Properties

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

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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: \( 2 \)
Twist minimal: no (minimal twist has level 405)
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.b.649.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.00000i q^{2} -2.00000 q^{4} +O(q^{10})\) \(q+2.00000i q^{2} -2.00000 q^{4} +5.00000 q^{11} -4.00000i q^{13} -4.00000 q^{16} -4.00000i q^{17} +5.00000 q^{19} +10.0000i q^{22} -6.00000i q^{23} +8.00000 q^{26} +5.00000 q^{29} -9.00000 q^{31} -8.00000i q^{32} +8.00000 q^{34} -10.0000i q^{37} +10.0000i q^{38} +7.00000 q^{41} +2.00000i q^{43} -10.0000 q^{44} +12.0000 q^{46} +2.00000i q^{47} +7.00000 q^{49} +8.00000i q^{52} -8.00000i q^{53} +10.0000i q^{58} +1.00000 q^{59} -2.00000 q^{61} -18.0000i q^{62} +8.00000 q^{64} +6.00000i q^{67} +8.00000i q^{68} +1.00000 q^{71} +8.00000i q^{73} +20.0000 q^{74} -10.0000 q^{76} -12.0000 q^{79} +14.0000i q^{82} -6.00000i q^{83} -4.00000 q^{86} +9.00000 q^{89} +12.0000i q^{92} -4.00000 q^{94} +14.0000i q^{97} +14.0000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 4 q^{4} + 10 q^{11} - 8 q^{16} + 10 q^{19} + 16 q^{26} + 10 q^{29} - 18 q^{31} + 16 q^{34} + 14 q^{41} - 20 q^{44} + 24 q^{46} + 14 q^{49} + 2 q^{59} - 4 q^{61} + 16 q^{64} + 2 q^{71} + 40 q^{74} - 20 q^{76} - 24 q^{79} - 8 q^{86} + 18 q^{89} - 8 q^{94}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.00000i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(3\) 0 0
\(4\) −2.00000 −1.00000
\(5\) 0 0
\(6\) 0 0
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 5.00000 1.50756 0.753778 0.657129i \(-0.228229\pi\)
0.753778 + 0.657129i \(0.228229\pi\)
\(12\) 0 0
\(13\) − 4.00000i − 1.10940i −0.832050 0.554700i \(-0.812833\pi\)
0.832050 0.554700i \(-0.187167\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −4.00000 −1.00000
\(17\) − 4.00000i − 0.970143i −0.874475 0.485071i \(-0.838794\pi\)
0.874475 0.485071i \(-0.161206\pi\)
\(18\) 0 0
\(19\) 5.00000 1.14708 0.573539 0.819178i \(-0.305570\pi\)
0.573539 + 0.819178i \(0.305570\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 10.0000i 2.13201i
\(23\) − 6.00000i − 1.25109i −0.780189 0.625543i \(-0.784877\pi\)
0.780189 0.625543i \(-0.215123\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 8.00000 1.56893
\(27\) 0 0
\(28\) 0 0
\(29\) 5.00000 0.928477 0.464238 0.885710i \(-0.346328\pi\)
0.464238 + 0.885710i \(0.346328\pi\)
\(30\) 0 0
\(31\) −9.00000 −1.61645 −0.808224 0.588875i \(-0.799571\pi\)
−0.808224 + 0.588875i \(0.799571\pi\)
\(32\) − 8.00000i − 1.41421i
\(33\) 0 0
\(34\) 8.00000 1.37199
\(35\) 0 0
\(36\) 0 0
\(37\) − 10.0000i − 1.64399i −0.569495 0.821995i \(-0.692861\pi\)
0.569495 0.821995i \(-0.307139\pi\)
\(38\) 10.0000i 1.62221i
\(39\) 0 0
\(40\) 0 0
\(41\) 7.00000 1.09322 0.546608 0.837389i \(-0.315919\pi\)
0.546608 + 0.837389i \(0.315919\pi\)
\(42\) 0 0
\(43\) 2.00000i 0.304997i 0.988304 + 0.152499i \(0.0487319\pi\)
−0.988304 + 0.152499i \(0.951268\pi\)
\(44\) −10.0000 −1.50756
\(45\) 0 0
\(46\) 12.0000 1.76930
\(47\) 2.00000i 0.291730i 0.989305 + 0.145865i \(0.0465965\pi\)
−0.989305 + 0.145865i \(0.953403\pi\)
\(48\) 0 0
\(49\) 7.00000 1.00000
\(50\) 0 0
\(51\) 0 0
\(52\) 8.00000i 1.10940i
\(53\) − 8.00000i − 1.09888i −0.835532 0.549442i \(-0.814840\pi\)
0.835532 0.549442i \(-0.185160\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 10.0000i 1.31306i
\(59\) 1.00000 0.130189 0.0650945 0.997879i \(-0.479265\pi\)
0.0650945 + 0.997879i \(0.479265\pi\)
\(60\) 0 0
\(61\) −2.00000 −0.256074 −0.128037 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) − 18.0000i − 2.28600i
\(63\) 0 0
\(64\) 8.00000 1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) 6.00000i 0.733017i 0.930415 + 0.366508i \(0.119447\pi\)
−0.930415 + 0.366508i \(0.880553\pi\)
\(68\) 8.00000i 0.970143i
\(69\) 0 0
\(70\) 0 0
\(71\) 1.00000 0.118678 0.0593391 0.998238i \(-0.481101\pi\)
0.0593391 + 0.998238i \(0.481101\pi\)
\(72\) 0 0
\(73\) 8.00000i 0.936329i 0.883641 + 0.468165i \(0.155085\pi\)
−0.883641 + 0.468165i \(0.844915\pi\)
\(74\) 20.0000 2.32495
\(75\) 0 0
\(76\) −10.0000 −1.14708
\(77\) 0 0
\(78\) 0 0
