Properties

Label 338.4.a.c.1.1
Level $338$
Weight $4$
Character 338.1
Self dual yes
Analytic conductor $19.943$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [338,4,Mod(1,338)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(338, 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("338.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 338 = 2 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 338.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: \(19.9426455819\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 26)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 338.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.00000 q^{2} +4.00000 q^{3} +4.00000 q^{4} +18.0000 q^{5} -8.00000 q^{6} -20.0000 q^{7} -8.00000 q^{8} -11.0000 q^{9} +O(q^{10})\) \(q-2.00000 q^{2} +4.00000 q^{3} +4.00000 q^{4} +18.0000 q^{5} -8.00000 q^{6} -20.0000 q^{7} -8.00000 q^{8} -11.0000 q^{9} -36.0000 q^{10} +48.0000 q^{11} +16.0000 q^{12} +40.0000 q^{14} +72.0000 q^{15} +16.0000 q^{16} +66.0000 q^{17} +22.0000 q^{18} +16.0000 q^{19} +72.0000 q^{20} -80.0000 q^{21} -96.0000 q^{22} +168.000 q^{23} -32.0000 q^{24} +199.000 q^{25} -152.000 q^{27} -80.0000 q^{28} +6.00000 q^{29} -144.000 q^{30} -20.0000 q^{31} -32.0000 q^{32} +192.000 q^{33} -132.000 q^{34} -360.000 q^{35} -44.0000 q^{36} -254.000 q^{37} -32.0000 q^{38} -144.000 q^{40} +390.000 q^{41} +160.000 q^{42} -124.000 q^{43} +192.000 q^{44} -198.000 q^{45} -336.000 q^{46} +468.000 q^{47} +64.0000 q^{48} +57.0000 q^{49} -398.000 q^{50} +264.000 q^{51} +558.000 q^{53} +304.000 q^{54} +864.000 q^{55} +160.000 q^{56} +64.0000 q^{57} -12.0000 q^{58} +96.0000 q^{59} +288.000 q^{60} -826.000 q^{61} +40.0000 q^{62} +220.000 q^{63} +64.0000 q^{64} -384.000 q^{66} +160.000 q^{67} +264.000 q^{68} +672.000 q^{69} +720.000 q^{70} +420.000 q^{71} +88.0000 q^{72} -362.000 q^{73} +508.000 q^{74} +796.000 q^{75} +64.0000 q^{76} -960.000 q^{77} +776.000 q^{79} +288.000 q^{80} -311.000 q^{81} -780.000 q^{82} -320.000 q^{84} +1188.00 q^{85} +248.000 q^{86} +24.0000 q^{87} -384.000 q^{88} -1626.00 q^{89} +396.000 q^{90} +672.000 q^{92} -80.0000 q^{93} -936.000 q^{94} +288.000 q^{95} -128.000 q^{96} +1294.00 q^{97} -114.000 q^{98} -528.000 q^{99} +O(q^{100})\)

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
\(3\) 4.00000 0.769800 0.384900 0.922958i \(-0.374236\pi\)
0.384900 + 0.922958i \(0.374236\pi\)
\(4\) 4.00000 0.500000
\(5\) 18.0000 1.60997 0.804984 0.593296i \(-0.202174\pi\)
0.804984 + 0.593296i \(0.202174\pi\)
\(6\) −8.00000 −0.544331
\(7\) −20.0000 −1.07990 −0.539949 0.841698i \(-0.681557\pi\)
−0.539949 + 0.841698i \(0.681557\pi\)
\(8\) −8.00000 −0.353553
\(9\) −11.0000 −0.407407
\(10\) −36.0000 −1.13842
\(11\) 48.0000 1.31569 0.657843 0.753155i \(-0.271469\pi\)
0.657843 + 0.753155i \(0.271469\pi\)
\(12\) 16.0000 0.384900
\(13\) 0 0
\(14\) 40.0000 0.763604
\(15\) 72.0000 1.23935
\(16\) 16.0000 0.250000
\(17\) 66.0000 0.941609 0.470804 0.882238i \(-0.343964\pi\)
0.470804 + 0.882238i \(0.343964\pi\)
\(18\) 22.0000 0.288081
\(19\) 16.0000 0.193192 0.0965961 0.995324i \(-0.469204\pi\)
0.0965961 + 0.995324i \(0.469204\pi\)
\(20\) 72.0000 0.804984
\(21\) −80.0000 −0.831306
\(22\) −96.0000 −0.930330
\(23\) 168.000 1.52306 0.761531 0.648129i \(-0.224448\pi\)
0.761531 + 0.648129i \(0.224448\pi\)
\(24\) −32.0000 −0.272166
\(25\) 199.000 1.59200
\(26\) 0 0
\(27\) −152.000 −1.08342
\(28\) −80.0000 −0.539949
\(29\) 6.00000 0.0384197 0.0192099 0.999815i \(-0.493885\pi\)
0.0192099 + 0.999815i \(0.493885\pi\)
\(30\) −144.000 −0.876356
\(31\) −20.0000 −0.115874 −0.0579372 0.998320i \(-0.518452\pi\)
−0.0579372 + 0.998320i \(0.518452\pi\)
\(32\) −32.0000 −0.176777
\(33\) 192.000 1.01282
\(34\) −132.000 −0.665818
\(35\) −360.000 −1.73860
\(36\) −44.0000 −0.203704
\(37\) −254.000 −1.12858 −0.564288 0.825578i \(-0.690849\pi\)
−0.564288 + 0.825578i \(0.690849\pi\)
\(38\) −32.0000 −0.136608
\(39\) 0 0
\(40\) −144.000 −0.569210
\(41\) 390.000 1.48556 0.742778 0.669538i \(-0.233508\pi\)
0.742778 + 0.669538i \(0.233508\pi\)
\(42\) 160.000 0.587822
\(43\) −124.000 −0.439763 −0.219882 0.975527i \(-0.570567\pi\)
−0.219882 + 0.975527i \(0.570567\pi\)
\(44\) 192.000 0.657843
\(45\) −198.000 −0.655913
\(46\) −336.000 −1.07697
\(47\) 468.000 1.45244 0.726221 0.687461i \(-0.241275\pi\)
0.726221 + 0.687461i \(0.241275\pi\)
\(48\) 64.0000 0.192450
\(49\) 57.0000 0.166181
\(50\) −398.000 −1.12571
\(51\) 264.000 0.724851
\(52\) 0 0
\(53\) 558.000 1.44617 0.723087 0.690757i \(-0.242723\pi\)
0.723087 + 0.690757i \(0.242723\pi\)
\(54\) 304.000 0.766096
\(55\) 864.000 2.11821
\(56\) 160.000 0.381802
\(57\) 64.0000 0.148719
\(58\) −12.0000 −0.0271668
\(59\) 96.0000 0.211833 0.105916 0.994375i \(-0.466222\pi\)
0.105916 + 0.994375i \(0.466222\pi\)
\(60\) 288.000 0.619677
\(61\) −826.000 −1.73375 −0.866873 0.498530i \(-0.833873\pi\)
−0.866873 + 0.498530i \(0.833873\pi\)
\(62\) 40.0000 0.0819356
\(63\) 220.000 0.439959
\(64\) 64.0000 0.125000
\(65\) 0 0
\(66\) −384.000 −0.716169
\(67\) 160.000 0.291748 0.145874 0.989303i \(-0.453401\pi\)
