Properties

Label 320.4.a.a.1.1
Level $320$
Weight $4$
Character 320.1
Self dual yes
Analytic conductor $18.881$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [320,4,Mod(1,320)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(320, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("320.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 320 = 2^{6} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 320.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: \(18.8806112018\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 40)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 320.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-10.0000 q^{3} +5.00000 q^{5} -18.0000 q^{7} +73.0000 q^{9} +O(q^{10})\) \(q-10.0000 q^{3} +5.00000 q^{5} -18.0000 q^{7} +73.0000 q^{9} +16.0000 q^{11} +6.00000 q^{13} -50.0000 q^{15} -6.00000 q^{17} +124.000 q^{19} +180.000 q^{21} +42.0000 q^{23} +25.0000 q^{25} -460.000 q^{27} -142.000 q^{29} -188.000 q^{31} -160.000 q^{33} -90.0000 q^{35} -202.000 q^{37} -60.0000 q^{39} +54.0000 q^{41} -66.0000 q^{43} +365.000 q^{45} +38.0000 q^{47} -19.0000 q^{49} +60.0000 q^{51} -738.000 q^{53} +80.0000 q^{55} -1240.00 q^{57} -564.000 q^{59} +262.000 q^{61} -1314.00 q^{63} +30.0000 q^{65} +554.000 q^{67} -420.000 q^{69} +140.000 q^{71} +882.000 q^{73} -250.000 q^{75} -288.000 q^{77} -1160.00 q^{79} +2629.00 q^{81} -642.000 q^{83} -30.0000 q^{85} +1420.00 q^{87} -854.000 q^{89} -108.000 q^{91} +1880.00 q^{93} +620.000 q^{95} -478.000 q^{97} +1168.00 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −10.0000 −1.92450 −0.962250 0.272166i \(-0.912260\pi\)
−0.962250 + 0.272166i \(0.912260\pi\)
\(4\) 0 0
\(5\) 5.00000 0.447214
\(6\) 0 0
\(7\) −18.0000 −0.971909 −0.485954 0.873984i \(-0.661528\pi\)
−0.485954 + 0.873984i \(0.661528\pi\)
\(8\) 0 0
\(9\) 73.0000 2.70370
\(10\) 0 0
\(11\) 16.0000 0.438562 0.219281 0.975662i \(-0.429629\pi\)
0.219281 + 0.975662i \(0.429629\pi\)
\(12\) 0 0
\(13\) 6.00000 0.128008 0.0640039 0.997950i \(-0.479613\pi\)
0.0640039 + 0.997950i \(0.479613\pi\)
\(14\) 0 0
\(15\) −50.0000 −0.860663
\(16\) 0 0
\(17\) −6.00000 −0.0856008 −0.0428004 0.999084i \(-0.513628\pi\)
−0.0428004 + 0.999084i \(0.513628\pi\)
\(18\) 0 0
\(19\) 124.000 1.49724 0.748620 0.663000i \(-0.230717\pi\)
0.748620 + 0.663000i \(0.230717\pi\)
\(20\) 0 0
\(21\) 180.000 1.87044
\(22\) 0 0
\(23\) 42.0000 0.380765 0.190383 0.981710i \(-0.439027\pi\)
0.190383 + 0.981710i \(0.439027\pi\)
\(24\) 0 0
\(25\) 25.0000 0.200000
\(26\) 0 0
\(27\) −460.000 −3.27878
\(28\) 0 0
\(29\) −142.000 −0.909267 −0.454633 0.890679i \(-0.650230\pi\)
−0.454633 + 0.890679i \(0.650230\pi\)
\(30\) 0 0
\(31\) −188.000 −1.08922 −0.544610 0.838690i \(-0.683322\pi\)
−0.544610 + 0.838690i \(0.683322\pi\)
\(32\) 0 0
\(33\) −160.000 −0.844013
\(34\) 0 0
\(35\) −90.0000 −0.434651
\(36\) 0 0
\(37\) −202.000 −0.897530 −0.448765 0.893650i \(-0.648136\pi\)
−0.448765 + 0.893650i \(0.648136\pi\)
\(38\) 0 0
\(39\) −60.0000 −0.246351
\(40\) 0 0
\(41\) 54.0000 0.205692 0.102846 0.994697i \(-0.467205\pi\)
0.102846 + 0.994697i \(0.467205\pi\)
\(42\) 0 0
\(43\) −66.0000 −0.234068 −0.117034 0.993128i \(-0.537339\pi\)
−0.117034 + 0.993128i \(0.537339\pi\)
\(44\) 0 0
\(45\) 365.000 1.20913
\(46\) 0 0
\(47\) 38.0000 0.117933 0.0589667 0.998260i \(-0.481219\pi\)
0.0589667 + 0.998260i \(0.481219\pi\)
\(48\) 0 0
\(49\) −19.0000 −0.0553936
\(50\) 0 0
\(51\) 60.0000 0.164739
\(52\) 0 0
\(53\) −738.000 −1.91268 −0.956341 0.292255i \(-0.905595\pi\)
−0.956341 + 0.292255i \(0.905595\pi\)
\(54\) 0 0
\(55\) 80.0000 0.196131
\(56\) 0 0
\(57\) −1240.00 −2.88144
\(58\) 0 0
\(59\) −564.000 −1.24452 −0.622259 0.782812i \(-0.713785\pi\)
−0.622259 + 0.782812i \(0.713785\pi\)
\(60\) 0 0
\(61\) 262.000 0.549929 0.274964 0.961454i \(-0.411334\pi\)
0.274964 + 0.961454i \(0.411334\pi\)
\(62\) 0 0
\(63\) −1314.00 −2.62775
\(64\) 0 0
\(65\) 30.0000 0.0572468
\(66\) 0 0
\(67\) 554.000 1.01018 0.505089 0.863067i \(-0.331460\pi\)
0.505089 + 0.863067i \(0.331460\pi\)
\(68\) 0 0
\(69\) −420.000 −0.732783
\(70\) 0 0
\(71\) 140.000 0.234013 0.117007 0.993131i \(-0.462670\pi\)
0.117007 + 0.993131i \(0.462670\pi\)
\(72\) 0 0
\(73\) 882.000 1.41411 0.707057 0.707157i \(-0.250023\pi\)
0.707057 + 0.707157i \(0.250023\pi\)
\(74\) 0 0
