Properties

Label 385.2.a.f.1.1
Level $385$
Weight $2$
Character 385.1
Self dual yes
Analytic conductor $3.074$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 385 = 5 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 385.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(3.07424047782\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.148.1
Defining polynomial: \(x^{3} - x^{2} - 3 x + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.48119\) of defining polynomial
Character \(\chi\) \(=\) 385.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.67513 q^{2} -2.48119 q^{3} +5.15633 q^{4} -1.00000 q^{5} +6.63752 q^{6} -1.00000 q^{7} -8.44358 q^{8} +3.15633 q^{9} +O(q^{10})\) \(q-2.67513 q^{2} -2.48119 q^{3} +5.15633 q^{4} -1.00000 q^{5} +6.63752 q^{6} -1.00000 q^{7} -8.44358 q^{8} +3.15633 q^{9} +2.67513 q^{10} -1.00000 q^{11} -12.7938 q^{12} +5.83146 q^{13} +2.67513 q^{14} +2.48119 q^{15} +12.2750 q^{16} +5.44358 q^{17} -8.44358 q^{18} -1.35026 q^{19} -5.15633 q^{20} +2.48119 q^{21} +2.67513 q^{22} -3.19394 q^{23} +20.9502 q^{24} +1.00000 q^{25} -15.5999 q^{26} -0.387873 q^{27} -5.15633 q^{28} -3.61213 q^{29} -6.63752 q^{30} -5.28726 q^{31} -15.9502 q^{32} +2.48119 q^{33} -14.5623 q^{34} +1.00000 q^{35} +16.2750 q^{36} -8.54420 q^{37} +3.61213 q^{38} -14.4690 q^{39} +8.44358 q^{40} -5.02539 q^{41} -6.63752 q^{42} +5.89446 q^{43} -5.15633 q^{44} -3.15633 q^{45} +8.54420 q^{46} -11.8315 q^{47} -30.4568 q^{48} +1.00000 q^{49} -2.67513 q^{50} -13.5066 q^{51} +30.0689 q^{52} -0.231548 q^{53} +1.03761 q^{54} +1.00000 q^{55} +8.44358 q^{56} +3.35026 q^{57} +9.66291 q^{58} +13.5999 q^{59} +12.7938 q^{60} -1.41327 q^{61} +14.1441 q^{62} -3.15633 q^{63} +18.1187 q^{64} -5.83146 q^{65} -6.63752 q^{66} -10.8568 q^{67} +28.0689 q^{68} +7.92478 q^{69} -2.67513 q^{70} -15.5369 q^{71} -26.6507 q^{72} -11.3684 q^{73} +22.8568 q^{74} -2.48119 q^{75} -6.96239 q^{76} +1.00000 q^{77} +38.7064 q^{78} +1.96968 q^{79} -12.2750 q^{80} -8.50659 q^{81} +13.4436 q^{82} +10.6253 q^{83} +12.7938 q^{84} -5.44358 q^{85} -15.7685 q^{86} +8.96239 q^{87} +8.44358 q^{88} +7.22425 q^{89} +8.44358 q^{90} -5.83146 q^{91} -16.4690 q^{92} +13.1187 q^{93} +31.6507 q^{94} +1.35026 q^{95} +39.5755 q^{96} -0.836381 q^{97} -2.67513 q^{98} -3.15633 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{2} - 2 q^{3} + 5 q^{4} - 3 q^{5} + 4 q^{6} - 3 q^{7} - 9 q^{8} - q^{9} + O(q^{10}) \) \( 3 q - 3 q^{2} - 2 q^{3} + 5 q^{4} - 3 q^{5} + 4 q^{6} - 3 q^{7} - 9 q^{8} - q^{9} + 3 q^{10} - 3 q^{11} - 12 q^{12} + 2 q^{13} + 3 q^{14} + 2 q^{15} + 5 q^{16} - 9 q^{18} + 6 q^{19} - 5 q^{20} + 2 q^{21} + 3 q^{22} - 10 q^{23} + 26 q^{24} + 3 q^{25} - 20 q^{26} - 2 q^{27} - 5 q^{28} - 10 q^{29} - 4 q^{30} - 10 q^{31} - 11 q^{32} + 2 q^{33} - 6 q^{34} + 3 q^{35} + 17 q^{36} - 16 q^{37} + 10 q^{38} - 12 q^{39} + 9 q^{40} - 4 q^{42} - 2 q^{43} - 5 q^{44} + q^{45} + 16 q^{46} - 20 q^{47} - 34 q^{48} + 3 q^{49} - 3 q^{50} - 20 q^{51} + 32 q^{52} - 12 q^{53} + 14 q^{54} + 3 q^{55} + 9 q^{56} - 2 q^{58} + 14 q^{59} + 12 q^{60} + 10 q^{61} + 6 q^{62} + q^{63} + 33 q^{64} - 2 q^{65} - 4 q^{66} - 2 q^{67} + 26 q^{68} + 2 q^{69} - 3 q^{70} - 24 q^{71} - 23 q^{72} + 4 q^{73} + 38 q^{74} - 2 q^{75} - 10 q^{76} + 3 q^{77} + 42 q^{78} + 8 q^{79} - 5 q^{80} - 5 q^{81} + 24 q^{82} - 10 q^{83} + 12 q^{84} - 36 q^{86} + 16 q^{87} + 9 q^{88} + 20 q^{89} + 9 q^{90} - 2 q^{91} - 18 q^{92} + 18 q^{93} + 38 q^{94} - 6 q^{95} + 40 q^{96} - 3 q^{98} + q^{99} + O(q^{100}) \)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.67513 −1.89160 −0.945802 0.324745i \(-0.894721\pi\)
−0.945802 + 0.324745i \(0.894721\pi\)
\(3\) −2.48119 −1.43252 −0.716259 0.697834i \(-0.754147\pi\)
−0.716259 + 0.697834i \(0.754147\pi\)
\(4\) 5.15633 2.57816
\(5\) −1.00000 −0.447214
\(6\) 6.63752 2.70976
\(7\) −1.00000 −0.377964
\(8\) −8.44358 −2.98526
\(9\) 3.15633 1.05211
\(10\) 2.67513 0.845951
\(11\) −1.00000 −0.301511
\(12\) −12.7938 −3.69326
\(13\) 5.83146 1.61735 0.808677 0.588252i \(-0.200184\pi\)
0.808677 + 0.588252i \(0.200184\pi\)
\(14\) 2.67513 0.714959
\(15\) 2.48119 0.640642
\(16\) 12.2750 3.06876
\(17\) 5.44358 1.32026 0.660131 0.751150i \(-0.270501\pi\)
0.660131 + 0.751150i \(0.270501\pi\)
\(18\) −8.44358 −1.99017
\(19\) −1.35026 −0.309771 −0.154886 0.987932i \(-0.549501\pi\)
−0.154886 + 0.987932i \(0.549501\pi\)
\(20\) −5.15633 −1.15299
\(21\) 2.48119 0.541441
\(22\) 2.67513 0.570340
\(23\) −3.19394 −0.665982 −0.332991 0.942930i \(-0.608058\pi\)
−0.332991 + 0.942930i \(0.608058\pi\)
\(24\) 20.9502 4.27644
\(25\) 1.00000 0.200000
\(26\) −15.5999 −3.05939
\(27\) −0.387873 −0.0746462
\(28\) −5.15633 −0.974454
\(29\) −3.61213 −0.670755 −0.335378 0.942084i \(-0.608864\pi\)
−0.335378 + 0.942084i \(0.608864\pi\)
\(30\) −6.63752 −1.21184
\(31\) −5.28726 −0.949620 −0.474810 0.880088i \(-0.657483\pi\)
−0.474810 + 0.880088i \(0.657483\pi\)
\(32\) −15.9502 −2.81962
\(33\) 2.48119 0.431920
\(34\) −14.5623 −2.49741
\(35\) 1.00000 0.169031
\(36\) 16.2750 2.71251
\(37\) −8.54420 −1.40466 −0.702329 0.711853i \(-0.747856\pi\)
−0.702329 + 0.711853i \(0.747856\pi\)
\(38\) 3.61213 0.585964
\(39\) −14.4690 −2.31689
\(40\) 8.44358 1.33505
\(41\) −5.02539 −0.784834 −0.392417 0.919787i \(-0.628361\pi\)
−0.392417 + 0.919787i \(0.628361\pi\)
