Properties

Label 1305.2.a.a.1.1
Level $1305$
Weight $2$
Character 1305.1
Self dual yes
Analytic conductor $10.420$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

Related objects

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1305,2,Mod(1,1305)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1305, 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("1305.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1305 = 3^{2} \cdot 5 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1305.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: \(10.4204774638\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 1305.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.00000 q^{2} +2.00000 q^{4} +1.00000 q^{5} -2.00000 q^{7} +O(q^{10})\) \(q-2.00000 q^{2} +2.00000 q^{4} +1.00000 q^{5} -2.00000 q^{7} -2.00000 q^{10} +3.00000 q^{11} -4.00000 q^{13} +4.00000 q^{14} -4.00000 q^{16} +2.00000 q^{17} -2.00000 q^{19} +2.00000 q^{20} -6.00000 q^{22} -5.00000 q^{23} +1.00000 q^{25} +8.00000 q^{26} -4.00000 q^{28} +1.00000 q^{29} +2.00000 q^{31} +8.00000 q^{32} -4.00000 q^{34} -2.00000 q^{35} -5.00000 q^{37} +4.00000 q^{38} +1.00000 q^{41} -1.00000 q^{43} +6.00000 q^{44} +10.0000 q^{46} -6.00000 q^{47} -3.00000 q^{49} -2.00000 q^{50} -8.00000 q^{52} -3.00000 q^{53} +3.00000 q^{55} -2.00000 q^{58} -4.00000 q^{59} +6.00000 q^{61} -4.00000 q^{62} -8.00000 q^{64} -4.00000 q^{65} +2.00000 q^{67} +4.00000 q^{68} +4.00000 q^{70} +12.0000 q^{71} +9.00000 q^{73} +10.0000 q^{74} -4.00000 q^{76} -6.00000 q^{77} -16.0000 q^{79} -4.00000 q^{80} -2.00000 q^{82} -7.00000 q^{83} +2.00000 q^{85} +2.00000 q^{86} +6.00000 q^{89} +8.00000 q^{91} -10.0000 q^{92} +12.0000 q^{94} -2.00000 q^{95} -13.0000 q^{97} +6.00000 q^{98} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.00000 −1.41421 −0.707107 0.707107i \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(3\) 0 0
\(4\) 2.00000 1.00000
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) −2.00000 −0.755929 −0.377964 0.925820i \(-0.623376\pi\)
−0.377964 + 0.925820i \(0.623376\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) −2.00000 −0.632456
\(11\) 3.00000 0.904534 0.452267 0.891883i \(-0.350615\pi\)
0.452267 + 0.891883i \(0.350615\pi\)
\(12\) 0 0
\(13\) −4.00000 −1.10940 −0.554700 0.832050i \(-0.687167\pi\)
−0.554700 + 0.832050i \(0.687167\pi\)
\(14\) 4.00000 1.06904
\(15\) 0 0
\(16\) −4.00000 −1.00000
\(17\) 2.00000 0.485071 0.242536 0.970143i \(-0.422021\pi\)
0.242536 + 0.970143i \(0.422021\pi\)
\(18\) 0 0
\(19\) −2.00000 −0.458831 −0.229416 0.973329i \(-0.573682\pi\)
−0.229416 + 0.973329i \(0.573682\pi\)
\(20\) 2.00000 0.447214
\(21\) 0 0
\(22\) −6.00000 −1.27920
\(23\) −5.00000 −1.04257 −0.521286 0.853382i \(-0.674548\pi\)
−0.521286 + 0.853382i \(0.674548\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 8.00000 1.56893
\(27\) 0 0
\(28\) −4.00000 −0.755929
\(29\) 1.00000 0.185695
\(30\) 0 0
\(31\) 2.00000 0.359211 0.179605 0.983739i \(-0.442518\pi\)
0.179605 + 0.983739i \(0.442518\pi\)
\(32\) 8.00000 1.41421
\(33\) 0 0
\(34\) −4.00000 −0.685994
\(35\) −2.00000 −0.338062
\(36\) 0 0
\(37\) −5.00000 −0.821995 −0.410997 0.911636i \(-0.634819\pi\)
−0.410997 + 0.911636i \(0.634819\pi\)
\(38\) 4.00000 0.648886
\(39\) 0 0
\(40\) 0 0
\(41\) 1.00000 0.156174 0.0780869 0.996947i \(-0.475119\pi\)
0.0780869 + 0.996947i \(0.475119\pi\)
\(42\) 0 0
\(43\) −1.00000 −0.152499 −0.0762493 0.997089i \(-0.524294\pi\)
−0.0762493 + 0.997089i \(0.524294\pi\)
\(44\) 6.00000 0.904534
\(45\) 0 0
\(46\) 10.0000 1.47442
\(47\) −6.00000 −0.875190 −0.437595 0.899172i \(-0.644170\pi\)
−0.437595 + 0.899172i \(0.644170\pi\)
\(48\) 0 0
\(49\) −3.00000 −0.428571
\(50\) −2.00000 −0.282843
\(51\) 0 0
\(52\) −8.00000 −1.10940
\(53\) −3.00000 −0.412082 −0.206041 0.978543i \(-0.566058\pi\)
−0.206041 + 0.978543i \(0.566058\pi\)
\(54\) 0 0
\(55\) 3.00000 0.404520
\(56\) 0 0
\(57\) 0 0
\(58\) −2.00000 −0.262613
\(59\) −4.00000 −0.520756 −0.260378 0.965507i \(-0.583847\pi\)
−0.260378 + 0.965507i \(0.583847\pi\)
\(60\) 0 0
\(61\) 6.00000 0.768221 0.384111 0.923287i \(-0.374508\pi\)
0.384111 + 0.923287i \(0.374508\pi\)
\(62\) −4.00000 −0.508001
\(63\) 0 0
\(64\) −8.00000 −1.00000
\(65\) −4.00000 −0.496139
\(66\) 0 0
\(67\) 2.00000 0.244339 0.122169 0.992509i \(-0.461015\pi\)
0.122169 + 0.992509i \(0.461015\pi\)
\(68\) 4.00000 0.485071
\(69\) 0 0