0.145874 + 0.989303i \(0.453401\pi\)
\(68\) 264.000 0.470804
\(69\) 672.000 1.17245
\(70\) 720.000 1.22938
\(71\) 420.000 0.702040 0.351020 0.936368i \(-0.385835\pi\)
0.351020 + 0.936368i \(0.385835\pi\)
\(72\) 88.0000 0.144040
\(73\) −362.000 −0.580396 −0.290198 0.956967i \(-0.593721\pi\)
−0.290198 + 0.956967i \(0.593721\pi\)
\(74\) 508.000 0.798024
\(75\) 796.000 1.22552
\(76\) 64.0000 0.0965961
\(77\) −960.000 −1.42081
\(78\) 0 0
\(79\) 776.000 1.10515 0.552575 0.833463i \(-0.313645\pi\)
0.552575 + 0.833463i \(0.313645\pi\)
\(80\) 288.000 0.402492
\(81\) −311.000 −0.426612
\(82\) −780.000 −1.05045
\(83\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) −320.000 −0.415653
\(85\) 1188.00 1.51596
\(86\) 248.000 0.310960
\(87\) 24.0000 0.0295755
\(88\) −384.000 −0.465165
\(89\) −1626.00 −1.93658 −0.968290 0.249828i \(-0.919626\pi\)
−0.968290 + 0.249828i \(0.919626\pi\)
\(90\) 396.000 0.463801
\(91\) 0 0
\(92\) 672.000 0.761531
\(93\) −80.0000 −0.0892001
\(94\) −936.000 −1.02703
\(95\) 288.000 0.311033
\(96\) −128.000 −0.136083
\(97\) 1294.00 1.35449 0.677246 0.735756i \(-0.263173\pi\)
0.677246 + 0.735756i \(0.263173\pi\)
\(98\) −114.000 −0.117508
\(99\) −528.000 −0.536020
\(100\) 796.000 0.796000
\(101\) 222.000 0.218711 0.109356 0.994003i \(-0.465121\pi\)
0.109356 + 0.994003i \(0.465121\pi\)
\(102\) −528.000 −0.512547
\(103\) 632.000 0.604590 0.302295 0.953214i \(-0.402247\pi\)
0.302295 + 0.953214i \(0.402247\pi\)
\(104\) 0 0
\(105\) −1440.00 −1.33838
\(106\) −1116.00 −1.02260
\(107\) −948.000 −0.856510 −0.428255 0.903658i \(-0.640872\pi\)
−0.428255 + 0.903658i \(0.640872\pi\)
\(108\) −608.000 −0.541711
\(109\) −758.000 −0.666085 −0.333042 0.942912i \(-0.608075\pi\)
−0.333042 + 0.942912i \(0.608075\pi\)
\(110\) −1728.00 −1.49780
\(111\) −1016.00 −0.868779
\(112\) −320.000 −0.269975
\(113\) 642.000 0.534463 0.267231 0.963632i \(-0.413891\pi\)
0.267231 + 0.963632i \(0.413891\pi\)
\(114\) −128.000 −0.105161
\(115\) 3024.00 2.45208
\(116\) 24.0000 0.0192099
\(117\) 0 0
\(118\) −192.000 −0.149788
\(119\) −1320.00 −1.01684
\(120\) −576.000 −0.438178
\(121\) 973.000 0.731029
\(122\) 1652.00 1.22594
\(123\) 1560.00 1.14358
\(124\) −80.0000 −0.0579372
\(125\) 1332.00 0.953102
\(126\) −440.000 −0.311098
\(127\) −880.000 −0.614861 −0.307431 0.951571i \(-0.599469\pi\)
−0.307431 + 0.951571i \(0.599469\pi\)
\(128\) −128.000 −0.0883883
\(129\) −496.000 −0.338530
\(130\) 0 0
\(131\) 324.000 0.216092 0.108046 0.994146i \(-0.465541\pi\)
0.108046 + 0.994146i \(0.465541\pi\)
\(132\) 768.000 0.506408
\(133\) −320.000 −0.208628
\(134\) −320.000 −0.206297
\(135\) −2736.00 −1.74428
\(136\) −528.000 −0.332909
\(137\) −1722.00 −1.07387 −0.536936 0.843623i \(-0.680418\pi\)
−0.536936 + 0.843623i \(0.680418\pi\)
\(138\) −1344.00 −0.829050
\(139\) −340.000 −0.207471 −0.103735 0.994605i \(-0.533079\pi\)
−0.103735 + 0.994605i \(0.533079\pi\)
\(140\) −1440.00 −0.869302
\(141\) 1872.00 1.11809
\(142\) −840.000 −0.496417
\(143\) 0 0
\(144\) −176.000 −0.101852
\(145\) 108.000 0.0618546
\(146\) 724.000 0.410402
\(147\) 228.000 0.127926
\(148\) −1016.00 −0.564288
\(149\) −750.000 −0.412365 −0.206183 0.978514i \(-0.566104\pi\)
−0.206183 + 0.978514i \(0.566104\pi\)
\(150\) −1592.00 −0.866575
\(151\) −1748.00 −0.942054 −0.471027 0.882119i \(-0.656117\pi\)
−0.471027 + 0.882119i \(0.656117\pi\)
\(152\) −128.000 −0.0683038
\(153\) −726.000 −0.383618
\(154\) 1920.00 1.00466
\(155\) −360.000 −0.186554
\(156\) 0 0
\(157\) 614.000 0.312118 0.156059 0.987748i \(-0.450121\pi\)
0.156059 + 0.987748i \(0.450121\pi\)
\(158\) −1552.00 −0.781459
\(159\) 2232.00 1.11326
\(160\) −576.000 −0.284605
\(161\) −3360.00 −1.64475
\(162\) 622.000 0.301660
\(163\) 808.000 0.388267 0.194133 0.980975i \(-0.437811\pi\)
0.194133 + 0.980975i \(0.437811\pi\)
\(164\) 1560.00 0.742778
\(165\) 3456.00 1.63060
\(166\) 0 0
\(167\) 2028.00 0.939709 0.469854 0.882744i \(-0.344306\pi\)
0.469854 + 0.882744i \(0.344306\pi\)
\(168\) 640.000 0.293911
\(169\) 0 0
\(170\) −2376.00 −1.07195
\(171\) −176.000 −0.0787079
\(172\) −496.000 −0.219882
\(173\) −1194.00 −0.524729 −0.262365 0.964969i \(-0.584502\pi\)
−0.262365 + 0.964969i \(0.584502\pi\)
\(174\) −48.0000 −0.0209130
\(175\) −3980.00 −1.71920
\(176\) 768.000 0.328921
\(177\) 384.000 0.163069
\(178\) 3252.00 1.36937
\(179\) −2820.00 −1.17752 −0.588762 0.808307i \(-0.700384\pi\)
−0.588762 + 0.808307i \(0.700384\pi\)
\(180\) −792.000 −0.327957
\(181\) −754.000 −0.309637 −0.154819 0.987943i \(-0.549479\pi\)
−0.154819 + 0.987943i \(0.549479\pi\)
\(182\) 0 0
\(183\) −3304.00 −1.33464
\(184\) −1344.00 −0.538484
\(185\) −4572.00 −1.81697
\(186\) 160.000 0.0630740
\(187\) 3168.00 1.23886
\(188\) 1872.00 0.726221
\(189\) 3040.00 1.16999
\(190\) −576.000 −0.219934
\(191\) −2328.00 −0.881928 −0.440964 0.897525i \(-0.645363\pi\)
−0.440964 + 0.897525i \(0.645363\pi\)
\(192\) 256.000 0.0962250
\(193\) −2450.00 −0.913756 −0.456878 0.889529i \(-0.651032\pi\)
−0.456878 + 0.889529i \(0.651032\pi\)
\(194\) −2588.00 −0.957771
\(195\) 0 0
\(196\) 228.000 0.0830904
\(197\) −4542.00 −1.64266 −0.821330 0.570453i \(-0.806768\pi\)
