Properties

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

$q$-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 \(0.0502938\pi\)
−0.987544 + 0.157346i \(0.949706\pi\)
\(3\) 0 0
\(4\) 1.80194 0.900969
\(5\) − 0.246980i − 0.110453i −0.998474 0.0552263i \(-0.982412\pi\)
0.998474 0.0552263i \(-0.0175880\pi\)
\(6\) 0 0
\(7\) 1.75302i 0.662579i 0.943529 + 0.331290i \(0.107484\pi\)
−0.943529 + 0.331290i \(0.892516\pi\)
\(8\) 1.69202i 0.598220i
\(9\) 0 0
\(10\) 0.109916 0.0347586
\(11\) − 5.65279i − 1.70438i −0.523232 0.852191i \(-0.675274\pi\)
0.523232 0.852191i \(-0.324726\pi\)
\(12\) 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 \(-0.778805\pi\)
0.768115 0.640311i \(-0.221195\pi\)
\(20\) − 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 \(-0.843118\pi\)
0.880983 0.473148i \(-0.156882\pi\)
\(32\) 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 \(0.0846720\pi\)
−0.964829 + 0.262879i \(0.915328\pi\)
\(38\) 2.48427 0.403002
\(39\) 0 0
\(40\) 0.417895 0.0660750
\(41\) 0.445042i 0.0695039i 0.999396 + 0.0347519i \(0.0110641\pi\)
−0.999396 + 0.0347519i \(0.988936\pi\)
\(42\) 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 \(0.163457\pi\)
−0.871024 + 0.491241i \(0.836543\pi\)
\(48\) 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 \(0.350631\pi\)
−0.452222 + 0.891905i \(0.649369\pi\)
\(60\) 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 \(-0.881373\pi\)
0.931355 0.364112i \(-0.118627\pi\)
\(68\) −6.85086 −0.830788
\(69\) 0 0
\(70\) 0.192685i 0.0230303i
\(71\) − 5.71917i − 0.678740i −0.940653 0.339370i \(-0.889786\pi\)
0.940653 0.339370i \(-0.110214\pi\)
\(72\) 0 0
\(73\) 7.35690i 0.861060i 0.902576 + 0.430530i \(0.141673\pi\)
−0.902576 + 0.430530i \(0.858327\pi\)
\(74\) −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 \(-0.811174\pi\)
0.829149 0.559028i \(-0.188826\pi\)
\(84\) 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 \(-0.997688\pi\)
0.999974 0.00726434i \(-0.00231233\pi\)
\(90\) 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 \(0.244589\pi\)
−0.719024 + 0.694986i \(0.755411\pi\)
\(98\) 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 \(-0.802378\pi\)
0.813386 0.581724i \(-0.197622\pi\)
\(110\) − 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 \(0.133732\pi\)
−0.913035 + 0.407881i \(0.866268\pi\)
\(138\) 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 \(-0.782919\pi\)
0.776326 0.630332i \(-0.217081\pi\)
\(150\) 0 0
\(151\) − 3.67456i − 0.299032i −0.988759 0.149516i \(-0.952228\pi\)
0.988759 0.149516i \(-0.0477715\pi\)
\(152\) 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 \(0.109756\pi\)
−0.941140 + 0.338017i \(0.890244\pi\)
\(164\) 0.801938i 0.0626208i
\(165\) 0 0
\(166\) 4.53319 0.351844
\(167\) − 9.46980i − 0.732795i −0.930458 0.366397i \(-0.880591\pi\)
0.930458 0.366397i \(-0.119409\pi\)
\(168\) 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 \(-0.892628\pi\)
0.943645 0.330959i \(-0.107372\pi\)
\(194\) −6.09246 −0.437413
\(195\) 0 0
\(196\) 7.07606 0.505433
\(197\) 4.11231i 0.292990i 0.989211 + 0.146495i \(0.0467992\pi\)
−0.989211 + 0.146495i \(0.953201\pi\)
\(198\) 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 \(-0.842272\pi\)
0.879723 0.475486i \(-0.157728\pi\)
\(224\) −8.15644 −0.544975
\(225\) 0 0
\(226\) 0.728857i 0.0484829i
\(227\) 16.0073i 1.06244i 0.847233 + 0.531221i \(0.178267\pi\)
−0.847233 + 0.531221i \(0.821733\pi\)
\(228\) 0 0
\(229\) 1.84117i 0.121668i 0.998148 + 0.0608339i \(0.0193760\pi\)
−0.998148 + 0.0608339i \(0.980624\pi\)
\(230\) 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 \(0.156556\pi\)
−0.881467 + 0.472246i \(0.843444\pi\)
\(240\) 0 0
\(241\) 8.63102i 0.555973i 0.960585 + 0.277987i \(0.0896671\pi\)
−0.960585 + 0.277987i \(0.910333\pi\)
\(242\) − 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 \(0.323398\pi\)
−0.526783 + 0.850000i \(0.676602\pi\)
\(272\) −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 \(-0.698189\pi\)
0.583172 0.812349i \(-0.301811\pi\)
\(282\) 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 \(-0.596836\pi\)
