Properties

Label 768.3.b.f.127.8
Level $768$
Weight $3$
Character 768.127
Analytic conductor $20.926$
Analytic rank $0$
Dimension $8$
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] = [768,3,Mod(127,768)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(768, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("768.127");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 768 = 2^{8} \cdot 3 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 768.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: \(20.9264843029\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\zeta_{24})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{14} \)
Twist minimal: no (minimal twist has level 384)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 127.8
Root \(-0.965926 + 0.258819i\) of defining polynomial
Character \(\chi\) \(=\) 768.127
Dual form 768.3.b.f.127.5

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.73205 q^{3} +4.29253i q^{5} +2.75787i q^{7} +3.00000 q^{9} +O(q^{10})\) \(q+1.73205 q^{3} +4.29253i q^{5} +2.75787i q^{7} +3.00000 q^{9} +13.7980 q^{11} -14.5266i q^{13} +7.43488i q^{15} +22.8673 q^{17} -16.0399 q^{19} +4.77678i q^{21} -17.1117i q^{23} +6.57420 q^{25} +5.19615 q^{27} +21.8667i q^{29} +38.6944i q^{31} +23.8988 q^{33} -11.8383 q^{35} +66.4204i q^{37} -25.1608i q^{39} -23.2392 q^{41} +47.9230 q^{43} +12.8776i q^{45} +14.8512i q^{47} +41.3941 q^{49} +39.6073 q^{51} +65.5589i q^{53} +59.2281i q^{55} -27.7819 q^{57} -65.8428 q^{59} +40.1123i q^{61} +8.27362i q^{63} +62.3559 q^{65} -74.8105 q^{67} -29.6383i q^{69} -122.681i q^{71} +144.904 q^{73} +11.3868 q^{75} +38.0530i q^{77} -128.657i q^{79} +9.00000 q^{81} -22.0417 q^{83} +98.1584i q^{85} +37.8743i q^{87} +122.075 q^{89} +40.0626 q^{91} +67.0208i q^{93} -68.8516i q^{95} -88.3072 q^{97} +41.3939 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 24 q^{9} + 32 q^{11} + 16 q^{17} + 96 q^{19} + 8 q^{25} - 96 q^{35} - 80 q^{41} + 224 q^{43} - 88 q^{49} + 96 q^{51} - 96 q^{57} - 512 q^{59} - 160 q^{65} + 16 q^{73} + 192 q^{75} + 72 q^{81} - 544 q^{83} + 240 q^{89} - 32 q^{91} + 400 q^{97} + 96 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/768\mathbb{Z}\right)^\times\).

\(n\) \(257\) \(511\) \(517\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.73205 0.577350
\(4\) 0 0
\(5\) 4.29253i 0.858506i 0.903184 + 0.429253i \(0.141223\pi\)
−0.903184 + 0.429253i \(0.858777\pi\)
\(6\) 0 0
\(7\) 2.75787i 0.393982i 0.980405 + 0.196991i \(0.0631170\pi\)
−0.980405 + 0.196991i \(0.936883\pi\)
\(8\) 0 0
\(9\) 3.00000 0.333333
\(10\) 0 0
\(11\) 13.7980 1.25436 0.627180 0.778874i \(-0.284209\pi\)
0.627180 + 0.778874i \(0.284209\pi\)
\(12\) 0 0
\(13\) − 14.5266i − 1.11743i −0.829359 0.558716i \(-0.811294\pi\)
0.829359 0.558716i \(-0.188706\pi\)
\(14\) 0 0
\(15\) 7.43488i 0.495659i
\(16\) 0 0
\(17\) 22.8673 1.34513 0.672567 0.740037i \(-0.265192\pi\)
0.672567 + 0.740037i \(0.265192\pi\)
\(18\) 0 0
\(19\) −16.0399 −0.844204 −0.422102 0.906548i \(-0.638708\pi\)
−0.422102 + 0.906548i \(0.638708\pi\)
\(20\) 0 0
\(21\) 4.77678i 0.227466i
\(22\) 0 0
\(23\) − 17.1117i − 0.743986i −0.928236 0.371993i \(-0.878675\pi\)
0.928236 0.371993i \(-0.121325\pi\)
\(24\) 0 0
\(25\) 6.57420 0.262968
\(26\) 0 0
\(27\) 5.19615 0.192450
\(28\) 0 0
\(29\) 21.8667i 0.754025i 0.926208 + 0.377013i \(0.123049\pi\)
−0.926208 + 0.377013i \(0.876951\pi\)
\(30\) 0 0
\(31\) 38.6944i 1.24821i 0.781341 + 0.624104i \(0.214536\pi\)
−0.781341 + 0.624104i \(0.785464\pi\)
\(32\) 0 0
\(33\) 23.8988 0.724205
\(34\) 0 0
\(35\) −11.8383 −0.338236
\(36\) 0 0
\(37\) 66.4204i 1.79515i 0.440866 + 0.897573i \(0.354671\pi\)
−0.440866 + 0.897573i \(0.645329\pi\)
\(38\) 0 0
\(39\) − 25.1608i − 0.645149i
\(40\) 0 0
\(41\) −23.2392 −0.566809 −0.283405 0.959000i \(-0.591464\pi\)
−0.283405 + 0.959000i \(0.591464\pi\)
\(42\) 0 0
\(43\) 47.9230 1.11449 0.557244 0.830349i \(-0.311859\pi\)
0.557244 + 0.830349i \(0.311859\pi\)
\(44\) 0 0
\(45\) 12.8776i 0.286169i
\(46\) 0 0
\(47\) 14.8512i 0.315983i 0.987441 + 0.157991i \(0.0505018\pi\)
−0.987441 + 0.157991i \(0.949498\pi\)
\(48\) 0 0
\(49\) 41.3941 0.844778
\(50\) 0 0
\(51\) 39.6073 0.776613
\(52\) 0 0
\(53\) 65.5589i 1.23696i 0.785800 + 0.618480i \(0.212251\pi\)
−0.785800 + 0.618480i \(0.787749\pi\)
\(54\) 0 0
\(55\) 59.2281i 1.07688i
\(56\) 0 0
\(57\) −27.7819 −0.487401
\(58\) 0 0
\(59\) −65.8428 −1.11598 −0.557990 0.829848i \(-0.688427\pi\)
