Properties

Label 7098.2.a.h.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

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -2.00000 q^{5} -1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -2.00000 q^{5} -1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +2.00000 q^{10} -4.00000 q^{11} +1.00000 q^{12} +1.00000 q^{14} -2.00000 q^{15} +1.00000 q^{16} +7.00000 q^{17} -1.00000 q^{18} +2.00000 q^{19} -2.00000 q^{20} -1.00000 q^{21} +4.00000 q^{22} -1.00000 q^{23} -1.00000 q^{24} -1.00000 q^{25} +1.00000 q^{27} -1.00000 q^{28} -2.00000 q^{29} +2.00000 q^{30} -9.00000 q^{31} -1.00000 q^{32} -4.00000 q^{33} -7.00000 q^{34} +2.00000 q^{35} +1.00000 q^{36} -2.00000 q^{37} -2.00000 q^{38} +2.00000 q^{40} +2.00000 q^{41} +1.00000 q^{42} -5.00000 q^{43} -4.00000 q^{44} -2.00000 q^{45} +1.00000 q^{46} +6.00000 q^{47} +1.00000 q^{48} +1.00000 q^{49} +1.00000 q^{50} +7.00000 q^{51} +3.00000 q^{53} -1.00000 q^{54} +8.00000 q^{55} +1.00000 q^{56} +2.00000 q^{57} +2.00000 q^{58} +15.0000 q^{59} -2.00000 q^{60} -7.00000 q^{61} +9.00000 q^{62} -1.00000 q^{63} +1.00000 q^{64} +4.00000 q^{66} -5.00000 q^{67} +7.00000 q^{68} -1.00000 q^{69} -2.00000 q^{70} +1.00000 q^{71} -1.00000 q^{72} +12.0000 q^{73} +2.00000 q^{74} -1.00000 q^{75} +2.00000 q^{76} +4.00000 q^{77} -4.00000 q^{79} -2.00000 q^{80} +1.00000 q^{81} -2.00000 q^{82} -1.00000 q^{83} -1.00000 q^{84} -14.0000 q^{85} +5.00000 q^{86} -2.00000 q^{87} +4.00000 q^{88} +3.00000 q^{89} +2.00000 q^{90} -1.00000 q^{92} -9.00000 q^{93} -6.00000 q^{94} -4.00000 q^{95} -1.00000 q^{96} -16.0000 q^{97} -1.00000 q^{98} -4.00000 q^{99} +O(q^{100})\)

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
\(5\) −2.00000 −0.894427 −0.447214 0.894427i \(-0.647584\pi\)
−0.447214 + 0.894427i \(0.647584\pi\)
\(6\) −1.00000 −0.408248
\(7\) −1.00000 −0.377964
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 2.00000 0.632456
\(11\) −4.00000 −1.20605 −0.603023 0.797724i \(-0.706037\pi\)
−0.603023 + 0.797724i \(0.706037\pi\)
\(12\) 1.00000 0.288675
\(13\) 0 0
\(14\) 1.00000 0.267261
\(15\) −2.00000 −0.516398
\(16\) 1.00000 0.250000
\(17\) 7.00000 1.69775 0.848875 0.528594i \(-0.177281\pi\)
0.848875 + 0.528594i \(0.177281\pi\)
\(18\) −1.00000 −0.235702
\(19\) 2.00000 0.458831 0.229416 0.973329i \(-0.426318\pi\)
0.229416 + 0.973329i \(0.426318\pi\)
\(20\) −2.00000 −0.447214
\(21\) −1.00000 −0.218218
\(22\) 4.00000 0.852803
\(23\) −1.00000 −0.208514 −0.104257 0.994550i \(-0.533247\pi\)
−0.104257 + 0.994550i \(0.533247\pi\)
\(24\) −1.00000 −0.204124
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) −1.00000 −0.188982
\(29\) −2.00000 −0.371391 −0.185695 0.982607i \(-0.559454\pi\)
−0.185695 + 0.982607i \(0.559454\pi\)
\(30\) 2.00000 0.365148
\(31\) −9.00000 −1.61645 −0.808224 0.588875i \(-0.799571\pi\)
−0.808224 + 0.588875i \(0.799571\pi\)
\(32\) −1.00000 −0.176777
\(33\) −4.00000 −0.696311
\(34\) −7.00000 −1.20049
\(35\) 2.00000 0.338062
\(36\) 1.00000 0.166667
\(37\) −2.00000 −0.328798 −0.164399 0.986394i \(-0.552568\pi\)
−0.164399 + 0.986394i \(0.552568\pi\)
\(38\) −2.00000 −0.324443
\(39\) 0 0
\(40\) 2.00000 0.316228
\(41\) 2.00000 0.312348 0.156174 0.987730i \(-0.450084\pi\)
0.156174 + 0.987730i \(0.450084\pi\)
\(42\) 1.00000 0.154303
\(43\) −5.00000 −0.762493 −0.381246 0.924473i \(-0.624505\pi\)
−0.381246 + 0.924473i \(0.624505\pi\)
\(44\) −4.00000 −0.603023
\(45\) −2.00000 −0.298142
\(46\) 1.00000 0.147442
\(47\) 6.00000 0.875190 0.437595 0.899172i \(-0.355830\pi\)
0.437595 + 0.899172i \(0.355830\pi\)
\(48\) 1.00000 0.144338
\(49\) 1.00000 0.142857
\(50\) 1.00000 0.141421
\(51\) 7.00000 0.980196
\(52\) 0 0
\(53\) 3.00000 0.412082 0.206041 0.978543i \(-0.433942\pi\)
0.206041 + 0.978543i \(0.433942\pi\)
\(54\) −1.00000 −0.136083
\(55\) 8.00000 1.07872
\(56\) 1.00000 0.133631
\(57\) 2.00000 0.264906
\(58\) 2.00000 0.262613
\(59\) 15.0000 1.95283 0.976417 0.215894i \(-0.0692665\pi\)
0.976417 + 0.215894i \(0.0692665\pi\)
\(60\) −2.00000 −0.258199
\(61\) −7.00000 −0.896258 −0.448129 0.893969i \(-0.647910\pi\)
−0.448129 + 0.893969i \(0.647910\pi\)
\(62\) 9.00000 1.14300
\(63\) −1.00000 −0.125988
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 4.00000 0.492366
\(67\) −5.00000 −0.610847 −0.305424 0.952217i \(-0.598798\pi\)
−0.305424 + 0.952217i \(0.598798\pi\)
\(68\) 7.00000 0.848875
\(69\) −1.00000 −0.120386
\(70\) −2.00000 −0.239046
\(71\) 1.00000 0.118678 0.0593391 0.998238i \(-0.481101\pi\)
0.0593391 + 0.998238i \(0.481101\pi\)
\(72\) −1.00000 −0.117851
\(73\) 12.0000 1.40449 0.702247 0.711934i \(-0.252180\pi\)
0.702247 + 0.711934i \(0.252180\pi\)
\(74\) 2.00000 0.232495
\(75\) −1.00000 −0.115470
\(76\) 2.00000 0.229416
\(77\) 4.00000 0.455842
\(78\) 0 0
\(79\) −4.00000 −0.450035 −0.225018 0.974355i \(-0.572244\pi\)
−0.225018 + 0.974355i \(0.572244\pi\)
\(80\) −2.00000 −0.223607
\(81\) 1.00000 0.111111
