Properties

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

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Newspace parameters

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

Newform invariants

Self dual: yes
Analytic conductor: \(6.58765816676\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.568.1
Defining polynomial: \( x^{3} - x^{2} - 6x - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-1.76156\) of defining polynomial
Character \(\chi\) \(=\) 825.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.76156 q^{2} +1.00000 q^{3} +5.62620 q^{4} +2.76156 q^{6} -1.86464 q^{7} +10.0140 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+2.76156 q^{2} +1.00000 q^{3} +5.62620 q^{4} +2.76156 q^{6} -1.86464 q^{7} +10.0140 q^{8} +1.00000 q^{9} +1.00000 q^{11} +5.62620 q^{12} -4.62620 q^{13} -5.14931 q^{14} +16.4017 q^{16} -2.49084 q^{17} +2.76156 q^{18} -5.38776 q^{19} -1.86464 q^{21} +2.76156 q^{22} -7.14931 q^{23} +10.0140 q^{24} -12.7755 q^{26} +1.00000 q^{27} -10.4908 q^{28} -3.52311 q^{29} +8.62620 q^{31} +25.2663 q^{32} +1.00000 q^{33} -6.87859 q^{34} +5.62620 q^{36} +8.87859 q^{37} -14.8786 q^{38} -4.62620 q^{39} -0.761557 q^{41} -5.14931 q^{42} +7.40171 q^{43} +5.62620 q^{44} -19.7432 q^{46} +0.373802 q^{47} +16.4017 q^{48} -3.52311 q^{49} -2.49084 q^{51} -26.0279 q^{52} +5.45856 q^{53} +2.76156 q^{54} -18.6724 q^{56} -5.38776 q^{57} -9.72928 q^{58} +5.14931 q^{59} +4.42003 q^{61} +23.8217 q^{62} -1.86464 q^{63} +36.9711 q^{64} +2.76156 q^{66} -11.9431 q^{67} -14.0140 q^{68} -7.14931 q^{69} +11.6262 q^{71} +10.0140 q^{72} -6.77551 q^{73} +24.5187 q^{74} -30.3126 q^{76} -1.86464 q^{77} -12.7755 q^{78} -6.01395 q^{79} +1.00000 q^{81} -2.10308 q^{82} -14.5693 q^{83} -10.4908 q^{84} +20.4402 q^{86} -3.52311 q^{87} +10.0140 q^{88} +9.04623 q^{89} +8.62620 q^{91} -40.2234 q^{92} +8.62620 q^{93} +1.03228 q^{94} +25.2663 q^{96} -16.3169 q^{97} -9.72928 q^{98} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 2 q^{2} + 3 q^{3} + 8 q^{4} + 2 q^{6} - 3 q^{7} + 6 q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 2 q^{2} + 3 q^{3} + 8 q^{4} + 2 q^{6} - 3 q^{7} + 6 q^{8} + 3 q^{9} + 3 q^{11} + 8 q^{12} - 5 q^{13} + 6 q^{14} + 10 q^{16} + 4 q^{17} + 2 q^{18} - q^{19} - 3 q^{21} + 2 q^{22} + 6 q^{24} - 8 q^{26} + 3 q^{27} - 20 q^{28} + 2 q^{29} + 17 q^{31} + 34 q^{32} + 3 q^{33} + 6 q^{34} + 8 q^{36} - 18 q^{38} - 5 q^{39} + 4 q^{41} + 6 q^{42} - 17 q^{43} + 8 q^{44} - 30 q^{46} + 10 q^{47} + 10 q^{48} + 2 q^{49} + 4 q^{51} - 30 q^{52} + 6 q^{53} + 2 q^{54} - 22 q^{56} - q^{57} - 24 q^{58} - 6 q^{59} - 3 q^{61} + 16 q^{62} - 3 q^{63} + 34 q^{64} + 2 q^{66} - 7 q^{67} - 18 q^{68} + 26 q^{71} + 6 q^{72} + 10 q^{73} + 14 q^{74} - 24 q^{76} - 3 q^{77} - 8 q^{78} + 6 q^{79} + 3 q^{81} - 10 q^{82} - 6 q^{83} - 20 q^{84} + 28 q^{86} + 2 q^{87} + 6 q^{88} + 2 q^{89} + 17 q^{91} - 26 q^{92} + 17 q^{93} + 2 q^{94} + 34 q^{96} - 29 q^{97} - 24 q^{98} + 3 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.76156 1.95272 0.976358 0.216160i \(-0.0693534\pi\)
0.976358 + 0.216160i \(0.0693534\pi\)
\(3\) 1.00000 0.577350
\(4\) 5.62620 2.81310
\(5\) 0 0
\(6\) 2.76156 1.12740
\(7\) −1.86464 −0.704768 −0.352384 0.935855i \(-0.614629\pi\)
−0.352384 + 0.935855i \(0.614629\pi\)
\(8\) 10.0140 3.54047
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) 5.62620 1.62414
\(13\) −4.62620 −1.28308 −0.641538 0.767091i \(-0.721703\pi\)
−0.641538 + 0.767091i \(0.721703\pi\)
\(14\) −5.14931 −1.37621
\(15\) 0 0
\(16\) 16.4017 4.10043
\(17\) −2.49084 −0.604117 −0.302059 0.953289i \(-0.597674\pi\)
−0.302059 + 0.953289i \(0.597674\pi\)
\(18\) 2.76156 0.650905
\(19\) −5.38776 −1.23604 −0.618018 0.786164i \(-0.712064\pi\)
−0.618018 + 0.786164i \(0.712064\pi\)
\(20\) 0 0
\(21\) −1.86464 −0.406898
\(22\) 2.76156 0.588766
\(23\) −7.14931 −1.49073 −0.745367 0.666654i \(-0.767726\pi\)
−0.745367 + 0.666654i \(0.767726\pi\)
\(24\) 10.0140 2.04409
\(25\) 0 0
\(26\) −12.7755 −2.50548
\(27\) 1.00000 0.192450
\(28\) −10.4908 −1.98258
\(29\) −3.52311 −0.654226 −0.327113 0.944985i \(-0.606076\pi\)
−0.327113 + 0.944985i \(0.606076\pi\)
\(30\) 0 0
\(31\) 8.62620 1.54931 0.774655 0.632384i \(-0.217923\pi\)
0.774655 + 0.632384i \(0.217923\pi\)
\(32\) 25.2663 4.46650
\(33\) 1.00000 0.174078
\(34\) −6.87859 −1.17967
\(35\) 0 0
\(36\) 5.62620 0.937700
\(37\) 8.87859 1.45963 0.729816 0.683644i \(-0.239606\pi\)
0.729816 + 0.683644i \(0.239606\pi\)
\(38\) −14.8786 −2.41363
\(39\) −4.62620 −0.740785
\(40\) 0 0
\(41\) −0.761557 −0.118935 −0.0594676 0.998230i \(-0.518940\pi\)
−0.0594676 + 0.998230i \(0.518940\pi\)
\(42\) −5.14931 −0.794556
\(43\) 7.40171 1.12875 0.564375 0.825519i \(-0.309117\pi\)
0.564375 + 0.825519i \(0.309117\pi\)
\(44\) 5.62620 0.848181
\(45\) 0 0
\(46\) −19.7432 −2.91098
\(47\) 0.373802 0.0545246 0.0272623 0.999628i \(-0.491321\pi\)
0.0272623 + 0.999628i \(0.491321\pi\)
