Properties

Label 225.6.a.g.1.1
Level $225$
Weight $6$
Character 225.1
Self dual yes
Analytic conductor $36.086$
Analytic rank $0$
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] = [225,6,Mod(1,225)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(225, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("225.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 225 = 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 225.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: \(36.0863594579\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 75)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 225.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+4.00000 q^{2} -16.0000 q^{4} +225.000 q^{7} -192.000 q^{8} +O(q^{10})\) \(q+4.00000 q^{2} -16.0000 q^{4} +225.000 q^{7} -192.000 q^{8} +434.000 q^{11} -613.000 q^{13} +900.000 q^{14} -256.000 q^{16} -878.000 q^{17} -731.000 q^{19} +1736.00 q^{22} +2850.00 q^{23} -2452.00 q^{26} -3600.00 q^{28} +7582.00 q^{29} +2175.00 q^{31} +5120.00 q^{32} -3512.00 q^{34} +9310.00 q^{37} -2924.00 q^{38} +12040.0 q^{41} +1121.00 q^{43} -6944.00 q^{44} +11400.0 q^{46} +29878.0 q^{47} +33818.0 q^{49} +9808.00 q^{52} +5740.00 q^{53} -43200.0 q^{56} +30328.0 q^{58} +5174.00 q^{59} -38717.0 q^{61} +8700.00 q^{62} +28672.0 q^{64} -31707.0 q^{67} +14048.0 q^{68} -64472.0 q^{71} +19790.0 q^{73} +37240.0 q^{74} +11696.0 q^{76} +97650.0 q^{77} -105000. q^{79} +48160.0 q^{82} -3318.00 q^{83} +4484.00 q^{86} -83328.0 q^{88} +65376.0 q^{89} -137925. q^{91} -45600.0 q^{92} +119512. q^{94} +89143.0 q^{97} +135272. q^{98} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 4.00000 0.707107 0.353553 0.935414i \(-0.384973\pi\)
0.353553 + 0.935414i \(0.384973\pi\)
\(3\) 0 0
\(4\) −16.0000 −0.500000
\(5\) 0 0
\(6\) 0 0
\(7\) 225.000 1.73555 0.867776 0.496956i \(-0.165549\pi\)
0.867776 + 0.496956i \(0.165549\pi\)
\(8\) −192.000 −1.06066
\(9\) 0 0
\(10\) 0 0
\(11\) 434.000 1.08145 0.540727 0.841198i \(-0.318149\pi\)
0.540727 + 0.841198i \(0.318149\pi\)
\(12\) 0 0
\(13\) −613.000 −1.00601 −0.503005 0.864284i \(-0.667772\pi\)
−0.503005 + 0.864284i \(0.667772\pi\)
\(14\) 900.000 1.22722
\(15\) 0 0
\(16\) −256.000 −0.250000
\(17\) −878.000 −0.736838 −0.368419 0.929660i \(-0.620101\pi\)
−0.368419 + 0.929660i \(0.620101\pi\)
\(18\) 0 0
\(19\) −731.000 −0.464551 −0.232275 0.972650i \(-0.574617\pi\)
−0.232275 + 0.972650i \(0.574617\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 1736.00 0.764703
\(23\) 2850.00 1.12338 0.561688 0.827349i \(-0.310152\pi\)
0.561688 + 0.827349i \(0.310152\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −2452.00 −0.711356
\(27\) 0 0
\(28\) −3600.00 −0.867776
\(29\) 7582.00 1.67413 0.837064 0.547105i \(-0.184270\pi\)
0.837064 + 0.547105i \(0.184270\pi\)
\(30\) 0 0
\(31\) 2175.00 0.406495 0.203247 0.979127i \(-0.434850\pi\)
0.203247 + 0.979127i \(0.434850\pi\)
\(32\) 5120.00 0.883883
\(33\) 0 0
\(34\) −3512.00 −0.521023
\(35\) 0 0
\(36\) 0 0
\(37\) 9310.00 1.11801 0.559005 0.829165i \(-0.311183\pi\)
0.559005 + 0.829165i \(0.311183\pi\)
\(38\) −2924.00 −0.328487
\(39\) 0 0
\(40\) 0 0
\(41\) 12040.0 1.11858 0.559290 0.828972i \(-0.311074\pi\)
0.559290 + 0.828972i \(0.311074\pi\)
\(42\) 0 0
\(43\) 1121.00 0.0924559 0.0462279 0.998931i \(-0.485280\pi\)
0.0462279 + 0.998931i \(0.485280\pi\)
\(44\) −6944.00 −0.540727
\(45\) 0 0
\(46\) 11400.0 0.794347
\(47\) 29878.0 1.97291 0.986454 0.164038i \(-0.0524518\pi\)
0.986454 + 0.164038i \(0.0524518\pi\)
\(48\) 0 0
\(49\) 33818.0 2.01214
\(50\) 0 0
\(51\) 0 0
\(52\) 9808.00 0.503005
\(53\) 5740.00 0.280687 0.140343 0.990103i \(-0.455179\pi\)
0.140343 + 0.990103i \(0.455179\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −43200.0 −1.84083
\(57\) 0 0
\(58\) 30328.0 1.18379
\(59\) 5174.00 0.193507 0.0967534 0.995308i \(-0.469154\pi\)
0.0967534 + 0.995308i \(0.469154\pi\)
\(60\) 0 0
\(61\) −38717.0 −1.33222 −0.666112 0.745852i \(-0.732043\pi\)
−0.666112 + 0.745852i \(0.732043\pi\)
\(62\) 8700.00 0.287435
\(63\) 0 0
\(64\) 28672.0 0.875000
\(65\) 0 0
\(66\) 0 0
\(67\) −31707.0 −0.862915 −0.431458 0.902133i \(-0.642001\pi\)
−0.431458 + 0.902133i \(0.642001\pi\)
\(68\) 14048.0 0.368419
\(69\) 0 0
\(70\) 0 0
\(71\) −64472.0 −1.51784 −0.758919 0.651185i \(-0.774272\pi\)
−0.758919 + 0.651185i \(0.774272\pi\)
\(72\) 0 0
\(73\) 19790.0 0.434649 0.217324 0.976099i \(-0.430267\pi\)
0.217324 + 0.976099i \(0.430267\pi\)
