Properties

Label 18.3.b.a.17.2
Level $18$
Weight $3$
Character 18.17
Analytic conductor $0.490$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 18 = 2 \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 18.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.490464475849\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-2}) \)
Defining polynomial: \( x^{2} + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 17.2
Root \(1.41421i\) of defining polynomial
Character \(\chi\) \(=\) 18.17
Dual form 18.3.b.a.17.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.41421i q^{2} -2.00000 q^{4} -4.24264i q^{5} -4.00000 q^{7} -2.82843i q^{8} +O(q^{10})\) \(q+1.41421i q^{2} -2.00000 q^{4} -4.24264i q^{5} -4.00000 q^{7} -2.82843i q^{8} +6.00000 q^{10} +16.9706i q^{11} +8.00000 q^{13} -5.65685i q^{14} +4.00000 q^{16} -12.7279i q^{17} -16.0000 q^{19} +8.48528i q^{20} -24.0000 q^{22} -16.9706i q^{23} +7.00000 q^{25} +11.3137i q^{26} +8.00000 q^{28} +4.24264i q^{29} +44.0000 q^{31} +5.65685i q^{32} +18.0000 q^{34} +16.9706i q^{35} -34.0000 q^{37} -22.6274i q^{38} -12.0000 q^{40} +46.6690i q^{41} -40.0000 q^{43} -33.9411i q^{44} +24.0000 q^{46} -84.8528i q^{47} -33.0000 q^{49} +9.89949i q^{50} -16.0000 q^{52} +38.1838i q^{53} +72.0000 q^{55} +11.3137i q^{56} -6.00000 q^{58} +33.9411i q^{59} +50.0000 q^{61} +62.2254i q^{62} -8.00000 q^{64} -33.9411i q^{65} +8.00000 q^{67} +25.4558i q^{68} -24.0000 q^{70} -50.9117i q^{71} -16.0000 q^{73} -48.0833i q^{74} +32.0000 q^{76} -67.8823i q^{77} -76.0000 q^{79} -16.9706i q^{80} -66.0000 q^{82} +118.794i q^{83} -54.0000 q^{85} -56.5685i q^{86} +48.0000 q^{88} +12.7279i q^{89} -32.0000 q^{91} +33.9411i q^{92} +120.000 q^{94} +67.8823i q^{95} +176.000 q^{97} -46.6690i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{4} - 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 4 q^{4} - 8 q^{7} + 12 q^{10} + 16 q^{13} + 8 q^{16} - 32 q^{19} - 48 q^{22} + 14 q^{25} + 16 q^{28} + 88 q^{31} + 36 q^{34} - 68 q^{37} - 24 q^{40} - 80 q^{43} + 48 q^{46} - 66 q^{49} - 32 q^{52} + 144 q^{55} - 12 q^{58} + 100 q^{61} - 16 q^{64} + 16 q^{67} - 48 q^{70} - 32 q^{73} + 64 q^{76} - 152 q^{79} - 132 q^{82} - 108 q^{85} + 96 q^{88} - 64 q^{91} + 240 q^{94} + 352 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/18\mathbb{Z}\right)^\times\).

\(n\) \(11\)
\(\chi(n)\) \(-1\)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.41421i 0.707107i
\(3\) 0 0
\(4\) −2.00000 −0.500000
\(5\) − 4.24264i − 0.848528i −0.905539 0.424264i \(-0.860533\pi\)
0.905539 0.424264i \(-0.139467\pi\)
\(6\) 0 0
\(7\) −4.00000 −0.571429 −0.285714 0.958315i \(-0.592231\pi\)
−0.285714 + 0.958315i \(0.592231\pi\)
\(8\) − 2.82843i − 0.353553i
\(9\) 0 0
\(10\) 6.00000 0.600000
\(11\) 16.9706i 1.54278i 0.636364 + 0.771389i \(0.280438\pi\)
−0.636364 + 0.771389i \(0.719562\pi\)
\(12\) 0 0
\(13\) 8.00000 0.615385 0.307692 0.951486i \(-0.400443\pi\)
0.307692 + 0.951486i \(0.400443\pi\)
\(14\) − 5.65685i − 0.404061i
\(15\) 0 0
\(16\) 4.00000 0.250000
\(17\) − 12.7279i − 0.748701i −0.927287 0.374351i \(-0.877866\pi\)
0.927287 0.374351i \(-0.122134\pi\)
\(18\) 0 0
\(19\) −16.0000 −0.842105 −0.421053 0.907036i \(-0.638339\pi\)
−0.421053 + 0.907036i \(0.638339\pi\)
\(20\) 8.48528i 0.424264i
\(21\) 0 0
\(22\) −24.0000 −1.09091
\(23\) − 16.9706i − 0.737851i −0.929459 0.368925i \(-0.879726\pi\)
0.929459 0.368925i \(-0.120274\pi\)
\(24\) 0 0
\(25\) 7.00000 0.280000
\(26\) 11.3137i 0.435143i
\(27\) 0 0
\(28\) 8.00000 0.285714
\(29\) 4.24264i 0.146298i 0.997321 + 0.0731490i \(0.0233049\pi\)
−0.997321 + 0.0731490i \(0.976695\pi\)
\(30\) 0 0
\(31\) 44.0000 1.41935 0.709677 0.704527i \(-0.248841\pi\)
0.709677 + 0.704527i \(0.248841\pi\)
\(32\) 5.65685i 0.176777i
\(33\) 0 0
\(34\) 18.0000 0.529412
\(35\) 16.9706i 0.484873i
\(36\) 0 0
\(37\) −34.0000 −0.918919 −0.459459 0.888199i \(-0.651957\pi\)
−0.459459 + 0.888199i \(0.651957\pi\)
\(38\) − 22.6274i − 0.595458i
\(39\) 0 0
\(40\) −12.0000 −0.300000
\(41\) 46.6690i 1.13827i 0.822244 + 0.569135i \(0.192722\pi\)
−0.822244 + 0.569135i \(0.807278\pi\)
\(42\) 0 0
\(43\) −40.0000 −0.930233 −0.465116 0.885250i \(-0.653987\pi\)
−0.465116 + 0.885250i \(0.653987\pi\)
\(44\) − 33.9411i − 0.771389i
\(45\) 0 0
\(46\) 24.0000 0.521739
\(47\) − 84.8528i − 1.80538i −0.430293 0.902690i \(-0.641590\pi\)
0.430293 0.902690i \(-0.358410\pi\)
\(48\) 0 0
\(49\) −33.0000 −0.673469
\(50\) 9.89949i 0.197990i
\(51\) 0 0
\(52\) −16.0000 −0.307692
\(53\) 38.1838i 0.720448i 0.932866 + 0.360224i \(0.117300\pi\)
−0.932866 + 0.360224i \(0.882700\pi\)
\(54\) 0 0
\(55\) 72.0000 1.30909
\(56\) 11.3137i 0.202031i
\(57\) 0 0
\(58\) −6.00000 −0.103448
\(59\) 33.9411i 0.575273i 0.957740 + 0.287637i \(0.0928695\pi\)
−0.957740 + 0.287637i \(0.907130\pi\)
\(60\) 0 0
\(61\) 50.0000 0.819672 0.409836 0.912159i \(-0.365586\pi\)
0.409836 + 0.912159i \(0.365586\pi\)
\(62\) 62.2254i 1.00364i
\(63\) 0 0
\(64\) −8.00000 −0.125000
\(65\) − 33.9411i − 0.522171i
\(66\) 0 0
\(67\) 8.00000 0.119403 0.0597015 0.998216i \(-0.480985\pi\)
