Properties

Label 225.6.a.a.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 3)
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)
 
\(f(q)\) \(=\) \(q-6.00000 q^{2} +4.00000 q^{4} +40.0000 q^{7} +168.000 q^{8} +O(q^{10})\) \(q-6.00000 q^{2} +4.00000 q^{4} +40.0000 q^{7} +168.000 q^{8} +564.000 q^{11} -638.000 q^{13} -240.000 q^{14} -1136.00 q^{16} +882.000 q^{17} -556.000 q^{19} -3384.00 q^{22} -840.000 q^{23} +3828.00 q^{26} +160.000 q^{28} -4638.00 q^{29} +4400.00 q^{31} +1440.00 q^{32} -5292.00 q^{34} +2410.00 q^{37} +3336.00 q^{38} +6870.00 q^{41} -9644.00 q^{43} +2256.00 q^{44} +5040.00 q^{46} -18672.0 q^{47} -15207.0 q^{49} -2552.00 q^{52} +33750.0 q^{53} +6720.00 q^{56} +27828.0 q^{58} +18084.0 q^{59} +39758.0 q^{61} -26400.0 q^{62} +27712.0 q^{64} +23068.0 q^{67} +3528.00 q^{68} +4248.00 q^{71} +41110.0 q^{73} -14460.0 q^{74} -2224.00 q^{76} +22560.0 q^{77} +21920.0 q^{79} -41220.0 q^{82} +82452.0 q^{83} +57864.0 q^{86} +94752.0 q^{88} +94086.0 q^{89} -25520.0 q^{91} -3360.00 q^{92} +112032. q^{94} -49442.0 q^{97} +91242.0 q^{98} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −6.00000 −1.06066 −0.530330 0.847791i \(-0.677932\pi\)
−0.530330 + 0.847791i \(0.677932\pi\)
\(3\) 0 0
\(4\) 4.00000 0.125000
\(5\) 0 0
\(6\) 0 0
\(7\) 40.0000 0.308542 0.154271 0.988029i \(-0.450697\pi\)
0.154271 + 0.988029i \(0.450697\pi\)
\(8\) 168.000 0.928078
\(9\) 0 0
\(10\) 0 0
\(11\) 564.000 1.40539 0.702696 0.711490i \(-0.251979\pi\)
0.702696 + 0.711490i \(0.251979\pi\)
\(12\) 0 0
\(13\) −638.000 −1.04704 −0.523519 0.852014i \(-0.675381\pi\)
−0.523519 + 0.852014i \(0.675381\pi\)
\(14\) −240.000 −0.327259
\(15\) 0 0
\(16\) −1136.00 −1.10938
\(17\) 882.000 0.740195 0.370098 0.928993i \(-0.379324\pi\)
0.370098 + 0.928993i \(0.379324\pi\)
\(18\) 0 0
\(19\) −556.000 −0.353338 −0.176669 0.984270i \(-0.556532\pi\)
−0.176669 + 0.984270i \(0.556532\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −3384.00 −1.49064
\(23\) −840.000 −0.331100 −0.165550 0.986201i \(-0.552940\pi\)
−0.165550 + 0.986201i \(0.552940\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 3828.00 1.11055
\(27\) 0 0
\(28\) 160.000 0.0385678
\(29\) −4638.00 −1.02408 −0.512042 0.858960i \(-0.671111\pi\)
−0.512042 + 0.858960i \(0.671111\pi\)
\(30\) 0 0
\(31\) 4400.00 0.822334 0.411167 0.911560i \(-0.365121\pi\)
0.411167 + 0.911560i \(0.365121\pi\)
\(32\) 1440.00 0.248592
\(33\) 0 0
\(34\) −5292.00 −0.785096
\(35\) 0 0
\(36\) 0 0
\(37\) 2410.00 0.289409 0.144705 0.989475i \(-0.453777\pi\)
0.144705 + 0.989475i \(0.453777\pi\)
\(38\) 3336.00 0.374772
\(39\) 0 0
\(40\) 0 0
\(41\) 6870.00 0.638259 0.319130 0.947711i \(-0.396609\pi\)
0.319130 + 0.947711i \(0.396609\pi\)
\(42\) 0 0
\(43\) −9644.00 −0.795401 −0.397700 0.917515i \(-0.630192\pi\)
−0.397700 + 0.917515i \(0.630192\pi\)
\(44\) 2256.00 0.175674
\(45\) 0 0
\(46\) 5040.00 0.351185
\(47\) −18672.0 −1.23295 −0.616476 0.787374i \(-0.711440\pi\)
−0.616476 + 0.787374i \(0.711440\pi\)
\(48\) 0 0
\(49\) −15207.0 −0.904802
\(50\) 0 0
\(51\) 0 0
\(52\) −2552.00 −0.130880
\(53\) 33750.0 1.65038 0.825190 0.564855i \(-0.191068\pi\)
0.825190 + 0.564855i \(0.191068\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 6720.00 0.286351
\(57\) 0 0
\(58\) 27828.0 1.08621
\(59\) 18084.0 0.676339 0.338170 0.941085i \(-0.390192\pi\)
0.338170 + 0.941085i \(0.390192\pi\)
\(60\) 0 0
\(61\) 39758.0 1.36804 0.684022 0.729462i \(-0.260229\pi\)
0.684022 + 0.729462i \(0.260229\pi\)
\(62\) −26400.0 −0.872217
\(63\) 0 0
\(64\) 27712.0 0.845703
\(65\) 0 0
\(66\) 0 0
\(67\) 23068.0 0.627802 0.313901 0.949456i \(-0.398364\pi\)
0.313901 + 0.949456i \(0.398364\pi\)
\(68\) 3528.00 0.0925244
\(69\) 0 0
\(70\) 0 0
\(71\) 4248.00 0.100009 0.0500044 0.998749i \(-0.484076\pi\)
0.0500044 + 0.998749i \(0.484076\pi\)
\(72\) 0 0
\(73\) 41110.0 0.902901 0.451451 0.892296i \(-0.350907\pi\)
0.451451 + 0.892296i \(0.350907\pi\)
\(74\) −14460.0 −0.306965
\(75\) 0 0
\(76\) −2224.00 −0.0441673
\(77\) 22560.0 0.433623
\(78\) 0 0
\(79\) 21920.0 0.395160 0.197580 0.980287i \(-0.436692\pi\)
0.197580 + 0.980287i \(0.436692\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −41220.0 −0.676976
\(83\) 82452.0 1.31373 0.656865 0.754008i \(-0.271882\pi\)
0.656865 + 0.754008i \(0.271882\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 57864.0 0.843650
\(87\) 0 0
\(88\) 94752.0 1.30431
\(89\) 94086.0 1.25907 0.629535 0.776972i \(-0.283245\pi\)
0.629535 + 0.776972i \(0.283245\pi\)
\(90\) 0 0
\(91\) −25520.0 −0.323056
\(92\) −3360.00 −0.0413875
\(93\) 0 0