\(74\) 37240.0 0.790552
\(75\) 0 0
\(76\) 11696.0 0.232275
\(77\) 97650.0 1.87692
\(78\) 0 0
\(79\) −105000. −1.89287 −0.946437 0.322889i \(-0.895346\pi\)
−0.946437 + 0.322889i \(0.895346\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 48160.0 0.790955
\(83\) −3318.00 −0.0528666 −0.0264333 0.999651i \(-0.508415\pi\)
−0.0264333 + 0.999651i \(0.508415\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 4484.00 0.0653762
\(87\) 0 0
\(88\) −83328.0 −1.14706
\(89\) 65376.0 0.874870 0.437435 0.899250i \(-0.355887\pi\)
0.437435 + 0.899250i \(0.355887\pi\)
\(90\) 0 0
\(91\) −137925. −1.74598
\(92\) −45600.0 −0.561688
\(93\) 0 0
\(94\) 119512. 1.39506
\(95\) 0 0
\(96\) 0 0
\(97\) 89143.0 0.961962 0.480981 0.876731i \(-0.340281\pi\)
0.480981 + 0.876731i \(0.340281\pi\)
\(98\) 135272. 1.42280
\(99\) 0 0
\(100\) 0 0
\(101\) 94644.0 0.923187 0.461593 0.887092i \(-0.347278\pi\)
0.461593 + 0.887092i \(0.347278\pi\)
\(102\) 0 0
\(103\) 113956. 1.05839 0.529193 0.848501i \(-0.322495\pi\)
0.529193 + 0.848501i \(0.322495\pi\)
\(104\) 117696. 1.06703
\(105\) 0 0
\(106\) 22960.0 0.198476
\(107\) −19212.0 −0.162223 −0.0811116 0.996705i \(-0.525847\pi\)
−0.0811116 + 0.996705i \(0.525847\pi\)
\(108\) 0 0
\(109\) −152791. −1.23178 −0.615888 0.787834i \(-0.711203\pi\)
−0.615888 + 0.787834i \(0.711203\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −57600.0 −0.433888
\(113\) 4012.00 0.0295573 0.0147787 0.999891i \(-0.495296\pi\)
0.0147787 + 0.999891i \(0.495296\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −121312. −0.837064
\(117\) 0 0
\(118\) 20696.0 0.136830
\(119\) −197550. −1.27882
\(120\) 0 0
\(121\) 27305.0 0.169543
\(122\) −154868. −0.942024
\(123\) 0 0
\(124\) −34800.0 −0.203247
\(125\) 0 0
\(126\) 0 0
\(127\) 36368.0 0.200083 0.100041 0.994983i \(-0.468102\pi\)
0.100041 + 0.994983i \(0.468102\pi\)
\(128\) −49152.0 −0.265165
\(129\) 0 0
\(130\) 0 0
\(131\) 43932.0 0.223667 0.111834 0.993727i \(-0.464328\pi\)
0.111834 + 0.993727i \(0.464328\pi\)
\(132\) 0 0
\(133\) −164475. −0.806252
\(134\) −126828. −0.610173
\(135\) 0 0
\(136\) 168576. 0.781535
\(137\) 185214. 0.843087 0.421544 0.906808i \(-0.361488\pi\)
0.421544 + 0.906808i \(0.361488\pi\)
\(138\) 0 0
\(139\) 115916. 0.508869 0.254435 0.967090i \(-0.418111\pi\)
0.254435 + 0.967090i \(0.418111\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −257888. −1.07327
\(143\) −266042. −1.08795
\(144\) 0 0
\(145\) 0 0
\(146\) 79160.0 0.307343
\(147\) 0 0
\(148\) −148960. −0.559005
\(149\) −33122.0 −0.122222 −0.0611112 0.998131i \(-0.519464\pi\)
−0.0611112 + 0.998131i \(0.519464\pi\)
\(150\) 0 0
\(151\) 431317. 1.53941 0.769705 0.638400i \(-0.220403\pi\)
0.769705 + 0.638400i \(0.220403\pi\)
\(152\) 140352. 0.492731
\(153\) 0 0
\(154\) 390600. 1.32718
\(155\) 0 0
\(156\) 0 0
\(157\) −40439.0 −0.130934 −0.0654668 0.997855i \(-0.520854\pi\)
−0.0654668 + 0.997855i \(0.520854\pi\)
\(158\) −420000. −1.33846
\(159\) 0 0
\(160\) 0 0
\(161\) 641250. 1.94968
\(162\) 0 0
\(163\) 116299. 0.342852 0.171426 0.985197i \(-0.445163\pi\)
0.171426 + 0.985197i \(0.445163\pi\)
\(164\) −192640. −0.559290
\(165\) 0 0
\(166\) −13272.0 −0.0373823
\(167\) −632700. −1.75552 −0.877762 0.479097i \(-0.840964\pi\)
−0.877762 + 0.479097i \(0.840964\pi\)
\(168\) 0 0
\(169\) 4476.00 0.0120552
\(170\) 0 0
\(171\) 0 0
\(172\) −17936.0 −0.0462279
\(173\) 108222. 0.274916 0.137458 0.990508i \(-0.456107\pi\)
0.137458 + 0.990508i \(0.456107\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −111104. −0.270363
\(177\) 0 0
\(178\) 261504. 0.618626
\(179\) −496570. −1.15837 −0.579186 0.815196i \(-0.696629\pi\)
−0.579186 + 0.815196i \(0.696629\pi\)
\(180\) 0 0
\(181\) 105365. 0.239056 0.119528 0.992831i \(-0.461862\pi\)
0.119528 + 0.992831i \(0.461862\pi\)
\(182\) −551700. −1.23460
\(183\) 0 0
\(184\) −547200. −1.19152
\(185\) 0 0
\(186\) 0 0
\(187\) −381052. −0.796857
\(188\) −478048. −0.986454
\(189\) 0 0
\(190\) 0 0
\(191\) −427346. −0.847610 −0.423805 0.905753i \(-0.639306\pi\)
−0.423805 + 0.905753i \(0.639306\pi\)
\(192\) 0 0
\(193\) −646427. −1.24918 −0.624592 0.780951i \(-0.714735\pi\)
−0.624592 + 0.780951i \(0.714735\pi\)
\(194\) 356572. 0.680210
\(195\) 0 0
\(196\) −541088. −1.00607
\(197\) −546954. −1.00412 −0.502060 0.864833i \(-0.667424\pi\)
−0.502060 + 0.864833i \(0.667424\pi\)
\(198\) 0 0
\(199\) −581783. −1.04143 −0.520713 0.853732i \(-0.674334\pi\)
