Properties

Label 162.3.b.a.161.3
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 161.3
Root \(1.22474 - 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 162.161
Dual form 162.3.b.a.161.2

$q$-expansion

\(f(q)\) \(=\) \(q+1.41421i q^{2} -2.00000 q^{4} -5.19615i q^{5} +6.34847 q^{7} -2.82843i q^{8} +O(q^{10})\) \(q+1.41421i q^{2} -2.00000 q^{4} -5.19615i q^{5} +6.34847 q^{7} -2.82843i q^{8} +7.34847 q^{10} -9.43879i q^{11} +19.6969 q^{13} +8.97809i q^{14} +4.00000 q^{16} -1.90702i q^{17} +4.69694 q^{19} +10.3923i q^{20} +13.3485 q^{22} +9.43879i q^{23} -2.00000 q^{25} +27.8557i q^{26} -12.6969 q^{28} +3.28913i q^{29} -41.0454 q^{31} +5.65685i q^{32} +2.69694 q^{34} -32.9876i q^{35} +17.3031 q^{37} +6.64247i q^{38} -14.6969 q^{40} -61.8289i q^{41} +0.954592 q^{43} +18.8776i q^{44} -13.3485 q^{46} +14.1100i q^{47} -8.69694 q^{49} -2.82843i q^{50} -39.3939 q^{52} -9.53512i q^{53} -49.0454 q^{55} -17.9562i q^{56} -4.65153 q^{58} +91.5274i q^{59} -75.0908 q^{61} -58.0470i q^{62} -8.00000 q^{64} -102.348i q^{65} +30.9546 q^{67} +3.81405i q^{68} +46.6515 q^{70} +85.9026i q^{71} -96.0908 q^{73} +24.4702i q^{74} -9.39388 q^{76} -59.9219i q^{77} +29.7423 q^{79} -20.7846i q^{80} +87.4393 q^{82} +87.9060i q^{83} -9.90918 q^{85} +1.35000i q^{86} -26.6969 q^{88} -41.3766i q^{89} +125.045 q^{91} -18.8776i q^{92} -19.9546 q^{94} -24.4060i q^{95} +95.8786 q^{97} -12.2993i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 8 q^{4} - 4 q^{7}+O(q^{10}) \) 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
\(7\) 6.34847 0.906924 0.453462 0.891276i \(-0.350189\pi\)
0.453462 + 0.891276i \(0.350189\pi\)
\(8\) − 2.82843i − 0.353553i
\(9\) 0 0
\(10\) 7.34847 0.734847
\(11\) − 9.43879i − 0.858072i −0.903287 0.429036i \(-0.858853\pi\)
0.903287 0.429036i \(-0.141147\pi\)
\(12\) 0 0
\(13\) 19.6969 1.51515 0.757575 0.652749i \(-0.226384\pi\)
0.757575 + 0.652749i \(0.226384\pi\)
\(14\) 8.97809i 0.641292i
\(15\) 0 0
\(16\) 4.00000 0.250000
\(17\) − 1.90702i − 0.112178i −0.998426 0.0560889i \(-0.982137\pi\)
0.998426 0.0560889i \(-0.0178630\pi\)
\(18\) 0 0
\(19\) 4.69694 0.247207 0.123604 0.992332i \(-0.460555\pi\)
0.123604 + 0.992332i \(0.460555\pi\)
\(20\) 10.3923i 0.519615i
\(21\) 0 0
\(22\) 13.3485 0.606749
\(23\) 9.43879i 0.410382i 0.978722 + 0.205191i \(0.0657816\pi\)
−0.978722 + 0.205191i \(0.934218\pi\)
\(24\) 0 0
\(25\) −2.00000 −0.0800000
\(26\) 27.8557i 1.07137i
\(27\) 0 0
\(28\) −12.6969 −0.453462
\(29\) 3.28913i 0.113418i 0.998391 + 0.0567091i \(0.0180608\pi\)
−0.998391 + 0.0567091i \(0.981939\pi\)
\(30\) 0 0
\(31\) −41.0454 −1.32405 −0.662023 0.749484i \(-0.730302\pi\)
−0.662023 + 0.749484i \(0.730302\pi\)
\(32\) 5.65685i 0.176777i
\(33\) 0 0
\(34\) 2.69694 0.0793217
\(35\) − 32.9876i − 0.942503i
\(36\) 0 0
\(37\) 17.3031 0.467650 0.233825 0.972279i \(-0.424876\pi\)
0.233825 + 0.972279i \(0.424876\pi\)
\(38\) 6.64247i 0.174802i
\(39\) 0 0
\(40\) −14.6969 −0.367423
\(41\) − 61.8289i − 1.50802i −0.656862 0.754011i \(-0.728116\pi\)
0.656862 0.754011i \(-0.271884\pi\)
\(42\) 0 0
\(43\) 0.954592 0.0221998 0.0110999 0.999938i \(-0.496467\pi\)
0.0110999 + 0.999938i \(0.496467\pi\)
\(44\) 18.8776i 0.429036i
\(45\) 0 0
\(46\) −13.3485 −0.290184
\(47\) 14.1100i 0.300213i 0.988670 + 0.150107i \(0.0479617\pi\)
−0.988670 + 0.150107i \(0.952038\pi\)
\(48\) 0 0
\(49\) −8.69694 −0.177489
\(50\) − 2.82843i − 0.0565685i
\(51\) 0 0
\(52\) −39.3939 −0.757575
\(53\) − 9.53512i − 0.179908i −0.995946 0.0899539i \(-0.971328\pi\)
0.995946 0.0899539i \(-0.0286720\pi\)
\(54\) 0 0
\(55\) −49.0454 −0.891735
\(56\) − 17.9562i − 0.320646i
\(57\) 0 0
\(58\) −4.65153 −0.0801988
\(59\) 91.5274i 1.55131i 0.631156 + 0.775656i \(0.282581\pi\)
−0.631156 + 0.775656i \(0.717419\pi\)
\(60\) 0 0
\(61\) −75.0908 −1.23100 −0.615498 0.788138i \(-0.711045\pi\)
−0.615498 + 0.788138i \(0.711045\pi\)
\(62\) − 58.0470i − 0.936241i
\(63\) 0 0
\(64\) −8.00000 −0.125000
\(65\) − 102.348i − 1.57459i
\(66\) 0 0
\(67\) 30.9546 0.462009 0.231004 0.972953i \(-0.425799\pi\)
0.231004 + 0.972953i \(0.425799\pi\)
\(68\) 3.81405i 0.0560889i
\(69\) 0 0
\(70\) 46.6515 0.666450
\(71\) 85.9026i 1.20990i 0.796265 + 0.604948i \(0.206806\pi\)
−0.796265 + 0.604948i \(0.793194\pi\)
\(72\) 0 0
\(73\) −96.0908 −1.31631 −0.658156 0.752881i \(-0.728663\pi\)
−0.658156 + 0.752881i \(0.728663\pi\)
\(74\) 24.4702i 0.330679i
\(75\) 0 0
\(76\) −9.39388 −0.123604
\(77\) − 59.9219i − 0.778206i
\(78\) 0 0
\(79\) 29.7423 0.376485 0.188243 0.982123i \(-0.439721\pi\)
0.188243 + 0.982123i \(0.439721\pi\)
\(80\) − 20.7846i − 0.259808i
\(81\) 0 0
\(82\) 87.4393 1.06633
\(83\) 87.9060i 1.05911i 0.848276 + 0.529554i \(0.177641\pi\)
−0.848276 + 0.529554i \(0.822359\pi\)
\(84\) 0 0
\(85\) −9.90918 −0.116579
\(86\) 1.35000i 0.0156976i
\(87\) 0 0
