Properties

Label 3234.2.a.u.1.1
Level $3234$
Weight $2$
Character 3234.1
Self dual yes
Analytic conductor $25.824$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3234 = 2 \cdot 3 \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3234.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(25.8236200137\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 3234.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -2.00000 q^{5} +1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -2.00000 q^{5} +1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} -2.00000 q^{10} +1.00000 q^{11} +1.00000 q^{12} -4.00000 q^{13} -2.00000 q^{15} +1.00000 q^{16} -6.00000 q^{17} +1.00000 q^{18} -2.00000 q^{19} -2.00000 q^{20} +1.00000 q^{22} +1.00000 q^{24} -1.00000 q^{25} -4.00000 q^{26} +1.00000 q^{27} -6.00000 q^{29} -2.00000 q^{30} -2.00000 q^{31} +1.00000 q^{32} +1.00000 q^{33} -6.00000 q^{34} +1.00000 q^{36} +2.00000 q^{37} -2.00000 q^{38} -4.00000 q^{39} -2.00000 q^{40} +2.00000 q^{41} -12.0000 q^{43} +1.00000 q^{44} -2.00000 q^{45} +6.00000 q^{47} +1.00000 q^{48} -1.00000 q^{50} -6.00000 q^{51} -4.00000 q^{52} +10.0000 q^{53} +1.00000 q^{54} -2.00000 q^{55} -2.00000 q^{57} -6.00000 q^{58} -8.00000 q^{59} -2.00000 q^{60} -4.00000 q^{61} -2.00000 q^{62} +1.00000 q^{64} +8.00000 q^{65} +1.00000 q^{66} -6.00000 q^{68} -8.00000 q^{71} +1.00000 q^{72} -6.00000 q^{73} +2.00000 q^{74} -1.00000 q^{75} -2.00000 q^{76} -4.00000 q^{78} +12.0000 q^{79} -2.00000 q^{80} +1.00000 q^{81} +2.00000 q^{82} -6.00000 q^{83} +12.0000 q^{85} -12.0000 q^{86} -6.00000 q^{87} +1.00000 q^{88} +12.0000 q^{89} -2.00000 q^{90} -2.00000 q^{93} +6.00000 q^{94} +4.00000 q^{95} +1.00000 q^{96} +1.00000 q^{99} +O(q^{100})\)

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
\(7\) 0 0
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) −2.00000 −0.632456
\(11\) 1.00000 0.301511
\(12\) 1.00000 0.288675
\(13\) −4.00000 −1.10940 −0.554700 0.832050i \(-0.687167\pi\)
−0.554700 + 0.832050i \(0.687167\pi\)
\(14\) 0 0
\(15\) −2.00000 −0.516398
\(16\) 1.00000 0.250000
\(17\) −6.00000 −1.45521 −0.727607 0.685994i \(-0.759367\pi\)
−0.727607 + 0.685994i \(0.759367\pi\)
\(18\) 1.00000 0.235702
\(19\) −2.00000 −0.458831 −0.229416 0.973329i \(-0.573682\pi\)
−0.229416 + 0.973329i \(0.573682\pi\)
\(20\) −2.00000 −0.447214
\(21\) 0 0
\(22\) 1.00000 0.213201
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 1.00000 0.204124
\(25\) −1.00000 −0.200000
\(26\) −4.00000 −0.784465
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −6.00000 −1.11417 −0.557086 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) −2.00000 −0.365148
\(31\) −2.00000 −0.359211 −0.179605 0.983739i \(-0.557482\pi\)
−0.179605 + 0.983739i \(0.557482\pi\)
\(32\) 1.00000 0.176777
\(33\) 1.00000 0.174078
\(34\) −6.00000 −1.02899
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 2.00000 0.328798 0.164399 0.986394i \(-0.447432\pi\)
0.164399 + 0.986394i \(0.447432\pi\)
\(38\) −2.00000 −0.324443
\(39\) −4.00000 −0.640513
\(40\) −2.00000 −0.316228
\(41\) 2.00000 0.312348 0.156174 0.987730i \(-0.450084\pi\)
0.156174 + 0.987730i \(0.450084\pi\)
\(42\) 0 0
\(43\) −12.0000 −1.82998 −0.914991 0.403473i \(-0.867803\pi\)
−0.914991 + 0.403473i \(0.867803\pi\)
\(44\) 1.00000 0.150756
\(45\) −2.00000 −0.298142
\(46\) 0 0
\(47\) 6.00000 0.875190 0.437595 0.899172i \(-0.355830\pi\)
0.437595 + 0.899172i \(0.355830\pi\)
\(48\) 1.00000 0.144338
\(49\) 0 0
\(50\) −1.00000 −0.141421
\(51\) −6.00000 −0.840168
\(52\) −4.00000 −0.554700
\(53\) 10.0000 1.37361 0.686803 0.726844i \(-0.259014\pi\)
0.686803 + 0.726844i \(0.259014\pi\)
\(54\) 1.00000 0.136083
\(55\) −2.00000 −0.269680
\(56\) 0 0
\(57\) −2.00000 −0.264906
\(58\) −6.00000 −0.787839
\(59\) −8.00000 −1.04151 −0.520756 0.853706i \(-0.674350\pi\)
−0.520756 + 0.853706i \(0.674350\pi\)
\(60\) −2.00000 −0.258199
\(61\) −4.00000 −0.512148 −0.256074 0.966657i \(-0.582429\pi\)
−0.256074 + 0.966657i \(0.582429\pi\)
\(62\) −2.00000 −0.254000
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 8.00000 0.992278
\(66\) 1.00000 0.123091
\(67\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(68\) −6.00000 −0.727607
\(69\) 0 0
\(70\) 0 0
\(71\) −8.00000 −0.949425 −0.474713 0.880141i \(-0.657448\pi\)
−0.474713 + 0.880141i \(0.657448\pi\)
\(72\) 1.00000 0.117851
\(73\) −6.00000 −0.702247 −0.351123 0.936329i \(-0.614200\pi\)
−0.351123 + 0.936329i \(0.614200\pi\)
\(74\) 2.00000 0.232495
\(75\) −1.00000 −0.115470
\(76\) −2.00000 −0.229416
\(77\) 0 0
\(78\) −4.00000 −0.452911
\(79\) 12.0000 1.35011 0.675053 0.737769i \(-0.264121\pi\)
