Properties

Label 756.4.f.e.377.3
Level $756$
Weight $4$
Character 756.377
Analytic conductor $44.605$
Analytic rank $0$
Dimension $16$
Inner twists $4$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [756,4,Mod(377,756)] 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("756.377"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(756, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 1])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 756 = 2^{2} \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 756.f (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [16,0,0,0,0,0,14] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(44.6054439643\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 4 x^{15} - 36 x^{14} - 780 x^{13} + 17052 x^{12} - 100012 x^{11} + 336938 x^{10} + \cdots + 1976222698452 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{8}\cdot 3^{26} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 377.3
Root \(5.48402 - 5.91781i\) of defining polynomial
Character \(\chi\) \(=\) 756.377
Dual form 756.4.f.e.377.4

$q$-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)
 
\(f(q)\) \(=\) \(q-13.6625 q^{5} +(-3.07615 - 18.2630i) q^{7} -10.2320i q^{11} -15.7462i q^{13} -52.8157 q^{17} -61.7090i q^{19} +46.2236i q^{23} +61.6631 q^{25} -26.5770i q^{29} -172.182i q^{31} +(42.0277 + 249.518i) q^{35} -66.8154 q^{37} -7.91347 q^{41} +264.488 q^{43} +136.842 q^{47} +(-324.075 + 112.359i) q^{49} +596.778i q^{53} +139.795i q^{55} +103.438 q^{59} +588.967i q^{61} +215.132i q^{65} -309.072 q^{67} +1030.43i q^{71} +478.042i q^{73} +(-186.867 + 31.4752i) q^{77} +54.8892 q^{79} -287.874 q^{83} +721.593 q^{85} -1006.71 q^{89} +(-287.573 + 48.4377i) q^{91} +843.097i q^{95} -59.9202i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 14 q^{7} + 388 q^{25} - 344 q^{37} - 404 q^{43} - 14 q^{49} + 1772 q^{67} - 1456 q^{79} - 4704 q^{85} + 2436 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/756\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(325\) \(379\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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
\(5\) −13.6625 −1.22201 −0.611004 0.791627i \(-0.709234\pi\)
−0.611004 + 0.791627i \(0.709234\pi\)
\(6\) 0 0
\(7\) −3.07615 18.2630i −0.166096 0.986110i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 10.2320i 0.280461i −0.990119 0.140230i \(-0.955216\pi\)
0.990119 0.140230i \(-0.0447843\pi\)
\(12\) 0 0
\(13\) 15.7462i 0.335940i −0.985792 0.167970i \(-0.946279\pi\)
0.985792 0.167970i \(-0.0537212\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −52.8157 −0.753511 −0.376756 0.926313i \(-0.622960\pi\)
−0.376756 + 0.926313i \(0.622960\pi\)
\(18\) 0 0
\(19\) 61.7090i 0.745106i −0.928011 0.372553i \(-0.878482\pi\)
0.928011 0.372553i \(-0.121518\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 46.2236i 0.419056i 0.977803 + 0.209528i \(0.0671928\pi\)
−0.977803 + 0.209528i \(0.932807\pi\)
\(24\) 0 0
\(25\) 61.6631 0.493305
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 26.5770i 0.170180i −0.996373 0.0850900i \(-0.972882\pi\)
0.996373 0.0850900i \(-0.0271178\pi\)
\(30\) 0 0
\(31\) 172.182i 0.997573i −0.866725 0.498787i \(-0.833779\pi\)
0.866725 0.498787i \(-0.166221\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 42.0277 + 249.518i 0.202971 + 1.20503i
\(36\) 0 0
\(37\) −66.8154 −0.296875 −0.148438 0.988922i \(-0.547424\pi\)
−0.148438 + 0.988922i \(0.547424\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −7.91347 −0.0301433 −0.0150717 0.999886i \(-0.504798\pi\)
−0.0150717 + 0.999886i \(0.504798\pi\)
\(42\) 0 0
\(43\) 264.488 0.938000 0.469000 0.883198i \(-0.344614\pi\)
0.469000 + 0.883198i \(0.344614\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 136.842 0.424691 0.212346 0.977195i \(-0.431890\pi\)
0.212346 + 0.977195i \(0.431890\pi\)
\(48\) 0 0
\(49\) −324.075 + 112.359i −0.944824 + 0.327578i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 596.778i 1.54668i 0.633994 + 0.773338i \(0.281414\pi\)
−0.633994 + 0.773338i \(0.718586\pi\)
\(54\) 0 0
\(55\) 139.795i 0.342725i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 103.438 0.228245 0.114123 0.993467i \(-0.463594\pi\)
0.114123 + 0.993467i \(0.463594\pi\)
