Properties

Label 825.2.a.l.1.2
Level $825$
Weight $2$
Character 825.1
Self dual yes
Analytic conductor $6.588$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

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

Newform invariants

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

Embedding invariants

Embedding label 1.2
Root \(0.311108\) of defining polynomial
Character \(\chi\) \(=\) 825.1

$q$-expansion

\(f(q)\) \(=\) \(q+0.311108 q^{2} +1.00000 q^{3} -1.90321 q^{4} +0.311108 q^{6} -0.903212 q^{7} -1.21432 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+0.311108 q^{2} +1.00000 q^{3} -1.90321 q^{4} +0.311108 q^{6} -0.903212 q^{7} -1.21432 q^{8} +1.00000 q^{9} +1.00000 q^{11} -1.90321 q^{12} +2.90321 q^{13} -0.280996 q^{14} +3.42864 q^{16} +2.28100 q^{17} +0.311108 q^{18} +2.42864 q^{19} -0.903212 q^{21} +0.311108 q^{22} +4.00000 q^{23} -1.21432 q^{24} +0.903212 q^{26} +1.00000 q^{27} +1.71900 q^{28} +7.05086 q^{29} -2.62222 q^{31} +3.49532 q^{32} +1.00000 q^{33} +0.709636 q^{34} -1.90321 q^{36} +5.80642 q^{37} +0.755569 q^{38} +2.90321 q^{39} -10.6637 q^{41} -0.280996 q^{42} +10.7096 q^{43} -1.90321 q^{44} +1.24443 q^{46} +0.949145 q^{47} +3.42864 q^{48} -6.18421 q^{49} +2.28100 q^{51} -5.52543 q^{52} +0.815792 q^{53} +0.311108 q^{54} +1.09679 q^{56} +2.42864 q^{57} +2.19358 q^{58} -1.67307 q^{59} -7.24443 q^{61} -0.815792 q^{62} -0.903212 q^{63} -5.76986 q^{64} +0.311108 q^{66} +12.8573 q^{67} -4.34122 q^{68} +4.00000 q^{69} +9.28592 q^{71} -1.21432 q^{72} -5.65878 q^{73} +1.80642 q^{74} -4.62222 q^{76} -0.903212 q^{77} +0.903212 q^{78} -16.5303 q^{79} +1.00000 q^{81} -3.31756 q^{82} +7.76049 q^{83} +1.71900 q^{84} +3.33185 q^{86} +7.05086 q^{87} -1.21432 q^{88} +6.13335 q^{89} -2.62222 q^{91} -7.61285 q^{92} -2.62222 q^{93} +0.295286 q^{94} +3.49532 q^{96} -12.4701 q^{97} -1.92396 q^{98} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + q^{2} + 3 q^{3} + q^{4} + q^{6} + 4 q^{7} + 3 q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + q^{2} + 3 q^{3} + q^{4} + q^{6} + 4 q^{7} + 3 q^{8} + 3 q^{9} + 3 q^{11} + q^{12} + 2 q^{13} + 6 q^{14} - 3 q^{16} + q^{18} - 6 q^{19} + 4 q^{21} + q^{22} + 12 q^{23} + 3 q^{24} - 4 q^{26} + 3 q^{27} + 12 q^{28} + 8 q^{29} - 8 q^{31} - 3 q^{32} + 3 q^{33} - 18 q^{34} + q^{36} + 4 q^{37} + 2 q^{38} + 2 q^{39} + 8 q^{41} + 6 q^{42} + 12 q^{43} + q^{44} + 4 q^{46} + 16 q^{47} - 3 q^{48} - 5 q^{49} - 10 q^{52} + 16 q^{53} + q^{54} + 10 q^{56} - 6 q^{57} + 20 q^{58} + 8 q^{59} - 22 q^{61} - 16 q^{62} + 4 q^{63} - 11 q^{64} + q^{66} + 12 q^{67} - 20 q^{68} + 12 q^{69} - 12 q^{71} + 3 q^{72} - 10 q^{73} - 8 q^{74} - 14 q^{76} + 4 q^{77} - 4 q^{78} - 10 q^{79} + 3 q^{81} + 4 q^{82} - 10 q^{83} + 12 q^{84} - 10 q^{86} + 8 q^{87} + 3 q^{88} + 18 q^{89} - 8 q^{91} + 4 q^{92} - 8 q^{93} - 12 q^{94} - 3 q^{96} + 16 q^{97} + 21 q^{98} + 3 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.311108 0.219986 0.109993 0.993932i \(-0.464917\pi\)
0.109993 + 0.993932i \(0.464917\pi\)
\(3\) 1.00000 0.577350
\(4\) −1.90321 −0.951606
\(5\) 0 0
\(6\) 0.311108 0.127009
\(7\) −0.903212 −0.341382 −0.170691 0.985325i \(-0.554600\pi\)
−0.170691 + 0.985325i \(0.554600\pi\)
\(8\) −1.21432 −0.429327
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) −1.90321 −0.549410
\(13\) 2.90321 0.805206 0.402603 0.915375i \(-0.368106\pi\)
0.402603 + 0.915375i \(0.368106\pi\)
\(14\) −0.280996 −0.0750994
\(15\) 0 0
\(16\) 3.42864 0.857160
\(17\) 2.28100 0.553223 0.276611 0.960982i \(-0.410789\pi\)
0.276611 + 0.960982i \(0.410789\pi\)
\(18\) 0.311108 0.0733288
\(19\) 2.42864 0.557168 0.278584 0.960412i \(-0.410135\pi\)
0.278584 + 0.960412i \(0.410135\pi\)
\(20\) 0 0
\(21\) −0.903212 −0.197097
\(22\) 0.311108 0.0663284
\(23\) 4.00000 0.834058 0.417029 0.908893i \(-0.363071\pi\)
0.417029 + 0.908893i \(0.363071\pi\)
\(24\) −1.21432 −0.247872
\(25\) 0 0
\(26\) 0.903212 0.177134
\(27\) 1.00000 0.192450
\(28\) 1.71900 0.324861
\(29\) 7.05086 1.30931 0.654655 0.755927i \(-0.272814\pi\)
0.654655 + 0.755927i \(0.272814\pi\)
\(30\) 0 0
\(31\) −2.62222 −0.470964 −0.235482 0.971879i \(-0.575667\pi\)
−0.235482 + 0.971879i \(0.575667\pi\)
\(32\) 3.49532 0.617890
\(33\) 1.00000 0.174078
\(34\) 0.709636 0.121702
\(35\) 0 0
\(36\) −1.90321 −0.317202
\(37\) 5.80642 0.954570 0.477285 0.878749i \(-0.341621\pi\)
0.477285 + 0.878749i \(0.341621\pi\)
\(38\) 0.755569 0.122569
\(39\) 2.90321 0.464886
\(40\) 0 0
\(41\) −10.6637 −1.66539 −0.832695 0.553731i \(-0.813203\pi\)
−0.832695 + 0.553731i \(0.813203\pi\)
\(42\) −0.280996 −0.0433587
\(43\) 10.7096 1.63320 0.816602 0.577201i \(-0.195855\pi\)
0.816602 + 0.577201i \(0.195855\pi\)
\(44\) −1.90321 −0.286920
\(45\) 0 0
\(46\) 1.24443 0.183481
\(47\) 0.949145 0.138447 0.0692235 0.997601i \(-0.477948\pi\)
0.0692235 + 0.997601i \(0.477948\pi\)
\(48\) 3.42864 0.494881
\(49\) −6.18421 −0.883458
\(50\) 0 0
\(51\) 2.28100 0.319403
\(52\) −5.52543 −0.766239
