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Properties

Label 2025.2.a.c.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 $1$

Related objects

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2025,2,Mod(1,2025)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2025, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2025.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2025 = 3^{4} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2025.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
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

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.00000 q^{4} -2.00000 q^{7} +O(q^{10})\) \(q-2.00000 q^{4} -2.00000 q^{7} -3.00000 q^{11} +4.00000 q^{13} +4.00000 q^{16} +6.00000 q^{17} -1.00000 q^{19} +6.00000 q^{23} +4.00000 q^{28} -9.00000 q^{29} -1.00000 q^{31} -8.00000 q^{37} +3.00000 q^{41} +4.00000 q^{43} +6.00000 q^{44} -12.0000 q^{47} -3.00000 q^{49} -8.00000 q^{52} -6.00000 q^{53} +3.00000 q^{59} -10.0000 q^{61} -8.00000 q^{64} -14.0000 q^{67} -12.0000 q^{68} -3.00000 q^{71} -2.00000 q^{73} +2.00000 q^{76} +6.00000 q^{77} -16.0000 q^{79} +12.0000 q^{83} +15.0000 q^{89} -8.00000 q^{91} -12.0000 q^{92} +4.00000 q^{97} +O(q^{100})\)

Display 100 coefficients 10 coefficients

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(3\) 0 0
\(4\) −2.00000 −1.00000
\(5\) 0 0
\(6\) 0 0
\(7\) −2.00000 −0.755929 −0.377964 − 0.925820i \(-0.623376\pi\)
−0.377964 + 0.925820i \(0.623376\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −3.00000 −0.904534 −0.452267 − 0.891883i \(-0.649385\pi\)
−0.452267 + 0.891883i \(0.649385\pi\)
\(12\) 0 0
\(13\) 4.00000 1.10940 0.554700 − 0.832050i \(-0.312833\pi\)
0.554700 + 0.832050i \(0.312833\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 4.00000 1.00000
\(17\) 6.00000 1.45521 0.727607 − 0.685994i \(-0.240633\pi\)
0.727607 + 0.685994i \(0.240633\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416 −0.114708 − 0.993399i \(-0.536593\pi\)
−0.114708 + 0.993399i \(0.536593\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 6.00000 1.25109 0.625543 − 0.780189i \(-0.284877\pi\)
0.625543 + 0.780189i \(0.284877\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 0 0
\(28\) 4.00000 0.755929
\(29\) −9.00000 −1.67126 −0.835629 − 0.549294i \(-0.814897\pi\)
−0.835629 + 0.549294i \(0.814897\pi\)
\(30\) 0 0
\(31\) −1.00000 −0.179605 −0.0898027 − 0.995960i \(-0.528624\pi\)
−0.0898027 + 0.995960i \(0.528624\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −8.00000 −1.31519 −0.657596 − 0.753371i \(-0.728427\pi\)
−0.657596 + 0.753371i \(0.728427\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 3.00000 0.468521 0.234261 − 0.972174i \(-0.424733\pi\)
0.234261 + 0.972174i \(0.424733\pi\)
\(42\) 0 0
\(43\) 4.00000 0.609994 0.304997 − 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) 6.00000 0.904534
\(45\) 0 0
\(46\) 0 0
\(47\) −12.0000 −1.75038 −0.875190 − 0.483779i \(-0.839264\pi\)
−0.875190 + 0.483779i \(0.839264\pi\)
\(48\) 0 0
\(49\) −3.00000 −0.428571
\(50\) 0 0
\(51\) 0 0
\(52\) −8.00000 −1.10940
\(53\) −6.00000 −0.824163 −0.412082 − 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 3.00000 0.390567 0.195283 − 0.980747i \(-0.437437\pi\)
0.195283 + 0.980747i \(0.437437\pi\)
\(60\) 0 0
\(61\) −10.0000 −1.28037 −0.640184 − 0.768221i \(-0.721142\pi\)
−0.640184 + 0.768221i \(0.721142\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) −8.00000 −1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) −14.0000 −1.71037 −0.855186 − 0.518321i \(-0.826557\pi\)
−0.855186 + 0.518321i \(0.826557\pi\)
\(68\) −12.0000 −1.45521
\(69\) 0 0
\(70\) 0 0
\(71\) −3.00000 −0.356034 −0.178017 − 0.984027i \(-0.556968\pi\)
−0.178017 + 0.984027i \(0.556968\pi\)
\(72\) 0 0
\(73\) −2.00000 −0.234082 −0.117041 − 0.993127i \(-0.537341\pi\)
