Properties

Label 825.2.c.g.199.5
Level $825$
Weight $2$
Character 825.199
Analytic conductor $6.588$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 825.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.58765816676\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.350464.1
Defining polynomial: \( x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 165)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 199.5
Root \(1.45161 - 1.45161i\) of defining polynomial
Character \(\chi\) \(=\) 825.199
Dual form 825.2.c.g.199.2

$q$-expansion

\(f(q)\) \(=\) \(q+1.90321i q^{2} -1.00000i q^{3} -1.62222 q^{4} +1.90321 q^{6} -4.42864i q^{7} +0.719004i q^{8} -1.00000 q^{9} +O(q^{10})\) \(q+1.90321i q^{2} -1.00000i q^{3} -1.62222 q^{4} +1.90321 q^{6} -4.42864i q^{7} +0.719004i q^{8} -1.00000 q^{9} +1.00000 q^{11} +1.62222i q^{12} +0.622216i q^{13} +8.42864 q^{14} -4.61285 q^{16} -5.18421i q^{17} -1.90321i q^{18} -7.05086 q^{19} -4.42864 q^{21} +1.90321i q^{22} -8.85728i q^{23} +0.719004 q^{24} -1.18421 q^{26} +1.00000i q^{27} +7.18421i q^{28} +7.80642 q^{29} +2.75557 q^{31} -7.34122i q^{32} -1.00000i q^{33} +9.86665 q^{34} +1.62222 q^{36} -2.00000i q^{37} -13.4193i q^{38} +0.622216 q^{39} -0.193576 q^{41} -8.42864i q^{42} -5.67307i q^{43} -1.62222 q^{44} +16.8573 q^{46} -2.75557i q^{47} +4.61285i q^{48} -12.6128 q^{49} -5.18421 q^{51} -1.00937i q^{52} +10.8573i q^{53} -1.90321 q^{54} +3.18421 q^{56} +7.05086i q^{57} +14.8573i q^{58} +4.85728 q^{59} +6.85728 q^{61} +5.24443i q^{62} +4.42864i q^{63} +4.74620 q^{64} +1.90321 q^{66} -1.24443i q^{67} +8.40990i q^{68} -8.85728 q^{69} +2.75557 q^{71} -0.719004i q^{72} -4.23506i q^{73} +3.80642 q^{74} +11.4380 q^{76} -4.42864i q^{77} +1.18421i q^{78} -8.56199 q^{79} +1.00000 q^{81} -0.368416i q^{82} -0.133353i q^{83} +7.18421 q^{84} +10.7971 q^{86} -7.80642i q^{87} +0.719004i q^{88} -5.61285 q^{89} +2.75557 q^{91} +14.3684i q^{92} -2.75557i q^{93} +5.24443 q^{94} -7.34122 q^{96} +7.24443i q^{97} -24.0049i q^{98} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 10 q^{4} - 2 q^{6} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 10 q^{4} - 2 q^{6} - 6 q^{9} + 6 q^{11} + 24 q^{14} + 26 q^{16} - 16 q^{19} + 18 q^{24} + 20 q^{26} + 20 q^{29} + 16 q^{31} + 60 q^{34} + 10 q^{36} + 4 q^{39} - 28 q^{41} - 10 q^{44} + 48 q^{46} - 22 q^{49} - 4 q^{51} + 2 q^{54} - 8 q^{56} - 24 q^{59} - 12 q^{61} - 26 q^{64} - 2 q^{66} + 16 q^{71} - 4 q^{74} + 96 q^{76} - 24 q^{79} + 6 q^{81} + 16 q^{84} - 16 q^{86} + 20 q^{89} + 16 q^{91} + 32 q^{94} - 58 q^{96} - 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/825\mathbb{Z}\right)^\times\).

\(n\) \(376\) \(551\) \(727\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.90321i 1.34577i 0.739745 + 0.672887i \(0.234946\pi\)
−0.739745 + 0.672887i \(0.765054\pi\)
\(3\) − 1.00000i − 0.577350i
\(4\) −1.62222 −0.811108
\(5\) 0 0
\(6\) 1.90321 0.776983
\(7\) − 4.42864i − 1.67387i −0.547304 0.836934i \(-0.684346\pi\)
0.547304 0.836934i \(-0.315654\pi\)
\(8\) 0.719004i 0.254206i
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) 1.62222i 0.468293i
\(13\) 0.622216i 0.172572i 0.996270 + 0.0862858i \(0.0274998\pi\)
−0.996270 + 0.0862858i \(0.972500\pi\)
\(14\) 8.42864 2.25265
\(15\) 0 0
\(16\) −4.61285 −1.15321
\(17\) − 5.18421i − 1.25736i −0.777666 0.628678i \(-0.783597\pi\)
0.777666 0.628678i \(-0.216403\pi\)
\(18\) − 1.90321i − 0.448591i
\(19\) −7.05086 −1.61758 −0.808789 0.588100i \(-0.799876\pi\)
−0.808789 + 0.588100i \(0.799876\pi\)
\(20\) 0 0
\(21\) −4.42864 −0.966408
\(22\) 1.90321i 0.405766i
\(23\) − 8.85728i − 1.84687i −0.383754 0.923435i \(-0.625369\pi\)
0.383754 0.923435i \(-0.374631\pi\)
\(24\) 0.719004 0.146766
\(25\) 0 0
\(26\) −1.18421 −0.232242
\(27\) 1.00000i 0.192450i
\(28\) 7.18421i 1.35769i
\(29\) 7.80642 1.44962 0.724808 0.688951i \(-0.241928\pi\)
0.724808 + 0.688951i \(0.241928\pi\)
\(30\) 0 0
\(31\) 2.75557 0.494915 0.247457 0.968899i \(-0.420405\pi\)
0.247457 + 0.968899i \(0.420405\pi\)
\(32\) − 7.34122i − 1.29776i
\(33\) − 1.00000i − 0.174078i
\(34\) 9.86665 1.69212
\(35\) 0 0
\(36\) 1.62222 0.270369
\(37\) − 2.00000i − 0.328798i −0.986394 0.164399i \(-0.947432\pi\)
0.986394 0.164399i \(-0.0525685\pi\)
\(38\) − 13.4193i − 2.17689i
\(39\) 0.622216 0.0996342
\(40\) 0 0
\(41\) −0.193576 −0.0302315 −0.0151158 0.999886i \(-0.504812\pi\)
−0.0151158 + 0.999886i \(0.504812\pi\)
\(42\) − 8.42864i − 1.30057i
\(43\) − 5.67307i − 0.865135i −0.901602 0.432568i \(-0.857608\pi\)
0.901602 0.432568i \(-0.142392\pi\)
\(44\) −1.62222 −0.244558
\(45\) 0 0
\(46\) 16.8573 2.48547
\(47\) − 2.75557i − 0.401941i −0.979597 0.200971i \(-0.935590\pi\)
0.979597 0.200971i \(-0.0644095\pi\)
\(48\) 4.61285i 0.665807i
\(49\) −12.6128 −1.80184
\(50\) 0 0
\(51\) −5.18421 −0.725934
\(52\) − 1.00937i − 0.139974i
\(53\) 10.8573i 1.49136i 0.666303 + 0.745681i \(0.267876\pi\)
−0.666303 + 0.745681i \(0.732124\pi\)
\(54\) −1.90321 −0.258994
\(55\) 0 0
\(56\) 3.18421 0.425508
\(57\) 7.05086i 0.933909i
\(58\) 14.8573i 1.95086i
\(59\) 4.85728 0.632364 0.316182 0.948699i \(-0.397599\pi\)
