Properties

Label 6080.2.a.bo.1.1
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 1.1
Root \(0.311108\) 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)
 
\(f(q)\) \(=\) \(q-2.90321 q^{3} -1.00000 q^{5} -4.42864 q^{7} +5.42864 q^{9} +O(q^{10})\) \(q-2.90321 q^{3} -1.00000 q^{5} -4.42864 q^{7} +5.42864 q^{9} +2.62222 q^{11} -0.474572 q^{13} +2.90321 q^{15} +5.05086 q^{17} +1.00000 q^{19} +12.8573 q^{21} -1.37778 q^{23} +1.00000 q^{25} -7.05086 q^{27} +7.80642 q^{29} +1.24443 q^{31} -7.61285 q^{33} +4.42864 q^{35} -4.47457 q^{37} +1.37778 q^{39} -5.05086 q^{41} -12.0415 q^{43} -5.42864 q^{45} -4.42864 q^{47} +12.6128 q^{49} -14.6637 q^{51} -7.52543 q^{53} -2.62222 q^{55} -2.90321 q^{57} +2.19358 q^{59} -3.67307 q^{61} -24.0415 q^{63} +0.474572 q^{65} +1.65878 q^{67} +4.00000 q^{69} +7.61285 q^{71} -3.80642 q^{73} -2.90321 q^{75} -11.6128 q^{77} -13.4193 q^{79} +4.18421 q^{81} +10.6222 q^{83} -5.05086 q^{85} -22.6637 q^{87} -12.6637 q^{89} +2.10171 q^{91} -3.61285 q^{93} -1.00000 q^{95} +17.8938 q^{97} +14.2351 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 2 q^{3} - 3 q^{5} + 3 q^{9}+O(q^{10}) \) 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
\(3\) −2.90321 −1.67617 −0.838085 0.545540i \(-0.816325\pi\)
−0.838085 + 0.545540i \(0.816325\pi\)
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −4.42864 −1.67387 −0.836934 0.547304i \(-0.815654\pi\)
−0.836934 + 0.547304i \(0.815654\pi\)
\(8\) 0 0
\(9\) 5.42864 1.80955
\(10\) 0 0
\(11\) 2.62222 0.790628 0.395314 0.918546i \(-0.370636\pi\)
0.395314 + 0.918546i \(0.370636\pi\)
\(12\) 0 0
\(13\) −0.474572 −0.131623 −0.0658114 0.997832i \(-0.520964\pi\)
−0.0658114 + 0.997832i \(0.520964\pi\)
\(14\) 0 0
\(15\) 2.90321 0.749606
\(16\) 0 0
\(17\) 5.05086 1.22501 0.612506 0.790466i \(-0.290161\pi\)
0.612506 + 0.790466i \(0.290161\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) 12.8573 2.80569
\(22\) 0 0
\(23\) −1.37778 −0.287288 −0.143644 0.989629i \(-0.545882\pi\)
−0.143644 + 0.989629i \(0.545882\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −7.05086 −1.35694
\(28\) 0 0
\(29\) 7.80642 1.44962 0.724808 0.688951i \(-0.241928\pi\)
0.724808 + 0.688951i \(0.241928\pi\)
\(30\) 0 0
\(31\) 1.24443 0.223506 0.111753 0.993736i \(-0.464353\pi\)
0.111753 + 0.993736i \(0.464353\pi\)
\(32\) 0 0
\(33\) −7.61285 −1.32523
\(34\) 0 0
\(35\) 4.42864 0.748577
\(36\) 0 0
\(37\) −4.47457 −0.735615 −0.367808 0.929902i \(-0.619891\pi\)
−0.367808 + 0.929902i \(0.619891\pi\)
\(38\) 0 0
\(39\) 1.37778 0.220622
\(40\) 0 0
\(41\) −5.05086 −0.788811 −0.394406 0.918936i \(-0.629049\pi\)
−0.394406 + 0.918936i \(0.629049\pi\)
\(42\) 0 0
\(43\) −12.0415 −1.83631 −0.918155 0.396222i \(-0.870321\pi\)
−0.918155 + 0.396222i \(0.870321\pi\)
\(44\) 0 0
\(45\) −5.42864 −0.809254
\(46\) 0 0
\(47\) −4.42864 −0.645983 −0.322992 0.946402i \(-0.604689\pi\)
−0.322992 + 0.946402i \(0.604689\pi\)
\(48\) 0 0
\(49\) 12.6128 1.80184
\(50\) 0 0
\(51\) −14.6637 −2.05333
\(52\) 0 0
\(53\) −7.52543 −1.03370 −0.516848 0.856077i \(-0.672895\pi\)
−0.516848 + 0.856077i \(0.672895\pi\)
\(54\) 0 0
\(55\) −2.62222 −0.353579
\(56\) 0 0
\(57\) −2.90321 −0.384540
\(58\) 0 0
\(59\) 2.19358 0.285579 0.142790 0.989753i \(-0.454393\pi\)
0.142790 + 0.989753i \(0.454393\pi\)
\(60\) 0 0
\(61\) −3.67307 −0.470289 −0.235144 0.971960i \(-0.575556\pi\)
−0.235144 + 0.971960i \(0.575556\pi\)
\(62\) 0 0
\(63\) −24.0415 −3.02894
\(64\) 0 0
\(65\) 0.474572 0.0588635
\(66\) 0 0
\(67\) 1.65878 0.202652 0.101326 0.994853i \(-0.467691\pi\)
0.101326 + 0.994853i \(0.467691\pi\)
\(68\) 0 0
\(69\) 4.00000 0.481543
\(70\) 0 0
\(71\) 7.61285 0.903479 0.451739 0.892150i \(-0.350804\pi\)
0.451739 + 0.892150i \(0.350804\pi\)
\(72\) 0 0
\(73\) −3.80642 −0.445508 −0.222754 0.974875i \(-0.571505\pi\)
−0.222754 + 0.974875i \(0.571505\pi\)
\(74\) 0 0
\(75\) −2.90321 −0.335234
\(76\) 0 0
\(77\) −11.6128 −1.32341
\(78\) 0 0
\(79\) −13.4193 −1.50979 −0.754893 0.655848i \(-0.772311\pi\)
−0.754893 + 0.655848i \(0.772311\pi\)
\(80\) 0 0
\(81\) 4.18421 0.464912
\(82\) 0 0
\(83\) 10.6222 1.16594 0.582970 0.812494i \(-0.301891\pi\)
0.582970 + 0.812494i \(0.301891\pi\)
\(84\) 0 0
