Properties

Label 162.3.b.a.161.1
Level $162$
Weight $3$
Character 162.161
Analytic conductor $4.414$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 162 = 2 \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 162.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.41418028264\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-3})\)
Defining polynomial: \( x^{4} - 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 18)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 161.1
Root \(-1.22474 + 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 162.161
Dual form 162.3.b.a.161.4

$q$-expansion

\(f(q)\) \(=\) \(q-1.41421i q^{2} -2.00000 q^{4} -5.19615i q^{5} -8.34847 q^{7} +2.82843i q^{8} +O(q^{10})\) \(q-1.41421i q^{2} -2.00000 q^{4} -5.19615i q^{5} -8.34847 q^{7} +2.82843i q^{8} -7.34847 q^{10} -0.953512i q^{11} -9.69694 q^{13} +11.8065i q^{14} +4.00000 q^{16} -18.8776i q^{17} -24.6969 q^{19} +10.3923i q^{20} -1.34847 q^{22} +0.953512i q^{23} -2.00000 q^{25} +13.7135i q^{26} +16.6969 q^{28} -13.6814i q^{29} +3.04541 q^{31} -5.65685i q^{32} -26.6969 q^{34} +43.3799i q^{35} +46.6969 q^{37} +34.9267i q^{38} +14.6969 q^{40} -10.9172i q^{41} +45.0454 q^{43} +1.90702i q^{44} +1.34847 q^{46} -45.2869i q^{47} +20.6969 q^{49} +2.82843i q^{50} +19.3939 q^{52} -94.3879i q^{53} -4.95459 q^{55} -23.6130i q^{56} -19.3485 q^{58} -18.7813i q^{59} +13.0908 q^{61} -4.30686i q^{62} -8.00000 q^{64} +50.3868i q^{65} +75.0454 q^{67} +37.7552i q^{68} +61.3485 q^{70} +18.0204i q^{71} -7.90918 q^{73} -66.0394i q^{74} +49.3939 q^{76} +7.96036i q^{77} -43.7423 q^{79} -20.7846i q^{80} -15.4393 q^{82} +130.332i q^{83} -98.0908 q^{85} -63.7038i q^{86} +2.69694 q^{88} +145.300i q^{89} +80.9546 q^{91} -1.90702i q^{92} -64.0454 q^{94} +128.329i q^{95} -109.879 q^{97} -29.2699i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 8 q^{4} - 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 8 q^{4} - 4 q^{7} + 20 q^{13} + 16 q^{16} - 40 q^{19} + 24 q^{22} - 8 q^{25} + 8 q^{28} - 76 q^{31} - 48 q^{34} + 128 q^{37} + 92 q^{43} - 24 q^{46} + 24 q^{49} - 40 q^{52} - 108 q^{55} - 48 q^{58} - 124 q^{61} - 32 q^{64} + 212 q^{67} + 216 q^{70} - 208 q^{73} + 80 q^{76} - 28 q^{79} + 144 q^{82} - 216 q^{85} - 48 q^{88} + 412 q^{91} - 168 q^{94} - 28 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/162\mathbb{Z}\right)^\times\).

\(n\) \(83\)
\(\chi(n)\) \(-1\)

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
\(5\) − 5.19615i − 1.03923i −0.854400 0.519615i \(-0.826075\pi\)
0.854400 0.519615i \(-0.173925\pi\)
\(6\) 0 0
\(7\) −8.34847 −1.19264 −0.596319 0.802747i \(-0.703371\pi\)
−0.596319 + 0.802747i \(0.703371\pi\)
\(8\) 2.82843i 0.353553i
\(9\) 0 0
\(10\) −7.34847 −0.734847
\(11\) − 0.953512i − 0.0866829i −0.999060 0.0433414i \(-0.986200\pi\)
0.999060 0.0433414i \(-0.0138003\pi\)
\(12\) 0 0
\(13\) −9.69694 −0.745918 −0.372959 0.927848i \(-0.621657\pi\)
−0.372959 + 0.927848i \(0.621657\pi\)
\(14\) 11.8065i 0.843323i
\(15\) 0 0
\(16\) 4.00000 0.250000
\(17\) − 18.8776i − 1.11045i −0.831701 0.555223i \(-0.812633\pi\)
0.831701 0.555223i \(-0.187367\pi\)
\(18\) 0 0
\(19\) −24.6969 −1.29984 −0.649919 0.760003i \(-0.725197\pi\)
−0.649919 + 0.760003i \(0.725197\pi\)
\(20\) 10.3923i 0.519615i
\(21\) 0 0
\(22\) −1.34847 −0.0612941
\(23\) 0.953512i 0.0414570i 0.999785 + 0.0207285i \(0.00659856\pi\)
−0.999785 + 0.0207285i \(0.993401\pi\)
\(24\) 0 0
\(25\) −2.00000 −0.0800000
\(26\) 13.7135i 0.527444i
\(27\) 0 0
\(28\) 16.6969 0.596319
\(29\) − 13.6814i − 0.471774i −0.971781 0.235887i \(-0.924201\pi\)
0.971781 0.235887i \(-0.0757995\pi\)
\(30\) 0 0
\(31\) 3.04541 0.0982390 0.0491195 0.998793i \(-0.484358\pi\)
0.0491195 + 0.998793i \(0.484358\pi\)
\(32\) − 5.65685i − 0.176777i
\(33\) 0 0
\(34\) −26.6969 −0.785204
\(35\) 43.3799i 1.23943i
\(36\) 0 0
\(37\) 46.6969 1.26208 0.631040 0.775751i \(-0.282628\pi\)
0.631040 + 0.775751i \(0.282628\pi\)
\(38\) 34.9267i 0.919125i
\(39\) 0 0
\(40\) 14.6969 0.367423
\(41\) − 10.9172i − 0.266274i −0.991098 0.133137i \(-0.957495\pi\)
0.991098 0.133137i \(-0.0425050\pi\)
\(42\) 0 0
\(43\) 45.0454 1.04757 0.523784 0.851851i \(-0.324520\pi\)
0.523784 + 0.851851i \(0.324520\pi\)
\(44\) 1.90702i 0.0433414i
\(45\) 0 0
\(46\) 1.34847 0.0293145
\(47\) − 45.2869i − 0.963552i −0.876294 0.481776i \(-0.839992\pi\)
0.876294 0.481776i \(-0.160008\pi\)
\(48\) 0 0
\(49\) 20.6969 0.422386
\(50\) 2.82843i 0.0565685i
\(51\) 0 0
\(52\) 19.3939 0.372959
\(53\) − 94.3879i − 1.78090i −0.455077 0.890452i \(-0.650388\pi\)
0.455077 0.890452i \(-0.349612\pi\)
\(54\) 0 0
\(55\) −4.95459 −0.0900835
\(56\) − 23.6130i − 0.421661i
\(57\) 0 0
\(58\) −19.3485 −0.333594
\(59\) − 18.7813i − 0.318326i −0.987252 0.159163i \(-0.949120\pi\)
0.987252 0.159163i \(-0.0508796\pi\)
\(60\) 0 0
\(61\) 13.0908 0.214604 0.107302 0.994226i \(-0.465779\pi\)
0.107302 + 0.994226i \(0.465779\pi\)
\(62\) − 4.30686i − 0.0694654i
