Properties

Label 2169.2.a.e.1.6
Level $2169$
Weight $2$
Character 2169.1
Self dual yes
Analytic conductor $17.320$
Analytic rank $0$
Dimension $7$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2169 = 3^{2} \cdot 241 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2169.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(17.3195521984\)
Analytic rank: \(0\)
Dimension: \(7\)
Coefficient field: 7.7.31056073.1
Defining polynomial: \(x^{7} - 3 x^{6} - 3 x^{5} + 11 x^{4} + x^{3} - 9 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 241)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.6
Root \(-0.911223\) of defining polynomial
Character \(\chi\) \(=\) 2169.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.91122 q^{2} +1.65278 q^{4} +2.25110 q^{5} +3.52970 q^{7} -0.663624 q^{8} +O(q^{10})\) \(q+1.91122 q^{2} +1.65278 q^{4} +2.25110 q^{5} +3.52970 q^{7} -0.663624 q^{8} +4.30235 q^{10} -0.515564 q^{11} -5.38098 q^{13} +6.74604 q^{14} -4.57388 q^{16} +4.16566 q^{17} +4.92935 q^{19} +3.72056 q^{20} -0.985358 q^{22} +7.69193 q^{23} +0.0674429 q^{25} -10.2843 q^{26} +5.83379 q^{28} +8.93755 q^{29} -4.43182 q^{31} -7.41447 q^{32} +7.96151 q^{34} +7.94569 q^{35} +5.99816 q^{37} +9.42110 q^{38} -1.49388 q^{40} +8.99946 q^{41} -1.66336 q^{43} -0.852112 q^{44} +14.7010 q^{46} -8.55484 q^{47} +5.45876 q^{49} +0.128898 q^{50} -8.89356 q^{52} -13.1736 q^{53} -1.16059 q^{55} -2.34239 q^{56} +17.0817 q^{58} -9.25521 q^{59} -10.4203 q^{61} -8.47019 q^{62} -5.02293 q^{64} -12.1131 q^{65} -3.91715 q^{67} +6.88490 q^{68} +15.1860 q^{70} +13.6724 q^{71} +11.5529 q^{73} +11.4638 q^{74} +8.14711 q^{76} -1.81979 q^{77} -1.43448 q^{79} -10.2963 q^{80} +17.2000 q^{82} -1.73047 q^{83} +9.37732 q^{85} -3.17906 q^{86} +0.342141 q^{88} +1.07999 q^{89} -18.9932 q^{91} +12.7130 q^{92} -16.3502 q^{94} +11.0965 q^{95} -16.0883 q^{97} +10.4329 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7q + 4q^{2} + 2q^{4} + 8q^{5} - 7q^{7} + 6q^{8} + O(q^{10}) \) \( 7q + 4q^{2} + 2q^{4} + 8q^{5} - 7q^{7} + 6q^{8} + 3q^{10} + 18q^{11} - q^{13} + 6q^{14} + 4q^{16} + 2q^{17} - 6q^{19} + 8q^{20} + 10q^{22} + 22q^{23} + 5q^{25} - 8q^{26} + 9q^{28} + 16q^{29} - 18q^{31} + 6q^{32} + 11q^{34} - 7q^{35} + 8q^{37} - 16q^{38} + 14q^{40} + 15q^{41} + 14q^{43} + 4q^{44} + 11q^{46} + 10q^{47} + 6q^{49} + 4q^{50} + 27q^{52} - 15q^{53} + 29q^{55} - 13q^{56} + 17q^{58} + 18q^{59} + 4q^{61} - 13q^{62} + 2q^{64} + 7q^{65} + 18q^{67} + 15q^{68} + 8q^{70} + 50q^{71} - 10q^{74} - 20q^{76} - 17q^{77} - 15q^{79} + 11q^{80} + 45q^{82} + 24q^{83} - 2q^{85} + 23q^{86} + 8q^{88} + 13q^{89} - 12q^{91} + 10q^{92} - 32q^{94} + 41q^{95} + q^{97} - 9q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.91122 1.35144 0.675720 0.737159i \(-0.263833\pi\)
0.675720 + 0.737159i \(0.263833\pi\)
\(3\) 0 0
\(4\) 1.65278 0.826388
\(5\) 2.25110 1.00672 0.503361 0.864076i \(-0.332097\pi\)
0.503361 + 0.864076i \(0.332097\pi\)
\(6\) 0 0
\(7\) 3.52970 1.33410 0.667050 0.745013i \(-0.267557\pi\)
0.667050 + 0.745013i \(0.267557\pi\)
\(8\) −0.663624 −0.234627
\(9\) 0 0
\(10\) 4.30235 1.36052
\(11\) −0.515564 −0.155448 −0.0777242 0.996975i \(-0.524765\pi\)
−0.0777242 + 0.996975i \(0.524765\pi\)
\(12\) 0 0
\(13\) −5.38098 −1.49242 −0.746208 0.665712i \(-0.768128\pi\)
−0.746208 + 0.665712i \(0.768128\pi\)
\(14\) 6.74604 1.80295
\(15\) 0 0
\(16\) −4.57388 −1.14347
\(17\) 4.16566 1.01032 0.505161 0.863025i \(-0.331433\pi\)
0.505161 + 0.863025i \(0.331433\pi\)
\(18\) 0 0
\(19\) 4.92935 1.13087 0.565436 0.824792i \(-0.308708\pi\)
0.565436 + 0.824792i \(0.308708\pi\)
\(20\) 3.72056 0.831942
\(21\) 0 0
\(22\) −0.985358 −0.210079
\(23\) 7.69193 1.60388 0.801940 0.597405i \(-0.203802\pi\)
0.801940 + 0.597405i \(0.203802\pi\)
\(24\) 0 0
\(25\) 0.0674429 0.0134886
\(26\) −10.2843 −2.01691
\(27\) 0 0
\(28\) 5.83379 1.10248
\(29\) 8.93755 1.65966 0.829831 0.558015i \(-0.188437\pi\)
0.829831 + 0.558015i \(0.188437\pi\)
\(30\) 0 0
\(31\) −4.43182 −0.795978 −0.397989 0.917390i \(-0.630292\pi\)
−0.397989 + 0.917390i \(0.630292\pi\)
\(32\) −7.41447 −1.31070
\(33\) 0 0
\(34\) 7.96151 1.36539
\(35\) 7.94569 1.34307
\(36\) 0 0
\(37\) 5.99816 0.986091 0.493046 0.870003i \(-0.335884\pi\)
0.493046 + 0.870003i \(0.335884\pi\)
\(38\) 9.42110 1.52830
\(39\) 0 0
\(40\) −1.49388 −0.236204
\(41\) 8.99946 1.40548 0.702740 0.711447i \(-0.251960\pi\)
0.702740 + 0.711447i \(0.251960\pi\)
\(42\) 0 0
\(43\) −1.66336 −0.253660 −0.126830 0.991924i \(-0.540480\pi\)
−0.126830 + 0.991924i \(0.540480\pi\)
\(44\) −0.852112 −0.128461
\(45\) 0 0
\(46\) 14.7010 2.16754
