Properties

Label 6080.2.a.by.1.2
Level $6080$
Weight $2$
Character 6080.1
Self dual yes
Analytic conductor $48.549$
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] = [6080,2,Mod(1,6080)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6080, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("6080.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6080 = 2^{6} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6080.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: \(48.5490444289\)
Analytic rank: \(0\)
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 95)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-1.48119\) of defining polynomial
Character \(\chi\) \(=\) 6080.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+0.806063 q^{3} -1.00000 q^{5} -3.35026 q^{7} -2.35026 q^{9} +O(q^{10})\) \(q+0.806063 q^{3} -1.00000 q^{5} -3.35026 q^{7} -2.35026 q^{9} +0.962389 q^{11} -6.15633 q^{13} -0.806063 q^{15} -6.31265 q^{17} -1.00000 q^{19} -2.70052 q^{21} +4.96239 q^{23} +1.00000 q^{25} -4.31265 q^{27} +3.61213 q^{29} +5.92478 q^{31} +0.775746 q^{33} +3.35026 q^{35} -10.1563 q^{37} -4.96239 q^{39} +6.31265 q^{41} -4.12601 q^{43} +2.35026 q^{45} -3.35026 q^{47} +4.22425 q^{49} -5.08840 q^{51} -1.84367 q^{53} -0.962389 q^{55} -0.806063 q^{57} -6.38787 q^{59} +11.2750 q^{61} +7.87399 q^{63} +6.15633 q^{65} -6.73084 q^{67} +4.00000 q^{69} +0.775746 q^{71} +0.387873 q^{73} +0.806063 q^{75} -3.22425 q^{77} +0.836381 q^{79} +3.57452 q^{81} -7.03761 q^{83} +6.31265 q^{85} +2.91160 q^{87} +7.08840 q^{89} +20.6253 q^{91} +4.77575 q^{93} +1.00000 q^{95} +10.9927 q^{97} -2.26187 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 2 q^{3} - 3 q^{5} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 2 q^{3} - 3 q^{5} + 3 q^{9} - 8 q^{11} - 8 q^{13} - 2 q^{15} + 2 q^{17} - 3 q^{19} + 12 q^{21} + 4 q^{23} + 3 q^{25} + 8 q^{27} + 10 q^{29} - 4 q^{31} + 4 q^{33} - 20 q^{37} - 4 q^{39} - 2 q^{41} - 4 q^{43} - 3 q^{45} + 11 q^{49} + 4 q^{51} - 16 q^{53} + 8 q^{55} - 2 q^{57} - 20 q^{59} + 2 q^{61} + 32 q^{63} + 8 q^{65} + 2 q^{67} + 12 q^{69} + 4 q^{71} + 2 q^{73} + 2 q^{75} - 8 q^{77} - q^{81} - 32 q^{83} - 2 q^{85} + 28 q^{87} + 2 q^{89} + 20 q^{91} + 16 q^{93} + 3 q^{95} + 20 q^{97} - 16 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0.806063 0.465381 0.232690 0.972551i \(-0.425247\pi\)
0.232690 + 0.972551i \(0.425247\pi\)
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −3.35026 −1.26628 −0.633140 0.774037i \(-0.718234\pi\)
−0.633140 + 0.774037i \(0.718234\pi\)
\(8\) 0 0
\(9\) −2.35026 −0.783421
\(10\) 0 0
\(11\) 0.962389 0.290171 0.145086 0.989419i \(-0.453654\pi\)
0.145086 + 0.989419i \(0.453654\pi\)
\(12\) 0 0
\(13\) −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
\(16\) 0 0
\(17\) −6.31265 −1.53104 −0.765521 0.643411i \(-0.777519\pi\)
−0.765521 + 0.643411i \(0.777519\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) −2.70052 −0.589303
\(22\) 0 0
\(23\) 4.96239 1.03473 0.517365 0.855765i \(-0.326913\pi\)
0.517365 + 0.855765i \(0.326913\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −4.31265 −0.829970
\(28\) 0 0
\(29\) 3.61213 0.670755 0.335378 0.942084i \(-0.391136\pi\)
0.335378 + 0.942084i \(0.391136\pi\)
\(30\) 0 0
\(31\) 5.92478 1.06412 0.532061 0.846706i \(-0.321418\pi\)
0.532061 + 0.846706i \(0.321418\pi\)
\(32\) 0 0
\(33\) 0.775746 0.135040
\(34\) 0 0
\(35\) 3.35026 0.566298
\(36\) 0 0
\(37\) −10.1563 −1.66969 −0.834845 0.550485i \(-0.814443\pi\)
−0.834845 + 0.550485i \(0.814443\pi\)
\(38\) 0 0
\(39\) −4.96239 −0.794618
\(40\) 0 0
\(41\) 6.31265 0.985870 0.492935 0.870066i \(-0.335924\pi\)
0.492935 + 0.870066i \(0.335924\pi\)
\(42\) 0 0
\(43\) −4.12601 −0.629210 −0.314605 0.949223i \(-0.601872\pi\)
−0.314605 + 0.949223i \(0.601872\pi\)
\(44\) 0 0
\(45\) 2.35026 0.350356
\(46\) 0 0
\(47\) −3.35026 −0.488686 −0.244343 0.969689i \(-0.578572\pi\)
−0.244343 + 0.969689i \(0.578572\pi\)
\(48\) 0 0
\(49\) 4.22425 0.603465
\(50\) 0 0
\(51\) −5.08840 −0.712518
\(52\) 0 0
\(53\) −1.84367 −0.253248 −0.126624 0.991951i \(-0.540414\pi\)
−0.126624 + 0.991951i \(0.540414\pi\)
\(54\) 0 0
\(55\) −0.962389 −0.129768
