Properties

Label 825.2.a.n.1.3
Level $825$
Weight $2$
Character 825.1
Self dual yes
Analytic conductor $6.588$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 825.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(6.58765816676\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.148.1
Defining polynomial: \( x^{3} - x^{2} - 3x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 165)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-1.48119\) of defining polynomial
Character \(\chi\) \(=\) 825.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.67513 q^{2} -1.00000 q^{3} +5.15633 q^{4} -2.67513 q^{6} +2.80606 q^{7} +8.44358 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+2.67513 q^{2} -1.00000 q^{3} +5.15633 q^{4} -2.67513 q^{6} +2.80606 q^{7} +8.44358 q^{8} +1.00000 q^{9} -1.00000 q^{11} -5.15633 q^{12} -5.11871 q^{13} +7.50659 q^{14} +12.2750 q^{16} +4.54420 q^{17} +2.67513 q^{18} -4.57452 q^{19} -2.80606 q^{21} -2.67513 q^{22} +4.00000 q^{23} -8.44358 q^{24} -13.6932 q^{26} -1.00000 q^{27} +14.4690 q^{28} -2.38787 q^{29} -0.962389 q^{31} +15.9502 q^{32} +1.00000 q^{33} +12.1563 q^{34} +5.15633 q^{36} +1.61213 q^{37} -12.2374 q^{38} +5.11871 q^{39} -2.38787 q^{41} -7.50659 q^{42} +2.80606 q^{43} -5.15633 q^{44} +10.7005 q^{46} -4.31265 q^{47} -12.2750 q^{48} +0.873992 q^{49} -4.54420 q^{51} -26.3938 q^{52} -6.57452 q^{53} -2.67513 q^{54} +23.6932 q^{56} +4.57452 q^{57} -6.38787 q^{58} -13.2750 q^{59} +7.92478 q^{61} -2.57452 q^{62} +2.80606 q^{63} +18.1187 q^{64} +2.67513 q^{66} +10.7005 q^{67} +23.4314 q^{68} -4.00000 q^{69} -7.35026 q^{71} +8.44358 q^{72} +6.41819 q^{73} +4.31265 q^{74} -23.5877 q^{76} -2.80606 q^{77} +13.6932 q^{78} +1.35026 q^{79} +1.00000 q^{81} -6.38787 q^{82} -0.806063 q^{83} -14.4690 q^{84} +7.50659 q^{86} +2.38787 q^{87} -8.44358 q^{88} -2.96239 q^{89} -14.3634 q^{91} +20.6253 q^{92} +0.962389 q^{93} -11.5369 q^{94} -15.9502 q^{96} -9.92478 q^{97} +2.33804 q^{98} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{2} - 3 q^{3} + 5 q^{4} - 3 q^{6} + 8 q^{7} + 9 q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 3 q^{2} - 3 q^{3} + 5 q^{4} - 3 q^{6} + 8 q^{7} + 9 q^{8} + 3 q^{9} - 3 q^{11} - 5 q^{12} + 6 q^{13} + 2 q^{14} + 5 q^{16} + 4 q^{17} + 3 q^{18} - 2 q^{19} - 8 q^{21} - 3 q^{22} + 12 q^{23} - 9 q^{24} - 8 q^{26} - 3 q^{27} + 12 q^{28} - 8 q^{29} + 8 q^{31} + 11 q^{32} + 3 q^{33} + 26 q^{34} + 5 q^{36} + 4 q^{37} + 6 q^{38} - 6 q^{39} - 8 q^{41} - 2 q^{42} + 8 q^{43} - 5 q^{44} + 12 q^{46} + 8 q^{47} - 5 q^{48} + 11 q^{49} - 4 q^{51} - 26 q^{52} - 8 q^{53} - 3 q^{54} + 38 q^{56} + 2 q^{57} - 20 q^{58} - 8 q^{59} + 2 q^{61} + 4 q^{62} + 8 q^{63} + 33 q^{64} + 3 q^{66} + 12 q^{67} + 28 q^{68} - 12 q^{69} - 12 q^{71} + 9 q^{72} + 18 q^{73} - 8 q^{74} - 18 q^{76} - 8 q^{77} + 8 q^{78} - 6 q^{79} + 3 q^{81} - 20 q^{82} - 2 q^{83} - 12 q^{84} + 2 q^{86} + 8 q^{87} - 9 q^{88} + 2 q^{89} + 8 q^{91} + 20 q^{92} - 8 q^{93} - 12 q^{94} - 11 q^{96} - 8 q^{97} - 29 q^{98} - 3 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.67513 1.89160 0.945802 0.324745i \(-0.105279\pi\)
0.945802 + 0.324745i \(0.105279\pi\)
\(3\) −1.00000 −0.577350
\(4\) 5.15633 2.57816
\(5\) 0 0
\(6\) −2.67513 −1.09212
\(7\) 2.80606 1.06059 0.530296 0.847812i \(-0.322081\pi\)
0.530296 + 0.847812i \(0.322081\pi\)
\(8\) 8.44358 2.98526
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) −5.15633 −1.48850
\(13\) −5.11871 −1.41968 −0.709838 0.704365i \(-0.751232\pi\)
−0.709838 + 0.704365i \(0.751232\pi\)
\(14\) 7.50659 2.00622
\(15\) 0 0
\(16\) 12.2750 3.06876
\(17\) 4.54420 1.10213 0.551065 0.834462i \(-0.314222\pi\)
0.551065 + 0.834462i \(0.314222\pi\)
\(18\) 2.67513 0.630534
\(19\) −4.57452 −1.04947 −0.524733 0.851267i \(-0.675835\pi\)
−0.524733 + 0.851267i \(0.675835\pi\)
\(20\) 0 0
\(21\) −2.80606 −0.612333
\(22\) −2.67513 −0.570340
\(23\) 4.00000 0.834058 0.417029 0.908893i \(-0.363071\pi\)
0.417029 + 0.908893i \(0.363071\pi\)
\(24\) −8.44358 −1.72354
\(25\) 0 0
\(26\) −13.6932 −2.68546
\(27\) −1.00000 −0.192450
\(28\) 14.4690 2.73438
\(29\) −2.38787 −0.443417 −0.221708 0.975113i \(-0.571163\pi\)
−0.221708 + 0.975113i \(0.571163\pi\)
\(30\) 0 0
\(31\) −0.962389 −0.172850 −0.0864250 0.996258i \(-0.527544\pi\)
−0.0864250 + 0.996258i \(0.527544\pi\)
\(32\) 15.9502 2.81962
\(33\) 1.00000 0.174078
\(34\) 12.1563 2.08479
\(35\) 0 0
\(36\) 5.15633 0.859388
\(37\) 1.61213 0.265032 0.132516 0.991181i \(-0.457694\pi\)
0.132516 + 0.991181i \(0.457694\pi\)
\(38\) −12.2374 −1.98517
\(39\) 5.11871 0.819650
\(40\) 0 0
\(41\) −2.38787 −0.372923 −0.186462 0.982462i \(-0.559702\pi\)
−0.186462 + 0.982462i \(0.559702\pi\)
\(42\) −7.50659 −1.15829
\(43\) 2.80606 0.427921 0.213960 0.976842i \(-0.431364\pi\)
0.213960 + 0.976842i \(0.431364\pi\)
\(44\) −5.15633 −0.777345
\(45\) 0 0
\(46\) 10.7005 1.57771
\(47\) −4.31265 −0.629065 −0.314532 0.949247i \(-0.601848\pi\)
−0.314532 + 0.949247i \(0.601848\pi\)
