Properties

Label 8048.2.a.u.1.13
Level $8048$
Weight $2$
Character 8048.1
Self dual yes
Analytic conductor $64.264$
Analytic rank $0$
Dimension $26$
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] = [8048,2,Mod(1,8048)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8048, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("8048.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8048 = 2^{4} \cdot 503 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8048.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: \(64.2636035467\)
Analytic rank: \(0\)
Dimension: \(26\)
Twist minimal: no (minimal twist has level 503)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.13
Character \(\chi\) \(=\) 8048.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.249551 q^{3} -3.41933 q^{5} +3.14640 q^{7} -2.93772 q^{9} +O(q^{10})\) \(q-0.249551 q^{3} -3.41933 q^{5} +3.14640 q^{7} -2.93772 q^{9} +4.41666 q^{11} +5.12764 q^{13} +0.853296 q^{15} +6.01726 q^{17} +5.70136 q^{19} -0.785187 q^{21} -3.81100 q^{23} +6.69180 q^{25} +1.48176 q^{27} -1.63013 q^{29} -3.23433 q^{31} -1.10218 q^{33} -10.7586 q^{35} +4.58346 q^{37} -1.27961 q^{39} +8.55052 q^{41} +7.40991 q^{43} +10.0450 q^{45} -0.286091 q^{47} +2.89984 q^{49} -1.50161 q^{51} -2.90591 q^{53} -15.1020 q^{55} -1.42278 q^{57} -5.10726 q^{59} +1.08325 q^{61} -9.24326 q^{63} -17.5331 q^{65} -2.78698 q^{67} +0.951037 q^{69} +1.27819 q^{71} +12.0666 q^{73} -1.66994 q^{75} +13.8966 q^{77} +14.0744 q^{79} +8.44340 q^{81} -10.3504 q^{83} -20.5750 q^{85} +0.406799 q^{87} -1.69526 q^{89} +16.1336 q^{91} +0.807131 q^{93} -19.4948 q^{95} -2.26933 q^{97} -12.9749 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 26 q - 4 q^{3} + 9 q^{5} - 11 q^{7} + 42 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 26 q - 4 q^{3} + 9 q^{5} - 11 q^{7} + 42 q^{9} + 17 q^{11} + 14 q^{13} - 18 q^{15} + 17 q^{17} + 22 q^{19} - 16 q^{21} - 27 q^{23} + 93 q^{25} - 31 q^{27} + 13 q^{29} - 26 q^{31} + 6 q^{33} + 22 q^{35} + 55 q^{37} + 15 q^{39} + 24 q^{41} - 20 q^{43} - 8 q^{45} + 25 q^{47} + 65 q^{49} - 7 q^{51} + 30 q^{53} - 25 q^{55} + 9 q^{57} + 26 q^{59} + 15 q^{61} + 19 q^{63} + 20 q^{65} + 20 q^{67} - 27 q^{69} + 35 q^{71} + 38 q^{73} - 2 q^{75} - 6 q^{77} - 21 q^{79} + 70 q^{81} + 48 q^{83} + 6 q^{85} + 9 q^{87} - 5 q^{89} + 24 q^{91} - 8 q^{93} - 43 q^{95} + 142 q^{97} + 50 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −0.249551 −0.144078 −0.0720391 0.997402i \(-0.522951\pi\)
−0.0720391 + 0.997402i \(0.522951\pi\)
\(4\) 0 0
\(5\) −3.41933 −1.52917 −0.764585 0.644523i \(-0.777056\pi\)
−0.764585 + 0.644523i \(0.777056\pi\)
\(6\) 0 0
\(7\) 3.14640 1.18923 0.594614 0.804011i \(-0.297305\pi\)
0.594614 + 0.804011i \(0.297305\pi\)
\(8\) 0 0
\(9\) −2.93772 −0.979241
\(10\) 0 0
\(11\) 4.41666 1.33167 0.665836 0.746098i \(-0.268075\pi\)
0.665836 + 0.746098i \(0.268075\pi\)
\(12\) 0 0
\(13\) 5.12764 1.42215 0.711076 0.703115i \(-0.248208\pi\)
0.711076 + 0.703115i \(0.248208\pi\)
\(14\) 0 0
\(15\) 0.853296 0.220320
\(16\) 0 0
\(17\) 6.01726 1.45940 0.729700 0.683767i \(-0.239660\pi\)
0.729700 + 0.683767i \(0.239660\pi\)
\(18\) 0 0
\(19\) 5.70136 1.30798 0.653991 0.756502i \(-0.273093\pi\)
0.653991 + 0.756502i \(0.273093\pi\)
\(20\) 0 0
\(21\) −0.785187 −0.171342
\(22\) 0 0
\(23\) −3.81100 −0.794648 −0.397324 0.917678i \(-0.630061\pi\)
−0.397324 + 0.917678i \(0.630061\pi\)
\(24\) 0 0
\(25\) 6.69180 1.33836
\(26\) 0 0
\(27\) 1.48176 0.285166
\(28\) 0 0
\(29\) −1.63013 −0.302707 −0.151353 0.988480i \(-0.548363\pi\)
−0.151353 + 0.988480i \(0.548363\pi\)
\(30\) 0 0
\(31\) −3.23433 −0.580904 −0.290452 0.956890i \(-0.593806\pi\)
−0.290452 + 0.956890i \(0.593806\pi\)
\(32\) 0 0
\(33\) −1.10218 −0.191865
\(34\) 0 0
\(35\) −10.7586 −1.81853
\(36\) 0 0
\(37\) 4.58346 0.753516 0.376758 0.926312i \(-0.377039\pi\)
0.376758 + 0.926312i \(0.377039\pi\)
\(38\) 0 0
\(39\) −1.27961 −0.204901
\(40\) 0 0
\(41\) 8.55052 1.33537 0.667684 0.744445i \(-0.267286\pi\)
0.667684 + 0.744445i \(0.267286\pi\)
\(42\) 0 0
\(43\) 7.40991 1.13000 0.565000 0.825091i \(-0.308876\pi\)
0.565000 + 0.825091i \(0.308876\pi\)
\(44\) 0 0
\(45\) 10.0450 1.49743
\(46\) 0 0
\(47\) −0.286091 −0.0417307 −0.0208653 0.999782i \(-0.506642\pi\)
−0.0208653 + 0.999782i \(0.506642\pi\)
