Properties

Label 224.2.x.a.27.1
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 27.1
Root \(-1.28897 + 0.581861i\) of defining polynomial
Character \(\chi\) \(=\) 224.27
Dual form 224.2.x.a.83.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.28897 + 0.581861i) q^{2} +(1.32288 - 1.50000i) q^{4} +(-1.87083 + 1.87083i) q^{7} +(-0.832353 + 2.70318i) q^{8} +(2.12132 - 2.12132i) q^{9} +O(q^{10})\) \(q+(-1.28897 + 0.581861i) q^{2} +(1.32288 - 1.50000i) q^{4} +(-1.87083 + 1.87083i) q^{7} +(-0.832353 + 2.70318i) q^{8} +(2.12132 - 2.12132i) q^{9} +(5.93079 + 2.45662i) q^{11} +(1.32288 - 3.50000i) q^{14} +(-0.500000 - 3.96863i) q^{16} +(-1.50000 + 3.96863i) q^{18} +(-9.07401 + 0.284400i) q^{22} +(6.64575 + 6.64575i) q^{23} +(3.53553 + 3.53553i) q^{25} +(0.331369 + 5.28112i) q^{28} +(-0.842738 - 2.03455i) q^{29} +(2.95367 + 4.82450i) q^{32} +(-0.375737 - 5.98822i) q^{36} +(-6.91184 - 2.86298i) q^{37} +(-12.1135 - 5.01756i) q^{43} +(11.5306 - 5.64639i) q^{44} +(-12.4331 - 4.69926i) q^{46} -7.00000i q^{49} +(-6.61438 - 2.50000i) q^{50} +(5.20612 - 12.5687i) q^{53} +(-3.50000 - 6.61438i) q^{56} +(2.27009 + 2.13211i) q^{58} +7.93725i q^{63} +(-6.61438 - 4.50000i) q^{64} +(9.02670 - 3.73898i) q^{67} +(-11.3137 + 11.3137i) q^{71} +(3.96863 + 7.50000i) q^{72} +(10.5750 - 0.331444i) q^{74} +(-15.6914 + 6.49959i) q^{77} -5.56812 q^{79} -9.00000i q^{81} +(18.5334 - 0.580878i) q^{86} +(-11.5772 + 13.9872i) q^{88} +(18.7601 - 1.17712i) q^{92} +(4.07303 + 9.02277i) q^{98} +(17.7924 - 7.36985i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.28897 + 0.581861i −0.911438 + 0.411438i
\(3\) 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
\(8\) −0.832353 + 2.70318i −0.294281 + 0.955719i
\(9\) 2.12132 2.12132i 0.707107 0.707107i
\(10\) 0 0
\(11\) 5.93079 + 2.45662i 1.78820 + 0.740697i 0.990478 + 0.137675i \(0.0439628\pi\)
0.797724 + 0.603023i \(0.206037\pi\)
\(12\) 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
\(22\) −9.07401 + 0.284400i −1.93459 + 0.0606342i
\(23\) 6.64575 + 6.64575i 1.38573 + 1.38573i 0.834058 + 0.551677i \(0.186012\pi\)
0.551677 + 0.834058i \(0.313988\pi\)
\(24\) 0 0
\(25\) 3.53553 + 3.53553i 0.707107 + 0.707107i
\(26\) 0 0
\(27\) 0 0
\(28\) 0.331369 + 5.28112i 0.0626229 + 0.998037i
\(29\) −0.842738 2.03455i −0.156493 0.377807i 0.826115 0.563502i \(-0.190546\pi\)
−0.982607 + 0.185695i \(0.940546\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 2.95367 + 4.82450i 0.522141 + 0.852859i
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) −0.375737 5.98822i −0.0626229 0.998037i
\(37\) −6.91184 2.86298i −1.13630 0.470671i −0.266382 0.963868i \(-0.585828\pi\)
−0.869918 + 0.493197i \(0.835828\pi\)
\(38\) 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
\(43\) −12.1135 5.01756i −1.84729 0.765171i −0.932295 0.361698i \(-0.882197\pi\)
−0.914991 0.403473i \(-0.867803\pi\)
\(44\) 11.5306 5.64639i 1.73831 0.851226i
\(45\) 0 0
\(46\) −12.4331 4.69926i −1.83316 0.692867i
\(47\) 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
\(53\) 5.20612 12.5687i 0.715116 1.72644i 0.0283132 0.999599i \(-0.490986\pi\)
0.686803 0.726844i \(-0.259014\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −3.50000 6.61438i −0.467707 0.883883i
\(57\) 0 0
\(58\) 2.27009 + 2.13211i 0.298077 + 0.279960i
\(59\) 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
\(67\) 9.02670 3.73898i 1.10279 0.456789i 0.244339 0.969690i \(-0.421429\pi\)
0.858448 + 0.512901i \(0.171429\pi\)
\(68\) 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 \(0.750000\pi\)
\(74\) 10.5750 0.331444i 1.22932 0.0385295i