\(75\) −250.000 −0.384900
\(76\) 0 0
\(77\) −288.000 −0.426242
\(78\) 0 0
\(79\) −1160.00 −1.65203 −0.826014 0.563650i \(-0.809397\pi\)
−0.826014 + 0.563650i \(0.809397\pi\)
\(80\) 0 0
\(81\) 2629.00 3.60631
\(82\) 0 0
\(83\) −642.000 −0.849020 −0.424510 0.905423i \(-0.639554\pi\)
−0.424510 + 0.905423i \(0.639554\pi\)
\(84\) 0 0
\(85\) −30.0000 −0.0382818
\(86\) 0 0
\(87\) 1420.00 1.74988
\(88\) 0 0
\(89\) −854.000 −1.01712 −0.508561 0.861026i \(-0.669822\pi\)
−0.508561 + 0.861026i \(0.669822\pi\)
\(90\) 0 0
\(91\) −108.000 −0.124412
\(92\) 0 0
\(93\) 1880.00 2.09620
\(94\) 0 0
\(95\) 620.000 0.669586
\(96\) 0 0
\(97\) −478.000 −0.500346 −0.250173 0.968201i \(-0.580487\pi\)
−0.250173 + 0.968201i \(0.580487\pi\)
\(98\) 0 0
\(99\) 1168.00 1.18574
\(100\) 0 0
\(101\) 1794.00 1.76742 0.883711 0.468033i \(-0.155037\pi\)
0.883711 + 0.468033i \(0.155037\pi\)
\(102\) 0 0
\(103\) 642.000 0.614157 0.307078 0.951684i \(-0.400649\pi\)
0.307078 + 0.951684i \(0.400649\pi\)
\(104\) 0 0
\(105\) 900.000 0.836486
\(106\) 0 0
\(107\) 850.000 0.767968 0.383984 0.923340i \(-0.374552\pi\)
0.383984 + 0.923340i \(0.374552\pi\)
\(108\) 0 0
\(109\) −666.000 −0.585241 −0.292620 0.956229i \(-0.594527\pi\)
−0.292620 + 0.956229i \(0.594527\pi\)
\(110\) 0 0
\(111\) 2020.00 1.72730
\(112\) 0 0
\(113\) −1446.00 −1.20379 −0.601895 0.798575i \(-0.705587\pi\)
−0.601895 + 0.798575i \(0.705587\pi\)
\(114\) 0 0
\(115\) 210.000 0.170283
\(116\) 0 0
\(117\) 438.000 0.346095
\(118\) 0 0
\(119\) 108.000 0.0831962
\(120\) 0 0
\(121\) −1075.00 −0.807663
\(122\) 0 0
\(123\) −540.000 −0.395855
\(124\) 0 0
\(125\) 125.000 0.0894427
\(126\) 0 0
\(127\) −1154.00 −0.806307 −0.403153 0.915132i \(-0.632086\pi\)
−0.403153 + 0.915132i \(0.632086\pi\)
\(128\) 0 0
\(129\) 660.000 0.450463
\(130\) 0 0
\(131\) 368.000 0.245437 0.122719 0.992441i \(-0.460839\pi\)
0.122719 + 0.992441i \(0.460839\pi\)
\(132\) 0 0
\(133\) −2232.00 −1.45518
\(134\) 0 0
\(135\) −2300.00 −1.46631
\(136\) 0 0
\(137\) −670.000 −0.417825 −0.208912 0.977934i \(-0.566992\pi\)
−0.208912 + 0.977934i \(0.566992\pi\)
\(138\) 0 0
\(139\) 572.000 0.349039 0.174519 0.984654i \(-0.444163\pi\)
0.174519 + 0.984654i \(0.444163\pi\)
\(140\) 0 0
\(141\) −380.000 −0.226963
\(142\) 0 0
\(143\) 96.0000 0.0561393
\(144\) 0 0
\(145\) −710.000 −0.406636
\(146\) 0 0
\(147\) 190.000 0.106605
\(148\) 0 0
\(149\) −1730.00 −0.951189 −0.475594 0.879665i \(-0.657767\pi\)
−0.475594 + 0.879665i \(0.657767\pi\)
\(150\) 0 0
\(151\) 1324.00 0.713547 0.356773 0.934191i \(-0.383877\pi\)
0.356773 + 0.934191i \(0.383877\pi\)
\(152\) 0 0
\(153\) −438.000 −0.231439
\(154\) 0 0
\(155\) −940.000 −0.487114
\(156\) 0 0
\(157\) −2946.00 −1.49756 −0.748778 0.662820i \(-0.769359\pi\)
−0.748778 + 0.662820i \(0.769359\pi\)
\(158\) 0 0
\(159\) 7380.00 3.68096
\(160\) 0 0
\(161\) −756.000 −0.370069
\(162\) 0 0
\(163\) −2098.00 −1.00815 −0.504074 0.863661i \(-0.668166\pi\)
−0.504074 + 0.863661i \(0.668166\pi\)
\(164\) 0 0
\(165\) −800.000 −0.377454
\(166\) 0 0
\(167\) −866.000 −0.401276 −0.200638 0.979665i \(-0.564301\pi\)
−0.200638 + 0.979665i \(0.564301\pi\)
\(168\) 0 0
\(169\) −2161.00 −0.983614
\(170\) 0 0
\(171\) 9052.00 4.04809
\(172\) 0 0
\(173\) 1678.00 0.737433 0.368717 0.929542i \(-0.379797\pi\)
0.368717 + 0.929542i \(0.379797\pi\)
\(174\) 0 0
\(175\) −450.000 −0.194382
\(176\) 0 0
\(177\) 5640.00 2.39508
\(178\) 0 0
\(179\) 1620.00 0.676450 0.338225 0.941065i \(-0.390174\pi\)
0.338225 + 0.941065i \(0.390174\pi\)
\(180\) 0 0
\(181\) −2510.00 −1.03076 −0.515378 0.856963i \(-0.672348\pi\)
−0.515378 + 0.856963i \(0.672348\pi\)
\(182\) 0 0
\(183\) −2620.00 −1.05834
\(184\) 0 0
\(185\) −1010.00 −0.401387
\(186\) 0 0
\(187\) −96.0000 −0.0375413
\(188\) 0 0
\(189\) 8280.00 3.18667
\(190\) 0 0
\(191\) −372.000 −0.140927 −0.0704633 0.997514i \(-0.522448\pi\)
−0.0704633 + 0.997514i \(0.522448\pi\)
\(192\) 0 0
\(193\) 2938.00 1.09576 0.547880 0.836557i \(-0.315435\pi\)
0.547880 + 0.836557i \(0.315435\pi\)
\(194\) 0 0
\(195\) −300.000 −0.110172
\(196\) 0 0
\(197\) −2234.00 −0.807949 −0.403974 0.914770i \(-0.632372\pi\)
−0.403974 + 0.914770i \(0.632372\pi\)
\(198\) 0 0
\(199\) −3048.00 −1.08576 −0.542882 0.839809i \(-0.682667\pi\)
−0.542882 + 0.839809i \(0.682667\pi\)
\(200\) 0 0
\(201\) −5540.00 −1.94409
\(202\) 0 0
