Properties

Label 1336.2.a.b.1.5
Level $1336$
Weight $2$
Character 1336.1
Self dual yes
Analytic conductor $10.668$
Analytic rank $1$
Dimension $7$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1336,2,Mod(1,1336)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1336, 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("1336.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1336 = 2^{3} \cdot 167 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1336.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: \(10.6680137100\)
Analytic rank: \(1\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{7} - x^{6} - 9x^{5} + 5x^{4} + 25x^{3} - 4x^{2} - 17x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(0.960446\) of defining polynomial
Character \(\chi\) \(=\) 1336.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.0395540 q^{3} +0.949625 q^{5} +1.23977 q^{7} -2.99844 q^{9} +O(q^{10})\) \(q-0.0395540 q^{3} +0.949625 q^{5} +1.23977 q^{7} -2.99844 q^{9} -3.02968 q^{11} -2.87052 q^{13} -0.0375615 q^{15} +0.912064 q^{17} -2.10358 q^{19} -0.0490380 q^{21} -3.82122 q^{23} -4.09821 q^{25} +0.237262 q^{27} -0.0768876 q^{29} -6.57708 q^{31} +0.119836 q^{33} +1.17732 q^{35} -1.49129 q^{37} +0.113540 q^{39} +10.0656 q^{41} +0.744801 q^{43} -2.84739 q^{45} -10.3191 q^{47} -5.46296 q^{49} -0.0360758 q^{51} +2.28582 q^{53} -2.87706 q^{55} +0.0832051 q^{57} -0.377610 q^{59} +8.54258 q^{61} -3.71738 q^{63} -2.72592 q^{65} +7.45388 q^{67} +0.151144 q^{69} -9.83565 q^{71} +14.8350 q^{73} +0.162101 q^{75} -3.75611 q^{77} +9.73971 q^{79} +8.98592 q^{81} -13.4173 q^{83} +0.866119 q^{85} +0.00304121 q^{87} -15.1009 q^{89} -3.55879 q^{91} +0.260150 q^{93} -1.99762 q^{95} -4.86636 q^{97} +9.08430 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q - 6 q^{3} - q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 7 q - 6 q^{3} - q^{7} + 3 q^{9} - 6 q^{11} - 2 q^{13} - 9 q^{15} - 9 q^{17} - 3 q^{19} + 7 q^{21} - 10 q^{23} + 3 q^{25} - 12 q^{27} - 5 q^{29} - 21 q^{31} + 8 q^{33} - 12 q^{35} + 19 q^{37} - 27 q^{39} - 22 q^{41} - 19 q^{43} + 13 q^{45} - 13 q^{47} - 14 q^{49} + 4 q^{51} + 5 q^{53} - 17 q^{55} + 5 q^{57} - 18 q^{59} + 26 q^{61} - 20 q^{63} - 20 q^{65} - 27 q^{67} - 3 q^{69} - 46 q^{71} - 25 q^{73} - 19 q^{75} - 19 q^{77} - 22 q^{79} - 9 q^{81} + q^{83} - 11 q^{85} + 9 q^{87} - 3 q^{89} + 33 q^{93} - 40 q^{95} + 11 q^{97} - 21 q^{99}+O(q^{100}) \) 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
\(3\) −0.0395540 −0.0228365 −0.0114183 0.999935i \(-0.503635\pi\)
−0.0114183 + 0.999935i \(0.503635\pi\)
\(4\) 0 0
\(5\) 0.949625 0.424685 0.212343 0.977195i \(-0.431891\pi\)
0.212343 + 0.977195i \(0.431891\pi\)
\(6\) 0 0
\(7\) 1.23977 0.468590 0.234295 0.972166i \(-0.424722\pi\)
0.234295 + 0.972166i \(0.424722\pi\)
\(8\) 0 0
\(9\) −2.99844 −0.999478
\(10\) 0 0
\(11\) −3.02968 −0.913483 −0.456741 0.889600i \(-0.650984\pi\)
−0.456741 + 0.889600i \(0.650984\pi\)
\(12\) 0 0
\(13\) −2.87052 −0.796138 −0.398069 0.917355i \(-0.630320\pi\)
−0.398069 + 0.917355i \(0.630320\pi\)
\(14\) 0 0
\(15\) −0.0375615 −0.00969833
\(16\) 0 0
\(17\) 0.912064 0.221208 0.110604 0.993865i \(-0.464721\pi\)
0.110604 + 0.993865i \(0.464721\pi\)
\(18\) 0 0
\(19\) −2.10358 −0.482595 −0.241298 0.970451i \(-0.577573\pi\)
−0.241298 + 0.970451i \(0.577573\pi\)
\(20\) 0 0
\(21\) −0.0490380 −0.0107010
\(22\) 0 0
\(23\) −3.82122 −0.796778 −0.398389 0.917216i \(-0.630431\pi\)
−0.398389 + 0.917216i \(0.630431\pi\)
\(24\) 0 0
\(25\) −4.09821 −0.819642
\(26\) 0 0
\(27\) 0.237262 0.0456611
\(28\) 0 0
\(29\) −0.0768876 −0.0142777 −0.00713883 0.999975i \(-0.502272\pi\)
−0.00713883 + 0.999975i \(0.502272\pi\)
\(30\) 0 0
\(31\) −6.57708 −1.18128 −0.590639 0.806936i \(-0.701124\pi\)
−0.590639 + 0.806936i \(0.701124\pi\)
\(32\) 0 0
\(33\) 0.119836 0.0208608
\(34\) 0 0
\(35\) 1.17732 0.199003
\(36\) 0 0
\(37\) −1.49129 −0.245166 −0.122583 0.992458i \(-0.539118\pi\)
−0.122583 + 0.992458i \(0.539118\pi\)
\(38\) 0 0
\(39\) 0.113540 0.0181810
\(40\) 0 0
\(41\) 10.0656 1.57199 0.785995 0.618233i \(-0.212151\pi\)
0.785995 + 0.618233i \(0.212151\pi\)
\(42\) 0 0
\(43\) 0.744801 0.113581 0.0567905 0.998386i \(-0.481913\pi\)
0.0567905 + 0.998386i \(0.481913\pi\)
\(44\) 0 0
\(45\) −2.84739 −0.424464
\(46\) 0 0
\(47\) −10.3191 −1.50520 −0.752598 0.658480i \(-0.771200\pi\)
