Properties

Label 14.4.a.b.1.1
Level $14$
Weight $4$
Character 14.1
Self dual yes
Analytic conductor $0.826$
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] = [14,4,Mod(1,14)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(14, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("14.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 14 = 2 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 14.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: \(0.826026740080\)
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 \(\chi\) \(=\) 14.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.00000 q^{2} -2.00000 q^{3} +4.00000 q^{4} -12.0000 q^{5} -4.00000 q^{6} +7.00000 q^{7} +8.00000 q^{8} -23.0000 q^{9} +O(q^{10})\) \(q+2.00000 q^{2} -2.00000 q^{3} +4.00000 q^{4} -12.0000 q^{5} -4.00000 q^{6} +7.00000 q^{7} +8.00000 q^{8} -23.0000 q^{9} -24.0000 q^{10} +48.0000 q^{11} -8.00000 q^{12} +56.0000 q^{13} +14.0000 q^{14} +24.0000 q^{15} +16.0000 q^{16} -114.000 q^{17} -46.0000 q^{18} +2.00000 q^{19} -48.0000 q^{20} -14.0000 q^{21} +96.0000 q^{22} -120.000 q^{23} -16.0000 q^{24} +19.0000 q^{25} +112.000 q^{26} +100.000 q^{27} +28.0000 q^{28} -54.0000 q^{29} +48.0000 q^{30} +236.000 q^{31} +32.0000 q^{32} -96.0000 q^{33} -228.000 q^{34} -84.0000 q^{35} -92.0000 q^{36} +146.000 q^{37} +4.00000 q^{38} -112.000 q^{39} -96.0000 q^{40} +126.000 q^{41} -28.0000 q^{42} -376.000 q^{43} +192.000 q^{44} +276.000 q^{45} -240.000 q^{46} -12.0000 q^{47} -32.0000 q^{48} +49.0000 q^{49} +38.0000 q^{50} +228.000 q^{51} +224.000 q^{52} +174.000 q^{53} +200.000 q^{54} -576.000 q^{55} +56.0000 q^{56} -4.00000 q^{57} -108.000 q^{58} +138.000 q^{59} +96.0000 q^{60} +380.000 q^{61} +472.000 q^{62} -161.000 q^{63} +64.0000 q^{64} -672.000 q^{65} -192.000 q^{66} -484.000 q^{67} -456.000 q^{68} +240.000 q^{69} -168.000 q^{70} +576.000 q^{71} -184.000 q^{72} -1150.00 q^{73} +292.000 q^{74} -38.0000 q^{75} +8.00000 q^{76} +336.000 q^{77} -224.000 q^{78} +776.000 q^{79} -192.000 q^{80} +421.000 q^{81} +252.000 q^{82} +378.000 q^{83} -56.0000 q^{84} +1368.00 q^{85} -752.000 q^{86} +108.000 q^{87} +384.000 q^{88} -390.000 q^{89} +552.000 q^{90} +392.000 q^{91} -480.000 q^{92} -472.000 q^{93} -24.0000 q^{94} -24.0000 q^{95} -64.0000 q^{96} -1330.00 q^{97} +98.0000 q^{98} -1104.00 q^{99} +O(q^{100})\)

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
\(3\) −2.00000 −0.384900 −0.192450 0.981307i \(-0.561643\pi\)
−0.192450 + 0.981307i \(0.561643\pi\)
\(4\) 4.00000 0.500000
\(5\) −12.0000 −1.07331 −0.536656 0.843801i \(-0.680313\pi\)
−0.536656 + 0.843801i \(0.680313\pi\)
\(6\) −4.00000 −0.272166
\(7\) 7.00000 0.377964
\(8\) 8.00000 0.353553
\(9\) −23.0000 −0.851852
\(10\) −24.0000 −0.758947
\(11\) 48.0000 1.31569 0.657843 0.753155i \(-0.271469\pi\)
0.657843 + 0.753155i \(0.271469\pi\)
\(12\) −8.00000 −0.192450
\(13\) 56.0000 1.19474 0.597369 0.801966i \(-0.296213\pi\)
0.597369 + 0.801966i \(0.296213\pi\)
\(14\) 14.0000 0.267261
\(15\) 24.0000 0.413118
\(16\) 16.0000 0.250000
\(17\) −114.000 −1.62642 −0.813208 0.581974i \(-0.802281\pi\)
−0.813208 + 0.581974i \(0.802281\pi\)
\(18\) −46.0000 −0.602350
\(19\) 2.00000 0.0241490 0.0120745 0.999927i \(-0.496156\pi\)
0.0120745 + 0.999927i \(0.496156\pi\)
\(20\) −48.0000 −0.536656
\(21\) −14.0000 −0.145479
\(22\) 96.0000 0.930330
\(23\) −120.000 −1.08790 −0.543951 0.839117i \(-0.683072\pi\)
−0.543951 + 0.839117i \(0.683072\pi\)
\(24\) −16.0000 −0.136083
\(25\) 19.0000 0.152000
\(26\) 112.000 0.844808
\(27\) 100.000 0.712778
\(28\) 28.0000 0.188982
\(29\) −54.0000 −0.345778 −0.172889 0.984941i \(-0.555310\pi\)
−0.172889 + 0.984941i \(0.555310\pi\)
\(30\) 48.0000 0.292119
\(31\) 236.000 1.36732 0.683659 0.729802i \(-0.260388\pi\)
0.683659 + 0.729802i \(0.260388\pi\)
\(32\) 32.0000 0.176777
\(33\) −96.0000 −0.506408
\(34\) −228.000 −1.15005
\(35\) −84.0000 −0.405674
\(36\) −92.0000 −0.425926
\(37\) 146.000 0.648710 0.324355 0.945936i \(-0.394853\pi\)
0.324355 + 0.945936i \(0.394853\pi\)
\(38\) 4.00000 0.0170759
\(39\) −112.000 −0.459855
\(40\) −96.0000 −0.379473
\(41\) 126.000 0.479949 0.239974 0.970779i \(-0.422861\pi\)
0.239974 + 0.970779i \(0.422861\pi\)
\(42\) −28.0000 −0.102869
\(43\) −376.000 −1.33348 −0.666738 0.745292i \(-0.732310\pi\)
−0.666738 + 0.745292i \(0.732310\pi\)
\(44\) 192.000 0.657843
\(45\) 276.000 0.914303
\(46\) −240.000 −0.769262
\(47\) −12.0000 −0.0372421 −0.0186211 0.999827i \(-0.505928\pi\)
−0.0186211 + 0.999827i \(0.505928\pi\)
\(48\) −32.0000 −0.0962250
\(49\) 49.0000 0.142857
\(50\) 38.0000 0.107480
\(51\) 228.000 0.626008
\(52\) 224.000 0.597369
\(53\) 174.000 0.450957 0.225479 0.974248i \(-0.427605\pi\)
0.225479 + 0.974248i \(0.427605\pi\)
\(54\) 200.000 0.504010
\(55\) −576.000 −1.41214
\(56\) 56.0000 0.133631
\(57\) −4.00000 −0.00929496
\(58\) −108.000 −0.244502
\(59\) 138.000 0.304510 0.152255 0.988341i \(-0.451347\pi\)
0.152255 + 0.988341i \(0.451347\pi\)
\(60\) 96.0000 0.206559
\(61\) 380.000 0.797607 0.398803 0.917036i \(-0.369426\pi\)
0.398803 + 0.917036i \(0.369426\pi\)
\(62\) 472.000 0.966840
\(63\) −161.000 −0.321970
\(64\) 64.0000 0.125000
\(65\) −672.000 −1.28233
\(66\) −192.000 −0.358084
\(67\) −484.000 −0.882537 −0.441269 0.897375i \(-0.645471\pi\)
−0.441269 + 0.897375i \(0.645471\pi\)