\(42\) −6.63752 −1.02419
\(43\) 5.89446 0.898897 0.449448 0.893306i \(-0.351621\pi\)
0.449448 + 0.893306i \(0.351621\pi\)
\(44\) −5.15633 −0.777345
\(45\) −3.15633 −0.470517
\(46\) 8.54420 1.25977
\(47\) −11.8315 −1.72580 −0.862898 0.505379i \(-0.831353\pi\)
−0.862898 + 0.505379i \(0.831353\pi\)
\(48\) −30.4568 −4.39605
\(49\) 1.00000 0.142857
\(50\) −2.67513 −0.378321
\(51\) −13.5066 −1.89130
\(52\) 30.0689 4.16980
\(53\) −0.231548 −0.0318056 −0.0159028 0.999874i \(-0.505062\pi\)
−0.0159028 + 0.999874i \(0.505062\pi\)
\(54\) 1.03761 0.141201
\(55\) 1.00000 0.134840
\(56\) 8.44358 1.12832
\(57\) 3.35026 0.443753
\(58\) 9.66291 1.26880
\(59\) 13.5999 1.77056 0.885279 0.465061i \(-0.153968\pi\)
0.885279 + 0.465061i \(0.153968\pi\)
\(60\) 12.7938 1.65168
\(61\) −1.41327 −0.180950 −0.0904751 0.995899i \(-0.528839\pi\)
−0.0904751 + 0.995899i \(0.528839\pi\)
\(62\) 14.1441 1.79630
\(63\) −3.15633 −0.397660
\(64\) 18.1187 2.26484
\(65\) −5.83146 −0.723303
\(66\) −6.63752 −0.817022
\(67\) −10.8568 −1.32638 −0.663188 0.748453i \(-0.730797\pi\)
−0.663188 + 0.748453i \(0.730797\pi\)
\(68\) 28.0689 3.40385
\(69\) 7.92478 0.954031
\(70\) −2.67513 −0.319739
\(71\) −15.5369 −1.84389 −0.921946 0.387319i \(-0.873401\pi\)
−0.921946 + 0.387319i \(0.873401\pi\)
\(72\) −26.6507 −3.14081
\(73\) −11.3684 −1.33057 −0.665283 0.746591i \(-0.731689\pi\)
−0.665283 + 0.746591i \(0.731689\pi\)
\(74\) 22.8568 2.65705
\(75\) −2.48119 −0.286504
\(76\) −6.96239 −0.798641
\(77\) 1.00000 0.113961
\(78\) 38.7064 4.38264
\(79\) 1.96968 0.221607 0.110803 0.993842i \(-0.464658\pi\)
0.110803 + 0.993842i \(0.464658\pi\)
\(80\) −12.2750 −1.37239
\(81\) −8.50659 −0.945176
\(82\) 13.4436 1.48460
\(83\) 10.6253 1.16628 0.583139 0.812372i \(-0.301824\pi\)
0.583139 + 0.812372i \(0.301824\pi\)
\(84\) 12.7938 1.39592
\(85\) −5.44358 −0.590439
\(86\) −15.7685 −1.70036
\(87\) 8.96239 0.960869
\(88\) 8.44358 0.900089
\(89\) 7.22425 0.765769 0.382885 0.923796i \(-0.374931\pi\)
0.382885 + 0.923796i \(0.374931\pi\)
\(90\) 8.44358 0.890032
\(91\) −5.83146 −0.611303
\(92\) −16.4690 −1.71701
\(93\) 13.1187 1.36035
\(94\) 31.6507 3.26452
\(95\) 1.35026 0.138534
\(96\) 39.5755 4.03915
\(97\) −0.836381 −0.0849216 −0.0424608 0.999098i \(-0.513520\pi\)
−0.0424608 + 0.999098i \(0.513520\pi\)
\(98\) −2.67513 −0.270229
\(99\) −3.15633 −0.317223
\(100\) 5.15633 0.515633
\(101\) 7.41327 0.737648 0.368824 0.929499i \(-0.379761\pi\)
0.368824 + 0.929499i \(0.379761\pi\)
\(102\) 36.1319 3.57759
\(103\) 4.21933 0.415743 0.207871 0.978156i \(-0.433346\pi\)
0.207871 + 0.978156i \(0.433346\pi\)
\(104\) −49.2384 −4.82822
\(105\) −2.48119 −0.242140
\(106\) 0.619421 0.0601635
\(107\) −11.5369 −1.11531 −0.557657 0.830071i \(-0.688300\pi\)
−0.557657 + 0.830071i \(0.688300\pi\)
\(108\) −2.00000 −0.192450
\(109\) −2.18664 −0.209442 −0.104721 0.994502i \(-0.533395\pi\)
−0.104721 + 0.994502i \(0.533395\pi\)
\(110\) −2.67513 −0.255064
\(111\) 21.1998 2.01220
\(112\) −12.2750 −1.15988
\(113\) −9.35026 −0.879599 −0.439799 0.898096i \(-0.644950\pi\)
−0.439799 + 0.898096i \(0.644950\pi\)
\(114\) −8.96239 −0.839405
\(115\) 3.19394 0.297836
\(116\) −18.6253 −1.72932
\(117\) 18.4060 1.70163
\(118\) −36.3815 −3.34919
\(119\) −5.44358 −0.499012
\(120\) −20.9502 −1.91248
\(121\) 1.00000 0.0909091
\(122\) 3.78067 0.342286
\(123\) 12.4690 1.12429
\(124\) −27.2628 −2.44827
\(125\) −1.00000 −0.0894427
\(126\) 8.44358 0.752214
\(127\) −16.9624 −1.50517 −0.752584 0.658496i \(-0.771193\pi\)
−0.752584 + 0.658496i \(0.771193\pi\)
\(128\) −16.5696 −1.46456
\(129\) −14.6253 −1.28769
\(130\) 15.5999 1.36820
\(131\) −9.92478 −0.867132 −0.433566 0.901122i \(-0.642745\pi\)
−0.433566 + 0.901122i \(0.642745\pi\)
\(132\) 12.7938 1.11356
\(133\) 1.35026 0.117083
\(134\) 29.0435 2.50898
\(135\) 0.387873 0.0333828
\(136\) −45.9633 −3.94132
\(137\) −10.9927 −0.939170 −0.469585 0.882887i \(-0.655596\pi\)
−0.469585 + 0.882887i \(0.655596\pi\)
\(138\) −21.1998 −1.80465
\(139\) 6.88717 0.584162 0.292081 0.956394i \(-0.405652\pi\)
0.292081 + 0.956394i \(0.405652\pi\)
\(140\) 5.15633 0.435789
\(141\) 29.3561 2.47223
\(142\) 41.5633 3.48791
\(143\) −5.83146 −0.487651
\(144\) 38.7440 3.22867
\(145\) 3.61213 0.299971
\(146\) 30.4119 2.51690
\(147\) −2.48119 −0.204645
\(148\) −44.0567 −3.62144
\(149\) 22.8119 1.86883 0.934414 0.356190i \(-0.115924\pi\)
0.934414 + 0.356190i \(0.115924\pi\)
\(150\) 6.63752 0.541951
\(151\) −3.24472 −0.264052 −0.132026 0.991246i \(-0.542148\pi\)
−0.132026 + 0.991246i \(0.542148\pi\)
\(152\) 11.4010 0.924747
\(153\) 17.1817 1.38906
\(154\) −2.67513 −0.215568
\(155\) 5.28726 0.424683
\(156\) −74.6067 −5.97332
\(157\) −5.42548 −0.433001 −0.216500 0.976283i \(-0.569464\pi\)
−0.216500 + 0.976283i \(0.569464\pi\)
\(158\) −5.26916 −0.419192
\(159\) 0.574515 0.0455620
\(160\) 15.9502 1.26097
\(161\) 3.19394 0.251717
\(162\) 22.7562 1.78790
\(163\) 3.38058 0.264787 0.132394 0.991197i \(-0.457734\pi\)
0.132394 + 0.991197i \(0.457734\pi\)
\(164\) −25.9126 −2.02343
\(165\) −2.48119 −0.193161
\(166\) −28.4241 −2.20614
\(167\) 11.2750 0.872489 0.436244 0.899828i \(-0.356308\pi\)
0.436244 + 0.899828i \(0.356308\pi\)
\(168\) −20.9502 −1.61634
\(169\) 21.0059 1.61584
\(170\) 14.5623 1.11688
\(171\) −4.26187 −0.325913
\(172\) 30.3938 2.31750
