Properties

Label 3675.2.a.l.1.1
Level $3675$
Weight $2$
Character 3675.1
Self dual yes
Analytic conductor $29.345$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3675 = 3 \cdot 5^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3675.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(29.3450227428\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 105)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 3675.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{3} -1.00000 q^{4} +1.00000 q^{6} -3.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{3} -1.00000 q^{4} +1.00000 q^{6} -3.00000 q^{8} +1.00000 q^{9} -6.00000 q^{11} -1.00000 q^{12} -2.00000 q^{13} -1.00000 q^{16} +4.00000 q^{17} +1.00000 q^{18} +6.00000 q^{19} -6.00000 q^{22} -3.00000 q^{24} -2.00000 q^{26} +1.00000 q^{27} -2.00000 q^{29} +10.0000 q^{31} +5.00000 q^{32} -6.00000 q^{33} +4.00000 q^{34} -1.00000 q^{36} +4.00000 q^{37} +6.00000 q^{38} -2.00000 q^{39} -2.00000 q^{41} +4.00000 q^{43} +6.00000 q^{44} -1.00000 q^{48} +4.00000 q^{51} +2.00000 q^{52} -6.00000 q^{53} +1.00000 q^{54} +6.00000 q^{57} -2.00000 q^{58} +8.00000 q^{59} +2.00000 q^{61} +10.0000 q^{62} +7.00000 q^{64} -6.00000 q^{66} +16.0000 q^{67} -4.00000 q^{68} +10.0000 q^{71} -3.00000 q^{72} -6.00000 q^{73} +4.00000 q^{74} -6.00000 q^{76} -2.00000 q^{78} +4.00000 q^{79} +1.00000 q^{81} -2.00000 q^{82} +8.00000 q^{83} +4.00000 q^{86} -2.00000 q^{87} +18.0000 q^{88} -6.00000 q^{89} +10.0000 q^{93} +5.00000 q^{96} -2.00000 q^{97} -6.00000 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107 0.353553 0.935414i \(-0.384973\pi\)
0.353553 + 0.935414i \(0.384973\pi\)
\(3\) 1.00000 0.577350
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) 1.00000 0.408248
\(7\) 0 0
\(8\) −3.00000 −1.06066
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −6.00000 −1.80907 −0.904534 0.426401i \(-0.859781\pi\)
−0.904534 + 0.426401i \(0.859781\pi\)
\(12\) −1.00000 −0.288675
\(13\) −2.00000 −0.554700 −0.277350 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −1.00000 −0.250000
\(17\) 4.00000 0.970143 0.485071 0.874475i \(-0.338794\pi\)
0.485071 + 0.874475i \(0.338794\pi\)
\(18\) 1.00000 0.235702
\(19\) 6.00000 1.37649 0.688247 0.725476i \(-0.258380\pi\)
0.688247 + 0.725476i \(0.258380\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −6.00000 −1.27920
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) −3.00000 −0.612372
\(25\) 0 0
\(26\) −2.00000 −0.392232
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −2.00000 −0.371391 −0.185695 0.982607i \(-0.559454\pi\)
−0.185695 + 0.982607i \(0.559454\pi\)
\(30\) 0 0
\(31\) 10.0000 1.79605 0.898027 0.439941i \(-0.145001\pi\)
0.898027 + 0.439941i \(0.145001\pi\)
\(32\) 5.00000 0.883883
\(33\) −6.00000 −1.04447
\(34\) 4.00000 0.685994
\(35\) 0 0
\(36\) −1.00000 −0.166667
\(37\) 4.00000 0.657596 0.328798 0.944400i \(-0.393356\pi\)
0.328798 + 0.944400i \(0.393356\pi\)
\(38\) 6.00000 0.973329
\(39\) −2.00000 −0.320256
\(40\) 0 0
\(41\) −2.00000 −0.312348 −0.156174 0.987730i \(-0.549916\pi\)
−0.156174 + 0.987730i \(0.549916\pi\)
\(42\) 0 0
\(43\) 4.00000 0.609994 0.304997 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) 6.00000 0.904534
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) −1.00000 −0.144338
\(49\) 0 0
\(50\) 0 0
\(51\) 4.00000 0.560112
\(52\) 2.00000 0.277350
\(53\) −6.00000 −0.824163 −0.412082 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 1.00000 0.136083
\(55\) 0 0
\(56\) 0 0
\(57\) 6.00000 0.794719
\(58\) −2.00000 −0.262613
\(59\) 8.00000 1.04151 0.520756 0.853706i \(-0.325650\pi\)
0.520756 + 0.853706i \(0.325650\pi\)
\(60\) 0 0
\(61\) 2.00000 0.256074 0.128037 0.991769i \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) 10.0000 1.27000
\(63\) 0 0
\(64\) 7.00000 0.875000
\(65\) 0 0
\(66\) −6.00000 −0.738549
\(67\) 16.0000 1.95471 0.977356 0.211604i \(-0.0678686\pi\)
0.977356 + 0.211604i \(0.0678686\pi\)
\(68\) −4.00000 −0.485071
\(69\) 0 0
\(70\) 0 0
\(71\) 10.0000 1.18678 0.593391 0.804914i \(-0.297789\pi\)
0.593391 + 0.804914i \(0.297789\pi\)
\(72\) −3.00000 −0.353553
\(73\) −6.00000 −0.702247 −0.351123 0.936329i \(-0.614200\pi\)
−0.351123 + 0.936329i \(0.614200\pi\)
\(74\) 4.00000 0.464991
\(75\) 0 0
\(76\) −6.00000 −0.688247
\(77\) 0 0
\(78\) −2.00000 −0.226455
\(79\) 4.00000 0.450035 0.225018 0.974355i \(-0.427756\pi\)
0.225018 + 0.974355i \(0.427756\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) −2.00000 −0.220863
\(83\) 8.00000 0.878114 0.439057 0.898459i \(-0.355313\pi\)