\(48\) 0 0
\(49\) 2.89984 0.414263
\(50\) 0 0
\(51\) −1.50161 −0.210268
\(52\) 0 0
\(53\) −2.90591 −0.399158 −0.199579 0.979882i \(-0.563957\pi\)
−0.199579 + 0.979882i \(0.563957\pi\)
\(54\) 0 0
\(55\) −15.1020 −2.03635
\(56\) 0 0
\(57\) −1.42278 −0.188452
\(58\) 0 0
\(59\) −5.10726 −0.664909 −0.332455 0.943119i \(-0.607877\pi\)
−0.332455 + 0.943119i \(0.607877\pi\)
\(60\) 0 0
\(61\) 1.08325 0.138696 0.0693480 0.997593i \(-0.477908\pi\)
0.0693480 + 0.997593i \(0.477908\pi\)
\(62\) 0 0
\(63\) −9.24326 −1.16454
\(64\) 0 0
\(65\) −17.5331 −2.17471
\(66\) 0 0
\(67\) −2.78698 −0.340484 −0.170242 0.985402i \(-0.554455\pi\)
−0.170242 + 0.985402i \(0.554455\pi\)
\(68\) 0 0
\(69\) 0.951037 0.114491
\(70\) 0 0
\(71\) 1.27819 0.151693 0.0758467 0.997119i \(-0.475834\pi\)
0.0758467 + 0.997119i \(0.475834\pi\)
\(72\) 0 0
\(73\) 12.0666 1.41229 0.706144 0.708068i \(-0.250433\pi\)
0.706144 + 0.708068i \(0.250433\pi\)
\(74\) 0 0
\(75\) −1.66994 −0.192829
\(76\) 0 0
\(77\) 13.8966 1.58366
\(78\) 0 0
\(79\) 14.0744 1.58349 0.791747 0.610849i \(-0.209172\pi\)
0.791747 + 0.610849i \(0.209172\pi\)
\(80\) 0 0
\(81\) 8.44340 0.938155
\(82\) 0 0
\(83\) −10.3504 −1.13610 −0.568051 0.822994i \(-0.692302\pi\)
−0.568051 + 0.822994i \(0.692302\pi\)
\(84\) 0 0
\(85\) −20.5750 −2.23167
\(86\) 0 0
\(87\) 0.406799 0.0436135
\(88\) 0 0
\(89\) −1.69526 −0.179697 −0.0898487 0.995955i \(-0.528638\pi\)
−0.0898487 + 0.995955i \(0.528638\pi\)
\(90\) 0 0
\(91\) 16.1336 1.69126
\(92\) 0 0
\(93\) 0.807131 0.0836956
\(94\) 0 0
\(95\) −19.4948 −2.00013
\(96\) 0 0
\(97\) −2.26933 −0.230415 −0.115208 0.993341i \(-0.536753\pi\)
−0.115208 + 0.993341i \(0.536753\pi\)
\(98\) 0 0
\(99\) −12.9749 −1.30403
\(100\) 0 0
\(101\) −0.603233 −0.0600240 −0.0300120 0.999550i \(-0.509555\pi\)
−0.0300120 + 0.999550i \(0.509555\pi\)
\(102\) 0 0
\(103\) −8.07973 −0.796119 −0.398060 0.917360i \(-0.630316\pi\)
−0.398060 + 0.917360i \(0.630316\pi\)
\(104\) 0 0
\(105\) 2.68481 0.262011
\(106\) 0 0
\(107\) −3.27015 −0.316137 −0.158069 0.987428i \(-0.550527\pi\)
−0.158069 + 0.987428i \(0.550527\pi\)
\(108\) 0 0
\(109\) −12.0790 −1.15696 −0.578479 0.815698i \(-0.696353\pi\)
−0.578479 + 0.815698i \(0.696353\pi\)
\(110\) 0 0
\(111\) −1.14381 −0.108565
\(112\) 0 0
\(113\) −8.19405 −0.770832 −0.385416 0.922743i \(-0.625942\pi\)
−0.385416 + 0.922743i \(0.625942\pi\)
\(114\) 0 0
\(115\) 13.0310 1.21515
\(116\) 0 0
\(117\) −15.0636 −1.39263
\(118\) 0 0
\(119\) 18.9327 1.73556
\(120\) 0 0
\(121\) 8.50687 0.773352
\(122\) 0 0
\(123\) −2.13379 −0.192397
\(124\) 0 0
\(125\) −5.78482 −0.517410
\(126\) 0 0
\(127\) −2.27060 −0.201483 −0.100742 0.994913i \(-0.532122\pi\)
−0.100742 + 0.994913i \(0.532122\pi\)
\(128\) 0 0
\(129\) −1.84915 −0.162809
\(130\) 0 0
\(131\) −16.9078 −1.47724 −0.738619 0.674123i \(-0.764522\pi\)
−0.738619 + 0.674123i \(0.764522\pi\)
\(132\) 0 0
\(133\) 17.9388 1.55549
\(134\) 0 0
\(135\) −5.06664 −0.436067
\(136\) 0 0
\(137\) 18.3040 1.56381 0.781907 0.623395i \(-0.214247\pi\)
0.781907 + 0.623395i \(0.214247\pi\)
\(138\) 0 0
\(139\) 7.33049 0.621764 0.310882 0.950449i \(-0.399376\pi\)
0.310882 + 0.950449i \(0.399376\pi\)
\(140\) 0 0
\(141\) 0.0713943 0.00601248
\(142\) 0 0
\(143\) 22.6470 1.89384
\(144\) 0 0
\(145\) 5.57394 0.462890
\(146\) 0 0
\(147\) −0.723658 −0.0596863
\(148\) 0 0
\(149\) −23.1496 −1.89649 −0.948243 0.317545i \(-0.897142\pi\)
−0.948243 + 0.317545i \(0.897142\pi\)
\(150\) 0 0
\(151\) −16.9903 −1.38265 −0.691326 0.722543i \(-0.742973\pi\)
−0.691326 + 0.722543i \(0.742973\pi\)
\(152\) 0 0
\(153\) −17.6771 −1.42911
\(154\) 0 0
\(155\) 11.0592 0.888300
\(156\) 0 0
\(157\) −9.29534 −0.741849 −0.370924 0.928663i \(-0.620959\pi\)
−0.370924 + 0.928663i \(0.620959\pi\)
\(158\) 0 0
\(159\) 0.725173 0.0575100
\(160\) 0 0
\(161\) −11.9909 −0.945017
\(162\) 0 0
\(163\) −19.5792 −1.53356 −0.766782 0.641908i \(-0.778143\pi\)
−0.766782 + 0.641908i \(0.778143\pi\)
\(164\) 0 0
\(165\) 3.76872 0.293394
\(166\) 0 0
\(167\) 4.52102 0.349847 0.174924 0.984582i \(-0.444032\pi\)
0.174924 + 0.984582i \(0.444032\pi\)
\(168\) 0 0
\(169\) 13.2927 1.02252
\(170\) 0 0
\(171\) −16.7490 −1.28083
\(172\) 0 0
\(173\) −5.04673 −0.383696 −0.191848 0.981425i \(-0.561448\pi\)
−0.191848 + 0.981425i \(0.561448\pi\)