\(173\) −8.98049 −0.682774 −0.341387 0.939923i \(-0.610897\pi\)
−0.341387 + 0.939923i \(0.610897\pi\)
\(174\) −23.9756 −1.81758
\(175\) −1.00000 −0.0755929
\(176\) −12.2750 −0.925266
\(177\) −33.7440 −2.53636
\(178\) −19.3258 −1.44853
\(179\) −26.2374 −1.96108 −0.980539 0.196326i \(-0.937099\pi\)
−0.980539 + 0.196326i \(0.937099\pi\)
\(180\) −16.2750 −1.21307
\(181\) −11.1998 −0.832476 −0.416238 0.909256i \(-0.636652\pi\)
−0.416238 + 0.909256i \(0.636652\pi\)
\(182\) 15.5999 1.15634
\(183\) 3.50659 0.259214
\(184\) 26.9683 1.98813
\(185\) 8.54420 0.628182
\(186\) −35.0943 −2.57324
\(187\) −5.44358 −0.398074
\(188\) −61.0068 −4.44938
\(189\) 0.387873 0.0282136
\(190\) −3.61213 −0.262051
\(191\) 11.1998 0.810390 0.405195 0.914230i \(-0.367204\pi\)
0.405195 + 0.914230i \(0.367204\pi\)
\(192\) −44.9560 −3.24442
\(193\) 0.604833 0.0435368 0.0217684 0.999763i \(-0.493070\pi\)
0.0217684 + 0.999763i \(0.493070\pi\)
\(194\) 2.23743 0.160638
\(195\) 14.4690 1.03614
\(196\) 5.15633 0.368309
\(197\) 15.3054 1.09046 0.545231 0.838286i \(-0.316442\pi\)
0.545231 + 0.838286i \(0.316442\pi\)
\(198\) 8.44358 0.600059
\(199\) −12.5623 −0.890518 −0.445259 0.895402i \(-0.646888\pi\)
−0.445259 + 0.895402i \(0.646888\pi\)
\(200\) −8.44358 −0.597051
\(201\) 26.9380 1.90006
\(202\) −19.8315 −1.39534
\(203\) 3.61213 0.253522
\(204\) −69.6444 −4.87608
\(205\) 5.02539 0.350989
\(206\) −11.2873 −0.786421
\(207\) −10.0811 −0.700685
\(208\) 71.5814 4.96327
\(209\) 1.35026 0.0933996
\(210\) 6.63752 0.458032
\(211\) −4.43866 −0.305570 −0.152785 0.988259i \(-0.548824\pi\)
−0.152785 + 0.988259i \(0.548824\pi\)
\(212\) −1.19394 −0.0819999
\(213\) 38.5501 2.64141
\(214\) 30.8627 2.10973
\(215\) −5.89446 −0.401999
\(216\) 3.27504 0.222838
\(217\) 5.28726 0.358922
\(218\) 5.84955 0.396182
\(219\) 28.2071 1.90606
\(220\) 5.15633 0.347639
\(221\) 31.7440 2.13533
\(222\) −56.7123 −3.80628
\(223\) −7.78067 −0.521032 −0.260516 0.965469i \(-0.583893\pi\)
−0.260516 + 0.965469i \(0.583893\pi\)
\(224\) 15.9502 1.06572
\(225\) 3.15633 0.210422
\(226\) 25.0132 1.66385
\(227\) −10.4485 −0.693492 −0.346746 0.937959i \(-0.612713\pi\)
−0.346746 + 0.937959i \(0.612713\pi\)
\(228\) 17.2750 1.14407
\(229\) −29.4518 −1.94623 −0.973116 0.230316i \(-0.926024\pi\)
−0.973116 + 0.230316i \(0.926024\pi\)
\(230\) −8.54420 −0.563388
\(231\) −2.48119 −0.163251
\(232\) 30.4993 2.00238
\(233\) −8.73084 −0.571976 −0.285988 0.958233i \(-0.592322\pi\)
−0.285988 + 0.958233i \(0.592322\pi\)
\(234\) −49.2384 −3.21881
\(235\) 11.8315 0.771799
\(236\) 70.1255 4.56478
\(237\) −4.88717 −0.317456
\(238\) 14.5623 0.943933
\(239\) −21.2144 −1.37225 −0.686123 0.727486i \(-0.740689\pi\)
−0.686123 + 0.727486i \(0.740689\pi\)
\(240\) 30.4568 1.96598
\(241\) −9.33804 −0.601516 −0.300758 0.953700i \(-0.597240\pi\)
−0.300758 + 0.953700i \(0.597240\pi\)
\(242\) −2.67513 −0.171964
\(243\) 22.2701 1.42863
\(244\) −7.28726 −0.466519
\(245\) −1.00000 −0.0638877
\(246\) −33.3561 −2.12671
\(247\) −7.87399 −0.501010
\(248\) 44.6434 2.83486
\(249\) −26.3634 −1.67071
\(250\) 2.67513 0.169190
\(251\) 1.87636 0.118435 0.0592174 0.998245i \(-0.481139\pi\)
0.0592174 + 0.998245i \(0.481139\pi\)
\(252\) −16.2750 −1.02523
\(253\) 3.19394 0.200801
\(254\) 45.3766 2.84718
\(255\) 13.5066 0.845815
\(256\) 8.08840 0.505525
\(257\) 27.1392 1.69290 0.846448 0.532472i \(-0.178737\pi\)
0.846448 + 0.532472i \(0.178737\pi\)
\(258\) 39.1246 2.43579
\(259\) 8.54420 0.530911
\(260\) −30.0689 −1.86479
\(261\) −11.4010 −0.705707
\(262\) 26.5501 1.64027
\(263\) 12.8119 0.790018 0.395009 0.918677i \(-0.370741\pi\)
0.395009 + 0.918677i \(0.370741\pi\)
\(264\) −20.9502 −1.28939
\(265\) 0.231548 0.0142239
\(266\) −3.61213 −0.221474
\(267\) −17.9248 −1.09698
\(268\) −55.9814 −3.41961
\(269\) −6.26187 −0.381793 −0.190896 0.981610i \(-0.561139\pi\)
−0.190896 + 0.981610i \(0.561139\pi\)
\(270\) −1.03761 −0.0631470
\(271\) −5.73813 −0.348567 −0.174283 0.984696i \(-0.555761\pi\)
−0.174283 + 0.984696i \(0.555761\pi\)
\(272\) 66.8202 4.05157
\(273\) 14.4690 0.875702
\(274\) 29.4069 1.77654
\(275\) −1.00000 −0.0603023
\(276\) 40.8627 2.45965
\(277\) −8.35756 −0.502157 −0.251078 0.967967i \(-0.580785\pi\)
−0.251078 + 0.967967i \(0.580785\pi\)
\(278\) −18.4241 −1.10500
\(279\) −16.6883 −0.999103
\(280\) −8.44358 −0.504601
\(281\) −8.44851 −0.503996 −0.251998 0.967728i \(-0.581088\pi\)
−0.251998 + 0.967728i \(0.581088\pi\)
\(282\) −78.5315 −4.67648
\(283\) 0.836381 0.0497177 0.0248588 0.999691i \(-0.492086\pi\)
0.0248588 + 0.999691i \(0.492086\pi\)
\(284\) −80.1133 −4.75385
\(285\) −3.35026 −0.198452
\(286\) 15.5999 0.922442
\(287\) 5.02539 0.296640
\(288\) −50.3439 −2.96654
\(289\) 12.6326 0.743094
\(290\) −9.66291 −0.567426
\(291\) 2.07522 0.121652
\(292\) −58.6190 −3.43042
\(293\) −2.71862 −0.158824 −0.0794118 0.996842i \(-0.525304\pi\)
−0.0794118 + 0.996842i \(0.525304\pi\)
\(294\) 6.63752 0.387108
\(295\) −13.5999 −0.791817
\(296\) 72.1436 4.19326
\(297\) 0.387873 0.0225067
\(298\) −61.0249 −3.53508
\(299\) −18.6253 −1.07713
\(300\) −12.7938 −0.738653
\(301\) −5.89446 −0.339751
\(302\) 8.68006 0.499481
\(303\) −18.3938 −1.05669
\(304\) −16.5745 −0.950614
\(305\) 1.41327 0.0809234
\(306\) −45.9633 −2.62755
\(307\) −8.36344 −0.477326 −0.238663 0.971102i \(-0.576709\pi\)
