Properties

Label 768.3.e.l.257.3
Level $768$
Weight $3$
Character 768.257
Analytic conductor $20.926$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [768,3,Mod(257,768)]
 
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([0, 0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("768.257");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 768 = 2^{8} \cdot 3 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 768.e (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: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{7})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 8x^{2} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 24)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 257.3
Root \(-1.16372i\) of defining polynomial
Character \(\chi\) \(=\) 768.257
Dual form 768.3.e.l.257.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(2.64575 - 1.41421i) q^{3} +5.65685i q^{5} +4.00000 q^{7} +(5.00000 - 7.48331i) q^{9} +O(q^{10})\) \(q+(2.64575 - 1.41421i) q^{3} +5.65685i q^{5} +4.00000 q^{7} +(5.00000 - 7.48331i) q^{9} +8.48528i q^{11} +10.5830 q^{13} +(8.00000 + 14.9666i) q^{15} -14.9666i q^{17} -5.29150 q^{19} +(10.5830 - 5.65685i) q^{21} +29.9333i q^{23} -7.00000 q^{25} +(2.64575 - 26.8701i) q^{27} +16.9706i q^{29} +4.00000 q^{31} +(12.0000 + 22.4499i) q^{33} +22.6274i q^{35} +52.9150 q^{37} +(28.0000 - 14.9666i) q^{39} +29.9333i q^{41} -5.29150 q^{43} +(42.3320 + 28.2843i) q^{45} -33.0000 q^{49} +(-21.1660 - 39.5980i) q^{51} +50.9117i q^{53} -48.0000 q^{55} +(-14.0000 + 7.48331i) q^{57} +48.0833i q^{59} +95.2470 q^{61} +(20.0000 - 29.9333i) q^{63} +59.8665i q^{65} +47.6235 q^{67} +(42.3320 + 79.1960i) q^{69} -89.7998i q^{71} +6.00000 q^{73} +(-18.5203 + 9.89949i) q^{75} +33.9411i q^{77} -124.000 q^{79} +(-31.0000 - 74.8331i) q^{81} +2.82843i q^{83} +84.6640 q^{85} +(24.0000 + 44.8999i) q^{87} -104.766i q^{89} +42.3320 q^{91} +(10.5830 - 5.65685i) q^{93} -29.9333i q^{95} +118.000 q^{97} +(63.4980 + 42.4264i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 16 q^{7} + 20 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 16 q^{7} + 20 q^{9} + 32 q^{15} - 28 q^{25} + 16 q^{31} + 48 q^{33} + 112 q^{39} - 132 q^{49} - 192 q^{55} - 56 q^{57} + 80 q^{63} + 24 q^{73} - 496 q^{79} - 124 q^{81} + 96 q^{87} + 472 q^{97}+O(q^{100}) \) 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
\(3\) 2.64575 1.41421i 0.881917 0.471405i
\(4\) 0 0
\(5\) 5.65685i 1.13137i 0.824621 + 0.565685i \(0.191388\pi\)
−0.824621 + 0.565685i \(0.808612\pi\)
\(6\) 0 0
\(7\) 4.00000 0.571429 0.285714 0.958315i \(-0.407769\pi\)
0.285714 + 0.958315i \(0.407769\pi\)
\(8\) 0 0
\(9\) 5.00000 7.48331i 0.555556 0.831479i
\(10\) 0 0
\(11\) 8.48528i 0.771389i 0.922627 + 0.385695i \(0.126038\pi\)
−0.922627 + 0.385695i \(0.873962\pi\)
\(12\) 0 0
\(13\) 10.5830 0.814077 0.407039 0.913411i \(-0.366561\pi\)
0.407039 + 0.913411i \(0.366561\pi\)
\(14\) 0 0
\(15\) 8.00000 + 14.9666i 0.533333 + 0.997775i
\(16\) 0 0
\(17\) 14.9666i 0.880390i −0.897902 0.440195i \(-0.854909\pi\)
0.897902 0.440195i \(-0.145091\pi\)
\(18\) 0 0
\(19\) −5.29150 −0.278500 −0.139250 0.990257i \(-0.544469\pi\)
−0.139250 + 0.990257i \(0.544469\pi\)
\(20\) 0 0
\(21\) 10.5830 5.65685i 0.503953 0.269374i
\(22\) 0 0
\(23\) 29.9333i 1.30145i 0.759315 + 0.650723i \(0.225534\pi\)
−0.759315 + 0.650723i \(0.774466\pi\)
\(24\) 0 0
\(25\) −7.00000 −0.280000
\(26\) 0 0
\(27\) 2.64575 26.8701i 0.0979908 0.995187i
\(28\) 0 0
\(29\) 16.9706i 0.585192i 0.956236 + 0.292596i \(0.0945191\pi\)
−0.956236 + 0.292596i \(0.905481\pi\)
\(30\) 0 0
\(31\) 4.00000 0.129032 0.0645161 0.997917i \(-0.479450\pi\)
0.0645161 + 0.997917i \(0.479450\pi\)
\(32\) 0 0
\(33\) 12.0000 + 22.4499i 0.363636 + 0.680301i
\(34\) 0 0
\(35\) 22.6274i 0.646498i
\(36\) 0 0
\(37\) 52.9150 1.43014 0.715068 0.699055i \(-0.246396\pi\)
0.715068 + 0.699055i \(0.246396\pi\)
\(38\) 0 0
\(39\) 28.0000 14.9666i 0.717949 0.383760i
\(40\) 0 0
\(41\) 29.9333i 0.730079i 0.930992 + 0.365040i \(0.118945\pi\)
−0.930992 + 0.365040i \(0.881055\pi\)
\(42\) 0 0
\(43\) −5.29150 −0.123058 −0.0615291 0.998105i \(-0.519598\pi\)
−0.0615291 + 0.998105i \(0.519598\pi\)
\(44\) 0 0
\(45\) 42.3320 + 28.2843i 0.940712 + 0.628539i
\(46\) 0 0
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 0 0
\(49\) −33.0000 −0.673469
\(50\) 0 0
\(51\) −21.1660 39.5980i −0.415020 0.776431i
\(52\) 0 0
\(53\) 50.9117i 0.960598i 0.877105 + 0.480299i \(0.159472\pi\)
−0.877105 + 0.480299i \(0.840528\pi\)
\(54\) 0 0
\(55\) −48.0000 −0.872727
\(56\) 0 0
\(57\) −14.0000 + 7.48331i −0.245614 + 0.131286i
\(58\) 0 0
\(59\) 48.0833i 0.814971i 0.913212 + 0.407485i \(0.133594\pi\)
−0.913212 + 0.407485i \(0.866406\pi\)
\(60\) 0 0
\(61\) 95.2470 1.56143 0.780714 0.624889i \(-0.214856\pi\)
