Properties

Label 2025.2.a.f.1.1
Level $2025$
Weight $2$
Character 2025.1
Self dual yes
Analytic conductor $16.170$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2025 = 3^{4} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2025.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(16.1697064093\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 405)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 2025.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.00000 q^{2} +2.00000 q^{4} +O(q^{10})\) \(q+2.00000 q^{2} +2.00000 q^{4} -5.00000 q^{11} -4.00000 q^{13} -4.00000 q^{16} -4.00000 q^{17} -5.00000 q^{19} -10.0000 q^{22} +6.00000 q^{23} -8.00000 q^{26} +5.00000 q^{29} -9.00000 q^{31} -8.00000 q^{32} -8.00000 q^{34} +10.0000 q^{37} -10.0000 q^{38} -7.00000 q^{41} +2.00000 q^{43} -10.0000 q^{44} +12.0000 q^{46} +2.00000 q^{47} -7.00000 q^{49} -8.00000 q^{52} +8.00000 q^{53} +10.0000 q^{58} +1.00000 q^{59} -2.00000 q^{61} -18.0000 q^{62} -8.00000 q^{64} -6.00000 q^{67} -8.00000 q^{68} -1.00000 q^{71} +8.00000 q^{73} +20.0000 q^{74} -10.0000 q^{76} +12.0000 q^{79} -14.0000 q^{82} +6.00000 q^{83} +4.00000 q^{86} +9.00000 q^{89} +12.0000 q^{92} +4.00000 q^{94} -14.0000 q^{97} -14.0000 q^{98} +O(q^{100})\)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.00000 1.41421 0.707107 0.707107i \(-0.250000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(3\) 0 0
\(4\) 2.00000 1.00000
\(5\) 0 0
\(6\) 0 0
\(7\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −5.00000 −1.50756 −0.753778 0.657129i \(-0.771771\pi\)
−0.753778 + 0.657129i \(0.771771\pi\)
\(12\) 0 0
\(13\) −4.00000 −1.10940 −0.554700 0.832050i \(-0.687167\pi\)
−0.554700 + 0.832050i \(0.687167\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −4.00000 −1.00000
\(17\) −4.00000 −0.970143 −0.485071 0.874475i \(-0.661206\pi\)
−0.485071 + 0.874475i \(0.661206\pi\)
\(18\) 0 0
\(19\) −5.00000 −1.14708 −0.573539 0.819178i \(-0.694430\pi\)
−0.573539 + 0.819178i \(0.694430\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −10.0000 −2.13201
\(23\) 6.00000 1.25109 0.625543 0.780189i \(-0.284877\pi\)
0.625543 + 0.780189i \(0.284877\pi\)
\(24\) 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 \(0.799571\pi\)
\(32\) −8.00000 −1.41421
\(33\) 0 0
\(34\) −8.00000 −1.37199
\(35\) 0 0
\(36\) 0 0
\(37\) 10.0000 1.64399 0.821995 0.569495i \(-0.192861\pi\)
0.821995 + 0.569495i \(0.192861\pi\)
\(38\) −10.0000 −1.62221
\(39\) 0 0
\(40\) 0 0
\(41\) −7.00000 −1.09322 −0.546608 0.837389i \(-0.684081\pi\)
−0.546608 + 0.837389i \(0.684081\pi\)
\(42\) 0 0
\(43\) 2.00000 0.304997 0.152499 0.988304i \(-0.451268\pi\)
0.152499 + 0.988304i \(0.451268\pi\)
\(44\) −10.0000 −1.50756
\(45\) 0 0
\(46\) 12.0000 1.76930
\(47\) 2.00000 0.291730 0.145865 0.989305i \(-0.453403\pi\)
0.145865 + 0.989305i \(0.453403\pi\)
\(48\) 0 0
\(49\) −7.00000 −1.00000
\(50\) 0 0
\(51\) 0 0
\(52\) −8.00000 −1.10940
\(53\) 8.00000 1.09888 0.549442 0.835532i \(-0.314840\pi\)
0.549442 + 0.835532i \(0.314840\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 10.0000 1.31306
\(59\) 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 \(0.540868\pi\)
\(62\) −18.0000 −2.28600
\(63\) 0 0
\(64\) −8.00000 −1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) −6.00000 −0.733017 −0.366508 0.930415i \(-0.619447\pi\)
−0.366508 + 0.930415i \(0.619447\pi\)
\(68\) −8.00000 −0.970143
\(69\) 0 0
\(70\) 0 0
\(71\) −1.00000 −0.118678 −0.0593391 0.998238i \(-0.518899\pi\)
−0.0593391 + 0.998238i \(0.518899\pi\)
\(72\) 0 0
\(73\) 8.00000 0.936329 0.468165 0.883641i \(-0.344915\pi\)
0.468165 + 0.883641i \(0.344915\pi\)
