Properties

Label 847.2.a.f.1.1
Level $847$
Weight $2$
Character 847.1
Self dual yes
Analytic conductor $6.763$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 847 = 7 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 847.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(6.76332905120\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
Defining polynomial: \(x^{2} - x - 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 77)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(1.61803\) of defining polynomial
Character \(\chi\) \(=\) 847.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.23607 q^{2} -1.23607 q^{3} +3.00000 q^{4} -2.00000 q^{5} +2.76393 q^{6} -1.00000 q^{7} -2.23607 q^{8} -1.47214 q^{9} +O(q^{10})\) \(q-2.23607 q^{2} -1.23607 q^{3} +3.00000 q^{4} -2.00000 q^{5} +2.76393 q^{6} -1.00000 q^{7} -2.23607 q^{8} -1.47214 q^{9} +4.47214 q^{10} -3.70820 q^{12} -3.23607 q^{13} +2.23607 q^{14} +2.47214 q^{15} -1.00000 q^{16} +3.23607 q^{17} +3.29180 q^{18} -6.47214 q^{19} -6.00000 q^{20} +1.23607 q^{21} +2.47214 q^{23} +2.76393 q^{24} -1.00000 q^{25} +7.23607 q^{26} +5.52786 q^{27} -3.00000 q^{28} -8.47214 q^{29} -5.52786 q^{30} -2.76393 q^{31} +6.70820 q^{32} -7.23607 q^{34} +2.00000 q^{35} -4.41641 q^{36} -8.47214 q^{37} +14.4721 q^{38} +4.00000 q^{39} +4.47214 q^{40} +11.2361 q^{41} -2.76393 q^{42} -8.00000 q^{43} +2.94427 q^{45} -5.52786 q^{46} +2.76393 q^{47} +1.23607 q^{48} +1.00000 q^{49} +2.23607 q^{50} -4.00000 q^{51} -9.70820 q^{52} -0.472136 q^{53} -12.3607 q^{54} +2.23607 q^{56} +8.00000 q^{57} +18.9443 q^{58} -1.23607 q^{59} +7.41641 q^{60} +7.23607 q^{61} +6.18034 q^{62} +1.47214 q^{63} -13.0000 q^{64} +6.47214 q^{65} +14.4721 q^{67} +9.70820 q^{68} -3.05573 q^{69} -4.47214 q^{70} -10.4721 q^{71} +3.29180 q^{72} +0.763932 q^{73} +18.9443 q^{74} +1.23607 q^{75} -19.4164 q^{76} -8.94427 q^{78} +8.94427 q^{79} +2.00000 q^{80} -2.41641 q^{81} -25.1246 q^{82} +11.4164 q^{83} +3.70820 q^{84} -6.47214 q^{85} +17.8885 q^{86} +10.4721 q^{87} +2.00000 q^{89} -6.58359 q^{90} +3.23607 q^{91} +7.41641 q^{92} +3.41641 q^{93} -6.18034 q^{94} +12.9443 q^{95} -8.29180 q^{96} +17.4164 q^{97} -2.23607 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} + 6 q^{4} - 4 q^{5} + 10 q^{6} - 2 q^{7} + 6 q^{9} + O(q^{10}) \) \( 2 q + 2 q^{3} + 6 q^{4} - 4 q^{5} + 10 q^{6} - 2 q^{7} + 6 q^{9} + 6 q^{12} - 2 q^{13} - 4 q^{15} - 2 q^{16} + 2 q^{17} + 20 q^{18} - 4 q^{19} - 12 q^{20} - 2 q^{21} - 4 q^{23} + 10 q^{24} - 2 q^{25} + 10 q^{26} + 20 q^{27} - 6 q^{28} - 8 q^{29} - 20 q^{30} - 10 q^{31} - 10 q^{34} + 4 q^{35} + 18 q^{36} - 8 q^{37} + 20 q^{38} + 8 q^{39} + 18 q^{41} - 10 q^{42} - 16 q^{43} - 12 q^{45} - 20 q^{46} + 10 q^{47} - 2 q^{48} + 2 q^{49} - 8 q^{51} - 6 q^{52} + 8 q^{53} + 20 q^{54} + 16 q^{57} + 20 q^{58} + 2 q^{59} - 12 q^{60} + 10 q^{61} - 10 q^{62} - 6 q^{63} - 26 q^{64} + 4 q^{65} + 20 q^{67} + 6 q^{68} - 24 q^{69} - 12 q^{71} + 20 q^{72} + 6 q^{73} + 20 q^{74} - 2 q^{75} - 12 q^{76} + 4 q^{80} + 22 q^{81} - 10 q^{82} - 4 q^{83} - 6 q^{84} - 4 q^{85} + 12 q^{87} + 4 q^{89} - 40 q^{90} + 2 q^{91} - 12 q^{92} - 20 q^{93} + 10 q^{94} + 8 q^{95} - 30 q^{96} + 8 q^{97} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.23607 −1.58114 −0.790569 0.612372i \(-0.790215\pi\)
−0.790569 + 0.612372i \(0.790215\pi\)
\(3\) −1.23607 −0.713644 −0.356822 0.934172i \(-0.616140\pi\)
−0.356822 + 0.934172i \(0.616140\pi\)
\(4\) 3.00000 1.50000
\(5\) −2.00000 −0.894427 −0.447214 0.894427i \(-0.647584\pi\)
−0.447214 + 0.894427i \(0.647584\pi\)
\(6\) 2.76393 1.12837
\(7\) −1.00000 −0.377964
\(8\) −2.23607 −0.790569
\(9\) −1.47214 −0.490712
\(10\) 4.47214 1.41421
\(11\) 0 0
\(12\) −3.70820 −1.07047
\(13\) −3.23607 −0.897524 −0.448762 0.893651i \(-0.648135\pi\)
−0.448762 + 0.893651i \(0.648135\pi\)
\(14\) 2.23607 0.597614
\(15\) 2.47214 0.638303
\(16\) −1.00000 −0.250000
\(17\) 3.23607 0.784862 0.392431 0.919781i \(-0.371634\pi\)
0.392431 + 0.919781i \(0.371634\pi\)
\(18\) 3.29180 0.775884
\(19\) −6.47214 −1.48481 −0.742405 0.669951i \(-0.766315\pi\)
−0.742405 + 0.669951i \(0.766315\pi\)
\(20\) −6.00000 −1.34164
\(21\) 1.23607 0.269732
\(22\) 0 0
\(23\) 2.47214 0.515476 0.257738 0.966215i \(-0.417023\pi\)
0.257738 + 0.966215i \(0.417023\pi\)
\(24\) 2.76393 0.564185
\(25\) −1.00000 −0.200000
\(26\) 7.23607 1.41911
\(27\) 5.52786 1.06384
\(28\) −3.00000 −0.566947
\(29\) −8.47214 −1.57324 −0.786618 0.617440i \(-0.788170\pi\)
−0.786618 + 0.617440i \(0.788170\pi\)
\(30\) −5.52786 −1.00925
\(31\) −2.76393 −0.496417 −0.248208 0.968707i \(-0.579842\pi\)
−0.248208 + 0.968707i \(0.579842\pi\)
\(32\) 6.70820 1.18585
\(33\) 0 0
\(34\) −7.23607 −1.24098
\(35\) 2.00000 0.338062
\(36\) −4.41641 −0.736068
\(37\) −8.47214 −1.39281 −0.696405 0.717649i \(-0.745218\pi\)
−0.696405 + 0.717649i \(0.745218\pi\)
\(38\) 14.4721 2.34769
\(39\) 4.00000 0.640513
\(40\) 4.47214 0.707107
\(41\) 11.2361 1.75478 0.877390 0.479779i \(-0.159283\pi\)
0.877390 + 0.479779i \(0.159283\pi\)
\(42\) −2.76393 −0.426484
\(43\) −8.00000 −1.21999 −0.609994 0.792406i \(-0.708828\pi\)
−0.609994 + 0.792406i \(0.708828\pi\)
\(44\) 0 0
\(45\) 2.94427 0.438906
\(46\) −5.52786 −0.815039
\(47\) 2.76393 0.403161 0.201580 0.979472i \(-0.435392\pi\)