0.0597015 + 0.998216i \(0.480985\pi\)
\(68\) 25.4558i 0.374351i
\(69\) 0 0
\(70\) −24.0000 −0.342857
\(71\) − 50.9117i − 0.717066i −0.933517 0.358533i \(-0.883277\pi\)
0.933517 0.358533i \(-0.116723\pi\)
\(72\) 0 0
\(73\) −16.0000 −0.219178 −0.109589 0.993977i \(-0.534953\pi\)
−0.109589 + 0.993977i \(0.534953\pi\)
\(74\) − 48.0833i − 0.649774i
\(75\) 0 0
\(76\) 32.0000 0.421053
\(77\) − 67.8823i − 0.881588i
\(78\) 0 0
\(79\) −76.0000 −0.962025 −0.481013 0.876714i \(-0.659731\pi\)
−0.481013 + 0.876714i \(0.659731\pi\)
\(80\) − 16.9706i − 0.212132i
\(81\) 0 0
\(82\) −66.0000 −0.804878
\(83\) 118.794i 1.43125i 0.698484 + 0.715626i \(0.253859\pi\)
−0.698484 + 0.715626i \(0.746141\pi\)
\(84\) 0 0
\(85\) −54.0000 −0.635294
\(86\) − 56.5685i − 0.657774i
\(87\) 0 0
\(88\) 48.0000 0.545455
\(89\) 12.7279i 0.143010i 0.997440 + 0.0715052i \(0.0227802\pi\)
−0.997440 + 0.0715052i \(0.977220\pi\)
\(90\) 0 0
\(91\) −32.0000 −0.351648
\(92\) 33.9411i 0.368925i
\(93\) 0 0
\(94\) 120.000 1.27660
\(95\) 67.8823i 0.714550i
\(96\) 0 0
\(97\) 176.000 1.81443 0.907216 0.420664i \(-0.138203\pi\)
0.907216 + 0.420664i \(0.138203\pi\)
\(98\) − 46.6690i − 0.476215i
\(99\) 0 0
\(100\) −14.0000 −0.140000
\(101\) 29.6985i 0.294044i 0.989133 + 0.147022i \(0.0469689\pi\)
−0.989133 + 0.147022i \(0.953031\pi\)
\(102\) 0 0
\(103\) −28.0000 −0.271845 −0.135922 0.990719i \(-0.543400\pi\)
−0.135922 + 0.990719i \(0.543400\pi\)
\(104\) − 22.6274i − 0.217571i
\(105\) 0 0
\(106\) −54.0000 −0.509434
\(107\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(108\) 0 0
\(109\) 56.0000 0.513761 0.256881 0.966443i \(-0.417305\pi\)
0.256881 + 0.966443i \(0.417305\pi\)
\(110\) 101.823i 0.925667i
\(111\) 0 0
\(112\) −16.0000 −0.142857
\(113\) − 156.978i − 1.38918i −0.719404 0.694592i \(-0.755585\pi\)
0.719404 0.694592i \(-0.244415\pi\)
\(114\) 0 0
\(115\) −72.0000 −0.626087
\(116\) − 8.48528i − 0.0731490i
\(117\) 0 0
\(118\) −48.0000 −0.406780
\(119\) 50.9117i 0.427829i
\(120\) 0 0
\(121\) −167.000 −1.38017
\(122\) 70.7107i 0.579596i
\(123\) 0 0
\(124\) −88.0000 −0.709677
\(125\) − 135.765i − 1.08612i
\(126\) 0 0
\(127\) 92.0000 0.724409 0.362205 0.932099i \(-0.382024\pi\)
0.362205 + 0.932099i \(0.382024\pi\)
\(128\) − 11.3137i − 0.0883883i
\(129\) 0 0
\(130\) 48.0000 0.369231
\(131\) − 169.706i − 1.29546i −0.761869 0.647731i \(-0.775718\pi\)
0.761869 0.647731i \(-0.224282\pi\)
\(132\) 0 0
\(133\) 64.0000 0.481203
\(134\) 11.3137i 0.0844307i
\(135\) 0 0
\(136\) −36.0000 −0.264706
\(137\) 156.978i 1.14582i 0.819617 + 0.572911i \(0.194186\pi\)
−0.819617 + 0.572911i \(0.805814\pi\)
\(138\) 0 0
\(139\) 152.000 1.09353 0.546763 0.837288i \(-0.315860\pi\)
0.546763 + 0.837288i \(0.315860\pi\)
\(140\) − 33.9411i − 0.242437i
\(141\) 0 0
\(142\) 72.0000 0.507042
\(143\) 135.765i 0.949402i
\(144\) 0 0
\(145\) 18.0000 0.124138
\(146\) − 22.6274i − 0.154982i
\(147\) 0 0
\(148\) 68.0000 0.459459
\(149\) 275.772i 1.85082i 0.378972 + 0.925408i \(0.376278\pi\)
−0.378972 + 0.925408i \(0.623722\pi\)
\(150\) 0 0
\(151\) −148.000 −0.980132 −0.490066 0.871685i \(-0.663027\pi\)
−0.490066 + 0.871685i \(0.663027\pi\)
\(152\) 45.2548i 0.297729i
\(153\) 0 0
\(154\) 96.0000 0.623377
\(155\) − 186.676i − 1.20436i
\(156\) 0 0
\(157\) −82.0000 −0.522293 −0.261146 0.965299i \(-0.584101\pi\)
−0.261146 + 0.965299i \(0.584101\pi\)
\(158\) − 107.480i − 0.680255i
\(159\) 0 0
\(160\) 24.0000 0.150000
\(161\) 67.8823i 0.421629i
\(162\) 0 0
\(163\) 56.0000 0.343558 0.171779 0.985135i \(-0.445048\pi\)
0.171779 + 0.985135i \(0.445048\pi\)
\(164\) − 93.3381i − 0.569135i
\(165\) 0 0
\(166\) −168.000 −1.01205
\(167\) 33.9411i 0.203240i 0.994823 + 0.101620i \(0.0324026\pi\)
−0.994823 + 0.101620i \(0.967597\pi\)
\(168\) 0 0
\(169\) −105.000 −0.621302
\(170\) − 76.3675i − 0.449221i
\(171\) 0 0
\(172\) 80.0000 0.465116
\(173\) − 173.948i − 1.00548i −0.864437 0.502741i \(-0.832325\pi\)
0.864437 0.502741i \(-0.167675\pi\)
\(174\) 0 0
\(175\) −28.0000 −0.160000
\(176\) 67.8823i 0.385695i
\(177\) 0 0
\(178\) −18.0000 −0.101124
\(179\) − 203.647i − 1.13769i −0.822444 0.568846i \(-0.807390\pi\)
0.822444 0.568846i \(-0.192610\pi\)
\(180\) 0 0
\(181\) −232.000 −1.28177 −0.640884 0.767638i \(-0.721432\pi\)
−0.640884 + 0.767638i \(0.721432\pi\)
\(182\) − 45.2548i − 0.248653i
\(183\) 0 0
\(184\) −48.0000 −0.260870
\(185\) 144.250i 0.779729i
\(186\) 0 0
\(187\) 216.000 1.15508
\(188\) 169.706i 0.902690i
\(189\) 0 0
\(190\) −96.0000 −0.505263
\(191\) − 33.9411i − 0.177702i −0.996045 0.0888511i \(-0.971680\pi\)
0.996045 0.0888511i \(-0.0283195\pi\)
\(192\) 0 0
\(193\) 206.000 1.06736 0.533679 0.845687i \(-0.320809\pi\)
0.533679 + 0.845687i \(0.320809\pi\)
\(194\) 248.902i 1.28300i
\(195\) 0 0
\(196\) 66.0000 0.336735
\(197\) 165.463i 0.839914i 0.907544 + 0.419957i \(0.137955\pi\)
−0.907544 + 0.419957i \(0.862045\pi\)
\(198\) 0 0
\(199\) 20.0000 0.100503 0.0502513 0.998737i \(-0.483998\pi\)
0.0502513 + 0.998737i \(0.483998\pi\)
\(200\) − 19.7990i − 0.0989949i
\(201\) 0 0
\(202\) −42.0000 −0.207921
\(203\) − 16.9706i − 0.0835988i
\(204\) 0 0