\(82\) −2.00000 −0.220863
\(83\) −1.00000 −0.109764 −0.0548821 0.998493i \(-0.517478\pi\)
−0.0548821 + 0.998493i \(0.517478\pi\)
\(84\) −1.00000 −0.109109
\(85\) −14.0000 −1.51851
\(86\) 5.00000 0.539164
\(87\) −2.00000 −0.214423
\(88\) 4.00000 0.426401
\(89\) 3.00000 0.317999 0.159000 0.987279i \(-0.449173\pi\)
0.159000 + 0.987279i \(0.449173\pi\)
\(90\) 2.00000 0.210819
\(91\) 0 0
\(92\) −1.00000 −0.104257
\(93\) −9.00000 −0.933257
\(94\) −6.00000 −0.618853
\(95\) −4.00000 −0.410391
\(96\) −1.00000 −0.102062
\(97\) −16.0000 −1.62455 −0.812277 0.583272i \(-0.801772\pi\)
−0.812277 + 0.583272i \(0.801772\pi\)
\(98\) −1.00000 −0.101015
\(99\) −4.00000 −0.402015
\(100\) −1.00000 −0.100000
\(101\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) −7.00000 −0.693103
\(103\) −11.0000 −1.08386 −0.541931 0.840423i \(-0.682307\pi\)
−0.541931 + 0.840423i \(0.682307\pi\)
\(104\) 0 0
\(105\) 2.00000 0.195180
\(106\) −3.00000 −0.291386
\(107\) −6.00000 −0.580042 −0.290021 0.957020i \(-0.593662\pi\)
−0.290021 + 0.957020i \(0.593662\pi\)
\(108\) 1.00000 0.0962250
\(109\) −16.0000 −1.53252 −0.766261 0.642529i \(-0.777885\pi\)
−0.766261 + 0.642529i \(0.777885\pi\)
\(110\) −8.00000 −0.762770
\(111\) −2.00000 −0.189832
\(112\) −1.00000 −0.0944911
\(113\) −8.00000 −0.752577 −0.376288 0.926503i \(-0.622800\pi\)
−0.376288 + 0.926503i \(0.622800\pi\)
\(114\) −2.00000 −0.187317
\(115\) 2.00000 0.186501
\(116\) −2.00000 −0.185695
\(117\) 0 0
\(118\) −15.0000 −1.38086
\(119\) −7.00000 −0.641689
\(120\) 2.00000 0.182574
\(121\) 5.00000 0.454545
\(122\) 7.00000 0.633750
\(123\) 2.00000 0.180334
\(124\) −9.00000 −0.808224
\(125\) 12.0000 1.07331
\(126\) 1.00000 0.0890871
\(127\) 16.0000 1.41977 0.709885 0.704317i \(-0.248747\pi\)
0.709885 + 0.704317i \(0.248747\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −5.00000 −0.440225
\(130\) 0 0
\(131\) 15.0000 1.31056 0.655278 0.755388i \(-0.272551\pi\)
0.655278 + 0.755388i \(0.272551\pi\)
\(132\) −4.00000 −0.348155
\(133\) −2.00000 −0.173422
\(134\) 5.00000 0.431934
\(135\) −2.00000 −0.172133
\(136\) −7.00000 −0.600245
\(137\) 4.00000 0.341743 0.170872 0.985293i \(-0.445342\pi\)
0.170872 + 0.985293i \(0.445342\pi\)
\(138\) 1.00000 0.0851257
\(139\) 16.0000 1.35710 0.678551 0.734553i \(-0.262608\pi\)
0.678551 + 0.734553i \(0.262608\pi\)
\(140\) 2.00000 0.169031
\(141\) 6.00000 0.505291
\(142\) −1.00000 −0.0839181
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) 4.00000 0.332182
\(146\) −12.0000 −0.993127
\(147\) 1.00000 0.0824786
\(148\) −2.00000 −0.164399
\(149\) 21.0000 1.72039 0.860194 0.509968i \(-0.170343\pi\)
0.860194 + 0.509968i \(0.170343\pi\)
\(150\) 1.00000 0.0816497
\(151\) 20.0000 1.62758 0.813788 0.581161i \(-0.197401\pi\)
0.813788 + 0.581161i \(0.197401\pi\)
\(152\) −2.00000 −0.162221
\(153\) 7.00000 0.565916
\(154\) −4.00000 −0.322329
\(155\) 18.0000 1.44579
\(156\) 0 0
\(157\) 10.0000 0.798087 0.399043 0.916932i \(-0.369342\pi\)
0.399043 + 0.916932i \(0.369342\pi\)
\(158\) 4.00000 0.318223
\(159\) 3.00000 0.237915
\(160\) 2.00000 0.158114
\(161\) 1.00000 0.0788110
\(162\) −1.00000 −0.0785674
\(163\) 21.0000 1.64485 0.822423 0.568876i \(-0.192621\pi\)
0.822423 + 0.568876i \(0.192621\pi\)
\(164\) 2.00000 0.156174
\(165\) 8.00000 0.622799
\(166\) 1.00000 0.0776151
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 1.00000 0.0771517
\(169\) 0 0
\(170\) 14.0000 1.07375
\(171\) 2.00000 0.152944
\(172\) −5.00000 −0.381246
\(173\) −16.0000 −1.21646 −0.608229 0.793762i \(-0.708120\pi\)
−0.608229 + 0.793762i \(0.708120\pi\)
\(174\) 2.00000 0.151620
\(175\) 1.00000 0.0755929
\(176\) −4.00000 −0.301511
\(177\) 15.0000 1.12747
\(178\) −3.00000 −0.224860
\(179\) 6.00000 0.448461 0.224231 0.974536i \(-0.428013\pi\)
0.224231 + 0.974536i \(0.428013\pi\)
\(180\) −2.00000 −0.149071
\(181\) −2.00000 −0.148659 −0.0743294 0.997234i \(-0.523682\pi\)
−0.0743294 + 0.997234i \(0.523682\pi\)
\(182\) 0 0
\(183\) −7.00000 −0.517455
\(184\) 1.00000 0.0737210
\(185\) 4.00000 0.294086
\(186\) 9.00000 0.659912
\(187\) −28.0000 −2.04756
\(188\) 6.00000 0.437595
\(189\) −1.00000 −0.0727393
\(190\) 4.00000 0.290191
\(191\) 3.00000 0.217072 0.108536 0.994092i \(-0.465384\pi\)
0.108536 + 0.994092i \(0.465384\pi\)
\(192\) 1.00000 0.0721688
\(193\) −22.0000 −1.58359 −0.791797 0.610784i \(-0.790854\pi\)
−0.791797 + 0.610784i \(0.790854\pi\)
\(194\) 16.0000 1.14873
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) −9.00000 −0.641223 −0.320612 0.947211i \(-0.603888\pi\)
−0.320612 + 0.947211i \(0.603888\pi\)
\(198\) 4.00000 0.284268
\(199\) −5.00000 −0.354441 −0.177220 0.984171i \(-0.556711\pi\)
−0.177220 + 0.984171i \(0.556711\pi\)
\(200\) 1.00000 0.0707107
\(201\) −5.00000 −0.352673
\(202\) 0 0
\(203\) 2.00000 0.140372
\(204\) 7.00000 0.490098
\(205\) −4.00000 −0.279372
\(206\) 11.0000 0.766406
\(207\) −1.00000 −0.0695048
