Properties

Label 405.2.a.j.1.2
Level $405$
Weight $2$
Character 405.1
Self dual yes
Analytic conductor $3.234$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 405 = 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 405.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(3.23394128186\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.564.1
Defining polynomial: \( x^{3} - x^{2} - 5x + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 45)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(0.571993\) of defining polynomial
Character \(\chi\) \(=\) 405.1

$q$-expansion

\(f(q)\) \(=\) \(q+0.571993 q^{2} -1.67282 q^{4} +1.00000 q^{5} +1.42801 q^{7} -2.10083 q^{8} +O(q^{10})\) \(q+0.571993 q^{2} -1.67282 q^{4} +1.00000 q^{5} +1.42801 q^{7} -2.10083 q^{8} +0.571993 q^{10} +2.67282 q^{11} +4.67282 q^{13} +0.816810 q^{14} +2.14399 q^{16} +2.67282 q^{17} +4.67282 q^{19} -1.67282 q^{20} +1.52884 q^{22} -5.91764 q^{23} +1.00000 q^{25} +2.67282 q^{26} -2.38880 q^{28} -9.48963 q^{29} +6.96080 q^{31} +5.42801 q^{32} +1.52884 q^{34} +1.42801 q^{35} -1.81681 q^{37} +2.67282 q^{38} -2.10083 q^{40} -1.47116 q^{41} +0.471163 q^{43} -4.47116 q^{44} -3.38485 q^{46} +6.95684 q^{47} -4.96080 q^{49} +0.571993 q^{50} -7.81681 q^{52} -1.14399 q^{53} +2.67282 q^{55} -3.00000 q^{56} -5.42801 q^{58} -1.14399 q^{59} -2.52884 q^{61} +3.98153 q^{62} -1.18319 q^{64} +4.67282 q^{65} -6.59046 q^{67} -4.47116 q^{68} +0.816810 q^{70} -12.8745 q^{71} -1.71203 q^{73} -1.03920 q^{74} -7.81681 q^{76} +3.81681 q^{77} -0.287973 q^{79} +2.14399 q^{80} -0.841495 q^{82} -4.28402 q^{83} +2.67282 q^{85} +0.269502 q^{86} -5.61515 q^{88} -3.00000 q^{89} +6.67282 q^{91} +9.89917 q^{92} +3.97927 q^{94} +4.67282 q^{95} +7.83528 q^{97} -2.83754 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + q^{2} + 5 q^{4} + 3 q^{5} + 5 q^{7} + 3 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + q^{2} + 5 q^{4} + 3 q^{5} + 5 q^{7} + 3 q^{8} + q^{10} - 2 q^{11} + 4 q^{13} - 9 q^{14} + 5 q^{16} - 2 q^{17} + 4 q^{19} + 5 q^{20} - 4 q^{22} + 3 q^{23} + 3 q^{25} - 2 q^{26} + 5 q^{28} - 7 q^{29} + 8 q^{31} + 17 q^{32} - 4 q^{34} + 5 q^{35} + 6 q^{37} - 2 q^{38} + 3 q^{40} - 13 q^{41} + 10 q^{43} - 22 q^{44} - 3 q^{46} + 13 q^{47} - 2 q^{49} + q^{50} - 12 q^{52} - 2 q^{53} - 2 q^{55} - 9 q^{56} - 17 q^{58} - 2 q^{59} + q^{61} + 42 q^{62} - 15 q^{64} + 4 q^{65} + 11 q^{67} - 22 q^{68} - 9 q^{70} - 10 q^{71} - 8 q^{73} - 16 q^{74} - 12 q^{76} + 2 q^{79} + 5 q^{80} - 29 q^{82} - 15 q^{83} - 2 q^{85} + 28 q^{86} - 24 q^{88} - 9 q^{89} + 10 q^{91} + 39 q^{92} - 31 q^{94} + 4 q^{95} - 18 q^{97} - 40 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.571993 0.404460 0.202230 0.979338i \(-0.435181\pi\)
0.202230 + 0.979338i \(0.435181\pi\)
\(3\) 0 0
\(4\) −1.67282 −0.836412
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 1.42801 0.539736 0.269868 0.962897i \(-0.413020\pi\)
0.269868 + 0.962897i \(0.413020\pi\)
\(8\) −2.10083 −0.742756
\(9\) 0 0
\(10\) 0.571993 0.180880
\(11\) 2.67282 0.805887 0.402943 0.915225i \(-0.367987\pi\)
0.402943 + 0.915225i \(0.367987\pi\)
\(12\) 0 0
\(13\) 4.67282 1.29601 0.648004 0.761637i \(-0.275604\pi\)
0.648004 + 0.761637i \(0.275604\pi\)
\(14\) 0.816810 0.218302
\(15\) 0 0
\(16\) 2.14399 0.535997
\(17\) 2.67282 0.648255 0.324127 0.946013i \(-0.394929\pi\)
0.324127 + 0.946013i \(0.394929\pi\)
\(18\) 0 0
\(19\) 4.67282 1.07202 0.536010 0.844212i \(-0.319931\pi\)
0.536010 + 0.844212i \(0.319931\pi\)
\(20\) −1.67282 −0.374055
\(21\) 0 0
\(22\) 1.52884 0.325949
\(23\) −5.91764 −1.23391 −0.616957 0.786997i \(-0.711635\pi\)
−0.616957 + 0.786997i \(0.711635\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 2.67282 0.524184
\(27\) 0 0
\(28\) −2.38880 −0.451441
\(29\) −9.48963 −1.76218 −0.881090 0.472948i \(-0.843190\pi\)
−0.881090 + 0.472948i \(0.843190\pi\)
\(30\) 0 0
\(31\) 6.96080 1.25020 0.625098 0.780546i \(-0.285059\pi\)
0.625098 + 0.780546i \(0.285059\pi\)
\(32\) 5.42801 0.959545
\(33\) 0 0
\(34\) 1.52884 0.262193
\(35\) 1.42801 0.241377
\(36\) 0 0
\(37\) −1.81681 −0.298682 −0.149341 0.988786i \(-0.547715\pi\)
−0.149341 + 0.988786i \(0.547715\pi\)
\(38\) 2.67282 0.433589
\(39\) 0 0
\(40\) −2.10083 −0.332170
\(41\) −1.47116 −0.229757 −0.114879 0.993380i \(-0.536648\pi\)
−0.114879 + 0.993380i \(0.536648\pi\)
\(42\) 0 0
\(43\) 0.471163 0.0718517 0.0359258 0.999354i \(-0.488562\pi\)
0.0359258 + 0.999354i \(0.488562\pi\)
\(44\) −4.47116 −0.674053
\(45\) 0 0
\(46\) −3.38485 −0.499069
\(47\) 6.95684 1.01476 0.507380 0.861722i \(-0.330614\pi\)
0.507380 + 0.861722i \(0.330614\pi\)
\(48\) 0 0
\(49\) −4.96080 −0.708685
\(50\) 0.571993 0.0808921
\(51\) 0 0
\(52\) −7.81681 −1.08400
\(53\) −1.14399 −0.157139 −0.0785693 0.996909i \(-0.525035\pi\)
−0.0785693 + 0.996909i \(0.525035\pi\)
\(54\) 0 0