\(48\) 16.4017 2.36738
\(49\) −3.52311 −0.503302
\(50\) 0 0
\(51\) −2.49084 −0.348787
\(52\) −26.0279 −3.60942
\(53\) 5.45856 0.749791 0.374896 0.927067i \(-0.377679\pi\)
0.374896 + 0.927067i \(0.377679\pi\)
\(54\) 2.76156 0.375800
\(55\) 0 0
\(56\) −18.6724 −2.49521
\(57\) −5.38776 −0.713626
\(58\) −9.72928 −1.27752
\(59\) 5.14931 0.670383 0.335192 0.942150i \(-0.391199\pi\)
0.335192 + 0.942150i \(0.391199\pi\)
\(60\) 0 0
\(61\) 4.42003 0.565927 0.282963 0.959131i \(-0.408682\pi\)
0.282963 + 0.959131i \(0.408682\pi\)
\(62\) 23.8217 3.02536
\(63\) −1.86464 −0.234923
\(64\) 36.9711 4.62138
\(65\) 0 0
\(66\) 2.76156 0.339924
\(67\) −11.9431 −1.45909 −0.729544 0.683934i \(-0.760268\pi\)
−0.729544 + 0.683934i \(0.760268\pi\)
\(68\) −14.0140 −1.69944
\(69\) −7.14931 −0.860676
\(70\) 0 0
\(71\) 11.6262 1.37978 0.689888 0.723916i \(-0.257660\pi\)
0.689888 + 0.723916i \(0.257660\pi\)
\(72\) 10.0140 1.18016
\(73\) −6.77551 −0.793014 −0.396507 0.918032i \(-0.629778\pi\)
−0.396507 + 0.918032i \(0.629778\pi\)
\(74\) 24.5187 2.85025
\(75\) 0 0
\(76\) −30.3126 −3.47709
\(77\) −1.86464 −0.212496
\(78\) −12.7755 −1.44654
\(79\) −6.01395 −0.676623 −0.338311 0.941034i \(-0.609856\pi\)
−0.338311 + 0.941034i \(0.609856\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) −2.10308 −0.232247
\(83\) −14.5693 −1.59919 −0.799597 0.600538i \(-0.794953\pi\)
−0.799597 + 0.600538i \(0.794953\pi\)
\(84\) −10.4908 −1.14464
\(85\) 0 0
\(86\) 20.4402 2.20413
\(87\) −3.52311 −0.377718
\(88\) 10.0140 1.06749
\(89\) 9.04623 0.958898 0.479449 0.877570i \(-0.340836\pi\)
0.479449 + 0.877570i \(0.340836\pi\)
\(90\) 0 0
\(91\) 8.62620 0.904271
\(92\) −40.2234 −4.19358
\(93\) 8.62620 0.894495
\(94\) 1.03228 0.106471
\(95\) 0 0
\(96\) 25.2663 2.57874
\(97\) −16.3169 −1.65673 −0.828367 0.560185i \(-0.810730\pi\)
−0.828367 + 0.560185i \(0.810730\pi\)
\(98\) −9.72928 −0.982806
\(99\) 1.00000 0.100504
\(100\) 0 0
\(101\) −1.03228 −0.102715 −0.0513576 0.998680i \(-0.516355\pi\)
−0.0513576 + 0.998680i \(0.516355\pi\)
\(102\) −6.87859 −0.681082
\(103\) −3.04623 −0.300154 −0.150077 0.988674i \(-0.547952\pi\)
−0.150077 + 0.988674i \(0.547952\pi\)
\(104\) −46.3265 −4.54269
\(105\) 0 0
\(106\) 15.0741 1.46413
\(107\) 8.50479 0.822189 0.411095 0.911593i \(-0.365147\pi\)
0.411095 + 0.911593i \(0.365147\pi\)
\(108\) 5.62620 0.541381
\(109\) −6.14931 −0.588997 −0.294499 0.955652i \(-0.595153\pi\)
−0.294499 + 0.955652i \(0.595153\pi\)
\(110\) 0 0
\(111\) 8.87859 0.842719
\(112\) −30.5833 −2.88985
\(113\) 6.00000 0.564433 0.282216 0.959351i \(-0.408930\pi\)
0.282216 + 0.959351i \(0.408930\pi\)
\(114\) −14.8786 −1.39351
\(115\) 0 0
\(116\) −19.8217 −1.84040
\(117\) −4.62620 −0.427692
\(118\) 14.2201 1.30907
\(119\) 4.64452 0.425762
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 12.2062 1.10509
\(123\) −0.761557 −0.0686673
\(124\) 48.5327 4.35837
\(125\) 0 0
\(126\) −5.14931 −0.458737
\(127\) 9.26635 0.822256 0.411128 0.911578i \(-0.365135\pi\)
0.411128 + 0.911578i \(0.365135\pi\)
\(128\) 51.5650 4.55774
\(129\) 7.40171 0.651684
\(130\) 0 0
\(131\) 4.06455 0.355121 0.177561 0.984110i \(-0.443179\pi\)
0.177561 + 0.984110i \(0.443179\pi\)
\(132\) 5.62620 0.489698
\(133\) 10.0462 0.871119
\(134\) −32.9817 −2.84918
\(135\) 0 0
\(136\) −24.9431 −2.13886
\(137\) −5.93545 −0.507100 −0.253550 0.967322i \(-0.581598\pi\)
−0.253550 + 0.967322i \(0.581598\pi\)
\(138\) −19.7432 −1.68066
\(139\) 6.71096 0.569216 0.284608 0.958644i \(-0.408137\pi\)
0.284608 + 0.958644i \(0.408137\pi\)
\(140\) 0 0
\(141\) 0.373802 0.0314798
\(142\) 32.1064 2.69431
\(143\) −4.62620 −0.386862
\(144\) 16.4017 1.36681
\(145\) 0 0
\(146\) −18.7110 −1.54853
\(147\) −3.52311 −0.290582
\(148\) 49.9527 4.10609
\(149\) 5.74324 0.470504 0.235252 0.971934i \(-0.424408\pi\)
0.235252 + 0.971934i \(0.424408\pi\)
\(150\) 0 0
\(151\) 10.5616 0.859495 0.429747 0.902949i \(-0.358603\pi\)
0.429747 + 0.902949i \(0.358603\pi\)
\(152\) −53.9527 −4.37614
\(153\) −2.49084 −0.201372
\(154\) −5.14931 −0.414943
\(155\) 0 0
\(156\) −26.0279 −2.08390
\(157\) −3.10308 −0.247653 −0.123827 0.992304i \(-0.539517\pi\)
−0.123827 + 0.992304i \(0.539517\pi\)
\(158\) −16.6079 −1.32125
\(159\) 5.45856 0.432892
\(160\) 0 0
\(161\) 13.3309 1.05062
\(162\) 2.76156 0.216968
\(163\) −15.6724 −1.22756 −0.613780 0.789477i \(-0.710352\pi\)
−0.613780 + 0.789477i \(0.710352\pi\)
\(164\) −4.28467 −0.334577
\(165\) 0 0
\(166\) −40.2341 −3.12277
\(167\) 8.98168 0.695023 0.347512 0.937676i \(-0.387027\pi\)
0.347512 + 0.937676i \(0.387027\pi\)
\(168\) −18.6724 −1.44061
\(169\) 8.40171 0.646285
\(170\) 0 0
\(171\) −5.38776 −0.412012
\(172\) 41.6435 3.17529
\(173\) 11.5092 0.875025 0.437513 0.899212i \(-0.355860\pi\)
0.437513 + 0.899212i \(0.355860\pi\)
\(174\) −9.72928 −0.737575
\(175\) 0 0
\(176\) 16.4017 1.23633
\(177\) 5.14931 0.387046
\(178\) 24.9817 1.87246
