Properties

Label 6048.2.a.bq.1.2
Level 6048
Weight 2
Character 6048.1
Self dual yes
Analytic conductor 48.294
Analytic rank 1
Dimension 4
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 6048 = 2^{5} \cdot 3^{3} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 6048.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(48.2935231425\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.25808.1
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-2.46810\)
Character \(\chi\) = 6048.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.34410 q^{5} -1.00000 q^{7} +O(q^{10})\) \(q-1.34410 q^{5} -1.00000 q^{7} +6.28030 q^{11} -4.43102 q^{13} -0.505178 q^{17} +6.19339 q^{19} -5.84928 q^{23} -3.19339 q^{25} -4.43102 q^{29} +6.62441 q^{31} +1.34410 q^{35} -8.44138 q^{37} +5.77512 q^{41} -7.44138 q^{43} -12.1296 q^{47} +1.00000 q^{49} +13.8724 q^{53} -8.44138 q^{55} -6.25719 q^{59} +1.01036 q^{61} +5.95575 q^{65} +14.8178 q^{67} +2.35446 q^{71} +11.8074 q^{73} -6.28030 q^{77} -12.3034 q^{79} -8.43102 q^{83} +0.679012 q^{85} +4.91308 q^{89} +4.43102 q^{91} -8.32455 q^{95} -5.37641 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 2q^{5} - 4q^{7} + O(q^{10}) \) \( 4q + 2q^{5} - 4q^{7} - 2q^{11} - 4q^{13} + 4q^{17} + 4q^{19} - 10q^{23} + 8q^{25} - 4q^{29} - 8q^{31} - 2q^{35} - 8q^{37} + 2q^{41} - 4q^{43} - 8q^{47} + 4q^{49} + 16q^{53} - 8q^{55} - 24q^{59} - 8q^{61} - 4q^{65} + 4q^{67} - 10q^{71} + 4q^{73} + 2q^{77} + 4q^{79} - 20q^{83} - 16q^{85} + 26q^{89} + 4q^{91} - 34q^{95} + 8q^{97} + O(q^{100}) \)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) −1.34410 −0.601101 −0.300551 0.953766i \(-0.597170\pi\)
−0.300551 + 0.953766i \(0.597170\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 6.28030 1.89358 0.946791 0.321848i \(-0.104304\pi\)
0.946791 + 0.321848i \(0.104304\pi\)
\(12\) 0 0
\(13\) −4.43102 −1.22894 −0.614472 0.788939i \(-0.710631\pi\)
−0.614472 + 0.788939i \(0.710631\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −0.505178 −0.122524 −0.0612619 0.998122i \(-0.519512\pi\)
−0.0612619 + 0.998122i \(0.519512\pi\)
\(18\) 0 0
\(19\) 6.19339 1.42086 0.710430 0.703768i \(-0.248500\pi\)
0.710430 + 0.703768i \(0.248500\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −5.84928 −1.21966 −0.609830 0.792532i \(-0.708762\pi\)
−0.609830 + 0.792532i \(0.708762\pi\)
\(24\) 0 0
\(25\) −3.19339 −0.638677
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −4.43102 −0.822820 −0.411410 0.911450i \(-0.634964\pi\)
−0.411410 + 0.911450i \(0.634964\pi\)
\(30\) 0 0
\(31\) 6.62441 1.18978 0.594889 0.803808i \(-0.297196\pi\)
0.594889 + 0.803808i \(0.297196\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 1.34410 0.227195
\(36\) 0 0
\(37\) −8.44138 −1.38775 −0.693877 0.720094i \(-0.744099\pi\)
−0.693877 + 0.720094i \(0.744099\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 5.77512 0.901923 0.450961 0.892543i \(-0.351081\pi\)
0.450961 + 0.892543i \(0.351081\pi\)
\(42\) 0 0
\(43\) −7.44138 −1.13480 −0.567400 0.823443i \(-0.692051\pi\)
−0.567400 + 0.823443i \(0.692051\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −12.1296 −1.76928 −0.884641 0.466273i \(-0.845596\pi\)
−0.884641 + 0.466273i \(0.845596\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 13.8724 1.90552 0.952760 0.303724i \(-0.0982300\pi\)
0.952760 + 0.303724i \(0.0982300\pi\)
\(54\) 0 0
\(55\) −8.44138 −1.13824
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −6.25719 −0.814616 −0.407308 0.913291i \(-0.633532\pi\)
−0.407308 + 0.913291i \(0.633532\pi\)
\(60\) 0 0
\(61\) 1.01036 0.129363 0.0646815 0.997906i \(-0.479397\pi\)
0.0646815 + 0.997906i \(0.479397\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 5.95575 0.738720
\(66\) 0 0
\(67\) 14.8178 1.81028 0.905141 0.425112i \(-0.139765\pi\)
0.905141 + 0.425112i \(0.139765\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 2.35446 0.279423 0.139712 0.990192i \(-0.455382\pi\)
0.139712 + 0.990192i \(0.455382\pi\)
\(72\) 0 0
\(73\) 11.8074 1.38196 0.690978 0.722876i \(-0.257180\pi\)
