Properties

Label 2352.3.m.h.1471.3
Level $2352$
Weight $3$
Character 2352.1471
Analytic conductor $64.087$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2352,3,Mod(1471,2352)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2352, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 0, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2352.1471");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2352 = 2^{4} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 2352.m (of order \(2\), degree \(1\), minimal)

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: no
Analytic conductor: \(64.0873581775\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-19})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 4x^{2} - 5x + 25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 336)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1471.3
Root \(-1.63746 - 1.52274i\) of defining polynomial
Character \(\chi\) \(=\) 2352.1471
Dual form 2352.3.m.h.1471.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.73205i q^{3} -3.27492 q^{5} -3.00000 q^{9} +O(q^{10})\) \(q+1.73205i q^{3} -3.27492 q^{5} -3.00000 q^{9} -18.6915i q^{11} -10.2749 q^{13} -5.67232i q^{15} +25.0997 q^{17} -30.9309i q^{19} +5.25370i q^{23} -14.2749 q^{25} -5.19615i q^{27} -42.0241 q^{29} +52.1342i q^{31} +32.3746 q^{33} +35.9244 q^{37} -17.7967i q^{39} +38.7492 q^{41} +31.5380i q^{43} +9.82475 q^{45} +24.2487i q^{47} +43.4739i q^{51} -65.6254 q^{53} +61.2130i q^{55} +53.5739 q^{57} -44.8449i q^{59} +63.0997 q^{61} +33.6495 q^{65} +22.2131i q^{67} -9.09967 q^{69} +97.5703i q^{71} -91.7251 q^{73} -24.7249i q^{75} +9.26733i q^{79} +9.00000 q^{81} +51.1976i q^{83} -82.1993 q^{85} -72.7879i q^{87} +53.4502 q^{89} -90.2990 q^{93} +101.296i q^{95} -129.522 q^{97} +56.0744i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{5} - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{5} - 12 q^{9} - 26 q^{13} + 40 q^{17} - 42 q^{25} - 2 q^{29} + 54 q^{33} + 38 q^{37} + 4 q^{41} - 6 q^{45} - 338 q^{53} + 18 q^{57} + 192 q^{61} + 44 q^{65} + 24 q^{69} - 382 q^{73} + 36 q^{81} - 208 q^{85} + 244 q^{89} - 180 q^{93} - 50 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2352\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(1471\) \(1765\) \(2257\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(1\)

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
\(3\) 1.73205i 0.577350i
\(4\) 0 0
\(5\) −3.27492 −0.654983 −0.327492 0.944854i \(-0.606203\pi\)
−0.327492 + 0.944854i \(0.606203\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) −3.00000 −0.333333
\(10\) 0 0
\(11\) − 18.6915i − 1.69923i −0.527407 0.849613i \(-0.676836\pi\)
0.527407 0.849613i \(-0.323164\pi\)
\(12\) 0 0
\(13\) −10.2749 −0.790378 −0.395189 0.918600i \(-0.629321\pi\)
−0.395189 + 0.918600i \(0.629321\pi\)
\(14\) 0 0
\(15\) − 5.67232i − 0.378155i
\(16\) 0 0
\(17\) 25.0997 1.47645 0.738226 0.674554i \(-0.235664\pi\)
0.738226 + 0.674554i \(0.235664\pi\)
\(18\) 0 0
\(19\) − 30.9309i − 1.62794i −0.580905 0.813972i \(-0.697301\pi\)
0.580905 0.813972i \(-0.302699\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 5.25370i 0.228422i 0.993457 + 0.114211i \(0.0364339\pi\)
−0.993457 + 0.114211i \(0.963566\pi\)
\(24\) 0 0
\(25\) −14.2749 −0.570997
\(26\) 0 0
\(27\) − 5.19615i − 0.192450i
\(28\) 0 0
\(29\) −42.0241 −1.44911 −0.724553 0.689219i \(-0.757954\pi\)
−0.724553 + 0.689219i \(0.757954\pi\)
\(30\) 0 0
\(31\) 52.1342i 1.68175i 0.541232 + 0.840873i \(0.317958\pi\)
−0.541232 + 0.840873i \(0.682042\pi\)
\(32\) 0 0
\(33\) 32.3746 0.981048
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 35.9244 0.970930 0.485465 0.874256i \(-0.338650\pi\)
0.485465 + 0.874256i \(0.338650\pi\)
\(38\) 0 0
\(39\) − 17.7967i − 0.456325i
\(40\) 0 0
\(41\) 38.7492 0.945102 0.472551 0.881303i \(-0.343333\pi\)
0.472551 + 0.881303i \(0.343333\pi\)
\(42\) 0 0
\(43\) 31.5380i 0.733442i 0.930331 + 0.366721i \(0.119520\pi\)
−0.930331 + 0.366721i \(0.880480\pi\)
\(44\) 0 0
\(45\) 9.82475 0.218328
\(46\) 0 0
\(47\) 24.2487i 0.515930i 0.966154 + 0.257965i \(0.0830519\pi\)
−0.966154 + 0.257965i \(0.916948\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 43.4739i 0.852429i
\(52\) 0 0
\(53\) −65.6254 −1.23822 −0.619108 0.785306i \(-0.712506\pi\)
−0.619108 + 0.785306i \(0.712506\pi\)
\(54\) 0 0
\(55\) 61.2130i 1.11296i
\(56\) 0 0
