Properties

Label 2475.2.c.n.199.2
Level $2475$
Weight $2$
Character 2475.199
Analytic conductor $19.763$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 2475 = 3^{2} \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2475.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(19.7629745003\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 165)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 199.2
Root \(-0.866025 + 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 2475.199
Dual form 2475.2.c.n.199.3

$q$-expansion

\(f(q)\) \(=\) \(q-1.73205i q^{2} -1.00000 q^{4} +2.00000i q^{7} -1.73205i q^{8} +O(q^{10})\) \(q-1.73205i q^{2} -1.00000 q^{4} +2.00000i q^{7} -1.73205i q^{8} +1.00000 q^{11} +1.46410i q^{13} +3.46410 q^{14} -5.00000 q^{16} +1.46410 q^{19} -1.73205i q^{22} -6.92820i q^{23} +2.53590 q^{26} -2.00000i q^{28} +3.46410 q^{29} +2.92820 q^{31} +5.19615i q^{32} +8.92820i q^{37} -2.53590i q^{38} +3.46410 q^{41} -8.92820i q^{43} -1.00000 q^{44} -12.0000 q^{46} -6.92820i q^{47} +3.00000 q^{49} -1.46410i q^{52} -12.9282i q^{53} +3.46410 q^{56} -6.00000i q^{58} +6.92820 q^{59} +2.00000 q^{61} -5.07180i q^{62} -1.00000 q^{64} +8.00000i q^{67} +13.8564 q^{71} -12.3923i q^{73} +15.4641 q^{74} -1.46410 q^{76} +2.00000i q^{77} +13.4641 q^{79} -6.00000i q^{82} +15.4641i q^{83} -15.4641 q^{86} -1.73205i q^{88} -12.9282 q^{89} -2.92820 q^{91} +6.92820i q^{92} -12.0000 q^{94} -10.0000i q^{97} -5.19615i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{4} + O(q^{10}) \) \( 4q - 4q^{4} + 4q^{11} - 20q^{16} - 8q^{19} + 24q^{26} - 16q^{31} - 4q^{44} - 48q^{46} + 12q^{49} + 8q^{61} - 4q^{64} + 48q^{74} + 8q^{76} + 40q^{79} - 48q^{86} - 24q^{89} + 16q^{91} - 48q^{94} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2475\mathbb{Z}\right)^\times\).

\(n\) \(551\) \(2026\) \(2377\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 1.73205i − 1.22474i −0.790569 0.612372i \(-0.790215\pi\)
0.790569 0.612372i \(-0.209785\pi\)
\(3\) 0 0
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) 0 0
\(7\) 2.00000i 0.755929i 0.925820 + 0.377964i \(0.123376\pi\)
−0.925820 + 0.377964i \(0.876624\pi\)
\(8\) − 1.73205i − 0.612372i
\(9\) 0 0
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) 0 0
\(13\) 1.46410i 0.406069i 0.979172 + 0.203034i \(0.0650803\pi\)
−0.979172 + 0.203034i \(0.934920\pi\)
\(14\) 3.46410 0.925820
\(15\) 0 0
\(16\) −5.00000 −1.25000
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) 0 0
\(19\) 1.46410 0.335888 0.167944 0.985797i \(-0.446287\pi\)
0.167944 + 0.985797i \(0.446287\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) − 1.73205i − 0.369274i
\(23\) − 6.92820i − 1.44463i −0.691564 0.722315i \(-0.743078\pi\)
0.691564 0.722315i \(-0.256922\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 2.53590 0.497331
\(27\) 0 0
\(28\) − 2.00000i − 0.377964i
\(29\) 3.46410 0.643268 0.321634 0.946864i \(-0.395768\pi\)
0.321634 + 0.946864i \(0.395768\pi\)
\(30\) 0 0
\(31\) 2.92820 0.525921 0.262960 0.964807i \(-0.415301\pi\)
0.262960 + 0.964807i \(0.415301\pi\)
\(32\) 5.19615i 0.918559i
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 8.92820i 1.46779i 0.679264 + 0.733894i \(0.262299\pi\)
−0.679264 + 0.733894i \(0.737701\pi\)
\(38\) − 2.53590i − 0.411377i
\(39\) 0 0
\(40\) 0 0
\(41\) 3.46410 0.541002 0.270501 0.962720i \(-0.412811\pi\)
0.270501 + 0.962720i \(0.412811\pi\)
\(42\) 0 0
\(43\) − 8.92820i − 1.36154i −0.732498 0.680769i \(-0.761646\pi\)
0.732498 0.680769i \(-0.238354\pi\)
\(44\) −1.00000 −0.150756
\(45\) 0 0
\(46\) −12.0000 −1.76930
\(47\) − 6.92820i − 1.01058i −0.862949 0.505291i \(-0.831385\pi\)
0.862949 0.505291i \(-0.168615\pi\)
\(48\) 0 0
\(49\) 3.00000 0.428571
\(50\) 0 0
\(51\) 0 0
\(52\) − 1.46410i − 0.203034i
\(53\) − 12.9282i − 1.77583i −0.460012 0.887913i \(-0.652155\pi\)
0.460012 0.887913i \(-0.347845\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 3.46410 0.462910
\(57\) 0 0
\(58\) − 6.00000i − 0.787839i
\(59\) 6.92820 0.901975 0.450988 0.892530i \(-0.351072\pi\)
0.450988 + 0.892530i \(0.351072\pi\)
\(60\) 0 0
\(61\) 2.00000 0.256074 0.128037 0.991769i \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) − 5.07180i − 0.644119i
\(63\) 0 0
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) 8.00000i 0.977356i 0.872464 + 0.488678i \(0.162521\pi\)
−0.872464 + 0.488678i \(0.837479\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 13.8564 1.64445 0.822226 0.569160i \(-0.192732\pi\)