\(55\) 2.67282 0.360403
\(56\) −3.00000 −0.400892
\(57\) 0 0
\(58\) −5.42801 −0.712732
\(59\) −1.14399 −0.148934 −0.0744672 0.997223i \(-0.523726\pi\)
−0.0744672 + 0.997223i \(0.523726\pi\)
\(60\) 0 0
\(61\) −2.52884 −0.323784 −0.161892 0.986808i \(-0.551760\pi\)
−0.161892 + 0.986808i \(0.551760\pi\)
\(62\) 3.98153 0.505655
\(63\) 0 0
\(64\) −1.18319 −0.147899
\(65\) 4.67282 0.579592
\(66\) 0 0
\(67\) −6.59046 −0.805153 −0.402577 0.915386i \(-0.631885\pi\)
−0.402577 + 0.915386i \(0.631885\pi\)
\(68\) −4.47116 −0.542208
\(69\) 0 0
\(70\) 0.816810 0.0976275
\(71\) −12.8745 −1.52792 −0.763960 0.645263i \(-0.776748\pi\)
−0.763960 + 0.645263i \(0.776748\pi\)
\(72\) 0 0
\(73\) −1.71203 −0.200378 −0.100189 0.994968i \(-0.531945\pi\)
−0.100189 + 0.994968i \(0.531945\pi\)
\(74\) −1.03920 −0.120805
\(75\) 0 0
\(76\) −7.81681 −0.896650
\(77\) 3.81681 0.434966
\(78\) 0 0
\(79\) −0.287973 −0.0323995 −0.0161998 0.999869i \(-0.505157\pi\)
−0.0161998 + 0.999869i \(0.505157\pi\)
\(80\) 2.14399 0.239705
\(81\) 0 0
\(82\) −0.841495 −0.0929276
\(83\) −4.28402 −0.470232 −0.235116 0.971967i \(-0.575547\pi\)
−0.235116 + 0.971967i \(0.575547\pi\)
\(84\) 0 0
\(85\) 2.67282 0.289908
\(86\) 0.269502 0.0290611
\(87\) 0 0
\(88\) −5.61515 −0.598577
\(89\) −3.00000 −0.317999 −0.159000 0.987279i \(-0.550827\pi\)
−0.159000 + 0.987279i \(0.550827\pi\)
\(90\) 0 0
\(91\) 6.67282 0.699502
\(92\) 9.89917 1.03206
\(93\) 0 0
\(94\) 3.97927 0.410430
\(95\) 4.67282 0.479422
\(96\) 0 0
\(97\) 7.83528 0.795552 0.397776 0.917483i \(-0.369782\pi\)
0.397776 + 0.917483i \(0.369782\pi\)
\(98\) −2.83754 −0.286635
\(99\) 0 0
\(100\) −1.67282 −0.167282
\(101\) −4.20166 −0.418081 −0.209040 0.977907i \(-0.567034\pi\)
−0.209040 + 0.977907i \(0.567034\pi\)
\(102\) 0 0
\(103\) −1.81681 −0.179016 −0.0895078 0.995986i \(-0.528529\pi\)
−0.0895078 + 0.995986i \(0.528529\pi\)
\(104\) −9.81681 −0.962617
\(105\) 0 0
\(106\) −0.654353 −0.0635563
\(107\) 11.9176 1.15212 0.576061 0.817407i \(-0.304589\pi\)
0.576061 + 0.817407i \(0.304589\pi\)
\(108\) 0 0
\(109\) −16.6521 −1.59498 −0.797491 0.603331i \(-0.793840\pi\)
−0.797491 + 0.603331i \(0.793840\pi\)
\(110\) 1.52884 0.145769
\(111\) 0 0
\(112\) 3.06163 0.289297
\(113\) 20.1233 1.89304 0.946518 0.322650i \(-0.104574\pi\)
0.946518 + 0.322650i \(0.104574\pi\)
\(114\) 0 0
\(115\) −5.91764 −0.551823
\(116\) 15.8745 1.47391
\(117\) 0 0
\(118\) −0.654353 −0.0602380
\(119\) 3.81681 0.349886
\(120\) 0 0
\(121\) −3.85601 −0.350547
\(122\) −1.44648 −0.130958
\(123\) 0 0
\(124\) −11.6442 −1.04568
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 2.18714 0.194078 0.0970388 0.995281i \(-0.469063\pi\)
0.0970388 + 0.995281i \(0.469063\pi\)
\(128\) −11.5328 −1.01936
\(129\) 0 0
\(130\) 2.67282 0.234422
\(131\) −6.00000 −0.524222 −0.262111 0.965038i \(-0.584419\pi\)
−0.262111 + 0.965038i \(0.584419\pi\)
\(132\) 0 0
\(133\) 6.67282 0.578607
\(134\) −3.76970 −0.325653
\(135\) 0 0
\(136\) −5.61515 −0.481495
\(137\) −10.2017 −0.871587 −0.435793 0.900047i \(-0.643532\pi\)
−0.435793 + 0.900047i \(0.643532\pi\)
\(138\) 0 0
\(139\) 8.00000 0.678551 0.339276 0.940687i \(-0.389818\pi\)
0.339276 + 0.940687i \(0.389818\pi\)
\(140\) −2.38880 −0.201891
\(141\) 0 0
\(142\) −7.36412 −0.617983
\(143\) 12.4896 1.04444
\(144\) 0 0
\(145\) −9.48963 −0.788071
\(146\) −0.979268 −0.0810448
\(147\) 0 0
\(148\) 3.03920 0.249821
\(149\) −20.0761 −1.64470 −0.822351 0.568981i \(-0.807338\pi\)
−0.822351 + 0.568981i \(0.807338\pi\)
\(150\) 0 0
\(151\) 3.03920 0.247327 0.123663 0.992324i \(-0.460536\pi\)
0.123663 + 0.992324i \(0.460536\pi\)
\(152\) −9.81681 −0.796248
\(153\) 0 0
\(154\) 2.18319 0.175926
\(155\) 6.96080 0.559105
\(156\) 0 0
\(157\) 0.201661 0.0160943 0.00804714 0.999968i \(-0.497438\pi\)
0.00804714 + 0.999968i \(0.497438\pi\)
\(158\) −0.164719 −0.0131043
\(159\) 0 0
\(160\) 5.42801 0.429122
\(161\) −8.45043 −0.665987
\(162\) 0 0
\(163\) 17.8168 1.39552 0.697760 0.716331i \(-0.254180\pi\)
0.697760 + 0.716331i \(0.254180\pi\)
\(164\) 2.46100 0.192172
\(165\) 0 0
\(166\) −2.45043 −0.190190
\(167\) 14.1008 1.09116 0.545578 0.838060i \(-0.316310\pi\)
0.545578 + 0.838060i \(0.316310\pi\)
\(168\) 0 0
\(169\) 8.83528 0.679637
\(170\) 1.52884 0.117256
\(171\) 0 0
\(172\) −0.788172 −0.0600976
\(173\) 4.36638 0.331970 0.165985 0.986128i \(-0.446920\pi\)
0.165985 + 0.986128i \(0.446920\pi\)
\(174\) 0 0
\(175\) 1.42801 0.107947
\(176\) 5.73050 0.431953
\(177\) 0 0
\(178\) −1.71598 −0.128618
\(179\) −15.1625 −1.13330 −0.566648 0.823960i \(-0.691760\pi\)
−0.566648 + 0.823960i \(0.691760\pi\)
\(180\) 0 0
\(181\) 3.20166 0.237978 0.118989 0.992896i \(-0.462035\pi\)
0.118989 + 0.992896i \(0.462035\pi\)
\(182\) 3.81681 0.282921
\(183\) 0 0
\(184\) 12.4320 0.916496
\(185\) −1.81681 −0.133575
\(186\) 0 0
\(187\) 7.14399 0.522420
\(188\) −11.6376 −0.848757