0.690978 + 0.722876i \(0.257180\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −6.28030 −0.715707
\(78\) 0 0
\(79\) −12.3034 −1.38424 −0.692121 0.721781i \(-0.743324\pi\)
−0.692121 + 0.721781i \(0.743324\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −8.43102 −0.925425 −0.462712 0.886508i \(-0.653124\pi\)
−0.462712 + 0.886508i \(0.653124\pi\)
\(84\) 0 0
\(85\) 0.679012 0.0736492
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 4.91308 0.520786 0.260393 0.965503i \(-0.416148\pi\)
0.260393 + 0.965503i \(0.416148\pi\)
\(90\) 0 0
\(91\) 4.43102 0.464497
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −8.32455 −0.854081
\(96\) 0 0
\(97\) −5.37641 −0.545892 −0.272946 0.962029i \(-0.587998\pi\)
−0.272946 + 0.962029i \(0.587998\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −9.88159 −0.983255 −0.491628 0.870806i \(-0.663598\pi\)
−0.491628 + 0.870806i \(0.663598\pi\)
\(102\) 0 0
\(103\) 15.4864 1.52592 0.762962 0.646443i \(-0.223744\pi\)
0.762962 + 0.646443i \(0.223744\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −4.80661 −0.464673 −0.232336 0.972635i \(-0.574637\pi\)
−0.232336 + 0.972635i \(0.574637\pi\)
\(108\) 0 0
\(109\) 6.62441 0.634503 0.317252 0.948341i \(-0.397240\pi\)
0.317252 + 0.948341i \(0.397240\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −9.31179 −0.875980 −0.437990 0.898980i \(-0.644309\pi\)
−0.437990 + 0.898980i \(0.644309\pi\)
\(114\) 0 0
\(115\) 7.86204 0.733139
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0.505178 0.0463096
\(120\) 0 0
\(121\) 28.4422 2.58565
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 11.0128 0.985011
\(126\) 0 0
\(127\) 5.44138 0.482844 0.241422 0.970420i \(-0.422386\pi\)
0.241422 + 0.970420i \(0.422386\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −15.1192 −1.32097 −0.660487 0.750838i \(-0.729650\pi\)
−0.660487 + 0.750838i \(0.729650\pi\)
\(132\) 0 0
\(133\) −6.19339 −0.537035
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −19.8281 −1.69403 −0.847017 0.531567i \(-0.821604\pi\)
−0.847017 + 0.531567i \(0.821604\pi\)
\(138\) 0 0
\(139\) −19.2488 −1.63266 −0.816331 0.577584i \(-0.803996\pi\)
−0.816331 + 0.577584i \(0.803996\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −27.8281 −2.32711
\(144\) 0 0
\(145\) 5.95575 0.494598
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −15.5502 −1.27393 −0.636963 0.770894i \(-0.719810\pi\)
−0.636963 + 0.770894i \(0.719810\pi\)
\(150\) 0 0
\(151\) 7.01036 0.570495 0.285247 0.958454i \(-0.407924\pi\)
0.285247 + 0.958454i \(0.407924\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −8.90389 −0.715177
\(156\) 0 0
\(157\) −4.06496 −0.324419 −0.162210 0.986756i \(-0.551862\pi\)
−0.162210 + 0.986756i \(0.551862\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 5.84928 0.460988
\(162\) 0 0
\(163\) −20.6902 −1.62058 −0.810290 0.586029i \(-0.800691\pi\)
−0.810290 + 0.586029i \(0.800691\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 9.93702 0.768950 0.384475 0.923135i \(-0.374383\pi\)
0.384475 + 0.923135i \(0.374383\pi\)
\(168\) 0 0
\(169\) 6.63394 0.510303
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −5.84928 −0.444713 −0.222356 0.974965i \(-0.571375\pi\)
−0.222356 + 0.974965i \(0.571375\pi\)
\(174\) 0 0
\(175\) 3.19339 0.241397
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −12.2037 −0.912151 −0.456075 0.889941i \(-0.650745\pi\)
−0.456075 + 0.889941i \(0.650745\pi\)
\(180\) 0 0
\(181\) 23.1212 1.71859 0.859293 0.511484i \(-0.170904\pi\)
0.859293 + 0.511484i \(0.170904\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 11.3461 0.834181
\(186\) 0 0
\(187\) −3.17267 −0.232009
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 1.33491 0.0965906 0.0482953 0.998833i \(-0.484621\pi\)
0.0482953 + 0.998833i \(0.484621\pi\)
\(192\) 0 0
\(193\) −9.24881 −0.665744 −0.332872 0.942972i \(-0.608018\pi\)
−0.332872 + 0.942972i \(0.608018\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −11.5502 −0.822921 −0.411461 0.911428i \(-0.634981\pi\)
