Properties

Label 3040.2.a.k.1.3
Level $3040$
Weight $2$
Character 3040.1
Self dual yes
Analytic conductor $24.275$
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] = [3040,2,Mod(1,3040)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3040, 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("3040.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3040 = 2^{5} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3040.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: \(24.2745222145\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.564.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 5x + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(2.51414\) of defining polynomial
Character \(\chi\) \(=\) 3040.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.51414 q^{3} -1.00000 q^{5} -1.32088 q^{7} -0.707389 q^{9} +O(q^{10})\) \(q+1.51414 q^{3} -1.00000 q^{5} -1.32088 q^{7} -0.707389 q^{9} -3.32088 q^{11} +1.80675 q^{13} -1.51414 q^{15} +2.00000 q^{17} +1.00000 q^{19} -2.00000 q^{21} +5.32088 q^{23} +1.00000 q^{25} -5.61350 q^{27} +5.02827 q^{29} +5.02827 q^{31} -5.02827 q^{33} +1.32088 q^{35} +0.193252 q^{37} +2.73566 q^{39} +0.971726 q^{41} +4.34916 q^{43} +0.707389 q^{45} -1.32088 q^{47} -5.25526 q^{49} +3.02827 q^{51} +3.80675 q^{53} +3.32088 q^{55} +1.51414 q^{57} +8.05655 q^{59} +12.9909 q^{61} +0.934380 q^{63} -1.80675 q^{65} -6.92892 q^{67} +8.05655 q^{69} +13.0283 q^{71} +5.02827 q^{73} +1.51414 q^{75} +4.38650 q^{77} -2.00000 q^{79} -6.37743 q^{81} -2.29261 q^{83} -2.00000 q^{85} +7.61350 q^{87} +13.0283 q^{89} -2.38650 q^{91} +7.61350 q^{93} -1.00000 q^{95} +1.16498 q^{97} +2.34916 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 2 q^{3} - 3 q^{5} + 4 q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 2 q^{3} - 3 q^{5} + 4 q^{7} + 3 q^{9} - 2 q^{11} + 4 q^{13} + 2 q^{15} + 6 q^{17} + 3 q^{19} - 6 q^{21} + 8 q^{23} + 3 q^{25} - 14 q^{27} + 2 q^{29} + 2 q^{31} - 2 q^{33} - 4 q^{35} + 2 q^{37} - 10 q^{39} + 16 q^{41} - 8 q^{43} - 3 q^{45} + 4 q^{47} + 3 q^{49} - 4 q^{51} + 10 q^{53} + 2 q^{55} - 2 q^{57} - 2 q^{59} + 2 q^{61} - 8 q^{63} - 4 q^{65} - 4 q^{67} - 2 q^{69} + 26 q^{71} + 2 q^{73} - 2 q^{75} + 16 q^{77} - 6 q^{79} + 15 q^{81} - 12 q^{83} - 6 q^{85} + 20 q^{87} + 26 q^{89} - 10 q^{91} + 20 q^{93} - 3 q^{95} + 18 q^{97} - 14 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.51414 0.874187 0.437094 0.899416i \(-0.356008\pi\)
0.437094 + 0.899416i \(0.356008\pi\)
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −1.32088 −0.499247 −0.249624 0.968343i \(-0.580307\pi\)
−0.249624 + 0.968343i \(0.580307\pi\)
\(8\) 0 0
\(9\) −0.707389 −0.235796
\(10\) 0 0
\(11\) −3.32088 −1.00128 −0.500642 0.865654i \(-0.666903\pi\)
−0.500642 + 0.865654i \(0.666903\pi\)
\(12\) 0 0
\(13\) 1.80675 0.501102 0.250551 0.968103i \(-0.419388\pi\)
0.250551 + 0.968103i \(0.419388\pi\)
\(14\) 0 0
\(15\) −1.51414 −0.390948
\(16\) 0 0
\(17\) 2.00000 0.485071 0.242536 0.970143i \(-0.422021\pi\)
0.242536 + 0.970143i \(0.422021\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) −2.00000 −0.436436
\(22\) 0 0
\(23\) 5.32088 1.10948 0.554741 0.832023i \(-0.312818\pi\)
0.554741 + 0.832023i \(0.312818\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −5.61350 −1.08032
\(28\) 0 0
\(29\) 5.02827 0.933727 0.466864 0.884329i \(-0.345384\pi\)
0.466864 + 0.884329i \(0.345384\pi\)
\(30\) 0 0
\(31\) 5.02827 0.903105 0.451552 0.892245i \(-0.350870\pi\)
0.451552 + 0.892245i \(0.350870\pi\)
\(32\) 0 0
\(33\) −5.02827 −0.875310
\(34\) 0 0
\(35\) 1.32088 0.223270
\(36\) 0 0
\(37\) 0.193252 0.0317705 0.0158853 0.999874i \(-0.494943\pi\)
0.0158853 + 0.999874i \(0.494943\pi\)
\(38\) 0 0
\(39\) 2.73566 0.438057
\(40\) 0 0
\(41\) 0.971726 0.151758 0.0758791 0.997117i \(-0.475824\pi\)
0.0758791 + 0.997117i \(0.475824\pi\)
\(42\) 0 0
\(43\) 4.34916 0.663240 0.331620 0.943413i \(-0.392405\pi\)
0.331620 + 0.943413i \(0.392405\pi\)
\(44\) 0 0
\(45\) 0.707389 0.105451
\(46\) 0 0
\(47\) −1.32088 −0.192671 −0.0963354 0.995349i \(-0.530712\pi\)
−0.0963354 + 0.995349i \(0.530712\pi\)
\(48\) 0 0
\(49\) −5.25526 −0.750752
\(50\) 0 0
\(51\) 3.02827 0.424043
\(52\) 0 0
\(53\) 3.80675 0.522897 0.261448 0.965217i \(-0.415800\pi\)
0.261448 + 0.965217i \(0.415800\pi\)
