Properties

Label 845.2.c.g.506.7
Level $845$
Weight $2$
Character 845.506
Analytic conductor $6.747$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 845 = 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 845.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.74735897080\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.22581504.2
Defining polynomial: \(x^{8} - 4 x^{7} + 5 x^{6} + 2 x^{5} - 11 x^{4} + 4 x^{3} + 20 x^{2} - 32 x + 16\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 65)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 506.7
Root \(1.20036 - 0.747754i\) of defining polynomial
Character \(\chi\) \(=\) 845.506
Dual form 845.2.c.g.506.2

$q$-expansion

\(f(q)\) \(=\) \(q+1.49551i q^{2} +0.0947876 q^{3} -0.236543 q^{4} +1.00000i q^{5} +0.141756i q^{6} +4.82684i q^{7} +2.63726i q^{8} -2.99102 q^{9} +O(q^{10})\) \(q+1.49551i q^{2} +0.0947876 q^{3} -0.236543 q^{4} +1.00000i q^{5} +0.141756i q^{6} +4.82684i q^{7} +2.63726i q^{8} -2.99102 q^{9} -1.49551 q^{10} -1.06939i q^{11} -0.0224214 q^{12} -7.21857 q^{14} +0.0947876i q^{15} -4.41713 q^{16} -3.55889 q^{17} -4.47309i q^{18} -5.73205i q^{19} -0.236543i q^{20} +0.457524i q^{21} +1.59928 q^{22} +7.08580 q^{23} +0.249980i q^{24} -1.00000 q^{25} -0.567874 q^{27} -1.14176i q^{28} +1.47309 q^{29} -0.141756 q^{30} +1.46410i q^{31} -1.33133i q^{32} -0.101365i q^{33} -5.32235i q^{34} -4.82684 q^{35} +0.707504 q^{36} -0.0253983i q^{37} +8.57233 q^{38} -2.63726 q^{40} -0.267949i q^{41} -0.684231 q^{42} -3.55889 q^{43} +0.252957i q^{44} -2.99102i q^{45} +10.5969i q^{46} +6.51793i q^{47} -0.418689 q^{48} -16.2984 q^{49} -1.49551i q^{50} -0.337339 q^{51} +0.991015 q^{53} -0.849260i q^{54} +1.06939 q^{55} -12.7296 q^{56} -0.543327i q^{57} +2.20301i q^{58} +8.72307i q^{59} -0.0224214i q^{60} +6.33734 q^{61} -2.18958 q^{62} -14.4371i q^{63} -6.84325 q^{64} +0.151592 q^{66} -5.17316i q^{67} +0.841831 q^{68} +0.671646 q^{69} -7.21857i q^{70} +7.76488i q^{71} -7.88809i q^{72} +10.1088i q^{73} +0.0379833 q^{74} -0.0947876 q^{75} +1.35588i q^{76} +5.16177 q^{77} +8.78347 q^{79} -4.41713i q^{80} +8.91922 q^{81} +0.400720 q^{82} +0.725474i q^{83} -0.108224i q^{84} -3.55889i q^{85} -5.32235i q^{86} +0.139630 q^{87} +2.82026 q^{88} +13.5065i q^{89} +4.47309 q^{90} -1.67610 q^{92} +0.138779i q^{93} -9.74761 q^{94} +5.73205 q^{95} -0.126194i q^{96} +3.43870i q^{97} -24.3743i q^{98} +3.19856i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 4q^{3} - 4q^{4} + 8q^{9} + O(q^{10}) \) \( 8q - 4q^{3} - 4q^{4} + 8q^{9} + 4q^{10} + 20q^{12} + 4q^{14} + 4q^{16} + 4q^{17} + 24q^{22} + 20q^{23} - 8q^{25} - 4q^{27} + 16q^{29} - 8q^{30} - 20q^{35} - 40q^{36} - 16q^{38} - 12q^{40} - 8q^{42} + 4q^{43} - 56q^{48} - 24q^{49} - 8q^{51} - 24q^{53} - 24q^{56} + 56q^{61} - 8q^{62} - 8q^{64} + 12q^{66} + 28q^{68} + 32q^{69} - 20q^{74} + 4q^{75} - 36q^{77} - 16q^{79} - 16q^{81} - 8q^{82} - 44q^{87} + 36q^{88} + 40q^{90} + 44q^{92} - 64q^{94} + 32q^{95} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/845\mathbb{Z}\right)^\times\).

\(n\) \(171\) \(677\)
\(\chi(n)\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.49551i 1.05748i 0.848783 + 0.528742i \(0.177336\pi\)
−0.848783 + 0.528742i \(0.822664\pi\)
\(3\) 0.0947876 0.0547256 0.0273628 0.999626i \(-0.491289\pi\)
0.0273628 + 0.999626i \(0.491289\pi\)
\(4\) −0.236543 −0.118272
\(5\) 1.00000i 0.447214i
\(6\) 0.141756i 0.0578715i
\(7\) 4.82684i 1.82437i 0.409775 + 0.912187i \(0.365607\pi\)
−0.409775 + 0.912187i \(0.634393\pi\)
\(8\) 2.63726i 0.932413i
\(9\) −2.99102 −0.997005
\(10\) −1.49551 −0.472921
\(11\) − 1.06939i − 0.322433i −0.986919 0.161217i \(-0.948458\pi\)
0.986919 0.161217i \(-0.0515417\pi\)
\(12\) −0.0224214 −0.00647249
\(13\) 0 0
\(14\) −7.21857 −1.92924
\(15\) 0.0947876i 0.0244740i
\(16\) −4.41713 −1.10428
\(17\) −3.55889 −0.863157 −0.431579 0.902075i \(-0.642043\pi\)
−0.431579 + 0.902075i \(0.642043\pi\)
\(18\) − 4.47309i − 1.05432i
\(19\) − 5.73205i − 1.31502i −0.753445 0.657511i \(-0.771609\pi\)
0.753445 0.657511i \(-0.228391\pi\)
\(20\) − 0.236543i − 0.0528927i
\(21\) 0.457524i 0.0998400i
\(22\) 1.59928 0.340968
\(23\) 7.08580 1.47749 0.738746 0.673984i \(-0.235418\pi\)
0.738746 + 0.673984i \(0.235418\pi\)
\(24\) 0.249980i 0.0510269i
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) −0.567874 −0.109287
\(28\) − 1.14176i − 0.215772i
\(29\) 1.47309 0.273545 0.136773 0.990602i \(-0.456327\pi\)
0.136773 + 0.990602i \(0.456327\pi\)
\(30\) −0.141756 −0.0258809
\(31\) 1.46410i 0.262960i 0.991319 + 0.131480i \(0.0419730\pi\)
−0.991319 + 0.131480i \(0.958027\pi\)
\(32\) − 1.33133i − 0.235348i
\(33\) − 0.101365i − 0.0176454i
\(34\) − 5.32235i − 0.912775i
\(35\) −4.82684 −0.815885
\(36\) 0.707504 0.117917
\(37\) − 0.0253983i − 0.00417545i −0.999998 0.00208772i \(-0.999335\pi\)
0.999998 0.00208772i \(-0.000664544\pi\)
\(38\) 8.57233 1.39061
\(39\) 0 0
\(40\) −2.63726 −0.416988
\(41\) − 0.267949i − 0.0418466i −0.999781 0.0209233i \(-0.993339\pi\)
0.999781 0.0209233i \(-0.00666058\pi\)
\(42\) −0.684231 −0.105579
\(43\) −3.55889 −0.542726 −0.271363 0.962477i \(-0.587474\pi\)
−0.271363 + 0.962477i \(0.587474\pi\)
\(44\) 0.252957i 0.0381347i
\(45\) − 2.99102i − 0.445874i
\(46\) 10.5969i 1.56242i
\(47\) 6.51793i 0.950738i 0.879787 + 0.475369i \(0.157685\pi\)
−0.879787 + 0.475369i \(0.842315\pi\)
\(48\) −0.418689 −0.0604326
\(49\) −16.2984 −2.32834
\(50\) − 1.49551i − 0.211497i
\(51\) −0.337339 −0.0472368
\(52\) 0 0
\(53\) 0.991015 0.136126 0.0680632 0.997681i \(-0.478318\pi\)
