Properties

Label 1815.2.a.y.1.1
Level $1815$
Weight $2$
Character 1815.1
Self dual yes
Analytic conductor $14.493$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1815 = 3 \cdot 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1815.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(14.4928479669\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.6.437199552.1
Defining polynomial: \(x^{6} - 13 x^{4} + 49 x^{2} - 48\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-2.63162\) of defining polynomial
Character \(\chi\) \(=\) 1815.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.63162 q^{2} -1.00000 q^{3} +4.92542 q^{4} +1.00000 q^{5} +2.63162 q^{6} +4.16741 q^{7} -7.69860 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-2.63162 q^{2} -1.00000 q^{3} +4.92542 q^{4} +1.00000 q^{5} +2.63162 q^{6} +4.16741 q^{7} -7.69860 q^{8} +1.00000 q^{9} -2.63162 q^{10} -4.92542 q^{12} -5.26324 q^{13} -10.9670 q^{14} -1.00000 q^{15} +10.4089 q^{16} -4.23450 q^{17} -2.63162 q^{18} +2.50245 q^{19} +4.92542 q^{20} -4.16741 q^{21} +5.48352 q^{23} +7.69860 q^{24} +1.00000 q^{25} +13.8508 q^{26} -1.00000 q^{27} +20.5263 q^{28} -5.26324 q^{29} +2.63162 q^{30} +10.1162 q^{31} -11.9952 q^{32} +11.1436 q^{34} +4.16741 q^{35} +4.92542 q^{36} -4.48352 q^{37} -6.58549 q^{38} +5.26324 q^{39} -7.69860 q^{40} +3.46410 q^{41} +10.9670 q^{42} +6.66986 q^{43} +1.00000 q^{45} -14.4305 q^{46} +4.21817 q^{47} -10.4089 q^{48} +10.3673 q^{49} -2.63162 q^{50} +4.23450 q^{51} -25.9237 q^{52} -3.63268 q^{53} +2.63162 q^{54} -32.0832 q^{56} -2.50245 q^{57} +13.8508 q^{58} -6.58549 q^{59} -4.92542 q^{60} +12.2585 q^{61} -26.6220 q^{62} +4.16741 q^{63} +10.7489 q^{64} -5.26324 q^{65} -4.48352 q^{67} -20.8567 q^{68} -5.48352 q^{69} -10.9670 q^{70} +1.26535 q^{71} -7.69860 q^{72} -4.55993 q^{73} +11.7989 q^{74} -1.00000 q^{75} +12.3256 q^{76} -13.8508 q^{78} +6.86112 q^{79} +10.4089 q^{80} +1.00000 q^{81} -9.11620 q^{82} +4.87072 q^{83} -20.5263 q^{84} -4.23450 q^{85} -17.5525 q^{86} +5.26324 q^{87} +15.1162 q^{89} -2.63162 q^{90} -21.9341 q^{91} +27.0087 q^{92} -10.1162 q^{93} -11.1006 q^{94} +2.50245 q^{95} +11.9952 q^{96} -16.1852 q^{97} -27.2829 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q - 6q^{3} + 14q^{4} + 6q^{5} + 6q^{9} + O(q^{10}) \) \( 6q - 6q^{3} + 14q^{4} + 6q^{5} + 6q^{9} - 14q^{12} + 8q^{14} - 6q^{15} + 10q^{16} + 14q^{20} - 4q^{23} + 6q^{25} + 52q^{26} - 6q^{27} + 18q^{31} + 26q^{34} + 14q^{36} + 10q^{37} - 20q^{38} - 8q^{42} + 6q^{45} - 10q^{48} + 68q^{49} - 16q^{53} - 76q^{56} + 52q^{58} - 20q^{59} - 14q^{60} + 16q^{64} + 10q^{67} + 4q^{69} + 8q^{70} - 4q^{71} - 6q^{75} - 52q^{78} + 10q^{80} + 6q^{81} - 12q^{82} - 12q^{86} + 48q^{89} + 16q^{91} + 30q^{92} - 18q^{93} + 2q^{97} + O(q^{100}) \)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.63162 −1.86084 −0.930418 0.366500i \(-0.880556\pi\)
−0.930418 + 0.366500i \(0.880556\pi\)
\(3\) −1.00000 −0.577350
\(4\) 4.92542 2.46271
\(5\) 1.00000 0.447214
\(6\) 2.63162 1.07435
\(7\) 4.16741 1.57513 0.787567 0.616229i \(-0.211341\pi\)
0.787567 + 0.616229i \(0.211341\pi\)
\(8\) −7.69860 −2.72187
\(9\) 1.00000 0.333333
\(10\) −2.63162 −0.832191
\(11\) 0 0
\(12\) −4.92542 −1.42185
\(13\) −5.26324 −1.45976 −0.729880 0.683575i \(-0.760424\pi\)
−0.729880 + 0.683575i \(0.760424\pi\)
\(14\) −10.9670 −2.93107
\(15\) −1.00000 −0.258199
\(16\) 10.4089 2.60224
\(17\) −4.23450 −1.02702 −0.513508 0.858085i \(-0.671655\pi\)
−0.513508 + 0.858085i \(0.671655\pi\)
\(18\) −2.63162 −0.620279
\(19\) 2.50245 0.574101 0.287051 0.957915i \(-0.407325\pi\)
0.287051 + 0.957915i \(0.407325\pi\)
\(20\) 4.92542 1.10136
\(21\) −4.16741 −0.909404
\(22\) 0 0
\(23\) 5.48352 1.14339 0.571697 0.820465i \(-0.306285\pi\)
0.571697 + 0.820465i \(0.306285\pi\)
\(24\) 7.69860 1.57147
\(25\) 1.00000 0.200000
\(26\) 13.8508 2.71637
\(27\) −1.00000 −0.192450
\(28\) 20.5263 3.87910
\(29\) −5.26324 −0.977359 −0.488680 0.872463i \(-0.662521\pi\)
−0.488680 + 0.872463i \(0.662521\pi\)
\(30\) 2.63162 0.480466
\(31\) 10.1162 1.81692 0.908461 0.417969i \(-0.137258\pi\)
0.908461 + 0.417969i \(0.137258\pi\)
\(32\) −11.9952 −2.12047
\(33\) 0 0
\(34\) 11.1436 1.91111
\(35\) 4.16741 0.704421
\(36\) 4.92542 0.820904
\(37\) −4.48352 −0.737087 −0.368543 0.929611i \(-0.620143\pi\)
−0.368543 + 0.929611i \(0.620143\pi\)
\(38\) −6.58549 −1.06831
\(39\) 5.26324 0.842793
\(40\) −7.69860 −1.21726
\(41\) 3.46410 0.541002 0.270501 0.962720i \(-0.412811\pi\)
0.270501 + 0.962720i \(0.412811\pi\)
\(42\) 10.9670 1.69225
\(43\) 6.66986 1.01714 0.508572 0.861019i \(-0.330174\pi\)
0.508572 + 0.861019i \(0.330174\pi\)
\(44\) 0 0
\(45\) 1.00000 0.149071
\(46\) −14.4305 −2.12767
\(47\) 4.21817 0.615283 0.307642 0.951502i \(-0.400460\pi\)
0.307642 + 0.951502i \(0.400460\pi\)
\(48\) −10.4089 −1.50240
\(49\) 10.3673 1.48105
\(50\) −2.63162 −0.372167
\(51\) 4.23450 0.592949
\(52\) −25.9237 −3.59497
\(53\) −3.63268 −0.498986 −0.249493 0.968377i \(-0.580264\pi\)
−0.249493 + 0.968377i \(0.580264\pi\)