−0.520713 + 0.853732i \(0.674334\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 378576. 0.652792
\(203\) 1.70595e6 2.90554
\(204\) 0 0
\(205\) 0 0
\(206\) 455824. 0.748392
\(207\) 0 0
\(208\) 156928. 0.251502
\(209\) −317254. −0.502390
\(210\) 0 0
\(211\) −441355. −0.682467 −0.341234 0.939978i \(-0.610845\pi\)
−0.341234 + 0.939978i \(0.610845\pi\)
\(212\) −91840.0 −0.140343
\(213\) 0 0
\(214\) −76848.0 −0.114709
\(215\) 0 0
\(216\) 0 0
\(217\) 489375. 0.705493
\(218\) −611164. −0.870997
\(219\) 0 0
\(220\) 0 0
\(221\) 538214. 0.741266
\(222\) 0 0
\(223\) 330457. 0.444993 0.222496 0.974934i \(-0.428579\pi\)
0.222496 + 0.974934i \(0.428579\pi\)
\(224\) 1.15200e6 1.53402
\(225\) 0 0
\(226\) 16048.0 0.0209002
\(227\) 1.12508e6 1.44917 0.724584 0.689186i \(-0.242032\pi\)
0.724584 + 0.689186i \(0.242032\pi\)
\(228\) 0 0
\(229\) −977379. −1.23161 −0.615807 0.787897i \(-0.711170\pi\)
−0.615807 + 0.787897i \(0.711170\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −1.45574e6 −1.77568
\(233\) 334584. 0.403753 0.201876 0.979411i \(-0.435296\pi\)
0.201876 + 0.979411i \(0.435296\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −82784.0 −0.0967534
\(237\) 0 0
\(238\) −790200. −0.904263
\(239\) −218764. −0.247731 −0.123866 0.992299i \(-0.539529\pi\)
−0.123866 + 0.992299i \(0.539529\pi\)
\(240\) 0 0
\(241\) 734041. 0.814100 0.407050 0.913406i \(-0.366557\pi\)
0.407050 + 0.913406i \(0.366557\pi\)
\(242\) 109220. 0.119885
\(243\) 0 0
\(244\) 619472. 0.666112
\(245\) 0 0
\(246\) 0 0
\(247\) 448103. 0.467343
\(248\) −417600. −0.431153
\(249\) 0 0
\(250\) 0 0
\(251\) −1.18381e6 −1.18604 −0.593019 0.805189i \(-0.702064\pi\)
−0.593019 + 0.805189i \(0.702064\pi\)
\(252\) 0 0
\(253\) 1.23690e6 1.21488
\(254\) 145472. 0.141480
\(255\) 0 0
\(256\) −1.11411e6 −1.06250
\(257\) −333732. −0.315185 −0.157592 0.987504i \(-0.550373\pi\)
−0.157592 + 0.987504i \(0.550373\pi\)
\(258\) 0 0
\(259\) 2.09475e6 1.94036
\(260\) 0 0
\(261\) 0 0
\(262\) 175728. 0.158157
\(263\) −731104. −0.651763 −0.325882 0.945411i \(-0.605661\pi\)
−0.325882 + 0.945411i \(0.605661\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −657900. −0.570106
\(267\) 0 0
\(268\) 507312. 0.431458
\(269\) −734254. −0.618679 −0.309340 0.950952i \(-0.600108\pi\)
−0.309340 + 0.950952i \(0.600108\pi\)
\(270\) 0 0
\(271\) 335056. 0.277137 0.138568 0.990353i \(-0.455750\pi\)
0.138568 + 0.990353i \(0.455750\pi\)
\(272\) 224768. 0.184210
\(273\) 0 0
\(274\) 740856. 0.596153
\(275\) 0 0
\(276\) 0 0
\(277\) 1.58260e6 1.23929 0.619643 0.784884i \(-0.287277\pi\)
0.619643 + 0.784884i \(0.287277\pi\)
\(278\) 463664. 0.359825
\(279\) 0 0
\(280\) 0 0
\(281\) −1.08713e6 −0.821329 −0.410665 0.911786i \(-0.634703\pi\)
−0.410665 + 0.911786i \(0.634703\pi\)
\(282\) 0 0
\(283\) 607407. 0.450831 0.225415 0.974263i \(-0.427626\pi\)
0.225415 + 0.974263i \(0.427626\pi\)
\(284\) 1.03155e6 0.758919
\(285\) 0 0
\(286\) −1.06417e6 −0.769299
\(287\) 2.70900e6 1.94135
\(288\) 0 0
\(289\) −648973. −0.457069
\(290\) 0 0
\(291\) 0 0
\(292\) −316640. −0.217324
\(293\) 1.23301e6 0.839066 0.419533 0.907740i \(-0.362194\pi\)
0.419533 + 0.907740i \(0.362194\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −1.78752e6 −1.18583
\(297\) 0 0
\(298\) −132488. −0.0864243
\(299\) −1.74705e6 −1.13013
\(300\) 0 0
\(301\) 252225. 0.160462
\(302\) 1.72527e6 1.08853
\(303\) 0 0
\(304\) 187136. 0.116138
\(305\) 0 0
\(306\) 0 0
\(307\) −562753. −0.340778 −0.170389 0.985377i \(-0.554502\pi\)
−0.170389 + 0.985377i \(0.554502\pi\)
\(308\) −1.56240e6 −0.938459
\(309\) 0 0
\(310\) 0 0
\(311\) 2.39645e6 1.40497 0.702487 0.711697i \(-0.252073\pi\)
0.702487 + 0.711697i \(0.252073\pi\)
\(312\) 0 0
\(313\) −1.52833e6 −0.881773 −0.440886 0.897563i \(-0.645336\pi\)
−0.440886 + 0.897563i \(0.645336\pi\)
\(314\) −161756. −0.0925841
\(315\) 0 0
\(316\) 1.68000e6 0.946437
\(317\) 1.00033e6 0.559106 0.279553 0.960130i \(-0.409814\pi\)
0.279553 + 0.960130i \(0.409814\pi\)
\(318\) 0 0
\(319\) 3.29059e6 1.81049
\(320\) 0 0
\(321\) 0 0
\(322\) 2.56500e6 1.37863
\(323\) 641818. 0.342299
\(324\) 0 0
\(325\) 0 0
\(326\) 465196. 0.242433
\(327\) 0 0
\(328\) −2.31168e6 −1.18643
\(329\) 6.72255e6 3.42408
\(330\) 0 0
\(331\) 2.29093e6 1.14932 0.574662 0.818391i \(-0.305134\pi\)
0.574662 + 0.818391i \(0.305134\pi\)
\(332\) 53088.0 0.0264333
\(333\) 0 0
\(334\) −2.53080e6 −1.24134
\(335\) 0 0
\(336\) 0 0
\(337\) −1.28904e6 −0.618290 −0.309145 0.951015i \(-0.600043\pi\)