\(74\) 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 \(-0.264121\pi\)
0.675053 + 0.737769i \(0.264121\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −14.0000 −1.54604
\(83\) 6.00000 0.658586 0.329293 0.944228i \(-0.393190\pi\)
0.329293 + 0.944228i \(0.393190\pi\)
\(84\) 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
\(92\) 12.0000 1.25109
\(93\) 0 0
\(94\) 4.00000 0.412568
\(95\) 0 0
\(96\) 0 0
\(97\) −14.0000 −1.42148 −0.710742 0.703452i \(-0.751641\pi\)
−0.710742 + 0.703452i \(0.751641\pi\)
\(98\) −14.0000 −1.41421
\(99\) 0 0
\(100\) 0 0
\(101\) 3.00000 0.298511 0.149256 0.988799i \(-0.452312\pi\)
0.149256 + 0.988799i \(0.452312\pi\)
\(102\) 0 0
\(103\) −2.00000 −0.197066 −0.0985329 0.995134i \(-0.531415\pi\)
−0.0985329 + 0.995134i \(0.531415\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 16.0000 1.55406
\(107\) −6.00000 −0.580042 −0.290021 0.957020i \(-0.593662\pi\)
−0.290021 + 0.957020i \(0.593662\pi\)
\(108\) 0 0
\(109\) −1.00000 −0.0957826 −0.0478913 0.998853i \(-0.515250\pi\)
−0.0478913 + 0.998853i \(0.515250\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −16.0000 −1.50515 −0.752577 0.658505i \(-0.771189\pi\)
−0.752577 + 0.658505i \(0.771189\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 10.0000 0.928477
\(117\) 0 0
\(118\) 2.00000 0.184115
\(119\) 0 0
\(120\) 0 0
\(121\) 14.0000 1.27273
\(122\) −4.00000 −0.362143
\(123\) 0 0
\(124\) −18.0000 −1.61645
\(125\) 0 0
\(126\) 0 0
\(127\) −16.0000 −1.41977 −0.709885 0.704317i \(-0.751253\pi\)
−0.709885 + 0.704317i \(0.751253\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −15.0000 −1.31056 −0.655278 0.755388i \(-0.727449\pi\)
−0.655278 + 0.755388i \(0.727449\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) −12.0000 −1.03664
\(135\) 0 0
\(136\) 0 0
\(137\) 12.0000 1.02523 0.512615 0.858619i \(-0.328677\pi\)
0.512615 + 0.858619i \(0.328677\pi\)
\(138\) 0 0
\(139\) −19.0000 −1.61156 −0.805779 0.592216i \(-0.798253\pi\)
−0.805779 + 0.592216i \(0.798253\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −2.00000 −0.167836
\(143\) 20.0000 1.67248
\(144\) 0 0
\(145\) 0 0
\(146\) 16.0000 1.32417
\(147\) 0 0
\(148\) 20.0000 1.64399
\(149\) 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
\(157\) −2.00000 −0.159617 −0.0798087 0.996810i \(-0.525431\pi\)
−0.0798087 + 0.996810i \(0.525431\pi\)
\(158\) 24.0000 1.90934
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −8.00000 −0.626608 −0.313304 0.949653i \(-0.601436\pi\)
−0.313304 + 0.949653i \(0.601436\pi\)
\(164\) −14.0000 −1.09322
\(165\) 0 0
\(166\) 12.0000 0.931381
\(167\) 12.0000 0.928588 0.464294 0.885681i \(-0.346308\pi\)
0.464294 + 0.885681i \(0.346308\pi\)
\(168\) 0 0
\(169\) 3.00000 0.230769
\(170\) 0 0
\(171\) 0 0
\(172\) 4.00000 0.304997
\(173\) −6.00000 −0.456172 −0.228086 0.973641i \(-0.573247\pi\)
−0.228086 + 0.973641i \(0.573247\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 20.0000 1.50756
\(177\) 0 0
\(178\) 18.0000 1.34916
\(179\) −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
\(187\) 20.0000 1.46254
\(188\) 4.00000 0.291730
\(189\) 0 0
\(190\) 0 0
\(191\) −1.00000 −0.0723575 −0.0361787 0.999345i \(-0.511519\pi\)
−0.0361787 + 0.999345i \(0.511519\pi\)
\(192\) 0 0
\(193\) −26.0000 −1.87152 −0.935760 0.352636i \(-0.885285\pi\)
−0.935760 + 0.352636i \(0.885285\pi\)
\(194\) −28.0000 −2.01028
\(195\) 0 0
\(196\) −14.0000 −1.00000
\(197\) 12.0000 0.854965 0.427482 0.904024i \(-0.359401\pi\)
0.427482 + 0.904024i \(0.359401\pi\)
\(198\) 0 0
\(199\) 16.0000 1.13421 0.567105 0.823646i \(-0.308063\pi\)
0.567105 + 0.823646i \(0.308063\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 6.00000 0.422159
