Properties

Label 1148.2.a.e.1.4
Level $1148$
Weight $2$
Character 1148.1
Self dual yes
Analytic conductor $9.167$
Analytic rank $0$
Dimension $5$
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] = [1148,2,Mod(1,1148)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1148, 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("1148.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1148 = 2^{2} \cdot 7 \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1148.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: \(9.16682615204\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: 5.5.1935333.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - 2x^{4} - 8x^{3} + 10x^{2} + 13x - 11 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(1.60064\) of defining polynomial
Character \(\chi\) \(=\) 1148.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.60064 q^{3} -2.47347 q^{5} +1.00000 q^{7} -0.437946 q^{9} +O(q^{10})\) \(q+1.60064 q^{3} -2.47347 q^{5} +1.00000 q^{7} -0.437946 q^{9} +1.48567 q^{11} +2.55292 q^{13} -3.95914 q^{15} +4.00686 q^{17} +1.95228 q^{19} +1.60064 q^{21} +3.60064 q^{23} +1.11804 q^{25} -5.50292 q^{27} +3.95914 q^{29} +4.47347 q^{31} +2.37802 q^{33} -2.47347 q^{35} +8.98552 q^{37} +4.08631 q^{39} +1.00000 q^{41} -2.00686 q^{43} +1.08324 q^{45} -0.268535 q^{47} +1.00000 q^{49} +6.41354 q^{51} +2.47347 q^{53} -3.67475 q^{55} +3.12490 q^{57} +2.37802 q^{59} +3.24959 q^{61} -0.437946 q^{63} -6.31456 q^{65} +2.16634 q^{67} +5.76334 q^{69} -14.7397 q^{71} +5.46127 q^{73} +1.78958 q^{75} +1.48567 q^{77} +13.2785 q^{79} -7.49437 q^{81} -8.73973 q^{83} -9.91083 q^{85} +6.33716 q^{87} +3.21533 q^{89} +2.55292 q^{91} +7.16042 q^{93} -4.82889 q^{95} -2.80193 q^{97} -0.650642 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + 2 q^{3} + 3 q^{5} + 5 q^{7} + 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q + 2 q^{3} + 3 q^{5} + 5 q^{7} + 5 q^{9} + 7 q^{13} + 3 q^{15} - 3 q^{17} + 10 q^{19} + 2 q^{21} + 12 q^{23} + 2 q^{25} + 14 q^{27} - 3 q^{29} + 7 q^{31} - 3 q^{33} + 3 q^{35} + q^{37} + 7 q^{39} + 5 q^{41} + 13 q^{43} - 3 q^{45} + 9 q^{47} + 5 q^{49} + 9 q^{51} - 3 q^{53} + 9 q^{55} - 11 q^{57} - 3 q^{59} + 16 q^{61} + 5 q^{63} + 3 q^{65} + 19 q^{67} + 24 q^{69} - 12 q^{71} + 4 q^{73} + 8 q^{75} + 28 q^{79} + 5 q^{81} + 18 q^{83} - 15 q^{85} - 6 q^{87} + 7 q^{91} + q^{93} + 3 q^{95} + 4 q^{97} - 33 q^{99}+O(q^{100}) \) 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
\(3\) 1.60064 0.924131 0.462066 0.886846i \(-0.347108\pi\)
0.462066 + 0.886846i \(0.347108\pi\)
\(4\) 0 0
\(5\) −2.47347 −1.10617 −0.553084 0.833125i \(-0.686549\pi\)
−0.553084 + 0.833125i \(0.686549\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) −0.437946 −0.145982
\(10\) 0 0
\(11\) 1.48567 0.447946 0.223973 0.974595i \(-0.428097\pi\)
0.223973 + 0.974595i \(0.428097\pi\)
\(12\) 0 0
\(13\) 2.55292 0.708052 0.354026 0.935235i \(-0.384812\pi\)
0.354026 + 0.935235i \(0.384812\pi\)
\(14\) 0 0
\(15\) −3.95914 −1.02224
\(16\) 0 0
\(17\) 4.00686 0.971806 0.485903 0.874013i \(-0.338491\pi\)
0.485903 + 0.874013i \(0.338491\pi\)
\(18\) 0 0
\(19\) 1.95228 0.447883 0.223942 0.974603i \(-0.428108\pi\)
0.223942 + 0.974603i \(0.428108\pi\)
\(20\) 0 0
\(21\) 1.60064 0.349289
\(22\) 0 0
\(23\) 3.60064 0.750786 0.375393 0.926866i \(-0.377508\pi\)
0.375393 + 0.926866i \(0.377508\pi\)
\(24\) 0 0
\(25\) 1.11804 0.223608
\(26\) 0 0
\(27\) −5.50292 −1.05904
\(28\) 0 0
\(29\) 3.95914 0.735193 0.367596 0.929985i \(-0.380181\pi\)
0.367596 + 0.929985i \(0.380181\pi\)
\(30\) 0 0
\(31\) 4.47347 0.803458 0.401729 0.915759i \(-0.368409\pi\)
0.401729 + 0.915759i \(0.368409\pi\)
\(32\) 0 0
\(33\) 2.37802 0.413961
\(34\) 0 0
\(35\) −2.47347 −0.418092
\(36\) 0 0
\(37\) 8.98552 1.47721 0.738605 0.674138i \(-0.235485\pi\)
0.738605 + 0.674138i \(0.235485\pi\)
\(38\) 0 0
\(39\) 4.08631 0.654333
\(40\) 0 0
\(41\) 1.00000 0.156174
\(42\) 0 0
\(43\) −2.00686 −0.306043 −0.153021 0.988223i \(-0.548900\pi\)
−0.153021 + 0.988223i \(0.548900\pi\)
\(44\) 0 0
\(45\) 1.08324 0.161480
\(46\) 0 0
\(47\) −0.268535 −0.0391699 −0.0195849 0.999808i \(-0.506234\pi\)
−0.0195849 + 0.999808i \(0.506234\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 6.41354 0.898076
\(52\) 0 0
