Properties

Label 4655.2.a.u.1.1
Level $4655$
Weight $2$
Character 4655.1
Self dual yes
Analytic conductor $37.170$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4655,2,Mod(1,4655)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4655, 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("4655.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4655 = 5 \cdot 7^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4655.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: \(37.1703621409\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.148.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 3x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 95)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.48119\) of defining polynomial
Character \(\chi\) \(=\) 4655.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.48119 q^{2} -0.806063 q^{3} +0.193937 q^{4} -1.00000 q^{5} +1.19394 q^{6} +2.67513 q^{8} -2.35026 q^{9} +O(q^{10})\) \(q-1.48119 q^{2} -0.806063 q^{3} +0.193937 q^{4} -1.00000 q^{5} +1.19394 q^{6} +2.67513 q^{8} -2.35026 q^{9} +1.48119 q^{10} +0.962389 q^{11} -0.156325 q^{12} -6.15633 q^{13} +0.806063 q^{15} -4.35026 q^{16} +6.31265 q^{17} +3.48119 q^{18} +1.00000 q^{19} -0.193937 q^{20} -1.42548 q^{22} -4.96239 q^{23} -2.15633 q^{24} +1.00000 q^{25} +9.11871 q^{26} +4.31265 q^{27} -3.61213 q^{29} -1.19394 q^{30} +5.92478 q^{31} +1.09332 q^{32} -0.775746 q^{33} -9.35026 q^{34} -0.455802 q^{36} +10.1563 q^{37} -1.48119 q^{38} +4.96239 q^{39} -2.67513 q^{40} -6.31265 q^{41} -4.12601 q^{43} +0.186642 q^{44} +2.35026 q^{45} +7.35026 q^{46} -3.35026 q^{47} +3.50659 q^{48} -1.48119 q^{50} -5.08840 q^{51} -1.19394 q^{52} +1.84367 q^{53} -6.38787 q^{54} -0.962389 q^{55} -0.806063 q^{57} +5.35026 q^{58} +6.38787 q^{59} +0.156325 q^{60} +11.2750 q^{61} -8.77575 q^{62} +7.08110 q^{64} +6.15633 q^{65} +1.14903 q^{66} -6.73084 q^{67} +1.22425 q^{68} +4.00000 q^{69} -0.775746 q^{71} -6.28726 q^{72} -0.387873 q^{73} -15.0435 q^{74} -0.806063 q^{75} +0.193937 q^{76} -7.35026 q^{78} -0.836381 q^{79} +4.35026 q^{80} +3.57452 q^{81} +9.35026 q^{82} +7.03761 q^{83} -6.31265 q^{85} +6.11142 q^{86} +2.91160 q^{87} +2.57452 q^{88} -7.08840 q^{89} -3.48119 q^{90} -0.962389 q^{92} -4.77575 q^{93} +4.96239 q^{94} -1.00000 q^{95} -0.881286 q^{96} -10.9927 q^{97} -2.26187 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + q^{2} - 2 q^{3} + q^{4} - 3 q^{5} + 4 q^{6} + 3 q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + q^{2} - 2 q^{3} + q^{4} - 3 q^{5} + 4 q^{6} + 3 q^{8} + 3 q^{9} - q^{10} - 8 q^{11} + 10 q^{12} - 8 q^{13} + 2 q^{15} - 3 q^{16} - 2 q^{17} + 5 q^{18} + 3 q^{19} - q^{20} - 16 q^{22} - 4 q^{23} + 4 q^{24} + 3 q^{25} + 6 q^{26} - 8 q^{27} - 10 q^{29} - 4 q^{30} - 4 q^{31} - 3 q^{32} - 4 q^{33} - 18 q^{34} - 11 q^{36} + 20 q^{37} + q^{38} + 4 q^{39} - 3 q^{40} + 2 q^{41} - 4 q^{43} - 12 q^{44} - 3 q^{45} + 12 q^{46} - 10 q^{48} + q^{50} + 4 q^{51} - 4 q^{52} + 16 q^{53} - 20 q^{54} + 8 q^{55} - 2 q^{57} + 6 q^{58} + 20 q^{59} - 10 q^{60} + 2 q^{61} - 28 q^{62} - 11 q^{64} + 8 q^{65} - 20 q^{66} + 2 q^{67} + 2 q^{68} + 12 q^{69} - 4 q^{71} - 13 q^{72} - 2 q^{73} - 2 q^{74} - 2 q^{75} + q^{76} - 12 q^{78} + 3 q^{80} - q^{81} + 18 q^{82} + 32 q^{83} + 2 q^{85} - 16 q^{86} + 28 q^{87} - 4 q^{88} - 2 q^{89} - 5 q^{90} + 8 q^{92} - 16 q^{93} + 4 q^{94} - 3 q^{95} - 24 q^{96} - 20 q^{97} - 16 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.48119 −1.04736 −0.523681 0.851914i \(-0.675442\pi\)
−0.523681 + 0.851914i \(0.675442\pi\)
\(3\) −0.806063 −0.465381 −0.232690 0.972551i \(-0.574753\pi\)
−0.232690 + 0.972551i \(0.574753\pi\)
\(4\) 0.193937 0.0969683
\(5\) −1.00000 −0.447214
\(6\) 1.19394 0.487423
\(7\) 0 0
\(8\) 2.67513 0.945802
\(9\) −2.35026 −0.783421
\(10\) 1.48119 0.468395
\(11\) 0.962389 0.290171 0.145086 0.989419i \(-0.453654\pi\)
0.145086 + 0.989419i \(0.453654\pi\)
\(12\) −0.156325 −0.0451272
\(13\) −6.15633 −1.70746 −0.853729 0.520718i \(-0.825664\pi\)
−0.853729 + 0.520718i \(0.825664\pi\)
\(14\) 0 0
\(15\) 0.806063 0.208125
\(16\) −4.35026 −1.08757
\(17\) 6.31265 1.53104 0.765521 0.643411i \(-0.222481\pi\)
0.765521 + 0.643411i \(0.222481\pi\)
\(18\) 3.48119 0.820525
\(19\) 1.00000 0.229416
\(20\) −0.193937 −0.0433655
\(21\) 0 0
\(22\) −1.42548 −0.303914
\(23\) −4.96239 −1.03473 −0.517365 0.855765i \(-0.673087\pi\)
−0.517365 + 0.855765i \(0.673087\pi\)
\(24\) −2.15633 −0.440158
\(25\) 1.00000 0.200000
\(26\) 9.11871 1.78833
\(27\) 4.31265 0.829970
\(28\) 0 0
\(29\) −3.61213 −0.670755 −0.335378 0.942084i \(-0.608864\pi\)
−0.335378 + 0.942084i \(0.608864\pi\)
\(30\) −1.19394 −0.217982
\(31\) 5.92478 1.06412 0.532061 0.846706i \(-0.321418\pi\)
0.532061 + 0.846706i \(0.321418\pi\)
\(32\) 1.09332 0.193274
\(33\) −0.775746 −0.135040
\(34\) −9.35026 −1.60356
\(35\) 0 0
\(36\) −0.455802 −0.0759669
\(37\) 10.1563 1.66969 0.834845 0.550485i \(-0.185557\pi\)
0.834845 + 0.550485i \(0.185557\pi\)
\(38\) −1.48119 −0.240281
\(39\) 4.96239 0.794618
\(40\) −2.67513 −0.422975
\(41\) −6.31265 −0.985870 −0.492935 0.870066i \(-0.664076\pi\)
−0.492935 + 0.870066i \(0.664076\pi\)
