Properties

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

Related objects

Downloads

Learn more

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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
Defining polynomial: \(x^{2} - 3\)
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 \(1.73205\) of defining polynomial
Character \(\chi\) \(=\) 825.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.73205 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.73205 q^{6} -2.00000 q^{7} -1.73205 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.73205 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.73205 q^{6} -2.00000 q^{7} -1.73205 q^{8} +1.00000 q^{9} -1.00000 q^{11} -1.00000 q^{12} -5.46410 q^{13} -3.46410 q^{14} -5.00000 q^{16} +1.73205 q^{18} +5.46410 q^{19} +2.00000 q^{21} -1.73205 q^{22} -6.92820 q^{23} +1.73205 q^{24} -9.46410 q^{26} -1.00000 q^{27} -2.00000 q^{28} -3.46410 q^{29} -10.9282 q^{31} -5.19615 q^{32} +1.00000 q^{33} +1.00000 q^{36} +4.92820 q^{37} +9.46410 q^{38} +5.46410 q^{39} +3.46410 q^{41} +3.46410 q^{42} +4.92820 q^{43} -1.00000 q^{44} -12.0000 q^{46} +6.92820 q^{47} +5.00000 q^{48} -3.00000 q^{49} -5.46410 q^{52} -0.928203 q^{53} -1.73205 q^{54} +3.46410 q^{56} -5.46410 q^{57} -6.00000 q^{58} -6.92820 q^{59} +2.00000 q^{61} -18.9282 q^{62} -2.00000 q^{63} +1.00000 q^{64} +1.73205 q^{66} -8.00000 q^{67} +6.92820 q^{69} +13.8564 q^{71} -1.73205 q^{72} +8.39230 q^{73} +8.53590 q^{74} +5.46410 q^{76} +2.00000 q^{77} +9.46410 q^{78} -6.53590 q^{79} +1.00000 q^{81} +6.00000 q^{82} -8.53590 q^{83} +2.00000 q^{84} +8.53590 q^{86} +3.46410 q^{87} +1.73205 q^{88} +0.928203 q^{89} +10.9282 q^{91} -6.92820 q^{92} +10.9282 q^{93} +12.0000 q^{94} +5.19615 q^{96} +10.0000 q^{97} -5.19615 q^{98} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{3} + 2q^{4} - 4q^{7} + 2q^{9} + O(q^{10}) \) \( 2q - 2q^{3} + 2q^{4} - 4q^{7} + 2q^{9} - 2q^{11} - 2q^{12} - 4q^{13} - 10q^{16} + 4q^{19} + 4q^{21} - 12q^{26} - 2q^{27} - 4q^{28} - 8q^{31} + 2q^{33} + 2q^{36} - 4q^{37} + 12q^{38} + 4q^{39} - 4q^{43} - 2q^{44} - 24q^{46} + 10q^{48} - 6q^{49} - 4q^{52} + 12q^{53} - 4q^{57} - 12q^{58} + 4q^{61} - 24q^{62} - 4q^{63} + 2q^{64} - 16q^{67} - 4q^{73} + 24q^{74} + 4q^{76} + 4q^{77} + 12q^{78} - 20q^{79} + 2q^{81} + 12q^{82} - 24q^{83} + 4q^{84} + 24q^{86} - 12q^{89} + 8q^{91} + 8q^{93} + 24q^{94} + 20q^{97} - 2q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.73205 1.22474 0.612372 0.790569i \(-0.290215\pi\)
0.612372 + 0.790569i \(0.290215\pi\)
\(3\) −1.00000 −0.577350
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) −1.73205 −0.707107
\(7\) −2.00000 −0.755929 −0.377964 0.925820i \(-0.623376\pi\)
−0.377964 + 0.925820i \(0.623376\pi\)
\(8\) −1.73205 −0.612372
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) −1.00000 −0.288675
\(13\) −5.46410 −1.51547 −0.757735 0.652563i \(-0.773694\pi\)
−0.757735 + 0.652563i \(0.773694\pi\)
\(14\) −3.46410 −0.925820
\(15\) 0 0
\(16\) −5.00000 −1.25000
\(17\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 1.73205 0.408248
\(19\) 5.46410 1.25355 0.626775 0.779200i \(-0.284374\pi\)
0.626775 + 0.779200i \(0.284374\pi\)
\(20\) 0 0
\(21\) 2.00000 0.436436
\(22\) −1.73205 −0.369274
\(23\) −6.92820 −1.44463 −0.722315 0.691564i \(-0.756922\pi\)
−0.722315 + 0.691564i \(0.756922\pi\)
\(24\) 1.73205 0.353553
\(25\) 0 0
\(26\) −9.46410 −1.85606
\(27\) −1.00000 −0.192450
\(28\) −2.00000 −0.377964
\(29\) −3.46410 −0.643268 −0.321634 0.946864i \(-0.604232\pi\)
−0.321634 + 0.946864i \(0.604232\pi\)
\(30\) 0 0
\(31\) −10.9282 −1.96276 −0.981382 0.192068i \(-0.938481\pi\)
−0.981382 + 0.192068i \(0.938481\pi\)
\(32\) −5.19615 −0.918559
\(33\) 1.00000 0.174078
\(34\) 0 0
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 4.92820 0.810192 0.405096 0.914274i \(-0.367238\pi\)
0.405096 + 0.914274i \(0.367238\pi\)
\(38\) 9.46410 1.53528
\(39\) 5.46410 0.874957
\(40\) 0 0
\(41\) 3.46410 0.541002 0.270501 0.962720i \(-0.412811\pi\)
0.270501 + 0.962720i \(0.412811\pi\)
\(42\) 3.46410 0.534522
\(43\) 4.92820 0.751544 0.375772 0.926712i \(-0.377378\pi\)
0.375772 + 0.926712i \(0.377378\pi\)
\(44\) −1.00000 −0.150756
\(45\) 0 0
\(46\) −12.0000 −1.76930
\(47\) 6.92820 1.01058 0.505291 0.862949i \(-0.331385\pi\)
0.505291 + 0.862949i \(0.331385\pi\)
\(48\) 5.00000 0.721688
\(49\) −3.00000 −0.428571
\(50\) 0 0
\(51\) 0 0
\(52\) −5.46410 −0.757735
\(53\) −0.928203 −0.127499 −0.0637493 0.997966i \(-0.520306\pi\)
−0.0637493 + 0.997966i \(0.520306\pi\)
\(54\) −1.73205 −0.235702
\(55\) 0 0
\(56\) 3.46410 0.462910
\(57\) −5.46410 −0.723738
\(58\) −6.00000 −0.787839
\(59\) −6.92820 −0.901975 −0.450988 0.892530i \(-0.648928\pi\)
−0.450988 + 0.892530i \(0.648928\pi\)
\(60\) 0 0