\(174\) 0 0
\(175\) 21.0551 1.59162
\(176\) 0 0
\(177\) 1.27452 0.0957989
\(178\) 0 0
\(179\) 14.0396 1.04937 0.524683 0.851297i \(-0.324184\pi\)
0.524683 + 0.851297i \(0.324184\pi\)
\(180\) 0 0
\(181\) 12.5186 0.930503 0.465251 0.885179i \(-0.345964\pi\)
0.465251 + 0.885179i \(0.345964\pi\)
\(182\) 0 0
\(183\) −0.270326 −0.0199831
\(184\) 0 0
\(185\) −15.6723 −1.15225
\(186\) 0 0
\(187\) 26.5762 1.94344
\(188\) 0 0
\(189\) 4.66222 0.339127
\(190\) 0 0
\(191\) −13.6964 −0.991039 −0.495520 0.868597i \(-0.665022\pi\)
−0.495520 + 0.868597i \(0.665022\pi\)
\(192\) 0 0
\(193\) 21.0423 1.51465 0.757327 0.653036i \(-0.226505\pi\)
0.757327 + 0.653036i \(0.226505\pi\)
\(194\) 0 0
\(195\) 4.37540 0.313329
\(196\) 0 0
\(197\) 7.22257 0.514587 0.257293 0.966333i \(-0.417169\pi\)
0.257293 + 0.966333i \(0.417169\pi\)
\(198\) 0 0
\(199\) −4.30012 −0.304828 −0.152414 0.988317i \(-0.548705\pi\)
−0.152414 + 0.988317i \(0.548705\pi\)
\(200\) 0 0
\(201\) 0.695494 0.0490563
\(202\) 0 0
\(203\) −5.12903 −0.359987
\(204\) 0 0
\(205\) −29.2370 −2.04200
\(206\) 0 0
\(207\) 11.1957 0.778152
\(208\) 0 0
\(209\) 25.1810 1.74180
\(210\) 0 0
\(211\) −5.14730 −0.354354 −0.177177 0.984179i \(-0.556697\pi\)
−0.177177 + 0.984179i \(0.556697\pi\)
\(212\) 0 0
\(213\) −0.318974 −0.0218557
\(214\) 0 0
\(215\) −25.3369 −1.72796
\(216\) 0 0
\(217\) −10.1765 −0.690827
\(218\) 0 0
\(219\) −3.01123 −0.203480
\(220\) 0 0
\(221\) 30.8544 2.07549
\(222\) 0 0
\(223\) 18.7155 1.25328 0.626642 0.779308i \(-0.284429\pi\)
0.626642 + 0.779308i \(0.284429\pi\)
\(224\) 0 0
\(225\) −19.6587 −1.31058
\(226\) 0 0
\(227\) 1.39574 0.0926385 0.0463193 0.998927i \(-0.485251\pi\)
0.0463193 + 0.998927i \(0.485251\pi\)
\(228\) 0 0
\(229\) 22.5174 1.48799 0.743997 0.668183i \(-0.232928\pi\)
0.743997 + 0.668183i \(0.232928\pi\)
\(230\) 0 0
\(231\) −3.46790 −0.228171
\(232\) 0 0
\(233\) −3.40361 −0.222978 −0.111489 0.993766i \(-0.535562\pi\)
−0.111489 + 0.993766i \(0.535562\pi\)
\(234\) 0 0
\(235\) 0.978239 0.0638133
\(236\) 0 0
\(237\) −3.51228 −0.228147
\(238\) 0 0
\(239\) 23.6423 1.52929 0.764647 0.644449i \(-0.222913\pi\)
0.764647 + 0.644449i \(0.222913\pi\)
\(240\) 0 0
\(241\) −14.3500 −0.924368 −0.462184 0.886784i \(-0.652934\pi\)
−0.462184 + 0.886784i \(0.652934\pi\)
\(242\) 0 0
\(243\) −6.55235 −0.420333
\(244\) 0 0
\(245\) −9.91551 −0.633479
\(246\) 0 0
\(247\) 29.2346 1.86015
\(248\) 0 0
\(249\) 2.58295 0.163688
\(250\) 0 0
\(251\) −4.21285 −0.265913 −0.132956 0.991122i \(-0.542447\pi\)
−0.132956 + 0.991122i \(0.542447\pi\)
\(252\) 0 0
\(253\) −16.8319 −1.05821
\(254\) 0 0
\(255\) 5.13451 0.321535
\(256\) 0 0
\(257\) 8.39309 0.523546 0.261773 0.965129i \(-0.415693\pi\)
0.261773 + 0.965129i \(0.415693\pi\)
\(258\) 0 0
\(259\) 14.4214 0.896102
\(260\) 0 0
\(261\) 4.78886 0.296423
\(262\) 0 0
\(263\) 5.71809 0.352592 0.176296 0.984337i \(-0.443588\pi\)
0.176296 + 0.984337i \(0.443588\pi\)
\(264\) 0 0
\(265\) 9.93627 0.610380
\(266\) 0 0
\(267\) 0.423054 0.0258905
\(268\) 0 0
\(269\) 25.0279 1.52598 0.762990 0.646410i \(-0.223731\pi\)
0.762990 + 0.646410i \(0.223731\pi\)
\(270\) 0 0
\(271\) 11.2035 0.680565 0.340282 0.940323i \(-0.389477\pi\)
0.340282 + 0.940323i \(0.389477\pi\)
\(272\) 0 0
\(273\) −4.02616 −0.243674
\(274\) 0 0
\(275\) 29.5554 1.78226
\(276\) 0 0
\(277\) 1.67194 0.100457 0.0502286 0.998738i \(-0.484005\pi\)
0.0502286 + 0.998738i \(0.484005\pi\)
\(278\) 0 0
\(279\) 9.50158 0.568845
\(280\) 0 0
\(281\) 9.69457 0.578330 0.289165 0.957279i \(-0.406622\pi\)
0.289165 + 0.957279i \(0.406622\pi\)
\(282\) 0 0
\(283\) 16.0644 0.954928 0.477464 0.878651i \(-0.341556\pi\)
0.477464 + 0.878651i \(0.341556\pi\)
\(284\) 0 0
\(285\) 4.86495 0.288175
\(286\) 0 0
\(287\) 26.9034 1.58806
\(288\) 0 0
\(289\) 19.2075 1.12985
\(290\) 0 0
\(291\) 0.566313 0.0331979
\(292\) 0 0
\(293\) −15.3639 −0.897567 −0.448783 0.893641i \(-0.648143\pi\)
−0.448783 + 0.893641i \(0.648143\pi\)
\(294\) 0 0
\(295\) 17.4634 1.01676
\(296\) 0 0
\(297\) 6.54445 0.379747
\(298\) 0 0
\(299\) −19.5414 −1.13011
\(300\) 0 0
\(301\) 23.3146 1.34383
\(302\) 0 0
\(303\) 0.150537 0.00864815
\(304\) 0 0
\(305\) −3.70399 −0.212090
\(306\) 0 0
\(307\) 31.9676 1.82449 0.912245 0.409645i \(-0.134348\pi\)
0.912245 + 0.409645i \(0.134348\pi\)