\(42\) 0 0
\(43\) −4.12601 −0.629210 −0.314605 0.949223i \(-0.601872\pi\)
−0.314605 + 0.949223i \(0.601872\pi\)
\(44\) 0.186642 0.0281374
\(45\) 2.35026 0.350356
\(46\) 7.35026 1.08374
\(47\) −3.35026 −0.488686 −0.244343 0.969689i \(-0.578572\pi\)
−0.244343 + 0.969689i \(0.578572\pi\)
\(48\) 3.50659 0.506132
\(49\) 0 0
\(50\) −1.48119 −0.209473
\(51\) −5.08840 −0.712518
\(52\) −1.19394 −0.165569
\(53\) 1.84367 0.253248 0.126624 0.991951i \(-0.459586\pi\)
0.126624 + 0.991951i \(0.459586\pi\)
\(54\) −6.38787 −0.869279
\(55\) −0.962389 −0.129768
\(56\) 0 0
\(57\) −0.806063 −0.106766
\(58\) 5.35026 0.702524
\(59\) 6.38787 0.831630 0.415815 0.909449i \(-0.363496\pi\)
0.415815 + 0.909449i \(0.363496\pi\)
\(60\) 0.156325 0.0201815
\(61\) 11.2750 1.44362 0.721810 0.692091i \(-0.243310\pi\)
0.721810 + 0.692091i \(0.243310\pi\)
\(62\) −8.77575 −1.11452
\(63\) 0 0
\(64\) 7.08110 0.885138
\(65\) 6.15633 0.763598
\(66\) 1.14903 0.141436
\(67\) −6.73084 −0.822303 −0.411152 0.911567i \(-0.634873\pi\)
−0.411152 + 0.911567i \(0.634873\pi\)
\(68\) 1.22425 0.148463
\(69\) 4.00000 0.481543
\(70\) 0 0
\(71\) −0.775746 −0.0920641 −0.0460321 0.998940i \(-0.514658\pi\)
−0.0460321 + 0.998940i \(0.514658\pi\)
\(72\) −6.28726 −0.740960
\(73\) −0.387873 −0.0453971 −0.0226986 0.999742i \(-0.507226\pi\)
−0.0226986 + 0.999742i \(0.507226\pi\)
\(74\) −15.0435 −1.74877
\(75\) −0.806063 −0.0930762
\(76\) 0.193937 0.0222460
\(77\) 0 0
\(78\) −7.35026 −0.832253
\(79\) −0.836381 −0.0941002 −0.0470501 0.998893i \(-0.514982\pi\)
−0.0470501 + 0.998893i \(0.514982\pi\)
\(80\) 4.35026 0.486374
\(81\) 3.57452 0.397168
\(82\) 9.35026 1.03256
\(83\) 7.03761 0.772478 0.386239 0.922399i \(-0.373774\pi\)
0.386239 + 0.922399i \(0.373774\pi\)
\(84\) 0 0
\(85\) −6.31265 −0.684703
\(86\) 6.11142 0.659011
\(87\) 2.91160 0.312157
\(88\) 2.57452 0.274444
\(89\) −7.08840 −0.751369 −0.375684 0.926748i \(-0.622592\pi\)
−0.375684 + 0.926748i \(0.622592\pi\)
\(90\) −3.48119 −0.366950
\(91\) 0 0
\(92\) −0.962389 −0.100336
\(93\) −4.77575 −0.495222
\(94\) 4.96239 0.511831
\(95\) −1.00000 −0.102598
\(96\) −0.881286 −0.0899459
\(97\) −10.9927 −1.11614 −0.558070 0.829794i \(-0.688458\pi\)
−0.558070 + 0.829794i \(0.688458\pi\)
\(98\) 0 0
\(99\) −2.26187 −0.227326
\(100\) 0.193937 0.0193937
\(101\) 2.64974 0.263659 0.131829 0.991272i \(-0.457915\pi\)
0.131829 + 0.991272i \(0.457915\pi\)
\(102\) 7.53690 0.746265
\(103\) 10.7308 1.05734 0.528671 0.848827i \(-0.322691\pi\)
0.528671 + 0.848827i \(0.322691\pi\)
\(104\) −16.4690 −1.61492
\(105\) 0 0
\(106\) −2.73084 −0.265243
\(107\) 4.80606 0.464620 0.232310 0.972642i \(-0.425372\pi\)
0.232310 + 0.972642i \(0.425372\pi\)
\(108\) 0.836381 0.0804808
\(109\) −2.77575 −0.265868 −0.132934 0.991125i \(-0.542440\pi\)
−0.132934 + 0.991125i \(0.542440\pi\)
\(110\) 1.42548 0.135915
\(111\) −8.18664 −0.777042
\(112\) 0 0
\(113\) 6.99271 0.657818 0.328909 0.944362i \(-0.393319\pi\)
0.328909 + 0.944362i \(0.393319\pi\)
\(114\) 1.19394 0.111822
\(115\) 4.96239 0.462745
\(116\) −0.700523 −0.0650420
\(117\) 14.4690 1.33766
\(118\) −9.46168 −0.871018
\(119\) 0 0
\(120\) 2.15633 0.196845
\(121\) −10.0738 −0.915801
\(122\) −16.7005 −1.51199
\(123\) 5.08840 0.458805
\(124\) 1.14903 0.103186
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 13.4314 1.19184 0.595920 0.803043i \(-0.296787\pi\)
0.595920 + 0.803043i \(0.296787\pi\)
\(128\) −12.6751 −1.12033
\(129\) 3.32582 0.292822
\(130\) −9.11871 −0.799764
\(131\) −20.6253 −1.80204 −0.901020 0.433777i \(-0.857181\pi\)
−0.901020 + 0.433777i \(0.857181\pi\)
\(132\) −0.150446 −0.0130946
\(133\) 0 0
\(134\) 9.96968 0.861249
\(135\) −4.31265 −0.371174
\(136\) 16.8872 1.44806
\(137\) 20.2374 1.72900 0.864500 0.502633i \(-0.167635\pi\)
0.864500 + 0.502633i \(0.167635\pi\)
\(138\) −5.92478 −0.504351
\(139\) 17.5877 1.49177 0.745884 0.666076i \(-0.232027\pi\)
0.745884 + 0.666076i \(0.232027\pi\)
\(140\) 0 0
\(141\) 2.70052 0.227425
\(142\) 1.14903 0.0964245
\(143\) −5.92478 −0.495455
\(144\) 10.2243 0.852021
\(145\) 3.61213 0.299971
\(146\) 0.574515 0.0475472
\(147\) 0 0
\(148\) 1.96968 0.161907
\(149\) −7.42548 −0.608319 −0.304160 0.952621i \(-0.598376\pi\)
−0.304160 + 0.952621i \(0.598376\pi\)
\(150\) 1.19394 0.0974845
\(151\) −1.61213 −0.131193 −0.0655965 0.997846i \(-0.520895\pi\)
−0.0655965 + 0.997846i \(0.520895\pi\)
\(152\) 2.67513 0.216982
\(153\) −14.8364 −1.19945
\(154\) 0 0
\(155\) −5.92478 −0.475890
\(156\) 0.962389 0.0770528
\(157\) −4.38787 −0.350190 −0.175095 0.984552i \(-0.556023\pi\)
−0.175095 + 0.984552i \(0.556023\pi\)
\(158\) 1.23884 0.0985570
\(159\) −1.48612 −0.117857
\(160\) −1.09332 −0.0864346
\(161\) 0 0
\(162\) −5.29455 −0.415979
\(163\) −0.649738 −0.0508914 −0.0254457 0.999676i \(-0.508100\pi\)
−0.0254457 + 0.999676i \(0.508100\pi\)
\(164\) −1.22425 −0.0955982
\(165\) 0.775746 0.0603918
\(166\) −10.4241 −0.809065
\(167\) 15.3561 1.18829 0.594147 0.804357i \(-0.297490\pi\)
0.594147 + 0.804357i \(0.297490\pi\)
\(168\) 0 0
\(169\) 24.9003 1.91541
\(170\) 9.35026 0.717132
\(171\) −2.35026 −0.179729
\(172\) −0.800184 −0.0610134
