Properties

Label 825.4.c.c.199.2
Level $825$
Weight $4$
Character 825.199
Analytic conductor $48.677$
Analytic rank $0$
Dimension $2$
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] = [825,4,Mod(199,825)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(825, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("825.199");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 825.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: \(48.6765757547\)
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 199.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 825.199
Dual form 825.4.c.c.199.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+4.00000i q^{2} +3.00000i q^{3} -8.00000 q^{4} -12.0000 q^{6} -21.0000i q^{7} -9.00000 q^{9} +O(q^{10})\) \(q+4.00000i q^{2} +3.00000i q^{3} -8.00000 q^{4} -12.0000 q^{6} -21.0000i q^{7} -9.00000 q^{9} +11.0000 q^{11} -24.0000i q^{12} -68.0000i q^{13} +84.0000 q^{14} -64.0000 q^{16} -21.0000i q^{17} -36.0000i q^{18} -125.000 q^{19} +63.0000 q^{21} +44.0000i q^{22} +137.000i q^{23} +272.000 q^{26} -27.0000i q^{27} +168.000i q^{28} +150.000 q^{29} +292.000 q^{31} -256.000i q^{32} +33.0000i q^{33} +84.0000 q^{34} +72.0000 q^{36} +349.000i q^{37} -500.000i q^{38} +204.000 q^{39} +497.000 q^{41} +252.000i q^{42} -208.000i q^{43} -88.0000 q^{44} -548.000 q^{46} +369.000i q^{47} -192.000i q^{48} -98.0000 q^{49} +63.0000 q^{51} +544.000i q^{52} +542.000i q^{53} +108.000 q^{54} -375.000i q^{57} +600.000i q^{58} -235.000 q^{59} +482.000 q^{61} +1168.00i q^{62} +189.000i q^{63} +512.000 q^{64} -132.000 q^{66} +734.000i q^{67} +168.000i q^{68} -411.000 q^{69} +587.000 q^{71} -518.000i q^{73} -1396.00 q^{74} +1000.00 q^{76} -231.000i q^{77} +816.000i q^{78} +1045.00 q^{79} +81.0000 q^{81} +1988.00i q^{82} -608.000i q^{83} -504.000 q^{84} +832.000 q^{86} +450.000i q^{87} +770.000 q^{89} -1428.00 q^{91} -1096.00i q^{92} +876.000i q^{93} -1476.00 q^{94} +768.000 q^{96} -1541.00i q^{97} -392.000i q^{98} -99.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 16 q^{4} - 24 q^{6} - 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 16 q^{4} - 24 q^{6} - 18 q^{9} + 22 q^{11} + 168 q^{14} - 128 q^{16} - 250 q^{19} + 126 q^{21} + 544 q^{26} + 300 q^{29} + 584 q^{31} + 168 q^{34} + 144 q^{36} + 408 q^{39} + 994 q^{41} - 176 q^{44} - 1096 q^{46} - 196 q^{49} + 126 q^{51} + 216 q^{54} - 470 q^{59} + 964 q^{61} + 1024 q^{64} - 264 q^{66} - 822 q^{69} + 1174 q^{71} - 2792 q^{74} + 2000 q^{76} + 2090 q^{79} + 162 q^{81} - 1008 q^{84} + 1664 q^{86} + 1540 q^{89} - 2856 q^{91} - 2952 q^{94} + 1536 q^{96} - 198 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/825\mathbb{Z}\right)^\times\).

\(n\) \(376\) \(551\) \(727\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 4.00000i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(3\) 3.00000i 0.577350i
\(4\) −8.00000 −1.00000
\(5\) 0 0
\(6\) −12.0000 −0.816497
\(7\) − 21.0000i − 1.13389i −0.823754 0.566947i \(-0.808125\pi\)
0.823754 0.566947i \(-0.191875\pi\)
\(8\) 0 0
\(9\) −9.00000 −0.333333
\(10\) 0 0
\(11\) 11.0000 0.301511
\(12\) − 24.0000i − 0.577350i
\(13\) − 68.0000i − 1.45075i −0.688352 0.725377i \(-0.741665\pi\)
0.688352 0.725377i \(-0.258335\pi\)
\(14\) 84.0000 1.60357
\(15\) 0 0
\(16\) −64.0000 −1.00000
\(17\) − 21.0000i − 0.299603i −0.988716 0.149801i \(-0.952137\pi\)
0.988716 0.149801i \(-0.0478634\pi\)
\(18\) − 36.0000i − 0.471405i
\(19\) −125.000 −1.50931 −0.754657 0.656119i \(-0.772197\pi\)
−0.754657 + 0.656119i \(0.772197\pi\)
\(20\) 0 0
\(21\) 63.0000 0.654654
\(22\) 44.0000i 0.426401i
\(23\) 137.000i 1.24202i 0.783802 + 0.621010i \(0.213278\pi\)
−0.783802 + 0.621010i \(0.786722\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 272.000 2.05168
\(27\) − 27.0000i − 0.192450i
\(28\) 168.000i 1.13389i
\(29\) 150.000 0.960493 0.480247 0.877134i \(-0.340547\pi\)
0.480247 + 0.877134i \(0.340547\pi\)
\(30\) 0 0
\(31\) 292.000 1.69177 0.845883 0.533368i \(-0.179074\pi\)
0.845883 + 0.533368i \(0.179074\pi\)
\(32\) − 256.000i − 1.41421i
\(33\) 33.0000i 0.174078i
\(34\) 84.0000 0.423702
\(35\) 0 0
\(36\) 72.0000 0.333333
\(37\) 349.000i 1.55068i 0.631543 + 0.775341i \(0.282422\pi\)
−0.631543 + 0.775341i \(0.717578\pi\)
\(38\) − 500.000i − 2.13449i
\(39\) 204.000 0.837593
\(40\) 0 0
\(41\) 497.000 1.89313 0.946565 0.322512i \(-0.104527\pi\)
0.946565 + 0.322512i \(0.104527\pi\)
\(42\) 252.000i 0.925820i
\(43\) − 208.000i − 0.737668i −0.929495 0.368834i \(-0.879757\pi\)
0.929495 0.368834i \(-0.120243\pi\)
\(44\) −88.0000 −0.301511
\(45\) 0 0
\(46\) −548.000 −1.75648
\(47\) 369.000i 1.14520i 0.819837 + 0.572598i \(0.194064\pi\)
−0.819837 + 0.572598i \(0.805936\pi\)
\(48\) − 192.000i − 0.577350i
\(49\) −98.0000 −0.285714
\(50\) 0 0
\(51\) 63.0000 0.172976
\(52\) 544.000i 1.45075i
\(53\) 542.000i 1.40471i 0.711829 + 0.702353i \(0.247867\pi\)
−0.711829 + 0.702353i \(0.752133\pi\)