\(56\) 0 0
\(57\) −0.806063 −0.106766
\(58\) 0 0
\(59\) −6.38787 −0.831630 −0.415815 0.909449i \(-0.636504\pi\)
−0.415815 + 0.909449i \(0.636504\pi\)
\(60\) 0 0
\(61\) 11.2750 1.44362 0.721810 0.692091i \(-0.243310\pi\)
0.721810 + 0.692091i \(0.243310\pi\)
\(62\) 0 0
\(63\) 7.87399 0.992030
\(64\) 0 0
\(65\) 6.15633 0.763598
\(66\) 0 0
\(67\) −6.73084 −0.822303 −0.411152 0.911567i \(-0.634873\pi\)
−0.411152 + 0.911567i \(0.634873\pi\)
\(68\) 0 0
\(69\) 4.00000 0.481543
\(70\) 0 0
\(71\) 0.775746 0.0920641 0.0460321 0.998940i \(-0.485342\pi\)
0.0460321 + 0.998940i \(0.485342\pi\)
\(72\) 0 0
\(73\) 0.387873 0.0453971 0.0226986 0.999742i \(-0.492774\pi\)
0.0226986 + 0.999742i \(0.492774\pi\)
\(74\) 0 0
\(75\) 0.806063 0.0930762
\(76\) 0 0
\(77\) −3.22425 −0.367438
\(78\) 0 0
\(79\) 0.836381 0.0941002 0.0470501 0.998893i \(-0.485018\pi\)
0.0470501 + 0.998893i \(0.485018\pi\)
\(80\) 0 0
\(81\) 3.57452 0.397168
\(82\) 0 0
\(83\) −7.03761 −0.772478 −0.386239 0.922399i \(-0.626226\pi\)
−0.386239 + 0.922399i \(0.626226\pi\)
\(84\) 0 0
\(85\) 6.31265 0.684703
\(86\) 0 0
\(87\) 2.91160 0.312157
\(88\) 0 0
\(89\) 7.08840 0.751369 0.375684 0.926748i \(-0.377408\pi\)
0.375684 + 0.926748i \(0.377408\pi\)
\(90\) 0 0
\(91\) 20.6253 2.16212
\(92\) 0 0
\(93\) 4.77575 0.495222
\(94\) 0 0
\(95\) 1.00000 0.102598
\(96\) 0 0
\(97\) 10.9927 1.11614 0.558070 0.829794i \(-0.311542\pi\)
0.558070 + 0.829794i \(0.311542\pi\)
\(98\) 0 0
\(99\) −2.26187 −0.227326
\(100\) 0 0
\(101\) 2.64974 0.263659 0.131829 0.991272i \(-0.457915\pi\)
0.131829 + 0.991272i \(0.457915\pi\)
\(102\) 0 0
\(103\) 10.7308 1.05734 0.528671 0.848827i \(-0.322691\pi\)
0.528671 + 0.848827i \(0.322691\pi\)
\(104\) 0 0
\(105\) 2.70052 0.263544
\(106\) 0 0
\(107\) 4.80606 0.464620 0.232310 0.972642i \(-0.425372\pi\)
0.232310 + 0.972642i \(0.425372\pi\)
\(108\) 0 0
\(109\) 2.77575 0.265868 0.132934 0.991125i \(-0.457560\pi\)
0.132934 + 0.991125i \(0.457560\pi\)
\(110\) 0 0
\(111\) −8.18664 −0.777042
\(112\) 0 0
\(113\) 6.99271 0.657818 0.328909 0.944362i \(-0.393319\pi\)
0.328909 + 0.944362i \(0.393319\pi\)
\(114\) 0 0
\(115\) −4.96239 −0.462745
\(116\) 0 0
\(117\) 14.4690 1.33766
\(118\) 0 0
\(119\) 21.1490 1.93873
\(120\) 0 0
\(121\) −10.0738 −0.915801
\(122\) 0 0
\(123\) 5.08840 0.458805
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −13.4314 −1.19184 −0.595920 0.803043i \(-0.703213\pi\)
−0.595920 + 0.803043i \(0.703213\pi\)
\(128\) 0 0
\(129\) −3.32582 −0.292822
\(130\) 0 0
\(131\) 20.6253 1.80204 0.901020 0.433777i \(-0.142819\pi\)
0.901020 + 0.433777i \(0.142819\pi\)
\(132\) 0 0
\(133\) 3.35026 0.290505
\(134\) 0 0
\(135\) 4.31265 0.371174
\(136\) 0 0
\(137\) 20.2374 1.72900 0.864500 0.502633i \(-0.167635\pi\)
0.864500 + 0.502633i \(0.167635\pi\)
\(138\) 0 0
\(139\) −17.5877 −1.49177 −0.745884 0.666076i \(-0.767973\pi\)
−0.745884 + 0.666076i \(0.767973\pi\)
\(140\) 0 0
\(141\) −2.70052 −0.227425
\(142\) 0 0
\(143\) −5.92478 −0.495455
\(144\) 0 0
\(145\) −3.61213 −0.299971
\(146\) 0 0
\(147\) 3.40502 0.280841
\(148\) 0 0
\(149\) 7.42548 0.608319 0.304160 0.952621i \(-0.401624\pi\)
0.304160 + 0.952621i \(0.401624\pi\)
\(150\) 0 0
\(151\) 1.61213 0.131193 0.0655965 0.997846i \(-0.479105\pi\)
0.0655965 + 0.997846i \(0.479105\pi\)
\(152\) 0 0
\(153\) 14.8364 1.19945
\(154\) 0 0
\(155\) −5.92478 −0.475890
\(156\) 0 0
\(157\) −4.38787 −0.350190 −0.175095 0.984552i \(-0.556023\pi\)
−0.175095 + 0.984552i \(0.556023\pi\)
\(158\) 0 0
\(159\) −1.48612 −0.117857
\(160\) 0 0
\(161\) −16.6253 −1.31026
\(162\) 0 0
\(163\) −0.649738 −0.0508914 −0.0254457 0.999676i \(-0.508100\pi\)
−0.0254457 + 0.999676i \(0.508100\pi\)
\(164\) 0 0
\(165\) −0.775746 −0.0603918
\(166\) 0 0
\(167\) 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
\(170\) 0 0
\(171\) 2.35026 0.179729
\(172\) 0 0
\(173\) −3.24472 −0.246692 −0.123346 0.992364i \(-0.539362\pi\)
−0.123346 + 0.992364i \(0.539362\pi\)
\(174\) 0 0
\(175\) −3.35026 −0.253256
\(176\) 0 0
\(177\) −5.14903 −0.387025
\(178\) 0 0
\(179\) 15.0132 1.12214 0.561069 0.827769i \(-0.310390\pi\)
0.561069 + 0.827769i \(0.310390\pi\)
\(180\) 0 0