0.0680632 + 0.997681i \(0.478318\pi\)
\(54\) − 0.849260i − 0.115570i
\(55\) 1.06939 0.144196
\(56\) −12.7296 −1.70107
\(57\) − 0.543327i − 0.0719655i
\(58\) 2.20301i 0.289270i
\(59\) 8.72307i 1.13565i 0.823151 + 0.567823i \(0.192214\pi\)
−0.823151 + 0.567823i \(0.807786\pi\)
\(60\) − 0.0224214i − 0.00289458i
\(61\) 6.33734 0.811413 0.405707 0.914003i \(-0.367026\pi\)
0.405707 + 0.914003i \(0.367026\pi\)
\(62\) −2.18958 −0.278076
\(63\) − 14.4371i − 1.81891i
\(64\) −6.84325 −0.855406
\(65\) 0 0
\(66\) 0.151592 0.0186597
\(67\) − 5.17316i − 0.632002i −0.948759 0.316001i \(-0.897660\pi\)
0.948759 0.316001i \(-0.102340\pi\)
\(68\) 0.841831 0.102087
\(69\) 0.671646 0.0808567
\(70\) − 7.21857i − 0.862785i
\(71\) 7.76488i 0.921521i 0.887524 + 0.460761i \(0.152423\pi\)
−0.887524 + 0.460761i \(0.847577\pi\)
\(72\) − 7.88809i − 0.929621i
\(73\) 10.1088i 1.18314i 0.806252 + 0.591572i \(0.201493\pi\)
−0.806252 + 0.591572i \(0.798507\pi\)
\(74\) 0.0379833 0.00441547
\(75\) −0.0947876 −0.0109451
\(76\) 1.35588i 0.155530i
\(77\) 5.16177 0.588238
\(78\) 0 0
\(79\) 8.78347 0.988218 0.494109 0.869400i \(-0.335494\pi\)
0.494109 + 0.869400i \(0.335494\pi\)
\(80\) − 4.41713i − 0.493851i
\(81\) 8.91922 0.991024
\(82\) 0.400720 0.0442521
\(83\) 0.725474i 0.0796311i 0.999207 + 0.0398155i \(0.0126770\pi\)
−0.999207 + 0.0398155i \(0.987323\pi\)
\(84\) − 0.108224i − 0.0118082i
\(85\) − 3.55889i − 0.386016i
\(86\) − 5.32235i − 0.573923i
\(87\) 0.139630 0.0149699
\(88\) 2.82026 0.300641
\(89\) 13.5065i 1.43169i 0.698259 + 0.715845i \(0.253958\pi\)
−0.698259 + 0.715845i \(0.746042\pi\)
\(90\) 4.47309 0.471505
\(91\) 0 0
\(92\) −1.67610 −0.174745
\(93\) 0.138779i 0.0143907i
\(94\) −9.74761 −1.00539
\(95\) 5.73205 0.588096
\(96\) − 0.126194i − 0.0128796i
\(97\) 3.43870i 0.349147i 0.984644 + 0.174574i \(0.0558547\pi\)
−0.984644 + 0.174574i \(0.944145\pi\)
\(98\) − 24.3743i − 2.46218i
\(99\) 3.19856i 0.321467i
\(100\) 0.236543 0.0236543
\(101\) −2.85527 −0.284110 −0.142055 0.989859i \(-0.545371\pi\)
−0.142055 + 0.989859i \(0.545371\pi\)
\(102\) − 0.504492i − 0.0499522i
\(103\) 5.54488 0.546354 0.273177 0.961964i \(-0.411926\pi\)
0.273177 + 0.961964i \(0.411926\pi\)
\(104\) 0 0
\(105\) −0.457524 −0.0446498
\(106\) 1.48207i 0.143951i
\(107\) −4.44111 −0.429338 −0.214669 0.976687i \(-0.568867\pi\)
−0.214669 + 0.976687i \(0.568867\pi\)
\(108\) 0.134327 0.0129256
\(109\) − 13.7804i − 1.31993i −0.751298 0.659963i \(-0.770572\pi\)
0.751298 0.659963i \(-0.229428\pi\)
\(110\) 1.59928i 0.152485i
\(111\) − 0.00240744i 0 0.000228504i
\(112\) − 21.3208i − 2.01463i
\(113\) −8.04399 −0.756715 −0.378358 0.925660i \(-0.623511\pi\)
−0.378358 + 0.925660i \(0.623511\pi\)
\(114\) 0.812550 0.0761023
\(115\) 7.08580i 0.660755i
\(116\) −0.348448 −0.0323526
\(117\) 0 0
\(118\) −13.0454 −1.20093
\(119\) − 17.1782i − 1.57472i
\(120\) −0.249980 −0.0228199
\(121\) 9.85641 0.896037
\(122\) 9.47754i 0.858056i
\(123\) − 0.0253983i − 0.00229008i
\(124\) − 0.346323i − 0.0311007i
\(125\) − 1.00000i − 0.0894427i
\(126\) 21.5909 1.92347
\(127\) −0.706653 −0.0627053 −0.0313526 0.999508i \(-0.509981\pi\)
−0.0313526 + 0.999508i \(0.509981\pi\)
\(128\) − 12.8968i − 1.13993i
\(129\) −0.337339 −0.0297010
\(130\) 0 0
\(131\) 6.26554 0.547423 0.273711 0.961812i \(-0.411749\pi\)
0.273711 + 0.961812i \(0.411749\pi\)
\(132\) 0.0239772i 0.00208694i
\(133\) 27.6677 2.39909
\(134\) 7.73650 0.668332
\(135\) − 0.567874i − 0.0488748i
\(136\) − 9.38573i − 0.804820i
\(137\) 16.3058i 1.39310i 0.717509 + 0.696549i \(0.245282\pi\)
−0.717509 + 0.696549i \(0.754718\pi\)
\(138\) 1.00445i 0.0855046i
\(139\) −6.82528 −0.578913 −0.289456 0.957191i \(-0.593475\pi\)
−0.289456 + 0.957191i \(0.593475\pi\)
\(140\) 1.14176 0.0964960
\(141\) 0.617819i 0.0520297i
\(142\) −11.6124 −0.974494
\(143\) 0 0
\(144\) 13.2117 1.10098
\(145\) 1.47309i 0.122333i
\(146\) −15.1178 −1.25116
\(147\) −1.54488 −0.127420
\(148\) 0.00600778i 0 0.000493837i
\(149\) 8.43955i 0.691395i 0.938346 + 0.345698i \(0.112358\pi\)
−0.938346 + 0.345698i \(0.887642\pi\)
\(150\) − 0.141756i − 0.0115743i
\(151\) − 1.37017i − 0.111503i −0.998445 0.0557513i \(-0.982245\pi\)
0.998445 0.0557513i \(-0.0177554\pi\)
\(152\) 15.1169 1.22614
\(153\) 10.6447 0.860572
\(154\) 7.71947i 0.622052i
\(155\) −1.46410 −0.117599
\(156\) 0 0
\(157\) 11.9700 0.955311 0.477656 0.878547i \(-0.341487\pi\)
0.477656 + 0.878547i \(0.341487\pi\)
\(158\) 13.1357i 1.04502i
\(159\) 0.0939360 0.00744961
\(160\) 1.33133 0.105251
\(161\) 34.2020i 2.69550i
\(162\) 13.3388i 1.04799i
\(163\) − 22.5713i − 1.76792i −0.467559 0.883962i \(-0.654866\pi\)
0.467559 0.883962i \(-0.345134\pi\)
\(164\) 0.0633815i 0.00494927i
\(165\) 0.101365 0.00789124
\(166\) −1.08495 −0.0842085
\(167\) 8.19700i 0.634303i 0.948375 + 0.317152i \(0.102726\pi\)
−0.948375 + 0.317152i \(0.897274\pi\)
\(168\) −1.20661 −0.0930922
\(169\) 0 0
\(170\) 5.32235 0.408205
\(171\) 17.1447i 1.31108i
\(172\) 0.841831 0.0641890
\(173\) 9.16772 0.697009 0.348505 0.937307i \(-0.386690\pi\)
0.348505 + 0.937307i \(0.386690\pi\)
\(174\) 0.208818i 0.0158305i
\(175\) − 4.82684i − 0.364875i
\(176\) 4.72364i 0.356057i
\(177\) 0.826838i 0.0621490i
\(178\) −20.1991 −1.51399
\(179\) 10.0370 0.750200 0.375100 0.926984i \(-0.377608\pi\)
0.375100 + 0.926984i \(0.377608\pi\)
\(180\) 0.707504i 0.0527342i
\(181\) −17.0238 −1.26537 −0.632686 0.774408i \(-0.718048\pi\)
−0.632686 + 0.774408i \(0.718048\pi\)
\(182\) 0 0
\(183\) 0.600701 0.0444051
\(184\) 18.6871i 1.37763i
\(185\) 0.0253983 0.00186732
\(186\) −0.207545 −0.0152179
