Properties

Label 6016.2.a.m.1.10
Level $6016$
Weight $2$
Character 6016.1
Self dual yes
Analytic conductor $48.038$
Analytic rank $1$
Dimension $13$
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] = [6016,2,Mod(1,6016)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6016, 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("6016.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6016 = 2^{7} \cdot 47 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6016.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: \(48.0380018560\)
Analytic rank: \(1\)
Dimension: \(13\)
Coefficient field: \(\mathbb{Q}[x]/(x^{13} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{13} - 4 x^{12} - 22 x^{11} + 96 x^{10} + 163 x^{9} - 840 x^{8} - 438 x^{7} + 3358 x^{6} + 5 x^{5} - 6230 x^{4} + 1488 x^{3} + 4720 x^{2} - 1440 x - 832 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.10
Root \(-1.49302\) of defining polynomial
Character \(\chi\) \(=\) 6016.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.49302 q^{3} +1.02737 q^{5} +3.87698 q^{7} -0.770878 q^{9} +O(q^{10})\) \(q+1.49302 q^{3} +1.02737 q^{5} +3.87698 q^{7} -0.770878 q^{9} -4.61496 q^{11} -4.62211 q^{13} +1.53389 q^{15} +2.47220 q^{17} -7.75700 q^{19} +5.78843 q^{21} +2.50449 q^{23} -3.94451 q^{25} -5.63001 q^{27} -4.13533 q^{29} +1.95319 q^{31} -6.89025 q^{33} +3.98309 q^{35} -1.50215 q^{37} -6.90092 q^{39} +6.64140 q^{41} +3.59172 q^{43} -0.791976 q^{45} +1.00000 q^{47} +8.03098 q^{49} +3.69105 q^{51} -10.4271 q^{53} -4.74127 q^{55} -11.5814 q^{57} -5.25042 q^{59} -13.7169 q^{61} -2.98868 q^{63} -4.74861 q^{65} +15.4380 q^{67} +3.73926 q^{69} -6.38702 q^{71} +2.34043 q^{73} -5.88926 q^{75} -17.8921 q^{77} -15.0550 q^{79} -6.09311 q^{81} +2.24845 q^{83} +2.53986 q^{85} -6.17414 q^{87} -7.54414 q^{89} -17.9198 q^{91} +2.91616 q^{93} -7.96930 q^{95} -3.03188 q^{97} +3.55757 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 13 q - 4 q^{3} - 6 q^{5} + 2 q^{7} + 21 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 13 q - 4 q^{3} - 6 q^{5} + 2 q^{7} + 21 q^{9} - 10 q^{11} - 4 q^{13} - 14 q^{15} + 10 q^{17} - 8 q^{19} - 10 q^{21} - 18 q^{23} + 23 q^{25} - 16 q^{27} - 14 q^{29} - 4 q^{31} + 14 q^{33} - 14 q^{35} - 16 q^{37} - 12 q^{39} + 10 q^{41} - 12 q^{43} - 10 q^{45} + 13 q^{47} + 9 q^{49} - 22 q^{51} - 26 q^{53} + 2 q^{55} + 20 q^{57} - 30 q^{59} - 18 q^{61} - 12 q^{63} - 4 q^{65} - 4 q^{67} - 2 q^{69} - 36 q^{71} + 10 q^{73} - 38 q^{75} - 42 q^{77} + 21 q^{81} - 12 q^{83} - 4 q^{85} - 6 q^{87} + 50 q^{89} + 4 q^{91} - 52 q^{93} - 8 q^{95} - 10 q^{97} - 34 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.49302 0.861998 0.430999 0.902352i \(-0.358161\pi\)
0.430999 + 0.902352i \(0.358161\pi\)
\(4\) 0 0
\(5\) 1.02737 0.459453 0.229727 0.973255i \(-0.426217\pi\)
0.229727 + 0.973255i \(0.426217\pi\)
\(6\) 0 0
\(7\) 3.87698 1.46536 0.732681 0.680573i \(-0.238269\pi\)
0.732681 + 0.680573i \(0.238269\pi\)
\(8\) 0 0
\(9\) −0.770878 −0.256959
\(10\) 0 0
\(11\) −4.61496 −1.39146 −0.695732 0.718301i \(-0.744920\pi\)
−0.695732 + 0.718301i \(0.744920\pi\)
\(12\) 0 0
\(13\) −4.62211 −1.28194 −0.640971 0.767565i \(-0.721468\pi\)
−0.640971 + 0.767565i \(0.721468\pi\)
\(14\) 0 0
\(15\) 1.53389 0.396048
\(16\) 0 0
\(17\) 2.47220 0.599596 0.299798 0.954003i \(-0.403081\pi\)
0.299798 + 0.954003i \(0.403081\pi\)
\(18\) 0 0
\(19\) −7.75700 −1.77958 −0.889789 0.456371i \(-0.849149\pi\)
−0.889789 + 0.456371i \(0.849149\pi\)
\(20\) 0 0
\(21\) 5.78843 1.26314
\(22\) 0 0
\(23\) 2.50449 0.522221 0.261111 0.965309i \(-0.415911\pi\)
0.261111 + 0.965309i \(0.415911\pi\)
\(24\) 0 0
\(25\) −3.94451 −0.788903
\(26\) 0 0
\(27\) −5.63001 −1.08350
\(28\) 0 0
\(29\) −4.13533 −0.767911 −0.383955 0.923352i \(-0.625438\pi\)
−0.383955 + 0.923352i \(0.625438\pi\)
\(30\) 0 0
\(31\) 1.95319 0.350804 0.175402 0.984497i \(-0.443878\pi\)
0.175402 + 0.984497i \(0.443878\pi\)
\(32\) 0 0
\(33\) −6.89025 −1.19944
\(34\) 0 0
\(35\) 3.98309 0.673265
\(36\) 0 0
\(37\) −1.50215 −0.246951 −0.123476 0.992348i \(-0.539404\pi\)
−0.123476 + 0.992348i \(0.539404\pi\)
\(38\) 0 0
\(39\) −6.90092 −1.10503
\(40\) 0 0
\(41\) 6.64140 1.03721 0.518606 0.855013i \(-0.326451\pi\)
0.518606 + 0.855013i \(0.326451\pi\)
\(42\) 0 0
\(43\) 3.59172 0.547732 0.273866 0.961768i \(-0.411697\pi\)