\(48\) −12.2750 −1.77175
\(49\) 0.873992 0.124856
\(50\) 0 0
\(51\) −4.54420 −0.636315
\(52\) −26.3938 −3.66015
\(53\) −6.57452 −0.903079 −0.451540 0.892251i \(-0.649125\pi\)
−0.451540 + 0.892251i \(0.649125\pi\)
\(54\) −2.67513 −0.364039
\(55\) 0 0
\(56\) 23.6932 3.16614
\(57\) 4.57452 0.605909
\(58\) −6.38787 −0.838769
\(59\) −13.2750 −1.72826 −0.864131 0.503266i \(-0.832132\pi\)
−0.864131 + 0.503266i \(0.832132\pi\)
\(60\) 0 0
\(61\) 7.92478 1.01466 0.507332 0.861751i \(-0.330632\pi\)
0.507332 + 0.861751i \(0.330632\pi\)
\(62\) −2.57452 −0.326964
\(63\) 2.80606 0.353531
\(64\) 18.1187 2.26484
\(65\) 0 0
\(66\) 2.67513 0.329286
\(67\) 10.7005 1.30728 0.653639 0.756807i \(-0.273242\pi\)
0.653639 + 0.756807i \(0.273242\pi\)
\(68\) 23.4314 2.84147
\(69\) −4.00000 −0.481543
\(70\) 0 0
\(71\) −7.35026 −0.872316 −0.436158 0.899870i \(-0.643661\pi\)
−0.436158 + 0.899870i \(0.643661\pi\)
\(72\) 8.44358 0.995086
\(73\) 6.41819 0.751192 0.375596 0.926783i \(-0.377438\pi\)
0.375596 + 0.926783i \(0.377438\pi\)
\(74\) 4.31265 0.501335
\(75\) 0 0
\(76\) −23.5877 −2.70569
\(77\) −2.80606 −0.319781
\(78\) 13.6932 1.55045
\(79\) 1.35026 0.151916 0.0759582 0.997111i \(-0.475798\pi\)
0.0759582 + 0.997111i \(0.475798\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) −6.38787 −0.705423
\(83\) −0.806063 −0.0884770 −0.0442385 0.999021i \(-0.514086\pi\)
−0.0442385 + 0.999021i \(0.514086\pi\)
\(84\) −14.4690 −1.57869
\(85\) 0 0
\(86\) 7.50659 0.809456
\(87\) 2.38787 0.256007
\(88\) −8.44358 −0.900089
\(89\) −2.96239 −0.314013 −0.157006 0.987598i \(-0.550184\pi\)
−0.157006 + 0.987598i \(0.550184\pi\)
\(90\) 0 0
\(91\) −14.3634 −1.50570
\(92\) 20.6253 2.15034
\(93\) 0.962389 0.0997950
\(94\) −11.5369 −1.18994
\(95\) 0 0
\(96\) −15.9502 −1.62791
\(97\) −9.92478 −1.00771 −0.503854 0.863789i \(-0.668085\pi\)
−0.503854 + 0.863789i \(0.668085\pi\)
\(98\) 2.33804 0.236178
\(99\) −1.00000 −0.100504
\(100\) 0 0
\(101\) −13.6121 −1.35446 −0.677229 0.735773i \(-0.736819\pi\)
−0.677229 + 0.735773i \(0.736819\pi\)
\(102\) −12.1563 −1.20366
\(103\) 16.3127 1.60733 0.803667 0.595080i \(-0.202880\pi\)
0.803667 + 0.595080i \(0.202880\pi\)
\(104\) −43.2203 −4.23810
\(105\) 0 0
\(106\) −17.5877 −1.70827
\(107\) −9.43136 −0.911764 −0.455882 0.890040i \(-0.650676\pi\)
−0.455882 + 0.890040i \(0.650676\pi\)
\(108\) −5.15633 −0.496168
\(109\) −15.4010 −1.47515 −0.737576 0.675264i \(-0.764030\pi\)
−0.737576 + 0.675264i \(0.764030\pi\)
\(110\) 0 0
\(111\) −1.61213 −0.153016
\(112\) 34.4445 3.25470
\(113\) −13.7381 −1.29238 −0.646188 0.763179i \(-0.723638\pi\)
−0.646188 + 0.763179i \(0.723638\pi\)
\(114\) 12.2374 1.14614
\(115\) 0 0
\(116\) −12.3127 −1.14320
\(117\) −5.11871 −0.473225
\(118\) −35.5125 −3.26919
\(119\) 12.7513 1.16891
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 21.1998 1.91934
\(123\) 2.38787 0.215307
\(124\) −4.96239 −0.445636
\(125\) 0 0
\(126\) 7.50659 0.668740
\(127\) 12.4182 1.10194 0.550968 0.834526i \(-0.314259\pi\)
0.550968 + 0.834526i \(0.314259\pi\)
\(128\) 16.5696 1.46456
\(129\) −2.80606 −0.247060
\(130\) 0 0
\(131\) 5.92478 0.517650 0.258825 0.965924i \(-0.416665\pi\)
0.258825 + 0.965924i \(0.416665\pi\)
\(132\) 5.15633 0.448800
\(133\) −12.8364 −1.11306
\(134\) 28.6253 2.47285
\(135\) 0 0
\(136\) 38.3693 3.29014
\(137\) 9.79877 0.837165 0.418583 0.908179i \(-0.362527\pi\)
0.418583 + 0.908179i \(0.362527\pi\)
\(138\) −10.7005 −0.910889
\(139\) −2.12601 −0.180326 −0.0901628 0.995927i \(-0.528739\pi\)
−0.0901628 + 0.995927i \(0.528739\pi\)
\(140\) 0 0
\(141\) 4.31265 0.363191
\(142\) −19.6629 −1.65007
\(143\) 5.11871 0.428048
\(144\) 12.2750 1.02292
\(145\) 0 0
\(146\) 17.1695 1.42096
\(147\) −0.873992 −0.0720856
\(148\) 8.31265 0.683296
\(149\) 8.31265 0.680999 0.340499 0.940245i \(-0.389404\pi\)
0.340499 + 0.940245i \(0.389404\pi\)
\(150\) 0 0
\(151\) −15.2750 −1.24307 −0.621533 0.783388i \(-0.713490\pi\)
−0.621533 + 0.783388i \(0.713490\pi\)
\(152\) −38.6253 −3.13293
\(153\) 4.54420 0.367377
\(154\) −7.50659 −0.604898
\(155\) 0 0
\(156\) 26.3938 2.11319
\(157\) 7.01317 0.559712 0.279856 0.960042i \(-0.409713\pi\)
0.279856 + 0.960042i \(0.409713\pi\)
\(158\) 3.61213 0.287365
\(159\) 6.57452 0.521393
\(160\) 0 0
\(161\) 11.2243 0.884595
\(162\) 2.67513 0.210178
\(163\) 6.76116 0.529575 0.264787 0.964307i \(-0.414698\pi\)
0.264787 + 0.964307i \(0.414698\pi\)
\(164\) −12.3127 −0.961456
\(165\) 0 0
\(166\) −2.15633 −0.167363
\(167\) −11.8192 −0.914600 −0.457300 0.889312i \(-0.651183\pi\)
−0.457300 + 0.889312i \(0.651183\pi\)
\(168\) −23.6932 −1.82797
\(169\) 13.2012 1.01548
\(170\) 0 0
\(171\) −4.57452 −0.349822
\(172\) 14.4690 1.10325
\(173\) 6.99271 0.531646 0.265823 0.964022i \(-0.414356\pi\)
0.265823 + 0.964022i \(0.414356\pi\)
\(174\) 6.38787 0.484263
\(175\) 0 0
\(176\) −12.2750 −0.925266
\(177\) 13.2750 0.997813
\(178\) −7.92478 −0.593987
\(179\) −2.70052 −0.201847 −0.100923 0.994894i \(-0.532180\pi\)
