↑

Properties

Label 720.4.a.n.1.1
Level $720$
Weight $4$
Character 720.1
Self dual yes
Analytic conductor $42.481$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [720,4,Mod(1,720)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("720.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(720, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 720 = 2^{4} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 720.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,0,0,0,-5,0,24,0,0,0,52] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(42.4813752041\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 15)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 720.1

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-5.00000 q^{5} +24.0000 q^{7} +52.0000 q^{11} +22.0000 q^{13} +14.0000 q^{17} +20.0000 q^{19} -168.000 q^{23} +25.0000 q^{25} -230.000 q^{29} +288.000 q^{31} -120.000 q^{35} -34.0000 q^{37} -122.000 q^{41} +188.000 q^{43} +256.000 q^{47} +233.000 q^{49} +338.000 q^{53} -260.000 q^{55} +100.000 q^{59} +742.000 q^{61} -110.000 q^{65} +84.0000 q^{67} -328.000 q^{71} -38.0000 q^{73} +1248.00 q^{77} +240.000 q^{79} +1212.00 q^{83} -70.0000 q^{85} -330.000 q^{89} +528.000 q^{91} -100.000 q^{95} +866.000 q^{97} +O(q^{100})\)

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
\(3\) 0 0
\(4\) 0 0
\(5\) −5.00000 −0.447214
\(6\) 0 0
\(7\) 24.0000 1.29588 0.647939 − 0.761692i \(-0.275631\pi\)
0.647939 + 0.761692i \(0.275631\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 52.0000 1.42533 0.712663 − 0.701506i \(-0.247489\pi\)
0.712663 + 0.701506i \(0.247489\pi\)
\(12\) 0 0
\(13\) 22.0000 0.469362 0.234681 − 0.972072i \(-0.424595\pi\)
0.234681 + 0.972072i \(0.424595\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 14.0000 0.199735 0.0998676 − 0.995001i \(-0.468158\pi\)
0.0998676 + 0.995001i \(0.468158\pi\)
\(18\) 0 0
\(19\) 20.0000 0.241490 0.120745 − 0.992684i \(-0.461472\pi\)
0.120745 + 0.992684i \(0.461472\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −168.000 −1.52306 −0.761531 − 0.648129i \(-0.775552\pi\)
−0.761531 + 0.648129i \(0.775552\pi\)
\(24\) 0 0
\(25\) 25.0000 0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −230.000 −1.47276 −0.736378 − 0.676570i \(-0.763465\pi\)
−0.736378 + 0.676570i \(0.763465\pi\)
\(30\) 0 0
\(31\) 288.000 1.66859 0.834296 − 0.551317i \(-0.185875\pi\)
0.834296 + 0.551317i \(0.185875\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −120.000 −0.579534
\(36\) 0 0
\(37\) −34.0000 −0.151069 −0.0755347 − 0.997143i \(-0.524066\pi\)
−0.0755347 + 0.997143i \(0.524066\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −122.000 −0.464712 −0.232356 − 0.972631i \(-0.574643\pi\)
−0.232356 + 0.972631i \(0.574643\pi\)
\(42\) 0 0
\(43\) 188.000 0.666738 0.333369 − 0.942796i \(-0.391815\pi\)
0.333369 + 0.942796i \(0.391815\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 256.000 0.794499 0.397249 − 0.917711i \(-0.369965\pi\)
0.397249 + 0.917711i \(0.369965\pi\)
\(48\) 0 0
\(49\) 233.000 0.679300
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 338.000 0.875998 0.437999 − 0.898976i \(-0.355687\pi\)
0.437999 + 0.898976i \(0.355687\pi\)
\(54\) 0 0
\(55\) −260.000 −0.637425
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 100.000 0.220659 0.110330 − 0.993895i \(-0.464809\pi\)
0.110330 + 0.993895i \(0.464809\pi\)
\(60\) 0 0
\(61\) 742.000 1.55743 0.778716 − 0.627376i \(-0.215871\pi\)
0.778716 + 0.627376i \(0.215871\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −110.000 −0.209905
\(66\) 0 0
\(67\) 84.0000 0.153168 0.0765838 − 0.997063i \(-0.475599\pi\)
0.0765838 + 0.997063i \(0.475599\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −328.000 −0.548260 −0.274130 − 0.961693i \(-0.588390\pi\)
−0.274130 + 0.961693i \(0.588390\pi\)
\(72\) 0 0
\(73\) −38.0000 −0.0609255 −0.0304628 − 0.999536i \(-0.509698\pi\)
−0.0304628 + 0.999536i \(0.509698\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 1248.00 1.84705
\(78\) 0 0
\(79\) 240.000 0.341799 0.170899 − 0.985288i \(-0.445333\pi\)
0.170899 + 0.985288i \(0.445333\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 1212.00 1.60282 0.801411 − 0.598114i \(-0.204083\pi\)