\(79\) −12.0000 −1.35011 −0.675053 0.737769i \(-0.735879\pi\)
−0.675053 + 0.737769i \(0.735879\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 14.0000i 1.54604i
\(83\) − 6.00000i − 0.658586i −0.944228 0.329293i \(-0.893190\pi\)
0.944228 0.329293i \(-0.106810\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −4.00000 −0.431331
\(87\) 0 0
\(88\) 0 0
\(89\) 9.00000 0.953998 0.476999 0.878904i \(-0.341725\pi\)
0.476999 + 0.878904i \(0.341725\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 12.0000i 1.25109i
\(93\) 0 0
\(94\) −4.00000 −0.412568
\(95\) 0 0
\(96\) 0 0
\(97\) 14.0000i 1.42148i 0.703452 + 0.710742i \(0.251641\pi\)
−0.703452 + 0.710742i \(0.748359\pi\)
\(98\) 14.0000i 1.41421i
\(99\) 0 0
\(100\) 0 0
\(101\) −3.00000 −0.298511 −0.149256 0.988799i \(-0.547688\pi\)
−0.149256 + 0.988799i \(0.547688\pi\)
\(102\) 0 0
\(103\) − 2.00000i − 0.197066i −0.995134 0.0985329i \(-0.968585\pi\)
0.995134 0.0985329i \(-0.0314150\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 16.0000 1.55406
\(107\) − 6.00000i − 0.580042i −0.957020 0.290021i \(-0.906338\pi\)
0.957020 0.290021i \(-0.0936623\pi\)
\(108\) 0 0
\(109\) 1.00000 0.0957826 0.0478913 0.998853i \(-0.484750\pi\)
0.0478913 + 0.998853i \(0.484750\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 16.0000i 1.50515i 0.658505 + 0.752577i \(0.271189\pi\)
−0.658505 + 0.752577i \(0.728811\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −10.0000 −0.928477
\(117\) 0 0
\(118\) 2.00000i 0.184115i
\(119\) 0 0
\(120\) 0 0
\(121\) 14.0000 1.27273
\(122\) − 4.00000i − 0.362143i
\(123\) 0 0
\(124\) 18.0000 1.61645
\(125\) 0 0
\(126\) 0 0
\(127\) 16.0000i 1.41977i 0.704317 + 0.709885i \(0.251253\pi\)
−0.704317 + 0.709885i \(0.748747\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 15.0000 1.31056 0.655278 0.755388i \(-0.272551\pi\)
0.655278 + 0.755388i \(0.272551\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) −12.0000 −1.03664
\(135\) 0 0
\(136\) 0 0
\(137\) 12.0000i 1.02523i 0.858619 + 0.512615i \(0.171323\pi\)
−0.858619 + 0.512615i \(0.828677\pi\)
\(138\) 0 0
\(139\) 19.0000 1.61156 0.805779 0.592216i \(-0.201747\pi\)
0.805779 + 0.592216i \(0.201747\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 2.00000i 0.167836i
\(143\) − 20.0000i − 1.67248i
\(144\) 0 0
\(145\) 0 0
\(146\) −16.0000 −1.32417
\(147\) 0 0
\(148\) 20.0000i 1.64399i
\(149\) 2.00000 0.163846 0.0819232 0.996639i \(-0.473894\pi\)
0.0819232 + 0.996639i \(0.473894\pi\)
\(150\) 0 0
\(151\) −5.00000 −0.406894 −0.203447 0.979086i \(-0.565214\pi\)
−0.203447 + 0.979086i \(0.565214\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 2.00000i 0.159617i 0.996810 + 0.0798087i \(0.0254309\pi\)
−0.996810 + 0.0798087i \(0.974569\pi\)
\(158\) − 24.0000i − 1.90934i
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) − 8.00000i − 0.626608i −0.949653 0.313304i \(-0.898564\pi\)
0.949653 0.313304i \(-0.101436\pi\)
\(164\) −14.0000 −1.09322
\(165\) 0 0
\(166\) 12.0000 0.931381
\(167\) 12.0000i 0.928588i 0.885681 + 0.464294i \(0.153692\pi\)
−0.885681 + 0.464294i \(0.846308\pi\)
\(168\) 0 0
\(169\) −3.00000 −0.230769
\(170\) 0 0
\(171\) 0 0
\(172\) − 4.00000i − 0.304997i
\(173\) 6.00000i 0.456172i 0.973641 + 0.228086i \(0.0732467\pi\)
−0.973641 + 0.228086i \(0.926753\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −20.0000 −1.50756
\(177\) 0 0
\(178\) 18.0000i 1.34916i
\(179\) −23.0000 −1.71910 −0.859550 0.511051i \(-0.829256\pi\)
−0.859550 + 0.511051i \(0.829256\pi\)
\(180\) 0 0
\(181\) −25.0000 −1.85824 −0.929118 0.369784i \(-0.879432\pi\)
−0.929118 + 0.369784i \(0.879432\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) − 20.0000i − 1.46254i
\(188\) − 4.00000i − 0.291730i
\(189\) 0 0
\(190\) 0 0
\(191\) 1.00000 0.0723575 0.0361787 0.999345i \(-0.488481\pi\)
0.0361787 + 0.999345i \(0.488481\pi\)
\(192\) 0 0
\(193\) − 26.0000i − 1.87152i −0.352636 0.935760i \(-0.614715\pi\)
0.352636 0.935760i \(-0.385285\pi\)
\(194\) −28.0000 −2.01028
\(195\) 0 0
\(196\) −14.0000 −1.00000
\(197\) 12.0000i 0.854965i 0.904024 + 0.427482i \(0.140599\pi\)
−0.904024 + 0.427482i \(0.859401\pi\)
\(198\) 0 0
\(199\) −16.0000 −1.13421 −0.567105 0.823646i \(-0.691937\pi\)
−0.567105 + 0.823646i \(0.691937\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) − 6.00000i − 0.422159i