−0.821330 + 0.570453i \(0.806768\pi\)
\(198\) 1056.00 0.379023
\(199\) −664.000 −0.236531 −0.118266 0.992982i \(-0.537733\pi\)
−0.118266 + 0.992982i \(0.537733\pi\)
\(200\) −1592.00 −0.562857
\(201\) 640.000 0.224588
\(202\) −444.000 −0.154652
\(203\) −120.000 −0.0414894
\(204\) 1056.00 0.362425
\(205\) 7020.00 2.39170
\(206\) −1264.00 −0.427510
\(207\) −1848.00 −0.620507
\(208\) 0 0
\(209\) 768.000 0.254180
\(210\) 2880.00 0.946376
\(211\) −4156.00 −1.35598 −0.677988 0.735073i \(-0.737148\pi\)
−0.677988 + 0.735073i \(0.737148\pi\)
\(212\) 2232.00 0.723087
\(213\) 1680.00 0.540431
\(214\) 1896.00 0.605644
\(215\) −2232.00 −0.708005
\(216\) 1216.00 0.383048
\(217\) 400.000 0.125133
\(218\) 1516.00 0.470993
\(219\) −1448.00 −0.446789
\(220\) 3456.00 1.05911
\(221\) 0 0
\(222\) 2032.00 0.614319
\(223\) 3292.00 0.988559 0.494279 0.869303i \(-0.335432\pi\)
0.494279 + 0.869303i \(0.335432\pi\)
\(224\) 640.000 0.190901
\(225\) −2189.00 −0.648593
\(226\) −1284.00 −0.377922
\(227\) −2352.00 −0.687699 −0.343850 0.939025i \(-0.611731\pi\)
−0.343850 + 0.939025i \(0.611731\pi\)
\(228\) 256.000 0.0743597
\(229\) −686.000 −0.197957 −0.0989785 0.995090i \(-0.531558\pi\)
−0.0989785 + 0.995090i \(0.531558\pi\)
\(230\) −6048.00 −1.73388
\(231\) −3840.00 −1.09374
\(232\) −48.0000 −0.0135834
\(233\) 1818.00 0.511164 0.255582 0.966787i \(-0.417733\pi\)
0.255582 + 0.966787i \(0.417733\pi\)
\(234\) 0 0
\(235\) 8424.00 2.33839
\(236\) 384.000 0.105916
\(237\) 3104.00 0.850745
\(238\) 2640.00 0.719016
\(239\) 540.000 0.146149 0.0730747 0.997326i \(-0.476719\pi\)
0.0730747 + 0.997326i \(0.476719\pi\)
\(240\) 1152.00 0.309839
\(241\) 862.000 0.230400 0.115200 0.993342i \(-0.463249\pi\)
0.115200 + 0.993342i \(0.463249\pi\)
\(242\) −1946.00 −0.516916
\(243\) 2860.00 0.755017
\(244\) −3304.00 −0.866873
\(245\) 1026.00 0.267546
\(246\) −3120.00 −0.808634
\(247\) 0 0
\(248\) 160.000 0.0409678
\(249\) 0 0
\(250\) −2664.00 −0.673945
\(251\) 4836.00 1.21612 0.608059 0.793892i \(-0.291948\pi\)
0.608059 + 0.793892i \(0.291948\pi\)
\(252\) 880.000 0.219979
\(253\) 8064.00 2.00387
\(254\) 1760.00 0.434773
\(255\) 4752.00 1.16699
\(256\) 256.000 0.0625000
\(257\) 1410.00 0.342231 0.171116 0.985251i \(-0.445263\pi\)
0.171116 + 0.985251i \(0.445263\pi\)
\(258\) 992.000 0.239377
\(259\) 5080.00 1.21875
\(260\) 0 0
\(261\) −66.0000 −0.0156525
\(262\) −648.000 −0.152800
\(263\) 8304.00 1.94695 0.973473 0.228804i \(-0.0734814\pi\)
0.973473 + 0.228804i \(0.0734814\pi\)
\(264\) −1536.00 −0.358084
\(265\) 10044.0 2.32829
\(266\) 640.000 0.147522
\(267\) −6504.00 −1.49078
\(268\) 640.000 0.145874
\(269\) −2634.00 −0.597018 −0.298509 0.954407i \(-0.596489\pi\)
−0.298509 + 0.954407i \(0.596489\pi\)
\(270\) 5472.00 1.23339
\(271\) −7436.00 −1.66681 −0.833404 0.552665i \(-0.813611\pi\)
−0.833404 + 0.552665i \(0.813611\pi\)
\(272\) 1056.00 0.235402
\(273\) 0 0
\(274\) 3444.00 0.759342
\(275\) 9552.00 2.09457
\(276\) 2688.00 0.586227
\(277\) −5074.00 −1.10060 −0.550302 0.834966i \(-0.685487\pi\)
−0.550302 + 0.834966i \(0.685487\pi\)
\(278\) 680.000 0.146704
\(279\) 220.000 0.0472081
\(280\) 2880.00 0.614689
\(281\) 1638.00 0.347740 0.173870 0.984769i \(-0.444373\pi\)
0.173870 + 0.984769i \(0.444373\pi\)
\(282\) −3744.00 −0.790610
\(283\) −4588.00 −0.963704 −0.481852 0.876253i \(-0.660036\pi\)
−0.481852 + 0.876253i \(0.660036\pi\)
\(284\) 1680.00 0.351020
\(285\) 1152.00 0.239434
\(286\) 0 0
\(287\) −7800.00 −1.60425
\(288\) 352.000 0.0720201
\(289\) −557.000 −0.113373
\(290\) −216.000 −0.0437378
\(291\) 5176.00 1.04269
\(292\) −1448.00 −0.290198
\(293\) 2850.00 0.568255 0.284128 0.958786i \(-0.408296\pi\)
0.284128 + 0.958786i \(0.408296\pi\)
\(294\) −456.000 −0.0904573
\(295\) 1728.00 0.341044
\(296\) 2032.00 0.399012
\(297\) −7296.00 −1.42544
\(298\) 1500.00 0.291586
\(299\) 0 0
\(300\) 3184.00 0.612761
\(301\) 2480.00 0.474900
\(302\) 3496.00 0.666133
\(303\) 888.000 0.168364
\(304\) 256.000 0.0482980
\(305\) −14868.0 −2.79128
\(306\) 1452.00 0.271259
\(307\) −8120.00 −1.50955 −0.754777 0.655982i \(-0.772255\pi\)
−0.754777 + 0.655982i \(0.772255\pi\)
\(308\) −3840.00 −0.710404
\(309\) 2528.00 0.465414
\(310\) 720.000 0.131914
\(311\) −3528.00 −0.643262 −0.321631 0.946865i \(-0.604231\pi\)
−0.321631 + 0.946865i \(0.604231\pi\)
\(312\) 0 0
\(313\) −6982.00 −1.26085 −0.630425 0.776250i \(-0.717119\pi\)
−0.630425 + 0.776250i \(0.717119\pi\)
\(314\) −1228.00 −0.220701
\(315\) 3960.00 0.708320
\(316\) 3104.00 0.552575
\(317\) −9270.00 −1.64245 −0.821223 0.570608i \(-0.806708\pi\)
−0.821223 + 0.570608i \(0.806708\pi\)
\(318\) −4464.00 −0.787197
\(319\) 288.000 0.0505483
\(320\) 1152.00 0.201246
\(321\) −3792.00 −0.659342
\(322\) 6720.00 1.16302
\(323\) 1056.00 0.181911
\(324\) −1244.00 −0.213306
\(325\) 0 0
\(326\) −1616.00 −0.274546
\(327\) −3032.00 −0.512752
\(328\) −3120.00 −0.525223
\(329\) −9360.00 −1.56849
\(330\) −6912.00 −1.15301
\(331\) 7720.00 1.28196 0.640981 0.767557i \(-0.278528\pi\)
0.640981 + 0.767557i \(0.278528\pi\)
\(332\) 0 0
\(333\) 2794.00 0.459791
\(334\) −4056.00 −0.664474