0.675053 + 0.737769i \(0.264121\pi\)
\(80\) −2.00000 −0.223607
\(81\) 1.00000 0.111111
\(82\) 2.00000 0.220863
\(83\) −6.00000 −0.658586 −0.329293 0.944228i \(-0.606810\pi\)
−0.329293 + 0.944228i \(0.606810\pi\)
\(84\) 0 0
\(85\) 12.0000 1.30158
\(86\) −12.0000 −1.29399
\(87\) −6.00000 −0.643268
\(88\) 1.00000 0.106600
\(89\) 12.0000 1.27200 0.635999 0.771690i \(-0.280588\pi\)
0.635999 + 0.771690i \(0.280588\pi\)
\(90\) −2.00000 −0.210819
\(91\) 0 0
\(92\) 0 0
\(93\) −2.00000 −0.207390
\(94\) 6.00000 0.618853
\(95\) 4.00000 0.410391
\(96\) 1.00000 0.102062
\(97\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(98\) 0 0
\(99\) 1.00000 0.100504
\(100\) −1.00000 −0.100000
\(101\) −12.0000 −1.19404 −0.597022 0.802225i \(-0.703650\pi\)
−0.597022 + 0.802225i \(0.703650\pi\)
\(102\) −6.00000 −0.594089
\(103\) −6.00000 −0.591198 −0.295599 0.955312i \(-0.595519\pi\)
−0.295599 + 0.955312i \(0.595519\pi\)
\(104\) −4.00000 −0.392232
\(105\) 0 0
\(106\) 10.0000 0.971286
\(107\) −12.0000 −1.16008 −0.580042 0.814587i \(-0.696964\pi\)
−0.580042 + 0.814587i \(0.696964\pi\)
\(108\) 1.00000 0.0962250
\(109\) −10.0000 −0.957826 −0.478913 0.877862i \(-0.658969\pi\)
−0.478913 + 0.877862i \(0.658969\pi\)
\(110\) −2.00000 −0.190693
\(111\) 2.00000 0.189832
\(112\) 0 0
\(113\) 14.0000 1.31701 0.658505 0.752577i \(-0.271189\pi\)
0.658505 + 0.752577i \(0.271189\pi\)
\(114\) −2.00000 −0.187317
\(115\) 0 0
\(116\) −6.00000 −0.557086
\(117\) −4.00000 −0.369800
\(118\) −8.00000 −0.736460
\(119\) 0 0
\(120\) −2.00000 −0.182574
\(121\) 1.00000 0.0909091
\(122\) −4.00000 −0.362143
\(123\) 2.00000 0.180334
\(124\) −2.00000 −0.179605
\(125\) 12.0000 1.07331
\(126\) 0 0
\(127\) 4.00000 0.354943 0.177471 0.984126i \(-0.443208\pi\)
0.177471 + 0.984126i \(0.443208\pi\)
\(128\) 1.00000 0.0883883
\(129\) −12.0000 −1.05654
\(130\) 8.00000 0.701646
\(131\) −2.00000 −0.174741 −0.0873704 0.996176i \(-0.527846\pi\)
−0.0873704 + 0.996176i \(0.527846\pi\)
\(132\) 1.00000 0.0870388
\(133\) 0 0
\(134\) 0 0
\(135\) −2.00000 −0.172133
\(136\) −6.00000 −0.514496
\(137\) 10.0000 0.854358 0.427179 0.904167i \(-0.359507\pi\)
0.427179 + 0.904167i \(0.359507\pi\)
\(138\) 0 0
\(139\) 10.0000 0.848189 0.424094 0.905618i \(-0.360592\pi\)
0.424094 + 0.905618i \(0.360592\pi\)
\(140\) 0 0
\(141\) 6.00000 0.505291
\(142\) −8.00000 −0.671345
\(143\) −4.00000 −0.334497
\(144\) 1.00000 0.0833333
\(145\) 12.0000 0.996546
\(146\) −6.00000 −0.496564
\(147\) 0 0
\(148\) 2.00000 0.164399
\(149\) 10.0000 0.819232 0.409616 0.912258i \(-0.365663\pi\)
0.409616 + 0.912258i \(0.365663\pi\)
\(150\) −1.00000 −0.0816497
\(151\) −16.0000 −1.30206 −0.651031 0.759051i \(-0.725663\pi\)
−0.651031 + 0.759051i \(0.725663\pi\)
\(152\) −2.00000 −0.162221
\(153\) −6.00000 −0.485071
\(154\) 0 0
\(155\) 4.00000 0.321288
\(156\) −4.00000 −0.320256
\(157\) −14.0000 −1.11732 −0.558661 0.829396i \(-0.688685\pi\)
−0.558661 + 0.829396i \(0.688685\pi\)
\(158\) 12.0000 0.954669
\(159\) 10.0000 0.793052
\(160\) −2.00000 −0.158114
\(161\) 0 0
\(162\) 1.00000 0.0785674
\(163\) −12.0000 −0.939913 −0.469956 0.882690i \(-0.655730\pi\)
−0.469956 + 0.882690i \(0.655730\pi\)
\(164\) 2.00000 0.156174
\(165\) −2.00000 −0.155700
\(166\) −6.00000 −0.465690
\(167\) −12.0000 −0.928588 −0.464294 0.885681i \(-0.653692\pi\)
−0.464294 + 0.885681i \(0.653692\pi\)
\(168\) 0 0
\(169\) 3.00000 0.230769
\(170\) 12.0000 0.920358
\(171\) −2.00000 −0.152944
\(172\) −12.0000 −0.914991
\(173\) 24.0000 1.82469 0.912343 0.409426i \(-0.134271\pi\)
0.912343 + 0.409426i \(0.134271\pi\)
\(174\) −6.00000 −0.454859
\(175\) 0 0
\(176\) 1.00000 0.0753778
\(177\) −8.00000 −0.601317
\(178\) 12.0000 0.899438
\(179\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) −2.00000 −0.149071
\(181\) −18.0000 −1.33793 −0.668965 0.743294i \(-0.733262\pi\)
−0.668965 + 0.743294i \(0.733262\pi\)
\(182\) 0 0
\(183\) −4.00000 −0.295689
\(184\) 0 0
\(185\) −4.00000 −0.294086
\(186\) −2.00000 −0.146647
\(187\) −6.00000 −0.438763
\(188\) 6.00000 0.437595
\(189\) 0 0
\(190\) 4.00000 0.290191
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) 1.00000 0.0721688
\(193\) −14.0000 −1.00774 −0.503871 0.863779i \(-0.668091\pi\)
−0.503871 + 0.863779i \(0.668091\pi\)
\(194\) 0 0
\(195\) 8.00000 0.572892
\(196\) 0 0
\(197\) −18.0000 −1.28245 −0.641223 0.767354i \(-0.721573\pi\)
−0.641223 + 0.767354i \(0.721573\pi\)
\(198\) 1.00000 0.0710669
\(199\) −14.0000 −0.992434 −0.496217 0.868199i \(-0.665278\pi\)
−0.496217 + 0.868199i \(0.665278\pi\)
\(200\) −1.00000 −0.0707107
\(201\) 0 0
\(202\) −12.0000 −0.844317
\(203\) 0 0
\(204\) −6.00000 −0.420084