\(187\) 3.80584i 0.278310i
\(188\) − 1.54177i − 0.112445i
\(189\) − 2.74104i − 0.199381i
\(190\) 8.57233i 0.621902i
\(191\) −3.87741 −0.280559 −0.140280 0.990112i \(-0.544800\pi\)
−0.140280 + 0.990112i \(0.544800\pi\)
\(192\) −0.648655 −0.0468127
\(193\) 1.25394i 0.0902608i 0.998981 + 0.0451304i \(0.0143703\pi\)
−0.998981 + 0.0451304i \(0.985630\pi\)
\(194\) −5.14261 −0.369218
\(195\) 0 0
\(196\) 3.85527 0.275376
\(197\) 15.2820i 1.08879i 0.838828 + 0.544397i \(0.183242\pi\)
−0.838828 + 0.544397i \(0.816758\pi\)
\(198\) −4.78347 −0.339946
\(199\) 13.2296 0.937822 0.468911 0.883246i \(-0.344647\pi\)
0.468911 + 0.883246i \(0.344647\pi\)
\(200\) − 2.63726i − 0.186483i
\(201\) − 0.490352i − 0.0345867i
\(202\) − 4.27007i − 0.300441i
\(203\) 7.11035i 0.499049i
\(204\) 0.0797951 0.00558678
\(205\) 0.267949 0.0187144
\(206\) 8.29242i 0.577760i
\(207\) −21.1937 −1.47307
\(208\) 0 0
\(209\) −6.12979 −0.424007
\(210\) − 0.684231i − 0.0472164i
\(211\) −4.81042 −0.331163 −0.165582 0.986196i \(-0.552950\pi\)
−0.165582 + 0.986196i \(0.552950\pi\)
\(212\) −0.234418 −0.0160999
\(213\) 0.736014i 0.0504309i
\(214\) − 6.64172i − 0.454018i
\(215\) − 3.55889i − 0.242714i
\(216\) − 1.49763i − 0.101901i
\(217\) −7.06698 −0.479738
\(218\) 20.6088 1.39580
\(219\) 0.958188i 0.0647484i
\(220\) −0.252957 −0.0170543
\(221\) 0 0
\(222\) 0.00360034 0.000241639 0
\(223\) − 14.7132i − 0.985271i −0.870236 0.492635i \(-0.836034\pi\)
0.870236 0.492635i \(-0.163966\pi\)
\(224\) 6.42612 0.429363
\(225\) 2.99102 0.199401
\(226\) − 12.0299i − 0.800214i
\(227\) − 14.9028i − 0.989134i −0.869140 0.494567i \(-0.835327\pi\)
0.869140 0.494567i \(-0.164673\pi\)
\(228\) 0.128520i 0.00851147i
\(229\) − 19.3074i − 1.27587i −0.770092 0.637933i \(-0.779790\pi\)
0.770092 0.637933i \(-0.220210\pi\)
\(230\) −10.5969 −0.698737
\(231\) 0.489272 0.0321917
\(232\) 3.88492i 0.255057i
\(233\) 21.1937 1.38845 0.694224 0.719759i \(-0.255748\pi\)
0.694224 + 0.719759i \(0.255748\pi\)
\(234\) 0 0
\(235\) −6.51793 −0.425183
\(236\) − 2.06338i − 0.134315i
\(237\) 0.832564 0.0540808
\(238\) 25.6901 1.66524
\(239\) 14.8971i 0.963612i 0.876278 + 0.481806i \(0.160019\pi\)
−0.876278 + 0.481806i \(0.839981\pi\)
\(240\) − 0.418689i − 0.0270263i
\(241\) 9.39168i 0.604971i 0.953154 + 0.302486i \(0.0978164\pi\)
−0.953154 + 0.302486i \(0.902184\pi\)
\(242\) 14.7403i 0.947544i
\(243\) 2.54905 0.163522
\(244\) −1.49905 −0.0959671
\(245\) − 16.2984i − 1.04126i
\(246\) 0.0379833 0.00242173
\(247\) 0 0
\(248\) −3.86122 −0.245188
\(249\) 0.0687659i 0.00435786i
\(250\) 1.49551 0.0945842
\(251\) −11.3163 −0.714281 −0.357140 0.934051i \(-0.616248\pi\)
−0.357140 + 0.934051i \(0.616248\pi\)
\(252\) 3.41501i 0.215125i
\(253\) − 7.57748i − 0.476392i
\(254\) − 1.05680i − 0.0663098i
\(255\) − 0.337339i − 0.0211250i
\(256\) 5.60076 0.350047
\(257\) 26.5319 1.65502 0.827508 0.561453i \(-0.189758\pi\)
0.827508 + 0.561453i \(0.189758\pi\)
\(258\) − 0.504492i − 0.0314083i
\(259\) 0.122593 0.00761758
\(260\) 0 0
\(261\) −4.40602 −0.272726
\(262\) 9.37017i 0.578891i
\(263\) −14.1408 −0.871957 −0.435979 0.899957i \(-0.643598\pi\)
−0.435979 + 0.899957i \(0.643598\pi\)
\(264\) 0.267326 0.0164528
\(265\) 0.991015i 0.0608776i
\(266\) 41.3772i 2.53700i
\(267\) 1.28025i 0.0783502i
\(268\) 1.22368i 0.0747479i
\(269\) 24.7745 1.51053 0.755264 0.655421i \(-0.227509\pi\)
0.755264 + 0.655421i \(0.227509\pi\)
\(270\) 0.849260 0.0516843
\(271\) 18.7171i 1.13698i 0.822689 + 0.568492i \(0.192473\pi\)
−0.822689 + 0.568492i \(0.807527\pi\)
\(272\) 15.7201 0.953170
\(273\) 0 0
\(274\) −24.3854 −1.47318
\(275\) 1.06939i 0.0644866i
\(276\) −0.158873 −0.00956305
\(277\) 22.6647 1.36179 0.680893 0.732382i \(-0.261592\pi\)
0.680893 + 0.732382i \(0.261592\pi\)
\(278\) − 10.2073i − 0.612191i
\(279\) − 4.37915i − 0.262173i
\(280\) − 12.7296i − 0.760742i
\(281\) − 27.8384i − 1.66070i −0.557241 0.830351i \(-0.688140\pi\)
0.557241 0.830351i \(-0.311860\pi\)
\(282\) −0.923953 −0.0550206
\(283\) −7.92007 −0.470799 −0.235400 0.971899i \(-0.575640\pi\)
−0.235400 + 0.971899i \(0.575640\pi\)
\(284\) − 1.83673i − 0.108990i
\(285\) 0.543327 0.0321839
\(286\) 0 0
\(287\) 1.29335 0.0763439
\(288\) 3.98203i 0.234643i
\(289\) −4.33431 −0.254959
\(290\) −2.20301 −0.129365
\(291\) 0.325946i 0.0191073i
\(292\) − 2.39117i − 0.139932i
\(293\) − 0.272971i − 0.0159471i −0.999968 0.00797356i \(-0.997462\pi\)
0.999968 0.00797356i \(-0.00253809\pi\)
\(294\) − 2.31038i − 0.134744i
\(295\) −8.72307 −0.507877
\(296\) 0.0669819 0.00389324
\(297\) 0.607278i 0.0352379i
\(298\) −12.6214 −0.731139
\(299\) 0 0
\(300\) 0.0224214 0.00129450
\(301\) − 17.1782i − 0.990134i
\(302\) 2.04909 0.117912
\(303\) −0.270644 −0.0155481
\(304\) 25.3192i 1.45216i
\(305\) 6.33734i 0.362875i
\(306\) 15.9192i 0.910041i
\(307\) 6.85224i 0.391078i 0.980696 + 0.195539i \(0.0626456\pi\)
−0.980696 + 0.195539i \(0.937354\pi\)
\(308\) −1.22098 −0.0695719
\(309\) 0.525586 0.0298995
\(310\) − 2.18958i − 0.124360i
\(311\) −10.6447 −0.603605 −0.301803 0.953370i \(-0.597588\pi\)
−0.301803 + 0.953370i \(0.597588\pi\)
\(312\) 0 0
\(313\) 17.8236 1.00745 0.503724 0.863865i \(-0.331963\pi\)
0.503724 + 0.863865i \(0.331963\pi\)
\(314\) 17.9012i 1.01023i
\(315\) 14.4371 0.813441
\(316\) −2.07767 −0.116878
\(317\) − 8.17161i − 0.458963i −0.973313 0.229482i \(-0.926297\pi\)
0.973313 0.229482i \(-0.0737031\pi\)
\(318\) 0.140482i 0.00787784i
\(319\) − 1.57530i − 0.0882000i
\(320\) − 6.84325i − 0.382549i
\(321\) −0.420962 −0.0234958
\(322\) −51.1494 −2.85044
\(323\) 20.3997i 1.13507i