−0.557990 + 0.829848i \(0.688427\pi\)
\(60\) 0 0
\(61\) 40.1123i 0.657579i 0.944403 + 0.328790i \(0.106641\pi\)
−0.944403 + 0.328790i \(0.893359\pi\)
\(62\) 0 0
\(63\) 8.27362i 0.131327i
\(64\) 0 0
\(65\) 62.3559 0.959321
\(66\) 0 0
\(67\) −74.8105 −1.11657 −0.558287 0.829648i \(-0.688541\pi\)
−0.558287 + 0.829648i \(0.688541\pi\)
\(68\) 0 0
\(69\) − 29.6383i − 0.429540i
\(70\) 0 0
\(71\) − 122.681i − 1.72790i −0.503578 0.863950i \(-0.667983\pi\)
0.503578 0.863950i \(-0.332017\pi\)
\(72\) 0 0
\(73\) 144.904 1.98498 0.992492 0.122312i \(-0.0390307\pi\)
0.992492 + 0.122312i \(0.0390307\pi\)
\(74\) 0 0
\(75\) 11.3868 0.151825
\(76\) 0 0
\(77\) 38.0530i 0.494195i
\(78\) 0 0
\(79\) − 128.657i − 1.62857i −0.580467 0.814284i \(-0.697130\pi\)
0.580467 0.814284i \(-0.302870\pi\)
\(80\) 0 0
\(81\) 9.00000 0.111111
\(82\) 0 0
\(83\) −22.0417 −0.265563 −0.132781 0.991145i \(-0.542391\pi\)
−0.132781 + 0.991145i \(0.542391\pi\)
\(84\) 0 0
\(85\) 98.1584i 1.15480i
\(86\) 0 0
\(87\) 37.8743i 0.435337i
\(88\) 0 0
\(89\) 122.075 1.37163 0.685813 0.727778i \(-0.259447\pi\)
0.685813 + 0.727778i \(0.259447\pi\)
\(90\) 0 0
\(91\) 40.0626 0.440248
\(92\) 0 0
\(93\) 67.0208i 0.720653i
\(94\) 0 0
\(95\) − 68.8516i − 0.724754i
\(96\) 0 0
\(97\) −88.3072 −0.910384 −0.455192 0.890393i \(-0.650429\pi\)
−0.455192 + 0.890393i \(0.650429\pi\)
\(98\) 0 0
\(99\) 41.3939 0.418120
\(100\) 0 0
\(101\) − 104.345i − 1.03312i −0.856251 0.516560i \(-0.827212\pi\)
0.856251 0.516560i \(-0.172788\pi\)
\(102\) 0 0
\(103\) 69.0609i 0.670494i 0.942130 + 0.335247i \(0.108820\pi\)
−0.942130 + 0.335247i \(0.891180\pi\)
\(104\) 0 0
\(105\) −20.5045 −0.195281
\(106\) 0 0
\(107\) −149.429 −1.39653 −0.698265 0.715839i \(-0.746044\pi\)
−0.698265 + 0.715839i \(0.746044\pi\)
\(108\) 0 0
\(109\) − 54.3341i − 0.498478i −0.968442 0.249239i \(-0.919820\pi\)
0.968442 0.249239i \(-0.0801805\pi\)
\(110\) 0 0
\(111\) 115.044i 1.03643i
\(112\) 0 0
\(113\) 6.32846 0.0560041 0.0280021 0.999608i \(-0.491086\pi\)
0.0280021 + 0.999608i \(0.491086\pi\)
\(114\) 0 0
\(115\) 73.4523 0.638716
\(116\) 0 0
\(117\) − 43.5798i − 0.372477i
\(118\) 0 0
\(119\) 63.0651i 0.529958i
\(120\) 0 0
\(121\) 69.3837 0.573419
\(122\) 0 0
\(123\) −40.2514 −0.327247
\(124\) 0 0
\(125\) 135.533i 1.08427i
\(126\) 0 0
\(127\) − 16.5228i − 0.130101i −0.997882 0.0650504i \(-0.979279\pi\)
0.997882 0.0650504i \(-0.0207208\pi\)
\(128\) 0 0
\(129\) 83.0050 0.643450
\(130\) 0 0
\(131\) 50.9799 0.389159 0.194580 0.980887i \(-0.437666\pi\)
0.194580 + 0.980887i \(0.437666\pi\)
\(132\) 0 0
\(133\) − 44.2360i − 0.332601i
\(134\) 0 0
\(135\) 22.3046i 0.165220i
\(136\) 0 0
\(137\) −10.0157 −0.0731070 −0.0365535 0.999332i \(-0.511638\pi\)
−0.0365535 + 0.999332i \(0.511638\pi\)
\(138\) 0 0
\(139\) 65.7088 0.472725 0.236363 0.971665i \(-0.424045\pi\)
0.236363 + 0.971665i \(0.424045\pi\)
\(140\) 0 0
\(141\) 25.7230i 0.182433i
\(142\) 0 0
\(143\) − 200.438i − 1.40166i
\(144\) 0 0
\(145\) −93.8635 −0.647335
\(146\) 0 0
\(147\) 71.6967 0.487733
\(148\) 0 0
\(149\) − 94.7716i − 0.636051i −0.948082 0.318026i \(-0.896980\pi\)
0.948082 0.318026i \(-0.103020\pi\)
\(150\) 0 0
\(151\) 269.769i 1.78655i 0.449508 + 0.893276i \(0.351599\pi\)
−0.449508 + 0.893276i \(0.648401\pi\)
\(152\) 0 0
\(153\) 68.6018 0.448378
\(154\) 0 0
\(155\) −166.097 −1.07159
\(156\) 0 0
\(157\) − 31.5058i − 0.200674i −0.994954 0.100337i \(-0.968008\pi\)
0.994954 0.100337i \(-0.0319921\pi\)
\(158\) 0 0
\(159\) 113.551i 0.714159i
\(160\) 0 0
\(161\) 47.1918 0.293117
\(162\) 0 0
\(163\) 201.159 1.23410 0.617051 0.786923i \(-0.288327\pi\)
0.617051 + 0.786923i \(0.288327\pi\)
\(164\) 0 0
\(165\) 102.586i 0.621734i
\(166\) 0 0
\(167\) − 266.393i − 1.59517i −0.603208 0.797584i \(-0.706111\pi\)
0.603208 0.797584i \(-0.293889\pi\)
\(168\) 0 0
\(169\) −42.0224 −0.248653
\(170\) 0 0
\(171\) −48.1196 −0.281401
\(172\) 0 0
\(173\) 43.8456i 0.253443i 0.991938 + 0.126721i \(0.0404454\pi\)
−0.991938 + 0.126721i \(0.959555\pi\)
\(174\) 0 0
\(175\) 18.1308i 0.103605i
\(176\) 0 0
\(177\) −114.043 −0.644311
\(178\) 0 0
\(179\) −275.778 −1.54066 −0.770330 0.637646i \(-0.779908\pi\)
−0.770330 + 0.637646i \(0.779908\pi\)
\(180\) 0 0
\(181\) − 79.0033i − 0.436482i −0.975895 0.218241i \(-0.929968\pi\)
0.975895 0.218241i \(-0.0700319\pi\)
\(182\) 0 0
\(183\) 69.4766i 0.379654i
\(184\) 0 0
\(185\) −285.111 −1.54114
\(186\) 0 0
\(187\) 315.522 1.68728
\(188\) 0 0
\(189\) 14.3303i 0.0758219i
\(190\) 0 0