\(54\) 0 0
\(55\) 3.32088 0.447788
\(56\) 0 0
\(57\) 1.51414 0.200552
\(58\) 0 0
\(59\) 8.05655 1.04887 0.524437 0.851450i \(-0.324276\pi\)
0.524437 + 0.851450i \(0.324276\pi\)
\(60\) 0 0
\(61\) 12.9909 1.66332 0.831659 0.555287i \(-0.187392\pi\)
0.831659 + 0.555287i \(0.187392\pi\)
\(62\) 0 0
\(63\) 0.934380 0.117721
\(64\) 0 0
\(65\) −1.80675 −0.224099
\(66\) 0 0
\(67\) −6.92892 −0.846502 −0.423251 0.906013i \(-0.639111\pi\)
−0.423251 + 0.906013i \(0.639111\pi\)
\(68\) 0 0
\(69\) 8.05655 0.969894
\(70\) 0 0
\(71\) 13.0283 1.54617 0.773086 0.634301i \(-0.218712\pi\)
0.773086 + 0.634301i \(0.218712\pi\)
\(72\) 0 0
\(73\) 5.02827 0.588515 0.294257 0.955726i \(-0.404928\pi\)
0.294257 + 0.955726i \(0.404928\pi\)
\(74\) 0 0
\(75\) 1.51414 0.174837
\(76\) 0 0
\(77\) 4.38650 0.499889
\(78\) 0 0
\(79\) −2.00000 −0.225018 −0.112509 0.993651i \(-0.535889\pi\)
−0.112509 + 0.993651i \(0.535889\pi\)
\(80\) 0 0
\(81\) −6.37743 −0.708604
\(82\) 0 0
\(83\) −2.29261 −0.251647 −0.125823 0.992053i \(-0.540157\pi\)
−0.125823 + 0.992053i \(0.540157\pi\)
\(84\) 0 0
\(85\) −2.00000 −0.216930
\(86\) 0 0
\(87\) 7.61350 0.816252
\(88\) 0 0
\(89\) 13.0283 1.38099 0.690497 0.723335i \(-0.257392\pi\)
0.690497 + 0.723335i \(0.257392\pi\)
\(90\) 0 0
\(91\) −2.38650 −0.250174
\(92\) 0 0
\(93\) 7.61350 0.789483
\(94\) 0 0
\(95\) −1.00000 −0.102598
\(96\) 0 0
\(97\) 1.16498 0.118286 0.0591428 0.998250i \(-0.481163\pi\)
0.0591428 + 0.998250i \(0.481163\pi\)
\(98\) 0 0
\(99\) 2.34916 0.236099
\(100\) 0 0
\(101\) −3.32088 −0.330440 −0.165220 0.986257i \(-0.552833\pi\)
−0.165220 + 0.986257i \(0.552833\pi\)
\(102\) 0 0
\(103\) 3.12763 0.308175 0.154087 0.988057i \(-0.450756\pi\)
0.154087 + 0.988057i \(0.450756\pi\)
\(104\) 0 0
\(105\) 2.00000 0.195180
\(106\) 0 0
\(107\) −9.18418 −0.887868 −0.443934 0.896059i \(-0.646418\pi\)
−0.443934 + 0.896059i \(0.646418\pi\)
\(108\) 0 0
\(109\) 7.41478 0.710207 0.355103 0.934827i \(-0.384446\pi\)
0.355103 + 0.934827i \(0.384446\pi\)
\(110\) 0 0
\(111\) 0.292611 0.0277734
\(112\) 0 0
\(113\) 6.77847 0.637665 0.318833 0.947811i \(-0.396709\pi\)
0.318833 + 0.947811i \(0.396709\pi\)
\(114\) 0 0
\(115\) −5.32088 −0.496175
\(116\) 0 0
\(117\) −1.27807 −0.118158
\(118\) 0 0
\(119\) −2.64177 −0.242171
\(120\) 0 0
\(121\) 0.0282739 0.00257035
\(122\) 0 0
\(123\) 1.47133 0.132665
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −1.12763 −0.100061 −0.0500306 0.998748i \(-0.515932\pi\)
−0.0500306 + 0.998748i \(0.515932\pi\)
\(128\) 0 0
\(129\) 6.58522 0.579796
\(130\) 0 0
\(131\) −6.44305 −0.562932 −0.281466 0.959571i \(-0.590821\pi\)
−0.281466 + 0.959571i \(0.590821\pi\)
\(132\) 0 0
\(133\) −1.32088 −0.114535
\(134\) 0 0
\(135\) 5.61350 0.483133
\(136\) 0 0
\(137\) 21.9253 1.87321 0.936603 0.350393i \(-0.113952\pi\)
0.936603 + 0.350393i \(0.113952\pi\)
\(138\) 0 0
\(139\) −6.73566 −0.571311 −0.285656 0.958332i \(-0.592211\pi\)
−0.285656 + 0.958332i \(0.592211\pi\)
\(140\) 0 0
\(141\) −2.00000 −0.168430
\(142\) 0 0
\(143\) −6.00000 −0.501745
\(144\) 0 0
\(145\) −5.02827 −0.417575
\(146\) 0 0
\(147\) −7.95719 −0.656298
\(148\) 0 0
\(149\) 20.9909 1.71964 0.859822 0.510594i \(-0.170574\pi\)
0.859822 + 0.510594i \(0.170574\pi\)
\(150\) 0 0
\(151\) −2.84049 −0.231155 −0.115578 0.993298i \(-0.536872\pi\)
−0.115578 + 0.993298i \(0.536872\pi\)
\(152\) 0 0
\(153\) −1.41478 −0.114378
\(154\) 0 0
\(155\) −5.02827 −0.403881
\(156\) 0 0
\(157\) 15.4713 1.23475 0.617373 0.786670i \(-0.288197\pi\)
0.617373 + 0.786670i \(0.288197\pi\)
\(158\) 0 0
\(159\) 5.76394 0.457110
\(160\) 0 0
\(161\) −7.02827 −0.553906
\(162\) 0 0
\(163\) −12.3492 −0.967261 −0.483630 0.875272i \(-0.660682\pi\)
−0.483630 + 0.875272i \(0.660682\pi\)
\(164\) 0 0
\(165\) 5.02827 0.391451
\(166\) 0 0
\(167\) −2.21245 −0.171205 −0.0856024 0.996329i \(-0.527281\pi\)
−0.0856024 + 0.996329i \(0.527281\pi\)
\(168\) 0 0
\(169\) −9.73566 −0.748897
\(170\) 0 0
\(171\) −0.707389 −0.0540954
\(172\) 0 0
\(173\) −2.50506 −0.190457 −0.0952283 0.995455i \(-0.530358\pi\)
−0.0952283 + 0.995455i \(0.530358\pi\)
\(174\) 0 0
\(175\) −1.32088 −0.0998495
\(176\) 0 0
\(177\) 12.1987 0.916912
\(178\) 0 0
\(179\) −6.00000 −0.448461 −0.224231 0.974536i \(-0.571987\pi\)
−0.224231 + 0.974536i \(0.571987\pi\)
\(180\) 0 0
