↑

Properties

Label 2646.2.a.y.1.1
Level $2646$
Weight $2$
Character 2646.1
Self dual yes
Analytic conductor $21.128$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2646,2,Mod(1,2646)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2646, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("2646.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2646 = 2 \cdot 3^{3} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2646.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: \(21.1284163748\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 378)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 2646.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{8} -5.00000 q^{13} +1.00000 q^{16} -3.00000 q^{17} -2.00000 q^{19} -9.00000 q^{23} -5.00000 q^{25} -5.00000 q^{26} -3.00000 q^{29} -5.00000 q^{31} +1.00000 q^{32} -3.00000 q^{34} +2.00000 q^{37} -2.00000 q^{38} +6.00000 q^{41} -1.00000 q^{43} -9.00000 q^{46} +6.00000 q^{47} -5.00000 q^{50} -5.00000 q^{52} +3.00000 q^{53} -3.00000 q^{58} +3.00000 q^{59} +10.0000 q^{61} -5.00000 q^{62} +1.00000 q^{64} -13.0000 q^{67} -3.00000 q^{68} +9.00000 q^{71} -2.00000 q^{73} +2.00000 q^{74} -2.00000 q^{76} -10.0000 q^{79} +6.00000 q^{82} +12.0000 q^{83} -1.00000 q^{86} -15.0000 q^{89} -9.00000 q^{92} +6.00000 q^{94} -8.00000 q^{97} +O(q^{100})\)

Display 100 coefficients 10 coefficients

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) 0 0
\(11\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) −5.00000 −1.38675 −0.693375 − 0.720577i \(-0.743877\pi\)
−0.693375 + 0.720577i \(0.743877\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −3.00000 −0.727607 −0.363803 − 0.931476i \(-0.618522\pi\)
−0.363803 + 0.931476i \(0.618522\pi\)
\(18\) 0 0
\(19\) −2.00000 −0.458831 −0.229416 − 0.973329i \(-0.573682\pi\)
−0.229416 + 0.973329i \(0.573682\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −9.00000 −1.87663 −0.938315 − 0.345782i \(-0.887614\pi\)
−0.938315 + 0.345782i \(0.887614\pi\)
\(24\) 0 0
\(25\) −5.00000 −1.00000
\(26\) −5.00000 −0.980581
\(27\) 0 0
\(28\) 0 0
\(29\) −3.00000 −0.557086 −0.278543 − 0.960424i \(-0.589851\pi\)
−0.278543 + 0.960424i \(0.589851\pi\)
\(30\) 0 0
\(31\) −5.00000 −0.898027 −0.449013 − 0.893525i \(-0.648224\pi\)
−0.449013 + 0.893525i \(0.648224\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) −3.00000 −0.514496
\(35\) 0 0
\(36\) 0 0
\(37\) 2.00000 0.328798 0.164399 − 0.986394i \(-0.447432\pi\)
0.164399 + 0.986394i \(0.447432\pi\)
\(38\) −2.00000 −0.324443
\(39\) 0 0
\(40\) 0 0
\(41\) 6.00000 0.937043 0.468521 − 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) 0 0
\(43\) −1.00000 −0.152499 −0.0762493 − 0.997089i \(-0.524294\pi\)
−0.0762493 + 0.997089i \(0.524294\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) −9.00000 −1.32698
\(47\) 6.00000 0.875190 0.437595 − 0.899172i \(-0.355830\pi\)
0.437595 + 0.899172i \(0.355830\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) −5.00000 −0.707107
\(51\) 0 0
\(52\) −5.00000 −0.693375
\(53\) 3.00000 0.412082 0.206041 − 0.978543i \(-0.433942\pi\)
0.206041 + 0.978543i \(0.433942\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) −3.00000 −0.393919
\(59\) 3.00000 0.390567 0.195283 − 0.980747i \(-0.437437\pi\)
0.195283 + 0.980747i \(0.437437\pi\)
\(60\) 0 0
\(61\) 10.0000 1.28037 0.640184 − 0.768221i \(-0.278858\pi\)
0.640184 + 0.768221i \(0.278858\pi\)
\(62\) −5.00000 −0.635001
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) −13.0000 −1.58820 −0.794101 − 0.607785i \(-0.792058\pi\)
−0.794101 + 0.607785i \(0.792058\pi\)
\(68\) −3.00000 −0.363803
\(69\) 0 0
\(70\) 0 0
\(71\) 9.00000 1.06810 0.534052 − 0.845452i \(-0.320669\pi\)
0.534052 + 0.845452i \(0.320669\pi\)
\(72\) 0 0
\(73\) −2.00000 −0.234082 −0.117041 − 0.993127i \(-0.537341\pi\)
−0.117041 + 0.993127i \(0.537341\pi\)
\(74\) 2.00000 0.232495
\(75\) 0 0
\(76\) −2.00000 −0.229416
\(77\) 0 0
\(78\) 0 0
\(79\) −10.0000 −1.12509 −0.562544 − 0.826767i \(-0.690177\pi\)
−0.562544 + 0.826767i \(0.690177\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 6.00000 0.662589
\(83\) 12.0000 1.31717 0.658586 − 0.752506i \(-0.271155\pi\)