\(75\) 0 0
\(76\) 0 0
\(77\) −15.6914 + 6.49959i −1.78820 + 0.740697i
\(78\) 0 0
\(79\) −5.56812 −0.626462 −0.313231 0.949677i \(-0.601411\pi\)
−0.313231 + 0.949677i \(0.601411\pi\)
\(80\) 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
\(86\) 18.5334 0.580878i 1.99851 0.0626376i
\(87\) 0 0
\(88\) −11.5772 + 13.9872i −1.23413 + 1.49104i
\(89\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 18.7601 1.17712i 1.95588 0.122724i
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) 4.07303 + 9.02277i 0.411438 + 0.911438i
\(99\) 17.7924 7.36985i 1.78820 0.740697i
\(100\) 9.98037 0.626229i 0.998037 0.0626229i
\(101\) 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
\(106\) 0.602707 + 19.2299i 0.0585401 + 1.86777i
\(107\) −15.2002 6.29615i −1.46946 0.608671i −0.502726 0.864446i \(-0.667670\pi\)
−0.966736 + 0.255774i \(0.917670\pi\)
\(108\) 0 0
\(109\) −11.6223 + 4.81412i −1.11321 + 0.461109i −0.862044 0.506834i \(-0.830816\pi\)
−0.251171 + 0.967943i \(0.580816\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 8.36004 + 6.48921i 0.789949 + 0.613172i
\(113\) 13.5524i 1.27490i 0.770490 + 0.637452i \(0.220012\pi\)
−0.770490 + 0.637452i \(0.779988\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −4.16666 1.42735i −0.386865 0.132526i
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 21.3612 + 21.3612i 1.94193 + 1.94193i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) −4.61838 10.2309i −0.411438 0.911438i
\(127\) 16.0000i 1.41977i −0.704317 0.709885i \(-0.748747\pi\)
0.704317 0.709885i \(-0.251253\pi\)
\(128\) 11.1441 + 1.95171i 0.985008 + 0.172508i
\(129\) 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
\(134\) −9.45956 + 10.0717i −0.817181 + 0.870063i
\(135\) 0 0
\(136\) 0 0
\(137\) 5.58301 5.58301i 0.476988 0.476988i −0.427179 0.904167i \(-0.640493\pi\)
0.904167 + 0.427179i \(0.140493\pi\)
\(138\) 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
\(144\) −9.47939 7.35807i −0.789949 0.613172i
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) −13.4380 + 6.58040i −1.10459 + 0.540905i
\(149\) −0.519832 + 1.25499i −0.0425863 + 0.102812i −0.943741 0.330684i \(-0.892720\pi\)
0.901155 + 0.433497i \(0.142720\pi\)
\(150\) 0 0
\(151\) 9.35425 + 9.35425i 0.761238 + 0.761238i 0.976546 0.215308i \(-0.0690756\pi\)
−0.215308 + 0.976546i \(0.569076\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 16.4439 17.5080i 1.32508 1.41083i
\(155\) 0 0
\(156\) 0 0
\(157\) 0 0 −0.382683 0.923880i \(-0.625000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(158\) 7.17712 3.23987i 0.570981 0.257750i
\(159\) 0 0
\(160\) 0 0
\(161\) −24.8661 −1.95973
\(162\) 5.23675 + 11.6007i 0.411438 + 0.911438i
\(163\) −6.47867 + 2.68355i −0.507449 + 0.210192i −0.621694 0.783260i \(-0.713555\pi\)
0.114245 + 0.993453i \(0.463555\pi\)
\(164\) 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
\(172\) −23.5510 + 11.5326i −1.79574 + 0.879352i
\(173\) 0 0 0.923880 0.382683i \(-0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(174\) 0 0
\(175\) −13.2288 −1.00000
\(176\) 6.78399 24.7654i 0.511363 1.86676i
\(177\) 0 0
\(178\) 0 0
\(179\) −2.46040 5.93993i −0.183899 0.443971i 0.804865 0.593458i \(-0.202238\pi\)
−0.988764 + 0.149487i \(0.952238\pi\)
\(180\) 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
\(184\) −23.4963 + 12.4331i −1.73217 + 0.916578i
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 24.3651i 1.76300i −0.472184 0.881500i \(-0.656534\pi\)
0.472184 0.881500i \(-0.343466\pi\)
\(192\) 0 0
\(193\) 2.23871 0.161146 0.0805728 0.996749i \(-0.474325\pi\)