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 4.00000 0.278693
\(207\) 0 0
\(208\) 16.0000i 1.10940i
\(209\) 25.0000 1.72929
\(210\) 0 0
\(211\) 11.0000 0.757271 0.378636 0.925546i \(-0.376393\pi\)
0.378636 + 0.925546i \(0.376393\pi\)
\(212\) 16.0000i 1.09888i
\(213\) 0 0
\(214\) 12.0000 0.820303
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 2.00000i 0.135457i
\(219\) 0 0
\(220\) 0 0
\(221\) −16.0000 −1.07628
\(222\) 0 0
\(223\) − 8.00000i − 0.535720i −0.963458 0.267860i \(-0.913684\pi\)
0.963458 0.267860i \(-0.0863164\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) −32.0000 −2.12861
\(227\) 4.00000i 0.265489i 0.991150 + 0.132745i \(0.0423790\pi\)
−0.991150 + 0.132745i \(0.957621\pi\)
\(228\) 0 0
\(229\) −6.00000 −0.396491 −0.198246 0.980152i \(-0.563524\pi\)
−0.198246 + 0.980152i \(0.563524\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) − 6.00000i − 0.393073i −0.980497 0.196537i \(-0.937031\pi\)
0.980497 0.196537i \(-0.0629694\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −2.00000 −0.130189
\(237\) 0 0
\(238\) 0 0
\(239\) 16.0000 1.03495 0.517477 0.855697i \(-0.326871\pi\)
0.517477 + 0.855697i \(0.326871\pi\)
\(240\) 0 0
\(241\) −11.0000 −0.708572 −0.354286 0.935137i \(-0.615276\pi\)
−0.354286 + 0.935137i \(0.615276\pi\)
\(242\) 28.0000i 1.79991i
\(243\) 0 0
\(244\) 4.00000 0.256074
\(245\) 0 0
\(246\) 0 0
\(247\) − 20.0000i − 1.27257i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −12.0000 −0.757433 −0.378717 0.925513i \(-0.623635\pi\)
−0.378717 + 0.925513i \(0.623635\pi\)
\(252\) 0 0
\(253\) − 30.0000i − 1.88608i
\(254\) −32.0000 −2.00786
\(255\) 0 0
\(256\) 16.0000 1.00000
\(257\) − 6.00000i − 0.374270i −0.982334 0.187135i \(-0.940080\pi\)
0.982334 0.187135i \(-0.0599201\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 30.0000i 1.85341i
\(263\) − 10.0000i − 0.616626i −0.951285 0.308313i \(-0.900236\pi\)
0.951285 0.308313i \(-0.0997645\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) − 12.0000i − 0.733017i
\(269\) 31.0000 1.89010 0.945052 0.326921i \(-0.106011\pi\)
0.945052 + 0.326921i \(0.106011\pi\)
\(270\) 0 0
\(271\) −8.00000 −0.485965 −0.242983 0.970031i \(-0.578126\pi\)
−0.242983 + 0.970031i \(0.578126\pi\)
\(272\) 16.0000i 0.970143i
\(273\) 0 0
\(274\) −24.0000 −1.44989
\(275\) 0 0
\(276\) 0 0
\(277\) − 18.0000i − 1.08152i −0.841178 0.540758i \(-0.818138\pi\)
0.841178 0.540758i \(-0.181862\pi\)
\(278\) 38.0000i 2.27909i
\(279\) 0 0
\(280\) 0 0
\(281\) 6.00000 0.357930 0.178965 0.983855i \(-0.442725\pi\)
0.178965 + 0.983855i \(0.442725\pi\)
\(282\) 0 0
\(283\) − 6.00000i − 0.356663i −0.983970 0.178331i \(-0.942930\pi\)
0.983970 0.178331i \(-0.0570699\pi\)
\(284\) −2.00000 −0.118678
\(285\) 0 0
\(286\) 40.0000 2.36525
\(287\) 0 0
\(288\) 0 0
\(289\) 1.00000 0.0588235
\(290\) 0 0
\(291\) 0 0
\(292\) − 16.0000i − 0.936329i
\(293\) − 18.0000i − 1.05157i −0.850617 0.525786i \(-0.823771\pi\)
0.850617 0.525786i \(-0.176229\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 4.00000i 0.231714i
\(299\) −24.0000 −1.38796
\(300\) 0 0
\(301\) 0 0
\(302\) − 10.0000i − 0.575435i
\(303\) 0 0
\(304\) −20.0000 −1.14708
\(305\) 0 0
\(306\) 0 0
\(307\) 10.0000i 0.570730i 0.958419 + 0.285365i \(0.0921148\pi\)
−0.958419 + 0.285365i \(0.907885\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −9.00000 −0.510343 −0.255172 0.966896i \(-0.582132\pi\)
−0.255172 + 0.966896i \(0.582132\pi\)
\(312\) 0 0
\(313\) 4.00000i 0.226093i 0.993590 + 0.113047i \(0.0360610\pi\)
−0.993590 + 0.113047i \(0.963939\pi\)
\(314\) −4.00000 −0.225733
\(315\) 0 0
\(316\) 24.0000 1.35011
\(317\) − 2.00000i − 0.112331i −0.998421 0.0561656i \(-0.982113\pi\)
0.998421 0.0561656i \(-0.0178875\pi\)
\(318\) 0 0
\(319\) 25.0000 1.39973
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) − 20.0000i − 1.11283i
\(324\) 0 0
\(325\) 0 0
\(326\) 16.0000 0.886158
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −21.0000 −1.15426 −0.577132 0.816651i \(-0.695828\pi\)
−0.577132 + 0.816651i \(0.695828\pi\)
\(332\) 12.0000i 0.658586i
\(333\) 0 0
\(334\) −24.0000 −1.31322
\(335\) 0 0
\(336\) 0 0
\(337\) − 8.00000i − 0.435788i −0.975972 0.217894i \(-0.930081\pi\)