\(54\) 2.63162 0.358118
\(55\) 0 0
\(56\) −32.0832 −4.28730
\(57\) −2.50245 −0.331457
\(58\) 13.8508 1.81871
\(59\) −6.58549 −0.857358 −0.428679 0.903457i \(-0.641021\pi\)
−0.428679 + 0.903457i \(0.641021\pi\)
\(60\) −4.92542 −0.635869
\(61\) 12.2585 1.56954 0.784772 0.619785i \(-0.212780\pi\)
0.784772 + 0.619785i \(0.212780\pi\)
\(62\) −26.6220 −3.38100
\(63\) 4.16741 0.525045
\(64\) 10.7489 1.34361
\(65\) −5.26324 −0.652825
\(66\) 0 0
\(67\) −4.48352 −0.547749 −0.273875 0.961765i \(-0.588305\pi\)
−0.273875 + 0.961765i \(0.588305\pi\)
\(68\) −20.8567 −2.52925
\(69\) −5.48352 −0.660138
\(70\) −10.9670 −1.31081
\(71\) 1.26535 0.150170 0.0750849 0.997177i \(-0.476077\pi\)
0.0750849 + 0.997177i \(0.476077\pi\)
\(72\) −7.69860 −0.907289
\(73\) −4.55993 −0.533699 −0.266850 0.963738i \(-0.585983\pi\)
−0.266850 + 0.963738i \(0.585983\pi\)
\(74\) 11.7989 1.37160
\(75\) −1.00000 −0.115470
\(76\) 12.3256 1.41385
\(77\) 0 0
\(78\) −13.8508 −1.56830
\(79\) 6.86112 0.771936 0.385968 0.922512i \(-0.373868\pi\)
0.385968 + 0.922512i \(0.373868\pi\)
\(80\) 10.4089 1.16376
\(81\) 1.00000 0.111111
\(82\) −9.11620 −1.00672
\(83\) 4.87072 0.534631 0.267316 0.963609i \(-0.413863\pi\)
0.267316 + 0.963609i \(0.413863\pi\)
\(84\) −20.5263 −2.23960
\(85\) −4.23450 −0.459296
\(86\) −17.5525 −1.89274
\(87\) 5.26324 0.564279
\(88\) 0 0
\(89\) 15.1162 1.60231 0.801157 0.598454i \(-0.204218\pi\)
0.801157 + 0.598454i \(0.204218\pi\)
\(90\) −2.63162 −0.277397
\(91\) −21.9341 −2.29932
\(92\) 27.0087 2.81585
\(93\) −10.1162 −1.04900
\(94\) −11.1006 −1.14494
\(95\) 2.50245 0.256746
\(96\) 11.9952 1.22425
\(97\) −16.1852 −1.64336 −0.821680 0.569949i \(-0.806963\pi\)
−0.821680 + 0.569949i \(0.806963\pi\)
\(98\) −27.2829 −2.75598
\(99\) 0 0
\(100\) 4.92542 0.492542
\(101\) 3.59828 0.358042 0.179021 0.983845i \(-0.442707\pi\)
0.179021 + 0.983845i \(0.442707\pi\)
\(102\) −11.1436 −1.10338
\(103\) 1.36732 0.134726 0.0673632 0.997729i \(-0.478541\pi\)
0.0673632 + 0.997729i \(0.478541\pi\)
\(104\) 40.5196 3.97327
\(105\) −4.16741 −0.406698
\(106\) 9.55982 0.928532
\(107\) 12.9618 1.25307 0.626534 0.779394i \(-0.284473\pi\)
0.626534 + 0.779394i \(0.284473\pi\)
\(108\) −4.92542 −0.473949
\(109\) 5.96655 0.571492 0.285746 0.958305i \(-0.407759\pi\)
0.285746 + 0.958305i \(0.407759\pi\)
\(110\) 0 0
\(111\) 4.48352 0.425557
\(112\) 43.3784 4.09887
\(113\) −6.06902 −0.570925 −0.285462 0.958390i \(-0.592147\pi\)
−0.285462 + 0.958390i \(0.592147\pi\)
\(114\) 6.58549 0.616788
\(115\) 5.48352 0.511341
\(116\) −25.9237 −2.40695
\(117\) −5.26324 −0.486587
\(118\) 17.3305 1.59540
\(119\) −17.6469 −1.61769
\(120\) 7.69860 0.702783
\(121\) 0 0
\(122\) −32.2598 −2.92066
\(123\) −3.46410 −0.312348
\(124\) 49.8266 4.47456
\(125\) 1.00000 0.0894427
\(126\) −10.9670 −0.977022
\(127\) 9.82317 0.871665 0.435833 0.900028i \(-0.356454\pi\)
0.435833 + 0.900028i \(0.356454\pi\)
\(128\) −4.29658 −0.379768
\(129\) −6.66986 −0.587248
\(130\) 13.8508 1.21480
\(131\) 3.46410 0.302660 0.151330 0.988483i \(-0.451644\pi\)
0.151330 + 0.988483i \(0.451644\pi\)
\(132\) 0 0
\(133\) 10.4287 0.904286
\(134\) 11.7989 1.01927
\(135\) −1.00000 −0.0860663
\(136\) 32.5997 2.79540
\(137\) 20.5997 1.75995 0.879976 0.475017i \(-0.157558\pi\)
0.879976 + 0.475017i \(0.157558\pi\)
\(138\) 14.4305 1.22841
\(139\) −17.6984 −1.50116 −0.750579 0.660781i \(-0.770225\pi\)
−0.750579 + 0.660781i \(0.770225\pi\)
\(140\) 20.5263 1.73479
\(141\) −4.21817 −0.355234
\(142\) −3.32993 −0.279441
\(143\) 0 0
\(144\) 10.4089 0.867412
\(145\) −5.26324 −0.437088
\(146\) 12.0000 0.993127
\(147\) −10.3673 −0.855082
\(148\) −22.0832 −1.81523
\(149\) −20.5263 −1.68158 −0.840789 0.541363i \(-0.817908\pi\)
−0.840789 + 0.541363i \(0.817908\pi\)
\(150\) 2.63162 0.214871
\(151\) −3.84198 −0.312656 −0.156328 0.987705i \(-0.549966\pi\)
−0.156328 + 0.987705i \(0.549966\pi\)
\(152\) −19.2654 −1.56263
\(153\) −4.23450 −0.342339
\(154\) 0 0
\(155\) 10.1162 0.812553
\(156\) 25.9237 2.07556
\(157\) −0.781830 −0.0623969 −0.0311984 0.999513i \(-0.509932\pi\)
−0.0311984 + 0.999513i \(0.509932\pi\)
\(158\) −18.0558 −1.43645
\(159\) 3.63268 0.288090
\(160\) −11.9952 −0.948303
\(161\) 22.8521 1.80100
\(162\) −2.63162 −0.206760
\(163\) −7.74887 −0.606939 −0.303469 0.952841i \(-0.598145\pi\)
−0.303469 + 0.952841i \(0.598145\pi\)
\(164\) 17.0622 1.33233
\(165\) 0 0
\(166\) −12.8179 −0.994861
\(167\) 14.8851 1.15185 0.575924 0.817504i \(-0.304643\pi\)
0.575924 + 0.817504i \(0.304643\pi\)
\(168\) 32.0832 2.47528
\(169\) 14.7017 1.13090
\(170\) 11.1436 0.854675
\(171\) 2.50245 0.191367
\(172\) 32.8519 2.50493
\(173\) 2.19166 0.166628 0.0833142 0.996523i \(-0.473449\pi\)
0.0833142 + 0.996523i \(0.473449\pi\)
\(174\) −13.8508 −1.05003
\(175\) 4.16741 0.315027
\(176\) 0 0
\(177\) 6.58549 0.494996
\(178\) −39.7801 −2.98164
\(179\) −13.8508 −1.03526 −0.517630 0.855604i \(-0.673186\pi\)
−0.517630 + 0.855604i \(0.673186\pi\)
\(180\) 4.92542 0.367119
\(181\) 7.36732 0.547609 0.273804 0.961785i \(-0.411718\pi\)
0.273804 + 0.961785i \(0.411718\pi\)
