Properties

Label 1350.4.c.ba.649.4
Level $1350$
Weight $4$
Character 1350.649
Analytic conductor $79.653$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1350,4,Mod(649,1350)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1350, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1350.649");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1350 = 2 \cdot 3^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1350.c (of order \(2\), degree \(1\), not minimal)

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: no
Analytic conductor: \(79.6525785077\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{21})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 11x^{2} + 25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 649.4
Root \(2.79129i\) of defining polynomial
Character \(\chi\) \(=\) 1350.649
Dual form 1350.4.c.ba.649.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.00000i q^{2} -4.00000 q^{4} +18.7477i q^{7} -8.00000i q^{8} +O(q^{10})\) \(q+2.00000i q^{2} -4.00000 q^{4} +18.7477i q^{7} -8.00000i q^{8} -39.9909 q^{11} -33.7477i q^{13} -37.4955 q^{14} +16.0000 q^{16} -53.7386i q^{17} -91.7386 q^{19} -79.9818i q^{22} +80.7386i q^{23} +67.4955 q^{26} -74.9909i q^{28} -141.234 q^{29} +264.955 q^{31} +32.0000i q^{32} +107.477 q^{34} +61.2432i q^{37} -183.477i q^{38} +314.973 q^{41} -236.261i q^{43} +159.964 q^{44} -161.477 q^{46} -243.261i q^{47} -8.47727 q^{49} +134.991i q^{52} -191.739i q^{53} +149.982 q^{56} -282.468i q^{58} +312.441 q^{59} -550.648 q^{61} +529.909i q^{62} -64.0000 q^{64} +571.270i q^{67} +214.955i q^{68} +183.784 q^{71} -125.027i q^{73} -122.486 q^{74} +366.955 q^{76} -749.739i q^{77} +429.216 q^{79} +629.945i q^{82} -1091.48i q^{83} +472.523 q^{86} +319.927i q^{88} -1233.59 q^{89} +632.693 q^{91} -322.955i q^{92} +486.523 q^{94} +1532.44i q^{97} -16.9545i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 16 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 16 q^{4} + 60 q^{11} - 40 q^{14} + 64 q^{16} - 92 q^{19} + 160 q^{26} - 180 q^{29} - 40 q^{31} - 120 q^{34} + 600 q^{41} - 240 q^{44} - 96 q^{46} + 516 q^{49} + 160 q^{56} - 180 q^{59} + 272 q^{61} - 256 q^{64} + 1560 q^{71} - 160 q^{74} + 368 q^{76} + 892 q^{79} + 2440 q^{86} - 1140 q^{89} + 1156 q^{91} + 2496 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1350\mathbb{Z}\right)^\times\).

\(n\) \(1001\) \(1027\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.00000i 0.707107i
\(3\) 0 0
\(4\) −4.00000 −0.500000
\(5\) 0 0
\(6\) 0 0
\(7\) 18.7477i 1.01228i 0.862451 + 0.506141i \(0.168928\pi\)
−0.862451 + 0.506141i \(0.831072\pi\)
\(8\) − 8.00000i − 0.353553i
\(9\) 0 0
\(10\) 0 0
\(11\) −39.9909 −1.09616 −0.548078 0.836427i \(-0.684640\pi\)
−0.548078 + 0.836427i \(0.684640\pi\)
\(12\) 0 0
\(13\) − 33.7477i − 0.719995i −0.932953 0.359998i \(-0.882778\pi\)
0.932953 0.359998i \(-0.117222\pi\)
\(14\) −37.4955 −0.715792
\(15\) 0 0
\(16\) 16.0000 0.250000
\(17\) − 53.7386i − 0.766678i −0.923608 0.383339i \(-0.874774\pi\)
0.923608 0.383339i \(-0.125226\pi\)
\(18\) 0 0
\(19\) −91.7386 −1.10770 −0.553850 0.832617i \(-0.686842\pi\)
−0.553850 + 0.832617i \(0.686842\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) − 79.9818i − 0.775099i
\(23\) 80.7386i 0.731964i 0.930622 + 0.365982i \(0.119267\pi\)
−0.930622 + 0.365982i \(0.880733\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 67.4955 0.509113
\(27\) 0 0
\(28\) − 74.9909i − 0.506141i
\(29\) −141.234 −0.904362 −0.452181 0.891926i \(-0.649354\pi\)
−0.452181 + 0.891926i \(0.649354\pi\)
\(30\) 0 0
\(31\) 264.955 1.53507 0.767536 0.641006i \(-0.221483\pi\)
0.767536 + 0.641006i \(0.221483\pi\)
\(32\) 32.0000i 0.176777i
\(33\) 0 0
\(34\) 107.477 0.542124
\(35\) 0 0
\(36\) 0 0
\(37\) 61.2432i 0.272117i 0.990701 + 0.136058i \(0.0434435\pi\)
−0.990701 + 0.136058i \(0.956557\pi\)
\(38\) − 183.477i − 0.783262i
\(39\) 0 0
\(40\) 0 0
\(41\) 314.973 1.19977 0.599884 0.800087i \(-0.295213\pi\)
0.599884 + 0.800087i \(0.295213\pi\)
\(42\) 0 0
\(43\) − 236.261i − 0.837896i −0.908010 0.418948i \(-0.862399\pi\)
0.908010 0.418948i \(-0.137601\pi\)
\(44\) 159.964 0.548078
\(45\) 0 0
\(46\) −161.477 −0.517577
\(47\) − 243.261i − 0.754964i −0.926017 0.377482i \(-0.876790\pi\)
0.926017 0.377482i \(-0.123210\pi\)
\(48\) 0 0
\(49\) −8.47727 −0.0247151
\(50\) 0 0
\(51\) 0 0
\(52\) 134.991i 0.359998i
\(53\) − 191.739i − 0.496931i −0.968641 0.248465i \(-0.920074\pi\)
0.968641 0.248465i \(-0.0799262\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 149.982 0.357896
\(57\) 0 0