\(85\) −5.05086 −0.547842
\(86\) 0 0
\(87\) −22.6637 −2.42980
\(88\) 0 0
\(89\) −12.6637 −1.34235 −0.671175 0.741299i \(-0.734210\pi\)
−0.671175 + 0.741299i \(0.734210\pi\)
\(90\) 0 0
\(91\) 2.10171 0.220319
\(92\) 0 0
\(93\) −3.61285 −0.374635
\(94\) 0 0
\(95\) −1.00000 −0.102598
\(96\) 0 0
\(97\) 17.8938 1.81684 0.908422 0.418054i \(-0.137288\pi\)
0.908422 + 0.418054i \(0.137288\pi\)
\(98\) 0 0
\(99\) 14.2351 1.43068
\(100\) 0 0
\(101\) 10.4286 1.03769 0.518844 0.854869i \(-0.326362\pi\)
0.518844 + 0.854869i \(0.326362\pi\)
\(102\) 0 0
\(103\) −5.65878 −0.557576 −0.278788 0.960353i \(-0.589933\pi\)
−0.278788 + 0.960353i \(0.589933\pi\)
\(104\) 0 0
\(105\) −12.8573 −1.25474
\(106\) 0 0
\(107\) −6.90321 −0.667359 −0.333679 0.942687i \(-0.608290\pi\)
−0.333679 + 0.942687i \(0.608290\pi\)
\(108\) 0 0
\(109\) −5.61285 −0.537613 −0.268807 0.963194i \(-0.586629\pi\)
−0.268807 + 0.963194i \(0.586629\pi\)
\(110\) 0 0
\(111\) 12.9906 1.23302
\(112\) 0 0
\(113\) 13.8938 1.30702 0.653511 0.756917i \(-0.273295\pi\)
0.653511 + 0.756917i \(0.273295\pi\)
\(114\) 0 0
\(115\) 1.37778 0.128479
\(116\) 0 0
\(117\) −2.57628 −0.238177
\(118\) 0 0
\(119\) −22.3684 −2.05051
\(120\) 0 0
\(121\) −4.12399 −0.374908
\(122\) 0 0
\(123\) 14.6637 1.32218
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −7.19850 −0.638763 −0.319382 0.947626i \(-0.603475\pi\)
−0.319382 + 0.947626i \(0.603475\pi\)
\(128\) 0 0
\(129\) 34.9590 3.07797
\(130\) 0 0
\(131\) 2.10171 0.183627 0.0918136 0.995776i \(-0.470734\pi\)
0.0918136 + 0.995776i \(0.470734\pi\)
\(132\) 0 0
\(133\) −4.42864 −0.384012
\(134\) 0 0
\(135\) 7.05086 0.606841
\(136\) 0 0
\(137\) 1.70471 0.145644 0.0728218 0.997345i \(-0.476800\pi\)
0.0728218 + 0.997345i \(0.476800\pi\)
\(138\) 0 0
\(139\) −8.72393 −0.739954 −0.369977 0.929041i \(-0.620634\pi\)
−0.369977 + 0.929041i \(0.620634\pi\)
\(140\) 0 0
\(141\) 12.8573 1.08278
\(142\) 0 0
\(143\) −1.24443 −0.104065
\(144\) 0 0
\(145\) −7.80642 −0.648288
\(146\) 0 0
\(147\) −36.6178 −3.02018
\(148\) 0 0
\(149\) 6.81579 0.558371 0.279186 0.960237i \(-0.409935\pi\)
0.279186 + 0.960237i \(0.409935\pi\)
\(150\) 0 0
\(151\) −5.80642 −0.472520 −0.236260 0.971690i \(-0.575922\pi\)
−0.236260 + 0.971690i \(0.575922\pi\)
\(152\) 0 0
\(153\) 27.4193 2.21672
\(154\) 0 0
\(155\) −1.24443 −0.0999551
\(156\) 0 0
\(157\) −0.193576 −0.0154491 −0.00772453 0.999970i \(-0.502459\pi\)
−0.00772453 + 0.999970i \(0.502459\pi\)
\(158\) 0 0
\(159\) 21.8479 1.73265
\(160\) 0 0
\(161\) 6.10171 0.480882
\(162\) 0 0
\(163\) 8.42864 0.660182 0.330091 0.943949i \(-0.392921\pi\)
0.330091 + 0.943949i \(0.392921\pi\)
\(164\) 0 0
\(165\) 7.61285 0.592659
\(166\) 0 0
\(167\) 12.4429 0.962863 0.481431 0.876484i \(-0.340117\pi\)
0.481431 + 0.876484i \(0.340117\pi\)
\(168\) 0 0
\(169\) −12.7748 −0.982675
\(170\) 0 0
\(171\) 5.42864 0.415138
\(172\) 0 0
\(173\) 22.1891 1.68701 0.843504 0.537123i \(-0.180489\pi\)
0.843504 + 0.537123i \(0.180489\pi\)
\(174\) 0 0
\(175\) −4.42864 −0.334774
\(176\) 0 0
\(177\) −6.36842 −0.478679
\(178\) 0 0
\(179\) 11.9081 0.890056 0.445028 0.895517i \(-0.353194\pi\)
0.445028 + 0.895517i \(0.353194\pi\)
\(180\) 0 0
\(181\) −17.6128 −1.30915 −0.654576 0.755996i \(-0.727153\pi\)
−0.654576 + 0.755996i \(0.727153\pi\)
\(182\) 0 0
\(183\) 10.6637 0.788284
\(184\) 0 0
\(185\) 4.47457 0.328977
\(186\) 0 0
\(187\) 13.2444 0.968529
\(188\) 0 0
\(189\) 31.2257 2.27134
\(190\) 0 0
\(191\) −0.266706 −0.0192982 −0.00964909 0.999953i \(-0.503071\pi\)
−0.00964909 + 0.999953i \(0.503071\pi\)
\(192\) 0 0
\(193\) 2.66815 0.192058 0.0960288 0.995379i \(-0.469386\pi\)
0.0960288 + 0.995379i \(0.469386\pi\)
\(194\) 0 0
\(195\) −1.37778 −0.0986652
\(196\) 0 0
\(197\) 5.34614 0.380897 0.190448 0.981697i \(-0.439006\pi\)
0.190448 + 0.981697i \(0.439006\pi\)
\(198\) 0 0
\(199\) 17.1240 1.21389 0.606944 0.794745i \(-0.292395\pi\)
0.606944 + 0.794745i \(0.292395\pi\)
\(200\) 0 0
\(201\) −4.81579 −0.339680
\(202\) 0 0
\(203\) −34.5718 −2.42647
\(204\) 0 0
\(205\) 5.05086 0.352767
\(206\) 0 0
\(207\) −7.47949 −0.519861
\(208\) 0 0
\(209\) 2.62222 0.181382
\(210\) 0 0
\(211\) −13.1526 −0.905460 −0.452730 0.891648i \(-0.649550\pi\)