\(181\) −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
\(184\) 0 0
\(185\) 10.1563 0.746708
\(186\) 0 0
\(187\) −6.07522 −0.444264
\(188\) 0 0
\(189\) 14.4485 1.05097
\(190\) 0 0
\(191\) 21.7743 1.57554 0.787768 0.615972i \(-0.211237\pi\)
0.787768 + 0.615972i \(0.211237\pi\)
\(192\) 0 0
\(193\) 12.5442 0.902951 0.451476 0.892283i \(-0.350898\pi\)
0.451476 + 0.892283i \(0.350898\pi\)
\(194\) 0 0
\(195\) 4.96239 0.355364
\(196\) 0 0
\(197\) −24.5501 −1.74912 −0.874560 0.484917i \(-0.838850\pi\)
−0.874560 + 0.484917i \(0.838850\pi\)
\(198\) 0 0
\(199\) −23.0738 −1.63566 −0.817829 0.575461i \(-0.804823\pi\)
−0.817829 + 0.575461i \(0.804823\pi\)
\(200\) 0 0
\(201\) −5.42548 −0.382684
\(202\) 0 0
\(203\) −12.1016 −0.849364
\(204\) 0 0
\(205\) −6.31265 −0.440895
\(206\) 0 0
\(207\) −11.6629 −0.810628
\(208\) 0 0
\(209\) −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 0
\(213\) 0.625301 0.0428449
\(214\) 0 0
\(215\) 4.12601 0.281391
\(216\) 0 0
\(217\) −19.8496 −1.34748
\(218\) 0 0
\(219\) 0.312650 0.0211270
\(220\) 0 0
\(221\) 38.8627 2.61419
\(222\) 0 0
\(223\) 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
\(226\) 0 0
\(227\) 4.80606 0.318990 0.159495 0.987199i \(-0.449013\pi\)
0.159495 + 0.987199i \(0.449013\pi\)
\(228\) 0 0
\(229\) −1.87399 −0.123837 −0.0619184 0.998081i \(-0.519722\pi\)
−0.0619184 + 0.998081i \(0.519722\pi\)
\(230\) 0 0
\(231\) −2.59895 −0.170999
\(232\) 0 0
\(233\) −11.1490 −0.730397 −0.365199 0.930930i \(-0.618999\pi\)
−0.365199 + 0.930930i \(0.618999\pi\)
\(234\) 0 0
\(235\) 3.35026 0.218547
\(236\) 0 0
\(237\) 0.674176 0.0437924
\(238\) 0 0
\(239\) −9.29948 −0.601533 −0.300767 0.953698i \(-0.597243\pi\)
−0.300767 + 0.953698i \(0.597243\pi\)
\(240\) 0 0
\(241\) −2.31265 −0.148971 −0.0744855 0.997222i \(-0.523731\pi\)
−0.0744855 + 0.997222i \(0.523731\pi\)
\(242\) 0 0
\(243\) 15.8192 1.01480
\(244\) 0 0
\(245\) −4.22425 −0.269878
\(246\) 0 0
\(247\) 6.15633 0.391718
\(248\) 0 0
\(249\) −5.67276 −0.359497
\(250\) 0 0
\(251\) 24.1016 1.52128 0.760639 0.649175i \(-0.224886\pi\)
0.760639 + 0.649175i \(0.224886\pi\)
\(252\) 0 0
\(253\) 4.77575 0.300249
\(254\) 0 0
\(255\) 5.08840 0.318648
\(256\) 0 0
\(257\) −13.3199 −0.830875 −0.415438 0.909622i \(-0.636372\pi\)
−0.415438 + 0.909622i \(0.636372\pi\)
\(258\) 0 0
\(259\) 34.0263 2.11429
\(260\) 0 0
\(261\) −8.48944 −0.525483
\(262\) 0 0
\(263\) −12.9624 −0.799295 −0.399648 0.916669i \(-0.630867\pi\)
−0.399648 + 0.916669i \(0.630867\pi\)
\(264\) 0 0
\(265\) 1.84367 0.113256
\(266\) 0 0
\(267\) 5.71370 0.349673
\(268\) 0 0
\(269\) 11.4010 0.695134 0.347567 0.937655i \(-0.387008\pi\)
0.347567 + 0.937655i \(0.387008\pi\)
\(270\) 0 0
\(271\) −16.8119 −1.02125 −0.510626 0.859803i \(-0.670586\pi\)
−0.510626 + 0.859803i \(0.670586\pi\)
\(272\) 0 0
\(273\) 16.6253 1.00621
\(274\) 0 0
\(275\) 0.962389 0.0580342
\(276\) 0 0
\(277\) 29.7889 1.78984 0.894921 0.446224i \(-0.147231\pi\)
0.894921 + 0.446224i \(0.147231\pi\)
\(278\) 0 0
\(279\) −13.9248 −0.833655
\(280\) 0 0
\(281\) −11.6121 −0.692721 −0.346361 0.938101i \(-0.612583\pi\)
−0.346361 + 0.938101i \(0.612583\pi\)
\(282\) 0 0
\(283\) 2.26187 0.134454 0.0672270 0.997738i \(-0.478585\pi\)
0.0672270 + 0.997738i \(0.478585\pi\)
\(284\) 0 0
\(285\) 0.806063 0.0477471
\(286\) 0 0
\(287\) −21.1490 −1.24839
\(288\) 0 0
\(289\) 22.8496 1.34409
\(290\) 0 0
\(291\) 8.86082 0.519430
\(292\) 0 0
\(293\) −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
\(296\) 0 0
\(297\) −4.15045 −0.240833
\(298\) 0 0
\(299\) −30.5501 −1.76676
\(300\) 0 0
\(301\) 13.8232 0.796756
\(302\) 0 0
\(303\) 2.13586 0.122702
\(304\) 0 0
\(305\) −11.2750 −0.645607
\(306\) 0 0
\(307\) 26.2071 1.49572 0.747859 0.663857i \(-0.231082\pi\)
0.747859 + 0.663857i \(0.231082\pi\)
\(308\) 0 0
\(309\) 8.64974 0.492066
\(310\) 0 0
\(311\) −6.51388 −0.369368 −0.184684 0.982798i \(-0.559126\pi\)
−0.184684 + 0.982798i \(0.559126\pi\)
\(312\) 0 0
\(313\) 16.0752 0.908625 0.454313 0.890842i \(-0.349885\pi\)
0.454313 + 0.890842i \(0.349885\pi\)
\(314\) 0 0
\(315\) −7.87399 −0.443649
\(316\) 0 0
\(317\) 5.69323 0.319764 0.159882 0.987136i \(-0.448889\pi\)
