Properties

Label 4028.2.a.e.1.7
Level $4028$
Weight $2$
Character 4028.1
Self dual yes
Analytic conductor $32.164$
Analytic rank $0$
Dimension $19$
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] = [4028,2,Mod(1,4028)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4028, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("4028.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4028 = 2^{2} \cdot 19 \cdot 53 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4028.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: \(32.1637419342\)
Analytic rank: \(0\)
Dimension: \(19\)
Coefficient field: \(\mathbb{Q}[x]/(x^{19} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{19} - 3 x^{18} - 35 x^{17} + 103 x^{16} + 501 x^{15} - 1437 x^{14} - 3775 x^{13} + 10450 x^{12} + 16076 x^{11} - 42255 x^{10} - 38701 x^{9} + 93907 x^{8} + 49522 x^{7} + \cdots - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.7
Root \(-1.13995\) of defining polynomial
Character \(\chi\) \(=\) 4028.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.13995 q^{3} -2.77941 q^{5} +1.98761 q^{7} -1.70052 q^{9} +O(q^{10})\) \(q-1.13995 q^{3} -2.77941 q^{5} +1.98761 q^{7} -1.70052 q^{9} -5.06951 q^{11} +4.33485 q^{13} +3.16838 q^{15} -2.00702 q^{17} +1.00000 q^{19} -2.26577 q^{21} -4.07948 q^{23} +2.72511 q^{25} +5.35835 q^{27} +0.0668313 q^{29} +1.97371 q^{31} +5.77898 q^{33} -5.52437 q^{35} -10.4620 q^{37} -4.94150 q^{39} -5.38189 q^{41} -9.33887 q^{43} +4.72644 q^{45} +2.29606 q^{47} -3.04941 q^{49} +2.28790 q^{51} +1.00000 q^{53} +14.0902 q^{55} -1.13995 q^{57} -0.209025 q^{59} +8.06405 q^{61} -3.37997 q^{63} -12.0483 q^{65} +6.60660 q^{67} +4.65039 q^{69} -11.7455 q^{71} +7.29444 q^{73} -3.10648 q^{75} -10.0762 q^{77} -7.32647 q^{79} -1.00668 q^{81} +16.9906 q^{83} +5.57833 q^{85} -0.0761842 q^{87} +4.39417 q^{89} +8.61598 q^{91} -2.24993 q^{93} -2.77941 q^{95} +3.09743 q^{97} +8.62080 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 19 q + 3 q^{3} + 3 q^{5} + 6 q^{7} + 22 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 19 q + 3 q^{3} + 3 q^{5} + 6 q^{7} + 22 q^{9} + 5 q^{11} + 25 q^{13} + 20 q^{15} - 7 q^{17} + 19 q^{19} + 2 q^{21} + 18 q^{23} + 22 q^{25} + 15 q^{27} + 4 q^{29} + 12 q^{31} - 8 q^{33} + 11 q^{35} + 19 q^{37} + 9 q^{39} - 9 q^{41} + 31 q^{43} - 2 q^{45} - 2 q^{47} + 7 q^{49} + 5 q^{51} + 19 q^{53} + 11 q^{55} + 3 q^{57} + 2 q^{59} + 6 q^{61} + 52 q^{63} - 6 q^{65} + 50 q^{67} - 7 q^{69} + 25 q^{71} - 5 q^{73} + 22 q^{75} - 14 q^{77} + 36 q^{79} + 11 q^{81} + 20 q^{83} + 5 q^{85} + 18 q^{87} + 9 q^{89} + 61 q^{91} + q^{93} + 3 q^{95} + 7 q^{97} + 48 q^{99}+O(q^{100}) \) 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
\(3\) −1.13995 −0.658149 −0.329075 0.944304i \(-0.606737\pi\)
−0.329075 + 0.944304i \(0.606737\pi\)
\(4\) 0 0
\(5\) −2.77941 −1.24299 −0.621494 0.783419i \(-0.713474\pi\)
−0.621494 + 0.783419i \(0.713474\pi\)
\(6\) 0 0
\(7\) 1.98761 0.751245 0.375623 0.926773i \(-0.377429\pi\)
0.375623 + 0.926773i \(0.377429\pi\)
\(8\) 0 0
\(9\) −1.70052 −0.566840
\(10\) 0 0
\(11\) −5.06951 −1.52851 −0.764257 0.644911i \(-0.776894\pi\)
−0.764257 + 0.644911i \(0.776894\pi\)
\(12\) 0 0
\(13\) 4.33485 1.20227 0.601135 0.799147i \(-0.294715\pi\)
0.601135 + 0.799147i \(0.294715\pi\)
\(14\) 0 0
\(15\) 3.16838 0.818072
\(16\) 0 0
\(17\) −2.00702 −0.486774 −0.243387 0.969929i \(-0.578259\pi\)
−0.243387 + 0.969929i \(0.578259\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) −2.26577 −0.494432
\(22\) 0 0
\(23\) −4.07948 −0.850630 −0.425315 0.905045i \(-0.639837\pi\)
−0.425315 + 0.905045i \(0.639837\pi\)
\(24\) 0 0
\(25\) 2.72511 0.545021
\(26\) 0 0
\(27\) 5.35835 1.03121
\(28\) 0 0
\(29\) 0.0668313 0.0124103 0.00620513 0.999981i \(-0.498025\pi\)
0.00620513 + 0.999981i \(0.498025\pi\)
\(30\) 0 0
\(31\) 1.97371 0.354489 0.177244 0.984167i \(-0.443282\pi\)
0.177244 + 0.984167i \(0.443282\pi\)
\(32\) 0 0
\(33\) 5.77898 1.00599
\(34\) 0 0
\(35\) −5.52437 −0.933790
\(36\) 0 0
\(37\) −10.4620 −1.71995 −0.859974 0.510339i \(-0.829520\pi\)
−0.859974 + 0.510339i \(0.829520\pi\)
\(38\) 0 0
\(39\) −4.94150 −0.791273
\(40\) 0 0
\(41\) −5.38189 −0.840510 −0.420255 0.907406i \(-0.638059\pi\)
−0.420255 + 0.907406i \(0.638059\pi\)
\(42\) 0 0