\(60\) 0 0
\(61\) 588.967i 1.23622i 0.786091 + 0.618111i \(0.212102\pi\)
−0.786091 + 0.618111i \(0.787898\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 215.132i 0.410521i
\(66\) 0 0
\(67\) −309.072 −0.563570 −0.281785 0.959478i \(-0.590926\pi\)
−0.281785 + 0.959478i \(0.590926\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 1030.43i 1.72239i 0.508271 + 0.861197i \(0.330285\pi\)
−0.508271 + 0.861197i \(0.669715\pi\)
\(72\) 0 0
\(73\) 478.042i 0.766447i 0.923656 + 0.383223i \(0.125186\pi\)
−0.923656 + 0.383223i \(0.874814\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −186.867 + 31.4752i −0.276565 + 0.0465835i
\(78\) 0 0
\(79\) 54.8892 0.0781710 0.0390855 0.999236i \(-0.487556\pi\)
0.0390855 + 0.999236i \(0.487556\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −287.874 −0.380702 −0.190351 0.981716i \(-0.560963\pi\)
−0.190351 + 0.981716i \(0.560963\pi\)
\(84\) 0 0
\(85\) 721.593 0.920797
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −1006.71 −1.19900 −0.599498 0.800376i \(-0.704633\pi\)
−0.599498 + 0.800376i \(0.704633\pi\)
\(90\) 0 0
\(91\) −287.573 + 48.4377i −0.331274 + 0.0557984i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 843.097i 0.910526i
\(96\) 0 0
\(97\) 59.9202i 0.0627214i −0.999508 0.0313607i \(-0.990016\pi\)
0.999508 0.0313607i \(-0.00998405\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −536.646 −0.528696 −0.264348 0.964427i \(-0.585157\pi\)
−0.264348 + 0.964427i \(0.585157\pi\)
\(102\) 0 0
\(103\) 982.064i 0.939472i −0.882807 0.469736i \(-0.844349\pi\)
0.882807 0.469736i \(-0.155651\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 1433.16i 1.29485i 0.762128 + 0.647426i \(0.224154\pi\)
−0.762128 + 0.647426i \(0.775846\pi\)
\(108\) 0 0
\(109\) −1184.49 −1.04085 −0.520427 0.853906i \(-0.674227\pi\)
−0.520427 + 0.853906i \(0.674227\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 1072.31i 0.892693i 0.894860 + 0.446346i \(0.147275\pi\)
−0.894860 + 0.446346i \(0.852725\pi\)
\(114\) 0 0
\(115\) 631.529i 0.512090i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 162.469 + 964.574i 0.125155 + 0.743045i
\(120\) 0 0
\(121\) 1226.31 0.921342
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 865.338 0.619186
\(126\) 0 0
\(127\) 2115.79 1.47831 0.739157 0.673533i \(-0.235224\pi\)
0.739157 + 0.673533i \(0.235224\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −592.183 −0.394956 −0.197478 0.980307i \(-0.563275\pi\)
−0.197478 + 0.980307i \(0.563275\pi\)
\(132\) 0 0
\(133\) −1126.99 + 189.826i −0.734756 + 0.123759i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 562.803i 0.350975i 0.984482 + 0.175487i \(0.0561501\pi\)
−0.984482 + 0.175487i \(0.943850\pi\)
\(138\) 0 0
\(139\) 1848.50i 1.12797i −0.825785 0.563986i \(-0.809267\pi\)
0.825785 0.563986i \(-0.190733\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −161.116 −0.0942179
\(144\) 0 0
\(145\) 363.107i 0.207961i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 1875.82i 1.03136i 0.856780 + 0.515682i \(0.172462\pi\)
−0.856780 + 0.515682i \(0.827538\pi\)
\(150\) 0 0
\(151\) −3006.73 −1.62042 −0.810212 0.586137i \(-0.800648\pi\)
−0.810212 + 0.586137i \(0.800648\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 2352.43i 1.21904i
\(156\) 0 0
\(157\) 1685.97i 0.857038i −0.903533 0.428519i \(-0.859036\pi\)
0.903533 0.428519i \(-0.140964\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 844.183 142.191i 0.413235 0.0696037i
\(162\) 0 0
\(163\) 794.673 0.381862 0.190931 0.981603i \(-0.438849\pi\)
0.190931 + 0.981603i \(0.438849\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 35.6499 0.0165190 0.00825950 0.999966i \(-0.497371\pi\)
0.00825950 + 0.999966i \(0.497371\pi\)
\(168\) 0 0
\(169\) 1949.06 0.887144
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 1297.51 0.570218 0.285109 0.958495i \(-0.407970\pi\)
0.285109 + 0.958495i \(0.407970\pi\)
\(174\) 0 0
\(175\) −189.685 1126.15i −0.0819361 0.486453i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 65.8037i 0.0274771i 0.999906 + 0.0137386i \(0.00437325\pi\)
−0.999906 + 0.0137386i \(0.995627\pi\)
\(180\) 0 0
\(181\) 2308.22i 0.947892i 0.880554 + 0.473946i \(0.157171\pi\)
−0.880554 + 0.473946i \(0.842829\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 912.864 0.362784