−0.452730 + 0.891648i \(0.649550\pi\)
\(212\) 0 0
\(213\) −22.1017 −1.51438
\(214\) 0 0
\(215\) 12.0415 0.821223
\(216\) 0 0
\(217\) −5.51114 −0.374120
\(218\) 0 0
\(219\) 11.0509 0.746748
\(220\) 0 0
\(221\) −2.39700 −0.161239
\(222\) 0 0
\(223\) −10.5161 −0.704207 −0.352104 0.935961i \(-0.614534\pi\)
−0.352104 + 0.935961i \(0.614534\pi\)
\(224\) 0 0
\(225\) 5.42864 0.361909
\(226\) 0 0
\(227\) −6.90321 −0.458182 −0.229091 0.973405i \(-0.573575\pi\)
−0.229091 + 0.973405i \(0.573575\pi\)
\(228\) 0 0
\(229\) −18.0415 −1.19222 −0.596108 0.802905i \(-0.703287\pi\)
−0.596108 + 0.802905i \(0.703287\pi\)
\(230\) 0 0
\(231\) 33.7146 2.21826
\(232\) 0 0
\(233\) −12.3684 −0.810282 −0.405141 0.914254i \(-0.632778\pi\)
−0.405141 + 0.914254i \(0.632778\pi\)
\(234\) 0 0
\(235\) 4.42864 0.288893
\(236\) 0 0
\(237\) 38.9590 2.53066
\(238\) 0 0
\(239\) 24.8573 1.60788 0.803942 0.594708i \(-0.202732\pi\)
0.803942 + 0.594708i \(0.202732\pi\)
\(240\) 0 0
\(241\) 9.05086 0.583017 0.291508 0.956568i \(-0.405843\pi\)
0.291508 + 0.956568i \(0.405843\pi\)
\(242\) 0 0
\(243\) 9.00492 0.577666
\(244\) 0 0
\(245\) −12.6128 −0.805805
\(246\) 0 0
\(247\) −0.474572 −0.0301963
\(248\) 0 0
\(249\) −30.8385 −1.95431
\(250\) 0 0
\(251\) 22.5718 1.42472 0.712361 0.701813i \(-0.247626\pi\)
0.712361 + 0.701813i \(0.247626\pi\)
\(252\) 0 0
\(253\) −3.61285 −0.227138
\(254\) 0 0
\(255\) 14.6637 0.918277
\(256\) 0 0
\(257\) 4.94470 0.308442 0.154221 0.988036i \(-0.450713\pi\)
0.154221 + 0.988036i \(0.450713\pi\)
\(258\) 0 0
\(259\) 19.8163 1.23132
\(260\) 0 0
\(261\) 42.3783 2.62315
\(262\) 0 0
\(263\) 9.37778 0.578259 0.289129 0.957290i \(-0.406634\pi\)
0.289129 + 0.957290i \(0.406634\pi\)
\(264\) 0 0
\(265\) 7.52543 0.462283
\(266\) 0 0
\(267\) 36.7654 2.25001
\(268\) 0 0
\(269\) −19.7146 −1.20202 −0.601009 0.799242i \(-0.705234\pi\)
−0.601009 + 0.799242i \(0.705234\pi\)
\(270\) 0 0
\(271\) −1.11108 −0.0674932 −0.0337466 0.999430i \(-0.510744\pi\)
−0.0337466 + 0.999430i \(0.510744\pi\)
\(272\) 0 0
\(273\) −6.10171 −0.369292
\(274\) 0 0
\(275\) 2.62222 0.158126
\(276\) 0 0
\(277\) −5.52098 −0.331724 −0.165862 0.986149i \(-0.553041\pi\)
−0.165862 + 0.986149i \(0.553041\pi\)
\(278\) 0 0
\(279\) 6.75557 0.404445
\(280\) 0 0
\(281\) −15.8064 −0.942932 −0.471466 0.881884i \(-0.656275\pi\)
−0.471466 + 0.881884i \(0.656275\pi\)
\(282\) 0 0
\(283\) −14.2351 −0.846187 −0.423093 0.906086i \(-0.639056\pi\)
−0.423093 + 0.906086i \(0.639056\pi\)
\(284\) 0 0
\(285\) 2.90321 0.171971
\(286\) 0 0
\(287\) 22.3684 1.32037
\(288\) 0 0
\(289\) 8.51114 0.500655
\(290\) 0 0
\(291\) −51.9496 −3.04534
\(292\) 0 0
\(293\) −7.52543 −0.439640 −0.219820 0.975540i \(-0.570547\pi\)
−0.219820 + 0.975540i \(0.570547\pi\)
\(294\) 0 0
\(295\) −2.19358 −0.127715
\(296\) 0 0
\(297\) −18.4889 −1.07283
\(298\) 0 0
\(299\) 0.653858 0.0378136
\(300\) 0 0
\(301\) 53.3274 3.07374
\(302\) 0 0
\(303\) −30.2766 −1.73934
\(304\) 0 0
\(305\) 3.67307 0.210319
\(306\) 0 0
\(307\) 2.81135 0.160452 0.0802260 0.996777i \(-0.474436\pi\)
0.0802260 + 0.996777i \(0.474436\pi\)
\(308\) 0 0
\(309\) 16.4286 0.934593
\(310\) 0 0
\(311\) −13.8479 −0.785243 −0.392621 0.919700i \(-0.628432\pi\)
−0.392621 + 0.919700i \(0.628432\pi\)
\(312\) 0 0
\(313\) 23.2444 1.31385 0.656926 0.753955i \(-0.271856\pi\)
0.656926 + 0.753955i \(0.271856\pi\)
\(314\) 0 0
\(315\) 24.0415 1.35458
\(316\) 0 0
\(317\) −2.96343 −0.166443 −0.0832215 0.996531i \(-0.526521\pi\)
−0.0832215 + 0.996531i \(0.526521\pi\)
\(318\) 0 0
\(319\) 20.4701 1.14611
\(320\) 0 0
\(321\) 20.0415 1.11861
\(322\) 0 0
\(323\) 5.05086 0.281037
\(324\) 0 0
\(325\) −0.474572 −0.0263245
\(326\) 0 0
\(327\) 16.2953 0.901131
\(328\) 0 0
\(329\) 19.6128 1.08129
\(330\) 0 0
\(331\) 0.949145 0.0521697 0.0260849 0.999660i \(-0.491696\pi\)
0.0260849 + 0.999660i \(0.491696\pi\)
\(332\) 0 0
\(333\) −24.2908 −1.33113
\(334\) 0 0
\(335\) −1.65878 −0.0906289
\(336\) 0 0
\(337\) 2.28100 0.124254 0.0621269 0.998068i \(-0.480212\pi\)
0.0621269 + 0.998068i \(0.480212\pi\)
\(338\) 0 0
\(339\) −40.3368 −2.19079
\(340\) 0 0
\(341\) 3.26317 0.176710
\(342\) 0 0
\(343\) −24.8573 −1.34217
\(344\) 0 0