\(63\) 0 0
\(64\) −8.00000 −0.125000
\(65\) 50.3868i 0.775181i
\(66\) 0 0
\(67\) 75.0454 1.12008 0.560040 0.828465i \(-0.310786\pi\)
0.560040 + 0.828465i \(0.310786\pi\)
\(68\) 37.7552i 0.555223i
\(69\) 0 0
\(70\) 61.3485 0.876407
\(71\) 18.0204i 0.253808i 0.991915 + 0.126904i \(0.0405041\pi\)
−0.991915 + 0.126904i \(0.959496\pi\)
\(72\) 0 0
\(73\) −7.90918 −0.108345 −0.0541725 0.998532i \(-0.517252\pi\)
−0.0541725 + 0.998532i \(0.517252\pi\)
\(74\) − 66.0394i − 0.892425i
\(75\) 0 0
\(76\) 49.3939 0.649919
\(77\) 7.96036i 0.103381i
\(78\) 0 0
\(79\) −43.7423 −0.553701 −0.276850 0.960913i \(-0.589291\pi\)
−0.276850 + 0.960913i \(0.589291\pi\)
\(80\) − 20.7846i − 0.259808i
\(81\) 0 0
\(82\) −15.4393 −0.188284
\(83\) 130.332i 1.57027i 0.619325 + 0.785135i \(0.287406\pi\)
−0.619325 + 0.785135i \(0.712594\pi\)
\(84\) 0 0
\(85\) −98.0908 −1.15401
\(86\) − 63.7038i − 0.740742i
\(87\) 0 0
\(88\) 2.69694 0.0306470
\(89\) 145.300i 1.63258i 0.577642 + 0.816290i \(0.303973\pi\)
−0.577642 + 0.816290i \(0.696027\pi\)
\(90\) 0 0
\(91\) 80.9546 0.889611
\(92\) − 1.90702i − 0.0207285i
\(93\) 0 0
\(94\) −64.0454 −0.681334
\(95\) 128.329i 1.35083i
\(96\) 0 0
\(97\) −109.879 −1.13277 −0.566384 0.824141i \(-0.691658\pi\)
−0.566384 + 0.824141i \(0.691658\pi\)
\(98\) − 29.2699i − 0.298672i
\(99\) 0 0
\(100\) 4.00000 0.0400000
\(101\) − 147.539i − 1.46078i −0.683030 0.730391i \(-0.739338\pi\)
0.683030 0.730391i \(-0.260662\pi\)
\(102\) 0 0
\(103\) −103.136 −1.00132 −0.500661 0.865643i \(-0.666910\pi\)
−0.500661 + 0.865643i \(0.666910\pi\)
\(104\) − 27.4271i − 0.263722i
\(105\) 0 0
\(106\) −133.485 −1.25929
\(107\) 36.0408i 0.336830i 0.985716 + 0.168415i \(0.0538649\pi\)
−0.985716 + 0.168415i \(0.946135\pi\)
\(108\) 0 0
\(109\) −148.272 −1.36030 −0.680149 0.733074i \(-0.738085\pi\)
−0.680149 + 0.733074i \(0.738085\pi\)
\(110\) 7.00685i 0.0636987i
\(111\) 0 0
\(112\) −33.3939 −0.298160
\(113\) − 171.088i − 1.51405i −0.653385 0.757025i \(-0.726652\pi\)
0.653385 0.757025i \(-0.273348\pi\)
\(114\) 0 0
\(115\) 4.95459 0.0430834
\(116\) 27.3629i 0.235887i
\(117\) 0 0
\(118\) −26.5607 −0.225091
\(119\) 157.599i 1.32436i
\(120\) 0 0
\(121\) 120.091 0.992486
\(122\) − 18.5132i − 0.151748i
\(123\) 0 0
\(124\) −6.09082 −0.0491195
\(125\) − 119.512i − 0.956092i
\(126\) 0 0
\(127\) −78.0908 −0.614888 −0.307444 0.951566i \(-0.599474\pi\)
−0.307444 + 0.951566i \(0.599474\pi\)
\(128\) 11.3137i 0.0883883i
\(129\) 0 0
\(130\) 71.2577 0.548136
\(131\) 234.063i 1.78674i 0.449323 + 0.893369i \(0.351665\pi\)
−0.449323 + 0.893369i \(0.648335\pi\)
\(132\) 0 0
\(133\) 206.182 1.55024
\(134\) − 106.130i − 0.792017i
\(135\) 0 0
\(136\) 53.3939 0.392602
\(137\) − 149.831i − 1.09366i −0.837244 0.546829i \(-0.815835\pi\)
0.837244 0.546829i \(-0.184165\pi\)
\(138\) 0 0
\(139\) −84.5301 −0.608130 −0.304065 0.952651i \(-0.598344\pi\)
−0.304065 + 0.952651i \(0.598344\pi\)
\(140\) − 86.7598i − 0.619713i
\(141\) 0 0
\(142\) 25.4847 0.179470
\(143\) 9.24614i 0.0646584i
\(144\) 0 0
\(145\) −71.0908 −0.490281
\(146\) 11.1853i 0.0766115i
\(147\) 0 0
\(148\) −93.3939 −0.631040
\(149\) 115.505i 0.775200i 0.921828 + 0.387600i \(0.126696\pi\)
−0.921828 + 0.387600i \(0.873304\pi\)
\(150\) 0 0
\(151\) −64.7730 −0.428960 −0.214480 0.976728i \(-0.568806\pi\)
−0.214480 + 0.976728i \(0.568806\pi\)
\(152\) − 69.8535i − 0.459562i
\(153\) 0 0
\(154\) 11.2577 0.0731016
\(155\) − 15.8244i − 0.102093i
\(156\) 0 0
\(157\) −20.8184 −0.132601 −0.0663005 0.997800i \(-0.521120\pi\)
−0.0663005 + 0.997800i \(0.521120\pi\)
\(158\) 61.8610i 0.391525i
\(159\) 0 0
\(160\) −29.3939 −0.183712
\(161\) − 7.96036i − 0.0494433i
\(162\) 0 0
\(163\) 133.060 0.816320 0.408160 0.912910i \(-0.366171\pi\)
0.408160 + 0.912910i \(0.366171\pi\)
\(164\) 21.8344i 0.133137i
\(165\) 0 0
\(166\) 184.318 1.11035
\(167\) − 294.510i − 1.76353i −0.471688 0.881765i \(-0.656355\pi\)
0.471688 0.881765i \(-0.343645\pi\)
\(168\) 0 0
\(169\) −74.9694 −0.443606
\(170\) 138.721i 0.816008i
\(171\) 0 0
\(172\) −90.0908 −0.523784
\(173\) 69.2644i 0.400372i 0.979758 + 0.200186i \(0.0641546\pi\)
−0.979758 + 0.200186i \(0.935845\pi\)
\(174\) 0 0
\(175\) 16.6969 0.0954111
\(176\) − 3.81405i − 0.0216707i
\(177\) 0 0
\(178\) 205.485 1.15441
\(179\) 47.4829i 0.265268i 0.991165 + 0.132634i \(0.0423435\pi\)
−0.991165 + 0.132634i \(0.957657\pi\)
\(180\) 0 0
\(181\) 242.879 1.34187 0.670935 0.741516i \(-0.265893\pi\)
0.670935 + 0.741516i \(0.265893\pi\)
\(182\) − 114.487i − 0.629050i
\(183\) 0 0
\(184\) −2.69694 −0.0146573
\(185\) − 242.644i − 1.31159i
\(186\) 0 0
\(187\) −18.0000 −0.0962567
\(188\) 90.5739i 0.481776i
\(189\) 0 0
\(190\) 181.485 0.955183
\(191\) − 7.53177i − 0.0394333i −0.999806 0.0197167i \(-0.993724\pi\)
0.999806 0.0197167i \(-0.00627642\pi\)
\(192\) 0 0
\(193\) 345.454 1.78992 0.894959 0.446149i \(-0.147205\pi\)
0.894959 + 0.446149i \(0.147205\pi\)
\(194\) 155.392i 0.800988i
\(195\) 0 0