0.316182 + 0.948699i \(0.397599\pi\)
\(60\) 0 0
\(61\) 6.85728 0.877985 0.438992 0.898491i \(-0.355336\pi\)
0.438992 + 0.898491i \(0.355336\pi\)
\(62\) 5.24443i 0.666043i
\(63\) 4.42864i 0.557956i
\(64\) 4.74620 0.593275
\(65\) 0 0
\(66\) 1.90321 0.234269
\(67\) − 1.24443i − 0.152031i −0.997107 0.0760157i \(-0.975780\pi\)
0.997107 0.0760157i \(-0.0242199\pi\)
\(68\) 8.40990i 1.01985i
\(69\) −8.85728 −1.06629
\(70\) 0 0
\(71\) 2.75557 0.327026 0.163513 0.986541i \(-0.447717\pi\)
0.163513 + 0.986541i \(0.447717\pi\)
\(72\) − 0.719004i − 0.0847354i
\(73\) − 4.23506i − 0.495677i −0.968801 0.247838i \(-0.920280\pi\)
0.968801 0.247838i \(-0.0797202\pi\)
\(74\) 3.80642 0.442488
\(75\) 0 0
\(76\) 11.4380 1.31203
\(77\) − 4.42864i − 0.504690i
\(78\) 1.18421i 0.134085i
\(79\) −8.56199 −0.963299 −0.481650 0.876364i \(-0.659962\pi\)
−0.481650 + 0.876364i \(0.659962\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) − 0.368416i − 0.0406848i
\(83\) − 0.133353i − 0.0146374i −0.999973 0.00731870i \(-0.997670\pi\)
0.999973 0.00731870i \(-0.00232964\pi\)
\(84\) 7.18421 0.783861
\(85\) 0 0
\(86\) 10.7971 1.16428
\(87\) − 7.80642i − 0.836936i
\(88\) 0.719004i 0.0766461i
\(89\) −5.61285 −0.594961 −0.297480 0.954728i \(-0.596146\pi\)
−0.297480 + 0.954728i \(0.596146\pi\)
\(90\) 0 0
\(91\) 2.75557 0.288862
\(92\) 14.3684i 1.49801i
\(93\) − 2.75557i − 0.285739i
\(94\) 5.24443 0.540922
\(95\) 0 0
\(96\) −7.34122 −0.749260
\(97\) 7.24443i 0.735561i 0.929913 + 0.367780i \(0.119882\pi\)
−0.929913 + 0.367780i \(0.880118\pi\)
\(98\) − 24.0049i − 2.42486i
\(99\) −1.00000 −0.100504
\(100\) 0 0
\(101\) 4.66370 0.464056 0.232028 0.972709i \(-0.425464\pi\)
0.232028 + 0.972709i \(0.425464\pi\)
\(102\) − 9.86665i − 0.976944i
\(103\) 11.6128i 1.14425i 0.820167 + 0.572124i \(0.193880\pi\)
−0.820167 + 0.572124i \(0.806120\pi\)
\(104\) −0.447375 −0.0438688
\(105\) 0 0
\(106\) −20.6637 −2.00704
\(107\) 2.62222i 0.253499i 0.991935 + 0.126750i \(0.0404545\pi\)
−0.991935 + 0.126750i \(0.959546\pi\)
\(108\) − 1.62222i − 0.156098i
\(109\) 19.7146 1.88831 0.944156 0.329499i \(-0.106880\pi\)
0.944156 + 0.329499i \(0.106880\pi\)
\(110\) 0 0
\(111\) −2.00000 −0.189832
\(112\) 20.4286i 1.93032i
\(113\) 6.00000i 0.564433i 0.959351 + 0.282216i \(0.0910696\pi\)
−0.959351 + 0.282216i \(0.908930\pi\)
\(114\) −13.4193 −1.25683
\(115\) 0 0
\(116\) −12.6637 −1.17580
\(117\) − 0.622216i − 0.0575239i
\(118\) 9.24443i 0.851019i
\(119\) −22.9590 −2.10465
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 13.0509i 1.18157i
\(123\) 0.193576i 0.0174542i
\(124\) −4.47013 −0.401429
\(125\) 0 0
\(126\) −8.42864 −0.750883
\(127\) − 15.1842i − 1.34738i −0.739014 0.673690i \(-0.764708\pi\)
0.739014 0.673690i \(-0.235292\pi\)
\(128\) − 5.64941i − 0.499342i
\(129\) −5.67307 −0.499486
\(130\) 0 0
\(131\) 1.24443 0.108726 0.0543632 0.998521i \(-0.482687\pi\)
0.0543632 + 0.998521i \(0.482687\pi\)
\(132\) 1.62222i 0.141196i
\(133\) 31.2257i 2.70761i
\(134\) 2.36842 0.204600
\(135\) 0 0
\(136\) 3.72746 0.319627
\(137\) − 0.488863i − 0.0417663i −0.999782 0.0208832i \(-0.993352\pi\)
0.999782 0.0208832i \(-0.00664780\pi\)
\(138\) − 16.8573i − 1.43499i
\(139\) −17.8064 −1.51032 −0.755161 0.655540i \(-0.772441\pi\)
−0.755161 + 0.655540i \(0.772441\pi\)
\(140\) 0 0
\(141\) −2.75557 −0.232061
\(142\) 5.24443i 0.440103i
\(143\) 0.622216i 0.0520323i
\(144\) 4.61285 0.384404
\(145\) 0 0
\(146\) 8.06022 0.667069
\(147\) 12.6128i 1.04029i
\(148\) 3.24443i 0.266691i
\(149\) 1.43801 0.117806 0.0589031 0.998264i \(-0.481240\pi\)
0.0589031 + 0.998264i \(0.481240\pi\)
\(150\) 0 0
\(151\) −12.1748 −0.990774 −0.495387 0.868672i \(-0.664974\pi\)
−0.495387 + 0.868672i \(0.664974\pi\)
\(152\) − 5.06959i − 0.411198i
\(153\) 5.18421i 0.419118i
\(154\) 8.42864 0.679199
\(155\) 0 0
\(156\) −1.00937 −0.0808141
\(157\) 18.4701i 1.47408i 0.675851 + 0.737038i \(0.263776\pi\)
−0.675851 + 0.737038i \(0.736224\pi\)
\(158\) − 16.2953i − 1.29638i
\(159\) 10.8573 0.861038
\(160\) 0 0
\(161\) −39.2257 −3.09142
\(162\) 1.90321i 0.149530i
\(163\) 10.1017i 0.791227i 0.918417 + 0.395614i \(0.129468\pi\)
−0.918417 + 0.395614i \(0.870532\pi\)
\(164\) 0.314022 0.0245210
\(165\) 0 0
\(166\) 0.253799 0.0196986
\(167\) − 16.3368i − 1.26418i −0.774896 0.632089i \(-0.782198\pi\)
0.774896 0.632089i \(-0.217802\pi\)
\(168\) − 3.18421i − 0.245667i
\(169\) 12.6128 0.970219
\(170\) 0 0
\(171\) 7.05086 0.539192
\(172\) 9.20294i 0.701718i
\(173\) 9.18421i 0.698262i 0.937074 + 0.349131i \(0.113523\pi\)
−0.937074 + 0.349131i \(0.886477\pi\)
\(174\) 14.8573 1.12633
\(175\) 0 0
\(176\) −4.61285 −0.347706
\(177\) − 4.85728i − 0.365095i
\(178\) − 10.6824i − 0.800683i
\(179\) 25.3274 1.89306 0.946530 0.322617i \(-0.104563\pi\)
0.946530 + 0.322617i \(0.104563\pi\)
\(180\) 0 0
\(181\) −13.6128 −1.01184 −0.505918 0.862582i \(-0.668846\pi\)
−0.505918 + 0.862582i \(0.668846\pi\)
\(182\) 5.24443i 0.388743i
\(183\) − 6.85728i − 0.506905i
\(184\) 6.36842 0.469486
\(185\) 0 0
\(186\) 5.24443 0.384540
\(187\) − 5.18421i − 0.379107i