\(208\) 0 0
\(209\) −8.00000 −0.553372
\(210\) −2.00000 −0.138013
\(211\) 4.00000 0.275371 0.137686 0.990476i \(-0.456034\pi\)
0.137686 + 0.990476i \(0.456034\pi\)
\(212\) 3.00000 0.206041
\(213\) 1.00000 0.0685189
\(214\) 6.00000 0.410152
\(215\) 10.0000 0.681994
\(216\) −1.00000 −0.0680414
\(217\) 9.00000 0.610960
\(218\) 16.0000 1.08366
\(219\) 12.0000 0.810885
\(220\) 8.00000 0.539360
\(221\) 0 0
\(222\) 2.00000 0.134231
\(223\) 3.00000 0.200895 0.100447 0.994942i \(-0.467973\pi\)
0.100447 + 0.994942i \(0.467973\pi\)
\(224\) 1.00000 0.0668153
\(225\) −1.00000 −0.0666667
\(226\) 8.00000 0.532152
\(227\) 28.0000 1.85843 0.929213 0.369546i \(-0.120487\pi\)
0.929213 + 0.369546i \(0.120487\pi\)
\(228\) 2.00000 0.132453
\(229\) 13.0000 0.859064 0.429532 0.903052i \(-0.358679\pi\)
0.429532 + 0.903052i \(0.358679\pi\)
\(230\) −2.00000 −0.131876
\(231\) 4.00000 0.263181
\(232\) 2.00000 0.131306
\(233\) −16.0000 −1.04819 −0.524097 0.851658i \(-0.675597\pi\)
−0.524097 + 0.851658i \(0.675597\pi\)
\(234\) 0 0
\(235\) −12.0000 −0.782794
\(236\) 15.0000 0.976417
\(237\) −4.00000 −0.259828
\(238\) 7.00000 0.453743
\(239\) −9.00000 −0.582162 −0.291081 0.956698i \(-0.594015\pi\)
−0.291081 + 0.956698i \(0.594015\pi\)
\(240\) −2.00000 −0.129099
\(241\) 14.0000 0.901819 0.450910 0.892570i \(-0.351100\pi\)
0.450910 + 0.892570i \(0.351100\pi\)
\(242\) −5.00000 −0.321412
\(243\) 1.00000 0.0641500
\(244\) −7.00000 −0.448129
\(245\) −2.00000 −0.127775
\(246\) −2.00000 −0.127515
\(247\) 0 0
\(248\) 9.00000 0.571501
\(249\) −1.00000 −0.0633724
\(250\) −12.0000 −0.758947
\(251\) 23.0000 1.45175 0.725874 0.687828i \(-0.241436\pi\)
0.725874 + 0.687828i \(0.241436\pi\)
\(252\) −1.00000 −0.0629941
\(253\) 4.00000 0.251478
\(254\) −16.0000 −1.00393
\(255\) −14.0000 −0.876714
\(256\) 1.00000 0.0625000
\(257\) 27.0000 1.68421 0.842107 0.539311i \(-0.181315\pi\)
0.842107 + 0.539311i \(0.181315\pi\)
\(258\) 5.00000 0.311286
\(259\) 2.00000 0.124274
\(260\) 0 0
\(261\) −2.00000 −0.123797
\(262\) −15.0000 −0.926703
\(263\) 4.00000 0.246651 0.123325 0.992366i \(-0.460644\pi\)
0.123325 + 0.992366i \(0.460644\pi\)
\(264\) 4.00000 0.246183
\(265\) −6.00000 −0.368577
\(266\) 2.00000 0.122628
\(267\) 3.00000 0.183597
\(268\) −5.00000 −0.305424
\(269\) 18.0000 1.09748 0.548740 0.835993i \(-0.315108\pi\)
0.548740 + 0.835993i \(0.315108\pi\)
\(270\) 2.00000 0.121716
\(271\) 9.00000 0.546711 0.273356 0.961913i \(-0.411866\pi\)
0.273356 + 0.961913i \(0.411866\pi\)
\(272\) 7.00000 0.424437
\(273\) 0 0
\(274\) −4.00000 −0.241649
\(275\) 4.00000 0.241209
\(276\) −1.00000 −0.0601929
\(277\) −8.00000 −0.480673 −0.240337 0.970690i \(-0.577258\pi\)
−0.240337 + 0.970690i \(0.577258\pi\)
\(278\) −16.0000 −0.959616
\(279\) −9.00000 −0.538816
\(280\) −2.00000 −0.119523
\(281\) −26.0000 −1.55103 −0.775515 0.631329i \(-0.782510\pi\)
−0.775515 + 0.631329i \(0.782510\pi\)
\(282\) −6.00000 −0.357295
\(283\) 14.0000 0.832214 0.416107 0.909316i \(-0.363394\pi\)
0.416107 + 0.909316i \(0.363394\pi\)
\(284\) 1.00000 0.0593391
\(285\) −4.00000 −0.236940
\(286\) 0 0
\(287\) −2.00000 −0.118056
\(288\) −1.00000 −0.0589256
\(289\) 32.0000 1.88235
\(290\) −4.00000 −0.234888
\(291\) −16.0000 −0.937937
\(292\) 12.0000 0.702247
\(293\) −16.0000 −0.934730 −0.467365 0.884064i \(-0.654797\pi\)
−0.467365 + 0.884064i \(0.654797\pi\)
\(294\) −1.00000 −0.0583212
\(295\) −30.0000 −1.74667
\(296\) 2.00000 0.116248
\(297\) −4.00000 −0.232104
\(298\) −21.0000 −1.21650
\(299\) 0 0
\(300\) −1.00000 −0.0577350
\(301\) 5.00000 0.288195
\(302\) −20.0000 −1.15087
\(303\) 0 0
\(304\) 2.00000 0.114708
\(305\) 14.0000 0.801638
\(306\) −7.00000 −0.400163
\(307\) 30.0000 1.71219 0.856095 0.516818i \(-0.172884\pi\)
0.856095 + 0.516818i \(0.172884\pi\)
\(308\) 4.00000 0.227921
\(309\) −11.0000 −0.625768
\(310\) −18.0000 −1.02233
\(311\) 30.0000 1.70114 0.850572 0.525859i \(-0.176256\pi\)
0.850572 + 0.525859i \(0.176256\pi\)
\(312\) 0 0
\(313\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(314\) −10.0000 −0.564333
\(315\) 2.00000 0.112687
\(316\) −4.00000 −0.225018
\(317\) 23.0000 1.29181 0.645904 0.763418i \(-0.276480\pi\)
0.645904 + 0.763418i \(0.276480\pi\)
\(318\) −3.00000 −0.168232
\(319\) 8.00000 0.447914
\(320\) −2.00000 −0.111803
\(321\) −6.00000 −0.334887
\(322\) −1.00000 −0.0557278
\(323\) 14.0000 0.778981
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) −21.0000 −1.16308
\(327\) −16.0000 −0.884802
\(328\) −2.00000 −0.110432
\(329\) −6.00000 −0.330791
\(330\) −8.00000 −0.440386
\(331\) 36.0000 1.97874 0.989369 0.145424i \(-0.0464545\pi\)
0.989369 + 0.145424i \(0.0464545\pi\)
\(332\) −1.00000 −0.0548821
\(333\) −2.00000 −0.109599
\(334\) 0 0
\(335\) 10.0000 0.546358
\(336\) −1.00000 −0.0545545
\(337\) 18.0000 0.980522 0.490261 0.871576i \(-0.336901\pi\)
0.490261 + 0.871576i \(0.336901\pi\)
\(338\) 0 0
