Properties

Label 224.2.f.a.223.6
Level $224$
Weight $2$
Character 224.223
Analytic conductor $1.789$
Analytic rank $0$
Dimension $8$
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] = [224,2,Mod(223,224)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(224, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("224.223");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 224 = 2^{5} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 224.f (of order \(2\), degree \(1\), 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: \(1.78864900528\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\zeta_{16})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{9} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 223.6
Root \(0.382683 - 0.923880i\) of defining polynomial
Character \(\chi\) \(=\) 224.223
Dual form 224.2.f.a.223.5

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.08239 q^{3} +2.61313i q^{5} +(2.61313 + 0.414214i) q^{7} -1.82843 q^{9} +O(q^{10})\) \(q+1.08239 q^{3} +2.61313i q^{5} +(2.61313 + 0.414214i) q^{7} -1.82843 q^{9} -2.00000i q^{11} +4.77791i q^{13} +2.82843i q^{15} -3.06147i q^{17} +4.14386 q^{19} +(2.82843 + 0.448342i) q^{21} -7.65685i q^{23} -1.82843 q^{25} -5.22625 q^{27} +3.65685 q^{29} -3.06147 q^{31} -2.16478i q^{33} +(-1.08239 + 6.82843i) q^{35} -7.65685 q^{37} +5.17157i q^{39} -9.55582i q^{41} +3.65685i q^{43} -4.77791i q^{45} -7.39104 q^{47} +(6.65685 + 2.16478i) q^{49} -3.31371i q^{51} -2.00000 q^{53} +5.22625 q^{55} +4.48528 q^{57} -8.47343 q^{59} -2.61313i q^{61} +(-4.77791 - 0.757359i) q^{63} -12.4853 q^{65} -15.6569i q^{67} -8.28772i q^{69} +8.82843i q^{71} +12.6173i q^{73} -1.97908 q^{75} +(0.828427 - 5.22625i) q^{77} -12.8284i q^{79} -0.171573 q^{81} +11.5349 q^{83} +8.00000 q^{85} +3.95815 q^{87} +2.16478i q^{89} +(-1.97908 + 12.4853i) q^{91} -3.31371 q^{93} +10.8284i q^{95} +13.5140i q^{97} +3.65685i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{9} + 8 q^{25} - 16 q^{29} - 16 q^{37} + 8 q^{49} - 16 q^{53} - 32 q^{57} - 32 q^{65} - 16 q^{77} - 24 q^{81} + 64 q^{85} + 64 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/224\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(129\) \(197\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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
\(3\) 1.08239 0.624919 0.312460 0.949931i \(-0.398847\pi\)
0.312460 + 0.949931i \(0.398847\pi\)
\(4\) 0 0
\(5\) 2.61313i 1.16863i 0.811529 + 0.584313i \(0.198636\pi\)
−0.811529 + 0.584313i \(0.801364\pi\)
\(6\) 0 0
\(7\) 2.61313 + 0.414214i 0.987669 + 0.156558i
\(8\) 0 0
\(9\) −1.82843 −0.609476
\(10\) 0 0
\(11\) 2.00000i 0.603023i −0.953463 0.301511i \(-0.902509\pi\)
0.953463 0.301511i \(-0.0974911\pi\)
\(12\) 0 0
\(13\) 4.77791i 1.32515i 0.748994 + 0.662577i \(0.230537\pi\)
−0.748994 + 0.662577i \(0.769463\pi\)
\(14\) 0 0
\(15\) 2.82843i 0.730297i
\(16\) 0 0
\(17\) 3.06147i 0.742515i −0.928530 0.371257i \(-0.878927\pi\)
0.928530 0.371257i \(-0.121073\pi\)
\(18\) 0 0
\(19\) 4.14386 0.950667 0.475333 0.879806i \(-0.342327\pi\)
0.475333 + 0.879806i \(0.342327\pi\)
\(20\) 0 0
\(21\) 2.82843 + 0.448342i 0.617213 + 0.0978361i
\(22\) 0 0
\(23\) 7.65685i 1.59656i −0.602284 0.798282i \(-0.705742\pi\)
0.602284 0.798282i \(-0.294258\pi\)
\(24\) 0 0
\(25\) −1.82843 −0.365685
\(26\) 0 0
\(27\) −5.22625 −1.00579
\(28\) 0 0
\(29\) 3.65685 0.679061 0.339530 0.940595i \(-0.389732\pi\)
0.339530 + 0.940595i \(0.389732\pi\)
\(30\) 0 0
\(31\) −3.06147 −0.549856 −0.274928 0.961465i \(-0.588654\pi\)
−0.274928 + 0.961465i \(0.588654\pi\)
\(32\) 0 0
\(33\) 2.16478i 0.376841i
\(34\) 0 0
\(35\) −1.08239 + 6.82843i −0.182958 + 1.15421i
\(36\) 0 0
\(37\) −7.65685 −1.25878 −0.629390 0.777090i \(-0.716695\pi\)
−0.629390 + 0.777090i \(0.716695\pi\)
\(38\) 0 0
\(39\) 5.17157i 0.828114i
\(40\) 0 0
\(41\) 9.55582i 1.49237i −0.665740 0.746184i \(-0.731884\pi\)
0.665740 0.746184i \(-0.268116\pi\)
\(42\) 0 0
\(43\) 3.65685i 0.557665i 0.960340 + 0.278833i \(0.0899474\pi\)
−0.960340 + 0.278833i \(0.910053\pi\)
\(44\) 0 0
\(45\) 4.77791i 0.712249i
\(46\) 0 0
\(47\) −7.39104 −1.07809 −0.539047 0.842276i \(-0.681215\pi\)
−0.539047 + 0.842276i \(0.681215\pi\)
\(48\) 0 0
\(49\) 6.65685 + 2.16478i 0.950979 + 0.309255i
\(50\) 0 0
\(51\) 3.31371i 0.464012i
\(52\) 0 0
\(53\) −2.00000 −0.274721 −0.137361 0.990521i \(-0.543862\pi\)
−0.137361 + 0.990521i \(0.543862\pi\)
\(54\) 0 0
\(55\) 5.22625 0.704708
\(56\) 0 0
\(57\) 4.48528 0.594090
\(58\) 0 0
\(59\) −8.47343 −1.10315 −0.551573 0.834126i \(-0.685972\pi\)
−0.551573 + 0.834126i \(0.685972\pi\)
\(60\) 0 0
\(61\) 2.61313i 0.334576i −0.985908 0.167288i \(-0.946499\pi\)
0.985908 0.167288i \(-0.0535011\pi\)
\(62\) 0 0