\(191\) 142.409i 0.745598i 0.927912 + 0.372799i \(0.121602\pi\)
−0.927912 + 0.372799i \(0.878398\pi\)
\(192\) 0 0
\(193\) −277.818 −1.43947 −0.719736 0.694248i \(-0.755737\pi\)
−0.719736 + 0.694248i \(0.755737\pi\)
\(194\) 0 0
\(195\) 108.004 0.553864
\(196\) 0 0
\(197\) 3.40848i 0.0173019i 0.999963 + 0.00865095i \(0.00275372\pi\)
−0.999963 + 0.00865095i \(0.997246\pi\)
\(198\) 0 0
\(199\) − 314.323i − 1.57951i −0.613421 0.789756i \(-0.710207\pi\)
0.613421 0.789756i \(-0.289793\pi\)
\(200\) 0 0
\(201\) −129.576 −0.644654
\(202\) 0 0
\(203\) −60.3057 −0.297072
\(204\) 0 0
\(205\) − 99.7548i − 0.486609i
\(206\) 0 0
\(207\) − 51.3350i − 0.247995i
\(208\) 0 0
\(209\) −221.317 −1.05894
\(210\) 0 0
\(211\) 1.54933 0.00734281 0.00367140 0.999993i \(-0.498831\pi\)
0.00367140 + 0.999993i \(0.498831\pi\)
\(212\) 0 0
\(213\) − 212.490i − 0.997603i
\(214\) 0 0
\(215\) 205.711i 0.956794i
\(216\) 0 0
\(217\) −106.714 −0.491772
\(218\) 0 0
\(219\) 250.981 1.14603
\(220\) 0 0
\(221\) − 332.184i − 1.50309i
\(222\) 0 0
\(223\) 148.964i 0.668001i 0.942573 + 0.334000i \(0.108399\pi\)
−0.942573 + 0.334000i \(0.891601\pi\)
\(224\) 0 0
\(225\) 19.7226 0.0876560
\(226\) 0 0
\(227\) −237.803 −1.04759 −0.523796 0.851844i \(-0.675485\pi\)
−0.523796 + 0.851844i \(0.675485\pi\)
\(228\) 0 0
\(229\) 70.0590i 0.305935i 0.988231 + 0.152967i \(0.0488829\pi\)
−0.988231 + 0.152967i \(0.951117\pi\)
\(230\) 0 0
\(231\) 65.9098i 0.285324i
\(232\) 0 0
\(233\) −7.18107 −0.0308200 −0.0154100 0.999881i \(-0.504905\pi\)
−0.0154100 + 0.999881i \(0.504905\pi\)
\(234\) 0 0
\(235\) −63.7491 −0.271273
\(236\) 0 0
\(237\) − 222.840i − 0.940254i
\(238\) 0 0
\(239\) − 216.620i − 0.906359i −0.891419 0.453180i \(-0.850290\pi\)
0.891419 0.453180i \(-0.149710\pi\)
\(240\) 0 0
\(241\) 385.001 1.59751 0.798757 0.601653i \(-0.205491\pi\)
0.798757 + 0.601653i \(0.205491\pi\)
\(242\) 0 0
\(243\) 15.5885 0.0641500
\(244\) 0 0
\(245\) 177.685i 0.725247i
\(246\) 0 0
\(247\) 233.005i 0.943340i
\(248\) 0 0
\(249\) −38.1774 −0.153323
\(250\) 0 0
\(251\) 85.4882 0.340590 0.170295 0.985393i \(-0.445528\pi\)
0.170295 + 0.985393i \(0.445528\pi\)
\(252\) 0 0
\(253\) − 236.106i − 0.933226i
\(254\) 0 0
\(255\) 170.015i 0.666727i
\(256\) 0 0
\(257\) −54.2981 −0.211277 −0.105638 0.994405i \(-0.533689\pi\)
−0.105638 + 0.994405i \(0.533689\pi\)
\(258\) 0 0
\(259\) −183.179 −0.707255
\(260\) 0 0
\(261\) 65.6002i 0.251342i
\(262\) 0 0
\(263\) − 501.827i − 1.90809i −0.299670 0.954043i \(-0.596877\pi\)
0.299670 0.954043i \(-0.403123\pi\)
\(264\) 0 0
\(265\) −281.413 −1.06194
\(266\) 0 0
\(267\) 211.440 0.791909
\(268\) 0 0
\(269\) − 284.911i − 1.05915i −0.848264 0.529574i \(-0.822352\pi\)
0.848264 0.529574i \(-0.177648\pi\)
\(270\) 0 0
\(271\) 241.190i 0.889998i 0.895531 + 0.444999i \(0.146796\pi\)
−0.895531 + 0.444999i \(0.853204\pi\)
\(272\) 0 0
\(273\) 69.3904 0.254177
\(274\) 0 0
\(275\) 90.7105 0.329856
\(276\) 0 0
\(277\) − 242.877i − 0.876813i −0.898777 0.438407i \(-0.855543\pi\)
0.898777 0.438407i \(-0.144457\pi\)
\(278\) 0 0
\(279\) 116.083i 0.416069i
\(280\) 0 0
\(281\) −105.341 −0.374880 −0.187440 0.982276i \(-0.560019\pi\)
−0.187440 + 0.982276i \(0.560019\pi\)
\(282\) 0 0
\(283\) −223.867 −0.791049 −0.395525 0.918455i \(-0.629437\pi\)
−0.395525 + 0.918455i \(0.629437\pi\)
\(284\) 0 0
\(285\) − 119.254i − 0.418437i
\(286\) 0 0
\(287\) − 64.0907i − 0.223313i
\(288\) 0 0
\(289\) 233.912 0.809384
\(290\) 0 0
\(291\) −152.953 −0.525610
\(292\) 0 0
\(293\) − 421.057i − 1.43705i −0.695499 0.718527i \(-0.744817\pi\)
0.695499 0.718527i \(-0.255183\pi\)
\(294\) 0 0
\(295\) − 282.632i − 0.958075i
\(296\) 0 0
\(297\) 71.6963 0.241402
\(298\) 0 0
\(299\) −248.575 −0.831353
\(300\) 0 0
\(301\) 132.166i 0.439088i
\(302\) 0 0
\(303\) − 180.731i − 0.596472i
\(304\) 0 0
\(305\) −172.183 −0.564536
\(306\) 0 0
\(307\) −122.865 −0.400211 −0.200106 0.979774i \(-0.564129\pi\)
−0.200106 + 0.979774i \(0.564129\pi\)
\(308\) 0 0
\(309\) 119.617i 0.387110i
\(310\) 0 0
\(311\) − 437.535i − 1.40686i −0.710762 0.703432i \(-0.751650\pi\)
0.710762 0.703432i \(-0.248350\pi\)
\(312\) 0 0
\(313\) −375.413 −1.19940 −0.599702 0.800223i \(-0.704714\pi\)
−0.599702 + 0.800223i \(0.704714\pi\)
\(314\) 0 0
\(315\) −35.5148 −0.112745
\(316\) 0 0
\(317\) − 388.867i − 1.22671i −0.789808 0.613355i \(-0.789820\pi\)
0.789808 0.613355i \(-0.210180\pi\)
\(318\) 0 0
\(319\) 301.716i 0.945819i
\(320\) 0 0
\(321\) −258.818 −0.806287
\(322\) 0 0
