Properties

Label 78.4.a.c.1.1
Level $78$
Weight $4$
Character 78.1
Self dual yes
Analytic conductor $4.602$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

Related objects

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [78,4,Mod(1,78)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(78, 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("78.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 78 = 2 \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 78.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: \(4.60214898045\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 78.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} +10.0000 q^{5} -6.00000 q^{6} -8.00000 q^{7} -8.00000 q^{8} +9.00000 q^{9} +O(q^{10})\) \(q-2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} +10.0000 q^{5} -6.00000 q^{6} -8.00000 q^{7} -8.00000 q^{8} +9.00000 q^{9} -20.0000 q^{10} +40.0000 q^{11} +12.0000 q^{12} +13.0000 q^{13} +16.0000 q^{14} +30.0000 q^{15} +16.0000 q^{16} +130.000 q^{17} -18.0000 q^{18} -20.0000 q^{19} +40.0000 q^{20} -24.0000 q^{21} -80.0000 q^{22} -24.0000 q^{24} -25.0000 q^{25} -26.0000 q^{26} +27.0000 q^{27} -32.0000 q^{28} -18.0000 q^{29} -60.0000 q^{30} -184.000 q^{31} -32.0000 q^{32} +120.000 q^{33} -260.000 q^{34} -80.0000 q^{35} +36.0000 q^{36} -74.0000 q^{37} +40.0000 q^{38} +39.0000 q^{39} -80.0000 q^{40} -362.000 q^{41} +48.0000 q^{42} +76.0000 q^{43} +160.000 q^{44} +90.0000 q^{45} -452.000 q^{47} +48.0000 q^{48} -279.000 q^{49} +50.0000 q^{50} +390.000 q^{51} +52.0000 q^{52} +382.000 q^{53} -54.0000 q^{54} +400.000 q^{55} +64.0000 q^{56} -60.0000 q^{57} +36.0000 q^{58} +464.000 q^{59} +120.000 q^{60} +358.000 q^{61} +368.000 q^{62} -72.0000 q^{63} +64.0000 q^{64} +130.000 q^{65} -240.000 q^{66} -700.000 q^{67} +520.000 q^{68} +160.000 q^{70} -748.000 q^{71} -72.0000 q^{72} +1058.00 q^{73} +148.000 q^{74} -75.0000 q^{75} -80.0000 q^{76} -320.000 q^{77} -78.0000 q^{78} -976.000 q^{79} +160.000 q^{80} +81.0000 q^{81} +724.000 q^{82} -1008.00 q^{83} -96.0000 q^{84} +1300.00 q^{85} -152.000 q^{86} -54.0000 q^{87} -320.000 q^{88} -386.000 q^{89} -180.000 q^{90} -104.000 q^{91} -552.000 q^{93} +904.000 q^{94} -200.000 q^{95} -96.0000 q^{96} -614.000 q^{97} +558.000 q^{98} +360.000 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.00000 −0.707107
\(3\) 3.00000 0.577350
\(4\) 4.00000 0.500000
\(5\) 10.0000 0.894427 0.447214 0.894427i \(-0.352416\pi\)
0.447214 + 0.894427i \(0.352416\pi\)
\(6\) −6.00000 −0.408248
\(7\) −8.00000 −0.431959 −0.215980 0.976398i \(-0.569295\pi\)
−0.215980 + 0.976398i \(0.569295\pi\)
\(8\) −8.00000 −0.353553
\(9\) 9.00000 0.333333
\(10\) −20.0000 −0.632456
\(11\) 40.0000 1.09640 0.548202 0.836346i \(-0.315312\pi\)
0.548202 + 0.836346i \(0.315312\pi\)
\(12\) 12.0000 0.288675
\(13\) 13.0000 0.277350
\(14\) 16.0000 0.305441
\(15\) 30.0000 0.516398
\(16\) 16.0000 0.250000
\(17\) 130.000 1.85468 0.927342 0.374215i \(-0.122088\pi\)
0.927342 + 0.374215i \(0.122088\pi\)
\(18\) −18.0000 −0.235702
\(19\) −20.0000 −0.241490 −0.120745 0.992684i \(-0.538528\pi\)
−0.120745 + 0.992684i \(0.538528\pi\)
\(20\) 40.0000 0.447214
\(21\) −24.0000 −0.249392
\(22\) −80.0000 −0.775275
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) −24.0000 −0.204124
\(25\) −25.0000 −0.200000
\(26\) −26.0000 −0.196116
\(27\) 27.0000 0.192450
\(28\) −32.0000 −0.215980
\(29\) −18.0000 −0.115259 −0.0576296 0.998338i \(-0.518354\pi\)
−0.0576296 + 0.998338i \(0.518354\pi\)
\(30\) −60.0000 −0.365148
\(31\) −184.000 −1.06604 −0.533022 0.846101i \(-0.678944\pi\)
−0.533022 + 0.846101i \(0.678944\pi\)
\(32\) −32.0000 −0.176777
\(33\) 120.000 0.633010
\(34\) −260.000 −1.31146
\(35\) −80.0000 −0.386356
\(36\) 36.0000 0.166667
\(37\) −74.0000 −0.328798 −0.164399 0.986394i \(-0.552568\pi\)
−0.164399 + 0.986394i \(0.552568\pi\)
\(38\) 40.0000 0.170759
\(39\) 39.0000 0.160128
\(40\) −80.0000 −0.316228
\(41\) −362.000 −1.37890 −0.689450 0.724333i \(-0.742148\pi\)
−0.689450 + 0.724333i \(0.742148\pi\)
\(42\) 48.0000 0.176347
\(43\) 76.0000 0.269532 0.134766 0.990877i \(-0.456972\pi\)
0.134766 + 0.990877i \(0.456972\pi\)
\(44\) 160.000 0.548202
\(45\) 90.0000 0.298142
\(46\) 0 0
\(47\) −452.000 −1.40279 −0.701393 0.712774i \(-0.747438\pi\)
−0.701393 + 0.712774i \(0.747438\pi\)
\(48\) 48.0000 0.144338
\(49\) −279.000 −0.813411
\(50\) 50.0000 0.141421
\(51\) 390.000 1.07080
\(52\) 52.0000 0.138675
\(53\) 382.000 0.990033 0.495016 0.868884i \(-0.335162\pi\)
0.495016 + 0.868884i \(0.335162\pi\)
\(54\) −54.0000 −0.136083
\(55\) 400.000 0.980654
\(56\) 64.0000 0.152721
\(57\) −60.0000 −0.139424
\(58\) 36.0000 0.0815005
\(59\) 464.000 1.02386 0.511929 0.859028i \(-0.328931\pi\)
0.511929 + 0.859028i \(0.328931\pi\)
\(60\) 120.000 0.258199
\(61\) 358.000 0.751430 0.375715 0.926735i \(-0.377397\pi\)
0.375715 + 0.926735i \(0.377397\pi\)
\(62\) 368.000 0.753807
\(63\) −72.0000 −0.143986
\(64\) 64.0000 0.125000
\(65\) 130.000 0.248069
\(66\) −240.000 −0.447605
\(67\) −700.000 −1.27640 −0.638199 0.769872i \(-0.720320\pi\)
−0.638199 + 0.769872i \(0.720320\pi\)