\(88\) −26.6969 −0.303374
\(89\) − 41.3766i − 0.464905i −0.972608 0.232453i \(-0.925325\pi\)
0.972608 0.232453i \(-0.0746751\pi\)
\(90\) 0 0
\(91\) 125.045 1.37413
\(92\) − 18.8776i − 0.205191i
\(93\) 0 0
\(94\) −19.9546 −0.212283
\(95\) − 24.4060i − 0.256905i
\(96\) 0 0
\(97\) 95.8786 0.988439 0.494219 0.869337i \(-0.335454\pi\)
0.494219 + 0.869337i \(0.335454\pi\)
\(98\) − 12.2993i − 0.125503i
\(99\) 0 0
\(100\) 4.00000 0.0400000
\(101\) 157.931i 1.56368i 0.623482 + 0.781838i \(0.285717\pi\)
−0.623482 + 0.781838i \(0.714283\pi\)
\(102\) 0 0
\(103\) 29.1362 0.282876 0.141438 0.989947i \(-0.454827\pi\)
0.141438 + 0.989947i \(0.454827\pi\)
\(104\) − 55.7114i − 0.535686i
\(105\) 0 0
\(106\) 13.4847 0.127214
\(107\) 171.805i 1.60566i 0.596210 + 0.802829i \(0.296673\pi\)
−0.596210 + 0.802829i \(0.703327\pi\)
\(108\) 0 0
\(109\) 116.272 1.06672 0.533360 0.845888i \(-0.320929\pi\)
0.533360 + 0.845888i \(0.320929\pi\)
\(110\) − 69.3607i − 0.630552i
\(111\) 0 0
\(112\) 25.3939 0.226731
\(113\) 202.265i 1.78995i 0.446114 + 0.894976i \(0.352808\pi\)
−0.446114 + 0.894976i \(0.647192\pi\)
\(114\) 0 0
\(115\) 49.0454 0.426482
\(116\) − 6.57826i − 0.0567091i
\(117\) 0 0
\(118\) −129.439 −1.09694
\(119\) − 12.1067i − 0.101737i
\(120\) 0 0
\(121\) 31.9092 0.263712
\(122\) − 106.194i − 0.870446i
\(123\) 0 0
\(124\) 82.0908 0.662023
\(125\) − 119.512i − 0.956092i
\(126\) 0 0
\(127\) 10.0908 0.0794552 0.0397276 0.999211i \(-0.487351\pi\)
0.0397276 + 0.999211i \(0.487351\pi\)
\(128\) − 11.3137i − 0.0883883i
\(129\) 0 0
\(130\) 144.742 1.11340
\(131\) 4.96021i 0.0378642i 0.999821 + 0.0189321i \(0.00602663\pi\)
−0.999821 + 0.0189321i \(0.993973\pi\)
\(132\) 0 0
\(133\) 29.8184 0.224198
\(134\) 43.7764i 0.326690i
\(135\) 0 0
\(136\) −5.39388 −0.0396609
\(137\) − 234.684i − 1.71302i −0.516129 0.856511i \(-0.672627\pi\)
0.516129 0.856511i \(-0.327373\pi\)
\(138\) 0 0
\(139\) 106.530 0.766404 0.383202 0.923665i \(-0.374821\pi\)
0.383202 + 0.923665i \(0.374821\pi\)
\(140\) 65.9752i 0.471252i
\(141\) 0 0
\(142\) −121.485 −0.855526
\(143\) − 185.915i − 1.30011i
\(144\) 0 0
\(145\) 17.0908 0.117868
\(146\) − 135.893i − 0.930774i
\(147\) 0 0
\(148\) −34.6061 −0.233825
\(149\) − 105.113i − 0.705453i −0.935726 0.352727i \(-0.885255\pi\)
0.935726 0.352727i \(-0.114745\pi\)
\(150\) 0 0
\(151\) −285.227 −1.88892 −0.944460 0.328625i \(-0.893415\pi\)
−0.944460 + 0.328625i \(0.893415\pi\)
\(152\) − 13.2849i − 0.0874010i
\(153\) 0 0
\(154\) 84.7423 0.550275
\(155\) 213.278i 1.37599i
\(156\) 0 0
\(157\) −197.182 −1.25593 −0.627967 0.778240i \(-0.716113\pi\)
−0.627967 + 0.778240i \(0.716113\pi\)
\(158\) 42.0620i 0.266215i
\(159\) 0 0
\(160\) 29.3939 0.183712
\(161\) 59.9219i 0.372186i
\(162\) 0 0
\(163\) −249.060 −1.52798 −0.763988 0.645230i \(-0.776762\pi\)
−0.763988 + 0.645230i \(0.776762\pi\)
\(164\) 123.658i 0.754011i
\(165\) 0 0
\(166\) −124.318 −0.748903
\(167\) − 48.4365i − 0.290039i −0.989429 0.145019i \(-0.953676\pi\)
0.989429 0.145019i \(-0.0463244\pi\)
\(168\) 0 0
\(169\) 218.969 1.29568
\(170\) − 14.0137i − 0.0824335i
\(171\) 0 0
\(172\) −1.90918 −0.0110999
\(173\) − 100.441i − 0.580585i −0.956938 0.290293i \(-0.906247\pi\)
0.956938 0.290293i \(-0.0937527\pi\)
\(174\) 0 0
\(175\) −12.6969 −0.0725539
\(176\) − 37.7552i − 0.214518i
\(177\) 0 0
\(178\) 58.5153 0.328738
\(179\) 285.071i 1.59257i 0.604919 + 0.796287i \(0.293206\pi\)
−0.604919 + 0.796287i \(0.706794\pi\)
\(180\) 0 0
\(181\) 37.1214 0.205091 0.102545 0.994728i \(-0.467301\pi\)
0.102545 + 0.994728i \(0.467301\pi\)
\(182\) 176.841i 0.971653i
\(183\) 0 0
\(184\) 26.6969 0.145092
\(185\) − 89.9093i − 0.485996i
\(186\) 0 0
\(187\) −18.0000 −0.0962567
\(188\) − 28.2201i − 0.150107i
\(189\) 0 0
\(190\) 34.5153 0.181660
\(191\) 17.9241i 0.0938433i 0.998899 + 0.0469217i \(0.0149411\pi\)
−0.998899 + 0.0469217i \(0.985059\pi\)
\(192\) 0 0
\(193\) −95.4541 −0.494581 −0.247290 0.968941i \(-0.579540\pi\)
−0.247290 + 0.968941i \(0.579540\pi\)
\(194\) 135.593i 0.698932i
\(195\) 0 0
\(196\) 17.3939 0.0887443
\(197\) − 160.363i − 0.814026i −0.913422 0.407013i \(-0.866570\pi\)
0.913422 0.407013i \(-0.133430\pi\)
\(198\) 0 0
\(199\) 6.51531 0.0327402 0.0163701 0.999866i \(-0.494789\pi\)
0.0163701 + 0.999866i \(0.494789\pi\)
\(200\) 5.65685i 0.0282843i
\(201\) 0 0
\(202\) −223.348 −1.10569
\(203\) 20.8809i 0.102862i
\(204\) 0 0
\(205\) −321.272 −1.56718
\(206\) 41.2048i 0.200024i
\(207\) 0 0
\(208\) 78.7878 0.378787
\(209\) − 44.3334i − 0.212122i
\(210\) 0 0
\(211\) −154.439 −0.731940 −0.365970 0.930627i \(-0.619263\pi\)
−0.365970 + 0.930627i \(0.619263\pi\)
\(212\) 19.0702i 0.0899539i
\(213\) 0 0
\(214\) −242.969 −1.13537
\(215\) − 4.96021i − 0.0230707i
\(216\) 0 0
\(217\) −260.576 −1.20081
\(218\) 164.434i 0.754285i
\(219\) 0 0
\(220\) 98.0908 0.445867