\(182\) 57.7222 4.27865
\(183\) −12.2585 −0.906177
\(184\) −42.2154 −3.11216
\(185\) −4.48352 −0.329635
\(186\) 26.6220 1.95202
\(187\) 0 0
\(188\) 20.7763 1.51527
\(189\) −4.16741 −0.303135
\(190\) −6.58549 −0.477762
\(191\) 0.679859 0.0491929 0.0245965 0.999697i \(-0.492170\pi\)
0.0245965 + 0.999697i \(0.492170\pi\)
\(192\) −10.7489 −0.775733
\(193\) 4.95245 0.356485 0.178242 0.983987i \(-0.442959\pi\)
0.178242 + 0.983987i \(0.442959\pi\)
\(194\) 42.5933 3.05802
\(195\) 5.26324 0.376908
\(196\) 51.0635 3.64739
\(197\) 3.59828 0.256367 0.128183 0.991750i \(-0.459085\pi\)
0.128183 + 0.991750i \(0.459085\pi\)
\(198\) 0 0
\(199\) 3.88380 0.275315 0.137658 0.990480i \(-0.456043\pi\)
0.137658 + 0.990480i \(0.456043\pi\)
\(200\) −7.69860 −0.544373
\(201\) 4.48352 0.316243
\(202\) −9.46929 −0.666257
\(203\) −21.9341 −1.53947
\(204\) 20.8567 1.46026
\(205\) 3.46410 0.241943
\(206\) −3.59828 −0.250704
\(207\) 5.48352 0.381131
\(208\) −54.7848 −3.79864
\(209\) 0 0
\(210\) 10.9670 0.756798
\(211\) 24.4500 1.68321 0.841603 0.540097i \(-0.181612\pi\)
0.841603 + 0.540097i \(0.181612\pi\)
\(212\) −17.8925 −1.22886
\(213\) −1.26535 −0.0867005
\(214\) −34.1106 −2.33176
\(215\) 6.66986 0.454881
\(216\) 7.69860 0.523823
\(217\) 42.1584 2.86190
\(218\) −15.7017 −1.06345
\(219\) 4.55993 0.308131
\(220\) 0 0
\(221\) 22.2872 1.49920
\(222\) −11.7989 −0.791892
\(223\) 1.59972 0.107125 0.0535626 0.998564i \(-0.482942\pi\)
0.0535626 + 0.998564i \(0.482942\pi\)
\(224\) −49.9889 −3.34002
\(225\) 1.00000 0.0666667
\(226\) 15.9713 1.06240
\(227\) 18.6176 1.23569 0.617847 0.786299i \(-0.288005\pi\)
0.617847 + 0.786299i \(0.288005\pi\)
\(228\) −12.3256 −0.816284
\(229\) 23.3344 1.54198 0.770989 0.636848i \(-0.219762\pi\)
0.770989 + 0.636848i \(0.219762\pi\)
\(230\) −14.4305 −0.951522
\(231\) 0 0
\(232\) 40.5196 2.66024
\(233\) 9.75608 0.639142 0.319571 0.947562i \(-0.396461\pi\)
0.319571 + 0.947562i \(0.396461\pi\)
\(234\) 13.8508 0.905458
\(235\) 4.21817 0.275163
\(236\) −32.4363 −2.11143
\(237\) −6.86112 −0.445677
\(238\) 46.4399 3.01025
\(239\) −8.59317 −0.555846 −0.277923 0.960603i \(-0.589646\pi\)
−0.277923 + 0.960603i \(0.589646\pi\)
\(240\) −10.4089 −0.671895
\(241\) −8.97105 −0.577876 −0.288938 0.957348i \(-0.593302\pi\)
−0.288938 + 0.957348i \(0.593302\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 60.3784 3.86533
\(245\) 10.3673 0.662344
\(246\) 9.11620 0.581228
\(247\) −13.1710 −0.838050
\(248\) −77.8806 −4.94542
\(249\) −4.87072 −0.308670
\(250\) −2.63162 −0.166438
\(251\) 5.32014 0.335804 0.167902 0.985804i \(-0.446301\pi\)
0.167902 + 0.985804i \(0.446301\pi\)
\(252\) 20.5263 1.29303
\(253\) 0 0
\(254\) −25.8508 −1.62203
\(255\) 4.23450 0.265175
\(256\) −10.1908 −0.636923
\(257\) 27.7707 1.73229 0.866145 0.499794i \(-0.166591\pi\)
0.866145 + 0.499794i \(0.166591\pi\)
\(258\) 17.5525 1.09277
\(259\) −18.6847 −1.16101
\(260\) −25.9237 −1.60772
\(261\) −5.26324 −0.325786
\(262\) −9.11620 −0.563201
\(263\) 5.89946 0.363776 0.181888 0.983319i \(-0.441779\pi\)
0.181888 + 0.983319i \(0.441779\pi\)
\(264\) 0 0
\(265\) −3.63268 −0.223154
\(266\) −27.4445 −1.68273
\(267\) −15.1162 −0.925096
\(268\) −22.0832 −1.34895
\(269\) −6.23240 −0.379996 −0.189998 0.981784i \(-0.560848\pi\)
−0.189998 + 0.981784i \(0.560848\pi\)
\(270\) 2.63162 0.160155
\(271\) −29.7657 −1.80814 −0.904068 0.427389i \(-0.859433\pi\)
−0.904068 + 0.427389i \(0.859433\pi\)
\(272\) −44.0767 −2.67254
\(273\) 21.9341 1.32751
\(274\) −54.2106 −3.27498
\(275\) 0 0
\(276\) −27.0087 −1.62573
\(277\) 9.29648 0.558571 0.279286 0.960208i \(-0.409902\pi\)
0.279286 + 0.960208i \(0.409902\pi\)
\(278\) 46.5754 2.79341
\(279\) 10.1162 0.605641
\(280\) −32.0832 −1.91734
\(281\) 11.5406 0.688453 0.344227 0.938887i \(-0.388141\pi\)
0.344227 + 0.938887i \(0.388141\pi\)
\(282\) 11.1006 0.661032
\(283\) −7.50735 −0.446265 −0.223133 0.974788i \(-0.571628\pi\)
−0.223133 + 0.974788i \(0.571628\pi\)
\(284\) 6.23240 0.369825
\(285\) −2.50245 −0.148232
\(286\) 0 0
\(287\) 14.4363 0.852150
\(288\) −11.9952 −0.706823
\(289\) 0.930985 0.0547638
\(290\) 13.8508 0.813350
\(291\) 16.1852 0.948794
\(292\) −22.4596 −1.31435
\(293\) −18.0909 −1.05688 −0.528441 0.848970i \(-0.677223\pi\)
−0.528441 + 0.848970i \(0.677223\pi\)
\(294\) 27.2829 1.59117
\(295\) −6.58549 −0.383422
\(296\) 34.5168 2.00625
\(297\) 0 0
\(298\) 54.0173 3.12914
\(299\) −28.8611 −1.66908
\(300\) −4.92542 −0.284369
\(301\) 27.7961 1.60214
\(302\) 10.1106 0.581802
\(303\) −3.59828 −0.206716
\(304\) 26.0478 1.49395
\(305\) 12.2585 0.701921
\(306\) 11.1436 0.637037
\(307\) 3.90907 0.223102 0.111551 0.993759i \(-0.464418\pi\)
0.111551 + 0.993759i \(0.464418\pi\)
\(308\) 0 0
\(309\) −1.36732 −0.0777843
\(310\) −26.6220 −1.51203
\(311\) 22.5196 1.27697 0.638484 0.769635i \(-0.279562\pi\)
0.638484 + 0.769635i \(0.279562\pi\)
\(312\) −40.5196 −2.29397
\(313\) 19.2542 1.08831 0.544157 0.838984i \(-0.316850\pi\)
0.544157 + 0.838984i \(0.316850\pi\)
\(314\) 2.05748 0.116110