0.159882 + 0.987136i \(0.448889\pi\)
\(318\) 0 0
\(319\) 3.47627 0.194634
\(320\) 0 0
\(321\) 3.87399 0.216225
\(322\) 0 0
\(323\) 6.31265 0.351245
\(324\) 0 0
\(325\) −6.15633 −0.341491
\(326\) 0 0
\(327\) 2.23743 0.123730
\(328\) 0 0
\(329\) 11.2243 0.618813
\(330\) 0 0
\(331\) −12.3127 −0.676764 −0.338382 0.941009i \(-0.609880\pi\)
−0.338382 + 0.941009i \(0.609880\pi\)
\(332\) 0 0
\(333\) 23.8700 1.30807
\(334\) 0 0
\(335\) 6.73084 0.367745
\(336\) 0 0
\(337\) 3.76845 0.205281 0.102640 0.994719i \(-0.467271\pi\)
0.102640 + 0.994719i \(0.467271\pi\)
\(338\) 0 0
\(339\) 5.63656 0.306136
\(340\) 0 0
\(341\) 5.70194 0.308777
\(342\) 0 0
\(343\) 9.29948 0.502125
\(344\) 0 0
\(345\) −4.00000 −0.215353
\(346\) 0 0
\(347\) −22.3634 −1.20053 −0.600266 0.799800i \(-0.704939\pi\)
−0.600266 + 0.799800i \(0.704939\pi\)
\(348\) 0 0
\(349\) 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
\(352\) 0 0
\(353\) 5.53690 0.294700 0.147350 0.989084i \(-0.452926\pi\)
0.147350 + 0.989084i \(0.452926\pi\)
\(354\) 0 0
\(355\) −0.775746 −0.0411723
\(356\) 0 0
\(357\) 17.0475 0.902247
\(358\) 0 0
\(359\) 10.3634 0.546961 0.273481 0.961878i \(-0.411825\pi\)
0.273481 + 0.961878i \(0.411825\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) −8.12013 −0.426196
\(364\) 0 0
\(365\) −0.387873 −0.0203022
\(366\) 0 0
\(367\) −3.35026 −0.174882 −0.0874411 0.996170i \(-0.527869\pi\)
−0.0874411 + 0.996170i \(0.527869\pi\)
\(368\) 0 0
\(369\) −14.8364 −0.772351
\(370\) 0 0
\(371\) 6.17679 0.320683
\(372\) 0 0
\(373\) 12.6048 0.652653 0.326327 0.945257i \(-0.394189\pi\)
0.326327 + 0.945257i \(0.394189\pi\)
\(374\) 0 0
\(375\) −0.806063 −0.0416249
\(376\) 0 0
\(377\) −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 0
\(381\) −10.8265 −0.554660
\(382\) 0 0
\(383\) −30.8324 −1.57546 −0.787731 0.616019i \(-0.788744\pi\)
−0.787731 + 0.616019i \(0.788744\pi\)
\(384\) 0 0
\(385\) 3.22425 0.164323
\(386\) 0 0
\(387\) 9.69720 0.492936
\(388\) 0 0
\(389\) 1.37470 0.0697000 0.0348500 0.999393i \(-0.488905\pi\)
0.0348500 + 0.999393i \(0.488905\pi\)
\(390\) 0 0
\(391\) −31.3258 −1.58422
\(392\) 0 0
\(393\) 16.6253 0.838635
\(394\) 0 0
\(395\) −0.836381 −0.0420829
\(396\) 0 0
\(397\) 9.38646 0.471093 0.235546 0.971863i \(-0.424312\pi\)
0.235546 + 0.971863i \(0.424312\pi\)
\(398\) 0 0
\(399\) 2.70052 0.135195
\(400\) 0 0
\(401\) 14.1016 0.704199 0.352099 0.935963i \(-0.385468\pi\)
0.352099 + 0.935963i \(0.385468\pi\)
\(402\) 0 0
\(403\) −36.4749 −1.81694
\(404\) 0 0
\(405\) −3.57452 −0.177619
\(406\) 0 0
\(407\) −9.77433 −0.484496
\(408\) 0 0
\(409\) 35.1490 1.73801 0.869004 0.494805i \(-0.164761\pi\)
0.869004 + 0.494805i \(0.164761\pi\)
\(410\) 0 0
\(411\) 16.3127 0.804644
\(412\) 0 0
\(413\) 21.4010 1.05308
\(414\) 0 0
\(415\) 7.03761 0.345463
\(416\) 0 0
\(417\) −14.1768 −0.694241
\(418\) 0 0
\(419\) 9.02776 0.441035 0.220518 0.975383i \(-0.429225\pi\)
0.220518 + 0.975383i \(0.429225\pi\)
\(420\) 0 0
\(421\) −15.2097 −0.741274 −0.370637 0.928778i \(-0.620861\pi\)
−0.370637 + 0.928778i \(0.620861\pi\)
\(422\) 0 0
\(423\) 7.87399 0.382847
\(424\) 0 0
\(425\) −6.31265 −0.306209
\(426\) 0 0
\(427\) −37.7743 −1.82803
\(428\) 0 0
\(429\) −4.77575 −0.230575
\(430\) 0 0
\(431\) −16.3127 −0.785753 −0.392876 0.919591i \(-0.628520\pi\)
−0.392876 + 0.919591i \(0.628520\pi\)
\(432\) 0 0
\(433\) 11.1432 0.535506 0.267753 0.963488i \(-0.413719\pi\)
0.267753 + 0.963488i \(0.413719\pi\)
\(434\) 0 0
\(435\) −2.91160 −0.139601
\(436\) 0 0
\(437\) −4.96239 −0.237383
\(438\) 0 0
\(439\) −27.3865 −1.30708 −0.653542 0.756890i \(-0.726718\pi\)
−0.653542 + 0.756890i \(0.726718\pi\)
\(440\) 0 0
\(441\) −9.92810 −0.472767
\(442\) 0 0
\(443\) 19.5125 0.927065 0.463533 0.886080i \(-0.346582\pi\)
0.463533 + 0.886080i \(0.346582\pi\)
\(444\) 0 0
\(445\) −7.08840 −0.336022
\(446\) 0 0
\(447\) 5.98541 0.283100
\(448\) 0 0
\(449\) −22.1016 −1.04304 −0.521519 0.853240i \(-0.674634\pi\)
−0.521519 + 0.853240i \(0.674634\pi\)
\(450\) 0 0
\(451\) 6.07522 0.286071
\(452\) 0 0
\(453\) 1.29948 0.0610547
\(454\) 0 0
\(455\) −20.6253 −0.966929
\(456\) 0 0
\(457\) −17.8496 −0.834967 −0.417483 0.908685i \(-0.637088\pi\)