\(57\) 53.5739 0.939893
\(58\) 0 0
\(59\) − 44.8449i − 0.760083i −0.924970 0.380041i \(-0.875910\pi\)
0.924970 0.380041i \(-0.124090\pi\)
\(60\) 0 0
\(61\) 63.0997 1.03442 0.517210 0.855858i \(-0.326970\pi\)
0.517210 + 0.855858i \(0.326970\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 33.6495 0.517685
\(66\) 0 0
\(67\) 22.2131i 0.331539i 0.986165 + 0.165770i \(0.0530108\pi\)
−0.986165 + 0.165770i \(0.946989\pi\)
\(68\) 0 0
\(69\) −9.09967 −0.131879
\(70\) 0 0
\(71\) 97.5703i 1.37423i 0.726549 + 0.687115i \(0.241123\pi\)
−0.726549 + 0.687115i \(0.758877\pi\)
\(72\) 0 0
\(73\) −91.7251 −1.25651 −0.628254 0.778008i \(-0.716230\pi\)
−0.628254 + 0.778008i \(0.716230\pi\)
\(74\) 0 0
\(75\) − 24.7249i − 0.329665i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 9.26733i 0.117308i 0.998278 + 0.0586540i \(0.0186809\pi\)
−0.998278 + 0.0586540i \(0.981319\pi\)
\(80\) 0 0
\(81\) 9.00000 0.111111
\(82\) 0 0
\(83\) 51.1976i 0.616839i 0.951250 + 0.308419i \(0.0998000\pi\)
−0.951250 + 0.308419i \(0.900200\pi\)
\(84\) 0 0
\(85\) −82.1993 −0.967051
\(86\) 0 0
\(87\) − 72.7879i − 0.836642i
\(88\) 0 0
\(89\) 53.4502 0.600564 0.300282 0.953851i \(-0.402919\pi\)
0.300282 + 0.953851i \(0.402919\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −90.2990 −0.970957
\(94\) 0 0
\(95\) 101.296i 1.06628i
\(96\) 0 0
\(97\) −129.522 −1.33528 −0.667641 0.744483i \(-0.732696\pi\)
−0.667641 + 0.744483i \(0.732696\pi\)
\(98\) 0 0
\(99\) 56.0744i 0.566408i
\(100\) 0 0
\(101\) −53.6977 −0.531660 −0.265830 0.964020i \(-0.585646\pi\)
−0.265830 + 0.964020i \(0.585646\pi\)
\(102\) 0 0
\(103\) 32.1134i 0.311781i 0.987774 + 0.155890i \(0.0498247\pi\)
−0.987774 + 0.155890i \(0.950175\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 57.2569i − 0.535112i −0.963542 0.267556i \(-0.913784\pi\)
0.963542 0.267556i \(-0.0862160\pi\)
\(108\) 0 0
\(109\) −141.375 −1.29701 −0.648507 0.761208i \(-0.724606\pi\)
−0.648507 + 0.761208i \(0.724606\pi\)
\(110\) 0 0
\(111\) 62.2229i 0.560567i
\(112\) 0 0
\(113\) −72.1512 −0.638506 −0.319253 0.947670i \(-0.603432\pi\)
−0.319253 + 0.947670i \(0.603432\pi\)
\(114\) 0 0
\(115\) − 17.2054i − 0.149612i
\(116\) 0 0
\(117\) 30.8248 0.263459
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −228.371 −1.88737
\(122\) 0 0
\(123\) 67.1155i 0.545655i
\(124\) 0 0
\(125\) 128.622 1.02898
\(126\) 0 0
\(127\) − 150.887i − 1.18809i −0.804433 0.594043i \(-0.797531\pi\)
0.804433 0.594043i \(-0.202469\pi\)
\(128\) 0 0
\(129\) −54.6254 −0.423453
\(130\) 0 0
\(131\) 140.741i 1.07436i 0.843469 + 0.537178i \(0.180510\pi\)
−0.843469 + 0.537178i \(0.819490\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 17.0170i 0.126052i
\(136\) 0 0
\(137\) −45.0515 −0.328843 −0.164421 0.986390i \(-0.552576\pi\)
−0.164421 + 0.986390i \(0.552576\pi\)
\(138\) 0 0
\(139\) − 32.2285i − 0.231860i −0.993257 0.115930i \(-0.963015\pi\)
0.993257 0.115930i \(-0.0369848\pi\)
\(140\) 0 0
\(141\) −42.0000 −0.297872
\(142\) 0 0
\(143\) 192.053i 1.34303i
\(144\) 0 0
\(145\) 137.625 0.949141
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 157.148 1.05468 0.527342 0.849653i \(-0.323189\pi\)
0.527342 + 0.849653i \(0.323189\pi\)
\(150\) 0 0
\(151\) 8.67607i 0.0574574i 0.999587 + 0.0287287i \(0.00914589\pi\)
−0.999587 + 0.0287287i \(0.990854\pi\)
\(152\) 0 0
\(153\) −75.2990 −0.492150
\(154\) 0 0
\(155\) − 170.735i − 1.10152i
\(156\) 0 0
\(157\) 110.048 0.700944 0.350472 0.936573i \(-0.386021\pi\)
0.350472 + 0.936573i \(0.386021\pi\)
\(158\) 0 0
\(159\) − 113.667i − 0.714884i
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) − 147.627i − 0.905688i −0.891590 0.452844i \(-0.850409\pi\)
0.891590 0.452844i \(-0.149591\pi\)
\(164\) 0 0
\(165\) −106.024 −0.642570
\(166\) 0 0
\(167\) − 60.5642i − 0.362660i −0.983422 0.181330i \(-0.941960\pi\)
0.983422 0.181330i \(-0.0580402\pi\)
\(168\) 0 0
\(169\) −63.4261 −0.375302
\(170\) 0 0
\(171\) 92.7928i 0.542648i
\(172\) 0 0
\(173\) 90.8522 0.525157 0.262578 0.964911i \(-0.415427\pi\)
0.262578 + 0.964911i \(0.415427\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 77.6736 0.438834
\(178\) 0 0
\(179\) 318.666i 1.78026i 0.455711 + 0.890128i \(0.349385\pi\)
−0.455711 + 0.890128i \(0.650615\pi\)
\(180\) 0 0
\(181\) −31.6769 −0.175011 −0.0875053 0.996164i \(-0.527889\pi\)
−0.0875053 + 0.996164i \(0.527889\pi\)
\(182\) 0 0
\(183\) 109.292i 0.597223i