0.780714 + 0.624889i \(0.214856\pi\)
\(62\) 0 0
\(63\) 20.0000 29.9333i 0.317460 0.475131i
\(64\) 0 0
\(65\) 59.8665i 0.921023i
\(66\) 0 0
\(67\) 47.6235 0.710799 0.355399 0.934715i \(-0.384345\pi\)
0.355399 + 0.934715i \(0.384345\pi\)
\(68\) 0 0
\(69\) 42.3320 + 79.1960i 0.613508 + 1.14777i
\(70\) 0 0
\(71\) 89.7998i 1.26479i −0.774648 0.632393i \(-0.782073\pi\)
0.774648 0.632393i \(-0.217927\pi\)
\(72\) 0 0
\(73\) 6.00000 0.0821918 0.0410959 0.999155i \(-0.486915\pi\)
0.0410959 + 0.999155i \(0.486915\pi\)
\(74\) 0 0
\(75\) −18.5203 + 9.89949i −0.246937 + 0.131993i
\(76\) 0 0
\(77\) 33.9411i 0.440794i
\(78\) 0 0
\(79\) −124.000 −1.56962 −0.784810 0.619736i \(-0.787240\pi\)
−0.784810 + 0.619736i \(0.787240\pi\)
\(80\) 0 0
\(81\) −31.0000 74.8331i −0.382716 0.923866i
\(82\) 0 0
\(83\) 2.82843i 0.0340774i 0.999855 + 0.0170387i \(0.00542385\pi\)
−0.999855 + 0.0170387i \(0.994576\pi\)
\(84\) 0 0
\(85\) 84.6640 0.996048
\(86\) 0 0
\(87\) 24.0000 + 44.8999i 0.275862 + 0.516091i
\(88\) 0 0
\(89\) 104.766i 1.17715i −0.808442 0.588575i \(-0.799689\pi\)
0.808442 0.588575i \(-0.200311\pi\)
\(90\) 0 0
\(91\) 42.3320 0.465187
\(92\) 0 0
\(93\) 10.5830 5.65685i 0.113796 0.0608264i
\(94\) 0 0
\(95\) 29.9333i 0.315087i
\(96\) 0 0
\(97\) 118.000 1.21649 0.608247 0.793747i \(-0.291873\pi\)
0.608247 + 0.793747i \(0.291873\pi\)
\(98\) 0 0
\(99\) 63.4980 + 42.4264i 0.641394 + 0.428550i
\(100\) 0 0
\(101\) 62.2254i 0.616093i −0.951371 0.308047i \(-0.900325\pi\)
0.951371 0.308047i \(-0.0996752\pi\)
\(102\) 0 0
\(103\) −108.000 −1.04854 −0.524272 0.851551i \(-0.675662\pi\)
−0.524272 + 0.851551i \(0.675662\pi\)
\(104\) 0 0
\(105\) 32.0000 + 59.8665i 0.304762 + 0.570157i
\(106\) 0 0
\(107\) 144.250i 1.34813i −0.738673 0.674064i \(-0.764547\pi\)
0.738673 0.674064i \(-0.235453\pi\)
\(108\) 0 0
\(109\) 52.9150 0.485459 0.242729 0.970094i \(-0.421957\pi\)
0.242729 + 0.970094i \(0.421957\pi\)
\(110\) 0 0
\(111\) 140.000 74.8331i 1.26126 0.674173i
\(112\) 0 0
\(113\) 89.7998i 0.794688i −0.917670 0.397344i \(-0.869932\pi\)
0.917670 0.397344i \(-0.130068\pi\)
\(114\) 0 0
\(115\) −169.328 −1.47242
\(116\) 0 0
\(117\) 52.9150 79.1960i 0.452265 0.676889i
\(118\) 0 0
\(119\) 59.8665i 0.503080i
\(120\) 0 0
\(121\) 49.0000 0.404959
\(122\) 0 0
\(123\) 42.3320 + 79.1960i 0.344163 + 0.643870i
\(124\) 0 0
\(125\) 101.823i 0.814587i
\(126\) 0 0
\(127\) −76.0000 −0.598425 −0.299213 0.954186i \(-0.596724\pi\)
−0.299213 + 0.954186i \(0.596724\pi\)
\(128\) 0 0
\(129\) −14.0000 + 7.48331i −0.108527 + 0.0580102i
\(130\) 0 0
\(131\) 14.1421i 0.107955i −0.998542 0.0539776i \(-0.982810\pi\)
0.998542 0.0539776i \(-0.0171900\pi\)
\(132\) 0 0
\(133\) −21.1660 −0.159143
\(134\) 0 0
\(135\) 152.000 + 14.9666i 1.12593 + 0.110864i
\(136\) 0 0
\(137\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(138\) 0 0
\(139\) −121.705 −0.875572 −0.437786 0.899079i \(-0.644237\pi\)
−0.437786 + 0.899079i \(0.644237\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 89.7998i 0.627970i
\(144\) 0 0
\(145\) −96.0000 −0.662069
\(146\) 0 0
\(147\) −87.3098 + 46.6690i −0.593944 + 0.317477i
\(148\) 0 0
\(149\) 186.676i 1.25286i 0.779478 + 0.626430i \(0.215485\pi\)
−0.779478 + 0.626430i \(0.784515\pi\)
\(150\) 0 0
\(151\) −60.0000 −0.397351 −0.198675 0.980065i \(-0.563664\pi\)
−0.198675 + 0.980065i \(0.563664\pi\)
\(152\) 0 0
\(153\) −112.000 74.8331i −0.732026 0.489106i
\(154\) 0 0
\(155\) 22.6274i 0.145983i
\(156\) 0 0
\(157\) −116.413 −0.741484 −0.370742 0.928736i \(-0.620897\pi\)
−0.370742 + 0.928736i \(0.620897\pi\)
\(158\) 0 0
\(159\) 72.0000 + 134.700i 0.452830 + 0.847168i
\(160\) 0 0
\(161\) 119.733i 0.743683i
\(162\) 0 0
\(163\) 291.033 1.78548 0.892738 0.450576i \(-0.148781\pi\)
0.892738 + 0.450576i \(0.148781\pi\)
\(164\) 0 0
\(165\) −126.996 + 67.8823i −0.769673 + 0.411408i
\(166\) 0 0
\(167\) 329.266i 1.97165i 0.167772 + 0.985826i \(0.446343\pi\)
−0.167772 + 0.985826i \(0.553657\pi\)
\(168\) 0 0
\(169\) −57.0000 −0.337278
\(170\) 0 0
\(171\) −26.4575 + 39.5980i −0.154722 + 0.231567i
\(172\) 0 0
\(173\) 186.676i 1.07905i −0.841969 0.539527i \(-0.818603\pi\)
0.841969 0.539527i \(-0.181397\pi\)
\(174\) 0 0
\(175\) −28.0000 −0.160000
\(176\) 0 0
\(177\) 68.0000 + 127.216i 0.384181 + 0.718736i
\(178\) 0 0
\(179\) 319.612i 1.78554i −0.450509 0.892772i \(-0.648758\pi\)
0.450509 0.892772i \(-0.351242\pi\)
\(180\) 0 0
\(181\) −116.413 −0.643166 −0.321583 0.946881i \(-0.604215\pi\)
−0.321583 + 0.946881i \(0.604215\pi\)
\(182\) 0 0
\(183\) 252.000 134.700i 1.37705 0.736064i
\(184\) 0 0
\(185\) 299.333i 1.61801i
\(186\) 0 0
\(187\) 126.996 0.679123
\(188\) 0 0
\(189\) 10.5830 107.480i 0.0559947 0.568678i
\(190\) 0 0
\(191\) 59.8665i 0.313437i 0.987643 + 0.156719i \(0.0500916\pi\)
−0.987643 + 0.156719i \(0.949908\pi\)
\(192\) 0 0
