Properties

Label 2025.2.b.d.649.2
Level $2025$
Weight $2$
Character 2025.649
Analytic conductor $16.170$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 2025 = 3^{4} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2025.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(16.1697064093\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 45)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 649.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 2025.649
Dual form 2025.2.b.d.649.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000i q^{2} +1.00000 q^{4} +3.00000i q^{7} +3.00000i q^{8} +O(q^{10})\) \(q+1.00000i q^{2} +1.00000 q^{4} +3.00000i q^{7} +3.00000i q^{8} +2.00000 q^{11} -2.00000i q^{13} -3.00000 q^{14} -1.00000 q^{16} +4.00000i q^{17} +8.00000 q^{19} +2.00000i q^{22} -3.00000i q^{23} +2.00000 q^{26} +3.00000i q^{28} -1.00000 q^{29} +5.00000i q^{32} -4.00000 q^{34} +4.00000i q^{37} +8.00000i q^{38} -5.00000 q^{41} -8.00000i q^{43} +2.00000 q^{44} +3.00000 q^{46} +7.00000i q^{47} -2.00000 q^{49} -2.00000i q^{52} +2.00000i q^{53} -9.00000 q^{56} -1.00000i q^{58} -14.0000 q^{59} +7.00000 q^{61} -7.00000 q^{64} +3.00000i q^{67} +4.00000i q^{68} -2.00000 q^{71} +4.00000i q^{73} -4.00000 q^{74} +8.00000 q^{76} +6.00000i q^{77} +6.00000 q^{79} -5.00000i q^{82} -9.00000i q^{83} +8.00000 q^{86} +6.00000i q^{88} -15.0000 q^{89} +6.00000 q^{91} -3.00000i q^{92} -7.00000 q^{94} -2.00000i q^{97} -2.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{4} + 4 q^{11} - 6 q^{14} - 2 q^{16} + 16 q^{19} + 4 q^{26} - 2 q^{29} - 8 q^{34} - 10 q^{41} + 4 q^{44} + 6 q^{46} - 4 q^{49} - 18 q^{56} - 28 q^{59} + 14 q^{61} - 14 q^{64} - 4 q^{71} - 8 q^{74} + 16 q^{76} + 12 q^{79} + 16 q^{86} - 30 q^{89} + 12 q^{91} - 14 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2025\mathbb{Z}\right)^\times\).

\(n\) \(326\) \(1702\)
\(\chi(n)\) \(1\) \(-1\)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000i 0.707107i 0.935414 + 0.353553i \(0.115027\pi\)
−0.935414 + 0.353553i \(0.884973\pi\)
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) 0 0
\(7\) 3.00000i 1.13389i 0.823754 + 0.566947i \(0.191875\pi\)
−0.823754 + 0.566947i \(0.808125\pi\)
\(8\) 3.00000i 1.06066i
\(9\) 0 0
\(10\) 0 0
\(11\) 2.00000 0.603023 0.301511 0.953463i \(-0.402509\pi\)
0.301511 + 0.953463i \(0.402509\pi\)
\(12\) 0 0
\(13\) − 2.00000i − 0.554700i −0.960769 0.277350i \(-0.910544\pi\)
0.960769 0.277350i \(-0.0894562\pi\)
\(14\) −3.00000 −0.801784
\(15\) 0 0
\(16\) −1.00000 −0.250000
\(17\) 4.00000i 0.970143i 0.874475 + 0.485071i \(0.161206\pi\)
−0.874475 + 0.485071i \(0.838794\pi\)
\(18\) 0 0
\(19\) 8.00000 1.83533 0.917663 0.397360i \(-0.130073\pi\)
0.917663 + 0.397360i \(0.130073\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 2.00000i 0.426401i
\(23\) − 3.00000i − 0.625543i −0.949828 0.312772i \(-0.898743\pi\)
0.949828 0.312772i \(-0.101257\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 2.00000 0.392232
\(27\) 0 0
\(28\) 3.00000i 0.566947i
\(29\) −1.00000 −0.185695 −0.0928477 0.995680i \(-0.529597\pi\)
−0.0928477 + 0.995680i \(0.529597\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 5.00000i 0.883883i
\(33\) 0 0
\(34\) −4.00000 −0.685994
\(35\) 0 0
\(36\) 0 0
\(37\) 4.00000i 0.657596i 0.944400 + 0.328798i \(0.106644\pi\)
−0.944400 + 0.328798i \(0.893356\pi\)
\(38\) 8.00000i 1.29777i
\(39\) 0 0
\(40\) 0 0
\(41\) −5.00000 −0.780869 −0.390434 0.920631i \(-0.627675\pi\)
−0.390434 + 0.920631i \(0.627675\pi\)
\(42\) 0 0
\(43\) − 8.00000i − 1.21999i −0.792406 0.609994i \(-0.791172\pi\)
0.792406 0.609994i \(-0.208828\pi\)
\(44\) 2.00000 0.301511
\(45\) 0 0
\(46\) 3.00000 0.442326
\(47\) 7.00000i 1.02105i 0.859861 + 0.510527i \(0.170550\pi\)
−0.859861 + 0.510527i \(0.829450\pi\)
\(48\) 0 0
\(49\) −2.00000 −0.285714
\(50\) 0 0
\(51\) 0 0
\(52\) − 2.00000i − 0.277350i
\(53\) 2.00000i 0.274721i 0.990521 + 0.137361i \(0.0438619\pi\)
−0.990521 + 0.137361i \(0.956138\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −9.00000 −1.20268
\(57\) 0 0
\(58\) − 1.00000i − 0.131306i
\(59\) −14.0000 −1.82264 −0.911322 0.411693i \(-0.864937\pi\)
−0.911322 + 0.411693i \(0.864937\pi\)
\(60\) 0 0
\(61\) 7.00000 0.896258 0.448129 0.893969i \(-0.352090\pi\)
0.448129 + 0.893969i \(0.352090\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) −7.00000 −0.875000
\(65\) 0 0
\(66\) 0 0
\(67\) 3.00000i 0.366508i 0.983066 + 0.183254i \(0.0586631\pi\)
−0.983066 + 0.183254i \(0.941337\pi\)
\(68\) 4.00000i 0.485071i
\(69\) 0 0
\(70\) 0 0
\(71\) −2.00000 −0.237356 −0.118678 0.992933i \(-0.537866\pi\)