\(339\) −8.00000 −0.434500
\(340\) −14.0000 −0.759257
\(341\) 36.0000 1.94951
\(342\) −2.00000 −0.108148
\(343\) −1.00000 −0.0539949
\(344\) 5.00000 0.269582
\(345\) 2.00000 0.107676
\(346\) 16.0000 0.860165
\(347\) 10.0000 0.536828 0.268414 0.963304i \(-0.413500\pi\)
0.268414 + 0.963304i \(0.413500\pi\)
\(348\) −2.00000 −0.107211
\(349\) −25.0000 −1.33822 −0.669110 0.743164i \(-0.733324\pi\)
−0.669110 + 0.743164i \(0.733324\pi\)
\(350\) −1.00000 −0.0534522
\(351\) 0 0
\(352\) 4.00000 0.213201
\(353\) −35.0000 −1.86286 −0.931431 0.363918i \(-0.881439\pi\)
−0.931431 + 0.363918i \(0.881439\pi\)
\(354\) −15.0000 −0.797241
\(355\) −2.00000 −0.106149
\(356\) 3.00000 0.159000
\(357\) −7.00000 −0.370479
\(358\) −6.00000 −0.317110
\(359\) −24.0000 −1.26667 −0.633336 0.773877i \(-0.718315\pi\)
−0.633336 + 0.773877i \(0.718315\pi\)
\(360\) 2.00000 0.105409
\(361\) −15.0000 −0.789474
\(362\) 2.00000 0.105118
\(363\) 5.00000 0.262432
\(364\) 0 0
\(365\) −24.0000 −1.25622
\(366\) 7.00000 0.365896
\(367\) −3.00000 −0.156599 −0.0782994 0.996930i \(-0.524949\pi\)
−0.0782994 + 0.996930i \(0.524949\pi\)
\(368\) −1.00000 −0.0521286
\(369\) 2.00000 0.104116
\(370\) −4.00000 −0.207950
\(371\) −3.00000 −0.155752
\(372\) −9.00000 −0.466628
\(373\) 4.00000 0.207112 0.103556 0.994624i \(-0.466978\pi\)
0.103556 + 0.994624i \(0.466978\pi\)
\(374\) 28.0000 1.44785
\(375\) 12.0000 0.619677
\(376\) −6.00000 −0.309426
\(377\) 0 0
\(378\) 1.00000 0.0514344
\(379\) −20.0000 −1.02733 −0.513665 0.857991i \(-0.671713\pi\)
−0.513665 + 0.857991i \(0.671713\pi\)
\(380\) −4.00000 −0.205196
\(381\) 16.0000 0.819705
\(382\) −3.00000 −0.153493
\(383\) 24.0000 1.22634 0.613171 0.789950i \(-0.289894\pi\)
0.613171 + 0.789950i \(0.289894\pi\)
\(384\) −1.00000 −0.0510310
\(385\) −8.00000 −0.407718
\(386\) 22.0000 1.11977
\(387\) −5.00000 −0.254164
\(388\) −16.0000 −0.812277
\(389\) −1.00000 −0.0507020 −0.0253510 0.999679i \(-0.508070\pi\)
−0.0253510 + 0.999679i \(0.508070\pi\)
\(390\) 0 0
\(391\) −7.00000 −0.354005
\(392\) −1.00000 −0.0505076
\(393\) 15.0000 0.756650
\(394\) 9.00000 0.453413
\(395\) 8.00000 0.402524
\(396\) −4.00000 −0.201008
\(397\) −1.00000 −0.0501886 −0.0250943 0.999685i \(-0.507989\pi\)
−0.0250943 + 0.999685i \(0.507989\pi\)
\(398\) 5.00000 0.250627
\(399\) −2.00000 −0.100125
\(400\) −1.00000 −0.0500000
\(401\) 32.0000 1.59800 0.799002 0.601329i \(-0.205362\pi\)
0.799002 + 0.601329i \(0.205362\pi\)
\(402\) 5.00000 0.249377
\(403\) 0 0
\(404\) 0 0
\(405\) −2.00000 −0.0993808
\(406\) −2.00000 −0.0992583
\(407\) 8.00000 0.396545
\(408\) −7.00000 −0.346552
\(409\) 14.0000 0.692255 0.346128 0.938187i \(-0.387496\pi\)
0.346128 + 0.938187i \(0.387496\pi\)
\(410\) 4.00000 0.197546
\(411\) 4.00000 0.197305
\(412\) −11.0000 −0.541931
\(413\) −15.0000 −0.738102
\(414\) 1.00000 0.0491473
\(415\) 2.00000 0.0981761
\(416\) 0 0
\(417\) 16.0000 0.783523
\(418\) 8.00000 0.391293
\(419\) −31.0000 −1.51445 −0.757225 0.653155i \(-0.773445\pi\)
−0.757225 + 0.653155i \(0.773445\pi\)
\(420\) 2.00000 0.0975900
\(421\) 34.0000 1.65706 0.828529 0.559946i \(-0.189178\pi\)
0.828529 + 0.559946i \(0.189178\pi\)
\(422\) −4.00000 −0.194717
\(423\) 6.00000 0.291730
\(424\) −3.00000 −0.145693
\(425\) −7.00000 −0.339550
\(426\) −1.00000 −0.0484502
\(427\) 7.00000 0.338754
\(428\) −6.00000 −0.290021
\(429\) 0 0
\(430\) −10.0000 −0.482243
\(431\) −1.00000 −0.0481683 −0.0240842 0.999710i \(-0.507667\pi\)
−0.0240842 + 0.999710i \(0.507667\pi\)
\(432\) 1.00000 0.0481125
\(433\) 12.0000 0.576683 0.288342 0.957528i \(-0.406896\pi\)
0.288342 + 0.957528i \(0.406896\pi\)
\(434\) −9.00000 −0.432014
\(435\) 4.00000 0.191785
\(436\) −16.0000 −0.766261
\(437\) −2.00000 −0.0956730
\(438\) −12.0000 −0.573382
\(439\) 4.00000 0.190910 0.0954548 0.995434i \(-0.469569\pi\)
0.0954548 + 0.995434i \(0.469569\pi\)
\(440\) −8.00000 −0.381385
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) −6.00000 −0.285069 −0.142534 0.989790i \(-0.545525\pi\)
−0.142534 + 0.989790i \(0.545525\pi\)
\(444\) −2.00000 −0.0949158
\(445\) −6.00000 −0.284427
\(446\) −3.00000 −0.142054
\(447\) 21.0000 0.993266
\(448\) −1.00000 −0.0472456
\(449\) 18.0000 0.849473 0.424736 0.905317i \(-0.360367\pi\)
0.424736 + 0.905317i \(0.360367\pi\)
\(450\) 1.00000 0.0471405
\(451\) −8.00000 −0.376705
\(452\) −8.00000 −0.376288
\(453\) 20.0000 0.939682
\(454\) −28.0000 −1.31411
\(455\) 0 0
\(456\) −2.00000 −0.0936586
\(457\) −9.00000 −0.421002 −0.210501 0.977594i \(-0.567510\pi\)
−0.210501 + 0.977594i \(0.567510\pi\)
\(458\) −13.0000 −0.607450
\(459\) 7.00000 0.326732
\(460\) 2.00000 0.0932505
\(461\) −20.0000 −0.931493 −0.465746 0.884918i \(-0.654214\pi\)
−0.465746 + 0.884918i \(0.654214\pi\)
\(462\) −4.00000 −0.186097
\(463\) 18.0000 0.836531 0.418265 0.908325i \(-0.362638\pi\)
0.418265 + 0.908325i \(0.362638\pi\)
\(464\) −2.00000 −0.0928477
