Properties

Label 6048.2.c.d.3025.11
Level 6048
Weight 2
Character 6048.3025
Analytic conductor 48.294
Analytic rank 0
Dimension 16
CM no
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 6048 = 2^{5} \cdot 3^{3} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 6048.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(48.2935231425\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\Q(\zeta_{40})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{12}\cdot 3^{8} \)
Twist minimal: no (minimal twist has level 1512)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 3025.11
Root \(0.453990 - 0.891007i\)
Character \(\chi\) = 6048.3025
Dual form 6048.2.c.d.3025.6

$q$-expansion

\(f(q)\) \(=\) \(q+1.60758i q^{5} +1.00000 q^{7} +O(q^{10})\) \(q+1.60758i q^{5} +1.00000 q^{7} +0.0549306i q^{11} -3.75621i q^{13} -3.16228 q^{17} +0.726543i q^{19} -7.77604 q^{23} +2.41570 q^{25} -2.75789i q^{29} +2.14475 q^{31} +1.60758i q^{35} -0.600848i q^{37} +4.53658 q^{41} +6.85004i q^{43} -0.744883 q^{47} +1.00000 q^{49} +10.5822i q^{53} -0.0883051 q^{55} +2.58013i q^{59} +9.80423i q^{61} +6.03841 q^{65} +12.2627i q^{67} -9.19025 q^{71} +3.85224 q^{73} +0.0549306i q^{77} -10.9057 q^{79} +13.1624i q^{83} -5.08361i q^{85} -7.90356 q^{89} -3.75621i q^{91} -1.16797 q^{95} +12.1803 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + 16q^{7} + O(q^{10}) \) \( 16q + 16q^{7} - 32q^{25} - 40q^{31} + 16q^{49} + 72q^{55} + 24q^{73} - 24q^{79} + 16q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/6048\mathbb{Z}\right)^\times\).

\(n\) \(2593\) \(3781\) \(3809\) \(4159\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(1\)

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 0
\(3\) 0 0
\(4\) 0 0
\(5\) 1.60758i 0.718930i 0.933158 + 0.359465i \(0.117041\pi\)
−0.933158 + 0.359465i \(0.882959\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0.0549306i 0.0165622i 0.999966 + 0.00828109i \(0.00263598\pi\)
−0.999966 + 0.00828109i \(0.997364\pi\)
\(12\) 0 0
\(13\) − 3.75621i − 1.04179i −0.853622 0.520893i \(-0.825599\pi\)
0.853622 0.520893i \(-0.174401\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −3.16228 −0.766965 −0.383482 0.923548i \(-0.625275\pi\)
−0.383482 + 0.923548i \(0.625275\pi\)
\(18\) 0 0
\(19\) 0.726543i 0.166680i 0.996521 + 0.0833401i \(0.0265588\pi\)
−0.996521 + 0.0833401i \(0.973441\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −7.77604 −1.62142 −0.810708 0.585450i \(-0.800918\pi\)
−0.810708 + 0.585450i \(0.800918\pi\)
\(24\) 0 0
\(25\) 2.41570 0.483139
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) − 2.75789i − 0.512128i −0.966660 0.256064i \(-0.917574\pi\)
0.966660 0.256064i \(-0.0824257\pi\)
\(30\) 0 0
\(31\) 2.14475 0.385208 0.192604 0.981277i \(-0.438307\pi\)
0.192604 + 0.981277i \(0.438307\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 1.60758i 0.271730i
\(36\) 0 0
\(37\) − 0.600848i − 0.0987788i −0.998780 0.0493894i \(-0.984272\pi\)
0.998780 0.0493894i \(-0.0157275\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 4.53658 0.708495 0.354248 0.935152i \(-0.384737\pi\)
0.354248 + 0.935152i \(0.384737\pi\)
\(42\) 0 0
\(43\) 6.85004i 1.04462i 0.852755 + 0.522311i \(0.174930\pi\)
−0.852755 + 0.522311i \(0.825070\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −0.744883 −0.108652 −0.0543261 0.998523i \(-0.517301\pi\)
−0.0543261 + 0.998523i \(0.517301\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 10.5822i 1.45358i 0.686860 + 0.726790i \(0.258989\pi\)
−0.686860 + 0.726790i \(0.741011\pi\)
\(54\) 0 0
\(55\) −0.0883051 −0.0119071
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 2.58013i 0.335905i 0.985795 + 0.167952i \(0.0537155\pi\)
−0.985795 + 0.167952i \(0.946285\pi\)
\(60\) 0 0
\(61\) 9.80423i 1.25530i 0.778495 + 0.627651i \(0.215984\pi\)
−0.778495 + 0.627651i \(0.784016\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 6.03841 0.748972
\(66\) 0 0
\(67\) 12.2627i 1.49813i 0.662496 + 0.749065i \(0.269497\pi\)
−0.662496 + 0.749065i \(0.730503\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −9.19025 −1.09068 −0.545341 0.838214i \(-0.683600\pi\)
−0.545341 + 0.838214i \(0.683600\pi\)