\(68\) 520.000 0.927342
\(69\) 0 0
\(70\) 160.000 0.273195
\(71\) −748.000 −1.25030 −0.625150 0.780505i \(-0.714962\pi\)
−0.625150 + 0.780505i \(0.714962\pi\)
\(72\) −72.0000 −0.117851
\(73\) 1058.00 1.69629 0.848147 0.529760i \(-0.177718\pi\)
0.848147 + 0.529760i \(0.177718\pi\)
\(74\) 148.000 0.232495
\(75\) −75.0000 −0.115470
\(76\) −80.0000 −0.120745
\(77\) −320.000 −0.473602
\(78\) −78.0000 −0.113228
\(79\) −976.000 −1.38998 −0.694991 0.719018i \(-0.744592\pi\)
−0.694991 + 0.719018i \(0.744592\pi\)
\(80\) 160.000 0.223607
\(81\) 81.0000 0.111111
\(82\) 724.000 0.975030
\(83\) −1008.00 −1.33304 −0.666520 0.745487i \(-0.732217\pi\)
−0.666520 + 0.745487i \(0.732217\pi\)
\(84\) −96.0000 −0.124696
\(85\) 1300.00 1.65888
\(86\) −152.000 −0.190588
\(87\) −54.0000 −0.0665449
\(88\) −320.000 −0.387638
\(89\) −386.000 −0.459729 −0.229865 0.973223i \(-0.573828\pi\)
−0.229865 + 0.973223i \(0.573828\pi\)
\(90\) −180.000 −0.210819
\(91\) −104.000 −0.119804
\(92\) 0 0
\(93\) −552.000 −0.615481
\(94\) 904.000 0.991920
\(95\) −200.000 −0.215995
\(96\) −96.0000 −0.102062
\(97\) −614.000 −0.642704 −0.321352 0.946960i \(-0.604137\pi\)
−0.321352 + 0.946960i \(0.604137\pi\)
\(98\) 558.000 0.575168
\(99\) 360.000 0.365468
\(100\) −100.000 −0.100000
\(101\) 518.000 0.510326 0.255163 0.966898i \(-0.417871\pi\)
0.255163 + 0.966898i \(0.417871\pi\)
\(102\) −780.000 −0.757172
\(103\) 112.000 0.107143 0.0535713 0.998564i \(-0.482940\pi\)
0.0535713 + 0.998564i \(0.482940\pi\)
\(104\) −104.000 −0.0980581
\(105\) −240.000 −0.223063
\(106\) −764.000 −0.700059
\(107\) −372.000 −0.336099 −0.168050 0.985779i \(-0.553747\pi\)
−0.168050 + 0.985779i \(0.553747\pi\)
\(108\) 108.000 0.0962250
\(109\) 934.000 0.820743 0.410371 0.911918i \(-0.365399\pi\)
0.410371 + 0.911918i \(0.365399\pi\)
\(110\) −800.000 −0.693427
\(111\) −222.000 −0.189832
\(112\) −128.000 −0.107990
\(113\) 1914.00 1.59340 0.796699 0.604376i \(-0.206578\pi\)
0.796699 + 0.604376i \(0.206578\pi\)
\(114\) 120.000 0.0985880
\(115\) 0 0
\(116\) −72.0000 −0.0576296
\(117\) 117.000 0.0924500
\(118\) −928.000 −0.723977
\(119\) −1040.00 −0.801148
\(120\) −240.000 −0.182574
\(121\) 269.000 0.202104
\(122\) −716.000 −0.531341
\(123\) −1086.00 −0.796108
\(124\) −736.000 −0.533022
\(125\) −1500.00 −1.07331
\(126\) 144.000 0.101814
\(127\) 1296.00 0.905523 0.452761 0.891632i \(-0.350439\pi\)
0.452761 + 0.891632i \(0.350439\pi\)
\(128\) −128.000 −0.0883883
\(129\) 228.000 0.155615
\(130\) −260.000 −0.175412
\(131\) −892.000 −0.594919 −0.297460 0.954734i \(-0.596139\pi\)
−0.297460 + 0.954734i \(0.596139\pi\)
\(132\) 480.000 0.316505
\(133\) 160.000 0.104314
\(134\) 1400.00 0.902549
\(135\) 270.000 0.172133
\(136\) −1040.00 −0.655730
\(137\) 2326.00 1.45054 0.725269 0.688466i \(-0.241716\pi\)
0.725269 + 0.688466i \(0.241716\pi\)
\(138\) 0 0
\(139\) 1932.00 1.17892 0.589461 0.807797i \(-0.299340\pi\)
0.589461 + 0.807797i \(0.299340\pi\)
\(140\) −320.000 −0.193178
\(141\) −1356.00 −0.809899
\(142\) 1496.00 0.884095
\(143\) 520.000 0.304088
\(144\) 144.000 0.0833333
\(145\) −180.000 −0.103091
\(146\) −2116.00 −1.19946
\(147\) −837.000 −0.469623
\(148\) −296.000 −0.164399
\(149\) 882.000 0.484941 0.242471 0.970159i \(-0.422042\pi\)
0.242471 + 0.970159i \(0.422042\pi\)
\(150\) 150.000 0.0816497
\(151\) −1776.00 −0.957145 −0.478572 0.878048i \(-0.658846\pi\)
−0.478572 + 0.878048i \(0.658846\pi\)
\(152\) 160.000 0.0853797
\(153\) 1170.00 0.618228
\(154\) 640.000 0.334887
\(155\) −1840.00 −0.953499
\(156\) 156.000 0.0800641
\(157\) −2410.00 −1.22509 −0.612544 0.790436i \(-0.709854\pi\)
−0.612544 + 0.790436i \(0.709854\pi\)
\(158\) 1952.00 0.982866
\(159\) 1146.00 0.571596
\(160\) −320.000 −0.158114
\(161\) 0 0
\(162\) −162.000 −0.0785674
\(163\) 3212.00 1.54346 0.771728 0.635953i \(-0.219393\pi\)
0.771728 + 0.635953i \(0.219393\pi\)
\(164\) −1448.00 −0.689450
\(165\) 1200.00 0.566181
\(166\) 2016.00 0.942602
\(167\) 1668.00 0.772896 0.386448 0.922311i \(-0.373702\pi\)
0.386448 + 0.922311i \(0.373702\pi\)
\(168\) 192.000 0.0881733
\(169\) 169.000 0.0769231
\(170\) −2600.00 −1.17301
\(171\) −180.000 −0.0804967
\(172\) 304.000 0.134766
\(173\) 3598.00 1.58122 0.790609 0.612321i \(-0.209764\pi\)
0.790609 + 0.612321i \(0.209764\pi\)
\(174\) 108.000 0.0470544
\(175\) 200.000 0.0863919
\(176\) 640.000 0.274101
\(177\) 1392.00 0.591125
\(178\) 772.000 0.325078
\(179\) 1068.00 0.445956 0.222978 0.974824i \(-0.428422\pi\)
0.222978 + 0.974824i \(0.428422\pi\)
\(180\) 360.000 0.149071
\(181\) −4786.00 −1.96542 −0.982709 0.185158i \(-0.940720\pi\)
−0.982709 + 0.185158i \(0.940720\pi\)
\(182\) 208.000 0.0847142
\(183\) 1074.00 0.433838
\(184\) 0 0
\(185\) −740.000 −0.294086
\(186\) 1104.00 0.435211
\(187\) 5200.00 2.03348
\(188\) −1808.00 −0.701393
\(189\) −216.000 −0.0831306
\(190\) 400.000 0.152732
\(191\) −1312.00 −0.497031 −0.248516 0.968628i \(-0.579943\pi\)
−0.248516 + 0.968628i \(0.579943\pi\)
\(192\) 192.000 0.0721688
\(193\) −350.000 −0.130537 −0.0652683 0.997868i \(-0.520790\pi\)
−0.0652683 + 0.997868i \(0.520790\pi\)
\(194\) 1228.00 0.454460
\(195\) 390.000 0.143223