−0.100923 + 0.994894i \(0.532180\pi\)
\(180\) 0 0
\(181\) −21.1998 −1.57577 −0.787885 0.615822i \(-0.788824\pi\)
−0.787885 + 0.615822i \(0.788824\pi\)
\(182\) −38.4241 −2.84818
\(183\) −7.92478 −0.585816
\(184\) 33.7743 2.48988
\(185\) 0 0
\(186\) 2.57452 0.188773
\(187\) −4.54420 −0.332305
\(188\) −22.2374 −1.62183
\(189\) −2.80606 −0.204111
\(190\) 0 0
\(191\) 13.1490 0.951430 0.475715 0.879599i \(-0.342189\pi\)
0.475715 + 0.879599i \(0.342189\pi\)
\(192\) −18.1187 −1.30761
\(193\) 5.89446 0.424293 0.212146 0.977238i \(-0.431955\pi\)
0.212146 + 0.977238i \(0.431955\pi\)
\(194\) −26.5501 −1.90618
\(195\) 0 0
\(196\) 4.50659 0.321899
\(197\) 20.3938 1.45299 0.726497 0.687169i \(-0.241147\pi\)
0.726497 + 0.687169i \(0.241147\pi\)
\(198\) −2.67513 −0.190113
\(199\) 17.4010 1.23353 0.616764 0.787148i \(-0.288443\pi\)
0.616764 + 0.787148i \(0.288443\pi\)
\(200\) 0 0
\(201\) −10.7005 −0.754757
\(202\) −36.4142 −2.56210
\(203\) −6.70052 −0.470285
\(204\) −23.4314 −1.64052
\(205\) 0 0
\(206\) 43.6385 3.04044
\(207\) 4.00000 0.278019
\(208\) −62.8324 −4.35664
\(209\) 4.57452 0.316426
\(210\) 0 0
\(211\) 18.1260 1.24785 0.623923 0.781486i \(-0.285538\pi\)
0.623923 + 0.781486i \(0.285538\pi\)
\(212\) −33.9003 −2.32828
\(213\) 7.35026 0.503632
\(214\) −25.2301 −1.72470
\(215\) 0 0
\(216\) −8.44358 −0.574513
\(217\) −2.70052 −0.183323
\(218\) −41.1998 −2.79040
\(219\) −6.41819 −0.433701
\(220\) 0 0
\(221\) −23.2605 −1.56467
\(222\) −4.31265 −0.289446
\(223\) 23.6385 1.58295 0.791475 0.611202i \(-0.209314\pi\)
0.791475 + 0.611202i \(0.209314\pi\)
\(224\) 44.7572 2.99047
\(225\) 0 0
\(226\) −36.7513 −2.44466
\(227\) −1.26916 −0.0842371 −0.0421185 0.999113i \(-0.513411\pi\)
−0.0421185 + 0.999113i \(0.513411\pi\)
\(228\) 23.5877 1.56213
\(229\) 16.1768 1.06899 0.534496 0.845171i \(-0.320501\pi\)
0.534496 + 0.845171i \(0.320501\pi\)
\(230\) 0 0
\(231\) 2.80606 0.184625
\(232\) −20.1622 −1.32371
\(233\) −18.4690 −1.20994 −0.604971 0.796247i \(-0.706815\pi\)
−0.604971 + 0.796247i \(0.706815\pi\)
\(234\) −13.6932 −0.895154
\(235\) 0 0
\(236\) −68.4504 −4.45574
\(237\) −1.35026 −0.0877089
\(238\) 34.1114 2.21111
\(239\) 19.3258 1.25008 0.625042 0.780591i \(-0.285082\pi\)
0.625042 + 0.780591i \(0.285082\pi\)
\(240\) 0 0
\(241\) 28.5501 1.83907 0.919536 0.393006i \(-0.128565\pi\)
0.919536 + 0.393006i \(0.128565\pi\)
\(242\) 2.67513 0.171964
\(243\) −1.00000 −0.0641500
\(244\) 40.8627 2.61597
\(245\) 0 0
\(246\) 6.38787 0.407276
\(247\) 23.4156 1.48990
\(248\) −8.12601 −0.516002
\(249\) 0.806063 0.0510822
\(250\) 0 0
\(251\) 23.1998 1.46436 0.732180 0.681112i \(-0.238503\pi\)
0.732180 + 0.681112i \(0.238503\pi\)
\(252\) 14.4690 0.911460
\(253\) −4.00000 −0.251478
\(254\) 33.2203 2.08443
\(255\) 0 0
\(256\) 8.08840 0.505525
\(257\) −10.8872 −0.679123 −0.339561 0.940584i \(-0.610279\pi\)
−0.339561 + 0.940584i \(0.610279\pi\)
\(258\) −7.50659 −0.467340
\(259\) 4.52373 0.281091
\(260\) 0 0
\(261\) −2.38787 −0.147806
\(262\) 15.8496 0.979189
\(263\) 9.11871 0.562284 0.281142 0.959666i \(-0.409287\pi\)
0.281142 + 0.959666i \(0.409287\pi\)
\(264\) 8.44358 0.519667
\(265\) 0 0
\(266\) −34.3390 −2.10546
\(267\) 2.96239 0.181295
\(268\) 55.1754 3.37037
\(269\) −11.2995 −0.688941 −0.344471 0.938797i \(-0.611942\pi\)
−0.344471 + 0.938797i \(0.611942\pi\)
\(270\) 0 0
\(271\) 25.1998 1.53078 0.765390 0.643567i \(-0.222546\pi\)
0.765390 + 0.643567i \(0.222546\pi\)
\(272\) 55.7802 3.38217
\(273\) 14.3634 0.869315
\(274\) 26.2130 1.58358
\(275\) 0 0
\(276\) −20.6253 −1.24150
\(277\) −2.41819 −0.145295 −0.0726475 0.997358i \(-0.523145\pi\)
−0.0726475 + 0.997358i \(0.523145\pi\)
\(278\) −5.68735 −0.341105
\(279\) −0.962389 −0.0576167
\(280\) 0 0
\(281\) 30.4894 1.81885 0.909424 0.415870i \(-0.136523\pi\)
0.909424 + 0.415870i \(0.136523\pi\)
\(282\) 11.5369 0.687013
\(283\) 8.35756 0.496805 0.248403 0.968657i \(-0.420094\pi\)
0.248403 + 0.968657i \(0.420094\pi\)
\(284\) −37.9003 −2.24897
\(285\) 0 0
\(286\) 13.6932 0.809698
\(287\) −6.70052 −0.395519
\(288\) 15.9502 0.939873
\(289\) 3.64974 0.214690
\(290\) 0 0
\(291\) 9.92478 0.581801
\(292\) 33.0943 1.93670
\(293\) −23.0943 −1.34918 −0.674591 0.738192i \(-0.735680\pi\)
−0.674591 + 0.738192i \(0.735680\pi\)
\(294\) −2.33804 −0.136357
\(295\) 0 0
\(296\) 13.6121 0.791189
\(297\) 1.00000 0.0580259
\(298\) 22.2374 1.28818
\(299\) −20.4749 −1.18409
\(300\) 0 0
\(301\) 7.87399 0.453849
\(302\) −40.8627 −2.35139
\(303\) 13.6121 0.781996
\(304\) −56.1524 −3.22056
\(305\) 0 0
\(306\) 12.1563 0.694931
\(307\) 2.65562 0.151564 0.0757821 0.997124i \(-0.475855\pi\)
0.0757821 + 0.997124i \(0.475855\pi\)
\(308\) −14.4690 −0.824446
\(309\) −16.3127 −0.927994
\(310\) 0 0
\(311\) −15.9756 −0.905891 −0.452946 0.891538i \(-0.649627\pi\)
−0.452946 + 0.891538i \(0.649627\pi\)
\(312\) 43.2203 2.44687
\(313\) −0.0606343 −0.00342726 −0.00171363 0.999999i \(-0.500545\pi\)