\(323\) −366.788 −1.13557
\(324\) 0 0
\(325\) − 95.5008i − 0.293849i
\(326\) 0 0
\(327\) − 94.1094i − 0.287796i
\(328\) 0 0
\(329\) −40.9577 −0.124491
\(330\) 0 0
\(331\) 15.4690 0.0467341 0.0233670 0.999727i \(-0.492561\pi\)
0.0233670 + 0.999727i \(0.492561\pi\)
\(332\) 0 0
\(333\) 199.261i 0.598382i
\(334\) 0 0
\(335\) − 321.126i − 0.958585i
\(336\) 0 0
\(337\) −88.0105 −0.261159 −0.130579 0.991438i \(-0.541684\pi\)
−0.130579 + 0.991438i \(0.541684\pi\)
\(338\) 0 0
\(339\) 10.9612 0.0323340
\(340\) 0 0
\(341\) 533.904i 1.56570i
\(342\) 0 0
\(343\) 249.296i 0.726810i
\(344\) 0 0
\(345\) 127.223 0.368763
\(346\) 0 0
\(347\) 183.589 0.529076 0.264538 0.964375i \(-0.414781\pi\)
0.264538 + 0.964375i \(0.414781\pi\)
\(348\) 0 0
\(349\) 242.556i 0.695003i 0.937679 + 0.347501i \(0.112970\pi\)
−0.937679 + 0.347501i \(0.887030\pi\)
\(350\) 0 0
\(351\) − 75.4825i − 0.215050i
\(352\) 0 0
\(353\) 10.2850 0.0291360 0.0145680 0.999894i \(-0.495363\pi\)
0.0145680 + 0.999894i \(0.495363\pi\)
\(354\) 0 0
\(355\) 526.611 1.48341
\(356\) 0 0
\(357\) 109.232i 0.305972i
\(358\) 0 0
\(359\) 137.976i 0.384334i 0.981362 + 0.192167i \(0.0615515\pi\)
−0.981362 + 0.192167i \(0.938448\pi\)
\(360\) 0 0
\(361\) −103.723 −0.287320
\(362\) 0 0
\(363\) 120.176 0.331063
\(364\) 0 0
\(365\) 622.004i 1.70412i
\(366\) 0 0
\(367\) 400.180i 1.09041i 0.838303 + 0.545205i \(0.183548\pi\)
−0.838303 + 0.545205i \(0.816452\pi\)
\(368\) 0 0
\(369\) −69.7175 −0.188936
\(370\) 0 0
\(371\) −180.803 −0.487340
\(372\) 0 0
\(373\) − 644.881i − 1.72890i −0.502717 0.864451i \(-0.667666\pi\)
0.502717 0.864451i \(-0.332334\pi\)
\(374\) 0 0
\(375\) 234.750i 0.626001i
\(376\) 0 0
\(377\) 317.649 0.842571
\(378\) 0 0
\(379\) −485.900 −1.28206 −0.641029 0.767517i \(-0.721492\pi\)
−0.641029 + 0.767517i \(0.721492\pi\)
\(380\) 0 0
\(381\) − 28.6183i − 0.0751137i
\(382\) 0 0
\(383\) − 168.828i − 0.440804i −0.975409 0.220402i \(-0.929263\pi\)
0.975409 0.220402i \(-0.0707370\pi\)
\(384\) 0 0
\(385\) −163.344 −0.424270
\(386\) 0 0
\(387\) 143.769 0.371496
\(388\) 0 0
\(389\) − 192.592i − 0.495095i −0.968876 0.247548i \(-0.920375\pi\)
0.968876 0.247548i \(-0.0796247\pi\)
\(390\) 0 0
\(391\) − 391.297i − 1.00076i
\(392\) 0 0
\(393\) 88.2998 0.224681
\(394\) 0 0
\(395\) 552.263 1.39813
\(396\) 0 0
\(397\) 62.0483i 0.156293i 0.996942 + 0.0781465i \(0.0249002\pi\)
−0.996942 + 0.0781465i \(0.975100\pi\)
\(398\) 0 0
\(399\) − 76.6189i − 0.192027i
\(400\) 0 0
\(401\) 353.066 0.880464 0.440232 0.897884i \(-0.354896\pi\)
0.440232 + 0.897884i \(0.354896\pi\)
\(402\) 0 0
\(403\) 562.099 1.39479
\(404\) 0 0
\(405\) 38.6328i 0.0953895i
\(406\) 0 0
\(407\) 916.466i 2.25176i
\(408\) 0 0
\(409\) 62.9725 0.153967 0.0769835 0.997032i \(-0.475471\pi\)
0.0769835 + 0.997032i \(0.475471\pi\)
\(410\) 0 0
\(411\) −17.3476 −0.0422084
\(412\) 0 0
\(413\) − 181.586i − 0.439676i
\(414\) 0 0
\(415\) − 94.6146i − 0.227987i
\(416\) 0 0
\(417\) 113.811 0.272928
\(418\) 0 0
\(419\) 398.741 0.951650 0.475825 0.879540i \(-0.342150\pi\)
0.475825 + 0.879540i \(0.342150\pi\)
\(420\) 0 0
\(421\) 533.059i 1.26617i 0.774081 + 0.633086i \(0.218212\pi\)
−0.774081 + 0.633086i \(0.781788\pi\)
\(422\) 0 0
\(423\) 44.5535i 0.105328i
\(424\) 0 0
\(425\) 150.334 0.353727
\(426\) 0 0
\(427\) −110.625 −0.259075
\(428\) 0 0
\(429\) − 347.168i − 0.809250i
\(430\) 0 0
\(431\) − 284.282i − 0.659587i −0.944053 0.329793i \(-0.893021\pi\)
0.944053 0.329793i \(-0.106979\pi\)
\(432\) 0 0
\(433\) −304.599 −0.703462 −0.351731 0.936101i \(-0.614407\pi\)
−0.351731 + 0.936101i \(0.614407\pi\)
\(434\) 0 0
\(435\) −162.576 −0.373739
\(436\) 0 0
\(437\) 274.469i 0.628075i
\(438\) 0 0
\(439\) 204.615i 0.466094i 0.972465 + 0.233047i \(0.0748696\pi\)
−0.972465 + 0.233047i \(0.925130\pi\)
\(440\) 0 0
\(441\) 124.182 0.281593
\(442\) 0 0
\(443\) −625.828 −1.41270 −0.706352 0.707861i \(-0.749660\pi\)
−0.706352 + 0.707861i \(0.749660\pi\)
\(444\) 0 0
\(445\) 524.009i 1.17755i
\(446\) 0 0
\(447\) − 164.149i − 0.367224i
\(448\) 0 0
\(449\) −557.532 −1.24172 −0.620860 0.783921i \(-0.713217\pi\)
−0.620860 + 0.783921i \(0.713217\pi\)
\(450\) 0 0
\(451\) −320.653 −0.710983
\(452\) 0 0
\(453\) 467.254i 1.03147i
\(454\) 0 0
\(455\) 171.970i 0.377955i
\(456\) 0 0
\(457\) 312.142 0.683024 0.341512 0.939877i \(-0.389061\pi\)
0.341512 + 0.939877i \(0.389061\pi\)
\(458\) 0 0
\(459\) 118.822 0.258871
\(460\) 0 0
\(461\) 603.187i 1.30843i 0.756308 + 0.654216i \(0.227001\pi\)