\(181\) −12.8296 −0.953613 −0.476807 0.879008i \(-0.658206\pi\)
−0.476807 + 0.879008i \(0.658206\pi\)
\(182\) 0 0
\(183\) 19.6700 1.45405
\(184\) 0 0
\(185\) −0.193252 −0.0142082
\(186\) 0 0
\(187\) −6.64177 −0.485694
\(188\) 0 0
\(189\) 7.41478 0.539346
\(190\) 0 0
\(191\) −11.9253 −0.862885 −0.431442 0.902140i \(-0.641995\pi\)
−0.431442 + 0.902140i \(0.641995\pi\)
\(192\) 0 0
\(193\) −2.06201 −0.148427 −0.0742134 0.997242i \(-0.523645\pi\)
−0.0742134 + 0.997242i \(0.523645\pi\)
\(194\) 0 0
\(195\) −2.73566 −0.195905
\(196\) 0 0
\(197\) −2.31181 −0.164710 −0.0823549 0.996603i \(-0.526244\pi\)
−0.0823549 + 0.996603i \(0.526244\pi\)
\(198\) 0 0
\(199\) 0.773010 0.0547972 0.0273986 0.999625i \(-0.491278\pi\)
0.0273986 + 0.999625i \(0.491278\pi\)
\(200\) 0 0
\(201\) −10.4913 −0.740001
\(202\) 0 0
\(203\) −6.64177 −0.466161
\(204\) 0 0
\(205\) −0.971726 −0.0678683
\(206\) 0 0
\(207\) −3.76394 −0.261612
\(208\) 0 0
\(209\) −3.32088 −0.229710
\(210\) 0 0
\(211\) 0.386505 0.0266081 0.0133040 0.999911i \(-0.495765\pi\)
0.0133040 + 0.999911i \(0.495765\pi\)
\(212\) 0 0
\(213\) 19.7266 1.35164
\(214\) 0 0
\(215\) −4.34916 −0.296610
\(216\) 0 0
\(217\) −6.64177 −0.450873
\(218\) 0 0
\(219\) 7.61350 0.514472
\(220\) 0 0
\(221\) 3.61350 0.243070
\(222\) 0 0
\(223\) 5.71285 0.382561 0.191280 0.981535i \(-0.438736\pi\)
0.191280 + 0.981535i \(0.438736\pi\)
\(224\) 0 0
\(225\) −0.707389 −0.0471593
\(226\) 0 0
\(227\) −17.3720 −1.15302 −0.576509 0.817091i \(-0.695585\pi\)
−0.576509 + 0.817091i \(0.695585\pi\)
\(228\) 0 0
\(229\) −8.79221 −0.581006 −0.290503 0.956874i \(-0.593823\pi\)
−0.290503 + 0.956874i \(0.593823\pi\)
\(230\) 0 0
\(231\) 6.64177 0.436996
\(232\) 0 0
\(233\) 20.3684 1.33438 0.667188 0.744890i \(-0.267498\pi\)
0.667188 + 0.744890i \(0.267498\pi\)
\(234\) 0 0
\(235\) 1.32088 0.0861650
\(236\) 0 0
\(237\) −3.02827 −0.196708
\(238\) 0 0
\(239\) 27.5388 1.78134 0.890669 0.454653i \(-0.150237\pi\)
0.890669 + 0.454653i \(0.150237\pi\)
\(240\) 0 0
\(241\) −3.61350 −0.232766 −0.116383 0.993204i \(-0.537130\pi\)
−0.116383 + 0.993204i \(0.537130\pi\)
\(242\) 0 0
\(243\) 7.18418 0.460865
\(244\) 0 0
\(245\) 5.25526 0.335747
\(246\) 0 0
\(247\) 1.80675 0.114961
\(248\) 0 0
\(249\) −3.47133 −0.219986
\(250\) 0 0
\(251\) −15.9253 −1.00520 −0.502598 0.864520i \(-0.667622\pi\)
−0.502598 + 0.864520i \(0.667622\pi\)
\(252\) 0 0
\(253\) −17.6700 −1.11091
\(254\) 0 0
\(255\) −3.02827 −0.189638
\(256\) 0 0
\(257\) 23.5333 1.46797 0.733985 0.679166i \(-0.237658\pi\)
0.733985 + 0.679166i \(0.237658\pi\)
\(258\) 0 0
\(259\) −0.255264 −0.0158613
\(260\) 0 0
\(261\) −3.55695 −0.220170
\(262\) 0 0
\(263\) −20.6610 −1.27401 −0.637005 0.770860i \(-0.719827\pi\)
−0.637005 + 0.770860i \(0.719827\pi\)
\(264\) 0 0
\(265\) −3.80675 −0.233847
\(266\) 0 0
\(267\) 19.7266 1.20725
\(268\) 0 0
\(269\) −18.5671 −1.13205 −0.566027 0.824386i \(-0.691520\pi\)
−0.566027 + 0.824386i \(0.691520\pi\)
\(270\) 0 0
\(271\) −14.1504 −0.859578 −0.429789 0.902929i \(-0.641412\pi\)
−0.429789 + 0.902929i \(0.641412\pi\)
\(272\) 0 0
\(273\) −3.61350 −0.218699
\(274\) 0 0
\(275\) −3.32088 −0.200257
\(276\) 0 0
\(277\) 10.5852 0.636004 0.318002 0.948090i \(-0.396988\pi\)
0.318002 + 0.948090i \(0.396988\pi\)
\(278\) 0 0
\(279\) −3.55695 −0.212949
\(280\) 0 0
\(281\) 19.1414 1.14188 0.570939 0.820992i \(-0.306579\pi\)
0.570939 + 0.820992i \(0.306579\pi\)
\(282\) 0 0
\(283\) 14.2926 0.849608 0.424804 0.905285i \(-0.360343\pi\)
0.424804 + 0.905285i \(0.360343\pi\)
\(284\) 0 0
\(285\) −1.51414 −0.0896897
\(286\) 0 0
\(287\) −1.28354 −0.0757649
\(288\) 0 0
\(289\) −13.0000 −0.764706
\(290\) 0 0
\(291\) 1.76394 0.103404
\(292\) 0 0
\(293\) −14.1751 −0.828119 −0.414059 0.910250i \(-0.635889\pi\)
−0.414059 + 0.910250i \(0.635889\pi\)
\(294\) 0 0
\(295\) −8.05655 −0.469070
\(296\) 0 0
\(297\) 18.6418 1.08171
\(298\) 0 0
\(299\) 9.61350 0.555963
\(300\) 0 0
\(301\) −5.74474 −0.331121
\(302\) 0 0
\(303\) −5.02827 −0.288867
\(304\) 0 0
\(305\) −12.9909 −0.743858
\(306\) 0 0
\(307\) −17.9572 −1.02487 −0.512435 0.858726i \(-0.671257\pi\)
−0.512435 + 0.858726i \(0.671257\pi\)
\(308\) 0 0
\(309\) 4.73566 0.269402
\(310\) 0 0
\(311\) −0.0938942 −0.00532425 −0.00266213 0.999996i \(-0.500847\pi\)
−0.00266213 + 0.999996i \(0.500847\pi\)
