Properties

Label 825.2.a.k.1.1
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: \(x^{3} - x^{2} - 3 x + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 165)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(0.311108\) of defining polynomial
Character \(\chi\) \(=\) 825.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.90321 q^{2} -1.00000 q^{3} +1.62222 q^{4} +1.90321 q^{6} +4.42864 q^{7} +0.719004 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.90321 q^{2} -1.00000 q^{3} +1.62222 q^{4} +1.90321 q^{6} +4.42864 q^{7} +0.719004 q^{8} +1.00000 q^{9} +1.00000 q^{11} -1.62222 q^{12} +0.622216 q^{13} -8.42864 q^{14} -4.61285 q^{16} +5.18421 q^{17} -1.90321 q^{18} +7.05086 q^{19} -4.42864 q^{21} -1.90321 q^{22} -8.85728 q^{23} -0.719004 q^{24} -1.18421 q^{26} -1.00000 q^{27} +7.18421 q^{28} -7.80642 q^{29} +2.75557 q^{31} +7.34122 q^{32} -1.00000 q^{33} -9.86665 q^{34} +1.62222 q^{36} +2.00000 q^{37} -13.4193 q^{38} -0.622216 q^{39} -0.193576 q^{41} +8.42864 q^{42} -5.67307 q^{43} +1.62222 q^{44} +16.8573 q^{46} +2.75557 q^{47} +4.61285 q^{48} +12.6128 q^{49} -5.18421 q^{51} +1.00937 q^{52} +10.8573 q^{53} +1.90321 q^{54} +3.18421 q^{56} -7.05086 q^{57} +14.8573 q^{58} -4.85728 q^{59} +6.85728 q^{61} -5.24443 q^{62} +4.42864 q^{63} -4.74620 q^{64} +1.90321 q^{66} +1.24443 q^{67} +8.40990 q^{68} +8.85728 q^{69} +2.75557 q^{71} +0.719004 q^{72} -4.23506 q^{73} -3.80642 q^{74} +11.4380 q^{76} +4.42864 q^{77} +1.18421 q^{78} +8.56199 q^{79} +1.00000 q^{81} +0.368416 q^{82} -0.133353 q^{83} -7.18421 q^{84} +10.7971 q^{86} +7.80642 q^{87} +0.719004 q^{88} +5.61285 q^{89} +2.75557 q^{91} -14.3684 q^{92} -2.75557 q^{93} -5.24443 q^{94} -7.34122 q^{96} -7.24443 q^{97} -24.0049 q^{98} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + q^{2} - 3 q^{3} + 5 q^{4} - q^{6} + 9 q^{8} + 3 q^{9} + O(q^{10}) \) \( 3 q + q^{2} - 3 q^{3} + 5 q^{4} - q^{6} + 9 q^{8} + 3 q^{9} + 3 q^{11} - 5 q^{12} + 2 q^{13} - 12 q^{14} + 13 q^{16} + 2 q^{17} + q^{18} + 8 q^{19} + q^{22} - 9 q^{24} + 10 q^{26} - 3 q^{27} + 8 q^{28} - 10 q^{29} + 8 q^{31} + 29 q^{32} - 3 q^{33} - 30 q^{34} + 5 q^{36} + 6 q^{37} - 2 q^{39} - 14 q^{41} + 12 q^{42} - 4 q^{43} + 5 q^{44} + 24 q^{46} + 8 q^{47} - 13 q^{48} + 11 q^{49} - 2 q^{51} + 30 q^{52} + 6 q^{53} - q^{54} - 4 q^{56} - 8 q^{57} + 18 q^{58} + 12 q^{59} - 6 q^{61} - 16 q^{62} + 13 q^{64} - q^{66} + 4 q^{67} - 42 q^{68} + 8 q^{71} + 9 q^{72} + 14 q^{73} + 2 q^{74} + 48 q^{76} - 10 q^{78} + 12 q^{79} + 3 q^{81} - 26 q^{82} - 8 q^{84} - 8 q^{86} + 10 q^{87} + 9 q^{88} - 10 q^{89} + 8 q^{91} - 16 q^{92} - 8 q^{93} - 16 q^{94} - 29 q^{96} - 22 q^{97} - 39 q^{98} + 3 q^{99} + O(q^{100}) \)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.90321 −1.34577 −0.672887 0.739745i \(-0.734946\pi\)
−0.672887 + 0.739745i \(0.734946\pi\)
\(3\) −1.00000 −0.577350
\(4\) 1.62222 0.811108
\(5\) 0 0
\(6\) 1.90321 0.776983
\(7\) 4.42864 1.67387 0.836934 0.547304i \(-0.184346\pi\)
0.836934 + 0.547304i \(0.184346\pi\)
\(8\) 0.719004 0.254206
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) −1.62222 −0.468293
\(13\) 0.622216 0.172572 0.0862858 0.996270i \(-0.472500\pi\)
0.0862858 + 0.996270i \(0.472500\pi\)
\(14\) −8.42864 −2.25265
\(15\) 0 0
\(16\) −4.61285 −1.15321
\(17\) 5.18421 1.25736 0.628678 0.777666i \(-0.283597\pi\)
0.628678 + 0.777666i \(0.283597\pi\)
\(18\) −1.90321 −0.448591
\(19\) 7.05086 1.61758 0.808789 0.588100i \(-0.200124\pi\)
0.808789 + 0.588100i \(0.200124\pi\)
\(20\) 0 0
\(21\) −4.42864 −0.966408
\(22\) −1.90321 −0.405766
\(23\) −8.85728 −1.84687 −0.923435 0.383754i \(-0.874631\pi\)
−0.923435 + 0.383754i \(0.874631\pi\)
\(24\) −0.719004 −0.146766
\(25\) 0 0
\(26\) −1.18421 −0.232242
\(27\) −1.00000 −0.192450
\(28\) 7.18421 1.35769
\(29\) −7.80642 −1.44962 −0.724808 0.688951i \(-0.758072\pi\)
−0.724808 + 0.688951i \(0.758072\pi\)
\(30\) 0 0
\(31\) 2.75557 0.494915 0.247457 0.968899i \(-0.420405\pi\)
0.247457 + 0.968899i \(0.420405\pi\)
\(32\) 7.34122 1.29776
\(33\) −1.00000 −0.174078
\(34\) −9.86665 −1.69212
\(35\) 0 0
\(36\) 1.62222 0.270369
\(37\) 2.00000 0.328798 0.164399 0.986394i \(-0.447432\pi\)
0.164399 + 0.986394i \(0.447432\pi\)
\(38\) −13.4193 −2.17689
\(39\) −0.622216 −0.0996342
\(40\) 0 0
\(41\) −0.193576 −0.0302315 −0.0151158 0.999886i \(-0.504812\pi\)
−0.0151158 + 0.999886i \(0.504812\pi\)
\(42\) 8.42864 1.30057
\(43\) −5.67307 −0.865135 −0.432568 0.901602i \(-0.642392\pi\)
−0.432568 + 0.901602i \(0.642392\pi\)
\(44\) 1.62222 0.244558
\(45\) 0 0
\(46\) 16.8573 2.48547
\(47\) 2.75557 0.401941 0.200971 0.979597i \(-0.435590\pi\)
0.200971 + 0.979597i \(0.435590\pi\)
\(48\) 4.61285 0.665807
\(49\) 12.6128 1.80184
\(50\) 0 0
\(51\) −5.18421 −0.725934
\(52\) 1.00937 0.139974
\(53\) 10.8573 1.49136 0.745681 0.666303i \(-0.232124\pi\)
0.745681 + 0.666303i \(0.232124\pi\)
\(54\) 1.90321 0.258994
\(55\) 0 0
\(56\) 3.18421 0.425508