0.201580 + 0.979472i \(0.435392\pi\)
\(48\) 1.23607 0.178411
\(49\) 1.00000 0.142857
\(50\) 2.23607 0.316228
\(51\) −4.00000 −0.560112
\(52\) −9.70820 −1.34629
\(53\) −0.472136 −0.0648529 −0.0324264 0.999474i \(-0.510323\pi\)
−0.0324264 + 0.999474i \(0.510323\pi\)
\(54\) −12.3607 −1.68208
\(55\) 0 0
\(56\) 2.23607 0.298807
\(57\) 8.00000 1.05963
\(58\) 18.9443 2.48750
\(59\) −1.23607 −0.160922 −0.0804612 0.996758i \(-0.525639\pi\)
−0.0804612 + 0.996758i \(0.525639\pi\)
\(60\) 7.41641 0.957454
\(61\) 7.23607 0.926484 0.463242 0.886232i \(-0.346686\pi\)
0.463242 + 0.886232i \(0.346686\pi\)
\(62\) 6.18034 0.784904
\(63\) 1.47214 0.185472
\(64\) −13.0000 −1.62500
\(65\) 6.47214 0.802770
\(66\) 0 0
\(67\) 14.4721 1.76805 0.884026 0.467437i \(-0.154823\pi\)
0.884026 + 0.467437i \(0.154823\pi\)
\(68\) 9.70820 1.17729
\(69\) −3.05573 −0.367866
\(70\) −4.47214 −0.534522
\(71\) −10.4721 −1.24281 −0.621407 0.783488i \(-0.713439\pi\)
−0.621407 + 0.783488i \(0.713439\pi\)
\(72\) 3.29180 0.387942
\(73\) 0.763932 0.0894115 0.0447057 0.999000i \(-0.485765\pi\)
0.0447057 + 0.999000i \(0.485765\pi\)
\(74\) 18.9443 2.20223
\(75\) 1.23607 0.142729
\(76\) −19.4164 −2.22721
\(77\) 0 0
\(78\) −8.94427 −1.01274
\(79\) 8.94427 1.00631 0.503155 0.864196i \(-0.332173\pi\)
0.503155 + 0.864196i \(0.332173\pi\)
\(80\) 2.00000 0.223607
\(81\) −2.41641 −0.268490
\(82\) −25.1246 −2.77455
\(83\) 11.4164 1.25311 0.626557 0.779376i \(-0.284464\pi\)
0.626557 + 0.779376i \(0.284464\pi\)
\(84\) 3.70820 0.404598
\(85\) −6.47214 −0.702002
\(86\) 17.8885 1.92897
\(87\) 10.4721 1.12273
\(88\) 0 0
\(89\) 2.00000 0.212000 0.106000 0.994366i \(-0.466196\pi\)
0.106000 + 0.994366i \(0.466196\pi\)
\(90\) −6.58359 −0.693972
\(91\) 3.23607 0.339232
\(92\) 7.41641 0.773214
\(93\) 3.41641 0.354265
\(94\) −6.18034 −0.637453
\(95\) 12.9443 1.32805
\(96\) −8.29180 −0.846278
\(97\) 17.4164 1.76837 0.884184 0.467139i \(-0.154715\pi\)
0.884184 + 0.467139i \(0.154715\pi\)
\(98\) −2.23607 −0.225877
\(99\) 0 0
\(100\) −3.00000 −0.300000
\(101\) 4.76393 0.474029 0.237014 0.971506i \(-0.423831\pi\)
0.237014 + 0.971506i \(0.423831\pi\)
\(102\) 8.94427 0.885615
\(103\) 7.70820 0.759512 0.379756 0.925087i \(-0.376008\pi\)
0.379756 + 0.925087i \(0.376008\pi\)
\(104\) 7.23607 0.709555
\(105\) −2.47214 −0.241256
\(106\) 1.05573 0.102541
\(107\) 4.00000 0.386695 0.193347 0.981130i \(-0.438066\pi\)
0.193347 + 0.981130i \(0.438066\pi\)
\(108\) 16.5836 1.59576
\(109\) 4.47214 0.428353 0.214176 0.976795i \(-0.431293\pi\)
0.214176 + 0.976795i \(0.431293\pi\)
\(110\) 0 0
\(111\) 10.4721 0.993971
\(112\) 1.00000 0.0944911
\(113\) 2.00000 0.188144 0.0940721 0.995565i \(-0.470012\pi\)
0.0940721 + 0.995565i \(0.470012\pi\)
\(114\) −17.8885 −1.67542
\(115\) −4.94427 −0.461056
\(116\) −25.4164 −2.35985
\(117\) 4.76393 0.440426
\(118\) 2.76393 0.254441
\(119\) −3.23607 −0.296650
\(120\) −5.52786 −0.504623
\(121\) 0 0
\(122\) −16.1803 −1.46490
\(123\) −13.8885 −1.25229
\(124\) −8.29180 −0.744625
\(125\) 12.0000 1.07331
\(126\) −3.29180 −0.293257
\(127\) −3.05573 −0.271152 −0.135576 0.990767i \(-0.543288\pi\)
−0.135576 + 0.990767i \(0.543288\pi\)
\(128\) 15.6525 1.38350
\(129\) 9.88854 0.870638
\(130\) −14.4721 −1.26929
\(131\) −21.8885 −1.91241 −0.956205 0.292696i \(-0.905448\pi\)
−0.956205 + 0.292696i \(0.905448\pi\)
\(132\) 0 0
\(133\) 6.47214 0.561205
\(134\) −32.3607 −2.79554
\(135\) −11.0557 −0.951526
\(136\) −7.23607 −0.620488
\(137\) 16.4721 1.40731 0.703655 0.710542i \(-0.251550\pi\)
0.703655 + 0.710542i \(0.251550\pi\)
\(138\) 6.83282 0.581648
\(139\) 1.52786 0.129592 0.0647959 0.997899i \(-0.479360\pi\)
0.0647959 + 0.997899i \(0.479360\pi\)
\(140\) 6.00000 0.507093
\(141\) −3.41641 −0.287713
\(142\) 23.4164 1.96506
\(143\) 0 0
\(144\) 1.47214 0.122678
\(145\) 16.9443 1.40715
\(146\) −1.70820 −0.141372
\(147\) −1.23607 −0.101949
\(148\) −25.4164 −2.08922
\(149\) −14.0000 −1.14692 −0.573462 0.819232i \(-0.694400\pi\)
−0.573462 + 0.819232i \(0.694400\pi\)
\(150\) −2.76393 −0.225674
\(151\) −8.94427 −0.727875 −0.363937 0.931423i \(-0.618568\pi\)
−0.363937 + 0.931423i \(0.618568\pi\)
\(152\) 14.4721 1.17385
\(153\) −4.76393 −0.385141
\(154\) 0 0
\(155\) 5.52786 0.444009
\(156\) 12.0000 0.960769
\(157\) 10.9443 0.873448 0.436724 0.899596i \(-0.356139\pi\)
0.436724 + 0.899596i \(0.356139\pi\)
\(158\) −20.0000 −1.59111
\(159\) 0.583592 0.0462819
\(160\) −13.4164 −1.06066
\(161\) −2.47214 −0.194832
\(162\) 5.40325 0.424520
\(163\) −3.41641 −0.267594 −0.133797 0.991009i \(-0.542717\pi\)
−0.133797 + 0.991009i \(0.542717\pi\)
\(164\) 33.7082 2.63217
\(165\) 0 0
\(166\) −25.5279 −1.98135
\(167\) −4.94427 −0.382599 −0.191300 0.981532i \(-0.561270\pi\)
−0.191300 + 0.981532i \(0.561270\pi\)
\(168\) −2.76393 −0.213242
\(169\) −2.52786 −0.194451
\(170\) 14.4721 1.10996
\(171\) 9.52786 0.728614
\(172\) −24.0000 −1.82998
\(173\) 12.7639 0.970424 0.485212 0.874397i \(-0.338742\pi\)
0.485212 + 0.874397i \(0.338742\pi\)
\(174\) −23.4164 −1.77519
\(175\) 1.00000 0.0755929
\(176\) 0 0
\(177\) 1.52786 0.114841
\(178\) −4.47214 −0.335201
\(179\) −8.94427 −0.668526 −0.334263 0.942480i \(-0.608487\pi\)
−0.334263 + 0.942480i \(0.608487\pi\)
\(180\) 8.83282 0.658359