−0.118678 + 0.992933i \(0.537866\pi\)
\(72\) 0 0
\(73\) 4.00000i 0.468165i 0.972217 + 0.234082i \(0.0752085\pi\)
−0.972217 + 0.234082i \(0.924791\pi\)
\(74\) −4.00000 −0.464991
\(75\) 0 0
\(76\) 8.00000 0.917663
\(77\) 6.00000i 0.683763i
\(78\) 0 0
\(79\) 6.00000 0.675053 0.337526 0.941316i \(-0.390410\pi\)
0.337526 + 0.941316i \(0.390410\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) − 5.00000i − 0.552158i
\(83\) − 9.00000i − 0.987878i −0.869496 0.493939i \(-0.835557\pi\)
0.869496 0.493939i \(-0.164443\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 8.00000 0.862662
\(87\) 0 0
\(88\) 6.00000i 0.639602i
\(89\) −15.0000 −1.59000 −0.794998 0.606612i \(-0.792528\pi\)
−0.794998 + 0.606612i \(0.792528\pi\)
\(90\) 0 0
\(91\) 6.00000 0.628971
\(92\) − 3.00000i − 0.312772i
\(93\) 0 0
\(94\) −7.00000 −0.721995
\(95\) 0 0
\(96\) 0 0
\(97\) − 2.00000i − 0.203069i −0.994832 0.101535i \(-0.967625\pi\)
0.994832 0.101535i \(-0.0323753\pi\)
\(98\) − 2.00000i − 0.202031i
\(99\) 0 0
\(100\) 0 0
\(101\) 18.0000 1.79107 0.895533 0.444994i \(-0.146794\pi\)
0.895533 + 0.444994i \(0.146794\pi\)
\(102\) 0 0
\(103\) 8.00000i 0.788263i 0.919054 + 0.394132i \(0.128955\pi\)
−0.919054 + 0.394132i \(0.871045\pi\)
\(104\) 6.00000 0.588348
\(105\) 0 0
\(106\) −2.00000 −0.194257
\(107\) 3.00000i 0.290021i 0.989430 + 0.145010i \(0.0463216\pi\)
−0.989430 + 0.145010i \(0.953678\pi\)
\(108\) 0 0
\(109\) −5.00000 −0.478913 −0.239457 0.970907i \(-0.576969\pi\)
−0.239457 + 0.970907i \(0.576969\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) − 3.00000i − 0.283473i
\(113\) 8.00000i 0.752577i 0.926503 + 0.376288i \(0.122800\pi\)
−0.926503 + 0.376288i \(0.877200\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −1.00000 −0.0928477
\(117\) 0 0
\(118\) − 14.0000i − 1.28880i
\(119\) −12.0000 −1.10004
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) 7.00000i 0.633750i
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 5.00000i 0.443678i 0.975083 + 0.221839i \(0.0712060\pi\)
−0.975083 + 0.221839i \(0.928794\pi\)
\(128\) 3.00000i 0.265165i
\(129\) 0 0
\(130\) 0 0
\(131\) 6.00000 0.524222 0.262111 0.965038i \(-0.415581\pi\)
0.262111 + 0.965038i \(0.415581\pi\)
\(132\) 0 0
\(133\) 24.0000i 2.08106i
\(134\) −3.00000 −0.259161
\(135\) 0 0
\(136\) −12.0000 −1.02899
\(137\) 12.0000i 1.02523i 0.858619 + 0.512615i \(0.171323\pi\)
−0.858619 + 0.512615i \(0.828677\pi\)
\(138\) 0 0
\(139\) 16.0000 1.35710 0.678551 0.734553i \(-0.262608\pi\)
0.678551 + 0.734553i \(0.262608\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) − 2.00000i − 0.167836i
\(143\) − 4.00000i − 0.334497i
\(144\) 0 0
\(145\) 0 0
\(146\) −4.00000 −0.331042
\(147\) 0 0
\(148\) 4.00000i 0.328798i
\(149\) 17.0000 1.39269 0.696347 0.717705i \(-0.254807\pi\)
0.696347 + 0.717705i \(0.254807\pi\)
\(150\) 0 0
\(151\) −2.00000 −0.162758 −0.0813788 0.996683i \(-0.525932\pi\)
−0.0813788 + 0.996683i \(0.525932\pi\)
\(152\) 24.0000i 1.94666i
\(153\) 0 0
\(154\) −6.00000 −0.483494
\(155\) 0 0
\(156\) 0 0
\(157\) − 14.0000i − 1.11732i −0.829396 0.558661i \(-0.811315\pi\)
0.829396 0.558661i \(-0.188685\pi\)
\(158\) 6.00000i 0.477334i
\(159\) 0 0
\(160\) 0 0
\(161\) 9.00000 0.709299
\(162\) 0 0
\(163\) − 4.00000i − 0.313304i −0.987654 0.156652i \(-0.949930\pi\)
0.987654 0.156652i \(-0.0500701\pi\)
\(164\) −5.00000 −0.390434
\(165\) 0 0
\(166\) 9.00000 0.698535
\(167\) − 9.00000i − 0.696441i −0.937413 0.348220i \(-0.886786\pi\)
0.937413 0.348220i \(-0.113214\pi\)
\(168\) 0 0
\(169\) 9.00000 0.692308
\(170\) 0 0
\(171\) 0 0
\(172\) − 8.00000i − 0.609994i
\(173\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −2.00000 −0.150756
\(177\) 0 0
\(178\) − 15.0000i − 1.12430i
\(179\) −2.00000 −0.149487 −0.0747435 0.997203i \(-0.523814\pi\)
−0.0747435 + 0.997203i \(0.523814\pi\)
\(180\) 0 0
\(181\) −7.00000 −0.520306 −0.260153 0.965567i \(-0.583773\pi\)
−0.260153 + 0.965567i \(0.583773\pi\)
\(182\) 6.00000i 0.444750i
\(183\) 0 0
\(184\) 9.00000 0.663489
\(185\) 0 0
\(186\) 0 0
\(187\) 8.00000i 0.585018i
\(188\) 7.00000i 0.510527i
\(189\) 0 0
\(190\) 0 0
\(191\) −8.00000 −0.578860 −0.289430 0.957199i \(-0.593466\pi\)
−0.289430 + 0.957199i \(0.593466\pi\)
\(192\) 0 0
\(193\) − 10.0000i − 0.719816i −0.932988 0.359908i \(-0.882808\pi\)
0.932988 0.359908i \(-0.117192\pi\)
\(194\) 2.00000 0.143592
\(195\) 0 0
\(196\) −2.00000 −0.142857
\(197\) − 12.0000i − 0.854965i −0.904024 0.427482i \(-0.859401\pi\)
0.904024 0.427482i \(-0.140599\pi\)
\(198\) 0 0
\(199\) −4.00000 −0.283552 −0.141776 0.989899i \(-0.545281\pi\)
