Properties

Label 912.2.f.h.113.9
Level $912$
Weight $2$
Character 912.113
Analytic conductor $7.282$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [912,2,Mod(113,912)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(912, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("912.113");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 912 = 2^{4} \cdot 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 912.f (of order \(2\), degree \(1\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.28235666434\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: 10.0.20322144469993472.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - x^{9} - x^{8} - 2x^{7} - 2x^{6} + 22x^{5} - 6x^{4} - 18x^{3} - 27x^{2} - 81x + 243 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 456)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 113.9
Root \(-1.69134 + 0.373340i\) of defining polynomial
Character \(\chi\) \(=\) 912.113
Dual form 912.2.f.h.113.10

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(1.69134 - 0.373340i) q^{3} +0.568907i q^{5} -0.718465 q^{7} +(2.72123 - 1.26289i) q^{9} +O(q^{10})\) \(q+(1.69134 - 0.373340i) q^{3} +0.568907i q^{5} -0.718465 q^{7} +(2.72123 - 1.26289i) q^{9} +4.78814i q^{11} +4.61037i q^{13} +(0.212396 + 0.962212i) q^{15} +7.10370i q^{17} +(-3.62497 + 2.42067i) q^{19} +(-1.21517 + 0.268232i) q^{21} -6.38946i q^{23} +4.67635 q^{25} +(4.13104 - 3.15191i) q^{27} +5.44247 q^{29} -7.28147i q^{31} +(1.78760 + 8.09835i) q^{33} -0.408739i q^{35} -5.99833i q^{37} +(1.72123 + 7.79768i) q^{39} +8.82514 q^{41} -3.08900 q^{43} +(0.718465 + 1.54813i) q^{45} +0.779123i q^{47} -6.48381 q^{49} +(2.65209 + 12.0147i) q^{51} +5.86726 q^{53} -2.72400 q^{55} +(-5.22730 + 5.44750i) q^{57} +8.45461 q^{59} -11.4893 q^{61} +(-1.95511 + 0.907340i) q^{63} -2.62287 q^{65} -3.87380i q^{67} +(-2.38544 - 10.8067i) q^{69} -7.30973 q^{71} -3.20305 q^{73} +(7.90927 - 1.74587i) q^{75} -3.44011i q^{77} +1.79886i q^{79} +(5.81023 - 6.87322i) q^{81} +1.28314i q^{83} -4.04134 q^{85} +(9.20504 - 2.03189i) q^{87} -0.138279 q^{89} -3.31239i q^{91} +(-2.71846 - 12.3154i) q^{93} +(-1.37713 - 2.06227i) q^{95} -9.25160i q^{97} +(6.04688 + 13.0296i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - q^{3} + 2 q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - q^{3} + 2 q^{7} + 3 q^{9} + 10 q^{15} - 2 q^{19} - 5 q^{21} - 14 q^{25} - 10 q^{27} + 6 q^{29} + 10 q^{33} - 7 q^{39} + 4 q^{41} - 20 q^{43} - 2 q^{45} + 16 q^{49} + q^{51} + 26 q^{53} + 12 q^{55} - 11 q^{57} + 2 q^{59} - 4 q^{61} + 17 q^{63} - 32 q^{65} + 27 q^{69} + 8 q^{71} - 26 q^{73} + 39 q^{75} + 23 q^{81} - 8 q^{85} - 13 q^{87} - 4 q^{89} - 18 q^{93} - 8 q^{95} - 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/912\mathbb{Z}\right)^\times\).

\(n\) \(97\) \(229\) \(305\) \(799\)
\(\chi(n)\) \(-1\) \(1\) \(-1\) \(1\)

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
\(3\) 1.69134 0.373340i 0.976493 0.215548i
\(4\) 0 0
\(5\) 0.568907i 0.254423i 0.991876 + 0.127211i \(0.0406027\pi\)
−0.991876 + 0.127211i \(0.959397\pi\)
\(6\) 0 0
\(7\) −0.718465 −0.271554 −0.135777 0.990739i \(-0.543353\pi\)
−0.135777 + 0.990739i \(0.543353\pi\)
\(8\) 0 0
\(9\) 2.72123 1.26289i 0.907078 0.420962i
\(10\) 0 0
\(11\) 4.78814i 1.44368i 0.692061 + 0.721839i \(0.256703\pi\)
−0.692061 + 0.721839i \(0.743297\pi\)
\(12\) 0 0
\(13\) 4.61037i 1.27869i 0.768922 + 0.639343i \(0.220793\pi\)
−0.768922 + 0.639343i \(0.779207\pi\)
\(14\) 0 0
\(15\) 0.212396 + 0.962212i 0.0548403 + 0.248442i
\(16\) 0 0
\(17\) 7.10370i 1.72290i 0.507843 + 0.861450i \(0.330443\pi\)
−0.507843 + 0.861450i \(0.669557\pi\)
\(18\) 0 0
\(19\) −3.62497 + 2.42067i −0.831624 + 0.555339i
\(20\) 0 0
\(21\) −1.21517 + 0.268232i −0.265171 + 0.0585330i
\(22\) 0 0
\(23\) 6.38946i 1.33229i −0.745820 0.666147i \(-0.767942\pi\)
0.745820 0.666147i \(-0.232058\pi\)
\(24\) 0 0
\(25\) 4.67635 0.935269
\(26\) 0 0
\(27\) 4.13104 3.15191i 0.795018 0.606586i
\(28\) 0 0
\(29\) 5.44247 1.01064 0.505321 0.862932i \(-0.331374\pi\)
0.505321 + 0.862932i \(0.331374\pi\)
\(30\) 0 0