−0.756308 + 0.654216i \(0.772999\pi\)
\(462\) 0 0
\(463\) − 707.866i − 1.52887i −0.644702 0.764434i \(-0.723019\pi\)
0.644702 0.764434i \(-0.276981\pi\)
\(464\) 0 0
\(465\) −287.689 −0.618685
\(466\) 0 0
\(467\) 195.220 0.418031 0.209015 0.977912i \(-0.432974\pi\)
0.209015 + 0.977912i \(0.432974\pi\)
\(468\) 0 0
\(469\) − 206.318i − 0.439910i
\(470\) 0 0
\(471\) − 54.5696i − 0.115859i
\(472\) 0 0
\(473\) 661.239 1.39797
\(474\) 0 0
\(475\) −105.449 −0.221998
\(476\) 0 0
\(477\) 196.677i 0.412320i
\(478\) 0 0
\(479\) 233.729i 0.487952i 0.969781 + 0.243976i \(0.0784517\pi\)
−0.969781 + 0.243976i \(0.921548\pi\)
\(480\) 0 0
\(481\) 964.863 2.00595
\(482\) 0 0
\(483\) 81.7387 0.169231
\(484\) 0 0
\(485\) − 379.061i − 0.781569i
\(486\) 0 0
\(487\) − 224.058i − 0.460078i −0.973181 0.230039i \(-0.926115\pi\)
0.973181 0.230039i \(-0.0738854\pi\)
\(488\) 0 0
\(489\) 348.417 0.712510
\(490\) 0 0
\(491\) −194.772 −0.396684 −0.198342 0.980133i \(-0.563556\pi\)
−0.198342 + 0.980133i \(0.563556\pi\)
\(492\) 0 0
\(493\) 500.032i 1.01426i
\(494\) 0 0
\(495\) 177.684i 0.358958i
\(496\) 0 0
\(497\) 338.339 0.680762
\(498\) 0 0
\(499\) 99.7462 0.199892 0.0999461 0.994993i \(-0.468133\pi\)
0.0999461 + 0.994993i \(0.468133\pi\)
\(500\) 0 0
\(501\) − 461.406i − 0.920970i
\(502\) 0 0
\(503\) 772.172i 1.53513i 0.640969 + 0.767567i \(0.278533\pi\)
−0.640969 + 0.767567i \(0.721467\pi\)
\(504\) 0 0
\(505\) 447.904 0.886939
\(506\) 0 0
\(507\) −72.7849 −0.143560
\(508\) 0 0
\(509\) 292.306i 0.574276i 0.957889 + 0.287138i \(0.0927038\pi\)
−0.957889 + 0.287138i \(0.907296\pi\)
\(510\) 0 0
\(511\) 399.627i 0.782048i
\(512\) 0 0
\(513\) −83.3456 −0.162467
\(514\) 0 0
\(515\) −296.446 −0.575623
\(516\) 0 0
\(517\) 204.916i 0.396356i
\(518\) 0 0
\(519\) 75.9428i 0.146325i
\(520\) 0 0
\(521\) 412.035 0.790853 0.395427 0.918498i \(-0.370597\pi\)
0.395427 + 0.918498i \(0.370597\pi\)
\(522\) 0 0
\(523\) 124.693 0.238420 0.119210 0.992869i \(-0.461964\pi\)
0.119210 + 0.992869i \(0.461964\pi\)
\(524\) 0 0
\(525\) 31.4035i 0.0598162i
\(526\) 0 0
\(527\) 884.836i 1.67901i
\(528\) 0 0
\(529\) 236.191 0.446486
\(530\) 0 0
\(531\) −197.528 −0.371993
\(532\) 0 0
\(533\) 337.586i 0.633370i
\(534\) 0 0
\(535\) − 641.427i − 1.19893i
\(536\) 0 0
\(537\) −477.662 −0.889500
\(538\) 0 0
\(539\) 571.154 1.05966
\(540\) 0 0
\(541\) 572.988i 1.05913i 0.848270 + 0.529564i \(0.177644\pi\)
−0.848270 + 0.529564i \(0.822356\pi\)
\(542\) 0 0
\(543\) − 136.838i − 0.252003i
\(544\) 0 0
\(545\) 233.231 0.427946
\(546\) 0 0
\(547\) 682.433 1.24759 0.623796 0.781587i \(-0.285590\pi\)
0.623796 + 0.781587i \(0.285590\pi\)
\(548\) 0 0
\(549\) 120.337i 0.219193i
\(550\) 0 0
\(551\) − 350.739i − 0.636551i
\(552\) 0 0
\(553\) 354.820 0.641627
\(554\) 0 0
\(555\) −493.828 −0.889779
\(556\) 0 0
\(557\) − 856.125i − 1.53703i −0.639832 0.768515i \(-0.720996\pi\)
0.639832 0.768515i \(-0.279004\pi\)
\(558\) 0 0
\(559\) − 696.158i − 1.24536i
\(560\) 0 0
\(561\) 546.499 0.974152
\(562\) 0 0
\(563\) 680.745 1.20914 0.604570 0.796552i \(-0.293345\pi\)
0.604570 + 0.796552i \(0.293345\pi\)
\(564\) 0 0
\(565\) 27.1651i 0.0480798i
\(566\) 0 0
\(567\) 24.8209i 0.0437758i
\(568\) 0 0
\(569\) −343.401 −0.603516 −0.301758 0.953384i \(-0.597574\pi\)
−0.301758 + 0.953384i \(0.597574\pi\)
\(570\) 0 0
\(571\) 329.137 0.576422 0.288211 0.957567i \(-0.406940\pi\)
0.288211 + 0.957567i \(0.406940\pi\)
\(572\) 0 0
\(573\) 246.660i 0.430471i
\(574\) 0 0
\(575\) − 112.495i − 0.195644i
\(576\) 0 0
\(577\) 734.738 1.27338 0.636688 0.771122i \(-0.280304\pi\)
0.636688 + 0.771122i \(0.280304\pi\)
\(578\) 0 0
\(579\) −481.195 −0.831079
\(580\) 0 0
\(581\) − 60.7883i − 0.104627i
\(582\) 0 0
\(583\) 904.579i 1.55159i
\(584\) 0 0
\(585\) 187.068 0.319774
\(586\) 0 0
\(587\) 433.815 0.739037 0.369519 0.929223i \(-0.379523\pi\)
0.369519 + 0.929223i \(0.379523\pi\)
\(588\) 0 0
\(589\) − 620.654i − 1.05374i
\(590\) 0 0
\(591\) 5.90365i 0.00998926i
\(592\) 0 0
\(593\) −425.075 −0.716822 −0.358411 0.933564i \(-0.616681\pi\)
−0.358411 + 0.933564i \(0.616681\pi\)
\(594\) 0 0
\(595\) −270.709 −0.454972
\(596\) 0 0
\(597\) − 544.424i − 0.911932i
\(598\) 0 0
\(599\) − 1048.73i − 1.75081i −0.483394 0.875403i \(-0.660596\pi\)
0.483394 0.875403i \(-0.339404\pi\)
\(600\) 0 0
\(601\) 478.816 0.796699 0.398349 0.917234i \(-0.369583\pi\)
0.398349 + 0.917234i \(0.369583\pi\)
\(602\) 0 0
\(603\) −224.431 −0.372191
\(604\) 0 0
\(605\) 297.831i 0.492283i