\(186\) 0 0
\(187\) 540.411i 0.211330i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 2207.89i 0.836424i 0.908349 + 0.418212i \(0.137343\pi\)
−0.908349 + 0.418212i \(0.862657\pi\)
\(192\) 0 0
\(193\) −403.746 −0.150582 −0.0752908 0.997162i \(-0.523989\pi\)
−0.0752908 + 0.997162i \(0.523989\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 2469.69i 0.893189i −0.894737 0.446594i \(-0.852637\pi\)
0.894737 0.446594i \(-0.147363\pi\)
\(198\) 0 0
\(199\) 4468.48i 1.59177i 0.605448 + 0.795885i \(0.292994\pi\)
−0.605448 + 0.795885i \(0.707006\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −485.375 + 81.7546i −0.167816 + 0.0282662i
\(204\) 0 0
\(205\) 108.117 0.0368354
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −631.407 −0.208973
\(210\) 0 0
\(211\) 2489.38 0.812210 0.406105 0.913826i \(-0.366887\pi\)
0.406105 + 0.913826i \(0.366887\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −3613.56 −1.14624
\(216\) 0 0
\(217\) −3144.56 + 529.657i −0.983717 + 0.165693i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 831.649i 0.253135i
\(222\) 0 0
\(223\) 5052.46i 1.51721i 0.651551 + 0.758605i \(0.274119\pi\)
−0.651551 + 0.758605i \(0.725881\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −5275.48 −1.54249 −0.771246 0.636537i \(-0.780366\pi\)
−0.771246 + 0.636537i \(0.780366\pi\)
\(228\) 0 0
\(229\) 2477.33i 0.714875i −0.933937 0.357438i \(-0.883650\pi\)
0.933937 0.357438i \(-0.116350\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 6566.31i 1.84624i −0.384516 0.923118i \(-0.625632\pi\)
0.384516 0.923118i \(-0.374368\pi\)
\(234\) 0 0
\(235\) −1869.60 −0.518977
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 2805.72i 0.759360i −0.925118 0.379680i \(-0.876034\pi\)
0.925118 0.379680i \(-0.123966\pi\)
\(240\) 0 0
\(241\) 1815.82i 0.485342i 0.970109 + 0.242671i \(0.0780236\pi\)
−0.970109 + 0.242671i \(0.921976\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 4427.66 1535.11i 1.15458 0.400303i
\(246\) 0 0
\(247\) −971.684 −0.250311
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 3410.13 0.857551 0.428776 0.903411i \(-0.358945\pi\)
0.428776 + 0.903411i \(0.358945\pi\)
\(252\) 0 0
\(253\) 472.961 0.117529
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −5607.99 −1.36116 −0.680578 0.732676i \(-0.738271\pi\)
−0.680578 + 0.732676i \(0.738271\pi\)
\(258\) 0 0
\(259\) 205.534 + 1220.25i 0.0493099 + 0.292752i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 1909.45i 0.447686i 0.974625 + 0.223843i \(0.0718603\pi\)
−0.974625 + 0.223843i \(0.928140\pi\)
\(264\) 0 0
\(265\) 8153.47i 1.89005i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 3112.24 0.705415 0.352708 0.935734i \(-0.385261\pi\)
0.352708 + 0.935734i \(0.385261\pi\)
\(270\) 0 0
\(271\) 5037.27i 1.12912i 0.825391 + 0.564562i \(0.190955\pi\)
−0.825391 + 0.564562i \(0.809045\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 630.938i 0.138353i
\(276\) 0 0
\(277\) 5634.96 1.22228 0.611141 0.791522i \(-0.290711\pi\)
0.611141 + 0.791522i \(0.290711\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 5695.98i 1.20923i −0.796518 0.604615i \(-0.793327\pi\)
0.796518 0.604615i \(-0.206673\pi\)
\(282\) 0 0
\(283\) 6805.18i 1.42942i −0.699421 0.714710i \(-0.746559\pi\)
0.699421 0.714710i \(-0.253441\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 24.3430 + 144.524i 0.00500669 + 0.0297246i
\(288\) 0 0
\(289\) −2123.50 −0.432221
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 6428.15 1.28169 0.640847 0.767669i \(-0.278583\pi\)
0.640847 + 0.767669i \(0.278583\pi\)
\(294\) 0 0
\(295\) −1413.22 −0.278918
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 727.848 0.140778
\(300\) 0 0
\(301\) −813.603 4830.34i −0.155798 0.924971i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 8046.75i 1.51067i
\(306\) 0 0
\(307\) 471.068i 0.0875742i 0.999041 + 0.0437871i \(0.0139423\pi\)
−0.999041 + 0.0437871i \(0.986058\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 6842.66 1.24763 0.623813 0.781573i \(-0.285583\pi\)
0.623813 + 0.781573i \(0.285583\pi\)
\(312\) 0 0
\(313\) 1020.33i 0.184258i −0.995747 0.0921290i \(-0.970633\pi\)