\(68\) −456.000 −0.813208
\(69\) 240.000 0.418733
\(70\) −168.000 −0.286855
\(71\) 576.000 0.962798 0.481399 0.876502i \(-0.340129\pi\)
0.481399 + 0.876502i \(0.340129\pi\)
\(72\) −184.000 −0.301175
\(73\) −1150.00 −1.84380 −0.921899 0.387429i \(-0.873363\pi\)
−0.921899 + 0.387429i \(0.873363\pi\)
\(74\) 292.000 0.458707
\(75\) −38.0000 −0.0585048
\(76\) 8.00000 0.0120745
\(77\) 336.000 0.497283
\(78\) −224.000 −0.325167
\(79\) 776.000 1.10515 0.552575 0.833463i \(-0.313645\pi\)
0.552575 + 0.833463i \(0.313645\pi\)
\(80\) −192.000 −0.268328
\(81\) 421.000 0.577503
\(82\) 252.000 0.339375
\(83\) 378.000 0.499890 0.249945 0.968260i \(-0.419587\pi\)
0.249945 + 0.968260i \(0.419587\pi\)
\(84\) −56.0000 −0.0727393
\(85\) 1368.00 1.74565
\(86\) −752.000 −0.942910
\(87\) 108.000 0.133090
\(88\) 384.000 0.465165
\(89\) −390.000 −0.464493 −0.232247 0.972657i \(-0.574608\pi\)
−0.232247 + 0.972657i \(0.574608\pi\)
\(90\) 552.000 0.646510
\(91\) 392.000 0.451569
\(92\) −480.000 −0.543951
\(93\) −472.000 −0.526281
\(94\) −24.0000 −0.0263342
\(95\) −24.0000 −0.0259195
\(96\) −64.0000 −0.0680414
\(97\) −1330.00 −1.39218 −0.696088 0.717957i \(-0.745078\pi\)
−0.696088 + 0.717957i \(0.745078\pi\)
\(98\) 98.0000 0.101015
\(99\) −1104.00 −1.12077
\(100\) 76.0000 0.0760000
\(101\) −1500.00 −1.47778 −0.738889 0.673827i \(-0.764649\pi\)
−0.738889 + 0.673827i \(0.764649\pi\)
\(102\) 456.000 0.442654
\(103\) 380.000 0.363520 0.181760 0.983343i \(-0.441821\pi\)
0.181760 + 0.983343i \(0.441821\pi\)
\(104\) 448.000 0.422404
\(105\) 168.000 0.156144
\(106\) 348.000 0.318875
\(107\) 636.000 0.574621 0.287310 0.957838i \(-0.407239\pi\)
0.287310 + 0.957838i \(0.407239\pi\)
\(108\) 400.000 0.356389
\(109\) 146.000 0.128296 0.0641480 0.997940i \(-0.479567\pi\)
0.0641480 + 0.997940i \(0.479567\pi\)
\(110\) −1152.00 −0.998535
\(111\) −292.000 −0.249688
\(112\) 112.000 0.0944911
\(113\) 198.000 0.164834 0.0824171 0.996598i \(-0.473736\pi\)
0.0824171 + 0.996598i \(0.473736\pi\)
\(114\) −8.00000 −0.00657253
\(115\) 1440.00 1.16766
\(116\) −216.000 −0.172889
\(117\) −1288.00 −1.01774
\(118\) 276.000 0.215321
\(119\) −798.000 −0.614727
\(120\) 192.000 0.146059
\(121\) 973.000 0.731029
\(122\) 760.000 0.563993
\(123\) −252.000 −0.184732
\(124\) 944.000 0.683659
\(125\) 1272.00 0.910169
\(126\) −322.000 −0.227667
\(127\) −376.000 −0.262713 −0.131357 0.991335i \(-0.541933\pi\)
−0.131357 + 0.991335i \(0.541933\pi\)
\(128\) 128.000 0.0883883
\(129\) 752.000 0.513255
\(130\) −1344.00 −0.906743
\(131\) 2130.00 1.42060 0.710301 0.703898i \(-0.248559\pi\)
0.710301 + 0.703898i \(0.248559\pi\)
\(132\) −384.000 −0.253204
\(133\) 14.0000 0.00912747
\(134\) −968.000 −0.624048
\(135\) −1200.00 −0.765034
\(136\) −912.000 −0.575025
\(137\) −78.0000 −0.0486423 −0.0243211 0.999704i \(-0.507742\pi\)
−0.0243211 + 0.999704i \(0.507742\pi\)
\(138\) 480.000 0.296089
\(139\) −2338.00 −1.42667 −0.713333 0.700825i \(-0.752815\pi\)
−0.713333 + 0.700825i \(0.752815\pi\)
\(140\) −336.000 −0.202837
\(141\) 24.0000 0.0143345
\(142\) 1152.00 0.680801
\(143\) 2688.00 1.57190
\(144\) −368.000 −0.212963
\(145\) 648.000 0.371127
\(146\) −2300.00 −1.30376
\(147\) −98.0000 −0.0549857
\(148\) 584.000 0.324355
\(149\) −1002.00 −0.550920 −0.275460 0.961313i \(-0.588830\pi\)
−0.275460 + 0.961313i \(0.588830\pi\)
\(150\) −76.0000 −0.0413692
\(151\) −2752.00 −1.48314 −0.741571 0.670874i \(-0.765919\pi\)
−0.741571 + 0.670874i \(0.765919\pi\)
\(152\) 16.0000 0.00853797
\(153\) 2622.00 1.38546
\(154\) 672.000 0.351632
\(155\) −2832.00 −1.46756
\(156\) −448.000 −0.229928
\(157\) −520.000 −0.264335 −0.132167 0.991227i \(-0.542194\pi\)
−0.132167 + 0.991227i \(0.542194\pi\)
\(158\) 1552.00 0.781459
\(159\) −348.000 −0.173574
\(160\) −384.000 −0.189737
\(161\) −840.000 −0.411188
\(162\) 842.000 0.408357
\(163\) 1280.00 0.615076 0.307538 0.951536i \(-0.400495\pi\)
0.307538 + 0.951536i \(0.400495\pi\)
\(164\) 504.000 0.239974
\(165\) 1152.00 0.543534
\(166\) 756.000 0.353476
\(167\) 1764.00 0.817380 0.408690 0.912673i \(-0.365986\pi\)
0.408690 + 0.912673i \(0.365986\pi\)
\(168\) −112.000 −0.0514344
\(169\) 939.000 0.427401
\(170\) 2736.00 1.23436
\(171\) −46.0000 −0.0205714
\(172\) −1504.00 −0.666738
\(173\) −768.000 −0.337514 −0.168757 0.985658i \(-0.553975\pi\)
−0.168757 + 0.985658i \(0.553975\pi\)
\(174\) 216.000 0.0941087
\(175\) 133.000 0.0574506
\(176\) 768.000 0.328921
\(177\) −276.000 −0.117206
\(178\) −780.000 −0.328446
\(179\) 1812.00 0.756621 0.378311 0.925679i \(-0.376505\pi\)
0.378311 + 0.925679i \(0.376505\pi\)
\(180\) 1104.00 0.457152
\(181\) −448.000 −0.183976 −0.0919878 0.995760i \(-0.529322\pi\)
−0.0919878 + 0.995760i \(0.529322\pi\)
\(182\) 784.000 0.319307
\(183\) −760.000 −0.306999
\(184\) −960.000 −0.384631
\(185\) −1752.00 −0.696268
\(186\) −944.000 −0.372137
\(187\) −5472.00 −2.13985
\(188\) −48.0000 −0.0186211
\(189\) 700.000 0.269405
\(190\) −48.0000 −0.0183278
\(191\) −2136.00 −0.809191 −0.404596 0.914496i \(-0.632588\pi\)
−0.404596 + 0.914496i \(0.632588\pi\)
\(192\) −128.000 −0.0481125
\(193\) 4430.00 1.65222 0.826110 0.563509i \(-0.190549\pi\)
0.826110 + 0.563509i \(0.190549\pi\)
\(194\) −2660.00 −0.984417
\(195\) 1344.00 0.493568
\(196\) 196.000 0.0714286
\(197\) 198.000 0.0716087 0.0358044 0.999359i \(-0.488601\pi\)
