Properties

Label 7098.2.a.bb.1.1
Level $7098$
Weight $2$
Character 7098.1
Self dual yes
Analytic conductor $56.678$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 7098 = 2 \cdot 3 \cdot 7 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7098.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(56.6778153547\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 546)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 7098.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +3.00000 q^{11} +1.00000 q^{12} +1.00000 q^{14} +1.00000 q^{16} +3.00000 q^{17} +1.00000 q^{18} +5.00000 q^{19} +1.00000 q^{21} +3.00000 q^{22} +6.00000 q^{23} +1.00000 q^{24} -5.00000 q^{25} +1.00000 q^{27} +1.00000 q^{28} -3.00000 q^{29} -4.00000 q^{31} +1.00000 q^{32} +3.00000 q^{33} +3.00000 q^{34} +1.00000 q^{36} -4.00000 q^{37} +5.00000 q^{38} -3.00000 q^{41} +1.00000 q^{42} +8.00000 q^{43} +3.00000 q^{44} +6.00000 q^{46} +9.00000 q^{47} +1.00000 q^{48} +1.00000 q^{49} -5.00000 q^{50} +3.00000 q^{51} -9.00000 q^{53} +1.00000 q^{54} +1.00000 q^{56} +5.00000 q^{57} -3.00000 q^{58} -6.00000 q^{59} +5.00000 q^{61} -4.00000 q^{62} +1.00000 q^{63} +1.00000 q^{64} +3.00000 q^{66} +14.0000 q^{67} +3.00000 q^{68} +6.00000 q^{69} -6.00000 q^{71} +1.00000 q^{72} -4.00000 q^{73} -4.00000 q^{74} -5.00000 q^{75} +5.00000 q^{76} +3.00000 q^{77} -1.00000 q^{79} +1.00000 q^{81} -3.00000 q^{82} -6.00000 q^{83} +1.00000 q^{84} +8.00000 q^{86} -3.00000 q^{87} +3.00000 q^{88} -9.00000 q^{89} +6.00000 q^{92} -4.00000 q^{93} +9.00000 q^{94} +1.00000 q^{96} +8.00000 q^{97} +1.00000 q^{98} +3.00000 q^{99} +O(q^{100})\)

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
\(3\) 1.00000 0.577350
\(4\) 1.00000 0.500000
\(5\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(6\) 1.00000 0.408248
\(7\) 1.00000 0.377964
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 3.00000 0.904534 0.452267 0.891883i \(-0.350615\pi\)
0.452267 + 0.891883i \(0.350615\pi\)
\(12\) 1.00000 0.288675
\(13\) 0 0
\(14\) 1.00000 0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 3.00000 0.727607 0.363803 0.931476i \(-0.381478\pi\)
0.363803 + 0.931476i \(0.381478\pi\)
\(18\) 1.00000 0.235702
\(19\) 5.00000 1.14708 0.573539 0.819178i \(-0.305570\pi\)
0.573539 + 0.819178i \(0.305570\pi\)
\(20\) 0 0
\(21\) 1.00000 0.218218
\(22\) 3.00000 0.639602
\(23\) 6.00000 1.25109 0.625543 0.780189i \(-0.284877\pi\)
0.625543 + 0.780189i \(0.284877\pi\)
\(24\) 1.00000 0.204124
\(25\) −5.00000 −1.00000
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 1.00000 0.188982
\(29\) −3.00000 −0.557086 −0.278543 0.960424i \(-0.589851\pi\)
−0.278543 + 0.960424i \(0.589851\pi\)
\(30\) 0 0
\(31\) −4.00000 −0.718421 −0.359211 0.933257i \(-0.616954\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) 1.00000 0.176777
\(33\) 3.00000 0.522233
\(34\) 3.00000 0.514496
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) −4.00000 −0.657596 −0.328798 0.944400i \(-0.606644\pi\)
−0.328798 + 0.944400i \(0.606644\pi\)
\(38\) 5.00000 0.811107
\(39\) 0 0
\(40\) 0 0
\(41\) −3.00000 −0.468521 −0.234261 0.972174i \(-0.575267\pi\)
−0.234261 + 0.972174i \(0.575267\pi\)
\(42\) 1.00000 0.154303
\(43\) 8.00000 1.21999 0.609994 0.792406i \(-0.291172\pi\)
0.609994 + 0.792406i \(0.291172\pi\)
\(44\) 3.00000 0.452267
\(45\) 0 0
\(46\) 6.00000 0.884652
\(47\) 9.00000 1.31278 0.656392 0.754420i \(-0.272082\pi\)
0.656392 + 0.754420i \(0.272082\pi\)
\(48\) 1.00000 0.144338
\(49\) 1.00000 0.142857
\(50\) −5.00000 −0.707107
\(51\) 3.00000 0.420084
\(52\) 0 0
\(53\) −9.00000 −1.23625 −0.618123 0.786082i \(-0.712106\pi\)
−0.618123 + 0.786082i \(0.712106\pi\)
\(54\) 1.00000 0.136083
\(55\) 0 0
\(56\) 1.00000 0.133631
\(57\) 5.00000 0.662266
\(58\) −3.00000 −0.393919
\(59\) −6.00000 −0.781133 −0.390567 0.920575i \(-0.627721\pi\)
−0.390567 + 0.920575i \(0.627721\pi\)
\(60\) 0 0
\(61\) 5.00000 0.640184 0.320092 0.947386i \(-0.396286\pi\)
0.320092 + 0.947386i \(0.396286\pi\)
\(62\) −4.00000 −0.508001
\(63\) 1.00000 0.125988
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 3.00000 0.369274
\(67\) 14.0000 1.71037 0.855186 0.518321i \(-0.173443\pi\)
0.855186 + 0.518321i \(0.173443\pi\)
\(68\) 3.00000 0.363803
\(69\) 6.00000 0.722315
\(70\) 0 0
\(71\) −6.00000 −0.712069 −0.356034 0.934473i \(-0.615871\pi\)
−0.356034 + 0.934473i \(0.615871\pi\)
\(72\) 1.00000 0.117851
\(73\) −4.00000 −0.468165 −0.234082 0.972217i \(-0.575209\pi\)
−0.234082 + 0.972217i \(0.575209\pi\)
\(74\) −4.00000 −0.464991
\(75\) −5.00000 −0.577350
\(76\) 5.00000 0.573539
\(77\) 3.00000 0.341882
\(78\) 0 0