\(58\) − 282.468i − 0.639481i
\(59\) 312.441 0.689430 0.344715 0.938707i \(-0.387976\pi\)
0.344715 + 0.938707i \(0.387976\pi\)
\(60\) 0 0
\(61\) −550.648 −1.15579 −0.577895 0.816111i \(-0.696126\pi\)
−0.577895 + 0.816111i \(0.696126\pi\)
\(62\) 529.909i 1.08546i
\(63\) 0 0
\(64\) −64.0000 −0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) 571.270i 1.04167i 0.853658 + 0.520834i \(0.174379\pi\)
−0.853658 + 0.520834i \(0.825621\pi\)
\(68\) 214.955i 0.383339i
\(69\) 0 0
\(70\) 0 0
\(71\) 183.784 0.307199 0.153600 0.988133i \(-0.450913\pi\)
0.153600 + 0.988133i \(0.450913\pi\)
\(72\) 0 0
\(73\) − 125.027i − 0.200457i −0.994964 0.100228i \(-0.968043\pi\)
0.994964 0.100228i \(-0.0319573\pi\)
\(74\) −122.486 −0.192416
\(75\) 0 0
\(76\) 366.955 0.553850
\(77\) − 749.739i − 1.10962i
\(78\) 0 0
\(79\) 429.216 0.611273 0.305636 0.952148i \(-0.401131\pi\)
0.305636 + 0.952148i \(0.401131\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 629.945i 0.848364i
\(83\) − 1091.48i − 1.44344i −0.692187 0.721718i \(-0.743353\pi\)
0.692187 0.721718i \(-0.256647\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 472.523 0.592482
\(87\) 0 0
\(88\) 319.927i 0.387550i
\(89\) −1233.59 −1.46922 −0.734610 0.678489i \(-0.762635\pi\)
−0.734610 + 0.678489i \(0.762635\pi\)
\(90\) 0 0
\(91\) 632.693 0.728838
\(92\) − 322.955i − 0.365982i
\(93\) 0 0
\(94\) 486.523 0.533840
\(95\) 0 0
\(96\) 0 0
\(97\) 1532.44i 1.60408i 0.597270 + 0.802040i \(0.296252\pi\)
−0.597270 + 0.802040i \(0.703748\pi\)
\(98\) − 16.9545i − 0.0174762i
\(99\) 0 0
\(100\) 0 0
\(101\) −358.639 −0.353326 −0.176663 0.984271i \(-0.556530\pi\)
−0.176663 + 0.984271i \(0.556530\pi\)
\(102\) 0 0
\(103\) 437.514i 0.418539i 0.977858 + 0.209269i \(0.0671086\pi\)
−0.977858 + 0.209269i \(0.932891\pi\)
\(104\) −269.982 −0.254557
\(105\) 0 0
\(106\) 383.477 0.351383
\(107\) − 922.818i − 0.833759i −0.908962 0.416879i \(-0.863124\pi\)
0.908962 0.416879i \(-0.136876\pi\)
\(108\) 0 0
\(109\) 1219.39 1.07152 0.535762 0.844369i \(-0.320025\pi\)
0.535762 + 0.844369i \(0.320025\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 299.964i 0.253071i
\(113\) − 463.045i − 0.385484i −0.981250 0.192742i \(-0.938262\pi\)
0.981250 0.192742i \(-0.0617380\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 564.936 0.452181
\(117\) 0 0
\(118\) 624.882i 0.487500i
\(119\) 1007.48 0.776095
\(120\) 0 0
\(121\) 268.273 0.201557
\(122\) − 1101.30i − 0.817267i
\(123\) 0 0
\(124\) −1059.82 −0.767536
\(125\) 0 0
\(126\) 0 0
\(127\) 207.677i 0.145105i 0.997365 + 0.0725527i \(0.0231145\pi\)
−0.997365 + 0.0725527i \(0.976885\pi\)
\(128\) − 128.000i − 0.0883883i
\(129\) 0 0
\(130\) 0 0
\(131\) 2567.34 1.71229 0.856144 0.516738i \(-0.172854\pi\)
0.856144 + 0.516738i \(0.172854\pi\)
\(132\) 0 0
\(133\) − 1719.89i − 1.12130i
\(134\) −1142.54 −0.736571
\(135\) 0 0
\(136\) −429.909 −0.271062
\(137\) − 15.9205i − 0.00992830i −0.999988 0.00496415i \(-0.998420\pi\)
0.999988 0.00496415i \(-0.00158014\pi\)
\(138\) 0 0
\(139\) 2839.65 1.73278 0.866388 0.499372i \(-0.166436\pi\)
0.866388 + 0.499372i \(0.166436\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 367.568i 0.217223i
\(143\) 1349.60i 0.789227i
\(144\) 0 0
\(145\) 0 0
\(146\) 250.055 0.141744
\(147\) 0 0
\(148\) − 244.973i − 0.136058i
\(149\) 202.377 0.111271 0.0556355 0.998451i \(-0.482282\pi\)
0.0556355 + 0.998451i \(0.482282\pi\)
\(150\) 0 0
\(151\) 3657.81 1.97131 0.985656 0.168767i \(-0.0539786\pi\)
0.985656 + 0.168767i \(0.0539786\pi\)
\(152\) 733.909i 0.391631i
\(153\) 0 0
\(154\) 1499.48 0.784619
\(155\) 0 0
\(156\) 0 0
\(157\) 2272.29i 1.15508i 0.816361 + 0.577542i \(0.195988\pi\)
−0.816361 + 0.577542i \(0.804012\pi\)
\(158\) 858.432i 0.432235i
\(159\) 0 0
\(160\) 0 0
\(161\) −1513.67 −0.740954
\(162\) 0 0
\(163\) − 2754.68i − 1.32370i −0.749636 0.661851i \(-0.769771\pi\)
0.749636 0.661851i \(-0.230229\pi\)
\(164\) −1259.89 −0.599884
\(165\) 0 0
\(166\) 2182.95 1.02066
\(167\) 1067.85i 0.494808i 0.968912 + 0.247404i \(0.0795774\pi\)
−0.968912 + 0.247404i \(0.920423\pi\)
\(168\) 0 0
\(169\) 1058.09 0.481607
\(170\) 0 0
\(171\) 0 0
\(172\) 945.045i 0.418948i
\(173\) − 2963.47i − 1.30236i −0.758924 0.651180i \(-0.774274\pi\)
0.758924 0.651180i \(-0.225726\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −639.855 −0.274039
\(177\) 0 0
\(178\) − 2467.19i − 1.03890i
\(179\) 3087.14 1.28907 0.644536 0.764574i \(-0.277051\pi\)
0.644536 + 0.764574i \(0.277051\pi\)
\(180\) 0 0
\(181\) 4080.03 1.67551 0.837753 0.546050i \(-0.183869\pi\)
0.837753 + 0.546050i \(0.183869\pi\)