0.299548 0.954081i \(-0.403164\pi\)
\(294\) 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 \(-0.798158\pi\)
0.805602 0.592457i \(-0.201842\pi\)
\(308\) 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 \(0.143129\pi\)
−0.900598 + 0.434653i \(0.856871\pi\)
\(318\) 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 \(-0.946666\pi\)
0.985996 0.166772i \(-0.0533343\pi\)
\(332\) − 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 \(-0.797527\pi\)
0.804426 0.594053i \(-0.202473\pi\)
\(350\) −3.85325 −0.205965
\(351\) 0 0
\(352\) 26.3013 1.40186
\(353\) − 5.07069i − 0.269885i −0.990853 0.134943i \(-0.956915\pi\)
0.990853 0.134943i \(-0.0430850\pi\)
\(354\) 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 \(0.145030\pi\)
−0.897986 + 0.440025i \(0.854970\pi\)
\(360\) 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 \(0.163797\pi\)
−0.870498 + 0.492172i \(0.836203\pi\)
\(380\) −2.48427 −0.127440
\(381\) 0 0
\(382\) 0.579121i 0.0296304i
\(383\) − 15.3884i − 0.786308i −0.919473 0.393154i \(-0.871384\pi\)
0.919473 0.393154i \(-0.128616\pi\)
\(384\) 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 \(0.266516\pi\)
−0.669482 + 0.742828i \(0.733484\pi\)
\(398\) 11.0248i 0.552621i
\(399\) 0 0
\(400\) 14.0804 0.704019
\(401\) 21.1032i 1.05384i 0.849914 + 0.526922i \(0.176654\pi\)
−0.849914 + 0.526922i \(0.823346\pi\)
\(402\) 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 \(0.349712\pi\)
−0.454796 + 0.890596i \(0.650288\pi\)
\(410\) 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 \(0.0162609\pi\)
−0.998695 + 0.0510628i \(0.983739\pi\)
\(422\) − 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 \(-0.977859\pi\)
0.997582 0.0695014i \(-0.0221408\pi\)
\(432\) 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 \(0.238042\pi\)
−0.733166 + 0.680050i \(0.761958\pi\)
\(450\) 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 \(0.138902\pi\)
−0.906290 + 0.422656i \(0.861098\pi\)
\(458\) −0.819396 −0.0382879
\(459\) 0 0
\(460\) − 3.71379i − 0.173156i
\(461\) − 7.56763i − 0.352460i −0.984349 0.176230i \(-0.943610\pi\)
0.984349 0.176230i \(-0.0563902\pi\)
\(462\) 0 0
\(463\) 35.3551i 1.64309i 0.570143 + 0.821545i \(0.306888\pi\)
−0.570143 + 0.821545i \(0.693112\pi\)
\(464\) 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 \(0.198187\pi\)
−0.812352 + 0.583167i \(0.801813\pi\)
\(480\) 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 \(-0.881862\pi\)
0.931914 0.362680i \(-0.118138\pi\)
\(488\) − 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 \(0.0577975\pi\)
−0.983560 + 0.180580i \(0.942203\pi\)
\(500\) − 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 \(0.115754\pi\)
−0.934604 + 0.355691i \(0.884246\pi\)
\(510\) 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 \(-0.731656\pi\)
0.665205 0.746660i \(-0.268344\pi\)
\(542\) −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 \(-0.824114\pi\)
0.851182 0.524871i \(-0.175886\pi\)
\(558\) 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 \(-0.976324\pi\)
0.997235 0.0743110i \(-0.0236758\pi\)
\(578\) − 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 \(-0.923982\pi\)
0.971618 0.236554i \(-0.0760180\pi\)
\(588\) 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 \(0.148118\pi\)
−0.893675 + 0.448715i \(0.851882\pi\)
\(594\) 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 \(-0.933734\pi\)
0.978409 0.206680i \(-0.0662659\pi\)
\(614\) 9.23968 0.372883
\(615\) 0 0
\(616\) 16.7670i 0.675563i
\(617\) 26.2828i 1.05810i 0.848590 + 0.529052i \(0.177452\pi\)
−0.848590 + 0.529052i \(0.822548\pi\)
\(618\) 0 0
\(619\) 29.0834i 1.16896i 0.811408 + 0.584479i \(0.198701\pi\)
−0.811408 + 0.584479i \(0.801299\pi\)
\(620\) −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 \(0.169074\pi\)
−0.862219 + 0.506535i \(0.830926\pi\)
\(632\) 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 \(0.225204\pi\)
−0.759990 + 0.649935i \(0.774796\pi\)
\(644\) 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 \(-0.932044\pi\)