−0.411461 + 0.911428i \(0.634981\pi\)
\(198\) 0 0
\(199\) −8.42066 −0.596925 −0.298463 0.954421i \(-0.596474\pi\)
−0.298463 + 0.954421i \(0.596474\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 4.43102 0.310997
\(204\) 0 0
\(205\) −7.76237 −0.542147
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 38.8963 2.69052
\(210\) 0 0
\(211\) −16.2385 −1.11790 −0.558951 0.829201i \(-0.688796\pi\)
−0.558951 + 0.829201i \(0.688796\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 10.0020 0.682129
\(216\) 0 0
\(217\) −6.62441 −0.449694
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 2.23846 0.150575
\(222\) 0 0
\(223\) 10.4414 0.699206 0.349603 0.936898i \(-0.386316\pi\)
0.349603 + 0.936898i \(0.386316\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −15.5502 −1.03211 −0.516053 0.856557i \(-0.672599\pi\)
−0.516053 + 0.856557i \(0.672599\pi\)
\(228\) 0 0
\(229\) −17.0104 −1.12408 −0.562038 0.827111i \(-0.689983\pi\)
−0.562038 + 0.827111i \(0.689983\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −14.9474 −0.979235 −0.489618 0.871937i \(-0.662864\pi\)
−0.489618 + 0.871937i \(0.662864\pi\)
\(234\) 0 0
\(235\) 16.3034 1.06352
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −6.33134 −0.409541 −0.204770 0.978810i \(-0.565645\pi\)
−0.204770 + 0.978810i \(0.565645\pi\)
\(240\) 0 0
\(241\) −2.06496 −0.133016 −0.0665080 0.997786i \(-0.521186\pi\)
−0.0665080 + 0.997786i \(0.521186\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −1.34410 −0.0858716
\(246\) 0 0
\(247\) −27.4430 −1.74616
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −19.7632 −1.24744 −0.623721 0.781647i \(-0.714380\pi\)
−0.623721 + 0.781647i \(0.714380\pi\)
\(252\) 0 0
\(253\) −36.7353 −2.30953
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −15.1423 −0.944553 −0.472277 0.881450i \(-0.656568\pi\)
−0.472277 + 0.881450i \(0.656568\pi\)
\(258\) 0 0
\(259\) 8.44138 0.524522
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 4.88516 0.301232 0.150616 0.988592i \(-0.451874\pi\)
0.150616 + 0.988592i \(0.451874\pi\)
\(264\) 0 0
\(265\) −18.6459 −1.14541
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 10.3637 0.631883 0.315942 0.948779i \(-0.397680\pi\)
0.315942 + 0.948779i \(0.397680\pi\)
\(270\) 0 0
\(271\) −5.52473 −0.335603 −0.167802 0.985821i \(-0.553667\pi\)
−0.167802 + 0.985821i \(0.553667\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −20.0554 −1.20939
\(276\) 0 0
\(277\) −27.1942 −1.63394 −0.816971 0.576679i \(-0.804348\pi\)
−0.816971 + 0.576679i \(0.804348\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −19.1192 −1.14056 −0.570279 0.821451i \(-0.693165\pi\)
−0.570279 + 0.821451i \(0.693165\pi\)
\(282\) 0 0
\(283\) 12.8620 0.764569 0.382284 0.924045i \(-0.375137\pi\)
0.382284 + 0.924045i \(0.375137\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −5.77512 −0.340895
\(288\) 0 0
\(289\) −16.7448 −0.984988
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −25.1304 −1.46813 −0.734067 0.679077i \(-0.762380\pi\)
−0.734067 + 0.679077i \(0.762380\pi\)
\(294\) 0 0
\(295\) 8.41031 0.489667
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 25.9183 1.49889
\(300\) 0 0
\(301\) 7.44138 0.428914
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −1.35802 −0.0777602
\(306\) 0 0
\(307\) 3.80661 0.217255 0.108627 0.994083i \(-0.465354\pi\)
0.108627 + 0.994083i \(0.465354\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 15.1838 0.860997 0.430499 0.902591i \(-0.358338\pi\)
0.430499 + 0.902591i \(0.358338\pi\)
\(312\) 0 0
\(313\) 24.4310 1.38092 0.690461 0.723369i \(-0.257408\pi\)
0.690461 + 0.723369i \(0.257408\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 34.3680 1.93030 0.965151 0.261695i \(-0.0842814\pi\)
0.965151 + 0.261695i \(0.0842814\pi\)
\(318\) 0 0
\(319\) −27.8281 −1.55808
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −3.12876 −0.174089
\(324\) 0 0
\(325\) 14.1500 0.784898
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 12.1296 0.668726