\(182\) 1265.39i 0.515366i
\(183\) 0 0
\(184\) 645.909 0.258788
\(185\) 0 0
\(186\) 0 0
\(187\) 2149.06i 0.840399i
\(188\) 973.045i 0.377482i
\(189\) 0 0
\(190\) 0 0
\(191\) 2672.01 1.01225 0.506126 0.862460i \(-0.331077\pi\)
0.506126 + 0.862460i \(0.331077\pi\)
\(192\) 0 0
\(193\) 4509.45i 1.68185i 0.541149 + 0.840927i \(0.317989\pi\)
−0.541149 + 0.840927i \(0.682011\pi\)
\(194\) −3064.88 −1.13426
\(195\) 0 0
\(196\) 33.9091 0.0123575
\(197\) 2251.82i 0.814393i 0.913341 + 0.407196i \(0.133494\pi\)
−0.913341 + 0.407196i \(0.866506\pi\)
\(198\) 0 0
\(199\) −1548.93 −0.551763 −0.275882 0.961192i \(-0.588970\pi\)
−0.275882 + 0.961192i \(0.588970\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) − 717.277i − 0.249839i
\(203\) − 2647.82i − 0.915470i
\(204\) 0 0
\(205\) 0 0
\(206\) −875.027 −0.295952
\(207\) 0 0
\(208\) − 539.964i − 0.179999i
\(209\) 3668.71 1.21421
\(210\) 0 0
\(211\) 2230.61 0.727781 0.363890 0.931442i \(-0.381448\pi\)
0.363890 + 0.931442i \(0.381448\pi\)
\(212\) 766.955i 0.248465i
\(213\) 0 0
\(214\) 1845.64 0.589557
\(215\) 0 0
\(216\) 0 0
\(217\) 4967.30i 1.55393i
\(218\) 2438.77i 0.757681i
\(219\) 0 0
\(220\) 0 0
\(221\) −1813.56 −0.552005
\(222\) 0 0
\(223\) − 2069.04i − 0.621314i −0.950522 0.310657i \(-0.899451\pi\)
0.950522 0.310657i \(-0.100549\pi\)
\(224\) −599.927 −0.178948
\(225\) 0 0
\(226\) 926.091 0.272578
\(227\) − 5240.45i − 1.53225i −0.642690 0.766126i \(-0.722182\pi\)
0.642690 0.766126i \(-0.277818\pi\)
\(228\) 0 0
\(229\) 3557.69 1.02663 0.513317 0.858199i \(-0.328417\pi\)
0.513317 + 0.858199i \(0.328417\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 1129.87i 0.319740i
\(233\) 35.4887i 0.00997828i 0.999988 + 0.00498914i \(0.00158810\pi\)
−0.999988 + 0.00498914i \(0.998412\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −1249.76 −0.344715
\(237\) 0 0
\(238\) 2014.95i 0.548782i
\(239\) −1170.62 −0.316826 −0.158413 0.987373i \(-0.550638\pi\)
−0.158413 + 0.987373i \(0.550638\pi\)
\(240\) 0 0
\(241\) −3319.50 −0.887252 −0.443626 0.896212i \(-0.646308\pi\)
−0.443626 + 0.896212i \(0.646308\pi\)
\(242\) 536.545i 0.142523i
\(243\) 0 0
\(244\) 2202.59 0.577895
\(245\) 0 0
\(246\) 0 0
\(247\) 3095.97i 0.797538i
\(248\) − 2119.64i − 0.542730i
\(249\) 0 0
\(250\) 0 0
\(251\) −5852.48 −1.47173 −0.735867 0.677126i \(-0.763225\pi\)
−0.735867 + 0.677126i \(0.763225\pi\)
\(252\) 0 0
\(253\) − 3228.81i − 0.802346i
\(254\) −415.355 −0.102605
\(255\) 0 0
\(256\) 256.000 0.0625000
\(257\) − 4239.62i − 1.02903i −0.857482 0.514515i \(-0.827972\pi\)
0.857482 0.514515i \(-0.172028\pi\)
\(258\) 0 0
\(259\) −1148.17 −0.275459
\(260\) 0 0
\(261\) 0 0
\(262\) 5134.68i 1.21077i
\(263\) 6702.60i 1.57148i 0.618555 + 0.785742i \(0.287719\pi\)
−0.618555 + 0.785742i \(0.712281\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 3439.78 0.792882
\(267\) 0 0
\(268\) − 2285.08i − 0.520834i
\(269\) 3182.94 0.721440 0.360720 0.932674i \(-0.382531\pi\)
0.360720 + 0.932674i \(0.382531\pi\)
\(270\) 0 0
\(271\) −2349.73 −0.526700 −0.263350 0.964700i \(-0.584827\pi\)
−0.263350 + 0.964700i \(0.584827\pi\)
\(272\) − 859.818i − 0.191670i
\(273\) 0 0
\(274\) 31.8409 0.00702037
\(275\) 0 0
\(276\) 0 0
\(277\) 6654.79i 1.44349i 0.692157 + 0.721747i \(0.256660\pi\)
−0.692157 + 0.721747i \(0.743340\pi\)
\(278\) 5679.30i 1.22526i
\(279\) 0 0
\(280\) 0 0
\(281\) −6245.88 −1.32597 −0.662986 0.748632i \(-0.730711\pi\)
−0.662986 + 0.748632i \(0.730711\pi\)
\(282\) 0 0
\(283\) − 2360.43i − 0.495806i −0.968785 0.247903i \(-0.920258\pi\)
0.968785 0.247903i \(-0.0797415\pi\)
\(284\) −735.136 −0.153600
\(285\) 0 0
\(286\) −2699.20 −0.558068
\(287\) 5905.02i 1.21450i
\(288\) 0 0
\(289\) 2025.16 0.412204
\(290\) 0 0
\(291\) 0 0
\(292\) 500.109i 0.100228i
\(293\) − 5834.08i − 1.16324i −0.813459 0.581622i \(-0.802418\pi\)
0.813459 0.581622i \(-0.197582\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 489.945 0.0962078
\(297\) 0 0
\(298\) 404.755i 0.0786805i
\(299\) 2724.75 0.527010
\(300\) 0 0
\(301\) 4429.36 0.848187
\(302\) 7315.61i 1.39393i
\(303\) 0 0
\(304\) −1467.82 −0.276925
\(305\) 0 0
\(306\) 0 0
\(307\) 1460.05i 0.271432i 0.990748 + 0.135716i \(0.0433335\pi\)
−0.990748 + 0.135716i \(0.956666\pi\)
\(308\) 2998.95i 0.554809i
\(309\) 0 0
\(310\) 0 0
\(311\) −4200.24 −0.765833 −0.382916 0.923783i \(-0.625080\pi\)
−0.382916 + 0.923783i \(0.625080\pi\)
\(312\) 0 0
\(313\) − 5890.15i − 1.06368i −0.846846 0.531838i \(-0.821501\pi\)
0.846846 0.531838i \(-0.178499\pi\)
\(314\) −4544.57 −0.816768
\(315\) 0 0