−0.00171363 + 0.999999i \(0.500545\pi\)
\(314\) 18.7612 1.05875
\(315\) 0 0
\(316\) 6.96239 0.391665
\(317\) 3.81336 0.214180 0.107090 0.994249i \(-0.465847\pi\)
0.107090 + 0.994249i \(0.465847\pi\)
\(318\) 17.5877 0.986269
\(319\) 2.38787 0.133695
\(320\) 0 0
\(321\) 9.43136 0.526407
\(322\) 30.0263 1.67330
\(323\) −20.7875 −1.15665
\(324\) 5.15633 0.286463
\(325\) 0 0
\(326\) 18.0870 1.00175
\(327\) 15.4010 0.851680
\(328\) −20.1622 −1.11327
\(329\) −12.1016 −0.667181
\(330\) 0 0
\(331\) 24.2882 1.33500 0.667500 0.744609i \(-0.267364\pi\)
0.667500 + 0.744609i \(0.267364\pi\)
\(332\) −4.15633 −0.228108
\(333\) 1.61213 0.0883440
\(334\) −31.6180 −1.73006
\(335\) 0 0
\(336\) −34.4445 −1.87910
\(337\) −22.5804 −1.23003 −0.615016 0.788514i \(-0.710851\pi\)
−0.615016 + 0.788514i \(0.710851\pi\)
\(338\) 35.3150 1.92088
\(339\) 13.7381 0.746153
\(340\) 0 0
\(341\) 0.962389 0.0521163
\(342\) −12.2374 −0.661724
\(343\) −17.1900 −0.928171
\(344\) 23.6932 1.27745
\(345\) 0 0
\(346\) 18.7064 1.00566
\(347\) 20.1925 1.08399 0.541996 0.840381i \(-0.317669\pi\)
0.541996 + 0.840381i \(0.317669\pi\)
\(348\) 12.3127 0.660027
\(349\) −27.2506 −1.45869 −0.729346 0.684145i \(-0.760175\pi\)
−0.729346 + 0.684145i \(0.760175\pi\)
\(350\) 0 0
\(351\) 5.11871 0.273217
\(352\) −15.9502 −0.850147
\(353\) 5.02302 0.267349 0.133674 0.991025i \(-0.457322\pi\)
0.133674 + 0.991025i \(0.457322\pi\)
\(354\) 35.5125 1.88747
\(355\) 0 0
\(356\) −15.2750 −0.809575
\(357\) −12.7513 −0.674871
\(358\) −7.22425 −0.381814
\(359\) −18.1768 −0.959334 −0.479667 0.877450i \(-0.659243\pi\)
−0.479667 + 0.877450i \(0.659243\pi\)
\(360\) 0 0
\(361\) 1.92619 0.101379
\(362\) −56.7123 −2.98073
\(363\) −1.00000 −0.0524864
\(364\) −74.0625 −3.88193
\(365\) 0 0
\(366\) −21.1998 −1.10813
\(367\) −8.56467 −0.447072 −0.223536 0.974696i \(-0.571760\pi\)
−0.223536 + 0.974696i \(0.571760\pi\)
\(368\) 49.1002 2.55952
\(369\) −2.38787 −0.124308
\(370\) 0 0
\(371\) −18.4485 −0.957799
\(372\) 4.96239 0.257288
\(373\) 7.81924 0.404865 0.202432 0.979296i \(-0.435115\pi\)
0.202432 + 0.979296i \(0.435115\pi\)
\(374\) −12.1563 −0.628589
\(375\) 0 0
\(376\) −36.4142 −1.87792
\(377\) 12.2228 0.629508
\(378\) −7.50659 −0.386097
\(379\) 1.14903 0.0590218 0.0295109 0.999564i \(-0.490605\pi\)
0.0295109 + 0.999564i \(0.490605\pi\)
\(380\) 0 0
\(381\) −12.4182 −0.636203
\(382\) 35.1754 1.79973
\(383\) 34.0870 1.74176 0.870882 0.491493i \(-0.163549\pi\)
0.870882 + 0.491493i \(0.163549\pi\)
\(384\) −16.5696 −0.845563
\(385\) 0 0
\(386\) 15.7685 0.802593
\(387\) 2.80606 0.142640
\(388\) −51.1754 −2.59804
\(389\) 23.7743 1.20541 0.602703 0.797965i \(-0.294090\pi\)
0.602703 + 0.797965i \(0.294090\pi\)
\(390\) 0 0
\(391\) 18.1768 0.919240
\(392\) 7.37962 0.372727
\(393\) −5.92478 −0.298865
\(394\) 54.5560 2.74849
\(395\) 0 0
\(396\) −5.15633 −0.259115
\(397\) −4.15045 −0.208305 −0.104152 0.994561i \(-0.533213\pi\)
−0.104152 + 0.994561i \(0.533213\pi\)
\(398\) 46.5501 2.33334
\(399\) 12.8364 0.642623
\(400\) 0 0
\(401\) −33.9149 −1.69363 −0.846815 0.531887i \(-0.821483\pi\)
−0.846815 + 0.531887i \(0.821483\pi\)
\(402\) −28.6253 −1.42770
\(403\) 4.92619 0.245391
\(404\) −70.1886 −3.49201
\(405\) 0 0
\(406\) −17.9248 −0.889592
\(407\) −1.61213 −0.0799102
\(408\) −38.3693 −1.89956
\(409\) 8.55008 0.422774 0.211387 0.977402i \(-0.432202\pi\)
0.211387 + 0.977402i \(0.432202\pi\)
\(410\) 0 0
\(411\) −9.79877 −0.483338
\(412\) 84.1133 4.14397
\(413\) −37.2506 −1.83298
\(414\) 10.7005 0.525902
\(415\) 0 0
\(416\) −81.6444 −4.00294
\(417\) 2.12601 0.104111
\(418\) 12.2374 0.598552
\(419\) 13.2995 0.649722 0.324861 0.945762i \(-0.394682\pi\)
0.324861 + 0.945762i \(0.394682\pi\)
\(420\) 0 0
\(421\) 33.3014 1.62301 0.811505 0.584345i \(-0.198649\pi\)
0.811505 + 0.584345i \(0.198649\pi\)
\(422\) 48.4894 2.36043
\(423\) −4.31265 −0.209688
\(424\) −55.5125 −2.69592
\(425\) 0 0
\(426\) 19.6629 0.952671
\(427\) 22.2374 1.07614
\(428\) −48.6312 −2.35068
\(429\) −5.11871 −0.247134
\(430\) 0 0
\(431\) 2.07522 0.0999600 0.0499800 0.998750i \(-0.484084\pi\)
0.0499800 + 0.998750i \(0.484084\pi\)
\(432\) −12.2750 −0.590583
\(433\) −37.4010 −1.79738 −0.898690 0.438585i \(-0.855480\pi\)
−0.898690 + 0.438585i \(0.855480\pi\)
\(434\) −7.22425 −0.346775
\(435\) 0 0
\(436\) −79.4128 −3.80318
\(437\) −18.2981 −0.875315
\(438\) −17.1695 −0.820390
\(439\) −6.02444 −0.287531 −0.143765 0.989612i \(-0.545921\pi\)
−0.143765 + 0.989612i \(0.545921\pi\)
\(440\) 0 0
\(441\) 0.873992 0.0416187
\(442\) −62.2247 −2.95973
\(443\) −10.1359 −0.481569 −0.240785 0.970579i \(-0.577405\pi\)
−0.240785 + 0.970579i \(0.577405\pi\)
\(444\) −8.31265 −0.394501
\(445\) 0 0
\(446\) 63.2360 2.99431
\(447\) −8.31265 −0.393175
\(448\) 50.8423 2.40207
\(449\) −11.4861 −0.542063 −0.271032 0.962570i \(-0.587365\pi\)
−0.271032 + 0.962570i \(0.587365\pi\)
\(450\) 0 0
\(451\) 2.38787 0.112441
\(452\) −70.8383 −3.33195
\(453\) 15.2750 0.717684