\(205\) −4.00000 −0.279372
\(206\) −6.00000 −0.418040
\(207\) 0 0
\(208\) −4.00000 −0.277350
\(209\) −2.00000 −0.138343
\(210\) 0 0
\(211\) 20.0000 1.37686 0.688428 0.725304i \(-0.258301\pi\)
0.688428 + 0.725304i \(0.258301\pi\)
\(212\) 10.0000 0.686803
\(213\) −8.00000 −0.548151
\(214\) −12.0000 −0.820303
\(215\) 24.0000 1.63679
\(216\) 1.00000 0.0680414
\(217\) 0 0
\(218\) −10.0000 −0.677285
\(219\) −6.00000 −0.405442
\(220\) −2.00000 −0.134840
\(221\) 24.0000 1.61441
\(222\) 2.00000 0.134231
\(223\) 14.0000 0.937509 0.468755 0.883328i \(-0.344703\pi\)
0.468755 + 0.883328i \(0.344703\pi\)
\(224\) 0 0
\(225\) −1.00000 −0.0666667
\(226\) 14.0000 0.931266
\(227\) 6.00000 0.398234 0.199117 0.979976i \(-0.436193\pi\)
0.199117 + 0.979976i \(0.436193\pi\)
\(228\) −2.00000 −0.132453
\(229\) 6.00000 0.396491 0.198246 0.980152i \(-0.436476\pi\)
0.198246 + 0.980152i \(0.436476\pi\)
\(230\) 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 \(0.175597\pi\)
\(234\) −4.00000 −0.261488
\(235\) −12.0000 −0.782794
\(236\) −8.00000 −0.520756
\(237\) 12.0000 0.779484
\(238\) 0 0
\(239\) −16.0000 −1.03495 −0.517477 0.855697i \(-0.673129\pi\)
−0.517477 + 0.855697i \(0.673129\pi\)
\(240\) −2.00000 −0.129099
\(241\) −22.0000 −1.41714 −0.708572 0.705638i \(-0.750660\pi\)
−0.708572 + 0.705638i \(0.750660\pi\)
\(242\) 1.00000 0.0642824
\(243\) 1.00000 0.0641500
\(244\) −4.00000 −0.256074
\(245\) 0 0
\(246\) 2.00000 0.127515
\(247\) 8.00000 0.509028
\(248\) −2.00000 −0.127000
\(249\) −6.00000 −0.380235
\(250\) 12.0000 0.758947
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 4.00000 0.250982
\(255\) 12.0000 0.751469
\(256\) 1.00000 0.0625000
\(257\) −20.0000 −1.24757 −0.623783 0.781598i \(-0.714405\pi\)
−0.623783 + 0.781598i \(0.714405\pi\)
\(258\) −12.0000 −0.747087
\(259\) 0 0
\(260\) 8.00000 0.496139
\(261\) −6.00000 −0.371391
\(262\) −2.00000 −0.123560
\(263\) 4.00000 0.246651 0.123325 0.992366i \(-0.460644\pi\)
0.123325 + 0.992366i \(0.460644\pi\)
\(264\) 1.00000 0.0615457
\(265\) −20.0000 −1.22859
\(266\) 0 0
\(267\) 12.0000 0.734388
\(268\) 0 0
\(269\) 6.00000 0.365826 0.182913 0.983129i \(-0.441447\pi\)
0.182913 + 0.983129i \(0.441447\pi\)
\(270\) −2.00000 −0.121716
\(271\) 28.0000 1.70088 0.850439 0.526073i \(-0.176336\pi\)
0.850439 + 0.526073i \(0.176336\pi\)
\(272\) −6.00000 −0.363803
\(273\) 0 0
\(274\) 10.0000 0.604122
\(275\) −1.00000 −0.0603023
\(276\) 0 0
\(277\) 10.0000 0.600842 0.300421 0.953807i \(-0.402873\pi\)
0.300421 + 0.953807i \(0.402873\pi\)
\(278\) 10.0000 0.599760
\(279\) −2.00000 −0.119737
\(280\) 0 0
\(281\) 2.00000 0.119310 0.0596550 0.998219i \(-0.481000\pi\)
0.0596550 + 0.998219i \(0.481000\pi\)
\(282\) 6.00000 0.357295
\(283\) −14.0000 −0.832214 −0.416107 0.909316i \(-0.636606\pi\)
−0.416107 + 0.909316i \(0.636606\pi\)
\(284\) −8.00000 −0.474713
\(285\) 4.00000 0.236940
\(286\) −4.00000 −0.236525
\(287\) 0 0
\(288\) 1.00000 0.0589256
\(289\) 19.0000 1.11765
\(290\) 12.0000 0.704664
\(291\) 0 0
\(292\) −6.00000 −0.351123
\(293\) −8.00000 −0.467365 −0.233682 0.972313i \(-0.575078\pi\)
−0.233682 + 0.972313i \(0.575078\pi\)
\(294\) 0 0
\(295\) 16.0000 0.931556
\(296\) 2.00000 0.116248
\(297\) 1.00000 0.0580259
\(298\) 10.0000 0.579284
\(299\) 0 0
\(300\) −1.00000 −0.0577350
\(301\) 0 0
\(302\) −16.0000 −0.920697
\(303\) −12.0000 −0.689382
\(304\) −2.00000 −0.114708
\(305\) 8.00000 0.458079
\(306\) −6.00000 −0.342997
\(307\) 26.0000 1.48390 0.741949 0.670456i \(-0.233902\pi\)
0.741949 + 0.670456i \(0.233902\pi\)
\(308\) 0 0
\(309\) −6.00000 −0.341328
\(310\) 4.00000 0.227185
\(311\) 34.0000 1.92796 0.963982 0.265969i \(-0.0856919\pi\)
0.963982 + 0.265969i \(0.0856919\pi\)
\(312\) −4.00000 −0.226455
\(313\) 20.0000 1.13047 0.565233 0.824931i \(-0.308786\pi\)
0.565233 + 0.824931i \(0.308786\pi\)
\(314\) −14.0000 −0.790066
\(315\) 0 0
\(316\) 12.0000 0.675053
\(317\) 18.0000 1.01098 0.505490 0.862832i \(-0.331312\pi\)
0.505490 + 0.862832i \(0.331312\pi\)
\(318\) 10.0000 0.560772
\(319\) −6.00000 −0.335936
\(320\) −2.00000 −0.111803
\(321\) −12.0000 −0.669775
\(322\) 0 0
\(323\) 12.0000 0.667698
\(324\) 1.00000 0.0555556
\(325\) 4.00000 0.221880
\(326\) −12.0000 −0.664619
\(327\) −10.0000 −0.553001
\(328\) 2.00000 0.110432
\(329\) 0 0
\(330\) −2.00000 −0.110096
\(331\) −24.0000 −1.31916 −0.659580 0.751635i \(-0.729266\pi\)
−0.659580 + 0.751635i \(0.729266\pi\)
\(332\) −6.00000 −0.329293
\(333\) 2.00000 0.109599
\(334\) −12.0000 −0.656611
\(335\) 0 0
\(336\) 0 0
\(337\) −26.0000 −1.41631 −0.708155 0.706057i \(-0.750472\pi\)
−0.708155 + 0.706057i \(0.750472\pi\)
\(338\) 3.00000 0.163178