\(196\) −1116.00 −0.406706
\(197\) −342.000 −0.123688 −0.0618439 0.998086i \(-0.519698\pi\)
−0.0618439 + 0.998086i \(0.519698\pi\)
\(198\) −720.000 −0.258425
\(199\) −3368.00 −1.19975 −0.599877 0.800092i \(-0.704784\pi\)
−0.599877 + 0.800092i \(0.704784\pi\)
\(200\) 200.000 0.0707107
\(201\) −2100.00 −0.736928
\(202\) −1036.00 −0.360855
\(203\) 144.000 0.0497873
\(204\) 1560.00 0.535401
\(205\) −3620.00 −1.23333
\(206\) −224.000 −0.0757613
\(207\) 0 0
\(208\) 208.000 0.0693375
\(209\) −800.000 −0.264771
\(210\) 480.000 0.157729
\(211\) −2004.00 −0.653844 −0.326922 0.945051i \(-0.606011\pi\)
−0.326922 + 0.945051i \(0.606011\pi\)
\(212\) 1528.00 0.495016
\(213\) −2244.00 −0.721861
\(214\) 744.000 0.237658
\(215\) 760.000 0.241077
\(216\) −216.000 −0.0680414
\(217\) 1472.00 0.460488
\(218\) −1868.00 −0.580353
\(219\) 3174.00 0.979356
\(220\) 1600.00 0.490327
\(221\) 1690.00 0.514397
\(222\) 444.000 0.134231
\(223\) −5608.00 −1.68403 −0.842017 0.539451i \(-0.818632\pi\)
−0.842017 + 0.539451i \(0.818632\pi\)
\(224\) 256.000 0.0763604
\(225\) −225.000 −0.0666667
\(226\) −3828.00 −1.12670
\(227\) −1928.00 −0.563726 −0.281863 0.959455i \(-0.590952\pi\)
−0.281863 + 0.959455i \(0.590952\pi\)
\(228\) −240.000 −0.0697122
\(229\) −3938.00 −1.13638 −0.568189 0.822898i \(-0.692356\pi\)
−0.568189 + 0.822898i \(0.692356\pi\)
\(230\) 0 0
\(231\) −960.000 −0.273434
\(232\) 144.000 0.0407503
\(233\) 2562.00 0.720353 0.360176 0.932884i \(-0.382717\pi\)
0.360176 + 0.932884i \(0.382717\pi\)
\(234\) −234.000 −0.0653720
\(235\) −4520.00 −1.25469
\(236\) 1856.00 0.511929
\(237\) −2928.00 −0.802506
\(238\) 2080.00 0.566497
\(239\) 7164.00 1.93891 0.969457 0.245260i \(-0.0788733\pi\)
0.969457 + 0.245260i \(0.0788733\pi\)
\(240\) 480.000 0.129099
\(241\) −6182.00 −1.65236 −0.826178 0.563410i \(-0.809489\pi\)
−0.826178 + 0.563410i \(0.809489\pi\)
\(242\) −538.000 −0.142909
\(243\) 243.000 0.0641500
\(244\) 1432.00 0.375715
\(245\) −2790.00 −0.727537
\(246\) 2172.00 0.562934
\(247\) −260.000 −0.0669773
\(248\) 1472.00 0.376904
\(249\) −3024.00 −0.769631
\(250\) 3000.00 0.758947
\(251\) −1396.00 −0.351055 −0.175527 0.984475i \(-0.556163\pi\)
−0.175527 + 0.984475i \(0.556163\pi\)
\(252\) −288.000 −0.0719932
\(253\) 0 0
\(254\) −2592.00 −0.640301
\(255\) 3900.00 0.957755
\(256\) 256.000 0.0625000
\(257\) 6906.00 1.67620 0.838102 0.545514i \(-0.183665\pi\)
0.838102 + 0.545514i \(0.183665\pi\)
\(258\) −456.000 −0.110036
\(259\) 592.000 0.142027
\(260\) 520.000 0.124035
\(261\) −162.000 −0.0384197
\(262\) 1784.00 0.420671
\(263\) −6848.00 −1.60557 −0.802787 0.596266i \(-0.796650\pi\)
−0.802787 + 0.596266i \(0.796650\pi\)
\(264\) −960.000 −0.223803
\(265\) 3820.00 0.885512
\(266\) −320.000 −0.0737611
\(267\) −1158.00 −0.265425
\(268\) −2800.00 −0.638199
\(269\) −6034.00 −1.36766 −0.683828 0.729643i \(-0.739686\pi\)
−0.683828 + 0.729643i \(0.739686\pi\)
\(270\) −540.000 −0.121716
\(271\) 4832.00 1.08311 0.541556 0.840665i \(-0.317836\pi\)
0.541556 + 0.840665i \(0.317836\pi\)
\(272\) 2080.00 0.463671
\(273\) −312.000 −0.0691689
\(274\) −4652.00 −1.02568
\(275\) −1000.00 −0.219281
\(276\) 0 0
\(277\) −4082.00 −0.885428 −0.442714 0.896663i \(-0.645984\pi\)
−0.442714 + 0.896663i \(0.645984\pi\)
\(278\) −3864.00 −0.833623
\(279\) −1656.00 −0.355348
\(280\) 640.000 0.136598
\(281\) 3350.00 0.711189 0.355595 0.934640i \(-0.384278\pi\)
0.355595 + 0.934640i \(0.384278\pi\)
\(282\) 2712.00 0.572685
\(283\) 7796.00 1.63754 0.818770 0.574121i \(-0.194656\pi\)
0.818770 + 0.574121i \(0.194656\pi\)
\(284\) −2992.00 −0.625150
\(285\) −600.000 −0.124705
\(286\) −1040.00 −0.215023
\(287\) 2896.00 0.595629
\(288\) −288.000 −0.0589256
\(289\) 11987.0 2.43985
\(290\) 360.000 0.0728963
\(291\) −1842.00 −0.371065
\(292\) 4232.00 0.848147
\(293\) 3922.00 0.781999 0.390999 0.920391i \(-0.372129\pi\)
0.390999 + 0.920391i \(0.372129\pi\)
\(294\) 1674.00 0.332074
\(295\) 4640.00 0.915767
\(296\) 592.000 0.116248
\(297\) 1080.00 0.211003
\(298\) −1764.00 −0.342905
\(299\) 0 0
\(300\) −300.000 −0.0577350
\(301\) −608.000 −0.116427
\(302\) 3552.00 0.676803
\(303\) 1554.00 0.294637
\(304\) −320.000 −0.0603726
\(305\) 3580.00 0.672099
\(306\) −2340.00 −0.437153
\(307\) 5956.00 1.10725 0.553627 0.832765i \(-0.313243\pi\)
0.553627 + 0.832765i \(0.313243\pi\)
\(308\) −1280.00 −0.236801
\(309\) 336.000 0.0618588
\(310\) 3680.00 0.674226
\(311\) 2352.00 0.428841 0.214421 0.976741i \(-0.431214\pi\)
0.214421 + 0.976741i \(0.431214\pi\)
\(312\) −312.000 −0.0566139
\(313\) 8442.00 1.52450 0.762252 0.647280i \(-0.224093\pi\)
0.762252 + 0.647280i \(0.224093\pi\)
\(314\) 4820.00 0.866269
\(315\) −720.000 −0.128785
\(316\) −3904.00 −0.694991
\(317\) −5550.00 −0.983341 −0.491670 0.870781i \(-0.663614\pi\)
−0.491670 + 0.870781i \(0.663614\pi\)
\(318\) −2292.00 −0.404179
\(319\) −720.000 −0.126371
\(320\) 640.000 0.111803
\(321\) −1116.00 −0.194047
\(322\) 0 0
\(323\) −2600.00 −0.447888
\(324\) 324.000 0.0555556
\(325\) −325.000 −0.0554700
\(326\) −6424.00 −1.09139
\(327\) 2802.00 0.473856
\(328\) 2896.00 0.487515
\(329\) 3616.00 0.605947
\(330\) −2400.00 −0.400350