\(72\) 0 0
\(73\) 3.85224 0.450870 0.225435 0.974258i \(-0.427620\pi\)
0.225435 + 0.974258i \(0.427620\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0.0549306i 0.00625992i
\(78\) 0 0
\(79\) −10.9057 −1.22698 −0.613491 0.789701i \(-0.710235\pi\)
−0.613491 + 0.789701i \(0.710235\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 13.1624i 1.44476i 0.691499 + 0.722378i \(0.256951\pi\)
−0.691499 + 0.722378i \(0.743049\pi\)
\(84\) 0 0
\(85\) − 5.08361i − 0.551394i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −7.90356 −0.837776 −0.418888 0.908038i \(-0.637580\pi\)
−0.418888 + 0.908038i \(0.637580\pi\)
\(90\) 0 0
\(91\) − 3.75621i − 0.393758i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −1.16797 −0.119832
\(96\) 0 0
\(97\) 12.1803 1.23673 0.618363 0.785893i \(-0.287796\pi\)
0.618363 + 0.785893i \(0.287796\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 4.35250i 0.433090i 0.976273 + 0.216545i \(0.0694788\pi\)
−0.976273 + 0.216545i \(0.930521\pi\)
\(102\) 0 0
\(103\) −17.5574 −1.72998 −0.864992 0.501785i \(-0.832677\pi\)
−0.864992 + 0.501785i \(0.832677\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 6.44944i − 0.623491i −0.950166 0.311745i \(-0.899086\pi\)
0.950166 0.311745i \(-0.100914\pi\)
\(108\) 0 0
\(109\) − 2.03559i − 0.194975i −0.995237 0.0974873i \(-0.968919\pi\)
0.995237 0.0974873i \(-0.0310805\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 2.61946 0.246418 0.123209 0.992381i \(-0.460681\pi\)
0.123209 + 0.992381i \(0.460681\pi\)
\(114\) 0 0
\(115\) − 12.5006i − 1.16569i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −3.16228 −0.289886
\(120\) 0 0
\(121\) 10.9970 0.999726
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 11.9213i 1.06627i
\(126\) 0 0
\(127\) 13.6733 1.21331 0.606656 0.794965i \(-0.292511\pi\)
0.606656 + 0.794965i \(0.292511\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 11.2592i 0.983721i 0.870674 + 0.491861i \(0.163683\pi\)
−0.870674 + 0.491861i \(0.836317\pi\)
\(132\) 0 0
\(133\) 0.726543i 0.0629992i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 1.00318 0.0857076 0.0428538 0.999081i \(-0.486355\pi\)
0.0428538 + 0.999081i \(0.486355\pi\)
\(138\) 0 0
\(139\) 2.60253i 0.220744i 0.993890 + 0.110372i \(0.0352042\pi\)
−0.993890 + 0.110372i \(0.964796\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0.206331 0.0172543
\(144\) 0 0
\(145\) 4.43352 0.368184
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 19.1514i 1.56895i 0.620163 + 0.784473i \(0.287066\pi\)
−0.620163 + 0.784473i \(0.712934\pi\)
\(150\) 0 0
\(151\) −15.8848 −1.29269 −0.646344 0.763046i \(-0.723703\pi\)
−0.646344 + 0.763046i \(0.723703\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 3.44784i 0.276937i
\(156\) 0 0
\(157\) − 14.7099i − 1.17398i −0.809595 0.586988i \(-0.800313\pi\)
0.809595 0.586988i \(-0.199687\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −7.77604 −0.612838
\(162\) 0 0
\(163\) − 9.69687i − 0.759517i −0.925086 0.379759i \(-0.876007\pi\)
0.925086 0.379759i \(-0.123993\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −2.57013 −0.198882 −0.0994412 0.995043i \(-0.531706\pi\)
−0.0994412 + 0.995043i \(0.531706\pi\)
\(168\) 0 0
\(169\) −1.10915 −0.0853193
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) − 1.69460i − 0.128838i −0.997923 0.0644192i \(-0.979481\pi\)
0.997923 0.0644192i \(-0.0205195\pi\)
\(174\) 0 0
\(175\) 2.41570 0.182609
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) − 11.1383i − 0.832518i −0.909246 0.416259i \(-0.863341\pi\)
0.909246 0.416259i \(-0.136659\pi\)
\(180\) 0 0
\(181\) 6.80203i 0.505591i 0.967520 + 0.252796i \(0.0813500\pi\)
−0.967520 + 0.252796i \(0.918650\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0.965909 0.0710151
\(186\) 0 0
\(187\) − 0.173706i − 0.0127026i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −15.6397 −1.13165 −0.565824 0.824526i \(-0.691442\pi\)
−0.565824 + 0.824526i \(0.691442\pi\)
\(192\) 0 0
\(193\) 19.5893 1.41007 0.705034 0.709174i \(-0.250932\pi\)
0.705034 + 0.709174i \(0.250932\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 7.29916i − 0.520044i −0.965603 0.260022i \(-0.916270\pi\)