\(339\) 14.0000 0.760376
\(340\) 12.0000 0.650791
\(341\) −2.00000 −0.108306
\(342\) −2.00000 −0.108148
\(343\) 0 0
\(344\) −12.0000 −0.646997
\(345\) 0 0
\(346\) 24.0000 1.29025
\(347\) 20.0000 1.07366 0.536828 0.843692i \(-0.319622\pi\)
0.536828 + 0.843692i \(0.319622\pi\)
\(348\) −6.00000 −0.321634
\(349\) 16.0000 0.856460 0.428230 0.903670i \(-0.359137\pi\)
0.428230 + 0.903670i \(0.359137\pi\)
\(350\) 0 0
\(351\) −4.00000 −0.213504
\(352\) 1.00000 0.0533002
\(353\) 12.0000 0.638696 0.319348 0.947638i \(-0.396536\pi\)
0.319348 + 0.947638i \(0.396536\pi\)
\(354\) −8.00000 −0.425195
\(355\) 16.0000 0.849192
\(356\) 12.0000 0.635999
\(357\) 0 0
\(358\) 0 0
\(359\) 20.0000 1.05556 0.527780 0.849381i \(-0.323025\pi\)
0.527780 + 0.849381i \(0.323025\pi\)
\(360\) −2.00000 −0.105409
\(361\) −15.0000 −0.789474
\(362\) −18.0000 −0.946059
\(363\) 1.00000 0.0524864
\(364\) 0 0
\(365\) 12.0000 0.628109
\(366\) −4.00000 −0.209083
\(367\) 22.0000 1.14839 0.574195 0.818718i \(-0.305315\pi\)
0.574195 + 0.818718i \(0.305315\pi\)
\(368\) 0 0
\(369\) 2.00000 0.104116
\(370\) −4.00000 −0.207950
\(371\) 0 0
\(372\) −2.00000 −0.103695
\(373\) 6.00000 0.310668 0.155334 0.987862i \(-0.450355\pi\)
0.155334 + 0.987862i \(0.450355\pi\)
\(374\) −6.00000 −0.310253
\(375\) 12.0000 0.619677
\(376\) 6.00000 0.309426
\(377\) 24.0000 1.23606
\(378\) 0 0
\(379\) −20.0000 −1.02733 −0.513665 0.857991i \(-0.671713\pi\)
−0.513665 + 0.857991i \(0.671713\pi\)
\(380\) 4.00000 0.205196
\(381\) 4.00000 0.204926
\(382\) 0 0
\(383\) −6.00000 −0.306586 −0.153293 0.988181i \(-0.548988\pi\)
−0.153293 + 0.988181i \(0.548988\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) −14.0000 −0.712581
\(387\) −12.0000 −0.609994
\(388\) 0 0
\(389\) 22.0000 1.11544 0.557722 0.830028i \(-0.311675\pi\)
0.557722 + 0.830028i \(0.311675\pi\)
\(390\) 8.00000 0.405096
\(391\) 0 0
\(392\) 0 0
\(393\) −2.00000 −0.100887
\(394\) −18.0000 −0.906827
\(395\) −24.0000 −1.20757
\(396\) 1.00000 0.0502519
\(397\) 34.0000 1.70641 0.853206 0.521575i \(-0.174655\pi\)
0.853206 + 0.521575i \(0.174655\pi\)
\(398\) −14.0000 −0.701757
\(399\) 0 0
\(400\) −1.00000 −0.0500000
\(401\) −30.0000 −1.49813 −0.749064 0.662497i \(-0.769497\pi\)
−0.749064 + 0.662497i \(0.769497\pi\)
\(402\) 0 0
\(403\) 8.00000 0.398508
\(404\) −12.0000 −0.597022
\(405\) −2.00000 −0.0993808
\(406\) 0 0
\(407\) 2.00000 0.0991363
\(408\) −6.00000 −0.297044
\(409\) −2.00000 −0.0988936 −0.0494468 0.998777i \(-0.515746\pi\)
−0.0494468 + 0.998777i \(0.515746\pi\)
\(410\) −4.00000 −0.197546
\(411\) 10.0000 0.493264
\(412\) −6.00000 −0.295599
\(413\) 0 0
\(414\) 0 0
\(415\) 12.0000 0.589057
\(416\) −4.00000 −0.196116
\(417\) 10.0000 0.489702
\(418\) −2.00000 −0.0978232
\(419\) −12.0000 −0.586238 −0.293119 0.956076i \(-0.594693\pi\)
−0.293119 + 0.956076i \(0.594693\pi\)
\(420\) 0 0
\(421\) 14.0000 0.682318 0.341159 0.940006i \(-0.389181\pi\)
0.341159 + 0.940006i \(0.389181\pi\)
\(422\) 20.0000 0.973585
\(423\) 6.00000 0.291730
\(424\) 10.0000 0.485643
\(425\) 6.00000 0.291043
\(426\) −8.00000 −0.387601
\(427\) 0 0
\(428\) −12.0000 −0.580042
\(429\) −4.00000 −0.193122
\(430\) 24.0000 1.15738
\(431\) 12.0000 0.578020 0.289010 0.957326i \(-0.406674\pi\)
0.289010 + 0.957326i \(0.406674\pi\)
\(432\) 1.00000 0.0481125
\(433\) −8.00000 −0.384455 −0.192228 0.981350i \(-0.561571\pi\)
−0.192228 + 0.981350i \(0.561571\pi\)
\(434\) 0 0
\(435\) 12.0000 0.575356
\(436\) −10.0000 −0.478913
\(437\) 0 0
\(438\) −6.00000 −0.286691
\(439\) −28.0000 −1.33637 −0.668184 0.743996i \(-0.732928\pi\)
−0.668184 + 0.743996i \(0.732928\pi\)
\(440\) −2.00000 −0.0953463
\(441\) 0 0
\(442\) 24.0000 1.14156
\(443\) 24.0000 1.14027 0.570137 0.821549i \(-0.306890\pi\)
0.570137 + 0.821549i \(0.306890\pi\)
\(444\) 2.00000 0.0949158
\(445\) −24.0000 −1.13771
\(446\) 14.0000 0.662919
\(447\) 10.0000 0.472984
\(448\) 0 0
\(449\) −18.0000 −0.849473 −0.424736 0.905317i \(-0.639633\pi\)
−0.424736 + 0.905317i \(0.639633\pi\)
\(450\) −1.00000 −0.0471405
\(451\) 2.00000 0.0941763
\(452\) 14.0000 0.658505
\(453\) −16.0000 −0.751746
\(454\) 6.00000 0.281594
\(455\) 0 0
\(456\) −2.00000 −0.0936586
\(457\) −6.00000 −0.280668 −0.140334 0.990104i \(-0.544818\pi\)
−0.140334 + 0.990104i \(0.544818\pi\)
\(458\) 6.00000 0.280362
\(459\) −6.00000 −0.280056
\(460\) 0 0
\(461\) −12.0000 −0.558896 −0.279448 0.960161i \(-0.590151\pi\)
−0.279448 + 0.960161i \(0.590151\pi\)
\(462\) 0 0
\(463\) 16.0000 0.743583 0.371792 0.928316i \(-0.378744\pi\)
0.371792 + 0.928316i \(0.378744\pi\)
\(464\) −6.00000 −0.278543
\(465\) 4.00000 0.185496
\(466\) 26.0000 1.20443