\(63\) −4.77791 0.757359i −0.601960 0.0954183i
\(64\) 0 0
\(65\) −12.4853 −1.54861
\(66\) 0 0
\(67\) 15.6569i 1.91279i −0.292078 0.956395i \(-0.594347\pi\)
0.292078 0.956395i \(-0.405653\pi\)
\(68\) 0 0
\(69\) 8.28772i 0.997724i
\(70\) 0 0
\(71\) 8.82843i 1.04774i 0.851798 + 0.523871i \(0.175513\pi\)
−0.851798 + 0.523871i \(0.824487\pi\)
\(72\) 0 0
\(73\) 12.6173i 1.47674i 0.674395 + 0.738371i \(0.264405\pi\)
−0.674395 + 0.738371i \(0.735595\pi\)
\(74\) 0 0
\(75\) −1.97908 −0.228524
\(76\) 0 0
\(77\) 0.828427 5.22625i 0.0944080 0.595587i
\(78\) 0 0
\(79\) 12.8284i 1.44331i −0.692252 0.721655i \(-0.743382\pi\)
0.692252 0.721655i \(-0.256618\pi\)
\(80\) 0 0
\(81\) −0.171573 −0.0190637
\(82\) 0 0
\(83\) 11.5349 1.26612 0.633060 0.774103i \(-0.281799\pi\)
0.633060 + 0.774103i \(0.281799\pi\)
\(84\) 0 0
\(85\) 8.00000 0.867722
\(86\) 0 0
\(87\) 3.95815 0.424358
\(88\) 0 0
\(89\) 2.16478i 0.229467i 0.993396 + 0.114733i \(0.0366014\pi\)
−0.993396 + 0.114733i \(0.963399\pi\)
\(90\) 0 0
\(91\) −1.97908 + 12.4853i −0.207463 + 1.30881i
\(92\) 0 0
\(93\) −3.31371 −0.343616
\(94\) 0 0
\(95\) 10.8284i 1.11097i
\(96\) 0 0
\(97\) 13.5140i 1.37214i 0.727537 + 0.686068i \(0.240665\pi\)
−0.727537 + 0.686068i \(0.759335\pi\)
\(98\) 0 0
\(99\) 3.65685i 0.367528i
\(100\) 0 0
\(101\) 5.67459i 0.564643i 0.959320 + 0.282322i \(0.0911045\pi\)
−0.959320 + 0.282322i \(0.908895\pi\)
\(102\) 0 0
\(103\) 16.9469 1.66982 0.834912 0.550384i \(-0.185519\pi\)
0.834912 + 0.550384i \(0.185519\pi\)
\(104\) 0 0
\(105\) −1.17157 + 7.39104i −0.114334 + 0.721291i
\(106\) 0 0
\(107\) 0.343146i 0.0331732i 0.999862 + 0.0165866i \(0.00527991\pi\)
−0.999862 + 0.0165866i \(0.994720\pi\)
\(108\) 0 0
\(109\) −5.31371 −0.508961 −0.254480 0.967078i \(-0.581904\pi\)
−0.254480 + 0.967078i \(0.581904\pi\)
\(110\) 0 0
\(111\) −8.28772 −0.786636
\(112\) 0 0
\(113\) 8.82843 0.830509 0.415254 0.909705i \(-0.363693\pi\)
0.415254 + 0.909705i \(0.363693\pi\)
\(114\) 0 0
\(115\) 20.0083 1.86579
\(116\) 0 0
\(117\) 8.73606i 0.807649i
\(118\) 0 0
\(119\) 1.26810 8.00000i 0.116247 0.733359i
\(120\) 0 0
\(121\) 7.00000 0.636364
\(122\) 0 0
\(123\) 10.3431i 0.932610i
\(124\) 0 0
\(125\) 8.28772i 0.741276i
\(126\) 0 0
\(127\) 3.65685i 0.324493i 0.986750 + 0.162247i \(0.0518740\pi\)
−0.986750 + 0.162247i \(0.948126\pi\)
\(128\) 0 0
\(129\) 3.95815i 0.348496i
\(130\) 0 0
\(131\) −13.6997 −1.19695 −0.598473 0.801143i \(-0.704226\pi\)
−0.598473 + 0.801143i \(0.704226\pi\)
\(132\) 0 0
\(133\) 10.8284 + 1.71644i 0.938944 + 0.148834i
\(134\) 0 0
\(135\) 13.6569i 1.17539i
\(136\) 0 0
\(137\) 2.00000 0.170872 0.0854358 0.996344i \(-0.472772\pi\)
0.0854358 + 0.996344i \(0.472772\pi\)
\(138\) 0 0
\(139\) 1.97908 0.167863 0.0839315 0.996472i \(-0.473252\pi\)
0.0839315 + 0.996472i \(0.473252\pi\)
\(140\) 0 0
\(141\) −8.00000 −0.673722
\(142\) 0 0
\(143\) 9.55582 0.799098
\(144\) 0 0
\(145\) 9.55582i 0.793568i
\(146\) 0 0
\(147\) 7.20533 + 2.34315i 0.594285 + 0.193259i
\(148\) 0 0
\(149\) −10.0000 −0.819232 −0.409616 0.912258i \(-0.634337\pi\)
−0.409616 + 0.912258i \(0.634337\pi\)
\(150\) 0 0
\(151\) 8.34315i 0.678956i 0.940614 + 0.339478i \(0.110250\pi\)
−0.940614 + 0.339478i \(0.889750\pi\)
\(152\) 0 0
\(153\) 5.59767i 0.452545i
\(154\) 0 0
\(155\) 8.00000i 0.642575i
\(156\) 0 0
\(157\) 5.67459i 0.452882i −0.974025 0.226441i \(-0.927291\pi\)
0.974025 0.226441i \(-0.0727090\pi\)
\(158\) 0 0
\(159\) −2.16478 −0.171679
\(160\) 0 0
\(161\) 3.17157 20.0083i 0.249955 1.57688i
\(162\) 0 0
\(163\) 17.3137i 1.35611i 0.735009 + 0.678057i \(0.237178\pi\)
−0.735009 + 0.678057i \(0.762822\pi\)
\(164\) 0 0
\(165\) 5.65685 0.440386
\(166\) 0 0
\(167\) −6.49435 −0.502548 −0.251274 0.967916i \(-0.580850\pi\)
−0.251274 + 0.967916i \(0.580850\pi\)
\(168\) 0 0
\(169\) −9.82843 −0.756033
\(170\) 0 0
\(171\) −7.57675 −0.579408
\(172\) 0 0
\(173\) 10.9008i 0.828776i 0.910100 + 0.414388i \(0.136004\pi\)
−0.910100 + 0.414388i \(0.863996\pi\)
\(174\) 0 0
\(175\) −4.77791 0.757359i −0.361176 0.0572510i
\(176\) 0 0
\(177\) −9.17157 −0.689378
\(178\) 0 0
\(179\) 2.00000i 0.149487i −0.997203 0.0747435i \(-0.976186\pi\)
0.997203 0.0747435i \(-0.0238138\pi\)
\(180\) 0 0
\(181\) 1.71644i 0.127582i −0.997963 0.0637911i \(-0.979681\pi\)
0.997963 0.0637911i \(-0.0203191\pi\)
\(182\) 0 0
\(183\) 2.82843i 0.209083i
\(184\) 0 0
\(185\) 20.0083i 1.47104i
\(186\) 0 0
\(187\) −6.12293 −0.447753
\(188\) 0 0
\(189\) −13.6569 2.16478i −0.993390 0.157465i
\(190\) 0 0
\(191\) 4.82843i 0.349373i −0.984624 0.174686i \(-0.944109\pi\)
0.984624 0.174686i \(-0.0558911\pi\)
\(192\) 0 0
\(193\) −10.4853 −0.754747 −0.377374 0.926061i \(-0.623173\pi\)
−0.377374 + 0.926061i \(0.623173\pi\)
\(194\) 0 0