−0.141776 + 0.989899i \(0.545281\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 18.0000i 1.26648i
\(203\) − 3.00000i − 0.210559i
\(204\) 0 0
\(205\) 0 0
\(206\) −8.00000 −0.557386
\(207\) 0 0
\(208\) 2.00000i 0.138675i
\(209\) 16.0000 1.10674
\(210\) 0 0
\(211\) −22.0000 −1.51454 −0.757271 0.653101i \(-0.773468\pi\)
−0.757271 + 0.653101i \(0.773468\pi\)
\(212\) 2.00000i 0.137361i
\(213\) 0 0
\(214\) −3.00000 −0.205076
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) − 5.00000i − 0.338643i
\(219\) 0 0
\(220\) 0 0
\(221\) 8.00000 0.538138
\(222\) 0 0
\(223\) − 19.0000i − 1.27233i −0.771551 0.636167i \(-0.780519\pi\)
0.771551 0.636167i \(-0.219481\pi\)
\(224\) −15.0000 −1.00223
\(225\) 0 0
\(226\) −8.00000 −0.532152
\(227\) − 4.00000i − 0.265489i −0.991150 0.132745i \(-0.957621\pi\)
0.991150 0.132745i \(-0.0423790\pi\)
\(228\) 0 0
\(229\) −15.0000 −0.991228 −0.495614 0.868543i \(-0.665057\pi\)
−0.495614 + 0.868543i \(0.665057\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) − 3.00000i − 0.196960i
\(233\) − 24.0000i − 1.57229i −0.618041 0.786146i \(-0.712073\pi\)
0.618041 0.786146i \(-0.287927\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −14.0000 −0.911322
\(237\) 0 0
\(238\) − 12.0000i − 0.777844i
\(239\) −8.00000 −0.517477 −0.258738 0.965947i \(-0.583307\pi\)
−0.258738 + 0.965947i \(0.583307\pi\)
\(240\) 0 0
\(241\) −11.0000 −0.708572 −0.354286 0.935137i \(-0.615276\pi\)
−0.354286 + 0.935137i \(0.615276\pi\)
\(242\) − 7.00000i − 0.449977i
\(243\) 0 0
\(244\) 7.00000 0.448129
\(245\) 0 0
\(246\) 0 0
\(247\) − 16.0000i − 1.01806i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 0 0
\(253\) − 6.00000i − 0.377217i
\(254\) −5.00000 −0.313728
\(255\) 0 0
\(256\) −17.0000 −1.06250
\(257\) − 6.00000i − 0.374270i −0.982334 0.187135i \(-0.940080\pi\)
0.982334 0.187135i \(-0.0599201\pi\)
\(258\) 0 0
\(259\) −12.0000 −0.745644
\(260\) 0 0
\(261\) 0 0
\(262\) 6.00000i 0.370681i
\(263\) 16.0000i 0.986602i 0.869859 + 0.493301i \(0.164210\pi\)
−0.869859 + 0.493301i \(0.835790\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −24.0000 −1.47153
\(267\) 0 0
\(268\) 3.00000i 0.183254i
\(269\) 25.0000 1.52428 0.762138 0.647414i \(-0.224150\pi\)
0.762138 + 0.647414i \(0.224150\pi\)
\(270\) 0 0
\(271\) −8.00000 −0.485965 −0.242983 0.970031i \(-0.578126\pi\)
−0.242983 + 0.970031i \(0.578126\pi\)
\(272\) − 4.00000i − 0.242536i
\(273\) 0 0
\(274\) −12.0000 −0.724947
\(275\) 0 0
\(276\) 0 0
\(277\) − 12.0000i − 0.721010i −0.932757 0.360505i \(-0.882604\pi\)
0.932757 0.360505i \(-0.117396\pi\)
\(278\) 16.0000i 0.959616i
\(279\) 0 0
\(280\) 0 0
\(281\) 15.0000 0.894825 0.447412 0.894328i \(-0.352346\pi\)
0.447412 + 0.894328i \(0.352346\pi\)
\(282\) 0 0
\(283\) 21.0000i 1.24832i 0.781296 + 0.624160i \(0.214559\pi\)
−0.781296 + 0.624160i \(0.785441\pi\)
\(284\) −2.00000 −0.118678
\(285\) 0 0
\(286\) 4.00000 0.236525
\(287\) − 15.0000i − 0.885422i
\(288\) 0 0
\(289\) 1.00000 0.0588235
\(290\) 0 0
\(291\) 0 0
\(292\) 4.00000i 0.234082i
\(293\) − 12.0000i − 0.701047i −0.936554 0.350524i \(-0.886004\pi\)
0.936554 0.350524i \(-0.113996\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −12.0000 −0.697486
\(297\) 0 0
\(298\) 17.0000i 0.984784i
\(299\) −6.00000 −0.346989
\(300\) 0 0
\(301\) 24.0000 1.38334
\(302\) − 2.00000i − 0.115087i
\(303\) 0 0
\(304\) −8.00000 −0.458831
\(305\) 0 0
\(306\) 0 0
\(307\) − 7.00000i − 0.399511i −0.979846 0.199756i \(-0.935985\pi\)
0.979846 0.199756i \(-0.0640148\pi\)
\(308\) 6.00000i 0.341882i
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) 14.0000i 0.791327i 0.918396 + 0.395663i \(0.129485\pi\)
−0.918396 + 0.395663i \(0.870515\pi\)
\(314\) 14.0000 0.790066
\(315\) 0 0
\(316\) 6.00000 0.337526
\(317\) − 34.0000i − 1.90963i −0.297200 0.954815i \(-0.596053\pi\)
0.297200 0.954815i \(-0.403947\pi\)
\(318\) 0 0
\(319\) −2.00000 −0.111979
\(320\) 0 0
\(321\) 0 0
\(322\) 9.00000i 0.501550i
\(323\) 32.0000i 1.78053i
\(324\) 0 0
\(325\) 0 0
\(326\) 4.00000 0.221540
\(327\) 0 0
\(328\) − 15.0000i − 0.828236i
\(329\) −21.0000 −1.15777
\(330\) 0 0
\(331\) −6.00000 −0.329790 −0.164895 0.986311i \(-0.552728\pi\)
−0.164895 + 0.986311i \(0.552728\pi\)
\(332\) − 9.00000i − 0.493939i
\(333\) 0 0
\(334\) 9.00000 0.492458
\(335\) 0 0
\(336\) 0 0
\(337\) 8.00000i 0.435788i 0.975972 + 0.217894i \(0.0699187\pi\)
−0.975972 + 0.217894i \(0.930081\pi\)
\(338\) 9.00000i 0.489535i
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 15.0000i 0.809924i
\(344\) 24.0000 1.29399
\(345\) 0 0
\(346\) 0 0
\(347\) 4.00000i 0.214731i 0.994220 + 0.107366i \(0.0342415\pi\)