\(43\) −9.33887 −1.42416 −0.712082 0.702097i \(-0.752247\pi\)
−0.712082 + 0.702097i \(0.752247\pi\)
\(44\) 0 0
\(45\) 4.72644 0.704575
\(46\) 0 0
\(47\) 2.29606 0.334915 0.167457 0.985879i \(-0.446444\pi\)
0.167457 + 0.985879i \(0.446444\pi\)
\(48\) 0 0
\(49\) −3.04941 −0.435630
\(50\) 0 0
\(51\) 2.28790 0.320370
\(52\) 0 0
\(53\) 1.00000 0.137361
\(54\) 0 0
\(55\) 14.0902 1.89993
\(56\) 0 0
\(57\) −1.13995 −0.150990
\(58\) 0 0
\(59\) −0.209025 −0.0272128 −0.0136064 0.999907i \(-0.504331\pi\)
−0.0136064 + 0.999907i \(0.504331\pi\)
\(60\) 0 0
\(61\) 8.06405 1.03250 0.516248 0.856439i \(-0.327328\pi\)
0.516248 + 0.856439i \(0.327328\pi\)
\(62\) 0 0
\(63\) −3.37997 −0.425836
\(64\) 0 0
\(65\) −12.0483 −1.49441
\(66\) 0 0
\(67\) 6.60660 0.807124 0.403562 0.914952i \(-0.367772\pi\)
0.403562 + 0.914952i \(0.367772\pi\)
\(68\) 0 0
\(69\) 4.65039 0.559841
\(70\) 0 0
\(71\) −11.7455 −1.39394 −0.696968 0.717102i \(-0.745468\pi\)
−0.696968 + 0.717102i \(0.745468\pi\)
\(72\) 0 0
\(73\) 7.29444 0.853750 0.426875 0.904311i \(-0.359615\pi\)
0.426875 + 0.904311i \(0.359615\pi\)
\(74\) 0 0
\(75\) −3.10648 −0.358705
\(76\) 0 0
\(77\) −10.0762 −1.14829
\(78\) 0 0
\(79\) −7.32647 −0.824293 −0.412146 0.911118i \(-0.635221\pi\)
−0.412146 + 0.911118i \(0.635221\pi\)
\(80\) 0 0
\(81\) −1.00668 −0.111853
\(82\) 0 0
\(83\) 16.9906 1.86497 0.932483 0.361214i \(-0.117638\pi\)
0.932483 + 0.361214i \(0.117638\pi\)
\(84\) 0 0
\(85\) 5.57833 0.605055
\(86\) 0 0
\(87\) −0.0761842 −0.00816780
\(88\) 0 0
\(89\) 4.39417 0.465781 0.232891 0.972503i \(-0.425182\pi\)
0.232891 + 0.972503i \(0.425182\pi\)
\(90\) 0 0
\(91\) 8.61598 0.903200
\(92\) 0 0
\(93\) −2.24993 −0.233306
\(94\) 0 0
\(95\) −2.77941 −0.285161
\(96\) 0 0
\(97\) 3.09743 0.314497 0.157248 0.987559i \(-0.449738\pi\)
0.157248 + 0.987559i \(0.449738\pi\)
\(98\) 0 0
\(99\) 8.62080 0.866423
\(100\) 0 0
\(101\) 10.4465 1.03947 0.519734 0.854328i \(-0.326031\pi\)
0.519734 + 0.854328i \(0.326031\pi\)
\(102\) 0 0
\(103\) −2.95839 −0.291499 −0.145749 0.989322i \(-0.546559\pi\)
−0.145749 + 0.989322i \(0.546559\pi\)
\(104\) 0 0
\(105\) 6.29750 0.614573
\(106\) 0 0
\(107\) 2.89044 0.279429 0.139715 0.990192i \(-0.455381\pi\)
0.139715 + 0.990192i \(0.455381\pi\)
\(108\) 0 0
\(109\) 3.26906 0.313120 0.156560 0.987668i \(-0.449960\pi\)
0.156560 + 0.987668i \(0.449960\pi\)
\(110\) 0 0
\(111\) 11.9262 1.13198
\(112\) 0 0
\(113\) −2.43144 −0.228731 −0.114365 0.993439i \(-0.536483\pi\)
−0.114365 + 0.993439i \(0.536483\pi\)
\(114\) 0 0
\(115\) 11.3385 1.05732
\(116\) 0 0
\(117\) −7.37149 −0.681494
\(118\) 0 0
\(119\) −3.98918 −0.365687
\(120\) 0 0
\(121\) 14.6999 1.33636
\(122\) 0 0
\(123\) 6.13507 0.553181
\(124\) 0 0
\(125\) 6.32286 0.565534
\(126\) 0 0
\(127\) −5.83797 −0.518036 −0.259018 0.965872i \(-0.583399\pi\)
−0.259018 + 0.965872i \(0.583399\pi\)
\(128\) 0 0
\(129\) 10.6458 0.937312
\(130\) 0 0
\(131\) −5.59502 −0.488839 −0.244420 0.969670i \(-0.578597\pi\)
−0.244420 + 0.969670i \(0.578597\pi\)
\(132\) 0 0
\(133\) 1.98761 0.172348
\(134\) 0 0
\(135\) −14.8930 −1.28179
\(136\) 0 0
\(137\) −9.71128 −0.829691 −0.414846 0.909892i \(-0.636164\pi\)
−0.414846 + 0.909892i \(0.636164\pi\)
\(138\) 0 0
\(139\) 12.6029 1.06896 0.534482 0.845180i \(-0.320507\pi\)
0.534482 + 0.845180i \(0.320507\pi\)
\(140\) 0 0
\(141\) −2.61739 −0.220424
\(142\) 0 0
\(143\) −21.9755 −1.83769
\(144\) 0 0
\(145\) −0.185751 −0.0154258
\(146\) 0 0
\(147\) 3.47617 0.286710
\(148\) 0 0
\(149\) 6.41435 0.525484 0.262742 0.964866i \(-0.415373\pi\)
0.262742 + 0.964866i \(0.415373\pi\)
\(150\) 0 0
\(151\) 1.91735 0.156032 0.0780159 0.996952i \(-0.475141\pi\)
0.0780159 + 0.996952i \(0.475141\pi\)
\(152\) 0 0
\(153\) 3.41298 0.275923
\(154\) 0 0
\(155\) −5.48574 −0.440626
\(156\) 0 0
\(157\) 8.62141 0.688064 0.344032 0.938958i \(-0.388207\pi\)
0.344032 + 0.938958i \(0.388207\pi\)
\(158\) 0 0
\(159\) −1.13995 −0.0904037
\(160\) 0 0
\(161\) −8.10841 −0.639032
\(162\) 0 0
\(163\) 12.1421 0.951044 0.475522 0.879704i \(-0.342259\pi\)
0.475522 + 0.879704i \(0.342259\pi\)
\(164\) 0 0
\(165\) −16.0621 −1.25044
\(166\) 0 0
\(167\) 2.51915 0.194938 0.0974690 0.995239i \(-0.468925\pi\)
0.0974690 + 0.995239i \(0.468925\pi\)
\(168\) 0 0
\(169\) 5.79089 0.445453
\(170\) 0 0
\(171\) −1.70052 −0.130042
\(172\) 0 0