\(315\) 4.16741 0.234807
\(316\) 33.7939 1.90106
\(317\) −8.36732 −0.469956 −0.234978 0.972001i \(-0.575502\pi\)
−0.234978 + 0.972001i \(0.575502\pi\)
\(318\) −9.55982 −0.536088
\(319\) 0 0
\(320\) 10.7489 0.600880
\(321\) −12.9618 −0.723459
\(322\) −60.1380 −3.35136
\(323\) −10.5966 −0.589612
\(324\) 4.92542 0.273635
\(325\) −5.26324 −0.291952
\(326\) 20.3921 1.12941
\(327\) −5.96655 −0.329951
\(328\) −26.6687 −1.47253
\(329\) 17.5789 0.969153
\(330\) 0 0
\(331\) −12.5525 −0.689950 −0.344975 0.938612i \(-0.612113\pi\)
−0.344975 + 0.938612i \(0.612113\pi\)
\(332\) 23.9904 1.31664
\(333\) −4.48352 −0.245696
\(334\) −39.1720 −2.14340
\(335\) −4.48352 −0.244961
\(336\) −43.3784 −2.36648
\(337\) 21.3638 1.16376 0.581879 0.813275i \(-0.302318\pi\)
0.581879 + 0.813275i \(0.302318\pi\)
\(338\) −38.6893 −2.10442
\(339\) 6.06902 0.329624
\(340\) −20.8567 −1.13111
\(341\) 0 0
\(342\) −6.58549 −0.356103
\(343\) 14.0330 0.757712
\(344\) −51.3486 −2.76853
\(345\) −5.48352 −0.295223
\(346\) −5.76760 −0.310068
\(347\) 8.22529 0.441557 0.220778 0.975324i \(-0.429140\pi\)
0.220778 + 0.975324i \(0.429140\pi\)
\(348\) 25.9237 1.38966
\(349\) −6.11536 −0.327348 −0.163674 0.986515i \(-0.552334\pi\)
−0.163674 + 0.986515i \(0.552334\pi\)
\(350\) −10.9670 −0.586213
\(351\) 5.26324 0.280931
\(352\) 0 0
\(353\) −23.0361 −1.22609 −0.613043 0.790050i \(-0.710055\pi\)
−0.613043 + 0.790050i \(0.710055\pi\)
\(354\) −17.3305 −0.921106
\(355\) 1.26535 0.0671579
\(356\) 74.4537 3.94604
\(357\) 17.6469 0.933973
\(358\) 36.4502 1.92645
\(359\) 15.2630 0.805552 0.402776 0.915299i \(-0.368045\pi\)
0.402776 + 0.915299i \(0.368045\pi\)
\(360\) −7.69860 −0.405752
\(361\) −12.7378 −0.670408
\(362\) −19.3880 −1.01901
\(363\) 0 0
\(364\) −108.035 −5.66255
\(365\) −4.55993 −0.238678
\(366\) 32.2598 1.68625
\(367\) −32.3816 −1.69030 −0.845152 0.534527i \(-0.820490\pi\)
−0.845152 + 0.534527i \(0.820490\pi\)
\(368\) 57.0777 2.97538
\(369\) 3.46410 0.180334
\(370\) 11.7989 0.613397
\(371\) −15.1389 −0.785970
\(372\) −49.8266 −2.58339
\(373\) 8.15821 0.422416 0.211208 0.977441i \(-0.432260\pi\)
0.211208 + 0.977441i \(0.432260\pi\)
\(374\) 0 0
\(375\) −1.00000 −0.0516398
\(376\) −32.4740 −1.67472
\(377\) 27.7017 1.42671
\(378\) 10.9670 0.564084
\(379\) 7.48352 0.384403 0.192201 0.981356i \(-0.438437\pi\)
0.192201 + 0.981356i \(0.438437\pi\)
\(380\) 12.3256 0.632291
\(381\) −9.82317 −0.503256
\(382\) −1.78913 −0.0915399
\(383\) −18.1380 −0.926810 −0.463405 0.886147i \(-0.653372\pi\)
−0.463405 + 0.886147i \(0.653372\pi\)
\(384\) 4.29658 0.219259
\(385\) 0 0
\(386\) −13.0330 −0.663360
\(387\) 6.66986 0.339048
\(388\) −79.7190 −4.04712
\(389\) 17.7849 0.901732 0.450866 0.892592i \(-0.351115\pi\)
0.450866 + 0.892592i \(0.351115\pi\)
\(390\) −13.8508 −0.701365
\(391\) −23.2200 −1.17428
\(392\) −79.8139 −4.03121
\(393\) −3.46410 −0.174741
\(394\) −9.46929 −0.477056
\(395\) 6.86112 0.345220
\(396\) 0 0
\(397\) −8.33437 −0.418290 −0.209145 0.977885i \(-0.567068\pi\)
−0.209145 + 0.977885i \(0.567068\pi\)
\(398\) −10.2207 −0.512317
\(399\) −10.4287 −0.522090
\(400\) 10.4089 0.520447
\(401\) −13.2654 −0.662440 −0.331220 0.943554i \(-0.607460\pi\)
−0.331220 + 0.943554i \(0.607460\pi\)
\(402\) −11.7989 −0.588477
\(403\) −53.2440 −2.65227
\(404\) 17.7230 0.881754
\(405\) 1.00000 0.0496904
\(406\) 57.7222 2.86470
\(407\) 0 0
\(408\) −32.5997 −1.61393
\(409\) −16.8609 −0.833718 −0.416859 0.908971i \(-0.636869\pi\)
−0.416859 + 0.908971i \(0.636869\pi\)
\(410\) −9.11620 −0.450217
\(411\) −20.5997 −1.01611
\(412\) 6.73465 0.331792
\(413\) −27.4445 −1.35045
\(414\) −14.4305 −0.709223
\(415\) 4.87072 0.239094
\(416\) 63.1335 3.09538
\(417\) 17.6984 0.866694
\(418\) 0 0
\(419\) 9.79606 0.478569 0.239284 0.970950i \(-0.423087\pi\)
0.239284 + 0.970950i \(0.423087\pi\)
\(420\) −20.5263 −1.00158
\(421\) −17.8651 −0.870690 −0.435345 0.900264i \(-0.643374\pi\)
−0.435345 + 0.900264i \(0.643374\pi\)
\(422\) −64.3430 −3.13217
\(423\) 4.21817 0.205094
\(424\) 27.9665 1.35817
\(425\) −4.23450 −0.205403
\(426\) 3.32993 0.161335
\(427\) 51.0863 2.47224
\(428\) 63.8425 3.08595
\(429\) 0 0
\(430\) −17.5525 −0.846459
\(431\) 37.1959 1.79166 0.895832 0.444393i \(-0.146580\pi\)
0.895832 + 0.444393i \(0.146580\pi\)
\(432\) −10.4089 −0.500801
\(433\) −10.1852 −0.489470 −0.244735 0.969590i \(-0.578701\pi\)
−0.244735 + 0.969590i \(0.578701\pi\)
\(434\) −110.945 −5.32552
\(435\) 5.26324 0.252353
\(436\) 29.3878 1.40742
\(437\) 13.7222 0.656423
\(438\) −12.0000 −0.573382
\(439\) −19.7134 −0.940870 −0.470435 0.882435i \(-0.655903\pi\)
−0.470435 + 0.882435i \(0.655903\pi\)
\(440\) 0 0
\(441\) 10.3673 0.493682
\(442\) −58.6514 −2.78976
\(443\) −26.4363 −1.25603 −0.628014 0.778202i \(-0.716132\pi\)
−0.628014 + 0.778202i \(0.716132\pi\)
\(444\) 22.0832 1.04802
\(445\) 15.1162 0.716577
\(446\) −4.20986 −0.199342
\(447\) 20.5263 0.970859
\(448\) 44.7950 2.11636
\(449\) −21.7017 −1.02417 −0.512083 0.858936i \(-0.671126\pi\)
−0.512083 + 0.858936i \(0.671126\pi\)
\(450\) −2.63162 −0.124056