0.0358044 + 0.999359i \(0.488601\pi\)
\(198\) −2208.00 −0.792504
\(199\) −2284.00 −0.813610 −0.406805 0.913515i \(-0.633357\pi\)
−0.406805 + 0.913515i \(0.633357\pi\)
\(200\) 152.000 0.0537401
\(201\) 968.000 0.339689
\(202\) −3000.00 −1.04495
\(203\) −378.000 −0.130692
\(204\) 912.000 0.313004
\(205\) −1512.00 −0.515135
\(206\) 760.000 0.257047
\(207\) 2760.00 0.926731
\(208\) 896.000 0.298685
\(209\) 96.0000 0.0317725
\(210\) 336.000 0.110410
\(211\) 4412.00 1.43950 0.719750 0.694233i \(-0.244256\pi\)
0.719750 + 0.694233i \(0.244256\pi\)
\(212\) 696.000 0.225479
\(213\) −1152.00 −0.370581
\(214\) 1272.00 0.406318
\(215\) 4512.00 1.43124
\(216\) 800.000 0.252005
\(217\) 1652.00 0.516798
\(218\) 292.000 0.0907190
\(219\) 2300.00 0.709679
\(220\) −2304.00 −0.706071
\(221\) −6384.00 −1.94314
\(222\) −584.000 −0.176556
\(223\) 2072.00 0.622204 0.311102 0.950377i \(-0.399302\pi\)
0.311102 + 0.950377i \(0.399302\pi\)
\(224\) 224.000 0.0668153
\(225\) −437.000 −0.129481
\(226\) 396.000 0.116555
\(227\) −366.000 −0.107014 −0.0535072 0.998567i \(-0.517040\pi\)
−0.0535072 + 0.998567i \(0.517040\pi\)
\(228\) −16.0000 −0.00464748
\(229\) −376.000 −0.108501 −0.0542506 0.998527i \(-0.517277\pi\)
−0.0542506 + 0.998527i \(0.517277\pi\)
\(230\) 2880.00 0.825659
\(231\) −672.000 −0.191404
\(232\) −432.000 −0.122251
\(233\) −2262.00 −0.636002 −0.318001 0.948090i \(-0.603012\pi\)
−0.318001 + 0.948090i \(0.603012\pi\)
\(234\) −2576.00 −0.719651
\(235\) 144.000 0.0399724
\(236\) 552.000 0.152255
\(237\) −1552.00 −0.425372
\(238\) −1596.00 −0.434678
\(239\) 2592.00 0.701517 0.350758 0.936466i \(-0.385924\pi\)
0.350758 + 0.936466i \(0.385924\pi\)
\(240\) 384.000 0.103280
\(241\) 110.000 0.0294013 0.0147007 0.999892i \(-0.495320\pi\)
0.0147007 + 0.999892i \(0.495320\pi\)
\(242\) 1946.00 0.516916
\(243\) −3542.00 −0.935059
\(244\) 1520.00 0.398803
\(245\) −588.000 −0.153330
\(246\) −504.000 −0.130625
\(247\) 112.000 0.0288518
\(248\) 1888.00 0.483420
\(249\) −756.000 −0.192408
\(250\) 2544.00 0.643587
\(251\) −1890.00 −0.475282 −0.237641 0.971353i \(-0.576374\pi\)
−0.237641 + 0.971353i \(0.576374\pi\)
\(252\) −644.000 −0.160985
\(253\) −5760.00 −1.43134
\(254\) −752.000 −0.185766
\(255\) −2736.00 −0.671902
\(256\) 256.000 0.0625000
\(257\) 2130.00 0.516987 0.258494 0.966013i \(-0.416774\pi\)
0.258494 + 0.966013i \(0.416774\pi\)
\(258\) 1504.00 0.362926
\(259\) 1022.00 0.245189
\(260\) −2688.00 −0.641164
\(261\) 1242.00 0.294551
\(262\) 4260.00 1.00452
\(263\) −4992.00 −1.17042 −0.585209 0.810883i \(-0.698988\pi\)
−0.585209 + 0.810883i \(0.698988\pi\)
\(264\) −768.000 −0.179042
\(265\) −2088.00 −0.484018
\(266\) 28.0000 0.00645410
\(267\) 780.000 0.178784
\(268\) −1936.00 −0.441269
\(269\) 6816.00 1.54490 0.772451 0.635074i \(-0.219030\pi\)
0.772451 + 0.635074i \(0.219030\pi\)
\(270\) −2400.00 −0.540961
\(271\) 8192.00 1.83627 0.918134 0.396270i \(-0.129696\pi\)
0.918134 + 0.396270i \(0.129696\pi\)
\(272\) −1824.00 −0.406604
\(273\) −784.000 −0.173809
\(274\) −156.000 −0.0343953
\(275\) 912.000 0.199984
\(276\) 960.000 0.209367
\(277\) 2414.00 0.523622 0.261811 0.965119i \(-0.415680\pi\)
0.261811 + 0.965119i \(0.415680\pi\)
\(278\) −4676.00 −1.00881
\(279\) −5428.00 −1.16475
\(280\) −672.000 −0.143427
\(281\) 1962.00 0.416524 0.208262 0.978073i \(-0.433219\pi\)
0.208262 + 0.978073i \(0.433219\pi\)
\(282\) 48.0000 0.0101360
\(283\) 5402.00 1.13468 0.567342 0.823482i \(-0.307972\pi\)
0.567342 + 0.823482i \(0.307972\pi\)
\(284\) 2304.00 0.481399
\(285\) 48.0000 0.00997640
\(286\) 5376.00 1.11150
\(287\) 882.000 0.181404
\(288\) −736.000 −0.150588
\(289\) 8083.00 1.64523
\(290\) 1296.00 0.262427
\(291\) 2660.00 0.535849
\(292\) −4600.00 −0.921899
\(293\) −4788.00 −0.954669 −0.477334 0.878722i \(-0.658397\pi\)
−0.477334 + 0.878722i \(0.658397\pi\)
\(294\) −196.000 −0.0388808
\(295\) −1656.00 −0.326834
\(296\) 1168.00 0.229353
\(297\) 4800.00 0.937792
\(298\) −2004.00 −0.389559
\(299\) −6720.00 −1.29976
\(300\) −152.000 −0.0292524
\(301\) −2632.00 −0.504007
\(302\) −5504.00 −1.04874
\(303\) 3000.00 0.568797
\(304\) 32.0000 0.00603726
\(305\) −4560.00 −0.856081
\(306\) 5244.00 0.979672
\(307\) −574.000 −0.106710 −0.0533549 0.998576i \(-0.516991\pi\)
−0.0533549 + 0.998576i \(0.516991\pi\)
\(308\) 1344.00 0.248641
\(309\) −760.000 −0.139919
\(310\) −5664.00 −1.03772
\(311\) −8808.00 −1.60597 −0.802984 0.596001i \(-0.796755\pi\)
−0.802984 + 0.596001i \(0.796755\pi\)
\(312\) −896.000 −0.162583
\(313\) −2770.00 −0.500223 −0.250111 0.968217i \(-0.580467\pi\)
−0.250111 + 0.968217i \(0.580467\pi\)
\(314\) −1040.00 −0.186913
\(315\) 1932.00 0.345574
\(316\) 3104.00 0.552575
\(317\) 7566.00 1.34053 0.670266 0.742121i \(-0.266180\pi\)
0.670266 + 0.742121i \(0.266180\pi\)
\(318\) −696.000 −0.122735
\(319\) −2592.00 −0.454935
\(320\) −768.000 −0.134164
\(321\) −1272.00 −0.221172
\(322\) −1680.00 −0.290754
\(323\) −228.000 −0.0392763
\(324\) 1684.00 0.288752
\(325\) 1064.00 0.181600
\(326\) 2560.00 0.434924
\(327\) −292.000 −0.0493812
\(328\) 1008.00 0.169687
\(329\) −84.0000 −0.0140762
\(330\) 2304.00 0.384336
\(331\) −11320.0 −1.87977 −0.939884 0.341493i \(-0.889068\pi\)
−0.939884 + 0.341493i \(0.889068\pi\)
\(332\) 1512.00 0.249945
\(333\) −3358.00 −0.552604
\(334\) 3528.00 0.577975
