Properties

Label 2025.2.a.b.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 45)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

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

$q$-expansion

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107 −0.353553 0.935414i \(-0.615027\pi\)
−0.353553 + 0.935414i \(0.615027\pi\)
\(3\) 0 0
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) 0 0
\(7\) 3.00000 1.13389 0.566947 0.823754i \(-0.308125\pi\)
0.566947 + 0.823754i \(0.308125\pi\)
\(8\) 3.00000 1.06066
\(9\) 0 0
\(10\) 0 0
\(11\) −2.00000 −0.603023 −0.301511 0.953463i \(-0.597491\pi\)
−0.301511 + 0.953463i \(0.597491\pi\)
\(12\) 0 0
\(13\) 2.00000 0.554700 0.277350 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) −3.00000 −0.801784
\(15\) 0 0
\(16\) −1.00000 −0.250000
\(17\) −4.00000 −0.970143 −0.485071 0.874475i \(-0.661206\pi\)
−0.485071 + 0.874475i \(0.661206\pi\)
\(18\) 0 0
\(19\) −8.00000 −1.83533 −0.917663 0.397360i \(-0.869927\pi\)
−0.917663 + 0.397360i \(0.869927\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 2.00000 0.426401
\(23\) −3.00000 −0.625543 −0.312772 0.949828i \(-0.601257\pi\)
−0.312772 + 0.949828i \(0.601257\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −2.00000 −0.392232
\(27\) 0 0
\(28\) −3.00000 −0.566947
\(29\) −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 \(0.5\pi\)
\(32\) −5.00000 −0.883883
\(33\) 0 0
\(34\) 4.00000 0.685994
\(35\) 0 0
\(36\) 0 0
\(37\) 4.00000 0.657596 0.328798 0.944400i \(-0.393356\pi\)
0.328798 + 0.944400i \(0.393356\pi\)
\(38\) 8.00000 1.29777
\(39\) 0 0
\(40\) 0 0
\(41\) 5.00000 0.780869 0.390434 0.920631i \(-0.372325\pi\)
0.390434 + 0.920631i \(0.372325\pi\)
\(42\) 0 0
\(43\) 8.00000 1.21999 0.609994 0.792406i \(-0.291172\pi\)
0.609994 + 0.792406i \(0.291172\pi\)
\(44\) 2.00000 0.301511
\(45\) 0 0
\(46\) 3.00000 0.442326
\(47\) −7.00000 −1.02105 −0.510527 0.859861i \(-0.670550\pi\)
−0.510527 + 0.859861i \(0.670550\pi\)
\(48\) 0 0
\(49\) 2.00000 0.285714
\(50\) 0 0
\(51\) 0 0
\(52\) −2.00000 −0.277350
\(53\) 2.00000 0.274721 0.137361 0.990521i \(-0.456138\pi\)
0.137361 + 0.990521i \(0.456138\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 9.00000 1.20268
\(57\) 0 0
\(58\) 1.00000 0.131306
\(59\) −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
\(67\) 3.00000 0.366508 0.183254 0.983066i \(-0.441337\pi\)
0.183254 + 0.983066i \(0.441337\pi\)
\(68\) 4.00000 0.485071
\(69\) 0 0
\(70\) 0 0
\(71\) 2.00000 0.237356 0.118678 0.992933i \(-0.462134\pi\)
0.118678 + 0.992933i \(0.462134\pi\)
\(72\) 0 0
\(73\) −4.00000 −0.468165 −0.234082 0.972217i \(-0.575209\pi\)
−0.234082 + 0.972217i \(0.575209\pi\)
\(74\) −4.00000 −0.464991
\(75\) 0 0
\(76\) 8.00000 0.917663
\(77\) −6.00000 −0.683763
\(78\) 0 0
\(79\) −6.00000 −0.675053 −0.337526 0.941316i \(-0.609590\pi\)
−0.337526 + 0.941316i \(0.609590\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −5.00000 −0.552158
\(83\) −9.00000 −0.987878 −0.493939 0.869496i \(-0.664443\pi\)
−0.493939 + 0.869496i \(0.664443\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −8.00000 −0.862662
\(87\) 0 0
\(88\) −6.00000 −0.639602
\(89\) −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
\(92\) 3.00000 0.312772
\(93\) 0 0
\(94\) 7.00000 0.721995
\(95\) 0 0
\(96\) 0 0
\(97\) −2.00000 −0.203069 −0.101535 0.994832i \(-0.532375\pi\)
−0.101535 + 0.994832i \(0.532375\pi\)
\(98\) −2.00000 −0.202031
\(99\) 0 0
\(100\) 0 0
\(101\) −18.0000 −1.79107 −0.895533 0.444994i \(-0.853206\pi\)
−0.895533 + 0.444994i \(0.853206\pi\)
\(102\) 0 0
\(103\) −8.00000 −0.788263 −0.394132 0.919054i \(-0.628955\pi\)
−0.394132 + 0.919054i \(0.628955\pi\)
\(104\) 6.00000 0.588348
\(105\) 0 0
\(106\) −2.00000 −0.194257
\(107\) −3.00000 −0.290021 −0.145010 0.989430i \(-0.546322\pi\)
−0.145010 + 0.989430i \(0.546322\pi\)
\(108\) 0 0
\(109\) 5.00000 0.478913 0.239457 0.970907i \(-0.423031\pi\)