\(53\) 2.47347 0.339757 0.169878 0.985465i \(-0.445662\pi\)
0.169878 + 0.985465i \(0.445662\pi\)
\(54\) 0 0
\(55\) −3.67475 −0.495503
\(56\) 0 0
\(57\) 3.12490 0.413903
\(58\) 0 0
\(59\) 2.37802 0.309592 0.154796 0.987946i \(-0.450528\pi\)
0.154796 + 0.987946i \(0.450528\pi\)
\(60\) 0 0
\(61\) 3.24959 0.416067 0.208034 0.978122i \(-0.433294\pi\)
0.208034 + 0.978122i \(0.433294\pi\)
\(62\) 0 0
\(63\) −0.437946 −0.0551760
\(64\) 0 0
\(65\) −6.31456 −0.783225
\(66\) 0 0
\(67\) 2.16634 0.264661 0.132331 0.991206i \(-0.457754\pi\)
0.132331 + 0.991206i \(0.457754\pi\)
\(68\) 0 0
\(69\) 5.76334 0.693824
\(70\) 0 0
\(71\) −14.7397 −1.74928 −0.874642 0.484770i \(-0.838903\pi\)
−0.874642 + 0.484770i \(0.838903\pi\)
\(72\) 0 0
\(73\) 5.46127 0.639193 0.319596 0.947554i \(-0.396453\pi\)
0.319596 + 0.947554i \(0.396453\pi\)
\(74\) 0 0
\(75\) 1.78958 0.206643
\(76\) 0 0
\(77\) 1.48567 0.169308
\(78\) 0 0
\(79\) 13.2785 1.49394 0.746972 0.664856i \(-0.231507\pi\)
0.746972 + 0.664856i \(0.231507\pi\)
\(80\) 0 0
\(81\) −7.49437 −0.832707
\(82\) 0 0
\(83\) −8.73973 −0.959309 −0.479655 0.877457i \(-0.659238\pi\)
−0.479655 + 0.877457i \(0.659238\pi\)
\(84\) 0 0
\(85\) −9.91083 −1.07498
\(86\) 0 0
\(87\) 6.33716 0.679415
\(88\) 0 0
\(89\) 3.21533 0.340824 0.170412 0.985373i \(-0.445490\pi\)
0.170412 + 0.985373i \(0.445490\pi\)
\(90\) 0 0
\(91\) 2.55292 0.267619
\(92\) 0 0
\(93\) 7.16042 0.742501
\(94\) 0 0
\(95\) −4.82889 −0.495434
\(96\) 0 0
\(97\) −2.80193 −0.284492 −0.142246 0.989831i \(-0.545432\pi\)
−0.142246 + 0.989831i \(0.545432\pi\)
\(98\) 0 0
\(99\) −0.650642 −0.0653920
\(100\) 0 0
\(101\) −5.48873 −0.546149 −0.273075 0.961993i \(-0.588041\pi\)
−0.273075 + 0.961993i \(0.588041\pi\)
\(102\) 0 0
\(103\) 7.72919 0.761579 0.380790 0.924662i \(-0.375652\pi\)
0.380790 + 0.924662i \(0.375652\pi\)
\(104\) 0 0
\(105\) −3.95914 −0.386372
\(106\) 0 0
\(107\) −7.79965 −0.754020 −0.377010 0.926209i \(-0.623048\pi\)
−0.377010 + 0.926209i \(0.623048\pi\)
\(108\) 0 0
\(109\) −4.71699 −0.451805 −0.225903 0.974150i \(-0.572533\pi\)
−0.225903 + 0.974150i \(0.572533\pi\)
\(110\) 0 0
\(111\) 14.3826 1.36514
\(112\) 0 0
\(113\) −15.6308 −1.47042 −0.735212 0.677837i \(-0.762917\pi\)
−0.735212 + 0.677837i \(0.762917\pi\)
\(114\) 0 0
\(115\) −8.90607 −0.830495
\(116\) 0 0
\(117\) −1.11804 −0.103363
\(118\) 0 0
\(119\) 4.00686 0.367308
\(120\) 0 0
\(121\) −8.79279 −0.799345
\(122\) 0 0
\(123\) 1.60064 0.144325
\(124\) 0 0
\(125\) 9.60190 0.858820
\(126\) 0 0
\(127\) −17.3227 −1.53714 −0.768570 0.639766i \(-0.779031\pi\)
−0.768570 + 0.639766i \(0.779031\pi\)
\(128\) 0 0
\(129\) −3.21226 −0.282824
\(130\) 0 0
\(131\) −16.4445 −1.43676 −0.718382 0.695649i \(-0.755117\pi\)
−0.718382 + 0.695649i \(0.755117\pi\)
\(132\) 0 0
\(133\) 1.95228 0.169284
\(134\) 0 0
\(135\) 13.6113 1.17147
\(136\) 0 0
\(137\) −15.2859 −1.30596 −0.652981 0.757374i \(-0.726482\pi\)
−0.652981 + 0.757374i \(0.726482\pi\)
\(138\) 0 0
\(139\) 10.5342 0.893497 0.446749 0.894659i \(-0.352582\pi\)
0.446749 + 0.894659i \(0.352582\pi\)
\(140\) 0 0
\(141\) −0.429829 −0.0361981
\(142\) 0 0
\(143\) 3.79279 0.317169
\(144\) 0 0
\(145\) −9.79279 −0.813247
\(146\) 0 0
\(147\) 1.60064 0.132019
\(148\) 0 0
\(149\) −1.88015 −0.154028 −0.0770141 0.997030i \(-0.524539\pi\)
−0.0770141 + 0.997030i \(0.524539\pi\)
\(150\) 0 0
\(151\) 18.6199 1.51526 0.757632 0.652682i \(-0.226356\pi\)
0.757632 + 0.652682i \(0.226356\pi\)
\(152\) 0 0
\(153\) −1.75479 −0.141866
\(154\) 0 0
\(155\) −11.0650 −0.888760
\(156\) 0 0
\(157\) 10.0357 0.800938 0.400469 0.916310i \(-0.368847\pi\)
0.400469 + 0.916310i \(0.368847\pi\)
\(158\) 0 0
\(159\) 3.95914 0.313980
\(160\) 0 0
\(161\) 3.60064 0.283770
\(162\) 0 0
\(163\) 10.2406 0.802105 0.401053 0.916055i \(-0.368644\pi\)
0.401053 + 0.916055i \(0.368644\pi\)
\(164\) 0 0
\(165\) −5.88196 −0.457910
\(166\) 0 0
\(167\) 12.6040 0.975327 0.487663 0.873032i \(-0.337849\pi\)
0.487663 + 0.873032i \(0.337849\pi\)
\(168\) 0 0
\(169\) −6.48260 −0.498662
\(170\) 0 0
\(171\) −0.854991 −0.0653828
\(172\) 0 0
\(173\) −3.49180 −0.265477 −0.132738 0.991151i \(-0.542377\pi\)
−0.132738 + 0.991151i \(0.542377\pi\)
\(174\) 0 0
\(175\) 1.11804 0.0845159
\(176\) 0 0
\(177\) 3.80636 0.286104
\(178\) 0 0
\(179\) 8.29016 0.619636 0.309818 0.950796i \(-0.399732\pi\)
