Properties

Label 224.2.x.a.27.2
Level $224$
Weight $2$
Character 224.27
Analytic conductor $1.789$
Analytic rank $0$
Dimension $8$
CM discriminant -7
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(27,224)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(224, base_ring=CyclotomicField(8))
 
chi = DirichletCharacter(H, H._module([4, 1, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("224.27");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 224 = 2^{5} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 224.x (of order \(8\), degree \(4\), 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: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{8})\)
Coefficient field: 8.0.157351936.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + x^{4} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{8}]$

Embedding invariants

Embedding label 27.2
Root \(0.581861 - 1.28897i\) of defining polynomial
Character \(\chi\) \(=\) 224.27
Dual form 224.2.x.a.83.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.581861 - 1.28897i) q^{2} +(-1.32288 - 1.50000i) q^{4} +(1.87083 - 1.87083i) q^{7} +(-2.70318 + 0.832353i) q^{8} +(2.12132 - 2.12132i) q^{9} +O(q^{10})\) \(q+(0.581861 - 1.28897i) q^{2} +(-1.32288 - 1.50000i) q^{4} +(1.87083 - 1.87083i) q^{7} +(-2.70318 + 0.832353i) q^{8} +(2.12132 - 2.12132i) q^{9} +(-3.10237 - 1.28504i) q^{11} +(-1.32288 - 3.50000i) q^{14} +(-0.500000 + 3.96863i) q^{16} +(-1.50000 - 3.96863i) q^{18} +(-3.46152 + 3.25113i) q^{22} +(1.35425 + 1.35425i) q^{23} +(3.53553 + 3.53553i) q^{25} +(-5.28112 - 0.331369i) q^{28} +(2.25695 + 5.44876i) q^{29} +(4.82450 + 2.95367i) q^{32} +(-5.98822 - 0.375737i) q^{36} +(11.1545 + 4.62034i) q^{37} +(-8.37181 - 3.46772i) q^{43} +(2.17648 + 6.35350i) q^{44} +(2.53357 - 0.957598i) q^{46} -7.00000i q^{49} +(6.61438 - 2.50000i) q^{50} +(-2.27719 + 5.49763i) q^{53} +(-3.50000 + 6.61438i) q^{56} +(8.33651 + 0.261285i) q^{58} -7.93725i q^{63} +(6.61438 - 4.50000i) q^{64} +(-2.19827 + 0.910554i) q^{67} +(-11.3137 + 11.3137i) q^{71} +(-3.96863 + 7.50000i) q^{72} +(12.4458 - 11.6894i) q^{74} +(-8.20809 + 3.39990i) q^{77} +16.8818 q^{79} -9.00000i q^{81} +(-9.34101 + 8.77327i) q^{86} +(9.45587 + 0.891439i) q^{88} +(0.239870 - 3.82288i) q^{92} +(-9.02277 - 4.07303i) q^{98} +(-9.30710 + 3.85513i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 4 q^{16} - 12 q^{18} - 36 q^{22} + 32 q^{23} - 48 q^{43} + 52 q^{44} + 40 q^{53} - 28 q^{56} + 16 q^{67} + 44 q^{74} - 56 q^{77} + 76 q^{92}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(127\) \(129\) \(197\)
\(\chi(n)\) \(-1\) \(-1\) \(e\left(\frac{1}{8}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.581861 1.28897i 0.411438 0.911438i
\(3\) 0 0 0.923880 0.382683i \(-0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(4\) −1.32288 1.50000i −0.661438 0.750000i
\(5\) 0 0 −0.923880 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(6\) 0 0
\(7\) 1.87083 1.87083i 0.707107 0.707107i
\(8\) −2.70318 + 0.832353i −0.955719 + 0.294281i
\(9\) 2.12132 2.12132i 0.707107 0.707107i
\(10\) 0 0
\(11\) −3.10237 1.28504i −0.935399 0.387455i −0.137675 0.990478i \(-0.543963\pi\)
−0.797724 + 0.603023i \(0.793963\pi\)
\(12\) 0 0
\(13\) 0 0 0.923880 0.382683i \(-0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(14\) −1.32288 3.50000i −0.353553 0.935414i
\(15\) 0 0
\(16\) −0.500000 + 3.96863i −0.125000 + 0.992157i
\(17\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) −1.50000 3.96863i −0.353553 0.935414i
\(19\) 0 0 −0.382683 0.923880i \(-0.625000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −3.46152 + 3.25113i −0.737999 + 0.693144i
\(23\) 1.35425 + 1.35425i 0.282380 + 0.282380i 0.834058 0.551677i \(-0.186012\pi\)
−0.551677 + 0.834058i \(0.686012\pi\)
\(24\) 0 0
\(25\) 3.53553 + 3.53553i 0.707107 + 0.707107i
\(26\) 0 0
\(27\) 0 0
\(28\) −5.28112 0.331369i −0.998037 0.0626229i
\(29\) 2.25695 + 5.44876i 0.419105 + 1.01181i 0.982607 + 0.185695i \(0.0594537\pi\)
−0.563502 + 0.826115i \(0.690546\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 4.82450 + 2.95367i 0.852859 + 0.522141i
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) −5.98822 0.375737i −0.998037 0.0626229i
\(37\) 11.1545 + 4.62034i 1.83379 + 0.759579i 0.963868 + 0.266382i \(0.0858282\pi\)