\(181\) −25.4164 −1.88919 −0.944593 0.328243i \(-0.893544\pi\)
−0.944593 + 0.328243i \(0.893544\pi\)
\(182\) −7.23607 −0.536373
\(183\) −8.94427 −0.661180
\(184\) −5.52786 −0.407520
\(185\) 16.9443 1.24577
\(186\) −7.63932 −0.560142
\(187\) 0 0
\(188\) 8.29180 0.604741
\(189\) −5.52786 −0.402093
\(190\) −28.9443 −2.09984
\(191\) −3.05573 −0.221105 −0.110552 0.993870i \(-0.535262\pi\)
−0.110552 + 0.993870i \(0.535262\pi\)
\(192\) 16.0689 1.15967
\(193\) −11.8885 −0.855756 −0.427878 0.903836i \(-0.640739\pi\)
−0.427878 + 0.903836i \(0.640739\pi\)
\(194\) −38.9443 −2.79604
\(195\) −8.00000 −0.572892
\(196\) 3.00000 0.214286
\(197\) 2.00000 0.142494 0.0712470 0.997459i \(-0.477302\pi\)
0.0712470 + 0.997459i \(0.477302\pi\)
\(198\) 0 0
\(199\) −2.18034 −0.154560 −0.0772801 0.997009i \(-0.524624\pi\)
−0.0772801 + 0.997009i \(0.524624\pi\)
\(200\) 2.23607 0.158114
\(201\) −17.8885 −1.26176
\(202\) −10.6525 −0.749506
\(203\) 8.47214 0.594627
\(204\) −12.0000 −0.840168
\(205\) −22.4721 −1.56952
\(206\) −17.2361 −1.20089
\(207\) −3.63932 −0.252950
\(208\) 3.23607 0.224381
\(209\) 0 0
\(210\) 5.52786 0.381459
\(211\) 13.8885 0.956127 0.478063 0.878325i \(-0.341339\pi\)
0.478063 + 0.878325i \(0.341339\pi\)
\(212\) −1.41641 −0.0972793
\(213\) 12.9443 0.886927
\(214\) −8.94427 −0.611418
\(215\) 16.0000 1.09119
\(216\) −12.3607 −0.841038
\(217\) 2.76393 0.187628
\(218\) −10.0000 −0.677285
\(219\) −0.944272 −0.0638080
\(220\) 0 0
\(221\) −10.4721 −0.704432
\(222\) −23.4164 −1.57161
\(223\) −10.1803 −0.681726 −0.340863 0.940113i \(-0.610719\pi\)
−0.340863 + 0.940113i \(0.610719\pi\)
\(224\) −6.70820 −0.448211
\(225\) 1.47214 0.0981424
\(226\) −4.47214 −0.297482
\(227\) −5.88854 −0.390836 −0.195418 0.980720i \(-0.562606\pi\)
−0.195418 + 0.980720i \(0.562606\pi\)
\(228\) 24.0000 1.58944
\(229\) −4.47214 −0.295527 −0.147764 0.989023i \(-0.547207\pi\)
−0.147764 + 0.989023i \(0.547207\pi\)
\(230\) 11.0557 0.728993
\(231\) 0 0
\(232\) 18.9443 1.24375
\(233\) −9.41641 −0.616889 −0.308445 0.951242i \(-0.599808\pi\)
−0.308445 + 0.951242i \(0.599808\pi\)
\(234\) −10.6525 −0.696374
\(235\) −5.52786 −0.360598
\(236\) −3.70820 −0.241384
\(237\) −11.0557 −0.718147
\(238\) 7.23607 0.469045
\(239\) 9.88854 0.639637 0.319818 0.947479i \(-0.396378\pi\)
0.319818 + 0.947479i \(0.396378\pi\)
\(240\) −2.47214 −0.159576
\(241\) 13.1246 0.845431 0.422715 0.906263i \(-0.361077\pi\)
0.422715 + 0.906263i \(0.361077\pi\)
\(242\) 0 0
\(243\) −13.5967 −0.872232
\(244\) 21.7082 1.38973
\(245\) −2.00000 −0.127775
\(246\) 31.0557 1.98004
\(247\) 20.9443 1.33265
\(248\) 6.18034 0.392452
\(249\) −14.1115 −0.894277
\(250\) −26.8328 −1.69706
\(251\) 4.29180 0.270896 0.135448 0.990784i \(-0.456753\pi\)
0.135448 + 0.990784i \(0.456753\pi\)
\(252\) 4.41641 0.278208
\(253\) 0 0
\(254\) 6.83282 0.428729
\(255\) 8.00000 0.500979
\(256\) −9.00000 −0.562500
\(257\) −6.00000 −0.374270 −0.187135 0.982334i \(-0.559920\pi\)
−0.187135 + 0.982334i \(0.559920\pi\)
\(258\) −22.1115 −1.37660
\(259\) 8.47214 0.526433
\(260\) 19.4164 1.20415
\(261\) 12.4721 0.772006
\(262\) 48.9443 3.02379
\(263\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(264\) 0 0
\(265\) 0.944272 0.0580062
\(266\) −14.4721 −0.887344
\(267\) −2.47214 −0.151292
\(268\) 43.4164 2.65208
\(269\) 13.4164 0.818013 0.409006 0.912532i \(-0.365875\pi\)
0.409006 + 0.912532i \(0.365875\pi\)
\(270\) 24.7214 1.50449
\(271\) 10.4721 0.636137 0.318068 0.948068i \(-0.396966\pi\)
0.318068 + 0.948068i \(0.396966\pi\)
\(272\) −3.23607 −0.196215
\(273\) −4.00000 −0.242091
\(274\) −36.8328 −2.22515
\(275\) 0 0
\(276\) −9.16718 −0.551800
\(277\) 19.8885 1.19499 0.597493 0.801874i \(-0.296163\pi\)
0.597493 + 0.801874i \(0.296163\pi\)
\(278\) −3.41641 −0.204903
\(279\) 4.06888 0.243598
\(280\) −4.47214 −0.267261
\(281\) 3.52786 0.210455 0.105227 0.994448i \(-0.466443\pi\)
0.105227 + 0.994448i \(0.466443\pi\)
\(282\) 7.63932 0.454915
\(283\) −29.8885 −1.77669 −0.888345 0.459177i \(-0.848144\pi\)
−0.888345 + 0.459177i \(0.848144\pi\)
\(284\) −31.4164 −1.86422
\(285\) −16.0000 −0.947758
\(286\) 0 0
\(287\) −11.2361 −0.663244
\(288\) −9.87539 −0.581913
\(289\) −6.52786 −0.383992
\(290\) −37.8885 −2.22489
\(291\) −21.5279 −1.26199
\(292\) 2.29180 0.134117
\(293\) 25.1246 1.46780 0.733898 0.679260i \(-0.237699\pi\)
0.733898 + 0.679260i \(0.237699\pi\)
\(294\) 2.76393 0.161196
\(295\) 2.47214 0.143933
\(296\) 18.9443 1.10111
\(297\) 0 0
\(298\) 31.3050 1.81345
\(299\) −8.00000 −0.462652
\(300\) 3.70820 0.214093
\(301\) 8.00000 0.461112
\(302\) 20.0000 1.15087
\(303\) −5.88854 −0.338288
\(304\) 6.47214 0.371202
\(305\) −14.4721 −0.828672
\(306\) 10.6525 0.608962
\(307\) 8.94427 0.510477 0.255238 0.966878i \(-0.417846\pi\)
0.255238 + 0.966878i \(0.417846\pi\)
\(308\) 0 0
\(309\) −9.52786 −0.542021
\(310\) −12.3607 −0.702039
\(311\) −8.29180 −0.470185 −0.235092 0.971973i \(-0.575539\pi\)
−0.235092 + 0.971973i \(0.575539\pi\)
\(312\) −8.94427 −0.506370
\(313\) 14.9443 0.844700 0.422350 0.906433i \(-0.361205\pi\)
0.422350 + 0.906433i \(0.361205\pi\)
\(314\) −24.4721 −1.38104
\(315\) −2.94427 −0.165891
\(316\) 26.8328 1.50946
\(317\) 14.0000 0.786318 0.393159 0.919470i \(-0.371382\pi\)
0.393159 + 0.919470i \(0.371382\pi\)
