Properties

Label 2025.4.a.a.1.1
Level $2025$
Weight $4$
Character 2025.1
Self dual yes
Analytic conductor $119.479$
Analytic rank $2$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2025,4,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, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2025.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2025 = 3^{4} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
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: \(119.478867762\)
Analytic rank: \(2\)
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)
 
\(f(q)\) \(=\) \(q-5.00000 q^{2} +17.0000 q^{4} -9.00000 q^{7} -45.0000 q^{8} +O(q^{10})\) \(q-5.00000 q^{2} +17.0000 q^{4} -9.00000 q^{7} -45.0000 q^{8} +8.00000 q^{11} -43.0000 q^{13} +45.0000 q^{14} +89.0000 q^{16} -122.000 q^{17} -59.0000 q^{19} -40.0000 q^{22} -213.000 q^{23} +215.000 q^{26} -153.000 q^{28} -224.000 q^{29} -36.0000 q^{31} -85.0000 q^{32} +610.000 q^{34} -206.000 q^{37} +295.000 q^{38} -413.000 q^{41} +392.000 q^{43} +136.000 q^{44} +1065.00 q^{46} -311.000 q^{47} -262.000 q^{49} -731.000 q^{52} -377.000 q^{53} +405.000 q^{56} +1120.00 q^{58} -337.000 q^{59} +40.0000 q^{61} +180.000 q^{62} -287.000 q^{64} -348.000 q^{67} -2074.00 q^{68} -62.0000 q^{71} +1214.00 q^{73} +1030.00 q^{74} -1003.00 q^{76} -72.0000 q^{77} -294.000 q^{79} +2065.00 q^{82} +534.000 q^{83} -1960.00 q^{86} -360.000 q^{88} +810.000 q^{89} +387.000 q^{91} -3621.00 q^{92} +1555.00 q^{94} +928.000 q^{97} +1310.00 q^{98} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −5.00000 −1.76777 −0.883883 0.467707i \(-0.845080\pi\)
−0.883883 + 0.467707i \(0.845080\pi\)
\(3\) 0 0
\(4\) 17.0000 2.12500
\(5\) 0 0
\(6\) 0 0
\(7\) −9.00000 −0.485954 −0.242977 0.970032i \(-0.578124\pi\)
−0.242977 + 0.970032i \(0.578124\pi\)
\(8\) −45.0000 −1.98874
\(9\) 0 0
\(10\) 0 0
\(11\) 8.00000 0.219281 0.109640 0.993971i \(-0.465030\pi\)
0.109640 + 0.993971i \(0.465030\pi\)
\(12\) 0 0
\(13\) −43.0000 −0.917389 −0.458694 0.888594i \(-0.651683\pi\)
−0.458694 + 0.888594i \(0.651683\pi\)
\(14\) 45.0000 0.859054
\(15\) 0 0
\(16\) 89.0000 1.39062
\(17\) −122.000 −1.74055 −0.870275 0.492566i \(-0.836059\pi\)
−0.870275 + 0.492566i \(0.836059\pi\)
\(18\) 0 0
\(19\) −59.0000 −0.712396 −0.356198 0.934410i \(-0.615927\pi\)
−0.356198 + 0.934410i \(0.615927\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −40.0000 −0.387638
\(23\) −213.000 −1.93102 −0.965512 0.260357i \(-0.916160\pi\)
−0.965512 + 0.260357i \(0.916160\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 215.000 1.62173
\(27\) 0 0
\(28\) −153.000 −1.03265
\(29\) −224.000 −1.43434 −0.717168 0.696900i \(-0.754562\pi\)
−0.717168 + 0.696900i \(0.754562\pi\)
\(30\) 0 0
\(31\) −36.0000 −0.208574 −0.104287 0.994547i \(-0.533256\pi\)
−0.104287 + 0.994547i \(0.533256\pi\)
\(32\) −85.0000 −0.469563
\(33\) 0 0
\(34\) 610.000 3.07689
\(35\) 0 0
\(36\) 0 0
\(37\) −206.000 −0.915302 −0.457651 0.889132i \(-0.651309\pi\)
−0.457651 + 0.889132i \(0.651309\pi\)
\(38\) 295.000 1.25935
\(39\) 0 0
\(40\) 0 0
\(41\) −413.000 −1.57316 −0.786582 0.617485i \(-0.788152\pi\)
−0.786582 + 0.617485i \(0.788152\pi\)
\(42\) 0 0
\(43\) 392.000 1.39022 0.695110 0.718904i \(-0.255356\pi\)
0.695110 + 0.718904i \(0.255356\pi\)
\(44\) 136.000 0.465972
\(45\) 0 0
\(46\) 1065.00 3.41360
\(47\) −311.000 −0.965192 −0.482596 0.875843i \(-0.660306\pi\)
−0.482596 + 0.875843i \(0.660306\pi\)
\(48\) 0 0
\(49\) −262.000 −0.763848
\(50\) 0 0
\(51\) 0 0
\(52\) −731.000 −1.94945
\(53\) −377.000 −0.977074 −0.488537 0.872543i \(-0.662469\pi\)
−0.488537 + 0.872543i \(0.662469\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 405.000 0.966436
\(57\) 0 0
\(58\) 1120.00 2.53557
\(59\) −337.000 −0.743621 −0.371811 0.928309i \(-0.621263\pi\)
−0.371811 + 0.928309i \(0.621263\pi\)
\(60\) 0 0
\(61\) 40.0000 0.0839586 0.0419793 0.999118i \(-0.486634\pi\)
0.0419793 + 0.999118i \(0.486634\pi\)
\(62\) 180.000 0.368710
\(63\) 0 0
\(64\) −287.000 −0.560547
\(65\) 0 0
\(66\) 0 0
\(67\) −348.000 −0.634552 −0.317276 0.948333i \(-0.602768\pi\)
−0.317276 + 0.948333i \(0.602768\pi\)
\(68\) −2074.00 −3.69867
\(69\) 0 0
\(70\) 0 0
\(71\) −62.0000 −0.103634 −0.0518172 0.998657i \(-0.516501\pi\)
−0.0518172 + 0.998657i \(0.516501\pi\)
\(72\) 0 0
\(73\) 1214.00 1.94641 0.973205 0.229939i \(-0.0738525\pi\)