\(54\) 108.000 0.272166
\(55\) 0 0
\(56\) 0 0
\(57\) − 375.000i − 0.871403i
\(58\) 600.000i 1.35834i
\(59\) −235.000 −0.518549 −0.259275 0.965804i \(-0.583483\pi\)
−0.259275 + 0.965804i \(0.583483\pi\)
\(60\) 0 0
\(61\) 482.000 1.01170 0.505851 0.862621i \(-0.331179\pi\)
0.505851 + 0.862621i \(0.331179\pi\)
\(62\) 1168.00i 2.39252i
\(63\) 189.000i 0.377964i
\(64\) 512.000 1.00000
\(65\) 0 0
\(66\) −132.000 −0.246183
\(67\) 734.000i 1.33839i 0.743085 + 0.669197i \(0.233362\pi\)
−0.743085 + 0.669197i \(0.766638\pi\)
\(68\) 168.000i 0.299603i
\(69\) −411.000 −0.717081
\(70\) 0 0
\(71\) 587.000 0.981184 0.490592 0.871389i \(-0.336781\pi\)
0.490592 + 0.871389i \(0.336781\pi\)
\(72\) 0 0
\(73\) − 518.000i − 0.830511i −0.909705 0.415256i \(-0.863692\pi\)
0.909705 0.415256i \(-0.136308\pi\)
\(74\) −1396.00 −2.19300
\(75\) 0 0
\(76\) 1000.00 1.50931
\(77\) − 231.000i − 0.341882i
\(78\) 816.000i 1.18454i
\(79\) 1045.00 1.48825 0.744125 0.668041i \(-0.232867\pi\)
0.744125 + 0.668041i \(0.232867\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 1988.00i 2.67729i
\(83\) − 608.000i − 0.804056i −0.915627 0.402028i \(-0.868305\pi\)
0.915627 0.402028i \(-0.131695\pi\)
\(84\) −504.000 −0.654654
\(85\) 0 0
\(86\) 832.000 1.04322
\(87\) 450.000i 0.554541i
\(88\) 0 0
\(89\) 770.000 0.917077 0.458538 0.888675i \(-0.348373\pi\)
0.458538 + 0.888675i \(0.348373\pi\)
\(90\) 0 0
\(91\) −1428.00 −1.64500
\(92\) − 1096.00i − 1.24202i
\(93\) 876.000i 0.976742i
\(94\) −1476.00 −1.61955
\(95\) 0 0
\(96\) 768.000 0.816497
\(97\) − 1541.00i − 1.61304i −0.591207 0.806520i \(-0.701348\pi\)
0.591207 0.806520i \(-0.298652\pi\)
\(98\) − 392.000i − 0.404061i
\(99\) −99.0000 −0.100504
\(100\) 0 0
\(101\) 827.000 0.814748 0.407374 0.913261i \(-0.366445\pi\)
0.407374 + 0.913261i \(0.366445\pi\)
\(102\) 252.000i 0.244625i
\(103\) − 248.000i − 0.237244i −0.992939 0.118622i \(-0.962152\pi\)
0.992939 0.118622i \(-0.0378477\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) −2168.00 −1.98655
\(107\) − 366.000i − 0.330678i −0.986237 0.165339i \(-0.947128\pi\)
0.986237 0.165339i \(-0.0528718\pi\)
\(108\) 216.000i 0.192450i
\(109\) −270.000 −0.237260 −0.118630 0.992939i \(-0.537850\pi\)
−0.118630 + 0.992939i \(0.537850\pi\)
\(110\) 0 0
\(111\) −1047.00 −0.895287
\(112\) 1344.00i 1.13389i
\(113\) 1002.00i 0.834161i 0.908869 + 0.417081i \(0.136947\pi\)
−0.908869 + 0.417081i \(0.863053\pi\)
\(114\) 1500.00 1.23235
\(115\) 0 0
\(116\) −1200.00 −0.960493
\(117\) 612.000i 0.483585i
\(118\) − 940.000i − 0.733339i
\(119\) −441.000 −0.339718
\(120\) 0 0
\(121\) 121.000 0.0909091
\(122\) 1928.00i 1.43076i
\(123\) 1491.00i 1.09300i
\(124\) −2336.00 −1.69177
\(125\) 0 0
\(126\) −756.000 −0.534522
\(127\) 469.000i 0.327693i 0.986486 + 0.163847i \(0.0523902\pi\)
−0.986486 + 0.163847i \(0.947610\pi\)
\(128\) 0 0
\(129\) 624.000 0.425893
\(130\) 0 0
\(131\) −408.000 −0.272115 −0.136058 0.990701i \(-0.543443\pi\)
−0.136058 + 0.990701i \(0.543443\pi\)
\(132\) − 264.000i − 0.174078i
\(133\) 2625.00i 1.71140i
\(134\) −2936.00 −1.89277
\(135\) 0 0
\(136\) 0 0
\(137\) − 2466.00i − 1.53784i −0.639343 0.768922i \(-0.720793\pi\)
0.639343 0.768922i \(-0.279207\pi\)
\(138\) − 1644.00i − 1.01411i
\(139\) −1020.00 −0.622412 −0.311206 0.950342i \(-0.600733\pi\)
−0.311206 + 0.950342i \(0.600733\pi\)
\(140\) 0 0
\(141\) −1107.00 −0.661179
\(142\) 2348.00i 1.38760i
\(143\) − 748.000i − 0.437419i
\(144\) 576.000 0.333333
\(145\) 0 0
\(146\) 2072.00 1.17452
\(147\) − 294.000i − 0.164957i
\(148\) − 2792.00i − 1.55068i
\(149\) −5.00000 −0.00274910 −0.00137455 0.999999i \(-0.500438\pi\)
−0.00137455 + 0.999999i \(0.500438\pi\)
\(150\) 0 0
\(151\) 452.000 0.243598 0.121799 0.992555i \(-0.461134\pi\)
0.121799 + 0.992555i \(0.461134\pi\)
\(152\) 0 0
\(153\) 189.000i 0.0998676i
\(154\) 924.000 0.483494
\(155\) 0 0
\(156\) −1632.00 −0.837593
\(157\) − 1766.00i − 0.897721i −0.893602 0.448860i \(-0.851830\pi\)
0.893602 0.448860i \(-0.148170\pi\)
\(158\) 4180.00i 2.10470i
\(159\) −1626.00 −0.811007
\(160\) 0 0
\(161\) 2877.00 1.40832
\(162\) 324.000i 0.157135i
\(163\) − 2068.00i − 0.993732i −0.867827 0.496866i \(-0.834484\pi\)
0.867827 0.496866i \(-0.165516\pi\)
\(164\) −3976.00 −1.89313
\(165\) 0 0
\(166\) 2432.00 1.13711
\(167\) − 3386.00i − 1.56896i −0.620153 0.784481i \(-0.712930\pi\)
0.620153 0.784481i \(-0.287070\pi\)
\(168\) 0 0
\(169\) −2427.00 −1.10469
\(170\) 0 0
\(171\) 1125.00 0.503105
\(172\) 1664.00i 0.737668i
\(173\) 117.000i 0.0514182i 0.999669 + 0.0257091i \(0.00818436\pi\)
−0.999669 + 0.0257091i \(0.991816\pi\)
\(174\) −1800.00 −0.784239
\(175\) 0 0
\(176\) −704.000 −0.301511
\(177\) − 705.000i − 0.299384i
\(178\) 3080.00i 1.29694i
\(179\) −2995.00 −1.25060 −0.625298 0.780386i \(-0.715023\pi\)
−0.625298 + 0.780386i \(0.715023\pi\)
\(180\) 0 0
\(181\) 4067.00 1.67015 0.835077 0.550134i \(-0.185423\pi\)
0.835077 + 0.550134i \(0.185423\pi\)