−0.309145 + 0.951015i \(0.600043\pi\)
\(338\) 17904.0 0.00852429
\(339\) 0 0
\(340\) 0 0
\(341\) 943950. 0.439605
\(342\) 0 0
\(343\) 3.82748e6 1.75662
\(344\) −215232. −0.0980643
\(345\) 0 0
\(346\) 432888. 0.194395
\(347\) 19102.0 0.00851638 0.00425819 0.999991i \(-0.498645\pi\)
0.00425819 + 0.999991i \(0.498645\pi\)
\(348\) 0 0
\(349\) −3.24690e6 −1.42694 −0.713470 0.700686i \(-0.752877\pi\)
−0.713470 + 0.700686i \(0.752877\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 2.22208e6 0.955879
\(353\) 4.28804e6 1.83156 0.915782 0.401677i \(-0.131572\pi\)
0.915782 + 0.401677i \(0.131572\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −1.04602e6 −0.437435
\(357\) 0 0
\(358\) −1.98628e6 −0.819092
\(359\) −88656.0 −0.0363055 −0.0181527 0.999835i \(-0.505779\pi\)
−0.0181527 + 0.999835i \(0.505779\pi\)
\(360\) 0 0
\(361\) −1.94174e6 −0.784192
\(362\) 421460. 0.169038
\(363\) 0 0
\(364\) 2.20680e6 0.872990
\(365\) 0 0
\(366\) 0 0
\(367\) 2.04184e6 0.791328 0.395664 0.918395i \(-0.370515\pi\)
0.395664 + 0.918395i \(0.370515\pi\)
\(368\) −729600. −0.280844
\(369\) 0 0
\(370\) 0 0
\(371\) 1.29150e6 0.487147
\(372\) 0 0
\(373\) 806321. 0.300079 0.150040 0.988680i \(-0.452060\pi\)
0.150040 + 0.988680i \(0.452060\pi\)
\(374\) −1.52421e6 −0.563463
\(375\) 0 0
\(376\) −5.73658e6 −2.09259
\(377\) −4.64777e6 −1.68419
\(378\) 0 0
\(379\) −2.41314e6 −0.862950 −0.431475 0.902125i \(-0.642007\pi\)
−0.431475 + 0.902125i \(0.642007\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −1.70938e6 −0.599351
\(383\) −1.45856e6 −0.508076 −0.254038 0.967194i \(-0.581759\pi\)
−0.254038 + 0.967194i \(0.581759\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −2.58571e6 −0.883306
\(387\) 0 0
\(388\) −1.42629e6 −0.480981
\(389\) −2.65618e6 −0.889985 −0.444993 0.895534i \(-0.646794\pi\)
−0.444993 + 0.895534i \(0.646794\pi\)
\(390\) 0 0
\(391\) −2.50230e6 −0.827747
\(392\) −6.49306e6 −2.13419
\(393\) 0 0
\(394\) −2.18782e6 −0.710019
\(395\) 0 0
\(396\) 0 0
\(397\) −1.68664e6 −0.537090 −0.268545 0.963267i \(-0.586543\pi\)
−0.268545 + 0.963267i \(0.586543\pi\)
\(398\) −2.32713e6 −0.736399
\(399\) 0 0
\(400\) 0 0
\(401\) 2.12533e6 0.660033 0.330017 0.943975i \(-0.392946\pi\)
0.330017 + 0.943975i \(0.392946\pi\)
\(402\) 0 0
\(403\) −1.33328e6 −0.408938
\(404\) −1.51430e6 −0.461593
\(405\) 0 0
\(406\) 6.82380e6 2.05452
\(407\) 4.04054e6 1.20908
\(408\) 0 0
\(409\) 1.97158e6 0.582782 0.291391 0.956604i \(-0.405882\pi\)
0.291391 + 0.956604i \(0.405882\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −1.82330e6 −0.529193
\(413\) 1.16415e6 0.335841
\(414\) 0 0
\(415\) 0 0
\(416\) −3.13856e6 −0.889195
\(417\) 0 0
\(418\) −1.26902e6 −0.355244
\(419\) −1.75744e6 −0.489042 −0.244521 0.969644i \(-0.578631\pi\)
−0.244521 + 0.969644i \(0.578631\pi\)
\(420\) 0 0
\(421\) −2.83947e6 −0.780785 −0.390392 0.920649i \(-0.627661\pi\)
−0.390392 + 0.920649i \(0.627661\pi\)
\(422\) −1.76542e6 −0.482577
\(423\) 0 0
\(424\) −1.10208e6 −0.297713
\(425\) 0 0
\(426\) 0 0
\(427\) −8.71132e6 −2.31214
\(428\) 307392. 0.0811116
\(429\) 0 0
\(430\) 0 0
\(431\) 4.69335e6 1.21700 0.608499 0.793555i \(-0.291772\pi\)
0.608499 + 0.793555i \(0.291772\pi\)
\(432\) 0 0
\(433\) 6.71343e6 1.72078 0.860389 0.509639i \(-0.170221\pi\)
0.860389 + 0.509639i \(0.170221\pi\)
\(434\) 1.95750e6 0.498859
\(435\) 0 0
\(436\) 2.44466e6 0.615888
\(437\) −2.08335e6 −0.521866
\(438\) 0 0
\(439\) 3.92022e6 0.970842 0.485421 0.874281i \(-0.338666\pi\)
0.485421 + 0.874281i \(0.338666\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 2.15286e6 0.524154
\(443\) −6.93031e6 −1.67781 −0.838906 0.544276i \(-0.816805\pi\)
−0.838906 + 0.544276i \(0.816805\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 1.32183e6 0.314657
\(447\) 0 0
\(448\) 6.45120e6 1.51861
\(449\) −4.38605e6 −1.02673 −0.513367 0.858169i \(-0.671602\pi\)
−0.513367 + 0.858169i \(0.671602\pi\)
\(450\) 0 0
\(451\) 5.22536e6 1.20969
\(452\) −64192.0 −0.0147787
\(453\) 0 0
\(454\) 4.50032e6 1.02472
\(455\) 0 0
\(456\) 0 0
\(457\) −2.41951e6 −0.541922 −0.270961 0.962590i \(-0.587341\pi\)
−0.270961 + 0.962590i \(0.587341\pi\)
\(458\) −3.90952e6 −0.870882
\(459\) 0 0
\(460\) 0 0
\(461\) 366228. 0.0802600 0.0401300 0.999194i \(-0.487223\pi\)
0.0401300 + 0.999194i \(0.487223\pi\)
\(462\) 0 0
\(463\) −6.62534e6 −1.43634 −0.718168 0.695870i \(-0.755019\pi\)
−0.718168 + 0.695870i \(0.755019\pi\)
\(464\) −1.94099e6 −0.418532