\(316\) −1716.86 −0.305636
\(317\) 1507.03i 0.267014i 0.991048 + 0.133507i \(0.0426239\pi\)
−0.991048 + 0.133507i \(0.957376\pi\)
\(318\) 0 0
\(319\) 5648.08 0.991322
\(320\) 0 0
\(321\) 0 0
\(322\) − 3027.33i − 0.523934i
\(323\) 4929.91i 0.849249i
\(324\) 0 0
\(325\) 0 0
\(326\) 5509.36 0.935998
\(327\) 0 0
\(328\) − 2519.78i − 0.424182i
\(329\) 4560.60 0.764237
\(330\) 0 0
\(331\) −3256.75 −0.540807 −0.270404 0.962747i \(-0.587157\pi\)
−0.270404 + 0.962747i \(0.587157\pi\)
\(332\) 4365.91i 0.721718i
\(333\) 0 0
\(334\) −2135.70 −0.349882
\(335\) 0 0
\(336\) 0 0
\(337\) − 7698.71i − 1.24444i −0.782843 0.622219i \(-0.786231\pi\)
0.782843 0.622219i \(-0.213769\pi\)
\(338\) 2116.18i 0.340548i
\(339\) 0 0
\(340\) 0 0
\(341\) −10595.8 −1.68268
\(342\) 0 0
\(343\) 6271.54i 0.987263i
\(344\) −1890.09 −0.296241
\(345\) 0 0
\(346\) 5926.93 0.920907
\(347\) − 9545.73i − 1.47678i −0.674376 0.738388i \(-0.735587\pi\)
0.674376 0.738388i \(-0.264413\pi\)
\(348\) 0 0
\(349\) 2926.48 0.448856 0.224428 0.974491i \(-0.427949\pi\)
0.224428 + 0.974491i \(0.427949\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) − 1279.71i − 0.193775i
\(353\) − 4679.69i − 0.705595i −0.935700 0.352797i \(-0.885231\pi\)
0.935700 0.352797i \(-0.114769\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 4934.37 0.734610
\(357\) 0 0
\(358\) 6174.28i 0.911511i
\(359\) 12252.8 1.80133 0.900667 0.434509i \(-0.143078\pi\)
0.900667 + 0.434509i \(0.143078\pi\)
\(360\) 0 0
\(361\) 1556.98 0.226998
\(362\) 8160.07i 1.18476i
\(363\) 0 0
\(364\) −2530.77 −0.364419
\(365\) 0 0
\(366\) 0 0
\(367\) 11666.1i 1.65931i 0.558280 + 0.829653i \(0.311462\pi\)
−0.558280 + 0.829653i \(0.688538\pi\)
\(368\) 1291.82i 0.182991i
\(369\) 0 0
\(370\) 0 0
\(371\) 3594.66 0.503034
\(372\) 0 0
\(373\) 1543.32i 0.214237i 0.994246 + 0.107118i \(0.0341623\pi\)
−0.994246 + 0.107118i \(0.965838\pi\)
\(374\) −4298.11 −0.594252
\(375\) 0 0
\(376\) −1946.09 −0.266920
\(377\) 4766.33i 0.651136i
\(378\) 0 0
\(379\) 1421.25 0.192625 0.0963123 0.995351i \(-0.469295\pi\)
0.0963123 + 0.995351i \(0.469295\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 5344.03i 0.715770i
\(383\) − 8785.42i − 1.17210i −0.810275 0.586050i \(-0.800682\pi\)
0.810275 0.586050i \(-0.199318\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −9018.91 −1.18925
\(387\) 0 0
\(388\) − 6129.76i − 0.802040i
\(389\) −5555.06 −0.724043 −0.362022 0.932170i \(-0.617913\pi\)
−0.362022 + 0.932170i \(0.617913\pi\)
\(390\) 0 0
\(391\) 4338.78 0.561181
\(392\) 67.8182i 0.00873810i
\(393\) 0 0
\(394\) −4503.64 −0.575863
\(395\) 0 0
\(396\) 0 0
\(397\) − 11483.1i − 1.45169i −0.687856 0.725847i \(-0.741448\pi\)
0.687856 0.725847i \(-0.258552\pi\)
\(398\) − 3097.86i − 0.390155i
\(399\) 0 0
\(400\) 0 0
\(401\) 12002.9 1.49475 0.747376 0.664401i \(-0.231313\pi\)
0.747376 + 0.664401i \(0.231313\pi\)
\(402\) 0 0
\(403\) − 8941.61i − 1.10524i
\(404\) 1434.55 0.176663
\(405\) 0 0
\(406\) 5295.64 0.647335
\(407\) − 2449.17i − 0.298282i
\(408\) 0 0
\(409\) −9770.32 −1.18120 −0.590600 0.806964i \(-0.701109\pi\)
−0.590600 + 0.806964i \(0.701109\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) − 1750.05i − 0.209269i
\(413\) 5857.56i 0.697897i
\(414\) 0 0
\(415\) 0 0
\(416\) 1079.93 0.127278
\(417\) 0 0
\(418\) 7337.42i 0.858577i
\(419\) −8055.79 −0.939262 −0.469631 0.882863i \(-0.655613\pi\)
−0.469631 + 0.882863i \(0.655613\pi\)
\(420\) 0 0
\(421\) 4967.99 0.575119 0.287559 0.957763i \(-0.407156\pi\)
0.287559 + 0.957763i \(0.407156\pi\)
\(422\) 4461.23i 0.514619i
\(423\) 0 0
\(424\) −1533.91 −0.175692
\(425\) 0 0
\(426\) 0 0
\(427\) − 10323.4i − 1.16999i
\(428\) 3691.27i 0.416879i
\(429\) 0 0
\(430\) 0 0
\(431\) −10826.5 −1.20997 −0.604984 0.796238i \(-0.706821\pi\)
−0.604984 + 0.796238i \(0.706821\pi\)
\(432\) 0 0
\(433\) − 2038.63i − 0.226259i −0.993580 0.113130i \(-0.963912\pi\)
0.993580 0.113130i \(-0.0360875\pi\)
\(434\) −9934.59 −1.09879
\(435\) 0 0
\(436\) −4877.55 −0.535762
\(437\) − 7406.85i − 0.810796i
\(438\) 0 0
\(439\) −11648.4 −1.26640 −0.633200 0.773988i \(-0.718259\pi\)
−0.633200 + 0.773988i \(0.718259\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) − 3627.11i − 0.390326i
\(443\) − 4156.59i − 0.445791i −0.974842 0.222896i \(-0.928449\pi\)
0.974842 0.222896i \(-0.0715509\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 4138.08 0.439336
\(447\) 0 0
\(448\) − 1199.85i − 0.126535i
\(449\) 14987.5 1.57529 0.787645 0.616129i \(-0.211300\pi\)
0.787645 + 0.616129i \(0.211300\pi\)
\(450\) 0 0
\(451\) −12596.0 −1.31513