\(79\) −1.00000 −0.112509 −0.0562544 0.998416i \(-0.517916\pi\)
−0.0562544 + 0.998416i \(0.517916\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) −3.00000 −0.331295
\(83\) −6.00000 −0.658586 −0.329293 0.944228i \(-0.606810\pi\)
−0.329293 + 0.944228i \(0.606810\pi\)
\(84\) 1.00000 0.109109
\(85\) 0 0
\(86\) 8.00000 0.862662
\(87\) −3.00000 −0.321634
\(88\) 3.00000 0.319801
\(89\) −9.00000 −0.953998 −0.476999 0.878904i \(-0.658275\pi\)
−0.476999 + 0.878904i \(0.658275\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 6.00000 0.625543
\(93\) −4.00000 −0.414781
\(94\) 9.00000 0.928279
\(95\) 0 0
\(96\) 1.00000 0.102062
\(97\) 8.00000 0.812277 0.406138 0.913812i \(-0.366875\pi\)
0.406138 + 0.913812i \(0.366875\pi\)
\(98\) 1.00000 0.101015
\(99\) 3.00000 0.301511
\(100\) −5.00000 −0.500000
\(101\) 6.00000 0.597022 0.298511 0.954406i \(-0.403510\pi\)
0.298511 + 0.954406i \(0.403510\pi\)
\(102\) 3.00000 0.297044
\(103\) 8.00000 0.788263 0.394132 0.919054i \(-0.371045\pi\)
0.394132 + 0.919054i \(0.371045\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) −9.00000 −0.874157
\(107\) 15.0000 1.45010 0.725052 0.688694i \(-0.241816\pi\)
0.725052 + 0.688694i \(0.241816\pi\)
\(108\) 1.00000 0.0962250
\(109\) −10.0000 −0.957826 −0.478913 0.877862i \(-0.658969\pi\)
−0.478913 + 0.877862i \(0.658969\pi\)
\(110\) 0 0
\(111\) −4.00000 −0.379663
\(112\) 1.00000 0.0944911
\(113\) −12.0000 −1.12887 −0.564433 0.825479i \(-0.690905\pi\)
−0.564433 + 0.825479i \(0.690905\pi\)
\(114\) 5.00000 0.468293
\(115\) 0 0
\(116\) −3.00000 −0.278543
\(117\) 0 0
\(118\) −6.00000 −0.552345
\(119\) 3.00000 0.275010
\(120\) 0 0
\(121\) −2.00000 −0.181818
\(122\) 5.00000 0.452679
\(123\) −3.00000 −0.270501
\(124\) −4.00000 −0.359211
\(125\) 0 0
\(126\) 1.00000 0.0890871
\(127\) 8.00000 0.709885 0.354943 0.934888i \(-0.384500\pi\)
0.354943 + 0.934888i \(0.384500\pi\)
\(128\) 1.00000 0.0883883
\(129\) 8.00000 0.704361
\(130\) 0 0
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 3.00000 0.261116
\(133\) 5.00000 0.433555
\(134\) 14.0000 1.20942
\(135\) 0 0
\(136\) 3.00000 0.257248
\(137\) −18.0000 −1.53784 −0.768922 0.639343i \(-0.779207\pi\)
−0.768922 + 0.639343i \(0.779207\pi\)
\(138\) 6.00000 0.510754
\(139\) −13.0000 −1.10265 −0.551323 0.834292i \(-0.685877\pi\)
−0.551323 + 0.834292i \(0.685877\pi\)
\(140\) 0 0
\(141\) 9.00000 0.757937
\(142\) −6.00000 −0.503509
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) −4.00000 −0.331042
\(147\) 1.00000 0.0824786
\(148\) −4.00000 −0.328798
\(149\) 6.00000 0.491539 0.245770 0.969328i \(-0.420959\pi\)
0.245770 + 0.969328i \(0.420959\pi\)
\(150\) −5.00000 −0.408248
\(151\) 17.0000 1.38344 0.691720 0.722166i \(-0.256853\pi\)
0.691720 + 0.722166i \(0.256853\pi\)
\(152\) 5.00000 0.405554
\(153\) 3.00000 0.242536
\(154\) 3.00000 0.241747
\(155\) 0 0
\(156\) 0 0
\(157\) −22.0000 −1.75579 −0.877896 0.478852i \(-0.841053\pi\)
−0.877896 + 0.478852i \(0.841053\pi\)
\(158\) −1.00000 −0.0795557
\(159\) −9.00000 −0.713746
\(160\) 0 0
\(161\) 6.00000 0.472866
\(162\) 1.00000 0.0785674
\(163\) −16.0000 −1.25322 −0.626608 0.779334i \(-0.715557\pi\)
−0.626608 + 0.779334i \(0.715557\pi\)
\(164\) −3.00000 −0.234261
\(165\) 0 0
\(166\) −6.00000 −0.465690
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 1.00000 0.0771517
\(169\) 0 0
\(170\) 0 0
\(171\) 5.00000 0.382360
\(172\) 8.00000 0.609994
\(173\) 18.0000 1.36851 0.684257 0.729241i \(-0.260127\pi\)
0.684257 + 0.729241i \(0.260127\pi\)
\(174\) −3.00000 −0.227429
\(175\) −5.00000 −0.377964
\(176\) 3.00000 0.226134
\(177\) −6.00000 −0.450988
\(178\) −9.00000 −0.674579
\(179\) 24.0000 1.79384 0.896922 0.442189i \(-0.145798\pi\)
0.896922 + 0.442189i \(0.145798\pi\)
\(180\) 0 0
\(181\) 17.0000 1.26360 0.631800 0.775131i \(-0.282316\pi\)
0.631800 + 0.775131i \(0.282316\pi\)
\(182\) 0 0
\(183\) 5.00000 0.369611
\(184\) 6.00000 0.442326
\(185\) 0 0
\(186\) −4.00000 −0.293294
\(187\) 9.00000 0.658145
\(188\) 9.00000 0.656392
\(189\) 1.00000 0.0727393
\(190\) 0 0
\(191\) 18.0000 1.30243 0.651217 0.758891i \(-0.274259\pi\)
0.651217 + 0.758891i \(0.274259\pi\)
\(192\) 1.00000 0.0721688
\(193\) −19.0000 −1.36765 −0.683825 0.729646i \(-0.739685\pi\)
−0.683825 + 0.729646i \(0.739685\pi\)
\(194\) 8.00000 0.574367
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) −9.00000 −0.641223 −0.320612 0.947211i \(-0.603888\pi\)
−0.320612 + 0.947211i \(0.603888\pi\)
\(198\) 3.00000 0.213201
\(199\) 20.0000 1.41776 0.708881 0.705328i \(-0.249200\pi\)
0.708881 + 0.705328i \(0.249200\pi\)
\(200\) −5.00000 −0.353553
\(201\) 14.0000 0.987484