0.973205 + 0.229939i \(0.0738525\pi\)
\(74\) 1030.00 1.61804
\(75\) 0 0
\(76\) −1003.00 −1.51384
\(77\) −72.0000 −0.106561
\(78\) 0 0
\(79\) −294.000 −0.418704 −0.209352 0.977840i \(-0.567135\pi\)
−0.209352 + 0.977840i \(0.567135\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 2065.00 2.78099
\(83\) 534.000 0.706194 0.353097 0.935587i \(-0.385129\pi\)
0.353097 + 0.935587i \(0.385129\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −1960.00 −2.45758
\(87\) 0 0
\(88\) −360.000 −0.436092
\(89\) 810.000 0.964717 0.482359 0.875974i \(-0.339780\pi\)
0.482359 + 0.875974i \(0.339780\pi\)
\(90\) 0 0
\(91\) 387.000 0.445809
\(92\) −3621.00 −4.10343
\(93\) 0 0
\(94\) 1555.00 1.70623
\(95\) 0 0
\(96\) 0 0
\(97\) 928.000 0.971383 0.485691 0.874130i \(-0.338568\pi\)
0.485691 + 0.874130i \(0.338568\pi\)
\(98\) 1310.00 1.35031
\(99\) 0 0
\(100\) 0 0
\(101\) 996.000 0.981245 0.490622 0.871372i \(-0.336769\pi\)
0.490622 + 0.871372i \(0.336769\pi\)
\(102\) 0 0
\(103\) 433.000 0.414221 0.207110 0.978318i \(-0.433594\pi\)
0.207110 + 0.978318i \(0.433594\pi\)
\(104\) 1935.00 1.82445
\(105\) 0 0
\(106\) 1885.00 1.72724
\(107\) −1686.00 −1.52329 −0.761644 0.647996i \(-0.775607\pi\)
−0.761644 + 0.647996i \(0.775607\pi\)
\(108\) 0 0
\(109\) 656.000 0.576453 0.288227 0.957562i \(-0.406934\pi\)
0.288227 + 0.957562i \(0.406934\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −801.000 −0.675780
\(113\) 1018.00 0.847481 0.423741 0.905784i \(-0.360717\pi\)
0.423741 + 0.905784i \(0.360717\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −3808.00 −3.04796
\(117\) 0 0
\(118\) 1685.00 1.31455
\(119\) 1098.00 0.845828
\(120\) 0 0
\(121\) −1267.00 −0.951916
\(122\) −200.000 −0.148419
\(123\) 0 0
\(124\) −612.000 −0.443220
\(125\) 0 0
\(126\) 0 0
\(127\) 1361.00 0.950939 0.475469 0.879732i \(-0.342278\pi\)
0.475469 + 0.879732i \(0.342278\pi\)
\(128\) 2115.00 1.46048
\(129\) 0 0
\(130\) 0 0
\(131\) −1911.00 −1.27454 −0.637270 0.770640i \(-0.719937\pi\)
−0.637270 + 0.770640i \(0.719937\pi\)
\(132\) 0 0
\(133\) 531.000 0.346192
\(134\) 1740.00 1.12174
\(135\) 0 0
\(136\) 5490.00 3.46150
\(137\) 654.000 0.407847 0.203923 0.978987i \(-0.434631\pi\)
0.203923 + 0.978987i \(0.434631\pi\)
\(138\) 0 0
\(139\) −733.000 −0.447282 −0.223641 0.974672i \(-0.571794\pi\)
−0.223641 + 0.974672i \(0.571794\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 310.000 0.183202
\(143\) −344.000 −0.201166
\(144\) 0 0
\(145\) 0 0
\(146\) −6070.00 −3.44080
\(147\) 0 0
\(148\) −3502.00 −1.94502
\(149\) 1126.00 0.619097 0.309549 0.950884i \(-0.399822\pi\)
0.309549 + 0.950884i \(0.399822\pi\)
\(150\) 0 0
\(151\) −2546.00 −1.37212 −0.686061 0.727544i \(-0.740662\pi\)
−0.686061 + 0.727544i \(0.740662\pi\)
\(152\) 2655.00 1.41677
\(153\) 0 0
\(154\) 360.000 0.188374
\(155\) 0 0
\(156\) 0 0
\(157\) −1223.00 −0.621694 −0.310847 0.950460i \(-0.600613\pi\)
−0.310847 + 0.950460i \(0.600613\pi\)
\(158\) 1470.00 0.740170
\(159\) 0 0
\(160\) 0 0
\(161\) 1917.00 0.938390
\(162\) 0 0
\(163\) −3176.00 −1.52616 −0.763078 0.646306i \(-0.776313\pi\)
−0.763078 + 0.646306i \(0.776313\pi\)
\(164\) −7021.00 −3.34298
\(165\) 0 0
\(166\) −2670.00 −1.24839
\(167\) 132.000 0.0611645 0.0305822 0.999532i \(-0.490264\pi\)
0.0305822 + 0.999532i \(0.490264\pi\)
\(168\) 0 0
\(169\) −348.000 −0.158398
\(170\) 0 0
\(171\) 0 0
\(172\) 6664.00 2.95422
\(173\) −993.000 −0.436395 −0.218198 0.975905i \(-0.570018\pi\)
−0.218198 + 0.975905i \(0.570018\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 712.000 0.304938
\(177\) 0 0
\(178\) −4050.00 −1.70540
\(179\) 3101.00 1.29486 0.647429 0.762126i \(-0.275844\pi\)
0.647429 + 0.762126i \(0.275844\pi\)
\(180\) 0 0
\(181\) 2846.00 1.16874 0.584369 0.811488i \(-0.301342\pi\)
0.584369 + 0.811488i \(0.301342\pi\)
\(182\) −1935.00 −0.788086
\(183\) 0 0
\(184\) 9585.00 3.84030
\(185\) 0 0
\(186\) 0 0
\(187\) −976.000 −0.381669
\(188\) −5287.00 −2.05103
\(189\) 0 0
\(190\) 0 0
\(191\) −3080.00 −1.16681 −0.583406 0.812181i \(-0.698280\pi\)
−0.583406 + 0.812181i \(0.698280\pi\)
\(192\) 0 0
\(193\) −2588.00 −0.965224 −0.482612 0.875834i \(-0.660312\pi\)
−0.482612 + 0.875834i \(0.660312\pi\)
\(194\) −4640.00 −1.71718
\(195\) 0 0
\(196\) −4454.00 −1.62318
\(197\) −1335.00 −0.482816 −0.241408 0.970424i \(-0.577609\pi\)