\(189\) 0 0
\(190\) 2.67282 0.193907
\(191\) −2.83754 −0.205317 −0.102659 0.994717i \(-0.532735\pi\)
−0.102659 + 0.994717i \(0.532735\pi\)
\(192\) 0 0
\(193\) −18.7882 −1.35240 −0.676201 0.736717i \(-0.736375\pi\)
−0.676201 + 0.736717i \(0.736375\pi\)
\(194\) 4.48173 0.321769
\(195\) 0 0
\(196\) 8.29854 0.592753
\(197\) 5.83528 0.415747 0.207873 0.978156i \(-0.433346\pi\)
0.207873 + 0.978156i \(0.433346\pi\)
\(198\) 0 0
\(199\) 13.0761 0.926943 0.463472 0.886112i \(-0.346604\pi\)
0.463472 + 0.886112i \(0.346604\pi\)
\(200\) −2.10083 −0.148551
\(201\) 0 0
\(202\) −2.40332 −0.169097
\(203\) −13.5513 −0.951112
\(204\) 0 0
\(205\) −1.47116 −0.102750
\(206\) −1.03920 −0.0724047
\(207\) 0 0
\(208\) 10.0185 0.694656
\(209\) 12.4896 0.863926
\(210\) 0 0
\(211\) 8.38485 0.577237 0.288618 0.957444i \(-0.406804\pi\)
0.288618 + 0.957444i \(0.406804\pi\)
\(212\) 1.91369 0.131433
\(213\) 0 0
\(214\) 6.81681 0.465988
\(215\) 0.471163 0.0321330
\(216\) 0 0
\(217\) 9.94006 0.674776
\(218\) −9.52488 −0.645107
\(219\) 0 0
\(220\) −4.47116 −0.301446
\(221\) 12.4896 0.840144
\(222\) 0 0
\(223\) 9.16641 0.613828 0.306914 0.951737i \(-0.400704\pi\)
0.306914 + 0.951737i \(0.400704\pi\)
\(224\) 7.75123 0.517901
\(225\) 0 0
\(226\) 11.5104 0.765658
\(227\) −2.67282 −0.177402 −0.0887008 0.996058i \(-0.528271\pi\)
−0.0887008 + 0.996058i \(0.528271\pi\)
\(228\) 0 0
\(229\) 2.54731 0.168331 0.0841654 0.996452i \(-0.473178\pi\)
0.0841654 + 0.996452i \(0.473178\pi\)
\(230\) −3.38485 −0.223190
\(231\) 0 0
\(232\) 19.9361 1.30887
\(233\) −6.22013 −0.407494 −0.203747 0.979024i \(-0.565312\pi\)
−0.203747 + 0.979024i \(0.565312\pi\)
\(234\) 0 0
\(235\) 6.95684 0.453814
\(236\) 1.91369 0.124570
\(237\) 0 0
\(238\) 2.18319 0.141515
\(239\) 8.12325 0.525450 0.262725 0.964871i \(-0.415379\pi\)
0.262725 + 0.964871i \(0.415379\pi\)
\(240\) 0 0
\(241\) −26.3641 −1.69826 −0.849131 0.528182i \(-0.822874\pi\)
−0.849131 + 0.528182i \(0.822874\pi\)
\(242\) −2.20561 −0.141782
\(243\) 0 0
\(244\) 4.23030 0.270817
\(245\) −4.96080 −0.316934
\(246\) 0 0
\(247\) 21.8353 1.38935
\(248\) −14.6235 −0.928590
\(249\) 0 0
\(250\) 0.571993 0.0361760
\(251\) −0.549569 −0.0346885 −0.0173443 0.999850i \(-0.505521\pi\)
−0.0173443 + 0.999850i \(0.505521\pi\)
\(252\) 0 0
\(253\) −15.8168 −0.994394
\(254\) 1.25103 0.0784967
\(255\) 0 0
\(256\) −4.23030 −0.264394
\(257\) −18.0000 −1.12281 −0.561405 0.827541i \(-0.689739\pi\)
−0.561405 + 0.827541i \(0.689739\pi\)
\(258\) 0 0
\(259\) −2.59442 −0.161209
\(260\) −7.81681 −0.484778
\(261\) 0 0
\(262\) −3.43196 −0.212027
\(263\) 11.8952 0.733490 0.366745 0.930321i \(-0.380472\pi\)
0.366745 + 0.930321i \(0.380472\pi\)
\(264\) 0 0
\(265\) −1.14399 −0.0702745
\(266\) 3.81681 0.234024
\(267\) 0 0
\(268\) 11.0247 0.673440
\(269\) 28.5737 1.74217 0.871084 0.491134i \(-0.163417\pi\)
0.871084 + 0.491134i \(0.163417\pi\)
\(270\) 0 0
\(271\) −23.3641 −1.41927 −0.709635 0.704570i \(-0.751140\pi\)
−0.709635 + 0.704570i \(0.751140\pi\)
\(272\) 5.73050 0.347462
\(273\) 0 0
\(274\) −5.83528 −0.352522
\(275\) 2.67282 0.161177
\(276\) 0 0
\(277\) −15.0761 −0.905838 −0.452919 0.891552i \(-0.649617\pi\)
−0.452919 + 0.891552i \(0.649617\pi\)
\(278\) 4.57595 0.274447
\(279\) 0 0
\(280\) −3.00000 −0.179284
\(281\) 6.65209 0.396831 0.198415 0.980118i \(-0.436421\pi\)
0.198415 + 0.980118i \(0.436421\pi\)
\(282\) 0 0
\(283\) 26.8969 1.59886 0.799428 0.600762i \(-0.205136\pi\)
0.799428 + 0.600762i \(0.205136\pi\)
\(284\) 21.5367 1.27797
\(285\) 0 0
\(286\) 7.14399 0.422433
\(287\) −2.10083 −0.124008
\(288\) 0 0
\(289\) −9.85601 −0.579765
\(290\) −5.42801 −0.318744
\(291\) 0 0
\(292\) 2.86392 0.167598
\(293\) 12.3849 0.723531 0.361765 0.932269i \(-0.382174\pi\)
0.361765 + 0.932269i \(0.382174\pi\)
\(294\) 0 0
\(295\) −1.14399 −0.0666055
\(296\) 3.81681 0.221848
\(297\) 0 0
\(298\) −11.4834 −0.665217
\(299\) −27.6521 −1.59916
\(300\) 0 0
\(301\) 0.672824 0.0387809
\(302\) 1.73840 0.100034
\(303\) 0 0
\(304\) 10.0185 0.574599
\(305\) −2.52884 −0.144801
\(306\) 0 0
\(307\) −2.49359 −0.142317 −0.0711583 0.997465i \(-0.522670\pi\)
−0.0711583 + 0.997465i \(0.522670\pi\)
\(308\) −6.38485 −0.363811
\(309\) 0 0
\(310\) 3.98153 0.226136
\(311\) −24.2201 −1.37340 −0.686699 0.726942i \(-0.740941\pi\)
−0.686699 + 0.726942i \(0.740941\pi\)
\(312\) 0 0
\(313\) −35.0841 −1.98307 −0.991534 0.129848i \(-0.958551\pi\)
−0.991534 + 0.129848i \(0.958551\pi\)
\(314\) 0.115349 0.00650950
\(315\) 0 0
\(316\) 0.481728 0.0270993
\(317\) −10.4712 −0.588119 −0.294060 0.955787i \(-0.595006\pi\)
−0.294060 + 0.955787i \(0.595006\pi\)
\(318\) 0 0
\(319\) −25.3641 −1.42012
\(320\) −1.18319 −0.0661423
\(321\) 0 0
\(322\) −4.83359 −0.269365
\(323\) 12.4896 0.694942
\(324\) 0 0
\(325\) 4.67282 0.259202
\(326\) 10.1911 0.564433
\(327\) 0 0
\(328\) 3.09066 0.170653