0.273866 + 0.961768i \(0.411697\pi\)
\(44\) 0 0
\(45\) −0.791976 −0.118061
\(46\) 0 0
\(47\) 1.00000 0.145865
\(48\) 0 0
\(49\) 8.03098 1.14728
\(50\) 0 0
\(51\) 3.69105 0.516851
\(52\) 0 0
\(53\) −10.4271 −1.43227 −0.716134 0.697963i \(-0.754090\pi\)
−0.716134 + 0.697963i \(0.754090\pi\)
\(54\) 0 0
\(55\) −4.74127 −0.639312
\(56\) 0 0
\(57\) −11.5814 −1.53399
\(58\) 0 0
\(59\) −5.25042 −0.683547 −0.341773 0.939782i \(-0.611028\pi\)
−0.341773 + 0.939782i \(0.611028\pi\)
\(60\) 0 0
\(61\) −13.7169 −1.75627 −0.878136 0.478412i \(-0.841213\pi\)
−0.878136 + 0.478412i \(0.841213\pi\)
\(62\) 0 0
\(63\) −2.98868 −0.376538
\(64\) 0 0
\(65\) −4.74861 −0.588992
\(66\) 0 0
\(67\) 15.4380 1.88605 0.943023 0.332727i \(-0.107969\pi\)
0.943023 + 0.332727i \(0.107969\pi\)
\(68\) 0 0
\(69\) 3.73926 0.450154
\(70\) 0 0
\(71\) −6.38702 −0.758000 −0.379000 0.925397i \(-0.623732\pi\)
−0.379000 + 0.925397i \(0.623732\pi\)
\(72\) 0 0
\(73\) 2.34043 0.273927 0.136963 0.990576i \(-0.456266\pi\)
0.136963 + 0.990576i \(0.456266\pi\)
\(74\) 0 0
\(75\) −5.88926 −0.680033
\(76\) 0 0
\(77\) −17.8921 −2.03900
\(78\) 0 0
\(79\) −15.0550 −1.69382 −0.846911 0.531735i \(-0.821540\pi\)
−0.846911 + 0.531735i \(0.821540\pi\)
\(80\) 0 0
\(81\) −6.09311 −0.677012
\(82\) 0 0
\(83\) 2.24845 0.246799 0.123399 0.992357i \(-0.460620\pi\)
0.123399 + 0.992357i \(0.460620\pi\)
\(84\) 0 0
\(85\) 2.53986 0.275486
\(86\) 0 0
\(87\) −6.17414 −0.661937
\(88\) 0 0
\(89\) −7.54414 −0.799678 −0.399839 0.916585i \(-0.630934\pi\)
−0.399839 + 0.916585i \(0.630934\pi\)
\(90\) 0 0
\(91\) −17.9198 −1.87851
\(92\) 0 0
\(93\) 2.91616 0.302392
\(94\) 0 0
\(95\) −7.96930 −0.817633
\(96\) 0 0
\(97\) −3.03188 −0.307841 −0.153921 0.988083i \(-0.549190\pi\)
−0.153921 + 0.988083i \(0.549190\pi\)
\(98\) 0 0
\(99\) 3.55757 0.357550
\(100\) 0 0
\(101\) 9.98951 0.993993 0.496997 0.867752i \(-0.334436\pi\)
0.496997 + 0.867752i \(0.334436\pi\)
\(102\) 0 0
\(103\) −1.03376 −0.101859 −0.0509296 0.998702i \(-0.516218\pi\)
−0.0509296 + 0.998702i \(0.516218\pi\)
\(104\) 0 0
\(105\) 5.94685 0.580353
\(106\) 0 0
\(107\) 10.1575 0.981963 0.490982 0.871170i \(-0.336638\pi\)
0.490982 + 0.871170i \(0.336638\pi\)
\(108\) 0 0
\(109\) 15.8459 1.51776 0.758880 0.651230i \(-0.225747\pi\)
0.758880 + 0.651230i \(0.225747\pi\)
\(110\) 0 0
\(111\) −2.24274 −0.212871
\(112\) 0 0
\(113\) −14.2294 −1.33859 −0.669293 0.742999i \(-0.733403\pi\)
−0.669293 + 0.742999i \(0.733403\pi\)
\(114\) 0 0
\(115\) 2.57303 0.239936
\(116\) 0 0
\(117\) 3.56308 0.329407
\(118\) 0 0
\(119\) 9.58467 0.878625
\(120\) 0 0
\(121\) 10.2979 0.936172
\(122\) 0 0
\(123\) 9.91577 0.894075
\(124\) 0 0
\(125\) −9.18931 −0.821917
\(126\) 0 0
\(127\) −18.4036 −1.63305 −0.816527 0.577308i \(-0.804103\pi\)
−0.816527 + 0.577308i \(0.804103\pi\)
\(128\) 0 0
\(129\) 5.36253 0.472144
\(130\) 0 0
\(131\) −8.07558 −0.705567 −0.352783 0.935705i \(-0.614765\pi\)
−0.352783 + 0.935705i \(0.614765\pi\)
\(132\) 0 0
\(133\) −30.0738 −2.60773
\(134\) 0 0
\(135\) −5.78410 −0.497816
\(136\) 0 0
\(137\) 12.6236 1.07850 0.539252 0.842144i \(-0.318707\pi\)
0.539252 + 0.842144i \(0.318707\pi\)
\(138\) 0 0
\(139\) −22.6425 −1.92051 −0.960257 0.279117i \(-0.909958\pi\)
−0.960257 + 0.279117i \(0.909958\pi\)
\(140\) 0 0
\(141\) 1.49302 0.125735
\(142\) 0 0
\(143\) 21.3309 1.78378
\(144\) 0 0
\(145\) −4.24850 −0.352819
\(146\) 0 0
\(147\) 11.9904 0.988956
\(148\) 0 0
\(149\) 1.57979 0.129422 0.0647109 0.997904i \(-0.479387\pi\)
0.0647109 + 0.997904i \(0.479387\pi\)
\(150\) 0 0
\(151\) −4.38147 −0.356559 −0.178279 0.983980i \(-0.557053\pi\)
−0.178279 + 0.983980i \(0.557053\pi\)
\(152\) 0 0
\(153\) −1.90576 −0.154072
\(154\) 0 0
\(155\) 2.00665 0.161178
\(156\) 0 0
\(157\) 13.4266 1.07156 0.535779 0.844358i \(-0.320018\pi\)
0.535779 + 0.844358i \(0.320018\pi\)
\(158\) 0 0
\(159\) −15.5679 −1.23461
\(160\) 0 0
\(161\) 9.70984 0.765243
\(162\) 0 0
\(163\) −8.08897 −0.633577 −0.316789 0.948496i \(-0.602605\pi\)
−0.316789 + 0.948496i \(0.602605\pi\)
\(164\) 0 0
\(165\) −7.07883 −0.551086
\(166\) 0 0
\(167\) −4.35268 −0.336820 −0.168410 0.985717i \(-0.553863\pi\)
−0.168410 + 0.985717i \(0.553863\pi\)
\(168\) 0 0
\(169\) 8.36390 0.643377
\(170\) 0 0
\(171\) 5.97971 0.457279
\(172\) 0 0
\(173\) 4.92311 0.374297 0.187149 0.982332i \(-0.440075\pi\)
