Properties

Label 3822.2.a.bs.1.2
Level $3822$
Weight $2$
Character 3822.1
Self dual yes
Analytic conductor $30.519$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3822 = 2 \cdot 3 \cdot 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3822.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(30.5188236525\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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 1.2
Root \(1.41421\) of defining polynomial
Character \(\chi\) \(=\) 3822.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.41421 q^{5} +1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.41421 q^{5} +1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +1.41421 q^{10} -2.41421 q^{11} +1.00000 q^{12} +1.00000 q^{13} +1.41421 q^{15} +1.00000 q^{16} -3.24264 q^{17} +1.00000 q^{18} +3.00000 q^{19} +1.41421 q^{20} -2.41421 q^{22} +5.07107 q^{23} +1.00000 q^{24} -3.00000 q^{25} +1.00000 q^{26} +1.00000 q^{27} +5.00000 q^{29} +1.41421 q^{30} +1.17157 q^{31} +1.00000 q^{32} -2.41421 q^{33} -3.24264 q^{34} +1.00000 q^{36} +9.07107 q^{37} +3.00000 q^{38} +1.00000 q^{39} +1.41421 q^{40} +7.41421 q^{41} -0.828427 q^{43} -2.41421 q^{44} +1.41421 q^{45} +5.07107 q^{46} +9.00000 q^{47} +1.00000 q^{48} -3.00000 q^{50} -3.24264 q^{51} +1.00000 q^{52} -10.3137 q^{53} +1.00000 q^{54} -3.41421 q^{55} +3.00000 q^{57} +5.00000 q^{58} +10.4142 q^{59} +1.41421 q^{60} +3.24264 q^{61} +1.17157 q^{62} +1.00000 q^{64} +1.41421 q^{65} -2.41421 q^{66} -3.34315 q^{67} -3.24264 q^{68} +5.07107 q^{69} -1.00000 q^{71} +1.00000 q^{72} +13.0711 q^{73} +9.07107 q^{74} -3.00000 q^{75} +3.00000 q^{76} +1.00000 q^{78} -9.31371 q^{79} +1.41421 q^{80} +1.00000 q^{81} +7.41421 q^{82} +3.65685 q^{83} -4.58579 q^{85} -0.828427 q^{86} +5.00000 q^{87} -2.41421 q^{88} +2.24264 q^{89} +1.41421 q^{90} +5.07107 q^{92} +1.17157 q^{93} +9.00000 q^{94} +4.24264 q^{95} +1.00000 q^{96} -11.8995 q^{97} -2.41421 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 2 q^{6} + 2 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 2 q^{6} + 2 q^{8} + 2 q^{9} - 2 q^{11} + 2 q^{12} + 2 q^{13} + 2 q^{16} + 2 q^{17} + 2 q^{18} + 6 q^{19} - 2 q^{22} - 4 q^{23} + 2 q^{24} - 6 q^{25} + 2 q^{26} + 2 q^{27} + 10 q^{29} + 8 q^{31} + 2 q^{32} - 2 q^{33} + 2 q^{34} + 2 q^{36} + 4 q^{37} + 6 q^{38} + 2 q^{39} + 12 q^{41} + 4 q^{43} - 2 q^{44} - 4 q^{46} + 18 q^{47} + 2 q^{48} - 6 q^{50} + 2 q^{51} + 2 q^{52} + 2 q^{53} + 2 q^{54} - 4 q^{55} + 6 q^{57} + 10 q^{58} + 18 q^{59} - 2 q^{61} + 8 q^{62} + 2 q^{64} - 2 q^{66} - 18 q^{67} + 2 q^{68} - 4 q^{69} - 2 q^{71} + 2 q^{72} + 12 q^{73} + 4 q^{74} - 6 q^{75} + 6 q^{76} + 2 q^{78} + 4 q^{79} + 2 q^{81} + 12 q^{82} - 4 q^{83} - 12 q^{85} + 4 q^{86} + 10 q^{87} - 2 q^{88} - 4 q^{89} - 4 q^{92} + 8 q^{93} + 18 q^{94} + 2 q^{96} - 4 q^{97} - 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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
\(5\) 1.41421 0.632456 0.316228 0.948683i \(-0.397584\pi\)
0.316228 + 0.948683i \(0.397584\pi\)
\(6\) 1.00000 0.408248
\(7\) 0 0
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 1.41421 0.447214
\(11\) −2.41421 −0.727913 −0.363956 0.931416i \(-0.618574\pi\)
−0.363956 + 0.931416i \(0.618574\pi\)
\(12\) 1.00000 0.288675
\(13\) 1.00000 0.277350
\(14\) 0 0
\(15\) 1.41421 0.365148
\(16\) 1.00000 0.250000
\(17\) −3.24264 −0.786456 −0.393228 0.919441i \(-0.628642\pi\)
−0.393228 + 0.919441i \(0.628642\pi\)
\(18\) 1.00000 0.235702
\(19\) 3.00000 0.688247 0.344124 0.938924i \(-0.388176\pi\)
0.344124 + 0.938924i \(0.388176\pi\)
\(20\) 1.41421 0.316228
\(21\) 0 0
\(22\) −2.41421 −0.514712
\(23\) 5.07107 1.05739 0.528695 0.848812i \(-0.322681\pi\)
0.528695 + 0.848812i \(0.322681\pi\)
\(24\) 1.00000 0.204124
\(25\) −3.00000 −0.600000
\(26\) 1.00000 0.196116
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 5.00000 0.928477 0.464238 0.885710i \(-0.346328\pi\)
0.464238 + 0.885710i \(0.346328\pi\)
\(30\) 1.41421 0.258199
\(31\) 1.17157 0.210421 0.105210 0.994450i \(-0.466448\pi\)
0.105210 + 0.994450i \(0.466448\pi\)
\(32\) 1.00000 0.176777
\(33\) −2.41421 −0.420261
\(34\) −3.24264 −0.556108
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 9.07107 1.49127 0.745637 0.666352i \(-0.232145\pi\)
0.745637 + 0.666352i \(0.232145\pi\)
\(38\) 3.00000 0.486664
\(39\) 1.00000 0.160128
\(40\) 1.41421 0.223607
\(41\) 7.41421 1.15791 0.578953 0.815361i \(-0.303462\pi\)
0.578953 + 0.815361i \(0.303462\pi\)
\(42\) 0 0
\(43\) −0.828427 −0.126334 −0.0631670 0.998003i \(-0.520120\pi\)
−0.0631670 + 0.998003i \(0.520120\pi\)
\(44\) −2.41421 −0.363956
\(45\) 1.41421 0.210819
\(46\) 5.07107 0.747688
\(47\) 9.00000 1.31278 0.656392 0.754420i \(-0.272082\pi\)
0.656392 + 0.754420i \(0.272082\pi\)
\(48\) 1.00000 0.144338
\(49\) 0 0
\(50\) −3.00000 −0.424264
\(51\) −3.24264 −0.454061
\(52\) 1.00000 0.138675
\(53\) −10.3137 −1.41670 −0.708348 0.705863i \(-0.750559\pi\)
−0.708348 + 0.705863i \(0.750559\pi\)