−0.417483 + 0.908685i \(0.637088\pi\)
\(458\) 0 0
\(459\) 27.2243 1.27072
\(460\) 0 0
\(461\) 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 \(-0.290121\pi\)
0.612606 + 0.790388i \(0.290121\pi\)
\(464\) 0 0
\(465\) −4.77575 −0.221470
\(466\) 0 0
\(467\) −6.78560 −0.314000 −0.157000 0.987599i \(-0.550182\pi\)
−0.157000 + 0.987599i \(0.550182\pi\)
\(468\) 0 0
\(469\) 22.5501 1.04127
\(470\) 0 0
\(471\) −3.53690 −0.162972
\(472\) 0 0
\(473\) −3.97082 −0.182579
\(474\) 0 0
\(475\) −1.00000 −0.0458831
\(476\) 0 0
\(477\) 4.33312 0.198400
\(478\) 0 0
\(479\) −12.7104 −0.580752 −0.290376 0.956913i \(-0.593780\pi\)
−0.290376 + 0.956913i \(0.593780\pi\)
\(480\) 0 0
\(481\) 62.5256 2.85092
\(482\) 0 0
\(483\) −13.4010 −0.609769
\(484\) 0 0
\(485\) −10.9927 −0.499153
\(486\) 0 0
\(487\) 15.7586 0.714090 0.357045 0.934087i \(-0.383784\pi\)
0.357045 + 0.934087i \(0.383784\pi\)
\(488\) 0 0
\(489\) −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 0
\(493\) −22.8021 −1.02695
\(494\) 0 0
\(495\) 2.26187 0.101663
\(496\) 0 0
\(497\) −2.59895 −0.116579
\(498\) 0 0
\(499\) −5.48612 −0.245592 −0.122796 0.992432i \(-0.539186\pi\)
−0.122796 + 0.992432i \(0.539186\pi\)
\(500\) 0 0
\(501\) 12.3780 0.553009
\(502\) 0 0
\(503\) 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
\(506\) 0 0
\(507\) 20.0713 0.891396
\(508\) 0 0
\(509\) −39.1900 −1.73706 −0.868532 0.495632i \(-0.834936\pi\)
−0.868532 + 0.495632i \(0.834936\pi\)
\(510\) 0 0
\(511\) −1.29948 −0.0574855
\(512\) 0 0
\(513\) 4.31265 0.190408
\(514\) 0 0
\(515\) −10.7308 −0.472857
\(516\) 0 0
\(517\) −3.22425 −0.141803
\(518\) 0 0
\(519\) −2.61545 −0.114806
\(520\) 0 0
\(521\) −17.7283 −0.776690 −0.388345 0.921514i \(-0.626953\pi\)
−0.388345 + 0.921514i \(0.626953\pi\)
\(522\) 0 0
\(523\) −40.7572 −1.78219 −0.891094 0.453819i \(-0.850061\pi\)
−0.891094 + 0.453819i \(0.850061\pi\)
\(524\) 0 0
\(525\) −2.70052 −0.117861
\(526\) 0 0
\(527\) −37.4010 −1.62922
\(528\) 0 0
\(529\) 1.62530 0.0706652
\(530\) 0 0
\(531\) 15.0132 0.651516
\(532\) 0 0
\(533\) −38.8627 −1.68333
\(534\) 0 0
\(535\) −4.80606 −0.207784
\(536\) 0 0
\(537\) 12.1016 0.522221
\(538\) 0 0
\(539\) 4.06537 0.175108
\(540\) 0 0
\(541\) −23.9003 −1.02756 −0.513778 0.857923i \(-0.671754\pi\)
−0.513778 + 0.857923i \(0.671754\pi\)
\(542\) 0 0
\(543\) −7.43533 −0.319081
\(544\) 0 0
\(545\) −2.77575 −0.118900
\(546\) 0 0
\(547\) 8.55405 0.365745 0.182872 0.983137i \(-0.441461\pi\)
0.182872 + 0.983137i \(0.441461\pi\)
\(548\) 0 0
\(549\) −26.4993 −1.13096
\(550\) 0 0
\(551\) −3.61213 −0.153882
\(552\) 0 0
\(553\) −2.80209 −0.119157
\(554\) 0 0
\(555\) 8.18664 0.347504
\(556\) 0 0
\(557\) 4.23743 0.179546 0.0897728 0.995962i \(-0.471386\pi\)
0.0897728 + 0.995962i \(0.471386\pi\)
\(558\) 0 0
\(559\) 25.4010 1.07435
\(560\) 0 0
\(561\) −4.89701 −0.206752
\(562\) 0 0
\(563\) 16.4934 0.695114 0.347557 0.937659i \(-0.387011\pi\)
0.347557 + 0.937659i \(0.387011\pi\)
\(564\) 0 0
\(565\) −6.99271 −0.294185
\(566\) 0 0
\(567\) −11.9756 −0.502926
\(568\) 0 0
\(569\) −10.0000 −0.419222 −0.209611 0.977785i \(-0.567220\pi\)
−0.209611 + 0.977785i \(0.567220\pi\)
\(570\) 0 0
\(571\) −26.2619 −1.09902 −0.549512 0.835486i \(-0.685186\pi\)
−0.549512 + 0.835486i \(0.685186\pi\)
\(572\) 0 0
\(573\) 17.5515 0.733224
\(574\) 0 0
\(575\) 4.96239 0.206946
\(576\) 0 0
\(577\) 30.1016 1.25314 0.626572 0.779363i \(-0.284457\pi\)
0.626572 + 0.779363i \(0.284457\pi\)
\(578\) 0 0
\(579\) 10.1114 0.420216
\(580\) 0 0
\(581\) 23.5778 0.978174
\(582\) 0 0
\(583\) −1.77433 −0.0734853
\(584\) 0 0
\(585\) −14.4690 −0.598219
\(586\) 0 0
\(587\) −35.1392 −1.45035 −0.725175 0.688565i \(-0.758241\pi\)
−0.725175 + 0.688565i \(0.758241\pi\)
\(588\) 0 0
\(589\) −5.92478 −0.244126
\(590\) 0 0
\(591\) −19.7889 −0.814007
\(592\) 0 0
\(593\) 34.3244 1.40953 0.704767 0.709439i \(-0.251051\pi\)
0.704767 + 0.709439i \(0.251051\pi\)
\(594\) 0 0
\(595\) −21.1490 −0.867026
\(596\) 0 0
\(597\) −18.5990 −0.761204
\(598\) 0 0
\(599\) −14.6107 −0.596978 −0.298489 0.954413i \(-0.596483\pi\)
−0.298489 + 0.954413i \(0.596483\pi\)
\(600\) 0 0
\(601\) 23.5633 0.961165 0.480583 0.876949i \(-0.340425\pi\)