\(345\) −4.00000 −0.215353
\(346\) 0 0
\(347\) −12.3368 −0.662273 −0.331136 0.943583i \(-0.607432\pi\)
−0.331136 + 0.943583i \(0.607432\pi\)
\(348\) 0 0
\(349\) 10.0000 0.535288 0.267644 0.963518i \(-0.413755\pi\)
0.267644 + 0.963518i \(0.413755\pi\)
\(350\) 0 0
\(351\) 3.34614 0.178604
\(352\) 0 0
\(353\) 2.56199 0.136361 0.0681806 0.997673i \(-0.478281\pi\)
0.0681806 + 0.997673i \(0.478281\pi\)
\(354\) 0 0
\(355\) −7.61285 −0.404048
\(356\) 0 0
\(357\) 64.9403 3.43700
\(358\) 0 0
\(359\) 24.3368 1.28445 0.642223 0.766518i \(-0.278012\pi\)
0.642223 + 0.766518i \(0.278012\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) 11.9728 0.628409
\(364\) 0 0
\(365\) 3.80642 0.199237
\(366\) 0 0
\(367\) −4.42864 −0.231173 −0.115587 0.993297i \(-0.536875\pi\)
−0.115587 + 0.993297i \(0.536875\pi\)
\(368\) 0 0
\(369\) −27.4193 −1.42739
\(370\) 0 0
\(371\) 33.3274 1.73027
\(372\) 0 0
\(373\) 23.7003 1.22715 0.613577 0.789635i \(-0.289730\pi\)
0.613577 + 0.789635i \(0.289730\pi\)
\(374\) 0 0
\(375\) 2.90321 0.149921
\(376\) 0 0
\(377\) −3.70471 −0.190802
\(378\) 0 0
\(379\) −8.20342 −0.421381 −0.210691 0.977553i \(-0.567571\pi\)
−0.210691 + 0.977553i \(0.567571\pi\)
\(380\) 0 0
\(381\) 20.8988 1.07068
\(382\) 0 0
\(383\) −20.9131 −1.06861 −0.534304 0.845293i \(-0.679426\pi\)
−0.534304 + 0.845293i \(0.679426\pi\)
\(384\) 0 0
\(385\) 11.6128 0.591846
\(386\) 0 0
\(387\) −65.3689 −3.32289
\(388\) 0 0
\(389\) 24.1017 1.22201 0.611003 0.791629i \(-0.290766\pi\)
0.611003 + 0.791629i \(0.290766\pi\)
\(390\) 0 0
\(391\) −6.95899 −0.351931
\(392\) 0 0
\(393\) −6.10171 −0.307791
\(394\) 0 0
\(395\) 13.4193 0.675197
\(396\) 0 0
\(397\) −7.92687 −0.397838 −0.198919 0.980016i \(-0.563743\pi\)
−0.198919 + 0.980016i \(0.563743\pi\)
\(398\) 0 0
\(399\) 12.8573 0.643669
\(400\) 0 0
\(401\) −32.5718 −1.62656 −0.813280 0.581873i \(-0.802320\pi\)
−0.813280 + 0.581873i \(0.802320\pi\)
\(402\) 0 0
\(403\) −0.590573 −0.0294185
\(404\) 0 0
\(405\) −4.18421 −0.207915
\(406\) 0 0
\(407\) −11.7333 −0.581598
\(408\) 0 0
\(409\) 36.3684 1.79830 0.899151 0.437638i \(-0.144185\pi\)
0.899151 + 0.437638i \(0.144185\pi\)
\(410\) 0 0
\(411\) −4.94914 −0.244123
\(412\) 0 0
\(413\) −9.71456 −0.478022
\(414\) 0 0
\(415\) −10.6222 −0.521424
\(416\) 0 0
\(417\) 25.3274 1.24029
\(418\) 0 0
\(419\) 31.6958 1.54844 0.774221 0.632915i \(-0.218142\pi\)
0.774221 + 0.632915i \(0.218142\pi\)
\(420\) 0 0
\(421\) −37.4005 −1.82279 −0.911395 0.411532i \(-0.864994\pi\)
−0.911395 + 0.411532i \(0.864994\pi\)
\(422\) 0 0
\(423\) −24.0415 −1.16894
\(424\) 0 0
\(425\) 5.05086 0.245002
\(426\) 0 0
\(427\) 16.2667 0.787201
\(428\) 0 0
\(429\) 3.61285 0.174430
\(430\) 0 0
\(431\) 4.94914 0.238392 0.119196 0.992871i \(-0.461968\pi\)
0.119196 + 0.992871i \(0.461968\pi\)
\(432\) 0 0
\(433\) 32.3827 1.55621 0.778107 0.628132i \(-0.216180\pi\)
0.778107 + 0.628132i \(0.216180\pi\)
\(434\) 0 0
\(435\) 22.6637 1.08664
\(436\) 0 0
\(437\) −1.37778 −0.0659084
\(438\) 0 0
\(439\) 10.0731 0.480764 0.240382 0.970678i \(-0.422727\pi\)
0.240382 + 0.970678i \(0.422727\pi\)
\(440\) 0 0
\(441\) 68.4706 3.26050
\(442\) 0 0
\(443\) 13.9684 0.663657 0.331828 0.943340i \(-0.392335\pi\)
0.331828 + 0.943340i \(0.392335\pi\)
\(444\) 0 0
\(445\) 12.6637 0.600317
\(446\) 0 0
\(447\) −19.7877 −0.935926
\(448\) 0 0
\(449\) 24.5718 1.15962 0.579808 0.814753i \(-0.303127\pi\)
0.579808 + 0.814753i \(0.303127\pi\)
\(450\) 0 0
\(451\) −13.2444 −0.623656
\(452\) 0 0
\(453\) 16.8573 0.792024
\(454\) 0 0
\(455\) −2.10171 −0.0985297
\(456\) 0 0
\(457\) −3.51114 −0.164244 −0.0821220 0.996622i \(-0.526170\pi\)
−0.0821220 + 0.996622i \(0.526170\pi\)
\(458\) 0 0
\(459\) −35.6128 −1.66227
\(460\) 0 0
\(461\) −10.2034 −0.475221 −0.237610 0.971361i \(-0.576364\pi\)
−0.237610 + 0.971361i \(0.576364\pi\)
\(462\) 0 0
\(463\) 8.33677 0.387443 0.193721 0.981057i \(-0.437944\pi\)
0.193721 + 0.981057i \(0.437944\pi\)
\(464\) 0 0
\(465\) 3.61285 0.167542
\(466\) 0 0
\(467\) 42.7052 1.97616 0.988080 0.153940i \(-0.0491961\pi\)
0.988080 + 0.153940i \(0.0491961\pi\)
\(468\) 0 0
\(469\) −7.34614 −0.339213
\(470\) 0 0
\(471\) 0.561993 0.0258953
\(472\) 0 0
\(473\) −31.5754 −1.45184