0.658586 + 0.752506i \(0.271155\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −1.00000 −0.107833
\(87\) 0 0
\(88\) 0 0
\(89\) −15.0000 −1.59000 −0.794998 − 0.606612i \(-0.792528\pi\)
−0.794998 + 0.606612i \(0.792528\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −9.00000 −0.938315
\(93\) 0 0
\(94\) 6.00000 0.618853
\(95\) 0 0
\(96\) 0 0
\(97\) −8.00000 −0.812277 −0.406138 − 0.913812i \(-0.633125\pi\)
−0.406138 + 0.913812i \(0.633125\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −5.00000 −0.500000
\(101\) −18.0000 −1.79107 −0.895533 − 0.444994i \(-0.853206\pi\)
−0.895533 + 0.444994i \(0.853206\pi\)
\(102\) 0 0
\(103\) 13.0000 1.28093 0.640464 − 0.767988i \(-0.278742\pi\)
0.640464 + 0.767988i \(0.278742\pi\)
\(104\) −5.00000 −0.490290
\(105\) 0 0
\(106\) 3.00000 0.291386
\(107\) −12.0000 −1.16008 −0.580042 − 0.814587i \(-0.696964\pi\)
−0.580042 + 0.814587i \(0.696964\pi\)
\(108\) 0 0
\(109\) −16.0000 −1.53252 −0.766261 − 0.642529i \(-0.777885\pi\)
−0.766261 + 0.642529i \(0.777885\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 12.0000 1.12887 0.564433 − 0.825479i \(-0.309095\pi\)
0.564433 + 0.825479i \(0.309095\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −3.00000 −0.278543
\(117\) 0 0
\(118\) 3.00000 0.276172
\(119\) 0 0
\(120\) 0 0
\(121\) −11.0000 −1.00000
\(122\) 10.0000 0.905357
\(123\) 0 0
\(124\) −5.00000 −0.449013
\(125\) 0 0
\(126\) 0 0
\(127\) 20.0000 1.77471 0.887357 − 0.461084i \(-0.152539\pi\)
0.887357 + 0.461084i \(0.152539\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) 0 0
\(131\) 9.00000 0.786334 0.393167 − 0.919467i \(-0.371379\pi\)
0.393167 + 0.919467i \(0.371379\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) −13.0000 −1.12303
\(135\) 0 0
\(136\) −3.00000 −0.257248
\(137\) 18.0000 1.53784 0.768922 − 0.639343i \(-0.220793\pi\)
0.768922 + 0.639343i \(0.220793\pi\)
\(138\) 0 0
\(139\) −14.0000 −1.18746 −0.593732 − 0.804663i \(-0.702346\pi\)
−0.593732 + 0.804663i \(0.702346\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 9.00000 0.755263
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) −2.00000 −0.165521
\(147\) 0 0
\(148\) 2.00000 0.164399
\(149\) 9.00000 0.737309 0.368654 − 0.929567i \(-0.379819\pi\)
0.368654 + 0.929567i \(0.379819\pi\)
\(150\) 0 0
\(151\) −10.0000 −0.813788 −0.406894 − 0.913475i \(-0.633388\pi\)
−0.406894 + 0.913475i \(0.633388\pi\)
\(152\) −2.00000 −0.162221
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −23.0000 −1.83560 −0.917800 − 0.397043i \(-0.870036\pi\)
−0.917800 + 0.397043i \(0.870036\pi\)
\(158\) −10.0000 −0.795557
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 11.0000 0.861586 0.430793 − 0.902451i \(-0.358234\pi\)
0.430793 + 0.902451i \(0.358234\pi\)
\(164\) 6.00000 0.468521
\(165\) 0 0
\(166\) 12.0000 0.931381
\(167\) −12.0000 −0.928588 −0.464294 − 0.885681i \(-0.653692\pi\)
−0.464294 + 0.885681i \(0.653692\pi\)
\(168\) 0 0
\(169\) 12.0000 0.923077
\(170\) 0 0
\(171\) 0 0
\(172\) −1.00000 −0.0762493
\(173\) 24.0000 1.82469 0.912343 − 0.409426i \(-0.134271\pi\)
0.912343 + 0.409426i \(0.134271\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) −15.0000 −1.12430
\(179\) −24.0000 −1.79384 −0.896922 − 0.442189i \(-0.854202\pi\)
−0.896922 + 0.442189i \(0.854202\pi\)
\(180\) 0 0
\(181\) 7.00000 0.520306 0.260153 − 0.965567i \(-0.416227\pi\)
0.260153 + 0.965567i \(0.416227\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −9.00000 −0.663489
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 6.00000 0.437595
\(189\) 0 0
\(190\) 0 0
\(191\) 12.0000 0.868290 0.434145 − 0.900843i \(-0.357051\pi\)
0.434145 + 0.900843i \(0.357051\pi\)
\(192\) 0 0
\(193\) 5.00000 0.359908 0.179954 − 0.983675i \(-0.442405\pi\)
0.179954 + 0.983675i \(0.442405\pi\)
\(194\) −8.00000 −0.574367
\(195\) 0 0
\(196\) 0 0
\(197\) 18.0000 1.28245 0.641223 − 0.767354i \(-0.278427\pi\)
0.641223 + 0.767354i \(0.278427\pi\)
\(198\) 0 0
\(199\) −11.0000 −0.779769 −0.389885 − 0.920864i \(-0.627485\pi\)
−0.389885 + 0.920864i \(0.627485\pi\)
\(200\) −5.00000 −0.353553
\(201\) 0 0
\(202\) −18.0000 −1.26648