\(70\) 4.00000 0.478091
\(71\) 12.0000 1.42414 0.712069 0.702109i \(-0.247758\pi\)
0.712069 + 0.702109i \(0.247758\pi\)
\(72\) 0 0
\(73\) 9.00000 1.05337 0.526685 0.850060i \(-0.323435\pi\)
0.526685 + 0.850060i \(0.323435\pi\)
\(74\) 10.0000 1.16248
\(75\) 0 0
\(76\) −4.00000 −0.458831
\(77\) −6.00000 −0.683763
\(78\) 0 0
\(79\) −16.0000 −1.80014 −0.900070 0.435745i \(-0.856485\pi\)
−0.900070 + 0.435745i \(0.856485\pi\)
\(80\) −4.00000 −0.447214
\(81\) 0 0
\(82\) −2.00000 −0.220863
\(83\) −7.00000 −0.768350 −0.384175 0.923260i \(-0.625514\pi\)
−0.384175 + 0.923260i \(0.625514\pi\)
\(84\) 0 0
\(85\) 2.00000 0.216930
\(86\) 2.00000 0.215666
\(87\) 0 0
\(88\) 0 0
\(89\) 6.00000 0.635999 0.317999 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 0 0
\(91\) 8.00000 0.838628
\(92\) −10.0000 −1.04257
\(93\) 0 0
\(94\) 12.0000 1.23771
\(95\) −2.00000 −0.205196
\(96\) 0 0
\(97\) −13.0000 −1.31995 −0.659975 0.751288i \(-0.729433\pi\)
−0.659975 + 0.751288i \(0.729433\pi\)
\(98\) 6.00000 0.606092
\(99\) 0 0
\(100\) 2.00000 0.200000
\(101\) −1.00000 −0.0995037 −0.0497519 0.998762i \(-0.515843\pi\)
−0.0497519 + 0.998762i \(0.515843\pi\)
\(102\) 0 0
\(103\) −12.0000 −1.18240 −0.591198 0.806527i \(-0.701345\pi\)
−0.591198 + 0.806527i \(0.701345\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 6.00000 0.582772
\(107\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(108\) 0 0
\(109\) −5.00000 −0.478913 −0.239457 0.970907i \(-0.576969\pi\)
−0.239457 + 0.970907i \(0.576969\pi\)
\(110\) −6.00000 −0.572078
\(111\) 0 0
\(112\) 8.00000 0.755929
\(113\) −8.00000 −0.752577 −0.376288 0.926503i \(-0.622800\pi\)
−0.376288 + 0.926503i \(0.622800\pi\)
\(114\) 0 0
\(115\) −5.00000 −0.466252
\(116\) 2.00000 0.185695
\(117\) 0 0
\(118\) 8.00000 0.736460
\(119\) −4.00000 −0.366679
\(120\) 0 0
\(121\) −2.00000 −0.181818
\(122\) −12.0000 −1.08643
\(123\) 0 0
\(124\) 4.00000 0.359211
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −7.00000 −0.621150 −0.310575 0.950549i \(-0.600522\pi\)
−0.310575 + 0.950549i \(0.600522\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 8.00000 0.701646
\(131\) −8.00000 −0.698963 −0.349482 0.936943i \(-0.613642\pi\)
−0.349482 + 0.936943i \(0.613642\pi\)
\(132\) 0 0
\(133\) 4.00000 0.346844
\(134\) −4.00000 −0.345547
\(135\) 0 0
\(136\) 0 0
\(137\) −12.0000 −1.02523 −0.512615 0.858619i \(-0.671323\pi\)
−0.512615 + 0.858619i \(0.671323\pi\)
\(138\) 0 0
\(139\) −15.0000 −1.27228 −0.636142 0.771572i \(-0.719471\pi\)
−0.636142 + 0.771572i \(0.719471\pi\)
\(140\) −4.00000 −0.338062
\(141\) 0 0
\(142\) −24.0000 −2.01404
\(143\) −12.0000 −1.00349
\(144\) 0 0
\(145\) 1.00000 0.0830455
\(146\) −18.0000 −1.48969
\(147\) 0 0
\(148\) −10.0000 −0.821995
\(149\) −20.0000 −1.63846 −0.819232 0.573462i \(-0.805600\pi\)
−0.819232 + 0.573462i \(0.805600\pi\)
\(150\) 0 0
\(151\) −7.00000 −0.569652 −0.284826 0.958579i \(-0.591936\pi\)
−0.284826 + 0.958579i \(0.591936\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 12.0000 0.966988
\(155\) 2.00000 0.160644
\(156\) 0 0
\(157\) −2.00000 −0.159617 −0.0798087 0.996810i \(-0.525431\pi\)
−0.0798087 + 0.996810i \(0.525431\pi\)
\(158\) 32.0000 2.54578
\(159\) 0 0
\(160\) 8.00000 0.632456
\(161\) 10.0000 0.788110
\(162\) 0 0
\(163\) 1.00000 0.0783260 0.0391630 0.999233i \(-0.487531\pi\)
0.0391630 + 0.999233i \(0.487531\pi\)
\(164\) 2.00000 0.156174
\(165\) 0 0
\(166\) 14.0000 1.08661
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 0 0
\(169\) 3.00000 0.230769
\(170\) −4.00000 −0.306786
\(171\) 0 0
\(172\) −2.00000 −0.152499
\(173\) −19.0000 −1.44454 −0.722272 0.691609i \(-0.756902\pi\)
−0.722272 + 0.691609i \(0.756902\pi\)
\(174\) 0 0
\(175\) −2.00000 −0.151186
\(176\) −12.0000 −0.904534
\(177\) 0 0
\(178\) −12.0000 −0.899438
\(179\) 24.0000 1.79384 0.896922 0.442189i \(-0.145798\pi\)
0.896922 + 0.442189i \(0.145798\pi\)
\(180\) 0 0
\(181\) 21.0000 1.56092 0.780459 0.625207i \(-0.214986\pi\)
0.780459 + 0.625207i \(0.214986\pi\)
\(182\) −16.0000 −1.18600
\(183\) 0 0
\(184\) 0 0
\(185\) −5.00000 −0.367607
\(186\) 0 0
\(187\) 6.00000 0.438763
\(188\) −12.0000 −0.875190
\(189\) 0 0
\(190\) 4.00000 0.290191
\(191\) −5.00000 −0.361787 −0.180894 0.983503i \(-0.557899\pi\)
−0.180894 + 0.983503i \(0.557899\pi\)
\(192\) 0 0
\(193\) −10.0000 −0.719816 −0.359908 0.932988i \(-0.617192\pi\)
−0.359908 + 0.932988i \(0.617192\pi\)
