Properties

Label 78.4.a.a.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)
 
\(f(q)\) \(=\) \(q-2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} -16.0000 q^{5} +6.00000 q^{6} +28.0000 q^{7} -8.00000 q^{8} +9.00000 q^{9} +O(q^{10})\) \(q-2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} -16.0000 q^{5} +6.00000 q^{6} +28.0000 q^{7} -8.00000 q^{8} +9.00000 q^{9} +32.0000 q^{10} +34.0000 q^{11} -12.0000 q^{12} -13.0000 q^{13} -56.0000 q^{14} +48.0000 q^{15} +16.0000 q^{16} +138.000 q^{17} -18.0000 q^{18} +108.000 q^{19} -64.0000 q^{20} -84.0000 q^{21} -68.0000 q^{22} -52.0000 q^{23} +24.0000 q^{24} +131.000 q^{25} +26.0000 q^{26} -27.0000 q^{27} +112.000 q^{28} -190.000 q^{29} -96.0000 q^{30} -176.000 q^{31} -32.0000 q^{32} -102.000 q^{33} -276.000 q^{34} -448.000 q^{35} +36.0000 q^{36} +342.000 q^{37} -216.000 q^{38} +39.0000 q^{39} +128.000 q^{40} +240.000 q^{41} +168.000 q^{42} -140.000 q^{43} +136.000 q^{44} -144.000 q^{45} +104.000 q^{46} +454.000 q^{47} -48.0000 q^{48} +441.000 q^{49} -262.000 q^{50} -414.000 q^{51} -52.0000 q^{52} +198.000 q^{53} +54.0000 q^{54} -544.000 q^{55} -224.000 q^{56} -324.000 q^{57} +380.000 q^{58} -154.000 q^{59} +192.000 q^{60} +34.0000 q^{61} +352.000 q^{62} +252.000 q^{63} +64.0000 q^{64} +208.000 q^{65} +204.000 q^{66} -656.000 q^{67} +552.000 q^{68} +156.000 q^{69} +896.000 q^{70} +550.000 q^{71} -72.0000 q^{72} +614.000 q^{73} -684.000 q^{74} -393.000 q^{75} +432.000 q^{76} +952.000 q^{77} -78.0000 q^{78} +8.00000 q^{79} -256.000 q^{80} +81.0000 q^{81} -480.000 q^{82} +762.000 q^{83} -336.000 q^{84} -2208.00 q^{85} +280.000 q^{86} +570.000 q^{87} -272.000 q^{88} -444.000 q^{89} +288.000 q^{90} -364.000 q^{91} -208.000 q^{92} +528.000 q^{93} -908.000 q^{94} -1728.00 q^{95} +96.0000 q^{96} +1022.00 q^{97} -882.000 q^{98} +306.000 q^{99} +O(q^{100})\)

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
\(5\) −16.0000 −1.43108 −0.715542 0.698570i \(-0.753820\pi\)
−0.715542 + 0.698570i \(0.753820\pi\)
\(6\) 6.00000 0.408248
\(7\) 28.0000 1.51186 0.755929 0.654654i \(-0.227186\pi\)
0.755929 + 0.654654i \(0.227186\pi\)
\(8\) −8.00000 −0.353553
\(9\) 9.00000 0.333333
\(10\) 32.0000 1.01193
\(11\) 34.0000 0.931944 0.465972 0.884799i \(-0.345705\pi\)
0.465972 + 0.884799i \(0.345705\pi\)
\(12\) −12.0000 −0.288675
\(13\) −13.0000 −0.277350
\(14\) −56.0000 −1.06904
\(15\) 48.0000 0.826236
\(16\) 16.0000 0.250000
\(17\) 138.000 1.96882 0.984409 0.175893i \(-0.0562813\pi\)
0.984409 + 0.175893i \(0.0562813\pi\)
\(18\) −18.0000 −0.235702
\(19\) 108.000 1.30405 0.652024 0.758199i \(-0.273920\pi\)
0.652024 + 0.758199i \(0.273920\pi\)
\(20\) −64.0000 −0.715542
\(21\) −84.0000 −0.872872
\(22\) −68.0000 −0.658984
\(23\) −52.0000 −0.471424 −0.235712 0.971823i \(-0.575742\pi\)
−0.235712 + 0.971823i \(0.575742\pi\)
\(24\) 24.0000 0.204124
\(25\) 131.000 1.04800
\(26\) 26.0000 0.196116
\(27\) −27.0000 −0.192450
\(28\) 112.000 0.755929
\(29\) −190.000 −1.21662 −0.608312 0.793698i \(-0.708153\pi\)
−0.608312 + 0.793698i \(0.708153\pi\)
\(30\) −96.0000 −0.584237
\(31\) −176.000 −1.01969 −0.509847 0.860265i \(-0.670298\pi\)
−0.509847 + 0.860265i \(0.670298\pi\)
\(32\) −32.0000 −0.176777
\(33\) −102.000 −0.538058
\(34\) −276.000 −1.39216
\(35\) −448.000 −2.16359
\(36\) 36.0000 0.166667
\(37\) 342.000 1.51958 0.759790 0.650169i \(-0.225302\pi\)
0.759790 + 0.650169i \(0.225302\pi\)
\(38\) −216.000 −0.922101
\(39\) 39.0000 0.160128
\(40\) 128.000 0.505964
\(41\) 240.000 0.914188 0.457094 0.889418i \(-0.348890\pi\)
0.457094 + 0.889418i \(0.348890\pi\)
\(42\) 168.000 0.617213
\(43\) −140.000 −0.496507 −0.248253 0.968695i \(-0.579857\pi\)
−0.248253 + 0.968695i \(0.579857\pi\)
\(44\) 136.000 0.465972
\(45\) −144.000 −0.477028
\(46\) 104.000 0.333347
\(47\) 454.000 1.40899 0.704497 0.709707i \(-0.251173\pi\)
0.704497 + 0.709707i \(0.251173\pi\)
\(48\) −48.0000 −0.144338
\(49\) 441.000 1.28571
\(50\) −262.000 −0.741048
\(51\) −414.000 −1.13670
\(52\) −52.0000 −0.138675
\(53\) 198.000 0.513158 0.256579 0.966523i \(-0.417405\pi\)
0.256579 + 0.966523i \(0.417405\pi\)
\(54\) 54.0000 0.136083
\(55\) −544.000 −1.33369
\(56\) −224.000 −0.534522
\(57\) −324.000 −0.752892
\(58\) 380.000 0.860284
\(59\) −154.000 −0.339815 −0.169908 0.985460i \(-0.554347\pi\)
−0.169908 + 0.985460i \(0.554347\pi\)
\(60\) 192.000 0.413118
\(61\) 34.0000 0.0713648 0.0356824 0.999363i \(-0.488640\pi\)
0.0356824 + 0.999363i \(0.488640\pi\)
\(62\) 352.000 0.721033
\(63\) 252.000 0.503953
\(64\) 64.0000 0.125000
\(65\) 208.000 0.396911
\(66\) 204.000 0.380465
\(67\) −656.000 −1.19617 −0.598083 0.801434i \(-0.704071\pi\)
−0.598083 + 0.801434i \(0.704071\pi\)
\(68\) 552.000 0.984409
\(69\) 156.000 0.272177
\(70\) 896.000 1.52989
\(71\) 550.000 0.919338 0.459669 0.888090i \(-0.347968\pi\)
0.459669 + 0.888090i \(0.347968\pi\)
\(72\) −72.0000 −0.117851
\(73\) 614.000 0.984428 0.492214 0.870474i \(-0.336188\pi\)
0.492214 + 0.870474i \(0.336188\pi\)
\(74\) −684.000 −1.07451
\(75\) −393.000 −0.605063
\(76\) 432.000 0.652024
\(77\) 952.000 1.40897
\(78\) −78.0000 −0.113228