\(312\) 0 0
\(313\) 26.4249 1.49362 0.746812 0.665035i \(-0.231583\pi\)
0.746812 + 0.665035i \(0.231583\pi\)
\(314\) 0 0
\(315\) −0.934380 −0.0526463
\(316\) 0 0
\(317\) −1.23245 −0.0692215 −0.0346108 0.999401i \(-0.511019\pi\)
−0.0346108 + 0.999401i \(0.511019\pi\)
\(318\) 0 0
\(319\) −16.6983 −0.934926
\(320\) 0 0
\(321\) −13.9061 −0.776163
\(322\) 0 0
\(323\) 2.00000 0.111283
\(324\) 0 0
\(325\) 1.80675 0.100220
\(326\) 0 0
\(327\) 11.2270 0.620854
\(328\) 0 0
\(329\) 1.74474 0.0961904
\(330\) 0 0
\(331\) −21.8578 −1.20142 −0.600708 0.799469i \(-0.705114\pi\)
−0.600708 + 0.799469i \(0.705114\pi\)
\(332\) 0 0
\(333\) −0.136705 −0.00749137
\(334\) 0 0
\(335\) 6.92892 0.378567
\(336\) 0 0
\(337\) 4.64723 0.253151 0.126575 0.991957i \(-0.459601\pi\)
0.126575 + 0.991957i \(0.459601\pi\)
\(338\) 0 0
\(339\) 10.2635 0.557439
\(340\) 0 0
\(341\) −16.6983 −0.904265
\(342\) 0 0
\(343\) 16.1878 0.874058
\(344\) 0 0
\(345\) −8.05655 −0.433750
\(346\) 0 0
\(347\) −13.9061 −0.746519 −0.373259 0.927727i \(-0.621760\pi\)
−0.373259 + 0.927727i \(0.621760\pi\)
\(348\) 0 0
\(349\) 18.4249 0.986263 0.493131 0.869955i \(-0.335852\pi\)
0.493131 + 0.869955i \(0.335852\pi\)
\(350\) 0 0
\(351\) −10.1422 −0.541349
\(352\) 0 0
\(353\) 36.1696 1.92512 0.962558 0.271076i \(-0.0873795\pi\)
0.962558 + 0.271076i \(0.0873795\pi\)
\(354\) 0 0
\(355\) −13.0283 −0.691469
\(356\) 0 0
\(357\) −4.00000 −0.211702
\(358\) 0 0
\(359\) 21.8880 1.15520 0.577601 0.816319i \(-0.303989\pi\)
0.577601 + 0.816319i \(0.303989\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) 0.0428105 0.00224697
\(364\) 0 0
\(365\) −5.02827 −0.263192
\(366\) 0 0
\(367\) 2.10482 0.109871 0.0549354 0.998490i \(-0.482505\pi\)
0.0549354 + 0.998490i \(0.482505\pi\)
\(368\) 0 0
\(369\) −0.687389 −0.0357840
\(370\) 0 0
\(371\) −5.02827 −0.261055
\(372\) 0 0
\(373\) −34.1751 −1.76952 −0.884760 0.466047i \(-0.845678\pi\)
−0.884760 + 0.466047i \(0.845678\pi\)
\(374\) 0 0
\(375\) −1.51414 −0.0781897
\(376\) 0 0
\(377\) 9.08482 0.467892
\(378\) 0 0
\(379\) −2.71646 −0.139535 −0.0697676 0.997563i \(-0.522226\pi\)
−0.0697676 + 0.997563i \(0.522226\pi\)
\(380\) 0 0
\(381\) −1.70739 −0.0874722
\(382\) 0 0
\(383\) 6.15591 0.314552 0.157276 0.987555i \(-0.449729\pi\)
0.157276 + 0.987555i \(0.449729\pi\)
\(384\) 0 0
\(385\) −4.38650 −0.223557
\(386\) 0 0
\(387\) −3.07655 −0.156390
\(388\) 0 0
\(389\) 11.4823 0.582173 0.291087 0.956697i \(-0.405983\pi\)
0.291087 + 0.956697i \(0.405983\pi\)
\(390\) 0 0
\(391\) 10.6418 0.538177
\(392\) 0 0
\(393\) −9.75566 −0.492108
\(394\) 0 0
\(395\) 2.00000 0.100631
\(396\) 0 0
\(397\) 8.06748 0.404895 0.202447 0.979293i \(-0.435110\pi\)
0.202447 + 0.979293i \(0.435110\pi\)
\(398\) 0 0
\(399\) −2.00000 −0.100125
\(400\) 0 0
\(401\) −34.8114 −1.73840 −0.869199 0.494462i \(-0.835365\pi\)
−0.869199 + 0.494462i \(0.835365\pi\)
\(402\) 0 0
\(403\) 9.08482 0.452547
\(404\) 0 0
\(405\) 6.37743 0.316897
\(406\) 0 0
\(407\) −0.641769 −0.0318113
\(408\) 0 0
\(409\) 0.773010 0.0382229 0.0191114 0.999817i \(-0.493916\pi\)
0.0191114 + 0.999817i \(0.493916\pi\)
\(410\) 0 0
\(411\) 33.1979 1.63753
\(412\) 0 0
\(413\) −10.6418 −0.523647
\(414\) 0 0
\(415\) 2.29261 0.112540
\(416\) 0 0
\(417\) −10.1987 −0.499433
\(418\) 0 0
\(419\) 21.0848 1.03006 0.515030 0.857172i \(-0.327781\pi\)
0.515030 + 0.857172i \(0.327781\pi\)
\(420\) 0 0
\(421\) −37.0101 −1.80376 −0.901882 0.431983i \(-0.857814\pi\)
−0.901882 + 0.431983i \(0.857814\pi\)
\(422\) 0 0
\(423\) 0.934380 0.0454311
\(424\) 0 0
\(425\) 2.00000 0.0970143
\(426\) 0 0
\(427\) −17.1595 −0.830407
\(428\) 0 0
\(429\) −9.08482 −0.438619
\(430\) 0 0
\(431\) 22.3684 1.07745 0.538723 0.842483i \(-0.318907\pi\)
0.538723 + 0.842483i \(0.318907\pi\)
\(432\) 0 0
\(433\) −11.6755 −0.561089 −0.280545 0.959841i \(-0.590515\pi\)
−0.280545 + 0.959841i \(0.590515\pi\)
\(434\) 0 0
\(435\) −7.61350 −0.365039
\(436\) 0 0
\(437\) 5.32088 0.254532
\(438\) 0 0
\(439\) 8.52867 0.407051 0.203526 0.979070i \(-0.434760\pi\)
0.203526 + 0.979070i \(0.434760\pi\)
\(440\) 0 0
\(441\) 3.71752 0.177025
\(442\) 0 0
\(443\) −15.9627 −0.758409 −0.379204 0.925313i \(-0.623802\pi\)
−0.379204 + 0.925313i \(0.623802\pi\)
\(444\) 0 0
\(445\) −13.0283 −0.617599
\(446\) 0 0
\(447\) 31.7831 1.50329
\(448\) 0 0