\(335\) 5808.00 0.947239
\(336\) −224.000 −0.0363696
\(337\) −4786.00 −0.773620 −0.386810 0.922159i \(-0.626423\pi\)
−0.386810 + 0.922159i \(0.626423\pi\)
\(338\) 1878.00 0.302218
\(339\) −396.000 −0.0634447
\(340\) 5472.00 0.872826
\(341\) 11328.0 1.79896
\(342\) −92.0000 −0.0145462
\(343\) 343.000 0.0539949
\(344\) −3008.00 −0.471455
\(345\) −2880.00 −0.449432
\(346\) −1536.00 −0.238659
\(347\) 12648.0 1.95672 0.978358 0.206921i \(-0.0663443\pi\)
0.978358 + 0.206921i \(0.0663443\pi\)
\(348\) 432.000 0.0665449
\(349\) 9632.00 1.47733 0.738666 0.674071i \(-0.235456\pi\)
0.738666 + 0.674071i \(0.235456\pi\)
\(350\) 266.000 0.0406237
\(351\) 5600.00 0.851584
\(352\) 1536.00 0.232583
\(353\) −3390.00 −0.511137 −0.255569 0.966791i \(-0.582263\pi\)
−0.255569 + 0.966791i \(0.582263\pi\)
\(354\) −552.000 −0.0828770
\(355\) −6912.00 −1.03338
\(356\) −1560.00 −0.232247
\(357\) 1596.00 0.236609
\(358\) 3624.00 0.535012
\(359\) −10704.0 −1.57364 −0.786818 0.617185i \(-0.788273\pi\)
−0.786818 + 0.617185i \(0.788273\pi\)
\(360\) 2208.00 0.323255
\(361\) −6855.00 −0.999417
\(362\) −896.000 −0.130090
\(363\) −1946.00 −0.281373
\(364\) 1568.00 0.225784
\(365\) 13800.0 1.97897
\(366\) −1520.00 −0.217081
\(367\) −8584.00 −1.22093 −0.610465 0.792043i \(-0.709017\pi\)
−0.610465 + 0.792043i \(0.709017\pi\)
\(368\) −1920.00 −0.271975
\(369\) −2898.00 −0.408845
\(370\) −3504.00 −0.492336
\(371\) 1218.00 0.170446
\(372\) −1888.00 −0.263140
\(373\) −2122.00 −0.294566 −0.147283 0.989094i \(-0.547053\pi\)
−0.147283 + 0.989094i \(0.547053\pi\)
\(374\) −10944.0 −1.51310
\(375\) −2544.00 −0.350324
\(376\) −96.0000 −0.0131671
\(377\) −3024.00 −0.413114
\(378\) 1400.00 0.190498
\(379\) −4912.00 −0.665732 −0.332866 0.942974i \(-0.608016\pi\)
−0.332866 + 0.942974i \(0.608016\pi\)
\(380\) −96.0000 −0.0129597
\(381\) 752.000 0.101118
\(382\) −4272.00 −0.572185
\(383\) 9060.00 1.20873 0.604366 0.796707i \(-0.293426\pi\)
0.604366 + 0.796707i \(0.293426\pi\)
\(384\) −256.000 −0.0340207
\(385\) −4032.00 −0.533740
\(386\) 8860.00 1.16830
\(387\) 8648.00 1.13592
\(388\) −5320.00 −0.696088
\(389\) 8994.00 1.17227 0.586136 0.810213i \(-0.300648\pi\)
0.586136 + 0.810213i \(0.300648\pi\)
\(390\) 2688.00 0.349006
\(391\) 13680.0 1.76938
\(392\) 392.000 0.0505076
\(393\) −4260.00 −0.546790
\(394\) 396.000 0.0506350
\(395\) −9312.00 −1.18617
\(396\) −4416.00 −0.560385
\(397\) −12976.0 −1.64042 −0.820210 0.572062i \(-0.806143\pi\)
−0.820210 + 0.572062i \(0.806143\pi\)
\(398\) −4568.00 −0.575309
\(399\) −28.0000 −0.00351317
\(400\) 304.000 0.0380000
\(401\) −3522.00 −0.438604 −0.219302 0.975657i \(-0.570378\pi\)
−0.219302 + 0.975657i \(0.570378\pi\)
\(402\) 1936.00 0.240196
\(403\) 13216.0 1.63359
\(404\) −6000.00 −0.738889
\(405\) −5052.00 −0.619842
\(406\) −756.000 −0.0924129
\(407\) 7008.00 0.853498
\(408\) 1824.00 0.221327
\(409\) 12710.0 1.53660 0.768300 0.640090i \(-0.221103\pi\)
0.768300 + 0.640090i \(0.221103\pi\)
\(410\) −3024.00 −0.364255
\(411\) 156.000 0.0187224
\(412\) 1520.00 0.181760
\(413\) 966.000 0.115094
\(414\) 5520.00 0.655298
\(415\) −4536.00 −0.536539
\(416\) 1792.00 0.211202
\(417\) 4676.00 0.549124
\(418\) 192.000 0.0224666
\(419\) 1638.00 0.190982 0.0954911 0.995430i \(-0.469558\pi\)
0.0954911 + 0.995430i \(0.469558\pi\)
\(420\) 672.000 0.0780720
\(421\) −12850.0 −1.48758 −0.743789 0.668414i \(-0.766973\pi\)
−0.743789 + 0.668414i \(0.766973\pi\)
\(422\) 8824.00 1.01788
\(423\) 276.000 0.0317248
\(424\) 1392.00 0.159437
\(425\) −2166.00 −0.247215
\(426\) −2304.00 −0.262040
\(427\) 2660.00 0.301467
\(428\) 2544.00 0.287310
\(429\) −5376.00 −0.605025
\(430\) 9024.00 1.01204
\(431\) −8016.00 −0.895863 −0.447932 0.894068i \(-0.647839\pi\)
−0.447932 + 0.894068i \(0.647839\pi\)
\(432\) 1600.00 0.178195
\(433\) 2198.00 0.243947 0.121974 0.992533i \(-0.461078\pi\)
0.121974 + 0.992533i \(0.461078\pi\)
\(434\) 3304.00 0.365431
\(435\) −1296.00 −0.142847
\(436\) 584.000 0.0641480
\(437\) −240.000 −0.0262718
\(438\) 4600.00 0.501818
\(439\) −376.000 −0.0408781 −0.0204391 0.999791i \(-0.506506\pi\)
−0.0204391 + 0.999791i \(0.506506\pi\)
\(440\) −4608.00 −0.499268
\(441\) −1127.00 −0.121693
\(442\) −12768.0 −1.37401
\(443\) 7188.00 0.770908 0.385454 0.922727i \(-0.374045\pi\)
0.385454 + 0.922727i \(0.374045\pi\)
\(444\) −1168.00 −0.124844
\(445\) 4680.00 0.498547
\(446\) 4144.00 0.439964
\(447\) 2004.00 0.212049
\(448\) 448.000 0.0472456
\(449\) −14670.0 −1.54192 −0.770958 0.636886i \(-0.780222\pi\)
−0.770958 + 0.636886i \(0.780222\pi\)
\(450\) −874.000 −0.0915572
\(451\) 6048.00 0.631462
\(452\) 792.000 0.0824171
\(453\) 5504.00 0.570862
\(454\) −732.000 −0.0756706
\(455\) −4704.00 −0.484675
\(456\) −32.0000 −0.00328627
\(457\) −5146.00 −0.526739 −0.263370 0.964695i \(-0.584834\pi\)
−0.263370 + 0.964695i \(0.584834\pi\)
\(458\) −752.000 −0.0767219
\(459\) −11400.0 −1.15927
\(460\) 5760.00 0.583829
\(461\) −1512.00 −0.152757 −0.0763784 0.997079i \(-0.524336\pi\)
−0.0763784 + 0.997079i \(0.524336\pi\)
\(462\) −1344.00 −0.135343
\(463\) 7184.00 0.721099 0.360549 0.932740i \(-0.382589\pi\)
0.360549 + 0.932740i \(0.382589\pi\)
\(464\) −864.000 −0.0864444
\(465\) 5664.00 0.564864
\(466\) −4524.00 −0.449722
\(467\) −16518.0 −1.63675 −0.818375 0.574685i \(-0.805125\pi\)
−0.818375 + 0.574685i \(0.805125\pi\)