\(184\) 0 0
\(185\) −117.650 −0.635943
\(186\) 0 0
\(187\) − 469.150i − 2.50882i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 241.011i 1.26184i 0.775849 + 0.630919i \(0.217322\pi\)
−0.775849 + 0.630919i \(0.782678\pi\)
\(192\) 0 0
\(193\) −341.447 −1.76915 −0.884577 0.466394i \(-0.845553\pi\)
−0.884577 + 0.466394i \(0.845553\pi\)
\(194\) 0 0
\(195\) 58.2826i 0.298885i
\(196\) 0 0
\(197\) −142.550 −0.723603 −0.361802 0.932255i \(-0.617838\pi\)
−0.361802 + 0.932255i \(0.617838\pi\)
\(198\) 0 0
\(199\) 5.25370i 0.0264005i 0.999913 + 0.0132002i \(0.00420189\pi\)
−0.999913 + 0.0132002i \(0.995798\pi\)
\(200\) 0 0
\(201\) −38.4743 −0.191414
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −126.900 −0.619026
\(206\) 0 0
\(207\) − 15.7611i − 0.0761405i
\(208\) 0 0
\(209\) −578.145 −2.76624
\(210\) 0 0
\(211\) 271.267i 1.28563i 0.766023 + 0.642814i \(0.222233\pi\)
−0.766023 + 0.642814i \(0.777767\pi\)
\(212\) 0 0
\(213\) −168.997 −0.793412
\(214\) 0 0
\(215\) − 103.284i − 0.480392i
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) − 158.873i − 0.725445i
\(220\) 0 0
\(221\) −257.897 −1.16695
\(222\) 0 0
\(223\) − 76.8907i − 0.344801i −0.985027 0.172401i \(-0.944848\pi\)
0.985027 0.172401i \(-0.0551524\pi\)
\(224\) 0 0
\(225\) 42.8248 0.190332
\(226\) 0 0
\(227\) 138.292i 0.609217i 0.952478 + 0.304608i \(0.0985256\pi\)
−0.952478 + 0.304608i \(0.901474\pi\)
\(228\) 0 0
\(229\) −134.371 −0.586774 −0.293387 0.955994i \(-0.594782\pi\)
−0.293387 + 0.955994i \(0.594782\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −71.8488 −0.308364 −0.154182 0.988042i \(-0.549274\pi\)
−0.154182 + 0.988042i \(0.549274\pi\)
\(234\) 0 0
\(235\) − 79.4125i − 0.337926i
\(236\) 0 0
\(237\) −16.0515 −0.0677278
\(238\) 0 0
\(239\) 331.308i 1.38623i 0.720829 + 0.693113i \(0.243761\pi\)
−0.720829 + 0.693113i \(0.756239\pi\)
\(240\) 0 0
\(241\) −106.526 −0.442016 −0.221008 0.975272i \(-0.570935\pi\)
−0.221008 + 0.975272i \(0.570935\pi\)
\(242\) 0 0
\(243\) 15.5885i 0.0641500i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 317.813i 1.28669i
\(248\) 0 0
\(249\) −88.6769 −0.356132
\(250\) 0 0
\(251\) 251.100i 1.00040i 0.865910 + 0.500199i \(0.166740\pi\)
−0.865910 + 0.500199i \(0.833260\pi\)
\(252\) 0 0
\(253\) 98.1993 0.388140
\(254\) 0 0
\(255\) − 142.373i − 0.558327i
\(256\) 0 0
\(257\) 406.399 1.58132 0.790659 0.612257i \(-0.209738\pi\)
0.790659 + 0.612257i \(0.209738\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 126.072 0.483036
\(262\) 0 0
\(263\) − 56.4298i − 0.214562i −0.994229 0.107281i \(-0.965786\pi\)
0.994229 0.107281i \(-0.0342144\pi\)
\(264\) 0 0
\(265\) 214.918 0.811011
\(266\) 0 0
\(267\) 92.5784i 0.346736i
\(268\) 0 0
\(269\) 136.471 0.507327 0.253663 0.967293i \(-0.418364\pi\)
0.253663 + 0.967293i \(0.418364\pi\)
\(270\) 0 0
\(271\) 0.617163i 0.00227735i 0.999999 + 0.00113868i \(0.000362452\pi\)
−0.999999 + 0.00113868i \(0.999638\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 266.819i 0.970252i
\(276\) 0 0
\(277\) 391.265 1.41251 0.706254 0.707958i \(-0.250383\pi\)
0.706254 + 0.707958i \(0.250383\pi\)
\(278\) 0 0
\(279\) − 156.402i − 0.560582i
\(280\) 0 0
\(281\) −534.248 −1.90124 −0.950618 0.310362i \(-0.899550\pi\)
−0.950618 + 0.310362i \(0.899550\pi\)
\(282\) 0 0
\(283\) − 64.0442i − 0.226304i −0.993578 0.113152i \(-0.963905\pi\)
0.993578 0.113152i \(-0.0360948\pi\)
\(284\) 0 0
\(285\) −175.450 −0.615615
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 340.993 1.17991
\(290\) 0 0
\(291\) − 224.339i − 0.770926i
\(292\) 0 0
\(293\) −88.6703 −0.302629 −0.151314 0.988486i \(-0.548351\pi\)
−0.151314 + 0.988486i \(0.548351\pi\)
\(294\) 0 0
\(295\) 146.863i 0.497841i
\(296\) 0 0
\(297\) −97.1238 −0.327016
\(298\) 0 0
\(299\) − 53.9813i − 0.180539i
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) − 93.0071i − 0.306954i
\(304\) 0 0
\(305\) −206.646 −0.677528
\(306\) 0 0
\(307\) − 147.004i − 0.478841i −0.970916 0.239421i \(-0.923043\pi\)
0.970916 0.239421i \(-0.0769575\pi\)
\(308\) 0 0
\(309\) −55.6221 −0.180007
\(310\) 0 0
\(311\) 184.204i 0.592297i 0.955142 + 0.296149i \(0.0957024\pi\)
−0.955142 + 0.296149i \(0.904298\pi\)
\(312\) 0 0
\(313\) 223.042 0.712593 0.356296 0.934373i \(-0.384039\pi\)
0.356296 + 0.934373i \(0.384039\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −620.718 −1.95810 −0.979051 0.203614i \(-0.934731\pi\)