0.239457 + 0.970907i \(0.423031\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −3.00000 −0.283473
\(113\) 8.00000 0.752577 0.376288 0.926503i \(-0.377200\pi\)
0.376288 + 0.926503i \(0.377200\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 1.00000 0.0928477
\(117\) 0 0
\(118\) 14.0000 1.28880
\(119\) −12.0000 −1.10004
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) −7.00000 −0.633750
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 5.00000 0.443678 0.221839 0.975083i \(-0.428794\pi\)
0.221839 + 0.975083i \(0.428794\pi\)
\(128\) 3.00000 0.265165
\(129\) 0 0
\(130\) 0 0
\(131\) −6.00000 −0.524222 −0.262111 0.965038i \(-0.584419\pi\)
−0.262111 + 0.965038i \(0.584419\pi\)
\(132\) 0 0
\(133\) −24.0000 −2.08106
\(134\) −3.00000 −0.259161
\(135\) 0 0
\(136\) −12.0000 −1.02899
\(137\) −12.0000 −1.02523 −0.512615 0.858619i \(-0.671323\pi\)
−0.512615 + 0.858619i \(0.671323\pi\)
\(138\) 0 0
\(139\) −16.0000 −1.35710 −0.678551 0.734553i \(-0.737392\pi\)
−0.678551 + 0.734553i \(0.737392\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −2.00000 −0.167836
\(143\) −4.00000 −0.334497
\(144\) 0 0
\(145\) 0 0
\(146\) 4.00000 0.331042
\(147\) 0 0
\(148\) −4.00000 −0.328798
\(149\) 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 \(0.525932\pi\)
\(152\) −24.0000 −1.94666
\(153\) 0 0
\(154\) 6.00000 0.483494
\(155\) 0 0
\(156\) 0 0
\(157\) −14.0000 −1.11732 −0.558661 0.829396i \(-0.688685\pi\)
−0.558661 + 0.829396i \(0.688685\pi\)
\(158\) 6.00000 0.477334
\(159\) 0 0
\(160\) 0 0
\(161\) −9.00000 −0.709299
\(162\) 0 0
\(163\) 4.00000 0.313304 0.156652 0.987654i \(-0.449930\pi\)
0.156652 + 0.987654i \(0.449930\pi\)
\(164\) −5.00000 −0.390434
\(165\) 0 0
\(166\) 9.00000 0.698535
\(167\) 9.00000 0.696441 0.348220 0.937413i \(-0.386786\pi\)
0.348220 + 0.937413i \(0.386786\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) 0 0
\(172\) −8.00000 −0.609994
\(173\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 2.00000 0.150756
\(177\) 0 0
\(178\) 15.0000 1.12430
\(179\) −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 \(0.583773\pi\)
\(182\) −6.00000 −0.444750
\(183\) 0 0
\(184\) −9.00000 −0.663489
\(185\) 0 0
\(186\) 0 0
\(187\) 8.00000 0.585018
\(188\) 7.00000 0.510527
\(189\) 0 0
\(190\) 0 0
\(191\) 8.00000 0.578860 0.289430 0.957199i \(-0.406534\pi\)
0.289430 + 0.957199i \(0.406534\pi\)
\(192\) 0 0
\(193\) 10.0000 0.719816 0.359908 0.932988i \(-0.382808\pi\)
0.359908 + 0.932988i \(0.382808\pi\)
\(194\) 2.00000 0.143592
\(195\) 0 0
\(196\) −2.00000 −0.142857
\(197\) 12.0000 0.854965 0.427482 0.904024i \(-0.359401\pi\)
0.427482 + 0.904024i \(0.359401\pi\)
\(198\) 0 0
\(199\) 4.00000 0.283552 0.141776 0.989899i \(-0.454719\pi\)
0.141776 + 0.989899i \(0.454719\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 18.0000 1.26648
\(203\) −3.00000 −0.210559
\(204\) 0 0
\(205\) 0 0
\(206\) 8.00000 0.557386
\(207\) 0 0
\(208\) −2.00000 −0.138675
\(209\) 16.0000 1.10674
\(210\) 0 0
\(211\) −22.0000 −1.51454 −0.757271 0.653101i \(-0.773468\pi\)
−0.757271 + 0.653101i \(0.773468\pi\)
\(212\) −2.00000 −0.137361
\(213\) 0 0
\(214\) 3.00000 0.205076
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) −5.00000 −0.338643
\(219\) 0 0
\(220\) 0 0
\(221\) −8.00000 −0.538138
\(222\) 0 0
\(223\) 19.0000 1.27233 0.636167 0.771551i \(-0.280519\pi\)
0.636167 + 0.771551i \(0.280519\pi\)
\(224\) −15.0000 −1.00223
\(225\) 0 0
\(226\) −8.00000 −0.532152
\(227\) 4.00000 0.265489 0.132745 0.991150i \(-0.457621\pi\)
0.132745 + 0.991150i \(0.457621\pi\)
\(228\) 0 0
\(229\) 15.0000 0.991228 0.495614 0.868543i \(-0.334943\pi\)
0.495614 + 0.868543i \(0.334943\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −3.00000 −0.196960
\(233\) −24.0000 −1.57229 −0.786146 0.618041i \(-0.787927\pi\)
−0.786146 + 0.618041i \(0.787927\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 14.0000 0.911322