\(205\) 198.000 0.965854
\(206\) − 39.5980i − 0.192223i
\(207\) 0 0
\(208\) 32.0000 0.153846
\(209\) − 271.529i − 1.29918i
\(210\) 0 0
\(211\) 296.000 1.40284 0.701422 0.712746i \(-0.252549\pi\)
0.701422 + 0.712746i \(0.252549\pi\)
\(212\) − 76.3675i − 0.360224i
\(213\) 0 0
\(214\) 0 0
\(215\) 169.706i 0.789328i
\(216\) 0 0
\(217\) −176.000 −0.811060
\(218\) 79.1960i 0.363284i
\(219\) 0 0
\(220\) −144.000 −0.654545
\(221\) − 101.823i − 0.460739i
\(222\) 0 0
\(223\) −436.000 −1.95516 −0.977578 0.210571i \(-0.932468\pi\)
−0.977578 + 0.210571i \(0.932468\pi\)
\(224\) − 22.6274i − 0.101015i
\(225\) 0 0
\(226\) 222.000 0.982301
\(227\) 16.9706i 0.0747602i 0.999301 + 0.0373801i \(0.0119012\pi\)
−0.999301 + 0.0373801i \(0.988099\pi\)
\(228\) 0 0
\(229\) 8.00000 0.0349345 0.0174672 0.999847i \(-0.494440\pi\)
0.0174672 + 0.999847i \(0.494440\pi\)
\(230\) − 101.823i − 0.442710i
\(231\) 0 0
\(232\) 12.0000 0.0517241
\(233\) − 12.7279i − 0.0546263i −0.999627 0.0273131i \(-0.991305\pi\)
0.999627 0.0273131i \(-0.00869512\pi\)
\(234\) 0 0
\(235\) −360.000 −1.53191
\(236\) − 67.8823i − 0.287637i
\(237\) 0 0
\(238\) −72.0000 −0.302521
\(239\) 135.765i 0.568052i 0.958817 + 0.284026i \(0.0916703\pi\)
−0.958817 + 0.284026i \(0.908330\pi\)
\(240\) 0 0
\(241\) 32.0000 0.132780 0.0663900 0.997794i \(-0.478852\pi\)
0.0663900 + 0.997794i \(0.478852\pi\)
\(242\) − 236.174i − 0.975924i
\(243\) 0 0
\(244\) −100.000 −0.409836
\(245\) 140.007i 0.571458i
\(246\) 0 0
\(247\) −128.000 −0.518219
\(248\) − 124.451i − 0.501818i
\(249\) 0 0
\(250\) 192.000 0.768000
\(251\) 50.9117i 0.202835i 0.994844 + 0.101418i \(0.0323379\pi\)
−0.994844 + 0.101418i \(0.967662\pi\)
\(252\) 0 0
\(253\) 288.000 1.13834
\(254\) 130.108i 0.512235i
\(255\) 0 0
\(256\) 16.0000 0.0625000
\(257\) − 182.434i − 0.709858i −0.934893 0.354929i \(-0.884505\pi\)
0.934893 0.354929i \(-0.115495\pi\)
\(258\) 0 0
\(259\) 136.000 0.525097
\(260\) 67.8823i 0.261086i
\(261\) 0 0
\(262\) 240.000 0.916031
\(263\) 373.352i 1.41959i 0.704408 + 0.709795i \(0.251213\pi\)
−0.704408 + 0.709795i \(0.748787\pi\)
\(264\) 0 0
\(265\) 162.000 0.611321
\(266\) 90.5097i 0.340262i
\(267\) 0 0
\(268\) −16.0000 −0.0597015
\(269\) − 343.654i − 1.27752i −0.769404 0.638762i \(-0.779447\pi\)
0.769404 0.638762i \(-0.220553\pi\)
\(270\) 0 0
\(271\) 380.000 1.40221 0.701107 0.713056i \(-0.252690\pi\)
0.701107 + 0.713056i \(0.252690\pi\)
\(272\) − 50.9117i − 0.187175i
\(273\) 0 0
\(274\) −222.000 −0.810219
\(275\) 118.794i 0.431978i
\(276\) 0 0
\(277\) −328.000 −1.18412 −0.592058 0.805896i \(-0.701684\pi\)
−0.592058 + 0.805896i \(0.701684\pi\)
\(278\) 214.960i 0.773239i
\(279\) 0 0
\(280\) 48.0000 0.171429
\(281\) 284.257i 1.01159i 0.862654 + 0.505795i \(0.168801\pi\)
−0.862654 + 0.505795i \(0.831199\pi\)
\(282\) 0 0
\(283\) −208.000 −0.734982 −0.367491 0.930027i \(-0.619783\pi\)
−0.367491 + 0.930027i \(0.619783\pi\)
\(284\) 101.823i 0.358533i
\(285\) 0 0
\(286\) −192.000 −0.671329
\(287\) − 186.676i − 0.650440i
\(288\) 0 0
\(289\) 127.000 0.439446
\(290\) 25.4558i 0.0877788i
\(291\) 0 0
\(292\) 32.0000 0.109589
\(293\) − 436.992i − 1.49144i −0.666259 0.745720i \(-0.732106\pi\)
0.666259 0.745720i \(-0.267894\pi\)
\(294\) 0 0
\(295\) 144.000 0.488136
\(296\) 96.1665i 0.324887i
\(297\) 0 0
\(298\) −390.000 −1.30872
\(299\) − 135.765i − 0.454062i
\(300\) 0 0
\(301\) 160.000 0.531561
\(302\) − 209.304i − 0.693058i
\(303\) 0 0
\(304\) −64.0000 −0.210526
\(305\) − 212.132i − 0.695515i
\(306\) 0 0
\(307\) −520.000 −1.69381 −0.846906 0.531743i \(-0.821537\pi\)
−0.846906 + 0.531743i \(0.821537\pi\)
\(308\) 135.765i 0.440794i
\(309\) 0 0
\(310\) 264.000 0.851613
\(311\) − 373.352i − 1.20049i −0.799816 0.600245i \(-0.795070\pi\)
0.799816 0.600245i \(-0.204930\pi\)
\(312\) 0 0
\(313\) −94.0000 −0.300319 −0.150160 0.988662i \(-0.547979\pi\)
−0.150160 + 0.988662i \(0.547979\pi\)
\(314\) − 115.966i − 0.369317i
\(315\) 0 0
\(316\) 152.000 0.481013
\(317\) 335.169i 1.05731i 0.848835 + 0.528657i \(0.177304\pi\)
−0.848835 + 0.528657i \(0.822696\pi\)
\(318\) 0 0
\(319\) −72.0000 −0.225705
\(320\) 33.9411i 0.106066i
\(321\) 0 0
\(322\) −96.0000 −0.298137
\(323\) 203.647i 0.630485i
\(324\) 0 0
\(325\) 56.0000 0.172308
\(326\) 79.1960i 0.242932i
\(327\) 0 0
\(328\) 132.000 0.402439
\(329\) 339.411i 1.03165i
\(330\) 0 0
\(331\) 536.000 1.61934 0.809668 0.586889i \(-0.199647\pi\)
0.809668 + 0.586889i \(0.199647\pi\)
\(332\) − 237.588i − 0.715626i
\(333\) 0 0
\(334\) −48.0000 −0.143713
\(335\) − 33.9411i − 0.101317i
\(336\) 0 0
\(337\) −208.000 −0.617211 −0.308605 0.951190i \(-0.599862\pi\)
−0.308605 + 0.951190i \(0.599862\pi\)
\(338\) − 148.492i − 0.439327i
\(339\) 0 0
\(340\) 108.000 0.317647
\(341\) 746.705i 2.18975i
\(342\) 0 0
\(343\) 328.000 0.956268
\(344\) 113.137i 0.328887i
\(345\) 0 0
\(346\) 246.000 0.710983
\(347\) 288.500i 0.831411i 0.909499 + 0.415705i \(0.136465\pi\)
−0.909499 + 0.415705i \(0.863535\pi\)
\(348\) 0 0
\(349\) −238.000 −0.681948 −0.340974 0.940073i \(-0.610757\pi\)
−0.340974 + 0.940073i \(0.610757\pi\)
\(350\) − 39.5980i − 0.113137i