\(53\) 0.815792 0.112058 0.0560288 0.998429i \(-0.482156\pi\)
0.0560288 + 0.998429i \(0.482156\pi\)
\(54\) 0.311108 0.0423364
\(55\) 0 0
\(56\) 1.09679 0.146564
\(57\) 2.42864 0.321681
\(58\) 2.19358 0.288031
\(59\) −1.67307 −0.217815 −0.108908 0.994052i \(-0.534735\pi\)
−0.108908 + 0.994052i \(0.534735\pi\)
\(60\) 0 0
\(61\) −7.24443 −0.927554 −0.463777 0.885952i \(-0.653506\pi\)
−0.463777 + 0.885952i \(0.653506\pi\)
\(62\) −0.815792 −0.103606
\(63\) −0.903212 −0.113794
\(64\) −5.76986 −0.721232
\(65\) 0 0
\(66\) 0.311108 0.0382947
\(67\) 12.8573 1.57077 0.785383 0.619010i \(-0.212466\pi\)
0.785383 + 0.619010i \(0.212466\pi\)
\(68\) −4.34122 −0.526450
\(69\) 4.00000 0.481543
\(70\) 0 0
\(71\) 9.28592 1.10204 0.551018 0.834493i \(-0.314240\pi\)
0.551018 + 0.834493i \(0.314240\pi\)
\(72\) −1.21432 −0.143109
\(73\) −5.65878 −0.662310 −0.331155 0.943576i \(-0.607438\pi\)
−0.331155 + 0.943576i \(0.607438\pi\)
\(74\) 1.80642 0.209993
\(75\) 0 0
\(76\) −4.62222 −0.530204
\(77\) −0.903212 −0.102931
\(78\) 0.903212 0.102269
\(79\) −16.5303 −1.85981 −0.929905 0.367800i \(-0.880111\pi\)
−0.929905 + 0.367800i \(0.880111\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) −3.31756 −0.366363
\(83\) 7.76049 0.851825 0.425912 0.904764i \(-0.359953\pi\)
0.425912 + 0.904764i \(0.359953\pi\)
\(84\) 1.71900 0.187559
\(85\) 0 0
\(86\) 3.33185 0.359283
\(87\) 7.05086 0.755931
\(88\) −1.21432 −0.129447
\(89\) 6.13335 0.650134 0.325067 0.945691i \(-0.394613\pi\)
0.325067 + 0.945691i \(0.394613\pi\)
\(90\) 0 0
\(91\) −2.62222 −0.274883
\(92\) −7.61285 −0.793694
\(93\) −2.62222 −0.271911
\(94\) 0.295286 0.0304565
\(95\) 0 0
\(96\) 3.49532 0.356739
\(97\) −12.4701 −1.26615 −0.633075 0.774091i \(-0.718207\pi\)
−0.633075 + 0.774091i \(0.718207\pi\)
\(98\) −1.92396 −0.194349
\(99\) 1.00000 0.100504
\(100\) 0 0
\(101\) 16.1748 1.60946 0.804728 0.593643i \(-0.202311\pi\)
0.804728 + 0.593643i \(0.202311\pi\)
\(102\) 0.709636 0.0702644
\(103\) −17.1526 −1.69009 −0.845046 0.534693i \(-0.820427\pi\)
−0.845046 + 0.534693i \(0.820427\pi\)
\(104\) −3.52543 −0.345697
\(105\) 0 0
\(106\) 0.253799 0.0246512
\(107\) −13.5669 −1.31156 −0.655782 0.754951i \(-0.727661\pi\)
−0.655782 + 0.754951i \(0.727661\pi\)
\(108\) −1.90321 −0.183137
\(109\) −10.0000 −0.957826 −0.478913 0.877862i \(-0.658969\pi\)
−0.478913 + 0.877862i \(0.658969\pi\)
\(110\) 0 0
\(111\) 5.80642 0.551121
\(112\) −3.09679 −0.292619
\(113\) 14.2351 1.33912 0.669561 0.742757i \(-0.266482\pi\)
0.669561 + 0.742757i \(0.266482\pi\)
\(114\) 0.755569 0.0707655
\(115\) 0 0
\(116\) −13.4193 −1.24595
\(117\) 2.90321 0.268402
\(118\) −0.520505 −0.0479164
\(119\) −2.06022 −0.188860
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) −2.25380 −0.204049
\(123\) −10.6637 −0.961514
\(124\) 4.99063 0.448172
\(125\) 0 0
\(126\) −0.280996 −0.0250331
\(127\) 11.0049 0.976529 0.488264 0.872696i \(-0.337630\pi\)
0.488264 + 0.872696i \(0.337630\pi\)
\(128\) −8.78568 −0.776552
\(129\) 10.7096 0.942931
\(130\) 0 0
\(131\) 1.24443 0.108726 0.0543632 0.998521i \(-0.482687\pi\)
0.0543632 + 0.998521i \(0.482687\pi\)
\(132\) −1.90321 −0.165653
\(133\) −2.19358 −0.190207
\(134\) 4.00000 0.345547
\(135\) 0 0
\(136\) −2.76986 −0.237513
\(137\) −4.42864 −0.378364 −0.189182 0.981942i \(-0.560584\pi\)
−0.189182 + 0.981942i \(0.560584\pi\)
\(138\) 1.24443 0.105933
\(139\) 0.917502 0.0778215 0.0389108 0.999243i \(-0.487611\pi\)
0.0389108 + 0.999243i \(0.487611\pi\)
\(140\) 0 0
\(141\) 0.949145 0.0799324
\(142\) 2.88892 0.242433
\(143\) 2.90321 0.242779
\(144\) 3.42864 0.285720
\(145\) 0 0
\(146\) −1.76049 −0.145699
\(147\) −6.18421 −0.510065
\(148\) −11.0509 −0.908375
\(149\) 2.19358 0.179705 0.0898524 0.995955i \(-0.471360\pi\)
0.0898524 + 0.995955i \(0.471360\pi\)
\(150\) 0 0
\(151\) −10.4286 −0.848671 −0.424335 0.905505i \(-0.639492\pi\)
−0.424335 + 0.905505i \(0.639492\pi\)
\(152\) −2.94914 −0.239207
\(153\) 2.28100 0.184408
\(154\) −0.280996 −0.0226433
\(155\) 0 0
\(156\) −5.52543 −0.442388
\(157\) 5.80642 0.463403 0.231702 0.972787i \(-0.425571\pi\)
0.231702 + 0.972787i \(0.425571\pi\)
\(158\) −5.14272 −0.409133
\(159\) 0.815792 0.0646965
\(160\) 0 0
\(161\) −3.61285 −0.284732
\(162\) 0.311108 0.0244429
\(163\) 11.0509 0.865570 0.432785 0.901497i \(-0.357531\pi\)
0.432785 + 0.901497i \(0.357531\pi\)
\(164\) 20.2953 1.58480
\(165\) 0 0
\(166\) 2.41435 0.187390
\(167\) 13.9541 1.07980 0.539899 0.841730i \(-0.318462\pi\)
0.539899 + 0.841730i \(0.318462\pi\)
\(168\) 1.09679 0.0846190
\(169\) −4.57136 −0.351643
\(170\) 0 0
\(171\) 2.42864 0.185723
\(172\) −20.3827 −1.55417
\(173\) −18.8430 −1.43261 −0.716303 0.697789i \(-0.754167\pi\)
−0.716303 + 0.697789i \(0.754167\pi\)
\(174\) 2.19358 0.166295
\(175\) 0 0
\(176\) 3.42864 0.258443
\(177\) −1.67307 −0.125756
\(178\) 1.90813 0.143021
\(179\) 4.85728 0.363050 0.181525 0.983386i \(-0.441897\pi\)
0.181525 + 0.983386i \(0.441897\pi\)
\(180\) 0 0
\(181\) −16.7971 −1.24852 −0.624258 0.781219i \(-0.714598\pi\)