\(188\) 4.47013i 0.326017i
\(189\) 4.42864 0.322136
\(190\) 0 0
\(191\) 6.10171 0.441504 0.220752 0.975330i \(-0.429149\pi\)
0.220752 + 0.975330i \(0.429149\pi\)
\(192\) − 4.74620i − 0.342528i
\(193\) − 18.3368i − 1.31991i −0.751305 0.659955i \(-0.770575\pi\)
0.751305 0.659955i \(-0.229425\pi\)
\(194\) −13.7877 −0.989898
\(195\) 0 0
\(196\) 20.4608 1.46148
\(197\) − 6.69535i − 0.477024i −0.971140 0.238512i \(-0.923340\pi\)
0.971140 0.238512i \(-0.0766596\pi\)
\(198\) − 1.90321i − 0.135255i
\(199\) −14.1017 −0.999644 −0.499822 0.866128i \(-0.666601\pi\)
−0.499822 + 0.866128i \(0.666601\pi\)
\(200\) 0 0
\(201\) −1.24443 −0.0877754
\(202\) 8.87601i 0.624514i
\(203\) − 34.5718i − 2.42647i
\(204\) 8.40990 0.588811
\(205\) 0 0
\(206\) −22.1017 −1.53990
\(207\) 8.85728i 0.615623i
\(208\) − 2.87019i − 0.199012i
\(209\) −7.05086 −0.487718
\(210\) 0 0
\(211\) −10.6637 −0.734120 −0.367060 0.930197i \(-0.619636\pi\)
−0.367060 + 0.930197i \(0.619636\pi\)
\(212\) − 17.6128i − 1.20966i
\(213\) − 2.75557i − 0.188808i
\(214\) −4.99063 −0.341153
\(215\) 0 0
\(216\) −0.719004 −0.0489220
\(217\) − 12.2034i − 0.828422i
\(218\) 37.5210i 2.54124i
\(219\) −4.23506 −0.286179
\(220\) 0 0
\(221\) 3.22570 0.216984
\(222\) − 3.80642i − 0.255470i
\(223\) − 8.85728i − 0.593127i −0.955013 0.296564i \(-0.904159\pi\)
0.955013 0.296564i \(-0.0958407\pi\)
\(224\) −32.5116 −2.17227
\(225\) 0 0
\(226\) −11.4193 −0.759599
\(227\) 13.3778i 0.887915i 0.896048 + 0.443957i \(0.146426\pi\)
−0.896048 + 0.443957i \(0.853574\pi\)
\(228\) − 11.4380i − 0.757501i
\(229\) −11.5111 −0.760677 −0.380339 0.924847i \(-0.624193\pi\)
−0.380339 + 0.924847i \(0.624193\pi\)
\(230\) 0 0
\(231\) −4.42864 −0.291383
\(232\) 5.61285i 0.368502i
\(233\) 4.32693i 0.283467i 0.989905 + 0.141733i \(0.0452675\pi\)
−0.989905 + 0.141733i \(0.954732\pi\)
\(234\) 1.18421 0.0774141
\(235\) 0 0
\(236\) −7.87955 −0.512915
\(237\) 8.56199i 0.556161i
\(238\) − 43.6958i − 2.83238i
\(239\) 3.34614 0.216444 0.108222 0.994127i \(-0.465484\pi\)
0.108222 + 0.994127i \(0.465484\pi\)
\(240\) 0 0
\(241\) −1.34614 −0.0867126 −0.0433563 0.999060i \(-0.513805\pi\)
−0.0433563 + 0.999060i \(0.513805\pi\)
\(242\) 1.90321i 0.122343i
\(243\) − 1.00000i − 0.0641500i
\(244\) −11.1240 −0.712140
\(245\) 0 0
\(246\) −0.368416 −0.0234894
\(247\) − 4.38715i − 0.279148i
\(248\) 1.98126i 0.125810i
\(249\) −0.133353 −0.00845091
\(250\) 0 0
\(251\) 22.7556 1.43632 0.718159 0.695879i \(-0.244985\pi\)
0.718159 + 0.695879i \(0.244985\pi\)
\(252\) − 7.18421i − 0.452563i
\(253\) − 8.85728i − 0.556852i
\(254\) 28.8988 1.81327
\(255\) 0 0
\(256\) 20.2444 1.26528
\(257\) − 6.85728i − 0.427745i −0.976862 0.213873i \(-0.931392\pi\)
0.976862 0.213873i \(-0.0686078\pi\)
\(258\) − 10.7971i − 0.672195i
\(259\) −8.85728 −0.550365
\(260\) 0 0
\(261\) −7.80642 −0.483206
\(262\) 2.36842i 0.146321i
\(263\) 29.5812i 1.82406i 0.410129 + 0.912028i \(0.365484\pi\)
−0.410129 + 0.912028i \(0.634516\pi\)
\(264\) 0.719004 0.0442516
\(265\) 0 0
\(266\) −59.4291 −3.64383
\(267\) 5.61285i 0.343501i
\(268\) 2.01874i 0.123314i
\(269\) −8.48886 −0.517575 −0.258788 0.965934i \(-0.583323\pi\)
−0.258788 + 0.965934i \(0.583323\pi\)
\(270\) 0 0
\(271\) 14.6637 0.890757 0.445378 0.895343i \(-0.353069\pi\)
0.445378 + 0.895343i \(0.353069\pi\)
\(272\) 23.9140i 1.45000i
\(273\) − 2.75557i − 0.166775i
\(274\) 0.930409 0.0562081
\(275\) 0 0
\(276\) 14.3684 0.864877
\(277\) 14.6035i 0.877438i 0.898624 + 0.438719i \(0.144568\pi\)
−0.898624 + 0.438719i \(0.855432\pi\)
\(278\) − 33.8894i − 2.03255i
\(279\) −2.75557 −0.164972
\(280\) 0 0
\(281\) −0.193576 −0.0115478 −0.00577389 0.999983i \(-0.501838\pi\)
−0.00577389 + 0.999983i \(0.501838\pi\)
\(282\) − 5.24443i − 0.312301i
\(283\) − 27.1842i − 1.61593i −0.589228 0.807967i \(-0.700568\pi\)
0.589228 0.807967i \(-0.299432\pi\)
\(284\) −4.47013 −0.265253
\(285\) 0 0
\(286\) −1.18421 −0.0700237
\(287\) 0.857279i 0.0506036i
\(288\) 7.34122i 0.432585i
\(289\) −9.87601 −0.580942
\(290\) 0 0
\(291\) 7.24443 0.424676
\(292\) 6.87019i 0.402047i
\(293\) 2.81579i 0.164500i 0.996612 + 0.0822502i \(0.0262106\pi\)
−0.996612 + 0.0822502i \(0.973789\pi\)
\(294\) −24.0049 −1.40000
\(295\) 0 0
\(296\) 1.43801 0.0835825
\(297\) 1.00000i 0.0580259i
\(298\) 2.73683i 0.158540i
\(299\) 5.51114 0.318717
\(300\) 0 0
\(301\) −25.1240 −1.44812
\(302\) − 23.1713i − 1.33336i
\(303\) − 4.66370i − 0.267923i
\(304\) 32.5245 1.86541
\(305\) 0 0
\(306\) −9.86665 −0.564039
\(307\) − 24.4286i − 1.39422i −0.716966 0.697108i \(-0.754470\pi\)
0.716966 0.697108i \(-0.245530\pi\)
\(308\) 7.18421i 0.409358i
\(309\) 11.6128 0.660632
\(310\) 0 0
\(311\) 19.8796 1.12727 0.563633 0.826025i \(-0.309403\pi\)
0.563633 + 0.826025i \(0.309403\pi\)
\(312\) 0.447375i 0.0253276i
\(313\) 15.7146i 0.888239i 0.895967 + 0.444120i \(0.146483\pi\)
−0.895967 + 0.444120i \(0.853517\pi\)
\(314\) −35.1526 −1.98377
\(315\) 0 0
\(316\) 13.8894 0.781340
\(317\) 16.4889i 0.926107i 0.886330 + 0.463053i \(0.153246\pi\)
−0.886330 + 0.463053i \(0.846754\pi\)
\(318\) 20.6637i 1.15876i
\(319\) 7.80642 0.437076