−0.994220 + 0.107366i \(0.965758\pi\)
\(348\) 0 0
\(349\) 5.00000 0.267644 0.133822 0.991005i \(-0.457275\pi\)
0.133822 + 0.991005i \(0.457275\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 10.0000i 0.533002i
\(353\) − 24.0000i − 1.27739i −0.769460 0.638696i \(-0.779474\pi\)
0.769460 0.638696i \(-0.220526\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −15.0000 −0.794998
\(357\) 0 0
\(358\) − 2.00000i − 0.105703i
\(359\) 24.0000 1.26667 0.633336 0.773877i \(-0.281685\pi\)
0.633336 + 0.773877i \(0.281685\pi\)
\(360\) 0 0
\(361\) 45.0000 2.36842
\(362\) − 7.00000i − 0.367912i
\(363\) 0 0
\(364\) 6.00000 0.314485
\(365\) 0 0
\(366\) 0 0
\(367\) − 24.0000i − 1.25279i −0.779506 0.626395i \(-0.784530\pi\)
0.779506 0.626395i \(-0.215470\pi\)
\(368\) 3.00000i 0.156386i
\(369\) 0 0
\(370\) 0 0
\(371\) −6.00000 −0.311504
\(372\) 0 0
\(373\) 10.0000i 0.517780i 0.965907 + 0.258890i \(0.0833568\pi\)
−0.965907 + 0.258890i \(0.916643\pi\)
\(374\) −8.00000 −0.413670
\(375\) 0 0
\(376\) −21.0000 −1.08299
\(377\) 2.00000i 0.103005i
\(378\) 0 0
\(379\) 26.0000 1.33553 0.667765 0.744372i \(-0.267251\pi\)
0.667765 + 0.744372i \(0.267251\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) − 8.00000i − 0.409316i
\(383\) − 36.0000i − 1.83951i −0.392488 0.919757i \(-0.628386\pi\)
0.392488 0.919757i \(-0.371614\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 10.0000 0.508987
\(387\) 0 0
\(388\) − 2.00000i − 0.101535i
\(389\) 33.0000 1.67317 0.836583 0.547840i \(-0.184550\pi\)
0.836583 + 0.547840i \(0.184550\pi\)
\(390\) 0 0
\(391\) 12.0000 0.606866
\(392\) − 6.00000i − 0.303046i
\(393\) 0 0
\(394\) 12.0000 0.604551
\(395\) 0 0
\(396\) 0 0
\(397\) − 34.0000i − 1.70641i −0.521575 0.853206i \(-0.674655\pi\)
0.521575 0.853206i \(-0.325345\pi\)
\(398\) − 4.00000i − 0.200502i
\(399\) 0 0
\(400\) 0 0
\(401\) 18.0000 0.898877 0.449439 0.893311i \(-0.351624\pi\)
0.449439 + 0.893311i \(0.351624\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 18.0000 0.895533
\(405\) 0 0
\(406\) 3.00000 0.148888
\(407\) 8.00000i 0.396545i
\(408\) 0 0
\(409\) −14.0000 −0.692255 −0.346128 0.938187i \(-0.612504\pi\)
−0.346128 + 0.938187i \(0.612504\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 8.00000i 0.394132i
\(413\) − 42.0000i − 2.06668i
\(414\) 0 0
\(415\) 0 0
\(416\) 10.0000 0.490290
\(417\) 0 0
\(418\) 16.0000i 0.782586i
\(419\) 26.0000 1.27018 0.635092 0.772437i \(-0.280962\pi\)
0.635092 + 0.772437i \(0.280962\pi\)
\(420\) 0 0
\(421\) 34.0000 1.65706 0.828529 0.559946i \(-0.189178\pi\)
0.828529 + 0.559946i \(0.189178\pi\)
\(422\) − 22.0000i − 1.07094i
\(423\) 0 0
\(424\) −6.00000 −0.291386
\(425\) 0 0
\(426\) 0 0
\(427\) 21.0000i 1.01626i
\(428\) 3.00000i 0.145010i
\(429\) 0 0
\(430\) 0 0
\(431\) 30.0000 1.44505 0.722525 0.691345i \(-0.242982\pi\)
0.722525 + 0.691345i \(0.242982\pi\)
\(432\) 0 0
\(433\) 28.0000i 1.34559i 0.739827 + 0.672797i \(0.234907\pi\)
−0.739827 + 0.672797i \(0.765093\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −5.00000 −0.239457
\(437\) − 24.0000i − 1.14808i
\(438\) 0 0
\(439\) −28.0000 −1.33637 −0.668184 0.743996i \(-0.732928\pi\)
−0.668184 + 0.743996i \(0.732928\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 8.00000i 0.380521i
\(443\) − 15.0000i − 0.712672i −0.934358 0.356336i \(-0.884026\pi\)
0.934358 0.356336i \(-0.115974\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 19.0000 0.899676
\(447\) 0 0
\(448\) − 21.0000i − 0.992157i
\(449\) −26.0000 −1.22702 −0.613508 0.789689i \(-0.710242\pi\)
−0.613508 + 0.789689i \(0.710242\pi\)
\(450\) 0 0
\(451\) −10.0000 −0.470882
\(452\) 8.00000i 0.376288i
\(453\) 0 0
\(454\) 4.00000 0.187729
\(455\) 0 0
\(456\) 0 0
\(457\) 20.0000i 0.935561i 0.883845 + 0.467780i \(0.154946\pi\)
−0.883845 + 0.467780i \(0.845054\pi\)
\(458\) − 15.0000i − 0.700904i
\(459\) 0 0
\(460\) 0 0
\(461\) −9.00000 −0.419172 −0.209586 0.977790i \(-0.567212\pi\)
−0.209586 + 0.977790i \(0.567212\pi\)
\(462\) 0 0
\(463\) − 36.0000i − 1.67306i −0.547920 0.836531i \(-0.684580\pi\)
0.547920 0.836531i \(-0.315420\pi\)
\(464\) 1.00000 0.0464238
\(465\) 0 0
\(466\) 24.0000 1.11178
\(467\) 20.0000i 0.925490i 0.886492 + 0.462745i \(0.153135\pi\)
−0.886492 + 0.462745i \(0.846865\pi\)
\(468\) 0 0
\(469\) −9.00000 −0.415581
\(470\) 0 0
\(471\) 0 0
\(472\) − 42.0000i − 1.93321i
\(473\) − 16.0000i − 0.735681i
\(474\) 0 0
\(475\) 0 0
\(476\) −12.0000 −0.550019
\(477\) 0 0
\(478\) − 8.00000i − 0.365911i
\(479\) −18.0000 −0.822441 −0.411220 0.911536i \(-0.634897\pi\)
−0.411220 + 0.911536i \(0.634897\pi\)
\(480\) 0 0