0.480583 + 0.876949i \(0.340425\pi\)
\(602\) 0 0
\(603\) 15.8192 0.644209
\(604\) 0 0
\(605\) 10.0738 0.409559
\(606\) 0 0
\(607\) 8.80606 0.357427 0.178714 0.983901i \(-0.442806\pi\)
0.178714 + 0.983901i \(0.442806\pi\)
\(608\) 0 0
\(609\) −9.75463 −0.395278
\(610\) 0 0
\(611\) 20.6253 0.834410
\(612\) 0 0
\(613\) −10.4142 −0.420626 −0.210313 0.977634i \(-0.567448\pi\)
−0.210313 + 0.977634i \(0.567448\pi\)
\(614\) 0 0
\(615\) −5.08840 −0.205184
\(616\) 0 0
\(617\) −17.2849 −0.695863 −0.347932 0.937520i \(-0.613116\pi\)
−0.347932 + 0.937520i \(0.613116\pi\)
\(618\) 0 0
\(619\) −10.6351 −0.427463 −0.213731 0.976892i \(-0.568562\pi\)
−0.213731 + 0.976892i \(0.568562\pi\)
\(620\) 0 0
\(621\) −21.4010 −0.858794
\(622\) 0 0
\(623\) −23.7480 −0.951443
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) −0.775746 −0.0309803
\(628\) 0 0
\(629\) 64.1133 2.55637
\(630\) 0 0
\(631\) 16.5599 0.659240 0.329620 0.944114i \(-0.393079\pi\)
0.329620 + 0.944114i \(0.393079\pi\)
\(632\) 0 0
\(633\) −16.8773 −0.670813
\(634\) 0 0
\(635\) 13.4314 0.533007
\(636\) 0 0
\(637\) −26.0059 −1.03039
\(638\) 0 0
\(639\) −1.82321 −0.0721249
\(640\) 0 0
\(641\) −16.7612 −0.662026 −0.331013 0.943626i \(-0.607390\pi\)
−0.331013 + 0.943626i \(0.607390\pi\)
\(642\) 0 0
\(643\) 5.73813 0.226290 0.113145 0.993578i \(-0.463908\pi\)
0.113145 + 0.993578i \(0.463908\pi\)
\(644\) 0 0
\(645\) 3.32582 0.130954
\(646\) 0 0
\(647\) 37.2144 1.46305 0.731525 0.681815i \(-0.238809\pi\)
0.731525 + 0.681815i \(0.238809\pi\)
\(648\) 0 0
\(649\) −6.14762 −0.241315
\(650\) 0 0
\(651\) −16.0000 −0.627089
\(652\) 0 0
\(653\) −11.7626 −0.460305 −0.230153 0.973155i \(-0.573923\pi\)
−0.230153 + 0.973155i \(0.573923\pi\)
\(654\) 0 0
\(655\) −20.6253 −0.805897
\(656\) 0 0
\(657\) −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 0
\(661\) 9.53690 0.370943 0.185471 0.982650i \(-0.440619\pi\)
0.185471 + 0.982650i \(0.440619\pi\)
\(662\) 0 0
\(663\) 31.3258 1.21659
\(664\) 0 0
\(665\) −3.35026 −0.129918
\(666\) 0 0
\(667\) 17.9248 0.694050
\(668\) 0 0
\(669\) 0.0244376 0.000944811 0
\(670\) 0 0
\(671\) 10.8510 0.418897
\(672\) 0 0
\(673\) 39.9307 1.53921 0.769607 0.638518i \(-0.220452\pi\)
0.769607 + 0.638518i \(0.220452\pi\)
\(674\) 0 0
\(675\) −4.31265 −0.165994
\(676\) 0 0
\(677\) 3.05334 0.117349 0.0586747 0.998277i \(-0.481313\pi\)
0.0586747 + 0.998277i \(0.481313\pi\)
\(678\) 0 0
\(679\) −36.8284 −1.41335
\(680\) 0 0
\(681\) 3.87399 0.148452
\(682\) 0 0
\(683\) 29.2692 1.11995 0.559977 0.828508i \(-0.310810\pi\)
0.559977 + 0.828508i \(0.310810\pi\)
\(684\) 0 0
\(685\) −20.2374 −0.773232
\(686\) 0 0
\(687\) −1.51056 −0.0576313
\(688\) 0 0
\(689\) 11.3503 0.432411
\(690\) 0 0
\(691\) 2.63515 0.100246 0.0501229 0.998743i \(-0.484039\pi\)
0.0501229 + 0.998743i \(0.484039\pi\)
\(692\) 0 0
\(693\) 7.57784 0.287858
\(694\) 0 0
\(695\) 17.5877 0.667139
\(696\) 0 0
\(697\) −39.8496 −1.50941
\(698\) 0 0
\(699\) −8.98683 −0.339913
\(700\) 0 0
\(701\) 25.0494 0.946102 0.473051 0.881035i \(-0.343153\pi\)
0.473051 + 0.881035i \(0.343153\pi\)
\(702\) 0 0
\(703\) 10.1563 0.383053
\(704\) 0 0
\(705\) 2.70052 0.101708
\(706\) 0 0
\(707\) −8.87732 −0.333866
\(708\) 0 0
\(709\) 41.6991 1.56604 0.783021 0.621995i \(-0.213677\pi\)
0.783021 + 0.621995i \(0.213677\pi\)
\(710\) 0 0
\(711\) −1.96571 −0.0737200
\(712\) 0 0
\(713\) 29.4010 1.10108
\(714\) 0 0
\(715\) 5.92478 0.221574
\(716\) 0 0
\(717\) −7.49597 −0.279942
\(718\) 0 0
\(719\) −30.6351 −1.14250 −0.571249 0.820777i \(-0.693541\pi\)
−0.571249 + 0.820777i \(0.693541\pi\)
\(720\) 0 0
\(721\) −35.9511 −1.33889
\(722\) 0 0
\(723\) −1.86414 −0.0693282
\(724\) 0 0
\(725\) 3.61213 0.134151
\(726\) 0 0
\(727\) −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 0
\(731\) 26.0460 0.963348
\(732\) 0 0
\(733\) −9.84955 −0.363802 −0.181901 0.983317i \(-0.558225\pi\)
−0.181901 + 0.983317i \(0.558225\pi\)
\(734\) 0 0
\(735\) −3.40502 −0.125596
\(736\) 0 0
\(737\) −6.47768 −0.238609
\(738\) 0 0
\(739\) −20.0000 −0.735712 −0.367856 0.929883i \(-0.619908\pi\)
−0.367856 + 0.929883i \(0.619908\pi\)
\(740\) 0 0
\(741\) 4.96239 0.182298
\(742\) 0 0