0.869918 + 0.493197i \(0.164172\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(42\) 0 0
\(43\) −8.37181 3.46772i −1.27669 0.528822i −0.361698 0.932295i \(-0.617803\pi\)
−0.914991 + 0.403473i \(0.867803\pi\)
\(44\) 2.17648 + 6.35350i 0.328117 + 0.957826i
\(45\) 0 0
\(46\) 2.53357 0.957598i 0.373554 0.141190i
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 0 0
\(49\) 7.00000i 1.00000i
\(50\) 6.61438 2.50000i 0.935414 0.353553i
\(51\) 0 0
\(52\) 0 0
\(53\) −2.27719 + 5.49763i −0.312796 + 0.755157i 0.686803 + 0.726844i \(0.259014\pi\)
−0.999599 + 0.0283132i \(0.990986\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −3.50000 + 6.61438i −0.467707 + 0.883883i
\(57\) 0 0
\(58\) 8.33651 + 0.261285i 1.09464 + 0.0343084i
\(59\) 0 0 0.382683 0.923880i \(-0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(60\) 0 0
\(61\) 0 0 −0.382683 0.923880i \(-0.625000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(62\) 0 0
\(63\) 7.93725i 1.00000i
\(64\) 6.61438 4.50000i 0.826797 0.562500i
\(65\) 0 0
\(66\) 0 0
\(67\) −2.19827 + 0.910554i −0.268562 + 0.111242i −0.512901 0.858448i \(-0.671429\pi\)
0.244339 + 0.969690i \(0.421429\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −11.3137 + 11.3137i −1.34269 + 1.34269i −0.449319 + 0.893372i \(0.648333\pi\)
−0.893372 + 0.449319i \(0.851667\pi\)
\(72\) −3.96863 + 7.50000i −0.467707 + 0.883883i
\(73\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(74\) 12.4458 11.6894i 1.44680 1.35886i
\(75\) 0 0
\(76\) 0 0
\(77\) −8.20809 + 3.39990i −0.935399 + 0.387455i
\(78\) 0 0
\(79\) 16.8818 1.89935 0.949677 0.313231i \(-0.101411\pi\)
0.949677 + 0.313231i \(0.101411\pi\)
\(80\) 0 0
\(81\) 9.00000i 1.00000i
\(82\) 0 0
\(83\) 0 0 −0.382683 0.923880i \(-0.625000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −9.34101 + 8.77327i −1.00727 + 0.946046i
\(87\) 0 0
\(88\) 9.45587 + 0.891439i 1.00800 + 0.0950278i
\(89\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0.239870 3.82288i 0.0250082 0.398562i
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) −9.02277 4.07303i −0.911438 0.411438i
\(99\) −9.30710 + 3.85513i −0.935399 + 0.387455i
\(100\) 0.626229 9.98037i 0.0626229 0.998037i
\(101\) 0 0 −0.923880 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(102\) 0 0
\(103\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 5.76125 + 6.13408i 0.559582 + 0.595795i
\(107\) −18.9419 7.84599i −1.83118 0.758501i −0.966736 0.255774i \(-0.917670\pi\)
−0.864446 0.502726i \(-0.832330\pi\)
\(108\) 0 0
\(109\) −19.1056 + 7.91381i −1.82999 + 0.758005i −0.862044 + 0.506834i \(0.830816\pi\)
−0.967943 + 0.251171i \(0.919184\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 6.48921 + 8.36004i 0.613172 + 0.789949i
\(113\) 16.3808i 1.54098i −0.637452 0.770490i \(-0.720012\pi\)
0.637452 0.770490i \(-0.279988\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 5.18748 10.5935i 0.481645 0.983579i
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 0.195169 + 0.195169i 0.0177427 + 0.0177427i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) −10.2309 4.61838i −0.911438 0.411438i
\(127\) 16.0000i 1.41977i −0.704317 0.709885i \(-0.748747\pi\)
0.704317 0.709885i \(-0.251253\pi\)
\(128\) −1.95171 11.1441i −0.172508 0.985008i
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 0.923880 0.382683i \(-0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) −0.105414 + 3.36332i −0.00910637 + 0.290546i
\(135\) 0 0
\(136\) 0 0
\(137\) −15.5830 + 15.5830i −1.33135 + 1.33135i −0.427179 + 0.904167i \(0.640493\pi\)
−0.904167 + 0.427179i \(0.859507\pi\)
\(138\) 0 0
\(139\) 0 0 −0.923880 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 8.00000 + 21.1660i 0.671345 + 1.77621i
\(143\) 0 0
\(144\) 7.35807 + 9.47939i 0.613172 + 0.789949i
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) −7.82548 22.8439i −0.643251 1.87775i
\(149\) 6.96348 16.8113i 0.570471 1.37724i −0.330684 0.943741i \(-0.607280\pi\)
0.901155 0.433497i \(-0.142720\pi\)
\(150\) 0 0
\(151\) 14.6458 + 14.6458i 1.19185 + 1.19185i 0.976546 + 0.215308i \(0.0690756\pi\)
0.215308 + 0.976546i \(0.430924\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) −0.393603 + 12.5582i −0.0317174 + 1.01197i
\(155\) 0 0
\(156\) 0 0
\(157\) 0 0 −0.382683 0.923880i \(-0.625000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(158\) 9.82288 21.7601i 0.781466 1.73114i