\(351\) 0 0
\(352\) −96.0000 −0.272727
\(353\) − 224.860i − 0.636997i −0.947923 0.318499i \(-0.896821\pi\)
0.947923 0.318499i \(-0.103179\pi\)
\(354\) 0 0
\(355\) −216.000 −0.608451
\(356\) − 25.4558i − 0.0715052i
\(357\) 0 0
\(358\) 288.000 0.804469
\(359\) − 560.029i − 1.55997i −0.625799 0.779984i \(-0.715227\pi\)
0.625799 0.779984i \(-0.284773\pi\)
\(360\) 0 0
\(361\) −105.000 −0.290859
\(362\) − 328.098i − 0.906347i
\(363\) 0 0
\(364\) 64.0000 0.175824
\(365\) 67.8823i 0.185979i
\(366\) 0 0
\(367\) 284.000 0.773842 0.386921 0.922113i \(-0.373539\pi\)
0.386921 + 0.922113i \(0.373539\pi\)
\(368\) − 67.8823i − 0.184463i
\(369\) 0 0
\(370\) −204.000 −0.551351
\(371\) − 152.735i − 0.411685i
\(372\) 0 0
\(373\) −190.000 −0.509383 −0.254692 0.967022i \(-0.581974\pi\)
−0.254692 + 0.967022i \(0.581974\pi\)
\(374\) 305.470i 0.816765i
\(375\) 0 0
\(376\) −240.000 −0.638298
\(377\) 33.9411i 0.0900295i
\(378\) 0 0
\(379\) −160.000 −0.422164 −0.211082 0.977468i \(-0.567699\pi\)
−0.211082 + 0.977468i \(0.567699\pi\)
\(380\) − 135.765i − 0.357275i
\(381\) 0 0
\(382\) 48.0000 0.125654
\(383\) − 271.529i − 0.708953i −0.935065 0.354477i \(-0.884659\pi\)
0.935065 0.354477i \(-0.115341\pi\)
\(384\) 0 0
\(385\) −288.000 −0.748052
\(386\) 291.328i 0.754736i
\(387\) 0 0
\(388\) −352.000 −0.907216
\(389\) − 403.051i − 1.03612i −0.855344 0.518060i \(-0.826654\pi\)
0.855344 0.518060i \(-0.173346\pi\)
\(390\) 0 0
\(391\) −216.000 −0.552430
\(392\) 93.3381i 0.238107i
\(393\) 0 0
\(394\) −234.000 −0.593909
\(395\) 322.441i 0.816306i
\(396\) 0 0
\(397\) 146.000 0.367758 0.183879 0.982949i \(-0.441135\pi\)
0.183879 + 0.982949i \(0.441135\pi\)
\(398\) 28.2843i 0.0710660i
\(399\) 0 0
\(400\) 28.0000 0.0700000
\(401\) 326.683i 0.814672i 0.913278 + 0.407336i \(0.133542\pi\)
−0.913278 + 0.407336i \(0.866458\pi\)
\(402\) 0 0
\(403\) 352.000 0.873449
\(404\) − 59.3970i − 0.147022i
\(405\) 0 0
\(406\) 24.0000 0.0591133
\(407\) − 576.999i − 1.41769i
\(408\) 0 0
\(409\) 368.000 0.899756 0.449878 0.893090i \(-0.351468\pi\)
0.449878 + 0.893090i \(0.351468\pi\)
\(410\) 280.014i 0.682962i
\(411\) 0 0
\(412\) 56.0000 0.135922
\(413\) − 135.765i − 0.328728i
\(414\) 0 0
\(415\) 504.000 1.21446
\(416\) 45.2548i 0.108786i
\(417\) 0 0
\(418\) 384.000 0.918660
\(419\) 390.323i 0.931558i 0.884901 + 0.465779i \(0.154226\pi\)
−0.884901 + 0.465779i \(0.845774\pi\)
\(420\) 0 0
\(421\) −40.0000 −0.0950119 −0.0475059 0.998871i \(-0.515127\pi\)
−0.0475059 + 0.998871i \(0.515127\pi\)
\(422\) 418.607i 0.991960i
\(423\) 0 0
\(424\) 108.000 0.254717
\(425\) − 89.0955i − 0.209636i
\(426\) 0 0
\(427\) −200.000 −0.468384
\(428\) 0 0
\(429\) 0 0
\(430\) −240.000 −0.558140
\(431\) − 152.735i − 0.354374i −0.984177 0.177187i \(-0.943300\pi\)
0.984177 0.177187i \(-0.0566997\pi\)
\(432\) 0 0
\(433\) 542.000 1.25173 0.625866 0.779931i \(-0.284746\pi\)
0.625866 + 0.779931i \(0.284746\pi\)
\(434\) − 248.902i − 0.573506i
\(435\) 0 0
\(436\) −112.000 −0.256881
\(437\) 271.529i 0.621348i
\(438\) 0 0
\(439\) −4.00000 −0.00911162 −0.00455581 0.999990i \(-0.501450\pi\)
−0.00455581 + 0.999990i \(0.501450\pi\)
\(440\) − 203.647i − 0.462834i
\(441\) 0 0
\(442\) 144.000 0.325792
\(443\) 322.441i 0.727857i 0.931427 + 0.363929i \(0.118565\pi\)
−0.931427 + 0.363929i \(0.881435\pi\)
\(444\) 0 0
\(445\) 54.0000 0.121348
\(446\) − 616.597i − 1.38250i
\(447\) 0 0
\(448\) 32.0000 0.0714286
\(449\) 216.375i 0.481904i 0.970537 + 0.240952i \(0.0774596\pi\)
−0.970537 + 0.240952i \(0.922540\pi\)
\(450\) 0 0
\(451\) −792.000 −1.75610
\(452\) 313.955i 0.694592i
\(453\) 0 0
\(454\) −24.0000 −0.0528634
\(455\) 135.765i 0.298384i
\(456\) 0 0
\(457\) −400.000 −0.875274 −0.437637 0.899152i \(-0.644184\pi\)
−0.437637 + 0.899152i \(0.644184\pi\)
\(458\) 11.3137i 0.0247024i
\(459\) 0 0
\(460\) 144.000 0.313043
\(461\) − 301.227i − 0.653422i −0.945124 0.326711i \(-0.894060\pi\)
0.945124 0.326711i \(-0.105940\pi\)
\(462\) 0 0
\(463\) −604.000 −1.30454 −0.652268 0.757989i \(-0.726182\pi\)
−0.652268 + 0.757989i \(0.726182\pi\)
\(464\) 16.9706i 0.0365745i
\(465\) 0 0
\(466\) 18.0000 0.0386266
\(467\) 356.382i 0.763130i 0.924342 + 0.381565i \(0.124615\pi\)
−0.924342 + 0.381565i \(0.875385\pi\)
\(468\) 0 0
\(469\) −32.0000 −0.0682303
\(470\) − 509.117i − 1.08323i
\(471\) 0 0
\(472\) 96.0000 0.203390
\(473\) − 678.823i − 1.43514i
\(474\) 0 0
\(475\) −112.000 −0.235789
\(476\) − 101.823i − 0.213915i
\(477\) 0 0
\(478\) −192.000 −0.401674
\(479\) 526.087i 1.09830i 0.835723 + 0.549152i \(0.185049\pi\)
−0.835723 + 0.549152i \(0.814951\pi\)
\(480\) 0 0
\(481\) −272.000 −0.565489
\(482\) 45.2548i 0.0938897i
\(483\) 0 0
\(484\) 334.000 0.690083
\(485\) − 746.705i − 1.53960i
\(486\) 0 0
\(487\) 596.000 1.22382 0.611910 0.790928i \(-0.290402\pi\)
0.611910 + 0.790928i \(0.290402\pi\)
\(488\) − 141.421i − 0.289798i
\(489\) 0 0
\(490\) −198.000 −0.404082
\(491\) − 271.529i − 0.553012i −0.961012 0.276506i \(-0.910823\pi\)
0.961012 0.276506i \(-0.0891766\pi\)
\(492\) 0 0
\(493\) 54.0000 0.109533
\(494\) − 181.019i − 0.366436i
\(495\) 0 0