\(454\) −3.39517 −0.159343
\(455\) 0 0
\(456\) 38.6253 1.80880
\(457\) −15.0435 −0.703705 −0.351852 0.936055i \(-0.614448\pi\)
−0.351852 + 0.936055i \(0.614448\pi\)
\(458\) 43.2750 2.02211
\(459\) −4.54420 −0.212105
\(460\) 0 0
\(461\) −26.2374 −1.22200 −0.610999 0.791631i \(-0.709232\pi\)
−0.610999 + 0.791631i \(0.709232\pi\)
\(462\) 7.50659 0.349238
\(463\) −5.46168 −0.253826 −0.126913 0.991914i \(-0.540507\pi\)
−0.126913 + 0.991914i \(0.540507\pi\)
\(464\) −29.3112 −1.36074
\(465\) 0 0
\(466\) −49.4069 −2.28873
\(467\) −2.70052 −0.124965 −0.0624827 0.998046i \(-0.519902\pi\)
−0.0624827 + 0.998046i \(0.519902\pi\)
\(468\) −26.3938 −1.22005
\(469\) 30.0263 1.38649
\(470\) 0 0
\(471\) −7.01317 −0.323150
\(472\) −112.089 −5.15931
\(473\) −2.80606 −0.129023
\(474\) −3.61213 −0.165910
\(475\) 0 0
\(476\) 65.7499 3.01364
\(477\) −6.57452 −0.301026
\(478\) 51.6991 2.36466
\(479\) 16.7757 0.766503 0.383252 0.923644i \(-0.374804\pi\)
0.383252 + 0.923644i \(0.374804\pi\)
\(480\) 0 0
\(481\) −8.25202 −0.376260
\(482\) 76.3752 3.47879
\(483\) −11.2243 −0.510721
\(484\) 5.15633 0.234378
\(485\) 0 0
\(486\) −2.67513 −0.121346
\(487\) −34.3996 −1.55880 −0.779398 0.626529i \(-0.784475\pi\)
−0.779398 + 0.626529i \(0.784475\pi\)
\(488\) 66.9135 3.02903
\(489\) −6.76116 −0.305750
\(490\) 0 0
\(491\) 14.9525 0.674799 0.337399 0.941362i \(-0.390453\pi\)
0.337399 + 0.941362i \(0.390453\pi\)
\(492\) 12.3127 0.555097
\(493\) −10.8510 −0.488703
\(494\) 62.6399 2.81830
\(495\) 0 0
\(496\) −11.8134 −0.530435
\(497\) −20.6253 −0.925171
\(498\) 2.15633 0.0966272
\(499\) −2.85097 −0.127627 −0.0638135 0.997962i \(-0.520326\pi\)
−0.0638135 + 0.997962i \(0.520326\pi\)
\(500\) 0 0
\(501\) 11.8192 0.528045
\(502\) 62.0625 2.76999
\(503\) 1.48024 0.0660006 0.0330003 0.999455i \(-0.489494\pi\)
0.0330003 + 0.999455i \(0.489494\pi\)
\(504\) 23.6932 1.05538
\(505\) 0 0
\(506\) −10.7005 −0.475696
\(507\) −13.2012 −0.586287
\(508\) 64.0322 2.84097
\(509\) −23.2995 −1.03273 −0.516366 0.856368i \(-0.672715\pi\)
−0.516366 + 0.856368i \(0.672715\pi\)
\(510\) 0 0
\(511\) 18.0098 0.796709
\(512\) −11.5017 −0.508306
\(513\) 4.57452 0.201970
\(514\) −29.1246 −1.28463
\(515\) 0 0
\(516\) −14.4690 −0.636961
\(517\) 4.31265 0.189670
\(518\) 12.1016 0.531712
\(519\) −6.99271 −0.306946
\(520\) 0 0
\(521\) −6.81194 −0.298437 −0.149218 0.988804i \(-0.547676\pi\)
−0.149218 + 0.988804i \(0.547676\pi\)
\(522\) −6.38787 −0.279590
\(523\) −16.0567 −0.702109 −0.351054 0.936355i \(-0.614177\pi\)
−0.351054 + 0.936355i \(0.614177\pi\)
\(524\) 30.5501 1.33459
\(525\) 0 0
\(526\) 24.3938 1.06362
\(527\) −4.37328 −0.190503
\(528\) 12.2750 0.534203
\(529\) −7.00000 −0.304348
\(530\) 0 0
\(531\) −13.2750 −0.576088
\(532\) −66.1886 −2.86964
\(533\) 12.2228 0.529430
\(534\) 7.92478 0.342939
\(535\) 0 0
\(536\) 90.3508 3.90256
\(537\) 2.70052 0.116536
\(538\) −30.2276 −1.30320
\(539\) −0.873992 −0.0376455
\(540\) 0 0
\(541\) 6.62530 0.284844 0.142422 0.989806i \(-0.454511\pi\)
0.142422 + 0.989806i \(0.454511\pi\)
\(542\) 67.4128 2.89563
\(543\) 21.1998 0.909771
\(544\) 72.4807 3.10759
\(545\) 0 0
\(546\) 38.4241 1.64440
\(547\) −5.75860 −0.246220 −0.123110 0.992393i \(-0.539287\pi\)
−0.123110 + 0.992393i \(0.539287\pi\)
\(548\) 50.5256 2.15835
\(549\) 7.92478 0.338221
\(550\) 0 0
\(551\) 10.9234 0.465351
\(552\) −33.7743 −1.43753
\(553\) 3.78892 0.161121
\(554\) −6.46898 −0.274840
\(555\) 0 0
\(556\) −10.9624 −0.464909
\(557\) 19.8700 0.841920 0.420960 0.907079i \(-0.361693\pi\)
0.420960 + 0.907079i \(0.361693\pi\)
\(558\) −2.57452 −0.108988
\(559\) −14.3634 −0.607509
\(560\) 0 0
\(561\) 4.54420 0.191856
\(562\) 81.5633 3.44054
\(563\) 5.83383 0.245866 0.122933 0.992415i \(-0.460770\pi\)
0.122933 + 0.992415i \(0.460770\pi\)
\(564\) 22.2374 0.936365
\(565\) 0 0
\(566\) 22.3576 0.939758
\(567\) 2.80606 0.117844
\(568\) −62.0625 −2.60409
\(569\) 46.7123 1.95828 0.979140 0.203185i \(-0.0651293\pi\)
0.979140 + 0.203185i \(0.0651293\pi\)
\(570\) 0 0
\(571\) −24.4241 −1.02212 −0.511058 0.859546i \(-0.670746\pi\)
−0.511058 + 0.859546i \(0.670746\pi\)
\(572\) 26.3938 1.10358
\(573\) −13.1490 −0.549309
\(574\) −17.9248 −0.748166
\(575\) 0 0
\(576\) 18.1187 0.754946
\(577\) 16.5647 0.689596 0.344798 0.938677i \(-0.387947\pi\)
0.344798 + 0.938677i \(0.387947\pi\)
\(578\) 9.76353 0.406109
\(579\) −5.89446 −0.244965
\(580\) 0 0
\(581\) −2.26187 −0.0938380
\(582\) 26.5501 1.10054
\(583\) 6.57452 0.272289
\(584\) 54.1925 2.24250
\(585\) 0 0
\(586\) −61.7802 −2.55212
\(587\) 27.3258 1.12786 0.563929 0.825823i \(-0.309289\pi\)
0.563929 + 0.825823i \(0.309289\pi\)
\(588\) −4.50659 −0.185849
\(589\) 4.40246 0.181400
\(590\) 0 0
\(591\) −20.3938 −0.838887
\(592\) 19.7889 0.813320
\(593\) −12.6048 −0.517618 −0.258809 0.965928i \(-0.583330\pi\)
−0.258809 + 0.965928i \(0.583330\pi\)
\(594\) 2.67513 0.109762
\(595\) 0 0
\(596\) 42.8627 1.75573
\(597\) −17.4010 −0.712177
\(598\) −54.7729 −2.23983
