Properties

Label 1521.2.b.m.1351.3
Level $1521$
Weight $2$
Character 1521.1351
Analytic conductor $12.145$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1521 = 3^{2} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1521.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(12.1452461474\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.153664.1
Defining polynomial: \(x^{6} + 5 x^{4} + 6 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 507)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1351.3
Root \(-0.445042i\) of defining polynomial
Character \(\chi\) \(=\) 1521.1351
Dual form 1521.2.b.m.1351.4

$q$-expansion

\(f(q)\) \(=\) \(q-0.445042i q^{2} +1.80194 q^{4} +0.246980i q^{5} -1.75302i q^{7} -1.69202i q^{8} +O(q^{10})\) \(q-0.445042i q^{2} +1.80194 q^{4} +0.246980i q^{5} -1.75302i q^{7} -1.69202i q^{8} +0.109916 q^{10} +5.65279i q^{11} -0.780167 q^{14} +2.85086 q^{16} -3.80194 q^{17} +5.58211i q^{19} +0.445042i q^{20} +2.51573 q^{22} +8.34481 q^{23} +4.93900 q^{25} -3.15883i q^{28} +5.93900 q^{29} +5.26875i q^{31} -4.65279i q^{32} +1.69202i q^{34} +0.432960 q^{35} -3.19806i q^{37} +2.48427 q^{38} +0.417895 q^{40} -0.445042i q^{41} -1.71379 q^{43} +10.1860i q^{44} -3.71379i q^{46} -6.73556i q^{47} +3.92692 q^{49} -2.19806i q^{50} +1.06100 q^{53} -1.39612 q^{55} -2.96615 q^{56} -2.64310i q^{58} -13.7017i q^{59} -8.51573 q^{61} +2.34481 q^{62} +3.63102 q^{64} +5.96077i q^{67} -6.85086 q^{68} -0.192685i q^{70} +5.71917i q^{71} -7.35690i q^{73} -1.42327 q^{74} +10.0586i q^{76} +9.90946 q^{77} +4.45473 q^{79} +0.704103i q^{80} -0.198062 q^{82} +10.1860i q^{83} -0.939001i q^{85} +0.762709i q^{86} +9.56465 q^{88} +0.137063i q^{89} +15.0368 q^{92} -2.99761 q^{94} -1.37867 q^{95} -13.6896i q^{97} -1.74764i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q + 2q^{4} + O(q^{10}) \) \( 6q + 2q^{4} + 2q^{10} - 2q^{14} - 10q^{16} - 14q^{17} - 10q^{22} + 4q^{23} + 10q^{25} + 16q^{29} - 36q^{35} + 40q^{38} + 14q^{40} + 6q^{43} - 34q^{49} + 26q^{53} - 26q^{55} + 14q^{56} - 26q^{61} - 32q^{62} - 8q^{64} - 14q^{68} - 14q^{74} - 30q^{77} - 18q^{79} - 10q^{82} + 14q^{88} + 34q^{92} + 64q^{94} + 6q^{95} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1521\mathbb{Z}\right)^\times\).

\(n\) \(677\) \(847\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 0.445042i − 0.314692i −0.987544 0.157346i \(-0.949706\pi\)
0.987544 0.157346i \(-0.0502938\pi\)
\(3\) 0 0
\(4\) 1.80194 0.900969
\(5\) 0.246980i 0.110453i 0.998474 + 0.0552263i \(0.0175880\pi\)
−0.998474 + 0.0552263i \(0.982412\pi\)
\(6\) 0 0
\(7\) − 1.75302i − 0.662579i −0.943529 0.331290i \(-0.892516\pi\)
0.943529 0.331290i \(-0.107484\pi\)
\(8\) − 1.69202i − 0.598220i
\(9\) 0 0
\(10\) 0.109916 0.0347586
\(11\) 5.65279i 1.70438i 0.523232 + 0.852191i \(0.324726\pi\)
−0.523232 + 0.852191i \(0.675274\pi\)
\(12\) 0 0
\(13\) 0 0
\(14\) −0.780167 −0.208509
\(15\) 0 0
\(16\) 2.85086 0.712714
\(17\) −3.80194 −0.922105 −0.461053 0.887373i \(-0.652528\pi\)
−0.461053 + 0.887373i \(0.652528\pi\)
\(18\) 0 0
\(19\) 5.58211i 1.28062i 0.768115 + 0.640311i \(0.221195\pi\)
−0.768115 + 0.640311i \(0.778805\pi\)
\(20\) 0.445042i 0.0995144i
\(21\) 0 0
\(22\) 2.51573 0.536355
\(23\) 8.34481 1.74001 0.870007 0.493039i \(-0.164114\pi\)
0.870007 + 0.493039i \(0.164114\pi\)
\(24\) 0 0
\(25\) 4.93900 0.987800
\(26\) 0 0
\(27\) 0 0
\(28\) − 3.15883i − 0.596963i
\(29\) 5.93900 1.10284 0.551422 0.834226i \(-0.314085\pi\)
0.551422 + 0.834226i \(0.314085\pi\)
\(30\) 0 0
\(31\) 5.26875i 0.946295i 0.880983 + 0.473148i \(0.156882\pi\)
−0.880983 + 0.473148i \(0.843118\pi\)
\(32\) − 4.65279i − 0.822505i
\(33\) 0 0
\(34\) 1.69202i 0.290179i
\(35\) 0.432960 0.0731836
\(36\) 0 0
\(37\) − 3.19806i − 0.525758i −0.964829 0.262879i \(-0.915328\pi\)
0.964829 0.262879i \(-0.0846720\pi\)
\(38\) 2.48427 0.403002
\(39\) 0 0
\(40\) 0.417895 0.0660750
\(41\) − 0.445042i − 0.0695039i −0.999396 0.0347519i \(-0.988936\pi\)
0.999396 0.0347519i \(-0.0110641\pi\)
\(42\) 0 0
\(43\) −1.71379 −0.261351 −0.130675 0.991425i \(-0.541715\pi\)
−0.130675 + 0.991425i \(0.541715\pi\)
\(44\) 10.1860i 1.53559i
\(45\) 0 0
\(46\) − 3.71379i − 0.547569i
\(47\) − 6.73556i − 0.982483i −0.871024 0.491241i \(-0.836543\pi\)
0.871024 0.491241i \(-0.163457\pi\)
\(48\) 0 0
\(49\) 3.92692 0.560988
\(50\) − 2.19806i − 0.310853i
\(51\) 0 0
\(52\) 0 0
\(53\) 1.06100 0.145739 0.0728697 0.997341i \(-0.476784\pi\)
0.0728697 + 0.997341i \(0.476784\pi\)
\(54\) 0 0
\(55\) −1.39612 −0.188253
\(56\) −2.96615 −0.396368
\(57\) 0 0
\(58\) − 2.64310i − 0.347057i
\(59\) − 13.7017i − 1.78381i −0.452222 0.891905i \(-0.649369\pi\)
0.452222 0.891905i \(-0.350631\pi\)
\(60\) 0 0
\(61\) −8.51573 −1.09033 −0.545164 0.838330i \(-0.683533\pi\)