0.965603 0.260022i \(-0.0837298\pi\)
\(198\) 0 0
\(199\) −13.8002 −0.978273 −0.489136 0.872207i \(-0.662688\pi\)
−0.489136 + 0.872207i \(0.662688\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) − 2.75789i − 0.193566i
\(204\) 0 0
\(205\) 7.29291i 0.509359i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −0.0399094 −0.00276059
\(210\) 0 0
\(211\) 5.25731i 0.361928i 0.983490 + 0.180964i \(0.0579218\pi\)
−0.983490 + 0.180964i \(0.942078\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −11.0120 −0.751011
\(216\) 0 0
\(217\) 2.14475 0.145595
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 11.8782i 0.799014i
\(222\) 0 0
\(223\) −4.96512 −0.332489 −0.166244 0.986085i \(-0.553164\pi\)
−0.166244 + 0.986085i \(0.553164\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 15.1773i 1.00735i 0.863893 + 0.503676i \(0.168019\pi\)
−0.863893 + 0.503676i \(0.831981\pi\)
\(228\) 0 0
\(229\) − 5.69636i − 0.376426i −0.982128 0.188213i \(-0.939730\pi\)
0.982128 0.188213i \(-0.0602695\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −16.5086 −1.08151 −0.540756 0.841179i \(-0.681862\pi\)
−0.540756 + 0.841179i \(0.681862\pi\)
\(234\) 0 0
\(235\) − 1.19746i − 0.0781134i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 15.1437 0.979564 0.489782 0.871845i \(-0.337076\pi\)
0.489782 + 0.871845i \(0.337076\pi\)
\(240\) 0 0
\(241\) −28.5047 −1.83615 −0.918075 0.396407i \(-0.870257\pi\)
−0.918075 + 0.396407i \(0.870257\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 1.60758i 0.102704i
\(246\) 0 0
\(247\) 2.72905 0.173645
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 7.18930i 0.453785i 0.973920 + 0.226892i \(0.0728566\pi\)
−0.973920 + 0.226892i \(0.927143\pi\)
\(252\) 0 0
\(253\) − 0.427142i − 0.0268542i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −30.4469 −1.89922 −0.949612 0.313427i \(-0.898523\pi\)
−0.949612 + 0.313427i \(0.898523\pi\)
\(258\) 0 0
\(259\) − 0.600848i − 0.0373349i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 8.28410 0.510820 0.255410 0.966833i \(-0.417790\pi\)
0.255410 + 0.966833i \(0.417790\pi\)
\(264\) 0 0
\(265\) −17.0117 −1.04502
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) − 7.16222i − 0.436689i −0.975872 0.218344i \(-0.929934\pi\)
0.975872 0.218344i \(-0.0700656\pi\)
\(270\) 0 0
\(271\) −12.4721 −0.757628 −0.378814 0.925473i \(-0.623668\pi\)
−0.378814 + 0.925473i \(0.623668\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0.132696i 0.00800184i
\(276\) 0 0
\(277\) − 8.91531i − 0.535669i −0.963465 0.267835i \(-0.913692\pi\)
0.963465 0.267835i \(-0.0863081\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −12.3577 −0.737198 −0.368599 0.929589i \(-0.620162\pi\)
−0.368599 + 0.929589i \(0.620162\pi\)
\(282\) 0 0
\(283\) 21.4245i 1.27356i 0.771047 + 0.636778i \(0.219733\pi\)
−0.771047 + 0.636778i \(0.780267\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 4.53658 0.267786
\(288\) 0 0
\(289\) −7.00000 −0.411765
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 0.0647976i 0.00378552i 0.999998 + 0.00189276i \(0.000602484\pi\)
−0.999998 + 0.00189276i \(0.999398\pi\)
\(294\) 0 0
\(295\) −4.14776 −0.241492
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 29.2085i 1.68917i
\(300\) 0 0
\(301\) 6.85004i 0.394830i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −15.7611 −0.902475
\(306\) 0 0
\(307\) − 15.8474i − 0.904460i −0.891901 0.452230i \(-0.850629\pi\)
0.891901 0.452230i \(-0.149371\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 13.0254 0.738602 0.369301 0.929310i \(-0.379597\pi\)
0.369301 + 0.929310i \(0.379597\pi\)
\(312\) 0 0
\(313\) 24.0265 1.35806 0.679030 0.734110i \(-0.262401\pi\)
0.679030 + 0.734110i \(0.262401\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 3.71438i 0.208620i 0.994545 + 0.104310i \(0.0332635\pi\)
−0.994545 + 0.104310i \(0.966737\pi\)
\(318\) 0 0
\(319\) 0.151493 0.00848195
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) − 2.29753i − 0.127838i
\(324\) 0 0
\(325\) − 9.07387i − 0.503328i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −0.744883 −0.0410667
\(330\) 0 0