0.801411 + 0.598114i \(0.204083\pi\)
\(84\) 0 0
\(85\) −70.0000 −0.0893243
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −330.000 −0.393033 −0.196516 − 0.980501i \(-0.562963\pi\)
−0.196516 + 0.980501i \(0.562963\pi\)
\(90\) 0 0
\(91\) 528.000 0.608236
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −100.000 −0.107998
\(96\) 0 0
\(97\) 866.000 0.906484 0.453242 − 0.891387i \(-0.350267\pi\)
0.453242 + 0.891387i \(0.350267\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 1218.00 1.19996 0.599978 − 0.800017i \(-0.295176\pi\)
0.599978 + 0.800017i \(0.295176\pi\)
\(102\) 0 0
\(103\) 88.0000 0.0841835 0.0420917 − 0.999114i \(-0.486598\pi\)
0.0420917 + 0.999114i \(0.486598\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 36.0000 0.0325257 0.0162629 − 0.999868i \(-0.494823\pi\)
0.0162629 + 0.999868i \(0.494823\pi\)
\(108\) 0 0
\(109\) −970.000 −0.852378 −0.426189 − 0.904634i \(-0.640144\pi\)
−0.426189 + 0.904634i \(0.640144\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −1042.00 −0.867461 −0.433731 − 0.901043i \(-0.642803\pi\)
−0.433731 + 0.901043i \(0.642803\pi\)
\(114\) 0 0
\(115\) 840.000 0.681134
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 336.000 0.258833
\(120\) 0 0
\(121\) 1373.00 1.03156
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −125.000 −0.0894427
\(126\) 0 0
\(127\) −1936.00 −1.35269 −0.676347 − 0.736583i \(-0.736438\pi\)
−0.676347 + 0.736583i \(0.736438\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 732.000 0.488207 0.244104 − 0.969749i \(-0.421506\pi\)
0.244104 + 0.969749i \(0.421506\pi\)
\(132\) 0 0
\(133\) 480.000 0.312942
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 2214.00 1.38069 0.690346 − 0.723479i \(-0.257458\pi\)
0.690346 + 0.723479i \(0.257458\pi\)
\(138\) 0 0
\(139\) −20.0000 −0.0122042 −0.00610208 − 0.999981i \(-0.501942\pi\)
−0.00610208 + 0.999981i \(0.501942\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 1144.00 0.668994
\(144\) 0 0
\(145\) 1150.00 0.658637
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 1330.00 0.731261 0.365630 − 0.930760i \(-0.380853\pi\)
0.365630 + 0.930760i \(0.380853\pi\)
\(150\) 0 0
\(151\) 1208.00 0.651031 0.325515 − 0.945537i \(-0.394462\pi\)
0.325515 + 0.945537i \(0.394462\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −1440.00 −0.746217
\(156\) 0 0
\(157\) −3514.00 −1.78629 −0.893146 − 0.449768i \(-0.851507\pi\)
−0.893146 + 0.449768i \(0.851507\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −4032.00 −1.97370
\(162\) 0 0
\(163\) 2068.00 0.993732 0.496866 − 0.867827i \(-0.334484\pi\)
0.496866 + 0.867827i \(0.334484\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −24.0000 −0.0111208 −0.00556041 − 0.999985i \(-0.501770\pi\)
−0.00556041 + 0.999985i \(0.501770\pi\)
\(168\) 0 0
\(169\) −1713.00 −0.779700
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 618.000 0.271593 0.135797 − 0.990737i \(-0.456641\pi\)
0.135797 + 0.990737i \(0.456641\pi\)
\(174\) 0 0
\(175\) 600.000 0.259176
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 3340.00 1.39466 0.697328 − 0.716752i \(-0.254372\pi\)
0.697328 + 0.716752i \(0.254372\pi\)
\(180\) 0 0
\(181\) −178.000 −0.0730974 −0.0365487 − 0.999332i \(-0.511636\pi\)
−0.0365487 + 0.999332i \(0.511636\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 170.000 0.0675603
\(186\) 0 0
\(187\) 728.000 0.284688
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −1888.00 −0.715240 −0.357620 − 0.933867i \(-0.616412\pi\)
−0.357620 + 0.933867i \(0.616412\pi\)
\(192\) 0 0
\(193\) 1922.00 0.716832 0.358416 − 0.933562i \(-0.383317\pi\)
0.358416 + 0.933562i \(0.383317\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −2526.00 −0.913554 −0.456777 − 0.889581i \(-0.650996\pi\)
−0.456777 + 0.889581i \(0.650996\pi\)
\(198\) 0 0
\(199\) 1160.00 0.413217 0.206609 − 0.978424i \(-0.433757\pi\)
0.206609 + 0.978424i \(0.433757\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −5520.00 −1.90851
\(204\) 0 0
\(205\) 610.000 0.207826
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 1040.00 0.344202
\(210\) 0 0
\(211\) 4468.00 1.45777 0.728886 − 0.684635i \(-0.240039\pi\)