\(196\) −41.3939 −0.211193
\(197\) 77.2247i 0.392004i 0.980604 + 0.196002i \(0.0627959\pi\)
−0.980604 + 0.196002i \(0.937204\pi\)
\(198\) 0 0
\(199\) 153.485 0.771280 0.385640 0.922649i \(-0.373981\pi\)
0.385640 + 0.922649i \(0.373981\pi\)
\(200\) − 5.65685i − 0.0282843i
\(201\) 0 0
\(202\) −208.652 −1.03293
\(203\) 114.219i 0.562655i
\(204\) 0 0
\(205\) −56.7276 −0.276720
\(206\) 145.857i 0.708042i
\(207\) 0 0
\(208\) −38.7878 −0.186480
\(209\) 23.5488i 0.112674i
\(210\) 0 0
\(211\) −51.5607 −0.244364 −0.122182 0.992508i \(-0.538989\pi\)
−0.122182 + 0.992508i \(0.538989\pi\)
\(212\) 188.776i 0.890452i
\(213\) 0 0
\(214\) 50.9694 0.238175
\(215\) − 234.063i − 1.08866i
\(216\) 0 0
\(217\) −25.4245 −0.117164
\(218\) 209.689i 0.961876i
\(219\) 0 0
\(220\) 9.90918 0.0450417
\(221\) 183.055i 0.828302i
\(222\) 0 0
\(223\) 313.227 1.40461 0.702303 0.711878i \(-0.252155\pi\)
0.702303 + 0.711878i \(0.252155\pi\)
\(224\) 47.2261i 0.210831i
\(225\) 0 0
\(226\) −241.955 −1.07060
\(227\) 76.2712i 0.335997i 0.985787 + 0.167998i \(0.0537303\pi\)
−0.985787 + 0.167998i \(0.946270\pi\)
\(228\) 0 0
\(229\) −121.545 −0.530764 −0.265382 0.964143i \(-0.585498\pi\)
−0.265382 + 0.964143i \(0.585498\pi\)
\(230\) − 7.00685i − 0.0304646i
\(231\) 0 0
\(232\) 38.6969 0.166797
\(233\) − 151.021i − 0.648157i −0.946030 0.324079i \(-0.894946\pi\)
0.946030 0.324079i \(-0.105054\pi\)
\(234\) 0 0
\(235\) −235.318 −1.00135
\(236\) 37.5625i 0.159163i
\(237\) 0 0
\(238\) 222.879 0.936465
\(239\) − 87.7133i − 0.367001i −0.983020 0.183501i \(-0.941257\pi\)
0.983020 0.183501i \(-0.0587430\pi\)
\(240\) 0 0
\(241\) 201.788 0.837294 0.418647 0.908149i \(-0.362505\pi\)
0.418647 + 0.908149i \(0.362505\pi\)
\(242\) − 169.834i − 0.701794i
\(243\) 0 0
\(244\) −26.1816 −0.107302
\(245\) − 107.544i − 0.438957i
\(246\) 0 0
\(247\) 239.485 0.969574
\(248\) 8.61371i 0.0347327i
\(249\) 0 0
\(250\) −169.015 −0.676059
\(251\) 52.6261i 0.209666i 0.994490 + 0.104833i \(0.0334307\pi\)
−0.994490 + 0.104833i \(0.966569\pi\)
\(252\) 0 0
\(253\) 0.909185 0.00359362
\(254\) 110.437i 0.434792i
\(255\) 0 0
\(256\) 16.0000 0.0625000
\(257\) − 80.7065i − 0.314033i −0.987596 0.157017i \(-0.949812\pi\)
0.987596 0.157017i \(-0.0501876\pi\)
\(258\) 0 0
\(259\) −389.848 −1.50520
\(260\) − 100.774i − 0.387591i
\(261\) 0 0
\(262\) 331.015 1.26342
\(263\) − 463.743i − 1.76328i −0.471920 0.881641i \(-0.656439\pi\)
0.471920 0.881641i \(-0.343561\pi\)
\(264\) 0 0
\(265\) −490.454 −1.85077
\(266\) − 291.585i − 1.09618i
\(267\) 0 0
\(268\) −150.091 −0.560040
\(269\) − 43.4762i − 0.161622i −0.996729 0.0808109i \(-0.974249\pi\)
0.996729 0.0808109i \(-0.0257510\pi\)
\(270\) 0 0
\(271\) −342.636 −1.26434 −0.632169 0.774830i \(-0.717835\pi\)
−0.632169 + 0.774830i \(0.717835\pi\)
\(272\) − 75.5103i − 0.277612i
\(273\) 0 0
\(274\) −211.893 −0.773333
\(275\) 1.90702i 0.00693463i
\(276\) 0 0
\(277\) −49.0000 −0.176895 −0.0884477 0.996081i \(-0.528191\pi\)
−0.0884477 + 0.996081i \(0.528191\pi\)
\(278\) 119.544i 0.430013i
\(279\) 0 0
\(280\) −122.697 −0.438203
\(281\) 20.6450i 0.0734697i 0.999325 + 0.0367349i \(0.0116957\pi\)
−0.999325 + 0.0367349i \(0.988304\pi\)
\(282\) 0 0
\(283\) 53.4087 0.188723 0.0943616 0.995538i \(-0.469919\pi\)
0.0943616 + 0.995538i \(0.469919\pi\)
\(284\) − 36.0408i − 0.126904i
\(285\) 0 0
\(286\) 13.0760 0.0457204
\(287\) 91.1421i 0.317568i
\(288\) 0 0
\(289\) −67.3633 −0.233091
\(290\) 100.538i 0.346681i
\(291\) 0 0
\(292\) 15.8184 0.0541725
\(293\) − 14.9239i − 0.0509349i −0.999676 0.0254674i \(-0.991893\pi\)
0.999676 0.0254674i \(-0.00810741\pi\)
\(294\) 0 0
\(295\) −97.5903 −0.330815
\(296\) 132.079i 0.446212i
\(297\) 0 0
\(298\) 163.348 0.548149
\(299\) − 9.24614i − 0.0309236i
\(300\) 0 0
\(301\) −376.060 −1.24937
\(302\) 91.6028i 0.303321i
\(303\) 0 0
\(304\) −98.7878 −0.324960
\(305\) − 68.0219i − 0.223023i
\(306\) 0 0
\(307\) 65.9092 0.214688 0.107344 0.994222i \(-0.465765\pi\)
0.107344 + 0.994222i \(0.465765\pi\)
\(308\) − 15.9207i − 0.0516907i
\(309\) 0 0
\(310\) −22.3791 −0.0721906
\(311\) − 250.176i − 0.804425i −0.915546 0.402213i \(-0.868241\pi\)
0.915546 0.402213i \(-0.131759\pi\)
\(312\) 0 0
\(313\) −426.394 −1.36228 −0.681140 0.732153i \(-0.738516\pi\)
−0.681140 + 0.732153i \(0.738516\pi\)
\(314\) 29.4416i 0.0937631i
\(315\) 0 0
\(316\) 87.4847 0.276850
\(317\) 463.979i 1.46366i 0.681489 + 0.731829i \(0.261333\pi\)
−0.681489 + 0.731829i \(0.738667\pi\)
\(318\) 0 0
\(319\) −13.0454 −0.0408947
\(320\) 41.5692i 0.129904i
\(321\) 0 0
\(322\) −11.2577 −0.0349617
\(323\) 466.219i 1.44340i
\(324\) 0 0
\(325\) 19.3939 0.0596735
\(326\) − 188.176i − 0.577226i
\(327\) 0 0
\(328\) 30.8786 0.0941420
\(329\) 378.077i 1.14917i
\(330\) 0 0
\(331\) 472.803 1.42841 0.714203 0.699938i \(-0.246789\pi\)
0.714203 + 0.699938i \(0.246789\pi\)
\(332\) − 260.665i − 0.785135i
\(333\) 0 0
\(334\) −416.499 −1.24700
\(335\) − 389.947i − 1.16402i
\(336\) 0 0