\(202\) 6.00000 0.422159
\(203\) −3.00000 −0.210559
\(204\) 3.00000 0.210042
\(205\) 0 0
\(206\) 8.00000 0.557386
\(207\) 6.00000 0.417029
\(208\) 0 0
\(209\) 15.0000 1.03757
\(210\) 0 0
\(211\) −10.0000 −0.688428 −0.344214 0.938891i \(-0.611855\pi\)
−0.344214 + 0.938891i \(0.611855\pi\)
\(212\) −9.00000 −0.618123
\(213\) −6.00000 −0.411113
\(214\) 15.0000 1.02538
\(215\) 0 0
\(216\) 1.00000 0.0680414
\(217\) −4.00000 −0.271538
\(218\) −10.0000 −0.677285
\(219\) −4.00000 −0.270295
\(220\) 0 0
\(221\) 0 0
\(222\) −4.00000 −0.268462
\(223\) 8.00000 0.535720 0.267860 0.963458i \(-0.413684\pi\)
0.267860 + 0.963458i \(0.413684\pi\)
\(224\) 1.00000 0.0668153
\(225\) −5.00000 −0.333333
\(226\) −12.0000 −0.798228
\(227\) −18.0000 −1.19470 −0.597351 0.801980i \(-0.703780\pi\)
−0.597351 + 0.801980i \(0.703780\pi\)
\(228\) 5.00000 0.331133
\(229\) 17.0000 1.12339 0.561696 0.827344i \(-0.310149\pi\)
0.561696 + 0.827344i \(0.310149\pi\)
\(230\) 0 0
\(231\) 3.00000 0.197386
\(232\) −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
\(236\) −6.00000 −0.390567
\(237\) −1.00000 −0.0649570
\(238\) 3.00000 0.194461
\(239\) 6.00000 0.388108 0.194054 0.980991i \(-0.437836\pi\)
0.194054 + 0.980991i \(0.437836\pi\)
\(240\) 0 0
\(241\) 8.00000 0.515325 0.257663 0.966235i \(-0.417048\pi\)
0.257663 + 0.966235i \(0.417048\pi\)
\(242\) −2.00000 −0.128565
\(243\) 1.00000 0.0641500
\(244\) 5.00000 0.320092
\(245\) 0 0
\(246\) −3.00000 −0.191273
\(247\) 0 0
\(248\) −4.00000 −0.254000
\(249\) −6.00000 −0.380235
\(250\) 0 0
\(251\) 30.0000 1.89358 0.946792 0.321847i \(-0.104304\pi\)
0.946792 + 0.321847i \(0.104304\pi\)
\(252\) 1.00000 0.0629941
\(253\) 18.0000 1.13165
\(254\) 8.00000 0.501965
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 15.0000 0.935674 0.467837 0.883815i \(-0.345033\pi\)
0.467837 + 0.883815i \(0.345033\pi\)
\(258\) 8.00000 0.498058
\(259\) −4.00000 −0.248548
\(260\) 0 0
\(261\) −3.00000 −0.185695
\(262\) 0 0
\(263\) 18.0000 1.10993 0.554964 0.831875i \(-0.312732\pi\)
0.554964 + 0.831875i \(0.312732\pi\)
\(264\) 3.00000 0.184637
\(265\) 0 0
\(266\) 5.00000 0.306570
\(267\) −9.00000 −0.550791
\(268\) 14.0000 0.855186
\(269\) 24.0000 1.46331 0.731653 0.681677i \(-0.238749\pi\)
0.731653 + 0.681677i \(0.238749\pi\)
\(270\) 0 0
\(271\) −28.0000 −1.70088 −0.850439 0.526073i \(-0.823664\pi\)
−0.850439 + 0.526073i \(0.823664\pi\)
\(272\) 3.00000 0.181902
\(273\) 0 0
\(274\) −18.0000 −1.08742
\(275\) −15.0000 −0.904534
\(276\) 6.00000 0.361158
\(277\) 2.00000 0.120168 0.0600842 0.998193i \(-0.480863\pi\)
0.0600842 + 0.998193i \(0.480863\pi\)
\(278\) −13.0000 −0.779688
\(279\) −4.00000 −0.239474
\(280\) 0 0
\(281\) 6.00000 0.357930 0.178965 0.983855i \(-0.442725\pi\)
0.178965 + 0.983855i \(0.442725\pi\)
\(282\) 9.00000 0.535942
\(283\) −4.00000 −0.237775 −0.118888 0.992908i \(-0.537933\pi\)
−0.118888 + 0.992908i \(0.537933\pi\)
\(284\) −6.00000 −0.356034
\(285\) 0 0
\(286\) 0 0
\(287\) −3.00000 −0.177084
\(288\) 1.00000 0.0589256
\(289\) −8.00000 −0.470588
\(290\) 0 0
\(291\) 8.00000 0.468968
\(292\) −4.00000 −0.234082
\(293\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(294\) 1.00000 0.0583212
\(295\) 0 0
\(296\) −4.00000 −0.232495
\(297\) 3.00000 0.174078
\(298\) 6.00000 0.347571
\(299\) 0 0
\(300\) −5.00000 −0.288675
\(301\) 8.00000 0.461112
\(302\) 17.0000 0.978240
\(303\) 6.00000 0.344691
\(304\) 5.00000 0.286770
\(305\) 0 0
\(306\) 3.00000 0.171499
\(307\) 11.0000 0.627803 0.313902 0.949456i \(-0.398364\pi\)
0.313902 + 0.949456i \(0.398364\pi\)
\(308\) 3.00000 0.170941
\(309\) 8.00000 0.455104
\(310\) 0 0
\(311\) −15.0000 −0.850572 −0.425286 0.905059i \(-0.639826\pi\)
−0.425286 + 0.905059i \(0.639826\pi\)
\(312\) 0 0
\(313\) −22.0000 −1.24351 −0.621757 0.783210i \(-0.713581\pi\)
−0.621757 + 0.783210i \(0.713581\pi\)
\(314\) −22.0000 −1.24153
\(315\) 0 0
\(316\) −1.00000 −0.0562544
\(317\) −6.00000 −0.336994 −0.168497 0.985702i \(-0.553891\pi\)
−0.168497 + 0.985702i \(0.553891\pi\)
\(318\) −9.00000 −0.504695
\(319\) −9.00000 −0.503903
\(320\) 0 0
\(321\) 15.0000 0.837218
\(322\) 6.00000 0.334367
\(323\) 15.0000 0.834622
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) −16.0000 −0.886158
\(327\) −10.0000 −0.553001
\(328\) −3.00000 −0.165647
\(329\) 9.00000 0.496186
\(330\) 0 0
\(331\) −10.0000 −0.549650 −0.274825 0.961494i \(-0.588620\pi\)
−0.274825 + 0.961494i \(0.588620\pi\)
\(332\) −6.00000 −0.329293
\(333\) −4.00000 −0.219199
\(334\) 0 0
\(335\) 0 0
\(336\) 1.00000 0.0545545
\(337\) −25.0000 −1.36184 −0.680918 0.732359i \(-0.738419\pi\)
−0.680918 + 0.732359i \(0.738419\pi\)