\(203\) 2556.00 0.883724
\(204\) 0 0
\(205\) 270.000 0.0919884
\(206\) 0 0
\(207\) 3066.00 1.02948
\(208\) 0 0
\(209\) 1984.00 0.656632
\(210\) 0 0
\(211\) −4896.00 −1.59741 −0.798707 0.601720i \(-0.794482\pi\)
−0.798707 + 0.601720i \(0.794482\pi\)
\(212\) 0 0
\(213\) −1400.00 −0.450359
\(214\) 0 0
\(215\) −330.000 −0.104678
\(216\) 0 0
\(217\) 3384.00 1.05862
\(218\) 0 0
\(219\) −8820.00 −2.72146
\(220\) 0 0
\(221\) −36.0000 −0.0109576
\(222\) 0 0
\(223\) −5302.00 −1.59214 −0.796072 0.605202i \(-0.793092\pi\)
−0.796072 + 0.605202i \(0.793092\pi\)
\(224\) 0 0
\(225\) 1825.00 0.540741
\(226\) 0 0
\(227\) 3778.00 1.10465 0.552323 0.833630i \(-0.313741\pi\)
0.552323 + 0.833630i \(0.313741\pi\)
\(228\) 0 0
\(229\) 3034.00 0.875513 0.437756 0.899094i \(-0.355773\pi\)
0.437756 + 0.899094i \(0.355773\pi\)
\(230\) 0 0
\(231\) 2880.00 0.820303
\(232\) 0 0
\(233\) −3478.00 −0.977903 −0.488951 0.872311i \(-0.662620\pi\)
−0.488951 + 0.872311i \(0.662620\pi\)
\(234\) 0 0
\(235\) 190.000 0.0527414
\(236\) 0 0
\(237\) 11600.0 3.17933
\(238\) 0 0
\(239\) 1560.00 0.422209 0.211105 0.977463i \(-0.432294\pi\)
0.211105 + 0.977463i \(0.432294\pi\)
\(240\) 0 0
\(241\) −3218.00 −0.860123 −0.430061 0.902800i \(-0.641508\pi\)
−0.430061 + 0.902800i \(0.641508\pi\)
\(242\) 0 0
\(243\) −13870.0 −3.66157
\(244\) 0 0
\(245\) −95.0000 −0.0247728
\(246\) 0 0
\(247\) 744.000 0.191658
\(248\) 0 0
\(249\) 6420.00 1.63394
\(250\) 0 0
\(251\) −688.000 −0.173013 −0.0865063 0.996251i \(-0.527570\pi\)
−0.0865063 + 0.996251i \(0.527570\pi\)
\(252\) 0 0
\(253\) 672.000 0.166989
\(254\) 0 0
\(255\) 300.000 0.0736734
\(256\) 0 0
\(257\) 2170.00 0.526696 0.263348 0.964701i \(-0.415173\pi\)
0.263348 + 0.964701i \(0.415173\pi\)
\(258\) 0 0
\(259\) 3636.00 0.872317
\(260\) 0 0
\(261\) −10366.0 −2.45839
\(262\) 0 0
\(263\) 2274.00 0.533159 0.266580 0.963813i \(-0.414106\pi\)
0.266580 + 0.963813i \(0.414106\pi\)
\(264\) 0 0
\(265\) −3690.00 −0.855377
\(266\) 0 0
\(267\) 8540.00 1.95745
\(268\) 0 0
\(269\) −7146.00 −1.61970 −0.809850 0.586637i \(-0.800452\pi\)
−0.809850 + 0.586637i \(0.800452\pi\)
\(270\) 0 0
\(271\) 2604.00 0.583696 0.291848 0.956465i \(-0.405730\pi\)
0.291848 + 0.956465i \(0.405730\pi\)
\(272\) 0 0
\(273\) 1080.00 0.239431
\(274\) 0 0
\(275\) 400.000 0.0877124
\(276\) 0 0
\(277\) 5150.00 1.11709 0.558544 0.829475i \(-0.311360\pi\)
0.558544 + 0.829475i \(0.311360\pi\)
\(278\) 0 0
\(279\) −13724.0 −2.94493
\(280\) 0 0
\(281\) 5270.00 1.11880 0.559398 0.828899i \(-0.311032\pi\)
0.559398 + 0.828899i \(0.311032\pi\)
\(282\) 0 0
\(283\) −3434.00 −0.721308 −0.360654 0.932700i \(-0.617446\pi\)
−0.360654 + 0.932700i \(0.617446\pi\)
\(284\) 0 0
\(285\) −6200.00 −1.28862
\(286\) 0 0
\(287\) −972.000 −0.199914
\(288\) 0 0
\(289\) −4877.00 −0.992673
\(290\) 0 0
\(291\) 4780.00 0.962916
\(292\) 0 0
\(293\) 9878.00 1.96955 0.984776 0.173826i \(-0.0556132\pi\)
0.984776 + 0.173826i \(0.0556132\pi\)
\(294\) 0 0
\(295\) −2820.00 −0.556565
\(296\) 0 0
\(297\) −7360.00 −1.43795
\(298\) 0 0
\(299\) 252.000 0.0487409
\(300\) 0 0
\(301\) 1188.00 0.227492
\(302\) 0 0
\(303\) −17940.0 −3.40141
\(304\) 0 0
\(305\) 1310.00 0.245936
\(306\) 0 0
\(307\) −8054.00 −1.49728 −0.748642 0.662975i \(-0.769294\pi\)
−0.748642 + 0.662975i \(0.769294\pi\)
\(308\) 0 0
\(309\) −6420.00 −1.18195
\(310\) 0 0
\(311\) 5492.00 1.00136 0.500680 0.865633i \(-0.333083\pi\)
0.500680 + 0.865633i \(0.333083\pi\)
\(312\) 0 0
\(313\) −422.000 −0.0762072 −0.0381036 0.999274i \(-0.512132\pi\)
−0.0381036 + 0.999274i \(0.512132\pi\)
\(314\) 0 0
\(315\) −6570.00 −1.17517
\(316\) 0 0
\(317\) −6194.00 −1.09744 −0.548722 0.836005i \(-0.684885\pi\)
−0.548722 + 0.836005i \(0.684885\pi\)
\(318\) 0 0
\(319\) −2272.00 −0.398770
\(320\) 0 0
\(321\) −8500.00 −1.47796
\(322\) 0 0
\(323\) −744.000 −0.128165
\(324\) 0 0
\(325\) 150.000 0.0256015
\(326\) 0 0
\(327\) 6660.00 1.12630
\(328\) 0 0
\(329\) −684.000 −0.114620
\(330\) 0 0
\(331\) −7688.00 −1.27665 −0.638324 0.769768i \(-0.720372\pi\)
−0.638324 + 0.769768i \(0.720372\pi\)
\(332\) 0 0
\(333\) −14746.0 −2.42665
\(334\) 0 0
\(335\) 2770.00 0.451765
\(336\) 0 0
\(337\) −1438.00 −0.232442 −0.116221 0.993223i \(-0.537078\pi\)
−0.116221 + 0.993223i \(0.537078\pi\)
\(338\) 0 0
\(339\) 14460.0 2.31669
\(340\) 0 0