\(496\) 176.000 0.354839
\(497\) 203.647i 0.409752i
\(498\) 0 0
\(499\) 224.000 0.448898 0.224449 0.974486i \(-0.427942\pi\)
0.224449 + 0.974486i \(0.427942\pi\)
\(500\) 271.529i 0.543058i
\(501\) 0 0
\(502\) −72.0000 −0.143426
\(503\) 865.499i 1.72067i 0.509726 + 0.860337i \(0.329747\pi\)
−0.509726 + 0.860337i \(0.670253\pi\)
\(504\) 0 0
\(505\) 126.000 0.249505
\(506\) 407.294i 0.804928i
\(507\) 0 0
\(508\) −184.000 −0.362205
\(509\) 479.418i 0.941883i 0.882164 + 0.470941i \(0.156086\pi\)
−0.882164 + 0.470941i \(0.843914\pi\)
\(510\) 0 0
\(511\) 64.0000 0.125245
\(512\) 22.6274i 0.0441942i
\(513\) 0 0
\(514\) 258.000 0.501946
\(515\) 118.794i 0.230668i
\(516\) 0 0
\(517\) 1440.00 2.78530
\(518\) 192.333i 0.371299i
\(519\) 0 0
\(520\) −96.0000 −0.184615
\(521\) − 521.845i − 1.00162i −0.865557 0.500811i \(-0.833035\pi\)
0.865557 0.500811i \(-0.166965\pi\)
\(522\) 0 0
\(523\) −736.000 −1.40727 −0.703633 0.710564i \(-0.748440\pi\)
−0.703633 + 0.710564i \(0.748440\pi\)
\(524\) 339.411i 0.647731i
\(525\) 0 0
\(526\) −528.000 −1.00380
\(527\) − 560.029i − 1.06267i
\(528\) 0 0
\(529\) 241.000 0.455577
\(530\) 229.103i 0.432269i
\(531\) 0 0
\(532\) −128.000 −0.240602
\(533\) 373.352i 0.700474i
\(534\) 0 0
\(535\) 0 0
\(536\) − 22.6274i − 0.0422153i
\(537\) 0 0
\(538\) 486.000 0.903346
\(539\) − 560.029i − 1.03901i
\(540\) 0 0
\(541\) −808.000 −1.49353 −0.746765 0.665088i \(-0.768394\pi\)
−0.746765 + 0.665088i \(0.768394\pi\)
\(542\) 537.401i 0.991515i
\(543\) 0 0
\(544\) 72.0000 0.132353
\(545\) − 237.588i − 0.435941i
\(546\) 0 0
\(547\) 536.000 0.979890 0.489945 0.871753i \(-0.337017\pi\)
0.489945 + 0.871753i \(0.337017\pi\)
\(548\) − 313.955i − 0.572911i
\(549\) 0 0
\(550\) −168.000 −0.305455
\(551\) − 67.8823i − 0.123198i
\(552\) 0 0
\(553\) 304.000 0.549729
\(554\) − 463.862i − 0.837296i
\(555\) 0 0
\(556\) −304.000 −0.546763
\(557\) − 165.463i − 0.297061i −0.988908 0.148531i \(-0.952546\pi\)
0.988908 0.148531i \(-0.0474543\pi\)
\(558\) 0 0
\(559\) −320.000 −0.572451
\(560\) 67.8823i 0.121218i
\(561\) 0 0
\(562\) −402.000 −0.715302
\(563\) − 322.441i − 0.572719i −0.958122 0.286359i \(-0.907555\pi\)
0.958122 0.286359i \(-0.0924451\pi\)
\(564\) 0 0
\(565\) −666.000 −1.17876
\(566\) − 294.156i − 0.519711i
\(567\) 0 0
\(568\) −144.000 −0.253521
\(569\) 156.978i 0.275883i 0.990440 + 0.137942i \(0.0440487\pi\)
−0.990440 + 0.137942i \(0.955951\pi\)
\(570\) 0 0
\(571\) 368.000 0.644483 0.322242 0.946657i \(-0.395564\pi\)
0.322242 + 0.946657i \(0.395564\pi\)
\(572\) − 271.529i − 0.474701i
\(573\) 0 0
\(574\) 264.000 0.459930
\(575\) − 118.794i − 0.206598i
\(576\) 0 0
\(577\) −142.000 −0.246101 −0.123050 0.992400i \(-0.539268\pi\)
−0.123050 + 0.992400i \(0.539268\pi\)
\(578\) 179.605i 0.310736i
\(579\) 0 0
\(580\) −36.0000 −0.0620690
\(581\) − 475.176i − 0.817858i
\(582\) 0 0
\(583\) −648.000 −1.11149
\(584\) 45.2548i 0.0774912i
\(585\) 0 0
\(586\) 618.000 1.05461
\(587\) 373.352i 0.636035i 0.948085 + 0.318017i \(0.103017\pi\)
−0.948085 + 0.318017i \(0.896983\pi\)
\(588\) 0 0
\(589\) −704.000 −1.19525
\(590\) 203.647i 0.345164i
\(591\) 0 0
\(592\) −136.000 −0.229730
\(593\) 1107.33i 1.86733i 0.358142 + 0.933667i \(0.383410\pi\)
−0.358142 + 0.933667i \(0.616590\pi\)
\(594\) 0 0
\(595\) 216.000 0.363025
\(596\) − 551.543i − 0.925408i
\(597\) 0 0
\(598\) 192.000 0.321070
\(599\) 797.616i 1.33158i 0.746139 + 0.665790i \(0.231905\pi\)
−0.746139 + 0.665790i \(0.768095\pi\)
\(600\) 0 0
\(601\) 158.000 0.262895 0.131448 0.991323i \(-0.458037\pi\)
0.131448 + 0.991323i \(0.458037\pi\)
\(602\) 226.274i 0.375871i
\(603\) 0 0
\(604\) 296.000 0.490066
\(605\) 708.521i 1.17111i
\(606\) 0 0
\(607\) 332.000 0.546952 0.273476 0.961879i \(-0.411827\pi\)
0.273476 + 0.961879i \(0.411827\pi\)
\(608\) − 90.5097i − 0.148865i
\(609\) 0 0
\(610\) 300.000 0.491803
\(611\) − 678.823i − 1.11100i
\(612\) 0 0
\(613\) 578.000 0.942904 0.471452 0.881892i \(-0.343730\pi\)
0.471452 + 0.881892i \(0.343730\pi\)
\(614\) − 735.391i − 1.19771i
\(615\) 0 0
\(616\) −192.000 −0.311688
\(617\) − 55.1543i − 0.0893911i −0.999001 0.0446956i \(-0.985768\pi\)
0.999001 0.0446956i \(-0.0142318\pi\)
\(618\) 0 0
\(619\) 896.000 1.44750 0.723748 0.690064i \(-0.242418\pi\)
0.723748 + 0.690064i \(0.242418\pi\)
\(620\) 373.352i 0.602181i
\(621\) 0 0
\(622\) 528.000 0.848875
\(623\) − 50.9117i − 0.0817202i
\(624\) 0 0
\(625\) −401.000 −0.641600
\(626\) − 132.936i − 0.212358i
\(627\) 0 0
\(628\) 164.000 0.261146
\(629\) 432.749i 0.687996i
\(630\) 0 0
\(631\) 20.0000 0.0316957 0.0158479 0.999874i \(-0.494955\pi\)
0.0158479 + 0.999874i \(0.494955\pi\)
\(632\) 214.960i 0.340127i
\(633\) 0 0
\(634\) −474.000 −0.747634
\(635\) − 390.323i − 0.614682i
\(636\) 0 0
\(637\) −264.000 −0.414443
\(638\) − 101.823i − 0.159598i
\(639\) 0 0
\(640\) −48.0000 −0.0750000
\(641\) 258.801i 0.403746i 0.979412 + 0.201873i \(0.0647028\pi\)
−0.979412 + 0.201873i \(0.935297\pi\)
\(642\) 0 0
\(643\) 728.000 1.13219 0.566096 0.824339i \(-0.308453\pi\)
0.566096 + 0.824339i \(0.308453\pi\)
\(644\) − 135.765i − 0.210814i