\(330\) 0 0
\(331\) −0.751188 −0.0412890 −0.0206445 0.999787i \(-0.506572\pi\)
−0.0206445 + 0.999787i \(0.506572\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −19.9166 −1.08816
\(336\) 0 0
\(337\) −5.86204 −0.319326 −0.159663 0.987172i \(-0.551041\pi\)
−0.159663 + 0.987172i \(0.551041\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 41.6033 2.25294
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −36.8756 −1.97959 −0.989794 0.142509i \(-0.954483\pi\)
−0.989794 + 0.142509i \(0.954483\pi\)
\(348\) 0 0
\(349\) −6.86204 −0.367317 −0.183658 0.982990i \(-0.558794\pi\)
−0.183658 + 0.982990i \(0.558794\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −13.9234 −0.741070 −0.370535 0.928819i \(-0.620826\pi\)
−0.370535 + 0.928819i \(0.620826\pi\)
\(354\) 0 0
\(355\) −3.16464 −0.167962
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −25.0658 −1.32292 −0.661461 0.749980i \(-0.730063\pi\)
−0.661461 + 0.749980i \(0.730063\pi\)
\(360\) 0 0
\(361\) 19.3580 1.01884
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −15.8704 −0.830696
\(366\) 0 0
\(367\) 2.05461 0.107250 0.0536248 0.998561i \(-0.482923\pi\)
0.0536248 + 0.998561i \(0.482923\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −13.8724 −0.720219
\(372\) 0 0
\(373\) 5.01118 0.259469 0.129734 0.991549i \(-0.458588\pi\)
0.129734 + 0.991549i \(0.458588\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 19.6339 1.01120
\(378\) 0 0
\(379\) 2.57934 0.132492 0.0662458 0.997803i \(-0.478898\pi\)
0.0662458 + 0.997803i \(0.478898\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −10.1738 −0.519859 −0.259929 0.965628i \(-0.583699\pi\)
−0.259929 + 0.965628i \(0.583699\pi\)
\(384\) 0 0
\(385\) 8.44138 0.430212
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 22.1316 1.12211 0.561057 0.827777i \(-0.310395\pi\)
0.561057 + 0.827777i \(0.310395\pi\)
\(390\) 0 0
\(391\) 2.95493 0.149437
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 16.5371 0.832070
\(396\) 0 0
\(397\) −14.4075 −0.723091 −0.361545 0.932354i \(-0.617751\pi\)
−0.361545 + 0.932354i \(0.617751\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −6.96611 −0.347871 −0.173935 0.984757i \(-0.555648\pi\)
−0.173935 + 0.984757i \(0.555648\pi\)
\(402\) 0 0
\(403\) −29.3529 −1.46217
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −53.0144 −2.62783
\(408\) 0 0
\(409\) 34.3425 1.69813 0.849064 0.528290i \(-0.177166\pi\)
0.849064 + 0.528290i \(0.177166\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 6.25719 0.307896
\(414\) 0 0
\(415\) 11.3322 0.556274
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 6.66982 0.325842 0.162921 0.986639i \(-0.447908\pi\)
0.162921 + 0.986639i \(0.447908\pi\)
\(420\) 0 0
\(421\) −19.9454 −0.972079 −0.486040 0.873937i \(-0.661559\pi\)
−0.486040 + 0.873937i \(0.661559\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 1.61323 0.0782531
\(426\) 0 0
\(427\) −1.01036 −0.0488946
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 5.84928 0.281750 0.140875 0.990027i \(-0.455008\pi\)
0.140875 + 0.990027i \(0.455008\pi\)
\(432\) 0 0
\(433\) 34.5626 1.66097 0.830486 0.557040i \(-0.188063\pi\)
0.830486 + 0.557040i \(0.188063\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −36.2269 −1.73297
\(438\) 0 0
\(439\) 27.7448 1.32419 0.662093 0.749421i \(-0.269668\pi\)
0.662093 + 0.749421i \(0.269668\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 4.47253 0.212496 0.106248 0.994340i \(-0.466116\pi\)
0.106248 + 0.994340i \(0.466116\pi\)
\(444\) 0 0
\(445\) −6.60369 −0.313045
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 7.03587 0.332043 0.166022 0.986122i \(-0.446908\pi\)
0.166022 + 0.986122i \(0.446908\pi\)
\(450\) 0 0
\(451\) 36.2695 1.70787
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −5.95575 −0.279210
\(456\) 0 0
\(457\) 23.7528 1.11111 0.555555 0.831480i \(-0.312506\pi\)
0.555555 + 0.831480i \(0.312506\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −12.0693 −0.562126 −0.281063 0.959689i \(-0.590687\pi\)