\(474\) 0 0
\(475\) 1.00000 0.0458831
\(476\) 0 0
\(477\) −40.8528 −1.87052
\(478\) 0 0
\(479\) 41.4608 1.89439 0.947195 0.320658i \(-0.103904\pi\)
0.947195 + 0.320658i \(0.103904\pi\)
\(480\) 0 0
\(481\) 2.12351 0.0968237
\(482\) 0 0
\(483\) −17.7146 −0.806040
\(484\) 0 0
\(485\) −17.8938 −0.812518
\(486\) 0 0
\(487\) 30.0370 1.36111 0.680554 0.732698i \(-0.261739\pi\)
0.680554 + 0.732698i \(0.261739\pi\)
\(488\) 0 0
\(489\) −24.4701 −1.10658
\(490\) 0 0
\(491\) −15.3461 −0.692562 −0.346281 0.938131i \(-0.612556\pi\)
−0.346281 + 0.938131i \(0.612556\pi\)
\(492\) 0 0
\(493\) 39.4291 1.77580
\(494\) 0 0
\(495\) −14.2351 −0.639819
\(496\) 0 0
\(497\) −33.7146 −1.51230
\(498\) 0 0
\(499\) 25.8479 1.15711 0.578556 0.815643i \(-0.303617\pi\)
0.578556 + 0.815643i \(0.303617\pi\)
\(500\) 0 0
\(501\) −36.1245 −1.61392
\(502\) 0 0
\(503\) −4.40006 −0.196189 −0.0980945 0.995177i \(-0.531275\pi\)
−0.0980945 + 0.995177i \(0.531275\pi\)
\(504\) 0 0
\(505\) −10.4286 −0.464068
\(506\) 0 0
\(507\) 37.0879 1.64713
\(508\) 0 0
\(509\) 27.2355 1.20719 0.603597 0.797290i \(-0.293734\pi\)
0.603597 + 0.797290i \(0.293734\pi\)
\(510\) 0 0
\(511\) 16.8573 0.745722
\(512\) 0 0
\(513\) −7.05086 −0.311303
\(514\) 0 0
\(515\) 5.65878 0.249356
\(516\) 0 0
\(517\) −11.6128 −0.510732
\(518\) 0 0
\(519\) −64.4197 −2.82771
\(520\) 0 0
\(521\) 38.5531 1.68904 0.844521 0.535522i \(-0.179885\pi\)
0.844521 + 0.535522i \(0.179885\pi\)
\(522\) 0 0
\(523\) −18.1575 −0.793971 −0.396986 0.917825i \(-0.629944\pi\)
−0.396986 + 0.917825i \(0.629944\pi\)
\(524\) 0 0
\(525\) 12.8573 0.561138
\(526\) 0 0
\(527\) 6.28544 0.273798
\(528\) 0 0
\(529\) −21.1017 −0.917466
\(530\) 0 0
\(531\) 11.9081 0.516769
\(532\) 0 0
\(533\) 2.39700 0.103825
\(534\) 0 0
\(535\) 6.90321 0.298452
\(536\) 0 0
\(537\) −34.5718 −1.49188
\(538\) 0 0
\(539\) 33.0736 1.42458
\(540\) 0 0
\(541\) 13.7748 0.592224 0.296112 0.955153i \(-0.404310\pi\)
0.296112 + 0.955153i \(0.404310\pi\)
\(542\) 0 0
\(543\) 51.1338 2.19436
\(544\) 0 0
\(545\) 5.61285 0.240428
\(546\) 0 0
\(547\) −42.9862 −1.83796 −0.918978 0.394308i \(-0.870984\pi\)
−0.918978 + 0.394308i \(0.870984\pi\)
\(548\) 0 0
\(549\) −19.9398 −0.851009
\(550\) 0 0
\(551\) 7.80642 0.332565
\(552\) 0 0
\(553\) 59.4291 2.52718
\(554\) 0 0
\(555\) −12.9906 −0.551422
\(556\) 0 0
\(557\) −14.2953 −0.605711 −0.302855 0.953037i \(-0.597940\pi\)
−0.302855 + 0.953037i \(0.597940\pi\)
\(558\) 0 0
\(559\) 5.71456 0.241700
\(560\) 0 0
\(561\) −38.4514 −1.62342
\(562\) 0 0
\(563\) −29.9541 −1.26241 −0.631207 0.775615i \(-0.717440\pi\)
−0.631207 + 0.775615i \(0.717440\pi\)
\(564\) 0 0
\(565\) −13.8938 −0.584518
\(566\) 0 0
\(567\) −18.5303 −0.778202
\(568\) 0 0
\(569\) −10.0000 −0.419222 −0.209611 0.977785i \(-0.567220\pi\)
−0.209611 + 0.977785i \(0.567220\pi\)
\(570\) 0 0
\(571\) 38.2351 1.60009 0.800044 0.599942i \(-0.204810\pi\)
0.800044 + 0.599942i \(0.204810\pi\)
\(572\) 0 0
\(573\) 0.774305 0.0323470
\(574\) 0 0
\(575\) −1.37778 −0.0574576
\(576\) 0 0
\(577\) −16.5718 −0.689895 −0.344947 0.938622i \(-0.612103\pi\)
−0.344947 + 0.938622i \(0.612103\pi\)
\(578\) 0 0
\(579\) −7.74620 −0.321921
\(580\) 0 0
\(581\) −47.0420 −1.95163
\(582\) 0 0
\(583\) −19.7333 −0.817270
\(584\) 0 0
\(585\) 2.57628 0.106516
\(586\) 0 0
\(587\) −7.94962 −0.328116 −0.164058 0.986451i \(-0.552458\pi\)
−0.164058 + 0.986451i \(0.552458\pi\)
\(588\) 0 0
\(589\) 1.24443 0.0512759
\(590\) 0 0
\(591\) −15.5210 −0.638448
\(592\) 0 0
\(593\) −17.0794 −0.701368 −0.350684 0.936494i \(-0.614051\pi\)
−0.350684 + 0.936494i \(0.614051\pi\)
\(594\) 0 0
\(595\) 22.3684 0.917016
\(596\) 0 0
\(597\) −49.7146 −2.03468
\(598\) 0 0
\(599\) 5.68598 0.232323 0.116161 0.993230i \(-0.462941\pi\)
0.116161 + 0.993230i \(0.462941\pi\)
\(600\) 0 0
\(601\) −33.2543 −1.35647 −0.678235 0.734845i \(-0.737255\pi\)
−0.678235 + 0.734845i \(0.737255\pi\)
\(602\) 0 0
\(603\) 9.00492 0.366709
\(604\) 0 0
\(605\) 4.12399 0.167664
\(606\) 0 0
\(607\) −10.9032 −0.442548 −0.221274 0.975212i \(-0.571021\pi\)
−0.221274 + 0.975212i \(0.571021\pi\)
\(608\) 0 0
\(609\) 100.369 4.06717
\(610\) 0 0
\(611\) 2.10171 0.0850261
\(612\) 0 0