0.822226 + 0.569160i \(0.192732\pi\)
\(72\) 0 0
\(73\) − 12.3923i − 1.45041i −0.688533 0.725205i \(-0.741745\pi\)
0.688533 0.725205i \(-0.258255\pi\)
\(74\) 15.4641 1.79767
\(75\) 0 0
\(76\) −1.46410 −0.167944
\(77\) 2.00000i 0.227921i
\(78\) 0 0
\(79\) 13.4641 1.51483 0.757415 0.652934i \(-0.226462\pi\)
0.757415 + 0.652934i \(0.226462\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) − 6.00000i − 0.662589i
\(83\) 15.4641i 1.69741i 0.528870 + 0.848703i \(0.322616\pi\)
−0.528870 + 0.848703i \(0.677384\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −15.4641 −1.66754
\(87\) 0 0
\(88\) − 1.73205i − 0.184637i
\(89\) −12.9282 −1.37039 −0.685193 0.728361i \(-0.740282\pi\)
−0.685193 + 0.728361i \(0.740282\pi\)
\(90\) 0 0
\(91\) −2.92820 −0.306959
\(92\) 6.92820i 0.722315i
\(93\) 0 0
\(94\) −12.0000 −1.23771
\(95\) 0 0
\(96\) 0 0
\(97\) − 10.0000i − 1.01535i −0.861550 0.507673i \(-0.830506\pi\)
0.861550 0.507673i \(-0.169494\pi\)
\(98\) − 5.19615i − 0.524891i
\(99\) 0 0
\(100\) 0 0
\(101\) −10.3923 −1.03407 −0.517036 0.855963i \(-0.672965\pi\)
−0.517036 + 0.855963i \(0.672965\pi\)
\(102\) 0 0
\(103\) − 8.00000i − 0.788263i −0.919054 0.394132i \(-0.871045\pi\)
0.919054 0.394132i \(-0.128955\pi\)
\(104\) 2.53590 0.248665
\(105\) 0 0
\(106\) −22.3923 −2.17493
\(107\) − 15.4641i − 1.49497i −0.664278 0.747486i \(-0.731261\pi\)
0.664278 0.747486i \(-0.268739\pi\)
\(108\) 0 0
\(109\) 10.0000 0.957826 0.478913 0.877862i \(-0.341031\pi\)
0.478913 + 0.877862i \(0.341031\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) − 10.0000i − 0.944911i
\(113\) 0.928203i 0.0873180i 0.999046 + 0.0436590i \(0.0139015\pi\)
−0.999046 + 0.0436590i \(0.986098\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −3.46410 −0.321634
\(117\) 0 0
\(118\) − 12.0000i − 1.10469i
\(119\) 0 0
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) − 3.46410i − 0.313625i
\(123\) 0 0
\(124\) −2.92820 −0.262960
\(125\) 0 0
\(126\) 0 0
\(127\) − 4.92820i − 0.437307i −0.975803 0.218654i \(-0.929834\pi\)
0.975803 0.218654i \(-0.0701665\pi\)
\(128\) 12.1244i 1.07165i
\(129\) 0 0
\(130\) 0 0
\(131\) −5.07180 −0.443125 −0.221562 0.975146i \(-0.571116\pi\)
−0.221562 + 0.975146i \(0.571116\pi\)
\(132\) 0 0
\(133\) 2.92820i 0.253907i
\(134\) 13.8564 1.19701
\(135\) 0 0
\(136\) 0 0
\(137\) 18.0000i 1.53784i 0.639343 + 0.768922i \(0.279207\pi\)
−0.639343 + 0.768922i \(0.720793\pi\)
\(138\) 0 0
\(139\) 8.39230 0.711826 0.355913 0.934519i \(-0.384170\pi\)
0.355913 + 0.934519i \(0.384170\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) − 24.0000i − 2.01404i
\(143\) 1.46410i 0.122434i
\(144\) 0 0
\(145\) 0 0
\(146\) −21.4641 −1.77638
\(147\) 0 0
\(148\) − 8.92820i − 0.733894i
\(149\) −8.53590 −0.699288 −0.349644 0.936883i \(-0.613697\pi\)
−0.349644 + 0.936883i \(0.613697\pi\)
\(150\) 0 0
\(151\) 0.392305 0.0319253 0.0159627 0.999873i \(-0.494919\pi\)
0.0159627 + 0.999873i \(0.494919\pi\)
\(152\) − 2.53590i − 0.205689i
\(153\) 0 0
\(154\) 3.46410 0.279145
\(155\) 0 0
\(156\) 0 0
\(157\) − 16.9282i − 1.35102i −0.737352 0.675509i \(-0.763924\pi\)
0.737352 0.675509i \(-0.236076\pi\)
\(158\) − 23.3205i − 1.85528i
\(159\) 0 0
\(160\) 0 0
\(161\) 13.8564 1.09204
\(162\) 0 0
\(163\) 17.8564i 1.39862i 0.714818 + 0.699311i \(0.246510\pi\)
−0.714818 + 0.699311i \(0.753490\pi\)
\(164\) −3.46410 −0.270501
\(165\) 0 0
\(166\) 26.7846 2.07889
\(167\) − 10.3923i − 0.804181i −0.915600 0.402090i \(-0.868284\pi\)
0.915600 0.402090i \(-0.131716\pi\)
\(168\) 0 0
\(169\) 10.8564 0.835108
\(170\) 0 0
\(171\) 0 0
\(172\) 8.92820i 0.680769i
\(173\) − 12.0000i − 0.912343i −0.889892 0.456172i \(-0.849220\pi\)
0.889892 0.456172i \(-0.150780\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −5.00000 −0.376889
\(177\) 0 0
\(178\) 22.3923i 1.67837i
\(179\) −6.92820 −0.517838 −0.258919 0.965899i \(-0.583366\pi\)
−0.258919 + 0.965899i \(0.583366\pi\)
\(180\) 0 0
\(181\) −11.8564 −0.881280 −0.440640 0.897684i \(-0.645248\pi\)
−0.440640 + 0.897684i \(0.645248\pi\)
\(182\) 5.07180i 0.375947i
\(183\) 0 0
\(184\) −12.0000 −0.884652
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 6.92820i 0.505291i
\(189\) 0 0
\(190\) 0 0
\(191\) 5.07180 0.366982 0.183491 0.983021i \(-0.441260\pi\)
0.183491 + 0.983021i \(0.441260\pi\)
\(192\) 0 0
\(193\) − 3.60770i − 0.259688i −0.991534 0.129844i \(-0.958552\pi\)
0.991534 0.129844i \(-0.0414476\pi\)
\(194\) −17.3205 −1.24354
\(195\) 0 0
\(196\) −3.00000 −0.214286