\(341\) −3008.00 −0.477690
\(342\) 0 0
\(343\) 6516.00 1.02575
\(344\) 0 0
\(345\) −2100.00 −0.327711
\(346\) 0 0
\(347\) −8838.00 −1.36729 −0.683644 0.729816i \(-0.739606\pi\)
−0.683644 + 0.729816i \(0.739606\pi\)
\(348\) 0 0
\(349\) 7810.00 1.19788 0.598939 0.800794i \(-0.295589\pi\)
0.598939 + 0.800794i \(0.295589\pi\)
\(350\) 0 0
\(351\) −2760.00 −0.419709
\(352\) 0 0
\(353\) 5906.00 0.890495 0.445247 0.895408i \(-0.353116\pi\)
0.445247 + 0.895408i \(0.353116\pi\)
\(354\) 0 0
\(355\) 700.000 0.104654
\(356\) 0 0
\(357\) −1080.00 −0.160111
\(358\) 0 0
\(359\) 8904.00 1.30901 0.654506 0.756057i \(-0.272877\pi\)
0.654506 + 0.756057i \(0.272877\pi\)
\(360\) 0 0
\(361\) 8517.00 1.24173
\(362\) 0 0
\(363\) 10750.0 1.55435
\(364\) 0 0
\(365\) 4410.00 0.632411
\(366\) 0 0
\(367\) −7370.00 −1.04826 −0.524129 0.851639i \(-0.675609\pi\)
−0.524129 + 0.851639i \(0.675609\pi\)
\(368\) 0 0
\(369\) 3942.00 0.556131
\(370\) 0 0
\(371\) 13284.0 1.85895
\(372\) 0 0
\(373\) 734.000 0.101890 0.0509451 0.998701i \(-0.483777\pi\)
0.0509451 + 0.998701i \(0.483777\pi\)
\(374\) 0 0
\(375\) −1250.00 −0.172133
\(376\) 0 0
\(377\) −852.000 −0.116393
\(378\) 0 0
\(379\) −10300.0 −1.39598 −0.697989 0.716109i \(-0.745921\pi\)
−0.697989 + 0.716109i \(0.745921\pi\)
\(380\) 0 0
\(381\) 11540.0 1.55174
\(382\) 0 0
\(383\) 2682.00 0.357817 0.178908 0.983866i \(-0.442743\pi\)
0.178908 + 0.983866i \(0.442743\pi\)
\(384\) 0 0
\(385\) −1440.00 −0.190621
\(386\) 0 0
\(387\) −4818.00 −0.632849
\(388\) 0 0
\(389\) −6114.00 −0.796895 −0.398447 0.917191i \(-0.630451\pi\)
−0.398447 + 0.917191i \(0.630451\pi\)
\(390\) 0 0
\(391\) −252.000 −0.0325938
\(392\) 0 0
\(393\) −3680.00 −0.472345
\(394\) 0 0
\(395\) −5800.00 −0.738809
\(396\) 0 0
\(397\) 7174.00 0.906934 0.453467 0.891273i \(-0.350187\pi\)
0.453467 + 0.891273i \(0.350187\pi\)
\(398\) 0 0
\(399\) 22320.0 2.80050
\(400\) 0 0
\(401\) 10498.0 1.30734 0.653672 0.756778i \(-0.273228\pi\)
0.653672 + 0.756778i \(0.273228\pi\)
\(402\) 0 0
\(403\) −1128.00 −0.139428
\(404\) 0 0
\(405\) 13145.0 1.61279
\(406\) 0 0
\(407\) −3232.00 −0.393622
\(408\) 0 0
\(409\) −1810.00 −0.218823 −0.109412 0.993997i \(-0.534897\pi\)
−0.109412 + 0.993997i \(0.534897\pi\)
\(410\) 0 0
\(411\) 6700.00 0.804104
\(412\) 0 0
\(413\) 10152.0 1.20956
\(414\) 0 0
\(415\) −3210.00 −0.379693
\(416\) 0 0
\(417\) −5720.00 −0.671726
\(418\) 0 0
\(419\) 3396.00 0.395956 0.197978 0.980206i \(-0.436563\pi\)
0.197978 + 0.980206i \(0.436563\pi\)
\(420\) 0 0
\(421\) 14974.0 1.73346 0.866732 0.498775i \(-0.166216\pi\)
0.866732 + 0.498775i \(0.166216\pi\)
\(422\) 0 0
\(423\) 2774.00 0.318857
\(424\) 0 0
\(425\) −150.000 −0.0171202
\(426\) 0 0
\(427\) −4716.00 −0.534481
\(428\) 0 0
\(429\) −960.000 −0.108040
\(430\) 0 0
\(431\) −13540.0 −1.51322 −0.756611 0.653865i \(-0.773146\pi\)
−0.756611 + 0.653865i \(0.773146\pi\)
\(432\) 0 0
\(433\) 15426.0 1.71207 0.856035 0.516918i \(-0.172921\pi\)
0.856035 + 0.516918i \(0.172921\pi\)
\(434\) 0 0
\(435\) 7100.00 0.782572
\(436\) 0 0
\(437\) 5208.00 0.570097
\(438\) 0 0
\(439\) −10472.0 −1.13850 −0.569250 0.822165i \(-0.692766\pi\)
−0.569250 + 0.822165i \(0.692766\pi\)
\(440\) 0 0
\(441\) −1387.00 −0.149768
\(442\) 0 0
\(443\) −722.000 −0.0774340 −0.0387170 0.999250i \(-0.512327\pi\)
−0.0387170 + 0.999250i \(0.512327\pi\)
\(444\) 0 0
\(445\) −4270.00 −0.454871
\(446\) 0 0
\(447\) 17300.0 1.83056
\(448\) 0 0
\(449\) −11898.0 −1.25056 −0.625280 0.780401i \(-0.715015\pi\)
−0.625280 + 0.780401i \(0.715015\pi\)
\(450\) 0 0
\(451\) 864.000 0.0902088
\(452\) 0 0
\(453\) −13240.0 −1.37322
\(454\) 0 0
\(455\) −540.000 −0.0556387
\(456\) 0 0
\(457\) −790.000 −0.0808635 −0.0404318 0.999182i \(-0.512873\pi\)
−0.0404318 + 0.999182i \(0.512873\pi\)
\(458\) 0 0
\(459\) 2760.00 0.280666
\(460\) 0 0
\(461\) 3418.00 0.345319 0.172660 0.984982i \(-0.444764\pi\)
0.172660 + 0.984982i \(0.444764\pi\)
\(462\) 0 0
\(463\) −7534.00 −0.756230 −0.378115 0.925759i \(-0.623428\pi\)
−0.378115 + 0.925759i \(0.623428\pi\)
\(464\) 0 0
\(465\) 9400.00 0.937451
\(466\) 0 0
\(467\) 14314.0 1.41836 0.709179 0.705029i \(-0.249066\pi\)
0.709179 + 0.705029i \(0.249066\pi\)
\(468\) 0 0
\(469\) −9972.00 −0.981800
\(470\) 0 0
\(471\) 29460.0 2.88205
\(472\) 0 0
\(473\) −1056.00 −0.102653
\(474\) 0 0
\(475\) 3100.00 0.299448