\(79\) 8.00000 0.0113933 0.00569665 0.999984i \(-0.498187\pi\)
0.00569665 + 0.999984i \(0.498187\pi\)
\(80\) −256.000 −0.357771
\(81\) 81.0000 0.111111
\(82\) −480.000 −0.646428
\(83\) 762.000 1.00772 0.503858 0.863787i \(-0.331914\pi\)
0.503858 + 0.863787i \(0.331914\pi\)
\(84\) −336.000 −0.436436
\(85\) −2208.00 −2.81754
\(86\) 280.000 0.351083
\(87\) 570.000 0.702419
\(88\) −272.000 −0.329492
\(89\) −444.000 −0.528808 −0.264404 0.964412i \(-0.585175\pi\)
−0.264404 + 0.964412i \(0.585175\pi\)
\(90\) 288.000 0.337310
\(91\) −364.000 −0.419314
\(92\) −208.000 −0.235712
\(93\) 528.000 0.588721
\(94\) −908.000 −0.996309
\(95\) −1728.00 −1.86620
\(96\) 96.0000 0.102062
\(97\) 1022.00 1.06978 0.534889 0.844923i \(-0.320354\pi\)
0.534889 + 0.844923i \(0.320354\pi\)
\(98\) −882.000 −0.909137
\(99\) 306.000 0.310648
\(100\) 524.000 0.524000
\(101\) −1190.00 −1.17237 −0.586185 0.810177i \(-0.699371\pi\)
−0.586185 + 0.810177i \(0.699371\pi\)
\(102\) 828.000 0.803767
\(103\) −224.000 −0.214285 −0.107143 0.994244i \(-0.534170\pi\)
−0.107143 + 0.994244i \(0.534170\pi\)
\(104\) 104.000 0.0980581
\(105\) 1344.00 1.24915
\(106\) −396.000 −0.362858
\(107\) −640.000 −0.578235 −0.289117 0.957294i \(-0.593362\pi\)
−0.289117 + 0.957294i \(0.593362\pi\)
\(108\) −108.000 −0.0962250
\(109\) −1934.00 −1.69948 −0.849741 0.527200i \(-0.823242\pi\)
−0.849741 + 0.527200i \(0.823242\pi\)
\(110\) 1088.00 0.943061
\(111\) −1026.00 −0.877330
\(112\) 448.000 0.377964
\(113\) −418.000 −0.347983 −0.173992 0.984747i \(-0.555667\pi\)
−0.173992 + 0.984747i \(0.555667\pi\)
\(114\) 648.000 0.532375
\(115\) 832.000 0.674647
\(116\) −760.000 −0.608312
\(117\) −117.000 −0.0924500
\(118\) 308.000 0.240286
\(119\) 3864.00 2.97657
\(120\) −384.000 −0.292119
\(121\) −175.000 −0.131480
\(122\) −68.0000 −0.0504625
\(123\) −720.000 −0.527807
\(124\) −704.000 −0.509847
\(125\) −96.0000 −0.0686920
\(126\) −504.000 −0.356348
\(127\) −1040.00 −0.726654 −0.363327 0.931662i \(-0.618359\pi\)
−0.363327 + 0.931662i \(0.618359\pi\)
\(128\) −128.000 −0.0883883
\(129\) 420.000 0.286658
\(130\) −416.000 −0.280659
\(131\) −568.000 −0.378827 −0.189414 0.981897i \(-0.560659\pi\)
−0.189414 + 0.981897i \(0.560659\pi\)
\(132\) −408.000 −0.269029
\(133\) 3024.00 1.97153
\(134\) 1312.00 0.845817
\(135\) 432.000 0.275412
\(136\) −1104.00 −0.696082
\(137\) 528.000 0.329271 0.164635 0.986355i \(-0.447355\pi\)
0.164635 + 0.986355i \(0.447355\pi\)
\(138\) −312.000 −0.192458
\(139\) −1556.00 −0.949483 −0.474742 0.880125i \(-0.657459\pi\)
−0.474742 + 0.880125i \(0.657459\pi\)
\(140\) −1792.00 −1.08180
\(141\) −1362.00 −0.813483
\(142\) −1100.00 −0.650070
\(143\) −442.000 −0.258475
\(144\) 144.000 0.0833333
\(145\) 3040.00 1.74109
\(146\) −1228.00 −0.696096
\(147\) −1323.00 −0.742307
\(148\) 1368.00 0.759790
\(149\) 1524.00 0.837926 0.418963 0.908003i \(-0.362394\pi\)
0.418963 + 0.908003i \(0.362394\pi\)
\(150\) 786.000 0.427844
\(151\) −3024.00 −1.62973 −0.814866 0.579649i \(-0.803190\pi\)
−0.814866 + 0.579649i \(0.803190\pi\)
\(152\) −864.000 −0.461050
\(153\) 1242.00 0.656273
\(154\) −1904.00 −0.996290
\(155\) 2816.00 1.45927
\(156\) 156.000 0.0800641
\(157\) 2198.00 1.11732 0.558661 0.829396i \(-0.311315\pi\)
0.558661 + 0.829396i \(0.311315\pi\)
\(158\) −16.0000 −0.00805628
\(159\) −594.000 −0.296272
\(160\) 512.000 0.252982
\(161\) −1456.00 −0.712726
\(162\) −162.000 −0.0785674
\(163\) −268.000 −0.128781 −0.0643907 0.997925i \(-0.520510\pi\)
−0.0643907 + 0.997925i \(0.520510\pi\)
\(164\) 960.000 0.457094
\(165\) 1632.00 0.770006
\(166\) −1524.00 −0.712562
\(167\) 702.000 0.325284 0.162642 0.986685i \(-0.447998\pi\)
0.162642 + 0.986685i \(0.447998\pi\)
\(168\) 672.000 0.308607
\(169\) 169.000 0.0769231
\(170\) 4416.00 1.99230
\(171\) 972.000 0.434682
\(172\) −560.000 −0.248253
\(173\) 2066.00 0.907948 0.453974 0.891015i \(-0.350006\pi\)
0.453974 + 0.891015i \(0.350006\pi\)
\(174\) −1140.00 −0.496685
\(175\) 3668.00 1.58443
\(176\) 544.000 0.232986
\(177\) 462.000 0.196192
\(178\) 888.000 0.373924
\(179\) −276.000 −0.115247 −0.0576235 0.998338i \(-0.518352\pi\)
−0.0576235 + 0.998338i \(0.518352\pi\)
\(180\) −576.000 −0.238514
\(181\) −3474.00 −1.42663 −0.713316 0.700843i \(-0.752808\pi\)
−0.713316 + 0.700843i \(0.752808\pi\)
\(182\) 728.000 0.296500
\(183\) −102.000 −0.0412025
\(184\) 416.000 0.166674
\(185\) −5472.00 −2.17465
\(186\) −1056.00 −0.416289
\(187\) 4692.00 1.83483
\(188\) 1816.00 0.704497
\(189\) −756.000 −0.290957
\(190\) 3456.00 1.31960
\(191\) −3920.00 −1.48503 −0.742516 0.669828i \(-0.766368\pi\)
−0.742516 + 0.669828i \(0.766368\pi\)
\(192\) −192.000 −0.0721688
\(193\) 2186.00 0.815294 0.407647 0.913140i \(-0.366349\pi\)
0.407647 + 0.913140i \(0.366349\pi\)
\(194\) −2044.00 −0.756447
\(195\) −624.000 −0.229157
\(196\) 1764.00 0.642857
\(197\) −1368.00 −0.494751 −0.247376 0.968920i \(-0.579568\pi\)
−0.247376 + 0.968920i \(0.579568\pi\)
\(198\) −612.000 −0.219661
\(199\) −1072.00 −0.381870 −0.190935 0.981603i \(-0.561152\pi\)
−0.190935 + 0.981603i \(0.561152\pi\)
\(200\) −1048.00 −0.370524
\(201\) 1968.00 0.690607
\(202\) 2380.00 0.828991
