Properties

Label 225.4.b.d.199.1
Level $225$
Weight $4$
Character 225.199
Analytic conductor $13.275$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [225,4,Mod(199,225)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(225, 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("225.199");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 225 = 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 225.b (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: \(13.2754297513\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 15)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 199.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 225.199
Dual form 225.4.b.d.199.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-3.00000i q^{2} -1.00000 q^{4} +20.0000i q^{7} -21.0000i q^{8} +O(q^{10})\) \(q-3.00000i q^{2} -1.00000 q^{4} +20.0000i q^{7} -21.0000i q^{8} +24.0000 q^{11} -74.0000i q^{13} +60.0000 q^{14} -71.0000 q^{16} -54.0000i q^{17} +124.000 q^{19} -72.0000i q^{22} -120.000i q^{23} -222.000 q^{26} -20.0000i q^{28} -78.0000 q^{29} +200.000 q^{31} +45.0000i q^{32} -162.000 q^{34} -70.0000i q^{37} -372.000i q^{38} -330.000 q^{41} -92.0000i q^{43} -24.0000 q^{44} -360.000 q^{46} +24.0000i q^{47} -57.0000 q^{49} +74.0000i q^{52} +450.000i q^{53} +420.000 q^{56} +234.000i q^{58} +24.0000 q^{59} -322.000 q^{61} -600.000i q^{62} -433.000 q^{64} -196.000i q^{67} +54.0000i q^{68} +288.000 q^{71} +430.000i q^{73} -210.000 q^{74} -124.000 q^{76} +480.000i q^{77} +520.000 q^{79} +990.000i q^{82} +156.000i q^{83} -276.000 q^{86} -504.000i q^{88} +1026.00 q^{89} +1480.00 q^{91} +120.000i q^{92} +72.0000 q^{94} -286.000i q^{97} +171.000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{4} + 48 q^{11} + 120 q^{14} - 142 q^{16} + 248 q^{19} - 444 q^{26} - 156 q^{29} + 400 q^{31} - 324 q^{34} - 660 q^{41} - 48 q^{44} - 720 q^{46} - 114 q^{49} + 840 q^{56} + 48 q^{59} - 644 q^{61} - 866 q^{64} + 576 q^{71} - 420 q^{74} - 248 q^{76} + 1040 q^{79} - 552 q^{86} + 2052 q^{89} + 2960 q^{91} + 144 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/225\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(127\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 3.00000i − 1.06066i −0.847791 0.530330i \(-0.822068\pi\)
0.847791 0.530330i \(-0.177932\pi\)
\(3\) 0 0
\(4\) −1.00000 −0.125000
\(5\) 0 0
\(6\) 0 0
\(7\) 20.0000i 1.07990i 0.841698 + 0.539949i \(0.181557\pi\)
−0.841698 + 0.539949i \(0.818443\pi\)
\(8\) − 21.0000i − 0.928078i
\(9\) 0 0
\(10\) 0 0
\(11\) 24.0000 0.657843 0.328921 0.944357i \(-0.393315\pi\)
0.328921 + 0.944357i \(0.393315\pi\)
\(12\) 0 0
\(13\) − 74.0000i − 1.57876i −0.613904 0.789381i \(-0.710402\pi\)
0.613904 0.789381i \(-0.289598\pi\)
\(14\) 60.0000 1.14541
\(15\) 0 0
\(16\) −71.0000 −1.10938
\(17\) − 54.0000i − 0.770407i −0.922832 0.385204i \(-0.874131\pi\)
0.922832 0.385204i \(-0.125869\pi\)
\(18\) 0 0
\(19\) 124.000 1.49724 0.748620 0.663000i \(-0.230717\pi\)
0.748620 + 0.663000i \(0.230717\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) − 72.0000i − 0.697748i
\(23\) − 120.000i − 1.08790i −0.839117 0.543951i \(-0.816928\pi\)
0.839117 0.543951i \(-0.183072\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −222.000 −1.67453
\(27\) 0 0
\(28\) − 20.0000i − 0.134987i
\(29\) −78.0000 −0.499456 −0.249728 0.968316i \(-0.580341\pi\)
−0.249728 + 0.968316i \(0.580341\pi\)
\(30\) 0 0
\(31\) 200.000 1.15874 0.579372 0.815063i \(-0.303298\pi\)
0.579372 + 0.815063i \(0.303298\pi\)
\(32\) 45.0000i 0.248592i
\(33\) 0 0
\(34\) −162.000 −0.817140
\(35\) 0 0
\(36\) 0 0
\(37\) − 70.0000i − 0.311025i −0.987834 0.155513i \(-0.950297\pi\)
0.987834 0.155513i \(-0.0497029\pi\)
\(38\) − 372.000i − 1.58806i
\(39\) 0 0
\(40\) 0 0
\(41\) −330.000 −1.25701 −0.628504 0.777806i \(-0.716332\pi\)
−0.628504 + 0.777806i \(0.716332\pi\)
\(42\) 0 0
\(43\) − 92.0000i − 0.326276i −0.986603 0.163138i \(-0.947838\pi\)
0.986603 0.163138i \(-0.0521616\pi\)
\(44\) −24.0000 −0.0822304
\(45\) 0 0
\(46\) −360.000 −1.15389
\(47\) 24.0000i 0.0744843i 0.999306 + 0.0372421i \(0.0118573\pi\)
−0.999306 + 0.0372421i \(0.988143\pi\)
\(48\) 0 0
\(49\) −57.0000 −0.166181
\(50\) 0 0
\(51\) 0 0
\(52\) 74.0000i 0.197345i
\(53\) 450.000i 1.16627i 0.812376 + 0.583134i \(0.198174\pi\)
−0.812376 + 0.583134i \(0.801826\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 420.000 1.00223
\(57\) 0 0
\(58\) 234.000i 0.529754i
\(59\) 24.0000 0.0529582 0.0264791 0.999649i \(-0.491570\pi\)
0.0264791 + 0.999649i \(0.491570\pi\)
\(60\) 0 0
\(61\) −322.000 −0.675867 −0.337933 0.941170i \(-0.609728\pi\)
−0.337933 + 0.941170i \(0.609728\pi\)