\(182\) − 5712.00i − 2.32638i
\(183\) 1446.00i 0.584106i
\(184\) 0 0
\(185\) 0 0
\(186\) −3504.00 −1.38132
\(187\) − 231.000i − 0.0903337i
\(188\) − 2952.00i − 1.14520i
\(189\) −567.000 −0.218218
\(190\) 0 0
\(191\) 3047.00 1.15431 0.577155 0.816635i \(-0.304163\pi\)
0.577155 + 0.816635i \(0.304163\pi\)
\(192\) 1536.00i 0.577350i
\(193\) 1232.00i 0.459489i 0.973251 + 0.229744i \(0.0737890\pi\)
−0.973251 + 0.229744i \(0.926211\pi\)
\(194\) 6164.00 2.28118
\(195\) 0 0
\(196\) 784.000 0.285714
\(197\) 4979.00i 1.80071i 0.435160 + 0.900353i \(0.356692\pi\)
−0.435160 + 0.900353i \(0.643308\pi\)
\(198\) − 396.000i − 0.142134i
\(199\) −600.000 −0.213733 −0.106867 0.994273i \(-0.534082\pi\)
−0.106867 + 0.994273i \(0.534082\pi\)
\(200\) 0 0
\(201\) −2202.00 −0.772722
\(202\) 3308.00i 1.15223i
\(203\) − 3150.00i − 1.08910i
\(204\) −504.000 −0.172976
\(205\) 0 0
\(206\) 992.000 0.335514
\(207\) − 1233.00i − 0.414007i
\(208\) 4352.00i 1.45075i
\(209\) −1375.00 −0.455075
\(210\) 0 0
\(211\) −2468.00 −0.805233 −0.402616 0.915369i \(-0.631899\pi\)
−0.402616 + 0.915369i \(0.631899\pi\)
\(212\) − 4336.00i − 1.40471i
\(213\) 1761.00i 0.566487i
\(214\) 1464.00 0.467649
\(215\) 0 0
\(216\) 0 0
\(217\) − 6132.00i − 1.91828i
\(218\) − 1080.00i − 0.335536i
\(219\) 1554.00 0.479496
\(220\) 0 0
\(221\) −1428.00 −0.434650
\(222\) − 4188.00i − 1.26613i
\(223\) 5392.00i 1.61917i 0.587002 + 0.809585i \(0.300308\pi\)
−0.587002 + 0.809585i \(0.699692\pi\)
\(224\) −5376.00 −1.60357
\(225\) 0 0
\(226\) −4008.00 −1.17968
\(227\) − 2366.00i − 0.691793i −0.938273 0.345896i \(-0.887575\pi\)
0.938273 0.345896i \(-0.112425\pi\)
\(228\) 3000.00i 0.871403i
\(229\) 4645.00 1.34039 0.670197 0.742183i \(-0.266210\pi\)
0.670197 + 0.742183i \(0.266210\pi\)
\(230\) 0 0
\(231\) 693.000 0.197386
\(232\) 0 0
\(233\) − 513.000i − 0.144239i −0.997396 0.0721196i \(-0.977024\pi\)
0.997396 0.0721196i \(-0.0229763\pi\)
\(234\) −2448.00 −0.683892
\(235\) 0 0
\(236\) 1880.00 0.518549
\(237\) 3135.00i 0.859241i
\(238\) − 1764.00i − 0.480433i
\(239\) −2690.00 −0.728040 −0.364020 0.931391i \(-0.618596\pi\)
−0.364020 + 0.931391i \(0.618596\pi\)
\(240\) 0 0
\(241\) −3728.00 −0.996438 −0.498219 0.867051i \(-0.666012\pi\)
−0.498219 + 0.867051i \(0.666012\pi\)
\(242\) 484.000i 0.128565i
\(243\) 243.000i 0.0641500i
\(244\) −3856.00 −1.01170
\(245\) 0 0
\(246\) −5964.00 −1.54573
\(247\) 8500.00i 2.18964i
\(248\) 0 0
\(249\) 1824.00 0.464222
\(250\) 0 0
\(251\) 2352.00 0.591462 0.295731 0.955271i \(-0.404437\pi\)
0.295731 + 0.955271i \(0.404437\pi\)
\(252\) − 1512.00i − 0.377964i
\(253\) 1507.00i 0.374483i
\(254\) −1876.00 −0.463428
\(255\) 0 0
\(256\) 4096.00 1.00000
\(257\) − 3846.00i − 0.933490i −0.884392 0.466745i \(-0.845427\pi\)
0.884392 0.466745i \(-0.154573\pi\)
\(258\) 2496.00i 0.602303i
\(259\) 7329.00 1.75831
\(260\) 0 0
\(261\) −1350.00 −0.320164
\(262\) − 1632.00i − 0.384829i
\(263\) 522.000i 0.122387i 0.998126 + 0.0611937i \(0.0194907\pi\)
−0.998126 + 0.0611937i \(0.980509\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −10500.0 −2.42029
\(267\) 2310.00i 0.529475i
\(268\) − 5872.00i − 1.33839i
\(269\) −4020.00 −0.911166 −0.455583 0.890193i \(-0.650569\pi\)
−0.455583 + 0.890193i \(0.650569\pi\)
\(270\) 0 0
\(271\) 6687.00 1.49892 0.749458 0.662052i \(-0.230314\pi\)
0.749458 + 0.662052i \(0.230314\pi\)
\(272\) 1344.00i 0.299603i
\(273\) − 4284.00i − 0.949742i
\(274\) 9864.00 2.17484
\(275\) 0 0
\(276\) 3288.00 0.717081
\(277\) − 3746.00i − 0.812546i −0.913752 0.406273i \(-0.866828\pi\)
0.913752 0.406273i \(-0.133172\pi\)
\(278\) − 4080.00i − 0.880224i
\(279\) −2628.00 −0.563922
\(280\) 0 0
\(281\) −5883.00 −1.24893 −0.624467 0.781051i \(-0.714684\pi\)
−0.624467 + 0.781051i \(0.714684\pi\)
\(282\) − 4428.00i − 0.935048i
\(283\) − 3943.00i − 0.828223i −0.910226 0.414111i \(-0.864092\pi\)
0.910226 0.414111i \(-0.135908\pi\)
\(284\) −4696.00 −0.981184
\(285\) 0 0
\(286\) 2992.00 0.618604
\(287\) − 10437.0i − 2.14661i
\(288\) 2304.00i 0.471405i
\(289\) 4472.00 0.910238
\(290\) 0 0
\(291\) 4623.00 0.931289
\(292\) 4144.00i 0.830511i
\(293\) 1487.00i 0.296490i 0.988951 + 0.148245i \(0.0473624\pi\)
−0.988951 + 0.148245i \(0.952638\pi\)
\(294\) 1176.00 0.233285
\(295\) 0 0
\(296\) 0 0
\(297\) − 297.000i − 0.0580259i
\(298\) − 20.0000i − 0.00388782i
\(299\) 9316.00 1.80187
\(300\) 0 0
\(301\) −4368.00 −0.836436
\(302\) 1808.00i 0.344499i
\(303\) 2481.00i 0.470395i
\(304\) 8000.00 1.50931
\(305\) 0 0
\(306\) −756.000 −0.141234
\(307\) 4844.00i 0.900527i 0.892896 + 0.450263i \(0.148670\pi\)
−0.892896 + 0.450263i \(0.851330\pi\)
\(308\) 1848.00i 0.341882i
\(309\) 744.000 0.136973
\(310\) 0 0
\(311\) 4632.00 0.844555 0.422278 0.906467i \(-0.361231\pi\)
0.422278 + 0.906467i \(0.361231\pi\)
\(312\) 0 0
\(313\) 8437.00i 1.52360i 0.647811 + 0.761801i \(0.275685\pi\)
−0.647811 + 0.761801i \(0.724315\pi\)