\(195\) −13.5140 −0.967756
\(196\) 0 0
\(197\) 3.65685 0.260540 0.130270 0.991479i \(-0.458416\pi\)
0.130270 + 0.991479i \(0.458416\pi\)
\(198\) 0 0
\(199\) −12.6173 −0.894416 −0.447208 0.894430i \(-0.647582\pi\)
−0.447208 + 0.894430i \(0.647582\pi\)
\(200\) 0 0
\(201\) 16.9469i 1.19534i
\(202\) 0 0
\(203\) 9.55582 + 1.51472i 0.670687 + 0.106312i
\(204\) 0 0
\(205\) 24.9706 1.74402
\(206\) 0 0
\(207\) 14.0000i 0.973067i
\(208\) 0 0
\(209\) 8.28772i 0.573274i
\(210\) 0 0
\(211\) 10.9706i 0.755245i −0.925960 0.377622i \(-0.876742\pi\)
0.925960 0.377622i \(-0.123258\pi\)
\(212\) 0 0
\(213\) 9.55582i 0.654754i
\(214\) 0 0
\(215\) −9.55582 −0.651702
\(216\) 0 0
\(217\) −8.00000 1.26810i −0.543075 0.0860843i
\(218\) 0 0
\(219\) 13.6569i 0.922845i
\(220\) 0 0
\(221\) 14.6274 0.983947
\(222\) 0 0
\(223\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(224\) 0 0
\(225\) 3.34315 0.222876
\(226\) 0 0
\(227\) −18.5545 −1.23151 −0.615753 0.787939i \(-0.711148\pi\)
−0.615753 + 0.787939i \(0.711148\pi\)
\(228\) 0 0
\(229\) 2.98454i 0.197224i −0.995126 0.0986121i \(-0.968560\pi\)
0.995126 0.0986121i \(-0.0314403\pi\)
\(230\) 0 0
\(231\) 0.896683 5.65685i 0.0589974 0.372194i
\(232\) 0 0
\(233\) 18.0000 1.17922 0.589610 0.807688i \(-0.299282\pi\)
0.589610 + 0.807688i \(0.299282\pi\)
\(234\) 0 0
\(235\) 19.3137i 1.25989i
\(236\) 0 0
\(237\) 13.8854i 0.901953i
\(238\) 0 0
\(239\) 8.34315i 0.539673i 0.962906 + 0.269837i \(0.0869697\pi\)
−0.962906 + 0.269837i \(0.913030\pi\)
\(240\) 0 0
\(241\) 23.9665i 1.54382i −0.635734 0.771908i \(-0.719302\pi\)
0.635734 0.771908i \(-0.280698\pi\)
\(242\) 0 0
\(243\) 15.4930 0.993879
\(244\) 0 0
\(245\) −5.65685 + 17.3952i −0.361403 + 1.11134i
\(246\) 0 0
\(247\) 19.7990i 1.25978i
\(248\) 0 0
\(249\) 12.4853 0.791223
\(250\) 0 0
\(251\) −1.08239 −0.0683200 −0.0341600 0.999416i \(-0.510876\pi\)
−0.0341600 + 0.999416i \(0.510876\pi\)
\(252\) 0 0
\(253\) −15.3137 −0.962765
\(254\) 0 0
\(255\) 8.65914 0.542256
\(256\) 0 0
\(257\) 6.12293i 0.381938i −0.981596 0.190969i \(-0.938837\pi\)
0.981596 0.190969i \(-0.0611630\pi\)
\(258\) 0 0
\(259\) −20.0083 3.17157i −1.24326 0.197072i
\(260\) 0 0
\(261\) −6.68629 −0.413871
\(262\) 0 0
\(263\) 4.14214i 0.255415i 0.991812 + 0.127708i \(0.0407619\pi\)
−0.991812 + 0.127708i \(0.959238\pi\)
\(264\) 0 0
\(265\) 5.22625i 0.321046i
\(266\) 0 0
\(267\) 2.34315i 0.143398i
\(268\) 0 0
\(269\) 13.4370i 0.819271i 0.912249 + 0.409636i \(0.134344\pi\)
−0.912249 + 0.409636i \(0.865656\pi\)
\(270\) 0 0
\(271\) 1.79337 0.108939 0.0544696 0.998515i \(-0.482653\pi\)
0.0544696 + 0.998515i \(0.482653\pi\)
\(272\) 0 0
\(273\) −2.14214 + 13.5140i −0.129648 + 0.817903i
\(274\) 0 0
\(275\) 3.65685i 0.220517i
\(276\) 0 0
\(277\) −26.9706 −1.62050 −0.810252 0.586082i \(-0.800670\pi\)
−0.810252 + 0.586082i \(0.800670\pi\)
\(278\) 0 0
\(279\) 5.59767 0.335124
\(280\) 0 0
\(281\) −28.6274 −1.70777 −0.853884 0.520463i \(-0.825759\pi\)
−0.853884 + 0.520463i \(0.825759\pi\)
\(282\) 0 0
\(283\) −1.08239 −0.0643415 −0.0321708 0.999482i \(-0.510242\pi\)
−0.0321708 + 0.999482i \(0.510242\pi\)
\(284\) 0 0
\(285\) 11.7206i 0.694269i
\(286\) 0 0
\(287\) 3.95815 24.9706i 0.233642 1.47397i
\(288\) 0 0
\(289\) 7.62742 0.448672
\(290\) 0 0
\(291\) 14.6274i 0.857474i
\(292\) 0 0
\(293\) 5.67459i 0.331513i 0.986167 + 0.165757i \(0.0530066\pi\)
−0.986167 + 0.165757i \(0.946993\pi\)
\(294\) 0 0
\(295\) 22.1421i 1.28916i
\(296\) 0 0
\(297\) 10.4525i 0.606516i
\(298\) 0 0
\(299\) 36.5838 2.11569
\(300\) 0 0
\(301\) −1.51472 + 9.55582i −0.0873069 + 0.550788i
\(302\) 0 0
\(303\) 6.14214i 0.352856i
\(304\) 0 0
\(305\) 6.82843 0.390995
\(306\) 0 0
\(307\) 11.5349 0.658331 0.329166 0.944272i \(-0.393233\pi\)
0.329166 + 0.944272i \(0.393233\pi\)
\(308\) 0 0
\(309\) 18.3431 1.04351
\(310\) 0 0
\(311\) −20.0083 −1.13457 −0.567284 0.823522i \(-0.692006\pi\)
−0.567284 + 0.823522i \(0.692006\pi\)
\(312\) 0 0
\(313\) 15.6788i 0.886216i 0.896468 + 0.443108i \(0.146124\pi\)
−0.896468 + 0.443108i \(0.853876\pi\)
\(314\) 0 0
\(315\) 1.97908 12.4853i 0.111508 0.703466i
\(316\) 0 0
\(317\) 27.6569 1.55336 0.776682 0.629893i \(-0.216901\pi\)
0.776682 + 0.629893i \(0.216901\pi\)
\(318\) 0 0
\(319\) 7.31371i 0.409489i
\(320\) 0 0
\(321\) 0.371418i 0.0207305i
\(322\) 0 0
\(323\) 12.6863i 0.705884i
\(324\) 0 0
\(325\) 8.73606i 0.484589i
\(326\) 0 0
\(327\) −5.75152 −0.318060
\(328\) 0 0
\(329\) −19.3137 3.06147i −1.06480 0.168784i
\(330\) 0 0
\(331\) 17.3137i 0.951647i 0.879541 + 0.475824i \(0.157850\pi\)
−0.879541 + 0.475824i \(0.842150\pi\)
\(332\) 0 0
\(333\) 14.0000 0.767195
\(334\) 0 0
\(335\) 40.9133 2.23533
\(336\) 0 0
\(337\) 8.82843 0.480915 0.240458 0.970660i \(-0.422703\pi\)