−0.624258 + 0.781219i \(0.714598\pi\)
\(182\) −0.815792 −0.0604705
\(183\) −7.24443 −0.535524
\(184\) −4.85728 −0.358083
\(185\) 0 0
\(186\) −0.815792 −0.0598168
\(187\) 2.28100 0.166803
\(188\) −1.80642 −0.131747
\(189\) −0.903212 −0.0656990
\(190\) 0 0
\(191\) 16.8573 1.21975 0.609875 0.792498i \(-0.291220\pi\)
0.609875 + 0.792498i \(0.291220\pi\)
\(192\) −5.76986 −0.416404
\(193\) −24.6178 −1.77203 −0.886013 0.463661i \(-0.846536\pi\)
−0.886013 + 0.463661i \(0.846536\pi\)
\(194\) −3.87955 −0.278536
\(195\) 0 0
\(196\) 11.7699 0.840704
\(197\) 14.8716 1.05956 0.529778 0.848137i \(-0.322275\pi\)
0.529778 + 0.848137i \(0.322275\pi\)
\(198\) 0.311108 0.0221095
\(199\) −11.2257 −0.795768 −0.397884 0.917436i \(-0.630255\pi\)
−0.397884 + 0.917436i \(0.630255\pi\)
\(200\) 0 0
\(201\) 12.8573 0.906883
\(202\) 5.03212 0.354059
\(203\) −6.36842 −0.446975
\(204\) −4.34122 −0.303946
\(205\) 0 0
\(206\) −5.33630 −0.371797
\(207\) 4.00000 0.278019
\(208\) 9.95407 0.690190
\(209\) 2.42864 0.167993
\(210\) 0 0
\(211\) −11.9398 −0.821968 −0.410984 0.911643i \(-0.634815\pi\)
−0.410984 + 0.911643i \(0.634815\pi\)
\(212\) −1.55262 −0.106635
\(213\) 9.28592 0.636261
\(214\) −4.22077 −0.288526
\(215\) 0 0
\(216\) −1.21432 −0.0826240
\(217\) 2.36842 0.160779
\(218\) −3.11108 −0.210709
\(219\) −5.65878 −0.382385
\(220\) 0 0
\(221\) 6.62222 0.445458
\(222\) 1.80642 0.121239
\(223\) 21.8064 1.46027 0.730133 0.683305i \(-0.239458\pi\)
0.730133 + 0.683305i \(0.239458\pi\)
\(224\) −3.15701 −0.210937
\(225\) 0 0
\(226\) 4.42864 0.294589
\(227\) 3.19850 0.212292 0.106146 0.994351i \(-0.466149\pi\)
0.106146 + 0.994351i \(0.466149\pi\)
\(228\) −4.62222 −0.306114
\(229\) 7.12399 0.470766 0.235383 0.971903i \(-0.424366\pi\)
0.235383 + 0.971903i \(0.424366\pi\)
\(230\) 0 0
\(231\) −0.903212 −0.0594270
\(232\) −8.56199 −0.562122
\(233\) −19.5254 −1.27915 −0.639577 0.768727i \(-0.720890\pi\)
−0.639577 + 0.768727i \(0.720890\pi\)
\(234\) 0.903212 0.0590448
\(235\) 0 0
\(236\) 3.18421 0.207274
\(237\) −16.5303 −1.07376
\(238\) −0.640951 −0.0415467
\(239\) −21.9813 −1.42185 −0.710925 0.703268i \(-0.751723\pi\)
−0.710925 + 0.703268i \(0.751723\pi\)
\(240\) 0 0
\(241\) 5.34614 0.344375 0.172188 0.985064i \(-0.444916\pi\)
0.172188 + 0.985064i \(0.444916\pi\)
\(242\) 0.311108 0.0199988
\(243\) 1.00000 0.0641500
\(244\) 13.7877 0.882666
\(245\) 0 0
\(246\) −3.31756 −0.211520
\(247\) 7.05086 0.448635
\(248\) 3.18421 0.202197
\(249\) 7.76049 0.491801
\(250\) 0 0
\(251\) −23.7748 −1.50065 −0.750325 0.661069i \(-0.770103\pi\)
−0.750325 + 0.661069i \(0.770103\pi\)
\(252\) 1.71900 0.108287
\(253\) 4.00000 0.251478
\(254\) 3.42372 0.214823
\(255\) 0 0
\(256\) 8.80642 0.550401
\(257\) −8.13335 −0.507345 −0.253672 0.967290i \(-0.581638\pi\)
−0.253672 + 0.967290i \(0.581638\pi\)
\(258\) 3.33185 0.207432
\(259\) −5.24443 −0.325873
\(260\) 0 0
\(261\) 7.05086 0.436437
\(262\) 0.387152 0.0239183
\(263\) 22.9032 1.41227 0.706136 0.708076i \(-0.250437\pi\)
0.706136 + 0.708076i \(0.250437\pi\)
\(264\) −1.21432 −0.0747362
\(265\) 0 0
\(266\) −0.682439 −0.0418430
\(267\) 6.13335 0.375355
\(268\) −24.4701 −1.49475
\(269\) 11.8350 0.721593 0.360796 0.932645i \(-0.382505\pi\)
0.360796 + 0.932645i \(0.382505\pi\)
\(270\) 0 0
\(271\) 14.8988 0.905036 0.452518 0.891755i \(-0.350526\pi\)
0.452518 + 0.891755i \(0.350526\pi\)
\(272\) 7.82071 0.474200
\(273\) −2.62222 −0.158704
\(274\) −1.37778 −0.0832350
\(275\) 0 0
\(276\) −7.61285 −0.458240
\(277\) −27.6686 −1.66245 −0.831223 0.555939i \(-0.812359\pi\)
−0.831223 + 0.555939i \(0.812359\pi\)
\(278\) 0.285442 0.0171197
\(279\) −2.62222 −0.156988
\(280\) 0 0
\(281\) 9.80642 0.585002 0.292501 0.956265i \(-0.405512\pi\)
0.292501 + 0.956265i \(0.405512\pi\)
\(282\) 0.295286 0.0175840
\(283\) −19.0049 −1.12973 −0.564863 0.825185i \(-0.691071\pi\)
−0.564863 + 0.825185i \(0.691071\pi\)
\(284\) −17.6731 −1.04870
\(285\) 0 0
\(286\) 0.903212 0.0534080
\(287\) 9.63158 0.568534
\(288\) 3.49532 0.205963
\(289\) −11.7971 −0.693944
\(290\) 0 0
\(291\) −12.4701 −0.731012
\(292\) 10.7699 0.630258
\(293\) −30.7511 −1.79650 −0.898250 0.439485i \(-0.855161\pi\)
−0.898250 + 0.439485i \(0.855161\pi\)
\(294\) −1.92396 −0.112207
\(295\) 0 0
\(296\) −7.05086 −0.409823
\(297\) 1.00000 0.0580259
\(298\) 0.682439 0.0395326
\(299\) 11.6128 0.671588
\(300\) 0 0
\(301\) −9.67307 −0.557547
\(302\) −3.24443 −0.186696
\(303\) 16.1748 0.929220
\(304\) 8.32693 0.477582
\(305\) 0 0
\(306\) 0.709636 0.0405672
\(307\) −13.4938 −0.770131 −0.385065 0.922889i \(-0.625821\pi\)
−0.385065 + 0.922889i \(0.625821\pi\)
\(308\) 1.71900 0.0979493
\(309\) −17.1526 −0.975775
\(310\) 0 0
\(311\) −17.5526 −0.995318 −0.497659 0.867373i \(-0.665807\pi\)
−0.497659 + 0.867373i \(0.665807\pi\)
\(312\) −3.52543 −0.199588
\(313\) 14.3970 0.813766 0.406883 0.913480i \(-0.366616\pi\)
0.406883 + 0.913480i \(0.366616\pi\)
\(314\) 1.80642 0.101942
\(315\) 0 0
\(316\) 31.4608 1.76981