\(331\) − 3.50059i − 0.192410i −0.995362 0.0962048i \(-0.969330\pi\)
0.995362 0.0962048i \(-0.0306704\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −19.7133 −1.07705
\(336\) 0 0
\(337\) −16.9917 −0.925595 −0.462797 0.886464i \(-0.653154\pi\)
−0.462797 + 0.886464i \(0.653154\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0.117812i 0.00637988i
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 36.6651i 1.96829i 0.177373 + 0.984144i \(0.443240\pi\)
−0.177373 + 0.984144i \(0.556760\pi\)
\(348\) 0 0
\(349\) 34.7887i 1.86220i 0.364770 + 0.931098i \(0.381148\pi\)
−0.364770 + 0.931098i \(0.618852\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −12.1913 −0.648876 −0.324438 0.945907i \(-0.605175\pi\)
−0.324438 + 0.945907i \(0.605175\pi\)
\(354\) 0 0
\(355\) − 14.7740i − 0.784125i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −33.4411 −1.76495 −0.882477 0.470355i \(-0.844126\pi\)
−0.882477 + 0.470355i \(0.844126\pi\)
\(360\) 0 0
\(361\) 18.4721 0.972218
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 6.19277i 0.324144i
\(366\) 0 0
\(367\) 14.4869 0.756212 0.378106 0.925762i \(-0.376575\pi\)
0.378106 + 0.925762i \(0.376575\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 10.5822i 0.549401i
\(372\) 0 0
\(373\) − 28.7612i − 1.48920i −0.667512 0.744599i \(-0.732641\pi\)
0.667512 0.744599i \(-0.267359\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −10.3592 −0.533528
\(378\) 0 0
\(379\) 16.1036i 0.827188i 0.910461 + 0.413594i \(0.135727\pi\)
−0.910461 + 0.413594i \(0.864273\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 25.7602 1.31629 0.658143 0.752893i \(-0.271342\pi\)
0.658143 + 0.752893i \(0.271342\pi\)
\(384\) 0 0
\(385\) −0.0883051 −0.00450045
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) − 9.82322i − 0.498057i −0.968496 0.249029i \(-0.919889\pi\)
0.968496 0.249029i \(-0.0801113\pi\)
\(390\) 0 0
\(391\) 24.5900 1.24357
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) − 17.5317i − 0.882115i
\(396\) 0 0
\(397\) − 24.1824i − 1.21368i −0.794824 0.606840i \(-0.792437\pi\)
0.794824 0.606840i \(-0.207563\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 23.4216 1.16962 0.584810 0.811170i \(-0.301169\pi\)
0.584810 + 0.811170i \(0.301169\pi\)
\(402\) 0 0
\(403\) − 8.05613i − 0.401304i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0.0330049 0.00163599
\(408\) 0 0
\(409\) −32.6813 −1.61599 −0.807994 0.589191i \(-0.799447\pi\)
−0.807994 + 0.589191i \(0.799447\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 2.58013i 0.126960i
\(414\) 0 0
\(415\) −21.1595 −1.03868
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 29.3981i 1.43619i 0.695945 + 0.718095i \(0.254986\pi\)
−0.695945 + 0.718095i \(0.745014\pi\)
\(420\) 0 0
\(421\) 1.59493i 0.0777319i 0.999244 + 0.0388660i \(0.0123745\pi\)
−0.999244 + 0.0388660i \(0.987625\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −7.63910 −0.370551
\(426\) 0 0
\(427\) 9.80423i 0.474460i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 10.7336 0.517020 0.258510 0.966009i \(-0.416768\pi\)
0.258510 + 0.966009i \(0.416768\pi\)
\(432\) 0 0
\(433\) −8.43352 −0.405289 −0.202645 0.979252i \(-0.564954\pi\)
−0.202645 + 0.979252i \(0.564954\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) − 5.64962i − 0.270258i
\(438\) 0 0
\(439\) 17.3452 0.827842 0.413921 0.910313i \(-0.364159\pi\)
0.413921 + 0.910313i \(0.364159\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 29.8824i 1.41976i 0.704325 + 0.709878i \(0.251250\pi\)
−0.704325 + 0.709878i \(0.748750\pi\)
\(444\) 0 0
\(445\) − 12.7056i − 0.602302i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 18.6303 0.879217 0.439608 0.898190i \(-0.355117\pi\)
0.439608 + 0.898190i \(0.355117\pi\)
\(450\) 0 0
\(451\) 0.249197i 0.0117342i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 6.03841 0.283085
\(456\) 0 0
\(457\) −6.63702 −0.310466 −0.155233 0.987878i \(-0.549613\pi\)
−0.155233 + 0.987878i \(0.549613\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 42.1500i 1.96312i 0.191155 + 0.981560i \(0.438777\pi\)
−0.191155 + 0.981560i \(0.561223\pi\)
\(462\) 0 0