0.728886 + 0.684635i \(0.240039\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −940.000 −0.298174
\(216\) 0 0
\(217\) 6912.00 2.16229
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 308.000 0.0937481
\(222\) 0 0
\(223\) −6032.00 −1.81136 −0.905678 − 0.423965i \(-0.860638\pi\)
−0.905678 + 0.423965i \(0.860638\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 2636.00 0.770738 0.385369 − 0.922763i \(-0.374074\pi\)
0.385369 + 0.922763i \(0.374074\pi\)
\(228\) 0 0
\(229\) 4830.00 1.39378 0.696889 − 0.717179i \(-0.254567\pi\)
0.696889 + 0.717179i \(0.254567\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −2682.00 −0.754093 −0.377046 − 0.926194i \(-0.623060\pi\)
−0.377046 + 0.926194i \(0.623060\pi\)
\(234\) 0 0
\(235\) −1280.00 −0.355311
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 2320.00 0.627901 0.313950 − 0.949439i \(-0.398347\pi\)
0.313950 + 0.949439i \(0.398347\pi\)
\(240\) 0 0
\(241\) 2002.00 0.535104 0.267552 − 0.963543i \(-0.413785\pi\)
0.267552 + 0.963543i \(0.413785\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −1165.00 −0.303792
\(246\) 0 0
\(247\) 440.000 0.113346
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 132.000 0.0331943 0.0165971 − 0.999862i \(-0.494717\pi\)
0.0165971 + 0.999862i \(0.494717\pi\)
\(252\) 0 0
\(253\) −8736.00 −2.17086
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 7614.00 1.84805 0.924024 − 0.382335i \(-0.124880\pi\)
0.924024 + 0.382335i \(0.124880\pi\)
\(258\) 0 0
\(259\) −816.000 −0.195767
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −4888.00 −1.14603 −0.573017 − 0.819543i \(-0.694227\pi\)
−0.573017 + 0.819543i \(0.694227\pi\)
\(264\) 0 0
\(265\) −1690.00 −0.391758
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −1270.00 −0.287856 −0.143928 − 0.989588i \(-0.545973\pi\)
−0.143928 + 0.989588i \(0.545973\pi\)
\(270\) 0 0
\(271\) −1072.00 −0.240293 −0.120146 − 0.992756i \(-0.538336\pi\)
−0.120146 + 0.992756i \(0.538336\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 1300.00 0.285065
\(276\) 0 0
\(277\) −5394.00 −1.17001 −0.585007 − 0.811028i \(-0.698908\pi\)
−0.585007 + 0.811028i \(0.698908\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −2442.00 −0.518425 −0.259213 − 0.965820i \(-0.583463\pi\)
−0.259213 + 0.965820i \(0.583463\pi\)
\(282\) 0 0
\(283\) −2772.00 −0.582255 −0.291128 − 0.956684i \(-0.594030\pi\)
−0.291128 + 0.956684i \(0.594030\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −2928.00 −0.602210
\(288\) 0 0
\(289\) −4717.00 −0.960106
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −4542.00 −0.905619 −0.452810 − 0.891607i \(-0.649578\pi\)
−0.452810 + 0.891607i \(0.649578\pi\)
\(294\) 0 0
\(295\) −500.000 −0.0986818
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −3696.00 −0.714867
\(300\) 0 0
\(301\) 4512.00 0.864011
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −3710.00 −0.696505
\(306\) 0 0
\(307\) −5116.00 −0.951093 −0.475546 − 0.879691i \(-0.657750\pi\)
−0.475546 + 0.879691i \(0.657750\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −2808.00 −0.511984 −0.255992 − 0.966679i \(-0.582402\pi\)
−0.255992 + 0.966679i \(0.582402\pi\)
\(312\) 0 0
\(313\) −7318.00 −1.32153 −0.660763 − 0.750594i \(-0.729767\pi\)
−0.660763 + 0.750594i \(0.729767\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −2246.00 −0.397943 −0.198971 − 0.980005i \(-0.563760\pi\)
−0.198971 + 0.980005i \(0.563760\pi\)
\(318\) 0 0
\(319\) −11960.0 −2.09916
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 280.000 0.0482341
\(324\) 0 0
\(325\) 550.000 0.0938723
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 6144.00 1.02957
\(330\) 0 0
\(331\) −1332.00 −0.221188 −0.110594 − 0.993866i \(-0.535275\pi\)
−0.110594 + 0.993866i \(0.535275\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −420.000 −0.0684987
\(336\) 0 0
\(337\) −11534.0 −1.86438 −0.932191 − 0.361966i \(-0.882106\pi\)
−0.932191 + 0.361966i \(0.882106\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 14976.0 2.37829
\(342\) 0 0
\(343\) −2640.00 −0.415588
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 11956.0 1.84966 0.924830 − 0.380382i \(-0.124207\pi\)