\(337\) 305.606 0.906843 0.453422 0.891296i \(-0.350203\pi\)
0.453422 + 0.891296i \(0.350203\pi\)
\(338\) 106.023i 0.313677i
\(339\) 0 0
\(340\) 196.182 0.577005
\(341\) − 2.90383i − 0.00851564i
\(342\) 0 0
\(343\) 236.287 0.688884
\(344\) 127.408i 0.370371i
\(345\) 0 0
\(346\) 97.9546 0.283106
\(347\) 133.675i 0.385229i 0.981274 + 0.192615i \(0.0616967\pi\)
−0.981274 + 0.192615i \(0.938303\pi\)
\(348\) 0 0
\(349\) −98.7571 −0.282972 −0.141486 0.989940i \(-0.545188\pi\)
−0.141486 + 0.989940i \(0.545188\pi\)
\(350\) − 23.6130i − 0.0674658i
\(351\) 0 0
\(352\) −5.39388 −0.0153235
\(353\) 326.115i 0.923839i 0.886922 + 0.461919i \(0.152839\pi\)
−0.886922 + 0.461919i \(0.847161\pi\)
\(354\) 0 0
\(355\) 93.6367 0.263765
\(356\) − 290.599i − 0.816290i
\(357\) 0 0
\(358\) 67.1510 0.187573
\(359\) − 418.736i − 1.16639i −0.812331 0.583197i \(-0.801801\pi\)
0.812331 0.583197i \(-0.198199\pi\)
\(360\) 0 0
\(361\) 248.939 0.689581
\(362\) − 343.482i − 0.948846i
\(363\) 0 0
\(364\) −161.909 −0.444805
\(365\) 41.0973i 0.112595i
\(366\) 0 0
\(367\) 187.227 0.510155 0.255078 0.966921i \(-0.417899\pi\)
0.255078 + 0.966921i \(0.417899\pi\)
\(368\) 3.81405i 0.0103643i
\(369\) 0 0
\(370\) −343.151 −0.927435
\(371\) 787.995i 2.12398i
\(372\) 0 0
\(373\) 451.030 1.20919 0.604597 0.796531i \(-0.293334\pi\)
0.604597 + 0.796531i \(0.293334\pi\)
\(374\) 25.4558i 0.0680638i
\(375\) 0 0
\(376\) 128.091 0.340667
\(377\) 132.668i 0.351905i
\(378\) 0 0
\(379\) −489.666 −1.29200 −0.645998 0.763339i \(-0.723558\pi\)
−0.645998 + 0.763339i \(0.723558\pi\)
\(380\) − 256.658i − 0.675416i
\(381\) 0 0
\(382\) −10.6515 −0.0278836
\(383\) 103.056i 0.269076i 0.990908 + 0.134538i \(0.0429550\pi\)
−0.990908 + 0.134538i \(0.957045\pi\)
\(384\) 0 0
\(385\) 41.3633 0.107437
\(386\) − 488.546i − 1.26566i
\(387\) 0 0
\(388\) 219.757 0.566384
\(389\) − 34.2734i − 0.0881064i −0.999029 0.0440532i \(-0.985973\pi\)
0.999029 0.0440532i \(-0.0140271\pi\)
\(390\) 0 0
\(391\) 18.0000 0.0460358
\(392\) 58.5398i 0.149336i
\(393\) 0 0
\(394\) 109.212 0.277188
\(395\) 227.292i 0.575423i
\(396\) 0 0
\(397\) 8.27245 0.0208374 0.0104187 0.999946i \(-0.496684\pi\)
0.0104187 + 0.999946i \(0.496684\pi\)
\(398\) − 217.060i − 0.545377i
\(399\) 0 0
\(400\) −8.00000 −0.0200000
\(401\) 414.117i 1.03271i 0.856374 + 0.516356i \(0.172712\pi\)
−0.856374 + 0.516356i \(0.827288\pi\)
\(402\) 0 0
\(403\) −29.5311 −0.0732782
\(404\) 295.078i 0.730391i
\(405\) 0 0
\(406\) 161.530 0.397857
\(407\) − 44.5261i − 0.109401i
\(408\) 0 0
\(409\) −326.212 −0.797585 −0.398792 0.917041i \(-0.630571\pi\)
−0.398792 + 0.917041i \(0.630571\pi\)
\(410\) 80.2249i 0.195670i
\(411\) 0 0
\(412\) 206.272 0.500661
\(413\) 156.795i 0.379648i
\(414\) 0 0
\(415\) 677.227 1.63187
\(416\) 54.8542i 0.131861i
\(417\) 0 0
\(418\) 33.3031 0.0796724
\(419\) − 540.775i − 1.29063i −0.763915 0.645317i \(-0.776725\pi\)
0.763915 0.645317i \(-0.223275\pi\)
\(420\) 0 0
\(421\) 283.697 0.673864 0.336932 0.941529i \(-0.390611\pi\)
0.336932 + 0.941529i \(0.390611\pi\)
\(422\) 72.9179i 0.172791i
\(423\) 0 0
\(424\) 266.969 0.629645
\(425\) 37.7552i 0.0888357i
\(426\) 0 0
\(427\) −109.288 −0.255944
\(428\) − 72.0816i − 0.168415i
\(429\) 0 0
\(430\) −331.015 −0.769802
\(431\) − 257.429i − 0.597282i −0.954365 0.298641i \(-0.903467\pi\)
0.954365 0.298641i \(-0.0965334\pi\)
\(432\) 0 0
\(433\) 476.272 1.09994 0.549968 0.835186i \(-0.314640\pi\)
0.549968 + 0.835186i \(0.314640\pi\)
\(434\) 35.9557i 0.0828471i
\(435\) 0 0
\(436\) 296.545 0.680149
\(437\) − 23.5488i − 0.0538875i
\(438\) 0 0
\(439\) −557.863 −1.27076 −0.635379 0.772200i \(-0.719156\pi\)
−0.635379 + 0.772200i \(0.719156\pi\)
\(440\) − 14.0137i − 0.0318493i
\(441\) 0 0
\(442\) 258.879 0.585698
\(443\) − 831.847i − 1.87776i −0.344248 0.938879i \(-0.611866\pi\)
0.344248 0.938879i \(-0.388134\pi\)
\(444\) 0 0
\(445\) 754.999 1.69663
\(446\) − 442.970i − 0.993206i
\(447\) 0 0
\(448\) 66.7878 0.149080
\(449\) 729.927i 1.62567i 0.582492 + 0.812836i \(0.302078\pi\)
−0.582492 + 0.812836i \(0.697922\pi\)
\(450\) 0 0
\(451\) −10.4097 −0.0230814
\(452\) 342.175i 0.757025i
\(453\) 0 0
\(454\) 107.864 0.237585
\(455\) − 420.652i − 0.924511i
\(456\) 0 0
\(457\) 709.636 1.55281 0.776407 0.630232i \(-0.217040\pi\)
0.776407 + 0.630232i \(0.217040\pi\)
\(458\) 171.890i 0.375307i
\(459\) 0 0
\(460\) −9.90918 −0.0215417
\(461\) 9.20285i 0.0199628i 0.999950 + 0.00998140i \(0.00317723\pi\)
−0.999950 + 0.00998140i \(0.996823\pi\)
\(462\) 0 0
\(463\) −55.1975 −0.119217 −0.0596085 0.998222i \(-0.518985\pi\)
−0.0596085 + 0.998222i \(0.518985\pi\)
\(464\) − 54.7257i − 0.117943i
\(465\) 0 0
\(466\) −213.576 −0.458317
\(467\) − 625.811i − 1.34007i −0.742331 0.670033i \(-0.766280\pi\)
0.742331 0.670033i \(-0.233720\pi\)
\(468\) 0 0
\(469\) −626.514 −1.33585
\(470\) 332.790i 0.708063i
\(471\) 0 0
\(472\) 53.1214 0.112545
\(473\) − 42.9513i − 0.0908062i
\(474\) 0 0
\(475\) 49.3939 0.103987