\(47\) −8.55484 −1.24785 −0.623926 0.781483i \(-0.714463\pi\)
−0.623926 + 0.781483i \(0.714463\pi\)
\(48\) 0 0
\(49\) 5.45876 0.779823
\(50\) 0.128898 0.0182290
\(51\) 0 0
\(52\) −8.89356 −1.23331
\(53\) −13.1736 −1.80954 −0.904769 0.425903i \(-0.859957\pi\)
−0.904769 + 0.425903i \(0.859957\pi\)
\(54\) 0 0
\(55\) −1.16059 −0.156493
\(56\) −2.34239 −0.313015
\(57\) 0 0
\(58\) 17.0817 2.24293
\(59\) −9.25521 −1.20493 −0.602463 0.798147i \(-0.705814\pi\)
−0.602463 + 0.798147i \(0.705814\pi\)
\(60\) 0 0
\(61\) −10.4203 −1.33418 −0.667090 0.744978i \(-0.732460\pi\)
−0.667090 + 0.744978i \(0.732460\pi\)
\(62\) −8.47019 −1.07572
\(63\) 0 0
\(64\) −5.02293 −0.627867
\(65\) −12.1131 −1.50245
\(66\) 0 0
\(67\) −3.91715 −0.478555 −0.239278 0.970951i \(-0.576911\pi\)
−0.239278 + 0.970951i \(0.576911\pi\)
\(68\) 6.88490 0.834917
\(69\) 0 0
\(70\) 15.1860 1.81507
\(71\) 13.6724 1.62262 0.811308 0.584619i \(-0.198756\pi\)
0.811308 + 0.584619i \(0.198756\pi\)
\(72\) 0 0
\(73\) 11.5529 1.35217 0.676083 0.736825i \(-0.263676\pi\)
0.676083 + 0.736825i \(0.263676\pi\)
\(74\) 11.4638 1.33264
\(75\) 0 0
\(76\) 8.14711 0.934538
\(77\) −1.81979 −0.207384
\(78\) 0 0
\(79\) −1.43448 −0.161391 −0.0806956 0.996739i \(-0.525714\pi\)
−0.0806956 + 0.996739i \(0.525714\pi\)
\(80\) −10.2963 −1.15116
\(81\) 0 0
\(82\) 17.2000 1.89942
\(83\) −1.73047 −0.189944 −0.0949720 0.995480i \(-0.530276\pi\)
−0.0949720 + 0.995480i \(0.530276\pi\)
\(84\) 0 0
\(85\) 9.37732 1.01711
\(86\) −3.17906 −0.342807
\(87\) 0 0
\(88\) 0.342141 0.0364724
\(89\) 1.07999 0.114478 0.0572392 0.998360i \(-0.481770\pi\)
0.0572392 + 0.998360i \(0.481770\pi\)
\(90\) 0 0
\(91\) −18.9932 −1.99103
\(92\) 12.7130 1.32543
\(93\) 0 0
\(94\) −16.3502 −1.68640
\(95\) 11.0965 1.13847
\(96\) 0 0
\(97\) −16.0883 −1.63352 −0.816759 0.576979i \(-0.804231\pi\)
−0.816759 + 0.576979i \(0.804231\pi\)
\(98\) 10.4329 1.05388
\(99\) 0 0
\(100\) 0.111468 0.0111468
\(101\) 1.23922 0.123307 0.0616536 0.998098i \(-0.480363\pi\)
0.0616536 + 0.998098i \(0.480363\pi\)
\(102\) 0 0
\(103\) 5.27633 0.519892 0.259946 0.965623i \(-0.416295\pi\)
0.259946 + 0.965623i \(0.416295\pi\)
\(104\) 3.57095 0.350161
\(105\) 0 0
\(106\) −25.1778 −2.44548
\(107\) −10.7822 −1.04236 −0.521178 0.853448i \(-0.674507\pi\)
−0.521178 + 0.853448i \(0.674507\pi\)
\(108\) 0 0
\(109\) −1.15255 −0.110394 −0.0551972 0.998475i \(-0.517579\pi\)
−0.0551972 + 0.998475i \(0.517579\pi\)
\(110\) −2.21814 −0.211491
\(111\) 0 0
\(112\) −16.1444 −1.52550
\(113\) 8.70999 0.819367 0.409684 0.912228i \(-0.365639\pi\)
0.409684 + 0.912228i \(0.365639\pi\)
\(114\) 0 0
\(115\) 17.3153 1.61466
\(116\) 14.7718 1.37152
\(117\) 0 0
\(118\) −17.6888 −1.62838
\(119\) 14.7035 1.34787
\(120\) 0 0
\(121\) −10.7342 −0.975836
\(122\) −19.9155 −1.80306
\(123\) 0 0
\(124\) −7.32480 −0.657786
\(125\) −11.1037 −0.993142
\(126\) 0 0
\(127\) −8.22063 −0.729463 −0.364732 0.931113i \(-0.618839\pi\)
−0.364732 + 0.931113i \(0.618839\pi\)
\(128\) 5.22899 0.462181
\(129\) 0 0
\(130\) −23.1509 −2.03047
\(131\) 7.30323 0.638086 0.319043 0.947740i \(-0.396638\pi\)
0.319043 + 0.947740i \(0.396638\pi\)
\(132\) 0 0
\(133\) 17.3991 1.50870
\(134\) −7.48654 −0.646739
\(135\) 0 0
\(136\) −2.76444 −0.237048
\(137\) −5.95317 −0.508614 −0.254307 0.967124i \(-0.581847\pi\)
−0.254307 + 0.967124i \(0.581847\pi\)
\(138\) 0 0
\(139\) 14.9650 1.26931 0.634656 0.772795i \(-0.281142\pi\)
0.634656 + 0.772795i \(0.281142\pi\)
\(140\) 13.1324 1.10989
\(141\) 0 0
\(142\) 26.1310 2.19287
\(143\) 2.77424 0.231994
\(144\) 0 0
\(145\) 20.1193 1.67082
\(146\) 22.0802 1.82737
\(147\) 0 0
\(148\) 9.91361 0.814894
\(149\) 0.130576 0.0106972 0.00534860 0.999986i \(-0.498297\pi\)
0.00534860 + 0.999986i \(0.498297\pi\)
\(150\) 0 0
\(151\) 0.276102 0.0224688 0.0112344 0.999937i \(-0.496424\pi\)
0.0112344 + 0.999937i \(0.496424\pi\)
\(152\) −3.27124 −0.265333
\(153\) 0 0
\(154\) −3.47802 −0.280267
\(155\) −9.97646 −0.801328
\(156\) 0 0
\(157\) 16.1044 1.28527 0.642636 0.766172i \(-0.277841\pi\)
0.642636 + 0.766172i \(0.277841\pi\)
\(158\) −2.74160 −0.218110
\(159\) 0 0
\(160\) −16.6907 −1.31952
\(161\) 27.1502 2.13973
\(162\) 0 0
\(163\) −18.5215 −1.45071 −0.725357 0.688373i \(-0.758325\pi\)
−0.725357 + 0.688373i \(0.758325\pi\)
\(164\) 14.8741 1.16147
\(165\) 0 0
\(166\) −3.30732 −0.256698
\(167\) −15.5017 −1.19956 −0.599779 0.800166i \(-0.704745\pi\)
−0.599779 + 0.800166i \(0.704745\pi\)
\(168\) 0 0
\(169\) 15.9550 1.22731
\(170\) 17.9221 1.37457
\(171\) 0 0
\(172\) −2.74916 −0.209622
\(173\) 0.366438 0.0278597 0.0139299 0.999903i \(-0.495566\pi\)