0.187149 + 0.982332i \(0.440075\pi\)
\(174\) 0 0
\(175\) −15.2928 −1.15603
\(176\) 0 0
\(177\) −7.83901 −0.589216
\(178\) 0 0
\(179\) −16.0083 −1.19652 −0.598260 0.801302i \(-0.704141\pi\)
−0.598260 + 0.801302i \(0.704141\pi\)
\(180\) 0 0
\(181\) 11.8600 0.881550 0.440775 0.897618i \(-0.354704\pi\)
0.440775 + 0.897618i \(0.354704\pi\)
\(182\) 0 0
\(183\) −20.4797 −1.51390
\(184\) 0 0
\(185\) −1.54326 −0.113462
\(186\) 0 0
\(187\) −11.4091 −0.834316
\(188\) 0 0
\(189\) −21.8275 −1.58771
\(190\) 0 0
\(191\) −1.29342 −0.0935889 −0.0467944 0.998905i \(-0.514901\pi\)
−0.0467944 + 0.998905i \(0.514901\pi\)
\(192\) 0 0
\(193\) 13.1614 0.947377 0.473688 0.880693i \(-0.342922\pi\)
0.473688 + 0.880693i \(0.342922\pi\)
\(194\) 0 0
\(195\) −7.08979 −0.507710
\(196\) 0 0
\(197\) −8.86306 −0.631467 −0.315734 0.948848i \(-0.602251\pi\)
−0.315734 + 0.948848i \(0.602251\pi\)
\(198\) 0 0
\(199\) 13.7508 0.974772 0.487386 0.873187i \(-0.337950\pi\)
0.487386 + 0.873187i \(0.337950\pi\)
\(200\) 0 0
\(201\) 23.0492 1.62577
\(202\) 0 0
\(203\) −16.0326 −1.12527
\(204\) 0 0
\(205\) 6.82316 0.476550
\(206\) 0 0
\(207\) −1.93065 −0.134190
\(208\) 0 0
\(209\) 35.7983 2.47622
\(210\) 0 0
\(211\) 28.0838 1.93337 0.966683 0.255975i \(-0.0823964\pi\)
0.966683 + 0.255975i \(0.0823964\pi\)
\(212\) 0 0
\(213\) −9.53598 −0.653395
\(214\) 0 0
\(215\) 3.69002 0.251657
\(216\) 0 0
\(217\) 7.57249 0.514054
\(218\) 0 0
\(219\) 3.49432 0.236124
\(220\) 0 0
\(221\) −11.4268 −0.768648
\(222\) 0 0
\(223\) −14.3016 −0.957706 −0.478853 0.877895i \(-0.658947\pi\)
−0.478853 + 0.877895i \(0.658947\pi\)
\(224\) 0 0
\(225\) 3.04074 0.202716
\(226\) 0 0
\(227\) −11.5207 −0.764655 −0.382328 0.924027i \(-0.624877\pi\)
−0.382328 + 0.924027i \(0.624877\pi\)
\(228\) 0 0
\(229\) 4.44069 0.293449 0.146725 0.989177i \(-0.453127\pi\)
0.146725 + 0.989177i \(0.453127\pi\)
\(230\) 0 0
\(231\) −26.7134 −1.75761
\(232\) 0 0
\(233\) 19.0449 1.24767 0.623835 0.781556i \(-0.285574\pi\)
0.623835 + 0.781556i \(0.285574\pi\)
\(234\) 0 0
\(235\) 1.02737 0.0670181
\(236\) 0 0
\(237\) −22.4775 −1.46007
\(238\) 0 0
\(239\) 4.40019 0.284625 0.142312 0.989822i \(-0.454546\pi\)
0.142312 + 0.989822i \(0.454546\pi\)
\(240\) 0 0
\(241\) 2.39750 0.154437 0.0772183 0.997014i \(-0.475396\pi\)
0.0772183 + 0.997014i \(0.475396\pi\)
\(242\) 0 0
\(243\) 7.79287 0.499913
\(244\) 0 0
\(245\) 8.25077 0.527123
\(246\) 0 0
\(247\) 35.8537 2.28132
\(248\) 0 0
\(249\) 3.35698 0.212740
\(250\) 0 0
\(251\) 30.8883 1.94965 0.974827 0.222965i \(-0.0715734\pi\)
0.974827 + 0.222965i \(0.0715734\pi\)
\(252\) 0 0
\(253\) −11.5581 −0.726652
\(254\) 0 0
\(255\) 3.79207 0.237469
\(256\) 0 0
\(257\) 17.3108 1.07982 0.539910 0.841723i \(-0.318458\pi\)
0.539910 + 0.841723i \(0.318458\pi\)
\(258\) 0 0
\(259\) −5.82379 −0.361873
\(260\) 0 0
\(261\) 3.18783 0.197322
\(262\) 0 0
\(263\) −10.8884 −0.671410 −0.335705 0.941967i \(-0.608975\pi\)
−0.335705 + 0.941967i \(0.608975\pi\)
\(264\) 0 0
\(265\) −10.7124 −0.658060
\(266\) 0 0
\(267\) −11.2636 −0.689321
\(268\) 0 0
\(269\) −3.00374 −0.183141 −0.0915707 0.995799i \(-0.529189\pi\)
−0.0915707 + 0.995799i \(0.529189\pi\)
\(270\) 0 0
\(271\) −22.3228 −1.35601 −0.678006 0.735056i \(-0.737156\pi\)
−0.678006 + 0.735056i \(0.737156\pi\)
\(272\) 0 0
\(273\) −26.7547 −1.61927
\(274\) 0 0
\(275\) 18.2038 1.09773
\(276\) 0 0
\(277\) 8.97377 0.539182 0.269591 0.962975i \(-0.413112\pi\)
0.269591 + 0.962975i \(0.413112\pi\)
\(278\) 0 0
\(279\) −1.50567 −0.0901423
\(280\) 0 0
\(281\) 0.000716445 0 4.27395e−5 0 2.13698e−5 1.00000i \(-0.499993\pi\)
2.13698e−5 1.00000i \(0.499993\pi\)
\(282\) 0 0
\(283\) 8.30795 0.493857 0.246928 0.969034i \(-0.420579\pi\)
0.246928 + 0.969034i \(0.420579\pi\)
\(284\) 0 0
\(285\) −11.8984 −0.704798
\(286\) 0 0
\(287\) 25.7486 1.51989
\(288\) 0 0
\(289\) −10.8882 −0.640484
\(290\) 0 0
\(291\) −4.52668 −0.265358
\(292\) 0 0
\(293\) 7.58037 0.442850 0.221425 0.975177i \(-0.428929\pi\)
0.221425 + 0.975177i \(0.428929\pi\)
\(294\) 0 0
\(295\) −5.39412 −0.314058
\(296\) 0 0
\(297\) 25.9823 1.50765
\(298\) 0 0
\(299\) −11.5760 −0.669458
\(300\) 0 0
\(301\) 13.9250 0.802626
\(302\) 0 0
\(303\) 14.9146 0.856820
\(304\) 0 0
\(305\) −14.0923 −0.806924
\(306\) 0 0
\(307\) −14.4123 −0.822556 −0.411278 0.911510i \(-0.634917\pi\)