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 13.0000 0.905753
\(207\) 0 0
\(208\) −5.00000 −0.346688
\(209\) 0 0
\(210\) 0 0
\(211\) −13.0000 −0.894957 −0.447478 − 0.894295i \(-0.647678\pi\)
−0.447478 + 0.894295i \(0.647678\pi\)
\(212\) 3.00000 0.206041
\(213\) 0 0
\(214\) −12.0000 −0.820303
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) −16.0000 −1.08366
\(219\) 0 0
\(220\) 0 0
\(221\) 15.0000 1.00901
\(222\) 0 0
\(223\) −8.00000 −0.535720 −0.267860 − 0.963458i \(-0.586316\pi\)
−0.267860 + 0.963458i \(0.586316\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 12.0000 0.798228
\(227\) −9.00000 −0.597351 −0.298675 − 0.954355i \(-0.596545\pi\)
−0.298675 + 0.954355i \(0.596545\pi\)
\(228\) 0 0
\(229\) −14.0000 −0.925146 −0.462573 − 0.886581i \(-0.653074\pi\)
−0.462573 + 0.886581i \(0.653074\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −3.00000 −0.196960
\(233\) −12.0000 −0.786146 −0.393073 − 0.919507i \(-0.628588\pi\)
−0.393073 + 0.919507i \(0.628588\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 3.00000 0.195283
\(237\) 0 0
\(238\) 0 0
\(239\) 12.0000 0.776215 0.388108 − 0.921614i \(-0.373129\pi\)
0.388108 + 0.921614i \(0.373129\pi\)
\(240\) 0 0
\(241\) 10.0000 0.644157 0.322078 − 0.946713i \(-0.395619\pi\)
0.322078 + 0.946713i \(0.395619\pi\)
\(242\) −11.0000 −0.707107
\(243\) 0 0
\(244\) 10.0000 0.640184
\(245\) 0 0
\(246\) 0 0
\(247\) 10.0000 0.636285
\(248\) −5.00000 −0.317500
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 20.0000 1.25491
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −18.0000 −1.12281 −0.561405 − 0.827541i \(-0.689739\pi\)
−0.561405 + 0.827541i \(0.689739\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 9.00000 0.556022
\(263\) 9.00000 0.554964 0.277482 − 0.960731i \(-0.410500\pi\)
0.277482 + 0.960731i \(0.410500\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) −13.0000 −0.794101
\(269\) 24.0000 1.46331 0.731653 − 0.681677i \(-0.238749\pi\)
0.731653 + 0.681677i \(0.238749\pi\)
\(270\) 0 0
\(271\) 7.00000 0.425220 0.212610 − 0.977137i \(-0.431804\pi\)
0.212610 + 0.977137i \(0.431804\pi\)
\(272\) −3.00000 −0.181902
\(273\) 0 0
\(274\) 18.0000 1.08742
\(275\) 0 0
\(276\) 0 0
\(277\) 8.00000 0.480673 0.240337 − 0.970690i \(-0.422742\pi\)
0.240337 + 0.970690i \(0.422742\pi\)
\(278\) −14.0000 −0.839664
\(279\) 0 0
\(280\) 0 0
\(281\) 6.00000 0.357930 0.178965 − 0.983855i \(-0.442725\pi\)
0.178965 + 0.983855i \(0.442725\pi\)
\(282\) 0 0
\(283\) 4.00000 0.237775 0.118888 − 0.992908i \(-0.462067\pi\)
0.118888 + 0.992908i \(0.462067\pi\)
\(284\) 9.00000 0.534052
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −8.00000 −0.470588
\(290\) 0 0
\(291\) 0 0
\(292\) −2.00000 −0.117041
\(293\) 6.00000 0.350524 0.175262 − 0.984522i \(-0.443923\pi\)
0.175262 + 0.984522i \(0.443923\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 2.00000 0.116248
\(297\) 0 0
\(298\) 9.00000 0.521356
\(299\) 45.0000 2.60242
\(300\) 0 0
\(301\) 0 0
\(302\) −10.0000 −0.575435
\(303\) 0 0
\(304\) −2.00000 −0.114708
\(305\) 0 0
\(306\) 0 0
\(307\) −2.00000 −0.114146 −0.0570730 − 0.998370i \(-0.518177\pi\)
−0.0570730 + 0.998370i \(0.518177\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −24.0000 −1.36092 −0.680458 − 0.732787i \(-0.738219\pi\)
−0.680458 + 0.732787i \(0.738219\pi\)
\(312\) 0 0
\(313\) 10.0000 0.565233 0.282617 − 0.959233i \(-0.408798\pi\)
0.282617 + 0.959233i \(0.408798\pi\)
\(314\) −23.0000 −1.29797
\(315\) 0 0
\(316\) −10.0000 −0.562544
\(317\) 30.0000 1.68497 0.842484 − 0.538721i \(-0.181092\pi\)
0.842484 + 0.538721i \(0.181092\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 6.00000 0.333849
\(324\) 0 0
\(325\) 25.0000 1.38675
\(326\) 11.0000 0.609234
\(327\) 0 0
\(328\) 6.00000 0.331295
\(329\) 0 0
\(330\) 0 0
\(331\) −19.0000 −1.04433 −0.522167 − 0.852843i \(-0.674876\pi\)
−0.522167 + 0.852843i \(0.674876\pi\)
\(332\) 12.0000 0.658586
\(333\) 0 0
\(334\) −12.0000 −0.656611
\(335\) 0 0
\(336\) 0 0
\(337\) −13.0000 −0.708155 −0.354078 − 0.935216i \(-0.615205\pi\)
−0.354078 + 0.935216i \(0.615205\pi\)
\(338\) 12.0000 0.652714