−0.117041 + 0.993127i \(0.537341\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 2.00000 0.229416
\(77\) 6.00000 0.683763
\(78\) 0 0
\(79\) −16.0000 −1.80014 −0.900070 − 0.435745i \(-0.856485\pi\)
−0.900070 + 0.435745i \(0.856485\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 12.0000 1.31717 0.658586 − 0.752506i \(-0.271155\pi\)
0.658586 + 0.752506i \(0.271155\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 15.0000 1.59000 0.794998 − 0.606612i \(-0.207472\pi\)
0.794998 + 0.606612i \(0.207472\pi\)
\(90\) 0 0
\(91\) −8.00000 −0.838628
\(92\) −12.0000 −1.25109
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 4.00000 0.406138 0.203069 − 0.979164i \(-0.434908\pi\)
0.203069 + 0.979164i \(0.434908\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 9.00000 0.895533 0.447767 − 0.894150i \(-0.352219\pi\)
0.447767 + 0.894150i \(0.352219\pi\)
\(102\) 0 0
\(103\) −14.0000 −1.37946 −0.689730 − 0.724066i \(-0.742271\pi\)
−0.689730 + 0.724066i \(0.742271\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 18.0000 1.74013 0.870063 − 0.492941i \(-0.164078\pi\)
0.870063 + 0.492941i \(0.164078\pi\)
\(108\) 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
\(112\) −8.00000 −0.755929
\(113\) −6.00000 −0.564433 −0.282216 − 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 18.0000 1.67126
\(117\) 0 0
\(118\) 0 0
\(119\) −12.0000 −1.10004
\(120\) 0 0
\(121\) −2.00000 −0.181818
\(122\) 0 0
\(123\) 0 0
\(124\) 2.00000 0.179605
\(125\) 0 0
\(126\) 0 0
\(127\) −2.00000 −0.177471 −0.0887357 − 0.996055i \(-0.528283\pi\)
−0.0887357 + 0.996055i \(0.528283\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −9.00000 −0.786334 −0.393167 − 0.919467i \(-0.628621\pi\)
−0.393167 + 0.919467i \(0.628621\pi\)
\(132\) 0 0
\(133\) 2.00000 0.173422
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −12.0000 −1.02523 −0.512615 − 0.858619i \(-0.671323\pi\)
−0.512615 + 0.858619i \(0.671323\pi\)
\(138\) 0 0
\(139\) −7.00000 −0.593732 −0.296866 − 0.954919i \(-0.595942\pi\)
−0.296866 + 0.954919i \(0.595942\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −12.0000 −1.00349
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 16.0000 1.31519
\(149\) 6.00000 0.491539 0.245770 − 0.969328i \(-0.420959\pi\)
0.245770 + 0.969328i \(0.420959\pi\)
\(150\) 0 0
\(151\) −1.00000 −0.0813788 −0.0406894 − 0.999172i \(-0.512955\pi\)
−0.0406894 + 0.999172i \(0.512955\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 16.0000 1.27694 0.638470 − 0.769647i \(-0.279568\pi\)
0.638470 + 0.769647i \(0.279568\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −12.0000 −0.945732
\(162\) 0 0
\(163\) −2.00000 −0.156652 −0.0783260 − 0.996928i \(-0.524958\pi\)
−0.0783260 + 0.996928i \(0.524958\pi\)
\(164\) −6.00000 −0.468521
\(165\) 0 0
\(166\) 0 0
\(167\) −6.00000 −0.464294 −0.232147 − 0.972681i \(-0.574575\pi\)
−0.232147 + 0.972681i \(0.574575\pi\)
\(168\) 0 0
\(169\) 3.00000 0.230769
\(170\) 0 0
\(171\) 0 0
\(172\) −8.00000 −0.609994
\(173\) 6.00000 0.456172 0.228086 − 0.973641i \(-0.426753\pi\)
0.228086 + 0.973641i \(0.426753\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −12.0000 −0.904534
\(177\) 0 0
\(178\) 0 0
\(179\) −9.00000 −0.672692 −0.336346 − 0.941739i \(-0.609191\pi\)
−0.336346 + 0.941739i \(0.609191\pi\)
\(180\) 0 0
\(181\) 11.0000 0.817624 0.408812 − 0.912619i \(-0.365943\pi\)
0.408812 + 0.912619i \(0.365943\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −18.0000 −1.31629
\(188\) 24.0000 1.75038
\(189\) 0 0
\(190\) 0 0
\(191\) −15.0000 −1.08536 −0.542681 − 0.839939i \(-0.682591\pi\)
−0.542681 + 0.839939i \(0.682591\pi\)
\(192\) 0 0
\(193\) 4.00000 0.287926 0.143963 − 0.989583i \(-0.454015\pi\)
0.143963 + 0.989583i \(0.454015\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 6.00000 0.428571
\(197\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(198\) 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
\(202\) 0 0