0.0139299 + 0.999903i \(0.495566\pi\)
\(174\) 0 0
\(175\) 0.238053 0.0179951
\(176\) 2.35813 0.177751
\(177\) 0 0
\(178\) 2.06410 0.154711
\(179\) −12.8719 −0.962090 −0.481045 0.876696i \(-0.659743\pi\)
−0.481045 + 0.876696i \(0.659743\pi\)
\(180\) 0 0
\(181\) −12.2125 −0.907746 −0.453873 0.891066i \(-0.649958\pi\)
−0.453873 + 0.891066i \(0.649958\pi\)
\(182\) −36.3003 −2.69076
\(183\) 0 0
\(184\) −5.10456 −0.376313
\(185\) 13.5024 0.992719
\(186\) 0 0
\(187\) −2.14767 −0.157053
\(188\) −14.1392 −1.03121
\(189\) 0 0
\(190\) 21.2078 1.53858
\(191\) 3.92400 0.283930 0.141965 0.989872i \(-0.454658\pi\)
0.141965 + 0.989872i \(0.454658\pi\)
\(192\) 0 0
\(193\) −1.69319 −0.121879 −0.0609393 0.998141i \(-0.519410\pi\)
−0.0609393 + 0.998141i \(0.519410\pi\)
\(194\) −30.7483 −2.20760
\(195\) 0 0
\(196\) 9.02210 0.644436
\(197\) 2.20382 0.157016 0.0785080 0.996913i \(-0.474984\pi\)
0.0785080 + 0.996913i \(0.474984\pi\)
\(198\) 0 0
\(199\) −15.5533 −1.10255 −0.551274 0.834324i \(-0.685858\pi\)
−0.551274 + 0.834324i \(0.685858\pi\)
\(200\) −0.0447567 −0.00316478
\(201\) 0 0
\(202\) 2.36843 0.166642
\(203\) 31.5468 2.21415
\(204\) 0 0
\(205\) 20.2587 1.41493
\(206\) 10.0842 0.702603
\(207\) 0 0
\(208\) 24.6120 1.70654
\(209\) −2.54140 −0.175792
\(210\) 0 0
\(211\) 6.55996 0.451606 0.225803 0.974173i \(-0.427499\pi\)
0.225803 + 0.974173i \(0.427499\pi\)
\(212\) −21.7731 −1.49538
\(213\) 0 0
\(214\) −20.6072 −1.40868
\(215\) −3.74439 −0.255365
\(216\) 0 0
\(217\) −15.6430 −1.06191
\(218\) −2.20278 −0.149191
\(219\) 0 0
\(220\) −1.91819 −0.129324
\(221\) −22.4154 −1.50782
\(222\) 0 0
\(223\) 6.53800 0.437817 0.218908 0.975745i \(-0.429750\pi\)
0.218908 + 0.975745i \(0.429750\pi\)
\(224\) −26.1708 −1.74861
\(225\) 0 0
\(226\) 16.6467 1.10732
\(227\) −16.1673 −1.07306 −0.536530 0.843881i \(-0.680265\pi\)
−0.536530 + 0.843881i \(0.680265\pi\)
\(228\) 0 0
\(229\) 0.0208412 0.00137722 0.000688612 1.00000i \(-0.499781\pi\)
0.000688612 1.00000i \(0.499781\pi\)
\(230\) 33.0934 2.18211
\(231\) 0 0
\(232\) −5.93118 −0.389401
\(233\) 0.651845 0.0427038 0.0213519 0.999772i \(-0.493203\pi\)
0.0213519 + 0.999772i \(0.493203\pi\)
\(234\) 0 0
\(235\) −19.2578 −1.25624
\(236\) −15.2968 −0.995736
\(237\) 0 0
\(238\) 28.1017 1.82156
\(239\) 8.99997 0.582160 0.291080 0.956699i \(-0.405985\pi\)
0.291080 + 0.956699i \(0.405985\pi\)
\(240\) 0 0
\(241\) −1.00000 −0.0644157
\(242\) −20.5154 −1.31878
\(243\) 0 0
\(244\) −17.2224 −1.10255
\(245\) 12.2882 0.785064
\(246\) 0 0
\(247\) −26.5248 −1.68773
\(248\) 2.94106 0.186758
\(249\) 0 0
\(250\) −21.2216 −1.34217
\(251\) −16.4986 −1.04138 −0.520691 0.853745i \(-0.674326\pi\)
−0.520691 + 0.853745i \(0.674326\pi\)
\(252\) 0 0
\(253\) −3.96569 −0.249321
\(254\) −15.7115 −0.985825
\(255\) 0 0
\(256\) 20.0396 1.25248
\(257\) −7.99029 −0.498420 −0.249210 0.968449i \(-0.580171\pi\)
−0.249210 + 0.968449i \(0.580171\pi\)
\(258\) 0 0
\(259\) 21.1717 1.31554
\(260\) −20.0203 −1.24160
\(261\) 0 0
\(262\) 13.9581 0.862335
\(263\) 5.47488 0.337595 0.168798 0.985651i \(-0.446012\pi\)
0.168798 + 0.985651i \(0.446012\pi\)
\(264\) 0 0
\(265\) −29.6551 −1.82170
\(266\) 33.2536 2.03891
\(267\) 0 0
\(268\) −6.47416 −0.395472
\(269\) 2.34697 0.143097 0.0715486 0.997437i \(-0.477206\pi\)
0.0715486 + 0.997437i \(0.477206\pi\)
\(270\) 0 0
\(271\) 1.60301 0.0973758 0.0486879 0.998814i \(-0.484496\pi\)
0.0486879 + 0.998814i \(0.484496\pi\)
\(272\) −19.0533 −1.15527
\(273\) 0 0
\(274\) −11.3778 −0.687360
\(275\) −0.0347711 −0.00209678
\(276\) 0 0
\(277\) −21.6775 −1.30247 −0.651237 0.758875i \(-0.725750\pi\)
−0.651237 + 0.758875i \(0.725750\pi\)
\(278\) 28.6014 1.71540
\(279\) 0 0
\(280\) −5.27296 −0.315119
\(281\) −26.9010 −1.60478 −0.802388 0.596802i \(-0.796438\pi\)
−0.802388 + 0.596802i \(0.796438\pi\)
\(282\) 0 0
\(283\) 3.89230 0.231373 0.115687 0.993286i \(-0.463093\pi\)
0.115687 + 0.993286i \(0.463093\pi\)
\(284\) 22.5974 1.34091
\(285\) 0 0
\(286\) 5.30220 0.313526
\(287\) 31.7654 1.87505
\(288\) 0 0
\(289\) 0.352752 0.0207501
\(290\) 38.4525 2.25801
\(291\) 0 0
\(292\) 19.0944 1.11741
\(293\) 5.57389 0.325630 0.162815 0.986657i \(-0.447943\pi\)
0.162815 + 0.986657i \(0.447943\pi\)
\(294\) 0 0
\(295\) −20.8344 −1.21302
\(296\) −3.98053 −0.231363
\(297\) 0 0
\(298\) 0.249560 0.0144566
\(299\) −41.3902 −2.39366
\(300\) 0 0
\(301\) −5.87116 −0.338408
\(302\) 0.527692 0.0303652
\(303\) 0 0
\(304\) −22.5463 −1.29312
\(305\) −23.4571 −1.34315
\(306\) 0 0
\(307\) 23.0795 1.31722 0.658608 0.752486i \(-0.271146\pi\)
0.658608 + 0.752486i \(0.271146\pi\)
\(308\) −3.00770 −0.171379
\(309\) 0 0