\(61\) 2.00000 0.256074 0.128037 0.991769i \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) −18.9282 −2.40388
\(63\) −2.00000 −0.251976
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 1.73205 0.213201
\(67\) −8.00000 −0.977356 −0.488678 0.872464i \(-0.662521\pi\)
−0.488678 + 0.872464i \(0.662521\pi\)
\(68\) 0 0
\(69\) 6.92820 0.834058
\(70\) 0 0
\(71\) 13.8564 1.64445 0.822226 0.569160i \(-0.192732\pi\)
0.822226 + 0.569160i \(0.192732\pi\)
\(72\) −1.73205 −0.204124
\(73\) 8.39230 0.982245 0.491122 0.871091i \(-0.336587\pi\)
0.491122 + 0.871091i \(0.336587\pi\)
\(74\) 8.53590 0.992278
\(75\) 0 0
\(76\) 5.46410 0.626775
\(77\) 2.00000 0.227921
\(78\) 9.46410 1.07160
\(79\) −6.53590 −0.735346 −0.367673 0.929955i \(-0.619845\pi\)
−0.367673 + 0.929955i \(0.619845\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 6.00000 0.662589
\(83\) −8.53590 −0.936937 −0.468468 0.883480i \(-0.655194\pi\)
−0.468468 + 0.883480i \(0.655194\pi\)
\(84\) 2.00000 0.218218
\(85\) 0 0
\(86\) 8.53590 0.920450
\(87\) 3.46410 0.371391
\(88\) 1.73205 0.184637
\(89\) 0.928203 0.0983893 0.0491947 0.998789i \(-0.484335\pi\)
0.0491947 + 0.998789i \(0.484335\pi\)
\(90\) 0 0
\(91\) 10.9282 1.14559
\(92\) −6.92820 −0.722315
\(93\) 10.9282 1.13320
\(94\) 12.0000 1.23771
\(95\) 0 0
\(96\) 5.19615 0.530330
\(97\) 10.0000 1.01535 0.507673 0.861550i \(-0.330506\pi\)
0.507673 + 0.861550i \(0.330506\pi\)
\(98\) −5.19615 −0.524891
\(99\) −1.00000 −0.100504
\(100\) 0 0
\(101\) −10.3923 −1.03407 −0.517036 0.855963i \(-0.672965\pi\)
−0.517036 + 0.855963i \(0.672965\pi\)
\(102\) 0 0
\(103\) −8.00000 −0.788263 −0.394132 0.919054i \(-0.628955\pi\)
−0.394132 + 0.919054i \(0.628955\pi\)
\(104\) 9.46410 0.928032
\(105\) 0 0
\(106\) −1.60770 −0.156153
\(107\) −8.53590 −0.825196 −0.412598 0.910913i \(-0.635379\pi\)
−0.412598 + 0.910913i \(0.635379\pi\)
\(108\) −1.00000 −0.0962250
\(109\) −10.0000 −0.957826 −0.478913 0.877862i \(-0.658969\pi\)
−0.478913 + 0.877862i \(0.658969\pi\)
\(110\) 0 0
\(111\) −4.92820 −0.467764
\(112\) 10.0000 0.944911
\(113\) 12.9282 1.21618 0.608092 0.793867i \(-0.291935\pi\)
0.608092 + 0.793867i \(0.291935\pi\)
\(114\) −9.46410 −0.886394
\(115\) 0 0
\(116\) −3.46410 −0.321634
\(117\) −5.46410 −0.505156
\(118\) −12.0000 −1.10469
\(119\) 0 0
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 3.46410 0.313625
\(123\) −3.46410 −0.312348
\(124\) −10.9282 −0.981382
\(125\) 0 0
\(126\) −3.46410 −0.308607
\(127\) −8.92820 −0.792250 −0.396125 0.918197i \(-0.629645\pi\)
−0.396125 + 0.918197i \(0.629645\pi\)
\(128\) 12.1244 1.07165
\(129\) −4.92820 −0.433904
\(130\) 0 0
\(131\) 18.9282 1.65376 0.826882 0.562375i \(-0.190112\pi\)
0.826882 + 0.562375i \(0.190112\pi\)
\(132\) 1.00000 0.0870388
\(133\) −10.9282 −0.947595
\(134\) −13.8564 −1.19701
\(135\) 0 0
\(136\) 0 0
\(137\) 18.0000 1.53784 0.768922 0.639343i \(-0.220793\pi\)
0.768922 + 0.639343i \(0.220793\pi\)
\(138\) 12.0000 1.02151
\(139\) 12.3923 1.05110 0.525551 0.850762i \(-0.323859\pi\)
0.525551 + 0.850762i \(0.323859\pi\)
\(140\) 0 0
\(141\) −6.92820 −0.583460
\(142\) 24.0000 2.01404
\(143\) 5.46410 0.456931
\(144\) −5.00000 −0.416667
\(145\) 0 0
\(146\) 14.5359 1.20300
\(147\) 3.00000 0.247436
\(148\) 4.92820 0.405096
\(149\) −15.4641 −1.26687 −0.633434 0.773796i \(-0.718355\pi\)
−0.633434 + 0.773796i \(0.718355\pi\)
\(150\) 0 0
\(151\) −20.3923 −1.65950 −0.829751 0.558134i \(-0.811518\pi\)
−0.829751 + 0.558134i \(0.811518\pi\)
\(152\) −9.46410 −0.767640
\(153\) 0 0
\(154\) 3.46410 0.279145
\(155\) 0 0
\(156\) 5.46410 0.437478
\(157\) 3.07180 0.245156 0.122578 0.992459i \(-0.460884\pi\)
0.122578 + 0.992459i \(0.460884\pi\)
\(158\) −11.3205 −0.900611
\(159\) 0.928203 0.0736113
\(160\) 0 0
\(161\) 13.8564 1.09204
\(162\) 1.73205 0.136083
\(163\) −9.85641 −0.772013 −0.386007 0.922496i \(-0.626146\pi\)
−0.386007 + 0.922496i \(0.626146\pi\)
\(164\) 3.46410 0.270501
\(165\) 0 0
\(166\) −14.7846 −1.14751
\(167\) 10.3923 0.804181 0.402090 0.915600i \(-0.368284\pi\)
0.402090 + 0.915600i \(0.368284\pi\)
\(168\) −3.46410 −0.267261
\(169\) 16.8564 1.29665
\(170\) 0 0
\(171\) 5.46410 0.417850
\(172\) 4.92820 0.375772
\(173\) 12.0000 0.912343 0.456172 0.889892i \(-0.349220\pi\)
0.456172 + 0.889892i \(0.349220\pi\)
\(174\) 6.00000 0.454859
\(175\) 0 0
\(176\) 5.00000 0.376889
\(177\) 6.92820 0.520756
\(178\) 1.60770 0.120502
\(179\) 6.92820 0.517838 0.258919 0.965899i \(-0.416634\pi\)
0.258919 + 0.965899i \(0.416634\pi\)
\(180\) 0 0
\(181\) 15.8564 1.17860 0.589299 0.807915i \(-0.299404\pi\)
0.589299 + 0.807915i \(0.299404\pi\)
\(182\) 18.9282 1.40305
\(183\) −2.00000 −0.147844
\(184\) 12.0000 0.884652
\(185\) 0 0
\(186\) 18.9282 1.38788
\(187\) 0 0
\(188\) 6.92820 0.505291
\(189\) 2.00000 0.145479
\(190\) 0 0
\(191\) −18.9282 −1.36960 −0.684798 0.728733i \(-0.740110\pi\)
−0.684798 + 0.728733i \(0.740110\pi\)