−0.241408 + 0.970424i \(0.577609\pi\)
\(198\) 0 0
\(199\) −5204.00 −1.85378 −0.926889 0.375336i \(-0.877527\pi\)
−0.926889 + 0.375336i \(0.877527\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) −4980.00 −1.73461
\(203\) 2016.00 0.697022
\(204\) 0 0
\(205\) 0 0
\(206\) −2165.00 −0.732246
\(207\) 0 0
\(208\) −3827.00 −1.27574
\(209\) −472.000 −0.156215
\(210\) 0 0
\(211\) 1637.00 0.534103 0.267051 0.963682i \(-0.413951\pi\)
0.267051 + 0.963682i \(0.413951\pi\)
\(212\) −6409.00 −2.07628
\(213\) 0 0
\(214\) 8430.00 2.69282
\(215\) 0 0
\(216\) 0 0
\(217\) 324.000 0.101357
\(218\) −3280.00 −1.01904
\(219\) 0 0
\(220\) 0 0
\(221\) 5246.00 1.59676
\(222\) 0 0
\(223\) 4480.00 1.34530 0.672652 0.739959i \(-0.265155\pi\)
0.672652 + 0.739959i \(0.265155\pi\)
\(224\) 765.000 0.228186
\(225\) 0 0
\(226\) −5090.00 −1.49815
\(227\) −3736.00 −1.09237 −0.546183 0.837666i \(-0.683920\pi\)
−0.546183 + 0.837666i \(0.683920\pi\)
\(228\) 0 0
\(229\) −1380.00 −0.398223 −0.199111 0.979977i \(-0.563806\pi\)
−0.199111 + 0.979977i \(0.563806\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 10080.0 2.85252
\(233\) −2904.00 −0.816512 −0.408256 0.912867i \(-0.633863\pi\)
−0.408256 + 0.912867i \(0.633863\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −5729.00 −1.58020
\(237\) 0 0
\(238\) −5490.00 −1.49523
\(239\) 5966.00 1.61468 0.807340 0.590087i \(-0.200906\pi\)
0.807340 + 0.590087i \(0.200906\pi\)
\(240\) 0 0
\(241\) −3218.00 −0.860123 −0.430061 0.902800i \(-0.641508\pi\)
−0.430061 + 0.902800i \(0.641508\pi\)
\(242\) 6335.00 1.68277
\(243\) 0 0
\(244\) 680.000 0.178412
\(245\) 0 0
\(246\) 0 0
\(247\) 2537.00 0.653544
\(248\) 1620.00 0.414799
\(249\) 0 0
\(250\) 0 0
\(251\) −6123.00 −1.53976 −0.769881 0.638187i \(-0.779685\pi\)
−0.769881 + 0.638187i \(0.779685\pi\)
\(252\) 0 0
\(253\) −1704.00 −0.423437
\(254\) −6805.00 −1.68104
\(255\) 0 0
\(256\) −8279.00 −2.02124
\(257\) −1398.00 −0.339318 −0.169659 0.985503i \(-0.554267\pi\)
−0.169659 + 0.985503i \(0.554267\pi\)
\(258\) 0 0
\(259\) 1854.00 0.444795
\(260\) 0 0
\(261\) 0 0
\(262\) 9555.00 2.25309
\(263\) −3211.00 −0.752847 −0.376423 0.926448i \(-0.622846\pi\)
−0.376423 + 0.926448i \(0.622846\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −2655.00 −0.611987
\(267\) 0 0
\(268\) −5916.00 −1.34842
\(269\) −4018.00 −0.910713 −0.455356 0.890309i \(-0.650488\pi\)
−0.455356 + 0.890309i \(0.650488\pi\)
\(270\) 0 0
\(271\) 2314.00 0.518692 0.259346 0.965784i \(-0.416493\pi\)
0.259346 + 0.965784i \(0.416493\pi\)
\(272\) −10858.0 −2.42045
\(273\) 0 0
\(274\) −3270.00 −0.720978
\(275\) 0 0
\(276\) 0 0
\(277\) 4347.00 0.942909 0.471455 0.881890i \(-0.343729\pi\)
0.471455 + 0.881890i \(0.343729\pi\)
\(278\) 3665.00 0.790691
\(279\) 0 0
\(280\) 0 0
\(281\) −1551.00 −0.329270 −0.164635 0.986355i \(-0.552645\pi\)
−0.164635 + 0.986355i \(0.552645\pi\)
\(282\) 0 0
\(283\) −4380.00 −0.920014 −0.460007 0.887915i \(-0.652153\pi\)
−0.460007 + 0.887915i \(0.652153\pi\)
\(284\) −1054.00 −0.220223
\(285\) 0 0
\(286\) 1720.00 0.355614
\(287\) 3717.00 0.764486
\(288\) 0 0
\(289\) 9971.00 2.02951
\(290\) 0 0
\(291\) 0 0
\(292\) 20638.0 4.13612
\(293\) 5049.00 1.00671 0.503354 0.864080i \(-0.332099\pi\)
0.503354 + 0.864080i \(0.332099\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 9270.00 1.82030
\(297\) 0 0
\(298\) −5630.00 −1.09442
\(299\) 9159.00 1.77150
\(300\) 0 0
\(301\) −3528.00 −0.675583
\(302\) 12730.0 2.42559
\(303\) 0 0
\(304\) −5251.00 −0.990676
\(305\) 0 0
\(306\) 0 0
\(307\) −5428.00 −1.00910 −0.504548 0.863384i \(-0.668341\pi\)
−0.504548 + 0.863384i \(0.668341\pi\)
\(308\) −1224.00 −0.226441
\(309\) 0 0
\(310\) 0 0
\(311\) 18.0000 0.00328195 0.00164097 0.999999i \(-0.499478\pi\)
0.00164097 + 0.999999i \(0.499478\pi\)
\(312\) 0 0
\(313\) 2116.00 0.382119 0.191060 0.981578i \(-0.438808\pi\)
0.191060 + 0.981578i \(0.438808\pi\)
\(314\) 6115.00 1.09901
\(315\) 0 0
\(316\) −4998.00 −0.889745
\(317\) 4415.00 0.782243 0.391122 0.920339i \(-0.372087\pi\)
0.391122 + 0.920339i \(0.372087\pi\)
\(318\) 0 0
\(319\) −1792.00 −0.314523
\(320\) 0 0
\(321\) 0 0
\(322\) −9585.00 −1.65885
\(323\) 7198.00 1.23996
\(324\) 0 0
\(325\) 0 0
\(326\) 15880.0 2.69789
\(327\) 0 0
\(328\) 18585.0 3.12861
\(329\) 2799.00 0.469039
\(330\) 0 0
\(331\) 3480.00 0.577879 0.288940 0.957347i \(-0.406697\pi\)