\(57\) −7.05086 −0.933909
\(58\) 14.8573 1.95086
\(59\) −4.85728 −0.632364 −0.316182 0.948699i \(-0.602401\pi\)
−0.316182 + 0.948699i \(0.602401\pi\)
\(60\) 0 0
\(61\) 6.85728 0.877985 0.438992 0.898491i \(-0.355336\pi\)
0.438992 + 0.898491i \(0.355336\pi\)
\(62\) −5.24443 −0.666043
\(63\) 4.42864 0.557956
\(64\) −4.74620 −0.593275
\(65\) 0 0
\(66\) 1.90321 0.234269
\(67\) 1.24443 0.152031 0.0760157 0.997107i \(-0.475780\pi\)
0.0760157 + 0.997107i \(0.475780\pi\)
\(68\) 8.40990 1.01985
\(69\) 8.85728 1.06629
\(70\) 0 0
\(71\) 2.75557 0.327026 0.163513 0.986541i \(-0.447717\pi\)
0.163513 + 0.986541i \(0.447717\pi\)
\(72\) 0.719004 0.0847354
\(73\) −4.23506 −0.495677 −0.247838 0.968801i \(-0.579720\pi\)
−0.247838 + 0.968801i \(0.579720\pi\)
\(74\) −3.80642 −0.442488
\(75\) 0 0
\(76\) 11.4380 1.31203
\(77\) 4.42864 0.504690
\(78\) 1.18421 0.134085
\(79\) 8.56199 0.963299 0.481650 0.876364i \(-0.340038\pi\)
0.481650 + 0.876364i \(0.340038\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0.368416 0.0406848
\(83\) −0.133353 −0.0146374 −0.00731870 0.999973i \(-0.502330\pi\)
−0.00731870 + 0.999973i \(0.502330\pi\)
\(84\) −7.18421 −0.783861
\(85\) 0 0
\(86\) 10.7971 1.16428
\(87\) 7.80642 0.836936
\(88\) 0.719004 0.0766461
\(89\) 5.61285 0.594961 0.297480 0.954728i \(-0.403854\pi\)
0.297480 + 0.954728i \(0.403854\pi\)
\(90\) 0 0
\(91\) 2.75557 0.288862
\(92\) −14.3684 −1.49801
\(93\) −2.75557 −0.285739
\(94\) −5.24443 −0.540922
\(95\) 0 0
\(96\) −7.34122 −0.749260
\(97\) −7.24443 −0.735561 −0.367780 0.929913i \(-0.619882\pi\)
−0.367780 + 0.929913i \(0.619882\pi\)
\(98\) −24.0049 −2.42486
\(99\) 1.00000 0.100504
\(100\) 0 0
\(101\) 4.66370 0.464056 0.232028 0.972709i \(-0.425464\pi\)
0.232028 + 0.972709i \(0.425464\pi\)
\(102\) 9.86665 0.976944
\(103\) 11.6128 1.14425 0.572124 0.820167i \(-0.306120\pi\)
0.572124 + 0.820167i \(0.306120\pi\)
\(104\) 0.447375 0.0438688
\(105\) 0 0
\(106\) −20.6637 −2.00704
\(107\) −2.62222 −0.253499 −0.126750 0.991935i \(-0.540454\pi\)
−0.126750 + 0.991935i \(0.540454\pi\)
\(108\) −1.62222 −0.156098
\(109\) −19.7146 −1.88831 −0.944156 0.329499i \(-0.893120\pi\)
−0.944156 + 0.329499i \(0.893120\pi\)
\(110\) 0 0
\(111\) −2.00000 −0.189832
\(112\) −20.4286 −1.93032
\(113\) 6.00000 0.564433 0.282216 0.959351i \(-0.408930\pi\)
0.282216 + 0.959351i \(0.408930\pi\)
\(114\) 13.4193 1.25683
\(115\) 0 0
\(116\) −12.6637 −1.17580
\(117\) 0.622216 0.0575239
\(118\) 9.24443 0.851019
\(119\) 22.9590 2.10465
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) −13.0509 −1.18157
\(123\) 0.193576 0.0174542
\(124\) 4.47013 0.401429
\(125\) 0 0
\(126\) −8.42864 −0.750883
\(127\) 15.1842 1.34738 0.673690 0.739014i \(-0.264708\pi\)
0.673690 + 0.739014i \(0.264708\pi\)
\(128\) −5.64941 −0.499342
\(129\) 5.67307 0.499486
\(130\) 0 0
\(131\) 1.24443 0.108726 0.0543632 0.998521i \(-0.482687\pi\)
0.0543632 + 0.998521i \(0.482687\pi\)
\(132\) −1.62222 −0.141196
\(133\) 31.2257 2.70761
\(134\) −2.36842 −0.204600
\(135\) 0 0
\(136\) 3.72746 0.319627
\(137\) 0.488863 0.0417663 0.0208832 0.999782i \(-0.493352\pi\)
0.0208832 + 0.999782i \(0.493352\pi\)
\(138\) −16.8573 −1.43499
\(139\) 17.8064 1.51032 0.755161 0.655540i \(-0.227559\pi\)
0.755161 + 0.655540i \(0.227559\pi\)
\(140\) 0 0
\(141\) −2.75557 −0.232061
\(142\) −5.24443 −0.440103
\(143\) 0.622216 0.0520323
\(144\) −4.61285 −0.384404
\(145\) 0 0
\(146\) 8.06022 0.667069
\(147\) −12.6128 −1.04029
\(148\) 3.24443 0.266691
\(149\) −1.43801 −0.117806 −0.0589031 0.998264i \(-0.518760\pi\)
−0.0589031 + 0.998264i \(0.518760\pi\)
\(150\) 0 0
\(151\) −12.1748 −0.990774 −0.495387 0.868672i \(-0.664974\pi\)
−0.495387 + 0.868672i \(0.664974\pi\)
\(152\) 5.06959 0.411198
\(153\) 5.18421 0.419118
\(154\) −8.42864 −0.679199
\(155\) 0 0
\(156\) −1.00937 −0.0808141
\(157\) −18.4701 −1.47408 −0.737038 0.675851i \(-0.763776\pi\)
−0.737038 + 0.675851i \(0.763776\pi\)
\(158\) −16.2953 −1.29638
\(159\) −10.8573 −0.861038
\(160\) 0 0
\(161\) −39.2257 −3.09142
\(162\) −1.90321 −0.149530
\(163\) 10.1017 0.791227 0.395614 0.918417i \(-0.370532\pi\)
0.395614 + 0.918417i \(0.370532\pi\)
\(164\) −0.314022 −0.0245210
\(165\) 0 0
\(166\) 0.253799 0.0196986
\(167\) 16.3368 1.26418 0.632089 0.774896i \(-0.282198\pi\)
0.632089 + 0.774896i \(0.282198\pi\)
\(168\) −3.18421 −0.245667
\(169\) −12.6128 −0.970219
\(170\) 0 0
\(171\) 7.05086 0.539192
\(172\) −9.20294 −0.701718
\(173\) 9.18421 0.698262 0.349131 0.937074i \(-0.386477\pi\)
0.349131 + 0.937074i \(0.386477\pi\)
\(174\) −14.8573 −1.12633
\(175\) 0 0
\(176\) −4.61285 −0.347706
\(177\) 4.85728 0.365095
\(178\) −10.6824 −0.800683
\(179\) −25.3274 −1.89306 −0.946530 0.322617i \(-0.895437\pi\)
−0.946530 + 0.322617i \(0.895437\pi\)
\(180\) 0 0
\(181\) −13.6128 −1.01184 −0.505918 0.862582i \(-0.668846\pi\)
−0.505918 + 0.862582i \(0.668846\pi\)
\(182\) −5.24443 −0.388743
\(183\) −6.85728 −0.506905