\(606\) 0 0
\(607\) − 18.7009i − 0.0308088i −0.999881 0.0154044i \(-0.995096\pi\)
0.999881 0.0154044i \(-0.00490356\pi\)
\(608\) 0 0
\(609\) −104.453 −0.171515
\(610\) 0 0
\(611\) 215.737 0.353089
\(612\) 0 0
\(613\) 314.964i 0.513807i 0.966437 + 0.256904i \(0.0827023\pi\)
−0.966437 + 0.256904i \(0.917298\pi\)
\(614\) 0 0
\(615\) − 172.780i − 0.280944i
\(616\) 0 0
\(617\) −786.590 −1.27486 −0.637431 0.770507i \(-0.720003\pi\)
−0.637431 + 0.770507i \(0.720003\pi\)
\(618\) 0 0
\(619\) −505.431 −0.816528 −0.408264 0.912864i \(-0.633866\pi\)
−0.408264 + 0.912864i \(0.633866\pi\)
\(620\) 0 0
\(621\) − 88.9148i − 0.143180i
\(622\) 0 0
\(623\) 336.667i 0.540396i
\(624\) 0 0
\(625\) −417.425 −0.667880
\(626\) 0 0
\(627\) −383.333 −0.611377
\(628\) 0 0
\(629\) 1518.85i 2.41471i
\(630\) 0 0
\(631\) − 682.834i − 1.08215i −0.840976 0.541073i \(-0.818018\pi\)
0.840976 0.541073i \(-0.181982\pi\)
\(632\) 0 0
\(633\) 2.68352 0.00423937
\(634\) 0 0
\(635\) 70.9246 0.111692
\(636\) 0 0
\(637\) − 601.316i − 0.943982i
\(638\) 0 0
\(639\) − 368.043i − 0.575967i
\(640\) 0 0
\(641\) 816.719 1.27413 0.637066 0.770809i \(-0.280148\pi\)
0.637066 + 0.770809i \(0.280148\pi\)
\(642\) 0 0
\(643\) −1164.05 −1.81034 −0.905172 0.425045i \(-0.860258\pi\)
−0.905172 + 0.425045i \(0.860258\pi\)
\(644\) 0 0
\(645\) 356.301i 0.552405i
\(646\) 0 0
\(647\) 471.519i 0.728778i 0.931247 + 0.364389i \(0.118722\pi\)
−0.931247 + 0.364389i \(0.881278\pi\)
\(648\) 0 0
\(649\) −908.496 −1.39984
\(650\) 0 0
\(651\) −184.835 −0.283924
\(652\) 0 0
\(653\) 1187.25i 1.81814i 0.416642 + 0.909071i \(0.363207\pi\)
−0.416642 + 0.909071i \(0.636793\pi\)
\(654\) 0 0
\(655\) 218.833i 0.334096i
\(656\) 0 0
\(657\) 434.711 0.661661
\(658\) 0 0
\(659\) 245.640 0.372747 0.186373 0.982479i \(-0.440327\pi\)
0.186373 + 0.982479i \(0.440327\pi\)
\(660\) 0 0
\(661\) 1244.70i 1.88305i 0.336936 + 0.941527i \(0.390609\pi\)
−0.336936 + 0.941527i \(0.609391\pi\)
\(662\) 0 0
\(663\) − 575.359i − 0.867812i
\(664\) 0 0
\(665\) 189.884 0.285540
\(666\) 0 0
\(667\) 374.176 0.560984
\(668\) 0 0
\(669\) 258.013i 0.385670i
\(670\) 0 0
\(671\) 553.468i 0.824841i
\(672\) 0 0
\(673\) −852.278 −1.26639 −0.633193 0.773994i \(-0.718256\pi\)
−0.633193 + 0.773994i \(0.718256\pi\)
\(674\) 0 0
\(675\) 34.1605 0.0506082
\(676\) 0 0
\(677\) − 555.765i − 0.820923i −0.911878 0.410461i \(-0.865368\pi\)
0.911878 0.410461i \(-0.134632\pi\)
\(678\) 0 0
\(679\) − 243.540i − 0.358675i
\(680\) 0 0
\(681\) −411.888 −0.604828
\(682\) 0 0
\(683\) −335.241 −0.490836 −0.245418 0.969417i \(-0.578925\pi\)
−0.245418 + 0.969417i \(0.578925\pi\)
\(684\) 0 0
\(685\) − 42.9925i − 0.0627628i
\(686\) 0 0
\(687\) 121.346i 0.176631i
\(688\) 0 0
\(689\) 952.349 1.38222
\(690\) 0 0
\(691\) −1274.20 −1.84400 −0.921999 0.387192i \(-0.873445\pi\)
−0.921999 + 0.387192i \(0.873445\pi\)
\(692\) 0 0
\(693\) 114.159i 0.164732i
\(694\) 0 0
\(695\) 282.057i 0.405837i
\(696\) 0 0
\(697\) −531.416 −0.762434
\(698\) 0 0
\(699\) −12.4380 −0.0177940
\(700\) 0 0
\(701\) − 676.056i − 0.964416i −0.876057 0.482208i \(-0.839835\pi\)
0.876057 0.482208i \(-0.160165\pi\)
\(702\) 0 0
\(703\) − 1065.37i − 1.51547i
\(704\) 0 0
\(705\) −110.417 −0.156619
\(706\) 0 0
\(707\) 287.771 0.407031
\(708\) 0 0
\(709\) − 62.3418i − 0.0879292i −0.999033 0.0439646i \(-0.986001\pi\)
0.999033 0.0439646i \(-0.0139989\pi\)
\(710\) 0 0
\(711\) − 385.971i − 0.542856i
\(712\) 0 0
\(713\) 662.127 0.928649
\(714\) 0 0
\(715\) 860.384 1.20333
\(716\) 0 0
\(717\) − 375.197i − 0.523287i
\(718\) 0 0
\(719\) 322.341i 0.448318i 0.974553 + 0.224159i \(0.0719636\pi\)
−0.974553 + 0.224159i \(0.928036\pi\)
\(720\) 0 0
\(721\) −190.461 −0.264163
\(722\) 0 0
\(723\) 666.841 0.922326
\(724\) 0 0
\(725\) 143.756i 0.198284i
\(726\) 0 0
\(727\) − 368.070i − 0.506287i −0.967429 0.253143i \(-0.918536\pi\)
0.967429 0.253143i \(-0.0814644\pi\)
\(728\) 0 0
\(729\) 27.0000 0.0370370
\(730\) 0 0
\(731\) 1095.87 1.49913
\(732\) 0 0
\(733\) − 430.587i − 0.587431i −0.955893 0.293715i \(-0.905108\pi\)
0.955893 0.293715i \(-0.0948918\pi\)
\(734\) 0 0
\(735\) 307.760i 0.418721i
\(736\) 0 0
\(737\) −1032.23 −1.40059
\(738\) 0 0
\(739\) −818.050 −1.10697 −0.553485 0.832859i \(-0.686702\pi\)
−0.553485 + 0.832859i \(0.686702\pi\)
\(740\) 0 0
\(741\) 403.576i 0.544638i
\(742\) 0 0
\(743\) 685.438i 0.922528i 0.887263 + 0.461264i \(0.152604\pi\)
−0.887263 + 0.461264i \(0.847396\pi\)
\(744\) 0 0
\(745\) 406.810 0.546053