\(743\) 15.7177 0.576625 0.288313 0.957536i \(-0.406906\pi\)
0.288313 + 0.957536i \(0.406906\pi\)
\(744\) 0 0
\(745\) −7.42548 −0.272049
\(746\) 0 0
\(747\) 16.5402 0.605175
\(748\) 0 0
\(749\) −16.1016 −0.588339
\(750\) 0 0
\(751\) 43.7400 1.59610 0.798048 0.602593i \(-0.205866\pi\)
0.798048 + 0.602593i \(0.205866\pi\)
\(752\) 0 0
\(753\) 19.4274 0.707974
\(754\) 0 0
\(755\) −1.61213 −0.0586713
\(756\) 0 0
\(757\) 15.7743 0.573328 0.286664 0.958031i \(-0.407454\pi\)
0.286664 + 0.958031i \(0.407454\pi\)
\(758\) 0 0
\(759\) 3.84955 0.139730
\(760\) 0 0
\(761\) 43.2262 1.56695 0.783474 0.621425i \(-0.213446\pi\)
0.783474 + 0.621425i \(0.213446\pi\)
\(762\) 0 0
\(763\) −9.29948 −0.336664
\(764\) 0 0
\(765\) −14.8364 −0.536410
\(766\) 0 0
\(767\) 39.3258 1.41997
\(768\) 0 0
\(769\) −19.1246 −0.689650 −0.344825 0.938667i \(-0.612062\pi\)
−0.344825 + 0.938667i \(0.612062\pi\)
\(770\) 0 0
\(771\) −10.7367 −0.386674
\(772\) 0 0
\(773\) −25.8846 −0.931005 −0.465502 0.885047i \(-0.654126\pi\)
−0.465502 + 0.885047i \(0.654126\pi\)
\(774\) 0 0
\(775\) 5.92478 0.212824
\(776\) 0 0
\(777\) 27.4274 0.983952
\(778\) 0 0
\(779\) −6.31265 −0.226174
\(780\) 0 0
\(781\) 0.746569 0.0267144
\(782\) 0 0
\(783\) −15.5778 −0.556707
\(784\) 0 0
\(785\) 4.38787 0.156610
\(786\) 0 0
\(787\) 19.9814 0.712261 0.356131 0.934436i \(-0.384096\pi\)
0.356131 + 0.934436i \(0.384096\pi\)
\(788\) 0 0
\(789\) −10.4485 −0.371977
\(790\) 0 0
\(791\) −23.4274 −0.832982
\(792\) 0 0
\(793\) −69.4128 −2.46492
\(794\) 0 0
\(795\) 1.48612 0.0527072
\(796\) 0 0
\(797\) −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
\(800\) 0 0
\(801\) −16.6596 −0.588638
\(802\) 0 0
\(803\) 0.373285 0.0131729
\(804\) 0 0
\(805\) 16.6253 0.585965
\(806\) 0 0
\(807\) 9.18997 0.323502
\(808\) 0 0
\(809\) −17.2243 −0.605573 −0.302786 0.953058i \(-0.597917\pi\)
−0.302786 + 0.953058i \(0.597917\pi\)
\(810\) 0 0
\(811\) −15.6267 −0.548728 −0.274364 0.961626i \(-0.588467\pi\)
−0.274364 + 0.961626i \(0.588467\pi\)
\(812\) 0 0
\(813\) −13.5515 −0.475272
\(814\) 0 0
\(815\) 0.649738 0.0227593
\(816\) 0 0
\(817\) 4.12601 0.144351
\(818\) 0 0
\(819\) −48.4749 −1.69385
\(820\) 0 0
\(821\) −39.2506 −1.36986 −0.684928 0.728611i \(-0.740166\pi\)
−0.684928 + 0.728611i \(0.740166\pi\)
\(822\) 0 0
\(823\) 45.5271 1.58697 0.793487 0.608588i \(-0.208264\pi\)
0.793487 + 0.608588i \(0.208264\pi\)
\(824\) 0 0
\(825\) 0.775746 0.0270080
\(826\) 0 0
\(827\) 17.0698 0.593576 0.296788 0.954943i \(-0.404084\pi\)
0.296788 + 0.954943i \(0.404084\pi\)
\(828\) 0 0
\(829\) −1.69911 −0.0590125 −0.0295062 0.999565i \(-0.509393\pi\)
−0.0295062 + 0.999565i \(0.509393\pi\)
\(830\) 0 0
\(831\) 24.0118 0.832959
\(832\) 0 0
\(833\) −26.6662 −0.923930
\(834\) 0 0
\(835\) −15.3561 −0.531421
\(836\) 0 0
\(837\) −25.5515 −0.883189
\(838\) 0 0
\(839\) 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
\(842\) 0 0
\(843\) −9.36011 −0.322379
\(844\) 0 0
\(845\) −24.9003 −0.856598
\(846\) 0 0
\(847\) 33.7499 1.15966
\(848\) 0 0
\(849\) 1.82321 0.0625723
\(850\) 0 0
\(851\) −50.3996 −1.72768
\(852\) 0 0
\(853\) 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
\(856\) 0 0
\(857\) −23.6180 −0.806776 −0.403388 0.915029i \(-0.632167\pi\)
−0.403388 + 0.915029i \(0.632167\pi\)
\(858\) 0 0
\(859\) 15.1754 0.517777 0.258889 0.965907i \(-0.416644\pi\)
0.258889 + 0.965907i \(0.416644\pi\)
\(860\) 0 0
\(861\) −17.0475 −0.580976
\(862\) 0 0
\(863\) −30.1055 −1.02480 −0.512402 0.858746i \(-0.671244\pi\)
−0.512402 + 0.858746i \(0.671244\pi\)
\(864\) 0 0
\(865\) 3.24472 0.110324
\(866\) 0 0
\(867\) 18.4182 0.625515
\(868\) 0 0
\(869\) 0.804923 0.0273051
\(870\) 0 0
\(871\) 41.4372 1.40405
\(872\) 0 0
\(873\) −25.8357 −0.874407
\(874\) 0 0
\(875\) 3.35026 0.113260
\(876\) 0 0
\(877\) 5.53102 0.186769 0.0933847 0.995630i \(-0.470231\pi\)
0.0933847 + 0.995630i \(0.470231\pi\)
\(878\) 0 0
\(879\) −1.48612 −0.0501255
\(880\) 0 0
\(881\) 20.8265 0.701664 0.350832 0.936438i \(-0.385899\pi\)
0.350832 + 0.936438i \(0.385899\pi\)
\(882\) 0 0
\(883\) 43.1509 1.45214 0.726072 0.687618i \(-0.241344\pi\)
0.726072 + 0.687618i \(0.241344\pi\)
\(884\) 0 0