\(173\) −21.0305 −1.59892 −0.799458 0.600722i \(-0.794880\pi\)
−0.799458 + 0.600722i \(0.794880\pi\)
\(174\) 0 0
\(175\) 5.41644 0.409445
\(176\) 0 0
\(177\) 0.238278 0.0179101
\(178\) 0 0
\(179\) 25.6751 1.91905 0.959524 0.281628i \(-0.0908744\pi\)
0.959524 + 0.281628i \(0.0908744\pi\)
\(180\) 0 0
\(181\) 7.05364 0.524293 0.262147 0.965028i \(-0.415570\pi\)
0.262147 + 0.965028i \(0.415570\pi\)
\(182\) 0 0
\(183\) −9.19260 −0.679537
\(184\) 0 0
\(185\) 29.0782 2.13787
\(186\) 0 0
\(187\) 10.1746 0.744042
\(188\) 0 0
\(189\) 10.6503 0.774695
\(190\) 0 0
\(191\) −2.95352 −0.213710 −0.106855 0.994275i \(-0.534078\pi\)
−0.106855 + 0.994275i \(0.534078\pi\)
\(192\) 0 0
\(193\) −21.0586 −1.51583 −0.757915 0.652353i \(-0.773782\pi\)
−0.757915 + 0.652353i \(0.773782\pi\)
\(194\) 0 0
\(195\) 13.7344 0.983543
\(196\) 0 0
\(197\) 2.53411 0.180548 0.0902741 0.995917i \(-0.471226\pi\)
0.0902741 + 0.995917i \(0.471226\pi\)
\(198\) 0 0
\(199\) 3.52575 0.249934 0.124967 0.992161i \(-0.460118\pi\)
0.124967 + 0.992161i \(0.460118\pi\)
\(200\) 0 0
\(201\) −7.53117 −0.531208
\(202\) 0 0
\(203\) 0.132834 0.00932315
\(204\) 0 0
\(205\) 14.9585 1.04474
\(206\) 0 0
\(207\) 6.93723 0.482171
\(208\) 0 0
\(209\) −5.06951 −0.350665
\(210\) 0 0
\(211\) 21.7194 1.49523 0.747613 0.664134i \(-0.231200\pi\)
0.747613 + 0.664134i \(0.231200\pi\)
\(212\) 0 0
\(213\) 13.3893 0.917418
\(214\) 0 0
\(215\) 25.9565 1.77022
\(216\) 0 0
\(217\) 3.92296 0.266308
\(218\) 0 0
\(219\) −8.31528 −0.561895
\(220\) 0 0
\(221\) −8.70013 −0.585234
\(222\) 0 0
\(223\) 25.9341 1.73667 0.868337 0.495974i \(-0.165189\pi\)
0.868337 + 0.495974i \(0.165189\pi\)
\(224\) 0 0
\(225\) −4.63409 −0.308940
\(226\) 0 0
\(227\) −10.7955 −0.716522 −0.358261 0.933622i \(-0.616630\pi\)
−0.358261 + 0.933622i \(0.616630\pi\)
\(228\) 0 0
\(229\) 25.4067 1.67892 0.839461 0.543419i \(-0.182871\pi\)
0.839461 + 0.543419i \(0.182871\pi\)
\(230\) 0 0
\(231\) 11.4863 0.755746
\(232\) 0 0
\(233\) −1.15393 −0.0755967 −0.0377983 0.999285i \(-0.512034\pi\)
−0.0377983 + 0.999285i \(0.512034\pi\)
\(234\) 0 0
\(235\) −6.38168 −0.416295
\(236\) 0 0
\(237\) 8.35180 0.542508
\(238\) 0 0
\(239\) −21.3725 −1.38247 −0.691235 0.722630i \(-0.742933\pi\)
−0.691235 + 0.722630i \(0.742933\pi\)
\(240\) 0 0
\(241\) 18.4746 1.19005 0.595026 0.803707i \(-0.297142\pi\)
0.595026 + 0.803707i \(0.297142\pi\)
\(242\) 0 0
\(243\) −14.9275 −0.957598
\(244\) 0 0
\(245\) 8.47556 0.541483
\(246\) 0 0
\(247\) 4.33485 0.275820
\(248\) 0 0
\(249\) −19.3684 −1.22743
\(250\) 0 0
\(251\) 0.441653 0.0278769 0.0139384 0.999903i \(-0.495563\pi\)
0.0139384 + 0.999903i \(0.495563\pi\)
\(252\) 0 0
\(253\) 20.6810 1.30020
\(254\) 0 0
\(255\) −6.35901 −0.398217
\(256\) 0 0
\(257\) −8.96951 −0.559503 −0.279751 0.960072i \(-0.590252\pi\)
−0.279751 + 0.960072i \(0.590252\pi\)
\(258\) 0 0
\(259\) −20.7944 −1.29210
\(260\) 0 0
\(261\) −0.113648 −0.00703463
\(262\) 0 0
\(263\) 23.7301 1.46326 0.731630 0.681702i \(-0.238760\pi\)
0.731630 + 0.681702i \(0.238760\pi\)
\(264\) 0 0
\(265\) −2.77941 −0.170738
\(266\) 0 0
\(267\) −5.00912 −0.306553
\(268\) 0 0
\(269\) 8.91159 0.543349 0.271675 0.962389i \(-0.412423\pi\)
0.271675 + 0.962389i \(0.412423\pi\)
\(270\) 0 0
\(271\) 4.32240 0.262567 0.131284 0.991345i \(-0.458090\pi\)
0.131284 + 0.991345i \(0.458090\pi\)
\(272\) 0 0
\(273\) −9.82177 −0.594440
\(274\) 0 0
\(275\) −13.8149 −0.833073
\(276\) 0 0
\(277\) −10.4927 −0.630448 −0.315224 0.949017i \(-0.602080\pi\)
−0.315224 + 0.949017i \(0.602080\pi\)
\(278\) 0 0
\(279\) −3.35633 −0.200938
\(280\) 0 0
\(281\) 22.8927 1.36566 0.682831 0.730576i \(-0.260749\pi\)
0.682831 + 0.730576i \(0.260749\pi\)
\(282\) 0 0
\(283\) 11.5026 0.683760 0.341880 0.939744i \(-0.388936\pi\)
0.341880 + 0.939744i \(0.388936\pi\)
\(284\) 0 0
\(285\) 3.16838 0.187679
\(286\) 0 0
\(287\) −10.6971 −0.631429
\(288\) 0 0
\(289\) −12.9719 −0.763051
\(290\) 0 0
\(291\) −3.53091 −0.206986
\(292\) 0 0
\(293\) −30.9918 −1.81056 −0.905279 0.424817i \(-0.860339\pi\)
−0.905279 + 0.424817i \(0.860339\pi\)
\(294\) 0 0
\(295\) 0.580967 0.0338252
\(296\) 0 0
\(297\) −27.1642 −1.57623
\(298\) 0 0
\(299\) −17.6839 −1.02269
\(300\) 0 0
\(301\) −18.5620 −1.06990
\(302\) 0 0
\(303\) −11.9085 −0.684126
\(304\) 0 0
\(305\) −22.4133 −1.28338
\(306\) 0 0