−0.238663 + 0.971102i \(0.576709\pi\)
\(308\) 5.15633 0.293809
\(309\) −10.4690 −0.595559
\(310\) −14.1441 −0.803331
\(311\) 4.43629 0.251559 0.125779 0.992058i \(-0.459857\pi\)
0.125779 + 0.992058i \(0.459857\pi\)
\(312\) 122.170 6.91651
\(313\) 29.7889 1.68377 0.841885 0.539658i \(-0.181446\pi\)
0.841885 + 0.539658i \(0.181446\pi\)
\(314\) 14.5139 0.819066
\(315\) 3.15633 0.177839
\(316\) 10.1563 0.571338
\(317\) 15.4010 0.865009 0.432504 0.901632i \(-0.357630\pi\)
0.432504 + 0.901632i \(0.357630\pi\)
\(318\) −1.53690 −0.0861853
\(319\) 3.61213 0.202240
\(320\) −18.1187 −1.01287
\(321\) 28.6253 1.59771
\(322\) −8.54420 −0.476150
\(323\) −7.35026 −0.408980
\(324\) −43.8627 −2.43682
\(325\) 5.83146 0.323471
\(326\) −9.04349 −0.500873
\(327\) 5.42548 0.300030
\(328\) 42.4323 2.34293
\(329\) 11.8315 0.652289
\(330\) 6.63752 0.365383
\(331\) 6.26187 0.344183 0.172092 0.985081i \(-0.444947\pi\)
0.172092 + 0.985081i \(0.444947\pi\)
\(332\) 54.7875 3.00685
\(333\) −26.9683 −1.47785
\(334\) −30.1622 −1.65040
\(335\) 10.8568 0.593173
\(336\) 30.4568 1.66155
\(337\) −15.8700 −0.864495 −0.432248 0.901755i \(-0.642279\pi\)
−0.432248 + 0.901755i \(0.642279\pi\)
\(338\) −56.1935 −3.05652
\(339\) 23.1998 1.26004
\(340\) −28.0689 −1.52225
\(341\) 5.28726 0.286321
\(342\) 11.4010 0.616498
\(343\) −1.00000 −0.0539949
\(344\) −49.7704 −2.68344
\(345\) −7.92478 −0.426656
\(346\) 24.0240 1.29154
\(347\) 6.79147 0.364585 0.182293 0.983244i \(-0.441648\pi\)
0.182293 + 0.983244i \(0.441648\pi\)
\(348\) 46.2130 2.47728
\(349\) 26.7489 1.43184 0.715919 0.698183i \(-0.246008\pi\)
0.715919 + 0.698183i \(0.246008\pi\)
\(350\) 2.67513 0.142992
\(351\) −2.26187 −0.120729
\(352\) 15.9502 0.850147
\(353\) −16.8627 −0.897512 −0.448756 0.893654i \(-0.648133\pi\)
−0.448756 + 0.893654i \(0.648133\pi\)
\(354\) 90.2697 4.79778
\(355\) 15.5369 0.824613
\(356\) 37.2506 1.97428
\(357\) 13.5066 0.714844
\(358\) 70.1886 3.70958
\(359\) −3.79289 −0.200181 −0.100091 0.994978i \(-0.531913\pi\)
−0.100091 + 0.994978i \(0.531913\pi\)
\(360\) 26.6507 1.40461
\(361\) −17.1768 −0.904042
\(362\) 29.9610 1.57471
\(363\) −2.48119 −0.130229
\(364\) −30.0689 −1.57604
\(365\) 11.3684 0.595047
\(366\) −9.38058 −0.490331
\(367\) −6.36977 −0.332500 −0.166250 0.986084i \(-0.553166\pi\)
−0.166250 + 0.986084i \(0.553166\pi\)
\(368\) −39.2057 −2.04374
\(369\) −15.8618 −0.825731
\(370\) −22.8568 −1.18827
\(371\) 0.231548 0.0120214
\(372\) 67.6444 3.50720
\(373\) −21.3317 −1.10451 −0.552257 0.833674i \(-0.686233\pi\)
−0.552257 + 0.833674i \(0.686233\pi\)
\(374\) 14.5623 0.752998
\(375\) 2.48119 0.128128
\(376\) 99.8999 5.15194
\(377\) −21.0640 −1.08485
\(378\) −1.03761 −0.0533690
\(379\) 24.7875 1.27325 0.636624 0.771174i \(-0.280330\pi\)
0.636624 + 0.771174i \(0.280330\pi\)
\(380\) 6.96239 0.357163
\(381\) 42.0870 2.15618
\(382\) −29.9610 −1.53294
\(383\) −5.45817 −0.278900 −0.139450 0.990229i \(-0.544533\pi\)
−0.139450 + 0.990229i \(0.544533\pi\)
\(384\) 41.1124 2.09801
\(385\) −1.00000 −0.0509647
\(386\) −1.61801 −0.0823544
\(387\) 18.6048 0.945737
\(388\) −4.31265 −0.218942
\(389\) −13.7235 −0.695811 −0.347906 0.937530i \(-0.613107\pi\)
−0.347906 + 0.937530i \(0.613107\pi\)
\(390\) −38.7064 −1.95997
\(391\) −17.3865 −0.879271
\(392\) −8.44358 −0.426465
\(393\) 24.6253 1.24218
\(394\) −40.9438 −2.06272
\(395\) −1.96968 −0.0991055
\(396\) −16.2750 −0.817851
\(397\) 2.11142 0.105969 0.0529846 0.998595i \(-0.483127\pi\)
0.0529846 + 0.998595i \(0.483127\pi\)
\(398\) 33.6058 1.68451
\(399\) −3.35026 −0.167723
\(400\) 12.2750 0.613752
\(401\) 19.1490 0.956257 0.478128 0.878290i \(-0.341315\pi\)
0.478128 + 0.878290i \(0.341315\pi\)
\(402\) −72.0625 −3.59415
\(403\) −30.8324 −1.53587
\(404\) 38.2252 1.90178
\(405\) 8.50659 0.422696
\(406\) −9.66291 −0.479562
\(407\) 8.54420 0.423520
\(408\) 114.044 5.64602
\(409\) −18.6883 −0.924077 −0.462039 0.886860i \(-0.652882\pi\)
−0.462039 + 0.886860i \(0.652882\pi\)
\(410\) −13.4436 −0.663931
\(411\) 27.2750 1.34538
\(412\) 21.7562 1.07185
\(413\) −13.5999 −0.669208
\(414\) 26.9683 1.32542
\(415\) −10.6253 −0.521575
\(416\) −93.0127 −4.56032
\(417\) −17.0884 −0.836822
\(418\) −3.61213 −0.176675
\(419\) −0.773377 −0.0377819 −0.0188910 0.999822i \(-0.506014\pi\)
−0.0188910 + 0.999822i \(0.506014\pi\)
\(420\) −12.7938 −0.624276
\(421\) −10.5198 −0.512702 −0.256351 0.966584i \(-0.582520\pi\)
−0.256351 + 0.966584i \(0.582520\pi\)
\(422\) 11.8740 0.578017
\(423\) −37.3439 −1.81572
\(424\) 1.95509 0.0949478
\(425\) 5.44358 0.264053
\(426\) −103.127 −4.99650
\(427\) 1.41327 0.0683927
\(428\) −59.4880 −2.87546
\(429\) 14.4690 0.698569
\(430\) 15.7685 0.760422
\(431\) −24.7308 −1.19124 −0.595621 0.803265i \(-0.703094\pi\)
−0.595621 + 0.803265i \(0.703094\pi\)
\(432\) −4.76116 −0.229071
\(433\) −18.5599 −0.891933 −0.445967 0.895050i \(-0.647140\pi\)
−0.445967 + 0.895050i \(0.647140\pi\)
\(434\) −14.1441 −0.678939
\(435\) −8.96239 −0.429714
\(436\) −11.2750 −0.539976
\(437\) 4.31265 0.206302
\(438\) −75.4577 −3.60551
\(439\) −1.42548 −0.0680347 −0.0340173 0.999421i \(-0.510830\pi\)
−0.0340173 + 0.999421i \(0.510830\pi\)
\(440\) −8.44358 −0.402532
\(441\) 3.15633 0.150301
\(442\) −84.9194 −4.03920
\(443\) 40.1925 1.90960 0.954802 0.297242i \(-0.0960668\pi\)