\(310\) −19.0672 −1.08295
\(311\) 14.9884 0.849916 0.424958 0.905213i \(-0.360289\pi\)
0.424958 + 0.905213i \(0.360289\pi\)
\(312\) 0 0
\(313\) −2.31082 −0.130615 −0.0653076 0.997865i \(-0.520803\pi\)
−0.0653076 + 0.997865i \(0.520803\pi\)
\(314\) 30.7791 1.73697
\(315\) 0 0
\(316\) −2.37087 −0.133372
\(317\) 18.3630 1.03137 0.515684 0.856779i \(-0.327538\pi\)
0.515684 + 0.856779i \(0.327538\pi\)
\(318\) 0 0
\(319\) −4.60788 −0.257992
\(320\) −11.3071 −0.632087
\(321\) 0 0
\(322\) 51.8901 2.89172
\(323\) 20.5340 1.14254
\(324\) 0 0
\(325\) −0.362909 −0.0201306
\(326\) −35.3987 −1.96055
\(327\) 0 0
\(328\) −5.97226 −0.329763
\(329\) −30.1960 −1.66476
\(330\) 0 0
\(331\) −6.47933 −0.356136 −0.178068 0.984018i \(-0.556985\pi\)
−0.178068 + 0.984018i \(0.556985\pi\)
\(332\) −2.86008 −0.156967
\(333\) 0 0
\(334\) −29.6272 −1.62113
\(335\) −8.81788 −0.481772
\(336\) 0 0
\(337\) −3.54257 −0.192976 −0.0964880 0.995334i \(-0.530761\pi\)
−0.0964880 + 0.995334i \(0.530761\pi\)
\(338\) 30.4936 1.65863
\(339\) 0 0
\(340\) 15.4986 0.840529
\(341\) 2.28489 0.123734
\(342\) 0 0
\(343\) −5.44011 −0.293739
\(344\) 1.10385 0.0595155
\(345\) 0 0
\(346\) 0.700344 0.0376507
\(347\) 14.4844 0.777566 0.388783 0.921329i \(-0.372896\pi\)
0.388783 + 0.921329i \(0.372896\pi\)
\(348\) 0 0
\(349\) −23.6287 −1.26482 −0.632408 0.774636i \(-0.717933\pi\)
−0.632408 + 0.774636i \(0.717933\pi\)
\(350\) 0.454972 0.0243193
\(351\) 0 0
\(352\) 3.82263 0.203747
\(353\) 18.2662 0.972209 0.486105 0.873901i \(-0.338417\pi\)
0.486105 + 0.873901i \(0.338417\pi\)
\(354\) 0 0
\(355\) 30.7779 1.63352
\(356\) 1.78498 0.0946036
\(357\) 0 0
\(358\) −24.6010 −1.30021
\(359\) 20.3758 1.07539 0.537697 0.843138i \(-0.319294\pi\)
0.537697 + 0.843138i \(0.319294\pi\)
\(360\) 0 0
\(361\) 5.29852 0.278870
\(362\) −23.3408 −1.22676
\(363\) 0 0
\(364\) −31.3916 −1.64536
\(365\) 26.0067 1.36125
\(366\) 0 0
\(367\) −1.76412 −0.0920862 −0.0460431 0.998939i \(-0.514661\pi\)
−0.0460431 + 0.998939i \(0.514661\pi\)
\(368\) −35.1820 −1.83399
\(369\) 0 0
\(370\) 25.8062 1.34160
\(371\) −46.4989 −2.41410
\(372\) 0 0
\(373\) −32.1736 −1.66589 −0.832943 0.553359i \(-0.813346\pi\)
−0.832943 + 0.553359i \(0.813346\pi\)
\(374\) −4.10467 −0.212247
\(375\) 0 0
\(376\) 5.67720 0.292779
\(377\) −48.0928 −2.47691
\(378\) 0 0
\(379\) −31.3680 −1.61126 −0.805632 0.592416i \(-0.798174\pi\)
−0.805632 + 0.592416i \(0.798174\pi\)
\(380\) 18.3399 0.940820
\(381\) 0 0
\(382\) 7.49963 0.383715
\(383\) −16.8623 −0.861624 −0.430812 0.902442i \(-0.641773\pi\)
−0.430812 + 0.902442i \(0.641773\pi\)
\(384\) 0 0
\(385\) −4.09651 −0.208778
\(386\) −3.23607 −0.164712
\(387\) 0 0
\(388\) −26.5903 −1.34992
\(389\) 0.816001 0.0413729 0.0206864 0.999786i \(-0.493415\pi\)
0.0206864 + 0.999786i \(0.493415\pi\)
\(390\) 0 0
\(391\) 32.0420 1.62043
\(392\) −3.62257 −0.182967
\(393\) 0 0
\(394\) 4.21200 0.212198
\(395\) −3.22915 −0.162476
\(396\) 0 0
\(397\) −21.8279 −1.09551 −0.547755 0.836639i \(-0.684517\pi\)
−0.547755 + 0.836639i \(0.684517\pi\)
\(398\) −29.7259 −1.49003
\(399\) 0 0
\(400\) −0.308476 −0.0154238
\(401\) 15.4857 0.773321 0.386661 0.922222i \(-0.373628\pi\)
0.386661 + 0.922222i \(0.373628\pi\)
\(402\) 0 0
\(403\) 23.8475 1.18793
\(404\) 2.04816 0.101900
\(405\) 0 0
\(406\) 60.2931 2.99229
\(407\) −3.09244 −0.153286
\(408\) 0 0
\(409\) 26.0377 1.28748 0.643740 0.765245i \(-0.277382\pi\)
0.643740 + 0.765245i \(0.277382\pi\)
\(410\) 38.7189 1.91219
\(411\) 0 0
\(412\) 8.72059 0.429633
\(413\) −32.6681 −1.60749
\(414\) 0 0
\(415\) −3.89546 −0.191221
\(416\) 39.8971 1.95612
\(417\) 0 0
\(418\) −4.85718 −0.237572
\(419\) 33.2204 1.62292 0.811462 0.584406i \(-0.198672\pi\)
0.811462 + 0.584406i \(0.198672\pi\)
\(420\) 0 0
\(421\) 2.90586 0.141623 0.0708115 0.997490i \(-0.477441\pi\)
0.0708115 + 0.997490i \(0.477441\pi\)
\(422\) 12.5375 0.610318
\(423\) 0 0
\(424\) 8.74235 0.424566
\(425\) 0.280944 0.0136278
\(426\) 0 0
\(427\) −36.7804 −1.77993
\(428\) −17.8206 −0.861390
\(429\) 0 0
\(430\) −7.15637 −0.345111
\(431\) 28.4892 1.37228 0.686139 0.727471i \(-0.259304\pi\)
0.686139 + 0.727471i \(0.259304\pi\)
\(432\) 0 0
\(433\) −28.9311 −1.39034 −0.695169 0.718846i \(-0.744671\pi\)
−0.695169 + 0.718846i \(0.744671\pi\)
\(434\) −29.8972 −1.43511
\(435\) 0 0
\(436\) −1.90491 −0.0912286
\(437\) 37.9163 1.81378
\(438\) 0 0
\(439\) −8.92156 −0.425803 −0.212901 0.977074i \(-0.568291\pi\)
−0.212901 + 0.977074i \(0.568291\pi\)
\(440\) 0.770193 0.0367175
\(441\) 0 0
\(442\) −42.8408 −2.03773
\(443\) −24.8216 −1.17931 −0.589656 0.807655i \(-0.700736\pi\)