\(308\) 0 0
\(309\) 2.01630 0.114703
\(310\) 0 0
\(311\) 21.3149 1.20866 0.604328 0.796736i \(-0.293442\pi\)
0.604328 + 0.796736i \(0.293442\pi\)
\(312\) 0 0
\(313\) 1.08215 0.0611667 0.0305833 0.999532i \(-0.490264\pi\)
0.0305833 + 0.999532i \(0.490264\pi\)
\(314\) 0 0
\(315\) 31.6057 1.78078
\(316\) 0 0
\(317\) −6.38550 −0.358645 −0.179323 0.983790i \(-0.557391\pi\)
−0.179323 + 0.983790i \(0.557391\pi\)
\(318\) 0 0
\(319\) −7.19971 −0.403106
\(320\) 0 0
\(321\) 0.816068 0.0455485
\(322\) 0 0
\(323\) 34.3066 1.90887
\(324\) 0 0
\(325\) 34.3132 1.90335
\(326\) 0 0
\(327\) 3.01432 0.166692
\(328\) 0 0
\(329\) −0.900157 −0.0496273
\(330\) 0 0
\(331\) 3.58779 0.197203 0.0986014 0.995127i \(-0.468563\pi\)
0.0986014 + 0.995127i \(0.468563\pi\)
\(332\) 0 0
\(333\) −13.4649 −0.737874
\(334\) 0 0
\(335\) 9.52960 0.520658
\(336\) 0 0
\(337\) 12.4395 0.677624 0.338812 0.940854i \(-0.389975\pi\)
0.338812 + 0.940854i \(0.389975\pi\)
\(338\) 0 0
\(339\) 2.04483 0.111060
\(340\) 0 0
\(341\) −14.2850 −0.773573
\(342\) 0 0
\(343\) −12.9007 −0.696575
\(344\) 0 0
\(345\) −3.25191 −0.175077
\(346\) 0 0
\(347\) −36.1358 −1.93987 −0.969936 0.243359i \(-0.921751\pi\)
−0.969936 + 0.243359i \(0.921751\pi\)
\(348\) 0 0
\(349\) 25.3208 1.35539 0.677695 0.735343i \(-0.262979\pi\)
0.677695 + 0.735343i \(0.262979\pi\)
\(350\) 0 0
\(351\) 7.59796 0.405549
\(352\) 0 0
\(353\) 29.7478 1.58332 0.791659 0.610963i \(-0.209218\pi\)
0.791659 + 0.610963i \(0.209218\pi\)
\(354\) 0 0
\(355\) −4.37056 −0.231965
\(356\) 0 0
\(357\) −4.72468 −0.250056
\(358\) 0 0
\(359\) −11.1156 −0.586658 −0.293329 0.956011i \(-0.594763\pi\)
−0.293329 + 0.956011i \(0.594763\pi\)
\(360\) 0 0
\(361\) 13.5055 0.710818
\(362\) 0 0
\(363\) −2.12290 −0.111423
\(364\) 0 0
\(365\) −41.2596 −2.15963
\(366\) 0 0
\(367\) 6.09458 0.318135 0.159067 0.987268i \(-0.449151\pi\)
0.159067 + 0.987268i \(0.449151\pi\)
\(368\) 0 0
\(369\) −25.1191 −1.30765
\(370\) 0 0
\(371\) −9.14317 −0.474690
\(372\) 0 0
\(373\) −23.6135 −1.22266 −0.611331 0.791375i \(-0.709365\pi\)
−0.611331 + 0.791375i \(0.709365\pi\)
\(374\) 0 0
\(375\) 1.44361 0.0745475
\(376\) 0 0
\(377\) −8.35870 −0.430495
\(378\) 0 0
\(379\) −34.4236 −1.76822 −0.884110 0.467278i \(-0.845235\pi\)
−0.884110 + 0.467278i \(0.845235\pi\)
\(380\) 0 0
\(381\) 0.566630 0.0290293
\(382\) 0 0
\(383\) −12.4911 −0.638267 −0.319133 0.947710i \(-0.603392\pi\)
−0.319133 + 0.947710i \(0.603392\pi\)
\(384\) 0 0
\(385\) −47.5170 −2.42169
\(386\) 0 0
\(387\) −21.7683 −1.10654
\(388\) 0 0
\(389\) 22.3367 1.13252 0.566258 0.824228i \(-0.308391\pi\)
0.566258 + 0.824228i \(0.308391\pi\)
\(390\) 0 0
\(391\) −22.9318 −1.15971
\(392\) 0 0
\(393\) 4.21935 0.212838
\(394\) 0 0
\(395\) −48.1250 −2.42143
\(396\) 0 0
\(397\) 35.4105 1.77720 0.888601 0.458681i \(-0.151678\pi\)
0.888601 + 0.458681i \(0.151678\pi\)
\(398\) 0 0
\(399\) −4.47664 −0.224112
\(400\) 0 0
\(401\) 36.2302 1.80925 0.904625 0.426208i \(-0.140151\pi\)
0.904625 + 0.426208i \(0.140151\pi\)
\(402\) 0 0
\(403\) −16.5845 −0.826133
\(404\) 0 0
\(405\) −28.8707 −1.43460
\(406\) 0 0
\(407\) 20.2436 1.00344
\(408\) 0 0
\(409\) 4.31877 0.213549 0.106775 0.994283i \(-0.465948\pi\)
0.106775 + 0.994283i \(0.465948\pi\)
\(410\) 0 0
\(411\) −4.56777 −0.225312
\(412\) 0 0
\(413\) −16.0695 −0.790729
\(414\) 0 0
\(415\) 35.3913 1.73729
\(416\) 0 0
\(417\) −1.82933 −0.0895827
\(418\) 0 0
\(419\) −36.0794 −1.76259 −0.881296 0.472564i \(-0.843329\pi\)
−0.881296 + 0.472564i \(0.843329\pi\)
\(420\) 0 0
\(421\) −7.25150 −0.353417 −0.176708 0.984263i \(-0.556545\pi\)
−0.176708 + 0.984263i \(0.556545\pi\)
\(422\) 0 0
\(423\) 0.840456 0.0408644
\(424\) 0 0
\(425\) 40.2663 1.95320
\(426\) 0 0
\(427\) 3.40834 0.164941
\(428\) 0 0
\(429\) −5.65159 −0.272861
\(430\) 0 0
\(431\) 11.2094 0.539939 0.269970 0.962869i \(-0.412986\pi\)
0.269970 + 0.962869i \(0.412986\pi\)
\(432\) 0 0
\(433\) 7.51226 0.361016 0.180508 0.983574i \(-0.442226\pi\)
0.180508 + 0.983574i \(0.442226\pi\)
\(434\) 0 0
\(435\) −1.39098 −0.0666924
\(436\) 0 0
\(437\) −21.7279 −1.03939
\(438\) 0 0
\(439\) −2.55983 −0.122174 −0.0610870 0.998132i \(-0.519457\pi\)
−0.0610870 + 0.998132i \(0.519457\pi\)
\(440\) 0 0
\(441\) −8.51894 −0.405664
\(442\) 0 0
\(443\) −0.773786 −0.0367637 −0.0183819 0.999831i \(-0.505851\pi\)