−0.979051 + 0.203614i \(0.934731\pi\)
\(318\) 0 0
\(319\) 785.492i 2.46236i
\(320\) 0 0
\(321\) 99.1719 0.308947
\(322\) 0 0
\(323\) − 776.356i − 2.40358i
\(324\) 0 0
\(325\) 146.674 0.451303
\(326\) 0 0
\(327\) − 244.868i − 0.748832i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) − 363.485i − 1.09814i −0.835776 0.549071i \(-0.814982\pi\)
0.835776 0.549071i \(-0.185018\pi\)
\(332\) 0 0
\(333\) −107.773 −0.323643
\(334\) 0 0
\(335\) − 72.7461i − 0.217153i
\(336\) 0 0
\(337\) −322.093 −0.955766 −0.477883 0.878424i \(-0.658596\pi\)
−0.477883 + 0.878424i \(0.658596\pi\)
\(338\) 0 0
\(339\) − 124.969i − 0.368642i
\(340\) 0 0
\(341\) 974.464 2.85767
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 29.8007 0.0863787
\(346\) 0 0
\(347\) 352.291i 1.01525i 0.861579 + 0.507624i \(0.169476\pi\)
−0.861579 + 0.507624i \(0.830524\pi\)
\(348\) 0 0
\(349\) −221.395 −0.634371 −0.317185 0.948364i \(-0.602738\pi\)
−0.317185 + 0.948364i \(0.602738\pi\)
\(350\) 0 0
\(351\) 53.3900i 0.152108i
\(352\) 0 0
\(353\) 57.7459 0.163586 0.0817930 0.996649i \(-0.473935\pi\)
0.0817930 + 0.996649i \(0.473935\pi\)
\(354\) 0 0
\(355\) − 319.535i − 0.900097i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) − 280.727i − 0.781971i −0.920397 0.390985i \(-0.872134\pi\)
0.920397 0.390985i \(-0.127866\pi\)
\(360\) 0 0
\(361\) −595.722 −1.65020
\(362\) 0 0
\(363\) − 395.551i − 1.08967i
\(364\) 0 0
\(365\) 300.392 0.822992
\(366\) 0 0
\(367\) − 92.7194i − 0.252641i −0.991989 0.126321i \(-0.959683\pi\)
0.991989 0.126321i \(-0.0403168\pi\)
\(368\) 0 0
\(369\) −116.248 −0.315034
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −156.873 −0.420571 −0.210285 0.977640i \(-0.567439\pi\)
−0.210285 + 0.977640i \(0.567439\pi\)
\(374\) 0 0
\(375\) 222.780i 0.594080i
\(376\) 0 0
\(377\) 431.794 1.14534
\(378\) 0 0
\(379\) 466.559i 1.23103i 0.788127 + 0.615513i \(0.211051\pi\)
−0.788127 + 0.615513i \(0.788949\pi\)
\(380\) 0 0
\(381\) 261.344 0.685942
\(382\) 0 0
\(383\) 496.089i 1.29527i 0.761950 + 0.647635i \(0.224242\pi\)
−0.761950 + 0.647635i \(0.775758\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) − 94.6140i − 0.244481i
\(388\) 0 0
\(389\) 648.296 1.66657 0.833285 0.552844i \(-0.186457\pi\)
0.833285 + 0.552844i \(0.186457\pi\)
\(390\) 0 0
\(391\) 131.866i 0.337253i
\(392\) 0 0
\(393\) −243.770 −0.620280
\(394\) 0 0
\(395\) − 30.3497i − 0.0768348i
\(396\) 0 0
\(397\) −312.419 −0.786951 −0.393475 0.919335i \(-0.628727\pi\)
−0.393475 + 0.919335i \(0.628727\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 32.0000 0.0798005 0.0399002 0.999204i \(-0.487296\pi\)
0.0399002 + 0.999204i \(0.487296\pi\)
\(402\) 0 0
\(403\) − 535.674i − 1.32922i
\(404\) 0 0
\(405\) −29.4743 −0.0727759
\(406\) 0 0
\(407\) − 671.480i − 1.64983i
\(408\) 0 0
\(409\) −610.739 −1.49325 −0.746625 0.665245i \(-0.768327\pi\)
−0.746625 + 0.665245i \(0.768327\pi\)
\(410\) 0 0
\(411\) − 78.0315i − 0.189858i
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) − 167.668i − 0.404019i
\(416\) 0 0
\(417\) 55.8214 0.133864
\(418\) 0 0
\(419\) 674.432i 1.60962i 0.593530 + 0.804812i \(0.297734\pi\)
−0.593530 + 0.804812i \(0.702266\pi\)
\(420\) 0 0
\(421\) −97.9792 −0.232730 −0.116365 0.993207i \(-0.537124\pi\)
−0.116365 + 0.993207i \(0.537124\pi\)
\(422\) 0 0
\(423\) − 72.7461i − 0.171977i
\(424\) 0 0
\(425\) −358.296 −0.843049
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) −332.646 −0.775399
\(430\) 0 0
\(431\) 2.45995i 0.00570755i 0.999996 + 0.00285377i \(0.000908385\pi\)
−0.999996 + 0.00285377i \(0.999092\pi\)
\(432\) 0 0
\(433\) −475.670 −1.09855 −0.549273 0.835643i \(-0.685095\pi\)
−0.549273 + 0.835643i \(0.685095\pi\)
\(434\) 0 0
\(435\) 238.374i 0.547987i
\(436\) 0 0
\(437\) 162.502 0.371857
\(438\) 0 0
\(439\) 137.046i 0.312179i 0.987743 + 0.156089i \(0.0498888\pi\)
−0.987743 + 0.156089i \(0.950111\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) − 161.033i − 0.363506i −0.983344 0.181753i \(-0.941823\pi\)
0.983344 0.181753i \(-0.0581772\pi\)
\(444\) 0 0
\(445\) −175.045 −0.393359
\(446\) 0 0
\(447\) 272.188i 0.608922i
\(448\) 0 0
\(449\) 279.643 0.622813 0.311406 0.950277i \(-0.399200\pi\)
0.311406 + 0.950277i \(0.399200\pi\)
\(450\) 0 0
\(451\) − 724.279i − 1.60594i
\(452\) 0 0
\(453\) −15.0274 −0.0331731
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 469.350 1.02703 0.513513 0.858082i \(-0.328344\pi\)
