Properties

Label 405.2.a.i.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 \(-0.564819\pi\)
−0.202230 + 0.979338i \(0.564819\pi\)
\(3\) 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 \(-0.632013\pi\)
−0.402943 + 0.915225i \(0.632013\pi\)
\(12\) 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 \(-0.605071\pi\)
−0.324127 + 0.946013i \(0.605071\pi\)
\(18\) 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 \(-0.288365\pi\)
0.616957 + 0.786997i \(0.288365\pi\)
\(24\) 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 \(-0.156810\pi\)
0.881090 + 0.472948i \(0.156810\pi\)
\(30\) 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 \(-0.463352\pi\)
0.114879 + 0.993380i \(0.463352\pi\)
\(42\) 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 \(-0.669386\pi\)
−0.507380 + 0.861722i \(0.669386\pi\)
\(48\) 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 \(-0.474965\pi\)
0.0785693 + 0.996909i \(0.474965\pi\)
\(54\) 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 \(-0.476274\pi\)
0.0744672 + 0.997223i \(0.476274\pi\)
\(60\) 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 \(-0.223252\pi\)
0.763960 + 0.645263i \(0.223252\pi\)
\(72\) 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 \(-0.424453\pi\)
0.235116 + 0.971967i \(0.424453\pi\)
\(84\) 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 \(-0.449173\pi\)
0.159000 + 0.987279i \(0.449173\pi\)
\(90\) 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 \(-0.432966\pi\)
0.209040 + 0.977907i \(0.432966\pi\)
\(102\) 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 \(-0.695411\pi\)
−0.576061 + 0.817407i \(0.695411\pi\)
\(108\) 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 \(-0.895426\pi\)
−0.946518 + 0.322650i \(0.895426\pi\)
\(114\) 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 \(-0.415581\pi\)
0.262111 + 0.965038i \(0.415581\pi\)
\(132\) 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 \(-0.356468\pi\)
0.435793 + 0.900047i \(0.356468\pi\)
\(138\) 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 \(-0.192662\pi\)
0.822351 + 0.568981i \(0.192662\pi\)
\(150\) 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 \(-0.683690\pi\)
−0.545578 + 0.838060i \(0.683690\pi\)
\(168\) 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 \(-0.553080\pi\)
−0.165985 + 0.986128i \(0.553080\pi\)
\(174\) 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 \(-0.308240\pi\)
0.566648 + 0.823960i \(0.308240\pi\)
\(180\) 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 \(-0.467265\pi\)
0.102659 + 0.994717i \(0.467265\pi\)
\(192\) 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 \(-0.566654\pi\)
−0.207873 + 0.978156i \(0.566654\pi\)
\(198\) 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 \(-0.471729\pi\)
0.0887008 + 0.996058i \(0.471729\pi\)
\(228\) 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 \(-0.434688\pi\)
0.203747 + 0.979024i \(0.434688\pi\)
\(234\) 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 \(-0.584621\pi\)
−0.262725 + 0.964871i \(0.584621\pi\)
\(240\) 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 \(-0.494479\pi\)
0.0173443 + 0.999850i \(0.494479\pi\)
\(252\) 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 \(-0.310261\pi\)
0.561405 + 0.827541i \(0.310261\pi\)
\(258\) 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 \(-0.619528\pi\)
−0.366745 + 0.930321i \(0.619528\pi\)
\(264\) 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 \(-0.836583\pi\)
−0.871084 + 0.491134i \(0.836583\pi\)
\(270\) 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 \(-0.563579\pi\)
−0.198415 + 0.980118i \(0.563579\pi\)
\(282\) 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 \(-0.617826\pi\)
−0.361765 + 0.932269i \(0.617826\pi\)
\(294\) 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 \(-0.259059\pi\)
0.686699 + 0.726942i \(0.259059\pi\)
\(312\) 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 \(-0.404994\pi\)
0.294060 + 0.955787i \(0.404994\pi\)