\(449\) 23.9253 1.12911 0.564553 0.825397i \(-0.309049\pi\)
0.564553 + 0.825397i \(0.309049\pi\)
\(450\) 0 0
\(451\) −3.22699 −0.151953
\(452\) 0 0
\(453\) −4.30088 −0.202073
\(454\) 0 0
\(455\) 2.38650 0.111881
\(456\) 0 0
\(457\) −17.7375 −0.829726 −0.414863 0.909884i \(-0.636171\pi\)
−0.414863 + 0.909884i \(0.636171\pi\)
\(458\) 0 0
\(459\) −11.2270 −0.524031
\(460\) 0 0
\(461\) −6.00000 −0.279448 −0.139724 0.990190i \(-0.544622\pi\)
−0.139724 + 0.990190i \(0.544622\pi\)
\(462\) 0 0
\(463\) 18.2070 0.846151 0.423075 0.906095i \(-0.360951\pi\)
0.423075 + 0.906095i \(0.360951\pi\)
\(464\) 0 0
\(465\) −7.61350 −0.353067
\(466\) 0 0
\(467\) −22.7922 −1.05470 −0.527349 0.849649i \(-0.676814\pi\)
−0.527349 + 0.849649i \(0.676814\pi\)
\(468\) 0 0
\(469\) 9.15230 0.422614
\(470\) 0 0
\(471\) 23.4257 1.07940
\(472\) 0 0
\(473\) −14.4431 −0.664092
\(474\) 0 0
\(475\) 1.00000 0.0458831
\(476\) 0 0
\(477\) −2.69285 −0.123297
\(478\) 0 0
\(479\) −26.7357 −1.22158 −0.610792 0.791791i \(-0.709149\pi\)
−0.610792 + 0.791791i \(0.709149\pi\)
\(480\) 0 0
\(481\) 0.349158 0.0159203
\(482\) 0 0
\(483\) −10.6418 −0.484217
\(484\) 0 0
\(485\) −1.16498 −0.0528990
\(486\) 0 0
\(487\) 25.7694 1.16772 0.583862 0.811853i \(-0.301541\pi\)
0.583862 + 0.811853i \(0.301541\pi\)
\(488\) 0 0
\(489\) −18.6983 −0.845567
\(490\) 0 0
\(491\) 26.3793 1.19048 0.595240 0.803548i \(-0.297057\pi\)
0.595240 + 0.803548i \(0.297057\pi\)
\(492\) 0 0
\(493\) 10.0565 0.452924
\(494\) 0 0
\(495\) −2.34916 −0.105587
\(496\) 0 0
\(497\) −17.2088 −0.771922
\(498\) 0 0
\(499\) 5.96265 0.266925 0.133463 0.991054i \(-0.457390\pi\)
0.133463 + 0.991054i \(0.457390\pi\)
\(500\) 0 0
\(501\) −3.34996 −0.149665
\(502\) 0 0
\(503\) −19.1896 −0.855624 −0.427812 0.903868i \(-0.640715\pi\)
−0.427812 + 0.903868i \(0.640715\pi\)
\(504\) 0 0
\(505\) 3.32088 0.147777
\(506\) 0 0
\(507\) −14.7411 −0.654676
\(508\) 0 0
\(509\) 23.6700 1.04916 0.524578 0.851362i \(-0.324223\pi\)
0.524578 + 0.851362i \(0.324223\pi\)
\(510\) 0 0
\(511\) −6.64177 −0.293815
\(512\) 0 0
\(513\) −5.61350 −0.247842
\(514\) 0 0
\(515\) −3.12763 −0.137820
\(516\) 0 0
\(517\) 4.38650 0.192918
\(518\) 0 0
\(519\) −3.79301 −0.166495
\(520\) 0 0
\(521\) −7.35823 −0.322370 −0.161185 0.986924i \(-0.551532\pi\)
−0.161185 + 0.986924i \(0.551532\pi\)
\(522\) 0 0
\(523\) 6.48586 0.283607 0.141803 0.989895i \(-0.454710\pi\)
0.141803 + 0.989895i \(0.454710\pi\)
\(524\) 0 0
\(525\) −2.00000 −0.0872872
\(526\) 0 0
\(527\) 10.0565 0.438070
\(528\) 0 0
\(529\) 5.31181 0.230948
\(530\) 0 0
\(531\) −5.69912 −0.247321
\(532\) 0 0
\(533\) 1.75566 0.0760462
\(534\) 0 0
\(535\) 9.18418 0.397067
\(536\) 0 0
\(537\) −9.08482 −0.392039
\(538\) 0 0
\(539\) 17.4521 0.751716
\(540\) 0 0
\(541\) −2.22513 −0.0956660 −0.0478330 0.998855i \(-0.515232\pi\)
−0.0478330 + 0.998855i \(0.515232\pi\)
\(542\) 0 0
\(543\) −19.4257 −0.833637
\(544\) 0 0
\(545\) −7.41478 −0.317614
\(546\) 0 0
\(547\) 22.1378 0.946542 0.473271 0.880917i \(-0.343073\pi\)
0.473271 + 0.880917i \(0.343073\pi\)
\(548\) 0 0
\(549\) −9.18964 −0.392204
\(550\) 0 0
\(551\) 5.02827 0.214212
\(552\) 0 0
\(553\) 2.64177 0.112339
\(554\) 0 0
\(555\) −0.292611 −0.0124206
\(556\) 0 0
\(557\) 31.8688 1.35032 0.675161 0.737670i \(-0.264074\pi\)
0.675161 + 0.737670i \(0.264074\pi\)
\(558\) 0 0
\(559\) 7.85783 0.332351
\(560\) 0 0
\(561\) −10.0565 −0.424588
\(562\) 0 0
\(563\) −23.9681 −1.01014 −0.505068 0.863080i \(-0.668533\pi\)
−0.505068 + 0.863080i \(0.668533\pi\)
\(564\) 0 0
\(565\) −6.77847 −0.285173
\(566\) 0 0
\(567\) 8.42385 0.353769
\(568\) 0 0
\(569\) 0.659914 0.0276650 0.0138325 0.999904i \(-0.495597\pi\)
0.0138325 + 0.999904i \(0.495597\pi\)
\(570\) 0 0
\(571\) 35.2462 1.47501 0.737504 0.675343i \(-0.236004\pi\)
0.737504 + 0.675343i \(0.236004\pi\)
\(572\) 0 0
\(573\) −18.0565 −0.754323
\(574\) 0 0
\(575\) 5.32088 0.221896
\(576\) 0 0
\(577\) 33.1414 1.37969 0.689847 0.723956i \(-0.257678\pi\)
0.689847 + 0.723956i \(0.257678\pi\)
\(578\) 0 0
\(579\) −3.12217 −0.129753
\(580\) 0 0
\(581\) 3.02827 0.125634
\(582\) 0 0
\(583\) −12.6418 −0.523569
\(584\) 0 0
\(585\) 1.27807 0.0528419
\(586\) 0 0
\(587\) −11.7639 −0.485550 −0.242775 0.970083i \(-0.578058\pi\)
−0.242775 + 0.970083i \(0.578058\pi\)
\(588\) 0 0