\(203\) −5320.00 −1.83936
\(204\) −1656.00 −0.568349
\(205\) −3840.00 −1.30828
\(206\) 448.000 0.151523
\(207\) −468.000 −0.157141
\(208\) −208.000 −0.0693375
\(209\) 3672.00 1.21530
\(210\) −2688.00 −0.883284
\(211\) 5444.00 1.77621 0.888105 0.459640i \(-0.152022\pi\)
0.888105 + 0.459640i \(0.152022\pi\)
\(212\) 792.000 0.256579
\(213\) −1650.00 −0.530780
\(214\) 1280.00 0.408874
\(215\) 2240.00 0.710543
\(216\) 216.000 0.0680414
\(217\) −4928.00 −1.54163
\(218\) 3868.00 1.20172
\(219\) −1842.00 −0.568360
\(220\) −2176.00 −0.666845
\(221\) −1794.00 −0.546052
\(222\) 2052.00 0.620366
\(223\) 96.0000 0.0288280 0.0144140 0.999896i \(-0.495412\pi\)
0.0144140 + 0.999896i \(0.495412\pi\)
\(224\) −896.000 −0.267261
\(225\) 1179.00 0.349333
\(226\) 836.000 0.246061
\(227\) 198.000 0.0578930 0.0289465 0.999581i \(-0.490785\pi\)
0.0289465 + 0.999581i \(0.490785\pi\)
\(228\) −1296.00 −0.376446
\(229\) 5922.00 1.70889 0.854447 0.519538i \(-0.173896\pi\)
0.854447 + 0.519538i \(0.173896\pi\)
\(230\) −1664.00 −0.477047
\(231\) −2856.00 −0.813468
\(232\) 1520.00 0.430142
\(233\) −5114.00 −1.43789 −0.718947 0.695065i \(-0.755376\pi\)
−0.718947 + 0.695065i \(0.755376\pi\)
\(234\) 234.000 0.0653720
\(235\) −7264.00 −2.01639
\(236\) −616.000 −0.169908
\(237\) −24.0000 −0.00657792
\(238\) −7728.00 −2.10476
\(239\) −5226.00 −1.41440 −0.707200 0.707013i \(-0.750042\pi\)
−0.707200 + 0.707013i \(0.750042\pi\)
\(240\) 768.000 0.206559
\(241\) −762.000 −0.203671 −0.101836 0.994801i \(-0.532472\pi\)
−0.101836 + 0.994801i \(0.532472\pi\)
\(242\) 350.000 0.0929705
\(243\) −243.000 −0.0641500
\(244\) 136.000 0.0356824
\(245\) −7056.00 −1.83996
\(246\) 1440.00 0.373216
\(247\) −1404.00 −0.361678
\(248\) 1408.00 0.360516
\(249\) −2286.00 −0.581805
\(250\) 192.000 0.0485726
\(251\) 3240.00 0.814769 0.407384 0.913257i \(-0.366441\pi\)
0.407384 + 0.913257i \(0.366441\pi\)
\(252\) 1008.00 0.251976
\(253\) −1768.00 −0.439341
\(254\) 2080.00 0.513822
\(255\) 6624.00 1.62671
\(256\) 256.000 0.0625000
\(257\) −1386.00 −0.336406 −0.168203 0.985752i \(-0.553796\pi\)
−0.168203 + 0.985752i \(0.553796\pi\)
\(258\) −840.000 −0.202698
\(259\) 9576.00 2.29739
\(260\) 832.000 0.198456
\(261\) −1710.00 −0.405542
\(262\) 1136.00 0.267871
\(263\) 3300.00 0.773714 0.386857 0.922140i \(-0.373561\pi\)
0.386857 + 0.922140i \(0.373561\pi\)
\(264\) 816.000 0.190232
\(265\) −3168.00 −0.734372
\(266\) −6048.00 −1.39409
\(267\) 1332.00 0.305307
\(268\) −2624.00 −0.598083
\(269\) 4290.00 0.972364 0.486182 0.873858i \(-0.338389\pi\)
0.486182 + 0.873858i \(0.338389\pi\)
\(270\) −864.000 −0.194746
\(271\) 2452.00 0.549625 0.274813 0.961498i \(-0.411384\pi\)
0.274813 + 0.961498i \(0.411384\pi\)
\(272\) 2208.00 0.492205
\(273\) 1092.00 0.242091
\(274\) −1056.00 −0.232830
\(275\) 4454.00 0.976677
\(276\) 624.000 0.136088
\(277\) −42.0000 −0.00911024 −0.00455512 0.999990i \(-0.501450\pi\)
−0.00455512 + 0.999990i \(0.501450\pi\)
\(278\) 3112.00 0.671386
\(279\) −1584.00 −0.339898
\(280\) 3584.00 0.764946
\(281\) −2288.00 −0.485732 −0.242866 0.970060i \(-0.578088\pi\)
−0.242866 + 0.970060i \(0.578088\pi\)
\(282\) 2724.00 0.575219
\(283\) 1156.00 0.242816 0.121408 0.992603i \(-0.461259\pi\)
0.121408 + 0.992603i \(0.461259\pi\)
\(284\) 2200.00 0.459669
\(285\) 5184.00 1.07745
\(286\) 884.000 0.182769
\(287\) 6720.00 1.38212
\(288\) −288.000 −0.0589256
\(289\) 14131.0 2.87625
\(290\) −6080.00 −1.23114
\(291\) −3066.00 −0.617636
\(292\) 2456.00 0.492214
\(293\) −8684.00 −1.73148 −0.865742 0.500491i \(-0.833153\pi\)
−0.865742 + 0.500491i \(0.833153\pi\)
\(294\) 2646.00 0.524891
\(295\) 2464.00 0.486304
\(296\) −2736.00 −0.537253
\(297\) −918.000 −0.179353
\(298\) −3048.00 −0.592503
\(299\) 676.000 0.130749
\(300\) −1572.00 −0.302532
\(301\) −3920.00 −0.750648
\(302\) 6048.00 1.15240
\(303\) 3570.00 0.676868
\(304\) 1728.00 0.326012
\(305\) −544.000 −0.102129
\(306\) −2484.00 −0.464055
\(307\) −7552.00 −1.40396 −0.701979 0.712197i \(-0.747700\pi\)
−0.701979 + 0.712197i \(0.747700\pi\)
\(308\) 3808.00 0.704484
\(309\) 672.000 0.123718
\(310\) −5632.00 −1.03186
\(311\) 2652.00 0.483541 0.241770 0.970334i \(-0.422272\pi\)
0.241770 + 0.970334i \(0.422272\pi\)
\(312\) −312.000 −0.0566139
\(313\) −4426.00 −0.799273 −0.399636 0.916674i \(-0.630864\pi\)
−0.399636 + 0.916674i \(0.630864\pi\)
\(314\) −4396.00 −0.790066
\(315\) −4032.00 −0.721198
\(316\) 32.0000 0.00569665
\(317\) −4944.00 −0.875971 −0.437985 0.898982i \(-0.644308\pi\)
−0.437985 + 0.898982i \(0.644308\pi\)
\(318\) 1188.00 0.209496
\(319\) −6460.00 −1.13383
\(320\) −1024.00 −0.178885
\(321\) 1920.00 0.333844
\(322\) 2912.00 0.503973
\(323\) 14904.0 2.56743
\(324\) 324.000 0.0555556
\(325\) −1703.00 −0.290663
\(326\) 536.000 0.0910623
\(327\) 5802.00 0.981197
\(328\) −1920.00 −0.323214
\(329\) 12712.0 2.13020
\(330\) −3264.00 −0.544477
\(331\) −6088.00 −1.01096 −0.505478 0.862839i \(-0.668684\pi\)
−0.505478 + 0.862839i \(0.668684\pi\)
\(332\) 3048.00 0.503858
\(333\) 3078.00 0.506527
\(334\) −1404.00 −0.230010
\(335\) 10496.0 1.71181
\(336\) −1344.00 −0.218218
\(337\) 6638.00 1.07298 0.536491 0.843906i \(-0.319750\pi\)
