298\pi\)
−0.209851 + 0.977733i \(0.567298\pi\)
\(150\) 0 0
\(151\) 4.87689 0.396876 0.198438 0.980113i \(-0.436413\pi\)
0.198438 + 0.980113i \(0.436413\pi\)
\(152\) 0 0
\(153\) −2.00000 −0.161690
\(154\) 0 0
\(155\) −3.12311 −0.250854
\(156\) 0 0
\(157\) 6.00000 0.478852 0.239426 0.970915i \(-0.423041\pi\)
0.239426 + 0.970915i \(0.423041\pi\)
\(158\) 0 0
\(159\) 3.80776 0.301975
\(160\) 0 0
\(161\) 2.93087 0.230985
\(162\) 0 0
\(163\) 20.2462 1.58581 0.792903 0.609348i \(-0.208569\pi\)
0.792903 + 0.609348i \(0.208569\pi\)
\(164\) 0 0
\(165\) 6.24621 0.486267
\(166\) 0 0
\(167\) 5.36932 0.415490 0.207745 0.978183i \(-0.433388\pi\)
0.207745 + 0.978183i \(0.433388\pi\)
\(168\) 0 0
\(169\) −10.5616 −0.812427
\(170\) 0 0
\(171\) 0.561553 0.0429430
\(172\) 0 0
\(173\) 11.1231 0.845674 0.422837 0.906206i \(-0.361034\pi\)
0.422837 + 0.906206i \(0.361034\pi\)
\(174\) 0 0
\(175\) 0.438447 0.0331435
\(176\) 0 0
\(177\) −12.1922 −0.916425
\(178\) 0 0
\(179\) −2.24621 −0.167890 −0.0839449 0.996470i \(-0.526752\pi\)
−0.0839449 + 0.996470i \(0.526752\pi\)
\(180\) 0 0
\(181\) 24.2462 1.80221 0.901103 0.433604i \(-0.142758\pi\)
0.901103 + 0.433604i \(0.142758\pi\)
\(182\) 0 0
\(183\) −20.4924 −1.51484
\(184\) 0 0
\(185\) −7.12311 −0.523701
\(186\) 0 0
\(187\) −14.2462 −1.04179
\(188\) 0 0
\(189\) −2.43845 −0.177371
\(190\) 0 0
\(191\) 16.6847 1.20726 0.603630 0.797265i \(-0.293721\pi\)
0.603630 + 0.797265i \(0.293721\pi\)
\(192\) 0 0
\(193\) 0.876894 0.0631202 0.0315601 0.999502i \(-0.489952\pi\)
0.0315601 + 0.999502i \(0.489952\pi\)
\(194\) 0 0
\(195\) 2.43845 0.174621
\(196\) 0 0
\(197\) −7.75379 −0.552435 −0.276217 0.961095i \(-0.589081\pi\)
−0.276217 + 0.961095i \(0.589081\pi\)
\(198\) 0 0
\(199\) −9.56155 −0.677801 −0.338900 0.940822i \(-0.610055\pi\)
−0.338900 + 0.940822i \(0.610055\pi\)
\(200\) 0 0
\(201\) −10.0540 −0.709153
\(202\) 0 0
\(203\) −3.31534 −0.232691
\(204\) 0 0
\(205\) 8.24621 0.575940
\(206\) 0 0
\(207\) −3.75379 −0.260906
\(208\) 0 0
\(209\) 4.00000 0.276686
\(210\) 0 0
\(211\) −14.0540 −0.967516 −0.483758 0.875202i \(-0.660728\pi\)
−0.483758 + 0.875202i \(0.660728\pi\)
\(212\) 0 0
\(213\) −12.4924 −0.855967
\(214\) 0 0
\(215\) 2.00000 0.136399
\(216\) 0 0
\(217\) 1.36932 0.0929553
\(218\) 0 0
\(219\) −9.06913 −0.612835
\(220\) 0 0
\(221\) −5.56155 −0.374111
\(222\) 0 0
\(223\) −9.36932 −0.627416 −0.313708 0.949520i \(-0.601571\pi\)
−0.313708 + 0.949520i \(0.601571\pi\)
\(224\) 0 0
\(225\) −0.561553 −0.0374369
\(226\) 0 0
\(227\) −16.6847 −1.10740 −0.553700 0.832716i \(-0.686784\pi\)
−0.553700 + 0.832716i \(0.686784\pi\)
\(228\) 0 0
\(229\) 20.7386 1.37045 0.685224 0.728333i \(-0.259704\pi\)
0.685224 + 0.728333i \(0.259704\pi\)
\(230\) 0 0
\(231\) −2.73863 −0.180189
\(232\) 0 0
\(233\) 6.49242 0.425333 0.212666 0.977125i \(-0.431785\pi\)
0.212666 + 0.977125i \(0.431785\pi\)
\(234\) 0 0
\(235\) 5.12311 0.334195
\(236\) 0 0
\(237\) −4.87689 −0.316788
\(238\) 0 0
\(239\) −23.8078 −1.54000 −0.769998 0.638046i \(-0.779743\pi\)
−0.769998 + 0.638046i \(0.779743\pi\)
\(240\) 0 0
\(241\) 27.8617 1.79473 0.897366 0.441287i \(-0.145478\pi\)
0.897366 + 0.441287i \(0.145478\pi\)
\(242\) 0 0
\(243\) 5.75379 0.369106
\(244\) 0 0
\(245\) 6.80776 0.434932
\(246\) 0 0
\(247\) 1.56155 0.0993592
\(248\) 0 0
\(249\) −21.8617 −1.38543
\(250\) 0 0