\(173\) −3.24472 −0.246692 −0.123346 0.992364i \(-0.539362\pi\)
−0.123346 + 0.992364i \(0.539362\pi\)
\(174\) −4.31265 −0.326941
\(175\) 0 0
\(176\) −4.18664 −0.315580
\(177\) −5.14903 −0.387025
\(178\) 10.4993 0.786955
\(179\) 15.0132 1.12214 0.561069 0.827769i \(-0.310390\pi\)
0.561069 + 0.827769i \(0.310390\pi\)
\(180\) 0.455802 0.0339735
\(181\) −9.22425 −0.685633 −0.342817 0.939402i \(-0.611381\pi\)
−0.342817 + 0.939402i \(0.611381\pi\)
\(182\) 0 0
\(183\) −9.08840 −0.671834
\(184\) −13.2750 −0.978649
\(185\) −10.1563 −0.746708
\(186\) 7.07381 0.518677
\(187\) 6.07522 0.444264
\(188\) −0.649738 −0.0473870
\(189\) 0 0
\(190\) 1.48119 0.107457
\(191\) −21.7743 −1.57554 −0.787768 0.615972i \(-0.788763\pi\)
−0.787768 + 0.615972i \(0.788763\pi\)
\(192\) −5.70782 −0.411926
\(193\) 12.5442 0.902951 0.451476 0.892283i \(-0.350898\pi\)
0.451476 + 0.892283i \(0.350898\pi\)
\(194\) 16.2823 1.16900
\(195\) −4.96239 −0.355364
\(196\) 0 0
\(197\) 24.5501 1.74912 0.874560 0.484917i \(-0.161150\pi\)
0.874560 + 0.484917i \(0.161150\pi\)
\(198\) 3.35026 0.238093
\(199\) −23.0738 −1.63566 −0.817829 0.575461i \(-0.804823\pi\)
−0.817829 + 0.575461i \(0.804823\pi\)
\(200\) 2.67513 0.189160
\(201\) 5.42548 0.382684
\(202\) −3.92478 −0.276146
\(203\) 0 0
\(204\) −0.986826 −0.0690917
\(205\) 6.31265 0.440895
\(206\) −15.8945 −1.10742
\(207\) 11.6629 0.810628
\(208\) 26.7816 1.85697
\(209\) 0.962389 0.0665698
\(210\) 0 0
\(211\) −20.9380 −1.44143 −0.720714 0.693233i \(-0.756186\pi\)
−0.720714 + 0.693233i \(0.756186\pi\)
\(212\) 0.357556 0.0245570
\(213\) 0.625301 0.0428449
\(214\) −7.11871 −0.486625
\(215\) 4.12601 0.281391
\(216\) 11.5369 0.784987
\(217\) 0 0
\(218\) 4.11142 0.278460
\(219\) 0.312650 0.0211270
\(220\) −0.186642 −0.0125834
\(221\) −38.8627 −2.61419
\(222\) 12.1260 0.813844
\(223\) 0.0303172 0.00203019 0.00101509 0.999999i \(-0.499677\pi\)
0.00101509 + 0.999999i \(0.499677\pi\)
\(224\) 0 0
\(225\) −2.35026 −0.156684
\(226\) −10.3576 −0.688974
\(227\) −4.80606 −0.318990 −0.159495 0.987199i \(-0.550987\pi\)
−0.159495 + 0.987199i \(0.550987\pi\)
\(228\) −0.156325 −0.0103529
\(229\) −1.87399 −0.123837 −0.0619184 0.998081i \(-0.519722\pi\)
−0.0619184 + 0.998081i \(0.519722\pi\)
\(230\) −7.35026 −0.484662
\(231\) 0 0
\(232\) −9.66291 −0.634401
\(233\) −11.1490 −0.730397 −0.365199 0.930930i \(-0.618999\pi\)
−0.365199 + 0.930930i \(0.618999\pi\)
\(234\) −21.4314 −1.40101
\(235\) 3.35026 0.218547
\(236\) 1.23884 0.0806418
\(237\) 0.674176 0.0437924
\(238\) 0 0
\(239\) 9.29948 0.601533 0.300767 0.953698i \(-0.402757\pi\)
0.300767 + 0.953698i \(0.402757\pi\)
\(240\) −3.50659 −0.226349
\(241\) 2.31265 0.148971 0.0744855 0.997222i \(-0.476269\pi\)
0.0744855 + 0.997222i \(0.476269\pi\)
\(242\) 14.9213 0.959175
\(243\) −15.8192 −1.01480
\(244\) 2.18664 0.139985
\(245\) 0 0
\(246\) −7.53690 −0.480535
\(247\) −6.15633 −0.391718
\(248\) 15.8496 1.00645
\(249\) −5.67276 −0.359497
\(250\) 1.48119 0.0936790
\(251\) −24.1016 −1.52128 −0.760639 0.649175i \(-0.775114\pi\)
−0.760639 + 0.649175i \(0.775114\pi\)
\(252\) 0 0
\(253\) −4.77575 −0.300249
\(254\) −19.8945 −1.24829
\(255\) 5.08840 0.318648
\(256\) 4.61213 0.288258
\(257\) 13.3199 0.830875 0.415438 0.909622i \(-0.363628\pi\)
0.415438 + 0.909622i \(0.363628\pi\)
\(258\) −4.92619 −0.306691
\(259\) 0 0
\(260\) 1.19394 0.0740448
\(261\) 8.48944 0.525483
\(262\) 30.5501 1.88739
\(263\) 12.9624 0.799295 0.399648 0.916669i \(-0.369133\pi\)
0.399648 + 0.916669i \(0.369133\pi\)
\(264\) −2.07522 −0.127721
\(265\) −1.84367 −0.113256
\(266\) 0 0
\(267\) 5.71370 0.349673
\(268\) −1.30536 −0.0797373
\(269\) 11.4010 0.695134 0.347567 0.937655i \(-0.387008\pi\)
0.347567 + 0.937655i \(0.387008\pi\)
\(270\) 6.38787 0.388754
\(271\) −16.8119 −1.02125 −0.510626 0.859803i \(-0.670586\pi\)
−0.510626 + 0.859803i \(0.670586\pi\)
\(272\) −27.4617 −1.66511
\(273\) 0 0
\(274\) −29.9756 −1.81089
\(275\) 0.962389 0.0580342
\(276\) 0.775746 0.0466944
\(277\) −29.7889 −1.78984 −0.894921 0.446224i \(-0.852769\pi\)
−0.894921 + 0.446224i \(0.852769\pi\)
\(278\) −26.0508 −1.56242
\(279\) −13.9248 −0.833655
\(280\) 0 0
\(281\) −11.6121 −0.692721 −0.346361 0.938101i \(-0.612583\pi\)
−0.346361 + 0.938101i \(0.612583\pi\)
\(282\) −4.00000 −0.238197
\(283\) −2.26187 −0.134454 −0.0672270 0.997738i \(-0.521415\pi\)
−0.0672270 + 0.997738i \(0.521415\pi\)
\(284\) −0.150446 −0.00892730
\(285\) 0.806063 0.0477471
\(286\) 8.77575 0.518921
\(287\) 0 0
\(288\) −2.56959 −0.151415
\(289\) 22.8496 1.34409
\(290\) −5.35026 −0.314178
\(291\) 8.86082 0.519430
\(292\) −0.0752228 −0.00440208
\(293\) −1.84367 −0.107709 −0.0538543 0.998549i \(-0.517151\pi\)
−0.0538543 + 0.998549i \(0.517151\pi\)
\(294\) 0 0
\(295\) −6.38787 −0.371916
\(296\) 27.1695 1.57920
\(297\) 4.15045 0.240833
\(298\) 10.9986 0.637131
\(299\) 30.5501 1.76676
\(300\) −0.156325 −0.00902544
\(301\) 0 0
\(302\) 2.38787 0.137407
\(303\) −2.13586 −0.122702
\(304\) −4.35026 −0.249505
\(305\) −11.2750 −0.645607
\(306\) 21.9756 1.25626
\(307\) −26.2071 −1.49572 −0.747859 0.663857i \(-0.768918\pi\)
−0.747859 + 0.663857i \(0.768918\pi\)
\(308\) 0 0
\(309\) −8.64974 −0.492066