\(237\) 0 0
\(238\) 12.0000 0.777844
\(239\) −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 \(0.615276\pi\)
\(242\) 7.00000 0.449977
\(243\) 0 0
\(244\) −7.00000 −0.448129
\(245\) 0 0
\(246\) 0 0
\(247\) −16.0000 −1.01806
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 0 0
\(253\) 6.00000 0.377217
\(254\) −5.00000 −0.313728
\(255\) 0 0
\(256\) −17.0000 −1.06250
\(257\) 6.00000 0.374270 0.187135 0.982334i \(-0.440080\pi\)
0.187135 + 0.982334i \(0.440080\pi\)
\(258\) 0 0
\(259\) 12.0000 0.745644
\(260\) 0 0
\(261\) 0 0
\(262\) 6.00000 0.370681
\(263\) 16.0000 0.986602 0.493301 0.869859i \(-0.335790\pi\)
0.493301 + 0.869859i \(0.335790\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 24.0000 1.47153
\(267\) 0 0
\(268\) −3.00000 −0.183254
\(269\) 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 \(0.578126\pi\)
\(272\) 4.00000 0.242536
\(273\) 0 0
\(274\) 12.0000 0.724947
\(275\) 0 0
\(276\) 0 0
\(277\) −12.0000 −0.721010 −0.360505 0.932757i \(-0.617396\pi\)
−0.360505 + 0.932757i \(0.617396\pi\)
\(278\) 16.0000 0.959616
\(279\) 0 0
\(280\) 0 0
\(281\) −15.0000 −0.894825 −0.447412 0.894328i \(-0.647654\pi\)
−0.447412 + 0.894328i \(0.647654\pi\)
\(282\) 0 0
\(283\) −21.0000 −1.24832 −0.624160 0.781296i \(-0.714559\pi\)
−0.624160 + 0.781296i \(0.714559\pi\)
\(284\) −2.00000 −0.118678
\(285\) 0 0
\(286\) 4.00000 0.236525
\(287\) 15.0000 0.885422
\(288\) 0 0
\(289\) −1.00000 −0.0588235
\(290\) 0 0
\(291\) 0 0
\(292\) 4.00000 0.234082
\(293\) −12.0000 −0.701047 −0.350524 0.936554i \(-0.613996\pi\)
−0.350524 + 0.936554i \(0.613996\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 12.0000 0.697486
\(297\) 0 0
\(298\) −17.0000 −0.984784
\(299\) −6.00000 −0.346989
\(300\) 0 0
\(301\) 24.0000 1.38334
\(302\) 2.00000 0.115087
\(303\) 0 0
\(304\) 8.00000 0.458831
\(305\) 0 0
\(306\) 0 0
\(307\) −7.00000 −0.399511 −0.199756 0.979846i \(-0.564015\pi\)
−0.199756 + 0.979846i \(0.564015\pi\)
\(308\) 6.00000 0.341882
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) −14.0000 −0.791327 −0.395663 0.918396i \(-0.629485\pi\)
−0.395663 + 0.918396i \(0.629485\pi\)
\(314\) 14.0000 0.790066
\(315\) 0 0
\(316\) 6.00000 0.337526
\(317\) 34.0000 1.90963 0.954815 0.297200i \(-0.0960529\pi\)
0.954815 + 0.297200i \(0.0960529\pi\)
\(318\) 0 0
\(319\) 2.00000 0.111979
\(320\) 0 0
\(321\) 0 0
\(322\) 9.00000 0.501550
\(323\) 32.0000 1.78053
\(324\) 0 0
\(325\) 0 0
\(326\) −4.00000 −0.221540
\(327\) 0 0
\(328\) 15.0000 0.828236
\(329\) −21.0000 −1.15777
\(330\) 0 0
\(331\) −6.00000 −0.329790 −0.164895 0.986311i \(-0.552728\pi\)
−0.164895 + 0.986311i \(0.552728\pi\)
\(332\) 9.00000 0.493939
\(333\) 0 0
\(334\) −9.00000 −0.492458
\(335\) 0 0
\(336\) 0 0
\(337\) 8.00000 0.435788 0.217894 0.975972i \(-0.430081\pi\)
0.217894 + 0.975972i \(0.430081\pi\)
\(338\) 9.00000 0.489535
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −15.0000 −0.809924
\(344\) 24.0000 1.29399
\(345\) 0 0
\(346\) 0 0
\(347\) −4.00000 −0.214731 −0.107366 0.994220i \(-0.534242\pi\)
−0.107366 + 0.994220i \(0.534242\pi\)
\(348\) 0 0
\(349\) −5.00000 −0.267644 −0.133822 0.991005i \(-0.542725\pi\)
−0.133822 + 0.991005i \(0.542725\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 10.0000 0.533002
\(353\) −24.0000 −1.27739 −0.638696 0.769460i \(-0.720526\pi\)
−0.638696 + 0.769460i \(0.720526\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 15.0000 0.794998
\(357\) 0 0
\(358\) 2.00000 0.105703
\(359\) 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
\(362\) 7.00000 0.367912
\(363\) 0 0
\(364\) −6.00000 −0.314485
\(365\) 0 0
\(366\) 0 0
\(367\) −24.0000 −1.25279 −0.626395 0.779506i \(-0.715470\pi\)
−0.626395 + 0.779506i \(0.715470\pi\)
\(368\) 3.00000 0.156386
\(369\) 0 0
\(370\) 0 0
\(371\) 6.00000 0.311504
\(372\) 0 0
\(373\) −10.0000 −0.517780 −0.258890 0.965907i \(-0.583357\pi\)
−0.258890 + 0.965907i \(0.583357\pi\)