\(465\) 0 0
\(466\) 1.33834e6 0.285496
\(467\) 3.38476e6 0.718183 0.359092 0.933302i \(-0.383087\pi\)
0.359092 + 0.933302i \(0.383087\pi\)
\(468\) 0 0
\(469\) −7.13408e6 −1.49763
\(470\) 0 0
\(471\) 0 0
\(472\) −993408. −0.205245
\(473\) 486514. 0.0999868
\(474\) 0 0
\(475\) 0 0
\(476\) 3.16080e6 0.639410
\(477\) 0 0
\(478\) −875056. −0.175173
\(479\) −6.61905e6 −1.31813 −0.659063 0.752088i \(-0.729047\pi\)
−0.659063 + 0.752088i \(0.729047\pi\)
\(480\) 0 0
\(481\) −5.70703e6 −1.12473
\(482\) 2.93616e6 0.575656
\(483\) 0 0
\(484\) −436880. −0.0847713
\(485\) 0 0
\(486\) 0 0
\(487\) 287383. 0.0549084 0.0274542 0.999623i \(-0.491260\pi\)
0.0274542 + 0.999623i \(0.491260\pi\)
\(488\) 7.43366e6 1.41304
\(489\) 0 0
\(490\) 0 0
\(491\) −6.51337e6 −1.21928 −0.609638 0.792680i \(-0.708685\pi\)
−0.609638 + 0.792680i \(0.708685\pi\)
\(492\) 0 0
\(493\) −6.65700e6 −1.23356
\(494\) 1.79241e6 0.330461
\(495\) 0 0
\(496\) −556800. −0.101624
\(497\) −1.45062e7 −2.63428
\(498\) 0 0
\(499\) −1.17037e6 −0.210413 −0.105206 0.994450i \(-0.533550\pi\)
−0.105206 + 0.994450i \(0.533550\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) −4.73525e6 −0.838655
\(503\) −1.40651e6 −0.247870 −0.123935 0.992290i \(-0.539551\pi\)
−0.123935 + 0.992290i \(0.539551\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 4.94760e6 0.859050
\(507\) 0 0
\(508\) −581888. −0.100041
\(509\) −3.65645e6 −0.625554 −0.312777 0.949827i \(-0.601259\pi\)
−0.312777 + 0.949827i \(0.601259\pi\)
\(510\) 0 0
\(511\) 4.45275e6 0.754356
\(512\) −2.88358e6 −0.486136
\(513\) 0 0
\(514\) −1.33493e6 −0.222869
\(515\) 0 0
\(516\) 0 0
\(517\) 1.29671e7 2.13361
\(518\) 8.37900e6 1.37204
\(519\) 0 0
\(520\) 0 0
\(521\) 2.70889e6 0.437217 0.218609 0.975813i \(-0.429848\pi\)
0.218609 + 0.975813i \(0.429848\pi\)
\(522\) 0 0
\(523\) 1.14014e7 1.82265 0.911326 0.411686i \(-0.135060\pi\)
0.911326 + 0.411686i \(0.135060\pi\)
\(524\) −702912. −0.111834
\(525\) 0 0
\(526\) −2.92442e6 −0.460866
\(527\) −1.90965e6 −0.299521
\(528\) 0 0
\(529\) 1.68616e6 0.261974
\(530\) 0 0
\(531\) 0 0
\(532\) 2.63160e6 0.403126
\(533\) −7.38052e6 −1.12530
\(534\) 0 0
\(535\) 0 0
\(536\) 6.08774e6 0.915260
\(537\) 0 0
\(538\) −2.93702e6 −0.437472
\(539\) 1.46770e7 2.17603
\(540\) 0 0
\(541\) 6.95050e6 1.02099 0.510497 0.859880i \(-0.329461\pi\)
0.510497 + 0.859880i \(0.329461\pi\)
\(542\) 1.34022e6 0.195965
\(543\) 0 0
\(544\) −4.49536e6 −0.651279
\(545\) 0 0
\(546\) 0 0
\(547\) 8.08532e6 1.15539 0.577695 0.816252i \(-0.303952\pi\)
0.577695 + 0.816252i \(0.303952\pi\)
\(548\) −2.96342e6 −0.421544
\(549\) 0 0
\(550\) 0 0
\(551\) −5.54244e6 −0.777718
\(552\) 0 0
\(553\) −2.36250e7 −3.28518
\(554\) 6.33040e6 0.876307
\(555\) 0 0
\(556\) −1.85466e6 −0.254435
\(557\) 1.46025e7 1.99429 0.997146 0.0754991i \(-0.0240550\pi\)
0.997146 + 0.0754991i \(0.0240550\pi\)
\(558\) 0 0
\(559\) −687173. −0.0930115
\(560\) 0 0
\(561\) 0 0
\(562\) −4.34854e6 −0.580767
\(563\) 3.54652e6 0.471554 0.235777 0.971807i \(-0.424236\pi\)
0.235777 + 0.971807i \(0.424236\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 2.42963e6 0.318786
\(567\) 0 0
\(568\) 1.23786e7 1.60991
\(569\) 1.42681e7 1.84750 0.923750 0.382996i \(-0.125108\pi\)
0.923750 + 0.382996i \(0.125108\pi\)
\(570\) 0 0
\(571\) −5.48534e6 −0.704066 −0.352033 0.935988i \(-0.614509\pi\)
−0.352033 + 0.935988i \(0.614509\pi\)
\(572\) 4.25667e6 0.543976
\(573\) 0 0
\(574\) 1.08360e7 1.37274
\(575\) 0 0
\(576\) 0 0
\(577\) 2.13253e6 0.266659 0.133329 0.991072i \(-0.457433\pi\)
0.133329 + 0.991072i \(0.457433\pi\)
\(578\) −2.59589e6 −0.323197
\(579\) 0 0
\(580\) 0 0
\(581\) −746550. −0.0917526
\(582\) 0 0
\(583\) 2.49116e6 0.303550
\(584\) −3.79968e6 −0.461015
\(585\) 0 0
\(586\) 4.93202e6 0.593309
\(587\) 2.39303e6 0.286650 0.143325 0.989676i \(-0.454221\pi\)
0.143325 + 0.989676i \(0.454221\pi\)
\(588\) 0 0
\(589\) −1.58992e6 −0.188838
\(590\) 0 0
\(591\) 0 0
\(592\) −2.38336e6 −0.279502
\(593\) 1.83478e6 0.214264 0.107132 0.994245i \(-0.465833\pi\)
0.107132 + 0.994245i \(0.465833\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 529952. 0.0611112
\(597\) 0 0
\(598\) −6.98820e6 −0.799121
\(599\) 126508. 0.0144062 0.00720312 0.999974i \(-0.497707\pi\)
0.00720312 + 0.999974i \(0.497707\pi\)
\(600\) 0 0
\(601\) 1.03791e7 1.17212 0.586059 0.810268i \(-0.300679\pi\)
0.586059 + 0.810268i \(0.300679\pi\)
\(602\) 1.00890e6 0.113464
\(603\) 0 0
\(604\) −6.90107e6 −0.769705