\(467\) −28.0000 −1.29569 −0.647843 0.761774i \(-0.724329\pi\)
−0.647843 + 0.761774i \(0.724329\pi\)
\(468\) −4.00000 −0.184900
\(469\) 0 0
\(470\) −12.0000 −0.553519
\(471\) −14.0000 −0.645086
\(472\) −8.00000 −0.368230
\(473\) −12.0000 −0.551761
\(474\) 12.0000 0.551178
\(475\) 2.00000 0.0917663
\(476\) 0 0
\(477\) 10.0000 0.457869
\(478\) −16.0000 −0.731823
\(479\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) −2.00000 −0.0912871
\(481\) −8.00000 −0.364769
\(482\) −22.0000 −1.00207
\(483\) 0 0
\(484\) 1.00000 0.0454545
\(485\) 0 0
\(486\) 1.00000 0.0453609
\(487\) −16.0000 −0.725029 −0.362515 0.931978i \(-0.618082\pi\)
−0.362515 + 0.931978i \(0.618082\pi\)
\(488\) −4.00000 −0.181071
\(489\) −12.0000 −0.542659
\(490\) 0 0
\(491\) −20.0000 −0.902587 −0.451294 0.892375i \(-0.649037\pi\)
−0.451294 + 0.892375i \(0.649037\pi\)
\(492\) 2.00000 0.0901670
\(493\) 36.0000 1.62136
\(494\) 8.00000 0.359937
\(495\) −2.00000 −0.0898933
\(496\) −2.00000 −0.0898027
\(497\) 0 0
\(498\) −6.00000 −0.268866
\(499\) −40.0000 −1.79065 −0.895323 0.445418i \(-0.853055\pi\)
−0.895323 + 0.445418i \(0.853055\pi\)
\(500\) 12.0000 0.536656
\(501\) −12.0000 −0.536120
\(502\) 0 0
\(503\) 28.0000 1.24846 0.624229 0.781241i \(-0.285413\pi\)
0.624229 + 0.781241i \(0.285413\pi\)
\(504\) 0 0
\(505\) 24.0000 1.06799
\(506\) 0 0
\(507\) 3.00000 0.133235
\(508\) 4.00000 0.177471
\(509\) 30.0000 1.32973 0.664863 0.746965i \(-0.268490\pi\)
0.664863 + 0.746965i \(0.268490\pi\)
\(510\) 12.0000 0.531369
\(511\) 0 0
\(512\) 1.00000 0.0441942
\(513\) −2.00000 −0.0883022
\(514\) −20.0000 −0.882162
\(515\) 12.0000 0.528783
\(516\) −12.0000 −0.528271
\(517\) 6.00000 0.263880
\(518\) 0 0
\(519\) 24.0000 1.05348
\(520\) 8.00000 0.350823
\(521\) −28.0000 −1.22670 −0.613351 0.789810i \(-0.710179\pi\)
−0.613351 + 0.789810i \(0.710179\pi\)
\(522\) −6.00000 −0.262613
\(523\) 2.00000 0.0874539 0.0437269 0.999044i \(-0.486077\pi\)
0.0437269 + 0.999044i \(0.486077\pi\)
\(524\) −2.00000 −0.0873704
\(525\) 0 0
\(526\) 4.00000 0.174408
\(527\) 12.0000 0.522728
\(528\) 1.00000 0.0435194
\(529\) −23.0000 −1.00000
\(530\) −20.0000 −0.868744
\(531\) −8.00000 −0.347170
\(532\) 0 0
\(533\) −8.00000 −0.346518
\(534\) 12.0000 0.519291
\(535\) 24.0000 1.03761
\(536\) 0 0
\(537\) 0 0
\(538\) 6.00000 0.258678
\(539\) 0 0
\(540\) −2.00000 −0.0860663
\(541\) −2.00000 −0.0859867 −0.0429934 0.999075i \(-0.513689\pi\)
−0.0429934 + 0.999075i \(0.513689\pi\)
\(542\) 28.0000 1.20270
\(543\) −18.0000 −0.772454
\(544\) −6.00000 −0.257248
\(545\) 20.0000 0.856706
\(546\) 0 0
\(547\) −20.0000 −0.855138 −0.427569 0.903983i \(-0.640630\pi\)
−0.427569 + 0.903983i \(0.640630\pi\)
\(548\) 10.0000 0.427179
\(549\) −4.00000 −0.170716
\(550\) −1.00000 −0.0426401
\(551\) 12.0000 0.511217
\(552\) 0 0
\(553\) 0 0
\(554\) 10.0000 0.424859
\(555\) −4.00000 −0.169791
\(556\) 10.0000 0.424094
\(557\) −26.0000 −1.10166 −0.550828 0.834619i \(-0.685688\pi\)
−0.550828 + 0.834619i \(0.685688\pi\)
\(558\) −2.00000 −0.0846668
\(559\) 48.0000 2.03018
\(560\) 0 0
\(561\) −6.00000 −0.253320
\(562\) 2.00000 0.0843649
\(563\) 14.0000 0.590030 0.295015 0.955493i \(-0.404675\pi\)
0.295015 + 0.955493i \(0.404675\pi\)
\(564\) 6.00000 0.252646
\(565\) −28.0000 −1.17797
\(566\) −14.0000 −0.588464
\(567\) 0 0
\(568\) −8.00000 −0.335673
\(569\) 34.0000 1.42535 0.712677 0.701492i \(-0.247483\pi\)
0.712677 + 0.701492i \(0.247483\pi\)
\(570\) 4.00000 0.167542
\(571\) 4.00000 0.167395 0.0836974 0.996491i \(-0.473327\pi\)
0.0836974 + 0.996491i \(0.473327\pi\)
\(572\) −4.00000 −0.167248
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) 28.0000 1.16566 0.582828 0.812596i \(-0.301946\pi\)
0.582828 + 0.812596i \(0.301946\pi\)
\(578\) 19.0000 0.790296
\(579\) −14.0000 −0.581820
\(580\) 12.0000 0.498273
\(581\) 0 0
\(582\) 0 0
\(583\) 10.0000 0.414158
\(584\) −6.00000 −0.248282
\(585\) 8.00000 0.330759
\(586\) −8.00000 −0.330477
\(587\) 24.0000 0.990586 0.495293 0.868726i \(-0.335061\pi\)
0.495293 + 0.868726i \(0.335061\pi\)
\(588\) 0 0
\(589\) 4.00000 0.164817
\(590\) 16.0000 0.658710
\(591\) −18.0000 −0.740421
\(592\) 2.00000 0.0821995
\(593\) −22.0000 −0.903432 −0.451716 0.892162i \(-0.649188\pi\)
−0.451716 + 0.892162i \(0.649188\pi\)
\(594\) 1.00000 0.0410305
\(595\) 0 0
\(596\) 10.0000 0.409616
\(597\) −14.0000 −0.572982
\(598\) 0 0
\(599\) −32.0000 −1.30748 −0.653742 0.756717i \(-0.726802\pi\)
−0.653742 + 0.756717i \(0.726802\pi\)
\(600\) −1.00000 −0.0408248
\(601\) 46.0000 1.87638 0.938190 0.346122i \(-0.112502\pi\)
0.938190 + 0.346122i \(0.112502\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −16.0000 −0.651031