\(159\) 0 0
\(160\) 0 0
\(161\) 5.06713 0.399346
\(162\) −11.6007 5.23675i −0.911438 0.411438i
\(163\) 20.6208 8.54142i 1.61515 0.669016i 0.621694 0.783260i \(-0.286445\pi\)
0.993453 + 0.114245i \(0.0364449\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(168\) 0 0
\(169\) 9.19239 9.19239i 0.707107 0.707107i
\(170\) 0 0
\(171\) 0 0
\(172\) 5.87329 + 17.1451i 0.447834 + 1.30730i
\(173\) 0 0 0.923880 0.382683i \(-0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(174\) 0 0
\(175\) 13.2288 1.00000
\(176\) 6.65104 11.6696i 0.501341 0.879630i
\(177\) 0 0
\(178\) 0 0
\(179\) 5.28883 + 12.7684i 0.395305 + 0.954352i 0.988764 + 0.149487i \(0.0477622\pi\)
−0.593458 + 0.804865i \(0.702238\pi\)
\(180\) 0 0
\(181\) 0 0 0.382683 0.923880i \(-0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −4.78799 2.53357i −0.352975 0.186777i
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 13.0514i 0.944369i 0.881500 + 0.472184i \(0.156534\pi\)
−0.881500 + 0.472184i \(0.843466\pi\)
\(192\) 0 0
\(193\) −27.6946 −1.99350 −0.996749 0.0805728i \(-0.974325\pi\)
−0.996749 + 0.0805728i \(0.974325\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) −10.5000 + 9.26013i −0.750000 + 0.661438i
\(197\) 0.159228 + 0.0659545i 0.0113445 + 0.00469906i 0.388348 0.921513i \(-0.373046\pi\)
−0.377004 + 0.926212i \(0.623046\pi\)
\(198\) −0.446304 + 14.2397i −0.0317174 + 1.01197i
\(199\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(200\) −12.5000 6.61438i −0.883883 0.467707i
\(201\) 0 0
\(202\) 0 0
\(203\) 14.4161 + 5.97133i 1.01181 + 0.419105i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 5.74559 0.399346
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 11.1115 + 26.8255i 0.764947 + 1.84675i 0.413057 + 0.910705i \(0.364461\pi\)
0.351890 + 0.936041i \(0.385539\pi\)
\(212\) 11.2589 3.85689i 0.773263 0.264892i
\(213\) 0 0
\(214\) −21.1348 + 19.8502i −1.44474 + 1.35693i
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) −0.916172 + 29.2313i −0.0620510 + 1.97979i
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(224\) 14.5516 3.50000i 0.972272 0.233854i
\(225\) 15.0000 1.00000
\(226\) −21.1144 9.53137i −1.40451 0.634018i
\(227\) 0 0 0.923880 0.382683i \(-0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(228\) 0 0
\(229\) 0 0 −0.923880 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −10.6362 12.8504i −0.698304 0.843671i
\(233\) −0.416995 + 0.416995i −0.0273182 + 0.0273182i −0.720634 0.693316i \(-0.756149\pi\)
0.693316 + 0.720634i \(0.256149\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −16.0000 −1.03495 −0.517477 0.855697i \(-0.673129\pi\)
−0.517477 + 0.855697i \(0.673129\pi\)
\(240\) 0 0
\(241\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(242\) 0.365128 0.138006i 0.0234713 0.00887133i
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 0.382683 0.923880i \(-0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(252\) −11.9059 + 10.5000i −0.750000 + 0.661438i
\(253\) −2.46111 5.94164i −0.154729 0.373548i
\(254\) −20.6235 9.30978i −1.29403 0.584147i
\(255\) 0 0
\(256\) −15.5000 3.96863i −0.968750 0.248039i
\(257\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(258\) 0 0
\(259\) 29.5120 12.2243i 1.83379 0.759579i
\(260\) 0 0
\(261\) 16.3463 + 6.77086i 1.01181 + 0.419105i
\(262\) 0 0
\(263\) −3.74166 + 3.74166i −0.230720 + 0.230720i −0.812993 0.582273i \(-0.802164\pi\)
0.582273 + 0.812993i \(0.302164\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 4.27387 + 2.09286i 0.261068 + 0.127842i
\(269\) 0 0 0.923880 0.382683i \(-0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 11.0188 + 29.1531i 0.665673 + 1.76121i
\(275\) −6.42521 15.5118i −0.387455 0.935399i
\(276\) 0 0
\(277\) −1.11400 + 2.68944i −0.0669339 + 0.161593i −0.953807 0.300421i \(-0.902873\pi\)
0.886873 + 0.462014i \(0.152873\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 23.5830 + 23.5830i 1.40684 + 1.40684i 0.775515 + 0.631329i \(0.217490\pi\)
0.631329 + 0.775515i \(0.282510\pi\)
\(282\) 0 0
\(283\) 0 0 0.382683 0.923880i \(-0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(284\) 31.9372 + 2.00393i 1.89512 + 0.118912i
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 16.5000 3.96863i 0.972272 0.233854i
\(289\) −17.0000 −1.00000