−0.411278 + 0.911510i \(0.634917\pi\)
\(308\) 0 0
\(309\) −1.54343 −0.0878024
\(310\) 0 0
\(311\) 23.6197 1.33935 0.669676 0.742654i \(-0.266433\pi\)
0.669676 + 0.742654i \(0.266433\pi\)
\(312\) 0 0
\(313\) 10.6649 0.602813 0.301407 0.953496i \(-0.402544\pi\)
0.301407 + 0.953496i \(0.402544\pi\)
\(314\) 0 0
\(315\) −3.07047 −0.173002
\(316\) 0 0
\(317\) −27.8725 −1.56548 −0.782739 0.622350i \(-0.786178\pi\)
−0.782739 + 0.622350i \(0.786178\pi\)
\(318\) 0 0
\(319\) 19.0844 1.06852
\(320\) 0 0
\(321\) 15.1654 0.846451
\(322\) 0 0
\(323\) −19.1769 −1.06703
\(324\) 0 0
\(325\) 18.2320 1.01133
\(326\) 0 0
\(327\) 23.6583 1.30831
\(328\) 0 0
\(329\) 3.87698 0.213745
\(330\) 0 0
\(331\) 27.0287 1.48563 0.742815 0.669496i \(-0.233490\pi\)
0.742815 + 0.669496i \(0.233490\pi\)
\(332\) 0 0
\(333\) 1.15797 0.0634564
\(334\) 0 0
\(335\) 15.8605 0.866550
\(336\) 0 0
\(337\) 14.7556 0.803788 0.401894 0.915686i \(-0.368352\pi\)
0.401894 + 0.915686i \(0.368352\pi\)
\(338\) 0 0
\(339\) −21.2448 −1.15386
\(340\) 0 0
\(341\) −9.01391 −0.488131
\(342\) 0 0
\(343\) 3.99709 0.215823
\(344\) 0 0
\(345\) 3.84160 0.206825
\(346\) 0 0
\(347\) −1.30950 −0.0702977 −0.0351489 0.999382i \(-0.511191\pi\)
−0.0351489 + 0.999382i \(0.511191\pi\)
\(348\) 0 0
\(349\) 0.680695 0.0364368 0.0182184 0.999834i \(-0.494201\pi\)
0.0182184 + 0.999834i \(0.494201\pi\)
\(350\) 0 0
\(351\) 26.0225 1.38898
\(352\) 0 0
\(353\) −12.0211 −0.639818 −0.319909 0.947448i \(-0.603652\pi\)
−0.319909 + 0.947448i \(0.603652\pi\)
\(354\) 0 0
\(355\) −6.56183 −0.348266
\(356\) 0 0
\(357\) 14.3101 0.757373
\(358\) 0 0
\(359\) −25.1639 −1.32810 −0.664051 0.747687i \(-0.731164\pi\)
−0.664051 + 0.747687i \(0.731164\pi\)
\(360\) 0 0
\(361\) 41.1711 2.16690
\(362\) 0 0
\(363\) 15.3750 0.806978
\(364\) 0 0
\(365\) 2.40449 0.125857
\(366\) 0 0
\(367\) −6.22700 −0.325047 −0.162523 0.986705i \(-0.551963\pi\)
−0.162523 + 0.986705i \(0.551963\pi\)
\(368\) 0 0
\(369\) −5.11971 −0.266521
\(370\) 0 0
\(371\) −40.4256 −2.09879
\(372\) 0 0
\(373\) 37.3793 1.93543 0.967713 0.252053i \(-0.0811059\pi\)
0.967713 + 0.252053i \(0.0811059\pi\)
\(374\) 0 0
\(375\) −13.7199 −0.708491
\(376\) 0 0
\(377\) 19.1139 0.984417
\(378\) 0 0
\(379\) −17.3685 −0.892160 −0.446080 0.894993i \(-0.647180\pi\)
−0.446080 + 0.894993i \(0.647180\pi\)
\(380\) 0 0
\(381\) −27.4770 −1.40769
\(382\) 0 0
\(383\) −28.4596 −1.45422 −0.727109 0.686522i \(-0.759136\pi\)
−0.727109 + 0.686522i \(0.759136\pi\)
\(384\) 0 0
\(385\) −18.3818 −0.936823
\(386\) 0 0
\(387\) −2.76878 −0.140745
\(388\) 0 0
\(389\) −27.0883 −1.37343 −0.686714 0.726927i \(-0.740948\pi\)
−0.686714 + 0.726927i \(0.740948\pi\)
\(390\) 0 0
\(391\) 6.19159 0.313122
\(392\) 0 0
\(393\) −12.0570 −0.608197
\(394\) 0 0
\(395\) −15.4670 −0.778231
\(396\) 0 0
\(397\) 13.1195 0.658450 0.329225 0.944252i \(-0.393213\pi\)
0.329225 + 0.944252i \(0.393213\pi\)
\(398\) 0 0
\(399\) −44.9009 −2.24785
\(400\) 0 0
\(401\) −33.8177 −1.68878 −0.844388 0.535732i \(-0.820036\pi\)
−0.844388 + 0.535732i \(0.820036\pi\)
\(402\) 0 0
\(403\) −9.02787 −0.449710
\(404\) 0 0
\(405\) −6.25987 −0.311055
\(406\) 0 0
\(407\) 6.93235 0.343624
\(408\) 0 0
\(409\) −15.5302 −0.767918 −0.383959 0.923350i \(-0.625440\pi\)
−0.383959 + 0.923350i \(0.625440\pi\)
\(410\) 0 0
\(411\) 18.8473 0.929669
\(412\) 0 0
\(413\) −20.3558 −1.00164
\(414\) 0 0
\(415\) 2.30998 0.113393
\(416\) 0 0
\(417\) −33.8058 −1.65548
\(418\) 0 0
\(419\) −31.3311 −1.53062 −0.765312 0.643660i \(-0.777415\pi\)
−0.765312 + 0.643660i \(0.777415\pi\)
\(420\) 0 0
\(421\) −3.17748 −0.154861 −0.0774305 0.996998i \(-0.524672\pi\)
−0.0774305 + 0.996998i \(0.524672\pi\)
\(422\) 0 0
\(423\) −0.770878 −0.0374814
\(424\) 0 0
\(425\) −9.75162 −0.473023
\(426\) 0 0
\(427\) −53.1802 −2.57357
\(428\) 0 0
\(429\) 31.8475 1.53761
\(430\) 0 0
\(431\) −27.6392 −1.33133 −0.665666 0.746249i \(-0.731853\pi\)
−0.665666 + 0.746249i \(0.731853\pi\)
\(432\) 0 0
\(433\) 31.8153 1.52894 0.764472 0.644657i \(-0.223000\pi\)
0.764472 + 0.644657i \(0.223000\pi\)
\(434\) 0 0
\(435\) −6.34312 −0.304129
\(436\) 0 0
\(437\) −19.4273 −0.929334
\(438\) 0 0
\(439\) 12.6534 0.603912 0.301956 0.953322i \(-0.402361\pi\)
0.301956 + 0.953322i \(0.402361\pi\)
\(440\) 0 0
\(441\) −6.19091 −0.294805
\(442\) 0 0