0.924830 + 0.380382i \(0.124207\pi\)
\(348\) 0 0
\(349\) 4870.00 0.746949 0.373474 − 0.927640i \(-0.378166\pi\)
0.373474 + 0.927640i \(0.378166\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −10722.0 −1.61664 −0.808321 − 0.588742i \(-0.799623\pi\)
−0.808321 + 0.588742i \(0.799623\pi\)
\(354\) 0 0
\(355\) 1640.00 0.245189
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 120.000 0.0176417 0.00882083 − 0.999961i \(-0.497192\pi\)
0.00882083 + 0.999961i \(0.497192\pi\)
\(360\) 0 0
\(361\) −6459.00 −0.941682
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 190.000 0.0272467
\(366\) 0 0
\(367\) −3936.00 −0.559830 −0.279915 − 0.960025i \(-0.590306\pi\)
−0.279915 + 0.960025i \(0.590306\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 8112.00 1.13519
\(372\) 0 0
\(373\) 3022.00 0.419499 0.209750 − 0.977755i \(-0.432735\pi\)
0.209750 + 0.977755i \(0.432735\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −5060.00 −0.691255
\(378\) 0 0
\(379\) 13340.0 1.80799 0.903997 − 0.427539i \(-0.140619\pi\)
0.903997 + 0.427539i \(0.140619\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −1008.00 −0.134481 −0.0672407 − 0.997737i \(-0.521420\pi\)
−0.0672407 + 0.997737i \(0.521420\pi\)
\(384\) 0 0
\(385\) −6240.00 −0.826026
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −9630.00 −1.25517 −0.627584 − 0.778549i \(-0.715956\pi\)
−0.627584 + 0.778549i \(0.715956\pi\)
\(390\) 0 0
\(391\) −2352.00 −0.304209
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −1200.00 −0.152857
\(396\) 0 0
\(397\) 7126.00 0.900866 0.450433 − 0.892810i \(-0.351270\pi\)
0.450433 + 0.892810i \(0.351270\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 8718.00 1.08568 0.542838 − 0.839837i \(-0.317350\pi\)
0.542838 + 0.839837i \(0.317350\pi\)
\(402\) 0 0
\(403\) 6336.00 0.783173
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −1768.00 −0.215323
\(408\) 0 0
\(409\) −10870.0 −1.31415 −0.657074 − 0.753826i \(-0.728206\pi\)
−0.657074 + 0.753826i \(0.728206\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 2400.00 0.285947
\(414\) 0 0
\(415\) −6060.00 −0.716804
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −9700.00 −1.13097 −0.565484 − 0.824759i \(-0.691311\pi\)
−0.565484 + 0.824759i \(0.691311\pi\)
\(420\) 0 0
\(421\) 862.000 0.0997893 0.0498947 − 0.998754i \(-0.484111\pi\)
0.0498947 + 0.998754i \(0.484111\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 350.000 0.0399470
\(426\) 0 0
\(427\) 17808.0 2.01824
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 15792.0 1.76490 0.882452 − 0.470402i \(-0.155891\pi\)
0.882452 + 0.470402i \(0.155891\pi\)
\(432\) 0 0
\(433\) 11602.0 1.28766 0.643830 − 0.765169i \(-0.277345\pi\)
0.643830 + 0.765169i \(0.277345\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −3360.00 −0.367805
\(438\) 0 0
\(439\) 440.000 0.0478361 0.0239181 − 0.999714i \(-0.492386\pi\)
0.0239181 + 0.999714i \(0.492386\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −10188.0 −1.09266 −0.546328 − 0.837571i \(-0.683975\pi\)
−0.546328 + 0.837571i \(0.683975\pi\)
\(444\) 0 0
\(445\) 1650.00 0.175770
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 13310.0 1.39897 0.699485 − 0.714647i \(-0.253413\pi\)
0.699485 + 0.714647i \(0.253413\pi\)
\(450\) 0 0
\(451\) −6344.00 −0.662367
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −2640.00 −0.272011
\(456\) 0 0
\(457\) 3226.00 0.330210 0.165105 − 0.986276i \(-0.447204\pi\)
0.165105 + 0.986276i \(0.447204\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −6582.00 −0.664977 −0.332488 − 0.943107i \(-0.607888\pi\)
−0.332488 + 0.943107i \(0.607888\pi\)
\(462\) 0 0
\(463\) −15072.0 −1.51286 −0.756431 − 0.654073i \(-0.773059\pi\)
−0.756431 + 0.654073i \(0.773059\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 476.000 0.0471663 0.0235831 − 0.999722i \(-0.492493\pi\)
0.0235831 + 0.999722i \(0.492493\pi\)
\(468\) 0 0
\(469\) 2016.00 0.198487
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 9776.00 0.950319
\(474\) 0 0
\(475\) 500.000 0.0482980
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −19680.0 −1.87725 −0.938624 − 0.344941i \(-0.887899\pi\)