\(94\) 112032. 1.30774
\(95\) 0 0
\(96\) 0 0
\(97\) −49442.0 −0.533540 −0.266770 0.963760i \(-0.585956\pi\)
−0.266770 + 0.963760i \(0.585956\pi\)
\(98\) 91242.0 0.959687
\(99\) 0 0
\(100\) 0 0
\(101\) 143034. 1.39520 0.697599 0.716488i \(-0.254252\pi\)
0.697599 + 0.716488i \(0.254252\pi\)
\(102\) 0 0
\(103\) −53144.0 −0.493584 −0.246792 0.969068i \(-0.579376\pi\)
−0.246792 + 0.969068i \(0.579376\pi\)
\(104\) −107184. −0.971732
\(105\) 0 0
\(106\) −202500. −1.75049
\(107\) 90828.0 0.766938 0.383469 0.923554i \(-0.374729\pi\)
0.383469 + 0.923554i \(0.374729\pi\)
\(108\) 0 0
\(109\) −61666.0 −0.497141 −0.248570 0.968614i \(-0.579961\pi\)
−0.248570 + 0.968614i \(0.579961\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −45440.0 −0.342289
\(113\) 10482.0 0.0772232 0.0386116 0.999254i \(-0.487706\pi\)
0.0386116 + 0.999254i \(0.487706\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −18552.0 −0.128011
\(117\) 0 0
\(118\) −108504. −0.717366
\(119\) 35280.0 0.228382
\(120\) 0 0
\(121\) 157045. 0.975126
\(122\) −238548. −1.45103
\(123\) 0 0
\(124\) 17600.0 0.102792
\(125\) 0 0
\(126\) 0 0
\(127\) 171088. 0.941261 0.470631 0.882330i \(-0.344026\pi\)
0.470631 + 0.882330i \(0.344026\pi\)
\(128\) −212352. −1.14560
\(129\) 0 0
\(130\) 0 0
\(131\) −258468. −1.31592 −0.657959 0.753054i \(-0.728580\pi\)
−0.657959 + 0.753054i \(0.728580\pi\)
\(132\) 0 0
\(133\) −22240.0 −0.109020
\(134\) −138408. −0.665885
\(135\) 0 0
\(136\) 148176. 0.686959
\(137\) 300234. 1.36665 0.683327 0.730113i \(-0.260532\pi\)
0.683327 + 0.730113i \(0.260532\pi\)
\(138\) 0 0
\(139\) −350164. −1.53721 −0.768607 0.639721i \(-0.779050\pi\)
−0.768607 + 0.639721i \(0.779050\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −25488.0 −0.106075
\(143\) −359832. −1.47150
\(144\) 0 0
\(145\) 0 0
\(146\) −246660. −0.957672
\(147\) 0 0
\(148\) 9640.00 0.0361762
\(149\) 105258. 0.388409 0.194205 0.980961i \(-0.437787\pi\)
0.194205 + 0.980961i \(0.437787\pi\)
\(150\) 0 0
\(151\) 396392. 1.41476 0.707380 0.706834i \(-0.249877\pi\)
0.707380 + 0.706834i \(0.249877\pi\)
\(152\) −93408.0 −0.327925
\(153\) 0 0
\(154\) −135360. −0.459927
\(155\) 0 0
\(156\) 0 0
\(157\) 137746. 0.445995 0.222997 0.974819i \(-0.428416\pi\)
0.222997 + 0.974819i \(0.428416\pi\)
\(158\) −131520. −0.419130
\(159\) 0 0
\(160\) 0 0
\(161\) −33600.0 −0.102159
\(162\) 0 0
\(163\) −352676. −1.03970 −0.519849 0.854258i \(-0.674012\pi\)
−0.519849 + 0.854258i \(0.674012\pi\)
\(164\) 27480.0 0.0797824
\(165\) 0 0
\(166\) −494712. −1.39342
\(167\) −217560. −0.603654 −0.301827 0.953363i \(-0.597596\pi\)
−0.301827 + 0.953363i \(0.597596\pi\)
\(168\) 0 0
\(169\) 35751.0 0.0962878
\(170\) 0 0
\(171\) 0 0
\(172\) −38576.0 −0.0994251
\(173\) −163698. −0.415842 −0.207921 0.978146i \(-0.566670\pi\)
−0.207921 + 0.978146i \(0.566670\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −640704. −1.55911
\(177\) 0 0
\(178\) −564516. −1.33545
\(179\) −358740. −0.836849 −0.418425 0.908252i \(-0.637418\pi\)
−0.418425 + 0.908252i \(0.637418\pi\)
\(180\) 0 0
\(181\) −507130. −1.15060 −0.575298 0.817944i \(-0.695114\pi\)
−0.575298 + 0.817944i \(0.695114\pi\)
\(182\) 153120. 0.342652
\(183\) 0 0
\(184\) −141120. −0.307287
\(185\) 0 0
\(186\) 0 0
\(187\) 497448. 1.04026
\(188\) −74688.0 −0.154119
\(189\) 0 0
\(190\) 0 0
\(191\) 648384. 1.28602 0.643012 0.765856i \(-0.277685\pi\)
0.643012 + 0.765856i \(0.277685\pi\)
\(192\) 0 0
\(193\) 27838.0 0.0537954 0.0268977 0.999638i \(-0.491437\pi\)
0.0268977 + 0.999638i \(0.491437\pi\)
\(194\) 296652. 0.565904
\(195\) 0 0
\(196\) −60828.0 −0.113100
\(197\) 611046. 1.12178 0.560891 0.827890i \(-0.310459\pi\)
0.560891 + 0.827890i \(0.310459\pi\)
\(198\) 0 0
\(199\) 879032. 1.57352 0.786760 0.617260i \(-0.211757\pi\)
0.786760 + 0.617260i \(0.211757\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) −858204. −1.47983
\(203\) −185520. −0.315973
\(204\) 0 0
\(205\) 0 0
\(206\) 318864. 0.523525
\(207\) 0 0
\(208\) 724768. 1.16156
\(209\) −313584. −0.496579
\(210\) 0 0
\(211\) 48500.0 0.0749956 0.0374978 0.999297i \(-0.488061\pi\)
0.0374978 + 0.999297i \(0.488061\pi\)
\(212\) 135000. 0.206298
\(213\) 0 0
\(214\) −544968. −0.813461
\(215\) 0 0
\(216\) 0 0
\(217\) 176000. 0.253725
\(218\) 369996. 0.527298
\(219\) 0 0
\(220\) 0 0
\(221\) −562716. −0.775012
\(222\) 0 0
\(223\) 999472. 1.34589 0.672943 0.739694i \(-0.265030\pi\)
0.672943 + 0.739694i \(0.265030\pi\)
\(224\) 57600.0 0.0767012
\(225\) 0 0
\(226\) −62892.0 −0.0819076
\(227\) 606180. 0.780795 0.390397 0.920646i \(-0.372338\pi\)