0.977298 0.211871i \(-0.0679558\pi\)
\(662\) 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 \(-0.864186\pi\)
0.910348 0.413844i \(-0.135814\pi\)
\(684\) 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 \(-0.984070\pi\)
0.998748 0.0500242i \(-0.0159298\pi\)
\(692\) 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 \(0.143545\pi\)
−0.900030 + 0.435829i \(0.856455\pi\)
\(710\) − 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 \(-0.886663\pi\)
0.937278 0.348582i \(-0.113337\pi\)
\(734\) − 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 \(0.336152\pi\)
−0.492312 + 0.870419i \(0.663848\pi\)
\(740\) 1.42327 0.0523205
\(741\) 0 0
\(742\) −0.827757 −0.0303879
\(743\) − 8.88769i − 0.326058i −0.986621 0.163029i \(-0.947874\pi\)
0.986621 0.163029i \(-0.0521264\pi\)
\(744\) 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 \(0.111178\pi\)
−0.939621 + 0.342218i \(0.888822\pi\)
\(762\) 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 \(0.0719790\pi\)
−0.974542 + 0.224207i \(0.928021\pi\)
\(770\) 1.08921 0.0392524
\(771\) 0 0
\(772\) − 16.5700i − 0.596368i
\(773\) 45.6746i 1.64280i 0.570353 + 0.821400i \(0.306807\pi\)
−0.570353 + 0.821400i \(0.693193\pi\)
\(774\) 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 \(-0.974336\pi\)
0.996752 0.0805374i \(-0.0256636\pi\)
\(788\) 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 \(-0.996484\pi\)
0.999939 0.0110454i \(-0.00351593\pi\)
\(812\) 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 \(0.218099\pi\)
−0.774307 + 0.632811i \(0.781901\pi\)
\(822\) 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 \(-0.768786\pi\)
0.747584 0.664167i \(-0.231214\pi\)
\(828\) 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 \(0.0255158\pi\)
−0.996789 + 0.0800744i \(0.974484\pi\)
\(840\) 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 \(0.101938\pi\)
−0.949157 + 0.314803i \(0.898062\pi\)
\(854\) 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 \(-0.738139\pi\)
0.680274 0.732958i \(-0.261861\pi\)
\(864\) 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 \(0.170128\pi\)
−0.860537 + 0.509387i \(0.829872\pi\)
\(878\) 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 \(-0.687166\pi\)
0.554697 0.832052i \(-0.312834\pi\)
\(930\) 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 \(-0.808624\pi\)
0.824643 0.565653i \(-0.191376\pi\)
\(942\) 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 \(0.0678112\pi\)
−0.977394 + 0.211428i \(0.932189\pi\)
\(948\) 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 \(-0.799315\pi\)
0.807751 0.589524i \(-0.200685\pi\)
\(968\) − 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 \(0.152973\pi\)
−0.886727 + 0.462293i \(0.847027\pi\)
\(978\) 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 \(0.100117\pi\)
−0.950942 + 0.309368i \(0.899883\pi\)
\(984\) 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.1351.4 6
3.2 odd 2 507.2.b.g.337.3 6
13.5 odd 4 1521.2.a.q.1.2 3
13.8 odd 4 1521.2.a.p.1.2 3
13.12 even 2 inner 1521.2.b.m.1351.3 6
39.2 even 12 507.2.e.k.22.2 6
39.5 even 4 507.2.a.j.1.2 3
39.8 even 4 507.2.a.k.1.2 yes 3
39.11 even 12 507.2.e.j.22.2 6
39.17 odd 6 507.2.j.h.361.4 12
39.20 even 12 507.2.e.j.484.2 6
39.23 odd 6 507.2.j.h.316.3 12
39.29 odd 6 507.2.j.h.316.4 12
39.32 even 12 507.2.e.k.484.2 6
39.35 odd 6 507.2.j.h.361.3 12
39.38 odd 2 507.2.b.g.337.4 6
156.47 odd 4 8112.2.a.cf.1.1 3
156.83 odd 4 8112.2.a.by.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
507.2.a.j.1.2 3 39.5 even 4
507.2.a.k.1.2 yes 3 39.8 even 4
507.2.b.g.337.3 6 3.2 odd 2
507.2.b.g.337.4 6 39.38 odd 2
507.2.e.j.22.2 6 39.11 even 12
507.2.e.j.484.2 6 39.20 even 12
507.2.e.k.22.2 6 39.2 even 12
507.2.e.k.484.2 6 39.32 even 12
507.2.j.h.316.3 12 39.23 odd 6
507.2.j.h.316.4 12 39.29 odd 6
507.2.j.h.361.3 12 39.35 odd 6
507.2.j.h.361.4 12 39.17 odd 6
1521.2.a.p.1.2 3 13.8 odd 4
1521.2.a.q.1.2 3 13.5 odd 4
1521.2.b.m.1351.3 6 13.12 even 2 inner
1521.2.b.m.1351.4 6 1.1 even 1 trivial
8112.2.a.by.1.3 3 156.83 odd 4
8112.2.a.cf.1.1 3 156.47 odd 4