\(192\) −1.00000 −0.0721688
\(193\) −24.3923 −1.75580 −0.877898 0.478847i \(-0.841055\pi\)
−0.877898 + 0.478847i \(0.841055\pi\)
\(194\) 17.3205 1.24354
\(195\) 0 0
\(196\) −3.00000 −0.214286
\(197\) 12.0000 0.854965 0.427482 0.904024i \(-0.359401\pi\)
0.427482 + 0.904024i \(0.359401\pi\)
\(198\) −1.73205 −0.123091
\(199\) −24.7846 −1.75693 −0.878467 0.477803i \(-0.841433\pi\)
−0.878467 + 0.477803i \(0.841433\pi\)
\(200\) 0 0
\(201\) 8.00000 0.564276
\(202\) −18.0000 −1.26648
\(203\) 6.92820 0.486265
\(204\) 0 0
\(205\) 0 0
\(206\) −13.8564 −0.965422
\(207\) −6.92820 −0.481543
\(208\) 27.3205 1.89434
\(209\) −5.46410 −0.377960
\(210\) 0 0
\(211\) −8.39230 −0.577750 −0.288875 0.957367i \(-0.593281\pi\)
−0.288875 + 0.957367i \(0.593281\pi\)
\(212\) −0.928203 −0.0637493
\(213\) −13.8564 −0.949425
\(214\) −14.7846 −1.01066
\(215\) 0 0
\(216\) 1.73205 0.117851
\(217\) 21.8564 1.48371
\(218\) −17.3205 −1.17309
\(219\) −8.39230 −0.567099
\(220\) 0 0
\(221\) 0 0
\(222\) −8.53590 −0.572892
\(223\) −9.85641 −0.660034 −0.330017 0.943975i \(-0.607054\pi\)
−0.330017 + 0.943975i \(0.607054\pi\)
\(224\) 10.3923 0.694365
\(225\) 0 0
\(226\) 22.3923 1.48951
\(227\) −15.4641 −1.02639 −0.513194 0.858272i \(-0.671538\pi\)
−0.513194 + 0.858272i \(0.671538\pi\)
\(228\) −5.46410 −0.361869
\(229\) −23.8564 −1.57648 −0.788238 0.615371i \(-0.789006\pi\)
−0.788238 + 0.615371i \(0.789006\pi\)
\(230\) 0 0
\(231\) −2.00000 −0.131590
\(232\) 6.00000 0.393919
\(233\) −12.0000 −0.786146 −0.393073 0.919507i \(-0.628588\pi\)
−0.393073 + 0.919507i \(0.628588\pi\)
\(234\) −9.46410 −0.618688
\(235\) 0 0
\(236\) −6.92820 −0.450988
\(237\) 6.53590 0.424552
\(238\) 0 0
\(239\) −12.0000 −0.776215 −0.388108 0.921614i \(-0.626871\pi\)
−0.388108 + 0.921614i \(0.626871\pi\)
\(240\) 0 0
\(241\) 0.143594 0.00924967 0.00462484 0.999989i \(-0.498528\pi\)
0.00462484 + 0.999989i \(0.498528\pi\)
\(242\) 1.73205 0.111340
\(243\) −1.00000 −0.0641500
\(244\) 2.00000 0.128037
\(245\) 0 0
\(246\) −6.00000 −0.382546
\(247\) −29.8564 −1.89972
\(248\) 18.9282 1.20194
\(249\) 8.53590 0.540941
\(250\) 0 0
\(251\) −1.85641 −0.117175 −0.0585877 0.998282i \(-0.518660\pi\)
−0.0585877 + 0.998282i \(0.518660\pi\)
\(252\) −2.00000 −0.125988
\(253\) 6.92820 0.435572
\(254\) −15.4641 −0.970304
\(255\) 0 0
\(256\) 19.0000 1.18750
\(257\) 19.8564 1.23861 0.619304 0.785151i \(-0.287415\pi\)
0.619304 + 0.785151i \(0.287415\pi\)
\(258\) −8.53590 −0.531422
\(259\) −9.85641 −0.612447
\(260\) 0 0
\(261\) −3.46410 −0.214423
\(262\) 32.7846 2.02544
\(263\) −20.5359 −1.26630 −0.633149 0.774030i \(-0.718238\pi\)
−0.633149 + 0.774030i \(0.718238\pi\)
\(264\) −1.73205 −0.106600
\(265\) 0 0
\(266\) −18.9282 −1.16056
\(267\) −0.928203 −0.0568051
\(268\) −8.00000 −0.488678
\(269\) 19.8564 1.21067 0.605333 0.795972i \(-0.293040\pi\)
0.605333 + 0.795972i \(0.293040\pi\)
\(270\) 0 0
\(271\) −11.6077 −0.705117 −0.352559 0.935790i \(-0.614688\pi\)
−0.352559 + 0.935790i \(0.614688\pi\)
\(272\) 0 0
\(273\) −10.9282 −0.661405
\(274\) 31.1769 1.88347
\(275\) 0 0
\(276\) 6.92820 0.417029
\(277\) −29.4641 −1.77033 −0.885163 0.465281i \(-0.845953\pi\)
−0.885163 + 0.465281i \(0.845953\pi\)
\(278\) 21.4641 1.28733
\(279\) −10.9282 −0.654254
\(280\) 0 0
\(281\) 3.46410 0.206651 0.103325 0.994648i \(-0.467052\pi\)
0.103325 + 0.994648i \(0.467052\pi\)
\(282\) −12.0000 −0.714590
\(283\) 4.92820 0.292951 0.146476 0.989214i \(-0.453207\pi\)
0.146476 + 0.989214i \(0.453207\pi\)
\(284\) 13.8564 0.822226
\(285\) 0 0
\(286\) 9.46410 0.559624
\(287\) −6.92820 −0.408959
\(288\) −5.19615 −0.306186
\(289\) −17.0000 −1.00000
\(290\) 0 0
\(291\) −10.0000 −0.586210
\(292\) 8.39230 0.491122
\(293\) 13.8564 0.809500 0.404750 0.914427i \(-0.367359\pi\)
0.404750 + 0.914427i \(0.367359\pi\)
\(294\) 5.19615 0.303046
\(295\) 0 0
\(296\) −8.53590 −0.496139
\(297\) 1.00000 0.0580259
\(298\) −26.7846 −1.55159
\(299\) 37.8564 2.18929
\(300\) 0 0
\(301\) −9.85641 −0.568114
\(302\) −35.3205 −2.03247
\(303\) 10.3923 0.597022
\(304\) −27.3205 −1.56694
\(305\) 0 0
\(306\) 0 0
\(307\) −14.0000 −0.799022 −0.399511 0.916728i \(-0.630820\pi\)
−0.399511 + 0.916728i \(0.630820\pi\)
\(308\) 2.00000 0.113961
\(309\) 8.00000 0.455104
\(310\) 0 0
\(311\) 5.07180 0.287595 0.143798 0.989607i \(-0.454069\pi\)
0.143798 + 0.989607i \(0.454069\pi\)
\(312\) −9.46410 −0.535799
\(313\) −20.9282 −1.18293 −0.591466 0.806330i \(-0.701451\pi\)
−0.591466 + 0.806330i \(0.701451\pi\)
\(314\) 5.32051 0.300254
\(315\) 0 0
\(316\) −6.53590 −0.367673
\(317\) 24.9282 1.40011 0.700054 0.714090i \(-0.253159\pi\)
0.700054 + 0.714090i \(0.253159\pi\)
\(318\) 1.60770 0.0901551
\(319\) 3.46410 0.193952
\(320\) 0 0
\(321\) 8.53590 0.476427
\(322\) 24.0000 1.33747
\(323\) 0 0
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) −17.0718 −0.945519