\(324\) −2.10978 −0.117210
\(325\) 0 0
\(326\) 33.7556 1.86955
\(327\) − 1.30621i − 0.0722338i
\(328\) 0.706653 0.0390184
\(329\) −31.4610 −1.73450
\(330\) 0.151592i 0.00834486i
\(331\) − 24.9395i − 1.37080i −0.728167 0.685400i \(-0.759627\pi\)
0.728167 0.685400i \(-0.240373\pi\)
\(332\) − 0.171606i − 0.00941809i
\(333\) 0.0759666i 0.00416294i
\(334\) −12.2587 −0.670765
\(335\) 5.17316 0.282640
\(336\) − 2.02095i − 0.110252i
\(337\) 19.6057 1.06799 0.533996 0.845487i \(-0.320690\pi\)
0.533996 + 0.845487i \(0.320690\pi\)
\(338\) 0 0
\(339\) −0.762471 −0.0414117
\(340\) 0.841831i 0.0456547i
\(341\) 1.56569 0.0847871
\(342\) −25.6400 −1.38645
\(343\) − 44.8817i − 2.42339i
\(344\) − 9.38573i − 0.506045i
\(345\) 0.671646i 0.0361602i
\(346\) 13.7104i 0.737076i
\(347\) 17.0810 0.916955 0.458478 0.888706i \(-0.348395\pi\)
0.458478 + 0.888706i \(0.348395\pi\)
\(348\) −0.0330286 −0.00177052
\(349\) − 28.3719i − 1.51871i −0.650674 0.759357i \(-0.725514\pi\)
0.650674 0.759357i \(-0.274486\pi\)
\(350\) 7.21857 0.385849
\(351\) 0 0
\(352\) −1.42371 −0.0758840
\(353\) − 21.2520i − 1.13113i −0.824704 0.565564i \(-0.808658\pi\)
0.824704 0.565564i \(-0.191342\pi\)
\(354\) −1.23654 −0.0657215
\(355\) −7.76488 −0.412117
\(356\) − 3.19488i − 0.169328i
\(357\) − 1.62828i − 0.0861776i
\(358\) 15.0104i 0.793325i
\(359\) 32.6519i 1.72330i 0.507502 + 0.861650i \(0.330569\pi\)
−0.507502 + 0.861650i \(0.669431\pi\)
\(360\) 7.88809 0.415739
\(361\) −13.8564 −0.729285
\(362\) − 25.4593i − 1.33811i
\(363\) 0.934265 0.0490362
\(364\) 0 0
\(365\) −10.1088 −0.529118
\(366\) 0.898353i 0.0469577i
\(367\) −5.91837 −0.308936 −0.154468 0.987998i \(-0.549366\pi\)
−0.154468 + 0.987998i \(0.549366\pi\)
\(368\) −31.2989 −1.63157
\(369\) 0.801440i 0.0417213i
\(370\) 0.0379833i 0.00197466i
\(371\) 4.78347i 0.248345i
\(372\) − 0.0328271i − 0.00170201i
\(373\) −13.3185 −0.689607 −0.344803 0.938675i \(-0.612054\pi\)
−0.344803 + 0.938675i \(0.612054\pi\)
\(374\) −5.69166 −0.294309
\(375\) − 0.0947876i − 0.00489481i
\(376\) −17.1895 −0.886480
\(377\) 0 0
\(378\) 4.09924 0.210842
\(379\) − 25.4186i − 1.30566i −0.757503 0.652832i \(-0.773581\pi\)
0.757503 0.652832i \(-0.226419\pi\)
\(380\) −1.35588 −0.0695550
\(381\) −0.0669819 −0.00343159
\(382\) − 5.79869i − 0.296687i
\(383\) 10.8268i 0.553226i 0.960981 + 0.276613i \(0.0892119\pi\)
−0.960981 + 0.276613i \(0.910788\pi\)
\(384\) − 1.22246i − 0.0623832i
\(385\) 5.16177i 0.263068i
\(386\) −1.87528 −0.0954493
\(387\) 10.6447 0.541100
\(388\) − 0.813402i − 0.0412942i
\(389\) 23.0370 1.16802 0.584011 0.811746i \(-0.301482\pi\)
0.584011 + 0.811746i \(0.301482\pi\)
\(390\) 0 0
\(391\) −25.2176 −1.27531
\(392\) − 42.9831i − 2.17097i
\(393\) 0.593896 0.0299581
\(394\) −22.8543 −1.15138
\(395\) 8.78347i 0.441944i
\(396\) − 0.756597i − 0.0380205i
\(397\) 21.0864i 1.05830i 0.848529 + 0.529149i \(0.177489\pi\)
−0.848529 + 0.529149i \(0.822511\pi\)
\(398\) 19.7850i 0.991731i
\(399\) 2.62255 0.131292
\(400\) 4.41713 0.220857
\(401\) − 19.7769i − 0.987611i −0.869572 0.493805i \(-0.835605\pi\)
0.869572 0.493805i \(-0.164395\pi\)
\(402\) 0.733324 0.0365749
\(403\) 0 0
\(404\) 0.675394 0.0336021
\(405\) 8.91922i 0.443200i
\(406\) −10.6336 −0.527736
\(407\) −0.0271606 −0.00134630
\(408\) − 0.889650i − 0.0440443i
\(409\) 31.8809i 1.57641i 0.615414 + 0.788204i \(0.288989\pi\)
−0.615414 + 0.788204i \(0.711011\pi\)
\(410\) 0.400720i 0.0197902i
\(411\) 1.54559i 0.0762382i
\(412\) −1.31160 −0.0646181
\(413\) −42.1048 −2.07184
\(414\) − 31.6954i − 1.55774i
\(415\) −0.725474 −0.0356121
\(416\) 0 0
\(417\) −0.646952 −0.0316814
\(418\) − 9.16715i − 0.448380i
\(419\) 30.7296 1.50124 0.750621 0.660733i \(-0.229755\pi\)
0.750621 + 0.660733i \(0.229755\pi\)
\(420\) 0.108224 0.00528080
\(421\) − 17.9820i − 0.876391i −0.898880 0.438195i \(-0.855618\pi\)
0.898880 0.438195i \(-0.144382\pi\)
\(422\) − 7.19403i − 0.350200i
\(423\) − 19.4952i − 0.947890i
\(424\) 2.61357i 0.126926i
\(425\) 3.55889 0.172631
\(426\) −1.10071 −0.0533298
\(427\) 30.5893i 1.48032i
\(428\) 1.05051 0.0507785
\(429\) 0 0
\(430\) 5.32235 0.256666
\(431\) − 4.89949i − 0.236000i −0.993014 0.118000i \(-0.962352\pi\)
0.993014 0.118000i \(-0.0376483\pi\)
\(432\) 2.50837 0.120684
\(433\) 19.2394 0.924588 0.462294 0.886727i \(-0.347026\pi\)
0.462294 + 0.886727i \(0.347026\pi\)
\(434\) − 10.5687i − 0.507315i
\(435\) 0.139630i 0.00669476i
\(436\) 3.25967i 0.156110i
\(437\) − 40.6162i − 1.94294i
\(438\) −1.43298 −0.0684703
\(439\) 8.55974 0.408534 0.204267 0.978915i \(-0.434519\pi\)
0.204267 + 0.978915i \(0.434519\pi\)
\(440\) 2.82026i 0.134451i
\(441\) 48.7487 2.32137
\(442\) 0 0
\(443\) −37.9652 −1.80378 −0.901891 0.431965i \(-0.857821\pi\)
−0.901891 + 0.431965i \(0.857821\pi\)
\(444\) 0 0.000569463i 0 2.70255e-5i
\(445\) −13.5065 −0.640271
\(446\) 22.0037 1.04191
\(447\) 0.799965i 0.0378370i
\(448\) − 33.0313i − 1.56058i
\(449\) − 26.7062i − 1.26035i −0.776455 0.630173i \(-0.782984\pi\)
0.776455 0.630173i \(-0.217016\pi\)
\(450\) 4.47309i 0.210863i
\(451\) −0.286542 −0.0134927
\(452\) 1.90275 0.0894979
\(453\) − 0.129875i − 0.00610205i
\(454\) 22.2873 1.04599
\(455\) 0 0
\(456\) 1.43290 0.0671016
\(457\) 4.27127i 0.199801i 0.994997 + 0.0999007i \(0.0318525\pi\)
−0.994997 + 0.0999007i \(0.968147\pi\)
\(458\) 28.8743 1.34921
\(459\) 2.02100 0.0943322
\(460\) − 1.67610i − 0.0781485i
\(461\) − 20.6423i − 0.961407i −0.876883 0.480704i \(-0.840381\pi\)
0.876883 0.480704i \(-0.159619\pi\)
\(462\) 0.731710i 0.0340422i
\(463\) − 32.1040i − 1.49200i −0.665947 0.745999i \(-0.731972\pi\)