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) −4.00000 −0.278693
\(207\) 0 0
\(208\) 16.0000 1.10940
\(209\) 25.0000 1.72929
\(210\) 0 0
\(211\) 11.0000 0.757271 0.378636 0.925546i \(-0.376393\pi\)
0.378636 + 0.925546i \(0.376393\pi\)
\(212\) 16.0000 1.09888
\(213\) 0 0
\(214\) −12.0000 −0.820303
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) −2.00000 −0.135457
\(219\) 0 0
\(220\) 0 0
\(221\) 16.0000 1.07628
\(222\) 0 0
\(223\) −8.00000 −0.535720 −0.267860 0.963458i \(-0.586316\pi\)
−0.267860 + 0.963458i \(0.586316\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) −32.0000 −2.12861
\(227\) 4.00000 0.265489 0.132745 0.991150i \(-0.457621\pi\)
0.132745 + 0.991150i \(0.457621\pi\)
\(228\) 0 0
\(229\) 6.00000 0.396491 0.198246 0.980152i \(-0.436476\pi\)
0.198246 + 0.980152i \(0.436476\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 6.00000 0.393073 0.196537 0.980497i \(-0.437031\pi\)
0.196537 + 0.980497i \(0.437031\pi\)
\(234\) 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 \(0.615276\pi\)
\(242\) 28.0000 1.79991
\(243\) 0 0
\(244\) −4.00000 −0.256074
\(245\) 0 0
\(246\) 0 0
\(247\) 20.0000 1.27257
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 12.0000 0.757433 0.378717 0.925513i \(-0.376365\pi\)
0.378717 + 0.925513i \(0.376365\pi\)
\(252\) 0 0
\(253\) −30.0000 −1.88608
\(254\) −32.0000 −2.00786
\(255\) 0 0
\(256\) 16.0000 1.00000
\(257\) −6.00000 −0.374270 −0.187135 0.982334i \(-0.559920\pi\)
−0.187135 + 0.982334i \(0.559920\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) −30.0000 −1.85341
\(263\) 10.0000 0.616626 0.308313 0.951285i \(-0.400236\pi\)
0.308313 + 0.951285i \(0.400236\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) −12.0000 −0.733017
\(269\) 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 \(0.578126\pi\)
\(272\) 16.0000 0.970143
\(273\) 0 0
\(274\) 24.0000 1.44989
\(275\) 0 0
\(276\) 0 0
\(277\) 18.0000 1.08152 0.540758 0.841178i \(-0.318138\pi\)
0.540758 + 0.841178i \(0.318138\pi\)
\(278\) −38.0000 −2.27909
\(279\) 0 0
\(280\) 0 0
\(281\) −6.00000 −0.357930 −0.178965 0.983855i \(-0.557275\pi\)
−0.178965 + 0.983855i \(0.557275\pi\)
\(282\) 0 0
\(283\) −6.00000 −0.356663 −0.178331 0.983970i \(-0.557070\pi\)
−0.178331 + 0.983970i \(0.557070\pi\)
\(284\) −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
\(292\) 16.0000 0.936329
\(293\) 18.0000 1.05157 0.525786 0.850617i \(-0.323771\pi\)
0.525786 + 0.850617i \(0.323771\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 4.00000 0.231714
\(299\) −24.0000 −1.38796
\(300\) 0 0
\(301\) 0 0
\(302\) −10.0000 −0.575435
\(303\) 0 0
\(304\) 20.0000 1.14708
\(305\) 0 0
\(306\) 0 0
\(307\) −10.0000 −0.570730 −0.285365 0.958419i \(-0.592115\pi\)
−0.285365 + 0.958419i \(0.592115\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 9.00000 0.510343 0.255172 0.966896i \(-0.417868\pi\)
0.255172 + 0.966896i \(0.417868\pi\)
\(312\) 0 0
\(313\) 4.00000 0.226093 0.113047 0.993590i \(-0.463939\pi\)
0.113047 + 0.993590i \(0.463939\pi\)
\(314\) −4.00000 −0.225733
\(315\) 0 0
\(316\) 24.0000 1.35011
\(317\) −2.00000 −0.112331 −0.0561656 0.998421i \(-0.517887\pi\)
−0.0561656 + 0.998421i \(0.517887\pi\)
\(318\) 0 0
\(319\) −25.0000 −1.39973
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 20.0000 1.11283
\(324\) 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 \(0.695828\pi\)
\(332\) 12.0000 0.658586
\(333\) 0 0
\(334\) 24.0000 1.31322
\(335\) 0 0
\(336\) 0 0
\(337\) 8.00000 0.435788 0.217894 0.975972i \(-0.430081\pi\)
0.217894 + 0.975972i \(0.430081\pi\)
\(338\) 6.00000 0.326357
\(339\) 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