\(317\) 29.4608 1.65468 0.827341 0.561701i \(-0.189853\pi\)
0.827341 + 0.561701i \(0.189853\pi\)
\(318\) 0.253799 0.0142324
\(319\) 7.05086 0.394772
\(320\) 0 0
\(321\) −13.5669 −0.757231
\(322\) −1.12399 −0.0626372
\(323\) 5.53972 0.308238
\(324\) −1.90321 −0.105734
\(325\) 0 0
\(326\) 3.43801 0.190414
\(327\) −10.0000 −0.553001
\(328\) 12.9491 0.714997
\(329\) −0.857279 −0.0472633
\(330\) 0 0
\(331\) −2.62222 −0.144130 −0.0720650 0.997400i \(-0.522959\pi\)
−0.0720650 + 0.997400i \(0.522959\pi\)
\(332\) −14.7699 −0.810601
\(333\) 5.80642 0.318190
\(334\) 4.34122 0.237541
\(335\) 0 0
\(336\) −3.09679 −0.168944
\(337\) −5.00492 −0.272635 −0.136318 0.990665i \(-0.543527\pi\)
−0.136318 + 0.990665i \(0.543527\pi\)
\(338\) −1.42219 −0.0773567
\(339\) 14.2351 0.773143
\(340\) 0 0
\(341\) −2.62222 −0.142001
\(342\) 0.755569 0.0408565
\(343\) 11.9081 0.642979
\(344\) −13.0049 −0.701178
\(345\) 0 0
\(346\) −5.86220 −0.315154
\(347\) −22.8113 −1.22458 −0.612289 0.790634i \(-0.709751\pi\)
−0.612289 + 0.790634i \(0.709751\pi\)
\(348\) −13.4193 −0.719348
\(349\) 21.2257 1.13619 0.568093 0.822965i \(-0.307681\pi\)
0.568093 + 0.822965i \(0.307681\pi\)
\(350\) 0 0
\(351\) 2.90321 0.154962
\(352\) 3.49532 0.186301
\(353\) 7.18421 0.382377 0.191188 0.981553i \(-0.438766\pi\)
0.191188 + 0.981553i \(0.438766\pi\)
\(354\) −0.520505 −0.0276645
\(355\) 0 0
\(356\) −11.6731 −0.618672
\(357\) −2.06022 −0.109039
\(358\) 1.51114 0.0798661
\(359\) −14.1017 −0.744260 −0.372130 0.928181i \(-0.621372\pi\)
−0.372130 + 0.928181i \(0.621372\pi\)
\(360\) 0 0
\(361\) −13.1017 −0.689564
\(362\) −5.22570 −0.274656
\(363\) 1.00000 0.0524864
\(364\) 4.99063 0.261580
\(365\) 0 0
\(366\) −2.25380 −0.117808
\(367\) −3.90813 −0.204003 −0.102001 0.994784i \(-0.532525\pi\)
−0.102001 + 0.994784i \(0.532525\pi\)
\(368\) 13.7146 0.714921
\(369\) −10.6637 −0.555130
\(370\) 0 0
\(371\) −0.736833 −0.0382545
\(372\) 4.99063 0.258752
\(373\) 12.9763 0.671890 0.335945 0.941882i \(-0.390944\pi\)
0.335945 + 0.941882i \(0.390944\pi\)
\(374\) 0.709636 0.0366944
\(375\) 0 0
\(376\) −1.15257 −0.0594390
\(377\) 20.4701 1.05427
\(378\) −0.280996 −0.0144529
\(379\) 36.0830 1.85346 0.926729 0.375731i \(-0.122608\pi\)
0.926729 + 0.375731i \(0.122608\pi\)
\(380\) 0 0
\(381\) 11.0049 0.563799
\(382\) 5.24443 0.268328
\(383\) 20.2953 1.03704 0.518520 0.855065i \(-0.326483\pi\)
0.518520 + 0.855065i \(0.326483\pi\)
\(384\) −8.78568 −0.448342
\(385\) 0 0
\(386\) −7.65878 −0.389822
\(387\) 10.7096 0.544401
\(388\) 23.7333 1.20488
\(389\) 30.4701 1.54490 0.772448 0.635078i \(-0.219032\pi\)
0.772448 + 0.635078i \(0.219032\pi\)
\(390\) 0 0
\(391\) 9.12399 0.461420
\(392\) 7.50961 0.379292
\(393\) 1.24443 0.0627733
\(394\) 4.62666 0.233088
\(395\) 0 0
\(396\) −1.90321 −0.0956400
\(397\) 4.97773 0.249825 0.124912 0.992168i \(-0.460135\pi\)
0.124912 + 0.992168i \(0.460135\pi\)
\(398\) −3.49240 −0.175058
\(399\) −2.19358 −0.109816
\(400\) 0 0
\(401\) −1.86665 −0.0932159 −0.0466079 0.998913i \(-0.514841\pi\)
−0.0466079 + 0.998913i \(0.514841\pi\)
\(402\) 4.00000 0.199502
\(403\) −7.61285 −0.379223
\(404\) −30.7841 −1.53157
\(405\) 0 0
\(406\) −1.98126 −0.0983285
\(407\) 5.80642 0.287814
\(408\) −2.76986 −0.137128
\(409\) −3.63158 −0.179570 −0.0897851 0.995961i \(-0.528618\pi\)
−0.0897851 + 0.995961i \(0.528618\pi\)
\(410\) 0 0
\(411\) −4.42864 −0.218449
\(412\) 32.6450 1.60830
\(413\) 1.51114 0.0743582
\(414\) 1.24443 0.0611605
\(415\) 0 0
\(416\) 10.1476 0.497529
\(417\) 0.917502 0.0449303
\(418\) 0.755569 0.0369561
\(419\) 4.85728 0.237294 0.118647 0.992937i \(-0.462144\pi\)
0.118647 + 0.992937i \(0.462144\pi\)
\(420\) 0 0
\(421\) 22.6321 1.10302 0.551510 0.834169i \(-0.314052\pi\)
0.551510 + 0.834169i \(0.314052\pi\)
\(422\) −3.71456 −0.180822
\(423\) 0.949145 0.0461490
\(424\) −0.990632 −0.0481093
\(425\) 0 0
\(426\) 2.88892 0.139969
\(427\) 6.54326 0.316650
\(428\) 25.8207 1.24809
\(429\) 2.90321 0.140168
\(430\) 0 0
\(431\) 1.24443 0.0599421 0.0299711 0.999551i \(-0.490458\pi\)
0.0299711 + 0.999551i \(0.490458\pi\)
\(432\) 3.42864 0.164960
\(433\) −16.0000 −0.768911 −0.384455 0.923144i \(-0.625611\pi\)
−0.384455 + 0.923144i \(0.625611\pi\)
\(434\) 0.736833 0.0353691
\(435\) 0 0
\(436\) 19.0321 0.911473
\(437\) 9.71456 0.464710
\(438\) −1.76049 −0.0841195
\(439\) −2.42864 −0.115913 −0.0579563 0.998319i \(-0.518458\pi\)
−0.0579563 + 0.998319i \(0.518458\pi\)
\(440\) 0 0
\(441\) −6.18421 −0.294486
\(442\) 2.06022 0.0979948
\(443\) 31.0509 1.47527 0.737635 0.675199i \(-0.235942\pi\)
0.737635 + 0.675199i \(0.235942\pi\)
\(444\) −11.0509 −0.524450
\(445\) 0 0
\(446\) 6.78415 0.321239
\(447\) 2.19358 0.103753
\(448\) 5.21141 0.246216
\(449\) −37.3590 −1.76308 −0.881541 0.472107i \(-0.843494\pi\)
−0.881541 + 0.472107i \(0.843494\pi\)
\(450\) 0 0
\(451\) −10.6637 −0.502134
\(452\) −27.0923 −1.27432
\(453\) −10.4286 −0.489980
\(454\) 0.995078 0.0467013
\(455\) 0 0
\(456\) −2.94914 −0.138106