0.309818 + 0.950796i \(0.399732\pi\)
\(180\) 0 0
\(181\) 1.06371 0.0790653 0.0395327 0.999218i \(-0.487413\pi\)
0.0395327 + 0.999218i \(0.487413\pi\)
\(182\) 0 0
\(183\) 5.20143 0.384501
\(184\) 0 0
\(185\) −22.2254 −1.63404
\(186\) 0 0
\(187\) 5.95286 0.435316
\(188\) 0 0
\(189\) −5.50292 −0.400279
\(190\) 0 0
\(191\) 2.79142 0.201980 0.100990 0.994887i \(-0.467799\pi\)
0.100990 + 0.994887i \(0.467799\pi\)
\(192\) 0 0
\(193\) 2.71894 0.195713 0.0978567 0.995201i \(-0.468801\pi\)
0.0978567 + 0.995201i \(0.468801\pi\)
\(194\) 0 0
\(195\) −10.1074 −0.723803
\(196\) 0 0
\(197\) −19.5379 −1.39201 −0.696007 0.718035i \(-0.745042\pi\)
−0.696007 + 0.718035i \(0.745042\pi\)
\(198\) 0 0
\(199\) 13.7435 0.974252 0.487126 0.873332i \(-0.338045\pi\)
0.487126 + 0.873332i \(0.338045\pi\)
\(200\) 0 0
\(201\) 3.46754 0.244582
\(202\) 0 0
\(203\) 3.95914 0.277877
\(204\) 0 0
\(205\) −2.47347 −0.172754
\(206\) 0 0
\(207\) −1.57689 −0.109601
\(208\) 0 0
\(209\) 2.90044 0.200627
\(210\) 0 0
\(211\) 9.66211 0.665167 0.332584 0.943074i \(-0.392080\pi\)
0.332584 + 0.943074i \(0.392080\pi\)
\(212\) 0 0
\(213\) −23.5930 −1.61657
\(214\) 0 0
\(215\) 4.96390 0.338535
\(216\) 0 0
\(217\) 4.47347 0.303679
\(218\) 0 0
\(219\) 8.74153 0.590698
\(220\) 0 0
\(221\) 10.2292 0.688089
\(222\) 0 0
\(223\) −14.3344 −0.959903 −0.479952 0.877295i \(-0.659346\pi\)
−0.479952 + 0.877295i \(0.659346\pi\)
\(224\) 0 0
\(225\) −0.489641 −0.0326427
\(226\) 0 0
\(227\) −2.39810 −0.159167 −0.0795837 0.996828i \(-0.525359\pi\)
−0.0795837 + 0.996828i \(0.525359\pi\)
\(228\) 0 0
\(229\) −4.26500 −0.281839 −0.140920 0.990021i \(-0.545006\pi\)
−0.140920 + 0.990021i \(0.545006\pi\)
\(230\) 0 0
\(231\) 2.37802 0.156462
\(232\) 0 0
\(233\) −12.8350 −0.840851 −0.420425 0.907327i \(-0.638119\pi\)
−0.420425 + 0.907327i \(0.638119\pi\)
\(234\) 0 0
\(235\) 0.664213 0.0433285
\(236\) 0 0
\(237\) 21.2541 1.38060
\(238\) 0 0
\(239\) −11.6337 −0.752524 −0.376262 0.926513i \(-0.622791\pi\)
−0.376262 + 0.926513i \(0.622791\pi\)
\(240\) 0 0
\(241\) −13.7984 −0.888835 −0.444417 0.895820i \(-0.646589\pi\)
−0.444417 + 0.895820i \(0.646589\pi\)
\(242\) 0 0
\(243\) 4.51296 0.289507
\(244\) 0 0
\(245\) −2.47347 −0.158024
\(246\) 0 0
\(247\) 4.98401 0.317125
\(248\) 0 0
\(249\) −13.9892 −0.886528
\(250\) 0 0
\(251\) 4.12548 0.260398 0.130199 0.991488i \(-0.458438\pi\)
0.130199 + 0.991488i \(0.458438\pi\)
\(252\) 0 0
\(253\) 5.34936 0.336311
\(254\) 0 0
\(255\) −15.8637 −0.993423
\(256\) 0 0
\(257\) −0.163921 −0.0102251 −0.00511255 0.999987i \(-0.501627\pi\)
−0.00511255 + 0.999987i \(0.501627\pi\)
\(258\) 0 0
\(259\) 8.98552 0.558333
\(260\) 0 0
\(261\) −1.73389 −0.107325
\(262\) 0 0
\(263\) 7.49577 0.462209 0.231104 0.972929i \(-0.425766\pi\)
0.231104 + 0.972929i \(0.425766\pi\)
\(264\) 0 0
\(265\) −6.11804 −0.375828
\(266\) 0 0
\(267\) 5.14659 0.314966
\(268\) 0 0
\(269\) 25.1315 1.53229 0.766146 0.642666i \(-0.222172\pi\)
0.766146 + 0.642666i \(0.222172\pi\)
\(270\) 0 0
\(271\) −17.2978 −1.05077 −0.525383 0.850866i \(-0.676078\pi\)
−0.525383 + 0.850866i \(0.676078\pi\)
\(272\) 0 0
\(273\) 4.08631 0.247315
\(274\) 0 0
\(275\) 1.66104 0.100164
\(276\) 0 0
\(277\) −14.7894 −0.888611 −0.444305 0.895875i \(-0.646549\pi\)
−0.444305 + 0.895875i \(0.646549\pi\)
\(278\) 0 0
\(279\) −1.95914 −0.117290
\(280\) 0 0
\(281\) −5.33132 −0.318040 −0.159020 0.987275i \(-0.550833\pi\)
−0.159020 + 0.987275i \(0.550833\pi\)
\(282\) 0 0
\(283\) 6.12743 0.364238 0.182119 0.983277i \(-0.441704\pi\)
0.182119 + 0.983277i \(0.441704\pi\)
\(284\) 0 0
\(285\) −7.72933 −0.457846
\(286\) 0 0
\(287\) 1.00000 0.0590281
\(288\) 0 0
\(289\) −0.945093 −0.0555937
\(290\) 0 0
\(291\) −4.48488 −0.262908
\(292\) 0 0
\(293\) 8.64544 0.505072 0.252536 0.967587i \(-0.418735\pi\)
0.252536 + 0.967587i \(0.418735\pi\)
\(294\) 0 0
\(295\) −5.88196 −0.342461
\(296\) 0 0
\(297\) −8.17551 −0.474391
\(298\) 0 0
\(299\) 9.19215 0.531596
\(300\) 0 0
\(301\) −2.00686 −0.115673
\(302\) 0 0
\(303\) −8.78550 −0.504714
\(304\) 0 0
\(305\) −8.03775 −0.460240
\(306\) 0 0
\(307\) −17.2043 −0.981902 −0.490951 0.871187i \(-0.663351\pi\)
−0.490951 + 0.871187i \(0.663351\pi\)
\(308\) 0 0
\(309\) 12.3717 0.703799
\(310\) 0 0
\(311\) −25.3526 −1.43761 −0.718807 0.695210i \(-0.755312\pi\)