0.536491 + 0.843906i \(0.319750\pi\)
\(338\) −338.000 −0.0543928
\(339\) 1254.00 0.200908
\(340\) −8832.00 −1.40877
\(341\) −5984.00 −0.950298
\(342\) −1944.00 −0.307367
\(343\) 2744.00 0.431959
\(344\) 1120.00 0.175542
\(345\) −2496.00 −0.389508
\(346\) −4132.00 −0.642016
\(347\) −2292.00 −0.354585 −0.177293 0.984158i \(-0.556734\pi\)
−0.177293 + 0.984158i \(0.556734\pi\)
\(348\) 2280.00 0.351209
\(349\) −9866.00 −1.51322 −0.756612 0.653865i \(-0.773147\pi\)
−0.756612 + 0.653865i \(0.773147\pi\)
\(350\) −7336.00 −1.12036
\(351\) 351.000 0.0533761
\(352\) −1088.00 −0.164746
\(353\) 2368.00 0.357042 0.178521 0.983936i \(-0.442869\pi\)
0.178521 + 0.983936i \(0.442869\pi\)
\(354\) −924.000 −0.138729
\(355\) −8800.00 −1.31565
\(356\) −1776.00 −0.264404
\(357\) −11592.0 −1.71853
\(358\) 552.000 0.0814919
\(359\) 5070.00 0.745360 0.372680 0.927960i \(-0.378439\pi\)
0.372680 + 0.927960i \(0.378439\pi\)
\(360\) 1152.00 0.168655
\(361\) 4805.00 0.700539
\(362\) 6948.00 1.00878
\(363\) 525.000 0.0759101
\(364\) −1456.00 −0.209657
\(365\) −9824.00 −1.40880
\(366\) 204.000 0.0291346
\(367\) −8584.00 −1.22093 −0.610465 0.792043i \(-0.709017\pi\)
−0.610465 + 0.792043i \(0.709017\pi\)
\(368\) −832.000 −0.117856
\(369\) 2160.00 0.304729
\(370\) 10944.0 1.53771
\(371\) 5544.00 0.775822
\(372\) 2112.00 0.294360
\(373\) 4994.00 0.693243 0.346621 0.938005i \(-0.387329\pi\)
0.346621 + 0.938005i \(0.387329\pi\)
\(374\) −9384.00 −1.29742
\(375\) 288.000 0.0396593
\(376\) −3632.00 −0.498155
\(377\) 2470.00 0.337431
\(378\) 1512.00 0.205738
\(379\) 1300.00 0.176191 0.0880957 0.996112i \(-0.471922\pi\)
0.0880957 + 0.996112i \(0.471922\pi\)
\(380\) −6912.00 −0.933100
\(381\) 3120.00 0.419534
\(382\) 7840.00 1.05008
\(383\) −4590.00 −0.612371 −0.306185 0.951972i \(-0.599053\pi\)
−0.306185 + 0.951972i \(0.599053\pi\)
\(384\) 384.000 0.0510310
\(385\) −15232.0 −2.01635
\(386\) −4372.00 −0.576500
\(387\) −1260.00 −0.165502
\(388\) 4088.00 0.534889
\(389\) 3510.00 0.457491 0.228746 0.973486i \(-0.426538\pi\)
0.228746 + 0.973486i \(0.426538\pi\)
\(390\) 1248.00 0.162038
\(391\) −7176.00 −0.928148
\(392\) −3528.00 −0.454569
\(393\) 1704.00 0.218716
\(394\) 2736.00 0.349842
\(395\) −128.000 −0.0163048
\(396\) 1224.00 0.155324
\(397\) 6230.00 0.787594 0.393797 0.919197i \(-0.371161\pi\)
0.393797 + 0.919197i \(0.371161\pi\)
\(398\) 2144.00 0.270023
\(399\) −9072.00 −1.13827
\(400\) 2096.00 0.262000
\(401\) −7500.00 −0.933995 −0.466998 0.884259i \(-0.654664\pi\)
−0.466998 + 0.884259i \(0.654664\pi\)
\(402\) −3936.00 −0.488333
\(403\) 2288.00 0.282812
\(404\) −4760.00 −0.586185
\(405\) −1296.00 −0.159009
\(406\) 10640.0 1.30063
\(407\) 11628.0 1.41616
\(408\) 3312.00 0.401883
\(409\) 8254.00 0.997883 0.498941 0.866636i \(-0.333722\pi\)
0.498941 + 0.866636i \(0.333722\pi\)
\(410\) 7680.00 0.925093
\(411\) −1584.00 −0.190105
\(412\) −896.000 −0.107143
\(413\) −4312.00 −0.513752
\(414\) 936.000 0.111116
\(415\) −12192.0 −1.44212
\(416\) 416.000 0.0490290
\(417\) 4668.00 0.548185
\(418\) −7344.00 −0.859346
\(419\) −14808.0 −1.72653 −0.863267 0.504747i \(-0.831586\pi\)
−0.863267 + 0.504747i \(0.831586\pi\)
\(420\) 5376.00 0.624576
\(421\) 10354.0 1.19863 0.599315 0.800513i \(-0.295440\pi\)
0.599315 + 0.800513i \(0.295440\pi\)
\(422\) −10888.0 −1.25597
\(423\) 4086.00 0.469665
\(424\) −1584.00 −0.181429
\(425\) 18078.0 2.06332
\(426\) 3300.00 0.375318
\(427\) 952.000 0.107893
\(428\) −2560.00 −0.289117
\(429\) 1326.00 0.149230
\(430\) −4480.00 −0.502430
\(431\) −15486.0 −1.73071 −0.865353 0.501163i \(-0.832906\pi\)
−0.865353 + 0.501163i \(0.832906\pi\)
\(432\) −432.000 −0.0481125
\(433\) −2018.00 −0.223970 −0.111985 0.993710i \(-0.535721\pi\)
−0.111985 + 0.993710i \(0.535721\pi\)
\(434\) 9856.00 1.09010
\(435\) −9120.00 −1.00522
\(436\) −7736.00 −0.849741
\(437\) −5616.00 −0.614759
\(438\) 3684.00 0.401891
\(439\) 8792.00 0.955853 0.477926 0.878400i \(-0.341389\pi\)
0.477926 + 0.878400i \(0.341389\pi\)
\(440\) 4352.00 0.471531
\(441\) 3969.00 0.428571
\(442\) 3588.00 0.386117
\(443\) −2760.00 −0.296008 −0.148004 0.988987i \(-0.547285\pi\)
−0.148004 + 0.988987i \(0.547285\pi\)
\(444\) −4104.00 −0.438665
\(445\) 7104.00 0.756768
\(446\) −192.000 −0.0203844
\(447\) −4572.00 −0.483777
\(448\) 1792.00 0.188982
\(449\) 9532.00 1.00188 0.500939 0.865483i \(-0.332988\pi\)
0.500939 + 0.865483i \(0.332988\pi\)
\(450\) −2358.00 −0.247016
\(451\) 8160.00 0.851972
\(452\) −1672.00 −0.173992
\(453\) 9072.00 0.940927
\(454\) −396.000 −0.0409366
\(455\) 5824.00 0.600073
\(456\) 2592.00 0.266188
\(457\) 12862.0 1.31654 0.658270 0.752782i \(-0.271288\pi\)
0.658270 + 0.752782i \(0.271288\pi\)
\(458\) −11844.0 −1.20837
\(459\) −3726.00 −0.378899
\(460\) 3328.00 0.337323
\(461\) −6744.00 −0.681344 −0.340672 0.940182i \(-0.610654\pi\)
−0.340672 + 0.940182i \(0.610654\pi\)
\(462\) 5712.00 0.575208
\(463\) −9572.00 −0.960796 −0.480398 0.877051i \(-0.659508\pi\)
−0.480398 + 0.877051i \(0.659508\pi\)
\(464\) −3040.00 −0.304156
\(465\) −8448.00 −0.842509
\(466\) 10228.0 1.01674
\(467\) 9104.00 0.902105 0.451052 0.892498i \(-0.351049\pi\)