\(605\) 0 0
\(606\) 0 0
\(607\) 366608. 0.0403859 0.0201930 0.999796i \(-0.493572\pi\)
0.0201930 + 0.999796i \(0.493572\pi\)
\(608\) −3.74272e6 −0.410609
\(609\) 0 0
\(610\) 0 0
\(611\) −1.83152e7 −1.98476
\(612\) 0 0
\(613\) 7.49272e6 0.805357 0.402678 0.915342i \(-0.368079\pi\)
0.402678 + 0.915342i \(0.368079\pi\)
\(614\) −2.25101e6 −0.240967
\(615\) 0 0
\(616\) −1.87488e7 −1.99077
\(617\) 2.23748e6 0.236618 0.118309 0.992977i \(-0.462253\pi\)
0.118309 + 0.992977i \(0.462253\pi\)
\(618\) 0 0
\(619\) 6.61648e6 0.694065 0.347033 0.937853i \(-0.387189\pi\)
0.347033 + 0.937853i \(0.387189\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 9.58582e6 0.993466
\(623\) 1.47096e7 1.51838
\(624\) 0 0
\(625\) 0 0
\(626\) −6.11332e6 −0.623507
\(627\) 0 0
\(628\) 647024. 0.0654668
\(629\) −8.17418e6 −0.823792
\(630\) 0 0
\(631\) −6.16481e6 −0.616377 −0.308189 0.951325i \(-0.599723\pi\)
−0.308189 + 0.951325i \(0.599723\pi\)
\(632\) 2.01600e7 2.00770
\(633\) 0 0
\(634\) 4.00131e6 0.395348
\(635\) 0 0
\(636\) 0 0
\(637\) −2.07304e7 −2.02423
\(638\) 1.31624e7 1.28021
\(639\) 0 0
\(640\) 0 0
\(641\) −4.79502e6 −0.460941 −0.230471 0.973079i \(-0.574027\pi\)
−0.230471 + 0.973079i \(0.574027\pi\)
\(642\) 0 0
\(643\) 1.20124e6 0.114578 0.0572890 0.998358i \(-0.481754\pi\)
0.0572890 + 0.998358i \(0.481754\pi\)
\(644\) −1.02600e7 −0.974839
\(645\) 0 0
\(646\) 2.56727e6 0.242042
\(647\) −1.01480e7 −0.953058 −0.476529 0.879159i \(-0.658105\pi\)
−0.476529 + 0.879159i \(0.658105\pi\)
\(648\) 0 0
\(649\) 2.24552e6 0.209269
\(650\) 0 0
\(651\) 0 0
\(652\) −1.86078e6 −0.171426
\(653\) −1.98614e7 −1.82275 −0.911375 0.411576i \(-0.864978\pi\)
−0.911375 + 0.411576i \(0.864978\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −3.08224e6 −0.279645
\(657\) 0 0
\(658\) 2.68902e7 2.42119
\(659\) 5.73308e6 0.514250 0.257125 0.966378i \(-0.417225\pi\)
0.257125 + 0.966378i \(0.417225\pi\)
\(660\) 0 0
\(661\) −7.92244e6 −0.705270 −0.352635 0.935761i \(-0.614714\pi\)
−0.352635 + 0.935761i \(0.614714\pi\)
\(662\) 9.16373e6 0.812694
\(663\) 0 0
\(664\) 637056. 0.0560735
\(665\) 0 0
\(666\) 0 0
\(667\) 2.16087e7 1.88068
\(668\) 1.01232e7 0.877762
\(669\) 0 0
\(670\) 0 0
\(671\) −1.68032e7 −1.44074
\(672\) 0 0
\(673\) 7.75353e6 0.659875 0.329938 0.944003i \(-0.392972\pi\)
0.329938 + 0.944003i \(0.392972\pi\)
\(674\) −5.15616e6 −0.437197
\(675\) 0 0
\(676\) −71616.0 −0.00602758
\(677\) −3.77777e6 −0.316785 −0.158392 0.987376i \(-0.550631\pi\)
−0.158392 + 0.987376i \(0.550631\pi\)
\(678\) 0 0
\(679\) 2.00572e7 1.66953
\(680\) 0 0
\(681\) 0 0
\(682\) 3.77580e6 0.310848
\(683\) −6.18290e6 −0.507155 −0.253578 0.967315i \(-0.581607\pi\)
−0.253578 + 0.967315i \(0.581607\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 1.53099e7 1.24212
\(687\) 0 0
\(688\) −286976. −0.0231140
\(689\) −3.51862e6 −0.282374
\(690\) 0 0
\(691\) −2.38475e7 −1.89997 −0.949986 0.312293i \(-0.898903\pi\)
−0.949986 + 0.312293i \(0.898903\pi\)
\(692\) −1.73155e6 −0.137458
\(693\) 0 0
\(694\) 76408.0 0.00602199
\(695\) 0 0
\(696\) 0 0
\(697\) −1.05711e7 −0.824212
\(698\) −1.29876e7 −1.00900
\(699\) 0 0
\(700\) 0 0
\(701\) 5.81240e6 0.446746 0.223373 0.974733i \(-0.428293\pi\)
0.223373 + 0.974733i \(0.428293\pi\)
\(702\) 0 0
\(703\) −6.80561e6 −0.519372
\(704\) 1.24436e7 0.946272
\(705\) 0 0
\(706\) 1.71522e7 1.29511
\(707\) 2.12949e7 1.60224
\(708\) 0 0
\(709\) −7.53133e6 −0.562673 −0.281337 0.959609i \(-0.590778\pi\)
−0.281337 + 0.959609i \(0.590778\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −1.25522e7 −0.927939
\(713\) 6.19875e6 0.456647
\(714\) 0 0
\(715\) 0 0
\(716\) 7.94512e6 0.579186
\(717\) 0 0
\(718\) −354624. −0.0256718
\(719\) 8.66228e6 0.624899 0.312450 0.949934i \(-0.398850\pi\)
0.312450 + 0.949934i \(0.398850\pi\)
\(720\) 0 0
\(721\) 2.56401e7 1.83688
\(722\) −7.76695e6 −0.554508
\(723\) 0 0
\(724\) −1.68584e6 −0.119528
\(725\) 0 0
\(726\) 0 0
\(727\) −2.94444e6 −0.206617 −0.103309 0.994649i \(-0.532943\pi\)
−0.103309 + 0.994649i \(0.532943\pi\)
\(728\) 2.64816e7 1.85189
\(729\) 0 0
\(730\) 0 0
\(731\) −984238. −0.0681250
\(732\) 0 0
\(733\) −1.92363e7 −1.32239 −0.661197 0.750213i \(-0.729951\pi\)
−0.661197 + 0.750213i \(0.729951\pi\)
\(734\) 8.16736e6 0.559553
\(735\) 0 0
\(736\) 1.45920e7 0.992934
\(737\) −1.37608e7 −0.933203
\(738\) 0 0
\(739\) −2.63909e7 −1.77764 −0.888819 0.458258i \(-0.848474\pi\)