0.954802 + 0.297242i \(0.0960668\pi\)
\(444\) 109.313 5.18777
\(445\) −7.22425 −0.342462
\(446\) 20.8143 0.985586
\(447\) −56.6009 −2.67713
\(448\) −18.1187 −0.856029
\(449\) 12.6556 0.597256 0.298628 0.954370i \(-0.403471\pi\)
0.298628 + 0.954370i \(0.403471\pi\)
\(450\) −8.44358 −0.398034
\(451\) 5.02539 0.236636
\(452\) −48.2130 −2.26775
\(453\) 8.05079 0.378259
\(454\) 27.9511 1.31181
\(455\) 5.83146 0.273383
\(456\) −28.2882 −1.32472
\(457\) −0.544198 −0.0254565 −0.0127283 0.999919i \(-0.504052\pi\)
−0.0127283 + 0.999919i \(0.504052\pi\)
\(458\) 78.7875 3.68150
\(459\) −2.11142 −0.0985526
\(460\) 16.4690 0.767870
\(461\) −11.5755 −0.539123 −0.269562 0.962983i \(-0.586879\pi\)
−0.269562 + 0.962983i \(0.586879\pi\)
\(462\) 6.63752 0.308805
\(463\) 23.7948 1.10584 0.552919 0.833235i \(-0.313514\pi\)
0.552919 + 0.833235i \(0.313514\pi\)
\(464\) −44.3390 −2.05839
\(465\) −13.1187 −0.608366
\(466\) 23.3561 1.08195
\(467\) −2.66784 −0.123453 −0.0617264 0.998093i \(-0.519661\pi\)
−0.0617264 + 0.998093i \(0.519661\pi\)
\(468\) 94.9072 4.38709
\(469\) 10.8568 0.501323
\(470\) −31.6507 −1.45994
\(471\) 13.4617 0.620282
\(472\) −114.832 −5.28557
\(473\) −5.89446 −0.271028
\(474\) 13.0738 0.600500
\(475\) −1.35026 −0.0619543
\(476\) −28.0689 −1.28654
\(477\) −0.730841 −0.0334629
\(478\) 56.7513 2.59574
\(479\) −10.7104 −0.489369 −0.244685 0.969603i \(-0.578684\pi\)
−0.244685 + 0.969603i \(0.578684\pi\)
\(480\) −39.5755 −1.80636
\(481\) −49.8251 −2.27183
\(482\) 24.9805 1.13783
\(483\) −7.92478 −0.360590
\(484\) 5.15633 0.234378
\(485\) 0.836381 0.0379781
\(486\) −59.5755 −2.70240
\(487\) 17.4314 0.789891 0.394945 0.918705i \(-0.370764\pi\)
0.394945 + 0.918705i \(0.370764\pi\)
\(488\) 11.9330 0.540183
\(489\) −8.38787 −0.379313
\(490\) 2.67513 0.120850
\(491\) −28.3693 −1.28029 −0.640145 0.768254i \(-0.721126\pi\)
−0.640145 + 0.768254i \(0.721126\pi\)
\(492\) 64.2941 2.89860
\(493\) −19.6629 −0.885573
\(494\) 21.0640 0.947712
\(495\) 3.15633 0.141866
\(496\) −64.9013 −2.91415
\(497\) 15.5369 0.696925
\(498\) 70.5256 3.16033
\(499\) 27.4763 1.23001 0.615003 0.788524i \(-0.289155\pi\)
0.615003 + 0.788524i \(0.289155\pi\)
\(500\) −5.15633 −0.230598
\(501\) −27.9756 −1.24986
\(502\) −5.01951 −0.224032
\(503\) 20.2981 0.905046 0.452523 0.891753i \(-0.350524\pi\)
0.452523 + 0.891753i \(0.350524\pi\)
\(504\) 26.6507 1.18712
\(505\) −7.41327 −0.329886
\(506\) −8.54420 −0.379836
\(507\) −52.1197 −2.31472
\(508\) −87.4636 −3.88057
\(509\) −24.2619 −1.07539 −0.537694 0.843140i \(-0.680704\pi\)
−0.537694 + 0.843140i \(0.680704\pi\)
\(510\) −36.1319 −1.59995
\(511\) 11.3684 0.502907
\(512\) 11.5017 0.508306
\(513\) 0.523730 0.0231233
\(514\) −72.6009 −3.20229
\(515\) −4.21933 −0.185926
\(516\) −75.4128 −3.31986
\(517\) 11.8315 0.520347
\(518\) −22.8568 −1.00427
\(519\) 22.2823 0.978086
\(520\) 49.2384 2.15925
\(521\) 2.20123 0.0964377 0.0482188 0.998837i \(-0.484646\pi\)
0.0482188 + 0.998837i \(0.484646\pi\)
\(522\) 30.4993 1.33492
\(523\) −22.1378 −0.968017 −0.484008 0.875063i \(-0.660820\pi\)
−0.484008 + 0.875063i \(0.660820\pi\)
\(524\) −51.1754 −2.23561
\(525\) 2.48119 0.108288
\(526\) −34.2736 −1.49440
\(527\) −28.7816 −1.25375
\(528\) 30.4568 1.32546
\(529\) −12.7988 −0.556468
\(530\) −0.619421 −0.0269059
\(531\) 42.9257 1.86282
\(532\) 6.96239 0.301858
\(533\) −29.3054 −1.26936
\(534\) 47.9511 2.07505
\(535\) 11.5369 0.498784
\(536\) 91.6707 3.95957
\(537\) 65.1002 2.80928
\(538\) 16.7513 0.722200
\(539\) −1.00000 −0.0430730
\(540\) 2.00000 0.0860663
\(541\) 23.0640 0.991597 0.495799 0.868438i \(-0.334875\pi\)
0.495799 + 0.868438i \(0.334875\pi\)
\(542\) 15.3503 0.659350
\(543\) 27.7889 1.19254
\(544\) −86.8261 −3.72264
\(545\) 2.18664 0.0936655
\(546\) −38.7064 −1.65648
\(547\) 21.3766 0.913998 0.456999 0.889467i \(-0.348924\pi\)
0.456999 + 0.889467i \(0.348924\pi\)
\(548\) −56.6820 −2.42133
\(549\) −4.46073 −0.190379
\(550\) 2.67513 0.114068
\(551\) 4.87732 0.207781
\(552\) −66.9135 −2.84803
\(553\) −1.96968 −0.0837594
\(554\) 22.3576 0.949882
\(555\) −21.1998 −0.899882
\(556\) 35.5125 1.50606
\(557\) −9.19394 −0.389560 −0.194780 0.980847i \(-0.562399\pi\)
−0.194780 + 0.980847i \(0.562399\pi\)
\(558\) 44.6434 1.88991
\(559\) 34.3733 1.45384
\(560\) 12.2750 0.518715
\(561\) 13.5066 0.570249
\(562\) 22.6009 0.953360
\(563\) −9.79877 −0.412969 −0.206484 0.978450i \(-0.566202\pi\)
−0.206484 + 0.978450i \(0.566202\pi\)
\(564\) 151.370 6.37382
\(565\) 9.35026 0.393368
\(566\) −2.23743 −0.0940461
\(567\) 8.50659 0.357243
\(568\) 131.187 5.50449
\(569\) −33.5125 −1.40492 −0.702458 0.711725i \(-0.747914\pi\)
−0.702458 + 0.711725i \(0.747914\pi\)
\(570\) 8.96239 0.375393
\(571\) 43.1392 1.80532 0.902659 0.430356i \(-0.141612\pi\)
0.902659 + 0.430356i \(0.141612\pi\)
\(572\) −30.0689 −1.25724
\(573\) −27.7889 −1.16090
\(574\) −13.4436 −0.561124
\(575\) −3.19394 −0.133196
\(576\) 57.1886 2.38286
\(577\) −14.8510 −0.618254 −0.309127 0.951021i \(-0.600037\pi\)
−0.309127 + 0.951021i \(0.600037\pi\)
\(578\) −33.7938 −1.40564
\(579\) −1.50071 −0.0623673
\(580\) 18.6253 0.773374
\(581\) −10.6253 −0.440812
\(582\) −5.55149 −0.230117
\(583\) 0.231548 0.00958974
\(584\) 95.9897 3.97208
\(585\) −18.4060 −0.760993
\(586\) 7.27267 0.300431
\(587\) 14.7938 0.610607 0.305304 0.952255i \(-0.401242\pi\)