\(179\) 10.3738 0.775374 0.387687 0.921791i \(-0.373274\pi\)
0.387687 + 0.921791i \(0.373274\pi\)
\(180\) 0 0
\(181\) −2.66473 −0.198068 −0.0990339 0.995084i \(-0.531575\pi\)
−0.0990339 + 0.995084i \(0.531575\pi\)
\(182\) 23.8217 1.76578
\(183\) 4.42003 0.326738
\(184\) −71.5929 −5.27790
\(185\) 0 0
\(186\) 23.8217 1.74669
\(187\) −2.49084 −0.182148
\(188\) 2.10308 0.153383
\(189\) −1.86464 −0.135633
\(190\) 0 0
\(191\) 14.6724 1.06166 0.530830 0.847478i \(-0.321880\pi\)
0.530830 + 0.847478i \(0.321880\pi\)
\(192\) 36.9711 2.66816
\(193\) −5.10308 −0.367328 −0.183664 0.982989i \(-0.558796\pi\)
−0.183664 + 0.982989i \(0.558796\pi\)
\(194\) −45.0602 −3.23513
\(195\) 0 0
\(196\) −19.8217 −1.41584
\(197\) −3.74324 −0.266694 −0.133347 0.991069i \(-0.542573\pi\)
−0.133347 + 0.991069i \(0.542573\pi\)
\(198\) 2.76156 0.196255
\(199\) −8.08476 −0.573114 −0.286557 0.958063i \(-0.592511\pi\)
−0.286557 + 0.958063i \(0.592511\pi\)
\(200\) 0 0
\(201\) −11.9431 −0.842404
\(202\) −2.85069 −0.200574
\(203\) 6.56934 0.461077
\(204\) −14.0140 −0.981173
\(205\) 0 0
\(206\) −8.41233 −0.586115
\(207\) −7.14931 −0.496912
\(208\) −75.8776 −5.26116
\(209\) −5.38776 −0.372679
\(210\) 0 0
\(211\) −15.9431 −1.09757 −0.548786 0.835963i \(-0.684910\pi\)
−0.548786 + 0.835963i \(0.684910\pi\)
\(212\) 30.7110 2.10924
\(213\) 11.6262 0.796614
\(214\) 23.4865 1.60550
\(215\) 0 0
\(216\) 10.0140 0.681363
\(217\) −16.0848 −1.09190
\(218\) −16.9817 −1.15014
\(219\) −6.77551 −0.457847
\(220\) 0 0
\(221\) 11.5231 0.775129
\(222\) 24.5187 1.64559
\(223\) 22.1772 1.48510 0.742548 0.669793i \(-0.233617\pi\)
0.742548 + 0.669793i \(0.233617\pi\)
\(224\) −47.1127 −3.14785
\(225\) 0 0
\(226\) 16.5693 1.10218
\(227\) −5.93545 −0.393950 −0.196975 0.980409i \(-0.563112\pi\)
−0.196975 + 0.980409i \(0.563112\pi\)
\(228\) −30.3126 −2.00750
\(229\) −8.31695 −0.549599 −0.274800 0.961502i \(-0.588612\pi\)
−0.274800 + 0.961502i \(0.588612\pi\)
\(230\) 0 0
\(231\) −1.86464 −0.122684
\(232\) −35.2803 −2.31627
\(233\) −28.5187 −1.86833 −0.934163 0.356848i \(-0.883852\pi\)
−0.934163 + 0.356848i \(0.883852\pi\)
\(234\) −12.7755 −0.835161
\(235\) 0 0
\(236\) 28.9711 1.88585
\(237\) −6.01395 −0.390648
\(238\) 12.8261 0.831393
\(239\) 24.0925 1.55841 0.779206 0.626768i \(-0.215623\pi\)
0.779206 + 0.626768i \(0.215623\pi\)
\(240\) 0 0
\(241\) −5.16763 −0.332877 −0.166438 0.986052i \(-0.553227\pi\)
−0.166438 + 0.986052i \(0.553227\pi\)
\(242\) 2.76156 0.177520
\(243\) 1.00000 0.0641500
\(244\) 24.8680 1.59201
\(245\) 0 0
\(246\) −2.10308 −0.134088
\(247\) 24.9248 1.58593
\(248\) 86.3823 5.48528
\(249\) −14.5693 −0.923295
\(250\) 0 0
\(251\) −9.25240 −0.584006 −0.292003 0.956417i \(-0.594322\pi\)
−0.292003 + 0.956417i \(0.594322\pi\)
\(252\) −10.4908 −0.660861
\(253\) −7.14931 −0.449473
\(254\) 25.5896 1.60563
\(255\) 0 0
\(256\) 68.4575 4.27860
\(257\) 25.2158 1.57292 0.786458 0.617644i \(-0.211913\pi\)
0.786458 + 0.617644i \(0.211913\pi\)
\(258\) 20.4402 1.27255
\(259\) −16.5554 −1.02870
\(260\) 0 0
\(261\) −3.52311 −0.218075
\(262\) 11.2245 0.693451
\(263\) −5.45856 −0.336589 −0.168295 0.985737i \(-0.553826\pi\)
−0.168295 + 0.985737i \(0.553826\pi\)
\(264\) 10.0140 0.616316
\(265\) 0 0
\(266\) 27.7432 1.70105
\(267\) 9.04623 0.553620
\(268\) −67.1945 −4.10456
\(269\) 17.0741 1.04103 0.520514 0.853853i \(-0.325740\pi\)
0.520514 + 0.853853i \(0.325740\pi\)
\(270\) 0 0
\(271\) 9.77988 0.594085 0.297043 0.954864i \(-0.404000\pi\)
0.297043 + 0.954864i \(0.404000\pi\)
\(272\) −40.8540 −2.47714
\(273\) 8.62620 0.522081
\(274\) −16.3911 −0.990221
\(275\) 0 0
\(276\) −40.2234 −2.42117
\(277\) −5.91524 −0.355412 −0.177706 0.984084i \(-0.556868\pi\)
−0.177706 + 0.984084i \(0.556868\pi\)
\(278\) 18.5327 1.11152
\(279\) 8.62620 0.516437
\(280\) 0 0
\(281\) −6.76156 −0.403361 −0.201680 0.979451i \(-0.564640\pi\)
−0.201680 + 0.979451i \(0.564640\pi\)
\(282\) 1.03228 0.0614711
\(283\) −8.81841 −0.524200 −0.262100 0.965041i \(-0.584415\pi\)
−0.262100 + 0.965041i \(0.584415\pi\)
\(284\) 65.4113 3.88145
\(285\) 0 0
\(286\) −12.7755 −0.755432
\(287\) 1.42003 0.0838218
\(288\) 25.2663 1.48883
\(289\) −10.7957 −0.635042
\(290\) 0 0
\(291\) −16.3169 −0.956516
\(292\) −38.1204 −2.23083
\(293\) 30.4908 1.78129 0.890647 0.454696i \(-0.150252\pi\)
0.890647 + 0.454696i \(0.150252\pi\)
\(294\) −9.72928 −0.567423
\(295\) 0 0
\(296\) 88.9098 5.16778
\(297\) 1.00000 0.0580259
\(298\) 15.8603 0.918761
\(299\) 33.0741 1.91273
\(300\) 0 0
\(301\) −13.8015 −0.795507
\(302\) 29.1666 1.67835
\(303\) −1.03228 −0.0593027
\(304\) −88.3684 −5.06827
\(305\) 0 0
\(306\) −6.87859 −0.393223
\(307\) −0.767815 −0.0438215 −0.0219107 0.999760i \(-0.506975\pi\)
−0.0219107 + 0.999760i \(0.506975\pi\)
\(308\) −10.4908 −0.597771
\(309\) −3.04623 −0.173294
\(310\) 0 0
\(311\) −33.8496 −1.91944 −0.959719 0.280963i \(-0.909346\pi\)
−0.959719 + 0.280963i \(0.909346\pi\)
\(312\) −46.3265 −2.62272