\(31\) 7.28147i 1.30779i −0.756585 0.653895i \(-0.773134\pi\)
0.756585 0.653895i \(-0.226866\pi\)
\(32\) 0 0
\(33\) 1.78760 + 8.09835i 0.311182 + 1.40974i
\(34\) 0 0
\(35\) 0.408739i 0.0690896i
\(36\) 0 0
\(37\) 5.99833i 0.986119i −0.869996 0.493059i \(-0.835879\pi\)
0.869996 0.493059i \(-0.164121\pi\)
\(38\) 0 0
\(39\) 1.72123 + 7.79768i 0.275618 + 1.24863i
\(40\) 0 0
\(41\) 8.82514 1.37826 0.689128 0.724640i \(-0.257994\pi\)
0.689128 + 0.724640i \(0.257994\pi\)
\(42\) 0 0
\(43\) −3.08900 −0.471068 −0.235534 0.971866i \(-0.575684\pi\)
−0.235534 + 0.971866i \(0.575684\pi\)
\(44\) 0 0
\(45\) 0.718465 + 1.54813i 0.107102 + 0.230781i
\(46\) 0 0
\(47\) 0.779123i 0.113647i 0.998384 + 0.0568234i \(0.0180972\pi\)
−0.998384 + 0.0568234i \(0.981903\pi\)
\(48\) 0 0
\(49\) −6.48381 −0.926258
\(50\) 0 0
\(51\) 2.65209 + 12.0147i 0.371368 + 1.68240i
\(52\) 0 0
\(53\) 5.86726 0.805930 0.402965 0.915215i \(-0.367980\pi\)
0.402965 + 0.915215i \(0.367980\pi\)
\(54\) 0 0
\(55\) −2.72400 −0.367305
\(56\) 0 0
\(57\) −5.22730 + 5.44750i −0.692373 + 0.721540i
\(58\) 0 0
\(59\) 8.45461 1.10070 0.550348 0.834935i \(-0.314495\pi\)
0.550348 + 0.834935i \(0.314495\pi\)
\(60\) 0 0
\(61\) −11.4893 −1.47106 −0.735530 0.677492i \(-0.763067\pi\)
−0.735530 + 0.677492i \(0.763067\pi\)
\(62\) 0 0
\(63\) −1.95511 + 0.907340i −0.246321 + 0.114314i
\(64\) 0 0
\(65\) −2.62287 −0.325327
\(66\) 0 0
\(67\) 3.87380i 0.473260i −0.971600 0.236630i \(-0.923957\pi\)
0.971600 0.236630i \(-0.0760430\pi\)
\(68\) 0 0
\(69\) −2.38544 10.8067i −0.287173 1.30098i
\(70\) 0 0
\(71\) −7.30973 −0.867505 −0.433753 0.901032i \(-0.642811\pi\)
−0.433753 + 0.901032i \(0.642811\pi\)
\(72\) 0 0
\(73\) −3.20305 −0.374889 −0.187444 0.982275i \(-0.560020\pi\)
−0.187444 + 0.982275i \(0.560020\pi\)
\(74\) 0 0
\(75\) 7.90927 1.74587i 0.913284 0.201595i
\(76\) 0 0
\(77\) 3.44011i 0.392037i
\(78\) 0 0
\(79\) 1.79886i 0.202387i 0.994867 + 0.101194i \(0.0322662\pi\)
−0.994867 + 0.101194i \(0.967734\pi\)
\(80\) 0 0
\(81\) 5.81023 6.87322i 0.645581 0.763691i
\(82\) 0 0
\(83\) 1.28314i 0.140843i 0.997517 + 0.0704217i \(0.0224345\pi\)
−0.997517 + 0.0704217i \(0.977566\pi\)
\(84\) 0 0
\(85\) −4.04134 −0.438345
\(86\) 0 0
\(87\) 9.20504 2.03189i 0.986884 0.217842i
\(88\) 0 0
\(89\) −0.138279 −0.0146576 −0.00732878 0.999973i \(-0.502333\pi\)
−0.00732878 + 0.999973i \(0.502333\pi\)
\(90\) 0 0
\(91\) 3.31239i 0.347232i
\(92\) 0 0
\(93\) −2.71846 12.3154i −0.281892 1.27705i
\(94\) 0 0
\(95\) −1.37713 2.06227i −0.141291 0.211584i
\(96\) 0 0
\(97\) 9.25160i 0.939358i −0.882837 0.469679i \(-0.844370\pi\)
0.882837 0.469679i \(-0.155630\pi\)
\(98\) 0 0
\(99\) 6.04688 + 13.0296i 0.607734 + 1.30953i
\(100\) 0 0
\(101\) 10.9095i 1.08553i 0.839884 + 0.542766i \(0.182623\pi\)
−0.839884 + 0.542766i \(0.817377\pi\)
\(102\) 0 0
\(103\) 18.8471i 1.85706i 0.371262 + 0.928528i \(0.378925\pi\)
−0.371262 + 0.928528i \(0.621075\pi\)
\(104\) 0 0
\(105\) −0.152599 0.691316i −0.0148921 0.0674655i
\(106\) 0 0
\(107\) −4.77088 −0.461219 −0.230609 0.973046i \(-0.574072\pi\)
−0.230609 + 0.973046i \(0.574072\pi\)
\(108\) 0 0
\(109\) 10.7040i 1.02525i −0.858611 0.512627i \(-0.828672\pi\)
0.858611 0.512627i \(-0.171328\pi\)
\(110\) 0 0
\(111\) −2.23942 10.1452i −0.212556 0.962938i
\(112\) 0 0
\(113\) −0.508865 −0.0478700 −0.0239350 0.999714i \(-0.507619\pi\)
−0.0239350 + 0.999714i \(0.507619\pi\)
\(114\) 0 0
\(115\) 3.63501 0.338966
\(116\) 0 0
\(117\) 5.82237 + 12.5459i 0.538278 + 1.15987i
\(118\) 0 0
\(119\) 5.10376i 0.467861i
\(120\) 0 0
\(121\) −11.9263 −1.08421
\(122\) 0 0
\(123\) 14.9263 3.29478i 1.34586 0.297080i
\(124\) 0 0
\(125\) 5.50494i 0.492376i
\(126\) 0 0
\(127\) 11.4815i 1.01882i −0.860524 0.509410i \(-0.829864\pi\)
0.860524 0.509410i \(-0.170136\pi\)
\(128\) 0 0
\(129\) −5.22453 + 1.15325i −0.459995 + 0.101538i
\(130\) 0 0
\(131\) 5.92595i 0.517753i 0.965910 + 0.258876i \(0.0833522\pi\)
−0.965910 + 0.258876i \(0.916648\pi\)
\(132\) 0 0
\(133\) 2.60441 1.73916i 0.225831 0.150805i
\(134\) 0 0
\(135\) 1.79314 + 2.35017i 0.154329 + 0.202271i
\(136\) 0 0
\(137\) 15.1420i 1.29367i −0.762632 0.646833i \(-0.776093\pi\)