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 0 0 −0.923880 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −33.9983 3.20515i −1.97611 0.186295i
\(297\) 0 0
\(298\) −17.6175 18.7576i −1.02055 1.08660i
\(299\) 0 0
\(300\) 0 0
\(301\) −22.1497 + 9.17472i −1.27669 + 0.528822i
\(302\) 27.3997 10.3561i 1.57668 0.595927i
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 0 0 −0.382683 0.923880i \(-0.625000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(308\) 15.9581 + 7.81449i 0.909299 + 0.445272i
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(312\) 0 0
\(313\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) −22.3326 25.3227i −1.25630 1.42452i
\(317\) −13.5707 32.7625i −0.762204 1.84012i −0.465004 0.885309i \(-0.653947\pi\)
−0.297200 0.954815i \(-0.596053\pi\)
\(318\) 0 0
\(319\) 19.8043i 1.10883i
\(320\) 0 0
\(321\) 0 0
\(322\) 2.94837 6.53137i 0.164306 0.363979i
\(323\) 0 0
\(324\) −13.5000 + 11.9059i −0.750000 + 0.661438i
\(325\) 0 0
\(326\) 0.988830 31.5495i 0.0547662 1.74736i
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −8.21134 3.40125i −0.451336 0.186950i 0.145424 0.989369i \(-0.453545\pi\)
−0.596760 + 0.802420i \(0.703545\pi\)
\(332\) 0 0
\(333\) 33.4634 13.8610i 1.83379 0.759579i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 6.24657i 0.340273i −0.985421 0.170136i \(-0.945579\pi\)
0.985421 0.170136i \(-0.0544208\pi\)
\(338\) −6.50000 17.1974i −0.353553 0.935414i
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −13.0958 13.0958i −0.707107 0.707107i
\(344\) 25.5169 + 2.40557i 1.37578 + 0.129700i
\(345\) 0 0
\(346\) 0 0
\(347\) 13.6816 33.0303i 0.734466 1.77316i 0.107366 0.994220i \(-0.465758\pi\)
0.627100 0.778938i \(-0.284242\pi\)
\(348\) 0 0
\(349\) 0 0 −0.382683 0.923880i \(-0.625000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(350\) 7.69730 17.0514i 0.411438 0.911438i
\(351\) 0 0
\(352\) −11.1718 15.3631i −0.595458 0.818854i
\(353\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 19.5354 + 0.612282i 1.03248 + 0.0323601i
\(359\) −22.5203 + 22.5203i −1.18857 + 1.18857i −0.211112 + 0.977462i \(0.567708\pi\)
−0.977462 + 0.211112i \(0.932292\pi\)
\(360\) 0 0
\(361\) −13.4350 + 13.4350i −0.707107 + 0.707107i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(368\) −6.05163 + 4.69738i −0.315463 + 0.244868i
\(369\) 0 0
\(370\) 0 0
\(371\) 6.02488 + 14.5454i 0.312796 + 0.755157i
\(372\) 0 0
\(373\) 12.4277 30.0031i 0.643482 1.55350i −0.178468 0.983946i \(-0.557114\pi\)
0.821951 0.569558i \(-0.192886\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −9.66710 + 23.3384i −0.496566 + 1.19882i 0.454756 + 0.890616i \(0.349726\pi\)
−0.951322 + 0.308199i \(0.900274\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 16.8229 + 7.59412i 0.860733 + 0.388549i
\(383\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −16.1144 + 35.6974i −0.820200 + 1.81695i
\(387\) −25.1154 + 10.4032i −1.27669 + 0.528822i
\(388\) 0 0
\(389\) −22.4682 9.30663i −1.13918 0.471865i −0.268290 0.963338i \(-0.586458\pi\)
−0.870893 + 0.491473i \(0.836458\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 5.82647 + 18.9223i 0.294281 + 0.955719i
\(393\) 0 0
\(394\) 0.177662 0.166864i 0.00895047 0.00840647i
\(395\) 0 0
\(396\) 18.0948 + 8.86080i 0.909299 + 0.445272i
\(397\) 0 0 0.923880 0.382683i \(-0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) −15.7990 + 12.2634i −0.789949 + 0.613172i
\(401\) 39.0083i 1.94798i 0.226592 + 0.973990i \(0.427242\pi\)
−0.226592 + 0.973990i \(0.572758\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 16.0850 15.1074i 0.798286 0.749766i
\(407\) −28.6680 28.6680i −1.42102 1.42102i
\(408\) 0 0
\(409\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 3.34313 7.40588i 0.164306 0.363979i
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 0.923880 0.382683i \(-0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(420\) 0 0
\(421\) 10.9674 + 4.54285i 0.534519 + 0.221405i 0.633581 0.773676i \(-0.281584\pi\)
−0.0990621 + 0.995081i \(0.531584\pi\)
\(422\) 41.0426 + 1.28637i 1.99792 + 0.0626193i
\(423\) 0 0
\(424\) 1.57970 16.7565i 0.0767169 0.813768i
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 13.2888 + 38.7921i 0.642338 + 1.87509i
\(429\) 0 0
\(430\) 0 0
\(431\) −26.4575 −1.27441 −0.637207 0.770693i \(-0.719910\pi\)
−0.637207 + 0.770693i \(0.719910\pi\)
\(432\) 0 0