0.390397 + 0.920646i \(0.372338\pi\)
\(228\) 0 0
\(229\) 1.35993e6 1.71367 0.856834 0.515593i \(-0.172428\pi\)
0.856834 + 0.515593i \(0.172428\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −779184. −0.950430
\(233\) −392886. −0.474107 −0.237054 0.971497i \(-0.576182\pi\)
−0.237054 + 0.971497i \(0.576182\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 72336.0 0.0845424
\(237\) 0 0
\(238\) −211680. −0.242235
\(239\) 1.32514e6 1.50060 0.750301 0.661096i \(-0.229908\pi\)
0.750301 + 0.661096i \(0.229908\pi\)
\(240\) 0 0
\(241\) −990094. −1.09808 −0.549040 0.835796i \(-0.685006\pi\)
−0.549040 + 0.835796i \(0.685006\pi\)
\(242\) −942270. −1.03428
\(243\) 0 0
\(244\) 159032. 0.171005
\(245\) 0 0
\(246\) 0 0
\(247\) 354728. 0.369959
\(248\) 739200. 0.763190
\(249\) 0 0
\(250\) 0 0
\(251\) −147132. −0.147409 −0.0737043 0.997280i \(-0.523482\pi\)
−0.0737043 + 0.997280i \(0.523482\pi\)
\(252\) 0 0
\(253\) −473760. −0.465326
\(254\) −1.02653e6 −0.998358
\(255\) 0 0
\(256\) 387328. 0.369385
\(257\) −483582. −0.456707 −0.228353 0.973578i \(-0.573334\pi\)
−0.228353 + 0.973578i \(0.573334\pi\)
\(258\) 0 0
\(259\) 96400.0 0.0892951
\(260\) 0 0
\(261\) 0 0
\(262\) 1.55081e6 1.39574
\(263\) 813576. 0.725285 0.362643 0.931928i \(-0.381875\pi\)
0.362643 + 0.931928i \(0.381875\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 133440. 0.115633
\(267\) 0 0
\(268\) 92272.0 0.0784753
\(269\) 461106. 0.388526 0.194263 0.980949i \(-0.437768\pi\)
0.194263 + 0.980949i \(0.437768\pi\)
\(270\) 0 0
\(271\) 1.67514e6 1.38556 0.692782 0.721147i \(-0.256385\pi\)
0.692782 + 0.721147i \(0.256385\pi\)
\(272\) −1.00195e6 −0.821154
\(273\) 0 0
\(274\) −1.80140e6 −1.44956
\(275\) 0 0
\(276\) 0 0
\(277\) −401126. −0.314110 −0.157055 0.987590i \(-0.550200\pi\)
−0.157055 + 0.987590i \(0.550200\pi\)
\(278\) 2.10098e6 1.63046
\(279\) 0 0
\(280\) 0 0
\(281\) 2.30977e6 1.74503 0.872514 0.488590i \(-0.162489\pi\)
0.872514 + 0.488590i \(0.162489\pi\)
\(282\) 0 0
\(283\) 1.12877e6 0.837800 0.418900 0.908032i \(-0.362416\pi\)
0.418900 + 0.908032i \(0.362416\pi\)
\(284\) 16992.0 0.0125011
\(285\) 0 0
\(286\) 2.15899e6 1.56076
\(287\) 274800. 0.196930
\(288\) 0 0
\(289\) −641933. −0.452111
\(290\) 0 0
\(291\) 0 0
\(292\) 164440. 0.112863
\(293\) −938874. −0.638908 −0.319454 0.947602i \(-0.603499\pi\)
−0.319454 + 0.947602i \(0.603499\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 404880. 0.268594
\(297\) 0 0
\(298\) −631548. −0.411970
\(299\) 535920. 0.346675
\(300\) 0 0
\(301\) −385760. −0.245415
\(302\) −2.37835e6 −1.50058
\(303\) 0 0
\(304\) 631616. 0.391985
\(305\) 0 0
\(306\) 0 0
\(307\) −692948. −0.419619 −0.209809 0.977742i \(-0.567284\pi\)
−0.209809 + 0.977742i \(0.567284\pi\)
\(308\) 90240.0 0.0542029
\(309\) 0 0
\(310\) 0 0
\(311\) −2.94310e6 −1.72545 −0.862727 0.505670i \(-0.831245\pi\)
−0.862727 + 0.505670i \(0.831245\pi\)
\(312\) 0 0
\(313\) −885146. −0.510686 −0.255343 0.966851i \(-0.582188\pi\)
−0.255343 + 0.966851i \(0.582188\pi\)
\(314\) −826476. −0.473049
\(315\) 0 0
\(316\) 87680.0 0.0493950
\(317\) 2.50880e6 1.40222 0.701112 0.713051i \(-0.252687\pi\)
0.701112 + 0.713051i \(0.252687\pi\)
\(318\) 0 0
\(319\) −2.61583e6 −1.43924
\(320\) 0 0
\(321\) 0 0
\(322\) 201600. 0.108355
\(323\) −490392. −0.261539
\(324\) 0 0
\(325\) 0 0
\(326\) 2.11606e6 1.10277
\(327\) 0 0
\(328\) 1.15416e6 0.592354
\(329\) −746880. −0.380418
\(330\) 0 0
\(331\) −216148. −0.108438 −0.0542190 0.998529i \(-0.517267\pi\)
−0.0542190 + 0.998529i \(0.517267\pi\)
\(332\) 329808. 0.164216
\(333\) 0 0
\(334\) 1.30536e6 0.640271
\(335\) 0 0
\(336\) 0 0
\(337\) −3.25263e6 −1.56012 −0.780062 0.625702i \(-0.784813\pi\)
−0.780062 + 0.625702i \(0.784813\pi\)
\(338\) −214506. −0.102129
\(339\) 0 0
\(340\) 0 0
\(341\) 2.48160e6 1.15570
\(342\) 0 0
\(343\) −1.28056e6 −0.587712
\(344\) −1.62019e6 −0.738194
\(345\) 0 0
\(346\) 982188. 0.441067
\(347\) −2.93207e6 −1.30723 −0.653613 0.756829i \(-0.726747\pi\)
−0.653613 + 0.756829i \(0.726747\pi\)
\(348\) 0 0
\(349\) 905198. 0.397814 0.198907 0.980018i \(-0.436261\pi\)
0.198907 + 0.980018i \(0.436261\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 812160. 0.349369
\(353\) 1.91786e6 0.819181 0.409590 0.912270i \(-0.365672\pi\)
0.409590 + 0.912270i \(0.365672\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 376344. 0.157384
\(357\) 0 0
\(358\) 2.15244e6 0.887613
\(359\) 2.43698e6 0.997968 0.498984 0.866611i \(-0.333707\pi\)
0.498984 + 0.866611i \(0.333707\pi\)
\(360\) 0 0
\(361\) −2.16696e6 −0.875152