\(374\) −8.00000 −0.413670
\(375\) 0 0
\(376\) −21.0000 −1.08299
\(377\) −2.00000 −0.103005
\(378\) 0 0
\(379\) −26.0000 −1.33553 −0.667765 0.744372i \(-0.732749\pi\)
−0.667765 + 0.744372i \(0.732749\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −8.00000 −0.409316
\(383\) −36.0000 −1.83951 −0.919757 0.392488i \(-0.871614\pi\)
−0.919757 + 0.392488i \(0.871614\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −10.0000 −0.508987
\(387\) 0 0
\(388\) 2.00000 0.101535
\(389\) 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
\(392\) 6.00000 0.303046
\(393\) 0 0
\(394\) −12.0000 −0.604551
\(395\) 0 0
\(396\) 0 0
\(397\) −34.0000 −1.70641 −0.853206 0.521575i \(-0.825345\pi\)
−0.853206 + 0.521575i \(0.825345\pi\)
\(398\) −4.00000 −0.200502
\(399\) 0 0
\(400\) 0 0
\(401\) −18.0000 −0.898877 −0.449439 0.893311i \(-0.648376\pi\)
−0.449439 + 0.893311i \(0.648376\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 18.0000 0.895533
\(405\) 0 0
\(406\) 3.00000 0.148888
\(407\) −8.00000 −0.396545
\(408\) 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
\(412\) 8.00000 0.394132
\(413\) −42.0000 −2.06668
\(414\) 0 0
\(415\) 0 0
\(416\) −10.0000 −0.490290
\(417\) 0 0
\(418\) −16.0000 −0.782586
\(419\) 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 \(0.189178\pi\)
\(422\) 22.0000 1.07094
\(423\) 0 0
\(424\) 6.00000 0.291386
\(425\) 0 0
\(426\) 0 0
\(427\) 21.0000 1.01626
\(428\) 3.00000 0.145010
\(429\) 0 0
\(430\) 0 0
\(431\) −30.0000 −1.44505 −0.722525 0.691345i \(-0.757018\pi\)
−0.722525 + 0.691345i \(0.757018\pi\)
\(432\) 0 0
\(433\) −28.0000 −1.34559 −0.672797 0.739827i \(-0.734907\pi\)
−0.672797 + 0.739827i \(0.734907\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −5.00000 −0.239457
\(437\) 24.0000 1.14808
\(438\) 0 0
\(439\) 28.0000 1.33637 0.668184 0.743996i \(-0.267072\pi\)
0.668184 + 0.743996i \(0.267072\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 8.00000 0.380521
\(443\) −15.0000 −0.712672 −0.356336 0.934358i \(-0.615974\pi\)
−0.356336 + 0.934358i \(0.615974\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −19.0000 −0.899676
\(447\) 0 0
\(448\) 21.0000 0.992157
\(449\) −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
\(452\) −8.00000 −0.376288
\(453\) 0 0
\(454\) −4.00000 −0.187729
\(455\) 0 0
\(456\) 0 0
\(457\) 20.0000 0.935561 0.467780 0.883845i \(-0.345054\pi\)
0.467780 + 0.883845i \(0.345054\pi\)
\(458\) −15.0000 −0.700904
\(459\) 0 0
\(460\) 0 0
\(461\) 9.00000 0.419172 0.209586 0.977790i \(-0.432788\pi\)
0.209586 + 0.977790i \(0.432788\pi\)
\(462\) 0 0
\(463\) 36.0000 1.67306 0.836531 0.547920i \(-0.184580\pi\)
0.836531 + 0.547920i \(0.184580\pi\)
\(464\) 1.00000 0.0464238
\(465\) 0 0
\(466\) 24.0000 1.11178
\(467\) −20.0000 −0.925490 −0.462745 0.886492i \(-0.653135\pi\)
−0.462745 + 0.886492i \(0.653135\pi\)
\(468\) 0 0
\(469\) 9.00000 0.415581
\(470\) 0 0
\(471\) 0 0
\(472\) −42.0000 −1.93321
\(473\) −16.0000 −0.735681
\(474\) 0 0
\(475\) 0 0
\(476\) 12.0000 0.550019
\(477\) 0 0
\(478\) 8.00000 0.365911
\(479\) −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
\(482\) 11.0000 0.501036
\(483\) 0 0
\(484\) 7.00000 0.318182
\(485\) 0 0
\(486\) 0 0
\(487\) 16.0000 0.725029 0.362515 0.931978i \(-0.381918\pi\)
0.362515 + 0.931978i \(0.381918\pi\)
\(488\) 21.0000 0.950625
\(489\) 0 0
\(490\) 0 0
\(491\) 20.0000 0.902587 0.451294 0.892375i \(-0.350963\pi\)
0.451294 + 0.892375i \(0.350963\pi\)
\(492\) 0 0
\(493\) 4.00000 0.180151
\(494\) 16.0000 0.719874
\(495\) 0 0
\(496\) 0 0
\(497\) 6.00000 0.269137
\(498\) 0 0
\(499\) −32.0000 −1.43252 −0.716258 0.697835i \(-0.754147\pi\)
−0.716258 + 0.697835i \(0.754147\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 7.00000 0.312115 0.156057 0.987748i \(-0.450122\pi\)
0.156057 + 0.987748i \(0.450122\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −6.00000 −0.266733
\(507\) 0 0
\(508\) −5.00000 −0.221839