−0.545164 + 0.838330i \(0.683533\pi\)
\(62\) 2.34481 0.297792
\(63\) 0 0
\(64\) 3.63102 0.453878
\(65\) 0 0
\(66\) 0 0
\(67\) 5.96077i 0.728224i 0.931355 + 0.364112i \(0.118627\pi\)
−0.931355 + 0.364112i \(0.881373\pi\)
\(68\) −6.85086 −0.830788
\(69\) 0 0
\(70\) − 0.192685i − 0.0230303i
\(71\) 5.71917i 0.678740i 0.940653 + 0.339370i \(0.110214\pi\)
−0.940653 + 0.339370i \(0.889786\pi\)
\(72\) 0 0
\(73\) − 7.35690i − 0.861060i −0.902576 0.430530i \(-0.858327\pi\)
0.902576 0.430530i \(-0.141673\pi\)
\(74\) −1.42327 −0.165452
\(75\) 0 0
\(76\) 10.0586i 1.15380i
\(77\) 9.90946 1.12929
\(78\) 0 0
\(79\) 4.45473 0.501196 0.250598 0.968091i \(-0.419373\pi\)
0.250598 + 0.968091i \(0.419373\pi\)
\(80\) 0.704103i 0.0787211i
\(81\) 0 0
\(82\) −0.198062 −0.0218723
\(83\) 10.1860i 1.11806i 0.829149 + 0.559028i \(0.188826\pi\)
−0.829149 + 0.559028i \(0.811174\pi\)
\(84\) 0 0
\(85\) − 0.939001i − 0.101849i
\(86\) 0.762709i 0.0822450i
\(87\) 0 0
\(88\) 9.56465 1.01959
\(89\) 0.137063i 0.0145287i 0.999974 + 0.00726434i \(0.00231233\pi\)
−0.999974 + 0.00726434i \(0.997688\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 15.0368 1.56770
\(93\) 0 0
\(94\) −2.99761 −0.309180
\(95\) −1.37867 −0.141448
\(96\) 0 0
\(97\) − 13.6896i − 1.38997i −0.719024 0.694986i \(-0.755411\pi\)
0.719024 0.694986i \(-0.244589\pi\)
\(98\) − 1.74764i − 0.176539i
\(99\) 0 0
\(100\) 8.89977 0.889977
\(101\) −5.41119 −0.538434 −0.269217 0.963080i \(-0.586765\pi\)
−0.269217 + 0.963080i \(0.586765\pi\)
\(102\) 0 0
\(103\) −13.7560 −1.35542 −0.677710 0.735330i \(-0.737027\pi\)
−0.677710 + 0.735330i \(0.737027\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) − 0.472189i − 0.0458630i
\(107\) 12.8170 1.23907 0.619533 0.784970i \(-0.287322\pi\)
0.619533 + 0.784970i \(0.287322\pi\)
\(108\) 0 0
\(109\) 12.1468i 1.16345i 0.813386 + 0.581724i \(0.197622\pi\)
−0.813386 + 0.581724i \(0.802378\pi\)
\(110\) 0.621334i 0.0592419i
\(111\) 0 0
\(112\) − 4.99761i − 0.472229i
\(113\) 1.63773 0.154064 0.0770322 0.997029i \(-0.475456\pi\)
0.0770322 + 0.997029i \(0.475456\pi\)
\(114\) 0 0
\(115\) 2.06100i 0.192189i
\(116\) 10.7017 0.993629
\(117\) 0 0
\(118\) −6.09783 −0.561351
\(119\) 6.66487i 0.610968i
\(120\) 0 0
\(121\) −20.9541 −1.90492
\(122\) 3.78986i 0.343117i
\(123\) 0 0
\(124\) 9.49396i 0.852583i
\(125\) 2.45473i 0.219558i
\(126\) 0 0
\(127\) −10.7995 −0.958305 −0.479152 0.877732i \(-0.659056\pi\)
−0.479152 + 0.877732i \(0.659056\pi\)
\(128\) − 10.9215i − 0.965337i
\(129\) 0 0
\(130\) 0 0
\(131\) −0.907542 −0.0792923 −0.0396462 0.999214i \(-0.512623\pi\)
−0.0396462 + 0.999214i \(0.512623\pi\)
\(132\) 0 0
\(133\) 9.78554 0.848514
\(134\) 2.65279 0.229166
\(135\) 0 0
\(136\) 6.43296i 0.551622i
\(137\) − 9.54825i − 0.815762i −0.913035 0.407881i \(-0.866268\pi\)
0.913035 0.407881i \(-0.133732\pi\)
\(138\) 0 0
\(139\) −4.09246 −0.347118 −0.173559 0.984823i \(-0.555527\pi\)
−0.173559 + 0.984823i \(0.555527\pi\)
\(140\) 0.780167 0.0659362
\(141\) 0 0
\(142\) 2.54527 0.213594
\(143\) 0 0
\(144\) 0 0
\(145\) 1.46681i 0.121812i
\(146\) −3.27413 −0.270969
\(147\) 0 0
\(148\) − 5.76271i − 0.473692i
\(149\) 15.3884i 1.26066i 0.776326 + 0.630332i \(0.217081\pi\)
−0.776326 + 0.630332i \(0.782919\pi\)
\(150\) 0 0
\(151\) 3.67456i 0.299032i 0.988759 + 0.149516i \(0.0477715\pi\)
−0.988759 + 0.149516i \(0.952228\pi\)
\(152\) 9.44504 0.766094
\(153\) 0 0
\(154\) − 4.41013i − 0.355378i
\(155\) −1.30127 −0.104521
\(156\) 0 0
\(157\) −4.87800 −0.389307 −0.194653 0.980872i \(-0.562358\pi\)
−0.194653 + 0.980872i \(0.562358\pi\)
\(158\) − 1.98254i − 0.157723i
\(159\) 0 0
\(160\) 1.14914 0.0908479
\(161\) − 14.6286i − 1.15290i
\(162\) 0 0
\(163\) − 8.63102i − 0.676034i −0.941140 0.338017i \(-0.890244\pi\)
0.941140 0.338017i \(-0.109756\pi\)
\(164\) − 0.801938i − 0.0626208i
\(165\) 0 0
\(166\) 4.53319 0.351844
\(167\) 9.46980i 0.732795i 0.930458 + 0.366397i \(0.119409\pi\)
−0.930458 + 0.366397i \(0.880591\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) −0.417895 −0.0320511
\(171\) 0 0
\(172\) −3.08815 −0.235469
\(173\) 4.77479 0.363021 0.181510 0.983389i \(-0.441901\pi\)
0.181510 + 0.983389i \(0.441901\pi\)
\(174\) 0 0
\(175\) − 8.65817i − 0.654496i
\(176\) 16.1153i 1.21474i
\(177\) 0 0
\(178\) 0.0609989 0.00457206
\(179\) 3.43535 0.256770 0.128385 0.991724i \(-0.459021\pi\)
0.128385 + 0.991724i \(0.459021\pi\)
\(180\) 0 0
\(181\) 13.4862 1.00242 0.501210 0.865326i \(-0.332888\pi\)
0.501210 + 0.865326i \(0.332888\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) − 14.1196i − 1.04091i
\(185\) 0.789856 0.0580714
\(186\) 0 0
\(187\) − 21.4916i − 1.57162i
\(188\) − 12.1371i − 0.885186i
\(189\) 0 0
\(190\) 0.613564i 0.0445126i
\(191\) 1.30127 0.0941569 0.0470784 0.998891i \(-0.485009\pi\)