0.439057 + 0.898459i \(0.355313\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 4.00000 0.431331
\(87\) −2.00000 −0.214423
\(88\) 18.0000 1.91881
\(89\) −6.00000 −0.635999 −0.317999 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 10.0000 1.03695
\(94\) 0 0
\(95\) 0 0
\(96\) 5.00000 0.510310
\(97\) −2.00000 −0.203069 −0.101535 0.994832i \(-0.532375\pi\)
−0.101535 + 0.994832i \(0.532375\pi\)
\(98\) 0 0
\(99\) −6.00000 −0.603023
\(100\) 0 0
\(101\) 6.00000 0.597022 0.298511 0.954406i \(-0.403510\pi\)
0.298511 + 0.954406i \(0.403510\pi\)
\(102\) 4.00000 0.396059
\(103\) 8.00000 0.788263 0.394132 0.919054i \(-0.371045\pi\)
0.394132 + 0.919054i \(0.371045\pi\)
\(104\) 6.00000 0.588348
\(105\) 0 0
\(106\) −6.00000 −0.582772
\(107\) −4.00000 −0.386695 −0.193347 0.981130i \(-0.561934\pi\)
−0.193347 + 0.981130i \(0.561934\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 2.00000 0.191565 0.0957826 0.995402i \(-0.469465\pi\)
0.0957826 + 0.995402i \(0.469465\pi\)
\(110\) 0 0
\(111\) 4.00000 0.379663
\(112\) 0 0
\(113\) −6.00000 −0.564433 −0.282216 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) 6.00000 0.561951
\(115\) 0 0
\(116\) 2.00000 0.185695
\(117\) −2.00000 −0.184900
\(118\) 8.00000 0.736460
\(119\) 0 0
\(120\) 0 0
\(121\) 25.0000 2.27273
\(122\) 2.00000 0.181071
\(123\) −2.00000 −0.180334
\(124\) −10.0000 −0.898027
\(125\) 0 0
\(126\) 0 0
\(127\) −20.0000 −1.77471 −0.887357 0.461084i \(-0.847461\pi\)
−0.887357 + 0.461084i \(0.847461\pi\)
\(128\) −3.00000 −0.265165
\(129\) 4.00000 0.352180
\(130\) 0 0
\(131\) −4.00000 −0.349482 −0.174741 0.984614i \(-0.555909\pi\)
−0.174741 + 0.984614i \(0.555909\pi\)
\(132\) 6.00000 0.522233
\(133\) 0 0
\(134\) 16.0000 1.38219
\(135\) 0 0
\(136\) −12.0000 −1.02899
\(137\) 6.00000 0.512615 0.256307 0.966595i \(-0.417494\pi\)
0.256307 + 0.966595i \(0.417494\pi\)
\(138\) 0 0
\(139\) −2.00000 −0.169638 −0.0848189 0.996396i \(-0.527031\pi\)
−0.0848189 + 0.996396i \(0.527031\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 10.0000 0.839181
\(143\) 12.0000 1.00349
\(144\) −1.00000 −0.0833333
\(145\) 0 0
\(146\) −6.00000 −0.496564
\(147\) 0 0
\(148\) −4.00000 −0.328798
\(149\) −14.0000 −1.14692 −0.573462 0.819232i \(-0.694400\pi\)
−0.573462 + 0.819232i \(0.694400\pi\)
\(150\) 0 0
\(151\) 8.00000 0.651031 0.325515 0.945537i \(-0.394462\pi\)
0.325515 + 0.945537i \(0.394462\pi\)
\(152\) −18.0000 −1.45999
\(153\) 4.00000 0.323381
\(154\) 0 0
\(155\) 0 0
\(156\) 2.00000 0.160128
\(157\) 18.0000 1.43656 0.718278 0.695756i \(-0.244931\pi\)
0.718278 + 0.695756i \(0.244931\pi\)
\(158\) 4.00000 0.318223
\(159\) −6.00000 −0.475831
\(160\) 0 0
\(161\) 0 0
\(162\) 1.00000 0.0785674
\(163\) −4.00000 −0.313304 −0.156652 0.987654i \(-0.550070\pi\)
−0.156652 + 0.987654i \(0.550070\pi\)
\(164\) 2.00000 0.156174
\(165\) 0 0
\(166\) 8.00000 0.620920
\(167\) 12.0000 0.928588 0.464294 0.885681i \(-0.346308\pi\)
0.464294 + 0.885681i \(0.346308\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) 6.00000 0.458831
\(172\) −4.00000 −0.304997
\(173\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(174\) −2.00000 −0.151620
\(175\) 0 0
\(176\) 6.00000 0.452267
\(177\) 8.00000 0.601317
\(178\) −6.00000 −0.449719
\(179\) −14.0000 −1.04641 −0.523205 0.852207i \(-0.675264\pi\)
−0.523205 + 0.852207i \(0.675264\pi\)
\(180\) 0 0
\(181\) 6.00000 0.445976 0.222988 0.974821i \(-0.428419\pi\)
0.222988 + 0.974821i \(0.428419\pi\)
\(182\) 0 0
\(183\) 2.00000 0.147844
\(184\) 0 0
\(185\) 0 0
\(186\) 10.0000 0.733236
\(187\) −24.0000 −1.75505
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −18.0000 −1.30243 −0.651217 0.758891i \(-0.725741\pi\)
−0.651217 + 0.758891i \(0.725741\pi\)
\(192\) 7.00000 0.505181
\(193\) 8.00000 0.575853 0.287926 0.957653i \(-0.407034\pi\)
0.287926 + 0.957653i \(0.407034\pi\)
\(194\) −2.00000 −0.143592
\(195\) 0 0
\(196\) 0 0
\(197\) −2.00000 −0.142494 −0.0712470 0.997459i \(-0.522698\pi\)
−0.0712470 + 0.997459i \(0.522698\pi\)
\(198\) −6.00000 −0.426401
\(199\) −14.0000 −0.992434 −0.496217 0.868199i \(-0.665278\pi\)
−0.496217 + 0.868199i \(0.665278\pi\)
\(200\) 0 0
\(201\) 16.0000 1.12855
\(202\) 6.00000 0.422159
\(203\) 0 0
\(204\) −4.00000 −0.280056
\(205\) 0 0
\(206\) 8.00000 0.557386
\(207\) 0 0
\(208\) 2.00000 0.138675
\(209\) −36.0000 −2.49017
\(210\) 0 0
\(211\) −16.0000 −1.10149 −0.550743 0.834675i \(-0.685655\pi\)
−0.550743 + 0.834675i \(0.685655\pi\)
\(212\) 6.00000 0.412082
\(213\) 10.0000 0.685189
\(214\) −4.00000 −0.273434
\(215\) 0 0
\(216\) −3.00000 −0.204124
\(217\) 0 0
\(218\) 2.00000 0.135457