0.240458 + 0.970660i \(0.422703\pi\)
\(338\) 0 0
\(339\) 9.55582 0.519001
\(340\) 0 0
\(341\) 6.12293i 0.331576i
\(342\) 0 0
\(343\) 16.4985 + 8.41421i 0.890836 + 0.454325i
\(344\) 0 0
\(345\) 21.6569 1.16597
\(346\) 0 0
\(347\) 19.6569i 1.05524i 0.849482 + 0.527618i \(0.176915\pi\)
−0.849482 + 0.527618i \(0.823085\pi\)
\(348\) 0 0
\(349\) 8.73606i 0.467631i −0.972281 0.233815i \(-0.924879\pi\)
0.972281 0.233815i \(-0.0751211\pi\)
\(350\) 0 0
\(351\) 24.9706i 1.33283i
\(352\) 0 0
\(353\) 29.5641i 1.57354i 0.617246 + 0.786770i \(0.288248\pi\)
−0.617246 + 0.786770i \(0.711752\pi\)
\(354\) 0 0
\(355\) −23.0698 −1.22442
\(356\) 0 0
\(357\) 1.37258 8.65914i 0.0726448 0.458290i
\(358\) 0 0
\(359\) 14.9706i 0.790116i 0.918656 + 0.395058i \(0.129276\pi\)
−0.918656 + 0.395058i \(0.870724\pi\)
\(360\) 0 0
\(361\) −1.82843 −0.0962330
\(362\) 0 0
\(363\) 7.57675 0.397676
\(364\) 0 0
\(365\) −32.9706 −1.72576
\(366\) 0 0
\(367\) −10.4525 −0.545616 −0.272808 0.962068i \(-0.587952\pi\)
−0.272808 + 0.962068i \(0.587952\pi\)
\(368\) 0 0
\(369\) 17.4721i 0.909562i
\(370\) 0 0
\(371\) −5.22625 0.828427i −0.271333 0.0430098i
\(372\) 0 0
\(373\) 6.97056 0.360922 0.180461 0.983582i \(-0.442241\pi\)
0.180461 + 0.983582i \(0.442241\pi\)
\(374\) 0 0
\(375\) 8.97056i 0.463238i
\(376\) 0 0
\(377\) 17.4721i 0.899860i
\(378\) 0 0
\(379\) 5.31371i 0.272947i −0.990644 0.136473i \(-0.956423\pi\)
0.990644 0.136473i \(-0.0435768\pi\)
\(380\) 0 0
\(381\) 3.95815i 0.202782i
\(382\) 0 0
\(383\) 11.7206 0.598895 0.299447 0.954113i \(-0.403198\pi\)
0.299447 + 0.954113i \(0.403198\pi\)
\(384\) 0 0
\(385\) 13.6569 + 2.16478i 0.696018 + 0.110328i
\(386\) 0 0
\(387\) 6.68629i 0.339883i
\(388\) 0 0
\(389\) 19.6569 0.996642 0.498321 0.866993i \(-0.333950\pi\)
0.498321 + 0.866993i \(0.333950\pi\)
\(390\) 0 0
\(391\) −23.4412 −1.18547
\(392\) 0 0
\(393\) −14.8284 −0.747995
\(394\) 0 0
\(395\) 33.5223 1.68669
\(396\) 0 0
\(397\) 33.9706i 1.70494i −0.522778 0.852469i \(-0.675104\pi\)
0.522778 0.852469i \(-0.324896\pi\)
\(398\) 0 0
\(399\) 11.7206 + 1.85786i 0.586764 + 0.0930096i
\(400\) 0 0
\(401\) −15.1716 −0.757632 −0.378816 0.925472i \(-0.623669\pi\)
−0.378816 + 0.925472i \(0.623669\pi\)
\(402\) 0 0
\(403\) 14.6274i 0.728644i
\(404\) 0 0
\(405\) 0.448342i 0.0222783i
\(406\) 0 0
\(407\) 15.3137i 0.759072i
\(408\) 0 0
\(409\) 0.896683i 0.0443381i 0.999754 + 0.0221691i \(0.00705721\pi\)
−0.999754 + 0.0221691i \(0.992943\pi\)
\(410\) 0 0
\(411\) 2.16478 0.106781
\(412\) 0 0
\(413\) −22.1421 3.50981i −1.08954 0.172706i
\(414\) 0 0
\(415\) 30.1421i 1.47962i
\(416\) 0 0
\(417\) 2.14214 0.104901
\(418\) 0 0
\(419\) −34.6047 −1.69055 −0.845275 0.534332i \(-0.820563\pi\)
−0.845275 + 0.534332i \(0.820563\pi\)
\(420\) 0 0
\(421\) −21.3137 −1.03877 −0.519383 0.854541i \(-0.673838\pi\)
−0.519383 + 0.854541i \(0.673838\pi\)
\(422\) 0 0
\(423\) 13.5140 0.657072
\(424\) 0 0
\(425\) 5.59767i 0.271527i
\(426\) 0 0
\(427\) 1.08239 6.82843i 0.0523806 0.330451i
\(428\) 0 0
\(429\) 10.3431 0.499372
\(430\) 0 0
\(431\) 12.3431i 0.594548i −0.954792 0.297274i \(-0.903922\pi\)
0.954792 0.297274i \(-0.0960775\pi\)
\(432\) 0 0
\(433\) 5.59767i 0.269007i −0.990913 0.134503i \(-0.957056\pi\)
0.990913 0.134503i \(-0.0429439\pi\)
\(434\) 0 0
\(435\) 10.3431i 0.495916i
\(436\) 0 0
\(437\) 31.7289i 1.51780i
\(438\) 0 0
\(439\) −22.5445 −1.07599 −0.537996 0.842948i \(-0.680818\pi\)
−0.537996 + 0.842948i \(0.680818\pi\)
\(440\) 0 0
\(441\) −12.1716 3.95815i −0.579599 0.188483i
\(442\) 0 0
\(443\) 10.0000i 0.475114i −0.971374 0.237557i \(-0.923653\pi\)
0.971374 0.237557i \(-0.0763467\pi\)
\(444\) 0 0
\(445\) −5.65685 −0.268161
\(446\) 0 0
\(447\) −10.8239 −0.511954
\(448\) 0 0
\(449\) 21.3137 1.00586 0.502928 0.864328i \(-0.332256\pi\)
0.502928 + 0.864328i \(0.332256\pi\)
\(450\) 0 0
\(451\) −19.1116 −0.899932
\(452\) 0 0
\(453\) 9.03056i 0.424293i
\(454\) 0 0
\(455\) −32.6256 5.17157i −1.52951 0.242447i
\(456\) 0 0
\(457\) 37.1127 1.73606 0.868029 0.496513i \(-0.165386\pi\)
0.868029 + 0.496513i \(0.165386\pi\)
\(458\) 0 0
\(459\) 16.0000i 0.746816i
\(460\) 0 0
\(461\) 10.3756i 0.483239i 0.970371 + 0.241619i \(0.0776786\pi\)
−0.970371 + 0.241619i \(0.922321\pi\)
\(462\) 0 0
\(463\) 16.1421i 0.750189i −0.926987 0.375094i \(-0.877610\pi\)
0.926987 0.375094i \(-0.122390\pi\)
\(464\) 0 0
\(465\) 8.65914i 0.401558i
\(466\) 0 0
\(467\) −8.10201 −0.374916 −0.187458 0.982273i \(-0.560025\pi\)
−0.187458 + 0.982273i \(0.560025\pi\)
\(468\) 0 0
\(469\) 6.48528 40.9133i 0.299462 1.88920i
\(470\) 0 0
\(471\) 6.14214i 0.283015i
\(472\) 0 0
\(473\) 7.31371 0.336285
\(474\) 0 0
\(475\) −7.57675 −0.347645
\(476\) 0 0
\(477\) 3.65685 0.167436
\(478\) 0 0