−0.888819 + 0.458258i \(0.848474\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 5.16600e6 0.344465
\(743\) −11516.0 −0.000765296 0 −0.000382648 1.00000i \(-0.500122\pi\)
−0.000382648 1.00000i \(0.500122\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 3.22528e6 0.212188
\(747\) 0 0
\(748\) 6.09683e6 0.398428
\(749\) −4.32270e6 −0.281547
\(750\) 0 0
\(751\) −2.44836e7 −1.58407 −0.792036 0.610474i \(-0.790979\pi\)
−0.792036 + 0.610474i \(0.790979\pi\)
\(752\) −7.64877e6 −0.493227
\(753\) 0 0
\(754\) −1.85911e7 −1.19090
\(755\) 0 0
\(756\) 0 0
\(757\) 1.01385e7 0.643031 0.321515 0.946904i \(-0.395808\pi\)
0.321515 + 0.946904i \(0.395808\pi\)
\(758\) −9.65258e6 −0.610197
\(759\) 0 0
\(760\) 0 0
\(761\) 1.34571e7 0.842343 0.421172 0.906981i \(-0.361619\pi\)
0.421172 + 0.906981i \(0.361619\pi\)
\(762\) 0 0
\(763\) −3.43780e7 −2.13781
\(764\) 6.83754e6 0.423805
\(765\) 0 0
\(766\) −5.83426e6 −0.359264
\(767\) −3.17166e6 −0.194670
\(768\) 0 0
\(769\) 2.13573e7 1.30236 0.651181 0.758923i \(-0.274274\pi\)
0.651181 + 0.758923i \(0.274274\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 1.03428e7 0.624592
\(773\) −2.55468e7 −1.53776 −0.768879 0.639394i \(-0.779185\pi\)
−0.768879 + 0.639394i \(0.779185\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −1.71155e7 −1.02031
\(777\) 0 0
\(778\) −1.06247e7 −0.629315
\(779\) −8.80124e6 −0.519637
\(780\) 0 0
\(781\) −2.79808e7 −1.64147
\(782\) −1.00092e7 −0.585305
\(783\) 0 0
\(784\) −8.65741e6 −0.503034
\(785\) 0 0
\(786\) 0 0
\(787\) −1.32668e7 −0.763535 −0.381768 0.924258i \(-0.624685\pi\)
−0.381768 + 0.924258i \(0.624685\pi\)
\(788\) 8.75126e6 0.502060
\(789\) 0 0
\(790\) 0 0
\(791\) 902700. 0.0512982
\(792\) 0 0
\(793\) 2.37335e7 1.34023
\(794\) −6.74657e6 −0.379780
\(795\) 0 0
\(796\) 9.30853e6 0.520713
\(797\) −1.24919e7 −0.696601 −0.348300 0.937383i \(-0.613241\pi\)
−0.348300 + 0.937383i \(0.613241\pi\)
\(798\) 0 0
\(799\) −2.62329e7 −1.45371
\(800\) 0 0
\(801\) 0 0
\(802\) 8.50133e6 0.466714
\(803\) 8.58886e6 0.470053
\(804\) 0 0
\(805\) 0 0
\(806\) −5.33310e6 −0.289163
\(807\) 0 0
\(808\) −1.81716e7 −0.979188
\(809\) 7.83514e6 0.420897 0.210448 0.977605i \(-0.432508\pi\)
0.210448 + 0.977605i \(0.432508\pi\)
\(810\) 0 0
\(811\) 3.04751e7 1.62702 0.813510 0.581551i \(-0.197554\pi\)
0.813510 + 0.581551i \(0.197554\pi\)
\(812\) −2.72952e7 −1.45277
\(813\) 0 0
\(814\) 1.61622e7 0.854945
\(815\) 0 0
\(816\) 0 0
\(817\) −819451. −0.0429505
\(818\) 7.88632e6 0.412089
\(819\) 0 0
\(820\) 0 0
\(821\) 2.90330e7 1.50326 0.751629 0.659586i \(-0.229268\pi\)
0.751629 + 0.659586i \(0.229268\pi\)
\(822\) 0 0
\(823\) 1.99935e7 1.02894 0.514468 0.857510i \(-0.327989\pi\)
0.514468 + 0.857510i \(0.327989\pi\)
\(824\) −2.18796e7 −1.12259
\(825\) 0 0
\(826\) 4.65660e6 0.237476
\(827\) −196388. −0.00998507 −0.00499254 0.999988i \(-0.501589\pi\)
−0.00499254 + 0.999988i \(0.501589\pi\)
\(828\) 0 0
\(829\) 7.05093e6 0.356336 0.178168 0.984000i \(-0.442983\pi\)
0.178168 + 0.984000i \(0.442983\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −1.75759e7 −0.880258
\(833\) −2.96922e7 −1.48262
\(834\) 0 0
\(835\) 0 0
\(836\) 5.07606e6 0.251195
\(837\) 0 0
\(838\) −7.02978e6 −0.345805
\(839\) 3.26118e7 1.59945 0.799724 0.600368i \(-0.204979\pi\)
0.799724 + 0.600368i \(0.204979\pi\)
\(840\) 0 0
\(841\) 3.69756e7 1.80271
\(842\) −1.13579e7 −0.552098
\(843\) 0 0
\(844\) 7.06168e6 0.341234
\(845\) 0 0
\(846\) 0 0
\(847\) 6.14362e6 0.294250
\(848\) −1.46944e6 −0.0701717
\(849\) 0 0
\(850\) 0 0
\(851\) 2.65335e7 1.25595
\(852\) 0 0
\(853\) 1.84687e7 0.869089 0.434545 0.900650i \(-0.356909\pi\)
0.434545 + 0.900650i \(0.356909\pi\)
\(854\) −3.48453e7 −1.63493
\(855\) 0 0
\(856\) 3.68870e6 0.172064
\(857\) −3.14310e7 −1.46186 −0.730930 0.682452i \(-0.760913\pi\)
−0.730930 + 0.682452i \(0.760913\pi\)
\(858\) 0 0
\(859\) 2.80767e7 1.29826 0.649132 0.760676i \(-0.275132\pi\)
0.649132 + 0.760676i \(0.275132\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 1.87734e7 0.860547
\(863\) −2.11951e7 −0.968744 −0.484372 0.874862i \(-0.660952\pi\)
−0.484372 + 0.874862i \(0.660952\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 2.68537e7 1.21677
\(867\) 0 0
\(868\) −7.83000e6 −0.352746
\(869\) −4.55700e7 −2.04706
\(870\) 0 0
\(871\) 1.94364e7 0.868101
\(872\) 2.93359e7 1.30650
\(873\) 0 0
\(874\) −8.33340e6 −0.369015
\(875\) 0 0
\(876\) 0 0
\(877\) −3.70911e7 −1.62843 −0.814217 0.580561i \(-0.802833\pi\)