\(329\) 9.93442 0.547702
\(330\) 0 0
\(331\) 16.7776 0.922181 0.461090 0.887353i \(-0.347458\pi\)
0.461090 + 0.887353i \(0.347458\pi\)
\(332\) 7.16641 0.393308
\(333\) 0 0
\(334\) 8.06558 0.441329
\(335\) −6.59046 −0.360076
\(336\) 0 0
\(337\) −27.1809 −1.48064 −0.740320 0.672255i \(-0.765326\pi\)
−0.740320 + 0.672255i \(0.765326\pi\)
\(338\) 5.05372 0.274886
\(339\) 0 0
\(340\) −4.47116 −0.242483
\(341\) 18.6050 1.00752
\(342\) 0 0
\(343\) −17.0801 −0.922239
\(344\) −0.989833 −0.0533682
\(345\) 0 0
\(346\) 2.49754 0.134269
\(347\) −23.5658 −1.26508 −0.632539 0.774529i \(-0.717987\pi\)
−0.632539 + 0.774529i \(0.717987\pi\)
\(348\) 0 0
\(349\) −10.7120 −0.573402 −0.286701 0.958020i \(-0.592559\pi\)
−0.286701 + 0.958020i \(0.592559\pi\)
\(350\) 0.816810 0.0436603
\(351\) 0 0
\(352\) 14.5081 0.773285
\(353\) 27.2672 1.45129 0.725644 0.688070i \(-0.241542\pi\)
0.725644 + 0.688070i \(0.241542\pi\)
\(354\) 0 0
\(355\) −12.8745 −0.683307
\(356\) 5.01847 0.265978
\(357\) 0 0
\(358\) −8.67282 −0.458373
\(359\) −10.6807 −0.563707 −0.281854 0.959457i \(-0.590949\pi\)
−0.281854 + 0.959457i \(0.590949\pi\)
\(360\) 0 0
\(361\) 2.83528 0.149225
\(362\) 1.83133 0.0962525
\(363\) 0 0
\(364\) −11.1625 −0.585072
\(365\) −1.71203 −0.0896116
\(366\) 0 0
\(367\) −8.47116 −0.442191 −0.221096 0.975252i \(-0.570963\pi\)
−0.221096 + 0.975252i \(0.570963\pi\)
\(368\) −12.6873 −0.661373
\(369\) 0 0
\(370\) −1.03920 −0.0540256
\(371\) −1.63362 −0.0848133
\(372\) 0 0
\(373\) 10.1233 0.524162 0.262081 0.965046i \(-0.415591\pi\)
0.262081 + 0.965046i \(0.415591\pi\)
\(374\) 4.08631 0.211298
\(375\) 0 0
\(376\) −14.6151 −0.753719
\(377\) −44.3434 −2.28380
\(378\) 0 0
\(379\) 11.9216 0.612371 0.306186 0.951972i \(-0.400947\pi\)
0.306186 + 0.951972i \(0.400947\pi\)
\(380\) −7.81681 −0.400994
\(381\) 0 0
\(382\) −1.62306 −0.0830427
\(383\) 9.81681 0.501616 0.250808 0.968037i \(-0.419304\pi\)
0.250808 + 0.968037i \(0.419304\pi\)
\(384\) 0 0
\(385\) 3.81681 0.194523
\(386\) −10.7467 −0.546993
\(387\) 0 0
\(388\) −13.1070 −0.665409
\(389\) 9.22013 0.467479 0.233740 0.972299i \(-0.424904\pi\)
0.233740 + 0.972299i \(0.424904\pi\)
\(390\) 0 0
\(391\) −15.8168 −0.799890
\(392\) 10.4218 0.526380
\(393\) 0 0
\(394\) 3.33774 0.168153
\(395\) −0.287973 −0.0144895
\(396\) 0 0
\(397\) −22.9793 −1.15330 −0.576648 0.816993i \(-0.695640\pi\)
−0.576648 + 0.816993i \(0.695640\pi\)
\(398\) 7.47947 0.374912
\(399\) 0 0
\(400\) 2.14399 0.107199
\(401\) −11.0656 −0.552589 −0.276294 0.961073i \(-0.589106\pi\)
−0.276294 + 0.961073i \(0.589106\pi\)
\(402\) 0 0
\(403\) 32.5266 1.62026
\(404\) 7.02864 0.349688
\(405\) 0 0
\(406\) −7.75123 −0.384687
\(407\) −4.85601 −0.240704
\(408\) 0 0
\(409\) −17.6336 −0.871926 −0.435963 0.899964i \(-0.643592\pi\)
−0.435963 + 0.899964i \(0.643592\pi\)
\(410\) −0.841495 −0.0415585
\(411\) 0 0
\(412\) 3.03920 0.149731
\(413\) −1.63362 −0.0803852
\(414\) 0 0
\(415\) −4.28402 −0.210294
\(416\) 25.3641 1.24358
\(417\) 0 0
\(418\) 7.14399 0.349424
\(419\) 37.0347 1.80926 0.904631 0.426195i \(-0.140146\pi\)
0.904631 + 0.426195i \(0.140146\pi\)
\(420\) 0 0
\(421\) 5.05767 0.246496 0.123248 0.992376i \(-0.460669\pi\)
0.123248 + 0.992376i \(0.460669\pi\)
\(422\) 4.79608 0.233469
\(423\) 0 0
\(424\) 2.40332 0.116716
\(425\) 2.67282 0.129651
\(426\) 0 0
\(427\) −3.61120 −0.174758
\(428\) −19.9361 −0.963648
\(429\) 0 0
\(430\) 0.269502 0.0129965
\(431\) −5.23030 −0.251935 −0.125967 0.992034i \(-0.540203\pi\)
−0.125967 + 0.992034i \(0.540203\pi\)
\(432\) 0 0
\(433\) 34.3434 1.65044 0.825219 0.564813i \(-0.191052\pi\)
0.825219 + 0.564813i \(0.191052\pi\)
\(434\) 5.68565 0.272920
\(435\) 0 0
\(436\) 27.8560 1.33406
\(437\) −27.6521 −1.32278
\(438\) 0 0
\(439\) −19.5473 −0.932942 −0.466471 0.884536i \(-0.654475\pi\)
−0.466471 + 0.884536i \(0.654475\pi\)
\(440\) −5.61515 −0.267692
\(441\) 0 0
\(442\) 7.14399 0.339805
\(443\) 10.5042 0.499067 0.249534 0.968366i \(-0.419723\pi\)
0.249534 + 0.968366i \(0.419723\pi\)
\(444\) 0 0
\(445\) −3.00000 −0.142214
\(446\) 5.24313 0.248269
\(447\) 0 0
\(448\) −1.68960 −0.0798262
\(449\) 22.8560 1.07864 0.539321 0.842100i \(-0.318681\pi\)
0.539321 + 0.842100i \(0.318681\pi\)
\(450\) 0 0
\(451\) −3.93216 −0.185158
\(452\) −33.6627 −1.58336
\(453\) 0 0
\(454\) −1.52884 −0.0717519
\(455\) 6.67282 0.312827
\(456\) 0 0
\(457\) −14.0264 −0.656126 −0.328063 0.944656i \(-0.606396\pi\)
−0.328063 + 0.944656i \(0.606396\pi\)
\(458\) 1.45704 0.0680832
\(459\) 0 0
\(460\) 9.89917 0.461551
\(461\) −24.1025 −1.12257 −0.561283 0.827624i \(-0.689692\pi\)
−0.561283 + 0.827624i \(0.689692\pi\)
\(462\) 0 0
\(463\) −33.9401 −1.57733 −0.788664 0.614824i \(-0.789227\pi\)
−0.788664 + 0.614824i \(0.789227\pi\)
\(464\) −20.3456 −0.944523
\(465\) 0 0
\(466\) −3.55787 −0.164815
\(467\) −27.3720 −1.26663 −0.633313 0.773896i \(-0.718305\pi\)