\(310\) 8.77575 0.498429
\(311\) −6.51388 −0.369368 −0.184684 0.982798i \(-0.559126\pi\)
−0.184684 + 0.982798i \(0.559126\pi\)
\(312\) 13.2750 0.751551
\(313\) −16.0752 −0.908625 −0.454313 0.890842i \(-0.650115\pi\)
−0.454313 + 0.890842i \(0.650115\pi\)
\(314\) 6.49929 0.366776
\(315\) 0 0
\(316\) −0.162205 −0.00912473
\(317\) −5.69323 −0.319764 −0.159882 0.987136i \(-0.551111\pi\)
−0.159882 + 0.987136i \(0.551111\pi\)
\(318\) 2.20123 0.123439
\(319\) −3.47627 −0.194634
\(320\) −7.08110 −0.395846
\(321\) −3.87399 −0.216225
\(322\) 0 0
\(323\) 6.31265 0.351245
\(324\) 0.693229 0.0385127
\(325\) −6.15633 −0.341491
\(326\) 0.962389 0.0533018
\(327\) 2.23743 0.123730
\(328\) −16.8872 −0.932438
\(329\) 0 0
\(330\) −1.14903 −0.0632521
\(331\) −12.3127 −0.676764 −0.338382 0.941009i \(-0.609880\pi\)
−0.338382 + 0.941009i \(0.609880\pi\)
\(332\) 1.36485 0.0749059
\(333\) −23.8700 −1.30807
\(334\) −22.7454 −1.24457
\(335\) 6.73084 0.367745
\(336\) 0 0
\(337\) 3.76845 0.205281 0.102640 0.994719i \(-0.467271\pi\)
0.102640 + 0.994719i \(0.467271\pi\)
\(338\) −36.8822 −2.00613
\(339\) −5.63656 −0.306136
\(340\) −1.22425 −0.0663945
\(341\) 5.70194 0.308777
\(342\) 3.48119 0.188241
\(343\) 0 0
\(344\) −11.0376 −0.595108
\(345\) −4.00000 −0.215353
\(346\) 4.80606 0.258376
\(347\) −22.3634 −1.20053 −0.600266 0.799800i \(-0.704939\pi\)
−0.600266 + 0.799800i \(0.704939\pi\)
\(348\) 0.564666 0.0302693
\(349\) 10.0000 0.535288 0.267644 0.963518i \(-0.413755\pi\)
0.267644 + 0.963518i \(0.413755\pi\)
\(350\) 0 0
\(351\) −26.5501 −1.41714
\(352\) 1.05220 0.0560824
\(353\) −5.53690 −0.294700 −0.147350 0.989084i \(-0.547074\pi\)
−0.147350 + 0.989084i \(0.547074\pi\)
\(354\) 7.62672 0.405355
\(355\) 0.775746 0.0411723
\(356\) −1.37470 −0.0728589
\(357\) 0 0
\(358\) −22.2374 −1.17528
\(359\) −10.3634 −0.546961 −0.273481 0.961878i \(-0.588175\pi\)
−0.273481 + 0.961878i \(0.588175\pi\)
\(360\) 6.28726 0.331368
\(361\) 1.00000 0.0526316
\(362\) 13.6629 0.718107
\(363\) 8.12013 0.426196
\(364\) 0 0
\(365\) 0.387873 0.0203022
\(366\) 13.4617 0.703653
\(367\) −3.35026 −0.174882 −0.0874411 0.996170i \(-0.527869\pi\)
−0.0874411 + 0.996170i \(0.527869\pi\)
\(368\) 21.5877 1.12534
\(369\) 14.8364 0.772351
\(370\) 15.0435 0.782074
\(371\) 0 0
\(372\) −0.926192 −0.0480208
\(373\) −12.6048 −0.652653 −0.326327 0.945257i \(-0.605811\pi\)
−0.326327 + 0.945257i \(0.605811\pi\)
\(374\) −8.99859 −0.465306
\(375\) 0.806063 0.0416249
\(376\) −8.96239 −0.462200
\(377\) 22.2374 1.14529
\(378\) 0 0
\(379\) −37.2506 −1.91343 −0.956717 0.291018i \(-0.906006\pi\)
−0.956717 + 0.291018i \(0.906006\pi\)
\(380\) −0.193937 −0.00994874
\(381\) −10.8265 −0.554660
\(382\) 32.2520 1.65016
\(383\) −30.8324 −1.57546 −0.787731 0.616019i \(-0.788744\pi\)
−0.787731 + 0.616019i \(0.788744\pi\)
\(384\) 10.2170 0.521382
\(385\) 0 0
\(386\) −18.5804 −0.945717
\(387\) 9.69720 0.492936
\(388\) −2.13189 −0.108230
\(389\) −1.37470 −0.0697000 −0.0348500 0.999393i \(-0.511095\pi\)
−0.0348500 + 0.999393i \(0.511095\pi\)
\(390\) 7.35026 0.372195
\(391\) −31.3258 −1.58422
\(392\) 0 0
\(393\) 16.6253 0.838635
\(394\) −36.3634 −1.83196
\(395\) 0.836381 0.0420829
\(396\) −0.438658 −0.0220434
\(397\) 9.38646 0.471093 0.235546 0.971863i \(-0.424312\pi\)
0.235546 + 0.971863i \(0.424312\pi\)
\(398\) 34.1768 1.71313
\(399\) 0 0
\(400\) −4.35026 −0.217513
\(401\) 14.1016 0.704199 0.352099 0.935963i \(-0.385468\pi\)
0.352099 + 0.935963i \(0.385468\pi\)
\(402\) −8.03620 −0.400809
\(403\) −36.4749 −1.81694
\(404\) 0.513881 0.0255665
\(405\) −3.57452 −0.177619
\(406\) 0 0
\(407\) 9.77433 0.484496
\(408\) −13.6121 −0.673901
\(409\) −35.1490 −1.73801 −0.869004 0.494805i \(-0.835239\pi\)
−0.869004 + 0.494805i \(0.835239\pi\)
\(410\) −9.35026 −0.461777
\(411\) −16.3127 −0.804644
\(412\) 2.08110 0.102529
\(413\) 0 0
\(414\) −17.2750 −0.849022
\(415\) −7.03761 −0.345463
\(416\) −6.73084 −0.330007
\(417\) −14.1768 −0.694241
\(418\) −1.42548 −0.0697227
\(419\) −9.02776 −0.441035 −0.220518 0.975383i \(-0.570775\pi\)
−0.220518 + 0.975383i \(0.570775\pi\)
\(420\) 0 0
\(421\) 15.2097 0.741274 0.370637 0.928778i \(-0.379139\pi\)
0.370637 + 0.928778i \(0.379139\pi\)
\(422\) 31.0132 1.50970
\(423\) 7.87399 0.382847
\(424\) 4.93207 0.239523
\(425\) 6.31265 0.306209
\(426\) −0.926192 −0.0448741
\(427\) 0 0
\(428\) 0.932071 0.0450534
\(429\) 4.77575 0.230575
\(430\) −6.11142 −0.294719
\(431\) 16.3127 0.785753 0.392876 0.919591i \(-0.371480\pi\)
0.392876 + 0.919591i \(0.371480\pi\)
\(432\) −18.7612 −0.902647
\(433\) −11.1432 −0.535506 −0.267753 0.963488i \(-0.586281\pi\)
−0.267753 + 0.963488i \(0.586281\pi\)
\(434\) 0 0
\(435\) −2.91160 −0.139601
\(436\) −0.538319 −0.0257808
\(437\) −4.96239 −0.237383
\(438\) −0.463096 −0.0221276
\(439\) −27.3865 −1.30708 −0.653542 0.756890i \(-0.726718\pi\)
−0.653542 + 0.756890i \(0.726718\pi\)
\(440\) −2.57452 −0.122735
\(441\) 0 0
\(442\) 57.5633 2.73800
\(443\) 19.5125 0.927065 0.463533 0.886080i \(-0.346582\pi\)
0.463533 + 0.886080i \(0.346582\pi\)
\(444\) −1.58769 −0.0753484
\(445\) 7.08840 0.336022
\(446\) −0.0449056 −0.00212634