0.288940 + 0.957347i \(0.406697\pi\)
\(332\) 9078.00 1.50066
\(333\) 0 0
\(334\) −660.000 −0.108125
\(335\) 0 0
\(336\) 0 0
\(337\) −6322.00 −1.02190 −0.510951 0.859610i \(-0.670707\pi\)
−0.510951 + 0.859610i \(0.670707\pi\)
\(338\) 1740.00 0.280010
\(339\) 0 0
\(340\) 0 0
\(341\) −288.000 −0.0457363
\(342\) 0 0
\(343\) 5445.00 0.857150
\(344\) −17640.0 −2.76478
\(345\) 0 0
\(346\) 4965.00 0.771445
\(347\) −10034.0 −1.55232 −0.776158 0.630539i \(-0.782834\pi\)
−0.776158 + 0.630539i \(0.782834\pi\)
\(348\) 0 0
\(349\) −2510.00 −0.384978 −0.192489 0.981299i \(-0.561656\pi\)
−0.192489 + 0.981299i \(0.561656\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −680.000 −0.102966
\(353\) 3726.00 0.561799 0.280899 0.959737i \(-0.409367\pi\)
0.280899 + 0.959737i \(0.409367\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 13770.0 2.05002
\(357\) 0 0
\(358\) −15505.0 −2.28901
\(359\) 10710.0 1.57452 0.787259 0.616622i \(-0.211499\pi\)
0.787259 + 0.616622i \(0.211499\pi\)
\(360\) 0 0
\(361\) −3378.00 −0.492492
\(362\) −14230.0 −2.06606
\(363\) 0 0
\(364\) 6579.00 0.947344
\(365\) 0 0
\(366\) 0 0
\(367\) 2160.00 0.307224 0.153612 0.988131i \(-0.450909\pi\)
0.153612 + 0.988131i \(0.450909\pi\)
\(368\) −18957.0 −2.68533
\(369\) 0 0
\(370\) 0 0
\(371\) 3393.00 0.474813
\(372\) 0 0
\(373\) −3394.00 −0.471138 −0.235569 0.971858i \(-0.575695\pi\)
−0.235569 + 0.971858i \(0.575695\pi\)
\(374\) 4880.00 0.674703
\(375\) 0 0
\(376\) 13995.0 1.91951
\(377\) 9632.00 1.31584
\(378\) 0 0
\(379\) 9031.00 1.22399 0.611994 0.790863i \(-0.290368\pi\)
0.611994 + 0.790863i \(0.290368\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 15400.0 2.06265
\(383\) −10305.0 −1.37483 −0.687416 0.726264i \(-0.741255\pi\)
−0.687416 + 0.726264i \(0.741255\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 12940.0 1.70629
\(387\) 0 0
\(388\) 15776.0 2.06419
\(389\) 3480.00 0.453581 0.226790 0.973944i \(-0.427177\pi\)
0.226790 + 0.973944i \(0.427177\pi\)
\(390\) 0 0
\(391\) 25986.0 3.36104
\(392\) 11790.0 1.51909
\(393\) 0 0
\(394\) 6675.00 0.853507
\(395\) 0 0
\(396\) 0 0
\(397\) −3706.00 −0.468511 −0.234255 0.972175i \(-0.575265\pi\)
−0.234255 + 0.972175i \(0.575265\pi\)
\(398\) 26020.0 3.27705
\(399\) 0 0
\(400\) 0 0
\(401\) −2679.00 −0.333623 −0.166812 0.985989i \(-0.553347\pi\)
−0.166812 + 0.985989i \(0.553347\pi\)
\(402\) 0 0
\(403\) 1548.00 0.191343
\(404\) 16932.0 2.08514
\(405\) 0 0
\(406\) −10080.0 −1.23217
\(407\) −1648.00 −0.200708
\(408\) 0 0
\(409\) −12499.0 −1.51109 −0.755545 0.655097i \(-0.772628\pi\)
−0.755545 + 0.655097i \(0.772628\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 7361.00 0.880220
\(413\) 3033.00 0.361366
\(414\) 0 0
\(415\) 0 0
\(416\) 3655.00 0.430772
\(417\) 0 0
\(418\) 2360.00 0.276152
\(419\) 928.000 0.108200 0.0541000 0.998536i \(-0.482771\pi\)
0.0541000 + 0.998536i \(0.482771\pi\)
\(420\) 0 0
\(421\) 7570.00 0.876340 0.438170 0.898892i \(-0.355627\pi\)
0.438170 + 0.898892i \(0.355627\pi\)
\(422\) −8185.00 −0.944170
\(423\) 0 0
\(424\) 16965.0 1.94314
\(425\) 0 0
\(426\) 0 0
\(427\) −360.000 −0.0408000
\(428\) −28662.0 −3.23699
\(429\) 0 0
\(430\) 0 0
\(431\) 2460.00 0.274928 0.137464 0.990507i \(-0.456105\pi\)
0.137464 + 0.990507i \(0.456105\pi\)
\(432\) 0 0
\(433\) −1648.00 −0.182905 −0.0914525 0.995809i \(-0.529151\pi\)
−0.0914525 + 0.995809i \(0.529151\pi\)
\(434\) −1620.00 −0.179176
\(435\) 0 0
\(436\) 11152.0 1.22496
\(437\) 12567.0 1.37565
\(438\) 0 0
\(439\) 15826.0 1.72058 0.860289 0.509807i \(-0.170283\pi\)
0.860289 + 0.509807i \(0.170283\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −26230.0 −2.82270
\(443\) 12774.0 1.37000 0.685001 0.728542i \(-0.259802\pi\)
0.685001 + 0.728542i \(0.259802\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −22400.0 −2.37819
\(447\) 0 0
\(448\) 2583.00 0.272400
\(449\) −8875.00 −0.932822 −0.466411 0.884568i \(-0.654453\pi\)
−0.466411 + 0.884568i \(0.654453\pi\)
\(450\) 0 0
\(451\) −3304.00 −0.344965
\(452\) 17306.0 1.80090
\(453\) 0 0
\(454\) 18680.0 1.93105
\(455\) 0 0
\(456\) 0 0
\(457\) −11524.0 −1.17958 −0.589792 0.807555i \(-0.700790\pi\)
−0.589792 + 0.807555i \(0.700790\pi\)
\(458\) 6900.00 0.703965
\(459\) 0 0
\(460\) 0 0
\(461\) 8544.00 0.863197 0.431598 0.902066i \(-0.357950\pi\)
0.431598 + 0.902066i \(0.357950\pi\)
\(462\) 0 0