\(194\) 26.0000 1.86669
\(195\) 0 0
\(196\) −6.00000 −0.428571
\(197\) 1.00000 0.0712470 0.0356235 0.999365i \(-0.488658\pi\)
0.0356235 + 0.999365i \(0.488658\pi\)
\(198\) 0 0
\(199\) −25.0000 −1.77220 −0.886102 0.463491i \(-0.846597\pi\)
−0.886102 + 0.463491i \(0.846597\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 2.00000 0.140720
\(203\) −2.00000 −0.140372
\(204\) 0 0
\(205\) 1.00000 0.0698430
\(206\) 24.0000 1.67216
\(207\) 0 0
\(208\) 16.0000 1.10940
\(209\) −6.00000 −0.415029
\(210\) 0 0
\(211\) 8.00000 0.550743 0.275371 0.961338i \(-0.411199\pi\)
0.275371 + 0.961338i \(0.411199\pi\)
\(212\) −6.00000 −0.412082
\(213\) 0 0
\(214\) 0 0
\(215\) −1.00000 −0.0681994
\(216\) 0 0
\(217\) −4.00000 −0.271538
\(218\) 10.0000 0.677285
\(219\) 0 0
\(220\) 6.00000 0.404520
\(221\) −8.00000 −0.538138
\(222\) 0 0
\(223\) −8.00000 −0.535720 −0.267860 0.963458i \(-0.586316\pi\)
−0.267860 + 0.963458i \(0.586316\pi\)
\(224\) −16.0000 −1.06904
\(225\) 0 0
\(226\) 16.0000 1.06430
\(227\) −13.0000 −0.862840 −0.431420 0.902151i \(-0.641987\pi\)
−0.431420 + 0.902151i \(0.641987\pi\)
\(228\) 0 0
\(229\) −6.00000 −0.396491 −0.198246 0.980152i \(-0.563524\pi\)
−0.198246 + 0.980152i \(0.563524\pi\)
\(230\) 10.0000 0.659380
\(231\) 0 0
\(232\) 0 0
\(233\) 27.0000 1.76883 0.884414 0.466702i \(-0.154558\pi\)
0.884414 + 0.466702i \(0.154558\pi\)
\(234\) 0 0
\(235\) −6.00000 −0.391397
\(236\) −8.00000 −0.520756
\(237\) 0 0
\(238\) 8.00000 0.518563
\(239\) 12.0000 0.776215 0.388108 0.921614i \(-0.373129\pi\)
0.388108 + 0.921614i \(0.373129\pi\)
\(240\) 0 0
\(241\) 25.0000 1.61039 0.805196 0.593009i \(-0.202060\pi\)
0.805196 + 0.593009i \(0.202060\pi\)
\(242\) 4.00000 0.257130
\(243\) 0 0
\(244\) 12.0000 0.768221
\(245\) −3.00000 −0.191663
\(246\) 0 0
\(247\) 8.00000 0.509028
\(248\) 0 0
\(249\) 0 0
\(250\) −2.00000 −0.126491
\(251\) −24.0000 −1.51487 −0.757433 0.652913i \(-0.773547\pi\)
−0.757433 + 0.652913i \(0.773547\pi\)
\(252\) 0 0
\(253\) −15.0000 −0.943042
\(254\) 14.0000 0.878438
\(255\) 0 0
\(256\) 16.0000 1.00000
\(257\) 27.0000 1.68421 0.842107 0.539311i \(-0.181315\pi\)
0.842107 + 0.539311i \(0.181315\pi\)
\(258\) 0 0
\(259\) 10.0000 0.621370
\(260\) −8.00000 −0.496139
\(261\) 0 0
\(262\) 16.0000 0.988483
\(263\) 6.00000 0.369976 0.184988 0.982741i \(-0.440775\pi\)
0.184988 + 0.982741i \(0.440775\pi\)
\(264\) 0 0
\(265\) −3.00000 −0.184289
\(266\) −8.00000 −0.490511
\(267\) 0 0
\(268\) 4.00000 0.244339
\(269\) 10.0000 0.609711 0.304855 0.952399i \(-0.401392\pi\)
0.304855 + 0.952399i \(0.401392\pi\)
\(270\) 0 0
\(271\) −8.00000 −0.485965 −0.242983 0.970031i \(-0.578126\pi\)
−0.242983 + 0.970031i \(0.578126\pi\)
\(272\) −8.00000 −0.485071
\(273\) 0 0
\(274\) 24.0000 1.44989
\(275\) 3.00000 0.180907
\(276\) 0 0
\(277\) 6.00000 0.360505 0.180253 0.983620i \(-0.442309\pi\)
0.180253 + 0.983620i \(0.442309\pi\)
\(278\) 30.0000 1.79928
\(279\) 0 0
\(280\) 0 0
\(281\) 8.00000 0.477240 0.238620 0.971113i \(-0.423305\pi\)
0.238620 + 0.971113i \(0.423305\pi\)
\(282\) 0 0
\(283\) 16.0000 0.951101 0.475551 0.879688i \(-0.342249\pi\)
0.475551 + 0.879688i \(0.342249\pi\)
\(284\) 24.0000 1.42414
\(285\) 0 0
\(286\) 24.0000 1.41915
\(287\) −2.00000 −0.118056
\(288\) 0 0
\(289\) −13.0000 −0.764706
\(290\) −2.00000 −0.117444
\(291\) 0 0
\(292\) 18.0000 1.05337
\(293\) 14.0000 0.817889 0.408944 0.912559i \(-0.365897\pi\)
0.408944 + 0.912559i \(0.365897\pi\)
\(294\) 0 0
\(295\) −4.00000 −0.232889
\(296\) 0 0
\(297\) 0 0
\(298\) 40.0000 2.31714
\(299\) 20.0000 1.15663
\(300\) 0 0
\(301\) 2.00000 0.115278
\(302\) 14.0000 0.805609
\(303\) 0 0
\(304\) 8.00000 0.458831
\(305\) 6.00000 0.343559
\(306\) 0 0
\(307\) −15.0000 −0.856095 −0.428048 0.903756i \(-0.640798\pi\)
−0.428048 + 0.903756i \(0.640798\pi\)
\(308\) −12.0000 −0.683763
\(309\) 0 0
\(310\) −4.00000 −0.227185
\(311\) −5.00000 −0.283524 −0.141762 0.989901i \(-0.545277\pi\)
−0.141762 + 0.989901i \(0.545277\pi\)
\(312\) 0 0
\(313\) 20.0000 1.13047 0.565233 0.824931i \(-0.308786\pi\)
0.565233 + 0.824931i \(0.308786\pi\)
\(314\) 4.00000 0.225733
\(315\) 0 0
\(316\) −32.0000 −1.80014
\(317\) −28.0000 −1.57264 −0.786318 0.617822i \(-0.788015\pi\)
−0.786318 + 0.617822i \(0.788015\pi\)
\(318\) 0 0
\(319\) 3.00000 0.167968
\(320\) −8.00000 −0.447214
\(321\) 0 0
\(322\) −20.0000 −1.11456
\(323\) −4.00000 −0.222566
\(324\) 0 0
\(325\) −4.00000 −0.221880