0.762632 0.646833i \(-0.223907\pi\)
\(138\) 0 0
\(139\) −5.36975 −0.455457 −0.227728 0.973725i \(-0.573130\pi\)
−0.227728 + 0.973725i \(0.573130\pi\)
\(140\) 0 0
\(141\) 0.290878 + 1.31776i 0.0244963 + 0.110975i
\(142\) 0 0
\(143\) −22.0751 −1.84601
\(144\) 0 0
\(145\) 3.09626i 0.257130i
\(146\) 0 0
\(147\) −10.9663 + 2.42067i −0.904485 + 0.199653i
\(148\) 0 0
\(149\) 2.63373i 0.215764i −0.994164 0.107882i \(-0.965593\pi\)
0.994164 0.107882i \(-0.0344069\pi\)
\(150\) 0 0
\(151\) 6.57038i 0.534690i −0.963601 0.267345i \(-0.913854\pi\)
0.963601 0.267345i \(-0.0861463\pi\)
\(152\) 0 0
\(153\) 8.97117 + 19.3308i 0.725276 + 1.56280i
\(154\) 0 0
\(155\) 4.14248 0.332732
\(156\) 0 0
\(157\) 12.9676 1.03493 0.517464 0.855705i \(-0.326876\pi\)
0.517464 + 0.855705i \(0.326876\pi\)
\(158\) 0 0
\(159\) 9.92351 2.19048i 0.786985 0.173717i
\(160\) 0 0
\(161\) 4.59060i 0.361790i
\(162\) 0 0
\(163\) 16.0938 1.26056 0.630280 0.776368i \(-0.282940\pi\)
0.630280 + 0.776368i \(0.282940\pi\)
\(164\) 0 0
\(165\) −4.60721 + 1.01698i −0.358670 + 0.0791718i
\(166\) 0 0
\(167\) −10.0243 −0.775702 −0.387851 0.921722i \(-0.626782\pi\)
−0.387851 + 0.921722i \(0.626782\pi\)
\(168\) 0 0
\(169\) −8.25547 −0.635036
\(170\) 0 0
\(171\) −6.80735 + 11.1651i −0.520571 + 0.853818i
\(172\) 0 0
\(173\) −15.1714 −1.15346 −0.576732 0.816933i \(-0.695672\pi\)
−0.576732 + 0.816933i \(0.695672\pi\)
\(174\) 0 0
\(175\) −3.35979 −0.253976
\(176\) 0 0
\(177\) 14.2996 3.15644i 1.07482 0.237253i
\(178\) 0 0
\(179\) 19.7288 1.47460 0.737299 0.675567i \(-0.236101\pi\)
0.737299 + 0.675567i \(0.236101\pi\)
\(180\) 0 0
\(181\) 19.4191i 1.44341i 0.692200 + 0.721705i \(0.256641\pi\)
−0.692200 + 0.721705i \(0.743359\pi\)
\(182\) 0 0
\(183\) −19.4323 + 4.28943i −1.43648 + 0.317084i
\(184\) 0 0
\(185\) 3.41249 0.250891
\(186\) 0 0
\(187\) −34.0135 −2.48731
\(188\) 0 0
\(189\) −2.96800 + 2.26454i −0.215890 + 0.164721i
\(190\) 0 0
\(191\) 15.8477i 1.14670i −0.819312 0.573348i \(-0.805644\pi\)
0.819312 0.573348i \(-0.194356\pi\)
\(192\) 0 0
\(193\) 8.50964i 0.612537i −0.951945 0.306269i \(-0.900919\pi\)
0.951945 0.306269i \(-0.0990805\pi\)
\(194\) 0 0
\(195\) −4.43615 + 0.979222i −0.317679 + 0.0701235i
\(196\) 0 0
\(197\) 14.0057i 0.997867i −0.866640 0.498933i \(-0.833725\pi\)
0.866640 0.498933i \(-0.166275\pi\)
\(198\) 0 0
\(199\) 19.6972 1.39629 0.698147 0.715954i \(-0.254008\pi\)
0.698147 + 0.715954i \(0.254008\pi\)
\(200\) 0 0
\(201\) −1.44625 6.55190i −0.102010 0.462136i
\(202\) 0 0
\(203\) −3.91022 −0.274444
\(204\) 0 0
\(205\) 5.02068i 0.350660i
\(206\) 0 0
\(207\) −8.06917 17.3872i −0.560846 1.20850i
\(208\) 0 0
\(209\) −11.5905 17.3568i −0.801731 1.20060i
\(210\) 0 0
\(211\) 4.30487i 0.296359i 0.988960 + 0.148180i \(0.0473414\pi\)
−0.988960 + 0.148180i \(0.952659\pi\)
\(212\) 0 0
\(213\) −12.3632 + 2.72901i −0.847113 + 0.186989i
\(214\) 0 0
\(215\) 1.75735i 0.119850i
\(216\) 0 0
\(217\) 5.23148i 0.355136i
\(218\) 0 0
\(219\) −5.41744 + 1.19583i −0.366077 + 0.0808066i
\(220\) 0 0
\(221\) −32.7506 −2.20305
\(222\) 0 0
\(223\) 20.8427i 1.39573i −0.716231 0.697864i \(-0.754134\pi\)
0.716231 0.697864i \(-0.245866\pi\)
\(224\) 0 0
\(225\) 12.7254 5.90570i 0.848362 0.393713i
\(226\) 0 0
\(227\) 9.88254 0.655927 0.327963 0.944690i \(-0.393638\pi\)
0.327963 + 0.944690i \(0.393638\pi\)
\(228\) 0 0
\(229\) 23.8112 1.57349 0.786745 0.617279i \(-0.211765\pi\)
0.786745 + 0.617279i \(0.211765\pi\)
\(230\) 0 0
\(231\) −1.28433 5.81838i −0.0845028 0.382821i
\(232\) 0 0
\(233\) 11.4784i 0.751974i 0.926625 + 0.375987i \(0.122696\pi\)
−0.926625 + 0.375987i \(0.877304\pi\)
\(234\) 0 0
\(235\) −0.443248 −0.0289143
\(236\) 0 0
\(237\) 0.671586 + 3.04247i 0.0436242 + 0.197630i
\(238\) 0 0
\(239\) 2.05755i 0.133092i 0.997783 + 0.0665459i \(0.0211979\pi\)
−0.997783 + 0.0665459i \(0.978802\pi\)
\(240\) 0 0
\(241\) 29.2663i 1.88521i −0.333914 0.942604i \(-0.608370\pi\)
0.333914 0.942604i \(-0.391630\pi\)
\(242\) 0 0
\(243\) 7.26101 13.7941i 0.465794 0.884893i
\(244\) 0 0
\(245\) 3.68868i 0.235661i
\(246\) 0 0
\(247\) −11.1602 16.7124i −0.710104 1.06339i
\(248\) 0 0