\(443\) −17.8285 −0.847060 −0.423530 0.905882i \(-0.639209\pi\)
−0.423530 + 0.905882i \(0.639209\pi\)
\(444\) 0 0
\(445\) −7.75062 −0.367414
\(446\) 0 0
\(447\) 2.35867 0.111561
\(448\) 0 0
\(449\) 19.0083 0.897055 0.448527 0.893769i \(-0.351949\pi\)
0.448527 + 0.893769i \(0.351949\pi\)
\(450\) 0 0
\(451\) −30.6498 −1.44324
\(452\) 0 0
\(453\) −6.54164 −0.307353
\(454\) 0 0
\(455\) −18.4103 −0.863087
\(456\) 0 0
\(457\) 7.57180 0.354194 0.177097 0.984193i \(-0.443329\pi\)
0.177097 + 0.984193i \(0.443329\pi\)
\(458\) 0 0
\(459\) −13.9185 −0.649660
\(460\) 0 0
\(461\) 2.04187 0.0950995 0.0475498 0.998869i \(-0.484859\pi\)
0.0475498 + 0.998869i \(0.484859\pi\)
\(462\) 0 0
\(463\) −4.29998 −0.199837 −0.0999184 0.994996i \(-0.531858\pi\)
−0.0999184 + 0.994996i \(0.531858\pi\)
\(464\) 0 0
\(465\) 2.99598 0.138935
\(466\) 0 0
\(467\) 22.6015 1.04587 0.522936 0.852372i \(-0.324837\pi\)
0.522936 + 0.852372i \(0.324837\pi\)
\(468\) 0 0
\(469\) 59.8527 2.76374
\(470\) 0 0
\(471\) 20.0462 0.923681
\(472\) 0 0
\(473\) −16.5757 −0.762150
\(474\) 0 0
\(475\) 30.5976 1.40391
\(476\) 0 0
\(477\) 8.03800 0.368035
\(478\) 0 0
\(479\) −3.92055 −0.179134 −0.0895672 0.995981i \(-0.528548\pi\)
−0.0895672 + 0.995981i \(0.528548\pi\)
\(480\) 0 0
\(481\) 6.94308 0.316577
\(482\) 0 0
\(483\) 14.4970 0.659638
\(484\) 0 0
\(485\) −3.11486 −0.141439
\(486\) 0 0
\(487\) 15.6655 0.709870 0.354935 0.934891i \(-0.384503\pi\)
0.354935 + 0.934891i \(0.384503\pi\)
\(488\) 0 0
\(489\) −12.0770 −0.546142
\(490\) 0 0
\(491\) 15.6893 0.708046 0.354023 0.935237i \(-0.384813\pi\)
0.354023 + 0.935237i \(0.384813\pi\)
\(492\) 0 0
\(493\) −10.2233 −0.460436
\(494\) 0 0
\(495\) 3.65494 0.164277
\(496\) 0 0
\(497\) −24.7624 −1.11074
\(498\) 0 0
\(499\) −23.5705 −1.05516 −0.527581 0.849505i \(-0.676901\pi\)
−0.527581 + 0.849505i \(0.676901\pi\)
\(500\) 0 0
\(501\) −6.49866 −0.290339
\(502\) 0 0
\(503\) −37.2356 −1.66025 −0.830126 0.557576i \(-0.811731\pi\)
−0.830126 + 0.557576i \(0.811731\pi\)
\(504\) 0 0
\(505\) 10.2629 0.456693
\(506\) 0 0
\(507\) 12.4875 0.554589
\(508\) 0 0
\(509\) 8.78382 0.389336 0.194668 0.980869i \(-0.437637\pi\)
0.194668 + 0.980869i \(0.437637\pi\)
\(510\) 0 0
\(511\) 9.07381 0.401402
\(512\) 0 0
\(513\) 43.6720 1.92817
\(514\) 0 0
\(515\) −1.06205 −0.0467995
\(516\) 0 0
\(517\) −4.61496 −0.202966
\(518\) 0 0
\(519\) 7.35033 0.322644
\(520\) 0 0
\(521\) 18.0774 0.791987 0.395994 0.918253i \(-0.370400\pi\)
0.395994 + 0.918253i \(0.370400\pi\)
\(522\) 0 0
\(523\) −17.3059 −0.756735 −0.378368 0.925655i \(-0.623515\pi\)
−0.378368 + 0.925655i \(0.623515\pi\)
\(524\) 0 0
\(525\) −22.8325 −0.996493
\(526\) 0 0
\(527\) 4.82868 0.210341
\(528\) 0 0
\(529\) −16.7275 −0.727285
\(530\) 0 0
\(531\) 4.04744 0.175644
\(532\) 0 0
\(533\) −30.6973 −1.32965
\(534\) 0 0
\(535\) 10.4355 0.451166
\(536\) 0 0
\(537\) −23.9009 −1.03140
\(538\) 0 0
\(539\) −37.0627 −1.59640
\(540\) 0 0
\(541\) 26.3028 1.13085 0.565423 0.824801i \(-0.308713\pi\)
0.565423 + 0.824801i \(0.308713\pi\)
\(542\) 0 0
\(543\) 17.7073 0.759894
\(544\) 0 0
\(545\) 16.2796 0.697340
\(546\) 0 0
\(547\) 6.26305 0.267789 0.133894 0.990996i \(-0.457252\pi\)
0.133894 + 0.990996i \(0.457252\pi\)
\(548\) 0 0
\(549\) 10.5741 0.451290
\(550\) 0 0
\(551\) 32.0777 1.36656
\(552\) 0 0
\(553\) −58.3680 −2.48206
\(554\) 0 0
\(555\) −2.30412 −0.0978044
\(556\) 0 0
\(557\) −23.5741 −0.998868 −0.499434 0.866352i \(-0.666459\pi\)
−0.499434 + 0.866352i \(0.666459\pi\)
\(558\) 0 0
\(559\) −16.6013 −0.702161
\(560\) 0 0
\(561\) −17.0341 −0.719179
\(562\) 0 0
\(563\) 15.1171 0.637110 0.318555 0.947904i \(-0.396802\pi\)
0.318555 + 0.947904i \(0.396802\pi\)
\(564\) 0 0
\(565\) −14.6188 −0.615017
\(566\) 0 0
\(567\) −23.6229 −0.992068
\(568\) 0 0
\(569\) 37.8408 1.58637 0.793184 0.608982i \(-0.208422\pi\)
0.793184 + 0.608982i \(0.208422\pi\)
\(570\) 0 0
\(571\) 23.5876 0.987110 0.493555 0.869715i \(-0.335697\pi\)
0.493555 + 0.869715i \(0.335697\pi\)
\(572\) 0 0
\(573\) −1.93111 −0.0806734
\(574\) 0 0
\(575\) −9.87898 −0.411982
\(576\) 0 0
\(577\) −10.0045 −0.416492 −0.208246 0.978076i \(-0.566776\pi\)
−0.208246 + 0.978076i \(0.566776\pi\)
\(578\) 0 0
\(579\) 19.6503 0.816637
\(580\) 0 0
\(581\) 8.71718 0.361650
\(582\) 0 0
\(583\) 48.1206 1.99295
\(584\) 0 0
\(585\) 3.66060 0.151347
\(586\) 0 0