0.451052 + 0.892498i \(0.351049\pi\)
\(468\) −468.000 −0.0462250
\(469\) −18368.0 −1.80843
\(470\) 14528.0 1.42580
\(471\) −6594.00 −0.645086
\(472\) 1232.00 0.120143
\(473\) −4760.00 −0.462717
\(474\) 48.0000 0.00465129
\(475\) 14148.0 1.36664
\(476\) 15456.0 1.48829
\(477\) 1782.00 0.171053
\(478\) 10452.0 1.00013
\(479\) −18870.0 −1.79998 −0.899992 0.435906i \(-0.856428\pi\)
−0.899992 + 0.435906i \(0.856428\pi\)
\(480\) −1536.00 −0.146059
\(481\) −4446.00 −0.421456
\(482\) 1524.00 0.144017
\(483\) 4368.00 0.411493
\(484\) −700.000 −0.0657400
\(485\) −16352.0 −1.53094
\(486\) 486.000 0.0453609
\(487\) −1744.00 −0.162276 −0.0811378 0.996703i \(-0.525855\pi\)
−0.0811378 + 0.996703i \(0.525855\pi\)
\(488\) −272.000 −0.0252313
\(489\) 804.000 0.0743520
\(490\) 14112.0 1.30105
\(491\) 13360.0 1.22796 0.613980 0.789322i \(-0.289568\pi\)
0.613980 + 0.789322i \(0.289568\pi\)
\(492\) −2880.00 −0.263903
\(493\) −26220.0 −2.39531
\(494\) 2808.00 0.255745
\(495\) −4896.00 −0.444563
\(496\) −2816.00 −0.254924
\(497\) 15400.0 1.38991
\(498\) 4572.00 0.411398
\(499\) 17368.0 1.55811 0.779057 0.626954i \(-0.215698\pi\)
0.779057 + 0.626954i \(0.215698\pi\)
\(500\) −384.000 −0.0343460
\(501\) −2106.00 −0.187803
\(502\) −6480.00 −0.576129
\(503\) −5828.00 −0.516616 −0.258308 0.966063i \(-0.583165\pi\)
−0.258308 + 0.966063i \(0.583165\pi\)
\(504\) −2016.00 −0.178174
\(505\) 19040.0 1.67776
\(506\) 3536.00 0.310661
\(507\) −507.000 −0.0444116
\(508\) −4160.00 −0.363327
\(509\) 10744.0 0.935598 0.467799 0.883835i \(-0.345047\pi\)
0.467799 + 0.883835i \(0.345047\pi\)
\(510\) −13248.0 −1.15026
\(511\) 17192.0 1.48832
\(512\) −512.000 −0.0441942
\(513\) −2916.00 −0.250964
\(514\) 2772.00 0.237875
\(515\) 3584.00 0.306660
\(516\) 1680.00 0.143329
\(517\) 15436.0 1.31310
\(518\) −19152.0 −1.62450
\(519\) −6198.00 −0.524204
\(520\) −1664.00 −0.140329
\(521\) −12234.0 −1.02875 −0.514377 0.857564i \(-0.671977\pi\)
−0.514377 + 0.857564i \(0.671977\pi\)
\(522\) 3420.00 0.286761
\(523\) 1812.00 0.151498 0.0757488 0.997127i \(-0.475865\pi\)
0.0757488 + 0.997127i \(0.475865\pi\)
\(524\) −2272.00 −0.189414
\(525\) −11004.0 −0.914769
\(526\) −6600.00 −0.547098
\(527\) −24288.0 −2.00759
\(528\) −1632.00 −0.134515
\(529\) −9463.00 −0.777760
\(530\) 6336.00 0.519280
\(531\) −1386.00 −0.113272
\(532\) 12096.0 0.985767
\(533\) −3120.00 −0.253550
\(534\) −2664.00 −0.215885
\(535\) 10240.0 0.827502
\(536\) 5248.00 0.422909
\(537\) 828.000 0.0665379
\(538\) −8580.00 −0.687565
\(539\) 14994.0 1.19821
\(540\) 1728.00 0.137706
\(541\) 6098.00 0.484609 0.242305 0.970200i \(-0.422097\pi\)
0.242305 + 0.970200i \(0.422097\pi\)
\(542\) −4904.00 −0.388644
\(543\) 10422.0 0.823666
\(544\) −4416.00 −0.348041
\(545\) 30944.0 2.43210
\(546\) −2184.00 −0.171184
\(547\) −18332.0 −1.43294 −0.716471 0.697616i \(-0.754244\pi\)
−0.716471 + 0.697616i \(0.754244\pi\)
\(548\) 2112.00 0.164635
\(549\) 306.000 0.0237883
\(550\) −8908.00 −0.690615
\(551\) −20520.0 −1.58654
\(552\) −1248.00 −0.0962290
\(553\) 224.000 0.0172250
\(554\) 84.0000 0.00644191
\(555\) 16416.0 1.25553
\(556\) −6224.00 −0.474742
\(557\) −20004.0 −1.52172 −0.760859 0.648917i \(-0.775222\pi\)
−0.760859 + 0.648917i \(0.775222\pi\)
\(558\) 3168.00 0.240344
\(559\) 1820.00 0.137706
\(560\) −7168.00 −0.540899
\(561\) −14076.0 −1.05934
\(562\) 4576.00 0.343464
\(563\) 10988.0 0.822538 0.411269 0.911514i \(-0.365086\pi\)
0.411269 + 0.911514i \(0.365086\pi\)
\(564\) −5448.00 −0.406741
\(565\) 6688.00 0.497993
\(566\) −2312.00 −0.171697
\(567\) 2268.00 0.167984
\(568\) −4400.00 −0.325035
\(569\) 11062.0 0.815014 0.407507 0.913202i \(-0.366398\pi\)
0.407507 + 0.913202i \(0.366398\pi\)
\(570\) −10368.0 −0.761873
\(571\) −708.000 −0.0518895 −0.0259447 0.999663i \(-0.508259\pi\)
−0.0259447 + 0.999663i \(0.508259\pi\)
\(572\) −1768.00 −0.129237
\(573\) 11760.0 0.857384
\(574\) −13440.0 −0.977308
\(575\) −6812.00 −0.494052
\(576\) 576.000 0.0416667
\(577\) −2094.00 −0.151082 −0.0755410 0.997143i \(-0.524068\pi\)
−0.0755410 + 0.997143i \(0.524068\pi\)
\(578\) −28262.0 −2.03381
\(579\) −6558.00 −0.470710
\(580\) 12160.0 0.870546
\(581\) 21336.0 1.52352
\(582\) 6132.00 0.436735
\(583\) 6732.00 0.478235
\(584\) −4912.00 −0.348048
\(585\) 1872.00 0.132304
\(586\) 17368.0 1.22434
\(587\) −17854.0 −1.25539 −0.627695 0.778460i \(-0.716001\pi\)
−0.627695 + 0.778460i \(0.716001\pi\)
\(588\) −5292.00 −0.371154
\(589\) −19008.0 −1.32973
\(590\) −4928.00 −0.343869
\(591\) 4104.00 0.285645
\(592\) 5472.00 0.379895
\(593\) 23948.0 1.65839 0.829196 0.558958i \(-0.188799\pi\)
0.829196 + 0.558958i \(0.188799\pi\)
\(594\) 1836.00 0.126822
\(595\) −61824.0 −4.25973
\(596\) 6096.00 0.418963
\(597\) 3216.00 0.220473
\(598\) −1352.00 −0.0924538
\(599\) −18068.0 −1.23245 −0.616226 0.787570i \(-0.711339\pi\)
−0.616226 + 0.787570i \(0.711339\pi\)
\(600\) 3144.00 0.213922
\(601\) 19942.0 1.35350 0.676748 0.736215i \(-0.263389\pi\)
0.676748 + 0.736215i \(0.263389\pi\)
\(602\) 7840.00 0.530788
\(603\) −5904.00 −0.398722
\(604\) −12096.0 −0.814866
\(605\) 2800.00 0.188159