\(463\) −2523.00 −0.253248 −0.126624 0.991951i \(-0.540414\pi\)
−0.126624 + 0.991951i \(0.540414\pi\)
\(464\) −19936.0 −1.99462
\(465\) 0 0
\(466\) 14520.0 1.44340
\(467\) −2902.00 −0.287556 −0.143778 0.989610i \(-0.545925\pi\)
−0.143778 + 0.989610i \(0.545925\pi\)
\(468\) 0 0
\(469\) 3132.00 0.308363
\(470\) 0 0
\(471\) 0 0
\(472\) 15165.0 1.47887
\(473\) 3136.00 0.304849
\(474\) 0 0
\(475\) 0 0
\(476\) 18666.0 1.79738
\(477\) 0 0
\(478\) −29830.0 −2.85438
\(479\) −4362.00 −0.416085 −0.208043 0.978120i \(-0.566709\pi\)
−0.208043 + 0.978120i \(0.566709\pi\)
\(480\) 0 0
\(481\) 8858.00 0.839688
\(482\) 16090.0 1.52050
\(483\) 0 0
\(484\) −21539.0 −2.02282
\(485\) 0 0
\(486\) 0 0
\(487\) −5723.00 −0.532513 −0.266257 0.963902i \(-0.585787\pi\)
−0.266257 + 0.963902i \(0.585787\pi\)
\(488\) −1800.00 −0.166972
\(489\) 0 0
\(490\) 0 0
\(491\) 13339.0 1.22603 0.613015 0.790071i \(-0.289957\pi\)
0.613015 + 0.790071i \(0.289957\pi\)
\(492\) 0 0
\(493\) 27328.0 2.49653
\(494\) −12685.0 −1.15531
\(495\) 0 0
\(496\) −3204.00 −0.290048
\(497\) 558.000 0.0503616
\(498\) 0 0
\(499\) −19637.0 −1.76167 −0.880835 0.473424i \(-0.843018\pi\)
−0.880835 + 0.473424i \(0.843018\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 30615.0 2.72194
\(503\) −5416.00 −0.480094 −0.240047 0.970761i \(-0.577163\pi\)
−0.240047 + 0.970761i \(0.577163\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 8520.00 0.748538
\(507\) 0 0
\(508\) 23137.0 2.02074
\(509\) 6110.00 0.532065 0.266032 0.963964i \(-0.414287\pi\)
0.266032 + 0.963964i \(0.414287\pi\)
\(510\) 0 0
\(511\) −10926.0 −0.945867
\(512\) 24475.0 2.11260
\(513\) 0 0
\(514\) 6990.00 0.599836
\(515\) 0 0
\(516\) 0 0
\(517\) −2488.00 −0.211648
\(518\) −9270.00 −0.786294
\(519\) 0 0
\(520\) 0 0
\(521\) −20375.0 −1.71333 −0.856665 0.515873i \(-0.827468\pi\)
−0.856665 + 0.515873i \(0.827468\pi\)
\(522\) 0 0
\(523\) 19010.0 1.58939 0.794693 0.607011i \(-0.207632\pi\)
0.794693 + 0.607011i \(0.207632\pi\)
\(524\) −32487.0 −2.70840
\(525\) 0 0
\(526\) 16055.0 1.33086
\(527\) 4392.00 0.363033
\(528\) 0 0
\(529\) 33202.0 2.72886
\(530\) 0 0
\(531\) 0 0
\(532\) 9027.00 0.735658
\(533\) 17759.0 1.44320
\(534\) 0 0
\(535\) 0 0
\(536\) 15660.0 1.26196
\(537\) 0 0
\(538\) 20090.0 1.60993
\(539\) −2096.00 −0.167497
\(540\) 0 0
\(541\) 3288.00 0.261298 0.130649 0.991429i \(-0.458294\pi\)
0.130649 + 0.991429i \(0.458294\pi\)
\(542\) −11570.0 −0.916926
\(543\) 0 0
\(544\) 10370.0 0.817298
\(545\) 0 0
\(546\) 0 0
\(547\) −3256.00 −0.254509 −0.127255 0.991870i \(-0.540617\pi\)
−0.127255 + 0.991870i \(0.540617\pi\)
\(548\) 11118.0 0.866674
\(549\) 0 0
\(550\) 0 0
\(551\) 13216.0 1.02182
\(552\) 0 0
\(553\) 2646.00 0.203471
\(554\) −21735.0 −1.66684
\(555\) 0 0
\(556\) −12461.0 −0.950475
\(557\) 213.000 0.0162031 0.00810153 0.999967i \(-0.497421\pi\)
0.00810153 + 0.999967i \(0.497421\pi\)
\(558\) 0 0
\(559\) −16856.0 −1.27537
\(560\) 0 0
\(561\) 0 0
\(562\) 7755.00 0.582073
\(563\) 17388.0 1.30163 0.650814 0.759237i \(-0.274428\pi\)
0.650814 + 0.759237i \(0.274428\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 21900.0 1.62637
\(567\) 0 0
\(568\) 2790.00 0.206102
\(569\) 4353.00 0.320716 0.160358 0.987059i \(-0.448735\pi\)
0.160358 + 0.987059i \(0.448735\pi\)
\(570\) 0 0
\(571\) −1924.00 −0.141010 −0.0705052 0.997511i \(-0.522461\pi\)
−0.0705052 + 0.997511i \(0.522461\pi\)
\(572\) −5848.00 −0.427478
\(573\) 0 0
\(574\) −18585.0 −1.35143
\(575\) 0 0
\(576\) 0 0
\(577\) −16832.0 −1.21443 −0.607214 0.794538i \(-0.707713\pi\)
−0.607214 + 0.794538i \(0.707713\pi\)
\(578\) −49855.0 −3.58771
\(579\) 0 0
\(580\) 0 0
\(581\) −4806.00 −0.343178
\(582\) 0 0
\(583\) −3016.00 −0.214254
\(584\) −54630.0 −3.87090
\(585\) 0 0
\(586\) −25245.0 −1.77963
\(587\) −2106.00 −0.148082 −0.0740408 0.997255i \(-0.523590\pi\)
−0.0740408 + 0.997255i \(0.523590\pi\)
\(588\) 0 0
\(589\) 2124.00 0.148587
\(590\) 0 0
\(591\) 0 0
\(592\) −18334.0 −1.27284
\(593\) 4694.00 0.325058 0.162529 0.986704i \(-0.448035\pi\)
0.162529 + 0.986704i \(0.448035\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 19142.0 1.31558
\(597\) 0 0
\(598\) −45795.0 −3.13160
\(599\) −13226.0 −0.902170 −0.451085 0.892481i \(-0.648963\pi\)
−0.451085 + 0.892481i \(0.648963\pi\)
\(600\) 0 0
\(601\) −13291.0 −0.902082 −0.451041 0.892503i \(-0.648947\pi\)