\(476\) 0 0
\(477\) −53874.0 −5.17132
\(478\) 0 0
\(479\) −7016.00 −0.669247 −0.334623 0.942352i \(-0.608609\pi\)
−0.334623 + 0.942352i \(0.608609\pi\)
\(480\) 0 0
\(481\) −1212.00 −0.114891
\(482\) 0 0
\(483\) 7560.00 0.712199
\(484\) 0 0
\(485\) −2390.00 −0.223761
\(486\) 0 0
\(487\) 15190.0 1.41340 0.706699 0.707515i \(-0.250184\pi\)
0.706699 + 0.707515i \(0.250184\pi\)
\(488\) 0 0
\(489\) 20980.0 1.94018
\(490\) 0 0
\(491\) 12624.0 1.16031 0.580156 0.814505i \(-0.302992\pi\)
0.580156 + 0.814505i \(0.302992\pi\)
\(492\) 0 0
\(493\) 852.000 0.0778340
\(494\) 0 0
\(495\) 5840.00 0.530280
\(496\) 0 0
\(497\) −2520.00 −0.227440
\(498\) 0 0
\(499\) 2492.00 0.223562 0.111781 0.993733i \(-0.464345\pi\)
0.111781 + 0.993733i \(0.464345\pi\)
\(500\) 0 0
\(501\) 8660.00 0.772256
\(502\) 0 0
\(503\) 11714.0 1.03837 0.519186 0.854661i \(-0.326235\pi\)
0.519186 + 0.854661i \(0.326235\pi\)
\(504\) 0 0
\(505\) 8970.00 0.790415
\(506\) 0 0
\(507\) 21610.0 1.89297
\(508\) 0 0
\(509\) 5618.00 0.489221 0.244610 0.969621i \(-0.421340\pi\)
0.244610 + 0.969621i \(0.421340\pi\)
\(510\) 0 0
\(511\) −15876.0 −1.37439
\(512\) 0 0
\(513\) −57040.0 −4.90912
\(514\) 0 0
\(515\) 3210.00 0.274659
\(516\) 0 0
\(517\) 608.000 0.0517211
\(518\) 0 0
\(519\) −16780.0 −1.41919
\(520\) 0 0
\(521\) 13770.0 1.15792 0.578958 0.815357i \(-0.303459\pi\)
0.578958 + 0.815357i \(0.303459\pi\)
\(522\) 0 0
\(523\) −6986.00 −0.584085 −0.292042 0.956405i \(-0.594335\pi\)
−0.292042 + 0.956405i \(0.594335\pi\)
\(524\) 0 0
\(525\) 4500.00 0.374088
\(526\) 0 0
\(527\) 1128.00 0.0932380
\(528\) 0 0
\(529\) −10403.0 −0.855018
\(530\) 0 0
\(531\) −41172.0 −3.36481
\(532\) 0 0
\(533\) 324.000 0.0263302
\(534\) 0 0
\(535\) 4250.00 0.343446
\(536\) 0 0
\(537\) −16200.0 −1.30183
\(538\) 0 0
\(539\) −304.000 −0.0242935
\(540\) 0 0
\(541\) −11958.0 −0.950304 −0.475152 0.879904i \(-0.657607\pi\)
−0.475152 + 0.879904i \(0.657607\pi\)
\(542\) 0 0
\(543\) 25100.0 1.98369
\(544\) 0 0
\(545\) −3330.00 −0.261728
\(546\) 0 0
\(547\) 4194.00 0.327829 0.163915 0.986475i \(-0.447588\pi\)
0.163915 + 0.986475i \(0.447588\pi\)
\(548\) 0 0
\(549\) 19126.0 1.48684
\(550\) 0 0
\(551\) −17608.0 −1.36139
\(552\) 0 0
\(553\) 20880.0 1.60562
\(554\) 0 0
\(555\) 10100.0 0.772470
\(556\) 0 0
\(557\) 5382.00 0.409412 0.204706 0.978823i \(-0.434376\pi\)
0.204706 + 0.978823i \(0.434376\pi\)
\(558\) 0 0
\(559\) −396.000 −0.0299625
\(560\) 0 0
\(561\) 960.000 0.0722482
\(562\) 0 0
\(563\) −15418.0 −1.15416 −0.577079 0.816688i \(-0.695808\pi\)
−0.577079 + 0.816688i \(0.695808\pi\)
\(564\) 0 0
\(565\) −7230.00 −0.538351
\(566\) 0 0
\(567\) −47322.0 −3.50500
\(568\) 0 0
\(569\) −5778.00 −0.425705 −0.212853 0.977084i \(-0.568275\pi\)
−0.212853 + 0.977084i \(0.568275\pi\)
\(570\) 0 0
\(571\) −6024.00 −0.441500 −0.220750 0.975330i \(-0.570851\pi\)
−0.220750 + 0.975330i \(0.570851\pi\)
\(572\) 0 0
\(573\) 3720.00 0.271213
\(574\) 0 0
\(575\) 1050.00 0.0761531
\(576\) 0 0
\(577\) 554.000 0.0399711 0.0199855 0.999800i \(-0.493638\pi\)
0.0199855 + 0.999800i \(0.493638\pi\)
\(578\) 0 0
\(579\) −29380.0 −2.10879
\(580\) 0 0
\(581\) 11556.0 0.825170
\(582\) 0 0
\(583\) −11808.0 −0.838829
\(584\) 0 0
\(585\) 2190.00 0.154778
\(586\) 0 0
\(587\) 2386.00 0.167770 0.0838848 0.996475i \(-0.473267\pi\)
0.0838848 + 0.996475i \(0.473267\pi\)
\(588\) 0 0
\(589\) −23312.0 −1.63082
\(590\) 0 0
\(591\) 22340.0 1.55490
\(592\) 0 0
\(593\) −846.000 −0.0585853 −0.0292926 0.999571i \(-0.509325\pi\)
−0.0292926 + 0.999571i \(0.509325\pi\)
\(594\) 0 0
\(595\) 540.000 0.0372065
\(596\) 0 0
\(597\) 30480.0 2.08955
\(598\) 0 0
\(599\) 22304.0 1.52140 0.760698 0.649105i \(-0.224857\pi\)
0.760698 + 0.649105i \(0.224857\pi\)
\(600\) 0 0
\(601\) 5510.00 0.373973 0.186986 0.982363i \(-0.440128\pi\)
0.186986 + 0.982363i \(0.440128\pi\)
\(602\) 0 0
\(603\) 40442.0 2.73122
\(604\) 0 0
\(605\) −5375.00 −0.361198
\(606\) 0 0
\(607\) −8234.00 −0.550589 −0.275295 0.961360i \(-0.588775\pi\)
−0.275295 + 0.961360i \(0.588775\pi\)
\(608\) 0 0
\(609\) −25560.0 −1.70073
\(610\) 0 0
\(611\) 228.000 0.0150964
\(612\) 0 0
\(613\) 1046.00 0.0689193 0.0344597 0.999406i \(-0.489029\pi\)
0.0344597 + 0.999406i \(0.489029\pi\)
\(614\) 0 0
\(615\) −2700.00 −0.177032
\(616\) 0 0
\(617\) −3862.00 −0.251991 −0.125995 0.992031i \(-0.540212\pi\)