\(606\) −7140.00 −0.478618
\(607\) 26376.0 1.76370 0.881852 0.471526i \(-0.156296\pi\)
0.881852 + 0.471526i \(0.156296\pi\)
\(608\) −3456.00 −0.230525
\(609\) 15960.0 1.06196
\(610\) 1088.00 0.0722161
\(611\) −5902.00 −0.390785
\(612\) 4968.00 0.328136
\(613\) −19426.0 −1.27995 −0.639975 0.768396i \(-0.721055\pi\)
−0.639975 + 0.768396i \(0.721055\pi\)
\(614\) 15104.0 0.992749
\(615\) 11520.0 0.755335
\(616\) −7616.00 −0.498145
\(617\) −8024.00 −0.523556 −0.261778 0.965128i \(-0.584309\pi\)
−0.261778 + 0.965128i \(0.584309\pi\)
\(618\) −1344.00 −0.0874816
\(619\) −20648.0 −1.34073 −0.670366 0.742031i \(-0.733863\pi\)
−0.670366 + 0.742031i \(0.733863\pi\)
\(620\) 11264.0 0.729634
\(621\) 1404.00 0.0907256
\(622\) −5304.00 −0.341915
\(623\) −12432.0 −0.799482
\(624\) 624.000 0.0400320
\(625\) −14839.0 −0.949696
\(626\) 8852.00 0.565171
\(627\) −11016.0 −0.701653
\(628\) 8792.00 0.558661
\(629\) 47196.0 2.99178
\(630\) 8064.00 0.509964
\(631\) 12280.0 0.774737 0.387369 0.921925i \(-0.373384\pi\)
0.387369 + 0.921925i \(0.373384\pi\)
\(632\) −64.0000 −0.00402814
\(633\) −16332.0 −1.02550
\(634\) 9888.00 0.619405
\(635\) 16640.0 1.03990
\(636\) −2376.00 −0.148136
\(637\) −5733.00 −0.356593
\(638\) 12920.0 0.801736
\(639\) 4950.00 0.306446
\(640\) 2048.00 0.126491
\(641\) −15878.0 −0.978383 −0.489191 0.872176i \(-0.662708\pi\)
−0.489191 + 0.872176i \(0.662708\pi\)
\(642\) −3840.00 −0.236063
\(643\) −21520.0 −1.31985 −0.659927 0.751330i \(-0.729413\pi\)
−0.659927 + 0.751330i \(0.729413\pi\)
\(644\) −5824.00 −0.356363
\(645\) −6720.00 −0.410232
\(646\) −29808.0 −1.81545
\(647\) 7312.00 0.444304 0.222152 0.975012i \(-0.428692\pi\)
0.222152 + 0.975012i \(0.428692\pi\)
\(648\) −648.000 −0.0392837
\(649\) −5236.00 −0.316689
\(650\) 3406.00 0.205530
\(651\) 14784.0 0.890062
\(652\) −1072.00 −0.0643907
\(653\) 3090.00 0.185178 0.0925889 0.995704i \(-0.470486\pi\)
0.0925889 + 0.995704i \(0.470486\pi\)
\(654\) −11604.0 −0.693811
\(655\) 9088.00 0.542134
\(656\) 3840.00 0.228547
\(657\) 5526.00 0.328143
\(658\) −25424.0 −1.50628
\(659\) −13428.0 −0.793749 −0.396875 0.917873i \(-0.629905\pi\)
−0.396875 + 0.917873i \(0.629905\pi\)
\(660\) 6528.00 0.385003
\(661\) 22598.0 1.32974 0.664872 0.746958i \(-0.268486\pi\)
0.664872 + 0.746958i \(0.268486\pi\)
\(662\) 12176.0 0.714854
\(663\) 5382.00 0.315263
\(664\) −6096.00 −0.356281
\(665\) −48384.0 −2.82143
\(666\) −6156.00 −0.358168
\(667\) 9880.00 0.573546
\(668\) 2808.00 0.162642
\(669\) −288.000 −0.0166438
\(670\) −20992.0 −1.21044
\(671\) 1156.00 0.0665080
\(672\) 2688.00 0.154303
\(673\) 6178.00 0.353855 0.176927 0.984224i \(-0.443384\pi\)
0.176927 + 0.984224i \(0.443384\pi\)
\(674\) −13276.0 −0.758713
\(675\) −3537.00 −0.201688
\(676\) 676.000 0.0384615
\(677\) 22398.0 1.27153 0.635764 0.771883i \(-0.280685\pi\)
0.635764 + 0.771883i \(0.280685\pi\)
\(678\) −2508.00 −0.142064
\(679\) 28616.0 1.61735
\(680\) 17664.0 0.996152
\(681\) −594.000 −0.0334246
\(682\) 11968.0 0.671962
\(683\) 11410.0 0.639226 0.319613 0.947548i \(-0.396447\pi\)
0.319613 + 0.947548i \(0.396447\pi\)
\(684\) 3888.00 0.217341
\(685\) −8448.00 −0.471214
\(686\) −5488.00 −0.305441
\(687\) −17766.0 −0.986631
\(688\) −2240.00 −0.124127
\(689\) −2574.00 −0.142325
\(690\) 4992.00 0.275423
\(691\) 32488.0 1.78857 0.894285 0.447498i \(-0.147685\pi\)
0.894285 + 0.447498i \(0.147685\pi\)
\(692\) 8264.00 0.453974
\(693\) 8568.00 0.469656
\(694\) 4584.00 0.250729
\(695\) 24896.0 1.35879
\(696\) −4560.00 −0.248342
\(697\) 33120.0 1.79987
\(698\) 19732.0 1.07001
\(699\) 15342.0 0.830168
\(700\) 14672.0 0.792214
\(701\) 5094.00 0.274462 0.137231 0.990539i \(-0.456180\pi\)
0.137231 + 0.990539i \(0.456180\pi\)
\(702\) −702.000 −0.0377426
\(703\) 36936.0 1.98160
\(704\) 2176.00 0.116493
\(705\) 21792.0 1.16416
\(706\) −4736.00 −0.252467
\(707\) −33320.0 −1.77246
\(708\) 1848.00 0.0980962
\(709\) 25418.0 1.34639 0.673197 0.739463i \(-0.264921\pi\)
0.673197 + 0.739463i \(0.264921\pi\)
\(710\) 17600.0 0.930305
\(711\) 72.0000 0.00379777
\(712\) 3552.00 0.186962
\(713\) 9152.00 0.480708
\(714\) 23184.0 1.21518
\(715\) 7072.00 0.369899
\(716\) −1104.00 −0.0576235
\(717\) 15678.0 0.816605
\(718\) −10140.0 −0.527049
\(719\) −20428.0 −1.05958 −0.529788 0.848130i \(-0.677729\pi\)
−0.529788 + 0.848130i \(0.677729\pi\)
\(720\) −2304.00 −0.119257
\(721\) −6272.00 −0.323969
\(722\) −9610.00 −0.495356
\(723\) 2286.00 0.117590
\(724\) −13896.0 −0.713316
\(725\) −24890.0 −1.27502
\(726\) −1050.00 −0.0536765
\(727\) −38336.0 −1.95571 −0.977857 0.209276i \(-0.932889\pi\)
−0.977857 + 0.209276i \(0.932889\pi\)
\(728\) 2912.00 0.148250
\(729\) 729.000 0.0370370
\(730\) 19648.0 0.996171
\(731\) −19320.0 −0.977532
\(732\) −408.000 −0.0206012
\(733\) −166.000 −0.00836473 −0.00418237 0.999991i \(-0.501331\pi\)
−0.00418237 + 0.999991i \(0.501331\pi\)
\(734\) 17168.0 0.863328
\(735\) 21168.0 1.06230
\(736\) 1664.00 0.0833368
\(737\) −22304.0 −1.11476
\(738\) −4320.00 −0.215476
\(739\) −25248.0 −1.25678 −0.628392 0.777897i \(-0.716286\pi\)
−0.628392 + 0.777897i \(0.716286\pi\)
\(740\) −21888.0 −1.08732
\(741\) 4212.00 0.208815