\(468\) −5152.00 −0.508870
\(469\) −3388.00 −0.333568
\(470\) 288.000 0.0282648
\(471\) 1040.00 0.101742
\(472\) 1104.00 0.107660
\(473\) −18048.0 −1.75444
\(474\) −3104.00 −0.300784
\(475\) 38.0000 0.00367065
\(476\) −3192.00 −0.307364
\(477\) −4002.00 −0.384149
\(478\) 5184.00 0.496047
\(479\) 10092.0 0.962662 0.481331 0.876539i \(-0.340153\pi\)
0.481331 + 0.876539i \(0.340153\pi\)
\(480\) 768.000 0.0730297
\(481\) 8176.00 0.775038
\(482\) 220.000 0.0207899
\(483\) 1680.00 0.158266
\(484\) 3892.00 0.365515
\(485\) 15960.0 1.49424
\(486\) −7084.00 −0.661187
\(487\) 7832.00 0.728751 0.364376 0.931252i \(-0.381282\pi\)
0.364376 + 0.931252i \(0.381282\pi\)
\(488\) 3040.00 0.281997
\(489\) −2560.00 −0.236743
\(490\) −1176.00 −0.108421
\(491\) −6732.00 −0.618759 −0.309380 0.950939i \(-0.600121\pi\)
−0.309380 + 0.950939i \(0.600121\pi\)
\(492\) −1008.00 −0.0923662
\(493\) 6156.00 0.562378
\(494\) 224.000 0.0204013
\(495\) 13248.0 1.20294
\(496\) 3776.00 0.341829
\(497\) 4032.00 0.363903
\(498\) −1512.00 −0.136053
\(499\) 18668.0 1.67474 0.837369 0.546638i \(-0.184093\pi\)
0.837369 + 0.546638i \(0.184093\pi\)
\(500\) 5088.00 0.455085
\(501\) −3528.00 −0.314610
\(502\) −3780.00 −0.336075
\(503\) −6048.00 −0.536117 −0.268059 0.963403i \(-0.586382\pi\)
−0.268059 + 0.963403i \(0.586382\pi\)
\(504\) −1288.00 −0.113833
\(505\) 18000.0 1.58612
\(506\) −11520.0 −1.01211
\(507\) −1878.00 −0.164507
\(508\) −1504.00 −0.131357
\(509\) 11328.0 0.986453 0.493227 0.869901i \(-0.335817\pi\)
0.493227 + 0.869901i \(0.335817\pi\)
\(510\) −5472.00 −0.475106
\(511\) −8050.00 −0.696890
\(512\) 512.000 0.0441942
\(513\) 200.000 0.0172129
\(514\) 4260.00 0.365565
\(515\) −4560.00 −0.390170
\(516\) 3008.00 0.256628
\(517\) −576.000 −0.0489989
\(518\) 2044.00 0.173375
\(519\) 1536.00 0.129909
\(520\) −5376.00 −0.453372
\(521\) −4146.00 −0.348636 −0.174318 0.984689i \(-0.555772\pi\)
−0.174318 + 0.984689i \(0.555772\pi\)
\(522\) 2484.00 0.208279
\(523\) −1006.00 −0.0841096 −0.0420548 0.999115i \(-0.513390\pi\)
−0.0420548 + 0.999115i \(0.513390\pi\)
\(524\) 8520.00 0.710301
\(525\) −266.000 −0.0221127
\(526\) −9984.00 −0.827610
\(527\) −26904.0 −2.22383
\(528\) −1536.00 −0.126602
\(529\) 2233.00 0.183529
\(530\) −4176.00 −0.342253
\(531\) −3174.00 −0.259397
\(532\) 56.0000 0.00456374
\(533\) 7056.00 0.573413
\(534\) 1560.00 0.126419
\(535\) −7632.00 −0.616748
\(536\) −3872.00 −0.312024
\(537\) −3624.00 −0.291224
\(538\) 13632.0 1.09241
\(539\) 2352.00 0.187955
\(540\) −4800.00 −0.382517
\(541\) −14722.0 −1.16996 −0.584980 0.811048i \(-0.698898\pi\)
−0.584980 + 0.811048i \(0.698898\pi\)
\(542\) 16384.0 1.29844
\(543\) 896.000 0.0708122
\(544\) −3648.00 −0.287512
\(545\) −1752.00 −0.137702
\(546\) −1568.00 −0.122901
\(547\) −13480.0 −1.05368 −0.526840 0.849964i \(-0.676623\pi\)
−0.526840 + 0.849964i \(0.676623\pi\)
\(548\) −312.000 −0.0243211
\(549\) −8740.00 −0.679443
\(550\) 1824.00 0.141410
\(551\) −108.000 −0.00835019
\(552\) 1920.00 0.148045
\(553\) 5432.00 0.417707
\(554\) 4828.00 0.370256
\(555\) 3504.00 0.267994
\(556\) −9352.00 −0.713333
\(557\) 6222.00 0.473312 0.236656 0.971594i \(-0.423949\pi\)
0.236656 + 0.971594i \(0.423949\pi\)
\(558\) −10856.0 −0.823604
\(559\) −21056.0 −1.59316
\(560\) −1344.00 −0.101419
\(561\) 10944.0 0.823629
\(562\) 3924.00 0.294527
\(563\) 4926.00 0.368750 0.184375 0.982856i \(-0.440974\pi\)
0.184375 + 0.982856i \(0.440974\pi\)
\(564\) 96.0000 0.00716725
\(565\) −2376.00 −0.176919
\(566\) 10804.0 0.802343
\(567\) 2947.00 0.218276
\(568\) 4608.00 0.340400
\(569\) 22182.0 1.63430 0.817151 0.576424i \(-0.195552\pi\)
0.817151 + 0.576424i \(0.195552\pi\)
\(570\) 96.0000 0.00705438
\(571\) 3296.00 0.241564 0.120782 0.992679i \(-0.461460\pi\)
0.120782 + 0.992679i \(0.461460\pi\)
\(572\) 10752.0 0.785951
\(573\) 4272.00 0.311458
\(574\) 1764.00 0.128272
\(575\) −2280.00 −0.165361
\(576\) −1472.00 −0.106481
\(577\) −24334.0 −1.75570 −0.877849 0.478938i \(-0.841022\pi\)
−0.877849 + 0.478938i \(0.841022\pi\)
\(578\) 16166.0 1.16335
\(579\) −8860.00 −0.635940
\(580\) 2592.00 0.185564
\(581\) 2646.00 0.188941
\(582\) 5320.00 0.378902
\(583\) 8352.00 0.593318
\(584\) −9200.00 −0.651881
\(585\) 15456.0 1.09235
\(586\) −9576.00 −0.675053
\(587\) 1638.00 0.115175 0.0575873 0.998340i \(-0.481659\pi\)
0.0575873 + 0.998340i \(0.481659\pi\)
\(588\) −392.000 −0.0274929
\(589\) 472.000 0.0330194
\(590\) −3312.00 −0.231107
\(591\) −396.000 −0.0275622
\(592\) 2336.00 0.162177
\(593\) −7446.00 −0.515633 −0.257817 0.966194i \(-0.583003\pi\)
−0.257817 + 0.966194i \(0.583003\pi\)
\(594\) 9600.00 0.663119
\(595\) 9576.00 0.659794
\(596\) −4008.00 −0.275460
\(597\) 4568.00 0.313159
\(598\) −13440.0 −0.919068
\(599\) −6504.00 −0.443650 −0.221825 0.975087i \(-0.571201\pi\)
−0.221825 + 0.975087i \(0.571201\pi\)
\(600\) −304.000 −0.0206846
\(601\) 16058.0 1.08988 0.544941 0.838474i \(-0.316552\pi\)
0.544941 + 0.838474i \(0.316552\pi\)
\(602\) −5264.00 −0.356386
\(603\) 11132.0 0.751791
\(604\) −11008.0 −0.741571
\(605\) −11676.0 −0.784623
\(606\) 6000.00 0.402200
\(607\) 10208.0 0.682586 0.341293 0.939957i \(-0.389135\pi\)
0.341293 + 0.939957i \(0.389135\pi\)
\(608\) 64.0000 0.00426898
\(609\) 756.000 0.0503032
\(610\) −9120.00 −0.605341
\(611\) −672.000 −0.0444946