\(219\) −6.00000 −0.405442
\(220\) 0 0
\(221\) −8.00000 −0.538138
\(222\) 4.00000 0.268462
\(223\) −24.0000 −1.60716 −0.803579 0.595198i \(-0.797074\pi\)
−0.803579 + 0.595198i \(0.797074\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) −6.00000 −0.399114
\(227\) 8.00000 0.530979 0.265489 0.964114i \(-0.414466\pi\)
0.265489 + 0.964114i \(0.414466\pi\)
\(228\) −6.00000 −0.397360
\(229\) 10.0000 0.660819 0.330409 0.943838i \(-0.392813\pi\)
0.330409 + 0.943838i \(0.392813\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 6.00000 0.393919
\(233\) 26.0000 1.70332 0.851658 0.524097i \(-0.175597\pi\)
0.851658 + 0.524097i \(0.175597\pi\)
\(234\) −2.00000 −0.130744
\(235\) 0 0
\(236\) −8.00000 −0.520756
\(237\) 4.00000 0.259828
\(238\) 0 0
\(239\) −6.00000 −0.388108 −0.194054 0.980991i \(-0.562164\pi\)
−0.194054 + 0.980991i \(0.562164\pi\)
\(240\) 0 0
\(241\) −22.0000 −1.41714 −0.708572 0.705638i \(-0.750660\pi\)
−0.708572 + 0.705638i \(0.750660\pi\)
\(242\) 25.0000 1.60706
\(243\) 1.00000 0.0641500
\(244\) −2.00000 −0.128037
\(245\) 0 0
\(246\) −2.00000 −0.127515
\(247\) −12.0000 −0.763542
\(248\) −30.0000 −1.90500
\(249\) 8.00000 0.506979
\(250\) 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
\(256\) −17.0000 −1.06250
\(257\) 16.0000 0.998053 0.499026 0.866587i \(-0.333691\pi\)
0.499026 + 0.866587i \(0.333691\pi\)
\(258\) 4.00000 0.249029
\(259\) 0 0
\(260\) 0 0
\(261\) −2.00000 −0.123797
\(262\) −4.00000 −0.247121
\(263\) 24.0000 1.47990 0.739952 0.672660i \(-0.234848\pi\)
0.739952 + 0.672660i \(0.234848\pi\)
\(264\) 18.0000 1.10782
\(265\) 0 0
\(266\) 0 0
\(267\) −6.00000 −0.367194
\(268\) −16.0000 −0.977356
\(269\) 14.0000 0.853595 0.426798 0.904347i \(-0.359642\pi\)
0.426798 + 0.904347i \(0.359642\pi\)
\(270\) 0 0
\(271\) 14.0000 0.850439 0.425220 0.905090i \(-0.360197\pi\)
0.425220 + 0.905090i \(0.360197\pi\)
\(272\) −4.00000 −0.242536
\(273\) 0 0
\(274\) 6.00000 0.362473
\(275\) 0 0
\(276\) 0 0
\(277\) 28.0000 1.68236 0.841178 0.540758i \(-0.181862\pi\)
0.841178 + 0.540758i \(0.181862\pi\)
\(278\) −2.00000 −0.119952
\(279\) 10.0000 0.598684
\(280\) 0 0
\(281\) −2.00000 −0.119310 −0.0596550 0.998219i \(-0.519000\pi\)
−0.0596550 + 0.998219i \(0.519000\pi\)
\(282\) 0 0
\(283\) 20.0000 1.18888 0.594438 0.804141i \(-0.297374\pi\)
0.594438 + 0.804141i \(0.297374\pi\)
\(284\) −10.0000 −0.593391
\(285\) 0 0
\(286\) 12.0000 0.709575
\(287\) 0 0
\(288\) 5.00000 0.294628
\(289\) −1.00000 −0.0588235
\(290\) 0 0
\(291\) −2.00000 −0.117242
\(292\) 6.00000 0.351123
\(293\) 24.0000 1.40209 0.701047 0.713115i \(-0.252716\pi\)
0.701047 + 0.713115i \(0.252716\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −12.0000 −0.697486
\(297\) −6.00000 −0.348155
\(298\) −14.0000 −0.810998
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) 8.00000 0.460348
\(303\) 6.00000 0.344691
\(304\) −6.00000 −0.344124
\(305\) 0 0
\(306\) 4.00000 0.228665
\(307\) 4.00000 0.228292 0.114146 0.993464i \(-0.463587\pi\)
0.114146 + 0.993464i \(0.463587\pi\)
\(308\) 0 0
\(309\) 8.00000 0.455104
\(310\) 0 0
\(311\) 24.0000 1.36092 0.680458 0.732787i \(-0.261781\pi\)
0.680458 + 0.732787i \(0.261781\pi\)
\(312\) 6.00000 0.339683
\(313\) 6.00000 0.339140 0.169570 0.985518i \(-0.445762\pi\)
0.169570 + 0.985518i \(0.445762\pi\)
\(314\) 18.0000 1.01580
\(315\) 0 0
\(316\) −4.00000 −0.225018
\(317\) 18.0000 1.01098 0.505490 0.862832i \(-0.331312\pi\)
0.505490 + 0.862832i \(0.331312\pi\)
\(318\) −6.00000 −0.336463
\(319\) 12.0000 0.671871
\(320\) 0 0
\(321\) −4.00000 −0.223258
\(322\) 0 0
\(323\) 24.0000 1.33540
\(324\) −1.00000 −0.0555556
\(325\) 0 0
\(326\) −4.00000 −0.221540
\(327\) 2.00000 0.110600
\(328\) 6.00000 0.331295
\(329\) 0 0
\(330\) 0 0
\(331\) −24.0000 −1.31916 −0.659580 0.751635i \(-0.729266\pi\)
−0.659580 + 0.751635i \(0.729266\pi\)
\(332\) −8.00000 −0.439057
\(333\) 4.00000 0.219199
\(334\) 12.0000 0.656611
\(335\) 0 0
\(336\) 0 0
\(337\) 24.0000 1.30736 0.653682 0.756770i \(-0.273224\pi\)
0.653682 + 0.756770i \(0.273224\pi\)
\(338\) −9.00000 −0.489535
\(339\) −6.00000 −0.325875
\(340\) 0 0
\(341\) −60.0000 −3.24918
\(342\) 6.00000 0.324443
\(343\) 0 0
\(344\) −12.0000 −0.646997
\(345\) 0 0
\(346\) 0 0
\(347\) −12.0000 −0.644194 −0.322097 0.946707i \(-0.604388\pi\)
−0.322097 + 0.946707i \(0.604388\pi\)
\(348\) 2.00000 0.107211
\(349\) 2.00000 0.107058 0.0535288 0.998566i \(-0.482953\pi\)
0.0535288 + 0.998566i \(0.482953\pi\)
\(350\) 0 0
\(351\) −2.00000 −0.106752
\(352\) −30.0000 −1.59901
\(353\) 20.0000 1.06449 0.532246 0.846590i \(-0.321348\pi\)