\(347\) 20.0000 1.07366 0.536828 0.843692i \(-0.319622\pi\)
0.536828 + 0.843692i \(0.319622\pi\)
\(348\) 0 0
\(349\) 13.0000 0.695874 0.347937 0.937518i \(-0.386882\pi\)
0.347937 + 0.937518i \(0.386882\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 40.0000 2.13201
\(353\) −18.0000 −0.958043 −0.479022 0.877803i \(-0.659008\pi\)
−0.479022 + 0.877803i \(0.659008\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 18.0000 0.953998
\(357\) 0 0
\(358\) −46.0000 −2.43118
\(359\) 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
\(362\) −50.0000 −2.62794
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −18.0000 −0.939592 −0.469796 0.882775i \(-0.655673\pi\)
−0.469796 + 0.882775i \(0.655673\pi\)
\(368\) −24.0000 −1.25109
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −16.0000 −0.828449 −0.414224 0.910175i \(-0.635947\pi\)
−0.414224 + 0.910175i \(0.635947\pi\)
\(374\) 40.0000 2.06835
\(375\) 0 0
\(376\) 0 0
\(377\) −20.0000 −1.03005
\(378\) 0 0
\(379\) 4.00000 0.205466 0.102733 0.994709i \(-0.467241\pi\)
0.102733 + 0.994709i \(0.467241\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −2.00000 −0.102329
\(383\) −36.0000 −1.83951 −0.919757 0.392488i \(-0.871614\pi\)
−0.919757 + 0.392488i \(0.871614\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −52.0000 −2.64673
\(387\) 0 0
\(388\) −28.0000 −1.42148
\(389\) −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
\(397\) 38.0000 1.90717 0.953583 0.301131i \(-0.0973643\pi\)
0.953583 + 0.301131i \(0.0973643\pi\)
\(398\) 32.0000 1.60402
\(399\) 0 0
\(400\) 0 0
\(401\) −30.0000 −1.49813 −0.749064 0.662497i \(-0.769497\pi\)
−0.749064 + 0.662497i \(0.769497\pi\)
\(402\) 0 0
\(403\) 36.0000 1.79329
\(404\) 6.00000 0.298511
\(405\) 0 0
\(406\) 0 0
\(407\) −50.0000 −2.47841
\(408\) 0 0
\(409\) 14.0000 0.692255 0.346128 0.938187i \(-0.387496\pi\)
0.346128 + 0.938187i \(0.387496\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −4.00000 −0.197066
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 32.0000 1.56893
\(417\) 0 0
\(418\) 50.0000 2.44558
\(419\) −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 \(0.397395\pi\)
\(422\) 22.0000 1.07094
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) −12.0000 −0.580042
\(429\) 0 0
\(430\) 0 0
\(431\) −3.00000 −0.144505 −0.0722525 0.997386i \(-0.523019\pi\)
−0.0722525 + 0.997386i \(0.523019\pi\)
\(432\) 0 0
\(433\) 26.0000 1.24948 0.624740 0.780833i \(-0.285205\pi\)
0.624740 + 0.780833i \(0.285205\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −2.00000 −0.0957826
\(437\) −30.0000 −1.43509
\(438\) 0 0
\(439\) −29.0000 −1.38409 −0.692047 0.721852i \(-0.743291\pi\)
−0.692047 + 0.721852i \(0.743291\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 32.0000 1.52208
\(443\) 6.00000 0.285069 0.142534 0.989790i \(-0.454475\pi\)
0.142534 + 0.989790i \(0.454475\pi\)
\(444\) 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
\(452\) −32.0000 −1.50515
\(453\) 0 0
\(454\) 8.00000 0.375459
\(455\) 0 0
\(456\) 0 0
\(457\) 38.0000 1.77757 0.888783 0.458329i \(-0.151552\pi\)
0.888783 + 0.458329i \(0.151552\pi\)
\(458\) 12.0000 0.560723
\(459\) 0 0
\(460\) 0 0
\(461\) 15.0000 0.698620 0.349310 0.937007i \(-0.386416\pi\)
0.349310 + 0.937007i \(0.386416\pi\)
\(462\) 0 0
\(463\) 6.00000 0.278844 0.139422 0.990233i \(-0.455476\pi\)
0.139422 + 0.990233i \(0.455476\pi\)
\(464\) −20.0000 −0.928477
\(465\) 0 0
\(466\) 12.0000 0.555889
\(467\) 4.00000 0.185098 0.0925490 0.995708i \(-0.470499\pi\)
0.0925490 + 0.995708i \(0.470499\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −10.0000 −0.459800
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) 32.0000 1.46365