\(481\) 8.00000 0.364769
\(482\) − 11.0000i − 0.501036i
\(483\) 0 0
\(484\) −7.00000 −0.318182
\(485\) 0 0
\(486\) 0 0
\(487\) 16.0000i 0.725029i 0.931978 + 0.362515i \(0.118082\pi\)
−0.931978 + 0.362515i \(0.881918\pi\)
\(488\) 21.0000i 0.950625i
\(489\) 0 0
\(490\) 0 0
\(491\) −20.0000 −0.902587 −0.451294 0.892375i \(-0.649037\pi\)
−0.451294 + 0.892375i \(0.649037\pi\)
\(492\) 0 0
\(493\) − 4.00000i − 0.180151i
\(494\) 16.0000 0.719874
\(495\) 0 0
\(496\) 0 0
\(497\) − 6.00000i − 0.269137i
\(498\) 0 0
\(499\) 32.0000 1.43252 0.716258 0.697835i \(-0.245853\pi\)
0.716258 + 0.697835i \(0.245853\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 7.00000i 0.312115i 0.987748 + 0.156057i \(0.0498784\pi\)
−0.987748 + 0.156057i \(0.950122\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 6.00000 0.266733
\(507\) 0 0
\(508\) 5.00000i 0.221839i
\(509\) 43.0000 1.90594 0.952971 0.303062i \(-0.0980090\pi\)
0.952971 + 0.303062i \(0.0980090\pi\)
\(510\) 0 0
\(511\) −12.0000 −0.530849
\(512\) − 11.0000i − 0.486136i
\(513\) 0 0
\(514\) 6.00000 0.264649
\(515\) 0 0
\(516\) 0 0
\(517\) 14.0000i 0.615719i
\(518\) − 12.0000i − 0.527250i
\(519\) 0 0
\(520\) 0 0
\(521\) −11.0000 −0.481919 −0.240959 0.970535i \(-0.577462\pi\)
−0.240959 + 0.970535i \(0.577462\pi\)
\(522\) 0 0
\(523\) − 29.0000i − 1.26808i −0.773300 0.634041i \(-0.781395\pi\)
0.773300 0.634041i \(-0.218605\pi\)
\(524\) 6.00000 0.262111
\(525\) 0 0
\(526\) −16.0000 −0.697633
\(527\) 0 0
\(528\) 0 0
\(529\) 14.0000 0.608696
\(530\) 0 0
\(531\) 0 0
\(532\) 24.0000i 1.04053i
\(533\) 10.0000i 0.433148i
\(534\) 0 0
\(535\) 0 0
\(536\) −9.00000 −0.388741
\(537\) 0 0
\(538\) 25.0000i 1.07783i
\(539\) −4.00000 −0.172292
\(540\) 0 0
\(541\) −39.0000 −1.67674 −0.838370 0.545101i \(-0.816491\pi\)
−0.838370 + 0.545101i \(0.816491\pi\)
\(542\) − 8.00000i − 0.343629i
\(543\) 0 0
\(544\) −20.0000 −0.857493
\(545\) 0 0
\(546\) 0 0
\(547\) 29.0000i 1.23995i 0.784621 + 0.619975i \(0.212857\pi\)
−0.784621 + 0.619975i \(0.787143\pi\)
\(548\) 12.0000i 0.512615i
\(549\) 0 0
\(550\) 0 0
\(551\) −8.00000 −0.340811
\(552\) 0 0
\(553\) 18.0000i 0.765438i
\(554\) 12.0000 0.509831
\(555\) 0 0
\(556\) 16.0000 0.678551
\(557\) − 30.0000i − 1.27114i −0.772043 0.635570i \(-0.780765\pi\)
0.772043 0.635570i \(-0.219235\pi\)
\(558\) 0 0
\(559\) −16.0000 −0.676728
\(560\) 0 0
\(561\) 0 0
\(562\) 15.0000i 0.632737i
\(563\) 21.0000i 0.885044i 0.896758 + 0.442522i \(0.145916\pi\)
−0.896758 + 0.442522i \(0.854084\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −21.0000 −0.882696
\(567\) 0 0
\(568\) − 6.00000i − 0.251754i
\(569\) −6.00000 −0.251533 −0.125767 0.992060i \(-0.540139\pi\)
−0.125767 + 0.992060i \(0.540139\pi\)
\(570\) 0 0
\(571\) 32.0000 1.33916 0.669579 0.742741i \(-0.266474\pi\)
0.669579 + 0.742741i \(0.266474\pi\)
\(572\) − 4.00000i − 0.167248i
\(573\) 0 0
\(574\) 15.0000 0.626088
\(575\) 0 0
\(576\) 0 0
\(577\) 10.0000i 0.416305i 0.978096 + 0.208153i \(0.0667451\pi\)
−0.978096 + 0.208153i \(0.933255\pi\)
\(578\) 1.00000i 0.0415945i
\(579\) 0 0
\(580\) 0 0
\(581\) 27.0000 1.12015
\(582\) 0 0
\(583\) 4.00000i 0.165663i
\(584\) −12.0000 −0.496564
\(585\) 0 0
\(586\) 12.0000 0.495715
\(587\) − 33.0000i − 1.36206i −0.732257 0.681028i \(-0.761533\pi\)
0.732257 0.681028i \(-0.238467\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) − 4.00000i − 0.164399i
\(593\) − 20.0000i − 0.821302i −0.911793 0.410651i \(-0.865302\pi\)
0.911793 0.410651i \(-0.134698\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 17.0000 0.696347
\(597\) 0 0
\(598\) − 6.00000i − 0.245358i
\(599\) −10.0000 −0.408589 −0.204294 0.978909i \(-0.565490\pi\)
−0.204294 + 0.978909i \(0.565490\pi\)
\(600\) 0 0
\(601\) 2.00000 0.0815817 0.0407909 0.999168i \(-0.487012\pi\)
0.0407909 + 0.999168i \(0.487012\pi\)
\(602\) 24.0000i 0.978167i
\(603\) 0 0
\(604\) −2.00000 −0.0813788
\(605\) 0 0
\(606\) 0 0
\(607\) 41.0000i 1.66414i 0.554672 + 0.832069i \(0.312844\pi\)
−0.554672 + 0.832069i \(0.687156\pi\)
\(608\) 40.0000i 1.62221i
\(609\) 0 0
\(610\) 0 0
\(611\) 14.0000 0.566379
\(612\) 0 0
\(613\) − 44.0000i − 1.77714i −0.458738 0.888572i \(-0.651698\pi\)
0.458738 0.888572i \(-0.348302\pi\)
\(614\) 7.00000 0.282497
\(615\) 0 0
\(616\) −18.0000 −0.725241
\(617\) − 36.0000i − 1.44931i −0.689114 0.724653i \(-0.742000\pi\)
0.689114 0.724653i \(-0.258000\pi\)
\(618\) 0 0
\(619\) 4.00000 0.160774 0.0803868 0.996764i \(-0.474384\pi\)
0.0803868 + 0.996764i \(0.474384\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) − 45.0000i − 1.80289i
\(624\) 0 0
\(625\) 0 0