\(54\) 1.00000 0.136083
\(55\) −3.41421 −0.460372
\(56\) 0 0
\(57\) 3.00000 0.397360
\(58\) 5.00000 0.656532
\(59\) 10.4142 1.35582 0.677908 0.735147i \(-0.262887\pi\)
0.677908 + 0.735147i \(0.262887\pi\)
\(60\) 1.41421 0.182574
\(61\) 3.24264 0.415178 0.207589 0.978216i \(-0.433438\pi\)
0.207589 + 0.978216i \(0.433438\pi\)
\(62\) 1.17157 0.148790
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 1.41421 0.175412
\(66\) −2.41421 −0.297169
\(67\) −3.34315 −0.408430 −0.204215 0.978926i \(-0.565464\pi\)
−0.204215 + 0.978926i \(0.565464\pi\)
\(68\) −3.24264 −0.393228
\(69\) 5.07107 0.610485
\(70\) 0 0
\(71\) −1.00000 −0.118678 −0.0593391 0.998238i \(-0.518899\pi\)
−0.0593391 + 0.998238i \(0.518899\pi\)
\(72\) 1.00000 0.117851
\(73\) 13.0711 1.52985 0.764926 0.644118i \(-0.222775\pi\)
0.764926 + 0.644118i \(0.222775\pi\)
\(74\) 9.07107 1.05449
\(75\) −3.00000 −0.346410
\(76\) 3.00000 0.344124
\(77\) 0 0
\(78\) 1.00000 0.113228
\(79\) −9.31371 −1.04787 −0.523937 0.851757i \(-0.675537\pi\)
−0.523937 + 0.851757i \(0.675537\pi\)
\(80\) 1.41421 0.158114
\(81\) 1.00000 0.111111
\(82\) 7.41421 0.818763
\(83\) 3.65685 0.401392 0.200696 0.979654i \(-0.435680\pi\)
0.200696 + 0.979654i \(0.435680\pi\)
\(84\) 0 0
\(85\) −4.58579 −0.497398
\(86\) −0.828427 −0.0893316
\(87\) 5.00000 0.536056
\(88\) −2.41421 −0.257356
\(89\) 2.24264 0.237719 0.118860 0.992911i \(-0.462076\pi\)
0.118860 + 0.992911i \(0.462076\pi\)
\(90\) 1.41421 0.149071
\(91\) 0 0
\(92\) 5.07107 0.528695
\(93\) 1.17157 0.121486
\(94\) 9.00000 0.928279
\(95\) 4.24264 0.435286
\(96\) 1.00000 0.102062
\(97\) −11.8995 −1.20821 −0.604105 0.796904i \(-0.706469\pi\)
−0.604105 + 0.796904i \(0.706469\pi\)
\(98\) 0 0
\(99\) −2.41421 −0.242638
\(100\) −3.00000 −0.300000
\(101\) −6.82843 −0.679454 −0.339727 0.940524i \(-0.610335\pi\)
−0.339727 + 0.940524i \(0.610335\pi\)
\(102\) −3.24264 −0.321069
\(103\) −10.4853 −1.03315 −0.516573 0.856243i \(-0.672792\pi\)
−0.516573 + 0.856243i \(0.672792\pi\)
\(104\) 1.00000 0.0980581
\(105\) 0 0
\(106\) −10.3137 −1.00176
\(107\) 2.34315 0.226520 0.113260 0.993565i \(-0.463871\pi\)
0.113260 + 0.993565i \(0.463871\pi\)
\(108\) 1.00000 0.0962250
\(109\) −9.65685 −0.924959 −0.462479 0.886630i \(-0.653040\pi\)
−0.462479 + 0.886630i \(0.653040\pi\)
\(110\) −3.41421 −0.325532
\(111\) 9.07107 0.860988
\(112\) 0 0
\(113\) 8.07107 0.759262 0.379631 0.925138i \(-0.376051\pi\)
0.379631 + 0.925138i \(0.376051\pi\)
\(114\) 3.00000 0.280976
\(115\) 7.17157 0.668753
\(116\) 5.00000 0.464238
\(117\) 1.00000 0.0924500
\(118\) 10.4142 0.958706
\(119\) 0 0
\(120\) 1.41421 0.129099
\(121\) −5.17157 −0.470143
\(122\) 3.24264 0.293575
\(123\) 7.41421 0.668517
\(124\) 1.17157 0.105210
\(125\) −11.3137 −1.01193
\(126\) 0 0
\(127\) −9.89949 −0.878438 −0.439219 0.898380i \(-0.644745\pi\)
−0.439219 + 0.898380i \(0.644745\pi\)
\(128\) 1.00000 0.0883883
\(129\) −0.828427 −0.0729389
\(130\) 1.41421 0.124035
\(131\) −9.89949 −0.864923 −0.432461 0.901652i \(-0.642355\pi\)
−0.432461 + 0.901652i \(0.642355\pi\)
\(132\) −2.41421 −0.210130
\(133\) 0 0
\(134\) −3.34315 −0.288804
\(135\) 1.41421 0.121716
\(136\) −3.24264 −0.278054
\(137\) 1.07107 0.0915075 0.0457537 0.998953i \(-0.485431\pi\)
0.0457537 + 0.998953i \(0.485431\pi\)
\(138\) 5.07107 0.431678
\(139\) 21.2132 1.79928 0.899640 0.436632i \(-0.143829\pi\)
0.899640 + 0.436632i \(0.143829\pi\)
\(140\) 0 0
\(141\) 9.00000 0.757937
\(142\) −1.00000 −0.0839181
\(143\) −2.41421 −0.201887
\(144\) 1.00000 0.0833333
\(145\) 7.07107 0.587220
\(146\) 13.0711 1.08177
\(147\) 0 0
\(148\) 9.07107 0.745637
\(149\) −3.89949 −0.319459 −0.159730 0.987161i \(-0.551062\pi\)
−0.159730 + 0.987161i \(0.551062\pi\)
\(150\) −3.00000 −0.244949
\(151\) −10.4142 −0.847497 −0.423748 0.905780i \(-0.639286\pi\)
−0.423748 + 0.905780i \(0.639286\pi\)
\(152\) 3.00000 0.243332
\(153\) −3.24264 −0.262152
\(154\) 0 0
\(155\) 1.65685 0.133082
\(156\) 1.00000 0.0800641
\(157\) 22.0711 1.76146 0.880731 0.473616i \(-0.157052\pi\)
0.880731 + 0.473616i \(0.157052\pi\)
\(158\) −9.31371 −0.740959
\(159\) −10.3137 −0.817930
\(160\) 1.41421 0.111803
\(161\) 0 0
\(162\) 1.00000 0.0785674
\(163\) −11.1421 −0.872719 −0.436360 0.899772i \(-0.643732\pi\)
−0.436360 + 0.899772i \(0.643732\pi\)
\(164\) 7.41421 0.578953
\(165\) −3.41421 −0.265796
\(166\) 3.65685 0.283827
\(167\) −10.3137 −0.798099 −0.399049 0.916929i \(-0.630660\pi\)
−0.399049 + 0.916929i \(0.630660\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) −4.58579 −0.351714
\(171\) 3.00000 0.229416
\(172\) −0.828427 −0.0631670
\(173\) −2.31371 −0.175908 −0.0879540 0.996125i \(-0.528033\pi\)
−0.0879540 + 0.996125i \(0.528033\pi\)
\(174\) 5.00000 0.379049
\(175\) 0 0
\(176\) −2.41421 −0.181978
\(177\) 10.4142 0.782780
\(178\) 2.24264 0.168093
\(179\) −23.6569 −1.76820 −0.884098 0.467301i \(-0.845226\pi\)
−0.884098 + 0.467301i \(0.845226\pi\)
\(180\) 1.41421 0.105409
\(181\) −14.4142 −1.07140 −0.535700 0.844408i \(-0.679952\pi\)