−0.589656 + 0.807655i \(0.700736\pi\)
\(444\) 0 0
\(445\) 2.43116 0.115248
\(446\) 12.4956 0.591683
\(447\) 0 0
\(448\) −17.7294 −0.837637
\(449\) −11.9254 −0.562793 −0.281397 0.959592i \(-0.590798\pi\)
−0.281397 + 0.959592i \(0.590798\pi\)
\(450\) 0 0
\(451\) −4.63980 −0.218480
\(452\) 14.3957 0.677115
\(453\) 0 0
\(454\) −30.8993 −1.45018
\(455\) −42.7557 −2.00442
\(456\) 0 0
\(457\) 8.44205 0.394902 0.197451 0.980313i \(-0.436734\pi\)
0.197451 + 0.980313i \(0.436734\pi\)
\(458\) 0.0398321 0.00186123
\(459\) 0 0
\(460\) 28.6183 1.33433
\(461\) −23.2925 −1.08484 −0.542419 0.840108i \(-0.682491\pi\)
−0.542419 + 0.840108i \(0.682491\pi\)
\(462\) 0 0
\(463\) 1.66578 0.0774152 0.0387076 0.999251i \(-0.487676\pi\)
0.0387076 + 0.999251i \(0.487676\pi\)
\(464\) −40.8793 −1.89777
\(465\) 0 0
\(466\) 1.24582 0.0577116
\(467\) 6.66428 0.308386 0.154193 0.988041i \(-0.450722\pi\)
0.154193 + 0.988041i \(0.450722\pi\)
\(468\) 0 0
\(469\) −13.8263 −0.638441
\(470\) −36.8059 −1.69773
\(471\) 0 0
\(472\) 6.14198 0.282708
\(473\) 0.857570 0.0394311
\(474\) 0 0
\(475\) 0.332450 0.0152538
\(476\) 24.3016 1.11386
\(477\) 0 0
\(478\) 17.2010 0.786753
\(479\) 13.6902 0.625523 0.312762 0.949832i \(-0.398746\pi\)
0.312762 + 0.949832i \(0.398746\pi\)
\(480\) 0 0
\(481\) −32.2760 −1.47166
\(482\) −1.91122 −0.0870538
\(483\) 0 0
\(484\) −17.7412 −0.806419
\(485\) −36.2163 −1.64450
\(486\) 0 0
\(487\) 31.9634 1.44840 0.724200 0.689590i \(-0.242209\pi\)
0.724200 + 0.689590i \(0.242209\pi\)
\(488\) 6.91515 0.313034
\(489\) 0 0
\(490\) 23.4855 1.06097
\(491\) 10.0550 0.453777 0.226889 0.973921i \(-0.427145\pi\)
0.226889 + 0.973921i \(0.427145\pi\)
\(492\) 0 0
\(493\) 37.2308 1.67679
\(494\) −50.6948 −2.28087
\(495\) 0 0
\(496\) 20.2706 0.910178
\(497\) 48.2594 2.16473
\(498\) 0 0
\(499\) 9.14295 0.409295 0.204647 0.978836i \(-0.434395\pi\)
0.204647 + 0.978836i \(0.434395\pi\)
\(500\) −18.3519 −0.820721
\(501\) 0 0
\(502\) −31.5325 −1.40736
\(503\) 27.2050 1.21301 0.606506 0.795079i \(-0.292571\pi\)
0.606506 + 0.795079i \(0.292571\pi\)
\(504\) 0 0
\(505\) 2.78961 0.124136
\(506\) −7.57931 −0.336941
\(507\) 0 0
\(508\) −13.5869 −0.602819
\(509\) −7.79598 −0.345551 −0.172775 0.984961i \(-0.555274\pi\)
−0.172775 + 0.984961i \(0.555274\pi\)
\(510\) 0 0
\(511\) 40.7783 1.80392
\(512\) 27.8422 1.23046
\(513\) 0 0
\(514\) −15.2712 −0.673585
\(515\) 11.8775 0.523387
\(516\) 0 0
\(517\) 4.41057 0.193977
\(518\) 40.4638 1.77788
\(519\) 0 0
\(520\) 8.03857 0.352514
\(521\) −32.6959 −1.43243 −0.716216 0.697878i \(-0.754128\pi\)
−0.716216 + 0.697878i \(0.754128\pi\)
\(522\) 0 0
\(523\) −7.80136 −0.341129 −0.170565 0.985346i \(-0.554559\pi\)
−0.170565 + 0.985346i \(0.554559\pi\)
\(524\) 12.0706 0.527307
\(525\) 0 0
\(526\) 10.4637 0.456240
\(527\) −18.4615 −0.804194
\(528\) 0 0
\(529\) 36.1658 1.57243
\(530\) −56.6776 −2.46192
\(531\) 0 0
\(532\) 28.7568 1.24677
\(533\) −48.4260 −2.09756
\(534\) 0 0
\(535\) −24.2718 −1.04936
\(536\) 2.59951 0.112282
\(537\) 0 0
\(538\) 4.48558 0.193387
\(539\) −2.81434 −0.121222
\(540\) 0 0
\(541\) 22.3971 0.962926 0.481463 0.876467i \(-0.340106\pi\)
0.481463 + 0.876467i \(0.340106\pi\)
\(542\) 3.06371 0.131597
\(543\) 0 0
\(544\) −30.8862 −1.32423
\(545\) −2.59451 −0.111136
\(546\) 0 0
\(547\) 37.2322 1.59193 0.795966 0.605341i \(-0.206963\pi\)
0.795966 + 0.605341i \(0.206963\pi\)
\(548\) −9.83925 −0.420312
\(549\) 0 0
\(550\) −0.0664554 −0.00283367
\(551\) 44.0563 1.87686
\(552\) 0 0
\(553\) −5.06326 −0.215312
\(554\) −41.4305 −1.76021
\(555\) 0 0
\(556\) 24.7337 1.04894
\(557\) 27.6457 1.17139 0.585693 0.810533i \(-0.300823\pi\)
0.585693 + 0.810533i \(0.300823\pi\)
\(558\) 0 0
\(559\) 8.95053 0.378567
\(560\) −36.3427 −1.53576
\(561\) 0 0
\(562\) −51.4138 −2.16876
\(563\) 23.3454 0.983891 0.491945 0.870626i \(-0.336286\pi\)
0.491945 + 0.870626i \(0.336286\pi\)
\(564\) 0 0
\(565\) 19.6070 0.824875
\(566\) 7.43905 0.312687
\(567\) 0 0
\(568\) −9.07334 −0.380709
\(569\) −2.61346 −0.109562 −0.0547811 0.998498i \(-0.517446\pi\)
−0.0547811 + 0.998498i \(0.517446\pi\)
\(570\) 0 0
\(571\) 16.9259 0.708326 0.354163 0.935184i \(-0.384766\pi\)
0.354163 + 0.935184i \(0.384766\pi\)
\(572\) 4.58520 0.191717
\(573\) 0 0
\(574\) 60.7107 2.53402
\(575\) 0.518766 0.0216340
\(576\) 0 0
\(577\) 4.67534 0.194637 0.0973185 0.995253i \(-0.468973\pi\)
0.0973185 + 0.995253i \(0.468973\pi\)
\(578\) 0.674188 0.0280425
\(579\) 0 0
\(580\) 33.2527 1.38074
\(581\) −6.10804 −0.253404
\(582\) 0 0
\(583\) 6.79185 0.281290
\(584\) −7.66680 −0.317254
\(585\) 0 0
\(586\) 10.6530 0.440069