−0.752598 + 0.658480i \(0.771200\pi\)
\(48\) 0 0
\(49\) −5.46296 −0.780423
\(50\) 0 0
\(51\) −0.0360758 −0.00505162
\(52\) 0 0
\(53\) 2.28582 0.313982 0.156991 0.987600i \(-0.449821\pi\)
0.156991 + 0.987600i \(0.449821\pi\)
\(54\) 0 0
\(55\) −2.87706 −0.387943
\(56\) 0 0
\(57\) 0.0832051 0.0110208
\(58\) 0 0
\(59\) −0.377610 −0.0491606 −0.0245803 0.999698i \(-0.507825\pi\)
−0.0245803 + 0.999698i \(0.507825\pi\)
\(60\) 0 0
\(61\) 8.54258 1.09377 0.546883 0.837209i \(-0.315814\pi\)
0.546883 + 0.837209i \(0.315814\pi\)
\(62\) 0 0
\(63\) −3.71738 −0.468346
\(64\) 0 0
\(65\) −2.72592 −0.338108
\(66\) 0 0
\(67\) 7.45388 0.910636 0.455318 0.890329i \(-0.349525\pi\)
0.455318 + 0.890329i \(0.349525\pi\)
\(68\) 0 0
\(69\) 0.151144 0.0181956
\(70\) 0 0
\(71\) −9.83565 −1.16728 −0.583638 0.812014i \(-0.698371\pi\)
−0.583638 + 0.812014i \(0.698371\pi\)
\(72\) 0 0
\(73\) 14.8350 1.73631 0.868153 0.496297i \(-0.165307\pi\)
0.868153 + 0.496297i \(0.165307\pi\)
\(74\) 0 0
\(75\) 0.162101 0.0187178
\(76\) 0 0
\(77\) −3.75611 −0.428049
\(78\) 0 0
\(79\) 9.73971 1.09580 0.547901 0.836543i \(-0.315427\pi\)
0.547901 + 0.836543i \(0.315427\pi\)
\(80\) 0 0
\(81\) 8.98592 0.998436
\(82\) 0 0
\(83\) −13.4173 −1.47275 −0.736373 0.676576i \(-0.763463\pi\)
−0.736373 + 0.676576i \(0.763463\pi\)
\(84\) 0 0
\(85\) 0.866119 0.0939438
\(86\) 0 0
\(87\) 0.00304121 0.000326052 0
\(88\) 0 0
\(89\) −15.1009 −1.60070 −0.800348 0.599536i \(-0.795352\pi\)
−0.800348 + 0.599536i \(0.795352\pi\)
\(90\) 0 0
\(91\) −3.55879 −0.373062
\(92\) 0 0
\(93\) 0.260150 0.0269763
\(94\) 0 0
\(95\) −1.99762 −0.204951
\(96\) 0 0
\(97\) −4.86636 −0.494104 −0.247052 0.969002i \(-0.579462\pi\)
−0.247052 + 0.969002i \(0.579462\pi\)
\(98\) 0 0
\(99\) 9.08430 0.913006
\(100\) 0 0
\(101\) 3.07427 0.305902 0.152951 0.988234i \(-0.451122\pi\)
0.152951 + 0.988234i \(0.451122\pi\)
\(102\) 0 0
\(103\) −16.8957 −1.66478 −0.832391 0.554189i \(-0.813028\pi\)
−0.832391 + 0.554189i \(0.813028\pi\)
\(104\) 0 0
\(105\) −0.0465677 −0.00454454
\(106\) 0 0
\(107\) −18.8737 −1.82459 −0.912297 0.409529i \(-0.865693\pi\)
−0.912297 + 0.409529i \(0.865693\pi\)
\(108\) 0 0
\(109\) −4.31618 −0.413415 −0.206707 0.978403i \(-0.566275\pi\)
−0.206707 + 0.978403i \(0.566275\pi\)
\(110\) 0 0
\(111\) 0.0589863 0.00559873
\(112\) 0 0
\(113\) −6.74376 −0.634400 −0.317200 0.948359i \(-0.602743\pi\)
−0.317200 + 0.948359i \(0.602743\pi\)
\(114\) 0 0
\(115\) −3.62872 −0.338380
\(116\) 0 0
\(117\) 8.60706 0.795723
\(118\) 0 0
\(119\) 1.13075 0.103656
\(120\) 0 0
\(121\) −1.82104 −0.165549
\(122\) 0 0
\(123\) −0.398136 −0.0358987
\(124\) 0 0
\(125\) −8.63989 −0.772776
\(126\) 0 0
\(127\) −0.331446 −0.0294111 −0.0147055 0.999892i \(-0.504681\pi\)
−0.0147055 + 0.999892i \(0.504681\pi\)
\(128\) 0 0
\(129\) −0.0294599 −0.00259380
\(130\) 0 0
\(131\) 14.9506 1.30624 0.653120 0.757255i \(-0.273460\pi\)
0.653120 + 0.757255i \(0.273460\pi\)
\(132\) 0 0
\(133\) −2.60796 −0.226139
\(134\) 0 0
\(135\) 0.225310 0.0193916
\(136\) 0 0
\(137\) 6.30018 0.538260 0.269130 0.963104i \(-0.413264\pi\)
0.269130 + 0.963104i \(0.413264\pi\)
\(138\) 0 0
\(139\) −19.7160 −1.67229 −0.836145 0.548509i \(-0.815196\pi\)
−0.836145 + 0.548509i \(0.815196\pi\)
\(140\) 0 0
\(141\) 0.408162 0.0343734
\(142\) 0 0
\(143\) 8.69675 0.727259
\(144\) 0 0
\(145\) −0.0730144 −0.00606352
\(146\) 0 0
\(147\) 0.216082 0.0178221
\(148\) 0 0
\(149\) 7.24667 0.593671 0.296835 0.954929i \(-0.404069\pi\)
0.296835 + 0.954929i \(0.404069\pi\)
\(150\) 0 0
\(151\) 12.0425 0.980007 0.490003 0.871721i \(-0.336996\pi\)
0.490003 + 0.871721i \(0.336996\pi\)
\(152\) 0 0
\(153\) −2.73476 −0.221093
\(154\) 0 0
\(155\) −6.24576 −0.501671
\(156\) 0 0
\(157\) −2.55988 −0.204301 −0.102150 0.994769i \(-0.532572\pi\)
−0.102150 + 0.994769i \(0.532572\pi\)
\(158\) 0 0
\(159\) −0.0904135 −0.00717026
\(160\) 0 0
\(161\) −4.73744 −0.373362
\(162\) 0 0
\(163\) 10.6944 0.837653 0.418827 0.908066i \(-0.362442\pi\)
0.418827 + 0.908066i \(0.362442\pi\)
\(164\) 0 0
\(165\) 0.113799 0.00885926
\(166\) 0 0
\(167\) −1.00000 −0.0773823
\(168\) 0 0
\(169\) −4.76013 −0.366164
\(170\) 0 0
\(171\) 6.30746 0.482343
\(172\) 0 0
\(173\) 23.1831 1.76258 0.881289 0.472577i \(-0.156676\pi\)
0.881289 + 0.472577i \(0.156676\pi\)
\(174\) 0 0
\(175\) −5.08085 −0.384076
\(176\) 0 0