\(338\) 0 0
\(339\) −12.0000 −0.651751
\(340\) 0 0
\(341\) −12.0000 −0.649836
\(342\) 5.00000 0.270369
\(343\) 1.00000 0.0539949
\(344\) 8.00000 0.431331
\(345\) 0 0
\(346\) 18.0000 0.967686
\(347\) −27.0000 −1.44944 −0.724718 0.689046i \(-0.758030\pi\)
−0.724718 + 0.689046i \(0.758030\pi\)
\(348\) −3.00000 −0.160817
\(349\) 2.00000 0.107058 0.0535288 0.998566i \(-0.482953\pi\)
0.0535288 + 0.998566i \(0.482953\pi\)
\(350\) −5.00000 −0.267261
\(351\) 0 0
\(352\) 3.00000 0.159901
\(353\) −18.0000 −0.958043 −0.479022 0.877803i \(-0.659008\pi\)
−0.479022 + 0.877803i \(0.659008\pi\)
\(354\) −6.00000 −0.318896
\(355\) 0 0
\(356\) −9.00000 −0.476999
\(357\) 3.00000 0.158777
\(358\) 24.0000 1.26844
\(359\) −12.0000 −0.633336 −0.316668 0.948536i \(-0.602564\pi\)
−0.316668 + 0.948536i \(0.602564\pi\)
\(360\) 0 0
\(361\) 6.00000 0.315789
\(362\) 17.0000 0.893500
\(363\) −2.00000 −0.104973
\(364\) 0 0
\(365\) 0 0
\(366\) 5.00000 0.261354
\(367\) 2.00000 0.104399 0.0521996 0.998637i \(-0.483377\pi\)
0.0521996 + 0.998637i \(0.483377\pi\)
\(368\) 6.00000 0.312772
\(369\) −3.00000 −0.156174
\(370\) 0 0
\(371\) −9.00000 −0.467257
\(372\) −4.00000 −0.207390
\(373\) 14.0000 0.724893 0.362446 0.932005i \(-0.381942\pi\)
0.362446 + 0.932005i \(0.381942\pi\)
\(374\) 9.00000 0.465379
\(375\) 0 0
\(376\) 9.00000 0.464140
\(377\) 0 0
\(378\) 1.00000 0.0514344
\(379\) −34.0000 −1.74646 −0.873231 0.487306i \(-0.837980\pi\)
−0.873231 + 0.487306i \(0.837980\pi\)
\(380\) 0 0
\(381\) 8.00000 0.409852
\(382\) 18.0000 0.920960
\(383\) 9.00000 0.459879 0.229939 0.973205i \(-0.426147\pi\)
0.229939 + 0.973205i \(0.426147\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) −19.0000 −0.967075
\(387\) 8.00000 0.406663
\(388\) 8.00000 0.406138
\(389\) −6.00000 −0.304212 −0.152106 0.988364i \(-0.548606\pi\)
−0.152106 + 0.988364i \(0.548606\pi\)
\(390\) 0 0
\(391\) 18.0000 0.910299
\(392\) 1.00000 0.0505076
\(393\) 0 0
\(394\) −9.00000 −0.453413
\(395\) 0 0
\(396\) 3.00000 0.150756
\(397\) −37.0000 −1.85698 −0.928488 0.371361i \(-0.878891\pi\)
−0.928488 + 0.371361i \(0.878891\pi\)
\(398\) 20.0000 1.00251
\(399\) 5.00000 0.250313
\(400\) −5.00000 −0.250000
\(401\) 30.0000 1.49813 0.749064 0.662497i \(-0.230503\pi\)
0.749064 + 0.662497i \(0.230503\pi\)
\(402\) 14.0000 0.698257
\(403\) 0 0
\(404\) 6.00000 0.298511
\(405\) 0 0
\(406\) −3.00000 −0.148888
\(407\) −12.0000 −0.594818
\(408\) 3.00000 0.148522
\(409\) 14.0000 0.692255 0.346128 0.938187i \(-0.387496\pi\)
0.346128 + 0.938187i \(0.387496\pi\)
\(410\) 0 0
\(411\) −18.0000 −0.887875
\(412\) 8.00000 0.394132
\(413\) −6.00000 −0.295241
\(414\) 6.00000 0.294884
\(415\) 0 0
\(416\) 0 0
\(417\) −13.0000 −0.636613
\(418\) 15.0000 0.733674
\(419\) 6.00000 0.293119 0.146560 0.989202i \(-0.453180\pi\)
0.146560 + 0.989202i \(0.453180\pi\)
\(420\) 0 0
\(421\) −10.0000 −0.487370 −0.243685 0.969854i \(-0.578356\pi\)
−0.243685 + 0.969854i \(0.578356\pi\)
\(422\) −10.0000 −0.486792
\(423\) 9.00000 0.437595
\(424\) −9.00000 −0.437079
\(425\) −15.0000 −0.727607
\(426\) −6.00000 −0.290701
\(427\) 5.00000 0.241967
\(428\) 15.0000 0.725052
\(429\) 0 0
\(430\) 0 0
\(431\) −30.0000 −1.44505 −0.722525 0.691345i \(-0.757018\pi\)
−0.722525 + 0.691345i \(0.757018\pi\)
\(432\) 1.00000 0.0481125
\(433\) −34.0000 −1.63394 −0.816968 0.576683i \(-0.804347\pi\)
−0.816968 + 0.576683i \(0.804347\pi\)
\(434\) −4.00000 −0.192006
\(435\) 0 0
\(436\) −10.0000 −0.478913
\(437\) 30.0000 1.43509
\(438\) −4.00000 −0.191127
\(439\) −4.00000 −0.190910 −0.0954548 0.995434i \(-0.530431\pi\)
−0.0954548 + 0.995434i \(0.530431\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) −9.00000 −0.427603 −0.213801 0.976877i \(-0.568585\pi\)
−0.213801 + 0.976877i \(0.568585\pi\)
\(444\) −4.00000 −0.189832
\(445\) 0 0
\(446\) 8.00000 0.378811
\(447\) 6.00000 0.283790
\(448\) 1.00000 0.0472456
\(449\) 12.0000 0.566315 0.283158 0.959073i \(-0.408618\pi\)
0.283158 + 0.959073i \(0.408618\pi\)
\(450\) −5.00000 −0.235702
\(451\) −9.00000 −0.423793
\(452\) −12.0000 −0.564433
\(453\) 17.0000 0.798730
\(454\) −18.0000 −0.844782
\(455\) 0 0
\(456\) 5.00000 0.234146
\(457\) −10.0000 −0.467780 −0.233890 0.972263i \(-0.575146\pi\)
−0.233890 + 0.972263i \(0.575146\pi\)
\(458\) 17.0000 0.794358
\(459\) 3.00000 0.140028
\(460\) 0 0
\(461\) −12.0000 −0.558896 −0.279448 0.960161i \(-0.590151\pi\)
−0.279448 + 0.960161i \(0.590151\pi\)
\(462\) 3.00000 0.139573
\(463\) 5.00000 0.232370 0.116185 0.993228i \(-0.462933\pi\)
0.116185 + 0.993228i \(0.462933\pi\)
\(464\) −3.00000 −0.139272
\(465\) 0 0
\(466\) −12.0000 −0.555889