\(326\) −2.00000 −0.110770
\(327\) 0 0
\(328\) 0 0
\(329\) 12.0000 0.661581
\(330\) 0 0
\(331\) −12.0000 −0.659580 −0.329790 0.944054i \(-0.606978\pi\)
−0.329790 + 0.944054i \(0.606978\pi\)
\(332\) −14.0000 −0.768350
\(333\) 0 0
\(334\) 0 0
\(335\) 2.00000 0.109272
\(336\) 0 0
\(337\) 14.0000 0.762629 0.381314 0.924445i \(-0.375472\pi\)
0.381314 + 0.924445i \(0.375472\pi\)
\(338\) −6.00000 −0.326357
\(339\) 0 0
\(340\) 4.00000 0.216930
\(341\) 6.00000 0.324918
\(342\) 0 0
\(343\) 20.0000 1.07990
\(344\) 0 0
\(345\) 0 0
\(346\) 38.0000 2.04289
\(347\) 3.00000 0.161048 0.0805242 0.996753i \(-0.474341\pi\)
0.0805242 + 0.996753i \(0.474341\pi\)
\(348\) 0 0
\(349\) 31.0000 1.65939 0.829696 0.558216i \(-0.188514\pi\)
0.829696 + 0.558216i \(0.188514\pi\)
\(350\) 4.00000 0.213809
\(351\) 0 0
\(352\) 24.0000 1.27920
\(353\) 14.0000 0.745145 0.372572 0.928003i \(-0.378476\pi\)
0.372572 + 0.928003i \(0.378476\pi\)
\(354\) 0 0
\(355\) 12.0000 0.636894
\(356\) 12.0000 0.635999
\(357\) 0 0
\(358\) −48.0000 −2.53688
\(359\) 35.0000 1.84723 0.923615 0.383322i \(-0.125220\pi\)
0.923615 + 0.383322i \(0.125220\pi\)
\(360\) 0 0
\(361\) −15.0000 −0.789474
\(362\) −42.0000 −2.20747
\(363\) 0 0
\(364\) 16.0000 0.838628
\(365\) 9.00000 0.471082
\(366\) 0 0
\(367\) −27.0000 −1.40939 −0.704694 0.709511i \(-0.748916\pi\)
−0.704694 + 0.709511i \(0.748916\pi\)
\(368\) 20.0000 1.04257
\(369\) 0 0
\(370\) 10.0000 0.519875
\(371\) 6.00000 0.311504
\(372\) 0 0
\(373\) 18.0000 0.932005 0.466002 0.884783i \(-0.345694\pi\)
0.466002 + 0.884783i \(0.345694\pi\)
\(374\) −12.0000 −0.620505
\(375\) 0 0
\(376\) 0 0
\(377\) −4.00000 −0.206010
\(378\) 0 0
\(379\) −6.00000 −0.308199 −0.154100 0.988055i \(-0.549248\pi\)
−0.154100 + 0.988055i \(0.549248\pi\)
\(380\) −4.00000 −0.205196
\(381\) 0 0
\(382\) 10.0000 0.511645
\(383\) −17.0000 −0.868659 −0.434330 0.900754i \(-0.643015\pi\)
−0.434330 + 0.900754i \(0.643015\pi\)
\(384\) 0 0
\(385\) −6.00000 −0.305788
\(386\) 20.0000 1.01797
\(387\) 0 0
\(388\) −26.0000 −1.31995
\(389\) 5.00000 0.253510 0.126755 0.991934i \(-0.459544\pi\)
0.126755 + 0.991934i \(0.459544\pi\)
\(390\) 0 0
\(391\) −10.0000 −0.505722
\(392\) 0 0
\(393\) 0 0
\(394\) −2.00000 −0.100759
\(395\) −16.0000 −0.805047
\(396\) 0 0
\(397\) −30.0000 −1.50566 −0.752828 0.658217i \(-0.771311\pi\)
−0.752828 + 0.658217i \(0.771311\pi\)
\(398\) 50.0000 2.50627
\(399\) 0 0
\(400\) −4.00000 −0.200000
\(401\) 8.00000 0.399501 0.199750 0.979847i \(-0.435987\pi\)
0.199750 + 0.979847i \(0.435987\pi\)
\(402\) 0 0
\(403\) −8.00000 −0.398508
\(404\) −2.00000 −0.0995037
\(405\) 0 0
\(406\) 4.00000 0.198517
\(407\) −15.0000 −0.743522
\(408\) 0 0
\(409\) 14.0000 0.692255 0.346128 0.938187i \(-0.387496\pi\)
0.346128 + 0.938187i \(0.387496\pi\)
\(410\) −2.00000 −0.0987730
\(411\) 0 0
\(412\) −24.0000 −1.18240
\(413\) 8.00000 0.393654
\(414\) 0 0
\(415\) −7.00000 −0.343616
\(416\) −32.0000 −1.56893
\(417\) 0 0
\(418\) 12.0000 0.586939
\(419\) −16.0000 −0.781651 −0.390826 0.920465i \(-0.627810\pi\)
−0.390826 + 0.920465i \(0.627810\pi\)
\(420\) 0 0
\(421\) −4.00000 −0.194948 −0.0974740 0.995238i \(-0.531076\pi\)
−0.0974740 + 0.995238i \(0.531076\pi\)
\(422\) −16.0000 −0.778868
\(423\) 0 0
\(424\) 0 0
\(425\) 2.00000 0.0970143
\(426\) 0 0
\(427\) −12.0000 −0.580721
\(428\) 0 0
\(429\) 0 0
\(430\) 2.00000 0.0964486
\(431\) 12.0000 0.578020 0.289010 0.957326i \(-0.406674\pi\)
0.289010 + 0.957326i \(0.406674\pi\)
\(432\) 0 0
\(433\) 17.0000 0.816968 0.408484 0.912766i \(-0.366058\pi\)
0.408484 + 0.912766i \(0.366058\pi\)
\(434\) 8.00000 0.384012
\(435\) 0 0
\(436\) −10.0000 −0.478913
\(437\) 10.0000 0.478365
\(438\) 0 0
\(439\) −24.0000 −1.14546 −0.572729 0.819745i \(-0.694115\pi\)
−0.572729 + 0.819745i \(0.694115\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 16.0000 0.761042
\(443\) 12.0000 0.570137 0.285069 0.958507i \(-0.407984\pi\)
0.285069 + 0.958507i \(0.407984\pi\)
\(444\) 0 0
\(445\) 6.00000 0.284427
\(446\) 16.0000 0.757622
\(447\) 0 0
\(448\) 16.0000 0.755929
\(449\) −13.0000 −0.613508 −0.306754 0.951789i \(-0.599243\pi\)
−0.306754 + 0.951789i \(0.599243\pi\)
\(450\) 0 0
\(451\) 3.00000 0.141264
\(452\) −16.0000 −0.752577
\(453\) 0 0
\(454\) 26.0000 1.22024
\(455\) 8.00000 0.375046
\(456\) 0 0
\(457\) 34.0000 1.59045 0.795226 0.606313i \(-0.207352\pi\)
0.795226 + 0.606313i \(0.207352\pi\)
\(458\) 12.0000 0.560723