\(314\) 7064.00 1.26957
\(315\) 0 0
\(316\) −8360.00 −1.48825
\(317\) − 3636.00i − 0.644221i −0.946702 0.322111i \(-0.895608\pi\)
0.946702 0.322111i \(-0.104392\pi\)
\(318\) − 6504.00i − 1.14694i
\(319\) 1650.00 0.289600
\(320\) 0 0
\(321\) 1098.00 0.190917
\(322\) 11508.0i 1.99166i
\(323\) 2625.00i 0.452195i
\(324\) −648.000 −0.111111
\(325\) 0 0
\(326\) 8272.00 1.40535
\(327\) − 810.000i − 0.136982i
\(328\) 0 0
\(329\) 7749.00 1.29853
\(330\) 0 0
\(331\) −758.000 −0.125871 −0.0629357 0.998018i \(-0.520046\pi\)
−0.0629357 + 0.998018i \(0.520046\pi\)
\(332\) 4864.00i 0.804056i
\(333\) − 3141.00i − 0.516894i
\(334\) 13544.0 2.21885
\(335\) 0 0
\(336\) −4032.00 −0.654654
\(337\) 7374.00i 1.19195i 0.803003 + 0.595975i \(0.203234\pi\)
−0.803003 + 0.595975i \(0.796766\pi\)
\(338\) − 9708.00i − 1.56227i
\(339\) −3006.00 −0.481603
\(340\) 0 0
\(341\) 3212.00 0.510087
\(342\) 4500.00i 0.711497i
\(343\) − 5145.00i − 0.809924i
\(344\) 0 0
\(345\) 0 0
\(346\) −468.000 −0.0727163
\(347\) 6524.00i 1.00930i 0.863324 + 0.504649i \(0.168378\pi\)
−0.863324 + 0.504649i \(0.831622\pi\)
\(348\) − 3600.00i − 0.554541i
\(349\) −3710.00 −0.569031 −0.284515 0.958671i \(-0.591833\pi\)
−0.284515 + 0.958671i \(0.591833\pi\)
\(350\) 0 0
\(351\) −1836.00 −0.279198
\(352\) − 2816.00i − 0.426401i
\(353\) 2832.00i 0.427003i 0.976943 + 0.213502i \(0.0684869\pi\)
−0.976943 + 0.213502i \(0.931513\pi\)
\(354\) 2820.00 0.423394
\(355\) 0 0
\(356\) −6160.00 −0.917077
\(357\) − 1323.00i − 0.196136i
\(358\) − 11980.0i − 1.76861i
\(359\) 7040.00 1.03498 0.517489 0.855690i \(-0.326867\pi\)
0.517489 + 0.855690i \(0.326867\pi\)
\(360\) 0 0
\(361\) 8766.00 1.27803
\(362\) 16268.0i 2.36195i
\(363\) 363.000i 0.0524864i
\(364\) 11424.0 1.64500
\(365\) 0 0
\(366\) −5784.00 −0.826051
\(367\) − 6206.00i − 0.882699i −0.897335 0.441350i \(-0.854500\pi\)
0.897335 0.441350i \(-0.145500\pi\)
\(368\) − 8768.00i − 1.24202i
\(369\) −4473.00 −0.631044
\(370\) 0 0
\(371\) 11382.0 1.59279
\(372\) − 7008.00i − 0.976742i
\(373\) 1962.00i 0.272355i 0.990684 + 0.136178i \(0.0434818\pi\)
−0.990684 + 0.136178i \(0.956518\pi\)
\(374\) 924.000 0.127751
\(375\) 0 0
\(376\) 0 0
\(377\) − 10200.0i − 1.39344i
\(378\) − 2268.00i − 0.308607i
\(379\) 7960.00 1.07883 0.539417 0.842039i \(-0.318645\pi\)
0.539417 + 0.842039i \(0.318645\pi\)
\(380\) 0 0
\(381\) −1407.00 −0.189194
\(382\) 12188.0i 1.63244i
\(383\) − 7188.00i − 0.958981i −0.877547 0.479490i \(-0.840822\pi\)
0.877547 0.479490i \(-0.159178\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −4928.00 −0.649815
\(387\) 1872.00i 0.245889i
\(388\) 12328.0i 1.61304i
\(389\) −7920.00 −1.03229 −0.516144 0.856502i \(-0.672633\pi\)
−0.516144 + 0.856502i \(0.672633\pi\)
\(390\) 0 0
\(391\) 2877.00 0.372113
\(392\) 0 0
\(393\) − 1224.00i − 0.157106i
\(394\) −19916.0 −2.54658
\(395\) 0 0
\(396\) 792.000 0.100504
\(397\) 9654.00i 1.22045i 0.792226 + 0.610227i \(0.208922\pi\)
−0.792226 + 0.610227i \(0.791078\pi\)
\(398\) − 2400.00i − 0.302264i
\(399\) −7875.00 −0.988078
\(400\) 0 0
\(401\) 1952.00 0.243088 0.121544 0.992586i \(-0.461215\pi\)
0.121544 + 0.992586i \(0.461215\pi\)
\(402\) − 8808.00i − 1.09279i
\(403\) − 19856.0i − 2.45434i
\(404\) −6616.00 −0.814748
\(405\) 0 0
\(406\) 12600.0 1.54022
\(407\) 3839.00i 0.467548i
\(408\) 0 0
\(409\) −9690.00 −1.17149 −0.585745 0.810495i \(-0.699198\pi\)
−0.585745 + 0.810495i \(0.699198\pi\)
\(410\) 0 0
\(411\) 7398.00 0.887875
\(412\) 1984.00i 0.237244i
\(413\) 4935.00i 0.587979i
\(414\) 4932.00 0.585494
\(415\) 0 0
\(416\) −17408.0 −2.05168
\(417\) − 3060.00i − 0.359350i
\(418\) − 5500.00i − 0.643574i
\(419\) 2935.00 0.342206 0.171103 0.985253i \(-0.445267\pi\)
0.171103 + 0.985253i \(0.445267\pi\)
\(420\) 0 0
\(421\) 12837.0 1.48607 0.743037 0.669250i \(-0.233385\pi\)
0.743037 + 0.669250i \(0.233385\pi\)
\(422\) − 9872.00i − 1.13877i
\(423\) − 3321.00i − 0.381732i
\(424\) 0 0
\(425\) 0 0
\(426\) −7044.00 −0.801134
\(427\) − 10122.0i − 1.14716i
\(428\) 2928.00i 0.330678i
\(429\) 2244.00 0.252544
\(430\) 0 0
\(431\) −6108.00 −0.682626 −0.341313 0.939950i \(-0.610872\pi\)
−0.341313 + 0.939950i \(0.610872\pi\)
\(432\) 1728.00i 0.192450i
\(433\) − 9278.00i − 1.02973i −0.857272 0.514864i \(-0.827842\pi\)
0.857272 0.514864i \(-0.172158\pi\)
\(434\) 24528.0 2.71286
\(435\) 0 0
\(436\) 2160.00 0.237260
\(437\) − 17125.0i − 1.87460i
\(438\) 6216.00i 0.678110i
\(439\) −2455.00 −0.266904 −0.133452 0.991055i \(-0.542606\pi\)
−0.133452 + 0.991055i \(0.542606\pi\)
\(440\) 0 0
\(441\) 882.000 0.0952381
\(442\) − 5712.00i − 0.614688i
\(443\) − 3503.00i − 0.375694i −0.982198 0.187847i \(-0.939849\pi\)
0.982198 0.187847i \(-0.0601510\pi\)
\(444\) 8376.00 0.895287
\(445\) 0 0
\(446\) −21568.0 −2.28985
\(447\) − 15.0000i − 0.00158719i
\(448\) − 10752.0i − 1.13389i
\(449\) 7630.00 0.801964 0.400982 0.916086i \(-0.368669\pi\)
0.400982 + 0.916086i \(0.368669\pi\)
\(450\) 0 0