0.975972 0.217894i \(-0.0699187\pi\)
\(338\) − 6.00000i − 0.326357i
\(339\) 0 0
\(340\) 0 0
\(341\) −45.0000 −2.43689
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) −12.0000 −0.645124
\(347\) 20.0000i 1.07366i 0.843692 + 0.536828i \(0.180378\pi\)
−0.843692 + 0.536828i \(0.819622\pi\)
\(348\) 0 0
\(349\) −13.0000 −0.695874 −0.347937 0.937518i \(-0.613118\pi\)
−0.347937 + 0.937518i \(0.613118\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) − 40.0000i − 2.13201i
\(353\) 18.0000i 0.958043i 0.877803 + 0.479022i \(0.159008\pi\)
−0.877803 + 0.479022i \(0.840992\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −18.0000 −0.953998
\(357\) 0 0
\(358\) − 46.0000i − 2.43118i
\(359\) 27.0000 1.42501 0.712503 0.701669i \(-0.247562\pi\)
0.712503 + 0.701669i \(0.247562\pi\)
\(360\) 0 0
\(361\) 6.00000 0.315789
\(362\) − 50.0000i − 2.62794i
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 18.0000i 0.939592i 0.882775 + 0.469796i \(0.155673\pi\)
−0.882775 + 0.469796i \(0.844327\pi\)
\(368\) 24.0000i 1.25109i
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) − 16.0000i − 0.828449i −0.910175 0.414224i \(-0.864053\pi\)
0.910175 0.414224i \(-0.135947\pi\)
\(374\) 40.0000 2.06835
\(375\) 0 0
\(376\) 0 0
\(377\) − 20.0000i − 1.03005i
\(378\) 0 0
\(379\) −4.00000 −0.205466 −0.102733 0.994709i \(-0.532759\pi\)
−0.102733 + 0.994709i \(0.532759\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 2.00000i 0.102329i
\(383\) 36.0000i 1.83951i 0.392488 + 0.919757i \(0.371614\pi\)
−0.392488 + 0.919757i \(0.628386\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 52.0000 2.64673
\(387\) 0 0
\(388\) − 28.0000i − 1.42148i
\(389\) −6.00000 −0.304212 −0.152106 0.988364i \(-0.548606\pi\)
−0.152106 + 0.988364i \(0.548606\pi\)
\(390\) 0 0
\(391\) −24.0000 −1.21373
\(392\) 0 0
\(393\) 0 0
\(394\) −24.0000 −1.20910
\(395\) 0 0
\(396\) 0 0
\(397\) − 38.0000i − 1.90717i −0.301131 0.953583i \(-0.597364\pi\)
0.301131 0.953583i \(-0.402636\pi\)
\(398\) − 32.0000i − 1.60402i
\(399\) 0 0
\(400\) 0 0
\(401\) 30.0000 1.49813 0.749064 0.662497i \(-0.230503\pi\)
0.749064 + 0.662497i \(0.230503\pi\)
\(402\) 0 0
\(403\) 36.0000i 1.79329i
\(404\) 6.00000 0.298511
\(405\) 0 0
\(406\) 0 0
\(407\) − 50.0000i − 2.47841i
\(408\) 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
\(412\) 4.00000i 0.197066i
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) −32.0000 −1.56893
\(417\) 0 0
\(418\) 50.0000i 2.44558i
\(419\) −16.0000 −0.781651 −0.390826 0.920465i \(-0.627810\pi\)
−0.390826 + 0.920465i \(0.627810\pi\)
\(420\) 0 0
\(421\) 13.0000 0.633581 0.316791 0.948495i \(-0.397395\pi\)
0.316791 + 0.948495i \(0.397395\pi\)
\(422\) 22.0000i 1.07094i
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 12.0000i 0.580042i
\(429\) 0 0
\(430\) 0 0
\(431\) 3.00000 0.144505 0.0722525 0.997386i \(-0.476981\pi\)
0.0722525 + 0.997386i \(0.476981\pi\)
\(432\) 0 0
\(433\) 26.0000i 1.24948i 0.780833 + 0.624740i \(0.214795\pi\)
−0.780833 + 0.624740i \(0.785205\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −2.00000 −0.0957826
\(437\) − 30.0000i − 1.43509i
\(438\) 0 0
\(439\) 29.0000 1.38409 0.692047 0.721852i \(-0.256709\pi\)
0.692047 + 0.721852i \(0.256709\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) − 32.0000i − 1.52208i
\(443\) − 6.00000i − 0.285069i −0.989790 0.142534i \(-0.954475\pi\)
0.989790 0.142534i \(-0.0455251\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 16.0000 0.757622
\(447\) 0 0
\(448\) 0 0
\(449\) −17.0000 −0.802280 −0.401140 0.916017i \(-0.631386\pi\)
−0.401140 + 0.916017i \(0.631386\pi\)
\(450\) 0 0
\(451\) 35.0000 1.64809
\(452\) − 32.0000i − 1.50515i
\(453\) 0 0
\(454\) −8.00000 −0.375459
\(455\) 0 0
\(456\) 0 0
\(457\) − 38.0000i − 1.77757i −0.458329 0.888783i \(-0.651552\pi\)
0.458329 0.888783i \(-0.348448\pi\)
\(458\) − 12.0000i − 0.560723i
\(459\) 0 0
\(460\) 0 0
\(461\) −15.0000 −0.698620 −0.349310 0.937007i \(-0.613584\pi\)
−0.349310 + 0.937007i \(0.613584\pi\)
\(462\) 0 0
\(463\) 6.00000i 0.278844i 0.990233 + 0.139422i \(0.0445244\pi\)
−0.990233 + 0.139422i \(0.955476\pi\)
\(464\) −20.0000 −0.928477
\(465\) 0 0
\(466\) 12.0000 0.555889
\(467\) 4.00000i 0.185098i 0.995708 + 0.0925490i \(0.0295015\pi\)