−0.633313 + 0.773896i \(0.718305\pi\)
\(468\) 0 0
\(469\) −9.41123 −0.434570
\(470\) 3.97927 0.183550
\(471\) 0 0
\(472\) 2.40332 0.110622
\(473\) 1.25934 0.0579043
\(474\) 0 0
\(475\) 4.67282 0.214404
\(476\) −6.38485 −0.292649
\(477\) 0 0
\(478\) 4.64645 0.212524
\(479\) 5.23030 0.238978 0.119489 0.992835i \(-0.461874\pi\)
0.119489 + 0.992835i \(0.461874\pi\)
\(480\) 0 0
\(481\) −8.48963 −0.387094
\(482\) −15.0801 −0.686880
\(483\) 0 0
\(484\) 6.45043 0.293201
\(485\) 7.83528 0.355782
\(486\) 0 0
\(487\) −24.0185 −1.08838 −0.544190 0.838962i \(-0.683163\pi\)
−0.544190 + 0.838962i \(0.683163\pi\)
\(488\) 5.31266 0.240493
\(489\) 0 0
\(490\) −2.83754 −0.128187
\(491\) 14.7776 0.666904 0.333452 0.942767i \(-0.391786\pi\)
0.333452 + 0.942767i \(0.391786\pi\)
\(492\) 0 0
\(493\) −25.3641 −1.14234
\(494\) 12.4896 0.561935
\(495\) 0 0
\(496\) 14.9239 0.670101
\(497\) −18.3849 −0.824673
\(498\) 0 0
\(499\) 24.8560 1.11271 0.556354 0.830945i \(-0.312200\pi\)
0.556354 + 0.830945i \(0.312200\pi\)
\(500\) −1.67282 −0.0748110
\(501\) 0 0
\(502\) −0.314350 −0.0140301
\(503\) 38.9154 1.73515 0.867576 0.497305i \(-0.165677\pi\)
0.867576 + 0.497305i \(0.165677\pi\)
\(504\) 0 0
\(505\) −4.20166 −0.186971
\(506\) −9.04711 −0.402193
\(507\) 0 0
\(508\) −3.65870 −0.162329
\(509\) 2.02073 0.0895674 0.0447837 0.998997i \(-0.485740\pi\)
0.0447837 + 0.998997i \(0.485740\pi\)
\(510\) 0 0
\(511\) −2.44479 −0.108151
\(512\) 20.6459 0.912428
\(513\) 0 0
\(514\) −10.2959 −0.454132
\(515\) −1.81681 −0.0800582
\(516\) 0 0
\(517\) 18.5944 0.817782
\(518\) −1.48399 −0.0652027
\(519\) 0 0
\(520\) −9.81681 −0.430496
\(521\) −23.0290 −1.00892 −0.504460 0.863435i \(-0.668308\pi\)
−0.504460 + 0.863435i \(0.668308\pi\)
\(522\) 0 0
\(523\) −41.1170 −1.79792 −0.898961 0.438028i \(-0.855677\pi\)
−0.898961 + 0.438028i \(0.855677\pi\)
\(524\) 10.0369 0.438466
\(525\) 0 0
\(526\) 6.80398 0.296668
\(527\) 18.6050 0.810446
\(528\) 0 0
\(529\) 12.0185 0.522542
\(530\) −0.654353 −0.0284233
\(531\) 0 0
\(532\) −11.1625 −0.483954
\(533\) −6.87448 −0.297767
\(534\) 0 0
\(535\) 11.9176 0.515245
\(536\) 13.8454 0.598032
\(537\) 0 0
\(538\) 16.3440 0.704638
\(539\) −13.2593 −0.571120
\(540\) 0 0
\(541\) 5.20957 0.223977 0.111988 0.993710i \(-0.464278\pi\)
0.111988 + 0.993710i \(0.464278\pi\)
\(542\) −13.3641 −0.574038
\(543\) 0 0
\(544\) 14.5081 0.622030
\(545\) −16.6521 −0.713297
\(546\) 0 0
\(547\) 40.0409 1.71203 0.856013 0.516955i \(-0.172935\pi\)
0.856013 + 0.516955i \(0.172935\pi\)
\(548\) 17.0656 0.729005
\(549\) 0 0
\(550\) 1.52884 0.0651898
\(551\) −44.3434 −1.88909
\(552\) 0 0
\(553\) −0.411227 −0.0174872
\(554\) −8.62345 −0.366375
\(555\) 0 0
\(556\) −13.3826 −0.567548
\(557\) −14.4033 −0.610288 −0.305144 0.952306i \(-0.598705\pi\)
−0.305144 + 0.952306i \(0.598705\pi\)
\(558\) 0 0
\(559\) 2.20166 0.0931203
\(560\) 3.06163 0.129377
\(561\) 0 0
\(562\) 3.80495 0.160502
\(563\) 29.3681 1.23772 0.618858 0.785503i \(-0.287595\pi\)
0.618858 + 0.785503i \(0.287595\pi\)
\(564\) 0 0
\(565\) 20.1233 0.846592
\(566\) 15.3849 0.646674
\(567\) 0 0
\(568\) 27.0471 1.13487
\(569\) 46.8066 1.96224 0.981118 0.193409i \(-0.0619543\pi\)
0.981118 + 0.193409i \(0.0619543\pi\)
\(570\) 0 0
\(571\) 20.0000 0.836974 0.418487 0.908223i \(-0.362561\pi\)
0.418487 + 0.908223i \(0.362561\pi\)
\(572\) −20.8930 −0.873578
\(573\) 0 0
\(574\) −1.20166 −0.0501564
\(575\) −5.91764 −0.246783
\(576\) 0 0
\(577\) 28.2386 1.17559 0.587794 0.809010i \(-0.299996\pi\)
0.587794 + 0.809010i \(0.299996\pi\)
\(578\) −5.63757 −0.234492
\(579\) 0 0
\(580\) 15.8745 0.659152
\(581\) −6.11761 −0.253801
\(582\) 0 0
\(583\) −3.05767 −0.126636
\(584\) 3.59668 0.148832
\(585\) 0 0
\(586\) 7.08405 0.292639
\(587\) 18.0824 0.746339 0.373169 0.927763i \(-0.378271\pi\)
0.373169 + 0.927763i \(0.378271\pi\)
\(588\) 0 0
\(589\) 32.5266 1.34023
\(590\) −0.654353 −0.0269393
\(591\) 0 0
\(592\) −3.89522 −0.160092
\(593\) −7.73840 −0.317778 −0.158889 0.987296i \(-0.550791\pi\)
−0.158889 + 0.987296i \(0.550791\pi\)
\(594\) 0 0
\(595\) 3.81681 0.156474
\(596\) 33.5839 1.37565
\(597\) 0 0
\(598\) −15.8168 −0.646797
\(599\) −27.9216 −1.14085 −0.570423 0.821351i \(-0.693221\pi\)
−0.570423 + 0.821351i \(0.693221\pi\)
\(600\) 0 0
\(601\) 38.4403 1.56801 0.784006 0.620754i \(-0.213173\pi\)
0.784006 + 0.620754i \(0.213173\pi\)
\(602\) 0.384851 0.0156853
\(603\) 0 0
\(604\) −5.08405 −0.206867
\(605\) −3.85601 −0.156769
\(606\) 0 0
\(607\) −0.639834 −0.0259701 −0.0129850 0.999916i \(-0.504133\pi\)
−0.0129850 + 0.999916i \(0.504133\pi\)
\(608\) 25.3641 1.02865
\(609\) 0 0
\(610\) −1.44648 −0.0585662
\(611\) 32.5081 1.31514
\(612\) 0 0
\(613\) 42.7467 1.72652 0.863262 0.504757i \(-0.168418\pi\)
0.863262 + 0.504757i \(0.168418\pi\)
\(614\) −1.42631 −0.0575614
\(615\) 0 0