\(451\) 5467.00 0.570800
\(452\) − 8016.00i − 0.834161i
\(453\) 1356.00i 0.140641i
\(454\) 9464.00 0.978343
\(455\) 0 0
\(456\) 0 0
\(457\) 7414.00i 0.758889i 0.925214 + 0.379445i \(0.123885\pi\)
−0.925214 + 0.379445i \(0.876115\pi\)
\(458\) 18580.0i 1.89560i
\(459\) −567.000 −0.0576586
\(460\) 0 0
\(461\) 4982.00 0.503329 0.251665 0.967814i \(-0.419022\pi\)
0.251665 + 0.967814i \(0.419022\pi\)
\(462\) 2772.00i 0.279145i
\(463\) 13422.0i 1.34724i 0.739077 + 0.673621i \(0.235262\pi\)
−0.739077 + 0.673621i \(0.764738\pi\)
\(464\) −9600.00 −0.960493
\(465\) 0 0
\(466\) 2052.00 0.203985
\(467\) 15804.0i 1.56600i 0.622022 + 0.783000i \(0.286311\pi\)
−0.622022 + 0.783000i \(0.713689\pi\)
\(468\) − 4896.00i − 0.483585i
\(469\) 15414.0 1.51760
\(470\) 0 0
\(471\) 5298.00 0.518299
\(472\) 0 0
\(473\) − 2288.00i − 0.222415i
\(474\) −12540.0 −1.21515
\(475\) 0 0
\(476\) 3528.00 0.339718
\(477\) − 4878.00i − 0.468235i
\(478\) − 10760.0i − 1.02960i
\(479\) 9060.00 0.864221 0.432111 0.901821i \(-0.357769\pi\)
0.432111 + 0.901821i \(0.357769\pi\)
\(480\) 0 0
\(481\) 23732.0 2.24966
\(482\) − 14912.0i − 1.40918i
\(483\) 8631.00i 0.813093i
\(484\) −968.000 −0.0909091
\(485\) 0 0
\(486\) −972.000 −0.0907218
\(487\) 2854.00i 0.265559i 0.991146 + 0.132779i \(0.0423902\pi\)
−0.991146 + 0.132779i \(0.957610\pi\)
\(488\) 0 0
\(489\) 6204.00 0.573731
\(490\) 0 0
\(491\) −2278.00 −0.209378 −0.104689 0.994505i \(-0.533385\pi\)
−0.104689 + 0.994505i \(0.533385\pi\)
\(492\) − 11928.0i − 1.09300i
\(493\) − 3150.00i − 0.287766i
\(494\) −34000.0 −3.09662
\(495\) 0 0
\(496\) −18688.0 −1.69177
\(497\) − 12327.0i − 1.11256i
\(498\) 7296.00i 0.656509i
\(499\) −13290.0 −1.19227 −0.596134 0.802885i \(-0.703297\pi\)
−0.596134 + 0.802885i \(0.703297\pi\)
\(500\) 0 0
\(501\) 10158.0 0.905840
\(502\) 9408.00i 0.836453i
\(503\) 10762.0i 0.953984i 0.878908 + 0.476992i \(0.158273\pi\)
−0.878908 + 0.476992i \(0.841727\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −6028.00 −0.529599
\(507\) − 7281.00i − 0.637792i
\(508\) − 3752.00i − 0.327693i
\(509\) 1570.00 0.136717 0.0683586 0.997661i \(-0.478224\pi\)
0.0683586 + 0.997661i \(0.478224\pi\)
\(510\) 0 0
\(511\) −10878.0 −0.941711
\(512\) 16384.0i 1.41421i
\(513\) 3375.00i 0.290468i
\(514\) 15384.0 1.32015
\(515\) 0 0
\(516\) −4992.00 −0.425893
\(517\) 4059.00i 0.345289i
\(518\) 29316.0i 2.48662i
\(519\) −351.000 −0.0296863
\(520\) 0 0
\(521\) −22638.0 −1.90363 −0.951813 0.306680i \(-0.900782\pi\)
−0.951813 + 0.306680i \(0.900782\pi\)
\(522\) − 5400.00i − 0.452781i
\(523\) − 10273.0i − 0.858904i −0.903090 0.429452i \(-0.858707\pi\)
0.903090 0.429452i \(-0.141293\pi\)
\(524\) 3264.00 0.272115
\(525\) 0 0
\(526\) −2088.00 −0.173082
\(527\) − 6132.00i − 0.506858i
\(528\) − 2112.00i − 0.174078i
\(529\) −6602.00 −0.542615
\(530\) 0 0
\(531\) 2115.00 0.172850
\(532\) − 21000.0i − 1.71140i
\(533\) − 33796.0i − 2.74647i
\(534\) −9240.00 −0.748790
\(535\) 0 0
\(536\) 0 0
\(537\) − 8985.00i − 0.722032i
\(538\) − 16080.0i − 1.28858i
\(539\) −1078.00 −0.0861461
\(540\) 0 0
\(541\) −6628.00 −0.526728 −0.263364 0.964697i \(-0.584832\pi\)
−0.263364 + 0.964697i \(0.584832\pi\)
\(542\) 26748.0i 2.11979i
\(543\) 12201.0i 0.964263i
\(544\) −5376.00 −0.423702
\(545\) 0 0
\(546\) 17136.0 1.34314
\(547\) − 1131.00i − 0.0884060i −0.999023 0.0442030i \(-0.985925\pi\)
0.999023 0.0442030i \(-0.0140748\pi\)
\(548\) 19728.0i 1.53784i
\(549\) −4338.00 −0.337234
\(550\) 0 0
\(551\) −18750.0 −1.44969
\(552\) 0 0
\(553\) − 21945.0i − 1.68752i
\(554\) 14984.0 1.14911
\(555\) 0 0
\(556\) 8160.00 0.622412
\(557\) 22954.0i 1.74613i 0.487607 + 0.873063i \(0.337870\pi\)
−0.487607 + 0.873063i \(0.662130\pi\)
\(558\) − 10512.0i − 0.797506i
\(559\) −14144.0 −1.07017
\(560\) 0 0
\(561\) 693.000 0.0521542
\(562\) − 23532.0i − 1.76626i
\(563\) 5532.00i 0.414114i 0.978329 + 0.207057i \(0.0663885\pi\)
−0.978329 + 0.207057i \(0.933611\pi\)
\(564\) 8856.00 0.661179
\(565\) 0 0
\(566\) 15772.0 1.17128
\(567\) − 1701.00i − 0.125988i
\(568\) 0 0
\(569\) 25225.0 1.85850 0.929250 0.369450i \(-0.120454\pi\)
0.929250 + 0.369450i \(0.120454\pi\)
\(570\) 0 0
\(571\) −2088.00 −0.153030 −0.0765150 0.997068i \(-0.524379\pi\)
−0.0765150 + 0.997068i \(0.524379\pi\)
\(572\) 5984.00i 0.437419i
\(573\) 9141.00i 0.666441i
\(574\) 41748.0 3.03576
\(575\) 0 0
\(576\) −4608.00 −0.333333
\(577\) − 7831.00i − 0.565007i −0.959266 0.282503i \(-0.908835\pi\)
0.959266 0.282503i \(-0.0911648\pi\)
\(578\) 17888.0i 1.28727i
\(579\) −3696.00 −0.265286
\(580\) 0 0
\(581\) −12768.0 −0.911714
\(582\) 18492.0i 1.31704i
\(583\) 5962.00i 0.423535i
\(584\) 0 0
\(585\) 0 0
\(586\) −5948.00 −0.419300
\(587\) 8199.00i 0.576506i 0.957554 + 0.288253i \(0.0930744\pi\)
−0.957554 + 0.288253i \(0.906926\pi\)
\(588\) 2352.00i 0.164957i
\(589\) −36500.0 −2.55341
\(590\) 0 0
\(591\) −14937.0 −1.03964
\(592\) − 22336.0i − 1.55068i