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) −1.00000 −0.0539164
\(345\) 0 0
\(346\) 24.0000 1.29025
\(347\) −30.0000 −1.61048 −0.805242 − 0.592946i \(-0.797965\pi\)
−0.805242 + 0.592946i \(0.797965\pi\)
\(348\) 0 0
\(349\) 1.00000 0.0535288 0.0267644 − 0.999642i \(-0.491480\pi\)
0.0267644 + 0.999642i \(0.491480\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 27.0000 1.43706 0.718532 − 0.695493i \(-0.244814\pi\)
0.718532 + 0.695493i \(0.244814\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −15.0000 −0.794998
\(357\) 0 0
\(358\) −24.0000 −1.26844
\(359\) −3.00000 −0.158334 −0.0791670 − 0.996861i \(-0.525226\pi\)
−0.0791670 + 0.996861i \(0.525226\pi\)
\(360\) 0 0
\(361\) −15.0000 −0.789474
\(362\) 7.00000 0.367912
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 19.0000 0.991792 0.495896 − 0.868382i \(-0.334840\pi\)
0.495896 + 0.868382i \(0.334840\pi\)
\(368\) −9.00000 −0.469157
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 14.0000 0.724893 0.362446 − 0.932005i \(-0.381942\pi\)
0.362446 + 0.932005i \(0.381942\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 6.00000 0.309426
\(377\) 15.0000 0.772539
\(378\) 0 0
\(379\) −16.0000 −0.821865 −0.410932 − 0.911666i \(-0.634797\pi\)
−0.410932 + 0.911666i \(0.634797\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 12.0000 0.613973
\(383\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 5.00000 0.254493
\(387\) 0 0
\(388\) −8.00000 −0.406138
\(389\) 18.0000 0.912636 0.456318 − 0.889817i \(-0.349168\pi\)
0.456318 + 0.889817i \(0.349168\pi\)
\(390\) 0 0
\(391\) 27.0000 1.36545
\(392\) 0 0
\(393\) 0 0
\(394\) 18.0000 0.906827
\(395\) 0 0
\(396\) 0 0
\(397\) −2.00000 −0.100377 −0.0501886 − 0.998740i \(-0.515982\pi\)
−0.0501886 + 0.998740i \(0.515982\pi\)
\(398\) −11.0000 −0.551380
\(399\) 0 0
\(400\) −5.00000 −0.250000
\(401\) −18.0000 −0.898877 −0.449439 − 0.893311i \(-0.648376\pi\)
−0.449439 + 0.893311i \(0.648376\pi\)
\(402\) 0 0
\(403\) 25.0000 1.24534
\(404\) −18.0000 −0.895533
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −14.0000 −0.692255 −0.346128 − 0.938187i \(-0.612504\pi\)
−0.346128 + 0.938187i \(0.612504\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 13.0000 0.640464
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) −5.00000 −0.245145
\(417\) 0 0
\(418\) 0 0
\(419\) 15.0000 0.732798 0.366399 − 0.930458i \(-0.380591\pi\)
0.366399 + 0.930458i \(0.380591\pi\)
\(420\) 0 0
\(421\) −10.0000 −0.487370 −0.243685 − 0.969854i \(-0.578356\pi\)
−0.243685 + 0.969854i \(0.578356\pi\)
\(422\) −13.0000 −0.632830
\(423\) 0 0
\(424\) 3.00000 0.145693
\(425\) 15.0000 0.727607
\(426\) 0 0
\(427\) 0 0
\(428\) −12.0000 −0.580042
\(429\) 0 0
\(430\) 0 0
\(431\) −24.0000 −1.15604 −0.578020 − 0.816023i \(-0.696174\pi\)
−0.578020 + 0.816023i \(0.696174\pi\)
\(432\) 0 0
\(433\) 16.0000 0.768911 0.384455 − 0.923144i \(-0.374389\pi\)
0.384455 + 0.923144i \(0.374389\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −16.0000 −0.766261
\(437\) 18.0000 0.861057
\(438\) 0 0
\(439\) −35.0000 −1.67046 −0.835229 − 0.549902i \(-0.814665\pi\)
−0.835229 + 0.549902i \(0.814665\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 15.0000 0.713477
\(443\) −6.00000 −0.285069 −0.142534 − 0.989790i \(-0.545525\pi\)
−0.142534 + 0.989790i \(0.545525\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −8.00000 −0.378811
\(447\) 0 0
\(448\) 0 0
\(449\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 12.0000 0.564433
\(453\) 0 0
\(454\) −9.00000 −0.422391
\(455\) 0 0
\(456\) 0 0
\(457\) 35.0000 1.63723 0.818615 − 0.574342i \(-0.194742\pi\)
0.818615 + 0.574342i \(0.194742\pi\)
\(458\) −14.0000 −0.654177
\(459\) 0 0
\(460\) 0 0
\(461\) 30.0000 1.39724 0.698620 − 0.715493i \(-0.253798\pi\)
0.698620 + 0.715493i \(0.253798\pi\)
\(462\) 0 0
\(463\) −22.0000 −1.02243 −0.511213 − 0.859454i \(-0.670804\pi\)
−0.511213 + 0.859454i \(0.670804\pi\)
\(464\) −3.00000 −0.139272
\(465\) 0 0
\(466\) −12.0000 −0.555889
\(467\) −24.0000 −1.11059 −0.555294 − 0.831654i \(-0.687394\pi\)
−0.555294 + 0.831654i \(0.687394\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 3.00000 0.138086
