Properties

Label 3822.2.a.bv.1.3
Level $3822$
Weight $2$
Character 3822.1
Self dual yes
Analytic conductor $30.519$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3822,2,Mod(1,3822)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3822, 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("3822.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3822 = 2 \cdot 3 \cdot 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3822.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: \(30.5188236525\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.2700.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 15x - 20 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 546)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-2.80560\) of defining polynomial
Character \(\chi\) \(=\) 3822.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +3.80560 q^{5} +1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +3.80560 q^{5} +1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} -3.80560 q^{10} +5.28822 q^{11} -1.00000 q^{12} +1.00000 q^{13} -3.80560 q^{15} +1.00000 q^{16} +6.15962 q^{17} -1.00000 q^{18} +5.48261 q^{19} +3.80560 q^{20} -5.28822 q^{22} +6.28822 q^{23} +1.00000 q^{24} +9.48261 q^{25} -1.00000 q^{26} -1.00000 q^{27} -3.00000 q^{29} +3.80560 q^{30} -8.48261 q^{31} -1.00000 q^{32} -5.28822 q^{33} -6.15962 q^{34} +1.00000 q^{36} +1.32299 q^{37} -5.48261 q^{38} -1.00000 q^{39} -3.80560 q^{40} +5.32299 q^{41} +1.61121 q^{43} +5.28822 q^{44} +3.80560 q^{45} -6.28822 q^{46} +7.09382 q^{47} -1.00000 q^{48} -9.48261 q^{50} -6.15962 q^{51} +1.00000 q^{52} -0.354020 q^{53} +1.00000 q^{54} +20.1248 q^{55} -5.48261 q^{57} +3.00000 q^{58} -1.67701 q^{59} -3.80560 q^{60} -9.77083 q^{61} +8.48261 q^{62} +1.00000 q^{64} +3.80560 q^{65} +5.28822 q^{66} -8.44784 q^{67} +6.15962 q^{68} -6.28822 q^{69} -15.4826 q^{71} -1.00000 q^{72} +1.71178 q^{73} -1.32299 q^{74} -9.48261 q^{75} +5.48261 q^{76} +1.00000 q^{78} +2.87141 q^{79} +3.80560 q^{80} +1.00000 q^{81} -5.32299 q^{82} -7.44784 q^{83} +23.4411 q^{85} -1.61121 q^{86} +3.00000 q^{87} -5.28822 q^{88} +10.6770 q^{89} -3.80560 q^{90} +6.28822 q^{92} +8.48261 q^{93} -7.09382 q^{94} +20.8646 q^{95} +1.00000 q^{96} -3.15962 q^{97} +5.28822 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{2} - 3 q^{3} + 3 q^{4} + 3 q^{5} + 3 q^{6} - 3 q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 3 q^{2} - 3 q^{3} + 3 q^{4} + 3 q^{5} + 3 q^{6} - 3 q^{8} + 3 q^{9} - 3 q^{10} - 3 q^{11} - 3 q^{12} + 3 q^{13} - 3 q^{15} + 3 q^{16} + 6 q^{17} - 3 q^{18} + 6 q^{19} + 3 q^{20} + 3 q^{22} + 3 q^{24} + 18 q^{25} - 3 q^{26} - 3 q^{27} - 9 q^{29} + 3 q^{30} - 15 q^{31} - 3 q^{32} + 3 q^{33} - 6 q^{34} + 3 q^{36} + 6 q^{37} - 6 q^{38} - 3 q^{39} - 3 q^{40} + 18 q^{41} - 12 q^{43} - 3 q^{44} + 3 q^{45} - 6 q^{47} - 3 q^{48} - 18 q^{50} - 6 q^{51} + 3 q^{52} + 3 q^{53} + 3 q^{54} + 27 q^{55} - 6 q^{57} + 9 q^{58} - 3 q^{59} - 3 q^{60} + 15 q^{62} + 3 q^{64} + 3 q^{65} - 3 q^{66} + 6 q^{67} + 6 q^{68} - 36 q^{71} - 3 q^{72} + 24 q^{73} - 6 q^{74} - 18 q^{75} + 6 q^{76} + 3 q^{78} + 15 q^{79} + 3 q^{80} + 3 q^{81} - 18 q^{82} + 9 q^{83} - 24 q^{85} + 12 q^{86} + 9 q^{87} + 3 q^{88} + 30 q^{89} - 3 q^{90} + 15 q^{93} + 6 q^{94} + 6 q^{95} + 3 q^{96} + 3 q^{97} - 3 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) −1.00000 −0.577350
\(4\) 1.00000 0.500000
\(5\) 3.80560 1.70192 0.850959 0.525233i \(-0.176022\pi\)
0.850959 + 0.525233i \(0.176022\pi\)
\(6\) 1.00000 0.408248
\(7\) 0 0
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) −3.80560 −1.20344
\(11\) 5.28822 1.59446 0.797229 0.603678i \(-0.206299\pi\)
0.797229 + 0.603678i \(0.206299\pi\)
\(12\) −1.00000 −0.288675
\(13\) 1.00000 0.277350
\(14\) 0 0
\(15\) −3.80560 −0.982602
\(16\) 1.00000 0.250000
\(17\) 6.15962 1.49393 0.746964 0.664864i \(-0.231511\pi\)
0.746964 + 0.664864i \(0.231511\pi\)
\(18\) −1.00000 −0.235702
\(19\) 5.48261 1.25780 0.628899 0.777487i \(-0.283506\pi\)
0.628899 + 0.777487i \(0.283506\pi\)
\(20\) 3.80560 0.850959
\(21\) 0 0
\(22\) −5.28822 −1.12745
\(23\) 6.28822 1.31118 0.655592 0.755115i \(-0.272419\pi\)
0.655592 + 0.755115i \(0.272419\pi\)
\(24\) 1.00000 0.204124
\(25\) 9.48261 1.89652
\(26\) −1.00000 −0.196116
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −3.00000 −0.557086 −0.278543 0.960424i \(-0.589851\pi\)
−0.278543 + 0.960424i \(0.589851\pi\)
\(30\) 3.80560 0.694805
\(31\) −8.48261 −1.52352 −0.761761 0.647858i \(-0.775665\pi\)
−0.761761 + 0.647858i \(0.775665\pi\)
\(32\) −1.00000 −0.176777
\(33\) −5.28822 −0.920560
\(34\) −6.15962 −1.05637
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 1.32299 0.217498 0.108749 0.994069i \(-0.465315\pi\)
0.108749 + 0.994069i \(0.465315\pi\)
\(38\) −5.48261 −0.889397
\(39\) −1.00000 −0.160128
\(40\) −3.80560 −0.601719
\(41\) 5.32299 0.831311 0.415656 0.909522i \(-0.363552\pi\)
0.415656 + 0.909522i \(0.363552\pi\)
\(42\) 0 0
\(43\) 1.61121 0.245707 0.122853 0.992425i \(-0.460796\pi\)
0.122853 + 0.992425i \(0.460796\pi\)