\(451\) 0 0
\(452\) −29.8925 −1.40602
\(453\) 3.84198 0.180512
\(454\) −48.9944 −2.29942
\(455\) −21.9341 −1.02829
\(456\) 19.2654 0.902183
\(457\) 11.7989 0.551930 0.275965 0.961168i \(-0.411003\pi\)
0.275965 + 0.961168i \(0.411003\pi\)
\(458\) −61.4072 −2.86937
\(459\) 4.23450 0.197650
\(460\) 27.0087 1.25929
\(461\) 31.0527 1.44627 0.723135 0.690706i \(-0.242700\pi\)
0.723135 + 0.690706i \(0.242700\pi\)
\(462\) 0 0
\(463\) 38.3704 1.78323 0.891613 0.452799i \(-0.149575\pi\)
0.891613 + 0.452799i \(0.149575\pi\)
\(464\) −54.7848 −2.54332
\(465\) −10.1162 −0.469127
\(466\) −25.6743 −1.18934
\(467\) 14.9528 0.691934 0.345967 0.938247i \(-0.387551\pi\)
0.345967 + 0.938247i \(0.387551\pi\)
\(468\) −25.9237 −1.19832
\(469\) −18.6847 −0.862779
\(470\) −11.1006 −0.512033
\(471\) 0.781830 0.0360248
\(472\) 50.6991 2.33361
\(473\) 0 0
\(474\) 18.0558 0.829333
\(475\) 2.50245 0.114820
\(476\) −86.9185 −3.98390
\(477\) −3.63268 −0.166329
\(478\) 22.6139 1.03434
\(479\) 12.1914 0.557041 0.278521 0.960430i \(-0.410156\pi\)
0.278521 + 0.960430i \(0.410156\pi\)
\(480\) 11.9952 0.547503
\(481\) 23.5979 1.07597
\(482\) 23.6084 1.07533
\(483\) −22.8521 −1.03981
\(484\) 0 0
\(485\) −16.1852 −0.734933
\(486\) 2.63162 0.119373
\(487\) 14.2872 0.647414 0.323707 0.946157i \(-0.395071\pi\)
0.323707 + 0.946157i \(0.395071\pi\)
\(488\) −94.3735 −4.27209
\(489\) 7.74887 0.350416
\(490\) −27.2829 −1.23251
\(491\) −26.0579 −1.17597 −0.587987 0.808870i \(-0.700079\pi\)
−0.587987 + 0.808870i \(0.700079\pi\)
\(492\) −17.0622 −0.769222
\(493\) 22.2872 1.00376
\(494\) 34.6610 1.55947
\(495\) 0 0
\(496\) 105.299 4.72806
\(497\) 5.27325 0.236537
\(498\) 12.8179 0.574383
\(499\) 28.8651 1.29218 0.646089 0.763262i \(-0.276403\pi\)
0.646089 + 0.763262i \(0.276403\pi\)
\(500\) 4.92542 0.220272
\(501\) −14.8851 −0.665019
\(502\) −14.0006 −0.624877
\(503\) 39.3729 1.75555 0.877776 0.479071i \(-0.159026\pi\)
0.877776 + 0.479071i \(0.159026\pi\)
\(504\) −32.0832 −1.42910
\(505\) 3.59828 0.160121
\(506\) 0 0
\(507\) −14.7017 −0.652925
\(508\) 48.3833 2.14666
\(509\) 20.0832 0.890174 0.445087 0.895487i \(-0.353173\pi\)
0.445087 + 0.895487i \(0.353173\pi\)
\(510\) −11.1436 −0.493447
\(511\) −19.0031 −0.840648
\(512\) 35.4114 1.56498
\(513\) −2.50245 −0.110486
\(514\) −73.0819 −3.22351
\(515\) 1.36732 0.0602515
\(516\) −32.8519 −1.44622
\(517\) 0 0
\(518\) 49.1710 2.16045
\(519\) −2.19166 −0.0962030
\(520\) 40.5196 1.77690
\(521\) −34.7520 −1.52251 −0.761256 0.648452i \(-0.775417\pi\)
−0.761256 + 0.648452i \(0.775417\pi\)
\(522\) 13.8508 0.606235
\(523\) 14.6939 0.642519 0.321260 0.946991i \(-0.395894\pi\)
0.321260 + 0.946991i \(0.395894\pi\)
\(524\) 17.0622 0.745364
\(525\) −4.16741 −0.181881
\(526\) −15.5251 −0.676928
\(527\) −42.8370 −1.86601
\(528\) 0 0
\(529\) 7.06902 0.307348
\(530\) 9.55982 0.415252
\(531\) −6.58549 −0.285786
\(532\) 51.3659 2.22700
\(533\) −18.2324 −0.789733
\(534\) 39.7801 1.72145
\(535\) 12.9618 0.560389
\(536\) 34.5168 1.49090
\(537\) 13.8508 0.597708
\(538\) 16.4013 0.707110
\(539\) 0 0
\(540\) −4.92542 −0.211956
\(541\) −3.32993 −0.143165 −0.0715824 0.997435i \(-0.522805\pi\)
−0.0715824 + 0.997435i \(0.522805\pi\)
\(542\) 78.3319 3.36464
\(543\) −7.36732 −0.316162
\(544\) 50.7936 2.17776
\(545\) 5.96655 0.255579
\(546\) −57.7222 −2.47028
\(547\) −27.4545 −1.17387 −0.586934 0.809635i \(-0.699665\pi\)
−0.586934 + 0.809635i \(0.699665\pi\)
\(548\) 101.462 4.33426
\(549\) 12.2585 0.523181
\(550\) 0 0
\(551\) −13.1710 −0.561103
\(552\) 42.2154 1.79681
\(553\) 28.5931 1.21590
\(554\) −24.4648 −1.03941
\(555\) 4.48352 0.190315
\(556\) −87.1720 −3.69692
\(557\) 26.8283 1.13675 0.568375 0.822770i \(-0.307572\pi\)
0.568375 + 0.822770i \(0.307572\pi\)
\(558\) −26.6220 −1.12700
\(559\) −35.1051 −1.48479
\(560\) 43.3784 1.83307
\(561\) 0 0
\(562\) −30.3704 −1.28110
\(563\) 36.9668 1.55797 0.778983 0.627045i \(-0.215736\pi\)
0.778983 + 0.627045i \(0.215736\pi\)
\(564\) −20.7763 −0.874839
\(565\) −6.06902 −0.255325
\(566\) 19.7565 0.830427
\(567\) 4.16741 0.175015
\(568\) −9.74145 −0.408742
\(569\) −32.4594 −1.36077 −0.680384 0.732856i \(-0.738187\pi\)
−0.680384 + 0.732856i \(0.738187\pi\)
\(570\) 6.58549 0.275836
\(571\) −18.0092 −0.753661 −0.376830 0.926282i \(-0.622986\pi\)
−0.376830 + 0.926282i \(0.622986\pi\)
\(572\) 0 0
\(573\) −0.679859 −0.0284015
\(574\) −37.9910 −1.58571
\(575\) 5.48352 0.228679
\(576\) 10.7489 0.447870
\(577\) 39.5997 1.64856 0.824279 0.566184i \(-0.191581\pi\)
0.824279 + 0.566184i \(0.191581\pi\)
\(578\) −2.45000 −0.101906
\(579\) −4.95245 −0.205817
\(580\) −25.9237 −1.07642
\(581\) 20.2983 0.842116
\(582\) −42.5933 −1.76555
\(583\) 0 0
\(584\) 35.1051 1.45266
\(585\) −5.26324 −0.217608
\(586\) 47.6084 1.96668
\(587\) −24.9813 −1.03109 −0.515544 0.856863i \(-0.672410\pi\)
−0.515544 + 0.856863i \(0.672410\pi\)
\(588\) −51.0635 −2.10582
\(589\) 25.3153 1.04310
\(590\) 17.3305 0.713486
\(591\) −3.59828 −0.148013
\(592\) −46.6687 −1.91807
\(593\) 1.92331 0.0789807 0.0394904 0.999220i \(-0.487427\pi\)