\(327\) 10.0000 0.553001
\(328\) −6.00000 −0.331295
\(329\) −13.8564 −0.763928
\(330\) 0 0
\(331\) 9.85641 0.541757 0.270879 0.962614i \(-0.412686\pi\)
0.270879 + 0.962614i \(0.412686\pi\)
\(332\) −8.53590 −0.468468
\(333\) 4.92820 0.270064
\(334\) 18.0000 0.984916
\(335\) 0 0
\(336\) −10.0000 −0.545545
\(337\) −33.1769 −1.80726 −0.903631 0.428312i \(-0.859108\pi\)
−0.903631 + 0.428312i \(0.859108\pi\)
\(338\) 29.1962 1.58806
\(339\) −12.9282 −0.702164
\(340\) 0 0
\(341\) 10.9282 0.591795
\(342\) 9.46410 0.511760
\(343\) 20.0000 1.07990
\(344\) −8.53590 −0.460225
\(345\) 0 0
\(346\) 20.7846 1.11739
\(347\) −22.3923 −1.20208 −0.601041 0.799218i \(-0.705247\pi\)
−0.601041 + 0.799218i \(0.705247\pi\)
\(348\) 3.46410 0.185695
\(349\) −8.14359 −0.435917 −0.217958 0.975958i \(-0.569940\pi\)
−0.217958 + 0.975958i \(0.569940\pi\)
\(350\) 0 0
\(351\) 5.46410 0.291652
\(352\) 5.19615 0.276956
\(353\) −12.9282 −0.688099 −0.344049 0.938952i \(-0.611799\pi\)
−0.344049 + 0.938952i \(0.611799\pi\)
\(354\) 12.0000 0.637793
\(355\) 0 0
\(356\) 0.928203 0.0491947
\(357\) 0 0
\(358\) 12.0000 0.634220
\(359\) −20.7846 −1.09697 −0.548485 0.836160i \(-0.684795\pi\)
−0.548485 + 0.836160i \(0.684795\pi\)
\(360\) 0 0
\(361\) 10.8564 0.571390
\(362\) 27.4641 1.44348
\(363\) −1.00000 −0.0524864
\(364\) 10.9282 0.572793
\(365\) 0 0
\(366\) −3.46410 −0.181071
\(367\) −20.0000 −1.04399 −0.521996 0.852948i \(-0.674812\pi\)
−0.521996 + 0.852948i \(0.674812\pi\)
\(368\) 34.6410 1.80579
\(369\) 3.46410 0.180334
\(370\) 0 0
\(371\) 1.85641 0.0963798
\(372\) 10.9282 0.566601
\(373\) 20.3923 1.05587 0.527937 0.849284i \(-0.322966\pi\)
0.527937 + 0.849284i \(0.322966\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) −12.0000 −0.618853
\(377\) 18.9282 0.974852
\(378\) 3.46410 0.178174
\(379\) −17.8564 −0.917222 −0.458611 0.888637i \(-0.651653\pi\)
−0.458611 + 0.888637i \(0.651653\pi\)
\(380\) 0 0
\(381\) 8.92820 0.457406
\(382\) −32.7846 −1.67741
\(383\) 13.8564 0.708029 0.354015 0.935240i \(-0.384816\pi\)
0.354015 + 0.935240i \(0.384816\pi\)
\(384\) −12.1244 −0.618718
\(385\) 0 0
\(386\) −42.2487 −2.15040
\(387\) 4.92820 0.250515
\(388\) 10.0000 0.507673
\(389\) 11.0718 0.561362 0.280681 0.959801i \(-0.409440\pi\)
0.280681 + 0.959801i \(0.409440\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 5.19615 0.262445
\(393\) −18.9282 −0.954802
\(394\) 20.7846 1.04711
\(395\) 0 0
\(396\) −1.00000 −0.0502519
\(397\) −2.00000 −0.100377 −0.0501886 0.998740i \(-0.515982\pi\)
−0.0501886 + 0.998740i \(0.515982\pi\)
\(398\) −42.9282 −2.15180
\(399\) 10.9282 0.547094
\(400\) 0 0
\(401\) −7.85641 −0.392330 −0.196165 0.980571i \(-0.562849\pi\)
−0.196165 + 0.980571i \(0.562849\pi\)
\(402\) 13.8564 0.691095
\(403\) 59.7128 2.97451
\(404\) −10.3923 −0.517036
\(405\) 0 0
\(406\) 12.0000 0.595550
\(407\) −4.92820 −0.244282
\(408\) 0 0
\(409\) −6.78461 −0.335477 −0.167739 0.985831i \(-0.553646\pi\)
−0.167739 + 0.985831i \(0.553646\pi\)
\(410\) 0 0
\(411\) −18.0000 −0.887875
\(412\) −8.00000 −0.394132
\(413\) 13.8564 0.681829
\(414\) −12.0000 −0.589768
\(415\) 0 0
\(416\) 28.3923 1.39205
\(417\) −12.3923 −0.606854
\(418\) −9.46410 −0.462904
\(419\) 30.9282 1.51094 0.755471 0.655182i \(-0.227408\pi\)
0.755471 + 0.655182i \(0.227408\pi\)
\(420\) 0 0
\(421\) 2.00000 0.0974740 0.0487370 0.998812i \(-0.484480\pi\)
0.0487370 + 0.998812i \(0.484480\pi\)
\(422\) −14.5359 −0.707596
\(423\) 6.92820 0.336861
\(424\) 1.60770 0.0780766
\(425\) 0 0
\(426\) −24.0000 −1.16280
\(427\) −4.00000 −0.193574
\(428\) −8.53590 −0.412598
\(429\) −5.46410 −0.263809
\(430\) 0 0
\(431\) 8.78461 0.423140 0.211570 0.977363i \(-0.432142\pi\)
0.211570 + 0.977363i \(0.432142\pi\)
\(432\) 5.00000 0.240563
\(433\) −0.143594 −0.00690067 −0.00345033 0.999994i \(-0.501098\pi\)
−0.00345033 + 0.999994i \(0.501098\pi\)
\(434\) 37.8564 1.81717
\(435\) 0 0
\(436\) −10.0000 −0.478913
\(437\) −37.8564 −1.81092
\(438\) −14.5359 −0.694552
\(439\) 33.1769 1.58345 0.791724 0.610879i \(-0.209184\pi\)
0.791724 + 0.610879i \(0.209184\pi\)
\(440\) 0 0
\(441\) −3.00000 −0.142857
\(442\) 0 0
\(443\) −12.0000 −0.570137 −0.285069 0.958507i \(-0.592016\pi\)
−0.285069 + 0.958507i \(0.592016\pi\)
\(444\) −4.92820 −0.233882
\(445\) 0 0
\(446\) −17.0718 −0.808373
\(447\) 15.4641 0.731427
\(448\) −2.00000 −0.0944911
\(449\) −26.7846 −1.26404 −0.632022 0.774950i \(-0.717775\pi\)
−0.632022 + 0.774950i \(0.717775\pi\)
\(450\) 0 0
\(451\) −3.46410 −0.163118
\(452\) 12.9282 0.608092
\(453\) 20.3923 0.958114
\(454\) −26.7846 −1.25706
\(455\) 0 0
\(456\) 9.46410 0.443197
\(457\) −12.3923 −0.579688 −0.289844 0.957074i \(-0.593603\pi\)
−0.289844 + 0.957074i \(0.593603\pi\)
\(458\) −41.3205 −1.93078
\(459\) 0 0
\(460\) 0 0
\(461\) 36.2487 1.68827 0.844135 0.536130i \(-0.180114\pi\)