\(335\) 2880.00 0.469705
\(336\) −1280.00 −0.207827
\(337\) −1726.00 −0.278995 −0.139497 0.990222i \(-0.544549\pi\)
−0.139497 + 0.990222i \(0.544549\pi\)
\(338\) 0 0
\(339\) 2568.00 0.411430
\(340\) 4752.00 0.757981
\(341\) −960.000 −0.152454
\(342\) 352.000 0.0556549
\(343\) 5720.00 0.900440
\(344\) 992.000 0.155480
\(345\) 12096.0 1.88761
\(346\) 2388.00 0.371040
\(347\) −4020.00 −0.621916 −0.310958 0.950424i \(-0.600650\pi\)
−0.310958 + 0.950424i \(0.600650\pi\)
\(348\) 96.0000 0.0147878
\(349\) −1910.00 −0.292951 −0.146476 0.989214i \(-0.546793\pi\)
−0.146476 + 0.989214i \(0.546793\pi\)
\(350\) 7960.00 1.21566
\(351\) 0 0
\(352\) −1536.00 −0.232583
\(353\) −5442.00 −0.820534 −0.410267 0.911965i \(-0.634564\pi\)
−0.410267 + 0.911965i \(0.634564\pi\)
\(354\) −768.000 −0.115307
\(355\) 7560.00 1.13026
\(356\) −6504.00 −0.968290
\(357\) −5280.00 −0.782765
\(358\) 5640.00 0.832635
\(359\) 9324.00 1.37076 0.685379 0.728187i \(-0.259637\pi\)
0.685379 + 0.728187i \(0.259637\pi\)
\(360\) 1584.00 0.231900
\(361\) −6603.00 −0.962677
\(362\) 1508.00 0.218947
\(363\) 3892.00 0.562747
\(364\) 0 0
\(365\) −6516.00 −0.934419
\(366\) 6608.00 0.943731
\(367\) 4520.00 0.642894 0.321447 0.946928i \(-0.395831\pi\)
0.321447 + 0.946928i \(0.395831\pi\)
\(368\) 2688.00 0.380765
\(369\) −4290.00 −0.605226
\(370\) 9144.00 1.28479
\(371\) −11160.0 −1.56172
\(372\) −320.000 −0.0446001
\(373\) −5938.00 −0.824284 −0.412142 0.911120i \(-0.635219\pi\)
−0.412142 + 0.911120i \(0.635219\pi\)
\(374\) −6336.00 −0.876007
\(375\) 5328.00 0.733698
\(376\) −3744.00 −0.513516
\(377\) 0 0
\(378\) −6080.00 −0.827305
\(379\) −2216.00 −0.300338 −0.150169 0.988660i \(-0.547982\pi\)
−0.150169 + 0.988660i \(0.547982\pi\)
\(380\) 1152.00 0.155517
\(381\) −3520.00 −0.473320
\(382\) 4656.00 0.623617
\(383\) −3828.00 −0.510709 −0.255355 0.966847i \(-0.582192\pi\)
−0.255355 + 0.966847i \(0.582192\pi\)
\(384\) −512.000 −0.0680414
\(385\) −17280.0 −2.28746
\(386\) 4900.00 0.646123
\(387\) 1364.00 0.179163
\(388\) 5176.00 0.677246
\(389\) 5022.00 0.654564 0.327282 0.944927i \(-0.393867\pi\)
0.327282 + 0.944927i \(0.393867\pi\)
\(390\) 0 0
\(391\) 11088.0 1.43413
\(392\) −456.000 −0.0587538
\(393\) 1296.00 0.166347
\(394\) 9084.00 1.16154
\(395\) 13968.0 1.77926
\(396\) −2112.00 −0.268010
\(397\) −6086.00 −0.769389 −0.384695 0.923044i \(-0.625693\pi\)
−0.384695 + 0.923044i \(0.625693\pi\)
\(398\) 1328.00 0.167253
\(399\) −1280.00 −0.160602
\(400\) 3184.00 0.398000
\(401\) −1122.00 −0.139726 −0.0698629 0.997557i \(-0.522256\pi\)
−0.0698629 + 0.997557i \(0.522256\pi\)
\(402\) −1280.00 −0.158807
\(403\) 0 0
\(404\) 888.000 0.109356
\(405\) −5598.00 −0.686832
\(406\) 240.000 0.0293374
\(407\) −12192.0 −1.48485
\(408\) −2112.00 −0.256273
\(409\) −362.000 −0.0437647 −0.0218823 0.999761i \(-0.506966\pi\)
−0.0218823 + 0.999761i \(0.506966\pi\)
\(410\) −14040.0 −1.69119
\(411\) −6888.00 −0.826667
\(412\) 2528.00 0.302295
\(413\) −1920.00 −0.228758
\(414\) 3696.00 0.438764
\(415\) 0 0
\(416\) 0 0
\(417\) −1360.00 −0.159711
\(418\) −1536.00 −0.179733
\(419\) −2316.00 −0.270033 −0.135017 0.990843i \(-0.543109\pi\)
−0.135017 + 0.990843i \(0.543109\pi\)
\(420\) −5760.00 −0.669189
\(421\) −5006.00 −0.579519 −0.289760 0.957099i \(-0.593575\pi\)
−0.289760 + 0.957099i \(0.593575\pi\)
\(422\) 8312.00 0.958820
\(423\) −5148.00 −0.591736
\(424\) −4464.00 −0.511300
\(425\) 13134.0 1.49904
\(426\) −3360.00 −0.382142
\(427\) 16520.0 1.87227
\(428\) −3792.00 −0.428255
\(429\) 0 0
\(430\) 4464.00 0.500635
\(431\) 11244.0 1.25662 0.628311 0.777962i \(-0.283746\pi\)
0.628311 + 0.777962i \(0.283746\pi\)
\(432\) −2432.00 −0.270856
\(433\) 13106.0 1.45458 0.727291 0.686329i \(-0.240779\pi\)
0.727291 + 0.686329i \(0.240779\pi\)
\(434\) −800.000 −0.0884821
\(435\) 432.000 0.0476157
\(436\) −3032.00 −0.333042
\(437\) 2688.00 0.294244
\(438\) 2896.00 0.315927
\(439\) −13480.0 −1.46552 −0.732762 0.680485i \(-0.761769\pi\)
−0.732762 + 0.680485i \(0.761769\pi\)
\(440\) −6912.00 −0.748902
\(441\) −627.000 −0.0677033
\(442\) 0 0
\(443\) 14508.0 1.55597 0.777986 0.628281i \(-0.216241\pi\)
0.777986 + 0.628281i \(0.216241\pi\)
\(444\) −4064.00 −0.434389
\(445\) −29268.0 −3.11783
\(446\) −6584.00 −0.699017
\(447\) −3000.00 −0.317439
\(448\) −1280.00 −0.134987
\(449\) 7566.00 0.795237 0.397619 0.917551i \(-0.369837\pi\)
0.397619 + 0.917551i \(0.369837\pi\)
\(450\) 4378.00 0.458624
\(451\) 18720.0 1.95452
\(452\) 2568.00 0.267231
\(453\) −6992.00 −0.725194
\(454\) 4704.00 0.486277
\(455\) 0 0
\(456\) −512.000 −0.0525803
\(457\) −5402.00 −0.552943 −0.276471 0.961022i \(-0.589165\pi\)
−0.276471 + 0.961022i \(0.589165\pi\)
\(458\) 1372.00 0.139977
\(459\) −10032.0 −1.02016
\(460\) 12096.0 1.22604
\(461\) 4650.00 0.469788 0.234894 0.972021i \(-0.424526\pi\)
0.234894 + 0.972021i \(0.424526\pi\)
\(462\) 7680.00 0.773389
\(463\) 17188.0 1.72526 0.862629 0.505838i \(-0.168817\pi\)
0.862629 + 0.505838i \(0.168817\pi\)
\(464\) 96.0000 0.00960493
\(465\) −1440.00 −0.143609
\(466\) −3636.00 −0.361447
\(467\) −14580.0 −1.44472 −0.722358 0.691520i \(-0.756941\pi\)