−0.535700 + 0.844408i \(0.679952\pi\)
\(182\) 0 0
\(183\) 3.24264 0.239703
\(184\) 5.07107 0.373844
\(185\) 12.8284 0.943165
\(186\) 1.17157 0.0859039
\(187\) 7.82843 0.572471
\(188\) 9.00000 0.656392
\(189\) 0 0
\(190\) 4.24264 0.307794
\(191\) 13.6569 0.988175 0.494088 0.869412i \(-0.335502\pi\)
0.494088 + 0.869412i \(0.335502\pi\)
\(192\) 1.00000 0.0721688
\(193\) −24.2426 −1.74502 −0.872512 0.488593i \(-0.837510\pi\)
−0.872512 + 0.488593i \(0.837510\pi\)
\(194\) −11.8995 −0.854334
\(195\) 1.41421 0.101274
\(196\) 0 0
\(197\) −21.5563 −1.53583 −0.767913 0.640554i \(-0.778705\pi\)
−0.767913 + 0.640554i \(0.778705\pi\)
\(198\) −2.41421 −0.171571
\(199\) 13.4142 0.950908 0.475454 0.879740i \(-0.342284\pi\)
0.475454 + 0.879740i \(0.342284\pi\)
\(200\) −3.00000 −0.212132
\(201\) −3.34315 −0.235807
\(202\) −6.82843 −0.480446
\(203\) 0 0
\(204\) −3.24264 −0.227030
\(205\) 10.4853 0.732324
\(206\) −10.4853 −0.730544
\(207\) 5.07107 0.352464
\(208\) 1.00000 0.0693375
\(209\) −7.24264 −0.500984
\(210\) 0 0
\(211\) 8.38478 0.577232 0.288616 0.957445i \(-0.406805\pi\)
0.288616 + 0.957445i \(0.406805\pi\)
\(212\) −10.3137 −0.708348
\(213\) −1.00000 −0.0685189
\(214\) 2.34315 0.160174
\(215\) −1.17157 −0.0799006
\(216\) 1.00000 0.0680414
\(217\) 0 0
\(218\) −9.65685 −0.654045
\(219\) 13.0711 0.883261
\(220\) −3.41421 −0.230186
\(221\) −3.24264 −0.218124
\(222\) 9.07107 0.608810
\(223\) −15.3848 −1.03024 −0.515120 0.857118i \(-0.672253\pi\)
−0.515120 + 0.857118i \(0.672253\pi\)
\(224\) 0 0
\(225\) −3.00000 −0.200000
\(226\) 8.07107 0.536879
\(227\) 2.14214 0.142179 0.0710893 0.997470i \(-0.477352\pi\)
0.0710893 + 0.997470i \(0.477352\pi\)
\(228\) 3.00000 0.198680
\(229\) −12.8284 −0.847726 −0.423863 0.905726i \(-0.639326\pi\)
−0.423863 + 0.905726i \(0.639326\pi\)
\(230\) 7.17157 0.472880
\(231\) 0 0
\(232\) 5.00000 0.328266
\(233\) 21.7279 1.42344 0.711722 0.702461i \(-0.247916\pi\)
0.711722 + 0.702461i \(0.247916\pi\)
\(234\) 1.00000 0.0653720
\(235\) 12.7279 0.830278
\(236\) 10.4142 0.677908
\(237\) −9.31371 −0.604990
\(238\) 0 0
\(239\) −19.4853 −1.26040 −0.630199 0.776434i \(-0.717027\pi\)
−0.630199 + 0.776434i \(0.717027\pi\)
\(240\) 1.41421 0.0912871
\(241\) 0.485281 0.0312597 0.0156299 0.999878i \(-0.495025\pi\)
0.0156299 + 0.999878i \(0.495025\pi\)
\(242\) −5.17157 −0.332441
\(243\) 1.00000 0.0641500
\(244\) 3.24264 0.207589
\(245\) 0 0
\(246\) 7.41421 0.472713
\(247\) 3.00000 0.190885
\(248\) 1.17157 0.0743950
\(249\) 3.65685 0.231744
\(250\) −11.3137 −0.715542
\(251\) 7.17157 0.452666 0.226333 0.974050i \(-0.427326\pi\)
0.226333 + 0.974050i \(0.427326\pi\)
\(252\) 0 0
\(253\) −12.2426 −0.769688
\(254\) −9.89949 −0.621150
\(255\) −4.58579 −0.287173
\(256\) 1.00000 0.0625000
\(257\) 17.7990 1.11027 0.555135 0.831760i \(-0.312666\pi\)
0.555135 + 0.831760i \(0.312666\pi\)
\(258\) −0.828427 −0.0515756
\(259\) 0 0
\(260\) 1.41421 0.0877058
\(261\) 5.00000 0.309492
\(262\) −9.89949 −0.611593
\(263\) 23.5563 1.45255 0.726273 0.687406i \(-0.241251\pi\)
0.726273 + 0.687406i \(0.241251\pi\)
\(264\) −2.41421 −0.148585
\(265\) −14.5858 −0.895998
\(266\) 0 0
\(267\) 2.24264 0.137247
\(268\) −3.34315 −0.204215
\(269\) −26.3137 −1.60438 −0.802188 0.597072i \(-0.796331\pi\)
−0.802188 + 0.597072i \(0.796331\pi\)
\(270\) 1.41421 0.0860663
\(271\) 21.3848 1.29903 0.649516 0.760348i \(-0.274971\pi\)
0.649516 + 0.760348i \(0.274971\pi\)
\(272\) −3.24264 −0.196614
\(273\) 0 0
\(274\) 1.07107 0.0647056
\(275\) 7.24264 0.436748
\(276\) 5.07107 0.305242
\(277\) −13.5858 −0.816291 −0.408145 0.912917i \(-0.633824\pi\)
−0.408145 + 0.912917i \(0.633824\pi\)
\(278\) 21.2132 1.27228
\(279\) 1.17157 0.0701402
\(280\) 0 0
\(281\) −15.3137 −0.913539 −0.456770 0.889585i \(-0.650994\pi\)
−0.456770 + 0.889585i \(0.650994\pi\)
\(282\) 9.00000 0.535942
\(283\) 2.92893 0.174107 0.0870535 0.996204i \(-0.472255\pi\)
0.0870535 + 0.996204i \(0.472255\pi\)
\(284\) −1.00000 −0.0593391
\(285\) 4.24264 0.251312
\(286\) −2.41421 −0.142755
\(287\) 0 0
\(288\) 1.00000 0.0589256
\(289\) −6.48528 −0.381487
\(290\) 7.07107 0.415227
\(291\) −11.8995 −0.697561
\(292\) 13.0711 0.764926
\(293\) 6.72792 0.393049 0.196525 0.980499i \(-0.437034\pi\)
0.196525 + 0.980499i \(0.437034\pi\)
\(294\) 0 0
\(295\) 14.7279 0.857493
\(296\) 9.07107 0.527245
\(297\) −2.41421 −0.140087
\(298\) −3.89949 −0.225892
\(299\) 5.07107 0.293267
\(300\) −3.00000 −0.173205
\(301\) 0 0
\(302\) −10.4142 −0.599271
\(303\) −6.82843 −0.392283
\(304\) 3.00000 0.172062
\(305\) 4.58579 0.262581
\(306\) −3.24264 −0.185369
\(307\) 25.6274 1.46263 0.731317 0.682038i \(-0.238906\pi\)
0.731317 + 0.682038i \(0.238906\pi\)
\(308\) 0 0
\(309\) −10.4853 −0.596487
\(310\) 1.65685 0.0941030
\(311\) −9.41421 −0.533831 −0.266916 0.963720i \(-0.586004\pi\)
−0.266916 + 0.963720i \(0.586004\pi\)
\(312\) 1.00000 0.0566139
\(313\) −10.1421 −0.573267 −0.286634 0.958040i \(-0.592536\pi\)
−0.286634 + 0.958040i \(0.592536\pi\)
\(314\) 22.0711 1.24554