0.665947 0.745999i \(-0.268028\pi\)
\(464\) −6.50682 −0.302071
\(465\) −0.138779 −0.00643571
\(466\) 31.6954i 1.46826i
\(467\) −23.3774 −1.08178 −0.540888 0.841095i \(-0.681912\pi\)
−0.540888 + 0.841095i \(0.681912\pi\)
\(468\) 0 0
\(469\) 24.9700 1.15301
\(470\) − 9.74761i − 0.449624i
\(471\) 1.13461 0.0522800
\(472\) −23.0050 −1.05889
\(473\) 3.80584i 0.174993i
\(474\) 1.24511i 0.0571896i
\(475\) 5.73205i 0.263005i
\(476\) 4.06338i 0.186245i
\(477\) −2.96414 −0.135719
\(478\) −22.2787 −1.01900
\(479\) − 5.17534i − 0.236467i −0.992986 0.118234i \(-0.962277\pi\)
0.992986 0.118234i \(-0.0377232\pi\)
\(480\) 0.126194 0.00575992
\(481\) 0 0
\(482\) −14.0453 −0.639747
\(483\) 3.24193i 0.147513i
\(484\) −2.33147 −0.105976
\(485\) −3.43870 −0.156143
\(486\) 3.81213i 0.172922i
\(487\) 30.7729i 1.39445i 0.716850 + 0.697227i \(0.245583\pi\)
−0.716850 + 0.697227i \(0.754417\pi\)
\(488\) 16.7132i 0.756572i
\(489\) − 2.13948i − 0.0967508i
\(490\) 24.3743 1.10112
\(491\) −35.7983 −1.61556 −0.807778 0.589487i \(-0.799330\pi\)
−0.807778 + 0.589487i \(0.799330\pi\)
\(492\) 0.00600778i 0 0.000270852i
\(493\) −5.24255 −0.236113
\(494\) 0 0
\(495\) −3.19856 −0.143765
\(496\) − 6.46713i − 0.290383i
\(497\) −37.4798 −1.68120
\(498\) −0.102840 −0.00460837
\(499\) − 28.8971i − 1.29361i −0.762655 0.646805i \(-0.776105\pi\)
0.762655 0.646805i \(-0.223895\pi\)
\(500\) 0.236543i 0.0105785i
\(501\) 0.776974i 0.0347126i
\(502\) − 16.9237i − 0.755340i
\(503\) −7.86321 −0.350603 −0.175302 0.984515i \(-0.556090\pi\)
−0.175302 + 0.984515i \(0.556090\pi\)
\(504\) 38.0746 1.69598
\(505\) − 2.85527i − 0.127058i
\(506\) 11.3322 0.503777
\(507\) 0 0
\(508\) 0.167154 0.00741625
\(509\) 28.0113i 1.24158i 0.783977 + 0.620790i \(0.213188\pi\)
−0.783977 + 0.620790i \(0.786812\pi\)
\(510\) 0.504492 0.0223393
\(511\) −48.7935 −2.15850
\(512\) − 17.4176i − 0.769757i
\(513\) 3.25508i 0.143715i
\(514\) 39.6787i 1.75015i
\(515\) 5.54488i 0.244337i
\(516\) 0.0797951 0.00351278
\(517\) 6.97020 0.306549
\(518\) 0.183339i 0.00805546i
\(519\) 0.868986 0.0381443
\(520\) 0 0
\(521\) −37.5609 −1.64557 −0.822786 0.568351i \(-0.807581\pi\)
−0.822786 + 0.568351i \(0.807581\pi\)
\(522\) − 6.58924i − 0.288403i
\(523\) −45.3106 −1.98129 −0.990647 0.136450i \(-0.956431\pi\)
−0.990647 + 0.136450i \(0.956431\pi\)
\(524\) −1.48207 −0.0647446
\(525\) − 0.457524i − 0.0199680i
\(526\) − 21.1476i − 0.922080i
\(527\) − 5.21058i − 0.226976i
\(528\) 0.447742i 0.0194855i
\(529\) 27.2086 1.18298
\(530\) −1.48207 −0.0643770
\(531\) − 26.0908i − 1.13225i
\(532\) −6.54460 −0.283744
\(533\) 0 0
\(534\) −1.91463 −0.0828540
\(535\) − 4.44111i − 0.192006i
\(536\) 13.6430 0.589287
\(537\) 0.951383 0.0410552
\(538\) 37.0504i 1.59736i
\(539\) 17.4293i 0.750733i
\(540\) 0.134327i 0.00578050i
\(541\) 19.7445i 0.848882i 0.905456 + 0.424441i \(0.139529\pi\)
−0.905456 + 0.424441i \(0.860471\pi\)
\(542\) −27.9916 −1.20234
\(543\) −1.61365 −0.0692483
\(544\) 4.73806i 0.203143i
\(545\) 13.7804 0.590289
\(546\) 0 0
\(547\) −11.8312 −0.505867 −0.252934 0.967484i \(-0.581395\pi\)
−0.252934 + 0.967484i \(0.581395\pi\)
\(548\) − 3.85702i − 0.164764i
\(549\) −18.9551 −0.808983
\(550\) −1.59928 −0.0681935
\(551\) − 8.44381i − 0.359718i
\(552\) 1.77131i 0.0753919i
\(553\) 42.3964i 1.80288i
\(554\) 33.8952i 1.44007i
\(555\) 0.00240744 0.000102190 0
\(556\) 1.61447 0.0684689
\(557\) − 4.04621i − 0.171443i −0.996319 0.0857217i \(-0.972680\pi\)
0.996319 0.0857217i \(-0.0273196\pi\)
\(558\) 6.54905 0.277244
\(559\) 0 0
\(560\) 21.3208 0.900968
\(561\) 0.360746i 0.0152307i
\(562\) 41.6326 1.75617
\(563\) −3.89926 −0.164334 −0.0821671 0.996619i \(-0.526184\pi\)
−0.0821671 + 0.996619i \(0.526184\pi\)
\(564\) − 0.146141i − 0.00615364i
\(565\) − 8.04399i − 0.338413i
\(566\) − 11.8445i − 0.497863i
\(567\) 43.0516i 1.80800i
\(568\) −20.4780 −0.859239
\(569\) 17.3356 0.726744 0.363372 0.931644i \(-0.381625\pi\)
0.363372 + 0.931644i \(0.381625\pi\)
\(570\) 0.812550i 0.0340340i
\(571\) −29.5118 −1.23503 −0.617515 0.786559i \(-0.711860\pi\)
−0.617515 + 0.786559i \(0.711860\pi\)
\(572\) 0 0
\(573\) −0.367530 −0.0153538
\(574\) 1.93421i 0.0807324i
\(575\) −7.08580 −0.295498
\(576\) 20.4683 0.852845
\(577\) 28.3684i 1.18099i 0.807041 + 0.590496i \(0.201068\pi\)
−0.807041 + 0.590496i \(0.798932\pi\)
\(578\) − 6.48199i − 0.269615i
\(579\) 0.118858i 0.00493958i
\(580\) − 0.348448i − 0.0144685i
\(581\) −3.50174 −0.145277
\(582\) −0.487455 −0.0202057
\(583\) − 1.05978i − 0.0438917i
\(584\) −26.6595 −1.10318
\(585\) 0 0
\(586\) 0.408230 0.0168638
\(587\) 34.3877i 1.41933i 0.704538 + 0.709667i \(0.251154\pi\)
−0.704538 + 0.709667i \(0.748846\pi\)
\(588\) 0.365432 0.0150701
\(589\) 8.39230 0.345799
\(590\) − 13.0454i − 0.537071i
\(591\) 1.44854i 0.0595850i
\(592\) 0.112187i 0.00461088i
\(593\) − 5.47612i − 0.224877i −0.993659 0.112439i \(-0.964134\pi\)
0.993659 0.112439i \(-0.0358662\pi\)
\(594\) −0.908189 −0.0372635
\(595\) 17.1782 0.704237
\(596\) − 1.99632i − 0.0817724i
\(597\) 1.25400 0.0513229
\(598\) 0 0
\(599\) 38.6039 1.57731 0.788657 0.614833i \(-0.210777\pi\)
0.788657 + 0.614833i \(0.210777\pi\)
\(600\) − 0.249980i − 0.0102054i
\(601\) 6.57896 0.268361 0.134181 0.990957i \(-0.457160\pi\)
0.134181 + 0.990957i \(0.457160\pi\)
\(602\) 25.6901 1.04705
\(603\) 15.4730i 0.630109i
\(604\) 0.324103i 0.0131876i
\(605\) 9.85641i 0.400720i
\(606\) − 0.404750i − 0.0164418i
\(607\) −16.7664 −0.680525 −0.340263 0.940330i \(-0.610516\pi\)
−0.340263 + 0.940330i \(0.610516\pi\)
\(608\) −7.63126 −0.309488