\(249\) 0.479049 + 2.17023i 0.0303585 + 0.137533i
\(250\) 0 0
\(251\) 7.57035i 0.477836i −0.971040 0.238918i \(-0.923207\pi\)
0.971040 0.238918i \(-0.0767927\pi\)
\(252\) 0 0
\(253\) 30.5936 1.92340
\(254\) 0 0
\(255\) −6.83526 + 1.50879i −0.428041 + 0.0944844i
\(256\) 0 0
\(257\) −16.5825 −1.03439 −0.517195 0.855867i \(-0.673024\pi\)
−0.517195 + 0.855867i \(0.673024\pi\)
\(258\) 0 0
\(259\) 4.30959i 0.267785i
\(260\) 0 0
\(261\) 14.8102 6.87322i 0.916730 0.425442i
\(262\) 0 0
\(263\) 7.76573i 0.478855i 0.970914 + 0.239428i \(0.0769598\pi\)
−0.970914 + 0.239428i \(0.923040\pi\)
\(264\) 0 0
\(265\) 3.33792i 0.205047i
\(266\) 0 0
\(267\) −0.233877 + 0.0516252i −0.0143130 + 0.00315941i
\(268\) 0 0
\(269\) 14.5185 0.885212 0.442606 0.896716i \(-0.354054\pi\)
0.442606 + 0.896716i \(0.354054\pi\)
\(270\) 0 0
\(271\) −30.8889 −1.87637 −0.938184 0.346137i \(-0.887493\pi\)
−0.938184 + 0.346137i \(0.887493\pi\)
\(272\) 0 0
\(273\) −1.23665 5.60236i −0.0748452 0.339070i
\(274\) 0 0
\(275\) 22.3910i 1.35023i
\(276\) 0 0
\(277\) −10.1351 −0.608959 −0.304479 0.952519i \(-0.598483\pi\)
−0.304479 + 0.952519i \(0.598483\pi\)
\(278\) 0 0
\(279\) −9.19567 19.8146i −0.550531 1.18627i
\(280\) 0 0
\(281\) 9.51541 0.567642 0.283821 0.958877i \(-0.408398\pi\)
0.283821 + 0.958877i \(0.408398\pi\)
\(282\) 0 0
\(283\) 8.16093 0.485117 0.242559 0.970137i \(-0.422013\pi\)
0.242559 + 0.970137i \(0.422013\pi\)
\(284\) 0 0
\(285\) −3.09912 2.97385i −0.183576 0.176156i
\(286\) 0 0
\(287\) −6.34055 −0.374271
\(288\) 0 0
\(289\) −33.4625 −1.96838
\(290\) 0 0
\(291\) −3.45399 15.6476i −0.202477 0.917276i
\(292\) 0 0
\(293\) 3.23864 0.189203 0.0946016 0.995515i \(-0.469842\pi\)
0.0946016 + 0.995515i \(0.469842\pi\)
\(294\) 0 0
\(295\) 4.80988i 0.280042i
\(296\) 0 0
\(297\) 15.0918 + 19.7800i 0.875715 + 1.14775i
\(298\) 0 0
\(299\) 29.4577 1.70359
\(300\) 0 0
\(301\) 2.21934 0.127920
\(302\) 0 0
\(303\) 4.07294 + 18.4516i 0.233984 + 1.06002i
\(304\) 0 0
\(305\) 6.53637i 0.374271i
\(306\) 0 0
\(307\) 0.189980i 0.0108427i −0.999985 0.00542135i \(-0.998274\pi\)
0.999985 0.00542135i \(-0.00172568\pi\)
\(308\) 0 0
\(309\) 7.03636 + 31.8767i 0.400285 + 1.81340i
\(310\) 0 0
\(311\) 5.58052i 0.316442i 0.987404 + 0.158221i \(0.0505759\pi\)
−0.987404 + 0.158221i \(0.949424\pi\)
\(312\) 0 0
\(313\) −17.9012 −1.01184 −0.505918 0.862581i \(-0.668846\pi\)
−0.505918 + 0.862581i \(0.668846\pi\)
\(314\) 0 0
\(315\) −0.516192 1.11228i −0.0290841 0.0626696i
\(316\) 0 0
\(317\) −22.1633 −1.24481 −0.622407 0.782694i \(-0.713845\pi\)
−0.622407 + 0.782694i \(0.713845\pi\)
\(318\) 0 0
\(319\) 26.0593i 1.45904i
\(320\) 0 0
\(321\) −8.06917 + 1.78116i −0.450377 + 0.0994148i
\(322\) 0 0
\(323\) −17.1957 25.7507i −0.956793 1.43280i
\(324\) 0 0
\(325\) 21.5597i 1.19591i
\(326\) 0 0
\(327\) −3.99622 18.1040i −0.220992 1.00115i
\(328\) 0 0
\(329\) 0.559773i 0.0308613i
\(330\) 0 0
\(331\) 25.9533i 1.42652i −0.700898 0.713262i \(-0.747217\pi\)
0.700898 0.713262i \(-0.252783\pi\)
\(332\) 0 0
\(333\) −7.57521 16.3229i −0.415119 0.894487i
\(334\) 0 0
\(335\) 2.20383 0.120408
\(336\) 0 0
\(337\) 29.0950i 1.58491i 0.609933 + 0.792453i \(0.291196\pi\)
−0.609933 + 0.792453i \(0.708804\pi\)
\(338\) 0 0
\(339\) −0.860661 + 0.189980i −0.0467447 + 0.0103183i
\(340\) 0 0
\(341\) 34.8647 1.88803
\(342\) 0 0
\(343\) 9.68764 0.523083
\(344\) 0 0
\(345\) 6.14802 1.35709i 0.330998 0.0730634i
\(346\) 0 0
\(347\) 6.14187i 0.329713i 0.986318 + 0.164856i \(0.0527161\pi\)
−0.986318 + 0.164856i \(0.947284\pi\)
\(348\) 0 0
\(349\) −8.91520 −0.477220 −0.238610 0.971116i \(-0.576692\pi\)
−0.238610 + 0.971116i \(0.576692\pi\)
\(350\) 0 0
\(351\) 14.5315 + 19.0456i 0.775632 + 1.01658i
\(352\) 0 0
\(353\) 21.9955i 1.17070i −0.810780 0.585352i \(-0.800957\pi\)
0.810780 0.585352i \(-0.199043\pi\)
\(354\) 0 0
\(355\) 4.15855i 0.220713i
\(356\) 0 0
\(357\) −1.90544 8.63216i −0.100846 0.456863i
\(358\) 0 0
\(359\) 4.63327i 0.244535i 0.992497 + 0.122267i \(0.0390165\pi\)
−0.992497 + 0.122267i \(0.960983\pi\)
\(360\) 0 0
\(361\) 7.28076 17.5497i 0.383198 0.923666i
\(362\) 0 0
\(363\) −20.1713 + 4.45256i −1.05872 + 0.233699i