0.305304 + 0.952255i \(0.401242\pi\)
\(588\) −12.7938 −0.527609
\(589\) 7.13918 0.294165
\(590\) 36.3815 1.49780
\(591\) −37.9756 −1.56211
\(592\) −104.880 −4.31056
\(593\) −27.4191 −1.12597 −0.562985 0.826467i \(-0.690347\pi\)
−0.562985 + 0.826467i \(0.690347\pi\)
\(594\) −1.03761 −0.0425737
\(595\) 5.44358 0.223165
\(596\) 117.626 4.81814
\(597\) 31.1695 1.27568
\(598\) 49.8251 2.03750
\(599\) 11.3258 0.462761 0.231380 0.972863i \(-0.425676\pi\)
0.231380 + 0.972863i \(0.425676\pi\)
\(600\) 20.9502 0.855287
\(601\) 15.5393 0.633860 0.316930 0.948449i \(-0.397348\pi\)
0.316930 + 0.948449i \(0.397348\pi\)
\(602\) 15.7685 0.642674
\(603\) −34.2677 −1.39549
\(604\) −16.7308 −0.680768
\(605\) −1.00000 −0.0406558
\(606\) 49.2057 1.99884
\(607\) 17.7235 0.719377 0.359688 0.933073i \(-0.382883\pi\)
0.359688 + 0.933073i \(0.382883\pi\)
\(608\) 21.5369 0.873437
\(609\) −8.96239 −0.363174
\(610\) −3.78067 −0.153075
\(611\) −68.9946 −2.79122
\(612\) 88.5945 3.58122
\(613\) 22.2941 0.900450 0.450225 0.892915i \(-0.351344\pi\)
0.450225 + 0.892915i \(0.351344\pi\)
\(614\) 22.3733 0.902912
\(615\) −12.4690 −0.502798
\(616\) −8.44358 −0.340202
\(617\) −30.9438 −1.24575 −0.622876 0.782321i \(-0.714036\pi\)
−0.622876 + 0.782321i \(0.714036\pi\)
\(618\) 28.0059 1.12656
\(619\) 32.4119 1.30274 0.651371 0.758759i \(-0.274194\pi\)
0.651371 + 0.758759i \(0.274194\pi\)
\(620\) 27.2628 1.09490
\(621\) 1.23884 0.0497130
\(622\) −11.8677 −0.475850
\(623\) −7.22425 −0.289434
\(624\) −177.607 −7.10998
\(625\) 1.00000 0.0400000
\(626\) −79.6893 −3.18502
\(627\) −3.35026 −0.133797
\(628\) −27.9756 −1.11635
\(629\) −46.5111 −1.85452
\(630\) −8.44358 −0.336400
\(631\) −27.3258 −1.08782 −0.543912 0.839142i \(-0.683057\pi\)
−0.543912 + 0.839142i \(0.683057\pi\)
\(632\) −16.6312 −0.661553
\(633\) 11.0132 0.437734
\(634\) −41.1998 −1.63625
\(635\) 16.9624 0.673132
\(636\) 2.96239 0.117466
\(637\) 5.83146 0.231051
\(638\) −9.66291 −0.382558
\(639\) −49.0395 −1.93997
\(640\) 16.5696 0.654971
\(641\) 19.4460 0.768069 0.384034 0.923319i \(-0.374534\pi\)
0.384034 + 0.923319i \(0.374534\pi\)
\(642\) −76.5764 −3.02223
\(643\) 5.29314 0.208741 0.104370 0.994538i \(-0.466717\pi\)
0.104370 + 0.994538i \(0.466717\pi\)
\(644\) 16.4690 0.648969
\(645\) 14.6253 0.575871
\(646\) 19.6629 0.773627
\(647\) 35.0966 1.37979 0.689896 0.723909i \(-0.257656\pi\)
0.689896 + 0.723909i \(0.257656\pi\)
\(648\) 71.8261 2.82159
\(649\) −13.5999 −0.533843
\(650\) −15.5999 −0.611879
\(651\) −13.1187 −0.514163
\(652\) 17.4314 0.682665
\(653\) 27.7988 1.08785 0.543925 0.839134i \(-0.316938\pi\)
0.543925 + 0.839134i \(0.316938\pi\)
\(654\) −14.5139 −0.567538
\(655\) 9.92478 0.387793
\(656\) −61.6869 −2.40847
\(657\) −35.8822 −1.39990
\(658\) −31.6507 −1.23387
\(659\) 19.6180 0.764209 0.382105 0.924119i \(-0.375199\pi\)
0.382105 + 0.924119i \(0.375199\pi\)
\(660\) −12.7938 −0.498000
\(661\) 21.5633 0.838713 0.419357 0.907822i \(-0.362256\pi\)
0.419357 + 0.907822i \(0.362256\pi\)
\(662\) −16.7513 −0.651058
\(663\) −78.7631 −3.05890
\(664\) −89.7156 −3.48164
\(665\) −1.35026 −0.0523609
\(666\) 72.1436 2.79551
\(667\) 11.5369 0.446711
\(668\) 58.1378 2.24942
\(669\) 19.3054 0.746388
\(670\) −29.0435 −1.12205
\(671\) 1.41327 0.0545585
\(672\) −39.5755 −1.52666
\(673\) −21.0679 −0.812109 −0.406054 0.913849i \(-0.633096\pi\)
−0.406054 + 0.913849i \(0.633096\pi\)
\(674\) 42.4544 1.63528
\(675\) −0.387873 −0.0149292
\(676\) 108.313 4.16589
\(677\) 34.5174 1.32661 0.663306 0.748349i \(-0.269153\pi\)
0.663306 + 0.748349i \(0.269153\pi\)
\(678\) −62.0625 −2.38350
\(679\) 0.836381 0.0320973
\(680\) 45.9633 1.76261
\(681\) 25.9248 0.993440
\(682\) −14.1441 −0.541606
\(683\) 33.7802 1.29256 0.646282 0.763099i \(-0.276323\pi\)
0.646282 + 0.763099i \(0.276323\pi\)
\(684\) −21.9756 −0.840257
\(685\) 10.9927 0.420010
\(686\) 2.67513 0.102137
\(687\) 73.0757 2.78801
\(688\) 72.3547 2.75850
\(689\) −1.35026 −0.0514409
\(690\) 21.1998 0.807063
\(691\) −13.8618 −0.527327 −0.263663 0.964615i \(-0.584931\pi\)
−0.263663 + 0.964615i \(0.584931\pi\)
\(692\) −46.3063 −1.76030
\(693\) 3.15633 0.119899
\(694\) −18.1681 −0.689651
\(695\) −6.88717 −0.261245
\(696\) −75.6747 −2.86844
\(697\) −27.3561 −1.03619
\(698\) −71.5569 −2.70847
\(699\) 21.6629 0.819367
\(700\) −5.15633 −0.194891
\(701\) 40.5256 1.53063 0.765316 0.643655i \(-0.222583\pi\)
0.765316 + 0.643655i \(0.222583\pi\)
\(702\) 6.05079 0.228372
\(703\) 11.5369 0.435123
\(704\) −18.1187 −0.682875
\(705\) −29.3561 −1.10562
\(706\) 45.1100 1.69774
\(707\) −7.41327 −0.278805
\(708\) −173.995 −6.53914
\(709\) −0.850969 −0.0319588 −0.0159794 0.999872i \(-0.505087\pi\)
−0.0159794 + 0.999872i \(0.505087\pi\)
\(710\) −41.5633 −1.55984
\(711\) 6.21696 0.233154
\(712\) −60.9986 −2.28602
\(713\) 16.8872 0.632429
\(714\) −36.1319 −1.35220
\(715\) 5.83146 0.218084
\(716\) −135.289 −5.05598
\(717\) 52.6371 1.96577
\(718\) 10.1465 0.378663
\(719\) 22.5769 0.841976 0.420988 0.907066i \(-0.361683\pi\)
0.420988 + 0.907066i \(0.361683\pi\)
\(720\) −38.7440 −1.44390
\(721\) −4.21933 −0.157136
\(722\) 45.9502 1.71009
\(723\) 23.1695 0.861683
\(724\) −57.7499 −2.14626
\(725\) −3.61213 −0.134151
\(726\) 6.63752 0.246341
\(727\) 12.5174 0.464244 0.232122 0.972687i \(-0.425433\pi\)