−0.125995 + 0.992031i \(0.540212\pi\)
\(618\) 0 0
\(619\) −13964.0 −0.906721 −0.453361 0.891327i \(-0.649775\pi\)
−0.453361 + 0.891327i \(0.649775\pi\)
\(620\) 0 0
\(621\) −19320.0 −1.24845
\(622\) 0 0
\(623\) 15372.0 0.988549
\(624\) 0 0
\(625\) 625.000 0.0400000
\(626\) 0 0
\(627\) −19840.0 −1.26369
\(628\) 0 0
\(629\) 1212.00 0.0768293
\(630\) 0 0
\(631\) −14884.0 −0.939022 −0.469511 0.882927i \(-0.655570\pi\)
−0.469511 + 0.882927i \(0.655570\pi\)
\(632\) 0 0
\(633\) 48960.0 3.07423
\(634\) 0 0
\(635\) −5770.00 −0.360591
\(636\) 0 0
\(637\) −114.000 −0.00709081
\(638\) 0 0
\(639\) 10220.0 0.632703
\(640\) 0 0
\(641\) 17838.0 1.09916 0.549578 0.835443i \(-0.314789\pi\)
0.549578 + 0.835443i \(0.314789\pi\)
\(642\) 0 0
\(643\) 7814.00 0.479244 0.239622 0.970866i \(-0.422976\pi\)
0.239622 + 0.970866i \(0.422976\pi\)
\(644\) 0 0
\(645\) 3300.00 0.201453
\(646\) 0 0
\(647\) 774.000 0.0470310 0.0235155 0.999723i \(-0.492514\pi\)
0.0235155 + 0.999723i \(0.492514\pi\)
\(648\) 0 0
\(649\) −9024.00 −0.545798
\(650\) 0 0
\(651\) −33840.0 −2.03732
\(652\) 0 0
\(653\) 23422.0 1.40364 0.701818 0.712357i \(-0.252372\pi\)
0.701818 + 0.712357i \(0.252372\pi\)
\(654\) 0 0
\(655\) 1840.00 0.109763
\(656\) 0 0
\(657\) 64386.0 3.82334
\(658\) 0 0
\(659\) 13508.0 0.798478 0.399239 0.916847i \(-0.369274\pi\)
0.399239 + 0.916847i \(0.369274\pi\)
\(660\) 0 0
\(661\) 6222.00 0.366124 0.183062 0.983101i \(-0.441399\pi\)
0.183062 + 0.983101i \(0.441399\pi\)
\(662\) 0 0
\(663\) 360.000 0.0210878
\(664\) 0 0
\(665\) −11160.0 −0.650776
\(666\) 0 0
\(667\) −5964.00 −0.346217
\(668\) 0 0
\(669\) 53020.0 3.06408
\(670\) 0 0
\(671\) 4192.00 0.241178
\(672\) 0 0
\(673\) −15566.0 −0.891568 −0.445784 0.895141i \(-0.647075\pi\)
−0.445784 + 0.895141i \(0.647075\pi\)
\(674\) 0 0
\(675\) −11500.0 −0.655756
\(676\) 0 0
\(677\) −2234.00 −0.126824 −0.0634118 0.997987i \(-0.520198\pi\)
−0.0634118 + 0.997987i \(0.520198\pi\)
\(678\) 0 0
\(679\) 8604.00 0.486290
\(680\) 0 0
\(681\) −37780.0 −2.12589
\(682\) 0 0
\(683\) −13282.0 −0.744102 −0.372051 0.928212i \(-0.621345\pi\)
−0.372051 + 0.928212i \(0.621345\pi\)
\(684\) 0 0
\(685\) −3350.00 −0.186857
\(686\) 0 0
\(687\) −30340.0 −1.68492
\(688\) 0 0
\(689\) −4428.00 −0.244838
\(690\) 0 0
\(691\) 27416.0 1.50934 0.754670 0.656105i \(-0.227797\pi\)
0.754670 + 0.656105i \(0.227797\pi\)
\(692\) 0 0
\(693\) −21024.0 −1.15243
\(694\) 0 0
\(695\) 2860.00 0.156095
\(696\) 0 0
\(697\) −324.000 −0.0176074
\(698\) 0 0
\(699\) 34780.0 1.88197
\(700\) 0 0
\(701\) −25626.0 −1.38071 −0.690357 0.723469i \(-0.742547\pi\)
−0.690357 + 0.723469i \(0.742547\pi\)
\(702\) 0 0
\(703\) −25048.0 −1.34382
\(704\) 0 0
\(705\) −1900.00 −0.101501
\(706\) 0 0
\(707\) −32292.0 −1.71777
\(708\) 0 0
\(709\) −11702.0 −0.619856 −0.309928 0.950760i \(-0.600305\pi\)
−0.309928 + 0.950760i \(0.600305\pi\)
\(710\) 0 0
\(711\) −84680.0 −4.46659
\(712\) 0 0
\(713\) −7896.00 −0.414737
\(714\) 0 0
\(715\) 480.000 0.0251063
\(716\) 0 0
\(717\) −15600.0 −0.812542
\(718\) 0 0
\(719\) −28008.0 −1.45274 −0.726371 0.687302i \(-0.758795\pi\)
−0.726371 + 0.687302i \(0.758795\pi\)
\(720\) 0 0
\(721\) −11556.0 −0.596904
\(722\) 0 0
\(723\) 32180.0 1.65531
\(724\) 0 0
\(725\) −3550.00 −0.181853
\(726\) 0 0
\(727\) −7682.00 −0.391898 −0.195949 0.980614i \(-0.562779\pi\)
−0.195949 + 0.980614i \(0.562779\pi\)
\(728\) 0 0
\(729\) 67717.0 3.44038
\(730\) 0 0
\(731\) 396.000 0.0200364
\(732\) 0 0
\(733\) 14270.0 0.719065 0.359532 0.933133i \(-0.382936\pi\)
0.359532 + 0.933133i \(0.382936\pi\)
\(734\) 0 0
\(735\) 950.000 0.0476752
\(736\) 0 0
\(737\) 8864.00 0.443025
\(738\) 0 0
\(739\) −29324.0 −1.45968 −0.729838 0.683620i \(-0.760405\pi\)
−0.729838 + 0.683620i \(0.760405\pi\)
\(740\) 0 0
\(741\) −7440.00 −0.368846
\(742\) 0 0
\(743\) 29258.0 1.44465 0.722323 0.691556i \(-0.243074\pi\)
0.722323 + 0.691556i \(0.243074\pi\)
\(744\) 0 0
\(745\) −8650.00 −0.425385
\(746\) 0 0
\(747\) −46866.0 −2.29550
\(748\) 0 0
\(749\) −15300.0 −0.746395
\(750\) 0 0
\(751\) 19076.0 0.926888 0.463444 0.886126i \(-0.346613\pi\)
0.463444 + 0.886126i \(0.346613\pi\)
\(752\) 0 0
\(753\) 6880.00 0.332963
\(754\) 0 0
\(755\) 6620.00 0.319108
\(756\) 0 0
\(757\) 22670.0 1.08845 0.544224 0.838940i \(-0.316824\pi\)
0.544224 + 0.838940i \(0.316824\pi\)