\(599\) 10.5990 0.433061 0.216531 0.976276i \(-0.430526\pi\)
0.216531 + 0.976276i \(0.430526\pi\)
\(600\) 0 0
\(601\) −22.4749 −0.916768 −0.458384 0.888754i \(-0.651572\pi\)
−0.458384 + 0.888754i \(0.651572\pi\)
\(602\) 21.0640 0.858503
\(603\) 10.7005 0.435759
\(604\) −78.7631 −3.20482
\(605\) 0 0
\(606\) 36.4142 1.47923
\(607\) 33.5183 1.36047 0.680234 0.732995i \(-0.261878\pi\)
0.680234 + 0.732995i \(0.261878\pi\)
\(608\) −72.9643 −2.95909
\(609\) 6.70052 0.271519
\(610\) 0 0
\(611\) 22.0752 0.893068
\(612\) 23.4314 0.947157
\(613\) 11.5672 0.467196 0.233598 0.972333i \(-0.424950\pi\)
0.233598 + 0.972333i \(0.424950\pi\)
\(614\) 7.10413 0.286699
\(615\) 0 0
\(616\) −23.6932 −0.954627
\(617\) −33.3357 −1.34204 −0.671022 0.741438i \(-0.734144\pi\)
−0.671022 + 0.741438i \(0.734144\pi\)
\(618\) −43.6385 −1.75540
\(619\) 4.43866 0.178405 0.0892024 0.996014i \(-0.471568\pi\)
0.0892024 + 0.996014i \(0.471568\pi\)
\(620\) 0 0
\(621\) −4.00000 −0.160514
\(622\) −42.7367 −1.71359
\(623\) −8.31265 −0.333039
\(624\) 62.8324 2.51531
\(625\) 0 0
\(626\) −0.162205 −0.00648301
\(627\) −4.57452 −0.182689
\(628\) 36.1622 1.44303
\(629\) 7.32582 0.292100
\(630\) 0 0
\(631\) −39.8496 −1.58639 −0.793193 0.608971i \(-0.791583\pi\)
−0.793193 + 0.608971i \(0.791583\pi\)
\(632\) 11.4010 0.453509
\(633\) −18.1260 −0.720444
\(634\) 10.2012 0.405143
\(635\) 0 0
\(636\) 33.9003 1.34424
\(637\) −4.47371 −0.177255
\(638\) 6.38787 0.252898
\(639\) −7.35026 −0.290772
\(640\) 0 0
\(641\) −3.88858 −0.153590 −0.0767948 0.997047i \(-0.524469\pi\)
−0.0767948 + 0.997047i \(0.524469\pi\)
\(642\) 25.2301 0.995754
\(643\) 40.9380 1.61444 0.807218 0.590254i \(-0.200972\pi\)
0.807218 + 0.590254i \(0.200972\pi\)
\(644\) 57.8759 2.28063
\(645\) 0 0
\(646\) −55.6093 −2.18792
\(647\) 1.76257 0.0692939 0.0346469 0.999400i \(-0.488969\pi\)
0.0346469 + 0.999400i \(0.488969\pi\)
\(648\) 8.44358 0.331695
\(649\) 13.2750 0.521091
\(650\) 0 0
\(651\) 2.70052 0.105842
\(652\) 34.8627 1.36533
\(653\) −7.03761 −0.275403 −0.137702 0.990474i \(-0.543971\pi\)
−0.137702 + 0.990474i \(0.543971\pi\)
\(654\) 41.1998 1.61104
\(655\) 0 0
\(656\) −29.3112 −1.14441
\(657\) 6.41819 0.250397
\(658\) −32.3733 −1.26204
\(659\) −12.6253 −0.491812 −0.245906 0.969294i \(-0.579085\pi\)
−0.245906 + 0.969294i \(0.579085\pi\)
\(660\) 0 0
\(661\) 13.2243 0.514364 0.257182 0.966363i \(-0.417206\pi\)
0.257182 + 0.966363i \(0.417206\pi\)
\(662\) 64.9741 2.52529
\(663\) 23.2605 0.903361
\(664\) −6.80606 −0.264126
\(665\) 0 0
\(666\) 4.31265 0.167112
\(667\) −9.55149 −0.369835
\(668\) −60.9438 −2.35799
\(669\) −23.6385 −0.913916
\(670\) 0 0
\(671\) −7.92478 −0.305933
\(672\) −44.7572 −1.72655
\(673\) 18.2677 0.704170 0.352085 0.935968i \(-0.385473\pi\)
0.352085 + 0.935968i \(0.385473\pi\)
\(674\) −60.4055 −2.32673
\(675\) 0 0
\(676\) 68.0698 2.61807
\(677\) −33.0191 −1.26903 −0.634513 0.772912i \(-0.718799\pi\)
−0.634513 + 0.772912i \(0.718799\pi\)
\(678\) 36.7513 1.41143
\(679\) −27.8496 −1.06877
\(680\) 0 0
\(681\) 1.26916 0.0486343
\(682\) 2.57452 0.0985833
\(683\) 30.8627 1.18093 0.590465 0.807063i \(-0.298944\pi\)
0.590465 + 0.807063i \(0.298944\pi\)
\(684\) −23.5877 −0.901898
\(685\) 0 0
\(686\) −45.9854 −1.75573
\(687\) −16.1768 −0.617183
\(688\) 34.4445 1.31319
\(689\) 33.6531 1.28208
\(690\) 0 0
\(691\) 27.6991 1.05372 0.526862 0.849951i \(-0.323369\pi\)
0.526862 + 0.849951i \(0.323369\pi\)
\(692\) 36.0567 1.37067
\(693\) −2.80606 −0.106594
\(694\) 54.0176 2.05048
\(695\) 0 0
\(696\) 20.1622 0.764246
\(697\) −10.8510 −0.411010
\(698\) −72.8989 −2.75927
\(699\) 18.4690 0.698561
\(700\) 0 0
\(701\) −38.9643 −1.47166 −0.735831 0.677166i \(-0.763208\pi\)
−0.735831 + 0.677166i \(0.763208\pi\)
\(702\) 13.6932 0.516818
\(703\) −7.37470 −0.278142
\(704\) −18.1187 −0.682875
\(705\) 0 0
\(706\) 13.4372 0.505717
\(707\) −38.1965 −1.43653
\(708\) 68.4504 2.57252
\(709\) −4.32250 −0.162335 −0.0811674 0.996700i \(-0.525865\pi\)
−0.0811674 + 0.996700i \(0.525865\pi\)
\(710\) 0 0
\(711\) 1.35026 0.0506388
\(712\) −25.0132 −0.937408
\(713\) −3.84955 −0.144167
\(714\) −34.1114 −1.27659
\(715\) 0 0
\(716\) −13.9248 −0.520393
\(717\) −19.3258 −0.721736
\(718\) −48.6253 −1.81468
\(719\) −38.3996 −1.43206 −0.716032 0.698067i \(-0.754044\pi\)
−0.716032 + 0.698067i \(0.754044\pi\)
\(720\) 0 0
\(721\) 45.7743 1.70473
\(722\) 5.15282 0.191768
\(723\) −28.5501 −1.06179
\(724\) −109.313 −4.06259
\(725\) 0 0
\(726\) −2.67513 −0.0992834
\(727\) −21.6728 −0.803798 −0.401899 0.915684i \(-0.631650\pi\)
−0.401899 + 0.915684i \(0.631650\pi\)
\(728\) −121.279 −4.49489
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 12.7513 0.471624
\(732\) −40.8627 −1.51033
\(733\) 25.0698 0.925976 0.462988 0.886365i \(-0.346777\pi\)
0.462988 + 0.886365i \(0.346777\pi\)
\(734\) −22.9116 −0.845683
\(735\) 0 0
\(736\) 63.8007 2.35172
\(737\) −10.7005 −0.394159
\(738\) −6.38787 −0.235141