\(203\) 18.0000 1.26335
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 16.0000 1.10940
\(209\) 3.00000 0.207514
\(210\) 0 0
\(211\) −13.0000 −0.894957 −0.447478 − 0.894295i \(-0.647678\pi\)
−0.447478 + 0.894295i \(0.647678\pi\)
\(212\) 12.0000 0.824163
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 2.00000 0.135769
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 24.0000 1.61441
\(222\) 0 0
\(223\) −20.0000 −1.33930 −0.669650 − 0.742677i \(-0.733556\pi\)
−0.669650 + 0.742677i \(0.733556\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 18.0000 1.19470 0.597351 − 0.801980i \(-0.296220\pi\)
0.597351 + 0.801980i \(0.296220\pi\)
\(228\) 0 0
\(229\) −10.0000 −0.660819 −0.330409 − 0.943838i \(-0.607187\pi\)
−0.330409 + 0.943838i \(0.607187\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −12.0000 −0.786146 −0.393073 − 0.919507i \(-0.628588\pi\)
−0.393073 + 0.919507i \(0.628588\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −6.00000 −0.390567
\(237\) 0 0
\(238\) 0 0
\(239\) −24.0000 −1.55243 −0.776215 − 0.630468i \(-0.782863\pi\)
−0.776215 + 0.630468i \(0.782863\pi\)
\(240\) 0 0
\(241\) 17.0000 1.09507 0.547533 − 0.836784i \(-0.315567\pi\)
0.547533 + 0.836784i \(0.315567\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 20.0000 1.28037
\(245\) 0 0
\(246\) 0 0
\(247\) −4.00000 −0.254514
\(248\) 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
\(253\) −18.0000 −1.13165
\(254\) 0 0
\(255\) 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
\(259\) 16.0000 0.994192
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −18.0000 −1.10993 −0.554964 − 0.831875i \(-0.687268\pi\)
−0.554964 + 0.831875i \(0.687268\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 28.0000 1.71037
\(269\) −3.00000 −0.182913 −0.0914566 − 0.995809i \(-0.529152\pi\)
−0.0914566 + 0.995809i \(0.529152\pi\)
\(270\) 0 0
\(271\) −16.0000 −0.971931 −0.485965 − 0.873978i \(-0.661532\pi\)
−0.485965 + 0.873978i \(0.661532\pi\)
\(272\) 24.0000 1.45521
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 16.0000 0.961347 0.480673 − 0.876900i \(-0.340392\pi\)
0.480673 + 0.876900i \(0.340392\pi\)
\(278\) 0 0
\(279\) 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
\(283\) 10.0000 0.594438 0.297219 − 0.954809i \(-0.403941\pi\)
0.297219 + 0.954809i \(0.403941\pi\)
\(284\) 6.00000 0.356034
\(285\) 0 0
\(286\) 0 0
\(287\) −6.00000 −0.354169
\(288\) 0 0
\(289\) 19.0000 1.11765
\(290\) 0 0
\(291\) 0 0
\(292\) 4.00000 0.234082
\(293\) 12.0000 0.701047 0.350524 − 0.936554i \(-0.386004\pi\)
0.350524 + 0.936554i \(0.386004\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 24.0000 1.38796
\(300\) 0 0
\(301\) −8.00000 −0.461112
\(302\) 0 0
\(303\) 0 0
\(304\) −4.00000 −0.229416
\(305\) 0 0
\(306\) 0 0
\(307\) 16.0000 0.913168 0.456584 − 0.889680i \(-0.349073\pi\)
0.456584 + 0.889680i \(0.349073\pi\)
\(308\) −12.0000 −0.683763
\(309\) 0 0
\(310\) 0 0
\(311\) 27.0000 1.53103 0.765515 − 0.643418i \(-0.222484\pi\)
0.765515 + 0.643418i \(0.222484\pi\)
\(312\) 0 0
\(313\) −8.00000 −0.452187 −0.226093 − 0.974106i \(-0.572595\pi\)
−0.226093 + 0.974106i \(0.572595\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 32.0000 1.80014
\(317\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(318\) 0 0
\(319\) 27.0000 1.51171
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −6.00000 −0.333849
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 24.0000 1.32316
\(330\) 0 0
\(331\) −25.0000 −1.37412 −0.687062 − 0.726599i \(-0.741100\pi\)
−0.687062 + 0.726599i \(0.741100\pi\)
\(332\) −24.0000 −1.31717
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 4.00000 0.217894 0.108947 − 0.994048i \(-0.465252\pi\)
0.108947 + 0.994048i \(0.465252\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 3.00000 0.162459
\(342\) 0 0
\(343\) 20.0000 1.07990