0.995747 0.0921290i \(-0.0293672\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 8891.25i 1.57534i −0.616099 0.787669i \(-0.711288\pi\)
0.616099 0.787669i \(-0.288712\pi\)
\(318\) 0 0
\(319\) −271.936 −0.0477288
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 3259.21i 0.561446i
\(324\) 0 0
\(325\) 970.962i 0.165721i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −420.947 2499.15i −0.0705396 0.418792i
\(330\) 0 0
\(331\) 1881.87 0.312498 0.156249 0.987718i \(-0.450060\pi\)
0.156249 + 0.987718i \(0.450060\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 4222.69 0.688687
\(336\) 0 0
\(337\) 1039.79 0.168074 0.0840372 0.996463i \(-0.473219\pi\)
0.0840372 + 0.996463i \(0.473219\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −1761.77 −0.279780
\(342\) 0 0
\(343\) 3048.92 + 5572.94i 0.479960 + 0.877291i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 8064.45i 1.24761i −0.781578 0.623807i \(-0.785585\pi\)
0.781578 0.623807i \(-0.214415\pi\)
\(348\) 0 0
\(349\) 408.871i 0.0627116i −0.999508 0.0313558i \(-0.990018\pi\)
0.999508 0.0313558i \(-0.00998249\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −9445.34 −1.42415 −0.712075 0.702104i \(-0.752244\pi\)
−0.712075 + 0.702104i \(0.752244\pi\)
\(354\) 0 0
\(355\) 14078.3i 2.10478i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 5770.25i 0.848306i 0.905590 + 0.424153i \(0.139428\pi\)
−0.905590 + 0.424153i \(0.860572\pi\)
\(360\) 0 0
\(361\) 3051.00 0.444817
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 6531.24i 0.936605i
\(366\) 0 0
\(367\) 6065.00i 0.862645i 0.902198 + 0.431322i \(0.141953\pi\)
−0.902198 + 0.431322i \(0.858047\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 10899.0 1835.78i 1.52519 0.256897i
\(372\) 0 0
\(373\) −2155.94 −0.299278 −0.149639 0.988741i \(-0.547811\pi\)
−0.149639 + 0.988741i \(0.547811\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −418.487 −0.0571702
\(378\) 0 0
\(379\) −7647.70 −1.03651 −0.518254 0.855227i \(-0.673418\pi\)
−0.518254 + 0.855227i \(0.673418\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −12433.2 −1.65877 −0.829383 0.558681i \(-0.811308\pi\)
−0.829383 + 0.558681i \(0.811308\pi\)
\(384\) 0 0
\(385\) 2553.07 430.028i 0.337965 0.0569254i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 14320.5i 1.86653i −0.359190 0.933264i \(-0.616947\pi\)
0.359190 0.933264i \(-0.383053\pi\)
\(390\) 0 0
\(391\) 2441.33i 0.315764i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −749.921 −0.0955257
\(396\) 0 0
\(397\) 9846.85i 1.24483i 0.782686 + 0.622417i \(0.213849\pi\)
−0.782686 + 0.622417i \(0.786151\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 9728.28i 1.21149i −0.795659 0.605744i \(-0.792875\pi\)
0.795659 0.605744i \(-0.207125\pi\)
\(402\) 0 0
\(403\) −2711.22 −0.335125
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 683.656i 0.0832619i
\(408\) 0 0
\(409\) 1606.02i 0.194163i −0.995276 0.0970817i \(-0.969049\pi\)
0.995276 0.0970817i \(-0.0309508\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −318.190 1889.09i −0.0379107 0.225075i
\(414\) 0 0
\(415\) 3933.07 0.465221
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 5853.73 0.682514 0.341257 0.939970i \(-0.389147\pi\)
0.341257 + 0.939970i \(0.389147\pi\)
\(420\) 0 0
\(421\) −13183.4 −1.52617 −0.763086 0.646297i \(-0.776317\pi\)
−0.763086 + 0.646297i \(0.776317\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −3256.78 −0.371711
\(426\) 0 0
\(427\) 10756.3 1811.75i 1.21905 0.205332i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 12848.9i 1.43599i 0.696050 + 0.717993i \(0.254939\pi\)
−0.696050 + 0.717993i \(0.745061\pi\)
\(432\) 0 0
\(433\) 12853.3i 1.42654i 0.700891 + 0.713268i \(0.252786\pi\)
−0.700891 + 0.713268i \(0.747214\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 2852.41 0.312241
\(438\) 0 0
\(439\) 13399.9i 1.45682i 0.685143 + 0.728408i \(0.259740\pi\)
−0.685143 + 0.728408i \(0.740260\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 4808.14i 0.515670i −0.966189 0.257835i \(-0.916991\pi\)
0.966189 0.257835i \(-0.0830090\pi\)
\(444\) 0 0
\(445\) 13754.1 1.46518
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 6631.03i 0.696966i −0.937315 0.348483i \(-0.886697\pi\)
0.937315 0.348483i \(-0.113303\pi\)
\(450\) 0 0