\(62\) − 600.000i − 1.22903i
\(63\) 0 0
\(64\) −433.000 −0.845703
\(65\) 0 0
\(66\) 0 0
\(67\) − 196.000i − 0.357391i −0.983904 0.178696i \(-0.942812\pi\)
0.983904 0.178696i \(-0.0571877\pi\)
\(68\) 54.0000i 0.0963009i
\(69\) 0 0
\(70\) 0 0
\(71\) 288.000 0.481399 0.240699 0.970600i \(-0.422623\pi\)
0.240699 + 0.970600i \(0.422623\pi\)
\(72\) 0 0
\(73\) 430.000i 0.689420i 0.938709 + 0.344710i \(0.112023\pi\)
−0.938709 + 0.344710i \(0.887977\pi\)
\(74\) −210.000 −0.329892
\(75\) 0 0
\(76\) −124.000 −0.187155
\(77\) 480.000i 0.710404i
\(78\) 0 0
\(79\) 520.000 0.740564 0.370282 0.928919i \(-0.379261\pi\)
0.370282 + 0.928919i \(0.379261\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 990.000i 1.33326i
\(83\) 156.000i 0.206304i 0.994666 + 0.103152i \(0.0328928\pi\)
−0.994666 + 0.103152i \(0.967107\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −276.000 −0.346068
\(87\) 0 0
\(88\) − 504.000i − 0.610529i
\(89\) 1026.00 1.22198 0.610988 0.791640i \(-0.290773\pi\)
0.610988 + 0.791640i \(0.290773\pi\)
\(90\) 0 0
\(91\) 1480.00 1.70490
\(92\) 120.000i 0.135988i
\(93\) 0 0
\(94\) 72.0000 0.0790025
\(95\) 0 0
\(96\) 0 0
\(97\) − 286.000i − 0.299370i −0.988734 0.149685i \(-0.952174\pi\)
0.988734 0.149685i \(-0.0478260\pi\)
\(98\) 171.000i 0.176261i
\(99\) 0 0
\(100\) 0 0
\(101\) 1734.00 1.70831 0.854156 0.520017i \(-0.174075\pi\)
0.854156 + 0.520017i \(0.174075\pi\)
\(102\) 0 0
\(103\) − 452.000i − 0.432397i −0.976349 0.216198i \(-0.930634\pi\)
0.976349 0.216198i \(-0.0693658\pi\)
\(104\) −1554.00 −1.46521
\(105\) 0 0
\(106\) 1350.00 1.23702
\(107\) 1404.00i 1.26850i 0.773127 + 0.634251i \(0.218692\pi\)
−0.773127 + 0.634251i \(0.781308\pi\)
\(108\) 0 0
\(109\) 1474.00 1.29526 0.647631 0.761954i \(-0.275760\pi\)
0.647631 + 0.761954i \(0.275760\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) − 1420.00i − 1.19801i
\(113\) 1086.00i 0.904091i 0.891995 + 0.452046i \(0.149306\pi\)
−0.891995 + 0.452046i \(0.850694\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 78.0000 0.0624321
\(117\) 0 0
\(118\) − 72.0000i − 0.0561707i
\(119\) 1080.00 0.831962
\(120\) 0 0
\(121\) −755.000 −0.567243
\(122\) 966.000i 0.716865i
\(123\) 0 0
\(124\) −200.000 −0.144843
\(125\) 0 0
\(126\) 0 0
\(127\) 1244.00i 0.869190i 0.900626 + 0.434595i \(0.143109\pi\)
−0.900626 + 0.434595i \(0.856891\pi\)
\(128\) 1659.00i 1.14560i
\(129\) 0 0
\(130\) 0 0
\(131\) −2328.00 −1.55266 −0.776329 0.630327i \(-0.782921\pi\)
−0.776329 + 0.630327i \(0.782921\pi\)
\(132\) 0 0
\(133\) 2480.00i 1.61687i
\(134\) −588.000 −0.379071
\(135\) 0 0
\(136\) −1134.00 −0.714998
\(137\) − 2118.00i − 1.32082i −0.750903 0.660412i \(-0.770382\pi\)
0.750903 0.660412i \(-0.229618\pi\)
\(138\) 0 0
\(139\) −2324.00 −1.41812 −0.709062 0.705147i \(-0.750881\pi\)
−0.709062 + 0.705147i \(0.750881\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) − 864.000i − 0.510600i
\(143\) − 1776.00i − 1.03858i
\(144\) 0 0
\(145\) 0 0
\(146\) 1290.00 0.731241
\(147\) 0 0
\(148\) 70.0000i 0.0388781i
\(149\) 258.000 0.141854 0.0709268 0.997482i \(-0.477404\pi\)
0.0709268 + 0.997482i \(0.477404\pi\)
\(150\) 0 0
\(151\) −808.000 −0.435458 −0.217729 0.976009i \(-0.569865\pi\)
−0.217729 + 0.976009i \(0.569865\pi\)
\(152\) − 2604.00i − 1.38955i
\(153\) 0 0
\(154\) 1440.00 0.753497
\(155\) 0 0
\(156\) 0 0
\(157\) 2378.00i 1.20882i 0.796673 + 0.604411i \(0.206592\pi\)
−0.796673 + 0.604411i \(0.793408\pi\)
\(158\) − 1560.00i − 0.785487i
\(159\) 0 0
\(160\) 0 0
\(161\) 2400.00 1.17482
\(162\) 0 0
\(163\) 52.0000i 0.0249874i 0.999922 + 0.0124937i \(0.00397698\pi\)
−0.999922 + 0.0124937i \(0.996023\pi\)
\(164\) 330.000 0.157126
\(165\) 0 0
\(166\) 468.000 0.218818
\(167\) 3720.00i 1.72373i 0.507141 + 0.861863i \(0.330702\pi\)
−0.507141 + 0.861863i \(0.669298\pi\)
\(168\) 0 0
\(169\) −3279.00 −1.49249
\(170\) 0 0
\(171\) 0 0
\(172\) 92.0000i 0.0407845i
\(173\) 426.000i 0.187215i 0.995609 + 0.0936075i \(0.0298399\pi\)
−0.995609 + 0.0936075i \(0.970160\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −1704.00 −0.729795
\(177\) 0 0
\(178\) − 3078.00i − 1.29610i
\(179\) −1440.00 −0.601289 −0.300644 0.953736i \(-0.597202\pi\)
−0.300644 + 0.953736i \(0.597202\pi\)
\(180\) 0 0
\(181\) −3130.00 −1.28537 −0.642683 0.766133i \(-0.722179\pi\)
−0.642683 + 0.766133i \(0.722179\pi\)
\(182\) − 4440.00i − 1.80832i
\(183\) 0 0
\(184\) −2520.00 −1.00966
\(185\) 0 0
\(186\) 0 0
\(187\) − 1296.00i − 0.506807i
\(188\) − 24.0000i − 0.00931053i
\(189\) 0 0
\(190\) 0 0
\(191\) −3576.00 −1.35471 −0.677357 0.735655i \(-0.736875\pi\)
−0.677357 + 0.735655i \(0.736875\pi\)
\(192\) 0 0