−0.281063 + 0.959689i \(0.590687\pi\)
\(462\) 0 0
\(463\) −2.86204 −0.133010 −0.0665052 0.997786i \(-0.521185\pi\)
−0.0665052 + 0.997786i \(0.521185\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −28.4773 −1.31777 −0.658885 0.752244i \(-0.728972\pi\)
−0.658885 + 0.752244i \(0.728972\pi\)
\(468\) 0 0
\(469\) −14.8178 −0.684222
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −46.7341 −2.14884
\(474\) 0 0
\(475\) −19.7779 −0.907471
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 23.3971 1.06904 0.534521 0.845155i \(-0.320492\pi\)
0.534521 + 0.845155i \(0.320492\pi\)
\(480\) 0 0
\(481\) 37.4039 1.70547
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 7.22646 0.328137
\(486\) 0 0
\(487\) −27.2931 −1.23677 −0.618383 0.785877i \(-0.712212\pi\)
−0.618383 + 0.785877i \(0.712212\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 11.9490 0.539249 0.269624 0.962966i \(-0.413100\pi\)
0.269624 + 0.962966i \(0.413100\pi\)
\(492\) 0 0
\(493\) 2.23846 0.100815
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −2.35446 −0.105612
\(498\) 0 0
\(499\) −11.4598 −0.513010 −0.256505 0.966543i \(-0.582571\pi\)
−0.256505 + 0.966543i \(0.582571\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −0.0649626 −0.00289654 −0.00144827 0.999999i \(-0.500461\pi\)
−0.00144827 + 0.999999i \(0.500461\pi\)
\(504\) 0 0
\(505\) 13.2819 0.591036
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 24.1200 1.06910 0.534551 0.845136i \(-0.320481\pi\)
0.534551 + 0.845136i \(0.320481\pi\)
\(510\) 0 0
\(511\) −11.8074 −0.522330
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −20.8154 −0.917236
\(516\) 0 0
\(517\) −76.1775 −3.35028
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 34.1734 1.49716 0.748582 0.663042i \(-0.230735\pi\)
0.748582 + 0.663042i \(0.230735\pi\)
\(522\) 0 0
\(523\) −14.2695 −0.623963 −0.311981 0.950088i \(-0.600993\pi\)
−0.311981 + 0.950088i \(0.600993\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −3.34651 −0.145776
\(528\) 0 0
\(529\) 11.2141 0.487570
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −25.5897 −1.10841
\(534\) 0 0
\(535\) 6.46059 0.279316
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 6.28030 0.270512
\(540\) 0 0
\(541\) 8.82815 0.379552 0.189776 0.981827i \(-0.439224\pi\)
0.189776 + 0.981827i \(0.439224\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −8.90389 −0.381401
\(546\) 0 0
\(547\) 13.6356 0.583015 0.291508 0.956568i \(-0.405843\pi\)
0.291508 + 0.956568i \(0.405843\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −27.4430 −1.16911
\(552\) 0 0
\(553\) 12.3034 0.523195
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 5.31179 0.225068 0.112534 0.993648i \(-0.464103\pi\)
0.112534 + 0.993648i \(0.464103\pi\)
\(558\) 0 0
\(559\) 32.9729 1.39460
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 3.91665 0.165067 0.0825335 0.996588i \(-0.473699\pi\)
0.0825335 + 0.996588i \(0.473699\pi\)
\(564\) 0 0
\(565\) 12.5160 0.526553
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −7.91631 −0.331869 −0.165934 0.986137i \(-0.553064\pi\)
−0.165934 + 0.986137i \(0.553064\pi\)
\(570\) 0 0
\(571\) 14.5144 0.607408 0.303704 0.952766i \(-0.401777\pi\)
0.303704 + 0.952766i \(0.401777\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 18.6790 0.778969
\(576\) 0 0
\(577\) 0.475270 0.0197857 0.00989287 0.999951i \(-0.496851\pi\)
0.00989287 + 0.999951i \(0.496851\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 8.43102 0.349778
\(582\) 0 0
\(583\) 87.1229 3.60826
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −20.0646 −0.828156 −0.414078 0.910242i \(-0.635896\pi\)
−0.414078 + 0.910242i \(0.635896\pi\)
\(588\) 0 0
\(589\) 41.0275 1.69051
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 37.2699 1.53049 0.765247 0.643737i \(-0.222617\pi\)
0.765247 + 0.643737i \(0.222617\pi\)
\(594\) 0 0
\(595\) −0.679012 −0.0278368
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −22.9059 −0.935908 −0.467954 0.883753i \(-0.655009\pi\)