\(447\) 5.98541 0.283100
\(448\) 0 0
\(449\) −22.1016 −1.04304 −0.521519 0.853240i \(-0.674634\pi\)
−0.521519 + 0.853240i \(0.674634\pi\)
\(450\) 3.48119 0.164105
\(451\) −6.07522 −0.286071
\(452\) 1.35614 0.0637875
\(453\) 1.29948 0.0610547
\(454\) 7.11871 0.334098
\(455\) 0 0
\(456\) −2.15633 −0.100979
\(457\) −17.8496 −0.834967 −0.417483 0.908685i \(-0.637088\pi\)
−0.417483 + 0.908685i \(0.637088\pi\)
\(458\) 2.77575 0.129702
\(459\) 27.2243 1.27072
\(460\) 0.962389 0.0448716
\(461\) 35.2506 1.64178 0.820892 0.571083i \(-0.193477\pi\)
0.820892 + 0.571083i \(0.193477\pi\)
\(462\) 0 0
\(463\) −26.3634 −1.22521 −0.612606 0.790388i \(-0.709879\pi\)
−0.612606 + 0.790388i \(0.709879\pi\)
\(464\) 15.7137 0.729490
\(465\) 4.77575 0.221470
\(466\) 16.5139 0.764991
\(467\) 6.78560 0.314000 0.157000 0.987599i \(-0.449818\pi\)
0.157000 + 0.987599i \(0.449818\pi\)
\(468\) 2.80606 0.129710
\(469\) 0 0
\(470\) −4.96239 −0.228898
\(471\) 3.53690 0.162972
\(472\) 17.0884 0.786557
\(473\) −3.97082 −0.182579
\(474\) −0.998585 −0.0458665
\(475\) 1.00000 0.0458831
\(476\) 0 0
\(477\) −4.33312 −0.198400
\(478\) −13.7743 −0.630023
\(479\) −12.7104 −0.580752 −0.290376 0.956913i \(-0.593780\pi\)
−0.290376 + 0.956913i \(0.593780\pi\)
\(480\) 0.881286 0.0402250
\(481\) −62.5256 −2.85092
\(482\) −3.42548 −0.156027
\(483\) 0 0
\(484\) −1.95368 −0.0888036
\(485\) 10.9927 0.499153
\(486\) 23.4314 1.06287
\(487\) −15.7586 −0.714090 −0.357045 0.934087i \(-0.616216\pi\)
−0.357045 + 0.934087i \(0.616216\pi\)
\(488\) 30.1622 1.36538
\(489\) 0.523730 0.0236839
\(490\) 0 0
\(491\) −14.5501 −0.656636 −0.328318 0.944567i \(-0.606482\pi\)
−0.328318 + 0.944567i \(0.606482\pi\)
\(492\) 0.986826 0.0444896
\(493\) −22.8021 −1.02695
\(494\) 9.11871 0.410270
\(495\) 2.26187 0.101663
\(496\) −25.7743 −1.15730
\(497\) 0 0
\(498\) 8.40246 0.376523
\(499\) −5.48612 −0.245592 −0.122796 0.992432i \(-0.539186\pi\)
−0.122796 + 0.992432i \(0.539186\pi\)
\(500\) −0.193937 −0.00867311
\(501\) −12.3780 −0.553009
\(502\) 35.6991 1.59333
\(503\) 36.6615 1.63466 0.817328 0.576173i \(-0.195455\pi\)
0.817328 + 0.576173i \(0.195455\pi\)
\(504\) 0 0
\(505\) −2.64974 −0.117912
\(506\) 7.07381 0.314469
\(507\) −20.0713 −0.891396
\(508\) 2.60483 0.115571
\(509\) −39.1900 −1.73706 −0.868532 0.495632i \(-0.834936\pi\)
−0.868532 + 0.495632i \(0.834936\pi\)
\(510\) −7.53690 −0.333740
\(511\) 0 0
\(512\) 18.5188 0.818423
\(513\) 4.31265 0.190408
\(514\) −19.7294 −0.870228
\(515\) −10.7308 −0.472857
\(516\) 0.644999 0.0283945
\(517\) −3.22425 −0.141803
\(518\) 0 0
\(519\) 2.61545 0.114806
\(520\) 16.4690 0.722212
\(521\) 17.7283 0.776690 0.388345 0.921514i \(-0.373047\pi\)
0.388345 + 0.921514i \(0.373047\pi\)
\(522\) −12.5745 −0.550372
\(523\) 40.7572 1.78219 0.891094 0.453819i \(-0.149939\pi\)
0.891094 + 0.453819i \(0.149939\pi\)
\(524\) −4.00000 −0.174741
\(525\) 0 0
\(526\) −19.1998 −0.837152
\(527\) 37.4010 1.62922
\(528\) 3.37470 0.146865
\(529\) 1.62530 0.0706652
\(530\) 2.73084 0.118620
\(531\) −15.0132 −0.651516
\(532\) 0 0
\(533\) 38.8627 1.68333
\(534\) −8.46310 −0.366234
\(535\) −4.80606 −0.207784
\(536\) −18.0059 −0.777736
\(537\) −12.1016 −0.522221
\(538\) −16.8872 −0.728057
\(539\) 0 0
\(540\) −0.836381 −0.0359921
\(541\) 23.9003 1.02756 0.513778 0.857923i \(-0.328246\pi\)
0.513778 + 0.857923i \(0.328246\pi\)
\(542\) 24.9018 1.06962
\(543\) 7.43533 0.319081
\(544\) 6.90175 0.295910
\(545\) 2.77575 0.118900
\(546\) 0 0
\(547\) 8.55405 0.365745 0.182872 0.983137i \(-0.441461\pi\)
0.182872 + 0.983137i \(0.441461\pi\)
\(548\) 3.92478 0.167658
\(549\) −26.4993 −1.13096
\(550\) −1.42548 −0.0607829
\(551\) −3.61213 −0.153882
\(552\) 10.7005 0.455445
\(553\) 0 0
\(554\) 44.1232 1.87461
\(555\) 8.18664 0.347504
\(556\) 3.41090 0.144654
\(557\) −4.23743 −0.179546 −0.0897728 0.995962i \(-0.528614\pi\)
−0.0897728 + 0.995962i \(0.528614\pi\)
\(558\) 20.6253 0.873139
\(559\) 25.4010 1.07435
\(560\) 0 0
\(561\) −4.89701 −0.206752
\(562\) 17.1998 0.725530
\(563\) −16.4934 −0.695114 −0.347557 0.937659i \(-0.612989\pi\)
−0.347557 + 0.937659i \(0.612989\pi\)
\(564\) 0.523730 0.0220530
\(565\) −6.99271 −0.294185
\(566\) 3.35026 0.140822
\(567\) 0 0
\(568\) −2.07522 −0.0870744
\(569\) −10.0000 −0.419222 −0.209611 0.977785i \(-0.567220\pi\)
−0.209611 + 0.977785i \(0.567220\pi\)
\(570\) −1.19394 −0.0500085
\(571\) −26.2619 −1.09902 −0.549512 0.835486i \(-0.685186\pi\)
−0.549512 + 0.835486i \(0.685186\pi\)
\(572\) −1.14903 −0.0480434
\(573\) 17.5515 0.733224
\(574\) 0 0
\(575\) −4.96239 −0.206946
\(576\) −16.6424 −0.693435
\(577\) −30.1016 −1.25314 −0.626572 0.779363i \(-0.715543\pi\)
−0.626572 + 0.779363i \(0.715543\pi\)
\(578\) −33.8446 −1.40775
\(579\) −10.1114 −0.420216
\(580\) 0.700523 0.0290877
\(581\) 0 0
\(582\) −13.1246 −0.544032
\(583\) 1.77433 0.0734853
\(584\) −1.03761 −0.0429367
\(585\) −14.4690 −0.598219
\(586\) 2.73084 0.112810
\(587\) 35.1392 1.45035 0.725175 0.688565i \(-0.241759\pi\)
0.725175 + 0.688565i \(0.241759\pi\)
\(588\) 0 0
\(589\) 5.92478 0.244126
\(590\) 9.46168 0.389531
\(591\) −19.7889 −0.814007
\(592\) −44.1827 −1.81590