\(746\) 0 0
\(747\) −66.1251 −0.0885209
\(748\) 0 0
\(749\) − 412.106i − 0.550208i
\(750\) 0 0
\(751\) − 104.336i − 0.138929i −0.997584 0.0694647i \(-0.977871\pi\)
0.997584 0.0694647i \(-0.0221291\pi\)
\(752\) 0 0
\(753\) 148.070 0.196640
\(754\) 0 0
\(755\) −1157.99 −1.53377
\(756\) 0 0
\(757\) − 539.949i − 0.713274i −0.934243 0.356637i \(-0.883923\pi\)
0.934243 0.356637i \(-0.116077\pi\)
\(758\) 0 0
\(759\) − 408.948i − 0.538798i
\(760\) 0 0
\(761\) −1192.21 −1.56663 −0.783317 0.621623i \(-0.786474\pi\)
−0.783317 + 0.621623i \(0.786474\pi\)
\(762\) 0 0
\(763\) 149.847 0.196391
\(764\) 0 0
\(765\) 294.475i 0.384935i
\(766\) 0 0
\(767\) 956.473i 1.24703i
\(768\) 0 0
\(769\) −321.506 −0.418083 −0.209042 0.977907i \(-0.567034\pi\)
−0.209042 + 0.977907i \(0.567034\pi\)
\(770\) 0 0
\(771\) −94.0471 −0.121981
\(772\) 0 0
\(773\) − 1453.29i − 1.88006i −0.341094 0.940029i \(-0.610798\pi\)
0.341094 0.940029i \(-0.389202\pi\)
\(774\) 0 0
\(775\) 254.385i 0.328239i
\(776\) 0 0
\(777\) −317.276 −0.408334
\(778\) 0 0
\(779\) 372.753 0.478502
\(780\) 0 0
\(781\) − 1692.75i − 2.16741i
\(782\) 0 0
\(783\) 113.623i 0.145112i
\(784\) 0 0
\(785\) 135.240 0.172280
\(786\) 0 0
\(787\) −231.223 −0.293802 −0.146901 0.989151i \(-0.546930\pi\)
−0.146901 + 0.989151i \(0.546930\pi\)
\(788\) 0 0
\(789\) − 869.189i − 1.10163i
\(790\) 0 0
\(791\) 17.4531i 0.0220646i
\(792\) 0 0
\(793\) 582.696 0.734800
\(794\) 0 0
\(795\) −487.422 −0.613110
\(796\) 0 0
\(797\) − 51.5068i − 0.0646259i −0.999478 0.0323129i \(-0.989713\pi\)
0.999478 0.0323129i \(-0.0102873\pi\)
\(798\) 0 0
\(799\) 339.606i 0.425039i
\(800\) 0 0
\(801\) 366.224 0.457209
\(802\) 0 0
\(803\) 1999.38 2.48988
\(804\) 0 0
\(805\) 202.572i 0.251643i
\(806\) 0 0
\(807\) − 493.480i − 0.611499i
\(808\) 0 0
\(809\) −580.914 −0.718065 −0.359032 0.933325i \(-0.616893\pi\)
−0.359032 + 0.933325i \(0.616893\pi\)
\(810\) 0 0
\(811\) −1351.98 −1.66706 −0.833529 0.552475i \(-0.813683\pi\)
−0.833529 + 0.552475i \(0.813683\pi\)
\(812\) 0 0
\(813\) 417.753i 0.513841i
\(814\) 0 0
\(815\) 863.480i 1.05948i
\(816\) 0 0
\(817\) −768.678 −0.940855
\(818\) 0 0
\(819\) 120.188 0.146749
\(820\) 0 0
\(821\) 179.511i 0.218650i 0.994006 + 0.109325i \(0.0348688\pi\)
−0.994006 + 0.109325i \(0.965131\pi\)
\(822\) 0 0
\(823\) 665.446i 0.808561i 0.914635 + 0.404281i \(0.132478\pi\)
−0.914635 + 0.404281i \(0.867522\pi\)
\(824\) 0 0
\(825\) 157.115 0.190443
\(826\) 0 0
\(827\) 901.166 1.08968 0.544840 0.838540i \(-0.316590\pi\)
0.544840 + 0.838540i \(0.316590\pi\)
\(828\) 0 0
\(829\) 183.151i 0.220930i 0.993880 + 0.110465i \(0.0352340\pi\)
−0.993880 + 0.110465i \(0.964766\pi\)
\(830\) 0 0
\(831\) − 420.676i − 0.506228i
\(832\) 0 0
\(833\) 946.571 1.13634
\(834\) 0 0
\(835\) 1143.50 1.36946
\(836\) 0 0
\(837\) 201.062i 0.240218i
\(838\) 0 0
\(839\) 846.082i 1.00844i 0.863575 + 0.504220i \(0.168220\pi\)
−0.863575 + 0.504220i \(0.831780\pi\)
\(840\) 0 0
\(841\) 362.846 0.431446
\(842\) 0 0
\(843\) −182.457 −0.216437
\(844\) 0 0
\(845\) − 180.382i − 0.213470i
\(846\) 0 0
\(847\) 191.351i 0.225917i
\(848\) 0 0
\(849\) −387.749 −0.456713
\(850\) 0 0
\(851\) 1136.56 1.33556
\(852\) 0 0
\(853\) − 540.635i − 0.633804i −0.948458 0.316902i \(-0.897357\pi\)
0.948458 0.316902i \(-0.102643\pi\)
\(854\) 0 0
\(855\) − 206.555i − 0.241585i
\(856\) 0 0
\(857\) −1333.41 −1.55590 −0.777952 0.628323i \(-0.783742\pi\)
−0.777952 + 0.628323i \(0.783742\pi\)
\(858\) 0 0
\(859\) −439.034 −0.511099 −0.255549 0.966796i \(-0.582256\pi\)
−0.255549 + 0.966796i \(0.582256\pi\)
\(860\) 0 0
\(861\) − 111.008i − 0.128930i
\(862\) 0 0
\(863\) − 1164.41i − 1.34926i −0.738155 0.674631i \(-0.764303\pi\)
0.738155 0.674631i \(-0.235697\pi\)
\(864\) 0 0
\(865\) −188.208 −0.217582
\(866\) 0 0
\(867\) 405.147 0.467298
\(868\) 0 0
\(869\) − 1775.20i − 2.04281i
\(870\) 0 0
\(871\) 1086.74i 1.24769i
\(872\) 0 0
\(873\) −264.922 −0.303461
\(874\) 0 0
\(875\) −373.783 −0.427181
\(876\) 0 0
\(877\) − 404.456i − 0.461181i −0.973051 0.230591i \(-0.925934\pi\)
0.973051 0.230591i \(-0.0740658\pi\)
\(878\) 0 0
\(879\) − 729.292i − 0.829684i
\(880\) 0 0
\(881\) −227.084 −0.257757 −0.128879 0.991660i \(-0.541138\pi\)
−0.128879 + 0.991660i \(0.541138\pi\)
\(882\) 0 0
\(883\) −673.730 −0.763001 −0.381501 0.924369i \(-0.624593\pi\)
−0.381501 + 0.924369i \(0.624593\pi\)
\(884\) 0 0
\(885\) − 489.533i − 0.553145i
\(886\) 0 0
\(887\) − 1370.95i − 1.54560i −0.634648 0.772801i \(-0.718855\pi\)