\(459\) 0 0
\(460\) −10.0000 −0.466252
\(461\) 21.0000 0.978068 0.489034 0.872265i \(-0.337349\pi\)
0.489034 + 0.872265i \(0.337349\pi\)
\(462\) 0 0
\(463\) −20.0000 −0.929479 −0.464739 0.885448i \(-0.653852\pi\)
−0.464739 + 0.885448i \(0.653852\pi\)
\(464\) −4.00000 −0.185695
\(465\) 0 0
\(466\) −54.0000 −2.50150
\(467\) 36.0000 1.66588 0.832941 0.553362i \(-0.186655\pi\)
0.832941 + 0.553362i \(0.186655\pi\)
\(468\) 0 0
\(469\) −4.00000 −0.184703
\(470\) 12.0000 0.553519
\(471\) 0 0
\(472\) 0 0
\(473\) −3.00000 −0.137940
\(474\) 0 0
\(475\) −2.00000 −0.0917663
\(476\) −8.00000 −0.366679
\(477\) 0 0
\(478\) −24.0000 −1.09773
\(479\) −4.00000 −0.182765 −0.0913823 0.995816i \(-0.529129\pi\)
−0.0913823 + 0.995816i \(0.529129\pi\)
\(480\) 0 0
\(481\) 20.0000 0.911922
\(482\) −50.0000 −2.27744
\(483\) 0 0
\(484\) −4.00000 −0.181818
\(485\) −13.0000 −0.590300
\(486\) 0 0
\(487\) −16.0000 −0.725029 −0.362515 0.931978i \(-0.618082\pi\)
−0.362515 + 0.931978i \(0.618082\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 6.00000 0.271052
\(491\) −28.0000 −1.26362 −0.631811 0.775122i \(-0.717688\pi\)
−0.631811 + 0.775122i \(0.717688\pi\)
\(492\) 0 0
\(493\) 2.00000 0.0900755
\(494\) −16.0000 −0.719874
\(495\) 0 0
\(496\) −8.00000 −0.359211
\(497\) −24.0000 −1.07655
\(498\) 0 0
\(499\) −36.0000 −1.61158 −0.805791 0.592200i \(-0.798259\pi\)
−0.805791 + 0.592200i \(0.798259\pi\)
\(500\) 2.00000 0.0894427
\(501\) 0 0
\(502\) 48.0000 2.14234
\(503\) 14.0000 0.624229 0.312115 0.950044i \(-0.398963\pi\)
0.312115 + 0.950044i \(0.398963\pi\)
\(504\) 0 0
\(505\) −1.00000 −0.0444994
\(506\) 30.0000 1.33366
\(507\) 0 0
\(508\) −14.0000 −0.621150
\(509\) 14.0000 0.620539 0.310270 0.950649i \(-0.399581\pi\)
0.310270 + 0.950649i \(0.399581\pi\)
\(510\) 0 0
\(511\) −18.0000 −0.796273
\(512\) −32.0000 −1.41421
\(513\) 0 0
\(514\) −54.0000 −2.38184
\(515\) −12.0000 −0.528783
\(516\) 0 0
\(517\) −18.0000 −0.791639
\(518\) −20.0000 −0.878750
\(519\) 0 0
\(520\) 0 0
\(521\) 18.0000 0.788594 0.394297 0.918983i \(-0.370988\pi\)
0.394297 + 0.918983i \(0.370988\pi\)
\(522\) 0 0
\(523\) 30.0000 1.31181 0.655904 0.754844i \(-0.272288\pi\)
0.655904 + 0.754844i \(0.272288\pi\)
\(524\) −16.0000 −0.698963
\(525\) 0 0
\(526\) −12.0000 −0.523225
\(527\) 4.00000 0.174243
\(528\) 0 0
\(529\) 2.00000 0.0869565
\(530\) 6.00000 0.260623
\(531\) 0 0
\(532\) 8.00000 0.346844
\(533\) −4.00000 −0.173259
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) −20.0000 −0.862261
\(539\) −9.00000 −0.387657
\(540\) 0 0
\(541\) −38.0000 −1.63375 −0.816874 0.576816i \(-0.804295\pi\)
−0.816874 + 0.576816i \(0.804295\pi\)
\(542\) 16.0000 0.687259
\(543\) 0 0
\(544\) 16.0000 0.685994
\(545\) −5.00000 −0.214176
\(546\) 0 0
\(547\) 46.0000 1.96682 0.983409 0.181402i \(-0.0580636\pi\)
0.983409 + 0.181402i \(0.0580636\pi\)
\(548\) −24.0000 −1.02523
\(549\) 0 0
\(550\) −6.00000 −0.255841
\(551\) −2.00000 −0.0852029
\(552\) 0 0
\(553\) 32.0000 1.36078
\(554\) −12.0000 −0.509831
\(555\) 0 0
\(556\) −30.0000 −1.27228
\(557\) 37.0000 1.56774 0.783870 0.620925i \(-0.213243\pi\)
0.783870 + 0.620925i \(0.213243\pi\)
\(558\) 0 0
\(559\) 4.00000 0.169182
\(560\) 8.00000 0.338062
\(561\) 0 0
\(562\) −16.0000 −0.674919
\(563\) −2.00000 −0.0842900 −0.0421450 0.999112i \(-0.513419\pi\)
−0.0421450 + 0.999112i \(0.513419\pi\)
\(564\) 0 0
\(565\) −8.00000 −0.336563
\(566\) −32.0000 −1.34506
\(567\) 0 0
\(568\) 0 0
\(569\) −38.0000 −1.59304 −0.796521 0.604610i \(-0.793329\pi\)
−0.796521 + 0.604610i \(0.793329\pi\)
\(570\) 0 0
\(571\) −11.0000 −0.460336 −0.230168 0.973151i \(-0.573928\pi\)
−0.230168 + 0.973151i \(0.573928\pi\)
\(572\) −24.0000 −1.00349
\(573\) 0 0
\(574\) 4.00000 0.166957
\(575\) −5.00000 −0.208514
\(576\) 0 0
\(577\) 22.0000 0.915872 0.457936 0.888985i \(-0.348589\pi\)
0.457936 + 0.888985i \(0.348589\pi\)
\(578\) 26.0000 1.08146
\(579\) 0 0
\(580\) 2.00000 0.0830455
\(581\) 14.0000 0.580818
\(582\) 0 0
\(583\) −9.00000 −0.372742
\(584\) 0 0
\(585\) 0 0
\(586\) −28.0000 −1.15667
\(587\) 12.0000 0.495293 0.247647 0.968850i \(-0.420343\pi\)
0.247647 + 0.968850i \(0.420343\pi\)
\(588\) 0 0
\(589\) −4.00000 −0.164817
\(590\) 8.00000 0.329355
\(591\) 0 0
\(592\) 20.0000 0.821995
\(593\) 2.00000 0.0821302 0.0410651 0.999156i \(-0.486925\pi\)
0.0410651 + 0.999156i \(0.486925\pi\)
\(594\) 0 0
\(595\) −4.00000 −0.163984