\(479\) 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
\(482\) −22.0000 −1.00207
\(483\) 0 0
\(484\) 28.0000 1.27273
\(485\) 0 0
\(486\) 0 0
\(487\) −8.00000 −0.362515 −0.181257 0.983436i \(-0.558017\pi\)
−0.181257 + 0.983436i \(0.558017\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −43.0000 −1.94056 −0.970281 0.241979i \(-0.922203\pi\)
−0.970281 + 0.241979i \(0.922203\pi\)
\(492\) 0 0
\(493\) −20.0000 −0.900755
\(494\) 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 \(-0.449920\pi\)
0.156682 + 0.987649i \(0.449920\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 24.0000 1.07117
\(503\) −20.0000 −0.891756 −0.445878 0.895094i \(-0.647108\pi\)
−0.445878 + 0.895094i \(0.647108\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −60.0000 −2.66733
\(507\) 0 0
\(508\) −32.0000 −1.41977
\(509\) −2.00000 −0.0886484 −0.0443242 0.999017i \(-0.514113\pi\)
−0.0443242 + 0.999017i \(0.514113\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 32.0000 1.41421
\(513\) 0 0
\(514\) −12.0000 −0.529297
\(515\) 0 0
\(516\) 0 0
\(517\) −10.0000 −0.439799
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −10.0000 −0.438108 −0.219054 0.975713i \(-0.570297\pi\)
−0.219054 + 0.975713i \(0.570297\pi\)
\(522\) 0 0
\(523\) −16.0000 −0.699631 −0.349816 0.936819i \(-0.613756\pi\)
−0.349816 + 0.936819i \(0.613756\pi\)
\(524\) −30.0000 −1.31056
\(525\) 0 0
\(526\) 20.0000 0.872041
\(527\) 36.0000 1.56818
\(528\) 0 0
\(529\) 13.0000 0.565217
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 28.0000 1.21281
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 62.0000 2.67301
\(539\) 35.0000 1.50756
\(540\) 0 0
\(541\) 3.00000 0.128980 0.0644900 0.997918i \(-0.479458\pi\)
0.0644900 + 0.997918i \(0.479458\pi\)
\(542\) −16.0000 −0.687259
\(543\) 0 0
\(544\) 32.0000 1.37199
\(545\) 0 0
\(546\) 0 0
\(547\) −28.0000 −1.19719 −0.598597 0.801050i \(-0.704275\pi\)
−0.598597 + 0.801050i \(0.704275\pi\)
\(548\) 24.0000 1.02523
\(549\) 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
\(557\) −24.0000 −1.01691 −0.508456 0.861088i \(-0.669784\pi\)
−0.508456 + 0.861088i \(0.669784\pi\)
\(558\) 0 0
\(559\) −8.00000 −0.338364
\(560\) 0 0
\(561\) 0 0
\(562\) −12.0000 −0.506189
\(563\) −18.0000 −0.758610 −0.379305 0.925272i \(-0.623837\pi\)
−0.379305 + 0.925272i \(0.623837\pi\)
\(564\) 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 \(0.466637\pi\)
\(572\) 40.0000 1.67248
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 16.0000 0.666089 0.333044 0.942911i \(-0.391924\pi\)
0.333044 + 0.942911i \(0.391924\pi\)
\(578\) −2.00000 −0.0831890
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −40.0000 −1.65663
\(584\) 0 0
\(585\) 0 0
\(586\) 36.0000 1.48715
\(587\) 18.0000 0.742940 0.371470 0.928445i \(-0.378854\pi\)
0.371470 + 0.928445i \(0.378854\pi\)
\(588\) 0 0
\(589\) 45.0000 1.85419
\(590\) 0 0
\(591\) 0 0
\(592\) −40.0000 −1.64399
\(593\) −14.0000 −0.574911 −0.287456 0.957794i \(-0.592809\pi\)
−0.287456 + 0.957794i \(0.592809\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 4.00000 0.163846
\(597\) 0 0
\(598\) −48.0000 −1.96287
\(599\) 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
\(607\) −34.0000 −1.38002 −0.690009 0.723801i \(-0.742393\pi\)
−0.690009 + 0.723801i \(0.742393\pi\)
\(608\) 40.0000 1.62221
\(609\) 0 0
\(610\) 0 0
\(611\) −8.00000 −0.323645
\(612\) 0 0
\(613\) −4.00000 −0.161558 −0.0807792 0.996732i \(-0.525741\pi\)
−0.0807792 + 0.996732i \(0.525741\pi\)
\(614\) −20.0000 −0.807134
\(615\) 0 0
\(616\) 0 0
\(617\) −24.0000 −0.966204 −0.483102 0.875564i \(-0.660490\pi\)
−0.483102 + 0.875564i \(0.660490\pi\)
\(618\) 0 0