\(593\) 9542.00i 0.660781i 0.943844 + 0.330390i \(0.107180\pi\)
−0.943844 + 0.330390i \(0.892820\pi\)
\(594\) 1188.00 0.0820610
\(595\) 0 0
\(596\) 40.0000 0.00274910
\(597\) − 1800.00i − 0.123399i
\(598\) 37264.0i 2.54822i
\(599\) −24705.0 −1.68517 −0.842587 0.538561i \(-0.818968\pi\)
−0.842587 + 0.538561i \(0.818968\pi\)
\(600\) 0 0
\(601\) 15452.0 1.04875 0.524376 0.851487i \(-0.324299\pi\)
0.524376 + 0.851487i \(0.324299\pi\)
\(602\) − 17472.0i − 1.18290i
\(603\) − 6606.00i − 0.446131i
\(604\) −3616.00 −0.243598
\(605\) 0 0
\(606\) −9924.00 −0.665239
\(607\) − 6176.00i − 0.412975i −0.978449 0.206488i \(-0.933797\pi\)
0.978449 0.206488i \(-0.0662034\pi\)
\(608\) 32000.0i 2.13449i
\(609\) 9450.00 0.628790
\(610\) 0 0
\(611\) 25092.0 1.66140
\(612\) − 1512.00i − 0.0998676i
\(613\) − 13198.0i − 0.869596i −0.900528 0.434798i \(-0.856820\pi\)
0.900528 0.434798i \(-0.143180\pi\)
\(614\) −19376.0 −1.27354
\(615\) 0 0
\(616\) 0 0
\(617\) − 19216.0i − 1.25382i −0.779092 0.626910i \(-0.784319\pi\)
0.779092 0.626910i \(-0.215681\pi\)
\(618\) 2976.00i 0.193709i
\(619\) −27700.0 −1.79864 −0.899319 0.437293i \(-0.855937\pi\)
−0.899319 + 0.437293i \(0.855937\pi\)
\(620\) 0 0
\(621\) 3699.00 0.239027
\(622\) 18528.0i 1.19438i
\(623\) − 16170.0i − 1.03987i
\(624\) −13056.0 −0.837593
\(625\) 0 0
\(626\) −33748.0 −2.15470
\(627\) − 4125.00i − 0.262738i
\(628\) 14128.0i 0.897721i
\(629\) 7329.00 0.464589
\(630\) 0 0
\(631\) −5108.00 −0.322260 −0.161130 0.986933i \(-0.551514\pi\)
−0.161130 + 0.986933i \(0.551514\pi\)
\(632\) 0 0
\(633\) − 7404.00i − 0.464901i
\(634\) 14544.0 0.911066
\(635\) 0 0
\(636\) 13008.0 0.811007
\(637\) 6664.00i 0.414501i
\(638\) 6600.00i 0.409556i
\(639\) −5283.00 −0.327061
\(640\) 0 0
\(641\) 322.000 0.0198412 0.00992062 0.999951i \(-0.496842\pi\)
0.00992062 + 0.999951i \(0.496842\pi\)
\(642\) 4392.00i 0.269998i
\(643\) 7432.00i 0.455816i 0.973683 + 0.227908i \(0.0731885\pi\)
−0.973683 + 0.227908i \(0.926812\pi\)
\(644\) −23016.0 −1.40832
\(645\) 0 0
\(646\) −10500.0 −0.639500
\(647\) 1409.00i 0.0856159i 0.999083 + 0.0428080i \(0.0136304\pi\)
−0.999083 + 0.0428080i \(0.986370\pi\)
\(648\) 0 0
\(649\) −2585.00 −0.156348
\(650\) 0 0
\(651\) 18396.0 1.10752
\(652\) 16544.0i 0.993732i
\(653\) − 21548.0i − 1.29133i −0.763621 0.645665i \(-0.776580\pi\)
0.763621 0.645665i \(-0.223420\pi\)
\(654\) 3240.00 0.193722
\(655\) 0 0
\(656\) −31808.0 −1.89313
\(657\) 4662.00i 0.276837i
\(658\) 30996.0i 1.83640i
\(659\) 13380.0 0.790912 0.395456 0.918485i \(-0.370587\pi\)
0.395456 + 0.918485i \(0.370587\pi\)
\(660\) 0 0
\(661\) 10907.0 0.641805 0.320903 0.947112i \(-0.396014\pi\)
0.320903 + 0.947112i \(0.396014\pi\)
\(662\) − 3032.00i − 0.178009i
\(663\) − 4284.00i − 0.250945i
\(664\) 0 0
\(665\) 0 0
\(666\) 12564.0 0.730999
\(667\) 20550.0i 1.19295i
\(668\) 27088.0i 1.56896i
\(669\) −16176.0 −0.934829
\(670\) 0 0
\(671\) 5302.00 0.305039
\(672\) − 16128.0i − 0.925820i
\(673\) 17522.0i 1.00360i 0.864983 + 0.501800i \(0.167329\pi\)
−0.864983 + 0.501800i \(0.832671\pi\)
\(674\) −29496.0 −1.68567
\(675\) 0 0
\(676\) 19416.0 1.10469
\(677\) − 8306.00i − 0.471530i −0.971810 0.235765i \(-0.924241\pi\)
0.971810 0.235765i \(-0.0757595\pi\)
\(678\) − 12024.0i − 0.681090i
\(679\) −32361.0 −1.82902
\(680\) 0 0
\(681\) 7098.00 0.399407
\(682\) 12848.0i 0.721371i
\(683\) 18427.0i 1.03234i 0.856486 + 0.516171i \(0.172643\pi\)
−0.856486 + 0.516171i \(0.827357\pi\)
\(684\) −9000.00 −0.503105
\(685\) 0 0
\(686\) 20580.0 1.14541
\(687\) 13935.0i 0.773877i
\(688\) 13312.0i 0.737668i
\(689\) 36856.0 2.03788
\(690\) 0 0
\(691\) −8278.00 −0.455731 −0.227865 0.973693i \(-0.573175\pi\)
−0.227865 + 0.973693i \(0.573175\pi\)
\(692\) − 936.000i − 0.0514182i
\(693\) 2079.00i 0.113961i
\(694\) −26096.0 −1.42736
\(695\) 0 0
\(696\) 0 0
\(697\) − 10437.0i − 0.567187i
\(698\) − 14840.0i − 0.804731i
\(699\) 1539.00 0.0832766
\(700\) 0 0
\(701\) −21923.0 −1.18120 −0.590599 0.806965i \(-0.701109\pi\)
−0.590599 + 0.806965i \(0.701109\pi\)
\(702\) − 7344.00i − 0.394845i
\(703\) − 43625.0i − 2.34047i
\(704\) 5632.00 0.301511
\(705\) 0 0
\(706\) −11328.0 −0.603874
\(707\) − 17367.0i − 0.923838i
\(708\) 5640.00i 0.299384i
\(709\) 11425.0 0.605183 0.302592 0.953120i \(-0.402148\pi\)
0.302592 + 0.953120i \(0.402148\pi\)
\(710\) 0 0
\(711\) −9405.00 −0.496083
\(712\) 0 0
\(713\) 40004.0i 2.10121i
\(714\) 5292.00 0.277378
\(715\) 0 0
\(716\) 23960.0 1.25060
\(717\) − 8070.00i − 0.420334i
\(718\) 28160.0i 1.46368i
\(719\) −7380.00 −0.382792 −0.191396 0.981513i \(-0.561301\pi\)
−0.191396 + 0.981513i \(0.561301\pi\)
\(720\) 0 0
\(721\) −5208.00 −0.269010
\(722\) 35064.0i 1.80741i
\(723\) − 11184.0i − 0.575294i
\(724\) −32536.0 −1.67015
\(725\) 0 0
\(726\) −1452.00 −0.0742270
\(727\) 10924.0i 0.557288i 0.960394 + 0.278644i \(0.0898850\pi\)
−0.960394 + 0.278644i \(0.910115\pi\)
\(728\) 0 0
\(729\) −729.000 −0.0370370
\(730\) 0 0