\(467\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(468\) 0 0
\(469\) 14.0000 0.646460
\(470\) 0 0
\(471\) −22.0000 −1.01371
\(472\) −6.00000 −0.276172
\(473\) 24.0000 1.10352
\(474\) −1.00000 −0.0459315
\(475\) −25.0000 −1.14708
\(476\) 3.00000 0.137505
\(477\) −9.00000 −0.412082
\(478\) 6.00000 0.274434
\(479\) 27.0000 1.23366 0.616831 0.787096i \(-0.288416\pi\)
0.616831 + 0.787096i \(0.288416\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 8.00000 0.364390
\(483\) 6.00000 0.273009
\(484\) −2.00000 −0.0909091
\(485\) 0 0
\(486\) 1.00000 0.0453609
\(487\) 23.0000 1.04223 0.521115 0.853487i \(-0.325516\pi\)
0.521115 + 0.853487i \(0.325516\pi\)
\(488\) 5.00000 0.226339
\(489\) −16.0000 −0.723545
\(490\) 0 0
\(491\) 36.0000 1.62466 0.812329 0.583200i \(-0.198200\pi\)
0.812329 + 0.583200i \(0.198200\pi\)
\(492\) −3.00000 −0.135250
\(493\) −9.00000 −0.405340
\(494\) 0 0
\(495\) 0 0
\(496\) −4.00000 −0.179605
\(497\) −6.00000 −0.269137
\(498\) −6.00000 −0.268866
\(499\) −40.0000 −1.79065 −0.895323 0.445418i \(-0.853055\pi\)
−0.895323 + 0.445418i \(0.853055\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 30.0000 1.33897
\(503\) 24.0000 1.07011 0.535054 0.844818i \(-0.320291\pi\)
0.535054 + 0.844818i \(0.320291\pi\)
\(504\) 1.00000 0.0445435
\(505\) 0 0
\(506\) 18.0000 0.800198
\(507\) 0 0
\(508\) 8.00000 0.354943
\(509\) 30.0000 1.32973 0.664863 0.746965i \(-0.268490\pi\)
0.664863 + 0.746965i \(0.268490\pi\)
\(510\) 0 0
\(511\) −4.00000 −0.176950
\(512\) 1.00000 0.0441942
\(513\) 5.00000 0.220755
\(514\) 15.0000 0.661622
\(515\) 0 0
\(516\) 8.00000 0.352180
\(517\) 27.0000 1.18746
\(518\) −4.00000 −0.175750
\(519\) 18.0000 0.790112
\(520\) 0 0
\(521\) 27.0000 1.18289 0.591446 0.806345i \(-0.298557\pi\)
0.591446 + 0.806345i \(0.298557\pi\)
\(522\) −3.00000 −0.131306
\(523\) 29.0000 1.26808 0.634041 0.773300i \(-0.281395\pi\)
0.634041 + 0.773300i \(0.281395\pi\)
\(524\) 0 0
\(525\) −5.00000 −0.218218
\(526\) 18.0000 0.784837
\(527\) −12.0000 −0.522728
\(528\) 3.00000 0.130558
\(529\) 13.0000 0.565217
\(530\) 0 0
\(531\) −6.00000 −0.260378
\(532\) 5.00000 0.216777
\(533\) 0 0
\(534\) −9.00000 −0.389468
\(535\) 0 0
\(536\) 14.0000 0.604708
\(537\) 24.0000 1.03568
\(538\) 24.0000 1.03471
\(539\) 3.00000 0.129219
\(540\) 0 0
\(541\) −10.0000 −0.429934 −0.214967 0.976621i \(-0.568964\pi\)
−0.214967 + 0.976621i \(0.568964\pi\)
\(542\) −28.0000 −1.20270
\(543\) 17.0000 0.729540
\(544\) 3.00000 0.128624
\(545\) 0 0
\(546\) 0 0
\(547\) −22.0000 −0.940652 −0.470326 0.882493i \(-0.655864\pi\)
−0.470326 + 0.882493i \(0.655864\pi\)
\(548\) −18.0000 −0.768922
\(549\) 5.00000 0.213395
\(550\) −15.0000 −0.639602
\(551\) −15.0000 −0.639021
\(552\) 6.00000 0.255377
\(553\) −1.00000 −0.0425243
\(554\) 2.00000 0.0849719
\(555\) 0 0
\(556\) −13.0000 −0.551323
\(557\) −15.0000 −0.635570 −0.317785 0.948163i \(-0.602939\pi\)
−0.317785 + 0.948163i \(0.602939\pi\)
\(558\) −4.00000 −0.169334
\(559\) 0 0
\(560\) 0 0
\(561\) 9.00000 0.379980
\(562\) 6.00000 0.253095
\(563\) 24.0000 1.01148 0.505740 0.862686i \(-0.331220\pi\)
0.505740 + 0.862686i \(0.331220\pi\)
\(564\) 9.00000 0.378968
\(565\) 0 0
\(566\) −4.00000 −0.168133
\(567\) 1.00000 0.0419961
\(568\) −6.00000 −0.251754
\(569\) −6.00000 −0.251533 −0.125767 0.992060i \(-0.540139\pi\)
−0.125767 + 0.992060i \(0.540139\pi\)
\(570\) 0 0
\(571\) 2.00000 0.0836974 0.0418487 0.999124i \(-0.486675\pi\)
0.0418487 + 0.999124i \(0.486675\pi\)
\(572\) 0 0
\(573\) 18.0000 0.751961
\(574\) −3.00000 −0.125218
\(575\) −30.0000 −1.25109
\(576\) 1.00000 0.0416667
\(577\) 2.00000 0.0832611 0.0416305 0.999133i \(-0.486745\pi\)
0.0416305 + 0.999133i \(0.486745\pi\)
\(578\) −8.00000 −0.332756
\(579\) −19.0000 −0.789613
\(580\) 0 0
\(581\) −6.00000 −0.248922
\(582\) 8.00000 0.331611
\(583\) −27.0000 −1.11823
\(584\) −4.00000 −0.165521
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(588\) 1.00000 0.0412393
\(589\) −20.0000 −0.824086
\(590\) 0 0
\(591\) −9.00000 −0.370211
\(592\) −4.00000 −0.164399
\(593\) −15.0000 −0.615976 −0.307988 0.951390i \(-0.599656\pi\)
−0.307988 + 0.951390i \(0.599656\pi\)
\(594\) 3.00000 0.123091
\(595\) 0 0
\(596\) 6.00000 0.245770
\(597\) 20.0000 0.818546
\(598\) 0 0
\(599\) 6.00000 0.245153 0.122577 0.992459i \(-0.460884\pi\)
0.122577 + 0.992459i \(0.460884\pi\)
\(600\) −5.00000 −0.204124
\(601\) −40.0000 −1.63163 −0.815817 0.578310i \(-0.803712\pi\)
−0.815817 + 0.578310i \(0.803712\pi\)
\(602\) 8.00000 0.326056
\(603\) 14.0000 0.570124
\(604\) 17.0000 0.691720
\(605\) 0 0
\(606\) 6.00000 0.243733