\(44\) 5.28822 0.797229
\(45\) 3.80560 0.567306
\(46\) −6.28822 −0.927147
\(47\) 7.09382 1.03474 0.517370 0.855762i \(-0.326911\pi\)
0.517370 + 0.855762i \(0.326911\pi\)
\(48\) −1.00000 −0.144338
\(49\) 0 0
\(50\) −9.48261 −1.34104
\(51\) −6.15962 −0.862520
\(52\) 1.00000 0.138675
\(53\) −0.354020 −0.0486284 −0.0243142 0.999704i \(-0.507740\pi\)
−0.0243142 + 0.999704i \(0.507740\pi\)
\(54\) 1.00000 0.136083
\(55\) 20.1248 2.71363
\(56\) 0 0
\(57\) −5.48261 −0.726190
\(58\) 3.00000 0.393919
\(59\) −1.67701 −0.218328 −0.109164 0.994024i \(-0.534817\pi\)
−0.109164 + 0.994024i \(0.534817\pi\)
\(60\) −3.80560 −0.491301
\(61\) −9.77083 −1.25103 −0.625513 0.780214i \(-0.715110\pi\)
−0.625513 + 0.780214i \(0.715110\pi\)
\(62\) 8.48261 1.07729
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 3.80560 0.472027
\(66\) 5.28822 0.650934
\(67\) −8.44784 −1.03207 −0.516034 0.856568i \(-0.672592\pi\)
−0.516034 + 0.856568i \(0.672592\pi\)
\(68\) 6.15962 0.746964
\(69\) −6.28822 −0.757012
\(70\) 0 0
\(71\) −15.4826 −1.83745 −0.918724 0.394900i \(-0.870779\pi\)
−0.918724 + 0.394900i \(0.870779\pi\)
\(72\) −1.00000 −0.117851
\(73\) 1.71178 0.200349 0.100175 0.994970i \(-0.468060\pi\)
0.100175 + 0.994970i \(0.468060\pi\)
\(74\) −1.32299 −0.153794
\(75\) −9.48261 −1.09496
\(76\) 5.48261 0.628899
\(77\) 0 0
\(78\) 1.00000 0.113228
\(79\) 2.87141 0.323059 0.161529 0.986868i \(-0.448357\pi\)
0.161529 + 0.986868i \(0.448357\pi\)
\(80\) 3.80560 0.425479
\(81\) 1.00000 0.111111
\(82\) −5.32299 −0.587826
\(83\) −7.44784 −0.817507 −0.408753 0.912645i \(-0.634036\pi\)
−0.408753 + 0.912645i \(0.634036\pi\)
\(84\) 0 0
\(85\) 23.4411 2.54254
\(86\) −1.61121 −0.173741
\(87\) 3.00000 0.321634
\(88\) −5.28822 −0.563726
\(89\) 10.6770 1.13176 0.565880 0.824487i \(-0.308536\pi\)
0.565880 + 0.824487i \(0.308536\pi\)
\(90\) −3.80560 −0.401146
\(91\) 0 0
\(92\) 6.28822 0.655592
\(93\) 8.48261 0.879606
\(94\) −7.09382 −0.731672
\(95\) 20.8646 2.14067
\(96\) 1.00000 0.102062
\(97\) −3.15962 −0.320811 −0.160406 0.987051i \(-0.551280\pi\)
−0.160406 + 0.987051i \(0.551280\pi\)
\(98\) 0 0
\(99\) 5.28822 0.531486
\(100\) 9.48261 0.948261
\(101\) −4.96523 −0.494058 −0.247029 0.969008i \(-0.579454\pi\)
−0.247029 + 0.969008i \(0.579454\pi\)
\(102\) 6.15962 0.609894
\(103\) −5.61121 −0.552889 −0.276444 0.961030i \(-0.589156\pi\)
−0.276444 + 0.961030i \(0.589156\pi\)
\(104\) −1.00000 −0.0980581
\(105\) 0 0
\(106\) 0.354020 0.0343855
\(107\) −17.0590 −1.64916 −0.824580 0.565745i \(-0.808589\pi\)
−0.824580 + 0.565745i \(0.808589\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 4.00000 0.383131 0.191565 0.981480i \(-0.438644\pi\)
0.191565 + 0.981480i \(0.438644\pi\)
\(110\) −20.1248 −1.91883
\(111\) −1.32299 −0.125573
\(112\) 0 0
\(113\) −5.45158 −0.512842 −0.256421 0.966565i \(-0.582543\pi\)
−0.256421 + 0.966565i \(0.582543\pi\)
\(114\) 5.48261 0.513494
\(115\) 23.9305 2.23153
\(116\) −3.00000 −0.278543
\(117\) 1.00000 0.0924500
\(118\) 1.67701 0.154381
\(119\) 0 0
\(120\) 3.80560 0.347402
\(121\) 16.9652 1.54229
\(122\) 9.77083 0.884609
\(123\) −5.32299 −0.479958
\(124\) −8.48261 −0.761761
\(125\) 17.0590 1.52581
\(126\) 0 0
\(127\) −10.1248 −0.898435 −0.449218 0.893422i \(-0.648297\pi\)
−0.449218 + 0.893422i \(0.648297\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −1.61121 −0.141859
\(130\) −3.80560 −0.333773
\(131\) 10.0628 0.879190 0.439595 0.898196i \(-0.355122\pi\)
0.439595 + 0.898196i \(0.355122\pi\)
\(132\) −5.28822 −0.460280
\(133\) 0 0
\(134\) 8.44784 0.729782
\(135\) −3.80560 −0.327534
\(136\) −6.15962 −0.528183
\(137\) −6.67701 −0.570455 −0.285228 0.958460i \(-0.592069\pi\)
−0.285228 + 0.958460i \(0.592069\pi\)
\(138\) 6.28822 0.535288
\(139\) −12.2882 −1.04227 −0.521136 0.853473i \(-0.674492\pi\)
−0.521136 + 0.853473i \(0.674492\pi\)
\(140\) 0 0
\(141\) −7.09382 −0.597407
\(142\) 15.4826 1.29927
\(143\) 5.28822 0.442223
\(144\) 1.00000 0.0833333
\(145\) −11.4168 −0.948114
\(146\) −1.71178 −0.141668
\(147\) 0 0
\(148\) 1.32299 0.108749
\(149\) −18.2882 −1.49823 −0.749115 0.662441i \(-0.769521\pi\)
−0.749115 + 0.662441i \(0.769521\pi\)
\(150\) 9.48261 0.774252
\(151\) −15.2882 −1.24414 −0.622069 0.782963i \(-0.713708\pi\)
−0.622069 + 0.782963i \(0.713708\pi\)
\(152\) −5.48261 −0.444699
\(153\) 6.15962 0.497976
\(154\) 0 0
\(155\) −32.2815 −2.59291
\(156\) −1.00000 −0.0800641
\(157\) 10.4168 0.831352 0.415676 0.909513i \(-0.363545\pi\)
0.415676 + 0.909513i \(0.363545\pi\)
\(158\) −2.87141 −0.228437
\(159\) 0.354020 0.0280756
\(160\) −3.80560 −0.300859
\(161\) 0 0
\(162\) −1.00000 −0.0785674
\(163\) −9.73980 −0.762880 −0.381440 0.924394i \(-0.624572\pi\)
−0.381440 + 0.924394i \(0.624572\pi\)
\(164\) 5.32299 0.415656
\(165\) −20.1248 −1.56672
\(166\) 7.44784 0.578064
\(167\) −8.70502 −0.673615 −0.336808 0.941574i \(-0.609347\pi\)
−0.336808 + 0.941574i \(0.609347\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) −23.4411 −1.79785
\(171\) 5.48261 0.419266
\(172\) 1.61121 0.122853