\(318\) −1.30495 −0.0731781
\(319\) 0 0
\(320\) 26.0000 1.45344
\(321\) −4.94427 −0.275962
\(322\) 5.52786 0.308056
\(323\) −20.9443 −1.16537
\(324\) −7.24922 −0.402735
\(325\) 3.23607 0.179505
\(326\) 7.63932 0.423103
\(327\) −5.52786 −0.305692
\(328\) −25.1246 −1.38727
\(329\) −2.76393 −0.152381
\(330\) 0 0
\(331\) −13.8885 −0.763383 −0.381692 0.924290i \(-0.624658\pi\)
−0.381692 + 0.924290i \(0.624658\pi\)
\(332\) 34.2492 1.87967
\(333\) 12.4721 0.683469
\(334\) 11.0557 0.604943
\(335\) −28.9443 −1.58139
\(336\) −1.23607 −0.0674330
\(337\) 11.5279 0.627963 0.313981 0.949429i \(-0.398337\pi\)
0.313981 + 0.949429i \(0.398337\pi\)
\(338\) 5.65248 0.307454
\(339\) −2.47214 −0.134268
\(340\) −19.4164 −1.05300
\(341\) 0 0
\(342\) −21.3050 −1.15204
\(343\) −1.00000 −0.0539949
\(344\) 17.8885 0.964486
\(345\) 6.11146 0.329030
\(346\) −28.5410 −1.53437
\(347\) −20.9443 −1.12435 −0.562174 0.827019i \(-0.690035\pi\)
−0.562174 + 0.827019i \(0.690035\pi\)
\(348\) 31.4164 1.68410
\(349\) 7.23607 0.387338 0.193669 0.981067i \(-0.437961\pi\)
0.193669 + 0.981067i \(0.437961\pi\)
\(350\) −2.23607 −0.119523
\(351\) −17.8885 −0.954820
\(352\) 0 0
\(353\) 19.8885 1.05856 0.529280 0.848447i \(-0.322462\pi\)
0.529280 + 0.848447i \(0.322462\pi\)
\(354\) −3.41641 −0.181580
\(355\) 20.9443 1.11161
\(356\) 6.00000 0.317999
\(357\) 4.00000 0.211702
\(358\) 20.0000 1.05703
\(359\) 24.9443 1.31651 0.658254 0.752796i \(-0.271295\pi\)
0.658254 + 0.752796i \(0.271295\pi\)
\(360\) −6.58359 −0.346986
\(361\) 22.8885 1.20466
\(362\) 56.8328 2.98707
\(363\) 0 0
\(364\) 9.70820 0.508848
\(365\) −1.52786 −0.0799721
\(366\) 20.0000 1.04542
\(367\) −23.1246 −1.20709 −0.603547 0.797327i \(-0.706247\pi\)
−0.603547 + 0.797327i \(0.706247\pi\)
\(368\) −2.47214 −0.128869
\(369\) −16.5410 −0.861091
\(370\) −37.8885 −1.96973
\(371\) 0.472136 0.0245121
\(372\) 10.2492 0.531397
\(373\) −6.00000 −0.310668 −0.155334 0.987862i \(-0.549645\pi\)
−0.155334 + 0.987862i \(0.549645\pi\)
\(374\) 0 0
\(375\) −14.8328 −0.765963
\(376\) −6.18034 −0.318727
\(377\) 27.4164 1.41202
\(378\) 12.3607 0.635765
\(379\) 37.3050 1.91623 0.958113 0.286389i \(-0.0924551\pi\)
0.958113 + 0.286389i \(0.0924551\pi\)
\(380\) 38.8328 1.99208
\(381\) 3.77709 0.193506
\(382\) 6.83282 0.349597
\(383\) −4.65248 −0.237730 −0.118865 0.992910i \(-0.537926\pi\)
−0.118865 + 0.992910i \(0.537926\pi\)
\(384\) −19.3475 −0.987324
\(385\) 0 0
\(386\) 26.5836 1.35307
\(387\) 11.7771 0.598663
\(388\) 52.2492 2.65255
\(389\) 15.8885 0.805581 0.402791 0.915292i \(-0.368040\pi\)
0.402791 + 0.915292i \(0.368040\pi\)
\(390\) 17.8885 0.905822
\(391\) 8.00000 0.404577
\(392\) −2.23607 −0.112938
\(393\) 27.0557 1.36478
\(394\) −4.47214 −0.225303
\(395\) −17.8885 −0.900070
\(396\) 0 0
\(397\) −35.8885 −1.80119 −0.900597 0.434655i \(-0.856870\pi\)
−0.900597 + 0.434655i \(0.856870\pi\)
\(398\) 4.87539 0.244381
\(399\) −8.00000 −0.400501
\(400\) 1.00000 0.0500000
\(401\) 22.9443 1.14578 0.572891 0.819631i \(-0.305822\pi\)
0.572891 + 0.819631i \(0.305822\pi\)
\(402\) 40.0000 1.99502
\(403\) 8.94427 0.445546
\(404\) 14.2918 0.711043
\(405\) 4.83282 0.240145
\(406\) −18.9443 −0.940188
\(407\) 0 0
\(408\) 8.94427 0.442807
\(409\) −9.12461 −0.451183 −0.225592 0.974222i \(-0.572431\pi\)
−0.225592 + 0.974222i \(0.572431\pi\)
\(410\) 50.2492 2.48163
\(411\) −20.3607 −1.00432
\(412\) 23.1246 1.13927
\(413\) 1.23607 0.0608229
\(414\) 8.13777 0.399949
\(415\) −22.8328 −1.12082
\(416\) −21.7082 −1.06433
\(417\) −1.88854 −0.0924824
\(418\) 0 0
\(419\) 24.6525 1.20435 0.602176 0.798363i \(-0.294300\pi\)
0.602176 + 0.798363i \(0.294300\pi\)
\(420\) −7.41641 −0.361884
\(421\) 22.3607 1.08979 0.544896 0.838503i \(-0.316569\pi\)
0.544896 + 0.838503i \(0.316569\pi\)
\(422\) −31.0557 −1.51177
\(423\) −4.06888 −0.197836
\(424\) 1.05573 0.0512707
\(425\) −3.23607 −0.156972
\(426\) −28.9443 −1.40235
\(427\) −7.23607 −0.350178
\(428\) 12.0000 0.580042
\(429\) 0 0
\(430\) −35.7771 −1.72532
\(431\) −12.0000 −0.578020 −0.289010 0.957326i \(-0.593326\pi\)
−0.289010 + 0.957326i \(0.593326\pi\)
\(432\) −5.52786 −0.265959
\(433\) −8.47214 −0.407145 −0.203572 0.979060i \(-0.565255\pi\)
−0.203572 + 0.979060i \(0.565255\pi\)
\(434\) −6.18034 −0.296666
\(435\) −20.9443 −1.00420
\(436\) 13.4164 0.642529
\(437\) −16.0000 −0.765384
\(438\) 2.11146 0.100889
\(439\) −10.4721 −0.499808 −0.249904 0.968271i \(-0.580399\pi\)
−0.249904 + 0.968271i \(0.580399\pi\)
\(440\) 0 0
\(441\) −1.47214 −0.0701017
\(442\) 23.4164 1.11380
\(443\) −24.9443 −1.18514 −0.592569 0.805520i \(-0.701886\pi\)
−0.592569 + 0.805520i \(0.701886\pi\)
\(444\) 31.4164 1.49096
\(445\) −4.00000 −0.189618
\(446\) 22.7639 1.07790
\(447\) 17.3050 0.818496
\(448\) 13.0000 0.614192
\(449\) −28.4721 −1.34368 −0.671842 0.740695i \(-0.734496\pi\)
−0.671842 + 0.740695i \(0.734496\pi\)
\(450\) −3.29180 −0.155177
\(451\) 0 0
\(452\) 6.00000 0.282216
\(453\) 11.0557 0.519443
\(454\) 13.1672 0.617967
\(455\) −6.47214 −0.303418
\(456\) −17.8885 −0.837708
\(457\) 28.8328 1.34874 0.674371 0.738393i \(-0.264415\pi\)
0.674371 + 0.738393i \(0.264415\pi\)
\(458\) 10.0000 0.467269
\(459\) 17.8885 0.834966
\(460\) −14.8328 −0.691584
\(461\) 12.1803 0.567295 0.283647 0.958929i \(-0.408455\pi\)