\(463\) 27.1209 1.26041 0.630207 0.776427i \(-0.282970\pi\)
0.630207 + 0.776427i \(0.282970\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 1.06443i 0.0492561i 0.999697 + 0.0246280i \(0.00784014\pi\)
−0.999697 + 0.0246280i \(0.992160\pi\)
\(468\) 0 0
\(469\) 12.2627i 0.566240i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −0.376277 −0.0173012
\(474\) 0 0
\(475\) 1.75511i 0.0805298i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 32.7777 1.49765 0.748825 0.662767i \(-0.230618\pi\)
0.748825 + 0.662767i \(0.230618\pi\)
\(480\) 0 0
\(481\) −2.25691 −0.102906
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 19.5808i 0.889120i
\(486\) 0 0
\(487\) 2.44329 0.110716 0.0553580 0.998467i \(-0.482370\pi\)
0.0553580 + 0.998467i \(0.482370\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) − 25.5978i − 1.15521i −0.816315 0.577606i \(-0.803987\pi\)
0.816315 0.577606i \(-0.196013\pi\)
\(492\) 0 0
\(493\) 8.72122i 0.392784i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −9.19025 −0.412239
\(498\) 0 0
\(499\) 7.96720i 0.356661i 0.983971 + 0.178330i \(0.0570696\pi\)
−0.983971 + 0.178330i \(0.942930\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −31.3158 −1.39630 −0.698151 0.715951i \(-0.745993\pi\)
−0.698151 + 0.715951i \(0.745993\pi\)
\(504\) 0 0
\(505\) −6.99698 −0.311362
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 40.7362i 1.80560i 0.430061 + 0.902800i \(0.358492\pi\)
−0.430061 + 0.902800i \(0.641508\pi\)
\(510\) 0 0
\(511\) 3.85224 0.170413
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) − 28.2249i − 1.24374i
\(516\) 0 0
\(517\) − 0.0409168i − 0.00179952i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −43.3880 −1.90086 −0.950432 0.310931i \(-0.899359\pi\)
−0.950432 + 0.310931i \(0.899359\pi\)
\(522\) 0 0
\(523\) 17.9726i 0.785887i 0.919563 + 0.392944i \(0.128543\pi\)
−0.919563 + 0.392944i \(0.871457\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −6.78228 −0.295441
\(528\) 0 0
\(529\) 37.4668 1.62899
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) − 17.0404i − 0.738101i
\(534\) 0 0
\(535\) 10.3680 0.448246
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0.0549306i 0.00236603i
\(540\) 0 0
\(541\) 11.5151i 0.495074i 0.968878 + 0.247537i \(0.0796212\pi\)
−0.968878 + 0.247537i \(0.920379\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 3.27238 0.140173
\(546\) 0 0
\(547\) − 14.5265i − 0.621107i −0.950556 0.310553i \(-0.899486\pi\)
0.950556 0.310553i \(-0.100514\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 2.00373 0.0853616
\(552\) 0 0
\(553\) −10.9057 −0.463756
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) − 44.6492i − 1.89185i −0.324387 0.945924i \(-0.605158\pi\)
0.324387 0.945924i \(-0.394842\pi\)
\(558\) 0 0
\(559\) 25.7302 1.08827
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) − 4.26356i − 0.179688i −0.995956 0.0898438i \(-0.971363\pi\)
0.995956 0.0898438i \(-0.0286368\pi\)
\(564\) 0 0
\(565\) 4.21098i 0.177157i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 38.6045 1.61838 0.809192 0.587544i \(-0.199905\pi\)
0.809192 + 0.587544i \(0.199905\pi\)
\(570\) 0 0
\(571\) 28.0556i 1.17409i 0.809554 + 0.587045i \(0.199709\pi\)
−0.809554 + 0.587045i \(0.800291\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −18.7845 −0.783370
\(576\) 0 0
\(577\) −2.46468 −0.102606 −0.0513029 0.998683i \(-0.516337\pi\)
−0.0513029 + 0.998683i \(0.516337\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 13.1624i 0.546066i
\(582\) 0 0
\(583\) −0.581287 −0.0240745
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) − 42.7523i − 1.76458i −0.470710 0.882288i \(-0.656002\pi\)
0.470710 0.882288i \(-0.343998\pi\)
\(588\) 0 0
\(589\) 1.55825i 0.0642065i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −17.7945 −0.730733 −0.365367 0.930864i \(-0.619056\pi\)
−0.365367 + 0.930864i \(0.619056\pi\)
\(594\) 0 0
\(595\) − 5.08361i − 0.208408i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −14.5626 −0.595011 −0.297506 0.954720i \(-0.596155\pi\)
−0.297506 + 0.954720i \(0.596155\pi\)
\(600\) 0 0
\(601\) −27.1497 −1.10746 −0.553730 0.832696i \(-0.686796\pi\)