\(193\) − 2666.00i − 0.994315i −0.867660 0.497158i \(-0.834377\pi\)
0.867660 0.497158i \(-0.165623\pi\)
\(194\) −858.000 −0.317530
\(195\) 0 0
\(196\) 57.0000 0.0207726
\(197\) 2718.00i 0.982992i 0.870880 + 0.491496i \(0.163550\pi\)
−0.870880 + 0.491496i \(0.836450\pi\)
\(198\) 0 0
\(199\) 3832.00 1.36504 0.682521 0.730866i \(-0.260884\pi\)
0.682521 + 0.730866i \(0.260884\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) − 5202.00i − 1.81194i
\(203\) − 1560.00i − 0.539362i
\(204\) 0 0
\(205\) 0 0
\(206\) −1356.00 −0.458626
\(207\) 0 0
\(208\) 5254.00i 1.75144i
\(209\) 2976.00 0.984948
\(210\) 0 0
\(211\) 1100.00 0.358896 0.179448 0.983767i \(-0.442569\pi\)
0.179448 + 0.983767i \(0.442569\pi\)
\(212\) − 450.000i − 0.145784i
\(213\) 0 0
\(214\) 4212.00 1.34545
\(215\) 0 0
\(216\) 0 0
\(217\) 4000.00i 1.25133i
\(218\) − 4422.00i − 1.37383i
\(219\) 0 0
\(220\) 0 0
\(221\) −3996.00 −1.21629
\(222\) 0 0
\(223\) − 1964.00i − 0.589772i −0.955532 0.294886i \(-0.904718\pi\)
0.955532 0.294886i \(-0.0952817\pi\)
\(224\) −900.000 −0.268454
\(225\) 0 0
\(226\) 3258.00 0.958933
\(227\) − 660.000i − 0.192977i −0.995334 0.0964884i \(-0.969239\pi\)
0.995334 0.0964884i \(-0.0307611\pi\)
\(228\) 0 0
\(229\) 1906.00 0.550009 0.275004 0.961443i \(-0.411321\pi\)
0.275004 + 0.961443i \(0.411321\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 1638.00i 0.463534i
\(233\) − 1458.00i − 0.409943i −0.978768 0.204972i \(-0.934290\pi\)
0.978768 0.204972i \(-0.0657102\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −24.0000 −0.00661978
\(237\) 0 0
\(238\) − 3240.00i − 0.882429i
\(239\) 1176.00 0.318281 0.159140 0.987256i \(-0.449128\pi\)
0.159140 + 0.987256i \(0.449128\pi\)
\(240\) 0 0
\(241\) 866.000 0.231469 0.115734 0.993280i \(-0.463078\pi\)
0.115734 + 0.993280i \(0.463078\pi\)
\(242\) 2265.00i 0.601652i
\(243\) 0 0
\(244\) 322.000 0.0844834
\(245\) 0 0
\(246\) 0 0
\(247\) − 9176.00i − 2.36379i
\(248\) − 4200.00i − 1.07540i
\(249\) 0 0
\(250\) 0 0
\(251\) −432.000 −0.108636 −0.0543179 0.998524i \(-0.517298\pi\)
−0.0543179 + 0.998524i \(0.517298\pi\)
\(252\) 0 0
\(253\) − 2880.00i − 0.715668i
\(254\) 3732.00 0.921915
\(255\) 0 0
\(256\) 1513.00 0.369385
\(257\) − 2526.00i − 0.613103i −0.951854 0.306552i \(-0.900825\pi\)
0.951854 0.306552i \(-0.0991752\pi\)
\(258\) 0 0
\(259\) 1400.00 0.335876
\(260\) 0 0
\(261\) 0 0
\(262\) 6984.00i 1.64684i
\(263\) 5448.00i 1.27733i 0.769484 + 0.638666i \(0.220513\pi\)
−0.769484 + 0.638666i \(0.779487\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 7440.00 1.71495
\(267\) 0 0
\(268\) 196.000i 0.0446739i
\(269\) −2574.00 −0.583418 −0.291709 0.956507i \(-0.594224\pi\)
−0.291709 + 0.956507i \(0.594224\pi\)
\(270\) 0 0
\(271\) −3184.00 −0.713706 −0.356853 0.934161i \(-0.616150\pi\)
−0.356853 + 0.934161i \(0.616150\pi\)
\(272\) 3834.00i 0.854671i
\(273\) 0 0
\(274\) −6354.00 −1.40095
\(275\) 0 0
\(276\) 0 0
\(277\) 3962.00i 0.859399i 0.902972 + 0.429699i \(0.141380\pi\)
−0.902972 + 0.429699i \(0.858620\pi\)
\(278\) 6972.00i 1.50415i
\(279\) 0 0
\(280\) 0 0
\(281\) 8286.00 1.75908 0.879540 0.475825i \(-0.157851\pi\)
0.879540 + 0.475825i \(0.157851\pi\)
\(282\) 0 0
\(283\) 2716.00i 0.570493i 0.958454 + 0.285246i \(0.0920754\pi\)
−0.958454 + 0.285246i \(0.907925\pi\)
\(284\) −288.000 −0.0601748
\(285\) 0 0
\(286\) −5328.00 −1.10158
\(287\) − 6600.00i − 1.35744i
\(288\) 0 0
\(289\) 1997.00 0.406473
\(290\) 0 0
\(291\) 0 0
\(292\) − 430.000i − 0.0861776i
\(293\) 6018.00i 1.19992i 0.800032 + 0.599958i \(0.204816\pi\)
−0.800032 + 0.599958i \(0.795184\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −1470.00 −0.288655
\(297\) 0 0
\(298\) − 774.000i − 0.150458i
\(299\) −8880.00 −1.71754
\(300\) 0 0
\(301\) 1840.00 0.352345
\(302\) 2424.00i 0.461873i
\(303\) 0 0
\(304\) −8804.00 −1.66100
\(305\) 0 0
\(306\) 0 0
\(307\) 9236.00i 1.71702i 0.512793 + 0.858512i \(0.328611\pi\)
−0.512793 + 0.858512i \(0.671389\pi\)
\(308\) − 480.000i − 0.0888004i
\(309\) 0 0
\(310\) 0 0
\(311\) −1536.00 −0.280060 −0.140030 0.990147i \(-0.544720\pi\)
−0.140030 + 0.990147i \(0.544720\pi\)
\(312\) 0 0
\(313\) 7342.00i 1.32586i 0.748681 + 0.662930i \(0.230687\pi\)
−0.748681 + 0.662930i \(0.769313\pi\)
\(314\) 7134.00 1.28215
\(315\) 0 0
\(316\) −520.000 −0.0925705
\(317\) 3894.00i 0.689933i 0.938615 + 0.344967i \(0.112110\pi\)
−0.938615 + 0.344967i \(0.887890\pi\)
\(318\) 0 0
\(319\) −1872.00 −0.328564
\(320\) 0 0
\(321\) 0 0
\(322\) − 7200.00i − 1.24609i
\(323\) − 6696.00i − 1.15348i
\(324\) 0 0
\(325\) 0 0
\(326\) 156.000 0.0265032
\(327\) 0 0
\(328\) 6930.00i 1.16660i
\(329\) −480.000 −0.0804354
\(330\) 0 0