0.0805728 + 0.996749i \(0.474325\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) −10.5000 9.26013i −0.750000 0.661438i
\(197\) 18.2255 + 7.54927i 1.29852 + 0.537863i 0.921513 0.388348i \(-0.126954\pi\)
0.377004 + 0.926212i \(0.376954\pi\)
\(198\) −18.6456 + 19.8522i −1.32508 + 1.41083i
\(199\) 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
\(203\) 5.38291 + 2.22968i 0.377807 + 0.156493i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 28.1955 1.95973
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) −7.59678 18.3403i −0.522984 1.26260i −0.936041 0.351890i \(-0.885539\pi\)
0.413057 0.910705i \(-0.364461\pi\)
\(212\) −11.9660 24.4360i −0.821827 1.67827i
\(213\) 0 0
\(214\) 23.2561 0.728898i 1.58975 0.0498264i
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 12.1796 12.9678i 0.824909 0.878291i
\(219\) 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
\(226\) −7.88562 17.4686i −0.524544 1.16200i
\(227\) 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
\(232\) 6.20121 0.584611i 0.407130 0.0383816i
\(233\) −21.5830 + 21.5830i −1.41395 + 1.41395i −0.693316 + 0.720634i \(0.743851\pi\)
−0.720634 + 0.693316i \(0.756149\pi\)
\(234\) 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 \(0.5\pi\)
\(242\) −39.9631 15.1046i −2.56893 0.970963i
\(243\) 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
\(253\) 23.0885 + 55.7406i 1.45156 + 3.50438i
\(254\) 9.30978 + 20.6235i 0.584147 + 1.29403i
\(255\) 0 0
\(256\) −15.5000 + 3.96863i −0.968750 + 0.248039i
\(257\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(258\) 0 0
\(259\) 18.2870 7.57473i 1.13630 0.470671i
\(260\) 0 0
\(261\) −6.10365 2.52822i −0.377807 0.156493i
\(262\) 0 0
\(263\) 3.74166 3.74166i 0.230720 0.230720i −0.582273 0.812993i \(-0.697836\pi\)
0.812993 + 0.582273i \(0.197836\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 6.33273 18.4863i 0.386833 1.12923i
\(269\) 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
\(274\) −3.94778 + 10.4448i −0.238494 + 0.630996i
\(275\) 12.2831 + 29.6540i 0.740697 + 1.78820i
\(276\) 0 0
\(277\) 8.18507 19.7605i 0.491793 1.18729i −0.462014 0.886873i \(-0.652873\pi\)
0.953807 0.300421i \(-0.0971271\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 2.41699 + 2.41699i 0.144186 + 0.144186i 0.775515 0.631329i \(-0.217490\pi\)
−0.631329 + 0.775515i \(0.717490\pi\)
\(282\) 0 0
\(283\) 0 0 0.382683 0.923880i \(-0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(284\) 2.00393 + 31.9372i 0.118912 + 1.89512i
\(285\) 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
\(296\) 13.4922 16.3009i 0.784220 0.947473i
\(297\) 0 0
\(298\) −0.0601804 1.92011i −0.00348615 0.111229i
\(299\) 0 0
\(300\) 0 0
\(301\) 32.0492 13.2752i 1.84729 0.765171i
\(302\) −17.5002 6.61445i −1.00702 0.380619i
\(303\) 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 \(0.375000\pi\)
\(308\) −11.0084 + 32.1353i −0.627261 + 1.83108i
\(309\) 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
\(316\) −7.36593 + 8.35218i −0.414366 + 0.469847i
\(317\) −10.4710 25.2792i −0.588108 1.41982i −0.885309 0.465004i \(-0.846053\pi\)
0.297200 0.954815i \(-0.403947\pi\)
\(318\) 0 0
\(319\) 14.1368i 0.791508i
\(320\) 0 0
\(321\) 0 0
\(322\) 32.0516 14.4686i 1.78617 0.806305i
\(323\) 0 0
\(324\) −13.5000 11.9059i −0.750000 0.661438i
\(325\) 0 0
\(326\) 6.78934 7.22870i 0.376027 0.400361i
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −17.2445 7.14291i −0.947844 0.392610i −0.145424 0.989369i \(-0.546455\pi\)
−0.802420 + 0.596760i \(0.796455\pi\)
\(332\) 0 0