\(758\) 0 0
\(759\) −6720.00 −0.321371
\(760\) 0 0
\(761\) −23206.0 −1.10541 −0.552705 0.833377i \(-0.686404\pi\)
−0.552705 + 0.833377i \(0.686404\pi\)
\(762\) 0 0
\(763\) 11988.0 0.568800
\(764\) 0 0
\(765\) −2190.00 −0.103503
\(766\) 0 0
\(767\) −3384.00 −0.159308
\(768\) 0 0
\(769\) −1854.00 −0.0869401 −0.0434701 0.999055i \(-0.513841\pi\)
−0.0434701 + 0.999055i \(0.513841\pi\)
\(770\) 0 0
\(771\) −21700.0 −1.01363
\(772\) 0 0
\(773\) −6474.00 −0.301234 −0.150617 0.988592i \(-0.548126\pi\)
−0.150617 + 0.988592i \(0.548126\pi\)
\(774\) 0 0
\(775\) −4700.00 −0.217844
\(776\) 0 0
\(777\) −36360.0 −1.67877
\(778\) 0 0
\(779\) 6696.00 0.307971
\(780\) 0 0
\(781\) 2240.00 0.102629
\(782\) 0 0
\(783\) 65320.0 2.98129
\(784\) 0 0
\(785\) −14730.0 −0.669728
\(786\) 0 0
\(787\) 20354.0 0.921908 0.460954 0.887424i \(-0.347507\pi\)
0.460954 + 0.887424i \(0.347507\pi\)
\(788\) 0 0
\(789\) −22740.0 −1.02607
\(790\) 0 0
\(791\) 26028.0 1.16997
\(792\) 0 0
\(793\) 1572.00 0.0703952
\(794\) 0 0
\(795\) 36900.0 1.64617
\(796\) 0 0
\(797\) 1886.00 0.0838213 0.0419106 0.999121i \(-0.486656\pi\)
0.0419106 + 0.999121i \(0.486656\pi\)
\(798\) 0 0
\(799\) −228.000 −0.0100952
\(800\) 0 0
\(801\) −62342.0 −2.75000
\(802\) 0 0
\(803\) 14112.0 0.620176
\(804\) 0 0
\(805\) −3780.00 −0.165500
\(806\) 0 0
\(807\) 71460.0 3.11711
\(808\) 0 0
\(809\) −9462.00 −0.411207 −0.205603 0.978635i \(-0.565916\pi\)
−0.205603 + 0.978635i \(0.565916\pi\)
\(810\) 0 0
\(811\) −24512.0 −1.06132 −0.530661 0.847584i \(-0.678056\pi\)
−0.530661 + 0.847584i \(0.678056\pi\)
\(812\) 0 0
\(813\) −26040.0 −1.12332
\(814\) 0 0
\(815\) −10490.0 −0.450857
\(816\) 0 0
\(817\) −8184.00 −0.350455
\(818\) 0 0
\(819\) −7884.00 −0.336373
\(820\) 0 0
\(821\) −36242.0 −1.54063 −0.770313 0.637666i \(-0.779900\pi\)
−0.770313 + 0.637666i \(0.779900\pi\)
\(822\) 0 0
\(823\) −17718.0 −0.750438 −0.375219 0.926936i \(-0.622433\pi\)
−0.375219 + 0.926936i \(0.622433\pi\)
\(824\) 0 0
\(825\) −4000.00 −0.168803
\(826\) 0 0
\(827\) −6726.00 −0.282812 −0.141406 0.989952i \(-0.545162\pi\)
−0.141406 + 0.989952i \(0.545162\pi\)
\(828\) 0 0
\(829\) −41722.0 −1.74797 −0.873984 0.485955i \(-0.838472\pi\)
−0.873984 + 0.485955i \(0.838472\pi\)
\(830\) 0 0
\(831\) −51500.0 −2.14984
\(832\) 0 0
\(833\) 114.000 0.00474174
\(834\) 0 0
\(835\) −4330.00 −0.179456
\(836\) 0 0
\(837\) 86480.0 3.57131
\(838\) 0 0
\(839\) 16720.0 0.688008 0.344004 0.938968i \(-0.388217\pi\)
0.344004 + 0.938968i \(0.388217\pi\)
\(840\) 0 0
\(841\) −4225.00 −0.173234
\(842\) 0 0
\(843\) −52700.0 −2.15313
\(844\) 0 0
\(845\) −10805.0 −0.439886
\(846\) 0 0
\(847\) 19350.0 0.784975
\(848\) 0 0
\(849\) 34340.0 1.38816
\(850\) 0 0
\(851\) −8484.00 −0.341748
\(852\) 0 0
\(853\) 33286.0 1.33610 0.668049 0.744118i \(-0.267130\pi\)
0.668049 + 0.744118i \(0.267130\pi\)
\(854\) 0 0
\(855\) 45260.0 1.81036
\(856\) 0 0
\(857\) 38978.0 1.55363 0.776816 0.629727i \(-0.216833\pi\)
0.776816 + 0.629727i \(0.216833\pi\)
\(858\) 0 0
\(859\) 1916.00 0.0761037 0.0380518 0.999276i \(-0.487885\pi\)
0.0380518 + 0.999276i \(0.487885\pi\)
\(860\) 0 0
\(861\) 9720.00 0.384735
\(862\) 0 0
\(863\) −2374.00 −0.0936407 −0.0468203 0.998903i \(-0.514909\pi\)
−0.0468203 + 0.998903i \(0.514909\pi\)
\(864\) 0 0
\(865\) 8390.00 0.329790
\(866\) 0 0
\(867\) 48770.0 1.91040
\(868\) 0 0
\(869\) −18560.0 −0.724517
\(870\) 0 0
\(871\) 3324.00 0.129310
\(872\) 0 0
\(873\) −34894.0 −1.35279
\(874\) 0 0
\(875\) −2250.00 −0.0869302
\(876\) 0 0
\(877\) −32722.0 −1.25991 −0.629956 0.776631i \(-0.716927\pi\)
−0.629956 + 0.776631i \(0.716927\pi\)
\(878\) 0 0
\(879\) −98780.0 −3.79041
\(880\) 0 0
\(881\) 5390.00 0.206122 0.103061 0.994675i \(-0.467136\pi\)
0.103061 + 0.994675i \(0.467136\pi\)
\(882\) 0 0
\(883\) 43238.0 1.64788 0.823938 0.566680i \(-0.191772\pi\)
0.823938 + 0.566680i \(0.191772\pi\)
\(884\) 0 0
\(885\) 28200.0 1.07111
\(886\) 0 0
\(887\) −11010.0 −0.416775 −0.208388 0.978046i \(-0.566822\pi\)
−0.208388 + 0.978046i \(0.566822\pi\)
\(888\) 0 0
\(889\) 20772.0 0.783656
\(890\) 0 0
\(891\) 42064.0 1.58159
\(892\) 0 0
\(893\) 4712.00 0.176575
\(894\) 0 0
\(895\) 8100.00 0.302517
\(896\) 0 0
\(897\) −2520.00 −0.0938020
\(898\) 0 0
\(899\) 26696.0 0.990391
\(900\) 0 0
\(901\) 4428.00 0.163727
\(902\) 0 0