\(451\) 80.9707i 0.00845401i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 3928.96 661.779i 0.404819 0.0681861i
\(456\) 0 0
\(457\) −12402.4 −1.26949 −0.634746 0.772721i \(-0.718895\pi\)
−0.634746 + 0.772721i \(0.718895\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 6950.99 0.702255 0.351128 0.936328i \(-0.385798\pi\)
0.351128 + 0.936328i \(0.385798\pi\)
\(462\) 0 0
\(463\) −14388.5 −1.44426 −0.722128 0.691759i \(-0.756836\pi\)
−0.722128 + 0.691759i \(0.756836\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −13922.6 −1.37957 −0.689786 0.724013i \(-0.742295\pi\)
−0.689786 + 0.724013i \(0.742295\pi\)
\(468\) 0 0
\(469\) 950.751 + 5644.59i 0.0936068 + 0.555742i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 2706.24i 0.263072i
\(474\) 0 0
\(475\) 3805.17i 0.367565i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −19333.3 −1.84418 −0.922090 0.386976i \(-0.873520\pi\)
−0.922090 + 0.386976i \(0.873520\pi\)
\(480\) 0 0
\(481\) 1052.09i 0.0997323i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 818.658i 0.0766460i
\(486\) 0 0
\(487\) −7812.00 −0.726890 −0.363445 0.931616i \(-0.618400\pi\)
−0.363445 + 0.931616i \(0.618400\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 13829.4i 1.27110i 0.772058 + 0.635552i \(0.219227\pi\)
−0.772058 + 0.635552i \(0.780773\pi\)
\(492\) 0 0
\(493\) 1403.68i 0.128232i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 18818.8 3169.77i 1.69847 0.286083i
\(498\) 0 0
\(499\) 12602.3 1.13057 0.565287 0.824894i \(-0.308766\pi\)
0.565287 + 0.824894i \(0.308766\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 13166.7 1.16715 0.583574 0.812060i \(-0.301654\pi\)
0.583574 + 0.812060i \(0.301654\pi\)
\(504\) 0 0
\(505\) 7331.92 0.646071
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −17280.3 −1.50479 −0.752395 0.658712i \(-0.771101\pi\)
−0.752395 + 0.658712i \(0.771101\pi\)
\(510\) 0 0
\(511\) 8730.49 1470.53i 0.755801 0.127304i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 13417.4i 1.14804i
\(516\) 0 0
\(517\) 1400.17i 0.119109i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 11014.5 0.926210 0.463105 0.886303i \(-0.346735\pi\)
0.463105 + 0.886303i \(0.346735\pi\)
\(522\) 0 0
\(523\) 5231.77i 0.437418i −0.975790 0.218709i \(-0.929816\pi\)
0.975790 0.218709i \(-0.0701845\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 9093.91i 0.751683i
\(528\) 0 0
\(529\) 10030.4 0.824392
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 124.607i 0.0101263i
\(534\) 0 0
\(535\) 19580.6i 1.58232i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 1149.66 + 3315.94i 0.0918728 + 0.264986i
\(540\) 0 0
\(541\) −8203.85 −0.651961 −0.325981 0.945376i \(-0.605694\pi\)
−0.325981 + 0.945376i \(0.605694\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 16183.0 1.27193
\(546\) 0 0
\(547\) −10187.8 −0.796342 −0.398171 0.917311i \(-0.630355\pi\)
−0.398171 + 0.917311i \(0.630355\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −1640.04 −0.126802
\(552\) 0 0
\(553\) −168.847 1002.44i −0.0129839 0.0770852i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 3644.61i 0.277248i −0.990345 0.138624i \(-0.955732\pi\)
0.990345 0.138624i \(-0.0442679\pi\)
\(558\) 0 0
\(559\) 4164.68i 0.315112i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 3529.23 0.264190 0.132095 0.991237i \(-0.457830\pi\)
0.132095 + 0.991237i \(0.457830\pi\)
\(564\) 0 0
\(565\) 14650.4i 1.09088i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 20979.7i 1.54572i 0.634575 + 0.772861i \(0.281175\pi\)
−0.634575 + 0.772861i \(0.718825\pi\)
\(570\) 0 0
\(571\) 3866.54 0.283379 0.141690 0.989911i \(-0.454747\pi\)
0.141690 + 0.989911i \(0.454747\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 2850.29i 0.206723i
\(576\) 0 0
\(577\) 8556.67i 0.617364i 0.951165 + 0.308682i \(0.0998878\pi\)
−0.951165 + 0.308682i \(0.900112\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 885.542 + 5257.44i 0.0632332 + 0.375414i
\(582\) 0 0
\(583\) 6106.24 0.433782
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 689.300 0.0484676 0.0242338 0.999706i \(-0.492285\pi\)
0.0242338 + 0.999706i \(0.492285\pi\)
\(588\) 0 0
\(589\) −10625.2 −0.743298
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −16487.3 −1.14174 −0.570869 0.821041i \(-0.693394\pi\)