\(173\) −12.0590 −0.916832 −0.458416 0.888738i \(-0.651583\pi\)
−0.458416 + 0.888738i \(0.651583\pi\)
\(174\) −3.00000 −0.227429
\(175\) 0 0
\(176\) 5.28822 0.398614
\(177\) 1.67701 0.126052
\(178\) −10.6770 −0.800276
\(179\) −4.25719 −0.318197 −0.159098 0.987263i \(-0.550859\pi\)
−0.159098 + 0.987263i \(0.550859\pi\)
\(180\) 3.80560 0.283653
\(181\) 12.1596 0.903818 0.451909 0.892064i \(-0.350743\pi\)
0.451909 + 0.892064i \(0.350743\pi\)
\(182\) 0 0
\(183\) 9.77083 0.722280
\(184\) −6.28822 −0.463573
\(185\) 5.03477 0.370164
\(186\) −8.48261 −0.621975
\(187\) 32.5734 2.38200
\(188\) 7.09382 0.517370
\(189\) 0 0
\(190\) −20.8646 −1.51368
\(191\) −24.1876 −1.75016 −0.875078 0.483982i \(-0.839190\pi\)
−0.875078 + 0.483982i \(0.839190\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 1.54842 0.111458 0.0557288 0.998446i \(-0.482252\pi\)
0.0557288 + 0.998446i \(0.482252\pi\)
\(194\) 3.15962 0.226848
\(195\) −3.80560 −0.272525
\(196\) 0 0
\(197\) −18.2882 −1.30298 −0.651491 0.758657i \(-0.725856\pi\)
−0.651491 + 0.758657i \(0.725856\pi\)
\(198\) −5.28822 −0.375817
\(199\) −26.2187 −1.85859 −0.929296 0.369336i \(-0.879585\pi\)
−0.929296 + 0.369336i \(0.879585\pi\)
\(200\) −9.48261 −0.670522
\(201\) 8.44784 0.595865
\(202\) 4.96523 0.349352
\(203\) 0 0
\(204\) −6.15962 −0.431260
\(205\) 20.2572 1.41482
\(206\) 5.61121 0.390951
\(207\) 6.28822 0.437061
\(208\) 1.00000 0.0693375
\(209\) 28.9932 2.00550
\(210\) 0 0
\(211\) −9.89942 −0.681504 −0.340752 0.940153i \(-0.610682\pi\)
−0.340752 + 0.940153i \(0.610682\pi\)
\(212\) −0.354020 −0.0243142
\(213\) 15.4826 1.06085
\(214\) 17.0590 1.16613
\(215\) 6.13161 0.418172
\(216\) 1.00000 0.0680414
\(217\) 0 0
\(218\) −4.00000 −0.270914
\(219\) −1.71178 −0.115672
\(220\) 20.1248 1.35682
\(221\) 6.15962 0.414341
\(222\) 1.32299 0.0887933
\(223\) 2.96897 0.198817 0.0994085 0.995047i \(-0.468305\pi\)
0.0994085 + 0.995047i \(0.468305\pi\)
\(224\) 0 0
\(225\) 9.48261 0.632174
\(226\) 5.45158 0.362634
\(227\) 18.4131 1.22212 0.611059 0.791585i \(-0.290744\pi\)
0.611059 + 0.791585i \(0.290744\pi\)
\(228\) −5.48261 −0.363095
\(229\) −1.42357 −0.0940721 −0.0470360 0.998893i \(-0.514978\pi\)
−0.0470360 + 0.998893i \(0.514978\pi\)
\(230\) −23.9305 −1.57793
\(231\) 0 0
\(232\) 3.00000 0.196960
\(233\) 10.7361 0.703342 0.351671 0.936124i \(-0.385614\pi\)
0.351671 + 0.936124i \(0.385614\pi\)
\(234\) −1.00000 −0.0653720
\(235\) 26.9963 1.76104
\(236\) −1.67701 −0.109164
\(237\) −2.87141 −0.186518
\(238\) 0 0
\(239\) 2.12859 0.137687 0.0688436 0.997627i \(-0.478069\pi\)
0.0688436 + 0.997627i \(0.478069\pi\)
\(240\) −3.80560 −0.245651
\(241\) 18.7398 1.20714 0.603568 0.797311i \(-0.293745\pi\)
0.603568 + 0.797311i \(0.293745\pi\)
\(242\) −16.9652 −1.09057
\(243\) −1.00000 −0.0641500
\(244\) −9.77083 −0.625513
\(245\) 0 0
\(246\) 5.32299 0.339381
\(247\) 5.48261 0.348850
\(248\) 8.48261 0.538646
\(249\) 7.44784 0.471988
\(250\) −17.0590 −1.07891
\(251\) −7.77457 −0.490727 −0.245363 0.969431i \(-0.578907\pi\)
−0.245363 + 0.969431i \(0.578907\pi\)
\(252\) 0 0
\(253\) 33.2534 2.09063
\(254\) 10.1248 0.635290
\(255\) −23.4411 −1.46794
\(256\) 1.00000 0.0625000
\(257\) 6.57643 0.410227 0.205113 0.978738i \(-0.434244\pi\)
0.205113 + 0.978738i \(0.434244\pi\)
\(258\) 1.61121 0.100309
\(259\) 0 0
\(260\) 3.80560 0.236013
\(261\) −3.00000 −0.185695
\(262\) −10.0628 −0.621681
\(263\) −20.4759 −1.26260 −0.631298 0.775541i \(-0.717477\pi\)
−0.631298 + 0.775541i \(0.717477\pi\)
\(264\) 5.28822 0.325467
\(265\) −1.34726 −0.0827615
\(266\) 0 0
\(267\) −10.6770 −0.653422
\(268\) −8.44784 −0.516034
\(269\) −24.1529 −1.47263 −0.736313 0.676641i \(-0.763435\pi\)
−0.736313 + 0.676641i \(0.763435\pi\)
\(270\) 3.80560 0.231602
\(271\) 30.4411 1.84916 0.924582 0.380983i \(-0.124414\pi\)
0.924582 + 0.380983i \(0.124414\pi\)
\(272\) 6.15962 0.373482
\(273\) 0 0
\(274\) 6.67701 0.403373
\(275\) 50.1461 3.02392
\(276\) −6.28822 −0.378506
\(277\) −9.12485 −0.548259 −0.274130 0.961693i \(-0.588390\pi\)
−0.274130 + 0.961693i \(0.588390\pi\)
\(278\) 12.2882 0.736998
\(279\) −8.48261 −0.507841
\(280\) 0 0
\(281\) −6.70804 −0.400168 −0.200084 0.979779i \(-0.564122\pi\)
−0.200084 + 0.979779i \(0.564122\pi\)
\(282\) 7.09382 0.422431
\(283\) 26.8646 1.59694 0.798469 0.602036i \(-0.205644\pi\)
0.798469 + 0.602036i \(0.205644\pi\)
\(284\) −15.4826 −0.918724
\(285\) −20.8646 −1.23592
\(286\) −5.28822 −0.312699
\(287\) 0 0
\(288\) −1.00000 −0.0589256
\(289\) 20.9410 1.23182
\(290\) 11.4168 0.670418
\(291\) 3.15962 0.185220
\(292\) 1.71178 0.100175
\(293\) 9.80560 0.572849 0.286425 0.958103i \(-0.407533\pi\)
0.286425 + 0.958103i \(0.407533\pi\)
\(294\) 0 0
\(295\) −6.38203 −0.371576
\(296\) −1.32299 −0.0768972
\(297\) −5.28822 −0.306853
\(298\) 18.2882 1.05941
\(299\) 6.28822 0.363657
\(300\) −9.48261 −0.547479
\(301\) 0 0
\(302\) 15.2882 0.879738
\(303\) 4.96523 0.285245
\(304\) 5.48261 0.314449
\(305\) −37.1839 −2.12914
\(306\) −6.15962 −0.352122
\(307\) 20.5174 1.17099 0.585495 0.810676i \(-0.300900\pi\)