\(251\) 22.2462 1.40417 0.702084 0.712094i \(-0.252253\pi\)
0.702084 + 0.712094i \(0.252253\pi\)
\(252\) 0 0
\(253\) −26.7386 −1.68104
\(254\) 0 0
\(255\) −5.56155 −0.348278
\(256\) 0 0
\(257\) −6.63068 −0.413611 −0.206805 0.978382i \(-0.566307\pi\)
−0.206805 + 0.978382i \(0.566307\pi\)
\(258\) 0 0
\(259\) 3.12311 0.194060
\(260\) 0 0
\(261\) 4.24621 0.262834
\(262\) 0 0
\(263\) 15.3693 0.947713 0.473856 0.880602i \(-0.342862\pi\)
0.473856 + 0.880602i \(0.342862\pi\)
\(264\) 0 0
\(265\) −2.43845 −0.149793
\(266\) 0 0
\(267\) −15.6155 −0.955655
\(268\) 0 0
\(269\) −7.75379 −0.472757 −0.236378 0.971661i \(-0.575960\pi\)
−0.236378 + 0.971661i \(0.575960\pi\)
\(270\) 0 0
\(271\) 25.1771 1.52940 0.764699 0.644387i \(-0.222887\pi\)
0.764699 + 0.644387i \(0.222887\pi\)
\(272\) 0 0
\(273\) −1.06913 −0.0647067
\(274\) 0 0
\(275\) −4.00000 −0.241209
\(276\) 0 0
\(277\) 21.6155 1.29875 0.649376 0.760468i \(-0.275030\pi\)
0.649376 + 0.760468i \(0.275030\pi\)
\(278\) 0 0
\(279\) −1.75379 −0.104997
\(280\) 0 0
\(281\) 7.75379 0.462552 0.231276 0.972888i \(-0.425710\pi\)
0.231276 + 0.972888i \(0.425710\pi\)
\(282\) 0 0
\(283\) −6.00000 −0.356663 −0.178331 0.983970i \(-0.557070\pi\)
−0.178331 + 0.983970i \(0.557070\pi\)
\(284\) 0 0
\(285\) 1.56155 0.0924984
\(286\) 0 0
\(287\) −3.61553 −0.213418
\(288\) 0 0
\(289\) −4.31534 −0.253844
\(290\) 0 0
\(291\) 11.1231 0.652048
\(292\) 0 0
\(293\) −22.9309 −1.33964 −0.669818 0.742525i \(-0.733628\pi\)
−0.669818 + 0.742525i \(0.733628\pi\)
\(294\) 0 0
\(295\) 7.80776 0.454586
\(296\) 0 0
\(297\) 22.2462 1.29086
\(298\) 0 0
\(299\) −10.4384 −0.603671
\(300\) 0 0
\(301\) −0.876894 −0.0505434
\(302\) 0 0
\(303\) 8.00000 0.459588
\(304\) 0 0
\(305\) 13.1231 0.751427
\(306\) 0 0
\(307\) −22.2462 −1.26966 −0.634829 0.772653i \(-0.718930\pi\)
−0.634829 + 0.772653i \(0.718930\pi\)
\(308\) 0 0
\(309\) −24.9848 −1.42134
\(310\) 0 0
\(311\) −9.56155 −0.542186 −0.271093 0.962553i \(-0.587385\pi\)
−0.271093 + 0.962553i \(0.587385\pi\)
\(312\) 0 0
\(313\) 15.1771 0.857859 0.428930 0.903338i \(-0.358891\pi\)
0.428930 + 0.903338i \(0.358891\pi\)
\(314\) 0 0
\(315\) 0.246211 0.0138724
\(316\) 0 0
\(317\) 9.56155 0.537030 0.268515 0.963275i \(-0.413467\pi\)
0.268515 + 0.963275i \(0.413467\pi\)
\(318\) 0 0
\(319\) 30.2462 1.69346
\(320\) 0 0
\(321\) 1.06913 0.0596730
\(322\) 0 0
\(323\) −3.56155 −0.198170
\(324\) 0 0
\(325\) −1.56155 −0.0866194
\(326\) 0 0
\(327\) 24.3002 1.34380
\(328\) 0 0
\(329\) −2.24621 −0.123838
\(330\) 0 0
\(331\) 7.80776 0.429154 0.214577 0.976707i \(-0.431163\pi\)
0.214577 + 0.976707i \(0.431163\pi\)
\(332\) 0 0
\(333\) −4.00000 −0.219199
\(334\) 0 0
\(335\) 6.43845 0.351770
\(336\) 0 0
\(337\) 23.1231 1.25960 0.629798 0.776759i \(-0.283138\pi\)
0.629798 + 0.776759i \(0.283138\pi\)
\(338\) 0 0
\(339\) 16.0000 0.869001
\(340\) 0 0
\(341\) −12.4924 −0.676503
\(342\) 0 0
\(343\) −6.05398 −0.326884
\(344\) 0 0
\(345\) −10.4384 −0.561987
\(346\) 0 0
\(347\) −18.0000 −0.966291 −0.483145 0.875540i \(-0.660506\pi\)
−0.483145 + 0.875540i \(0.660506\pi\)
\(348\) 0 0
\(349\) −2.00000 −0.107058 −0.0535288 0.998566i \(-0.517047\pi\)
−0.0535288 + 0.998566i \(0.517047\pi\)
\(350\) 0 0
\(351\) 8.68466 0.463553
\(352\) 0 0
\(353\) 25.4233 1.35315 0.676573 0.736376i \(-0.263464\pi\)
0.676573 + 0.736376i \(0.263464\pi\)