\(742\) −11088.0 −0.548589
\(743\) −4442.00 −0.219329 −0.109664 0.993969i \(-0.534978\pi\)
−0.109664 + 0.993969i \(0.534978\pi\)
\(744\) −4224.00 −0.208144
\(745\) −24384.0 −1.19914
\(746\) −9988.00 −0.490197
\(747\) 6858.00 0.335905
\(748\) 18768.0 0.917414
\(749\) −17920.0 −0.874209
\(750\) −576.000 −0.0280434
\(751\) −19848.0 −0.964399 −0.482200 0.876061i \(-0.660162\pi\)
−0.482200 + 0.876061i \(0.660162\pi\)
\(752\) 7264.00 0.352248
\(753\) −9720.00 −0.470407
\(754\) −4940.00 −0.238600
\(755\) 48384.0 2.33228
\(756\) −3024.00 −0.145479
\(757\) −29166.0 −1.40034 −0.700169 0.713977i \(-0.746892\pi\)
−0.700169 + 0.713977i \(0.746892\pi\)
\(758\) −2600.00 −0.124586
\(759\) 5304.00 0.253653
\(760\) 13824.0 0.659802
\(761\) −6240.00 −0.297240 −0.148620 0.988894i \(-0.547483\pi\)
−0.148620 + 0.988894i \(0.547483\pi\)
\(762\) −6240.00 −0.296655
\(763\) −54152.0 −2.56938
\(764\) −15680.0 −0.742516
\(765\) −19872.0 −0.939181
\(766\) 9180.00 0.433012
\(767\) 2002.00 0.0942478
\(768\) −768.000 −0.0360844
\(769\) −39750.0 −1.86401 −0.932004 0.362449i \(-0.881941\pi\)
−0.932004 + 0.362449i \(0.881941\pi\)
\(770\) 30464.0 1.42577
\(771\) 4158.00 0.194224
\(772\) 8744.00 0.407647
\(773\) 9764.00 0.454317 0.227158 0.973858i \(-0.427057\pi\)
0.227158 + 0.973858i \(0.427057\pi\)
\(774\) 2520.00 0.117028
\(775\) −23056.0 −1.06864
\(776\) −8176.00 −0.378223
\(777\) −28728.0 −1.32640
\(778\) −7020.00 −0.323495
\(779\) 25920.0 1.19214
\(780\) −2496.00 −0.114578
\(781\) 18700.0 0.856772
\(782\) 14352.0 0.656300
\(783\) 5130.00 0.234140
\(784\) 7056.00 0.321429
\(785\) −35168.0 −1.59898
\(786\) −3408.00 −0.154656
\(787\) −36016.0 −1.63130 −0.815649 0.578547i \(-0.803620\pi\)
−0.815649 + 0.578547i \(0.803620\pi\)
\(788\) −5472.00 −0.247376
\(789\) −9900.00 −0.446704
\(790\) 256.000 0.0115292
\(791\) −11704.0 −0.526102
\(792\) −2448.00 −0.109831
\(793\) −442.000 −0.0197930
\(794\) −12460.0 −0.556913
\(795\) 9504.00 0.423990
\(796\) −4288.00 −0.190935
\(797\) −22290.0 −0.990655 −0.495328 0.868706i \(-0.664952\pi\)
−0.495328 + 0.868706i \(0.664952\pi\)
\(798\) 18144.0 0.804875
\(799\) 62652.0 2.77405
\(800\) −4192.00 −0.185262
\(801\) −3996.00 −0.176269
\(802\) 15000.0 0.660434
\(803\) 20876.0 0.917432
\(804\) 7872.00 0.345304
\(805\) 23296.0 1.01997
\(806\) −4576.00 −0.199979
\(807\) −12870.0 −0.561395
\(808\) 9520.00 0.414496
\(809\) −25578.0 −1.11159 −0.555794 0.831320i \(-0.687586\pi\)
−0.555794 + 0.831320i \(0.687586\pi\)
\(810\) 2592.00 0.112437
\(811\) 29900.0 1.29461 0.647306 0.762230i \(-0.275895\pi\)
0.647306 + 0.762230i \(0.275895\pi\)
\(812\) −21280.0 −0.919682
\(813\) −7356.00 −0.317326
\(814\) −23256.0 −1.00138
\(815\) 4288.00 0.184297
\(816\) −6624.00 −0.284174
\(817\) −15120.0 −0.647469
\(818\) −16508.0 −0.705610
\(819\) −3276.00 −0.139771
\(820\) −15360.0 −0.654140
\(821\) 16412.0 0.697665 0.348832 0.937185i \(-0.386578\pi\)
0.348832 + 0.937185i \(0.386578\pi\)
\(822\) 3168.00 0.134424
\(823\) 18552.0 0.785762 0.392881 0.919589i \(-0.371478\pi\)
0.392881 + 0.919589i \(0.371478\pi\)
\(824\) 1792.00 0.0757613
\(825\) −13362.0 −0.563885
\(826\) 8624.00 0.363278
\(827\) −28662.0 −1.20517 −0.602585 0.798055i \(-0.705863\pi\)
−0.602585 + 0.798055i \(0.705863\pi\)
\(828\) −1872.00 −0.0785706
\(829\) −3686.00 −0.154427 −0.0772136 0.997015i \(-0.524602\pi\)
−0.0772136 + 0.997015i \(0.524602\pi\)
\(830\) 24384.0 1.01974
\(831\) 126.000 0.00525980
\(832\) −832.000 −0.0346688
\(833\) 60858.0 2.53134
\(834\) −9336.00 −0.387625
\(835\) −11232.0 −0.465508
\(836\) 14688.0 0.607650
\(837\) 4752.00 0.196240
\(838\) 29616.0 1.22084
\(839\) 13370.0 0.550159 0.275080 0.961421i \(-0.411296\pi\)
0.275080 + 0.961421i \(0.411296\pi\)
\(840\) −10752.0 −0.441642
\(841\) 11711.0 0.480175
\(842\) −20708.0 −0.847559
\(843\) 6864.00 0.280437
\(844\) 21776.0 0.888105
\(845\) −2704.00 −0.110083
\(846\) −8172.00 −0.332103
\(847\) −4900.00 −0.198779
\(848\) 3168.00 0.128290
\(849\) −3468.00 −0.140190
\(850\) −36156.0 −1.45899
\(851\) −17784.0 −0.716366
\(852\) −6600.00 −0.265390
\(853\) 11398.0 0.457515 0.228757 0.973483i \(-0.426534\pi\)
0.228757 + 0.973483i \(0.426534\pi\)
\(854\) −1904.00 −0.0762922
\(855\) −15552.0 −0.622067
\(856\) 5120.00 0.204437
\(857\) 7990.00 0.318475 0.159238 0.987240i \(-0.449096\pi\)
0.159238 + 0.987240i \(0.449096\pi\)
\(858\) −2652.00 −0.105522
\(859\) 7652.00 0.303938 0.151969 0.988385i \(-0.451439\pi\)
0.151969 + 0.988385i \(0.451439\pi\)
\(860\) 8960.00 0.355271
\(861\) −20160.0 −0.797969
\(862\) 30972.0 1.22379
\(863\) −1022.00 −0.0403120 −0.0201560 0.999797i \(-0.506416\pi\)
−0.0201560 + 0.999797i \(0.506416\pi\)
\(864\) 864.000 0.0340207
\(865\) −33056.0 −1.29935
\(866\) 4036.00 0.158371
\(867\) −42393.0 −1.66060
\(868\) −19712.0 −0.770817
\(869\) 272.000 0.0106179
\(870\) 18240.0 0.710798
\(871\) 8528.00 0.331757
\(872\) 15472.0 0.600858
\(873\) 9198.00 0.356592
\(874\) 11232.0 0.434700
\(875\) −2688.00 −0.103853
\(876\) −7368.00 −0.284180
\(877\) −15546.0 −0.598576 −0.299288 0.954163i \(-0.596749\pi\)
−0.299288 + 0.954163i \(0.596749\pi\)
\(878\) −17584.0 −0.675890