\(476\) − 315.198i − 0.662180i
\(477\) 0 0
\(478\) −124.045 −0.259509
\(479\) − 309.294i − 0.645708i −0.946449 0.322854i \(-0.895358\pi\)
0.946449 0.322854i \(-0.104642\pi\)
\(480\) 0 0
\(481\) −452.817 −0.941408
\(482\) − 285.371i − 0.592056i
\(483\) 0 0
\(484\) −240.182 −0.496243
\(485\) 570.946i 1.17721i
\(486\) 0 0
\(487\) −28.3337 −0.0581800 −0.0290900 0.999577i \(-0.509261\pi\)
−0.0290900 + 0.999577i \(0.509261\pi\)
\(488\) 37.0264i 0.0758738i
\(489\) 0 0
\(490\) −152.091 −0.310389
\(491\) 949.697i 1.93421i 0.254379 + 0.967105i \(0.418129\pi\)
−0.254379 + 0.967105i \(0.581871\pi\)
\(492\) 0 0
\(493\) −258.272 −0.523879
\(494\) − 338.682i − 0.685592i
\(495\) 0 0
\(496\) 12.1816 0.0245597
\(497\) − 150.443i − 0.302702i
\(498\) 0 0
\(499\) −560.226 −1.12270 −0.561349 0.827579i \(-0.689717\pi\)
−0.561349 + 0.827579i \(0.689717\pi\)
\(500\) 239.023i 0.478046i
\(501\) 0 0
\(502\) 74.4245 0.148256
\(503\) − 897.832i − 1.78495i −0.451094 0.892477i \(-0.648966\pi\)
0.451094 0.892477i \(-0.351034\pi\)
\(504\) 0 0
\(505\) −766.635 −1.51809
\(506\) − 1.28578i − 0.00254107i
\(507\) 0 0
\(508\) 156.182 0.307444
\(509\) − 196.823i − 0.386685i −0.981131 0.193343i \(-0.938067\pi\)
0.981131 0.193343i \(-0.0619329\pi\)
\(510\) 0 0
\(511\) 66.0296 0.129216
\(512\) − 22.6274i − 0.0441942i
\(513\) 0 0
\(514\) −114.136 −0.222055
\(515\) 535.912i 1.04060i
\(516\) 0 0
\(517\) −43.1816 −0.0835235
\(518\) 551.328i 1.06434i
\(519\) 0 0
\(520\) −142.515 −0.274068
\(521\) − 375.837i − 0.721377i −0.932686 0.360688i \(-0.882542\pi\)
0.932686 0.360688i \(-0.117458\pi\)
\(522\) 0 0
\(523\) 91.1827 0.174345 0.0871727 0.996193i \(-0.472217\pi\)
0.0871727 + 0.996193i \(0.472217\pi\)
\(524\) − 468.126i − 0.893369i
\(525\) 0 0
\(526\) −655.832 −1.24683
\(527\) − 57.4899i − 0.109089i
\(528\) 0 0
\(529\) 528.091 0.998281
\(530\) 693.607i 1.30869i
\(531\) 0 0
\(532\) −412.363 −0.775119
\(533\) 105.864i 0.198618i
\(534\) 0 0
\(535\) 187.273 0.350044
\(536\) 212.260i 0.396008i
\(537\) 0 0
\(538\) −61.4847 −0.114284
\(539\) − 19.7348i − 0.0366137i
\(540\) 0 0
\(541\) −38.8490 −0.0718096 −0.0359048 0.999355i \(-0.511431\pi\)
−0.0359048 + 0.999355i \(0.511431\pi\)
\(542\) 484.560i 0.894022i
\(543\) 0 0
\(544\) −106.788 −0.196301
\(545\) 770.446i 1.41366i
\(546\) 0 0
\(547\) −466.044 −0.852001 −0.426000 0.904723i \(-0.640078\pi\)
−0.426000 + 0.904723i \(0.640078\pi\)
\(548\) 299.662i 0.546829i
\(549\) 0 0
\(550\) 2.69694 0.00490352
\(551\) 337.890i 0.613230i
\(552\) 0 0
\(553\) 365.182 0.660365
\(554\) 69.2965i 0.125084i
\(555\) 0 0
\(556\) 169.060 0.304065
\(557\) 695.042i 1.24783i 0.781492 + 0.623916i \(0.214459\pi\)
−0.781492 + 0.623916i \(0.785541\pi\)
\(558\) 0 0
\(559\) −436.803 −0.781400
\(560\) 173.520i 0.309857i
\(561\) 0 0
\(562\) 29.1964 0.0519509
\(563\) − 547.074i − 0.971713i −0.874039 0.485857i \(-0.838508\pi\)
0.874039 0.485857i \(-0.161492\pi\)
\(564\) 0 0
\(565\) −888.998 −1.57345
\(566\) − 75.5313i − 0.133447i
\(567\) 0 0
\(568\) −50.9694 −0.0897348
\(569\) − 249.362i − 0.438247i −0.975697 0.219123i \(-0.929680\pi\)
0.975697 0.219123i \(-0.0703197\pi\)
\(570\) 0 0
\(571\) 73.8332 0.129305 0.0646525 0.997908i \(-0.479406\pi\)
0.0646525 + 0.997908i \(0.479406\pi\)
\(572\) − 18.4923i − 0.0323292i
\(573\) 0 0
\(574\) 128.894 0.224555
\(575\) − 1.90702i − 0.00331656i
\(576\) 0 0
\(577\) −43.9092 −0.0760991 −0.0380496 0.999276i \(-0.512114\pi\)
−0.0380496 + 0.999276i \(0.512114\pi\)
\(578\) 95.2660i 0.164820i
\(579\) 0 0
\(580\) 142.182 0.245141
\(581\) − 1088.08i − 1.87276i
\(582\) 0 0
\(583\) −90.0000 −0.154374
\(584\) − 22.3706i − 0.0383057i
\(585\) 0 0
\(586\) −21.1056 −0.0360164
\(587\) 440.387i 0.750234i 0.926978 + 0.375117i \(0.122397\pi\)
−0.926978 + 0.375117i \(0.877603\pi\)
\(588\) 0 0
\(589\) −75.2122 −0.127695
\(590\) 138.014i 0.233921i
\(591\) 0 0
\(592\) 186.788 0.315520
\(593\) − 347.232i − 0.585551i −0.956181 0.292776i \(-0.905421\pi\)
0.956181 0.292776i \(-0.0945789\pi\)
\(594\) 0 0
\(595\) 818.908 1.37632
\(596\) − 231.010i − 0.387600i
\(597\) 0 0
\(598\) −13.0760 −0.0218663
\(599\) − 789.911i − 1.31872i −0.751829 0.659359i \(-0.770828\pi\)
0.751829 0.659359i \(-0.229172\pi\)
\(600\) 0 0
\(601\) −706.909 −1.17622 −0.588111 0.808780i \(-0.700128\pi\)
−0.588111 + 0.808780i \(0.700128\pi\)
\(602\) 531.829i 0.883438i
\(603\) 0 0
\(604\) 129.546 0.214480
\(605\) − 624.010i − 1.03142i
\(606\) 0 0
\(607\) −1193.26 −1.96583 −0.982913 0.184070i \(-0.941073\pi\)
−0.982913 + 0.184070i \(0.941073\pi\)
\(608\) 139.707i 0.229781i
\(609\) 0 0
\(610\) −96.1975 −0.157701
\(611\) 439.145i 0.718731i
\(612\) 0 0
\(613\) 629.181 1.02640 0.513198 0.858270i \(-0.328461\pi\)
0.513198 + 0.858270i \(0.328461\pi\)
\(614\) − 93.2097i − 0.151807i
\(615\) 0 0
\(616\) −22.5153 −0.0365508
\(617\) 192.730i 0.312366i 0.987728 + 0.156183i \(0.0499189\pi\)
−0.987728 + 0.156183i \(0.950081\pi\)
\(618\) 0 0