\(364\) 0 0
\(365\) 1.82224i 0.0953803i
\(366\) 0 0
\(367\) −19.7087 −1.02878 −0.514392 0.857555i \(-0.671983\pi\)
−0.514392 + 0.857555i \(0.671983\pi\)
\(368\) 0 0
\(369\) 24.0153 11.1452i 1.25019 0.580194i
\(370\) 0 0
\(371\) −4.21542 −0.218854
\(372\) 0 0
\(373\) 13.5397i 0.701057i −0.936552 0.350529i \(-0.886002\pi\)
0.936552 0.350529i \(-0.113998\pi\)
\(374\) 0 0
\(375\) 2.05521 + 9.31070i 0.106131 + 0.480802i
\(376\) 0 0
\(377\) 25.0918i 1.29229i
\(378\) 0 0
\(379\) 26.4548i 1.35889i 0.733726 + 0.679445i \(0.237780\pi\)
−0.733726 + 0.679445i \(0.762220\pi\)
\(380\) 0 0
\(381\) −4.28651 19.4191i −0.219605 0.994871i
\(382\) 0 0
\(383\) 24.7319 1.26374 0.631871 0.775074i \(-0.282287\pi\)
0.631871 + 0.775074i \(0.282287\pi\)
\(384\) 0 0
\(385\) 1.95710 0.0997431
\(386\) 0 0
\(387\) −8.40589 + 3.90106i −0.427295 + 0.198302i
\(388\) 0 0
\(389\) 2.52420i 0.127982i −0.997950 0.0639910i \(-0.979617\pi\)
0.997950 0.0639910i \(-0.0203829\pi\)
\(390\) 0 0
\(391\) 45.3888 2.29541
\(392\) 0 0
\(393\) 2.21240 + 10.0228i 0.111601 + 0.505582i
\(394\) 0 0
\(395\) −1.02338 −0.0514919
\(396\) 0 0
\(397\) 12.4474 0.624719 0.312360 0.949964i \(-0.398881\pi\)
0.312360 + 0.949964i \(0.398881\pi\)
\(398\) 0 0
\(399\) 3.75563 3.91384i 0.188017 0.195937i
\(400\) 0 0
\(401\) −18.1965 −0.908691 −0.454345 0.890826i \(-0.650127\pi\)
−0.454345 + 0.890826i \(0.650127\pi\)
\(402\) 0 0
\(403\) 33.5702 1.67225
\(404\) 0 0
\(405\) 3.91022 + 3.30548i 0.194300 + 0.164251i
\(406\) 0 0
\(407\) 28.7208 1.42364
\(408\) 0 0
\(409\) 23.2016i 1.14724i −0.819120 0.573622i \(-0.805538\pi\)
0.819120 0.573622i \(-0.194462\pi\)
\(410\) 0 0
\(411\) −5.65310 25.6101i −0.278847 1.26326i
\(412\) 0 0
\(413\) −6.07434 −0.298899
\(414\) 0 0
\(415\) −0.729989 −0.0358337
\(416\) 0 0
\(417\) −9.08206 + 2.00474i −0.444750 + 0.0981728i
\(418\) 0 0
\(419\) 30.2544i 1.47803i 0.673692 + 0.739013i \(0.264708\pi\)
−0.673692 + 0.739013i \(0.735292\pi\)
\(420\) 0 0
\(421\) 8.79979i 0.428875i 0.976738 + 0.214438i \(0.0687919\pi\)
−0.976738 + 0.214438i \(0.931208\pi\)
\(422\) 0 0
\(423\) 0.983945 + 2.12018i 0.0478410 + 0.103087i
\(424\) 0 0
\(425\) 33.2193i 1.61137i
\(426\) 0 0
\(427\) 8.25469 0.399473
\(428\) 0 0
\(429\) −37.3364 + 8.24151i −1.80262 + 0.397904i
\(430\) 0 0
\(431\) −13.6900 −0.659424 −0.329712 0.944082i \(-0.606952\pi\)
−0.329712 + 0.944082i \(0.606952\pi\)
\(432\) 0 0
\(433\) 8.50964i 0.408947i −0.978872 0.204474i \(-0.934452\pi\)
0.978872 0.204474i \(-0.0655482\pi\)
\(434\) 0 0
\(435\) 1.15596 + 5.23681i 0.0554239 + 0.251086i
\(436\) 0 0
\(437\) 15.4667 + 23.1616i 0.739875 + 1.10797i
\(438\) 0 0
\(439\) 7.21224i 0.344221i 0.985078 + 0.172111i \(0.0550587\pi\)
−0.985078 + 0.172111i \(0.944941\pi\)
\(440\) 0 0
\(441\) −17.6440 + 8.18832i −0.840189 + 0.389920i
\(442\) 0 0
\(443\) 37.3326i 1.77372i 0.462034 + 0.886862i \(0.347120\pi\)
−0.462034 + 0.886862i \(0.652880\pi\)
\(444\) 0 0
\(445\) 0.0786680i 0.00372922i
\(446\) 0 0
\(447\) −0.983278 4.45453i −0.0465075 0.210692i
\(448\) 0 0
\(449\) −9.67813 −0.456739 −0.228370 0.973574i \(-0.573339\pi\)
−0.228370 + 0.973574i \(0.573339\pi\)
\(450\) 0 0
\(451\) 42.2560i 1.98976i
\(452\) 0 0
\(453\) −2.45298 11.1127i −0.115251 0.522121i
\(454\) 0 0
\(455\) 1.88444 0.0883438
\(456\) 0 0
\(457\) 6.99502 0.327213 0.163607 0.986526i \(-0.447687\pi\)
0.163607 + 0.986526i \(0.447687\pi\)
\(458\) 0 0
\(459\) 22.3902 + 29.3456i 1.04509 + 1.36974i
\(460\) 0 0
\(461\) 38.3788i 1.78748i −0.448584 0.893741i \(-0.648072\pi\)
0.448584 0.893741i \(-0.351928\pi\)
\(462\) 0 0
\(463\) 22.3500 1.03869 0.519346 0.854564i \(-0.326175\pi\)
0.519346 + 0.854564i \(0.326175\pi\)
\(464\) 0 0
\(465\) 7.00632 1.54655i 0.324910 0.0717197i
\(466\) 0 0
\(467\) 0.0209668i 0.000970227i 1.00000 0.000485113i \(0.000154416\pi\)
−1.00000 0.000485113i \(0.999846\pi\)
\(468\) 0 0
\(469\) 2.78319i 0.128516i
\(470\) 0 0
\(471\) 21.9326 4.84133i 1.01060 0.223077i
\(472\) 0 0
\(473\) 14.7906i 0.680070i
\(474\) 0 0
\(475\) −16.9516 + 11.3199i −0.777792 + 0.519391i
\(476\) 0 0
\(477\) 15.9662 7.40969i 0.731042 0.339266i
\(478\) 0 0