\(616\) −8.01847 −0.323073
\(617\) 21.1025 0.849556 0.424778 0.905298i \(-0.360352\pi\)
0.424778 + 0.905298i \(0.360352\pi\)
\(618\) 0 0
\(619\) −13.6521 −0.548724 −0.274362 0.961626i \(-0.588467\pi\)
−0.274362 + 0.961626i \(0.588467\pi\)
\(620\) −11.6442 −0.467642
\(621\) 0 0
\(622\) −13.8538 −0.555485
\(623\) −4.28402 −0.171636
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) −20.0678 −0.802072
\(627\) 0 0
\(628\) −0.337343 −0.0134615
\(629\) −4.85601 −0.193622
\(630\) 0 0
\(631\) 33.2593 1.32403 0.662017 0.749489i \(-0.269701\pi\)
0.662017 + 0.749489i \(0.269701\pi\)
\(632\) 0.604983 0.0240649
\(633\) 0 0
\(634\) −5.98943 −0.237871
\(635\) 2.18714 0.0867941
\(636\) 0 0
\(637\) −23.1809 −0.918462
\(638\) −14.5081 −0.574381
\(639\) 0 0
\(640\) −11.5328 −0.455874
\(641\) 26.2857 1.03822 0.519112 0.854706i \(-0.326263\pi\)
0.519112 + 0.854706i \(0.326263\pi\)
\(642\) 0 0
\(643\) 20.5826 0.811697 0.405848 0.913940i \(-0.366976\pi\)
0.405848 + 0.913940i \(0.366976\pi\)
\(644\) 14.1361 0.557040
\(645\) 0 0
\(646\) 7.14399 0.281076
\(647\) 23.2527 0.914159 0.457079 0.889426i \(-0.348896\pi\)
0.457079 + 0.889426i \(0.348896\pi\)
\(648\) 0 0
\(649\) −3.05767 −0.120024
\(650\) 2.67282 0.104837
\(651\) 0 0
\(652\) −29.8044 −1.16723
\(653\) 18.7591 0.734102 0.367051 0.930201i \(-0.380367\pi\)
0.367051 + 0.930201i \(0.380367\pi\)
\(654\) 0 0
\(655\) −6.00000 −0.234439
\(656\) −3.15415 −0.123149
\(657\) 0 0
\(658\) 5.68242 0.221524
\(659\) −0.280067 −0.0109099 −0.00545494 0.999985i \(-0.501736\pi\)
−0.00545494 + 0.999985i \(0.501736\pi\)
\(660\) 0 0
\(661\) −39.7859 −1.54749 −0.773746 0.633496i \(-0.781619\pi\)
−0.773746 + 0.633496i \(0.781619\pi\)
\(662\) 9.59668 0.372985
\(663\) 0 0
\(664\) 9.00000 0.349268
\(665\) 6.67282 0.258761
\(666\) 0 0
\(667\) 56.1562 2.17438
\(668\) −23.5882 −0.912655
\(669\) 0 0
\(670\) −3.76970 −0.145636
\(671\) −6.75914 −0.260934
\(672\) 0 0
\(673\) 33.5288 1.29244 0.646221 0.763150i \(-0.276348\pi\)
0.646221 + 0.763150i \(0.276348\pi\)
\(674\) −15.5473 −0.598860
\(675\) 0 0
\(676\) −14.7799 −0.568456
\(677\) 27.4874 1.05643 0.528213 0.849112i \(-0.322862\pi\)
0.528213 + 0.849112i \(0.322862\pi\)
\(678\) 0 0
\(679\) 11.1888 0.429388
\(680\) −5.61515 −0.215331
\(681\) 0 0
\(682\) 10.6419 0.407500
\(683\) −34.5865 −1.32342 −0.661708 0.749762i \(-0.730168\pi\)
−0.661708 + 0.749762i \(0.730168\pi\)
\(684\) 0 0
\(685\) −10.2017 −0.389785
\(686\) −9.76970 −0.373009
\(687\) 0 0
\(688\) 1.01017 0.0385122
\(689\) −5.34565 −0.203653
\(690\) 0 0
\(691\) 40.7282 1.54938 0.774688 0.632344i \(-0.217907\pi\)
0.774688 + 0.632344i \(0.217907\pi\)
\(692\) −7.30418 −0.277663
\(693\) 0 0
\(694\) −13.4795 −0.511674
\(695\) 8.00000 0.303457
\(696\) 0 0
\(697\) −3.93216 −0.148941
\(698\) −6.12721 −0.231918
\(699\) 0 0
\(700\) −2.38880 −0.0902883
\(701\) −19.4712 −0.735416 −0.367708 0.929941i \(-0.619857\pi\)
−0.367708 + 0.929941i \(0.619857\pi\)
\(702\) 0 0
\(703\) −8.48963 −0.320193
\(704\) −3.16246 −0.119190
\(705\) 0 0
\(706\) 15.5967 0.586989
\(707\) −6.00000 −0.225653
\(708\) 0 0
\(709\) 15.0863 0.566578 0.283289 0.959035i \(-0.408574\pi\)
0.283289 + 0.959035i \(0.408574\pi\)
\(710\) −7.36412 −0.276370
\(711\) 0 0
\(712\) 6.30249 0.236196
\(713\) −41.1915 −1.54263
\(714\) 0 0
\(715\) 12.4896 0.467086
\(716\) 25.3641 0.947902
\(717\) 0 0
\(718\) −6.10931 −0.227997
\(719\) −3.43196 −0.127990 −0.0639952 0.997950i \(-0.520384\pi\)
−0.0639952 + 0.997950i \(0.520384\pi\)
\(720\) 0 0
\(721\) −2.59442 −0.0966211
\(722\) 1.62176 0.0603557
\(723\) 0 0
\(724\) −5.35581 −0.199047
\(725\) −9.48963 −0.352436
\(726\) 0 0
\(727\) −35.7714 −1.32669 −0.663344 0.748315i \(-0.730863\pi\)
−0.663344 + 0.748315i \(0.730863\pi\)
\(728\) −14.0185 −0.519559
\(729\) 0 0
\(730\) −0.979268 −0.0362443
\(731\) 1.25934 0.0465782
\(732\) 0 0
\(733\) −22.0000 −0.812589 −0.406294 0.913742i \(-0.633179\pi\)
−0.406294 + 0.913742i \(0.633179\pi\)
\(734\) −4.84545 −0.178849
\(735\) 0 0
\(736\) −32.1210 −1.18400
\(737\) −17.6151 −0.648862
\(738\) 0 0
\(739\) 6.08631 0.223889 0.111944 0.993714i \(-0.464292\pi\)
0.111944 + 0.993714i \(0.464292\pi\)
\(740\) 3.03920 0.111723
\(741\) 0 0
\(742\) −0.934420 −0.0343036
\(743\) −25.5019 −0.935574 −0.467787 0.883841i \(-0.654949\pi\)
−0.467787 + 0.883841i \(0.654949\pi\)
\(744\) 0 0
\(745\) −20.0761 −0.735533
\(746\) 5.79043 0.212003
\(747\) 0 0
\(748\) −11.9506 −0.436958
\(749\) 17.0185 0.621841
\(750\) 0 0
\(751\) −18.3928 −0.671161 −0.335581 0.942011i \(-0.608932\pi\)
−0.335581 + 0.942011i \(0.608932\pi\)
\(752\) 14.9154 0.543908
\(753\) 0 0
\(754\) −25.3641 −0.923707
\(755\) 3.03920 0.110608
\(756\) 0 0
\(757\) −41.8986 −1.52283 −0.761415 0.648264i \(-0.775495\pi\)
−0.761415 + 0.648264i \(0.775495\pi\)
\(758\) 6.81907 0.247680
\(759\) 0 0
\(760\) −9.81681 −0.356093