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −18.0000 −0.966291 −0.483145 − 0.875540i \(-0.660506\pi\)
−0.483145 + 0.875540i \(0.660506\pi\)
\(348\) 0 0
\(349\) 5.00000 0.267644 0.133822 − 0.991005i \(-0.457275\pi\)
0.133822 + 0.991005i \(0.457275\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −12.0000 −0.638696 −0.319348 − 0.947638i \(-0.603464\pi\)
−0.319348 + 0.947638i \(0.603464\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −30.0000 −1.59000
\(357\) 0 0
\(358\) 0 0
\(359\) −15.0000 −0.791670 −0.395835 − 0.918322i \(-0.629545\pi\)
−0.395835 + 0.918322i \(0.629545\pi\)
\(360\) 0 0
\(361\) −18.0000 −0.947368
\(362\) 0 0
\(363\) 0 0
\(364\) 16.0000 0.838628
\(365\) 0 0
\(366\) 0 0
\(367\) 10.0000 0.521996 0.260998 − 0.965339i \(-0.415948\pi\)
0.260998 + 0.965339i \(0.415948\pi\)
\(368\) 24.0000 1.25109
\(369\) 0 0
\(370\) 0 0
\(371\) 12.0000 0.623009
\(372\) 0 0
\(373\) −38.0000 −1.96757 −0.983783 − 0.179364i \(-0.942596\pi\)
−0.983783 + 0.179364i \(0.942596\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −36.0000 −1.85409
\(378\) 0 0
\(379\) −16.0000 −0.821865 −0.410932 − 0.911666i \(-0.634797\pi\)
−0.410932 + 0.911666i \(0.634797\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) −8.00000 −0.406138
\(389\) 30.0000 1.52106 0.760530 − 0.649303i \(-0.224939\pi\)
0.760530 + 0.649303i \(0.224939\pi\)
\(390\) 0 0
\(391\) 36.0000 1.82060
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 22.0000 1.10415 0.552074 − 0.833795i \(-0.313837\pi\)
0.552074 + 0.833795i \(0.313837\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −6.00000 −0.299626 −0.149813 − 0.988714i \(-0.547867\pi\)
−0.149813 + 0.988714i \(0.547867\pi\)
\(402\) 0 0
\(403\) −4.00000 −0.199254
\(404\) −18.0000 −0.895533
\(405\) 0 0
\(406\) 0 0
\(407\) 24.0000 1.18964
\(408\) 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
\(412\) 28.0000 1.37946
\(413\) −6.00000 −0.295241
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −12.0000 −0.586238 −0.293119 − 0.956076i \(-0.594693\pi\)
−0.293119 + 0.956076i \(0.594693\pi\)
\(420\) 0 0
\(421\) 5.00000 0.243685 0.121843 − 0.992549i \(-0.461120\pi\)
0.121843 + 0.992549i \(0.461120\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 20.0000 0.967868
\(428\) −36.0000 −1.74013
\(429\) 0 0
\(430\) 0 0
\(431\) 15.0000 0.722525 0.361262 − 0.932464i \(-0.382346\pi\)
0.361262 + 0.932464i \(0.382346\pi\)
\(432\) 0 0
\(433\) −26.0000 −1.24948 −0.624740 − 0.780833i \(-0.714795\pi\)
−0.624740 + 0.780833i \(0.714795\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 2.00000 0.0957826
\(437\) −6.00000 −0.287019
\(438\) 0 0
\(439\) −1.00000 −0.0477274 −0.0238637 − 0.999715i \(-0.507597\pi\)
−0.0238637 + 0.999715i \(0.507597\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 12.0000 0.570137 0.285069 − 0.958507i \(-0.407984\pi\)
0.285069 + 0.958507i \(0.407984\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 16.0000 0.755929
\(449\) −15.0000 −0.707894 −0.353947 − 0.935266i \(-0.615161\pi\)
−0.353947 + 0.935266i \(0.615161\pi\)
\(450\) 0 0
\(451\) −9.00000 −0.423793
\(452\) 12.0000 0.564433
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 28.0000 1.30978 0.654892 − 0.755722i \(-0.272714\pi\)
0.654892 + 0.755722i \(0.272714\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 21.0000 0.978068 0.489034 − 0.872265i \(-0.337349\pi\)
0.489034 + 0.872265i \(0.337349\pi\)
\(462\) 0 0
\(463\) −26.0000 −1.20832 −0.604161 − 0.796862i \(-0.706492\pi\)
−0.604161 + 0.796862i \(0.706492\pi\)
\(464\) −36.0000 −1.67126
\(465\) 0 0
\(466\) 0 0
\(467\) 6.00000 0.277647 0.138823 − 0.990317i \(-0.455668\pi\)
0.138823 + 0.990317i \(0.455668\pi\)
\(468\) 0 0
\(469\) 28.0000 1.29292
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −12.0000 −0.551761
\(474\) 0 0
\(475\) 0 0
\(476\) 24.0000 1.10004
\(477\) 0 0
\(478\) 0 0
\(479\) 33.0000 1.50781 0.753904 − 0.656984i \(-0.228168\pi\)
0.753904 + 0.656984i \(0.228168\pi\)