\(587\) −2.48325 −0.102495 −0.0512473 0.998686i \(-0.516320\pi\)
−0.0512473 + 0.998686i \(0.516320\pi\)
\(588\) 0 0
\(589\) −15.1509 −0.624283
\(590\) 0 0
\(591\) −13.2328 −0.544323
\(592\) 0 0
\(593\) −6.16410 −0.253129 −0.126565 0.991958i \(-0.540395\pi\)
−0.126565 + 0.991958i \(0.540395\pi\)
\(594\) 0 0
\(595\) 9.84698 0.403687
\(596\) 0 0
\(597\) 20.5304 0.840251
\(598\) 0 0
\(599\) 46.4563 1.89815 0.949077 0.315044i \(-0.102019\pi\)
0.949077 + 0.315044i \(0.102019\pi\)
\(600\) 0 0
\(601\) 32.0275 1.30643 0.653215 0.757172i \(-0.273420\pi\)
0.653215 + 0.757172i \(0.273420\pi\)
\(602\) 0 0
\(603\) −11.9008 −0.484637
\(604\) 0 0
\(605\) 10.5797 0.430127
\(606\) 0 0
\(607\) −41.7903 −1.69621 −0.848107 0.529825i \(-0.822258\pi\)
−0.848107 + 0.529825i \(0.822258\pi\)
\(608\) 0 0
\(609\) −23.9370 −0.969977
\(610\) 0 0
\(611\) −4.62211 −0.186991
\(612\) 0 0
\(613\) −24.2969 −0.981343 −0.490672 0.871345i \(-0.663249\pi\)
−0.490672 + 0.871345i \(0.663249\pi\)
\(614\) 0 0
\(615\) 10.1871 0.410785
\(616\) 0 0
\(617\) 47.6303 1.91752 0.958761 0.284212i \(-0.0917319\pi\)
0.958761 + 0.284212i \(0.0917319\pi\)
\(618\) 0 0
\(619\) −28.1133 −1.12997 −0.564984 0.825102i \(-0.691118\pi\)
−0.564984 + 0.825102i \(0.691118\pi\)
\(620\) 0 0
\(621\) −14.1003 −0.565825
\(622\) 0 0
\(623\) −29.2485 −1.17182
\(624\) 0 0
\(625\) 10.2818 0.411271
\(626\) 0 0
\(627\) 53.4477 2.13450
\(628\) 0 0
\(629\) −3.71360 −0.148071
\(630\) 0 0
\(631\) −12.0142 −0.478277 −0.239138 0.970986i \(-0.576865\pi\)
−0.239138 + 0.970986i \(0.576865\pi\)
\(632\) 0 0
\(633\) 41.9298 1.66656
\(634\) 0 0
\(635\) −18.9072 −0.750311
\(636\) 0 0
\(637\) −37.1201 −1.47075
\(638\) 0 0
\(639\) 4.92362 0.194775
\(640\) 0 0
\(641\) −24.1072 −0.952175 −0.476088 0.879398i \(-0.657945\pi\)
−0.476088 + 0.879398i \(0.657945\pi\)
\(642\) 0 0
\(643\) −11.8324 −0.466623 −0.233312 0.972402i \(-0.574956\pi\)
−0.233312 + 0.972402i \(0.574956\pi\)
\(644\) 0 0
\(645\) 5.50929 0.216928
\(646\) 0 0
\(647\) −31.4896 −1.23798 −0.618991 0.785398i \(-0.712458\pi\)
−0.618991 + 0.785398i \(0.712458\pi\)
\(648\) 0 0
\(649\) 24.2305 0.951131
\(650\) 0 0
\(651\) 11.3059 0.443114
\(652\) 0 0
\(653\) −21.8703 −0.855849 −0.427925 0.903814i \(-0.640755\pi\)
−0.427925 + 0.903814i \(0.640755\pi\)
\(654\) 0 0
\(655\) −8.29659 −0.324175
\(656\) 0 0
\(657\) −1.80419 −0.0703881
\(658\) 0 0
\(659\) −17.6467 −0.687418 −0.343709 0.939076i \(-0.611683\pi\)
−0.343709 + 0.939076i \(0.611683\pi\)
\(660\) 0 0
\(661\) 20.0644 0.780415 0.390208 0.920727i \(-0.372403\pi\)
0.390208 + 0.920727i \(0.372403\pi\)
\(662\) 0 0
\(663\) −17.0604 −0.662573
\(664\) 0 0
\(665\) −30.8968 −1.19813
\(666\) 0 0
\(667\) −10.3569 −0.401019
\(668\) 0 0
\(669\) −21.3526 −0.825541
\(670\) 0 0
\(671\) 63.3031 2.44379
\(672\) 0 0
\(673\) −51.7534 −1.99495 −0.997474 0.0710349i \(-0.977370\pi\)
−0.997474 + 0.0710349i \(0.977370\pi\)
\(674\) 0 0
\(675\) 22.2077 0.854774
\(676\) 0 0
\(677\) −22.3687 −0.859700 −0.429850 0.902900i \(-0.641433\pi\)
−0.429850 + 0.902900i \(0.641433\pi\)
\(678\) 0 0
\(679\) −11.7546 −0.451098
\(680\) 0 0
\(681\) −17.2007 −0.659131
\(682\) 0 0
\(683\) −19.3522 −0.740491 −0.370245 0.928934i \(-0.620726\pi\)
−0.370245 + 0.928934i \(0.620726\pi\)
\(684\) 0 0
\(685\) 12.9691 0.495522
\(686\) 0 0
\(687\) 6.63006 0.252953
\(688\) 0 0
\(689\) 48.1951 1.83609
\(690\) 0 0
\(691\) −4.24792 −0.161598 −0.0807992 0.996730i \(-0.525747\pi\)
−0.0807992 + 0.996730i \(0.525747\pi\)
\(692\) 0 0
\(693\) 13.7926 0.523939
\(694\) 0 0
\(695\) −23.2622 −0.882386
\(696\) 0 0
\(697\) 16.4189 0.621908
\(698\) 0 0
\(699\) 28.4345 1.07549
\(700\) 0 0
\(701\) −11.6059 −0.438349 −0.219174 0.975686i \(-0.570336\pi\)
−0.219174 + 0.975686i \(0.570336\pi\)
\(702\) 0 0
\(703\) 11.6521 0.439469
\(704\) 0 0
\(705\) 1.53389 0.0577695
\(706\) 0 0
\(707\) 38.7291 1.45656
\(708\) 0 0
\(709\) 46.3715 1.74152 0.870759 0.491711i \(-0.163628\pi\)
0.870759 + 0.491711i \(0.163628\pi\)
\(710\) 0 0
\(711\) 11.6056 0.435243
\(712\) 0 0
\(713\) 4.89174 0.183197
\(714\) 0 0
\(715\) 21.9147 0.819562
\(716\) 0 0
\(717\) 6.56959 0.245346
\(718\) 0 0
\(719\) −9.77255 −0.364455 −0.182227 0.983256i \(-0.558331\pi\)
−0.182227 + 0.983256i \(0.558331\pi\)
\(720\) 0 0
\(721\) −4.00786 −0.149260
\(722\) 0 0
\(723\) 3.57953 0.133124
\(724\) 0 0
\(725\) 16.3119 0.605807