\(885\) 5.14903 0.173083
\(886\) 0 0
\(887\) 44.5461 1.49571 0.747856 0.663861i \(-0.231083\pi\)
0.747856 + 0.663861i \(0.231083\pi\)
\(888\) 0 0
\(889\) 44.9986 1.50920
\(890\) 0 0
\(891\) 3.44007 0.115247
\(892\) 0 0
\(893\) 3.35026 0.112112
\(894\) 0 0
\(895\) −15.0132 −0.501835
\(896\) 0 0
\(897\) −24.6253 −0.822215
\(898\) 0 0
\(899\) 21.4010 0.713765
\(900\) 0 0
\(901\) 11.6385 0.387734
\(902\) 0 0
\(903\) 11.1424 0.370795
\(904\) 0 0
\(905\) 9.22425 0.306625
\(906\) 0 0
\(907\) 53.7558 1.78493 0.892466 0.451115i \(-0.148974\pi\)
0.892466 + 0.451115i \(0.148974\pi\)
\(908\) 0 0
\(909\) −6.22758 −0.206556
\(910\) 0 0
\(911\) 2.28630 0.0757486 0.0378743 0.999283i \(-0.487941\pi\)
0.0378743 + 0.999283i \(0.487941\pi\)
\(912\) 0 0
\(913\) −6.77292 −0.224151
\(914\) 0 0
\(915\) −9.08840 −0.300453
\(916\) 0 0
\(917\) −69.1002 −2.28189
\(918\) 0 0
\(919\) 34.8510 1.14963 0.574814 0.818284i \(-0.305075\pi\)
0.574814 + 0.818284i \(0.305075\pi\)
\(920\) 0 0
\(921\) 21.1246 0.696079
\(922\) 0 0
\(923\) −4.77575 −0.157196
\(924\) 0 0
\(925\) −10.1563 −0.333938
\(926\) 0 0
\(927\) −25.2203 −0.828343
\(928\) 0 0
\(929\) 41.6991 1.36810 0.684052 0.729434i \(-0.260216\pi\)
0.684052 + 0.729434i \(0.260216\pi\)
\(930\) 0 0
\(931\) −4.22425 −0.138444
\(932\) 0 0
\(933\) −5.25060 −0.171897
\(934\) 0 0
\(935\) 6.07522 0.198681
\(936\) 0 0
\(937\) 21.9102 0.715775 0.357887 0.933765i \(-0.383497\pi\)
0.357887 + 0.933765i \(0.383497\pi\)
\(938\) 0 0
\(939\) 12.9576 0.422857
\(940\) 0 0
\(941\) 3.55149 0.115775 0.0578877 0.998323i \(-0.481563\pi\)
0.0578877 + 0.998323i \(0.481563\pi\)
\(942\) 0 0
\(943\) 31.3258 1.02011
\(944\) 0 0
\(945\) −14.4485 −0.470010
\(946\) 0 0
\(947\) 38.3634 1.24664 0.623322 0.781965i \(-0.285783\pi\)
0.623322 + 0.781965i \(0.285783\pi\)
\(948\) 0 0
\(949\) −2.38787 −0.0775136
\(950\) 0 0
\(951\) 4.58910 0.148812
\(952\) 0 0
\(953\) −22.8714 −0.740879 −0.370439 0.928857i \(-0.620793\pi\)
−0.370439 + 0.928857i \(0.620793\pi\)
\(954\) 0 0
\(955\) −21.7743 −0.704601
\(956\) 0 0
\(957\) 2.80209 0.0905788
\(958\) 0 0
\(959\) −67.8007 −2.18940
\(960\) 0 0
\(961\) 4.10299 0.132354
\(962\) 0 0
\(963\) −11.2955 −0.363993
\(964\) 0 0
\(965\) −12.5442 −0.403812
\(966\) 0 0
\(967\) −27.6629 −0.889579 −0.444790 0.895635i \(-0.646722\pi\)
−0.444790 + 0.895635i \(0.646722\pi\)
\(968\) 0 0
\(969\) 5.08840 0.163463
\(970\) 0 0
\(971\) −42.1768 −1.35352 −0.676759 0.736205i \(-0.736616\pi\)
−0.676759 + 0.736205i \(0.736616\pi\)
\(972\) 0 0
\(973\) 58.9234 1.88900
\(974\) 0 0
\(975\) −4.96239 −0.158924
\(976\) 0 0
\(977\) 0.856849 0.0274130 0.0137065 0.999906i \(-0.495637\pi\)
0.0137065 + 0.999906i \(0.495637\pi\)
\(978\) 0 0
\(979\) 6.82179 0.218025
\(980\) 0 0
\(981\) −6.52373 −0.208287
\(982\) 0 0
\(983\) −45.2809 −1.44424 −0.722119 0.691769i \(-0.756831\pi\)
−0.722119 + 0.691769i \(0.756831\pi\)
\(984\) 0 0
\(985\) 24.5501 0.782231
\(986\) 0 0
\(987\) 9.04746 0.287984
\(988\) 0 0
\(989\) −20.4749 −0.651063
\(990\) 0 0
\(991\) −47.0132 −1.49342 −0.746711 0.665148i \(-0.768368\pi\)
−0.746711 + 0.665148i \(0.768368\pi\)
\(992\) 0 0
\(993\) −9.92478 −0.314953
\(994\) 0 0
\(995\) 23.0738 0.731489
\(996\) 0 0
\(997\) −13.6873 −0.433483 −0.216741 0.976229i \(-0.569543\pi\)
−0.216741 + 0.976229i \(0.569543\pi\)
\(998\) 0 0
\(999\) 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 6080.2.a.by.1.2 3
4.3 odd 2 6080.2.a.bo.1.2 3
8.3 odd 2 95.2.a.a.1.1 3
8.5 even 2 1520.2.a.p.1.2 3
24.11 even 2 855.2.a.i.1.3 3
40.3 even 4 475.2.b.d.324.5 6
40.19 odd 2 475.2.a.f.1.3 3
40.27 even 4 475.2.b.d.324.2 6
40.29 even 2 7600.2.a.bx.1.2 3
56.27 even 2 4655.2.a.u.1.1 3
120.59 even 2 4275.2.a.bk.1.1 3
152.75 even 2 1805.2.a.f.1.3 3
760.379 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 8.3 odd 2
475.2.a.f.1.3 3 40.19 odd 2
475.2.b.d.324.2 6 40.27 even 4
475.2.b.d.324.5 6 40.3 even 4
855.2.a.i.1.3 3 24.11 even 2
1520.2.a.p.1.2 3 8.5 even 2
1805.2.a.f.1.3 3 152.75 even 2
4275.2.a.bk.1.1 3 120.59 even 2
4655.2.a.u.1.1 3 56.27 even 2
6080.2.a.bo.1.2 3 4.3 odd 2
6080.2.a.by.1.2 3 1.1 even 1 trivial
7600.2.a.bx.1.2 3 40.29 even 2
9025.2.a.bb.1.1 3 760.379 even 2