\(333\) −20.7355 + 8.58893i −1.13630 + 0.470671i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 36.1798i 1.97084i −0.170136 0.985421i \(-0.554421\pi\)
0.170136 0.985421i \(-0.445579\pi\)
\(338\) −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
\(344\) 23.6461 28.5685i 1.27491 1.54031i
\(345\) 0 0
\(346\) 0 0
\(347\) −12.5100 + 30.2018i −0.671573 + 1.62132i 0.107366 + 0.994220i \(0.465758\pi\)
−0.778938 + 0.627100i \(0.784242\pi\)
\(348\) 0 0
\(349\) 0 0 −0.382683 0.923880i \(-0.625000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(350\) 17.0514 7.69730i 0.911438 0.411438i
\(351\) 0 0
\(352\) 5.66568 + 35.8692i 0.301982 + 1.91183i
\(353\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 6.62759 + 6.22477i 0.350279 + 0.328989i
\(359\) 14.5203 14.5203i 0.766350 0.766350i −0.211112 0.977462i \(-0.567708\pi\)
0.977462 + 0.211112i \(0.0677085\pi\)
\(360\) 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 \(\pi\)
\(368\) 23.0516 29.6974i 1.20165 1.54808i
\(369\) 0 0
\(370\) 0 0
\(371\) 13.7741 + 33.2536i 0.715116 + 1.72644i
\(372\) 0 0
\(373\) 3.12864 7.55320i 0.161995 0.391090i −0.821951 0.569558i \(-0.807114\pi\)
0.983946 + 0.178468i \(0.0571142\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 1.18182 2.85316i 0.0607059 0.146557i −0.890616 0.454756i \(-0.849726\pi\)
0.951322 + 0.308199i \(0.0997264\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 14.1771 + 31.4059i 0.725365 + 1.60686i
\(383\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −2.88562 + 1.30262i −0.146874 + 0.0663014i
\(387\) −36.3404 + 15.0527i −1.84729 + 0.765171i
\(388\) 0 0
\(389\) −4.40187 1.82331i −0.223184 0.0924457i 0.268290 0.963338i \(-0.413542\pi\)
−0.491473 + 0.870893i \(0.663542\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 18.9223 + 5.82647i 0.955719 + 0.294281i
\(393\) 0 0
\(394\) −27.8848 + 0.873970i −1.40481 + 0.0440300i
\(395\) 0 0
\(396\) 12.4823 36.4380i 0.627261 1.83108i
\(397\) 0 0 0.923880 0.382683i \(-0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 12.2634 15.7990i 0.613172 0.789949i
\(401\) 9.07500i 0.453184i 0.973990 + 0.226592i \(0.0727584\pi\)
−0.973990 + 0.226592i \(0.927242\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) −8.23577 + 0.258127i −0.408734 + 0.0128106i
\(407\) −33.9595 33.9595i −1.68331 1.68331i
\(408\) 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
\(414\) −36.3431 + 16.4059i −1.78617 + 0.806305i
\(415\) 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
\(421\) 33.4174 + 13.8419i 1.62866 + 0.674614i 0.995081 0.0990621i \(-0.0315842\pi\)
0.633581 + 0.773676i \(0.281584\pi\)
\(422\) 20.4635 + 19.2197i 0.996147 + 0.935602i
\(423\) 0 0
\(424\) 29.6421 + 24.5347i 1.43955 + 1.19151i
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) −29.5522 + 14.4713i −1.42846 + 0.699499i
\(429\) 0 0
\(430\) 0 0
\(431\) 26.4575 1.27441 0.637207 0.770693i \(-0.280090\pi\)
0.637207 + 0.770693i \(0.280090\pi\)
\(432\) 0 0
\(433\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −8.15369 + 23.8019i −0.390491 + 1.13991i
\(437\) 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
\(443\) −12.4955 + 30.1669i −0.593680 + 1.43327i 0.286244 + 0.958157i \(0.407593\pi\)
−0.879924 + 0.475114i \(0.842407\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 20.7931 3.95564i 0.982382 0.186886i
\(449\) −42.3320 −1.99777 −0.998886 0.0471929i \(-0.984972\pi\)
−0.998886 + 0.0471929i \(0.984972\pi\)
\(450\) −19.3345 + 8.72791i −0.911438 + 0.411438i
\(451\) 0 0
\(452\) 20.3286 + 17.9282i 0.956178 + 0.843270i
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 29.9333 29.9333i 1.40022 1.40022i 0.600878 0.799341i \(-0.294818\pi\)