\(193\) −102.000 −0.528497 −0.264249 0.964455i \(-0.585124\pi\)
−0.264249 + 0.964455i \(0.585124\pi\)
\(194\) 0 0
\(195\) 84.6640 + 158.392i 0.434175 + 0.812266i
\(196\) 0 0
\(197\) 243.245i 1.23474i −0.786671 0.617372i \(-0.788197\pi\)
0.786671 0.617372i \(-0.211803\pi\)
\(198\) 0 0
\(199\) −188.000 −0.944724 −0.472362 0.881405i \(-0.656598\pi\)
−0.472362 + 0.881405i \(0.656598\pi\)
\(200\) 0 0
\(201\) 126.000 67.3498i 0.626866 0.335074i
\(202\) 0 0
\(203\) 67.8823i 0.334395i
\(204\) 0 0
\(205\) −169.328 −0.825991
\(206\) 0 0
\(207\) 224.000 + 149.666i 1.08213 + 0.723026i
\(208\) 0 0
\(209\) 44.8999i 0.214832i
\(210\) 0 0
\(211\) −248.701 −1.17868 −0.589338 0.807887i \(-0.700611\pi\)
−0.589338 + 0.807887i \(0.700611\pi\)
\(212\) 0 0
\(213\) −126.996 237.588i −0.596226 1.11544i
\(214\) 0 0
\(215\) 29.9333i 0.139224i
\(216\) 0 0
\(217\) 16.0000 0.0737327
\(218\) 0 0
\(219\) 15.8745 8.48528i 0.0724863 0.0387456i
\(220\) 0 0
\(221\) 158.392i 0.716706i
\(222\) 0 0
\(223\) −188.000 −0.843049 −0.421525 0.906817i \(-0.638505\pi\)
−0.421525 + 0.906817i \(0.638505\pi\)
\(224\) 0 0
\(225\) −35.0000 + 52.3832i −0.155556 + 0.232814i
\(226\) 0 0
\(227\) 387.495i 1.70702i 0.521073 + 0.853512i \(0.325532\pi\)
−0.521073 + 0.853512i \(0.674468\pi\)
\(228\) 0 0
\(229\) −243.409 −1.06292 −0.531461 0.847083i \(-0.678357\pi\)
−0.531461 + 0.847083i \(0.678357\pi\)
\(230\) 0 0
\(231\) 48.0000 + 89.7998i 0.207792 + 0.388744i
\(232\) 0 0
\(233\) 104.766i 0.449641i −0.974400 0.224821i \(-0.927820\pi\)
0.974400 0.224821i \(-0.0721796\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −328.073 + 175.362i −1.38427 + 0.739926i
\(238\) 0 0
\(239\) 359.199i 1.50293i −0.659776 0.751463i \(-0.729349\pi\)
0.659776 0.751463i \(-0.270651\pi\)
\(240\) 0 0
\(241\) 122.000 0.506224 0.253112 0.967437i \(-0.418546\pi\)
0.253112 + 0.967437i \(0.418546\pi\)
\(242\) 0 0
\(243\) −187.848 154.149i −0.773038 0.634359i
\(244\) 0 0
\(245\) 186.676i 0.761944i
\(246\) 0 0
\(247\) −56.0000 −0.226721
\(248\) 0 0
\(249\) 4.00000 + 7.48331i 0.0160643 + 0.0300535i
\(250\) 0 0
\(251\) 161.220i 0.642312i −0.947026 0.321156i \(-0.895929\pi\)
0.947026 0.321156i \(-0.104071\pi\)
\(252\) 0 0
\(253\) −253.992 −1.00392
\(254\) 0 0
\(255\) 224.000 119.733i 0.878431 0.469541i
\(256\) 0 0
\(257\) 448.999i 1.74708i −0.486755 0.873539i \(-0.661819\pi\)
0.486755 0.873539i \(-0.338181\pi\)
\(258\) 0 0
\(259\) 211.660 0.817220
\(260\) 0 0
\(261\) 126.996 + 84.8528i 0.486575 + 0.325107i
\(262\) 0 0
\(263\) 209.533i 0.796703i −0.917233 0.398351i \(-0.869582\pi\)
0.917233 0.398351i \(-0.130418\pi\)
\(264\) 0 0
\(265\) −288.000 −1.08679
\(266\) 0 0
\(267\) −148.162 277.186i −0.554914 1.03815i
\(268\) 0 0
\(269\) 50.9117i 0.189263i −0.995512 0.0946314i \(-0.969833\pi\)
0.995512 0.0946314i \(-0.0301672\pi\)
\(270\) 0 0
\(271\) −348.000 −1.28413 −0.642066 0.766649i \(-0.721923\pi\)
−0.642066 + 0.766649i \(0.721923\pi\)
\(272\) 0 0
\(273\) 112.000 59.8665i 0.410256 0.219291i
\(274\) 0 0
\(275\) 59.3970i 0.215989i
\(276\) 0 0
\(277\) −243.409 −0.878733 −0.439367 0.898308i \(-0.644797\pi\)
−0.439367 + 0.898308i \(0.644797\pi\)
\(278\) 0 0
\(279\) 20.0000 29.9333i 0.0716846 0.107288i
\(280\) 0 0
\(281\) 314.299i 1.11850i −0.828998 0.559251i \(-0.811089\pi\)
0.828998 0.559251i \(-0.188911\pi\)
\(282\) 0 0
\(283\) 89.9555 0.317864 0.158932 0.987290i \(-0.449195\pi\)
0.158932 + 0.987290i \(0.449195\pi\)
\(284\) 0 0
\(285\) −42.3320 79.1960i −0.148533 0.277881i
\(286\) 0 0
\(287\) 119.733i 0.417188i
\(288\) 0 0
\(289\) 65.0000 0.224913
\(290\) 0 0
\(291\) 312.199 166.877i 1.07285 0.573461i
\(292\) 0 0
\(293\) 16.9706i 0.0579200i −0.999581 0.0289600i \(-0.990780\pi\)
0.999581 0.0289600i \(-0.00921954\pi\)
\(294\) 0 0
\(295\) −272.000 −0.922034
\(296\) 0 0
\(297\) 228.000 + 22.4499i 0.767677 + 0.0755890i
\(298\) 0 0
\(299\) 316.784i 1.05948i
\(300\) 0 0
\(301\) −21.1660 −0.0703190
\(302\) 0 0
\(303\) −88.0000 164.633i −0.290429 0.543343i
\(304\) 0 0
\(305\) 538.799i 1.76655i
\(306\) 0 0
\(307\) −460.361 −1.49955 −0.749773 0.661695i \(-0.769837\pi\)
−0.749773 + 0.661695i \(0.769837\pi\)
\(308\) 0 0
\(309\) −285.741 + 152.735i −0.924729 + 0.494288i
\(310\) 0 0
\(311\) 149.666i 0.481242i −0.970619 0.240621i \(-0.922649\pi\)
0.970619 0.240621i \(-0.0773511\pi\)
\(312\) 0 0
\(313\) 562.000 1.79553 0.897764 0.440478i \(-0.145191\pi\)
0.897764 + 0.440478i \(0.145191\pi\)
\(314\) 0 0
\(315\) 169.328 + 113.137i 0.537549 + 0.359165i
\(316\) 0 0
\(317\) 5.65685i 0.0178450i −0.999960 0.00892248i \(-0.997160\pi\)
0.999960 0.00892248i \(-0.00284015\pi\)
\(318\) 0 0
\(319\) −144.000 −0.451411
\(320\) 0 0
\(321\) −204.000 381.649i −0.635514 1.18894i
\(322\) 0 0
\(323\) 79.1960i 0.245189i
\(324\) 0 0
\(325\) −74.0810 −0.227942
\(326\) 0 0