\(589\) 5.02827 0.207186
\(590\) 0 0
\(591\) −3.50040 −0.143987
\(592\) 0 0
\(593\) 5.34009 0.219291 0.109646 0.993971i \(-0.465028\pi\)
0.109646 + 0.993971i \(0.465028\pi\)
\(594\) 0 0
\(595\) 2.64177 0.108302
\(596\) 0 0
\(597\) 1.17044 0.0479030
\(598\) 0 0
\(599\) 4.64177 0.189658 0.0948288 0.995494i \(-0.469770\pi\)
0.0948288 + 0.995494i \(0.469770\pi\)
\(600\) 0 0
\(601\) −18.4431 −0.752308 −0.376154 0.926557i \(-0.622754\pi\)
−0.376154 + 0.926557i \(0.622754\pi\)
\(602\) 0 0
\(603\) 4.90144 0.199602
\(604\) 0 0
\(605\) −0.0282739 −0.00114950
\(606\) 0 0
\(607\) −2.92892 −0.118881 −0.0594405 0.998232i \(-0.518932\pi\)
−0.0594405 + 0.998232i \(0.518932\pi\)
\(608\) 0 0
\(609\) −10.0565 −0.407512
\(610\) 0 0
\(611\) −2.38650 −0.0965477
\(612\) 0 0
\(613\) −7.48225 −0.302205 −0.151103 0.988518i \(-0.548282\pi\)
−0.151103 + 0.988518i \(0.548282\pi\)
\(614\) 0 0
\(615\) −1.47133 −0.0593296
\(616\) 0 0
\(617\) −5.53880 −0.222984 −0.111492 0.993765i \(-0.535563\pi\)
−0.111492 + 0.993765i \(0.535563\pi\)
\(618\) 0 0
\(619\) −13.3027 −0.534682 −0.267341 0.963602i \(-0.586145\pi\)
−0.267341 + 0.963602i \(0.586145\pi\)
\(620\) 0 0
\(621\) −29.8688 −1.19859
\(622\) 0 0
\(623\) −17.2088 −0.689458
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) −5.02827 −0.200810
\(628\) 0 0
\(629\) 0.386505 0.0154110
\(630\) 0 0
\(631\) 31.2462 1.24389 0.621946 0.783060i \(-0.286342\pi\)
0.621946 + 0.783060i \(0.286342\pi\)
\(632\) 0 0
\(633\) 0.585221 0.0232605
\(634\) 0 0
\(635\) 1.12763 0.0447487
\(636\) 0 0
\(637\) −9.49494 −0.376203
\(638\) 0 0
\(639\) −9.21606 −0.364582
\(640\) 0 0
\(641\) 17.0101 0.671860 0.335930 0.941887i \(-0.390949\pi\)
0.335930 + 0.941887i \(0.390949\pi\)
\(642\) 0 0
\(643\) 20.0757 0.791710 0.395855 0.918313i \(-0.370448\pi\)
0.395855 + 0.918313i \(0.370448\pi\)
\(644\) 0 0
\(645\) −6.58522 −0.259293
\(646\) 0 0
\(647\) 23.6508 0.929811 0.464905 0.885360i \(-0.346088\pi\)
0.464905 + 0.885360i \(0.346088\pi\)
\(648\) 0 0
\(649\) −26.7549 −1.05022
\(650\) 0 0
\(651\) −10.0565 −0.394147
\(652\) 0 0
\(653\) 12.3684 0.484011 0.242006 0.970275i \(-0.422195\pi\)
0.242006 + 0.970275i \(0.422195\pi\)
\(654\) 0 0
\(655\) 6.44305 0.251751
\(656\) 0 0
\(657\) −3.55695 −0.138770
\(658\) 0 0
\(659\) −14.7730 −0.575475 −0.287737 0.957709i \(-0.592903\pi\)
−0.287737 + 0.957709i \(0.592903\pi\)
\(660\) 0 0
\(661\) −41.7831 −1.62518 −0.812588 0.582839i \(-0.801942\pi\)
−0.812588 + 0.582839i \(0.801942\pi\)
\(662\) 0 0
\(663\) 5.47133 0.212489
\(664\) 0 0
\(665\) 1.32088 0.0512217
\(666\) 0 0
\(667\) 26.7549 1.03595
\(668\) 0 0
\(669\) 8.65004 0.334430
\(670\) 0 0
\(671\) −43.1414 −1.66545
\(672\) 0 0
\(673\) −29.1359 −1.12311 −0.561553 0.827441i \(-0.689796\pi\)
−0.561553 + 0.827441i \(0.689796\pi\)
\(674\) 0 0
\(675\) −5.61350 −0.216064
\(676\) 0 0
\(677\) −10.9481 −0.420770 −0.210385 0.977619i \(-0.567472\pi\)
−0.210385 + 0.977619i \(0.567472\pi\)
\(678\) 0 0
\(679\) −1.53880 −0.0590538
\(680\) 0 0
\(681\) −26.3035 −1.00795
\(682\) 0 0
\(683\) 36.2799 1.38821 0.694106 0.719872i \(-0.255800\pi\)
0.694106 + 0.719872i \(0.255800\pi\)
\(684\) 0 0
\(685\) −21.9253 −0.837723
\(686\) 0 0
\(687\) −13.3126 −0.507908
\(688\) 0 0
\(689\) 6.87783 0.262025
\(690\) 0 0
\(691\) −31.5471 −1.20011 −0.600054 0.799960i \(-0.704854\pi\)
−0.600054 + 0.799960i \(0.704854\pi\)
\(692\) 0 0
\(693\) −3.10297 −0.117872
\(694\) 0 0
\(695\) 6.73566 0.255498
\(696\) 0 0
\(697\) 1.94345 0.0736135
\(698\) 0 0
\(699\) 30.8405 1.16649
\(700\) 0 0
\(701\) −51.9336 −1.96150 −0.980752 0.195257i \(-0.937446\pi\)
−0.980752 + 0.195257i \(0.937446\pi\)
\(702\) 0 0
\(703\) 0.193252 0.00728865
\(704\) 0 0
\(705\) 2.00000 0.0753244
\(706\) 0 0
\(707\) 4.38650 0.164971
\(708\) 0 0
\(709\) −35.5953 −1.33681 −0.668406 0.743797i \(-0.733023\pi\)
−0.668406 + 0.743797i \(0.733023\pi\)
\(710\) 0 0
\(711\) 1.41478 0.0530583
\(712\) 0 0
\(713\) 26.7549 1.00198
\(714\) 0 0
\(715\) 6.00000 0.224387
\(716\) 0 0
\(717\) 41.6975 1.55722
\(718\) 0 0
\(719\) −26.2635 −0.979465 −0.489732 0.871873i \(-0.662906\pi\)
−0.489732 + 0.871873i \(0.662906\pi\)
\(720\) 0 0
\(721\) −4.13124 −0.153855
\(722\) 0 0
\(723\) −5.47133 −0.203481
\(724\) 0 0
\(725\) 5.02827 0.186745
\(726\) 0 0
\(727\) 22.4804 0.833752 0.416876 0.908963i \(-0.363125\pi\)