\(626\) −14.0000 −0.559553
\(627\) 0 0
\(628\) − 14.0000i − 0.558661i
\(629\) −16.0000 −0.637962
\(630\) 0 0
\(631\) 16.0000 0.636950 0.318475 0.947931i \(-0.396829\pi\)
0.318475 + 0.947931i \(0.396829\pi\)
\(632\) 18.0000i 0.716002i
\(633\) 0 0
\(634\) 34.0000 1.35031
\(635\) 0 0
\(636\) 0 0
\(637\) 4.00000i 0.158486i
\(638\) − 2.00000i − 0.0791808i
\(639\) 0 0
\(640\) 0 0
\(641\) −33.0000 −1.30342 −0.651711 0.758468i \(-0.725948\pi\)
−0.651711 + 0.758468i \(0.725948\pi\)
\(642\) 0 0
\(643\) − 9.00000i − 0.354925i −0.984128 0.177463i \(-0.943211\pi\)
0.984128 0.177463i \(-0.0567889\pi\)
\(644\) 9.00000 0.354650
\(645\) 0 0
\(646\) −32.0000 −1.25902
\(647\) − 17.0000i − 0.668339i −0.942513 0.334169i \(-0.891544\pi\)
0.942513 0.334169i \(-0.108456\pi\)
\(648\) 0 0
\(649\) −28.0000 −1.09910
\(650\) 0 0
\(651\) 0 0
\(652\) − 4.00000i − 0.156652i
\(653\) 4.00000i 0.156532i 0.996933 + 0.0782660i \(0.0249384\pi\)
−0.996933 + 0.0782660i \(0.975062\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 5.00000 0.195217
\(657\) 0 0
\(658\) − 21.0000i − 0.818665i
\(659\) −8.00000 −0.311636 −0.155818 0.987786i \(-0.549801\pi\)
−0.155818 + 0.987786i \(0.549801\pi\)
\(660\) 0 0
\(661\) −14.0000 −0.544537 −0.272268 0.962221i \(-0.587774\pi\)
−0.272268 + 0.962221i \(0.587774\pi\)
\(662\) − 6.00000i − 0.233197i
\(663\) 0 0
\(664\) 27.0000 1.04780
\(665\) 0 0
\(666\) 0 0
\(667\) 3.00000i 0.116160i
\(668\) − 9.00000i − 0.348220i
\(669\) 0 0
\(670\) 0 0
\(671\) 14.0000 0.540464
\(672\) 0 0
\(673\) 6.00000i 0.231283i 0.993291 + 0.115642i \(0.0368924\pi\)
−0.993291 + 0.115642i \(0.963108\pi\)
\(674\) −8.00000 −0.308148
\(675\) 0 0
\(676\) 9.00000 0.346154
\(677\) − 42.0000i − 1.61419i −0.590421 0.807096i \(-0.701038\pi\)
0.590421 0.807096i \(-0.298962\pi\)
\(678\) 0 0
\(679\) 6.00000 0.230259
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) − 12.0000i − 0.459167i −0.973289 0.229584i \(-0.926264\pi\)
0.973289 0.229584i \(-0.0737364\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −15.0000 −0.572703
\(687\) 0 0
\(688\) 8.00000i 0.304997i
\(689\) 4.00000 0.152388
\(690\) 0 0
\(691\) −14.0000 −0.532585 −0.266293 0.963892i \(-0.585799\pi\)
−0.266293 + 0.963892i \(0.585799\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) −4.00000 −0.151838
\(695\) 0 0
\(696\) 0 0
\(697\) − 20.0000i − 0.757554i
\(698\) 5.00000i 0.189253i
\(699\) 0 0
\(700\) 0 0
\(701\) −23.0000 −0.868698 −0.434349 0.900745i \(-0.643022\pi\)
−0.434349 + 0.900745i \(0.643022\pi\)
\(702\) 0 0
\(703\) 32.0000i 1.20690i
\(704\) −14.0000 −0.527645
\(705\) 0 0
\(706\) 24.0000 0.903252
\(707\) 54.0000i 2.03088i
\(708\) 0 0
\(709\) 41.0000 1.53979 0.769894 0.638172i \(-0.220309\pi\)
0.769894 + 0.638172i \(0.220309\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) − 45.0000i − 1.68645i
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) −2.00000 −0.0747435
\(717\) 0 0
\(718\) 24.0000i 0.895672i
\(719\) 6.00000 0.223762 0.111881 0.993722i \(-0.464312\pi\)
0.111881 + 0.993722i \(0.464312\pi\)
\(720\) 0 0
\(721\) −24.0000 −0.893807
\(722\) 45.0000i 1.67473i
\(723\) 0 0
\(724\) −7.00000 −0.260153
\(725\) 0 0
\(726\) 0 0
\(727\) − 23.0000i − 0.853023i −0.904482 0.426511i \(-0.859742\pi\)
0.904482 0.426511i \(-0.140258\pi\)
\(728\) 18.0000i 0.667124i
\(729\) 0 0
\(730\) 0 0
\(731\) 32.0000 1.18356
\(732\) 0 0
\(733\) − 34.0000i − 1.25582i −0.778287 0.627909i \(-0.783911\pi\)
0.778287 0.627909i \(-0.216089\pi\)
\(734\) 24.0000 0.885856
\(735\) 0 0
\(736\) 15.0000 0.552907
\(737\) 6.00000i 0.221013i
\(738\) 0 0
\(739\) 2.00000 0.0735712 0.0367856 0.999323i \(-0.488288\pi\)
0.0367856 + 0.999323i \(0.488288\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) − 6.00000i − 0.220267i
\(743\) 29.0000i 1.06391i 0.846774 + 0.531953i \(0.178542\pi\)
−0.846774 + 0.531953i \(0.821458\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −10.0000 −0.366126
\(747\) 0 0
\(748\) 8.00000i 0.292509i
\(749\) −9.00000 −0.328853
\(750\) 0 0
\(751\) 10.0000 0.364905 0.182453 0.983215i \(-0.441596\pi\)
0.182453 + 0.983215i \(0.441596\pi\)
\(752\) − 7.00000i − 0.255264i
\(753\) 0 0
\(754\) −2.00000 −0.0728357
\(755\) 0 0
\(756\) 0 0
\(757\) 26.0000i 0.944986i 0.881334 + 0.472493i \(0.156646\pi\)
−0.881334 + 0.472493i \(0.843354\pi\)
\(758\) 26.0000i 0.944363i
\(759\) 0 0
\(760\) 0 0
\(761\) −15.0000 −0.543750 −0.271875 0.962333i \(-0.587644\pi\)
−0.271875 + 0.962333i \(0.587644\pi\)
\(762\) 0 0
\(763\) − 15.0000i − 0.543036i
\(764\) −8.00000 −0.289430
\(765\) 0 0
\(766\) 36.0000 1.30073
\(767\) 28.0000i 1.01102i
\(768\) 0 0