\(327\) 140.000 74.8331i 0.428135 0.228848i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −418.029 −1.26293 −0.631463 0.775406i \(-0.717545\pi\)
−0.631463 + 0.775406i \(0.717545\pi\)
\(332\) 0 0
\(333\) 264.575 395.980i 0.794520 1.18913i
\(334\) 0 0
\(335\) 269.399i 0.804177i
\(336\) 0 0
\(337\) −50.0000 −0.148368 −0.0741840 0.997245i \(-0.523635\pi\)
−0.0741840 + 0.997245i \(0.523635\pi\)
\(338\) 0 0
\(339\) −126.996 237.588i −0.374620 0.700849i
\(340\) 0 0
\(341\) 33.9411i 0.0995341i
\(342\) 0 0
\(343\) −328.000 −0.956268
\(344\) 0 0
\(345\) −448.000 + 239.466i −1.29855 + 0.694105i
\(346\) 0 0
\(347\) 359.210i 1.03519i 0.855626 + 0.517594i \(0.173172\pi\)
−0.855626 + 0.517594i \(0.826828\pi\)
\(348\) 0 0
\(349\) −116.413 −0.333562 −0.166781 0.985994i \(-0.553337\pi\)
−0.166781 + 0.985994i \(0.553337\pi\)
\(350\) 0 0
\(351\) 28.0000 284.366i 0.0797721 0.810159i
\(352\) 0 0
\(353\) 179.600i 0.508781i 0.967102 + 0.254390i \(0.0818748\pi\)
−0.967102 + 0.254390i \(0.918125\pi\)
\(354\) 0 0
\(355\) 507.984 1.43094
\(356\) 0 0
\(357\) −84.6640 158.392i −0.237154 0.443675i
\(358\) 0 0
\(359\) 329.266i 0.917175i −0.888649 0.458588i \(-0.848356\pi\)
0.888649 0.458588i \(-0.151644\pi\)
\(360\) 0 0
\(361\) −333.000 −0.922438
\(362\) 0 0
\(363\) 129.642 69.2965i 0.357140 0.190899i
\(364\) 0 0
\(365\) 33.9411i 0.0929894i
\(366\) 0 0
\(367\) 228.000 0.621253 0.310627 0.950532i \(-0.399461\pi\)
0.310627 + 0.950532i \(0.399461\pi\)
\(368\) 0 0
\(369\) 224.000 + 149.666i 0.607046 + 0.405600i
\(370\) 0 0
\(371\) 203.647i 0.548913i
\(372\) 0 0
\(373\) 137.579 0.368845 0.184422 0.982847i \(-0.440959\pi\)
0.184422 + 0.982847i \(0.440959\pi\)
\(374\) 0 0
\(375\) 144.000 + 269.399i 0.384000 + 0.718398i
\(376\) 0 0
\(377\) 179.600i 0.476391i
\(378\) 0 0
\(379\) 164.037 0.432814 0.216407 0.976303i \(-0.430566\pi\)
0.216407 + 0.976303i \(0.430566\pi\)
\(380\) 0 0
\(381\) −201.077 + 107.480i −0.527761 + 0.282100i
\(382\) 0 0
\(383\) 179.600i 0.468928i −0.972125 0.234464i \(-0.924666\pi\)
0.972125 0.234464i \(-0.0753336\pi\)
\(384\) 0 0
\(385\) −192.000 −0.498701
\(386\) 0 0
\(387\) −26.4575 + 39.5980i −0.0683657 + 0.102320i
\(388\) 0 0
\(389\) 96.1665i 0.247215i 0.992331 + 0.123607i \(0.0394463\pi\)
−0.992331 + 0.123607i \(0.960554\pi\)
\(390\) 0 0
\(391\) 448.000 1.14578
\(392\) 0 0
\(393\) −20.0000 37.4166i −0.0508906 0.0952076i
\(394\) 0 0
\(395\) 701.450i 1.77582i
\(396\) 0 0
\(397\) 52.9150 0.133287 0.0666436 0.997777i \(-0.478771\pi\)
0.0666436 + 0.997777i \(0.478771\pi\)
\(398\) 0 0
\(399\) −56.0000 + 29.9333i −0.140351 + 0.0750207i
\(400\) 0 0
\(401\) 134.700i 0.335909i 0.985795 + 0.167955i \(0.0537162\pi\)
−0.985795 + 0.167955i \(0.946284\pi\)
\(402\) 0 0
\(403\) 42.3320 0.105042
\(404\) 0 0
\(405\) 423.320 175.362i 1.04524 0.432994i
\(406\) 0 0
\(407\) 448.999i 1.10319i
\(408\) 0 0
\(409\) 158.000 0.386308 0.193154 0.981168i \(-0.438128\pi\)
0.193154 + 0.981168i \(0.438128\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 192.333i 0.465697i
\(414\) 0 0
\(415\) −16.0000 −0.0385542
\(416\) 0 0
\(417\) −322.000 + 172.116i −0.772182 + 0.412749i
\(418\) 0 0
\(419\) 330.926i 0.789799i 0.918724 + 0.394900i \(0.129221\pi\)
−0.918724 + 0.394900i \(0.870779\pi\)
\(420\) 0 0
\(421\) 518.567 1.23175 0.615876 0.787843i \(-0.288802\pi\)
0.615876 + 0.787843i \(0.288802\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 104.766i 0.246509i
\(426\) 0 0
\(427\) 380.988 0.892244
\(428\) 0 0
\(429\) 126.996 + 237.588i 0.296028 + 0.553818i
\(430\) 0 0
\(431\) 359.199i 0.833409i 0.909042 + 0.416704i \(0.136815\pi\)
−0.909042 + 0.416704i \(0.863185\pi\)
\(432\) 0 0
\(433\) 86.0000 0.198614 0.0993072 0.995057i \(-0.468337\pi\)
0.0993072 + 0.995057i \(0.468337\pi\)
\(434\) 0 0
\(435\) −253.992 + 135.765i −0.583890 + 0.312102i
\(436\) 0 0
\(437\) 158.392i 0.362453i
\(438\) 0 0
\(439\) −156.000 −0.355353 −0.177677 0.984089i \(-0.556858\pi\)
−0.177677 + 0.984089i \(0.556858\pi\)
\(440\) 0 0
\(441\) −165.000 + 246.949i −0.374150 + 0.559976i
\(442\) 0 0
\(443\) 212.132i 0.478853i 0.970914 + 0.239427i \(0.0769595\pi\)
−0.970914 + 0.239427i \(0.923041\pi\)
\(444\) 0 0
\(445\) 592.648 1.33179
\(446\) 0 0
\(447\) 264.000 + 493.899i 0.590604 + 1.10492i
\(448\) 0 0
\(449\) 493.899i 1.10000i −0.835166 0.549999i \(-0.814628\pi\)
0.835166 0.549999i \(-0.185372\pi\)
\(450\) 0 0
\(451\) −253.992 −0.563175
\(452\) 0 0
\(453\) −158.745 + 84.8528i −0.350431 + 0.187313i
\(454\) 0 0
\(455\) 239.466i 0.526299i
\(456\) 0 0
\(457\) −194.000 −0.424508 −0.212254 0.977215i \(-0.568080\pi\)
−0.212254 + 0.977215i \(0.568080\pi\)
\(458\) 0 0
\(459\) −402.154 39.5980i −0.876153 0.0862701i
\(460\) 0 0
\(461\) 560.029i 1.21481i 0.794391 + 0.607406i \(0.207790\pi\)
−0.794391 + 0.607406i \(0.792210\pi\)
\(462\) 0 0
\(463\) 404.000 0.872570 0.436285 0.899808i \(-0.356294\pi\)