\(457\) 8.73822 0.408757 0.204378 0.978892i \(-0.434483\pi\)
0.204378 + 0.978892i \(0.434483\pi\)
\(458\) 2.21633 0.103562
\(459\) 2.28100 0.106468
\(460\) 0 0
\(461\) 31.7877 1.48050 0.740250 0.672332i \(-0.234707\pi\)
0.740250 + 0.672332i \(0.234707\pi\)
\(462\) −0.280996 −0.0130731
\(463\) 12.0919 0.561957 0.280978 0.959714i \(-0.409341\pi\)
0.280978 + 0.959714i \(0.409341\pi\)
\(464\) 24.1748 1.12229
\(465\) 0 0
\(466\) −6.07451 −0.281396
\(467\) −15.3461 −0.710135 −0.355067 0.934841i \(-0.615542\pi\)
−0.355067 + 0.934841i \(0.615542\pi\)
\(468\) −5.52543 −0.255413
\(469\) −11.6128 −0.536231
\(470\) 0 0
\(471\) 5.80642 0.267546
\(472\) 2.03164 0.0935139
\(473\) 10.7096 0.492430
\(474\) −5.14272 −0.236213
\(475\) 0 0
\(476\) 3.92104 0.179721
\(477\) 0.815792 0.0373525
\(478\) −6.83854 −0.312788
\(479\) 5.89829 0.269500 0.134750 0.990880i \(-0.456977\pi\)
0.134750 + 0.990880i \(0.456977\pi\)
\(480\) 0 0
\(481\) 16.8573 0.768626
\(482\) 1.66323 0.0757579
\(483\) −3.61285 −0.164390
\(484\) −1.90321 −0.0865096
\(485\) 0 0
\(486\) 0.311108 0.0141121
\(487\) 31.3461 1.42043 0.710215 0.703985i \(-0.248598\pi\)
0.710215 + 0.703985i \(0.248598\pi\)
\(488\) 8.79706 0.398224
\(489\) 11.0509 0.499737
\(490\) 0 0
\(491\) −8.00000 −0.361035 −0.180517 0.983572i \(-0.557777\pi\)
−0.180517 + 0.983572i \(0.557777\pi\)
\(492\) 20.2953 0.914982
\(493\) 16.0830 0.724341
\(494\) 2.19358 0.0986937
\(495\) 0 0
\(496\) −8.99063 −0.403691
\(497\) −8.38715 −0.376215
\(498\) 2.41435 0.108190
\(499\) 15.1427 0.677881 0.338941 0.940808i \(-0.389931\pi\)
0.338941 + 0.940808i \(0.389931\pi\)
\(500\) 0 0
\(501\) 13.9541 0.623422
\(502\) −7.39652 −0.330123
\(503\) 26.0370 1.16093 0.580467 0.814284i \(-0.302870\pi\)
0.580467 + 0.814284i \(0.302870\pi\)
\(504\) 1.09679 0.0488548
\(505\) 0 0
\(506\) 1.24443 0.0553217
\(507\) −4.57136 −0.203021
\(508\) −20.9447 −0.929271
\(509\) −24.5718 −1.08913 −0.544564 0.838719i \(-0.683305\pi\)
−0.544564 + 0.838719i \(0.683305\pi\)
\(510\) 0 0
\(511\) 5.11108 0.226101
\(512\) 20.3111 0.897633
\(513\) 2.42864 0.107227
\(514\) −2.53035 −0.111609
\(515\) 0 0
\(516\) −20.3827 −0.897299
\(517\) 0.949145 0.0417433
\(518\) −1.63158 −0.0716877
\(519\) −18.8430 −0.827115
\(520\) 0 0
\(521\) −4.88892 −0.214188 −0.107094 0.994249i \(-0.534155\pi\)
−0.107094 + 0.994249i \(0.534155\pi\)
\(522\) 2.19358 0.0960102
\(523\) −4.22077 −0.184562 −0.0922808 0.995733i \(-0.529416\pi\)
−0.0922808 + 0.995733i \(0.529416\pi\)
\(524\) −2.36842 −0.103465
\(525\) 0 0
\(526\) 7.12537 0.310681
\(527\) −5.98126 −0.260548
\(528\) 3.42864 0.149212
\(529\) −7.00000 −0.304348
\(530\) 0 0
\(531\) −1.67307 −0.0726051
\(532\) 4.17484 0.181002
\(533\) −30.9590 −1.34098
\(534\) 1.90813 0.0825730
\(535\) 0 0
\(536\) −15.6128 −0.674372
\(537\) 4.85728 0.209607
\(538\) 3.68196 0.158741
\(539\) −6.18421 −0.266373
\(540\) 0 0
\(541\) −13.6128 −0.585262 −0.292631 0.956225i \(-0.594531\pi\)
−0.292631 + 0.956225i \(0.594531\pi\)
\(542\) 4.63512 0.199096
\(543\) −16.7971 −0.720831
\(544\) 7.97280 0.341831
\(545\) 0 0
\(546\) −0.815792 −0.0349127
\(547\) −7.48394 −0.319990 −0.159995 0.987118i \(-0.551148\pi\)
−0.159995 + 0.987118i \(0.551148\pi\)
\(548\) 8.42864 0.360054
\(549\) −7.24443 −0.309185
\(550\) 0 0
\(551\) 17.1240 0.729506
\(552\) −4.85728 −0.206740
\(553\) 14.9304 0.634906
\(554\) −8.60793 −0.365716
\(555\) 0 0
\(556\) −1.74620 −0.0740554
\(557\) −32.2908 −1.36821 −0.684103 0.729385i \(-0.739806\pi\)
−0.684103 + 0.729385i \(0.739806\pi\)
\(558\) −0.815792 −0.0345352
\(559\) 31.0923 1.31507
\(560\) 0 0
\(561\) 2.28100 0.0963037
\(562\) 3.05086 0.128693
\(563\) −7.49378 −0.315825 −0.157913 0.987453i \(-0.550476\pi\)
−0.157913 + 0.987453i \(0.550476\pi\)
\(564\) −1.80642 −0.0760642
\(565\) 0 0
\(566\) −5.91258 −0.248524
\(567\) −0.903212 −0.0379313
\(568\) −11.2761 −0.473134
\(569\) −12.9491 −0.542856 −0.271428 0.962459i \(-0.587496\pi\)
−0.271428 + 0.962459i \(0.587496\pi\)
\(570\) 0 0
\(571\) −15.2859 −0.639696 −0.319848 0.947469i \(-0.603632\pi\)
−0.319848 + 0.947469i \(0.603632\pi\)
\(572\) −5.52543 −0.231030
\(573\) 16.8573 0.704223
\(574\) 2.99646 0.125070
\(575\) 0 0
\(576\) −5.76986 −0.240411
\(577\) −28.4415 −1.18404 −0.592019 0.805924i \(-0.701669\pi\)
−0.592019 + 0.805924i \(0.701669\pi\)
\(578\) −3.67016 −0.152658
\(579\) −24.6178 −1.02308
\(580\) 0 0
\(581\) −7.00937 −0.290798
\(582\) −3.87955 −0.160813
\(583\) 0.815792 0.0337866
\(584\) 6.87157 0.284348
\(585\) 0 0
\(586\) −9.56691 −0.395206
\(587\) 8.47013 0.349600 0.174800 0.984604i \(-0.444072\pi\)
0.174800 + 0.984604i \(0.444072\pi\)
\(588\) 11.7699 0.485381
\(589\) −6.36842 −0.262406
\(590\) 0 0
\(591\) 14.8716 0.611735
\(592\) 19.9081 0.818219
\(593\) −26.5763 −1.09136 −0.545679 0.837995i \(-0.683728\pi\)
−0.545679 + 0.837995i \(0.683728\pi\)
\(594\) 0.311108 0.0127649
\(595\) 0 0
\(596\) −4.17484 −0.171008
\(597\) −11.2257 −0.459437
\(598\) 3.61285 0.147740
\(599\) −8.77430 −0.358508 −0.179254 0.983803i \(-0.557368\pi\)