−0.938624 + 0.344941i \(0.887899\pi\)
\(480\) 0 0
\(481\) −748.000 −0.0709062
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −4330.00 −0.405392
\(486\) 0 0
\(487\) 5944.00 0.553077 0.276538 − 0.961003i \(-0.410813\pi\)
0.276538 + 0.961003i \(0.410813\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 10772.0 0.990089 0.495044 − 0.868868i \(-0.335152\pi\)
0.495044 + 0.868868i \(0.335152\pi\)
\(492\) 0 0
\(493\) −3220.00 −0.294161
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −7872.00 −0.710478
\(498\) 0 0
\(499\) −8140.00 −0.730253 −0.365127 − 0.930958i \(-0.618974\pi\)
−0.365127 + 0.930958i \(0.618974\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −13768.0 −1.22045 −0.610223 − 0.792229i \(-0.708920\pi\)
−0.610223 + 0.792229i \(0.708920\pi\)
\(504\) 0 0
\(505\) −6090.00 −0.536637
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −22150.0 −1.92884 −0.964422 − 0.264368i \(-0.914837\pi\)
−0.964422 + 0.264368i \(0.914837\pi\)
\(510\) 0 0
\(511\) −912.000 −0.0789521
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −440.000 −0.0376480
\(516\) 0 0
\(517\) 13312.0 1.13242
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −1562.00 −0.131348 −0.0656741 − 0.997841i \(-0.520920\pi\)
−0.0656741 + 0.997841i \(0.520920\pi\)
\(522\) 0 0
\(523\) 668.000 0.0558501 0.0279250 − 0.999610i \(-0.491110\pi\)
0.0279250 + 0.999610i \(0.491110\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 4032.00 0.333276
\(528\) 0 0
\(529\) 16057.0 1.31972
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −2684.00 −0.218118
\(534\) 0 0
\(535\) −180.000 −0.0145459
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 12116.0 0.968225
\(540\) 0 0
\(541\) −6138.00 −0.487788 −0.243894 − 0.969802i \(-0.578425\pi\)
−0.243894 + 0.969802i \(0.578425\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 4850.00 0.381195
\(546\) 0 0
\(547\) 10484.0 0.819494 0.409747 − 0.912199i \(-0.365617\pi\)
0.409747 + 0.912199i \(0.365617\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −4600.00 −0.355656
\(552\) 0 0
\(553\) 5760.00 0.442930
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −3606.00 −0.274311 −0.137155 − 0.990550i \(-0.543796\pi\)
−0.137155 + 0.990550i \(0.543796\pi\)
\(558\) 0 0
\(559\) 4136.00 0.312941
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 12252.0 0.917159 0.458579 − 0.888654i \(-0.348359\pi\)
0.458579 + 0.888654i \(0.348359\pi\)
\(564\) 0 0
\(565\) 5210.00 0.387940
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 14550.0 1.07200 0.536000 − 0.844218i \(-0.319935\pi\)
0.536000 + 0.844218i \(0.319935\pi\)
\(570\) 0 0
\(571\) 25468.0 1.86655 0.933277 − 0.359157i \(-0.116936\pi\)
0.933277 + 0.359157i \(0.116936\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −4200.00 −0.304612
\(576\) 0 0
\(577\) 12866.0 0.928282 0.464141 − 0.885761i \(-0.346363\pi\)
0.464141 + 0.885761i \(0.346363\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 29088.0 2.07706
\(582\) 0 0
\(583\) 17576.0 1.24858
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −14844.0 −1.04374 −0.521872 − 0.853024i \(-0.674766\pi\)
−0.521872 + 0.853024i \(0.674766\pi\)
\(588\) 0 0
\(589\) 5760.00 0.402948
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −20402.0 −1.41283 −0.706416 − 0.707797i \(-0.749689\pi\)
−0.706416 + 0.707797i \(0.749689\pi\)
\(594\) 0 0
\(595\) −1680.00 −0.115753
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 10760.0 0.733959 0.366980 − 0.930229i \(-0.380392\pi\)
0.366980 + 0.930229i \(0.380392\pi\)
\(600\) 0 0
\(601\) 14282.0 0.969343 0.484671 − 0.874696i \(-0.338939\pi\)
0.484671 + 0.874696i \(0.338939\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −6865.00 −0.461326
\(606\) 0 0
\(607\) −11056.0 −0.739290 −0.369645 − 0.929173i \(-0.620521\pi\)
−0.369645 + 0.929173i \(0.620521\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 5632.00 0.372907
\(612\) 0 0
\(613\) −16418.0 −1.08176 −0.540878 − 0.841101i \(-0.681908\pi\)
−0.540878 + 0.841101i \(0.681908\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 10374.0 0.676891 0.338445 − 0.940986i \(-0.390099\pi\)
0.338445 + 0.940986i \(0.390099\pi\)
\(618\) 0 0