0.0470784 + 0.998891i \(0.485009\pi\)
\(192\) 0 0
\(193\) 9.19567i 0.661919i 0.943645 + 0.330959i \(0.107372\pi\)
−0.943645 + 0.330959i \(0.892628\pi\)
\(194\) −6.09246 −0.437413
\(195\) 0 0
\(196\) 7.07606 0.505433
\(197\) − 4.11231i − 0.292990i −0.989211 0.146495i \(-0.953201\pi\)
0.989211 0.146495i \(-0.0467992\pi\)
\(198\) 0 0
\(199\) 24.7724 1.75607 0.878034 0.478598i \(-0.158855\pi\)
0.878034 + 0.478598i \(0.158855\pi\)
\(200\) − 8.35690i − 0.590922i
\(201\) 0 0
\(202\) 2.40821i 0.169441i
\(203\) − 10.4112i − 0.730722i
\(204\) 0 0
\(205\) 0.109916 0.00767688
\(206\) 6.12200i 0.426540i
\(207\) 0 0
\(208\) 0 0
\(209\) −31.5545 −2.18267
\(210\) 0 0
\(211\) −5.93900 −0.408858 −0.204429 0.978881i \(-0.565534\pi\)
−0.204429 + 0.978881i \(0.565534\pi\)
\(212\) 1.91185 0.131307
\(213\) 0 0
\(214\) − 5.70410i − 0.389924i
\(215\) − 0.423272i − 0.0288669i
\(216\) 0 0
\(217\) 9.23623 0.626996
\(218\) 5.40581 0.366128
\(219\) 0 0
\(220\) −2.51573 −0.169610
\(221\) 0 0
\(222\) 0 0
\(223\) 14.2010i 0.950972i 0.879723 + 0.475486i \(0.157728\pi\)
−0.879723 + 0.475486i \(0.842272\pi\)
\(224\) −8.15644 −0.544975
\(225\) 0 0
\(226\) − 0.728857i − 0.0484829i
\(227\) − 16.0073i − 1.06244i −0.847233 0.531221i \(-0.821733\pi\)
0.847233 0.531221i \(-0.178267\pi\)
\(228\) 0 0
\(229\) − 1.84117i − 0.121668i −0.998148 0.0608339i \(-0.980624\pi\)
0.998148 0.0608339i \(-0.0193760\pi\)
\(230\) 0.917231 0.0604804
\(231\) 0 0
\(232\) − 10.0489i − 0.659744i
\(233\) −23.4252 −1.53464 −0.767318 0.641267i \(-0.778409\pi\)
−0.767318 + 0.641267i \(0.778409\pi\)
\(234\) 0 0
\(235\) 1.66355 0.108518
\(236\) − 24.6896i − 1.60716i
\(237\) 0 0
\(238\) 2.96615 0.192267
\(239\) − 14.6015i − 0.944491i −0.881467 0.472246i \(-0.843444\pi\)
0.881467 0.472246i \(-0.156556\pi\)
\(240\) 0 0
\(241\) − 8.63102i − 0.555973i −0.960585 0.277987i \(-0.910333\pi\)
0.960585 0.277987i \(-0.0896671\pi\)
\(242\) 9.32544i 0.599462i
\(243\) 0 0
\(244\) −15.3448 −0.982351
\(245\) 0.969869i 0.0619627i
\(246\) 0 0
\(247\) 0 0
\(248\) 8.91484 0.566093
\(249\) 0 0
\(250\) 1.09246 0.0690931
\(251\) −3.80194 −0.239976 −0.119988 0.992775i \(-0.538286\pi\)
−0.119988 + 0.992775i \(0.538286\pi\)
\(252\) 0 0
\(253\) 47.1715i 2.96565i
\(254\) 4.80625i 0.301571i
\(255\) 0 0
\(256\) 2.40150 0.150094
\(257\) −20.5961 −1.28475 −0.642375 0.766391i \(-0.722051\pi\)
−0.642375 + 0.766391i \(0.722051\pi\)
\(258\) 0 0
\(259\) −5.60627 −0.348357
\(260\) 0 0
\(261\) 0 0
\(262\) 0.403894i 0.0249527i
\(263\) −0.332733 −0.0205172 −0.0102586 0.999947i \(-0.503265\pi\)
−0.0102586 + 0.999947i \(0.503265\pi\)
\(264\) 0 0
\(265\) 0.262045i 0.0160973i
\(266\) − 4.35498i − 0.267021i
\(267\) 0 0
\(268\) 10.7409i 0.656107i
\(269\) −27.3032 −1.66471 −0.832353 0.554247i \(-0.813006\pi\)
−0.832353 + 0.554247i \(0.813006\pi\)
\(270\) 0 0
\(271\) − 27.9855i − 1.70000i −0.526783 0.850000i \(-0.676602\pi\)
0.526783 0.850000i \(-0.323398\pi\)
\(272\) −10.8388 −0.657197
\(273\) 0 0
\(274\) −4.24937 −0.256714
\(275\) 27.9191i 1.68359i
\(276\) 0 0
\(277\) 2.10321 0.126370 0.0631849 0.998002i \(-0.479874\pi\)
0.0631849 + 0.998002i \(0.479874\pi\)
\(278\) 1.82132i 0.109235i
\(279\) 0 0
\(280\) − 0.732578i − 0.0437799i
\(281\) 27.2349i 1.62470i 0.583172 + 0.812349i \(0.301811\pi\)
−0.583172 + 0.812349i \(0.698189\pi\)
\(282\) 0 0
\(283\) −5.28382 −0.314090 −0.157045 0.987591i \(-0.550197\pi\)
−0.157045 + 0.987591i \(0.550197\pi\)
\(284\) 10.3056i 0.611524i
\(285\) 0 0
\(286\) 0 0
\(287\) −0.780167 −0.0460518
\(288\) 0 0
\(289\) −2.54527 −0.149722
\(290\) 0.652793 0.0383333
\(291\) 0 0
\(292\) − 13.2567i − 0.775788i
\(293\) 32.6625i 1.90816i 0.299548 + 0.954081i \(0.403164\pi\)
−0.299548 + 0.954081i \(0.596836\pi\)
\(294\) 0 0
\(295\) 3.38404 0.197027
\(296\) −5.41119 −0.314519
\(297\) 0 0
\(298\) 6.84846 0.396721
\(299\) 0 0
\(300\) 0 0
\(301\) 3.00431i 0.173166i
\(302\) 1.63533 0.0941029
\(303\) 0 0
\(304\) 15.9138i 0.912717i
\(305\) − 2.10321i − 0.120430i
\(306\) 0 0
\(307\) 20.7614i 1.18491i 0.805602 + 0.592457i \(0.201842\pi\)
−0.805602 + 0.592457i \(0.798158\pi\)
\(308\) 17.8562 1.01745
\(309\) 0 0
\(310\) 0.579121i 0.0328919i
\(311\) −11.3013 −0.640836 −0.320418 0.947276i \(-0.603823\pi\)
−0.320418 + 0.947276i \(0.603823\pi\)
\(312\) 0 0
\(313\) −4.27173 −0.241453 −0.120726 0.992686i \(-0.538522\pi\)
−0.120726 + 0.992686i \(0.538522\pi\)
\(314\) 2.17092i 0.122512i
\(315\) 0 0
\(316\) 8.02715 0.451562
\(317\) − 15.4776i − 0.869307i −0.900598 0.434653i \(-0.856871\pi\)
0.900598 0.434653i \(-0.143129\pi\)
\(318\) 0 0
\(319\) 33.5719i 1.87967i
\(320\) 0.896789i 0.0501320i
\(321\) 0 0
\(322\) −6.51035 −0.362808
\(323\) − 21.2228i − 1.18087i
\(324\) 0 0
\(325\) 0 0
\(326\) −3.84117 −0.212743