−0.467954 + 0.883753i \(0.655009\pi\)
\(600\) 0 0
\(601\) 6.55862 0.267532 0.133766 0.991013i \(-0.457293\pi\)
0.133766 + 0.991013i \(0.457293\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −38.2293 −1.55424
\(606\) 0 0
\(607\) −24.4960 −0.994261 −0.497131 0.867676i \(-0.665613\pi\)
−0.497131 + 0.867676i \(0.665613\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 53.7464 2.17435
\(612\) 0 0
\(613\) −20.6037 −0.832175 −0.416088 0.909325i \(-0.636599\pi\)
−0.416088 + 0.909325i \(0.636599\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −26.2572 −1.05707 −0.528537 0.848910i \(-0.677259\pi\)
−0.528537 + 0.848910i \(0.677259\pi\)
\(618\) 0 0
\(619\) −27.9729 −1.12433 −0.562163 0.827027i \(-0.690031\pi\)
−0.562163 + 0.827027i \(0.690031\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −4.91308 −0.196839
\(624\) 0 0
\(625\) 1.16464 0.0465855
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 4.26440 0.170033
\(630\) 0 0
\(631\) −13.5690 −0.540173 −0.270086 0.962836i \(-0.587052\pi\)
−0.270086 + 0.962836i \(0.587052\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −7.31377 −0.290238
\(636\) 0 0
\(637\) −4.43102 −0.175563
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −6.53989 −0.258310 −0.129155 0.991624i \(-0.541227\pi\)
−0.129155 + 0.991624i \(0.541227\pi\)
\(642\) 0 0
\(643\) 25.2755 0.996768 0.498384 0.866956i \(-0.333927\pi\)
0.498384 + 0.866956i \(0.333927\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −12.3664 −0.486173 −0.243087 0.970005i \(-0.578160\pi\)
−0.243087 + 0.970005i \(0.578160\pi\)
\(648\) 0 0
\(649\) −39.2970 −1.54254
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 40.4996 1.58487 0.792436 0.609955i \(-0.208813\pi\)
0.792436 + 0.609955i \(0.208813\pi\)
\(654\) 0 0
\(655\) 20.3218 0.794039
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 23.4015 0.911593 0.455797 0.890084i \(-0.349354\pi\)
0.455797 + 0.890084i \(0.349354\pi\)
\(660\) 0 0
\(661\) −10.1926 −0.396445 −0.198222 0.980157i \(-0.563517\pi\)
−0.198222 + 0.980157i \(0.563517\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 8.32455 0.322812
\(666\) 0 0
\(667\) 25.9183 1.00356
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 6.34534 0.244959
\(672\) 0 0
\(673\) 15.2281 0.587000 0.293500 0.955959i \(-0.405180\pi\)
0.293500 + 0.955959i \(0.405180\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 38.0993 1.46427 0.732137 0.681158i \(-0.238523\pi\)
0.732137 + 0.681158i \(0.238523\pi\)
\(678\) 0 0
\(679\) 5.37641 0.206328
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 35.8329 1.37111 0.685553 0.728022i \(-0.259560\pi\)
0.685553 + 0.728022i \(0.259560\pi\)
\(684\) 0 0
\(685\) 26.6511 1.01829
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −61.4689 −2.34178
\(690\) 0 0
\(691\) −48.8844 −1.85965 −0.929825 0.368002i \(-0.880042\pi\)
−0.929825 + 0.368002i \(0.880042\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 25.8724 0.981396
\(696\) 0 0
\(697\) −2.91747 −0.110507
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −14.0857 −0.532009 −0.266004 0.963972i \(-0.585704\pi\)
−0.266004 + 0.963972i \(0.585704\pi\)
\(702\) 0 0
\(703\) −52.2807 −1.97180
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 9.88159 0.371636
\(708\) 0 0
\(709\) −3.85086 −0.144622 −0.0723111 0.997382i \(-0.523037\pi\)
−0.0723111 + 0.997382i \(0.523037\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −38.7480 −1.45112
\(714\) 0 0
\(715\) 37.4039 1.39883
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −0.792273 −0.0295468 −0.0147734 0.999891i \(-0.504703\pi\)
−0.0147734 + 0.999891i \(0.504703\pi\)
\(720\) 0 0
\(721\) −15.4864 −0.576745
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 14.1500 0.525516
\(726\) 0 0
\(727\) 5.11725 0.189788 0.0948941 0.995487i \(-0.469749\pi\)
0.0948941 + 0.995487i \(0.469749\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 3.75922 0.139040
\(732\) 0 0
\(733\) 1.71730 0.0634297 0.0317149 0.999497i \(-0.489903\pi\)