\(596\) −40.0000 −1.63846
\(597\) 0 0
\(598\) −40.0000 −1.63572
\(599\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) 0 0
\(601\) −22.0000 −0.897399 −0.448699 0.893683i \(-0.648113\pi\)
−0.448699 + 0.893683i \(0.648113\pi\)
\(602\) −4.00000 −0.163028
\(603\) 0 0
\(604\) −14.0000 −0.569652
\(605\) −2.00000 −0.0813116
\(606\) 0 0
\(607\) 40.0000 1.62355 0.811775 0.583970i \(-0.198502\pi\)
0.811775 + 0.583970i \(0.198502\pi\)
\(608\) −16.0000 −0.648886
\(609\) 0 0
\(610\) −12.0000 −0.485866
\(611\) 24.0000 0.970936
\(612\) 0 0
\(613\) 12.0000 0.484675 0.242338 0.970192i \(-0.422086\pi\)
0.242338 + 0.970192i \(0.422086\pi\)
\(614\) 30.0000 1.21070
\(615\) 0 0
\(616\) 0 0
\(617\) 18.0000 0.724653 0.362326 0.932051i \(-0.381983\pi\)
0.362326 + 0.932051i \(0.381983\pi\)
\(618\) 0 0
\(619\) −16.0000 −0.643094 −0.321547 0.946894i \(-0.604203\pi\)
−0.321547 + 0.946894i \(0.604203\pi\)
\(620\) 4.00000 0.160644
\(621\) 0 0
\(622\) 10.0000 0.400963
\(623\) −12.0000 −0.480770
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) −40.0000 −1.59872
\(627\) 0 0
\(628\) −4.00000 −0.159617
\(629\) −10.0000 −0.398726
\(630\) 0 0
\(631\) −8.00000 −0.318475 −0.159237 0.987240i \(-0.550904\pi\)
−0.159237 + 0.987240i \(0.550904\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 56.0000 2.22404
\(635\) −7.00000 −0.277787
\(636\) 0 0
\(637\) 12.0000 0.475457
\(638\) −6.00000 −0.237542
\(639\) 0 0
\(640\) 0 0
\(641\) −5.00000 −0.197488 −0.0987441 0.995113i \(-0.531483\pi\)
−0.0987441 + 0.995113i \(0.531483\pi\)
\(642\) 0 0
\(643\) −8.00000 −0.315489 −0.157745 0.987480i \(-0.550422\pi\)
−0.157745 + 0.987480i \(0.550422\pi\)
\(644\) 20.0000 0.788110
\(645\) 0 0
\(646\) 8.00000 0.314756
\(647\) −27.0000 −1.06148 −0.530740 0.847535i \(-0.678086\pi\)
−0.530740 + 0.847535i \(0.678086\pi\)
\(648\) 0 0
\(649\) −12.0000 −0.471041
\(650\) 8.00000 0.313786
\(651\) 0 0
\(652\) 2.00000 0.0783260
\(653\) −14.0000 −0.547862 −0.273931 0.961749i \(-0.588324\pi\)
−0.273931 + 0.961749i \(0.588324\pi\)
\(654\) 0 0
\(655\) −8.00000 −0.312586
\(656\) −4.00000 −0.156174
\(657\) 0 0
\(658\) −24.0000 −0.935617
\(659\) −1.00000 −0.0389545 −0.0194772 0.999810i \(-0.506200\pi\)
−0.0194772 + 0.999810i \(0.506200\pi\)
\(660\) 0 0
\(661\) −13.0000 −0.505641 −0.252821 0.967513i \(-0.581358\pi\)
−0.252821 + 0.967513i \(0.581358\pi\)
\(662\) 24.0000 0.932786
\(663\) 0 0
\(664\) 0 0
\(665\) 4.00000 0.155113
\(666\) 0 0
\(667\) −5.00000 −0.193601
\(668\) 0 0
\(669\) 0 0
\(670\) −4.00000 −0.154533
\(671\) 18.0000 0.694882
\(672\) 0 0
\(673\) 10.0000 0.385472 0.192736 0.981251i \(-0.438264\pi\)
0.192736 + 0.981251i \(0.438264\pi\)
\(674\) −28.0000 −1.07852
\(675\) 0 0
\(676\) 6.00000 0.230769
\(677\) 6.00000 0.230599 0.115299 0.993331i \(-0.463217\pi\)
0.115299 + 0.993331i \(0.463217\pi\)
\(678\) 0 0
\(679\) 26.0000 0.997788
\(680\) 0 0
\(681\) 0 0
\(682\) −12.0000 −0.459504
\(683\) −33.0000 −1.26271 −0.631355 0.775494i \(-0.717501\pi\)
−0.631355 + 0.775494i \(0.717501\pi\)
\(684\) 0 0
\(685\) −12.0000 −0.458496
\(686\) −40.0000 −1.52721
\(687\) 0 0
\(688\) 4.00000 0.152499
\(689\) 12.0000 0.457164
\(690\) 0 0
\(691\) 28.0000 1.06517 0.532585 0.846376i \(-0.321221\pi\)
0.532585 + 0.846376i \(0.321221\pi\)
\(692\) −38.0000 −1.44454
\(693\) 0 0
\(694\) −6.00000 −0.227757
\(695\) −15.0000 −0.568982
\(696\) 0 0
\(697\) 2.00000 0.0757554
\(698\) −62.0000 −2.34673
\(699\) 0 0
\(700\) −4.00000 −0.151186
\(701\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(702\) 0 0
\(703\) 10.0000 0.377157
\(704\) −24.0000 −0.904534
\(705\) 0 0
\(706\) −28.0000 −1.05379
\(707\) 2.00000 0.0752177
\(708\) 0 0
\(709\) −21.0000 −0.788672 −0.394336 0.918966i \(-0.629025\pi\)
−0.394336 + 0.918966i \(0.629025\pi\)
\(710\) −24.0000 −0.900704
\(711\) 0 0
\(712\) 0 0
\(713\) −10.0000 −0.374503
\(714\) 0 0
\(715\) −12.0000 −0.448775
\(716\) 48.0000 1.79384
\(717\) 0 0
\(718\) −70.0000 −2.61238
\(719\) 24.0000 0.895049 0.447524 0.894272i \(-0.352306\pi\)
0.447524 + 0.894272i \(0.352306\pi\)
\(720\) 0 0
\(721\) 24.0000 0.893807
\(722\) 30.0000 1.11648
\(723\) 0 0
\(724\) 42.0000 1.56092
\(725\) 1.00000 0.0371391
\(726\) 0 0
\(727\) −8.00000 −0.296704 −0.148352 0.988935i \(-0.547397\pi\)
−0.148352 + 0.988935i \(0.547397\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) −18.0000 −0.666210
\(731\) −2.00000 −0.0739727
\(732\) 0 0