\(619\) −28.0000 −1.12542 −0.562708 0.826656i \(-0.690240\pi\)
−0.562708 + 0.826656i \(0.690240\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 18.0000 0.721734
\(623\) 0 0
\(624\) 0 0
\(625\) 0 0
\(626\) 8.00000 0.319744
\(627\) 0 0
\(628\) −4.00000 −0.159617
\(629\) −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
\(637\) 28.0000 1.10940
\(638\) −50.0000 −1.97952
\(639\) 0 0
\(640\) 0 0
\(641\) −3.00000 −0.118493 −0.0592464 0.998243i \(-0.518870\pi\)
−0.0592464 + 0.998243i \(0.518870\pi\)
\(642\) 0 0
\(643\) −6.00000 −0.236617 −0.118308 0.992977i \(-0.537747\pi\)
−0.118308 + 0.992977i \(0.537747\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 40.0000 1.57378
\(647\) −10.0000 −0.393141 −0.196570 0.980490i \(-0.562980\pi\)
−0.196570 + 0.980490i \(0.562980\pi\)
\(648\) 0 0
\(649\) −5.00000 −0.196267
\(650\) 0 0
\(651\) 0 0
\(652\) −16.0000 −0.626608
\(653\) −8.00000 −0.313064 −0.156532 0.987673i \(-0.550031\pi\)
−0.156532 + 0.987673i \(0.550031\pi\)
\(654\) 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 \(0.338385\pi\)
\(662\) −42.0000 −1.63238
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 30.0000 1.16160
\(668\) 24.0000 0.928588
\(669\) 0 0
\(670\) 0 0
\(671\) 10.0000 0.386046
\(672\) 0 0
\(673\) −42.0000 −1.61898 −0.809491 0.587133i \(-0.800257\pi\)
−0.809491 + 0.587133i \(0.800257\pi\)
\(674\) 16.0000 0.616297
\(675\) 0 0
\(676\) 6.00000 0.230769
\(677\) 6.00000 0.230599 0.115299 0.993331i \(-0.463217\pi\)
0.115299 + 0.993331i \(0.463217\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 90.0000 3.44628
\(683\) 48.0000 1.83667 0.918334 0.395805i \(-0.129534\pi\)
0.918334 + 0.395805i \(0.129534\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) −8.00000 −0.304997
\(689\) −32.0000 −1.21910
\(690\) 0 0
\(691\) 4.00000 0.152167 0.0760836 0.997101i \(-0.475758\pi\)
0.0760836 + 0.997101i \(0.475758\pi\)
\(692\) −12.0000 −0.456172
\(693\) 0 0
\(694\) 40.0000 1.51838
\(695\) 0 0
\(696\) 0 0
\(697\) 28.0000 1.06058
\(698\) 26.0000 0.984115
\(699\) 0 0
\(700\) 0 0
\(701\) −19.0000 −0.717620 −0.358810 0.933411i \(-0.616817\pi\)
−0.358810 + 0.933411i \(0.616817\pi\)
\(702\) 0 0
\(703\) −50.0000 −1.88579
\(704\) 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 \(-0.364441\pi\)
0.413114 + 0.910679i \(0.364441\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −54.0000 −2.02232
\(714\) 0 0
\(715\) 0 0
\(716\) −46.0000 −1.71910
\(717\) 0 0
\(718\) 54.0000 2.01526
\(719\) −27.0000 −1.00693 −0.503465 0.864016i \(-0.667942\pi\)
−0.503465 + 0.864016i \(0.667942\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 12.0000 0.446594
\(723\) 0 0
\(724\) −50.0000 −1.85824
\(725\) 0 0
\(726\) 0 0
\(727\) −44.0000 −1.63187 −0.815935 0.578144i \(-0.803777\pi\)
−0.815935 + 0.578144i \(0.803777\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −8.00000 −0.295891
\(732\) 0 0
\(733\) 46.0000 1.69905 0.849524 0.527549i \(-0.176889\pi\)
0.849524 + 0.527549i \(0.176889\pi\)
\(734\) −36.0000 −1.32878
\(735\) 0 0
\(736\) −48.0000 −1.76930
\(737\) 30.0000 1.10506
\(738\) 0 0
\(739\) −35.0000 −1.28750 −0.643748 0.765238i \(-0.722621\pi\)
−0.643748 + 0.765238i \(0.722621\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −4.00000 −0.146746 −0.0733729 0.997305i \(-0.523376\pi\)
−0.0733729 + 0.997305i \(0.523376\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −32.0000 −1.17160
\(747\) 0 0
\(748\) 40.0000 1.46254
\(749\) 0 0
\(750\) 0 0
\(751\) −32.0000 −1.16770 −0.583848 0.811863i \(-0.698454\pi\)
−0.583848 + 0.811863i \(0.698454\pi\)
\(752\) −8.00000 −0.291730
\(753\) 0 0
\(754\) −40.0000 −1.45671
\(755\) 0 0
\(756\) 0 0