\(605\) −2.00000 −0.0813116
\(606\) −12.0000 −0.487467
\(607\) 8.00000 0.324710 0.162355 0.986732i \(-0.448091\pi\)
0.162355 + 0.986732i \(0.448091\pi\)
\(608\) −2.00000 −0.0811107
\(609\) 0 0
\(610\) 8.00000 0.323911
\(611\) −24.0000 −0.970936
\(612\) −6.00000 −0.242536
\(613\) −6.00000 −0.242338 −0.121169 0.992632i \(-0.538664\pi\)
−0.121169 + 0.992632i \(0.538664\pi\)
\(614\) 26.0000 1.04927
\(615\) −4.00000 −0.161296
\(616\) 0 0
\(617\) −18.0000 −0.724653 −0.362326 0.932051i \(-0.618017\pi\)
−0.362326 + 0.932051i \(0.618017\pi\)
\(618\) −6.00000 −0.241355
\(619\) 4.00000 0.160774 0.0803868 0.996764i \(-0.474384\pi\)
0.0803868 + 0.996764i \(0.474384\pi\)
\(620\) 4.00000 0.160644
\(621\) 0 0
\(622\) 34.0000 1.36328
\(623\) 0 0
\(624\) −4.00000 −0.160128
\(625\) −19.0000 −0.760000
\(626\) 20.0000 0.799361
\(627\) −2.00000 −0.0798723
\(628\) −14.0000 −0.558661
\(629\) −12.0000 −0.478471
\(630\) 0 0
\(631\) −40.0000 −1.59237 −0.796187 0.605050i \(-0.793153\pi\)
−0.796187 + 0.605050i \(0.793153\pi\)
\(632\) 12.0000 0.477334
\(633\) 20.0000 0.794929
\(634\) 18.0000 0.714871
\(635\) −8.00000 −0.317470
\(636\) 10.0000 0.396526
\(637\) 0 0
\(638\) −6.00000 −0.237542
\(639\) −8.00000 −0.316475
\(640\) −2.00000 −0.0790569
\(641\) 6.00000 0.236986 0.118493 0.992955i \(-0.462194\pi\)
0.118493 + 0.992955i \(0.462194\pi\)
\(642\) −12.0000 −0.473602
\(643\) −40.0000 −1.57745 −0.788723 0.614749i \(-0.789257\pi\)
−0.788723 + 0.614749i \(0.789257\pi\)
\(644\) 0 0
\(645\) 24.0000 0.944999
\(646\) 12.0000 0.472134
\(647\) 10.0000 0.393141 0.196570 0.980490i \(-0.437020\pi\)
0.196570 + 0.980490i \(0.437020\pi\)
\(648\) 1.00000 0.0392837
\(649\) −8.00000 −0.314027
\(650\) 4.00000 0.156893
\(651\) 0 0
\(652\) −12.0000 −0.469956
\(653\) −18.0000 −0.704394 −0.352197 0.935926i \(-0.614565\pi\)
−0.352197 + 0.935926i \(0.614565\pi\)
\(654\) −10.0000 −0.391031
\(655\) 4.00000 0.156293
\(656\) 2.00000 0.0780869
\(657\) −6.00000 −0.234082
\(658\) 0 0
\(659\) −12.0000 −0.467454 −0.233727 0.972302i \(-0.575092\pi\)
−0.233727 + 0.972302i \(0.575092\pi\)
\(660\) −2.00000 −0.0778499
\(661\) −18.0000 −0.700119 −0.350059 0.936727i \(-0.613839\pi\)
−0.350059 + 0.936727i \(0.613839\pi\)
\(662\) −24.0000 −0.932786
\(663\) 24.0000 0.932083
\(664\) −6.00000 −0.232845
\(665\) 0 0
\(666\) 2.00000 0.0774984
\(667\) 0 0
\(668\) −12.0000 −0.464294
\(669\) 14.0000 0.541271
\(670\) 0 0
\(671\) −4.00000 −0.154418
\(672\) 0 0
\(673\) −18.0000 −0.693849 −0.346925 0.937893i \(-0.612774\pi\)
−0.346925 + 0.937893i \(0.612774\pi\)
\(674\) −26.0000 −1.00148
\(675\) −1.00000 −0.0384900
\(676\) 3.00000 0.115385
\(677\) 48.0000 1.84479 0.922395 0.386248i \(-0.126229\pi\)
0.922395 + 0.386248i \(0.126229\pi\)
\(678\) 14.0000 0.537667
\(679\) 0 0
\(680\) 12.0000 0.460179
\(681\) 6.00000 0.229920
\(682\) −2.00000 −0.0765840
\(683\) 16.0000 0.612223 0.306111 0.951996i \(-0.400972\pi\)
0.306111 + 0.951996i \(0.400972\pi\)
\(684\) −2.00000 −0.0764719
\(685\) −20.0000 −0.764161
\(686\) 0 0
\(687\) 6.00000 0.228914
\(688\) −12.0000 −0.457496
\(689\) −40.0000 −1.52388
\(690\) 0 0
\(691\) −40.0000 −1.52167 −0.760836 0.648944i \(-0.775211\pi\)
−0.760836 + 0.648944i \(0.775211\pi\)
\(692\) 24.0000 0.912343
\(693\) 0 0
\(694\) 20.0000 0.759190
\(695\) −20.0000 −0.758643
\(696\) −6.00000 −0.227429
\(697\) −12.0000 −0.454532
\(698\) 16.0000 0.605609
\(699\) 26.0000 0.983410
\(700\) 0 0
\(701\) 14.0000 0.528773 0.264386 0.964417i \(-0.414831\pi\)
0.264386 + 0.964417i \(0.414831\pi\)
\(702\) −4.00000 −0.150970
\(703\) −4.00000 −0.150863
\(704\) 1.00000 0.0376889
\(705\) −12.0000 −0.451946
\(706\) 12.0000 0.451626
\(707\) 0 0
\(708\) −8.00000 −0.300658
\(709\) −38.0000 −1.42712 −0.713560 0.700594i \(-0.752918\pi\)
−0.713560 + 0.700594i \(0.752918\pi\)
\(710\) 16.0000 0.600469
\(711\) 12.0000 0.450035
\(712\) 12.0000 0.449719
\(713\) 0 0
\(714\) 0 0
\(715\) 8.00000 0.299183
\(716\) 0 0
\(717\) −16.0000 −0.597531
\(718\) 20.0000 0.746393
\(719\) −46.0000 −1.71551 −0.857755 0.514058i \(-0.828142\pi\)
−0.857755 + 0.514058i \(0.828142\pi\)
\(720\) −2.00000 −0.0745356
\(721\) 0 0
\(722\) −15.0000 −0.558242
\(723\) −22.0000 −0.818189
\(724\) −18.0000 −0.668965
\(725\) 6.00000 0.222834
\(726\) 1.00000 0.0371135
\(727\) 30.0000 1.11264 0.556319 0.830969i \(-0.312213\pi\)
0.556319 + 0.830969i \(0.312213\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 12.0000 0.444140
\(731\) 72.0000 2.66302
\(732\) −4.00000 −0.147844
\(733\) −12.0000 −0.443230 −0.221615 0.975134i \(-0.571133\pi\)
−0.221615 + 0.975134i \(0.571133\pi\)
\(734\) 22.0000 0.812035