\(509\) 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
\(512\) 11.0000 0.486136
\(513\) 0 0
\(514\) −6.00000 −0.264649
\(515\) 0 0
\(516\) 0 0
\(517\) 14.0000 0.615719
\(518\) −12.0000 −0.527250
\(519\) 0 0
\(520\) 0 0
\(521\) 11.0000 0.481919 0.240959 0.970535i \(-0.422538\pi\)
0.240959 + 0.970535i \(0.422538\pi\)
\(522\) 0 0
\(523\) 29.0000 1.26808 0.634041 0.773300i \(-0.281395\pi\)
0.634041 + 0.773300i \(0.281395\pi\)
\(524\) 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
\(532\) 24.0000 1.04053
\(533\) 10.0000 0.433148
\(534\) 0 0
\(535\) 0 0
\(536\) 9.00000 0.388741
\(537\) 0 0
\(538\) −25.0000 −1.07783
\(539\) −4.00000 −0.172292
\(540\) 0 0
\(541\) −39.0000 −1.67674 −0.838370 0.545101i \(-0.816491\pi\)
−0.838370 + 0.545101i \(0.816491\pi\)
\(542\) 8.00000 0.343629
\(543\) 0 0
\(544\) 20.0000 0.857493
\(545\) 0 0
\(546\) 0 0
\(547\) 29.0000 1.23995 0.619975 0.784621i \(-0.287143\pi\)
0.619975 + 0.784621i \(0.287143\pi\)
\(548\) 12.0000 0.512615
\(549\) 0 0
\(550\) 0 0
\(551\) 8.00000 0.340811
\(552\) 0 0
\(553\) −18.0000 −0.765438
\(554\) 12.0000 0.509831
\(555\) 0 0
\(556\) 16.0000 0.678551
\(557\) 30.0000 1.27114 0.635570 0.772043i \(-0.280765\pi\)
0.635570 + 0.772043i \(0.280765\pi\)
\(558\) 0 0
\(559\) 16.0000 0.676728
\(560\) 0 0
\(561\) 0 0
\(562\) 15.0000 0.632737
\(563\) 21.0000 0.885044 0.442522 0.896758i \(-0.354084\pi\)
0.442522 + 0.896758i \(0.354084\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 21.0000 0.882696
\(567\) 0 0
\(568\) 6.00000 0.251754
\(569\) −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 \(0.266474\pi\)
\(572\) 4.00000 0.167248
\(573\) 0 0
\(574\) −15.0000 −0.626088
\(575\) 0 0
\(576\) 0 0
\(577\) 10.0000 0.416305 0.208153 0.978096i \(-0.433255\pi\)
0.208153 + 0.978096i \(0.433255\pi\)
\(578\) 1.00000 0.0415945
\(579\) 0 0
\(580\) 0 0
\(581\) −27.0000 −1.12015
\(582\) 0 0
\(583\) −4.00000 −0.165663
\(584\) −12.0000 −0.496564
\(585\) 0 0
\(586\) 12.0000 0.495715
\(587\) 33.0000 1.36206 0.681028 0.732257i \(-0.261533\pi\)
0.681028 + 0.732257i \(0.261533\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) −4.00000 −0.164399
\(593\) −20.0000 −0.821302 −0.410651 0.911793i \(-0.634698\pi\)
−0.410651 + 0.911793i \(0.634698\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −17.0000 −0.696347
\(597\) 0 0
\(598\) 6.00000 0.245358
\(599\) −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 \(0.487012\pi\)
\(602\) −24.0000 −0.978167
\(603\) 0 0
\(604\) 2.00000 0.0813788
\(605\) 0 0
\(606\) 0 0
\(607\) 41.0000 1.66414 0.832069 0.554672i \(-0.187156\pi\)
0.832069 + 0.554672i \(0.187156\pi\)
\(608\) 40.0000 1.62221
\(609\) 0 0
\(610\) 0 0
\(611\) −14.0000 −0.566379
\(612\) 0 0
\(613\) 44.0000 1.77714 0.888572 0.458738i \(-0.151698\pi\)
0.888572 + 0.458738i \(0.151698\pi\)
\(614\) 7.00000 0.282497
\(615\) 0 0
\(616\) −18.0000 −0.725241
\(617\) 36.0000 1.44931 0.724653 0.689114i \(-0.242000\pi\)
0.724653 + 0.689114i \(0.242000\pi\)
\(618\) 0 0
\(619\) −4.00000 −0.160774 −0.0803868 0.996764i \(-0.525616\pi\)
−0.0803868 + 0.996764i \(0.525616\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −45.0000 −1.80289
\(624\) 0 0
\(625\) 0 0
\(626\) 14.0000 0.559553
\(627\) 0 0
\(628\) 14.0000 0.558661
\(629\) −16.0000 −0.637962
\(630\) 0 0
\(631\) 16.0000 0.636950 0.318475 0.947931i \(-0.396829\pi\)
0.318475 + 0.947931i \(0.396829\pi\)
\(632\) −18.0000 −0.716002
\(633\) 0 0
\(634\) −34.0000 −1.35031
\(635\) 0 0
\(636\) 0 0
\(637\) 4.00000 0.158486
\(638\) −2.00000 −0.0791808
\(639\) 0 0
\(640\) 0 0
\(641\) 33.0000 1.30342 0.651711 0.758468i \(-0.274052\pi\)
0.651711 + 0.758468i \(0.274052\pi\)
\(642\) 0 0
\(643\) 9.00000 0.354925 0.177463 0.984128i \(-0.443211\pi\)
0.177463 + 0.984128i \(0.443211\pi\)
\(644\) 9.00000 0.354650
\(645\) 0 0
\(646\) −32.0000 −1.25902
\(647\) 17.0000 0.668339 0.334169 0.942513i \(-0.391544\pi\)