\(757\) −28.0000 −1.01768 −0.508839 0.860862i \(-0.669925\pi\)
−0.508839 + 0.860862i \(0.669925\pi\)
\(758\) 8.00000 0.290573
\(759\) 0 0
\(760\) 0 0
\(761\) 27.0000 0.978749 0.489375 0.872074i \(-0.337225\pi\)
0.489375 + 0.872074i \(0.337225\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) −2.00000 −0.0723575
\(765\) 0 0
\(766\) −72.0000 −2.60147
\(767\) −4.00000 −0.144432
\(768\) 0 0
\(769\) 5.00000 0.180305 0.0901523 0.995928i \(-0.471265\pi\)
0.0901523 + 0.995928i \(0.471265\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −52.0000 −1.87152
\(773\) 18.0000 0.647415 0.323708 0.946157i \(-0.395071\pi\)
0.323708 + 0.946157i \(0.395071\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) −12.0000 −0.430221
\(779\) 35.0000 1.25401
\(780\) 0 0
\(781\) 5.00000 0.178914
\(782\) −48.0000 −1.71648
\(783\) 0 0
\(784\) 28.0000 1.00000
\(785\) 0 0
\(786\) 0 0
\(787\) −4.00000 −0.142585 −0.0712923 0.997455i \(-0.522712\pi\)
−0.0712923 + 0.997455i \(0.522712\pi\)
\(788\) 24.0000 0.854965
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 8.00000 0.284088
\(794\) 76.0000 2.69714
\(795\) 0 0
\(796\) 32.0000 1.13421
\(797\) 4.00000 0.141687 0.0708436 0.997487i \(-0.477431\pi\)
0.0708436 + 0.997487i \(0.477431\pi\)
\(798\) 0 0
\(799\) −8.00000 −0.283020
\(800\) 0 0
\(801\) 0 0
\(802\) −60.0000 −2.11867
\(803\) −40.0000 −1.41157
\(804\) 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
\(817\) −10.0000 −0.349856
\(818\) 28.0000 0.978997
\(819\) 0 0
\(820\) 0 0
\(821\) −3.00000 −0.104701 −0.0523504 0.998629i \(-0.516671\pi\)
−0.0523504 + 0.998629i \(0.516671\pi\)
\(822\) 0 0
\(823\) 46.0000 1.60346 0.801730 0.597687i \(-0.203913\pi\)
0.801730 + 0.597687i \(0.203913\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 44.0000 1.53003 0.765015 0.644013i \(-0.222732\pi\)
0.765015 + 0.644013i \(0.222732\pi\)
\(828\) 0 0
\(829\) 27.0000 0.937749 0.468874 0.883265i \(-0.344660\pi\)
0.468874 + 0.883265i \(0.344660\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 32.0000 1.10940
\(833\) 28.0000 0.970143
\(834\) 0 0
\(835\) 0 0
\(836\) 50.0000 1.72929
\(837\) 0 0
\(838\) −32.0000 −1.10542
\(839\) −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
\(842\) 26.0000 0.896019
\(843\) 0 0
\(844\) 22.0000 0.757271
\(845\) 0 0
\(846\) 0 0
\(847\) 0 0
\(848\) −32.0000 −1.09888
\(849\) 0 0
\(850\) 0 0
\(851\) 60.0000 2.05677
\(852\) 0 0
\(853\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 32.0000 1.09310 0.546550 0.837427i \(-0.315941\pi\)
0.546550 + 0.837427i \(0.315941\pi\)
\(858\) 0 0
\(859\) 55.0000 1.87658 0.938288 0.345855i \(-0.112411\pi\)
0.938288 + 0.345855i \(0.112411\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −6.00000 −0.204361
\(863\) −44.0000 −1.49778 −0.748889 0.662696i \(-0.769412\pi\)
−0.748889 + 0.662696i \(0.769412\pi\)
\(864\) 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
\(877\) 24.0000 0.810422 0.405211 0.914223i \(-0.367198\pi\)
0.405211 + 0.914223i \(0.367198\pi\)
\(878\) −58.0000 −1.95741
\(879\) 0 0
\(880\) 0 0
\(881\) 25.0000 0.842271 0.421136 0.906998i \(-0.361632\pi\)
0.421136 + 0.906998i \(0.361632\pi\)
\(882\) 0 0
\(883\) 52.0000 1.74994 0.874970 0.484178i \(-0.160881\pi\)
0.874970 + 0.484178i \(0.160881\pi\)
\(884\) 32.0000 1.07628
\(885\) 0 0
\(886\) 12.0000 0.403148
\(887\) 6.00000 0.201460 0.100730 0.994914i \(-0.467882\pi\)
0.100730 + 0.994914i \(0.467882\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) −16.0000 −0.535720
\(893\) −10.0000 −0.334637
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) −34.0000 −1.13459
\(899\) −45.0000 −1.50083
\(900\) 0 0
\(901\) −32.0000 −1.06607
\(902\) 70.0000 2.33075