\(331\) 140.000 0.0232480 0.0116240 0.999932i \(-0.496300\pi\)
0.0116240 + 0.999932i \(0.496300\pi\)
\(332\) −4032.00 −0.666520
\(333\) −666.000 −0.109599
\(334\) −3336.00 −0.546520
\(335\) −7000.00 −1.14164
\(336\) −384.000 −0.0623480
\(337\) −6174.00 −0.997980 −0.498990 0.866608i \(-0.666296\pi\)
−0.498990 + 0.866608i \(0.666296\pi\)
\(338\) −338.000 −0.0543928
\(339\) 5742.00 0.919949
\(340\) 5200.00 0.829440
\(341\) −7360.00 −1.16882
\(342\) 360.000 0.0569198
\(343\) 4976.00 0.783320
\(344\) −608.000 −0.0952941
\(345\) 0 0
\(346\) −7196.00 −1.11809
\(347\) −2988.00 −0.462260 −0.231130 0.972923i \(-0.574242\pi\)
−0.231130 + 0.972923i \(0.574242\pi\)
\(348\) −216.000 −0.0332725
\(349\) −162.000 −0.0248472 −0.0124236 0.999923i \(-0.503955\pi\)
−0.0124236 + 0.999923i \(0.503955\pi\)
\(350\) −400.000 −0.0610883
\(351\) 351.000 0.0533761
\(352\) −1280.00 −0.193819
\(353\) −10754.0 −1.62147 −0.810733 0.585416i \(-0.800931\pi\)
−0.810733 + 0.585416i \(0.800931\pi\)
\(354\) −2784.00 −0.417989
\(355\) −7480.00 −1.11830
\(356\) −1544.00 −0.229865
\(357\) −3120.00 −0.462543
\(358\) −2136.00 −0.315338
\(359\) 3588.00 0.527486 0.263743 0.964593i \(-0.415043\pi\)
0.263743 + 0.964593i \(0.415043\pi\)
\(360\) −720.000 −0.105409
\(361\) −6459.00 −0.941682
\(362\) 9572.00 1.38976
\(363\) 807.000 0.116685
\(364\) −416.000 −0.0599020
\(365\) 10580.0 1.51721
\(366\) −2148.00 −0.306770
\(367\) 11272.0 1.60325 0.801626 0.597826i \(-0.203968\pi\)
0.801626 + 0.597826i \(0.203968\pi\)
\(368\) 0 0
\(369\) −3258.00 −0.459633
\(370\) 1480.00 0.207950
\(371\) −3056.00 −0.427654
\(372\) −2208.00 −0.307741
\(373\) −10914.0 −1.51503 −0.757514 0.652819i \(-0.773586\pi\)
−0.757514 + 0.652819i \(0.773586\pi\)
\(374\) −10400.0 −1.43789
\(375\) −4500.00 −0.619677
\(376\) 3616.00 0.495960
\(377\) −234.000 −0.0319671
\(378\) 432.000 0.0587822
\(379\) 8100.00 1.09781 0.548904 0.835886i \(-0.315045\pi\)
0.548904 + 0.835886i \(0.315045\pi\)
\(380\) −800.000 −0.107998
\(381\) 3888.00 0.522804
\(382\) 2624.00 0.351454
\(383\) 6180.00 0.824499 0.412250 0.911071i \(-0.364743\pi\)
0.412250 + 0.911071i \(0.364743\pi\)
\(384\) −384.000 −0.0510310
\(385\) −3200.00 −0.423603
\(386\) 700.000 0.0923033
\(387\) 684.000 0.0898441
\(388\) −2456.00 −0.321352
\(389\) −7522.00 −0.980413 −0.490206 0.871606i \(-0.663079\pi\)
−0.490206 + 0.871606i \(0.663079\pi\)
\(390\) −780.000 −0.101274
\(391\) 0 0
\(392\) 2232.00 0.287584
\(393\) −2676.00 −0.343477
\(394\) 684.000 0.0874605
\(395\) −9760.00 −1.24324
\(396\) 1440.00 0.182734
\(397\) 6078.00 0.768378 0.384189 0.923254i \(-0.374481\pi\)
0.384189 + 0.923254i \(0.374481\pi\)
\(398\) 6736.00 0.848355
\(399\) 480.000 0.0602257
\(400\) −400.000 −0.0500000
\(401\) 1830.00 0.227895 0.113947 0.993487i \(-0.463650\pi\)
0.113947 + 0.993487i \(0.463650\pi\)
\(402\) 4200.00 0.521087
\(403\) −2392.00 −0.295668
\(404\) 2072.00 0.255163
\(405\) 810.000 0.0993808
\(406\) −288.000 −0.0352049
\(407\) −2960.00 −0.360496
\(408\) −3120.00 −0.378586
\(409\) 12434.0 1.50323 0.751616 0.659601i \(-0.229275\pi\)
0.751616 + 0.659601i \(0.229275\pi\)
\(410\) 7240.00 0.872093
\(411\) 6978.00 0.837468
\(412\) 448.000 0.0535713
\(413\) −3712.00 −0.442265
\(414\) 0 0
\(415\) −10080.0 −1.19231
\(416\) −416.000 −0.0490290
\(417\) 5796.00 0.680651
\(418\) 1600.00 0.187221
\(419\) −14188.0 −1.65425 −0.827123 0.562021i \(-0.810024\pi\)
−0.827123 + 0.562021i \(0.810024\pi\)
\(420\) −960.000 −0.111531
\(421\) 8638.00 0.999977 0.499989 0.866032i \(-0.333338\pi\)
0.499989 + 0.866032i \(0.333338\pi\)
\(422\) 4008.00 0.462337
\(423\) −4068.00 −0.467596
\(424\) −3056.00 −0.350029
\(425\) −3250.00 −0.370937
\(426\) 4488.00 0.510433
\(427\) −2864.00 −0.324587
\(428\) −1488.00 −0.168050
\(429\) 1560.00 0.175565
\(430\) −1520.00 −0.170467
\(431\) 4292.00 0.479671 0.239836 0.970813i \(-0.422906\pi\)
0.239836 + 0.970813i \(0.422906\pi\)
\(432\) 432.000 0.0481125
\(433\) −5982.00 −0.663918 −0.331959 0.943294i \(-0.607710\pi\)
−0.331959 + 0.943294i \(0.607710\pi\)
\(434\) −2944.00 −0.325614
\(435\) −540.000 −0.0595196
\(436\) 3736.00 0.410371
\(437\) 0 0
\(438\) −6348.00 −0.692510
\(439\) 256.000 0.0278319 0.0139160 0.999903i \(-0.495570\pi\)
0.0139160 + 0.999903i \(0.495570\pi\)
\(440\) −3200.00 −0.346714
\(441\) −2511.00 −0.271137
\(442\) −3380.00 −0.363733
\(443\) 12556.0 1.34662 0.673311 0.739359i \(-0.264872\pi\)
0.673311 + 0.739359i \(0.264872\pi\)
\(444\) −888.000 −0.0949158
\(445\) −3860.00 −0.411194
\(446\) 11216.0 1.19079
\(447\) 2646.00 0.279981
\(448\) −512.000 −0.0539949
\(449\) 5574.00 0.585865 0.292932 0.956133i \(-0.405369\pi\)
0.292932 + 0.956133i \(0.405369\pi\)
\(450\) 450.000 0.0471405
\(451\) −14480.0 −1.51183
\(452\) 7656.00 0.796699
\(453\) −5328.00 −0.552608
\(454\) 3856.00 0.398615
\(455\) −1040.00 −0.107156
\(456\) 480.000 0.0492940
\(457\) 1266.00 0.129586 0.0647932 0.997899i \(-0.479361\pi\)
0.0647932 + 0.997899i \(0.479361\pi\)
\(458\) 7876.00 0.803540
\(459\) 3510.00 0.356934
\(460\) 0 0
\(461\) 7554.00 0.763178 0.381589 0.924332i \(-0.375377\pi\)
0.381589 + 0.924332i \(0.375377\pi\)