\(480\) 0 0
\(481\) −32.0000 −1.45907
\(482\) 0 0
\(483\) 0 0
\(484\) 4.00000 0.181818
\(485\) 0 0
\(486\) 0 0
\(487\) 16.0000 0.725029 0.362515 − 0.931978i \(-0.381918\pi\)
0.362515 + 0.931978i \(0.381918\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −9.00000 −0.406164 −0.203082 − 0.979162i \(-0.565096\pi\)
−0.203082 + 0.979162i \(0.565096\pi\)
\(492\) 0 0
\(493\) −54.0000 −2.43204
\(494\) 0 0
\(495\) 0 0
\(496\) −4.00000 −0.179605
\(497\) 6.00000 0.269137
\(498\) 0 0
\(499\) −13.0000 −0.581960 −0.290980 − 0.956729i \(-0.593981\pi\)
−0.290980 + 0.956729i \(0.593981\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 12.0000 0.535054 0.267527 − 0.963550i \(-0.413794\pi\)
0.267527 + 0.963550i \(0.413794\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) 4.00000 0.177471
\(509\) 30.0000 1.32973 0.664863 − 0.746965i \(-0.268490\pi\)
0.664863 + 0.746965i \(0.268490\pi\)
\(510\) 0 0
\(511\) 4.00000 0.176950
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 36.0000 1.58328
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −30.0000 −1.31432 −0.657162 − 0.753749i \(-0.728243\pi\)
−0.657162 + 0.753749i \(0.728243\pi\)
\(522\) 0 0
\(523\) −20.0000 −0.874539 −0.437269 − 0.899331i \(-0.644054\pi\)
−0.437269 + 0.899331i \(0.644054\pi\)
\(524\) 18.0000 0.786334
\(525\) 0 0
\(526\) 0 0
\(527\) −6.00000 −0.261364
\(528\) 0 0
\(529\) 13.0000 0.565217
\(530\) 0 0
\(531\) 0 0
\(532\) −4.00000 −0.173422
\(533\) 12.0000 0.519778
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 9.00000 0.387657
\(540\) 0 0
\(541\) 23.0000 0.988847 0.494424 − 0.869221i \(-0.335379\pi\)
0.494424 + 0.869221i \(0.335379\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −8.00000 −0.342055 −0.171028 − 0.985266i \(-0.554709\pi\)
−0.171028 + 0.985266i \(0.554709\pi\)
\(548\) 24.0000 1.02523
\(549\) 0 0
\(550\) 0 0
\(551\) 9.00000 0.383413
\(552\) 0 0
\(553\) 32.0000 1.36078
\(554\) 0 0
\(555\) 0 0
\(556\) 14.0000 0.593732
\(557\) −24.0000 −1.01691 −0.508456 − 0.861088i \(-0.669784\pi\)
−0.508456 + 0.861088i \(0.669784\pi\)
\(558\) 0 0
\(559\) 16.0000 0.676728
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 24.0000 1.01148 0.505740 − 0.862686i \(-0.331220\pi\)
0.505740 + 0.862686i \(0.331220\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −27.0000 −1.13190 −0.565949 − 0.824440i \(-0.691490\pi\)
−0.565949 + 0.824440i \(0.691490\pi\)
\(570\) 0 0
\(571\) −19.0000 −0.795125 −0.397563 − 0.917575i \(-0.630144\pi\)
−0.397563 + 0.917575i \(0.630144\pi\)
\(572\) 24.0000 1.00349
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 34.0000 1.41544 0.707719 − 0.706494i \(-0.249724\pi\)
0.707719 + 0.706494i \(0.249724\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −24.0000 −0.995688
\(582\) 0 0
\(583\) 18.0000 0.745484
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 48.0000 1.98117 0.990586 − 0.136892i \(-0.0437113\pi\)
0.990586 + 0.136892i \(0.0437113\pi\)
\(588\) 0 0
\(589\) 1.00000 0.0412043
\(590\) 0 0
\(591\) 0 0
\(592\) −32.0000 −1.31519
\(593\) −18.0000 −0.739171 −0.369586 − 0.929197i \(-0.620500\pi\)
−0.369586 + 0.929197i \(0.620500\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −12.0000 −0.491539
\(597\) 0 0
\(598\) 0 0
\(599\) −21.0000 −0.858037 −0.429018 − 0.903296i \(-0.641140\pi\)
−0.429018 + 0.903296i \(0.641140\pi\)
\(600\) 0 0
\(601\) −31.0000 −1.26452 −0.632258 − 0.774758i \(-0.717872\pi\)
−0.632258 + 0.774758i \(0.717872\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 2.00000 0.0813788
\(605\) 0 0
\(606\) 0 0
\(607\) −20.0000 −0.811775 −0.405887 − 0.913923i \(-0.633038\pi\)
−0.405887 + 0.913923i \(0.633038\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −48.0000 −1.94187
\(612\) 0 0
\(613\) 16.0000 0.646234 0.323117 − 0.946359i \(-0.395269\pi\)
0.323117 + 0.946359i \(0.395269\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 24.0000 0.966204 0.483102 − 0.875564i \(-0.339510\pi\)