−0.0183819 + 0.999831i \(0.505851\pi\)
\(444\) 0 0
\(445\) 5.79665 0.274788
\(446\) 0 0
\(447\) 5.77699 0.273242
\(448\) 0 0
\(449\) 26.3342 1.24279 0.621395 0.783497i \(-0.286566\pi\)
0.621395 + 0.783497i \(0.286566\pi\)
\(450\) 0 0
\(451\) 37.7647 1.77827
\(452\) 0 0
\(453\) 4.23995 0.199210
\(454\) 0 0
\(455\) −55.1661 −2.58623
\(456\) 0 0
\(457\) −4.22931 −0.197839 −0.0989193 0.995095i \(-0.531539\pi\)
−0.0989193 + 0.995095i \(0.531539\pi\)
\(458\) 0 0
\(459\) 8.91616 0.416171
\(460\) 0 0
\(461\) 18.2410 0.849568 0.424784 0.905295i \(-0.360350\pi\)
0.424784 + 0.905295i \(0.360350\pi\)
\(462\) 0 0
\(463\) −26.8683 −1.24868 −0.624339 0.781154i \(-0.714632\pi\)
−0.624339 + 0.781154i \(0.714632\pi\)
\(464\) 0 0
\(465\) −2.75984 −0.127985
\(466\) 0 0
\(467\) 34.0159 1.57407 0.787034 0.616910i \(-0.211616\pi\)
0.787034 + 0.616910i \(0.211616\pi\)
\(468\) 0 0
\(469\) −8.76896 −0.404913
\(470\) 0 0
\(471\) 2.31966 0.106884
\(472\) 0 0
\(473\) 32.7271 1.50479
\(474\) 0 0
\(475\) 38.1524 1.75055
\(476\) 0 0
\(477\) 8.53677 0.390872
\(478\) 0 0
\(479\) 14.9326 0.682286 0.341143 0.940011i \(-0.389186\pi\)
0.341143 + 0.940011i \(0.389186\pi\)
\(480\) 0 0
\(481\) 23.5023 1.07161
\(482\) 0 0
\(483\) 2.99235 0.136156
\(484\) 0 0
\(485\) 7.75958 0.352344
\(486\) 0 0
\(487\) −22.9187 −1.03854 −0.519272 0.854609i \(-0.673797\pi\)
−0.519272 + 0.854609i \(0.673797\pi\)
\(488\) 0 0
\(489\) 4.88601 0.220953
\(490\) 0 0
\(491\) 8.62408 0.389199 0.194600 0.980883i \(-0.437659\pi\)
0.194600 + 0.980883i \(0.437659\pi\)
\(492\) 0 0
\(493\) −9.80890 −0.441771
\(494\) 0 0
\(495\) 44.3655 1.99408
\(496\) 0 0
\(497\) 4.02171 0.180398
\(498\) 0 0
\(499\) −6.70698 −0.300245 −0.150123 0.988667i \(-0.547967\pi\)
−0.150123 + 0.988667i \(0.547967\pi\)
\(500\) 0 0
\(501\) −1.12822 −0.0504053
\(502\) 0 0
\(503\) −1.00000 −0.0445878
\(504\) 0 0
\(505\) 2.06265 0.0917868
\(506\) 0 0
\(507\) −3.31721 −0.147322
\(508\) 0 0
\(509\) −36.2859 −1.60834 −0.804172 0.594396i \(-0.797391\pi\)
−0.804172 + 0.594396i \(0.797391\pi\)
\(510\) 0 0
\(511\) 37.9664 1.67953
\(512\) 0 0
\(513\) 8.44808 0.372992
\(514\) 0 0
\(515\) 27.6272 1.21740
\(516\) 0 0
\(517\) −1.26357 −0.0555716
\(518\) 0 0
\(519\) 1.25942 0.0552822
\(520\) 0 0
\(521\) −13.9946 −0.613113 −0.306556 0.951853i \(-0.599177\pi\)
−0.306556 + 0.951853i \(0.599177\pi\)
\(522\) 0 0
\(523\) −14.9203 −0.652418 −0.326209 0.945298i \(-0.605771\pi\)
−0.326209 + 0.945298i \(0.605771\pi\)
\(524\) 0 0
\(525\) −5.25432 −0.229317
\(526\) 0 0
\(527\) −19.4618 −0.847771
\(528\) 0 0
\(529\) −8.47631 −0.368535
\(530\) 0 0
\(531\) 15.0037 0.651107
\(532\) 0 0
\(533\) 43.8440 1.89910
\(534\) 0 0
\(535\) 11.1817 0.483427
\(536\) 0 0
\(537\) −3.50359 −0.151191
\(538\) 0 0
\(539\) 12.8076 0.551663
\(540\) 0 0
\(541\) −6.25747 −0.269030 −0.134515 0.990912i \(-0.542948\pi\)
−0.134515 + 0.990912i \(0.542948\pi\)
\(542\) 0 0
\(543\) −3.12404 −0.134065
\(544\) 0 0
\(545\) 41.3020 1.76918
\(546\) 0 0
\(547\) −2.52078 −0.107781 −0.0538903 0.998547i \(-0.517162\pi\)
−0.0538903 + 0.998547i \(0.517162\pi\)
\(548\) 0 0
\(549\) −3.18229 −0.135817
\(550\) 0 0
\(551\) −9.29394 −0.395935
\(552\) 0 0
\(553\) 44.2837 1.88313
\(554\) 0 0
\(555\) 3.91105 0.166015
\(556\) 0 0
\(557\) −43.5318 −1.84450 −0.922251 0.386592i \(-0.873652\pi\)
−0.922251 + 0.386592i \(0.873652\pi\)
\(558\) 0 0
\(559\) 37.9954 1.60703
\(560\) 0 0
\(561\) −6.63211 −0.280008
\(562\) 0 0
\(563\) −32.9369 −1.38812 −0.694062 0.719915i \(-0.744181\pi\)
−0.694062 + 0.719915i \(0.744181\pi\)
\(564\) 0 0
\(565\) 28.0181 1.17873
\(566\) 0 0
\(567\) 26.5663 1.11568
\(568\) 0 0
\(569\) 30.6741 1.28593 0.642963 0.765897i \(-0.277705\pi\)
0.642963 + 0.765897i \(0.277705\pi\)
\(570\) 0 0
\(571\) 13.7285 0.574518 0.287259 0.957853i \(-0.407256\pi\)
0.287259 + 0.957853i \(0.407256\pi\)
\(572\) 0 0
\(573\) 3.41796 0.142787
\(574\) 0 0
\(575\) −25.5024 −1.06352
\(576\) 0 0
\(577\) 4.27924 0.178147 0.0890735 0.996025i \(-0.471609\pi\)
0.0890735 + 0.996025i \(0.471609\pi\)
\(578\) 0 0
\(579\) −5.25111 −0.218229
\(580\) 0 0
\(581\) −32.5664 −1.35108
\(582\) 0 0
\(583\) −12.8344 −0.531548
\(584\) 0 0
\(585\) 51.5074 2.12957
\(586\) 0 0
\(587\) −27.5066 −1.13532 −0.567660 0.823263i \(-0.692151\pi\)