\(609\) 0.673973i 0.0273108i
\(610\) −9.47754 −0.383734
\(611\) 0 0
\(612\) −2.51793 −0.101781
\(613\) 28.7789i 1.16237i 0.813772 + 0.581184i \(0.197410\pi\)
−0.813772 + 0.581184i \(0.802590\pi\)
\(614\) −10.2476 −0.413558
\(615\) 0.0253983 0.00102416
\(616\) 13.6129i 0.548481i
\(617\) − 37.3291i − 1.50281i −0.659840 0.751406i \(-0.729376\pi\)
0.659840 0.751406i \(-0.270624\pi\)
\(618\) 0.786018i 0.0316183i
\(619\) − 12.7535i − 0.512606i −0.966597 0.256303i \(-0.917496\pi\)
0.966597 0.256303i \(-0.0825045\pi\)
\(620\) 0.346323 0.0139087
\(621\) −4.02384 −0.161471
\(622\) − 15.9192i − 0.638303i
\(623\) −65.1939 −2.61194
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 26.6553i 1.06536i
\(627\) −0.581028 −0.0232040
\(628\) −2.83143 −0.112986
\(629\) 0.0903896i 0.00360407i
\(630\) 21.5909i 0.860201i
\(631\) − 28.7242i − 1.14349i −0.820430 0.571746i \(-0.806266\pi\)
0.820430 0.571746i \(-0.193734\pi\)
\(632\) 23.1643i 0.921427i
\(633\) −0.455969 −0.0181231
\(634\) 12.2207 0.485346
\(635\) − 0.706653i − 0.0280427i
\(636\) −0.0222199 −0.000881077 0
\(637\) 0 0
\(638\) 2.35588 0.0932701
\(639\) − 23.2249i − 0.918762i
\(640\) 12.8968 0.509791
\(641\) 22.3970 0.884630 0.442315 0.896860i \(-0.354157\pi\)
0.442315 + 0.896860i \(0.354157\pi\)
\(642\) − 0.629552i − 0.0248464i
\(643\) − 14.1642i − 0.558581i −0.960207 0.279290i \(-0.909901\pi\)
0.960207 0.279290i \(-0.0900992\pi\)
\(644\) − 8.09025i − 0.318801i
\(645\) − 0.337339i − 0.0132827i
\(646\) −30.5080 −1.20032
\(647\) 23.6097 0.928193 0.464096 0.885785i \(-0.346379\pi\)
0.464096 + 0.885785i \(0.346379\pi\)
\(648\) 23.5223i 0.924044i
\(649\) 9.32835 0.366170
\(650\) 0 0
\(651\) −0.669862 −0.0262540
\(652\) 5.33910i 0.209095i
\(653\) 33.2765 1.30221 0.651105 0.758987i \(-0.274306\pi\)
0.651105 + 0.758987i \(0.274306\pi\)
\(654\) 1.95345 0.0763861
\(655\) 6.26554i 0.244815i
\(656\) 1.18357i 0.0462105i
\(657\) − 30.2356i − 1.17960i
\(658\) − 47.0502i − 1.83421i
\(659\) 23.0908 0.899491 0.449745 0.893157i \(-0.351515\pi\)
0.449745 + 0.893157i \(0.351515\pi\)
\(660\) −0.0239772 −0.000933310 0
\(661\) 13.4365i 0.522620i 0.965255 + 0.261310i \(0.0841545\pi\)
−0.965255 + 0.261310i \(0.915845\pi\)
\(662\) 37.2972 1.44960
\(663\) 0 0
\(664\) −1.91326 −0.0742491
\(665\) 27.6677i 1.07291i
\(666\) −0.113609 −0.00440224
\(667\) 10.4380 0.404161
\(668\) − 1.93895i − 0.0750200i
\(669\) − 1.39463i − 0.0539196i
\(670\) 7.73650i 0.298887i
\(671\) − 6.77708i − 0.261626i
\(672\) 0.609116 0.0234972
\(673\) 1.94524 0.0749835 0.0374918 0.999297i \(-0.488063\pi\)
0.0374918 + 0.999297i \(0.488063\pi\)
\(674\) 29.3205i 1.12938i
\(675\) 0.567874 0.0218575
\(676\) 0 0
\(677\) 24.8683 0.955768 0.477884 0.878423i \(-0.341404\pi\)
0.477884 + 0.878423i \(0.341404\pi\)
\(678\) − 1.14028i − 0.0437922i
\(679\) −16.5981 −0.636975
\(680\) 9.38573 0.359926
\(681\) − 1.41260i − 0.0541310i
\(682\) 2.34151i 0.0896610i
\(683\) 14.6221i 0.559500i 0.960073 + 0.279750i \(0.0902517\pi\)
−0.960073 + 0.279750i \(0.909748\pi\)
\(684\) − 4.05545i − 0.155064i
\(685\) −16.3058 −0.623013
\(686\) 67.1210 2.56269
\(687\) − 1.83010i − 0.0698226i
\(688\) 15.7201 0.599323
\(689\) 0 0
\(690\) −1.00445 −0.0382388
\(691\) − 3.52451i − 0.134079i −0.997750 0.0670393i \(-0.978645\pi\)
0.997750 0.0670393i \(-0.0213553\pi\)
\(692\) −2.16856 −0.0824364
\(693\) −15.4389 −0.586477
\(694\) 25.5447i 0.969665i
\(695\) − 6.82528i − 0.258898i
\(696\) 0.368242i 0.0139582i
\(697\) 0.953601i 0.0361202i
\(698\) 42.4304 1.60602
\(699\) 2.00890 0.0759837
\(700\) 1.14176i 0.0431543i
\(701\) 1.53457 0.0579599 0.0289800 0.999580i \(-0.490774\pi\)
0.0289800 + 0.999580i \(0.490774\pi\)
\(702\) 0 0
\(703\) −0.145584 −0.00549081
\(704\) 7.31810i 0.275811i
\(705\) −0.617819 −0.0232684
\(706\) 31.7825 1.19615
\(707\) − 13.7819i − 0.518322i
\(708\) − 0.195583i − 0.00735046i
\(709\) − 14.0052i − 0.525978i −0.964799 0.262989i \(-0.915292\pi\)
0.964799 0.262989i \(-0.0847083\pi\)
\(710\) − 11.6124i − 0.435807i
\(711\) −26.2715 −0.985258
\(712\) −35.6203 −1.33493
\(713\) 10.3743i 0.388522i
\(714\) 2.43510 0.0911314
\(715\) 0 0
\(716\) −2.37418 −0.0887274
\(717\) 1.41206i 0.0527343i
\(718\) −48.8312 −1.82236
\(719\) −22.4761 −0.838218 −0.419109 0.907936i \(-0.637657\pi\)
−0.419109 + 0.907936i \(0.637657\pi\)
\(720\) 13.2117i 0.492372i
\(721\) 26.7643i 0.996753i
\(722\) − 20.7224i − 0.771206i
\(723\) 0.890215i 0.0331074i
\(724\) 4.02687 0.149658
\(725\) −1.47309 −0.0547091
\(726\) 1.39720i 0.0518550i
\(727\) 10.3421 0.383566 0.191783 0.981437i \(-0.438573\pi\)
0.191783 + 0.981437i \(0.438573\pi\)
\(728\) 0 0
\(729\) −26.5160 −0.982075
\(730\) − 15.1178i − 0.559534i
\(731\) 12.6657 0.468458
\(732\) −0.142092 −0.00525186
\(733\) 27.3533i 1.01032i 0.863026 + 0.505159i \(0.168566\pi\)
−0.863026 + 0.505159i \(0.831434\pi\)
\(734\) − 8.85096i − 0.326695i
\(735\) − 1.54488i − 0.0569839i
\(736\) − 9.43355i − 0.347725i
\(737\) −5.53212 −0.203778
\(738\) −1.19856 −0.0441196
\(739\) 13.4517i 0.494827i 0.968910 + 0.247413i \(0.0795806\pi\)
−0.968910 + 0.247413i \(0.920419\pi\)
\(740\) −0.00600778 −0.000220851 0
\(741\) 0 0
\(742\) −7.15372 −0.262621
\(743\) 16.3926i 0.601388i 0.953721 + 0.300694i \(0.0972182\pi\)
−0.953721 + 0.300694i \(0.902782\pi\)
\(744\) −0.365996 −0.0134181
\(745\) −8.43955 −0.309201
\(746\) − 19.9179i − 0.729248i
\(747\) − 2.16990i − 0.0793926i
\(748\) − 0.900245i − 0.0329162i
\(749\) − 21.4365i − 0.783274i
\(750\) 0.141756 0.00517618
\(751\) 27.6655 1.00953 0.504764 0.863257i \(-0.331579\pi\)
0.504764 + 0.863257i \(0.331579\pi\)