\(318\) 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 \(-0.282013\pi\)
0.632539 + 0.774529i \(0.282013\pi\)
\(348\) 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 \(-0.758458\pi\)
−0.725644 + 0.688070i \(0.758458\pi\)
\(354\) 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 \(-0.409051\pi\)
0.281854 + 0.959457i \(0.409051\pi\)
\(360\) 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 \(-0.580696\pi\)
−0.250808 + 0.968037i \(0.580696\pi\)
\(384\) 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 \(-0.575096\pi\)
−0.233740 + 0.972299i \(0.575096\pi\)
\(390\) 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 \(-0.410894\pi\)
0.276294 + 0.961073i \(0.410894\pi\)
\(402\) 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 \(-0.859854\pi\)
−0.904631 + 0.426195i \(0.859854\pi\)
\(420\) 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 \(-0.459797\pi\)
0.125967 + 0.992034i \(0.459797\pi\)
\(432\) 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 \(-0.580277\pi\)
−0.249534 + 0.968366i \(0.580277\pi\)
\(444\) 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 \(-0.681319\pi\)
−0.539321 + 0.842100i \(0.681319\pi\)
\(450\) 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 \(-0.310308\pi\)
0.561283 + 0.827624i \(0.310308\pi\)
\(462\) 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 \(-0.281695\pi\)
0.633313 + 0.773896i \(0.281695\pi\)
\(468\) 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 \(-0.538126\pi\)
−0.119489 + 0.992835i \(0.538126\pi\)
\(480\) 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 \(-0.608214\pi\)
−0.333452 + 0.942767i \(0.608214\pi\)
\(492\) 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 \(-0.834323\pi\)
−0.867576 + 0.497305i \(0.834323\pi\)
\(504\) 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 \(-0.514260\pi\)
−0.0447837 + 0.998997i \(0.514260\pi\)
\(510\) 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 \(-0.331692\pi\)
0.504460 + 0.863435i \(0.331692\pi\)
\(522\) 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 \(-0.401295\pi\)
0.305144 + 0.952306i \(0.401295\pi\)
\(558\) 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 \(-0.712405\pi\)
−0.618858 + 0.785503i \(0.712405\pi\)
\(564\) 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 \(-0.938046\pi\)
−0.981118 + 0.193409i \(0.938046\pi\)
\(570\) 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 \(-0.621729\pi\)
−0.373169 + 0.927763i \(0.621729\pi\)
\(588\) 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 \(-0.449209\pi\)
0.158889 + 0.987296i \(0.449209\pi\)
\(594\) 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 \(-0.306779\pi\)
0.570423 + 0.821351i \(0.306779\pi\)
\(600\) 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 \(-0.639648\pi\)
−0.424778 + 0.905298i \(0.639648\pi\)
\(618\) 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 \(-0.673737\pi\)
−0.519112 + 0.854706i \(0.673737\pi\)
\(642\) 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 \(-0.651104\pi\)
−0.457079 + 0.889426i \(0.651104\pi\)
\(648\) 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 \(-0.619633\pi\)
−0.367051 + 0.930201i \(0.619633\pi\)
\(654\) 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 \(-0.498264\pi\)
0.00545494 + 0.999985i \(0.498264\pi\)
\(660\) 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 \(-0.677138\pi\)
−0.528213 + 0.849112i \(0.677138\pi\)
\(678\) 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 \(-0.269832\pi\)
0.661708 + 0.749762i \(0.269832\pi\)
\(684\) 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 \(-0.380143\pi\)
0.367708 + 0.929941i \(0.380143\pi\)
\(702\) 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 \(-0.479616\pi\)
0.0639952 + 0.997950i \(0.479616\pi\)
\(720\) 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 \(-0.345051\pi\)
0.467787 + 0.883841i \(0.345051\pi\)
\(744\) 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 \(-0.453849\pi\)
0.144481 + 0.989508i \(0.453849\pi\)