0.585495 + 0.810676i \(0.300900\pi\)
\(308\) 0 0
\(309\) 5.61121 0.319210
\(310\) 32.2815 1.83346
\(311\) −22.5454 −1.27843 −0.639216 0.769027i \(-0.720741\pi\)
−0.639216 + 0.769027i \(0.720741\pi\)
\(312\) 1.00000 0.0566139
\(313\) 32.0938 1.81405 0.907025 0.421077i \(-0.138348\pi\)
0.907025 + 0.421077i \(0.138348\pi\)
\(314\) −10.4168 −0.587855
\(315\) 0 0
\(316\) 2.87141 0.161529
\(317\) 31.4478 1.76629 0.883143 0.469103i \(-0.155423\pi\)
0.883143 + 0.469103i \(0.155423\pi\)
\(318\) −0.354020 −0.0198525
\(319\) −15.8646 −0.888250
\(320\) 3.80560 0.212740
\(321\) 17.0590 0.952143
\(322\) 0 0
\(323\) 33.7708 1.87906
\(324\) 1.00000 0.0555556
\(325\) 9.48261 0.526001
\(326\) 9.73980 0.539438
\(327\) −4.00000 −0.221201
\(328\) −5.32299 −0.293913
\(329\) 0 0
\(330\) 20.1248 1.10784
\(331\) 16.8957 0.928670 0.464335 0.885660i \(-0.346293\pi\)
0.464335 + 0.885660i \(0.346293\pi\)
\(332\) −7.44784 −0.408753
\(333\) 1.32299 0.0724994
\(334\) 8.70502 0.476318
\(335\) −32.1491 −1.75649
\(336\) 0 0
\(337\) −7.25719 −0.395324 −0.197662 0.980270i \(-0.563335\pi\)
−0.197662 + 0.980270i \(0.563335\pi\)
\(338\) −1.00000 −0.0543928
\(339\) 5.45158 0.296089
\(340\) 23.4411 1.27127
\(341\) −44.8579 −2.42919
\(342\) −5.48261 −0.296466
\(343\) 0 0
\(344\) −1.61121 −0.0868704
\(345\) −23.9305 −1.28837
\(346\) 12.0590 0.648298
\(347\) 8.03103 0.431128 0.215564 0.976490i \(-0.430841\pi\)
0.215564 + 0.976490i \(0.430841\pi\)
\(348\) 3.00000 0.160817
\(349\) −9.25344 −0.495325 −0.247663 0.968846i \(-0.579662\pi\)
−0.247663 + 0.968846i \(0.579662\pi\)
\(350\) 0 0
\(351\) −1.00000 −0.0533761
\(352\) −5.28822 −0.281863
\(353\) 12.7080 0.676381 0.338190 0.941078i \(-0.390185\pi\)
0.338190 + 0.941078i \(0.390185\pi\)
\(354\) −1.67701 −0.0891321
\(355\) −58.9207 −3.12718
\(356\) 10.6770 0.565880
\(357\) 0 0
\(358\) 4.25719 0.224999
\(359\) −18.0000 −0.950004 −0.475002 0.879985i \(-0.657553\pi\)
−0.475002 + 0.879985i \(0.657553\pi\)
\(360\) −3.80560 −0.200573
\(361\) 11.0590 0.582055
\(362\) −12.1596 −0.639096
\(363\) −16.9652 −0.890443
\(364\) 0 0
\(365\) 6.51437 0.340978
\(366\) −9.77083 −0.510729
\(367\) 27.0590 1.41247 0.706235 0.707977i \(-0.250392\pi\)
0.706235 + 0.707977i \(0.250392\pi\)
\(368\) 6.28822 0.327796
\(369\) 5.32299 0.277104
\(370\) −5.03477 −0.261745
\(371\) 0 0
\(372\) 8.48261 0.439803
\(373\) 3.70877 0.192033 0.0960164 0.995380i \(-0.469390\pi\)
0.0960164 + 0.995380i \(0.469390\pi\)
\(374\) −32.5734 −1.68433
\(375\) −17.0590 −0.880925
\(376\) −7.09382 −0.365836
\(377\) −3.00000 −0.154508
\(378\) 0 0
\(379\) −19.2224 −0.987389 −0.493694 0.869635i \(-0.664354\pi\)
−0.493694 + 0.869635i \(0.664354\pi\)
\(380\) 20.8646 1.07033
\(381\) 10.1248 0.518712
\(382\) 24.1876 1.23755
\(383\) 25.2224 1.28881 0.644403 0.764686i \(-0.277106\pi\)
0.644403 + 0.764686i \(0.277106\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) −1.54842 −0.0788124
\(387\) 1.61121 0.0819022
\(388\) −3.15962 −0.160406
\(389\) 18.1286 0.919156 0.459578 0.888137i \(-0.348001\pi\)
0.459578 + 0.888137i \(0.348001\pi\)
\(390\) 3.80560 0.192704
\(391\) 38.7330 1.95881
\(392\) 0 0
\(393\) −10.0628 −0.507601
\(394\) 18.2882 0.921347
\(395\) 10.9274 0.549819
\(396\) 5.28822 0.265743
\(397\) −14.6075 −0.733127 −0.366564 0.930393i \(-0.619466\pi\)
−0.366564 + 0.930393i \(0.619466\pi\)
\(398\) 26.2187 1.31422
\(399\) 0 0
\(400\) 9.48261 0.474131
\(401\) −21.2534 −1.06135 −0.530673 0.847577i \(-0.678061\pi\)
−0.530673 + 0.847577i \(0.678061\pi\)
\(402\) −8.44784 −0.421340
\(403\) −8.48261 −0.422549
\(404\) −4.96523 −0.247029
\(405\) 3.80560 0.189102
\(406\) 0 0
\(407\) 6.99626 0.346792
\(408\) 6.15962 0.304947
\(409\) −15.9932 −0.790815 −0.395407 0.918506i \(-0.629397\pi\)
−0.395407 + 0.918506i \(0.629397\pi\)
\(410\) −20.2572 −1.00043
\(411\) 6.67701 0.329353
\(412\) −5.61121 −0.276444
\(413\) 0 0
\(414\) −6.28822 −0.309049
\(415\) −28.3435 −1.39133
\(416\) −1.00000 −0.0490290
\(417\) 12.2882 0.601757
\(418\) −28.9932 −1.41811
\(419\) −5.61121 −0.274125 −0.137063 0.990562i \(-0.543766\pi\)
−0.137063 + 0.990562i \(0.543766\pi\)
\(420\) 0 0
\(421\) 17.5802 0.856805 0.428403 0.903588i \(-0.359076\pi\)
0.428403 + 0.903588i \(0.359076\pi\)
\(422\) 9.89942 0.481896
\(423\) 7.09382 0.344913
\(424\) 0.354020 0.0171927
\(425\) 58.4093 2.83327
\(426\) −15.4826 −0.750135
\(427\) 0 0
\(428\) −17.0590 −0.824580
\(429\) −5.28822 −0.255317
\(430\) −6.13161 −0.295692
\(431\) 17.2224 0.829574 0.414787 0.909918i \(-0.363856\pi\)
0.414787 + 0.909918i \(0.363856\pi\)
\(432\) −1.00000 −0.0481125
\(433\) 35.0243 1.68316 0.841580 0.540133i \(-0.181626\pi\)
0.841580 + 0.540133i \(0.181626\pi\)
\(434\) 0 0
\(435\) 11.4168 0.547394
\(436\) 4.00000 0.191565
\(437\) 34.4759 1.64920
\(438\) 1.71178 0.0817922
\(439\) −14.5136 −0.692698 −0.346349 0.938106i \(-0.612579\pi\)
−0.346349 + 0.938106i \(0.612579\pi\)
\(440\) −20.1248 −0.959415
\(441\) 0 0
\(442\) −6.15962 −0.292983
\(443\) 10.3820 0.493265 0.246633 0.969109i \(-0.420676\pi\)
0.246633 + 0.969109i \(0.420676\pi\)