0.844135 + 0.536130i \(0.180114\pi\)
\(462\) −3.46410 −0.161165
\(463\) 28.0000 1.30127 0.650635 0.759390i \(-0.274503\pi\)
0.650635 + 0.759390i \(0.274503\pi\)
\(464\) 17.3205 0.804084
\(465\) 0 0
\(466\) −20.7846 −0.962828
\(467\) −5.07180 −0.234695 −0.117347 0.993091i \(-0.537439\pi\)
−0.117347 + 0.993091i \(0.537439\pi\)
\(468\) −5.46410 −0.252578
\(469\) 16.0000 0.738811
\(470\) 0 0
\(471\) −3.07180 −0.141541
\(472\) 12.0000 0.552345
\(473\) −4.92820 −0.226599
\(474\) 11.3205 0.519968
\(475\) 0 0
\(476\) 0 0
\(477\) −0.928203 −0.0424995
\(478\) −20.7846 −0.950666
\(479\) 12.0000 0.548294 0.274147 0.961688i \(-0.411605\pi\)
0.274147 + 0.961688i \(0.411605\pi\)
\(480\) 0 0
\(481\) −26.9282 −1.22782
\(482\) 0.248711 0.0113285
\(483\) −13.8564 −0.630488
\(484\) 1.00000 0.0454545
\(485\) 0 0
\(486\) −1.73205 −0.0785674
\(487\) 31.7128 1.43704 0.718522 0.695504i \(-0.244819\pi\)
0.718522 + 0.695504i \(0.244819\pi\)
\(488\) −3.46410 −0.156813
\(489\) 9.85641 0.445722
\(490\) 0 0
\(491\) −30.9282 −1.39577 −0.697885 0.716210i \(-0.745875\pi\)
−0.697885 + 0.716210i \(0.745875\pi\)
\(492\) −3.46410 −0.156174
\(493\) 0 0
\(494\) −51.7128 −2.32667
\(495\) 0 0
\(496\) 54.6410 2.45345
\(497\) −27.7128 −1.24309
\(498\) 14.7846 0.662514
\(499\) 28.7846 1.28858 0.644288 0.764783i \(-0.277154\pi\)
0.644288 + 0.764783i \(0.277154\pi\)
\(500\) 0 0
\(501\) −10.3923 −0.464294
\(502\) −3.21539 −0.143510
\(503\) −31.1769 −1.39011 −0.695055 0.718957i \(-0.744620\pi\)
−0.695055 + 0.718957i \(0.744620\pi\)
\(504\) 3.46410 0.154303
\(505\) 0 0
\(506\) 12.0000 0.533465
\(507\) −16.8564 −0.748619
\(508\) −8.92820 −0.396125
\(509\) 19.8564 0.880120 0.440060 0.897968i \(-0.354957\pi\)
0.440060 + 0.897968i \(0.354957\pi\)
\(510\) 0 0
\(511\) −16.7846 −0.742507
\(512\) 8.66025 0.382733
\(513\) −5.46410 −0.241246
\(514\) 34.3923 1.51698
\(515\) 0 0
\(516\) −4.92820 −0.216952
\(517\) −6.92820 −0.304702
\(518\) −17.0718 −0.750092
\(519\) −12.0000 −0.526742
\(520\) 0 0
\(521\) 6.00000 0.262865 0.131432 0.991325i \(-0.458042\pi\)
0.131432 + 0.991325i \(0.458042\pi\)
\(522\) −6.00000 −0.262613
\(523\) 22.0000 0.961993 0.480996 0.876723i \(-0.340275\pi\)
0.480996 + 0.876723i \(0.340275\pi\)
\(524\) 18.9282 0.826882
\(525\) 0 0
\(526\) −35.5692 −1.55089
\(527\) 0 0
\(528\) −5.00000 −0.217597
\(529\) 25.0000 1.08696
\(530\) 0 0
\(531\) −6.92820 −0.300658
\(532\) −10.9282 −0.473798
\(533\) −18.9282 −0.819871
\(534\) −1.60770 −0.0695718
\(535\) 0 0
\(536\) 13.8564 0.598506
\(537\) −6.92820 −0.298974
\(538\) 34.3923 1.48276
\(539\) 3.00000 0.129219
\(540\) 0 0
\(541\) 27.8564 1.19764 0.598820 0.800883i \(-0.295636\pi\)
0.598820 + 0.800883i \(0.295636\pi\)
\(542\) −20.1051 −0.863589
\(543\) −15.8564 −0.680464
\(544\) 0 0
\(545\) 0 0
\(546\) −18.9282 −0.810052
\(547\) −2.00000 −0.0855138 −0.0427569 0.999086i \(-0.513614\pi\)
−0.0427569 + 0.999086i \(0.513614\pi\)
\(548\) 18.0000 0.768922
\(549\) 2.00000 0.0853579
\(550\) 0 0
\(551\) −18.9282 −0.806369
\(552\) −12.0000 −0.510754
\(553\) 13.0718 0.555869
\(554\) −51.0333 −2.16820
\(555\) 0 0
\(556\) 12.3923 0.525551
\(557\) −3.21539 −0.136240 −0.0681202 0.997677i \(-0.521700\pi\)
−0.0681202 + 0.997677i \(0.521700\pi\)
\(558\) −18.9282 −0.801295
\(559\) −26.9282 −1.13894
\(560\) 0 0
\(561\) 0 0
\(562\) 6.00000 0.253095
\(563\) −10.3923 −0.437983 −0.218992 0.975727i \(-0.570277\pi\)
−0.218992 + 0.975727i \(0.570277\pi\)
\(564\) −6.92820 −0.291730
\(565\) 0 0
\(566\) 8.53590 0.358791
\(567\) −2.00000 −0.0839921
\(568\) −24.0000 −1.00702
\(569\) 5.32051 0.223047 0.111524 0.993762i \(-0.464427\pi\)
0.111524 + 0.993762i \(0.464427\pi\)
\(570\) 0 0
\(571\) 3.60770 0.150977 0.0754887 0.997147i \(-0.475948\pi\)
0.0754887 + 0.997147i \(0.475948\pi\)
\(572\) 5.46410 0.228466
\(573\) 18.9282 0.790737
\(574\) −12.0000 −0.500870
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) 18.7846 0.782014 0.391007 0.920388i \(-0.372127\pi\)
0.391007 + 0.920388i \(0.372127\pi\)
\(578\) −29.4449 −1.22474
\(579\) 24.3923 1.01371
\(580\) 0 0
\(581\) 17.0718 0.708257
\(582\) −17.3205 −0.717958
\(583\) 0.928203 0.0384422
\(584\) −14.5359 −0.601500
\(585\) 0 0
\(586\) 24.0000 0.991431
\(587\) 18.9282 0.781251 0.390625 0.920550i \(-0.372259\pi\)
0.390625 + 0.920550i \(0.372259\pi\)
\(588\) 3.00000 0.123718
\(589\) −59.7128 −2.46042
\(590\) 0 0
\(591\) −12.0000 −0.493614
\(592\) −24.6410 −1.01274
\(593\) −8.78461 −0.360741 −0.180370 0.983599i \(-0.557730\pi\)
−0.180370 + 0.983599i \(0.557730\pi\)
\(594\) 1.73205 0.0710669
\(595\) 0 0
\(596\) −15.4641 −0.633434
\(597\) 24.7846 1.01437
\(598\) 65.5692 2.68132
\(599\) −37.8564 −1.54677 −0.773385 0.633936i \(-0.781438\pi\)
−0.773385 + 0.633936i \(0.781438\pi\)
\(600\) 0 0
\(601\) −32.6410 −1.33145 −0.665727 0.746195i \(-0.731879\pi\)