−0.179254 + 0.983803i \(0.557368\pi\)
\(600\) 0 0
\(601\) −41.8163 −1.70572 −0.852861 0.522139i \(-0.825134\pi\)
−0.852861 + 0.522139i \(0.825134\pi\)
\(602\) −3.00937 −0.122653
\(603\) 12.8573 0.523589
\(604\) 19.8479 0.807600
\(605\) 0 0
\(606\) 5.03212 0.204416
\(607\) −29.9353 −1.21504 −0.607519 0.794305i \(-0.707835\pi\)
−0.607519 + 0.794305i \(0.707835\pi\)
\(608\) 8.48886 0.344269
\(609\) −6.36842 −0.258061
\(610\) 0 0
\(611\) 2.75557 0.111478
\(612\) −4.34122 −0.175483
\(613\) −26.1289 −1.05534 −0.527668 0.849450i \(-0.676934\pi\)
−0.527668 + 0.849450i \(0.676934\pi\)
\(614\) −4.19802 −0.169418
\(615\) 0 0
\(616\) 1.09679 0.0441909
\(617\) 3.66323 0.147476 0.0737380 0.997278i \(-0.476507\pi\)
0.0737380 + 0.997278i \(0.476507\pi\)
\(618\) −5.33630 −0.214657
\(619\) 43.2958 1.74020 0.870102 0.492872i \(-0.164053\pi\)
0.870102 + 0.492872i \(0.164053\pi\)
\(620\) 0 0
\(621\) 4.00000 0.160514
\(622\) −5.46076 −0.218956
\(623\) −5.53972 −0.221944
\(624\) 9.95407 0.398482
\(625\) 0 0
\(626\) 4.47902 0.179018
\(627\) 2.42864 0.0969905
\(628\) −11.0509 −0.440977
\(629\) 13.2444 0.528090
\(630\) 0 0
\(631\) 8.97773 0.357398 0.178699 0.983904i \(-0.442811\pi\)
0.178699 + 0.983904i \(0.442811\pi\)
\(632\) 20.0731 0.798466
\(633\) −11.9398 −0.474564
\(634\) 9.16547 0.364007
\(635\) 0 0
\(636\) −1.55262 −0.0615656
\(637\) −17.9541 −0.711366
\(638\) 2.19358 0.0868445
\(639\) 9.28592 0.367345
\(640\) 0 0
\(641\) 9.21279 0.363883 0.181942 0.983309i \(-0.441762\pi\)
0.181942 + 0.983309i \(0.441762\pi\)
\(642\) −4.22077 −0.166581
\(643\) 16.3783 0.645896 0.322948 0.946417i \(-0.395326\pi\)
0.322948 + 0.946417i \(0.395326\pi\)
\(644\) 6.87601 0.270953
\(645\) 0 0
\(646\) 1.72345 0.0678082
\(647\) −9.80642 −0.385530 −0.192765 0.981245i \(-0.561746\pi\)
−0.192765 + 0.981245i \(0.561746\pi\)
\(648\) −1.21432 −0.0477030
\(649\) −1.67307 −0.0656738
\(650\) 0 0
\(651\) 2.36842 0.0928256
\(652\) −21.0321 −0.823681
\(653\) −33.0736 −1.29427 −0.647135 0.762375i \(-0.724033\pi\)
−0.647135 + 0.762375i \(0.724033\pi\)
\(654\) −3.11108 −0.121653
\(655\) 0 0
\(656\) −36.5620 −1.42751
\(657\) −5.65878 −0.220770
\(658\) −0.266706 −0.0103973
\(659\) 34.1017 1.32841 0.664207 0.747549i \(-0.268769\pi\)
0.664207 + 0.747549i \(0.268769\pi\)
\(660\) 0 0
\(661\) −5.40943 −0.210402 −0.105201 0.994451i \(-0.533549\pi\)
−0.105201 + 0.994451i \(0.533549\pi\)
\(662\) −0.815792 −0.0317066
\(663\) 6.62222 0.257186
\(664\) −9.42372 −0.365711
\(665\) 0 0
\(666\) 1.80642 0.0699975
\(667\) 28.2034 1.09204
\(668\) −26.5575 −1.02754
\(669\) 21.8064 0.843085
\(670\) 0 0
\(671\) −7.24443 −0.279668
\(672\) −3.15701 −0.121784
\(673\) −24.1476 −0.930823 −0.465412 0.885094i \(-0.654094\pi\)
−0.465412 + 0.885094i \(0.654094\pi\)
\(674\) −1.55707 −0.0599761
\(675\) 0 0
\(676\) 8.70027 0.334626
\(677\) −26.2810 −1.01006 −0.505030 0.863102i \(-0.668519\pi\)
−0.505030 + 0.863102i \(0.668519\pi\)
\(678\) 4.42864 0.170081
\(679\) 11.2632 0.432241
\(680\) 0 0
\(681\) 3.19850 0.122567
\(682\) −0.815792 −0.0312383
\(683\) −15.3176 −0.586110 −0.293055 0.956096i \(-0.594672\pi\)
−0.293055 + 0.956096i \(0.594672\pi\)
\(684\) −4.62222 −0.176735
\(685\) 0 0
\(686\) 3.70471 0.141447
\(687\) 7.12399 0.271797
\(688\) 36.7195 1.39992
\(689\) 2.36842 0.0902295
\(690\) 0 0
\(691\) 15.0223 0.571474 0.285737 0.958308i \(-0.407762\pi\)
0.285737 + 0.958308i \(0.407762\pi\)
\(692\) 35.8622 1.36328
\(693\) −0.903212 −0.0343102
\(694\) −7.09679 −0.269390
\(695\) 0 0
\(696\) −8.56199 −0.324541
\(697\) −24.3239 −0.921332
\(698\) 6.60348 0.249945
\(699\) −19.5254 −0.738519
\(700\) 0 0
\(701\) −19.9081 −0.751920 −0.375960 0.926636i \(-0.622687\pi\)
−0.375960 + 0.926636i \(0.622687\pi\)
\(702\) 0.903212 0.0340895
\(703\) 14.1017 0.531856
\(704\) −5.76986 −0.217460
\(705\) 0 0
\(706\) 2.23506 0.0841177
\(707\) −14.6093 −0.549440
\(708\) 3.18421 0.119670
\(709\) −13.5081 −0.507306 −0.253653 0.967295i \(-0.581632\pi\)
−0.253653 + 0.967295i \(0.581632\pi\)
\(710\) 0 0
\(711\) −16.5303 −0.619937
\(712\) −7.44785 −0.279120
\(713\) −10.4889 −0.392811
\(714\) −0.640951 −0.0239870
\(715\) 0 0
\(716\) −9.24443 −0.345481
\(717\) −21.9813 −0.820905
\(718\) −4.38715 −0.163727
\(719\) −16.0830 −0.599794 −0.299897 0.953972i \(-0.596952\pi\)
−0.299897 + 0.953972i \(0.596952\pi\)
\(720\) 0 0
\(721\) 15.4924 0.576967
\(722\) −4.07604 −0.151695
\(723\) 5.34614 0.198825
\(724\) 31.9684 1.18809
\(725\) 0 0
\(726\) 0.311108 0.0115463
\(727\) 23.6128 0.875752 0.437876 0.899035i \(-0.355731\pi\)
0.437876 + 0.899035i \(0.355731\pi\)
\(728\) 3.18421 0.118015
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 24.4286 0.903526
\(732\) 13.7877 0.509608
\(733\) −30.0459 −1.10977 −0.554886 0.831926i \(-0.687238\pi\)
−0.554886 + 0.831926i \(0.687238\pi\)
\(734\) −1.21585 −0.0448779
\(735\) 0 0
\(736\) 13.9813 0.515356
\(737\) 12.8573 0.473604
\(738\) −3.31756 −0.122121