0.436285 + 0.899808i \(0.356294\pi\)
\(464\) 0 0
\(465\) 32.0000 + 59.8665i 0.0688172 + 0.128745i
\(466\) 0 0
\(467\) 664.680i 1.42330i −0.702535 0.711649i \(-0.747949\pi\)
0.702535 0.711649i \(-0.252051\pi\)
\(468\) 0 0
\(469\) 190.494 0.406171
\(470\) 0 0
\(471\) −308.000 + 164.633i −0.653928 + 0.349539i
\(472\) 0 0
\(473\) 44.8999i 0.0949258i
\(474\) 0 0
\(475\) 37.0405 0.0779800
\(476\) 0 0
\(477\) 380.988 + 254.558i 0.798717 + 0.533665i
\(478\) 0 0
\(479\) 239.466i 0.499929i −0.968255 0.249965i \(-0.919581\pi\)
0.968255 0.249965i \(-0.0804190\pi\)
\(480\) 0 0
\(481\) 560.000 1.16424
\(482\) 0 0
\(483\) 169.328 + 316.784i 0.350576 + 0.655867i
\(484\) 0 0
\(485\) 667.509i 1.37631i
\(486\) 0 0
\(487\) 500.000 1.02669 0.513347 0.858181i \(-0.328405\pi\)
0.513347 + 0.858181i \(0.328405\pi\)
\(488\) 0 0
\(489\) 770.000 411.582i 1.57464 0.841682i
\(490\) 0 0
\(491\) 115.966i 0.236182i 0.993003 + 0.118091i \(0.0376775\pi\)
−0.993003 + 0.118091i \(0.962322\pi\)
\(492\) 0 0
\(493\) 253.992 0.515197
\(494\) 0 0
\(495\) −240.000 + 359.199i −0.484848 + 0.725655i
\(496\) 0 0
\(497\) 359.199i 0.722735i
\(498\) 0 0
\(499\) −333.365 −0.668065 −0.334033 0.942561i \(-0.608410\pi\)
−0.334033 + 0.942561i \(0.608410\pi\)
\(500\) 0 0
\(501\) 465.652 + 871.156i 0.929446 + 1.73883i
\(502\) 0 0
\(503\) 448.999i 0.892642i 0.894873 + 0.446321i \(0.147266\pi\)
−0.894873 + 0.446321i \(0.852734\pi\)
\(504\) 0 0
\(505\) 352.000 0.697030
\(506\) 0 0
\(507\) −150.808 + 80.6102i −0.297451 + 0.158994i
\(508\) 0 0
\(509\) 322.441i 0.633479i −0.948513 0.316739i \(-0.897412\pi\)
0.948513 0.316739i \(-0.102588\pi\)
\(510\) 0 0
\(511\) 24.0000 0.0469667
\(512\) 0 0
\(513\) −14.0000 + 142.183i −0.0272904 + 0.277160i
\(514\) 0 0
\(515\) 610.940i 1.18629i
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) −264.000 493.899i −0.508671 0.951635i
\(520\) 0 0
\(521\) 179.600i 0.344721i −0.985034 0.172360i \(-0.944861\pi\)
0.985034 0.172360i \(-0.0551394\pi\)
\(522\) 0 0
\(523\) −555.608 −1.06235 −0.531174 0.847263i \(-0.678249\pi\)
−0.531174 + 0.847263i \(0.678249\pi\)
\(524\) 0 0
\(525\) −74.0810 + 39.5980i −0.141107 + 0.0754247i
\(526\) 0 0
\(527\) 59.8665i 0.113599i
\(528\) 0 0
\(529\) −367.000 −0.693762
\(530\) 0 0
\(531\) 359.822 + 240.416i 0.677631 + 0.452761i
\(532\) 0 0
\(533\) 316.784i 0.594341i
\(534\) 0 0
\(535\) 816.000 1.52523
\(536\) 0 0
\(537\) −452.000 845.615i −0.841713 1.57470i
\(538\) 0 0
\(539\) 280.014i 0.519507i
\(540\) 0 0
\(541\) 772.559 1.42802 0.714011 0.700135i \(-0.246877\pi\)
0.714011 + 0.700135i \(0.246877\pi\)
\(542\) 0 0
\(543\) −308.000 + 164.633i −0.567219 + 0.303191i
\(544\) 0 0
\(545\) 299.333i 0.549234i
\(546\) 0 0
\(547\) 502.693 0.919000 0.459500 0.888178i \(-0.348029\pi\)
0.459500 + 0.888178i \(0.348029\pi\)
\(548\) 0 0
\(549\) 476.235 712.764i 0.867459 1.29829i
\(550\) 0 0
\(551\) 89.7998i 0.162976i
\(552\) 0 0
\(553\) −496.000 −0.896926
\(554\) 0 0
\(555\) 423.320 + 791.960i 0.762739 + 1.42695i
\(556\) 0 0
\(557\) 526.087i 0.944502i −0.881464 0.472251i \(-0.843442\pi\)
0.881464 0.472251i \(-0.156558\pi\)
\(558\) 0 0
\(559\) −56.0000 −0.100179
\(560\) 0 0
\(561\) 336.000 179.600i 0.598930 0.320142i
\(562\) 0 0
\(563\) 8.48528i 0.0150715i −0.999972 0.00753577i \(-0.997601\pi\)
0.999972 0.00753577i \(-0.00239873\pi\)
\(564\) 0 0
\(565\) 507.984 0.899087
\(566\) 0 0
\(567\) −124.000 299.333i −0.218695 0.527923i
\(568\) 0 0
\(569\) 359.199i 0.631281i 0.948879 + 0.315641i \(0.102219\pi\)
−0.948879 + 0.315641i \(0.897781\pi\)
\(570\) 0 0
\(571\) 5.29150 0.00926708 0.00463354 0.999989i \(-0.498525\pi\)
0.00463354 + 0.999989i \(0.498525\pi\)
\(572\) 0 0
\(573\) 84.6640 + 158.392i 0.147756 + 0.276426i
\(574\) 0 0
\(575\) 209.533i 0.364405i
\(576\) 0 0
\(577\) −18.0000 −0.0311958 −0.0155979 0.999878i \(-0.504965\pi\)
−0.0155979 + 0.999878i \(0.504965\pi\)
\(578\) 0 0
\(579\) −269.867 + 144.250i −0.466091 + 0.249136i
\(580\) 0 0
\(581\) 11.3137i 0.0194728i
\(582\) 0 0
\(583\) −432.000 −0.740995
\(584\) 0 0
\(585\) 448.000 + 299.333i 0.765812 + 0.511680i
\(586\) 0 0
\(587\) 121.622i 0.207193i −0.994619 0.103597i \(-0.966965\pi\)
0.994619 0.103597i \(-0.0330351\pi\)
\(588\) 0 0
\(589\) −21.1660 −0.0359355
\(590\) 0 0
\(591\) −344.000 643.565i −0.582064 1.08894i
\(592\) 0 0
\(593\) 718.398i 1.21146i −0.795669 0.605732i \(-0.792880\pi\)
0.795669 0.605732i \(-0.207120\pi\)
\(594\) 0 0
\(595\) 338.656 0.569170
\(596\) 0 0
\(597\) −497.401 + 265.872i −0.833168 + 0.445347i
\(598\) 0 0
\(599\) 688.465i 1.14936i −0.818379 0.574679i \(-0.805127\pi\)
0.818379 0.574679i \(-0.194873\pi\)
\(600\) 0 0
\(601\) 358.000 0.595674 0.297837 0.954617i \(-0.403735\pi\)
0.297837 + 0.954617i \(0.403735\pi\)
\(602\) 0 0
\(603\) 238.118 356.382i 0.394888 0.591015i
\(604\) 0 0