\(313\) 8.11078 0.458448 0.229224 0.973374i \(-0.426381\pi\)
0.229224 + 0.973374i \(0.426381\pi\)
\(314\) −8.56934 −0.483596
\(315\) 0 0
\(316\) −33.8357 −1.90341
\(317\) −8.06455 −0.452950 −0.226475 0.974017i \(-0.572720\pi\)
−0.226475 + 0.974017i \(0.572720\pi\)
\(318\) 15.0741 0.845316
\(319\) −3.52311 −0.197257
\(320\) 0 0
\(321\) 8.50479 0.474691
\(322\) 36.8140 2.05157
\(323\) 13.4200 0.746710
\(324\) 5.62620 0.312567
\(325\) 0 0
\(326\) −43.2803 −2.39707
\(327\) −6.14931 −0.340058
\(328\) −7.62620 −0.421086
\(329\) −0.697006 −0.0384272
\(330\) 0 0
\(331\) 22.4769 1.23544 0.617721 0.786398i \(-0.288056\pi\)
0.617721 + 0.786398i \(0.288056\pi\)
\(332\) −81.9700 −4.49869
\(333\) 8.87859 0.486544
\(334\) 24.8034 1.35718
\(335\) 0 0
\(336\) −30.5833 −1.66846
\(337\) −6.32757 −0.344685 −0.172342 0.985037i \(-0.555134\pi\)
−0.172342 + 0.985037i \(0.555134\pi\)
\(338\) 23.2018 1.26201
\(339\) 6.00000 0.325875
\(340\) 0 0
\(341\) 8.62620 0.467135
\(342\) −14.8786 −0.804542
\(343\) 19.6218 1.05948
\(344\) 74.1204 3.99630
\(345\) 0 0
\(346\) 31.7832 1.70868
\(347\) 0.178261 0.00956954 0.00478477 0.999989i \(-0.498477\pi\)
0.00478477 + 0.999989i \(0.498477\pi\)
\(348\) −19.8217 −1.06256
\(349\) −19.7293 −1.05608 −0.528042 0.849218i \(-0.677074\pi\)
−0.528042 + 0.849218i \(0.677074\pi\)
\(350\) 0 0
\(351\) −4.62620 −0.246928
\(352\) 25.2663 1.34670
\(353\) 19.5231 1.03911 0.519555 0.854437i \(-0.326098\pi\)
0.519555 + 0.854437i \(0.326098\pi\)
\(354\) 14.2201 0.755791
\(355\) 0 0
\(356\) 50.8959 2.69748
\(357\) 4.64452 0.245814
\(358\) 28.6478 1.51409
\(359\) 35.6926 1.88379 0.941893 0.335914i \(-0.109045\pi\)
0.941893 + 0.335914i \(0.109045\pi\)
\(360\) 0 0
\(361\) 10.0279 0.527785
\(362\) −7.35881 −0.386770
\(363\) 1.00000 0.0524864
\(364\) 48.5327 2.54380
\(365\) 0 0
\(366\) 12.2062 0.638027
\(367\) −20.1127 −1.04987 −0.524936 0.851141i \(-0.675911\pi\)
−0.524936 + 0.851141i \(0.675911\pi\)
\(368\) −117.261 −6.11265
\(369\) −0.761557 −0.0396451
\(370\) 0 0
\(371\) −10.1783 −0.528429
\(372\) 48.5327 2.51630
\(373\) 2.56165 0.132637 0.0663185 0.997799i \(-0.478875\pi\)
0.0663185 + 0.997799i \(0.478875\pi\)
\(374\) −6.87859 −0.355684
\(375\) 0 0
\(376\) 3.74324 0.193043
\(377\) 16.2986 0.839422
\(378\) −5.14931 −0.264852
\(379\) 23.2601 1.19479 0.597395 0.801947i \(-0.296202\pi\)
0.597395 + 0.801947i \(0.296202\pi\)
\(380\) 0 0
\(381\) 9.26635 0.474729
\(382\) 40.5187 2.07312
\(383\) −25.7938 −1.31800 −0.659002 0.752142i \(-0.729021\pi\)
−0.659002 + 0.752142i \(0.729021\pi\)
\(384\) 51.5650 2.63141
\(385\) 0 0
\(386\) −14.0925 −0.717287
\(387\) 7.40171 0.376250
\(388\) −91.8024 −4.66056
\(389\) 2.33527 0.118403 0.0592014 0.998246i \(-0.481145\pi\)
0.0592014 + 0.998246i \(0.481145\pi\)
\(390\) 0 0
\(391\) 17.8078 0.900578
\(392\) −35.2803 −1.78192
\(393\) 4.06455 0.205029
\(394\) −10.3372 −0.520778
\(395\) 0 0
\(396\) 5.62620 0.282727
\(397\) −17.1955 −0.863019 −0.431510 0.902108i \(-0.642019\pi\)
−0.431510 + 0.902108i \(0.642019\pi\)
\(398\) −22.3265 −1.11913
\(399\) 10.0462 0.502941
\(400\) 0 0
\(401\) −26.5693 −1.32681 −0.663405 0.748261i \(-0.730889\pi\)
−0.663405 + 0.748261i \(0.730889\pi\)
\(402\) −32.9817 −1.64498
\(403\) −39.9065 −1.98788
\(404\) −5.80779 −0.288948
\(405\) 0 0
\(406\) 18.1416 0.900353
\(407\) 8.87859 0.440096
\(408\) −24.9431 −1.23487
\(409\) −20.4200 −1.00971 −0.504853 0.863205i \(-0.668453\pi\)
−0.504853 + 0.863205i \(0.668453\pi\)
\(410\) 0 0
\(411\) −5.93545 −0.292774
\(412\) −17.1387 −0.844362
\(413\) −9.60162 −0.472465
\(414\) −19.7432 −0.970327
\(415\) 0 0
\(416\) −116.887 −5.73086
\(417\) 6.71096 0.328637
\(418\) −14.8786 −0.727736
\(419\) −13.5896 −0.663893 −0.331947 0.943298i \(-0.607705\pi\)
−0.331947 + 0.943298i \(0.607705\pi\)
\(420\) 0 0
\(421\) −39.6175 −1.93084 −0.965418 0.260705i \(-0.916045\pi\)
−0.965418 + 0.260705i \(0.916045\pi\)
\(422\) −44.0279 −2.14324
\(423\) 0.373802 0.0181749
\(424\) 54.6618 2.65461
\(425\) 0 0
\(426\) 32.1064 1.55556
\(427\) −8.24177 −0.398847
\(428\) 47.8496 2.31290
\(429\) −4.62620 −0.223355
\(430\) 0 0
\(431\) −6.56934 −0.316434 −0.158217 0.987404i \(-0.550575\pi\)
−0.158217 + 0.987404i \(0.550575\pi\)
\(432\) 16.4017 0.789128
\(433\) 36.4113 1.74982 0.874908 0.484290i \(-0.160922\pi\)
0.874908 + 0.484290i \(0.160922\pi\)
\(434\) −44.4190 −2.13218
\(435\) 0 0
\(436\) −34.5972 −1.65691
\(437\) 38.5187 1.84260
\(438\) −18.7110 −0.894044
\(439\) 25.3878 1.21169 0.605846 0.795582i \(-0.292835\pi\)
0.605846 + 0.795582i \(0.292835\pi\)
\(440\) 0 0
\(441\) −3.52311 −0.167767
\(442\) 31.8217 1.51361
\(443\) 20.9065 0.993298 0.496649 0.867952i \(-0.334564\pi\)
0.496649 + 0.867952i \(0.334564\pi\)
\(444\) 49.9527 2.37065
\(445\) 0 0
\(446\) 61.2437 2.89997
\(447\) 5.74324 0.271646
\(448\) −68.9377 −3.25700
\(449\) −34.9450 −1.64916 −0.824579 0.565747i \(-0.808588\pi\)