\(731\) −4368.00 −0.221007
\(732\) − 11568.0i − 0.584106i
\(733\) − 13578.0i − 0.684195i −0.939664 0.342097i \(-0.888863\pi\)
0.939664 0.342097i \(-0.111137\pi\)
\(734\) 24824.0 1.24833
\(735\) 0 0
\(736\) 35072.0 1.75648
\(737\) 8074.00i 0.403541i
\(738\) − 17892.0i − 0.892430i
\(739\) −2875.00 −0.143110 −0.0715552 0.997437i \(-0.522796\pi\)
−0.0715552 + 0.997437i \(0.522796\pi\)
\(740\) 0 0
\(741\) −25500.0 −1.26419
\(742\) 45528.0i 2.25254i
\(743\) − 9568.00i − 0.472431i −0.971701 0.236215i \(-0.924093\pi\)
0.971701 0.236215i \(-0.0759071\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −7848.00 −0.385168
\(747\) 5472.00i 0.268019i
\(748\) 1848.00i 0.0903337i
\(749\) −7686.00 −0.374954
\(750\) 0 0
\(751\) −35048.0 −1.70296 −0.851478 0.524391i \(-0.824293\pi\)
−0.851478 + 0.524391i \(0.824293\pi\)
\(752\) − 23616.0i − 1.14520i
\(753\) 7056.00i 0.341481i
\(754\) 40800.0 1.97062
\(755\) 0 0
\(756\) 4536.00 0.218218
\(757\) − 8226.00i − 0.394953i −0.980308 0.197476i \(-0.936725\pi\)
0.980308 0.197476i \(-0.0632745\pi\)
\(758\) 31840.0i 1.52570i
\(759\) −4521.00 −0.216208
\(760\) 0 0
\(761\) −6818.00 −0.324773 −0.162387 0.986727i \(-0.551919\pi\)
−0.162387 + 0.986727i \(0.551919\pi\)
\(762\) − 5628.00i − 0.267560i
\(763\) 5670.00i 0.269027i
\(764\) −24376.0 −1.15431
\(765\) 0 0
\(766\) 28752.0 1.35620
\(767\) 15980.0i 0.752287i
\(768\) 12288.0i 0.577350i
\(769\) −29390.0 −1.37819 −0.689097 0.724670i \(-0.741992\pi\)
−0.689097 + 0.724670i \(0.741992\pi\)
\(770\) 0 0
\(771\) 11538.0 0.538951
\(772\) − 9856.00i − 0.459489i
\(773\) − 34358.0i − 1.59867i −0.600886 0.799335i \(-0.705185\pi\)
0.600886 0.799335i \(-0.294815\pi\)
\(774\) −7488.00 −0.347740
\(775\) 0 0
\(776\) 0 0
\(777\) 21987.0i 1.01516i
\(778\) − 31680.0i − 1.45988i
\(779\) −62125.0 −2.85733
\(780\) 0 0
\(781\) 6457.00 0.295838
\(782\) 11508.0i 0.526247i
\(783\) − 4050.00i − 0.184847i
\(784\) 6272.00 0.285714
\(785\) 0 0
\(786\) 4896.00 0.222181
\(787\) − 14291.0i − 0.647292i −0.946178 0.323646i \(-0.895091\pi\)
0.946178 0.323646i \(-0.104909\pi\)
\(788\) − 39832.0i − 1.80071i
\(789\) −1566.00 −0.0706604
\(790\) 0 0
\(791\) 21042.0 0.945850
\(792\) 0 0
\(793\) − 32776.0i − 1.46773i
\(794\) −38616.0 −1.72598
\(795\) 0 0
\(796\) 4800.00 0.213733
\(797\) − 11576.0i − 0.514483i −0.966347 0.257242i \(-0.917186\pi\)
0.966347 0.257242i \(-0.0828136\pi\)
\(798\) − 31500.0i − 1.39735i
\(799\) 7749.00 0.343104
\(800\) 0 0
\(801\) −6930.00 −0.305692
\(802\) 7808.00i 0.343778i
\(803\) − 5698.00i − 0.250409i
\(804\) 17616.0 0.772722
\(805\) 0 0
\(806\) 79424.0 3.47096
\(807\) − 12060.0i − 0.526062i
\(808\) 0 0
\(809\) 12825.0 0.557358 0.278679 0.960384i \(-0.410103\pi\)
0.278679 + 0.960384i \(0.410103\pi\)
\(810\) 0 0
\(811\) −36843.0 −1.59523 −0.797616 0.603166i \(-0.793906\pi\)
−0.797616 + 0.603166i \(0.793906\pi\)
\(812\) 25200.0i 1.08910i
\(813\) 20061.0i 0.865400i
\(814\) −15356.0 −0.661213
\(815\) 0 0
\(816\) −4032.00 −0.172976
\(817\) 26000.0i 1.11337i
\(818\) − 38760.0i − 1.65674i
\(819\) 12852.0 0.548334
\(820\) 0 0
\(821\) 43962.0 1.86880 0.934400 0.356226i \(-0.115937\pi\)
0.934400 + 0.356226i \(0.115937\pi\)
\(822\) 29592.0i 1.25564i
\(823\) 33522.0i 1.41981i 0.704298 + 0.709905i \(0.251262\pi\)
−0.704298 + 0.709905i \(0.748738\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) −19740.0 −0.831528
\(827\) 1704.00i 0.0716492i 0.999358 + 0.0358246i \(0.0114058\pi\)
−0.999358 + 0.0358246i \(0.988594\pi\)
\(828\) 9864.00i 0.414007i
\(829\) −1750.00 −0.0733173 −0.0366586 0.999328i \(-0.511671\pi\)
−0.0366586 + 0.999328i \(0.511671\pi\)
\(830\) 0 0
\(831\) 11238.0 0.469124
\(832\) − 34816.0i − 1.45075i
\(833\) 2058.00i 0.0856008i
\(834\) 12240.0 0.508197
\(835\) 0 0
\(836\) 11000.0 0.455075
\(837\) − 7884.00i − 0.325581i
\(838\) 11740.0i 0.483952i
\(839\) 15260.0 0.627931 0.313965 0.949434i \(-0.398342\pi\)
0.313965 + 0.949434i \(0.398342\pi\)
\(840\) 0 0
\(841\) −1889.00 −0.0774530
\(842\) 51348.0i 2.10163i
\(843\) − 17649.0i − 0.721072i
\(844\) 19744.0 0.805233
\(845\) 0 0
\(846\) 13284.0 0.539850
\(847\) − 2541.00i − 0.103081i
\(848\) − 34688.0i − 1.40471i
\(849\) 11829.0 0.478175
\(850\) 0 0
\(851\) −47813.0 −1.92598
\(852\) − 14088.0i − 0.566487i
\(853\) − 878.000i − 0.0352428i −0.999845 0.0176214i \(-0.994391\pi\)
0.999845 0.0176214i \(-0.00560936\pi\)
\(854\) 40488.0 1.62233
\(855\) 0 0
\(856\) 0 0
\(857\) 35019.0i 1.39583i 0.716181 + 0.697915i \(0.245889\pi\)
−0.716181 + 0.697915i \(0.754111\pi\)
\(858\) 8976.00i 0.357151i
\(859\) 1280.00 0.0508417 0.0254209 0.999677i \(-0.491907\pi\)
0.0254209 + 0.999677i \(0.491907\pi\)
\(860\) 0 0
\(861\) 31311.0 1.23934
\(862\) − 24432.0i − 0.965380i
\(863\) − 16888.0i − 0.666135i −0.942903 0.333067i \(-0.891916\pi\)
0.942903 0.333067i \(-0.108084\pi\)
\(864\) −6912.00 −0.272166
\(865\) 0 0
\(866\) 37112.0 1.45626
\(867\) 13416.0i 0.525526i
\(868\) 49056.0i 1.91828i