\(221\) − 37.5625i − 0.169966i
\(222\) 0 0
\(223\) 92.7730 0.416022 0.208011 0.978126i \(-0.433301\pi\)
0.208011 + 0.978126i \(0.433301\pi\)
\(224\) 35.9124i 0.160323i
\(225\) 0 0
\(226\) −286.045 −1.26569
\(227\) − 169.802i − 0.748026i −0.927423 0.374013i \(-0.877981\pi\)
0.927423 0.374013i \(-0.122019\pi\)
\(228\) 0 0
\(229\) 407.545 1.77967 0.889836 0.456280i \(-0.150819\pi\)
0.889836 + 0.456280i \(0.150819\pi\)
\(230\) 69.3607i 0.301568i
\(231\) 0 0
\(232\) 9.30306 0.0400994
\(233\) − 15.2562i − 0.0654772i −0.999464 0.0327386i \(-0.989577\pi\)
0.999464 0.0327386i \(-0.0104229\pi\)
\(234\) 0 0
\(235\) 73.3179 0.311991
\(236\) − 183.055i − 0.775656i
\(237\) 0 0
\(238\) 17.1214 0.0719388
\(239\) 56.5364i 0.236554i 0.992981 + 0.118277i \(0.0377371\pi\)
−0.992981 + 0.118277i \(0.962263\pi\)
\(240\) 0 0
\(241\) 84.2122 0.349428 0.174714 0.984619i \(-0.444100\pi\)
0.174714 + 0.984619i \(0.444100\pi\)
\(242\) 45.1264i 0.186473i
\(243\) 0 0
\(244\) 150.182 0.615498
\(245\) 45.1906i 0.184452i
\(246\) 0 0
\(247\) 92.5153 0.374556
\(248\) 116.094i 0.468121i
\(249\) 0 0
\(250\) 169.015 0.676059
\(251\) − 218.903i − 0.872123i −0.899917 0.436062i \(-0.856373\pi\)
0.899917 0.436062i \(-0.143627\pi\)
\(252\) 0 0
\(253\) 89.0908 0.352138
\(254\) 14.2706i 0.0561833i
\(255\) 0 0
\(256\) 16.0000 0.0625000
\(257\) − 12.8242i − 0.0498998i −0.999689 0.0249499i \(-0.992057\pi\)
0.999689 0.0249499i \(-0.00794262\pi\)
\(258\) 0 0
\(259\) 109.848 0.424123
\(260\) 204.697i 0.787295i
\(261\) 0 0
\(262\) −7.01479 −0.0267740
\(263\) − 336.464i − 1.27933i −0.768653 0.639666i \(-0.779073\pi\)
0.768653 0.639666i \(-0.220927\pi\)
\(264\) 0 0
\(265\) −49.5459 −0.186966
\(266\) 42.1695i 0.158532i
\(267\) 0 0
\(268\) −61.9092 −0.231004
\(269\) − 60.4468i − 0.224709i −0.993668 0.112355i \(-0.964161\pi\)
0.993668 0.112355i \(-0.0358393\pi\)
\(270\) 0 0
\(271\) 274.636 1.01342 0.506708 0.862118i \(-0.330862\pi\)
0.506708 + 0.862118i \(0.330862\pi\)
\(272\) − 7.62809i − 0.0280445i
\(273\) 0 0
\(274\) 331.893 1.21129
\(275\) 18.8776i 0.0686458i
\(276\) 0 0
\(277\) −49.0000 −0.176895 −0.0884477 0.996081i \(-0.528191\pi\)
−0.0884477 + 0.996081i \(0.528191\pi\)
\(278\) 150.656i 0.541929i
\(279\) 0 0
\(280\) −93.3031 −0.333225
\(281\) 343.086i 1.22095i 0.792037 + 0.610473i \(0.209021\pi\)
−0.792037 + 0.610473i \(0.790979\pi\)
\(282\) 0 0
\(283\) −343.409 −1.21346 −0.606729 0.794909i \(-0.707519\pi\)
−0.606729 + 0.794909i \(0.707519\pi\)
\(284\) − 171.805i − 0.604948i
\(285\) 0 0
\(286\) 262.924 0.919315
\(287\) − 392.519i − 1.36766i
\(288\) 0 0
\(289\) 285.363 0.987416
\(290\) 24.1701i 0.0833450i
\(291\) 0 0
\(292\) 192.182 0.658156
\(293\) − 286.453i − 0.977655i −0.872380 0.488828i \(-0.837425\pi\)
0.872380 0.488828i \(-0.162575\pi\)
\(294\) 0 0
\(295\) 475.590 1.61217
\(296\) − 48.9404i − 0.165339i
\(297\) 0 0
\(298\) 148.652 0.498831
\(299\) 185.915i 0.621790i
\(300\) 0 0
\(301\) 6.06020 0.0201336
\(302\) − 403.372i − 1.33567i
\(303\) 0 0
\(304\) 18.7878 0.0618018
\(305\) 390.183i 1.27929i
\(306\) 0 0
\(307\) 154.091 0.501924 0.250962 0.967997i \(-0.419253\pi\)
0.250962 + 0.967997i \(0.419253\pi\)
\(308\) 119.844i 0.389103i
\(309\) 0 0
\(310\) −301.621 −0.972971
\(311\) − 71.9853i − 0.231464i −0.993280 0.115732i \(-0.963079\pi\)
0.993280 0.115732i \(-0.0369214\pi\)
\(312\) 0 0
\(313\) −367.606 −1.17446 −0.587230 0.809420i \(-0.699782\pi\)
−0.587230 + 0.809420i \(0.699782\pi\)
\(314\) − 278.857i − 0.888079i
\(315\) 0 0
\(316\) −59.4847 −0.188243
\(317\) 107.597i 0.339424i 0.985494 + 0.169712i \(0.0542838\pi\)
−0.985494 + 0.169712i \(0.945716\pi\)
\(318\) 0 0
\(319\) 31.0454 0.0973210
\(320\) 41.5692i 0.129904i
\(321\) 0 0
\(322\) −84.7423 −0.263175
\(323\) − 8.95717i − 0.0277312i
\(324\) 0 0
\(325\) −39.3939 −0.121212
\(326\) − 352.224i − 1.08044i
\(327\) 0 0
\(328\) −174.879 −0.533166
\(329\) 89.5771i 0.272271i
\(330\) 0 0
\(331\) 17.1975 0.0519561 0.0259780 0.999663i \(-0.491730\pi\)
0.0259780 + 0.999663i \(0.491730\pi\)
\(332\) − 175.812i − 0.529554i
\(333\) 0 0
\(334\) 68.4995 0.205088
\(335\) − 160.845i − 0.480134i
\(336\) 0 0
\(337\) 364.394 1.08129 0.540644 0.841252i \(-0.318181\pi\)
0.540644 + 0.841252i \(0.318181\pi\)
\(338\) 309.669i 0.916182i
\(339\) 0 0
\(340\) 19.8184 0.0582893
\(341\) 387.419i 1.13613i
\(342\) 0 0
\(343\) −366.287 −1.06789
\(344\) − 2.69999i − 0.00784882i
\(345\) 0 0
\(346\) 142.045 0.410536
\(347\) 583.394i 1.68125i 0.541616 + 0.840626i \(0.317813\pi\)
−0.541616 + 0.840626i \(0.682187\pi\)
\(348\) 0 0
\(349\) 312.757 0.896152 0.448076 0.893995i \(-0.352109\pi\)
0.448076 + 0.893995i \(0.352109\pi\)
\(350\) − 17.9562i − 0.0513034i
\(351\) 0 0
\(352\) 53.3939 0.151687
\(353\) 37.6156i 0.106560i 0.998580 + 0.0532798i \(0.0169675\pi\)
−0.998580 + 0.0532798i \(0.983032\pi\)