\(605\) 277.186i 0.458158i
\(606\) 0 0
\(607\) 884.000 1.45634 0.728171 0.685395i \(-0.240371\pi\)
0.728171 + 0.685395i \(0.240371\pi\)
\(608\) 0 0
\(609\) 96.0000 + 179.600i 0.157635 + 0.294909i
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 391.571 0.638778 0.319389 0.947624i \(-0.396522\pi\)
0.319389 + 0.947624i \(0.396522\pi\)
\(614\) 0 0
\(615\) −448.000 + 239.466i −0.728455 + 0.389376i
\(616\) 0 0
\(617\) 104.766i 0.169800i 0.996389 + 0.0848998i \(0.0270570\pi\)
−0.996389 + 0.0848998i \(0.972943\pi\)
\(618\) 0 0
\(619\) 682.604 1.10275 0.551376 0.834257i \(-0.314103\pi\)
0.551376 + 0.834257i \(0.314103\pi\)
\(620\) 0 0
\(621\) 804.308 + 79.1960i 1.29518 + 0.127530i
\(622\) 0 0
\(623\) 419.066i 0.672658i
\(624\) 0 0
\(625\) −751.000 −1.20160
\(626\) 0 0
\(627\) −63.4980 118.794i −0.101273 0.189464i
\(628\) 0 0
\(629\) 791.960i 1.25908i
\(630\) 0 0
\(631\) −428.000 −0.678288 −0.339144 0.940734i \(-0.610137\pi\)
−0.339144 + 0.940734i \(0.610137\pi\)
\(632\) 0 0
\(633\) −658.000 + 351.716i −1.03949 + 0.555633i
\(634\) 0 0
\(635\) 429.921i 0.677041i
\(636\) 0 0
\(637\) −349.239 −0.548256
\(638\) 0 0
\(639\) −672.000 448.999i −1.05164 0.702659i
\(640\) 0 0
\(641\) 793.231i 1.23749i 0.785592 + 0.618745i \(0.212359\pi\)
−0.785592 + 0.618745i \(0.787641\pi\)
\(642\) 0 0
\(643\) −851.932 −1.32493 −0.662467 0.749092i \(-0.730490\pi\)
−0.662467 + 0.749092i \(0.730490\pi\)
\(644\) 0 0
\(645\) −42.3320 79.1960i −0.0656310 0.122784i
\(646\) 0 0
\(647\) 448.999i 0.693970i 0.937871 + 0.346985i \(0.112795\pi\)
−0.937871 + 0.346985i \(0.887205\pi\)
\(648\) 0 0
\(649\) −408.000 −0.628659
\(650\) 0 0
\(651\) 42.3320 22.6274i 0.0650261 0.0347579i
\(652\) 0 0
\(653\) 1103.09i 1.68926i 0.535351 + 0.844630i \(0.320179\pi\)
−0.535351 + 0.844630i \(0.679821\pi\)
\(654\) 0 0
\(655\) 80.0000 0.122137
\(656\) 0 0
\(657\) 30.0000 44.8999i 0.0456621 0.0683408i
\(658\) 0 0
\(659\) 924.896i 1.40348i 0.712431 + 0.701742i \(0.247594\pi\)
−0.712431 + 0.701742i \(0.752406\pi\)
\(660\) 0 0
\(661\) 433.903 0.656435 0.328217 0.944602i \(-0.393552\pi\)
0.328217 + 0.944602i \(0.393552\pi\)
\(662\) 0 0
\(663\) −224.000 419.066i −0.337858 0.632075i
\(664\) 0 0
\(665\) 119.733i 0.180050i
\(666\) 0 0
\(667\) −507.984 −0.761596
\(668\) 0 0
\(669\) −497.401 + 265.872i −0.743500 + 0.397417i
\(670\) 0 0
\(671\) 808.198i 1.20447i
\(672\) 0 0
\(673\) 566.000 0.841010 0.420505 0.907290i \(-0.361853\pi\)
0.420505 + 0.907290i \(0.361853\pi\)
\(674\) 0 0
\(675\) −18.5203 + 188.090i −0.0274374 + 0.278652i
\(676\) 0 0
\(677\) 797.616i 1.17816i 0.808074 + 0.589082i \(0.200510\pi\)
−0.808074 + 0.589082i \(0.799490\pi\)
\(678\) 0 0
\(679\) 472.000 0.695140
\(680\) 0 0
\(681\) 548.000 + 1025.21i 0.804699 + 1.50545i
\(682\) 0 0
\(683\) 404.465i 0.592189i 0.955159 + 0.296094i \(0.0956843\pi\)
−0.955159 + 0.296094i \(0.904316\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −644.000 + 344.232i −0.937409 + 0.501066i
\(688\) 0 0
\(689\) 538.799i 0.782001i
\(690\) 0 0
\(691\) 121.705 0.176128 0.0880641 0.996115i \(-0.471932\pi\)
0.0880641 + 0.996115i \(0.471932\pi\)
\(692\) 0 0
\(693\) 253.992 + 169.706i 0.366511 + 0.244885i
\(694\) 0 0
\(695\) 688.465i 0.990597i
\(696\) 0 0
\(697\) 448.000 0.642755
\(698\) 0 0
\(699\) −148.162 277.186i −0.211963 0.396546i
\(700\) 0 0
\(701\) 333.754i 0.476112i 0.971251 + 0.238056i \(0.0765102\pi\)
−0.971251 + 0.238056i \(0.923490\pi\)
\(702\) 0 0
\(703\) −280.000 −0.398293
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 248.902i 0.352053i
\(708\) 0 0
\(709\) −370.405 −0.522433 −0.261217 0.965280i \(-0.584124\pi\)
−0.261217 + 0.965280i \(0.584124\pi\)
\(710\) 0 0
\(711\) −620.000 + 927.931i −0.872011 + 1.30511i
\(712\) 0 0
\(713\) 119.733i 0.167929i
\(714\) 0 0
\(715\) −507.984 −0.710467
\(716\) 0 0
\(717\) −507.984 950.352i −0.708486 1.32546i
\(718\) 0 0
\(719\) 1017.73i 1.41548i −0.706473 0.707740i \(-0.749715\pi\)
0.706473 0.707740i \(-0.250285\pi\)
\(720\) 0 0
\(721\) −432.000 −0.599168
\(722\) 0 0
\(723\) 322.782 172.534i 0.446448 0.238636i
\(724\) 0 0
\(725\) 118.794i 0.163854i
\(726\) 0 0
\(727\) 1140.00 1.56809 0.784044 0.620705i \(-0.213154\pi\)
0.784044 + 0.620705i \(0.213154\pi\)
\(728\) 0 0
\(729\) −715.000 142.183i −0.980796 0.195038i
\(730\) 0 0
\(731\) 79.1960i 0.108339i
\(732\) 0 0
\(733\) −370.405 −0.505328 −0.252664 0.967554i \(-0.581307\pi\)
−0.252664 + 0.967554i \(0.581307\pi\)
\(734\) 0 0
\(735\) −264.000 493.899i −0.359184 0.671971i
\(736\) 0 0
\(737\) 404.099i 0.548303i
\(738\) 0 0
\(739\) 5.29150 0.00716036 0.00358018 0.999994i \(-0.498860\pi\)
0.00358018 + 0.999994i \(0.498860\pi\)
\(740\) 0 0
\(741\) −148.162 + 79.1960i −0.199949 + 0.106877i
\(742\) 0 0
\(743\) 748.331i 1.00718i −0.863944 0.503588i \(-0.832013\pi\)
0.863944 0.503588i \(-0.167987\pi\)