\(452\) 1852.18i 0.192742i
\(453\) 0 0
\(454\) 10480.9 1.08347
\(455\) 0 0
\(456\) 0 0
\(457\) − 13215.5i − 1.35272i −0.736571 0.676361i \(-0.763556\pi\)
0.736571 0.676361i \(-0.236444\pi\)
\(458\) 7115.39i 0.725939i
\(459\) 0 0
\(460\) 0 0
\(461\) −6365.55 −0.643109 −0.321554 0.946891i \(-0.604205\pi\)
−0.321554 + 0.946891i \(0.604205\pi\)
\(462\) 0 0
\(463\) 13954.3i 1.40068i 0.713811 + 0.700338i \(0.246967\pi\)
−0.713811 + 0.700338i \(0.753033\pi\)
\(464\) −2259.75 −0.226091
\(465\) 0 0
\(466\) −70.9773 −0.00705571
\(467\) − 15289.1i − 1.51498i −0.652848 0.757489i \(-0.726426\pi\)
0.652848 0.757489i \(-0.273574\pi\)
\(468\) 0 0
\(469\) −10710.0 −1.05446
\(470\) 0 0
\(471\) 0 0
\(472\) − 2499.53i − 0.243750i
\(473\) 9448.31i 0.918464i
\(474\) 0 0
\(475\) 0 0
\(476\) −4029.91 −0.388047
\(477\) 0 0
\(478\) − 2341.25i − 0.224030i
\(479\) 8268.35 0.788706 0.394353 0.918959i \(-0.370969\pi\)
0.394353 + 0.918959i \(0.370969\pi\)
\(480\) 0 0
\(481\) 2066.82 0.195923
\(482\) − 6639.00i − 0.627382i
\(483\) 0 0
\(484\) −1073.09 −0.100779
\(485\) 0 0
\(486\) 0 0
\(487\) − 7144.78i − 0.664806i −0.943137 0.332403i \(-0.892141\pi\)
0.943137 0.332403i \(-0.107859\pi\)
\(488\) 4405.18i 0.408634i
\(489\) 0 0
\(490\) 0 0
\(491\) 19527.2 1.79481 0.897405 0.441209i \(-0.145450\pi\)
0.897405 + 0.441209i \(0.145450\pi\)
\(492\) 0 0
\(493\) 7589.73i 0.693355i
\(494\) −6191.94 −0.563945
\(495\) 0 0
\(496\) 4239.27 0.383768
\(497\) 3445.53i 0.310972i
\(498\) 0 0
\(499\) 12923.7 1.15941 0.579704 0.814827i \(-0.303168\pi\)
0.579704 + 0.814827i \(0.303168\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) − 11705.0i − 1.04067i
\(503\) − 14125.0i − 1.25209i −0.779787 0.626045i \(-0.784672\pi\)
0.779787 0.626045i \(-0.215328\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 6457.62 0.567345
\(507\) 0 0
\(508\) − 830.709i − 0.0725527i
\(509\) 13955.7 1.21527 0.607636 0.794215i \(-0.292118\pi\)
0.607636 + 0.794215i \(0.292118\pi\)
\(510\) 0 0
\(511\) 2343.98 0.202919
\(512\) 512.000i 0.0441942i
\(513\) 0 0
\(514\) 8479.25 0.727634
\(515\) 0 0
\(516\) 0 0
\(517\) 9728.24i 0.827558i
\(518\) − 2296.34i − 0.194779i
\(519\) 0 0
\(520\) 0 0
\(521\) −19825.1 −1.66709 −0.833544 0.552452i \(-0.813692\pi\)
−0.833544 + 0.552452i \(0.813692\pi\)
\(522\) 0 0
\(523\) 14258.7i 1.19214i 0.802933 + 0.596070i \(0.203272\pi\)
−0.802933 + 0.596070i \(0.796728\pi\)
\(524\) −10269.4 −0.856144
\(525\) 0 0
\(526\) −13405.2 −1.11121
\(527\) − 14238.3i − 1.17691i
\(528\) 0 0
\(529\) 5648.27 0.464229
\(530\) 0 0
\(531\) 0 0
\(532\) 6879.56i 0.560652i
\(533\) − 10629.6i − 0.863827i
\(534\) 0 0
\(535\) 0 0
\(536\) 4570.16 0.368285
\(537\) 0 0
\(538\) 6365.88i 0.510135i
\(539\) 339.014 0.0270916
\(540\) 0 0
\(541\) −19037.6 −1.51292 −0.756462 0.654037i \(-0.773074\pi\)
−0.756462 + 0.654037i \(0.773074\pi\)
\(542\) − 4699.45i − 0.372433i
\(543\) 0 0
\(544\) 1719.64 0.135531
\(545\) 0 0
\(546\) 0 0
\(547\) − 11176.9i − 0.873657i −0.899545 0.436828i \(-0.856102\pi\)
0.899545 0.436828i \(-0.143898\pi\)
\(548\) 63.6819i 0.00496415i
\(549\) 0 0
\(550\) 0 0
\(551\) 12956.6 1.00176
\(552\) 0 0
\(553\) 8046.82i 0.618781i
\(554\) −13309.6 −1.02070
\(555\) 0 0
\(556\) −11358.6 −0.866388
\(557\) 19583.4i 1.48972i 0.667219 + 0.744862i \(0.267485\pi\)
−0.667219 + 0.744862i \(0.732515\pi\)
\(558\) 0 0
\(559\) −7973.28 −0.603281
\(560\) 0 0
\(561\) 0 0
\(562\) − 12491.8i − 0.937603i
\(563\) − 26341.1i − 1.97184i −0.167230 0.985918i \(-0.553482\pi\)
0.167230 0.985918i \(-0.446518\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 4720.87 0.350588
\(567\) 0 0
\(568\) − 1470.27i − 0.108611i
\(569\) −5794.46 −0.426918 −0.213459 0.976952i \(-0.568473\pi\)
−0.213459 + 0.976952i \(0.568473\pi\)
\(570\) 0 0
\(571\) −17512.7 −1.28351 −0.641756 0.766909i \(-0.721794\pi\)
−0.641756 + 0.766909i \(0.721794\pi\)
\(572\) − 5398.41i − 0.394613i
\(573\) 0 0
\(574\) −11810.0 −0.858784
\(575\) 0 0
\(576\) 0 0
\(577\) − 7908.96i − 0.570631i −0.958434 0.285316i \(-0.907902\pi\)
0.958434 0.285316i \(-0.0920984\pi\)
\(578\) 4050.32i 0.291472i
\(579\) 0 0
\(580\) 0 0
\(581\) 20462.7 1.46116
\(582\) 0 0
\(583\) 7667.80i 0.544713i
\(584\) −1000.22 −0.0708721
\(585\) 0 0
\(586\) 11668.2 0.822538
\(587\) 769.500i 0.0541067i 0.999634 + 0.0270534i \(0.00861241\pi\)
−0.999634 + 0.0270534i \(0.991388\pi\)
\(588\) 0 0
\(589\) −24306.6 −1.70040
\(590\) 0 0
\(591\) 0 0
\(592\) 979.891i 0.0680292i
\(593\) 13608.5i 0.942385i 0.882030 + 0.471192i \(0.156176\pi\)
−0.882030 + 0.471192i \(0.843824\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −809.509 −0.0556355