0.799341 0.600878i \(-0.205182\pi\)
\(458\) 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
\(463\) −17.0593 −0.792813 −0.396406 0.918075i \(-0.629743\pi\)
−0.396406 + 0.918075i \(0.629743\pi\)
\(464\) −7.65300 + 4.36179i −0.355282 + 0.202491i
\(465\) 0 0
\(466\) 15.2615 40.3781i 0.706975 1.87048i
\(467\) 0 0 −0.382683 0.923880i \(-0.625000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(468\) 0 0
\(469\) −9.89241 + 23.8824i −0.456789 + 1.10279i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −59.5163 59.5163i −2.73656 2.73656i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −15.6184 37.7061i −0.715116 1.72644i
\(478\) 20.6235 9.30978i 0.943296 0.425819i
\(479\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 60.3000 3.78358i 2.74091 0.171981i
\(485\) 0 0
\(486\) 0 0
\(487\) −30.5203 + 30.5203i −1.38300 + 1.38300i −0.543772 + 0.839233i \(0.683004\pi\)
−0.839233 + 0.543772i \(0.816996\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 39.4272 + 16.3313i 1.77932 + 0.737020i 0.992846 + 0.119401i \(0.0380974\pi\)
0.786478 + 0.617619i \(0.211903\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 42.3320i 1.89885i
\(498\) 0 0
\(499\) −8.85331 21.3738i −0.396329 0.956822i −0.988529 0.151031i \(-0.951741\pi\)
0.592200 0.805791i \(-0.298259\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(504\) −21.4558 6.60659i −0.955719 0.294281i
\(505\) 0 0
\(506\) −62.1937 58.4136i −2.76485 2.59680i
\(507\) 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
\(512\) 17.6698 14.1343i 0.780903 0.624653i
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) −19.1639 + 20.4041i −0.842015 + 0.896504i
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(522\) 9.33848 0.292689i 0.408734 0.0128106i
\(523\) 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
\(529\) 65.3320i 2.84052i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 2.59375 + 27.5130i 0.112033 + 1.18838i
\(537\) 0 0
\(538\) 0 0
\(539\) 17.1963 41.5156i 0.740697 1.78820i
\(540\) 0 0
\(541\) −6.24579 15.0787i −0.268527 0.648282i 0.730887 0.682498i \(-0.239107\pi\)
−0.999414 + 0.0342160i \(0.989107\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −29.1061 + 12.0561i −1.24449 + 0.515483i −0.905114 0.425169i \(-0.860215\pi\)
−0.339372 + 0.940652i \(0.610215\pi\)
\(548\) −0.988886 15.7601i −0.0422431 0.673239i
\(549\) 0 0
\(550\) −33.0870 31.0760i −1.41083 1.32508i
\(551\) 0 0
\(552\) 0 0
\(553\) 10.4170 10.4170i 0.442976 0.442976i
\(554\) 0.947576 + 30.2332i 0.0402587 + 1.28449i
\(555\) 0 0
\(556\) 0 0
\(557\) 43.0051 17.8133i 1.82219 0.754774i 0.847644 0.530566i \(-0.178020\pi\)
0.974541 0.224208i \(-0.0719796\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) −4.52178 1.70907i −0.190740 0.0720929i
\(563\) 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 \(0.0974392\pi\)
0.301356 + 0.953512i \(0.402561\pi\)
\(570\) 0 0
\(571\) −17.4232 + 42.0634i −0.729140 + 1.76030i −0.0836974 + 0.996491i \(0.526673\pi\)
−0.645443 + 0.763809i \(0.723327\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 46.9926i 1.95973i
\(576\) −23.5772 + 4.48527i −0.982382 + 0.186886i
\(577\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(578\) 21.9125 9.89164i 0.911438 0.411438i
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 61.7529 61.7529i 2.55754 2.55754i
\(584\) 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
\(592\) −7.90617 + 28.8620i −0.324942 + 1.18622i
\(593\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 1.19481 + 2.43994i 0.0489411 + 0.0999437i
\(597\) 0 0
\(598\) 0 0