\(197\) 12.0000i 0.854965i 0.904024 + 0.427482i \(0.140599\pi\)
−0.904024 + 0.427482i \(0.859401\pi\)
\(198\) 0 0
\(199\) −16.7846 −1.18983 −0.594915 0.803789i \(-0.702814\pi\)
−0.594915 + 0.803789i \(0.702814\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 18.0000i 1.26648i
\(203\) 6.92820i 0.486265i
\(204\) 0 0
\(205\) 0 0
\(206\) −13.8564 −0.965422
\(207\) 0 0
\(208\) − 7.32051i − 0.507586i
\(209\) 1.46410 0.101274
\(210\) 0 0
\(211\) 12.3923 0.853121 0.426561 0.904459i \(-0.359725\pi\)
0.426561 + 0.904459i \(0.359725\pi\)
\(212\) 12.9282i 0.887913i
\(213\) 0 0
\(214\) −26.7846 −1.83096
\(215\) 0 0
\(216\) 0 0
\(217\) 5.85641i 0.397559i
\(218\) − 17.3205i − 1.17309i
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 17.8564i 1.19575i 0.801588 + 0.597877i \(0.203989\pi\)
−0.801588 + 0.597877i \(0.796011\pi\)
\(224\) −10.3923 −0.694365
\(225\) 0 0
\(226\) 1.60770 0.106942
\(227\) − 8.53590i − 0.566547i −0.959039 0.283274i \(-0.908580\pi\)
0.959039 0.283274i \(-0.0914205\pi\)
\(228\) 0 0
\(229\) −3.85641 −0.254839 −0.127419 0.991849i \(-0.540669\pi\)
−0.127419 + 0.991849i \(0.540669\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) − 6.00000i − 0.393919i
\(233\) 12.0000i 0.786146i 0.919507 + 0.393073i \(0.128588\pi\)
−0.919507 + 0.393073i \(0.871412\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −6.92820 −0.450988
\(237\) 0 0
\(238\) 0 0
\(239\) −12.0000 −0.776215 −0.388108 0.921614i \(-0.626871\pi\)
−0.388108 + 0.921614i \(0.626871\pi\)
\(240\) 0 0
\(241\) 27.8564 1.79439 0.897194 0.441636i \(-0.145602\pi\)
0.897194 + 0.441636i \(0.145602\pi\)
\(242\) − 1.73205i − 0.111340i
\(243\) 0 0
\(244\) −2.00000 −0.128037
\(245\) 0 0
\(246\) 0 0
\(247\) 2.14359i 0.136394i
\(248\) − 5.07180i − 0.322059i
\(249\) 0 0
\(250\) 0 0
\(251\) −25.8564 −1.63204 −0.816021 0.578022i \(-0.803825\pi\)
−0.816021 + 0.578022i \(0.803825\pi\)
\(252\) 0 0
\(253\) − 6.92820i − 0.435572i
\(254\) −8.53590 −0.535590
\(255\) 0 0
\(256\) 19.0000 1.18750
\(257\) − 7.85641i − 0.490069i −0.969514 0.245035i \(-0.921201\pi\)
0.969514 0.245035i \(-0.0787993\pi\)
\(258\) 0 0
\(259\) −17.8564 −1.10954
\(260\) 0 0
\(261\) 0 0
\(262\) 8.78461i 0.542715i
\(263\) 27.4641i 1.69351i 0.531984 + 0.846755i \(0.321447\pi\)
−0.531984 + 0.846755i \(0.678553\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 5.07180 0.310972
\(267\) 0 0
\(268\) − 8.00000i − 0.488678i
\(269\) −7.85641 −0.479014 −0.239507 0.970895i \(-0.576986\pi\)
−0.239507 + 0.970895i \(0.576986\pi\)
\(270\) 0 0
\(271\) −32.3923 −1.96769 −0.983846 0.179016i \(-0.942709\pi\)
−0.983846 + 0.179016i \(0.942709\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 31.1769 1.88347
\(275\) 0 0
\(276\) 0 0
\(277\) 22.5359i 1.35405i 0.735960 + 0.677025i \(0.236731\pi\)
−0.735960 + 0.677025i \(0.763269\pi\)
\(278\) − 14.5359i − 0.871805i
\(279\) 0 0
\(280\) 0 0
\(281\) 3.46410 0.206651 0.103325 0.994648i \(-0.467052\pi\)
0.103325 + 0.994648i \(0.467052\pi\)
\(282\) 0 0
\(283\) − 8.92820i − 0.530727i −0.964148 0.265363i \(-0.914508\pi\)
0.964148 0.265363i \(-0.0854919\pi\)
\(284\) −13.8564 −0.822226
\(285\) 0 0
\(286\) 2.53590 0.149951
\(287\) 6.92820i 0.408959i
\(288\) 0 0
\(289\) 17.0000 1.00000
\(290\) 0 0
\(291\) 0 0
\(292\) 12.3923i 0.725205i
\(293\) 13.8564i 0.809500i 0.914427 + 0.404750i \(0.132641\pi\)
−0.914427 + 0.404750i \(0.867359\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 15.4641 0.898833
\(297\) 0 0
\(298\) 14.7846i 0.856449i
\(299\) 10.1436 0.586619
\(300\) 0 0
\(301\) 17.8564 1.02923
\(302\) − 0.679492i − 0.0391004i
\(303\) 0 0
\(304\) −7.32051 −0.419860
\(305\) 0 0
\(306\) 0 0
\(307\) 14.0000i 0.799022i 0.916728 + 0.399511i \(0.130820\pi\)
−0.916728 + 0.399511i \(0.869180\pi\)
\(308\) − 2.00000i − 0.113961i
\(309\) 0 0
\(310\) 0 0
\(311\) −18.9282 −1.07332 −0.536660 0.843799i \(-0.680314\pi\)
−0.536660 + 0.843799i \(0.680314\pi\)
\(312\) 0 0
\(313\) − 7.07180i − 0.399722i −0.979824 0.199861i \(-0.935951\pi\)
0.979824 0.199861i \(-0.0640490\pi\)
\(314\) −29.3205 −1.65465
\(315\) 0 0
\(316\) −13.4641 −0.757415
\(317\) 11.0718i 0.621854i 0.950434 + 0.310927i \(0.100639\pi\)
−0.950434 + 0.310927i \(0.899361\pi\)
\(318\) 0 0
\(319\) 3.46410 0.193952
\(320\) 0 0
\(321\) 0 0
\(322\) − 24.0000i − 1.33747i
\(323\) 0 0
\(324\) 0 0
\(325\) 0 0
\(326\) 30.9282 1.71295
\(327\) 0 0
\(328\) − 6.00000i − 0.331295i
\(329\) 13.8564 0.763928
\(330\) 0 0
\(331\) −17.8564 −0.981477 −0.490738 0.871307i \(-0.663273\pi\)
−0.490738 + 0.871307i \(0.663273\pi\)