0.334169 + 0.942513i \(0.391544\pi\)
\(648\) 0 0
\(649\) 28.0000 1.09910
\(650\) 0 0
\(651\) 0 0
\(652\) −4.00000 −0.156652
\(653\) 4.00000 0.156532 0.0782660 0.996933i \(-0.475062\pi\)
0.0782660 + 0.996933i \(0.475062\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −5.00000 −0.195217
\(657\) 0 0
\(658\) 21.0000 0.818665
\(659\) −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 \(0.587774\pi\)
\(662\) 6.00000 0.233197
\(663\) 0 0
\(664\) −27.0000 −1.04780
\(665\) 0 0
\(666\) 0 0
\(667\) 3.00000 0.116160
\(668\) −9.00000 −0.348220
\(669\) 0 0
\(670\) 0 0
\(671\) −14.0000 −0.540464
\(672\) 0 0
\(673\) −6.00000 −0.231283 −0.115642 0.993291i \(-0.536892\pi\)
−0.115642 + 0.993291i \(0.536892\pi\)
\(674\) −8.00000 −0.308148
\(675\) 0 0
\(676\) 9.00000 0.346154
\(677\) 42.0000 1.61419 0.807096 0.590421i \(-0.201038\pi\)
0.807096 + 0.590421i \(0.201038\pi\)
\(678\) 0 0
\(679\) −6.00000 −0.230259
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −12.0000 −0.459167 −0.229584 0.973289i \(-0.573736\pi\)
−0.229584 + 0.973289i \(0.573736\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 15.0000 0.572703
\(687\) 0 0
\(688\) −8.00000 −0.304997
\(689\) 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
\(697\) −20.0000 −0.757554
\(698\) 5.00000 0.189253
\(699\) 0 0
\(700\) 0 0
\(701\) 23.0000 0.868698 0.434349 0.900745i \(-0.356978\pi\)
0.434349 + 0.900745i \(0.356978\pi\)
\(702\) 0 0
\(703\) −32.0000 −1.20690
\(704\) −14.0000 −0.527645
\(705\) 0 0
\(706\) 24.0000 0.903252
\(707\) −54.0000 −2.03088
\(708\) 0 0
\(709\) −41.0000 −1.53979 −0.769894 0.638172i \(-0.779691\pi\)
−0.769894 + 0.638172i \(0.779691\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −45.0000 −1.68645
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 2.00000 0.0747435
\(717\) 0 0
\(718\) −24.0000 −0.895672
\(719\) 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
\(722\) −45.0000 −1.67473
\(723\) 0 0
\(724\) 7.00000 0.260153
\(725\) 0 0
\(726\) 0 0
\(727\) −23.0000 −0.853023 −0.426511 0.904482i \(-0.640258\pi\)
−0.426511 + 0.904482i \(0.640258\pi\)
\(728\) 18.0000 0.667124
\(729\) 0 0
\(730\) 0 0
\(731\) −32.0000 −1.18356
\(732\) 0 0
\(733\) 34.0000 1.25582 0.627909 0.778287i \(-0.283911\pi\)
0.627909 + 0.778287i \(0.283911\pi\)
\(734\) 24.0000 0.885856
\(735\) 0 0
\(736\) 15.0000 0.552907
\(737\) −6.00000 −0.221013
\(738\) 0 0
\(739\) −2.00000 −0.0735712 −0.0367856 0.999323i \(-0.511712\pi\)
−0.0367856 + 0.999323i \(0.511712\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −6.00000 −0.220267
\(743\) 29.0000 1.06391 0.531953 0.846774i \(-0.321458\pi\)
0.531953 + 0.846774i \(0.321458\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 10.0000 0.366126
\(747\) 0 0
\(748\) −8.00000 −0.292509
\(749\) −9.00000 −0.328853
\(750\) 0 0
\(751\) 10.0000 0.364905 0.182453 0.983215i \(-0.441596\pi\)
0.182453 + 0.983215i \(0.441596\pi\)
\(752\) 7.00000 0.255264
\(753\) 0 0
\(754\) 2.00000 0.0728357
\(755\) 0 0
\(756\) 0 0
\(757\) 26.0000 0.944986 0.472493 0.881334i \(-0.343354\pi\)
0.472493 + 0.881334i \(0.343354\pi\)
\(758\) 26.0000 0.944363
\(759\) 0 0
\(760\) 0 0
\(761\) 15.0000 0.543750 0.271875 0.962333i \(-0.412356\pi\)
0.271875 + 0.962333i \(0.412356\pi\)
\(762\) 0 0
\(763\) 15.0000 0.543036
\(764\) −8.00000 −0.289430
\(765\) 0 0
\(766\) 36.0000 1.30073
\(767\) −28.0000 −1.01102
\(768\) 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
\(772\) −10.0000 −0.359908
\(773\) 24.0000 0.863220 0.431610 0.902060i \(-0.357946\pi\)
0.431610 + 0.902060i \(0.357946\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −6.00000 −0.215387
\(777\) 0 0
\(778\) −33.0000 −1.18311
\(779\) −40.0000 −1.43315
\(780\) 0 0
\(781\) −4.00000 −0.143131
\(782\) −12.0000 −0.429119
\(783\) 0 0
\(784\) −2.00000 −0.0714286
\(785\) 0 0
\(786\) 0 0
\(787\) −28.0000 −0.998092 −0.499046 0.866575i \(-0.666316\pi\)
−0.499046 + 0.866575i \(0.666316\pi\)