\(607\) −16.0000 −0.649420 −0.324710 0.945814i \(-0.605267\pi\)
−0.324710 + 0.945814i \(0.605267\pi\)
\(608\) 5.00000 0.202777
\(609\) −3.00000 −0.121566
\(610\) 0 0
\(611\) 0 0
\(612\) 3.00000 0.121268
\(613\) 8.00000 0.323117 0.161558 0.986863i \(-0.448348\pi\)
0.161558 + 0.986863i \(0.448348\pi\)
\(614\) 11.0000 0.443924
\(615\) 0 0
\(616\) 3.00000 0.120873
\(617\) −18.0000 −0.724653 −0.362326 0.932051i \(-0.618017\pi\)
−0.362326 + 0.932051i \(0.618017\pi\)
\(618\) 8.00000 0.321807
\(619\) −25.0000 −1.00483 −0.502417 0.864625i \(-0.667556\pi\)
−0.502417 + 0.864625i \(0.667556\pi\)
\(620\) 0 0
\(621\) 6.00000 0.240772
\(622\) −15.0000 −0.601445
\(623\) −9.00000 −0.360577
\(624\) 0 0
\(625\) 25.0000 1.00000
\(626\) −22.0000 −0.879297
\(627\) 15.0000 0.599042
\(628\) −22.0000 −0.877896
\(629\) −12.0000 −0.478471
\(630\) 0 0
\(631\) 23.0000 0.915616 0.457808 0.889051i \(-0.348635\pi\)
0.457808 + 0.889051i \(0.348635\pi\)
\(632\) −1.00000 −0.0397779
\(633\) −10.0000 −0.397464
\(634\) −6.00000 −0.238290
\(635\) 0 0
\(636\) −9.00000 −0.356873
\(637\) 0 0
\(638\) −9.00000 −0.356313
\(639\) −6.00000 −0.237356
\(640\) 0 0
\(641\) 6.00000 0.236986 0.118493 0.992955i \(-0.462194\pi\)
0.118493 + 0.992955i \(0.462194\pi\)
\(642\) 15.0000 0.592003
\(643\) 17.0000 0.670415 0.335207 0.942144i \(-0.391194\pi\)
0.335207 + 0.942144i \(0.391194\pi\)
\(644\) 6.00000 0.236433
\(645\) 0 0
\(646\) 15.0000 0.590167
\(647\) −45.0000 −1.76913 −0.884566 0.466415i \(-0.845546\pi\)
−0.884566 + 0.466415i \(0.845546\pi\)
\(648\) 1.00000 0.0392837
\(649\) −18.0000 −0.706562
\(650\) 0 0
\(651\) −4.00000 −0.156772
\(652\) −16.0000 −0.626608
\(653\) −33.0000 −1.29139 −0.645695 0.763596i \(-0.723432\pi\)
−0.645695 + 0.763596i \(0.723432\pi\)
\(654\) −10.0000 −0.391031
\(655\) 0 0
\(656\) −3.00000 −0.117130
\(657\) −4.00000 −0.156055
\(658\) 9.00000 0.350857
\(659\) −15.0000 −0.584317 −0.292159 0.956370i \(-0.594373\pi\)
−0.292159 + 0.956370i \(0.594373\pi\)
\(660\) 0 0
\(661\) −22.0000 −0.855701 −0.427850 0.903850i \(-0.640729\pi\)
−0.427850 + 0.903850i \(0.640729\pi\)
\(662\) −10.0000 −0.388661
\(663\) 0 0
\(664\) −6.00000 −0.232845
\(665\) 0 0
\(666\) −4.00000 −0.154997
\(667\) −18.0000 −0.696963
\(668\) 0 0
\(669\) 8.00000 0.309298
\(670\) 0 0
\(671\) 15.0000 0.579069
\(672\) 1.00000 0.0385758
\(673\) −1.00000 −0.0385472 −0.0192736 0.999814i \(-0.506135\pi\)
−0.0192736 + 0.999814i \(0.506135\pi\)
\(674\) −25.0000 −0.962964
\(675\) −5.00000 −0.192450
\(676\) 0 0
\(677\) 6.00000 0.230599 0.115299 0.993331i \(-0.463217\pi\)
0.115299 + 0.993331i \(0.463217\pi\)
\(678\) −12.0000 −0.460857
\(679\) 8.00000 0.307012
\(680\) 0 0
\(681\) −18.0000 −0.689761
\(682\) −12.0000 −0.459504
\(683\) 12.0000 0.459167 0.229584 0.973289i \(-0.426264\pi\)
0.229584 + 0.973289i \(0.426264\pi\)
\(684\) 5.00000 0.191180
\(685\) 0 0
\(686\) 1.00000 0.0381802
\(687\) 17.0000 0.648590
\(688\) 8.00000 0.304997
\(689\) 0 0
\(690\) 0 0
\(691\) 8.00000 0.304334 0.152167 0.988355i \(-0.451375\pi\)
0.152167 + 0.988355i \(0.451375\pi\)
\(692\) 18.0000 0.684257
\(693\) 3.00000 0.113961
\(694\) −27.0000 −1.02491
\(695\) 0 0
\(696\) −3.00000 −0.113715
\(697\) −9.00000 −0.340899
\(698\) 2.00000 0.0757011
\(699\) −12.0000 −0.453882
\(700\) −5.00000 −0.188982
\(701\) −15.0000 −0.566542 −0.283271 0.959040i \(-0.591420\pi\)
−0.283271 + 0.959040i \(0.591420\pi\)
\(702\) 0 0
\(703\) −20.0000 −0.754314
\(704\) 3.00000 0.113067
\(705\) 0 0
\(706\) −18.0000 −0.677439
\(707\) 6.00000 0.225653
\(708\) −6.00000 −0.225494
\(709\) 2.00000 0.0751116 0.0375558 0.999295i \(-0.488043\pi\)
0.0375558 + 0.999295i \(0.488043\pi\)
\(710\) 0 0
\(711\) −1.00000 −0.0375029
\(712\) −9.00000 −0.337289
\(713\) −24.0000 −0.898807
\(714\) 3.00000 0.112272
\(715\) 0 0
\(716\) 24.0000 0.896922
\(717\) 6.00000 0.224074
\(718\) −12.0000 −0.447836
\(719\) 27.0000 1.00693 0.503465 0.864016i \(-0.332058\pi\)
0.503465 + 0.864016i \(0.332058\pi\)
\(720\) 0 0
\(721\) 8.00000 0.297936
\(722\) 6.00000 0.223297
\(723\) 8.00000 0.297523
\(724\) 17.0000 0.631800
\(725\) 15.0000 0.557086
\(726\) −2.00000 −0.0742270
\(727\) −28.0000 −1.03846 −0.519231 0.854634i \(-0.673782\pi\)
−0.519231 + 0.854634i \(0.673782\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 24.0000 0.887672
\(732\) 5.00000 0.184805
\(733\) −49.0000 −1.80986 −0.904928 0.425564i \(-0.860076\pi\)
−0.904928 + 0.425564i \(0.860076\pi\)
\(734\) 2.00000 0.0738213
\(735\) 0 0
\(736\) 6.00000 0.221163
\(737\) 42.0000 1.54709
\(738\) −3.00000 −0.110432
\(739\) −10.0000 −0.367856 −0.183928 0.982940i \(-0.558881\pi\)