\(612\) 10488.0 0.692732
\(613\) −14974.0 −0.986614 −0.493307 0.869855i \(-0.664212\pi\)
−0.493307 + 0.869855i \(0.664212\pi\)
\(614\) −1148.00 −0.0754552
\(615\) 3024.00 0.198276
\(616\) 2688.00 0.175816
\(617\) 7254.00 0.473314 0.236657 0.971593i \(-0.423948\pi\)
0.236657 + 0.971593i \(0.423948\pi\)
\(618\) −1520.00 −0.0989375
\(619\) 12458.0 0.808933 0.404466 0.914553i \(-0.367457\pi\)
0.404466 + 0.914553i \(0.367457\pi\)
\(620\) −11328.0 −0.733780
\(621\) −12000.0 −0.775432
\(622\) −17616.0 −1.13559
\(623\) −2730.00 −0.175562
\(624\) −1792.00 −0.114964
\(625\) −17639.0 −1.12890
\(626\) −5540.00 −0.353711
\(627\) −192.000 −0.0122293
\(628\) −2080.00 −0.132167
\(629\) −16644.0 −1.05507
\(630\) 3864.00 0.244358
\(631\) 28352.0 1.78871 0.894354 0.447359i \(-0.147635\pi\)
0.894354 + 0.447359i \(0.147635\pi\)
\(632\) 6208.00 0.390729
\(633\) −8824.00 −0.554064
\(634\) 15132.0 0.947900
\(635\) 4512.00 0.281974
\(636\) −1392.00 −0.0867868
\(637\) 2744.00 0.170677
\(638\) −5184.00 −0.321687
\(639\) −13248.0 −0.820161
\(640\) −1536.00 −0.0948683
\(641\) 27390.0 1.68774 0.843869 0.536549i \(-0.180272\pi\)
0.843869 + 0.536549i \(0.180272\pi\)
\(642\) −2544.00 −0.156392
\(643\) −21490.0 −1.31801 −0.659007 0.752137i \(-0.729023\pi\)
−0.659007 + 0.752137i \(0.729023\pi\)
\(644\) −3360.00 −0.205594
\(645\) −9024.00 −0.550883
\(646\) −456.000 −0.0277726
\(647\) 17652.0 1.07260 0.536300 0.844028i \(-0.319822\pi\)
0.536300 + 0.844028i \(0.319822\pi\)
\(648\) 3368.00 0.204178
\(649\) 6624.00 0.400639
\(650\) 2128.00 0.128411
\(651\) −3304.00 −0.198915
\(652\) 5120.00 0.307538
\(653\) −4782.00 −0.286576 −0.143288 0.989681i \(-0.545768\pi\)
−0.143288 + 0.989681i \(0.545768\pi\)
\(654\) −584.000 −0.0349177
\(655\) −25560.0 −1.52475
\(656\) 2016.00 0.119987
\(657\) 26450.0 1.57064
\(658\) −168.000 −0.00995338
\(659\) −27144.0 −1.60452 −0.802261 0.596973i \(-0.796370\pi\)
−0.802261 + 0.596973i \(0.796370\pi\)
\(660\) 4608.00 0.271767
\(661\) −11860.0 −0.697883 −0.348941 0.937145i \(-0.613459\pi\)
−0.348941 + 0.937145i \(0.613459\pi\)
\(662\) −22640.0 −1.32920
\(663\) 12768.0 0.747916
\(664\) 3024.00 0.176738
\(665\) −168.000 −0.00979663
\(666\) −6716.00 −0.390750
\(667\) 6480.00 0.376172
\(668\) 7056.00 0.408690
\(669\) −4144.00 −0.239486
\(670\) 11616.0 0.669799
\(671\) 18240.0 1.04940
\(672\) −448.000 −0.0257172
\(673\) 5546.00 0.317656 0.158828 0.987306i \(-0.449228\pi\)
0.158828 + 0.987306i \(0.449228\pi\)
\(674\) −9572.00 −0.547032
\(675\) 1900.00 0.108342
\(676\) 3756.00 0.213701
\(677\) −14880.0 −0.844734 −0.422367 0.906425i \(-0.638801\pi\)
−0.422367 + 0.906425i \(0.638801\pi\)
\(678\) −792.000 −0.0448622
\(679\) −9310.00 −0.526193
\(680\) 10944.0 0.617181
\(681\) 732.000 0.0411899
\(682\) 22656.0 1.27206
\(683\) 20964.0 1.17447 0.587237 0.809415i \(-0.300216\pi\)
0.587237 + 0.809415i \(0.300216\pi\)
\(684\) −184.000 −0.0102857
\(685\) 936.000 0.0522084
\(686\) 686.000 0.0381802
\(687\) 752.000 0.0417621
\(688\) −6016.00 −0.333369
\(689\) 9744.00 0.538776
\(690\) −5760.00 −0.317796
\(691\) 13106.0 0.721528 0.360764 0.932657i \(-0.382516\pi\)
0.360764 + 0.932657i \(0.382516\pi\)
\(692\) −3072.00 −0.168757
\(693\) −7728.00 −0.423611
\(694\) 25296.0 1.38361
\(695\) 28056.0 1.53126
\(696\) 864.000 0.0470544
\(697\) −14364.0 −0.780596
\(698\) 19264.0 1.04463
\(699\) 4524.00 0.244797
\(700\) 532.000 0.0287253
\(701\) −4590.00 −0.247307 −0.123653 0.992325i \(-0.539461\pi\)
−0.123653 + 0.992325i \(0.539461\pi\)
\(702\) 11200.0 0.602161
\(703\) 292.000 0.0156657
\(704\) 3072.00 0.164461
\(705\) −288.000 −0.0153854
\(706\) −6780.00 −0.361429
\(707\) −10500.0 −0.558548
\(708\) −1104.00 −0.0586029
\(709\) −862.000 −0.0456602 −0.0228301 0.999739i \(-0.507268\pi\)
−0.0228301 + 0.999739i \(0.507268\pi\)
\(710\) −13824.0 −0.730712
\(711\) −17848.0 −0.941424
\(712\) −3120.00 −0.164223
\(713\) −28320.0 −1.48751
\(714\) 3192.00 0.167308
\(715\) −32256.0 −1.68714
\(716\) 7248.00 0.378311
\(717\) −5184.00 −0.270014
\(718\) −21408.0 −1.11273
\(719\) −3540.00 −0.183616 −0.0918079 0.995777i \(-0.529265\pi\)
−0.0918079 + 0.995777i \(0.529265\pi\)
\(720\) 4416.00 0.228576
\(721\) 2660.00 0.137397
\(722\) −13710.0 −0.706694
\(723\) −220.000 −0.0113166
\(724\) −1792.00 −0.0919878
\(725\) −1026.00 −0.0525582
\(726\) −3892.00 −0.198961
\(727\) −4228.00 −0.215692 −0.107846 0.994168i \(-0.534395\pi\)
−0.107846 + 0.994168i \(0.534395\pi\)
\(728\) 3136.00 0.159654
\(729\) −4283.00 −0.217599
\(730\) 27600.0 1.39934
\(731\) 42864.0 2.16879
\(732\) −3040.00 −0.153499
\(733\) 5420.00 0.273114 0.136557 0.990632i \(-0.456396\pi\)
0.136557 + 0.990632i \(0.456396\pi\)
\(734\) −17168.0 −0.863328
\(735\) 1176.00 0.0590169
\(736\) −3840.00 −0.192316
\(737\) −23232.0 −1.16114
\(738\) −5796.00 −0.289097
\(739\) 1280.00 0.0637152 0.0318576 0.999492i \(-0.489858\pi\)
0.0318576 + 0.999492i \(0.489858\pi\)
\(740\) −7008.00 −0.348134
\(741\) −224.000 −0.0111051
\(742\) 2436.00 0.120523
\(743\) −35712.0 −1.76332 −0.881660 0.471886i \(-0.843573\pi\)
−0.881660 + 0.471886i \(0.843573\pi\)
\(744\) −3776.00 −0.186068
\(745\) 12024.0 0.591309
\(746\) −4244.00 −0.208289
\(747\) −8694.00 −0.425832
\(748\) −21888.0 −1.06993
\(749\) 4452.00 0.217186
\(750\) −5088.00 −0.247717
\(751\) 24464.0 1.18869 0.594344 0.804211i \(-0.297412\pi\)