\(444\) −1.32299 −0.0627863
\(445\) 40.6325 1.92616
\(446\) −2.96897 −0.140585
\(447\) 18.2882 0.865003
\(448\) 0 0
\(449\) 31.0833 1.46691 0.733456 0.679737i \(-0.237906\pi\)
0.733456 + 0.679737i \(0.237906\pi\)
\(450\) −9.48261 −0.447015
\(451\) 28.1491 1.32549
\(452\) −5.45158 −0.256421
\(453\) 15.2882 0.718303
\(454\) −18.4131 −0.864168
\(455\) 0 0
\(456\) 5.48261 0.256747
\(457\) 5.41681 0.253388 0.126694 0.991942i \(-0.459563\pi\)
0.126694 + 0.991942i \(0.459563\pi\)
\(458\) 1.42357 0.0665190
\(459\) −6.15962 −0.287507
\(460\) 23.9305 1.11576
\(461\) 9.54166 0.444399 0.222200 0.975001i \(-0.428676\pi\)
0.222200 + 0.975001i \(0.428676\pi\)
\(462\) 0 0
\(463\) −27.2224 −1.26513 −0.632566 0.774506i \(-0.717998\pi\)
−0.632566 + 0.774506i \(0.717998\pi\)
\(464\) −3.00000 −0.139272
\(465\) 32.2815 1.49702
\(466\) −10.7361 −0.497338
\(467\) 13.0658 0.604613 0.302307 0.953211i \(-0.402243\pi\)
0.302307 + 0.953211i \(0.402243\pi\)
\(468\) 1.00000 0.0462250
\(469\) 0 0
\(470\) −26.9963 −1.24524
\(471\) −10.4168 −0.479981
\(472\) 1.67701 0.0771906
\(473\) 8.52040 0.391769
\(474\) 2.87141 0.131888
\(475\) 51.9895 2.38544
\(476\) 0 0
\(477\) −0.354020 −0.0162095
\(478\) −2.12859 −0.0973596
\(479\) −25.2119 −1.15196 −0.575981 0.817463i \(-0.695380\pi\)
−0.575981 + 0.817463i \(0.695380\pi\)
\(480\) 3.80560 0.173701
\(481\) 1.32299 0.0603231
\(482\) −18.7398 −0.853574
\(483\) 0 0
\(484\) 16.9652 0.771147
\(485\) −12.0243 −0.545994
\(486\) 1.00000 0.0453609
\(487\) 13.9342 0.631419 0.315709 0.948856i \(-0.397758\pi\)
0.315709 + 0.948856i \(0.397758\pi\)
\(488\) 9.77083 0.442305
\(489\) 9.73980 0.440449
\(490\) 0 0
\(491\) −12.3510 −0.557393 −0.278697 0.960379i \(-0.589902\pi\)
−0.278697 + 0.960379i \(0.589902\pi\)
\(492\) −5.32299 −0.239979
\(493\) −18.4789 −0.832246
\(494\) −5.48261 −0.246674
\(495\) 20.1248 0.904545
\(496\) −8.48261 −0.380881
\(497\) 0 0
\(498\) −7.44784 −0.333746
\(499\) −34.7641 −1.55625 −0.778127 0.628107i \(-0.783830\pi\)
−0.778127 + 0.628107i \(0.783830\pi\)
\(500\) 17.0590 0.762904
\(501\) 8.70502 0.388912
\(502\) 7.77457 0.346996
\(503\) −5.68075 −0.253292 −0.126646 0.991948i \(-0.540421\pi\)
−0.126646 + 0.991948i \(0.540421\pi\)
\(504\) 0 0
\(505\) −18.8957 −0.840847
\(506\) −33.2534 −1.47830
\(507\) −1.00000 −0.0444116
\(508\) −10.1248 −0.449218
\(509\) −9.38578 −0.416017 −0.208009 0.978127i \(-0.566698\pi\)
−0.208009 + 0.978127i \(0.566698\pi\)
\(510\) 23.4411 1.03799
\(511\) 0 0
\(512\) −1.00000 −0.0441942
\(513\) −5.48261 −0.242063
\(514\) −6.57643 −0.290074
\(515\) −21.3540 −0.940971
\(516\) −1.61121 −0.0709294
\(517\) 37.5136 1.64985
\(518\) 0 0
\(519\) 12.0590 0.529333
\(520\) −3.80560 −0.166887
\(521\) −42.0485 −1.84218 −0.921090 0.389350i \(-0.872700\pi\)
−0.921090 + 0.389350i \(0.872700\pi\)
\(522\) 3.00000 0.131306
\(523\) 20.8336 0.910990 0.455495 0.890238i \(-0.349462\pi\)
0.455495 + 0.890238i \(0.349462\pi\)
\(524\) 10.0628 0.439595
\(525\) 0 0
\(526\) 20.4759 0.892790
\(527\) −52.2497 −2.27603
\(528\) −5.28822 −0.230140
\(529\) 16.5417 0.719202
\(530\) 1.34726 0.0585212
\(531\) −1.67701 −0.0727760
\(532\) 0 0
\(533\) 5.32299 0.230564
\(534\) 10.6770 0.462039
\(535\) −64.9199 −2.80673
\(536\) 8.44784 0.364891
\(537\) 4.25719 0.183711
\(538\) 24.1529 1.04130
\(539\) 0 0
\(540\) −3.80560 −0.163767
\(541\) 16.0310 0.689228 0.344614 0.938745i \(-0.388010\pi\)
0.344614 + 0.938745i \(0.388010\pi\)
\(542\) −30.4411 −1.30756
\(543\) −12.1596 −0.521819
\(544\) −6.15962 −0.264092
\(545\) 15.2224 0.652056
\(546\) 0 0
\(547\) 23.2534 0.994245 0.497123 0.867680i \(-0.334390\pi\)
0.497123 + 0.867680i \(0.334390\pi\)
\(548\) −6.67701 −0.285228
\(549\) −9.77083 −0.417009
\(550\) −50.1461 −2.13824
\(551\) −16.4478 −0.700701
\(552\) 6.28822 0.267644
\(553\) 0 0
\(554\) 9.12485 0.387678
\(555\) −5.03477 −0.213714
\(556\) −12.2882 −0.521136
\(557\) 35.0590 1.48550 0.742750 0.669569i \(-0.233521\pi\)
0.742750 + 0.669569i \(0.233521\pi\)
\(558\) 8.48261 0.359098
\(559\) 1.61121 0.0681467
\(560\) 0 0
\(561\) −32.5734 −1.37525
\(562\) 6.70804 0.282962
\(563\) −27.5174 −1.15972 −0.579860 0.814716i \(-0.696893\pi\)
−0.579860 + 0.814716i \(0.696893\pi\)
\(564\) −7.09382 −0.298704
\(565\) −20.7466 −0.872814
\(566\) −26.8646 −1.12921
\(567\) 0 0
\(568\) 15.4826 0.649636
\(569\) −18.0280 −0.755774 −0.377887 0.925852i \(-0.623349\pi\)
−0.377887 + 0.925852i \(0.623349\pi\)
\(570\) 20.8646 0.873924
\(571\) 5.29196 0.221462 0.110731 0.993850i \(-0.464681\pi\)
0.110731 + 0.993850i \(0.464681\pi\)
\(572\) 5.28822 0.221111
\(573\) 24.1876 1.01045
\(574\) 0 0
\(575\) 59.6287 2.48669
\(576\) 1.00000 0.0416667
\(577\) −8.16337 −0.339845 −0.169923 0.985457i \(-0.554352\pi\)
−0.169923 + 0.985457i \(0.554352\pi\)
\(578\) −20.9410 −0.871029
\(579\) −1.54842 −0.0643500
\(580\) −11.4168 −0.474057
\(581\) 0 0
\(582\) −3.15962 −0.130971
\(583\) −1.87214 −0.0775359
\(584\) −1.71178 −0.0708341
\(585\) 3.80560 0.157342
\(586\) −9.80560 −0.405066
\(587\) 14.4546 0.596605 0.298303 0.954471i \(-0.403580\pi\)