−0.567660 + 0.823263i \(0.692151\pi\)
\(588\) 0 0
\(589\) −18.4401 −0.759812
\(590\) 0 0
\(591\) −1.80240 −0.0741407
\(592\) 0 0
\(593\) −11.7697 −0.483322 −0.241661 0.970361i \(-0.577692\pi\)
−0.241661 + 0.970361i \(0.577692\pi\)
\(594\) 0 0
\(595\) −64.7372 −2.65397
\(596\) 0 0
\(597\) 1.07310 0.0439190
\(598\) 0 0
\(599\) 18.1565 0.741853 0.370926 0.928662i \(-0.379040\pi\)
0.370926 + 0.928662i \(0.379040\pi\)
\(600\) 0 0
\(601\) 11.6836 0.476584 0.238292 0.971194i \(-0.423413\pi\)
0.238292 + 0.971194i \(0.423413\pi\)
\(602\) 0 0
\(603\) 8.18739 0.333416
\(604\) 0 0
\(605\) −29.0878 −1.18259
\(606\) 0 0
\(607\) 9.86359 0.400351 0.200175 0.979760i \(-0.435849\pi\)
0.200175 + 0.979760i \(0.435849\pi\)
\(608\) 0 0
\(609\) 1.27995 0.0518664
\(610\) 0 0
\(611\) −1.46697 −0.0593473
\(612\) 0 0
\(613\) −14.5408 −0.587298 −0.293649 0.955913i \(-0.594870\pi\)
−0.293649 + 0.955913i \(0.594870\pi\)
\(614\) 0 0
\(615\) 7.29613 0.294208
\(616\) 0 0
\(617\) 19.1178 0.769655 0.384828 0.922988i \(-0.374261\pi\)
0.384828 + 0.922988i \(0.374261\pi\)
\(618\) 0 0
\(619\) 4.05831 0.163117 0.0815586 0.996669i \(-0.474010\pi\)
0.0815586 + 0.996669i \(0.474010\pi\)
\(620\) 0 0
\(621\) −5.64700 −0.226606
\(622\) 0 0
\(623\) −5.33397 −0.213701
\(624\) 0 0
\(625\) −13.6788 −0.547152
\(626\) 0 0
\(627\) −6.28393 −0.250956
\(628\) 0 0
\(629\) 27.5799 1.09968
\(630\) 0 0
\(631\) 22.4761 0.894760 0.447380 0.894344i \(-0.352357\pi\)
0.447380 + 0.894344i \(0.352357\pi\)
\(632\) 0 0
\(633\) 1.28451 0.0510548
\(634\) 0 0
\(635\) 7.76392 0.308102
\(636\) 0 0
\(637\) 14.8694 0.589145
\(638\) 0 0
\(639\) −3.75498 −0.148545
\(640\) 0 0
\(641\) 11.8624 0.468536 0.234268 0.972172i \(-0.424731\pi\)
0.234268 + 0.972172i \(0.424731\pi\)
\(642\) 0 0
\(643\) −29.5003 −1.16338 −0.581689 0.813412i \(-0.697608\pi\)
−0.581689 + 0.813412i \(0.697608\pi\)
\(644\) 0 0
\(645\) 6.32285 0.248962
\(646\) 0 0
\(647\) −13.3929 −0.526531 −0.263265 0.964723i \(-0.584800\pi\)
−0.263265 + 0.964723i \(0.584800\pi\)
\(648\) 0 0
\(649\) −22.5570 −0.885441
\(650\) 0 0
\(651\) 2.53956 0.0995331
\(652\) 0 0
\(653\) −19.8275 −0.775911 −0.387955 0.921678i \(-0.626819\pi\)
−0.387955 + 0.921678i \(0.626819\pi\)
\(654\) 0 0
\(655\) 57.8132 2.25895
\(656\) 0 0
\(657\) −35.4483 −1.38297
\(658\) 0 0
\(659\) 1.34099 0.0522376 0.0261188 0.999659i \(-0.491685\pi\)
0.0261188 + 0.999659i \(0.491685\pi\)
\(660\) 0 0
\(661\) 36.0535 1.40232 0.701160 0.713004i \(-0.252666\pi\)
0.701160 + 0.713004i \(0.252666\pi\)
\(662\) 0 0
\(663\) −7.69973 −0.299033
\(664\) 0 0
\(665\) −61.3386 −2.37861
\(666\) 0 0
\(667\) 6.21241 0.240545
\(668\) 0 0
\(669\) −4.67047 −0.180571
\(670\) 0 0
\(671\) 4.78435 0.184698
\(672\) 0 0
\(673\) −20.6803 −0.797167 −0.398583 0.917132i \(-0.630498\pi\)
−0.398583 + 0.917132i \(0.630498\pi\)
\(674\) 0 0
\(675\) 9.91567 0.381654
\(676\) 0 0
\(677\) −42.7381 −1.64256 −0.821280 0.570526i \(-0.806740\pi\)
−0.821280 + 0.570526i \(0.806740\pi\)
\(678\) 0 0
\(679\) −7.14022 −0.274017
\(680\) 0 0
\(681\) −0.348308 −0.0133472
\(682\) 0 0
\(683\) 4.23750 0.162143 0.0810716 0.996708i \(-0.474166\pi\)
0.0810716 + 0.996708i \(0.474166\pi\)
\(684\) 0 0
\(685\) −62.5873 −2.39134
\(686\) 0 0
\(687\) −5.61924 −0.214387
\(688\) 0 0
\(689\) −14.9005 −0.567663
\(690\) 0 0
\(691\) −5.29484 −0.201425 −0.100713 0.994916i \(-0.532112\pi\)
−0.100713 + 0.994916i \(0.532112\pi\)
\(692\) 0 0
\(693\) −40.8243 −1.55079
\(694\) 0 0
\(695\) −25.0653 −0.950783
\(696\) 0 0
\(697\) 51.4507 1.94884
\(698\) 0 0
\(699\) 0.849375 0.0321263
\(700\) 0 0
\(701\) −19.9675 −0.754163 −0.377082 0.926180i \(-0.623072\pi\)
−0.377082 + 0.926180i \(0.623072\pi\)
\(702\) 0 0
\(703\) 26.1320 0.985586
\(704\) 0 0
\(705\) −0.244120 −0.00919410
\(706\) 0 0
\(707\) −1.89801 −0.0713822
\(708\) 0 0
\(709\) −3.83463 −0.144012 −0.0720062 0.997404i \(-0.522940\pi\)
−0.0720062 + 0.997404i \(0.522940\pi\)
\(710\) 0 0
\(711\) −41.3467 −1.55062
\(712\) 0 0
\(713\) 12.3260 0.461614
\(714\) 0 0
\(715\) −77.4377 −2.89600
\(716\) 0 0
\(717\) −5.89996 −0.220338
\(718\) 0 0
\(719\) 6.93826 0.258753 0.129377 0.991596i \(-0.458702\pi\)
0.129377 + 0.991596i \(0.458702\pi\)
\(720\) 0 0
\(721\) −25.4221 −0.946767
\(722\) 0 0
\(723\) 3.58107 0.133181
\(724\) 0 0
\(725\) −10.9085 −0.405131