\(331\) 3692.00 0.613084 0.306542 0.951857i \(-0.400828\pi\)
0.306542 + 0.951857i \(0.400828\pi\)
\(332\) − 156.000i − 0.0257880i
\(333\) 0 0
\(334\) 11160.0 1.82829
\(335\) 0 0
\(336\) 0 0
\(337\) − 8998.00i − 1.45446i −0.686395 0.727229i \(-0.740808\pi\)
0.686395 0.727229i \(-0.259192\pi\)
\(338\) 9837.00i 1.58302i
\(339\) 0 0
\(340\) 0 0
\(341\) 4800.00 0.762271
\(342\) 0 0
\(343\) 5720.00i 0.900440i
\(344\) −1932.00 −0.302809
\(345\) 0 0
\(346\) 1278.00 0.198571
\(347\) − 5244.00i − 0.811276i −0.914034 0.405638i \(-0.867049\pi\)
0.914034 0.405638i \(-0.132951\pi\)
\(348\) 0 0
\(349\) −6302.00 −0.966585 −0.483293 0.875459i \(-0.660559\pi\)
−0.483293 + 0.875459i \(0.660559\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 1080.00i 0.163535i
\(353\) 3414.00i 0.514756i 0.966311 + 0.257378i \(0.0828586\pi\)
−0.966311 + 0.257378i \(0.917141\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −1026.00 −0.152747
\(357\) 0 0
\(358\) 4320.00i 0.637763i
\(359\) 4824.00 0.709195 0.354597 0.935019i \(-0.384618\pi\)
0.354597 + 0.935019i \(0.384618\pi\)
\(360\) 0 0
\(361\) 8517.00 1.24173
\(362\) 9390.00i 1.36334i
\(363\) 0 0
\(364\) −1480.00 −0.213113
\(365\) 0 0
\(366\) 0 0
\(367\) − 3508.00i − 0.498954i −0.968381 0.249477i \(-0.919741\pi\)
0.968381 0.249477i \(-0.0802587\pi\)
\(368\) 8520.00i 1.20689i
\(369\) 0 0
\(370\) 0 0
\(371\) −9000.00 −1.25945
\(372\) 0 0
\(373\) − 10802.0i − 1.49948i −0.661732 0.749740i \(-0.730178\pi\)
0.661732 0.749740i \(-0.269822\pi\)
\(374\) −3888.00 −0.537550
\(375\) 0 0
\(376\) 504.000 0.0691272
\(377\) 5772.00i 0.788523i
\(378\) 0 0
\(379\) −1460.00 −0.197876 −0.0989382 0.995094i \(-0.531545\pi\)
−0.0989382 + 0.995094i \(0.531545\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 10728.0i 1.43689i
\(383\) − 4872.00i − 0.649994i −0.945715 0.324997i \(-0.894637\pi\)
0.945715 0.324997i \(-0.105363\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −7998.00 −1.05463
\(387\) 0 0
\(388\) 286.000i 0.0374213i
\(389\) −14046.0 −1.83075 −0.915373 0.402606i \(-0.868104\pi\)
−0.915373 + 0.402606i \(0.868104\pi\)
\(390\) 0 0
\(391\) −6480.00 −0.838127
\(392\) 1197.00i 0.154229i
\(393\) 0 0
\(394\) 8154.00 1.04262
\(395\) 0 0
\(396\) 0 0
\(397\) − 2734.00i − 0.345631i −0.984954 0.172816i \(-0.944714\pi\)
0.984954 0.172816i \(-0.0552864\pi\)
\(398\) − 11496.0i − 1.44785i
\(399\) 0 0
\(400\) 0 0
\(401\) 15942.0 1.98530 0.992650 0.121019i \(-0.0386161\pi\)
0.992650 + 0.121019i \(0.0386161\pi\)
\(402\) 0 0
\(403\) − 14800.0i − 1.82938i
\(404\) −1734.00 −0.213539
\(405\) 0 0
\(406\) −4680.00 −0.572080
\(407\) − 1680.00i − 0.204606i
\(408\) 0 0
\(409\) −8714.00 −1.05350 −0.526748 0.850022i \(-0.676589\pi\)
−0.526748 + 0.850022i \(0.676589\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 452.000i 0.0540496i
\(413\) 480.000i 0.0571895i
\(414\) 0 0
\(415\) 0 0
\(416\) 3330.00 0.392468
\(417\) 0 0
\(418\) − 8928.00i − 1.04470i
\(419\) 11976.0 1.39634 0.698169 0.715933i \(-0.253998\pi\)
0.698169 + 0.715933i \(0.253998\pi\)
\(420\) 0 0
\(421\) 11054.0 1.27967 0.639833 0.768514i \(-0.279004\pi\)
0.639833 + 0.768514i \(0.279004\pi\)
\(422\) − 3300.00i − 0.380667i
\(423\) 0 0
\(424\) 9450.00 1.08239
\(425\) 0 0
\(426\) 0 0
\(427\) − 6440.00i − 0.729868i
\(428\) − 1404.00i − 0.158563i
\(429\) 0 0
\(430\) 0 0
\(431\) −720.000 −0.0804668 −0.0402334 0.999190i \(-0.512810\pi\)
−0.0402334 + 0.999190i \(0.512810\pi\)
\(432\) 0 0
\(433\) 15622.0i 1.73382i 0.498462 + 0.866912i \(0.333898\pi\)
−0.498462 + 0.866912i \(0.666102\pi\)
\(434\) 12000.0 1.32723
\(435\) 0 0
\(436\) −1474.00 −0.161908
\(437\) − 14880.0i − 1.62885i
\(438\) 0 0
\(439\) 9880.00 1.07414 0.537069 0.843538i \(-0.319531\pi\)
0.537069 + 0.843538i \(0.319531\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 11988.0i 1.29007i
\(443\) − 16116.0i − 1.72843i −0.503123 0.864215i \(-0.667816\pi\)
0.503123 0.864215i \(-0.332184\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −5892.00 −0.625548
\(447\) 0 0
\(448\) − 8660.00i − 0.913274i
\(449\) 9018.00 0.947852 0.473926 0.880565i \(-0.342836\pi\)
0.473926 + 0.880565i \(0.342836\pi\)
\(450\) 0 0
\(451\) −7920.00 −0.826914
\(452\) − 1086.00i − 0.113011i
\(453\) 0 0
\(454\) −1980.00 −0.204683
\(455\) 0 0
\(456\) 0 0
\(457\) − 3670.00i − 0.375657i −0.982202 0.187829i \(-0.939855\pi\)
0.982202 0.187829i \(-0.0601450\pi\)
\(458\) − 5718.00i − 0.583372i
\(459\) 0 0
\(460\) 0 0
\(461\) −17562.0 −1.77428 −0.887141 0.461499i \(-0.847312\pi\)
−0.887141 + 0.461499i \(0.847312\pi\)
\(462\) 0 0
\(463\) − 1172.00i − 0.117640i −0.998269 0.0588202i \(-0.981266\pi\)
0.998269 0.0588202i \(-0.0187338\pi\)