\(645\) 0 0
\(646\) −288.000 −0.445820
\(647\) − 458.205i − 0.708200i −0.935208 0.354100i \(-0.884787\pi\)
0.935208 0.354100i \(-0.115213\pi\)
\(648\) 0 0
\(649\) −576.000 −0.887519
\(650\) 79.1960i 0.121840i
\(651\) 0 0
\(652\) −112.000 −0.171779
\(653\) 301.227i 0.461298i 0.973037 + 0.230649i \(0.0740849\pi\)
−0.973037 + 0.230649i \(0.925915\pi\)
\(654\) 0 0
\(655\) −720.000 −1.09924
\(656\) 186.676i 0.284567i
\(657\) 0 0
\(658\) −480.000 −0.729483
\(659\) − 1052.17i − 1.59662i −0.602244 0.798312i \(-0.705727\pi\)
0.602244 0.798312i \(-0.294273\pi\)
\(660\) 0 0
\(661\) 62.0000 0.0937973 0.0468986 0.998900i \(-0.485066\pi\)
0.0468986 + 0.998900i \(0.485066\pi\)
\(662\) 758.018i 1.14504i
\(663\) 0 0
\(664\) 336.000 0.506024
\(665\) − 271.529i − 0.408314i
\(666\) 0 0
\(667\) 72.0000 0.107946
\(668\) − 67.8823i − 0.101620i
\(669\) 0 0
\(670\) 48.0000 0.0716418
\(671\) 848.528i 1.26457i
\(672\) 0 0
\(673\) −670.000 −0.995542 −0.497771 0.867308i \(-0.665848\pi\)
−0.497771 + 0.867308i \(0.665848\pi\)
\(674\) − 294.156i − 0.436434i
\(675\) 0 0
\(676\) 210.000 0.310651
\(677\) − 1294.01i − 1.91138i −0.294372 0.955691i \(-0.595111\pi\)
0.294372 0.955691i \(-0.404889\pi\)
\(678\) 0 0
\(679\) −704.000 −1.03682
\(680\) 152.735i 0.224610i
\(681\) 0 0
\(682\) −1056.00 −1.54839
\(683\) − 560.029i − 0.819954i −0.912096 0.409977i \(-0.865537\pi\)
0.912096 0.409977i \(-0.134463\pi\)
\(684\) 0 0
\(685\) 666.000 0.972263
\(686\) 463.862i 0.676184i
\(687\) 0 0
\(688\) −160.000 −0.232558
\(689\) 305.470i 0.443353i
\(690\) 0 0
\(691\) −40.0000 −0.0578871 −0.0289436 0.999581i \(-0.509214\pi\)
−0.0289436 + 0.999581i \(0.509214\pi\)
\(692\) 347.897i 0.502741i
\(693\) 0 0
\(694\) −408.000 −0.587896
\(695\) − 644.881i − 0.927887i
\(696\) 0 0
\(697\) 594.000 0.852224
\(698\) − 336.583i − 0.482210i
\(699\) 0 0
\(700\) 56.0000 0.0800000
\(701\) 954.594i 1.36176i 0.732395 + 0.680880i \(0.238403\pi\)
−0.732395 + 0.680880i \(0.761597\pi\)
\(702\) 0 0
\(703\) 544.000 0.773826
\(704\) − 135.765i − 0.192847i
\(705\) 0 0
\(706\) 318.000 0.450425
\(707\) − 118.794i − 0.168025i
\(708\) 0 0
\(709\) 968.000 1.36530 0.682652 0.730744i \(-0.260827\pi\)
0.682652 + 0.730744i \(0.260827\pi\)
\(710\) − 305.470i − 0.430240i
\(711\) 0 0
\(712\) 36.0000 0.0505618
\(713\) − 746.705i − 1.04727i
\(714\) 0 0
\(715\) 576.000 0.805594
\(716\) 407.294i 0.568846i
\(717\) 0 0
\(718\) 792.000 1.10306
\(719\) 1170.97i 1.62861i 0.580439 + 0.814304i \(0.302881\pi\)
−0.580439 + 0.814304i \(0.697119\pi\)
\(720\) 0 0
\(721\) 112.000 0.155340
\(722\) − 148.492i − 0.205668i
\(723\) 0 0
\(724\) 464.000 0.640884
\(725\) 29.6985i 0.0409634i
\(726\) 0 0
\(727\) −508.000 −0.698762 −0.349381 0.936981i \(-0.613608\pi\)
−0.349381 + 0.936981i \(0.613608\pi\)
\(728\) 90.5097i 0.124326i
\(729\) 0 0
\(730\) −96.0000 −0.131507
\(731\) 509.117i 0.696466i
\(732\) 0 0
\(733\) −1144.00 −1.56071 −0.780355 0.625337i \(-0.784961\pi\)
−0.780355 + 0.625337i \(0.784961\pi\)
\(734\) 401.637i 0.547189i
\(735\) 0 0
\(736\) 96.0000 0.130435
\(737\) 135.765i 0.184212i
\(738\) 0 0
\(739\) −304.000 −0.411367 −0.205683 0.978619i \(-0.565942\pi\)
−0.205683 + 0.978619i \(0.565942\pi\)
\(740\) − 288.500i − 0.389864i
\(741\) 0 0
\(742\) 216.000 0.291105
\(743\) 848.528i 1.14203i 0.820940 + 0.571015i \(0.193450\pi\)
−0.820940 + 0.571015i \(0.806550\pi\)
\(744\) 0 0
\(745\) 1170.00 1.57047
\(746\) − 268.701i − 0.360188i
\(747\) 0 0
\(748\) −432.000 −0.577540
\(749\) 0 0
\(750\) 0 0
\(751\) 188.000 0.250333 0.125166 0.992136i \(-0.460054\pi\)
0.125166 + 0.992136i \(0.460054\pi\)
\(752\) − 339.411i − 0.451345i
\(753\) 0 0
\(754\) −48.0000 −0.0636605
\(755\) 627.911i 0.831670i
\(756\) 0 0
\(757\) −1240.00 −1.63804 −0.819022 0.573761i \(-0.805484\pi\)
−0.819022 + 0.573761i \(0.805484\pi\)
\(758\) − 226.274i − 0.298515i
\(759\) 0 0
\(760\) 192.000 0.252632
\(761\) − 156.978i − 0.206278i −0.994667 0.103139i \(-0.967111\pi\)
0.994667 0.103139i \(-0.0328887\pi\)
\(762\) 0 0
\(763\) −224.000 −0.293578
\(764\) 67.8823i 0.0888511i
\(765\) 0 0
\(766\) 384.000 0.501305
\(767\) 271.529i 0.354014i
\(768\) 0 0
\(769\) −910.000 −1.18336 −0.591678 0.806175i \(-0.701534\pi\)
−0.591678 + 0.806175i \(0.701534\pi\)
\(770\) − 407.294i − 0.528953i
\(771\) 0 0
\(772\) −412.000 −0.533679
\(773\) − 1387.34i − 1.79475i −0.441266 0.897376i \(-0.645471\pi\)
0.441266 0.897376i \(-0.354529\pi\)
\(774\) 0 0
\(775\) 308.000 0.397419
\(776\) − 497.803i − 0.641499i
\(777\) 0 0
\(778\) 570.000 0.732648
\(779\) − 746.705i − 0.958543i
\(780\) 0 0
\(781\) 864.000 1.10627
\(782\) − 305.470i − 0.390627i
\(783\) 0 0
\(784\) −132.000 −0.168367
\(785\) 347.897i 0.443180i
\(786\) 0 0
\(787\) −1360.00 −1.72808 −0.864041 0.503422i \(-0.832074\pi\)
−0.864041 + 0.503422i \(0.832074\pi\)
\(788\) − 330.926i − 0.419957i
\(789\) 0 0
\(790\) −456.000 −0.577215
\(791\) 627.911i 0.793819i
\(792\) 0 0
\(793\) 400.000 0.504414
\(794\) 206.475i 0.260044i
\(795\) 0 0
\(796\) −40.0000 −0.0502513
\(797\) − 106.066i − 0.133082i −0.997784 0.0665408i \(-0.978804\pi\)
0.997784 0.0665408i \(-0.0211963\pi\)