0.232122 + 0.972687i \(0.425433\pi\)
\(728\) 49.2384 1.82490
\(729\) −29.7367 −1.10136
\(730\) −30.4119 −1.12559
\(731\) 32.0870 1.18678
\(732\) 18.0811 0.668297
\(733\) 16.6678 0.615641 0.307820 0.951445i \(-0.400400\pi\)
0.307820 + 0.951445i \(0.400400\pi\)
\(734\) 17.0400 0.628957
\(735\) 2.48119 0.0915202
\(736\) 50.9438 1.87781
\(737\) 10.8568 0.399917
\(738\) 42.4323 1.56196
\(739\) 42.7005 1.57076 0.785382 0.619011i \(-0.212467\pi\)
0.785382 + 0.619011i \(0.212467\pi\)
\(740\) 44.0567 1.61956
\(741\) 19.5369 0.717706
\(742\) −0.619421 −0.0227397
\(743\) −19.6873 −0.722259 −0.361129 0.932516i \(-0.617609\pi\)
−0.361129 + 0.932516i \(0.617609\pi\)
\(744\) −110.769 −4.06099
\(745\) −22.8119 −0.835765
\(746\) 57.0651 2.08930
\(747\) 33.5369 1.22705
\(748\) −28.0689 −1.02630
\(749\) 11.5369 0.421549
\(750\) −6.63752 −0.242368
\(751\) −5.85940 −0.213813 −0.106906 0.994269i \(-0.534094\pi\)
−0.106906 + 0.994269i \(0.534094\pi\)
\(752\) −145.232 −5.29605
\(753\) −4.65562 −0.169660
\(754\) 56.3488 2.05210
\(755\) 3.24472 0.118088
\(756\) 2.00000 0.0727393
\(757\) −40.5863 −1.47513 −0.737567 0.675274i \(-0.764025\pi\)
−0.737567 + 0.675274i \(0.764025\pi\)
\(758\) −66.3098 −2.40848
\(759\) −7.92478 −0.287651
\(760\) −11.4010 −0.413559
\(761\) 21.8472 0.791960 0.395980 0.918259i \(-0.370405\pi\)
0.395980 + 0.918259i \(0.370405\pi\)
\(762\) −112.588 −4.07864
\(763\) 2.18664 0.0791618
\(764\) 57.7499 2.08932
\(765\) −17.1817 −0.621206
\(766\) 14.6013 0.527567
\(767\) 79.3073 2.86362
\(768\) −20.0689 −0.724173
\(769\) 45.2892 1.63317 0.816585 0.577226i \(-0.195865\pi\)
0.816585 + 0.577226i \(0.195865\pi\)
\(770\) 2.67513 0.0964050
\(771\) −67.3376 −2.42510
\(772\) 3.11871 0.112245
\(773\) −33.8153 −1.21625 −0.608125 0.793841i \(-0.708078\pi\)
−0.608125 + 0.793841i \(0.708078\pi\)
\(774\) −49.7704 −1.78896
\(775\) −5.28726 −0.189924
\(776\) 7.06205 0.253513
\(777\) −21.1998 −0.760539
\(778\) 36.7123 1.31620
\(779\) 6.78560 0.243119
\(780\) 74.6067 2.67135
\(781\) 15.5369 0.555954
\(782\) 46.5111 1.66323
\(783\) 1.40105 0.0500693
\(784\) 12.2750 0.438394
\(785\) 5.42548 0.193644
\(786\) −65.8759 −2.34972
\(787\) 1.27504 0.0454502 0.0227251 0.999742i \(-0.492766\pi\)
0.0227251 + 0.999742i \(0.492766\pi\)
\(788\) 78.9194 2.81139
\(789\) −31.7889 −1.13172
\(790\) 5.26916 0.187468
\(791\) 9.35026 0.332457
\(792\) 26.6507 0.946991
\(793\) −8.24140 −0.292661
\(794\) −5.64832 −0.200452
\(795\) −0.574515 −0.0203760
\(796\) −64.7753 −2.29590
\(797\) −42.5256 −1.50634 −0.753168 0.657829i \(-0.771475\pi\)
−0.753168 + 0.657829i \(0.771475\pi\)
\(798\) 8.96239 0.317265
\(799\) −64.4055 −2.27850
\(800\) −15.9502 −0.563924
\(801\) 22.8021 0.805672
\(802\) −51.2262 −1.80886
\(803\) 11.3684 0.401181
\(804\) 138.901 4.89865
\(805\) −3.19394 −0.112571
\(806\) 82.4807 2.90526
\(807\) 15.5369 0.546925
\(808\) −62.5945 −2.20207
\(809\) −14.7151 −0.517356 −0.258678 0.965964i \(-0.583287\pi\)
−0.258678 + 0.965964i \(0.583287\pi\)
\(810\) −22.7562 −0.799573
\(811\) 51.7743 1.81804 0.909021 0.416750i \(-0.136831\pi\)
0.909021 + 0.416750i \(0.136831\pi\)
\(812\) 18.6253 0.653620
\(813\) 14.2374 0.499328
\(814\) −22.8568 −0.801132
\(815\) −3.38058 −0.118417
\(816\) −165.794 −5.80395
\(817\) −7.95906 −0.278452
\(818\) 49.9937 1.74799
\(819\) −18.4060 −0.643157
\(820\) 25.9126 0.904906
\(821\) 2.64974 0.0924765 0.0462383 0.998930i \(-0.485277\pi\)
0.0462383 + 0.998930i \(0.485277\pi\)
\(822\) −72.9643 −2.54492
\(823\) 5.76845 0.201076 0.100538 0.994933i \(-0.467944\pi\)
0.100538 + 0.994933i \(0.467944\pi\)
\(824\) −35.6263 −1.24110
\(825\) 2.48119 0.0863841
\(826\) 36.3815 1.26588
\(827\) −13.4920 −0.469163 −0.234581 0.972096i \(-0.575372\pi\)
−0.234581 + 0.972096i \(0.575372\pi\)
\(828\) −51.9814 −1.80648
\(829\) −4.70052 −0.163256 −0.0816280 0.996663i \(-0.526012\pi\)
−0.0816280 + 0.996663i \(0.526012\pi\)
\(830\) 28.4241 0.986614
\(831\) 20.7367 0.719349
\(832\) 105.658 3.66305
\(833\) 5.44358 0.188609
\(834\) 45.7137 1.58294
\(835\) −11.2750 −0.390189
\(836\) 6.96239 0.240799
\(837\) 2.05079 0.0708855
\(838\) 2.06888 0.0714684
\(839\) −38.8045 −1.33968 −0.669839 0.742506i \(-0.733637\pi\)
−0.669839 + 0.742506i \(0.733637\pi\)
\(840\) 20.9502 0.722850
\(841\) −15.9525 −0.550088
\(842\) 28.1417 0.969828
\(843\) 20.9624 0.721983
\(844\) −22.8872 −0.787809
\(845\) −21.0059 −0.722624
\(846\) 99.8999 3.43463
\(847\) −1.00000 −0.0343604
\(848\) −2.84226 −0.0976036
\(849\) −2.07522 −0.0712215
\(850\) −14.5623 −0.499483
\(851\) 27.2896 0.935476
\(852\) 198.777 6.80998
\(853\) 20.6824 0.708153 0.354076 0.935217i \(-0.384795\pi\)
0.354076 + 0.935217i \(0.384795\pi\)
\(854\) −3.78067 −0.129372
\(855\) 4.26187 0.145753
\(856\) 97.4128 3.32950
\(857\) 26.3453 0.899940 0.449970 0.893044i \(-0.351435\pi\)
0.449970 + 0.893044i \(0.351435\pi\)
\(858\) −38.7064 −1.32141
\(859\) −8.51151 −0.290409 −0.145205 0.989402i \(-0.546384\pi\)
−0.145205 + 0.989402i \(0.546384\pi\)
\(860\) −30.3938 −1.03642
\(861\) −12.4690 −0.424942
\(862\) 66.1582 2.25336
\(863\) 7.56722 0.257591 0.128796 0.991671i \(-0.458889\pi\)
0.128796 + 0.991671i \(0.458889\pi\)
\(864\) 6.18664 0.210474
\(865\) 8.98049 0.305346
\(866\) 49.6502 1.68718
\(867\) −31.3439 −1.06450
\(868\) 27.2628 0.925360
\(869\) −1.96968 −0.0668169