\(473\) 0 0
\(474\) 0 0
\(475\) 10.0000 0.458831
\(476\) 0 0
\(477\) 0 0
\(478\) 12.0000 0.548867
\(479\) −36.0000 −1.64488 −0.822441 − 0.568850i \(-0.807388\pi\)
−0.822441 + 0.568850i \(0.807388\pi\)
\(480\) 0 0
\(481\) −10.0000 −0.455961
\(482\) 10.0000 0.455488
\(483\) 0 0
\(484\) −11.0000 −0.500000
\(485\) 0 0
\(486\) 0 0
\(487\) 38.0000 1.72194 0.860972 − 0.508652i \(-0.169856\pi\)
0.860972 + 0.508652i \(0.169856\pi\)
\(488\) 10.0000 0.452679
\(489\) 0 0
\(490\) 0 0
\(491\) 12.0000 0.541552 0.270776 − 0.962642i \(-0.412720\pi\)
0.270776 + 0.962642i \(0.412720\pi\)
\(492\) 0 0
\(493\) 9.00000 0.405340
\(494\) 10.0000 0.449921
\(495\) 0 0
\(496\) −5.00000 −0.224507
\(497\) 0 0
\(498\) 0 0
\(499\) −4.00000 −0.179065 −0.0895323 − 0.995984i \(-0.528537\pi\)
−0.0895323 + 0.995984i \(0.528537\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −36.0000 −1.60516 −0.802580 − 0.596544i \(-0.796540\pi\)
−0.802580 + 0.596544i \(0.796540\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) 20.0000 0.887357
\(509\) 18.0000 0.797836 0.398918 − 0.916987i \(-0.369386\pi\)
0.398918 + 0.916987i \(0.369386\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) −18.0000 −0.793946
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −3.00000 −0.131432 −0.0657162 − 0.997838i \(-0.520933\pi\)
−0.0657162 + 0.997838i \(0.520933\pi\)
\(522\) 0 0
\(523\) −20.0000 −0.874539 −0.437269 − 0.899331i \(-0.644054\pi\)
−0.437269 + 0.899331i \(0.644054\pi\)
\(524\) 9.00000 0.393167
\(525\) 0 0
\(526\) 9.00000 0.392419
\(527\) 15.0000 0.653410
\(528\) 0 0
\(529\) 58.0000 2.52174
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −30.0000 −1.29944
\(534\) 0 0
\(535\) 0 0
\(536\) −13.0000 −0.561514
\(537\) 0 0
\(538\) 24.0000 1.03471
\(539\) 0 0
\(540\) 0 0
\(541\) −16.0000 −0.687894 −0.343947 − 0.938989i \(-0.611764\pi\)
−0.343947 + 0.938989i \(0.611764\pi\)
\(542\) 7.00000 0.300676
\(543\) 0 0
\(544\) −3.00000 −0.128624
\(545\) 0 0
\(546\) 0 0
\(547\) 44.0000 1.88130 0.940652 − 0.339372i \(-0.110215\pi\)
0.940652 + 0.339372i \(0.110215\pi\)
\(548\) 18.0000 0.768922
\(549\) 0 0
\(550\) 0 0
\(551\) 6.00000 0.255609
\(552\) 0 0
\(553\) 0 0
\(554\) 8.00000 0.339887
\(555\) 0 0
\(556\) −14.0000 −0.593732
\(557\) 21.0000 0.889799 0.444899 − 0.895581i \(-0.353239\pi\)
0.444899 + 0.895581i \(0.353239\pi\)
\(558\) 0 0
\(559\) 5.00000 0.211477
\(560\) 0 0
\(561\) 0 0
\(562\) 6.00000 0.253095
\(563\) 3.00000 0.126435 0.0632175 − 0.998000i \(-0.479864\pi\)
0.0632175 + 0.998000i \(0.479864\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 4.00000 0.168133
\(567\) 0 0
\(568\) 9.00000 0.377632
\(569\) −24.0000 −1.00613 −0.503066 − 0.864248i \(-0.667795\pi\)
−0.503066 + 0.864248i \(0.667795\pi\)
\(570\) 0 0
\(571\) 5.00000 0.209243 0.104622 − 0.994512i \(-0.466637\pi\)
0.104622 + 0.994512i \(0.466637\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 45.0000 1.87663
\(576\) 0 0
\(577\) 34.0000 1.41544 0.707719 − 0.706494i \(-0.249724\pi\)
0.707719 + 0.706494i \(0.249724\pi\)
\(578\) −8.00000 −0.332756
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) −2.00000 −0.0827606
\(585\) 0 0
\(586\) 6.00000 0.247858
\(587\) −33.0000 −1.36206 −0.681028 − 0.732257i \(-0.738467\pi\)
−0.681028 + 0.732257i \(0.738467\pi\)
\(588\) 0 0
\(589\) 10.0000 0.412043
\(590\) 0 0
\(591\) 0 0
\(592\) 2.00000 0.0821995
\(593\) −42.0000 −1.72473 −0.862367 − 0.506284i \(-0.831019\pi\)
−0.862367 + 0.506284i \(0.831019\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 9.00000 0.368654
\(597\) 0 0
\(598\) 45.0000 1.84019
\(599\) −3.00000 −0.122577 −0.0612883 − 0.998120i \(-0.519521\pi\)
−0.0612883 + 0.998120i \(0.519521\pi\)
\(600\) 0 0
\(601\) −8.00000 −0.326327 −0.163163 − 0.986599i \(-0.552170\pi\)
−0.163163 + 0.986599i \(0.552170\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −10.0000 −0.406894
\(605\) 0 0
\(606\) 0 0
\(607\) −5.00000 −0.202944 −0.101472 − 0.994838i \(-0.532355\pi\)
−0.101472 + 0.994838i \(0.532355\pi\)
\(608\) −2.00000 −0.0811107
\(609\) 0 0
\(610\) 0 0
\(611\) −30.0000 −1.21367
\(612\) 0 0
\(613\) 38.0000 1.53481 0.767403 − 0.641165i \(-0.221549\pi\)