\(332\) − 15.4641i − 0.848703i
\(333\) 0 0
\(334\) −18.0000 −0.984916
\(335\) 0 0
\(336\) 0 0
\(337\) − 29.1769i − 1.58937i −0.607023 0.794684i \(-0.707637\pi\)
0.607023 0.794684i \(-0.292363\pi\)
\(338\) − 18.8038i − 1.02279i
\(339\) 0 0
\(340\) 0 0
\(341\) 2.92820 0.158571
\(342\) 0 0
\(343\) 20.0000i 1.07990i
\(344\) −15.4641 −0.833768
\(345\) 0 0
\(346\) −20.7846 −1.11739
\(347\) − 1.60770i − 0.0863056i −0.999068 0.0431528i \(-0.986260\pi\)
0.999068 0.0431528i \(-0.0137402\pi\)
\(348\) 0 0
\(349\) 35.8564 1.91935 0.959675 0.281113i \(-0.0907035\pi\)
0.959675 + 0.281113i \(0.0907035\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 5.19615i 0.276956i
\(353\) − 0.928203i − 0.0494033i −0.999695 0.0247016i \(-0.992136\pi\)
0.999695 0.0247016i \(-0.00786357\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 12.9282 0.685193
\(357\) 0 0
\(358\) 12.0000i 0.634220i
\(359\) 20.7846 1.09697 0.548485 0.836160i \(-0.315205\pi\)
0.548485 + 0.836160i \(0.315205\pi\)
\(360\) 0 0
\(361\) −16.8564 −0.887179
\(362\) 20.5359i 1.07934i
\(363\) 0 0
\(364\) 2.92820 0.153480
\(365\) 0 0
\(366\) 0 0
\(367\) 20.0000i 1.04399i 0.852948 + 0.521996i \(0.174812\pi\)
−0.852948 + 0.521996i \(0.825188\pi\)
\(368\) 34.6410i 1.80579i
\(369\) 0 0
\(370\) 0 0
\(371\) 25.8564 1.34240
\(372\) 0 0
\(373\) − 0.392305i − 0.0203128i −0.999948 0.0101564i \(-0.996767\pi\)
0.999948 0.0101564i \(-0.00323293\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) −12.0000 −0.618853
\(377\) 5.07180i 0.261211i
\(378\) 0 0
\(379\) −9.85641 −0.506290 −0.253145 0.967428i \(-0.581465\pi\)
−0.253145 + 0.967428i \(0.581465\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) − 8.78461i − 0.449460i
\(383\) 13.8564i 0.708029i 0.935240 + 0.354015i \(0.115184\pi\)
−0.935240 + 0.354015i \(0.884816\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −6.24871 −0.318051
\(387\) 0 0
\(388\) 10.0000i 0.507673i
\(389\) 24.9282 1.26391 0.631955 0.775005i \(-0.282253\pi\)
0.631955 + 0.775005i \(0.282253\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) − 5.19615i − 0.262445i
\(393\) 0 0
\(394\) 20.7846 1.04711
\(395\) 0 0
\(396\) 0 0
\(397\) 2.00000i 0.100377i 0.998740 + 0.0501886i \(0.0159822\pi\)
−0.998740 + 0.0501886i \(0.984018\pi\)
\(398\) 29.0718i 1.45724i
\(399\) 0 0
\(400\) 0 0
\(401\) −19.8564 −0.991582 −0.495791 0.868442i \(-0.665122\pi\)
−0.495791 + 0.868442i \(0.665122\pi\)
\(402\) 0 0
\(403\) 4.28719i 0.213560i
\(404\) 10.3923 0.517036
\(405\) 0 0
\(406\) 12.0000 0.595550
\(407\) 8.92820i 0.442555i
\(408\) 0 0
\(409\) −34.7846 −1.71999 −0.859994 0.510304i \(-0.829533\pi\)
−0.859994 + 0.510304i \(0.829533\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 8.00000i 0.394132i
\(413\) 13.8564i 0.681829i
\(414\) 0 0
\(415\) 0 0
\(416\) −7.60770 −0.372998
\(417\) 0 0
\(418\) − 2.53590i − 0.124035i
\(419\) 17.0718 0.834012 0.417006 0.908904i \(-0.363079\pi\)
0.417006 + 0.908904i \(0.363079\pi\)
\(420\) 0 0
\(421\) 2.00000 0.0974740 0.0487370 0.998812i \(-0.484480\pi\)
0.0487370 + 0.998812i \(0.484480\pi\)
\(422\) − 21.4641i − 1.04486i
\(423\) 0 0
\(424\) −22.3923 −1.08747
\(425\) 0 0
\(426\) 0 0
\(427\) 4.00000i 0.193574i
\(428\) 15.4641i 0.747486i
\(429\) 0 0
\(430\) 0 0
\(431\) 32.7846 1.57918 0.789590 0.613635i \(-0.210294\pi\)
0.789590 + 0.613635i \(0.210294\pi\)
\(432\) 0 0
\(433\) − 27.8564i − 1.33869i −0.742950 0.669347i \(-0.766574\pi\)
0.742950 0.669347i \(-0.233426\pi\)
\(434\) 10.1436 0.486908
\(435\) 0 0
\(436\) −10.0000 −0.478913
\(437\) − 10.1436i − 0.485234i
\(438\) 0 0
\(439\) 29.1769 1.39254 0.696269 0.717781i \(-0.254842\pi\)
0.696269 + 0.717781i \(0.254842\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 12.0000i 0.570137i 0.958507 + 0.285069i \(0.0920164\pi\)
−0.958507 + 0.285069i \(0.907984\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 30.9282 1.46449
\(447\) 0 0
\(448\) − 2.00000i − 0.0944911i
\(449\) 14.7846 0.697729 0.348864 0.937173i \(-0.386567\pi\)
0.348864 + 0.937173i \(0.386567\pi\)
\(450\) 0 0
\(451\) 3.46410 0.163118
\(452\) − 0.928203i − 0.0436590i
\(453\) 0 0
\(454\) −14.7846 −0.693876
\(455\) 0 0
\(456\) 0 0
\(457\) − 8.39230i − 0.392575i −0.980546 0.196288i \(-0.937111\pi\)
0.980546 0.196288i \(-0.0628887\pi\)
\(458\) 6.67949i 0.312112i
\(459\) 0 0
\(460\) 0 0
\(461\) 12.2487 0.570479 0.285240 0.958456i \(-0.407927\pi\)
0.285240 + 0.958456i \(0.407927\pi\)
\(462\) 0 0
\(463\) 28.0000i 1.30127i 0.759390 + 0.650635i \(0.225497\pi\)