\(327\) 0 0
\(328\) −0.753020 −0.0415786
\(329\) −11.8076 −0.650973
\(330\) 0 0
\(331\) 6.06829i 0.333544i 0.985996 + 0.166772i \(0.0533343\pi\)
−0.985996 + 0.166772i \(0.946666\pi\)
\(332\) 18.3545i 1.00733i
\(333\) 0 0
\(334\) 4.21446 0.230605
\(335\) −1.47219 −0.0804343
\(336\) 0 0
\(337\) −12.1239 −0.660432 −0.330216 0.943905i \(-0.607122\pi\)
−0.330216 + 0.943905i \(0.607122\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) − 1.69202i − 0.0917627i
\(341\) −29.7832 −1.61285
\(342\) 0 0
\(343\) − 19.1551i − 1.03428i
\(344\) 2.89977i 0.156345i
\(345\) 0 0
\(346\) − 2.12498i − 0.114240i
\(347\) −23.1497 −1.24274 −0.621371 0.783516i \(-0.713424\pi\)
−0.621371 + 0.783516i \(0.713424\pi\)
\(348\) 0 0
\(349\) 22.1957i 1.18811i 0.804426 + 0.594053i \(0.202473\pi\)
−0.804426 + 0.594053i \(0.797527\pi\)
\(350\) −3.85325 −0.205965
\(351\) 0 0
\(352\) 26.3013 1.40186
\(353\) 5.07069i 0.269885i 0.990853 + 0.134943i \(0.0430850\pi\)
−0.990853 + 0.134943i \(0.956915\pi\)
\(354\) 0 0
\(355\) −1.41252 −0.0749687
\(356\) 0.246980i 0.0130899i
\(357\) 0 0
\(358\) − 1.52888i − 0.0808036i
\(359\) − 16.6746i − 0.880050i −0.897986 0.440025i \(-0.854970\pi\)
0.897986 0.440025i \(-0.145030\pi\)
\(360\) 0 0
\(361\) −12.1599 −0.639995
\(362\) − 6.00192i − 0.315454i
\(363\) 0 0
\(364\) 0 0
\(365\) 1.81700 0.0951063
\(366\) 0 0
\(367\) −1.17928 −0.0615577 −0.0307789 0.999526i \(-0.509799\pi\)
−0.0307789 + 0.999526i \(0.509799\pi\)
\(368\) 23.7899 1.24013
\(369\) 0 0
\(370\) − 0.351519i − 0.0182746i
\(371\) − 1.85995i − 0.0965639i
\(372\) 0 0
\(373\) −30.0925 −1.55813 −0.779064 0.626944i \(-0.784305\pi\)
−0.779064 + 0.626944i \(0.784305\pi\)
\(374\) −9.56465 −0.494576
\(375\) 0 0
\(376\) −11.3967 −0.587741
\(377\) 0 0
\(378\) 0 0
\(379\) − 19.1631i − 0.984345i −0.870498 0.492172i \(-0.836203\pi\)
0.870498 0.492172i \(-0.163797\pi\)
\(380\) −2.48427 −0.127440
\(381\) 0 0
\(382\) − 0.579121i − 0.0296304i
\(383\) 15.3884i 0.786308i 0.919473 + 0.393154i \(0.128616\pi\)
−0.919473 + 0.393154i \(0.871384\pi\)
\(384\) 0 0
\(385\) 2.44743i 0.124733i
\(386\) 4.09246 0.208301
\(387\) 0 0
\(388\) − 24.6679i − 1.25232i
\(389\) −24.0315 −1.21844 −0.609222 0.793000i \(-0.708518\pi\)
−0.609222 + 0.793000i \(0.708518\pi\)
\(390\) 0 0
\(391\) −31.7265 −1.60448
\(392\) − 6.64443i − 0.335594i
\(393\) 0 0
\(394\) −1.83015 −0.0922016
\(395\) 1.10023i 0.0553585i
\(396\) 0 0
\(397\) − 29.6015i − 1.48566i −0.669482 0.742828i \(-0.733484\pi\)
0.669482 0.742828i \(-0.266516\pi\)
\(398\) − 11.0248i − 0.552621i
\(399\) 0 0
\(400\) 14.0804 0.704019
\(401\) − 21.1032i − 1.05384i −0.849914 0.526922i \(-0.823346\pi\)
0.849914 0.526922i \(-0.176654\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) −9.75063 −0.485112
\(405\) 0 0
\(406\) −4.63342 −0.229953
\(407\) 18.0780 0.896092
\(408\) 0 0
\(409\) − 36.0224i − 1.78119i −0.454796 0.890596i \(-0.650288\pi\)
0.454796 0.890596i \(-0.349712\pi\)
\(410\) − 0.0489173i − 0.00241586i
\(411\) 0 0
\(412\) −24.7875 −1.22119
\(413\) −24.0194 −1.18192
\(414\) 0 0
\(415\) −2.51573 −0.123492
\(416\) 0 0
\(417\) 0 0
\(418\) 14.0431i 0.686869i
\(419\) −5.96854 −0.291582 −0.145791 0.989315i \(-0.546573\pi\)
−0.145791 + 0.989315i \(0.546573\pi\)
\(420\) 0 0
\(421\) − 2.09544i − 0.102126i −0.998695 0.0510628i \(-0.983739\pi\)
0.998695 0.0510628i \(-0.0162609\pi\)
\(422\) 2.64310i 0.128664i
\(423\) 0 0
\(424\) − 1.79523i − 0.0871842i
\(425\) −18.7778 −0.910856
\(426\) 0 0
\(427\) 14.9282i 0.722429i
\(428\) 23.0954 1.11636
\(429\) 0 0
\(430\) −0.188374 −0.00908418
\(431\) 2.88577i 0.139003i 0.997582 + 0.0695014i \(0.0221408\pi\)
−0.997582 + 0.0695014i \(0.977859\pi\)
\(432\) 0 0
\(433\) 12.6485 0.607847 0.303924 0.952696i \(-0.401703\pi\)
0.303924 + 0.952696i \(0.401703\pi\)
\(434\) − 4.11051i − 0.197311i
\(435\) 0 0
\(436\) 21.8877i 1.04823i
\(437\) 46.5816i 2.22830i
\(438\) 0 0
\(439\) 10.9269 0.521513 0.260757 0.965405i \(-0.416028\pi\)
0.260757 + 0.965405i \(0.416028\pi\)
\(440\) 2.36227i 0.112617i
\(441\) 0 0
\(442\) 0 0
\(443\) 19.2403 0.914133 0.457067 0.889433i \(-0.348900\pi\)
0.457067 + 0.889433i \(0.348900\pi\)
\(444\) 0 0
\(445\) −0.0338518 −0.00160473
\(446\) 6.32006 0.299264
\(447\) 0 0
\(448\) − 6.36526i − 0.300730i
\(449\) − 28.8200i − 1.36010i −0.733166 0.680050i \(-0.761958\pi\)
0.733166 0.680050i \(-0.238042\pi\)
\(450\) 0 0
\(451\) 2.51573 0.118461
\(452\) 2.95108 0.138807
\(453\) 0 0
\(454\) −7.12392 −0.334342
\(455\) 0 0
\(456\) 0 0
\(457\) − 18.0707i − 0.845311i −0.906290 0.422656i \(-0.861098\pi\)
0.906290 0.422656i \(-0.138902\pi\)
\(458\) −0.819396 −0.0382879
\(459\) 0 0
\(460\) 3.71379i 0.173156i
\(461\) 7.56763i 0.352460i 0.984349 + 0.176230i \(0.0563902\pi\)
−0.984349 + 0.176230i \(0.943610\pi\)