\(613\) 47.6227 1.92346 0.961731 0.273995i \(-0.0883451\pi\)
0.961731 + 0.273995i \(0.0883451\pi\)
\(614\) 0 0
\(615\) −14.6637 −0.591298
\(616\) 0 0
\(617\) −46.6450 −1.87786 −0.938928 0.344114i \(-0.888179\pi\)
−0.938928 + 0.344114i \(0.888179\pi\)
\(618\) 0 0
\(619\) 32.2163 1.29488 0.647442 0.762115i \(-0.275839\pi\)
0.647442 + 0.762115i \(0.275839\pi\)
\(620\) 0 0
\(621\) 9.71456 0.389832
\(622\) 0 0
\(623\) 56.0830 2.24692
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) −7.61285 −0.304028
\(628\) 0 0
\(629\) −22.6004 −0.901138
\(630\) 0 0
\(631\) −30.9719 −1.23297 −0.616486 0.787366i \(-0.711444\pi\)
−0.616486 + 0.787366i \(0.711444\pi\)
\(632\) 0 0
\(633\) 38.1847 1.51770
\(634\) 0 0
\(635\) 7.19850 0.285664
\(636\) 0 0
\(637\) −5.98571 −0.237162
\(638\) 0 0
\(639\) 41.3274 1.63489
\(640\) 0 0
\(641\) −22.1748 −0.875854 −0.437927 0.899011i \(-0.644287\pi\)
−0.437927 + 0.899011i \(0.644287\pi\)
\(642\) 0 0
\(643\) 6.23506 0.245887 0.122943 0.992414i \(-0.460767\pi\)
0.122943 + 0.992414i \(0.460767\pi\)
\(644\) 0 0
\(645\) −34.9590 −1.37651
\(646\) 0 0
\(647\) −1.29481 −0.0509042 −0.0254521 0.999676i \(-0.508103\pi\)
−0.0254521 + 0.999676i \(0.508103\pi\)
\(648\) 0 0
\(649\) 5.75203 0.225787
\(650\) 0 0
\(651\) 16.0000 0.627089
\(652\) 0 0
\(653\) −30.2953 −1.18555 −0.592773 0.805370i \(-0.701967\pi\)
−0.592773 + 0.805370i \(0.701967\pi\)
\(654\) 0 0
\(655\) −2.10171 −0.0821206
\(656\) 0 0
\(657\) −20.6637 −0.806168
\(658\) 0 0
\(659\) 4.17484 0.162629 0.0813143 0.996689i \(-0.474088\pi\)
0.0813143 + 0.996689i \(0.474088\pi\)
\(660\) 0 0
\(661\) 6.56199 0.255232 0.127616 0.991824i \(-0.459268\pi\)
0.127616 + 0.991824i \(0.459268\pi\)
\(662\) 0 0
\(663\) 6.95899 0.270265
\(664\) 0 0
\(665\) 4.42864 0.171735
\(666\) 0 0
\(667\) −10.7556 −0.416457
\(668\) 0 0
\(669\) 30.5303 1.18037
\(670\) 0 0
\(671\) −9.63158 −0.371823
\(672\) 0 0
\(673\) 12.7413 0.491140 0.245570 0.969379i \(-0.421025\pi\)
0.245570 + 0.969379i \(0.421025\pi\)
\(674\) 0 0
\(675\) −7.05086 −0.271388
\(676\) 0 0
\(677\) 30.9260 1.18858 0.594291 0.804250i \(-0.297433\pi\)
0.594291 + 0.804250i \(0.297433\pi\)
\(678\) 0 0
\(679\) −79.2454 −3.04116
\(680\) 0 0
\(681\) 20.0415 0.767991
\(682\) 0 0
\(683\) −34.3412 −1.31403 −0.657015 0.753877i \(-0.728181\pi\)
−0.657015 + 0.753877i \(0.728181\pi\)
\(684\) 0 0
\(685\) −1.70471 −0.0651338
\(686\) 0 0
\(687\) 52.3783 1.99836
\(688\) 0 0
\(689\) 3.57136 0.136058
\(690\) 0 0
\(691\) −24.2163 −0.921233 −0.460616 0.887599i \(-0.652372\pi\)
−0.460616 + 0.887599i \(0.652372\pi\)
\(692\) 0 0
\(693\) −63.0420 −2.39477
\(694\) 0 0
\(695\) 8.72393 0.330917
\(696\) 0 0
\(697\) −25.5111 −0.966303
\(698\) 0 0
\(699\) 35.9081 1.35817
\(700\) 0 0
\(701\) −11.4064 −0.430812 −0.215406 0.976525i \(-0.569108\pi\)
−0.215406 + 0.976525i \(0.569108\pi\)
\(702\) 0 0
\(703\) −4.47457 −0.168762
\(704\) 0 0
\(705\) −12.8573 −0.484233
\(706\) 0 0
\(707\) −46.1847 −1.73695
\(708\) 0 0
\(709\) 13.0223 0.489062 0.244531 0.969642i \(-0.421366\pi\)
0.244531 + 0.969642i \(0.421366\pi\)
\(710\) 0 0
\(711\) −72.8484 −2.73203
\(712\) 0 0
\(713\) −1.71456 −0.0642107
\(714\) 0 0
\(715\) 1.24443 0.0465391
\(716\) 0 0
\(717\) −72.1659 −2.69509
\(718\) 0 0
\(719\) 52.2163 1.94734 0.973670 0.227961i \(-0.0732059\pi\)
0.973670 + 0.227961i \(0.0732059\pi\)
\(720\) 0 0
\(721\) 25.0607 0.933309
\(722\) 0 0
\(723\) −26.2766 −0.977235
\(724\) 0 0
\(725\) 7.80642 0.289923
\(726\) 0 0
\(727\) 14.0602 0.521465 0.260732 0.965411i \(-0.416036\pi\)
0.260732 + 0.965411i \(0.416036\pi\)
\(728\) 0 0
\(729\) −38.6958 −1.43318
\(730\) 0 0
\(731\) −60.8198 −2.24950
\(732\) 0 0
\(733\) 4.48886 0.165800 0.0829000 0.996558i \(-0.473582\pi\)
0.0829000 + 0.996558i \(0.473582\pi\)
\(734\) 0 0
\(735\) 36.6178 1.35067
\(736\) 0 0
\(737\) 4.34968 0.160223
\(738\) 0 0
\(739\) 20.0000 0.735712 0.367856 0.929883i \(-0.380092\pi\)
0.367856 + 0.929883i \(0.380092\pi\)
\(740\) 0 0
\(741\) 1.37778 0.0506142
\(742\) 0 0
\(743\) −37.5669 −1.37820 −0.689098 0.724668i \(-0.741993\pi\)
−0.689098 + 0.724668i \(0.741993\pi\)
\(744\) 0 0
\(745\) −6.81579 −0.249711
\(746\) 0 0
\(747\) 57.6642 2.10982
\(748\) 0 0