\(739\) −18.9018 −0.695312 −0.347656 0.937622i \(-0.613022\pi\)
−0.347656 + 0.937622i \(0.613022\pi\)
\(740\) 0 0
\(741\) −23.4156 −0.860195
\(742\) −49.3522 −1.81178
\(743\) 43.6688 1.60205 0.801026 0.598629i \(-0.204288\pi\)
0.801026 + 0.598629i \(0.204288\pi\)
\(744\) 8.12601 0.297914
\(745\) 0 0
\(746\) 20.9175 0.765843
\(747\) −0.806063 −0.0294923
\(748\) −23.4314 −0.856736
\(749\) −26.4650 −0.967010
\(750\) 0 0
\(751\) −47.0541 −1.71703 −0.858514 0.512789i \(-0.828612\pi\)
−0.858514 + 0.512789i \(0.828612\pi\)
\(752\) −52.9380 −1.93045
\(753\) −23.1998 −0.845448
\(754\) 32.6977 1.19078
\(755\) 0 0
\(756\) −14.4690 −0.526232
\(757\) −26.9116 −0.978119 −0.489059 0.872250i \(-0.662660\pi\)
−0.489059 + 0.872250i \(0.662660\pi\)
\(758\) 3.07381 0.111646
\(759\) 4.00000 0.145191
\(760\) 0 0
\(761\) 16.6859 0.604865 0.302432 0.953171i \(-0.402201\pi\)
0.302432 + 0.953171i \(0.402201\pi\)
\(762\) −33.2203 −1.20344
\(763\) −43.2163 −1.56454
\(764\) 67.8007 2.45294
\(765\) 0 0
\(766\) 91.1871 3.29473
\(767\) 67.9511 2.45357
\(768\) −8.08840 −0.291865
\(769\) 23.2995 0.840201 0.420100 0.907478i \(-0.361995\pi\)
0.420100 + 0.907478i \(0.361995\pi\)
\(770\) 0 0
\(771\) 10.8872 0.392092
\(772\) 30.3938 1.09390
\(773\) −1.63656 −0.0588631 −0.0294316 0.999567i \(-0.509370\pi\)
−0.0294316 + 0.999567i \(0.509370\pi\)
\(774\) 7.50659 0.269819
\(775\) 0 0
\(776\) −83.8007 −3.00827
\(777\) −4.52373 −0.162288
\(778\) 63.5994 2.28015
\(779\) 10.9234 0.391370
\(780\) 0 0
\(781\) 7.35026 0.263013
\(782\) 48.6253 1.73884
\(783\) 2.38787 0.0853356
\(784\) 10.7283 0.383153
\(785\) 0 0
\(786\) −15.8496 −0.565335
\(787\) −2.09095 −0.0745344 −0.0372672 0.999305i \(-0.511865\pi\)
−0.0372672 + 0.999305i \(0.511865\pi\)
\(788\) 105.157 3.74606
\(789\) −9.11871 −0.324635
\(790\) 0 0
\(791\) −38.5501 −1.37068
\(792\) −8.44358 −0.300030
\(793\) −40.5647 −1.44049
\(794\) −11.1030 −0.394030
\(795\) 0 0
\(796\) 89.7255 3.18023
\(797\) −26.8872 −0.952392 −0.476196 0.879339i \(-0.657985\pi\)
−0.476196 + 0.879339i \(0.657985\pi\)
\(798\) 34.3390 1.21559
\(799\) −19.5975 −0.693311
\(800\) 0 0
\(801\) −2.96239 −0.104671
\(802\) −90.7269 −3.20368
\(803\) −6.41819 −0.226493
\(804\) −55.1754 −1.94589
\(805\) 0 0
\(806\) 13.1782 0.464183
\(807\) 11.2995 0.397760
\(808\) −114.935 −4.04340
\(809\) −45.4128 −1.59663 −0.798315 0.602241i \(-0.794275\pi\)
−0.798315 + 0.602241i \(0.794275\pi\)
\(810\) 0 0
\(811\) 34.3488 1.20615 0.603076 0.797684i \(-0.293942\pi\)
0.603076 + 0.797684i \(0.293942\pi\)
\(812\) −34.5501 −1.21247
\(813\) −25.1998 −0.883796
\(814\) −4.31265 −0.151158
\(815\) 0 0
\(816\) −55.7802 −1.95270
\(817\) −12.8364 −0.449088
\(818\) 22.8726 0.799721
\(819\) −14.3634 −0.501899
\(820\) 0 0
\(821\) 18.7612 0.654769 0.327384 0.944891i \(-0.393833\pi\)
0.327384 + 0.944891i \(0.393833\pi\)
\(822\) −26.2130 −0.914283
\(823\) −33.0592 −1.15237 −0.576186 0.817319i \(-0.695460\pi\)
−0.576186 + 0.817319i \(0.695460\pi\)
\(824\) 137.737 4.79830
\(825\) 0 0
\(826\) −99.6502 −3.46728
\(827\) −10.8813 −0.378379 −0.189190 0.981941i \(-0.560586\pi\)
−0.189190 + 0.981941i \(0.560586\pi\)
\(828\) 20.6253 0.716779
\(829\) 42.7221 1.48380 0.741900 0.670510i \(-0.233925\pi\)
0.741900 + 0.670510i \(0.233925\pi\)
\(830\) 0 0
\(831\) 2.41819 0.0838861
\(832\) −92.7445 −3.21534
\(833\) 3.97159 0.137608
\(834\) 5.68735 0.196937
\(835\) 0 0
\(836\) 23.5877 0.815797
\(837\) 0.962389 0.0332650
\(838\) 35.5778 1.22902
\(839\) −21.1735 −0.730989 −0.365495 0.930813i \(-0.619100\pi\)
−0.365495 + 0.930813i \(0.619100\pi\)
\(840\) 0 0
\(841\) −23.2981 −0.803381
\(842\) 89.0856 3.07009
\(843\) −30.4894 −1.05011
\(844\) 93.4636 3.21715
\(845\) 0 0
\(846\) −11.5369 −0.396647
\(847\) 2.80606 0.0964175
\(848\) −80.7024 −2.77133
\(849\) −8.35756 −0.286831
\(850\) 0 0
\(851\) 6.44851 0.221052
\(852\) 37.9003 1.29844
\(853\) 54.4709 1.86505 0.932524 0.361109i \(-0.117602\pi\)
0.932524 + 0.361109i \(0.117602\pi\)
\(854\) 59.4880 2.03564
\(855\) 0 0
\(856\) −79.6345 −2.72185
\(857\) 38.5705 1.31754 0.658772 0.752342i \(-0.271076\pi\)
0.658772 + 0.752342i \(0.271076\pi\)
\(858\) −13.6932 −0.467479
\(859\) −18.5139 −0.631685 −0.315843 0.948812i \(-0.602287\pi\)
−0.315843 + 0.948812i \(0.602287\pi\)
\(860\) 0 0
\(861\) 6.70052 0.228353
\(862\) 5.55149 0.189085
\(863\) 22.5383 0.767213 0.383607 0.923497i \(-0.374682\pi\)
0.383607 + 0.923497i \(0.374682\pi\)
\(864\) −15.9502 −0.542636
\(865\) 0 0
\(866\) −100.053 −3.39993
\(867\) −3.64974 −0.123952
\(868\) −13.9248 −0.472638
\(869\) −1.35026 −0.0458045
\(870\) 0 0
\(871\) −54.7729 −1.85591
\(872\) −130.040 −4.40371
\(873\) −9.92478 −0.335903
\(874\) −48.9497 −1.65575
\(875\) 0 0
\(876\) −33.0943 −1.11815
\(877\) 18.9419 0.639623 0.319812 0.947481i \(-0.396380\pi\)
0.319812 + 0.947481i \(0.396380\pi\)
\(878\) −16.1162 −0.543894
\(879\) 23.0943 0.778951
\(880\) 0 0
\(881\) −20.0263 −0.674705 −0.337352 0.941378i \(-0.609531\pi\)