−0.665727 + 0.746195i \(0.731879\pi\)
\(602\) −17.0718 −0.695794
\(603\) −8.00000 −0.325785
\(604\) −20.3923 −0.829751
\(605\) 0 0
\(606\) 18.0000 0.731200
\(607\) 18.7846 0.762444 0.381222 0.924484i \(-0.375503\pi\)
0.381222 + 0.924484i \(0.375503\pi\)
\(608\) −28.3923 −1.15146
\(609\) −6.92820 −0.280745
\(610\) 0 0
\(611\) −37.8564 −1.53151
\(612\) 0 0
\(613\) 20.3923 0.823637 0.411819 0.911266i \(-0.364894\pi\)
0.411819 + 0.911266i \(0.364894\pi\)
\(614\) −24.2487 −0.978598
\(615\) 0 0
\(616\) −3.46410 −0.139573
\(617\) 36.9282 1.48667 0.743337 0.668917i \(-0.233242\pi\)
0.743337 + 0.668917i \(0.233242\pi\)
\(618\) 13.8564 0.557386
\(619\) 20.0000 0.803868 0.401934 0.915669i \(-0.368338\pi\)
0.401934 + 0.915669i \(0.368338\pi\)
\(620\) 0 0
\(621\) 6.92820 0.278019
\(622\) 8.78461 0.352231
\(623\) −1.85641 −0.0743754
\(624\) −27.3205 −1.09370
\(625\) 0 0
\(626\) −36.2487 −1.44879
\(627\) 5.46410 0.218215
\(628\) 3.07180 0.122578
\(629\) 0 0
\(630\) 0 0
\(631\) −21.0718 −0.838855 −0.419427 0.907789i \(-0.637769\pi\)
−0.419427 + 0.907789i \(0.637769\pi\)
\(632\) 11.3205 0.450306
\(633\) 8.39230 0.333564
\(634\) 43.1769 1.71477
\(635\) 0 0
\(636\) 0.928203 0.0368057
\(637\) 16.3923 0.649487
\(638\) 6.00000 0.237542
\(639\) 13.8564 0.548151
\(640\) 0 0
\(641\) −12.9282 −0.510633 −0.255317 0.966857i \(-0.582180\pi\)
−0.255317 + 0.966857i \(0.582180\pi\)
\(642\) 14.7846 0.583502
\(643\) −37.5692 −1.48159 −0.740793 0.671734i \(-0.765550\pi\)
−0.740793 + 0.671734i \(0.765550\pi\)
\(644\) 13.8564 0.546019
\(645\) 0 0
\(646\) 0 0
\(647\) −27.7128 −1.08950 −0.544752 0.838597i \(-0.683376\pi\)
−0.544752 + 0.838597i \(0.683376\pi\)
\(648\) −1.73205 −0.0680414
\(649\) 6.92820 0.271956
\(650\) 0 0
\(651\) −21.8564 −0.856620
\(652\) −9.85641 −0.386007
\(653\) −19.8564 −0.777041 −0.388521 0.921440i \(-0.627014\pi\)
−0.388521 + 0.921440i \(0.627014\pi\)
\(654\) 17.3205 0.677285
\(655\) 0 0
\(656\) −17.3205 −0.676252
\(657\) 8.39230 0.327415
\(658\) −24.0000 −0.935617
\(659\) −15.7128 −0.612084 −0.306042 0.952018i \(-0.599005\pi\)
−0.306042 + 0.952018i \(0.599005\pi\)
\(660\) 0 0
\(661\) 14.0000 0.544537 0.272268 0.962221i \(-0.412226\pi\)
0.272268 + 0.962221i \(0.412226\pi\)
\(662\) 17.0718 0.663514
\(663\) 0 0
\(664\) 14.7846 0.573754
\(665\) 0 0
\(666\) 8.53590 0.330759
\(667\) 24.0000 0.929284
\(668\) 10.3923 0.402090
\(669\) 9.85641 0.381071
\(670\) 0 0
\(671\) −2.00000 −0.0772091
\(672\) −10.3923 −0.400892
\(673\) 3.32051 0.127996 0.0639981 0.997950i \(-0.479615\pi\)
0.0639981 + 0.997950i \(0.479615\pi\)
\(674\) −57.4641 −2.21343
\(675\) 0 0
\(676\) 16.8564 0.648323
\(677\) −8.78461 −0.337620 −0.168810 0.985649i \(-0.553992\pi\)
−0.168810 + 0.985649i \(0.553992\pi\)
\(678\) −22.3923 −0.859971
\(679\) −20.0000 −0.767530
\(680\) 0 0
\(681\) 15.4641 0.592586
\(682\) 18.9282 0.724798
\(683\) 32.7846 1.25447 0.627234 0.778831i \(-0.284187\pi\)
0.627234 + 0.778831i \(0.284187\pi\)
\(684\) 5.46410 0.208925
\(685\) 0 0
\(686\) 34.6410 1.32260
\(687\) 23.8564 0.910179
\(688\) −24.6410 −0.939430
\(689\) 5.07180 0.193220
\(690\) 0 0
\(691\) 47.7128 1.81508 0.907540 0.419965i \(-0.137958\pi\)
0.907540 + 0.419965i \(0.137958\pi\)
\(692\) 12.0000 0.456172
\(693\) 2.00000 0.0759737
\(694\) −38.7846 −1.47224
\(695\) 0 0
\(696\) −6.00000 −0.227429
\(697\) 0 0
\(698\) −14.1051 −0.533887
\(699\) 12.0000 0.453882
\(700\) 0 0
\(701\) 39.4641 1.49054 0.745269 0.666764i \(-0.232321\pi\)
0.745269 + 0.666764i \(0.232321\pi\)
\(702\) 9.46410 0.357199
\(703\) 26.9282 1.01562
\(704\) −1.00000 −0.0376889
\(705\) 0 0
\(706\) −22.3923 −0.842746
\(707\) 20.7846 0.781686
\(708\) 6.92820 0.260378
\(709\) −11.8564 −0.445277 −0.222638 0.974901i \(-0.571467\pi\)
−0.222638 + 0.974901i \(0.571467\pi\)
\(710\) 0 0
\(711\) −6.53590 −0.245115
\(712\) −1.60770 −0.0602509
\(713\) 75.7128 2.83547
\(714\) 0 0
\(715\) 0 0
\(716\) 6.92820 0.258919
\(717\) 12.0000 0.448148
\(718\) −36.0000 −1.34351
\(719\) −5.07180 −0.189146 −0.0945731 0.995518i \(-0.530149\pi\)
−0.0945731 + 0.995518i \(0.530149\pi\)
\(720\) 0 0
\(721\) 16.0000 0.595871
\(722\) 18.8038 0.699807
\(723\) −0.143594 −0.00534030
\(724\) 15.8564 0.589299
\(725\) 0 0
\(726\) −1.73205 −0.0642824
\(727\) −8.00000 −0.296704 −0.148352 0.988935i \(-0.547397\pi\)
−0.148352 + 0.988935i \(0.547397\pi\)
\(728\) −18.9282 −0.701526
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 0 0
\(732\) −2.00000 −0.0739221
\(733\) −53.9615 −1.99311 −0.996557 0.0829082i \(-0.973579\pi\)
−0.996557 + 0.0829082i \(0.973579\pi\)
\(734\) −34.6410 −1.27862
\(735\) 0 0
\(736\) 36.0000 1.32698
\(737\) 8.00000 0.294684
\(738\) 6.00000 0.220863
\(739\) 17.4641 0.642427 0.321214 0.947007i \(-0.395909\pi\)
0.321214 + 0.947007i \(0.395909\pi\)
\(740\) 0 0
\(741\) 29.8564 1.09680