\(464\) 5538.00 0.554084
\(465\) 0 0
\(466\) −4374.00 −0.434810
\(467\) − 6876.00i − 0.681335i −0.940184 0.340667i \(-0.889347\pi\)
0.940184 0.340667i \(-0.110653\pi\)
\(468\) 0 0
\(469\) 3920.00 0.385946
\(470\) 0 0
\(471\) 0 0
\(472\) − 504.000i − 0.0491493i
\(473\) − 2208.00i − 0.214638i
\(474\) 0 0
\(475\) 0 0
\(476\) −1080.00 −0.103995
\(477\) 0 0
\(478\) − 3528.00i − 0.337588i
\(479\) 2280.00 0.217486 0.108743 0.994070i \(-0.465317\pi\)
0.108743 + 0.994070i \(0.465317\pi\)
\(480\) 0 0
\(481\) −5180.00 −0.491035
\(482\) − 2598.00i − 0.245510i
\(483\) 0 0
\(484\) 755.000 0.0709053
\(485\) 0 0
\(486\) 0 0
\(487\) − 3076.00i − 0.286215i −0.989707 0.143108i \(-0.954290\pi\)
0.989707 0.143108i \(-0.0457095\pi\)
\(488\) 6762.00i 0.627257i
\(489\) 0 0
\(490\) 0 0
\(491\) 18912.0 1.73826 0.869131 0.494582i \(-0.164679\pi\)
0.869131 + 0.494582i \(0.164679\pi\)
\(492\) 0 0
\(493\) 4212.00i 0.384785i
\(494\) −27528.0 −2.50717
\(495\) 0 0
\(496\) −14200.0 −1.28548
\(497\) 5760.00i 0.519862i
\(498\) 0 0
\(499\) −9956.00 −0.893170 −0.446585 0.894741i \(-0.647360\pi\)
−0.446585 + 0.894741i \(0.647360\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 1296.00i 0.115226i
\(503\) − 10656.0i − 0.944588i −0.881441 0.472294i \(-0.843426\pi\)
0.881441 0.472294i \(-0.156574\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −8640.00 −0.759081
\(507\) 0 0
\(508\) − 1244.00i − 0.108649i
\(509\) −2766.00 −0.240866 −0.120433 0.992721i \(-0.538428\pi\)
−0.120433 + 0.992721i \(0.538428\pi\)
\(510\) 0 0
\(511\) −8600.00 −0.744504
\(512\) 8733.00i 0.753804i
\(513\) 0 0
\(514\) −7578.00 −0.650294
\(515\) 0 0
\(516\) 0 0
\(517\) 576.000i 0.0489989i
\(518\) − 4200.00i − 0.356250i
\(519\) 0 0
\(520\) 0 0
\(521\) −10530.0 −0.885466 −0.442733 0.896654i \(-0.645991\pi\)
−0.442733 + 0.896654i \(0.645991\pi\)
\(522\) 0 0
\(523\) − 12692.0i − 1.06115i −0.847637 0.530576i \(-0.821976\pi\)
0.847637 0.530576i \(-0.178024\pi\)
\(524\) 2328.00 0.194082
\(525\) 0 0
\(526\) 16344.0 1.35481
\(527\) − 10800.0i − 0.892705i
\(528\) 0 0
\(529\) −2233.00 −0.183529
\(530\) 0 0
\(531\) 0 0
\(532\) − 2480.00i − 0.202108i
\(533\) 24420.0i 1.98452i
\(534\) 0 0
\(535\) 0 0
\(536\) −4116.00 −0.331687
\(537\) 0 0
\(538\) 7722.00i 0.618809i
\(539\) −1368.00 −0.109321
\(540\) 0 0
\(541\) 18110.0 1.43920 0.719602 0.694386i \(-0.244324\pi\)
0.719602 + 0.694386i \(0.244324\pi\)
\(542\) 9552.00i 0.756999i
\(543\) 0 0
\(544\) 2430.00 0.191517
\(545\) 0 0
\(546\) 0 0
\(547\) 3620.00i 0.282962i 0.989941 + 0.141481i \(0.0451864\pi\)
−0.989941 + 0.141481i \(0.954814\pi\)
\(548\) 2118.00i 0.165103i
\(549\) 0 0
\(550\) 0 0
\(551\) −9672.00 −0.747806
\(552\) 0 0
\(553\) 10400.0i 0.799734i
\(554\) 11886.0 0.911530
\(555\) 0 0
\(556\) 2324.00 0.177265
\(557\) 14166.0i 1.07762i 0.842428 + 0.538809i \(0.181125\pi\)
−0.842428 + 0.538809i \(0.818875\pi\)
\(558\) 0 0
\(559\) −6808.00 −0.515112
\(560\) 0 0
\(561\) 0 0
\(562\) − 24858.0i − 1.86579i
\(563\) − 13404.0i − 1.00339i −0.865043 0.501697i \(-0.832709\pi\)
0.865043 0.501697i \(-0.167291\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 8148.00 0.605099
\(567\) 0 0
\(568\) − 6048.00i − 0.446775i
\(569\) −18654.0 −1.37437 −0.687185 0.726483i \(-0.741154\pi\)
−0.687185 + 0.726483i \(0.741154\pi\)
\(570\) 0 0
\(571\) −7684.00 −0.563162 −0.281581 0.959537i \(-0.590859\pi\)
−0.281581 + 0.959537i \(0.590859\pi\)
\(572\) 1776.00i 0.129822i
\(573\) 0 0
\(574\) −19800.0 −1.43978
\(575\) 0 0
\(576\) 0 0
\(577\) − 1726.00i − 0.124531i −0.998060 0.0622654i \(-0.980167\pi\)
0.998060 0.0622654i \(-0.0198325\pi\)
\(578\) − 5991.00i − 0.431129i
\(579\) 0 0
\(580\) 0 0
\(581\) −3120.00 −0.222787
\(582\) 0 0
\(583\) 10800.0i 0.767222i
\(584\) 9030.00 0.639836
\(585\) 0 0
\(586\) 18054.0 1.27270
\(587\) − 10596.0i − 0.745049i −0.928022 0.372524i \(-0.878492\pi\)
0.928022 0.372524i \(-0.121508\pi\)
\(588\) 0 0
\(589\) 24800.0 1.73492
\(590\) 0 0
\(591\) 0 0
\(592\) 4970.00i 0.345043i
\(593\) 2862.00i 0.198193i 0.995078 + 0.0990963i \(0.0315952\pi\)
−0.995078 + 0.0990963i \(0.968405\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −258.000 −0.0177317
\(597\) 0 0
\(598\) 26640.0i 1.82172i
\(599\) −23592.0 −1.60925 −0.804627 0.593781i \(-0.797635\pi\)
−0.804627 + 0.593781i \(0.797635\pi\)
\(600\) 0 0
\(601\) −9574.00 −0.649803 −0.324902 0.945748i \(-0.605331\pi\)
−0.324902 + 0.945748i \(0.605331\pi\)
\(602\) − 5520.00i − 0.373718i
\(603\) 0 0
\(604\) 808.000 0.0544322
\(605\) 0 0
\(606\) 0 0
\(607\) 17444.0i 1.16644i 0.812314 + 0.583221i \(0.198208\pi\)
−0.812314 + 0.583221i \(0.801792\pi\)
\(608\) 5580.00i 0.372202i