\(597\) 0 0
\(598\) 5449.49i 0.372653i
\(599\) 1242.90 0.0847805 0.0423902 0.999101i \(-0.486503\pi\)
0.0423902 + 0.999101i \(0.486503\pi\)
\(600\) 0 0
\(601\) −8879.95 −0.602697 −0.301349 0.953514i \(-0.597437\pi\)
−0.301349 + 0.953514i \(0.597437\pi\)
\(602\) 8858.73i 0.599759i
\(603\) 0 0
\(604\) −14631.2 −0.985656
\(605\) 0 0
\(606\) 0 0
\(607\) − 17859.3i − 1.19421i −0.802163 0.597106i \(-0.796317\pi\)
0.802163 0.597106i \(-0.203683\pi\)
\(608\) − 2935.64i − 0.195815i
\(609\) 0 0
\(610\) 0 0
\(611\) −8209.52 −0.543570
\(612\) 0 0
\(613\) 8241.30i 0.543006i 0.962438 + 0.271503i \(0.0875207\pi\)
−0.962438 + 0.271503i \(0.912479\pi\)
\(614\) −2920.11 −0.191932
\(615\) 0 0
\(616\) −5997.91 −0.392309
\(617\) − 6854.93i − 0.447276i −0.974672 0.223638i \(-0.928207\pi\)
0.974672 0.223638i \(-0.0717933\pi\)
\(618\) 0 0
\(619\) 1857.93 0.120641 0.0603203 0.998179i \(-0.480788\pi\)
0.0603203 + 0.998179i \(0.480788\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) − 8400.49i − 0.541525i
\(623\) − 23127.1i − 1.48727i
\(624\) 0 0
\(625\) 0 0
\(626\) 11780.3 0.752133
\(627\) 0 0
\(628\) − 9089.15i − 0.577542i
\(629\) 3291.12 0.208626
\(630\) 0 0
\(631\) −8079.78 −0.509748 −0.254874 0.966974i \(-0.582034\pi\)
−0.254874 + 0.966974i \(0.582034\pi\)
\(632\) − 3433.73i − 0.216118i
\(633\) 0 0
\(634\) −3014.07 −0.188807
\(635\) 0 0
\(636\) 0 0
\(637\) 286.089i 0.0177947i
\(638\) 11296.2i 0.700971i
\(639\) 0 0
\(640\) 0 0
\(641\) 21810.4 1.34393 0.671966 0.740582i \(-0.265450\pi\)
0.671966 + 0.740582i \(0.265450\pi\)
\(642\) 0 0
\(643\) − 10487.6i − 0.643220i −0.946872 0.321610i \(-0.895776\pi\)
0.946872 0.321610i \(-0.104224\pi\)
\(644\) 6054.66 0.370477
\(645\) 0 0
\(646\) −9859.82 −0.600510
\(647\) 23644.5i 1.43672i 0.695669 + 0.718362i \(0.255108\pi\)
−0.695669 + 0.718362i \(0.744892\pi\)
\(648\) 0 0
\(649\) −12494.8 −0.755722
\(650\) 0 0
\(651\) 0 0
\(652\) 11018.7i 0.661851i
\(653\) − 9685.08i − 0.580408i −0.956965 0.290204i \(-0.906277\pi\)
0.956965 0.290204i \(-0.0937232\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 5039.56 0.299942
\(657\) 0 0
\(658\) 9121.20i 0.540397i
\(659\) −15248.7 −0.901374 −0.450687 0.892682i \(-0.648821\pi\)
−0.450687 + 0.892682i \(0.648821\pi\)
\(660\) 0 0
\(661\) −1423.58 −0.0837683 −0.0418841 0.999122i \(-0.513336\pi\)
−0.0418841 + 0.999122i \(0.513336\pi\)
\(662\) − 6513.50i − 0.382408i
\(663\) 0 0
\(664\) −8731.82 −0.510332
\(665\) 0 0
\(666\) 0 0
\(667\) − 11403.0i − 0.661961i
\(668\) − 4271.41i − 0.247404i
\(669\) 0 0
\(670\) 0 0
\(671\) 22020.9 1.26693
\(672\) 0 0
\(673\) − 16652.6i − 0.953805i −0.878956 0.476902i \(-0.841760\pi\)
0.878956 0.476902i \(-0.158240\pi\)
\(674\) 15397.4 0.879950
\(675\) 0 0
\(676\) −4232.36 −0.240804
\(677\) − 30296.0i − 1.71989i −0.510384 0.859947i \(-0.670497\pi\)
0.510384 0.859947i \(-0.329503\pi\)
\(678\) 0 0
\(679\) −28729.8 −1.62378
\(680\) 0 0
\(681\) 0 0
\(682\) − 21191.5i − 1.18983i
\(683\) 21943.5i 1.22935i 0.788780 + 0.614675i \(0.210713\pi\)
−0.788780 + 0.614675i \(0.789287\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −12543.1 −0.698101
\(687\) 0 0
\(688\) − 3780.18i − 0.209474i
\(689\) −6470.74 −0.357788
\(690\) 0 0
\(691\) 16153.5 0.889300 0.444650 0.895704i \(-0.353328\pi\)
0.444650 + 0.895704i \(0.353328\pi\)
\(692\) 11853.9i 0.651180i
\(693\) 0 0
\(694\) 19091.5 1.04424
\(695\) 0 0
\(696\) 0 0
\(697\) − 16926.2i − 0.919836i
\(698\) 5852.95i 0.317389i
\(699\) 0 0
\(700\) 0 0
\(701\) −31969.9 −1.72252 −0.861261 0.508163i \(-0.830325\pi\)
−0.861261 + 0.508163i \(0.830325\pi\)
\(702\) 0 0
\(703\) − 5618.37i − 0.301423i
\(704\) 2559.42 0.137019
\(705\) 0 0
\(706\) 9359.39 0.498931
\(707\) − 6723.66i − 0.357665i
\(708\) 0 0
\(709\) 27367.1 1.44964 0.724819 0.688939i \(-0.241923\pi\)
0.724819 + 0.688939i \(0.241923\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 9868.75i 0.519448i
\(713\) 21392.1i 1.12362i
\(714\) 0 0
\(715\) 0 0
\(716\) −12348.6 −0.644536
\(717\) 0 0
\(718\) 24505.6i 1.27374i
\(719\) 3379.36 0.175283 0.0876417 0.996152i \(-0.472067\pi\)
0.0876417 + 0.996152i \(0.472067\pi\)
\(720\) 0 0
\(721\) −8202.39 −0.423679
\(722\) 3113.95i 0.160512i
\(723\) 0 0
\(724\) −16320.1 −0.837753
\(725\) 0 0
\(726\) 0 0
\(727\) 2831.94i 0.144472i 0.997388 + 0.0722358i \(0.0230134\pi\)
−0.997388 + 0.0722358i \(0.976987\pi\)
\(728\) − 5061.55i − 0.257683i
\(729\) 0 0
\(730\) 0 0
\(731\) −12696.4 −0.642397
\(732\) 0 0
\(733\) − 25298.0i − 1.27476i −0.770548 0.637382i \(-0.780017\pi\)