\(479\) 30.0894 1.37482 0.687410 0.726269i \(-0.258748\pi\)
0.687410 + 0.726269i \(0.258748\pi\)
\(480\) 0 0
\(481\) 36.5838i 1.66808i
\(482\) 0 0
\(483\) 3.43289 21.6569i 0.156202 0.985421i
\(484\) 0 0
\(485\) −35.3137 −1.60351
\(486\) 0 0
\(487\) 30.2843i 1.37231i −0.727455 0.686156i \(-0.759297\pi\)
0.727455 0.686156i \(-0.240703\pi\)
\(488\) 0 0
\(489\) 18.7402i 0.847462i
\(490\) 0 0
\(491\) 30.2843i 1.36671i −0.730086 0.683355i \(-0.760520\pi\)
0.730086 0.683355i \(-0.239480\pi\)
\(492\) 0 0
\(493\) 11.1953i 0.504213i
\(494\) 0 0
\(495\) −9.55582 −0.429502
\(496\) 0 0
\(497\) −3.65685 + 23.0698i −0.164032 + 1.03482i
\(498\) 0 0
\(499\) 17.3137i 0.775068i 0.921855 + 0.387534i \(0.126673\pi\)
−0.921855 + 0.387534i \(0.873327\pi\)
\(500\) 0 0
\(501\) −7.02944 −0.314052
\(502\) 0 0
\(503\) −17.4721 −0.779043 −0.389522 0.921017i \(-0.627360\pi\)
−0.389522 + 0.921017i \(0.627360\pi\)
\(504\) 0 0
\(505\) −14.8284 −0.659856
\(506\) 0 0
\(507\) −10.6382 −0.472460
\(508\) 0 0
\(509\) 26.0543i 1.15484i −0.816448 0.577419i \(-0.804060\pi\)
0.816448 0.577419i \(-0.195940\pi\)
\(510\) 0 0
\(511\) −5.22625 + 32.9706i −0.231196 + 1.45853i
\(512\) 0 0
\(513\) −21.6569 −0.956173
\(514\) 0 0
\(515\) 44.2843i 1.95140i
\(516\) 0 0
\(517\) 14.7821i 0.650115i
\(518\) 0 0
\(519\) 11.7990i 0.517918i
\(520\) 0 0
\(521\) 0.896683i 0.0392844i −0.999807 0.0196422i \(-0.993747\pi\)
0.999807 0.0196422i \(-0.00625271\pi\)
\(522\) 0 0
\(523\) −18.9259 −0.827573 −0.413787 0.910374i \(-0.635794\pi\)
−0.413787 + 0.910374i \(0.635794\pi\)
\(524\) 0 0
\(525\) −5.17157 0.819760i −0.225706 0.0357773i
\(526\) 0 0
\(527\) 9.37258i 0.408276i
\(528\) 0 0
\(529\) −35.6274 −1.54902
\(530\) 0 0
\(531\) 15.4930 0.672341
\(532\) 0 0
\(533\) 45.6569 1.97762
\(534\) 0 0
\(535\) −0.896683 −0.0387670
\(536\) 0 0
\(537\) 2.16478i 0.0934173i
\(538\) 0 0
\(539\) 4.32957 13.3137i 0.186488 0.573462i
\(540\) 0 0
\(541\) 25.3137 1.08832 0.544161 0.838981i \(-0.316848\pi\)
0.544161 + 0.838981i \(0.316848\pi\)
\(542\) 0 0
\(543\) 1.85786i 0.0797286i
\(544\) 0 0
\(545\) 13.8854i 0.594785i
\(546\) 0 0
\(547\) 33.3137i 1.42439i 0.701981 + 0.712196i \(0.252299\pi\)
−0.701981 + 0.712196i \(0.747701\pi\)
\(548\) 0 0
\(549\) 4.77791i 0.203916i
\(550\) 0 0
\(551\) 15.1535 0.645560
\(552\) 0 0
\(553\) 5.31371 33.5223i 0.225962 1.42551i
\(554\) 0 0
\(555\) 21.6569i 0.919282i
\(556\) 0 0
\(557\) −5.31371 −0.225149 −0.112575 0.993643i \(-0.535910\pi\)
−0.112575 + 0.993643i \(0.535910\pi\)
\(558\) 0 0
\(559\) −17.4721 −0.738992
\(560\) 0 0
\(561\) −6.62742 −0.279810
\(562\) 0 0
\(563\) −4.51528 −0.190296 −0.0951481 0.995463i \(-0.530332\pi\)
−0.0951481 + 0.995463i \(0.530332\pi\)
\(564\) 0 0
\(565\) 23.0698i 0.970553i
\(566\) 0 0
\(567\) −0.448342 0.0710678i −0.0188286 0.00298457i
\(568\) 0 0
\(569\) 7.85786 0.329419 0.164709 0.986342i \(-0.447331\pi\)
0.164709 + 0.986342i \(0.447331\pi\)
\(570\) 0 0
\(571\) 8.34315i 0.349150i 0.984644 + 0.174575i \(0.0558551\pi\)
−0.984644 + 0.174575i \(0.944145\pi\)
\(572\) 0 0
\(573\) 5.22625i 0.218330i
\(574\) 0 0
\(575\) 14.0000i 0.583840i
\(576\) 0 0
\(577\) 35.6871i 1.48567i 0.669473 + 0.742836i \(0.266520\pi\)
−0.669473 + 0.742836i \(0.733480\pi\)
\(578\) 0 0
\(579\) −11.3492 −0.471656
\(580\) 0 0
\(581\) 30.1421 + 4.77791i 1.25051 + 0.198221i
\(582\) 0 0
\(583\) 4.00000i 0.165663i
\(584\) 0 0
\(585\) 22.8284 0.943839
\(586\) 0 0
\(587\) 29.0070 1.19725 0.598624 0.801030i \(-0.295714\pi\)
0.598624 + 0.801030i \(0.295714\pi\)
\(588\) 0 0
\(589\) −12.6863 −0.522730
\(590\) 0 0
\(591\) 3.95815 0.162817
\(592\) 0 0
\(593\) 1.79337i 0.0736447i 0.999322 + 0.0368224i \(0.0117236\pi\)
−0.999322 + 0.0368224i \(0.988276\pi\)
\(594\) 0 0
\(595\) 20.9050 + 3.31371i 0.857022 + 0.135849i
\(596\) 0 0
\(597\) −13.6569 −0.558938
\(598\) 0 0
\(599\) 31.1716i 1.27364i −0.771014 0.636818i \(-0.780250\pi\)
0.771014 0.636818i \(-0.219750\pi\)
\(600\) 0 0
\(601\) 42.1814i 1.72062i −0.509774 0.860308i \(-0.670271\pi\)
0.509774 0.860308i \(-0.329729\pi\)
\(602\) 0 0
\(603\) 28.6274i 1.16580i
\(604\) 0 0
\(605\) 18.2919i 0.743671i
\(606\) 0 0
\(607\) 17.3183 0.702927 0.351464 0.936202i \(-0.385684\pi\)
0.351464 + 0.936202i \(0.385684\pi\)
\(608\) 0 0
\(609\) 10.3431 + 1.63952i 0.419125 + 0.0664367i
\(610\) 0 0
\(611\) 35.3137i 1.42864i
\(612\) 0 0
\(613\) −5.31371 −0.214619 −0.107309 0.994226i \(-0.534224\pi\)
−0.107309 + 0.994226i \(0.534224\pi\)
\(614\) 0 0
\(615\) 27.0279 1.08987
\(616\) 0 0
\(617\) 13.1127 0.527897 0.263949 0.964537i \(-0.414975\pi\)
0.263949 + 0.964537i \(0.414975\pi\)
\(618\) 0 0
\(619\) −2.87576 −0.115586 −0.0577932 0.998329i \(-0.518406\pi\)
−0.0577932 + 0.998329i \(0.518406\pi\)
\(620\) 0 0
\(621\) 40.0166i 1.60581i
\(622\) 0 0