\(462\) 1920.00 0.193347
\(463\) −6752.00 −0.677737 −0.338868 0.940834i \(-0.610044\pi\)
−0.338868 + 0.940834i \(0.610044\pi\)
\(464\) −288.000 −0.0288148
\(465\) −5520.00 −0.550503
\(466\) −5124.00 −0.509366
\(467\) 7924.00 0.785180 0.392590 0.919714i \(-0.371579\pi\)
0.392590 + 0.919714i \(0.371579\pi\)
\(468\) 468.000 0.0462250
\(469\) 5600.00 0.551352
\(470\) 9040.00 0.887200
\(471\) −7230.00 −0.707305
\(472\) −3712.00 −0.361989
\(473\) 3040.00 0.295517
\(474\) 5856.00 0.567458
\(475\) 500.000 0.0482980
\(476\) −4160.00 −0.400574
\(477\) 3438.00 0.330011
\(478\) −14328.0 −1.37102
\(479\) −11084.0 −1.05729 −0.528644 0.848844i \(-0.677299\pi\)
−0.528644 + 0.848844i \(0.677299\pi\)
\(480\) −960.000 −0.0912871
\(481\) −962.000 −0.0911922
\(482\) 12364.0 1.16839
\(483\) 0 0
\(484\) 1076.00 0.101052
\(485\) −6140.00 −0.574852
\(486\) −486.000 −0.0453609
\(487\) 4432.00 0.412388 0.206194 0.978511i \(-0.433892\pi\)
0.206194 + 0.978511i \(0.433892\pi\)
\(488\) −2864.00 −0.265670
\(489\) 9636.00 0.891114
\(490\) 5580.00 0.514446
\(491\) −1140.00 −0.104781 −0.0523905 0.998627i \(-0.516684\pi\)
−0.0523905 + 0.998627i \(0.516684\pi\)
\(492\) −4344.00 −0.398054
\(493\) −2340.00 −0.213769
\(494\) 520.000 0.0473601
\(495\) 3600.00 0.326885
\(496\) −2944.00 −0.266511
\(497\) 5984.00 0.540079
\(498\) 6048.00 0.544212
\(499\) 1764.00 0.158251 0.0791257 0.996865i \(-0.474787\pi\)
0.0791257 + 0.996865i \(0.474787\pi\)
\(500\) −6000.00 −0.536656
\(501\) 5004.00 0.446232
\(502\) 2792.00 0.248233
\(503\) 16976.0 1.50482 0.752408 0.658697i \(-0.228892\pi\)
0.752408 + 0.658697i \(0.228892\pi\)
\(504\) 576.000 0.0509069
\(505\) 5180.00 0.456449
\(506\) 0 0
\(507\) 507.000 0.0444116
\(508\) 5184.00 0.452761
\(509\) 9474.00 0.825005 0.412503 0.910956i \(-0.364655\pi\)
0.412503 + 0.910956i \(0.364655\pi\)
\(510\) −7800.00 −0.677235
\(511\) −8464.00 −0.732731
\(512\) −512.000 −0.0441942
\(513\) −540.000 −0.0464748
\(514\) −13812.0 −1.18526
\(515\) 1120.00 0.0958313
\(516\) 912.000 0.0778073
\(517\) −18080.0 −1.53802
\(518\) −1184.00 −0.100429
\(519\) 10794.0 0.912917
\(520\) −1040.00 −0.0877058
\(521\) 14114.0 1.18684 0.593422 0.804892i \(-0.297777\pi\)
0.593422 + 0.804892i \(0.297777\pi\)
\(522\) 324.000 0.0271668
\(523\) 20284.0 1.69590 0.847952 0.530074i \(-0.177836\pi\)
0.847952 + 0.530074i \(0.177836\pi\)
\(524\) −3568.00 −0.297460
\(525\) 600.000 0.0498784
\(526\) 13696.0 1.13531
\(527\) −23920.0 −1.97718
\(528\) 1920.00 0.158252
\(529\) −12167.0 −1.00000
\(530\) −7640.00 −0.626152
\(531\) 4176.00 0.341286
\(532\) 640.000 0.0521570
\(533\) −4706.00 −0.382438
\(534\) 2316.00 0.187684
\(535\) −3720.00 −0.300616
\(536\) 5600.00 0.451275
\(537\) 3204.00 0.257473
\(538\) 12068.0 0.967079
\(539\) −11160.0 −0.891828
\(540\) 1080.00 0.0860663
\(541\) −14362.0 −1.14135 −0.570675 0.821176i \(-0.693318\pi\)
−0.570675 + 0.821176i \(0.693318\pi\)
\(542\) −9664.00 −0.765875
\(543\) −14358.0 −1.13473
\(544\) −4160.00 −0.327865
\(545\) 9340.00 0.734095
\(546\) 624.000 0.0489098
\(547\) −20956.0 −1.63805 −0.819025 0.573757i \(-0.805485\pi\)
−0.819025 + 0.573757i \(0.805485\pi\)
\(548\) 9304.00 0.725269
\(549\) 3222.00 0.250477
\(550\) 2000.00 0.155055
\(551\) 360.000 0.0278340
\(552\) 0 0
\(553\) 7808.00 0.600416
\(554\) 8164.00 0.626092
\(555\) −2220.00 −0.169791
\(556\) 7728.00 0.589461
\(557\) −4134.00 −0.314476 −0.157238 0.987561i \(-0.550259\pi\)
−0.157238 + 0.987561i \(0.550259\pi\)
\(558\) 3312.00 0.251269
\(559\) 988.000 0.0747548
\(560\) −1280.00 −0.0965891
\(561\) 15600.0 1.17403
\(562\) −6700.00 −0.502887
\(563\) −16228.0 −1.21479 −0.607397 0.794399i \(-0.707786\pi\)
−0.607397 + 0.794399i \(0.707786\pi\)
\(564\) −5424.00 −0.404950
\(565\) 19140.0 1.42518
\(566\) −15592.0 −1.15792
\(567\) −648.000 −0.0479955
\(568\) 5984.00 0.442048
\(569\) 2514.00 0.185224 0.0926119 0.995702i \(-0.470478\pi\)
0.0926119 + 0.995702i \(0.470478\pi\)
\(570\) 1200.00 0.0881798
\(571\) −11612.0 −0.851046 −0.425523 0.904948i \(-0.639910\pi\)
−0.425523 + 0.904948i \(0.639910\pi\)
\(572\) 2080.00 0.152044
\(573\) −3936.00 −0.286961
\(574\) −5792.00 −0.421173
\(575\) 0 0
\(576\) 576.000 0.0416667
\(577\) 6354.00 0.458441 0.229221 0.973375i \(-0.426382\pi\)
0.229221 + 0.973375i \(0.426382\pi\)
\(578\) −23974.0 −1.72524
\(579\) −1050.00 −0.0753653
\(580\) −720.000 −0.0515455
\(581\) 8064.00 0.575819
\(582\) 3684.00 0.262383
\(583\) 15280.0 1.08548
\(584\) −8464.00 −0.599731
\(585\) 1170.00 0.0826898
\(586\) −7844.00 −0.552957
\(587\) −13240.0 −0.930960 −0.465480 0.885059i \(-0.654118\pi\)
−0.465480 + 0.885059i \(0.654118\pi\)
\(588\) −3348.00 −0.234812
\(589\) 3680.00 0.257439
\(590\) −9280.00 −0.647545
\(591\) −1026.00 −0.0714112
\(592\) −1184.00 −0.0821995
\(593\) −1146.00 −0.0793602 −0.0396801 0.999212i \(-0.512634\pi\)
−0.0396801 + 0.999212i \(0.512634\pi\)
\(594\) −2160.00 −0.149202
\(595\) −10400.0 −0.716569
\(596\) 3528.00 0.242471
\(597\) −10104.0 −0.692679
\(598\) 0 0
\(599\) 10464.0 0.713769 0.356884 0.934149i \(-0.383839\pi\)
0.356884 + 0.934149i \(0.383839\pi\)
\(600\) 600.000 0.0408248