−0.824579 + 0.565747i \(0.808588\pi\)
\(450\) 0 0
\(451\) −0.761557 −0.0358603
\(452\) 33.7572 1.58780
\(453\) 10.5616 0.496229
\(454\) −16.3911 −0.769272
\(455\) 0 0
\(456\) −53.9527 −2.52657
\(457\) −7.31695 −0.342272 −0.171136 0.985247i \(-0.554744\pi\)
−0.171136 + 0.985247i \(0.554744\pi\)
\(458\) −22.9677 −1.07321
\(459\) −2.49084 −0.116262
\(460\) 0 0
\(461\) −14.0279 −0.653345 −0.326672 0.945138i \(-0.605927\pi\)
−0.326672 + 0.945138i \(0.605927\pi\)
\(462\) −5.14931 −0.239568
\(463\) 4.33527 0.201477 0.100739 0.994913i \(-0.467879\pi\)
0.100739 + 0.994913i \(0.467879\pi\)
\(464\) −57.7851 −2.68261
\(465\) 0 0
\(466\) −78.7561 −3.64831
\(467\) 3.70138 0.171279 0.0856396 0.996326i \(-0.472707\pi\)
0.0856396 + 0.996326i \(0.472707\pi\)
\(468\) −26.0279 −1.20314
\(469\) 22.2697 1.02832
\(470\) 0 0
\(471\) −3.10308 −0.142983
\(472\) 51.5650 2.37347
\(473\) 7.40171 0.340331
\(474\) −16.6079 −0.762825
\(475\) 0 0
\(476\) 26.1310 1.19771
\(477\) 5.45856 0.249930
\(478\) 66.5327 3.04313
\(479\) −22.8680 −1.04486 −0.522432 0.852681i \(-0.674975\pi\)
−0.522432 + 0.852681i \(0.674975\pi\)
\(480\) 0 0
\(481\) −41.0741 −1.87282
\(482\) −14.2707 −0.650013
\(483\) 13.3309 0.606577
\(484\) 5.62620 0.255736
\(485\) 0 0
\(486\) 2.76156 0.125267
\(487\) −5.42962 −0.246039 −0.123020 0.992404i \(-0.539258\pi\)
−0.123020 + 0.992404i \(0.539258\pi\)
\(488\) 44.2620 2.00365
\(489\) −15.6724 −0.708732
\(490\) 0 0
\(491\) 21.2803 0.960367 0.480183 0.877168i \(-0.340570\pi\)
0.480183 + 0.877168i \(0.340570\pi\)
\(492\) −4.28467 −0.193168
\(493\) 8.77551 0.395229
\(494\) 68.8313 3.09687
\(495\) 0 0
\(496\) 141.484 6.35284
\(497\) −21.6787 −0.972422
\(498\) −40.2341 −1.80293
\(499\) −16.0077 −0.716603 −0.358301 0.933606i \(-0.616644\pi\)
−0.358301 + 0.933606i \(0.616644\pi\)
\(500\) 0 0
\(501\) 8.98168 0.401272
\(502\) −25.5510 −1.14040
\(503\) 12.0925 0.539176 0.269588 0.962976i \(-0.413112\pi\)
0.269588 + 0.962976i \(0.413112\pi\)
\(504\) −18.6724 −0.831736
\(505\) 0 0
\(506\) −19.7432 −0.877694
\(507\) 8.40171 0.373133
\(508\) 52.1343 2.31309
\(509\) 14.7110 0.652052 0.326026 0.945361i \(-0.394290\pi\)
0.326026 + 0.945361i \(0.394290\pi\)
\(510\) 0 0
\(511\) 12.6339 0.558891
\(512\) 85.9194 3.79714
\(513\) −5.38776 −0.237875
\(514\) 69.6347 3.07146
\(515\) 0 0
\(516\) 41.6435 1.83325
\(517\) 0.373802 0.0164398
\(518\) −45.7187 −2.00876
\(519\) 11.5092 0.505196
\(520\) 0 0
\(521\) −17.2803 −0.757064 −0.378532 0.925588i \(-0.623571\pi\)
−0.378532 + 0.925588i \(0.623571\pi\)
\(522\) −9.72928 −0.425839
\(523\) −42.7326 −1.86857 −0.934283 0.356532i \(-0.883959\pi\)
−0.934283 + 0.356532i \(0.883959\pi\)
\(524\) 22.8680 0.998992
\(525\) 0 0
\(526\) −15.0741 −0.657264
\(527\) −21.4865 −0.935965
\(528\) 16.4017 0.713793
\(529\) 28.1127 1.22229
\(530\) 0 0
\(531\) 5.14931 0.223461
\(532\) 56.5221 2.45054
\(533\) 3.52311 0.152603
\(534\) 24.9817 1.08106
\(535\) 0 0
\(536\) −119.598 −5.16585
\(537\) 10.3738 0.447663
\(538\) 47.1512 2.03283
\(539\) −3.52311 −0.151751
\(540\) 0 0
\(541\) 8.69075 0.373644 0.186822 0.982394i \(-0.440181\pi\)
0.186822 + 0.982394i \(0.440181\pi\)
\(542\) 27.0077 1.16008
\(543\) −2.66473 −0.114355
\(544\) −62.9344 −2.69829
\(545\) 0 0
\(546\) 23.8217 1.01948
\(547\) 44.1064 1.88585 0.942927 0.333000i \(-0.108061\pi\)
0.942927 + 0.333000i \(0.108061\pi\)
\(548\) −33.3940 −1.42652
\(549\) 4.42003 0.188642
\(550\) 0 0
\(551\) 18.9817 0.808647
\(552\) −71.5929 −3.04720
\(553\) 11.2139 0.476862
\(554\) −16.3353 −0.694019
\(555\) 0 0
\(556\) 37.7572 1.60126
\(557\) 7.69264 0.325948 0.162974 0.986630i \(-0.447891\pi\)
0.162974 + 0.986630i \(0.447891\pi\)
\(558\) 23.8217 1.00845
\(559\) −34.2418 −1.44827
\(560\) 0 0
\(561\) −2.49084 −0.105163
\(562\) −18.6724 −0.787649
\(563\) −9.25240 −0.389942 −0.194971 0.980809i \(-0.562461\pi\)
−0.194971 + 0.980809i \(0.562461\pi\)
\(564\) 2.10308 0.0885558
\(565\) 0 0
\(566\) −24.3525 −1.02361
\(567\) −1.86464 −0.0783076
\(568\) 116.424 4.88505
\(569\) 4.96772 0.208258 0.104129 0.994564i \(-0.466795\pi\)
0.104129 + 0.994564i \(0.466795\pi\)
\(570\) 0 0
\(571\) 6.02021 0.251938 0.125969 0.992034i \(-0.459796\pi\)
0.125969 + 0.992034i \(0.459796\pi\)
\(572\) −26.0279 −1.08828
\(573\) 14.6724 0.612949
\(574\) 3.92150 0.163680
\(575\) 0 0
\(576\) 36.9711 1.54046
\(577\) −8.25240 −0.343552 −0.171776 0.985136i \(-0.554950\pi\)
−0.171776 + 0.985136i \(0.554950\pi\)
\(578\) −29.8130 −1.24006
\(579\) −5.10308 −0.212077
\(580\) 0 0
\(581\) 27.1666 1.12706
\(582\) −45.0602 −1.86780
\(583\) 5.45856 0.226071
\(584\) −67.8496 −2.80764
\(585\) 0 0
\(586\) 84.2022 3.47836
\(587\) 11.4846 0.474019 0.237010 0.971507i \(-0.423833\pi\)
0.237010 + 0.971507i \(0.423833\pi\)
\(588\) −19.8217 −0.817435
\(589\) −46.4758 −1.91500
\(590\) 0 0
\(591\) −3.74324 −0.153976
\(592\) 145.624 5.98511
\(593\) 39.0375 1.60308 0.801539 0.597943i \(-0.204015\pi\)