0.483102 + 0.875564i \(0.339510\pi\)
\(618\) 0 0
\(619\) 44.0000 1.76851 0.884255 − 0.467005i \(-0.154667\pi\)
0.884255 + 0.467005i \(0.154667\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −30.0000 −1.20192
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 0 0
\(628\) −32.0000 −1.27694
\(629\) −48.0000 −1.91389
\(630\) 0 0
\(631\) −25.0000 −0.995234 −0.497617 − 0.867397i \(-0.665792\pi\)
−0.497617 + 0.867397i \(0.665792\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −12.0000 −0.475457
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −33.0000 −1.30342 −0.651711 − 0.758468i \(-0.725948\pi\)
−0.651711 + 0.758468i \(0.725948\pi\)
\(642\) 0 0
\(643\) 4.00000 0.157745 0.0788723 − 0.996885i \(-0.474868\pi\)
0.0788723 + 0.996885i \(0.474868\pi\)
\(644\) 24.0000 0.945732
\(645\) 0 0
\(646\) 0 0
\(647\) −6.00000 −0.235884 −0.117942 − 0.993020i \(-0.537630\pi\)
−0.117942 + 0.993020i \(0.537630\pi\)
\(648\) 0 0
\(649\) −9.00000 −0.353281
\(650\) 0 0
\(651\) 0 0
\(652\) 4.00000 0.156652
\(653\) 24.0000 0.939193 0.469596 − 0.882881i \(-0.344399\pi\)
0.469596 + 0.882881i \(0.344399\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 12.0000 0.468521
\(657\) 0 0
\(658\) 0 0
\(659\) −24.0000 −0.934907 −0.467454 − 0.884018i \(-0.654829\pi\)
−0.467454 + 0.884018i \(0.654829\pi\)
\(660\) 0 0
\(661\) 5.00000 0.194477 0.0972387 − 0.995261i \(-0.468999\pi\)
0.0972387 + 0.995261i \(0.468999\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −54.0000 −2.09089
\(668\) 12.0000 0.464294
\(669\) 0 0
\(670\) 0 0
\(671\) 30.0000 1.15814
\(672\) 0 0
\(673\) 34.0000 1.31060 0.655302 − 0.755367i \(-0.272541\pi\)
0.655302 + 0.755367i \(0.272541\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) −6.00000 −0.230769
\(677\) −30.0000 −1.15299 −0.576497 − 0.817099i \(-0.695581\pi\)
−0.576497 + 0.817099i \(0.695581\pi\)
\(678\) 0 0
\(679\) −8.00000 −0.307012
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 12.0000 0.459167 0.229584 − 0.973289i \(-0.426264\pi\)
0.229584 + 0.973289i \(0.426264\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 16.0000 0.609994
\(689\) −24.0000 −0.914327
\(690\) 0 0
\(691\) 44.0000 1.67384 0.836919 − 0.547326i \(-0.184354\pi\)
0.836919 + 0.547326i \(0.184354\pi\)
\(692\) −12.0000 −0.456172
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 18.0000 0.681799
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 15.0000 0.566542 0.283271 − 0.959040i \(-0.408580\pi\)
0.283271 + 0.959040i \(0.408580\pi\)
\(702\) 0 0
\(703\) 8.00000 0.301726
\(704\) 24.0000 0.904534
\(705\) 0 0
\(706\) 0 0
\(707\) −18.0000 −0.676960
\(708\) 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
\(713\) −6.00000 −0.224702
\(714\) 0 0
\(715\) 0 0
\(716\) 18.0000 0.672692
\(717\) 0 0
\(718\) 0 0
\(719\) 27.0000 1.00693 0.503465 − 0.864016i \(-0.332058\pi\)
0.503465 + 0.864016i \(0.332058\pi\)
\(720\) 0 0
\(721\) 28.0000 1.04277
\(722\) 0 0
\(723\) 0 0
\(724\) −22.0000 −0.817624
\(725\) 0 0
\(726\) 0 0
\(727\) 28.0000 1.03846 0.519231 − 0.854634i \(-0.326218\pi\)
0.519231 + 0.854634i \(0.326218\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 24.0000 0.887672
\(732\) 0 0
\(733\) −14.0000 −0.517102 −0.258551 − 0.965998i \(-0.583245\pi\)
−0.258551 + 0.965998i \(0.583245\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 42.0000 1.54709
\(738\) 0 0
\(739\) −7.00000 −0.257499 −0.128750 − 0.991677i \(-0.541096\pi\)
−0.128750 + 0.991677i \(0.541096\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 18.0000 0.660356 0.330178 − 0.943919i \(-0.392891\pi\)
0.330178 + 0.943919i \(0.392891\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 36.0000 1.31629
\(749\) −36.0000 −1.31541
\(750\) 0 0
\(751\) −16.0000 −0.583848 −0.291924 − 0.956441i \(-0.594295\pi\)
−0.291924 + 0.956441i \(0.594295\pi\)
\(752\) −48.0000 −1.75038