\(870\) 23.9756 0.812848
\(871\) −63.3112 −2.14522
\(872\) 18.4631 0.625239
\(873\) −2.63989 −0.0893467
\(874\) −11.5369 −0.390242
\(875\) 1.00000 0.0338062
\(876\) 145.445 4.91413
\(877\) 17.2955 0.584028 0.292014 0.956414i \(-0.405675\pi\)
0.292014 + 0.956414i \(0.405675\pi\)
\(878\) 3.81336 0.128695
\(879\) 6.74543 0.227518
\(880\) 12.2750 0.413791
\(881\) 20.4504 0.688992 0.344496 0.938788i \(-0.388050\pi\)
0.344496 + 0.938788i \(0.388050\pi\)
\(882\) −8.44358 −0.284310
\(883\) −49.6589 −1.67116 −0.835578 0.549371i \(-0.814867\pi\)
−0.835578 + 0.549371i \(0.814867\pi\)
\(884\) 163.682 5.50524
\(885\) 33.7440 1.13429
\(886\) −107.520 −3.61221
\(887\) −47.1100 −1.58180 −0.790900 0.611946i \(-0.790387\pi\)
−0.790900 + 0.611946i \(0.790387\pi\)
\(888\) −179.002 −6.00693
\(889\) 16.9624 0.568900
\(890\) 19.3258 0.647803
\(891\) 8.50659 0.284981
\(892\) −40.1197 −1.34331
\(893\) 15.9756 0.534602
\(894\) 151.415 5.06407
\(895\) 26.2374 0.877020
\(896\) 16.5696 0.553551
\(897\) 46.2130 1.54301
\(898\) −33.8554 −1.12977
\(899\) 19.0982 0.636962
\(900\) 16.2750 0.542501
\(901\) −1.26045 −0.0419917
\(902\) −13.4436 −0.447622
\(903\) 14.6253 0.486700
\(904\) 78.9497 2.62583
\(905\) 11.1998 0.372294
\(906\) −21.5369 −0.715516
\(907\) 14.4591 0.480107 0.240054 0.970760i \(-0.422835\pi\)
0.240054 + 0.970760i \(0.422835\pi\)
\(908\) −53.8759 −1.78793
\(909\) 23.3987 0.776085
\(910\) −15.5999 −0.517132
\(911\) −31.5369 −1.04486 −0.522432 0.852681i \(-0.674975\pi\)
−0.522432 + 0.852681i \(0.674975\pi\)
\(912\) 41.1246 1.36177
\(913\) −10.6253 −0.351646
\(914\) 1.45580 0.0481536
\(915\) −3.50659 −0.115924
\(916\) −151.863 −5.01770
\(917\) 9.92478 0.327745
\(918\) 5.64832 0.186422
\(919\) 5.26328 0.173620 0.0868098 0.996225i \(-0.472333\pi\)
0.0868098 + 0.996225i \(0.472333\pi\)
\(920\) −26.9683 −0.889117
\(921\) 20.7513 0.683779
\(922\) 30.9659 1.01981
\(923\) −90.6028 −2.98223
\(924\) −12.7938 −0.420887
\(925\) −8.54420 −0.280932
\(926\) −63.6542 −2.09181
\(927\) 13.3176 0.437407
\(928\) 57.6140 1.89127
\(929\) 26.0508 0.854699 0.427349 0.904087i \(-0.359447\pi\)
0.427349 + 0.904087i \(0.359447\pi\)
\(930\) 35.0943 1.15079
\(931\) −1.35026 −0.0442530
\(932\) −45.0191 −1.47465
\(933\) −11.0073 −0.360363
\(934\) 7.13681 0.233524
\(935\) 5.44358 0.178024
\(936\) −155.412 −5.07981
\(937\) −29.3439 −0.958624 −0.479312 0.877645i \(-0.659114\pi\)
−0.479312 + 0.877645i \(0.659114\pi\)
\(938\) −29.0435 −0.948304
\(939\) −73.9121 −2.41203
\(940\) 61.0068 1.98982
\(941\) −28.6375 −0.933556 −0.466778 0.884374i \(-0.654585\pi\)
−0.466778 + 0.884374i \(0.654585\pi\)
\(942\) −36.0118 −1.17333
\(943\) 16.0508 0.522685
\(944\) 166.939 5.43341
\(945\) −0.387873 −0.0126175
\(946\) 15.7685 0.512677
\(947\) −52.8178 −1.71635 −0.858174 0.513358i \(-0.828401\pi\)
−0.858174 + 0.513358i \(0.828401\pi\)
\(948\) −25.1998 −0.818452
\(949\) −66.2941 −2.15200
\(950\) 3.61213 0.117193
\(951\) −38.2130 −1.23914
\(952\) 45.9633 1.48968
\(953\) 37.1939 1.20483 0.602415 0.798183i \(-0.294205\pi\)
0.602415 + 0.798183i \(0.294205\pi\)
\(954\) 1.95509 0.0632985
\(955\) −11.1998 −0.362418
\(956\) −109.388 −3.53787
\(957\) −8.96239 −0.289713
\(958\) 28.6516 0.925693
\(959\) 10.9927 0.354973
\(960\) 44.9560 1.45095
\(961\) −3.04491 −0.0982228
\(962\) 133.289 4.29740
\(963\) −36.4142 −1.17343
\(964\) −48.1500 −1.55081
\(965\) −0.604833 −0.0194703
\(966\) 21.1998 0.682093
\(967\) 4.07125 0.130923 0.0654613 0.997855i \(-0.479148\pi\)
0.0654613 + 0.997855i \(0.479148\pi\)
\(968\) −8.44358 −0.271387
\(969\) 18.2374 0.585871
\(970\) −2.23743 −0.0718395
\(971\) 0.773377 0.0248188 0.0124094 0.999923i \(-0.496050\pi\)
0.0124094 + 0.999923i \(0.496050\pi\)
\(972\) 114.832 3.68324
\(973\) −6.88717 −0.220792
\(974\) −46.6312 −1.49416
\(975\) −14.4690 −0.463378
\(976\) −17.3479 −0.555292
\(977\) −37.8740 −1.21170 −0.605848 0.795580i \(-0.707166\pi\)
−0.605848 + 0.795580i \(0.707166\pi\)
\(978\) 22.4387 0.717509
\(979\) −7.22425 −0.230888
\(980\) −5.15633 −0.164713
\(981\) −6.90175 −0.220356
\(982\) 75.8916 2.42180
\(983\) −15.5794 −0.496907 −0.248453 0.968644i \(-0.579922\pi\)
−0.248453 + 0.968644i \(0.579922\pi\)
\(984\) −105.283 −3.35629
\(985\) −15.3054 −0.487669
\(986\) 52.6009 1.67515
\(987\) −29.3561 −0.934416
\(988\) −40.6009 −1.29169
\(989\) −18.8265 −0.598649
\(990\) −8.44358 −0.268355
\(991\) 27.0982 0.860804 0.430402 0.902637i \(-0.358372\pi\)
0.430402 + 0.902637i \(0.358372\pi\)
\(992\) 84.3327 2.67756
\(993\) −15.5369 −0.493049
\(994\) −41.5633 −1.31831
\(995\) 12.5623 0.398252
\(996\) −135.938 −4.30737
\(997\) 50.4060 1.59637 0.798187 0.602410i \(-0.205793\pi\)
0.798187 + 0.602410i \(0.205793\pi\)
\(998\) −73.5026 −2.32668
\(999\) 3.31406 0.104852
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 385.2.a.f.1.1 3
3.2 odd 2 3465.2.a.bh.1.3 3
4.3 odd 2 6160.2.a.bn.1.3 3
5.2 odd 4 1925.2.b.n.1849.1 6
5.3 odd 4 1925.2.b.n.1849.6 6
5.4 even 2 1925.2.a.v.1.3 3
7.6 odd 2 2695.2.a.g.1.1 3
11.10 odd 2 4235.2.a.q.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
385.2.a.f.1.1 3 1.1 even 1 trivial
1925.2.a.v.1.3 3 5.4 even 2
1925.2.b.n.1849.1 6 5.2 odd 4
1925.2.b.n.1849.6 6 5.3 odd 4
2695.2.a.g.1.1 3 7.6 odd 2
3465.2.a.bh.1.3 3 3.2 odd 2
4235.2.a.q.1.3 3 11.10 odd 2
6160.2.a.bn.1.3 3 4.3 odd 2