\(869\) 11495.0 0.448724
\(870\) 0 0
\(871\) 49912.0 1.94168
\(872\) 0 0
\(873\) 13869.0i 0.537680i
\(874\) 68500.0 2.65108
\(875\) 0 0
\(876\) −12432.0 −0.479496
\(877\) − 29836.0i − 1.14879i −0.818578 0.574396i \(-0.805237\pi\)
0.818578 0.574396i \(-0.194763\pi\)
\(878\) − 9820.00i − 0.377459i
\(879\) −4461.00 −0.171178
\(880\) 0 0
\(881\) 29292.0 1.12017 0.560087 0.828434i \(-0.310768\pi\)
0.560087 + 0.828434i \(0.310768\pi\)
\(882\) 3528.00i 0.134687i
\(883\) 6532.00i 0.248946i 0.992223 + 0.124473i \(0.0397240\pi\)
−0.992223 + 0.124473i \(0.960276\pi\)
\(884\) 11424.0 0.434650
\(885\) 0 0
\(886\) 14012.0 0.531312
\(887\) − 20476.0i − 0.775103i −0.921848 0.387552i \(-0.873321\pi\)
0.921848 0.387552i \(-0.126679\pi\)
\(888\) 0 0
\(889\) 9849.00 0.371569
\(890\) 0 0
\(891\) 891.000 0.0335013
\(892\) − 43136.0i − 1.61917i
\(893\) − 46125.0i − 1.72846i
\(894\) 60.0000 0.00224463
\(895\) 0 0
\(896\) 0 0
\(897\) 27948.0i 1.04031i
\(898\) 30520.0i 1.13415i
\(899\) 43800.0 1.62493
\(900\) 0 0
\(901\) 11382.0 0.420854
\(902\) 21868.0i 0.807234i
\(903\) − 13104.0i − 0.482917i
\(904\) 0 0
\(905\) 0 0
\(906\) −5424.00 −0.198897
\(907\) 51914.0i 1.90052i 0.311450 + 0.950262i \(0.399185\pi\)
−0.311450 + 0.950262i \(0.600815\pi\)
\(908\) 18928.0i 0.691793i
\(909\) −7443.00 −0.271583
\(910\) 0 0
\(911\) −41893.0 −1.52358 −0.761788 0.647827i \(-0.775678\pi\)
−0.761788 + 0.647827i \(0.775678\pi\)
\(912\) 24000.0i 0.871403i
\(913\) − 6688.00i − 0.242432i
\(914\) −29656.0 −1.07323
\(915\) 0 0
\(916\) −37160.0 −1.34039
\(917\) 8568.00i 0.308550i
\(918\) − 2268.00i − 0.0815416i
\(919\) −495.000 −0.0177677 −0.00888386 0.999961i \(-0.502828\pi\)
−0.00888386 + 0.999961i \(0.502828\pi\)
\(920\) 0 0
\(921\) −14532.0 −0.519919
\(922\) 19928.0i 0.711815i
\(923\) − 39916.0i − 1.42346i
\(924\) −5544.00 −0.197386
\(925\) 0 0
\(926\) −53688.0 −1.90529
\(927\) 2232.00i 0.0790814i
\(928\) − 38400.0i − 1.35834i
\(929\) 16310.0 0.576010 0.288005 0.957629i \(-0.407008\pi\)
0.288005 + 0.957629i \(0.407008\pi\)
\(930\) 0 0
\(931\) 12250.0 0.431233
\(932\) 4104.00i 0.144239i
\(933\) 13896.0i 0.487604i
\(934\) −63216.0 −2.21466
\(935\) 0 0
\(936\) 0 0
\(937\) 18744.0i 0.653511i 0.945109 + 0.326755i \(0.105955\pi\)
−0.945109 + 0.326755i \(0.894045\pi\)
\(938\) 61656.0i 2.14620i
\(939\) −25311.0 −0.879652
\(940\) 0 0
\(941\) −25553.0 −0.885233 −0.442616 0.896711i \(-0.645950\pi\)
−0.442616 + 0.896711i \(0.645950\pi\)
\(942\) 21192.0i 0.732986i
\(943\) 68089.0i 2.35131i
\(944\) 15040.0 0.518549
\(945\) 0 0
\(946\) 9152.00 0.314542
\(947\) 6879.00i 0.236048i 0.993011 + 0.118024i \(0.0376560\pi\)
−0.993011 + 0.118024i \(0.962344\pi\)
\(948\) − 25080.0i − 0.859241i
\(949\) −35224.0 −1.20487
\(950\) 0 0
\(951\) 10908.0 0.371941
\(952\) 0 0
\(953\) 13677.0i 0.464891i 0.972609 + 0.232446i \(0.0746728\pi\)
−0.972609 + 0.232446i \(0.925327\pi\)
\(954\) 19512.0 0.662185
\(955\) 0 0
\(956\) 21520.0 0.728040
\(957\) 4950.00i 0.167200i
\(958\) 36240.0i 1.22219i
\(959\) −51786.0 −1.74375
\(960\) 0 0
\(961\) 55473.0 1.86207
\(962\) 94928.0i 3.18150i
\(963\) 3294.00i 0.110226i
\(964\) 29824.0 0.996438
\(965\) 0 0
\(966\) −34524.0 −1.14989
\(967\) 26984.0i 0.897360i 0.893693 + 0.448680i \(0.148106\pi\)
−0.893693 + 0.448680i \(0.851894\pi\)
\(968\) 0 0
\(969\) −7875.00 −0.261075
\(970\) 0 0
\(971\) 41937.0 1.38602 0.693008 0.720929i \(-0.256285\pi\)
0.693008 + 0.720929i \(0.256285\pi\)
\(972\) − 1944.00i − 0.0641500i
\(973\) 21420.0i 0.705749i
\(974\) −11416.0 −0.375557
\(975\) 0 0
\(976\) −30848.0 −1.01170
\(977\) 13504.0i 0.442202i 0.975251 + 0.221101i \(0.0709650\pi\)
−0.975251 + 0.221101i \(0.929035\pi\)
\(978\) 24816.0i 0.811379i
\(979\) 8470.00 0.276509
\(980\) 0 0
\(981\) 2430.00 0.0790866
\(982\) − 9112.00i − 0.296106i
\(983\) − 33353.0i − 1.08219i −0.840961 0.541096i \(-0.818009\pi\)
0.840961 0.541096i \(-0.181991\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 12600.0 0.406963
\(987\) 23247.0i 0.749706i
\(988\) − 68000.0i − 2.18964i
\(989\) 28496.0 0.916198
\(990\) 0 0
\(991\) −16978.0 −0.544222 −0.272111 0.962266i \(-0.587722\pi\)
−0.272111 + 0.962266i \(0.587722\pi\)
\(992\) − 74752.0i − 2.39252i
\(993\) − 2274.00i − 0.0726719i
\(994\) 49308.0 1.57340
\(995\) 0 0
\(996\) −14592.0 −0.464222
\(997\) 2714.00i 0.0862119i 0.999071 + 0.0431059i \(0.0137253\pi\)
−0.999071 + 0.0431059i \(0.986275\pi\)
\(998\) − 53160.0i − 1.68612i
\(999\) 9423.00 0.298429
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 825.4.c.c.199.2 2
5.2 odd 4 825.4.a.b.1.1 1
5.3 odd 4 825.4.a.h.1.1 yes 1
5.4 even 2 inner 825.4.c.c.199.1 2
15.2 even 4 2475.4.a.j.1.1 1
15.8 even 4 2475.4.a.c.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
825.4.a.b.1.1 1 5.2 odd 4
825.4.a.h.1.1 yes 1 5.3 odd 4
825.4.c.c.199.1 2 5.4 even 2 inner
825.4.c.c.199.2 2 1.1 even 1 trivial
2475.4.a.c.1.1 1 15.8 even 4
2475.4.a.j.1.1 1 15.2 even 4