\(619\) 5260.00 0.341546 0.170773 − 0.985310i \(-0.445373\pi\)
0.170773 + 0.985310i \(0.445373\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −7920.00 −0.509323
\(624\) 0 0
\(625\) 625.000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −476.000 −0.0301739
\(630\) 0 0
\(631\) −21352.0 −1.34708 −0.673542 − 0.739149i \(-0.735228\pi\)
−0.673542 + 0.739149i \(0.735228\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 9680.00 0.604943
\(636\) 0 0
\(637\) 5126.00 0.318838
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 29118.0 1.79422 0.897108 − 0.441812i \(-0.145664\pi\)
0.897108 + 0.441812i \(0.145664\pi\)
\(642\) 0 0
\(643\) −5772.00 −0.354005 −0.177003 − 0.984210i \(-0.556640\pi\)
−0.177003 + 0.984210i \(0.556640\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −14264.0 −0.866732 −0.433366 − 0.901218i \(-0.642674\pi\)
−0.433366 + 0.901218i \(0.642674\pi\)
\(648\) 0 0
\(649\) 5200.00 0.314511
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −6902.00 −0.413623 −0.206812 − 0.978381i \(-0.566309\pi\)
−0.206812 + 0.978381i \(0.566309\pi\)
\(654\) 0 0
\(655\) −3660.00 −0.218333
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 20140.0 1.19051 0.595253 − 0.803539i \(-0.297052\pi\)
0.595253 + 0.803539i \(0.297052\pi\)
\(660\) 0 0
\(661\) −3218.00 −0.189358 −0.0946790 − 0.995508i \(-0.530182\pi\)
−0.0946790 + 0.995508i \(0.530182\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −2400.00 −0.139952
\(666\) 0 0
\(667\) 38640.0 2.24310
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 38584.0 2.21985
\(672\) 0 0
\(673\) −7518.00 −0.430606 −0.215303 − 0.976547i \(-0.569074\pi\)
−0.215303 + 0.976547i \(0.569074\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 18114.0 1.02833 0.514164 − 0.857692i \(-0.328102\pi\)
0.514164 + 0.857692i \(0.328102\pi\)
\(678\) 0 0
\(679\) 20784.0 1.17469
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −23868.0 −1.33716 −0.668582 − 0.743638i \(-0.733099\pi\)
−0.668582 + 0.743638i \(0.733099\pi\)
\(684\) 0 0
\(685\) −11070.0 −0.617464
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 7436.00 0.411160
\(690\) 0 0
\(691\) −172.000 −0.00946916 −0.00473458 − 0.999989i \(-0.501507\pi\)
−0.00473458 + 0.999989i \(0.501507\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 100.000 0.00545787
\(696\) 0 0
\(697\) −1708.00 −0.0928194
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 22138.0 1.19278 0.596391 − 0.802694i \(-0.296601\pi\)
0.596391 + 0.802694i \(0.296601\pi\)
\(702\) 0 0
\(703\) −680.000 −0.0364818
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 29232.0 1.55500
\(708\) 0 0
\(709\) 3070.00 0.162618 0.0813091 − 0.996689i \(-0.474090\pi\)
0.0813091 + 0.996689i \(0.474090\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −48384.0 −2.54137
\(714\) 0 0
\(715\) −5720.00 −0.299183
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 15600.0 0.809154 0.404577 − 0.914504i \(-0.367419\pi\)
0.404577 + 0.914504i \(0.367419\pi\)
\(720\) 0 0
\(721\) 2112.00 0.109092
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −5750.00 −0.294551
\(726\) 0 0
\(727\) −20696.0 −1.05581 −0.527904 − 0.849304i \(-0.677022\pi\)
−0.527904 + 0.849304i \(0.677022\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 2632.00 0.133171
\(732\) 0 0
\(733\) −30778.0 −1.55090 −0.775451 − 0.631408i \(-0.782478\pi\)
−0.775451 + 0.631408i \(0.782478\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 4368.00 0.218314
\(738\) 0 0
\(739\) −11740.0 −0.584388 −0.292194 − 0.956359i \(-0.594385\pi\)
−0.292194 + 0.956359i \(0.594385\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 2632.00 0.129958 0.0649789 − 0.997887i \(-0.479302\pi\)
0.0649789 + 0.997887i \(0.479302\pi\)
\(744\) 0 0
\(745\) −6650.00 −0.327030
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 864.000 0.0421494
\(750\) 0 0
\(751\) 20528.0 0.997440 0.498720 − 0.866763i \(-0.333804\pi\)
0.498720 + 0.866763i \(0.333804\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −6040.00 −0.291150
\(756\) 0 0
\(757\) 21646.0 1.03928 0.519642 − 0.854384i \(-0.326066\pi\)
0.519642 + 0.854384i \(0.326066\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −18282.0 −0.870857 −0.435428 − 0.900223i \(-0.643403\pi\)