\(599\) −26.1916 26.1916i −1.07016 1.07016i −0.997346 0.0728143i \(-0.976802\pi\)
−0.0728143 0.997346i \(-0.523198\pi\)
\(600\) 0 0
\(601\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(602\) −33.5861 + 35.7595i −1.36887 + 1.45745i
\(603\) 11.2169 27.0801i 0.456789 1.10279i
\(604\) 26.4059 1.65687i 1.07444 0.0674169i
\(605\) 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
\(613\) −21.2101 8.78549i −0.856666 0.354843i −0.0892633 0.996008i \(-0.528451\pi\)
−0.767403 + 0.641165i \(0.778451\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) −4.50880 47.8267i −0.181665 1.92699i
\(617\) −29.9333 + 29.9333i −1.20507 + 1.20507i −0.232462 + 0.972605i \(0.574678\pi\)
−0.972605 + 0.232462i \(0.925322\pi\)
\(618\) 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 \(0.646829\pi\)
\(632\) 4.63464 15.0516i 0.184356 0.598722i
\(633\) 0 0
\(634\) 28.2057 + 26.4914i 1.12019 + 1.05211i
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 8.22564 + 18.2219i 0.325656 + 0.721410i
\(639\) 48.0000i 1.89885i
\(640\) 0 0
\(641\) 47.4935 1.87588 0.937941 0.346795i \(-0.112730\pi\)
0.937941 + 0.346795i \(0.112730\pi\)
\(642\) 0 0
\(643\) 0 0 0.923880 0.382683i \(-0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(644\) −32.8948 + 37.2992i −1.29624 + 1.46979i
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(648\) 24.3286 + 7.49117i 0.955719 + 0.294281i
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) −4.54515 + 13.2680i −0.178002 + 0.519616i
\(653\) 38.9360 16.1278i 1.52368 0.631131i 0.545358 0.838203i \(-0.316394\pi\)
0.978326 + 0.207072i \(0.0663936\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 2.91049 + 7.02655i 0.113377 + 0.273716i 0.970375 0.241604i \(-0.0776734\pi\)
−0.856998 + 0.515319i \(0.827673\pi\)
\(660\) 0 0
\(661\) 0 0 0.382683 0.923880i \(-0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(662\) 26.3838 0.826926i 1.02543 0.0321394i
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 21.7299 23.1360i 0.842015 0.896504i
\(667\) 7.92049 19.1217i 0.306682 0.740397i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 42.3320 1.63178 0.815890 0.578208i \(-0.196248\pi\)
0.815890 + 0.578208i \(0.196248\pi\)
\(674\) 21.0516 + 46.6346i 0.810879 + 1.79630i
\(675\) 0 0
\(676\) −1.62820 25.9490i −0.0626229 0.998037i
\(677\) 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
\(683\) 42.5139 + 17.6099i 1.62675 + 0.673822i 0.994862 0.101237i \(-0.0322800\pi\)
0.631889 + 0.775059i \(0.282280\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −24.5000 9.26013i −0.935414 0.353553i
\(687\) 0 0
\(688\) −13.8561 + 50.5826i −0.528259 + 1.92844i
\(689\) 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
\(693\) −19.4988 + 47.0742i −0.740697 + 1.78820i
\(694\) −1.44827 46.2083i −0.0549756 1.75404i
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) −17.5000 + 19.8431i −0.661438 + 0.750000i
\(701\) 19.0012 + 45.8729i 0.717665 + 1.73260i 0.679895 + 0.733309i \(0.262025\pi\)
0.0377695 + 0.999286i \(0.487975\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) −28.1737 42.9376i −1.06184 1.61827i
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −23.8296 9.87055i −0.894940 0.370696i −0.112667 0.993633i \(-0.535939\pi\)
−0.782272 + 0.622937i \(0.785939\pi\)
\(710\) 0 0
\(711\) −11.8118 + 11.8118i −0.442976 + 0.442976i
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) −12.1647 4.16719i −0.454616 0.155735i
\(717\) 0 0
\(718\) −10.2674 + 27.1649i −0.383175 + 1.01379i
\(719\) 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
\(725\) 4.21369 10.1728i 0.156493 0.377807i
