Properties

Label 1305.2.a.g.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 \(-0.250000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(3\) 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 \(-0.649385\pi\)
−0.452267 + 0.891883i \(0.649385\pi\)
\(12\) 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 \(-0.577979\pi\)
−0.242536 + 0.970143i \(0.577979\pi\)
\(18\) 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 \(-0.325452\pi\)
0.521286 + 0.853382i \(0.325452\pi\)
\(24\) 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 \(-0.524881\pi\)
−0.0780869 + 0.996947i \(0.524881\pi\)
\(42\) 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 \(-0.355830\pi\)
0.437595 + 0.899172i \(0.355830\pi\)
\(48\) 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 \(-0.433942\pi\)
0.206041 + 0.978543i \(0.433942\pi\)
\(54\) 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 \(-0.416153\pi\)
0.260378 + 0.965507i \(0.416153\pi\)
\(60\) 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 \(-0.752242\pi\)
−0.712069 + 0.702109i \(0.752242\pi\)
\(72\) 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 \(-0.374486\pi\)
0.384175 + 0.923260i \(0.374486\pi\)
\(84\) 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 \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) 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 \(-0.484157\pi\)
0.0497519 + 0.998762i \(0.484157\pi\)
\(102\) 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 \(-0.377200\pi\)
0.376288 + 0.926503i \(0.377200\pi\)
\(114\) 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 \(-0.386358\pi\)
0.349482 + 0.936943i \(0.386358\pi\)
\(132\) 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 \(-0.328677\pi\)
0.512615 + 0.858619i \(0.328677\pi\)
\(138\) 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 \(-0.194400\pi\)
0.819232 + 0.573462i \(0.194400\pi\)
\(150\) 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 \(-0.243098\pi\)
0.722272 + 0.691609i \(0.243098\pi\)
\(174\) 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 \(-0.854202\pi\)
−0.896922 + 0.442189i \(0.854202\pi\)
\(180\) 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 \(-0.442101\pi\)
0.180894 + 0.983503i \(0.442101\pi\)
\(192\) 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 \(-0.511342\pi\)
−0.0356235 + 0.999365i \(0.511342\pi\)
\(198\) 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 \(-0.358013\pi\)
0.431420 + 0.902151i \(0.358013\pi\)
\(228\) 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 \(-0.845442\pi\)
−0.884414 + 0.466702i \(0.845442\pi\)
\(234\) 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 \(-0.626871\pi\)
−0.388108 + 0.921614i \(0.626871\pi\)
\(240\) 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 \(-0.226453\pi\)
0.757433 + 0.652913i \(0.226453\pi\)
\(252\) 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 \(-0.818685\pi\)
−0.842107 + 0.539311i \(0.818685\pi\)
\(258\) 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 \(-0.559225\pi\)
−0.184988 + 0.982741i \(0.559225\pi\)
\(264\) 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 \(-0.598608\pi\)
−0.304855 + 0.952399i \(0.598608\pi\)
\(270\) 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 \(-0.576695\pi\)
−0.238620 + 0.971113i \(0.576695\pi\)
\(282\) 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 \(-0.634103\pi\)
−0.408944 + 0.912559i \(0.634103\pi\)
\(294\) 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 \(-0.454723\pi\)
0.141762 + 0.989901i \(0.454723\pi\)
\(312\) 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 \(-0.211985\pi\)
0.786318 + 0.617822i \(0.211985\pi\)
\(318\) 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 \(-0.525659\pi\)
−0.0805242 + 0.996753i \(0.525659\pi\)
\(348\) 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 \(-0.621524\pi\)
−0.372572 + 0.928003i \(0.621524\pi\)
\(354\) 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 \(-0.874780\pi\)
−0.923615 + 0.383322i \(0.874780\pi\)
\(360\) 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 \(-0.356985\pi\)
0.434330 + 0.900754i \(0.356985\pi\)
\(384\) 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 \(-0.540456\pi\)
−0.126755 + 0.991934i \(0.540456\pi\)
\(390\) 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 \(-0.564013\pi\)
−0.199750 + 0.979847i \(0.564013\pi\)
\(402\) 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 \(-0.372190\pi\)
0.390826 + 0.920465i \(0.372190\pi\)
\(420\) 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 \(-0.593326\pi\)
−0.289010 + 0.957326i \(0.593326\pi\)
\(432\) 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 \(-0.592016\pi\)
−0.285069 + 0.958507i \(0.592016\pi\)
\(444\) 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 \(-0.400757\pi\)
0.306754 + 0.951789i \(0.400757\pi\)
\(450\) 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 \(-0.662651\pi\)