−0.451041 + 0.892503i \(0.648947\pi\)
\(602\) 17640.0 1.19427
\(603\) 0 0
\(604\) −43282.0 −2.91576
\(605\) 0 0
\(606\) 0 0
\(607\) −6040.00 −0.403881 −0.201941 0.979398i \(-0.564725\pi\)
−0.201941 + 0.979398i \(0.564725\pi\)
\(608\) 5015.00 0.334515
\(609\) 0 0
\(610\) 0 0
\(611\) 13373.0 0.885456
\(612\) 0 0
\(613\) 23.0000 0.00151543 0.000757717 1.00000i \(-0.499759\pi\)
0.000757717 1.00000i \(0.499759\pi\)
\(614\) 27140.0 1.78385
\(615\) 0 0
\(616\) 3240.00 0.211921
\(617\) −3018.00 −0.196921 −0.0984604 0.995141i \(-0.531392\pi\)
−0.0984604 + 0.995141i \(0.531392\pi\)
\(618\) 0 0
\(619\) −9439.00 −0.612901 −0.306450 0.951887i \(-0.599141\pi\)
−0.306450 + 0.951887i \(0.599141\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) −90.0000 −0.00580172
\(623\) −7290.00 −0.468808
\(624\) 0 0
\(625\) 0 0
\(626\) −10580.0 −0.675498
\(627\) 0 0
\(628\) −20791.0 −1.32110
\(629\) 25132.0 1.59313
\(630\) 0 0
\(631\) −9800.00 −0.618275 −0.309138 0.951017i \(-0.600040\pi\)
−0.309138 + 0.951017i \(0.600040\pi\)
\(632\) 13230.0 0.832692
\(633\) 0 0
\(634\) −22075.0 −1.38282
\(635\) 0 0
\(636\) 0 0
\(637\) 11266.0 0.700746
\(638\) 8960.00 0.556003
\(639\) 0 0
\(640\) 0 0
\(641\) −23442.0 −1.44447 −0.722233 0.691649i \(-0.756884\pi\)
−0.722233 + 0.691649i \(0.756884\pi\)
\(642\) 0 0
\(643\) −31308.0 −1.92017 −0.960083 0.279715i \(-0.909760\pi\)
−0.960083 + 0.279715i \(0.909760\pi\)
\(644\) 32589.0 1.99408
\(645\) 0 0
\(646\) −35990.0 −2.19196
\(647\) 712.000 0.0432637 0.0216318 0.999766i \(-0.493114\pi\)
0.0216318 + 0.999766i \(0.493114\pi\)
\(648\) 0 0
\(649\) −2696.00 −0.163062
\(650\) 0 0
\(651\) 0 0
\(652\) −53992.0 −3.24308
\(653\) 31478.0 1.88642 0.943208 0.332203i \(-0.107792\pi\)
0.943208 + 0.332203i \(0.107792\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −36757.0 −2.18768
\(657\) 0 0
\(658\) −13995.0 −0.829152
\(659\) 16121.0 0.952936 0.476468 0.879192i \(-0.341917\pi\)
0.476468 + 0.879192i \(0.341917\pi\)
\(660\) 0 0
\(661\) −19160.0 −1.12744 −0.563720 0.825966i \(-0.690630\pi\)
−0.563720 + 0.825966i \(0.690630\pi\)
\(662\) −17400.0 −1.02156
\(663\) 0 0
\(664\) −24030.0 −1.40444
\(665\) 0 0
\(666\) 0 0
\(667\) 47712.0 2.76974
\(668\) 2244.00 0.129975
\(669\) 0 0
\(670\) 0 0
\(671\) 320.000 0.0184105
\(672\) 0 0
\(673\) 13422.0 0.768767 0.384383 0.923174i \(-0.374414\pi\)
0.384383 + 0.923174i \(0.374414\pi\)
\(674\) 31610.0 1.80649
\(675\) 0 0
\(676\) −5916.00 −0.336595
\(677\) 25905.0 1.47062 0.735310 0.677731i \(-0.237036\pi\)
0.735310 + 0.677731i \(0.237036\pi\)
\(678\) 0 0
\(679\) −8352.00 −0.472048
\(680\) 0 0
\(681\) 0 0
\(682\) 1440.00 0.0808511
\(683\) −9246.00 −0.517992 −0.258996 0.965878i \(-0.583392\pi\)
−0.258996 + 0.965878i \(0.583392\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −27225.0 −1.51524
\(687\) 0 0
\(688\) 34888.0 1.93327
\(689\) 16211.0 0.896357
\(690\) 0 0
\(691\) 25039.0 1.37848 0.689239 0.724534i \(-0.257945\pi\)
0.689239 + 0.724534i \(0.257945\pi\)
\(692\) −16881.0 −0.927340
\(693\) 0 0
\(694\) 50170.0 2.74413
\(695\) 0 0
\(696\) 0 0
\(697\) 50386.0 2.73817
\(698\) 12550.0 0.680551
\(699\) 0 0
\(700\) 0 0
\(701\) −32930.0 −1.77425 −0.887125 0.461530i \(-0.847301\pi\)
−0.887125 + 0.461530i \(0.847301\pi\)
\(702\) 0 0
\(703\) 12154.0 0.652058
\(704\) −2296.00 −0.122917
\(705\) 0 0
\(706\) −18630.0 −0.993129
\(707\) −8964.00 −0.476840
\(708\) 0 0
\(709\) 1882.00 0.0996897 0.0498448 0.998757i \(-0.484127\pi\)
0.0498448 + 0.998757i \(0.484127\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −36450.0 −1.91857
\(713\) 7668.00 0.402761
\(714\) 0 0
\(715\) 0 0
\(716\) 52717.0 2.75157
\(717\) 0 0
\(718\) −53550.0 −2.78338
\(719\) 1962.00 0.101767 0.0508833 0.998705i \(-0.483796\pi\)
0.0508833 + 0.998705i \(0.483796\pi\)
\(720\) 0 0
\(721\) −3897.00 −0.201292
\(722\) 16890.0 0.870610
\(723\) 0 0
\(724\) 48382.0 2.48357
\(725\) 0 0
\(726\) 0 0
\(727\) 13741.0 0.700998 0.350499 0.936563i \(-0.386012\pi\)
0.350499 + 0.936563i \(0.386012\pi\)
\(728\) −17415.0 −0.886597
\(729\) 0 0
\(730\) 0 0
\(731\) −47824.0 −2.41975
\(732\) 0 0
\(733\) 32458.0 1.63556 0.817779 0.575533i \(-0.195205\pi\)
0.817779 + 0.575533i \(0.195205\pi\)
\(734\) −10800.0 −0.543100
\(735\) 0 0
\(736\) 18105.0 0.906738
\(737\) −2784.00 −0.139145
\(738\) 0 0
\(739\) 19612.0 0.976237 0.488118 0.872777i \(-0.337683\pi\)