−0.553730 + 0.832696i \(0.686796\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 17.6785i 0.718733i
\(606\) 0 0
\(607\) −10.8708 −0.441231 −0.220616 0.975361i \(-0.570807\pi\)
−0.220616 + 0.975361i \(0.570807\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 2.79794i 0.113192i
\(612\) 0 0
\(613\) 26.1387i 1.05573i 0.849327 + 0.527866i \(0.177008\pi\)
−0.849327 + 0.527866i \(0.822992\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 42.5464 1.71285 0.856427 0.516269i \(-0.172679\pi\)
0.856427 + 0.516269i \(0.172679\pi\)
\(618\) 0 0
\(619\) − 41.0012i − 1.64798i −0.566606 0.823989i \(-0.691744\pi\)
0.566606 0.823989i \(-0.308256\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −7.90356 −0.316649
\(624\) 0 0
\(625\) −7.08594 −0.283437
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 1.90005i 0.0757599i
\(630\) 0 0
\(631\) 29.8045 1.18650 0.593249 0.805019i \(-0.297845\pi\)
0.593249 + 0.805019i \(0.297845\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 21.9809i 0.872286i
\(636\) 0 0
\(637\) − 3.75621i − 0.148827i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −18.1768 −0.717941 −0.358971 0.933349i \(-0.616872\pi\)
−0.358971 + 0.933349i \(0.616872\pi\)
\(642\) 0 0
\(643\) − 20.8490i − 0.822203i −0.911590 0.411101i \(-0.865144\pi\)
0.911590 0.411101i \(-0.134856\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −18.7706 −0.737948 −0.368974 0.929440i \(-0.620291\pi\)
−0.368974 + 0.929440i \(0.620291\pi\)
\(648\) 0 0
\(649\) −0.141728 −0.00556332
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 25.5220i 0.998751i 0.866386 + 0.499376i \(0.166437\pi\)
−0.866386 + 0.499376i \(0.833563\pi\)
\(654\) 0 0
\(655\) −18.1000 −0.707227
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) − 18.8249i − 0.733314i −0.930356 0.366657i \(-0.880502\pi\)
0.930356 0.366657i \(-0.119498\pi\)
\(660\) 0 0
\(661\) 25.1764i 0.979247i 0.871934 + 0.489623i \(0.162866\pi\)
−0.871934 + 0.489623i \(0.837134\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −1.16797 −0.0452921
\(666\) 0 0
\(667\) 21.4455i 0.830372i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −0.538552 −0.0207906
\(672\) 0 0
\(673\) 8.29322 0.319680 0.159840 0.987143i \(-0.448902\pi\)
0.159840 + 0.987143i \(0.448902\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) − 11.6825i − 0.448993i −0.974475 0.224497i \(-0.927926\pi\)
0.974475 0.224497i \(-0.0720738\pi\)
\(678\) 0 0
\(679\) 12.1803 0.467439
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 22.9388i 0.877727i 0.898554 + 0.438864i \(0.144619\pi\)
−0.898554 + 0.438864i \(0.855381\pi\)
\(684\) 0 0
\(685\) 1.61269i 0.0616178i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 39.7491 1.51432
\(690\) 0 0
\(691\) − 41.2800i − 1.57036i −0.619265 0.785182i \(-0.712569\pi\)
0.619265 0.785182i \(-0.287431\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −4.18377 −0.158699
\(696\) 0 0
\(697\) −14.3459 −0.543391
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) − 22.9596i − 0.867174i −0.901112 0.433587i \(-0.857248\pi\)
0.901112 0.433587i \(-0.142752\pi\)
\(702\) 0 0
\(703\) 0.436542 0.0164645
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 4.35250i 0.163693i
\(708\) 0 0
\(709\) 22.0831i 0.829348i 0.909970 + 0.414674i \(0.136104\pi\)
−0.909970 + 0.414674i \(0.863896\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −16.6776 −0.624582
\(714\) 0 0
\(715\) 0.331693i 0.0124046i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −29.4594 −1.09865 −0.549325 0.835609i \(-0.685115\pi\)
−0.549325 + 0.835609i \(0.685115\pi\)
\(720\) 0 0
\(721\) −17.5574 −0.653873
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) − 6.66223i − 0.247429i
\(726\) 0 0
\(727\) 25.5893 0.949054 0.474527 0.880241i \(-0.342619\pi\)
0.474527 + 0.880241i \(0.342619\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) − 21.6617i − 0.801189i
\(732\) 0 0
\(733\) 24.3587i 0.899711i 0.893101 + 0.449855i \(0.148524\pi\)
−0.893101 + 0.449855i \(0.851476\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −0.673598 −0.0248123
\(738\) 0 0
\(739\) − 18.8064i − 0.691805i −0.938270 0.345903i \(-0.887573\pi\)