0.634648 0.772801i \(-0.281145\pi\)
\(888\) 0 0
\(889\) 45.5678 0.0512574
\(890\) 0 0
\(891\) 124.182 0.139373
\(892\) 0 0
\(893\) − 238.211i − 0.266754i
\(894\) 0 0
\(895\) − 1183.79i − 1.32267i
\(896\) 0 0
\(897\) −430.544 −0.479982
\(898\) 0 0
\(899\) −846.121 −0.941180
\(900\) 0 0
\(901\) 1499.15i 1.66388i
\(902\) 0 0
\(903\) 228.917i 0.253508i
\(904\) 0 0
\(905\) 339.124 0.374723
\(906\) 0 0
\(907\) 1469.46 1.62013 0.810063 0.586342i \(-0.199433\pi\)
0.810063 + 0.586342i \(0.199433\pi\)
\(908\) 0 0
\(909\) − 313.035i − 0.344373i
\(910\) 0 0
\(911\) 871.203i 0.956315i 0.878274 + 0.478157i \(0.158695\pi\)
−0.878274 + 0.478157i \(0.841305\pi\)
\(912\) 0 0
\(913\) −304.131 −0.333111
\(914\) 0 0
\(915\) −298.230 −0.325935
\(916\) 0 0
\(917\) 140.596i 0.153322i
\(918\) 0 0
\(919\) − 256.361i − 0.278957i −0.990225 0.139478i \(-0.955457\pi\)
0.990225 0.139478i \(-0.0445426\pi\)
\(920\) 0 0
\(921\) −212.808 −0.231062
\(922\) 0 0
\(923\) −1782.14 −1.93081
\(924\) 0 0
\(925\) 436.661i 0.472066i
\(926\) 0 0
\(927\) 207.183i 0.223498i
\(928\) 0 0
\(929\) −1399.93 −1.50692 −0.753462 0.657491i \(-0.771618\pi\)
−0.753462 + 0.657491i \(0.771618\pi\)
\(930\) 0 0
\(931\) −663.956 −0.713165
\(932\) 0 0
\(933\) − 757.832i − 0.812253i
\(934\) 0 0
\(935\) 1354.39i 1.44854i
\(936\) 0 0
\(937\) −365.610 −0.390192 −0.195096 0.980784i \(-0.562502\pi\)
−0.195096 + 0.980784i \(0.562502\pi\)
\(938\) 0 0
\(939\) −650.235 −0.692476
\(940\) 0 0
\(941\) 407.498i 0.433048i 0.976277 + 0.216524i \(0.0694720\pi\)
−0.976277 + 0.216524i \(0.930528\pi\)
\(942\) 0 0
\(943\) 397.661i 0.421698i
\(944\) 0 0
\(945\) −61.5134 −0.0650935
\(946\) 0 0
\(947\) 1150.64 1.21503 0.607516 0.794307i \(-0.292166\pi\)
0.607516 + 0.794307i \(0.292166\pi\)
\(948\) 0 0
\(949\) − 2104.96i − 2.21808i
\(950\) 0 0
\(951\) − 673.537i − 0.708241i
\(952\) 0 0
\(953\) 1341.53 1.40769 0.703844 0.710355i \(-0.251466\pi\)
0.703844 + 0.710355i \(0.251466\pi\)
\(954\) 0 0
\(955\) −611.295 −0.640100
\(956\) 0 0
\(957\) 522.588i 0.546069i
\(958\) 0 0
\(959\) − 27.6219i − 0.0288029i
\(960\) 0 0
\(961\) −536.260 −0.558023
\(962\) 0 0
\(963\) −448.286 −0.465510
\(964\) 0 0
\(965\) − 1192.54i − 1.23579i
\(966\) 0 0
\(967\) 1227.25i 1.26914i 0.772867 + 0.634568i \(0.218822\pi\)
−0.772867 + 0.634568i \(0.781178\pi\)
\(968\) 0 0
\(969\) −635.295 −0.655620
\(970\) 0 0
\(971\) 1864.31 1.91999 0.959994 0.280020i \(-0.0903413\pi\)
0.959994 + 0.280020i \(0.0903413\pi\)
\(972\) 0 0
\(973\) 181.217i 0.186245i
\(974\) 0 0
\(975\) − 165.412i − 0.169654i
\(976\) 0 0
\(977\) −1209.56 −1.23803 −0.619017 0.785378i \(-0.712469\pi\)
−0.619017 + 0.785378i \(0.712469\pi\)
\(978\) 0 0
\(979\) 1684.38 1.72051
\(980\) 0 0
\(981\) − 163.002i − 0.166159i
\(982\) 0 0
\(983\) − 1018.26i − 1.03587i −0.855420 0.517935i \(-0.826701\pi\)
0.855420 0.517935i \(-0.173299\pi\)
\(984\) 0 0
\(985\) −14.6310 −0.0148538
\(986\) 0 0
\(987\) −70.9408 −0.0718752
\(988\) 0 0
\(989\) − 820.042i − 0.829163i
\(990\) 0 0
\(991\) − 1627.52i − 1.64230i −0.570712 0.821150i \(-0.693333\pi\)
0.570712 0.821150i \(-0.306667\pi\)
\(992\) 0 0
\(993\) 26.7931 0.0269819
\(994\) 0 0
\(995\) 1349.24 1.35602
\(996\) 0 0
\(997\) 540.192i 0.541817i 0.962605 + 0.270909i \(0.0873241\pi\)
−0.962605 + 0.270909i \(0.912676\pi\)
\(998\) 0 0
\(999\) 345.131i 0.345476i
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 768.3.b.f.127.8 8
3.2 odd 2 2304.3.b.q.127.2 8
4.3 odd 2 768.3.b.e.127.4 8
8.3 odd 2 inner 768.3.b.f.127.5 8
8.5 even 2 768.3.b.e.127.1 8
12.11 even 2 2304.3.b.t.127.2 8
16.3 odd 4 384.3.g.a.127.4 8
16.5 even 4 384.3.g.b.127.1 yes 8
16.11 odd 4 384.3.g.b.127.5 yes 8
16.13 even 4 384.3.g.a.127.8 yes 8
24.5 odd 2 2304.3.b.t.127.7 8
24.11 even 2 2304.3.b.q.127.7 8
48.5 odd 4 1152.3.g.c.127.7 8
48.11 even 4 1152.3.g.c.127.8 8
48.29 odd 4 1152.3.g.f.127.1 8
48.35 even 4 1152.3.g.f.127.2 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
384.3.g.a.127.4 8 16.3 odd 4
384.3.g.a.127.8 yes 8 16.13 even 4
384.3.g.b.127.1 yes 8 16.5 even 4
384.3.g.b.127.5 yes 8 16.11 odd 4
768.3.b.e.127.1 8 8.5 even 2
768.3.b.e.127.4 8 4.3 odd 2
768.3.b.f.127.5 8 8.3 odd 2 inner
768.3.b.f.127.8 8 1.1 even 1 trivial
1152.3.g.c.127.7 8 48.5 odd 4
1152.3.g.c.127.8 8 48.11 even 4
1152.3.g.f.127.1 8 48.29 odd 4
1152.3.g.f.127.2 8 48.35 even 4
2304.3.b.q.127.2 8 3.2 odd 2
2304.3.b.q.127.7 8 24.11 even 2
2304.3.b.t.127.2 8 12.11 even 2
2304.3.b.t.127.7 8 24.5 odd 2