0.513513 + 0.858082i \(0.328344\pi\)
\(458\) 0 0
\(459\) − 130.422i − 0.284143i
\(460\) 0 0
\(461\) −234.440 −0.508547 −0.254274 0.967132i \(-0.581836\pi\)
−0.254274 + 0.967132i \(0.581836\pi\)
\(462\) 0 0
\(463\) 515.120i 1.11257i 0.830992 + 0.556285i \(0.187774\pi\)
−0.830992 + 0.556285i \(0.812226\pi\)
\(464\) 0 0
\(465\) 295.722 0.635961
\(466\) 0 0
\(467\) 140.500i 0.300857i 0.988621 + 0.150429i \(0.0480654\pi\)
−0.988621 + 0.150429i \(0.951935\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 190.609i 0.404690i
\(472\) 0 0
\(473\) 589.492 1.24628
\(474\) 0 0
\(475\) 441.536i 0.929550i
\(476\) 0 0
\(477\) 196.876 0.412738
\(478\) 0 0
\(479\) 907.091i 1.89372i 0.321648 + 0.946859i \(0.395763\pi\)
−0.321648 + 0.946859i \(0.604237\pi\)
\(480\) 0 0
\(481\) −369.120 −0.767402
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 424.175 0.874588
\(486\) 0 0
\(487\) − 290.006i − 0.595495i −0.954645 0.297748i \(-0.903765\pi\)
0.954645 0.297748i \(-0.0962354\pi\)
\(488\) 0 0
\(489\) 255.698 0.522899
\(490\) 0 0
\(491\) − 353.412i − 0.719779i −0.932995 0.359890i \(-0.882814\pi\)
0.932995 0.359890i \(-0.117186\pi\)
\(492\) 0 0
\(493\) −1054.79 −2.13953
\(494\) 0 0
\(495\) − 183.639i − 0.370988i
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) − 328.562i − 0.658440i −0.944253 0.329220i \(-0.893214\pi\)
0.944253 0.329220i \(-0.106786\pi\)
\(500\) 0 0
\(501\) 104.900 0.209382
\(502\) 0 0
\(503\) − 791.625i − 1.57381i −0.617076 0.786904i \(-0.711683\pi\)
0.617076 0.786904i \(-0.288317\pi\)
\(504\) 0 0
\(505\) 175.855 0.348229
\(506\) 0 0
\(507\) − 109.857i − 0.216681i
\(508\) 0 0
\(509\) 435.275 0.855157 0.427579 0.903978i \(-0.359367\pi\)
0.427579 + 0.903978i \(0.359367\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) −160.722 −0.313298
\(514\) 0 0
\(515\) − 105.169i − 0.204211i
\(516\) 0 0
\(517\) 453.244 0.876681
\(518\) 0 0
\(519\) 157.361i 0.303200i
\(520\) 0 0
\(521\) 258.756 0.496652 0.248326 0.968676i \(-0.420120\pi\)
0.248326 + 0.968676i \(0.420120\pi\)
\(522\) 0 0
\(523\) 934.757i 1.78730i 0.448767 + 0.893649i \(0.351863\pi\)
−0.448767 + 0.893649i \(0.648137\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 1308.55i 2.48302i
\(528\) 0 0
\(529\) 501.399 0.947824
\(530\) 0 0
\(531\) 134.535i 0.253361i
\(532\) 0 0
\(533\) −398.145 −0.746988
\(534\) 0 0
\(535\) 187.512i 0.350489i
\(536\) 0 0
\(537\) −551.945 −1.02783
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 106.914 0.197624 0.0988119 0.995106i \(-0.468496\pi\)
0.0988119 + 0.995106i \(0.468496\pi\)
\(542\) 0 0
\(543\) − 54.8660i − 0.101042i
\(544\) 0 0
\(545\) 462.990 0.849523
\(546\) 0 0
\(547\) 33.5534i 0.0613408i 0.999530 + 0.0306704i \(0.00976422\pi\)
−0.999530 + 0.0306704i \(0.990236\pi\)
\(548\) 0 0
\(549\) −189.299 −0.344807
\(550\) 0 0
\(551\) 1299.84i 2.35906i
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) − 203.775i − 0.367162i
\(556\) 0 0
\(557\) −969.908 −1.74131 −0.870653 0.491897i \(-0.836304\pi\)
−0.870653 + 0.491897i \(0.836304\pi\)
\(558\) 0 0
\(559\) − 324.050i − 0.579696i
\(560\) 0 0
\(561\) 812.591 1.44847
\(562\) 0 0
\(563\) − 114.619i − 0.203586i −0.994806 0.101793i \(-0.967542\pi\)
0.994806 0.101793i \(-0.0324579\pi\)
\(564\) 0 0
\(565\) 236.289 0.418211
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 304.550 0.535237 0.267618 0.963525i \(-0.413763\pi\)
0.267618 + 0.963525i \(0.413763\pi\)
\(570\) 0 0
\(571\) − 819.417i − 1.43506i −0.696530 0.717528i \(-0.745274\pi\)
0.696530 0.717528i \(-0.254726\pi\)
\(572\) 0 0
\(573\) −417.444 −0.728523
\(574\) 0 0
\(575\) − 74.9961i − 0.130428i
\(576\) 0 0
\(577\) 1035.79 1.79513 0.897566 0.440881i \(-0.145334\pi\)
0.897566 + 0.440881i \(0.145334\pi\)
\(578\) 0 0
\(579\) − 591.403i − 1.02142i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 1226.64i 2.10401i
\(584\) 0 0
\(585\) −100.949 −0.172562
\(586\) 0 0
\(587\) 991.224i 1.68863i 0.535850 + 0.844313i \(0.319991\pi\)
−0.535850 + 0.844313i \(0.680009\pi\)
\(588\) 0 0
\(589\) 1612.56 2.73779
\(590\) 0 0
\(591\) − 246.904i − 0.417773i
\(592\) 0 0
\(593\) −546.736 −0.921983 −0.460992 0.887405i \(-0.652506\pi\)
−0.460992 + 0.887405i \(0.652506\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −9.09967 −0.0152423
\(598\) 0 0
\(599\) 449.726i 0.750795i 0.926864 + 0.375397i \(0.122494\pi\)