−0.718807 + 0.695210i \(0.755312\pi\)
\(312\) 0 0
\(313\) −9.22280 −0.521303 −0.260652 0.965433i \(-0.583937\pi\)
−0.260652 + 0.965433i \(0.583937\pi\)
\(314\) 0 0
\(315\) 1.08324 0.0610339
\(316\) 0 0
\(317\) −2.21363 −0.124330 −0.0621649 0.998066i \(-0.519800\pi\)
−0.0621649 + 0.998066i \(0.519800\pi\)
\(318\) 0 0
\(319\) 5.88196 0.329327
\(320\) 0 0
\(321\) −12.4844 −0.696814
\(322\) 0 0
\(323\) 7.82250 0.435255
\(324\) 0 0
\(325\) 2.85427 0.158326
\(326\) 0 0
\(327\) −7.55021 −0.417527
\(328\) 0 0
\(329\) −0.268535 −0.0148048
\(330\) 0 0
\(331\) 8.62038 0.473819 0.236909 0.971532i \(-0.423866\pi\)
0.236909 + 0.971532i \(0.423866\pi\)
\(332\) 0 0
\(333\) −3.93517 −0.215646
\(334\) 0 0
\(335\) −5.35838 −0.292760
\(336\) 0 0
\(337\) 1.79536 0.0977994 0.0488997 0.998804i \(-0.484429\pi\)
0.0488997 + 0.998804i \(0.484429\pi\)
\(338\) 0 0
\(339\) −25.0193 −1.35886
\(340\) 0 0
\(341\) 6.64609 0.359906
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) −14.2554 −0.767486
\(346\) 0 0
\(347\) −9.35563 −0.502237 −0.251118 0.967956i \(-0.580798\pi\)
−0.251118 + 0.967956i \(0.580798\pi\)
\(348\) 0 0
\(349\) −22.5763 −1.20848 −0.604240 0.796802i \(-0.706523\pi\)
−0.604240 + 0.796802i \(0.706523\pi\)
\(350\) 0 0
\(351\) −14.0485 −0.749854
\(352\) 0 0
\(353\) −17.2479 −0.918014 −0.459007 0.888433i \(-0.651795\pi\)
−0.459007 + 0.888433i \(0.651795\pi\)
\(354\) 0 0
\(355\) 36.4582 1.93500
\(356\) 0 0
\(357\) 6.41354 0.339441
\(358\) 0 0
\(359\) −12.7449 −0.672649 −0.336325 0.941746i \(-0.609184\pi\)
−0.336325 + 0.941746i \(0.609184\pi\)
\(360\) 0 0
\(361\) −15.1886 −0.799401
\(362\) 0 0
\(363\) −14.0741 −0.738699
\(364\) 0 0
\(365\) −13.5083 −0.707055
\(366\) 0 0
\(367\) −8.57951 −0.447847 −0.223923 0.974607i \(-0.571887\pi\)
−0.223923 + 0.974607i \(0.571887\pi\)
\(368\) 0 0
\(369\) −0.437946 −0.0227985
\(370\) 0 0
\(371\) 2.47347 0.128416
\(372\) 0 0
\(373\) −15.1282 −0.783309 −0.391655 0.920112i \(-0.628097\pi\)
−0.391655 + 0.920112i \(0.628097\pi\)
\(374\) 0 0
\(375\) 15.3692 0.793662
\(376\) 0 0
\(377\) 10.1074 0.520555
\(378\) 0 0
\(379\) −12.7972 −0.657349 −0.328675 0.944443i \(-0.606602\pi\)
−0.328675 + 0.944443i \(0.606602\pi\)
\(380\) 0 0
\(381\) −27.7274 −1.42052
\(382\) 0 0
\(383\) 9.89743 0.505735 0.252868 0.967501i \(-0.418626\pi\)
0.252868 + 0.967501i \(0.418626\pi\)
\(384\) 0 0
\(385\) −3.67475 −0.187283
\(386\) 0 0
\(387\) 0.878894 0.0446767
\(388\) 0 0
\(389\) 9.07916 0.460332 0.230166 0.973151i \(-0.426073\pi\)
0.230166 + 0.973151i \(0.426073\pi\)
\(390\) 0 0
\(391\) 14.4273 0.729618
\(392\) 0 0
\(393\) −26.3218 −1.32776
\(394\) 0 0
\(395\) −32.8438 −1.65255
\(396\) 0 0
\(397\) 23.0487 1.15678 0.578389 0.815761i \(-0.303681\pi\)
0.578389 + 0.815761i \(0.303681\pi\)
\(398\) 0 0
\(399\) 3.12490 0.156441
\(400\) 0 0
\(401\) −7.89679 −0.394347 −0.197173 0.980369i \(-0.563176\pi\)
−0.197173 + 0.980369i \(0.563176\pi\)
\(402\) 0 0
\(403\) 11.4204 0.568891
\(404\) 0 0
\(405\) 18.5371 0.921114
\(406\) 0 0
\(407\) 13.3495 0.661710
\(408\) 0 0
\(409\) 19.0087 0.939921 0.469961 0.882687i \(-0.344268\pi\)
0.469961 + 0.882687i \(0.344268\pi\)
\(410\) 0 0
\(411\) −24.4672 −1.20688
\(412\) 0 0
\(413\) 2.37802 0.117015
\(414\) 0 0
\(415\) 21.6174 1.06116
\(416\) 0 0
\(417\) 16.8614 0.825709
\(418\) 0 0
\(419\) 27.2980 1.33359 0.666796 0.745240i \(-0.267665\pi\)
0.666796 + 0.745240i \(0.267665\pi\)
\(420\) 0 0
\(421\) −30.2599 −1.47478 −0.737389 0.675469i \(-0.763941\pi\)
−0.737389 + 0.675469i \(0.763941\pi\)
\(422\) 0 0
\(423\) 0.117604 0.00571809
\(424\) 0 0
\(425\) 4.47983 0.217303
\(426\) 0 0
\(427\) 3.24959 0.157259
\(428\) 0 0
\(429\) 6.07090 0.293106
\(430\) 0 0
\(431\) −37.4875 −1.80571 −0.902855 0.429944i \(-0.858533\pi\)
−0.902855 + 0.429944i \(0.858533\pi\)
\(432\) 0 0
\(433\) 35.5823 1.70998 0.854989 0.518646i \(-0.173564\pi\)
0.854989 + 0.518646i \(0.173564\pi\)
\(434\) 0 0
\(435\) −15.6748 −0.751547
\(436\) 0 0
\(437\) 7.02945 0.336264
\(438\) 0 0
\(439\) 34.4665 1.64500 0.822498 0.568768i \(-0.192580\pi\)
0.822498 + 0.568768i \(0.192580\pi\)
\(440\) 0 0
\(441\) −0.437946 −0.0208546
\(442\) 0 0
\(443\) 3.20825 0.152429 0.0762143 0.997091i \(-0.475717\pi\)
0.0762143 + 0.997091i \(0.475717\pi\)
\(444\) 0 0