0.298303 + 0.954471i \(0.403580\pi\)
\(588\) 0 0
\(589\) −46.5069 −1.91628
\(590\) 6.38203 0.262744
\(591\) 18.2882 0.752277
\(592\) 1.32299 0.0543746
\(593\) 3.96897 0.162986 0.0814930 0.996674i \(-0.474031\pi\)
0.0814930 + 0.996674i \(0.474031\pi\)
\(594\) 5.28822 0.216978
\(595\) 0 0
\(596\) −18.2882 −0.749115
\(597\) 26.2187 1.07306
\(598\) −6.28822 −0.257144
\(599\) −21.2534 −0.868392 −0.434196 0.900818i \(-0.642967\pi\)
−0.434196 + 0.900818i \(0.642967\pi\)
\(600\) 9.48261 0.387126
\(601\) −27.2497 −1.11154 −0.555769 0.831337i \(-0.687576\pi\)
−0.555769 + 0.831337i \(0.687576\pi\)
\(602\) 0 0
\(603\) −8.44784 −0.344023
\(604\) −15.2882 −0.622069
\(605\) 64.5629 2.62486
\(606\) −4.96523 −0.201699
\(607\) −9.92369 −0.402790 −0.201395 0.979510i \(-0.564548\pi\)
−0.201395 + 0.979510i \(0.564548\pi\)
\(608\) −5.48261 −0.222349
\(609\) 0 0
\(610\) 37.1839 1.50553
\(611\) 7.09382 0.286985
\(612\) 6.15962 0.248988
\(613\) −9.86839 −0.398581 −0.199290 0.979940i \(-0.563864\pi\)
−0.199290 + 0.979940i \(0.563864\pi\)
\(614\) −20.5174 −0.828014
\(615\) −20.2572 −0.816849
\(616\) 0 0
\(617\) −41.9865 −1.69031 −0.845156 0.534520i \(-0.820493\pi\)
−0.845156 + 0.534520i \(0.820493\pi\)
\(618\) −5.61121 −0.225716
\(619\) −25.0348 −1.00623 −0.503116 0.864219i \(-0.667813\pi\)
−0.503116 + 0.864219i \(0.667813\pi\)
\(620\) −32.2815 −1.29645
\(621\) −6.28822 −0.252337
\(622\) 22.5454 0.903988
\(623\) 0 0
\(624\) −1.00000 −0.0400320
\(625\) 17.5069 0.700275
\(626\) −32.0938 −1.28273
\(627\) −28.9932 −1.15788
\(628\) 10.4168 0.415676
\(629\) 8.14912 0.324927
\(630\) 0 0
\(631\) −4.55216 −0.181219 −0.0906093 0.995887i \(-0.528881\pi\)
−0.0906093 + 0.995887i \(0.528881\pi\)
\(632\) −2.87141 −0.114218
\(633\) 9.89942 0.393467
\(634\) −31.4478 −1.24895
\(635\) −38.5312 −1.52906
\(636\) 0.354020 0.0140378
\(637\) 0 0
\(638\) 15.8646 0.628087
\(639\) −15.4826 −0.612483
\(640\) −3.80560 −0.150430
\(641\) 33.1529 1.30946 0.654730 0.755863i \(-0.272782\pi\)
0.654730 + 0.755863i \(0.272782\pi\)
\(642\) −17.0590 −0.673267
\(643\) −42.0590 −1.65865 −0.829323 0.558769i \(-0.811274\pi\)
−0.829323 + 0.558769i \(0.811274\pi\)
\(644\) 0 0
\(645\) −6.13161 −0.241432
\(646\) −33.7708 −1.32870
\(647\) −16.2572 −0.639136 −0.319568 0.947563i \(-0.603538\pi\)
−0.319568 + 0.947563i \(0.603538\pi\)
\(648\) −1.00000 −0.0392837
\(649\) −8.86839 −0.348115
\(650\) −9.48261 −0.371939
\(651\) 0 0
\(652\) −9.73980 −0.381440
\(653\) 30.9895 1.21271 0.606356 0.795193i \(-0.292631\pi\)
0.606356 + 0.795193i \(0.292631\pi\)
\(654\) 4.00000 0.156412
\(655\) 38.2950 1.49631
\(656\) 5.32299 0.207828
\(657\) 1.71178 0.0667831
\(658\) 0 0
\(659\) 3.25344 0.126736 0.0633680 0.997990i \(-0.479816\pi\)
0.0633680 + 0.997990i \(0.479816\pi\)
\(660\) −20.1248 −0.783359
\(661\) −4.87214 −0.189504 −0.0947520 0.995501i \(-0.530206\pi\)
−0.0947520 + 0.995501i \(0.530206\pi\)
\(662\) −16.8957 −0.656669
\(663\) −6.15962 −0.239220
\(664\) 7.44784 0.289032
\(665\) 0 0
\(666\) −1.32299 −0.0512648
\(667\) −18.8646 −0.730442
\(668\) −8.70502 −0.336808
\(669\) −2.96897 −0.114787
\(670\) 32.1491 1.24203
\(671\) −51.6703 −1.99471
\(672\) 0 0
\(673\) 16.8714 0.650345 0.325172 0.945655i \(-0.394578\pi\)
0.325172 + 0.945655i \(0.394578\pi\)
\(674\) 7.25719 0.279536
\(675\) −9.48261 −0.364986
\(676\) 1.00000 0.0384615
\(677\) 30.7988 1.18370 0.591848 0.806050i \(-0.298399\pi\)
0.591848 + 0.806050i \(0.298399\pi\)
\(678\) −5.45158 −0.209367
\(679\) 0 0
\(680\) −23.4411 −0.898924
\(681\) −18.4131 −0.705590
\(682\) 44.8579 1.71770
\(683\) 36.3435 1.39065 0.695323 0.718697i \(-0.255261\pi\)
0.695323 + 0.718697i \(0.255261\pi\)
\(684\) 5.48261 0.209633
\(685\) −25.4100 −0.970868
\(686\) 0 0
\(687\) 1.42357 0.0543125
\(688\) 1.61121 0.0614266
\(689\) −0.354020 −0.0134871
\(690\) 23.9305 0.911017
\(691\) 38.9547 1.48191 0.740954 0.671556i \(-0.234374\pi\)
0.740954 + 0.671556i \(0.234374\pi\)
\(692\) −12.0590 −0.458416
\(693\) 0 0
\(694\) −8.03103 −0.304854
\(695\) −46.7641 −1.77386
\(696\) −3.00000 −0.113715
\(697\) 32.7876 1.24192
\(698\) 9.25344 0.350248
\(699\) −10.7361 −0.406075
\(700\) 0 0
\(701\) 40.7323 1.53844 0.769219 0.638985i \(-0.220645\pi\)
0.769219 + 0.638985i \(0.220645\pi\)
\(702\) 1.00000 0.0377426
\(703\) 7.25344 0.273569
\(704\) 5.28822 0.199307
\(705\) −26.9963 −1.01674
\(706\) −12.7080 −0.478273
\(707\) 0 0
\(708\) 1.67701 0.0630259
\(709\) −23.1218 −0.868359 −0.434179 0.900826i \(-0.642962\pi\)
−0.434179 + 0.900826i \(0.642962\pi\)
\(710\) 58.9207 2.21125
\(711\) 2.87141 0.107686
\(712\) −10.6770 −0.400138
\(713\) −53.3405 −1.99762
\(714\) 0 0
\(715\) 20.1248 0.752627
\(716\) −4.25719 −0.159098
\(717\) −2.12859 −0.0794938
\(718\) 18.0000 0.671754
\(719\) −26.3753 −0.983632 −0.491816 0.870699i \(-0.663667\pi\)
−0.491816 + 0.870699i \(0.663667\pi\)
\(720\) 3.80560 0.141826
\(721\) 0 0
\(722\) −11.0590 −0.411575
\(723\) −18.7398 −0.696941
\(724\) 12.1596 0.451909
\(725\) −28.4478 −1.05653
\(726\) 16.9652 0.629639