\(433\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 37.1451 + 18.1894i 1.77893 + 0.871116i
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(440\) 0 0
\(441\) −14.8492 14.8492i −0.707107 0.707107i
\(442\) 0 0
\(443\) −1.64661 + 3.97527i −0.0782328 + 0.188871i −0.958157 0.286244i \(-0.907593\pi\)
0.879924 + 0.475114i \(0.157593\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 3.95564 20.7931i 0.186886 0.982382i
\(449\) 42.3320 1.99777 0.998886 0.0471929i \(-0.0150276\pi\)
0.998886 + 0.0471929i \(0.0150276\pi\)
\(450\) 8.72791 19.3345i 0.411438 0.911438i
\(451\) 0 0
\(452\) −24.5713 + 21.6698i −1.15574 + 1.01926i
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −29.9333 + 29.9333i −1.40022 + 1.40022i −0.600878 + 0.799341i \(0.705182\pi\)
−0.799341 + 0.600878i \(0.794818\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 0.923880 0.382683i \(-0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(462\) 0 0
\(463\) −39.5092 −1.83615 −0.918075 0.396406i \(-0.870257\pi\)
−0.918075 + 0.396406i \(0.870257\pi\)
\(464\) −22.7526 + 6.23262i −1.05626 + 0.289342i
\(465\) 0 0
\(466\) 0.294860 + 0.780126i 0.0136591 + 0.0361386i
\(467\) 0 0 −0.382683 0.923880i \(-0.625000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(468\) 0 0
\(469\) −2.40910 + 5.81608i −0.111242 + 0.268562i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 21.5163 + 21.5163i 0.989319 + 0.989319i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 6.83157 + 16.4929i 0.312796 + 0.755157i
\(478\) −9.30978 + 20.6235i −0.425819 + 0.943296i
\(479\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 0.0345692 0.550939i 0.00157133 0.0250427i
\(485\) 0 0
\(486\) 0 0
\(487\) 6.52026 6.52026i 0.295461 0.295461i −0.543772 0.839233i \(-0.683004\pi\)
0.839233 + 0.543772i \(0.183004\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 35.6855 + 14.7814i 1.61046 + 0.667076i 0.992846 0.119401i \(-0.0380974\pi\)
0.617619 + 0.786478i \(0.288097\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 42.3320i 1.89885i
\(498\) 0 0
\(499\) −16.6025 40.0821i −0.743232 1.79432i −0.592200 0.805791i \(-0.701741\pi\)
−0.151031 0.988529i \(-0.548259\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(504\) 6.60659 + 21.4558i 0.294281 + 0.955719i
\(505\) 0 0
\(506\) −9.09061 0.284920i −0.404127 0.0126662i
\(507\) 0 0
\(508\) −24.0000 + 21.1660i −1.06483 + 0.939090i
\(509\) 0 0 −0.382683 0.923880i \(-0.625000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −14.1343 + 17.6698i −0.624653 + 0.780903i
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 1.41519 45.1528i 0.0621798 1.98390i
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(522\) 18.2387 17.1301i 0.798286 0.749766i
\(523\) 0 0 −0.923880 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 2.64575 + 7.00000i 0.115360 + 0.305215i
\(527\) 0 0
\(528\) 0 0
\(529\) 19.3320i 0.840523i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 5.18443 4.29113i 0.223933 0.185349i
\(537\) 0 0
\(538\) 0 0
\(539\) −8.99530 + 21.7166i −0.387455 + 0.935399i
\(540\) 0 0
\(541\) 16.2042 + 39.1203i 0.696671 + 1.68191i 0.730887 + 0.682498i \(0.239107\pi\)
−0.0342160 + 0.999414i \(0.510893\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −2.00661 + 0.831165i −0.0857964 + 0.0355380i −0.425169 0.905114i \(-0.639785\pi\)
0.339372 + 0.940652i \(0.389785\pi\)
\(548\) 43.9889 + 2.76013i 1.87911 + 0.117907i
\(549\) 0 0
\(550\) −23.7328 0.743839i −1.01197 0.0317174i
\(551\) 0 0
\(552\) 0 0
\(553\) 31.5830 31.5830i 1.34305 1.34305i
\(554\) 2.81840 + 3.00079i 0.119743 + 0.127491i
\(555\) 0 0
\(556\) 0 0
\(557\) 35.5218 14.7136i 1.50511 0.623436i 0.530566 0.847644i \(-0.321980\pi\)
0.974541 + 0.224208i \(0.0719796\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 44.1198 16.6757i 1.86108 0.703422i
\(563\) 0 0 −0.382683 0.923880i \(-0.625000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −16.8375 16.8375i −0.707107 0.707107i
\(568\) 21.1660 40.0000i 0.888106 1.67836i
\(569\) −29.9333 29.9333i −1.25487 1.25487i −0.953512 0.301356i \(-0.902561\pi\)
−0.301356 0.953512i \(-0.597439\pi\)
\(570\) 0 0
\(571\) 16.2517 39.2350i 0.680111 1.64193i −0.0836974 0.996491i \(-0.526673\pi\)
0.763809 0.645443i \(-0.223327\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 9.57598i 0.399346i