0.938270 0.345903i \(-0.112427\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 29.4039 1.07873 0.539363 0.842074i \(-0.318665\pi\)
0.539363 + 0.842074i \(0.318665\pi\)
\(744\) 0 0
\(745\) −30.7874 −1.12796
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) − 6.44944i − 0.235657i
\(750\) 0 0
\(751\) 16.9057 0.616896 0.308448 0.951241i \(-0.400190\pi\)
0.308448 + 0.951241i \(0.400190\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) − 25.5361i − 0.929353i
\(756\) 0 0
\(757\) 30.3657i 1.10366i 0.833957 + 0.551830i \(0.186070\pi\)
−0.833957 + 0.551830i \(0.813930\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 18.0960 0.655979 0.327990 0.944681i \(-0.393629\pi\)
0.327990 + 0.944681i \(0.393629\pi\)
\(762\) 0 0
\(763\) − 2.03559i − 0.0736935i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 9.69153 0.349941
\(768\) 0 0
\(769\) 15.4944 0.558743 0.279371 0.960183i \(-0.409874\pi\)
0.279371 + 0.960183i \(0.409874\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) − 33.1186i − 1.19119i −0.803284 0.595597i \(-0.796916\pi\)
0.803284 0.595597i \(-0.203084\pi\)
\(774\) 0 0
\(775\) 5.18105 0.186109
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 3.29602i 0.118092i
\(780\) 0 0
\(781\) − 0.504826i − 0.0180641i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 23.6473 0.844008
\(786\) 0 0
\(787\) − 13.8484i − 0.493641i −0.969061 0.246820i \(-0.920614\pi\)
0.969061 0.246820i \(-0.0793858\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 2.61946 0.0931372
\(792\) 0 0
\(793\) 36.8268 1.30776
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 27.5510i − 0.975908i −0.872869 0.487954i \(-0.837743\pi\)
0.872869 0.487954i \(-0.162257\pi\)
\(798\) 0 0
\(799\) 2.35553 0.0833325
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 0.211606i 0.00746740i
\(804\) 0 0
\(805\) − 12.5006i − 0.440588i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −0.181117 −0.00636775 −0.00318388 0.999995i \(-0.501013\pi\)
−0.00318388 + 0.999995i \(0.501013\pi\)
\(810\) 0 0
\(811\) 23.7508i 0.834003i 0.908906 + 0.417002i \(0.136919\pi\)
−0.908906 + 0.417002i \(0.863081\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 15.5885 0.546040
\(816\) 0 0
\(817\) −4.97685 −0.174118
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) − 8.16381i − 0.284919i −0.989801 0.142460i \(-0.954499\pi\)
0.989801 0.142460i \(-0.0455011\pi\)
\(822\) 0 0
\(823\) −39.6984 −1.38380 −0.691900 0.721993i \(-0.743226\pi\)
−0.691900 + 0.721993i \(0.743226\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 46.0390i − 1.60093i −0.599377 0.800467i \(-0.704585\pi\)
0.599377 0.800467i \(-0.295415\pi\)
\(828\) 0 0
\(829\) 14.1434i 0.491220i 0.969369 + 0.245610i \(0.0789882\pi\)
−0.969369 + 0.245610i \(0.921012\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −3.16228 −0.109566
\(834\) 0 0
\(835\) − 4.13168i − 0.142983i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −31.0217 −1.07099 −0.535494 0.844539i \(-0.679875\pi\)
−0.535494 + 0.844539i \(0.679875\pi\)
\(840\) 0 0
\(841\) 21.3940 0.737725
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) − 1.78305i − 0.0613386i
\(846\) 0 0
\(847\) 10.9970 0.377861
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 4.67222i 0.160162i
\(852\) 0 0
\(853\) − 36.2812i − 1.24224i −0.783714 0.621121i \(-0.786677\pi\)
0.783714 0.621121i \(-0.213323\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 27.6532 0.944616 0.472308 0.881434i \(-0.343421\pi\)
0.472308 + 0.881434i \(0.343421\pi\)
\(858\) 0 0
\(859\) 27.0733i 0.923729i 0.886950 + 0.461865i \(0.152819\pi\)
−0.886950 + 0.461865i \(0.847181\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −8.98203 −0.305752 −0.152876 0.988245i \(-0.548853\pi\)
−0.152876 + 0.988245i \(0.548853\pi\)
\(864\) 0 0
\(865\) 2.72421 0.0926258
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) − 0.599054i − 0.0203215i
\(870\) 0 0
\(871\) 46.0614 1.56073
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 11.9213i 0.403014i
\(876\) 0 0
\(877\) − 27.9552i − 0.943980i −0.881604 0.471990i \(-0.843536\pi\)
0.881604 0.471990i \(-0.156464\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −33.0213 −1.11252 −0.556258 0.831010i \(-0.687763\pi\)