−0.722358 + 0.691520i \(0.756941\pi\)
\(468\) 0 0
\(469\) −3200.00 −0.315058
\(470\) −16848.0 −1.65349
\(471\) 2456.00 0.240269
\(472\) −768.000 −0.0748942
\(473\) −5952.00 −0.578590
\(474\) −6208.00 −0.601567
\(475\) 3184.00 0.307562
\(476\) −5280.00 −0.508421
\(477\) −6138.00 −0.589182
\(478\) −1080.00 −0.103343
\(479\) 2100.00 0.200316 0.100158 0.994972i \(-0.468065\pi\)
0.100158 + 0.994972i \(0.468065\pi\)
\(480\) −2304.00 −0.219089
\(481\) 0 0
\(482\) −1724.00 −0.162917
\(483\) −13440.0 −1.26613
\(484\) 3892.00 0.365515
\(485\) 23292.0 2.18069
\(486\) −5720.00 −0.533878
\(487\) 12004.0 1.11695 0.558473 0.829522i \(-0.311387\pi\)
0.558473 + 0.829522i \(0.311387\pi\)
\(488\) 6608.00 0.612972
\(489\) 3232.00 0.298888
\(490\) −2052.00 −0.189183
\(491\) 13236.0 1.21656 0.608281 0.793721i \(-0.291859\pi\)
0.608281 + 0.793721i \(0.291859\pi\)
\(492\) 6240.00 0.571791
\(493\) 396.000 0.0361764
\(494\) 0 0
\(495\) −9504.00 −0.862976
\(496\) −320.000 −0.0289686
\(497\) −8400.00 −0.758132
\(498\) 0 0
\(499\) −18560.0 −1.66505 −0.832525 0.553988i \(-0.813105\pi\)
−0.832525 + 0.553988i \(0.813105\pi\)
\(500\) 5328.00 0.476551
\(501\) 8112.00 0.723388
\(502\) −9672.00 −0.859925
\(503\) 12432.0 1.10202 0.551009 0.834499i \(-0.314243\pi\)
0.551009 + 0.834499i \(0.314243\pi\)
\(504\) −1760.00 −0.155549
\(505\) 3996.00 0.352118
\(506\) −16128.0 −1.41695
\(507\) 0 0
\(508\) −3520.00 −0.307431
\(509\) 7914.00 0.689159 0.344579 0.938757i \(-0.388022\pi\)
0.344579 + 0.938757i \(0.388022\pi\)
\(510\) −9504.00 −0.825185
\(511\) 7240.00 0.626769
\(512\) −512.000 −0.0441942
\(513\) −2432.00 −0.209309
\(514\) −2820.00 −0.241994
\(515\) 11376.0 0.973372
\(516\) −1984.00 −0.169265
\(517\) 22464.0 1.91096
\(518\) −10160.0 −0.861785
\(519\) −4776.00 −0.403937
\(520\) 0 0
\(521\) −14742.0 −1.23965 −0.619826 0.784739i \(-0.712797\pi\)
−0.619826 + 0.784739i \(0.712797\pi\)
\(522\) 132.000 0.0110680
\(523\) −2500.00 −0.209020 −0.104510 0.994524i \(-0.533327\pi\)
−0.104510 + 0.994524i \(0.533327\pi\)
\(524\) 1296.00 0.108046
\(525\) −15920.0 −1.32344
\(526\) −16608.0 −1.37670
\(527\) −1320.00 −0.109108
\(528\) 3072.00 0.253204
\(529\) 16057.0 1.31972
\(530\) −20088.0 −1.64635
\(531\) −1056.00 −0.0863023
\(532\) −1280.00 −0.104314
\(533\) 0 0
\(534\) 13008.0 1.05414
\(535\) −17064.0 −1.37896
\(536\) −1280.00 −0.103148
\(537\) −11280.0 −0.906458
\(538\) 5268.00 0.422155
\(539\) 2736.00 0.218642
\(540\) −10944.0 −0.872138
\(541\) −17894.0 −1.42204 −0.711020 0.703172i \(-0.751766\pi\)
−0.711020 + 0.703172i \(0.751766\pi\)
\(542\) 14872.0 1.17861
\(543\) −3016.00 −0.238359
\(544\) −2112.00 −0.166455
\(545\) −13644.0 −1.07238
\(546\) 0 0
\(547\) 17444.0 1.36353 0.681766 0.731571i \(-0.261212\pi\)
0.681766 + 0.731571i \(0.261212\pi\)
\(548\) −6888.00 −0.536936
\(549\) 9086.00 0.706341
\(550\) −19104.0 −1.48109
\(551\) 96.0000 0.00742239
\(552\) −5376.00 −0.414525
\(553\) −15520.0 −1.19345
\(554\) 10148.0 0.778244
\(555\) −18288.0 −1.39871
\(556\) −1360.00 −0.103735
\(557\) 1002.00 0.0762228 0.0381114 0.999273i \(-0.487866\pi\)
0.0381114 + 0.999273i \(0.487866\pi\)
\(558\) −440.000 −0.0333812
\(559\) 0 0
\(560\) −5760.00 −0.434651
\(561\) 12672.0 0.953676
\(562\) −3276.00 −0.245889
\(563\) −4740.00 −0.354826 −0.177413 0.984136i \(-0.556773\pi\)
−0.177413 + 0.984136i \(0.556773\pi\)
\(564\) 7488.00 0.559046
\(565\) 11556.0 0.860468
\(566\) 9176.00 0.681442
\(567\) 6220.00 0.460697
\(568\) −3360.00 −0.248209
\(569\) 8682.00 0.639663 0.319832 0.947474i \(-0.396374\pi\)
0.319832 + 0.947474i \(0.396374\pi\)
\(570\) −2304.00 −0.169305
\(571\) 5492.00 0.402510 0.201255 0.979539i \(-0.435498\pi\)
0.201255 + 0.979539i \(0.435498\pi\)
\(572\) 0 0
\(573\) −9312.00 −0.678908
\(574\) 15600.0 1.13438
\(575\) 33432.0 2.42471
\(576\) −704.000 −0.0509259
\(577\) 17278.0 1.24661 0.623304 0.781980i \(-0.285790\pi\)
0.623304 + 0.781980i \(0.285790\pi\)
\(578\) 1114.00 0.0801666
\(579\) −9800.00 −0.703410
\(580\) 432.000 0.0309273
\(581\) 0 0
\(582\) −10352.0 −0.737292
\(583\) 26784.0 1.90271
\(584\) 2896.00 0.205201
\(585\) 0 0
\(586\) −5700.00 −0.401817
\(587\) 15240.0 1.07159 0.535794 0.844349i \(-0.320012\pi\)
0.535794 + 0.844349i \(0.320012\pi\)
\(588\) 912.000 0.0639630
\(589\) −320.000 −0.0223860
\(590\) −3456.00 −0.241155
\(591\) −18168.0 −1.26452
\(592\) −4064.00 −0.282144
\(593\) 9198.00 0.636959 0.318479 0.947930i \(-0.396828\pi\)
0.318479 + 0.947930i \(0.396828\pi\)
\(594\) 14592.0 1.00794
\(595\) −23760.0 −1.63708
\(596\) −3000.00 −0.206183
\(597\) −2656.00 −0.182082
\(598\) 0 0
\(599\) 7200.00 0.491125 0.245563 0.969381i \(-0.421027\pi\)
0.245563 + 0.969381i \(0.421027\pi\)
\(600\) −6368.00 −0.433288
\(601\) −14470.0 −0.982103 −0.491051 0.871131i \(-0.663387\pi\)
−0.491051 + 0.871131i \(0.663387\pi\)
\(602\) −4960.00 −0.335805
\(603\) −1760.00 −0.118860
\(604\) −6992.00 −0.471027
\(605\) 17514.0 1.17693
\(606\) −1776.00 −0.119051
\(607\) −20824.0 −1.39245 −0.696227 0.717821i \(-0.745139\pi\)
−0.696227 + 0.717821i \(0.745139\pi\)
\(608\) −512.000 −0.0341519
\(609\) −480.000 −0.0319386
\(610\) 29736.0 1.97373