\(576\) 4.48527 23.5772i 0.186886 0.982382i
\(577\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(578\) −9.89164 + 21.9125i −0.411438 + 0.911438i
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 14.1294 14.1294i 0.585178 0.585178i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 −0.923880 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) −23.9136 + 41.9578i −0.982844 + 1.72446i
\(593\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −34.4288 + 11.7941i −1.41026 + 0.483104i
\(597\) 0 0
\(598\) 0 0
\(599\) 26.1916 + 26.1916i 1.07016 + 1.07016i 0.997346 + 0.0728143i \(0.0231980\pi\)
0.0728143 + 0.997346i \(0.476802\pi\)
\(600\) 0 0
\(601\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(602\) −1.06215 + 33.8887i −0.0432899 + 1.38120i
\(603\) −2.73166 + 6.59482i −0.111242 + 0.268562i
\(604\) 2.59412 41.3431i 0.105553 1.68223i
\(605\) 0 0
\(606\) 0 0
\(607\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −43.6600 18.0846i −1.76341 0.730429i −0.996008 0.0892633i \(-0.971549\pi\)
−0.767403 0.641165i \(-0.778451\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 19.3580 16.0226i 0.779958 0.645568i
\(617\) 29.9333 29.9333i 1.20507 1.20507i 0.232462 0.972605i \(-0.425322\pi\)
0.972605 0.232462i \(-0.0746782\pi\)
\(618\) 0 0
\(619\) 0 0 −0.923880 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 25.0000i 1.00000i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 11.3137 + 11.3137i 0.450392 + 0.450392i 0.895484 0.445093i \(-0.146829\pi\)
−0.445093 + 0.895484i \(0.646829\pi\)
\(632\) −45.6346 + 14.0516i −1.81525 + 0.558944i
\(633\) 0 0
\(634\) −50.1260 1.57106i −1.99076 0.0623948i
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) −25.5272 11.5234i −1.01063 0.456215i
\(639\) 48.0000i 1.89885i
\(640\) 0 0
\(641\) 17.5603 0.693589 0.346795 0.937941i \(-0.387270\pi\)
0.346795 + 0.937941i \(0.387270\pi\)
\(642\) 0 0
\(643\) 0 0 0.923880 0.382683i \(-0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(644\) −6.70319 7.60070i −0.264143 0.299510i
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(648\) 7.49117 + 24.3286i 0.294281 + 0.955719i
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) −40.0909 19.6320i −1.57008 0.768848i
\(653\) 46.4193 19.2275i 1.81653 0.752431i 0.838203 0.545358i \(-0.183606\pi\)
0.978326 0.207072i \(-0.0663936\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −15.7978 38.1392i −0.615395 1.48569i −0.856998 0.515319i \(-0.827673\pi\)
0.241604 0.970375i \(-0.422327\pi\)
\(660\) 0 0
\(661\) 0 0 0.382683 0.923880i \(-0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(662\) −9.16196 + 8.60510i −0.356090 + 0.334447i
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 1.60467 51.1985i 0.0621798 1.98390i
\(667\) −4.32251 + 10.4355i −0.167368 + 0.404062i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −42.3320 −1.63178 −0.815890 0.578208i \(-0.803752\pi\)
−0.815890 + 0.578208i \(0.803752\pi\)
\(674\) −8.05163 3.63464i −0.310137 0.140001i
\(675\) 0 0
\(676\) −25.9490 1.62820i −0.998037 0.0626229i
\(677\) 0 0 −0.923880 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 46.2556 + 19.1597i 1.76992 + 0.733126i 0.994862 + 0.101237i \(0.0322800\pi\)
0.775059 + 0.631889i \(0.217720\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −24.5000 + 9.26013i −0.935414 + 0.353553i
\(687\) 0 0
\(688\) 17.9480 31.4907i 0.684261 1.20057i
\(689\) 0 0
\(690\) 0 0
\(691\) 0 0 −0.382683 0.923880i \(-0.625000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(692\) 0 0
\(693\) −10.1997 + 24.6243i −0.387455 + 0.935399i
\(694\) −34.6142 36.8542i −1.31394 1.39896i
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) −17.5000 19.8431i −0.661438 0.750000i
\(701\) −18.4154 44.4587i −0.695540 1.67918i −0.733309 0.679895i \(-0.762025\pi\)
0.0377695 0.999286i \(-0.487975\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) −26.3029 + 5.46089i −0.991328 + 0.205815i
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 13.5870 + 5.62791i 0.510269 + 0.211360i 0.622937 0.782272i \(-0.285939\pi\)
−0.112667 + 0.993633i \(0.535939\pi\)
\(710\) 0 0
\(711\) 35.8118 35.8118i 1.34305 1.34305i
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 12.1561 24.8242i 0.454294 0.927723i
\(717\) 0 0
\(718\) 15.9242 + 42.1315i 0.594287 + 1.57234i
\(719\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 9.50000 + 25.1346i 0.353553 + 0.935414i