\(744\) 0 0
\(745\) −1056.00 −1.41745
\(746\) 0 0
\(747\) 21.1660 + 14.1421i 0.0283347 + 0.0189319i
\(748\) 0 0
\(749\) 576.999i 0.770359i
\(750\) 0 0
\(751\) 1108.00 1.47537 0.737683 0.675147i \(-0.235920\pi\)
0.737683 + 0.675147i \(0.235920\pi\)
\(752\) 0 0
\(753\) −228.000 426.549i −0.302789 0.566466i
\(754\) 0 0
\(755\) 339.411i 0.449551i
\(756\) 0 0
\(757\) −1047.72 −1.38404 −0.692019 0.721879i \(-0.743279\pi\)
−0.692019 + 0.721879i \(0.743279\pi\)
\(758\) 0 0
\(759\) −672.000 + 359.199i −0.885375 + 0.473253i
\(760\) 0 0
\(761\) 1287.13i 1.69137i −0.533685 0.845683i \(-0.679193\pi\)
0.533685 0.845683i \(-0.320807\pi\)
\(762\) 0 0
\(763\) 211.660 0.277405
\(764\) 0 0
\(765\) 423.320 633.568i 0.553360 0.828193i
\(766\) 0 0
\(767\) 508.865i 0.663449i
\(768\) 0 0
\(769\) 538.000 0.699610 0.349805 0.936823i \(-0.386248\pi\)
0.349805 + 0.936823i \(0.386248\pi\)
\(770\) 0 0
\(771\) −634.980 1187.94i −0.823580 1.54078i
\(772\) 0 0
\(773\) 62.2254i 0.0804986i −0.999190 0.0402493i \(-0.987185\pi\)
0.999190 0.0402493i \(-0.0128152\pi\)
\(774\) 0 0
\(775\) −28.0000 −0.0361290
\(776\) 0 0
\(777\) 560.000 299.333i 0.720721 0.385241i
\(778\) 0 0
\(779\) 158.392i 0.203327i
\(780\) 0 0
\(781\) 761.976 0.975642
\(782\) 0 0
\(783\) 456.000 + 44.8999i 0.582375 + 0.0573434i
\(784\) 0 0
\(785\) 658.532i 0.838894i
\(786\) 0 0
\(787\) 1476.33 1.87589 0.937947 0.346778i \(-0.112724\pi\)
0.937947 + 0.346778i \(0.112724\pi\)
\(788\) 0 0
\(789\) −296.324 554.372i −0.375569 0.702626i
\(790\) 0 0
\(791\) 359.199i 0.454108i
\(792\) 0 0
\(793\) 1008.00 1.27112
\(794\) 0 0
\(795\) −761.976 + 407.294i −0.958461 + 0.512319i
\(796\) 0 0
\(797\) 1250.16i 1.56859i −0.620389 0.784294i \(-0.713025\pi\)
0.620389 0.784294i \(-0.286975\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) −784.000 523.832i −0.978777 0.653973i
\(802\) 0 0
\(803\) 50.9117i 0.0634019i
\(804\) 0 0
\(805\) −677.312 −0.841382
\(806\) 0 0
\(807\) −72.0000 134.700i −0.0892193 0.166914i
\(808\) 0 0
\(809\) 448.999i 0.555005i 0.960725 + 0.277502i \(0.0895066\pi\)
−0.960725 + 0.277502i \(0.910493\pi\)
\(810\) 0 0
\(811\) 1391.67 1.71599 0.857993 0.513661i \(-0.171711\pi\)
0.857993 + 0.513661i \(0.171711\pi\)
\(812\) 0 0
\(813\) −920.721 + 492.146i −1.13250 + 0.605346i
\(814\) 0 0
\(815\) 1646.33i 2.02004i
\(816\) 0 0
\(817\) 28.0000 0.0342717
\(818\) 0 0
\(819\) 211.660 316.784i 0.258437 0.386793i
\(820\) 0 0
\(821\) 469.519i 0.571887i −0.958247 0.285943i \(-0.907693\pi\)
0.958247 0.285943i \(-0.0923069\pi\)
\(822\) 0 0
\(823\) 1268.00 1.54070 0.770352 0.637618i \(-0.220080\pi\)
0.770352 + 0.637618i \(0.220080\pi\)
\(824\) 0 0
\(825\) −84.0000 157.150i −0.101818 0.190484i
\(826\) 0 0
\(827\) 1145.51i 1.38514i 0.721349 + 0.692571i \(0.243522\pi\)
−0.721349 + 0.692571i \(0.756478\pi\)
\(828\) 0 0
\(829\) −1217.05 −1.46809 −0.734044 0.679101i \(-0.762370\pi\)
−0.734044 + 0.679101i \(0.762370\pi\)
\(830\) 0 0
\(831\) −644.000 + 344.232i −0.774970 + 0.414239i
\(832\) 0 0
\(833\) 493.899i 0.592916i
\(834\) 0 0
\(835\) −1862.61 −2.23067
\(836\) 0 0
\(837\) 10.5830 107.480i 0.0126440 0.128411i
\(838\) 0 0
\(839\) 1227.26i 1.46277i 0.681965 + 0.731385i \(0.261126\pi\)
−0.681965 + 0.731385i \(0.738874\pi\)
\(840\) 0 0
\(841\) 553.000 0.657551
\(842\) 0 0
\(843\) −444.486 831.558i −0.527267 0.986427i
\(844\) 0 0
\(845\) 322.441i 0.381587i
\(846\) 0 0
\(847\) 196.000 0.231405
\(848\) 0 0
\(849\) 238.000 127.216i 0.280330 0.149843i
\(850\) 0 0
\(851\) 1583.92i 1.86124i
\(852\) 0 0
\(853\) −1047.72 −1.22827 −0.614137 0.789200i \(-0.710496\pi\)
−0.614137 + 0.789200i \(0.710496\pi\)
\(854\) 0 0
\(855\) −224.000 149.666i −0.261988 0.175048i
\(856\) 0 0
\(857\) 359.199i 0.419135i 0.977794 + 0.209568i \(0.0672057\pi\)
−0.977794 + 0.209568i \(0.932794\pi\)
\(858\) 0 0
\(859\) −513.276 −0.597527 −0.298764 0.954327i \(-0.596574\pi\)
−0.298764 + 0.954327i \(0.596574\pi\)
\(860\) 0 0
\(861\) 169.328 + 316.784i 0.196664 + 0.367925i
\(862\) 0 0
\(863\) 538.799i 0.624332i −0.950028 0.312166i \(-0.898945\pi\)
0.950028 0.312166i \(-0.101055\pi\)
\(864\) 0 0
\(865\) 1056.00 1.22081
\(866\) 0 0
\(867\) 171.974 91.9239i 0.198355 0.106025i
\(868\) 0 0
\(869\) 1052.17i 1.21079i
\(870\) 0 0
\(871\) 504.000 0.578645
\(872\) 0 0
\(873\) 590.000 883.031i 0.675830 1.01149i
\(874\) 0 0
\(875\) 407.294i 0.465478i
\(876\) 0 0
\(877\) −1344.04 −1.53254 −0.766272 0.642516i \(-0.777891\pi\)
−0.766272 + 0.642516i \(0.777891\pi\)
\(878\) 0 0
\(879\) −24.0000 44.8999i −0.0273038 0.0510806i
\(880\) 0 0
\(881\) 748.331i 0.849411i −0.905331 0.424706i \(-0.860378\pi\)
0.905331 0.424706i \(-0.139622\pi\)
\(882\) 0 0
\(883\) 418.029 0.473419 0.236709 0.971581i \(-0.423931\pi\)
0.236709 + 0.971581i \(0.423931\pi\)
\(884\) 0 0
\(885\) −719.644 + 384.666i −0.813157 + 0.434651i