\(315\) 0 0
\(316\) −9.31371 −0.523937
\(317\) −4.82843 −0.271191 −0.135596 0.990764i \(-0.543295\pi\)
−0.135596 + 0.990764i \(0.543295\pi\)
\(318\) −10.3137 −0.578364
\(319\) −12.0711 −0.675850
\(320\) 1.41421 0.0790569
\(321\) 2.34315 0.130782
\(322\) 0 0
\(323\) −9.72792 −0.541276
\(324\) 1.00000 0.0555556
\(325\) −3.00000 −0.166410
\(326\) −11.1421 −0.617106
\(327\) −9.65685 −0.534025
\(328\) 7.41421 0.409381
\(329\) 0 0
\(330\) −3.41421 −0.187946
\(331\) −5.51472 −0.303116 −0.151558 0.988448i \(-0.548429\pi\)
−0.151558 + 0.988448i \(0.548429\pi\)
\(332\) 3.65685 0.200696
\(333\) 9.07107 0.497091
\(334\) −10.3137 −0.564341
\(335\) −4.72792 −0.258314
\(336\) 0 0
\(337\) 17.3431 0.944741 0.472371 0.881400i \(-0.343398\pi\)
0.472371 + 0.881400i \(0.343398\pi\)
\(338\) 1.00000 0.0543928
\(339\) 8.07107 0.438360
\(340\) −4.58579 −0.248699
\(341\) −2.82843 −0.153168
\(342\) 3.00000 0.162221
\(343\) 0 0
\(344\) −0.828427 −0.0446658
\(345\) 7.17157 0.386105
\(346\) −2.31371 −0.124386
\(347\) 14.2426 0.764585 0.382293 0.924041i \(-0.375135\pi\)
0.382293 + 0.924041i \(0.375135\pi\)
\(348\) 5.00000 0.268028
\(349\) 2.58579 0.138414 0.0692070 0.997602i \(-0.477953\pi\)
0.0692070 + 0.997602i \(0.477953\pi\)
\(350\) 0 0
\(351\) 1.00000 0.0533761
\(352\) −2.41421 −0.128678
\(353\) −32.6274 −1.73658 −0.868291 0.496055i \(-0.834781\pi\)
−0.868291 + 0.496055i \(0.834781\pi\)
\(354\) 10.4142 0.553509
\(355\) −1.41421 −0.0750587
\(356\) 2.24264 0.118860
\(357\) 0 0
\(358\) −23.6569 −1.25030
\(359\) 35.4558 1.87129 0.935644 0.352945i \(-0.114820\pi\)
0.935644 + 0.352945i \(0.114820\pi\)
\(360\) 1.41421 0.0745356
\(361\) −10.0000 −0.526316
\(362\) −14.4142 −0.757594
\(363\) −5.17157 −0.271437
\(364\) 0 0
\(365\) 18.4853 0.967564
\(366\) 3.24264 0.169496
\(367\) 14.0000 0.730794 0.365397 0.930852i \(-0.380933\pi\)
0.365397 + 0.930852i \(0.380933\pi\)
\(368\) 5.07107 0.264348
\(369\) 7.41421 0.385969
\(370\) 12.8284 0.666918
\(371\) 0 0
\(372\) 1.17157 0.0607432
\(373\) −30.8995 −1.59992 −0.799958 0.600057i \(-0.795145\pi\)
−0.799958 + 0.600057i \(0.795145\pi\)
\(374\) 7.82843 0.404798
\(375\) −11.3137 −0.584237
\(376\) 9.00000 0.464140
\(377\) 5.00000 0.257513
\(378\) 0 0
\(379\) −22.3431 −1.14769 −0.573845 0.818964i \(-0.694549\pi\)
−0.573845 + 0.818964i \(0.694549\pi\)
\(380\) 4.24264 0.217643
\(381\) −9.89949 −0.507166
\(382\) 13.6569 0.698745
\(383\) −10.4853 −0.535773 −0.267886 0.963450i \(-0.586325\pi\)
−0.267886 + 0.963450i \(0.586325\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) −24.2426 −1.23392
\(387\) −0.828427 −0.0421113
\(388\) −11.8995 −0.604105
\(389\) 23.0000 1.16615 0.583073 0.812420i \(-0.301850\pi\)
0.583073 + 0.812420i \(0.301850\pi\)
\(390\) 1.41421 0.0716115
\(391\) −16.4437 −0.831591
\(392\) 0 0
\(393\) −9.89949 −0.499363
\(394\) −21.5563 −1.08599
\(395\) −13.1716 −0.662734
\(396\) −2.41421 −0.121319
\(397\) −28.2426 −1.41746 −0.708729 0.705481i \(-0.750731\pi\)
−0.708729 + 0.705481i \(0.750731\pi\)
\(398\) 13.4142 0.672394
\(399\) 0 0
\(400\) −3.00000 −0.150000
\(401\) 4.92893 0.246139 0.123070 0.992398i \(-0.460726\pi\)
0.123070 + 0.992398i \(0.460726\pi\)
\(402\) −3.34315 −0.166741
\(403\) 1.17157 0.0583602
\(404\) −6.82843 −0.339727
\(405\) 1.41421 0.0702728
\(406\) 0 0
\(407\) −21.8995 −1.08552
\(408\) −3.24264 −0.160535
\(409\) −5.41421 −0.267716 −0.133858 0.991001i \(-0.542737\pi\)
−0.133858 + 0.991001i \(0.542737\pi\)
\(410\) 10.4853 0.517831
\(411\) 1.07107 0.0528319
\(412\) −10.4853 −0.516573
\(413\) 0 0
\(414\) 5.07107 0.249229
\(415\) 5.17157 0.253863
\(416\) 1.00000 0.0490290
\(417\) 21.2132 1.03882
\(418\) −7.24264 −0.354249
\(419\) 23.4558 1.14589 0.572946 0.819593i \(-0.305800\pi\)
0.572946 + 0.819593i \(0.305800\pi\)
\(420\) 0 0
\(421\) 13.4142 0.653769 0.326884 0.945064i \(-0.394001\pi\)
0.326884 + 0.945064i \(0.394001\pi\)
\(422\) 8.38478 0.408165
\(423\) 9.00000 0.437595
\(424\) −10.3137 −0.500878
\(425\) 9.72792 0.471874
\(426\) −1.00000 −0.0484502
\(427\) 0 0
\(428\) 2.34315 0.113260
\(429\) −2.41421 −0.116559
\(430\) −1.17157 −0.0564983
\(431\) −2.48528 −0.119712 −0.0598559 0.998207i \(-0.519064\pi\)
−0.0598559 + 0.998207i \(0.519064\pi\)
\(432\) 1.00000 0.0481125
\(433\) −19.0000 −0.913082 −0.456541 0.889702i \(-0.650912\pi\)
−0.456541 + 0.889702i \(0.650912\pi\)
\(434\) 0 0
\(435\) 7.07107 0.339032
\(436\) −9.65685 −0.462479
\(437\) 15.2132 0.727746
\(438\) 13.0711 0.624560
\(439\) −13.4142 −0.640225 −0.320113 0.947379i \(-0.603721\pi\)
−0.320113 + 0.947379i \(0.603721\pi\)
\(440\) −3.41421 −0.162766
\(441\) 0 0
\(442\) −3.24264 −0.154237
\(443\) −2.72792 −0.129607 −0.0648037 0.997898i \(-0.520642\pi\)
−0.0648037 + 0.997898i \(0.520642\pi\)
\(444\) 9.07107 0.430494
\(445\) 3.17157 0.150347
\(446\) −15.3848 −0.728490
\(447\) −3.89949 −0.184440
\(448\) 0 0
\(449\) 33.9411 1.60178 0.800890 0.598811i \(-0.204360\pi\)
0.800890 + 0.598811i \(0.204360\pi\)
\(450\) −3.00000 −0.141421