\(320\) 0 0
\(321\) 2.62222 0.146358
\(322\) − 74.6548i − 4.16035i
\(323\) 36.5531i 2.03387i
\(324\) −1.62222 −0.0901231
\(325\) 0 0
\(326\) −19.2257 −1.06481
\(327\) − 19.7146i − 1.09022i
\(328\) − 0.139182i − 0.00768504i
\(329\) −12.2034 −0.672796
\(330\) 0 0
\(331\) 15.3461 0.843500 0.421750 0.906712i \(-0.361416\pi\)
0.421750 + 0.906712i \(0.361416\pi\)
\(332\) 0.216327i 0.0118725i
\(333\) 2.00000i 0.109599i
\(334\) 31.0923 1.70130
\(335\) 0 0
\(336\) 20.4286 1.11447
\(337\) 28.2351i 1.53806i 0.639212 + 0.769031i \(0.279261\pi\)
−0.639212 + 0.769031i \(0.720739\pi\)
\(338\) 24.0049i 1.30570i
\(339\) 6.00000 0.325875
\(340\) 0 0
\(341\) 2.75557 0.149222
\(342\) 13.4193i 0.725631i
\(343\) 24.8573i 1.34217i
\(344\) 4.07896 0.219923
\(345\) 0 0
\(346\) −17.4795 −0.939703
\(347\) 2.62222i 0.140768i 0.997520 + 0.0703840i \(0.0224224\pi\)
−0.997520 + 0.0703840i \(0.977578\pi\)
\(348\) 12.6637i 0.678846i
\(349\) −5.14272 −0.275284 −0.137642 0.990482i \(-0.543952\pi\)
−0.137642 + 0.990482i \(0.543952\pi\)
\(350\) 0 0
\(351\) −0.622216 −0.0332114
\(352\) − 7.34122i − 0.391288i
\(353\) 9.34614i 0.497445i 0.968575 + 0.248722i \(0.0800107\pi\)
−0.968575 + 0.248722i \(0.919989\pi\)
\(354\) 9.24443 0.491336
\(355\) 0 0
\(356\) 9.10525 0.482577
\(357\) 22.9590i 1.21512i
\(358\) 48.2034i 2.54763i
\(359\) −10.7556 −0.567657 −0.283829 0.958875i \(-0.591605\pi\)
−0.283829 + 0.958875i \(0.591605\pi\)
\(360\) 0 0
\(361\) 30.7146 1.61656
\(362\) − 25.9081i − 1.36170i
\(363\) − 1.00000i − 0.0524864i
\(364\) −4.47013 −0.234298
\(365\) 0 0
\(366\) 13.0509 0.682179
\(367\) 33.7975i 1.76422i 0.471046 + 0.882108i \(0.343876\pi\)
−0.471046 + 0.882108i \(0.656124\pi\)
\(368\) 40.8573i 2.12983i
\(369\) 0.193576 0.0100772
\(370\) 0 0
\(371\) 48.0830 2.49634
\(372\) 4.47013i 0.231765i
\(373\) − 33.9496i − 1.75784i −0.476965 0.878922i \(-0.658263\pi\)
0.476965 0.878922i \(-0.341737\pi\)
\(374\) 9.86665 0.510192
\(375\) 0 0
\(376\) 1.98126 0.102176
\(377\) 4.85728i 0.250163i
\(378\) 8.42864i 0.433522i
\(379\) 20.0000 1.02733 0.513665 0.857991i \(-0.328287\pi\)
0.513665 + 0.857991i \(0.328287\pi\)
\(380\) 0 0
\(381\) −15.1842 −0.777911
\(382\) 11.6128i 0.594165i
\(383\) 14.6351i 0.747820i 0.927465 + 0.373910i \(0.121983\pi\)
−0.927465 + 0.373910i \(0.878017\pi\)
\(384\) −5.64941 −0.288295
\(385\) 0 0
\(386\) 34.8988 1.77630
\(387\) 5.67307i 0.288378i
\(388\) − 11.7520i − 0.596619i
\(389\) 5.61285 0.284583 0.142291 0.989825i \(-0.454553\pi\)
0.142291 + 0.989825i \(0.454553\pi\)
\(390\) 0 0
\(391\) −45.9180 −2.32217
\(392\) − 9.06868i − 0.458038i
\(393\) − 1.24443i − 0.0627733i
\(394\) 12.7427 0.641966
\(395\) 0 0
\(396\) 1.62222 0.0815194
\(397\) − 12.7556i − 0.640184i −0.947387 0.320092i \(-0.896286\pi\)
0.947387 0.320092i \(-0.103714\pi\)
\(398\) − 26.8385i − 1.34529i
\(399\) 31.2257 1.56324
\(400\) 0 0
\(401\) 2.00000 0.0998752 0.0499376 0.998752i \(-0.484098\pi\)
0.0499376 + 0.998752i \(0.484098\pi\)
\(402\) − 2.36842i − 0.118126i
\(403\) 1.71456i 0.0854082i
\(404\) −7.56553 −0.376399
\(405\) 0 0
\(406\) 65.7975 3.26548
\(407\) − 2.00000i − 0.0991363i
\(408\) − 3.72746i − 0.184537i
\(409\) 7.12399 0.352258 0.176129 0.984367i \(-0.443642\pi\)
0.176129 + 0.984367i \(0.443642\pi\)
\(410\) 0 0
\(411\) −0.488863 −0.0241138
\(412\) − 18.8385i − 0.928108i
\(413\) − 21.5111i − 1.05849i
\(414\) −16.8573 −0.828490
\(415\) 0 0
\(416\) 4.56782 0.223956
\(417\) 17.8064i 0.871984i
\(418\) − 13.4193i − 0.656358i
\(419\) −15.6128 −0.762738 −0.381369 0.924423i \(-0.624547\pi\)
−0.381369 + 0.924423i \(0.624547\pi\)
\(420\) 0 0
\(421\) 7.89829 0.384939 0.192470 0.981303i \(-0.438350\pi\)
0.192470 + 0.981303i \(0.438350\pi\)
\(422\) − 20.2953i − 0.987959i
\(423\) 2.75557i 0.133980i
\(424\) −7.80642 −0.379113
\(425\) 0 0
\(426\) 5.24443 0.254094
\(427\) − 30.3684i − 1.46963i
\(428\) − 4.25380i − 0.205615i
\(429\) 0.622216 0.0300409
\(430\) 0 0
\(431\) 34.3051 1.65242 0.826210 0.563362i \(-0.190492\pi\)
0.826210 + 0.563362i \(0.190492\pi\)
\(432\) − 4.61285i − 0.221936i
\(433\) − 14.4701i − 0.695390i −0.937608 0.347695i \(-0.886964\pi\)
0.937608 0.347695i \(-0.113036\pi\)
\(434\) 23.2257 1.11487
\(435\) 0 0
\(436\) −31.9813 −1.53162
\(437\) 62.4514i 2.98746i
\(438\) − 8.06022i − 0.385132i
\(439\) 19.3176 0.921977 0.460988 0.887406i \(-0.347495\pi\)
0.460988 + 0.887406i \(0.347495\pi\)
\(440\) 0 0
\(441\) 12.6128 0.600612
\(442\) 6.13918i 0.292011i
\(443\) 13.1240i 0.623539i 0.950158 + 0.311770i \(0.100922\pi\)
−0.950158 + 0.311770i \(0.899078\pi\)
\(444\) 3.24443 0.153974
\(445\) 0 0
\(446\) 16.8573 0.798215
\(447\) − 1.43801i − 0.0680154i
\(448\) − 21.0192i − 0.993064i
\(449\) 32.3051 1.52457 0.762287 0.647240i \(-0.224077\pi\)
0.762287 + 0.647240i \(0.224077\pi\)
\(450\) 0 0
\(451\) −0.193576 −0.00911514
\(452\) − 9.73329i − 0.457816i
\(453\) 12.1748i 0.572024i
\(454\) −25.4608 −1.19493
\(455\) 0 0
\(456\) −5.06959 −0.237405
\(457\) − 23.4608i − 1.09745i −0.836004 0.548724i \(-0.815114\pi\)
0.836004 0.548724i \(-0.184886\pi\)
\(458\) − 21.9081i − 1.02370i