−0.183928 + 0.982940i \(0.558881\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −9.00000 −0.330400
\(743\) −24.0000 −0.880475 −0.440237 0.897881i \(-0.645106\pi\)
−0.440237 + 0.897881i \(0.645106\pi\)
\(744\) −4.00000 −0.146647
\(745\) 0 0
\(746\) 14.0000 0.512576
\(747\) −6.00000 −0.219529
\(748\) 9.00000 0.329073
\(749\) 15.0000 0.548088
\(750\) 0 0
\(751\) 11.0000 0.401396 0.200698 0.979653i \(-0.435679\pi\)
0.200698 + 0.979653i \(0.435679\pi\)
\(752\) 9.00000 0.328196
\(753\) 30.0000 1.09326
\(754\) 0 0
\(755\) 0 0
\(756\) 1.00000 0.0363696
\(757\) 2.00000 0.0726912 0.0363456 0.999339i \(-0.488428\pi\)
0.0363456 + 0.999339i \(0.488428\pi\)
\(758\) −34.0000 −1.23494
\(759\) 18.0000 0.653359
\(760\) 0 0
\(761\) −42.0000 −1.52250 −0.761249 0.648459i \(-0.775414\pi\)
−0.761249 + 0.648459i \(0.775414\pi\)
\(762\) 8.00000 0.289809
\(763\) −10.0000 −0.362024
\(764\) 18.0000 0.651217
\(765\) 0 0
\(766\) 9.00000 0.325183
\(767\) 0 0
\(768\) 1.00000 0.0360844
\(769\) 44.0000 1.58668 0.793340 0.608778i \(-0.208340\pi\)
0.793340 + 0.608778i \(0.208340\pi\)
\(770\) 0 0
\(771\) 15.0000 0.540212
\(772\) −19.0000 −0.683825
\(773\) 18.0000 0.647415 0.323708 0.946157i \(-0.395071\pi\)
0.323708 + 0.946157i \(0.395071\pi\)
\(774\) 8.00000 0.287554
\(775\) 20.0000 0.718421
\(776\) 8.00000 0.287183
\(777\) −4.00000 −0.143499
\(778\) −6.00000 −0.215110
\(779\) −15.0000 −0.537431
\(780\) 0 0
\(781\) −18.0000 −0.644091
\(782\) 18.0000 0.643679
\(783\) −3.00000 −0.107211
\(784\) 1.00000 0.0357143
\(785\) 0 0
\(786\) 0 0
\(787\) 11.0000 0.392108 0.196054 0.980593i \(-0.437187\pi\)
0.196054 + 0.980593i \(0.437187\pi\)
\(788\) −9.00000 −0.320612
\(789\) 18.0000 0.640817
\(790\) 0 0
\(791\) −12.0000 −0.426671
\(792\) 3.00000 0.106600
\(793\) 0 0
\(794\) −37.0000 −1.31308
\(795\) 0 0
\(796\) 20.0000 0.708881
\(797\) −6.00000 −0.212531 −0.106265 0.994338i \(-0.533889\pi\)
−0.106265 + 0.994338i \(0.533889\pi\)
\(798\) 5.00000 0.176998
\(799\) 27.0000 0.955191
\(800\) −5.00000 −0.176777
\(801\) −9.00000 −0.317999
\(802\) 30.0000 1.05934
\(803\) −12.0000 −0.423471
\(804\) 14.0000 0.493742
\(805\) 0 0
\(806\) 0 0
\(807\) 24.0000 0.844840
\(808\) 6.00000 0.211079
\(809\) −42.0000 −1.47664 −0.738321 0.674450i \(-0.764381\pi\)
−0.738321 + 0.674450i \(0.764381\pi\)
\(810\) 0 0
\(811\) −40.0000 −1.40459 −0.702295 0.711886i \(-0.747841\pi\)
−0.702295 + 0.711886i \(0.747841\pi\)
\(812\) −3.00000 −0.105279
\(813\) −28.0000 −0.982003
\(814\) −12.0000 −0.420600
\(815\) 0 0
\(816\) 3.00000 0.105021
\(817\) 40.0000 1.39942
\(818\) 14.0000 0.489499
\(819\) 0 0
\(820\) 0 0
\(821\) −45.0000 −1.57051 −0.785255 0.619172i \(-0.787468\pi\)
−0.785255 + 0.619172i \(0.787468\pi\)
\(822\) −18.0000 −0.627822
\(823\) 32.0000 1.11545 0.557725 0.830026i \(-0.311674\pi\)
0.557725 + 0.830026i \(0.311674\pi\)
\(824\) 8.00000 0.278693
\(825\) −15.0000 −0.522233
\(826\) −6.00000 −0.208767
\(827\) −12.0000 −0.417281 −0.208640 0.977992i \(-0.566904\pi\)
−0.208640 + 0.977992i \(0.566904\pi\)
\(828\) 6.00000 0.208514
\(829\) −25.0000 −0.868286 −0.434143 0.900844i \(-0.642949\pi\)
−0.434143 + 0.900844i \(0.642949\pi\)
\(830\) 0 0
\(831\) 2.00000 0.0693792
\(832\) 0 0
\(833\) 3.00000 0.103944
\(834\) −13.0000 −0.450153
\(835\) 0 0
\(836\) 15.0000 0.518786
\(837\) −4.00000 −0.138260
\(838\) 6.00000 0.207267
\(839\) 36.0000 1.24286 0.621429 0.783470i \(-0.286552\pi\)
0.621429 + 0.783470i \(0.286552\pi\)
\(840\) 0 0
\(841\) −20.0000 −0.689655
\(842\) −10.0000 −0.344623
\(843\) 6.00000 0.206651
\(844\) −10.0000 −0.344214
\(845\) 0 0
\(846\) 9.00000 0.309426
\(847\) −2.00000 −0.0687208
\(848\) −9.00000 −0.309061
\(849\) −4.00000 −0.137280
\(850\) −15.0000 −0.514496
\(851\) −24.0000 −0.822709
\(852\) −6.00000 −0.205557
\(853\) −37.0000 −1.26686 −0.633428 0.773802i \(-0.718353\pi\)
−0.633428 + 0.773802i \(0.718353\pi\)
\(854\) 5.00000 0.171096
\(855\) 0 0
\(856\) 15.0000 0.512689
\(857\) 18.0000 0.614868 0.307434 0.951569i \(-0.400530\pi\)
0.307434 + 0.951569i \(0.400530\pi\)
\(858\) 0 0
\(859\) 5.00000 0.170598 0.0852989 0.996355i \(-0.472815\pi\)
0.0852989 + 0.996355i \(0.472815\pi\)
\(860\) 0 0
\(861\) −3.00000 −0.102240
\(862\) −30.0000 −1.02180
\(863\) −30.0000 −1.02121 −0.510606 0.859815i \(-0.670579\pi\)
−0.510606 + 0.859815i \(0.670579\pi\)
\(864\) 1.00000 0.0340207
\(865\) 0 0
\(866\) −34.0000 −1.15537
\(867\) −8.00000 −0.271694
\(868\) −4.00000 −0.135769
\(869\) −3.00000 −0.101768
\(870\) 0 0
\(871\) 0 0
\(872\) −10.0000 −0.338643
\(873\) 8.00000 0.270759
\(874\) 30.0000 1.01477
\(875\) 0 0
\(876\) −4.00000 −0.135147
\(877\) −22.0000 −0.742887 −0.371444 0.928456i \(-0.621137\pi\)