0.283647 + 0.958929i \(0.408455\pi\)
\(462\) 0 0
\(463\) −5.52786 −0.256902 −0.128451 0.991716i \(-0.541000\pi\)
−0.128451 + 0.991716i \(0.541000\pi\)
\(464\) 8.47214 0.393309
\(465\) −6.83282 −0.316864
\(466\) 21.0557 0.975388
\(467\) 24.0689 1.11378 0.556888 0.830588i \(-0.311995\pi\)
0.556888 + 0.830588i \(0.311995\pi\)
\(468\) 14.2918 0.660639
\(469\) −14.4721 −0.668261
\(470\) 12.3607 0.570156
\(471\) −13.5279 −0.623331
\(472\) 2.76393 0.127220
\(473\) 0 0
\(474\) 24.7214 1.13549
\(475\) 6.47214 0.296962
\(476\) −9.70820 −0.444975
\(477\) 0.695048 0.0318241
\(478\) −22.1115 −1.01135
\(479\) −13.5279 −0.618104 −0.309052 0.951045i \(-0.600012\pi\)
−0.309052 + 0.951045i \(0.600012\pi\)
\(480\) 16.5836 0.756934
\(481\) 27.4164 1.25008
\(482\) −29.3475 −1.33674
\(483\) 3.05573 0.139040
\(484\) 0 0
\(485\) −34.8328 −1.58168
\(486\) 30.4033 1.37912
\(487\) −36.3607 −1.64766 −0.823830 0.566837i \(-0.808167\pi\)
−0.823830 + 0.566837i \(0.808167\pi\)
\(488\) −16.1803 −0.732450
\(489\) 4.22291 0.190967
\(490\) 4.47214 0.202031
\(491\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(492\) −41.6656 −1.87843
\(493\) −27.4164 −1.23477
\(494\) −46.8328 −2.10711
\(495\) 0 0
\(496\) 2.76393 0.124104
\(497\) 10.4721 0.469739
\(498\) 31.5542 1.41398
\(499\) 1.52786 0.0683966 0.0341983 0.999415i \(-0.489112\pi\)
0.0341983 + 0.999415i \(0.489112\pi\)
\(500\) 36.0000 1.60997
\(501\) 6.11146 0.273040
\(502\) −9.59675 −0.428324
\(503\) −23.4164 −1.04409 −0.522043 0.852919i \(-0.674830\pi\)
−0.522043 + 0.852919i \(0.674830\pi\)
\(504\) −3.29180 −0.146628
\(505\) −9.52786 −0.423984
\(506\) 0 0
\(507\) 3.12461 0.138769
\(508\) −9.16718 −0.406728
\(509\) 40.4721 1.79390 0.896948 0.442136i \(-0.145779\pi\)
0.896948 + 0.442136i \(0.145779\pi\)
\(510\) −17.8885 −0.792118
\(511\) −0.763932 −0.0337944
\(512\) −11.1803 −0.494106
\(513\) −35.7771 −1.57960
\(514\) 13.4164 0.591772
\(515\) −15.4164 −0.679328
\(516\) 29.6656 1.30596
\(517\) 0 0
\(518\) −18.9443 −0.832364
\(519\) −15.7771 −0.692537
\(520\) −14.4721 −0.634645
\(521\) 30.3607 1.33013 0.665063 0.746787i \(-0.268405\pi\)
0.665063 + 0.746787i \(0.268405\pi\)
\(522\) −27.8885 −1.22065
\(523\) −44.0000 −1.92399 −0.961993 0.273075i \(-0.911959\pi\)
−0.961993 + 0.273075i \(0.911959\pi\)
\(524\) −65.6656 −2.86862
\(525\) −1.23607 −0.0539464
\(526\) 0 0
\(527\) −8.94427 −0.389619
\(528\) 0 0
\(529\) −16.8885 −0.734285
\(530\) −2.11146 −0.0917158
\(531\) 1.81966 0.0789665
\(532\) 19.4164 0.841808
\(533\) −36.3607 −1.57496
\(534\) 5.52786 0.239214
\(535\) −8.00000 −0.345870
\(536\) −32.3607 −1.39777
\(537\) 11.0557 0.477090
\(538\) −30.0000 −1.29339
\(539\) 0 0
\(540\) −33.1672 −1.42729
\(541\) −20.8328 −0.895673 −0.447836 0.894116i \(-0.647805\pi\)
−0.447836 + 0.894116i \(0.647805\pi\)
\(542\) −23.4164 −1.00582
\(543\) 31.4164 1.34821
\(544\) 21.7082 0.930732
\(545\) −8.94427 −0.383131
\(546\) 8.94427 0.382780
\(547\) −28.0000 −1.19719 −0.598597 0.801050i \(-0.704275\pi\)
−0.598597 + 0.801050i \(0.704275\pi\)
\(548\) 49.4164 2.11096
\(549\) −10.6525 −0.454637
\(550\) 0 0
\(551\) 54.8328 2.33596
\(552\) 6.83282 0.290824
\(553\) −8.94427 −0.380349
\(554\) −44.4721 −1.88944
\(555\) −20.9443 −0.889035
\(556\) 4.58359 0.194388
\(557\) 38.9443 1.65012 0.825061 0.565044i \(-0.191141\pi\)
0.825061 + 0.565044i \(0.191141\pi\)
\(558\) −9.09830 −0.385162
\(559\) 25.8885 1.09497
\(560\) −2.00000 −0.0845154
\(561\) 0 0
\(562\) −7.88854 −0.332758
\(563\) −12.5836 −0.530335 −0.265168 0.964202i \(-0.585427\pi\)
−0.265168 + 0.964202i \(0.585427\pi\)
\(564\) −10.2492 −0.431570
\(565\) −4.00000 −0.168281
\(566\) 66.8328 2.80919
\(567\) 2.41641 0.101480
\(568\) 23.4164 0.982531
\(569\) −7.52786 −0.315584 −0.157792 0.987472i \(-0.550438\pi\)
−0.157792 + 0.987472i \(0.550438\pi\)
\(570\) 35.7771 1.49854
\(571\) 15.0557 0.630063 0.315031 0.949081i \(-0.397985\pi\)
0.315031 + 0.949081i \(0.397985\pi\)
\(572\) 0 0
\(573\) 3.77709 0.157790
\(574\) 25.1246 1.04868
\(575\) −2.47214 −0.103095
\(576\) 19.1378 0.797407
\(577\) −19.5279 −0.812956 −0.406478 0.913661i \(-0.633243\pi\)
−0.406478 + 0.913661i \(0.633243\pi\)
\(578\) 14.5967 0.607145
\(579\) 14.6950 0.610705
\(580\) 50.8328 2.11072
\(581\) −11.4164 −0.473632
\(582\) 48.1378 1.99537
\(583\) 0 0
\(584\) −1.70820 −0.0706860
\(585\) −9.52786 −0.393929
\(586\) −56.1803 −2.32079
\(587\) 27.1246 1.11955 0.559776 0.828644i \(-0.310887\pi\)
0.559776 + 0.828644i \(0.310887\pi\)
\(588\) −3.70820 −0.152924
\(589\) 17.8885 0.737085
\(590\) −5.52786 −0.227579
\(591\) −2.47214 −0.101690
\(592\) 8.47214 0.348203
\(593\) 45.7082 1.87701 0.938505 0.345264i \(-0.112211\pi\)
0.938505 + 0.345264i \(0.112211\pi\)
\(594\) 0 0
\(595\) 6.47214 0.265332
\(596\) −42.0000 −1.72039
\(597\) 2.69505 0.110301
\(598\) 17.8885 0.731517
\(599\) −23.4164 −0.956768 −0.478384 0.878151i \(-0.658777\pi\)
−0.478384 + 0.878151i \(0.658777\pi\)
\(600\) −2.76393 −0.112837
\(601\) 37.1246 1.51434 0.757172 0.653215i \(-0.226580\pi\)
0.757172 + 0.653215i \(0.226580\pi\)
\(602\) −17.8885 −0.729083
\(603\) −21.3050 −0.867605
\(604\) −26.8328 −1.09181
\(605\) 0 0
\(606\) 13.1672 0.534880
\(607\) −12.9443 −0.525392 −0.262696 0.964879i \(-0.584612\pi\)