0.770548 0.637382i \(-0.219983\pi\)
\(734\) −23332.2 −1.17331
\(735\) 0 0
\(736\) −2583.64 −0.129394
\(737\) − 22845.6i − 1.14183i
\(738\) 0 0
\(739\) −7545.60 −0.375601 −0.187801 0.982207i \(-0.560136\pi\)
−0.187801 + 0.982207i \(0.560136\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 7189.33i 0.355699i
\(743\) − 26471.0i − 1.30704i −0.756911 0.653518i \(-0.773293\pi\)
0.756911 0.653518i \(-0.226707\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −3086.65 −0.151488
\(747\) 0 0
\(748\) − 8596.23i − 0.420199i
\(749\) 17300.7 0.843999
\(750\) 0 0
\(751\) −20997.8 −1.02027 −0.510133 0.860095i \(-0.670404\pi\)
−0.510133 + 0.860095i \(0.670404\pi\)
\(752\) − 3892.18i − 0.188741i
\(753\) 0 0
\(754\) −9532.66 −0.460423
\(755\) 0 0
\(756\) 0 0
\(757\) − 37735.2i − 1.81177i −0.423524 0.905885i \(-0.639207\pi\)
0.423524 0.905885i \(-0.360793\pi\)
\(758\) 2842.50i 0.136206i
\(759\) 0 0
\(760\) 0 0
\(761\) 4449.85 0.211967 0.105984 0.994368i \(-0.466201\pi\)
0.105984 + 0.994368i \(0.466201\pi\)
\(762\) 0 0
\(763\) 22860.7i 1.08468i
\(764\) −10688.1 −0.506126
\(765\) 0 0
\(766\) 17570.8 0.828799
\(767\) − 10544.2i − 0.496386i
\(768\) 0 0
\(769\) −6269.70 −0.294007 −0.147003 0.989136i \(-0.546963\pi\)
−0.147003 + 0.989136i \(0.546963\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) − 18037.8i − 0.840927i
\(773\) 23242.7i 1.08148i 0.841190 + 0.540739i \(0.181855\pi\)
−0.841190 + 0.540739i \(0.818145\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 12259.5 0.567128
\(777\) 0 0
\(778\) − 11110.1i − 0.511976i
\(779\) −28895.2 −1.32898
\(780\) 0 0
\(781\) −7349.69 −0.336738
\(782\) 8677.57i 0.396815i
\(783\) 0 0
\(784\) −135.636 −0.00617877
\(785\) 0 0
\(786\) 0 0
\(787\) − 32584.4i − 1.47587i −0.674873 0.737934i \(-0.735802\pi\)
0.674873 0.737934i \(-0.264198\pi\)
\(788\) − 9007.27i − 0.407196i
\(789\) 0 0
\(790\) 0 0
\(791\) 8681.05 0.390218
\(792\) 0 0
\(793\) 18583.1i 0.832163i
\(794\) 22966.3 1.02650
\(795\) 0 0
\(796\) 6195.73 0.275882
\(797\) − 6338.31i − 0.281699i −0.990031 0.140850i \(-0.955017\pi\)
0.990031 0.140850i \(-0.0449834\pi\)
\(798\) 0 0
\(799\) −13072.5 −0.578815
\(800\) 0 0
\(801\) 0 0
\(802\) 24005.8i 1.05695i
\(803\) 4999.95i 0.219732i
\(804\) 0 0
\(805\) 0 0
\(806\) 17883.2 0.781526
\(807\) 0 0
\(808\) 2869.11i 0.124919i
\(809\) 17364.4 0.754635 0.377317 0.926084i \(-0.376847\pi\)
0.377317 + 0.926084i \(0.376847\pi\)
\(810\) 0 0
\(811\) 8146.53 0.352729 0.176365 0.984325i \(-0.443566\pi\)
0.176365 + 0.984325i \(0.443566\pi\)
\(812\) 10591.3i 0.457735i
\(813\) 0 0
\(814\) 4898.34 0.210917
\(815\) 0 0
\(816\) 0 0
\(817\) 21674.3i 0.928137i
\(818\) − 19540.6i − 0.835235i
\(819\) 0 0
\(820\) 0 0
\(821\) −17433.3 −0.741078 −0.370539 0.928817i \(-0.620827\pi\)
−0.370539 + 0.928817i \(0.620827\pi\)
\(822\) 0 0
\(823\) − 29692.7i − 1.25762i −0.777559 0.628810i \(-0.783542\pi\)
0.777559 0.628810i \(-0.216458\pi\)
\(824\) 3500.11 0.147976
\(825\) 0 0
\(826\) −11715.1 −0.493488
\(827\) 9694.27i 0.407621i 0.979010 + 0.203811i \(0.0653327\pi\)
−0.979010 + 0.203811i \(0.934667\pi\)
\(828\) 0 0
\(829\) 10975.2 0.459814 0.229907 0.973213i \(-0.426158\pi\)
0.229907 + 0.973213i \(0.426158\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 2159.85i 0.0899994i
\(833\) 455.557i 0.0189485i
\(834\) 0 0
\(835\) 0 0
\(836\) −14674.8 −0.607105
\(837\) 0 0
\(838\) − 16111.6i − 0.664159i
\(839\) 5314.17 0.218671 0.109336 0.994005i \(-0.465128\pi\)
0.109336 + 0.994005i \(0.465128\pi\)
\(840\) 0 0
\(841\) −4441.93 −0.182128
\(842\) 9935.98i 0.406670i
\(843\) 0 0
\(844\) −8922.45 −0.363890
\(845\) 0 0
\(846\) 0 0
\(847\) 5029.50i 0.204033i
\(848\) − 3067.82i − 0.124233i
\(849\) 0 0
\(850\) 0 0
\(851\) −4944.69 −0.199180
\(852\) 0 0
\(853\) 12904.3i 0.517978i 0.965880 + 0.258989i \(0.0833893\pi\)
−0.965880 + 0.258989i \(0.916611\pi\)
\(854\) 20646.8 0.827305
\(855\) 0 0
\(856\) −7382.55 −0.294778
\(857\) 33810.1i 1.34764i 0.738893 + 0.673822i \(0.235349\pi\)
−0.738893 + 0.673822i \(0.764651\pi\)
\(858\) 0 0
\(859\) 13512.0 0.536698 0.268349 0.963322i \(-0.413522\pi\)
0.268349 + 0.963322i \(0.413522\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) − 21653.1i − 0.855577i
\(863\) 8599.41i 0.339197i 0.985513 + 0.169599i \(0.0542471\pi\)
−0.985513 + 0.169599i \(0.945753\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 4077.25 0.159989
\(867\) 0 0
\(868\) − 19869.2i − 0.776963i
\(869\) −17164.7 −0.670050
\(870\) 0 0
\(871\) 19279.1 0.749996
\(872\) − 9755.09i − 0.378841i
\(873\) 0 0
\(874\) 14813.7 0.573319