\(727\) −6.73980 −0.249965 −0.124983 0.992159i \(-0.539888\pi\)
−0.124983 + 0.992159i \(0.539888\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) −6.51437 −0.241108
\(731\) 9.92442 0.367068
\(732\) 9.77083 0.361140
\(733\) 23.9615 0.885038 0.442519 0.896759i \(-0.354085\pi\)
0.442519 + 0.896759i \(0.354085\pi\)
\(734\) −27.0590 −0.998768
\(735\) 0 0
\(736\) −6.28822 −0.231787
\(737\) −44.6740 −1.64559
\(738\) −5.32299 −0.195942
\(739\) 5.93794 0.218431 0.109215 0.994018i \(-0.465166\pi\)
0.109215 + 0.994018i \(0.465166\pi\)
\(740\) 5.03477 0.185082
\(741\) −5.48261 −0.201409
\(742\) 0 0
\(743\) −22.3783 −0.820980 −0.410490 0.911865i \(-0.634642\pi\)
−0.410490 + 0.911865i \(0.634642\pi\)
\(744\) −8.48261 −0.310988
\(745\) −69.5977 −2.54986
\(746\) −3.70877 −0.135788
\(747\) −7.44784 −0.272502
\(748\) 32.5734 1.19100
\(749\) 0 0
\(750\) 17.0590 0.622908
\(751\) 41.2467 1.50511 0.752556 0.658528i \(-0.228821\pi\)
0.752556 + 0.658528i \(0.228821\pi\)
\(752\) 7.09382 0.258685
\(753\) 7.77457 0.283321
\(754\) 3.00000 0.109254
\(755\) −58.1809 −2.11742
\(756\) 0 0
\(757\) −41.9585 −1.52501 −0.762503 0.646984i \(-0.776030\pi\)
−0.762503 + 0.646984i \(0.776030\pi\)
\(758\) 19.2224 0.698189
\(759\) −33.2534 −1.20702
\(760\) −20.8646 −0.756840
\(761\) 19.1604 0.694562 0.347281 0.937761i \(-0.387105\pi\)
0.347281 + 0.937761i \(0.387105\pi\)
\(762\) −10.1248 −0.366785
\(763\) 0 0
\(764\) −24.1876 −0.875078
\(765\) 23.4411 0.847514
\(766\) −25.2224 −0.911323
\(767\) −1.67701 −0.0605533
\(768\) −1.00000 −0.0360844
\(769\) 31.7981 1.14667 0.573335 0.819321i \(-0.305649\pi\)
0.573335 + 0.819321i \(0.305649\pi\)
\(770\) 0 0
\(771\) −6.57643 −0.236844
\(772\) 1.54842 0.0557288
\(773\) −50.3753 −1.81187 −0.905936 0.423414i \(-0.860832\pi\)
−0.905936 + 0.423414i \(0.860832\pi\)
\(774\) −1.61121 −0.0579136
\(775\) −80.4373 −2.88939
\(776\) 3.15962 0.113424
\(777\) 0 0
\(778\) −18.1286 −0.649942
\(779\) 29.1839 1.04562
\(780\) −3.80560 −0.136262
\(781\) −81.8754 −2.92973
\(782\) −38.7330 −1.38509
\(783\) 3.00000 0.107211
\(784\) 0 0
\(785\) 39.6422 1.41489
\(786\) 10.0628 0.358928
\(787\) −22.0590 −0.786320 −0.393160 0.919470i \(-0.628618\pi\)
−0.393160 + 0.919470i \(0.628618\pi\)
\(788\) −18.2882 −0.651491
\(789\) 20.4759 0.728960
\(790\) −10.9274 −0.388781
\(791\) 0 0
\(792\) −5.28822 −0.187909
\(793\) −9.77083 −0.346972
\(794\) 14.6075 0.518399
\(795\) 1.34726 0.0477824
\(796\) −26.2187 −0.929296
\(797\) −52.5312 −1.86075 −0.930374 0.366611i \(-0.880518\pi\)
−0.930374 + 0.366611i \(0.880518\pi\)
\(798\) 0 0
\(799\) 43.6952 1.54583
\(800\) −9.48261 −0.335261
\(801\) 10.6770 0.377254
\(802\) 21.2534 0.750485
\(803\) 9.05228 0.319448
\(804\) 8.44784 0.297932
\(805\) 0 0
\(806\) 8.48261 0.298787
\(807\) 24.1529 0.850221
\(808\) 4.96523 0.174676
\(809\) −46.9858 −1.65193 −0.825966 0.563721i \(-0.809369\pi\)
−0.825966 + 0.563721i \(0.809369\pi\)
\(810\) −3.80560 −0.133715
\(811\) 1.81236 0.0636407 0.0318203 0.999494i \(-0.489870\pi\)
0.0318203 + 0.999494i \(0.489870\pi\)
\(812\) 0 0
\(813\) −30.4411 −1.06762
\(814\) −6.99626 −0.245219
\(815\) −37.0658 −1.29836
\(816\) −6.15962 −0.215630
\(817\) 8.83362 0.309049
\(818\) 15.9932 0.559191
\(819\) 0 0
\(820\) 20.2572 0.707412
\(821\) −35.1521 −1.22682 −0.613409 0.789765i \(-0.710202\pi\)
−0.613409 + 0.789765i \(0.710202\pi\)
\(822\) −6.67701 −0.232887
\(823\) 34.6770 1.20876 0.604382 0.796694i \(-0.293420\pi\)
0.604382 + 0.796694i \(0.293420\pi\)
\(824\) 5.61121 0.195476
\(825\) −50.1461 −1.74586
\(826\) 0 0
\(827\) 19.4138 0.675084 0.337542 0.941311i \(-0.390405\pi\)
0.337542 + 0.941311i \(0.390405\pi\)
\(828\) 6.28822 0.218531
\(829\) 41.8964 1.45512 0.727561 0.686043i \(-0.240654\pi\)
0.727561 + 0.686043i \(0.240654\pi\)
\(830\) 28.3435 0.983818
\(831\) 9.12485 0.316537
\(832\) 1.00000 0.0346688
\(833\) 0 0
\(834\) −12.2882 −0.425506
\(835\) −33.1279 −1.14644
\(836\) 28.9932 1.00275
\(837\) 8.48261 0.293202
\(838\) 5.61121 0.193836
\(839\) 15.9895 0.552019 0.276009 0.961155i \(-0.410988\pi\)
0.276009 + 0.961155i \(0.410988\pi\)
\(840\) 0 0
\(841\) −20.0000 −0.689655
\(842\) −17.5802 −0.605853
\(843\) 6.70804 0.231037
\(844\) −9.89942 −0.340752
\(845\) 3.80560 0.130917
\(846\) −7.09382 −0.243891
\(847\) 0 0
\(848\) −0.354020 −0.0121571
\(849\) −26.8646 −0.921992
\(850\) −58.4093 −2.00342
\(851\) 8.31925 0.285180
\(852\) 15.4826 0.530426
\(853\) −44.2747 −1.51594 −0.757968 0.652291i \(-0.773808\pi\)
−0.757968 + 0.652291i \(0.773808\pi\)
\(854\) 0 0
\(855\) 20.8646 0.713556
\(856\) 17.0590 0.583066
\(857\) 23.8329 0.814116 0.407058 0.913402i \(-0.366555\pi\)
0.407058 + 0.913402i \(0.366555\pi\)
\(858\) 5.28822 0.180537
\(859\) −14.6460 −0.499714 −0.249857 0.968283i \(-0.580384\pi\)
−0.249857 + 0.968283i \(0.580384\pi\)
\(860\) 6.13161 0.209086
\(861\) 0 0
\(862\) −17.2224 −0.586598
\(863\) −21.1529 −0.720052 −0.360026 0.932942i \(-0.617232\pi\)
−0.360026 + 0.932942i \(0.617232\pi\)
\(864\) 1.00000 0.0340207
\(865\) −45.8919 −1.56037
\(866\) −35.0243 −1.19017
\(867\) −20.9410 −0.711192