\(609\) 0 0
\(610\) 0 0
\(611\) 1776.00 0.117593
\(612\) 0 0
\(613\) 2374.00i 0.156419i 0.996937 + 0.0782096i \(0.0249203\pi\)
−0.996937 + 0.0782096i \(0.975080\pi\)
\(614\) 27708.0 1.82118
\(615\) 0 0
\(616\) 10080.0 0.659310
\(617\) 12162.0i 0.793555i 0.917915 + 0.396778i \(0.129872\pi\)
−0.917915 + 0.396778i \(0.870128\pi\)
\(618\) 0 0
\(619\) −8804.00 −0.571668 −0.285834 0.958279i \(-0.592271\pi\)
−0.285834 + 0.958279i \(0.592271\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 4608.00i 0.297048i
\(623\) 20520.0i 1.31961i
\(624\) 0 0
\(625\) 0 0
\(626\) 22026.0 1.40629
\(627\) 0 0
\(628\) − 2378.00i − 0.151103i
\(629\) −3780.00 −0.239616
\(630\) 0 0
\(631\) −12688.0 −0.800478 −0.400239 0.916411i \(-0.631073\pi\)
−0.400239 + 0.916411i \(0.631073\pi\)
\(632\) − 10920.0i − 0.687301i
\(633\) 0 0
\(634\) 11682.0 0.731785
\(635\) 0 0
\(636\) 0 0
\(637\) 4218.00i 0.262360i
\(638\) 5616.00i 0.348495i
\(639\) 0 0
\(640\) 0 0
\(641\) 9150.00 0.563812 0.281906 0.959442i \(-0.409033\pi\)
0.281906 + 0.959442i \(0.409033\pi\)
\(642\) 0 0
\(643\) − 25292.0i − 1.55120i −0.631227 0.775598i \(-0.717448\pi\)
0.631227 0.775598i \(-0.282552\pi\)
\(644\) −2400.00 −0.146853
\(645\) 0 0
\(646\) −20088.0 −1.22345
\(647\) 2736.00i 0.166249i 0.996539 + 0.0831246i \(0.0264900\pi\)
−0.996539 + 0.0831246i \(0.973510\pi\)
\(648\) 0 0
\(649\) 576.000 0.0348382
\(650\) 0 0
\(651\) 0 0
\(652\) − 52.0000i − 0.00312343i
\(653\) 22218.0i 1.33148i 0.746183 + 0.665741i \(0.231884\pi\)
−0.746183 + 0.665741i \(0.768116\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 23430.0 1.39449
\(657\) 0 0
\(658\) 1440.00i 0.0853147i
\(659\) 14520.0 0.858299 0.429149 0.903234i \(-0.358813\pi\)
0.429149 + 0.903234i \(0.358813\pi\)
\(660\) 0 0
\(661\) −10618.0 −0.624799 −0.312400 0.949951i \(-0.601133\pi\)
−0.312400 + 0.949951i \(0.601133\pi\)
\(662\) − 11076.0i − 0.650273i
\(663\) 0 0
\(664\) 3276.00 0.191466
\(665\) 0 0
\(666\) 0 0
\(667\) 9360.00i 0.543359i
\(668\) − 3720.00i − 0.215466i
\(669\) 0 0
\(670\) 0 0
\(671\) −7728.00 −0.444614
\(672\) 0 0
\(673\) − 1370.00i − 0.0784690i −0.999230 0.0392345i \(-0.987508\pi\)
0.999230 0.0392345i \(-0.0124919\pi\)
\(674\) −26994.0 −1.54269
\(675\) 0 0
\(676\) 3279.00 0.186561
\(677\) 13758.0i 0.781038i 0.920595 + 0.390519i \(0.127704\pi\)
−0.920595 + 0.390519i \(0.872296\pi\)
\(678\) 0 0
\(679\) 5720.00 0.323289
\(680\) 0 0
\(681\) 0 0
\(682\) − 14400.0i − 0.808511i
\(683\) 11988.0i 0.671608i 0.941932 + 0.335804i \(0.109008\pi\)
−0.941932 + 0.335804i \(0.890992\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 17160.0 0.955061
\(687\) 0 0
\(688\) 6532.00i 0.361962i
\(689\) 33300.0 1.84126
\(690\) 0 0
\(691\) 32996.0 1.81654 0.908268 0.418388i \(-0.137405\pi\)
0.908268 + 0.418388i \(0.137405\pi\)
\(692\) − 426.000i − 0.0234019i
\(693\) 0 0
\(694\) −15732.0 −0.860488
\(695\) 0 0
\(696\) 0 0
\(697\) 17820.0i 0.968408i
\(698\) 18906.0i 1.02522i
\(699\) 0 0
\(700\) 0 0
\(701\) 25902.0 1.39558 0.697792 0.716300i \(-0.254166\pi\)
0.697792 + 0.716300i \(0.254166\pi\)
\(702\) 0 0
\(703\) − 8680.00i − 0.465679i
\(704\) −10392.0 −0.556340
\(705\) 0 0
\(706\) 10242.0 0.545981
\(707\) 34680.0i 1.84480i
\(708\) 0 0
\(709\) 27394.0 1.45106 0.725531 0.688189i \(-0.241594\pi\)
0.725531 + 0.688189i \(0.241594\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) − 21546.0i − 1.13409i
\(713\) − 24000.0i − 1.26060i
\(714\) 0 0
\(715\) 0 0
\(716\) 1440.00 0.0751611
\(717\) 0 0
\(718\) − 14472.0i − 0.752215i
\(719\) 34848.0 1.80753 0.903763 0.428033i \(-0.140793\pi\)
0.903763 + 0.428033i \(0.140793\pi\)
\(720\) 0 0
\(721\) 9040.00 0.466945
\(722\) − 25551.0i − 1.31705i
\(723\) 0 0
\(724\) 3130.00 0.160671
\(725\) 0 0
\(726\) 0 0
\(727\) 28028.0i 1.42985i 0.699201 + 0.714925i \(0.253539\pi\)
−0.699201 + 0.714925i \(0.746461\pi\)
\(728\) − 31080.0i − 1.58228i
\(729\) 0 0
\(730\) 0 0
\(731\) −4968.00 −0.251365
\(732\) 0 0
\(733\) − 18002.0i − 0.907120i −0.891226 0.453560i \(-0.850154\pi\)
0.891226 0.453560i \(-0.149846\pi\)
\(734\) −10524.0 −0.529221
\(735\) 0 0
\(736\) 5400.00 0.270444
\(737\) − 4704.00i − 0.235107i
\(738\) 0 0
\(739\) −15284.0 −0.760800 −0.380400 0.924822i \(-0.624214\pi\)
−0.380400 + 0.924822i \(0.624214\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 27000.0i 1.33585i
\(743\) − 18768.0i − 0.926691i −0.886178 0.463345i \(-0.846649\pi\)
0.886178 0.463345i \(-0.153351\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −32406.0 −1.59044
\(747\) 0 0
\(748\) 1296.00i 0.0633509i
\(749\) −28080.0 −1.36985
\(750\) 0 0
\(751\) 8696.00 0.422532 0.211266 0.977429i \(-0.432241\pi\)