0.801539 + 0.597943i \(0.204015\pi\)
\(594\) 2.76156 0.113308
\(595\) 0 0
\(596\) 32.3126 1.32357
\(597\) −8.08476 −0.330887
\(598\) 91.3361 3.73501
\(599\) −30.1589 −1.23226 −0.616130 0.787645i \(-0.711300\pi\)
−0.616130 + 0.787645i \(0.711300\pi\)
\(600\) 0 0
\(601\) 19.1310 0.780369 0.390185 0.920737i \(-0.372411\pi\)
0.390185 + 0.920737i \(0.372411\pi\)
\(602\) −38.1137 −1.55340
\(603\) −11.9431 −0.486362
\(604\) 59.4219 2.41784
\(605\) 0 0
\(606\) −2.85069 −0.115801
\(607\) −4.20617 −0.170723 −0.0853615 0.996350i \(-0.527205\pi\)
−0.0853615 + 0.996350i \(0.527205\pi\)
\(608\) −136.129 −5.52076
\(609\) 6.56934 0.266203
\(610\) 0 0
\(611\) −1.72928 −0.0699593
\(612\) −14.0140 −0.566480
\(613\) −5.45856 −0.220469 −0.110235 0.993906i \(-0.535160\pi\)
−0.110235 + 0.993906i \(0.535160\pi\)
\(614\) −2.12036 −0.0855709
\(615\) 0 0
\(616\) −18.6724 −0.752334
\(617\) −30.1974 −1.21570 −0.607851 0.794051i \(-0.707968\pi\)
−0.607851 + 0.794051i \(0.707968\pi\)
\(618\) −8.41233 −0.338394
\(619\) 23.1955 0.932308 0.466154 0.884704i \(-0.345639\pi\)
0.466154 + 0.884704i \(0.345639\pi\)
\(620\) 0 0
\(621\) −7.14931 −0.286892
\(622\) −93.4777 −3.74812
\(623\) −16.8680 −0.675801
\(624\) −75.8776 −3.03753
\(625\) 0 0
\(626\) 22.3984 0.895219
\(627\) −5.38776 −0.215166
\(628\) −17.4586 −0.696673
\(629\) −22.1151 −0.881789
\(630\) 0 0
\(631\) −3.19554 −0.127212 −0.0636062 0.997975i \(-0.520260\pi\)
−0.0636062 + 0.997975i \(0.520260\pi\)
\(632\) −60.2234 −2.39556
\(633\) −15.9431 −0.633683
\(634\) −22.2707 −0.884483
\(635\) 0 0
\(636\) 30.7110 1.21777
\(637\) 16.2986 0.645775
\(638\) −9.72928 −0.385186
\(639\) 11.6262 0.459925
\(640\) 0 0
\(641\) 28.1974 1.11373 0.556866 0.830603i \(-0.312004\pi\)
0.556866 + 0.830603i \(0.312004\pi\)
\(642\) 23.4865 0.926937
\(643\) −17.9634 −0.708406 −0.354203 0.935169i \(-0.615248\pi\)
−0.354203 + 0.935169i \(0.615248\pi\)
\(644\) 75.0023 2.95550
\(645\) 0 0
\(646\) 37.0602 1.45811
\(647\) 40.9990 1.61184 0.805918 0.592028i \(-0.201672\pi\)
0.805918 + 0.592028i \(0.201672\pi\)
\(648\) 10.0140 0.393385
\(649\) 5.14931 0.202128
\(650\) 0 0
\(651\) −16.0848 −0.630412
\(652\) −88.1762 −3.45325
\(653\) −11.6926 −0.457568 −0.228784 0.973477i \(-0.573475\pi\)
−0.228784 + 0.973477i \(0.573475\pi\)
\(654\) −16.9817 −0.664036
\(655\) 0 0
\(656\) −12.4908 −0.487685
\(657\) −6.77551 −0.264338
\(658\) −1.92482 −0.0750374
\(659\) −1.72928 −0.0673633 −0.0336816 0.999433i \(-0.510723\pi\)
−0.0336816 + 0.999433i \(0.510723\pi\)
\(660\) 0 0
\(661\) 0.232185 0.00903096 0.00451548 0.999990i \(-0.498563\pi\)
0.00451548 + 0.999990i \(0.498563\pi\)
\(662\) 62.0712 2.41247
\(663\) 11.5231 0.447521
\(664\) −145.897 −5.66189
\(665\) 0 0
\(666\) 24.5187 0.950082
\(667\) 25.1878 0.975277
\(668\) 50.5327 1.95517
\(669\) 22.1772 0.857421
\(670\) 0 0
\(671\) 4.42003 0.170633
\(672\) −47.1127 −1.81741
\(673\) −33.5144 −1.29188 −0.645942 0.763386i \(-0.723535\pi\)
−0.645942 + 0.763386i \(0.723535\pi\)
\(674\) −17.4740 −0.673072
\(675\) 0 0
\(676\) 47.2697 1.81806
\(677\) 13.0183 0.500335 0.250167 0.968203i \(-0.419514\pi\)
0.250167 + 0.968203i \(0.419514\pi\)
\(678\) 16.5693 0.636342
\(679\) 30.4252 1.16761
\(680\) 0 0
\(681\) −5.93545 −0.227447
\(682\) 23.8217 0.912182
\(683\) 4.54333 0.173846 0.0869228 0.996215i \(-0.472297\pi\)
0.0869228 + 0.996215i \(0.472297\pi\)
\(684\) −30.3126 −1.15903
\(685\) 0 0
\(686\) 54.1868 2.06886
\(687\) −8.31695 −0.317311
\(688\) 121.401 4.62836
\(689\) −25.2524 −0.962040
\(690\) 0 0
\(691\) −19.8988 −0.756986 −0.378493 0.925604i \(-0.623558\pi\)
−0.378493 + 0.925604i \(0.623558\pi\)
\(692\) 64.7528 2.46153
\(693\) −1.86464 −0.0708319
\(694\) 0.492277 0.0186866
\(695\) 0 0
\(696\) −35.2803 −1.33730
\(697\) 1.89692 0.0718508
\(698\) −54.4835 −2.06223
\(699\) −28.5187 −1.07868
\(700\) 0 0
\(701\) −15.8444 −0.598436 −0.299218 0.954185i \(-0.596726\pi\)
−0.299218 + 0.954185i \(0.596726\pi\)
\(702\) −12.7755 −0.482181
\(703\) −47.8357 −1.80416
\(704\) 36.9711 1.39340
\(705\) 0 0
\(706\) 53.9142 2.02909
\(707\) 1.92482 0.0723904
\(708\) 28.9711 1.08880
\(709\) 8.82174 0.331307 0.165654 0.986184i \(-0.447027\pi\)
0.165654 + 0.986184i \(0.447027\pi\)
\(710\) 0 0
\(711\) −6.01395 −0.225541
\(712\) 90.5885 3.39495
\(713\) −61.6714 −2.30961
\(714\) 12.8261 0.480005
\(715\) 0 0
\(716\) 58.3651 2.18120
\(717\) 24.0925 0.899749
\(718\) 98.5673 3.67850
\(719\) 36.7668 1.37117 0.685585 0.727993i \(-0.259547\pi\)
0.685585 + 0.727993i \(0.259547\pi\)
\(720\) 0 0
\(721\) 5.68012 0.211539
\(722\) 27.6926 1.03061
\(723\) −5.16763 −0.192186
\(724\) −14.9923 −0.557185
\(725\) 0 0
\(726\) 2.76156 0.102491
\(727\) 5.27261 0.195550 0.0977751 0.995209i \(-0.468827\pi\)
0.0977751 + 0.995209i \(0.468827\pi\)
\(728\) 86.3823 3.20154
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −18.4365 −0.681897
\(732\) 24.8680 0.919147
\(733\) −37.9508 −1.40175 −0.700873 0.713286i \(-0.747206\pi\)