0.594344 + 0.804211i \(0.297412\pi\)
\(752\) −192.000 −0.00931053
\(753\) 3780.00 0.182936
\(754\) −6048.00 −0.292116
\(755\) 33024.0 1.59188
\(756\) 2800.00 0.134702
\(757\) 30242.0 1.45200 0.726000 0.687695i \(-0.241377\pi\)
0.726000 + 0.687695i \(0.241377\pi\)
\(758\) −9824.00 −0.470744
\(759\) 11520.0 0.550922
\(760\) −192.000 −0.00916391
\(761\) −2154.00 −0.102605 −0.0513025 0.998683i \(-0.516337\pi\)
−0.0513025 + 0.998683i \(0.516337\pi\)
\(762\) 1504.00 0.0715015
\(763\) 1022.00 0.0484913
\(764\) −8544.00 −0.404596
\(765\) −31464.0 −1.48704
\(766\) 18120.0 0.854703
\(767\) 7728.00 0.363810
\(768\) −512.000 −0.0240563
\(769\) 10262.0 0.481219 0.240609 0.970622i \(-0.422653\pi\)
0.240609 + 0.970622i \(0.422653\pi\)
\(770\) −8064.00 −0.377411
\(771\) −4260.00 −0.198989
\(772\) 17720.0 0.826110
\(773\) 9084.00 0.422676 0.211338 0.977413i \(-0.432218\pi\)
0.211338 + 0.977413i \(0.432218\pi\)
\(774\) 17296.0 0.803219
\(775\) 4484.00 0.207832
\(776\) −10640.0 −0.492208
\(777\) −2044.00 −0.0943733
\(778\) 17988.0 0.828922
\(779\) 252.000 0.0115903
\(780\) 5376.00 0.246784
\(781\) 27648.0 1.26674
\(782\) 27360.0 1.25114
\(783\) −5400.00 −0.246463
\(784\) 784.000 0.0357143
\(785\) 6240.00 0.283714
\(786\) −8520.00 −0.386639
\(787\) −19798.0 −0.896725 −0.448362 0.893852i \(-0.647993\pi\)
−0.448362 + 0.893852i \(0.647993\pi\)
\(788\) 792.000 0.0358044
\(789\) 9984.00 0.450494
\(790\) −18624.0 −0.838750
\(791\) 1386.00 0.0623015
\(792\) −8832.00 −0.396252
\(793\) 21280.0 0.952932
\(794\) −25952.0 −1.15995
\(795\) 4176.00 0.186299
\(796\) −9136.00 −0.406805
\(797\) 30240.0 1.34398 0.671992 0.740558i \(-0.265439\pi\)
0.671992 + 0.740558i \(0.265439\pi\)
\(798\) −56.0000 −0.00248418
\(799\) 1368.00 0.0605712
\(800\) 608.000 0.0268701
\(801\) 8970.00 0.395680
\(802\) −7044.00 −0.310140
\(803\) −55200.0 −2.42586
\(804\) 3872.00 0.169844
\(805\) 10080.0 0.441333
\(806\) 26432.0 1.15512
\(807\) −13632.0 −0.594633
\(808\) −12000.0 −0.522473
\(809\) −2346.00 −0.101954 −0.0509771 0.998700i \(-0.516234\pi\)
−0.0509771 + 0.998700i \(0.516234\pi\)
\(810\) −10104.0 −0.438294
\(811\) −29806.0 −1.29054 −0.645271 0.763953i \(-0.723256\pi\)
−0.645271 + 0.763953i \(0.723256\pi\)
\(812\) −1512.00 −0.0653458
\(813\) −16384.0 −0.706780
\(814\) 14016.0 0.603514
\(815\) −15360.0 −0.660169
\(816\) 3648.00 0.156502
\(817\) −752.000 −0.0322021
\(818\) 25420.0 1.08654
\(819\) −9016.00 −0.384670
\(820\) −6048.00 −0.257567
\(821\) −1506.00 −0.0640192 −0.0320096 0.999488i \(-0.510191\pi\)
−0.0320096 + 0.999488i \(0.510191\pi\)
\(822\) 312.000 0.0132387
\(823\) −20392.0 −0.863694 −0.431847 0.901947i \(-0.642138\pi\)
−0.431847 + 0.901947i \(0.642138\pi\)
\(824\) 3040.00 0.128524
\(825\) −1824.00 −0.0769740
\(826\) 1932.00 0.0813836
\(827\) 36108.0 1.51826 0.759128 0.650941i \(-0.225626\pi\)
0.759128 + 0.650941i \(0.225626\pi\)
\(828\) 11040.0 0.463365
\(829\) −13876.0 −0.581343 −0.290672 0.956823i \(-0.593879\pi\)
−0.290672 + 0.956823i \(0.593879\pi\)
\(830\) −9072.00 −0.379390
\(831\) −4828.00 −0.201542
\(832\) 3584.00 0.149342
\(833\) −5586.00 −0.232345
\(834\) 9352.00 0.388289
\(835\) −21168.0 −0.877304
\(836\) 384.000 0.0158863
\(837\) 23600.0 0.974594
\(838\) 3276.00 0.135045
\(839\) 23436.0 0.964363 0.482182 0.876071i \(-0.339845\pi\)
0.482182 + 0.876071i \(0.339845\pi\)
\(840\) 1344.00 0.0552052
\(841\) −21473.0 −0.880438
\(842\) −25700.0 −1.05188
\(843\) −3924.00 −0.160320
\(844\) 17648.0 0.719750
\(845\) −11268.0 −0.458735
\(846\) 552.000 0.0224328
\(847\) 6811.00 0.276303
\(848\) 2784.00 0.112739
\(849\) −10804.0 −0.436740
\(850\) −4332.00 −0.174807
\(851\) −17520.0 −0.705732
\(852\) −4608.00 −0.185290
\(853\) 8120.00 0.325936 0.162968 0.986631i \(-0.447893\pi\)
0.162968 + 0.986631i \(0.447893\pi\)
\(854\) 5320.00 0.213169
\(855\) 552.000 0.0220795
\(856\) 5088.00 0.203159
\(857\) −50010.0 −1.99336 −0.996680 0.0814218i \(-0.974054\pi\)
−0.996680 + 0.0814218i \(0.974054\pi\)
\(858\) −10752.0 −0.427817
\(859\) 34526.0 1.37138 0.685688 0.727896i \(-0.259501\pi\)
0.685688 + 0.727896i \(0.259501\pi\)
\(860\) 18048.0 0.715618
\(861\) −1764.00 −0.0698223
\(862\) −16032.0 −0.633471
\(863\) −17256.0 −0.680650 −0.340325 0.940308i \(-0.610537\pi\)
−0.340325 + 0.940308i \(0.610537\pi\)
\(864\) 3200.00 0.126003
\(865\) 9216.00 0.362258
\(866\) 4396.00 0.172497
\(867\) −16166.0 −0.633248
\(868\) 6608.00 0.258399
\(869\) 37248.0 1.45403
\(870\) −2592.00 −0.101008
\(871\) −27104.0 −1.05440
\(872\) 1168.00 0.0453595
\(873\) 30590.0 1.18593
\(874\) −480.000 −0.0185769
\(875\) 8904.00 0.344012
\(876\) 9200.00 0.354839
\(877\) 8714.00 0.335520 0.167760 0.985828i \(-0.446347\pi\)
0.167760 + 0.985828i \(0.446347\pi\)
\(878\) −752.000 −0.0289052
\(879\) 9576.00 0.367452
\(880\) −9216.00 −0.353036
\(881\) −22806.0 −0.872138 −0.436069 0.899913i \(-0.643630\pi\)
−0.436069 + 0.899913i \(0.643630\pi\)
\(882\) −2254.00 −0.0860500
\(883\) 40196.0 1.53194 0.765970 0.642876i \(-0.222259\pi\)
0.765970 + 0.642876i \(0.222259\pi\)
\(884\) −25536.0 −0.971571
\(885\) 3312.00 0.125798
\(886\) 14376.0 0.545114
\(887\) 40812.0 1.54491 0.772454 0.635071i \(-0.219029\pi\)
0.772454 + 0.635071i \(0.219029\pi\)
\(888\) −2336.00 −0.0882782
\(889\) −2632.00 −0.0992963
\(890\) 9360.00 0.352526
\(891\) 20208.0 0.759813