\(726\) 0 0
\(727\) −7.76248 −0.287894 −0.143947 0.989585i \(-0.545980\pi\)
−0.143947 + 0.989585i \(0.545980\pi\)
\(728\) 0 0
\(729\) 29.9143 1.10794
\(730\) 0 0
\(731\) 8.87945 0.328418
\(732\) 0 0
\(733\) 32.8115 1.21192 0.605961 0.795494i \(-0.292789\pi\)
0.605961 + 0.795494i \(0.292789\pi\)
\(734\) 0 0
\(735\) 12.3186 0.454379
\(736\) 0 0
\(737\) −71.2456 −2.62437
\(738\) 0 0
\(739\) −12.9710 −0.477146 −0.238573 0.971125i \(-0.576680\pi\)
−0.238573 + 0.971125i \(0.576680\pi\)
\(740\) 0 0
\(741\) 53.5305 1.96649
\(742\) 0 0
\(743\) −14.8415 −0.544483 −0.272241 0.962229i \(-0.587765\pi\)
−0.272241 + 0.962229i \(0.587765\pi\)
\(744\) 0 0
\(745\) 1.62303 0.0594632
\(746\) 0 0
\(747\) −1.73328 −0.0634173
\(748\) 0 0
\(749\) 39.3805 1.43893
\(750\) 0 0
\(751\) 48.3413 1.76400 0.882000 0.471250i \(-0.156197\pi\)
0.882000 + 0.471250i \(0.156197\pi\)
\(752\) 0 0
\(753\) 46.1170 1.68060
\(754\) 0 0
\(755\) −4.50138 −0.163822
\(756\) 0 0
\(757\) −49.4744 −1.79818 −0.899089 0.437767i \(-0.855770\pi\)
−0.899089 + 0.437767i \(0.855770\pi\)
\(758\) 0 0
\(759\) −17.2565 −0.626373
\(760\) 0 0
\(761\) −52.9920 −1.92096 −0.960480 0.278350i \(-0.910213\pi\)
−0.960480 + 0.278350i \(0.910213\pi\)
\(762\) 0 0
\(763\) 61.4342 2.22407
\(764\) 0 0
\(765\) −1.95792 −0.0707888
\(766\) 0 0
\(767\) 24.2680 0.876268
\(768\) 0 0
\(769\) −25.5342 −0.920786 −0.460393 0.887715i \(-0.652291\pi\)
−0.460393 + 0.887715i \(0.652291\pi\)
\(770\) 0 0
\(771\) 25.8455 0.930803
\(772\) 0 0
\(773\) 15.0208 0.540260 0.270130 0.962824i \(-0.412933\pi\)
0.270130 + 0.962824i \(0.412933\pi\)
\(774\) 0 0
\(775\) −7.70440 −0.276750
\(776\) 0 0
\(777\) −8.69506 −0.311934
\(778\) 0 0
\(779\) −51.5174 −1.84580
\(780\) 0 0
\(781\) 29.4759 1.05473
\(782\) 0 0
\(783\) 23.2819 0.832029
\(784\) 0 0
\(785\) 13.7941 0.492331
\(786\) 0 0
\(787\) 20.7354 0.739136 0.369568 0.929204i \(-0.379506\pi\)
0.369568 + 0.929204i \(0.379506\pi\)
\(788\) 0 0
\(789\) −16.2567 −0.578754
\(790\) 0 0
\(791\) −55.1670 −1.96151
\(792\) 0 0
\(793\) 63.4011 2.25144
\(794\) 0 0
\(795\) −15.9939 −0.567247
\(796\) 0 0
\(797\) −22.5536 −0.798890 −0.399445 0.916757i \(-0.630797\pi\)
−0.399445 + 0.916757i \(0.630797\pi\)
\(798\) 0 0
\(799\) 2.47220 0.0874601
\(800\) 0 0
\(801\) 5.81562 0.205485
\(802\) 0 0
\(803\) −10.8010 −0.381159
\(804\) 0 0
\(805\) 9.97559 0.351593
\(806\) 0 0
\(807\) −4.48466 −0.157868
\(808\) 0 0
\(809\) 31.0902 1.09307 0.546537 0.837435i \(-0.315946\pi\)
0.546537 + 0.837435i \(0.315946\pi\)
\(810\) 0 0
\(811\) 51.3496 1.80313 0.901564 0.432647i \(-0.142420\pi\)
0.901564 + 0.432647i \(0.142420\pi\)
\(812\) 0 0
\(813\) −33.3285 −1.16888
\(814\) 0 0
\(815\) −8.31035 −0.291099
\(816\) 0 0
\(817\) −27.8610 −0.974733
\(818\) 0 0
\(819\) 13.8140 0.482700
\(820\) 0 0
\(821\) 46.0087 1.60571 0.802857 0.596171i \(-0.203312\pi\)
0.802857 + 0.596171i \(0.203312\pi\)
\(822\) 0 0
\(823\) −13.9780 −0.487243 −0.243622 0.969870i \(-0.578336\pi\)
−0.243622 + 0.969870i \(0.578336\pi\)
\(824\) 0 0
\(825\) 27.1787 0.946241
\(826\) 0 0
\(827\) 7.26566 0.252652 0.126326 0.991989i \(-0.459681\pi\)
0.126326 + 0.991989i \(0.459681\pi\)
\(828\) 0 0
\(829\) 7.19459 0.249878 0.124939 0.992164i \(-0.460126\pi\)
0.124939 + 0.992164i \(0.460126\pi\)
\(830\) 0 0
\(831\) 13.3981 0.464774
\(832\) 0 0
\(833\) 19.8542 0.687906
\(834\) 0 0
\(835\) −4.47180 −0.154753
\(836\) 0 0
\(837\) −10.9965 −0.380095
\(838\) 0 0
\(839\) 10.4530 0.360879 0.180439 0.983586i \(-0.442248\pi\)
0.180439 + 0.983586i \(0.442248\pi\)
\(840\) 0 0
\(841\) −11.8991 −0.410313
\(842\) 0 0
\(843\) 0.00106967 3.68414e−5 0
\(844\) 0 0
\(845\) 8.59280 0.295601
\(846\) 0 0
\(847\) 39.9247 1.37183
\(848\) 0 0
\(849\) 12.4040 0.425703
\(850\) 0 0
\(851\) −3.76210 −0.128963
\(852\) 0 0
\(853\) 2.43910 0.0835131 0.0417566 0.999128i \(-0.486705\pi\)
0.0417566 + 0.999128i \(0.486705\pi\)
\(854\) 0 0
\(855\) 6.14336 0.210098
\(856\) 0 0
\(857\) −39.5827 −1.35212 −0.676060 0.736847i \(-0.736314\pi\)
−0.676060 + 0.736847i \(0.736314\pi\)
\(858\) 0 0
\(859\) 40.6196 1.38592 0.692962 0.720975i \(-0.256306\pi\)
0.692962 + 0.720975i \(0.256306\pi\)
\(860\) 0 0
\(861\) 38.4432 1.31014
\(862\) 0 0
\(863\) −30.4721 −1.03728 −0.518642 0.854992i \(-0.673562\pi\)
−0.518642 + 0.854992i \(0.673562\pi\)
\(864\) 0 0