\(886\) 0 0
\(887\) 1167.40i 1.31612i 0.752966 + 0.658059i \(0.228622\pi\)
−0.752966 + 0.658059i \(0.771378\pi\)
\(888\) 0 0
\(889\) −304.000 −0.341957
\(890\) 0 0
\(891\) 634.980 263.044i 0.712660 0.295223i
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 1808.00 2.02011
\(896\) 0 0
\(897\) 448.000 + 838.131i 0.499443 + 0.934372i
\(898\) 0 0
\(899\) 67.8823i 0.0755086i
\(900\) 0 0
\(901\) 761.976 0.845701
\(902\) 0 0
\(903\) −56.0000 + 29.9333i −0.0620155 + 0.0331487i
\(904\) 0 0
\(905\) 658.532i 0.727659i
\(906\) 0 0
\(907\) −597.940 −0.659250 −0.329625 0.944112i \(-0.606922\pi\)
−0.329625 + 0.944112i \(0.606922\pi\)
\(908\) 0 0
\(909\) −465.652 311.127i −0.512269 0.342274i
\(910\) 0 0
\(911\) 359.199i 0.394291i −0.980374 0.197146i \(-0.936833\pi\)
0.980374 0.197146i \(-0.0631671\pi\)
\(912\) 0 0
\(913\) −24.0000 −0.0262870
\(914\) 0 0
\(915\) 761.976 + 1425.53i 0.832761 + 1.55795i
\(916\) 0 0
\(917\) 56.5685i 0.0616887i
\(918\) 0 0
\(919\) −780.000 −0.848749 −0.424374 0.905487i \(-0.639506\pi\)
−0.424374 + 0.905487i \(0.639506\pi\)
\(920\) 0 0
\(921\) −1218.00 + 651.048i −1.32248 + 0.706893i
\(922\) 0 0
\(923\) 950.352i 1.02963i
\(924\) 0 0
\(925\) −370.405 −0.400438
\(926\) 0 0
\(927\) −540.000 + 808.198i −0.582524 + 0.871842i
\(928\) 0 0
\(929\) 1361.96i 1.46605i −0.680200 0.733027i \(-0.738107\pi\)
0.680200 0.733027i \(-0.261893\pi\)
\(930\) 0 0
\(931\) 174.620 0.187561
\(932\) 0 0
\(933\) −211.660 395.980i −0.226860 0.424416i
\(934\) 0 0
\(935\) 718.398i 0.768340i
\(936\) 0 0
\(937\) −566.000 −0.604055 −0.302028 0.953299i \(-0.597664\pi\)
−0.302028 + 0.953299i \(0.597664\pi\)
\(938\) 0 0
\(939\) 1486.91 794.788i 1.58351 0.846420i
\(940\) 0 0
\(941\) 424.264i 0.450865i 0.974259 + 0.225433i \(0.0723795\pi\)
−0.974259 + 0.225433i \(0.927620\pi\)
\(942\) 0 0
\(943\) −896.000 −0.950159
\(944\) 0 0
\(945\) 608.000 + 59.8665i 0.643386 + 0.0633508i
\(946\) 0 0
\(947\) 466.690i 0.492809i −0.969167 0.246405i \(-0.920751\pi\)
0.969167 0.246405i \(-0.0792492\pi\)
\(948\) 0 0
\(949\) 63.4980 0.0669105
\(950\) 0 0
\(951\) −8.00000 14.9666i −0.00841220 0.0157378i
\(952\) 0 0
\(953\) 897.998i 0.942285i 0.882057 + 0.471143i \(0.156158\pi\)
−0.882057 + 0.471143i \(0.843842\pi\)
\(954\) 0 0
\(955\) −338.656 −0.354614
\(956\) 0 0
\(957\) −380.988 + 203.647i −0.398107 + 0.212797i
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −945.000 −0.983351
\(962\) 0 0
\(963\) −1079.47 721.249i −1.12094 0.748960i
\(964\) 0 0
\(965\) 576.999i 0.597927i
\(966\) 0 0
\(967\) 692.000 0.715615 0.357808 0.933795i \(-0.383524\pi\)
0.357808 + 0.933795i \(0.383524\pi\)
\(968\) 0 0
\(969\) 112.000 + 209.533i 0.115583 + 0.216236i
\(970\) 0 0
\(971\) 449.720i 0.463151i 0.972817 + 0.231576i \(0.0743881\pi\)
−0.972817 + 0.231576i \(0.925612\pi\)
\(972\) 0 0
\(973\) −486.818 −0.500327
\(974\) 0 0
\(975\) −196.000 + 104.766i −0.201026 + 0.107453i
\(976\) 0 0
\(977\) 883.031i 0.903819i 0.892064 + 0.451909i \(0.149257\pi\)
−0.892064 + 0.451909i \(0.850743\pi\)
\(978\) 0 0
\(979\) 888.972 0.908041
\(980\) 0 0
\(981\) 264.575 395.980i 0.269699 0.403649i
\(982\) 0 0
\(983\) 329.266i 0.334960i 0.985875 + 0.167480i \(0.0535630\pi\)
−0.985875 + 0.167480i \(0.946437\pi\)
\(984\) 0 0
\(985\) 1376.00 1.39695
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 158.392i 0.160154i
\(990\) 0 0
\(991\) 340.000 0.343088 0.171544 0.985176i \(-0.445124\pi\)
0.171544 + 0.985176i \(0.445124\pi\)
\(992\) 0 0
\(993\) −1106.00 + 591.182i −1.11380 + 0.595349i
\(994\) 0 0
\(995\) 1063.49i 1.06883i
\(996\) 0 0
\(997\) −1555.70 −1.56038 −0.780191 0.625541i \(-0.784878\pi\)
−0.780191 + 0.625541i \(0.784878\pi\)
\(998\) 0 0
\(999\) 140.000 1421.83i 0.140140 1.42325i
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.e.l.257.3 4
3.2 odd 2 inner 768.3.e.l.257.4 4
4.3 odd 2 768.3.e.i.257.2 4
8.3 odd 2 768.3.e.i.257.3 4
8.5 even 2 inner 768.3.e.l.257.2 4
12.11 even 2 768.3.e.i.257.1 4
16.3 odd 4 24.3.h.c.5.1 4
16.5 even 4 96.3.h.c.17.1 4
16.11 odd 4 24.3.h.c.5.3 yes 4
16.13 even 4 96.3.h.c.17.4 4
24.5 odd 2 inner 768.3.e.l.257.1 4
24.11 even 2 768.3.e.i.257.4 4
48.5 odd 4 96.3.h.c.17.3 4
48.11 even 4 24.3.h.c.5.2 yes 4
48.29 odd 4 96.3.h.c.17.2 4
48.35 even 4 24.3.h.c.5.4 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
24.3.h.c.5.1 4 16.3 odd 4
24.3.h.c.5.2 yes 4 48.11 even 4
24.3.h.c.5.3 yes 4 16.11 odd 4
24.3.h.c.5.4 yes 4 48.35 even 4
96.3.h.c.17.1 4 16.5 even 4
96.3.h.c.17.2 4 48.29 odd 4
96.3.h.c.17.3 4 48.5 odd 4
96.3.h.c.17.4 4 16.13 even 4
768.3.e.i.257.1 4 12.11 even 2
768.3.e.i.257.2 4 4.3 odd 2
768.3.e.i.257.3 4 8.3 odd 2
768.3.e.i.257.4 4 24.11 even 2
768.3.e.l.257.1 4 24.5 odd 2 inner
768.3.e.l.257.2 4 8.5 even 2 inner
768.3.e.l.257.3 4 1.1 even 1 trivial
768.3.e.l.257.4 4 3.2 odd 2 inner