0.416876 + 0.908963i \(0.363125\pi\)
\(728\) 0 0
\(729\) 30.0101 1.11149
\(730\) 0 0
\(731\) 8.69832 0.321719
\(732\) 0 0
\(733\) 28.1806 1.04087 0.520437 0.853900i \(-0.325769\pi\)
0.520437 + 0.853900i \(0.325769\pi\)
\(734\) 0 0
\(735\) 7.95719 0.293505
\(736\) 0 0
\(737\) 23.0101 0.847589
\(738\) 0 0
\(739\) −35.8506 −1.31879 −0.659393 0.751798i \(-0.729187\pi\)
−0.659393 + 0.751798i \(0.729187\pi\)
\(740\) 0 0
\(741\) 2.73566 0.100497
\(742\) 0 0
\(743\) 33.5525 1.23092 0.615462 0.788167i \(-0.288970\pi\)
0.615462 + 0.788167i \(0.288970\pi\)
\(744\) 0 0
\(745\) −20.9909 −0.769048
\(746\) 0 0
\(747\) 1.62177 0.0593374
\(748\) 0 0
\(749\) 12.1312 0.443266
\(750\) 0 0
\(751\) 11.4257 0.416930 0.208465 0.978030i \(-0.433153\pi\)
0.208465 + 0.978030i \(0.433153\pi\)
\(752\) 0 0
\(753\) −24.1131 −0.878730
\(754\) 0 0
\(755\) 2.84049 0.103376
\(756\) 0 0
\(757\) 13.9144 0.505727 0.252863 0.967502i \(-0.418628\pi\)
0.252863 + 0.967502i \(0.418628\pi\)
\(758\) 0 0
\(759\) −26.7549 −0.971140
\(760\) 0 0
\(761\) 1.30274 0.0472243 0.0236121 0.999721i \(-0.492483\pi\)
0.0236121 + 0.999721i \(0.492483\pi\)
\(762\) 0 0
\(763\) −9.79407 −0.354569
\(764\) 0 0
\(765\) 1.41478 0.0511514
\(766\) 0 0
\(767\) 14.5561 0.525592
\(768\) 0 0
\(769\) 16.5297 0.596077 0.298039 0.954554i \(-0.403668\pi\)
0.298039 + 0.954554i \(0.403668\pi\)
\(770\) 0 0
\(771\) 35.6327 1.28328
\(772\) 0 0
\(773\) 17.1084 0.615347 0.307674 0.951492i \(-0.400449\pi\)
0.307674 + 0.951492i \(0.400449\pi\)
\(774\) 0 0
\(775\) 5.02827 0.180621
\(776\) 0 0
\(777\) −0.386505 −0.0138658
\(778\) 0 0
\(779\) 0.971726 0.0348157
\(780\) 0 0
\(781\) −43.2654 −1.54816
\(782\) 0 0
\(783\) −28.2262 −1.00872
\(784\) 0 0
\(785\) −15.4713 −0.552195
\(786\) 0 0
\(787\) −20.4293 −0.728226 −0.364113 0.931355i \(-0.618628\pi\)
−0.364113 + 0.931355i \(0.618628\pi\)
\(788\) 0 0
\(789\) −31.2835 −1.11372
\(790\) 0 0
\(791\) −8.95358 −0.318353
\(792\) 0 0
\(793\) 23.4713 0.833491
\(794\) 0 0
\(795\) −5.76394 −0.204426
\(796\) 0 0
\(797\) 4.69285 0.166229 0.0831147 0.996540i \(-0.473513\pi\)
0.0831147 + 0.996540i \(0.473513\pi\)
\(798\) 0 0
\(799\) −2.64177 −0.0934591
\(800\) 0 0
\(801\) −9.21606 −0.325634
\(802\) 0 0
\(803\) −16.6983 −0.589271
\(804\) 0 0
\(805\) 7.02827 0.247714
\(806\) 0 0
\(807\) −28.1131 −0.989628
\(808\) 0 0
\(809\) −24.7658 −0.870719 −0.435359 0.900257i \(-0.643379\pi\)
−0.435359 + 0.900257i \(0.643379\pi\)
\(810\) 0 0
\(811\) −48.8789 −1.71637 −0.858185 0.513341i \(-0.828408\pi\)
−0.858185 + 0.513341i \(0.828408\pi\)
\(812\) 0 0
\(813\) −21.4257 −0.751432
\(814\) 0 0
\(815\) 12.3492 0.432572
\(816\) 0 0
\(817\) 4.34916 0.152158
\(818\) 0 0
\(819\) 1.68819 0.0589901
\(820\) 0 0
\(821\) 17.0283 0.594291 0.297145 0.954832i \(-0.403965\pi\)
0.297145 + 0.954832i \(0.403965\pi\)
\(822\) 0 0
\(823\) −53.8205 −1.87606 −0.938032 0.346548i \(-0.887354\pi\)
−0.938032 + 0.346548i \(0.887354\pi\)
\(824\) 0 0
\(825\) −5.02827 −0.175062
\(826\) 0 0
\(827\) −23.1951 −0.806573 −0.403286 0.915074i \(-0.632132\pi\)
−0.403286 + 0.915074i \(0.632132\pi\)
\(828\) 0 0
\(829\) −17.3582 −0.602876 −0.301438 0.953486i \(-0.597467\pi\)
−0.301438 + 0.953486i \(0.597467\pi\)
\(830\) 0 0
\(831\) 16.0275 0.555987
\(832\) 0 0
\(833\) −10.5105 −0.364168
\(834\) 0 0
\(835\) 2.21245 0.0765651
\(836\) 0 0
\(837\) −28.2262 −0.975640
\(838\) 0 0
\(839\) −45.4148 −1.56789 −0.783946 0.620829i \(-0.786796\pi\)
−0.783946 + 0.620829i \(0.786796\pi\)
\(840\) 0 0
\(841\) −3.71646 −0.128154
\(842\) 0 0
\(843\) 28.9827 0.998216
\(844\) 0 0
\(845\) 9.73566 0.334917
\(846\) 0 0
\(847\) −0.0373465 −0.00128324
\(848\) 0 0
\(849\) 21.6410 0.742716
\(850\) 0 0
\(851\) 1.02827 0.0352488
\(852\) 0 0
\(853\) −28.0950 −0.961953 −0.480976 0.876734i \(-0.659718\pi\)
−0.480976 + 0.876734i \(0.659718\pi\)
\(854\) 0 0
\(855\) 0.707389 0.0241922
\(856\) 0 0
\(857\) −34.6000 −1.18191 −0.590957 0.806703i \(-0.701250\pi\)
−0.590957 + 0.806703i \(0.701250\pi\)
\(858\) 0 0
\(859\) 5.74474 0.196008 0.0980039 0.995186i \(-0.468754\pi\)
0.0980039 + 0.995186i \(0.468754\pi\)
\(860\) 0 0
\(861\) −1.94345 −0.0662327
\(862\) 0 0
\(863\) −4.03188 −0.137247 −0.0686234 0.997643i \(-0.521861\pi\)
−0.0686234 + 0.997643i \(0.521861\pi\)
\(864\) 0 0
\(865\) 2.50506 0.0851747