\(479\) 37.0140i 1.69121i 0.533807 + 0.845606i \(0.320761\pi\)
−0.533807 + 0.845606i \(0.679239\pi\)
\(480\) 0 0
\(481\) 27.6545 1.26094
\(482\) 0 0
\(483\) 1.71386 + 7.76425i 0.0779831 + 0.353286i
\(484\) 0 0
\(485\) 5.26330 0.238994
\(486\) 0 0
\(487\) 21.9954i 0.996709i 0.866973 + 0.498354i \(0.166062\pi\)
−0.866973 + 0.498354i \(0.833938\pi\)
\(488\) 0 0
\(489\) 27.2200 6.00845i 1.23093 0.271711i
\(490\) 0 0
\(491\) 26.0141i 1.17400i −0.809588 0.586999i \(-0.800309\pi\)
0.809588 0.586999i \(-0.199691\pi\)
\(492\) 0 0
\(493\) 38.6616i 1.74123i
\(494\) 0 0
\(495\) −7.41265 + 3.44011i −0.333174 + 0.154621i
\(496\) 0 0
\(497\) 5.25178 0.235575
\(498\) 0 0
\(499\) 6.20204 0.277642 0.138821 0.990318i \(-0.455669\pi\)
0.138821 + 0.990318i \(0.455669\pi\)
\(500\) 0 0
\(501\) −16.9544 + 3.74246i −0.757468 + 0.167201i
\(502\) 0 0
\(503\) 11.4208i 0.509227i −0.967043 0.254613i \(-0.918052\pi\)
0.967043 0.254613i \(-0.0819482\pi\)
\(504\) 0 0
\(505\) −6.20647 −0.276184
\(506\) 0 0
\(507\) −13.9628 + 3.08210i −0.620109 + 0.136881i
\(508\) 0 0
\(509\) 0.235716 0.0104479 0.00522397 0.999986i \(-0.498337\pi\)
0.00522397 + 0.999986i \(0.498337\pi\)
\(510\) 0 0
\(511\) 2.30128 0.101803
\(512\) 0 0
\(513\) −7.34514 + 21.4254i −0.324296 + 0.945956i
\(514\) 0 0
\(515\) −10.7222 −0.472477
\(516\) 0 0
\(517\) −3.73055 −0.164069
\(518\) 0 0
\(519\) −25.6600 + 5.66411i −1.12635 + 0.248627i
\(520\) 0 0
\(521\) 39.2131 1.71796 0.858978 0.512012i \(-0.171100\pi\)
0.858978 + 0.512012i \(0.171100\pi\)
\(522\) 0 0
\(523\) 16.7721i 0.733393i −0.930341 0.366697i \(-0.880489\pi\)
0.930341 0.366697i \(-0.119511\pi\)
\(524\) 0 0
\(525\) −5.68253 + 1.25434i −0.248006 + 0.0547441i
\(526\) 0 0
\(527\) 51.7254 2.25319
\(528\) 0 0
\(529\) −17.8252 −0.775008
\(530\) 0 0
\(531\) 23.0070 10.6772i 0.998417 0.463352i
\(532\) 0 0
\(533\) 40.6871i 1.76235i
\(534\) 0 0
\(535\) 2.71419i 0.117345i
\(536\) 0 0
\(537\) 33.3680 7.36554i 1.43993 0.317846i
\(538\) 0 0
\(539\) 31.0454i 1.33722i
\(540\) 0 0
\(541\) −0.978406 −0.0420649 −0.0210325 0.999779i \(-0.506695\pi\)
−0.0210325 + 0.999779i \(0.506695\pi\)
\(542\) 0 0
\(543\) 7.24993 + 32.8442i 0.311124 + 1.40948i
\(544\) 0 0
\(545\) 6.08956 0.260848
\(546\) 0 0
\(547\) 18.8731i 0.806956i −0.914989 0.403478i \(-0.867801\pi\)
0.914989 0.403478i \(-0.132199\pi\)
\(548\) 0 0
\(549\) −31.2652 + 14.5097i −1.33437 + 0.619261i
\(550\) 0 0
\(551\) −19.7288 + 13.1744i −0.840474 + 0.561248i
\(552\) 0 0
\(553\) 1.29242i 0.0549591i
\(554\) 0 0
\(555\) 5.77166 1.27402i 0.244993 0.0540791i
\(556\) 0 0
\(557\) 9.53419i 0.403977i 0.979388 + 0.201988i \(0.0647403\pi\)
−0.979388 + 0.201988i \(0.935260\pi\)
\(558\) 0 0
\(559\) 14.2414i 0.602348i
\(560\) 0 0
\(561\) −57.5282 + 12.6986i −2.42884 + 0.536135i
\(562\) 0 0
\(563\) −11.5947 −0.488658 −0.244329 0.969692i \(-0.578568\pi\)
−0.244329 + 0.969692i \(0.578568\pi\)
\(564\) 0 0
\(565\) 0.289496i 0.0121792i
\(566\) 0 0
\(567\) −4.17445 + 4.93817i −0.175310 + 0.207384i
\(568\) 0 0
\(569\) 46.1820 1.93605 0.968026 0.250851i \(-0.0807105\pi\)
0.968026 + 0.250851i \(0.0807105\pi\)
\(570\) 0 0
\(571\) 7.24993 0.303400 0.151700 0.988427i \(-0.451525\pi\)
0.151700 + 0.988427i \(0.451525\pi\)
\(572\) 0 0
\(573\) −5.91657 26.8037i −0.247168 1.11974i
\(574\) 0 0
\(575\) 29.8793i 1.24605i
\(576\) 0 0
\(577\) 6.17967 0.257263 0.128632 0.991692i \(-0.458942\pi\)
0.128632 + 0.991692i \(0.458942\pi\)
\(578\) 0 0
\(579\) −3.17699 14.3927i −0.132031 0.598138i
\(580\) 0 0
\(581\) 0.921893i 0.0382466i
\(582\) 0 0
\(583\) 28.0933i 1.16350i
\(584\) 0 0
\(585\) −7.13744 + 3.31239i −0.295097 + 0.136950i
\(586\) 0 0
\(587\) 15.4820i 0.639010i −0.947585 0.319505i \(-0.896483\pi\)
0.947585 0.319505i \(-0.103517\pi\)
\(588\) 0 0
\(589\) 17.6260 + 26.3951i 0.726267 + 1.08759i
\(590\) 0 0
\(591\) −5.22890 23.6884i −0.215088 0.974410i
\(592\) 0 0
\(593\) 2.88034i 0.118281i −0.998250 0.0591406i \(-0.981164\pi\)
0.998250 0.0591406i \(-0.0188360\pi\)
\(594\) 0 0
\(595\) 2.90356 0.119034
\(596\) 0 0
\(597\) 33.3145 7.35374i 1.36347 0.300969i
\(598\) 0 0
\(599\) −7.66338 −0.313117 −0.156559 0.987669i \(-0.550040\pi\)