0.0317149 + 0.999497i \(0.489903\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 93.0602 3.42792
\(738\) 0 0
\(739\) 16.1276 0.593263 0.296632 0.954992i \(-0.404137\pi\)
0.296632 + 0.954992i \(0.404137\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 3.10565 0.113935 0.0569676 0.998376i \(-0.481857\pi\)
0.0569676 + 0.998376i \(0.481857\pi\)
\(744\) 0 0
\(745\) 20.9011 0.765759
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 4.80661 0.175630
\(750\) 0 0
\(751\) 21.9581 0.801261 0.400631 0.916240i \(-0.368791\pi\)
0.400631 + 0.916240i \(0.368791\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −9.42265 −0.342925
\(756\) 0 0
\(757\) −53.3597 −1.93939 −0.969695 0.244319i \(-0.921436\pi\)
−0.969695 + 0.244319i \(0.921436\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 9.34883 0.338895 0.169447 0.985539i \(-0.445802\pi\)
0.169447 + 0.985539i \(0.445802\pi\)
\(762\) 0 0
\(763\) −6.62441 −0.239820
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 27.7257 1.00112
\(768\) 0 0
\(769\) 7.22131 0.260407 0.130204 0.991487i \(-0.458437\pi\)
0.130204 + 0.991487i \(0.458437\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 5.69177 0.204719 0.102359 0.994747i \(-0.467361\pi\)
0.102359 + 0.994747i \(0.467361\pi\)
\(774\) 0 0
\(775\) −21.1543 −0.759884
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 35.7676 1.28151
\(780\) 0 0
\(781\) 14.7867 0.529111
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 5.46373 0.195009
\(786\) 0 0
\(787\) −15.3787 −0.548193 −0.274096 0.961702i \(-0.588379\pi\)
−0.274096 + 0.961702i \(0.588379\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 9.31179 0.331089
\(792\) 0 0
\(793\) −4.47691 −0.158980
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 35.0211 1.24051 0.620256 0.784400i \(-0.287029\pi\)
0.620256 + 0.784400i \(0.287029\pi\)
\(798\) 0 0
\(799\) 6.12760 0.216779
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 74.1543 2.61685
\(804\) 0 0
\(805\) −7.86204 −0.277101
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 17.6591 0.620862 0.310431 0.950596i \(-0.399527\pi\)
0.310431 + 0.950596i \(0.399527\pi\)
\(810\) 0 0
\(811\) 4.91150 0.172466 0.0862331 0.996275i \(-0.472517\pi\)
0.0862331 + 0.996275i \(0.472517\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 27.8098 0.974133
\(816\) 0 0
\(817\) −46.0873 −1.61239
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −13.6359 −0.475897 −0.237949 0.971278i \(-0.576475\pi\)
−0.237949 + 0.971278i \(0.576475\pi\)
\(822\) 0 0
\(823\) 6.77354 0.236111 0.118055 0.993007i \(-0.462334\pi\)
0.118055 + 0.993007i \(0.462334\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −51.7161 −1.79835 −0.899173 0.437594i \(-0.855831\pi\)
−0.899173 + 0.437594i \(0.855831\pi\)
\(828\) 0 0
\(829\) −27.2046 −0.944854 −0.472427 0.881370i \(-0.656622\pi\)
−0.472427 + 0.881370i \(0.656622\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −0.505178 −0.0175034
\(834\) 0 0
\(835\) −13.3564 −0.462217
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −0.431020 −0.0148805 −0.00744024 0.999972i \(-0.502368\pi\)
−0.00744024 + 0.999972i \(0.502368\pi\)
\(840\) 0 0
\(841\) −9.36606 −0.322968
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −8.91671 −0.306744
\(846\) 0 0
\(847\) −28.4422 −0.977285
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 49.3760 1.69259
\(852\) 0 0
\(853\) −23.3971 −0.801102 −0.400551 0.916274i \(-0.631181\pi\)
−0.400551 + 0.916274i \(0.631181\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −3.23523 −0.110513 −0.0552567 0.998472i \(-0.517598\pi\)
−0.0552567 + 0.998472i \(0.517598\pi\)
\(858\) 0 0
\(859\) 54.3042 1.85284 0.926418 0.376496i \(-0.122871\pi\)
0.926418 + 0.376496i \(0.122871\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −45.5611 −1.55092 −0.775459 0.631398i \(-0.782481\pi\)
−0.775459 + 0.631398i \(0.782481\pi\)
\(864\) 0 0
\(865\) 7.86204 0.267317
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −77.2692 −2.62118
\(870\) 0 0
\(871\) −65.6579 −2.22473