\(762\) 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 \(-0.794393\pi\)
−0.798538 + 0.601944i \(0.794393\pi\)
\(774\) 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 \(-0.298714\pi\)
0.591049 + 0.806636i \(0.298714\pi\)
\(798\) 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 \(-0.671456\pi\)
−0.512973 + 0.858404i \(0.671456\pi\)
\(810\) 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 \(-0.364175\pi\)
0.413876 + 0.910333i \(0.364175\pi\)
\(822\) 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 \(-0.862395\pi\)
−0.908005 + 0.418960i \(0.862395\pi\)
\(828\) 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 \(-0.357729\pi\)
0.432225 + 0.901766i \(0.357729\pi\)
\(840\) 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 \(-0.397128\pi\)
0.317587 + 0.948229i \(0.397128\pi\)
\(858\) 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 \(-0.341516\pi\)
0.477576 + 0.878590i \(0.341516\pi\)
\(864\) 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 \(-0.471551\pi\)
0.0892561 + 0.996009i \(0.471551\pi\)
\(882\) 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 \(-0.744988\pi\)
−0.695885 + 0.718154i \(0.744988\pi\)
\(888\) 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 \(-0.712377\pi\)
−0.618789 + 0.785557i \(0.712377\pi\)
\(912\) 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 \(-0.211458\pi\)
0.787340 + 0.616519i \(0.211458\pi\)
\(930\) 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 \(-0.730082\pi\)
−0.661504 + 0.749941i \(0.730082\pi\)
\(942\) 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 \(-0.663443\pi\)
−0.491204 + 0.871045i \(0.663443\pi\)
\(948\) 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 \(-0.512934\pi\)
−0.0406227 + 0.999175i \(0.512934\pi\)
\(954\) 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 \(-0.618952\pi\)
−0.365061 + 0.930984i \(0.618952\pi\)
\(972\) 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 \(-0.716251\pi\)
−0.628304 + 0.777968i \(0.716251\pi\)
\(978\) 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 \(-0.681147\pi\)
−0.538865 + 0.842392i \(0.681147\pi\)
\(984\) 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.i.1.2 3
3.2 odd 2 405.2.a.j.1.2 3
4.3 odd 2 6480.2.a.bs.1.2 3
5.2 odd 4 2025.2.b.m.649.3 6
5.3 odd 4 2025.2.b.m.649.4 6
5.4 even 2 2025.2.a.o.1.2 3
9.2 odd 6 45.2.e.b.31.2 yes 6
9.4 even 3 135.2.e.b.46.2 6
9.5 odd 6 45.2.e.b.16.2 6
9.7 even 3 135.2.e.b.91.2 6
12.11 even 2 6480.2.a.bv.1.2 3
15.2 even 4 2025.2.b.l.649.4 6
15.8 even 4 2025.2.b.l.649.3 6
15.14 odd 2 2025.2.a.n.1.2 3
36.7 odd 6 2160.2.q.k.1441.2 6
36.11 even 6 720.2.q.i.481.2 6
36.23 even 6 720.2.q.i.241.2 6
36.31 odd 6 2160.2.q.k.721.2 6
45.2 even 12 225.2.k.b.49.4 12
45.4 even 6 675.2.e.b.451.2 6
45.7 odd 12 675.2.k.b.199.3 12
45.13 odd 12 675.2.k.b.424.3 12
45.14 odd 6 225.2.e.b.151.2 6
45.22 odd 12 675.2.k.b.424.4 12
45.23 even 12 225.2.k.b.124.4 12
45.29 odd 6 225.2.e.b.76.2 6
45.32 even 12 225.2.k.b.124.3 12
45.34 even 6 675.2.e.b.226.2 6
45.38 even 12 225.2.k.b.49.3 12
45.43 odd 12 675.2.k.b.199.4 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
45.2.e.b.16.2 6 9.5 odd 6
45.2.e.b.31.2 yes 6 9.2 odd 6
135.2.e.b.46.2 6 9.4 even 3
135.2.e.b.91.2 6 9.7 even 3
225.2.e.b.76.2 6 45.29 odd 6
225.2.e.b.151.2 6 45.14 odd 6
225.2.k.b.49.3 12 45.38 even 12
225.2.k.b.49.4 12 45.2 even 12
225.2.k.b.124.3 12 45.32 even 12
225.2.k.b.124.4 12 45.23 even 12
405.2.a.i.1.2 3 1.1 even 1 trivial
405.2.a.j.1.2 3 3.2 odd 2
675.2.e.b.226.2 6 45.34 even 6
675.2.e.b.451.2 6 45.4 even 6
675.2.k.b.199.3 12 45.7 odd 12
675.2.k.b.199.4 12 45.43 odd 12
675.2.k.b.424.3 12 45.13 odd 12
675.2.k.b.424.4 12 45.22 odd 12
720.2.q.i.241.2 6 36.23 even 6
720.2.q.i.481.2 6 36.11 even 6
2025.2.a.n.1.2 3 15.14 odd 2
2025.2.a.o.1.2 3 5.4 even 2
2025.2.b.l.649.3 6 15.8 even 4
2025.2.b.l.649.4 6 15.2 even 4
2025.2.b.m.649.3 6 5.2 odd 4
2025.2.b.m.649.4 6 5.3 odd 4
2160.2.q.k.721.2 6 36.31 odd 6
2160.2.q.k.1441.2 6 36.7 odd 6
6480.2.a.bs.1.2 3 4.3 odd 2
6480.2.a.bv.1.2 3 12.11 even 2