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −20.0000 −0.726912 −0.363456 − 0.931611i \(-0.618403\pi\)
−0.363456 + 0.931611i \(0.618403\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 33.0000 1.19625 0.598125 − 0.801403i \(-0.295913\pi\)
0.598125 + 0.801403i \(0.295913\pi\)
\(762\) 0 0
\(763\) 2.00000 0.0724049
\(764\) 30.0000 1.08536
\(765\) 0 0
\(766\) 0 0
\(767\) 12.0000 0.433295
\(768\) 0 0
\(769\) 41.0000 1.47850 0.739249 − 0.673432i \(-0.235181\pi\)
0.739249 + 0.673432i \(0.235181\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −8.00000 −0.287926
\(773\) −24.0000 −0.863220 −0.431610 − 0.902060i \(-0.642054\pi\)
−0.431610 + 0.902060i \(0.642054\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −3.00000 −0.107486
\(780\) 0 0
\(781\) 9.00000 0.322045
\(782\) 0 0
\(783\) 0 0
\(784\) −12.0000 −0.428571
\(785\) 0 0
\(786\) 0 0
\(787\) −14.0000 −0.499046 −0.249523 − 0.968369i \(-0.580274\pi\)
−0.249523 + 0.968369i \(0.580274\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 12.0000 0.426671
\(792\) 0 0
\(793\) −40.0000 −1.42044
\(794\) 0 0
\(795\) 0 0
\(796\) 32.0000 1.13421
\(797\) 24.0000 0.850124 0.425062 − 0.905164i \(-0.360252\pi\)
0.425062 + 0.905164i \(0.360252\pi\)
\(798\) 0 0
\(799\) −72.0000 −2.54718
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 6.00000 0.211735
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 15.0000 0.527372 0.263686 − 0.964609i \(-0.415062\pi\)
0.263686 + 0.964609i \(0.415062\pi\)
\(810\) 0 0
\(811\) 23.0000 0.807639 0.403820 − 0.914839i \(-0.367682\pi\)
0.403820 + 0.914839i \(0.367682\pi\)
\(812\) −36.0000 −1.26335
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −4.00000 −0.139942
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −45.0000 −1.57051 −0.785255 − 0.619172i \(-0.787468\pi\)
−0.785255 + 0.619172i \(0.787468\pi\)
\(822\) 0 0
\(823\) 22.0000 0.766872 0.383436 − 0.923567i \(-0.374741\pi\)
0.383436 + 0.923567i \(0.374741\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 36.0000 1.25184 0.625921 − 0.779886i \(-0.284723\pi\)
0.625921 + 0.779886i \(0.284723\pi\)
\(828\) 0 0
\(829\) −13.0000 −0.451509 −0.225754 − 0.974184i \(-0.572485\pi\)
−0.225754 + 0.974184i \(0.572485\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −32.0000 −1.10940
\(833\) −18.0000 −0.623663
\(834\) 0 0
\(835\) 0 0
\(836\) −6.00000 −0.207514
\(837\) 0 0
\(838\) 0 0
\(839\) 45.0000 1.55357 0.776786 − 0.629764i \(-0.216849\pi\)
0.776786 + 0.629764i \(0.216849\pi\)
\(840\) 0 0
\(841\) 52.0000 1.79310
\(842\) 0 0
\(843\) 0 0
\(844\) 26.0000 0.894957
\(845\) 0 0
\(846\) 0 0
\(847\) 4.00000 0.137442
\(848\) −24.0000 −0.824163
\(849\) 0 0
\(850\) 0 0
\(851\) −48.0000 −1.64542
\(852\) 0 0
\(853\) 46.0000 1.57501 0.787505 − 0.616308i \(-0.211372\pi\)
0.787505 + 0.616308i \(0.211372\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −48.0000 −1.63965 −0.819824 − 0.572615i \(-0.805929\pi\)
−0.819824 + 0.572615i \(0.805929\pi\)
\(858\) 0 0
\(859\) −25.0000 −0.852989 −0.426494 − 0.904490i \(-0.640252\pi\)
−0.426494 + 0.904490i \(0.640252\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −30.0000 −1.02121 −0.510606 − 0.859815i \(-0.670579\pi\)
−0.510606 + 0.859815i \(0.670579\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 0 0
\(868\) −4.00000 −0.135769
\(869\) 48.0000 1.62829
\(870\) 0 0
\(871\) −56.0000 −1.89749
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −44.0000 −1.48577 −0.742887 − 0.669417i \(-0.766544\pi\)
−0.742887 + 0.669417i \(0.766544\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 39.0000 1.31394 0.656972 − 0.753915i \(-0.271837\pi\)
0.656972 + 0.753915i \(0.271837\pi\)
\(882\) 0 0
\(883\) 46.0000 1.54802 0.774012 − 0.633171i \(-0.218247\pi\)
0.774012 + 0.633171i \(0.218247\pi\)
\(884\) −48.0000 −1.61441
\(885\) 0 0
\(886\) 0 0
\(887\) −36.0000 −1.20876 −0.604381 − 0.796696i \(-0.706579\pi\)