\(307\) 18.9957 1.08414 0.542072 0.840332i \(-0.317640\pi\)
0.542072 + 0.840332i \(0.317640\pi\)
\(308\) 0 0
\(309\) 3.37241 0.191850
\(310\) 0 0
\(311\) 32.2956 1.83131 0.915657 0.401960i \(-0.131671\pi\)
0.915657 + 0.401960i \(0.131671\pi\)
\(312\) 0 0
\(313\) 3.05599 0.172735 0.0863673 0.996263i \(-0.472474\pi\)
0.0863673 + 0.996263i \(0.472474\pi\)
\(314\) 0 0
\(315\) 9.39430 0.529309
\(316\) 0 0
\(317\) −2.55848 −0.143698 −0.0718492 0.997416i \(-0.522890\pi\)
−0.0718492 + 0.997416i \(0.522890\pi\)
\(318\) 0 0
\(319\) −0.338802 −0.0189693
\(320\) 0 0
\(321\) −3.29495 −0.183906
\(322\) 0 0
\(323\) −2.00702 −0.111674
\(324\) 0 0
\(325\) 11.8129 0.655262
\(326\) 0 0
\(327\) −3.72656 −0.206079
\(328\) 0 0
\(329\) 4.56367 0.251603
\(330\) 0 0
\(331\) 27.5751 1.51567 0.757833 0.652448i \(-0.226258\pi\)
0.757833 + 0.652448i \(0.226258\pi\)
\(332\) 0 0
\(333\) 17.7909 0.974934
\(334\) 0 0
\(335\) −18.3624 −1.00325
\(336\) 0 0
\(337\) 22.7108 1.23713 0.618567 0.785732i \(-0.287714\pi\)
0.618567 + 0.785732i \(0.287714\pi\)
\(338\) 0 0
\(339\) 2.77171 0.150539
\(340\) 0 0
\(341\) −10.0057 −0.541841
\(342\) 0 0
\(343\) −19.9743 −1.07851
\(344\) 0 0
\(345\) −12.9253 −0.695877
\(346\) 0 0
\(347\) 32.1150 1.72402 0.862011 0.506889i \(-0.169205\pi\)
0.862011 + 0.506889i \(0.169205\pi\)
\(348\) 0 0
\(349\) −16.9106 −0.905205 −0.452602 0.891712i \(-0.649504\pi\)
−0.452602 + 0.891712i \(0.649504\pi\)
\(350\) 0 0
\(351\) 23.2276 1.23980
\(352\) 0 0
\(353\) 9.79709 0.521447 0.260723 0.965414i \(-0.416039\pi\)
0.260723 + 0.965414i \(0.416039\pi\)
\(354\) 0 0
\(355\) 32.6456 1.73265
\(356\) 0 0
\(357\) 4.54745 0.240677
\(358\) 0 0
\(359\) 14.8739 0.785012 0.392506 0.919749i \(-0.371608\pi\)
0.392506 + 0.919749i \(0.371608\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) −16.7572 −0.879522
\(364\) 0 0
\(365\) −20.2742 −1.06120
\(366\) 0 0
\(367\) −5.63231 −0.294004 −0.147002 0.989136i \(-0.546962\pi\)
−0.147002 + 0.989136i \(0.546962\pi\)
\(368\) 0 0
\(369\) 9.15200 0.476434
\(370\) 0 0
\(371\) 1.98761 0.103192
\(372\) 0 0
\(373\) 28.6421 1.48303 0.741515 0.670937i \(-0.234108\pi\)
0.741515 + 0.670937i \(0.234108\pi\)
\(374\) 0 0
\(375\) −7.20773 −0.372206
\(376\) 0 0
\(377\) 0.289703 0.0149205
\(378\) 0 0
\(379\) −4.83077 −0.248140 −0.124070 0.992273i \(-0.539595\pi\)
−0.124070 + 0.992273i \(0.539595\pi\)
\(380\) 0 0
\(381\) 6.65498 0.340945
\(382\) 0 0
\(383\) −19.4181 −0.992219 −0.496110 0.868260i \(-0.665239\pi\)
−0.496110 + 0.868260i \(0.665239\pi\)
\(384\) 0 0
\(385\) 28.0059 1.42731
\(386\) 0 0
\(387\) 15.8809 0.807272
\(388\) 0 0
\(389\) −0.905582 −0.0459148 −0.0229574 0.999736i \(-0.507308\pi\)
−0.0229574 + 0.999736i \(0.507308\pi\)
\(390\) 0 0
\(391\) 8.18760 0.414065
\(392\) 0 0
\(393\) 6.37803 0.321729
\(394\) 0 0
\(395\) 20.3633 1.02459
\(396\) 0 0
\(397\) −21.7294 −1.09057 −0.545284 0.838252i \(-0.683578\pi\)
−0.545284 + 0.838252i \(0.683578\pi\)
\(398\) 0 0
\(399\) −2.26577 −0.113430
\(400\) 0 0
\(401\) 14.5204 0.725116 0.362558 0.931961i \(-0.381903\pi\)
0.362558 + 0.931961i \(0.381903\pi\)
\(402\) 0 0
\(403\) 8.55573 0.426191
\(404\) 0 0
\(405\) 2.79797 0.139032
\(406\) 0 0
\(407\) 53.0374 2.62896
\(408\) 0 0
\(409\) 14.7946 0.731548 0.365774 0.930704i \(-0.380804\pi\)
0.365774 + 0.930704i \(0.380804\pi\)
\(410\) 0 0
\(411\) 11.0704 0.546060
\(412\) 0 0
\(413\) −0.415461 −0.0204435
\(414\) 0 0
\(415\) −47.2239 −2.31813
\(416\) 0 0
\(417\) −14.3667 −0.703538
\(418\) 0 0
\(419\) −6.50447 −0.317764 −0.158882 0.987298i \(-0.550789\pi\)
−0.158882 + 0.987298i \(0.550789\pi\)
\(420\) 0 0
\(421\) −31.6865 −1.54431 −0.772153 0.635437i \(-0.780820\pi\)
−0.772153 + 0.635437i \(0.780820\pi\)
\(422\) 0 0
\(423\) −3.90449 −0.189843
\(424\) 0 0
\(425\) −5.46935 −0.265302
\(426\) 0 0
\(427\) 16.0282 0.775658
\(428\) 0 0
\(429\) 25.0510 1.20947
\(430\) 0 0
\(431\) −6.76860 −0.326032 −0.163016 0.986623i \(-0.552122\pi\)
−0.163016 + 0.986623i \(0.552122\pi\)
\(432\) 0 0
\(433\) 30.2217 1.45236 0.726182 0.687502i \(-0.241293\pi\)
0.726182 + 0.687502i \(0.241293\pi\)
\(434\) 0 0
\(435\) 0.211747 0.0101525
\(436\) 0 0
\(437\) −4.07948 −0.195148
\(438\) 0 0
\(439\) −12.2835 −0.586260 −0.293130 0.956073i \(-0.594697\pi\)
−0.293130 + 0.956073i \(0.594697\pi\)
\(440\) 0 0