\(587\) 3.78940 0.156405 0.0782027 0.996937i \(-0.475082\pi\)
0.0782027 + 0.996937i \(0.475082\pi\)
\(588\) 0 0
\(589\) −21.8460 −0.900148
\(590\) −39.8192 −1.63933
\(591\) 0 0
\(592\) −27.4349 −1.12757
\(593\) 6.81434 0.279831 0.139916 0.990163i \(-0.455317\pi\)
0.139916 + 0.990163i \(0.455317\pi\)
\(594\) 0 0
\(595\) 33.0991 1.35693
\(596\) 0.215813 0.00884004
\(597\) 0 0
\(598\) −79.1059 −3.23488
\(599\) 39.7739 1.62512 0.812559 0.582879i \(-0.198074\pi\)
0.812559 + 0.582879i \(0.198074\pi\)
\(600\) 0 0
\(601\) 27.4256 1.11871 0.559356 0.828927i \(-0.311048\pi\)
0.559356 + 0.828927i \(0.311048\pi\)
\(602\) −11.2211 −0.457338
\(603\) 0 0
\(604\) 0.456334 0.0185680
\(605\) −24.1637 −0.982395
\(606\) 0 0
\(607\) 20.6168 0.836811 0.418405 0.908260i \(-0.362589\pi\)
0.418405 + 0.908260i \(0.362589\pi\)
\(608\) −36.5485 −1.48224
\(609\) 0 0
\(610\) −44.8317 −1.81518
\(611\) 46.0335 1.86232
\(612\) 0 0
\(613\) −12.9127 −0.521540 −0.260770 0.965401i \(-0.583976\pi\)
−0.260770 + 0.965401i \(0.583976\pi\)
\(614\) 44.1100 1.78014
\(615\) 0 0
\(616\) 1.20765 0.0486578
\(617\) −26.3856 −1.06225 −0.531123 0.847295i \(-0.678230\pi\)
−0.531123 + 0.847295i \(0.678230\pi\)
\(618\) 0 0
\(619\) −2.30850 −0.0927866 −0.0463933 0.998923i \(-0.514773\pi\)
−0.0463933 + 0.998923i \(0.514773\pi\)
\(620\) −16.4888 −0.662208
\(621\) 0 0
\(622\) 28.6462 1.14861
\(623\) 3.81203 0.152726
\(624\) 0 0
\(625\) −25.3327 −1.01331
\(626\) −4.41649 −0.176519
\(627\) 0 0
\(628\) 26.6170 1.06213
\(629\) 24.9863 0.996270
\(630\) 0 0
\(631\) −8.60820 −0.342687 −0.171343 0.985211i \(-0.554811\pi\)
−0.171343 + 0.985211i \(0.554811\pi\)
\(632\) 0.951953 0.0378667
\(633\) 0 0
\(634\) 35.0958 1.39383
\(635\) −18.5054 −0.734366
\(636\) 0 0
\(637\) −29.3735 −1.16382
\(638\) −8.80669 −0.348660
\(639\) 0 0
\(640\) 11.7710 0.465288
\(641\) 11.7479 0.464014 0.232007 0.972714i \(-0.425471\pi\)
0.232007 + 0.972714i \(0.425471\pi\)
\(642\) 0 0
\(643\) 15.0808 0.594729 0.297364 0.954764i \(-0.403892\pi\)
0.297364 + 0.954764i \(0.403892\pi\)
\(644\) 44.8732 1.76825
\(645\) 0 0
\(646\) 39.2451 1.54408
\(647\) 5.68014 0.223309 0.111655 0.993747i \(-0.464385\pi\)
0.111655 + 0.993747i \(0.464385\pi\)
\(648\) 0 0
\(649\) 4.77166 0.187304
\(650\) −0.693600 −0.0272052
\(651\) 0 0
\(652\) −30.6118 −1.19885
\(653\) 48.9941 1.91729 0.958644 0.284608i \(-0.0918633\pi\)
0.958644 + 0.284608i \(0.0918633\pi\)
\(654\) 0 0
\(655\) 16.4403 0.642375
\(656\) −41.1625 −1.60713
\(657\) 0 0
\(658\) −57.7113 −2.24982
\(659\) 23.5789 0.918505 0.459252 0.888306i \(-0.348117\pi\)
0.459252 + 0.888306i \(0.348117\pi\)
\(660\) 0 0
\(661\) −38.6322 −1.50262 −0.751310 0.659949i \(-0.770578\pi\)
−0.751310 + 0.659949i \(0.770578\pi\)
\(662\) −12.3835 −0.481296
\(663\) 0 0
\(664\) 1.14838 0.0445659
\(665\) 39.1671 1.51884
\(666\) 0 0
\(667\) 68.7470 2.66190
\(668\) −25.6208 −0.991300
\(669\) 0 0
\(670\) −16.8529 −0.651086
\(671\) 5.37232 0.207396
\(672\) 0 0
\(673\) −37.4530 −1.44371 −0.721853 0.692047i \(-0.756709\pi\)
−0.721853 + 0.692047i \(0.756709\pi\)
\(674\) −6.77064 −0.260795
\(675\) 0 0
\(676\) 26.3700 1.01423
\(677\) −16.0616 −0.617297 −0.308649 0.951176i \(-0.599877\pi\)
−0.308649 + 0.951176i \(0.599877\pi\)
\(678\) 0 0
\(679\) −56.7868 −2.17928
\(680\) −6.22302 −0.238642
\(681\) 0 0
\(682\) 4.36693 0.167218
\(683\) −13.5819 −0.519696 −0.259848 0.965649i \(-0.583673\pi\)
−0.259848 + 0.965649i \(0.583673\pi\)
\(684\) 0 0
\(685\) −13.4012 −0.512032
\(686\) −10.3973 −0.396970
\(687\) 0 0
\(688\) 7.60803 0.290053
\(689\) 70.8871 2.70058
\(690\) 0 0
\(691\) −5.72328 −0.217724 −0.108862 0.994057i \(-0.534721\pi\)
−0.108862 + 0.994057i \(0.534721\pi\)
\(692\) 0.605639 0.0230229
\(693\) 0 0
\(694\) 27.6830 1.05083
\(695\) 33.6876 1.27784
\(696\) 0 0
\(697\) 37.4887 1.41999
\(698\) −45.1597 −1.70932
\(699\) 0 0
\(700\) 0.393448 0.0148709
\(701\) 20.2921 0.766421 0.383211 0.923661i \(-0.374818\pi\)
0.383211 + 0.923661i \(0.374818\pi\)
\(702\) 0 0
\(703\) 29.5670 1.11514
\(704\) 2.58964 0.0976009
\(705\) 0 0
\(706\) 34.9107 1.31388
\(707\) 4.37408 0.164504
\(708\) 0 0
\(709\) −31.5398 −1.18450 −0.592251 0.805754i \(-0.701761\pi\)
−0.592251 + 0.805754i \(0.701761\pi\)
\(710\) 58.8235 2.20761
\(711\) 0 0
\(712\) −0.716706 −0.0268597
\(713\) −34.0892 −1.27665
\(714\) 0 0
\(715\) 6.24509 0.233553
\(716\) −21.2743 −0.795059
\(717\) 0 0
\(718\) 38.9427 1.45333
\(719\) 11.0676 0.412753 0.206376 0.978473i \(-0.433833\pi\)
0.206376 + 0.978473i \(0.433833\pi\)
\(720\) 0 0
\(721\) 18.6238 0.693588
\(722\) 10.1267 0.376875
\(723\) 0 0
\(724\) −20.1845 −0.750150
\(725\) 0.602774 0.0223865