\(726\) 0 0
\(727\) −46.7607 −1.73426 −0.867129 0.498084i \(-0.834037\pi\)
−0.867129 + 0.498084i \(0.834037\pi\)
\(728\) 0 0
\(729\) −23.6950 −0.877594
\(730\) 0 0
\(731\) 44.5874 1.64912
\(732\) 0 0
\(733\) −8.50440 −0.314117 −0.157059 0.987589i \(-0.550201\pi\)
−0.157059 + 0.987589i \(0.550201\pi\)
\(734\) 0 0
\(735\) 2.47442 0.0912705
\(736\) 0 0
\(737\) −12.3091 −0.453413
\(738\) 0 0
\(739\) 19.9848 0.735153 0.367577 0.929993i \(-0.380188\pi\)
0.367577 + 0.929993i \(0.380188\pi\)
\(740\) 0 0
\(741\) −7.29551 −0.268007
\(742\) 0 0
\(743\) −1.20328 −0.0441442 −0.0220721 0.999756i \(-0.507026\pi\)
−0.0220721 + 0.999756i \(0.507026\pi\)
\(744\) 0 0
\(745\) 79.1559 2.90005
\(746\) 0 0
\(747\) 30.4066 1.11252
\(748\) 0 0
\(749\) −10.2892 −0.375959
\(750\) 0 0
\(751\) −20.4312 −0.745546 −0.372773 0.927923i \(-0.621593\pi\)
−0.372773 + 0.927923i \(0.621593\pi\)
\(752\) 0 0
\(753\) 1.05132 0.0383122
\(754\) 0 0
\(755\) 58.0955 2.11431
\(756\) 0 0
\(757\) 30.1282 1.09503 0.547514 0.836796i \(-0.315574\pi\)
0.547514 + 0.836796i \(0.315574\pi\)
\(758\) 0 0
\(759\) 4.20041 0.152465
\(760\) 0 0
\(761\) −2.50437 −0.0907832 −0.0453916 0.998969i \(-0.514454\pi\)
−0.0453916 + 0.998969i \(0.514454\pi\)
\(762\) 0 0
\(763\) −38.0053 −1.37589
\(764\) 0 0
\(765\) 60.4437 2.18535
\(766\) 0 0
\(767\) −26.1882 −0.945602
\(768\) 0 0
\(769\) 31.8242 1.14761 0.573805 0.818992i \(-0.305467\pi\)
0.573805 + 0.818992i \(0.305467\pi\)
\(770\) 0 0
\(771\) −2.09450 −0.0754317
\(772\) 0 0
\(773\) −31.8710 −1.14632 −0.573160 0.819443i \(-0.694283\pi\)
−0.573160 + 0.819443i \(0.694283\pi\)
\(774\) 0 0
\(775\) −21.6435 −0.777458
\(776\) 0 0
\(777\) −3.59887 −0.129109
\(778\) 0 0
\(779\) 48.7496 1.74664
\(780\) 0 0
\(781\) 5.64534 0.202006
\(782\) 0 0
\(783\) −2.41546 −0.0863216
\(784\) 0 0
\(785\) 31.7838 1.13441
\(786\) 0 0
\(787\) 6.79776 0.242314 0.121157 0.992633i \(-0.461340\pi\)
0.121157 + 0.992633i \(0.461340\pi\)
\(788\) 0 0
\(789\) −1.42695 −0.0508009
\(790\) 0 0
\(791\) −25.7818 −0.916694
\(792\) 0 0
\(793\) 5.55452 0.197247
\(794\) 0 0
\(795\) −2.47960 −0.0879425
\(796\) 0 0
\(797\) −8.34934 −0.295749 −0.147874 0.989006i \(-0.547243\pi\)
−0.147874 + 0.989006i \(0.547243\pi\)
\(798\) 0 0
\(799\) −1.72148 −0.0609018
\(800\) 0 0
\(801\) 4.98021 0.175967
\(802\) 0 0
\(803\) 53.2940 1.88071
\(804\) 0 0
\(805\) 41.0009 1.44509
\(806\) 0 0
\(807\) −6.24574 −0.219861
\(808\) 0 0
\(809\) 5.99656 0.210828 0.105414 0.994428i \(-0.466383\pi\)
0.105414 + 0.994428i \(0.466383\pi\)
\(810\) 0 0
\(811\) 4.49461 0.157827 0.0789135 0.996881i \(-0.474855\pi\)
0.0789135 + 0.996881i \(0.474855\pi\)
\(812\) 0 0
\(813\) −2.79584 −0.0980546
\(814\) 0 0
\(815\) 66.9478 2.34508
\(816\) 0 0
\(817\) 42.2466 1.47802
\(818\) 0 0
\(819\) −47.3961 −1.65615
\(820\) 0 0
\(821\) −12.8301 −0.447775 −0.223887 0.974615i \(-0.571875\pi\)
−0.223887 + 0.974615i \(0.571875\pi\)
\(822\) 0 0
\(823\) −32.0580 −1.11747 −0.558736 0.829346i \(-0.688713\pi\)
−0.558736 + 0.829346i \(0.688713\pi\)
\(824\) 0 0
\(825\) −7.37557 −0.256785
\(826\) 0 0
\(827\) −13.8056 −0.480067 −0.240033 0.970765i \(-0.577158\pi\)
−0.240033 + 0.970765i \(0.577158\pi\)
\(828\) 0 0
\(829\) 27.6122 0.959012 0.479506 0.877539i \(-0.340816\pi\)
0.479506 + 0.877539i \(0.340816\pi\)
\(830\) 0 0
\(831\) −0.417234 −0.0144737
\(832\) 0 0
\(833\) 17.4491 0.604576
\(834\) 0 0
\(835\) −15.4588 −0.534975
\(836\) 0 0
\(837\) −4.79252 −0.165654
\(838\) 0 0
\(839\) −52.2002 −1.80215 −0.901075 0.433664i \(-0.857221\pi\)
−0.901075 + 0.433664i \(0.857221\pi\)
\(840\) 0 0
\(841\) −26.3427 −0.908369
\(842\) 0 0
\(843\) −2.41929 −0.0833247
\(844\) 0 0
\(845\) −45.4521 −1.56360
\(846\) 0 0
\(847\) 26.7660 0.919692
\(848\) 0 0
\(849\) −4.00888 −0.137584
\(850\) 0 0
\(851\) −17.4675 −0.598780
\(852\) 0 0
\(853\) −35.1534 −1.20363 −0.601815 0.798635i \(-0.705556\pi\)
−0.601815 + 0.798635i \(0.705556\pi\)
\(854\) 0 0
\(855\) 57.2704 1.95861
\(856\) 0 0
\(857\) 29.5507 1.00943 0.504717 0.863285i \(-0.331597\pi\)
0.504717 + 0.863285i \(0.331597\pi\)
\(858\) 0 0
\(859\) 47.5173 1.62127 0.810634 0.585553i \(-0.199122\pi\)
0.810634 + 0.585553i \(0.199122\pi\)
\(860\) 0 0
\(861\) −6.71376 −0.228804
\(862\) 0 0
\(863\) 33.3256 1.13442 0.567208 0.823575i \(-0.308024\pi\)