\(798\) 0 0
\(799\) −1080.00 −1.35169
\(800\) 39.5980i 0.0494975i
\(801\) 0 0
\(802\) −462.000 −0.576060
\(803\) − 271.529i − 0.338143i
\(804\) 0 0
\(805\) 288.000 0.357764
\(806\) 497.803i 0.617622i
\(807\) 0 0
\(808\) 84.0000 0.103960
\(809\) − 1107.33i − 1.36876i −0.729124 0.684381i \(-0.760072\pi\)
0.729124 0.684381i \(-0.239928\pi\)
\(810\) 0 0
\(811\) −160.000 −0.197287 −0.0986436 0.995123i \(-0.531450\pi\)
−0.0986436 + 0.995123i \(0.531450\pi\)
\(812\) 33.9411i 0.0417994i
\(813\) 0 0
\(814\) 816.000 1.00246
\(815\) − 237.588i − 0.291519i
\(816\) 0 0
\(817\) 640.000 0.783354
\(818\) 520.431i 0.636223i
\(819\) 0 0
\(820\) −396.000 −0.482927
\(821\) 436.992i 0.532268i 0.963936 + 0.266134i \(0.0857464\pi\)
−0.963936 + 0.266134i \(0.914254\pi\)
\(822\) 0 0
\(823\) 332.000 0.403402 0.201701 0.979447i \(-0.435353\pi\)
0.201701 + 0.979447i \(0.435353\pi\)
\(824\) 79.1960i 0.0961116i
\(825\) 0 0
\(826\) 192.000 0.232446
\(827\) 101.823i 0.123124i 0.998103 + 0.0615619i \(0.0196082\pi\)
−0.998103 + 0.0615619i \(0.980392\pi\)
\(828\) 0 0
\(829\) 632.000 0.762364 0.381182 0.924500i \(-0.375517\pi\)
0.381182 + 0.924500i \(0.375517\pi\)
\(830\) 712.764i 0.858751i
\(831\) 0 0
\(832\) −64.0000 −0.0769231
\(833\) 420.021i 0.504227i
\(834\) 0 0
\(835\) 144.000 0.172455
\(836\) 543.058i 0.649591i
\(837\) 0 0
\(838\) −552.000 −0.658711
\(839\) 729.734i 0.869767i 0.900487 + 0.434883i \(0.143210\pi\)
−0.900487 + 0.434883i \(0.856790\pi\)
\(840\) 0 0
\(841\) 823.000 0.978597
\(842\) − 56.5685i − 0.0671835i
\(843\) 0 0
\(844\) −592.000 −0.701422
\(845\) 445.477i 0.527192i
\(846\) 0 0
\(847\) 668.000 0.788666
\(848\) 152.735i 0.180112i
\(849\) 0 0
\(850\) 126.000 0.148235
\(851\) 576.999i 0.678025i
\(852\) 0 0
\(853\) 446.000 0.522860 0.261430 0.965222i \(-0.415806\pi\)
0.261430 + 0.965222i \(0.415806\pi\)
\(854\) − 282.843i − 0.331198i
\(855\) 0 0
\(856\) 0 0
\(857\) − 428.507i − 0.500008i −0.968245 0.250004i \(-0.919568\pi\)
0.968245 0.250004i \(-0.0804319\pi\)
\(858\) 0 0
\(859\) 728.000 0.847497 0.423749 0.905780i \(-0.360714\pi\)
0.423749 + 0.905780i \(0.360714\pi\)
\(860\) − 339.411i − 0.394664i
\(861\) 0 0
\(862\) 216.000 0.250580
\(863\) − 916.410i − 1.06189i −0.847407 0.530945i \(-0.821837\pi\)
0.847407 0.530945i \(-0.178163\pi\)
\(864\) 0 0
\(865\) −738.000 −0.853179
\(866\) 766.504i 0.885108i
\(867\) 0 0
\(868\) 352.000 0.405530
\(869\) − 1289.76i − 1.48419i
\(870\) 0 0
\(871\) 64.0000 0.0734788
\(872\) − 158.392i − 0.181642i
\(873\) 0 0
\(874\) −384.000 −0.439359
\(875\) 543.058i 0.620638i
\(876\) 0 0
\(877\) −910.000 −1.03763 −0.518814 0.854887i \(-0.673626\pi\)
−0.518814 + 0.854887i \(0.673626\pi\)
\(878\) − 5.65685i − 0.00644289i
\(879\) 0 0
\(880\) 288.000 0.327273
\(881\) − 929.138i − 1.05464i −0.849667 0.527320i \(-0.823197\pi\)
0.849667 0.527320i \(-0.176803\pi\)
\(882\) 0 0
\(883\) 1064.00 1.20498 0.602492 0.798125i \(-0.294175\pi\)
0.602492 + 0.798125i \(0.294175\pi\)
\(884\) 203.647i 0.230370i
\(885\) 0 0
\(886\) −456.000 −0.514673
\(887\) − 1391.59i − 1.56887i −0.620212 0.784434i \(-0.712953\pi\)
0.620212 0.784434i \(-0.287047\pi\)
\(888\) 0 0
\(889\) −368.000 −0.413948
\(890\) 76.3675i 0.0858062i
\(891\) 0 0
\(892\) 872.000 0.977578
\(893\) 1357.65i 1.52032i
\(894\) 0 0
\(895\) −864.000 −0.965363
\(896\) 45.2548i 0.0505076i
\(897\) 0 0
\(898\) −306.000 −0.340757
\(899\) 186.676i 0.207649i
\(900\) 0 0
\(901\) 486.000 0.539401
\(902\) − 1120.06i − 1.24175i
\(903\) 0 0
\(904\) −444.000 −0.491150
\(905\) 984.293i 1.08762i
\(906\) 0 0
\(907\) −1768.00 −1.94928 −0.974642 0.223771i \(-0.928163\pi\)
−0.974642 + 0.223771i \(0.928163\pi\)
\(908\) − 33.9411i − 0.0373801i
\(909\) 0 0
\(910\) −192.000 −0.210989
\(911\) − 237.588i − 0.260799i −0.991462 0.130399i \(-0.958374\pi\)
0.991462 0.130399i \(-0.0416260\pi\)
\(912\) 0 0
\(913\) −2016.00 −2.20811
\(914\) − 565.685i − 0.618912i
\(915\) 0 0
\(916\) −16.0000 −0.0174672
\(917\) 678.823i 0.740264i
\(918\) 0 0
\(919\) 380.000 0.413493 0.206746 0.978395i \(-0.433712\pi\)
0.206746 + 0.978395i \(0.433712\pi\)
\(920\) 203.647i 0.221355i
\(921\) 0 0
\(922\) 426.000 0.462039
\(923\) − 407.294i − 0.441271i
\(924\) 0 0
\(925\) −238.000 −0.257297
\(926\) − 854.185i − 0.922446i
\(927\) 0 0
\(928\) −24.0000 −0.0258621
\(929\) 666.095i 0.717002i 0.933529 + 0.358501i \(0.116712\pi\)
−0.933529 + 0.358501i \(0.883288\pi\)
\(930\) 0 0
\(931\) 528.000 0.567132
\(932\) 25.4558i 0.0273131i
\(933\) 0 0
\(934\) −504.000 −0.539615
\(935\) − 916.410i − 0.980118i
\(936\) 0 0
\(937\) −178.000 −0.189968 −0.0949840 0.995479i \(-0.530280\pi\)
−0.0949840 + 0.995479i \(0.530280\pi\)
\(938\) − 45.2548i − 0.0482461i
\(939\) 0 0
\(940\) 720.000 0.765957
\(941\) − 436.992i − 0.464391i −0.972669 0.232196i \(-0.925409\pi\)
0.972669 0.232196i \(-0.0745909\pi\)
\(942\) 0 0
\(943\) 792.000 0.839873
\(944\) 135.765i 0.143818i
\(945\) 0 0
\(946\) 960.000 1.01480
\(947\) 1798.88i 1.89956i 0.312924 + 0.949778i \(0.398691\pi\)
−0.312924 + 0.949778i \(0.601309\pi\)
\(948\) 0 0
\(949\) −128.000 −0.134879