\(868\) 0 0
\(869\) 15.1846 0.515103
\(870\) −11.4168 −0.387066
\(871\) −8.44784 −0.286244
\(872\) −4.00000 −0.135457
\(873\) −3.15962 −0.106937
\(874\) −34.4759 −1.16616
\(875\) 0 0
\(876\) −1.71178 −0.0578358
\(877\) 6.22616 0.210242 0.105121 0.994459i \(-0.466477\pi\)
0.105121 + 0.994459i \(0.466477\pi\)
\(878\) 14.5136 0.489812
\(879\) −9.80560 −0.330735
\(880\) 20.1248 0.678409
\(881\) 46.9517 1.58184 0.790922 0.611918i \(-0.209602\pi\)
0.790922 + 0.611918i \(0.209602\pi\)
\(882\) 0 0
\(883\) −29.1143 −0.979776 −0.489888 0.871785i \(-0.662962\pi\)
−0.489888 + 0.871785i \(0.662962\pi\)
\(884\) 6.15962 0.207171
\(885\) 6.38203 0.214530
\(886\) −10.3820 −0.348791
\(887\) 43.6037 1.46407 0.732035 0.681267i \(-0.238571\pi\)
0.732035 + 0.681267i \(0.238571\pi\)
\(888\) 1.32299 0.0443966
\(889\) 0 0
\(890\) −40.6325 −1.36200
\(891\) 5.28822 0.177162
\(892\) 2.96897 0.0994085
\(893\) 38.8927 1.30149
\(894\) −18.2882 −0.611649
\(895\) −16.2012 −0.541545
\(896\) 0 0
\(897\) −6.28822 −0.209957
\(898\) −31.0833 −1.03726
\(899\) 25.4478 0.848733
\(900\) 9.48261 0.316087
\(901\) −2.18063 −0.0726473
\(902\) −28.1491 −0.937263
\(903\) 0 0
\(904\) 5.45158 0.181317
\(905\) 46.2747 1.53822
\(906\) −15.2882 −0.507917
\(907\) 2.47585 0.0822093 0.0411047 0.999155i \(-0.486912\pi\)
0.0411047 + 0.999155i \(0.486912\pi\)
\(908\) 18.4131 0.611059
\(909\) −4.96523 −0.164686
\(910\) 0 0
\(911\) 46.0000 1.52405 0.762024 0.647549i \(-0.224206\pi\)
0.762024 + 0.647549i \(0.224206\pi\)
\(912\) −5.48261 −0.181547
\(913\) −39.3858 −1.30348
\(914\) −5.41681 −0.179172
\(915\) 37.1839 1.22926
\(916\) −1.42357 −0.0470360
\(917\) 0 0
\(918\) 6.15962 0.203298
\(919\) 35.5417 1.17241 0.586206 0.810162i \(-0.300621\pi\)
0.586206 + 0.810162i \(0.300621\pi\)
\(920\) −23.9305 −0.788964
\(921\) −20.5174 −0.676071
\(922\) −9.54166 −0.314238
\(923\) −15.4826 −0.509616
\(924\) 0 0
\(925\) 12.5454 0.412490
\(926\) 27.2224 0.894584
\(927\) −5.61121 −0.184296
\(928\) 3.00000 0.0984798
\(929\) 14.1876 0.465481 0.232741 0.972539i \(-0.425231\pi\)
0.232741 + 0.972539i \(0.425231\pi\)
\(930\) −32.2815 −1.05855
\(931\) 0 0
\(932\) 10.7361 0.351671
\(933\) 22.5454 0.738103
\(934\) −13.0658 −0.427526
\(935\) 123.961 4.05397
\(936\) −1.00000 −0.0326860
\(937\) −2.09683 −0.0685006 −0.0342503 0.999413i \(-0.510904\pi\)
−0.0342503 + 0.999413i \(0.510904\pi\)
\(938\) 0 0
\(939\) −32.0938 −1.04734
\(940\) 26.9963 0.880521
\(941\) −7.90618 −0.257734 −0.128867 0.991662i \(-0.541134\pi\)
−0.128867 + 0.991662i \(0.541134\pi\)
\(942\) 10.4168 0.339398
\(943\) 33.4721 1.09000
\(944\) −1.67701 −0.0545820
\(945\) 0 0
\(946\) −8.52040 −0.277022
\(947\) −36.8541 −1.19760 −0.598799 0.800899i \(-0.704355\pi\)
−0.598799 + 0.800899i \(0.704355\pi\)
\(948\) −2.87141 −0.0932590
\(949\) 1.71178 0.0555669
\(950\) −51.9895 −1.68676
\(951\) −31.4478 −1.01977
\(952\) 0 0
\(953\) 21.6527 0.701401 0.350701 0.936488i \(-0.385944\pi\)
0.350701 + 0.936488i \(0.385944\pi\)
\(954\) 0.354020 0.0114618
\(955\) −92.0485 −2.97862
\(956\) 2.12859 0.0688436
\(957\) 15.8646 0.512831
\(958\) 25.2119 0.814560
\(959\) 0 0
\(960\) −3.80560 −0.122825
\(961\) 40.9547 1.32112
\(962\) −1.32299 −0.0426549
\(963\) −17.0590 −0.549720
\(964\) 18.7398 0.603568
\(965\) 5.89266 0.189691
\(966\) 0 0
\(967\) 7.21867 0.232137 0.116068 0.993241i \(-0.462971\pi\)
0.116068 + 0.993241i \(0.462971\pi\)
\(968\) −16.9652 −0.545283
\(969\) −33.7708 −1.08488
\(970\) 12.0243 0.386076
\(971\) −27.7981 −0.892084 −0.446042 0.895012i \(-0.647167\pi\)
−0.446042 + 0.895012i \(0.647167\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 0 0
\(974\) −13.9342 −0.446480
\(975\) −9.48261 −0.303687
\(976\) −9.77083 −0.312757
\(977\) −6.57643 −0.210399 −0.105199 0.994451i \(-0.533548\pi\)
−0.105199 + 0.994451i \(0.533548\pi\)
\(978\) −9.73980 −0.311444
\(979\) 56.4623 1.80454
\(980\) 0 0
\(981\) 4.00000 0.127710
\(982\) 12.3510 0.394136
\(983\) 10.4478 0.333234 0.166617 0.986022i \(-0.446716\pi\)
0.166617 + 0.986022i \(0.446716\pi\)
\(984\) 5.32299 0.169691
\(985\) −69.5977 −2.21757
\(986\) 18.4789 0.588487
\(987\) 0 0
\(988\) 5.48261 0.174425
\(989\) 10.1316 0.322166
\(990\) −20.1248 −0.639610
\(991\) 10.9895 0.349093 0.174546 0.984649i \(-0.444154\pi\)
0.174546 + 0.984649i \(0.444154\pi\)
\(992\) 8.48261 0.269323
\(993\) −16.8957 −0.536168
\(994\) 0 0
\(995\) −99.7778 −3.16317
\(996\) 7.44784 0.235994
\(997\) 44.2157 1.40032 0.700162 0.713984i \(-0.253111\pi\)
0.700162 + 0.713984i \(0.253111\pi\)
\(998\) 34.7641 1.10044
\(999\) −1.32299 −0.0418575
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 3822.2.a.bv.1.3 3
7.2 even 3 546.2.i.k.235.1 yes 6
7.4 even 3 546.2.i.k.79.1 6
7.6 odd 2 3822.2.a.bw.1.1 3
21.2 odd 6 1638.2.j.q.235.3 6
21.11 odd 6 1638.2.j.q.1171.3 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
546.2.i.k.79.1 6 7.4 even 3
546.2.i.k.235.1 yes 6 7.2 even 3
1638.2.j.q.235.3 6 21.2 odd 6
1638.2.j.q.1171.3 6 21.11 odd 6
3822.2.a.bv.1.3 3 1.1 even 1 trivial
3822.2.a.bw.1.1 3 7.6 odd 2