\(465\) 18.0000 0.834730
\(466\) 16.0000 0.741186
\(467\) 37.0000 1.71216 0.856078 0.516847i \(-0.172894\pi\)
0.856078 + 0.516847i \(0.172894\pi\)
\(468\) 0 0
\(469\) 5.00000 0.230879
\(470\) 12.0000 0.553519
\(471\) 10.0000 0.460776
\(472\) −15.0000 −0.690431
\(473\) 20.0000 0.919601
\(474\) 4.00000 0.183726
\(475\) −2.00000 −0.0917663
\(476\) −7.00000 −0.320844
\(477\) 3.00000 0.137361
\(478\) 9.00000 0.411650
\(479\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 2.00000 0.0912871
\(481\) 0 0
\(482\) −14.0000 −0.637683
\(483\) 1.00000 0.0455016
\(484\) 5.00000 0.227273
\(485\) 32.0000 1.45305
\(486\) −1.00000 −0.0453609
\(487\) −28.0000 −1.26880 −0.634401 0.773004i \(-0.718753\pi\)
−0.634401 + 0.773004i \(0.718753\pi\)
\(488\) 7.00000 0.316875
\(489\) 21.0000 0.949653
\(490\) 2.00000 0.0903508
\(491\) −8.00000 −0.361035 −0.180517 0.983572i \(-0.557777\pi\)
−0.180517 + 0.983572i \(0.557777\pi\)
\(492\) 2.00000 0.0901670
\(493\) −14.0000 −0.630528
\(494\) 0 0
\(495\) 8.00000 0.359573
\(496\) −9.00000 −0.404112
\(497\) −1.00000 −0.0448561
\(498\) 1.00000 0.0448111
\(499\) 7.00000 0.313363 0.156682 0.987649i \(-0.449920\pi\)
0.156682 + 0.987649i \(0.449920\pi\)
\(500\) 12.0000 0.536656
\(501\) 0 0
\(502\) −23.0000 −1.02654
\(503\) −4.00000 −0.178351 −0.0891756 0.996016i \(-0.528423\pi\)
−0.0891756 + 0.996016i \(0.528423\pi\)
\(504\) 1.00000 0.0445435
\(505\) 0 0
\(506\) −4.00000 −0.177822
\(507\) 0 0
\(508\) 16.0000 0.709885
\(509\) −12.0000 −0.531891 −0.265945 0.963988i \(-0.585684\pi\)
−0.265945 + 0.963988i \(0.585684\pi\)
\(510\) 14.0000 0.619930
\(511\) −12.0000 −0.530849
\(512\) −1.00000 −0.0441942
\(513\) 2.00000 0.0883022
\(514\) −27.0000 −1.19092
\(515\) 22.0000 0.969436
\(516\) −5.00000 −0.220113
\(517\) −24.0000 −1.05552
\(518\) −2.00000 −0.0878750
\(519\) −16.0000 −0.702322
\(520\) 0 0
\(521\) −42.0000 −1.84005 −0.920027 0.391856i \(-0.871833\pi\)
−0.920027 + 0.391856i \(0.871833\pi\)
\(522\) 2.00000 0.0875376
\(523\) −24.0000 −1.04945 −0.524723 0.851273i \(-0.675831\pi\)
−0.524723 + 0.851273i \(0.675831\pi\)
\(524\) 15.0000 0.655278
\(525\) 1.00000 0.0436436
\(526\) −4.00000 −0.174408
\(527\) −63.0000 −2.74432
\(528\) −4.00000 −0.174078
\(529\) −22.0000 −0.956522
\(530\) 6.00000 0.260623
\(531\) 15.0000 0.650945
\(532\) −2.00000 −0.0867110
\(533\) 0 0
\(534\) −3.00000 −0.129823
\(535\) 12.0000 0.518805
\(536\) 5.00000 0.215967
\(537\) 6.00000 0.258919
\(538\) −18.0000 −0.776035
\(539\) −4.00000 −0.172292
\(540\) −2.00000 −0.0860663
\(541\) 34.0000 1.46177 0.730887 0.682498i \(-0.239107\pi\)
0.730887 + 0.682498i \(0.239107\pi\)
\(542\) −9.00000 −0.386583
\(543\) −2.00000 −0.0858282
\(544\) −7.00000 −0.300123
\(545\) 32.0000 1.37073
\(546\) 0 0
\(547\) −4.00000 −0.171028 −0.0855138 0.996337i \(-0.527253\pi\)
−0.0855138 + 0.996337i \(0.527253\pi\)
\(548\) 4.00000 0.170872
\(549\) −7.00000 −0.298753
\(550\) −4.00000 −0.170561
\(551\) −4.00000 −0.170406
\(552\) 1.00000 0.0425628
\(553\) 4.00000 0.170097
\(554\) 8.00000 0.339887
\(555\) 4.00000 0.169791
\(556\) 16.0000 0.678551
\(557\) 13.0000 0.550828 0.275414 0.961326i \(-0.411185\pi\)
0.275414 + 0.961326i \(0.411185\pi\)
\(558\) 9.00000 0.381000
\(559\) 0 0
\(560\) 2.00000 0.0845154
\(561\) −28.0000 −1.18216
\(562\) 26.0000 1.09674
\(563\) −40.0000 −1.68580 −0.842900 0.538071i \(-0.819153\pi\)
−0.842900 + 0.538071i \(0.819153\pi\)
\(564\) 6.00000 0.252646
\(565\) 16.0000 0.673125
\(566\) −14.0000 −0.588464
\(567\) −1.00000 −0.0419961
\(568\) −1.00000 −0.0419591
\(569\) 26.0000 1.08998 0.544988 0.838444i \(-0.316534\pi\)
0.544988 + 0.838444i \(0.316534\pi\)
\(570\) 4.00000 0.167542
\(571\) 23.0000 0.962520 0.481260 0.876578i \(-0.340179\pi\)
0.481260 + 0.876578i \(0.340179\pi\)
\(572\) 0 0
\(573\) 3.00000 0.125327
\(574\) 2.00000 0.0834784
\(575\) 1.00000 0.0417029
\(576\) 1.00000 0.0416667
\(577\) 6.00000 0.249783 0.124892 0.992170i \(-0.460142\pi\)
0.124892 + 0.992170i \(0.460142\pi\)
\(578\) −32.0000 −1.33102
\(579\) −22.0000 −0.914289
\(580\) 4.00000 0.166091
\(581\) 1.00000 0.0414870
\(582\) 16.0000 0.663221
\(583\) −12.0000 −0.496989
\(584\) −12.0000 −0.496564
\(585\) 0 0
\(586\) 16.0000 0.660954
\(587\) 45.0000 1.85735 0.928674 0.370896i \(-0.120949\pi\)
0.928674 + 0.370896i \(0.120949\pi\)
\(588\) 1.00000 0.0412393
\(589\) −18.0000 −0.741677
\(590\) 30.0000 1.23508
\(591\) −9.00000 −0.370211
\(592\) −2.00000 −0.0821995
\(593\) 29.0000 1.19089 0.595444 0.803397i \(-0.296976\pi\)
0.595444 + 0.803397i \(0.296976\pi\)
\(594\) 4.00000 0.164122
\(595\) 14.0000 0.573944
\(596\) 21.0000 0.860194
\(597\) −5.00000 −0.204636
\(598\) 0 0
\(599\) 9.00000 0.367730 0.183865 0.982952i \(-0.441139\pi\)
0.183865 + 0.982952i \(0.441139\pi\)
\(600\) 1.00000 0.0408248
\(601\) 38.0000 1.55005 0.775026 0.631929i \(-0.217737\pi\)
0.775026 + 0.631929i \(0.217737\pi\)
\(602\) −5.00000 −0.203785