\(451\) −17.8995 −0.842854
\(452\) 8.07107 0.379631
\(453\) −10.4142 −0.489302
\(454\) 2.14214 0.100535
\(455\) 0 0
\(456\) 3.00000 0.140488
\(457\) 27.2132 1.27298 0.636490 0.771285i \(-0.280386\pi\)
0.636490 + 0.771285i \(0.280386\pi\)
\(458\) −12.8284 −0.599433
\(459\) −3.24264 −0.151354
\(460\) 7.17157 0.334376
\(461\) 17.4558 0.813000 0.406500 0.913651i \(-0.366749\pi\)
0.406500 + 0.913651i \(0.366749\pi\)
\(462\) 0 0
\(463\) 29.9411 1.39148 0.695741 0.718293i \(-0.255076\pi\)
0.695741 + 0.718293i \(0.255076\pi\)
\(464\) 5.00000 0.232119
\(465\) 1.65685 0.0768348
\(466\) 21.7279 1.00653
\(467\) 8.24264 0.381424 0.190712 0.981646i \(-0.438920\pi\)
0.190712 + 0.981646i \(0.438920\pi\)
\(468\) 1.00000 0.0462250
\(469\) 0 0
\(470\) 12.7279 0.587095
\(471\) 22.0711 1.01698
\(472\) 10.4142 0.479353
\(473\) 2.00000 0.0919601
\(474\) −9.31371 −0.427793
\(475\) −9.00000 −0.412948
\(476\) 0 0
\(477\) −10.3137 −0.472232
\(478\) −19.4853 −0.891236
\(479\) −2.85786 −0.130579 −0.0652896 0.997866i \(-0.520797\pi\)
−0.0652896 + 0.997866i \(0.520797\pi\)
\(480\) 1.41421 0.0645497
\(481\) 9.07107 0.413605
\(482\) 0.485281 0.0221040
\(483\) 0 0
\(484\) −5.17157 −0.235071
\(485\) −16.8284 −0.764140
\(486\) 1.00000 0.0453609
\(487\) 1.72792 0.0782996 0.0391498 0.999233i \(-0.487535\pi\)
0.0391498 + 0.999233i \(0.487535\pi\)
\(488\) 3.24264 0.146787
\(489\) −11.1421 −0.503865
\(490\) 0 0
\(491\) 30.2843 1.36671 0.683355 0.730086i \(-0.260520\pi\)
0.683355 + 0.730086i \(0.260520\pi\)
\(492\) 7.41421 0.334259
\(493\) −16.2132 −0.730206
\(494\) 3.00000 0.134976
\(495\) −3.41421 −0.153457
\(496\) 1.17157 0.0526052
\(497\) 0 0
\(498\) 3.65685 0.163868
\(499\) −30.0000 −1.34298 −0.671492 0.741012i \(-0.734346\pi\)
−0.671492 + 0.741012i \(0.734346\pi\)
\(500\) −11.3137 −0.505964
\(501\) −10.3137 −0.460783
\(502\) 7.17157 0.320083
\(503\) −3.79899 −0.169389 −0.0846943 0.996407i \(-0.526991\pi\)
−0.0846943 + 0.996407i \(0.526991\pi\)
\(504\) 0 0
\(505\) −9.65685 −0.429724
\(506\) −12.2426 −0.544252
\(507\) 1.00000 0.0444116
\(508\) −9.89949 −0.439219
\(509\) −20.9706 −0.929504 −0.464752 0.885441i \(-0.653856\pi\)
−0.464752 + 0.885441i \(0.653856\pi\)
\(510\) −4.58579 −0.203062
\(511\) 0 0
\(512\) 1.00000 0.0441942
\(513\) 3.00000 0.132453
\(514\) 17.7990 0.785080
\(515\) −14.8284 −0.653419
\(516\) −0.828427 −0.0364695
\(517\) −21.7279 −0.955593
\(518\) 0 0
\(519\) −2.31371 −0.101561
\(520\) 1.41421 0.0620174
\(521\) −14.0000 −0.613351 −0.306676 0.951814i \(-0.599217\pi\)
−0.306676 + 0.951814i \(0.599217\pi\)
\(522\) 5.00000 0.218844
\(523\) −32.1421 −1.40548 −0.702739 0.711448i \(-0.748040\pi\)
−0.702739 + 0.711448i \(0.748040\pi\)
\(524\) −9.89949 −0.432461
\(525\) 0 0
\(526\) 23.5563 1.02711
\(527\) −3.79899 −0.165487
\(528\) −2.41421 −0.105065
\(529\) 2.71573 0.118075
\(530\) −14.5858 −0.633566
\(531\) 10.4142 0.451938
\(532\) 0 0
\(533\) 7.41421 0.321145
\(534\) 2.24264 0.0970486
\(535\) 3.31371 0.143264
\(536\) −3.34315 −0.144402
\(537\) −23.6569 −1.02087
\(538\) −26.3137 −1.13446
\(539\) 0 0
\(540\) 1.41421 0.0608581
\(541\) −40.8701 −1.75714 −0.878571 0.477613i \(-0.841502\pi\)
−0.878571 + 0.477613i \(0.841502\pi\)
\(542\) 21.3848 0.918555
\(543\) −14.4142 −0.618573
\(544\) −3.24264 −0.139027
\(545\) −13.6569 −0.584995
\(546\) 0 0
\(547\) −14.2426 −0.608971 −0.304486 0.952517i \(-0.598485\pi\)
−0.304486 + 0.952517i \(0.598485\pi\)
\(548\) 1.07107 0.0457537
\(549\) 3.24264 0.138393
\(550\) 7.24264 0.308827
\(551\) 15.0000 0.639021
\(552\) 5.07107 0.215839
\(553\) 0 0
\(554\) −13.5858 −0.577205
\(555\) 12.8284 0.544536
\(556\) 21.2132 0.899640
\(557\) 22.9706 0.973294 0.486647 0.873599i \(-0.338220\pi\)
0.486647 + 0.873599i \(0.338220\pi\)
\(558\) 1.17157 0.0495966
\(559\) −0.828427 −0.0350387
\(560\) 0 0
\(561\) 7.82843 0.330516
\(562\) −15.3137 −0.645970
\(563\) 16.4853 0.694772 0.347386 0.937722i \(-0.387069\pi\)
0.347386 + 0.937722i \(0.387069\pi\)
\(564\) 9.00000 0.378968
\(565\) 11.4142 0.480200
\(566\) 2.92893 0.123112
\(567\) 0 0
\(568\) −1.00000 −0.0419591
\(569\) −28.4142 −1.19119 −0.595593 0.803286i \(-0.703083\pi\)
−0.595593 + 0.803286i \(0.703083\pi\)
\(570\) 4.24264 0.177705
\(571\) 9.65685 0.404127 0.202063 0.979372i \(-0.435235\pi\)
0.202063 + 0.979372i \(0.435235\pi\)
\(572\) −2.41421 −0.100943
\(573\) 13.6569 0.570523
\(574\) 0 0
\(575\) −15.2132 −0.634434
\(576\) 1.00000 0.0416667
\(577\) −18.0000 −0.749350 −0.374675 0.927156i \(-0.622246\pi\)
−0.374675 + 0.927156i \(0.622246\pi\)
\(578\) −6.48528 −0.269752
\(579\) −24.2426 −1.00749
\(580\) 7.07107 0.293610
\(581\) 0 0
\(582\) −11.8995 −0.493250
\(583\) 24.8995 1.03123
\(584\) 13.0711 0.540885
\(585\) 1.41421 0.0584705
\(586\) 6.72792 0.277928
\(587\) 2.75736 0.113808 0.0569042 0.998380i \(-0.481877\pi\)
0.0569042 + 0.998380i \(0.481877\pi\)
\(588\) 0 0
\(589\) 3.51472 0.144821
\(590\) 14.7279 0.606339
\(591\) −21.5563 −0.886710
\(592\) 9.07107 0.372819