\(184\) −6.36842 −0.469486
\(185\) 0 0
\(186\) 5.24443 0.384540
\(187\) 5.18421 0.379107
\(188\) 4.47013 0.326017
\(189\) −4.42864 −0.322136
\(190\) 0 0
\(191\) 6.10171 0.441504 0.220752 0.975330i \(-0.429149\pi\)
0.220752 + 0.975330i \(0.429149\pi\)
\(192\) 4.74620 0.342528
\(193\) −18.3368 −1.31991 −0.659955 0.751305i \(-0.729425\pi\)
−0.659955 + 0.751305i \(0.729425\pi\)
\(194\) 13.7877 0.989898
\(195\) 0 0
\(196\) 20.4608 1.46148
\(197\) 6.69535 0.477024 0.238512 0.971140i \(-0.423340\pi\)
0.238512 + 0.971140i \(0.423340\pi\)
\(198\) −1.90321 −0.135255
\(199\) 14.1017 0.999644 0.499822 0.866128i \(-0.333399\pi\)
0.499822 + 0.866128i \(0.333399\pi\)
\(200\) 0 0
\(201\) −1.24443 −0.0877754
\(202\) −8.87601 −0.624514
\(203\) −34.5718 −2.42647
\(204\) −8.40990 −0.588811
\(205\) 0 0
\(206\) −22.1017 −1.53990
\(207\) −8.85728 −0.615623
\(208\) −2.87019 −0.199012
\(209\) 7.05086 0.487718
\(210\) 0 0
\(211\) −10.6637 −0.734120 −0.367060 0.930197i \(-0.619636\pi\)
−0.367060 + 0.930197i \(0.619636\pi\)
\(212\) 17.6128 1.20966
\(213\) −2.75557 −0.188808
\(214\) 4.99063 0.341153
\(215\) 0 0
\(216\) −0.719004 −0.0489220
\(217\) 12.2034 0.828422
\(218\) 37.5210 2.54124
\(219\) 4.23506 0.286179
\(220\) 0 0
\(221\) 3.22570 0.216984
\(222\) 3.80642 0.255470
\(223\) −8.85728 −0.593127 −0.296564 0.955013i \(-0.595841\pi\)
−0.296564 + 0.955013i \(0.595841\pi\)
\(224\) 32.5116 2.17227
\(225\) 0 0
\(226\) −11.4193 −0.759599
\(227\) −13.3778 −0.887915 −0.443957 0.896048i \(-0.646426\pi\)
−0.443957 + 0.896048i \(0.646426\pi\)
\(228\) −11.4380 −0.757501
\(229\) 11.5111 0.760677 0.380339 0.924847i \(-0.375807\pi\)
0.380339 + 0.924847i \(0.375807\pi\)
\(230\) 0 0
\(231\) −4.42864 −0.291383
\(232\) −5.61285 −0.368502
\(233\) 4.32693 0.283467 0.141733 0.989905i \(-0.454732\pi\)
0.141733 + 0.989905i \(0.454732\pi\)
\(234\) −1.18421 −0.0774141
\(235\) 0 0
\(236\) −7.87955 −0.512915
\(237\) −8.56199 −0.556161
\(238\) −43.6958 −2.83238
\(239\) −3.34614 −0.216444 −0.108222 0.994127i \(-0.534516\pi\)
−0.108222 + 0.994127i \(0.534516\pi\)
\(240\) 0 0
\(241\) −1.34614 −0.0867126 −0.0433563 0.999060i \(-0.513805\pi\)
−0.0433563 + 0.999060i \(0.513805\pi\)
\(242\) −1.90321 −0.122343
\(243\) −1.00000 −0.0641500
\(244\) 11.1240 0.712140
\(245\) 0 0
\(246\) −0.368416 −0.0234894
\(247\) 4.38715 0.279148
\(248\) 1.98126 0.125810
\(249\) 0.133353 0.00845091
\(250\) 0 0
\(251\) 22.7556 1.43632 0.718159 0.695879i \(-0.244985\pi\)
0.718159 + 0.695879i \(0.244985\pi\)
\(252\) 7.18421 0.452563
\(253\) −8.85728 −0.556852
\(254\) −28.8988 −1.81327
\(255\) 0 0
\(256\) 20.2444 1.26528
\(257\) 6.85728 0.427745 0.213873 0.976862i \(-0.431392\pi\)
0.213873 + 0.976862i \(0.431392\pi\)
\(258\) −10.7971 −0.672195
\(259\) 8.85728 0.550365
\(260\) 0 0
\(261\) −7.80642 −0.483206
\(262\) −2.36842 −0.146321
\(263\) 29.5812 1.82406 0.912028 0.410129i \(-0.134516\pi\)
0.912028 + 0.410129i \(0.134516\pi\)
\(264\) −0.719004 −0.0442516
\(265\) 0 0
\(266\) −59.4291 −3.64383
\(267\) −5.61285 −0.343501
\(268\) 2.01874 0.123314
\(269\) 8.48886 0.517575 0.258788 0.965934i \(-0.416677\pi\)
0.258788 + 0.965934i \(0.416677\pi\)
\(270\) 0 0
\(271\) 14.6637 0.890757 0.445378 0.895343i \(-0.353069\pi\)
0.445378 + 0.895343i \(0.353069\pi\)
\(272\) −23.9140 −1.45000
\(273\) −2.75557 −0.166775
\(274\) −0.930409 −0.0562081
\(275\) 0 0
\(276\) 14.3684 0.864877
\(277\) −14.6035 −0.877438 −0.438719 0.898624i \(-0.644568\pi\)
−0.438719 + 0.898624i \(0.644568\pi\)
\(278\) −33.8894 −2.03255
\(279\) 2.75557 0.164972
\(280\) 0 0
\(281\) −0.193576 −0.0115478 −0.00577389 0.999983i \(-0.501838\pi\)
−0.00577389 + 0.999983i \(0.501838\pi\)
\(282\) 5.24443 0.312301
\(283\) −27.1842 −1.61593 −0.807967 0.589228i \(-0.799432\pi\)
−0.807967 + 0.589228i \(0.799432\pi\)
\(284\) 4.47013 0.265253
\(285\) 0 0
\(286\) −1.18421 −0.0700237
\(287\) −0.857279 −0.0506036
\(288\) 7.34122 0.432585
\(289\) 9.87601 0.580942
\(290\) 0 0
\(291\) 7.24443 0.424676
\(292\) −6.87019 −0.402047
\(293\) 2.81579 0.164500 0.0822502 0.996612i \(-0.473789\pi\)
0.0822502 + 0.996612i \(0.473789\pi\)
\(294\) 24.0049 1.40000
\(295\) 0 0
\(296\) 1.43801 0.0835825
\(297\) −1.00000 −0.0580259
\(298\) 2.73683 0.158540
\(299\) −5.51114 −0.318717
\(300\) 0 0
\(301\) −25.1240 −1.44812
\(302\) 23.1713 1.33336
\(303\) −4.66370 −0.267923
\(304\) −32.5245 −1.86541
\(305\) 0 0
\(306\) −9.86665 −0.564039
\(307\) 24.4286 1.39422 0.697108 0.716966i \(-0.254470\pi\)
0.697108 + 0.716966i \(0.254470\pi\)
\(308\) 7.18421 0.409358
\(309\) −11.6128 −0.660632
\(310\) 0 0
\(311\) 19.8796 1.12727 0.563633 0.826025i \(-0.309403\pi\)
0.563633 + 0.826025i \(0.309403\pi\)
\(312\) −0.447375 −0.0253276
\(313\) 15.7146 0.888239 0.444120 0.895967i \(-0.353517\pi\)
0.444120 + 0.895967i \(0.353517\pi\)
\(314\) 35.1526 1.98377
\(315\) 0 0
\(316\) 13.8894 0.781340
\(317\) −16.4889 −0.926107 −0.463053 0.886330i \(-0.653246\pi\)