\(441\) 5.18558 0.246933
\(442\) 0 0
\(443\) −28.7324 −1.36512 −0.682558 0.730831i \(-0.739133\pi\)
−0.682558 + 0.730831i \(0.739133\pi\)
\(444\) 0 0
\(445\) −12.2132 −0.578961
\(446\) 0 0
\(447\) −7.31202 −0.345847
\(448\) 0 0
\(449\) −14.1318 −0.666919 −0.333459 0.942764i \(-0.608216\pi\)
−0.333459 + 0.942764i \(0.608216\pi\)
\(450\) 0 0
\(451\) 27.2835 1.28473
\(452\) 0 0
\(453\) −2.18568 −0.102692
\(454\) 0 0
\(455\) −23.9473 −1.12267
\(456\) 0 0
\(457\) −34.0631 −1.59340 −0.796702 0.604372i \(-0.793424\pi\)
−0.796702 + 0.604372i \(0.793424\pi\)
\(458\) 0 0
\(459\) −10.7543 −0.501969
\(460\) 0 0
\(461\) −9.63906 −0.448936 −0.224468 0.974482i \(-0.572064\pi\)
−0.224468 + 0.974482i \(0.572064\pi\)
\(462\) 0 0
\(463\) −4.06975 −0.189137 −0.0945686 0.995518i \(-0.530147\pi\)
−0.0945686 + 0.995518i \(0.530147\pi\)
\(464\) 0 0
\(465\) 6.25346 0.289997
\(466\) 0 0
\(467\) 29.4052 1.36071 0.680355 0.732883i \(-0.261826\pi\)
0.680355 + 0.732883i \(0.261826\pi\)
\(468\) 0 0
\(469\) 13.1313 0.606348
\(470\) 0 0
\(471\) −9.82796 −0.452849
\(472\) 0 0
\(473\) 47.3435 2.17686
\(474\) 0 0
\(475\) 2.72511 0.125036
\(476\) 0 0
\(477\) −1.70052 −0.0778614
\(478\) 0 0
\(479\) 34.1649 1.56103 0.780516 0.625136i \(-0.214956\pi\)
0.780516 + 0.625136i \(0.214956\pi\)
\(480\) 0 0
\(481\) −45.3513 −2.06784
\(482\) 0 0
\(483\) 9.24316 0.420578
\(484\) 0 0
\(485\) −8.60902 −0.390916
\(486\) 0 0
\(487\) 18.6269 0.844064 0.422032 0.906581i \(-0.361317\pi\)
0.422032 + 0.906581i \(0.361317\pi\)
\(488\) 0 0
\(489\) −13.8414 −0.625929
\(490\) 0 0
\(491\) −23.9630 −1.08144 −0.540718 0.841204i \(-0.681847\pi\)
−0.540718 + 0.841204i \(0.681847\pi\)
\(492\) 0 0
\(493\) −0.134132 −0.00604100
\(494\) 0 0
\(495\) −23.9607 −1.07695
\(496\) 0 0
\(497\) −23.3455 −1.04719
\(498\) 0 0
\(499\) −34.5389 −1.54617 −0.773087 0.634300i \(-0.781288\pi\)
−0.773087 + 0.634300i \(0.781288\pi\)
\(500\) 0 0
\(501\) −2.87170 −0.128298
\(502\) 0 0
\(503\) 4.99944 0.222914 0.111457 0.993769i \(-0.464448\pi\)
0.111457 + 0.993769i \(0.464448\pi\)
\(504\) 0 0
\(505\) −29.0352 −1.29205
\(506\) 0 0
\(507\) −6.60131 −0.293175
\(508\) 0 0
\(509\) −1.22713 −0.0543914 −0.0271957 0.999630i \(-0.508658\pi\)
−0.0271957 + 0.999630i \(0.508658\pi\)
\(510\) 0 0
\(511\) 14.4985 0.641376
\(512\) 0 0
\(513\) 5.35835 0.236577
\(514\) 0 0
\(515\) 8.22257 0.362330
\(516\) 0 0
\(517\) −11.6399 −0.511922
\(518\) 0 0
\(519\) 23.9736 1.05233
\(520\) 0 0
\(521\) −28.8537 −1.26410 −0.632051 0.774927i \(-0.717787\pi\)
−0.632051 + 0.774927i \(0.717787\pi\)
\(522\) 0 0
\(523\) −25.6998 −1.12378 −0.561888 0.827213i \(-0.689925\pi\)
−0.561888 + 0.827213i \(0.689925\pi\)
\(524\) 0 0
\(525\) −6.17446 −0.269476
\(526\) 0 0
\(527\) −3.96128 −0.172556
\(528\) 0 0
\(529\) −6.35786 −0.276429
\(530\) 0 0
\(531\) 0.355452 0.0154253
\(532\) 0 0
\(533\) −23.3297 −1.01052
\(534\) 0 0
\(535\) −8.03371 −0.347328
\(536\) 0 0
\(537\) −29.2683 −1.26302
\(538\) 0 0
\(539\) 15.4590 0.665867
\(540\) 0 0
\(541\) −27.5450 −1.18425 −0.592127 0.805845i \(-0.701712\pi\)
−0.592127 + 0.805845i \(0.701712\pi\)
\(542\) 0 0
\(543\) −8.04079 −0.345063
\(544\) 0 0
\(545\) −9.08606 −0.389204
\(546\) 0 0
\(547\) 1.21661 0.0520183 0.0260092 0.999662i \(-0.491720\pi\)
0.0260092 + 0.999662i \(0.491720\pi\)
\(548\) 0 0
\(549\) −13.7131 −0.585260
\(550\) 0 0
\(551\) 0.0668313 0.00284711
\(552\) 0 0
\(553\) −14.5622 −0.619246
\(554\) 0 0
\(555\) −33.1477 −1.40704
\(556\) 0 0
\(557\) −18.0693 −0.765621 −0.382810 0.923827i \(-0.625044\pi\)
−0.382810 + 0.923827i \(0.625044\pi\)
\(558\) 0 0
\(559\) −40.4825 −1.71223
\(560\) 0 0
\(561\) −11.5985 −0.489691
\(562\) 0 0
\(563\) −27.8866 −1.17528 −0.587640 0.809123i \(-0.699943\pi\)
−0.587640 + 0.809123i \(0.699943\pi\)
\(564\) 0 0
\(565\) 6.75796 0.284310
\(566\) 0 0
\(567\) −2.00088 −0.0840291
\(568\) 0 0
\(569\) 25.1878 1.05593 0.527964 0.849266i \(-0.322955\pi\)
0.527964 + 0.849266i \(0.322955\pi\)
\(570\) 0 0
\(571\) 7.05094 0.295073 0.147536 0.989057i \(-0.452866\pi\)
0.147536 + 0.989057i \(0.452866\pi\)
\(572\) 0 0
\(573\) 3.36686 0.140653
\(574\) 0 0
\(575\) −11.1170 −0.463611
\(576\) 0 0
\(577\) −15.5720 −0.648271 −0.324136 0.946011i \(-0.605073\pi\)
−0.324136 + 0.946011i \(0.605073\pi\)
\(578\) 0 0
\(579\) 24.0057 0.997642
\(580\) 0 0