\(445\) −7.95300 −0.377009
\(446\) 0 0
\(447\) −3.00945 −0.142342
\(448\) 0 0
\(449\) −24.5493 −1.15855 −0.579276 0.815131i \(-0.696665\pi\)
−0.579276 + 0.815131i \(0.696665\pi\)
\(450\) 0 0
\(451\) 1.48567 0.0699574
\(452\) 0 0
\(453\) 29.8038 1.40030
\(454\) 0 0
\(455\) −6.31456 −0.296031
\(456\) 0 0
\(457\) −38.5083 −1.80134 −0.900670 0.434504i \(-0.856924\pi\)
−0.900670 + 0.434504i \(0.856924\pi\)
\(458\) 0 0
\(459\) −22.0494 −1.02918
\(460\) 0 0
\(461\) 37.2933 1.73692 0.868460 0.495759i \(-0.165110\pi\)
0.868460 + 0.495759i \(0.165110\pi\)
\(462\) 0 0
\(463\) 24.1601 1.12282 0.561408 0.827539i \(-0.310260\pi\)
0.561408 + 0.827539i \(0.310260\pi\)
\(464\) 0 0
\(465\) −17.7111 −0.821331
\(466\) 0 0
\(467\) 8.64767 0.400166 0.200083 0.979779i \(-0.435879\pi\)
0.200083 + 0.979779i \(0.435879\pi\)
\(468\) 0 0
\(469\) 2.16634 0.100032
\(470\) 0 0
\(471\) 16.0636 0.740172
\(472\) 0 0
\(473\) −2.98152 −0.137091
\(474\) 0 0
\(475\) 2.18272 0.100150
\(476\) 0 0
\(477\) −1.08324 −0.0495983
\(478\) 0 0
\(479\) 10.3240 0.471718 0.235859 0.971787i \(-0.424210\pi\)
0.235859 + 0.971787i \(0.424210\pi\)
\(480\) 0 0
\(481\) 22.9393 1.04594
\(482\) 0 0
\(483\) 5.76334 0.262241
\(484\) 0 0
\(485\) 6.93047 0.314696
\(486\) 0 0
\(487\) 35.2407 1.59691 0.798454 0.602056i \(-0.205651\pi\)
0.798454 + 0.602056i \(0.205651\pi\)
\(488\) 0 0
\(489\) 16.3915 0.741251
\(490\) 0 0
\(491\) 36.8607 1.66350 0.831751 0.555149i \(-0.187339\pi\)
0.831751 + 0.555149i \(0.187339\pi\)
\(492\) 0 0
\(493\) 15.8637 0.714465
\(494\) 0 0
\(495\) 1.60934 0.0723345
\(496\) 0 0
\(497\) −14.7397 −0.661167
\(498\) 0 0
\(499\) 33.4943 1.49941 0.749706 0.661771i \(-0.230195\pi\)
0.749706 + 0.661771i \(0.230195\pi\)
\(500\) 0 0
\(501\) 20.1745 0.901330
\(502\) 0 0
\(503\) 24.8665 1.10874 0.554370 0.832270i \(-0.312959\pi\)
0.554370 + 0.832270i \(0.312959\pi\)
\(504\) 0 0
\(505\) 13.5762 0.604133
\(506\) 0 0
\(507\) −10.3763 −0.460829
\(508\) 0 0
\(509\) 6.15050 0.272616 0.136308 0.990667i \(-0.456476\pi\)
0.136308 + 0.990667i \(0.456476\pi\)
\(510\) 0 0
\(511\) 5.46127 0.241592
\(512\) 0 0
\(513\) −10.7432 −0.474325
\(514\) 0 0
\(515\) −19.1179 −0.842435
\(516\) 0 0
\(517\) −0.398954 −0.0175460
\(518\) 0 0
\(519\) −5.58912 −0.245335
\(520\) 0 0
\(521\) 28.8734 1.26497 0.632483 0.774574i \(-0.282036\pi\)
0.632483 + 0.774574i \(0.282036\pi\)
\(522\) 0 0
\(523\) 13.9944 0.611931 0.305965 0.952043i \(-0.401021\pi\)
0.305965 + 0.952043i \(0.401021\pi\)
\(524\) 0 0
\(525\) 1.78958 0.0781037
\(526\) 0 0
\(527\) 17.9245 0.780805
\(528\) 0 0
\(529\) −10.0354 −0.436321
\(530\) 0 0
\(531\) −1.04144 −0.0451948
\(532\) 0 0
\(533\) 2.55292 0.110579
\(534\) 0 0
\(535\) 19.2922 0.834073
\(536\) 0 0
\(537\) 13.2696 0.572625
\(538\) 0 0
\(539\) 1.48567 0.0639922
\(540\) 0 0
\(541\) −8.42199 −0.362090 −0.181045 0.983475i \(-0.557948\pi\)
−0.181045 + 0.983475i \(0.557948\pi\)
\(542\) 0 0
\(543\) 1.70263 0.0730667
\(544\) 0 0
\(545\) 11.6673 0.499773
\(546\) 0 0
\(547\) −7.38048 −0.315566 −0.157783 0.987474i \(-0.550435\pi\)
−0.157783 + 0.987474i \(0.550435\pi\)
\(548\) 0 0
\(549\) −1.42314 −0.0607383
\(550\) 0 0
\(551\) 7.72933 0.329281
\(552\) 0 0
\(553\) 13.2785 0.564657
\(554\) 0 0
\(555\) −35.5749 −1.51007
\(556\) 0 0
\(557\) 21.7185 0.920242 0.460121 0.887856i \(-0.347806\pi\)
0.460121 + 0.887856i \(0.347806\pi\)
\(558\) 0 0
\(559\) −5.12335 −0.216694
\(560\) 0 0
\(561\) 9.52840 0.402289
\(562\) 0 0
\(563\) −12.3047 −0.518581 −0.259291 0.965799i \(-0.583489\pi\)
−0.259291 + 0.965799i \(0.583489\pi\)
\(564\) 0 0
\(565\) 38.6623 1.62654
\(566\) 0 0
\(567\) −7.49437 −0.314734
\(568\) 0 0
\(569\) 16.4711 0.690504 0.345252 0.938510i \(-0.387794\pi\)
0.345252 + 0.938510i \(0.387794\pi\)
\(570\) 0 0
\(571\) −7.46234 −0.312289 −0.156145 0.987734i \(-0.549907\pi\)
−0.156145 + 0.987734i \(0.549907\pi\)
\(572\) 0 0
\(573\) 4.46806 0.186656
\(574\) 0 0
\(575\) 4.02566 0.167882
\(576\) 0 0
\(577\) −22.2121 −0.924703 −0.462352 0.886697i \(-0.652994\pi\)
−0.462352 + 0.886697i \(0.652994\pi\)
\(578\) 0 0
\(579\) 4.35204 0.180865
\(580\) 0 0
\(581\) −8.73973 −0.362585
\(582\) 0 0
\(583\) 3.67475 0.152193
\(584\) 0 0
\(585\) 2.76543 0.114337
\(586\) 0 0
\(587\) −29.9587 −1.23653 −0.618264 0.785971i \(-0.712164\pi\)