−0.262696 + 0.964879i \(0.584612\pi\)
\(608\) −43.4164 −1.76077
\(609\) −10.4721 −0.424352
\(610\) 32.3607 1.31025
\(611\) −8.94427 −0.361847
\(612\) −14.2918 −0.577712
\(613\) −15.3050 −0.618161 −0.309081 0.951036i \(-0.600021\pi\)
−0.309081 + 0.951036i \(0.600021\pi\)
\(614\) −20.0000 −0.807134
\(615\) 27.7771 1.12008
\(616\) 0 0
\(617\) 6.58359 0.265045 0.132523 0.991180i \(-0.457692\pi\)
0.132523 + 0.991180i \(0.457692\pi\)
\(618\) 21.3050 0.857011
\(619\) 11.1246 0.447136 0.223568 0.974688i \(-0.428230\pi\)
0.223568 + 0.974688i \(0.428230\pi\)
\(620\) 16.5836 0.666013
\(621\) 13.6656 0.548383
\(622\) 18.5410 0.743427
\(623\) −2.00000 −0.0801283
\(624\) −4.00000 −0.160128
\(625\) −19.0000 −0.760000
\(626\) −33.4164 −1.33559
\(627\) 0 0
\(628\) 32.8328 1.31017
\(629\) −27.4164 −1.09316
\(630\) 6.58359 0.262297
\(631\) −24.0000 −0.955425 −0.477712 0.878516i \(-0.658534\pi\)
−0.477712 + 0.878516i \(0.658534\pi\)
\(632\) −20.0000 −0.795557
\(633\) −17.1672 −0.682334
\(634\) −31.3050 −1.24328
\(635\) 6.11146 0.242526
\(636\) 1.75078 0.0694228
\(637\) −3.23607 −0.128218
\(638\) 0 0
\(639\) 15.4164 0.609864
\(640\) −31.3050 −1.23744
\(641\) −15.5279 −0.613314 −0.306657 0.951820i \(-0.599210\pi\)
−0.306657 + 0.951820i \(0.599210\pi\)
\(642\) 11.0557 0.436335
\(643\) 11.1246 0.438712 0.219356 0.975645i \(-0.429604\pi\)
0.219356 + 0.975645i \(0.429604\pi\)
\(644\) −7.41641 −0.292247
\(645\) −19.7771 −0.778722
\(646\) 46.8328 1.84261
\(647\) 36.0689 1.41801 0.709007 0.705201i \(-0.249143\pi\)
0.709007 + 0.705201i \(0.249143\pi\)
\(648\) 5.40325 0.212260
\(649\) 0 0
\(650\) −7.23607 −0.283822
\(651\) −3.41641 −0.133900
\(652\) −10.2492 −0.401391
\(653\) 25.0557 0.980506 0.490253 0.871580i \(-0.336904\pi\)
0.490253 + 0.871580i \(0.336904\pi\)
\(654\) 12.3607 0.483341
\(655\) 43.7771 1.71051
\(656\) −11.2361 −0.438695
\(657\) −1.12461 −0.0438753
\(658\) 6.18034 0.240935
\(659\) 17.8885 0.696839 0.348419 0.937339i \(-0.386719\pi\)
0.348419 + 0.937339i \(0.386719\pi\)
\(660\) 0 0
\(661\) −40.8328 −1.58821 −0.794106 0.607779i \(-0.792061\pi\)
−0.794106 + 0.607779i \(0.792061\pi\)
\(662\) 31.0557 1.20702
\(663\) 12.9443 0.502714
\(664\) −25.5279 −0.990673
\(665\) −12.9443 −0.501957
\(666\) −27.8885 −1.08066
\(667\) −20.9443 −0.810965
\(668\) −14.8328 −0.573899
\(669\) 12.5836 0.486510
\(670\) 64.7214 2.50040
\(671\) 0 0
\(672\) 8.29180 0.319863
\(673\) 21.4164 0.825542 0.412771 0.910835i \(-0.364561\pi\)
0.412771 + 0.910835i \(0.364561\pi\)
\(674\) −25.7771 −0.992896
\(675\) −5.52786 −0.212768
\(676\) −7.58359 −0.291677
\(677\) 9.70820 0.373117 0.186558 0.982444i \(-0.440267\pi\)
0.186558 + 0.982444i \(0.440267\pi\)
\(678\) 5.52786 0.212296
\(679\) −17.4164 −0.668380
\(680\) 14.4721 0.554981
\(681\) 7.27864 0.278918
\(682\) 0 0
\(683\) −5.88854 −0.225319 −0.112659 0.993634i \(-0.535937\pi\)
−0.112659 + 0.993634i \(0.535937\pi\)
\(684\) 28.5836 1.09292
\(685\) −32.9443 −1.25874
\(686\) 2.23607 0.0853735
\(687\) 5.52786 0.210901
\(688\) 8.00000 0.304997
\(689\) 1.52786 0.0582070
\(690\) −13.6656 −0.520242
\(691\) 18.5410 0.705334 0.352667 0.935749i \(-0.385275\pi\)
0.352667 + 0.935749i \(0.385275\pi\)
\(692\) 38.2918 1.45564
\(693\) 0 0
\(694\) 46.8328 1.77775
\(695\) −3.05573 −0.115910
\(696\) −23.4164 −0.887597
\(697\) 36.3607 1.37726
\(698\) −16.1803 −0.612435
\(699\) 11.6393 0.440240
\(700\) 3.00000 0.113389
\(701\) 15.5279 0.586479 0.293240 0.956039i \(-0.405267\pi\)
0.293240 + 0.956039i \(0.405267\pi\)
\(702\) 40.0000 1.50970
\(703\) 54.8328 2.06806
\(704\) 0 0
\(705\) 6.83282 0.257339
\(706\) −44.4721 −1.67373
\(707\) −4.76393 −0.179166
\(708\) 4.58359 0.172262
\(709\) −14.9443 −0.561244 −0.280622 0.959818i \(-0.590541\pi\)
−0.280622 + 0.959818i \(0.590541\pi\)
\(710\) −46.8328 −1.75760
\(711\) −13.1672 −0.493808
\(712\) −4.47214 −0.167600
\(713\) −6.83282 −0.255891
\(714\) −8.94427 −0.334731
\(715\) 0 0
\(716\) −26.8328 −1.00279
\(717\) −12.2229 −0.456473
\(718\) −55.7771 −2.08158
\(719\) −51.4853 −1.92008 −0.960039 0.279867i \(-0.909710\pi\)
−0.960039 + 0.279867i \(0.909710\pi\)
\(720\) −2.94427 −0.109727
\(721\) −7.70820 −0.287069
\(722\) −51.1803 −1.90474
\(723\) −16.2229 −0.603337
\(724\) −76.2492 −2.83378
\(725\) 8.47214 0.314647
\(726\) 0 0
\(727\) 25.0132 0.927687 0.463843 0.885917i \(-0.346470\pi\)
0.463843 + 0.885917i \(0.346470\pi\)
\(728\) −7.23607 −0.268187
\(729\) 24.0557 0.890953
\(730\) 3.41641 0.126447
\(731\) −25.8885 −0.957522
\(732\) −26.8328 −0.991769
\(733\) −8.76393 −0.323703 −0.161852 0.986815i \(-0.551747\pi\)
−0.161852 + 0.986815i \(0.551747\pi\)
\(734\) 51.7082 1.90858
\(735\) 2.47214 0.0911861
\(736\) 16.5836 0.611279
\(737\) 0 0
\(738\) 36.9868 1.36150
\(739\) −24.9443 −0.917590 −0.458795 0.888542i \(-0.651719\pi\)
−0.458795 + 0.888542i \(0.651719\pi\)
\(740\) 50.8328 1.86865
\(741\) −25.8885 −0.951039
\(742\) −1.05573 −0.0387570
\(743\) −1.88854 −0.0692840 −0.0346420 0.999400i \(-0.511029\pi\)
−0.0346420 + 0.999400i \(0.511029\pi\)
\(744\) −7.63932 −0.280071
\(745\) 28.0000 1.02584
\(746\) 13.4164 0.491210
\(747\) −16.8065 −0.614918
\(748\) 0 0
\(749\) −4.00000 −0.146157
\(750\) 33.1672 1.21109
\(751\) 29.5279 1.07749 0.538744 0.842470i \(-0.318899\pi\)