\(593\) −34.3244 −1.40953 −0.704767 0.709439i \(-0.748949\pi\)
−0.704767 + 0.709439i \(0.748949\pi\)
\(594\) −6.14762 −0.252240
\(595\) 0 0
\(596\) −1.44007 −0.0589877
\(597\) 18.5990 0.761204
\(598\) −45.2506 −1.85043
\(599\) 14.6107 0.596978 0.298489 0.954413i \(-0.403517\pi\)
0.298489 + 0.954413i \(0.403517\pi\)
\(600\) −2.15633 −0.0880316
\(601\) −23.5633 −0.961165 −0.480583 0.876949i \(-0.659575\pi\)
−0.480583 + 0.876949i \(0.659575\pi\)
\(602\) 0 0
\(603\) 15.8192 0.644209
\(604\) −0.312650 −0.0127216
\(605\) 10.0738 0.409559
\(606\) 3.16362 0.128513
\(607\) 8.80606 0.357427 0.178714 0.983901i \(-0.442806\pi\)
0.178714 + 0.983901i \(0.442806\pi\)
\(608\) 1.09332 0.0443400
\(609\) 0 0
\(610\) 16.7005 0.676184
\(611\) 20.6253 0.834410
\(612\) −2.87732 −0.116309
\(613\) 10.4142 0.420626 0.210313 0.977634i \(-0.432552\pi\)
0.210313 + 0.977634i \(0.432552\pi\)
\(614\) 38.8178 1.56656
\(615\) −5.08840 −0.205184
\(616\) 0 0
\(617\) −17.2849 −0.695863 −0.347932 0.937520i \(-0.613116\pi\)
−0.347932 + 0.937520i \(0.613116\pi\)
\(618\) 12.8119 0.515372
\(619\) 10.6351 0.427463 0.213731 0.976892i \(-0.431438\pi\)
0.213731 + 0.976892i \(0.431438\pi\)
\(620\) −1.14903 −0.0461462
\(621\) −21.4010 −0.858794
\(622\) 9.64832 0.386863
\(623\) 0 0
\(624\) −21.5877 −0.864199
\(625\) 1.00000 0.0400000
\(626\) 23.8105 0.951660
\(627\) −0.775746 −0.0309803
\(628\) −0.850969 −0.0339574
\(629\) 64.1133 2.55637
\(630\) 0 0
\(631\) −16.5599 −0.659240 −0.329620 0.944114i \(-0.606921\pi\)
−0.329620 + 0.944114i \(0.606921\pi\)
\(632\) −2.23743 −0.0890001
\(633\) 16.8773 0.670813
\(634\) 8.43278 0.334908
\(635\) −13.4314 −0.533007
\(636\) −0.288213 −0.0114284
\(637\) 0 0
\(638\) 5.14903 0.203852
\(639\) 1.82321 0.0721249
\(640\) 12.6751 0.501029
\(641\) −16.7612 −0.662026 −0.331013 0.943626i \(-0.607390\pi\)
−0.331013 + 0.943626i \(0.607390\pi\)
\(642\) 5.73813 0.226466
\(643\) −5.73813 −0.226290 −0.113145 0.993578i \(-0.536092\pi\)
−0.113145 + 0.993578i \(0.536092\pi\)
\(644\) 0 0
\(645\) −3.32582 −0.130954
\(646\) −9.35026 −0.367881
\(647\) 37.2144 1.46305 0.731525 0.681815i \(-0.238809\pi\)
0.731525 + 0.681815i \(0.238809\pi\)
\(648\) 9.56230 0.375642
\(649\) 6.14762 0.241315
\(650\) 9.11871 0.357665
\(651\) 0 0
\(652\) −0.126008 −0.00493485
\(653\) 11.7626 0.460305 0.230153 0.973155i \(-0.426077\pi\)
0.230153 + 0.973155i \(0.426077\pi\)
\(654\) −3.31406 −0.129590
\(655\) 20.6253 0.805897
\(656\) 27.4617 1.07220
\(657\) 0.911603 0.0355650
\(658\) 0 0
\(659\) 1.23884 0.0482584 0.0241292 0.999709i \(-0.492319\pi\)
0.0241292 + 0.999709i \(0.492319\pi\)
\(660\) 0.150446 0.00585609
\(661\) 9.53690 0.370943 0.185471 0.982650i \(-0.440619\pi\)
0.185471 + 0.982650i \(0.440619\pi\)
\(662\) 18.2374 0.708818
\(663\) 31.3258 1.21659
\(664\) 18.8265 0.730611
\(665\) 0 0
\(666\) 35.3561 1.37002
\(667\) 17.9248 0.694050
\(668\) 2.97812 0.115227
\(669\) −0.0244376 −0.000944811 0
\(670\) −9.96968 −0.385162
\(671\) 10.8510 0.418897
\(672\) 0 0
\(673\) 39.9307 1.53921 0.769607 0.638518i \(-0.220452\pi\)
0.769607 + 0.638518i \(0.220452\pi\)
\(674\) −5.58181 −0.215003
\(675\) 4.31265 0.165994
\(676\) 4.82909 0.185734
\(677\) 3.05334 0.117349 0.0586747 0.998277i \(-0.481313\pi\)
0.0586747 + 0.998277i \(0.481313\pi\)
\(678\) 8.34885 0.320636
\(679\) 0 0
\(680\) −16.8872 −0.647593
\(681\) 3.87399 0.148452
\(682\) −8.44568 −0.323402
\(683\) 29.2692 1.11995 0.559977 0.828508i \(-0.310810\pi\)
0.559977 + 0.828508i \(0.310810\pi\)
\(684\) −0.455802 −0.0174280
\(685\) −20.2374 −0.773232
\(686\) 0 0
\(687\) 1.51056 0.0576313
\(688\) 17.9492 0.684307
\(689\) −11.3503 −0.432411
\(690\) 5.92478 0.225552
\(691\) −2.63515 −0.100246 −0.0501229 0.998743i \(-0.515961\pi\)
−0.0501229 + 0.998743i \(0.515961\pi\)
\(692\) −0.629270 −0.0239213
\(693\) 0 0
\(694\) 33.1246 1.25739
\(695\) −17.5877 −0.667139
\(696\) 7.78892 0.295238
\(697\) −39.8496 −1.50941
\(698\) −14.8119 −0.560640
\(699\) 8.98683 0.339913
\(700\) 0 0
\(701\) −25.0494 −0.946102 −0.473051 0.881035i \(-0.656847\pi\)
−0.473051 + 0.881035i \(0.656847\pi\)
\(702\) 39.3258 1.48426
\(703\) 10.1563 0.383053
\(704\) 6.81477 0.256841
\(705\) −2.70052 −0.101708
\(706\) 8.20123 0.308657
\(707\) 0 0
\(708\) −0.998585 −0.0375291
\(709\) −41.6991 −1.56604 −0.783021 0.621995i \(-0.786323\pi\)
−0.783021 + 0.621995i \(0.786323\pi\)
\(710\) −1.14903 −0.0431224
\(711\) 1.96571 0.0737200
\(712\) −18.9624 −0.710646
\(713\) −29.4010 −1.10108
\(714\) 0 0
\(715\) 5.92478 0.221574
\(716\) 2.91160 0.108812
\(717\) −7.49597 −0.279942
\(718\) 15.3503 0.572867
\(719\) −30.6351 −1.14250 −0.571249 0.820777i \(-0.693541\pi\)
−0.571249 + 0.820777i \(0.693541\pi\)
\(720\) −10.2243 −0.381035
\(721\) 0 0
\(722\) −1.48119 −0.0551243
\(723\) −1.86414 −0.0693282
\(724\) −1.78892 −0.0664847
\(725\) −3.61213 −0.134151
\(726\) −12.0275 −0.446382
\(727\) −7.50071 −0.278186 −0.139093 0.990279i \(-0.544419\pi\)
−0.139093 + 0.990279i \(0.544419\pi\)
\(728\) 0 0
\(729\) 2.02776 0.0751023
\(730\) −0.574515 −0.0212638
\(731\) −26.0460 −0.963348
\(732\) −1.76257 −0.0651466
\(733\) −9.84955 −0.363802 −0.181901 0.983317i \(-0.558225\pi\)