\(752\) − 28.7906i − 1.04988i
\(753\) −1.07265 −0.0390895
\(754\) 0 0
\(755\) 1.37017 0.0498654
\(756\) 0.648373i 0.0235811i
\(757\) 22.9978 0.835870 0.417935 0.908477i \(-0.362754\pi\)
0.417935 + 0.908477i \(0.362754\pi\)
\(758\) 38.0136 1.38072
\(759\) − 0.718251i − 0.0260709i
\(760\) 15.1169i 0.548349i
\(761\) − 7.66442i − 0.277835i −0.990304 0.138918i \(-0.955638\pi\)
0.990304 0.138918i \(-0.0443623\pi\)
\(762\) − 0.100172i − 0.00362885i
\(763\) 66.5160 2.40804
\(764\) 0.917174 0.0331822
\(765\) 10.6447i 0.384860i
\(766\) −16.1916 −0.585027
\(767\) 0 0
\(768\) 0.530882 0.0191566
\(769\) 7.23095i 0.260755i 0.991464 + 0.130377i \(0.0416189\pi\)
−0.991464 + 0.130377i \(0.958381\pi\)
\(770\) −7.71947 −0.278190
\(771\) 2.51490 0.0905718
\(772\) − 0.296612i − 0.0106753i
\(773\) 33.5995i 1.20849i 0.796798 + 0.604246i \(0.206525\pi\)
−0.796798 + 0.604246i \(0.793475\pi\)
\(774\) 15.9192i 0.572204i
\(775\) − 1.46410i − 0.0525921i
\(776\) −9.06877 −0.325550
\(777\) 0.0116203 0.000416877 0
\(778\) 34.4520i 1.23516i
\(779\) −1.53590 −0.0550293
\(780\) 0 0
\(781\) 8.30368 0.297129
\(782\) − 37.7131i − 1.34862i
\(783\) −0.836527 −0.0298950
\(784\) 71.9921 2.57115
\(785\) 11.9700i 0.427228i
\(786\) 0.888175i 0.0316802i
\(787\) − 2.97168i − 0.105929i −0.998596 0.0529645i \(-0.983133\pi\)
0.998596 0.0529645i \(-0.0168670\pi\)
\(788\) − 3.61484i − 0.128773i
\(789\) −1.34037 −0.0477184
\(790\) −13.1357 −0.467349
\(791\) − 38.8270i − 1.38053i
\(792\) −8.43544 −0.299740
\(793\) 0 0
\(794\) −31.5349 −1.11913
\(795\) 0.0939360i 0.00333156i
\(796\) −3.12937 −0.110918
\(797\) 22.5751 0.799650 0.399825 0.916591i \(-0.369071\pi\)
0.399825 + 0.916591i \(0.369071\pi\)
\(798\) 3.92205i 0.138839i
\(799\) − 23.1966i − 0.820636i
\(800\) 1.33133i 0.0470696i
\(801\) − 40.3983i − 1.42740i
\(802\) 29.5765 1.04438
\(803\) 10.8102 0.381485
\(804\) 0.115989i 0.00409063i
\(805\) −34.2020 −1.20546
\(806\) 0 0
\(807\) 2.34831 0.0826646
\(808\) − 7.53009i − 0.264908i
\(809\) 13.6584 0.480204 0.240102 0.970748i \(-0.422819\pi\)
0.240102 + 0.970748i \(0.422819\pi\)
\(810\) −13.3388 −0.468676
\(811\) − 14.1147i − 0.495636i −0.968807 0.247818i \(-0.920287\pi\)
0.968807 0.247818i \(-0.0797135\pi\)
\(812\) − 1.68190i − 0.0590233i
\(813\) 1.77415i 0.0622222i
\(814\) − 0.0406189i − 0.00142369i
\(815\) 22.5713 0.790640
\(816\) 1.49007 0.0521629
\(817\) 20.3997i 0.713696i
\(818\) −47.6781 −1.66703
\(819\) 0 0
\(820\) −0.0633815 −0.00221338
\(821\) − 1.58562i − 0.0553383i −0.999617 0.0276692i \(-0.991192\pi\)
0.999617 0.0276692i \(-0.00880850\pi\)
\(822\) −2.31144 −0.0806206
\(823\) −18.5648 −0.647127 −0.323563 0.946206i \(-0.604881\pi\)
−0.323563 + 0.946206i \(0.604881\pi\)
\(824\) 14.6233i 0.509427i
\(825\) 0.101365i 0.00352907i
\(826\) − 62.9681i − 2.19094i
\(827\) 9.01023i 0.313316i 0.987653 + 0.156658i \(0.0500721\pi\)
−0.987653 + 0.156658i \(0.949928\pi\)
\(828\) 5.01324 0.174222
\(829\) −47.1177 −1.63647 −0.818233 0.574887i \(-0.805046\pi\)
−0.818233 + 0.574887i \(0.805046\pi\)
\(830\) − 1.08495i − 0.0376592i
\(831\) 2.14833 0.0745247
\(832\) 0 0
\(833\) 58.0041 2.00972
\(834\) − 0.967522i − 0.0335025i
\(835\) −8.19700 −0.283669
\(836\) 1.44996 0.0501479
\(837\) − 0.831425i − 0.0287383i
\(838\) 45.9564i 1.58754i
\(839\) − 53.4766i − 1.84622i −0.384541 0.923108i \(-0.625640\pi\)
0.384541 0.923108i \(-0.374360\pi\)
\(840\) − 1.20661i − 0.0416321i
\(841\) −26.8300 −0.925173
\(842\) 26.8923 0.926769
\(843\) − 2.63874i − 0.0908830i
\(844\) 1.13787 0.0391672
\(845\) 0 0
\(846\) 29.1553 1.00238
\(847\) 47.5753i 1.63471i
\(848\) −4.37745 −0.150322
\(849\) −0.750724 −0.0257648
\(850\) 5.32235i 0.182555i
\(851\) − 0.179967i − 0.00616919i
\(852\) − 0.174099i − 0.00596454i
\(853\) 27.7756i 0.951019i 0.879711 + 0.475510i \(0.157736\pi\)
−0.879711 + 0.475510i \(0.842264\pi\)
\(854\) −45.7465 −1.56541
\(855\) −17.1447 −0.586335
\(856\) − 11.7124i − 0.400321i
\(857\) −53.6917 −1.83407 −0.917037 0.398801i \(-0.869426\pi\)
−0.917037 + 0.398801i \(0.869426\pi\)
\(858\) 0 0
\(859\) 2.08958 0.0712955 0.0356477 0.999364i \(-0.488651\pi\)
0.0356477 + 0.999364i \(0.488651\pi\)
\(860\) 0.841831i 0.0287062i
\(861\) 0.122593 0.00417797
\(862\) 7.32722 0.249566
\(863\) − 1.75413i − 0.0597113i −0.999554 0.0298557i \(-0.990495\pi\)
0.999554 0.0298557i \(-0.00950476\pi\)
\(864\) 0.756028i 0.0257206i
\(865\) 9.16772i 0.311712i
\(866\) 28.7727i 0.977737i
\(867\) −0.410839 −0.0139528
\(868\) 1.67165 0.0567394
\(869\) − 9.39295i − 0.318634i
\(870\) −0.208818 −0.00707960
\(871\) 0 0
\(872\) 36.3426 1.23072
\(873\) − 10.2852i − 0.348102i
\(874\) 60.7418 2.05462
\(875\) 4.82684 0.163177
\(876\) − 0.226653i − 0.00765789i
\(877\) 21.5672i 0.728272i 0.931346 + 0.364136i \(0.118636\pi\)
−0.931346 + 0.364136i \(0.881364\pi\)
\(878\) 12.8012i 0.432018i
\(879\) − 0.0258742i 0 0.000872716i
\(880\) −4.72364 −0.159234
\(881\) −25.0263 −0.843158 −0.421579 0.906792i \(-0.638524\pi\)
−0.421579 + 0.906792i \(0.638524\pi\)
\(882\) 72.9040i 2.45481i
\(883\) 48.7832 1.64169 0.820843 0.571154i \(-0.193504\pi\)
0.820843 + 0.571154i \(0.193504\pi\)
\(884\) 0 0
\(885\) −0.826838 −0.0277939
\(886\) − 56.7772i − 1.90747i
\(887\) −33.7933 −1.13467 −0.567334 0.823488i \(-0.692025\pi\)
−0.567334 + 0.823488i \(0.692025\pi\)
\(888\) 0.00634905 0.000213060 0
\(889\) − 3.41090i − 0.114398i
\(890\) − 20.1991i − 0.677076i
\(891\) − 9.53812i − 0.319539i
\(892\) 3.48031i 0.116530i
\(893\) 37.3611 1.25024
\(894\) −1.19635 −0.0400121
\(895\) 10.0370i 0.335500i
\(896\) 62.2508 2.07965
\(897\) 0 0