\(354\) 0 0
\(355\) 446.363 1.25736
\(356\) 82.7531i 0.232453i
\(357\) 0 0
\(358\) −403.151 −1.12612
\(359\) 294.028i 0.819019i 0.912306 + 0.409510i \(0.134300\pi\)
−0.912306 + 0.409510i \(0.865700\pi\)
\(360\) 0 0
\(361\) −338.939 −0.938889
\(362\) 52.4976i 0.145021i
\(363\) 0 0
\(364\) −250.091 −0.687063
\(365\) 499.303i 1.36795i
\(366\) 0 0
\(367\) −33.2270 −0.0905369 −0.0452684 0.998975i \(-0.514414\pi\)
−0.0452684 + 0.998975i \(0.514414\pi\)
\(368\) 37.7552i 0.102596i
\(369\) 0 0
\(370\) 127.151 0.343651
\(371\) − 60.5334i − 0.163163i
\(372\) 0 0
\(373\) −225.030 −0.603296 −0.301648 0.953419i \(-0.597537\pi\)
−0.301648 + 0.953419i \(0.597537\pi\)
\(374\) − 25.4558i − 0.0680638i
\(375\) 0 0
\(376\) 39.9092 0.106141
\(377\) 64.7858i 0.171846i
\(378\) 0 0
\(379\) −166.334 −0.438875 −0.219438 0.975627i \(-0.570422\pi\)
−0.219438 + 0.975627i \(0.570422\pi\)
\(380\) 48.8120i 0.128453i
\(381\) 0 0
\(382\) −25.3485 −0.0663572
\(383\) − 736.987i − 1.92425i −0.272613 0.962124i \(-0.587888\pi\)
0.272613 0.962124i \(-0.412112\pi\)
\(384\) 0 0
\(385\) −311.363 −0.808736
\(386\) − 134.992i − 0.349721i
\(387\) 0 0
\(388\) −191.757 −0.494219
\(389\) 169.373i 0.435407i 0.976015 + 0.217704i \(0.0698566\pi\)
−0.976015 + 0.217704i \(0.930143\pi\)
\(390\) 0 0
\(391\) 18.0000 0.0460358
\(392\) 24.5987i 0.0627517i
\(393\) 0 0
\(394\) 226.788 0.575603
\(395\) − 154.546i − 0.391255i
\(396\) 0 0
\(397\) −256.272 −0.645523 −0.322761 0.946480i \(-0.604611\pi\)
−0.322761 + 0.946480i \(0.604611\pi\)
\(398\) 9.21404i 0.0231508i
\(399\) 0 0
\(400\) −8.00000 −0.0200000
\(401\) 261.382i 0.651826i 0.945400 + 0.325913i \(0.105672\pi\)
−0.945400 + 0.325913i \(0.894328\pi\)
\(402\) 0 0
\(403\) −808.469 −2.00613
\(404\) − 315.862i − 0.781838i
\(405\) 0 0
\(406\) −29.5301 −0.0727342
\(407\) − 163.320i − 0.401278i
\(408\) 0 0
\(409\) −443.788 −1.08506 −0.542528 0.840038i \(-0.682533\pi\)
−0.542528 + 0.840038i \(0.682533\pi\)
\(410\) − 454.348i − 1.10817i
\(411\) 0 0
\(412\) −58.2724 −0.141438
\(413\) 581.059i 1.40692i
\(414\) 0 0
\(415\) 456.773 1.10066
\(416\) 111.423i 0.267843i
\(417\) 0 0
\(418\) 62.6969 0.149993
\(419\) 10.7679i 0.0256990i 0.999917 + 0.0128495i \(0.00409023\pi\)
−0.999917 + 0.0128495i \(0.995910\pi\)
\(420\) 0 0
\(421\) 254.303 0.604045 0.302023 0.953301i \(-0.402338\pi\)
0.302023 + 0.953301i \(0.402338\pi\)
\(422\) − 218.410i − 0.517560i
\(423\) 0 0
\(424\) −26.9694 −0.0636070
\(425\) 3.81405i 0.00897423i
\(426\) 0 0
\(427\) −476.712 −1.11642
\(428\) − 343.611i − 0.802829i
\(429\) 0 0
\(430\) 7.01479 0.0163135
\(431\) − 698.663i − 1.62103i −0.585719 0.810514i \(-0.699188\pi\)
0.585719 0.810514i \(-0.300812\pi\)
\(432\) 0 0
\(433\) 211.728 0.488978 0.244489 0.969652i \(-0.421380\pi\)
0.244489 + 0.969652i \(0.421380\pi\)
\(434\) − 368.509i − 0.849100i
\(435\) 0 0
\(436\) −232.545 −0.533360
\(437\) 44.3334i 0.101449i
\(438\) 0 0
\(439\) 279.863 0.637501 0.318750 0.947839i \(-0.396737\pi\)
0.318750 + 0.947839i \(0.396737\pi\)
\(440\) 138.721i 0.315276i
\(441\) 0 0
\(442\) 53.1214 0.120184
\(443\) 551.254i 1.24437i 0.782872 + 0.622183i \(0.213754\pi\)
−0.782872 + 0.622183i \(0.786246\pi\)
\(444\) 0 0
\(445\) −214.999 −0.483144
\(446\) 131.201i 0.294172i
\(447\) 0 0
\(448\) −50.7878 −0.113366
\(449\) − 542.865i − 1.20905i −0.796585 0.604527i \(-0.793362\pi\)
0.796585 0.604527i \(-0.206638\pi\)
\(450\) 0 0
\(451\) −583.590 −1.29399
\(452\) − 404.529i − 0.894976i
\(453\) 0 0
\(454\) 240.136 0.528934
\(455\) − 649.755i − 1.42803i
\(456\) 0 0
\(457\) 92.3643 0.202110 0.101055 0.994881i \(-0.467778\pi\)
0.101055 + 0.994881i \(0.467778\pi\)
\(458\) 576.356i 1.25842i
\(459\) 0 0
\(460\) −98.0908 −0.213241
\(461\) 229.820i 0.498525i 0.968436 + 0.249263i \(0.0801882\pi\)
−0.968436 + 0.249263i \(0.919812\pi\)
\(462\) 0 0
\(463\) −510.803 −1.10325 −0.551623 0.834094i \(-0.685991\pi\)
−0.551623 + 0.834094i \(0.685991\pi\)
\(464\) 13.1565i 0.0283546i
\(465\) 0 0
\(466\) 21.5755 0.0462994
\(467\) 833.657i 1.78513i 0.450915 + 0.892567i \(0.351098\pi\)
−0.450915 + 0.892567i \(0.648902\pi\)
\(468\) 0 0
\(469\) 196.514 0.419007
\(470\) 103.687i 0.220611i
\(471\) 0 0
\(472\) 258.879 0.548472
\(473\) − 9.01020i − 0.0190490i
\(474\) 0 0
\(475\) −9.39388 −0.0197766
\(476\) 24.2134i 0.0508684i
\(477\) 0 0
\(478\) −79.9546 −0.167269
\(479\) − 657.190i − 1.37201i −0.727599 0.686003i \(-0.759364\pi\)
0.727599 0.686003i \(-0.240636\pi\)
\(480\) 0 0
\(481\) 340.817 0.708560
\(482\) 119.094i 0.247083i
\(483\) 0 0
\(484\) −63.8184 −0.131856
\(485\) − 498.200i − 1.02722i
\(486\) 0 0
\(487\) −351.666 −0.722107 −0.361054 0.932545i \(-0.617583\pi\)
−0.361054 + 0.932545i \(0.617583\pi\)
\(488\) 212.389i 0.435223i
\(489\) 0 0
\(490\) −63.9092 −0.130427
\(491\) 245.418i 0.499834i 0.968267 + 0.249917i \(0.0804033\pi\)