−0.618264 + 0.785971i \(0.712164\pi\)
\(588\) 0 0
\(589\) 8.73345 0.359856
\(590\) 0 0
\(591\) −31.2731 −1.28640
\(592\) 0 0
\(593\) −6.60881 −0.271391 −0.135696 0.990751i \(-0.543327\pi\)
−0.135696 + 0.990751i \(0.543327\pi\)
\(594\) 0 0
\(595\) −9.91083 −0.406304
\(596\) 0 0
\(597\) 21.9984 0.900337
\(598\) 0 0
\(599\) −16.6199 −0.679071 −0.339535 0.940593i \(-0.610270\pi\)
−0.339535 + 0.940593i \(0.610270\pi\)
\(600\) 0 0
\(601\) 2.81524 0.114836 0.0574180 0.998350i \(-0.481713\pi\)
0.0574180 + 0.998350i \(0.481713\pi\)
\(602\) 0 0
\(603\) −0.948741 −0.0386357
\(604\) 0 0
\(605\) 21.7487 0.884210
\(606\) 0 0
\(607\) −4.70796 −0.191090 −0.0955452 0.995425i \(-0.530459\pi\)
−0.0955452 + 0.995425i \(0.530459\pi\)
\(608\) 0 0
\(609\) 6.33716 0.256795
\(610\) 0 0
\(611\) −0.685549 −0.0277343
\(612\) 0 0
\(613\) −33.5857 −1.35651 −0.678257 0.734825i \(-0.737264\pi\)
−0.678257 + 0.734825i \(0.737264\pi\)
\(614\) 0 0
\(615\) −3.95914 −0.159648
\(616\) 0 0
\(617\) 10.1069 0.406887 0.203443 0.979087i \(-0.434787\pi\)
0.203443 + 0.979087i \(0.434787\pi\)
\(618\) 0 0
\(619\) −15.2098 −0.611334 −0.305667 0.952139i \(-0.598879\pi\)
−0.305667 + 0.952139i \(0.598879\pi\)
\(620\) 0 0
\(621\) −19.8140 −0.795110
\(622\) 0 0
\(623\) 3.21533 0.128819
\(624\) 0 0
\(625\) −29.3402 −1.17361
\(626\) 0 0
\(627\) 4.64256 0.185406
\(628\) 0 0
\(629\) 36.0037 1.43556
\(630\) 0 0
\(631\) −29.8417 −1.18798 −0.593990 0.804473i \(-0.702448\pi\)
−0.593990 + 0.804473i \(0.702448\pi\)
\(632\) 0 0
\(633\) 15.4656 0.614702
\(634\) 0 0
\(635\) 42.8471 1.70033
\(636\) 0 0
\(637\) 2.55292 0.101150
\(638\) 0 0
\(639\) 6.45520 0.255364
\(640\) 0 0
\(641\) −19.7674 −0.780767 −0.390384 0.920652i \(-0.627658\pi\)
−0.390384 + 0.920652i \(0.627658\pi\)
\(642\) 0 0
\(643\) −7.55248 −0.297841 −0.148920 0.988849i \(-0.547580\pi\)
−0.148920 + 0.988849i \(0.547580\pi\)
\(644\) 0 0
\(645\) 7.94542 0.312851
\(646\) 0 0
\(647\) 23.3795 0.919144 0.459572 0.888140i \(-0.348003\pi\)
0.459572 + 0.888140i \(0.348003\pi\)
\(648\) 0 0
\(649\) 3.53295 0.138680
\(650\) 0 0
\(651\) 7.16042 0.280639
\(652\) 0 0
\(653\) 43.9764 1.72093 0.860464 0.509511i \(-0.170174\pi\)
0.860464 + 0.509511i \(0.170174\pi\)
\(654\) 0 0
\(655\) 40.6750 1.58930
\(656\) 0 0
\(657\) −2.39174 −0.0933106
\(658\) 0 0
\(659\) −24.3823 −0.949801 −0.474901 0.880039i \(-0.657516\pi\)
−0.474901 + 0.880039i \(0.657516\pi\)
\(660\) 0 0
\(661\) 14.0954 0.548249 0.274125 0.961694i \(-0.411612\pi\)
0.274125 + 0.961694i \(0.411612\pi\)
\(662\) 0 0
\(663\) 16.3733 0.635885
\(664\) 0 0
\(665\) −4.82889 −0.187256
\(666\) 0 0
\(667\) 14.2554 0.551972
\(668\) 0 0
\(669\) −22.9443 −0.887076
\(670\) 0 0
\(671\) 4.82781 0.186376
\(672\) 0 0
\(673\) 39.8755 1.53709 0.768544 0.639797i \(-0.220982\pi\)
0.768544 + 0.639797i \(0.220982\pi\)
\(674\) 0 0
\(675\) −6.15248 −0.236809
\(676\) 0 0
\(677\) −32.1023 −1.23379 −0.616896 0.787045i \(-0.711610\pi\)
−0.616896 + 0.787045i \(0.711610\pi\)
\(678\) 0 0
\(679\) −2.80193 −0.107528
\(680\) 0 0
\(681\) −3.83850 −0.147092
\(682\) 0 0
\(683\) 17.5202 0.670391 0.335195 0.942149i \(-0.391198\pi\)
0.335195 + 0.942149i \(0.391198\pi\)
\(684\) 0 0
\(685\) 37.8092 1.44461
\(686\) 0 0
\(687\) −6.82674 −0.260456
\(688\) 0 0
\(689\) 6.31456 0.240566
\(690\) 0 0
\(691\) 4.08090 0.155245 0.0776225 0.996983i \(-0.475267\pi\)
0.0776225 + 0.996983i \(0.475267\pi\)
\(692\) 0 0
\(693\) −0.650642 −0.0247158
\(694\) 0 0
\(695\) −26.0559 −0.988358
\(696\) 0 0
\(697\) 4.00686 0.151771
\(698\) 0 0
\(699\) −20.5443 −0.777056
\(700\) 0 0
\(701\) −20.6923 −0.781536 −0.390768 0.920489i \(-0.627790\pi\)
−0.390768 + 0.920489i \(0.627790\pi\)
\(702\) 0 0
\(703\) 17.5422 0.661618
\(704\) 0 0
\(705\) 1.06317 0.0400412
\(706\) 0 0
\(707\) −5.48873 −0.206425
\(708\) 0 0
\(709\) 33.7685 1.26820 0.634102 0.773250i \(-0.281370\pi\)
0.634102 + 0.773250i \(0.281370\pi\)
\(710\) 0 0
\(711\) −5.81524 −0.218089
\(712\) 0 0
\(713\) 16.1074 0.603225
\(714\) 0 0
\(715\) −9.38134 −0.350842
\(716\) 0 0
\(717\) −18.6215 −0.695431
\(718\) 0 0
\(719\) −36.2970 −1.35365 −0.676825 0.736144i \(-0.736645\pi\)
−0.676825 + 0.736144i \(0.736645\pi\)
\(720\) 0 0
\(721\) 7.72919 0.287850
\(722\) 0 0
\(723\) −22.0863 −0.821400
\(724\) 0 0
\(725\) 4.42647 0.164395
\(726\) 0 0