\(354\) 0 0
\(355\) 8.00000 0.424596
\(356\) 0 0
\(357\) 2.43845 0.129056
\(358\) 0 0
\(359\) −22.9309 −1.21025 −0.605123 0.796132i \(-0.706876\pi\)
−0.605123 + 0.796132i \(0.706876\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) 7.80776 0.409801
\(364\) 0 0
\(365\) 5.80776 0.303992
\(366\) 0 0
\(367\) 26.8769 1.40296 0.701481 0.712688i \(-0.252522\pi\)
0.701481 + 0.712688i \(0.252522\pi\)
\(368\) 0 0
\(369\) 4.63068 0.241064
\(370\) 0 0
\(371\) 1.06913 0.0555065
\(372\) 0 0
\(373\) −4.19224 −0.217066 −0.108533 0.994093i \(-0.534615\pi\)
−0.108533 + 0.994093i \(0.534615\pi\)
\(374\) 0 0
\(375\) −1.56155 −0.0806382
\(376\) 0 0
\(377\) 11.8078 0.608131
\(378\) 0 0
\(379\) 20.6847 1.06250 0.531250 0.847215i \(-0.321723\pi\)
0.531250 + 0.847215i \(0.321723\pi\)
\(380\) 0 0
\(381\) 14.6307 0.749553
\(382\) 0 0
\(383\) −22.7386 −1.16189 −0.580945 0.813943i \(-0.697317\pi\)
−0.580945 + 0.813943i \(0.697317\pi\)
\(384\) 0 0
\(385\) 1.75379 0.0893814
\(386\) 0 0
\(387\) 1.12311 0.0570907
\(388\) 0 0
\(389\) −1.12311 −0.0569437 −0.0284719 0.999595i \(-0.509064\pi\)
−0.0284719 + 0.999595i \(0.509064\pi\)
\(390\) 0 0
\(391\) 23.8078 1.20401
\(392\) 0 0
\(393\) −24.9848 −1.26032
\(394\) 0 0
\(395\) 3.12311 0.157140
\(396\) 0 0
\(397\) −31.8617 −1.59909 −0.799547 0.600603i \(-0.794927\pi\)
−0.799547 + 0.600603i \(0.794927\pi\)
\(398\) 0 0
\(399\) −0.684658 −0.0342758
\(400\) 0 0
\(401\) 28.7386 1.43514 0.717569 0.696487i \(-0.245255\pi\)
0.717569 + 0.696487i \(0.245255\pi\)
\(402\) 0 0
\(403\) −4.87689 −0.242935
\(404\) 0 0
\(405\) 7.00000 0.347833
\(406\) 0 0
\(407\) −28.4924 −1.41232
\(408\) 0 0
\(409\) −24.7386 −1.22325 −0.611623 0.791149i \(-0.709483\pi\)
−0.611623 + 0.791149i \(0.709483\pi\)
\(410\) 0 0
\(411\) 27.8078 1.37166
\(412\) 0 0
\(413\) −3.42329 −0.168449
\(414\) 0 0
\(415\) 14.0000 0.687233
\(416\) 0 0
\(417\) −22.2462 −1.08940
\(418\) 0 0
\(419\) −31.1231 −1.52046 −0.760232 0.649652i \(-0.774915\pi\)
−0.760232 + 0.649652i \(0.774915\pi\)
\(420\) 0 0
\(421\) −34.3002 −1.67169 −0.835844 0.548966i \(-0.815021\pi\)
−0.835844 + 0.548966i \(0.815021\pi\)
\(422\) 0 0
\(423\) 2.87689 0.139879
\(424\) 0 0
\(425\) 3.56155 0.172761
\(426\) 0 0
\(427\) −5.75379 −0.278445
\(428\) 0 0
\(429\) 9.75379 0.470917
\(430\) 0 0
\(431\) −12.4924 −0.601739 −0.300869 0.953665i \(-0.597277\pi\)
−0.300869 + 0.953665i \(0.597277\pi\)
\(432\) 0 0
\(433\) −4.00000 −0.192228 −0.0961139 0.995370i \(-0.530641\pi\)
−0.0961139 + 0.995370i \(0.530641\pi\)
\(434\) 0 0
\(435\) 11.8078 0.566139
\(436\) 0 0
\(437\) −6.68466 −0.319771
\(438\) 0 0
\(439\) 6.63068 0.316465 0.158233 0.987402i \(-0.449420\pi\)
0.158233 + 0.987402i \(0.449420\pi\)
\(440\) 0 0
\(441\) 3.82292 0.182044
\(442\) 0 0
\(443\) −29.6155 −1.40708 −0.703538 0.710658i \(-0.748398\pi\)
−0.703538 + 0.710658i \(0.748398\pi\)
\(444\) 0 0
\(445\) 10.0000 0.474045
\(446\) 0 0
\(447\) −8.00000 −0.378387
\(448\) 0 0
\(449\) −32.7386 −1.54503 −0.772516 0.634996i \(-0.781002\pi\)
−0.772516 + 0.634996i \(0.781002\pi\)
\(450\) 0 0
\(451\) 32.9848 1.55320
\(452\) 0 0
\(453\) 7.61553 0.357809
\(454\) 0 0
\(455\) 0.684658 0.0320973
\(456\) 0 0
\(457\) 12.4384 0.581846 0.290923 0.956746i \(-0.406038\pi\)
0.290923 + 0.956746i \(0.406038\pi\)
\(458\) 0 0
\(459\) −19.8078 −0.924547
\(460\) 0 0