−0.156559 + 0.987669i \(0.550040\pi\)
\(600\) 0 0
\(601\) 16.3265i 0.665971i −0.942932 0.332985i \(-0.891944\pi\)
0.942932 0.332985i \(-0.108056\pi\)
\(602\) 0 0
\(603\) −4.89218 10.5415i −0.199225 0.429284i
\(604\) 0 0
\(605\) 6.78494i 0.275847i
\(606\) 0 0
\(607\) 24.5202i 0.995246i −0.867394 0.497623i \(-0.834206\pi\)
0.867394 0.497623i \(-0.165794\pi\)
\(608\) 0 0
\(609\) −6.61350 + 1.45984i −0.267992 + 0.0591558i
\(610\) 0 0
\(611\) −3.59204 −0.145319
\(612\) 0 0
\(613\) 8.07345 0.326084 0.163042 0.986619i \(-0.447869\pi\)
0.163042 + 0.986619i \(0.447869\pi\)
\(614\) 0 0
\(615\) 1.87442 + 8.49166i 0.0755840 + 0.342417i
\(616\) 0 0
\(617\) 0.770572i 0.0310220i 0.999880 + 0.0155110i \(0.00493751\pi\)
−0.999880 + 0.0155110i \(0.995062\pi\)
\(618\) 0 0
\(619\) 8.83275 0.355018 0.177509 0.984119i \(-0.443196\pi\)
0.177509 + 0.984119i \(0.443196\pi\)
\(620\) 0 0
\(621\) −20.1390 26.3951i −0.808151 1.05920i
\(622\) 0 0
\(623\) 0.0993487 0.00398032
\(624\) 0 0
\(625\) 20.2499 0.809997
\(626\) 0 0
\(627\) −26.0834 25.0291i −1.04167 0.999564i
\(628\) 0 0
\(629\) 42.6103 1.69898
\(630\) 0 0
\(631\) −12.5460 −0.499449 −0.249724 0.968317i \(-0.580340\pi\)
−0.249724 + 0.968317i \(0.580340\pi\)
\(632\) 0 0
\(633\) 1.60718 + 7.28098i 0.0638797 + 0.289393i
\(634\) 0 0
\(635\) 6.53191 0.259211
\(636\) 0 0
\(637\) 29.8927i 1.18439i
\(638\) 0 0
\(639\) −19.8915 + 9.23136i −0.786895 + 0.365187i
\(640\) 0 0
\(641\) −27.0121 −1.06692 −0.533458 0.845827i \(-0.679108\pi\)
−0.533458 + 0.845827i \(0.679108\pi\)
\(642\) 0 0
\(643\) −23.8987 −0.942473 −0.471236 0.882007i \(-0.656192\pi\)
−0.471236 + 0.882007i \(0.656192\pi\)
\(644\) 0 0
\(645\) −0.656090 2.97227i −0.0258335 0.117033i
\(646\) 0 0
\(647\) 6.16500i 0.242371i 0.992630 + 0.121186i \(0.0386696\pi\)
−0.992630 + 0.121186i \(0.961330\pi\)
\(648\) 0 0
\(649\) 40.4818i 1.58905i
\(650\) 0 0
\(651\) 1.95312 + 8.84819i 0.0765489 + 0.346788i
\(652\) 0 0
\(653\) 20.7295i 0.811208i −0.914049 0.405604i \(-0.867061\pi\)
0.914049 0.405604i \(-0.132939\pi\)
\(654\) 0 0
\(655\) −3.37131 −0.131728
\(656\) 0 0
\(657\) −8.71626 + 4.04509i −0.340054 + 0.157814i
\(658\) 0 0
\(659\) −18.5728 −0.723494 −0.361747 0.932276i \(-0.617820\pi\)
−0.361747 + 0.932276i \(0.617820\pi\)
\(660\) 0 0
\(661\) 0.259282i 0.0100849i −0.999987 0.00504244i \(-0.998395\pi\)
0.999987 0.00504244i \(-0.00160507\pi\)
\(662\) 0 0
\(663\) −55.3923 + 12.2271i −2.15126 + 0.474862i
\(664\) 0 0
\(665\) 0.989421 + 1.48167i 0.0383681 + 0.0574565i
\(666\) 0 0
\(667\) 34.7744i 1.34647i
\(668\) 0 0
\(669\) −7.78140 35.2519i −0.300846 1.36292i
\(670\) 0 0
\(671\) 55.0126i 2.12374i
\(672\) 0 0
\(673\) 16.8663i 0.650148i 0.945689 + 0.325074i \(0.105389\pi\)
−0.945689 + 0.325074i \(0.894611\pi\)
\(674\) 0 0
\(675\) 19.3181 14.7394i 0.743556 0.567321i
\(676\) 0 0
\(677\) −43.9992 −1.69103 −0.845513 0.533955i \(-0.820705\pi\)
−0.845513 + 0.533955i \(0.820705\pi\)
\(678\) 0 0
\(679\) 6.64695i 0.255086i
\(680\) 0 0
\(681\) 16.7147 3.68955i 0.640508 0.141384i
\(682\) 0 0
\(683\) −30.9790 −1.18538 −0.592689 0.805432i \(-0.701933\pi\)
−0.592689 + 0.805432i \(0.701933\pi\)
\(684\) 0 0
\(685\) 8.61436 0.329138
\(686\) 0 0
\(687\) 40.2728 8.88968i 1.53650 0.339162i
\(688\) 0 0
\(689\) 27.0502i 1.03053i
\(690\) 0 0
\(691\) −22.6698 −0.862400 −0.431200 0.902256i \(-0.641910\pi\)
−0.431200 + 0.902256i \(0.641910\pi\)
\(692\) 0 0
\(693\) −4.34447 9.36134i −0.165033 0.355608i
\(694\) 0 0
\(695\) 3.05489i 0.115879i
\(696\) 0 0
\(697\) 62.6911i 2.37460i
\(698\) 0 0
\(699\) 4.28534 + 19.4138i 0.162086 + 0.734297i
\(700\) 0 0
\(701\) 5.35960i 0.202429i 0.994865 + 0.101215i \(0.0322729\pi\)
−0.994865 + 0.101215i \(0.967727\pi\)
\(702\) 0 0
\(703\) 14.5199 + 21.7437i 0.547630 + 0.820080i
\(704\) 0 0
\(705\) −0.749682 + 0.165482i −0.0282347 + 0.00623243i
\(706\) 0 0
\(707\) 7.83807i 0.294781i
\(708\) 0 0
\(709\) −1.53225 −0.0575447 −0.0287724 0.999586i \(-0.509160\pi\)
−0.0287724 + 0.999586i \(0.509160\pi\)
\(710\) 0 0
\(711\) 2.27175 + 4.89511i 0.0851975 + 0.183581i
\(712\) 0 0
\(713\) −46.5247 −1.74236
\(714\) 0 0