\(362\) 3.04278e6 1.22039
\(363\) 0 0
\(364\) −102080. −0.0403819
\(365\) 0 0
\(366\) 0 0
\(367\) 984064. 0.381380 0.190690 0.981650i \(-0.438927\pi\)
0.190690 + 0.981650i \(0.438927\pi\)
\(368\) 954240. 0.367314
\(369\) 0 0
\(370\) 0 0
\(371\) 1.35000e6 0.509212
\(372\) 0 0
\(373\) −1.70365e6 −0.634029 −0.317015 0.948421i \(-0.602680\pi\)
−0.317015 + 0.948421i \(0.602680\pi\)
\(374\) −2.98469e6 −1.10337
\(375\) 0 0
\(376\) −3.13690e6 −1.14428
\(377\) 2.95904e6 1.07225
\(378\) 0 0
\(379\) 2.75654e6 0.985749 0.492874 0.870100i \(-0.335946\pi\)
0.492874 + 0.870100i \(0.335946\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −3.89030e6 −1.36403
\(383\) 456576. 0.159044 0.0795218 0.996833i \(-0.474661\pi\)
0.0795218 + 0.996833i \(0.474661\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −167028. −0.0570586
\(387\) 0 0
\(388\) −197768. −0.0666925
\(389\) 2.00639e6 0.672268 0.336134 0.941814i \(-0.390881\pi\)
0.336134 + 0.941814i \(0.390881\pi\)
\(390\) 0 0
\(391\) −740880. −0.245079
\(392\) −2.55478e6 −0.839726
\(393\) 0 0
\(394\) −3.66628e6 −1.18983
\(395\) 0 0
\(396\) 0 0
\(397\) 5.77040e6 1.83751 0.918755 0.394828i \(-0.129196\pi\)
0.918755 + 0.394828i \(0.129196\pi\)
\(398\) −5.27419e6 −1.66897
\(399\) 0 0
\(400\) 0 0
\(401\) −3.00626e6 −0.933610 −0.466805 0.884360i \(-0.654595\pi\)
−0.466805 + 0.884360i \(0.654595\pi\)
\(402\) 0 0
\(403\) −2.80720e6 −0.861015
\(404\) 572136. 0.174400
\(405\) 0 0
\(406\) 1.11312e6 0.335140
\(407\) 1.35924e6 0.406734
\(408\) 0 0
\(409\) 1.53363e6 0.453327 0.226663 0.973973i \(-0.427218\pi\)
0.226663 + 0.973973i \(0.427218\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −212576. −0.0616980
\(413\) 723360. 0.208679
\(414\) 0 0
\(415\) 0 0
\(416\) −918720. −0.260285
\(417\) 0 0
\(418\) 1.88150e6 0.526701
\(419\) 3.87376e6 1.07795 0.538973 0.842323i \(-0.318812\pi\)
0.538973 + 0.842323i \(0.318812\pi\)
\(420\) 0 0
\(421\) −1.33307e6 −0.366561 −0.183281 0.983061i \(-0.558672\pi\)
−0.183281 + 0.983061i \(0.558672\pi\)
\(422\) −291000. −0.0795448
\(423\) 0 0
\(424\) 5.67000e6 1.53168
\(425\) 0 0
\(426\) 0 0
\(427\) 1.59032e6 0.422099
\(428\) 363312. 0.0958673
\(429\) 0 0
\(430\) 0 0
\(431\) −6.45192e6 −1.67300 −0.836500 0.547967i \(-0.815402\pi\)
−0.836500 + 0.547967i \(0.815402\pi\)
\(432\) 0 0
\(433\) 4.16577e6 1.06777 0.533883 0.845558i \(-0.320732\pi\)
0.533883 + 0.845558i \(0.320732\pi\)
\(434\) −1.05600e6 −0.269116
\(435\) 0 0
\(436\) −246664. −0.0621426
\(437\) 467040. 0.116990
\(438\) 0 0
\(439\) 792680. 0.196307 0.0981537 0.995171i \(-0.468706\pi\)
0.0981537 + 0.995171i \(0.468706\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 3.37630e6 0.822025
\(443\) −1.39981e6 −0.338891 −0.169446 0.985540i \(-0.554198\pi\)
−0.169446 + 0.985540i \(0.554198\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −5.99683e6 −1.42753
\(447\) 0 0
\(448\) 1.10848e6 0.260935
\(449\) −2.99248e6 −0.700512 −0.350256 0.936654i \(-0.613905\pi\)
−0.350256 + 0.936654i \(0.613905\pi\)
\(450\) 0 0
\(451\) 3.87468e6 0.897004
\(452\) 41928.0 0.00965291
\(453\) 0 0
\(454\) −3.63708e6 −0.828158
\(455\) 0 0
\(456\) 0 0
\(457\) −6.29969e6 −1.41101 −0.705503 0.708707i \(-0.749279\pi\)
−0.705503 + 0.708707i \(0.749279\pi\)
\(458\) −8.15956e6 −1.81762
\(459\) 0 0
\(460\) 0 0
\(461\) −3.40318e6 −0.745818 −0.372909 0.927868i \(-0.621640\pi\)
−0.372909 + 0.927868i \(0.621640\pi\)
\(462\) 0 0
\(463\) 2.23034e6 0.483524 0.241762 0.970336i \(-0.422275\pi\)
0.241762 + 0.970336i \(0.422275\pi\)
\(464\) 5.26877e6 1.13609
\(465\) 0 0
\(466\) 2.35732e6 0.502867
\(467\) −6.51409e6 −1.38217 −0.691085 0.722773i \(-0.742867\pi\)
−0.691085 + 0.722773i \(0.742867\pi\)
\(468\) 0 0
\(469\) 922720. 0.193704
\(470\) 0 0
\(471\) 0 0
\(472\) 3.03811e6 0.627695
\(473\) −5.43922e6 −1.11785
\(474\) 0 0
\(475\) 0 0
\(476\) 141120. 0.0285477
\(477\) 0 0
\(478\) −7.95082e6 −1.59163
\(479\) −2.39232e6 −0.476410 −0.238205 0.971215i \(-0.576559\pi\)
−0.238205 + 0.971215i \(0.576559\pi\)
\(480\) 0 0
\(481\) −1.53758e6 −0.303023
\(482\) 5.94056e6 1.16469
\(483\) 0 0
\(484\) 628180. 0.121891
\(485\) 0 0
\(486\) 0 0
\(487\) 6.13089e6 1.17139 0.585694 0.810532i \(-0.300822\pi\)
0.585694 + 0.810532i \(0.300822\pi\)
\(488\) 6.67934e6 1.26965
\(489\) 0 0
\(490\) 0 0
\(491\) 1.23589e6 0.231354 0.115677 0.993287i \(-0.463096\pi\)
0.115677 + 0.993287i \(0.463096\pi\)
\(492\) 0 0
\(493\) −4.09072e6 −0.758022
\(494\) −2.12837e6 −0.392400
\(495\) 0 0
\(496\) −4.99840e6 −0.912277
\(497\) 169920. 0.0308570
\(498\) 0 0
\(499\) −9.85496e6 −1.77175 −0.885877 0.463921i \(-0.846442\pi\)