0.767403 + 0.641165i \(0.221549\pi\)
\(614\) −2.00000 −0.0807134
\(615\) 0 0
\(616\) 0 0
\(617\) −42.0000 −1.69086 −0.845428 − 0.534089i \(-0.820655\pi\)
−0.845428 + 0.534089i \(0.820655\pi\)
\(618\) 0 0
\(619\) 10.0000 0.401934 0.200967 − 0.979598i \(-0.435592\pi\)
0.200967 + 0.979598i \(0.435592\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) −24.0000 −0.962312
\(623\) 0 0
\(624\) 0 0
\(625\) 25.0000 1.00000
\(626\) 10.0000 0.399680
\(627\) 0 0
\(628\) −23.0000 −0.917800
\(629\) −6.00000 −0.239236
\(630\) 0 0
\(631\) −16.0000 −0.636950 −0.318475 − 0.947931i \(-0.603171\pi\)
−0.318475 + 0.947931i \(0.603171\pi\)
\(632\) −10.0000 −0.397779
\(633\) 0 0
\(634\) 30.0000 1.19145
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 36.0000 1.42191 0.710957 − 0.703235i \(-0.248262\pi\)
0.710957 + 0.703235i \(0.248262\pi\)
\(642\) 0 0
\(643\) 40.0000 1.57745 0.788723 − 0.614749i \(-0.210743\pi\)
0.788723 + 0.614749i \(0.210743\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 6.00000 0.236067
\(647\) −30.0000 −1.17942 −0.589711 − 0.807614i \(-0.700758\pi\)
−0.589711 + 0.807614i \(0.700758\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 25.0000 0.980581
\(651\) 0 0
\(652\) 11.0000 0.430793
\(653\) −9.00000 −0.352197 −0.176099 − 0.984373i \(-0.556348\pi\)
−0.176099 + 0.984373i \(0.556348\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 6.00000 0.234261
\(657\) 0 0
\(658\) 0 0
\(659\) −12.0000 −0.467454 −0.233727 − 0.972302i \(-0.575092\pi\)
−0.233727 + 0.972302i \(0.575092\pi\)
\(660\) 0 0
\(661\) −14.0000 −0.544537 −0.272268 − 0.962221i \(-0.587774\pi\)
−0.272268 + 0.962221i \(0.587774\pi\)
\(662\) −19.0000 −0.738456
\(663\) 0 0
\(664\) 12.0000 0.465690
\(665\) 0 0
\(666\) 0 0
\(667\) 27.0000 1.04544
\(668\) −12.0000 −0.464294
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −1.00000 −0.0385472 −0.0192736 − 0.999814i \(-0.506135\pi\)
−0.0192736 + 0.999814i \(0.506135\pi\)
\(674\) −13.0000 −0.500741
\(675\) 0 0
\(676\) 12.0000 0.461538
\(677\) −12.0000 −0.461197 −0.230599 − 0.973049i \(-0.574068\pi\)
−0.230599 + 0.973049i \(0.574068\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 12.0000 0.459167 0.229584 − 0.973289i \(-0.426264\pi\)
0.229584 + 0.973289i \(0.426264\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) −1.00000 −0.0381246
\(689\) −15.0000 −0.571454
\(690\) 0 0
\(691\) 28.0000 1.06517 0.532585 − 0.846376i \(-0.321221\pi\)
0.532585 + 0.846376i \(0.321221\pi\)
\(692\) 24.0000 0.912343
\(693\) 0 0
\(694\) −30.0000 −1.13878
\(695\) 0 0
\(696\) 0 0
\(697\) −18.0000 −0.681799
\(698\) 1.00000 0.0378506
\(699\) 0 0
\(700\) 0 0
\(701\) −18.0000 −0.679851 −0.339925 − 0.940452i \(-0.610402\pi\)
−0.339925 + 0.940452i \(0.610402\pi\)
\(702\) 0 0
\(703\) −4.00000 −0.150863
\(704\) 0 0
\(705\) 0 0
\(706\) 27.0000 1.01616
\(707\) 0 0
\(708\) 0 0
\(709\) −28.0000 −1.05156 −0.525781 − 0.850620i \(-0.676227\pi\)
−0.525781 + 0.850620i \(0.676227\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −15.0000 −0.562149
\(713\) 45.0000 1.68526
\(714\) 0 0
\(715\) 0 0
\(716\) −24.0000 −0.896922
\(717\) 0 0
\(718\) −3.00000 −0.111959
\(719\) −24.0000 −0.895049 −0.447524 − 0.894272i \(-0.647694\pi\)
−0.447524 + 0.894272i \(0.647694\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −15.0000 −0.558242
\(723\) 0 0
\(724\) 7.00000 0.260153
\(725\) 15.0000 0.557086
\(726\) 0 0
\(727\) −17.0000 −0.630495 −0.315248 − 0.949009i \(-0.602088\pi\)
−0.315248 + 0.949009i \(0.602088\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 3.00000 0.110959
\(732\) 0 0
\(733\) −23.0000 −0.849524 −0.424762 − 0.905305i \(-0.639642\pi\)
−0.424762 + 0.905305i \(0.639642\pi\)
\(734\) 19.0000 0.701303
\(735\) 0 0
\(736\) −9.00000 −0.331744
\(737\) 0 0
\(738\) 0 0
\(739\) 20.0000 0.735712 0.367856 − 0.929883i \(-0.380092\pi\)
0.367856 + 0.929883i \(0.380092\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 3.00000 0.110059 0.0550297 − 0.998485i \(-0.482475\pi\)
0.0550297 + 0.998485i \(0.482475\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 14.0000 0.512576