\(733\) 42.0000 1.55131 0.775653 0.631160i \(-0.217421\pi\)
0.775653 + 0.631160i \(0.217421\pi\)
\(734\) 54.0000 1.99318
\(735\) 0 0
\(736\) −40.0000 −1.47442
\(737\) 6.00000 0.221013
\(738\) 0 0
\(739\) −50.0000 −1.83928 −0.919640 0.392763i \(-0.871519\pi\)
−0.919640 + 0.392763i \(0.871519\pi\)
\(740\) −10.0000 −0.367607
\(741\) 0 0
\(742\) −12.0000 −0.440534
\(743\) −12.0000 −0.440237 −0.220119 0.975473i \(-0.570644\pi\)
−0.220119 + 0.975473i \(0.570644\pi\)
\(744\) 0 0
\(745\) −20.0000 −0.732743
\(746\) −36.0000 −1.31805
\(747\) 0 0
\(748\) 12.0000 0.438763
\(749\) 0 0
\(750\) 0 0
\(751\) −8.00000 −0.291924 −0.145962 0.989290i \(-0.546628\pi\)
−0.145962 + 0.989290i \(0.546628\pi\)
\(752\) 24.0000 0.875190
\(753\) 0 0
\(754\) 8.00000 0.291343
\(755\) −7.00000 −0.254756
\(756\) 0 0
\(757\) −47.0000 −1.70824 −0.854122 0.520073i \(-0.825905\pi\)
−0.854122 + 0.520073i \(0.825905\pi\)
\(758\) 12.0000 0.435860
\(759\) 0 0
\(760\) 0 0
\(761\) 10.0000 0.362500 0.181250 0.983437i \(-0.441986\pi\)
0.181250 + 0.983437i \(0.441986\pi\)
\(762\) 0 0
\(763\) 10.0000 0.362024
\(764\) −10.0000 −0.361787
\(765\) 0 0
\(766\) 34.0000 1.22847
\(767\) 16.0000 0.577727
\(768\) 0 0
\(769\) 14.0000 0.504853 0.252426 0.967616i \(-0.418771\pi\)
0.252426 + 0.967616i \(0.418771\pi\)
\(770\) 12.0000 0.432450
\(771\) 0 0
\(772\) −20.0000 −0.719816
\(773\) −46.0000 −1.65451 −0.827253 0.561830i \(-0.810097\pi\)
−0.827253 + 0.561830i \(0.810097\pi\)
\(774\) 0 0
\(775\) 2.00000 0.0718421
\(776\) 0 0
\(777\) 0 0
\(778\) −10.0000 −0.358517
\(779\) −2.00000 −0.0716574
\(780\) 0 0
\(781\) 36.0000 1.28818
\(782\) 20.0000 0.715199
\(783\) 0 0
\(784\) 12.0000 0.428571
\(785\) −2.00000 −0.0713831
\(786\) 0 0
\(787\) 42.0000 1.49714 0.748569 0.663057i \(-0.230741\pi\)
0.748569 + 0.663057i \(0.230741\pi\)
\(788\) 2.00000 0.0712470
\(789\) 0 0
\(790\) 32.0000 1.13851
\(791\) 16.0000 0.568895
\(792\) 0 0
\(793\) −24.0000 −0.852265
\(794\) 60.0000 2.12932
\(795\) 0 0
\(796\) −50.0000 −1.77220
\(797\) −2.00000 −0.0708436 −0.0354218 0.999372i \(-0.511277\pi\)
−0.0354218 + 0.999372i \(0.511277\pi\)
\(798\) 0 0
\(799\) −12.0000 −0.424529
\(800\) 8.00000 0.282843
\(801\) 0 0
\(802\) −16.0000 −0.564980
\(803\) 27.0000 0.952809
\(804\) 0 0
\(805\) 10.0000 0.352454
\(806\) 16.0000 0.563576
\(807\) 0 0
\(808\) 0 0
\(809\) −19.0000 −0.668004 −0.334002 0.942572i \(-0.608399\pi\)
−0.334002 + 0.942572i \(0.608399\pi\)
\(810\) 0 0
\(811\) −41.0000 −1.43970 −0.719852 0.694127i \(-0.755791\pi\)
−0.719852 + 0.694127i \(0.755791\pi\)
\(812\) −4.00000 −0.140372
\(813\) 0 0
\(814\) 30.0000 1.05150
\(815\) 1.00000 0.0350285
\(816\) 0 0
\(817\) 2.00000 0.0699711
\(818\) −28.0000 −0.978997
\(819\) 0 0
\(820\) 2.00000 0.0698430
\(821\) 20.0000 0.698005 0.349002 0.937122i \(-0.386521\pi\)
0.349002 + 0.937122i \(0.386521\pi\)
\(822\) 0 0
\(823\) −32.0000 −1.11545 −0.557725 0.830026i \(-0.688326\pi\)
−0.557725 + 0.830026i \(0.688326\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) −16.0000 −0.556711
\(827\) 24.0000 0.834562 0.417281 0.908778i \(-0.362983\pi\)
0.417281 + 0.908778i \(0.362983\pi\)
\(828\) 0 0
\(829\) 28.0000 0.972480 0.486240 0.873825i \(-0.338368\pi\)
0.486240 + 0.873825i \(0.338368\pi\)
\(830\) 14.0000 0.485947
\(831\) 0 0
\(832\) 32.0000 1.10940
\(833\) −6.00000 −0.207888
\(834\) 0 0
\(835\) 0 0
\(836\) −12.0000 −0.415029
\(837\) 0 0
\(838\) 32.0000 1.10542
\(839\) 8.00000 0.276191 0.138095 0.990419i \(-0.455902\pi\)
0.138095 + 0.990419i \(0.455902\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) 8.00000 0.275698
\(843\) 0 0
\(844\) 16.0000 0.550743
\(845\) 3.00000 0.103203
\(846\) 0 0
\(847\) 4.00000 0.137442
\(848\) 12.0000 0.412082
\(849\) 0 0
\(850\) −4.00000 −0.137199
\(851\) 25.0000 0.856989
\(852\) 0 0
\(853\) 5.00000 0.171197 0.0855984 0.996330i \(-0.472720\pi\)
0.0855984 + 0.996330i \(0.472720\pi\)
\(854\) 24.0000 0.821263
\(855\) 0 0
\(856\) 0 0
\(857\) −19.0000 −0.649028 −0.324514 0.945881i \(-0.605201\pi\)
−0.324514 + 0.945881i \(0.605201\pi\)
\(858\) 0 0
\(859\) 30.0000 1.02359 0.511793 0.859109i \(-0.328981\pi\)
0.511793 + 0.859109i \(0.328981\pi\)
\(860\) −2.00000 −0.0681994
\(861\) 0 0
\(862\) −24.0000 −0.817443
\(863\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(864\) 0 0
\(865\) −19.0000 −0.646019
\(866\) −34.0000 −1.15537
\(867\) 0 0
\(868\) −8.00000 −0.271538
\(869\) −48.0000 −1.62829
\(870\) 0 0