0.532246 + 0.846590i \(0.321348\pi\)
\(354\) 8.00000 0.425195
\(355\) 0 0
\(356\) 6.00000 0.317999
\(357\) 0 0
\(358\) −14.0000 −0.739923
\(359\) 22.0000 1.16112 0.580558 0.814219i \(-0.302835\pi\)
0.580558 + 0.814219i \(0.302835\pi\)
\(360\) 0 0
\(361\) 17.0000 0.894737
\(362\) 6.00000 0.315353
\(363\) 25.0000 1.31216
\(364\) 0 0
\(365\) 0 0
\(366\) 2.00000 0.104542
\(367\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(368\) 0 0
\(369\) −2.00000 −0.104116
\(370\) 0 0
\(371\) 0 0
\(372\) −10.0000 −0.518476
\(373\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(374\) −24.0000 −1.24101
\(375\) 0 0
\(376\) 0 0
\(377\) 4.00000 0.206010
\(378\) 0 0
\(379\) −28.0000 −1.43826 −0.719132 0.694874i \(-0.755460\pi\)
−0.719132 + 0.694874i \(0.755460\pi\)
\(380\) 0 0
\(381\) −20.0000 −1.02463
\(382\) −18.0000 −0.920960
\(383\) −20.0000 −1.02195 −0.510976 0.859595i \(-0.670716\pi\)
−0.510976 + 0.859595i \(0.670716\pi\)
\(384\) −3.00000 −0.153093
\(385\) 0 0
\(386\) 8.00000 0.407189
\(387\) 4.00000 0.203331
\(388\) 2.00000 0.101535
\(389\) 26.0000 1.31825 0.659126 0.752032i \(-0.270926\pi\)
0.659126 + 0.752032i \(0.270926\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) −4.00000 −0.201773
\(394\) −2.00000 −0.100759
\(395\) 0 0
\(396\) 6.00000 0.301511
\(397\) 22.0000 1.10415 0.552074 0.833795i \(-0.313837\pi\)
0.552074 + 0.833795i \(0.313837\pi\)
\(398\) −14.0000 −0.701757
\(399\) 0 0
\(400\) 0 0
\(401\) −30.0000 −1.49813 −0.749064 0.662497i \(-0.769497\pi\)
−0.749064 + 0.662497i \(0.769497\pi\)
\(402\) 16.0000 0.798007
\(403\) −20.0000 −0.996271
\(404\) −6.00000 −0.298511
\(405\) 0 0
\(406\) 0 0
\(407\) −24.0000 −1.18964
\(408\) −12.0000 −0.594089
\(409\) 22.0000 1.08783 0.543915 0.839140i \(-0.316941\pi\)
0.543915 + 0.839140i \(0.316941\pi\)
\(410\) 0 0
\(411\) 6.00000 0.295958
\(412\) −8.00000 −0.394132
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) −10.0000 −0.490290
\(417\) −2.00000 −0.0979404
\(418\) −36.0000 −1.76082
\(419\) 12.0000 0.586238 0.293119 0.956076i \(-0.405307\pi\)
0.293119 + 0.956076i \(0.405307\pi\)
\(420\) 0 0
\(421\) 18.0000 0.877266 0.438633 0.898666i \(-0.355463\pi\)
0.438633 + 0.898666i \(0.355463\pi\)
\(422\) −16.0000 −0.778868
\(423\) 0 0
\(424\) 18.0000 0.874157
\(425\) 0 0
\(426\) 10.0000 0.484502
\(427\) 0 0
\(428\) 4.00000 0.193347
\(429\) 12.0000 0.579365
\(430\) 0 0
\(431\) −14.0000 −0.674356 −0.337178 0.941441i \(-0.609472\pi\)
−0.337178 + 0.941441i \(0.609472\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −34.0000 −1.63394 −0.816968 0.576683i \(-0.804347\pi\)
−0.816968 + 0.576683i \(0.804347\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −2.00000 −0.0957826
\(437\) 0 0
\(438\) −6.00000 −0.286691
\(439\) −6.00000 −0.286364 −0.143182 0.989696i \(-0.545733\pi\)
−0.143182 + 0.989696i \(0.545733\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −8.00000 −0.380521
\(443\) 4.00000 0.190046 0.0950229 0.995475i \(-0.469708\pi\)
0.0950229 + 0.995475i \(0.469708\pi\)
\(444\) −4.00000 −0.189832
\(445\) 0 0
\(446\) −24.0000 −1.13643
\(447\) −14.0000 −0.662177
\(448\) 0 0
\(449\) 10.0000 0.471929 0.235965 0.971762i \(-0.424175\pi\)
0.235965 + 0.971762i \(0.424175\pi\)
\(450\) 0 0
\(451\) 12.0000 0.565058
\(452\) 6.00000 0.282216
\(453\) 8.00000 0.375873
\(454\) 8.00000 0.375459
\(455\) 0 0
\(456\) −18.0000 −0.842927
\(457\) 20.0000 0.935561 0.467780 0.883845i \(-0.345054\pi\)
0.467780 + 0.883845i \(0.345054\pi\)
\(458\) 10.0000 0.467269
\(459\) 4.00000 0.186704
\(460\) 0 0
\(461\) −30.0000 −1.39724 −0.698620 0.715493i \(-0.746202\pi\)
−0.698620 + 0.715493i \(0.746202\pi\)
\(462\) 0 0
\(463\) −36.0000 −1.67306 −0.836531 0.547920i \(-0.815420\pi\)
−0.836531 + 0.547920i \(0.815420\pi\)
\(464\) 2.00000 0.0928477
\(465\) 0 0
\(466\) 26.0000 1.20443
\(467\) −24.0000 −1.11059 −0.555294 0.831654i \(-0.687394\pi\)
−0.555294 + 0.831654i \(0.687394\pi\)
\(468\) 2.00000 0.0924500
\(469\) 0 0
\(470\) 0 0
\(471\) 18.0000 0.829396
\(472\) −24.0000 −1.10469
\(473\) −24.0000 −1.10352
\(474\) 4.00000 0.183726
\(475\) 0 0
\(476\) 0 0
\(477\) −6.00000 −0.274721
\(478\) −6.00000 −0.274434
\(479\) −24.0000 −1.09659 −0.548294 0.836286i \(-0.684723\pi\)
−0.548294 + 0.836286i \(0.684723\pi\)
\(480\) 0 0
\(481\) −8.00000 −0.364769
\(482\) −22.0000 −1.00207
\(483\) 0 0
\(484\) −25.0000 −1.13636
\(485\) 0 0
\(486\) 1.00000 0.0453609
\(487\) −12.0000 −0.543772 −0.271886 0.962329i \(-0.587647\pi\)
−0.271886 + 0.962329i \(0.587647\pi\)
\(488\) −6.00000 −0.271607
\(489\) −4.00000 −0.180886
\(490\) 0 0