\(619\) −152.955 −0.247100 −0.123550 0.992338i \(-0.539428\pi\)
−0.123550 + 0.992338i \(0.539428\pi\)
\(620\) 31.6488i 0.0510465i
\(621\) 0 0
\(622\) −353.803 −0.568814
\(623\) − 1213.03i − 1.94708i
\(624\) 0 0
\(625\) −671.000 −1.07360
\(626\) 603.012i 0.963278i
\(627\) 0 0
\(628\) 41.6367 0.0663005
\(629\) − 881.525i − 1.40147i
\(630\) 0 0
\(631\) 44.8786 0.0711229 0.0355615 0.999367i \(-0.488678\pi\)
0.0355615 + 0.999367i \(0.488678\pi\)
\(632\) − 123.722i − 0.195763i
\(633\) 0 0
\(634\) 656.166 1.03496
\(635\) 405.772i 0.639011i
\(636\) 0 0
\(637\) −200.697 −0.315066
\(638\) 18.4490i 0.0289169i
\(639\) 0 0
\(640\) 58.7878 0.0918559
\(641\) 241.455i 0.376685i 0.982103 + 0.188342i \(0.0603115\pi\)
−0.982103 + 0.188342i \(0.939689\pi\)
\(642\) 0 0
\(643\) 791.409 1.23081 0.615403 0.788212i \(-0.288993\pi\)
0.615403 + 0.788212i \(0.288993\pi\)
\(644\) 15.9207i 0.0247216i
\(645\) 0 0
\(646\) 659.333 1.02064
\(647\) 294.028i 0.454448i 0.973842 + 0.227224i \(0.0729650\pi\)
−0.973842 + 0.227224i \(0.927035\pi\)
\(648\) 0 0
\(649\) −17.9082 −0.0275935
\(650\) − 27.4271i − 0.0421955i
\(651\) 0 0
\(652\) −266.120 −0.408160
\(653\) − 768.313i − 1.17659i −0.808647 0.588295i \(-0.799799\pi\)
0.808647 0.588295i \(-0.200201\pi\)
\(654\) 0 0
\(655\) 1216.23 1.85683
\(656\) − 43.6689i − 0.0665684i
\(657\) 0 0
\(658\) 534.681 0.812585
\(659\) 430.939i 0.653928i 0.945037 + 0.326964i \(0.106026\pi\)
−0.945037 + 0.326964i \(0.893974\pi\)
\(660\) 0 0
\(661\) 1012.27 1.53142 0.765712 0.643183i \(-0.222387\pi\)
0.765712 + 0.643183i \(0.222387\pi\)
\(662\) − 668.644i − 1.01004i
\(663\) 0 0
\(664\) −368.636 −0.555174
\(665\) − 1071.35i − 1.61105i
\(666\) 0 0
\(667\) 13.0454 0.0195583
\(668\) 589.019i 0.881765i
\(669\) 0 0
\(670\) −551.469 −0.823088
\(671\) − 12.4822i − 0.0186025i
\(672\) 0 0
\(673\) 563.211 0.836867 0.418433 0.908248i \(-0.362579\pi\)
0.418433 + 0.908248i \(0.362579\pi\)
\(674\) − 432.192i − 0.641235i
\(675\) 0 0
\(676\) 149.939 0.221803
\(677\) 350.136i 0.517187i 0.965986 + 0.258594i \(0.0832591\pi\)
−0.965986 + 0.258594i \(0.916741\pi\)
\(678\) 0 0
\(679\) 917.318 1.35098
\(680\) − 277.443i − 0.408004i
\(681\) 0 0
\(682\) −4.10664 −0.00602146
\(683\) 502.818i 0.736190i 0.929788 + 0.368095i \(0.119990\pi\)
−0.929788 + 0.368095i \(0.880010\pi\)
\(684\) 0 0
\(685\) −778.546 −1.13656
\(686\) − 334.161i − 0.487115i
\(687\) 0 0
\(688\) 180.182 0.261892
\(689\) 915.274i 1.32841i
\(690\) 0 0
\(691\) 376.319 0.544600 0.272300 0.962212i \(-0.412216\pi\)
0.272300 + 0.962212i \(0.412216\pi\)
\(692\) − 138.529i − 0.200186i
\(693\) 0 0
\(694\) 189.044 0.272398
\(695\) 439.231i 0.631987i
\(696\) 0 0
\(697\) −206.091 −0.295683
\(698\) 139.664i 0.200091i
\(699\) 0 0
\(700\) −33.3939 −0.0477055
\(701\) 489.681i 0.698546i 0.937021 + 0.349273i \(0.113571\pi\)
−0.937021 + 0.349273i \(0.886429\pi\)
\(702\) 0 0
\(703\) −1153.27 −1.64050
\(704\) 7.62809i 0.0108354i
\(705\) 0 0
\(706\) 461.196 0.653253
\(707\) 1231.72i 1.74218i
\(708\) 0 0
\(709\) 474.029 0.668588 0.334294 0.942469i \(-0.391502\pi\)
0.334294 + 0.942469i \(0.391502\pi\)
\(710\) − 132.422i − 0.186510i
\(711\) 0 0
\(712\) −410.969 −0.577204
\(713\) 2.90383i 0.00407270i
\(714\) 0 0
\(715\) 48.0444 0.0671949
\(716\) − 94.9659i − 0.132634i
\(717\) 0 0
\(718\) −592.182 −0.824766
\(719\) − 108.122i − 0.150379i −0.997169 0.0751894i \(-0.976044\pi\)
0.997169 0.0751894i \(-0.0239561\pi\)
\(720\) 0 0
\(721\) 861.030 1.19422
\(722\) − 352.053i − 0.487607i
\(723\) 0 0
\(724\) −485.757 −0.670935
\(725\) 27.3629i 0.0377419i
\(726\) 0 0
\(727\) −444.591 −0.611542 −0.305771 0.952105i \(-0.598914\pi\)
−0.305771 + 0.952105i \(0.598914\pi\)
\(728\) 228.974i 0.314525i
\(729\) 0 0
\(730\) 58.1204 0.0796170
\(731\) − 850.349i − 1.16327i
\(732\) 0 0
\(733\) −716.362 −0.977302 −0.488651 0.872479i \(-0.662511\pi\)
−0.488651 + 0.872479i \(0.662511\pi\)
\(734\) − 264.779i − 0.360734i
\(735\) 0 0
\(736\) 5.39388 0.00732864
\(737\) − 71.5567i − 0.0970918i
\(738\) 0 0
\(739\) 933.362 1.26301 0.631504 0.775373i \(-0.282438\pi\)
0.631504 + 0.775373i \(0.282438\pi\)
\(740\) 485.289i 0.655796i
\(741\) 0 0
\(742\) 1114.39 1.50188
\(743\) − 15.9110i − 0.0214145i −0.999943 0.0107073i \(-0.996592\pi\)
0.999943 0.0107073i \(-0.00340829\pi\)
\(744\) 0 0
\(745\) 600.181 0.805612
\(746\) − 637.852i − 0.855030i
\(747\) 0 0
\(748\) 36.0000 0.0481283
\(749\) − 300.885i − 0.401716i
\(750\) 0 0
\(751\) 809.831 1.07834 0.539169 0.842198i \(-0.318739\pi\)
0.539169 + 0.842198i \(0.318739\pi\)
\(752\) − 181.148i − 0.240888i
\(753\) 0 0
\(754\) 187.621 0.248834
\(755\) 336.570i 0.445788i
\(756\) 0 0
\(757\) 689.637 0.911013 0.455506 0.890232i \(-0.349458\pi\)
0.455506 + 0.890232i \(0.349458\pi\)
\(758\) 692.493i 0.913579i
\(759\) 0 0
\(760\) −362.969 −0.477591
\(761\) − 953.082i − 1.25241i −0.779659 0.626204i \(-0.784608\pi\)
0.779659 0.626204i \(-0.215392\pi\)
\(762\) 0 0
\(763\) 1237.85 1.62234
\(764\) 15.0635i 0.0197167i