\(462\) 0 0
\(463\) − 35.3551i − 1.64309i −0.570143 0.821545i \(-0.693112\pi\)
0.570143 0.821545i \(-0.306888\pi\)
\(464\) 16.9312 0.786013
\(465\) 0 0
\(466\) 10.4252i 0.482938i
\(467\) 13.0000 0.601568 0.300784 0.953692i \(-0.402752\pi\)
0.300784 + 0.953692i \(0.402752\pi\)
\(468\) 0 0
\(469\) 10.4494 0.482506
\(470\) − 0.740348i − 0.0341497i
\(471\) 0 0
\(472\) −23.1836 −1.06711
\(473\) − 9.68771i − 0.445441i
\(474\) 0 0
\(475\) 27.5700i 1.26500i
\(476\) 12.0097i 0.550463i
\(477\) 0 0
\(478\) −6.49827 −0.297224
\(479\) − 25.5265i − 1.16633i −0.812352 0.583167i \(-0.801813\pi\)
0.812352 0.583167i \(-0.198187\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) −3.84117 −0.174960
\(483\) 0 0
\(484\) −37.7579 −1.71627
\(485\) 3.38106 0.153526
\(486\) 0 0
\(487\) 16.0073i 0.725360i 0.931914 + 0.362680i \(0.118138\pi\)
−0.931914 + 0.362680i \(0.881862\pi\)
\(488\) 14.4088i 0.652256i
\(489\) 0 0
\(490\) 0.431632 0.0194992
\(491\) 20.7385 0.935917 0.467959 0.883750i \(-0.344990\pi\)
0.467959 + 0.883750i \(0.344990\pi\)
\(492\) 0 0
\(493\) −22.5797 −1.01694
\(494\) 0 0
\(495\) 0 0
\(496\) 15.0204i 0.674438i
\(497\) 10.0258 0.449719
\(498\) 0 0
\(499\) − 8.06770i − 0.361160i −0.983560 0.180580i \(-0.942203\pi\)
0.983560 0.180580i \(-0.0577975\pi\)
\(500\) 4.42327i 0.197815i
\(501\) 0 0
\(502\) 1.69202i 0.0755186i
\(503\) 30.2422 1.34843 0.674216 0.738534i \(-0.264482\pi\)
0.674216 + 0.738534i \(0.264482\pi\)
\(504\) 0 0
\(505\) − 1.33645i − 0.0594714i
\(506\) 20.9933 0.933266
\(507\) 0 0
\(508\) −19.4601 −0.863403
\(509\) − 16.0495i − 0.711382i −0.934604 0.355691i \(-0.884246\pi\)
0.934604 0.355691i \(-0.115754\pi\)
\(510\) 0 0
\(511\) −12.8968 −0.570520
\(512\) − 22.9119i − 1.01257i
\(513\) 0 0
\(514\) 9.16613i 0.404301i
\(515\) − 3.39745i − 0.149710i
\(516\) 0 0
\(517\) 38.0747 1.67452
\(518\) 2.49502i 0.109625i
\(519\) 0 0
\(520\) 0 0
\(521\) 2.69309 0.117986 0.0589931 0.998258i \(-0.481211\pi\)
0.0589931 + 0.998258i \(0.481211\pi\)
\(522\) 0 0
\(523\) 35.3957 1.54774 0.773872 0.633342i \(-0.218317\pi\)
0.773872 + 0.633342i \(0.218317\pi\)
\(524\) −1.63533 −0.0714399
\(525\) 0 0
\(526\) 0.148080i 0.00645659i
\(527\) − 20.0315i − 0.872584i
\(528\) 0 0
\(529\) 46.6359 2.02765
\(530\) 0.116621 0.00506569
\(531\) 0 0
\(532\) 17.6329 0.764485
\(533\) 0 0
\(534\) 0 0
\(535\) 3.16554i 0.136858i
\(536\) 10.0858 0.435638
\(537\) 0 0
\(538\) 12.1511i 0.523870i
\(539\) 22.1981i 0.956138i
\(540\) 0 0
\(541\) 34.7338i 1.49332i 0.665205 + 0.746660i \(0.268344\pi\)
−0.665205 + 0.746660i \(0.731656\pi\)
\(542\) −12.4547 −0.534976
\(543\) 0 0
\(544\) 17.6896i 0.758437i
\(545\) −3.00000 −0.128506
\(546\) 0 0
\(547\) −26.1183 −1.11674 −0.558368 0.829593i \(-0.688572\pi\)
−0.558368 + 0.829593i \(0.688572\pi\)
\(548\) − 17.2054i − 0.734976i
\(549\) 0 0
\(550\) 12.4252 0.529812
\(551\) 33.1521i 1.41233i
\(552\) 0 0
\(553\) − 7.80923i − 0.332082i
\(554\) − 0.936017i − 0.0397676i
\(555\) 0 0
\(556\) −7.37435 −0.312742
\(557\) 24.7748i 1.04974i 0.851182 + 0.524871i \(0.175886\pi\)
−0.851182 + 0.524871i \(0.824114\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 1.23431 0.0521590
\(561\) 0 0
\(562\) 12.1207 0.511280
\(563\) 5.26098 0.221724 0.110862 0.993836i \(-0.464639\pi\)
0.110862 + 0.993836i \(0.464639\pi\)
\(564\) 0 0
\(565\) 0.404485i 0.0170168i
\(566\) 2.35152i 0.0988417i
\(567\) 0 0
\(568\) 9.67696 0.406036
\(569\) −33.7458 −1.41470 −0.707350 0.706864i \(-0.750109\pi\)
−0.707350 + 0.706864i \(0.750109\pi\)
\(570\) 0 0
\(571\) −23.0887 −0.966234 −0.483117 0.875556i \(-0.660495\pi\)
−0.483117 + 0.875556i \(0.660495\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0.347207i 0.0144921i
\(575\) 41.2150 1.71879
\(576\) 0 0
\(577\) 3.57002i 0.148622i 0.997235 + 0.0743110i \(0.0236758\pi\)
−0.997235 + 0.0743110i \(0.976324\pi\)
\(578\) 1.13275i 0.0471162i
\(579\) 0 0
\(580\) 2.64310i 0.109749i
\(581\) 17.8562 0.740801
\(582\) 0 0
\(583\) 5.99761i 0.248396i
\(584\) −12.4480 −0.515103
\(585\) 0 0
\(586\) 14.5362 0.600484
\(587\) 11.4625i 0.473108i 0.971618 + 0.236554i \(0.0760180\pi\)
−0.971618 + 0.236554i \(0.923982\pi\)
\(588\) 0 0
\(589\) −29.4107 −1.21185
\(590\) − 1.50604i − 0.0620027i
\(591\) 0 0
\(592\) − 9.11721i − 0.374715i
\(593\) − 21.8538i − 0.897430i −0.893675 0.448715i \(-0.851882\pi\)
0.893675 0.448715i \(-0.148118\pi\)
\(594\) 0 0
\(595\) −1.64609 −0.0674830
\(596\) 27.7289i 1.13582i
\(597\) 0 0
\(598\) 0 0
\(599\) −27.0573 −1.10553 −0.552765 0.833337i \(-0.686427\pi\)
−0.552765 + 0.833337i \(0.686427\pi\)
\(600\) 0 0
\(601\) 10.8780 0.443723 0.221861 0.975078i \(-0.428787\pi\)
0.221861 + 0.975078i \(0.428787\pi\)
\(602\) 1.33704 0.0544939
\(603\) 0 0
\(604\) 6.62133i 0.269418i
\(605\) − 5.17523i − 0.210403i