\(726\) 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
\(736\) −12.4331 + 51.6918i −0.458289 + 1.90539i
\(737\) 62.7207 2.31035
\(738\) 0 0
\(739\) −4.83504 + 2.00274i −0.177860 + 0.0736719i −0.469836 0.882754i \(-0.655687\pi\)
0.291977 + 0.956425i \(0.405687\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −37.1034 34.8483i −1.36211 1.27932i
\(743\) 1.47974 1.47974i 0.0542864 0.0542864i −0.679442 0.733729i \(-0.737778\pi\)
0.733729 + 0.679442i \(0.237778\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0.362199 + 11.5563i 0.0132610 + 0.423105i
\(747\) 0 0
\(748\) 0 0
\(749\) 40.2161 16.6580i 1.46946 0.608671i
\(750\) 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
\(757\) −4.16646 + 10.0587i −0.151433 + 0.365591i −0.981332 0.192323i \(-0.938398\pi\)
0.829899 + 0.557913i \(0.188398\pi\)
\(758\) 0.136818 + 4.36528i 0.00496944 + 0.158554i
\(759\) 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
\(763\) 12.7370 30.7497i 0.461109 1.11321i
\(764\) −36.5477 32.2321i −1.32225 1.16611i
\(765\) 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
\(772\) 2.96153 3.35806i 0.106588 0.120859i
\(773\) 0 0 −0.923880 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(774\) 38.0830 40.5475i 1.36887 1.45745i
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 6.73478 0.211083i 0.241454 0.00756769i
\(779\) 0 0
\(780\) 0 0
\(781\) −94.8927 + 39.3058i −3.39553 + 1.40647i
\(782\) 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 \(0.375000\pi\)
\(788\) 35.4340 17.3516i 1.26229 0.618124i
\(789\) 0 0
\(790\) 0 0
\(791\) −25.3542 25.3542i −0.901493 0.901493i
\(792\) 5.11249 + 54.2303i 0.181665 + 1.92699i
\(793\) 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
\(802\) −5.28039 11.6974i −0.186457 0.413049i
\(803\) 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 \(0.517149\pi\)
−0.998549 + 0.0538482i \(0.982851\pi\)
\(810\) 0 0
\(811\) 0 0 −0.923880 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(812\) 10.4654 5.12479i 0.367265 0.179845i
\(813\) 0 0
\(814\) 63.5323 + 24.0130i 2.22681 + 0.841654i
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −15.5274 + 37.4865i −0.541910 + 1.30829i 0.381464 + 0.924384i \(0.375420\pi\)
−0.923374 + 0.383903i \(0.874580\pi\)
\(822\) 0 0
\(823\) −33.6749 33.6749i −1.17383 1.17383i −0.981288 0.192546i \(-0.938326\pi\)
−0.192546 0.981288i \(-0.561674\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 10.1319 24.4605i 0.352320 0.850577i −0.644013 0.765015i \(-0.722732\pi\)
0.996333 0.0855616i \(-0.0272685\pi\)
\(828\) 37.2992 42.2933i 1.29624 1.46979i
\(829\) 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
\(841\) 17.0769 17.0769i 0.588859 0.588859i
\(842\) −51.1280 + 1.60246i −1.76199 + 0.0552246i
\(843\) 0 0
\(844\) −37.5600 12.8667i −1.29287 0.442890i
\(845\) 0 0
\(846\) 0 0
\(847\) −79.9262 −2.74630
\(848\) −52.4835 14.3768i −1.80229 0.493702i
\(849\) 0 0
\(850\) 0 0
\(851\) −26.9077 64.9610i −0.922385 2.22683i
\(852\) 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
\(856\) 29.6716 35.8484i 1.01415 1.22527i
\(857\) 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
\(862\) −34.1029 + 15.3946i −1.16155 + 0.524342i
\(863\) 35.5014i 1.20848i −0.796802 0.604240i \(-0.793477\pi\)
0.796802 0.604240i \(-0.206523\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −33.0234 13.6787i −1.12024 0.464019i
\(870\) 0 0
\(871\) 0 0
\(872\) −3.33957 35.4242i −0.113092 1.19962i
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 9.42181 3.90264i 0.318152 0.131783i −0.217892 0.975973i \(-0.569918\pi\)
0.536044 + 0.844190i \(0.319918\pi\)