−0.926864 + 0.375397i \(0.877506\pi\)
\(600\) 0 0
\(601\) 392.106 0.652423 0.326212 0.945297i \(-0.394228\pi\)
0.326212 + 0.945297i \(0.394228\pi\)
\(602\) 0 0
\(603\) − 66.6394i − 0.110513i
\(604\) 0 0
\(605\) 747.897 1.23619
\(606\) 0 0
\(607\) − 758.684i − 1.24989i −0.780668 0.624946i \(-0.785121\pi\)
0.780668 0.624946i \(-0.214879\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) − 249.154i − 0.407780i
\(612\) 0 0
\(613\) −435.341 −0.710180 −0.355090 0.934832i \(-0.615550\pi\)
−0.355090 + 0.934832i \(0.615550\pi\)
\(614\) 0 0
\(615\) − 219.798i − 0.357395i
\(616\) 0 0
\(617\) −145.189 −0.235315 −0.117658 0.993054i \(-0.537539\pi\)
−0.117658 + 0.993054i \(0.537539\pi\)
\(618\) 0 0
\(619\) 510.839i 0.825264i 0.910898 + 0.412632i \(0.135390\pi\)
−0.910898 + 0.412632i \(0.864610\pi\)
\(620\) 0 0
\(621\) 27.2990 0.0439598
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −64.3538 −0.102966
\(626\) 0 0
\(627\) − 1001.38i − 1.59709i
\(628\) 0 0
\(629\) 901.691 1.43353
\(630\) 0 0
\(631\) − 345.666i − 0.547807i −0.961757 0.273904i \(-0.911685\pi\)
0.961757 0.273904i \(-0.0883150\pi\)
\(632\) 0 0
\(633\) −469.849 −0.742257
\(634\) 0 0
\(635\) 494.142i 0.778177i
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) − 292.711i − 0.458076i
\(640\) 0 0
\(641\) 904.145 1.41052 0.705261 0.708948i \(-0.250830\pi\)
0.705261 + 0.708948i \(0.250830\pi\)
\(642\) 0 0
\(643\) 265.621i 0.413096i 0.978436 + 0.206548i \(0.0662230\pi\)
−0.978436 + 0.206548i \(0.933777\pi\)
\(644\) 0 0
\(645\) 178.894 0.277355
\(646\) 0 0
\(647\) 354.271i 0.547559i 0.961793 + 0.273779i \(0.0882738\pi\)
−0.961793 + 0.273779i \(0.911726\pi\)
\(648\) 0 0
\(649\) −838.217 −1.29155
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 780.966 1.19597 0.597983 0.801509i \(-0.295969\pi\)
0.597983 + 0.801509i \(0.295969\pi\)
\(654\) 0 0
\(655\) − 460.914i − 0.703685i
\(656\) 0 0
\(657\) 275.175 0.418836
\(658\) 0 0
\(659\) 309.905i 0.470265i 0.971963 + 0.235133i \(0.0755524\pi\)
−0.971963 + 0.235133i \(0.924448\pi\)
\(660\) 0 0
\(661\) 690.117 1.04405 0.522025 0.852930i \(-0.325177\pi\)
0.522025 + 0.852930i \(0.325177\pi\)
\(662\) 0 0
\(663\) − 446.691i − 0.673742i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) − 220.782i − 0.331007i
\(668\) 0 0
\(669\) 133.179 0.199071
\(670\) 0 0
\(671\) − 1179.43i − 1.75771i
\(672\) 0 0
\(673\) 714.238 1.06127 0.530637 0.847599i \(-0.321953\pi\)
0.530637 + 0.847599i \(0.321953\pi\)
\(674\) 0 0
\(675\) 74.1746i 0.109888i
\(676\) 0 0
\(677\) −1178.41 −1.74063 −0.870317 0.492492i \(-0.836086\pi\)
−0.870317 + 0.492492i \(0.836086\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) −239.529 −0.351731
\(682\) 0 0
\(683\) − 845.501i − 1.23792i −0.785421 0.618961i \(-0.787554\pi\)
0.785421 0.618961i \(-0.212446\pi\)
\(684\) 0 0
\(685\) 147.540 0.215387
\(686\) 0 0
\(687\) − 232.738i − 0.338774i
\(688\) 0 0
\(689\) 674.296 0.978658
\(690\) 0 0
\(691\) − 1006.44i − 1.45649i −0.685316 0.728245i \(-0.740336\pi\)
0.685316 0.728245i \(-0.259664\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 105.546i 0.151864i
\(696\) 0 0
\(697\) 972.591 1.39540
\(698\) 0 0
\(699\) − 124.446i − 0.178034i
\(700\) 0 0
\(701\) −1047.00 −1.49358 −0.746791 0.665059i \(-0.768406\pi\)
−0.746791 + 0.665059i \(0.768406\pi\)
\(702\) 0 0
\(703\) − 1111.18i − 1.58062i
\(704\) 0 0
\(705\) 137.547 0.195101
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −1328.97 −1.87443 −0.937215 0.348753i \(-0.886605\pi\)
−0.937215 + 0.348753i \(0.886605\pi\)
\(710\) 0 0
\(711\) − 27.8020i − 0.0391027i
\(712\) 0 0
\(713\) −273.897 −0.384147
\(714\) 0 0
\(715\) − 628.959i − 0.879663i
\(716\) 0 0
\(717\) −573.842 −0.800338
\(718\) 0 0
\(719\) 1301.17i 1.80970i 0.425732 + 0.904849i \(0.360016\pi\)
−0.425732 + 0.904849i \(0.639984\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) − 184.508i − 0.255198i
\(724\) 0 0
\(725\) 599.890 0.827435
\(726\) 0 0
\(727\) − 951.043i − 1.30817i −0.756419 0.654087i \(-0.773053\pi\)
0.756419 0.654087i \(-0.226947\pi\)
\(728\) 0 0
\(729\) −27.0000 −0.0370370
\(730\) 0 0
\(731\) 791.593i 1.08289i
\(732\) 0 0
\(733\) −730.625 −0.996760 −0.498380 0.866959i \(-0.666072\pi\)
−0.498380 + 0.866959i \(0.666072\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 415.196 0.563360
\(738\) 0 0
\(739\) 227.599i 0.307983i 0.988072 + 0.153991i \(0.0492128\pi\)