0.538744 + 0.842470i \(0.318899\pi\)
\(752\) −2.76393 −0.100790
\(753\) −5.30495 −0.193323
\(754\) −61.3050 −2.23259
\(755\) 17.8885 0.651031
\(756\) −16.5836 −0.603139
\(757\) 15.8885 0.577479 0.288739 0.957408i \(-0.406764\pi\)
0.288739 + 0.957408i \(0.406764\pi\)
\(758\) −83.4164 −3.02982
\(759\) 0 0
\(760\) −28.9443 −1.04992
\(761\) 31.5967 1.14538 0.572691 0.819772i \(-0.305900\pi\)
0.572691 + 0.819772i \(0.305900\pi\)
\(762\) −8.44582 −0.305960
\(763\) −4.47214 −0.161902
\(764\) −9.16718 −0.331657
\(765\) 9.52786 0.344481
\(766\) 10.4033 0.375885
\(767\) 4.00000 0.144432
\(768\) 11.1246 0.401425
\(769\) −18.2918 −0.659619 −0.329810 0.944047i \(-0.606985\pi\)
−0.329810 + 0.944047i \(0.606985\pi\)
\(770\) 0 0
\(771\) 7.41641 0.267095
\(772\) −35.6656 −1.28363
\(773\) −38.3607 −1.37974 −0.689869 0.723934i \(-0.742332\pi\)
−0.689869 + 0.723934i \(0.742332\pi\)
\(774\) −26.3344 −0.946569
\(775\) 2.76393 0.0992834
\(776\) −38.9443 −1.39802
\(777\) −10.4721 −0.375686
\(778\) −35.5279 −1.27374
\(779\) −72.7214 −2.60551
\(780\) −24.0000 −0.859338
\(781\) 0 0
\(782\) −17.8885 −0.639693
\(783\) −46.8328 −1.67367
\(784\) −1.00000 −0.0357143
\(785\) −21.8885 −0.781236
\(786\) −60.4984 −2.15791
\(787\) 43.4164 1.54763 0.773814 0.633413i \(-0.218347\pi\)
0.773814 + 0.633413i \(0.218347\pi\)
\(788\) 6.00000 0.213741
\(789\) 0 0
\(790\) 40.0000 1.42314
\(791\) −2.00000 −0.0711118
\(792\) 0 0
\(793\) −23.4164 −0.831541
\(794\) 80.2492 2.84794
\(795\) −1.16718 −0.0413958
\(796\) −6.54102 −0.231840
\(797\) −14.9443 −0.529353 −0.264677 0.964337i \(-0.585265\pi\)
−0.264677 + 0.964337i \(0.585265\pi\)
\(798\) 17.8885 0.633248
\(799\) 8.94427 0.316426
\(800\) −6.70820 −0.237171
\(801\) −2.94427 −0.104031
\(802\) −51.3050 −1.81164
\(803\) 0 0
\(804\) −53.6656 −1.89264
\(805\) 4.94427 0.174263
\(806\) −20.0000 −0.704470
\(807\) −16.5836 −0.583770
\(808\) −10.6525 −0.374753
\(809\) −21.0557 −0.740280 −0.370140 0.928976i \(-0.620690\pi\)
−0.370140 + 0.928976i \(0.620690\pi\)
\(810\) −10.8065 −0.379702
\(811\) 34.8328 1.22315 0.611573 0.791188i \(-0.290537\pi\)
0.611573 + 0.791188i \(0.290537\pi\)
\(812\) 25.4164 0.891941
\(813\) −12.9443 −0.453975
\(814\) 0 0
\(815\) 6.83282 0.239343
\(816\) 4.00000 0.140028
\(817\) 51.7771 1.81145
\(818\) 20.4033 0.713383
\(819\) −4.76393 −0.166465
\(820\) −67.4164 −2.35428
\(821\) −44.8328 −1.56468 −0.782338 0.622854i \(-0.785973\pi\)
−0.782338 + 0.622854i \(0.785973\pi\)
\(822\) 45.5279 1.58797
\(823\) −14.1115 −0.491894 −0.245947 0.969283i \(-0.579099\pi\)
−0.245947 + 0.969283i \(0.579099\pi\)
\(824\) −17.2361 −0.600447
\(825\) 0 0
\(826\) −2.76393 −0.0961695
\(827\) 12.9443 0.450116 0.225058 0.974345i \(-0.427743\pi\)
0.225058 + 0.974345i \(0.427743\pi\)
\(828\) −10.9180 −0.379425
\(829\) 36.8328 1.27926 0.639628 0.768684i \(-0.279088\pi\)
0.639628 + 0.768684i \(0.279088\pi\)
\(830\) 51.0557 1.77217
\(831\) −24.5836 −0.852795
\(832\) 42.0689 1.45848
\(833\) 3.23607 0.112123
\(834\) 4.22291 0.146227
\(835\) 9.88854 0.342207
\(836\) 0 0
\(837\) −15.2786 −0.528107
\(838\) −55.1246 −1.90425
\(839\) 44.0689 1.52143 0.760713 0.649088i \(-0.224849\pi\)
0.760713 + 0.649088i \(0.224849\pi\)
\(840\) 5.52786 0.190729
\(841\) 42.7771 1.47507
\(842\) −50.0000 −1.72311
\(843\) −4.36068 −0.150190
\(844\) 41.6656 1.43419
\(845\) 5.05573 0.173922
\(846\) 9.09830 0.312806
\(847\) 0 0
\(848\) 0.472136 0.0162132
\(849\) 36.9443 1.26792
\(850\) 7.23607 0.248195
\(851\) −20.9443 −0.717960
\(852\) 38.8328 1.33039
\(853\) 30.6525 1.04952 0.524760 0.851250i \(-0.324155\pi\)
0.524760 + 0.851250i \(0.324155\pi\)
\(854\) 16.1803 0.553680
\(855\) −19.0557 −0.651692
\(856\) −8.94427 −0.305709
\(857\) −15.2361 −0.520454 −0.260227 0.965547i \(-0.583797\pi\)
−0.260227 + 0.965547i \(0.583797\pi\)
\(858\) 0 0
\(859\) −26.5410 −0.905568 −0.452784 0.891620i \(-0.649569\pi\)
−0.452784 + 0.891620i \(0.649569\pi\)
\(860\) 48.0000 1.63679
\(861\) 13.8885 0.473320
\(862\) 26.8328 0.913929
\(863\) −3.05573 −0.104018 −0.0520091 0.998647i \(-0.516562\pi\)
−0.0520091 + 0.998647i \(0.516562\pi\)
\(864\) 37.0820 1.26156
\(865\) −25.5279 −0.867973
\(866\) 18.9443 0.643753
\(867\) 8.06888 0.274034
\(868\) 8.29180 0.281442
\(869\) 0 0
\(870\) 46.8328 1.58778
\(871\) −46.8328 −1.58687
\(872\) −10.0000 −0.338643
\(873\) −25.6393 −0.867760
\(874\) 35.7771 1.21018
\(875\) −12.0000 −0.405674
\(876\) −2.83282 −0.0957120
\(877\) −14.5836 −0.492453 −0.246226 0.969212i \(-0.579191\pi\)
−0.246226 + 0.969212i \(0.579191\pi\)
\(878\) 23.4164 0.790265
\(879\) −31.0557 −1.04748
\(880\) 0 0
\(881\) 2.58359 0.0870434 0.0435217 0.999052i \(-0.486142\pi\)
0.0435217 + 0.999052i \(0.486142\pi\)
\(882\) 3.29180 0.110841
\(883\) −8.94427 −0.300999 −0.150499 0.988610i \(-0.548088\pi\)
−0.150499 + 0.988610i \(0.548088\pi\)
\(884\) −31.4164 −1.05665
\(885\) −3.05573 −0.102717
\(886\) 55.7771 1.87387
\(887\) 4.36068 0.146417 0.0732086 0.997317i \(-0.476676\pi\)
0.0732086 + 0.997317i \(0.476676\pi\)
\(888\) −23.4164 −0.785803
\(889\) 3.05573 0.102486
\(890\) 8.94427 0.299813
\(891\) 0 0
\(892\) −30.5410 −1.02259
\(893\) −17.8885 −0.598617
\(894\) −38.6950 −1.29416
\(895\) 17.8885 0.597948
\(896\) −15.6525 −0.522913
\(897\) 9.88854 0.330169
\(898\) 63.6656 2.12455