0.488118 + 0.872777i \(0.337683\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −16965.0 −0.839359
\(743\) 36736.0 1.81388 0.906940 0.421259i \(-0.138412\pi\)
0.906940 + 0.421259i \(0.138412\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 16970.0 0.832863
\(747\) 0 0
\(748\) −16592.0 −0.811048
\(749\) 15174.0 0.740248
\(750\) 0 0
\(751\) −3746.00 −0.182015 −0.0910076 0.995850i \(-0.529009\pi\)
−0.0910076 + 0.995850i \(0.529009\pi\)
\(752\) −27679.0 −1.34222
\(753\) 0 0
\(754\) −48160.0 −2.32611
\(755\) 0 0
\(756\) 0 0
\(757\) −5725.00 −0.274873 −0.137436 0.990511i \(-0.543886\pi\)
−0.137436 + 0.990511i \(0.543886\pi\)
\(758\) −45155.0 −2.16372
\(759\) 0 0
\(760\) 0 0
\(761\) −37323.0 −1.77787 −0.888934 0.458035i \(-0.848553\pi\)
−0.888934 + 0.458035i \(0.848553\pi\)
\(762\) 0 0
\(763\) −5904.00 −0.280130
\(764\) −52360.0 −2.47947
\(765\) 0 0
\(766\) 51525.0 2.43038
\(767\) 14491.0 0.682190
\(768\) 0 0
\(769\) −24586.0 −1.15292 −0.576459 0.817126i \(-0.695566\pi\)
−0.576459 + 0.817126i \(0.695566\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −43996.0 −2.05110
\(773\) −3078.00 −0.143219 −0.0716093 0.997433i \(-0.522813\pi\)
−0.0716093 + 0.997433i \(0.522813\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −41760.0 −1.93183
\(777\) 0 0
\(778\) −17400.0 −0.801825
\(779\) 24367.0 1.12072
\(780\) 0 0
\(781\) −496.000 −0.0227251
\(782\) −129930. −5.94154
\(783\) 0 0
\(784\) −23318.0 −1.06223
\(785\) 0 0
\(786\) 0 0
\(787\) −41038.0 −1.85876 −0.929382 0.369120i \(-0.879659\pi\)
−0.929382 + 0.369120i \(0.879659\pi\)
\(788\) −22695.0 −1.02598
\(789\) 0 0
\(790\) 0 0
\(791\) −9162.00 −0.411837
\(792\) 0 0
\(793\) −1720.00 −0.0770227
\(794\) 18530.0 0.828218
\(795\) 0 0
\(796\) −88468.0 −3.93928
\(797\) 9362.00 0.416084 0.208042 0.978120i \(-0.433291\pi\)
0.208042 + 0.978120i \(0.433291\pi\)
\(798\) 0 0
\(799\) 37942.0 1.67996
\(800\) 0 0
\(801\) 0 0
\(802\) 13395.0 0.589768
\(803\) 9712.00 0.426811
\(804\) 0 0
\(805\) 0 0
\(806\) −7740.00 −0.338250
\(807\) 0 0
\(808\) −44820.0 −1.95144
\(809\) 45115.0 1.96064 0.980321 0.197411i \(-0.0632535\pi\)
0.980321 + 0.197411i \(0.0632535\pi\)
\(810\) 0 0
\(811\) −13512.0 −0.585044 −0.292522 0.956259i \(-0.594494\pi\)
−0.292522 + 0.956259i \(0.594494\pi\)
\(812\) 34272.0 1.48117
\(813\) 0 0
\(814\) 8240.00 0.354806
\(815\) 0 0
\(816\) 0 0
\(817\) −23128.0 −0.990387
\(818\) 62495.0 2.67125
\(819\) 0 0
\(820\) 0 0
\(821\) 4530.00 0.192568 0.0962839 0.995354i \(-0.469304\pi\)
0.0962839 + 0.995354i \(0.469304\pi\)
\(822\) 0 0
\(823\) −30884.0 −1.30808 −0.654039 0.756461i \(-0.726927\pi\)
−0.654039 + 0.756461i \(0.726927\pi\)
\(824\) −19485.0 −0.823777
\(825\) 0 0
\(826\) −15165.0 −0.638811
\(827\) 12088.0 0.508272 0.254136 0.967168i \(-0.418209\pi\)
0.254136 + 0.967168i \(0.418209\pi\)
\(828\) 0 0
\(829\) −14112.0 −0.591230 −0.295615 0.955307i \(-0.595525\pi\)
−0.295615 + 0.955307i \(0.595525\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 12341.0 0.514239
\(833\) 31964.0 1.32952
\(834\) 0 0
\(835\) 0 0
\(836\) −8024.00 −0.331957
\(837\) 0 0
\(838\) −4640.00 −0.191272
\(839\) −3860.00 −0.158834 −0.0794172 0.996841i \(-0.525306\pi\)
−0.0794172 + 0.996841i \(0.525306\pi\)
\(840\) 0 0
\(841\) 25787.0 1.05732
\(842\) −37850.0 −1.54917
\(843\) 0 0
\(844\) 27829.0 1.13497
\(845\) 0 0
\(846\) 0 0
\(847\) 11403.0 0.462588
\(848\) −33553.0 −1.35874
\(849\) 0 0
\(850\) 0 0
\(851\) 43878.0 1.76747
\(852\) 0 0
\(853\) −2862.00 −0.114880 −0.0574402 0.998349i \(-0.518294\pi\)
−0.0574402 + 0.998349i \(0.518294\pi\)
\(854\) 1800.00 0.0721250
\(855\) 0 0
\(856\) 75870.0 3.02942
\(857\) −32534.0 −1.29678 −0.648390 0.761308i \(-0.724557\pi\)
−0.648390 + 0.761308i \(0.724557\pi\)
\(858\) 0 0
\(859\) −500.000 −0.0198600 −0.00993002 0.999951i \(-0.503161\pi\)
−0.00993002 + 0.999951i \(0.503161\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −12300.0 −0.486009
\(863\) −36595.0 −1.44346 −0.721731 0.692173i \(-0.756653\pi\)
−0.721731 + 0.692173i \(0.756653\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 8240.00 0.323333
\(867\) 0 0
\(868\) 5508.00 0.215384
\(869\) −2352.00 −0.0918137
\(870\) 0 0
\(871\) 14964.0 0.582131
\(872\) −29520.0 −1.14641
\(873\) 0 0
\(874\) −62835.0 −2.43184
\(875\) 0 0
\(876\) 0 0
\(877\) −40143.0 −1.54565 −0.772824 0.634621i \(-0.781156\pi\)