\(177\) 0.0149360 0.00112266
\(178\) 0 0
\(179\) −6.85130 −0.512090 −0.256045 0.966665i \(-0.582420\pi\)
−0.256045 + 0.966665i \(0.582420\pi\)
\(180\) 0 0
\(181\) 2.93347 0.218043 0.109022 0.994039i \(-0.465228\pi\)
0.109022 + 0.994039i \(0.465228\pi\)
\(182\) 0 0
\(183\) −0.337893 −0.0249778
\(184\) 0 0
\(185\) −1.41616 −0.104118
\(186\) 0 0
\(187\) −2.76326 −0.202070
\(188\) 0 0
\(189\) 0.294151 0.0213963
\(190\) 0 0
\(191\) 9.40287 0.680368 0.340184 0.940359i \(-0.389511\pi\)
0.340184 + 0.940359i \(0.389511\pi\)
\(192\) 0 0
\(193\) 21.3656 1.53793 0.768963 0.639293i \(-0.220773\pi\)
0.768963 + 0.639293i \(0.220773\pi\)
\(194\) 0 0
\(195\) 0.107821 0.00772121
\(196\) 0 0
\(197\) 8.49612 0.605324 0.302662 0.953098i \(-0.402125\pi\)
0.302662 + 0.953098i \(0.402125\pi\)
\(198\) 0 0
\(199\) 1.14532 0.0811897 0.0405949 0.999176i \(-0.487075\pi\)
0.0405949 + 0.999176i \(0.487075\pi\)
\(200\) 0 0
\(201\) −0.294831 −0.0207958
\(202\) 0 0
\(203\) −0.0953231 −0.00669037
\(204\) 0 0
\(205\) 9.55859 0.667601
\(206\) 0 0
\(207\) 11.4577 0.796363
\(208\) 0 0
\(209\) 6.37318 0.440842
\(210\) 0 0
\(211\) 22.7779 1.56810 0.784048 0.620701i \(-0.213152\pi\)
0.784048 + 0.620701i \(0.213152\pi\)
\(212\) 0 0
\(213\) 0.389039 0.0266565
\(214\) 0 0
\(215\) 0.707282 0.0482362
\(216\) 0 0
\(217\) −8.15408 −0.553535
\(218\) 0 0
\(219\) −0.586783 −0.0396511
\(220\) 0 0
\(221\) −2.61810 −0.176112
\(222\) 0 0
\(223\) −0.652262 −0.0436787 −0.0218393 0.999761i \(-0.506952\pi\)
−0.0218393 + 0.999761i \(0.506952\pi\)
\(224\) 0 0
\(225\) 12.2882 0.819215
\(226\) 0 0
\(227\) 0.999405 0.0663329 0.0331664 0.999450i \(-0.489441\pi\)
0.0331664 + 0.999450i \(0.489441\pi\)
\(228\) 0 0
\(229\) 16.0664 1.06170 0.530850 0.847466i \(-0.321873\pi\)
0.530850 + 0.847466i \(0.321873\pi\)
\(230\) 0 0
\(231\) 0.148569 0.00977514
\(232\) 0 0
\(233\) −23.9924 −1.57179 −0.785896 0.618359i \(-0.787798\pi\)
−0.785896 + 0.618359i \(0.787798\pi\)
\(234\) 0 0
\(235\) −9.79928 −0.639235
\(236\) 0 0
\(237\) −0.385244 −0.0250243
\(238\) 0 0
\(239\) −1.81344 −0.117302 −0.0586509 0.998279i \(-0.518680\pi\)
−0.0586509 + 0.998279i \(0.518680\pi\)
\(240\) 0 0
\(241\) −6.60250 −0.425304 −0.212652 0.977128i \(-0.568210\pi\)
−0.212652 + 0.977128i \(0.568210\pi\)
\(242\) 0 0
\(243\) −1.06722 −0.0684619
\(244\) 0 0
\(245\) −5.18777 −0.331434
\(246\) 0 0
\(247\) 6.03837 0.384212
\(248\) 0 0
\(249\) 0.530710 0.0336324
\(250\) 0 0
\(251\) −23.2755 −1.46913 −0.734567 0.678536i \(-0.762615\pi\)
−0.734567 + 0.678536i \(0.762615\pi\)
\(252\) 0 0
\(253\) 11.5771 0.727843
\(254\) 0 0
\(255\) −0.0342585 −0.00214535
\(256\) 0 0
\(257\) 2.40517 0.150030 0.0750151 0.997182i \(-0.476100\pi\)
0.0750151 + 0.997182i \(0.476100\pi\)
\(258\) 0 0
\(259\) −1.84886 −0.114882
\(260\) 0 0
\(261\) 0.230543 0.0142702
\(262\) 0 0
\(263\) −14.8866 −0.917945 −0.458973 0.888450i \(-0.651782\pi\)
−0.458973 + 0.888450i \(0.651782\pi\)
\(264\) 0 0
\(265\) 2.17068 0.133344
\(266\) 0 0
\(267\) 0.597302 0.0365543
\(268\) 0 0
\(269\) 18.3302 1.11761 0.558805 0.829299i \(-0.311260\pi\)
0.558805 + 0.829299i \(0.311260\pi\)
\(270\) 0 0
\(271\) 16.4803 1.00111 0.500553 0.865706i \(-0.333130\pi\)
0.500553 + 0.865706i \(0.333130\pi\)
\(272\) 0 0
\(273\) 0.140764 0.00851944
\(274\) 0 0
\(275\) 12.4163 0.748729
\(276\) 0 0
\(277\) 19.7791 1.18841 0.594205 0.804313i \(-0.297467\pi\)
0.594205 + 0.804313i \(0.297467\pi\)
\(278\) 0 0
\(279\) 19.7209 1.18066
\(280\) 0 0
\(281\) 18.3136 1.09250 0.546250 0.837622i \(-0.316055\pi\)
0.546250 + 0.837622i \(0.316055\pi\)
\(282\) 0 0
\(283\) −8.52286 −0.506631 −0.253316 0.967384i \(-0.581521\pi\)
−0.253316 + 0.967384i \(0.581521\pi\)
\(284\) 0 0
\(285\) 0.0790137 0.00468037
\(286\) 0 0
\(287\) 12.4791 0.736619
\(288\) 0 0
\(289\) −16.1681 −0.951067
\(290\) 0 0
\(291\) 0.192484 0.0112836
\(292\) 0 0
\(293\) −28.5538 −1.66813 −0.834066 0.551665i \(-0.813993\pi\)
−0.834066 + 0.551665i \(0.813993\pi\)
\(294\) 0 0
\(295\) −0.358588 −0.0208778
\(296\) 0 0
\(297\) −0.718828 −0.0417106
\(298\) 0 0
\(299\) 10.9689 0.634346
\(300\) 0 0
\(301\) 0.923384 0.0532230
\(302\) 0 0
\(303\) −0.121600 −0.00698573
\(304\) 0 0
\(305\) 8.11225 0.464506
\(306\) 0 0
\(307\) 13.0965 0.747455 0.373728 0.927538i \(-0.378079\pi\)
0.373728 + 0.927538i \(0.378079\pi\)
\(308\) 0 0
\(309\) 0.668292 0.0380178