\(623\) −0.896683 + 5.65685i −0.0359248 + 0.226637i
\(624\) 0 0
\(625\) −30.7990 −1.23196
\(626\) 0 0
\(627\) 8.97056i 0.358250i
\(628\) 0 0
\(629\) 23.4412i 0.934662i
\(630\) 0 0
\(631\) 26.4853i 1.05436i −0.849753 0.527181i \(-0.823249\pi\)
0.849753 0.527181i \(-0.176751\pi\)
\(632\) 0 0
\(633\) 11.8745i 0.471967i
\(634\) 0 0
\(635\) −9.55582 −0.379211
\(636\) 0 0
\(637\) −10.3431 + 31.8059i −0.409810 + 1.26019i
\(638\) 0 0
\(639\) 16.1421i 0.638573i
\(640\) 0 0
\(641\) 28.1421 1.11155 0.555774 0.831334i \(-0.312422\pi\)
0.555774 + 0.831334i \(0.312422\pi\)
\(642\) 0 0
\(643\) 17.1326 0.675642 0.337821 0.941210i \(-0.390310\pi\)
0.337821 + 0.941210i \(0.390310\pi\)
\(644\) 0 0
\(645\) −10.3431 −0.407261
\(646\) 0 0
\(647\) 4.70099 0.184815 0.0924074 0.995721i \(-0.470544\pi\)
0.0924074 + 0.995721i \(0.470544\pi\)
\(648\) 0 0
\(649\) 16.9469i 0.665222i
\(650\) 0 0
\(651\) −8.65914 1.37258i −0.339378 0.0537958i
\(652\) 0 0
\(653\) 2.68629 0.105123 0.0525614 0.998618i \(-0.483261\pi\)
0.0525614 + 0.998618i \(0.483261\pi\)
\(654\) 0 0
\(655\) 35.7990i 1.39878i
\(656\) 0 0
\(657\) 23.0698i 0.900038i
\(658\) 0 0
\(659\) 17.0294i 0.663373i −0.943390 0.331686i \(-0.892382\pi\)
0.943390 0.331686i \(-0.107618\pi\)
\(660\) 0 0
\(661\) 30.9092i 1.20223i 0.799164 + 0.601114i \(0.205276\pi\)
−0.799164 + 0.601114i \(0.794724\pi\)
\(662\) 0 0
\(663\) 15.8326 0.614887
\(664\) 0 0
\(665\) −4.48528 + 28.2960i −0.173932 + 1.09727i
\(666\) 0 0
\(667\) 28.0000i 1.08416i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −5.22625 −0.201757
\(672\) 0 0
\(673\) 6.68629 0.257738 0.128869 0.991662i \(-0.458865\pi\)
0.128869 + 0.991662i \(0.458865\pi\)
\(674\) 0 0
\(675\) 9.55582 0.367804
\(676\) 0 0
\(677\) 24.4148i 0.938338i −0.883109 0.469169i \(-0.844554\pi\)
0.883109 0.469169i \(-0.155446\pi\)
\(678\) 0 0
\(679\) −5.59767 + 35.3137i −0.214819 + 1.35522i
\(680\) 0 0
\(681\) −20.0833 −0.769592
\(682\) 0 0
\(683\) 43.6569i 1.67048i 0.549883 + 0.835242i \(0.314672\pi\)
−0.549883 + 0.835242i \(0.685328\pi\)
\(684\) 0 0
\(685\) 5.22625i 0.199685i
\(686\) 0 0
\(687\) 3.23045i 0.123249i
\(688\) 0 0
\(689\) 9.55582i 0.364048i
\(690\) 0 0
\(691\) 32.4399 1.23407 0.617036 0.786935i \(-0.288333\pi\)
0.617036 + 0.786935i \(0.288333\pi\)
\(692\) 0 0
\(693\) −1.51472 + 9.55582i −0.0575394 + 0.362996i
\(694\) 0 0
\(695\) 5.17157i 0.196169i
\(696\) 0 0
\(697\) −29.2548 −1.10811
\(698\) 0 0
\(699\) 19.4831 0.736917
\(700\) 0 0
\(701\) 9.31371 0.351774 0.175887 0.984410i \(-0.443721\pi\)
0.175887 + 0.984410i \(0.443721\pi\)
\(702\) 0 0
\(703\) −31.7289 −1.19668
\(704\) 0 0
\(705\) 20.9050i 0.787328i
\(706\) 0 0
\(707\) −2.35049 + 14.8284i −0.0883994 + 0.557680i
\(708\) 0 0
\(709\) −18.0000 −0.676004 −0.338002 0.941145i \(-0.609751\pi\)
−0.338002 + 0.941145i \(0.609751\pi\)
\(710\) 0 0
\(711\) 23.4558i 0.879663i
\(712\) 0 0
\(713\) 23.4412i 0.877880i
\(714\) 0 0
\(715\) 24.9706i 0.933846i
\(716\) 0 0
\(717\) 9.03056i 0.337252i
\(718\) 0 0
\(719\) 34.4190 1.28361 0.641806 0.766867i \(-0.278186\pi\)
0.641806 + 0.766867i \(0.278186\pi\)
\(720\) 0 0
\(721\) 44.2843 + 7.01962i 1.64923 + 0.261424i
\(722\) 0 0
\(723\) 25.9411i 0.964761i
\(724\) 0 0
\(725\) −6.68629 −0.248323
\(726\) 0 0
\(727\) 42.1814 1.56442 0.782211 0.623013i \(-0.214092\pi\)
0.782211 + 0.623013i \(0.214092\pi\)
\(728\) 0 0
\(729\) 17.2843 0.640158
\(730\) 0 0
\(731\) 11.1953 0.414075
\(732\) 0 0
\(733\) 24.4148i 0.901782i 0.892579 + 0.450891i \(0.148894\pi\)
−0.892579 + 0.450891i \(0.851106\pi\)
\(734\) 0 0
\(735\) −6.12293 + 18.8284i −0.225848 + 0.694497i
\(736\) 0 0
\(737\) −31.3137 −1.15346
\(738\) 0 0
\(739\) 16.6274i 0.611649i −0.952088 0.305825i \(-0.901068\pi\)
0.952088 0.305825i \(-0.0989322\pi\)
\(740\) 0 0
\(741\) 21.4303i 0.787261i
\(742\) 0 0
\(743\) 24.3431i 0.893063i 0.894768 + 0.446532i \(0.147341\pi\)
−0.894768 + 0.446532i \(0.852659\pi\)
\(744\) 0 0
\(745\) 26.1313i 0.957375i
\(746\) 0 0
\(747\) −21.0907 −0.771669
\(748\) 0 0
\(749\) −0.142136 + 0.896683i −0.00519352 + 0.0327641i
\(750\) 0 0
\(751\) 14.9706i 0.546284i 0.961974 + 0.273142i \(0.0880628\pi\)
−0.961974 + 0.273142i \(0.911937\pi\)
\(752\) 0 0
\(753\) −1.17157 −0.0426945
\(754\) 0 0
\(755\) −21.8017 −0.793445
\(756\) 0 0
\(757\) 49.3137 1.79234 0.896169 0.443714i \(-0.146339\pi\)
0.896169 + 0.443714i \(0.146339\pi\)
\(758\) 0 0
\(759\) −16.5754 −0.601650
\(760\) 0 0
\(761\) 7.76245i 0.281389i −0.990053 0.140694i \(-0.955067\pi\)
0.990053 0.140694i \(-0.0449335\pi\)
\(762\) 0 0
\(763\) −13.8854 2.20101i −0.502685 0.0796819i
\(764\) 0 0
\(765\) −14.6274 −0.528855
\(766\) 0 0
\(767\) 40.4853i 1.46184i
\(768\) 0 0
\(769\) 13.5140i 0.487326i −0.969860 0.243663i \(-0.921651\pi\)