\(726\) 0 0
\(727\) −49.5631 −1.83819 −0.919096 0.394033i \(-0.871079\pi\)
−0.919096 + 0.394033i \(0.871079\pi\)
\(728\) 12.6044 0.467149
\(729\) 0 0
\(730\) 49.7047 1.83965
\(731\) −6.92901 −0.256279
\(732\) 0 0
\(733\) −2.44508 −0.0903110 −0.0451555 0.998980i \(-0.514378\pi\)
−0.0451555 + 0.998980i \(0.514378\pi\)
\(734\) −3.37162 −0.124449
\(735\) 0 0
\(736\) −57.0316 −2.10221
\(737\) 2.01954 0.0743907
\(738\) 0 0
\(739\) 36.9067 1.35764 0.678818 0.734307i \(-0.262492\pi\)
0.678818 + 0.734307i \(0.262492\pi\)
\(740\) 22.3165 0.820371
\(741\) 0 0
\(742\) −88.8699 −3.26252
\(743\) −40.6188 −1.49016 −0.745079 0.666976i \(-0.767588\pi\)
−0.745079 + 0.666976i \(0.767588\pi\)
\(744\) 0 0
\(745\) 0.293940 0.0107691
\(746\) −61.4909 −2.25134
\(747\) 0 0
\(748\) −3.54961 −0.129787
\(749\) −38.0579 −1.39061
\(750\) 0 0
\(751\) −14.0406 −0.512349 −0.256174 0.966631i \(-0.582462\pi\)
−0.256174 + 0.966631i \(0.582462\pi\)
\(752\) 39.1289 1.42688
\(753\) 0 0
\(754\) −91.9161 −3.34739
\(755\) 0.621532 0.0226199
\(756\) 0 0
\(757\) 24.6936 0.897505 0.448752 0.893656i \(-0.351869\pi\)
0.448752 + 0.893656i \(0.351869\pi\)
\(758\) −59.9512 −2.17753
\(759\) 0 0
\(760\) −7.36388 −0.267116
\(761\) −12.6083 −0.457051 −0.228525 0.973538i \(-0.573390\pi\)
−0.228525 + 0.973538i \(0.573390\pi\)
\(762\) 0 0
\(763\) −4.06816 −0.147277
\(764\) 6.48548 0.234636
\(765\) 0 0
\(766\) −32.2276 −1.16443
\(767\) 49.8021 1.79825
\(768\) 0 0
\(769\) 42.1413 1.51965 0.759827 0.650125i \(-0.225283\pi\)
0.759827 + 0.650125i \(0.225283\pi\)
\(770\) −7.82936 −0.282150
\(771\) 0 0
\(772\) −2.79847 −0.100719
\(773\) 9.62659 0.346244 0.173122 0.984900i \(-0.444614\pi\)
0.173122 + 0.984900i \(0.444614\pi\)
\(774\) 0 0
\(775\) −0.298895 −0.0107366
\(776\) 10.6766 0.383267
\(777\) 0 0
\(778\) 1.55956 0.0559129
\(779\) 44.3615 1.58942
\(780\) 0 0
\(781\) −7.04900 −0.252233
\(782\) 61.2394 2.18992
\(783\) 0 0
\(784\) −24.9677 −0.891705
\(785\) 36.2526 1.29391
\(786\) 0 0
\(787\) 50.4413 1.79804 0.899019 0.437909i \(-0.144281\pi\)
0.899019 + 0.437909i \(0.144281\pi\)
\(788\) 3.64243 0.129756
\(789\) 0 0
\(790\) −6.17162 −0.219576
\(791\) 30.7436 1.09312
\(792\) 0 0
\(793\) 56.0713 1.99115
\(794\) −41.7179 −1.48051
\(795\) 0 0
\(796\) −25.7062 −0.911131
\(797\) −34.3043 −1.21512 −0.607561 0.794273i \(-0.707852\pi\)
−0.607561 + 0.794273i \(0.707852\pi\)
\(798\) 0 0
\(799\) −35.6366 −1.26073
\(800\) −0.500053 −0.0176795
\(801\) 0 0
\(802\) 29.5967 1.04510
\(803\) −5.95627 −0.210192
\(804\) 0 0
\(805\) 61.1177 2.15412
\(806\) 45.5780 1.60542
\(807\) 0 0
\(808\) −0.822378 −0.0289312
\(809\) 43.9886 1.54656 0.773279 0.634066i \(-0.218615\pi\)
0.773279 + 0.634066i \(0.218615\pi\)
\(810\) 0 0
\(811\) −43.0270 −1.51088 −0.755440 0.655217i \(-0.772577\pi\)
−0.755440 + 0.655217i \(0.772577\pi\)
\(812\) 52.1398 1.82975
\(813\) 0 0
\(814\) −5.91034 −0.207157
\(815\) −41.6937 −1.46047
\(816\) 0 0
\(817\) −8.19930 −0.286857
\(818\) 49.7638 1.73995
\(819\) 0 0
\(820\) 33.4830 1.16928
\(821\) −18.7704 −0.655090 −0.327545 0.944836i \(-0.606221\pi\)
−0.327545 + 0.944836i \(0.606221\pi\)
\(822\) 0 0
\(823\) 27.1588 0.946695 0.473348 0.880876i \(-0.343045\pi\)
0.473348 + 0.880876i \(0.343045\pi\)
\(824\) −3.50150 −0.121981
\(825\) 0 0
\(826\) −62.4360 −2.17243
\(827\) −41.5157 −1.44364 −0.721821 0.692079i \(-0.756695\pi\)
−0.721821 + 0.692079i \(0.756695\pi\)
\(828\) 0 0
\(829\) −30.6349 −1.06399 −0.531997 0.846746i \(-0.678558\pi\)
−0.531997 + 0.846746i \(0.678558\pi\)
\(830\) −7.44510 −0.258423
\(831\) 0 0
\(832\) 27.0283 0.937039
\(833\) 22.7394 0.787872
\(834\) 0 0
\(835\) −34.8959 −1.20762
\(836\) −4.20036 −0.145272
\(837\) 0 0
\(838\) 63.4916 2.19328
\(839\) −30.2456 −1.04419 −0.522097 0.852886i \(-0.674850\pi\)
−0.522097 + 0.852886i \(0.674850\pi\)
\(840\) 0 0
\(841\) 50.8798 1.75447
\(842\) 5.55375 0.191395
\(843\) 0 0
\(844\) 10.8421 0.373202
\(845\) 35.9163 1.23556
\(846\) 0 0
\(847\) −37.8884 −1.30186
\(848\) 60.2547 2.06915
\(849\) 0 0
\(850\) 0.536947 0.0184171
\(851\) 46.1374 1.58157
\(852\) 0 0
\(853\) 42.7734 1.46453 0.732267 0.681018i \(-0.238462\pi\)
0.732267 + 0.681018i \(0.238462\pi\)
\(854\) −70.2956 −2.40546
\(855\) 0 0
\(856\) 7.15534 0.244564
\(857\) 18.8048 0.642361 0.321181 0.947018i \(-0.395920\pi\)
0.321181 + 0.947018i \(0.395920\pi\)
\(858\) 0 0
\(859\) −8.22676 −0.280693 −0.140347 0.990102i \(-0.544822\pi\)
−0.140347 + 0.990102i \(0.544822\pi\)
\(860\) −6.18864 −0.211031
\(861\) 0 0
\(862\) 54.4493 1.85455
\(863\) 41.8464 1.42447 0.712235 0.701941i \(-0.247683\pi\)
0.712235 + 0.701941i \(0.247683\pi\)