−0.885877 + 0.463921i \(0.846442\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 882792. 0.156350
\(503\) 1.16777e6 0.205796 0.102898 0.994692i \(-0.467188\pi\)
0.102898 + 0.994692i \(0.467188\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 2.84256e6 0.493552
\(507\) 0 0
\(508\) 684352. 0.117658
\(509\) −1.04941e6 −0.179535 −0.0897675 0.995963i \(-0.528612\pi\)
−0.0897675 + 0.995963i \(0.528612\pi\)
\(510\) 0 0
\(511\) 1.64440e6 0.278583
\(512\) 4.47130e6 0.753804
\(513\) 0 0
\(514\) 2.90149e6 0.484411
\(515\) 0 0
\(516\) 0 0
\(517\) −1.05310e7 −1.73278
\(518\) −578400. −0.0947118
\(519\) 0 0
\(520\) 0 0
\(521\) 9.61407e6 1.55172 0.775859 0.630906i \(-0.217317\pi\)
0.775859 + 0.630906i \(0.217317\pi\)
\(522\) 0 0
\(523\) −6.96148e6 −1.11288 −0.556439 0.830888i \(-0.687833\pi\)
−0.556439 + 0.830888i \(0.687833\pi\)
\(524\) −1.03387e6 −0.164490
\(525\) 0 0
\(526\) −4.88146e6 −0.769281
\(527\) 3.88080e6 0.608688
\(528\) 0 0
\(529\) −5.73074e6 −0.890373
\(530\) 0 0
\(531\) 0 0
\(532\) −88960.0 −0.0136275
\(533\) −4.38306e6 −0.668281
\(534\) 0 0
\(535\) 0 0
\(536\) 3.87542e6 0.582649
\(537\) 0 0
\(538\) −2.76664e6 −0.412094
\(539\) −8.57675e6 −1.27160
\(540\) 0 0
\(541\) −712690. −0.104691 −0.0523453 0.998629i \(-0.516670\pi\)
−0.0523453 + 0.998629i \(0.516670\pi\)
\(542\) −1.00508e7 −1.46961
\(543\) 0 0
\(544\) 1.27008e6 0.184007
\(545\) 0 0
\(546\) 0 0
\(547\) 3.62614e6 0.518175 0.259087 0.965854i \(-0.416578\pi\)
0.259087 + 0.965854i \(0.416578\pi\)
\(548\) 1.20094e6 0.170832
\(549\) 0 0
\(550\) 0 0
\(551\) 2.57873e6 0.361848
\(552\) 0 0
\(553\) 876800. 0.121924
\(554\) 2.40676e6 0.333164
\(555\) 0 0
\(556\) −1.40066e6 −0.192152
\(557\) 4.84846e6 0.662165 0.331082 0.943602i \(-0.392586\pi\)
0.331082 + 0.943602i \(0.392586\pi\)
\(558\) 0 0
\(559\) 6.15287e6 0.832815
\(560\) 0 0
\(561\) 0 0
\(562\) −1.38586e7 −1.85088
\(563\) 8.50405e6 1.13072 0.565360 0.824844i \(-0.308737\pi\)
0.565360 + 0.824844i \(0.308737\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −6.77263e6 −0.888621
\(567\) 0 0
\(568\) 713664. 0.0928160
\(569\) −362874. −0.0469867 −0.0234934 0.999724i \(-0.507479\pi\)
−0.0234934 + 0.999724i \(0.507479\pi\)
\(570\) 0 0
\(571\) 4.11024e6 0.527566 0.263783 0.964582i \(-0.415030\pi\)
0.263783 + 0.964582i \(0.415030\pi\)
\(572\) −1.43933e6 −0.183937
\(573\) 0 0
\(574\) −1.64880e6 −0.208876
\(575\) 0 0
\(576\) 0 0
\(577\) 7.87680e6 0.984941 0.492470 0.870329i \(-0.336094\pi\)
0.492470 + 0.870329i \(0.336094\pi\)
\(578\) 3.85160e6 0.479536
\(579\) 0 0
\(580\) 0 0
\(581\) 3.29808e6 0.405341
\(582\) 0 0
\(583\) 1.90350e7 2.31943
\(584\) 6.90648e6 0.837963
\(585\) 0 0
\(586\) 5.63324e6 0.677664
\(587\) 603948. 0.0723443 0.0361721 0.999346i \(-0.488484\pi\)
0.0361721 + 0.999346i \(0.488484\pi\)
\(588\) 0 0
\(589\) −2.44640e6 −0.290562
\(590\) 0 0
\(591\) 0 0
\(592\) −2.73776e6 −0.321064
\(593\) −5.39077e6 −0.629526 −0.314763 0.949170i \(-0.601925\pi\)
−0.314763 + 0.949170i \(0.601925\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 421032. 0.0485511
\(597\) 0 0
\(598\) −3.21552e6 −0.367704
\(599\) −4.27999e6 −0.487389 −0.243695 0.969852i \(-0.578359\pi\)
−0.243695 + 0.969852i \(0.578359\pi\)
\(600\) 0 0
\(601\) 1.02483e6 0.115735 0.0578674 0.998324i \(-0.481570\pi\)
0.0578674 + 0.998324i \(0.481570\pi\)
\(602\) 2.31456e6 0.260302
\(603\) 0 0
\(604\) 1.58557e6 0.176845
\(605\) 0 0
\(606\) 0 0
\(607\) −1.24342e7 −1.36976 −0.684882 0.728654i \(-0.740146\pi\)
−0.684882 + 0.728654i \(0.740146\pi\)
\(608\) −800640. −0.0878372
\(609\) 0 0
\(610\) 0 0
\(611\) 1.19127e7 1.29095
\(612\) 0 0
\(613\) −4.21506e6 −0.453057 −0.226528 0.974005i \(-0.572738\pi\)
−0.226528 + 0.974005i \(0.572738\pi\)
\(614\) 4.15769e6 0.445073
\(615\) 0 0
\(616\) 3.79008e6 0.402436
\(617\) −4.40665e6 −0.466010 −0.233005 0.972476i \(-0.574856\pi\)
−0.233005 + 0.972476i \(0.574856\pi\)
\(618\) 0 0
\(619\) 4.80168e6 0.503693 0.251847 0.967767i \(-0.418962\pi\)
0.251847 + 0.967767i \(0.418962\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 1.76586e7 1.83012
\(623\) 3.76344e6 0.388477
\(624\) 0 0
\(625\) 0 0
\(626\) 5.31088e6 0.541664
\(627\) 0 0
\(628\) 550984. 0.0557494
\(629\) 2.12562e6 0.214220
\(630\) 0 0
\(631\) 8.30727e6 0.830587 0.415293 0.909688i \(-0.363679\pi\)
0.415293 + 0.909688i \(0.363679\pi\)
\(632\) 3.68256e6 0.366739
\(633\) 0 0
\(634\) −1.50528e7 −1.48728
\(635\) 0 0
\(636\) 0 0
\(637\) 9.70207e6 0.947361
\(638\) 1.56950e7 1.52654
\(639\) 0 0
\(640\) 0 0