\(788\) −12.0000 −0.427482
\(789\) 0 0
\(790\) 0 0
\(791\) 24.0000 0.853342
\(792\) 0 0
\(793\) 14.0000 0.497155
\(794\) 34.0000 1.20661
\(795\) 0 0
\(796\) −4.00000 −0.141776
\(797\) −26.0000 −0.920967 −0.460484 0.887668i \(-0.652324\pi\)
−0.460484 + 0.887668i \(0.652324\pi\)
\(798\) 0 0
\(799\) 28.0000 0.990569
\(800\) 0 0
\(801\) 0 0
\(802\) 18.0000 0.635602
\(803\) 8.00000 0.282314
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) −54.0000 −1.89971
\(809\) 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 \(0.763951\pi\)
\(812\) 3.00000 0.105279
\(813\) 0 0
\(814\) 8.00000 0.280400
\(815\) 0 0
\(816\) 0 0
\(817\) −64.0000 −2.23908
\(818\) −14.0000 −0.489499
\(819\) 0 0
\(820\) 0 0
\(821\) 15.0000 0.523504 0.261752 0.965135i \(-0.415700\pi\)
0.261752 + 0.965135i \(0.415700\pi\)
\(822\) 0 0
\(823\) −53.0000 −1.84746 −0.923732 0.383040i \(-0.874877\pi\)
−0.923732 + 0.383040i \(0.874877\pi\)
\(824\) −24.0000 −0.836080
\(825\) 0 0
\(826\) 42.0000 1.46137
\(827\) −37.0000 −1.28662 −0.643308 0.765607i \(-0.722439\pi\)
−0.643308 + 0.765607i \(0.722439\pi\)
\(828\) 0 0
\(829\) −3.00000 −0.104194 −0.0520972 0.998642i \(-0.516591\pi\)
−0.0520972 + 0.998642i \(0.516591\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 14.0000 0.485363
\(833\) −8.00000 −0.277184
\(834\) 0 0
\(835\) 0 0
\(836\) −16.0000 −0.553372
\(837\) 0 0
\(838\) −26.0000 −0.898155
\(839\) −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
\(842\) −34.0000 −1.17172
\(843\) 0 0
\(844\) 22.0000 0.757271
\(845\) 0 0
\(846\) 0 0
\(847\) −21.0000 −0.721569
\(848\) −2.00000 −0.0686803
\(849\) 0 0
\(850\) 0 0
\(851\) −12.0000 −0.411355
\(852\) 0 0
\(853\) 54.0000 1.84892 0.924462 0.381273i \(-0.124514\pi\)
0.924462 + 0.381273i \(0.124514\pi\)
\(854\) −21.0000 −0.718605
\(855\) 0 0
\(856\) −9.00000 −0.307614
\(857\) −10.0000 −0.341593 −0.170797 0.985306i \(-0.554634\pi\)
−0.170797 + 0.985306i \(0.554634\pi\)
\(858\) 0 0
\(859\) 22.0000 0.750630 0.375315 0.926897i \(-0.377534\pi\)
0.375315 + 0.926897i \(0.377534\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 30.0000 1.02180
\(863\) −17.0000 −0.578687 −0.289343 0.957225i \(-0.593437\pi\)
−0.289343 + 0.957225i \(0.593437\pi\)
\(864\) 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
\(872\) 15.0000 0.507964
\(873\) 0 0
\(874\) −24.0000 −0.811812
\(875\) 0 0
\(876\) 0 0
\(877\) 18.0000 0.607817 0.303908 0.952701i \(-0.401708\pi\)
0.303908 + 0.952701i \(0.401708\pi\)
\(878\) −28.0000 −0.944954
\(879\) 0 0
\(880\) 0 0
\(881\) −35.0000 −1.17918 −0.589590 0.807703i \(-0.700711\pi\)
−0.589590 + 0.807703i \(0.700711\pi\)
\(882\) 0 0
\(883\) −23.0000 −0.774012 −0.387006 0.922077i \(-0.626491\pi\)
−0.387006 + 0.922077i \(0.626491\pi\)
\(884\) 8.00000 0.269069
\(885\) 0 0
\(886\) 15.0000 0.503935
\(887\) −36.0000 −1.20876 −0.604381 0.796696i \(-0.706579\pi\)
−0.604381 + 0.796696i \(0.706579\pi\)
\(888\) 0 0
\(889\) 15.0000 0.503084
\(890\) 0 0
\(891\) 0 0
\(892\) −19.0000 −0.636167
\(893\) 56.0000 1.87397
\(894\) 0 0
\(895\) 0 0
\(896\) 9.00000 0.300669
\(897\) 0 0
\(898\) 26.0000 0.867631
\(899\) 0 0
\(900\) 0 0
\(901\) −8.00000 −0.266519
\(902\) 10.0000 0.332964
\(903\) 0 0
\(904\) 24.0000 0.798228
\(905\) 0 0
\(906\) 0 0
\(907\) 51.0000 1.69343 0.846714 0.532049i \(-0.178578\pi\)
0.846714 + 0.532049i \(0.178578\pi\)
\(908\) −4.00000 −0.132745
\(909\) 0 0
\(910\) 0 0
\(911\) −50.0000 −1.65657 −0.828287 0.560304i \(-0.810684\pi\)
−0.828287 + 0.560304i \(0.810684\pi\)
\(912\) 0 0
\(913\) 18.0000 0.595713
\(914\) −20.0000 −0.661541
\(915\) 0 0
\(916\) −15.0000 −0.495614
\(917\) −18.0000 −0.594412
\(918\) 0 0
\(919\) 10.0000 0.329870 0.164935 0.986304i \(-0.447259\pi\)
0.164935 + 0.986304i \(0.447259\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −9.00000 −0.296399
\(923\) 4.00000 0.131662
\(924\) 0 0
\(925\) 0 0
\(926\) −36.0000 −1.18303
\(927\) 0 0