−0.181901 + 0.983317i \(0.558225\pi\)
\(734\) 4.96239 0.183165
\(735\) 0 0
\(736\) −5.42548 −0.199986
\(737\) −6.47768 −0.238609
\(738\) −21.9756 −0.808932
\(739\) −20.0000 −0.735712 −0.367856 0.929883i \(-0.619908\pi\)
−0.367856 + 0.929883i \(0.619908\pi\)
\(740\) −1.96968 −0.0724070
\(741\) 4.96239 0.182298
\(742\) 0 0
\(743\) −15.7177 −0.576625 −0.288313 0.957536i \(-0.593094\pi\)
−0.288313 + 0.957536i \(0.593094\pi\)
\(744\) −12.7757 −0.468382
\(745\) 7.42548 0.272049
\(746\) 18.6702 0.683565
\(747\) −16.5402 −0.605175
\(748\) 1.17821 0.0430795
\(749\) 0 0
\(750\) −1.19394 −0.0435964
\(751\) −43.7400 −1.59610 −0.798048 0.602593i \(-0.794134\pi\)
−0.798048 + 0.602593i \(0.794134\pi\)
\(752\) 14.5745 0.531478
\(753\) 19.4274 0.707974
\(754\) −32.9380 −1.19953
\(755\) 1.61213 0.0586713
\(756\) 0 0
\(757\) −15.7743 −0.573328 −0.286664 0.958031i \(-0.592546\pi\)
−0.286664 + 0.958031i \(0.592546\pi\)
\(758\) 55.1754 2.00406
\(759\) 3.84955 0.139730
\(760\) −2.67513 −0.0970372
\(761\) −43.2262 −1.56695 −0.783474 0.621425i \(-0.786554\pi\)
−0.783474 + 0.621425i \(0.786554\pi\)
\(762\) 16.0362 0.580930
\(763\) 0 0
\(764\) −4.22284 −0.152777
\(765\) 14.8364 0.536410
\(766\) 45.6688 1.65008
\(767\) −39.3258 −1.41997
\(768\) −3.71767 −0.134150
\(769\) 19.1246 0.689650 0.344825 0.938667i \(-0.387938\pi\)
0.344825 + 0.938667i \(0.387938\pi\)
\(770\) 0 0
\(771\) −10.7367 −0.386674
\(772\) 2.43278 0.0875576
\(773\) −25.8846 −0.931005 −0.465502 0.885047i \(-0.654126\pi\)
−0.465502 + 0.885047i \(0.654126\pi\)
\(774\) −14.3634 −0.516283
\(775\) 5.92478 0.212824
\(776\) −29.4069 −1.05565
\(777\) 0 0
\(778\) 2.03620 0.0730012
\(779\) −6.31265 −0.226174
\(780\) −0.962389 −0.0344590
\(781\) −0.746569 −0.0267144
\(782\) 46.3996 1.65925
\(783\) −15.5778 −0.556707
\(784\) 0 0
\(785\) 4.38787 0.156610
\(786\) −24.6253 −0.878355
\(787\) −19.9814 −0.712261 −0.356131 0.934436i \(-0.615904\pi\)
−0.356131 + 0.934436i \(0.615904\pi\)
\(788\) 4.76116 0.169609
\(789\) −10.4485 −0.371977
\(790\) −1.23884 −0.0440760
\(791\) 0 0
\(792\) −6.05079 −0.215005
\(793\) −69.4128 −2.46492
\(794\) −13.9032 −0.493405
\(795\) 1.48612 0.0527072
\(796\) −4.47486 −0.158607
\(797\) −28.6458 −1.01469 −0.507343 0.861744i \(-0.669372\pi\)
−0.507343 + 0.861744i \(0.669372\pi\)
\(798\) 0 0
\(799\) −21.1490 −0.748199
\(800\) 1.09332 0.0386547
\(801\) 16.6596 0.588638
\(802\) −20.8872 −0.737551
\(803\) −0.373285 −0.0131729
\(804\) 1.05220 0.0371082
\(805\) 0 0
\(806\) 54.0263 1.90300
\(807\) −9.18997 −0.323502
\(808\) 7.08840 0.249369
\(809\) −17.2243 −0.605573 −0.302786 0.953058i \(-0.597917\pi\)
−0.302786 + 0.953058i \(0.597917\pi\)
\(810\) 5.29455 0.186032
\(811\) 15.6267 0.548728 0.274364 0.961626i \(-0.411533\pi\)
0.274364 + 0.961626i \(0.411533\pi\)
\(812\) 0 0
\(813\) 13.5515 0.475272
\(814\) −14.4777 −0.507443
\(815\) 0.649738 0.0227593
\(816\) 22.1359 0.774910
\(817\) −4.12601 −0.144351
\(818\) 52.0625 1.82032
\(819\) 0 0
\(820\) 1.22425 0.0427528
\(821\) 39.2506 1.36986 0.684928 0.728611i \(-0.259834\pi\)
0.684928 + 0.728611i \(0.259834\pi\)
\(822\) 24.1622 0.842754
\(823\) −45.5271 −1.58697 −0.793487 0.608588i \(-0.791736\pi\)
−0.793487 + 0.608588i \(0.791736\pi\)
\(824\) 28.7064 1.00003
\(825\) −0.775746 −0.0270080
\(826\) 0 0
\(827\) 17.0698 0.593576 0.296788 0.954943i \(-0.404084\pi\)
0.296788 + 0.954943i \(0.404084\pi\)
\(828\) 2.26187 0.0786052
\(829\) −1.69911 −0.0590125 −0.0295062 0.999565i \(-0.509393\pi\)
−0.0295062 + 0.999565i \(0.509393\pi\)
\(830\) 10.4241 0.361825
\(831\) 24.0118 0.832959
\(832\) −43.5936 −1.51134
\(833\) 0 0
\(834\) 20.9986 0.727122
\(835\) −15.3561 −0.531421
\(836\) 0.186642 0.00645516
\(837\) 25.5515 0.883189
\(838\) 13.3719 0.461924
\(839\) 50.5910 1.74660 0.873298 0.487187i \(-0.161977\pi\)
0.873298 + 0.487187i \(0.161977\pi\)
\(840\) 0 0
\(841\) −15.9525 −0.550088
\(842\) −22.5285 −0.776382
\(843\) 9.36011 0.322379
\(844\) −4.06063 −0.139773
\(845\) −24.9003 −0.856598
\(846\) −11.6629 −0.400979
\(847\) 0 0
\(848\) −8.02047 −0.275424
\(849\) 1.82321 0.0625723
\(850\) −9.35026 −0.320711
\(851\) −50.3996 −1.72768
\(852\) 0.121269 0.00415460
\(853\) 22.5237 0.771198 0.385599 0.922667i \(-0.373995\pi\)
0.385599 + 0.922667i \(0.373995\pi\)
\(854\) 0 0
\(855\) 2.35026 0.0803773
\(856\) 12.8568 0.439438
\(857\) 23.6180 0.806776 0.403388 0.915029i \(-0.367833\pi\)
0.403388 + 0.915029i \(0.367833\pi\)
\(858\) −7.07381 −0.241496
\(859\) −15.1754 −0.517777 −0.258889 0.965907i \(-0.583356\pi\)
−0.258889 + 0.965907i \(0.583356\pi\)
\(860\) 0.800184 0.0272860
\(861\) 0 0
\(862\) −24.1622 −0.822968
\(863\) 30.1055 1.02480 0.512402 0.858746i \(-0.328756\pi\)
0.512402 + 0.858746i \(0.328756\pi\)
\(864\) 4.71511 0.160411
\(865\) 3.24472 0.110324
\(866\) 16.5052 0.560869
\(867\) −18.4182 −0.625515
\(868\) 0 0
\(869\) −0.804923 −0.0273051
\(870\) 4.31265 0.146213
\(871\) 41.4372 1.40405
\(872\) −7.42548 −0.251459
\(873\) 25.8357 0.874407
\(874\) 7.35026 0.248626
\(875\) 0 0
\(876\) 0.0606343 0.00204864
\(877\) −5.53102 −0.186769 −0.0933847 0.995630i \(-0.529769\pi\)
−0.0933847 + 0.995630i \(0.529769\pi\)
\(878\) 40.5647 1.36899