−0.463053 + 0.886330i \(0.653246\pi\)
\(318\) 20.6637 1.15876
\(319\) −7.80642 −0.437076
\(320\) 0 0
\(321\) 2.62222 0.146358
\(322\) 74.6548 4.16035
\(323\) 36.5531 2.03387
\(324\) 1.62222 0.0901231
\(325\) 0 0
\(326\) −19.2257 −1.06481
\(327\) 19.7146 1.09022
\(328\) −0.139182 −0.00768504
\(329\) 12.2034 0.672796
\(330\) 0 0
\(331\) 15.3461 0.843500 0.421750 0.906712i \(-0.361416\pi\)
0.421750 + 0.906712i \(0.361416\pi\)
\(332\) −0.216327 −0.0118725
\(333\) 2.00000 0.109599
\(334\) −31.0923 −1.70130
\(335\) 0 0
\(336\) 20.4286 1.11447
\(337\) −28.2351 −1.53806 −0.769031 0.639212i \(-0.779261\pi\)
−0.769031 + 0.639212i \(0.779261\pi\)
\(338\) 24.0049 1.30570
\(339\) −6.00000 −0.325875
\(340\) 0 0
\(341\) 2.75557 0.149222
\(342\) −13.4193 −0.725631
\(343\) 24.8573 1.34217
\(344\) −4.07896 −0.219923
\(345\) 0 0
\(346\) −17.4795 −0.939703
\(347\) −2.62222 −0.140768 −0.0703840 0.997520i \(-0.522422\pi\)
−0.0703840 + 0.997520i \(0.522422\pi\)
\(348\) 12.6637 0.678846
\(349\) 5.14272 0.275284 0.137642 0.990482i \(-0.456048\pi\)
0.137642 + 0.990482i \(0.456048\pi\)
\(350\) 0 0
\(351\) −0.622216 −0.0332114
\(352\) 7.34122 0.391288
\(353\) 9.34614 0.497445 0.248722 0.968575i \(-0.419989\pi\)
0.248722 + 0.968575i \(0.419989\pi\)
\(354\) −9.24443 −0.491336
\(355\) 0 0
\(356\) 9.10525 0.482577
\(357\) −22.9590 −1.21512
\(358\) 48.2034 2.54763
\(359\) 10.7556 0.567657 0.283829 0.958875i \(-0.408395\pi\)
0.283829 + 0.958875i \(0.408395\pi\)
\(360\) 0 0
\(361\) 30.7146 1.61656
\(362\) 25.9081 1.36170
\(363\) −1.00000 −0.0524864
\(364\) 4.47013 0.234298
\(365\) 0 0
\(366\) 13.0509 0.682179
\(367\) −33.7975 −1.76422 −0.882108 0.471046i \(-0.843876\pi\)
−0.882108 + 0.471046i \(0.843876\pi\)
\(368\) 40.8573 2.12983
\(369\) −0.193576 −0.0100772
\(370\) 0 0
\(371\) 48.0830 2.49634
\(372\) −4.47013 −0.231765
\(373\) −33.9496 −1.75784 −0.878922 0.476965i \(-0.841737\pi\)
−0.878922 + 0.476965i \(0.841737\pi\)
\(374\) −9.86665 −0.510192
\(375\) 0 0
\(376\) 1.98126 0.102176
\(377\) −4.85728 −0.250163
\(378\) 8.42864 0.433522
\(379\) −20.0000 −1.02733 −0.513665 0.857991i \(-0.671713\pi\)
−0.513665 + 0.857991i \(0.671713\pi\)
\(380\) 0 0
\(381\) −15.1842 −0.777911
\(382\) −11.6128 −0.594165
\(383\) 14.6351 0.747820 0.373910 0.927465i \(-0.378017\pi\)
0.373910 + 0.927465i \(0.378017\pi\)
\(384\) 5.64941 0.288295
\(385\) 0 0
\(386\) 34.8988 1.77630
\(387\) −5.67307 −0.288378
\(388\) −11.7520 −0.596619
\(389\) −5.61285 −0.284583 −0.142291 0.989825i \(-0.545447\pi\)
−0.142291 + 0.989825i \(0.545447\pi\)
\(390\) 0 0
\(391\) −45.9180 −2.32217
\(392\) 9.06868 0.458038
\(393\) −1.24443 −0.0627733
\(394\) −12.7427 −0.641966
\(395\) 0 0
\(396\) 1.62222 0.0815194
\(397\) 12.7556 0.640184 0.320092 0.947387i \(-0.396286\pi\)
0.320092 + 0.947387i \(0.396286\pi\)
\(398\) −26.8385 −1.34529
\(399\) −31.2257 −1.56324
\(400\) 0 0
\(401\) 2.00000 0.0998752 0.0499376 0.998752i \(-0.484098\pi\)
0.0499376 + 0.998752i \(0.484098\pi\)
\(402\) 2.36842 0.118126
\(403\) 1.71456 0.0854082
\(404\) 7.56553 0.376399
\(405\) 0 0
\(406\) 65.7975 3.26548
\(407\) 2.00000 0.0991363
\(408\) −3.72746 −0.184537
\(409\) −7.12399 −0.352258 −0.176129 0.984367i \(-0.556358\pi\)
−0.176129 + 0.984367i \(0.556358\pi\)
\(410\) 0 0
\(411\) −0.488863 −0.0241138
\(412\) 18.8385 0.928108
\(413\) −21.5111 −1.05849
\(414\) 16.8573 0.828490
\(415\) 0 0
\(416\) 4.56782 0.223956
\(417\) −17.8064 −0.871984
\(418\) −13.4193 −0.656358
\(419\) 15.6128 0.762738 0.381369 0.924423i \(-0.375453\pi\)
0.381369 + 0.924423i \(0.375453\pi\)
\(420\) 0 0
\(421\) 7.89829 0.384939 0.192470 0.981303i \(-0.438350\pi\)
0.192470 + 0.981303i \(0.438350\pi\)
\(422\) 20.2953 0.987959
\(423\) 2.75557 0.133980
\(424\) 7.80642 0.379113
\(425\) 0 0
\(426\) 5.24443 0.254094
\(427\) 30.3684 1.46963
\(428\) −4.25380 −0.205615
\(429\) −0.622216 −0.0300409
\(430\) 0 0
\(431\) 34.3051 1.65242 0.826210 0.563362i \(-0.190492\pi\)
0.826210 + 0.563362i \(0.190492\pi\)
\(432\) 4.61285 0.221936
\(433\) −14.4701 −0.695390 −0.347695 0.937608i \(-0.613036\pi\)
−0.347695 + 0.937608i \(0.613036\pi\)
\(434\) −23.2257 −1.11487
\(435\) 0 0
\(436\) −31.9813 −1.53162
\(437\) −62.4514 −2.98746
\(438\) −8.06022 −0.385132
\(439\) −19.3176 −0.921977 −0.460988 0.887406i \(-0.652505\pi\)
−0.460988 + 0.887406i \(0.652505\pi\)
\(440\) 0 0
\(441\) 12.6128 0.600612
\(442\) −6.13918 −0.292011
\(443\) 13.1240 0.623539 0.311770 0.950158i \(-0.399078\pi\)
0.311770 + 0.950158i \(0.399078\pi\)
\(444\) −3.24443 −0.153974
\(445\) 0 0
\(446\) 16.8573 0.798215
\(447\) 1.43801 0.0680154
\(448\) −21.0192 −0.993064
\(449\) −32.3051 −1.52457 −0.762287 0.647240i \(-0.775923\pi\)
−0.762287 + 0.647240i \(0.775923\pi\)
\(450\) 0 0
\(451\) −0.193576 −0.00911514
\(452\) 9.73329 0.457816
\(453\) 12.1748 0.572024
\(454\) 25.4608 1.19493
\(455\) 0 0
\(456\) −5.06959 −0.237405
\(457\) 23.4608 1.09745 0.548724 0.836004i \(-0.315114\pi\)