\(928\) 5.00000 0.164133
\(929\) 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
\(932\) 24.0000 0.786146
\(933\) 0 0
\(934\) 20.0000 0.654420
\(935\) 0 0
\(936\) 0 0
\(937\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(938\) −9.00000 −0.293860
\(939\) 0 0
\(940\) 0 0
\(941\) −7.00000 −0.228193 −0.114097 0.993470i \(-0.536397\pi\)
−0.114097 + 0.993470i \(0.536397\pi\)
\(942\) 0 0
\(943\) −15.0000 −0.488467
\(944\) 14.0000 0.455661
\(945\) 0 0
\(946\) 16.0000 0.520205
\(947\) −57.0000 −1.85225 −0.926126 0.377215i \(-0.876882\pi\)
−0.926126 + 0.377215i \(0.876882\pi\)
\(948\) 0 0
\(949\) −8.00000 −0.259691
\(950\) 0 0
\(951\) 0 0
\(952\) −36.0000 −1.16677
\(953\) 26.0000 0.842223 0.421111 0.907009i \(-0.361640\pi\)
0.421111 + 0.907009i \(0.361640\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 8.00000 0.258738
\(957\) 0 0
\(958\) 18.0000 0.581554
\(959\) −36.0000 −1.16250
\(960\) 0 0
\(961\) −31.0000 −1.00000
\(962\) −8.00000 −0.257930
\(963\) 0 0
\(964\) 11.0000 0.354286
\(965\) 0 0
\(966\) 0 0
\(967\) −41.0000 −1.31847 −0.659236 0.751936i \(-0.729120\pi\)
−0.659236 + 0.751936i \(0.729120\pi\)
\(968\) −21.0000 −0.674966
\(969\) 0 0
\(970\) 0 0
\(971\) −36.0000 −1.15529 −0.577647 0.816286i \(-0.696029\pi\)
−0.577647 + 0.816286i \(0.696029\pi\)
\(972\) 0 0
\(973\) −48.0000 −1.53881
\(974\) −16.0000 −0.512673
\(975\) 0 0
\(976\) −7.00000 −0.224065
\(977\) 38.0000 1.21573 0.607864 0.794041i \(-0.292027\pi\)
0.607864 + 0.794041i \(0.292027\pi\)
\(978\) 0 0
\(979\) 30.0000 0.958804
\(980\) 0 0
\(981\) 0 0
\(982\) −20.0000 −0.638226
\(983\) −3.00000 −0.0956851 −0.0478426 0.998855i \(-0.515235\pi\)
−0.0478426 + 0.998855i \(0.515235\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) −4.00000 −0.127386
\(987\) 0 0
\(988\) 16.0000 0.509028
\(989\) −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
\(997\) 18.0000 0.570066 0.285033 0.958518i \(-0.407995\pi\)
0.285033 + 0.958518i \(0.407995\pi\)
\(998\) 32.0000 1.01294
\(999\) 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.b.1.1 1
3.2 odd 2 2025.2.a.e.1.1 1
5.2 odd 4 2025.2.b.c.649.1 2
5.3 odd 4 2025.2.b.c.649.2 2
5.4 even 2 405.2.a.e.1.1 1
9.2 odd 6 675.2.e.a.226.1 2
9.4 even 3 225.2.e.a.151.1 2
9.5 odd 6 675.2.e.a.451.1 2
9.7 even 3 225.2.e.a.76.1 2
15.2 even 4 2025.2.b.d.649.2 2
15.8 even 4 2025.2.b.d.649.1 2
15.14 odd 2 405.2.a.b.1.1 1
20.19 odd 2 6480.2.a.k.1.1 1
45.2 even 12 675.2.k.a.199.2 4
45.4 even 6 45.2.e.a.16.1 2
45.7 odd 12 225.2.k.a.49.1 4
45.13 odd 12 225.2.k.a.124.1 4
45.14 odd 6 135.2.e.a.46.1 2
45.22 odd 12 225.2.k.a.124.2 4
45.23 even 12 675.2.k.a.424.2 4
45.29 odd 6 135.2.e.a.91.1 2
45.32 even 12 675.2.k.a.424.1 4
45.34 even 6 45.2.e.a.31.1 yes 2
45.38 even 12 675.2.k.a.199.1 4
45.43 odd 12 225.2.k.a.49.2 4
60.59 even 2 6480.2.a.x.1.1 1
180.59 even 6 2160.2.q.a.721.1 2
180.79 odd 6 720.2.q.d.481.1 2
180.119 even 6 2160.2.q.a.1441.1 2
180.139 odd 6 720.2.q.d.241.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
45.2.e.a.16.1 2 45.4 even 6
45.2.e.a.31.1 yes 2 45.34 even 6
135.2.e.a.46.1 2 45.14 odd 6
135.2.e.a.91.1 2 45.29 odd 6
225.2.e.a.76.1 2 9.7 even 3
225.2.e.a.151.1 2 9.4 even 3
225.2.k.a.49.1 4 45.7 odd 12
225.2.k.a.49.2 4 45.43 odd 12
225.2.k.a.124.1 4 45.13 odd 12
225.2.k.a.124.2 4 45.22 odd 12
405.2.a.b.1.1 1 15.14 odd 2
405.2.a.e.1.1 1 5.4 even 2
675.2.e.a.226.1 2 9.2 odd 6
675.2.e.a.451.1 2 9.5 odd 6
675.2.k.a.199.1 4 45.38 even 12
675.2.k.a.199.2 4 45.2 even 12
675.2.k.a.424.1 4 45.32 even 12
675.2.k.a.424.2 4 45.23 even 12
720.2.q.d.241.1 2 180.139 odd 6
720.2.q.d.481.1 2 180.79 odd 6
2025.2.a.b.1.1 1 1.1 even 1 trivial
2025.2.a.e.1.1 1 3.2 odd 2
2025.2.b.c.649.1 2 5.2 odd 4
2025.2.b.c.649.2 2 5.3 odd 4
2025.2.b.d.649.1 2 15.8 even 4
2025.2.b.d.649.2 2 15.2 even 4
2160.2.q.a.721.1 2 180.59 even 6
2160.2.q.a.1441.1 2 180.119 even 6
6480.2.a.k.1.1 1 20.19 odd 2
6480.2.a.x.1.1 1 60.59 even 2