0.0394904 + 0.999220i \(0.487427\pi\)
\(594\) 0 0
\(595\) −17.6469 −0.723453
\(596\) −101.101 −4.14124
\(597\) −3.88380 −0.158953
\(598\) 75.9514 3.10588
\(599\) 19.9889 0.816723 0.408362 0.912820i \(-0.366100\pi\)
0.408362 + 0.912820i \(0.366100\pi\)
\(600\) 7.69860 0.314294
\(601\) 0.176619 0.00720444 0.00360222 0.999994i \(-0.498853\pi\)
0.00360222 + 0.999994i \(0.498853\pi\)
\(602\) −73.1487 −2.98132
\(603\) −4.48352 −0.182583
\(604\) −18.9234 −0.769982
\(605\) 0 0
\(606\) 9.46929 0.384664
\(607\) −11.8089 −0.479310 −0.239655 0.970858i \(-0.577034\pi\)
−0.239655 + 0.970858i \(0.577034\pi\)
\(608\) −30.0173 −1.21736
\(609\) 21.9341 0.888814
\(610\) −32.2598 −1.30616
\(611\) −22.2012 −0.898166
\(612\) −20.8567 −0.843082
\(613\) 13.4114 0.541683 0.270841 0.962624i \(-0.412698\pi\)
0.270841 + 0.962624i \(0.412698\pi\)
\(614\) −10.2872 −0.415157
\(615\) −3.46410 −0.139686
\(616\) 0 0
\(617\) −27.9341 −1.12458 −0.562292 0.826939i \(-0.690080\pi\)
−0.562292 + 0.826939i \(0.690080\pi\)
\(618\) 3.59828 0.144744
\(619\) 38.2324 1.53669 0.768345 0.640036i \(-0.221081\pi\)
0.768345 + 0.640036i \(0.221081\pi\)
\(620\) 49.8266 2.00108
\(621\) −5.48352 −0.220046
\(622\) −59.2630 −2.37623
\(623\) 62.9954 2.52386
\(624\) 54.7848 2.19315
\(625\) 1.00000 0.0400000
\(626\) −50.6698 −2.02517
\(627\) 0 0
\(628\) −3.85085 −0.153665
\(629\) 18.9855 0.757000
\(630\) −10.9670 −0.436938
\(631\) 0.952817 0.0379310 0.0189655 0.999820i \(-0.493963\pi\)
0.0189655 + 0.999820i \(0.493963\pi\)
\(632\) −52.8210 −2.10111
\(633\) −24.4500 −0.971799
\(634\) 22.0196 0.874511
\(635\) 9.82317 0.389821
\(636\) 17.8925 0.709482
\(637\) −54.5657 −2.16197
\(638\) 0 0
\(639\) 1.26535 0.0500566
\(640\) −4.29658 −0.169837
\(641\) 4.47592 0.176788 0.0883941 0.996086i \(-0.471827\pi\)
0.0883941 + 0.996086i \(0.471827\pi\)
\(642\) 34.1106 1.34624
\(643\) 21.8980 0.863574 0.431787 0.901976i \(-0.357883\pi\)
0.431787 + 0.901976i \(0.357883\pi\)
\(644\) 112.556 4.43534
\(645\) −6.66986 −0.262626
\(646\) 27.8863 1.09717
\(647\) 21.2796 0.836587 0.418293 0.908312i \(-0.362628\pi\)
0.418293 + 0.908312i \(0.362628\pi\)
\(648\) −7.69860 −0.302430
\(649\) 0 0
\(650\) 13.8508 0.543275
\(651\) −42.1584 −1.65232
\(652\) −38.1665 −1.49471
\(653\) 20.5307 0.803429 0.401714 0.915765i \(-0.368414\pi\)
0.401714 + 0.915765i \(0.368414\pi\)
\(654\) 15.7017 0.613985
\(655\) 3.46410 0.135354
\(656\) 36.0576 1.40781
\(657\) −4.55993 −0.177900
\(658\) −46.2609 −1.80344
\(659\) −18.9855 −0.739569 −0.369784 0.929118i \(-0.620568\pi\)
−0.369784 + 0.929118i \(0.620568\pi\)
\(660\) 0 0
\(661\) −2.86507 −0.111438 −0.0557192 0.998446i \(-0.517745\pi\)
−0.0557192 + 0.998446i \(0.517745\pi\)
\(662\) 33.0335 1.28388
\(663\) −22.2872 −0.865563
\(664\) −37.4977 −1.45519
\(665\) 10.4287 0.404409
\(666\) 11.7989 0.457199
\(667\) −28.8611 −1.11751
\(668\) 73.3156 2.83667
\(669\) −1.59972 −0.0618488
\(670\) 11.7989 0.455832
\(671\) 0 0
\(672\) 49.9889 1.92836
\(673\) −49.6033 −1.91207 −0.956033 0.293261i \(-0.905260\pi\)
−0.956033 + 0.293261i \(0.905260\pi\)
\(674\) −56.2213 −2.16556
\(675\) −1.00000 −0.0384900
\(676\) 72.4120 2.78508
\(677\) −34.3927 −1.32182 −0.660909 0.750466i \(-0.729829\pi\)
−0.660909 + 0.750466i \(0.729829\pi\)
\(678\) −15.9713 −0.613376
\(679\) −67.4505 −2.58851
\(680\) 32.5997 1.25014
\(681\) −18.6176 −0.713428
\(682\) 0 0
\(683\) 8.20394 0.313915 0.156958 0.987605i \(-0.449831\pi\)
0.156958 + 0.987605i \(0.449831\pi\)
\(684\) 12.3256 0.471282
\(685\) 20.5997 0.787075
\(686\) −36.9296 −1.40998
\(687\) −23.3344 −0.890262
\(688\) 69.4262 2.64685
\(689\) 19.1196 0.728401
\(690\) 14.4305 0.549361
\(691\) −21.0832 −0.802044 −0.401022 0.916068i \(-0.631345\pi\)
−0.401022 + 0.916068i \(0.631345\pi\)
\(692\) 10.7948 0.410358
\(693\) 0 0
\(694\) −21.6458 −0.821665
\(695\) −17.6984 −0.671338
\(696\) −40.5196 −1.53589
\(697\) −14.6687 −0.555618
\(698\) 16.0933 0.609141
\(699\) −9.75608 −0.369009
\(700\) 20.5263 0.775820
\(701\) 22.0671 0.833461 0.416731 0.909030i \(-0.363176\pi\)
0.416731 + 0.909030i \(0.363176\pi\)
\(702\) −13.8508 −0.522766
\(703\) −11.2198 −0.423162
\(704\) 0 0
\(705\) −4.21817 −0.158865
\(706\) 60.6222 2.28154
\(707\) 14.9955 0.563964
\(708\) 32.4363 1.21903
\(709\) −39.7992 −1.49469 −0.747344 0.664437i \(-0.768671\pi\)
−0.747344 + 0.664437i \(0.768671\pi\)
\(710\) −3.32993 −0.124970
\(711\) 6.86112 0.257312
\(712\) −116.374 −4.36128
\(713\) 55.4724 2.07746
\(714\) −46.4399 −1.73797
\(715\) 0 0
\(716\) −68.2213 −2.54955
\(717\) 8.59317 0.320918
\(718\) −40.1665 −1.49900
\(719\) 12.1380 0.452672 0.226336 0.974049i \(-0.427325\pi\)
0.226336 + 0.974049i \(0.427325\pi\)
\(720\) 10.4089 0.387918
\(721\) 5.69820 0.212212
\(722\) 33.5209 1.24752
\(723\) 8.97105 0.333637
\(724\) 36.2872 1.34860
\(725\) −5.26324 −0.195472
\(726\) 0 0
\(727\) 38.9670 1.44521 0.722604 0.691262i \(-0.242945\pi\)
0.722604 + 0.691262i \(0.242945\pi\)
\(728\) 168.862 6.25843
\(729\) 1.00000 0.0370370
\(730\) 12.0000 0.444140