\(593\) −31.4142 −1.29003 −0.645014 0.764171i \(-0.723148\pi\)
−0.645014 + 0.764171i \(0.723148\pi\)
\(594\) −2.41421 −0.0990564
\(595\) 0 0
\(596\) −3.89949 −0.159730
\(597\) 13.4142 0.549007
\(598\) 5.07107 0.207371
\(599\) 38.8701 1.58819 0.794094 0.607795i \(-0.207946\pi\)
0.794094 + 0.607795i \(0.207946\pi\)
\(600\) −3.00000 −0.122474
\(601\) −12.1716 −0.496489 −0.248244 0.968697i \(-0.579854\pi\)
−0.248244 + 0.968697i \(0.579854\pi\)
\(602\) 0 0
\(603\) −3.34315 −0.136143
\(604\) −10.4142 −0.423748
\(605\) −7.31371 −0.297345
\(606\) −6.82843 −0.277386
\(607\) 12.9289 0.524769 0.262385 0.964963i \(-0.415491\pi\)
0.262385 + 0.964963i \(0.415491\pi\)
\(608\) 3.00000 0.121666
\(609\) 0 0
\(610\) 4.58579 0.185673
\(611\) 9.00000 0.364101
\(612\) −3.24264 −0.131076
\(613\) 3.79899 0.153440 0.0767199 0.997053i \(-0.475555\pi\)
0.0767199 + 0.997053i \(0.475555\pi\)
\(614\) 25.6274 1.03424
\(615\) 10.4853 0.422807
\(616\) 0 0
\(617\) 10.4853 0.422122 0.211061 0.977473i \(-0.432308\pi\)
0.211061 + 0.977473i \(0.432308\pi\)
\(618\) −10.4853 −0.421780
\(619\) −7.45584 −0.299676 −0.149838 0.988711i \(-0.547875\pi\)
−0.149838 + 0.988711i \(0.547875\pi\)
\(620\) 1.65685 0.0665409
\(621\) 5.07107 0.203495
\(622\) −9.41421 −0.377476
\(623\) 0 0
\(624\) 1.00000 0.0400320
\(625\) −1.00000 −0.0400000
\(626\) −10.1421 −0.405361
\(627\) −7.24264 −0.289243
\(628\) 22.0711 0.880731
\(629\) −29.4142 −1.17282
\(630\) 0 0
\(631\) −8.62742 −0.343452 −0.171726 0.985145i \(-0.554934\pi\)
−0.171726 + 0.985145i \(0.554934\pi\)
\(632\) −9.31371 −0.370479
\(633\) 8.38478 0.333265
\(634\) −4.82843 −0.191761
\(635\) −14.0000 −0.555573
\(636\) −10.3137 −0.408965
\(637\) 0 0
\(638\) −12.0711 −0.477898
\(639\) −1.00000 −0.0395594
\(640\) 1.41421 0.0559017
\(641\) 38.8284 1.53363 0.766815 0.641868i \(-0.221840\pi\)
0.766815 + 0.641868i \(0.221840\pi\)
\(642\) 2.34315 0.0924766
\(643\) 13.6863 0.539735 0.269867 0.962898i \(-0.413020\pi\)
0.269867 + 0.962898i \(0.413020\pi\)
\(644\) 0 0
\(645\) −1.17157 −0.0461306
\(646\) −9.72792 −0.382740
\(647\) −16.6274 −0.653691 −0.326846 0.945078i \(-0.605986\pi\)
−0.326846 + 0.945078i \(0.605986\pi\)
\(648\) 1.00000 0.0392837
\(649\) −25.1421 −0.986915
\(650\) −3.00000 −0.117670
\(651\) 0 0
\(652\) −11.1421 −0.436360
\(653\) −47.1127 −1.84366 −0.921831 0.387592i \(-0.873307\pi\)
−0.921831 + 0.387592i \(0.873307\pi\)
\(654\) −9.65685 −0.377613
\(655\) −14.0000 −0.547025
\(656\) 7.41421 0.289476
\(657\) 13.0711 0.509951
\(658\) 0 0
\(659\) 5.27208 0.205371 0.102685 0.994714i \(-0.467256\pi\)
0.102685 + 0.994714i \(0.467256\pi\)
\(660\) −3.41421 −0.132898
\(661\) 17.5563 0.682863 0.341431 0.939907i \(-0.389088\pi\)
0.341431 + 0.939907i \(0.389088\pi\)
\(662\) −5.51472 −0.214336
\(663\) −3.24264 −0.125934
\(664\) 3.65685 0.141913
\(665\) 0 0
\(666\) 9.07107 0.351497
\(667\) 25.3553 0.981763
\(668\) −10.3137 −0.399049
\(669\) −15.3848 −0.594810
\(670\) −4.72792 −0.182656
\(671\) −7.82843 −0.302213
\(672\) 0 0
\(673\) −6.82843 −0.263217 −0.131608 0.991302i \(-0.542014\pi\)
−0.131608 + 0.991302i \(0.542014\pi\)
\(674\) 17.3431 0.668033
\(675\) −3.00000 −0.115470
\(676\) 1.00000 0.0384615
\(677\) 33.8284 1.30013 0.650066 0.759878i \(-0.274741\pi\)
0.650066 + 0.759878i \(0.274741\pi\)
\(678\) 8.07107 0.309967
\(679\) 0 0
\(680\) −4.58579 −0.175857
\(681\) 2.14214 0.0820868
\(682\) −2.82843 −0.108306
\(683\) −45.4558 −1.73932 −0.869660 0.493652i \(-0.835662\pi\)
−0.869660 + 0.493652i \(0.835662\pi\)
\(684\) 3.00000 0.114708
\(685\) 1.51472 0.0578744
\(686\) 0 0
\(687\) −12.8284 −0.489435
\(688\) −0.828427 −0.0315835
\(689\) −10.3137 −0.392921
\(690\) 7.17157 0.273017
\(691\) −17.1421 −0.652118 −0.326059 0.945349i \(-0.605721\pi\)
−0.326059 + 0.945349i \(0.605721\pi\)
\(692\) −2.31371 −0.0879540
\(693\) 0 0
\(694\) 14.2426 0.540643
\(695\) 30.0000 1.13796
\(696\) 5.00000 0.189525
\(697\) −24.0416 −0.910642
\(698\) 2.58579 0.0978735
\(699\) 21.7279 0.821825
\(700\) 0 0
\(701\) 31.5147 1.19029 0.595147 0.803617i \(-0.297094\pi\)
0.595147 + 0.803617i \(0.297094\pi\)
\(702\) 1.00000 0.0377426
\(703\) 27.2132 1.02637
\(704\) −2.41421 −0.0909891
\(705\) 12.7279 0.479361
\(706\) −32.6274 −1.22795
\(707\) 0 0
\(708\) 10.4142 0.391390
\(709\) 16.0416 0.602456 0.301228 0.953552i \(-0.402603\pi\)
0.301228 + 0.953552i \(0.402603\pi\)
\(710\) −1.41421 −0.0530745
\(711\) −9.31371 −0.349291
\(712\) 2.24264 0.0840465
\(713\) 5.94113 0.222497
\(714\) 0 0
\(715\) −3.41421 −0.127684
\(716\) −23.6569 −0.884098
\(717\) −19.4853 −0.727691
\(718\) 35.4558 1.32320
\(719\) 34.2843 1.27859 0.639294 0.768963i \(-0.279227\pi\)
0.639294 + 0.768963i \(0.279227\pi\)
\(720\) 1.41421 0.0527046
\(721\) 0 0
\(722\) −10.0000 −0.372161
\(723\) 0.485281 0.0180478
\(724\) −14.4142 −0.535700
\(725\) −15.0000 −0.557086
\(726\) −5.17157 −0.191935
\(727\) −33.1716 −1.23027 −0.615133 0.788424i \(-0.710898\pi\)
−0.615133 + 0.788424i \(0.710898\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 18.4853 0.684171