−0.968267 + 0.249917i \(0.919597\pi\)
\(492\) 0 0
\(493\) 6.27245 0.0127230
\(494\) 130.836i 0.264851i
\(495\) 0 0
\(496\) −164.182 −0.331011
\(497\) 545.350i 1.09728i
\(498\) 0 0
\(499\) 630.226 1.26298 0.631489 0.775385i \(-0.282444\pi\)
0.631489 + 0.775385i \(0.282444\pi\)
\(500\) 239.023i 0.478046i
\(501\) 0 0
\(502\) 309.576 0.616684
\(503\) − 286.891i − 0.570360i −0.958474 0.285180i \(-0.907947\pi\)
0.958474 0.285180i \(-0.0920534\pi\)
\(504\) 0 0
\(505\) 820.635 1.62502
\(506\) 125.993i 0.248999i
\(507\) 0 0
\(508\) −20.1816 −0.0397276
\(509\) 872.323i 1.71380i 0.515485 + 0.856898i \(0.327612\pi\)
−0.515485 + 0.856898i \(0.672388\pi\)
\(510\) 0 0
\(511\) −610.030 −1.19380
\(512\) 22.6274i 0.0441942i
\(513\) 0 0
\(514\) 18.1362 0.0352845
\(515\) − 151.396i − 0.293973i
\(516\) 0 0
\(517\) 133.182 0.257605
\(518\) 155.348i 0.299901i
\(519\) 0 0
\(520\) −289.485 −0.556701
\(521\) − 206.132i − 0.395646i −0.980238 0.197823i \(-0.936613\pi\)
0.980238 0.197823i \(-0.0633872\pi\)
\(522\) 0 0
\(523\) 884.817 1.69181 0.845906 0.533333i \(-0.179061\pi\)
0.845906 + 0.533333i \(0.179061\pi\)
\(524\) − 9.92041i − 0.0189321i
\(525\) 0 0
\(526\) 475.832 0.904624
\(527\) 78.2746i 0.148529i
\(528\) 0 0
\(529\) 439.909 0.831586
\(530\) − 70.0685i − 0.132205i
\(531\) 0 0
\(532\) −59.6367 −0.112099
\(533\) − 1217.84i − 2.28488i
\(534\) 0 0
\(535\) 892.727 1.66865
\(536\) − 87.5528i − 0.163345i
\(537\) 0 0
\(538\) 85.4847 0.158893
\(539\) 82.0886i 0.152298i
\(540\) 0 0
\(541\) −509.151 −0.941129 −0.470565 0.882365i \(-0.655950\pi\)
−0.470565 + 0.882365i \(0.655950\pi\)
\(542\) 388.394i 0.716593i
\(543\) 0 0
\(544\) 10.7878 0.0198304
\(545\) − 604.169i − 1.10857i
\(546\) 0 0
\(547\) 548.044 1.00191 0.500955 0.865474i \(-0.332982\pi\)
0.500955 + 0.865474i \(0.332982\pi\)
\(548\) 469.368i 0.856511i
\(549\) 0 0
\(550\) −26.6969 −0.0485399
\(551\) 15.4488i 0.0280378i
\(552\) 0 0
\(553\) 188.818 0.341444
\(554\) − 69.2965i − 0.125084i
\(555\) 0 0
\(556\) −213.060 −0.383202
\(557\) 406.542i 0.729879i 0.931031 + 0.364939i \(0.118910\pi\)
−0.931031 + 0.364939i \(0.881090\pi\)
\(558\) 0 0
\(559\) 18.8025 0.0336360
\(560\) − 131.950i − 0.235626i
\(561\) 0 0
\(562\) −485.196 −0.863339
\(563\) − 606.471i − 1.07721i −0.842557 0.538607i \(-0.818951\pi\)
0.842557 0.538607i \(-0.181049\pi\)
\(564\) 0 0
\(565\) 1051.00 1.86017
\(566\) − 485.653i − 0.858045i
\(567\) 0 0
\(568\) 242.969 0.427763
\(569\) 259.755i 0.456511i 0.973601 + 0.228255i \(0.0733021\pi\)
−0.973601 + 0.228255i \(0.926698\pi\)
\(570\) 0 0
\(571\) −87.8332 −0.153823 −0.0769117 0.997038i \(-0.524506\pi\)
−0.0769117 + 0.997038i \(0.524506\pi\)
\(572\) 371.831i 0.650054i
\(573\) 0 0
\(574\) 555.106 0.967083
\(575\) − 18.8776i − 0.0328306i
\(576\) 0 0
\(577\) −132.091 −0.228927 −0.114463 0.993427i \(-0.536515\pi\)
−0.114463 + 0.993427i \(0.536515\pi\)
\(578\) 403.565i 0.698209i
\(579\) 0 0
\(580\) −34.1816 −0.0589338
\(581\) 558.069i 0.960531i
\(582\) 0 0
\(583\) −90.0000 −0.154374
\(584\) 271.786i 0.465387i
\(585\) 0 0
\(586\) 405.106 0.691306
\(587\) 567.666i 0.967064i 0.875327 + 0.483532i \(0.160646\pi\)
−0.875327 + 0.483532i \(0.839354\pi\)
\(588\) 0 0
\(589\) −192.788 −0.327314
\(590\) 672.586i 1.13998i
\(591\) 0 0
\(592\) 69.2122 0.116913
\(593\) 77.0321i 0.129902i 0.997888 + 0.0649512i \(0.0206892\pi\)
−0.997888 + 0.0649512i \(0.979311\pi\)
\(594\) 0 0
\(595\) −62.9082 −0.105728
\(596\) 210.225i 0.352727i
\(597\) 0 0
\(598\) −262.924 −0.439672
\(599\) − 883.250i − 1.47454i −0.675598 0.737270i \(-0.736115\pi\)
0.675598 0.737270i \(-0.263885\pi\)
\(600\) 0 0
\(601\) −795.091 −1.32295 −0.661473 0.749969i \(-0.730068\pi\)
−0.661473 + 0.749969i \(0.730068\pi\)
\(602\) 8.57042i 0.0142366i
\(603\) 0 0
\(604\) 570.454 0.944460
\(605\) − 165.805i − 0.274058i
\(606\) 0 0
\(607\) −296.743 −0.488869 −0.244434 0.969666i \(-0.578602\pi\)
−0.244434 + 0.969666i \(0.578602\pi\)
\(608\) 26.5699i 0.0437005i
\(609\) 0 0
\(610\) −551.803 −0.904594
\(611\) 277.924i 0.454868i
\(612\) 0 0
\(613\) −517.181 −0.843688 −0.421844 0.906668i \(-0.638617\pi\)
−0.421844 + 0.906668i \(0.638617\pi\)
\(614\) 217.917i 0.354914i
\(615\) 0 0
\(616\) −169.485 −0.275137
\(617\) − 265.476i − 0.430268i −0.976584 0.215134i \(-0.930981\pi\)
0.976584 0.215134i \(-0.0690189\pi\)
\(618\) 0 0
\(619\) −197.045 −0.318329 −0.159164 0.987252i \(-0.550880\pi\)
−0.159164 + 0.987252i \(0.550880\pi\)
\(620\) − 426.556i − 0.687994i
\(621\) 0 0
\(622\) 101.803 0.163670
\(623\) − 262.678i − 0.421634i
\(624\) 0 0
\(625\) −671.000 −1.07360
\(626\) − 519.874i − 0.830469i
\(627\) 0 0
\(628\) 394.363 0.627967
\(629\) − 32.9973i − 0.0524600i
\(630\) 0 0
\(631\) −160.879 −0.254958 −0.127479 0.991841i \(-0.540689\pi\)
−0.127479 + 0.991841i \(0.540689\pi\)