\(727\) −22.1013 −0.819691 −0.409846 0.912155i \(-0.634417\pi\)
−0.409846 + 0.912155i \(0.634417\pi\)
\(728\) 0 0
\(729\) 29.7067 1.10025
\(730\) 0 0
\(731\) −8.04119 −0.297414
\(732\) 0 0
\(733\) 5.75923 0.212722 0.106361 0.994328i \(-0.466080\pi\)
0.106361 + 0.994328i \(0.466080\pi\)
\(734\) 0 0
\(735\) −3.95914 −0.146035
\(736\) 0 0
\(737\) 3.21847 0.118554
\(738\) 0 0
\(739\) −15.2456 −0.560820 −0.280410 0.959880i \(-0.590470\pi\)
−0.280410 + 0.959880i \(0.590470\pi\)
\(740\) 0 0
\(741\) 7.97761 0.293065
\(742\) 0 0
\(743\) −7.62428 −0.279708 −0.139854 0.990172i \(-0.544663\pi\)
−0.139854 + 0.990172i \(0.544663\pi\)
\(744\) 0 0
\(745\) 4.65050 0.170381
\(746\) 0 0
\(747\) 3.82752 0.140042
\(748\) 0 0
\(749\) −7.79965 −0.284993
\(750\) 0 0
\(751\) −11.6531 −0.425228 −0.212614 0.977136i \(-0.568198\pi\)
−0.212614 + 0.977136i \(0.568198\pi\)
\(752\) 0 0
\(753\) 6.60342 0.240642
\(754\) 0 0
\(755\) −46.0557 −1.67614
\(756\) 0 0
\(757\) −13.5722 −0.493288 −0.246644 0.969106i \(-0.579328\pi\)
−0.246644 + 0.969106i \(0.579328\pi\)
\(758\) 0 0
\(759\) 8.56241 0.310796
\(760\) 0 0
\(761\) −37.3416 −1.35363 −0.676815 0.736153i \(-0.736640\pi\)
−0.676815 + 0.736153i \(0.736640\pi\)
\(762\) 0 0
\(763\) −4.71699 −0.170766
\(764\) 0 0
\(765\) 4.34040 0.156928
\(766\) 0 0
\(767\) 6.07090 0.219207
\(768\) 0 0
\(769\) 35.1074 1.26601 0.633003 0.774149i \(-0.281822\pi\)
0.633003 + 0.774149i \(0.281822\pi\)
\(770\) 0 0
\(771\) −0.262378 −0.00944932
\(772\) 0 0
\(773\) −18.5320 −0.666550 −0.333275 0.942830i \(-0.608154\pi\)
−0.333275 + 0.942830i \(0.608154\pi\)
\(774\) 0 0
\(775\) 5.00151 0.179660
\(776\) 0 0
\(777\) 14.3826 0.515973
\(778\) 0 0
\(779\) 1.95228 0.0699476
\(780\) 0 0
\(781\) −21.8983 −0.783584
\(782\) 0 0
\(783\) −21.7868 −0.778597
\(784\) 0 0
\(785\) −24.8230 −0.885972
\(786\) 0 0
\(787\) 32.4031 1.15505 0.577524 0.816374i \(-0.304019\pi\)
0.577524 + 0.816374i \(0.304019\pi\)
\(788\) 0 0
\(789\) 11.9980 0.427142
\(790\) 0 0
\(791\) −15.6308 −0.555768
\(792\) 0 0
\(793\) 8.29594 0.294597
\(794\) 0 0
\(795\) −9.79279 −0.347314
\(796\) 0 0
\(797\) −22.8991 −0.811129 −0.405565 0.914066i \(-0.632925\pi\)
−0.405565 + 0.914066i \(0.632925\pi\)
\(798\) 0 0
\(799\) −1.07598 −0.0380655
\(800\) 0 0
\(801\) −1.40814 −0.0497541
\(802\) 0 0
\(803\) 8.11363 0.286324
\(804\) 0 0
\(805\) −8.90607 −0.313898
\(806\) 0 0
\(807\) 40.2265 1.41604
\(808\) 0 0
\(809\) −11.4414 −0.402259 −0.201129 0.979565i \(-0.564461\pi\)
−0.201129 + 0.979565i \(0.564461\pi\)
\(810\) 0 0
\(811\) −6.26979 −0.220162 −0.110081 0.993923i \(-0.535111\pi\)
−0.110081 + 0.993923i \(0.535111\pi\)
\(812\) 0 0
\(813\) −27.6876 −0.971046
\(814\) 0 0
\(815\) −25.3298 −0.887264
\(816\) 0 0
\(817\) −3.91794 −0.137071
\(818\) 0 0
\(819\) −1.11804 −0.0390675
\(820\) 0 0
\(821\) −7.74037 −0.270141 −0.135070 0.990836i \(-0.543126\pi\)
−0.135070 + 0.990836i \(0.543126\pi\)
\(822\) 0 0
\(823\) −25.5868 −0.891900 −0.445950 0.895058i \(-0.647134\pi\)
−0.445950 + 0.895058i \(0.647134\pi\)
\(824\) 0 0
\(825\) 2.65872 0.0925649
\(826\) 0 0
\(827\) 21.3085 0.740969 0.370484 0.928839i \(-0.379192\pi\)
0.370484 + 0.928839i \(0.379192\pi\)
\(828\) 0 0
\(829\) 35.5631 1.23516 0.617579 0.786509i \(-0.288113\pi\)
0.617579 + 0.786509i \(0.288113\pi\)
\(830\) 0 0
\(831\) −23.6726 −0.821193
\(832\) 0 0
\(833\) 4.00686 0.138829
\(834\) 0 0
\(835\) −31.1756 −1.07888
\(836\) 0 0
\(837\) −24.6171 −0.850893
\(838\) 0 0
\(839\) 44.7213 1.54395 0.771976 0.635652i \(-0.219269\pi\)
0.771976 + 0.635652i \(0.219269\pi\)
\(840\) 0 0
\(841\) −13.3252 −0.459491
\(842\) 0 0
\(843\) −8.53353 −0.293910
\(844\) 0 0
\(845\) 16.0345 0.551604
\(846\) 0 0
\(847\) −8.79279 −0.302124
\(848\) 0 0
\(849\) 9.80782 0.336604
\(850\) 0 0
\(851\) 32.3536 1.10907
\(852\) 0 0
\(853\) −51.1574 −1.75160 −0.875798 0.482679i \(-0.839664\pi\)
−0.875798 + 0.482679i \(0.839664\pi\)
\(854\) 0 0
\(855\) 2.11479 0.0723244
\(856\) 0 0
\(857\) 32.0528 1.09490 0.547452 0.836837i \(-0.315598\pi\)
0.547452 + 0.836837i \(0.315598\pi\)
\(858\) 0 0
\(859\) 24.0209 0.819584 0.409792 0.912179i \(-0.365601\pi\)
0.409792 + 0.912179i \(0.365601\pi\)
\(860\) 0 0
\(861\) 1.60064 0.0545497
\(862\) 0 0
\(863\) 17.0819 0.581476 0.290738 0.956803i \(-0.406099\pi\)