−0.570869 + 0.821041i \(0.693394\pi\)
\(594\) 0 0
\(595\) −2219.73 13178.5i −0.152941 0.908007i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 25662.6i 1.75050i 0.483675 + 0.875248i \(0.339302\pi\)
−0.483675 + 0.875248i \(0.660698\pi\)
\(600\) 0 0
\(601\) 2857.59i 0.193949i −0.995287 0.0969745i \(-0.969083\pi\)
0.995287 0.0969745i \(-0.0309165\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −16754.4 −1.12589
\(606\) 0 0
\(607\) 3907.10i 0.261259i −0.991431 0.130629i \(-0.958300\pi\)
0.991431 0.130629i \(-0.0416998\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 2154.75i 0.142671i
\(612\) 0 0
\(613\) 12098.4 0.797145 0.398572 0.917137i \(-0.369506\pi\)
0.398572 + 0.917137i \(0.369506\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 8194.00i 0.534648i −0.963607 0.267324i \(-0.913861\pi\)
0.963607 0.267324i \(-0.0861394\pi\)
\(618\) 0 0
\(619\) 6486.57i 0.421191i 0.977573 + 0.210596i \(0.0675403\pi\)
−0.977573 + 0.210596i \(0.932460\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 3096.77 + 18385.5i 0.199149 + 1.18234i
\(624\) 0 0
\(625\) −19530.5 −1.24996
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 3528.90 0.223699
\(630\) 0 0
\(631\) 4594.51 0.289865 0.144932 0.989442i \(-0.453704\pi\)
0.144932 + 0.989442i \(0.453704\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −28906.9 −1.80651
\(636\) 0 0
\(637\) 1769.24 + 5102.95i 0.110047 + 0.317404i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 12005.3i 0.739751i 0.929081 + 0.369875i \(0.120600\pi\)
−0.929081 + 0.369875i \(0.879400\pi\)
\(642\) 0 0
\(643\) 3617.96i 0.221895i −0.993826 0.110948i \(-0.964611\pi\)
0.993826 0.110948i \(-0.0353886\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −11293.2 −0.686217 −0.343109 0.939296i \(-0.611480\pi\)
−0.343109 + 0.939296i \(0.611480\pi\)
\(648\) 0 0
\(649\) 1058.38i 0.0640139i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 8362.09i 0.501124i 0.968101 + 0.250562i \(0.0806154\pi\)
−0.968101 + 0.250562i \(0.919385\pi\)
\(654\) 0 0
\(655\) 8090.68 0.482640
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 5804.98i 0.343141i 0.985172 + 0.171570i \(0.0548841\pi\)
−0.985172 + 0.171570i \(0.945116\pi\)
\(660\) 0 0
\(661\) 15522.3i 0.913383i 0.889625 + 0.456691i \(0.150966\pi\)
−0.889625 + 0.456691i \(0.849034\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 15397.5 2593.49i 0.897878 0.151235i
\(666\) 0 0
\(667\) 1228.48 0.0713150
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 6026.32 0.346712
\(672\) 0 0
\(673\) 7294.62 0.417811 0.208906 0.977936i \(-0.433010\pi\)
0.208906 + 0.977936i \(0.433010\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −20845.4 −1.18339 −0.591693 0.806164i \(-0.701540\pi\)
−0.591693 + 0.806164i \(0.701540\pi\)
\(678\) 0 0
\(679\) −1094.32 + 184.323i −0.0618501 + 0.0104178i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 7625.94i 0.427231i −0.976918 0.213615i \(-0.931476\pi\)
0.976918 0.213615i \(-0.0685239\pi\)
\(684\) 0 0
\(685\) 7689.28i 0.428894i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 9397.01 0.519590
\(690\) 0 0
\(691\) 7613.20i 0.419131i 0.977795 + 0.209566i \(0.0672050\pi\)
−0.977795 + 0.209566i \(0.932795\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 25255.1i 1.37839i
\(696\) 0 0
\(697\) 417.955 0.0227133
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 13841.2i 0.745756i 0.927880 + 0.372878i \(0.121629\pi\)
−0.927880 + 0.372878i \(0.878371\pi\)
\(702\) 0 0
\(703\) 4123.11i 0.221204i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 1650.80 + 9800.78i 0.0878144 + 0.521352i
\(708\) 0 0
\(709\) 1122.62 0.0594651 0.0297326 0.999558i \(-0.490534\pi\)
0.0297326 + 0.999558i \(0.490534\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 7958.87 0.418039
\(714\) 0 0
\(715\) 2201.24 0.115135
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −32986.5 −1.71097 −0.855485 0.517827i \(-0.826741\pi\)
−0.855485 + 0.517827i \(0.826741\pi\)
\(720\) 0 0
\(721\) −17935.4 + 3020.97i −0.926422 + 0.156043i
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 1638.82i 0.0839506i
\(726\) 0 0
\(727\) 16174.0i 0.825116i 0.910931 + 0.412558i \(0.135365\pi\)