\(459\) 5.18421 0.241978
\(460\) 0 0
\(461\) −28.8671 −1.34448 −0.672238 0.740335i \(-0.734667\pi\)
−0.672238 + 0.740335i \(0.734667\pi\)
\(462\) − 8.42864i − 0.392136i
\(463\) 19.3461i 0.899091i 0.893257 + 0.449546i \(0.148414\pi\)
−0.893257 + 0.449546i \(0.851586\pi\)
\(464\) −36.0098 −1.67172
\(465\) 0 0
\(466\) −8.23506 −0.381482
\(467\) 3.14272i 0.145428i 0.997353 + 0.0727139i \(0.0231660\pi\)
−0.997353 + 0.0727139i \(0.976834\pi\)
\(468\) 1.00937i 0.0466580i
\(469\) −5.51114 −0.254481
\(470\) 0 0
\(471\) 18.4701 0.851059
\(472\) 3.49240i 0.160751i
\(473\) − 5.67307i − 0.260848i
\(474\) −16.2953 −0.748467
\(475\) 0 0
\(476\) 37.2444 1.70710
\(477\) − 10.8573i − 0.497121i
\(478\) 6.36842i 0.291285i
\(479\) 24.8573 1.13576 0.567879 0.823112i \(-0.307764\pi\)
0.567879 + 0.823112i \(0.307764\pi\)
\(480\) 0 0
\(481\) 1.24443 0.0567412
\(482\) − 2.56199i − 0.116696i
\(483\) 39.2257i 1.78483i
\(484\) −1.62222 −0.0737371
\(485\) 0 0
\(486\) 1.90321 0.0863314
\(487\) − 11.5299i − 0.522468i −0.965275 0.261234i \(-0.915871\pi\)
0.965275 0.261234i \(-0.0841295\pi\)
\(488\) 4.93041i 0.223189i
\(489\) 10.1017 0.456815
\(490\) 0 0
\(491\) 16.3872 0.739542 0.369771 0.929123i \(-0.379436\pi\)
0.369771 + 0.929123i \(0.379436\pi\)
\(492\) − 0.314022i − 0.0141572i
\(493\) − 40.4701i − 1.82268i
\(494\) 8.34968 0.375670
\(495\) 0 0
\(496\) −12.7110 −0.570742
\(497\) − 12.2034i − 0.547398i
\(498\) − 0.253799i − 0.0113730i
\(499\) 25.3274 1.13381 0.566905 0.823783i \(-0.308141\pi\)
0.566905 + 0.823783i \(0.308141\pi\)
\(500\) 0 0
\(501\) −16.3368 −0.729873
\(502\) 43.3087i 1.93296i
\(503\) − 19.0923i − 0.851285i −0.904891 0.425643i \(-0.860048\pi\)
0.904891 0.425643i \(-0.139952\pi\)
\(504\) −3.18421 −0.141836
\(505\) 0 0
\(506\) 16.8573 0.749397
\(507\) − 12.6128i − 0.560156i
\(508\) 24.6321i 1.09287i
\(509\) 32.4514 1.43838 0.719191 0.694812i \(-0.244512\pi\)
0.719191 + 0.694812i \(0.244512\pi\)
\(510\) 0 0
\(511\) −18.7556 −0.829698
\(512\) 27.2306i 1.20343i
\(513\) − 7.05086i − 0.311303i
\(514\) 13.0509 0.575649
\(515\) 0 0
\(516\) 9.20294 0.405137
\(517\) − 2.75557i − 0.121190i
\(518\) − 16.8573i − 0.740666i
\(519\) 9.18421 0.403142
\(520\) 0 0
\(521\) −29.2257 −1.28040 −0.640200 0.768208i \(-0.721149\pi\)
−0.640200 + 0.768208i \(0.721149\pi\)
\(522\) − 14.8573i − 0.650285i
\(523\) − 6.71408i − 0.293586i −0.989167 0.146793i \(-0.953105\pi\)
0.989167 0.146793i \(-0.0468952\pi\)
\(524\) −2.01874 −0.0881889
\(525\) 0 0
\(526\) −56.2993 −2.45477
\(527\) − 14.2854i − 0.622284i
\(528\) 4.61285i 0.200748i
\(529\) −55.4514 −2.41093
\(530\) 0 0
\(531\) −4.85728 −0.210788
\(532\) − 50.6548i − 2.19616i
\(533\) − 0.120446i − 0.00521710i
\(534\) −10.6824 −0.462274
\(535\) 0 0
\(536\) 0.894751 0.0386473
\(537\) − 25.3274i − 1.09296i
\(538\) − 16.1561i − 0.696539i
\(539\) −12.6128 −0.543274
\(540\) 0 0
\(541\) −18.0000 −0.773880 −0.386940 0.922105i \(-0.626468\pi\)
−0.386940 + 0.922105i \(0.626468\pi\)
\(542\) 27.9081i 1.19876i
\(543\) 13.6128i 0.584183i
\(544\) −38.0584 −1.63174
\(545\) 0 0
\(546\) 5.24443 0.224441
\(547\) 41.3689i 1.76881i 0.466724 + 0.884403i \(0.345434\pi\)
−0.466724 + 0.884403i \(0.654566\pi\)
\(548\) 0.793040i 0.0338770i
\(549\) −6.85728 −0.292662
\(550\) 0 0
\(551\) −55.0420 −2.34487
\(552\) − 6.36842i − 0.271058i
\(553\) 37.9180i 1.61244i
\(554\) −27.7935 −1.18083
\(555\) 0 0
\(556\) 28.8859 1.22503
\(557\) − 20.7971i − 0.881200i −0.897704 0.440600i \(-0.854766\pi\)
0.897704 0.440600i \(-0.145234\pi\)
\(558\) − 5.24443i − 0.222014i
\(559\) 3.52987 0.149298
\(560\) 0 0
\(561\) −5.18421 −0.218877
\(562\) − 0.368416i − 0.0155407i
\(563\) − 37.7275i − 1.59002i −0.606594 0.795012i \(-0.707465\pi\)
0.606594 0.795012i \(-0.292535\pi\)
\(564\) 4.47013 0.188226
\(565\) 0 0
\(566\) 51.7373 2.17468
\(567\) − 4.42864i − 0.185985i
\(568\) 1.98126i 0.0831320i
\(569\) 7.33630 0.307554 0.153777 0.988106i \(-0.450856\pi\)
0.153777 + 0.988106i \(0.450856\pi\)
\(570\) 0 0
\(571\) 36.6450 1.53354 0.766772 0.641919i \(-0.221862\pi\)
0.766772 + 0.641919i \(0.221862\pi\)
\(572\) − 1.00937i − 0.0422038i
\(573\) − 6.10171i − 0.254903i
\(574\) −1.63158 −0.0681010
\(575\) 0 0
\(576\) −4.74620 −0.197758
\(577\) 4.22216i 0.175771i 0.996131 + 0.0878853i \(0.0280109\pi\)
−0.996131 + 0.0878853i \(0.971989\pi\)
\(578\) − 18.7961i − 0.781817i
\(579\) −18.3368 −0.762050
\(580\) 0 0
\(581\) −0.590573 −0.0245011
\(582\) 13.7877i 0.571518i
\(583\) 10.8573i 0.449663i
\(584\) 3.04503 0.126004
\(585\) 0 0
\(586\) −5.35905 −0.221380
\(587\) 34.3684i 1.41854i 0.704939 + 0.709268i \(0.250974\pi\)
−0.704939 + 0.709268i \(0.749026\pi\)
\(588\) − 20.4608i − 0.843787i
\(589\) −19.4291 −0.800563
\(590\) 0 0
\(591\) −6.69535 −0.275410
\(592\) 9.22570i 0.379174i
\(593\) − 27.9398i − 1.14735i −0.819083 0.573675i \(-0.805517\pi\)
0.819083 0.573675i \(-0.194483\pi\)
\(594\) −1.90321 −0.0780897
\(595\) 0 0
\(596\) −2.33276 −0.0955535
\(597\) 14.1017i 0.577145i
\(598\) 10.4889i 0.428921i
\(599\) 31.2257 1.27585 0.637924 0.770100i \(-0.279794\pi\)
0.637924 + 0.770100i \(0.279794\pi\)