\(491\) 10.0000 0.451294 0.225647 0.974209i \(-0.427550\pi\)
0.225647 + 0.974209i \(0.427550\pi\)
\(492\) 2.00000 0.0901670
\(493\) −8.00000 −0.360302
\(494\) −12.0000 −0.539906
\(495\) 0 0
\(496\) −10.0000 −0.449013
\(497\) 0 0
\(498\) 8.00000 0.358489
\(499\) 4.00000 0.179065 0.0895323 0.995984i \(-0.471463\pi\)
0.0895323 + 0.995984i \(0.471463\pi\)
\(500\) 0 0
\(501\) 12.0000 0.536120
\(502\) 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
\(507\) −9.00000 −0.399704
\(508\) 20.0000 0.887357
\(509\) −30.0000 −1.32973 −0.664863 0.746965i \(-0.731510\pi\)
−0.664863 + 0.746965i \(0.731510\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −11.0000 −0.486136
\(513\) 6.00000 0.264906
\(514\) 16.0000 0.705730
\(515\) 0 0
\(516\) −4.00000 −0.176090
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −38.0000 −1.66481 −0.832405 0.554168i \(-0.813037\pi\)
−0.832405 + 0.554168i \(0.813037\pi\)
\(522\) −2.00000 −0.0875376
\(523\) 20.0000 0.874539 0.437269 0.899331i \(-0.355946\pi\)
0.437269 + 0.899331i \(0.355946\pi\)
\(524\) 4.00000 0.174741
\(525\) 0 0
\(526\) 24.0000 1.04645
\(527\) 40.0000 1.74243
\(528\) 6.00000 0.261116
\(529\) −23.0000 −1.00000
\(530\) 0 0
\(531\) 8.00000 0.347170
\(532\) 0 0
\(533\) 4.00000 0.173259
\(534\) −6.00000 −0.259645
\(535\) 0 0
\(536\) −48.0000 −2.07328
\(537\) −14.0000 −0.604145
\(538\) 14.0000 0.603583
\(539\) 0 0
\(540\) 0 0
\(541\) 6.00000 0.257960 0.128980 0.991647i \(-0.458830\pi\)
0.128980 + 0.991647i \(0.458830\pi\)
\(542\) 14.0000 0.601351
\(543\) 6.00000 0.257485
\(544\) 20.0000 0.857493
\(545\) 0 0
\(546\) 0 0
\(547\) 16.0000 0.684111 0.342055 0.939680i \(-0.388877\pi\)
0.342055 + 0.939680i \(0.388877\pi\)
\(548\) −6.00000 −0.256307
\(549\) 2.00000 0.0853579
\(550\) 0 0
\(551\) −12.0000 −0.511217
\(552\) 0 0
\(553\) 0 0
\(554\) 28.0000 1.18961
\(555\) 0 0
\(556\) 2.00000 0.0848189
\(557\) −38.0000 −1.61011 −0.805056 0.593199i \(-0.797865\pi\)
−0.805056 + 0.593199i \(0.797865\pi\)
\(558\) 10.0000 0.423334
\(559\) −8.00000 −0.338364
\(560\) 0 0
\(561\) −24.0000 −1.01328
\(562\) −2.00000 −0.0843649
\(563\) −36.0000 −1.51722 −0.758610 0.651546i \(-0.774121\pi\)
−0.758610 + 0.651546i \(0.774121\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 20.0000 0.840663
\(567\) 0 0
\(568\) −30.0000 −1.25877
\(569\) −30.0000 −1.25767 −0.628833 0.777541i \(-0.716467\pi\)
−0.628833 + 0.777541i \(0.716467\pi\)
\(570\) 0 0
\(571\) 36.0000 1.50655 0.753277 0.657704i \(-0.228472\pi\)
0.753277 + 0.657704i \(0.228472\pi\)
\(572\) −12.0000 −0.501745
\(573\) −18.0000 −0.751961
\(574\) 0 0
\(575\) 0 0
\(576\) 7.00000 0.291667
\(577\) 14.0000 0.582828 0.291414 0.956597i \(-0.405874\pi\)
0.291414 + 0.956597i \(0.405874\pi\)
\(578\) −1.00000 −0.0415945
\(579\) 8.00000 0.332469
\(580\) 0 0
\(581\) 0 0
\(582\) −2.00000 −0.0829027
\(583\) 36.0000 1.49097
\(584\) 18.0000 0.744845
\(585\) 0 0
\(586\) 24.0000 0.991431
\(587\) −12.0000 −0.495293 −0.247647 0.968850i \(-0.579657\pi\)
−0.247647 + 0.968850i \(0.579657\pi\)
\(588\) 0 0
\(589\) 60.0000 2.47226
\(590\) 0 0
\(591\) −2.00000 −0.0822690
\(592\) −4.00000 −0.164399
\(593\) 44.0000 1.80686 0.903432 0.428732i \(-0.141040\pi\)
0.903432 + 0.428732i \(0.141040\pi\)
\(594\) −6.00000 −0.246183
\(595\) 0 0
\(596\) 14.0000 0.573462
\(597\) −14.0000 −0.572982
\(598\) 0 0
\(599\) −2.00000 −0.0817178 −0.0408589 0.999165i \(-0.513009\pi\)
−0.0408589 + 0.999165i \(0.513009\pi\)
\(600\) 0 0
\(601\) 26.0000 1.06056 0.530281 0.847822i \(-0.322086\pi\)
0.530281 + 0.847822i \(0.322086\pi\)
\(602\) 0 0
\(603\) 16.0000 0.651570
\(604\) −8.00000 −0.325515
\(605\) 0 0
\(606\) 6.00000 0.243733
\(607\) 24.0000 0.974130 0.487065 0.873366i \(-0.338067\pi\)
0.487065 + 0.873366i \(0.338067\pi\)
\(608\) 30.0000 1.21666
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) −4.00000 −0.161690
\(613\) −4.00000 −0.161558 −0.0807792 0.996732i \(-0.525741\pi\)
−0.0807792 + 0.996732i \(0.525741\pi\)
\(614\) 4.00000 0.161427
\(615\) 0 0
\(616\) 0 0
\(617\) 26.0000 1.04672 0.523360 0.852111i \(-0.324678\pi\)
0.523360 + 0.852111i \(0.324678\pi\)
\(618\) 8.00000 0.321807
\(619\) −10.0000 −0.401934 −0.200967 0.979598i \(-0.564408\pi\)
−0.200967 + 0.979598i \(0.564408\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 24.0000 0.962312
\(623\) 0 0
\(624\) 2.00000 0.0800641
\(625\) 0 0
\(626\) 6.00000 0.239808
\(627\) −36.0000 −1.43770
\(628\) −18.0000 −0.718278
\(629\) 16.0000 0.637962
\(630\) 0 0
\(631\) −20.0000 −0.796187 −0.398094 0.917345i \(-0.630328\pi\)
−0.398094 + 0.917345i \(0.630328\pi\)
\(632\) −12.0000 −0.477334