\(866\) 0 0
\(867\) −19.6838 −0.668496
\(868\) 0 0
\(869\) 6.64177 0.225307
\(870\) 0 0
\(871\) −12.5188 −0.424183
\(872\) 0 0
\(873\) −0.824093 −0.0278913
\(874\) 0 0
\(875\) 1.32088 0.0446540
\(876\) 0 0
\(877\) 38.9590 1.31555 0.657777 0.753213i \(-0.271497\pi\)
0.657777 + 0.753213i \(0.271497\pi\)
\(878\) 0 0
\(879\) −21.4631 −0.723931
\(880\) 0 0
\(881\) −11.7074 −0.394432 −0.197216 0.980360i \(-0.563190\pi\)
−0.197216 + 0.980360i \(0.563190\pi\)
\(882\) 0 0
\(883\) −28.3382 −0.953657 −0.476829 0.878996i \(-0.658214\pi\)
−0.476829 + 0.878996i \(0.658214\pi\)
\(884\) 0 0
\(885\) −12.1987 −0.410055
\(886\) 0 0
\(887\) 3.13856 0.105383 0.0526913 0.998611i \(-0.483220\pi\)
0.0526913 + 0.998611i \(0.483220\pi\)
\(888\) 0 0
\(889\) 1.48947 0.0499553
\(890\) 0 0
\(891\) 21.1787 0.709514
\(892\) 0 0
\(893\) −1.32088 −0.0442017
\(894\) 0 0
\(895\) 6.00000 0.200558
\(896\) 0 0
\(897\) 14.5561 0.486016
\(898\) 0 0
\(899\) 25.2835 0.843253
\(900\) 0 0
\(901\) 7.61350 0.253642
\(902\) 0 0
\(903\) −8.69832 −0.289462
\(904\) 0 0
\(905\) 12.8296 0.426469
\(906\) 0 0
\(907\) −13.1842 −0.437774 −0.218887 0.975750i \(-0.570243\pi\)
−0.218887 + 0.975750i \(0.570243\pi\)
\(908\) 0 0
\(909\) 2.34916 0.0779167
\(910\) 0 0
\(911\) −43.6519 −1.44625 −0.723126 0.690716i \(-0.757295\pi\)
−0.723126 + 0.690716i \(0.757295\pi\)
\(912\) 0 0
\(913\) 7.61350 0.251970
\(914\) 0 0
\(915\) −19.6700 −0.650272
\(916\) 0 0
\(917\) 8.51053 0.281042
\(918\) 0 0
\(919\) 19.8013 0.653184 0.326592 0.945165i \(-0.394100\pi\)
0.326592 + 0.945165i \(0.394100\pi\)
\(920\) 0 0
\(921\) −27.1896 −0.895929
\(922\) 0 0
\(923\) 23.5388 0.774789
\(924\) 0 0
\(925\) 0.193252 0.00635410
\(926\) 0 0
\(927\) −2.21245 −0.0726665
\(928\) 0 0
\(929\) 30.0000 0.984268 0.492134 0.870519i \(-0.336217\pi\)
0.492134 + 0.870519i \(0.336217\pi\)
\(930\) 0 0
\(931\) −5.25526 −0.172234
\(932\) 0 0
\(933\) −0.142169 −0.00465439
\(934\) 0 0
\(935\) 6.64177 0.217209
\(936\) 0 0
\(937\) −31.9709 −1.04444 −0.522222 0.852809i \(-0.674897\pi\)
−0.522222 + 0.852809i \(0.674897\pi\)
\(938\) 0 0
\(939\) 40.0109 1.30571
\(940\) 0 0
\(941\) −38.8114 −1.26522 −0.632608 0.774472i \(-0.718016\pi\)
−0.632608 + 0.774472i \(0.718016\pi\)
\(942\) 0 0
\(943\) 5.17044 0.168373
\(944\) 0 0
\(945\) −7.41478 −0.241203
\(946\) 0 0
\(947\) 32.9619 1.07112 0.535558 0.844498i \(-0.320101\pi\)
0.535558 + 0.844498i \(0.320101\pi\)
\(948\) 0 0
\(949\) 9.08482 0.294906
\(950\) 0 0
\(951\) −1.86610 −0.0605126
\(952\) 0 0
\(953\) −6.82409 −0.221054 −0.110527 0.993873i \(-0.535254\pi\)
−0.110527 + 0.993873i \(0.535254\pi\)
\(954\) 0 0
\(955\) 11.9253 0.385894
\(956\) 0 0
\(957\) −25.2835 −0.817301
\(958\) 0 0
\(959\) −28.9608 −0.935193
\(960\) 0 0
\(961\) −5.71646 −0.184402
\(962\) 0 0
\(963\) 6.49679 0.209356
\(964\) 0 0
\(965\) 2.06201 0.0663785
\(966\) 0 0
\(967\) 17.7458 0.570666 0.285333 0.958428i \(-0.407896\pi\)
0.285333 + 0.958428i \(0.407896\pi\)
\(968\) 0 0
\(969\) 3.02827 0.0972822
\(970\) 0 0
\(971\) 31.1595 0.999956 0.499978 0.866038i \(-0.333341\pi\)
0.499978 + 0.866038i \(0.333341\pi\)
\(972\) 0 0
\(973\) 8.89703 0.285226
\(974\) 0 0
\(975\) 2.73566 0.0876113
\(976\) 0 0
\(977\) −14.3063 −0.457701 −0.228850 0.973462i \(-0.573497\pi\)
−0.228850 + 0.973462i \(0.573497\pi\)
\(978\) 0 0
\(979\) −43.2654 −1.38277
\(980\) 0 0
\(981\) −5.24514 −0.167464
\(982\) 0 0
\(983\) 48.2509 1.53896 0.769482 0.638669i \(-0.220515\pi\)
0.769482 + 0.638669i \(0.220515\pi\)
\(984\) 0 0
\(985\) 2.31181 0.0736605
\(986\) 0 0
\(987\) 2.64177 0.0840884
\(988\) 0 0
\(989\) 23.1414 0.735853
\(990\) 0 0
\(991\) −6.04562 −0.192045 −0.0960227 0.995379i \(-0.530612\pi\)
−0.0960227 + 0.995379i \(0.530612\pi\)
\(992\) 0 0
\(993\) −33.0957 −1.05026
\(994\) 0 0
\(995\) −0.773010 −0.0245061
\(996\) 0 0
\(997\) −23.9072 −0.757147 −0.378574 0.925571i \(-0.623585\pi\)
−0.378574 + 0.925571i \(0.623585\pi\)
\(998\) 0 0
\(999\) −1.08482 −0.0343222
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 3040.2.a.k.1.3 3
4.3 odd 2 3040.2.a.n.1.1 yes 3
8.3 odd 2 6080.2.a.bp.1.3 3
8.5 even 2 6080.2.a.bz.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3040.2.a.k.1.3 3 1.1 even 1 trivial
3040.2.a.n.1.1 yes 3 4.3 odd 2
6080.2.a.bp.1.3 3 8.3 odd 2
6080.2.a.bz.1.1 3 8.5 even 2