\(871\) −8.00000 −0.271070
\(872\) 0 0
\(873\) 0 0
\(874\) −20.0000 −0.676510
\(875\) −2.00000 −0.0676123
\(876\) 0 0
\(877\) 28.0000 0.945493 0.472746 0.881199i \(-0.343263\pi\)
0.472746 + 0.881199i \(0.343263\pi\)
\(878\) 48.0000 1.61992
\(879\) 0 0
\(880\) −12.0000 −0.404520
\(881\) 3.00000 0.101073 0.0505363 0.998722i \(-0.483907\pi\)
0.0505363 + 0.998722i \(0.483907\pi\)
\(882\) 0 0
\(883\) 22.0000 0.740359 0.370179 0.928960i \(-0.379296\pi\)
0.370179 + 0.928960i \(0.379296\pi\)
\(884\) −16.0000 −0.538138
\(885\) 0 0
\(886\) −24.0000 −0.806296
\(887\) −4.00000 −0.134307 −0.0671534 0.997743i \(-0.521392\pi\)
−0.0671534 + 0.997743i \(0.521392\pi\)
\(888\) 0 0
\(889\) 14.0000 0.469545
\(890\) −12.0000 −0.402241
\(891\) 0 0
\(892\) −16.0000 −0.535720
\(893\) 12.0000 0.401565
\(894\) 0 0
\(895\) 24.0000 0.802232
\(896\) 0 0
\(897\) 0 0
\(898\) 26.0000 0.867631
\(899\) 2.00000 0.0667037
\(900\) 0 0
\(901\) −6.00000 −0.199889
\(902\) −6.00000 −0.199778
\(903\) 0 0
\(904\) 0 0
\(905\) 21.0000 0.698064
\(906\) 0 0
\(907\) −3.00000 −0.0996134 −0.0498067 0.998759i \(-0.515861\pi\)
−0.0498067 + 0.998759i \(0.515861\pi\)
\(908\) −26.0000 −0.862840
\(909\) 0 0
\(910\) −16.0000 −0.530395
\(911\) 49.0000 1.62344 0.811721 0.584045i \(-0.198531\pi\)
0.811721 + 0.584045i \(0.198531\pi\)
\(912\) 0 0
\(913\) −21.0000 −0.694999
\(914\) −68.0000 −2.24924
\(915\) 0 0
\(916\) −12.0000 −0.396491
\(917\) 16.0000 0.528367
\(918\) 0 0
\(919\) −32.0000 −1.05558 −0.527791 0.849374i \(-0.676980\pi\)
−0.527791 + 0.849374i \(0.676980\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −42.0000 −1.38320
\(923\) −48.0000 −1.57994
\(924\) 0 0
\(925\) −5.00000 −0.164399
\(926\) 40.0000 1.31448
\(927\) 0 0
\(928\) 8.00000 0.262613
\(929\) −38.0000 −1.24674 −0.623370 0.781927i \(-0.714237\pi\)
−0.623370 + 0.781927i \(0.714237\pi\)
\(930\) 0 0
\(931\) 6.00000 0.196642
\(932\) 54.0000 1.76883
\(933\) 0 0
\(934\) −72.0000 −2.35591
\(935\) 6.00000 0.196221
\(936\) 0 0
\(937\) 36.0000 1.17607 0.588034 0.808836i \(-0.299902\pi\)
0.588034 + 0.808836i \(0.299902\pi\)
\(938\) 8.00000 0.261209
\(939\) 0 0
\(940\) −12.0000 −0.391397
\(941\) −10.0000 −0.325991 −0.162995 0.986627i \(-0.552116\pi\)
−0.162995 + 0.986627i \(0.552116\pi\)
\(942\) 0 0
\(943\) −5.00000 −0.162822
\(944\) 16.0000 0.520756
\(945\) 0 0
\(946\) 6.00000 0.195077
\(947\) 24.0000 0.779895 0.389948 0.920837i \(-0.372493\pi\)
0.389948 + 0.920837i \(0.372493\pi\)
\(948\) 0 0
\(949\) −36.0000 −1.16861
\(950\) 4.00000 0.129777
\(951\) 0 0
\(952\) 0 0
\(953\) −54.0000 −1.74923 −0.874616 0.484817i \(-0.838886\pi\)
−0.874616 + 0.484817i \(0.838886\pi\)
\(954\) 0 0
\(955\) −5.00000 −0.161796
\(956\) 24.0000 0.776215
\(957\) 0 0
\(958\) 8.00000 0.258468
\(959\) 24.0000 0.775000
\(960\) 0 0
\(961\) −27.0000 −0.870968
\(962\) −40.0000 −1.28965
\(963\) 0 0
\(964\) 50.0000 1.61039
\(965\) −10.0000 −0.321911
\(966\) 0 0
\(967\) −47.0000 −1.51142 −0.755709 0.654907i \(-0.772708\pi\)
−0.755709 + 0.654907i \(0.772708\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 26.0000 0.834810
\(971\) 27.0000 0.866471 0.433236 0.901281i \(-0.357372\pi\)
0.433236 + 0.901281i \(0.357372\pi\)
\(972\) 0 0
\(973\) 30.0000 0.961756
\(974\) 32.0000 1.02535
\(975\) 0 0
\(976\) −24.0000 −0.768221
\(977\) 45.0000 1.43968 0.719839 0.694141i \(-0.244216\pi\)
0.719839 + 0.694141i \(0.244216\pi\)
\(978\) 0 0
\(979\) 18.0000 0.575282
\(980\) −6.00000 −0.191663
\(981\) 0 0
\(982\) 56.0000 1.78703
\(983\) 48.0000 1.53096 0.765481 0.643458i \(-0.222501\pi\)
0.765481 + 0.643458i \(0.222501\pi\)
\(984\) 0 0
\(985\) 1.00000 0.0318626
\(986\) −4.00000 −0.127386
\(987\) 0 0
\(988\) 16.0000 0.509028
\(989\) 5.00000 0.158991
\(990\) 0 0
\(991\) 19.0000 0.603555 0.301777 0.953378i \(-0.402420\pi\)
0.301777 + 0.953378i \(0.402420\pi\)
\(992\) 16.0000 0.508001
\(993\) 0 0
\(994\) 48.0000 1.52247
\(995\) −25.0000 −0.792553
\(996\) 0 0
\(997\) 59.0000 1.86855 0.934274 0.356555i \(-0.116049\pi\)
0.934274 + 0.356555i \(0.116049\pi\)
\(998\) 72.0000 2.27912
\(999\) 0 0
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 1305.2.a.a.1.1 1
3.2 odd 2 1305.2.a.g.1.1 yes 1
5.4 even 2 6525.2.a.m.1.1 1
15.14 odd 2 6525.2.a.a.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1305.2.a.a.1.1 1 1.1 even 1 trivial
1305.2.a.g.1.1 yes 1 3.2 odd 2
6525.2.a.a.1.1 1 15.14 odd 2
6525.2.a.m.1.1 1 5.4 even 2