\(739\) −24.4099 −0.897933 −0.448966 0.893549i \(-0.648208\pi\)
−0.448966 + 0.893549i \(0.648208\pi\)
\(740\) 0 0
\(741\) 7.05086 0.259020
\(742\) −0.229234 −0.00841546
\(743\) −33.1798 −1.21725 −0.608624 0.793459i \(-0.708278\pi\)
−0.608624 + 0.793459i \(0.708278\pi\)
\(744\) 3.18421 0.116739
\(745\) 0 0
\(746\) 4.03704 0.147807
\(747\) 7.76049 0.283942
\(748\) −4.34122 −0.158731
\(749\) 12.2538 0.447744
\(750\) 0 0
\(751\) −22.5718 −0.823658 −0.411829 0.911261i \(-0.635110\pi\)
−0.411829 + 0.911261i \(0.635110\pi\)
\(752\) 3.25428 0.118671
\(753\) −23.7748 −0.866401
\(754\) 6.36842 0.231924
\(755\) 0 0
\(756\) 1.71900 0.0625196
\(757\) −4.94914 −0.179880 −0.0899399 0.995947i \(-0.528667\pi\)
−0.0899399 + 0.995947i \(0.528667\pi\)
\(758\) 11.2257 0.407736
\(759\) 4.00000 0.145191
\(760\) 0 0
\(761\) 14.6637 0.531559 0.265779 0.964034i \(-0.414371\pi\)
0.265779 + 0.964034i \(0.414371\pi\)
\(762\) 3.42372 0.124028
\(763\) 9.03212 0.326985
\(764\) −32.0830 −1.16072
\(765\) 0 0
\(766\) 6.31402 0.228135
\(767\) −4.85728 −0.175386
\(768\) 8.80642 0.317774
\(769\) −44.5718 −1.60730 −0.803651 0.595101i \(-0.797112\pi\)
−0.803651 + 0.595101i \(0.797112\pi\)
\(770\) 0 0
\(771\) −8.13335 −0.292916
\(772\) 46.8528 1.68627
\(773\) −17.3145 −0.622759 −0.311380 0.950286i \(-0.600791\pi\)
−0.311380 + 0.950286i \(0.600791\pi\)
\(774\) 3.33185 0.119761
\(775\) 0 0
\(776\) 15.1427 0.543592
\(777\) −5.24443 −0.188143
\(778\) 9.47949 0.339856
\(779\) −25.8983 −0.927903
\(780\) 0 0
\(781\) 9.28592 0.332276
\(782\) 2.83854 0.101506
\(783\) 7.05086 0.251977
\(784\) −21.2034 −0.757265
\(785\) 0 0
\(786\) 0.387152 0.0138093
\(787\) −36.5161 −1.30166 −0.650828 0.759225i \(-0.725578\pi\)
−0.650828 + 0.759225i \(0.725578\pi\)
\(788\) −28.3037 −1.00828
\(789\) 22.9032 0.815376
\(790\) 0 0
\(791\) −12.8573 −0.457152
\(792\) −1.21432 −0.0431490
\(793\) −21.0321 −0.746872
\(794\) 1.54861 0.0549581
\(795\) 0 0
\(796\) 21.3649 0.757258
\(797\) 14.3180 0.507171 0.253585 0.967313i \(-0.418390\pi\)
0.253585 + 0.967313i \(0.418390\pi\)
\(798\) −0.682439 −0.0241581
\(799\) 2.16500 0.0765921
\(800\) 0 0
\(801\) 6.13335 0.216711
\(802\) −0.580728 −0.0205062
\(803\) −5.65878 −0.199694
\(804\) −24.4701 −0.862995
\(805\) 0 0
\(806\) −2.36842 −0.0834239
\(807\) 11.8350 0.416612
\(808\) −19.6414 −0.690983
\(809\) 32.0544 1.12697 0.563486 0.826125i \(-0.309460\pi\)
0.563486 + 0.826125i \(0.309460\pi\)
\(810\) 0 0
\(811\) −8.44738 −0.296627 −0.148314 0.988940i \(-0.547385\pi\)
−0.148314 + 0.988940i \(0.547385\pi\)
\(812\) 12.1204 0.425344
\(813\) 14.8988 0.522523
\(814\) 1.80642 0.0633151
\(815\) 0 0
\(816\) 7.82071 0.273780
\(817\) 26.0098 0.909969
\(818\) −1.12981 −0.0395030
\(819\) −2.62222 −0.0916276
\(820\) 0 0
\(821\) 17.2159 0.600837 0.300419 0.953807i \(-0.402874\pi\)
0.300419 + 0.953807i \(0.402874\pi\)
\(822\) −1.37778 −0.0480557
\(823\) −12.7654 −0.444974 −0.222487 0.974936i \(-0.571418\pi\)
−0.222487 + 0.974936i \(0.571418\pi\)
\(824\) 20.8287 0.725602
\(825\) 0 0
\(826\) 0.470127 0.0163578
\(827\) −8.70964 −0.302864 −0.151432 0.988468i \(-0.548388\pi\)
−0.151432 + 0.988468i \(0.548388\pi\)
\(828\) −7.61285 −0.264565
\(829\) −8.32693 −0.289206 −0.144603 0.989490i \(-0.546191\pi\)
−0.144603 + 0.989490i \(0.546191\pi\)
\(830\) 0 0
\(831\) −27.6686 −0.959814
\(832\) −16.7511 −0.580741
\(833\) −14.1062 −0.488749
\(834\) 0.285442 0.00988405
\(835\) 0 0
\(836\) −4.62222 −0.159863
\(837\) −2.62222 −0.0906370
\(838\) 1.51114 0.0522014
\(839\) 12.8988 0.445315 0.222657 0.974897i \(-0.428527\pi\)
0.222657 + 0.974897i \(0.428527\pi\)
\(840\) 0 0
\(841\) 20.7146 0.714295
\(842\) 7.04101 0.242649
\(843\) 9.80642 0.337751
\(844\) 22.7239 0.782190
\(845\) 0 0
\(846\) 0.295286 0.0101522
\(847\) −0.903212 −0.0310347
\(848\) 2.79706 0.0960513
\(849\) −19.0049 −0.652247
\(850\) 0 0
\(851\) 23.2257 0.796167
\(852\) −17.6731 −0.605469
\(853\) 19.6686 0.673441 0.336720 0.941605i \(-0.390682\pi\)
0.336720 + 0.941605i \(0.390682\pi\)
\(854\) 2.03566 0.0696588
\(855\) 0 0
\(856\) 16.4746 0.563089
\(857\) −31.8207 −1.08697 −0.543487 0.839417i \(-0.682896\pi\)
−0.543487 + 0.839417i \(0.682896\pi\)
\(858\) 0.903212 0.0308351
\(859\) 27.8292 0.949519 0.474760 0.880116i \(-0.342535\pi\)
0.474760 + 0.880116i \(0.342535\pi\)
\(860\) 0 0
\(861\) 9.63158 0.328243
\(862\) 0.387152 0.0131865
\(863\) 4.82870 0.164371 0.0821854 0.996617i \(-0.473810\pi\)
0.0821854 + 0.996617i \(0.473810\pi\)
\(864\) 3.49532 0.118913
\(865\) 0 0
\(866\) −4.97773 −0.169150
\(867\) −11.7971 −0.400649
\(868\) −4.50760 −0.152998
\(869\) −16.5303 −0.560754
\(870\) 0 0
\(871\) 37.3274 1.26479
\(872\) 12.1432 0.411221
\(873\) −12.4701 −0.422050
\(874\) 3.02227 0.102230
\(875\) 0 0
\(876\) 10.7699 0.363880
\(877\) −21.9826 −0.742301 −0.371151 0.928573i \(-0.621037\pi\)
−0.371151 + 0.928573i \(0.621037\pi\)
\(878\) −0.755569 −0.0254992
\(879\) −30.7511 −1.03721
\(880\) 0 0
\(881\) −12.1017 −0.407717 −0.203858 0.979000i \(-0.565348\pi\)