\(632\) − 84.1241i − 0.133108i
\(633\) 0 0
\(634\) −152.166 −0.240009
\(635\) − 52.4334i − 0.0825723i
\(636\) 0 0
\(637\) −171.303 −0.268922
\(638\) 43.9048i 0.0688164i
\(639\) 0 0
\(640\) −58.7878 −0.0918559
\(641\) 309.337i 0.482585i 0.970452 + 0.241293i \(0.0775714\pi\)
−0.970452 + 0.241293i \(0.922429\pi\)
\(642\) 0 0
\(643\) 394.591 0.613672 0.306836 0.951762i \(-0.400730\pi\)
0.306836 + 0.951762i \(0.400730\pi\)
\(644\) − 119.844i − 0.186093i
\(645\) 0 0
\(646\) 12.6674 0.0196089
\(647\) − 418.736i − 0.647196i −0.946195 0.323598i \(-0.895108\pi\)
0.946195 0.323598i \(-0.104892\pi\)
\(648\) 0 0
\(649\) 863.908 1.33114
\(650\) − 55.7114i − 0.0857098i
\(651\) 0 0
\(652\) 498.120 0.763988
\(653\) − 530.725i − 0.812749i −0.913707 0.406375i \(-0.866793\pi\)
0.913707 0.406375i \(-0.133207\pi\)
\(654\) 0 0
\(655\) 25.7740 0.0393496
\(656\) − 247.316i − 0.377006i
\(657\) 0 0
\(658\) −126.681 −0.192524
\(659\) − 358.193i − 0.543539i −0.962362 0.271770i \(-0.912391\pi\)
0.962362 0.271770i \(-0.0876089\pi\)
\(660\) 0 0
\(661\) −222.271 −0.336265 −0.168133 0.985764i \(-0.553774\pi\)
−0.168133 + 0.985764i \(0.553774\pi\)
\(662\) 24.3209i 0.0367385i
\(663\) 0 0
\(664\) 248.636 0.374451
\(665\) − 154.941i − 0.232994i
\(666\) 0 0
\(667\) −31.0454 −0.0465448
\(668\) 96.8729i 0.145019i
\(669\) 0 0
\(670\) 227.469 0.339506
\(671\) 708.767i 1.05628i
\(672\) 0 0
\(673\) −289.211 −0.429734 −0.214867 0.976643i \(-0.568932\pi\)
−0.214867 + 0.976643i \(0.568932\pi\)
\(674\) 515.331i 0.764586i
\(675\) 0 0
\(676\) −437.939 −0.647838
\(677\) − 464.451i − 0.686043i −0.939328 0.343022i \(-0.888550\pi\)
0.939328 0.343022i \(-0.111450\pi\)
\(678\) 0 0
\(679\) 608.682 0.896439
\(680\) 28.0274i 0.0412168i
\(681\) 0 0
\(682\) −547.893 −0.803363
\(683\) − 1126.36i − 1.64913i −0.565767 0.824565i \(-0.691420\pi\)
0.565767 0.824565i \(-0.308580\pi\)
\(684\) 0 0
\(685\) −1219.45 −1.78022
\(686\) − 518.008i − 0.755114i
\(687\) 0 0
\(688\) 3.81837 0.00554996
\(689\) − 187.813i − 0.272587i
\(690\) 0 0
\(691\) 1037.68 1.50171 0.750855 0.660467i \(-0.229642\pi\)
0.750855 + 0.660467i \(0.229642\pi\)
\(692\) 200.883i 0.290293i
\(693\) 0 0
\(694\) −825.044 −1.18882
\(695\) − 553.547i − 0.796470i
\(696\) 0 0
\(697\) −117.909 −0.169167
\(698\) 442.305i 0.633675i
\(699\) 0 0
\(700\) 25.3939 0.0362770
\(701\) 778.180i 1.11010i 0.831817 + 0.555050i \(0.187301\pi\)
−0.831817 + 0.555050i \(0.812699\pi\)
\(702\) 0 0
\(703\) 81.2714 0.115607
\(704\) 75.5103i 0.107259i
\(705\) 0 0
\(706\) −53.1964 −0.0753490
\(707\) 1002.62i 1.41814i
\(708\) 0 0
\(709\) −1172.03 −1.65307 −0.826536 0.562883i \(-0.809692\pi\)
−0.826536 + 0.562883i \(0.809692\pi\)
\(710\) 631.253i 0.889089i
\(711\) 0 0
\(712\) −117.031 −0.164369
\(713\) − 387.419i − 0.543365i
\(714\) 0 0
\(715\) −966.044 −1.35111
\(716\) − 570.142i − 0.796287i
\(717\) 0 0
\(718\) −415.818 −0.579134
\(719\) − 515.416i − 0.716851i −0.933558 0.358426i \(-0.883314\pi\)
0.933558 0.358426i \(-0.116686\pi\)
\(720\) 0 0
\(721\) 184.970 0.256547
\(722\) − 479.332i − 0.663894i
\(723\) 0 0
\(724\) −74.2429 −0.102545
\(725\) − 6.57826i − 0.00907346i
\(726\) 0 0
\(727\) −841.409 −1.15737 −0.578685 0.815551i \(-0.696434\pi\)
−0.578685 + 0.815551i \(0.696434\pi\)
\(728\) − 353.682i − 0.485827i
\(729\) 0 0
\(730\) −706.120 −0.967288
\(731\) − 1.82043i − 0.00249033i
\(732\) 0 0
\(733\) 606.362 0.827234 0.413617 0.910451i \(-0.364265\pi\)
0.413617 + 0.910451i \(0.364265\pi\)
\(734\) − 46.9901i − 0.0640192i
\(735\) 0 0
\(736\) −53.3939 −0.0725460
\(737\) − 292.174i − 0.396437i
\(738\) 0 0
\(739\) −389.362 −0.526877 −0.263439 0.964676i \(-0.584857\pi\)
−0.263439 + 0.964676i \(0.584857\pi\)
\(740\) 179.819i 0.242998i
\(741\) 0 0
\(742\) 85.6072 0.115374
\(743\) 1044.75i 1.40612i 0.711129 + 0.703061i \(0.248184\pi\)
−0.711129 + 0.703061i \(0.751816\pi\)
\(744\) 0 0
\(745\) −546.181 −0.733128
\(746\) − 318.240i − 0.426595i
\(747\) 0 0
\(748\) 36.0000 0.0481283
\(749\) 1090.70i 1.45621i
\(750\) 0 0
\(751\) −1291.83 −1.72015 −0.860074 0.510169i \(-0.829583\pi\)
−0.860074 + 0.510169i \(0.829583\pi\)
\(752\) 56.4401i 0.0750533i
\(753\) 0 0
\(754\) −91.6209 −0.121513
\(755\) 1482.08i 1.96302i
\(756\) 0 0
\(757\) 1042.36 1.37697 0.688483 0.725252i \(-0.258277\pi\)
0.688483 + 0.725252i \(0.258277\pi\)
\(758\) − 235.231i − 0.310332i
\(759\) 0 0
\(760\) −69.0306 −0.0908298
\(761\) − 325.171i − 0.427295i −0.976911 0.213647i \(-0.931466\pi\)
0.976911 0.213647i \(-0.0685344\pi\)
\(762\) 0 0
\(763\) 738.152 0.967434
\(764\) − 35.8481i − 0.0469217i
\(765\) 0 0
\(766\) 1042.26 1.36065
\(767\) 1802.81i 2.35047i
\(768\) 0 0
\(769\) 342.696 0.445638 0.222819 0.974860i \(-0.428474\pi\)
0.222819 + 0.974860i \(0.428474\pi\)
\(770\) − 440.334i − 0.571863i
\(771\) 0 0
\(772\) 190.908 0.247290