−0.337352 + 0.941378i \(0.609531\pi\)
\(882\) 2.33804 0.0787260
\(883\) 22.2981 0.750390 0.375195 0.926946i \(-0.377576\pi\)
0.375195 + 0.926946i \(0.377576\pi\)
\(884\) −119.938 −4.03397
\(885\) 0 0
\(886\) −27.1147 −0.910938
\(887\) 3.10413 0.104226 0.0521132 0.998641i \(-0.483404\pi\)
0.0521132 + 0.998641i \(0.483404\pi\)
\(888\) −13.6121 −0.456793
\(889\) 34.8462 1.16871
\(890\) 0 0
\(891\) −1.00000 −0.0335013
\(892\) 121.888 4.08110
\(893\) 19.7283 0.660182
\(894\) −22.2374 −0.743731
\(895\) 0 0
\(896\) 46.4953 1.55330
\(897\) 20.4749 0.683636
\(898\) −30.7269 −1.02537
\(899\) 2.29806 0.0766447
\(900\) 0 0
\(901\) −29.8759 −0.995311
\(902\) 6.38787 0.212693
\(903\) −7.87399 −0.262030
\(904\) −115.999 −3.85807
\(905\) 0 0
\(906\) 40.8627 1.35757
\(907\) 10.8627 0.360691 0.180345 0.983603i \(-0.442278\pi\)
0.180345 + 0.983603i \(0.442278\pi\)
\(908\) −6.54420 −0.217177
\(909\) −13.6121 −0.451486
\(910\) 0 0
\(911\) −5.82321 −0.192931 −0.0964657 0.995336i \(-0.530754\pi\)
−0.0964657 + 0.995336i \(0.530754\pi\)
\(912\) 56.1524 1.85939
\(913\) 0.806063 0.0266768
\(914\) −40.2433 −1.33113
\(915\) 0 0
\(916\) 83.4128 2.75604
\(917\) 16.6253 0.549016
\(918\) −12.1563 −0.401219
\(919\) −31.2750 −1.03167 −0.515834 0.856688i \(-0.672518\pi\)
−0.515834 + 0.856688i \(0.672518\pi\)
\(920\) 0 0
\(921\) −2.65562 −0.0875056
\(922\) −70.1886 −2.31154
\(923\) 37.6239 1.23841
\(924\) 14.4690 0.475994
\(925\) 0 0
\(926\) −14.6107 −0.480138
\(927\) 16.3127 0.535778
\(928\) −38.0870 −1.25027
\(929\) −21.2243 −0.696345 −0.348173 0.937430i \(-0.613198\pi\)
−0.348173 + 0.937430i \(0.613198\pi\)
\(930\) 0 0
\(931\) −3.99809 −0.131032
\(932\) −95.2320 −3.11943
\(933\) 15.9756 0.523016
\(934\) −7.22425 −0.236385
\(935\) 0 0
\(936\) −43.2203 −1.41270
\(937\) 16.4328 0.536835 0.268418 0.963303i \(-0.413499\pi\)
0.268418 + 0.963303i \(0.413499\pi\)
\(938\) 80.3244 2.62268
\(939\) 0.0606343 0.00197873
\(940\) 0 0
\(941\) 5.86414 0.191166 0.0955828 0.995421i \(-0.469529\pi\)
0.0955828 + 0.995421i \(0.469529\pi\)
\(942\) −18.7612 −0.611272
\(943\) −9.55149 −0.311039
\(944\) −162.952 −5.30362
\(945\) 0 0
\(946\) −7.50659 −0.244060
\(947\) −38.7415 −1.25893 −0.629464 0.777030i \(-0.716726\pi\)
−0.629464 + 0.777030i \(0.716726\pi\)
\(948\) −6.96239 −0.226128
\(949\) −32.8529 −1.06645
\(950\) 0 0
\(951\) −3.81336 −0.123657
\(952\) 107.667 3.48950
\(953\) −6.09569 −0.197459 −0.0987294 0.995114i \(-0.531478\pi\)
−0.0987294 + 0.995114i \(0.531478\pi\)
\(954\) −17.5877 −0.569422
\(955\) 0 0
\(956\) 99.6502 3.22292
\(957\) −2.38787 −0.0771890
\(958\) 44.8773 1.44992
\(959\) 27.4960 0.887891
\(960\) 0 0
\(961\) −30.0738 −0.970123
\(962\) −22.0752 −0.711734
\(963\) −9.43136 −0.303921
\(964\) 147.213 4.74143
\(965\) 0 0
\(966\) −30.0263 −0.966082
\(967\) 37.9697 1.22102 0.610511 0.792008i \(-0.290964\pi\)
0.610511 + 0.792008i \(0.290964\pi\)
\(968\) 8.44358 0.271387
\(969\) 20.7875 0.667791
\(970\) 0 0
\(971\) −60.8773 −1.95365 −0.976823 0.214049i \(-0.931335\pi\)
−0.976823 + 0.214049i \(0.931335\pi\)
\(972\) −5.15633 −0.165389
\(973\) −5.96571 −0.191252
\(974\) −92.0235 −2.94862
\(975\) 0 0
\(976\) 97.2769 3.11376
\(977\) 21.3963 0.684529 0.342264 0.939604i \(-0.388806\pi\)
0.342264 + 0.939604i \(0.388806\pi\)
\(978\) −18.0870 −0.578358
\(979\) 2.96239 0.0946784
\(980\) 0 0
\(981\) −15.4010 −0.491718
\(982\) 40.0000 1.27645
\(983\) 18.8919 0.602558 0.301279 0.953536i \(-0.402586\pi\)
0.301279 + 0.953536i \(0.402586\pi\)
\(984\) 20.1622 0.642748
\(985\) 0 0
\(986\) −29.0278 −0.924432
\(987\) 12.1016 0.385197
\(988\) 120.739 3.84121
\(989\) 11.2243 0.356911
\(990\) 0 0
\(991\) −1.33567 −0.0424291 −0.0212145 0.999775i \(-0.506753\pi\)
−0.0212145 + 0.999775i \(0.506753\pi\)
\(992\) −15.3503 −0.487371
\(993\) −24.2882 −0.770763
\(994\) −55.1754 −1.75006
\(995\) 0 0
\(996\) 4.15633 0.131698
\(997\) −48.9076 −1.54892 −0.774460 0.632623i \(-0.781978\pi\)
−0.774460 + 0.632623i \(0.781978\pi\)
\(998\) −7.62672 −0.241419
\(999\) −1.61213 −0.0510054
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 825.2.a.n.1.3 3
3.2 odd 2 2475.2.a.y.1.1 3
5.2 odd 4 165.2.c.a.34.6 yes 6
5.3 odd 4 165.2.c.a.34.1 6
5.4 even 2 825.2.a.h.1.1 3
11.10 odd 2 9075.2.a.cc.1.1 3
15.2 even 4 495.2.c.d.199.1 6
15.8 even 4 495.2.c.d.199.6 6
15.14 odd 2 2475.2.a.be.1.3 3
20.3 even 4 2640.2.d.i.529.4 6
20.7 even 4 2640.2.d.i.529.1 6
55.32 even 4 1815.2.c.d.364.1 6
55.43 even 4 1815.2.c.d.364.6 6
55.54 odd 2 9075.2.a.ck.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
165.2.c.a.34.1 6 5.3 odd 4
165.2.c.a.34.6 yes 6 5.2 odd 4
495.2.c.d.199.1 6 15.2 even 4
495.2.c.d.199.6 6 15.8 even 4
825.2.a.h.1.1 3 5.4 even 2
825.2.a.n.1.3 3 1.1 even 1 trivial
1815.2.c.d.364.1 6 55.32 even 4
1815.2.c.d.364.6 6 55.43 even 4
2475.2.a.y.1.1 3 3.2 odd 2
2475.2.a.be.1.3 3 15.14 odd 2
2640.2.d.i.529.1 6 20.7 even 4
2640.2.d.i.529.4 6 20.3 even 4
9075.2.a.cc.1.1 3 11.10 odd 2
9075.2.a.ck.1.3 3 55.54 odd 2