0.548724 + 0.836004i \(0.315114\pi\)
\(458\) −21.9081 −1.02370
\(459\) −5.18421 −0.241978
\(460\) 0 0
\(461\) −28.8671 −1.34448 −0.672238 0.740335i \(-0.734667\pi\)
−0.672238 + 0.740335i \(0.734667\pi\)
\(462\) 8.42864 0.392136
\(463\) 19.3461 0.899091 0.449546 0.893257i \(-0.351586\pi\)
0.449546 + 0.893257i \(0.351586\pi\)
\(464\) 36.0098 1.67172
\(465\) 0 0
\(466\) −8.23506 −0.381482
\(467\) −3.14272 −0.145428 −0.0727139 0.997353i \(-0.523166\pi\)
−0.0727139 + 0.997353i \(0.523166\pi\)
\(468\) 1.00937 0.0466580
\(469\) 5.51114 0.254481
\(470\) 0 0
\(471\) 18.4701 0.851059
\(472\) −3.49240 −0.160751
\(473\) −5.67307 −0.260848
\(474\) 16.2953 0.748467
\(475\) 0 0
\(476\) 37.2444 1.70710
\(477\) 10.8573 0.497121
\(478\) 6.36842 0.291285
\(479\) −24.8573 −1.13576 −0.567879 0.823112i \(-0.692236\pi\)
−0.567879 + 0.823112i \(0.692236\pi\)
\(480\) 0 0
\(481\) 1.24443 0.0567412
\(482\) 2.56199 0.116696
\(483\) 39.2257 1.78483
\(484\) 1.62222 0.0737371
\(485\) 0 0
\(486\) 1.90321 0.0863314
\(487\) 11.5299 0.522468 0.261234 0.965275i \(-0.415871\pi\)
0.261234 + 0.965275i \(0.415871\pi\)
\(488\) 4.93041 0.223189
\(489\) −10.1017 −0.456815
\(490\) 0 0
\(491\) 16.3872 0.739542 0.369771 0.929123i \(-0.379436\pi\)
0.369771 + 0.929123i \(0.379436\pi\)
\(492\) 0.314022 0.0141572
\(493\) −40.4701 −1.82268
\(494\) −8.34968 −0.375670
\(495\) 0 0
\(496\) −12.7110 −0.570742
\(497\) 12.2034 0.547398
\(498\) −0.253799 −0.0113730
\(499\) −25.3274 −1.13381 −0.566905 0.823783i \(-0.691859\pi\)
−0.566905 + 0.823783i \(0.691859\pi\)
\(500\) 0 0
\(501\) −16.3368 −0.729873
\(502\) −43.3087 −1.93296
\(503\) −19.0923 −0.851285 −0.425643 0.904891i \(-0.639952\pi\)
−0.425643 + 0.904891i \(0.639952\pi\)
\(504\) 3.18421 0.141836
\(505\) 0 0
\(506\) 16.8573 0.749397
\(507\) 12.6128 0.560156
\(508\) 24.6321 1.09287
\(509\) −32.4514 −1.43838 −0.719191 0.694812i \(-0.755488\pi\)
−0.719191 + 0.694812i \(0.755488\pi\)
\(510\) 0 0
\(511\) −18.7556 −0.829698
\(512\) −27.2306 −1.20343
\(513\) −7.05086 −0.311303
\(514\) −13.0509 −0.575649
\(515\) 0 0
\(516\) 9.20294 0.405137
\(517\) 2.75557 0.121190
\(518\) −16.8573 −0.740666
\(519\) −9.18421 −0.403142
\(520\) 0 0
\(521\) −29.2257 −1.28040 −0.640200 0.768208i \(-0.721149\pi\)
−0.640200 + 0.768208i \(0.721149\pi\)
\(522\) 14.8573 0.650285
\(523\) −6.71408 −0.293586 −0.146793 0.989167i \(-0.546895\pi\)
−0.146793 + 0.989167i \(0.546895\pi\)
\(524\) 2.01874 0.0881889
\(525\) 0 0
\(526\) −56.2993 −2.45477
\(527\) 14.2854 0.622284
\(528\) 4.61285 0.200748
\(529\) 55.4514 2.41093
\(530\) 0 0
\(531\) −4.85728 −0.210788
\(532\) 50.6548 2.19616
\(533\) −0.120446 −0.00521710
\(534\) 10.6824 0.462274
\(535\) 0 0
\(536\) 0.894751 0.0386473
\(537\) 25.3274 1.09296
\(538\) −16.1561 −0.696539
\(539\) 12.6128 0.543274
\(540\) 0 0
\(541\) −18.0000 −0.773880 −0.386940 0.922105i \(-0.626468\pi\)
−0.386940 + 0.922105i \(0.626468\pi\)
\(542\) −27.9081 −1.19876
\(543\) 13.6128 0.584183
\(544\) 38.0584 1.63174
\(545\) 0 0
\(546\) 5.24443 0.224441
\(547\) −41.3689 −1.76881 −0.884403 0.466724i \(-0.845434\pi\)
−0.884403 + 0.466724i \(0.845434\pi\)
\(548\) 0.793040 0.0338770
\(549\) 6.85728 0.292662
\(550\) 0 0
\(551\) −55.0420 −2.34487
\(552\) 6.36842 0.271058
\(553\) 37.9180 1.61244
\(554\) 27.7935 1.18083
\(555\) 0 0
\(556\) 28.8859 1.22503
\(557\) 20.7971 0.881200 0.440600 0.897704i \(-0.354766\pi\)
0.440600 + 0.897704i \(0.354766\pi\)
\(558\) −5.24443 −0.222014
\(559\) −3.52987 −0.149298
\(560\) 0 0
\(561\) −5.18421 −0.218877
\(562\) 0.368416 0.0155407
\(563\) −37.7275 −1.59002 −0.795012 0.606594i \(-0.792535\pi\)
−0.795012 + 0.606594i \(0.792535\pi\)
\(564\) −4.47013 −0.188226
\(565\) 0 0
\(566\) 51.7373 2.17468
\(567\) 4.42864 0.185985
\(568\) 1.98126 0.0831320
\(569\) −7.33630 −0.307554 −0.153777 0.988106i \(-0.549144\pi\)
−0.153777 + 0.988106i \(0.549144\pi\)
\(570\) 0 0
\(571\) 36.6450 1.53354 0.766772 0.641919i \(-0.221862\pi\)
0.766772 + 0.641919i \(0.221862\pi\)
\(572\) 1.00937 0.0422038
\(573\) −6.10171 −0.254903
\(574\) 1.63158 0.0681010
\(575\) 0 0
\(576\) −4.74620 −0.197758
\(577\) −4.22216 −0.175771 −0.0878853 0.996131i \(-0.528011\pi\)
−0.0878853 + 0.996131i \(0.528011\pi\)
\(578\) −18.7961 −0.781817
\(579\) 18.3368 0.762050
\(580\) 0 0
\(581\) −0.590573 −0.0245011
\(582\) −13.7877 −0.571518
\(583\) 10.8573 0.449663
\(584\) −3.04503 −0.126004
\(585\) 0 0
\(586\) −5.35905 −0.221380
\(587\) −34.3684 −1.41854 −0.709268 0.704939i \(-0.750974\pi\)
−0.709268 + 0.704939i \(0.750974\pi\)
\(588\) −20.4608 −0.843787
\(589\) 19.4291 0.800563
\(590\) 0 0
\(591\) −6.69535 −0.275410
\(592\) −9.22570 −0.379174
\(593\) −27.9398 −1.14735 −0.573675 0.819083i \(-0.694483\pi\)
−0.573675 + 0.819083i \(0.694483\pi\)
\(594\) 1.90321 0.0780897
\(595\) 0 0
\(596\) −2.33276 −0.0955535
\(597\) −14.1017 −0.577145
\(598\) 10.4889 0.428921
\(599\) −31.2257 −1.27585 −0.637924 0.770100i \(-0.720206\pi\)