\(875\) 0 0
\(876\) 0 0
\(877\) 35722.7i 1.37545i 0.725971 + 0.687725i \(0.241390\pi\)
−0.725971 + 0.687725i \(0.758610\pi\)
\(878\) − 23296.9i − 0.895480i
\(879\) 0 0
\(880\) 0 0
\(881\) −44841.2 −1.71480 −0.857400 0.514650i \(-0.827922\pi\)
−0.857400 + 0.514650i \(0.827922\pi\)
\(882\) 0 0
\(883\) − 22688.4i − 0.864694i −0.901707 0.432347i \(-0.857685\pi\)
0.901707 0.432347i \(-0.142315\pi\)
\(884\) 7254.23 0.276002
\(885\) 0 0
\(886\) 8313.18 0.315222
\(887\) 14717.2i 0.557110i 0.960420 + 0.278555i \(0.0898554\pi\)
−0.960420 + 0.278555i \(0.910145\pi\)
\(888\) 0 0
\(889\) −3893.48 −0.146888
\(890\) 0 0
\(891\) 0 0
\(892\) 8276.15i 0.310657i
\(893\) 22316.5i 0.836273i
\(894\) 0 0
\(895\) 0 0
\(896\) 2399.71 0.0894739
\(897\) 0 0
\(898\) 29975.1i 1.11390i
\(899\) −37420.6 −1.38826
\(900\) 0 0
\(901\) −10303.8 −0.380986
\(902\) − 25192.1i − 0.929939i
\(903\) 0 0
\(904\) −3704.36 −0.136289
\(905\) 0 0
\(906\) 0 0
\(907\) 43973.3i 1.60982i 0.593394 + 0.804912i \(0.297787\pi\)
−0.593394 + 0.804912i \(0.702213\pi\)
\(908\) 20961.8i 0.766126i
\(909\) 0 0
\(910\) 0 0
\(911\) 46506.3 1.69135 0.845677 0.533695i \(-0.179197\pi\)
0.845677 + 0.533695i \(0.179197\pi\)
\(912\) 0 0
\(913\) 43649.2i 1.58223i
\(914\) 26430.9 0.956518
\(915\) 0 0
\(916\) −14230.8 −0.513317
\(917\) 48131.8i 1.73332i
\(918\) 0 0
\(919\) −22676.1 −0.813945 −0.406973 0.913440i \(-0.633416\pi\)
−0.406973 + 0.913440i \(0.633416\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) − 12731.1i − 0.454747i
\(923\) − 6202.30i − 0.221182i
\(924\) 0 0
\(925\) 0 0
\(926\) −27908.7 −0.990427
\(927\) 0 0
\(928\) − 4519.49i − 0.159870i
\(929\) 31583.2 1.11541 0.557703 0.830040i \(-0.311683\pi\)
0.557703 + 0.830040i \(0.311683\pi\)
\(930\) 0 0
\(931\) 777.693 0.0273769
\(932\) − 141.955i − 0.00498914i
\(933\) 0 0
\(934\) 30578.2 1.07125
\(935\) 0 0
\(936\) 0 0
\(937\) − 6038.94i − 0.210548i −0.994443 0.105274i \(-0.966428\pi\)
0.994443 0.105274i \(-0.0335720\pi\)
\(938\) − 21420.0i − 0.745618i
\(939\) 0 0
\(940\) 0 0
\(941\) −40751.5 −1.41175 −0.705877 0.708335i \(-0.749447\pi\)
−0.705877 + 0.708335i \(0.749447\pi\)
\(942\) 0 0
\(943\) 25430.5i 0.878187i
\(944\) 4999.05 0.172357
\(945\) 0 0
\(946\) −18896.6 −0.649452
\(947\) − 18283.2i − 0.627377i −0.949526 0.313688i \(-0.898435\pi\)
0.949526 0.313688i \(-0.101565\pi\)
\(948\) 0 0
\(949\) −4219.39 −0.144328
\(950\) 0 0
\(951\) 0 0
\(952\) − 8059.82i − 0.274391i
\(953\) 31486.4i 1.07025i 0.844774 + 0.535123i \(0.179735\pi\)
−0.844774 + 0.535123i \(0.820265\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 4682.50 0.158413
\(957\) 0 0
\(958\) 16536.7i 0.557700i
\(959\) 298.473 0.0100502
\(960\) 0 0
\(961\) 40409.9 1.35645
\(962\) 4133.64i 0.138538i
\(963\) 0 0
\(964\) 13278.0 0.443626
\(965\) 0 0
\(966\) 0 0
\(967\) 30578.2i 1.01689i 0.861096 + 0.508443i \(0.169779\pi\)
−0.861096 + 0.508443i \(0.830221\pi\)
\(968\) − 2146.18i − 0.0712613i
\(969\) 0 0
\(970\) 0 0
\(971\) −8854.87 −0.292653 −0.146327 0.989236i \(-0.546745\pi\)
−0.146327 + 0.989236i \(0.546745\pi\)
\(972\) 0 0
\(973\) 53236.9i 1.75406i
\(974\) 14289.6 0.470089
\(975\) 0 0
\(976\) −8810.36 −0.288948
\(977\) 24452.9i 0.800736i 0.916355 + 0.400368i \(0.131118\pi\)
−0.916355 + 0.400368i \(0.868882\pi\)
\(978\) 0 0
\(979\) 49332.5 1.61049
\(980\) 0 0
\(981\) 0 0
\(982\) 39054.4i 1.26912i
\(983\) − 5512.42i − 0.178860i −0.995993 0.0894298i \(-0.971496\pi\)
0.995993 0.0894298i \(-0.0285045\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) −15179.5 −0.490276
\(987\) 0 0
\(988\) − 12383.9i − 0.398769i
\(989\) 19075.4 0.613309
\(990\) 0 0
\(991\) 22724.4 0.728420 0.364210 0.931317i \(-0.381339\pi\)
0.364210 + 0.931317i \(0.381339\pi\)
\(992\) 8478.55i 0.271365i
\(993\) 0 0
\(994\) −6891.07 −0.219891
\(995\) 0 0
\(996\) 0 0
\(997\) 10218.0i 0.324582i 0.986743 + 0.162291i \(0.0518883\pi\)
−0.986743 + 0.162291i \(0.948112\pi\)
\(998\) 25847.4i 0.819825i
\(999\) 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 1350.4.c.ba.649.4 4
3.2 odd 2 1350.4.c.v.649.2 4
5.2 odd 4 1350.4.a.be.1.1 2
5.3 odd 4 1350.4.a.bn.1.2 yes 2
5.4 even 2 inner 1350.4.c.ba.649.1 4
15.2 even 4 1350.4.a.bl.1.1 yes 2
15.8 even 4 1350.4.a.bg.1.2 yes 2
15.14 odd 2 1350.4.c.v.649.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1350.4.a.be.1.1 2 5.2 odd 4
1350.4.a.bg.1.2 yes 2 15.8 even 4
1350.4.a.bl.1.1 yes 2 15.2 even 4
1350.4.a.bn.1.2 yes 2 5.3 odd 4
1350.4.c.v.649.2 4 3.2 odd 2
1350.4.c.v.649.3 4 15.14 odd 2
1350.4.c.ba.649.1 4 5.4 even 2 inner
1350.4.c.ba.649.4 4 1.1 even 1 trivial