\(899\) 23.4164 0.780981
\(900\) 4.41641 0.147214
\(901\) −1.52786 −0.0509005
\(902\) 0 0
\(903\) −9.88854 −0.329070
\(904\) −4.47214 −0.148741
\(905\) 50.8328 1.68974
\(906\) −24.7214 −0.821312
\(907\) 22.4721 0.746175 0.373088 0.927796i \(-0.378299\pi\)
0.373088 + 0.927796i \(0.378299\pi\)
\(908\) −17.6656 −0.586255
\(909\) −7.01316 −0.232612
\(910\) 14.4721 0.479747
\(911\) 42.4721 1.40716 0.703582 0.710614i \(-0.251583\pi\)
0.703582 + 0.710614i \(0.251583\pi\)
\(912\) −8.00000 −0.264906
\(913\) 0 0
\(914\) −64.4721 −2.13255
\(915\) 17.8885 0.591377
\(916\) −13.4164 −0.443291
\(917\) 21.8885 0.722823
\(918\) −40.0000 −1.32020
\(919\) −41.8885 −1.38178 −0.690888 0.722962i \(-0.742780\pi\)
−0.690888 + 0.722962i \(0.742780\pi\)
\(920\) 11.0557 0.364497
\(921\) −11.0557 −0.364299
\(922\) −27.2361 −0.896972
\(923\) 33.8885 1.11546
\(924\) 0 0
\(925\) 8.47214 0.278562
\(926\) 12.3607 0.406197
\(927\) −11.3475 −0.372702
\(928\) −56.8328 −1.86563
\(929\) −52.2492 −1.71424 −0.857121 0.515116i \(-0.827749\pi\)
−0.857121 + 0.515116i \(0.827749\pi\)
\(930\) 15.2786 0.501006
\(931\) −6.47214 −0.212116
\(932\) −28.2492 −0.925334
\(933\) 10.2492 0.335545
\(934\) −53.8197 −1.76103
\(935\) 0 0
\(936\) −10.6525 −0.348187
\(937\) 10.6525 0.348001 0.174001 0.984746i \(-0.444331\pi\)
0.174001 + 0.984746i \(0.444331\pi\)
\(938\) 32.3607 1.05661
\(939\) −18.4721 −0.602815
\(940\) −16.5836 −0.540897
\(941\) −7.59675 −0.247647 −0.123823 0.992304i \(-0.539516\pi\)
−0.123823 + 0.992304i \(0.539516\pi\)
\(942\) 30.2492 0.985573
\(943\) 27.7771 0.904546
\(944\) 1.23607 0.0402306
\(945\) 11.0557 0.359643
\(946\) 0 0
\(947\) −5.16718 −0.167911 −0.0839555 0.996470i \(-0.526755\pi\)
−0.0839555 + 0.996470i \(0.526755\pi\)
\(948\) −33.1672 −1.07722
\(949\) −2.47214 −0.0802489
\(950\) −14.4721 −0.469538
\(951\) −17.3050 −0.561152
\(952\) 7.23607 0.234522
\(953\) −22.9443 −0.743238 −0.371619 0.928385i \(-0.621197\pi\)
−0.371619 + 0.928385i \(0.621197\pi\)
\(954\) −1.55418 −0.0503183
\(955\) 6.11146 0.197762
\(956\) 29.6656 0.959455
\(957\) 0 0
\(958\) 30.2492 0.977308
\(959\) −16.4721 −0.531913
\(960\) −32.1378 −1.03724
\(961\) −23.3607 −0.753570
\(962\) −61.3050 −1.97655
\(963\) −5.88854 −0.189756
\(964\) 39.3738 1.26815
\(965\) 23.7771 0.765412
\(966\) −6.83282 −0.219842
\(967\) 13.8885 0.446625 0.223313 0.974747i \(-0.428313\pi\)
0.223313 + 0.974747i \(0.428313\pi\)
\(968\) 0 0
\(969\) 25.8885 0.831660
\(970\) 77.8885 2.50085
\(971\) 11.1246 0.357006 0.178503 0.983939i \(-0.442875\pi\)
0.178503 + 0.983939i \(0.442875\pi\)
\(972\) −40.7902 −1.30835
\(973\) −1.52786 −0.0489811
\(974\) 81.3050 2.60518
\(975\) −4.00000 −0.128103
\(976\) −7.23607 −0.231621
\(977\) 22.9443 0.734052 0.367026 0.930211i \(-0.380376\pi\)
0.367026 + 0.930211i \(0.380376\pi\)
\(978\) −9.44272 −0.301945
\(979\) 0 0
\(980\) −6.00000 −0.191663
\(981\) −6.58359 −0.210198
\(982\) 0 0
\(983\) 21.8197 0.695939 0.347970 0.937506i \(-0.386871\pi\)
0.347970 + 0.937506i \(0.386871\pi\)
\(984\) 31.0557 0.990020
\(985\) −4.00000 −0.127451
\(986\) 61.3050 1.95235
\(987\) 3.41641 0.108745
\(988\) 62.8328 1.99898
\(989\) −19.7771 −0.628875
\(990\) 0 0
\(991\) 54.2492 1.72328 0.861642 0.507517i \(-0.169437\pi\)
0.861642 + 0.507517i \(0.169437\pi\)
\(992\) −18.5410 −0.588678
\(993\) 17.1672 0.544784
\(994\) −23.4164 −0.742723
\(995\) 4.36068 0.138243
\(996\) −42.3344 −1.34142
\(997\) −1.34752 −0.0426765 −0.0213383 0.999772i \(-0.506793\pi\)
−0.0213383 + 0.999772i \(0.506793\pi\)
\(998\) −3.41641 −0.108145
\(999\) −46.8328 −1.48172
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 847.2.a.f.1.1 2
3.2 odd 2 7623.2.a.bl.1.2 2
7.6 odd 2 5929.2.a.m.1.1 2
11.2 odd 10 847.2.f.a.323.1 4
11.3 even 5 847.2.f.b.372.1 4
11.4 even 5 847.2.f.b.148.1 4
11.5 even 5 847.2.f.m.729.1 4
11.6 odd 10 847.2.f.a.729.1 4
11.7 odd 10 847.2.f.n.148.1 4
11.8 odd 10 847.2.f.n.372.1 4
11.9 even 5 847.2.f.m.323.1 4
11.10 odd 2 77.2.a.d.1.2 2
33.32 even 2 693.2.a.h.1.1 2
44.43 even 2 1232.2.a.m.1.2 2
55.32 even 4 1925.2.b.h.1849.3 4
55.43 even 4 1925.2.b.h.1849.2 4
55.54 odd 2 1925.2.a.r.1.1 2
77.10 even 6 539.2.e.j.177.1 4
77.32 odd 6 539.2.e.i.177.1 4
77.54 even 6 539.2.e.j.67.1 4
77.65 odd 6 539.2.e.i.67.1 4
77.76 even 2 539.2.a.f.1.2 2
88.21 odd 2 4928.2.a.bm.1.2 2
88.43 even 2 4928.2.a.bv.1.1 2
231.230 odd 2 4851.2.a.y.1.1 2
308.307 odd 2 8624.2.a.ce.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
77.2.a.d.1.2 2 11.10 odd 2
539.2.a.f.1.2 2 77.76 even 2
539.2.e.i.67.1 4 77.65 odd 6
539.2.e.i.177.1 4 77.32 odd 6
539.2.e.j.67.1 4 77.54 even 6
539.2.e.j.177.1 4 77.10 even 6
693.2.a.h.1.1 2 33.32 even 2
847.2.a.f.1.1 2 1.1 even 1 trivial
847.2.f.a.323.1 4 11.2 odd 10
847.2.f.a.729.1 4 11.6 odd 10
847.2.f.b.148.1 4 11.4 even 5
847.2.f.b.372.1 4 11.3 even 5
847.2.f.m.323.1 4 11.9 even 5
847.2.f.m.729.1 4 11.5 even 5
847.2.f.n.148.1 4 11.7 odd 10
847.2.f.n.372.1 4 11.8 odd 10
1232.2.a.m.1.2 2 44.43 even 2
1925.2.a.r.1.1 2 55.54 odd 2
1925.2.b.h.1849.2 4 55.43 even 4
1925.2.b.h.1849.3 4 55.32 even 4
4851.2.a.y.1.1 2 231.230 odd 2
4928.2.a.bm.1.2 2 88.21 odd 2
4928.2.a.bv.1.1 2 88.43 even 2
5929.2.a.m.1.1 2 7.6 odd 2
7623.2.a.bl.1.2 2 3.2 odd 2
8624.2.a.ce.1.1 2 308.307 odd 2