−0.700873 + 0.713286i \(0.747206\pi\)
\(734\) −55.5423 −2.05010
\(735\) 0 0
\(736\) −180.637 −6.65837
\(737\) −11.9431 −0.439931
\(738\) −2.10308 −0.0774156
\(739\) 37.2943 1.37189 0.685946 0.727653i \(-0.259389\pi\)
0.685946 + 0.727653i \(0.259389\pi\)
\(740\) 0 0
\(741\) 24.9248 0.915636
\(742\) −28.1078 −1.03187
\(743\) −22.5693 −0.827989 −0.413994 0.910279i \(-0.635867\pi\)
−0.413994 + 0.910279i \(0.635867\pi\)
\(744\) 86.3823 3.16693
\(745\) 0 0
\(746\) 7.07414 0.259002
\(747\) −14.5693 −0.533064
\(748\) −14.0140 −0.512401
\(749\) −15.8584 −0.579453
\(750\) 0 0
\(751\) −9.55102 −0.348522 −0.174261 0.984700i \(-0.555754\pi\)
−0.174261 + 0.984700i \(0.555754\pi\)
\(752\) 6.13099 0.223574
\(753\) −9.25240 −0.337176
\(754\) 45.0096 1.63915
\(755\) 0 0
\(756\) −10.4908 −0.381548
\(757\) −30.3188 −1.10196 −0.550978 0.834519i \(-0.685745\pi\)
−0.550978 + 0.834519i \(0.685745\pi\)
\(758\) 64.2341 2.33309
\(759\) −7.14931 −0.259504
\(760\) 0 0
\(761\) −1.43066 −0.0518613 −0.0259306 0.999664i \(-0.508255\pi\)
−0.0259306 + 0.999664i \(0.508255\pi\)
\(762\) 25.5896 0.927012
\(763\) 11.4663 0.415106
\(764\) 82.5500 2.98655
\(765\) 0 0
\(766\) −71.2311 −2.57369
\(767\) −23.8217 −0.860153
\(768\) 68.4575 2.47025
\(769\) −35.4017 −1.27662 −0.638309 0.769780i \(-0.720366\pi\)
−0.638309 + 0.769780i \(0.720366\pi\)
\(770\) 0 0
\(771\) 25.2158 0.908123
\(772\) −28.7110 −1.03333
\(773\) 13.4094 0.482303 0.241151 0.970488i \(-0.422475\pi\)
0.241151 + 0.970488i \(0.422475\pi\)
\(774\) 20.4402 0.734709
\(775\) 0 0
\(776\) −163.397 −5.86562
\(777\) −16.5554 −0.593921
\(778\) 6.44898 0.231207
\(779\) 4.10308 0.147008
\(780\) 0 0
\(781\) 11.6262 0.416018
\(782\) 49.1772 1.75857
\(783\) −3.52311 −0.125906
\(784\) −57.7851 −2.06375
\(785\) 0 0
\(786\) 11.2245 0.400364
\(787\) 37.3232 1.33043 0.665214 0.746653i \(-0.268340\pi\)
0.665214 + 0.746653i \(0.268340\pi\)
\(788\) −21.0602 −0.750238
\(789\) −5.45856 −0.194330
\(790\) 0 0
\(791\) −11.1878 −0.397794
\(792\) 10.0140 0.355830
\(793\) −20.4479 −0.726128
\(794\) −47.4865 −1.68523
\(795\) 0 0
\(796\) −45.4865 −1.61223
\(797\) 8.05581 0.285352 0.142676 0.989769i \(-0.454429\pi\)
0.142676 + 0.989769i \(0.454429\pi\)
\(798\) 27.7432 0.982100
\(799\) −0.931080 −0.0329393
\(800\) 0 0
\(801\) 9.04623 0.319633
\(802\) −73.3728 −2.59088
\(803\) −6.77551 −0.239103
\(804\) −67.1945 −2.36977
\(805\) 0 0
\(806\) −110.204 −3.88177
\(807\) 17.0741 0.601038
\(808\) −10.3372 −0.363660
\(809\) −42.3771 −1.48990 −0.744950 0.667120i \(-0.767527\pi\)
−0.744950 + 0.667120i \(0.767527\pi\)
\(810\) 0 0
\(811\) −21.3878 −0.751026 −0.375513 0.926817i \(-0.622533\pi\)
−0.375513 + 0.926817i \(0.622533\pi\)
\(812\) 36.9604 1.29706
\(813\) 9.77988 0.342995
\(814\) 24.5187 0.859382
\(815\) 0 0
\(816\) −40.8540 −1.43018
\(817\) −39.8786 −1.39518
\(818\) −56.3911 −1.97167
\(819\) 8.62620 0.301424
\(820\) 0 0
\(821\) −47.8776 −1.67094 −0.835469 0.549537i \(-0.814804\pi\)
−0.835469 + 0.549537i \(0.814804\pi\)
\(822\) −16.3911 −0.571705
\(823\) −16.3555 −0.570116 −0.285058 0.958510i \(-0.592013\pi\)
−0.285058 + 0.958510i \(0.592013\pi\)
\(824\) −30.5048 −1.06268
\(825\) 0 0
\(826\) −26.5154 −0.922589
\(827\) 44.7110 1.55475 0.777376 0.629036i \(-0.216550\pi\)
0.777376 + 0.629036i \(0.216550\pi\)
\(828\) −40.2234 −1.39786
\(829\) −38.5972 −1.34054 −0.670269 0.742118i \(-0.733821\pi\)
−0.670269 + 0.742118i \(0.733821\pi\)
\(830\) 0 0
\(831\) −5.91524 −0.205197
\(832\) −171.035 −5.92959
\(833\) 8.77551 0.304053
\(834\) 18.5327 0.641735
\(835\) 0 0
\(836\) −30.3126 −1.04838
\(837\) 8.62620 0.298165
\(838\) −37.5283 −1.29639
\(839\) −0.710960 −0.0245451 −0.0122725 0.999925i \(-0.503907\pi\)
−0.0122725 + 0.999925i \(0.503907\pi\)
\(840\) 0 0
\(841\) −16.5877 −0.571988
\(842\) −109.406 −3.77038
\(843\) −6.76156 −0.232880
\(844\) −89.6993 −3.08758
\(845\) 0 0
\(846\) 1.03228 0.0354904
\(847\) −1.86464 −0.0640698
\(848\) 89.5298 3.07446
\(849\) −8.81841 −0.302647
\(850\) 0 0
\(851\) −63.4758 −2.17592
\(852\) 65.4113 2.24095
\(853\) −52.7187 −1.80505 −0.902526 0.430635i \(-0.858290\pi\)
−0.902526 + 0.430635i \(0.858290\pi\)
\(854\) −22.7601 −0.778835
\(855\) 0 0
\(856\) 85.1666 2.91093
\(857\) −16.7124 −0.570885 −0.285442 0.958396i \(-0.592141\pi\)
−0.285442 + 0.958396i \(0.592141\pi\)
\(858\) −12.7755 −0.436149
\(859\) −41.9142 −1.43009 −0.715047 0.699076i \(-0.753595\pi\)
−0.715047 + 0.699076i \(0.753595\pi\)
\(860\) 0 0
\(861\) 1.42003 0.0483945
\(862\) −18.1416 −0.617906
\(863\) 41.4219 1.41002 0.705009 0.709198i \(-0.250943\pi\)
0.705009 + 0.709198i \(0.250943\pi\)
\(864\) 25.2663 0.859579
\(865\) 0 0
\(866\) 100.552 3.41689
\(867\) −10.7957 −0.366642
\(868\) −90.4961 −3.07164
\(869\) −6.01395 −0.204009
\(870\) 0 0
\(871\) 55.2514 1.87212
\(872\) −61.5789 −2.08533
\(873\) −16.3169 −0.552245
\(874\) 106.372 3.59808
\(875\) 0 0
\(876\) −38.1204 −1.28797