\(773\) − 532.579i − 0.688977i −0.938791 0.344488i \(-0.888052\pi\)
0.938791 0.344488i \(-0.111948\pi\)
\(774\) 0 0
\(775\) 82.0908 0.105924
\(776\) − 271.186i − 0.349466i
\(777\) 0 0
\(778\) −239.530 −0.307879
\(779\) − 290.407i − 0.372794i
\(780\) 0 0
\(781\) 810.817 1.03818
\(782\) 25.4558i 0.0325522i
\(783\) 0 0
\(784\) −34.7878 −0.0443721
\(785\) 1024.59i 1.30520i
\(786\) 0 0
\(787\) −103.954 −0.132088 −0.0660442 0.997817i \(-0.521038\pi\)
−0.0660442 + 0.997817i \(0.521038\pi\)
\(788\) 320.726i 0.407013i
\(789\) 0 0
\(790\) 218.561 0.276659
\(791\) 1284.07i 1.62335i
\(792\) 0 0
\(793\) −1479.06 −1.86514
\(794\) − 362.424i − 0.456453i
\(795\) 0 0
\(796\) −13.0306 −0.0163701
\(797\) − 1104.28i − 1.38554i −0.721158 0.692770i \(-0.756390\pi\)
0.721158 0.692770i \(-0.243610\pi\)
\(798\) 0 0
\(799\) 26.9082 0.0336773
\(800\) − 11.3137i − 0.0141421i
\(801\) 0 0
\(802\) −369.650 −0.460911
\(803\) 906.981i 1.12949i
\(804\) 0 0
\(805\) 311.363 0.386787
\(806\) − 1143.35i − 1.41855i
\(807\) 0 0
\(808\) 446.697 0.552843
\(809\) − 256.465i − 0.317015i −0.987358 0.158508i \(-0.949332\pi\)
0.987358 0.158508i \(-0.0506683\pi\)
\(810\) 0 0
\(811\) 735.362 0.906735 0.453368 0.891324i \(-0.350222\pi\)
0.453368 + 0.891324i \(0.350222\pi\)
\(812\) − 41.7619i − 0.0514309i
\(813\) 0 0
\(814\) 230.969 0.283746
\(815\) 1294.15i 1.58792i
\(816\) 0 0
\(817\) 4.48366 0.00548796
\(818\) − 627.611i − 0.767250i
\(819\) 0 0
\(820\) 642.545 0.783591
\(821\) − 1245.29i − 1.51680i −0.651792 0.758398i \(-0.725982\pi\)
0.651792 0.758398i \(-0.274018\pi\)
\(822\) 0 0
\(823\) −1542.26 −1.87395 −0.936973 0.349402i \(-0.886385\pi\)
−0.936973 + 0.349402i \(0.886385\pi\)
\(824\) − 82.4097i − 0.100012i
\(825\) 0 0
\(826\) −821.741 −0.994844
\(827\) 955.707i 1.15563i 0.816167 + 0.577815i \(0.196095\pi\)
−0.816167 + 0.577815i \(0.803905\pi\)
\(828\) 0 0
\(829\) 1082.88 1.30625 0.653123 0.757252i \(-0.273458\pi\)
0.653123 + 0.757252i \(0.273458\pi\)
\(830\) 645.975i 0.778283i
\(831\) 0 0
\(832\) −157.576 −0.189394
\(833\) 16.5853i 0.0199103i
\(834\) 0 0
\(835\) −251.683 −0.301417
\(836\) 88.6669i 0.106061i
\(837\) 0 0
\(838\) −15.2281 −0.0181719
\(839\) − 1043.59i − 1.24385i −0.783075 0.621927i \(-0.786350\pi\)
0.783075 0.621927i \(-0.213650\pi\)
\(840\) 0 0
\(841\) 830.182 0.987136
\(842\) 359.639i 0.427125i
\(843\) 0 0
\(844\) 308.879 0.365970
\(845\) − 1137.80i − 1.34651i
\(846\) 0 0
\(847\) 202.574 0.239167
\(848\) − 38.1405i − 0.0449770i
\(849\) 0 0
\(850\) −5.39388 −0.00634574
\(851\) 163.320i 0.191915i
\(852\) 0 0
\(853\) 473.817 0.555472 0.277736 0.960657i \(-0.410416\pi\)
0.277736 + 0.960657i \(0.410416\pi\)
\(854\) − 674.172i − 0.789429i
\(855\) 0 0
\(856\) 485.939 0.567685
\(857\) − 916.762i − 1.06973i −0.844936 0.534867i \(-0.820362\pi\)
0.844936 0.534867i \(-0.179638\pi\)
\(858\) 0 0
\(859\) 957.802 1.11502 0.557510 0.830170i \(-0.311757\pi\)
0.557510 + 0.830170i \(0.311757\pi\)
\(860\) 9.92041i 0.0115354i
\(861\) 0 0
\(862\) 988.059 1.14624
\(863\) − 524.200i − 0.607416i −0.952765 0.303708i \(-0.901775\pi\)
0.952765 0.303708i \(-0.0982247\pi\)
\(864\) 0 0
\(865\) −521.908 −0.603362
\(866\) 299.428i 0.345760i
\(867\) 0 0
\(868\) 521.151 0.600404
\(869\) − 280.732i − 0.323052i
\(870\) 0 0
\(871\) 609.711 0.700012
\(872\) − 328.868i − 0.377142i
\(873\) 0 0
\(874\) −62.6969 −0.0717356
\(875\) − 758.715i − 0.867103i
\(876\) 0 0
\(877\) 1007.76 1.14909 0.574547 0.818471i \(-0.305178\pi\)
0.574547 + 0.818471i \(0.305178\pi\)
\(878\) 395.786i 0.450781i
\(879\) 0 0
\(880\) −196.182 −0.222934
\(881\) 1536.71i 1.74428i 0.489254 + 0.872141i \(0.337269\pi\)
−0.489254 + 0.872141i \(0.662731\pi\)
\(882\) 0 0
\(883\) −294.213 −0.333197 −0.166599 0.986025i \(-0.553278\pi\)
−0.166599 + 0.986025i \(0.553278\pi\)
\(884\) 75.1250i 0.0849831i
\(885\) 0 0
\(886\) −779.591 −0.879900
\(887\) − 574.803i − 0.648031i −0.946052 0.324015i \(-0.894967\pi\)
0.946052 0.324015i \(-0.105033\pi\)
\(888\) 0 0
\(889\) 64.0612 0.0720599
\(890\) − 304.054i − 0.341634i
\(891\) 0 0
\(892\) −185.546 −0.208011
\(893\) 66.2739i 0.0742149i
\(894\) 0 0
\(895\) 1481.27 1.65505
\(896\) − 71.8247i − 0.0801615i
\(897\) 0 0
\(898\) 767.728 0.854930
\(899\) − 135.004i − 0.150171i
\(900\) 0 0
\(901\) −18.1837 −0.0201817
\(902\) − 825.321i − 0.914990i
\(903\) 0 0
\(904\) 572.091 0.632844
\(905\) − 192.889i − 0.213137i
\(906\) 0 0
\(907\) −510.074 −0.562375 −0.281187 0.959653i \(-0.590728\pi\)
−0.281187 + 0.959653i \(0.590728\pi\)
\(908\) 339.604i 0.374013i
\(909\) 0 0
\(910\) 918.892 1.00977
\(911\) 927.371i 1.01797i 0.860775 + 0.508985i \(0.169979\pi\)
−0.860775 + 0.508985i \(0.830021\pi\)
\(912\) 0 0
\(913\) 829.727 0.908791
\(914\) 130.623i 0.142913i
\(915\) 0 0
\(916\) −815.090 −0.889836
\(917\) 31.4897i 0.0343399i
\(918\) 0 0