−0.910931 + 0.412558i \(0.864635\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −13969.1 −0.706794
\(732\) 0 0
\(733\) 23202.0i 1.16915i 0.811340 + 0.584574i \(0.198738\pi\)
−0.811340 + 0.584574i \(0.801262\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 3162.43i 0.158059i
\(738\) 0 0
\(739\) −28250.5 −1.40624 −0.703120 0.711071i \(-0.748210\pi\)
−0.703120 + 0.711071i \(0.748210\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 30939.7i 1.52768i 0.645404 + 0.763841i \(0.276689\pi\)
−0.645404 + 0.763841i \(0.723311\pi\)
\(744\) 0 0
\(745\) 25628.4i 1.26034i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 26173.9 4408.62i 1.27687 0.215070i
\(750\) 0 0
\(751\) 6241.46 0.303268 0.151634 0.988437i \(-0.451546\pi\)
0.151634 + 0.988437i \(0.451546\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 41079.3 1.98017
\(756\) 0 0
\(757\) 20696.0 0.993672 0.496836 0.867844i \(-0.334495\pi\)
0.496836 + 0.867844i \(0.334495\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −21392.7 −1.01903 −0.509516 0.860461i \(-0.670176\pi\)
−0.509516 + 0.860461i \(0.670176\pi\)
\(762\) 0 0
\(763\) 3643.65 + 21632.3i 0.172882 + 1.02640i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 1628.76i 0.0766768i
\(768\) 0 0
\(769\) 25114.3i 1.17769i 0.808246 + 0.588845i \(0.200417\pi\)
−0.808246 + 0.588845i \(0.799583\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −7411.13 −0.344838 −0.172419 0.985024i \(-0.555158\pi\)
−0.172419 + 0.985024i \(0.555158\pi\)
\(774\) 0 0
\(775\) 10617.3i 0.492108i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 488.332i 0.0224600i
\(780\) 0 0
\(781\) 10543.4 0.483064
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 23034.5i 1.04731i
\(786\) 0 0
\(787\) 28731.4i 1.30135i 0.759355 + 0.650676i \(0.225515\pi\)
−0.759355 + 0.650676i \(0.774485\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 19583.6 3298.58i 0.880293 0.148273i
\(792\) 0 0
\(793\) 9274.02 0.415296
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 12556.5 0.558062 0.279031 0.960282i \(-0.409987\pi\)
0.279031 + 0.960282i \(0.409987\pi\)
\(798\) 0 0
\(799\) −7227.42 −0.320010
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 4891.33 0.214958
\(804\) 0 0
\(805\) −11533.6 + 1942.68i −0.504977 + 0.0850563i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 24449.2i 1.06253i 0.847205 + 0.531266i \(0.178284\pi\)
−0.847205 + 0.531266i \(0.821716\pi\)
\(810\) 0 0
\(811\) 2328.38i 0.100815i 0.998729 + 0.0504073i \(0.0160519\pi\)
−0.998729 + 0.0504073i \(0.983948\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −10857.2 −0.466639
\(816\) 0 0
\(817\) 16321.3i 0.698909i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 24291.6i 1.03262i −0.856401 0.516311i \(-0.827305\pi\)
0.856401 0.516311i \(-0.172695\pi\)
\(822\) 0 0
\(823\) −30865.5 −1.30730 −0.653648 0.756799i \(-0.726762\pi\)
−0.653648 + 0.756799i \(0.726762\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 14505.4i 0.609920i −0.952365 0.304960i \(-0.901357\pi\)
0.952365 0.304960i \(-0.0986430\pi\)
\(828\) 0 0
\(829\) 13689.4i 0.573527i −0.958001 0.286764i \(-0.907421\pi\)
0.958001 0.286764i \(-0.0925794\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 17116.2 5934.34i 0.711936 0.246834i
\(834\) 0 0
\(835\) −487.066 −0.0201864
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 17721.3 0.729210 0.364605 0.931162i \(-0.381204\pi\)
0.364605 + 0.931162i \(0.381204\pi\)
\(840\) 0 0
\(841\) 23682.7 0.971039
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −26628.9 −1.08410
\(846\) 0 0
\(847\) −3772.30 22396.0i −0.153031 0.908544i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 3088.45i 0.124407i
\(852\) 0 0
\(853\) 32915.4i 1.32122i −0.750728 0.660611i \(-0.770297\pi\)
0.750728 0.660611i \(-0.229703\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −198.537 −0.00791353 −0.00395676 0.999992i \(-0.501259\pi\)
−0.00395676 + 0.999992i \(0.501259\pi\)
\(858\) 0 0
\(859\) 17797.3i 0.706909i 0.935452 + 0.353455i \(0.114993\pi\)
−0.935452 + 0.353455i \(0.885007\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 19544.9i 0.770934i 0.922722 + 0.385467i \(0.125960\pi\)
−0.922722 + 0.385467i \(0.874040\pi\)
\(864\) 0 0
\(865\) −17727.2 −0.696812
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 561.626i 0.0219239i