0.567208 + 0.823575i \(0.308024\pi\)
\(864\) 0 0
\(865\) 17.2564 0.586736
\(866\) 0 0
\(867\) −4.79324 −0.162787
\(868\) 0 0
\(869\) 62.1618 2.10869
\(870\) 0 0
\(871\) −14.2906 −0.484220
\(872\) 0 0
\(873\) 6.66666 0.225632
\(874\) 0 0
\(875\) −18.2014 −0.615318
\(876\) 0 0
\(877\) −13.4900 −0.455526 −0.227763 0.973717i \(-0.573141\pi\)
−0.227763 + 0.973717i \(0.573141\pi\)
\(878\) 0 0
\(879\) 3.83407 0.129320
\(880\) 0 0
\(881\) 22.4577 0.756619 0.378309 0.925679i \(-0.376506\pi\)
0.378309 + 0.925679i \(0.376506\pi\)
\(882\) 0 0
\(883\) −27.1656 −0.914194 −0.457097 0.889417i \(-0.651111\pi\)
−0.457097 + 0.889417i \(0.651111\pi\)
\(884\) 0 0
\(885\) −4.35801 −0.146493
\(886\) 0 0
\(887\) −44.0317 −1.47844 −0.739219 0.673465i \(-0.764805\pi\)
−0.739219 + 0.673465i \(0.764805\pi\)
\(888\) 0 0
\(889\) −7.14422 −0.239609
\(890\) 0 0
\(891\) 37.2916 1.24932
\(892\) 0 0
\(893\) −1.63111 −0.0545830
\(894\) 0 0
\(895\) −48.0059 −1.60466
\(896\) 0 0
\(897\) 4.87658 0.162824
\(898\) 0 0
\(899\) 5.27237 0.175844
\(900\) 0 0
\(901\) −17.4856 −0.582531
\(902\) 0 0
\(903\) −5.81817 −0.193616
\(904\) 0 0
\(905\) −42.8053 −1.42290
\(906\) 0 0
\(907\) 40.3798 1.34079 0.670395 0.742005i \(-0.266125\pi\)
0.670395 + 0.742005i \(0.266125\pi\)
\(908\) 0 0
\(909\) 1.77213 0.0587779
\(910\) 0 0
\(911\) −28.5817 −0.946953 −0.473477 0.880806i \(-0.657001\pi\)
−0.473477 + 0.880806i \(0.657001\pi\)
\(912\) 0 0
\(913\) −45.7141 −1.51292
\(914\) 0 0
\(915\) 0.924334 0.0305575
\(916\) 0 0
\(917\) −53.1986 −1.75677
\(918\) 0 0
\(919\) 17.6365 0.581773 0.290886 0.956758i \(-0.406050\pi\)
0.290886 + 0.956758i \(0.406050\pi\)
\(920\) 0 0
\(921\) −7.97755 −0.262869
\(922\) 0 0
\(923\) 6.55411 0.215731
\(924\) 0 0
\(925\) 30.6716 1.00848
\(926\) 0 0
\(927\) 23.7360 0.779593
\(928\) 0 0
\(929\) 12.6827 0.416104 0.208052 0.978118i \(-0.433288\pi\)
0.208052 + 0.978118i \(0.433288\pi\)
\(930\) 0 0
\(931\) 16.5331 0.541849
\(932\) 0 0
\(933\) −5.31914 −0.174141
\(934\) 0 0
\(935\) −90.8727 −2.97186
\(936\) 0 0
\(937\) −45.9762 −1.50198 −0.750988 0.660315i \(-0.770423\pi\)
−0.750988 + 0.660315i \(0.770423\pi\)
\(938\) 0 0
\(939\) −0.270051 −0.00881279
\(940\) 0 0
\(941\) −11.6920 −0.381147 −0.190573 0.981673i \(-0.561035\pi\)
−0.190573 + 0.981673i \(0.561035\pi\)
\(942\) 0 0
\(943\) −32.5860 −1.06115
\(944\) 0 0
\(945\) −15.9417 −0.518583
\(946\) 0 0
\(947\) 27.5772 0.896138 0.448069 0.893999i \(-0.352112\pi\)
0.448069 + 0.893999i \(0.352112\pi\)
\(948\) 0 0
\(949\) 61.8732 2.00849
\(950\) 0 0
\(951\) 1.59351 0.0516730
\(952\) 0 0
\(953\) 38.5211 1.24782 0.623910 0.781496i \(-0.285543\pi\)
0.623910 + 0.781496i \(0.285543\pi\)
\(954\) 0 0
\(955\) 46.8326 1.51547
\(956\) 0 0
\(957\) 1.79669 0.0580789
\(958\) 0 0
\(959\) 57.5917 1.85973
\(960\) 0 0
\(961\) −20.5391 −0.662551
\(962\) 0 0
\(963\) 9.60679 0.309575
\(964\) 0 0
\(965\) −71.9504 −2.31616
\(966\) 0 0
\(967\) −3.89304 −0.125192 −0.0625958 0.998039i \(-0.519938\pi\)
−0.0625958 + 0.998039i \(0.519938\pi\)
\(968\) 0 0
\(969\) −8.56124 −0.275027
\(970\) 0 0
\(971\) 29.7208 0.953786 0.476893 0.878961i \(-0.341763\pi\)
0.476893 + 0.878961i \(0.341763\pi\)
\(972\) 0 0
\(973\) 23.0647 0.739419
\(974\) 0 0
\(975\) −8.56288 −0.274232
\(976\) 0 0
\(977\) 6.60092 0.211182 0.105591 0.994410i \(-0.466327\pi\)
0.105591 + 0.994410i \(0.466327\pi\)
\(978\) 0 0
\(979\) −7.48739 −0.239298
\(980\) 0 0
\(981\) 35.4847 1.13294
\(982\) 0 0
\(983\) −20.2962 −0.647350 −0.323675 0.946168i \(-0.604918\pi\)
−0.323675 + 0.946168i \(0.604918\pi\)
\(984\) 0 0
\(985\) −24.6963 −0.786890
\(986\) 0 0
\(987\) 0.224635 0.00715021
\(988\) 0 0
\(989\) −28.2391 −0.897953
\(990\) 0 0
\(991\) −15.9316 −0.506083 −0.253041 0.967455i \(-0.581431\pi\)
−0.253041 + 0.967455i \(0.581431\pi\)
\(992\) 0 0
\(993\) −0.895336 −0.0284126
\(994\) 0 0
\(995\) 14.7035 0.466133
\(996\) 0 0
\(997\) 31.3449 0.992702 0.496351 0.868122i \(-0.334673\pi\)
0.496351 + 0.868122i \(0.334673\pi\)
\(998\) 0 0
\(999\) 6.79160 0.214877
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 8048.2.a.u.1.13 26
4.3 odd 2 503.2.a.f.1.1 26
12.11 even 2 4527.2.a.o.1.26 26
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
503.2.a.f.1.1 26 4.3 odd 2
4527.2.a.o.1.26 26 12.11 even 2
8048.2.a.u.1.13 26 1.1 even 1 trivial