\(461\) 7.75379 0.361130 0.180565 0.983563i \(-0.442207\pi\)
0.180565 + 0.983563i \(0.442207\pi\)
\(462\) 0 0
\(463\) 35.8617 1.66664 0.833318 0.552794i \(-0.186438\pi\)
0.833318 + 0.552794i \(0.186438\pi\)
\(464\) 0 0
\(465\) −4.87689 −0.226161
\(466\) 0 0
\(467\) 27.8617 1.28929 0.644644 0.764483i \(-0.277006\pi\)
0.644644 + 0.764483i \(0.277006\pi\)
\(468\) 0 0
\(469\) −2.82292 −0.130350
\(470\) 0 0
\(471\) 9.36932 0.431715
\(472\) 0 0
\(473\) 8.00000 0.367840
\(474\) 0 0
\(475\) −1.00000 −0.0458831
\(476\) 0 0
\(477\) −1.36932 −0.0626967
\(478\) 0 0
\(479\) −24.0000 −1.09659 −0.548294 0.836286i \(-0.684723\pi\)
−0.548294 + 0.836286i \(0.684723\pi\)
\(480\) 0 0
\(481\) −11.1231 −0.507170
\(482\) 0 0
\(483\) 4.57671 0.208247
\(484\) 0 0
\(485\) −7.12311 −0.323444
\(486\) 0 0
\(487\) 2.63068 0.119208 0.0596038 0.998222i \(-0.481016\pi\)
0.0596038 + 0.998222i \(0.481016\pi\)
\(488\) 0 0
\(489\) 31.6155 1.42970
\(490\) 0 0
\(491\) −11.1231 −0.501979 −0.250989 0.967990i \(-0.580756\pi\)
−0.250989 + 0.967990i \(0.580756\pi\)
\(492\) 0 0
\(493\) −26.9309 −1.21291
\(494\) 0 0
\(495\) −2.24621 −0.100960
\(496\) 0 0
\(497\) −3.50758 −0.157336
\(498\) 0 0
\(499\) 20.8769 0.934578 0.467289 0.884105i \(-0.345231\pi\)
0.467289 + 0.884105i \(0.345231\pi\)
\(500\) 0 0
\(501\) 8.38447 0.374591
\(502\) 0 0
\(503\) 25.4233 1.13357 0.566784 0.823866i \(-0.308187\pi\)
0.566784 + 0.823866i \(0.308187\pi\)
\(504\) 0 0
\(505\) −5.12311 −0.227975
\(506\) 0 0
\(507\) −16.4924 −0.732454
\(508\) 0 0
\(509\) −2.00000 −0.0886484 −0.0443242 0.999017i \(-0.514113\pi\)
−0.0443242 + 0.999017i \(0.514113\pi\)
\(510\) 0 0
\(511\) −2.54640 −0.112646
\(512\) 0 0
\(513\) 5.56155 0.245549
\(514\) 0 0
\(515\) 16.0000 0.705044
\(516\) 0 0
\(517\) 20.4924 0.901256
\(518\) 0 0
\(519\) 17.3693 0.762428
\(520\) 0 0
\(521\) −1.12311 −0.0492042 −0.0246021 0.999697i \(-0.507832\pi\)
−0.0246021 + 0.999697i \(0.507832\pi\)
\(522\) 0 0
\(523\) 14.9309 0.652881 0.326441 0.945218i \(-0.394151\pi\)
0.326441 + 0.945218i \(0.394151\pi\)
\(524\) 0 0
\(525\) 0.684658 0.0298809
\(526\) 0 0
\(527\) 11.1231 0.484530
\(528\) 0 0
\(529\) 21.6847 0.942811
\(530\) 0 0
\(531\) 4.38447 0.190270
\(532\) 0 0
\(533\) 12.8769 0.557760
\(534\) 0 0
\(535\) −0.684658 −0.0296004
\(536\) 0 0
\(537\) −3.50758 −0.151363
\(538\) 0 0
\(539\) 27.2311 1.17292
\(540\) 0 0
\(541\) −25.1231 −1.08013 −0.540063 0.841624i \(-0.681600\pi\)
−0.540063 + 0.841624i \(0.681600\pi\)
\(542\) 0 0
\(543\) 37.8617 1.62480
\(544\) 0 0
\(545\) −15.5616 −0.666584
\(546\) 0 0
\(547\) −38.2462 −1.63529 −0.817645 0.575723i \(-0.804721\pi\)
−0.817645 + 0.575723i \(0.804721\pi\)
\(548\) 0 0
\(549\) 7.36932 0.314515
\(550\) 0 0
\(551\) 7.56155 0.322133
\(552\) 0 0
\(553\) −1.36932 −0.0582293
\(554\) 0 0
\(555\) −11.1231 −0.472150
\(556\) 0 0
\(557\) 4.63068 0.196208 0.0981042 0.995176i \(-0.468722\pi\)
0.0981042 + 0.995176i \(0.468722\pi\)
\(558\) 0 0
\(559\) 3.12311 0.132093
\(560\) 0 0
\(561\) −22.2462 −0.939236
\(562\) 0 0
\(563\) −18.7386 −0.789739 −0.394870 0.918737i \(-0.629210\pi\)
−0.394870 + 0.918737i \(0.629210\pi\)
\(564\) 0 0
\(565\) −10.2462 −0.431061
\(566\) 0 0
\(567\) −3.06913 −0.128891
\(568\) 0 0
\(569\) −9.12311 −0.382460 −0.191230 0.981545i \(-0.561248\pi\)
−0.191230 + 0.981545i \(0.561248\pi\)
\(570\) 0 0
\(571\) 2.63068 0.110091 0.0550453 0.998484i \(-0.482470\pi\)