−0.988072 + 0.153991i \(0.950787\pi\)
\(740\) 0 0
\(741\) −550.468 −0.742871
\(742\) 0 0
\(743\) − 1129.94i − 1.52078i −0.649470 0.760388i \(-0.725009\pi\)
0.649470 0.760388i \(-0.274991\pi\)
\(744\) 0 0
\(745\) −514.646 −0.690800
\(746\) 0 0
\(747\) − 153.593i − 0.205613i
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) − 991.605i − 1.32038i −0.751099 0.660190i \(-0.770476\pi\)
0.751099 0.660190i \(-0.229524\pi\)
\(752\) 0 0
\(753\) −434.918 −0.577580
\(754\) 0 0
\(755\) − 28.4134i − 0.0376337i
\(756\) 0 0
\(757\) −373.286 −0.493112 −0.246556 0.969129i \(-0.579299\pi\)
−0.246556 + 0.969129i \(0.579299\pi\)
\(758\) 0 0
\(759\) 170.086i 0.224093i
\(760\) 0 0
\(761\) −226.213 −0.297257 −0.148629 0.988893i \(-0.547486\pi\)
−0.148629 + 0.988893i \(0.547486\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 246.598 0.322350
\(766\) 0 0
\(767\) 460.777i 0.600753i
\(768\) 0 0
\(769\) −100.836 −0.131126 −0.0655628 0.997848i \(-0.520884\pi\)
−0.0655628 + 0.997848i \(0.520884\pi\)
\(770\) 0 0
\(771\) 703.903i 0.912974i
\(772\) 0 0
\(773\) 71.6079 0.0926364 0.0463182 0.998927i \(-0.485251\pi\)
0.0463182 + 0.998927i \(0.485251\pi\)
\(774\) 0 0
\(775\) − 744.211i − 0.960272i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) − 1198.55i − 1.53857i
\(780\) 0 0
\(781\) 1823.73 2.33512
\(782\) 0 0
\(783\) 218.364i 0.278881i
\(784\) 0 0
\(785\) −360.399 −0.459107
\(786\) 0 0
\(787\) − 484.534i − 0.615672i −0.951439 0.307836i \(-0.900395\pi\)
0.951439 0.307836i \(-0.0996048\pi\)
\(788\) 0 0
\(789\) 97.7392 0.123877
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −648.344 −0.817584
\(794\) 0 0
\(795\) 372.249i 0.468237i
\(796\) 0 0
\(797\) 189.275 0.237484 0.118742 0.992925i \(-0.462114\pi\)
0.118742 + 0.992925i \(0.462114\pi\)
\(798\) 0 0
\(799\) 608.635i 0.761745i
\(800\) 0 0
\(801\) −160.350 −0.200188
\(802\) 0 0
\(803\) 1714.48i 2.13509i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 236.375i 0.292905i
\(808\) 0 0
\(809\) 1117.95 1.38189 0.690947 0.722906i \(-0.257194\pi\)
0.690947 + 0.722906i \(0.257194\pi\)
\(810\) 0 0
\(811\) 706.070i 0.870616i 0.900282 + 0.435308i \(0.143361\pi\)
−0.900282 + 0.435308i \(0.856639\pi\)
\(812\) 0 0
\(813\) −1.06896 −0.00131483
\(814\) 0 0
\(815\) 483.467i 0.593211i
\(816\) 0 0
\(817\) 975.499 1.19400
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 848.423 1.03340 0.516701 0.856166i \(-0.327160\pi\)
0.516701 + 0.856166i \(0.327160\pi\)
\(822\) 0 0
\(823\) − 33.5938i − 0.0408187i −0.999792 0.0204093i \(-0.993503\pi\)
0.999792 0.0204093i \(-0.00649694\pi\)
\(824\) 0 0
\(825\) −462.145 −0.560175
\(826\) 0 0
\(827\) − 1246.59i − 1.50737i −0.657236 0.753685i \(-0.728275\pi\)
0.657236 0.753685i \(-0.271725\pi\)
\(828\) 0 0
\(829\) 769.107 0.927753 0.463876 0.885900i \(-0.346458\pi\)
0.463876 + 0.885900i \(0.346458\pi\)
\(830\) 0 0
\(831\) 677.691i 0.815512i
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 198.343i 0.237536i
\(836\) 0 0
\(837\) 270.897 0.323652
\(838\) 0 0
\(839\) 244.162i 0.291015i 0.989357 + 0.145508i \(0.0464815\pi\)
−0.989357 + 0.145508i \(0.953519\pi\)
\(840\) 0 0
\(841\) 925.024 1.09991
\(842\) 0 0
\(843\) − 925.344i − 1.09768i
\(844\) 0 0
\(845\) 207.715 0.245817
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 110.928 0.130657
\(850\) 0 0
\(851\) 188.736i 0.221781i
\(852\) 0 0
\(853\) 594.722 0.697212 0.348606 0.937269i \(-0.386655\pi\)
0.348606 + 0.937269i \(0.386655\pi\)
\(854\) 0 0
\(855\) − 303.889i − 0.355425i
\(856\) 0 0
\(857\) −302.344 −0.352793 −0.176397 0.984319i \(-0.556444\pi\)
−0.176397 + 0.984319i \(0.556444\pi\)
\(858\) 0 0
\(859\) 585.966i 0.682148i 0.940036 + 0.341074i \(0.110791\pi\)
−0.940036 + 0.341074i \(0.889209\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 43.5487i 0.0504619i 0.999682 + 0.0252310i \(0.00803212\pi\)
−0.999682 + 0.0252310i \(0.991968\pi\)
\(864\) 0 0
\(865\) −297.533 −0.343969
\(866\) 0 0
\(867\) 590.618i 0.681220i
\(868\) 0 0
\(869\) 173.220 0.199333
\(870\) 0 0
\(871\) − 228.238i − 0.262041i
\(872\) 0 0
\(873\) 388.567 0.445094
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 1404.03 1.60094 0.800472 0.599370i \(-0.204582\pi\)
0.800472 + 0.599370i \(0.204582\pi\)
\(878\) 0 0
\(879\) − 153.581i − 0.174723i
\(880\) 0 0
\(881\) 507.842 0.576438 0.288219 0.957564i \(-0.406937\pi\)
0.288219 + 0.957564i \(0.406937\pi\)
\(882\) 0 0