\(600\) 0 0
\(601\) −8.75557 −0.357147 −0.178574 0.983927i \(-0.557148\pi\)
−0.178574 + 0.983927i \(0.557148\pi\)
\(602\) − 47.8163i − 1.94885i
\(603\) 1.24443i 0.0506772i
\(604\) 19.7502 0.803625
\(605\) 0 0
\(606\) 8.87601 0.360563
\(607\) − 15.1842i − 0.616308i −0.951336 0.308154i \(-0.900289\pi\)
0.951336 0.308154i \(-0.0997112\pi\)
\(608\) 51.7619i 2.09922i
\(609\) −34.5718 −1.40092
\(610\) 0 0
\(611\) 1.71456 0.0693636
\(612\) − 8.40990i − 0.339950i
\(613\) − 42.7239i − 1.72560i −0.505543 0.862802i \(-0.668708\pi\)
0.505543 0.862802i \(-0.331292\pi\)
\(614\) 46.4929 1.87630
\(615\) 0 0
\(616\) 3.18421 0.128295
\(617\) − 3.51114i − 0.141353i −0.997499 0.0706765i \(-0.977484\pi\)
0.997499 0.0706765i \(-0.0225158\pi\)
\(618\) 22.1017i 0.889061i
\(619\) −17.5941 −0.707167 −0.353584 0.935403i \(-0.615037\pi\)
−0.353584 + 0.935403i \(0.615037\pi\)
\(620\) 0 0
\(621\) 8.85728 0.355430
\(622\) 37.8350i 1.51705i
\(623\) 24.8573i 0.995886i
\(624\) −2.87019 −0.114899
\(625\) 0 0
\(626\) −29.9081 −1.19537
\(627\) 7.05086i 0.281584i
\(628\) − 29.9625i − 1.19564i
\(629\) −10.3684 −0.413416
\(630\) 0 0
\(631\) 15.8163 0.629636 0.314818 0.949152i \(-0.398057\pi\)
0.314818 + 0.949152i \(0.398057\pi\)
\(632\) − 6.15610i − 0.244877i
\(633\) 10.6637i 0.423844i
\(634\) −31.3818 −1.24633
\(635\) 0 0
\(636\) −17.6128 −0.698395
\(637\) − 7.84791i − 0.310946i
\(638\) 14.8573i 0.588205i
\(639\) −2.75557 −0.109009
\(640\) 0 0
\(641\) 25.8163 1.01968 0.509841 0.860269i \(-0.329704\pi\)
0.509841 + 0.860269i \(0.329704\pi\)
\(642\) 4.99063i 0.196965i
\(643\) − 18.1017i − 0.713862i −0.934131 0.356931i \(-0.883823\pi\)
0.934131 0.356931i \(-0.116177\pi\)
\(644\) 63.6325 2.50747
\(645\) 0 0
\(646\) −69.5683 −2.73713
\(647\) − 47.0420i − 1.84941i −0.380684 0.924705i \(-0.624311\pi\)
0.380684 0.924705i \(-0.375689\pi\)
\(648\) 0.719004i 0.0282451i
\(649\) 4.85728 0.190665
\(650\) 0 0
\(651\) −12.2034 −0.478290
\(652\) − 16.3872i − 0.641770i
\(653\) − 30.0830i − 1.17724i −0.808411 0.588619i \(-0.799672\pi\)
0.808411 0.588619i \(-0.200328\pi\)
\(654\) 37.5210 1.46719
\(655\) 0 0
\(656\) 0.892937 0.0348633
\(657\) 4.23506i 0.165226i
\(658\) − 23.2257i − 0.905432i
\(659\) 10.2854 0.400664 0.200332 0.979728i \(-0.435798\pi\)
0.200332 + 0.979728i \(0.435798\pi\)
\(660\) 0 0
\(661\) −27.7146 −1.07797 −0.538986 0.842315i \(-0.681192\pi\)
−0.538986 + 0.842315i \(0.681192\pi\)
\(662\) 29.2070i 1.13516i
\(663\) − 3.22570i − 0.125276i
\(664\) 0.0958814 0.00372092
\(665\) 0 0
\(666\) −3.80642 −0.147496
\(667\) − 69.1437i − 2.67725i
\(668\) 26.5018i 1.02538i
\(669\) −8.85728 −0.342442
\(670\) 0 0
\(671\) 6.85728 0.264722
\(672\) 32.5116i 1.25416i
\(673\) 9.86665i 0.380331i 0.981752 + 0.190166i \(0.0609025\pi\)
−0.981752 + 0.190166i \(0.939097\pi\)
\(674\) −53.7373 −2.06988
\(675\) 0 0
\(676\) −20.4608 −0.786952
\(677\) − 5.65433i − 0.217314i −0.994079 0.108657i \(-0.965345\pi\)
0.994079 0.108657i \(-0.0346550\pi\)
\(678\) 11.4193i 0.438554i
\(679\) 32.0830 1.23123
\(680\) 0 0
\(681\) 13.3778 0.512638
\(682\) 5.24443i 0.200820i
\(683\) − 34.1847i − 1.30804i −0.756477 0.654020i \(-0.773081\pi\)
0.756477 0.654020i \(-0.226919\pi\)
\(684\) −11.4380 −0.437343
\(685\) 0 0
\(686\) −47.3087 −1.80625
\(687\) 11.5111i 0.439177i
\(688\) 26.1690i 0.997684i
\(689\) −6.75557 −0.257367
\(690\) 0 0
\(691\) −19.2257 −0.731380 −0.365690 0.930737i \(-0.619167\pi\)
−0.365690 + 0.930737i \(0.619167\pi\)
\(692\) − 14.8988i − 0.566366i
\(693\) 4.42864i 0.168230i
\(694\) −4.99063 −0.189442
\(695\) 0 0
\(696\) 5.61285 0.212754
\(697\) 1.00354i 0.0380118i
\(698\) − 9.78769i − 0.370469i
\(699\) 4.32693 0.163659
\(700\) 0 0
\(701\) −29.9081 −1.12961 −0.564807 0.825223i \(-0.691050\pi\)
−0.564807 + 0.825223i \(0.691050\pi\)
\(702\) − 1.18421i − 0.0446951i
\(703\) 14.1017i 0.531856i
\(704\) 4.74620 0.178879
\(705\) 0 0
\(706\) −17.7877 −0.669448
\(707\) − 20.6539i − 0.776768i
\(708\) 7.87955i 0.296132i
\(709\) 15.3274 0.575633 0.287816 0.957686i \(-0.407071\pi\)
0.287816 + 0.957686i \(0.407071\pi\)
\(710\) 0 0
\(711\) 8.56199 0.321100
\(712\) − 4.03566i − 0.151243i
\(713\) − 24.4068i − 0.914043i
\(714\) −43.6958 −1.63528
\(715\) 0 0
\(716\) −41.0865 −1.53548
\(717\) − 3.34614i − 0.124964i
\(718\) − 20.4701i − 0.763938i
\(719\) −23.8163 −0.888197 −0.444098 0.895978i \(-0.646476\pi\)
−0.444098 + 0.895978i \(0.646476\pi\)
\(720\) 0 0
\(721\) 51.4291 1.91532
\(722\) 58.4563i 2.17552i
\(723\) 1.34614i 0.0500635i
\(724\) 22.0830 0.820707
\(725\) 0 0
\(726\) 1.90321 0.0706348
\(727\) − 32.9403i − 1.22169i −0.791752 0.610843i \(-0.790831\pi\)
0.791752 0.610843i \(-0.209169\pi\)
\(728\) 1.98126i 0.0734305i
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) −29.4104 −1.08778
\(732\) 11.1240i 0.411154i
\(733\) 29.8666i 1.10315i 0.834125 + 0.551575i \(0.185973\pi\)
−0.834125 + 0.551575i \(0.814027\pi\)
\(734\) −64.3239 −2.37424
\(735\) 0 0
\(736\) −65.0232 −2.39679
\(737\) − 1.24443i − 0.0458392i
\(738\) 0.368416i 0.0135616i
\(739\) 5.06959 0.186488 0.0932440 0.995643i \(-0.470276\pi\)