−0.637924 + 0.770100i \(0.720206\pi\)
\(600\) 0 0
\(601\) −8.75557 −0.357147 −0.178574 0.983927i \(-0.557148\pi\)
−0.178574 + 0.983927i \(0.557148\pi\)
\(602\) 47.8163 1.94885
\(603\) 1.24443 0.0506772
\(604\) −19.7502 −0.803625
\(605\) 0 0
\(606\) 8.87601 0.360563
\(607\) 15.1842 0.616308 0.308154 0.951336i \(-0.400289\pi\)
0.308154 + 0.951336i \(0.400289\pi\)
\(608\) 51.7619 2.09922
\(609\) 34.5718 1.40092
\(610\) 0 0
\(611\) 1.71456 0.0693636
\(612\) 8.40990 0.339950
\(613\) −42.7239 −1.72560 −0.862802 0.505543i \(-0.831292\pi\)
−0.862802 + 0.505543i \(0.831292\pi\)
\(614\) −46.4929 −1.87630
\(615\) 0 0
\(616\) 3.18421 0.128295
\(617\) 3.51114 0.141353 0.0706765 0.997499i \(-0.477484\pi\)
0.0706765 + 0.997499i \(0.477484\pi\)
\(618\) 22.1017 0.889061
\(619\) 17.5941 0.707167 0.353584 0.935403i \(-0.384963\pi\)
0.353584 + 0.935403i \(0.384963\pi\)
\(620\) 0 0
\(621\) 8.85728 0.355430
\(622\) −37.8350 −1.51705
\(623\) 24.8573 0.995886
\(624\) 2.87019 0.114899
\(625\) 0 0
\(626\) −29.9081 −1.19537
\(627\) −7.05086 −0.281584
\(628\) −29.9625 −1.19564
\(629\) 10.3684 0.413416
\(630\) 0 0
\(631\) 15.8163 0.629636 0.314818 0.949152i \(-0.398057\pi\)
0.314818 + 0.949152i \(0.398057\pi\)
\(632\) 6.15610 0.244877
\(633\) 10.6637 0.423844
\(634\) 31.3818 1.24633
\(635\) 0 0
\(636\) −17.6128 −0.698395
\(637\) 7.84791 0.310946
\(638\) 14.8573 0.588205
\(639\) 2.75557 0.109009
\(640\) 0 0
\(641\) 25.8163 1.01968 0.509841 0.860269i \(-0.329704\pi\)
0.509841 + 0.860269i \(0.329704\pi\)
\(642\) −4.99063 −0.196965
\(643\) −18.1017 −0.713862 −0.356931 0.934131i \(-0.616177\pi\)
−0.356931 + 0.934131i \(0.616177\pi\)
\(644\) −63.6325 −2.50747
\(645\) 0 0
\(646\) −69.5683 −2.73713
\(647\) 47.0420 1.84941 0.924705 0.380684i \(-0.124311\pi\)
0.924705 + 0.380684i \(0.124311\pi\)
\(648\) 0.719004 0.0282451
\(649\) −4.85728 −0.190665
\(650\) 0 0
\(651\) −12.2034 −0.478290
\(652\) 16.3872 0.641770
\(653\) −30.0830 −1.17724 −0.588619 0.808411i \(-0.700328\pi\)
−0.588619 + 0.808411i \(0.700328\pi\)
\(654\) −37.5210 −1.46719
\(655\) 0 0
\(656\) 0.892937 0.0348633
\(657\) −4.23506 −0.165226
\(658\) −23.2257 −0.905432
\(659\) −10.2854 −0.400664 −0.200332 0.979728i \(-0.564202\pi\)
−0.200332 + 0.979728i \(0.564202\pi\)
\(660\) 0 0
\(661\) −27.7146 −1.07797 −0.538986 0.842315i \(-0.681192\pi\)
−0.538986 + 0.842315i \(0.681192\pi\)
\(662\) −29.2070 −1.13516
\(663\) −3.22570 −0.125276
\(664\) −0.0958814 −0.00372092
\(665\) 0 0
\(666\) −3.80642 −0.147496
\(667\) 69.1437 2.67725
\(668\) 26.5018 1.02538
\(669\) 8.85728 0.342442
\(670\) 0 0
\(671\) 6.85728 0.264722
\(672\) −32.5116 −1.25416
\(673\) 9.86665 0.380331 0.190166 0.981752i \(-0.439097\pi\)
0.190166 + 0.981752i \(0.439097\pi\)
\(674\) 53.7373 2.06988
\(675\) 0 0
\(676\) −20.4608 −0.786952
\(677\) 5.65433 0.217314 0.108657 0.994079i \(-0.465345\pi\)
0.108657 + 0.994079i \(0.465345\pi\)
\(678\) 11.4193 0.438554
\(679\) −32.0830 −1.23123
\(680\) 0 0
\(681\) 13.3778 0.512638
\(682\) −5.24443 −0.200820
\(683\) −34.1847 −1.30804 −0.654020 0.756477i \(-0.726919\pi\)
−0.654020 + 0.756477i \(0.726919\pi\)
\(684\) 11.4380 0.437343
\(685\) 0 0
\(686\) −47.3087 −1.80625
\(687\) −11.5111 −0.439177
\(688\) 26.1690 0.997684
\(689\) 6.75557 0.257367
\(690\) 0 0
\(691\) −19.2257 −0.731380 −0.365690 0.930737i \(-0.619167\pi\)
−0.365690 + 0.930737i \(0.619167\pi\)
\(692\) 14.8988 0.566366
\(693\) 4.42864 0.168230
\(694\) 4.99063 0.189442
\(695\) 0 0
\(696\) 5.61285 0.212754
\(697\) −1.00354 −0.0380118
\(698\) −9.78769 −0.370469
\(699\) −4.32693 −0.163659
\(700\) 0 0
\(701\) −29.9081 −1.12961 −0.564807 0.825223i \(-0.691050\pi\)
−0.564807 + 0.825223i \(0.691050\pi\)
\(702\) 1.18421 0.0446951
\(703\) 14.1017 0.531856
\(704\) −4.74620 −0.178879
\(705\) 0 0
\(706\) −17.7877 −0.669448
\(707\) 20.6539 0.776768
\(708\) 7.87955 0.296132
\(709\) −15.3274 −0.575633 −0.287816 0.957686i \(-0.592929\pi\)
−0.287816 + 0.957686i \(0.592929\pi\)
\(710\) 0 0
\(711\) 8.56199 0.321100
\(712\) 4.03566 0.151243
\(713\) −24.4068 −0.914043
\(714\) 43.6958 1.63528
\(715\) 0 0
\(716\) −41.0865 −1.53548
\(717\) 3.34614 0.124964
\(718\) −20.4701 −0.763938
\(719\) 23.8163 0.888197 0.444098 0.895978i \(-0.353524\pi\)
0.444098 + 0.895978i \(0.353524\pi\)
\(720\) 0 0
\(721\) 51.4291 1.91532
\(722\) −58.4563 −2.17552
\(723\) 1.34614 0.0500635
\(724\) −22.0830 −0.820707
\(725\) 0 0
\(726\) 1.90321 0.0706348
\(727\) 32.9403 1.22169 0.610843 0.791752i \(-0.290831\pi\)
0.610843 + 0.791752i \(0.290831\pi\)
\(728\) 1.98126 0.0734305
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −29.4104 −1.08778
\(732\) −11.1240 −0.411154
\(733\) 29.8666 1.10315 0.551575 0.834125i \(-0.314027\pi\)
0.551575 + 0.834125i \(0.314027\pi\)
\(734\) 64.3239 2.37424
\(735\) 0 0
\(736\) −65.0232 −2.39679
\(737\) 1.24443 0.0458392
\(738\) 0.368416 0.0135616
\(739\) −5.06959 −0.186488 −0.0932440 0.995643i \(-0.529724\pi\)