\(870\) 0 0
\(871\) 4866.72i 0.189326i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −2661.91 15803.7i −0.102844 0.610585i
\(876\) 0 0
\(877\) 977.650 0.0376430 0.0188215 0.999823i \(-0.494009\pi\)
0.0188215 + 0.999823i \(0.494009\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −6419.94 −0.245509 −0.122754 0.992437i \(-0.539173\pi\)
−0.122754 + 0.992437i \(0.539173\pi\)
\(882\) 0 0
\(883\) 44260.8 1.68686 0.843429 0.537241i \(-0.180533\pi\)
0.843429 + 0.537241i \(0.180533\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 2960.18 0.112055 0.0560277 0.998429i \(-0.482156\pi\)
0.0560277 + 0.998429i \(0.482156\pi\)
\(888\) 0 0
\(889\) −6508.48 38640.7i −0.245542 1.45778i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 8444.40i 0.316440i
\(894\) 0 0
\(895\) 899.041i 0.0335773i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −4576.07 −0.169767
\(900\) 0 0
\(901\) 31519.3i 1.16544i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 31536.0i 1.15833i
\(906\) 0 0
\(907\) −28177.0 −1.03154 −0.515768 0.856728i \(-0.672493\pi\)
−0.515768 + 0.856728i \(0.672493\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 18518.8i 0.673495i −0.941595 0.336747i \(-0.890673\pi\)
0.941595 0.336747i \(-0.109327\pi\)
\(912\) 0 0
\(913\) 2945.53i 0.106772i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 1821.64 + 10815.0i 0.0656007 + 0.389470i
\(918\) 0 0
\(919\) 39516.9 1.41843 0.709217 0.704990i \(-0.249049\pi\)
0.709217 + 0.704990i \(0.249049\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 16225.5 0.578621
\(924\) 0 0
\(925\) −4120.05 −0.146450
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −35210.4 −1.24351 −0.621753 0.783213i \(-0.713579\pi\)
−0.621753 + 0.783213i \(0.713579\pi\)
\(930\) 0 0
\(931\) 6933.58 + 19998.3i 0.244080 + 0.703994i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 7383.35i 0.258247i
\(936\) 0 0
\(937\) 25099.1i 0.875081i −0.899199 0.437541i \(-0.855850\pi\)
0.899199 0.437541i \(-0.144150\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −21420.0 −0.742055 −0.371027 0.928622i \(-0.620994\pi\)
−0.371027 + 0.928622i \(0.620994\pi\)
\(942\) 0 0
\(943\) 365.789i 0.0126317i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 45264.3i 1.55321i −0.629986 0.776607i \(-0.716939\pi\)
0.629986 0.776607i \(-0.283061\pi\)
\(948\) 0 0
\(949\) 7527.37 0.257480
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 5430.75i 0.184595i 0.995731 + 0.0922977i \(0.0294211\pi\)
−0.995731 + 0.0922977i \(0.970579\pi\)
\(954\) 0 0
\(955\) 30165.2i 1.02212i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 10278.5 1731.26i 0.346099 0.0582956i
\(960\) 0 0
\(961\) 144.401 0.00484715
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 5516.17 0.184012
\(966\) 0 0
\(967\) 11336.0 0.376982 0.188491 0.982075i \(-0.439640\pi\)
0.188491 + 0.982075i \(0.439640\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −43570.7 −1.44001 −0.720006 0.693968i \(-0.755861\pi\)
−0.720006 + 0.693968i \(0.755861\pi\)
\(972\) 0 0
\(973\) −33759.2 + 5686.26i −1.11230 + 0.187352i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 44248.7i 1.44897i −0.689291 0.724484i \(-0.742078\pi\)
0.689291 0.724484i \(-0.257922\pi\)
\(978\) 0 0
\(979\) 10300.6i 0.336271i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 16774.5 0.544275 0.272137 0.962258i \(-0.412269\pi\)
0.272137 + 0.962258i \(0.412269\pi\)
\(984\) 0 0
\(985\) 33742.1i 1.09148i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 12225.6i 0.393075i
\(990\) 0 0
\(991\) −37053.5 −1.18773 −0.593866 0.804564i \(-0.702399\pi\)
−0.593866 + 0.804564i \(0.702399\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 61050.5i 1.94516i
\(996\) 0 0
\(997\) 36680.2i 1.16517i −0.812770 0.582585i \(-0.802041\pi\)
0.812770 0.582585i \(-0.197959\pi\)
\(998\) 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 756.4.f.e.377.3 16
3.2 odd 2 inner 756.4.f.e.377.13 yes 16
7.6 odd 2 inner 756.4.f.e.377.14 yes 16
21.20 even 2 inner 756.4.f.e.377.4 yes 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
756.4.f.e.377.3 16 1.1 even 1 trivial
756.4.f.e.377.4 yes 16 21.20 even 2 inner
756.4.f.e.377.13 yes 16 3.2 odd 2 inner
756.4.f.e.377.14 yes 16 7.6 odd 2 inner