\(919\) −1240.63 −1.34998 −0.674991 0.737826i \(-0.735853\pi\)
−0.674991 + 0.737826i \(0.735853\pi\)
\(920\) − 138.721i − 0.150784i
\(921\) 0 0
\(922\) −325.015 −0.352511
\(923\) 1692.02i 1.83317i
\(924\) 0 0
\(925\) −34.6061 −0.0374120
\(926\) − 722.384i − 0.780112i
\(927\) 0 0
\(928\) −18.6061 −0.0200497
\(929\) − 338.992i − 0.364900i −0.983215 0.182450i \(-0.941597\pi\)
0.983215 0.182450i \(-0.0584028\pi\)
\(930\) 0 0
\(931\) −40.8490 −0.0438765
\(932\) 30.5124i 0.0327386i
\(933\) 0 0
\(934\) −1178.97 −1.26228
\(935\) 93.5307i 0.100033i
\(936\) 0 0
\(937\) 1322.21 1.41111 0.705556 0.708655i \(-0.250698\pi\)
0.705556 + 0.708655i \(0.250698\pi\)
\(938\) 277.913i 0.296283i
\(939\) 0 0
\(940\) −146.636 −0.155995
\(941\) − 359.093i − 0.381608i −0.981628 0.190804i \(-0.938891\pi\)
0.981628 0.190804i \(-0.0611095\pi\)
\(942\) 0 0
\(943\) 583.590 0.618866
\(944\) 366.110i 0.387828i
\(945\) 0 0
\(946\) 12.7423 0.0134697
\(947\) 775.792i 0.819210i 0.912263 + 0.409605i \(0.134333\pi\)
−0.912263 + 0.409605i \(0.865667\pi\)
\(948\) 0 0
\(949\) −1892.69 −1.99441
\(950\) − 13.2849i − 0.0139842i
\(951\) 0 0
\(952\) −34.2429 −0.0359694
\(953\) − 465.082i − 0.488019i −0.969773 0.244010i \(-0.921537\pi\)
0.969773 0.244010i \(-0.0784628\pi\)
\(954\) 0 0
\(955\) 93.1362 0.0975248
\(956\) − 113.073i − 0.118277i
\(957\) 0 0
\(958\) 929.408 0.970154
\(959\) − 1489.88i − 1.55358i
\(960\) 0 0
\(961\) 723.725 0.753096
\(962\) 481.989i 0.501028i
\(963\) 0 0
\(964\) −168.424 −0.174714
\(965\) 495.994i 0.513983i
\(966\) 0 0
\(967\) 1224.23 1.26600 0.633002 0.774150i \(-0.281822\pi\)
0.633002 + 0.774150i \(0.281822\pi\)
\(968\) − 90.2528i − 0.0932364i
\(969\) 0 0
\(970\) 704.561 0.726351
\(971\) − 658.702i − 0.678375i −0.940719 0.339188i \(-0.889848\pi\)
0.940719 0.339188i \(-0.110152\pi\)
\(972\) 0 0
\(973\) 676.303 0.695070
\(974\) − 497.331i − 0.510607i
\(975\) 0 0
\(976\) −300.363 −0.307749
\(977\) 1518.34i 1.55408i 0.629449 + 0.777042i \(0.283281\pi\)
−0.629449 + 0.777042i \(0.716719\pi\)
\(978\) 0 0
\(979\) −390.545 −0.398922
\(980\) − 90.3812i − 0.0922258i
\(981\) 0 0
\(982\) −347.074 −0.353436
\(983\) − 827.840i − 0.842157i −0.907024 0.421078i \(-0.861652\pi\)
0.907024 0.421078i \(-0.138348\pi\)
\(984\) 0 0
\(985\) −833.271 −0.845961
\(986\) 8.87058i 0.00899653i
\(987\) 0 0
\(988\) −185.031 −0.187278
\(989\) 9.01020i 0.00911041i
\(990\) 0 0
\(991\) 429.546 0.433447 0.216723 0.976233i \(-0.430463\pi\)
0.216723 + 0.976233i \(0.430463\pi\)
\(992\) − 232.188i − 0.234060i
\(993\) 0 0
\(994\) −771.242 −0.775897
\(995\) − 33.8545i − 0.0340247i
\(996\) 0 0
\(997\) −694.998 −0.697089 −0.348545 0.937292i \(-0.613324\pi\)
−0.348545 + 0.937292i \(0.613324\pi\)
\(998\) 891.274i 0.893060i
\(999\) 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.161.3 4
3.2 odd 2 inner 162.3.b.a.161.2 4
4.3 odd 2 1296.3.e.g.161.1 4
9.2 odd 6 18.3.d.a.5.1 4
9.4 even 3 18.3.d.a.11.1 yes 4
9.5 odd 6 54.3.d.a.35.2 4
9.7 even 3 54.3.d.a.17.2 4
12.11 even 2 1296.3.e.g.161.3 4
36.7 odd 6 432.3.q.d.17.2 4
36.11 even 6 144.3.q.c.113.1 4
36.23 even 6 432.3.q.d.305.2 4
36.31 odd 6 144.3.q.c.65.1 4
45.2 even 12 450.3.k.a.149.2 8
45.4 even 6 450.3.i.b.101.2 4
45.7 odd 12 1350.3.k.a.449.3 8
45.13 odd 12 450.3.k.a.299.2 8
45.14 odd 6 1350.3.i.b.251.1 4
45.22 odd 12 450.3.k.a.299.3 8
45.23 even 12 1350.3.k.a.899.3 8
45.29 odd 6 450.3.i.b.401.2 4
45.32 even 12 1350.3.k.a.899.2 8
45.34 even 6 1350.3.i.b.1151.1 4
45.38 even 12 450.3.k.a.149.3 8
45.43 odd 12 1350.3.k.a.449.2 8
72.5 odd 6 1728.3.q.d.1601.1 4
72.11 even 6 576.3.q.e.257.2 4
72.13 even 6 576.3.q.f.65.1 4
72.29 odd 6 576.3.q.f.257.1 4
72.43 odd 6 1728.3.q.c.449.2 4
72.59 even 6 1728.3.q.c.1601.2 4
72.61 even 6 1728.3.q.d.449.1 4
72.67 odd 6 576.3.q.e.65.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
18.3.d.a.5.1 4 9.2 odd 6
18.3.d.a.11.1 yes 4 9.4 even 3
54.3.d.a.17.2 4 9.7 even 3
54.3.d.a.35.2 4 9.5 odd 6
144.3.q.c.65.1 4 36.31 odd 6
144.3.q.c.113.1 4 36.11 even 6
162.3.b.a.161.2 4 3.2 odd 2 inner
162.3.b.a.161.3 4 1.1 even 1 trivial
432.3.q.d.17.2 4 36.7 odd 6
432.3.q.d.305.2 4 36.23 even 6
450.3.i.b.101.2 4 45.4 even 6
450.3.i.b.401.2 4 45.29 odd 6
450.3.k.a.149.2 8 45.2 even 12
450.3.k.a.149.3 8 45.38 even 12
450.3.k.a.299.2 8 45.13 odd 12
450.3.k.a.299.3 8 45.22 odd 12
576.3.q.e.65.2 4 72.67 odd 6
576.3.q.e.257.2 4 72.11 even 6
576.3.q.f.65.1 4 72.13 even 6
576.3.q.f.257.1 4 72.29 odd 6
1296.3.e.g.161.1 4 4.3 odd 2
1296.3.e.g.161.3 4 12.11 even 2
1350.3.i.b.251.1 4 45.14 odd 6
1350.3.i.b.1151.1 4 45.34 even 6
1350.3.k.a.449.2 8 45.43 odd 12
1350.3.k.a.449.3 8 45.7 odd 12
1350.3.k.a.899.2 8 45.32 even 12
1350.3.k.a.899.3 8 45.23 even 12
1728.3.q.c.449.2 4 72.43 odd 6
1728.3.q.c.1601.2 4 72.59 even 6
1728.3.q.d.449.1 4 72.61 even 6
1728.3.q.d.1601.1 4 72.5 odd 6