\(749\) 30.5718 1.11707
\(750\) 0 0
\(751\) 52.5817 1.91873 0.959366 0.282163i \(-0.0910520\pi\)
0.959366 + 0.282163i \(0.0910520\pi\)
\(752\) 0 0
\(753\) −65.5308 −2.38808
\(754\) 0 0
\(755\) 5.80642 0.211317
\(756\) 0 0
\(757\) −5.73329 −0.208380 −0.104190 0.994557i \(-0.533225\pi\)
−0.104190 + 0.994557i \(0.533225\pi\)
\(758\) 0 0
\(759\) 10.4889 0.380722
\(760\) 0 0
\(761\) −32.7338 −1.18660 −0.593299 0.804982i \(-0.702175\pi\)
−0.593299 + 0.804982i \(0.702175\pi\)
\(762\) 0 0
\(763\) 24.8573 0.899894
\(764\) 0 0
\(765\) −27.4193 −0.991346
\(766\) 0 0
\(767\) −1.04101 −0.0375887
\(768\) 0 0
\(769\) 10.1619 0.366449 0.183224 0.983071i \(-0.441347\pi\)
0.183224 + 0.983071i \(0.441347\pi\)
\(770\) 0 0
\(771\) −14.3555 −0.517001
\(772\) 0 0
\(773\) 36.0785 1.29765 0.648827 0.760936i \(-0.275260\pi\)
0.648827 + 0.760936i \(0.275260\pi\)
\(774\) 0 0
\(775\) 1.24443 0.0447013
\(776\) 0 0
\(777\) −57.5308 −2.06391
\(778\) 0 0
\(779\) −5.05086 −0.180966
\(780\) 0 0
\(781\) 19.9625 0.714315
\(782\) 0 0
\(783\) −55.0420 −1.96704
\(784\) 0 0
\(785\) 0.193576 0.00690903
\(786\) 0 0
\(787\) 30.5446 1.08880 0.544399 0.838826i \(-0.316758\pi\)
0.544399 + 0.838826i \(0.316758\pi\)
\(788\) 0 0
\(789\) −27.2257 −0.969260
\(790\) 0 0
\(791\) −61.5308 −2.18778
\(792\) 0 0
\(793\) 1.74314 0.0619007
\(794\) 0 0
\(795\) −21.8479 −0.774866
\(796\) 0 0
\(797\) 27.9037 0.988399 0.494200 0.869348i \(-0.335461\pi\)
0.494200 + 0.869348i \(0.335461\pi\)
\(798\) 0 0
\(799\) −22.3684 −0.791338
\(800\) 0 0
\(801\) −68.7467 −2.42904
\(802\) 0 0
\(803\) −9.98126 −0.352231
\(804\) 0 0
\(805\) −6.10171 −0.215057
\(806\) 0 0
\(807\) 57.2355 2.01479
\(808\) 0 0
\(809\) −25.6128 −0.900500 −0.450250 0.892903i \(-0.648665\pi\)
−0.450250 + 0.892903i \(0.648665\pi\)
\(810\) 0 0
\(811\) 6.01874 0.211346 0.105673 0.994401i \(-0.466300\pi\)
0.105673 + 0.994401i \(0.466300\pi\)
\(812\) 0 0
\(813\) 3.22570 0.113130
\(814\) 0 0
\(815\) −8.42864 −0.295242
\(816\) 0 0
\(817\) −12.0415 −0.421278
\(818\) 0 0
\(819\) 11.4094 0.398678
\(820\) 0 0
\(821\) 6.20342 0.216501 0.108250 0.994124i \(-0.465475\pi\)
0.108250 + 0.994124i \(0.465475\pi\)
\(822\) 0 0
\(823\) 1.75605 0.0612119 0.0306059 0.999532i \(-0.490256\pi\)
0.0306059 + 0.999532i \(0.490256\pi\)
\(824\) 0 0
\(825\) −7.61285 −0.265045
\(826\) 0 0
\(827\) 53.2083 1.85024 0.925118 0.379680i \(-0.123966\pi\)
0.925118 + 0.379680i \(0.123966\pi\)
\(828\) 0 0
\(829\) 26.9777 0.936975 0.468488 0.883470i \(-0.344799\pi\)
0.468488 + 0.883470i \(0.344799\pi\)
\(830\) 0 0
\(831\) 16.0286 0.556025
\(832\) 0 0
\(833\) 63.7057 2.20727
\(834\) 0 0
\(835\) −12.4429 −0.430605
\(836\) 0 0
\(837\) −8.77430 −0.303284
\(838\) 0 0
\(839\) 46.9501 1.62090 0.810449 0.585810i \(-0.199223\pi\)
0.810449 + 0.585810i \(0.199223\pi\)
\(840\) 0 0
\(841\) 31.9403 1.10139
\(842\) 0 0
\(843\) 45.8894 1.58051
\(844\) 0 0
\(845\) 12.7748 0.439466
\(846\) 0 0
\(847\) 18.2636 0.627546
\(848\) 0 0
\(849\) 41.3274 1.41835
\(850\) 0 0
\(851\) 6.16500 0.211333
\(852\) 0 0
\(853\) 46.4701 1.59111 0.795553 0.605883i \(-0.207180\pi\)
0.795553 + 0.605883i \(0.207180\pi\)
\(854\) 0 0
\(855\) −5.42864 −0.185656
\(856\) 0 0
\(857\) −7.79213 −0.266174 −0.133087 0.991104i \(-0.542489\pi\)
−0.133087 + 0.991104i \(0.542489\pi\)
\(858\) 0 0
\(859\) 37.4479 1.27770 0.638852 0.769330i \(-0.279410\pi\)
0.638852 + 0.769330i \(0.279410\pi\)
\(860\) 0 0
\(861\) −64.9403 −2.21316
\(862\) 0 0
\(863\) 47.7605 1.62579 0.812893 0.582413i \(-0.197891\pi\)
0.812893 + 0.582413i \(0.197891\pi\)
\(864\) 0 0
\(865\) −22.1891 −0.754453
\(866\) 0 0
\(867\) −24.7096 −0.839183
\(868\) 0 0
\(869\) −35.1882 −1.19368
\(870\) 0 0
\(871\) −0.787212 −0.0266736
\(872\) 0 0
\(873\) 97.1392 3.28766
\(874\) 0 0
\(875\) 4.42864 0.149715
\(876\) 0 0
\(877\) 22.5763 0.762347 0.381173 0.924504i \(-0.375520\pi\)
0.381173 + 0.924504i \(0.375520\pi\)
\(878\) 0 0
\(879\) 21.8479 0.736912
\(880\) 0 0
\(881\) −10.8988 −0.367189 −0.183594 0.983002i \(-0.558773\pi\)
−0.183594 + 0.983002i \(0.558773\pi\)
\(882\) 0 0
\(883\) 39.9782 1.34537 0.672687 0.739927i \(-0.265140\pi\)
0.672687 + 0.739927i \(0.265140\pi\)
\(884\) 0 0
\(885\) 6.36842 0.214072