\(731\) −28.2435 −1.04462
\(732\) −60.3784 −2.23165
\(733\) 5.65576 0.208900 0.104450 0.994530i \(-0.466692\pi\)
0.104450 + 0.994530i \(0.466692\pi\)
\(734\) 85.2159 3.14538
\(735\) −10.3673 −0.382404
\(736\) −65.7759 −2.42453
\(737\) 0 0
\(738\) −9.11620 −0.335572
\(739\) −28.8040 −1.05957 −0.529786 0.848131i \(-0.677728\pi\)
−0.529786 + 0.848131i \(0.677728\pi\)
\(740\) −22.0832 −0.811796
\(741\) 13.1710 0.483848
\(742\) 39.8397 1.46256
\(743\) −43.2688 −1.58738 −0.793690 0.608323i \(-0.791843\pi\)
−0.793690 + 0.608323i \(0.791843\pi\)
\(744\) 77.8806 2.85524
\(745\) −20.5263 −0.752024
\(746\) −21.4693 −0.786047
\(747\) 4.87072 0.178210
\(748\) 0 0
\(749\) 54.0173 1.97375
\(750\) 2.63162 0.0960932
\(751\) 7.58549 0.276799 0.138399 0.990377i \(-0.455804\pi\)
0.138399 + 0.990377i \(0.455804\pi\)
\(752\) 43.9067 1.60111
\(753\) −5.32014 −0.193877
\(754\) −72.9003 −2.65487
\(755\) −3.84198 −0.139824
\(756\) −20.5263 −0.746533
\(757\) 42.1741 1.53284 0.766422 0.642338i \(-0.222035\pi\)
0.766422 + 0.642338i \(0.222035\pi\)
\(758\) −19.6938 −0.715310
\(759\) 0 0
\(760\) −19.2654 −0.698828
\(761\) 6.15318 0.223052 0.111526 0.993761i \(-0.464426\pi\)
0.111526 + 0.993761i \(0.464426\pi\)
\(762\) 25.8508 0.936477
\(763\) 24.8651 0.900176
\(764\) 3.34860 0.121148
\(765\) −4.23450 −0.153099
\(766\) 47.7324 1.72464
\(767\) 34.6610 1.25154
\(768\) 10.1908 0.367728
\(769\) 20.4592 0.737777 0.368888 0.929474i \(-0.379738\pi\)
0.368888 + 0.929474i \(0.379738\pi\)
\(770\) 0 0
\(771\) −27.7707 −1.00014
\(772\) 24.3929 0.877919
\(773\) 34.2355 1.23137 0.615683 0.787994i \(-0.288880\pi\)
0.615683 + 0.787994i \(0.288880\pi\)
\(774\) −17.5525 −0.630913
\(775\) 10.1162 0.363385
\(776\) 124.604 4.47301
\(777\) 18.6847 0.670309
\(778\) −46.8032 −1.67798
\(779\) 8.66874 0.310590
\(780\) 25.9237 0.928217
\(781\) 0 0
\(782\) 61.1061 2.18515
\(783\) 5.26324 0.188093
\(784\) 107.913 3.85403
\(785\) −0.781830 −0.0279047
\(786\) 9.11620 0.325164
\(787\) −4.73655 −0.168840 −0.0844199 0.996430i \(-0.526904\pi\)
−0.0844199 + 0.996430i \(0.526904\pi\)
\(788\) 17.7230 0.631357
\(789\) −5.89946 −0.210026
\(790\) −18.0558 −0.642398
\(791\) −25.2921 −0.899283
\(792\) 0 0
\(793\) −64.5196 −2.29116
\(794\) 21.9329 0.778369
\(795\) 3.63268 0.128838
\(796\) 19.1294 0.678022
\(797\) −20.5307 −0.727235 −0.363617 0.931548i \(-0.618458\pi\)
−0.363617 + 0.931548i \(0.618458\pi\)
\(798\) 27.4445 0.971523
\(799\) −17.8618 −0.631906
\(800\) −11.9952 −0.424094
\(801\) 15.1162 0.534105
\(802\) 34.9094 1.23269
\(803\) 0 0
\(804\) 22.0832 0.778816
\(805\) 22.8521 0.805431
\(806\) 140.118 4.93544
\(807\) 6.23240 0.219391
\(808\) −27.7017 −0.974542
\(809\) −48.8707 −1.71820 −0.859101 0.511807i \(-0.828976\pi\)
−0.859101 + 0.511807i \(0.828976\pi\)
\(810\) −2.63162 −0.0924657
\(811\) −10.9468 −0.384394 −0.192197 0.981356i \(-0.561561\pi\)
−0.192197 + 0.981356i \(0.561561\pi\)
\(812\) −108.035 −3.79127
\(813\) 29.7657 1.04393
\(814\) 0 0
\(815\) −7.74887 −0.271431
\(816\) 44.0767 1.54299
\(817\) 16.6910 0.583944
\(818\) 44.3715 1.55141
\(819\) −21.9341 −0.766439
\(820\) 17.0622 0.595837
\(821\) −26.8136 −0.935802 −0.467901 0.883781i \(-0.654990\pi\)
−0.467901 + 0.883781i \(0.654990\pi\)
\(822\) 54.2106 1.89081
\(823\) −20.1020 −0.700711 −0.350355 0.936617i \(-0.613939\pi\)
−0.350355 + 0.936617i \(0.613939\pi\)
\(824\) −10.5265 −0.366707
\(825\) 0 0
\(826\) 72.2234 2.51297
\(827\) 23.1104 0.803629 0.401814 0.915721i \(-0.368380\pi\)
0.401814 + 0.915721i \(0.368380\pi\)
\(828\) 27.0087 0.938616
\(829\) −41.0721 −1.42649 −0.713247 0.700913i \(-0.752776\pi\)
−0.713247 + 0.700913i \(0.752776\pi\)
\(830\) −12.8179 −0.444915
\(831\) −9.29648 −0.322491
\(832\) −56.5739 −1.96135
\(833\) −43.9004 −1.52106
\(834\) −46.5754 −1.61278
\(835\) 14.8851 0.515122
\(836\) 0 0
\(837\) −10.1162 −0.349667
\(838\) −25.7795 −0.890538
\(839\) −7.40338 −0.255593 −0.127797 0.991800i \(-0.540790\pi\)
−0.127797 + 0.991800i \(0.540790\pi\)
\(840\) 32.0832 1.10698
\(841\) −1.29831 −0.0447693
\(842\) 47.0141 1.62021
\(843\) −11.5406 −0.397479
\(844\) 120.426 4.14525
\(845\) 14.7017 0.505754
\(846\) −11.1006 −0.381647
\(847\) 0 0
\(848\) −37.8123 −1.29848
\(849\) 7.50735 0.257651
\(850\) 11.1436 0.382222
\(851\) −24.5855 −0.842780
\(852\) −6.23240 −0.213518
\(853\) −26.3686 −0.902845 −0.451423 0.892310i \(-0.649083\pi\)
−0.451423 + 0.892310i \(0.649083\pi\)
\(854\) −134.440 −4.60044
\(855\) 2.50245 0.0855819
\(856\) −99.7880 −3.41068
\(857\) 38.3588 1.31031 0.655156 0.755493i \(-0.272603\pi\)
0.655156 + 0.755493i \(0.272603\pi\)
\(858\) 0 0
\(859\) 2.53831 0.0866060 0.0433030 0.999062i \(-0.486212\pi\)
0.0433030 + 0.999062i \(0.486212\pi\)
\(860\) 32.8519 1.12024
\(861\) −14.4363 −0.491989
\(862\) −97.8855 −3.33399
\(863\) −12.0000 −0.408485 −0.204242 0.978920i \(-0.565473\pi\)
−0.204242 + 0.978920i \(0.565473\pi\)
\(864\) 11.9952 0.408084
\(865\) 2.19166 0.0745185
\(866\) 26.8036 0.910824
\(867\) −0.930985 −0.0316179
\(868\) 207.648 7.04803
\(869\) 0 0
\(870\) −13.8508 −0.469588