\(310\) 0 0
\(311\) −7.66348 −0.434556 −0.217278 0.976110i \(-0.569718\pi\)
−0.217278 + 0.976110i \(0.569718\pi\)
\(312\) 0 0
\(313\) 9.47042 0.535300 0.267650 0.963516i \(-0.413753\pi\)
0.267650 + 0.963516i \(0.413753\pi\)
\(314\) 0 0
\(315\) −3.53012 −0.198900
\(316\) 0 0
\(317\) −9.08696 −0.510374 −0.255187 0.966892i \(-0.582137\pi\)
−0.255187 + 0.966892i \(0.582137\pi\)
\(318\) 0 0
\(319\) 0.232945 0.0130424
\(320\) 0 0
\(321\) 0.746532 0.0416674
\(322\) 0 0
\(323\) −1.91860 −0.106754
\(324\) 0 0
\(325\) 11.7640 0.652549
\(326\) 0 0
\(327\) 0.170722 0.00944095
\(328\) 0 0
\(329\) −12.7933 −0.705320
\(330\) 0 0
\(331\) −15.0330 −0.826287 −0.413143 0.910666i \(-0.635569\pi\)
−0.413143 + 0.910666i \(0.635569\pi\)
\(332\) 0 0
\(333\) 4.47153 0.245038
\(334\) 0 0
\(335\) 7.07839 0.386734
\(336\) 0 0
\(337\) 5.26580 0.286847 0.143423 0.989661i \(-0.454189\pi\)
0.143423 + 0.989661i \(0.454189\pi\)
\(338\) 0 0
\(339\) 0.266743 0.0144875
\(340\) 0 0
\(341\) 19.9264 1.07908
\(342\) 0 0
\(343\) −15.4512 −0.834289
\(344\) 0 0
\(345\) 0.143530 0.00772742
\(346\) 0 0
\(347\) 24.9180 1.33767 0.668835 0.743411i \(-0.266793\pi\)
0.668835 + 0.743411i \(0.266793\pi\)
\(348\) 0 0
\(349\) 16.6911 0.893452 0.446726 0.894671i \(-0.352590\pi\)
0.446726 + 0.894671i \(0.352590\pi\)
\(350\) 0 0
\(351\) −0.681065 −0.0363526
\(352\) 0 0
\(353\) −20.5946 −1.09614 −0.548069 0.836433i \(-0.684637\pi\)
−0.548069 + 0.836433i \(0.684637\pi\)
\(354\) 0 0
\(355\) −9.34018 −0.495725
\(356\) 0 0
\(357\) −0.0447258 −0.00236714
\(358\) 0 0
\(359\) −9.96132 −0.525738 −0.262869 0.964831i \(-0.584669\pi\)
−0.262869 + 0.964831i \(0.584669\pi\)
\(360\) 0 0
\(361\) −14.5749 −0.767102
\(362\) 0 0
\(363\) 0.0720295 0.00378057
\(364\) 0 0
\(365\) 14.0877 0.737383
\(366\) 0 0
\(367\) −0.377347 −0.0196973 −0.00984867 0.999952i \(-0.503135\pi\)
−0.00984867 + 0.999952i \(0.503135\pi\)
\(368\) 0 0
\(369\) −30.1812 −1.57117
\(370\) 0 0
\(371\) 2.83390 0.147129
\(372\) 0 0
\(373\) 11.7304 0.607376 0.303688 0.952772i \(-0.401782\pi\)
0.303688 + 0.952772i \(0.401782\pi\)
\(374\) 0 0
\(375\) 0.341742 0.0176475
\(376\) 0 0
\(377\) 0.220707 0.0113670
\(378\) 0 0
\(379\) 25.2301 1.29598 0.647992 0.761647i \(-0.275609\pi\)
0.647992 + 0.761647i \(0.275609\pi\)
\(380\) 0 0
\(381\) 0.0131100 0.000671647 0
\(382\) 0 0
\(383\) 2.43389 0.124366 0.0621829 0.998065i \(-0.480194\pi\)
0.0621829 + 0.998065i \(0.480194\pi\)
\(384\) 0 0
\(385\) −3.56690 −0.181786
\(386\) 0 0
\(387\) −2.23324 −0.113522
\(388\) 0 0
\(389\) 4.53105 0.229734 0.114867 0.993381i \(-0.463356\pi\)
0.114867 + 0.993381i \(0.463356\pi\)
\(390\) 0 0
\(391\) −3.48519 −0.176254
\(392\) 0 0
\(393\) −0.591356 −0.0298300
\(394\) 0 0
\(395\) 9.24907 0.465371
\(396\) 0 0
\(397\) 32.6635 1.63933 0.819667 0.572840i \(-0.194158\pi\)
0.819667 + 0.572840i \(0.194158\pi\)
\(398\) 0 0
\(399\) 0.103155 0.00516423
\(400\) 0 0
\(401\) −15.3745 −0.767765 −0.383883 0.923382i \(-0.625413\pi\)
−0.383883 + 0.923382i \(0.625413\pi\)
\(402\) 0 0
\(403\) 18.8796 0.940460
\(404\) 0 0
\(405\) 8.53326 0.424021
\(406\) 0 0
\(407\) 4.51812 0.223955
\(408\) 0 0
\(409\) −37.5233 −1.85541 −0.927704 0.373316i \(-0.878221\pi\)
−0.927704 + 0.373316i \(0.878221\pi\)
\(410\) 0 0
\(411\) −0.249197 −0.0122920
\(412\) 0 0
\(413\) −0.468150 −0.0230362
\(414\) 0 0
\(415\) −12.7415 −0.625453
\(416\) 0 0
\(417\) 0.779847 0.0381893
\(418\) 0 0
\(419\) −3.28954 −0.160705 −0.0803523 0.996767i \(-0.525605\pi\)
−0.0803523 + 0.996767i \(0.525605\pi\)
\(420\) 0 0
\(421\) 38.1287 1.85828 0.929140 0.369727i \(-0.120549\pi\)
0.929140 + 0.369727i \(0.120549\pi\)
\(422\) 0 0
\(423\) 30.9412 1.50441
\(424\) 0 0
\(425\) −3.73783 −0.181311
\(426\) 0 0
\(427\) 10.5909 0.512528
\(428\) 0 0
\(429\) −0.343991 −0.0166080
\(430\) 0 0
\(431\) −20.7620 −1.00007 −0.500034 0.866006i \(-0.666679\pi\)
−0.500034 + 0.866006i \(0.666679\pi\)
\(432\) 0 0
\(433\) −14.2343 −0.684057 −0.342029 0.939690i \(-0.611114\pi\)
−0.342029 + 0.939690i \(0.611114\pi\)
\(434\) 0 0
\(435\) 0.00288801 0.000138470 0
\(436\) 0 0
\(437\) 8.03824 0.384521
\(438\) 0 0
\(439\) 9.58593 0.457511 0.228756 0.973484i \(-0.426534\pi\)
0.228756 + 0.973484i \(0.426534\pi\)
\(440\) 0 0
\(441\) 16.3803 0.780016
\(442\) 0 0
\(443\) −3.25114 −0.154466 −0.0772331 0.997013i \(-0.524609\pi\)