\(606\) 0 0
\(607\) −29.6359 −1.20289 −0.601443 0.798916i \(-0.705407\pi\)
−0.601443 + 0.798916i \(0.705407\pi\)
\(608\) 25.9724 1.05332
\(609\) 0 0
\(610\) −0.936017 −0.0378982
\(611\) 0 0
\(612\) 0 0
\(613\) 10.2343i 0.413360i 0.978409 + 0.206680i \(0.0662659\pi\)
−0.978409 + 0.206680i \(0.933734\pi\)
\(614\) 9.23968 0.372883
\(615\) 0 0
\(616\) − 16.7670i − 0.675563i
\(617\) − 26.2828i − 1.05810i −0.848590 0.529052i \(-0.822548\pi\)
0.848590 0.529052i \(-0.177452\pi\)
\(618\) 0 0
\(619\) − 29.0834i − 1.16896i −0.811408 0.584479i \(-0.801299\pi\)
0.811408 0.584479i \(-0.198701\pi\)
\(620\) −2.34481 −0.0941700
\(621\) 0 0
\(622\) 5.02954i 0.201666i
\(623\) 0.240275 0.00962641
\(624\) 0 0
\(625\) 24.0887 0.963549
\(626\) 1.90110i 0.0759833i
\(627\) 0 0
\(628\) −8.78986 −0.350753
\(629\) 12.1588i 0.484804i
\(630\) 0 0
\(631\) − 25.4480i − 1.01307i −0.862219 0.506535i \(-0.830926\pi\)
0.862219 0.506535i \(-0.169074\pi\)
\(632\) − 7.53750i − 0.299826i
\(633\) 0 0
\(634\) −6.88816 −0.273564
\(635\) − 2.66727i − 0.105847i
\(636\) 0 0
\(637\) 0 0
\(638\) 14.9409 0.591517
\(639\) 0 0
\(640\) 2.69740 0.106624
\(641\) −26.7409 −1.05620 −0.528102 0.849181i \(-0.677096\pi\)
−0.528102 + 0.849181i \(0.677096\pi\)
\(642\) 0 0
\(643\) − 32.9614i − 1.29987i −0.759990 0.649935i \(-0.774796\pi\)
0.759990 0.649935i \(-0.225204\pi\)
\(644\) − 26.3599i − 1.03872i
\(645\) 0 0
\(646\) −9.44504 −0.371610
\(647\) 34.4946 1.35612 0.678060 0.735006i \(-0.262821\pi\)
0.678060 + 0.735006i \(0.262821\pi\)
\(648\) 0 0
\(649\) 77.4529 3.04029
\(650\) 0 0
\(651\) 0 0
\(652\) − 15.5526i − 0.609085i
\(653\) −36.1517 −1.41472 −0.707362 0.706852i \(-0.750115\pi\)
−0.707362 + 0.706852i \(0.750115\pi\)
\(654\) 0 0
\(655\) − 0.224144i − 0.00875805i
\(656\) − 1.26875i − 0.0495364i
\(657\) 0 0
\(658\) 5.25487i 0.204856i
\(659\) 6.81700 0.265553 0.132776 0.991146i \(-0.457611\pi\)
0.132776 + 0.991146i \(0.457611\pi\)
\(660\) 0 0
\(661\) 10.8944i 0.423743i 0.977298 + 0.211871i \(0.0679558\pi\)
−0.977298 + 0.211871i \(0.932044\pi\)
\(662\) 2.70065 0.104964
\(663\) 0 0
\(664\) 17.2349 0.668844
\(665\) 2.41683i 0.0937206i
\(666\) 0 0
\(667\) 49.5599 1.91897
\(668\) 17.0640i 0.660225i
\(669\) 0 0
\(670\) 0.655186i 0.0253120i
\(671\) − 48.1377i − 1.85833i
\(672\) 0 0
\(673\) −20.7385 −0.799412 −0.399706 0.916643i \(-0.630888\pi\)
−0.399706 + 0.916643i \(0.630888\pi\)
\(674\) 5.39565i 0.207833i
\(675\) 0 0
\(676\) 0 0
\(677\) 25.5786 0.983067 0.491534 0.870859i \(-0.336436\pi\)
0.491534 + 0.870859i \(0.336436\pi\)
\(678\) 0 0
\(679\) −23.9982 −0.920966
\(680\) −1.58881 −0.0609281
\(681\) 0 0
\(682\) 13.2547i 0.507551i
\(683\) 21.6310i 0.827688i 0.910348 + 0.413844i \(0.135814\pi\)
−0.910348 + 0.413844i \(0.864186\pi\)
\(684\) 0 0
\(685\) 2.35822 0.0901031
\(686\) −8.52483 −0.325479
\(687\) 0 0
\(688\) −4.88577 −0.186268
\(689\) 0 0
\(690\) 0 0
\(691\) 2.62996i 0.100048i 0.998748 + 0.0500242i \(0.0159298\pi\)
−0.998748 + 0.0500242i \(0.984070\pi\)
\(692\) 8.60388 0.327070
\(693\) 0 0
\(694\) 10.3026i 0.391081i
\(695\) − 1.01075i − 0.0383401i
\(696\) 0 0
\(697\) 1.69202i 0.0640899i
\(698\) 9.87800 0.373888
\(699\) 0 0
\(700\) − 15.6015i − 0.589681i
\(701\) 40.0925 1.51427 0.757136 0.653258i \(-0.226598\pi\)
0.757136 + 0.653258i \(0.226598\pi\)
\(702\) 0 0
\(703\) 17.8519 0.673298
\(704\) 20.5254i 0.773581i
\(705\) 0 0
\(706\) 2.25667 0.0849308
\(707\) 9.48593i 0.356755i
\(708\) 0 0
\(709\) − 23.2097i − 0.871657i −0.900030 0.435829i \(-0.856455\pi\)
0.900030 0.435829i \(-0.143545\pi\)
\(710\) 0.628630i 0.0235921i
\(711\) 0 0
\(712\) 0.231914 0.00869135
\(713\) 43.9667i 1.64657i
\(714\) 0 0
\(715\) 0 0
\(716\) 6.19029 0.231342
\(717\) 0 0
\(718\) −7.42088 −0.276945
\(719\) −26.0146 −0.970181 −0.485090 0.874464i \(-0.661213\pi\)
−0.485090 + 0.874464i \(0.661213\pi\)
\(720\) 0 0
\(721\) 24.1146i 0.898073i
\(722\) 5.41166i 0.201401i
\(723\) 0 0
\(724\) 24.3013 0.903150
\(725\) 29.3327 1.08939
\(726\) 0 0
\(727\) 16.5472 0.613701 0.306851 0.951758i \(-0.400725\pi\)
0.306851 + 0.951758i \(0.400725\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) − 0.808643i − 0.0299292i
\(731\) 6.51573 0.240993
\(732\) 0 0
\(733\) 18.8750i 0.697165i 0.937278 + 0.348582i \(0.113337\pi\)
−0.937278 + 0.348582i \(0.886663\pi\)
\(734\) 0.524827i 0.0193717i
\(735\) 0 0
\(736\) − 38.8267i − 1.43117i
\(737\) −33.6950 −1.24117
\(738\) 0 0
\(739\) − 47.3239i − 1.74084i −0.492312 0.870419i \(-0.663848\pi\)
0.492312 0.870419i \(-0.336152\pi\)
\(740\) 1.42327 0.0523205
\(741\) 0 0
\(742\) −0.827757 −0.0303879
\(743\) 8.88769i 0.326058i 0.986621 + 0.163029i \(0.0521264\pi\)
−0.986621 + 0.163029i \(0.947874\pi\)
\(744\) 0 0
\(745\) −3.80061 −0.139244
\(746\) 13.3924i 0.490331i