0.290738 + 0.956803i \(0.406099\pi\)
\(864\) 0 0
\(865\) 8.63685 0.293662
\(866\) 0 0
\(867\) −1.51276 −0.0513759
\(868\) 0 0
\(869\) 19.7274 0.669206
\(870\) 0 0
\(871\) 5.53050 0.187394
\(872\) 0 0
\(873\) 1.22709 0.0415307
\(874\) 0 0
\(875\) 9.60190 0.324604
\(876\) 0 0
\(877\) 36.8241 1.24346 0.621730 0.783231i \(-0.286430\pi\)
0.621730 + 0.783231i \(0.286430\pi\)
\(878\) 0 0
\(879\) 13.8383 0.466753
\(880\) 0 0
\(881\) −20.6341 −0.695180 −0.347590 0.937647i \(-0.613000\pi\)
−0.347590 + 0.937647i \(0.613000\pi\)
\(882\) 0 0
\(883\) 27.6324 0.929903 0.464952 0.885336i \(-0.346072\pi\)
0.464952 + 0.885336i \(0.346072\pi\)
\(884\) 0 0
\(885\) −9.41491 −0.316479
\(886\) 0 0
\(887\) −3.00146 −0.100779 −0.0503896 0.998730i \(-0.516046\pi\)
−0.0503896 + 0.998730i \(0.516046\pi\)
\(888\) 0 0
\(889\) −17.3227 −0.580984
\(890\) 0 0
\(891\) −11.1341 −0.373008
\(892\) 0 0
\(893\) −0.524255 −0.0175435
\(894\) 0 0
\(895\) −20.5054 −0.685421
\(896\) 0 0
\(897\) 14.7133 0.491264
\(898\) 0 0
\(899\) 17.7111 0.590697
\(900\) 0 0
\(901\) 9.91083 0.330178
\(902\) 0 0
\(903\) −3.21226 −0.106897
\(904\) 0 0
\(905\) −2.63106 −0.0874595
\(906\) 0 0
\(907\) 41.6479 1.38290 0.691448 0.722426i \(-0.256973\pi\)
0.691448 + 0.722426i \(0.256973\pi\)
\(908\) 0 0
\(909\) 2.40377 0.0797279
\(910\) 0 0
\(911\) −32.5434 −1.07821 −0.539106 0.842238i \(-0.681238\pi\)
−0.539106 + 0.842238i \(0.681238\pi\)
\(912\) 0 0
\(913\) −12.9843 −0.429719
\(914\) 0 0
\(915\) −12.8656 −0.425322
\(916\) 0 0
\(917\) −16.4445 −0.543046
\(918\) 0 0
\(919\) −3.91249 −0.129061 −0.0645306 0.997916i \(-0.520555\pi\)
−0.0645306 + 0.997916i \(0.520555\pi\)
\(920\) 0 0
\(921\) −27.5379 −0.907406
\(922\) 0 0
\(923\) −37.6293 −1.23858
\(924\) 0 0
\(925\) 10.0462 0.330316
\(926\) 0 0
\(927\) −3.38496 −0.111177
\(928\) 0 0
\(929\) −24.6363 −0.808291 −0.404145 0.914695i \(-0.632431\pi\)
−0.404145 + 0.914695i \(0.632431\pi\)
\(930\) 0 0
\(931\) 1.95228 0.0639833
\(932\) 0 0
\(933\) −40.5804 −1.32854
\(934\) 0 0
\(935\) −14.7242 −0.481533
\(936\) 0 0
\(937\) −23.8006 −0.777530 −0.388765 0.921337i \(-0.627098\pi\)
−0.388765 + 0.921337i \(0.627098\pi\)
\(938\) 0 0
\(939\) −14.7624 −0.481753
\(940\) 0 0
\(941\) 33.7599 1.10054 0.550271 0.834986i \(-0.314524\pi\)
0.550271 + 0.834986i \(0.314524\pi\)
\(942\) 0 0
\(943\) 3.60064 0.117253
\(944\) 0 0
\(945\) 13.6113 0.442775
\(946\) 0 0
\(947\) −26.7783 −0.870178 −0.435089 0.900387i \(-0.643283\pi\)
−0.435089 + 0.900387i \(0.643283\pi\)
\(948\) 0 0
\(949\) 13.9422 0.452582
\(950\) 0 0
\(951\) −3.54323 −0.114897
\(952\) 0 0
\(953\) −3.41776 −0.110712 −0.0553560 0.998467i \(-0.517629\pi\)
−0.0553560 + 0.998467i \(0.517629\pi\)
\(954\) 0 0
\(955\) −6.90449 −0.223424
\(956\) 0 0
\(957\) 9.41491 0.304341
\(958\) 0 0
\(959\) −15.2859 −0.493607
\(960\) 0 0
\(961\) −10.9881 −0.354455
\(962\) 0 0
\(963\) 3.41582 0.110073
\(964\) 0 0
\(965\) −6.72520 −0.216492
\(966\) 0 0
\(967\) 9.23940 0.297119 0.148560 0.988903i \(-0.452536\pi\)
0.148560 + 0.988903i \(0.452536\pi\)
\(968\) 0 0
\(969\) 12.5210 0.402233
\(970\) 0 0
\(971\) 19.2701 0.618406 0.309203 0.950996i \(-0.399938\pi\)
0.309203 + 0.950996i \(0.399938\pi\)
\(972\) 0 0
\(973\) 10.5342 0.337710
\(974\) 0 0
\(975\) 4.56866 0.146314
\(976\) 0 0
\(977\) −9.89539 −0.316582 −0.158291 0.987393i \(-0.550598\pi\)
−0.158291 + 0.987393i \(0.550598\pi\)
\(978\) 0 0
\(979\) 4.77691 0.152671
\(980\) 0 0
\(981\) 2.06578 0.0659554
\(982\) 0 0
\(983\) 1.81719 0.0579594 0.0289797 0.999580i \(-0.490774\pi\)
0.0289797 + 0.999580i \(0.490774\pi\)
\(984\) 0 0
\(985\) 48.3263 1.53980
\(986\) 0 0
\(987\) −0.429829 −0.0136816
\(988\) 0 0
\(989\) −7.22598 −0.229773
\(990\) 0 0
\(991\) 6.96926 0.221386 0.110693 0.993855i \(-0.464693\pi\)
0.110693 + 0.993855i \(0.464693\pi\)
\(992\) 0 0
\(993\) 13.7981 0.437871
\(994\) 0 0
\(995\) −33.9941 −1.07769
\(996\) 0 0
\(997\) 31.3816 0.993865 0.496932 0.867789i \(-0.334460\pi\)
0.496932 + 0.867789i \(0.334460\pi\)
\(998\) 0 0
\(999\) −49.4466 −1.56442
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 1148.2.a.e.1.4 5
4.3 odd 2 4592.2.a.bd.1.2 5
7.6 odd 2 8036.2.a.j.1.2 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1148.2.a.e.1.4 5 1.1 even 1 trivial
4592.2.a.bd.1.2 5 4.3 odd 2
8036.2.a.j.1.2 5 7.6 odd 2