−0.0772331 + 0.997013i \(0.524609\pi\)
\(444\) 0 0
\(445\) −14.3402 −0.679792
\(446\) 0 0
\(447\) −0.286635 −0.0135574
\(448\) 0 0
\(449\) −25.7412 −1.21480 −0.607401 0.794395i \(-0.707788\pi\)
−0.607401 + 0.794395i \(0.707788\pi\)
\(450\) 0 0
\(451\) −30.4957 −1.43599
\(452\) 0 0
\(453\) −0.476330 −0.0223799
\(454\) 0 0
\(455\) −3.37952 −0.158434
\(456\) 0 0
\(457\) −6.60084 −0.308774 −0.154387 0.988010i \(-0.549340\pi\)
−0.154387 + 0.988010i \(0.549340\pi\)
\(458\) 0 0
\(459\) 0.216398 0.0101006
\(460\) 0 0
\(461\) −27.5339 −1.28238 −0.641190 0.767382i \(-0.721559\pi\)
−0.641190 + 0.767382i \(0.721559\pi\)
\(462\) 0 0
\(463\) −20.2087 −0.939179 −0.469589 0.882885i \(-0.655598\pi\)
−0.469589 + 0.882885i \(0.655598\pi\)
\(464\) 0 0
\(465\) 0.247045 0.0114564
\(466\) 0 0
\(467\) 16.3548 0.756808 0.378404 0.925640i \(-0.376473\pi\)
0.378404 + 0.925640i \(0.376473\pi\)
\(468\) 0 0
\(469\) 9.24112 0.426715
\(470\) 0 0
\(471\) 0.101254 0.00466552
\(472\) 0 0
\(473\) −2.25651 −0.103754
\(474\) 0 0
\(475\) 8.62093 0.395555
\(476\) 0 0
\(477\) −6.85390 −0.313818
\(478\) 0 0
\(479\) 26.4284 1.20755 0.603773 0.797156i \(-0.293663\pi\)
0.603773 + 0.797156i \(0.293663\pi\)
\(480\) 0 0
\(481\) 4.28076 0.195186
\(482\) 0 0
\(483\) 0.187385 0.00852629
\(484\) 0 0
\(485\) −4.62122 −0.209839
\(486\) 0 0
\(487\) 21.5253 0.975407 0.487703 0.873009i \(-0.337835\pi\)
0.487703 + 0.873009i \(0.337835\pi\)
\(488\) 0 0
\(489\) −0.423008 −0.0191291
\(490\) 0 0
\(491\) −6.06918 −0.273898 −0.136949 0.990578i \(-0.543730\pi\)
−0.136949 + 0.990578i \(0.543730\pi\)
\(492\) 0 0
\(493\) −0.0701264 −0.00315833
\(494\) 0 0
\(495\) 8.62668 0.387740
\(496\) 0 0
\(497\) −12.1940 −0.546974
\(498\) 0 0
\(499\) 6.64230 0.297350 0.148675 0.988886i \(-0.452499\pi\)
0.148675 + 0.988886i \(0.452499\pi\)
\(500\) 0 0
\(501\) 0.0395540 0.00176714
\(502\) 0 0
\(503\) 17.8481 0.795806 0.397903 0.917427i \(-0.369738\pi\)
0.397903 + 0.917427i \(0.369738\pi\)
\(504\) 0 0
\(505\) 2.91941 0.129912
\(506\) 0 0
\(507\) 0.188282 0.00836190
\(508\) 0 0
\(509\) −42.7605 −1.89533 −0.947663 0.319271i \(-0.896562\pi\)
−0.947663 + 0.319271i \(0.896562\pi\)
\(510\) 0 0
\(511\) 18.3920 0.813615
\(512\) 0 0
\(513\) −0.499100 −0.0220358
\(514\) 0 0
\(515\) −16.0446 −0.707008
\(516\) 0 0
\(517\) 31.2636 1.37497
\(518\) 0 0
\(519\) −0.916984 −0.0402511
\(520\) 0 0
\(521\) −21.5909 −0.945913 −0.472957 0.881086i \(-0.656813\pi\)
−0.472957 + 0.881086i \(0.656813\pi\)
\(522\) 0 0
\(523\) 9.67558 0.423084 0.211542 0.977369i \(-0.432152\pi\)
0.211542 + 0.977369i \(0.432152\pi\)
\(524\) 0 0
\(525\) 0.200968 0.00877096
\(526\) 0 0
\(527\) −5.99871 −0.261308
\(528\) 0 0
\(529\) −8.39831 −0.365144
\(530\) 0 0
\(531\) 1.13224 0.0491350
\(532\) 0 0
\(533\) −28.8936 −1.25152
\(534\) 0 0
\(535\) −17.9230 −0.774879
\(536\) 0 0
\(537\) 0.270996 0.0116943
\(538\) 0 0
\(539\) 16.5510 0.712903
\(540\) 0 0
\(541\) −22.5143 −0.967965 −0.483982 0.875078i \(-0.660810\pi\)
−0.483982 + 0.875078i \(0.660810\pi\)
\(542\) 0 0
\(543\) −0.116030 −0.00497934
\(544\) 0 0
\(545\) −4.09875 −0.175571
\(546\) 0 0
\(547\) 4.84060 0.206969 0.103485 0.994631i \(-0.467001\pi\)
0.103485 + 0.994631i \(0.467001\pi\)
\(548\) 0 0
\(549\) −25.6144 −1.09319
\(550\) 0 0
\(551\) 0.161739 0.00689033
\(552\) 0 0
\(553\) 12.0750 0.513482
\(554\) 0 0
\(555\) 0.0560149 0.00237770
\(556\) 0 0
\(557\) −31.5882 −1.33843 −0.669217 0.743067i \(-0.733370\pi\)
−0.669217 + 0.743067i \(0.733370\pi\)
\(558\) 0 0
\(559\) −2.13796 −0.0904263
\(560\) 0 0
\(561\) 0.109298 0.00461457
\(562\) 0 0
\(563\) 40.3929 1.70236 0.851178 0.524877i \(-0.175889\pi\)
0.851178 + 0.524877i \(0.175889\pi\)
\(564\) 0 0
\(565\) −6.40405 −0.269420
\(566\) 0 0
\(567\) 11.1405 0.467857
\(568\) 0 0
\(569\) −3.53972 −0.148393 −0.0741964 0.997244i \(-0.523639\pi\)
−0.0741964 + 0.997244i \(0.523639\pi\)
\(570\) 0 0
\(571\) −21.3498 −0.893460 −0.446730 0.894669i \(-0.647412\pi\)
−0.446730 + 0.894669i \(0.647412\pi\)
\(572\) 0 0
\(573\) −0.371921 −0.0155372
\(574\) 0 0
\(575\) 15.6601 0.653073
\(576\) 0 0
\(577\) 7.47187 0.311058 0.155529 0.987831i \(-0.450292\pi\)
0.155529 + 0.987831i \(0.450292\pi\)
\(578\) 0 0
\(579\) −0.845093 −0.0351209
\(580\) 0 0
\(581\) −16.6345 −0.690114
\(582\) 0 0
\(583\) −6.92532 −0.286817
\(584\) 0 0
\(585\) 8.17348 0.337932
\(586\) 0 0