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −11.0128 −0.372299
\(876\) 0 0
\(877\) 18.3661 0.620178 0.310089 0.950708i \(-0.399641\pi\)
0.310089 + 0.950708i \(0.399641\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −5.02394 −0.169261 −0.0846304 0.996412i \(-0.526971\pi\)
−0.0846304 + 0.996412i \(0.526971\pi\)
\(882\) 0 0
\(883\) −14.7344 −0.495853 −0.247927 0.968779i \(-0.579749\pi\)
−0.247927 + 0.968779i \(0.579749\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 11.4621 0.384859 0.192430 0.981311i \(-0.438363\pi\)
0.192430 + 0.981311i \(0.438363\pi\)
\(888\) 0 0
\(889\) −5.44138 −0.182498
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −75.1232 −2.51390
\(894\) 0 0
\(895\) 16.4031 0.548295
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −29.3529 −0.978973
\(900\) 0 0
\(901\) −7.00803 −0.233471
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −31.0773 −1.03304
\(906\) 0 0
\(907\) −21.9809 −0.729865 −0.364932 0.931034i \(-0.618908\pi\)
−0.364932 + 0.931034i \(0.618908\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 0.676689 0.0224197 0.0112099 0.999937i \(-0.496432\pi\)
0.0112099 + 0.999937i \(0.496432\pi\)
\(912\) 0 0
\(913\) −52.9494 −1.75237
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 15.1192 0.499281
\(918\) 0 0
\(919\) 22.0717 0.728080 0.364040 0.931383i \(-0.381397\pi\)
0.364040 + 0.931383i \(0.381397\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −10.4327 −0.343395
\(924\) 0 0
\(925\) 26.9566 0.886327
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −28.5091 −0.935355 −0.467677 0.883899i \(-0.654909\pi\)
−0.467677 + 0.883899i \(0.654909\pi\)
\(930\) 0 0
\(931\) 6.19339 0.202980
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 4.26440 0.139461
\(936\) 0 0
\(937\) 31.8046 1.03901 0.519506 0.854467i \(-0.326116\pi\)
0.519506 + 0.854467i \(0.326116\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −34.6320 −1.12897 −0.564486 0.825443i \(-0.690926\pi\)
−0.564486 + 0.825443i \(0.690926\pi\)
\(942\) 0 0
\(943\) −33.7803 −1.10004
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −33.9920 −1.10459 −0.552296 0.833648i \(-0.686248\pi\)
−0.552296 + 0.833648i \(0.686248\pi\)
\(948\) 0 0
\(949\) −52.3190 −1.69835
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −6.41031 −0.207650 −0.103825 0.994596i \(-0.533108\pi\)
−0.103825 + 0.994596i \(0.533108\pi\)
\(954\) 0 0
\(955\) −1.79426 −0.0580607
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 19.8281 0.640284
\(960\) 0 0
\(961\) 12.8828 0.415573
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 12.4314 0.400180
\(966\) 0 0
\(967\) 31.1028 1.00020 0.500100 0.865968i \(-0.333297\pi\)
0.500100 + 0.865968i \(0.333297\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −38.9701 −1.25061 −0.625305 0.780381i \(-0.715025\pi\)
−0.625305 + 0.780381i \(0.715025\pi\)
\(972\) 0 0
\(973\) 19.2488 0.617089
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −45.6356 −1.46001 −0.730006 0.683441i \(-0.760483\pi\)
−0.730006 + 0.683441i \(0.760483\pi\)
\(978\) 0 0
\(979\) 30.8556 0.986151
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 30.1089 0.960324 0.480162 0.877180i \(-0.340578\pi\)
0.480162 + 0.877180i \(0.340578\pi\)
\(984\) 0 0
\(985\) 15.5247 0.494659
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 43.5267 1.38407
\(990\) 0 0
\(991\) 0.927003 0.0294472 0.0147236 0.999892i \(-0.495313\pi\)
0.0147236 + 0.999892i \(0.495313\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 11.3182 0.358812
\(996\) 0 0
\(997\) 26.7735 0.847927 0.423963 0.905679i \(-0.360639\pi\)
0.423963 + 0.905679i \(0.360639\pi\)
\(998\) 0 0
\(999\) 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 6048.2.a.bq.1.2 yes 4
3.2 odd 2 6048.2.a.bl.1.3 4
4.3 odd 2 6048.2.a.bu.1.2 yes 4
12.11 even 2 6048.2.a.bn.1.3 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6048.2.a.bl.1.3 4 3.2 odd 2
6048.2.a.bn.1.3 yes 4 12.11 even 2
6048.2.a.bq.1.2 yes 4 1.1 even 1 trivial
6048.2.a.bu.1.2 yes 4 4.3 odd 2