−0.556258 + 0.831010i \(0.687763\pi\)
\(882\) 0 0
\(883\) − 19.1451i − 0.644283i −0.946692 0.322141i \(-0.895597\pi\)
0.946692 0.322141i \(-0.104403\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −8.99598 −0.302056 −0.151028 0.988530i \(-0.548258\pi\)
−0.151028 + 0.988530i \(0.548258\pi\)
\(888\) 0 0
\(889\) 13.6733 0.458588
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) − 0.541189i − 0.0181102i
\(894\) 0 0
\(895\) 17.9057 0.598522
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) − 5.91498i − 0.197276i
\(900\) 0 0
\(901\) − 33.4639i − 1.11484i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −10.9348 −0.363485
\(906\) 0 0
\(907\) 53.5978i 1.77969i 0.456267 + 0.889843i \(0.349186\pi\)
−0.456267 + 0.889843i \(0.650814\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 47.1958 1.56367 0.781833 0.623487i \(-0.214285\pi\)
0.781833 + 0.623487i \(0.214285\pi\)
\(912\) 0 0
\(913\) −0.723015 −0.0239283
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 11.2592i 0.371812i
\(918\) 0 0
\(919\) −22.6942 −0.748612 −0.374306 0.927305i \(-0.622119\pi\)
−0.374306 + 0.927305i \(0.622119\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 34.5206i 1.13626i
\(924\) 0 0
\(925\) − 1.45147i − 0.0477239i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 31.0001 1.01708 0.508541 0.861038i \(-0.330185\pi\)
0.508541 + 0.861038i \(0.330185\pi\)
\(930\) 0 0
\(931\) 0.726543i 0.0238115i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0.279245 0.00913230
\(936\) 0 0
\(937\) −49.0011 −1.60080 −0.800398 0.599469i \(-0.795378\pi\)
−0.800398 + 0.599469i \(0.795378\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) − 49.2485i − 1.60546i −0.596345 0.802728i \(-0.703381\pi\)
0.596345 0.802728i \(-0.296619\pi\)
\(942\) 0 0
\(943\) −35.2766 −1.14877
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 9.28008i − 0.301562i −0.988567 0.150781i \(-0.951821\pi\)
0.988567 0.150781i \(-0.0481788\pi\)
\(948\) 0 0
\(949\) − 14.4698i − 0.469711i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 50.1855 1.62567 0.812834 0.582496i \(-0.197924\pi\)
0.812834 + 0.582496i \(0.197924\pi\)
\(954\) 0 0
\(955\) − 25.1420i − 0.813576i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 1.00318 0.0323944
\(960\) 0 0
\(961\) −26.4001 −0.851615
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 31.4913i 1.01374i
\(966\) 0 0
\(967\) 7.81736 0.251389 0.125695 0.992069i \(-0.459884\pi\)
0.125695 + 0.992069i \(0.459884\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 7.60460i 0.244043i 0.992527 + 0.122022i \(0.0389377\pi\)
−0.992527 + 0.122022i \(0.961062\pi\)
\(972\) 0 0
\(973\) 2.60253i 0.0834333i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 37.9179 1.21310 0.606551 0.795044i \(-0.292553\pi\)
0.606551 + 0.795044i \(0.292553\pi\)
\(978\) 0 0
\(979\) − 0.434147i − 0.0138754i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 33.7230 1.07560 0.537798 0.843074i \(-0.319256\pi\)
0.537798 + 0.843074i \(0.319256\pi\)
\(984\) 0 0
\(985\) 11.7340 0.373875
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) − 53.2662i − 1.69377i
\(990\) 0 0
\(991\) −46.3137 −1.47120 −0.735602 0.677414i \(-0.763101\pi\)
−0.735602 + 0.677414i \(0.763101\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) − 22.1849i − 0.703310i
\(996\) 0 0
\(997\) 5.20068i 0.164707i 0.996603 + 0.0823535i \(0.0262437\pi\)
−0.996603 + 0.0823535i \(0.973756\pi\)
\(998\) 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.c.d.3025.11 16
3.2 odd 2 inner 6048.2.c.d.3025.5 16
4.3 odd 2 1512.2.c.d.757.4 yes 16
8.3 odd 2 1512.2.c.d.757.1 16
8.5 even 2 inner 6048.2.c.d.3025.6 16
12.11 even 2 1512.2.c.d.757.13 yes 16
24.5 odd 2 inner 6048.2.c.d.3025.12 16
24.11 even 2 1512.2.c.d.757.16 yes 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1512.2.c.d.757.1 16 8.3 odd 2
1512.2.c.d.757.4 yes 16 4.3 odd 2
1512.2.c.d.757.13 yes 16 12.11 even 2
1512.2.c.d.757.16 yes 16 24.11 even 2
6048.2.c.d.3025.5 16 3.2 odd 2 inner
6048.2.c.d.3025.6 16 8.5 even 2 inner
6048.2.c.d.3025.11 16 1.1 even 1 trivial
6048.2.c.d.3025.12 16 24.5 odd 2 inner