0.211266 + 0.977429i \(0.432241\pi\)
\(752\) − 1704.00i − 0.0826310i
\(753\) 0 0
\(754\) 17316.0 0.836355
\(755\) 0 0
\(756\) 0 0
\(757\) − 38662.0i − 1.85627i −0.372247 0.928134i \(-0.621413\pi\)
0.372247 0.928134i \(-0.378587\pi\)
\(758\) 4380.00i 0.209880i
\(759\) 0 0
\(760\) 0 0
\(761\) −23874.0 −1.13723 −0.568615 0.822604i \(-0.692521\pi\)
−0.568615 + 0.822604i \(0.692521\pi\)
\(762\) 0 0
\(763\) 29480.0i 1.39875i
\(764\) 3576.00 0.169339
\(765\) 0 0
\(766\) −14616.0 −0.689422
\(767\) − 1776.00i − 0.0836084i
\(768\) 0 0
\(769\) −23618.0 −1.10753 −0.553763 0.832675i \(-0.686808\pi\)
−0.553763 + 0.832675i \(0.686808\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 2666.00i 0.124289i
\(773\) 11538.0i 0.536860i 0.963299 + 0.268430i \(0.0865049\pi\)
−0.963299 + 0.268430i \(0.913495\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −6006.00 −0.277839
\(777\) 0 0
\(778\) 42138.0i 1.94180i
\(779\) −40920.0 −1.88204
\(780\) 0 0
\(781\) 6912.00 0.316685
\(782\) 19440.0i 0.888968i
\(783\) 0 0
\(784\) 4047.00 0.184357
\(785\) 0 0
\(786\) 0 0
\(787\) − 14884.0i − 0.674152i −0.941478 0.337076i \(-0.890562\pi\)
0.941478 0.337076i \(-0.109438\pi\)
\(788\) − 2718.00i − 0.122874i
\(789\) 0 0
\(790\) 0 0
\(791\) −21720.0 −0.976327
\(792\) 0 0
\(793\) 23828.0i 1.06703i
\(794\) −8202.00 −0.366597
\(795\) 0 0
\(796\) −3832.00 −0.170630
\(797\) 11334.0i 0.503728i 0.967763 + 0.251864i \(0.0810435\pi\)
−0.967763 + 0.251864i \(0.918957\pi\)
\(798\) 0 0
\(799\) 1296.00 0.0573832
\(800\) 0 0
\(801\) 0 0
\(802\) − 47826.0i − 2.10573i
\(803\) 10320.0i 0.453530i
\(804\) 0 0
\(805\) 0 0
\(806\) −44400.0 −1.94035
\(807\) 0 0
\(808\) − 36414.0i − 1.58545i
\(809\) 44730.0 1.94391 0.971955 0.235167i \(-0.0755638\pi\)
0.971955 + 0.235167i \(0.0755638\pi\)
\(810\) 0 0
\(811\) −42748.0 −1.85091 −0.925453 0.378862i \(-0.876316\pi\)
−0.925453 + 0.378862i \(0.876316\pi\)
\(812\) 1560.00i 0.0674203i
\(813\) 0 0
\(814\) −5040.00 −0.217017
\(815\) 0 0
\(816\) 0 0
\(817\) − 11408.0i − 0.488513i
\(818\) 26142.0i 1.11740i
\(819\) 0 0
\(820\) 0 0
\(821\) 31686.0 1.34695 0.673477 0.739208i \(-0.264800\pi\)
0.673477 + 0.739208i \(0.264800\pi\)
\(822\) 0 0
\(823\) − 11036.0i − 0.467425i −0.972306 0.233713i \(-0.924913\pi\)
0.972306 0.233713i \(-0.0750875\pi\)
\(824\) −9492.00 −0.401298
\(825\) 0 0
\(826\) 1440.00 0.0606586
\(827\) − 25884.0i − 1.08836i −0.838968 0.544181i \(-0.816841\pi\)
0.838968 0.544181i \(-0.183159\pi\)
\(828\) 0 0
\(829\) −15950.0 −0.668234 −0.334117 0.942532i \(-0.608438\pi\)
−0.334117 + 0.942532i \(0.608438\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 32042.0i 1.33516i
\(833\) 3078.00i 0.128027i
\(834\) 0 0
\(835\) 0 0
\(836\) −2976.00 −0.123119
\(837\) 0 0
\(838\) − 35928.0i − 1.48104i
\(839\) 13800.0 0.567853 0.283927 0.958846i \(-0.408363\pi\)
0.283927 + 0.958846i \(0.408363\pi\)
\(840\) 0 0
\(841\) −18305.0 −0.750543
\(842\) − 33162.0i − 1.35729i
\(843\) 0 0
\(844\) −1100.00 −0.0448620
\(845\) 0 0
\(846\) 0 0
\(847\) − 15100.0i − 0.612565i
\(848\) − 31950.0i − 1.29383i
\(849\) 0 0
\(850\) 0 0
\(851\) −8400.00 −0.338365
\(852\) 0 0
\(853\) 27862.0i 1.11838i 0.829040 + 0.559189i \(0.188887\pi\)
−0.829040 + 0.559189i \(0.811113\pi\)
\(854\) −19320.0 −0.774141
\(855\) 0 0
\(856\) 29484.0 1.17727
\(857\) 7314.00i 0.291530i 0.989319 + 0.145765i \(0.0465644\pi\)
−0.989319 + 0.145765i \(0.953436\pi\)
\(858\) 0 0
\(859\) 28780.0 1.14314 0.571572 0.820552i \(-0.306334\pi\)
0.571572 + 0.820552i \(0.306334\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 2160.00i 0.0853479i
\(863\) − 32688.0i − 1.28935i −0.764455 0.644677i \(-0.776992\pi\)
0.764455 0.644677i \(-0.223008\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 46866.0 1.83900
\(867\) 0 0
\(868\) − 4000.00i − 0.156416i
\(869\) 12480.0 0.487175
\(870\) 0 0
\(871\) −14504.0 −0.564236
\(872\) − 30954.0i − 1.20210i
\(873\) 0 0
\(874\) −44640.0 −1.72766
\(875\) 0 0
\(876\) 0 0
\(877\) 36650.0i 1.41115i 0.708633 + 0.705577i \(0.249312\pi\)
−0.708633 + 0.705577i \(0.750688\pi\)
\(878\) − 29640.0i − 1.13930i
\(879\) 0 0
\(880\) 0 0
\(881\) 2646.00 0.101187 0.0505936 0.998719i \(-0.483889\pi\)
0.0505936 + 0.998719i \(0.483889\pi\)
\(882\) 0 0
\(883\) − 10892.0i − 0.415113i −0.978223 0.207557i \(-0.933449\pi\)
0.978223 0.207557i \(-0.0665511\pi\)
\(884\) 3996.00 0.152036
\(885\) 0 0
\(886\) −48348.0 −1.83328
\(887\) 43464.0i 1.64530i 0.568550 + 0.822648i \(0.307504\pi\)
−0.568550 + 0.822648i \(0.692496\pi\)
\(888\) 0 0
\(889\) −24880.0 −0.938637
\(890\) 0 0
\(891\) 0 0
\(892\) 1964.00i 0.0737215i
\(893\) 2976.00i 0.111521i
\(894\) 0 0
\(895\) 0 0