\(603\) −5.00000 −0.203616
\(604\) 20.0000 0.813788
\(605\) −10.0000 −0.406558
\(606\) 0 0
\(607\) 47.0000 1.90767 0.953836 0.300329i \(-0.0970966\pi\)
0.953836 + 0.300329i \(0.0970966\pi\)
\(608\) −2.00000 −0.0811107
\(609\) 2.00000 0.0810441
\(610\) −14.0000 −0.566843
\(611\) 0 0
\(612\) 7.00000 0.282958
\(613\) 14.0000 0.565455 0.282727 0.959200i \(-0.408761\pi\)
0.282727 + 0.959200i \(0.408761\pi\)
\(614\) −30.0000 −1.21070
\(615\) −4.00000 −0.161296
\(616\) −4.00000 −0.161165
\(617\) −20.0000 −0.805170 −0.402585 0.915383i \(-0.631888\pi\)
−0.402585 + 0.915383i \(0.631888\pi\)
\(618\) 11.0000 0.442485
\(619\) −22.0000 −0.884255 −0.442127 0.896952i \(-0.645776\pi\)
−0.442127 + 0.896952i \(0.645776\pi\)
\(620\) 18.0000 0.722897
\(621\) −1.00000 −0.0401286
\(622\) −30.0000 −1.20289
\(623\) −3.00000 −0.120192
\(624\) 0 0
\(625\) −19.0000 −0.760000
\(626\) 0 0
\(627\) −8.00000 −0.319489
\(628\) 10.0000 0.399043
\(629\) −14.0000 −0.558217
\(630\) −2.00000 −0.0796819
\(631\) −20.0000 −0.796187 −0.398094 0.917345i \(-0.630328\pi\)
−0.398094 + 0.917345i \(0.630328\pi\)
\(632\) 4.00000 0.159111
\(633\) 4.00000 0.158986
\(634\) −23.0000 −0.913447
\(635\) −32.0000 −1.26988
\(636\) 3.00000 0.118958
\(637\) 0 0
\(638\) −8.00000 −0.316723
\(639\) 1.00000 0.0395594
\(640\) 2.00000 0.0790569
\(641\) 32.0000 1.26392 0.631962 0.774999i \(-0.282250\pi\)
0.631962 + 0.774999i \(0.282250\pi\)
\(642\) 6.00000 0.236801
\(643\) 14.0000 0.552106 0.276053 0.961142i \(-0.410973\pi\)
0.276053 + 0.961142i \(0.410973\pi\)
\(644\) 1.00000 0.0394055
\(645\) 10.0000 0.393750
\(646\) −14.0000 −0.550823
\(647\) −6.00000 −0.235884 −0.117942 0.993020i \(-0.537630\pi\)
−0.117942 + 0.993020i \(0.537630\pi\)
\(648\) −1.00000 −0.0392837
\(649\) −60.0000 −2.35521
\(650\) 0 0
\(651\) 9.00000 0.352738
\(652\) 21.0000 0.822423
\(653\) 27.0000 1.05659 0.528296 0.849060i \(-0.322831\pi\)
0.528296 + 0.849060i \(0.322831\pi\)
\(654\) 16.0000 0.625650
\(655\) −30.0000 −1.17220
\(656\) 2.00000 0.0780869
\(657\) 12.0000 0.468165
\(658\) 6.00000 0.233904
\(659\) −6.00000 −0.233727 −0.116863 0.993148i \(-0.537284\pi\)
−0.116863 + 0.993148i \(0.537284\pi\)
\(660\) 8.00000 0.311400
\(661\) −35.0000 −1.36134 −0.680671 0.732589i \(-0.738312\pi\)
−0.680671 + 0.732589i \(0.738312\pi\)
\(662\) −36.0000 −1.39918
\(663\) 0 0
\(664\) 1.00000 0.0388075
\(665\) 4.00000 0.155113
\(666\) 2.00000 0.0774984
\(667\) 2.00000 0.0774403
\(668\) 0 0
\(669\) 3.00000 0.115987
\(670\) −10.0000 −0.386334
\(671\) 28.0000 1.08093
\(672\) 1.00000 0.0385758
\(673\) −33.0000 −1.27206 −0.636028 0.771666i \(-0.719424\pi\)
−0.636028 + 0.771666i \(0.719424\pi\)
\(674\) −18.0000 −0.693334
\(675\) −1.00000 −0.0384900
\(676\) 0 0
\(677\) 28.0000 1.07613 0.538064 0.842904i \(-0.319156\pi\)
0.538064 + 0.842904i \(0.319156\pi\)
\(678\) 8.00000 0.307238
\(679\) 16.0000 0.614024
\(680\) 14.0000 0.536875
\(681\) 28.0000 1.07296
\(682\) −36.0000 −1.37851
\(683\) 22.0000 0.841807 0.420903 0.907106i \(-0.361713\pi\)
0.420903 + 0.907106i \(0.361713\pi\)
\(684\) 2.00000 0.0764719
\(685\) −8.00000 −0.305664
\(686\) 1.00000 0.0381802
\(687\) 13.0000 0.495981
\(688\) −5.00000 −0.190623
\(689\) 0 0
\(690\) −2.00000 −0.0761387
\(691\) −44.0000 −1.67384 −0.836919 0.547326i \(-0.815646\pi\)
−0.836919 + 0.547326i \(0.815646\pi\)
\(692\) −16.0000 −0.608229
\(693\) 4.00000 0.151947
\(694\) −10.0000 −0.379595
\(695\) −32.0000 −1.21383
\(696\) 2.00000 0.0758098
\(697\) 14.0000 0.530288
\(698\) 25.0000 0.946264
\(699\) −16.0000 −0.605176
\(700\) 1.00000 0.0377964
\(701\) −15.0000 −0.566542 −0.283271 0.959040i \(-0.591420\pi\)
−0.283271 + 0.959040i \(0.591420\pi\)
\(702\) 0 0
\(703\) −4.00000 −0.150863
\(704\) −4.00000 −0.150756
\(705\) −12.0000 −0.451946
\(706\) 35.0000 1.31724
\(707\) 0 0
\(708\) 15.0000 0.563735
\(709\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(710\) 2.00000 0.0750587
\(711\) −4.00000 −0.150012
\(712\) −3.00000 −0.112430
\(713\) 9.00000 0.337053
\(714\) 7.00000 0.261968
\(715\) 0 0
\(716\) 6.00000 0.224231
\(717\) −9.00000 −0.336111
\(718\) 24.0000 0.895672
\(719\) 20.0000 0.745874 0.372937 0.927857i \(-0.378351\pi\)
0.372937 + 0.927857i \(0.378351\pi\)
\(720\) −2.00000 −0.0745356
\(721\) 11.0000 0.409661
\(722\) 15.0000 0.558242
\(723\) 14.0000 0.520666
\(724\) −2.00000 −0.0743294
\(725\) 2.00000 0.0742781
\(726\) −5.00000 −0.185567
\(727\) −21.0000 −0.778847 −0.389423 0.921059i \(-0.627326\pi\)
−0.389423 + 0.921059i \(0.627326\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 24.0000 0.888280
\(731\) −35.0000 −1.29452
\(732\) −7.00000 −0.258727
\(733\) −21.0000 −0.775653 −0.387826 0.921732i \(-0.626774\pi\)
−0.387826 + 0.921732i \(0.626774\pi\)
\(734\) 3.00000 0.110732
\(735\) −2.00000 −0.0737711
\(736\) 1.00000 0.0368605
\(737\) 20.0000 0.736709
\(738\) −2.00000 −0.0736210
\(739\) 49.0000 1.80249 0.901247 0.433306i \(-0.142653\pi\)