\(765\) 0 0
\(766\) 145.743 0.190266
\(767\) 182.121i 0.237446i
\(768\) 0 0
\(769\) −656.696 −0.853961 −0.426980 0.904261i \(-0.640423\pi\)
−0.426980 + 0.904261i \(0.640423\pi\)
\(770\) − 58.4965i − 0.0759695i
\(771\) 0 0
\(772\) −690.908 −0.894959
\(773\) − 278.021i − 0.359665i −0.983697 0.179832i \(-0.942445\pi\)
0.983697 0.179832i \(-0.0575555\pi\)
\(774\) 0 0
\(775\) −6.09082 −0.00785912
\(776\) − 310.784i − 0.400494i
\(777\) 0 0
\(778\) −48.4699 −0.0623006
\(779\) 269.622i 0.346113i
\(780\) 0 0
\(781\) 17.1827 0.0220008
\(782\) − 25.4558i − 0.0325522i
\(783\) 0 0
\(784\) 82.7878 0.105597
\(785\) 108.175i 0.137803i
\(786\) 0 0
\(787\) 821.954 1.04441 0.522207 0.852819i \(-0.325109\pi\)
0.522207 + 0.852819i \(0.325109\pi\)
\(788\) − 154.449i − 0.196002i
\(789\) 0 0
\(790\) 321.439 0.406885
\(791\) 1428.32i 1.80572i
\(792\) 0 0
\(793\) −126.941 −0.160077
\(794\) − 11.6990i − 0.0147343i
\(795\) 0 0
\(796\) −306.969 −0.385640
\(797\) 1322.51i 1.65937i 0.558235 + 0.829683i \(0.311479\pi\)
−0.558235 + 0.829683i \(0.688521\pi\)
\(798\) 0 0
\(799\) −854.908 −1.06997
\(800\) 11.3137i 0.0141421i
\(801\) 0 0
\(802\) 585.650 0.730238
\(803\) 7.54150i 0.00939166i
\(804\) 0 0
\(805\) −41.3633 −0.0513829
\(806\) 41.7633i 0.0518155i
\(807\) 0 0
\(808\) 417.303 0.516464
\(809\) 235.681i 0.291324i 0.989334 + 0.145662i \(0.0465311\pi\)
−0.989334 + 0.145662i \(0.953469\pi\)
\(810\) 0 0
\(811\) −587.362 −0.724244 −0.362122 0.932131i \(-0.617948\pi\)
−0.362122 + 0.932131i \(0.617948\pi\)
\(812\) − 228.438i − 0.281328i
\(813\) 0 0
\(814\) −62.9694 −0.0773580
\(815\) − 691.401i − 0.848345i
\(816\) 0 0
\(817\) −1112.48 −1.36167
\(818\) 461.334i 0.563978i
\(819\) 0 0
\(820\) 113.455 0.138360
\(821\) 943.913i 1.14971i 0.818255 + 0.574856i \(0.194942\pi\)
−0.818255 + 0.574856i \(0.805058\pi\)
\(822\) 0 0
\(823\) −1615.74 −1.96323 −0.981617 0.190859i \(-0.938873\pi\)
−0.981617 + 0.190859i \(0.938873\pi\)
\(824\) − 291.713i − 0.354021i
\(825\) 0 0
\(826\) 221.741 0.268452
\(827\) 582.354i 0.704177i 0.935967 + 0.352088i \(0.114528\pi\)
−0.935967 + 0.352088i \(0.885472\pi\)
\(828\) 0 0
\(829\) 877.121 1.05805 0.529024 0.848607i \(-0.322558\pi\)
0.529024 + 0.848607i \(0.322558\pi\)
\(830\) − 957.744i − 1.15391i
\(831\) 0 0
\(832\) 77.5755 0.0932398
\(833\) − 390.708i − 0.469038i
\(834\) 0 0
\(835\) −1530.32 −1.83271
\(836\) − 47.0976i − 0.0563369i
\(837\) 0 0
\(838\) −764.772 −0.912616
\(839\) 1137.12i 1.35533i 0.735370 + 0.677666i \(0.237009\pi\)
−0.735370 + 0.677666i \(0.762991\pi\)
\(840\) 0 0
\(841\) 653.818 0.777430
\(842\) − 401.208i − 0.476494i
\(843\) 0 0
\(844\) 103.121 0.122182
\(845\) 389.552i 0.461009i
\(846\) 0 0
\(847\) −1002.57 −1.18368
\(848\) − 377.552i − 0.445226i
\(849\) 0 0
\(850\) 53.3939 0.0628163
\(851\) 44.5261i 0.0523221i
\(852\) 0 0
\(853\) −319.817 −0.374932 −0.187466 0.982271i \(-0.560028\pi\)
−0.187466 + 0.982271i \(0.560028\pi\)
\(854\) 154.557i 0.180980i
\(855\) 0 0
\(856\) −101.939 −0.119087
\(857\) − 797.968i − 0.931118i −0.885017 0.465559i \(-0.845853\pi\)
0.885017 0.465559i \(-0.154147\pi\)
\(858\) 0 0
\(859\) −467.802 −0.544588 −0.272294 0.962214i \(-0.587782\pi\)
−0.272294 + 0.962214i \(0.587782\pi\)
\(860\) 468.126i 0.544332i
\(861\) 0 0
\(862\) −364.059 −0.422342
\(863\) − 1304.85i − 1.51199i −0.654578 0.755994i \(-0.727154\pi\)
0.654578 0.755994i \(-0.272846\pi\)
\(864\) 0 0
\(865\) 359.908 0.416079
\(866\) − 673.551i − 0.777772i
\(867\) 0 0
\(868\) 50.8490 0.0585818
\(869\) 41.7088i 0.0479964i
\(870\) 0 0
\(871\) −727.711 −0.835489
\(872\) − 419.378i − 0.480938i
\(873\) 0 0
\(874\) −33.3031 −0.0381042
\(875\) 997.738i 1.14027i
\(876\) 0 0
\(877\) −373.756 −0.426176 −0.213088 0.977033i \(-0.568352\pi\)
−0.213088 + 0.977033i \(0.568352\pi\)
\(878\) 788.937i 0.898562i
\(879\) 0 0
\(880\) −19.8184 −0.0225209
\(881\) 229.979i 0.261043i 0.991445 + 0.130522i \(0.0416652\pi\)
−0.991445 + 0.130522i \(0.958335\pi\)
\(882\) 0 0
\(883\) −1381.79 −1.56488 −0.782439 0.622728i \(-0.786024\pi\)
−0.782439 + 0.622728i \(0.786024\pi\)
\(884\) − 366.110i − 0.414151i
\(885\) 0 0
\(886\) −1176.41 −1.32778
\(887\) 876.180i 0.987802i 0.869518 + 0.493901i \(0.164429\pi\)
−0.869518 + 0.493901i \(0.835571\pi\)
\(888\) 0 0
\(889\) 651.939 0.733339
\(890\) − 1067.73i − 1.19970i
\(891\) 0 0
\(892\) −626.454 −0.702303
\(893\) 1118.45i 1.25246i
\(894\) 0 0
\(895\) 246.729 0.275674
\(896\) − 94.4521i − 0.105415i
\(897\) 0 0
\(898\) 1032.27 1.14952
\(899\) − 41.6655i − 0.0463465i
\(900\) 0 0
\(901\) −1781.82 −1.97760
\(902\) 14.7215i 0.0163210i
\(903\) 0 0
\(904\) 483.909 0.535298
\(905\) − 1262.03i − 1.39451i
\(906\) 0 0
\(907\) 1180.07 1.30107 0.650537 0.759475i \(-0.274544\pi\)
0.650537 + 0.759475i \(0.274544\pi\)
\(908\) − 152.542i − 0.167998i
\(909\) 0 0
\(910\) −594.892 −0.653728
\(911\) − 1270.32i − 1.39442i −0.716867 0.697210i \(-0.754424\pi\)
0.716867 0.697210i \(-0.245576\pi\)
\(912\) 0 0