0.0550453 + 0.998484i \(0.482470\pi\)
\(572\) 0 0
\(573\) 26.0540 1.08842
\(574\) 0 0
\(575\) 6.68466 0.278770
\(576\) 0 0
\(577\) −33.4233 −1.39143 −0.695715 0.718318i \(-0.744912\pi\)
−0.695715 + 0.718318i \(0.744912\pi\)
\(578\) 0 0
\(579\) 1.36932 0.0569069
\(580\) 0 0
\(581\) −6.13826 −0.254658
\(582\) 0 0
\(583\) −9.75379 −0.403961
\(584\) 0 0
\(585\) −0.876894 −0.0362551
\(586\) 0 0
\(587\) −32.2462 −1.33094 −0.665472 0.746423i \(-0.731770\pi\)
−0.665472 + 0.746423i \(0.731770\pi\)
\(588\) 0 0
\(589\) −3.12311 −0.128685
\(590\) 0 0
\(591\) −12.1080 −0.498055
\(592\) 0 0
\(593\) 8.24621 0.338631 0.169316 0.985562i \(-0.445844\pi\)
0.169316 + 0.985562i \(0.445844\pi\)
\(594\) 0 0
\(595\) −1.56155 −0.0640174
\(596\) 0 0
\(597\) −14.9309 −0.611080
\(598\) 0 0
\(599\) −31.6155 −1.29178 −0.645888 0.763432i \(-0.723513\pi\)
−0.645888 + 0.763432i \(0.723513\pi\)
\(600\) 0 0
\(601\) 0.738634 0.0301295 0.0150647 0.999887i \(-0.495205\pi\)
0.0150647 + 0.999887i \(0.495205\pi\)
\(602\) 0 0
\(603\) 3.61553 0.147236
\(604\) 0 0
\(605\) −5.00000 −0.203279
\(606\) 0 0
\(607\) −0.876894 −0.0355920 −0.0177960 0.999842i \(-0.505665\pi\)
−0.0177960 + 0.999842i \(0.505665\pi\)
\(608\) 0 0
\(609\) −5.17708 −0.209786
\(610\) 0 0
\(611\) 8.00000 0.323645
\(612\) 0 0
\(613\) −16.6307 −0.671707 −0.335853 0.941914i \(-0.609025\pi\)
−0.335853 + 0.941914i \(0.609025\pi\)
\(614\) 0 0
\(615\) 12.8769 0.519246
\(616\) 0 0
\(617\) −42.9848 −1.73050 −0.865252 0.501337i \(-0.832842\pi\)
−0.865252 + 0.501337i \(0.832842\pi\)
\(618\) 0 0
\(619\) 14.2462 0.572604 0.286302 0.958139i \(-0.407574\pi\)
0.286302 + 0.958139i \(0.407574\pi\)
\(620\) 0 0
\(621\) −37.1771 −1.49186
\(622\) 0 0
\(623\) −4.38447 −0.175660
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 6.24621 0.249450
\(628\) 0 0
\(629\) 25.3693 1.01154
\(630\) 0 0
\(631\) 22.7386 0.905211 0.452605 0.891711i \(-0.350495\pi\)
0.452605 + 0.891711i \(0.350495\pi\)
\(632\) 0 0
\(633\) −21.9460 −0.872276
\(634\) 0 0
\(635\) −9.36932 −0.371810
\(636\) 0 0
\(637\) 10.6307 0.421203
\(638\) 0 0
\(639\) 4.49242 0.177717
\(640\) 0 0
\(641\) 1.12311 0.0443600 0.0221800 0.999754i \(-0.492939\pi\)
0.0221800 + 0.999754i \(0.492939\pi\)
\(642\) 0 0
\(643\) −18.4924 −0.729270 −0.364635 0.931151i \(-0.618806\pi\)
−0.364635 + 0.931151i \(0.618806\pi\)
\(644\) 0 0
\(645\) 3.12311 0.122972
\(646\) 0 0
\(647\) −5.31534 −0.208968 −0.104484 0.994527i \(-0.533319\pi\)
−0.104484 + 0.994527i \(0.533319\pi\)
\(648\) 0 0
\(649\) 31.2311 1.22593
\(650\) 0 0
\(651\) 2.13826 0.0838050
\(652\) 0 0
\(653\) 20.6307 0.807341 0.403671 0.914904i \(-0.367734\pi\)
0.403671 + 0.914904i \(0.367734\pi\)
\(654\) 0 0
\(655\) 16.0000 0.625172
\(656\) 0 0
\(657\) 3.26137 0.127238
\(658\) 0 0
\(659\) 47.8078 1.86233 0.931163 0.364603i \(-0.118795\pi\)
0.931163 + 0.364603i \(0.118795\pi\)
\(660\) 0 0
\(661\) −8.93087 −0.347371 −0.173685 0.984801i \(-0.555568\pi\)
−0.173685 + 0.984801i \(0.555568\pi\)
\(662\) 0 0
\(663\) −8.68466 −0.337284
\(664\) 0 0
\(665\) 0.438447 0.0170023
\(666\) 0 0
\(667\) −50.5464 −1.95716
\(668\) 0 0
\(669\) −14.6307 −0.565655
\(670\) 0 0
\(671\) 52.4924 2.02645
\(672\) 0 0
\(673\) −3.50758 −0.135207 −0.0676036 0.997712i \(-0.521535\pi\)
−0.0676036 + 0.997712i \(0.521535\pi\)
\(674\) 0 0
\(675\) −5.56155 −0.214064