\(611\) 0 0
\(612\) −2904.00 −0.191809
\(613\) −8606.00 −0.567036 −0.283518 0.958967i \(-0.591502\pi\)
−0.283518 + 0.958967i \(0.591502\pi\)
\(614\) 16240.0 1.06742
\(615\) 28080.0 1.84113
\(616\) 7680.00 0.502331
\(617\) 9654.00 0.629912 0.314956 0.949106i \(-0.398010\pi\)
0.314956 + 0.949106i \(0.398010\pi\)
\(618\) −5056.00 −0.329097
\(619\) −14384.0 −0.933993 −0.466997 0.884259i \(-0.654664\pi\)
−0.466997 + 0.884259i \(0.654664\pi\)
\(620\) −1440.00 −0.0932771
\(621\) −25536.0 −1.65012
\(622\) 7056.00 0.454855
\(623\) 32520.0 2.09131
\(624\) 0 0
\(625\) −899.000 −0.0575360
\(626\) 13964.0 0.891555
\(627\) 3072.00 0.195668
\(628\) 2456.00 0.156059
\(629\) −16764.0 −1.06268
\(630\) −7920.00 −0.500858
\(631\) −1460.00 −0.0921104 −0.0460552 0.998939i \(-0.514665\pi\)
−0.0460552 + 0.998939i \(0.514665\pi\)
\(632\) −6208.00 −0.390729
\(633\) −16624.0 −1.04383
\(634\) 18540.0 1.16138
\(635\) −15840.0 −0.989907
\(636\) 8928.00 0.556632
\(637\) 0 0
\(638\) −576.000 −0.0357430
\(639\) −4620.00 −0.286016
\(640\) −2304.00 −0.142302
\(641\) −12462.0 −0.767893 −0.383946 0.923355i \(-0.625435\pi\)
−0.383946 + 0.923355i \(0.625435\pi\)
\(642\) 7584.00 0.466225
\(643\) 9952.00 0.610371 0.305186 0.952293i \(-0.401282\pi\)
0.305186 + 0.952293i \(0.401282\pi\)
\(644\) −13440.0 −0.822376
\(645\) −8928.00 −0.545023
\(646\) −2112.00 −0.128631
\(647\) 26088.0 1.58520 0.792601 0.609741i \(-0.208727\pi\)
0.792601 + 0.609741i \(0.208727\pi\)
\(648\) 2488.00 0.150830
\(649\) 4608.00 0.278705
\(650\) 0 0
\(651\) 1600.00 0.0963271
\(652\) 3232.00 0.194133
\(653\) 3894.00 0.233360 0.116680 0.993170i \(-0.462775\pi\)
0.116680 + 0.993170i \(0.462775\pi\)
\(654\) 6064.00 0.362571
\(655\) 5832.00 0.347901
\(656\) 6240.00 0.371389
\(657\) 3982.00 0.236458
\(658\) 18720.0 1.10909
\(659\) −23820.0 −1.40804 −0.704018 0.710182i \(-0.748612\pi\)
−0.704018 + 0.710182i \(0.748612\pi\)
\(660\) 13824.0 0.815301
\(661\) −7742.00 −0.455566 −0.227783 0.973712i \(-0.573148\pi\)
−0.227783 + 0.973712i \(0.573148\pi\)
\(662\) −15440.0 −0.906484
\(663\) 0 0
\(664\) 0 0
\(665\) −5760.00 −0.335885
\(666\) −5588.00 −0.325121
\(667\) 1008.00 0.0585156
\(668\) 8112.00 0.469854
\(669\) 13168.0 0.760993
\(670\) −5760.00 −0.332132
\(671\) −39648.0 −2.28106
\(672\) 2560.00 0.146956
\(673\) 21170.0 1.21255 0.606273 0.795257i \(-0.292664\pi\)
0.606273 + 0.795257i \(0.292664\pi\)
\(674\) 3452.00 0.197279
\(675\) −30248.0 −1.72481
\(676\) 0 0
\(677\) 17982.0 1.02083 0.510417 0.859927i \(-0.329491\pi\)
0.510417 + 0.859927i \(0.329491\pi\)
\(678\) −5136.00 −0.290925
\(679\) −25880.0 −1.46271
\(680\) −9504.00 −0.535973
\(681\) −9408.00 −0.529391
\(682\) 1920.00 0.107801
\(683\) 17520.0 0.981529 0.490764 0.871292i \(-0.336718\pi\)
0.490764 + 0.871292i \(0.336718\pi\)
\(684\) −704.000 −0.0393540
\(685\) −30996.0 −1.72890
\(686\) −11440.0 −0.636707
\(687\) −2744.00 −0.152387
\(688\) −1984.00 −0.109941
\(689\) 0 0
\(690\) −24192.0 −1.33474
\(691\) 28096.0 1.54678 0.773388 0.633933i \(-0.218560\pi\)
0.773388 + 0.633933i \(0.218560\pi\)
\(692\) −4776.00 −0.262365
\(693\) 10560.0 0.578847
\(694\) 8040.00 0.439761
\(695\) −6120.00 −0.334021
\(696\) −192.000 −0.0104565
\(697\) 25740.0 1.39881
\(698\) 3820.00 0.207148
\(699\) 7272.00 0.393494
\(700\) −15920.0 −0.859599
\(701\) 18342.0 0.988256 0.494128 0.869389i \(-0.335487\pi\)
0.494128 + 0.869389i \(0.335487\pi\)
\(702\) 0 0
\(703\) −4064.00 −0.218032
\(704\) 3072.00 0.164461
\(705\) 33696.0 1.80009
\(706\) 10884.0 0.580205
\(707\) −4440.00 −0.236186
\(708\) 1536.00 0.0815345
\(709\) 37330.0 1.97737 0.988687 0.149996i \(-0.0479261\pi\)
0.988687 + 0.149996i \(0.0479261\pi\)
\(710\) −15120.0 −0.799216
\(711\) −8536.00 −0.450246
\(712\) 13008.0 0.684685
\(713\) −3360.00 −0.176484
\(714\) 10560.0 0.553499
\(715\) 0 0
\(716\) −11280.0 −0.588762
\(717\) 2160.00 0.112506
\(718\) −18648.0 −0.969272
\(719\) 4800.00 0.248971 0.124485 0.992221i \(-0.460272\pi\)
0.124485 + 0.992221i \(0.460272\pi\)
\(720\) −3168.00 −0.163978
\(721\) −12640.0 −0.652896
\(722\) 13206.0 0.680715
\(723\) 3448.00 0.177362
\(724\) −3016.00 −0.154819
\(725\) 1194.00 0.0611642
\(726\) −7784.00 −0.397922
\(727\) 23960.0 1.22232 0.611160 0.791507i \(-0.290703\pi\)
0.611160 + 0.791507i \(0.290703\pi\)
\(728\) 0 0
\(729\) 19837.0 1.00782
\(730\) 13032.0 0.660734
\(731\) −8184.00 −0.414085
\(732\) −13216.0 −0.667319
\(733\) 21418.0 1.07925 0.539626 0.841905i \(-0.318566\pi\)
0.539626 + 0.841905i \(0.318566\pi\)
\(734\) −9040.00 −0.454595
\(735\) 4104.00 0.205957
\(736\) −5376.00 −0.269242
\(737\) 7680.00 0.383849
\(738\) 8580.00 0.427960
\(739\) −5384.00 −0.268002 −0.134001 0.990981i \(-0.542783\pi\)
−0.134001 + 0.990981i \(0.542783\pi\)
\(740\) −18288.0 −0.908487
\(741\) 0 0
\(742\) 22320.0 1.10430
\(743\) 1524.00 0.0752492 0.0376246 0.999292i \(-0.488021\pi\)
0.0376246 + 0.999292i \(0.488021\pi\)
\(744\) 640.000 0.0315370
\(745\) −13500.0 −0.663895
\(746\) 11876.0 0.582857
\(747\) 0 0
\(748\) 12672.0 0.619431
\(749\) 18960.0 0.924944
\(750\) −10656.0 −0.518803
\(751\) −19312.0 −0.938355 −0.469178 0.883104i \(-0.655450\pi\)