\(723\) 0 0
\(724\) 0 0
\(725\) −11.2848 + 27.2438i −0.419105 + 1.01181i
\(726\) 0 0
\(727\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(728\) 0 0
\(729\) −19.0919 19.0919i −0.707107 0.707107i
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 0 0 −0.382683 0.923880i \(-0.625000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 2.53357 + 10.5336i 0.0933885 + 0.388273i
\(737\) 7.98995 0.294314
\(738\) 0 0
\(739\) −31.9345 + 13.2277i −1.17473 + 0.486589i −0.882754 0.469836i \(-0.844313\pi\)
−0.291977 + 0.956425i \(0.594313\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 22.2541 + 0.697494i 0.816975 + 0.0256058i
\(743\) 38.5203 38.5203i 1.41317 1.41317i 0.679442 0.733729i \(-0.262222\pi\)
0.733729 0.679442i \(-0.237778\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −31.4419 33.4766i −1.15117 1.22566i
\(747\) 0 0
\(748\) 0 0
\(749\) −50.1155 + 20.7585i −1.83118 + 0.758501i
\(750\) 0 0
\(751\) 48.0000 1.75154 0.875772 0.482724i \(-0.160353\pi\)
0.875772 + 0.482724i \(0.160353\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −11.6498 + 28.1250i −0.423418 + 1.02222i 0.557913 + 0.829899i \(0.311602\pi\)
−0.981332 + 0.192323i \(0.938398\pi\)
\(758\) 24.4576 + 26.0403i 0.888340 + 0.945827i
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(762\) 0 0
\(763\) −20.9380 + 50.5487i −0.758005 + 1.82999i
\(764\) 19.5771 17.2654i 0.708276 0.624641i
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 36.6364 + 41.5418i 1.31857 + 1.49512i
\(773\) 0 0 −0.923880 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(774\) −1.20436 + 38.4262i −0.0432899 + 1.38120i
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) −25.0693 + 23.5456i −0.898778 + 0.844151i
\(779\) 0 0
\(780\) 0 0
\(781\) 49.6379 20.5607i 1.77618 0.735719i
\(782\) 0 0
\(783\) 0 0
\(784\) 27.7804 + 3.50000i 0.992157 + 0.125000i
\(785\) 0 0
\(786\) 0 0
\(787\) 0 0 −0.382683 0.923880i \(-0.625000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(788\) −0.111707 0.326092i −0.00397941 0.0116165i
\(789\) 0 0
\(790\) 0 0
\(791\) −30.6458 30.6458i −1.08964 1.08964i
\(792\) 21.9500 18.1679i 0.779958 0.645568i
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 0 0 −0.382683 0.923880i \(-0.625000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 6.61438 + 27.5000i 0.233854 + 0.972272i
\(801\) 0 0
\(802\) 50.2804 + 22.6974i 1.77546 + 0.801472i
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 29.9333 29.9333i 1.05240 1.05240i 0.0538482 0.998549i \(-0.482851\pi\)
0.998549 0.0538482i \(-0.0171487\pi\)
\(810\) 0 0
\(811\) 0 0 −0.923880 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(812\) −10.1137 29.5234i −0.354920 1.03607i
\(813\) 0 0
\(814\) −53.6328 + 20.2713i −1.87983 + 0.710509i
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −0.0289484 + 0.0698877i −0.00101031 + 0.00243910i −0.924384 0.381464i \(-0.875420\pi\)
0.923374 + 0.383903i \(0.125420\pi\)
\(822\) 0 0
\(823\) 33.6749 + 33.6749i 1.17383 + 1.17383i 0.981288 + 0.192546i \(0.0616744\pi\)
0.192546 + 0.981288i \(0.438326\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 20.9808 50.6522i 0.729574 1.76135i 0.0855616 0.996333i \(-0.472732\pi\)
0.644013 0.765015i \(-0.277268\pi\)
\(828\) −7.60070 8.61839i −0.264143 0.299510i
\(829\) 0 0 −0.382683 0.923880i \(-0.625000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(840\) 0 0
\(841\) −4.08910 + 4.08910i −0.141004 + 0.141004i
\(842\) 12.2371 11.4933i 0.421718 0.396087i
\(843\) 0 0
\(844\) 25.5392 52.1541i 0.879095 1.79522i
\(845\) 0 0
\(846\) 0 0
\(847\) 0.730257 0.0250919
\(848\) −20.6794 11.7861i −0.710135 0.404738i
\(849\) 0 0
\(850\) 0 0
\(851\) 8.84886 + 21.3630i 0.303335 + 0.732315i
\(852\) 0 0
\(853\) 0 0 0.382683 0.923880i \(-0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 57.7340 + 5.44280i 1.97331 + 0.186031i
\(857\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(858\) 0 0
\(859\) 0 0 0.382683 0.923880i \(-0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −15.3946 + 34.1029i −0.524342 + 1.16155i
\(863\) 46.8151i 1.59360i 0.604240 + 0.796802i \(0.293477\pi\)
−0.604240 + 0.796802i \(0.706523\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −52.3736 21.6939i −1.77665 0.735914i
\(870\) 0 0
\(871\) 0 0
\(872\) 45.0589 37.2951i 1.52589 1.26297i