\(878\) 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
\(883\) −22.3365 53.9250i −0.751682 1.81472i −0.549766 0.835319i \(-0.685283\pi\)
−0.201916 0.979403i \(-0.564717\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −1.44659 46.1548i −0.0485992 1.55060i
\(887\) 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
\(891\) 22.1095 53.3771i 0.740697 1.78820i
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) −24.5000 + 17.1974i −0.818488 + 0.574524i
\(897\) 0 0
\(898\) 54.5646 24.6314i 1.82084 0.821959i
\(899\) 0 0
\(900\) 19.8431 22.5000i 0.661438 0.750000i
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) −36.6346 11.2804i −1.21845 0.375180i
\(905\) 0 0
\(906\) 0 0
\(907\) −16.6966 6.91597i −0.554402 0.229641i 0.0878507 0.996134i \(-0.472000\pi\)
−0.642253 + 0.766493i \(0.722000\pi\)
\(908\) 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
\(925\) −14.3149 34.5592i −0.470671 1.13630i
\(926\) 21.9889 9.92614i 0.722600 0.326193i
\(927\) 0 0
\(928\) 7.32652 10.0752i 0.240505 0.330734i
\(929\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 3.82288 + 60.9261i 0.125222 + 1.99570i
\(933\) 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 \(0.750000\pi\)
\(938\) −1.14523 36.5397i −0.0373932 1.19306i
\(939\) 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
\(946\) 111.345 + 42.0844i 3.62013 + 1.36828i
\(947\) 17.6502 + 42.6113i 0.573554 + 1.38468i 0.898510 + 0.438953i \(0.144650\pi\)
−0.324956 + 0.945729i \(0.605350\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −18.4170 18.4170i −0.596585 0.596585i 0.342817 0.939402i \(-0.388619\pi\)
−0.939402 + 0.342817i \(0.888619\pi\)
\(954\) 42.0713 + 39.5142i 1.36211 + 1.27932i
\(955\) 0 0
\(956\) −21.1660 + 24.0000i −0.684558 + 0.776215i
\(957\) 0 0
\(958\) 0 0
\(959\) 20.8897i 0.674563i
\(960\) 0 0
\(961\) −31.0000 −1.00000
\(962\) 0 0
\(963\) −45.6007 + 18.8884i −1.46946 + 0.608671i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −43.8118 + 43.8118i −1.40889 + 1.40889i −0.643157 + 0.765735i \(0.722376\pi\)
−0.765735 + 0.643157i \(0.777624\pi\)
\(968\) −75.5232 + 39.9631i −2.42741 + 1.28446i
\(969\) 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
\(974\) 21.5811 57.0982i 0.691502 1.82954i
\(975\) 0 0
\(976\) 0 0
\(977\) 42.3320i 1.35432i 0.735835 + 0.677161i \(0.236790\pi\)
−0.735835 + 0.677161i \(0.763210\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) −14.4423 + 34.8669i −0.461109 + 1.11321i
\(982\) −60.3229 + 1.89065i −1.92498 + 0.0603332i
\(983\) 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
\(989\) −47.1576 113.849i −1.49953 3.62017i
\(990\) 0 0
\(991\) 58.1288i 1.84652i 0.384173 + 0.923261i \(0.374487\pi\)
−0.384173 + 0.923261i \(0.625513\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 24.6314 + 54.5646i 0.781259 + 1.73068i
\(995\) 0 0
\(996\) 0 0
\(997\) 0 0 −0.923880 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(998\) 23.8482 + 22.3987i 0.754901 + 0.709019i
\(999\) 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.27.1 8
4.3 odd 2 896.2.x.a.783.2 8
7.6 odd 2 CM 224.2.x.a.27.1 8
28.27 even 2 896.2.x.a.783.2 8
32.13 even 8 896.2.x.a.111.2 8
32.19 odd 8 inner 224.2.x.a.83.1 yes 8
224.13 odd 8 896.2.x.a.111.2 8
224.83 even 8 inner 224.2.x.a.83.1 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
224.2.x.a.27.1 8 1.1 even 1 trivial
224.2.x.a.27.1 8 7.6 odd 2 CM
224.2.x.a.83.1 yes 8 32.19 odd 8 inner
224.2.x.a.83.1 yes 8 224.83 even 8 inner
896.2.x.a.111.2 8 32.13 even 8
896.2.x.a.111.2 8 224.13 odd 8
896.2.x.a.783.2 8 4.3 odd 2
896.2.x.a.783.2 8 28.27 even 2