−0.435428 + 0.900223i \(0.643403\pi\)
\(762\) 0 0
\(763\) −23280.0 −1.10458
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 2200.00 0.103569
\(768\) 0 0
\(769\) −24190.0 −1.13435 −0.567174 − 0.823598i \(-0.691963\pi\)
−0.567174 + 0.823598i \(0.691963\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 25698.0 1.19572 0.597861 − 0.801600i \(-0.296018\pi\)
0.597861 + 0.801600i \(0.296018\pi\)
\(774\) 0 0
\(775\) 7200.00 0.333718
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −2440.00 −0.112223
\(780\) 0 0
\(781\) −17056.0 −0.781449
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 17570.0 0.798854
\(786\) 0 0
\(787\) −33436.0 −1.51444 −0.757220 − 0.653160i \(-0.773443\pi\)
−0.757220 + 0.653160i \(0.773443\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −25008.0 −1.12412
\(792\) 0 0
\(793\) 16324.0 0.730999
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 37594.0 1.67083 0.835413 − 0.549623i \(-0.185229\pi\)
0.835413 + 0.549623i \(0.185229\pi\)
\(798\) 0 0
\(799\) 3584.00 0.158689
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −1976.00 −0.0868388
\(804\) 0 0
\(805\) 20160.0 0.882667
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −4730.00 −0.205560 −0.102780 − 0.994704i \(-0.532774\pi\)
−0.102780 + 0.994704i \(0.532774\pi\)
\(810\) 0 0
\(811\) 8748.00 0.378772 0.189386 − 0.981903i \(-0.439350\pi\)
0.189386 + 0.981903i \(0.439350\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −10340.0 −0.444410
\(816\) 0 0
\(817\) 3760.00 0.161011
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −44142.0 −1.87645 −0.938226 − 0.346024i \(-0.887532\pi\)
−0.938226 + 0.346024i \(0.887532\pi\)
\(822\) 0 0
\(823\) −3992.00 −0.169079 −0.0845397 − 0.996420i \(-0.526942\pi\)
−0.0845397 + 0.996420i \(0.526942\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −14444.0 −0.607336 −0.303668 − 0.952778i \(-0.598211\pi\)
−0.303668 + 0.952778i \(0.598211\pi\)
\(828\) 0 0
\(829\) 42150.0 1.76590 0.882949 − 0.469468i \(-0.155554\pi\)
0.882949 + 0.469468i \(0.155554\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 3262.00 0.135680
\(834\) 0 0
\(835\) 120.000 0.00497338
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 13400.0 0.551394 0.275697 − 0.961245i \(-0.411091\pi\)
0.275697 + 0.961245i \(0.411091\pi\)
\(840\) 0 0
\(841\) 28511.0 1.16901
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 8565.00 0.348692
\(846\) 0 0
\(847\) 32952.0 1.33677
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 5712.00 0.230088
\(852\) 0 0
\(853\) −8658.00 −0.347531 −0.173766 − 0.984787i \(-0.555594\pi\)
−0.173766 + 0.984787i \(0.555594\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −42826.0 −1.70701 −0.853505 − 0.521084i \(-0.825528\pi\)
−0.853505 + 0.521084i \(0.825528\pi\)
\(858\) 0 0
\(859\) 35900.0 1.42595 0.712976 − 0.701189i \(-0.247347\pi\)
0.712976 + 0.701189i \(0.247347\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −3088.00 −0.121804 −0.0609019 − 0.998144i \(-0.519398\pi\)
−0.0609019 + 0.998144i \(0.519398\pi\)
\(864\) 0 0
\(865\) −3090.00 −0.121460
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 12480.0 0.487175
\(870\) 0 0
\(871\) 1848.00 0.0718910
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −3000.00 −0.115907
\(876\) 0 0
\(877\) −35274.0 −1.35817 −0.679087 − 0.734058i \(-0.737624\pi\)
−0.679087 + 0.734058i \(0.737624\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −25042.0 −0.957646 −0.478823 − 0.877911i \(-0.658936\pi\)
−0.478823 + 0.877911i \(0.658936\pi\)
\(882\) 0 0
\(883\) −12572.0 −0.479141 −0.239570 − 0.970879i \(-0.577007\pi\)
−0.239570 + 0.970879i \(0.577007\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −21864.0 −0.827645 −0.413823 − 0.910358i \(-0.635807\pi\)
−0.413823 + 0.910358i \(0.635807\pi\)
\(888\) 0 0
\(889\) −46464.0 −1.75293
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 5120.00 0.191864
\(894\) 0 0
\(895\) −16700.0 −0.623709
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −66240.0 −2.45743
\(900\) 0 0
\(901\) 4732.00 0.174968
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 890.000 0.0326902