\(761\) −7.97136 −0.288962 −0.144481 0.989508i \(-0.546151\pi\)
−0.144481 + 0.989508i \(0.546151\pi\)
\(762\) 0 0
\(763\) −23.7793 −0.860868
\(764\) 4.74671 0.171730
\(765\) 0 0
\(766\) 5.61515 0.202884
\(767\) −5.34565 −0.193020
\(768\) 0 0
\(769\) 6.02864 0.217398 0.108699 0.994075i \(-0.465331\pi\)
0.108699 + 0.994075i \(0.465331\pi\)
\(770\) 2.18319 0.0786767
\(771\) 0 0
\(772\) 31.4293 1.13117
\(773\) 44.4033 1.59708 0.798538 0.601944i \(-0.205607\pi\)
0.798538 + 0.601944i \(0.205607\pi\)
\(774\) 0 0
\(775\) 6.96080 0.250039
\(776\) −16.4606 −0.590901
\(777\) 0 0
\(778\) 5.27385 0.189077
\(779\) −6.87448 −0.246304
\(780\) 0 0
\(781\) −34.4112 −1.23133
\(782\) −9.04711 −0.323524
\(783\) 0 0
\(784\) −10.6359 −0.379853
\(785\) 0.201661 0.00719758
\(786\) 0 0
\(787\) 16.8375 0.600194 0.300097 0.953909i \(-0.402981\pi\)
0.300097 + 0.953909i \(0.402981\pi\)
\(788\) −9.76140 −0.347735
\(789\) 0 0
\(790\) −0.164719 −0.00586043
\(791\) 28.7361 1.02174
\(792\) 0 0
\(793\) −11.8168 −0.419627
\(794\) −13.1440 −0.466463
\(795\) 0 0
\(796\) −21.8741 −0.775306
\(797\) −33.3720 −1.18210 −0.591049 0.806636i \(-0.701286\pi\)
−0.591049 + 0.806636i \(0.701286\pi\)
\(798\) 0 0
\(799\) 18.5944 0.657823
\(800\) 5.42801 0.191909
\(801\) 0 0
\(802\) −6.32944 −0.223500
\(803\) −4.57595 −0.161482
\(804\) 0 0
\(805\) −8.45043 −0.297839
\(806\) 18.6050 0.655333
\(807\) 0 0
\(808\) 8.82698 0.310532
\(809\) 29.1809 1.02595 0.512973 0.858404i \(-0.328544\pi\)
0.512973 + 0.858404i \(0.328544\pi\)
\(810\) 0 0
\(811\) −15.5552 −0.546217 −0.273109 0.961983i \(-0.588052\pi\)
−0.273109 + 0.961983i \(0.588052\pi\)
\(812\) 22.6689 0.795521
\(813\) 0 0
\(814\) −2.77761 −0.0973551
\(815\) 17.8168 0.624096
\(816\) 0 0
\(817\) 2.20166 0.0770264
\(818\) −10.0863 −0.352660
\(819\) 0 0
\(820\) 2.46100 0.0859417
\(821\) −23.7177 −0.827752 −0.413876 0.910333i \(-0.635825\pi\)
−0.413876 + 0.910333i \(0.635825\pi\)
\(822\) 0 0
\(823\) 19.3681 0.675129 0.337564 0.941302i \(-0.390397\pi\)
0.337564 + 0.941302i \(0.390397\pi\)
\(824\) 3.81681 0.132965
\(825\) 0 0
\(826\) −0.934420 −0.0325126
\(827\) 52.2241 1.81601 0.908005 0.418960i \(-0.137605\pi\)
0.908005 + 0.418960i \(0.137605\pi\)
\(828\) 0 0
\(829\) −26.6442 −0.925391 −0.462695 0.886517i \(-0.653118\pi\)
−0.462695 + 0.886517i \(0.653118\pi\)
\(830\) −2.45043 −0.0850557
\(831\) 0 0
\(832\) −5.52884 −0.191678
\(833\) −13.2593 −0.459409
\(834\) 0 0
\(835\) 14.1008 0.487979
\(836\) −20.8930 −0.722598
\(837\) 0 0
\(838\) 21.1836 0.731775
\(839\) −25.0392 −0.864449 −0.432225 0.901766i \(-0.642271\pi\)
−0.432225 + 0.901766i \(0.642271\pi\)
\(840\) 0 0
\(841\) 61.0532 2.10528
\(842\) 2.89296 0.0996978
\(843\) 0 0
\(844\) −14.0264 −0.482808
\(845\) 8.83528 0.303943
\(846\) 0 0
\(847\) −5.50641 −0.189203
\(848\) −2.45269 −0.0842258
\(849\) 0 0
\(850\) 1.52884 0.0524387
\(851\) 10.7512 0.368547
\(852\) 0 0
\(853\) −10.8745 −0.372335 −0.186168 0.982518i \(-0.559607\pi\)
−0.186168 + 0.982518i \(0.559607\pi\)
\(854\) −2.06558 −0.0706827
\(855\) 0 0
\(856\) −25.0369 −0.855745
\(857\) −18.5944 −0.635173 −0.317587 0.948229i \(-0.602872\pi\)
−0.317587 + 0.948229i \(0.602872\pi\)
\(858\) 0 0
\(859\) −4.66492 −0.159165 −0.0795825 0.996828i \(-0.525359\pi\)
−0.0795825 + 0.996828i \(0.525359\pi\)
\(860\) −0.788172 −0.0268765
\(861\) 0 0
\(862\) −2.99170 −0.101898
\(863\) −28.0594 −0.955152 −0.477576 0.878590i \(-0.658484\pi\)
−0.477576 + 0.878590i \(0.658484\pi\)
\(864\) 0 0
\(865\) 4.36638 0.148461
\(866\) 19.6442 0.667537
\(867\) 0 0
\(868\) −16.6280 −0.564390
\(869\) −0.769701 −0.0261103
\(870\) 0 0
\(871\) −30.7961 −1.04349
\(872\) 34.9832 1.18468
\(873\) 0 0
\(874\) −15.8168 −0.535012
\(875\) 1.42801 0.0482754
\(876\) 0 0
\(877\) 34.6683 1.17067 0.585333 0.810793i \(-0.300964\pi\)
0.585333 + 0.810793i \(0.300964\pi\)
\(878\) −11.1809 −0.377338
\(879\) 0 0
\(880\) 5.73050 0.193175
\(881\) −5.29854 −0.178512 −0.0892561 0.996009i \(-0.528449\pi\)
−0.0892561 + 0.996009i \(0.528449\pi\)
\(882\) 0 0
\(883\) −10.3025 −0.346706 −0.173353 0.984860i \(-0.555460\pi\)
−0.173353 + 0.984860i \(0.555460\pi\)
\(884\) −20.8930 −0.702706
\(885\) 0 0
\(886\) 6.00830 0.201853
\(887\) 41.4504 1.39177 0.695885 0.718154i \(-0.255012\pi\)
0.695885 + 0.718154i \(0.255012\pi\)
\(888\) 0 0
\(889\) 3.12325 0.104751
\(890\) −1.71598 −0.0575198
\(891\) 0 0
\(892\) −15.3338 −0.513413
\(893\) 32.5081 1.08784
\(894\) 0 0
\(895\) −15.1625 −0.506825
\(896\) −16.4689 −0.550187
\(897\) 0 0
\(898\) 13.0735 0.436268
\(899\) −66.0554 −2.20307
\(900\) 0 0
\(901\) −3.05767 −0.101866
\(902\) −2.24917 −0.0748891
\(903\) 0 0
\(904\) −42.2755 −1.40606
\(905\) 3.20166 0.106427
\(906\) 0 0
\(907\) −16.7921 −0.557573 −0.278787 0.960353i \(-0.589932\pi\)
−0.278787 + 0.960353i \(0.589932\pi\)
\(908\) 4.47116 0.148381
\(909\) 0 0