0.901247 + 0.433306i \(0.142653\pi\)
\(740\) 4.00000 0.147043
\(741\) 0 0
\(742\) 3.00000 0.110133
\(743\) −41.0000 −1.50414 −0.752072 0.659081i \(-0.770945\pi\)
−0.752072 + 0.659081i \(0.770945\pi\)
\(744\) 9.00000 0.329956
\(745\) −42.0000 −1.53876
\(746\) −4.00000 −0.146450
\(747\) −1.00000 −0.0365881
\(748\) −28.0000 −1.02378
\(749\) 6.00000 0.219235
\(750\) −12.0000 −0.438178
\(751\) −26.0000 −0.948753 −0.474377 0.880322i \(-0.657327\pi\)
−0.474377 + 0.880322i \(0.657327\pi\)
\(752\) 6.00000 0.218797
\(753\) 23.0000 0.838167
\(754\) 0 0
\(755\) −40.0000 −1.45575
\(756\) −1.00000 −0.0363696
\(757\) 46.0000 1.67190 0.835949 0.548807i \(-0.184918\pi\)
0.835949 + 0.548807i \(0.184918\pi\)
\(758\) 20.0000 0.726433
\(759\) 4.00000 0.145191
\(760\) 4.00000 0.145095
\(761\) 10.0000 0.362500 0.181250 0.983437i \(-0.441986\pi\)
0.181250 + 0.983437i \(0.441986\pi\)
\(762\) −16.0000 −0.579619
\(763\) 16.0000 0.579239
\(764\) 3.00000 0.108536
\(765\) −14.0000 −0.506171
\(766\) −24.0000 −0.867155
\(767\) 0 0
\(768\) 1.00000 0.0360844
\(769\) 4.00000 0.144244 0.0721218 0.997396i \(-0.477023\pi\)
0.0721218 + 0.997396i \(0.477023\pi\)
\(770\) 8.00000 0.288300
\(771\) 27.0000 0.972381
\(772\) −22.0000 −0.791797
\(773\) −10.0000 −0.359675 −0.179838 0.983696i \(-0.557557\pi\)
−0.179838 + 0.983696i \(0.557557\pi\)
\(774\) 5.00000 0.179721
\(775\) 9.00000 0.323290
\(776\) 16.0000 0.574367
\(777\) 2.00000 0.0717496
\(778\) 1.00000 0.0358517
\(779\) 4.00000 0.143315
\(780\) 0 0
\(781\) −4.00000 −0.143131
\(782\) 7.00000 0.250319
\(783\) −2.00000 −0.0714742
\(784\) 1.00000 0.0357143
\(785\) −20.0000 −0.713831
\(786\) −15.0000 −0.535032
\(787\) 32.0000 1.14068 0.570338 0.821410i \(-0.306812\pi\)
0.570338 + 0.821410i \(0.306812\pi\)
\(788\) −9.00000 −0.320612
\(789\) 4.00000 0.142404
\(790\) −8.00000 −0.284627
\(791\) 8.00000 0.284447
\(792\) 4.00000 0.142134
\(793\) 0 0
\(794\) 1.00000 0.0354887
\(795\) −6.00000 −0.212798
\(796\) −5.00000 −0.177220
\(797\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(798\) 2.00000 0.0707992
\(799\) 42.0000 1.48585
\(800\) 1.00000 0.0353553
\(801\) 3.00000 0.106000
\(802\) −32.0000 −1.12996
\(803\) −48.0000 −1.69388
\(804\) −5.00000 −0.176336
\(805\) −2.00000 −0.0704907
\(806\) 0 0
\(807\) 18.0000 0.633630
\(808\) 0 0
\(809\) 24.0000 0.843795 0.421898 0.906644i \(-0.361364\pi\)
0.421898 + 0.906644i \(0.361364\pi\)
\(810\) 2.00000 0.0702728
\(811\) −4.00000 −0.140459 −0.0702295 0.997531i \(-0.522373\pi\)
−0.0702295 + 0.997531i \(0.522373\pi\)
\(812\) 2.00000 0.0701862
\(813\) 9.00000 0.315644
\(814\) −8.00000 −0.280400
\(815\) −42.0000 −1.47120
\(816\) 7.00000 0.245049
\(817\) −10.0000 −0.349856
\(818\) −14.0000 −0.489499
\(819\) 0 0
\(820\) −4.00000 −0.139686
\(821\) 43.0000 1.50071 0.750355 0.661035i \(-0.229882\pi\)
0.750355 + 0.661035i \(0.229882\pi\)
\(822\) −4.00000 −0.139516
\(823\) −20.0000 −0.697156 −0.348578 0.937280i \(-0.613335\pi\)
−0.348578 + 0.937280i \(0.613335\pi\)
\(824\) 11.0000 0.383203
\(825\) 4.00000 0.139262
\(826\) 15.0000 0.521917
\(827\) −8.00000 −0.278187 −0.139094 0.990279i \(-0.544419\pi\)
−0.139094 + 0.990279i \(0.544419\pi\)
\(828\) −1.00000 −0.0347524
\(829\) 42.0000 1.45872 0.729360 0.684130i \(-0.239818\pi\)
0.729360 + 0.684130i \(0.239818\pi\)
\(830\) −2.00000 −0.0694210
\(831\) −8.00000 −0.277517
\(832\) 0 0
\(833\) 7.00000 0.242536
\(834\) −16.0000 −0.554035
\(835\) 0 0
\(836\) −8.00000 −0.276686
\(837\) −9.00000 −0.311086
\(838\) 31.0000 1.07088
\(839\) 18.0000 0.621429 0.310715 0.950503i \(-0.399432\pi\)
0.310715 + 0.950503i \(0.399432\pi\)
\(840\) −2.00000 −0.0690066
\(841\) −25.0000 −0.862069
\(842\) −34.0000 −1.17172
\(843\) −26.0000 −0.895488
\(844\) 4.00000 0.137686
\(845\) 0 0
\(846\) −6.00000 −0.206284
\(847\) −5.00000 −0.171802
\(848\) 3.00000 0.103020
\(849\) 14.0000 0.480479
\(850\) 7.00000 0.240098
\(851\) 2.00000 0.0685591
\(852\) 1.00000 0.0342594
\(853\) −29.0000 −0.992941 −0.496471 0.868054i \(-0.665371\pi\)
−0.496471 + 0.868054i \(0.665371\pi\)
\(854\) −7.00000 −0.239535
\(855\) −4.00000 −0.136797
\(856\) 6.00000 0.205076
\(857\) 6.00000 0.204956 0.102478 0.994735i \(-0.467323\pi\)
0.102478 + 0.994735i \(0.467323\pi\)
\(858\) 0 0
\(859\) −14.0000 −0.477674 −0.238837 0.971060i \(-0.576766\pi\)
−0.238837 + 0.971060i \(0.576766\pi\)
\(860\) 10.0000 0.340997
\(861\) −2.00000 −0.0681598
\(862\) 1.00000 0.0340601
\(863\) 48.0000 1.63394 0.816970 0.576681i \(-0.195652\pi\)
0.816970 + 0.576681i \(0.195652\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 32.0000 1.08803
\(866\) −12.0000 −0.407777
\(867\) 32.0000 1.08678
\(868\) 9.00000 0.305480
\(869\) 16.0000 0.542763
\(870\) −4.00000 −0.135613
\(871\) 0 0
\(872\) 16.0000 0.541828
\(873\) −16.0000 −0.541518
\(874\) 2.00000 0.0676510
\(875\) −12.0000 −0.405674
\(876\) 12.0000 0.405442