\(587\) −6.32749 −0.261163 −0.130582 0.991438i \(-0.541685\pi\)
−0.130582 + 0.991438i \(0.541685\pi\)
\(588\) 0 0
\(589\) 13.8354 0.570079
\(590\) 0 0
\(591\) −0.336056 −0.0138235
\(592\) 0 0
\(593\) −24.4067 −1.00227 −0.501133 0.865371i \(-0.667083\pi\)
−0.501133 + 0.865371i \(0.667083\pi\)
\(594\) 0 0
\(595\) 1.07379 0.0440211
\(596\) 0 0
\(597\) −0.0453021 −0.00185409
\(598\) 0 0
\(599\) −24.0464 −0.982510 −0.491255 0.871016i \(-0.663462\pi\)
−0.491255 + 0.871016i \(0.663462\pi\)
\(600\) 0 0
\(601\) −23.6360 −0.964131 −0.482065 0.876135i \(-0.660113\pi\)
−0.482065 + 0.876135i \(0.660113\pi\)
\(602\) 0 0
\(603\) −22.3500 −0.910162
\(604\) 0 0
\(605\) −1.72931 −0.0703064
\(606\) 0 0
\(607\) −21.4194 −0.869387 −0.434693 0.900579i \(-0.643143\pi\)
−0.434693 + 0.900579i \(0.643143\pi\)
\(608\) 0 0
\(609\) 0.00377041 0.000152785 0
\(610\) 0 0
\(611\) 29.6212 1.19834
\(612\) 0 0
\(613\) −15.6553 −0.632312 −0.316156 0.948707i \(-0.602392\pi\)
−0.316156 + 0.948707i \(0.602392\pi\)
\(614\) 0 0
\(615\) −0.378080 −0.0152457
\(616\) 0 0
\(617\) 11.3465 0.456793 0.228397 0.973568i \(-0.426652\pi\)
0.228397 + 0.973568i \(0.426652\pi\)
\(618\) 0 0
\(619\) −3.48960 −0.140259 −0.0701294 0.997538i \(-0.522341\pi\)
−0.0701294 + 0.997538i \(0.522341\pi\)
\(620\) 0 0
\(621\) −0.906629 −0.0363818
\(622\) 0 0
\(623\) −18.7217 −0.750070
\(624\) 0 0
\(625\) 12.2864 0.491456
\(626\) 0 0
\(627\) −0.252085 −0.0100673
\(628\) 0 0
\(629\) −1.36015 −0.0542327
\(630\) 0 0
\(631\) 1.47130 0.0585717 0.0292858 0.999571i \(-0.490677\pi\)
0.0292858 + 0.999571i \(0.490677\pi\)
\(632\) 0 0
\(633\) −0.900957 −0.0358098
\(634\) 0 0
\(635\) −0.314750 −0.0124905
\(636\) 0 0
\(637\) 15.6815 0.621325
\(638\) 0 0
\(639\) 29.4916 1.16667
\(640\) 0 0
\(641\) 30.6705 1.21141 0.605706 0.795688i \(-0.292891\pi\)
0.605706 + 0.795688i \(0.292891\pi\)
\(642\) 0 0
\(643\) −9.60725 −0.378873 −0.189436 0.981893i \(-0.560666\pi\)
−0.189436 + 0.981893i \(0.560666\pi\)
\(644\) 0 0
\(645\) −0.0279758 −0.00110155
\(646\) 0 0
\(647\) −11.7775 −0.463023 −0.231512 0.972832i \(-0.574367\pi\)
−0.231512 + 0.972832i \(0.574367\pi\)
\(648\) 0 0
\(649\) 1.14404 0.0449074
\(650\) 0 0
\(651\) 0.322526 0.0126408
\(652\) 0 0
\(653\) 19.0014 0.743582 0.371791 0.928316i \(-0.378744\pi\)
0.371791 + 0.928316i \(0.378744\pi\)
\(654\) 0 0
\(655\) 14.1975 0.554741
\(656\) 0 0
\(657\) −44.4818 −1.73540
\(658\) 0 0
\(659\) −18.2717 −0.711766 −0.355883 0.934531i \(-0.615820\pi\)
−0.355883 + 0.934531i \(0.615820\pi\)
\(660\) 0 0
\(661\) −6.75810 −0.262860 −0.131430 0.991325i \(-0.541957\pi\)
−0.131430 + 0.991325i \(0.541957\pi\)
\(662\) 0 0
\(663\) 0.103556 0.00402179
\(664\) 0 0
\(665\) −2.47659 −0.0960380
\(666\) 0 0
\(667\) 0.293804 0.0113761
\(668\) 0 0
\(669\) 0.0257996 0.000997468 0
\(670\) 0 0
\(671\) −25.8813 −0.999136
\(672\) 0 0
\(673\) 1.78348 0.0687482 0.0343741 0.999409i \(-0.489056\pi\)
0.0343741 + 0.999409i \(0.489056\pi\)
\(674\) 0 0
\(675\) −0.972350 −0.0374258
\(676\) 0 0
\(677\) −26.7045 −1.02634 −0.513169 0.858287i \(-0.671529\pi\)
−0.513169 + 0.858287i \(0.671529\pi\)
\(678\) 0 0
\(679\) −6.03318 −0.231532
\(680\) 0 0
\(681\) −0.0395305 −0.00151481
\(682\) 0 0
\(683\) −38.9664 −1.49101 −0.745505 0.666500i \(-0.767792\pi\)
−0.745505 + 0.666500i \(0.767792\pi\)
\(684\) 0 0
\(685\) 5.98281 0.228591
\(686\) 0 0
\(687\) −0.635491 −0.0242455
\(688\) 0 0
\(689\) −6.56150 −0.249973
\(690\) 0 0
\(691\) 6.79132 0.258354 0.129177 0.991622i \(-0.458766\pi\)
0.129177 + 0.991622i \(0.458766\pi\)
\(692\) 0 0
\(693\) 11.2625 0.427826
\(694\) 0 0
\(695\) −18.7228 −0.710197
\(696\) 0 0
\(697\) 9.18051 0.347737
\(698\) 0 0
\(699\) 0.948994 0.0358942
\(700\) 0 0
\(701\) −9.87709 −0.373052 −0.186526 0.982450i \(-0.559723\pi\)
−0.186526 + 0.982450i \(0.559723\pi\)
\(702\) 0 0
\(703\) 3.13704 0.118316
\(704\) 0 0
\(705\) 0.387601 0.0145979
\(706\) 0 0
\(707\) 3.81140 0.143342
\(708\) 0 0
\(709\) −12.4489 −0.467530 −0.233765 0.972293i \(-0.575105\pi\)
−0.233765 + 0.972293i \(0.575105\pi\)
\(710\) 0 0
\(711\) −29.2039 −1.09523
\(712\) 0 0
\(713\) 25.1324 0.941217
\(714\) 0 0
\(715\) 8.25865 0.308856
\(716\) 0 0
\(717\) 0.0717289 0.00267877
\(718\) 0 0
\(719\) 6.66173 0.248441 0.124220 0.992255i \(-0.460357\pi\)
0.124220 + 0.992255i \(0.460357\pi\)
\(720\) 0 0
\(721\) −20.9468 −0.780100
\(722\) 0 0