0.0932440 + 0.995643i \(0.470276\pi\)
\(740\) 0 0
\(741\) −4.38715 −0.161166
\(742\) 91.5121i 3.35951i
\(743\) − 22.4385i − 0.823188i −0.911367 0.411594i \(-0.864972\pi\)
0.911367 0.411594i \(-0.135028\pi\)
\(744\) 1.98126 0.0726367
\(745\) 0 0
\(746\) 64.6133 2.36566
\(747\) 0.133353i 0.00487913i
\(748\) 8.40990i 0.307497i
\(749\) 11.6128 0.424324
\(750\) 0 0
\(751\) −6.63512 −0.242119 −0.121060 0.992645i \(-0.538629\pi\)
−0.121060 + 0.992645i \(0.538629\pi\)
\(752\) 12.7110i 0.463523i
\(753\) − 22.7556i − 0.829259i
\(754\) −9.24443 −0.336662
\(755\) 0 0
\(756\) −7.18421 −0.261287
\(757\) 8.75557i 0.318227i 0.987260 + 0.159113i \(0.0508635\pi\)
−0.987260 + 0.159113i \(0.949136\pi\)
\(758\) 38.0642i 1.38256i
\(759\) −8.85728 −0.321499
\(760\) 0 0
\(761\) 3.15257 0.114280 0.0571402 0.998366i \(-0.481802\pi\)
0.0571402 + 0.998366i \(0.481802\pi\)
\(762\) − 28.8988i − 1.04689i
\(763\) − 87.3087i − 3.16079i
\(764\) −9.89829 −0.358108
\(765\) 0 0
\(766\) −27.8537 −1.00640
\(767\) 3.02227i 0.109128i
\(768\) − 20.2444i − 0.730508i
\(769\) 28.9590 1.04429 0.522144 0.852857i \(-0.325132\pi\)
0.522144 + 0.852857i \(0.325132\pi\)
\(770\) 0 0
\(771\) −6.85728 −0.246959
\(772\) 29.7462i 1.07059i
\(773\) − 29.1427i − 1.04819i −0.851660 0.524095i \(-0.824404\pi\)
0.851660 0.524095i \(-0.175596\pi\)
\(774\) −10.7971 −0.388092
\(775\) 0 0
\(776\) −5.20877 −0.186984
\(777\) 8.85728i 0.317753i
\(778\) 10.6824i 0.382984i
\(779\) 1.36488 0.0489018
\(780\) 0 0
\(781\) 2.75557 0.0986020
\(782\) − 87.3916i − 3.12512i
\(783\) 7.80642i 0.278979i
\(784\) 58.1811 2.07790
\(785\) 0 0
\(786\) 2.36842 0.0844786
\(787\) − 11.2672i − 0.401632i −0.979629 0.200816i \(-0.935641\pi\)
0.979629 0.200816i \(-0.0643593\pi\)
\(788\) 10.8613i 0.386918i
\(789\) 29.5812 1.05312
\(790\) 0 0
\(791\) 26.5718 0.944786
\(792\) − 0.719004i − 0.0255487i
\(793\) 4.26671i 0.151515i
\(794\) 24.2766 0.861543
\(795\) 0 0
\(796\) 22.8760 0.810819
\(797\) 41.9625i 1.48639i 0.669075 + 0.743195i \(0.266690\pi\)
−0.669075 + 0.743195i \(0.733310\pi\)
\(798\) 59.4291i 2.10377i
\(799\) −14.2854 −0.505383
\(800\) 0 0
\(801\) 5.61285 0.198320
\(802\) 3.80642i 0.134409i
\(803\) − 4.23506i − 0.149452i
\(804\) 2.01874 0.0711953
\(805\) 0 0
\(806\) −3.26317 −0.114940
\(807\) 8.48886i 0.298822i
\(808\) 3.35322i 0.117966i
\(809\) 27.8064 0.977622 0.488811 0.872390i \(-0.337431\pi\)
0.488811 + 0.872390i \(0.337431\pi\)
\(810\) 0 0
\(811\) 6.78415 0.238224 0.119112 0.992881i \(-0.461995\pi\)
0.119112 + 0.992881i \(0.461995\pi\)
\(812\) 56.0830i 1.96813i
\(813\) − 14.6637i − 0.514279i
\(814\) 3.80642 0.133415
\(815\) 0 0
\(816\) 23.9140 0.837156
\(817\) 40.0000i 1.39942i
\(818\) 13.5585i 0.474060i
\(819\) −2.75557 −0.0962874
\(820\) 0 0
\(821\) 3.62269 0.126433 0.0632164 0.998000i \(-0.479864\pi\)
0.0632164 + 0.998000i \(0.479864\pi\)
\(822\) − 0.930409i − 0.0324517i
\(823\) − 42.0642i − 1.46627i −0.680085 0.733134i \(-0.738057\pi\)
0.680085 0.733134i \(-0.261943\pi\)
\(824\) −8.34968 −0.290875
\(825\) 0 0
\(826\) 40.9403 1.42449
\(827\) 30.8256i 1.07191i 0.844246 + 0.535956i \(0.180049\pi\)
−0.844246 + 0.535956i \(0.819951\pi\)
\(828\) − 14.3684i − 0.499337i
\(829\) −7.12399 −0.247426 −0.123713 0.992318i \(-0.539480\pi\)
−0.123713 + 0.992318i \(0.539480\pi\)
\(830\) 0 0
\(831\) 14.6035 0.506589
\(832\) 2.95316i 0.102382i
\(833\) 65.3876i 2.26555i
\(834\) −33.8894 −1.17349
\(835\) 0 0
\(836\) 11.4380 0.395592
\(837\) 2.75557i 0.0952464i
\(838\) − 29.7146i − 1.02647i
\(839\) −3.34614 −0.115522 −0.0577608 0.998330i \(-0.518396\pi\)
−0.0577608 + 0.998330i \(0.518396\pi\)
\(840\) 0 0
\(841\) 31.9403 1.10139
\(842\) 15.0321i 0.518041i
\(843\) 0.193576i 0.00666712i
\(844\) 17.2988 0.595450
\(845\) 0 0
\(846\) −5.24443 −0.180307
\(847\) − 4.42864i − 0.152170i
\(848\) − 50.0830i − 1.71986i
\(849\) −27.1842 −0.932960
\(850\) 0 0
\(851\) −17.7146 −0.607247
\(852\) 4.47013i 0.153144i
\(853\) 26.4197i 0.904595i 0.891867 + 0.452297i \(0.149395\pi\)
−0.891867 + 0.452297i \(0.850605\pi\)
\(854\) 57.7975 1.97779
\(855\) 0 0
\(856\) −1.88538 −0.0644411
\(857\) 38.7783i 1.32464i 0.749220 + 0.662321i \(0.230429\pi\)
−0.749220 + 0.662321i \(0.769571\pi\)
\(858\) 1.18421i 0.0404282i
\(859\) 27.3087 0.931760 0.465880 0.884848i \(-0.345738\pi\)
0.465880 + 0.884848i \(0.345738\pi\)
\(860\) 0 0
\(861\) 0.857279 0.0292160
\(862\) 65.2899i 2.22378i
\(863\) 49.5308i 1.68605i 0.537875 + 0.843024i \(0.319227\pi\)
−0.537875 + 0.843024i \(0.680773\pi\)
\(864\) 7.34122 0.249753
\(865\) 0 0
\(866\) 27.5397 0.935838
\(867\) 9.87601i 0.335407i
\(868\) 19.7966i 0.671940i
\(869\) −8.56199 −0.290446
\(870\) 0 0
\(871\) 0.774305 0.0262363
\(872\) 14.1748i 0.480021i
\(873\) − 7.24443i − 0.245187i
\(874\) −118.858 −4.02044
\(875\) 0 0
\(876\) 6.87019 0.232122
\(877\) − 4.50177i − 0.152014i −0.997107 0.0760070i \(-0.975783\pi\)
0.997107 0.0760070i \(-0.0242171\pi\)
\(878\) 36.7654i 1.24077i
\(879\) 2.81579 0.0949743
\(880\) 0 0
\(881\) −15.1240 −0.509540 −0.254770 0.967002i \(-0.582000\pi\)
−0.254770 + 0.967002i \(0.582000\pi\)
\(882\) 24.0049i 0.808288i