0.969860 0.243663i \(-0.0783491\pi\)
\(770\) 0 0
\(771\) 6.62742i 0.238681i
\(772\) 0 0
\(773\) 26.9510i 0.969361i −0.874691 0.484680i \(-0.838936\pi\)
0.874691 0.484680i \(-0.161064\pi\)
\(774\) 0 0
\(775\) 5.59767 0.201074
\(776\) 0 0
\(777\) −21.6569 3.43289i −0.776935 0.123154i
\(778\) 0 0
\(779\) 39.5980i 1.41874i
\(780\) 0 0
\(781\) 17.6569 0.631812
\(782\) 0 0
\(783\) −19.1116 −0.682994
\(784\) 0 0
\(785\) 14.8284 0.529249
\(786\) 0 0
\(787\) −4.51528 −0.160952 −0.0804761 0.996757i \(-0.525644\pi\)
−0.0804761 + 0.996757i \(0.525644\pi\)
\(788\) 0 0
\(789\) 4.48342i 0.159614i
\(790\) 0 0
\(791\) 23.0698 + 3.65685i 0.820267 + 0.130023i
\(792\) 0 0
\(793\) 12.4853 0.443365
\(794\) 0 0
\(795\) 5.65685i 0.200628i
\(796\) 0 0
\(797\) 41.3617i 1.46511i −0.680710 0.732553i \(-0.738329\pi\)
0.680710 0.732553i \(-0.261671\pi\)
\(798\) 0 0
\(799\) 22.6274i 0.800500i
\(800\) 0 0
\(801\) 3.95815i 0.139854i
\(802\) 0 0
\(803\) 25.2346 0.890509
\(804\) 0 0
\(805\) 52.2843 + 8.28772i 1.84278 + 0.292104i
\(806\) 0 0
\(807\) 14.5442i 0.511979i
\(808\) 0 0
\(809\) −17.5147 −0.615785 −0.307892 0.951421i \(-0.599624\pi\)
−0.307892 + 0.951421i \(0.599624\pi\)
\(810\) 0 0
\(811\) −38.5628 −1.35412 −0.677062 0.735926i \(-0.736747\pi\)
−0.677062 + 0.735926i \(0.736747\pi\)
\(812\) 0 0
\(813\) 1.94113 0.0680782
\(814\) 0 0
\(815\) −45.2429 −1.58479
\(816\) 0 0
\(817\) 15.1535i 0.530154i
\(818\) 0 0
\(819\) 3.61859 22.8284i 0.126444 0.797690i
\(820\) 0 0
\(821\) −49.5980 −1.73098 −0.865491 0.500925i \(-0.832993\pi\)
−0.865491 + 0.500925i \(0.832993\pi\)
\(822\) 0 0
\(823\) 25.1127i 0.875374i −0.899127 0.437687i \(-0.855798\pi\)
0.899127 0.437687i \(-0.144202\pi\)
\(824\) 0 0
\(825\) 3.95815i 0.137805i
\(826\) 0 0
\(827\) 43.2548i 1.50412i 0.659096 + 0.752059i \(0.270939\pi\)
−0.659096 + 0.752059i \(0.729061\pi\)
\(828\) 0 0
\(829\) 6.19986i 0.215330i −0.994187 0.107665i \(-0.965663\pi\)
0.994187 0.107665i \(-0.0343374\pi\)
\(830\) 0 0
\(831\) −29.1927 −1.01268
\(832\) 0 0
\(833\) 6.62742 20.3797i 0.229626 0.706116i
\(834\) 0 0
\(835\) 16.9706i 0.587291i
\(836\) 0 0
\(837\) 16.0000 0.553041
\(838\) 0 0
\(839\) −42.1814 −1.45626 −0.728132 0.685437i \(-0.759611\pi\)
−0.728132 + 0.685437i \(0.759611\pi\)
\(840\) 0 0
\(841\) −15.6274 −0.538876
\(842\) 0 0
\(843\) −30.9861 −1.06722
\(844\) 0 0
\(845\) 25.6829i 0.883519i
\(846\) 0 0
\(847\) 18.2919 + 2.89949i 0.628516 + 0.0996278i
\(848\) 0 0
\(849\) −1.17157 −0.0402083
\(850\) 0 0
\(851\) 58.6274i 2.00972i
\(852\) 0 0
\(853\) 31.8059i 1.08901i −0.838757 0.544506i \(-0.816717\pi\)
0.838757 0.544506i \(-0.183283\pi\)
\(854\) 0 0
\(855\) 19.7990i 0.677111i
\(856\) 0 0
\(857\) 7.01962i 0.239786i 0.992787 + 0.119893i \(0.0382551\pi\)
−0.992787 + 0.119893i \(0.961745\pi\)
\(858\) 0 0
\(859\) 2.50434 0.0854470 0.0427235 0.999087i \(-0.486397\pi\)
0.0427235 + 0.999087i \(0.486397\pi\)
\(860\) 0 0
\(861\) 4.28427 27.0279i 0.146008 0.921110i
\(862\) 0 0
\(863\) 27.4558i 0.934608i −0.884097 0.467304i \(-0.845225\pi\)
0.884097 0.467304i \(-0.154775\pi\)
\(864\) 0 0
\(865\) −28.4853 −0.968529
\(866\) 0 0
\(867\) 8.25586 0.280384
\(868\) 0 0
\(869\) −25.6569 −0.870349
\(870\) 0 0
\(871\) 74.8070 2.53474
\(872\) 0 0
\(873\) 24.7093i 0.836283i
\(874\) 0 0
\(875\) −3.43289 + 21.6569i −0.116053 + 0.732135i
\(876\) 0 0
\(877\) 0.343146 0.0115872 0.00579360 0.999983i \(-0.498156\pi\)
0.00579360 + 0.999983i \(0.498156\pi\)
\(878\) 0 0
\(879\) 6.14214i 0.207169i
\(880\) 0 0
\(881\) 27.7708i 0.935621i 0.883829 + 0.467811i \(0.154957\pi\)
−0.883829 + 0.467811i \(0.845043\pi\)
\(882\) 0 0
\(883\) 24.3431i 0.819212i 0.912263 + 0.409606i \(0.134334\pi\)
−0.912263 + 0.409606i \(0.865666\pi\)
\(884\) 0 0
\(885\) 23.9665i 0.805624i
\(886\) 0 0
\(887\) −55.1701 −1.85243 −0.926216 0.376993i \(-0.876958\pi\)
−0.926216 + 0.376993i \(0.876958\pi\)
\(888\) 0 0
\(889\) −1.51472 + 9.55582i −0.0508020 + 0.320492i
\(890\) 0 0
\(891\) 0.343146i 0.0114958i
\(892\) 0 0
\(893\) −30.6274 −1.02491
\(894\) 0 0
\(895\) 5.22625 0.174694
\(896\) 0 0
\(897\) 39.5980 1.32214
\(898\) 0 0
\(899\) −11.1953 −0.373386
\(900\) 0 0
\(901\) 6.12293i 0.203985i
\(902\) 0 0
\(903\) −1.63952 + 10.3431i −0.0545598 + 0.344198i
\(904\) 0 0
\(905\) 4.48528 0.149096
\(906\) 0 0
\(907\) 26.0000i 0.863316i −0.902037 0.431658i \(-0.857929\pi\)
0.902037 0.431658i \(-0.142071\pi\)
\(908\) 0 0
\(909\) 10.3756i 0.344136i
\(910\) 0 0
\(911\) 24.3431i 0.806524i 0.915084 + 0.403262i \(0.132124\pi\)
−0.915084 + 0.403262i \(0.867876\pi\)
\(912\) 0 0
\(913\) 23.0698i 0.763499i
\(914\) 0 0
\(915\) 7.39104 0.244340
\(916\) 0 0
\(917\) −35.7990 5.67459i −1.18219 0.187392i
\(918\) 0 0
\(919\) 29.7990i 0.982978i −0.870884 0.491489i \(-0.836453\pi\)