\(898\) 39.9394 1.33279
\(899\) 2.15675i 0.0719316i
\(900\) −0.707504 −0.0235835
\(901\) −3.52691 −0.117499
\(902\) − 0.428526i − 0.0142683i
\(903\) − 1.62828i − 0.0541857i
\(904\) − 21.2141i − 0.705571i
\(905\) − 17.0238i − 0.565892i
\(906\) 0.194229 0.00645281
\(907\) 34.6270 1.14977 0.574885 0.818234i \(-0.305047\pi\)
0.574885 + 0.818234i \(0.305047\pi\)
\(908\) 3.52516i 0.116986i
\(909\) 8.54015 0.283259
\(910\) 0 0
\(911\) 31.1865 1.03326 0.516628 0.856210i \(-0.327187\pi\)
0.516628 + 0.856210i \(0.327187\pi\)
\(912\) 2.39995i 0.0794703i
\(913\) 0.775814 0.0256757
\(914\) −6.38771 −0.211287
\(915\) 0.600701i 0.0198586i
\(916\) 4.56702i 0.150899i
\(917\) 30.2428i 0.998704i
\(918\) 3.02242i 0.0997548i
\(919\) −51.9220 −1.71275 −0.856374 0.516356i \(-0.827288\pi\)
−0.856374 + 0.516356i \(0.827288\pi\)
\(920\) −18.6871 −0.616096
\(921\) 0.649507i 0.0214020i
\(922\) 30.8707 1.01667
\(923\) 0 0
\(924\) −0.115734 −0.00380736
\(925\) 0.0253983i 0 0.000835090i
\(926\) 48.0117 1.57776
\(927\) −16.5848 −0.544717
\(928\) − 1.96117i − 0.0643784i
\(929\) 20.4915i 0.672304i 0.941808 + 0.336152i \(0.109126\pi\)
−0.941808 + 0.336152i \(0.890874\pi\)
\(930\) − 0.207545i − 0.00680565i
\(931\) 93.4231i 3.06182i
\(932\) −5.01324 −0.164214
\(933\) −1.00898 −0.0330327
\(934\) − 34.9610i − 1.14396i
\(935\) −3.80584 −0.124464
\(936\) 0 0
\(937\) −39.6806 −1.29631 −0.648154 0.761510i \(-0.724459\pi\)
−0.648154 + 0.761510i \(0.724459\pi\)
\(938\) 37.3428i 1.21929i
\(939\) 1.68945 0.0551333
\(940\) 1.54177 0.0502870
\(941\) − 19.6189i − 0.639557i −0.947492 0.319779i \(-0.896391\pi\)
0.947492 0.319779i \(-0.103609\pi\)
\(942\) 1.69682i 0.0552853i
\(943\) − 1.89864i − 0.0618281i
\(944\) − 38.5309i − 1.25408i
\(945\) 2.74104 0.0891659
\(946\) −5.69166 −0.185052
\(947\) − 57.2124i − 1.85915i −0.368631 0.929576i \(-0.620173\pi\)
0.368631 0.929576i \(-0.379827\pi\)
\(948\) −0.196937 −0.00639623
\(949\) 0 0
\(950\) −8.57233 −0.278123
\(951\) − 0.774567i − 0.0251170i
\(952\) 45.3034 1.46829
\(953\) −27.4770 −0.890066 −0.445033 0.895514i \(-0.646808\pi\)
−0.445033 + 0.895514i \(0.646808\pi\)
\(954\) − 4.43290i − 0.143520i
\(955\) − 3.87741i − 0.125470i
\(956\) − 3.52380i − 0.113968i
\(957\) − 0.149319i − 0.00482680i
\(958\) 7.73976 0.250060
\(959\) −78.7055 −2.54153
\(960\) − 0.648655i − 0.0209353i
\(961\) 28.8564 0.930852
\(962\) 0 0
\(963\) 13.2834 0.428053
\(964\) − 2.22154i − 0.0715509i
\(965\) −1.25394 −0.0403659
\(966\) −4.84833 −0.155992
\(967\) − 10.3643i − 0.333293i −0.986017 0.166647i \(-0.946706\pi\)
0.986017 0.166647i \(-0.0532939\pi\)
\(968\) 25.9939i 0.835477i
\(969\) 1.93364i 0.0621175i
\(970\) − 5.14261i − 0.165119i
\(971\) 41.7515 1.33987 0.669935 0.742420i \(-0.266322\pi\)
0.669935 + 0.742420i \(0.266322\pi\)
\(972\) −0.602961 −0.0193400
\(973\) − 32.9445i − 1.05615i
\(974\) −46.0212 −1.47461
\(975\) 0 0
\(976\) −27.9929 −0.896030
\(977\) − 13.7938i − 0.441303i −0.975353 0.220652i \(-0.929182\pi\)
0.975353 0.220652i \(-0.0708184\pi\)
\(978\) 3.19961 0.102312
\(979\) 14.4437 0.461624
\(980\) 3.85527i 0.123152i
\(981\) 41.2175i 1.31597i
\(982\) − 53.5367i − 1.70842i
\(983\) − 37.9997i − 1.21200i −0.795463 0.606002i \(-0.792773\pi\)
0.795463 0.606002i \(-0.207227\pi\)
\(984\) 0.0669819 0.00213530
\(985\) −15.2820 −0.486924
\(986\) − 7.84028i − 0.249685i
\(987\) −2.98211 −0.0949217
\(988\) 0 0
\(989\) −25.2176 −0.801873
\(990\) − 4.78347i − 0.152029i
\(991\) −52.5530 −1.66940 −0.834700 0.550705i \(-0.814359\pi\)
−0.834700 + 0.550705i \(0.814359\pi\)
\(992\) 1.94920 0.0618873
\(993\) − 2.36396i − 0.0750179i
\(994\) − 56.0513i − 1.77784i
\(995\) 13.2296i 0.419407i
\(996\) − 0.0162661i 0 0.000515411i
\(997\) 18.5899 0.588749 0.294375 0.955690i \(-0.404889\pi\)
0.294375 + 0.955690i \(0.404889\pi\)
\(998\) 43.2158 1.36797
\(999\) 0.0144230i 0 0.000456324i
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 845.2.c.g.506.7 8
13.2 odd 12 845.2.e.n.191.1 8
13.3 even 3 845.2.m.g.316.1 8
13.4 even 6 845.2.m.g.361.1 8
13.5 odd 4 845.2.a.l.1.4 4
13.6 odd 12 845.2.e.n.146.1 8
13.7 odd 12 845.2.e.m.146.4 8
13.8 odd 4 845.2.a.m.1.1 4
13.9 even 3 65.2.m.a.36.4 8
13.10 even 6 65.2.m.a.56.4 yes 8
13.11 odd 12 845.2.e.m.191.4 8
13.12 even 2 inner 845.2.c.g.506.2 8
39.5 even 4 7605.2.a.cj.1.1 4
39.8 even 4 7605.2.a.cf.1.4 4
39.23 odd 6 585.2.bu.c.316.1 8
39.35 odd 6 585.2.bu.c.361.1 8
52.23 odd 6 1040.2.da.b.641.3 8
52.35 odd 6 1040.2.da.b.881.3 8
65.9 even 6 325.2.n.d.101.1 8
65.22 odd 12 325.2.m.b.49.4 8
65.23 odd 12 325.2.m.b.199.4 8
65.34 odd 4 4225.2.a.bi.1.4 4
65.44 odd 4 4225.2.a.bl.1.1 4
65.48 odd 12 325.2.m.c.49.1 8
65.49 even 6 325.2.n.d.251.1 8
65.62 odd 12 325.2.m.c.199.1 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
65.2.m.a.36.4 8 13.9 even 3
65.2.m.a.56.4 yes 8 13.10 even 6
325.2.m.b.49.4 8 65.22 odd 12
325.2.m.b.199.4 8 65.23 odd 12
325.2.m.c.49.1 8 65.48 odd 12
325.2.m.c.199.1 8 65.62 odd 12
325.2.n.d.101.1 8 65.9 even 6
325.2.n.d.251.1 8 65.49 even 6
585.2.bu.c.316.1 8 39.23 odd 6
585.2.bu.c.361.1 8 39.35 odd 6
845.2.a.l.1.4 4 13.5 odd 4
845.2.a.m.1.1 4 13.8 odd 4
845.2.c.g.506.2 8 13.12 even 2 inner
845.2.c.g.506.7 8 1.1 even 1 trivial
845.2.e.m.146.4 8 13.7 odd 12
845.2.e.m.191.4 8 13.11 odd 12
845.2.e.n.146.1 8 13.6 odd 12
845.2.e.n.191.1 8 13.2 odd 12
845.2.m.g.316.1 8 13.3 even 3
845.2.m.g.361.1 8 13.4 even 6
1040.2.da.b.641.3 8 52.23 odd 6
1040.2.da.b.881.3 8 52.35 odd 6
4225.2.a.bi.1.4 4 65.34 odd 4
4225.2.a.bl.1.1 4 65.44 odd 4
7605.2.a.cf.1.4 4 39.8 even 4
7605.2.a.cj.1.1 4 39.5 even 4