\(879\) 26052.0 0.999673
\(880\) −8704.00 −0.333422
\(881\) 11310.0 0.432513 0.216256 0.976337i \(-0.430615\pi\)
0.216256 + 0.976337i \(0.430615\pi\)
\(882\) −7938.00 −0.303046
\(883\) 17260.0 0.657809 0.328904 0.944363i \(-0.393321\pi\)
0.328904 + 0.944363i \(0.393321\pi\)
\(884\) −7176.00 −0.273026
\(885\) −7392.00 −0.280768
\(886\) 5520.00 0.209309
\(887\) 832.000 0.0314947 0.0157474 0.999876i \(-0.494987\pi\)
0.0157474 + 0.999876i \(0.494987\pi\)
\(888\) 8208.00 0.310183
\(889\) −29120.0 −1.09860
\(890\) −14208.0 −0.535116
\(891\) 2754.00 0.103549
\(892\) 384.000 0.0144140
\(893\) 49032.0 1.83739
\(894\) 9144.00 0.342082
\(895\) 4416.00 0.164928
\(896\) −3584.00 −0.133631
\(897\) −2028.00 −0.0754882
\(898\) −19064.0 −0.708434
\(899\) 33440.0 1.24059
\(900\) 4716.00 0.174667
\(901\) 27324.0 1.01032
\(902\) −16320.0 −0.602435
\(903\) 11760.0 0.433387
\(904\) 3344.00 0.123031
\(905\) 55584.0 2.04163
\(906\) −18144.0 −0.665336
\(907\) 31740.0 1.16197 0.580986 0.813913i \(-0.302667\pi\)
0.580986 + 0.813913i \(0.302667\pi\)
\(908\) 792.000 0.0289465
\(909\) −10710.0 −0.390790
\(910\) −11648.0 −0.424316
\(911\) −23568.0 −0.857127 −0.428563 0.903512i \(-0.640980\pi\)
−0.428563 + 0.903512i \(0.640980\pi\)
\(912\) −5184.00 −0.188223
\(913\) 25908.0 0.939134
\(914\) −25724.0 −0.930935
\(915\) 1632.00 0.0589642
\(916\) 23688.0 0.854447
\(917\) −15904.0 −0.572733
\(918\) 7452.00 0.267922
\(919\) −18864.0 −0.677112 −0.338556 0.940946i \(-0.609938\pi\)
−0.338556 + 0.940946i \(0.609938\pi\)
\(920\) −6656.00 −0.238524
\(921\) 22656.0 0.810576
\(922\) 13488.0 0.481783
\(923\) −7150.00 −0.254978
\(924\) −11424.0 −0.406734
\(925\) 44802.0 1.59252
\(926\) 19144.0 0.679385
\(927\) −2016.00 −0.0714284
\(928\) 6080.00 0.215071
\(929\) −19536.0 −0.689941 −0.344971 0.938613i \(-0.612111\pi\)
−0.344971 + 0.938613i \(0.612111\pi\)
\(930\) 16896.0 0.595744
\(931\) 47628.0 1.67663
\(932\) −20456.0 −0.718947
\(933\) −7956.00 −0.279172
\(934\) −18208.0 −0.637884
\(935\) −75072.0 −2.62579
\(936\) 936.000 0.0326860
\(937\) 18174.0 0.633638 0.316819 0.948486i \(-0.397385\pi\)
0.316819 + 0.948486i \(0.397385\pi\)
\(938\) 36736.0 1.27876
\(939\) 13278.0 0.461460
\(940\) −29056.0 −1.00819
\(941\) 51172.0 1.77275 0.886376 0.462966i \(-0.153215\pi\)
0.886376 + 0.462966i \(0.153215\pi\)
\(942\) 13188.0 0.456145
\(943\) −12480.0 −0.430970
\(944\) −2464.00 −0.0849538
\(945\) 12096.0 0.416384
\(946\) 9520.00 0.327190
\(947\) 3726.00 0.127855 0.0639275 0.997955i \(-0.479637\pi\)
0.0639275 + 0.997955i \(0.479637\pi\)
\(948\) −96.0000 −0.00328896
\(949\) −7982.00 −0.273031
\(950\) −28296.0 −0.966362
\(951\) 14832.0 0.505742
\(952\) −30912.0 −1.05238
\(953\) −40498.0 −1.37656 −0.688279 0.725447i \(-0.741633\pi\)
−0.688279 + 0.725447i \(0.741633\pi\)
\(954\) −3564.00 −0.120953
\(955\) 62720.0 2.12521
\(956\) −20904.0 −0.707200
\(957\) 19380.0 0.654615
\(958\) 37740.0 1.27278
\(959\) 14784.0 0.497810
\(960\) 3072.00 0.103280
\(961\) 1185.00 0.0397771
\(962\) 8892.00 0.298014
\(963\) −5760.00 −0.192745
\(964\) −3048.00 −0.101836
\(965\) −34976.0 −1.16675
\(966\) −8736.00 −0.290969
\(967\) 28568.0 0.950036 0.475018 0.879976i \(-0.342442\pi\)
0.475018 + 0.879976i \(0.342442\pi\)
\(968\) 1400.00 0.0464852
\(969\) −44712.0 −1.48231
\(970\) 32704.0 1.08254
\(971\) −8676.00 −0.286742 −0.143371 0.989669i \(-0.545794\pi\)
−0.143371 + 0.989669i \(0.545794\pi\)
\(972\) −972.000 −0.0320750
\(973\) −43568.0 −1.43548
\(974\) 3488.00 0.114746
\(975\) 5109.00 0.167814
\(976\) 544.000 0.0178412
\(977\) −2796.00 −0.0915578 −0.0457789 0.998952i \(-0.514577\pi\)
−0.0457789 + 0.998952i \(0.514577\pi\)
\(978\) −1608.00 −0.0525748
\(979\) −15096.0 −0.492819
\(980\) −28224.0 −0.919982
\(981\) −17406.0 −0.566494
\(982\) −26720.0 −0.868299
\(983\) −406.000 −0.0131733 −0.00658667 0.999978i \(-0.502097\pi\)
−0.00658667 + 0.999978i \(0.502097\pi\)
\(984\) 5760.00 0.186608
\(985\) 21888.0 0.708030
\(986\) 52440.0 1.69374
\(987\) −38136.0 −1.22987
\(988\) −5616.00 −0.180839
\(989\) 7280.00 0.234065
\(990\) 9792.00 0.314354
\(991\) −23232.0 −0.744691 −0.372346 0.928094i \(-0.621446\pi\)
−0.372346 + 0.928094i \(0.621446\pi\)
\(992\) 5632.00 0.180258
\(993\) 18264.0 0.583676
\(994\) −30800.0 −0.982814
\(995\) 17152.0 0.546487
\(996\) −9144.00 −0.290902
\(997\) 6110.00 0.194088 0.0970440 0.995280i \(-0.469061\pi\)
0.0970440 + 0.995280i \(0.469061\pi\)
\(998\) −34736.0 −1.10175
\(999\) −9234.00 −0.292443
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.a.1.1 1
3.2 odd 2 234.4.a.k.1.1 1
4.3 odd 2 624.4.a.f.1.1 1
5.4 even 2 1950.4.a.o.1.1 1
8.3 odd 2 2496.4.a.g.1.1 1
8.5 even 2 2496.4.a.q.1.1 1
12.11 even 2 1872.4.a.o.1.1 1
13.5 odd 4 1014.4.b.a.337.2 2
13.8 odd 4 1014.4.b.a.337.1 2
13.12 even 2 1014.4.a.i.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
78.4.a.a.1.1 1 1.1 even 1 trivial
234.4.a.k.1.1 1 3.2 odd 2
624.4.a.f.1.1 1 4.3 odd 2
1014.4.a.i.1.1 1 13.12 even 2
1014.4.b.a.337.1 2 13.8 odd 4
1014.4.b.a.337.2 2 13.5 odd 4
1872.4.a.o.1.1 1 12.11 even 2
1950.4.a.o.1.1 1 5.4 even 2
2496.4.a.g.1.1 1 8.3 odd 2
2496.4.a.q.1.1 1 8.5 even 2