\(896\) −33180.0 −1.23713
\(897\) 0 0
\(898\) − 27054.0i − 1.00535i
\(899\) −15600.0 −0.578742
\(900\) 0 0
\(901\) 24300.0 0.898502
\(902\) 23760.0i 0.877075i
\(903\) 0 0
\(904\) 22806.0 0.839067
\(905\) 0 0
\(906\) 0 0
\(907\) − 14884.0i − 0.544890i −0.962171 0.272445i \(-0.912168\pi\)
0.962171 0.272445i \(-0.0878323\pi\)
\(908\) 660.000i 0.0241221i
\(909\) 0 0
\(910\) 0 0
\(911\) 1248.00 0.0453876 0.0226938 0.999742i \(-0.492776\pi\)
0.0226938 + 0.999742i \(0.492776\pi\)
\(912\) 0 0
\(913\) 3744.00i 0.135716i
\(914\) −11010.0 −0.398445
\(915\) 0 0
\(916\) −1906.00 −0.0687511
\(917\) − 46560.0i − 1.67671i
\(918\) 0 0
\(919\) 6640.00 0.238339 0.119169 0.992874i \(-0.461977\pi\)
0.119169 + 0.992874i \(0.461977\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 52686.0i 1.88191i
\(923\) − 21312.0i − 0.760014i
\(924\) 0 0
\(925\) 0 0
\(926\) −3516.00 −0.124776
\(927\) 0 0
\(928\) − 3510.00i − 0.124161i
\(929\) 29946.0 1.05758 0.528792 0.848751i \(-0.322645\pi\)
0.528792 + 0.848751i \(0.322645\pi\)
\(930\) 0 0
\(931\) −7068.00 −0.248812
\(932\) 1458.00i 0.0512429i
\(933\) 0 0
\(934\) −20628.0 −0.722665
\(935\) 0 0
\(936\) 0 0
\(937\) 45002.0i 1.56900i 0.620130 + 0.784499i \(0.287080\pi\)
−0.620130 + 0.784499i \(0.712920\pi\)
\(938\) − 11760.0i − 0.409358i
\(939\) 0 0
\(940\) 0 0
\(941\) −6090.00 −0.210976 −0.105488 0.994421i \(-0.533640\pi\)
−0.105488 + 0.994421i \(0.533640\pi\)
\(942\) 0 0
\(943\) 39600.0i 1.36750i
\(944\) −1704.00 −0.0587505
\(945\) 0 0
\(946\) −6624.00 −0.227658
\(947\) − 56388.0i − 1.93491i −0.253035 0.967457i \(-0.581429\pi\)
0.253035 0.967457i \(-0.418571\pi\)
\(948\) 0 0
\(949\) 31820.0 1.08843
\(950\) 0 0
\(951\) 0 0
\(952\) − 22680.0i − 0.772125i
\(953\) 10854.0i 0.368936i 0.982839 + 0.184468i \(0.0590561\pi\)
−0.982839 + 0.184468i \(0.940944\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −1176.00 −0.0397851
\(957\) 0 0
\(958\) − 6840.00i − 0.230679i
\(959\) 42360.0 1.42636
\(960\) 0 0
\(961\) 10209.0 0.342687
\(962\) 15540.0i 0.520821i
\(963\) 0 0
\(964\) −866.000 −0.0289336
\(965\) 0 0
\(966\) 0 0
\(967\) − 42316.0i − 1.40723i −0.710582 0.703615i \(-0.751568\pi\)
0.710582 0.703615i \(-0.248432\pi\)
\(968\) 15855.0i 0.526445i
\(969\) 0 0
\(970\) 0 0
\(971\) −24480.0 −0.809063 −0.404532 0.914524i \(-0.632565\pi\)
−0.404532 + 0.914524i \(0.632565\pi\)
\(972\) 0 0
\(973\) − 46480.0i − 1.53143i
\(974\) −9228.00 −0.303577
\(975\) 0 0
\(976\) 22862.0 0.749790
\(977\) 6906.00i 0.226144i 0.993587 + 0.113072i \(0.0360690\pi\)
−0.993587 + 0.113072i \(0.963931\pi\)
\(978\) 0 0
\(979\) 24624.0 0.803868
\(980\) 0 0
\(981\) 0 0
\(982\) − 56736.0i − 1.84371i
\(983\) 6960.00i 0.225829i 0.993605 + 0.112914i \(0.0360186\pi\)
−0.993605 + 0.112914i \(0.963981\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 12636.0 0.408126
\(987\) 0 0
\(988\) 9176.00i 0.295473i
\(989\) −11040.0 −0.354956
\(990\) 0 0
\(991\) 47792.0 1.53195 0.765975 0.642870i \(-0.222256\pi\)
0.765975 + 0.642870i \(0.222256\pi\)
\(992\) 9000.00i 0.288055i
\(993\) 0 0
\(994\) 17280.0 0.551397
\(995\) 0 0
\(996\) 0 0
\(997\) 9938.00i 0.315687i 0.987464 + 0.157843i \(0.0504541\pi\)
−0.987464 + 0.157843i \(0.949546\pi\)
\(998\) 29868.0i 0.947350i
\(999\) 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 225.4.b.d.199.1 2
3.2 odd 2 75.4.b.a.49.2 2
5.2 odd 4 225.4.a.g.1.1 1
5.3 odd 4 45.4.a.b.1.1 1
5.4 even 2 inner 225.4.b.d.199.2 2
12.11 even 2 1200.4.f.m.49.1 2
15.2 even 4 75.4.a.a.1.1 1
15.8 even 4 15.4.a.b.1.1 1
15.14 odd 2 75.4.b.a.49.1 2
20.3 even 4 720.4.a.r.1.1 1
35.13 even 4 2205.4.a.c.1.1 1
45.13 odd 12 405.4.e.k.136.1 2
45.23 even 12 405.4.e.d.136.1 2
45.38 even 12 405.4.e.d.271.1 2
45.43 odd 12 405.4.e.k.271.1 2
60.23 odd 4 240.4.a.f.1.1 1
60.47 odd 4 1200.4.a.o.1.1 1
60.59 even 2 1200.4.f.m.49.2 2
105.83 odd 4 735.4.a.i.1.1 1
120.53 even 4 960.4.a.bi.1.1 1
120.83 odd 4 960.4.a.l.1.1 1
165.98 odd 4 1815.4.a.a.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
15.4.a.b.1.1 1 15.8 even 4
45.4.a.b.1.1 1 5.3 odd 4
75.4.a.a.1.1 1 15.2 even 4
75.4.b.a.49.1 2 15.14 odd 2
75.4.b.a.49.2 2 3.2 odd 2
225.4.a.g.1.1 1 5.2 odd 4
225.4.b.d.199.1 2 1.1 even 1 trivial
225.4.b.d.199.2 2 5.4 even 2 inner
240.4.a.f.1.1 1 60.23 odd 4
405.4.e.d.136.1 2 45.23 even 12
405.4.e.d.271.1 2 45.38 even 12
405.4.e.k.136.1 2 45.13 odd 12
405.4.e.k.271.1 2 45.43 odd 12
720.4.a.r.1.1 1 20.3 even 4
735.4.a.i.1.1 1 105.83 odd 4
960.4.a.l.1.1 1 120.83 odd 4
960.4.a.bi.1.1 1 120.53 even 4
1200.4.a.o.1.1 1 60.47 odd 4
1200.4.f.m.49.1 2 12.11 even 2
1200.4.f.m.49.2 2 60.59 even 2
1815.4.a.a.1.1 1 165.98 odd 4
2205.4.a.c.1.1 1 35.13 even 4