\(641\) −1.76956e7 −1.70107 −0.850534 0.525921i \(-0.823721\pi\)
−0.850534 + 0.525921i \(0.823721\pi\)
\(642\) 0 0
\(643\) 1.28394e7 1.22466 0.612330 0.790602i \(-0.290232\pi\)
0.612330 + 0.790602i \(0.290232\pi\)
\(644\) −134400. −0.0127698
\(645\) 0 0
\(646\) 2.94235e6 0.277404
\(647\) −2.08468e7 −1.95785 −0.978924 0.204226i \(-0.934532\pi\)
−0.978924 + 0.204226i \(0.934532\pi\)
\(648\) 0 0
\(649\) 1.01994e7 0.950521
\(650\) 0 0
\(651\) 0 0
\(652\) −1.41070e6 −0.129962
\(653\) 1.29632e7 1.18968 0.594841 0.803843i \(-0.297215\pi\)
0.594841 + 0.803843i \(0.297215\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −7.80432e6 −0.708069
\(657\) 0 0
\(658\) 4.48128e6 0.403494
\(659\) 5.66862e6 0.508468 0.254234 0.967143i \(-0.418177\pi\)
0.254234 + 0.967143i \(0.418177\pi\)
\(660\) 0 0
\(661\) −3.11430e6 −0.277240 −0.138620 0.990346i \(-0.544267\pi\)
−0.138620 + 0.990346i \(0.544267\pi\)
\(662\) 1.29689e6 0.115016
\(663\) 0 0
\(664\) 1.38519e7 1.21924
\(665\) 0 0
\(666\) 0 0
\(667\) 3.89592e6 0.339075
\(668\) −870240. −0.0754567
\(669\) 0 0
\(670\) 0 0
\(671\) 2.24235e7 1.92264
\(672\) 0 0
\(673\) −105890. −0.00901192 −0.00450596 0.999990i \(-0.501434\pi\)
−0.00450596 + 0.999990i \(0.501434\pi\)
\(674\) 1.95158e7 1.65476
\(675\) 0 0
\(676\) 143004. 0.0120360
\(677\) −1.60910e7 −1.34931 −0.674656 0.738132i \(-0.735708\pi\)
−0.674656 + 0.738132i \(0.735708\pi\)
\(678\) 0 0
\(679\) −1.97768e6 −0.164620
\(680\) 0 0
\(681\) 0 0
\(682\) −1.48896e7 −1.22581
\(683\) 1.60780e7 1.31880 0.659402 0.751791i \(-0.270810\pi\)
0.659402 + 0.751791i \(0.270810\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 7.68336e6 0.623363
\(687\) 0 0
\(688\) 1.09556e7 0.882398
\(689\) −2.15325e7 −1.72801
\(690\) 0 0
\(691\) −165964. −0.0132227 −0.00661133 0.999978i \(-0.502104\pi\)
−0.00661133 + 0.999978i \(0.502104\pi\)
\(692\) −654792. −0.0519802
\(693\) 0 0
\(694\) 1.75924e7 1.38652
\(695\) 0 0
\(696\) 0 0
\(697\) 6.05934e6 0.472436
\(698\) −5.43119e6 −0.421945
\(699\) 0 0
\(700\) 0 0
\(701\) −1.77248e7 −1.36234 −0.681171 0.732124i \(-0.738529\pi\)
−0.681171 + 0.732124i \(0.738529\pi\)
\(702\) 0 0
\(703\) −1.33996e6 −0.102259
\(704\) 1.56296e7 1.18854
\(705\) 0 0
\(706\) −1.15071e7 −0.868872
\(707\) 5.72136e6 0.430478
\(708\) 0 0
\(709\) −1.06023e7 −0.792112 −0.396056 0.918226i \(-0.629621\pi\)
−0.396056 + 0.918226i \(0.629621\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 1.58064e7 1.16852
\(713\) −3.69600e6 −0.272275
\(714\) 0 0
\(715\) 0 0
\(716\) −1.43496e6 −0.104606
\(717\) 0 0
\(718\) −1.46219e7 −1.05850
\(719\) −9.03211e6 −0.651579 −0.325790 0.945442i \(-0.605630\pi\)
−0.325790 + 0.945442i \(0.605630\pi\)
\(720\) 0 0
\(721\) −2.12576e6 −0.152292
\(722\) 1.30018e7 0.928239
\(723\) 0 0
\(724\) −2.02852e6 −0.143825
\(725\) 0 0
\(726\) 0 0
\(727\) −1.87575e7 −1.31625 −0.658127 0.752907i \(-0.728651\pi\)
−0.658127 + 0.752907i \(0.728651\pi\)
\(728\) −4.28736e6 −0.299821
\(729\) 0 0
\(730\) 0 0
\(731\) −8.50601e6 −0.588752
\(732\) 0 0
\(733\) 1.17773e7 0.809626 0.404813 0.914399i \(-0.367337\pi\)
0.404813 + 0.914399i \(0.367337\pi\)
\(734\) −5.90438e6 −0.404515
\(735\) 0 0
\(736\) −1.20960e6 −0.0823090
\(737\) 1.30104e7 0.882308
\(738\) 0 0
\(739\) 5.88948e6 0.396703 0.198352 0.980131i \(-0.436441\pi\)
0.198352 + 0.980131i \(0.436441\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −8.10000e6 −0.540101
\(743\) −1.00476e7 −0.667712 −0.333856 0.942624i \(-0.608350\pi\)
−0.333856 + 0.942624i \(0.608350\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 1.02219e7 0.672490
\(747\) 0 0
\(748\) 1.98979e6 0.130033
\(749\) 3.63312e6 0.236633
\(750\) 0 0
\(751\) 4.81530e6 0.311547 0.155773 0.987793i \(-0.450213\pi\)
0.155773 + 0.987793i \(0.450213\pi\)
\(752\) 2.12114e7 1.36781
\(753\) 0 0
\(754\) −1.77543e7 −1.13730
\(755\) 0 0
\(756\) 0 0
\(757\) −3.12973e6 −0.198503 −0.0992516 0.995062i \(-0.531645\pi\)
−0.0992516 + 0.995062i \(0.531645\pi\)
\(758\) −1.65392e7 −1.04554
\(759\) 0 0
\(760\) 0 0
\(761\) 1.17773e7 0.737197 0.368599 0.929589i \(-0.379838\pi\)
0.368599 + 0.929589i \(0.379838\pi\)
\(762\) 0 0
\(763\) −2.46664e6 −0.153389
\(764\) 2.59354e6 0.160753
\(765\) 0 0
\(766\) −2.73946e6 −0.168691
\(767\) −1.15376e7 −0.708152
\(768\) 0 0
\(769\) −1.49376e6 −0.0910887 −0.0455443 0.998962i \(-0.514502\pi\)
−0.0455443 + 0.998962i \(0.514502\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 111352. 0.00672442
\(773\) −2.25125e7 −1.35511 −0.677555 0.735472i \(-0.736960\pi\)
−0.677555 + 0.735472i \(0.736960\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −8.30626e6 −0.495166