−0.469178 + 0.883104i \(0.655450\pi\)
\(752\) 7488.00 0.363111
\(753\) 19344.0 0.936168
\(754\) 0 0
\(755\) −31464.0 −1.51668
\(756\) 12160.0 0.584993
\(757\) 35246.0 1.69226 0.846128 0.532980i \(-0.178928\pi\)
0.846128 + 0.532980i \(0.178928\pi\)
\(758\) 4432.00 0.212371
\(759\) 32256.0 1.54258
\(760\) −2304.00 −0.109967
\(761\) −12522.0 −0.596481 −0.298241 0.954491i \(-0.596400\pi\)
−0.298241 + 0.954491i \(0.596400\pi\)
\(762\) 7040.00 0.334688
\(763\) 15160.0 0.719304
\(764\) −9312.00 −0.440964
\(765\) −13068.0 −0.617614
\(766\) 7656.00 0.361126
\(767\) 0 0
\(768\) 1024.00 0.0481125
\(769\) −24050.0 −1.12778 −0.563892 0.825849i \(-0.690696\pi\)
−0.563892 + 0.825849i \(0.690696\pi\)
\(770\) 34560.0 1.61748
\(771\) 5640.00 0.263450
\(772\) −9800.00 −0.456878
\(773\) −25806.0 −1.20075 −0.600373 0.799720i \(-0.704981\pi\)
−0.600373 + 0.799720i \(0.704981\pi\)
\(774\) −2728.00 −0.126687
\(775\) −3980.00 −0.184472
\(776\) −10352.0 −0.478885
\(777\) 20320.0 0.938193
\(778\) −10044.0 −0.462847
\(779\) 6240.00 0.286998
\(780\) 0 0
\(781\) 20160.0 0.923664
\(782\) −22176.0 −1.01408
\(783\) −912.000 −0.0416248
\(784\) 912.000 0.0415452
\(785\) 11052.0 0.502500
\(786\) −2592.00 −0.117625
\(787\) −18632.0 −0.843912 −0.421956 0.906616i \(-0.638656\pi\)
−0.421956 + 0.906616i \(0.638656\pi\)
\(788\) −18168.0 −0.821330
\(789\) 33216.0 1.49876
\(790\) −27936.0 −1.25812
\(791\) −12840.0 −0.577165
\(792\) 4224.00 0.189512
\(793\) 0 0
\(794\) 12172.0 0.544040
\(795\) 40176.0 1.79232
\(796\) −2656.00 −0.118266
\(797\) −16314.0 −0.725058 −0.362529 0.931972i \(-0.618087\pi\)
−0.362529 + 0.931972i \(0.618087\pi\)
\(798\) 2560.00 0.113563
\(799\) 30888.0 1.36763
\(800\) −6368.00 −0.281428
\(801\) 17886.0 0.788977
\(802\) 2244.00 0.0988010
\(803\) −17376.0 −0.763619
\(804\) 2560.00 0.112294
\(805\) −60480.0 −2.64800
\(806\) 0 0
\(807\) −10536.0 −0.459585
\(808\) −1776.00 −0.0773261
\(809\) −4278.00 −0.185917 −0.0929583 0.995670i \(-0.529632\pi\)
−0.0929583 + 0.995670i \(0.529632\pi\)
\(810\) 11196.0 0.485663
\(811\) −18632.0 −0.806730 −0.403365 0.915039i \(-0.632159\pi\)
−0.403365 + 0.915039i \(0.632159\pi\)
\(812\) −480.000 −0.0207447
\(813\) −29744.0 −1.28311
\(814\) 24384.0 1.04995
\(815\) 14544.0 0.625097
\(816\) 4224.00 0.181213
\(817\) −1984.00 −0.0849588
\(818\) 724.000 0.0309463
\(819\) 0 0
\(820\) 28080.0 1.19585
\(821\) 46434.0 1.97388 0.986941 0.161080i \(-0.0514976\pi\)
0.986941 + 0.161080i \(0.0514976\pi\)
\(822\) 13776.0 0.584542
\(823\) 24968.0 1.05751 0.528754 0.848775i \(-0.322659\pi\)
0.528754 + 0.848775i \(0.322659\pi\)
\(824\) −5056.00 −0.213755
\(825\) 38208.0 1.61240
\(826\) 3840.00 0.161756
\(827\) 14112.0 0.593376 0.296688 0.954974i \(-0.404118\pi\)
0.296688 + 0.954974i \(0.404118\pi\)
\(828\) −7392.00 −0.310253
\(829\) 37190.0 1.55810 0.779048 0.626964i \(-0.215703\pi\)
0.779048 + 0.626964i \(0.215703\pi\)
\(830\) 0 0
\(831\) −20296.0 −0.847245
\(832\) 0 0
\(833\) 3762.00 0.156477
\(834\) 2720.00 0.112933
\(835\) 36504.0 1.51290
\(836\) 3072.00 0.127090
\(837\) 3040.00 0.125541
\(838\) 4632.00 0.190942
\(839\) −1380.00 −0.0567853 −0.0283927 0.999597i \(-0.509039\pi\)
−0.0283927 + 0.999597i \(0.509039\pi\)
\(840\) 11520.0 0.473188
\(841\) −24353.0 −0.998524
\(842\) 10012.0 0.409782
\(843\) 6552.00 0.267690
\(844\) −16624.0 −0.677988
\(845\) 0 0
\(846\) 10296.0 0.418421
\(847\) −19460.0 −0.789437
\(848\) 8928.00 0.361543
\(849\) −18352.0 −0.741860
\(850\) −26268.0 −1.05998
\(851\) −42672.0 −1.71889
\(852\) 6720.00 0.270215
\(853\) −5150.00 −0.206721 −0.103360 0.994644i \(-0.532959\pi\)
−0.103360 + 0.994644i \(0.532959\pi\)
\(854\) −33040.0 −1.32389
\(855\) −3168.00 −0.126717
\(856\) 7584.00 0.302822
\(857\) 23562.0 0.939163 0.469581 0.882889i \(-0.344405\pi\)
0.469581 + 0.882889i \(0.344405\pi\)
\(858\) 0 0
\(859\) −34612.0 −1.37479 −0.687396 0.726283i \(-0.741246\pi\)
−0.687396 + 0.726283i \(0.741246\pi\)
\(860\) −8928.00 −0.354003
\(861\) −31200.0 −1.23495
\(862\) −22488.0 −0.888566
\(863\) 14940.0 0.589297 0.294649 0.955606i \(-0.404797\pi\)
0.294649 + 0.955606i \(0.404797\pi\)
\(864\) 4864.00 0.191524
\(865\) −21492.0 −0.844798
\(866\) −26212.0 −1.02855
\(867\) −2228.00 −0.0872743
\(868\) 1600.00 0.0625663
\(869\) 37248.0 1.45403
\(870\) −864.000 −0.0336694
\(871\) 0 0
\(872\) 6064.00 0.235497
\(873\) −14234.0 −0.551830
\(874\) −5376.00 −0.208062
\(875\) −26640.0 −1.02925
\(876\) −5792.00 −0.223394
\(877\) −17030.0 −0.655715 −0.327858 0.944727i \(-0.606327\pi\)
−0.327858 + 0.944727i \(0.606327\pi\)
\(878\) 26960.0 1.03628
\(879\) 11400.0 0.437443
\(880\) 13824.0 0.529553
\(881\) −27246.0 −1.04193 −0.520965 0.853578i \(-0.674428\pi\)
−0.520965 + 0.853578i \(0.674428\pi\)
\(882\) 1254.00 0.0478734
\(883\) −8260.00 −0.314803 −0.157402 0.987535i \(-0.550312\pi\)
−0.157402 + 0.987535i \(0.550312\pi\)
\(884\) 0 0
\(885\) 6912.00 0.262536
\(886\) −29016.0 −1.10024
\(887\) −43392.0 −1.64257 −0.821286 0.570517i \(-0.806743\pi\)
−0.821286 + 0.570517i \(0.806743\pi\)
\(888\) 8128.00 0.307160
\(889\) 17600.0 0.663988
\(890\) 58536.0 2.20464