\(871\) 23.5979 0.799583
\(872\) −45.9341 −1.55552
\(873\) −16.1852 −0.547786
\(874\) −36.1117 −1.22150
\(875\) 4.16741 0.140884
\(876\) 22.4596 0.758839
\(877\) 32.1386 1.08524 0.542621 0.839978i \(-0.317432\pi\)
0.542621 + 0.839978i \(0.317432\pi\)
\(878\) 51.8782 1.75081
\(879\) 18.0909 0.610191
\(880\) 0 0
\(881\) 47.1051 1.58701 0.793505 0.608564i \(-0.208254\pi\)
0.793505 + 0.608564i \(0.208254\pi\)
\(882\) −27.2829 −0.918661
\(883\) −19.9528 −0.671466 −0.335733 0.941957i \(-0.608984\pi\)
−0.335733 + 0.941957i \(0.608984\pi\)
\(884\) 109.774 3.69209
\(885\) 6.58549 0.221369
\(886\) 69.5704 2.33726
\(887\) 48.6316 1.63289 0.816445 0.577424i \(-0.195942\pi\)
0.816445 + 0.577424i \(0.195942\pi\)
\(888\) −34.5168 −1.15831
\(889\) 40.9372 1.37299
\(890\) −39.7801 −1.33343
\(891\) 0 0
\(892\) 7.87930 0.263819
\(893\) 10.5558 0.353235
\(894\) −54.0173 −1.80661
\(895\) −13.8508 −0.462983
\(896\) −17.9056 −0.598185
\(897\) 28.8611 0.963644
\(898\) 57.1106 1.90581
\(899\) −53.2440 −1.77579
\(900\) 4.92542 0.164181
\(901\) 15.3826 0.512468
\(902\) 0 0
\(903\) −27.7961 −0.924995
\(904\) 46.7229 1.55398
\(905\) 7.36732 0.244898
\(906\) −10.1106 −0.335903
\(907\) 40.0361 1.32938 0.664688 0.747121i \(-0.268565\pi\)
0.664688 + 0.747121i \(0.268565\pi\)
\(908\) 91.6995 3.04316
\(909\) 3.59828 0.119347
\(910\) 57.7222 1.91347
\(911\) −22.1492 −0.733834 −0.366917 0.930254i \(-0.619587\pi\)
−0.366917 + 0.930254i \(0.619587\pi\)
\(912\) −26.0478 −0.862530
\(913\) 0 0
\(914\) −31.0503 −1.02705
\(915\) −12.2585 −0.405255
\(916\) 114.932 3.79745
\(917\) 14.4363 0.476730
\(918\) −11.1436 −0.367793
\(919\) −24.4453 −0.806377 −0.403189 0.915117i \(-0.632098\pi\)
−0.403189 + 0.915117i \(0.632098\pi\)
\(920\) −42.2154 −1.39180
\(921\) −3.90907 −0.128808
\(922\) −81.7190 −2.69127
\(923\) −6.65985 −0.219212
\(924\) 0 0
\(925\) −4.48352 −0.147417
\(926\) −100.976 −3.31829
\(927\) 1.36732 0.0449088
\(928\) 63.1335 2.07246
\(929\) −57.2716 −1.87902 −0.939509 0.342523i \(-0.888719\pi\)
−0.939509 + 0.342523i \(0.888719\pi\)
\(930\) 26.6220 0.872969
\(931\) 25.9437 0.850270
\(932\) 48.0528 1.57402
\(933\) −22.5196 −0.737258
\(934\) −39.3501 −1.28758
\(935\) 0 0
\(936\) 40.5196 1.32442
\(937\) −51.0099 −1.66642 −0.833210 0.552957i \(-0.813500\pi\)
−0.833210 + 0.552957i \(0.813500\pi\)
\(938\) 49.1710 1.60549
\(939\) −19.2542 −0.628338
\(940\) 20.7763 0.677647
\(941\) −15.1681 −0.494467 −0.247233 0.968956i \(-0.579521\pi\)
−0.247233 + 0.968956i \(0.579521\pi\)
\(942\) −2.05748 −0.0670363
\(943\) 18.9955 0.618578
\(944\) −68.5480 −2.23105
\(945\) −4.16741 −0.135566
\(946\) 0 0
\(947\) −55.9199 −1.81715 −0.908576 0.417720i \(-0.862829\pi\)
−0.908576 + 0.417720i \(0.862829\pi\)
\(948\) −33.7939 −1.09757
\(949\) 24.0000 0.779073
\(950\) −6.58549 −0.213662
\(951\) 8.36732 0.271329
\(952\) 135.856 4.40313
\(953\) −32.7177 −1.05983 −0.529915 0.848051i \(-0.677776\pi\)
−0.529915 + 0.848051i \(0.677776\pi\)
\(954\) 9.55982 0.309511
\(955\) 0.679859 0.0219997
\(956\) −42.3250 −1.36889
\(957\) 0 0
\(958\) −32.0832 −1.03656
\(959\) 85.8475 2.77216
\(960\) −10.7489 −0.346918
\(961\) 71.3375 2.30121
\(962\) −62.1006 −2.00220
\(963\) 12.9618 0.417689
\(964\) −44.1862 −1.42314
\(965\) 4.95245 0.159425
\(966\) 60.1380 1.93491
\(967\) −53.9473 −1.73483 −0.867414 0.497587i \(-0.834219\pi\)
−0.867414 + 0.497587i \(0.834219\pi\)
\(968\) 0 0
\(969\) 10.5966 0.340412
\(970\) 42.5933 1.36759
\(971\) 4.14915 0.133153 0.0665763 0.997781i \(-0.478792\pi\)
0.0665763 + 0.997781i \(0.478792\pi\)
\(972\) −4.92542 −0.157983
\(973\) −73.7565 −2.36452
\(974\) −37.5984 −1.20473
\(975\) 5.26324 0.168559
\(976\) 127.598 4.08432
\(977\) −13.5668 −0.434039 −0.217020 0.976167i \(-0.569634\pi\)
−0.217020 + 0.976167i \(0.569634\pi\)
\(978\) −20.3921 −0.652067
\(979\) 0 0
\(980\) 51.0635 1.63116
\(981\) 5.96655 0.190497
\(982\) 68.5744 2.18830
\(983\) 36.7489 1.17211 0.586054 0.810272i \(-0.300681\pi\)
0.586054 + 0.810272i \(0.300681\pi\)
\(984\) 26.6687 0.850168
\(985\) 3.59828 0.114651
\(986\) −58.6514 −1.86784
\(987\) −17.5789 −0.559541
\(988\) −64.8727 −2.06387
\(989\) 36.5743 1.16300
\(990\) 0 0
\(991\) −48.6830 −1.54647 −0.773233 0.634122i \(-0.781362\pi\)
−0.773233 + 0.634122i \(0.781362\pi\)
\(992\) −121.346 −3.85273
\(993\) 12.5525 0.398343
\(994\) −13.8772 −0.440157
\(995\) 3.88380 0.123125
\(996\) −23.9904 −0.760164
\(997\) −41.7558 −1.32242 −0.661210 0.750200i \(-0.729957\pi\)
−0.661210 + 0.750200i \(0.729957\pi\)
\(998\) −75.9619 −2.40453
\(999\) 4.48352 0.141852
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 1815.2.a.y.1.1 6
3.2 odd 2 5445.2.a.bz.1.6 6
5.4 even 2 9075.2.a.dq.1.6 6
11.10 odd 2 inner 1815.2.a.y.1.6 yes 6
33.32 even 2 5445.2.a.bz.1.1 6
55.54 odd 2 9075.2.a.dq.1.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1815.2.a.y.1.1 6 1.1 even 1 trivial
1815.2.a.y.1.6 yes 6 11.10 odd 2 inner
5445.2.a.bz.1.1 6 33.32 even 2
5445.2.a.bz.1.6 6 3.2 odd 2
9075.2.a.dq.1.1 6 55.54 odd 2
9075.2.a.dq.1.6 6 5.4 even 2