\(903\) −11880.0 −0.437809
\(904\) 0 0
\(905\) −12550.0 −0.460968
\(906\) 0 0
\(907\) 74.0000 0.00270907 0.00135454 0.999999i \(-0.499569\pi\)
0.00135454 + 0.999999i \(0.499569\pi\)
\(908\) 0 0
\(909\) 130962. 4.77859
\(910\) 0 0
\(911\) 17460.0 0.634990 0.317495 0.948260i \(-0.397158\pi\)
0.317495 + 0.948260i \(0.397158\pi\)
\(912\) 0 0
\(913\) −10272.0 −0.372348
\(914\) 0 0
\(915\) −13100.0 −0.473303
\(916\) 0 0
\(917\) −6624.00 −0.238543
\(918\) 0 0
\(919\) 17072.0 0.612789 0.306395 0.951905i \(-0.400877\pi\)
0.306395 + 0.951905i \(0.400877\pi\)
\(920\) 0 0
\(921\) 80540.0 2.88152
\(922\) 0 0
\(923\) 840.000 0.0299555
\(924\) 0 0
\(925\) −5050.00 −0.179506
\(926\) 0 0
\(927\) 46866.0 1.66050
\(928\) 0 0
\(929\) −14826.0 −0.523601 −0.261800 0.965122i \(-0.584316\pi\)
−0.261800 + 0.965122i \(0.584316\pi\)
\(930\) 0 0
\(931\) −2356.00 −0.0829375
\(932\) 0 0
\(933\) −54920.0 −1.92712
\(934\) 0 0
\(935\) −480.000 −0.0167890
\(936\) 0 0
\(937\) 3354.00 0.116937 0.0584687 0.998289i \(-0.481378\pi\)
0.0584687 + 0.998289i \(0.481378\pi\)
\(938\) 0 0
\(939\) 4220.00 0.146661
\(940\) 0 0
\(941\) 15434.0 0.534680 0.267340 0.963602i \(-0.413855\pi\)
0.267340 + 0.963602i \(0.413855\pi\)
\(942\) 0 0
\(943\) 2268.00 0.0783205
\(944\) 0 0
\(945\) 41400.0 1.42512
\(946\) 0 0
\(947\) 9306.00 0.319329 0.159664 0.987171i \(-0.448959\pi\)
0.159664 + 0.987171i \(0.448959\pi\)
\(948\) 0 0
\(949\) 5292.00 0.181017
\(950\) 0 0
\(951\) 61940.0 2.11203
\(952\) 0 0
\(953\) 12202.0 0.414755 0.207378 0.978261i \(-0.433507\pi\)
0.207378 + 0.978261i \(0.433507\pi\)
\(954\) 0 0
\(955\) −1860.00 −0.0630243
\(956\) 0 0
\(957\) 22720.0 0.767433
\(958\) 0 0
\(959\) 12060.0 0.406087
\(960\) 0 0
\(961\) 5553.00 0.186399
\(962\) 0 0
\(963\) 62050.0 2.07636
\(964\) 0 0
\(965\) 14690.0 0.490039
\(966\) 0 0
\(967\) 17478.0 0.581235 0.290618 0.956839i \(-0.406139\pi\)
0.290618 + 0.956839i \(0.406139\pi\)
\(968\) 0 0
\(969\) 7440.00 0.246653
\(970\) 0 0
\(971\) −10920.0 −0.360906 −0.180453 0.983584i \(-0.557756\pi\)
−0.180453 + 0.983584i \(0.557756\pi\)
\(972\) 0 0
\(973\) −10296.0 −0.339234
\(974\) 0 0
\(975\) −1500.00 −0.0492702
\(976\) 0 0
\(977\) 10834.0 0.354770 0.177385 0.984142i \(-0.443236\pi\)
0.177385 + 0.984142i \(0.443236\pi\)
\(978\) 0 0
\(979\) −13664.0 −0.446071
\(980\) 0 0
\(981\) −48618.0 −1.58232
\(982\) 0 0
\(983\) −36862.0 −1.19605 −0.598024 0.801478i \(-0.704047\pi\)
−0.598024 + 0.801478i \(0.704047\pi\)
\(984\) 0 0
\(985\) −11170.0 −0.361326
\(986\) 0 0
\(987\) 6840.00 0.220587
\(988\) 0 0
\(989\) −2772.00 −0.0891248
\(990\) 0 0
\(991\) 5380.00 0.172453 0.0862267 0.996276i \(-0.472519\pi\)
0.0862267 + 0.996276i \(0.472519\pi\)
\(992\) 0 0
\(993\) 76880.0 2.45691
\(994\) 0 0
\(995\) −15240.0 −0.485568
\(996\) 0 0
\(997\) −31266.0 −0.993184 −0.496592 0.867984i \(-0.665415\pi\)
−0.496592 + 0.867984i \(0.665415\pi\)
\(998\) 0 0
\(999\) 92920.0 2.94280
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 320.4.a.a.1.1 1
4.3 odd 2 320.4.a.n.1.1 1
5.4 even 2 1600.4.a.ca.1.1 1
8.3 odd 2 80.4.a.a.1.1 1
8.5 even 2 40.4.a.c.1.1 1
16.3 odd 4 1280.4.d.b.641.2 2
16.5 even 4 1280.4.d.o.641.2 2
16.11 odd 4 1280.4.d.b.641.1 2
16.13 even 4 1280.4.d.o.641.1 2
20.19 odd 2 1600.4.a.a.1.1 1
24.5 odd 2 360.4.a.i.1.1 1
24.11 even 2 720.4.a.ba.1.1 1
40.3 even 4 400.4.c.a.49.1 2
40.13 odd 4 200.4.c.a.49.2 2
40.19 odd 2 400.4.a.u.1.1 1
40.27 even 4 400.4.c.a.49.2 2
40.29 even 2 200.4.a.a.1.1 1
40.37 odd 4 200.4.c.a.49.1 2
56.13 odd 2 1960.4.a.a.1.1 1
120.29 odd 2 1800.4.a.bd.1.1 1
120.53 even 4 1800.4.f.n.649.2 2
120.77 even 4 1800.4.f.n.649.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
40.4.a.c.1.1 1 8.5 even 2
80.4.a.a.1.1 1 8.3 odd 2
200.4.a.a.1.1 1 40.29 even 2
200.4.c.a.49.1 2 40.37 odd 4
200.4.c.a.49.2 2 40.13 odd 4
320.4.a.a.1.1 1 1.1 even 1 trivial
320.4.a.n.1.1 1 4.3 odd 2
360.4.a.i.1.1 1 24.5 odd 2
400.4.a.u.1.1 1 40.19 odd 2
400.4.c.a.49.1 2 40.3 even 4
400.4.c.a.49.2 2 40.27 even 4
720.4.a.ba.1.1 1 24.11 even 2
1280.4.d.b.641.1 2 16.11 odd 4
1280.4.d.b.641.2 2 16.3 odd 4
1280.4.d.o.641.1 2 16.13 even 4
1280.4.d.o.641.2 2 16.5 even 4
1600.4.a.a.1.1 1 20.19 odd 2
1600.4.a.ca.1.1 1 5.4 even 2
1800.4.a.bd.1.1 1 120.29 odd 2
1800.4.f.n.649.1 2 120.77 even 4
1800.4.f.n.649.2 2 120.53 even 4
1960.4.a.a.1.1 1 56.13 odd 2