−0.772824 + 0.634621i \(0.781156\pi\)
\(878\) −79130.0 −3.04158
\(879\) 0 0
\(880\) 0 0
\(881\) −16414.0 −0.627698 −0.313849 0.949473i \(-0.601619\pi\)
−0.313849 + 0.949473i \(0.601619\pi\)
\(882\) 0 0
\(883\) 33478.0 1.27591 0.637953 0.770076i \(-0.279782\pi\)
0.637953 + 0.770076i \(0.279782\pi\)
\(884\) 89182.0 3.39312
\(885\) 0 0
\(886\) −63870.0 −2.42184
\(887\) −3633.00 −0.137524 −0.0687622 0.997633i \(-0.521905\pi\)
−0.0687622 + 0.997633i \(0.521905\pi\)
\(888\) 0 0
\(889\) −12249.0 −0.462113
\(890\) 0 0
\(891\) 0 0
\(892\) 76160.0 2.85877
\(893\) 18349.0 0.687599
\(894\) 0 0
\(895\) 0 0
\(896\) −19035.0 −0.709726
\(897\) 0 0
\(898\) 44375.0 1.64901
\(899\) 8064.00 0.299165
\(900\) 0 0
\(901\) 45994.0 1.70065
\(902\) 16520.0 0.609818
\(903\) 0 0
\(904\) −45810.0 −1.68542
\(905\) 0 0
\(906\) 0 0
\(907\) −20466.0 −0.749242 −0.374621 0.927178i \(-0.622227\pi\)
−0.374621 + 0.927178i \(0.622227\pi\)
\(908\) −63512.0 −2.32128
\(909\) 0 0
\(910\) 0 0
\(911\) 15074.0 0.548215 0.274108 0.961699i \(-0.411618\pi\)
0.274108 + 0.961699i \(0.411618\pi\)
\(912\) 0 0
\(913\) 4272.00 0.154855
\(914\) 57620.0 2.08523
\(915\) 0 0
\(916\) −23460.0 −0.846223
\(917\) 17199.0 0.619369
\(918\) 0 0
\(919\) −36848.0 −1.32264 −0.661318 0.750105i \(-0.730003\pi\)
−0.661318 + 0.750105i \(0.730003\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −42720.0 −1.52593
\(923\) 2666.00 0.0950731
\(924\) 0 0
\(925\) 0 0
\(926\) 12615.0 0.447683
\(927\) 0 0
\(928\) 19040.0 0.673511
\(929\) −35174.0 −1.24222 −0.621110 0.783724i \(-0.713318\pi\)
−0.621110 + 0.783724i \(0.713318\pi\)
\(930\) 0 0
\(931\) 15458.0 0.544163
\(932\) −49368.0 −1.73509
\(933\) 0 0
\(934\) 14510.0 0.508332
\(935\) 0 0
\(936\) 0 0
\(937\) 10092.0 0.351858 0.175929 0.984403i \(-0.443707\pi\)
0.175929 + 0.984403i \(0.443707\pi\)
\(938\) −15660.0 −0.545114
\(939\) 0 0
\(940\) 0 0
\(941\) 12910.0 0.447241 0.223621 0.974676i \(-0.428212\pi\)
0.223621 + 0.974676i \(0.428212\pi\)
\(942\) 0 0
\(943\) 87969.0 3.03782
\(944\) −29993.0 −1.03410
\(945\) 0 0
\(946\) −15680.0 −0.538901
\(947\) −48060.0 −1.64914 −0.824572 0.565756i \(-0.808584\pi\)
−0.824572 + 0.565756i \(0.808584\pi\)
\(948\) 0 0
\(949\) −52202.0 −1.78561
\(950\) 0 0
\(951\) 0 0
\(952\) −49410.0 −1.68213
\(953\) 6316.00 0.214686 0.107343 0.994222i \(-0.465766\pi\)
0.107343 + 0.994222i \(0.465766\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 101422. 3.43119
\(957\) 0 0
\(958\) 21810.0 0.735542
\(959\) −5886.00 −0.198195
\(960\) 0 0
\(961\) −28495.0 −0.956497
\(962\) −44290.0 −1.48437
\(963\) 0 0
\(964\) −54706.0 −1.82776
\(965\) 0 0
\(966\) 0 0
\(967\) −7760.00 −0.258061 −0.129030 0.991641i \(-0.541186\pi\)
−0.129030 + 0.991641i \(0.541186\pi\)
\(968\) 57015.0 1.89311
\(969\) 0 0
\(970\) 0 0
\(971\) −22437.0 −0.741542 −0.370771 0.928724i \(-0.620907\pi\)
−0.370771 + 0.928724i \(0.620907\pi\)
\(972\) 0 0
\(973\) 6597.00 0.217359
\(974\) 28615.0 0.941359
\(975\) 0 0
\(976\) 3560.00 0.116755
\(977\) −3506.00 −0.114807 −0.0574037 0.998351i \(-0.518282\pi\)
−0.0574037 + 0.998351i \(0.518282\pi\)
\(978\) 0 0
\(979\) 6480.00 0.211544
\(980\) 0 0
\(981\) 0 0
\(982\) −66695.0 −2.16734
\(983\) −30648.0 −0.994425 −0.497212 0.867629i \(-0.665643\pi\)
−0.497212 + 0.867629i \(0.665643\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) −136640. −4.41329
\(987\) 0 0
\(988\) 43129.0 1.38878
\(989\) −83496.0 −2.68455
\(990\) 0 0
\(991\) 19232.0 0.616473 0.308236 0.951310i \(-0.400261\pi\)
0.308236 + 0.951310i \(0.400261\pi\)
\(992\) 3060.00 0.0979386
\(993\) 0 0
\(994\) −2790.00 −0.0890276
\(995\) 0 0
\(996\) 0 0
\(997\) −38415.0 −1.22028 −0.610138 0.792295i \(-0.708886\pi\)
−0.610138 + 0.792295i \(0.708886\pi\)
\(998\) 98185.0 3.11422
\(999\) 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 2025.4.a.a.1.1 1
3.2 odd 2 2025.4.a.f.1.1 1
5.4 even 2 405.4.a.b.1.1 yes 1
15.14 odd 2 405.4.a.a.1.1 1
45.4 even 6 405.4.e.a.136.1 2
45.14 odd 6 405.4.e.m.136.1 2
45.29 odd 6 405.4.e.m.271.1 2
45.34 even 6 405.4.e.a.271.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
405.4.a.a.1.1 1 15.14 odd 2
405.4.a.b.1.1 yes 1 5.4 even 2
405.4.e.a.136.1 2 45.4 even 6
405.4.e.a.271.1 2 45.34 even 6
405.4.e.m.136.1 2 45.14 odd 6
405.4.e.m.271.1 2 45.29 odd 6
2025.4.a.a.1.1 1 1.1 even 1 trivial
2025.4.a.f.1.1 1 3.2 odd 2