\(769\) −5.00000 −0.180305 −0.0901523 0.995928i \(-0.528735\pi\)
−0.0901523 + 0.995928i \(0.528735\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) − 10.0000i − 0.359908i
\(773\) 24.0000i 0.863220i 0.902060 + 0.431610i \(0.142054\pi\)
−0.902060 + 0.431610i \(0.857946\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 6.00000 0.215387
\(777\) 0 0
\(778\) 33.0000i 1.18311i
\(779\) −40.0000 −1.43315
\(780\) 0 0
\(781\) −4.00000 −0.143131
\(782\) 12.0000i 0.429119i
\(783\) 0 0
\(784\) 2.00000 0.0714286
\(785\) 0 0
\(786\) 0 0
\(787\) − 28.0000i − 0.998092i −0.866575 0.499046i \(-0.833684\pi\)
0.866575 0.499046i \(-0.166316\pi\)
\(788\) − 12.0000i − 0.427482i
\(789\) 0 0
\(790\) 0 0
\(791\) −24.0000 −0.853342
\(792\) 0 0
\(793\) − 14.0000i − 0.497155i
\(794\) 34.0000 1.20661
\(795\) 0 0
\(796\) −4.00000 −0.141776
\(797\) 26.0000i 0.920967i 0.887668 + 0.460484i \(0.152324\pi\)
−0.887668 + 0.460484i \(0.847676\pi\)
\(798\) 0 0
\(799\) −28.0000 −0.990569
\(800\) 0 0
\(801\) 0 0
\(802\) 18.0000i 0.635602i
\(803\) 8.00000i 0.282314i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 54.0000i 1.89971i
\(809\) 26.0000 0.914111 0.457056 0.889438i \(-0.348904\pi\)
0.457056 + 0.889438i \(0.348904\pi\)
\(810\) 0 0
\(811\) −42.0000 −1.47482 −0.737410 0.675446i \(-0.763951\pi\)
−0.737410 + 0.675446i \(0.763951\pi\)
\(812\) − 3.00000i − 0.105279i
\(813\) 0 0
\(814\) −8.00000 −0.280400
\(815\) 0 0
\(816\) 0 0
\(817\) − 64.0000i − 2.23908i
\(818\) − 14.0000i − 0.489499i
\(819\) 0 0
\(820\) 0 0
\(821\) −15.0000 −0.523504 −0.261752 0.965135i \(-0.584300\pi\)
−0.261752 + 0.965135i \(0.584300\pi\)
\(822\) 0 0
\(823\) 53.0000i 1.84746i 0.383040 + 0.923732i \(0.374877\pi\)
−0.383040 + 0.923732i \(0.625123\pi\)
\(824\) −24.0000 −0.836080
\(825\) 0 0
\(826\) 42.0000 1.46137
\(827\) 37.0000i 1.28662i 0.765607 + 0.643308i \(0.222439\pi\)
−0.765607 + 0.643308i \(0.777561\pi\)
\(828\) 0 0
\(829\) 3.00000 0.104194 0.0520972 0.998642i \(-0.483409\pi\)
0.0520972 + 0.998642i \(0.483409\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 14.0000i 0.485363i
\(833\) − 8.00000i − 0.277184i
\(834\) 0 0
\(835\) 0 0
\(836\) 16.0000 0.553372
\(837\) 0 0
\(838\) 26.0000i 0.898155i
\(839\) −40.0000 −1.38095 −0.690477 0.723355i \(-0.742599\pi\)
−0.690477 + 0.723355i \(0.742599\pi\)
\(840\) 0 0
\(841\) −28.0000 −0.965517
\(842\) 34.0000i 1.17172i
\(843\) 0 0
\(844\) −22.0000 −0.757271
\(845\) 0 0
\(846\) 0 0
\(847\) − 21.0000i − 0.721569i
\(848\) − 2.00000i − 0.0686803i
\(849\) 0 0
\(850\) 0 0
\(851\) 12.0000 0.411355
\(852\) 0 0
\(853\) − 54.0000i − 1.84892i −0.381273 0.924462i \(-0.624514\pi\)
0.381273 0.924462i \(-0.375486\pi\)
\(854\) −21.0000 −0.718605
\(855\) 0 0
\(856\) −9.00000 −0.307614
\(857\) 10.0000i 0.341593i 0.985306 + 0.170797i \(0.0546341\pi\)
−0.985306 + 0.170797i \(0.945366\pi\)
\(858\) 0 0
\(859\) −22.0000 −0.750630 −0.375315 0.926897i \(-0.622466\pi\)
−0.375315 + 0.926897i \(0.622466\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 30.0000i 1.02180i
\(863\) − 17.0000i − 0.578687i −0.957225 0.289343i \(-0.906563\pi\)
0.957225 0.289343i \(-0.0934369\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) −28.0000 −0.951479
\(867\) 0 0
\(868\) 0 0
\(869\) 12.0000 0.407072
\(870\) 0 0
\(871\) 6.00000 0.203302
\(872\) − 15.0000i − 0.507964i
\(873\) 0 0
\(874\) 24.0000 0.811812
\(875\) 0 0
\(876\) 0 0
\(877\) 18.0000i 0.607817i 0.952701 + 0.303908i \(0.0982917\pi\)
−0.952701 + 0.303908i \(0.901708\pi\)
\(878\) − 28.0000i − 0.944954i
\(879\) 0 0
\(880\) 0 0
\(881\) 35.0000 1.17918 0.589590 0.807703i \(-0.299289\pi\)
0.589590 + 0.807703i \(0.299289\pi\)
\(882\) 0 0
\(883\) 23.0000i 0.774012i 0.922077 + 0.387006i \(0.126491\pi\)
−0.922077 + 0.387006i \(0.873509\pi\)
\(884\) 8.00000 0.269069
\(885\) 0 0
\(886\) 15.0000 0.503935
\(887\) 36.0000i 1.20876i 0.796696 + 0.604381i \(0.206579\pi\)
−0.796696 + 0.604381i \(0.793421\pi\)
\(888\) 0 0
\(889\) −15.0000 −0.503084
\(890\) 0 0
\(891\) 0 0
\(892\) − 19.0000i − 0.636167i
\(893\) 56.0000i 1.87397i
\(894\) 0 0
\(895\) 0 0
\(896\) −9.00000 −0.300669
\(897\) 0 0
\(898\) − 26.0000i − 0.867631i
\(899\) 0 0
\(900\) 0 0
\(901\) −8.00000 −0.266519
\(902\) − 10.0000i − 0.332964i
\(903\) 0 0
\(904\) −24.0000 −0.798228
\(905\) 0 0
\(906\) 0 0
\(907\) 51.0000i 1.69343i 0.532049 + 0.846714i \(0.321422\pi\)
−0.532049 + 0.846714i \(0.678578\pi\)
\(908\) − 4.00000i − 0.132745i
\(909\) 0 0
\(910\) 0 0
\(911\) 50.0000 1.65657 0.828287 0.560304i \(-0.189316\pi\)
0.828287 + 0.560304i \(0.189316\pi\)
\(912\) 0 0
\(913\) − 18.0000i − 0.595713i
\(914\) −20.0000 −0.661541