\(883\) − 438.774i − 0.496913i −0.968643 0.248457i \(-0.920077\pi\)
0.968643 0.248457i \(-0.0799233\pi\)
\(884\) 0 0
\(885\) −254.375 −0.287429
\(886\) 0 0
\(887\) 205.616i 0.231811i 0.993260 + 0.115905i \(0.0369770\pi\)
−0.993260 + 0.115905i \(0.963023\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) − 168.223i − 0.188803i
\(892\) 0 0
\(893\) 750.035 0.839905
\(894\) 0 0
\(895\) − 1043.60i − 1.16604i
\(896\) 0 0
\(897\) 93.4983 0.104234
\(898\) 0 0
\(899\) − 2190.89i − 2.43703i
\(900\) 0 0
\(901\) −1647.18 −1.82816
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 103.739 0.114629
\(906\) 0 0
\(907\) − 703.490i − 0.775623i −0.921739 0.387812i \(-0.873231\pi\)
0.921739 0.387812i \(-0.126769\pi\)
\(908\) 0 0
\(909\) 161.093 0.177220
\(910\) 0 0
\(911\) 372.699i 0.409110i 0.978855 + 0.204555i \(0.0655747\pi\)
−0.978855 + 0.204555i \(0.934425\pi\)
\(912\) 0 0
\(913\) 956.959 1.04815
\(914\) 0 0
\(915\) − 357.922i − 0.391171i
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) − 295.584i − 0.321636i −0.986984 0.160818i \(-0.948587\pi\)
0.986984 0.160818i \(-0.0514133\pi\)
\(920\) 0 0
\(921\) 254.619 0.276459
\(922\) 0 0
\(923\) − 1002.53i − 1.08616i
\(924\) 0 0
\(925\) −512.818 −0.554398
\(926\) 0 0
\(927\) − 96.3403i − 0.103927i
\(928\) 0 0
\(929\) 1694.58 1.82409 0.912044 0.410092i \(-0.134503\pi\)
0.912044 + 0.410092i \(0.134503\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) −319.051 −0.341963
\(934\) 0 0
\(935\) 1536.43i 1.64324i
\(936\) 0 0
\(937\) 246.691 0.263278 0.131639 0.991298i \(-0.457976\pi\)
0.131639 + 0.991298i \(0.457976\pi\)
\(938\) 0 0
\(939\) 386.319i 0.411416i
\(940\) 0 0
\(941\) −327.715 −0.348263 −0.174131 0.984722i \(-0.555712\pi\)
−0.174131 + 0.984722i \(0.555712\pi\)
\(942\) 0 0
\(943\) 203.576i 0.215882i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 9.26160i 0.00977993i 0.999988 + 0.00488997i \(0.00155653\pi\)
−0.999988 + 0.00488997i \(0.998443\pi\)
\(948\) 0 0
\(949\) 942.468 0.993117
\(950\) 0 0
\(951\) − 1075.12i − 1.13051i
\(952\) 0 0
\(953\) 1300.02 1.36414 0.682068 0.731289i \(-0.261081\pi\)
0.682068 + 0.731289i \(0.261081\pi\)
\(954\) 0 0
\(955\) − 789.292i − 0.826483i
\(956\) 0 0
\(957\) −1360.51 −1.42164
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −1756.97 −1.82827
\(962\) 0 0
\(963\) 171.771i 0.178371i
\(964\) 0 0
\(965\) 1118.21 1.15877
\(966\) 0 0
\(967\) 669.581i 0.692432i 0.938155 + 0.346216i \(0.112534\pi\)
−0.938155 + 0.346216i \(0.887466\pi\)
\(968\) 0 0
\(969\) 1344.69 1.38771
\(970\) 0 0
\(971\) − 1245.39i − 1.28258i −0.767297 0.641292i \(-0.778399\pi\)
0.767297 0.641292i \(-0.221601\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 254.046i 0.260560i
\(976\) 0 0
\(977\) 216.515 0.221612 0.110806 0.993842i \(-0.464657\pi\)
0.110806 + 0.993842i \(0.464657\pi\)
\(978\) 0 0
\(979\) − 999.062i − 1.02049i
\(980\) 0 0
\(981\) 424.124 0.432338
\(982\) 0 0
\(983\) − 818.489i − 0.832644i −0.909217 0.416322i \(-0.863319\pi\)
0.909217 0.416322i \(-0.136681\pi\)
\(984\) 0 0
\(985\) 466.839 0.473948
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −165.691 −0.167534
\(990\) 0 0
\(991\) 711.643i 0.718106i 0.933317 + 0.359053i \(0.116900\pi\)
−0.933317 + 0.359053i \(0.883100\pi\)
\(992\) 0 0
\(993\) 629.574 0.634012
\(994\) 0 0
\(995\) − 17.2054i − 0.0172919i
\(996\) 0 0
\(997\) −1089.16 −1.09243 −0.546216 0.837644i \(-0.683932\pi\)
−0.546216 + 0.837644i \(0.683932\pi\)
\(998\) 0 0
\(999\) − 186.669i − 0.186856i
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 2352.3.m.h.1471.3 4
4.3 odd 2 inner 2352.3.m.h.1471.1 4
7.3 odd 6 336.3.be.a.79.1 4
7.5 odd 6 336.3.be.b.319.1 yes 4
7.6 odd 2 2352.3.m.g.1471.2 4
21.5 even 6 1008.3.cd.f.991.2 4
21.17 even 6 1008.3.cd.g.415.2 4
28.3 even 6 336.3.be.b.79.1 yes 4
28.19 even 6 336.3.be.a.319.1 yes 4
28.27 even 2 2352.3.m.g.1471.4 4
84.47 odd 6 1008.3.cd.g.991.2 4
84.59 odd 6 1008.3.cd.f.415.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
336.3.be.a.79.1 4 7.3 odd 6
336.3.be.a.319.1 yes 4 28.19 even 6
336.3.be.b.79.1 yes 4 28.3 even 6
336.3.be.b.319.1 yes 4 7.5 odd 6
1008.3.cd.f.415.2 4 84.59 odd 6
1008.3.cd.f.991.2 4 21.5 even 6
1008.3.cd.g.415.2 4 21.17 even 6
1008.3.cd.g.991.2 4 84.47 odd 6
2352.3.m.g.1471.2 4 7.6 odd 2
2352.3.m.g.1471.4 4 28.27 even 2
2352.3.m.h.1471.1 4 4.3 odd 2 inner
2352.3.m.h.1471.3 4 1.1 even 1 trivial