\(723\) 0.261155 0.00971247
\(724\) 0 0
\(725\) 0.315102 0.0117026
\(726\) 0 0
\(727\) −25.7916 −0.956556 −0.478278 0.878209i \(-0.658739\pi\)
−0.478278 + 0.878209i \(0.658739\pi\)
\(728\) 0 0
\(729\) −26.9156 −0.996872
\(730\) 0 0
\(731\) 0.679306 0.0251250
\(732\) 0 0
\(733\) 42.2454 1.56037 0.780184 0.625550i \(-0.215125\pi\)
0.780184 + 0.625550i \(0.215125\pi\)
\(734\) 0 0
\(735\) 0.205197 0.00756880
\(736\) 0 0
\(737\) −22.5829 −0.831851
\(738\) 0 0
\(739\) −19.7987 −0.728308 −0.364154 0.931339i \(-0.618642\pi\)
−0.364154 + 0.931339i \(0.618642\pi\)
\(740\) 0 0
\(741\) −0.238842 −0.00877407
\(742\) 0 0
\(743\) 7.45381 0.273454 0.136727 0.990609i \(-0.456342\pi\)
0.136727 + 0.990609i \(0.456342\pi\)
\(744\) 0 0
\(745\) 6.88162 0.252123
\(746\) 0 0
\(747\) 40.2311 1.47198
\(748\) 0 0
\(749\) −23.3992 −0.854987
\(750\) 0 0
\(751\) 54.5391 1.99016 0.995080 0.0990702i \(-0.0315868\pi\)
0.995080 + 0.0990702i \(0.0315868\pi\)
\(752\) 0 0
\(753\) 0.920637 0.0335499
\(754\) 0 0
\(755\) 11.4359 0.416194
\(756\) 0 0
\(757\) 41.9960 1.52637 0.763185 0.646180i \(-0.223635\pi\)
0.763185 + 0.646180i \(0.223635\pi\)
\(758\) 0 0
\(759\) −0.457919 −0.0166214
\(760\) 0 0
\(761\) −23.0099 −0.834108 −0.417054 0.908882i \(-0.636937\pi\)
−0.417054 + 0.908882i \(0.636937\pi\)
\(762\) 0 0
\(763\) −5.35108 −0.193722
\(764\) 0 0
\(765\) −2.59700 −0.0938948
\(766\) 0 0
\(767\) 1.08394 0.0391387
\(768\) 0 0
\(769\) −28.0412 −1.01119 −0.505596 0.862770i \(-0.668727\pi\)
−0.505596 + 0.862770i \(0.668727\pi\)
\(770\) 0 0
\(771\) −0.0951339 −0.00342616
\(772\) 0 0
\(773\) 4.49512 0.161678 0.0808390 0.996727i \(-0.474240\pi\)
0.0808390 + 0.996727i \(0.474240\pi\)
\(774\) 0 0
\(775\) 26.9542 0.968225
\(776\) 0 0
\(777\) 0.0731296 0.00262351
\(778\) 0 0
\(779\) −21.1739 −0.758634
\(780\) 0 0
\(781\) 29.7989 1.06629
\(782\) 0 0
\(783\) −0.0182425 −0.000651934 0
\(784\) 0 0
\(785\) −2.43093 −0.0867636
\(786\) 0 0
\(787\) 22.4824 0.801411 0.400706 0.916207i \(-0.368765\pi\)
0.400706 + 0.916207i \(0.368765\pi\)
\(788\) 0 0
\(789\) 0.588823 0.0209627
\(790\) 0 0
\(791\) −8.36074 −0.297274
\(792\) 0 0
\(793\) −24.5216 −0.870789
\(794\) 0 0
\(795\) −0.0858589 −0.00304510
\(796\) 0 0
\(797\) −10.5407 −0.373369 −0.186685 0.982420i \(-0.559774\pi\)
−0.186685 + 0.982420i \(0.559774\pi\)
\(798\) 0 0
\(799\) −9.41168 −0.332961
\(800\) 0 0
\(801\) 45.2792 1.59986
\(802\) 0 0
\(803\) −44.9453 −1.58608
\(804\) 0 0
\(805\) −4.49879 −0.158562
\(806\) 0 0
\(807\) −0.725031 −0.0255223
\(808\) 0 0
\(809\) 33.3965 1.17416 0.587079 0.809530i \(-0.300278\pi\)
0.587079 + 0.809530i \(0.300278\pi\)
\(810\) 0 0
\(811\) 18.2379 0.640419 0.320209 0.947347i \(-0.396247\pi\)
0.320209 + 0.947347i \(0.396247\pi\)
\(812\) 0 0
\(813\) −0.651861 −0.0228618
\(814\) 0 0
\(815\) 10.1557 0.355739
\(816\) 0 0
\(817\) −1.56675 −0.0548137
\(818\) 0 0
\(819\) 10.6708 0.372868
\(820\) 0 0
\(821\) −49.4067 −1.72431 −0.862154 0.506647i \(-0.830885\pi\)
−0.862154 + 0.506647i \(0.830885\pi\)
\(822\) 0 0
\(823\) −8.48844 −0.295888 −0.147944 0.988996i \(-0.547266\pi\)
−0.147944 + 0.988996i \(0.547266\pi\)
\(824\) 0 0
\(825\) −0.491113 −0.0170984
\(826\) 0 0
\(827\) −22.7054 −0.789545 −0.394772 0.918779i \(-0.629177\pi\)
−0.394772 + 0.918779i \(0.629177\pi\)
\(828\) 0 0
\(829\) 35.1609 1.22119 0.610594 0.791944i \(-0.290931\pi\)
0.610594 + 0.791944i \(0.290931\pi\)
\(830\) 0 0
\(831\) −0.782342 −0.0271392
\(832\) 0 0
\(833\) −4.98257 −0.172636
\(834\) 0 0
\(835\) −0.949625 −0.0328631
\(836\) 0 0
\(837\) −1.56049 −0.0539384
\(838\) 0 0
\(839\) −42.6085 −1.47101 −0.735504 0.677520i \(-0.763055\pi\)
−0.735504 + 0.677520i \(0.763055\pi\)
\(840\) 0 0
\(841\) −28.9941 −0.999796
\(842\) 0 0
\(843\) −0.724378 −0.0249489
\(844\) 0 0
\(845\) −4.52034 −0.155504
\(846\) 0 0
\(847\) −2.25768 −0.0775748
\(848\) 0 0
\(849\) 0.337113 0.0115697
\(850\) 0 0
\(851\) 5.69852 0.195343
\(852\) 0 0
\(853\) −32.5614 −1.11488 −0.557441 0.830217i \(-0.688217\pi\)
−0.557441 + 0.830217i \(0.688217\pi\)
\(854\) 0 0
\(855\) 5.98972 0.204844
\(856\) 0 0
\(857\) 10.4053 0.355439 0.177720 0.984081i \(-0.443128\pi\)
0.177720 + 0.984081i \(0.443128\pi\)
\(858\) 0 0
\(859\) −34.6257 −1.18141 −0.590706 0.806887i \(-0.701151\pi\)
−0.590706 + 0.806887i \(0.701151\pi\)
\(860\) 0 0
\(861\) −0.493599 −0.0168218