−0.604381 + 0.796696i \(0.706579\pi\)
\(888\) 0 0
\(889\) 4.00000 0.134156
\(890\) 0 0
\(891\) 0 0
\(892\) 40.0000 1.33930
\(893\) 12.0000 0.401565
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 9.00000 0.300167
\(900\) 0 0
\(901\) −36.0000 −1.19933
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −38.0000 −1.26177 −0.630885 − 0.775877i \(-0.717308\pi\)
−0.630885 + 0.775877i \(0.717308\pi\)
\(908\) −36.0000 −1.19470
\(909\) 0 0
\(910\) 0 0
\(911\) −9.00000 −0.298183 −0.149092 − 0.988823i \(-0.547635\pi\)
−0.149092 + 0.988823i \(0.547635\pi\)
\(912\) 0 0
\(913\) −36.0000 −1.19143
\(914\) 0 0
\(915\) 0 0
\(916\) 20.0000 0.660819
\(917\) 18.0000 0.594412
\(918\) 0 0
\(919\) −7.00000 −0.230909 −0.115454 − 0.993313i \(-0.536832\pi\)
−0.115454 + 0.993313i \(0.536832\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −12.0000 −0.394985
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −39.0000 −1.27955 −0.639774 − 0.768563i \(-0.720972\pi\)
−0.639774 + 0.768563i \(0.720972\pi\)
\(930\) 0 0
\(931\) 3.00000 0.0983210
\(932\) 24.0000 0.786146
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −14.0000 −0.457360 −0.228680 − 0.973502i \(-0.573441\pi\)
−0.228680 + 0.973502i \(0.573441\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 30.0000 0.977972 0.488986 − 0.872292i \(-0.337367\pi\)
0.488986 + 0.872292i \(0.337367\pi\)
\(942\) 0 0
\(943\) 18.0000 0.586161
\(944\) 12.0000 0.390567
\(945\) 0 0
\(946\) 0 0
\(947\) −30.0000 −0.974869 −0.487435 − 0.873160i \(-0.662067\pi\)
−0.487435 + 0.873160i \(0.662067\pi\)
\(948\) 0 0
\(949\) −8.00000 −0.259691
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 48.0000 1.55487 0.777436 − 0.628962i \(-0.216520\pi\)
0.777436 + 0.628962i \(0.216520\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 48.0000 1.55243
\(957\) 0 0
\(958\) 0 0
\(959\) 24.0000 0.775000
\(960\) 0 0
\(961\) −30.0000 −0.967742
\(962\) 0 0
\(963\) 0 0
\(964\) −34.0000 −1.09507
\(965\) 0 0
\(966\) 0 0
\(967\) −50.0000 −1.60789 −0.803946 − 0.594703i \(-0.797270\pi\)
−0.803946 + 0.594703i \(0.797270\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −3.00000 −0.0962746 −0.0481373 − 0.998841i \(-0.515328\pi\)
−0.0481373 + 0.998841i \(0.515328\pi\)
\(972\) 0 0
\(973\) 14.0000 0.448819
\(974\) 0 0
\(975\) 0 0
\(976\) −40.0000 −1.28037
\(977\) 48.0000 1.53566 0.767828 − 0.640656i \(-0.221338\pi\)
0.767828 + 0.640656i \(0.221338\pi\)
\(978\) 0 0
\(979\) −45.0000 −1.43821
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 6.00000 0.191370 0.0956851 − 0.995412i \(-0.469496\pi\)
0.0956851 + 0.995412i \(0.469496\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 8.00000 0.254514
\(989\) 24.0000 0.763156
\(990\) 0 0
\(991\) 47.0000 1.49300 0.746502 − 0.665383i \(-0.231732\pi\)
0.746502 + 0.665383i \(0.231732\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 10.0000 0.316703 0.158352 − 0.987383i \(-0.449382\pi\)
0.158352 + 0.987383i \(0.449382\pi\)
\(998\) 0 0
\(999\) 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.c.1.1 1
3.2 odd 2 2025.2.a.d.1.1 1
5.2 odd 4 2025.2.b.e.649.1 2
5.3 odd 4 2025.2.b.e.649.2 2
5.4 even 2 405.2.a.c.1.1 ✓ 1
15.2 even 4 2025.2.b.f.649.1 2
15.8 even 4 2025.2.b.f.649.2 2
15.14 odd 2 405.2.a.d.1.1 yes 1
20.19 odd 2 6480.2.a.c.1.1 1
45.4 even 6 405.2.e.e.136.1 2
45.14 odd 6 405.2.e.d.136.1 2
45.29 odd 6 405.2.e.d.271.1 2
45.34 even 6 405.2.e.e.271.1 2
60.59 even 2 6480.2.a.o.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
405.2.a.c.1.1 ✓ 1 5.4 even 2
405.2.a.d.1.1 yes 1 15.14 odd 2
405.2.e.d.136.1 2 45.14 odd 6
405.2.e.d.271.1 2 45.29 odd 6
405.2.e.e.136.1 2 45.4 even 6
405.2.e.e.271.1 2 45.34 even 6
2025.2.a.c.1.1 1 1.1 even 1 trivial
2025.2.a.d.1.1 1 3.2 odd 2
2025.2.b.e.649.1 2 5.2 odd 4
2025.2.b.e.649.2 2 5.3 odd 4
2025.2.b.f.649.1 2 15.2 even 4
2025.2.b.f.649.2 2 15.8 even 4
6480.2.a.c.1.1 1 20.19 odd 2
6480.2.a.o.1.1 1 60.59 even 2