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 18.0000 0.597680 0.298840 0.954303i \(-0.403400\pi\)
0.298840 + 0.954303i \(0.403400\pi\)
\(908\) 8.00000 0.265489
\(909\) 0 0
\(910\) 0 0
\(911\) 49.0000 1.62344 0.811721 0.584045i \(-0.198531\pi\)
0.811721 + 0.584045i \(0.198531\pi\)
\(912\) 0 0
\(913\) −30.0000 −0.992855
\(914\) 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 \(-0.558072\pi\)
−0.181428 + 0.983404i \(0.558072\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 30.0000 0.987997
\(923\) 4.00000 0.131662
\(924\) 0 0
\(925\) 0 0
\(926\) 12.0000 0.394344
\(927\) 0 0
\(928\) −40.0000 −1.31306
\(929\) −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
\(932\) 12.0000 0.393073
\(933\) 0 0
\(934\) 8.00000 0.261768
\(935\) 0 0
\(936\) 0 0
\(937\) −42.0000 −1.37208 −0.686040 0.727564i \(-0.740653\pi\)
−0.686040 + 0.727564i \(0.740653\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −46.0000 −1.49956 −0.749779 0.661689i \(-0.769840\pi\)
−0.749779 + 0.661689i \(0.769840\pi\)
\(942\) 0 0
\(943\) −42.0000 −1.36771
\(944\) −4.00000 −0.130189
\(945\) 0 0
\(946\) −20.0000 −0.650256
\(947\) 18.0000 0.584921 0.292461 0.956278i \(-0.405526\pi\)
0.292461 + 0.956278i \(0.405526\pi\)
\(948\) 0 0
\(949\) −32.0000 −1.03876
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −16.0000 −0.518291 −0.259145 0.965838i \(-0.583441\pi\)
−0.259145 + 0.965838i \(0.583441\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 32.0000 1.03495
\(957\) 0 0
\(958\) 30.0000 0.969256
\(959\) 0 0
\(960\) 0 0
\(961\) 50.0000 1.61290
\(962\) −80.0000 −2.57930
\(963\) 0 0
\(964\) −22.0000 −0.708572
\(965\) 0 0
\(966\) 0 0
\(967\) 40.0000 1.28631 0.643157 0.765735i \(-0.277624\pi\)
0.643157 + 0.765735i \(0.277624\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 27.0000 0.866471 0.433236 0.901281i \(-0.357372\pi\)
0.433236 + 0.901281i \(0.357372\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) −16.0000 −0.512673
\(975\) 0 0
\(976\) 8.00000 0.256074
\(977\) −28.0000 −0.895799 −0.447900 0.894084i \(-0.647828\pi\)
−0.447900 + 0.894084i \(0.647828\pi\)
\(978\) 0 0
\(979\) −45.0000 −1.43821
\(980\) 0 0
\(981\) 0 0
\(982\) −86.0000 −2.74437
\(983\) 48.0000 1.53096 0.765481 0.643458i \(-0.222501\pi\)
0.765481 + 0.643458i \(0.222501\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) −40.0000 −1.27386
\(987\) 0 0
\(988\) 40.0000 1.27257
\(989\) 12.0000 0.381578
\(990\) 0 0
\(991\) −1.00000 −0.0317660 −0.0158830 0.999874i \(-0.505056\pi\)
−0.0158830 + 0.999874i \(0.505056\pi\)
\(992\) 72.0000 2.28600
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −48.0000 −1.52018 −0.760088 0.649821i \(-0.774844\pi\)
−0.760088 + 0.649821i \(0.774844\pi\)
\(998\) 14.0000 0.443162
\(999\) 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.a.f.1.1 1
3.2 odd 2 2025.2.a.a.1.1 1
5.2 odd 4 2025.2.b.a.649.2 2
5.3 odd 4 2025.2.b.a.649.1 2
5.4 even 2 405.2.a.a.1.1 1
15.2 even 4 2025.2.b.b.649.1 2
15.8 even 4 2025.2.b.b.649.2 2
15.14 odd 2 405.2.a.f.1.1 yes 1
20.19 odd 2 6480.2.a.f.1.1 1
45.4 even 6 405.2.e.g.136.1 2
45.14 odd 6 405.2.e.a.136.1 2
45.29 odd 6 405.2.e.a.271.1 2
45.34 even 6 405.2.e.g.271.1 2
60.59 even 2 6480.2.a.r.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
405.2.a.a.1.1 1 5.4 even 2
405.2.a.f.1.1 yes 1 15.14 odd 2
405.2.e.a.136.1 2 45.14 odd 6
405.2.e.a.271.1 2 45.29 odd 6
405.2.e.g.136.1 2 45.4 even 6
405.2.e.g.271.1 2 45.34 even 6
2025.2.a.a.1.1 1 3.2 odd 2
2025.2.a.f.1.1 1 1.1 even 1 trivial
2025.2.b.a.649.1 2 5.3 odd 4
2025.2.b.a.649.2 2 5.2 odd 4
2025.2.b.b.649.1 2 15.2 even 4
2025.2.b.b.649.2 2 15.8 even 4
6480.2.a.f.1.1 1 20.19 odd 2
6480.2.a.r.1.1 1 60.59 even 2