\(601\) 6650.00 0.451346 0.225673 0.974203i \(-0.427542\pi\)
0.225673 + 0.974203i \(0.427542\pi\)
\(602\) 1216.00 0.0823263
\(603\) −6300.00 −0.425466
\(604\) −7104.00 −0.478572
\(605\) 2690.00 0.180767
\(606\) −3108.00 −0.208340
\(607\) −6664.00 −0.445607 −0.222803 0.974863i \(-0.571521\pi\)
−0.222803 + 0.974863i \(0.571521\pi\)
\(608\) 640.000 0.0426898
\(609\) 432.000 0.0287447
\(610\) −7160.00 −0.475246
\(611\) −5876.00 −0.389063
\(612\) 4680.00 0.309114
\(613\) 2134.00 0.140606 0.0703030 0.997526i \(-0.477603\pi\)
0.0703030 + 0.997526i \(0.477603\pi\)
\(614\) −11912.0 −0.782947
\(615\) −10860.0 −0.712061
\(616\) 2560.00 0.167444
\(617\) −714.000 −0.0465876 −0.0232938 0.999729i \(-0.507415\pi\)
−0.0232938 + 0.999729i \(0.507415\pi\)
\(618\) −672.000 −0.0437408
\(619\) 29228.0 1.89786 0.948928 0.315494i \(-0.102170\pi\)
0.948928 + 0.315494i \(0.102170\pi\)
\(620\) −7360.00 −0.476750
\(621\) 0 0
\(622\) −4704.00 −0.303237
\(623\) 3088.00 0.198584
\(624\) 624.000 0.0400320
\(625\) −11875.0 −0.760000
\(626\) −16884.0 −1.07799
\(627\) −2400.00 −0.152866
\(628\) −9640.00 −0.612544
\(629\) −9620.00 −0.609816
\(630\) 1440.00 0.0910650
\(631\) −13536.0 −0.853977 −0.426989 0.904257i \(-0.640426\pi\)
−0.426989 + 0.904257i \(0.640426\pi\)
\(632\) 7808.00 0.491433
\(633\) −6012.00 −0.377497
\(634\) 11100.0 0.695327
\(635\) 12960.0 0.809924
\(636\) 4584.00 0.285798
\(637\) −3627.00 −0.225600
\(638\) 1440.00 0.0893576
\(639\) −6732.00 −0.416767
\(640\) −1280.00 −0.0790569
\(641\) 17218.0 1.06095 0.530476 0.847700i \(-0.322013\pi\)
0.530476 + 0.847700i \(0.322013\pi\)
\(642\) 2232.00 0.137212
\(643\) 15044.0 0.922671 0.461335 0.887226i \(-0.347370\pi\)
0.461335 + 0.887226i \(0.347370\pi\)
\(644\) 0 0
\(645\) 2280.00 0.139186
\(646\) 5200.00 0.316705
\(647\) 25176.0 1.52978 0.764892 0.644158i \(-0.222792\pi\)
0.764892 + 0.644158i \(0.222792\pi\)
\(648\) −648.000 −0.0392837
\(649\) 18560.0 1.12256
\(650\) 650.000 0.0392232
\(651\) 4416.00 0.265863
\(652\) 12848.0 0.771728
\(653\) −16034.0 −0.960887 −0.480443 0.877026i \(-0.659524\pi\)
−0.480443 + 0.877026i \(0.659524\pi\)
\(654\) −5604.00 −0.335067
\(655\) −8920.00 −0.532112
\(656\) −5792.00 −0.344725
\(657\) 9522.00 0.565432
\(658\) −7232.00 −0.428469
\(659\) 25356.0 1.49883 0.749415 0.662100i \(-0.230335\pi\)
0.749415 + 0.662100i \(0.230335\pi\)
\(660\) 4800.00 0.283091
\(661\) 18310.0 1.07742 0.538711 0.842490i \(-0.318911\pi\)
0.538711 + 0.842490i \(0.318911\pi\)
\(662\) −280.000 −0.0164388
\(663\) 5070.00 0.296987
\(664\) 8064.00 0.471301
\(665\) 1600.00 0.0933013
\(666\) 1332.00 0.0774984
\(667\) 0 0
\(668\) 6672.00 0.386448
\(669\) −16824.0 −0.972277
\(670\) 14000.0 0.807264
\(671\) 14320.0 0.823871
\(672\) 768.000 0.0440867
\(673\) 24802.0 1.42057 0.710287 0.703912i \(-0.248565\pi\)
0.710287 + 0.703912i \(0.248565\pi\)
\(674\) 12348.0 0.705678
\(675\) −675.000 −0.0384900
\(676\) 676.000 0.0384615
\(677\) −22706.0 −1.28901 −0.644507 0.764598i \(-0.722937\pi\)
−0.644507 + 0.764598i \(0.722937\pi\)
\(678\) −11484.0 −0.650502
\(679\) 4912.00 0.277622
\(680\) −10400.0 −0.586503
\(681\) −5784.00 −0.325467
\(682\) 14720.0 0.826478
\(683\) −14792.0 −0.828697 −0.414349 0.910118i \(-0.635991\pi\)
−0.414349 + 0.910118i \(0.635991\pi\)
\(684\) −720.000 −0.0402484
\(685\) 23260.0 1.29740
\(686\) −9952.00 −0.553891
\(687\) −11814.0 −0.656088
\(688\) 1216.00 0.0673831
\(689\) 4966.00 0.274586
\(690\) 0 0
\(691\) −1148.00 −0.0632011 −0.0316006 0.999501i \(-0.510060\pi\)
−0.0316006 + 0.999501i \(0.510060\pi\)
\(692\) 14392.0 0.790609
\(693\) −2880.00 −0.157867
\(694\) 5976.00 0.326867
\(695\) 19320.0 1.05446
\(696\) 432.000 0.0235272
\(697\) −47060.0 −2.55742
\(698\) 324.000 0.0175696
\(699\) 7686.00 0.415896
\(700\) 800.000 0.0431959
\(701\) 14870.0 0.801187 0.400594 0.916256i \(-0.368804\pi\)
0.400594 + 0.916256i \(0.368804\pi\)
\(702\) −702.000 −0.0377426
\(703\) 1480.00 0.0794015
\(704\) 2560.00 0.137051
\(705\) −13560.0 −0.724396
\(706\) 21508.0 1.14655
\(707\) −4144.00 −0.220440
\(708\) 5568.00 0.295563
\(709\) −6354.00 −0.336572 −0.168286 0.985738i \(-0.553823\pi\)
−0.168286 + 0.985738i \(0.553823\pi\)
\(710\) 14960.0 0.790759
\(711\) −8784.00 −0.463327
\(712\) 3088.00 0.162539
\(713\) 0 0
\(714\) 6240.00 0.327067
\(715\) 5200.00 0.271985
\(716\) 4272.00 0.222978
\(717\) 21492.0 1.11943
\(718\) −7176.00 −0.372989
\(719\) 9288.00 0.481758 0.240879 0.970555i \(-0.422564\pi\)
0.240879 + 0.970555i \(0.422564\pi\)
\(720\) 1440.00 0.0745356
\(721\) −896.000 −0.0462813
\(722\) 12918.0 0.665870
\(723\) −18546.0 −0.953988
\(724\) −19144.0 −0.982709
\(725\) 450.000 0.0230518
\(726\) −1614.00 −0.0825085
\(727\) −21544.0 −1.09907 −0.549534 0.835471i \(-0.685195\pi\)
−0.549534 + 0.835471i \(0.685195\pi\)
\(728\) 832.000 0.0423571
\(729\) 729.000 0.0370370
\(730\) −21160.0 −1.07283
\(731\) 9880.00 0.499897
\(732\) 4296.00 0.216919
\(733\) 19990.0 1.00730 0.503648 0.863909i \(-0.331991\pi\)
0.503648 + 0.863909i \(0.331991\pi\)
\(734\) −22544.0 −1.13367
\(735\) −8370.00 −0.420044
\(736\) 0 0
\(737\) −28000.0 −1.39945
\(738\) 6516.00 0.325010