\(862\) 0 0
\(863\) −0.611644 −0.0208206 −0.0104103 0.999946i \(-0.503314\pi\)
−0.0104103 + 0.999946i \(0.503314\pi\)
\(864\) 0 0
\(865\) 22.0153 0.748541
\(866\) 0 0
\(867\) 0.639514 0.0217190
\(868\) 0 0
\(869\) −29.5082 −1.00100
\(870\) 0 0
\(871\) −21.3965 −0.724993
\(872\) 0 0
\(873\) 14.5915 0.493846
\(874\) 0 0
\(875\) −10.7115 −0.362115
\(876\) 0 0
\(877\) −24.3415 −0.821953 −0.410976 0.911646i \(-0.634812\pi\)
−0.410976 + 0.911646i \(0.634812\pi\)
\(878\) 0 0
\(879\) 1.12942 0.0380943
\(880\) 0 0
\(881\) −50.9917 −1.71795 −0.858977 0.512014i \(-0.828900\pi\)
−0.858977 + 0.512014i \(0.828900\pi\)
\(882\) 0 0
\(883\) −52.7274 −1.77442 −0.887210 0.461366i \(-0.847360\pi\)
−0.887210 + 0.461366i \(0.847360\pi\)
\(884\) 0 0
\(885\) 0.0141836 0.000476776 0
\(886\) 0 0
\(887\) 16.3444 0.548791 0.274395 0.961617i \(-0.411522\pi\)
0.274395 + 0.961617i \(0.411522\pi\)
\(888\) 0 0
\(889\) −0.410918 −0.0137817
\(890\) 0 0
\(891\) −27.2245 −0.912054
\(892\) 0 0
\(893\) 21.7071 0.726400
\(894\) 0 0
\(895\) −6.50617 −0.217477
\(896\) 0 0
\(897\) −0.433862 −0.0144862
\(898\) 0 0
\(899\) 0.505696 0.0168659
\(900\) 0 0
\(901\) 2.08482 0.0694554
\(902\) 0 0
\(903\) −0.0365235 −0.00121543
\(904\) 0 0
\(905\) 2.78570 0.0925997
\(906\) 0 0
\(907\) 41.2683 1.37029 0.685146 0.728406i \(-0.259738\pi\)
0.685146 + 0.728406i \(0.259738\pi\)
\(908\) 0 0
\(909\) −9.21801 −0.305742
\(910\) 0 0
\(911\) 14.1161 0.467687 0.233843 0.972274i \(-0.424870\pi\)
0.233843 + 0.972274i \(0.424870\pi\)
\(912\) 0 0
\(913\) 40.6503 1.34533
\(914\) 0 0
\(915\) −0.320872 −0.0106077
\(916\) 0 0
\(917\) 18.5353 0.612091
\(918\) 0 0
\(919\) −20.5244 −0.677036 −0.338518 0.940960i \(-0.609926\pi\)
−0.338518 + 0.940960i \(0.609926\pi\)
\(920\) 0 0
\(921\) −0.518018 −0.0170693
\(922\) 0 0
\(923\) 28.2334 0.929314
\(924\) 0 0
\(925\) 6.11161 0.200948
\(926\) 0 0
\(927\) 50.6606 1.66391
\(928\) 0 0
\(929\) −21.5819 −0.708079 −0.354040 0.935230i \(-0.615192\pi\)
−0.354040 + 0.935230i \(0.615192\pi\)
\(930\) 0 0
\(931\) 11.4918 0.376628
\(932\) 0 0
\(933\) 0.303121 0.00992374
\(934\) 0 0
\(935\) −2.62406 −0.0858160
\(936\) 0 0
\(937\) −24.8393 −0.811465 −0.405732 0.913992i \(-0.632984\pi\)
−0.405732 + 0.913992i \(0.632984\pi\)
\(938\) 0 0
\(939\) −0.374593 −0.0122244
\(940\) 0 0
\(941\) 22.3993 0.730197 0.365099 0.930969i \(-0.381035\pi\)
0.365099 + 0.930969i \(0.381035\pi\)
\(942\) 0 0
\(943\) −38.4630 −1.25253
\(944\) 0 0
\(945\) 0.279333 0.00908671
\(946\) 0 0
\(947\) 48.8252 1.58661 0.793304 0.608826i \(-0.208359\pi\)
0.793304 + 0.608826i \(0.208359\pi\)
\(948\) 0 0
\(949\) −42.5841 −1.38234
\(950\) 0 0
\(951\) 0.359425 0.0116552
\(952\) 0 0
\(953\) −33.2303 −1.07644 −0.538218 0.842806i \(-0.680902\pi\)
−0.538218 + 0.842806i \(0.680902\pi\)
\(954\) 0 0
\(955\) 8.92920 0.288942
\(956\) 0 0
\(957\) −0.00921390 −0.000297843 0
\(958\) 0 0
\(959\) 7.81078 0.252223
\(960\) 0 0
\(961\) 12.2579 0.395417
\(962\) 0 0
\(963\) 56.5917 1.82364
\(964\) 0 0
\(965\) 20.2893 0.653135
\(966\) 0 0
\(967\) −37.5138 −1.20636 −0.603181 0.797604i \(-0.706100\pi\)
−0.603181 + 0.797604i \(0.706100\pi\)
\(968\) 0 0
\(969\) 0.0758884 0.00243789
\(970\) 0 0
\(971\) −37.5961 −1.20652 −0.603259 0.797546i \(-0.706131\pi\)
−0.603259 + 0.797546i \(0.706131\pi\)
\(972\) 0 0
\(973\) −24.4434 −0.783618
\(974\) 0 0
\(975\) −0.465313 −0.0149019
\(976\) 0 0
\(977\) 50.9736 1.63079 0.815395 0.578904i \(-0.196520\pi\)
0.815395 + 0.578904i \(0.196520\pi\)
\(978\) 0 0
\(979\) 45.7510 1.46221
\(980\) 0 0
\(981\) 12.9418 0.413199
\(982\) 0 0
\(983\) 40.1745 1.28137 0.640684 0.767804i \(-0.278651\pi\)
0.640684 + 0.767804i \(0.278651\pi\)
\(984\) 0 0
\(985\) 8.06813 0.257072
\(986\) 0 0
\(987\) 0.506028 0.0161070
\(988\) 0 0
\(989\) −2.84604 −0.0904990
\(990\) 0 0
\(991\) −32.2318 −1.02388 −0.511938 0.859022i \(-0.671072\pi\)
−0.511938 + 0.859022i \(0.671072\pi\)
\(992\) 0 0
\(993\) 0.594614 0.0188695
\(994\) 0 0
\(995\) 1.08763 0.0344801
\(996\) 0 0
\(997\) 31.6267 1.00163 0.500813 0.865555i \(-0.333034\pi\)
0.500813 + 0.865555i \(0.333034\pi\)
\(998\) 0 0
\(999\) −0.353826 −0.0111945
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 1336.2.a.b.1.5 7
4.3 odd 2 2672.2.a.l.1.3 7
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1336.2.a.b.1.5 7 1.1 even 1 trivial
2672.2.a.l.1.3 7 4.3 odd 2