\(742\) 3.21539 0.118041
\(743\) −25.6077 −0.939455 −0.469728 0.882811i \(-0.655648\pi\)
−0.469728 + 0.882811i \(0.655648\pi\)
\(744\) −18.9282 −0.693942
\(745\) 0 0
\(746\) 35.3205 1.29318
\(747\) −8.53590 −0.312312
\(748\) 0 0
\(749\) 17.0718 0.623790
\(750\) 0 0
\(751\) 26.9282 0.982624 0.491312 0.870984i \(-0.336517\pi\)
0.491312 + 0.870984i \(0.336517\pi\)
\(752\) −34.6410 −1.26323
\(753\) 1.85641 0.0676512
\(754\) 32.7846 1.19395
\(755\) 0 0
\(756\) 2.00000 0.0727393
\(757\) −34.7846 −1.26427 −0.632134 0.774859i \(-0.717821\pi\)
−0.632134 + 0.774859i \(0.717821\pi\)
\(758\) −30.9282 −1.12336
\(759\) −6.92820 −0.251478
\(760\) 0 0
\(761\) −32.5359 −1.17943 −0.589713 0.807613i \(-0.700759\pi\)
−0.589713 + 0.807613i \(0.700759\pi\)
\(762\) 15.4641 0.560205
\(763\) 20.0000 0.724049
\(764\) −18.9282 −0.684798
\(765\) 0 0
\(766\) 24.0000 0.867155
\(767\) 37.8564 1.36692
\(768\) −19.0000 −0.685603
\(769\) 50.4974 1.82098 0.910492 0.413527i \(-0.135703\pi\)
0.910492 + 0.413527i \(0.135703\pi\)
\(770\) 0 0
\(771\) −19.8564 −0.715111
\(772\) −24.3923 −0.877898
\(773\) 4.14359 0.149035 0.0745174 0.997220i \(-0.476258\pi\)
0.0745174 + 0.997220i \(0.476258\pi\)
\(774\) 8.53590 0.306817
\(775\) 0 0
\(776\) −17.3205 −0.621770
\(777\) 9.85641 0.353597
\(778\) 19.1769 0.687526
\(779\) 18.9282 0.678173
\(780\) 0 0
\(781\) −13.8564 −0.495821
\(782\) 0 0
\(783\) 3.46410 0.123797
\(784\) 15.0000 0.535714
\(785\) 0 0
\(786\) −32.7846 −1.16939
\(787\) −22.7846 −0.812184 −0.406092 0.913832i \(-0.633109\pi\)
−0.406092 + 0.913832i \(0.633109\pi\)
\(788\) 12.0000 0.427482
\(789\) 20.5359 0.731097
\(790\) 0 0
\(791\) −25.8564 −0.919348
\(792\) 1.73205 0.0615457
\(793\) −10.9282 −0.388072
\(794\) −3.46410 −0.122936
\(795\) 0 0
\(796\) −24.7846 −0.878467
\(797\) 52.6410 1.86464 0.932320 0.361634i \(-0.117781\pi\)
0.932320 + 0.361634i \(0.117781\pi\)
\(798\) 18.9282 0.670051
\(799\) 0 0
\(800\) 0 0
\(801\) 0.928203 0.0327964
\(802\) −13.6077 −0.480504
\(803\) −8.39230 −0.296158
\(804\) 8.00000 0.282138
\(805\) 0 0
\(806\) 103.426 3.64301
\(807\) −19.8564 −0.698979
\(808\) 18.0000 0.633238
\(809\) −15.4641 −0.543689 −0.271844 0.962341i \(-0.587634\pi\)
−0.271844 + 0.962341i \(0.587634\pi\)
\(810\) 0 0
\(811\) 12.3923 0.435153 0.217576 0.976043i \(-0.430185\pi\)
0.217576 + 0.976043i \(0.430185\pi\)
\(812\) 6.92820 0.243132
\(813\) 11.6077 0.407100
\(814\) −8.53590 −0.299183
\(815\) 0 0
\(816\) 0 0
\(817\) 26.9282 0.942099
\(818\) −11.7513 −0.410874
\(819\) 10.9282 0.381862
\(820\) 0 0
\(821\) 20.5359 0.716708 0.358354 0.933586i \(-0.383338\pi\)
0.358354 + 0.933586i \(0.383338\pi\)
\(822\) −31.1769 −1.08742
\(823\) 33.5692 1.17015 0.585075 0.810979i \(-0.301065\pi\)
0.585075 + 0.810979i \(0.301065\pi\)
\(824\) 13.8564 0.482711
\(825\) 0 0
\(826\) 24.0000 0.835067
\(827\) −22.3923 −0.778657 −0.389328 0.921099i \(-0.627293\pi\)
−0.389328 + 0.921099i \(0.627293\pi\)
\(828\) −6.92820 −0.240772
\(829\) 29.7128 1.03197 0.515984 0.856598i \(-0.327426\pi\)
0.515984 + 0.856598i \(0.327426\pi\)
\(830\) 0 0
\(831\) 29.4641 1.02210
\(832\) −5.46410 −0.189434
\(833\) 0 0
\(834\) −21.4641 −0.743241
\(835\) 0 0
\(836\) −5.46410 −0.188980
\(837\) 10.9282 0.377734
\(838\) 53.5692 1.85052
\(839\) 56.7846 1.96042 0.980211 0.197954i \(-0.0634298\pi\)
0.980211 + 0.197954i \(0.0634298\pi\)
\(840\) 0 0
\(841\) −17.0000 −0.586207
\(842\) 3.46410 0.119381
\(843\) −3.46410 −0.119310
\(844\) −8.39230 −0.288875
\(845\) 0 0
\(846\) 12.0000 0.412568
\(847\) −2.00000 −0.0687208
\(848\) 4.64102 0.159373
\(849\) −4.92820 −0.169135
\(850\) 0 0
\(851\) −34.1436 −1.17043
\(852\) −13.8564 −0.474713
\(853\) −3.60770 −0.123525 −0.0617626 0.998091i \(-0.519672\pi\)
−0.0617626 + 0.998091i \(0.519672\pi\)
\(854\) −6.92820 −0.237078
\(855\) 0 0
\(856\) 14.7846 0.505328
\(857\) 37.8564 1.29315 0.646575 0.762850i \(-0.276201\pi\)
0.646575 + 0.762850i \(0.276201\pi\)
\(858\) −9.46410 −0.323099
\(859\) −7.71281 −0.263158 −0.131579 0.991306i \(-0.542005\pi\)
−0.131579 + 0.991306i \(0.542005\pi\)
\(860\) 0 0
\(861\) 6.92820 0.236113
\(862\) 15.2154 0.518238
\(863\) 37.8564 1.28865 0.644324 0.764753i \(-0.277139\pi\)
0.644324 + 0.764753i \(0.277139\pi\)
\(864\) 5.19615 0.176777
\(865\) 0 0
\(866\) −0.248711 −0.00845155
\(867\) 17.0000 0.577350
\(868\) 21.8564 0.741855
\(869\) 6.53590 0.221715
\(870\) 0 0
\(871\) 43.7128 1.48115
\(872\) 17.3205 0.586546
\(873\) 10.0000 0.338449
\(874\) −65.5692 −2.21791
\(875\) 0 0
\(876\) −8.39230 −0.283550
\(877\) 34.2487 1.15650 0.578248 0.815861i \(-0.303736\pi\)
0.578248 + 0.815861i \(0.303736\pi\)
\(878\) 57.4641 1.93932
\(879\) −13.8564 −0.467365
\(880\) 0 0
\(881\) −0.928203 −0.0312720 −0.0156360 0.999878i \(-0.504977\pi\)
−0.0156360 + 0.999878i \(0.504977\pi\)
\(882\) −5.19615 −0.174964