\(581\) 33.7708 1.40105
\(582\) 0 0
\(583\) −5.06951 −0.209958
\(584\) 0 0
\(585\) 20.4884 0.847090
\(586\) 0 0
\(587\) 37.5421 1.54953 0.774765 0.632249i \(-0.217868\pi\)
0.774765 + 0.632249i \(0.217868\pi\)
\(588\) 0 0
\(589\) 1.97371 0.0813253
\(590\) 0 0
\(591\) −2.88876 −0.118828
\(592\) 0 0
\(593\) 16.7456 0.687660 0.343830 0.939032i \(-0.388276\pi\)
0.343830 + 0.939032i \(0.388276\pi\)
\(594\) 0 0
\(595\) 11.0875 0.454545
\(596\) 0 0
\(597\) −4.01917 −0.164494
\(598\) 0 0
\(599\) 16.3428 0.667750 0.333875 0.942617i \(-0.391644\pi\)
0.333875 + 0.942617i \(0.391644\pi\)
\(600\) 0 0
\(601\) −38.1231 −1.55507 −0.777537 0.628837i \(-0.783531\pi\)
−0.777537 + 0.628837i \(0.783531\pi\)
\(602\) 0 0
\(603\) −11.2346 −0.457510
\(604\) 0 0
\(605\) −40.8571 −1.66108
\(606\) 0 0
\(607\) −20.5423 −0.833785 −0.416892 0.908956i \(-0.636881\pi\)
−0.416892 + 0.908956i \(0.636881\pi\)
\(608\) 0 0
\(609\) −0.151424 −0.00613602
\(610\) 0 0
\(611\) 9.95306 0.402658
\(612\) 0 0
\(613\) −23.4726 −0.948049 −0.474024 0.880512i \(-0.657199\pi\)
−0.474024 + 0.880512i \(0.657199\pi\)
\(614\) 0 0
\(615\) −17.0519 −0.687597
\(616\) 0 0
\(617\) −38.2521 −1.53997 −0.769985 0.638062i \(-0.779736\pi\)
−0.769985 + 0.638062i \(0.779736\pi\)
\(618\) 0 0
\(619\) 6.47558 0.260276 0.130138 0.991496i \(-0.458458\pi\)
0.130138 + 0.991496i \(0.458458\pi\)
\(620\) 0 0
\(621\) −21.8593 −0.877182
\(622\) 0 0
\(623\) 8.73389 0.349916
\(624\) 0 0
\(625\) −31.1993 −1.24797
\(626\) 0 0
\(627\) 5.77898 0.230790
\(628\) 0 0
\(629\) 20.9975 0.837226
\(630\) 0 0
\(631\) 38.6841 1.53999 0.769995 0.638050i \(-0.220259\pi\)
0.769995 + 0.638050i \(0.220259\pi\)
\(632\) 0 0
\(633\) −24.7590 −0.984082
\(634\) 0 0
\(635\) 16.2261 0.643913
\(636\) 0 0
\(637\) −13.2187 −0.523745
\(638\) 0 0
\(639\) 19.9735 0.790139
\(640\) 0 0
\(641\) −0.270432 −0.0106814 −0.00534071 0.999986i \(-0.501700\pi\)
−0.00534071 + 0.999986i \(0.501700\pi\)
\(642\) 0 0
\(643\) 25.3941 1.00145 0.500723 0.865608i \(-0.333068\pi\)
0.500723 + 0.865608i \(0.333068\pi\)
\(644\) 0 0
\(645\) −29.5891 −1.16507
\(646\) 0 0
\(647\) −2.15234 −0.0846173 −0.0423086 0.999105i \(-0.513471\pi\)
−0.0423086 + 0.999105i \(0.513471\pi\)
\(648\) 0 0
\(649\) 1.05966 0.0415952
\(650\) 0 0
\(651\) −4.47197 −0.175270
\(652\) 0 0
\(653\) −15.3821 −0.601949 −0.300975 0.953632i \(-0.597312\pi\)
−0.300975 + 0.953632i \(0.597312\pi\)
\(654\) 0 0
\(655\) 15.5508 0.607622
\(656\) 0 0
\(657\) −12.4043 −0.483939
\(658\) 0 0
\(659\) 44.9048 1.74924 0.874622 0.484806i \(-0.161110\pi\)
0.874622 + 0.484806i \(0.161110\pi\)
\(660\) 0 0
\(661\) 8.93215 0.347420 0.173710 0.984797i \(-0.444424\pi\)
0.173710 + 0.984797i \(0.444424\pi\)
\(662\) 0 0
\(663\) 9.91770 0.385171
\(664\) 0 0
\(665\) −5.52437 −0.214226
\(666\) 0 0
\(667\) −0.272637 −0.0105565
\(668\) 0 0
\(669\) −29.5635 −1.14299
\(670\) 0 0
\(671\) −40.8808 −1.57819
\(672\) 0 0
\(673\) −11.1820 −0.431033 −0.215517 0.976500i \(-0.569144\pi\)
−0.215517 + 0.976500i \(0.569144\pi\)
\(674\) 0 0
\(675\) 14.6021 0.562033
\(676\) 0 0
\(677\) −9.17661 −0.352686 −0.176343 0.984329i \(-0.556427\pi\)
−0.176343 + 0.984329i \(0.556427\pi\)
\(678\) 0 0
\(679\) 6.15648 0.236264
\(680\) 0 0
\(681\) 12.3063 0.471578
\(682\) 0 0
\(683\) −3.50041 −0.133939 −0.0669697 0.997755i \(-0.521333\pi\)
−0.0669697 + 0.997755i \(0.521333\pi\)
\(684\) 0 0
\(685\) 26.9916 1.03130
\(686\) 0 0
\(687\) −28.9623 −1.10498
\(688\) 0 0
\(689\) 4.33485 0.165144
\(690\) 0 0
\(691\) −33.1390 −1.26067 −0.630333 0.776325i \(-0.717082\pi\)
−0.630333 + 0.776325i \(0.717082\pi\)
\(692\) 0 0
\(693\) 17.1348 0.650896
\(694\) 0 0
\(695\) −35.0286 −1.32871
\(696\) 0 0
\(697\) 10.8016 0.409139
\(698\) 0 0
\(699\) 1.31542 0.0497539
\(700\) 0 0
\(701\) 17.5170 0.661609 0.330805 0.943699i \(-0.392680\pi\)
0.330805 + 0.943699i \(0.392680\pi\)
\(702\) 0 0
\(703\) −10.4620 −0.394583
\(704\) 0 0
\(705\) 7.27478 0.273984
\(706\) 0 0
\(707\) 20.7636 0.780896
\(708\) 0 0
\(709\) −22.9394 −0.861506 −0.430753 0.902470i \(-0.641752\pi\)
−0.430753 + 0.902470i \(0.641752\pi\)
\(710\) 0 0
\(711\) 12.4588 0.467242
\(712\) 0 0
\(713\) −8.05171 −0.301539
\(714\) 0 0
\(715\) 61.0790 2.28422
\(716\) 0 0
\(717\) 24.3635 0.909872
\(718\) 0 0
\(719\) −14.3926 −0.536754 −0.268377 0.963314i \(-0.586487\pi\)