−0.814217 + 0.580561i \(0.802833\pi\)
\(878\) 1.56809e7 0.686489
\(879\) 0 0
\(880\) 0 0
\(881\) 9.16386e6 0.397776 0.198888 0.980022i \(-0.436267\pi\)
0.198888 + 0.980022i \(0.436267\pi\)
\(882\) 0 0
\(883\) −1.36396e7 −0.588706 −0.294353 0.955697i \(-0.595104\pi\)
−0.294353 + 0.955697i \(0.595104\pi\)
\(884\) −8.61142e6 −0.370633
\(885\) 0 0
\(886\) −2.77212e7 −1.18639
\(887\) 2.90401e7 1.23934 0.619669 0.784863i \(-0.287267\pi\)
0.619669 + 0.784863i \(0.287267\pi\)
\(888\) 0 0
\(889\) 8.18280e6 0.347254
\(890\) 0 0
\(891\) 0 0
\(892\) −5.28731e6 −0.222496
\(893\) −2.18408e7 −0.916516
\(894\) 0 0
\(895\) 0 0
\(896\) −1.10592e7 −0.460207
\(897\) 0 0
\(898\) −1.75442e7 −0.726011
\(899\) 1.64908e7 0.680525
\(900\) 0 0
\(901\) −5.03972e6 −0.206821
\(902\) 2.09014e7 0.855382
\(903\) 0 0
\(904\) −770304. −0.0313503
\(905\) 0 0
\(906\) 0 0
\(907\) −2.12100e7 −0.856095 −0.428048 0.903756i \(-0.640798\pi\)
−0.428048 + 0.903756i \(0.640798\pi\)
\(908\) −1.80013e7 −0.724584
\(909\) 0 0
\(910\) 0 0
\(911\) −1.95267e7 −0.779531 −0.389766 0.920914i \(-0.627444\pi\)
−0.389766 + 0.920914i \(0.627444\pi\)
\(912\) 0 0
\(913\) −1.44001e6 −0.0571728
\(914\) −9.67804e6 −0.383197
\(915\) 0 0
\(916\) 1.56381e7 0.615807
\(917\) 9.88470e6 0.388186
\(918\) 0 0
\(919\) −2.85930e7 −1.11679 −0.558395 0.829575i \(-0.688583\pi\)
−0.558395 + 0.829575i \(0.688583\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 1.46491e6 0.0567524
\(923\) 3.95213e7 1.52696
\(924\) 0 0
\(925\) 0 0
\(926\) −2.65014e7 −1.01564
\(927\) 0 0
\(928\) 3.88198e7 1.47973
\(929\) −2.38769e7 −0.907692 −0.453846 0.891080i \(-0.649948\pi\)
−0.453846 + 0.891080i \(0.649948\pi\)
\(930\) 0 0
\(931\) −2.47210e7 −0.934741
\(932\) −5.35334e6 −0.201876
\(933\) 0 0
\(934\) 1.35390e7 0.507832
\(935\) 0 0
\(936\) 0 0
\(937\) −3.46672e7 −1.28994 −0.644970 0.764208i \(-0.723130\pi\)
−0.644970 + 0.764208i \(0.723130\pi\)
\(938\) −2.85363e7 −1.05899
\(939\) 0 0
\(940\) 0 0
\(941\) −4.64188e7 −1.70891 −0.854455 0.519525i \(-0.826109\pi\)
−0.854455 + 0.519525i \(0.826109\pi\)
\(942\) 0 0
\(943\) 3.43140e7 1.25659
\(944\) −1.32454e6 −0.0483767
\(945\) 0 0
\(946\) 1.94606e6 0.0707013
\(947\) −2.10829e7 −0.763934 −0.381967 0.924176i \(-0.624753\pi\)
−0.381967 + 0.924176i \(0.624753\pi\)
\(948\) 0 0
\(949\) −1.21313e7 −0.437261
\(950\) 0 0
\(951\) 0 0
\(952\) 3.79296e7 1.35639
\(953\) 3.67438e7 1.31055 0.655273 0.755392i \(-0.272554\pi\)
0.655273 + 0.755392i \(0.272554\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 3.50022e6 0.123866
\(957\) 0 0
\(958\) −2.64762e7 −0.932056
\(959\) 4.16732e7 1.46322
\(960\) 0 0
\(961\) −2.38985e7 −0.834762
\(962\) −2.28281e7 −0.795303
\(963\) 0 0
\(964\) −1.17447e7 −0.407050
\(965\) 0 0
\(966\) 0 0
\(967\) 3.76234e7 1.29388 0.646938 0.762543i \(-0.276049\pi\)
0.646938 + 0.762543i \(0.276049\pi\)
\(968\) −5.24256e6 −0.179827
\(969\) 0 0
\(970\) 0 0
\(971\) 4.89205e7 1.66511 0.832555 0.553943i \(-0.186877\pi\)
0.832555 + 0.553943i \(0.186877\pi\)
\(972\) 0 0
\(973\) 2.60811e7 0.883169
\(974\) 1.14953e6 0.0388261
\(975\) 0 0
\(976\) 9.91155e6 0.333056
\(977\) −2.84450e7 −0.953386 −0.476693 0.879070i \(-0.658165\pi\)
−0.476693 + 0.879070i \(0.658165\pi\)
\(978\) 0 0
\(979\) 2.83732e7 0.946131
\(980\) 0 0
\(981\) 0 0
\(982\) −2.60535e7 −0.862158
\(983\) 4.51611e7 1.49067 0.745334 0.666692i \(-0.232290\pi\)
0.745334 + 0.666692i \(0.232290\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) −2.66280e7 −0.872260
\(987\) 0 0
\(988\) −7.16965e6 −0.233671
\(989\) 3.19485e6 0.103863
\(990\) 0 0
\(991\) −2.85517e7 −0.923524 −0.461762 0.887004i \(-0.652783\pi\)
−0.461762 + 0.887004i \(0.652783\pi\)
\(992\) 1.11360e7 0.359294
\(993\) 0 0
\(994\) −5.80248e7 −1.86272
\(995\) 0 0
\(996\) 0 0
\(997\) −3.26527e7 −1.04035 −0.520177 0.854058i \(-0.674134\pi\)
−0.520177 + 0.854058i \(0.674134\pi\)
\(998\) −4.68148e6 −0.148784
\(999\) 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 225.6.a.g.1.1 1
3.2 odd 2 75.6.a.b.1.1 1
5.2 odd 4 225.6.b.c.199.2 2
5.3 odd 4 225.6.b.c.199.1 2
5.4 even 2 225.6.a.b.1.1 1
15.2 even 4 75.6.b.c.49.1 2
15.8 even 4 75.6.b.c.49.2 2
15.14 odd 2 75.6.a.d.1.1 yes 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
75.6.a.b.1.1 1 3.2 odd 2
75.6.a.d.1.1 yes 1 15.14 odd 2
75.6.b.c.49.1 2 15.2 even 4
75.6.b.c.49.2 2 15.8 even 4
225.6.a.b.1.1 1 5.4 even 2
225.6.a.g.1.1 1 1.1 even 1 trivial
225.6.b.c.199.1 2 5.3 odd 4
225.6.b.c.199.2 2 5.2 odd 4