\(879\) 1.48612 0.0501255
\(880\) 4.18664 0.141132
\(881\) −20.8265 −0.701664 −0.350832 0.936438i \(-0.614101\pi\)
−0.350832 + 0.936438i \(0.614101\pi\)
\(882\) 0 0
\(883\) 43.1509 1.45214 0.726072 0.687618i \(-0.241344\pi\)
0.726072 + 0.687618i \(0.241344\pi\)
\(884\) −7.53690 −0.253494
\(885\) 5.14903 0.173083
\(886\) −28.9018 −0.970973
\(887\) 44.5461 1.49571 0.747856 0.663861i \(-0.231083\pi\)
0.747856 + 0.663861i \(0.231083\pi\)
\(888\) −21.9003 −0.734927
\(889\) 0 0
\(890\) −10.4993 −0.351937
\(891\) 3.44007 0.115247
\(892\) 0.00587961 0.000196864 0
\(893\) −3.35026 −0.112112
\(894\) −8.86556 −0.296509
\(895\) −15.0132 −0.501835
\(896\) 0 0
\(897\) −24.6253 −0.822215
\(898\) 32.7367 1.09244
\(899\) −21.4010 −0.713765
\(900\) −0.455802 −0.0151934
\(901\) 11.6385 0.387734
\(902\) 8.99859 0.299620
\(903\) 0 0
\(904\) 18.7064 0.622166
\(905\) 9.22425 0.306625
\(906\) −1.92478 −0.0639464
\(907\) 53.7558 1.78493 0.892466 0.451115i \(-0.148974\pi\)
0.892466 + 0.451115i \(0.148974\pi\)
\(908\) −0.932071 −0.0309319
\(909\) −6.22758 −0.206556
\(910\) 0 0
\(911\) −2.28630 −0.0757486 −0.0378743 0.999283i \(-0.512059\pi\)
−0.0378743 + 0.999283i \(0.512059\pi\)
\(912\) 3.50659 0.116115
\(913\) 6.77292 0.224151
\(914\) 26.4387 0.874513
\(915\) 9.08840 0.300453
\(916\) −0.363436 −0.0120082
\(917\) 0 0
\(918\) −40.3244 −1.33090
\(919\) −34.8510 −1.14963 −0.574814 0.818284i \(-0.694925\pi\)
−0.574814 + 0.818284i \(0.694925\pi\)
\(920\) 13.2750 0.437665
\(921\) 21.1246 0.696079
\(922\) −52.2130 −1.71954
\(923\) 4.77575 0.157196
\(924\) 0 0
\(925\) 10.1563 0.333938
\(926\) 39.0494 1.28324
\(927\) −25.2203 −0.828343
\(928\) −3.94921 −0.129639
\(929\) −41.6991 −1.36810 −0.684052 0.729434i \(-0.739784\pi\)
−0.684052 + 0.729434i \(0.739784\pi\)
\(930\) −7.07381 −0.231959
\(931\) 0 0
\(932\) −2.16220 −0.0708254
\(933\) 5.25060 0.171897
\(934\) −10.0508 −0.328872
\(935\) −6.07522 −0.198681
\(936\) 38.7064 1.26516
\(937\) −21.9102 −0.715775 −0.357887 0.933765i \(-0.616503\pi\)
−0.357887 + 0.933765i \(0.616503\pi\)
\(938\) 0 0
\(939\) 12.9576 0.422857
\(940\) 0.649738 0.0211921
\(941\) 3.55149 0.115775 0.0578877 0.998323i \(-0.481563\pi\)
0.0578877 + 0.998323i \(0.481563\pi\)
\(942\) −5.23884 −0.170691
\(943\) 31.3258 1.02011
\(944\) −27.7889 −0.904452
\(945\) 0 0
\(946\) 5.88156 0.191226
\(947\) 38.3634 1.24664 0.623322 0.781965i \(-0.285783\pi\)
0.623322 + 0.781965i \(0.285783\pi\)
\(948\) 0.130747 0.00424648
\(949\) 2.38787 0.0775136
\(950\) −1.48119 −0.0480563
\(951\) 4.58910 0.148812
\(952\) 0 0
\(953\) −22.8714 −0.740879 −0.370439 0.928857i \(-0.620793\pi\)
−0.370439 + 0.928857i \(0.620793\pi\)
\(954\) 6.41819 0.207797
\(955\) 21.7743 0.704601
\(956\) 1.80351 0.0583296
\(957\) 2.80209 0.0905788
\(958\) 18.8265 0.608258
\(959\) 0 0
\(960\) 5.70782 0.184219
\(961\) 4.10299 0.132354
\(962\) 92.6126 2.98595
\(963\) −11.2955 −0.363993
\(964\) 0.448507 0.0144455
\(965\) −12.5442 −0.403812
\(966\) 0 0
\(967\) 27.6629 0.889579 0.444790 0.895635i \(-0.353278\pi\)
0.444790 + 0.895635i \(0.353278\pi\)
\(968\) −26.9488 −0.866166
\(969\) −5.08840 −0.163463
\(970\) −16.2823 −0.522794
\(971\) 42.1768 1.35352 0.676759 0.736205i \(-0.263384\pi\)
0.676759 + 0.736205i \(0.263384\pi\)
\(972\) −3.06793 −0.0984039
\(973\) 0 0
\(974\) 23.3416 0.747912
\(975\) 4.96239 0.158924
\(976\) −49.0494 −1.57003
\(977\) 0.856849 0.0274130 0.0137065 0.999906i \(-0.495637\pi\)
0.0137065 + 0.999906i \(0.495637\pi\)
\(978\) −0.775746 −0.0248056
\(979\) −6.82179 −0.218025
\(980\) 0 0
\(981\) 6.52373 0.208287
\(982\) 21.5515 0.687736
\(983\) −45.2809 −1.44424 −0.722119 0.691769i \(-0.756831\pi\)
−0.722119 + 0.691769i \(0.756831\pi\)
\(984\) 13.6121 0.433939
\(985\) −24.5501 −0.782231
\(986\) 33.7743 1.07559
\(987\) 0 0
\(988\) −1.19394 −0.0379842
\(989\) 20.4749 0.651063
\(990\) −3.35026 −0.106478
\(991\) 47.0132 1.49342 0.746711 0.665148i \(-0.231632\pi\)
0.746711 + 0.665148i \(0.231632\pi\)
\(992\) 6.47768 0.205667
\(993\) 9.92478 0.314953
\(994\) 0 0
\(995\) 23.0738 0.731489
\(996\) −1.10016 −0.0348598
\(997\) −13.6873 −0.433483 −0.216741 0.976229i \(-0.569543\pi\)
−0.216741 + 0.976229i \(0.569543\pi\)
\(998\) 8.12601 0.257224
\(999\) 43.8007 1.38579
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 4655.2.a.u.1.1 3
7.6 odd 2 95.2.a.a.1.1 3
21.20 even 2 855.2.a.i.1.3 3
28.27 even 2 1520.2.a.p.1.2 3
35.13 even 4 475.2.b.d.324.5 6
35.27 even 4 475.2.b.d.324.2 6
35.34 odd 2 475.2.a.f.1.3 3
56.13 odd 2 6080.2.a.bo.1.2 3
56.27 even 2 6080.2.a.by.1.2 3
105.104 even 2 4275.2.a.bk.1.1 3
133.132 even 2 1805.2.a.f.1.3 3
140.139 even 2 7600.2.a.bx.1.2 3
665.664 even 2 9025.2.a.bb.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
95.2.a.a.1.1 3 7.6 odd 2
475.2.a.f.1.3 3 35.34 odd 2
475.2.b.d.324.2 6 35.27 even 4
475.2.b.d.324.5 6 35.13 even 4
855.2.a.i.1.3 3 21.20 even 2
1520.2.a.p.1.2 3 28.27 even 2
1805.2.a.f.1.3 3 133.132 even 2
4275.2.a.bk.1.1 3 105.104 even 2
4655.2.a.u.1.1 3 1.1 even 1 trivial
6080.2.a.bo.1.2 3 56.13 odd 2
6080.2.a.by.1.2 3 56.27 even 2
7600.2.a.bx.1.2 3 140.139 even 2
9025.2.a.bb.1.1 3 665.664 even 2