−0.995708 + 0.0925490i \(0.970499\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 10.0000i 0.459800i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) 32.0000i 1.46365i
\(479\) 15.0000 0.685367 0.342684 0.939451i \(-0.388664\pi\)
0.342684 + 0.939451i \(0.388664\pi\)
\(480\) 0 0
\(481\) −40.0000 −1.82384
\(482\) − 22.0000i − 1.00207i
\(483\) 0 0
\(484\) −28.0000 −1.27273
\(485\) 0 0
\(486\) 0 0
\(487\) 8.00000i 0.362515i 0.983436 + 0.181257i \(0.0580167\pi\)
−0.983436 + 0.181257i \(0.941983\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 43.0000 1.94056 0.970281 0.241979i \(-0.0777966\pi\)
0.970281 + 0.241979i \(0.0777966\pi\)
\(492\) 0 0
\(493\) − 20.0000i − 0.900755i
\(494\) 40.0000 1.79969
\(495\) 0 0
\(496\) 36.0000 1.61645
\(497\) 0 0
\(498\) 0 0
\(499\) −7.00000 −0.313363 −0.156682 0.987649i \(-0.550080\pi\)
−0.156682 + 0.987649i \(0.550080\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) − 24.0000i − 1.07117i
\(503\) 20.0000i 0.891756i 0.895094 + 0.445878i \(0.147108\pi\)
−0.895094 + 0.445878i \(0.852892\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 60.0000 2.66733
\(507\) 0 0
\(508\) − 32.0000i − 1.41977i
\(509\) −2.00000 −0.0886484 −0.0443242 0.999017i \(-0.514113\pi\)
−0.0443242 + 0.999017i \(0.514113\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 32.0000i 1.41421i
\(513\) 0 0
\(514\) 12.0000 0.529297
\(515\) 0 0
\(516\) 0 0
\(517\) 10.0000i 0.439799i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 10.0000 0.438108 0.219054 0.975713i \(-0.429703\pi\)
0.219054 + 0.975713i \(0.429703\pi\)
\(522\) 0 0
\(523\) − 16.0000i − 0.699631i −0.936819 0.349816i \(-0.886244\pi\)
0.936819 0.349816i \(-0.113756\pi\)
\(524\) −30.0000 −1.31056
\(525\) 0 0
\(526\) 20.0000 0.872041
\(527\) 36.0000i 1.56818i
\(528\) 0 0
\(529\) −13.0000 −0.565217
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) − 28.0000i − 1.21281i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 62.0000i 2.67301i
\(539\) 35.0000 1.50756
\(540\) 0 0
\(541\) 3.00000 0.128980 0.0644900 0.997918i \(-0.479458\pi\)
0.0644900 + 0.997918i \(0.479458\pi\)
\(542\) − 16.0000i − 0.687259i
\(543\) 0 0
\(544\) −32.0000 −1.37199
\(545\) 0 0
\(546\) 0 0
\(547\) 28.0000i 1.19719i 0.801050 + 0.598597i \(0.204275\pi\)
−0.801050 + 0.598597i \(0.795725\pi\)
\(548\) − 24.0000i − 1.02523i
\(549\) 0 0
\(550\) 0 0
\(551\) 25.0000 1.06504
\(552\) 0 0
\(553\) 0 0
\(554\) 36.0000 1.52949
\(555\) 0 0
\(556\) −38.0000 −1.61156
\(557\) − 24.0000i − 1.01691i −0.861088 0.508456i \(-0.830216\pi\)
0.861088 0.508456i \(-0.169784\pi\)
\(558\) 0 0
\(559\) 8.00000 0.338364
\(560\) 0 0
\(561\) 0 0
\(562\) 12.0000i 0.506189i
\(563\) 18.0000i 0.758610i 0.925272 + 0.379305i \(0.123837\pi\)
−0.925272 + 0.379305i \(0.876163\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 12.0000 0.504398
\(567\) 0 0
\(568\) 0 0
\(569\) 3.00000 0.125767 0.0628833 0.998021i \(-0.479970\pi\)
0.0628833 + 0.998021i \(0.479970\pi\)
\(570\) 0 0
\(571\) 5.00000 0.209243 0.104622 0.994512i \(-0.466637\pi\)
0.104622 + 0.994512i \(0.466637\pi\)
\(572\) 40.0000i 1.67248i
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) − 16.0000i − 0.666089i −0.942911 0.333044i \(-0.891924\pi\)
0.942911 0.333044i \(-0.108076\pi\)
\(578\) 2.00000i 0.0831890i
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) − 40.0000i − 1.65663i
\(584\) 0 0
\(585\) 0 0
\(586\) 36.0000 1.48715
\(587\) 18.0000i 0.742940i 0.928445 + 0.371470i \(0.121146\pi\)
−0.928445 + 0.371470i \(0.878854\pi\)
\(588\) 0 0
\(589\) −45.0000 −1.85419
\(590\) 0 0
\(591\) 0 0
\(592\) 40.0000i 1.64399i
\(593\) 14.0000i 0.574911i 0.957794 + 0.287456i \(0.0928094\pi\)
−0.957794 + 0.287456i \(0.907191\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −4.00000 −0.163846
\(597\) 0 0
\(598\) − 48.0000i − 1.96287i
\(599\) 17.0000 0.694601 0.347301 0.937754i \(-0.387098\pi\)
0.347301 + 0.937754i \(0.387098\pi\)
\(600\) 0 0
\(601\) −19.0000 −0.775026 −0.387513 0.921864i \(-0.626666\pi\)
−0.387513 + 0.921864i \(0.626666\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 10.0000 0.406894
\(605\) 0 0
\(606\) 0 0
\(607\) 34.0000i 1.38002i 0.723801 + 0.690009i \(0.242393\pi\)
−0.723801 + 0.690009i \(0.757607\pi\)
\(608\) − 40.0000i − 1.62221i