\(676\) 0 0
\(677\) 35.8078 1.37620 0.688102 0.725614i \(-0.258444\pi\)
0.688102 + 0.725614i \(0.258444\pi\)
\(678\) 0 0
\(679\) 3.12311 0.119854
\(680\) 0 0
\(681\) −26.0540 −0.998391
\(682\) 0 0
\(683\) 17.7538 0.679330 0.339665 0.940547i \(-0.389686\pi\)
0.339665 + 0.940547i \(0.389686\pi\)
\(684\) 0 0
\(685\) −17.8078 −0.680400
\(686\) 0 0
\(687\) 32.3845 1.23554
\(688\) 0 0
\(689\) −3.80776 −0.145064
\(690\) 0 0
\(691\) 40.1080 1.52578 0.762889 0.646529i \(-0.223780\pi\)
0.762889 + 0.646529i \(0.223780\pi\)
\(692\) 0 0
\(693\) 0.984845 0.0374112
\(694\) 0 0
\(695\) 14.2462 0.540390
\(696\) 0 0
\(697\) −29.3693 −1.11244
\(698\) 0 0
\(699\) 10.1383 0.383464
\(700\) 0 0
\(701\) 2.87689 0.108659 0.0543294 0.998523i \(-0.482698\pi\)
0.0543294 + 0.998523i \(0.482698\pi\)
\(702\) 0 0
\(703\) −7.12311 −0.268653
\(704\) 0 0
\(705\) 8.00000 0.301297
\(706\) 0 0
\(707\) 2.24621 0.0844775
\(708\) 0 0
\(709\) −26.0000 −0.976450 −0.488225 0.872718i \(-0.662356\pi\)
−0.488225 + 0.872718i \(0.662356\pi\)
\(710\) 0 0
\(711\) 1.75379 0.0657722
\(712\) 0 0
\(713\) 20.8769 0.781846
\(714\) 0 0
\(715\) −6.24621 −0.233595
\(716\) 0 0
\(717\) −37.1771 −1.38840
\(718\) 0 0
\(719\) 27.4233 1.02272 0.511358 0.859368i \(-0.329143\pi\)
0.511358 + 0.859368i \(0.329143\pi\)
\(720\) 0 0
\(721\) −7.01515 −0.261258
\(722\) 0 0
\(723\) 43.5076 1.61806
\(724\) 0 0
\(725\) −7.56155 −0.280829
\(726\) 0 0
\(727\) −1.31534 −0.0487833 −0.0243917 0.999702i \(-0.507765\pi\)
−0.0243917 + 0.999702i \(0.507765\pi\)
\(728\) 0 0
\(729\) 29.9848 1.11055
\(730\) 0 0
\(731\) −7.12311 −0.263458
\(732\) 0 0
\(733\) −45.6155 −1.68485 −0.842424 0.538815i \(-0.818872\pi\)
−0.842424 + 0.538815i \(0.818872\pi\)
\(734\) 0 0
\(735\) 10.6307 0.392119
\(736\) 0 0
\(737\) 25.7538 0.948653
\(738\) 0 0
\(739\) 8.87689 0.326542 0.163271 0.986581i \(-0.447796\pi\)
0.163271 + 0.986581i \(0.447796\pi\)
\(740\) 0 0
\(741\) 2.43845 0.0895786
\(742\) 0 0
\(743\) −10.6307 −0.390002 −0.195001 0.980803i \(-0.562471\pi\)
−0.195001 + 0.980803i \(0.562471\pi\)
\(744\) 0 0
\(745\) 5.12311 0.187696
\(746\) 0 0
\(747\) 7.86174 0.287646
\(748\) 0 0
\(749\) 0.300187 0.0109686
\(750\) 0 0
\(751\) 6.63068 0.241957 0.120979 0.992655i \(-0.461397\pi\)
0.120979 + 0.992655i \(0.461397\pi\)
\(752\) 0 0
\(753\) 34.7386 1.26595
\(754\) 0 0
\(755\) −4.87689 −0.177488
\(756\) 0 0
\(757\) 36.7386 1.33529 0.667644 0.744481i \(-0.267303\pi\)
0.667644 + 0.744481i \(0.267303\pi\)
\(758\) 0 0
\(759\) −41.7538 −1.51557
\(760\) 0 0
\(761\) 30.6847 1.11232 0.556159 0.831076i \(-0.312275\pi\)
0.556159 + 0.831076i \(0.312275\pi\)
\(762\) 0 0
\(763\) 6.82292 0.247006
\(764\) 0 0
\(765\) 2.00000 0.0723102
\(766\) 0 0
\(767\) 12.1922 0.440236
\(768\) 0 0
\(769\) −7.17708 −0.258812 −0.129406 0.991592i \(-0.541307\pi\)
−0.129406 + 0.991592i \(0.541307\pi\)
\(770\) 0 0
\(771\) −10.3542 −0.372896
\(772\) 0 0
\(773\) −25.1771 −0.905557 −0.452778 0.891623i \(-0.649567\pi\)
−0.452778 + 0.891623i \(0.649567\pi\)
\(774\) 0 0
\(775\) 3.12311 0.112185
\(776\) 0 0
\(777\) 4.87689 0.174958
\(778\) 0 0
\(779\) 8.24621 0.295451
\(780\) 0 0
\(781\) 32.0000 1.14505
\(782\) 0 0
\(783\) 42.0540 1.50289
\(784\) 0 0
\(785\) −6.00000 −0.214149
\(786\) 0 0
\(787\) 2.05398 0.0732163 0.0366082 0.999330i \(-0.488345\pi\)
0.0366082 + 0.999330i \(0.488345\pi\)