\(891\) −14928.0 −0.561287
\(892\) 13168.0 0.494279
\(893\) 7488.00 0.280601
\(894\) 6000.00 0.224463
\(895\) −50760.0 −1.89578
\(896\) 2560.00 0.0954504
\(897\) 0 0
\(898\) −15132.0 −0.562318
\(899\) −120.000 −0.00445186
\(900\) −8756.00 −0.324296
\(901\) 36828.0 1.36173
\(902\) −37440.0 −1.38206
\(903\) 9920.00 0.365578
\(904\) −5136.00 −0.188961
\(905\) −13572.0 −0.498507
\(906\) 13984.0 0.512789
\(907\) 1028.00 0.0376342 0.0188171 0.999823i \(-0.494010\pi\)
0.0188171 + 0.999823i \(0.494010\pi\)
\(908\) −9408.00 −0.343850
\(909\) −2442.00 −0.0891045
\(910\) 0 0
\(911\) −21816.0 −0.793410 −0.396705 0.917946i \(-0.629846\pi\)
−0.396705 + 0.917946i \(0.629846\pi\)
\(912\) 1024.00 0.0371799
\(913\) 0 0
\(914\) 10804.0 0.390990
\(915\) −59472.0 −2.14873
\(916\) −2744.00 −0.0989785
\(917\) −6480.00 −0.233357
\(918\) 20064.0 0.721362
\(919\) 42752.0 1.53456 0.767279 0.641314i \(-0.221610\pi\)
0.767279 + 0.641314i \(0.221610\pi\)
\(920\) −24192.0 −0.866942
\(921\) −32480.0 −1.16205
\(922\) −9300.00 −0.332190
\(923\) 0 0
\(924\) −15360.0 −0.546869
\(925\) −50546.0 −1.79669
\(926\) −34376.0 −1.21994
\(927\) −6952.00 −0.246315
\(928\) −192.000 −0.00679171
\(929\) −24978.0 −0.882133 −0.441067 0.897474i \(-0.645400\pi\)
−0.441067 + 0.897474i \(0.645400\pi\)
\(930\) 2880.00 0.101547
\(931\) 912.000 0.0321048
\(932\) 7272.00 0.255582
\(933\) −14112.0 −0.495183
\(934\) 29160.0 1.02157
\(935\) 57024.0 1.99453
\(936\) 0 0
\(937\) 33914.0 1.18241 0.591207 0.806520i \(-0.298652\pi\)
0.591207 + 0.806520i \(0.298652\pi\)
\(938\) 6400.00 0.222780
\(939\) −27928.0 −0.970603
\(940\) 33696.0 1.16919
\(941\) 8442.00 0.292456 0.146228 0.989251i \(-0.453287\pi\)
0.146228 + 0.989251i \(0.453287\pi\)
\(942\) −4912.00 −0.169896
\(943\) 65520.0 2.26259
\(944\) 1536.00 0.0529582
\(945\) 54720.0 1.88364
\(946\) 11904.0 0.409125
\(947\) 43176.0 1.48155 0.740777 0.671751i \(-0.234458\pi\)
0.740777 + 0.671751i \(0.234458\pi\)
\(948\) 12416.0 0.425372
\(949\) 0 0
\(950\) −6368.00 −0.217479
\(951\) −37080.0 −1.26435
\(952\) 10560.0 0.359508
\(953\) −43926.0 −1.49308 −0.746539 0.665342i \(-0.768286\pi\)
−0.746539 + 0.665342i \(0.768286\pi\)
\(954\) 12276.0 0.416614
\(955\) −41904.0 −1.41988
\(956\) 2160.00 0.0730747
\(957\) 1152.00 0.0389121
\(958\) −4200.00 −0.141645
\(959\) 34440.0 1.15967
\(960\) 4608.00 0.154919
\(961\) −29391.0 −0.986573
\(962\) 0 0
\(963\) 10428.0 0.348949
\(964\) 3448.00 0.115200
\(965\) −44100.0 −1.47112
\(966\) 26880.0 0.895290
\(967\) 11572.0 0.384830 0.192415 0.981314i \(-0.438368\pi\)
0.192415 + 0.981314i \(0.438368\pi\)
\(968\) −7784.00 −0.258458
\(969\) 4224.00 0.140036
\(970\) −46584.0 −1.54198
\(971\) 14412.0 0.476316 0.238158 0.971226i \(-0.423456\pi\)
0.238158 + 0.971226i \(0.423456\pi\)
\(972\) 11440.0 0.377508
\(973\) 6800.00 0.224047
\(974\) −24008.0 −0.789801
\(975\) 0 0
\(976\) −13216.0 −0.433436
\(977\) −25602.0 −0.838363 −0.419181 0.907902i \(-0.637683\pi\)
−0.419181 + 0.907902i \(0.637683\pi\)
\(978\) −6464.00 −0.211346
\(979\) −78048.0 −2.54793
\(980\) 4104.00 0.133773
\(981\) 8338.00 0.271368
\(982\) −26472.0 −0.860240
\(983\) −32148.0 −1.04309 −0.521547 0.853222i \(-0.674645\pi\)
−0.521547 + 0.853222i \(0.674645\pi\)
\(984\) −12480.0 −0.404317
\(985\) −81756.0 −2.64463
\(986\) −792.000 −0.0255805
\(987\) −37440.0 −1.20742
\(988\) 0 0
\(989\) −20832.0 −0.669787
\(990\) 19008.0 0.610216
\(991\) −9736.00 −0.312083 −0.156041 0.987751i \(-0.549873\pi\)
−0.156041 + 0.987751i \(0.549873\pi\)
\(992\) 640.000 0.0204839
\(993\) 30880.0 0.986855
\(994\) 16800.0 0.536080
\(995\) −11952.0 −0.380808
\(996\) 0 0
\(997\) 6878.00 0.218484 0.109242 0.994015i \(-0.465158\pi\)
0.109242 + 0.994015i \(0.465158\pi\)
\(998\) 37120.0 1.17737
\(999\) 38608.0 1.22273
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 338.4.a.c.1.1 1
13.2 odd 12 338.4.e.a.147.1 4
13.3 even 3 338.4.c.e.191.1 2
13.4 even 6 338.4.c.a.315.1 2
13.5 odd 4 338.4.b.d.337.2 2
13.6 odd 12 338.4.e.a.23.2 4
13.7 odd 12 338.4.e.a.23.1 4
13.8 odd 4 338.4.b.d.337.1 2
13.9 even 3 338.4.c.e.315.1 2
13.10 even 6 338.4.c.a.191.1 2
13.11 odd 12 338.4.e.a.147.2 4
13.12 even 2 26.4.a.c.1.1 1
39.38 odd 2 234.4.a.e.1.1 1
52.51 odd 2 208.4.a.b.1.1 1
65.12 odd 4 650.4.b.f.599.2 2
65.38 odd 4 650.4.b.f.599.1 2
65.64 even 2 650.4.a.b.1.1 1
91.90 odd 2 1274.4.a.d.1.1 1
104.51 odd 2 832.4.a.o.1.1 1
104.77 even 2 832.4.a.d.1.1 1
156.155 even 2 1872.4.a.q.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
26.4.a.c.1.1 1 13.12 even 2
208.4.a.b.1.1 1 52.51 odd 2
234.4.a.e.1.1 1 39.38 odd 2
338.4.a.c.1.1 1 1.1 even 1 trivial
338.4.b.d.337.1 2 13.8 odd 4
338.4.b.d.337.2 2 13.5 odd 4
338.4.c.a.191.1 2 13.10 even 6
338.4.c.a.315.1 2 13.4 even 6
338.4.c.e.191.1 2 13.3 even 3
338.4.c.e.315.1 2 13.9 even 3
338.4.e.a.23.1 4 13.7 odd 12
338.4.e.a.23.2 4 13.6 odd 12
338.4.e.a.147.1 4 13.2 odd 12
338.4.e.a.147.2 4 13.11 odd 12
650.4.a.b.1.1 1 65.64 even 2
650.4.b.f.599.1 2 65.38 odd 4
650.4.b.f.599.2 2 65.12 odd 4
832.4.a.d.1.1 1 104.77 even 2
832.4.a.o.1.1 1 104.51 odd 2
1274.4.a.d.1.1 1 91.90 odd 2
1872.4.a.q.1.1 1 156.155 even 2