\(609\) 0 0
\(610\) 0 0
\(611\) 8.00000 0.323645
\(612\) 0 0
\(613\) − 4.00000i − 0.161558i −0.996732 0.0807792i \(-0.974259\pi\)
0.996732 0.0807792i \(-0.0257409\pi\)
\(614\) −20.0000 −0.807134
\(615\) 0 0
\(616\) 0 0
\(617\) − 24.0000i − 0.966204i −0.875564 0.483102i \(-0.839510\pi\)
0.875564 0.483102i \(-0.160490\pi\)
\(618\) 0 0
\(619\) 28.0000 1.12542 0.562708 0.826656i \(-0.309760\pi\)
0.562708 + 0.826656i \(0.309760\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) − 18.0000i − 0.721734i
\(623\) 0 0
\(624\) 0 0
\(625\) 0 0
\(626\) −8.00000 −0.319744
\(627\) 0 0
\(628\) − 4.00000i − 0.159617i
\(629\) −40.0000 −1.59490
\(630\) 0 0
\(631\) −17.0000 −0.676759 −0.338380 0.941010i \(-0.609879\pi\)
−0.338380 + 0.941010i \(0.609879\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 4.00000 0.158860
\(635\) 0 0
\(636\) 0 0
\(637\) − 28.0000i − 1.10940i
\(638\) 50.0000i 1.97952i
\(639\) 0 0
\(640\) 0 0
\(641\) 3.00000 0.118493 0.0592464 0.998243i \(-0.481130\pi\)
0.0592464 + 0.998243i \(0.481130\pi\)
\(642\) 0 0
\(643\) − 6.00000i − 0.236617i −0.992977 0.118308i \(-0.962253\pi\)
0.992977 0.118308i \(-0.0377472\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 40.0000 1.57378
\(647\) − 10.0000i − 0.393141i −0.980490 0.196570i \(-0.937020\pi\)
0.980490 0.196570i \(-0.0629804\pi\)
\(648\) 0 0
\(649\) 5.00000 0.196267
\(650\) 0 0
\(651\) 0 0
\(652\) 16.0000i 0.626608i
\(653\) 8.00000i 0.313064i 0.987673 + 0.156532i \(0.0500315\pi\)
−0.987673 + 0.156532i \(0.949969\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −28.0000 −1.09322
\(657\) 0 0
\(658\) 0 0
\(659\) −44.0000 −1.71400 −0.856998 0.515319i \(-0.827673\pi\)
−0.856998 + 0.515319i \(0.827673\pi\)
\(660\) 0 0
\(661\) 25.0000 0.972387 0.486194 0.873851i \(-0.338385\pi\)
0.486194 + 0.873851i \(0.338385\pi\)
\(662\) − 42.0000i − 1.63238i
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) − 30.0000i − 1.16160i
\(668\) − 24.0000i − 0.928588i
\(669\) 0 0
\(670\) 0 0
\(671\) −10.0000 −0.386046
\(672\) 0 0
\(673\) − 42.0000i − 1.61898i −0.587133 0.809491i \(-0.699743\pi\)
0.587133 0.809491i \(-0.300257\pi\)
\(674\) 16.0000 0.616297
\(675\) 0 0
\(676\) 6.00000 0.230769
\(677\) 6.00000i 0.230599i 0.993331 + 0.115299i \(0.0367827\pi\)
−0.993331 + 0.115299i \(0.963217\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) − 90.0000i − 3.44628i
\(683\) − 48.0000i − 1.83667i −0.395805 0.918334i \(-0.629534\pi\)
0.395805 0.918334i \(-0.370466\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) − 8.00000i − 0.304997i
\(689\) −32.0000 −1.21910
\(690\) 0 0
\(691\) 4.00000 0.152167 0.0760836 0.997101i \(-0.475758\pi\)
0.0760836 + 0.997101i \(0.475758\pi\)
\(692\) − 12.0000i − 0.456172i
\(693\) 0 0
\(694\) −40.0000 −1.51838
\(695\) 0 0
\(696\) 0 0
\(697\) − 28.0000i − 1.06058i
\(698\) − 26.0000i − 0.984115i
\(699\) 0 0
\(700\) 0 0
\(701\) 19.0000 0.717620 0.358810 0.933411i \(-0.383183\pi\)
0.358810 + 0.933411i \(0.383183\pi\)
\(702\) 0 0
\(703\) − 50.0000i − 1.88579i
\(704\) 40.0000 1.50756
\(705\) 0 0
\(706\) −36.0000 −1.35488
\(707\) 0 0
\(708\) 0 0
\(709\) −22.0000 −0.826227 −0.413114 0.910679i \(-0.635559\pi\)
−0.413114 + 0.910679i \(0.635559\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 54.0000i 2.02232i
\(714\) 0 0
\(715\) 0 0
\(716\) 46.0000 1.71910
\(717\) 0 0
\(718\) 54.0000i 2.01526i
\(719\) −27.0000 −1.00693 −0.503465 0.864016i \(-0.667942\pi\)
−0.503465 + 0.864016i \(0.667942\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 12.0000i 0.446594i
\(723\) 0 0
\(724\) 50.0000 1.85824
\(725\) 0 0
\(726\) 0 0
\(727\) 44.0000i 1.63187i 0.578144 + 0.815935i \(0.303777\pi\)
−0.578144 + 0.815935i \(0.696223\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 8.00000 0.295891
\(732\) 0 0
\(733\) 46.0000i 1.69905i 0.527549 + 0.849524i \(0.323111\pi\)
−0.527549 + 0.849524i \(0.676889\pi\)
\(734\) −36.0000 −1.32878
\(735\) 0 0
\(736\) −48.0000 −1.76930
\(737\) 30.0000i 1.10506i
\(738\) 0 0
\(739\) 35.0000 1.28750 0.643748 0.765238i \(-0.277379\pi\)
0.643748 + 0.765238i \(0.277379\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 4.00000i 0.146746i 0.997305 + 0.0733729i \(0.0233763\pi\)
−0.997305 + 0.0733729i \(0.976624\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 32.0000 1.17160