\(883\) 30.2480i 1.01793i 0.860789 + 0.508963i \(0.169971\pi\)
−0.860789 + 0.508963i \(0.830029\pi\)
\(884\) −5.23277 −0.175997
\(885\) 0 0
\(886\) −24.9777 −0.839143
\(887\) 57.1941i 1.92039i 0.279333 + 0.960194i \(0.409887\pi\)
−0.279333 + 0.960194i \(0.590113\pi\)
\(888\) − 1.43801i − 0.0482564i
\(889\) −67.2454 −2.25534
\(890\) 0 0
\(891\) 1.00000 0.0335013
\(892\) 14.3684i 0.481090i
\(893\) 19.4291i 0.650171i
\(894\) 2.73683 0.0915334
\(895\) 0 0
\(896\) −25.0192 −0.835833
\(897\) − 5.51114i − 0.184012i
\(898\) 61.4835i 2.05173i
\(899\) 21.5111 0.717437
\(900\) 0 0
\(901\) 56.2864 1.87517
\(902\) − 0.368416i − 0.0122669i
\(903\) 25.1240i 0.836074i
\(904\) −4.31402 −0.143482
\(905\) 0 0
\(906\) −23.1713 −0.769815
\(907\) − 53.2641i − 1.76861i −0.466913 0.884303i \(-0.654634\pi\)
0.466913 0.884303i \(-0.345366\pi\)
\(908\) − 21.7017i − 0.720195i
\(909\) −4.66370 −0.154685
\(910\) 0 0
\(911\) −0.590573 −0.0195665 −0.00978327 0.999952i \(-0.503114\pi\)
−0.00978327 + 0.999952i \(0.503114\pi\)
\(912\) − 32.5245i − 1.07699i
\(913\) − 0.133353i − 0.00441334i
\(914\) 44.6508 1.47692
\(915\) 0 0
\(916\) 18.6735 0.616991
\(917\) − 5.51114i − 0.181994i
\(918\) 9.86665i 0.325648i
\(919\) −55.8707 −1.84300 −0.921502 0.388375i \(-0.873037\pi\)
−0.921502 + 0.388375i \(0.873037\pi\)
\(920\) 0 0
\(921\) −24.4286 −0.804951
\(922\) − 54.9403i − 1.80936i
\(923\) 1.71456i 0.0564354i
\(924\) 7.18421 0.236343
\(925\) 0 0
\(926\) −36.8198 −1.20997
\(927\) − 11.6128i − 0.381416i
\(928\) − 57.3087i − 1.88125i
\(929\) −15.3274 −0.502876 −0.251438 0.967873i \(-0.580903\pi\)
−0.251438 + 0.967873i \(0.580903\pi\)
\(930\) 0 0
\(931\) 88.9314 2.91461
\(932\) − 7.01921i − 0.229922i
\(933\) − 19.8796i − 0.650827i
\(934\) −5.98126 −0.195713
\(935\) 0 0
\(936\) 0.447375 0.0146229
\(937\) − 27.8479i − 0.909752i −0.890555 0.454876i \(-0.849684\pi\)
0.890555 0.454876i \(-0.150316\pi\)
\(938\) − 10.4889i − 0.342474i
\(939\) 15.7146 0.512825
\(940\) 0 0
\(941\) 10.4157 0.339543 0.169772 0.985483i \(-0.445697\pi\)
0.169772 + 0.985483i \(0.445697\pi\)
\(942\) 35.1526i 1.14533i
\(943\) 1.71456i 0.0558337i
\(944\) −22.4059 −0.729250
\(945\) 0 0
\(946\) 10.7971 0.351043
\(947\) 8.47013i 0.275242i 0.990485 + 0.137621i \(0.0439456\pi\)
−0.990485 + 0.137621i \(0.956054\pi\)
\(948\) − 13.8894i − 0.451107i
\(949\) 2.63512 0.0855397
\(950\) 0 0
\(951\) 16.4889 0.534688
\(952\) − 16.5076i − 0.535014i
\(953\) 8.71408i 0.282277i 0.989990 + 0.141138i \(0.0450763\pi\)
−0.989990 + 0.141138i \(0.954924\pi\)
\(954\) 20.6637 0.669012
\(955\) 0 0
\(956\) −5.42816 −0.175559
\(957\) − 7.80642i − 0.252346i
\(958\) 47.3087i 1.52847i
\(959\) −2.16500 −0.0699114
\(960\) 0 0
\(961\) −23.4068 −0.755059
\(962\) 2.36842i 0.0763608i
\(963\) − 2.62222i − 0.0844997i
\(964\) 2.18373 0.0703333
\(965\) 0 0
\(966\) −74.6548 −2.40198
\(967\) 44.2449i 1.42282i 0.702777 + 0.711410i \(0.251943\pi\)
−0.702777 + 0.711410i \(0.748057\pi\)
\(968\) 0.719004i 0.0231097i
\(969\) 36.5531 1.17425
\(970\) 0 0
\(971\) −57.1437 −1.83383 −0.916914 0.399085i \(-0.869328\pi\)
−0.916914 + 0.399085i \(0.869328\pi\)
\(972\) 1.62222i 0.0520326i
\(973\) 78.8582i 2.52808i
\(974\) 21.9438 0.703124
\(975\) 0 0
\(976\) −31.6316 −1.01250
\(977\) − 16.2480i − 0.519819i −0.965633 0.259909i \(-0.916307\pi\)
0.965633 0.259909i \(-0.0836927\pi\)
\(978\) 19.2257i 0.614770i
\(979\) −5.61285 −0.179387
\(980\) 0 0
\(981\) −19.7146 −0.629437
\(982\) 31.1882i 0.995256i
\(983\) − 1.12399i − 0.0358496i −0.999839 0.0179248i \(-0.994294\pi\)
0.999839 0.0179248i \(-0.00570594\pi\)
\(984\) −0.139182 −0.00443696
\(985\) 0 0
\(986\) 77.0232 2.45292
\(987\) 12.2034i 0.388439i
\(988\) 7.11691i 0.226419i
\(989\) −50.2480 −1.59779
\(990\) 0 0
\(991\) −53.6513 −1.70429 −0.852144 0.523307i \(-0.824698\pi\)
−0.852144 + 0.523307i \(0.824698\pi\)
\(992\) − 20.2292i − 0.642279i
\(993\) − 15.3461i − 0.486995i
\(994\) 23.2257 0.736674
\(995\) 0 0
\(996\) 0.216327 0.00685460
\(997\) − 35.7275i − 1.13150i −0.824577 0.565750i \(-0.808587\pi\)
0.824577 0.565750i \(-0.191413\pi\)
\(998\) 48.2034i 1.52585i
\(999\) 2.00000 0.0632772
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 825.2.c.g.199.5 6
3.2 odd 2 2475.2.c.r.199.2 6
5.2 odd 4 825.2.a.k.1.1 3
5.3 odd 4 165.2.a.c.1.3 3
5.4 even 2 inner 825.2.c.g.199.2 6
15.2 even 4 2475.2.a.bb.1.3 3
15.8 even 4 495.2.a.e.1.1 3
15.14 odd 2 2475.2.c.r.199.5 6
20.3 even 4 2640.2.a.be.1.3 3
35.13 even 4 8085.2.a.bk.1.3 3
55.32 even 4 9075.2.a.cf.1.3 3
55.43 even 4 1815.2.a.m.1.1 3
60.23 odd 4 7920.2.a.cj.1.3 3
165.98 odd 4 5445.2.a.z.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
165.2.a.c.1.3 3 5.3 odd 4
495.2.a.e.1.1 3 15.8 even 4
825.2.a.k.1.1 3 5.2 odd 4
825.2.c.g.199.2 6 5.4 even 2 inner
825.2.c.g.199.5 6 1.1 even 1 trivial
1815.2.a.m.1.1 3 55.43 even 4
2475.2.a.bb.1.3 3 15.2 even 4
2475.2.c.r.199.2 6 3.2 odd 2
2475.2.c.r.199.5 6 15.14 odd 2
2640.2.a.be.1.3 3 20.3 even 4
5445.2.a.z.1.3 3 165.98 odd 4
7920.2.a.cj.1.3 3 60.23 odd 4
8085.2.a.bk.1.3 3 35.13 even 4
9075.2.a.cf.1.3 3 55.32 even 4