\(731\) 2.68629 0.0993561
\(732\) 3.24264 0.119851
\(733\) 53.0122 1.95805 0.979025 0.203740i \(-0.0653097\pi\)
0.979025 + 0.203740i \(0.0653097\pi\)
\(734\) 14.0000 0.516749
\(735\) 0 0
\(736\) 5.07107 0.186922
\(737\) 8.07107 0.297302
\(738\) 7.41421 0.272921
\(739\) −26.9706 −0.992128 −0.496064 0.868286i \(-0.665222\pi\)
−0.496064 + 0.868286i \(0.665222\pi\)
\(740\) 12.8284 0.471582
\(741\) 3.00000 0.110208
\(742\) 0 0
\(743\) −5.48528 −0.201235 −0.100618 0.994925i \(-0.532082\pi\)
−0.100618 + 0.994925i \(0.532082\pi\)
\(744\) 1.17157 0.0429519
\(745\) −5.51472 −0.202044
\(746\) −30.8995 −1.13131
\(747\) 3.65685 0.133797
\(748\) 7.82843 0.286236
\(749\) 0 0
\(750\) −11.3137 −0.413118
\(751\) −45.6569 −1.66604 −0.833021 0.553241i \(-0.813391\pi\)
−0.833021 + 0.553241i \(0.813391\pi\)
\(752\) 9.00000 0.328196
\(753\) 7.17157 0.261347
\(754\) 5.00000 0.182089
\(755\) −14.7279 −0.536004
\(756\) 0 0
\(757\) 54.5563 1.98288 0.991442 0.130547i \(-0.0416734\pi\)
0.991442 + 0.130547i \(0.0416734\pi\)
\(758\) −22.3431 −0.811540
\(759\) −12.2426 −0.444380
\(760\) 4.24264 0.153897
\(761\) −46.1421 −1.67265 −0.836326 0.548233i \(-0.815301\pi\)
−0.836326 + 0.548233i \(0.815301\pi\)
\(762\) −9.89949 −0.358621
\(763\) 0 0
\(764\) 13.6569 0.494088
\(765\) −4.58579 −0.165799
\(766\) −10.4853 −0.378849
\(767\) 10.4142 0.376035
\(768\) 1.00000 0.0360844
\(769\) −21.0711 −0.759842 −0.379921 0.925019i \(-0.624049\pi\)
−0.379921 + 0.925019i \(0.624049\pi\)
\(770\) 0 0
\(771\) 17.7990 0.641015
\(772\) −24.2426 −0.872512
\(773\) 49.5980 1.78392 0.891958 0.452119i \(-0.149332\pi\)
0.891958 + 0.452119i \(0.149332\pi\)
\(774\) −0.828427 −0.0297772
\(775\) −3.51472 −0.126252
\(776\) −11.8995 −0.427167
\(777\) 0 0
\(778\) 23.0000 0.824590
\(779\) 22.2426 0.796925
\(780\) 1.41421 0.0506370
\(781\) 2.41421 0.0863874
\(782\) −16.4437 −0.588024
\(783\) 5.00000 0.178685
\(784\) 0 0
\(785\) 31.2132 1.11405
\(786\) −9.89949 −0.353103
\(787\) −13.8284 −0.492930 −0.246465 0.969152i \(-0.579269\pi\)
−0.246465 + 0.969152i \(0.579269\pi\)
\(788\) −21.5563 −0.767913
\(789\) 23.5563 0.838628
\(790\) −13.1716 −0.468624
\(791\) 0 0
\(792\) −2.41421 −0.0857853
\(793\) 3.24264 0.115150
\(794\) −28.2426 −1.00229
\(795\) −14.5858 −0.517305
\(796\) 13.4142 0.475454
\(797\) −3.02944 −0.107308 −0.0536541 0.998560i \(-0.517087\pi\)
−0.0536541 + 0.998560i \(0.517087\pi\)
\(798\) 0 0
\(799\) −29.1838 −1.03245
\(800\) −3.00000 −0.106066
\(801\) 2.24264 0.0792398
\(802\) 4.92893 0.174047
\(803\) −31.5563 −1.11360
\(804\) −3.34315 −0.117904
\(805\) 0 0
\(806\) 1.17157 0.0412669
\(807\) −26.3137 −0.926286
\(808\) −6.82843 −0.240223
\(809\) −13.0416 −0.458519 −0.229260 0.973365i \(-0.573630\pi\)
−0.229260 + 0.973365i \(0.573630\pi\)
\(810\) 1.41421 0.0496904
\(811\) −46.7696 −1.64230 −0.821151 0.570712i \(-0.806667\pi\)
−0.821151 + 0.570712i \(0.806667\pi\)
\(812\) 0 0
\(813\) 21.3848 0.749997
\(814\) −21.8995 −0.767577
\(815\) −15.7574 −0.551956
\(816\) −3.24264 −0.113515
\(817\) −2.48528 −0.0869490
\(818\) −5.41421 −0.189304
\(819\) 0 0
\(820\) 10.4853 0.366162
\(821\) −46.3848 −1.61884 −0.809420 0.587230i \(-0.800218\pi\)
−0.809420 + 0.587230i \(0.800218\pi\)
\(822\) 1.07107 0.0373578
\(823\) −27.8995 −0.972515 −0.486258 0.873815i \(-0.661638\pi\)
−0.486258 + 0.873815i \(0.661638\pi\)
\(824\) −10.4853 −0.365272
\(825\) 7.24264 0.252156
\(826\) 0 0
\(827\) −51.5269 −1.79177 −0.895883 0.444290i \(-0.853456\pi\)
−0.895883 + 0.444290i \(0.853456\pi\)
\(828\) 5.07107 0.176232
\(829\) −40.6985 −1.41352 −0.706758 0.707455i \(-0.749843\pi\)
−0.706758 + 0.707455i \(0.749843\pi\)
\(830\) 5.17157 0.179508
\(831\) −13.5858 −0.471286
\(832\) 1.00000 0.0346688
\(833\) 0 0
\(834\) 21.2132 0.734553
\(835\) −14.5858 −0.504762
\(836\) −7.24264 −0.250492
\(837\) 1.17157 0.0404955
\(838\) 23.4558 0.810269
\(839\) 19.4853 0.672707 0.336353 0.941736i \(-0.390806\pi\)
0.336353 + 0.941736i \(0.390806\pi\)
\(840\) 0 0
\(841\) −4.00000 −0.137931
\(842\) 13.4142 0.462284
\(843\) −15.3137 −0.527432
\(844\) 8.38478 0.288616
\(845\) 1.41421 0.0486504
\(846\) 9.00000 0.309426
\(847\) 0 0
\(848\) −10.3137 −0.354174
\(849\) 2.92893 0.100521
\(850\) 9.72792 0.333665
\(851\) 46.0000 1.57686
\(852\) −1.00000 −0.0342594
\(853\) 41.6985 1.42773 0.713864 0.700284i \(-0.246943\pi\)
0.713864 + 0.700284i \(0.246943\pi\)
\(854\) 0 0
\(855\) 4.24264 0.145095
\(856\) 2.34315 0.0800871
\(857\) −14.6152 −0.499247 −0.249623 0.968343i \(-0.580307\pi\)
−0.249623 + 0.968343i \(0.580307\pi\)
\(858\) −2.41421 −0.0824199
\(859\) 39.7990 1.35792 0.678962 0.734173i \(-0.262430\pi\)
0.678962 + 0.734173i \(0.262430\pi\)
\(860\) −1.17157 −0.0399503
\(861\) 0 0
\(862\) −2.48528 −0.0846490
\(863\) 4.00000 0.136162 0.0680808 0.997680i \(-0.478312\pi\)
0.0680808 + 0.997680i \(0.478312\pi\)
\(864\) 1.00000 0.0340207
\(865\) −3.27208 −0.111254
\(866\) −19.0000 −0.645646
\(867\) −6.48528 −0.220252
\(868\) 0 0
\(869\) 22.4853 0.762761