\(892\) 8288.00 0.311102
\(893\) −24.0000 −0.000899361 0
\(894\) 4008.00 0.149941
\(895\) −21744.0 −0.812091
\(896\) 896.000 0.0334077
\(897\) 13440.0 0.500277
\(898\) −29340.0 −1.09030
\(899\) −12744.0 −0.472788
\(900\) −1748.00 −0.0647407
\(901\) −19836.0 −0.733444
\(902\) 12096.0 0.446511
\(903\) 5264.00 0.193992
\(904\) 1584.00 0.0582777
\(905\) 5376.00 0.197463
\(906\) 11008.0 0.403660
\(907\) −13588.0 −0.497444 −0.248722 0.968575i \(-0.580011\pi\)
−0.248722 + 0.968575i \(0.580011\pi\)
\(908\) −1464.00 −0.0535072
\(909\) 34500.0 1.25885
\(910\) −9408.00 −0.342717
\(911\) −47304.0 −1.72036 −0.860182 0.509987i \(-0.829650\pi\)
−0.860182 + 0.509987i \(0.829650\pi\)
\(912\) −64.0000 −0.00232374
\(913\) 18144.0 0.657699
\(914\) −10292.0 −0.372461
\(915\) 9120.00 0.329506
\(916\) −1504.00 −0.0542506
\(917\) 14910.0 0.536937
\(918\) −22800.0 −0.819730
\(919\) 1784.00 0.0640356 0.0320178 0.999487i \(-0.489807\pi\)
0.0320178 + 0.999487i \(0.489807\pi\)
\(920\) 11520.0 0.412830
\(921\) 1148.00 0.0410726
\(922\) −3024.00 −0.108015
\(923\) 32256.0 1.15029
\(924\) −2688.00 −0.0957021
\(925\) 2774.00 0.0986038
\(926\) 14368.0 0.509894
\(927\) −8740.00 −0.309665
\(928\) −1728.00 −0.0611254
\(929\) −35922.0 −1.26864 −0.634318 0.773072i \(-0.718719\pi\)
−0.634318 + 0.773072i \(0.718719\pi\)
\(930\) 11328.0 0.399419
\(931\) 98.0000 0.00344986
\(932\) −9048.00 −0.318001
\(933\) 17616.0 0.618137
\(934\) −33036.0 −1.15736
\(935\) 65664.0 2.29673
\(936\) −10304.0 −0.359826
\(937\) −26782.0 −0.933756 −0.466878 0.884322i \(-0.654621\pi\)
−0.466878 + 0.884322i \(0.654621\pi\)
\(938\) −6776.00 −0.235868
\(939\) 5540.00 0.192536
\(940\) 576.000 0.0199862
\(941\) 4044.00 0.140096 0.0700482 0.997544i \(-0.477685\pi\)
0.0700482 + 0.997544i \(0.477685\pi\)
\(942\) 2080.00 0.0719427
\(943\) −15120.0 −0.522137
\(944\) 2208.00 0.0761274
\(945\) −8400.00 −0.289156
\(946\) −36096.0 −1.24057
\(947\) −2136.00 −0.0732953 −0.0366477 0.999328i \(-0.511668\pi\)
−0.0366477 + 0.999328i \(0.511668\pi\)
\(948\) −6208.00 −0.212686
\(949\) −64400.0 −2.20286
\(950\) 76.0000 0.00259554
\(951\) −15132.0 −0.515971
\(952\) −6384.00 −0.217339
\(953\) −15174.0 −0.515776 −0.257888 0.966175i \(-0.583026\pi\)
−0.257888 + 0.966175i \(0.583026\pi\)
\(954\) −8004.00 −0.271634
\(955\) 25632.0 0.868515
\(956\) 10368.0 0.350758
\(957\) 5184.00 0.175104
\(958\) 20184.0 0.680705
\(959\) −546.000 −0.0183850
\(960\) 1536.00 0.0516398
\(961\) 25905.0 0.869558
\(962\) 16352.0 0.548035
\(963\) −14628.0 −0.489492
\(964\) 440.000 0.0147007
\(965\) −53160.0 −1.77335
\(966\) 3360.00 0.111911
\(967\) 25832.0 0.859050 0.429525 0.903055i \(-0.358681\pi\)
0.429525 + 0.903055i \(0.358681\pi\)
\(968\) 7784.00 0.258458
\(969\) 456.000 0.0151175
\(970\) 31920.0 1.05659
\(971\) −37686.0 −1.24552 −0.622761 0.782412i \(-0.713989\pi\)
−0.622761 + 0.782412i \(0.713989\pi\)
\(972\) −14168.0 −0.467530
\(973\) −16366.0 −0.539229
\(974\) 15664.0 0.515305
\(975\) −2128.00 −0.0698980
\(976\) 6080.00 0.199402
\(977\) −54006.0 −1.76848 −0.884240 0.467033i \(-0.845323\pi\)
−0.884240 + 0.467033i \(0.845323\pi\)
\(978\) −5120.00 −0.167402
\(979\) −18720.0 −0.611127
\(980\) −2352.00 −0.0766652
\(981\) −3358.00 −0.109289
\(982\) −13464.0 −0.437529
\(983\) 33276.0 1.07969 0.539847 0.841763i \(-0.318482\pi\)
0.539847 + 0.841763i \(0.318482\pi\)
\(984\) −2016.00 −0.0653127
\(985\) −2376.00 −0.0768585
\(986\) 12312.0 0.397661
\(987\) 168.000 0.00541793
\(988\) 448.000 0.0144259
\(989\) 45120.0 1.45069
\(990\) 26496.0 0.850604
\(991\) −3760.00 −0.120525 −0.0602625 0.998183i \(-0.519194\pi\)
−0.0602625 + 0.998183i \(0.519194\pi\)
\(992\) 7552.00 0.241710
\(993\) 22640.0 0.723523
\(994\) 8064.00 0.257318
\(995\) 27408.0 0.873258
\(996\) −3024.00 −0.0962039
\(997\) 36524.0 1.16021 0.580104 0.814543i \(-0.303012\pi\)
0.580104 + 0.814543i \(0.303012\pi\)
\(998\) 37336.0 1.18422
\(999\) 14600.0 0.462386
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 14.4.a.b.1.1 1
3.2 odd 2 126.4.a.d.1.1 1
4.3 odd 2 112.4.a.e.1.1 1
5.2 odd 4 350.4.c.g.99.2 2
5.3 odd 4 350.4.c.g.99.1 2
5.4 even 2 350.4.a.f.1.1 1
7.2 even 3 98.4.c.c.67.1 2
7.3 odd 6 98.4.c.b.79.1 2
7.4 even 3 98.4.c.c.79.1 2
7.5 odd 6 98.4.c.b.67.1 2
7.6 odd 2 98.4.a.e.1.1 1
8.3 odd 2 448.4.a.g.1.1 1
8.5 even 2 448.4.a.k.1.1 1
11.10 odd 2 1694.4.a.b.1.1 1
12.11 even 2 1008.4.a.r.1.1 1
13.12 even 2 2366.4.a.c.1.1 1
21.2 odd 6 882.4.g.p.361.1 2
21.5 even 6 882.4.g.v.361.1 2
21.11 odd 6 882.4.g.p.667.1 2
21.17 even 6 882.4.g.v.667.1 2
21.20 even 2 882.4.a.b.1.1 1
28.27 even 2 784.4.a.h.1.1 1
35.34 odd 2 2450.4.a.i.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
14.4.a.b.1.1 1 1.1 even 1 trivial
98.4.a.e.1.1 1 7.6 odd 2
98.4.c.b.67.1 2 7.5 odd 6
98.4.c.b.79.1 2 7.3 odd 6
98.4.c.c.67.1 2 7.2 even 3
98.4.c.c.79.1 2 7.4 even 3
112.4.a.e.1.1 1 4.3 odd 2
126.4.a.d.1.1 1 3.2 odd 2
350.4.a.f.1.1 1 5.4 even 2
350.4.c.g.99.1 2 5.3 odd 4
350.4.c.g.99.2 2 5.2 odd 4
448.4.a.g.1.1 1 8.3 odd 2
448.4.a.k.1.1 1 8.5 even 2
784.4.a.h.1.1 1 28.27 even 2
882.4.a.b.1.1 1 21.20 even 2
882.4.g.p.361.1 2 21.2 odd 6
882.4.g.p.667.1 2 21.11 odd 6
882.4.g.v.361.1 2 21.5 even 6
882.4.g.v.667.1 2 21.17 even 6
1008.4.a.r.1.1 1 12.11 even 2
1694.4.a.b.1.1 1 11.10 odd 2
2366.4.a.c.1.1 1 13.12 even 2
2450.4.a.i.1.1 1 35.34 odd 2