\(877\) 26.0000 0.877958 0.438979 0.898497i \(-0.355340\pi\)
0.438979 + 0.898497i \(0.355340\pi\)
\(878\) −4.00000 −0.134993
\(879\) −16.0000 −0.539667
\(880\) 8.00000 0.269680
\(881\) −47.0000 −1.58347 −0.791735 0.610865i \(-0.790822\pi\)
−0.791735 + 0.610865i \(0.790822\pi\)
\(882\) −1.00000 −0.0336718
\(883\) −7.00000 −0.235569 −0.117784 0.993039i \(-0.537579\pi\)
−0.117784 + 0.993039i \(0.537579\pi\)
\(884\) 0 0
\(885\) −30.0000 −1.00844
\(886\) 6.00000 0.201574
\(887\) 2.00000 0.0671534 0.0335767 0.999436i \(-0.489310\pi\)
0.0335767 + 0.999436i \(0.489310\pi\)
\(888\) 2.00000 0.0671156
\(889\) −16.0000 −0.536623
\(890\) 6.00000 0.201120
\(891\) −4.00000 −0.134005
\(892\) 3.00000 0.100447
\(893\) 12.0000 0.401565
\(894\) −21.0000 −0.702345
\(895\) −12.0000 −0.401116
\(896\) 1.00000 0.0334077
\(897\) 0 0
\(898\) −18.0000 −0.600668
\(899\) 18.0000 0.600334
\(900\) −1.00000 −0.0333333
\(901\) 21.0000 0.699611
\(902\) 8.00000 0.266371
\(903\) 5.00000 0.166390
\(904\) 8.00000 0.266076
\(905\) 4.00000 0.132964
\(906\) −20.0000 −0.664455
\(907\) 39.0000 1.29497 0.647487 0.762077i \(-0.275820\pi\)
0.647487 + 0.762077i \(0.275820\pi\)
\(908\) 28.0000 0.929213
\(909\) 0 0
\(910\) 0 0
\(911\) −24.0000 −0.795155 −0.397578 0.917568i \(-0.630149\pi\)
−0.397578 + 0.917568i \(0.630149\pi\)
\(912\) 2.00000 0.0662266
\(913\) 4.00000 0.132381
\(914\) 9.00000 0.297694
\(915\) 14.0000 0.462826
\(916\) 13.0000 0.429532
\(917\) −15.0000 −0.495344
\(918\) −7.00000 −0.231034
\(919\) −46.0000 −1.51740 −0.758700 0.651440i \(-0.774165\pi\)
−0.758700 + 0.651440i \(0.774165\pi\)
\(920\) −2.00000 −0.0659380
\(921\) 30.0000 0.988534
\(922\) 20.0000 0.658665
\(923\) 0 0
\(924\) 4.00000 0.131590
\(925\) 2.00000 0.0657596
\(926\) −18.0000 −0.591517
\(927\) −11.0000 −0.361287
\(928\) 2.00000 0.0656532
\(929\) −27.0000 −0.885841 −0.442921 0.896561i \(-0.646058\pi\)
−0.442921 + 0.896561i \(0.646058\pi\)
\(930\) −18.0000 −0.590243
\(931\) 2.00000 0.0655474
\(932\) −16.0000 −0.524097
\(933\) 30.0000 0.982156
\(934\) −37.0000 −1.21068
\(935\) 56.0000 1.83140
\(936\) 0 0
\(937\) −18.0000 −0.588034 −0.294017 0.955800i \(-0.594992\pi\)
−0.294017 + 0.955800i \(0.594992\pi\)
\(938\) −5.00000 −0.163256
\(939\) 0 0
\(940\) −12.0000 −0.391397
\(941\) −28.0000 −0.912774 −0.456387 0.889781i \(-0.650857\pi\)
−0.456387 + 0.889781i \(0.650857\pi\)
\(942\) −10.0000 −0.325818
\(943\) −2.00000 −0.0651290
\(944\) 15.0000 0.488208
\(945\) 2.00000 0.0650600
\(946\) −20.0000 −0.650256
\(947\) 4.00000 0.129983 0.0649913 0.997886i \(-0.479298\pi\)
0.0649913 + 0.997886i \(0.479298\pi\)
\(948\) −4.00000 −0.129914
\(949\) 0 0
\(950\) 2.00000 0.0648886
\(951\) 23.0000 0.745826
\(952\) 7.00000 0.226871
\(953\) 14.0000 0.453504 0.226752 0.973952i \(-0.427189\pi\)
0.226752 + 0.973952i \(0.427189\pi\)
\(954\) −3.00000 −0.0971286
\(955\) −6.00000 −0.194155
\(956\) −9.00000 −0.291081
\(957\) 8.00000 0.258603
\(958\) 0 0
\(959\) −4.00000 −0.129167
\(960\) −2.00000 −0.0645497
\(961\) 50.0000 1.61290
\(962\) 0 0
\(963\) −6.00000 −0.193347
\(964\) 14.0000 0.450910
\(965\) 44.0000 1.41641
\(966\) −1.00000 −0.0321745
\(967\) 18.0000 0.578841 0.289420 0.957202i \(-0.406537\pi\)
0.289420 + 0.957202i \(0.406537\pi\)
\(968\) −5.00000 −0.160706
\(969\) 14.0000 0.449745
\(970\) −32.0000 −1.02746
\(971\) 15.0000 0.481373 0.240686 0.970603i \(-0.422627\pi\)
0.240686 + 0.970603i \(0.422627\pi\)
\(972\) 1.00000 0.0320750
\(973\) −16.0000 −0.512936
\(974\) 28.0000 0.897178
\(975\) 0 0
\(976\) −7.00000 −0.224065
\(977\) −48.0000 −1.53566 −0.767828 0.640656i \(-0.778662\pi\)
−0.767828 + 0.640656i \(0.778662\pi\)
\(978\) −21.0000 −0.671506
\(979\) −12.0000 −0.383522
\(980\) −2.00000 −0.0638877
\(981\) −16.0000 −0.510841
\(982\) 8.00000 0.255290
\(983\) −46.0000 −1.46717 −0.733586 0.679597i \(-0.762155\pi\)
−0.733586 + 0.679597i \(0.762155\pi\)
\(984\) −2.00000 −0.0637577
\(985\) 18.0000 0.573528
\(986\) 14.0000 0.445851
\(987\) −6.00000 −0.190982
\(988\) 0 0
\(989\) 5.00000 0.158991
\(990\) −8.00000 −0.254257
\(991\) 18.0000 0.571789 0.285894 0.958261i \(-0.407709\pi\)
0.285894 + 0.958261i \(0.407709\pi\)
\(992\) 9.00000 0.285750
\(993\) 36.0000 1.14243
\(994\) 1.00000 0.0317181
\(995\) 10.0000 0.317021
\(996\) −1.00000 −0.0316862
\(997\) −35.0000 −1.10846 −0.554231 0.832363i \(-0.686987\pi\)
−0.554231 + 0.832363i \(0.686987\pi\)
\(998\) −7.00000 −0.221581
\(999\) −2.00000 −0.0632772
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.h.1.1 1
13.3 even 3 546.2.l.d.295.1 yes 2
13.9 even 3 546.2.l.d.211.1 2
13.12 even 2 7098.2.a.bf.1.1 1
39.29 odd 6 1638.2.r.k.1387.1 2
39.35 odd 6 1638.2.r.k.757.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
546.2.l.d.211.1 2 13.9 even 3
546.2.l.d.295.1 yes 2 13.3 even 3
1638.2.r.k.757.1 2 39.35 odd 6
1638.2.r.k.1387.1 2 39.29 odd 6
7098.2.a.h.1.1 1 1.1 even 1 trivial
7098.2.a.bf.1.1 1 13.12 even 2