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 2.00000 0.0736210
\(739\) −4.00000 −0.147142 −0.0735712 0.997290i \(-0.523440\pi\)
−0.0735712 + 0.997290i \(0.523440\pi\)
\(740\) −4.00000 −0.147043
\(741\) 8.00000 0.293887
\(742\) 0 0
\(743\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(744\) −2.00000 −0.0733236
\(745\) −20.0000 −0.732743
\(746\) 6.00000 0.219676
\(747\) −6.00000 −0.219529
\(748\) −6.00000 −0.219382
\(749\) 0 0
\(750\) 12.0000 0.438178
\(751\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(752\) 6.00000 0.218797
\(753\) 0 0
\(754\) 24.0000 0.874028
\(755\) 32.0000 1.16460
\(756\) 0 0
\(757\) −30.0000 −1.09037 −0.545184 0.838316i \(-0.683540\pi\)
−0.545184 + 0.838316i \(0.683540\pi\)
\(758\) −20.0000 −0.726433
\(759\) 0 0
\(760\) 4.00000 0.145095
\(761\) −34.0000 −1.23250 −0.616250 0.787551i \(-0.711349\pi\)
−0.616250 + 0.787551i \(0.711349\pi\)
\(762\) 4.00000 0.144905
\(763\) 0 0
\(764\) 0 0
\(765\) 12.0000 0.433861
\(766\) −6.00000 −0.216789
\(767\) 32.0000 1.15545
\(768\) 1.00000 0.0360844
\(769\) 14.0000 0.504853 0.252426 0.967616i \(-0.418771\pi\)
0.252426 + 0.967616i \(0.418771\pi\)
\(770\) 0 0
\(771\) −20.0000 −0.720282
\(772\) −14.0000 −0.503871
\(773\) 26.0000 0.935155 0.467578 0.883952i \(-0.345127\pi\)
0.467578 + 0.883952i \(0.345127\pi\)
\(774\) −12.0000 −0.431331
\(775\) 2.00000 0.0718421
\(776\) 0 0
\(777\) 0 0
\(778\) 22.0000 0.788738
\(779\) −4.00000 −0.143315
\(780\) 8.00000 0.286446
\(781\) −8.00000 −0.286263
\(782\) 0 0
\(783\) −6.00000 −0.214423
\(784\) 0 0
\(785\) 28.0000 0.999363
\(786\) −2.00000 −0.0713376
\(787\) 38.0000 1.35455 0.677277 0.735728i \(-0.263160\pi\)
0.677277 + 0.735728i \(0.263160\pi\)
\(788\) −18.0000 −0.641223
\(789\) 4.00000 0.142404
\(790\) −24.0000 −0.853882
\(791\) 0 0
\(792\) 1.00000 0.0355335
\(793\) 16.0000 0.568177
\(794\) 34.0000 1.20661
\(795\) −20.0000 −0.709327
\(796\) −14.0000 −0.496217
\(797\) −6.00000 −0.212531 −0.106265 0.994338i \(-0.533889\pi\)
−0.106265 + 0.994338i \(0.533889\pi\)
\(798\) 0 0
\(799\) −36.0000 −1.27359
\(800\) −1.00000 −0.0353553
\(801\) 12.0000 0.423999
\(802\) −30.0000 −1.05934
\(803\) −6.00000 −0.211735
\(804\) 0 0
\(805\) 0 0
\(806\) 8.00000 0.281788
\(807\) 6.00000 0.211210
\(808\) −12.0000 −0.422159
\(809\) 14.0000 0.492214 0.246107 0.969243i \(-0.420849\pi\)
0.246107 + 0.969243i \(0.420849\pi\)
\(810\) −2.00000 −0.0702728
\(811\) 34.0000 1.19390 0.596951 0.802278i \(-0.296379\pi\)
0.596951 + 0.802278i \(0.296379\pi\)
\(812\) 0 0
\(813\) 28.0000 0.982003
\(814\) 2.00000 0.0701000
\(815\) 24.0000 0.840683
\(816\) −6.00000 −0.210042
\(817\) 24.0000 0.839654
\(818\) −2.00000 −0.0699284
\(819\) 0 0
\(820\) −4.00000 −0.139686
\(821\) 42.0000 1.46581 0.732905 0.680331i \(-0.238164\pi\)
0.732905 + 0.680331i \(0.238164\pi\)
\(822\) 10.0000 0.348790
\(823\) 40.0000 1.39431 0.697156 0.716919i \(-0.254448\pi\)
0.697156 + 0.716919i \(0.254448\pi\)
\(824\) −6.00000 −0.209020
\(825\) −1.00000 −0.0348155
\(826\) 0 0
\(827\) −36.0000 −1.25184 −0.625921 0.779886i \(-0.715277\pi\)
−0.625921 + 0.779886i \(0.715277\pi\)
\(828\) 0 0
\(829\) −10.0000 −0.347314 −0.173657 0.984806i \(-0.555558\pi\)
−0.173657 + 0.984806i \(0.555558\pi\)
\(830\) 12.0000 0.416526
\(831\) 10.0000 0.346896
\(832\) −4.00000 −0.138675
\(833\) 0 0
\(834\) 10.0000 0.346272
\(835\) 24.0000 0.830554
\(836\) −2.00000 −0.0691714
\(837\) −2.00000 −0.0691301
\(838\) −12.0000 −0.414533
\(839\) 42.0000 1.45000 0.725001 0.688748i \(-0.241839\pi\)
0.725001 + 0.688748i \(0.241839\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) 14.0000 0.482472
\(843\) 2.00000 0.0688837
\(844\) 20.0000 0.688428
\(845\) −6.00000 −0.206406
\(846\) 6.00000 0.206284
\(847\) 0 0
\(848\) 10.0000 0.343401
\(849\) −14.0000 −0.480479
\(850\) 6.00000 0.205798
\(851\) 0 0
\(852\) −8.00000 −0.274075
\(853\) −36.0000 −1.23262 −0.616308 0.787505i \(-0.711372\pi\)
−0.616308 + 0.787505i \(0.711372\pi\)
\(854\) 0 0
\(855\) 4.00000 0.136797
\(856\) −12.0000 −0.410152
\(857\) −42.0000 −1.43469 −0.717346 0.696717i \(-0.754643\pi\)
−0.717346 + 0.696717i \(0.754643\pi\)
\(858\) −4.00000 −0.136558
\(859\) 8.00000 0.272956 0.136478 0.990643i \(-0.456422\pi\)
0.136478 + 0.990643i \(0.456422\pi\)
\(860\) 24.0000 0.818393
\(861\) 0 0
\(862\) 12.0000 0.408722
\(863\) 56.0000 1.90626 0.953131 0.302558i \(-0.0978405\pi\)
0.953131 + 0.302558i \(0.0978405\pi\)
\(864\) 1.00000 0.0340207
\(865\) −48.0000 −1.63205
\(866\) −8.00000 −0.271851
\(867\) 19.0000 0.645274
\(868\) 0 0
\(869\) 12.0000 0.407072
\(870\) 12.0000 0.406838
\(871\) 0 0
\(872\) −10.0000 −0.338643