−0.371444 + 0.928456i \(0.621137\pi\)
\(878\) −4.00000 −0.134993
\(879\) 0 0
\(880\) 0 0
\(881\) 42.0000 1.41502 0.707508 0.706705i \(-0.249819\pi\)
0.707508 + 0.706705i \(0.249819\pi\)
\(882\) 1.00000 0.0336718
\(883\) −52.0000 −1.74994 −0.874970 0.484178i \(-0.839119\pi\)
−0.874970 + 0.484178i \(0.839119\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −9.00000 −0.302361
\(887\) 3.00000 0.100730 0.0503651 0.998731i \(-0.483962\pi\)
0.0503651 + 0.998731i \(0.483962\pi\)
\(888\) −4.00000 −0.134231
\(889\) 8.00000 0.268311
\(890\) 0 0
\(891\) 3.00000 0.100504
\(892\) 8.00000 0.267860
\(893\) 45.0000 1.50587
\(894\) 6.00000 0.200670
\(895\) 0 0
\(896\) 1.00000 0.0334077
\(897\) 0 0
\(898\) 12.0000 0.400445
\(899\) 12.0000 0.400222
\(900\) −5.00000 −0.166667
\(901\) −27.0000 −0.899500
\(902\) −9.00000 −0.299667
\(903\) 8.00000 0.266223
\(904\) −12.0000 −0.399114
\(905\) 0 0
\(906\) 17.0000 0.564787
\(907\) 50.0000 1.66022 0.830111 0.557598i \(-0.188277\pi\)
0.830111 + 0.557598i \(0.188277\pi\)
\(908\) −18.0000 −0.597351
\(909\) 6.00000 0.199007
\(910\) 0 0
\(911\) 6.00000 0.198789 0.0993944 0.995048i \(-0.468309\pi\)
0.0993944 + 0.995048i \(0.468309\pi\)
\(912\) 5.00000 0.165567
\(913\) −18.0000 −0.595713
\(914\) −10.0000 −0.330771
\(915\) 0 0
\(916\) 17.0000 0.561696
\(917\) 0 0
\(918\) 3.00000 0.0990148
\(919\) −7.00000 −0.230909 −0.115454 0.993313i \(-0.536832\pi\)
−0.115454 + 0.993313i \(0.536832\pi\)
\(920\) 0 0
\(921\) 11.0000 0.362462
\(922\) −12.0000 −0.395199
\(923\) 0 0
\(924\) 3.00000 0.0986928
\(925\) 20.0000 0.657596
\(926\) 5.00000 0.164310
\(927\) 8.00000 0.262754
\(928\) −3.00000 −0.0984798
\(929\) −39.0000 −1.27955 −0.639774 0.768563i \(-0.720972\pi\)
−0.639774 + 0.768563i \(0.720972\pi\)
\(930\) 0 0
\(931\) 5.00000 0.163868
\(932\) −12.0000 −0.393073
\(933\) −15.0000 −0.491078
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 32.0000 1.04539 0.522697 0.852518i \(-0.324926\pi\)
0.522697 + 0.852518i \(0.324926\pi\)
\(938\) 14.0000 0.457116
\(939\) −22.0000 −0.717943
\(940\) 0 0
\(941\) 12.0000 0.391189 0.195594 0.980685i \(-0.437336\pi\)
0.195594 + 0.980685i \(0.437336\pi\)
\(942\) −22.0000 −0.716799
\(943\) −18.0000 −0.586161
\(944\) −6.00000 −0.195283
\(945\) 0 0
\(946\) 24.0000 0.780307
\(947\) −27.0000 −0.877382 −0.438691 0.898638i \(-0.644558\pi\)
−0.438691 + 0.898638i \(0.644558\pi\)
\(948\) −1.00000 −0.0324785
\(949\) 0 0
\(950\) −25.0000 −0.811107
\(951\) −6.00000 −0.194563
\(952\) 3.00000 0.0972306
\(953\) −36.0000 −1.16615 −0.583077 0.812417i \(-0.698151\pi\)
−0.583077 + 0.812417i \(0.698151\pi\)
\(954\) −9.00000 −0.291386
\(955\) 0 0
\(956\) 6.00000 0.194054
\(957\) −9.00000 −0.290929
\(958\) 27.0000 0.872330
\(959\) −18.0000 −0.581250
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) 0 0
\(963\) 15.0000 0.483368
\(964\) 8.00000 0.257663
\(965\) 0 0
\(966\) 6.00000 0.193047
\(967\) 32.0000 1.02905 0.514525 0.857475i \(-0.327968\pi\)
0.514525 + 0.857475i \(0.327968\pi\)
\(968\) −2.00000 −0.0642824
\(969\) 15.0000 0.481869
\(970\) 0 0
\(971\) 42.0000 1.34784 0.673922 0.738802i \(-0.264608\pi\)
0.673922 + 0.738802i \(0.264608\pi\)
\(972\) 1.00000 0.0320750
\(973\) −13.0000 −0.416761
\(974\) 23.0000 0.736968
\(975\) 0 0
\(976\) 5.00000 0.160046
\(977\) −24.0000 −0.767828 −0.383914 0.923369i \(-0.625424\pi\)
−0.383914 + 0.923369i \(0.625424\pi\)
\(978\) −16.0000 −0.511624
\(979\) −27.0000 −0.862924
\(980\) 0 0
\(981\) −10.0000 −0.319275
\(982\) 36.0000 1.14881
\(983\) −48.0000 −1.53096 −0.765481 0.643458i \(-0.777499\pi\)
−0.765481 + 0.643458i \(0.777499\pi\)
\(984\) −3.00000 −0.0956365
\(985\) 0 0
\(986\) −9.00000 −0.286618
\(987\) 9.00000 0.286473
\(988\) 0 0
\(989\) 48.0000 1.52631
\(990\) 0 0
\(991\) −43.0000 −1.36594 −0.682970 0.730446i \(-0.739312\pi\)
−0.682970 + 0.730446i \(0.739312\pi\)
\(992\) −4.00000 −0.127000
\(993\) −10.0000 −0.317340
\(994\) −6.00000 −0.190308
\(995\) 0 0
\(996\) −6.00000 −0.190117
\(997\) −7.00000 −0.221692 −0.110846 0.993838i \(-0.535356\pi\)
−0.110846 + 0.993838i \(0.535356\pi\)
\(998\) −40.0000 −1.26618
\(999\) −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 7098.2.a.bb.1.1 1
13.3 even 3 546.2.l.a.295.1 yes 2
13.9 even 3 546.2.l.a.211.1 2
13.12 even 2 7098.2.a.l.1.1 1
39.29 odd 6 1638.2.r.r.1387.1 2
39.35 odd 6 1638.2.r.r.757.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
546.2.l.a.211.1 2 13.9 even 3
546.2.l.a.295.1 yes 2 13.3 even 3
1638.2.r.r.757.1 2 39.35 odd 6
1638.2.r.r.1387.1 2 39.29 odd 6
7098.2.a.l.1.1 1 13.12 even 2
7098.2.a.bb.1.1 1 1.1 even 1 trivial