0.870884 0.491489i \(-0.163547\pi\)
\(920\) 0 0
\(921\) 12.4853 0.411404
\(922\) 0 0
\(923\) −42.1814 −1.38842
\(924\) 0 0
\(925\) 14.0000 0.460317
\(926\) 0 0
\(927\) −30.9861 −1.01772
\(928\) 0 0
\(929\) 57.8602i 1.89833i 0.314777 + 0.949166i \(0.398070\pi\)
−0.314777 + 0.949166i \(0.601930\pi\)
\(930\) 0 0
\(931\) 27.5851 + 8.97056i 0.904064 + 0.293998i
\(932\) 0 0
\(933\) −21.6569 −0.709014
\(934\) 0 0
\(935\) 16.0000i 0.523256i
\(936\) 0 0
\(937\) 44.7176i 1.46086i 0.682987 + 0.730431i \(0.260681\pi\)
−0.682987 + 0.730431i \(0.739319\pi\)
\(938\) 0 0
\(939\) 16.9706i 0.553813i
\(940\) 0 0
\(941\) 24.7862i 0.808008i −0.914757 0.404004i \(-0.867618\pi\)
0.914757 0.404004i \(-0.132382\pi\)
\(942\) 0 0
\(943\) −73.1675 −2.38266
\(944\) 0 0
\(945\) 5.65685 35.6871i 0.184017 1.16090i
\(946\) 0 0
\(947\) 28.3431i 0.921028i −0.887652 0.460514i \(-0.847665\pi\)
0.887652 0.460514i \(-0.152335\pi\)
\(948\) 0 0
\(949\) −60.2843 −1.95691
\(950\) 0 0
\(951\) 29.9356 0.970727
\(952\) 0 0
\(953\) 0.627417 0.0203240 0.0101620 0.999948i \(-0.496765\pi\)
0.0101620 + 0.999948i \(0.496765\pi\)
\(954\) 0 0
\(955\) 12.6173 0.408286
\(956\) 0 0
\(957\) 7.91630i 0.255898i
\(958\) 0 0
\(959\) 5.22625 + 0.828427i 0.168764 + 0.0267513i
\(960\) 0 0
\(961\) −21.6274 −0.697659
\(962\) 0 0
\(963\) 0.627417i 0.0202182i
\(964\) 0 0
\(965\) 27.3994i 0.882017i
\(966\) 0 0
\(967\) 17.0294i 0.547630i −0.961782 0.273815i \(-0.911714\pi\)
0.961782 0.273815i \(-0.0882856\pi\)
\(968\) 0 0
\(969\) 13.7315i 0.441121i
\(970\) 0 0
\(971\) −4.14386 −0.132983 −0.0664914 0.997787i \(-0.521180\pi\)
−0.0664914 + 0.997787i \(0.521180\pi\)
\(972\) 0 0
\(973\) 5.17157 + 0.819760i 0.165793 + 0.0262803i
\(974\) 0 0
\(975\) 9.45584i 0.302829i
\(976\) 0 0
\(977\) 16.6274 0.531958 0.265979 0.963979i \(-0.414305\pi\)
0.265979 + 0.963979i \(0.414305\pi\)
\(978\) 0 0
\(979\) 4.32957 0.138374
\(980\) 0 0
\(981\) 9.71573 0.310199
\(982\) 0 0
\(983\) 33.5223 1.06920 0.534598 0.845107i \(-0.320463\pi\)
0.534598 + 0.845107i \(0.320463\pi\)
\(984\) 0 0
\(985\) 9.55582i 0.304474i
\(986\) 0 0
\(987\) −20.9050 3.31371i −0.665414 0.105477i
\(988\) 0 0
\(989\) 28.0000 0.890348
\(990\) 0 0
\(991\) 14.4853i 0.460140i 0.973174 + 0.230070i \(0.0738955\pi\)
−0.973174 + 0.230070i \(0.926104\pi\)
\(992\) 0 0
\(993\) 18.7402i 0.594703i
\(994\) 0 0
\(995\) 32.9706i 1.04524i
\(996\) 0 0
\(997\) 17.3952i 0.550911i 0.961314 + 0.275456i \(0.0888287\pi\)
−0.961314 + 0.275456i \(0.911171\pi\)
\(998\) 0 0
\(999\) 40.0166 1.26607
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 224.2.f.a.223.6 yes 8
3.2 odd 2 2016.2.b.b.1567.2 8
4.3 odd 2 inner 224.2.f.a.223.4 yes 8
7.2 even 3 1568.2.p.a.31.4 16
7.3 odd 6 1568.2.p.a.607.6 16
7.4 even 3 1568.2.p.a.607.3 16
7.5 odd 6 1568.2.p.a.31.5 16
7.6 odd 2 inner 224.2.f.a.223.3 8
8.3 odd 2 448.2.f.d.447.5 8
8.5 even 2 448.2.f.d.447.3 8
12.11 even 2 2016.2.b.b.1567.1 8
16.3 odd 4 1792.2.e.g.895.4 8
16.5 even 4 1792.2.e.g.895.3 8
16.11 odd 4 1792.2.e.f.895.5 8
16.13 even 4 1792.2.e.f.895.6 8
21.20 even 2 2016.2.b.b.1567.7 8
24.5 odd 2 4032.2.b.p.3583.8 8
24.11 even 2 4032.2.b.p.3583.7 8
28.3 even 6 1568.2.p.a.607.4 16
28.11 odd 6 1568.2.p.a.607.5 16
28.19 even 6 1568.2.p.a.31.3 16
28.23 odd 6 1568.2.p.a.31.6 16
28.27 even 2 inner 224.2.f.a.223.5 yes 8
56.13 odd 2 448.2.f.d.447.6 8
56.27 even 2 448.2.f.d.447.4 8
84.83 odd 2 2016.2.b.b.1567.8 8
112.13 odd 4 1792.2.e.f.895.3 8
112.27 even 4 1792.2.e.f.895.4 8
112.69 odd 4 1792.2.e.g.895.6 8
112.83 even 4 1792.2.e.g.895.5 8
168.83 odd 2 4032.2.b.p.3583.2 8
168.125 even 2 4032.2.b.p.3583.1 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
224.2.f.a.223.3 8 7.6 odd 2 inner
224.2.f.a.223.4 yes 8 4.3 odd 2 inner
224.2.f.a.223.5 yes 8 28.27 even 2 inner
224.2.f.a.223.6 yes 8 1.1 even 1 trivial
448.2.f.d.447.3 8 8.5 even 2
448.2.f.d.447.4 8 56.27 even 2
448.2.f.d.447.5 8 8.3 odd 2
448.2.f.d.447.6 8 56.13 odd 2
1568.2.p.a.31.3 16 28.19 even 6
1568.2.p.a.31.4 16 7.2 even 3
1568.2.p.a.31.5 16 7.5 odd 6
1568.2.p.a.31.6 16 28.23 odd 6
1568.2.p.a.607.3 16 7.4 even 3
1568.2.p.a.607.4 16 28.3 even 6
1568.2.p.a.607.5 16 28.11 odd 6
1568.2.p.a.607.6 16 7.3 odd 6
1792.2.e.f.895.3 8 112.13 odd 4
1792.2.e.f.895.4 8 112.27 even 4
1792.2.e.f.895.5 8 16.11 odd 4
1792.2.e.f.895.6 8 16.13 even 4
1792.2.e.g.895.3 8 16.5 even 4
1792.2.e.g.895.4 8 16.3 odd 4
1792.2.e.g.895.5 8 112.83 even 4
1792.2.e.g.895.6 8 112.69 odd 4
2016.2.b.b.1567.1 8 12.11 even 2
2016.2.b.b.1567.2 8 3.2 odd 2
2016.2.b.b.1567.7 8 21.20 even 2
2016.2.b.b.1567.8 8 84.83 odd 2
4032.2.b.p.3583.1 8 168.125 even 2
4032.2.b.p.3583.2 8 168.83 odd 2
4032.2.b.p.3583.7 8 24.11 even 2
4032.2.b.p.3583.8 8 24.5 odd 2