\(865\) 5.05785 0.171972
\(866\) 0 0
\(867\) −16.2564 −0.552096
\(868\) 0 0
\(869\) 69.4783 2.35689
\(870\) 0 0
\(871\) −71.3559 −2.41780
\(872\) 0 0
\(873\) 2.33721 0.0791027
\(874\) 0 0
\(875\) −35.6268 −1.20441
\(876\) 0 0
\(877\) 39.2267 1.32459 0.662295 0.749243i \(-0.269583\pi\)
0.662295 + 0.749243i \(0.269583\pi\)
\(878\) 0 0
\(879\) 11.3177 0.381736
\(880\) 0 0
\(881\) 30.4238 1.02500 0.512501 0.858687i \(-0.328719\pi\)
0.512501 + 0.858687i \(0.328719\pi\)
\(882\) 0 0
\(883\) 30.3525 1.02144 0.510721 0.859747i \(-0.329379\pi\)
0.510721 + 0.859747i \(0.329379\pi\)
\(884\) 0 0
\(885\) −8.05355 −0.270717
\(886\) 0 0
\(887\) −8.20845 −0.275613 −0.137806 0.990459i \(-0.544005\pi\)
−0.137806 + 0.990459i \(0.544005\pi\)
\(888\) 0 0
\(889\) −71.3503 −2.39301
\(890\) 0 0
\(891\) 28.1195 0.942038
\(892\) 0 0
\(893\) −7.75700 −0.259578
\(894\) 0 0
\(895\) −16.4465 −0.549745
\(896\) 0 0
\(897\) −17.2833 −0.577071
\(898\) 0 0
\(899\) −8.07709 −0.269386
\(900\) 0 0
\(901\) −25.7778 −0.858783
\(902\) 0 0
\(903\) 20.7904 0.691862
\(904\) 0 0
\(905\) 12.1846 0.405031
\(906\) 0 0
\(907\) 9.22770 0.306401 0.153200 0.988195i \(-0.451042\pi\)
0.153200 + 0.988195i \(0.451042\pi\)
\(908\) 0 0
\(909\) −7.70069 −0.255416
\(910\) 0 0
\(911\) −17.7976 −0.589660 −0.294830 0.955550i \(-0.595263\pi\)
−0.294830 + 0.955550i \(0.595263\pi\)
\(912\) 0 0
\(913\) −10.3765 −0.343412
\(914\) 0 0
\(915\) −21.0402 −0.695567
\(916\) 0 0
\(917\) −31.3089 −1.03391
\(918\) 0 0
\(919\) 50.1369 1.65386 0.826931 0.562303i \(-0.190085\pi\)
0.826931 + 0.562303i \(0.190085\pi\)
\(920\) 0 0
\(921\) −21.5180 −0.709041
\(922\) 0 0
\(923\) 29.5215 0.971713
\(924\) 0 0
\(925\) 5.92523 0.194821
\(926\) 0 0
\(927\) 0.796901 0.0261737
\(928\) 0 0
\(929\) −52.7399 −1.73034 −0.865170 0.501479i \(-0.832790\pi\)
−0.865170 + 0.501479i \(0.832790\pi\)
\(930\) 0 0
\(931\) −62.2963 −2.04168
\(932\) 0 0
\(933\) 35.2648 1.15452
\(934\) 0 0
\(935\) −11.7214 −0.383329
\(936\) 0 0
\(937\) −13.7772 −0.450080 −0.225040 0.974349i \(-0.572251\pi\)
−0.225040 + 0.974349i \(0.572251\pi\)
\(938\) 0 0
\(939\) 15.9229 0.519624
\(940\) 0 0
\(941\) 22.5400 0.734784 0.367392 0.930066i \(-0.380251\pi\)
0.367392 + 0.930066i \(0.380251\pi\)
\(942\) 0 0
\(943\) 16.6333 0.541654
\(944\) 0 0
\(945\) −22.4248 −0.729480
\(946\) 0 0
\(947\) −57.7961 −1.87812 −0.939060 0.343753i \(-0.888302\pi\)
−0.939060 + 0.343753i \(0.888302\pi\)
\(948\) 0 0
\(949\) −10.8177 −0.351158
\(950\) 0 0
\(951\) −41.6144 −1.34944
\(952\) 0 0
\(953\) −20.1887 −0.653977 −0.326989 0.945028i \(-0.606034\pi\)
−0.326989 + 0.945028i \(0.606034\pi\)
\(954\) 0 0
\(955\) −1.32882 −0.0429997
\(956\) 0 0
\(957\) 28.4934 0.921062
\(958\) 0 0
\(959\) 48.9413 1.58040
\(960\) 0 0
\(961\) −27.1850 −0.876937
\(962\) 0 0
\(963\) −7.83020 −0.252325
\(964\) 0 0
\(965\) 13.5216 0.435275
\(966\) 0 0
\(967\) −31.1154 −1.00060 −0.500301 0.865851i \(-0.666777\pi\)
−0.500301 + 0.865851i \(0.666777\pi\)
\(968\) 0 0
\(969\) −28.6315 −0.919777
\(970\) 0 0
\(971\) −27.4477 −0.880840 −0.440420 0.897792i \(-0.645170\pi\)
−0.440420 + 0.897792i \(0.645170\pi\)
\(972\) 0 0
\(973\) −87.7846 −2.81425
\(974\) 0 0
\(975\) 27.2208 0.871763
\(976\) 0 0
\(977\) −3.65464 −0.116922 −0.0584611 0.998290i \(-0.518619\pi\)
−0.0584611 + 0.998290i \(0.518619\pi\)
\(978\) 0 0
\(979\) 34.8160 1.11272
\(980\) 0 0
\(981\) −12.2152 −0.390003
\(982\) 0 0
\(983\) −12.0031 −0.382841 −0.191420 0.981508i \(-0.561309\pi\)
−0.191420 + 0.981508i \(0.561309\pi\)
\(984\) 0 0
\(985\) −9.10563 −0.290129
\(986\) 0 0
\(987\) 5.78843 0.184248
\(988\) 0 0
\(989\) 8.99542 0.286038
\(990\) 0 0
\(991\) −34.9133 −1.10906 −0.554529 0.832165i \(-0.687101\pi\)
−0.554529 + 0.832165i \(0.687101\pi\)
\(992\) 0 0
\(993\) 40.3545 1.28061
\(994\) 0 0
\(995\) 14.1272 0.447862
\(996\) 0 0
\(997\) −38.2808 −1.21236 −0.606182 0.795326i \(-0.707300\pi\)
−0.606182 + 0.795326i \(0.707300\pi\)
\(998\) 0 0
\(999\) 8.45710 0.267571
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 6016.2.a.m.1.10 13
4.3 odd 2 6016.2.a.o.1.4 yes 13
8.3 odd 2 6016.2.a.n.1.10 yes 13
8.5 even 2 6016.2.a.p.1.4 yes 13
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6016.2.a.m.1.10 13 1.1 even 1 trivial
6016.2.a.n.1.10 yes 13 8.3 odd 2
6016.2.a.o.1.4 yes 13 4.3 odd 2
6016.2.a.p.1.4 yes 13 8.5 even 2