\(633\) −16.0000 −0.635943
\(634\) 18.0000 0.714871
\(635\) 0 0
\(636\) 6.00000 0.237915
\(637\) 0 0
\(638\) 12.0000 0.475085
\(639\) 10.0000 0.395594
\(640\) 0 0
\(641\) −2.00000 −0.0789953 −0.0394976 0.999220i \(-0.512576\pi\)
−0.0394976 + 0.999220i \(0.512576\pi\)
\(642\) −4.00000 −0.157867
\(643\) −20.0000 −0.788723 −0.394362 0.918955i \(-0.629034\pi\)
−0.394362 + 0.918955i \(0.629034\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 24.0000 0.944267
\(647\) −20.0000 −0.786281 −0.393141 0.919478i \(-0.628611\pi\)
−0.393141 + 0.919478i \(0.628611\pi\)
\(648\) −3.00000 −0.117851
\(649\) −48.0000 −1.88416
\(650\) 0 0
\(651\) 0 0
\(652\) 4.00000 0.156652
\(653\) −2.00000 −0.0782660 −0.0391330 0.999234i \(-0.512460\pi\)
−0.0391330 + 0.999234i \(0.512460\pi\)
\(654\) 2.00000 0.0782062
\(655\) 0 0
\(656\) 2.00000 0.0780869
\(657\) −6.00000 −0.234082
\(658\) 0 0
\(659\) −18.0000 −0.701180 −0.350590 0.936529i \(-0.614019\pi\)
−0.350590 + 0.936529i \(0.614019\pi\)
\(660\) 0 0
\(661\) 18.0000 0.700119 0.350059 0.936727i \(-0.386161\pi\)
0.350059 + 0.936727i \(0.386161\pi\)
\(662\) −24.0000 −0.932786
\(663\) −8.00000 −0.310694
\(664\) −24.0000 −0.931381
\(665\) 0 0
\(666\) 4.00000 0.154997
\(667\) 0 0
\(668\) −12.0000 −0.464294
\(669\) −24.0000 −0.927894
\(670\) 0 0
\(671\) −12.0000 −0.463255
\(672\) 0 0
\(673\) 36.0000 1.38770 0.693849 0.720121i \(-0.255914\pi\)
0.693849 + 0.720121i \(0.255914\pi\)
\(674\) 24.0000 0.924445
\(675\) 0 0
\(676\) 9.00000 0.346154
\(677\) 32.0000 1.22986 0.614930 0.788582i \(-0.289184\pi\)
0.614930 + 0.788582i \(0.289184\pi\)
\(678\) −6.00000 −0.230429
\(679\) 0 0
\(680\) 0 0
\(681\) 8.00000 0.306561
\(682\) −60.0000 −2.29752
\(683\) −28.0000 −1.07139 −0.535695 0.844411i \(-0.679950\pi\)
−0.535695 + 0.844411i \(0.679950\pi\)
\(684\) −6.00000 −0.229416
\(685\) 0 0
\(686\) 0 0
\(687\) 10.0000 0.381524
\(688\) −4.00000 −0.152499
\(689\) 12.0000 0.457164
\(690\) 0 0
\(691\) 50.0000 1.90209 0.951045 0.309053i \(-0.100012\pi\)
0.951045 + 0.309053i \(0.100012\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) −12.0000 −0.455514
\(695\) 0 0
\(696\) 6.00000 0.227429
\(697\) −8.00000 −0.303022
\(698\) 2.00000 0.0757011
\(699\) 26.0000 0.983410
\(700\) 0 0
\(701\) −10.0000 −0.377695 −0.188847 0.982006i \(-0.560475\pi\)
−0.188847 + 0.982006i \(0.560475\pi\)
\(702\) −2.00000 −0.0754851
\(703\) 24.0000 0.905177
\(704\) −42.0000 −1.58293
\(705\) 0 0
\(706\) 20.0000 0.752710
\(707\) 0 0
\(708\) −8.00000 −0.300658
\(709\) 26.0000 0.976450 0.488225 0.872718i \(-0.337644\pi\)
0.488225 + 0.872718i \(0.337644\pi\)
\(710\) 0 0
\(711\) 4.00000 0.150012
\(712\) 18.0000 0.674579
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 14.0000 0.523205
\(717\) −6.00000 −0.224074
\(718\) 22.0000 0.821033
\(719\) −12.0000 −0.447524 −0.223762 0.974644i \(-0.571834\pi\)
−0.223762 + 0.974644i \(0.571834\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 17.0000 0.632674
\(723\) −22.0000 −0.818189
\(724\) −6.00000 −0.222988
\(725\) 0 0
\(726\) 25.0000 0.927837
\(727\) −40.0000 −1.48352 −0.741759 0.670667i \(-0.766008\pi\)
−0.741759 + 0.670667i \(0.766008\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 16.0000 0.591781
\(732\) −2.00000 −0.0739221
\(733\) 22.0000 0.812589 0.406294 0.913742i \(-0.366821\pi\)
0.406294 + 0.913742i \(0.366821\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −96.0000 −3.53621
\(738\) −2.00000 −0.0736210
\(739\) −44.0000 −1.61857 −0.809283 0.587419i \(-0.800144\pi\)
−0.809283 + 0.587419i \(0.800144\pi\)
\(740\) 0 0
\(741\) −12.0000 −0.440831
\(742\) 0 0
\(743\) −40.0000 −1.46746 −0.733729 0.679442i \(-0.762222\pi\)
−0.733729 + 0.679442i \(0.762222\pi\)
\(744\) −30.0000 −1.09985
\(745\) 0 0
\(746\) 0 0
\(747\) 8.00000 0.292705
\(748\) 24.0000 0.877527
\(749\) 0 0
\(750\) 0 0
\(751\) −12.0000 −0.437886 −0.218943 0.975738i \(-0.570261\pi\)
−0.218943 + 0.975738i \(0.570261\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 4.00000 0.145671
\(755\) 0 0
\(756\) 0 0
\(757\) 40.0000 1.45382 0.726912 0.686730i \(-0.240955\pi\)
0.726912 + 0.686730i \(0.240955\pi\)
\(758\) −28.0000 −1.01701
\(759\) 0 0
\(760\) 0 0
\(761\) 22.0000 0.797499 0.398750 0.917060i \(-0.369444\pi\)
0.398750 + 0.917060i \(0.369444\pi\)
\(762\) −20.0000 −0.724524
\(763\) 0 0
\(764\) 18.0000 0.651217
\(765\) 0 0
\(766\) −20.0000 −0.722629
\(767\) −16.0000 −0.577727
\(768\) −17.0000 −0.613435
\(769\) 18.0000 0.649097 0.324548 0.945869i \(-0.394788\pi\)
0.324548 + 0.945869i \(0.394788\pi\)
\(770\) 0 0
\(771\) 16.0000 0.576226
\(772\) −8.00000 −0.287926