\(913\) 124.273 0.136116
\(914\) − 1003.58i − 1.09800i
\(915\) 0 0
\(916\) 243.090 0.265382
\(917\) − 1954.07i − 2.13093i
\(918\) 0 0
\(919\) 1316.63 1.43268 0.716340 0.697751i \(-0.245816\pi\)
0.716340 + 0.697751i \(0.245816\pi\)
\(920\) 14.0137i 0.0152323i
\(921\) 0 0
\(922\) 13.0148 0.0141158
\(923\) − 174.743i − 0.189320i
\(924\) 0 0
\(925\) −93.3939 −0.100966
\(926\) 78.0610i 0.0842991i
\(927\) 0 0
\(928\) −77.3939 −0.0833986
\(929\) − 627.492i − 0.675449i −0.941245 0.337724i \(-0.890343\pi\)
0.941245 0.337724i \(-0.109657\pi\)
\(930\) 0 0
\(931\) −511.151 −0.549034
\(932\) 302.041i 0.324079i
\(933\) 0 0
\(934\) −885.031 −0.947570
\(935\) 93.5307i 0.100033i
\(936\) 0 0
\(937\) 469.789 0.501375 0.250688 0.968068i \(-0.419343\pi\)
0.250688 + 0.968068i \(0.419343\pi\)
\(938\) 886.025i 0.944590i
\(939\) 0 0
\(940\) 470.636 0.500676
\(941\) 930.670i 0.989022i 0.869171 + 0.494511i \(0.164653\pi\)
−0.869171 + 0.494511i \(0.835347\pi\)
\(942\) 0 0
\(943\) 10.4097 0.0110389
\(944\) − 75.1250i − 0.0795816i
\(945\) 0 0
\(946\) −60.7423 −0.0642097
\(947\) 3.63113i 0.00383435i 0.999998 + 0.00191717i \(0.000610256\pi\)
−0.999998 + 0.00191717i \(0.999390\pi\)
\(948\) 0 0
\(949\) 76.6949 0.0808165
\(950\) − 69.8535i − 0.0735300i
\(951\) 0 0
\(952\) −445.757 −0.468232
\(953\) − 719.641i − 0.755132i −0.925983 0.377566i \(-0.876761\pi\)
0.925983 0.377566i \(-0.123239\pi\)
\(954\) 0 0
\(955\) −39.1362 −0.0409803
\(956\) 175.427i 0.183501i
\(957\) 0 0
\(958\) −437.408 −0.456584
\(959\) 1250.86i 1.30434i
\(960\) 0 0
\(961\) −951.725 −0.990349
\(962\) 640.380i 0.665676i
\(963\) 0 0
\(964\) −403.576 −0.418647
\(965\) − 1795.03i − 1.86014i
\(966\) 0 0
\(967\) 33.7740 0.0349266 0.0174633 0.999848i \(-0.494441\pi\)
0.0174633 + 0.999848i \(0.494441\pi\)
\(968\) 339.668i 0.350897i
\(969\) 0 0
\(970\) 807.439 0.832412
\(971\) 970.472i 0.999456i 0.866182 + 0.499728i \(0.166567\pi\)
−0.866182 + 0.499728i \(0.833433\pi\)
\(972\) 0 0
\(973\) 705.697 0.725279
\(974\) 40.0699i 0.0411395i
\(975\) 0 0
\(976\) 52.3633 0.0536509
\(977\) − 1570.30i − 1.60727i −0.595123 0.803635i \(-0.702897\pi\)
0.595123 0.803635i \(-0.297103\pi\)
\(978\) 0 0
\(979\) 138.545 0.141517
\(980\) 215.089i 0.219478i
\(981\) 0 0
\(982\) 1343.07 1.36769
\(983\) 775.878i 0.789296i 0.918832 + 0.394648i \(0.129134\pi\)
−0.918832 + 0.394648i \(0.870866\pi\)
\(984\) 0 0
\(985\) 401.271 0.407382
\(986\) 365.252i 0.370439i
\(987\) 0 0
\(988\) −478.969 −0.484787
\(989\) 42.9513i 0.0434290i
\(990\) 0 0
\(991\) 870.454 0.878359 0.439180 0.898399i \(-0.355269\pi\)
0.439180 + 0.898399i \(0.355269\pi\)
\(992\) − 17.2274i − 0.0173664i
\(993\) 0 0
\(994\) −212.758 −0.214042
\(995\) − 797.530i − 0.801538i
\(996\) 0 0
\(997\) 1245.00 1.24874 0.624372 0.781127i \(-0.285355\pi\)
0.624372 + 0.781127i \(0.285355\pi\)
\(998\) 792.279i 0.793867i
\(999\) 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 162.3.b.a.161.1 4
3.2 odd 2 inner 162.3.b.a.161.4 4
4.3 odd 2 1296.3.e.g.161.2 4
9.2 odd 6 18.3.d.a.5.2 4
9.4 even 3 18.3.d.a.11.2 yes 4
9.5 odd 6 54.3.d.a.35.1 4
9.7 even 3 54.3.d.a.17.1 4
12.11 even 2 1296.3.e.g.161.4 4
36.7 odd 6 432.3.q.d.17.1 4
36.11 even 6 144.3.q.c.113.2 4
36.23 even 6 432.3.q.d.305.1 4
36.31 odd 6 144.3.q.c.65.2 4
45.2 even 12 450.3.k.a.149.4 8
45.4 even 6 450.3.i.b.101.1 4
45.7 odd 12 1350.3.k.a.449.1 8
45.13 odd 12 450.3.k.a.299.4 8
45.14 odd 6 1350.3.i.b.251.2 4
45.22 odd 12 450.3.k.a.299.1 8
45.23 even 12 1350.3.k.a.899.1 8
45.29 odd 6 450.3.i.b.401.1 4
45.32 even 12 1350.3.k.a.899.4 8
45.34 even 6 1350.3.i.b.1151.2 4
45.38 even 12 450.3.k.a.149.1 8
45.43 odd 12 1350.3.k.a.449.4 8
72.5 odd 6 1728.3.q.d.1601.2 4
72.11 even 6 576.3.q.e.257.1 4
72.13 even 6 576.3.q.f.65.2 4
72.29 odd 6 576.3.q.f.257.2 4
72.43 odd 6 1728.3.q.c.449.1 4
72.59 even 6 1728.3.q.c.1601.1 4
72.61 even 6 1728.3.q.d.449.2 4
72.67 odd 6 576.3.q.e.65.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
18.3.d.a.5.2 4 9.2 odd 6
18.3.d.a.11.2 yes 4 9.4 even 3
54.3.d.a.17.1 4 9.7 even 3
54.3.d.a.35.1 4 9.5 odd 6
144.3.q.c.65.2 4 36.31 odd 6
144.3.q.c.113.2 4 36.11 even 6
162.3.b.a.161.1 4 1.1 even 1 trivial
162.3.b.a.161.4 4 3.2 odd 2 inner
432.3.q.d.17.1 4 36.7 odd 6
432.3.q.d.305.1 4 36.23 even 6
450.3.i.b.101.1 4 45.4 even 6
450.3.i.b.401.1 4 45.29 odd 6
450.3.k.a.149.1 8 45.38 even 12
450.3.k.a.149.4 8 45.2 even 12
450.3.k.a.299.1 8 45.22 odd 12
450.3.k.a.299.4 8 45.13 odd 12
576.3.q.e.65.1 4 72.67 odd 6
576.3.q.e.257.1 4 72.11 even 6
576.3.q.f.65.2 4 72.13 even 6
576.3.q.f.257.2 4 72.29 odd 6
1296.3.e.g.161.2 4 4.3 odd 2
1296.3.e.g.161.4 4 12.11 even 2
1350.3.i.b.251.2 4 45.14 odd 6
1350.3.i.b.1151.2 4 45.34 even 6
1350.3.k.a.449.1 8 45.7 odd 12
1350.3.k.a.449.4 8 45.43 odd 12
1350.3.k.a.899.1 8 45.23 even 12
1350.3.k.a.899.4 8 45.32 even 12
1728.3.q.c.449.1 4 72.43 odd 6
1728.3.q.c.1601.1 4 72.59 even 6
1728.3.q.d.449.2 4 72.61 even 6
1728.3.q.d.1601.2 4 72.5 odd 6