−0.759390 + 0.650635i \(0.774503\pi\)
\(464\) −17.3205 −0.804084
\(465\) 0 0
\(466\) 20.7846 0.962828
\(467\) − 18.9282i − 0.875893i −0.899001 0.437946i \(-0.855706\pi\)
0.899001 0.437946i \(-0.144294\pi\)
\(468\) 0 0
\(469\) −16.0000 −0.738811
\(470\) 0 0
\(471\) 0 0
\(472\) − 12.0000i − 0.552345i
\(473\) − 8.92820i − 0.410519i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) 20.7846i 0.950666i
\(479\) 12.0000 0.548294 0.274147 0.961688i \(-0.411605\pi\)
0.274147 + 0.961688i \(0.411605\pi\)
\(480\) 0 0
\(481\) −13.0718 −0.596023
\(482\) − 48.2487i − 2.19767i
\(483\) 0 0
\(484\) −1.00000 −0.0454545
\(485\) 0 0
\(486\) 0 0
\(487\) 23.7128i 1.07453i 0.843413 + 0.537265i \(0.180543\pi\)
−0.843413 + 0.537265i \(0.819457\pi\)
\(488\) − 3.46410i − 0.156813i
\(489\) 0 0
\(490\) 0 0
\(491\) 17.0718 0.770439 0.385220 0.922825i \(-0.374126\pi\)
0.385220 + 0.922825i \(0.374126\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 3.71281 0.167047
\(495\) 0 0
\(496\) −14.6410 −0.657401
\(497\) 27.7128i 1.24309i
\(498\) 0 0
\(499\) 12.7846 0.572318 0.286159 0.958182i \(-0.407621\pi\)
0.286159 + 0.958182i \(0.407621\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 44.7846i 1.99883i
\(503\) − 31.1769i − 1.39011i −0.718957 0.695055i \(-0.755380\pi\)
0.718957 0.695055i \(-0.244620\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −12.0000 −0.533465
\(507\) 0 0
\(508\) 4.92820i 0.218654i
\(509\) −7.85641 −0.348229 −0.174115 0.984725i \(-0.555706\pi\)
−0.174115 + 0.984725i \(0.555706\pi\)
\(510\) 0 0
\(511\) 24.7846 1.09641
\(512\) − 8.66025i − 0.382733i
\(513\) 0 0
\(514\) −13.6077 −0.600210
\(515\) 0 0
\(516\) 0 0
\(517\) − 6.92820i − 0.304702i
\(518\) 30.9282i 1.35891i
\(519\) 0 0
\(520\) 0 0
\(521\) −6.00000 −0.262865 −0.131432 0.991325i \(-0.541958\pi\)
−0.131432 + 0.991325i \(0.541958\pi\)
\(522\) 0 0
\(523\) 22.0000i 0.961993i 0.876723 + 0.480996i \(0.159725\pi\)
−0.876723 + 0.480996i \(0.840275\pi\)
\(524\) 5.07180 0.221562
\(525\) 0 0
\(526\) 47.5692 2.07412
\(527\) 0 0
\(528\) 0 0
\(529\) −25.0000 −1.08696
\(530\) 0 0
\(531\) 0 0
\(532\) − 2.92820i − 0.126954i
\(533\) 5.07180i 0.219684i
\(534\) 0 0
\(535\) 0 0
\(536\) 13.8564 0.598506
\(537\) 0 0
\(538\) 13.6077i 0.586669i
\(539\) 3.00000 0.129219
\(540\) 0 0
\(541\) 0.143594 0.00617357 0.00308678 0.999995i \(-0.499017\pi\)
0.00308678 + 0.999995i \(0.499017\pi\)
\(542\) 56.1051i 2.40992i
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 2.00000i 0.0855138i 0.999086 + 0.0427569i \(0.0136141\pi\)
−0.999086 + 0.0427569i \(0.986386\pi\)
\(548\) − 18.0000i − 0.768922i
\(549\) 0 0
\(550\) 0 0
\(551\) 5.07180 0.216066
\(552\) 0 0
\(553\) 26.9282i 1.14510i
\(554\) 39.0333 1.65837
\(555\) 0 0
\(556\) −8.39230 −0.355913
\(557\) − 44.7846i − 1.89758i −0.315900 0.948792i \(-0.602306\pi\)
0.315900 0.948792i \(-0.397694\pi\)
\(558\) 0 0
\(559\) 13.0718 0.552878
\(560\) 0 0
\(561\) 0 0
\(562\) − 6.00000i − 0.253095i
\(563\) − 10.3923i − 0.437983i −0.975727 0.218992i \(-0.929723\pi\)
0.975727 0.218992i \(-0.0702768\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −15.4641 −0.650005
\(567\) 0 0
\(568\) − 24.0000i − 1.00702i
\(569\) −29.3205 −1.22918 −0.614590 0.788847i \(-0.710678\pi\)
−0.614590 + 0.788847i \(0.710678\pi\)
\(570\) 0 0
\(571\) 24.3923 1.02079 0.510393 0.859941i \(-0.329500\pi\)
0.510393 + 0.859941i \(0.329500\pi\)
\(572\) − 1.46410i − 0.0612172i
\(573\) 0 0
\(574\) 12.0000 0.500870
\(575\) 0 0
\(576\) 0 0
\(577\) 22.7846i 0.948536i 0.880381 + 0.474268i \(0.157287\pi\)
−0.880381 + 0.474268i \(0.842713\pi\)
\(578\) − 29.4449i − 1.22474i
\(579\) 0 0
\(580\) 0 0
\(581\) −30.9282 −1.28312
\(582\) 0 0
\(583\) − 12.9282i − 0.535431i
\(584\) −21.4641 −0.888191
\(585\) 0 0
\(586\) 24.0000 0.991431
\(587\) 5.07180i 0.209335i 0.994507 + 0.104668i \(0.0333779\pi\)
−0.994507 + 0.104668i \(0.966622\pi\)
\(588\) 0 0
\(589\) 4.28719 0.176650
\(590\) 0 0
\(591\) 0 0
\(592\) − 44.6410i − 1.83473i
\(593\) − 32.7846i − 1.34630i −0.739505 0.673151i \(-0.764940\pi\)
0.739505 0.673151i \(-0.235060\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 8.53590 0.349644
\(597\) 0 0
\(598\) − 17.5692i − 0.718459i
\(599\) −10.1436 −0.414456 −0.207228 0.978293i \(-0.566444\pi\)
−0.207228 + 0.978293i \(0.566444\pi\)
\(600\) 0 0
\(601\) 36.6410 1.49462 0.747309 0.664477i \(-0.231345\pi\)
0.747309 + 0.664477i \(0.231345\pi\)
\(602\) − 30.9282i − 1.26054i
\(603\) 0 0
\(604\) −0.392305 −0.0159627
\(605\) 0 0
\(606\) 0 0