\(773\) 24.0000 0.863220 0.431610 0.902060i \(-0.357946\pi\)
0.431610 + 0.902060i \(0.357946\pi\)
\(774\) 4.00000 0.143777
\(775\) 0 0
\(776\) 6.00000 0.215387
\(777\) 0 0
\(778\) 26.0000 0.932145
\(779\) −12.0000 −0.429945
\(780\) 0 0
\(781\) −60.0000 −2.14697
\(782\) 0 0
\(783\) −2.00000 −0.0714742
\(784\) 0 0
\(785\) 0 0
\(786\) −4.00000 −0.142675
\(787\) −4.00000 −0.142585 −0.0712923 0.997455i \(-0.522712\pi\)
−0.0712923 + 0.997455i \(0.522712\pi\)
\(788\) 2.00000 0.0712470
\(789\) 24.0000 0.854423
\(790\) 0 0
\(791\) 0 0
\(792\) 18.0000 0.639602
\(793\) −4.00000 −0.142044
\(794\) 22.0000 0.780751
\(795\) 0 0
\(796\) 14.0000 0.496217
\(797\) −16.0000 −0.566749 −0.283375 0.959009i \(-0.591454\pi\)
−0.283375 + 0.959009i \(0.591454\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) −6.00000 −0.212000
\(802\) −30.0000 −1.05934
\(803\) 36.0000 1.27041
\(804\) −16.0000 −0.564276
\(805\) 0 0
\(806\) −20.0000 −0.704470
\(807\) 14.0000 0.492823
\(808\) −18.0000 −0.633238
\(809\) −18.0000 −0.632846 −0.316423 0.948618i \(-0.602482\pi\)
−0.316423 + 0.948618i \(0.602482\pi\)
\(810\) 0 0
\(811\) −34.0000 −1.19390 −0.596951 0.802278i \(-0.703621\pi\)
−0.596951 + 0.802278i \(0.703621\pi\)
\(812\) 0 0
\(813\) 14.0000 0.491001
\(814\) −24.0000 −0.841200
\(815\) 0 0
\(816\) −4.00000 −0.140028
\(817\) 24.0000 0.839654
\(818\) 22.0000 0.769212
\(819\) 0 0
\(820\) 0 0
\(821\) −2.00000 −0.0698005 −0.0349002 0.999391i \(-0.511111\pi\)
−0.0349002 + 0.999391i \(0.511111\pi\)
\(822\) 6.00000 0.209274
\(823\) −44.0000 −1.53374 −0.766872 0.641800i \(-0.778188\pi\)
−0.766872 + 0.641800i \(0.778188\pi\)
\(824\) −24.0000 −0.836080
\(825\) 0 0
\(826\) 0 0
\(827\) 12.0000 0.417281 0.208640 0.977992i \(-0.433096\pi\)
0.208640 + 0.977992i \(0.433096\pi\)
\(828\) 0 0
\(829\) 6.00000 0.208389 0.104194 0.994557i \(-0.466774\pi\)
0.104194 + 0.994557i \(0.466774\pi\)
\(830\) 0 0
\(831\) 28.0000 0.971309
\(832\) −14.0000 −0.485363
\(833\) 0 0
\(834\) −2.00000 −0.0692543
\(835\) 0 0
\(836\) 36.0000 1.24509
\(837\) 10.0000 0.345651
\(838\) 12.0000 0.414533
\(839\) −12.0000 −0.414286 −0.207143 0.978311i \(-0.566417\pi\)
−0.207143 + 0.978311i \(0.566417\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) 18.0000 0.620321
\(843\) −2.00000 −0.0688837
\(844\) 16.0000 0.550743
\(845\) 0 0
\(846\) 0 0
\(847\) 0 0
\(848\) 6.00000 0.206041
\(849\) 20.0000 0.686398
\(850\) 0 0
\(851\) 0 0
\(852\) −10.0000 −0.342594
\(853\) 30.0000 1.02718 0.513590 0.858036i \(-0.328315\pi\)
0.513590 + 0.858036i \(0.328315\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 12.0000 0.410152
\(857\) −24.0000 −0.819824 −0.409912 0.912125i \(-0.634441\pi\)
−0.409912 + 0.912125i \(0.634441\pi\)
\(858\) 12.0000 0.409673
\(859\) 26.0000 0.887109 0.443554 0.896248i \(-0.353717\pi\)
0.443554 + 0.896248i \(0.353717\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −14.0000 −0.476842
\(863\) 24.0000 0.816970 0.408485 0.912765i \(-0.366057\pi\)
0.408485 + 0.912765i \(0.366057\pi\)
\(864\) 5.00000 0.170103
\(865\) 0 0
\(866\) −34.0000 −1.15537
\(867\) −1.00000 −0.0339618
\(868\) 0 0
\(869\) −24.0000 −0.814144
\(870\) 0 0
\(871\) −32.0000 −1.08428
\(872\) −6.00000 −0.203186
\(873\) −2.00000 −0.0676897
\(874\) 0 0
\(875\) 0 0
\(876\) 6.00000 0.202721
\(877\) 4.00000 0.135070 0.0675352 0.997717i \(-0.478487\pi\)
0.0675352 + 0.997717i \(0.478487\pi\)
\(878\) −6.00000 −0.202490
\(879\) 24.0000 0.809500
\(880\) 0 0
\(881\) −14.0000 −0.471672 −0.235836 0.971793i \(-0.575783\pi\)
−0.235836 + 0.971793i \(0.575783\pi\)
\(882\) 0 0
\(883\) 48.0000 1.61533 0.807664 0.589643i \(-0.200731\pi\)
0.807664 + 0.589643i \(0.200731\pi\)
\(884\) 8.00000 0.269069
\(885\) 0 0
\(886\) 4.00000 0.134383
\(887\) 8.00000 0.268614 0.134307 0.990940i \(-0.457119\pi\)
0.134307 + 0.990940i \(0.457119\pi\)
\(888\) −12.0000 −0.402694
\(889\) 0 0
\(890\) 0 0
\(891\) −6.00000 −0.201008
\(892\) 24.0000 0.803579
\(893\) 0 0
\(894\) −14.0000 −0.468230
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 10.0000 0.333704
\(899\) −20.0000 −0.667037
\(900\) 0 0
\(901\) −24.0000 −0.799556
\(902\) 12.0000 0.399556
\(903\) 0 0
\(904\) 18.0000 0.598671
\(905\) 0 0
\(906\) 8.00000 0.265782
\(907\) −16.0000 −0.531271 −0.265636 0.964073i \(-0.585582\pi\)
−0.265636 + 0.964073i \(0.585582\pi\)
\(908\) −8.00000 −0.265489
\(909\) 6.00000 0.199007
\(910\) 0 0
\(911\) 6.00000 0.198789 0.0993944 0.995048i \(-0.468309\pi\)
0.0993944 + 0.995048i \(0.468309\pi\)
\(912\) −6.00000 −0.198680
\(913\) −48.0000 −1.58857
\(914\) 20.0000 0.661541
\(915\) 0 0