\(906\) 0 0
\(907\) −31236.0 −1.14352 −0.571761 − 0.820420i \(-0.693740\pi\)
−0.571761 + 0.820420i \(0.693740\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 8272.00 0.300838 0.150419 − 0.988622i \(-0.451938\pi\)
0.150419 + 0.988622i \(0.451938\pi\)
\(912\) 0 0
\(913\) 63024.0 2.28455
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 17568.0 0.632657
\(918\) 0 0
\(919\) −20200.0 −0.725067 −0.362533 − 0.931971i \(-0.618088\pi\)
−0.362533 + 0.931971i \(0.618088\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −7216.00 −0.257332
\(924\) 0 0
\(925\) −850.000 −0.0302139
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −31010.0 −1.09516 −0.547581 − 0.836753i \(-0.684451\pi\)
−0.547581 + 0.836753i \(0.684451\pi\)
\(930\) 0 0
\(931\) 4660.00 0.164044
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −3640.00 −0.127316
\(936\) 0 0
\(937\) −39174.0 −1.36580 −0.682902 − 0.730510i \(-0.739283\pi\)
−0.682902 + 0.730510i \(0.739283\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 4138.00 0.143353 0.0716764 − 0.997428i \(-0.477165\pi\)
0.0716764 + 0.997428i \(0.477165\pi\)
\(942\) 0 0
\(943\) 20496.0 0.707785
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 23676.0 0.812425 0.406213 − 0.913779i \(-0.366849\pi\)
0.406213 + 0.913779i \(0.366849\pi\)
\(948\) 0 0
\(949\) −836.000 −0.0285961
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −18922.0 −0.643173 −0.321586 − 0.946880i \(-0.604216\pi\)
−0.321586 + 0.946880i \(0.604216\pi\)
\(954\) 0 0
\(955\) 9440.00 0.319865
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 53136.0 1.78921
\(960\) 0 0
\(961\) 53153.0 1.78420
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −9610.00 −0.320577
\(966\) 0 0
\(967\) −39656.0 −1.31877 −0.659385 − 0.751805i \(-0.729183\pi\)
−0.659385 + 0.751805i \(0.729183\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −33228.0 −1.09818 −0.549092 − 0.835762i \(-0.685026\pi\)
−0.549092 + 0.835762i \(0.685026\pi\)
\(972\) 0 0
\(973\) −480.000 −0.0158151
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 974.000 0.0318946 0.0159473 − 0.999873i \(-0.494924\pi\)
0.0159473 + 0.999873i \(0.494924\pi\)
\(978\) 0 0
\(979\) −17160.0 −0.560200
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −13608.0 −0.441534 −0.220767 − 0.975327i \(-0.570856\pi\)
−0.220767 + 0.975327i \(0.570856\pi\)
\(984\) 0 0
\(985\) 12630.0 0.408554
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −31584.0 −1.01548
\(990\) 0 0
\(991\) −13472.0 −0.431839 −0.215919 − 0.976411i \(-0.569275\pi\)
−0.215919 + 0.976411i \(0.569275\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −5800.00 −0.184796
\(996\) 0 0
\(997\) −3234.00 −0.102730 −0.0513650 − 0.998680i \(-0.516357\pi\)
−0.0513650 + 0.998680i \(0.516357\pi\)
\(998\) 0 0
\(999\) 0 0
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 720.4.a.n.1.1 1
3.2 odd 2 240.4.a.e.1.1 1
4.3 odd 2 45.4.a.c.1.1 1
12.11 even 2 15.4.a.a.1.1 ✓ 1
15.2 even 4 1200.4.f.b.49.2 2
15.8 even 4 1200.4.f.b.49.1 2
15.14 odd 2 1200.4.a.t.1.1 1
20.3 even 4 225.4.b.e.199.2 2
20.7 even 4 225.4.b.e.199.1 2
20.19 odd 2 225.4.a.f.1.1 1
24.5 odd 2 960.4.a.ba.1.1 1
24.11 even 2 960.4.a.b.1.1 1
28.27 even 2 2205.4.a.l.1.1 1
36.7 odd 6 405.4.e.i.271.1 2
36.11 even 6 405.4.e.g.271.1 2
36.23 even 6 405.4.e.g.136.1 2
36.31 odd 6 405.4.e.i.136.1 2
60.23 odd 4 75.4.b.b.49.1 2
60.47 odd 4 75.4.b.b.49.2 2
60.59 even 2 75.4.a.b.1.1 1
84.83 odd 2 735.4.a.e.1.1 1
132.131 odd 2 1815.4.a.e.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
15.4.a.a.1.1 ✓ 1 12.11 even 2
45.4.a.c.1.1 1 4.3 odd 2
75.4.a.b.1.1 1 60.59 even 2
75.4.b.b.49.1 2 60.23 odd 4
75.4.b.b.49.2 2 60.47 odd 4
225.4.a.f.1.1 1 20.19 odd 2
225.4.b.e.199.1 2 20.7 even 4
225.4.b.e.199.2 2 20.3 even 4
240.4.a.e.1.1 1 3.2 odd 2
405.4.e.g.136.1 2 36.23 even 6
405.4.e.g.271.1 2 36.11 even 6
405.4.e.i.136.1 2 36.31 odd 6
405.4.e.i.271.1 2 36.7 odd 6
720.4.a.n.1.1 1 1.1 even 1 trivial
735.4.a.e.1.1 1 84.83 odd 2
960.4.a.b.1.1 1 24.11 even 2
960.4.a.ba.1.1 1 24.5 odd 2
1200.4.a.t.1.1 1 15.14 odd 2
1200.4.f.b.49.1 2 15.8 even 4
1200.4.f.b.49.2 2 15.2 even 4
1815.4.a.e.1.1 1 132.131 odd 2
2205.4.a.l.1.1 1 28.27 even 2