\(886\) 0 0
\(887\) 49.6785 1.66804 0.834020 0.551734i \(-0.186034\pi\)
0.834020 + 0.551734i \(0.186034\pi\)
\(888\) 0 0
\(889\) 31.8796 1.06921
\(890\) 0 0
\(891\) 10.9719 0.367572
\(892\) 0 0
\(893\) −4.42864 −0.148199
\(894\) 0 0
\(895\) −11.9081 −0.398045
\(896\) 0 0
\(897\) −1.89829 −0.0633821
\(898\) 0 0
\(899\) 9.71456 0.323999
\(900\) 0 0
\(901\) −38.0098 −1.26629
\(902\) 0 0
\(903\) −154.821 −5.15211
\(904\) 0 0
\(905\) 17.6128 0.585471
\(906\) 0 0
\(907\) 18.2779 0.606909 0.303454 0.952846i \(-0.401860\pi\)
0.303454 + 0.952846i \(0.401860\pi\)
\(908\) 0 0
\(909\) 56.6133 1.87775
\(910\) 0 0
\(911\) −44.7654 −1.48314 −0.741572 0.670873i \(-0.765920\pi\)
−0.741572 + 0.670873i \(0.765920\pi\)
\(912\) 0 0
\(913\) 27.8537 0.921824
\(914\) 0 0
\(915\) −10.6637 −0.352531
\(916\) 0 0
\(917\) −9.30772 −0.307368
\(918\) 0 0
\(919\) −33.6316 −1.10940 −0.554702 0.832049i \(-0.687168\pi\)
−0.554702 + 0.832049i \(0.687168\pi\)
\(920\) 0 0
\(921\) −8.16193 −0.268945
\(922\) 0 0
\(923\) −3.61285 −0.118918
\(924\) 0 0
\(925\) −4.47457 −0.147123
\(926\) 0 0
\(927\) −30.7195 −1.00896
\(928\) 0 0
\(929\) 13.0223 0.427247 0.213623 0.976916i \(-0.431473\pi\)
0.213623 + 0.976916i \(0.431473\pi\)
\(930\) 0 0
\(931\) 12.6128 0.413369
\(932\) 0 0
\(933\) 40.2034 1.31620
\(934\) 0 0
\(935\) −13.2444 −0.433139
\(936\) 0 0
\(937\) 28.5433 0.932468 0.466234 0.884662i \(-0.345611\pi\)
0.466234 + 0.884662i \(0.345611\pi\)
\(938\) 0 0
\(939\) −67.4835 −2.20224
\(940\) 0 0
\(941\) −13.2257 −0.431145 −0.215573 0.976488i \(-0.569162\pi\)
−0.215573 + 0.976488i \(0.569162\pi\)
\(942\) 0 0
\(943\) 6.95899 0.226616
\(944\) 0 0
\(945\) −31.2257 −1.01577
\(946\) 0 0
\(947\) −3.66323 −0.119039 −0.0595194 0.998227i \(-0.518957\pi\)
−0.0595194 + 0.998227i \(0.518957\pi\)
\(948\) 0 0
\(949\) 1.80642 0.0586390
\(950\) 0 0
\(951\) 8.60348 0.278987
\(952\) 0 0
\(953\) 12.1704 0.394238 0.197119 0.980380i \(-0.436842\pi\)
0.197119 + 0.980380i \(0.436842\pi\)
\(954\) 0 0
\(955\) 0.266706 0.00863041
\(956\) 0 0
\(957\) −59.4291 −1.92107
\(958\) 0 0
\(959\) −7.54956 −0.243788
\(960\) 0 0
\(961\) −29.4514 −0.950045
\(962\) 0 0
\(963\) −37.4750 −1.20762
\(964\) 0 0
\(965\) −2.66815 −0.0858907
\(966\) 0 0
\(967\) 8.52051 0.274001 0.137000 0.990571i \(-0.456254\pi\)
0.137000 + 0.990571i \(0.456254\pi\)
\(968\) 0 0
\(969\) −14.6637 −0.471066
\(970\) 0 0
\(971\) 2.67259 0.0857676 0.0428838 0.999080i \(-0.486345\pi\)
0.0428838 + 0.999080i \(0.486345\pi\)
\(972\) 0 0
\(973\) 38.6351 1.23859
\(974\) 0 0
\(975\) 1.37778 0.0441244
\(976\) 0 0
\(977\) −20.3827 −0.652101 −0.326050 0.945352i \(-0.605718\pi\)
−0.326050 + 0.945352i \(0.605718\pi\)
\(978\) 0 0
\(979\) −33.2070 −1.06130
\(980\) 0 0
\(981\) −30.4701 −0.972836
\(982\) 0 0
\(983\) 10.3126 0.328922 0.164461 0.986384i \(-0.447412\pi\)
0.164461 + 0.986384i \(0.447412\pi\)
\(984\) 0 0
\(985\) −5.34614 −0.170342
\(986\) 0 0
\(987\) −56.9403 −1.81243
\(988\) 0 0
\(989\) 16.5906 0.527550
\(990\) 0 0
\(991\) 20.0919 0.638239 0.319120 0.947714i \(-0.396613\pi\)
0.319120 + 0.947714i \(0.396613\pi\)
\(992\) 0 0
\(993\) −2.75557 −0.0874453
\(994\) 0 0
\(995\) −17.1240 −0.542867
\(996\) 0 0
\(997\) −25.0509 −0.793369 −0.396684 0.917955i \(-0.629839\pi\)
−0.396684 + 0.917955i \(0.629839\pi\)
\(998\) 0 0
\(999\) 31.5496 0.998184
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.bo.1.1 3
4.3 odd 2 6080.2.a.by.1.3 3
8.3 odd 2 1520.2.a.p.1.1 3
8.5 even 2 95.2.a.a.1.2 3
24.5 odd 2 855.2.a.i.1.2 3
40.13 odd 4 475.2.b.d.324.3 6
40.19 odd 2 7600.2.a.bx.1.3 3
40.29 even 2 475.2.a.f.1.2 3
40.37 odd 4 475.2.b.d.324.4 6
56.13 odd 2 4655.2.a.u.1.2 3
120.29 odd 2 4275.2.a.bk.1.2 3
152.37 odd 2 1805.2.a.f.1.2 3
760.189 odd 2 9025.2.a.bb.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
95.2.a.a.1.2 3 8.5 even 2
475.2.a.f.1.2 3 40.29 even 2
475.2.b.d.324.3 6 40.13 odd 4
475.2.b.d.324.4 6 40.37 odd 4
855.2.a.i.1.2 3 24.5 odd 2
1520.2.a.p.1.1 3 8.3 odd 2
1805.2.a.f.1.2 3 152.37 odd 2
4275.2.a.bk.1.2 3 120.29 odd 2
4655.2.a.u.1.2 3 56.13 odd 2
6080.2.a.bo.1.1 3 1.1 even 1 trivial
6080.2.a.by.1.3 3 4.3 odd 2
7600.2.a.bx.1.3 3 40.19 odd 2
9025.2.a.bb.1.2 3 760.189 odd 2