\(788\) 0 0
\(789\) 24.0000 0.854423
\(790\) 0 0
\(791\) 4.49242 0.159732
\(792\) 0 0
\(793\) 20.4924 0.727707
\(794\) 0 0
\(795\) −3.80776 −0.135047
\(796\) 0 0
\(797\) 4.19224 0.148497 0.0742483 0.997240i \(-0.476344\pi\)
0.0742483 + 0.997240i \(0.476344\pi\)
\(798\) 0 0
\(799\) −18.2462 −0.645505
\(800\) 0 0
\(801\) 5.61553 0.198415
\(802\) 0 0
\(803\) 23.2311 0.819806
\(804\) 0 0
\(805\) −2.93087 −0.103300
\(806\) 0 0
\(807\) −12.1080 −0.426220
\(808\) 0 0
\(809\) 11.0691 0.389170 0.194585 0.980886i \(-0.437664\pi\)
0.194585 + 0.980886i \(0.437664\pi\)
\(810\) 0 0
\(811\) 55.0388 1.93267 0.966337 0.257279i \(-0.0828259\pi\)
0.966337 + 0.257279i \(0.0828259\pi\)
\(812\) 0 0
\(813\) 39.3153 1.37885
\(814\) 0 0
\(815\) −20.2462 −0.709194
\(816\) 0 0
\(817\) 2.00000 0.0699711
\(818\) 0 0
\(819\) 0.384472 0.0134345
\(820\) 0 0
\(821\) 13.1231 0.458000 0.229000 0.973426i \(-0.426454\pi\)
0.229000 + 0.973426i \(0.426454\pi\)
\(822\) 0 0
\(823\) −41.4233 −1.44393 −0.721963 0.691932i \(-0.756760\pi\)
−0.721963 + 0.691932i \(0.756760\pi\)
\(824\) 0 0
\(825\) −6.24621 −0.217465
\(826\) 0 0
\(827\) −1.94602 −0.0676699 −0.0338350 0.999427i \(-0.510772\pi\)
−0.0338350 + 0.999427i \(0.510772\pi\)
\(828\) 0 0
\(829\) 47.5616 1.65188 0.825941 0.563757i \(-0.190645\pi\)
0.825941 + 0.563757i \(0.190645\pi\)
\(830\) 0 0
\(831\) 33.7538 1.17091
\(832\) 0 0
\(833\) −24.2462 −0.840081
\(834\) 0 0
\(835\) −5.36932 −0.185813
\(836\) 0 0
\(837\) −17.3693 −0.600371
\(838\) 0 0
\(839\) −47.2311 −1.63060 −0.815299 0.579041i \(-0.803427\pi\)
−0.815299 + 0.579041i \(0.803427\pi\)
\(840\) 0 0
\(841\) 28.1771 0.971623
\(842\) 0 0
\(843\) 12.1080 0.417020
\(844\) 0 0
\(845\) 10.5616 0.363328
\(846\) 0 0
\(847\) 2.19224 0.0753261
\(848\) 0 0
\(849\) −9.36932 −0.321554
\(850\) 0 0
\(851\) 47.6155 1.63224
\(852\) 0 0
\(853\) 4.73863 0.162248 0.0811239 0.996704i \(-0.474149\pi\)
0.0811239 + 0.996704i \(0.474149\pi\)
\(854\) 0 0
\(855\) −0.561553 −0.0192047
\(856\) 0 0
\(857\) 40.9848 1.40002 0.700008 0.714135i \(-0.253180\pi\)
0.700008 + 0.714135i \(0.253180\pi\)
\(858\) 0 0
\(859\) −54.7386 −1.86766 −0.933829 0.357720i \(-0.883554\pi\)
−0.933829 + 0.357720i \(0.883554\pi\)
\(860\) 0 0
\(861\) −5.64584 −0.192410
\(862\) 0 0
\(863\) −49.4773 −1.68423 −0.842113 0.539301i \(-0.818688\pi\)
−0.842113 + 0.539301i \(0.818688\pi\)
\(864\) 0 0
\(865\) −11.1231 −0.378197
\(866\) 0 0
\(867\) −6.73863 −0.228856
\(868\) 0 0
\(869\) 12.4924 0.423776
\(870\) 0 0
\(871\) 10.0540 0.340666
\(872\) 0 0
\(873\) −4.00000 −0.135379
\(874\) 0 0
\(875\) −0.438447 −0.0148222
\(876\) 0 0
\(877\) −4.19224 −0.141562 −0.0707809 0.997492i \(-0.522549\pi\)
−0.0707809 + 0.997492i \(0.522549\pi\)
\(878\) 0 0
\(879\) −35.8078 −1.20777
\(880\) 0 0
\(881\) 6.49242 0.218735 0.109368 0.994001i \(-0.465117\pi\)
0.109368 + 0.994001i \(0.465117\pi\)
\(882\) 0 0
\(883\) 36.2462 1.21978 0.609891 0.792485i \(-0.291213\pi\)
0.609891 + 0.792485i \(0.291213\pi\)
\(884\) 0 0
\(885\) 12.1922 0.409838
\(886\) 0 0
\(887\) −24.0000 −0.805841 −0.402921 0.915235i \(-0.632005\pi\)
−0.402921 + 0.915235i \(0.632005\pi\)
\(888\) 0 0
\(889\) 4.10795 0.137776
\(890\) 0 0
\(891\) 28.0000 0.938035
\(892\) 0 0
\(893\) 5.12311 0.171438
\(894\) 0 0
\(895\) 2.24621 0.0750826
\(896\) 0 0
\(897\) −16.3002 −0.544247
\(898\) 0 0