−0.203858 + 0.979000i \(0.565348\pi\)
\(882\) −1.92396 −0.0647830
\(883\) −8.73683 −0.294018 −0.147009 0.989135i \(-0.546965\pi\)
−0.147009 + 0.989135i \(0.546965\pi\)
\(884\) −12.6035 −0.423901
\(885\) 0 0
\(886\) 9.66016 0.324540
\(887\) 19.8524 0.666577 0.333288 0.942825i \(-0.391842\pi\)
0.333288 + 0.942825i \(0.391842\pi\)
\(888\) −7.05086 −0.236611
\(889\) −9.93978 −0.333369
\(890\) 0 0
\(891\) 1.00000 0.0335013
\(892\) −41.5022 −1.38960
\(893\) 2.30513 0.0771383
\(894\) 0.682439 0.0228242
\(895\) 0 0
\(896\) 7.93533 0.265101
\(897\) 11.6128 0.387742
\(898\) −11.6227 −0.387854
\(899\) −18.4889 −0.616638
\(900\) 0 0
\(901\) 1.86082 0.0619928
\(902\) −3.31756 −0.110463
\(903\) −9.67307 −0.321900
\(904\) −17.2859 −0.574921
\(905\) 0 0
\(906\) −3.24443 −0.107789
\(907\) −32.8287 −1.09006 −0.545030 0.838417i \(-0.683482\pi\)
−0.545030 + 0.838417i \(0.683482\pi\)
\(908\) −6.08742 −0.202018
\(909\) 16.1748 0.536486
\(910\) 0 0
\(911\) −16.3497 −0.541689 −0.270845 0.962623i \(-0.587303\pi\)
−0.270845 + 0.962623i \(0.587303\pi\)
\(912\) 8.32693 0.275732
\(913\) 7.76049 0.256835
\(914\) 2.71853 0.0899209
\(915\) 0 0
\(916\) −13.5585 −0.447984
\(917\) −1.12399 −0.0371173
\(918\) 0.709636 0.0234215
\(919\) 20.0228 0.660490 0.330245 0.943895i \(-0.392869\pi\)
0.330245 + 0.943895i \(0.392869\pi\)
\(920\) 0 0
\(921\) −13.4938 −0.444635
\(922\) 9.88940 0.325690
\(923\) 26.9590 0.887366
\(924\) 1.71900 0.0565511
\(925\) 0 0
\(926\) 3.76187 0.123623
\(927\) −17.1526 −0.563364
\(928\) 24.6450 0.809011
\(929\) −43.5308 −1.42820 −0.714100 0.700044i \(-0.753164\pi\)
−0.714100 + 0.700044i \(0.753164\pi\)
\(930\) 0 0
\(931\) −15.0192 −0.492235
\(932\) 37.1610 1.21725
\(933\) −17.5526 −0.574647
\(934\) −4.77430 −0.156220
\(935\) 0 0
\(936\) −3.52543 −0.115232
\(937\) 43.4563 1.41966 0.709828 0.704375i \(-0.248773\pi\)
0.709828 + 0.704375i \(0.248773\pi\)
\(938\) −3.61285 −0.117964
\(939\) 14.3970 0.469828
\(940\) 0 0
\(941\) −23.7244 −0.773393 −0.386697 0.922207i \(-0.626384\pi\)
−0.386697 + 0.922207i \(0.626384\pi\)
\(942\) 1.80642 0.0588565
\(943\) −42.6548 −1.38903
\(944\) −5.73636 −0.186703
\(945\) 0 0
\(946\) 3.33185 0.108328
\(947\) 11.7047 0.380352 0.190176 0.981750i \(-0.439094\pi\)
0.190176 + 0.981750i \(0.439094\pi\)
\(948\) 31.4608 1.02180
\(949\) −16.4286 −0.533296
\(950\) 0 0
\(951\) 29.4608 0.955331
\(952\) 2.50177 0.0810828
\(953\) 46.1258 1.49416 0.747081 0.664733i \(-0.231455\pi\)
0.747081 + 0.664733i \(0.231455\pi\)
\(954\) 0.253799 0.00821705
\(955\) 0 0
\(956\) 41.8350 1.35304
\(957\) 7.05086 0.227922
\(958\) 1.83500 0.0592863
\(959\) 4.00000 0.129167
\(960\) 0 0
\(961\) −24.1240 −0.778193
\(962\) 5.24443 0.169087
\(963\) −13.5669 −0.437188
\(964\) −10.1748 −0.327710
\(965\) 0 0
\(966\) −1.12399 −0.0361636
\(967\) 17.0495 0.548274 0.274137 0.961691i \(-0.411608\pi\)
0.274137 + 0.961691i \(0.411608\pi\)
\(968\) −1.21432 −0.0390297
\(969\) 5.53972 0.177961
\(970\) 0 0
\(971\) −58.1847 −1.86724 −0.933618 0.358271i \(-0.883366\pi\)
−0.933618 + 0.358271i \(0.883366\pi\)
\(972\) −1.90321 −0.0610456
\(973\) −0.828699 −0.0265669
\(974\) 9.75203 0.312475
\(975\) 0 0
\(976\) −24.8385 −0.795062
\(977\) −51.7373 −1.65522 −0.827612 0.561301i \(-0.810301\pi\)
−0.827612 + 0.561301i \(0.810301\pi\)
\(978\) 3.43801 0.109935
\(979\) 6.13335 0.196023
\(980\) 0 0
\(981\) −10.0000 −0.319275
\(982\) −2.48886 −0.0794228
\(983\) −26.3970 −0.841933 −0.420967 0.907076i \(-0.638309\pi\)
−0.420967 + 0.907076i \(0.638309\pi\)
\(984\) 12.9491 0.412804
\(985\) 0 0
\(986\) 5.00354 0.159345
\(987\) −0.857279 −0.0272875
\(988\) −13.4193 −0.426924
\(989\) 42.8385 1.36219
\(990\) 0 0
\(991\) −23.0923 −0.733552 −0.366776 0.930309i \(-0.619539\pi\)
−0.366776 + 0.930309i \(0.619539\pi\)
\(992\) −9.16547 −0.291004
\(993\) −2.62222 −0.0832135
\(994\) −2.60931 −0.0827622
\(995\) 0 0
\(996\) −14.7699 −0.468001
\(997\) 12.9131 0.408961 0.204480 0.978871i \(-0.434450\pi\)
0.204480 + 0.978871i \(0.434450\pi\)
\(998\) 4.71102 0.149125
\(999\) 5.80642 0.183707
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.l.1.2 3
3.2 odd 2 2475.2.a.ba.1.2 3
5.2 odd 4 165.2.c.b.34.4 yes 6
5.3 odd 4 165.2.c.b.34.3 6
5.4 even 2 825.2.a.j.1.2 3
11.10 odd 2 9075.2.a.cg.1.2 3
15.2 even 4 495.2.c.e.199.3 6
15.8 even 4 495.2.c.e.199.4 6
15.14 odd 2 2475.2.a.bc.1.2 3
20.3 even 4 2640.2.d.h.529.2 6
20.7 even 4 2640.2.d.h.529.5 6
55.32 even 4 1815.2.c.e.364.3 6
55.43 even 4 1815.2.c.e.364.4 6
55.54 odd 2 9075.2.a.ch.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
165.2.c.b.34.3 6 5.3 odd 4
165.2.c.b.34.4 yes 6 5.2 odd 4
495.2.c.e.199.3 6 15.2 even 4
495.2.c.e.199.4 6 15.8 even 4
825.2.a.j.1.2 3 5.4 even 2
825.2.a.l.1.2 3 1.1 even 1 trivial
1815.2.c.e.364.3 6 55.32 even 4
1815.2.c.e.364.4 6 55.43 even 4
2475.2.a.ba.1.2 3 3.2 odd 2
2475.2.a.bc.1.2 3 15.14 odd 2
2640.2.d.h.529.2 6 20.3 even 4
2640.2.d.h.529.5 6 20.7 even 4
9075.2.a.cg.1.2 3 11.10 odd 2
9075.2.a.ch.1.2 3 55.54 odd 2