−0.0932440 + 0.995643i \(0.529724\pi\)
\(740\) 0 0
\(741\) −4.38715 −0.161166
\(742\) −91.5121 −3.35951
\(743\) −22.4385 −0.823188 −0.411594 0.911367i \(-0.635028\pi\)
−0.411594 + 0.911367i \(0.635028\pi\)
\(744\) −1.98126 −0.0726367
\(745\) 0 0
\(746\) 64.6133 2.36566
\(747\) −0.133353 −0.00487913
\(748\) 8.40990 0.307497
\(749\) −11.6128 −0.424324
\(750\) 0 0
\(751\) −6.63512 −0.242119 −0.121060 0.992645i \(-0.538629\pi\)
−0.121060 + 0.992645i \(0.538629\pi\)
\(752\) −12.7110 −0.463523
\(753\) −22.7556 −0.829259
\(754\) 9.24443 0.336662
\(755\) 0 0
\(756\) −7.18421 −0.261287
\(757\) −8.75557 −0.318227 −0.159113 0.987260i \(-0.550864\pi\)
−0.159113 + 0.987260i \(0.550864\pi\)
\(758\) 38.0642 1.38256
\(759\) 8.85728 0.321499
\(760\) 0 0
\(761\) 3.15257 0.114280 0.0571402 0.998366i \(-0.481802\pi\)
0.0571402 + 0.998366i \(0.481802\pi\)
\(762\) 28.8988 1.04689
\(763\) −87.3087 −3.16079
\(764\) 9.89829 0.358108
\(765\) 0 0
\(766\) −27.8537 −1.00640
\(767\) −3.02227 −0.109128
\(768\) −20.2444 −0.730508
\(769\) −28.9590 −1.04429 −0.522144 0.852857i \(-0.674868\pi\)
−0.522144 + 0.852857i \(0.674868\pi\)
\(770\) 0 0
\(771\) −6.85728 −0.246959
\(772\) −29.7462 −1.07059
\(773\) −29.1427 −1.04819 −0.524095 0.851660i \(-0.675596\pi\)
−0.524095 + 0.851660i \(0.675596\pi\)
\(774\) 10.7971 0.388092
\(775\) 0 0
\(776\) −5.20877 −0.186984
\(777\) −8.85728 −0.317753
\(778\) 10.6824 0.382984
\(779\) −1.36488 −0.0489018
\(780\) 0 0
\(781\) 2.75557 0.0986020
\(782\) 87.3916 3.12512
\(783\) 7.80642 0.278979
\(784\) −58.1811 −2.07790
\(785\) 0 0
\(786\) 2.36842 0.0844786
\(787\) 11.2672 0.401632 0.200816 0.979629i \(-0.435641\pi\)
0.200816 + 0.979629i \(0.435641\pi\)
\(788\) 10.8613 0.386918
\(789\) −29.5812 −1.05312
\(790\) 0 0
\(791\) 26.5718 0.944786
\(792\) 0.719004 0.0255487
\(793\) 4.26671 0.151515
\(794\) −24.2766 −0.861543
\(795\) 0 0
\(796\) 22.8760 0.810819
\(797\) −41.9625 −1.48639 −0.743195 0.669075i \(-0.766690\pi\)
−0.743195 + 0.669075i \(0.766690\pi\)
\(798\) 59.4291 2.10377
\(799\) 14.2854 0.505383
\(800\) 0 0
\(801\) 5.61285 0.198320
\(802\) −3.80642 −0.134409
\(803\) −4.23506 −0.149452
\(804\) −2.01874 −0.0711953
\(805\) 0 0
\(806\) −3.26317 −0.114940
\(807\) −8.48886 −0.298822
\(808\) 3.35322 0.117966
\(809\) −27.8064 −0.977622 −0.488811 0.872390i \(-0.662569\pi\)
−0.488811 + 0.872390i \(0.662569\pi\)
\(810\) 0 0
\(811\) 6.78415 0.238224 0.119112 0.992881i \(-0.461995\pi\)
0.119112 + 0.992881i \(0.461995\pi\)
\(812\) −56.0830 −1.96813
\(813\) −14.6637 −0.514279
\(814\) −3.80642 −0.133415
\(815\) 0 0
\(816\) 23.9140 0.837156
\(817\) −40.0000 −1.39942
\(818\) 13.5585 0.474060
\(819\) 2.75557 0.0962874
\(820\) 0 0
\(821\) 3.62269 0.126433 0.0632164 0.998000i \(-0.479864\pi\)
0.0632164 + 0.998000i \(0.479864\pi\)
\(822\) 0.930409 0.0324517
\(823\) −42.0642 −1.46627 −0.733134 0.680085i \(-0.761943\pi\)
−0.733134 + 0.680085i \(0.761943\pi\)
\(824\) 8.34968 0.290875
\(825\) 0 0
\(826\) 40.9403 1.42449
\(827\) −30.8256 −1.07191 −0.535956 0.844246i \(-0.680049\pi\)
−0.535956 + 0.844246i \(0.680049\pi\)
\(828\) −14.3684 −0.499337
\(829\) 7.12399 0.247426 0.123713 0.992318i \(-0.460520\pi\)
0.123713 + 0.992318i \(0.460520\pi\)
\(830\) 0 0
\(831\) 14.6035 0.506589
\(832\) −2.95316 −0.102382
\(833\) 65.3876 2.26555
\(834\) 33.8894 1.17349
\(835\) 0 0
\(836\) 11.4380 0.395592
\(837\) −2.75557 −0.0952464
\(838\) −29.7146 −1.02647
\(839\) 3.34614 0.115522 0.0577608 0.998330i \(-0.481604\pi\)
0.0577608 + 0.998330i \(0.481604\pi\)
\(840\) 0 0
\(841\) 31.9403 1.10139
\(842\) −15.0321 −0.518041
\(843\) 0.193576 0.00666712
\(844\) −17.2988 −0.595450
\(845\) 0 0
\(846\) −5.24443 −0.180307
\(847\) 4.42864 0.152170
\(848\) −50.0830 −1.71986
\(849\) 27.1842 0.932960
\(850\) 0 0
\(851\) −17.7146 −0.607247
\(852\) −4.47013 −0.153144
\(853\) 26.4197 0.904595 0.452297 0.891867i \(-0.350605\pi\)
0.452297 + 0.891867i \(0.350605\pi\)
\(854\) −57.7975 −1.97779
\(855\) 0 0
\(856\) −1.88538 −0.0644411
\(857\) −38.7783 −1.32464 −0.662321 0.749220i \(-0.730429\pi\)
−0.662321 + 0.749220i \(0.730429\pi\)
\(858\) 1.18421 0.0404282
\(859\) −27.3087 −0.931760 −0.465880 0.884848i \(-0.654262\pi\)
−0.465880 + 0.884848i \(0.654262\pi\)
\(860\) 0 0
\(861\) 0.857279 0.0292160
\(862\) −65.2899 −2.22378
\(863\) 49.5308 1.68605 0.843024 0.537875i \(-0.180773\pi\)
0.843024 + 0.537875i \(0.180773\pi\)
\(864\) −7.34122 −0.249753
\(865\) 0 0
\(866\) 27.5397 0.935838
\(867\) −9.87601 −0.335407
\(868\) 19.7966 0.671940
\(869\) 8.56199 0.290446
\(870\) 0 0
\(871\) 0.774305 0.0262363
\(872\) −14.1748 −0.480021
\(873\) −7.24443 −0.245187
\(874\) 118.858 4.02044
\(875\) 0 0
\(876\) 6.87019 0.232122
\(877\) 4.50177 0.152014 0.0760070 0.997107i \(-0.475783\pi\)
0.0760070 + 0.997107i \(0.475783\pi\)
\(878\) 36.7654 1.24077
\(879\) −2.81579 −0.0949743
\(880\) 0 0
\(881\) −15.1240 −0.509540 −0.254770 0.967002i \(-0.582000\pi\)