\(916\) −10.0000 −0.330409
\(917\) 0 0
\(918\) 4.00000 0.132020
\(919\) 28.0000 0.923635 0.461817 0.886975i \(-0.347198\pi\)
0.461817 + 0.886975i \(0.347198\pi\)
\(920\) 0 0
\(921\) 4.00000 0.131804
\(922\) −30.0000 −0.987997
\(923\) −20.0000 −0.658308
\(924\) 0 0
\(925\) 0 0
\(926\) −36.0000 −1.18303
\(927\) 8.00000 0.262754
\(928\) −10.0000 −0.328266
\(929\) 14.0000 0.459325 0.229663 0.973270i \(-0.426238\pi\)
0.229663 + 0.973270i \(0.426238\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −26.0000 −0.851658
\(933\) 24.0000 0.785725
\(934\) −24.0000 −0.785304
\(935\) 0 0
\(936\) 6.00000 0.196116
\(937\) 2.00000 0.0653372 0.0326686 0.999466i \(-0.489599\pi\)
0.0326686 + 0.999466i \(0.489599\pi\)
\(938\) 0 0
\(939\) 6.00000 0.195803
\(940\) 0 0
\(941\) 42.0000 1.36916 0.684580 0.728937i \(-0.259985\pi\)
0.684580 + 0.728937i \(0.259985\pi\)
\(942\) 18.0000 0.586472
\(943\) 0 0
\(944\) −8.00000 −0.260378
\(945\) 0 0
\(946\) −24.0000 −0.780307
\(947\) 36.0000 1.16984 0.584921 0.811090i \(-0.301125\pi\)
0.584921 + 0.811090i \(0.301125\pi\)
\(948\) −4.00000 −0.129914
\(949\) 12.0000 0.389536
\(950\) 0 0
\(951\) 18.0000 0.583690
\(952\) 0 0
\(953\) −6.00000 −0.194359 −0.0971795 0.995267i \(-0.530982\pi\)
−0.0971795 + 0.995267i \(0.530982\pi\)
\(954\) −6.00000 −0.194257
\(955\) 0 0
\(956\) 6.00000 0.194054
\(957\) 12.0000 0.387905
\(958\) −24.0000 −0.775405
\(959\) 0 0
\(960\) 0 0
\(961\) 69.0000 2.22581
\(962\) −8.00000 −0.257930
\(963\) −4.00000 −0.128898
\(964\) 22.0000 0.708572
\(965\) 0 0
\(966\) 0 0
\(967\) −40.0000 −1.28631 −0.643157 0.765735i \(-0.722376\pi\)
−0.643157 + 0.765735i \(0.722376\pi\)
\(968\) −75.0000 −2.41059
\(969\) 24.0000 0.770991
\(970\) 0 0
\(971\) 8.00000 0.256732 0.128366 0.991727i \(-0.459027\pi\)
0.128366 + 0.991727i \(0.459027\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 0 0
\(974\) −12.0000 −0.384505
\(975\) 0 0
\(976\) −2.00000 −0.0640184
\(977\) −30.0000 −0.959785 −0.479893 0.877327i \(-0.659324\pi\)
−0.479893 + 0.877327i \(0.659324\pi\)
\(978\) −4.00000 −0.127906
\(979\) 36.0000 1.15056
\(980\) 0 0
\(981\) 2.00000 0.0638551
\(982\) 10.0000 0.319113
\(983\) 24.0000 0.765481 0.382741 0.923856i \(-0.374980\pi\)
0.382741 + 0.923856i \(0.374980\pi\)
\(984\) 6.00000 0.191273
\(985\) 0 0
\(986\) −8.00000 −0.254772
\(987\) 0 0
\(988\) 12.0000 0.381771
\(989\) 0 0
\(990\) 0 0
\(991\) 4.00000 0.127064 0.0635321 0.997980i \(-0.479763\pi\)
0.0635321 + 0.997980i \(0.479763\pi\)
\(992\) 50.0000 1.58750
\(993\) −24.0000 −0.761617
\(994\) 0 0
\(995\) 0 0
\(996\) −8.00000 −0.253490
\(997\) 14.0000 0.443384 0.221692 0.975117i \(-0.428842\pi\)
0.221692 + 0.975117i \(0.428842\pi\)
\(998\) 4.00000 0.126618
\(999\) 4.00000 0.126554
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 3675.2.a.l.1.1 1
5.2 odd 4 735.2.d.a.589.2 2
5.3 odd 4 735.2.d.a.589.1 2
5.4 even 2 3675.2.a.d.1.1 1
7.6 odd 2 525.2.a.c.1.1 1
15.2 even 4 2205.2.d.f.1324.1 2
15.8 even 4 2205.2.d.f.1324.2 2
21.20 even 2 1575.2.a.e.1.1 1
28.27 even 2 8400.2.a.ch.1.1 1
35.2 odd 12 735.2.q.b.214.2 4
35.3 even 12 735.2.q.a.79.2 4
35.12 even 12 735.2.q.a.214.2 4
35.13 even 4 105.2.d.a.64.1 2
35.17 even 12 735.2.q.a.79.1 4
35.18 odd 12 735.2.q.b.79.2 4
35.23 odd 12 735.2.q.b.214.1 4
35.27 even 4 105.2.d.a.64.2 yes 2
35.32 odd 12 735.2.q.b.79.1 4
35.33 even 12 735.2.q.a.214.1 4
35.34 odd 2 525.2.a.b.1.1 1
105.62 odd 4 315.2.d.c.64.1 2
105.83 odd 4 315.2.d.c.64.2 2
105.104 even 2 1575.2.a.i.1.1 1
140.27 odd 4 1680.2.t.f.1009.1 2
140.83 odd 4 1680.2.t.f.1009.2 2
140.139 even 2 8400.2.a.bj.1.1 1
420.83 even 4 5040.2.t.e.1009.1 2
420.167 even 4 5040.2.t.e.1009.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
105.2.d.a.64.1 2 35.13 even 4
105.2.d.a.64.2 yes 2 35.27 even 4
315.2.d.c.64.1 2 105.62 odd 4
315.2.d.c.64.2 2 105.83 odd 4
525.2.a.b.1.1 1 35.34 odd 2
525.2.a.c.1.1 1 7.6 odd 2
735.2.d.a.589.1 2 5.3 odd 4
735.2.d.a.589.2 2 5.2 odd 4
735.2.q.a.79.1 4 35.17 even 12
735.2.q.a.79.2 4 35.3 even 12
735.2.q.a.214.1 4 35.33 even 12
735.2.q.a.214.2 4 35.12 even 12
735.2.q.b.79.1 4 35.32 odd 12
735.2.q.b.79.2 4 35.18 odd 12
735.2.q.b.214.1 4 35.23 odd 12
735.2.q.b.214.2 4 35.2 odd 12
1575.2.a.e.1.1 1 21.20 even 2
1575.2.a.i.1.1 1 105.104 even 2
1680.2.t.f.1009.1 2 140.27 odd 4
1680.2.t.f.1009.2 2 140.83 odd 4
2205.2.d.f.1324.1 2 15.2 even 4
2205.2.d.f.1324.2 2 15.8 even 4
3675.2.a.d.1.1 1 5.4 even 2
3675.2.a.l.1.1 1 1.1 even 1 trivial
5040.2.t.e.1009.1 2 420.83 even 4
5040.2.t.e.1009.2 2 420.167 even 4
8400.2.a.bj.1.1 1 140.139 even 2
8400.2.a.ch.1.1 1 28.27 even 2