\(777\) 0 0
\(778\) −1.20384e7 −0.713048
\(779\) −3.81972e6 −0.225521
\(780\) 0 0
\(781\) 2.39587e6 0.140552
\(782\) 4.44528e6 0.259945
\(783\) 0 0
\(784\) 1.72752e7 1.00376
\(785\) 0 0
\(786\) 0 0
\(787\) −1.19547e7 −0.688022 −0.344011 0.938966i \(-0.611786\pi\)
−0.344011 + 0.938966i \(0.611786\pi\)
\(788\) 2.44418e6 0.140223
\(789\) 0 0
\(790\) 0 0
\(791\) 419280. 0.0238266
\(792\) 0 0
\(793\) −2.53656e7 −1.43239
\(794\) −3.46224e7 −1.94897
\(795\) 0 0
\(796\) 3.51613e6 0.196690
\(797\) 540798. 0.0301571 0.0150785 0.999886i \(-0.495200\pi\)
0.0150785 + 0.999886i \(0.495200\pi\)
\(798\) 0 0
\(799\) −1.64687e7 −0.912625
\(800\) 0 0
\(801\) 0 0
\(802\) 1.80375e7 0.990243
\(803\) 2.31860e7 1.26893
\(804\) 0 0
\(805\) 0 0
\(806\) 1.68432e7 0.913244
\(807\) 0 0
\(808\) 2.40297e7 1.29485
\(809\) 6.14223e6 0.329955 0.164978 0.986297i \(-0.447245\pi\)
0.164978 + 0.986297i \(0.447245\pi\)
\(810\) 0 0
\(811\) −3.16734e7 −1.69100 −0.845499 0.533977i \(-0.820697\pi\)
−0.845499 + 0.533977i \(0.820697\pi\)
\(812\) −742080. −0.0394967
\(813\) 0 0
\(814\) −8.15544e6 −0.431406
\(815\) 0 0
\(816\) 0 0
\(817\) 5.36206e6 0.281046
\(818\) −9.20176e6 −0.480825
\(819\) 0 0
\(820\) 0 0
\(821\) −2.66175e7 −1.37819 −0.689095 0.724671i \(-0.741992\pi\)
−0.689095 + 0.724671i \(0.741992\pi\)
\(822\) 0 0
\(823\) −3.62817e7 −1.86719 −0.933593 0.358335i \(-0.883345\pi\)
−0.933593 + 0.358335i \(0.883345\pi\)
\(824\) −8.92819e6 −0.458084
\(825\) 0 0
\(826\) −4.34016e6 −0.221338
\(827\) 1.09033e6 0.0554364 0.0277182 0.999616i \(-0.491176\pi\)
0.0277182 + 0.999616i \(0.491176\pi\)
\(828\) 0 0
\(829\) −1.03016e7 −0.520620 −0.260310 0.965525i \(-0.583825\pi\)
−0.260310 + 0.965525i \(0.583825\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −1.76803e7 −0.885483
\(833\) −1.34126e7 −0.669730
\(834\) 0 0
\(835\) 0 0
\(836\) −1.25434e6 −0.0620724
\(837\) 0 0
\(838\) −2.32425e7 −1.14333
\(839\) 1.96134e7 0.961940 0.480970 0.876737i \(-0.340285\pi\)
0.480970 + 0.876737i \(0.340285\pi\)
\(840\) 0 0
\(841\) 999895. 0.0487489
\(842\) 7.99840e6 0.388797
\(843\) 0 0
\(844\) 194000. 0.00937445
\(845\) 0 0
\(846\) 0 0
\(847\) 6.28180e6 0.300868
\(848\) −3.83400e7 −1.83089
\(849\) 0 0
\(850\) 0 0
\(851\) −2.02440e6 −0.0958236
\(852\) 0 0
\(853\) −3.27565e7 −1.54143 −0.770717 0.637178i \(-0.780102\pi\)
−0.770717 + 0.637178i \(0.780102\pi\)
\(854\) −9.54192e6 −0.447704
\(855\) 0 0
\(856\) 1.52591e7 0.711778
\(857\) −2.57953e7 −1.19974 −0.599872 0.800096i \(-0.704782\pi\)
−0.599872 + 0.800096i \(0.704782\pi\)
\(858\) 0 0
\(859\) −1.98548e7 −0.918085 −0.459043 0.888414i \(-0.651807\pi\)
−0.459043 + 0.888414i \(0.651807\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 3.87115e7 1.77448
\(863\) −673056. −0.0307627 −0.0153813 0.999882i \(-0.504896\pi\)
−0.0153813 + 0.999882i \(0.504896\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) −2.49946e7 −1.13254
\(867\) 0 0
\(868\) 704000. 0.0317156
\(869\) 1.23629e7 0.555354
\(870\) 0 0
\(871\) −1.47174e7 −0.657333
\(872\) −1.03599e7 −0.461385
\(873\) 0 0
\(874\) −2.80224e6 −0.124087
\(875\) 0 0
\(876\) 0 0
\(877\) −5.32115e6 −0.233618 −0.116809 0.993154i \(-0.537267\pi\)
−0.116809 + 0.993154i \(0.537267\pi\)
\(878\) −4.75608e6 −0.208215
\(879\) 0 0
\(880\) 0 0
\(881\) −2.78891e7 −1.21058 −0.605291 0.796004i \(-0.706943\pi\)
−0.605291 + 0.796004i \(0.706943\pi\)
\(882\) 0 0
\(883\) 2.83786e7 1.22487 0.612435 0.790521i \(-0.290190\pi\)
0.612435 + 0.790521i \(0.290190\pi\)
\(884\) −2.25086e6 −0.0968765
\(885\) 0 0
\(886\) 8.39887e6 0.359448
\(887\) 4.22678e7 1.80385 0.901925 0.431893i \(-0.142154\pi\)
0.901925 + 0.431893i \(0.142154\pi\)
\(888\) 0 0
\(889\) 6.84352e6 0.290419
\(890\) 0 0
\(891\) 0 0
\(892\) 3.99789e6 0.168236
\(893\) 1.03816e7 0.435649
\(894\) 0 0
\(895\) 0 0
\(896\) −8.49408e6 −0.353465
\(897\) 0 0
\(898\) 1.79549e7 0.743005
\(899\) −2.04072e7 −0.842140
\(900\) 0 0
\(901\) 2.97675e7 1.22160
\(902\) −2.32481e7 −0.951417
\(903\) 0 0
\(904\) 1.76098e6 0.0716692
\(905\) 0 0
\(906\) 0 0
\(907\) −3.19526e7 −1.28970 −0.644849 0.764310i \(-0.723080\pi\)
−0.644849 + 0.764310i \(0.723080\pi\)
\(908\) 2.42472e6 0.0975994
\(909\) 0 0
\(910\) 0 0
\(911\) 1.16429e7 0.464800 0.232400 0.972620i \(-0.425342\pi\)
0.232400 + 0.972620i \(0.425342\pi\)
\(912\) 0 0
\(913\) 4.65029e7 1.84630
\(914\) 3.77981e7 1.49660
\(915\) 0 0
\(916\) 5.43970e6 0.214208
\(917\) −1.03387e7 −0.406016
\(918\) 0 0
\(919\) 1.39844e6 0.0546204 0.0273102 0.999627i \(-0.491306\pi\)