\(747\) 0 0
\(748\) − 38.7265i − 1.41598i
\(749\) − 22.4685i − 0.820980i
\(750\) 0 0
\(751\) −0.710808 −0.0259377 −0.0129689 0.999916i \(-0.504128\pi\)
−0.0129689 + 0.999916i \(0.504128\pi\)
\(752\) − 19.2021i − 0.700229i
\(753\) 0 0
\(754\) 0 0
\(755\) −0.907542 −0.0330288
\(756\) 0 0
\(757\) 9.78554 0.355662 0.177831 0.984061i \(-0.443092\pi\)
0.177831 + 0.984061i \(0.443092\pi\)
\(758\) −8.52840 −0.309766
\(759\) 0 0
\(760\) 2.33273i 0.0846171i
\(761\) − 18.8810i − 0.684435i −0.939621 0.342218i \(-0.888822\pi\)
0.939621 0.342218i \(-0.111178\pi\)
\(762\) 0 0
\(763\) 21.2935 0.770877
\(764\) 2.34481 0.0848324
\(765\) 0 0
\(766\) 6.84846 0.247445
\(767\) 0 0
\(768\) 0 0
\(769\) − 12.4349i − 0.448413i −0.974542 0.224207i \(-0.928021\pi\)
0.974542 0.224207i \(-0.0719790\pi\)
\(770\) 1.08921 0.0392524
\(771\) 0 0
\(772\) 16.5700i 0.596368i
\(773\) − 45.6746i − 1.64280i −0.570353 0.821400i \(-0.693193\pi\)
0.570353 0.821400i \(-0.306807\pi\)
\(774\) 0 0
\(775\) 26.0224i 0.934751i
\(776\) −23.1631 −0.831508
\(777\) 0 0
\(778\) 10.6950i 0.383435i
\(779\) 2.48427 0.0890082
\(780\) 0 0
\(781\) −32.3293 −1.15683
\(782\) 14.1196i 0.504916i
\(783\) 0 0
\(784\) 11.1951 0.399824
\(785\) − 1.20477i − 0.0430000i
\(786\) 0 0
\(787\) 4.51871i 0.161075i 0.996752 + 0.0805374i \(0.0256636\pi\)
−0.996752 + 0.0805374i \(0.974336\pi\)
\(788\) − 7.41013i − 0.263975i
\(789\) 0 0
\(790\) 0.489647 0.0174209
\(791\) − 2.87097i − 0.102080i
\(792\) 0 0
\(793\) 0 0
\(794\) −13.1739 −0.467524
\(795\) 0 0
\(796\) 44.6383 1.58216
\(797\) −28.7391 −1.01799 −0.508996 0.860769i \(-0.669983\pi\)
−0.508996 + 0.860769i \(0.669983\pi\)
\(798\) 0 0
\(799\) 25.6082i 0.905953i
\(800\) − 22.9801i − 0.812471i
\(801\) 0 0
\(802\) −9.39181 −0.331636
\(803\) 41.5870 1.46757
\(804\) 0 0
\(805\) 3.61297 0.127341
\(806\) 0 0
\(807\) 0 0
\(808\) 9.15585i 0.322102i
\(809\) −5.42891 −0.190870 −0.0954352 0.995436i \(-0.530424\pi\)
−0.0954352 + 0.995436i \(0.530424\pi\)
\(810\) 0 0
\(811\) 0.629104i 0.0220908i 0.999939 + 0.0110454i \(0.00351593\pi\)
−0.999939 + 0.0110454i \(0.996484\pi\)
\(812\) − 18.7603i − 0.658358i
\(813\) 0 0
\(814\) − 8.04546i − 0.281993i
\(815\) 2.13169 0.0746697
\(816\) 0 0
\(817\) − 9.56657i − 0.334692i
\(818\) −16.0315 −0.560527
\(819\) 0 0
\(820\) 0.198062 0.00691663
\(821\) − 36.2640i − 1.26562i −0.774307 0.632811i \(-0.781901\pi\)
0.774307 0.632811i \(-0.218099\pi\)
\(822\) 0 0
\(823\) −41.7396 −1.45495 −0.727476 0.686133i \(-0.759307\pi\)
−0.727476 + 0.686133i \(0.759307\pi\)
\(824\) 23.2755i 0.810839i
\(825\) 0 0
\(826\) 10.6896i 0.371940i
\(827\) 38.1997i 1.32833i 0.747584 + 0.664167i \(0.231214\pi\)
−0.747584 + 0.664167i \(0.768786\pi\)
\(828\) 0 0
\(829\) −15.9788 −0.554967 −0.277484 0.960730i \(-0.589500\pi\)
−0.277484 + 0.960730i \(0.589500\pi\)
\(830\) 1.11960i 0.0388621i
\(831\) 0 0
\(832\) 0 0
\(833\) −14.9299 −0.517290
\(834\) 0 0
\(835\) −2.33885 −0.0809391
\(836\) −56.8592 −1.96652
\(837\) 0 0
\(838\) 2.65625i 0.0917587i
\(839\) − 4.63879i − 0.160149i −0.996789 0.0800744i \(-0.974484\pi\)
0.996789 0.0800744i \(-0.0255158\pi\)
\(840\) 0 0
\(841\) 6.27173 0.216267
\(842\) −0.932559 −0.0321381
\(843\) 0 0
\(844\) −10.7017 −0.368368
\(845\) 0 0
\(846\) 0 0
\(847\) 36.7329i 1.26216i
\(848\) 3.02475 0.103870
\(849\) 0 0
\(850\) 8.35690i 0.286639i
\(851\) − 26.6872i − 0.914827i
\(852\) 0 0
\(853\) − 18.3884i − 0.629605i −0.949157 0.314803i \(-0.898062\pi\)
0.949157 0.314803i \(-0.101938\pi\)
\(854\) 6.64370 0.227343
\(855\) 0 0
\(856\) − 21.6866i − 0.741234i
\(857\) 28.1849 0.962778 0.481389 0.876507i \(-0.340132\pi\)
0.481389 + 0.876507i \(0.340132\pi\)
\(858\) 0 0
\(859\) 33.3957 1.13944 0.569722 0.821837i \(-0.307051\pi\)
0.569722 + 0.821837i \(0.307051\pi\)
\(860\) − 0.762709i − 0.0260082i
\(861\) 0 0
\(862\) 1.28429 0.0437431
\(863\) 43.0640i 1.46592i 0.680274 + 0.732958i \(0.261861\pi\)
−0.680274 + 0.732958i \(0.738139\pi\)
\(864\) 0 0
\(865\) 1.17928i 0.0400966i
\(866\) − 5.62910i − 0.191285i
\(867\) 0 0
\(868\) 16.6431 0.564904
\(869\) 25.1817i 0.854230i
\(870\) 0 0
\(871\) 0 0
\(872\) 20.5526 0.695998
\(873\) 0 0
\(874\) 20.7308 0.701229
\(875\) 4.30319 0.145474
\(876\) 0 0
\(877\) − 30.1702i − 1.01877i −0.860537 0.509387i \(-0.829872\pi\)
0.860537 0.509387i \(-0.170128\pi\)
\(878\) − 4.86294i − 0.164116i
\(879\) 0 0
\(880\) −3.98015 −0.134171
\(881\) −3.56273 −0.120031 −0.0600157 0.998197i \(-0.519115\pi\)
−0.0600157 + 0.998197i \(0.519115\pi\)
\(882\) 0 0
\(883\) 10.2088 0.343554 0.171777 0.985136i \(-0.445049\pi\)
0.171777 + 0.985136i \(0.445049\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) − 8.56273i − 0.287670i
\(887\) 9.33645 0.313487 0.156744 0.987639i \(-0.449900\pi\)
0.156744 + 0.987639i \(0.449900\pi\)