\(950\) − 158.392i − 0.166728i
\(951\) 0 0
\(952\) 144.000 0.151261
\(953\) 1310.98i 1.37563i 0.725886 + 0.687815i \(0.241430\pi\)
−0.725886 + 0.687815i \(0.758570\pi\)
\(954\) 0 0
\(955\) −144.000 −0.150785
\(956\) − 271.529i − 0.284026i
\(957\) 0 0
\(958\) −744.000 −0.776618
\(959\) − 627.911i − 0.654756i
\(960\) 0 0
\(961\) 975.000 1.01457
\(962\) − 384.666i − 0.399861i
\(963\) 0 0
\(964\) −64.0000 −0.0663900
\(965\) − 873.984i − 0.905683i
\(966\) 0 0
\(967\) 1700.00 1.75801 0.879007 0.476808i \(-0.158206\pi\)
0.879007 + 0.476808i \(0.158206\pi\)
\(968\) 472.347i 0.487962i
\(969\) 0 0
\(970\) 1056.00 1.08866
\(971\) 458.205i 0.471890i 0.971766 + 0.235945i \(0.0758185\pi\)
−0.971766 + 0.235945i \(0.924181\pi\)
\(972\) 0 0
\(973\) −608.000 −0.624872
\(974\) 842.871i 0.865371i
\(975\) 0 0
\(976\) 200.000 0.204918
\(977\) 759.433i 0.777311i 0.921383 + 0.388655i \(0.127060\pi\)
−0.921383 + 0.388655i \(0.872940\pi\)
\(978\) 0 0
\(979\) −216.000 −0.220633
\(980\) − 280.014i − 0.285729i
\(981\) 0 0
\(982\) 384.000 0.391039
\(983\) − 1052.17i − 1.07037i −0.844734 0.535186i \(-0.820242\pi\)
0.844734 0.535186i \(-0.179758\pi\)
\(984\) 0 0
\(985\) 702.000 0.712690
\(986\) 76.3675i 0.0774519i
\(987\) 0 0
\(988\) 256.000 0.259109
\(989\) 678.823i 0.686373i
\(990\) 0 0
\(991\) −772.000 −0.779011 −0.389506 0.921024i \(-0.627354\pi\)
−0.389506 + 0.921024i \(0.627354\pi\)
\(992\) 248.902i 0.250909i
\(993\) 0 0
\(994\) −288.000 −0.289738
\(995\) − 84.8528i − 0.0852792i
\(996\) 0 0
\(997\) 194.000 0.194584 0.0972919 0.995256i \(-0.468982\pi\)
0.0972919 + 0.995256i \(0.468982\pi\)
\(998\) 316.784i 0.317419i
\(999\) 0 0
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 18.3.b.a.17.2 yes 2
3.2 odd 2 inner 18.3.b.a.17.1 2
4.3 odd 2 144.3.e.b.17.1 2
5.2 odd 4 450.3.b.b.449.1 4
5.3 odd 4 450.3.b.b.449.4 4
5.4 even 2 450.3.d.f.251.1 2
7.2 even 3 882.3.s.b.557.2 4
7.3 odd 6 882.3.s.d.863.1 4
7.4 even 3 882.3.s.b.863.1 4
7.5 odd 6 882.3.s.d.557.2 4
7.6 odd 2 882.3.b.a.197.2 2
8.3 odd 2 576.3.e.f.449.2 2
8.5 even 2 576.3.e.c.449.2 2
9.2 odd 6 162.3.d.b.53.1 4
9.4 even 3 162.3.d.b.107.1 4
9.5 odd 6 162.3.d.b.107.2 4
9.7 even 3 162.3.d.b.53.2 4
11.10 odd 2 2178.3.c.d.485.1 2
12.11 even 2 144.3.e.b.17.2 2
13.5 odd 4 3042.3.d.a.3041.3 4
13.8 odd 4 3042.3.d.a.3041.2 4
13.12 even 2 3042.3.c.e.1691.1 2
15.2 even 4 450.3.b.b.449.3 4
15.8 even 4 450.3.b.b.449.2 4
15.14 odd 2 450.3.d.f.251.2 2
16.3 odd 4 2304.3.h.c.2177.2 4
16.5 even 4 2304.3.h.f.2177.3 4
16.11 odd 4 2304.3.h.c.2177.3 4
16.13 even 4 2304.3.h.f.2177.2 4
20.3 even 4 3600.3.c.b.449.1 4
20.7 even 4 3600.3.c.b.449.3 4
20.19 odd 2 3600.3.l.d.1601.1 2
21.2 odd 6 882.3.s.b.557.1 4
21.5 even 6 882.3.s.d.557.1 4
21.11 odd 6 882.3.s.b.863.2 4
21.17 even 6 882.3.s.d.863.2 4
21.20 even 2 882.3.b.a.197.1 2
24.5 odd 2 576.3.e.c.449.1 2
24.11 even 2 576.3.e.f.449.1 2
33.32 even 2 2178.3.c.d.485.2 2
36.7 odd 6 1296.3.q.f.1025.2 4
36.11 even 6 1296.3.q.f.1025.1 4
36.23 even 6 1296.3.q.f.593.2 4
36.31 odd 6 1296.3.q.f.593.1 4
39.5 even 4 3042.3.d.a.3041.1 4
39.8 even 4 3042.3.d.a.3041.4 4
39.38 odd 2 3042.3.c.e.1691.2 2
48.5 odd 4 2304.3.h.f.2177.1 4
48.11 even 4 2304.3.h.c.2177.1 4
48.29 odd 4 2304.3.h.f.2177.4 4
48.35 even 4 2304.3.h.c.2177.4 4
60.23 odd 4 3600.3.c.b.449.2 4
60.47 odd 4 3600.3.c.b.449.4 4
60.59 even 2 3600.3.l.d.1601.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
18.3.b.a.17.1 2 3.2 odd 2 inner
18.3.b.a.17.2 yes 2 1.1 even 1 trivial
144.3.e.b.17.1 2 4.3 odd 2
144.3.e.b.17.2 2 12.11 even 2
162.3.d.b.53.1 4 9.2 odd 6
162.3.d.b.53.2 4 9.7 even 3
162.3.d.b.107.1 4 9.4 even 3
162.3.d.b.107.2 4 9.5 odd 6
450.3.b.b.449.1 4 5.2 odd 4
450.3.b.b.449.2 4 15.8 even 4
450.3.b.b.449.3 4 15.2 even 4
450.3.b.b.449.4 4 5.3 odd 4
450.3.d.f.251.1 2 5.4 even 2
450.3.d.f.251.2 2 15.14 odd 2
576.3.e.c.449.1 2 24.5 odd 2
576.3.e.c.449.2 2 8.5 even 2
576.3.e.f.449.1 2 24.11 even 2
576.3.e.f.449.2 2 8.3 odd 2
882.3.b.a.197.1 2 21.20 even 2
882.3.b.a.197.2 2 7.6 odd 2
882.3.s.b.557.1 4 21.2 odd 6
882.3.s.b.557.2 4 7.2 even 3
882.3.s.b.863.1 4 7.4 even 3
882.3.s.b.863.2 4 21.11 odd 6
882.3.s.d.557.1 4 21.5 even 6
882.3.s.d.557.2 4 7.5 odd 6
882.3.s.d.863.1 4 7.3 odd 6
882.3.s.d.863.2 4 21.17 even 6
1296.3.q.f.593.1 4 36.31 odd 6
1296.3.q.f.593.2 4 36.23 even 6
1296.3.q.f.1025.1 4 36.11 even 6
1296.3.q.f.1025.2 4 36.7 odd 6
2178.3.c.d.485.1 2 11.10 odd 2
2178.3.c.d.485.2 2 33.32 even 2
2304.3.h.c.2177.1 4 48.11 even 4
2304.3.h.c.2177.2 4 16.3 odd 4
2304.3.h.c.2177.3 4 16.11 odd 4
2304.3.h.c.2177.4 4 48.35 even 4
2304.3.h.f.2177.1 4 48.5 odd 4
2304.3.h.f.2177.2 4 16.13 even 4
2304.3.h.f.2177.3 4 16.5 even 4
2304.3.h.f.2177.4 4 48.29 odd 4
3042.3.c.e.1691.1 2 13.12 even 2
3042.3.c.e.1691.2 2 39.38 odd 2
3042.3.d.a.3041.1 4 39.5 even 4
3042.3.d.a.3041.2 4 13.8 odd 4
3042.3.d.a.3041.3 4 13.5 odd 4
3042.3.d.a.3041.4 4 39.8 even 4
3600.3.c.b.449.1 4 20.3 even 4
3600.3.c.b.449.2 4 60.23 odd 4
3600.3.c.b.449.3 4 20.7 even 4
3600.3.c.b.449.4 4 60.47 odd 4
3600.3.l.d.1601.1 2 20.19 odd 2
3600.3.l.d.1601.2 2 60.59 even 2