\(883\) 4.00000 0.134611 0.0673054 0.997732i \(-0.478560\pi\)
0.0673054 + 0.997732i \(0.478560\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −20.7846 −0.698273
\(887\) −12.2487 −0.411271 −0.205636 0.978629i \(-0.565926\pi\)
−0.205636 + 0.978629i \(0.565926\pi\)
\(888\) 8.53590 0.286446
\(889\) 17.8564 0.598885
\(890\) 0 0
\(891\) −1.00000 −0.0335013
\(892\) −9.85641 −0.330017
\(893\) 37.8564 1.26682
\(894\) 26.7846 0.895811
\(895\) 0 0
\(896\) −24.2487 −0.810093
\(897\) −37.8564 −1.26399
\(898\) −46.3923 −1.54813
\(899\) 37.8564 1.26258
\(900\) 0 0
\(901\) 0 0
\(902\) −6.00000 −0.199778
\(903\) 9.85641 0.328001
\(904\) −22.3923 −0.744757
\(905\) 0 0
\(906\) 35.3205 1.17345
\(907\) −18.1436 −0.602448 −0.301224 0.953553i \(-0.597395\pi\)
−0.301224 + 0.953553i \(0.597395\pi\)
\(908\) −15.4641 −0.513194
\(909\) −10.3923 −0.344691
\(910\) 0 0
\(911\) 18.9282 0.627119 0.313560 0.949568i \(-0.398478\pi\)
0.313560 + 0.949568i \(0.398478\pi\)
\(912\) 27.3205 0.904672
\(913\) 8.53590 0.282497
\(914\) −21.4641 −0.709969
\(915\) 0 0
\(916\) −23.8564 −0.788238
\(917\) −37.8564 −1.25013
\(918\) 0 0
\(919\) −32.3923 −1.06852 −0.534262 0.845319i \(-0.679410\pi\)
−0.534262 + 0.845319i \(0.679410\pi\)
\(920\) 0 0
\(921\) 14.0000 0.461316
\(922\) 62.7846 2.06770
\(923\) −75.7128 −2.49212
\(924\) −2.00000 −0.0657952
\(925\) 0 0
\(926\) 48.4974 1.59372
\(927\) −8.00000 −0.262754
\(928\) 18.0000 0.590879
\(929\) 2.78461 0.0913601 0.0456800 0.998956i \(-0.485455\pi\)
0.0456800 + 0.998956i \(0.485455\pi\)
\(930\) 0 0
\(931\) −16.3923 −0.537236
\(932\) −12.0000 −0.393073
\(933\) −5.07180 −0.166043
\(934\) −8.78461 −0.287441
\(935\) 0 0
\(936\) 9.46410 0.309344
\(937\) 20.3923 0.666188 0.333094 0.942894i \(-0.391907\pi\)
0.333094 + 0.942894i \(0.391907\pi\)
\(938\) 27.7128 0.904855
\(939\) 20.9282 0.682966
\(940\) 0 0
\(941\) 27.4641 0.895304 0.447652 0.894208i \(-0.352260\pi\)
0.447652 + 0.894208i \(0.352260\pi\)
\(942\) −5.32051 −0.173352
\(943\) −24.0000 −0.781548
\(944\) 34.6410 1.12747
\(945\) 0 0
\(946\) −8.53590 −0.277526
\(947\) 18.9282 0.615084 0.307542 0.951535i \(-0.400494\pi\)
0.307542 + 0.951535i \(0.400494\pi\)
\(948\) 6.53590 0.212276
\(949\) −45.8564 −1.48856
\(950\) 0 0
\(951\) −24.9282 −0.808352
\(952\) 0 0
\(953\) 3.21539 0.104157 0.0520784 0.998643i \(-0.483415\pi\)
0.0520784 + 0.998643i \(0.483415\pi\)
\(954\) −1.60770 −0.0520511
\(955\) 0 0
\(956\) −12.0000 −0.388108
\(957\) −3.46410 −0.111979
\(958\) 20.7846 0.671520
\(959\) −36.0000 −1.16250
\(960\) 0 0
\(961\) 88.4256 2.85244
\(962\) −46.6410 −1.50377
\(963\) −8.53590 −0.275065
\(964\) 0.143594 0.00462484
\(965\) 0 0
\(966\) −24.0000 −0.772187
\(967\) −22.7846 −0.732704 −0.366352 0.930476i \(-0.619393\pi\)
−0.366352 + 0.930476i \(0.619393\pi\)
\(968\) −1.73205 −0.0556702
\(969\) 0 0
\(970\) 0 0
\(971\) 25.8564 0.829772 0.414886 0.909873i \(-0.363822\pi\)
0.414886 + 0.909873i \(0.363822\pi\)
\(972\) −1.00000 −0.0320750
\(973\) −24.7846 −0.794558
\(974\) 54.9282 1.76001
\(975\) 0 0
\(976\) −10.0000 −0.320092
\(977\) −47.5692 −1.52187 −0.760937 0.648826i \(-0.775260\pi\)
−0.760937 + 0.648826i \(0.775260\pi\)
\(978\) 17.0718 0.545896
\(979\) −0.928203 −0.0296655
\(980\) 0 0
\(981\) −10.0000 −0.319275
\(982\) −53.5692 −1.70946
\(983\) −24.0000 −0.765481 −0.382741 0.923856i \(-0.625020\pi\)
−0.382741 + 0.923856i \(0.625020\pi\)
\(984\) 6.00000 0.191273
\(985\) 0 0
\(986\) 0 0
\(987\) 13.8564 0.441054
\(988\) −29.8564 −0.949859
\(989\) −34.1436 −1.08570
\(990\) 0 0
\(991\) −7.21539 −0.229204 −0.114602 0.993411i \(-0.536559\pi\)
−0.114602 + 0.993411i \(0.536559\pi\)
\(992\) 56.7846 1.80291
\(993\) −9.85641 −0.312784
\(994\) −48.0000 −1.52247
\(995\) 0 0
\(996\) 8.53590 0.270470
\(997\) −27.6077 −0.874344 −0.437172 0.899378i \(-0.644020\pi\)
−0.437172 + 0.899378i \(0.644020\pi\)
\(998\) 49.8564 1.57818
\(999\) −4.92820 −0.155921
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.e.1.2 2
3.2 odd 2 2475.2.a.r.1.1 2
5.2 odd 4 825.2.c.c.199.4 4
5.3 odd 4 825.2.c.c.199.1 4
5.4 even 2 165.2.a.b.1.1 2
11.10 odd 2 9075.2.a.bh.1.1 2
15.2 even 4 2475.2.c.n.199.1 4
15.8 even 4 2475.2.c.n.199.4 4
15.14 odd 2 495.2.a.c.1.2 2
20.19 odd 2 2640.2.a.x.1.2 2
35.34 odd 2 8085.2.a.bd.1.1 2
55.54 odd 2 1815.2.a.i.1.2 2
60.59 even 2 7920.2.a.bz.1.2 2
165.164 even 2 5445.2.a.s.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
165.2.a.b.1.1 2 5.4 even 2
495.2.a.c.1.2 2 15.14 odd 2
825.2.a.e.1.2 2 1.1 even 1 trivial
825.2.c.c.199.1 4 5.3 odd 4
825.2.c.c.199.4 4 5.2 odd 4
1815.2.a.i.1.2 2 55.54 odd 2
2475.2.a.r.1.1 2 3.2 odd 2
2475.2.c.n.199.1 4 15.2 even 4
2475.2.c.n.199.4 4 15.8 even 4
2640.2.a.x.1.2 2 20.19 odd 2
5445.2.a.s.1.1 2 165.164 even 2
7920.2.a.bz.1.2 2 60.59 even 2
8085.2.a.bd.1.1 2 35.34 odd 2
9075.2.a.bh.1.1 2 11.10 odd 2