\(739\) 532.000 0.0264816 0.0132408 0.999912i \(-0.495785\pi\)
0.0132408 + 0.999912i \(0.495785\pi\)
\(740\) −2960.00 −0.147043
\(741\) −780.000 −0.0386694
\(742\) 6112.00 0.302397
\(743\) −25452.0 −1.25672 −0.628360 0.777922i \(-0.716274\pi\)
−0.628360 + 0.777922i \(0.716274\pi\)
\(744\) 4416.00 0.217605
\(745\) 8820.00 0.433745
\(746\) 21828.0 1.07129
\(747\) −9072.00 −0.444347
\(748\) 20800.0 1.01674
\(749\) 2976.00 0.145181
\(750\) 9000.00 0.438178
\(751\) 6440.00 0.312915 0.156457 0.987685i \(-0.449993\pi\)
0.156457 + 0.987685i \(0.449993\pi\)
\(752\) −7232.00 −0.350697
\(753\) −4188.00 −0.202682
\(754\) 468.000 0.0226042
\(755\) −17760.0 −0.856096
\(756\) −864.000 −0.0415653
\(757\) −786.000 −0.0377380 −0.0188690 0.999822i \(-0.506007\pi\)
−0.0188690 + 0.999822i \(0.506007\pi\)
\(758\) −16200.0 −0.776267
\(759\) 0 0
\(760\) 1600.00 0.0763659
\(761\) −1498.00 −0.0713567 −0.0356784 0.999363i \(-0.511359\pi\)
−0.0356784 + 0.999363i \(0.511359\pi\)
\(762\) −7776.00 −0.369678
\(763\) −7472.00 −0.354528
\(764\) −5248.00 −0.248516
\(765\) 11700.0 0.552960
\(766\) −12360.0 −0.583009
\(767\) 6032.00 0.283967
\(768\) 768.000 0.0360844
\(769\) 14738.0 0.691113 0.345556 0.938398i \(-0.387690\pi\)
0.345556 + 0.938398i \(0.387690\pi\)
\(770\) 6400.00 0.299532
\(771\) 20718.0 0.967757
\(772\) −1400.00 −0.0652683
\(773\) −3822.00 −0.177837 −0.0889184 0.996039i \(-0.528341\pi\)
−0.0889184 + 0.996039i \(0.528341\pi\)
\(774\) −1368.00 −0.0635294
\(775\) 4600.00 0.213209
\(776\) 4912.00 0.227230
\(777\) 1776.00 0.0819995
\(778\) 15044.0 0.693256
\(779\) 7240.00 0.332991
\(780\) 1560.00 0.0716115
\(781\) −29920.0 −1.37083
\(782\) 0 0
\(783\) −486.000 −0.0221816
\(784\) −4464.00 −0.203353
\(785\) −24100.0 −1.09575
\(786\) 5352.00 0.242875
\(787\) −11900.0 −0.538995 −0.269498 0.963001i \(-0.586858\pi\)
−0.269498 + 0.963001i \(0.586858\pi\)
\(788\) −1368.00 −0.0618439
\(789\) −20544.0 −0.926978
\(790\) 19520.0 0.879102
\(791\) −15312.0 −0.688283
\(792\) −2880.00 −0.129213
\(793\) 4654.00 0.208409
\(794\) −12156.0 −0.543325
\(795\) 11460.0 0.511251
\(796\) −13472.0 −0.599877
\(797\) −21274.0 −0.945500 −0.472750 0.881197i \(-0.656739\pi\)
−0.472750 + 0.881197i \(0.656739\pi\)
\(798\) −960.000 −0.0425860
\(799\) −58760.0 −2.60173
\(800\) 800.000 0.0353553
\(801\) −3474.00 −0.153243
\(802\) −3660.00 −0.161146
\(803\) 42320.0 1.85983
\(804\) −8400.00 −0.368464
\(805\) 0 0
\(806\) 4784.00 0.209069
\(807\) −18102.0 −0.789617
\(808\) −4144.00 −0.180427
\(809\) −27566.0 −1.19798 −0.598992 0.800755i \(-0.704432\pi\)
−0.598992 + 0.800755i \(0.704432\pi\)
\(810\) −1620.00 −0.0702728
\(811\) −11244.0 −0.486844 −0.243422 0.969921i \(-0.578270\pi\)
−0.243422 + 0.969921i \(0.578270\pi\)
\(812\) 576.000 0.0248936
\(813\) 14496.0 0.625334
\(814\) 5920.00 0.254909
\(815\) 32120.0 1.38051
\(816\) 6240.00 0.267701
\(817\) −1520.00 −0.0650894
\(818\) −24868.0 −1.06295
\(819\) −936.000 −0.0399347
\(820\) −14480.0 −0.616663
\(821\) 13554.0 0.576173 0.288086 0.957604i \(-0.406981\pi\)
0.288086 + 0.957604i \(0.406981\pi\)
\(822\) −13956.0 −0.592179
\(823\) 14384.0 0.609228 0.304614 0.952476i \(-0.401473\pi\)
0.304614 + 0.952476i \(0.401473\pi\)
\(824\) −896.000 −0.0378806
\(825\) −3000.00 −0.126602
\(826\) 7424.00 0.312729
\(827\) −2488.00 −0.104615 −0.0523073 0.998631i \(-0.516658\pi\)
−0.0523073 + 0.998631i \(0.516658\pi\)
\(828\) 0 0
\(829\) −20858.0 −0.873858 −0.436929 0.899496i \(-0.643934\pi\)
−0.436929 + 0.899496i \(0.643934\pi\)
\(830\) 20160.0 0.843089
\(831\) −12246.0 −0.511202
\(832\) 832.000 0.0346688
\(833\) −36270.0 −1.50862
\(834\) −11592.0 −0.481293
\(835\) 16680.0 0.691300
\(836\) −3200.00 −0.132386
\(837\) −4968.00 −0.205160
\(838\) 28376.0 1.16973
\(839\) 23116.0 0.951195 0.475598 0.879663i \(-0.342232\pi\)
0.475598 + 0.879663i \(0.342232\pi\)
\(840\) 1920.00 0.0788646
\(841\) −24065.0 −0.986715
\(842\) −17276.0 −0.707091
\(843\) 10050.0 0.410605
\(844\) −8016.00 −0.326922
\(845\) 1690.00 0.0688021
\(846\) 8136.00 0.330640
\(847\) −2152.00 −0.0873006
\(848\) 6112.00 0.247508
\(849\) 23388.0 0.945435
\(850\) 6500.00 0.262292
\(851\) 0 0
\(852\) −8976.00 −0.360930
\(853\) 934.000 0.0374907 0.0187453 0.999824i \(-0.494033\pi\)
0.0187453 + 0.999824i \(0.494033\pi\)
\(854\) 5728.00 0.229518
\(855\) −1800.00 −0.0719985
\(856\) 2976.00 0.118829
\(857\) 12642.0 0.503900 0.251950 0.967740i \(-0.418928\pi\)
0.251950 + 0.967740i \(0.418928\pi\)
\(858\) −3120.00 −0.124143
\(859\) −22796.0 −0.905459 −0.452730 0.891648i \(-0.649550\pi\)
−0.452730 + 0.891648i \(0.649550\pi\)
\(860\) 3040.00 0.120539
\(861\) 8688.00 0.343886
\(862\) −8584.00 −0.339179
\(863\) −76.0000 −0.00299776 −0.00149888 0.999999i \(-0.500477\pi\)
−0.00149888 + 0.999999i \(0.500477\pi\)
\(864\) −864.000 −0.0340207
\(865\) 35980.0 1.41429
\(866\) 11964.0 0.469461
\(867\) 35961.0 1.40865
\(868\) 5888.00 0.230244
\(869\) −39040.0 −1.52398
\(870\) 1080.00 0.0420867
\(871\) −9100.00 −0.354009
\(872\) −7472.00 −0.290176
\(873\) −5526.00 −0.214235
\(874\) 0 0
\(875\) 12000.0 0.463627
\(876\) 12696.0 0.489678
\(877\) −46130.0 −1.77617 −0.888084 0.459681i \(-0.847964\pi\)