\(607\) 22.7846i 0.924799i 0.886672 + 0.462399i \(0.153011\pi\)
−0.886672 + 0.462399i \(0.846989\pi\)
\(608\) 7.60770i 0.308533i
\(609\) 0 0
\(610\) 0 0
\(611\) 10.1436 0.410366
\(612\) 0 0
\(613\) − 0.392305i − 0.0158450i −0.999969 0.00792252i \(-0.997478\pi\)
0.999969 0.00792252i \(-0.00252184\pi\)
\(614\) 24.2487 0.978598
\(615\) 0 0
\(616\) 3.46410 0.139573
\(617\) 23.0718i 0.928836i 0.885616 + 0.464418i \(0.153736\pi\)
−0.885616 + 0.464418i \(0.846264\pi\)
\(618\) 0 0
\(619\) −20.0000 −0.803868 −0.401934 0.915669i \(-0.631662\pi\)
−0.401934 + 0.915669i \(0.631662\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 32.7846i 1.31454i
\(623\) − 25.8564i − 1.03592i
\(624\) 0 0
\(625\) 0 0
\(626\) −12.2487 −0.489557
\(627\) 0 0
\(628\) 16.9282i 0.675509i
\(629\) 0 0
\(630\) 0 0
\(631\) −34.9282 −1.39047 −0.695235 0.718783i \(-0.744700\pi\)
−0.695235 + 0.718783i \(0.744700\pi\)
\(632\) − 23.3205i − 0.927640i
\(633\) 0 0
\(634\) 19.1769 0.761613
\(635\) 0 0
\(636\) 0 0
\(637\) 4.39230i 0.174029i
\(638\) − 6.00000i − 0.237542i
\(639\) 0 0
\(640\) 0 0
\(641\) −0.928203 −0.0366618 −0.0183309 0.999832i \(-0.505835\pi\)
−0.0183309 + 0.999832i \(0.505835\pi\)
\(642\) 0 0
\(643\) 45.5692i 1.79707i 0.438897 + 0.898537i \(0.355369\pi\)
−0.438897 + 0.898537i \(0.644631\pi\)
\(644\) −13.8564 −0.546019
\(645\) 0 0
\(646\) 0 0
\(647\) 27.7128i 1.08950i 0.838597 + 0.544752i \(0.183376\pi\)
−0.838597 + 0.544752i \(0.816624\pi\)
\(648\) 0 0
\(649\) 6.92820 0.271956
\(650\) 0 0
\(651\) 0 0
\(652\) − 17.8564i − 0.699311i
\(653\) − 7.85641i − 0.307445i −0.988114 0.153722i \(-0.950874\pi\)
0.988114 0.153722i \(-0.0491262\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −17.3205 −0.676252
\(657\) 0 0
\(658\) − 24.0000i − 0.935617i
\(659\) 39.7128 1.54699 0.773496 0.633801i \(-0.218506\pi\)
0.773496 + 0.633801i \(0.218506\pi\)
\(660\) 0 0
\(661\) 14.0000 0.544537 0.272268 0.962221i \(-0.412226\pi\)
0.272268 + 0.962221i \(0.412226\pi\)
\(662\) 30.9282i 1.20206i
\(663\) 0 0
\(664\) 26.7846 1.03944
\(665\) 0 0
\(666\) 0 0
\(667\) − 24.0000i − 0.929284i
\(668\) 10.3923i 0.402090i
\(669\) 0 0
\(670\) 0 0
\(671\) 2.00000 0.0772091
\(672\) 0 0
\(673\) − 31.3205i − 1.20732i −0.797243 0.603658i \(-0.793709\pi\)
0.797243 0.603658i \(-0.206291\pi\)
\(674\) −50.5359 −1.94657
\(675\) 0 0
\(676\) −10.8564 −0.417554
\(677\) 32.7846i 1.26001i 0.776589 + 0.630007i \(0.216948\pi\)
−0.776589 + 0.630007i \(0.783052\pi\)
\(678\) 0 0
\(679\) 20.0000 0.767530
\(680\) 0 0
\(681\) 0 0
\(682\) − 5.07180i − 0.194209i
\(683\) 8.78461i 0.336134i 0.985776 + 0.168067i \(0.0537525\pi\)
−0.985776 + 0.168067i \(0.946248\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 34.6410 1.32260
\(687\) 0 0
\(688\) 44.6410i 1.70192i
\(689\) 18.9282 0.721107
\(690\) 0 0
\(691\) −7.71281 −0.293409 −0.146705 0.989180i \(-0.546867\pi\)
−0.146705 + 0.989180i \(0.546867\pi\)
\(692\) 12.0000i 0.456172i
\(693\) 0 0
\(694\) −2.78461 −0.105702
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) − 62.1051i − 2.35071i
\(699\) 0 0
\(700\) 0 0
\(701\) −32.5359 −1.22886 −0.614432 0.788970i \(-0.710615\pi\)
−0.614432 + 0.788970i \(0.710615\pi\)
\(702\) 0 0
\(703\) 13.0718i 0.493012i
\(704\) −1.00000 −0.0376889
\(705\) 0 0
\(706\) −1.60770 −0.0605064
\(707\) − 20.7846i − 0.781686i
\(708\) 0 0
\(709\) −15.8564 −0.595500 −0.297750 0.954644i \(-0.596236\pi\)
−0.297750 + 0.954644i \(0.596236\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 22.3923i 0.839187i
\(713\) − 20.2872i − 0.759761i
\(714\) 0 0
\(715\) 0 0
\(716\) 6.92820 0.258919
\(717\) 0 0
\(718\) − 36.0000i − 1.34351i
\(719\) −18.9282 −0.705903 −0.352951 0.935642i \(-0.614822\pi\)
−0.352951 + 0.935642i \(0.614822\pi\)
\(720\) 0 0
\(721\) 16.0000 0.595871
\(722\) 29.1962i 1.08657i
\(723\) 0 0
\(724\) 11.8564 0.440640
\(725\) 0 0
\(726\) 0 0
\(727\) 8.00000i 0.296704i 0.988935 + 0.148352i \(0.0473968\pi\)
−0.988935 + 0.148352i \(0.952603\pi\)
\(728\) 5.07180i 0.187973i
\(729\) 0 0
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 49.9615i 1.84537i 0.385553 + 0.922686i \(0.374011\pi\)
−0.385553 + 0.922686i \(0.625989\pi\)
\(734\) 34.6410 1.27862
\(735\) 0 0
\(736\) 36.0000 1.32698
\(737\) 8.00000i 0.294684i
\(738\) 0 0
\(739\) −10.5359 −0.387569 −0.193785 0.981044i \(-0.562076\pi\)
−0.193785 + 0.981044i \(0.562076\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) − 44.7846i − 1.64409i
\(743\) 46.3923i 1.70197i 0.525191 + 0.850984i \(0.323994\pi\)