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −44.7771 + 18.5473i −1.51202 + 0.626298i −0.975973 0.217892i \(-0.930082\pi\)
−0.536044 + 0.844190i \(0.680082\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(882\) −27.7804 + 10.5000i −0.935414 + 0.353553i
\(883\) 18.8218 + 45.4397i 0.633403 + 1.52917i 0.835319 + 0.549766i \(0.185283\pi\)
−0.201916 + 0.979403i \(0.564717\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 4.16589 + 4.43548i 0.139956 + 0.149013i
\(887\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(888\) 0 0
\(889\) −29.9333 29.9333i −1.00393 1.00393i
\(890\) 0 0
\(891\) −11.5654 + 27.9213i −0.387455 + 0.935399i
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) −24.5000 17.1974i −0.818488 0.574524i
\(897\) 0 0
\(898\) 24.6314 54.5646i 0.821959 1.82084i
\(899\) 0 0
\(900\) −19.8431 22.5000i −0.661438 0.750000i
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 13.6346 + 44.2804i 0.453481 + 1.47274i
\(905\) 0 0
\(906\) 0 0
\(907\) −25.7298 10.6576i −0.854343 0.353881i −0.0878507 0.996134i \(-0.528000\pi\)
−0.766493 + 0.642253i \(0.778000\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 16.0000 0.530104 0.265052 0.964234i \(-0.414611\pi\)
0.265052 + 0.964234i \(0.414611\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 21.1660 + 56.0000i 0.700109 + 1.85232i
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 33.9411 + 33.9411i 1.11961 + 1.11961i 0.991798 + 0.127817i \(0.0407969\pi\)
0.127817 + 0.991798i \(0.459203\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 23.1017 + 55.7724i 0.759579 + 1.83379i
\(926\) −22.9889 + 50.9261i −0.755462 + 1.67354i
\(927\) 0 0
\(928\) −5.20520 + 32.9539i −0.170869 + 1.08176i
\(929\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 1.17712 + 0.0738599i 0.0385580 + 0.00241936i
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(938\) 6.09498 + 6.48940i 0.199008 + 0.211887i
\(939\) 0 0
\(940\) 0 0
\(941\) 0 0 0.923880 0.382683i \(-0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 40.2532 15.2143i 1.30875 0.494659i
\(947\) −23.5080 56.7534i −0.763909 1.84424i −0.438953 0.898510i \(-0.644650\pi\)
−0.324956 0.945729i \(-0.605350\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −39.5830 39.5830i −1.28222 1.28222i −0.939402 0.342817i \(-0.888619\pi\)
−0.342817 0.939402i \(-0.611381\pi\)
\(954\) 25.2338 + 0.790884i 0.816975 + 0.0256058i
\(955\) 0 0
\(956\) 21.1660 + 24.0000i 0.684558 + 0.776215i
\(957\) 0 0
\(958\) 0 0
\(959\) 58.3063i 1.88281i
\(960\) 0 0
\(961\) −31.0000 −1.00000
\(962\) 0 0
\(963\) −56.8257 + 23.5380i −1.83118 + 0.758501i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 3.81176 3.81176i 0.122578 0.122578i −0.643157 0.765735i \(-0.722376\pi\)
0.765735 + 0.643157i \(0.222376\pi\)
\(968\) −0.690028 0.365128i −0.0221783 0.0117357i
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 −0.923880 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) −4.61052 12.1983i −0.147731 0.390858i
\(975\) 0 0
\(976\) 0 0
\(977\) 42.3320i 1.35432i −0.735835 0.677161i \(-0.763210\pi\)
0.735835 0.677161i \(-0.236790\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) −23.7414 + 57.3169i −0.758005 + 1.82999i
\(982\) 39.8168 37.3967i 1.27060 1.19338i
\(983\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −6.64136 16.0337i −0.211183 0.509841i
\(990\) 0 0
\(991\) 24.1877i 0.768347i −0.923261 0.384173i \(-0.874487\pi\)
0.923261 0.384173i \(-0.125513\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 54.5646 + 24.6314i 1.73068 + 0.781259i
\(995\) 0 0
\(996\) 0 0
\(997\) 0 0 −0.923880 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(998\) −61.3249 1.92206i −1.94120 0.0608416i
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 224.2.x.a.27.2 8
4.3 odd 2 896.2.x.a.783.1 8
7.6 odd 2 CM 224.2.x.a.27.2 8
28.27 even 2 896.2.x.a.783.1 8
32.13 even 8 896.2.x.a.111.1 8
32.19 odd 8 inner 224.2.x.a.83.2 yes 8
224.13 odd 8 896.2.x.a.111.1 8
224.83 even 8 inner 224.2.x.a.83.2 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
224.2.x.a.27.2 8 1.1 even 1 trivial
224.2.x.a.27.2 8 7.6 odd 2 CM
224.2.x.a.83.2 yes 8 32.19 odd 8 inner
224.2.x.a.83.2 yes 8 224.83 even 8 inner
896.2.x.a.111.1 8 32.13 even 8
896.2.x.a.111.1 8 224.13 odd 8
896.2.x.a.783.1 8 4.3 odd 2
896.2.x.a.783.1 8 28.27 even 2