−0.489034 + 0.872265i \(0.662651\pi\)
\(462\) 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 \(-0.813345\pi\)
−0.832941 + 0.553362i \(0.813345\pi\)
\(468\) 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 \(-0.470871\pi\)
0.0913823 + 0.995816i \(0.470871\pi\)
\(480\) 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 \(-0.282312\pi\)
0.631811 + 0.775122i \(0.282312\pi\)
\(492\) 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 \(-0.601037\pi\)
−0.312115 + 0.950044i \(0.601037\pi\)
\(504\) 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 \(-0.600419\pi\)
−0.310270 + 0.950649i \(0.600419\pi\)
\(510\) 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 \(-0.629012\pi\)
−0.394297 + 0.918983i \(0.629012\pi\)
\(522\) 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 \(-0.786757\pi\)
−0.783870 + 0.620925i \(0.786757\pi\)
\(558\) 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 \(-0.486581\pi\)
0.0421450 + 0.999112i \(0.486581\pi\)
\(564\) 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 \(-0.206671\pi\)
0.796521 + 0.604610i \(0.206671\pi\)
\(570\) 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 \(-0.579657\pi\)
−0.247647 + 0.968850i \(0.579657\pi\)
\(588\) 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 \(-0.513075\pi\)
−0.0410651 + 0.999156i \(0.513075\pi\)
\(594\) 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 \(-0.618017\pi\)
−0.362326 + 0.932051i \(0.618017\pi\)
\(618\) 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 \(-0.468517\pi\)
0.0987441 + 0.995113i \(0.468517\pi\)
\(642\) 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 \(-0.321914\pi\)
0.530740 + 0.847535i \(0.321914\pi\)
\(648\) 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 \(-0.411676\pi\)
0.273931 + 0.961749i \(0.411676\pi\)
\(654\) 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 \(-0.493800\pi\)
0.0194772 + 0.999810i \(0.493800\pi\)
\(660\) 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 \(-0.536783\pi\)
−0.115299 + 0.993331i \(0.536783\pi\)
\(678\) 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 \(-0.282499\pi\)
0.631355 + 0.775494i \(0.282499\pi\)
\(684\) 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 \(-0.647694\pi\)
−0.447524 + 0.894272i \(0.647694\pi\)
\(720\) 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 \(-0.429356\pi\)
0.220119 + 0.975473i \(0.429356\pi\)
\(744\) 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 \(-0.558014\pi\)
−0.181250 + 0.983437i \(0.558014\pi\)
\(762\) 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 \(-0.189903\pi\)
0.827253 + 0.561830i \(0.189903\pi\)
\(774\) 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 \(-0.488723\pi\)
0.0354218 + 0.999372i \(0.488723\pi\)
\(798\) 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 \(-0.391601\pi\)
0.334002 + 0.942572i \(0.391601\pi\)
\(810\) 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 \(-0.613479\pi\)
−0.349002 + 0.937122i \(0.613479\pi\)
\(822\) 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 \(-0.637017\pi\)
−0.417281 + 0.908778i \(0.637017\pi\)
\(828\) 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 \(-0.544098\pi\)
−0.138095 + 0.990419i \(0.544098\pi\)
\(840\) 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 \(-0.394799\pi\)
0.324514 + 0.945881i \(0.394799\pi\)
\(858\) 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 \(-0.516093\pi\)
−0.0505363 + 0.998722i \(0.516093\pi\)
\(882\) 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 \(-0.478608\pi\)
0.0671534 + 0.997743i \(0.478608\pi\)
\(888\) 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 \(-0.801469\pi\)
−0.811721 + 0.584045i \(0.801469\pi\)
\(912\) 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 \(-0.285763\pi\)
0.623370 + 0.781927i \(0.285763\pi\)
\(930\) 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 \(-0.447884\pi\)
0.162995 + 0.986627i \(0.447884\pi\)
\(942\) 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 \(-0.627507\pi\)
−0.389948 + 0.920837i \(0.627507\pi\)
\(948\) 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 \(-0.161114\pi\)
0.874616 + 0.484817i \(0.161114\pi\)
\(954\) 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 \(-0.642628\pi\)
−0.433236 + 0.901281i \(0.642628\pi\)
\(972\) 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 \(-0.755784\pi\)
−0.719839 + 0.694141i \(0.755784\pi\)
\(978\) 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 \(-0.777499\pi\)
−0.765481 + 0.643458i \(0.777499\pi\)
\(984\) 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.g.1.1 yes 1
3.2 odd 2 1305.2.a.a.1.1 1
5.4 even 2 6525.2.a.a.1.1 1
15.14 odd 2 6525.2.a.m.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1305.2.a.a.1.1 1 3.2 odd 2
1305.2.a.g.1.1 yes 1 1.1 even 1 trivial
6525.2.a.a.1.1 1 5.4 even 2
6525.2.a.m.1.1 1 15.14 odd 2