\(747\) 0 0
\(748\) 40.0000i 1.46254i
\(749\) 0 0
\(750\) 0 0
\(751\) −32.0000 −1.16770 −0.583848 0.811863i \(-0.698454\pi\)
−0.583848 + 0.811863i \(0.698454\pi\)
\(752\) − 8.00000i − 0.291730i
\(753\) 0 0
\(754\) 40.0000 1.45671
\(755\) 0 0
\(756\) 0 0
\(757\) 28.0000i 1.01768i 0.860862 + 0.508839i \(0.169925\pi\)
−0.860862 + 0.508839i \(0.830075\pi\)
\(758\) − 8.00000i − 0.290573i
\(759\) 0 0
\(760\) 0 0
\(761\) −27.0000 −0.978749 −0.489375 0.872074i \(-0.662775\pi\)
−0.489375 + 0.872074i \(0.662775\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) −2.00000 −0.0723575
\(765\) 0 0
\(766\) −72.0000 −2.60147
\(767\) − 4.00000i − 0.144432i
\(768\) 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
\(772\) 52.0000i 1.87152i
\(773\) − 18.0000i − 0.647415i −0.946157 0.323708i \(-0.895071\pi\)
0.946157 0.323708i \(-0.104929\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) − 12.0000i − 0.430221i
\(779\) 35.0000 1.25401
\(780\) 0 0
\(781\) 5.00000 0.178914
\(782\) − 48.0000i − 1.71648i
\(783\) 0 0
\(784\) −28.0000 −1.00000
\(785\) 0 0
\(786\) 0 0
\(787\) 4.00000i 0.142585i 0.997455 + 0.0712923i \(0.0227123\pi\)
−0.997455 + 0.0712923i \(0.977288\pi\)
\(788\) − 24.0000i − 0.854965i
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 8.00000i 0.284088i
\(794\) 76.0000 2.69714
\(795\) 0 0
\(796\) 32.0000 1.13421
\(797\) 4.00000i 0.141687i 0.997487 + 0.0708436i \(0.0225691\pi\)
−0.997487 + 0.0708436i \(0.977431\pi\)
\(798\) 0 0
\(799\) 8.00000 0.283020
\(800\) 0 0
\(801\) 0 0
\(802\) 60.0000i 2.11867i
\(803\) 40.0000i 1.41157i
\(804\) 0 0
\(805\) 0 0
\(806\) −72.0000 −2.53609
\(807\) 0 0
\(808\) 0 0
\(809\) −7.00000 −0.246107 −0.123053 0.992400i \(-0.539269\pi\)
−0.123053 + 0.992400i \(0.539269\pi\)
\(810\) 0 0
\(811\) −21.0000 −0.737410 −0.368705 0.929547i \(-0.620199\pi\)
−0.368705 + 0.929547i \(0.620199\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 100.000 3.50500
\(815\) 0 0
\(816\) 0 0
\(817\) 10.0000i 0.349856i
\(818\) − 28.0000i − 0.978997i
\(819\) 0 0
\(820\) 0 0
\(821\) 3.00000 0.104701 0.0523504 0.998629i \(-0.483329\pi\)
0.0523504 + 0.998629i \(0.483329\pi\)
\(822\) 0 0
\(823\) 46.0000i 1.60346i 0.597687 + 0.801730i \(0.296087\pi\)
−0.597687 + 0.801730i \(0.703913\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 44.0000i 1.53003i 0.644013 + 0.765015i \(0.277268\pi\)
−0.644013 + 0.765015i \(0.722732\pi\)
\(828\) 0 0
\(829\) −27.0000 −0.937749 −0.468874 0.883265i \(-0.655340\pi\)
−0.468874 + 0.883265i \(0.655340\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) − 32.0000i − 1.10940i
\(833\) − 28.0000i − 0.970143i
\(834\) 0 0
\(835\) 0 0
\(836\) −50.0000 −1.72929
\(837\) 0 0
\(838\) − 32.0000i − 1.10542i
\(839\) −13.0000 −0.448810 −0.224405 0.974496i \(-0.572044\pi\)
−0.224405 + 0.974496i \(0.572044\pi\)
\(840\) 0 0
\(841\) −4.00000 −0.137931
\(842\) 26.0000i 0.896019i
\(843\) 0 0
\(844\) −22.0000 −0.757271
\(845\) 0 0
\(846\) 0 0
\(847\) 0 0
\(848\) 32.0000i 1.09888i
\(849\) 0 0
\(850\) 0 0
\(851\) −60.0000 −2.05677
\(852\) 0 0
\(853\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 32.0000i 1.09310i 0.837427 + 0.546550i \(0.184059\pi\)
−0.837427 + 0.546550i \(0.815941\pi\)
\(858\) 0 0
\(859\) −55.0000 −1.87658 −0.938288 0.345855i \(-0.887589\pi\)
−0.938288 + 0.345855i \(0.887589\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 6.00000i 0.204361i
\(863\) 44.0000i 1.49778i 0.662696 + 0.748889i \(0.269412\pi\)
−0.662696 + 0.748889i \(0.730588\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) −52.0000 −1.76703
\(867\) 0 0
\(868\) 0 0
\(869\) −60.0000 −2.03536
\(870\) 0 0
\(871\) 24.0000 0.813209
\(872\) 0 0
\(873\) 0 0
\(874\) 60.0000 2.02953
\(875\) 0 0
\(876\) 0 0
\(877\) − 24.0000i − 0.810422i −0.914223 0.405211i \(-0.867198\pi\)
0.914223 0.405211i \(-0.132802\pi\)
\(878\) 58.0000i 1.95741i
\(879\) 0 0
\(880\) 0 0
\(881\) −25.0000 −0.842271 −0.421136 0.906998i \(-0.638368\pi\)
−0.421136 + 0.906998i \(0.638368\pi\)
\(882\) 0 0
\(883\) 52.0000i 1.74994i 0.484178 + 0.874970i \(0.339119\pi\)
−0.484178 + 0.874970i \(0.660881\pi\)
\(884\) 32.0000 1.07628
\(885\) 0 0
\(886\) 12.0000 0.403148
\(887\) 6.00000i 0.201460i 0.994914 + 0.100730i \(0.0321179\pi\)