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) −6.00000 −0.202721
\(877\) −2.00000 −0.0675352 −0.0337676 0.999430i \(-0.510751\pi\)
−0.0337676 + 0.999430i \(0.510751\pi\)
\(878\) −28.0000 −0.944954
\(879\) −8.00000 −0.269833
\(880\) −2.00000 −0.0674200
\(881\) −24.0000 −0.808581 −0.404290 0.914631i \(-0.632481\pi\)
−0.404290 + 0.914631i \(0.632481\pi\)
\(882\) 0 0
\(883\) −52.0000 −1.74994 −0.874970 0.484178i \(-0.839119\pi\)
−0.874970 + 0.484178i \(0.839119\pi\)
\(884\) 24.0000 0.807207
\(885\) 16.0000 0.537834
\(886\) 24.0000 0.806296
\(887\) −40.0000 −1.34307 −0.671534 0.740973i \(-0.734364\pi\)
−0.671534 + 0.740973i \(0.734364\pi\)
\(888\) 2.00000 0.0671156
\(889\) 0 0
\(890\) −24.0000 −0.804482
\(891\) 1.00000 0.0335013
\(892\) 14.0000 0.468755
\(893\) −12.0000 −0.401565
\(894\) 10.0000 0.334450
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) −18.0000 −0.600668
\(899\) 12.0000 0.400222
\(900\) −1.00000 −0.0333333
\(901\) −60.0000 −1.99889
\(902\) 2.00000 0.0665927
\(903\) 0 0
\(904\) 14.0000 0.465633
\(905\) 36.0000 1.19668
\(906\) −16.0000 −0.531564
\(907\) −8.00000 −0.265636 −0.132818 0.991140i \(-0.542403\pi\)
−0.132818 + 0.991140i \(0.542403\pi\)
\(908\) 6.00000 0.199117
\(909\) −12.0000 −0.398015
\(910\) 0 0
\(911\) 40.0000 1.32526 0.662630 0.748947i \(-0.269440\pi\)
0.662630 + 0.748947i \(0.269440\pi\)
\(912\) −2.00000 −0.0662266
\(913\) −6.00000 −0.198571
\(914\) −6.00000 −0.198462
\(915\) 8.00000 0.264472
\(916\) 6.00000 0.198246
\(917\) 0 0
\(918\) −6.00000 −0.198030
\(919\) 40.0000 1.31948 0.659739 0.751495i \(-0.270667\pi\)
0.659739 + 0.751495i \(0.270667\pi\)
\(920\) 0 0
\(921\) 26.0000 0.856729
\(922\) −12.0000 −0.395199
\(923\) 32.0000 1.05329
\(924\) 0 0
\(925\) −2.00000 −0.0657596
\(926\) 16.0000 0.525793
\(927\) −6.00000 −0.197066
\(928\) −6.00000 −0.196960
\(929\) 44.0000 1.44359 0.721797 0.692105i \(-0.243317\pi\)
0.721797 + 0.692105i \(0.243317\pi\)
\(930\) 4.00000 0.131165
\(931\) 0 0
\(932\) 26.0000 0.851658
\(933\) 34.0000 1.11311
\(934\) −28.0000 −0.916188
\(935\) 12.0000 0.392442
\(936\) −4.00000 −0.130744
\(937\) −38.0000 −1.24141 −0.620703 0.784046i \(-0.713153\pi\)
−0.620703 + 0.784046i \(0.713153\pi\)
\(938\) 0 0
\(939\) 20.0000 0.652675
\(940\) −12.0000 −0.391397
\(941\) −48.0000 −1.56476 −0.782378 0.622804i \(-0.785993\pi\)
−0.782378 + 0.622804i \(0.785993\pi\)
\(942\) −14.0000 −0.456145
\(943\) 0 0
\(944\) −8.00000 −0.260378
\(945\) 0 0
\(946\) −12.0000 −0.390154
\(947\) 12.0000 0.389948 0.194974 0.980808i \(-0.437538\pi\)
0.194974 + 0.980808i \(0.437538\pi\)
\(948\) 12.0000 0.389742
\(949\) 24.0000 0.779073
\(950\) 2.00000 0.0648886
\(951\) 18.0000 0.583690
\(952\) 0 0
\(953\) −2.00000 −0.0647864 −0.0323932 0.999475i \(-0.510313\pi\)
−0.0323932 + 0.999475i \(0.510313\pi\)
\(954\) 10.0000 0.323762
\(955\) 0 0
\(956\) −16.0000 −0.517477
\(957\) −6.00000 −0.193952
\(958\) 0 0
\(959\) 0 0
\(960\) −2.00000 −0.0645497
\(961\) −27.0000 −0.870968
\(962\) −8.00000 −0.257930
\(963\) −12.0000 −0.386695
\(964\) −22.0000 −0.708572
\(965\) 28.0000 0.901352
\(966\) 0 0
\(967\) 8.00000 0.257263 0.128631 0.991692i \(-0.458942\pi\)
0.128631 + 0.991692i \(0.458942\pi\)
\(968\) 1.00000 0.0321412
\(969\) 12.0000 0.385496
\(970\) 0 0
\(971\) −60.0000 −1.92549 −0.962746 0.270408i \(-0.912841\pi\)
−0.962746 + 0.270408i \(0.912841\pi\)
\(972\) 1.00000 0.0320750
\(973\) 0 0
\(974\) −16.0000 −0.512673
\(975\) 4.00000 0.128103
\(976\) −4.00000 −0.128037
\(977\) −2.00000 −0.0639857 −0.0319928 0.999488i \(-0.510185\pi\)
−0.0319928 + 0.999488i \(0.510185\pi\)
\(978\) −12.0000 −0.383718
\(979\) 12.0000 0.383522
\(980\) 0 0
\(981\) −10.0000 −0.319275
\(982\) −20.0000 −0.638226
\(983\) 38.0000 1.21201 0.606006 0.795460i \(-0.292771\pi\)
0.606006 + 0.795460i \(0.292771\pi\)
\(984\) 2.00000 0.0637577
\(985\) 36.0000 1.14706
\(986\) 36.0000 1.14647
\(987\) 0 0
\(988\) 8.00000 0.254514
\(989\) 0 0
\(990\) −2.00000 −0.0635642
\(991\) −40.0000 −1.27064 −0.635321 0.772248i \(-0.719132\pi\)
−0.635321 + 0.772248i \(0.719132\pi\)
\(992\) −2.00000 −0.0635001
\(993\) −24.0000 −0.761617
\(994\) 0 0
\(995\) 28.0000 0.887660
\(996\) −6.00000 −0.190117
\(997\) 8.00000 0.253363 0.126681 0.991943i \(-0.459567\pi\)
0.126681 + 0.991943i \(0.459567\pi\)
\(998\) −40.0000 −1.26618
\(999\) 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 3234.2.a.u.1.1 yes 1
3.2 odd 2 9702.2.a.q.1.1 1
7.6 odd 2 3234.2.a.r.1.1 1
21.20 even 2 9702.2.a.g.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3234.2.a.r.1.1 1 7.6 odd 2
3234.2.a.u.1.1 yes 1 1.1 even 1 trivial
9702.2.a.g.1.1 1 21.20 even 2
9702.2.a.q.1.1 1 3.2 odd 2