\(910\) 3.81681 0.126526
\(911\) 37.3536 1.23758 0.618789 0.785557i \(-0.287623\pi\)
0.618789 + 0.785557i \(0.287623\pi\)
\(912\) 0 0
\(913\) −11.4504 −0.378954
\(914\) −8.02299 −0.265377
\(915\) 0 0
\(916\) −4.26120 −0.140794
\(917\) −8.56804 −0.282942
\(918\) 0 0
\(919\) 37.1316 1.22486 0.612429 0.790526i \(-0.290193\pi\)
0.612429 + 0.790526i \(0.290193\pi\)
\(920\) 12.4320 0.409870
\(921\) 0 0
\(922\) −13.7865 −0.454034
\(923\) −60.1602 −1.98020
\(924\) 0 0
\(925\) −1.81681 −0.0597364
\(926\) −19.4135 −0.637967
\(927\) 0 0
\(928\) −51.5098 −1.69089
\(929\) −47.9955 −1.57468 −0.787340 0.616519i \(-0.788542\pi\)
−0.787340 + 0.616519i \(0.788542\pi\)
\(930\) 0 0
\(931\) −23.1809 −0.759724
\(932\) 10.4052 0.340833
\(933\) 0 0
\(934\) −15.6566 −0.512300
\(935\) 7.14399 0.233633
\(936\) 0 0
\(937\) 22.0079 0.718967 0.359483 0.933151i \(-0.382953\pi\)
0.359483 + 0.933151i \(0.382953\pi\)
\(938\) −5.38316 −0.175766
\(939\) 0 0
\(940\) −11.6376 −0.379576
\(941\) 40.5843 1.32301 0.661504 0.749941i \(-0.269918\pi\)
0.661504 + 0.749941i \(0.269918\pi\)
\(942\) 0 0
\(943\) 8.70581 0.283500
\(944\) −2.45269 −0.0798283
\(945\) 0 0
\(946\) 0.720331 0.0234200
\(947\) 30.2320 0.982408 0.491204 0.871045i \(-0.336557\pi\)
0.491204 + 0.871045i \(0.336557\pi\)
\(948\) 0 0
\(949\) −8.00000 −0.259691
\(950\) 2.67282 0.0867179
\(951\) 0 0
\(952\) −8.01847 −0.259880
\(953\) 2.50811 0.0812455 0.0406227 0.999175i \(-0.487066\pi\)
0.0406227 + 0.999175i \(0.487066\pi\)
\(954\) 0 0
\(955\) −2.83754 −0.0918207
\(956\) −13.5888 −0.439492
\(957\) 0 0
\(958\) 2.99170 0.0966573
\(959\) −14.5680 −0.470427
\(960\) 0 0
\(961\) 17.4527 0.562990
\(962\) −4.85601 −0.156564
\(963\) 0 0
\(964\) 44.1025 1.42045
\(965\) −18.7882 −0.604813
\(966\) 0 0
\(967\) −21.7529 −0.699527 −0.349763 0.936838i \(-0.613738\pi\)
−0.349763 + 0.936838i \(0.613738\pi\)
\(968\) 8.10083 0.260371
\(969\) 0 0
\(970\) 4.48173 0.143900
\(971\) 22.7512 0.730122 0.365061 0.930984i \(-0.381048\pi\)
0.365061 + 0.930984i \(0.381048\pi\)
\(972\) 0 0
\(973\) 11.4241 0.366238
\(974\) −13.7384 −0.440207
\(975\) 0 0
\(976\) −5.42179 −0.173547
\(977\) 39.2778 1.25661 0.628304 0.777968i \(-0.283749\pi\)
0.628304 + 0.777968i \(0.283749\pi\)
\(978\) 0 0
\(979\) −8.01847 −0.256271
\(980\) 8.29854 0.265087
\(981\) 0 0
\(982\) 8.45269 0.269736
\(983\) 33.7899 1.07773 0.538865 0.842392i \(-0.318853\pi\)
0.538865 + 0.842392i \(0.318853\pi\)
\(984\) 0 0
\(985\) 5.83528 0.185928
\(986\) −14.5081 −0.462032
\(987\) 0 0
\(988\) −36.5266 −1.16207
\(989\) −2.78817 −0.0886587
\(990\) 0 0
\(991\) −23.7983 −0.755979 −0.377990 0.925810i \(-0.623384\pi\)
−0.377990 + 0.925810i \(0.623384\pi\)
\(992\) 37.7833 1.19962
\(993\) 0 0
\(994\) −10.5160 −0.333548
\(995\) 13.0761 0.414542
\(996\) 0 0
\(997\) 51.6785 1.63667 0.818337 0.574739i \(-0.194896\pi\)
0.818337 + 0.574739i \(0.194896\pi\)
\(998\) 14.2175 0.450046
\(999\) 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 405.2.a.j.1.2 3
3.2 odd 2 405.2.a.i.1.2 3
4.3 odd 2 6480.2.a.bv.1.2 3
5.2 odd 4 2025.2.b.l.649.4 6
5.3 odd 4 2025.2.b.l.649.3 6
5.4 even 2 2025.2.a.n.1.2 3
9.2 odd 6 135.2.e.b.91.2 6
9.4 even 3 45.2.e.b.16.2 6
9.5 odd 6 135.2.e.b.46.2 6
9.7 even 3 45.2.e.b.31.2 yes 6
12.11 even 2 6480.2.a.bs.1.2 3
15.2 even 4 2025.2.b.m.649.3 6
15.8 even 4 2025.2.b.m.649.4 6
15.14 odd 2 2025.2.a.o.1.2 3
36.7 odd 6 720.2.q.i.481.2 6
36.11 even 6 2160.2.q.k.1441.2 6
36.23 even 6 2160.2.q.k.721.2 6
36.31 odd 6 720.2.q.i.241.2 6
45.2 even 12 675.2.k.b.199.3 12
45.4 even 6 225.2.e.b.151.2 6
45.7 odd 12 225.2.k.b.49.4 12
45.13 odd 12 225.2.k.b.124.4 12
45.14 odd 6 675.2.e.b.451.2 6
45.22 odd 12 225.2.k.b.124.3 12
45.23 even 12 675.2.k.b.424.3 12
45.29 odd 6 675.2.e.b.226.2 6
45.32 even 12 675.2.k.b.424.4 12
45.34 even 6 225.2.e.b.76.2 6
45.38 even 12 675.2.k.b.199.4 12
45.43 odd 12 225.2.k.b.49.3 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
45.2.e.b.16.2 6 9.4 even 3
45.2.e.b.31.2 yes 6 9.7 even 3
135.2.e.b.46.2 6 9.5 odd 6
135.2.e.b.91.2 6 9.2 odd 6
225.2.e.b.76.2 6 45.34 even 6
225.2.e.b.151.2 6 45.4 even 6
225.2.k.b.49.3 12 45.43 odd 12
225.2.k.b.49.4 12 45.7 odd 12
225.2.k.b.124.3 12 45.22 odd 12
225.2.k.b.124.4 12 45.13 odd 12
405.2.a.i.1.2 3 3.2 odd 2
405.2.a.j.1.2 3 1.1 even 1 trivial
675.2.e.b.226.2 6 45.29 odd 6
675.2.e.b.451.2 6 45.14 odd 6
675.2.k.b.199.3 12 45.2 even 12
675.2.k.b.199.4 12 45.38 even 12
675.2.k.b.424.3 12 45.23 even 12
675.2.k.b.424.4 12 45.32 even 12
720.2.q.i.241.2 6 36.31 odd 6
720.2.q.i.481.2 6 36.7 odd 6
2025.2.a.n.1.2 3 5.4 even 2
2025.2.a.o.1.2 3 15.14 odd 2
2025.2.b.l.649.3 6 5.3 odd 4
2025.2.b.l.649.4 6 5.2 odd 4
2025.2.b.m.649.3 6 15.2 even 4
2025.2.b.m.649.4 6 15.8 even 4
2160.2.q.k.721.2 6 36.23 even 6
2160.2.q.k.1441.2 6 36.11 even 6
6480.2.a.bs.1.2 3 12.11 even 2
6480.2.a.bv.1.2 3 4.3 odd 2