\(888\) 0 0
\(889\) 18.9318i 0.634953i
\(890\) 0.0150655i 0 0.000504996i
\(891\) 0 0
\(892\) 25.5894i 0.856797i
\(893\) 37.5986 1.25819
\(894\) 0 0
\(895\) 0.848462i 0.0283610i
\(896\) −19.1457 −0.639613
\(897\) 0 0
\(898\) −12.8261 −0.428013
\(899\) 31.2911i 1.04362i
\(900\) 0 0
\(901\) −4.03385 −0.134387
\(902\) − 1.11960i − 0.0372788i
\(903\) 0 0
\(904\) − 2.77107i − 0.0921644i
\(905\) 3.33081i 0.110720i
\(906\) 0 0
\(907\) 41.0804 1.36405 0.682026 0.731328i \(-0.261099\pi\)
0.682026 + 0.731328i \(0.261099\pi\)
\(908\) − 28.8442i − 0.957227i
\(909\) 0 0
\(910\) 0 0
\(911\) 18.9705 0.628519 0.314260 0.949337i \(-0.398244\pi\)
0.314260 + 0.949337i \(0.398244\pi\)
\(912\) 0 0
\(913\) −57.5792 −1.90559
\(914\) −8.04221 −0.266013
\(915\) 0 0
\(916\) − 3.31767i − 0.109619i
\(917\) 1.59094i 0.0525375i
\(918\) 0 0
\(919\) 29.0019 0.956685 0.478343 0.878173i \(-0.341238\pi\)
0.478343 + 0.878173i \(0.341238\pi\)
\(920\) 3.48725 0.114971
\(921\) 0 0
\(922\) 3.36791 0.110916
\(923\) 0 0
\(924\) 0 0
\(925\) − 15.7952i − 0.519344i
\(926\) −15.7345 −0.517068
\(927\) 0 0
\(928\) − 27.6329i − 0.907096i
\(929\) 50.7211i 1.66410i 0.554697 + 0.832052i \(0.312834\pi\)
−0.554697 + 0.832052i \(0.687166\pi\)
\(930\) 0 0
\(931\) 21.9205i 0.718415i
\(932\) −42.2107 −1.38266
\(933\) 0 0
\(934\) − 5.78554i − 0.189309i
\(935\) 5.30798 0.173589
\(936\) 0 0
\(937\) 51.3051 1.67606 0.838032 0.545620i \(-0.183706\pi\)
0.838032 + 0.545620i \(0.183706\pi\)
\(938\) − 4.65040i − 0.151841i
\(939\) 0 0
\(940\) 2.99761 0.0977712
\(941\) 34.7036i 1.13131i 0.824643 + 0.565653i \(0.191376\pi\)
−0.824643 + 0.565653i \(0.808624\pi\)
\(942\) 0 0
\(943\) − 3.71379i − 0.120938i
\(944\) − 39.0616i − 1.27135i
\(945\) 0 0
\(946\) −4.31144 −0.140177
\(947\) − 13.0127i − 0.422855i −0.977394 0.211428i \(-0.932189\pi\)
0.977394 0.211428i \(-0.0678112\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 12.2698 0.398085
\(951\) 0 0
\(952\) 11.2771 0.365493
\(953\) 26.0151 0.842711 0.421355 0.906896i \(-0.361555\pi\)
0.421355 + 0.906896i \(0.361555\pi\)
\(954\) 0 0
\(955\) 0.321388i 0.0103999i
\(956\) − 26.3110i − 0.850957i
\(957\) 0 0
\(958\) −11.3604 −0.367036
\(959\) −16.7383 −0.540507
\(960\) 0 0
\(961\) 3.24027 0.104525
\(962\) 0 0
\(963\) 0 0
\(964\) − 15.5526i − 0.500914i
\(965\) −2.27114 −0.0731107
\(966\) 0 0
\(967\) 36.6644i 1.17905i 0.807751 + 0.589524i \(0.200685\pi\)
−0.807751 + 0.589524i \(0.799315\pi\)
\(968\) 35.4547i 1.13956i
\(969\) 0 0
\(970\) − 1.50471i − 0.0483134i
\(971\) 37.8465 1.21455 0.607277 0.794490i \(-0.292262\pi\)
0.607277 + 0.794490i \(0.292262\pi\)
\(972\) 0 0
\(973\) 7.17416i 0.229993i
\(974\) 7.12392 0.228265
\(975\) 0 0
\(976\) −24.2771 −0.777091
\(977\) − 28.8998i − 0.924586i −0.886727 0.462293i \(-0.847027\pi\)
0.886727 0.462293i \(-0.152973\pi\)
\(978\) 0 0
\(979\) −0.774791 −0.0247624
\(980\) 1.74764i 0.0558264i
\(981\) 0 0
\(982\) − 9.22952i − 0.294526i
\(983\) − 19.3991i − 0.618735i −0.950942 0.309368i \(-0.899883\pi\)
0.950942 0.309368i \(-0.100117\pi\)
\(984\) 0 0
\(985\) 1.01566 0.0323615
\(986\) 10.0489i 0.320023i
\(987\) 0 0
\(988\) 0 0
\(989\) −14.3013 −0.454754
\(990\) 0 0
\(991\) −5.18300 −0.164643 −0.0823217 0.996606i \(-0.526233\pi\)
−0.0823217 + 0.996606i \(0.526233\pi\)
\(992\) 24.5144 0.778333
\(993\) 0 0
\(994\) − 4.46191i − 0.141523i
\(995\) 6.11828i 0.193962i
\(996\) 0 0
\(997\) −49.3642 −1.56338 −0.781690 0.623667i \(-0.785642\pi\)
−0.781690 + 0.623667i \(0.785642\pi\)
\(998\) −3.59047 −0.113654
\(999\) 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 1521.2.b.m.1351.3 6
3.2 odd 2 507.2.b.g.337.4 6
13.5 odd 4 1521.2.a.p.1.2 3
13.8 odd 4 1521.2.a.q.1.2 3
13.12 even 2 inner 1521.2.b.m.1351.4 6
39.2 even 12 507.2.e.j.22.2 6
39.5 even 4 507.2.a.k.1.2 yes 3
39.8 even 4 507.2.a.j.1.2 3
39.11 even 12 507.2.e.k.22.2 6
39.17 odd 6 507.2.j.h.361.3 12
39.20 even 12 507.2.e.k.484.2 6
39.23 odd 6 507.2.j.h.316.4 12
39.29 odd 6 507.2.j.h.316.3 12
39.32 even 12 507.2.e.j.484.2 6
39.35 odd 6 507.2.j.h.361.4 12
39.38 odd 2 507.2.b.g.337.3 6
156.47 odd 4 8112.2.a.by.1.3 3
156.83 odd 4 8112.2.a.cf.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
507.2.a.j.1.2 3 39.8 even 4
507.2.a.k.1.2 yes 3 39.5 even 4
507.2.b.g.337.3 6 39.38 odd 2
507.2.b.g.337.4 6 3.2 odd 2
507.2.e.j.22.2 6 39.2 even 12
507.2.e.j.484.2 6 39.32 even 12
507.2.e.k.22.2 6 39.11 even 12
507.2.e.k.484.2 6 39.20 even 12
507.2.j.h.316.3 12 39.29 odd 6
507.2.j.h.316.4 12 39.23 odd 6
507.2.j.h.361.3 12 39.17 odd 6
507.2.j.h.361.4 12 39.35 odd 6
1521.2.a.p.1.2 3 13.5 odd 4
1521.2.a.q.1.2 3 13.8 odd 4
1521.2.b.m.1351.3 6 1.1 even 1 trivial
1521.2.b.m.1351.4 6 13.12 even 2 inner
8112.2.a.by.1.3 3 156.47 odd 4
8112.2.a.cf.1.1 3 156.83 odd 4