\(870\) 7.07107 0.239732
\(871\) −3.34315 −0.113278
\(872\) −9.65685 −0.327022
\(873\) −11.8995 −0.402737
\(874\) 15.2132 0.514594
\(875\) 0 0
\(876\) 13.0711 0.441630
\(877\) 20.5858 0.695133 0.347566 0.937655i \(-0.387008\pi\)
0.347566 + 0.937655i \(0.387008\pi\)
\(878\) −13.4142 −0.452708
\(879\) 6.72792 0.226927
\(880\) −3.41421 −0.115093
\(881\) 21.1127 0.711305 0.355652 0.934618i \(-0.384259\pi\)
0.355652 + 0.934618i \(0.384259\pi\)
\(882\) 0 0
\(883\) −15.0711 −0.507182 −0.253591 0.967312i \(-0.581612\pi\)
−0.253591 + 0.967312i \(0.581612\pi\)
\(884\) −3.24264 −0.109062
\(885\) 14.7279 0.495074
\(886\) −2.72792 −0.0916463
\(887\) 18.6863 0.627424 0.313712 0.949518i \(-0.398427\pi\)
0.313712 + 0.949518i \(0.398427\pi\)
\(888\) 9.07107 0.304405
\(889\) 0 0
\(890\) 3.17157 0.106311
\(891\) −2.41421 −0.0808792
\(892\) −15.3848 −0.515120
\(893\) 27.0000 0.903521
\(894\) −3.89949 −0.130419
\(895\) −33.4558 −1.11831
\(896\) 0 0
\(897\) 5.07107 0.169318
\(898\) 33.9411 1.13263
\(899\) 5.85786 0.195371
\(900\) −3.00000 −0.100000
\(901\) 33.4437 1.11417
\(902\) −17.8995 −0.595988
\(903\) 0 0
\(904\) 8.07107 0.268440
\(905\) −20.3848 −0.677613
\(906\) −10.4142 −0.345989
\(907\) −51.8995 −1.72329 −0.861647 0.507508i \(-0.830567\pi\)
−0.861647 + 0.507508i \(0.830567\pi\)
\(908\) 2.14214 0.0710893
\(909\) −6.82843 −0.226485
\(910\) 0 0
\(911\) 54.0000 1.78910 0.894550 0.446968i \(-0.147496\pi\)
0.894550 + 0.446968i \(0.147496\pi\)
\(912\) 3.00000 0.0993399
\(913\) −8.82843 −0.292178
\(914\) 27.2132 0.900133
\(915\) 4.58579 0.151601
\(916\) −12.8284 −0.423863
\(917\) 0 0
\(918\) −3.24264 −0.107023
\(919\) 13.7990 0.455187 0.227593 0.973756i \(-0.426914\pi\)
0.227593 + 0.973756i \(0.426914\pi\)
\(920\) 7.17157 0.236440
\(921\) 25.6274 0.844452
\(922\) 17.4558 0.574878
\(923\) −1.00000 −0.0329154
\(924\) 0 0
\(925\) −27.2132 −0.894765
\(926\) 29.9411 0.983926
\(927\) −10.4853 −0.344382
\(928\) 5.00000 0.164133
\(929\) −38.2843 −1.25607 −0.628033 0.778187i \(-0.716140\pi\)
−0.628033 + 0.778187i \(0.716140\pi\)
\(930\) 1.65685 0.0543304
\(931\) 0 0
\(932\) 21.7279 0.711722
\(933\) −9.41421 −0.308208
\(934\) 8.24264 0.269707
\(935\) 11.0711 0.362063
\(936\) 1.00000 0.0326860
\(937\) −38.5980 −1.26094 −0.630471 0.776213i \(-0.717138\pi\)
−0.630471 + 0.776213i \(0.717138\pi\)
\(938\) 0 0
\(939\) −10.1421 −0.330976
\(940\) 12.7279 0.415139
\(941\) 12.9706 0.422828 0.211414 0.977397i \(-0.432193\pi\)
0.211414 + 0.977397i \(0.432193\pi\)
\(942\) 22.0711 0.719114
\(943\) 37.5980 1.22436
\(944\) 10.4142 0.338954
\(945\) 0 0
\(946\) 2.00000 0.0650256
\(947\) 8.75736 0.284576 0.142288 0.989825i \(-0.454554\pi\)
0.142288 + 0.989825i \(0.454554\pi\)
\(948\) −9.31371 −0.302495
\(949\) 13.0711 0.424305
\(950\) −9.00000 −0.291999
\(951\) −4.82843 −0.156572
\(952\) 0 0
\(953\) 32.2132 1.04349 0.521744 0.853102i \(-0.325282\pi\)
0.521744 + 0.853102i \(0.325282\pi\)
\(954\) −10.3137 −0.333919
\(955\) 19.3137 0.624977
\(956\) −19.4853 −0.630199
\(957\) −12.0711 −0.390202
\(958\) −2.85786 −0.0923334
\(959\) 0 0
\(960\) 1.41421 0.0456435
\(961\) −29.6274 −0.955723
\(962\) 9.07107 0.292463
\(963\) 2.34315 0.0755068
\(964\) 0.485281 0.0156299
\(965\) −34.2843 −1.10365
\(966\) 0 0
\(967\) −27.3848 −0.880635 −0.440318 0.897842i \(-0.645134\pi\)
−0.440318 + 0.897842i \(0.645134\pi\)
\(968\) −5.17157 −0.166221
\(969\) −9.72792 −0.312506
\(970\) −16.8284 −0.540328
\(971\) 26.4437 0.848617 0.424309 0.905518i \(-0.360517\pi\)
0.424309 + 0.905518i \(0.360517\pi\)
\(972\) 1.00000 0.0320750
\(973\) 0 0
\(974\) 1.72792 0.0553662
\(975\) −3.00000 −0.0960769
\(976\) 3.24264 0.103794
\(977\) −14.4853 −0.463425 −0.231713 0.972784i \(-0.574433\pi\)
−0.231713 + 0.972784i \(0.574433\pi\)
\(978\) −11.1421 −0.356286
\(979\) −5.41421 −0.173039
\(980\) 0 0
\(981\) −9.65685 −0.308320
\(982\) 30.2843 0.966410
\(983\) −27.8284 −0.887589 −0.443794 0.896129i \(-0.646368\pi\)
−0.443794 + 0.896129i \(0.646368\pi\)
\(984\) 7.41421 0.236356
\(985\) −30.4853 −0.971342
\(986\) −16.2132 −0.516334
\(987\) 0 0
\(988\) 3.00000 0.0954427
\(989\) −4.20101 −0.133584
\(990\) −3.41421 −0.108511
\(991\) 15.3137 0.486456 0.243228 0.969969i \(-0.421794\pi\)
0.243228 + 0.969969i \(0.421794\pi\)
\(992\) 1.17157 0.0371975
\(993\) −5.51472 −0.175004
\(994\) 0 0
\(995\) 18.9706 0.601407
\(996\) 3.65685 0.115872
\(997\) −52.6985 −1.66898 −0.834489 0.551025i \(-0.814237\pi\)
−0.834489 + 0.551025i \(0.814237\pi\)
\(998\) −30.0000 −0.949633
\(999\) 9.07107 0.286996
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 3822.2.a.bs.1.2 2
7.2 even 3 546.2.i.h.235.1 yes 4
7.4 even 3 546.2.i.h.79.1 4
7.6 odd 2 3822.2.a.bp.1.1 2
21.2 odd 6 1638.2.j.n.235.2 4
21.11 odd 6 1638.2.j.n.1171.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
546.2.i.h.79.1 4 7.4 even 3
546.2.i.h.235.1 yes 4 7.2 even 3
1638.2.j.n.235.2 4 21.2 odd 6
1638.2.j.n.1171.2 4 21.11 odd 6
3822.2.a.bp.1.1 2 7.6 odd 2
3822.2.a.bs.1.2 2 1.1 even 1 trivial