−0.268377 + 0.963314i \(0.586487\pi\)
\(720\) 0 0
\(721\) −5.88012 −0.218987
\(722\) 0 0
\(723\) −21.0600 −0.783231
\(724\) 0 0
\(725\) 0.182122 0.00676385
\(726\) 0 0
\(727\) 13.5404 0.502186 0.251093 0.967963i \(-0.419210\pi\)
0.251093 + 0.967963i \(0.419210\pi\)
\(728\) 0 0
\(729\) 20.0366 0.742095
\(730\) 0 0
\(731\) 18.7433 0.693246
\(732\) 0 0
\(733\) 4.11689 0.152061 0.0760304 0.997105i \(-0.475775\pi\)
0.0760304 + 0.997105i \(0.475775\pi\)
\(734\) 0 0
\(735\) −9.66169 −0.356377
\(736\) 0 0
\(737\) −33.4922 −1.23370
\(738\) 0 0
\(739\) 47.4171 1.74427 0.872133 0.489269i \(-0.162736\pi\)
0.872133 + 0.489269i \(0.162736\pi\)
\(740\) 0 0
\(741\) −4.94150 −0.181530
\(742\) 0 0
\(743\) 13.9829 0.512984 0.256492 0.966546i \(-0.417433\pi\)
0.256492 + 0.966546i \(0.417433\pi\)
\(744\) 0 0
\(745\) −17.8281 −0.653170
\(746\) 0 0
\(747\) −28.8929 −1.05714
\(748\) 0 0
\(749\) 5.74507 0.209920
\(750\) 0 0
\(751\) 33.8396 1.23482 0.617412 0.786640i \(-0.288181\pi\)
0.617412 + 0.786640i \(0.288181\pi\)
\(752\) 0 0
\(753\) −0.503461 −0.0183471
\(754\) 0 0
\(755\) −5.32910 −0.193946
\(756\) 0 0
\(757\) 18.1805 0.660782 0.330391 0.943844i \(-0.392819\pi\)
0.330391 + 0.943844i \(0.392819\pi\)
\(758\) 0 0
\(759\) −23.5752 −0.855726
\(760\) 0 0
\(761\) −25.0272 −0.907236 −0.453618 0.891196i \(-0.649867\pi\)
−0.453618 + 0.891196i \(0.649867\pi\)
\(762\) 0 0
\(763\) 6.49762 0.235230
\(764\) 0 0
\(765\) −9.48606 −0.342969
\(766\) 0 0
\(767\) −0.906093 −0.0327171
\(768\) 0 0
\(769\) −52.3351 −1.88725 −0.943626 0.331013i \(-0.892610\pi\)
−0.943626 + 0.331013i \(0.892610\pi\)
\(770\) 0 0
\(771\) 10.2248 0.368236
\(772\) 0 0
\(773\) −43.6396 −1.56961 −0.784804 0.619744i \(-0.787236\pi\)
−0.784804 + 0.619744i \(0.787236\pi\)
\(774\) 0 0
\(775\) 5.37857 0.193204
\(776\) 0 0
\(777\) 23.7046 0.850396
\(778\) 0 0
\(779\) −5.38189 −0.192826
\(780\) 0 0
\(781\) 59.5440 2.13065
\(782\) 0 0
\(783\) 0.358105 0.0127976
\(784\) 0 0
\(785\) −23.9624 −0.855256
\(786\) 0 0
\(787\) 16.4911 0.587845 0.293922 0.955829i \(-0.405039\pi\)
0.293922 + 0.955829i \(0.405039\pi\)
\(788\) 0 0
\(789\) −27.0510 −0.963043
\(790\) 0 0
\(791\) −4.83275 −0.171833
\(792\) 0 0
\(793\) 34.9564 1.24134
\(794\) 0 0
\(795\) 3.16838 0.112371
\(796\) 0 0
\(797\) 32.8373 1.16316 0.581578 0.813490i \(-0.302435\pi\)
0.581578 + 0.813490i \(0.302435\pi\)
\(798\) 0 0
\(799\) −4.60824 −0.163028
\(800\) 0 0
\(801\) −7.47237 −0.264023
\(802\) 0 0
\(803\) −36.9792 −1.30497
\(804\) 0 0
\(805\) 22.5366 0.794309
\(806\) 0 0
\(807\) −10.1587 −0.357605
\(808\) 0 0
\(809\) −37.3229 −1.31220 −0.656101 0.754673i \(-0.727796\pi\)
−0.656101 + 0.754673i \(0.727796\pi\)
\(810\) 0 0
\(811\) 21.8831 0.768419 0.384209 0.923246i \(-0.374474\pi\)
0.384209 + 0.923246i \(0.374474\pi\)
\(812\) 0 0
\(813\) −4.92731 −0.172808
\(814\) 0 0
\(815\) −33.7479 −1.18214
\(816\) 0 0
\(817\) −9.33887 −0.326726
\(818\) 0 0
\(819\) −14.6516 −0.511970
\(820\) 0 0
\(821\) 15.3194 0.534650 0.267325 0.963606i \(-0.413860\pi\)
0.267325 + 0.963606i \(0.413860\pi\)
\(822\) 0 0
\(823\) 22.8197 0.795444 0.397722 0.917506i \(-0.369801\pi\)
0.397722 + 0.917506i \(0.369801\pi\)
\(824\) 0 0
\(825\) 15.7483 0.548286
\(826\) 0 0
\(827\) 34.3269 1.19366 0.596832 0.802366i \(-0.296426\pi\)
0.596832 + 0.802366i \(0.296426\pi\)
\(828\) 0 0
\(829\) 3.89533 0.135291 0.0676453 0.997709i \(-0.478451\pi\)
0.0676453 + 0.997709i \(0.478451\pi\)
\(830\) 0 0
\(831\) 11.9612 0.414929
\(832\) 0 0
\(833\) 6.12024 0.212054
\(834\) 0 0
\(835\) −7.00176 −0.242306
\(836\) 0 0
\(837\) 10.5758 0.365554
\(838\) 0 0
\(839\) −44.0342 −1.52023 −0.760115 0.649788i \(-0.774858\pi\)
−0.760115 + 0.649788i \(0.774858\pi\)
\(840\) 0 0
\(841\) −28.9955 −0.999846
\(842\) 0 0
\(843\) −26.0964 −0.898809
\(844\) 0 0
\(845\) −16.0952 −0.553693
\(846\) 0 0
\(847\) 29.2177 1.00393
\(848\) 0 0
\(849\) −13.1124 −0.450016
\(850\) 0 0
\(851\) 42.6796 1.46304
\(852\) 0 0
\(853\) 26.2699 0.899463 0.449732 0.893164i \(-0.351520\pi\)
0.449732 + 0.893164i \(0.351520\pi\)
\(854\) 0 0
\(855\) 4.72644 0.161641
\(856\) 0 0
\(857\) 28.6281 0.977919 0.488959 0.872307i \(-0.337377\pi\)
0.488959 + 0.872307i \(0.337377\pi\)
\(858\) 0 0
\(859\) 33.6100 1.14676 0.573380 0.819290i \(-0.305632\pi\)
0.573380 + 0.819290i \(0.305632\pi\)