−0.994914 + 0.100730i \(0.967882\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) 16.0000i 0.535720i
\(893\) 10.0000i 0.334637i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) − 34.0000i − 1.13459i
\(899\) −45.0000 −1.50083
\(900\) 0 0
\(901\) −32.0000 −1.06607
\(902\) 70.0000i 2.33075i
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) − 18.0000i − 0.597680i −0.954303 0.298840i \(-0.903400\pi\)
0.954303 0.298840i \(-0.0965997\pi\)
\(908\) − 8.00000i − 0.265489i
\(909\) 0 0
\(910\) 0 0
\(911\) −49.0000 −1.62344 −0.811721 0.584045i \(-0.801469\pi\)
−0.811721 + 0.584045i \(0.801469\pi\)
\(912\) 0 0
\(913\) − 30.0000i − 0.992855i
\(914\) 76.0000 2.51386
\(915\) 0 0
\(916\) 12.0000 0.396491
\(917\) 0 0
\(918\) 0 0
\(919\) 11.0000 0.362857 0.181428 0.983404i \(-0.441928\pi\)
0.181428 + 0.983404i \(0.441928\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) − 30.0000i − 0.987997i
\(923\) − 4.00000i − 0.131662i
\(924\) 0 0
\(925\) 0 0
\(926\) −12.0000 −0.394344
\(927\) 0 0
\(928\) − 40.0000i − 1.31306i
\(929\) −1.00000 −0.0328089 −0.0164045 0.999865i \(-0.505222\pi\)
−0.0164045 + 0.999865i \(0.505222\pi\)
\(930\) 0 0
\(931\) 35.0000 1.14708
\(932\) 12.0000i 0.393073i
\(933\) 0 0
\(934\) −8.00000 −0.261768
\(935\) 0 0
\(936\) 0 0
\(937\) 42.0000i 1.37208i 0.727564 + 0.686040i \(0.240653\pi\)
−0.727564 + 0.686040i \(0.759347\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 46.0000 1.49956 0.749779 0.661689i \(-0.230160\pi\)
0.749779 + 0.661689i \(0.230160\pi\)
\(942\) 0 0
\(943\) − 42.0000i − 1.36771i
\(944\) −4.00000 −0.130189
\(945\) 0 0
\(946\) −20.0000 −0.650256
\(947\) 18.0000i 0.584921i 0.956278 + 0.292461i \(0.0944741\pi\)
−0.956278 + 0.292461i \(0.905526\pi\)
\(948\) 0 0
\(949\) 32.0000 1.03876
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 16.0000i 0.518291i 0.965838 + 0.259145i \(0.0834409\pi\)
−0.965838 + 0.259145i \(0.916559\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −32.0000 −1.03495
\(957\) 0 0
\(958\) 30.0000i 0.969256i
\(959\) 0 0
\(960\) 0 0
\(961\) 50.0000 1.61290
\(962\) − 80.0000i − 2.57930i
\(963\) 0 0
\(964\) 22.0000 0.708572
\(965\) 0 0
\(966\) 0 0
\(967\) − 40.0000i − 1.28631i −0.765735 0.643157i \(-0.777624\pi\)
0.765735 0.643157i \(-0.222376\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −27.0000 −0.866471 −0.433236 0.901281i \(-0.642628\pi\)
−0.433236 + 0.901281i \(0.642628\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) −16.0000 −0.512673
\(975\) 0 0
\(976\) 8.00000 0.256074
\(977\) − 28.0000i − 0.895799i −0.894084 0.447900i \(-0.852172\pi\)
0.894084 0.447900i \(-0.147828\pi\)
\(978\) 0 0
\(979\) 45.0000 1.43821
\(980\) 0 0
\(981\) 0 0
\(982\) 86.0000i 2.74437i
\(983\) − 48.0000i − 1.53096i −0.643458 0.765481i \(-0.722501\pi\)
0.643458 0.765481i \(-0.277499\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 40.0000 1.27386
\(987\) 0 0
\(988\) 40.0000i 1.27257i
\(989\) 12.0000 0.381578
\(990\) 0 0
\(991\) −1.00000 −0.0317660 −0.0158830 0.999874i \(-0.505056\pi\)
−0.0158830 + 0.999874i \(0.505056\pi\)
\(992\) 72.0000i 2.28600i
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 48.0000i 1.52018i 0.649821 + 0.760088i \(0.274844\pi\)
−0.649821 + 0.760088i \(0.725156\pi\)
\(998\) − 14.0000i − 0.443162i
\(999\) 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.b.649.2 2
3.2 odd 2 2025.2.b.a.649.1 2
5.2 odd 4 2025.2.a.a.1.1 1
5.3 odd 4 405.2.a.f.1.1 yes 1
5.4 even 2 inner 2025.2.b.b.649.1 2
15.2 even 4 2025.2.a.f.1.1 1
15.8 even 4 405.2.a.a.1.1 1
15.14 odd 2 2025.2.b.a.649.2 2
20.3 even 4 6480.2.a.r.1.1 1
45.13 odd 12 405.2.e.a.136.1 2
45.23 even 12 405.2.e.g.136.1 2
45.38 even 12 405.2.e.g.271.1 2
45.43 odd 12 405.2.e.a.271.1 2
60.23 odd 4 6480.2.a.f.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
405.2.a.a.1.1 1 15.8 even 4
405.2.a.f.1.1 yes 1 5.3 odd 4
405.2.e.a.136.1 2 45.13 odd 12
405.2.e.a.271.1 2 45.43 odd 12
405.2.e.g.136.1 2 45.23 even 12
405.2.e.g.271.1 2 45.38 even 12
2025.2.a.a.1.1 1 5.2 odd 4
2025.2.a.f.1.1 1 15.2 even 4
2025.2.b.a.649.1 2 3.2 odd 2
2025.2.b.a.649.2 2 15.14 odd 2
2025.2.b.b.649.1 2 5.4 even 2 inner
2025.2.b.b.649.2 2 1.1 even 1 trivial
6480.2.a.f.1.1 1 60.23 odd 4
6480.2.a.r.1.1 1 20.3 even 4