−0.888084 + 0.459681i \(0.847964\pi\)
\(878\) −512.000 −0.0196801
\(879\) 11766.0 0.451487
\(880\) 6400.00 0.245164
\(881\) 6682.00 0.255530 0.127765 0.991804i \(-0.459220\pi\)
0.127765 + 0.991804i \(0.459220\pi\)
\(882\) 5022.00 0.191723
\(883\) 47404.0 1.80665 0.903325 0.428957i \(-0.141119\pi\)
0.903325 + 0.428957i \(0.141119\pi\)
\(884\) 6760.00 0.257198
\(885\) 13920.0 0.528718
\(886\) −25112.0 −0.952206
\(887\) 33672.0 1.27463 0.637314 0.770604i \(-0.280045\pi\)
0.637314 + 0.770604i \(0.280045\pi\)
\(888\) 1776.00 0.0671156
\(889\) −10368.0 −0.391149
\(890\) 7720.00 0.290758
\(891\) 3240.00 0.121823
\(892\) −22432.0 −0.842017
\(893\) 9040.00 0.338759
\(894\) −5292.00 −0.197976
\(895\) 10680.0 0.398875
\(896\) 1024.00 0.0381802
\(897\) 0 0
\(898\) −11148.0 −0.414269
\(899\) 3312.00 0.122871
\(900\) −900.000 −0.0333333
\(901\) 49660.0 1.83620
\(902\) 28960.0 1.06903
\(903\) −1824.00 −0.0672192
\(904\) −15312.0 −0.563351
\(905\) −47860.0 −1.75792
\(906\) 10656.0 0.390753
\(907\) −14540.0 −0.532296 −0.266148 0.963932i \(-0.585751\pi\)
−0.266148 + 0.963932i \(0.585751\pi\)
\(908\) −7712.00 −0.281863
\(909\) 4662.00 0.170109
\(910\) 2080.00 0.0757707
\(911\) −7840.00 −0.285127 −0.142564 0.989786i \(-0.545535\pi\)
−0.142564 + 0.989786i \(0.545535\pi\)
\(912\) −960.000 −0.0348561
\(913\) −40320.0 −1.46155
\(914\) −2532.00 −0.0916314
\(915\) 10740.0 0.388037
\(916\) −15752.0 −0.568189
\(917\) 7136.00 0.256981
\(918\) −7020.00 −0.252391
\(919\) 47720.0 1.71288 0.856440 0.516246i \(-0.172671\pi\)
0.856440 + 0.516246i \(0.172671\pi\)
\(920\) 0 0
\(921\) 17868.0 0.639273
\(922\) −15108.0 −0.539648
\(923\) −9724.00 −0.346771
\(924\) −3840.00 −0.136717
\(925\) 1850.00 0.0657596
\(926\) 13504.0 0.479232
\(927\) 1008.00 0.0357142
\(928\) 576.000 0.0203751
\(929\) 7502.00 0.264944 0.132472 0.991187i \(-0.457709\pi\)
0.132472 + 0.991187i \(0.457709\pi\)
\(930\) 11040.0 0.389264
\(931\) 5580.00 0.196431
\(932\) 10248.0 0.360176
\(933\) 7056.00 0.247592
\(934\) −15848.0 −0.555206
\(935\) 52000.0 1.81880
\(936\) −936.000 −0.0326860
\(937\) 22058.0 0.769054 0.384527 0.923114i \(-0.374365\pi\)
0.384527 + 0.923114i \(0.374365\pi\)
\(938\) −11200.0 −0.389865
\(939\) 25326.0 0.880173
\(940\) −18080.0 −0.627345
\(941\) 23338.0 0.808498 0.404249 0.914649i \(-0.367533\pi\)
0.404249 + 0.914649i \(0.367533\pi\)
\(942\) 14460.0 0.500140
\(943\) 0 0
\(944\) 7424.00 0.255965
\(945\) −2160.00 −0.0743543
\(946\) −6080.00 −0.208962
\(947\) −30488.0 −1.04617 −0.523087 0.852279i \(-0.675220\pi\)
−0.523087 + 0.852279i \(0.675220\pi\)
\(948\) −11712.0 −0.401253
\(949\) 13754.0 0.470468
\(950\) −1000.00 −0.0341519
\(951\) −16650.0 −0.567732
\(952\) 8320.00 0.283249
\(953\) 9522.00 0.323660 0.161830 0.986819i \(-0.448260\pi\)
0.161830 + 0.986819i \(0.448260\pi\)
\(954\) −6876.00 −0.233353
\(955\) −13120.0 −0.444558
\(956\) 28656.0 0.969457
\(957\) −2160.00 −0.0729602
\(958\) 22168.0 0.747615
\(959\) −18608.0 −0.626573
\(960\) 1920.00 0.0645497
\(961\) 4065.00 0.136451
\(962\) 1924.00 0.0644826
\(963\) −3348.00 −0.112033
\(964\) −24728.0 −0.826178
\(965\) −3500.00 −0.116755
\(966\) 0 0
\(967\) −7616.00 −0.253272 −0.126636 0.991949i \(-0.540418\pi\)
−0.126636 + 0.991949i \(0.540418\pi\)
\(968\) −2152.00 −0.0714544
\(969\) −7800.00 −0.258588
\(970\) 12280.0 0.406481
\(971\) 51316.0 1.69599 0.847996 0.530002i \(-0.177809\pi\)
0.847996 + 0.530002i \(0.177809\pi\)
\(972\) 972.000 0.0320750
\(973\) −15456.0 −0.509246
\(974\) −8864.00 −0.291603
\(975\) −975.000 −0.0320256
\(976\) 5728.00 0.187857
\(977\) −48666.0 −1.59362 −0.796808 0.604232i \(-0.793480\pi\)
−0.796808 + 0.604232i \(0.793480\pi\)
\(978\) −19272.0 −0.630113
\(979\) −15440.0 −0.504050
\(980\) −11160.0 −0.363768
\(981\) 8406.00 0.273581
\(982\) 2280.00 0.0740914
\(983\) 17388.0 0.564182 0.282091 0.959388i \(-0.408972\pi\)
0.282091 + 0.959388i \(0.408972\pi\)
\(984\) 8688.00 0.281467
\(985\) −3420.00 −0.110630
\(986\) 4680.00 0.151158
\(987\) 10848.0 0.349844
\(988\) −1040.00 −0.0334887
\(989\) 0 0
\(990\) −7200.00 −0.231142
\(991\) 11496.0 0.368499 0.184249 0.982880i \(-0.441015\pi\)
0.184249 + 0.982880i \(0.441015\pi\)
\(992\) 5888.00 0.188452
\(993\) 420.000 0.0134223
\(994\) −11968.0 −0.381893
\(995\) −33680.0 −1.07309
\(996\) −12096.0 −0.384816
\(997\) 48862.0 1.55213 0.776066 0.630652i \(-0.217212\pi\)
0.776066 + 0.630652i \(0.217212\pi\)
\(998\) −3528.00 −0.111901
\(999\) −1998.00 −0.0632772
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 78.4.a.c.1.1 1
3.2 odd 2 234.4.a.h.1.1 1
4.3 odd 2 624.4.a.d.1.1 1
5.4 even 2 1950.4.a.l.1.1 1
8.3 odd 2 2496.4.a.j.1.1 1
8.5 even 2 2496.4.a.a.1.1 1
12.11 even 2 1872.4.a.d.1.1 1
13.5 odd 4 1014.4.b.h.337.2 2
13.8 odd 4 1014.4.b.h.337.1 2
13.12 even 2 1014.4.a.j.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
78.4.a.c.1.1 1 1.1 even 1 trivial
234.4.a.h.1.1 1 3.2 odd 2
624.4.a.d.1.1 1 4.3 odd 2
1014.4.a.j.1.1 1 13.12 even 2
1014.4.b.h.337.1 2 13.8 odd 4
1014.4.b.h.337.2 2 13.5 odd 4
1872.4.a.d.1.1 1 12.11 even 2
1950.4.a.l.1.1 1 5.4 even 2
2496.4.a.a.1.1 1 8.5 even 2
2496.4.a.j.1.1 1 8.3 odd 2