\(915\) 0 0
\(916\) −15.0000 −0.495614
\(917\) 18.0000i 0.594412i
\(918\) 0 0
\(919\) −10.0000 −0.329870 −0.164935 0.986304i \(-0.552741\pi\)
−0.164935 + 0.986304i \(0.552741\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) − 9.00000i − 0.296399i
\(923\) 4.00000i 0.131662i
\(924\) 0 0
\(925\) 0 0
\(926\) 36.0000 1.18303
\(927\) 0 0
\(928\) − 5.00000i − 0.164133i
\(929\) 14.0000 0.459325 0.229663 0.973270i \(-0.426238\pi\)
0.229663 + 0.973270i \(0.426238\pi\)
\(930\) 0 0
\(931\) −16.0000 −0.524379
\(932\) − 24.0000i − 0.786146i
\(933\) 0 0
\(934\) −20.0000 −0.654420
\(935\) 0 0
\(936\) 0 0
\(937\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(938\) − 9.00000i − 0.293860i
\(939\) 0 0
\(940\) 0 0
\(941\) 7.00000 0.228193 0.114097 0.993470i \(-0.463603\pi\)
0.114097 + 0.993470i \(0.463603\pi\)
\(942\) 0 0
\(943\) 15.0000i 0.488467i
\(944\) 14.0000 0.455661
\(945\) 0 0
\(946\) 16.0000 0.520205
\(947\) 57.0000i 1.85225i 0.377215 + 0.926126i \(0.376882\pi\)
−0.377215 + 0.926126i \(0.623118\pi\)
\(948\) 0 0
\(949\) 8.00000 0.259691
\(950\) 0 0
\(951\) 0 0
\(952\) − 36.0000i − 1.16677i
\(953\) 26.0000i 0.842223i 0.907009 + 0.421111i \(0.138360\pi\)
−0.907009 + 0.421111i \(0.861640\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −8.00000 −0.258738
\(957\) 0 0
\(958\) − 18.0000i − 0.581554i
\(959\) −36.0000 −1.16250
\(960\) 0 0
\(961\) −31.0000 −1.00000
\(962\) 8.00000i 0.257930i
\(963\) 0 0
\(964\) −11.0000 −0.354286
\(965\) 0 0
\(966\) 0 0
\(967\) − 41.0000i − 1.31847i −0.751936 0.659236i \(-0.770880\pi\)
0.751936 0.659236i \(-0.229120\pi\)
\(968\) − 21.0000i − 0.674966i
\(969\) 0 0
\(970\) 0 0
\(971\) 36.0000 1.15529 0.577647 0.816286i \(-0.303971\pi\)
0.577647 + 0.816286i \(0.303971\pi\)
\(972\) 0 0
\(973\) 48.0000i 1.53881i
\(974\) −16.0000 −0.512673
\(975\) 0 0
\(976\) −7.00000 −0.224065
\(977\) − 38.0000i − 1.21573i −0.794041 0.607864i \(-0.792027\pi\)
0.794041 0.607864i \(-0.207973\pi\)
\(978\) 0 0
\(979\) −30.0000 −0.958804
\(980\) 0 0
\(981\) 0 0
\(982\) − 20.0000i − 0.638226i
\(983\) − 3.00000i − 0.0956851i −0.998855 0.0478426i \(-0.984765\pi\)
0.998855 0.0478426i \(-0.0152346\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 4.00000 0.127386
\(987\) 0 0
\(988\) − 16.0000i − 0.509028i
\(989\) −24.0000 −0.763156
\(990\) 0 0
\(991\) 26.0000 0.825917 0.412959 0.910750i \(-0.364495\pi\)
0.412959 + 0.910750i \(0.364495\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 6.00000 0.190308
\(995\) 0 0
\(996\) 0 0
\(997\) 18.0000i 0.570066i 0.958518 + 0.285033i \(0.0920045\pi\)
−0.958518 + 0.285033i \(0.907995\pi\)
\(998\) 32.0000i 1.01294i
\(999\) 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 2025.2.b.d.649.2 2
3.2 odd 2 2025.2.b.c.649.1 2
5.2 odd 4 405.2.a.b.1.1 1
5.3 odd 4 2025.2.a.e.1.1 1
5.4 even 2 inner 2025.2.b.d.649.1 2
9.2 odd 6 225.2.k.a.49.1 4
9.4 even 3 675.2.k.a.424.1 4
9.5 odd 6 225.2.k.a.124.2 4
9.7 even 3 675.2.k.a.199.2 4
15.2 even 4 405.2.a.e.1.1 1
15.8 even 4 2025.2.a.b.1.1 1
15.14 odd 2 2025.2.b.c.649.2 2
20.7 even 4 6480.2.a.x.1.1 1
45.2 even 12 45.2.e.a.31.1 yes 2
45.4 even 6 675.2.k.a.424.2 4
45.7 odd 12 135.2.e.a.91.1 2
45.13 odd 12 675.2.e.a.451.1 2
45.14 odd 6 225.2.k.a.124.1 4
45.22 odd 12 135.2.e.a.46.1 2
45.23 even 12 225.2.e.a.151.1 2
45.29 odd 6 225.2.k.a.49.2 4
45.32 even 12 45.2.e.a.16.1 2
45.34 even 6 675.2.k.a.199.1 4
45.38 even 12 225.2.e.a.76.1 2
45.43 odd 12 675.2.e.a.226.1 2
60.47 odd 4 6480.2.a.k.1.1 1
180.7 even 12 2160.2.q.a.1441.1 2
180.47 odd 12 720.2.q.d.481.1 2
180.67 even 12 2160.2.q.a.721.1 2
180.167 odd 12 720.2.q.d.241.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
45.2.e.a.16.1 2 45.32 even 12
45.2.e.a.31.1 yes 2 45.2 even 12
135.2.e.a.46.1 2 45.22 odd 12
135.2.e.a.91.1 2 45.7 odd 12
225.2.e.a.76.1 2 45.38 even 12
225.2.e.a.151.1 2 45.23 even 12
225.2.k.a.49.1 4 9.2 odd 6
225.2.k.a.49.2 4 45.29 odd 6
225.2.k.a.124.1 4 45.14 odd 6
225.2.k.a.124.2 4 9.5 odd 6
405.2.a.b.1.1 1 5.2 odd 4
405.2.a.e.1.1 1 15.2 even 4
675.2.e.a.226.1 2 45.43 odd 12
675.2.e.a.451.1 2 45.13 odd 12
675.2.k.a.199.1 4 45.34 even 6
675.2.k.a.199.2 4 9.7 even 3
675.2.k.a.424.1 4 9.4 even 3
675.2.k.a.424.2 4 45.4 even 6
720.2.q.d.241.1 2 180.167 odd 12
720.2.q.d.481.1 2 180.47 odd 12
2025.2.a.b.1.1 1 15.8 even 4
2025.2.a.e.1.1 1 5.3 odd 4
2025.2.b.c.649.1 2 3.2 odd 2
2025.2.b.c.649.2 2 15.14 odd 2
2025.2.b.d.649.1 2 5.4 even 2 inner
2025.2.b.d.649.2 2 1.1 even 1 trivial
2160.2.q.a.721.1 2 180.67 even 12
2160.2.q.a.1441.1 2 180.7 even 12
6480.2.a.k.1.1 1 60.47 odd 4
6480.2.a.x.1.1 1 20.7 even 4