−0.525191 + 0.850984i \(0.676006\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −0.679492 −0.0248780
\(747\) 0 0
\(748\) 0 0
\(749\) 30.9282 1.13009
\(750\) 0 0
\(751\) 13.0718 0.476997 0.238498 0.971143i \(-0.423345\pi\)
0.238498 + 0.971143i \(0.423345\pi\)
\(752\) 34.6410i 1.26323i
\(753\) 0 0
\(754\) 8.78461 0.319917
\(755\) 0 0
\(756\) 0 0
\(757\) − 6.78461i − 0.246591i −0.992370 0.123295i \(-0.960654\pi\)
0.992370 0.123295i \(-0.0393463\pi\)
\(758\) 17.0718i 0.620076i
\(759\) 0 0
\(760\) 0 0
\(761\) 39.4641 1.43057 0.715286 0.698832i \(-0.246296\pi\)
0.715286 + 0.698832i \(0.246296\pi\)
\(762\) 0 0
\(763\) 20.0000i 0.724049i
\(764\) −5.07180 −0.183491
\(765\) 0 0
\(766\) 24.0000 0.867155
\(767\) 10.1436i 0.366264i
\(768\) 0 0
\(769\) 46.4974 1.67674 0.838370 0.545102i \(-0.183509\pi\)
0.838370 + 0.545102i \(0.183509\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 3.60770i 0.129844i
\(773\) − 31.8564i − 1.14580i −0.819627 0.572898i \(-0.805819\pi\)
0.819627 0.572898i \(-0.194181\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −17.3205 −0.621770
\(777\) 0 0
\(778\) − 43.1769i − 1.54797i
\(779\) 5.07180 0.181716
\(780\) 0 0
\(781\) 13.8564 0.495821
\(782\) 0 0
\(783\) 0 0
\(784\) −15.0000 −0.535714
\(785\) 0 0
\(786\) 0 0
\(787\) − 18.7846i − 0.669599i −0.942289 0.334800i \(-0.891331\pi\)
0.942289 0.334800i \(-0.108669\pi\)
\(788\) − 12.0000i − 0.427482i
\(789\) 0 0
\(790\) 0 0
\(791\) −1.85641 −0.0660062
\(792\) 0 0
\(793\) 2.92820i 0.103984i
\(794\) 3.46410 0.122936
\(795\) 0 0
\(796\) 16.7846 0.594915
\(797\) − 16.6410i − 0.589455i −0.955581 0.294728i \(-0.904771\pi\)
0.955581 0.294728i \(-0.0952289\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) 34.3923i 1.21443i
\(803\) − 12.3923i − 0.437315i
\(804\) 0 0
\(805\) 0 0
\(806\) 7.42563 0.261557
\(807\) 0 0
\(808\) 18.0000i 0.633238i
\(809\) −8.53590 −0.300106 −0.150053 0.988678i \(-0.547944\pi\)
−0.150053 + 0.988678i \(0.547944\pi\)
\(810\) 0 0
\(811\) −8.39230 −0.294694 −0.147347 0.989085i \(-0.547073\pi\)
−0.147347 + 0.989085i \(0.547073\pi\)
\(812\) − 6.92820i − 0.243132i
\(813\) 0 0
\(814\) 15.4641 0.542016
\(815\) 0 0
\(816\) 0 0
\(817\) − 13.0718i − 0.457324i
\(818\) 60.2487i 2.10655i
\(819\) 0 0
\(820\) 0 0
\(821\) −27.4641 −0.958504 −0.479252 0.877677i \(-0.659092\pi\)
−0.479252 + 0.877677i \(0.659092\pi\)
\(822\) 0 0
\(823\) − 49.5692i − 1.72787i −0.503600 0.863937i \(-0.667991\pi\)
0.503600 0.863937i \(-0.332009\pi\)
\(824\) −13.8564 −0.482711
\(825\) 0 0
\(826\) 24.0000 0.835067
\(827\) − 1.60770i − 0.0559050i −0.999609 0.0279525i \(-0.991101\pi\)
0.999609 0.0279525i \(-0.00889872\pi\)
\(828\) 0 0
\(829\) 25.7128 0.893043 0.446521 0.894773i \(-0.352663\pi\)
0.446521 + 0.894773i \(0.352663\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) − 1.46410i − 0.0507586i
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) −1.46410 −0.0506370
\(837\) 0 0
\(838\) − 29.5692i − 1.02145i
\(839\) 15.2154 0.525294 0.262647 0.964892i \(-0.415405\pi\)
0.262647 + 0.964892i \(0.415405\pi\)
\(840\) 0 0
\(841\) −17.0000 −0.586207
\(842\) − 3.46410i − 0.119381i
\(843\) 0 0
\(844\) −12.3923 −0.426561
\(845\) 0 0
\(846\) 0 0
\(847\) 2.00000i 0.0687208i
\(848\) 64.6410i 2.21978i
\(849\) 0 0
\(850\) 0 0
\(851\) 61.8564 2.12041
\(852\) 0 0
\(853\) − 24.3923i − 0.835177i −0.908636 0.417588i \(-0.862875\pi\)
0.908636 0.417588i \(-0.137125\pi\)
\(854\) 6.92820 0.237078
\(855\) 0 0
\(856\) −26.7846 −0.915479
\(857\) 10.1436i 0.346499i 0.984878 + 0.173249i \(0.0554266\pi\)
−0.984878 + 0.173249i \(0.944573\pi\)
\(858\) 0 0
\(859\) −47.7128 −1.62794 −0.813970 0.580907i \(-0.802698\pi\)
−0.813970 + 0.580907i \(0.802698\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) − 56.7846i − 1.93409i
\(863\) − 10.1436i − 0.345292i −0.984984 0.172646i \(-0.944768\pi\)
0.984984 0.172646i \(-0.0552317\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) −48.2487 −1.63956
\(867\) 0 0
\(868\) − 5.85641i − 0.198779i
\(869\) 13.4641 0.456738
\(870\) 0 0
\(871\) −11.7128 −0.396874
\(872\) − 17.3205i − 0.586546i
\(873\) 0 0
\(874\) −17.5692 −0.594288
\(875\) 0 0
\(876\) 0 0
\(877\) 14.2487i 0.481145i 0.970631 + 0.240572i \(0.0773351\pi\)
−0.970631 + 0.240572i \(0.922665\pi\)
\(878\) − 50.5359i − 1.70550i
\(879\) 0 0
\(880\) 0 0
\(881\) −12.9282 −0.435562 −0.217781 0.975998i \(-0.569882\pi\)
−0.217781 + 0.975998i \(0.569882\pi\)
\(882\) 0 0
\(883\) 4.00000i 0.134611i 0.997732 + 0.0673054i \(0.0214402\pi\)