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 14.0000 0.510867 0.255434 − 0.966827i \(-0.417782\pi\)
0.255434 + 0.966827i \(0.417782\pi\)
\(752\) 6.00000 0.218797
\(753\) 0 0
\(754\) 15.0000 0.546268
\(755\) 0 0
\(756\) 0 0
\(757\) −16.0000 −0.581530 −0.290765 − 0.956795i \(-0.593910\pi\)
−0.290765 + 0.956795i \(0.593910\pi\)
\(758\) −16.0000 −0.581146
\(759\) 0 0
\(760\) 0 0
\(761\) −9.00000 −0.326250 −0.163125 − 0.986605i \(-0.552157\pi\)
−0.163125 + 0.986605i \(0.552157\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 12.0000 0.434145
\(765\) 0 0
\(766\) 0 0
\(767\) −15.0000 −0.541619
\(768\) 0 0
\(769\) 40.0000 1.44244 0.721218 − 0.692708i \(-0.243582\pi\)
0.721218 + 0.692708i \(0.243582\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 5.00000 0.179954
\(773\) 24.0000 0.863220 0.431610 − 0.902060i \(-0.357946\pi\)
0.431610 + 0.902060i \(0.357946\pi\)
\(774\) 0 0
\(775\) 25.0000 0.898027
\(776\) −8.00000 −0.287183
\(777\) 0 0
\(778\) 18.0000 0.645331
\(779\) −12.0000 −0.429945
\(780\) 0 0
\(781\) 0 0
\(782\) 27.0000 0.965518
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −14.0000 −0.499046 −0.249523 − 0.968369i \(-0.580274\pi\)
−0.249523 + 0.968369i \(0.580274\pi\)
\(788\) 18.0000 0.641223
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −50.0000 −1.77555
\(794\) −2.00000 −0.0709773
\(795\) 0 0
\(796\) −11.0000 −0.389885
\(797\) 6.00000 0.212531 0.106265 − 0.994338i \(-0.466111\pi\)
0.106265 + 0.994338i \(0.466111\pi\)
\(798\) 0 0
\(799\) −18.0000 −0.636794
\(800\) −5.00000 −0.176777
\(801\) 0 0
\(802\) −18.0000 −0.635602
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 25.0000 0.880587
\(807\) 0 0
\(808\) −18.0000 −0.633238
\(809\) −24.0000 −0.843795 −0.421898 − 0.906644i \(-0.638636\pi\)
−0.421898 + 0.906644i \(0.638636\pi\)
\(810\) 0 0
\(811\) −2.00000 −0.0702295 −0.0351147 − 0.999383i \(-0.511180\pi\)
−0.0351147 + 0.999383i \(0.511180\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 2.00000 0.0699711
\(818\) −14.0000 −0.489499
\(819\) 0 0
\(820\) 0 0
\(821\) 39.0000 1.36111 0.680555 − 0.732697i \(-0.261739\pi\)
0.680555 + 0.732697i \(0.261739\pi\)
\(822\) 0 0
\(823\) −22.0000 −0.766872 −0.383436 − 0.923567i \(-0.625259\pi\)
−0.383436 + 0.923567i \(0.625259\pi\)
\(824\) 13.0000 0.452876
\(825\) 0 0
\(826\) 0 0
\(827\) −18.0000 −0.625921 −0.312961 − 0.949766i \(-0.601321\pi\)
−0.312961 + 0.949766i \(0.601321\pi\)
\(828\) 0 0
\(829\) −2.00000 −0.0694629 −0.0347314 − 0.999397i \(-0.511058\pi\)
−0.0347314 + 0.999397i \(0.511058\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −5.00000 −0.173344
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 15.0000 0.518166
\(839\) −42.0000 −1.45000 −0.725001 − 0.688748i \(-0.758161\pi\)
−0.725001 + 0.688748i \(0.758161\pi\)
\(840\) 0 0
\(841\) −20.0000 −0.689655
\(842\) −10.0000 −0.344623
\(843\) 0 0
\(844\) −13.0000 −0.447478
\(845\) 0 0
\(846\) 0 0
\(847\) 0 0
\(848\) 3.00000 0.103020
\(849\) 0 0
\(850\) 15.0000 0.514496
\(851\) −18.0000 −0.617032
\(852\) 0 0
\(853\) 37.0000 1.26686 0.633428 − 0.773802i \(-0.281647\pi\)
0.633428 + 0.773802i \(0.281647\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −12.0000 −0.410152
\(857\) −9.00000 −0.307434 −0.153717 − 0.988115i \(-0.549124\pi\)
−0.153717 + 0.988115i \(0.549124\pi\)
\(858\) 0 0
\(859\) −14.0000 −0.477674 −0.238837 − 0.971060i \(-0.576766\pi\)
−0.238837 + 0.971060i \(0.576766\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −24.0000 −0.817443
\(863\) 33.0000 1.12333 0.561667 − 0.827364i \(-0.310160\pi\)
0.561667 + 0.827364i \(0.310160\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 16.0000 0.543702
\(867\) 0 0
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 65.0000 2.20244
\(872\) −16.0000 −0.541828
\(873\) 0 0
\(874\) 18.0000 0.608859
\(875\) 0 0
\(876\) 0 0
\(877\) 32.0000 1.08056 0.540282 − 0.841484i \(-0.318318\pi\)
0.540282 + 0.841484i \(0.318318\pi\)
\(878\) −35.0000 −1.18119
\(879\) 0 0
\(880\) 0 0
\(881\) 27.0000 0.909653 0.454827 − 0.890580i \(-0.349701\pi\)
0.454827 + 0.890580i \(0.349701\pi\)
\(882\) 0 0
\(883\) −7.00000 −0.235569 −0.117784 − 0.993039i \(-0.537579\pi\)