\(899\) −23.6155 −0.787622
\(900\) 0 0
\(901\) 8.68466 0.289328
\(902\) 0 0
\(903\) −1.36932 −0.0455680
\(904\) 0 0
\(905\) −24.2462 −0.805971
\(906\) 0 0
\(907\) 19.4233 0.644940 0.322470 0.946580i \(-0.395487\pi\)
0.322470 + 0.946580i \(0.395487\pi\)
\(908\) 0 0
\(909\) −2.87689 −0.0954206
\(910\) 0 0
\(911\) 26.7386 0.885890 0.442945 0.896549i \(-0.353934\pi\)
0.442945 + 0.896549i \(0.353934\pi\)
\(912\) 0 0
\(913\) 56.0000 1.85333
\(914\) 0 0
\(915\) 20.4924 0.677459
\(916\) 0 0
\(917\) −7.01515 −0.231661
\(918\) 0 0
\(919\) −0.192236 −0.00634128 −0.00317064 0.999995i \(-0.501009\pi\)
−0.00317064 + 0.999995i \(0.501009\pi\)
\(920\) 0 0
\(921\) −34.7386 −1.14468
\(922\) 0 0
\(923\) 12.4924 0.411193
\(924\) 0 0
\(925\) 7.12311 0.234206
\(926\) 0 0
\(927\) 8.98485 0.295101
\(928\) 0 0
\(929\) 32.0540 1.05166 0.525828 0.850591i \(-0.323755\pi\)
0.525828 + 0.850591i \(0.323755\pi\)
\(930\) 0 0
\(931\) 6.80776 0.223115
\(932\) 0 0
\(933\) −14.9309 −0.488815
\(934\) 0 0
\(935\) 14.2462 0.465901
\(936\) 0 0
\(937\) −20.0540 −0.655135 −0.327567 0.944828i \(-0.606229\pi\)
−0.327567 + 0.944828i \(0.606229\pi\)
\(938\) 0 0
\(939\) 23.6998 0.773414
\(940\) 0 0
\(941\) 26.6847 0.869895 0.434948 0.900456i \(-0.356767\pi\)
0.434948 + 0.900456i \(0.356767\pi\)
\(942\) 0 0
\(943\) −55.1231 −1.79506
\(944\) 0 0
\(945\) 2.43845 0.0793227
\(946\) 0 0
\(947\) −8.63068 −0.280460 −0.140230 0.990119i \(-0.544784\pi\)
−0.140230 + 0.990119i \(0.544784\pi\)
\(948\) 0 0
\(949\) 9.06913 0.294396
\(950\) 0 0
\(951\) 14.9309 0.484167
\(952\) 0 0
\(953\) −31.2311 −1.01167 −0.505837 0.862629i \(-0.668816\pi\)
−0.505837 + 0.862629i \(0.668816\pi\)
\(954\) 0 0
\(955\) −16.6847 −0.539903
\(956\) 0 0
\(957\) 47.2311 1.52676
\(958\) 0 0
\(959\) 7.80776 0.252126
\(960\) 0 0
\(961\) −21.2462 −0.685362
\(962\) 0 0
\(963\) −0.384472 −0.0123894
\(964\) 0 0
\(965\) −0.876894 −0.0282282
\(966\) 0 0
\(967\) 30.8769 0.992934 0.496467 0.868056i \(-0.334630\pi\)
0.496467 + 0.868056i \(0.334630\pi\)
\(968\) 0 0
\(969\) −5.56155 −0.178663
\(970\) 0 0
\(971\) −52.9848 −1.70036 −0.850182 0.526488i \(-0.823508\pi\)
−0.850182 + 0.526488i \(0.823508\pi\)
\(972\) 0 0
\(973\) −6.24621 −0.200244
\(974\) 0 0
\(975\) −2.43845 −0.0780928
\(976\) 0 0
\(977\) −3.12311 −0.0999170 −0.0499585 0.998751i \(-0.515909\pi\)
−0.0499585 + 0.998751i \(0.515909\pi\)
\(978\) 0 0
\(979\) 40.0000 1.27841
\(980\) 0 0
\(981\) −8.73863 −0.279003
\(982\) 0 0
\(983\) −11.6155 −0.370478 −0.185239 0.982694i \(-0.559306\pi\)
−0.185239 + 0.982694i \(0.559306\pi\)
\(984\) 0 0
\(985\) 7.75379 0.247056
\(986\) 0 0
\(987\) −3.50758 −0.111647
\(988\) 0 0
\(989\) −13.3693 −0.425120
\(990\) 0 0
\(991\) 12.8769 0.409048 0.204524 0.978862i \(-0.434435\pi\)
0.204524 + 0.978862i \(0.434435\pi\)
\(992\) 0 0
\(993\) 12.1922 0.386909
\(994\) 0 0
\(995\) 9.56155 0.303122
\(996\) 0 0
\(997\) −8.24621 −0.261160 −0.130580 0.991438i \(-0.541684\pi\)
−0.130580 + 0.991438i \(0.541684\pi\)
\(998\) 0 0
\(999\) −39.6155 −1.25338
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.e.1.2 2
4.3 odd 2 3040.2.a.h.1.1 yes 2
8.3 odd 2 6080.2.a.bc.1.2 2
8.5 even 2 6080.2.a.bi.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3040.2.a.e.1.2 2 1.1 even 1 trivial
3040.2.a.h.1.1 yes 2 4.3 odd 2
6080.2.a.bc.1.2 2 8.3 odd 2
6080.2.a.bi.1.1 2 8.5 even 2