−0.997732 + 0.0673054i \(0.978560\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 20.7846 0.698273
\(887\) 36.2487i 1.21711i 0.793511 + 0.608556i \(0.208251\pi\)
−0.793511 + 0.608556i \(0.791749\pi\)
\(888\) 0 0
\(889\) 9.85641 0.330573
\(890\) 0 0
\(891\) 0 0
\(892\) − 17.8564i − 0.597877i
\(893\) − 10.1436i − 0.339442i
\(894\) 0 0
\(895\) 0 0
\(896\) −24.2487 −0.810093
\(897\) 0 0
\(898\) − 25.6077i − 0.854540i
\(899\) 10.1436 0.338308
\(900\) 0 0
\(901\) 0 0
\(902\) − 6.00000i − 0.199778i
\(903\) 0 0
\(904\) 1.60770 0.0534711
\(905\) 0 0
\(906\) 0 0
\(907\) 45.8564i 1.52264i 0.648378 + 0.761318i \(0.275448\pi\)
−0.648378 + 0.761318i \(0.724552\pi\)
\(908\) 8.53590i 0.283274i
\(909\) 0 0
\(910\) 0 0
\(911\) −5.07180 −0.168036 −0.0840181 0.996464i \(-0.526775\pi\)
−0.0840181 + 0.996464i \(0.526775\pi\)
\(912\) 0 0
\(913\) 15.4641i 0.511787i
\(914\) −14.5359 −0.480805
\(915\) 0 0
\(916\) 3.85641 0.127419
\(917\) − 10.1436i − 0.334971i
\(918\) 0 0
\(919\) 11.6077 0.382903 0.191451 0.981502i \(-0.438681\pi\)
0.191451 + 0.981502i \(0.438681\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) − 21.2154i − 0.698692i
\(923\) 20.2872i 0.667761i
\(924\) 0 0
\(925\) 0 0
\(926\) 48.4974 1.59372
\(927\) 0 0
\(928\) 18.0000i 0.590879i
\(929\) −38.7846 −1.27248 −0.636241 0.771490i \(-0.719512\pi\)
−0.636241 + 0.771490i \(0.719512\pi\)
\(930\) 0 0
\(931\) 4.39230 0.143952
\(932\) − 12.0000i − 0.393073i
\(933\) 0 0
\(934\) −32.7846 −1.07275
\(935\) 0 0
\(936\) 0 0
\(937\) 0.392305i 0.0128160i 0.999979 + 0.00640802i \(0.00203975\pi\)
−0.999979 + 0.00640802i \(0.997960\pi\)
\(938\) 27.7128i 0.904855i
\(939\) 0 0
\(940\) 0 0
\(941\) −20.5359 −0.669451 −0.334726 0.942316i \(-0.608644\pi\)
−0.334726 + 0.942316i \(0.608644\pi\)
\(942\) 0 0
\(943\) − 24.0000i − 0.781548i
\(944\) −34.6410 −1.12747
\(945\) 0 0
\(946\) −15.4641 −0.502781
\(947\) 5.07180i 0.164811i 0.996599 + 0.0824056i \(0.0262603\pi\)
−0.996599 + 0.0824056i \(0.973740\pi\)
\(948\) 0 0
\(949\) 18.1436 0.588966
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) − 44.7846i − 1.45072i −0.688372 0.725358i \(-0.741674\pi\)
0.688372 0.725358i \(-0.258326\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 12.0000 0.388108
\(957\) 0 0
\(958\) − 20.7846i − 0.671520i
\(959\) −36.0000 −1.16250
\(960\) 0 0
\(961\) −22.4256 −0.723407
\(962\) 22.6410i 0.729976i
\(963\) 0 0
\(964\) −27.8564 −0.897194
\(965\) 0 0
\(966\) 0 0
\(967\) − 18.7846i − 0.604072i −0.953296 0.302036i \(-0.902334\pi\)
0.953296 0.302036i \(-0.0976663\pi\)
\(968\) − 1.73205i − 0.0556702i
\(969\) 0 0
\(970\) 0 0
\(971\) 1.85641 0.0595749 0.0297875 0.999556i \(-0.490517\pi\)
0.0297875 + 0.999556i \(0.490517\pi\)
\(972\) 0 0
\(973\) 16.7846i 0.538090i
\(974\) 41.0718 1.31603
\(975\) 0 0
\(976\) −10.0000 −0.320092
\(977\) 35.5692i 1.13796i 0.822351 + 0.568980i \(0.192662\pi\)
−0.822351 + 0.568980i \(0.807338\pi\)
\(978\) 0 0
\(979\) −12.9282 −0.413187
\(980\) 0 0
\(981\) 0 0
\(982\) − 29.5692i − 0.943592i
\(983\) 24.0000i 0.765481i 0.923856 + 0.382741i \(0.125020\pi\)
−0.923856 + 0.382741i \(0.874980\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) − 2.14359i − 0.0681968i
\(989\) −61.8564 −1.96692
\(990\) 0 0
\(991\) −48.7846 −1.54969 −0.774847 0.632149i \(-0.782173\pi\)
−0.774847 + 0.632149i \(0.782173\pi\)
\(992\) 15.2154i 0.483089i
\(993\) 0 0
\(994\) 48.0000 1.52247
\(995\) 0 0
\(996\) 0 0
\(997\) 48.3923i 1.53260i 0.642483 + 0.766300i \(0.277904\pi\)
−0.642483 + 0.766300i \(0.722096\pi\)
\(998\) − 22.1436i − 0.700943i
\(999\) 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 2475.2.c.n.199.2 4
3.2 odd 2 825.2.c.c.199.3 4
5.2 odd 4 2475.2.a.r.1.2 2
5.3 odd 4 495.2.a.c.1.1 2
5.4 even 2 inner 2475.2.c.n.199.3 4
15.2 even 4 825.2.a.e.1.1 2
15.8 even 4 165.2.a.b.1.2 2
15.14 odd 2 825.2.c.c.199.2 4
20.3 even 4 7920.2.a.bz.1.1 2
55.43 even 4 5445.2.a.s.1.2 2
60.23 odd 4 2640.2.a.x.1.1 2
105.83 odd 4 8085.2.a.bd.1.2 2
165.32 odd 4 9075.2.a.bh.1.2 2
165.98 odd 4 1815.2.a.i.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
165.2.a.b.1.2 2 15.8 even 4
495.2.a.c.1.1 2 5.3 odd 4
825.2.a.e.1.1 2 15.2 even 4
825.2.c.c.199.2 4 15.14 odd 2
825.2.c.c.199.3 4 3.2 odd 2
1815.2.a.i.1.1 2 165.98 odd 4
2475.2.a.r.1.2 2 5.2 odd 4
2475.2.c.n.199.2 4 1.1 even 1 trivial
2475.2.c.n.199.3 4 5.4 even 2 inner
2640.2.a.x.1.1 2 60.23 odd 4
5445.2.a.s.1.2 2 55.43 even 4
7920.2.a.bz.1.1 2 20.3 even 4
8085.2.a.bd.1.2 2 105.83 odd 4
9075.2.a.bh.1.2 2 165.32 odd 4