−0.117784 + 0.993039i \(0.537579\pi\)
\(884\) 15.0000 0.504505
\(885\) 0 0
\(886\) −6.00000 −0.201574
\(887\) −36.0000 −1.20876 −0.604381 − 0.796696i \(-0.706579\pi\)
−0.604381 + 0.796696i \(0.706579\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) −8.00000 −0.267860
\(893\) −12.0000 −0.401565
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 15.0000 0.500278
\(900\) 0 0
\(901\) −9.00000 −0.299833
\(902\) 0 0
\(903\) 0 0
\(904\) 12.0000 0.399114
\(905\) 0 0
\(906\) 0 0
\(907\) −28.0000 −0.929725 −0.464862 − 0.885383i \(-0.653896\pi\)
−0.464862 + 0.885383i \(0.653896\pi\)
\(908\) −9.00000 −0.298675
\(909\) 0 0
\(910\) 0 0
\(911\) 24.0000 0.795155 0.397578 − 0.917568i \(-0.369851\pi\)
0.397578 + 0.917568i \(0.369851\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 35.0000 1.15770
\(915\) 0 0
\(916\) −14.0000 −0.462573
\(917\) 0 0
\(918\) 0 0
\(919\) 2.00000 0.0659739 0.0329870 − 0.999456i \(-0.489498\pi\)
0.0329870 + 0.999456i \(0.489498\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 30.0000 0.987997
\(923\) −45.0000 −1.48119
\(924\) 0 0
\(925\) −10.0000 −0.328798
\(926\) −22.0000 −0.722965
\(927\) 0 0
\(928\) −3.00000 −0.0984798
\(929\) −30.0000 −0.984268 −0.492134 − 0.870519i \(-0.663783\pi\)
−0.492134 + 0.870519i \(0.663783\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −12.0000 −0.393073
\(933\) 0 0
\(934\) −24.0000 −0.785304
\(935\) 0 0
\(936\) 0 0
\(937\) −56.0000 −1.82944 −0.914720 − 0.404088i \(-0.867589\pi\)
−0.914720 + 0.404088i \(0.867589\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 30.0000 0.977972 0.488986 − 0.872292i \(-0.337367\pi\)
0.488986 + 0.872292i \(0.337367\pi\)
\(942\) 0 0
\(943\) −54.0000 −1.75848
\(944\) 3.00000 0.0976417
\(945\) 0 0
\(946\) 0 0
\(947\) −6.00000 −0.194974 −0.0974869 − 0.995237i \(-0.531080\pi\)
−0.0974869 + 0.995237i \(0.531080\pi\)
\(948\) 0 0
\(949\) 10.0000 0.324614
\(950\) 10.0000 0.324443
\(951\) 0 0
\(952\) 0 0
\(953\) −36.0000 −1.16615 −0.583077 − 0.812417i \(-0.698151\pi\)
−0.583077 + 0.812417i \(0.698151\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 12.0000 0.388108
\(957\) 0 0
\(958\) −36.0000 −1.16311
\(959\) 0 0
\(960\) 0 0
\(961\) −6.00000 −0.193548
\(962\) −10.0000 −0.322413
\(963\) 0 0
\(964\) 10.0000 0.322078
\(965\) 0 0
\(966\) 0 0
\(967\) −4.00000 −0.128631 −0.0643157 − 0.997930i \(-0.520486\pi\)
−0.0643157 + 0.997930i \(0.520486\pi\)
\(968\) −11.0000 −0.353553
\(969\) 0 0
\(970\) 0 0
\(971\) 57.0000 1.82922 0.914609 − 0.404341i \(-0.132499\pi\)
0.914609 + 0.404341i \(0.132499\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 38.0000 1.21760
\(975\) 0 0
\(976\) 10.0000 0.320092
\(977\) 42.0000 1.34370 0.671850 − 0.740688i \(-0.265500\pi\)
0.671850 + 0.740688i \(0.265500\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 0 0
\(982\) 12.0000 0.382935
\(983\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 9.00000 0.286618
\(987\) 0 0
\(988\) 10.0000 0.318142
\(989\) 9.00000 0.286183
\(990\) 0 0
\(991\) 2.00000 0.0635321 0.0317660 − 0.999495i \(-0.489887\pi\)
0.0317660 + 0.999495i \(0.489887\pi\)
\(992\) −5.00000 −0.158750
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 19.0000 0.601736 0.300868 − 0.953666i \(-0.402724\pi\)
0.300868 + 0.953666i \(0.402724\pi\)
\(998\) −4.00000 −0.126618
\(999\) 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 2646.2.a.y.1.1 1
3.2 odd 2 2646.2.a.f.1.1 1
7.6 odd 2 378.2.a.e.1.1 yes 1
21.20 even 2 378.2.a.d.1.1 ✓ 1
28.27 even 2 3024.2.a.p.1.1 1
35.34 odd 2 9450.2.a.l.1.1 1
63.13 odd 6 1134.2.f.e.379.1 2
63.20 even 6 1134.2.f.k.757.1 2
63.34 odd 6 1134.2.f.e.757.1 2
63.41 even 6 1134.2.f.k.379.1 2
84.83 odd 2 3024.2.a.o.1.1 1
105.104 even 2 9450.2.a.cl.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
378.2.a.d.1.1 ✓ 1 21.20 even 2
378.2.a.e.1.1 yes 1 7.6 odd 2
1134.2.f.e.379.1 2 63.13 odd 6
1134.2.f.e.757.1 2 63.34 odd 6
1134.2.f.k.379.1 2 63.41 even 6
1134.2.f.k.757.1 2 63.20 even 6
2646.2.a.f.1.1 1 3.2 odd 2
2646.2.a.y.1.1 1 1.1 even 1 trivial
3024.2.a.o.1.1 1 84.83 odd 2
3024.2.a.p.1.1 1 28.27 even 2
9450.2.a.l.1.1 1 35.34 odd 2
9450.2.a.cl.1.1 1 105.104 even 2