−0.254770 + 0.967002i \(0.582000\pi\)
\(882\) −24.0049 −0.808288
\(883\) 30.2480 1.01793 0.508963 0.860789i \(-0.330029\pi\)
0.508963 + 0.860789i \(0.330029\pi\)
\(884\) 5.23277 0.175997
\(885\) 0 0
\(886\) −24.9777 −0.839143
\(887\) −57.1941 −1.92039 −0.960194 0.279333i \(-0.909887\pi\)
−0.960194 + 0.279333i \(0.909887\pi\)
\(888\) −1.43801 −0.0482564
\(889\) 67.2454 2.25534
\(890\) 0 0
\(891\) 1.00000 0.0335013
\(892\) −14.3684 −0.481090
\(893\) 19.4291 0.650171
\(894\) −2.73683 −0.0915334
\(895\) 0 0
\(896\) −25.0192 −0.835833
\(897\) 5.51114 0.184012
\(898\) 61.4835 2.05173
\(899\) −21.5111 −0.717437
\(900\) 0 0
\(901\) 56.2864 1.87517
\(902\) 0.368416 0.0122669
\(903\) 25.1240 0.836074
\(904\) 4.31402 0.143482
\(905\) 0 0
\(906\) −23.1713 −0.769815
\(907\) 53.2641 1.76861 0.884303 0.466913i \(-0.154634\pi\)
0.884303 + 0.466913i \(0.154634\pi\)
\(908\) −21.7017 −0.720195
\(909\) 4.66370 0.154685
\(910\) 0 0
\(911\) −0.590573 −0.0195665 −0.00978327 0.999952i \(-0.503114\pi\)
−0.00978327 + 0.999952i \(0.503114\pi\)
\(912\) 32.5245 1.07699
\(913\) −0.133353 −0.00441334
\(914\) −44.6508 −1.47692
\(915\) 0 0
\(916\) 18.6735 0.616991
\(917\) 5.51114 0.181994
\(918\) 9.86665 0.325648
\(919\) 55.8707 1.84300 0.921502 0.388375i \(-0.126963\pi\)
0.921502 + 0.388375i \(0.126963\pi\)
\(920\) 0 0
\(921\) −24.4286 −0.804951
\(922\) 54.9403 1.80936
\(923\) 1.71456 0.0564354
\(924\) −7.18421 −0.236343
\(925\) 0 0
\(926\) −36.8198 −1.20997
\(927\) 11.6128 0.381416
\(928\) −57.3087 −1.88125
\(929\) 15.3274 0.502876 0.251438 0.967873i \(-0.419097\pi\)
0.251438 + 0.967873i \(0.419097\pi\)
\(930\) 0 0
\(931\) 88.9314 2.91461
\(932\) 7.01921 0.229922
\(933\) −19.8796 −0.650827
\(934\) 5.98126 0.195713
\(935\) 0 0
\(936\) 0.447375 0.0146229
\(937\) 27.8479 0.909752 0.454876 0.890555i \(-0.349684\pi\)
0.454876 + 0.890555i \(0.349684\pi\)
\(938\) −10.4889 −0.342474
\(939\) −15.7146 −0.512825
\(940\) 0 0
\(941\) 10.4157 0.339543 0.169772 0.985483i \(-0.445697\pi\)
0.169772 + 0.985483i \(0.445697\pi\)
\(942\) −35.1526 −1.14533
\(943\) 1.71456 0.0558337
\(944\) 22.4059 0.729250
\(945\) 0 0
\(946\) 10.7971 0.351043
\(947\) −8.47013 −0.275242 −0.137621 0.990485i \(-0.543946\pi\)
−0.137621 + 0.990485i \(0.543946\pi\)
\(948\) −13.8894 −0.451107
\(949\) −2.63512 −0.0855397
\(950\) 0 0
\(951\) 16.4889 0.534688
\(952\) 16.5076 0.535014
\(953\) 8.71408 0.282277 0.141138 0.989990i \(-0.454924\pi\)
0.141138 + 0.989990i \(0.454924\pi\)
\(954\) −20.6637 −0.669012
\(955\) 0 0
\(956\) −5.42816 −0.175559
\(957\) 7.80642 0.252346
\(958\) 47.3087 1.52847
\(959\) 2.16500 0.0699114
\(960\) 0 0
\(961\) −23.4068 −0.755059
\(962\) −2.36842 −0.0763608
\(963\) −2.62222 −0.0844997
\(964\) −2.18373 −0.0703333
\(965\) 0 0
\(966\) −74.6548 −2.40198
\(967\) −44.2449 −1.42282 −0.711410 0.702777i \(-0.751943\pi\)
−0.711410 + 0.702777i \(0.751943\pi\)
\(968\) 0.719004 0.0231097
\(969\) −36.5531 −1.17425
\(970\) 0 0
\(971\) −57.1437 −1.83383 −0.916914 0.399085i \(-0.869328\pi\)
−0.916914 + 0.399085i \(0.869328\pi\)
\(972\) −1.62222 −0.0520326
\(973\) 78.8582 2.52808
\(974\) −21.9438 −0.703124
\(975\) 0 0
\(976\) −31.6316 −1.01250
\(977\) 16.2480 0.519819 0.259909 0.965633i \(-0.416307\pi\)
0.259909 + 0.965633i \(0.416307\pi\)
\(978\) 19.2257 0.614770
\(979\) 5.61285 0.179387
\(980\) 0 0
\(981\) −19.7146 −0.629437
\(982\) −31.1882 −0.995256
\(983\) −1.12399 −0.0358496 −0.0179248 0.999839i \(-0.505706\pi\)
−0.0179248 + 0.999839i \(0.505706\pi\)
\(984\) 0.139182 0.00443696
\(985\) 0 0
\(986\) 77.0232 2.45292
\(987\) −12.2034 −0.388439
\(988\) 7.11691 0.226419
\(989\) 50.2480 1.59779
\(990\) 0 0
\(991\) −53.6513 −1.70429 −0.852144 0.523307i \(-0.824698\pi\)
−0.852144 + 0.523307i \(0.824698\pi\)
\(992\) 20.2292 0.642279
\(993\) −15.3461 −0.486995
\(994\) −23.2257 −0.736674
\(995\) 0 0
\(996\) 0.216327 0.00685460
\(997\) 35.7275 1.13150 0.565750 0.824577i \(-0.308587\pi\)
0.565750 + 0.824577i \(0.308587\pi\)
\(998\) 48.2034 1.52585
\(999\) −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.a.k.1.1 3
3.2 odd 2 2475.2.a.bb.1.3 3
5.2 odd 4 825.2.c.g.199.2 6
5.3 odd 4 825.2.c.g.199.5 6
5.4 even 2 165.2.a.c.1.3 3
11.10 odd 2 9075.2.a.cf.1.3 3
15.2 even 4 2475.2.c.r.199.5 6
15.8 even 4 2475.2.c.r.199.2 6
15.14 odd 2 495.2.a.e.1.1 3
20.19 odd 2 2640.2.a.be.1.3 3
35.34 odd 2 8085.2.a.bk.1.3 3
55.54 odd 2 1815.2.a.m.1.1 3
60.59 even 2 7920.2.a.cj.1.3 3
165.164 even 2 5445.2.a.z.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
165.2.a.c.1.3 3 5.4 even 2
495.2.a.e.1.1 3 15.14 odd 2
825.2.a.k.1.1 3 1.1 even 1 trivial
825.2.c.g.199.2 6 5.2 odd 4
825.2.c.g.199.5 6 5.3 odd 4
1815.2.a.m.1.1 3 55.54 odd 2
2475.2.a.bb.1.3 3 3.2 odd 2
2475.2.c.r.199.2 6 15.8 even 4
2475.2.c.r.199.5 6 15.2 even 4
2640.2.a.be.1.3 3 20.19 odd 2
5445.2.a.z.1.3 3 165.164 even 2
7920.2.a.cj.1.3 3 60.59 even 2
8085.2.a.bk.1.3 3 35.34 odd 2
9075.2.a.cf.1.3 3 11.10 odd 2