0.0273102 + 0.999627i \(0.491306\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 2.04191e7 0.791059
\(923\) −2.71022e6 −0.104713
\(924\) 0 0
\(925\) 0 0
\(926\) −1.33820e7 −0.512854
\(927\) 0 0
\(928\) −6.67872e6 −0.254579
\(929\) 1.66792e7 0.634067 0.317033 0.948414i \(-0.397313\pi\)
0.317033 + 0.948414i \(0.397313\pi\)
\(930\) 0 0
\(931\) 8.45509e6 0.319701
\(932\) −1.57154e6 −0.0592634
\(933\) 0 0
\(934\) 3.90846e7 1.46601
\(935\) 0 0
\(936\) 0 0
\(937\) 2.47956e7 0.922625 0.461312 0.887238i \(-0.347379\pi\)
0.461312 + 0.887238i \(0.347379\pi\)
\(938\) −5.53632e6 −0.205454
\(939\) 0 0
\(940\) 0 0
\(941\) −2.79574e7 −1.02925 −0.514627 0.857414i \(-0.672070\pi\)
−0.514627 + 0.857414i \(0.672070\pi\)
\(942\) 0 0
\(943\) −5.77080e6 −0.211328
\(944\) −2.05434e7 −0.750314
\(945\) 0 0
\(946\) 3.26353e7 1.18566
\(947\) 7.64936e6 0.277173 0.138586 0.990350i \(-0.455744\pi\)
0.138586 + 0.990350i \(0.455744\pi\)
\(948\) 0 0
\(949\) −2.62282e7 −0.945372
\(950\) 0 0
\(951\) 0 0
\(952\) 5.92704e6 0.211956
\(953\) −4.62179e7 −1.64846 −0.824228 0.566257i \(-0.808391\pi\)
−0.824228 + 0.566257i \(0.808391\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 5.30054e6 0.187575
\(957\) 0 0
\(958\) 1.43539e7 0.505309
\(959\) 1.20094e7 0.421671
\(960\) 0 0
\(961\) −9.26915e6 −0.323766
\(962\) 9.22548e6 0.321404
\(963\) 0 0
\(964\) −3.96038e6 −0.137260
\(965\) 0 0
\(966\) 0 0
\(967\) −2.08557e7 −0.717229 −0.358615 0.933486i \(-0.616751\pi\)
−0.358615 + 0.933486i \(0.616751\pi\)
\(968\) 2.63836e7 0.904993
\(969\) 0 0
\(970\) 0 0
\(971\) 4.58152e7 1.55941 0.779707 0.626144i \(-0.215368\pi\)
0.779707 + 0.626144i \(0.215368\pi\)
\(972\) 0 0
\(973\) −1.40066e7 −0.474296
\(974\) −3.67853e7 −1.24245
\(975\) 0 0
\(976\) −4.51651e7 −1.51767
\(977\) −1.09544e6 −0.0367157 −0.0183578 0.999831i \(-0.505844\pi\)
−0.0183578 + 0.999831i \(0.505844\pi\)
\(978\) 0 0
\(979\) 5.30645e7 1.76949
\(980\) 0 0
\(981\) 0 0
\(982\) −7.41535e6 −0.245388
\(983\) 5.25817e7 1.73561 0.867803 0.496909i \(-0.165532\pi\)
0.867803 + 0.496909i \(0.165532\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 2.45443e7 0.804004
\(987\) 0 0
\(988\) 1.41891e6 0.0462448
\(989\) 8.10096e6 0.263358
\(990\) 0 0
\(991\) −4.90389e7 −1.58620 −0.793098 0.609094i \(-0.791533\pi\)
−0.793098 + 0.609094i \(0.791533\pi\)
\(992\) 6.33600e6 0.204426
\(993\) 0 0
\(994\) −1.01952e6 −0.0327288
\(995\) 0 0
\(996\) 0 0
\(997\) −3.05461e6 −0.0973237 −0.0486618 0.998815i \(-0.515496\pi\)
−0.0486618 + 0.998815i \(0.515496\pi\)
\(998\) 5.91297e7 1.87923
\(999\) 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.a.1.1 1
3.2 odd 2 75.6.a.e.1.1 1
5.2 odd 4 225.6.b.b.199.1 2
5.3 odd 4 225.6.b.b.199.2 2
5.4 even 2 9.6.a.a.1.1 1
15.2 even 4 75.6.b.b.49.2 2
15.8 even 4 75.6.b.b.49.1 2
15.14 odd 2 3.6.a.a.1.1 1
20.19 odd 2 144.6.a.f.1.1 1
35.34 odd 2 441.6.a.i.1.1 1
40.19 odd 2 576.6.a.t.1.1 1
40.29 even 2 576.6.a.s.1.1 1
45.4 even 6 81.6.c.a.55.1 2
45.14 odd 6 81.6.c.c.55.1 2
45.29 odd 6 81.6.c.c.28.1 2
45.34 even 6 81.6.c.a.28.1 2
55.54 odd 2 1089.6.a.b.1.1 1
60.59 even 2 48.6.a.a.1.1 1
105.44 odd 6 147.6.e.h.67.1 2
105.59 even 6 147.6.e.k.79.1 2
105.74 odd 6 147.6.e.h.79.1 2
105.89 even 6 147.6.e.k.67.1 2
105.104 even 2 147.6.a.a.1.1 1
120.29 odd 2 192.6.a.d.1.1 1
120.59 even 2 192.6.a.l.1.1 1
165.164 even 2 363.6.a.d.1.1 1
195.194 odd 2 507.6.a.b.1.1 1
240.29 odd 4 768.6.d.k.385.2 2
240.59 even 4 768.6.d.h.385.2 2
240.149 odd 4 768.6.d.k.385.1 2
240.179 even 4 768.6.d.h.385.1 2
255.254 odd 2 867.6.a.a.1.1 1
285.284 even 2 1083.6.a.c.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3.6.a.a.1.1 1 15.14 odd 2
9.6.a.a.1.1 1 5.4 even 2
48.6.a.a.1.1 1 60.59 even 2
75.6.a.e.1.1 1 3.2 odd 2
75.6.b.b.49.1 2 15.8 even 4
75.6.b.b.49.2 2 15.2 even 4
81.6.c.a.28.1 2 45.34 even 6
81.6.c.a.55.1 2 45.4 even 6
81.6.c.c.28.1 2 45.29 odd 6
81.6.c.c.55.1 2 45.14 odd 6
144.6.a.f.1.1 1 20.19 odd 2
147.6.a.a.1.1 1 105.104 even 2
147.6.e.h.67.1 2 105.44 odd 6
147.6.e.h.79.1 2 105.74 odd 6
147.6.e.k.67.1 2 105.89 even 6
147.6.e.k.79.1 2 105.59 even 6
192.6.a.d.1.1 1 120.29 odd 2
192.6.a.l.1.1 1 120.59 even 2
225.6.a.a.1.1 1 1.1 even 1 trivial
225.6.b.b.199.1 2 5.2 odd 4
225.6.b.b.199.2 2 5.3 odd 4
363.6.a.d.1.1 1 165.164 even 2
441.6.a.i.1.1 1 35.34 odd 2
507.6.a.b.1.1 1 195.194 odd 2
576.6.a.s.1.1 1 40.29 even 2
576.6.a.t.1.1 1 40.19 odd 2
768.6.d.h.385.1 2 240.179 even 4
768.6.d.h.385.2 2 240.59 even 4
768.6.d.k.385.1 2 240.149 odd 4
768.6.d.k.385.2 2 240.29 odd 4
867.6.a.a.1.1 1 255.254 odd 2
1083.6.a.c.1.1 1 285.284 even 2
1089.6.a.b.1.1 1 55.54 odd 2