\(877\) 12.0848 0.408073 0.204037 0.978963i \(-0.434594\pi\)
0.204037 + 0.978963i \(0.434594\pi\)
\(878\) 70.1097 2.36609
\(879\) 30.4908 1.02843
\(880\) 0 0
\(881\) −31.8130 −1.07181 −0.535904 0.844279i \(-0.680029\pi\)
−0.535904 + 0.844279i \(0.680029\pi\)
\(882\) −9.72928 −0.327602
\(883\) 20.9894 0.706349 0.353174 0.935558i \(-0.385102\pi\)
0.353174 + 0.935558i \(0.385102\pi\)
\(884\) 64.8313 2.18051
\(885\) 0 0
\(886\) 57.7345 1.93963
\(887\) −49.2032 −1.65208 −0.826042 0.563609i \(-0.809412\pi\)
−0.826042 + 0.563609i \(0.809412\pi\)
\(888\) 88.9098 2.98362
\(889\) −17.2784 −0.579499
\(890\) 0 0
\(891\) 1.00000 0.0335013
\(892\) 124.773 4.17772
\(893\) −2.01395 −0.0673944
\(894\) 15.8603 0.530447
\(895\) 0 0
\(896\) −96.1502 −3.21215
\(897\) 33.0741 1.10431
\(898\) −96.5027 −3.22034
\(899\) −30.3911 −1.01360
\(900\) 0 0
\(901\) −13.5964 −0.452962
\(902\) −2.10308 −0.0700250
\(903\) −13.8015 −0.459286
\(904\) 60.0837 1.99835
\(905\) 0 0
\(906\) 29.1666 0.968995
\(907\) 24.8526 0.825216 0.412608 0.910909i \(-0.364618\pi\)
0.412608 + 0.910909i \(0.364618\pi\)
\(908\) −33.3940 −1.10822
\(909\) −1.03228 −0.0342384
\(910\) 0 0
\(911\) 45.4758 1.50668 0.753341 0.657630i \(-0.228441\pi\)
0.753341 + 0.657630i \(0.228441\pi\)
\(912\) −88.3684 −2.92617
\(913\) −14.5693 −0.482175
\(914\) −20.2062 −0.668361
\(915\) 0 0
\(916\) −46.7928 −1.54608
\(917\) −7.57893 −0.250278
\(918\) −6.87859 −0.227027
\(919\) −15.7355 −0.519068 −0.259534 0.965734i \(-0.583569\pi\)
−0.259534 + 0.965734i \(0.583569\pi\)
\(920\) 0 0
\(921\) −0.767815 −0.0253004
\(922\) −38.7389 −1.27580
\(923\) −53.7851 −1.77036
\(924\) −10.4908 −0.345123
\(925\) 0 0
\(926\) 11.9721 0.393427
\(927\) −3.04623 −0.100051
\(928\) −89.0162 −2.92210
\(929\) −2.06455 −0.0677357 −0.0338679 0.999426i \(-0.510783\pi\)
−0.0338679 + 0.999426i \(0.510783\pi\)
\(930\) 0 0
\(931\) 18.9817 0.622099
\(932\) −160.452 −5.25578
\(933\) −33.8496 −1.10819
\(934\) 10.2216 0.334460
\(935\) 0 0
\(936\) −46.3265 −1.51423
\(937\) −40.1493 −1.31162 −0.655810 0.754926i \(-0.727673\pi\)
−0.655810 + 0.754926i \(0.727673\pi\)
\(938\) 61.4990 2.00801
\(939\) 8.11078 0.264685
\(940\) 0 0
\(941\) 26.4050 0.860780 0.430390 0.902643i \(-0.358376\pi\)
0.430390 + 0.902643i \(0.358376\pi\)
\(942\) −8.56934 −0.279204
\(943\) 5.44461 0.177301
\(944\) 84.4575 2.74886
\(945\) 0 0
\(946\) 20.4402 0.664570
\(947\) −8.67243 −0.281816 −0.140908 0.990023i \(-0.545002\pi\)
−0.140908 + 0.990023i \(0.545002\pi\)
\(948\) −33.8357 −1.09893
\(949\) 31.3449 1.01750
\(950\) 0 0
\(951\) −8.06455 −0.261511
\(952\) 46.5100 1.50740
\(953\) 26.8540 0.869887 0.434943 0.900458i \(-0.356768\pi\)
0.434943 + 0.900458i \(0.356768\pi\)
\(954\) 15.0741 0.488043
\(955\) 0 0
\(956\) 135.549 4.38397
\(957\) −3.52311 −0.113886
\(958\) −63.1512 −2.04032
\(959\) 11.0675 0.357388
\(960\) 0 0
\(961\) 43.4113 1.40036
\(962\) −113.429 −3.65708
\(963\) 8.50479 0.274063
\(964\) −29.0741 −0.936415
\(965\) 0 0
\(966\) 36.8140 1.18447
\(967\) 55.8496 1.79600 0.898002 0.439992i \(-0.145019\pi\)
0.898002 + 0.439992i \(0.145019\pi\)
\(968\) 10.0140 0.321861
\(969\) 13.4200 0.431113
\(970\) 0 0
\(971\) 30.0664 0.964878 0.482439 0.875930i \(-0.339751\pi\)
0.482439 + 0.875930i \(0.339751\pi\)
\(972\) 5.62620 0.180460
\(973\) −12.5135 −0.401165
\(974\) −14.9942 −0.480445
\(975\) 0 0
\(976\) 72.4961 2.32054
\(977\) −15.7014 −0.502331 −0.251166 0.967944i \(-0.580814\pi\)
−0.251166 + 0.967944i \(0.580814\pi\)
\(978\) −43.2803 −1.38395
\(979\) 9.04623 0.289119
\(980\) 0 0
\(981\) −6.14931 −0.196332
\(982\) 58.7668 1.87532
\(983\) 51.9894 1.65820 0.829102 0.559098i \(-0.188852\pi\)
0.829102 + 0.559098i \(0.188852\pi\)
\(984\) −7.62620 −0.243114
\(985\) 0 0
\(986\) 24.2341 0.771770
\(987\) −0.697006 −0.0221860
\(988\) 140.232 4.46137
\(989\) −52.9171 −1.68267
\(990\) 0 0
\(991\) 27.6445 0.878157 0.439079 0.898449i \(-0.355305\pi\)
0.439079 + 0.898449i \(0.355305\pi\)
\(992\) 217.953 6.92000
\(993\) 22.4769 0.713282
\(994\) −59.8669 −1.89886
\(995\) 0 0
\(996\) −81.9700 −2.59732
\(997\) −40.5048 −1.28280 −0.641400 0.767207i \(-0.721646\pi\)
−0.641400 + 0.767207i \(0.721646\pi\)
\(998\) −44.2062 −1.39932
\(999\) 8.87859 0.280906
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.m.1.3 yes 3
3.2 odd 2 2475.2.a.z.1.1 3
5.2 odd 4 825.2.c.f.199.6 6
5.3 odd 4 825.2.c.f.199.1 6
5.4 even 2 825.2.a.i.1.1 3
11.10 odd 2 9075.2.a.cd.1.1 3
15.2 even 4 2475.2.c.q.199.1 6
15.8 even 4 2475.2.c.q.199.6 6
15.14 odd 2 2475.2.a.bd.1.3 3
55.54 odd 2 9075.2.a.cj.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
825.2.a.i.1.1 3 5.4 even 2
825.2.a.m.1.3 yes 3 1.1 even 1 trivial
825.2.c.f.199.1 6 5.3 odd 4
825.2.c.f.199.6 6 5.2 odd 4
2475.2.a.z.1.1 3 3.2 odd 2
2475.2.a.bd.1.3 3 15.14 odd 2
2475.2.c.q.199.1 6 15.2 even 4
2475.2.c.q.199.6 6 15.8 even 4
9075.2.a.cd.1.1 3 11.10 odd 2
9075.2.a.cj.1.3 3 55.54 odd 2