\(864\) 0 0
\(865\) 0.824887 0.0280470
\(866\) −55.2937 −1.87896
\(867\) 0 0
\(868\) −25.8543 −0.877552
\(869\) 0.739564 0.0250880
\(870\) 0 0
\(871\) 21.0781 0.714204
\(872\) 0.764862 0.0259015
\(873\) 0 0
\(874\) 72.4664 2.45121
\(875\) −39.1926 −1.32495
\(876\) 0 0
\(877\) −6.11483 −0.206483 −0.103242 0.994656i \(-0.532921\pi\)
−0.103242 + 0.994656i \(0.532921\pi\)
\(878\) −17.0511 −0.575446
\(879\) 0 0
\(880\) 5.30838 0.178946
\(881\) 13.5157 0.455355 0.227678 0.973737i \(-0.426887\pi\)
0.227678 + 0.973737i \(0.426887\pi\)
\(882\) 0 0
\(883\) 38.5489 1.29727 0.648637 0.761098i \(-0.275339\pi\)
0.648637 + 0.761098i \(0.275339\pi\)
\(884\) −37.0476 −1.24604
\(885\) 0 0
\(886\) −47.4397 −1.59377
\(887\) −40.8386 −1.37123 −0.685613 0.727966i \(-0.740466\pi\)
−0.685613 + 0.727966i \(0.740466\pi\)
\(888\) 0 0
\(889\) −29.0163 −0.973177
\(890\) 4.64649 0.155751
\(891\) 0 0
\(892\) 10.8058 0.361806
\(893\) −42.1698 −1.41116
\(894\) 0 0
\(895\) −28.9759 −0.968557
\(896\) 18.4567 0.616596
\(897\) 0 0
\(898\) −22.7921 −0.760581
\(899\) −39.6096 −1.32105
\(900\) 0 0
\(901\) −54.8769 −1.82822
\(902\) −8.86770 −0.295262
\(903\) 0 0
\(904\) −5.78016 −0.192245
\(905\) −27.4915 −0.913848
\(906\) 0 0
\(907\) −34.3321 −1.13998 −0.569989 0.821652i \(-0.693053\pi\)
−0.569989 + 0.821652i \(0.693053\pi\)
\(908\) −26.7209 −0.886763
\(909\) 0 0
\(910\) −81.7156 −2.70885
\(911\) −49.0410 −1.62480 −0.812400 0.583100i \(-0.801840\pi\)
−0.812400 + 0.583100i \(0.801840\pi\)
\(912\) 0 0
\(913\) 0.892169 0.0295265
\(914\) 16.1346 0.533686
\(915\) 0 0
\(916\) 0.0344458 0.00113812
\(917\) 25.7782 0.851271
\(918\) 0 0
\(919\) 48.7329 1.60755 0.803775 0.594934i \(-0.202822\pi\)
0.803775 + 0.594934i \(0.202822\pi\)
\(920\) −11.4909 −0.378842
\(921\) 0 0
\(922\) −44.5171 −1.46609
\(923\) −73.5710 −2.42162
\(924\) 0 0
\(925\) 0.404533 0.0133010
\(926\) 3.18367 0.104622
\(927\) 0 0
\(928\) −66.2672 −2.17533
\(929\) −52.9031 −1.73569 −0.867847 0.496832i \(-0.834496\pi\)
−0.867847 + 0.496832i \(0.834496\pi\)
\(930\) 0 0
\(931\) 26.9081 0.881879
\(932\) 1.07735 0.0352899
\(933\) 0 0
\(934\) 12.7369 0.416765
\(935\) −4.83461 −0.158109
\(936\) 0 0
\(937\) −16.1310 −0.526976 −0.263488 0.964663i \(-0.584873\pi\)
−0.263488 + 0.964663i \(0.584873\pi\)
\(938\) −26.4252 −0.862814
\(939\) 0 0
\(940\) −31.8288 −1.03814
\(941\) −10.7225 −0.349544 −0.174772 0.984609i \(-0.555919\pi\)
−0.174772 + 0.984609i \(0.555919\pi\)
\(942\) 0 0
\(943\) 69.2233 2.25422
\(944\) 42.3323 1.37780
\(945\) 0 0
\(946\) 1.63901 0.0532887
\(947\) 21.9984 0.714851 0.357426 0.933942i \(-0.383655\pi\)
0.357426 + 0.933942i \(0.383655\pi\)
\(948\) 0 0
\(949\) −62.1660 −2.01800
\(950\) 0.635386 0.0206146
\(951\) 0 0
\(952\) −9.75762 −0.316246
\(953\) 40.5214 1.31262 0.656309 0.754492i \(-0.272117\pi\)
0.656309 + 0.754492i \(0.272117\pi\)
\(954\) 0 0
\(955\) 8.83330 0.285839
\(956\) 14.8749 0.481090
\(957\) 0 0
\(958\) 26.1651 0.845356
\(959\) −21.0129 −0.678542
\(960\) 0 0
\(961\) −11.3590 −0.366419
\(962\) −61.6867 −1.98886
\(963\) 0 0
\(964\) −1.65278 −0.0532323
\(965\) −3.81154 −0.122698
\(966\) 0 0
\(967\) −46.5645 −1.49741 −0.748706 0.662902i \(-0.769324\pi\)
−0.748706 + 0.662902i \(0.769324\pi\)
\(968\) 7.12347 0.228957
\(969\) 0 0
\(970\) −69.2175 −2.22244
\(971\) −2.03099 −0.0651775 −0.0325887 0.999469i \(-0.510375\pi\)
−0.0325887 + 0.999469i \(0.510375\pi\)
\(972\) 0 0
\(973\) 52.8218 1.69339
\(974\) 61.0892 1.95743
\(975\) 0 0
\(976\) 47.6611 1.52560
\(977\) 45.7170 1.46262 0.731308 0.682047i \(-0.238910\pi\)
0.731308 + 0.682047i \(0.238910\pi\)
\(978\) 0 0
\(979\) −0.556803 −0.0177955
\(980\) 20.3096 0.648767
\(981\) 0 0
\(982\) 19.2174 0.613252
\(983\) 47.0190 1.49967 0.749837 0.661623i \(-0.230132\pi\)
0.749837 + 0.661623i \(0.230132\pi\)
\(984\) 0 0
\(985\) 4.96103 0.158071
\(986\) 71.1564 2.26608
\(987\) 0 0
\(988\) −43.8395 −1.39472
\(989\) −12.7945 −0.406841
\(990\) 0 0
\(991\) 54.4626 1.73006 0.865031 0.501718i \(-0.167298\pi\)
0.865031 + 0.501718i \(0.167298\pi\)
\(992\) 32.8596 1.04329
\(993\) 0 0
\(994\) 92.2346 2.92550
\(995\) −35.0121 −1.10996
\(996\) 0 0
\(997\) 13.5868 0.430297 0.215149 0.976581i \(-0.430976\pi\)
0.215149 + 0.976581i \(0.430976\pi\)
\(998\) 17.4742 0.553137
\(999\) 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 2169.2.a.e.1.6 7
3.2 odd 2 241.2.a.a.1.2 7
12.11 even 2 3856.2.a.j.1.3 7
15.14 odd 2 6025.2.a.f.1.6 7
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
241.2.a.a.1.2 7 3.2 odd 2
2169.2.a.e.1.6 7 1.1 even 1 trivial
3856.2.a.j.1.3 7 12.11 even 2
6025.2.a.f.1.6 7 15.14 odd 2