\(860\) 0 0
\(861\) 12.1941 0.415575
\(862\) 0 0
\(863\) −10.0673 −0.342696 −0.171348 0.985211i \(-0.554812\pi\)
−0.171348 + 0.985211i \(0.554812\pi\)
\(864\) 0 0
\(865\) 58.4522 1.98743
\(866\) 0 0
\(867\) 14.7872 0.502201
\(868\) 0 0
\(869\) 37.1416 1.25994
\(870\) 0 0
\(871\) 28.6386 0.970381
\(872\) 0 0
\(873\) −5.26724 −0.178269
\(874\) 0 0
\(875\) 12.5674 0.424855
\(876\) 0 0
\(877\) −16.4279 −0.554732 −0.277366 0.960764i \(-0.589461\pi\)
−0.277366 + 0.960764i \(0.589461\pi\)
\(878\) 0 0
\(879\) 35.3290 1.19162
\(880\) 0 0
\(881\) −43.3567 −1.46072 −0.730362 0.683060i \(-0.760649\pi\)
−0.730362 + 0.683060i \(0.760649\pi\)
\(882\) 0 0
\(883\) 24.2453 0.815920 0.407960 0.913000i \(-0.366240\pi\)
0.407960 + 0.913000i \(0.366240\pi\)
\(884\) 0 0
\(885\) −0.662272 −0.0222620
\(886\) 0 0
\(887\) 11.5718 0.388544 0.194272 0.980948i \(-0.437766\pi\)
0.194272 + 0.980948i \(0.437766\pi\)
\(888\) 0 0
\(889\) −11.6036 −0.389172
\(890\) 0 0
\(891\) 5.10336 0.170969
\(892\) 0 0
\(893\) 2.29606 0.0768347
\(894\) 0 0
\(895\) −71.3616 −2.38535
\(896\) 0 0
\(897\) 20.1587 0.673080
\(898\) 0 0
\(899\) 0.131906 0.00439930
\(900\) 0 0
\(901\) −2.00702 −0.0668636
\(902\) 0 0
\(903\) 21.1597 0.704151
\(904\) 0 0
\(905\) −19.6049 −0.651691
\(906\) 0 0
\(907\) −16.4306 −0.545568 −0.272784 0.962075i \(-0.587944\pi\)
−0.272784 + 0.962075i \(0.587944\pi\)
\(908\) 0 0
\(909\) −17.7645 −0.589212
\(910\) 0 0
\(911\) 20.2515 0.670963 0.335482 0.942047i \(-0.391101\pi\)
0.335482 + 0.942047i \(0.391101\pi\)
\(912\) 0 0
\(913\) −86.1342 −2.85063
\(914\) 0 0
\(915\) 25.5500 0.844656
\(916\) 0 0
\(917\) −11.1207 −0.367238
\(918\) 0 0
\(919\) −29.8886 −0.985934 −0.492967 0.870048i \(-0.664088\pi\)
−0.492967 + 0.870048i \(0.664088\pi\)
\(920\) 0 0
\(921\) −21.6542 −0.713529
\(922\) 0 0
\(923\) −50.9150 −1.67589
\(924\) 0 0
\(925\) −28.5101 −0.937407
\(926\) 0 0
\(927\) 5.03080 0.165233
\(928\) 0 0
\(929\) 33.8483 1.11053 0.555263 0.831675i \(-0.312618\pi\)
0.555263 + 0.831675i \(0.312618\pi\)
\(930\) 0 0
\(931\) −3.04941 −0.0999404
\(932\) 0 0
\(933\) −36.8153 −1.20528
\(934\) 0 0
\(935\) −28.2794 −0.924836
\(936\) 0 0
\(937\) 33.3480 1.08943 0.544715 0.838621i \(-0.316638\pi\)
0.544715 + 0.838621i \(0.316638\pi\)
\(938\) 0 0
\(939\) −3.48367 −0.113685
\(940\) 0 0
\(941\) 6.45689 0.210489 0.105244 0.994446i \(-0.466438\pi\)
0.105244 + 0.994446i \(0.466438\pi\)
\(942\) 0 0
\(943\) 21.9553 0.714963
\(944\) 0 0
\(945\) −29.6015 −0.962937
\(946\) 0 0
\(947\) −1.36475 −0.0443485 −0.0221742 0.999754i \(-0.507059\pi\)
−0.0221742 + 0.999754i \(0.507059\pi\)
\(948\) 0 0
\(949\) 31.6203 1.02644
\(950\) 0 0
\(951\) 2.91653 0.0945749
\(952\) 0 0
\(953\) 7.51534 0.243446 0.121723 0.992564i \(-0.461158\pi\)
0.121723 + 0.992564i \(0.461158\pi\)
\(954\) 0 0
\(955\) 8.20905 0.265639
\(956\) 0 0
\(957\) 0.386216 0.0124846
\(958\) 0 0
\(959\) −19.3022 −0.623302
\(960\) 0 0
\(961\) −27.1045 −0.874338
\(962\) 0 0
\(963\) −4.91525 −0.158392
\(964\) 0 0
\(965\) 58.5304 1.88416
\(966\) 0 0
\(967\) 40.4244 1.29996 0.649981 0.759950i \(-0.274777\pi\)
0.649981 + 0.759950i \(0.274777\pi\)
\(968\) 0 0
\(969\) 2.28790 0.0734980
\(970\) 0 0
\(971\) −32.3407 −1.03786 −0.518932 0.854816i \(-0.673670\pi\)
−0.518932 + 0.854816i \(0.673670\pi\)
\(972\) 0 0
\(973\) 25.0496 0.803055
\(974\) 0 0
\(975\) −13.4661 −0.431260
\(976\) 0 0
\(977\) 41.2386 1.31934 0.659670 0.751556i \(-0.270696\pi\)
0.659670 + 0.751556i \(0.270696\pi\)
\(978\) 0 0
\(979\) −22.2763 −0.711953
\(980\) 0 0
\(981\) −5.55911 −0.177489
\(982\) 0 0
\(983\) 57.1516 1.82285 0.911426 0.411465i \(-0.134983\pi\)
0.911426 + 0.411465i \(0.134983\pi\)
\(984\) 0 0
\(985\) −7.04334 −0.224419
\(986\) 0 0
\(987\) −5.20234 −0.165592
\(988\) 0 0
\(989\) 38.0977 1.21144
\(990\) 0 0
\(991\) 0.489719 0.0155564 0.00777822 0.999970i \(-0.497524\pi\)
0.00777822 + 0.999970i \(0.497524\pi\)
\(992\) 0 0
\(993\) −31.4342 −0.997535
\(994\) 0 0
\(995\) −9.79950 −0.310665
\(996\) 0 0
\(997\) 19.7390 0.625142 0.312571 0.949894i \(-0.398810\pi\)
0.312571 + 0.949894i \(0.398810\pi\)
\(998\) 0 0
\(999\) −56.0592 −1.77363
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 4028.2.a.e.1.7 19
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4028.2.a.e.1.7 19 1.1 even 1 trivial