\(715\) 12.5587i 0.469667i
\(716\) 0 0
\(717\) 0.768165 + 3.48001i 0.0286877 + 0.129963i
\(718\) 0 0
\(719\) 29.8942i 1.11486i 0.830223 + 0.557432i \(0.188213\pi\)
−0.830223 + 0.557432i \(0.811787\pi\)
\(720\) 0 0
\(721\) 13.5409i 0.504291i
\(722\) 0 0
\(723\) −10.9263 49.4991i −0.406353 1.84089i
\(724\) 0 0
\(725\) 25.4509 0.945221
\(726\) 0 0
\(727\) −44.6663 −1.65658 −0.828291 0.560298i \(-0.810687\pi\)
−0.828291 + 0.560298i \(0.810687\pi\)
\(728\) 0 0
\(729\) 7.13090 26.0413i 0.264107 0.964493i
\(730\) 0 0
\(731\) 21.9433i 0.811603i
\(732\) 0 0
\(733\) 17.9257 0.662101 0.331051 0.943613i \(-0.392597\pi\)
0.331051 + 0.943613i \(0.392597\pi\)
\(734\) 0 0
\(735\) −1.37713 6.23880i −0.0507963 0.230122i
\(736\) 0 0
\(737\) 18.5483 0.683236
\(738\) 0 0
\(739\) 33.0669 1.21639 0.608193 0.793789i \(-0.291895\pi\)
0.608193 + 0.793789i \(0.291895\pi\)
\(740\) 0 0
\(741\) −25.1150 24.0998i −0.922622 0.885328i
\(742\) 0 0
\(743\) −4.63266 −0.169956 −0.0849778 0.996383i \(-0.527082\pi\)
−0.0849778 + 0.996383i \(0.527082\pi\)
\(744\) 0 0
\(745\) 1.49835 0.0548952
\(746\) 0 0
\(747\) 1.62047 + 3.49173i 0.0592897 + 0.127756i
\(748\) 0 0
\(749\) 3.42771 0.125246
\(750\) 0 0
\(751\) 27.7388i 1.01220i 0.862474 + 0.506101i \(0.168914\pi\)
−0.862474 + 0.506101i \(0.831086\pi\)
\(752\) 0 0
\(753\) −2.82631 12.8040i −0.102997 0.466604i
\(754\) 0 0
\(755\) 3.73793 0.136037
\(756\) 0 0
\(757\) 34.3596 1.24882 0.624411 0.781096i \(-0.285339\pi\)
0.624411 + 0.781096i \(0.285339\pi\)
\(758\) 0 0
\(759\) 51.7441 11.4218i 1.87819 0.414586i
\(760\) 0 0
\(761\) 5.53916i 0.200795i −0.994947 0.100397i \(-0.967989\pi\)
0.994947 0.100397i \(-0.0320114\pi\)
\(762\) 0 0
\(763\) 7.69043i 0.278412i
\(764\) 0 0
\(765\) −10.9974 + 5.10376i −0.397613 + 0.184527i
\(766\) 0 0
\(767\) 38.9788i 1.40744i
\(768\) 0 0
\(769\) −16.7969 −0.605713 −0.302857 0.953036i \(-0.597940\pi\)
−0.302857 + 0.953036i \(0.597940\pi\)
\(770\) 0 0
\(771\) −28.0467 + 6.19093i −1.01008 + 0.222961i
\(772\) 0 0
\(773\) −30.3092 −1.09015 −0.545073 0.838389i \(-0.683498\pi\)
−0.545073 + 0.838389i \(0.683498\pi\)
\(774\) 0 0
\(775\) 34.0507i 1.22314i
\(776\) 0 0
\(777\) 1.60894 + 7.28896i 0.0577204 + 0.261490i
\(778\) 0 0
\(779\) −31.9908 + 21.3627i −1.14619 + 0.765399i
\(780\) 0 0
\(781\) 35.0000i 1.25240i
\(782\) 0 0
\(783\) 22.4830 17.1542i 0.803478 0.613040i
\(784\) 0 0
\(785\) 7.37736i 0.263309i
\(786\) 0 0
\(787\) 48.2737i 1.72077i 0.509643 + 0.860386i \(0.329778\pi\)
−0.509643 + 0.860386i \(0.670222\pi\)
\(788\) 0 0
\(789\) 2.89926 + 13.1345i 0.103216 + 0.467599i
\(790\) 0 0
\(791\) 0.365601 0.0129993
\(792\) 0 0
\(793\) 52.9701i 1.88102i
\(794\) 0 0
\(795\) 1.24618 + 5.64555i 0.0441975 + 0.200227i
\(796\) 0 0
\(797\) 10.8341 0.383763 0.191882 0.981418i \(-0.438541\pi\)
0.191882 + 0.981418i \(0.438541\pi\)
\(798\) 0 0
\(799\) −5.53466 −0.195802
\(800\) 0 0
\(801\) −0.376290 + 0.174631i −0.0132956 + 0.00617028i
\(802\) 0 0
\(803\) 15.3367i 0.541219i
\(804\) 0 0
\(805\) −2.61162 −0.0920476
\(806\) 0 0
\(807\) 24.5557 5.42036i 0.864403 0.190806i
\(808\) 0 0
\(809\) 38.1880i 1.34262i 0.741177 + 0.671310i \(0.234268\pi\)
−0.741177 + 0.671310i \(0.765732\pi\)
\(810\) 0 0
\(811\) 16.0919i 0.565063i −0.959258 0.282531i \(-0.908826\pi\)
0.959258 0.282531i \(-0.0911741\pi\)
\(812\) 0 0
\(813\) −52.2435 + 11.5321i −1.83226 + 0.404447i
\(814\) 0 0
\(815\) 9.15584i 0.320715i
\(816\) 0 0
\(817\) 11.1975 7.47743i 0.391751 0.261602i
\(818\) 0 0
\(819\) −4.18317 9.01378i −0.146172 0.314967i
\(820\) 0 0
\(821\) 20.8593i 0.727994i 0.931400 + 0.363997i \(0.118588\pi\)
−0.931400 + 0.363997i \(0.881412\pi\)
\(822\) 0 0
\(823\) −5.11462 −0.178284 −0.0891422 0.996019i \(-0.528413\pi\)
−0.0891422 + 0.996019i \(0.528413\pi\)
\(824\) 0 0
\(825\) 8.35946 + 37.8707i 0.291039 + 1.31849i
\(826\) 0 0
\(827\) −53.1212 −1.84721 −0.923603 0.383350i \(-0.874770\pi\)
−0.923603 + 0.383350i \(0.874770\pi\)
\(828\) 0 0
\(829\) 21.8202i 0.757848i 0.925428 + 0.378924i \(0.123706\pi\)
−0.925428 + 0.378924i \(0.876294\pi\)
\(830\) 0 0
\(831\) −17.1419 + 3.78384i −0.594644 + 0.131260i
\(832\) 0 0
\(833\) 46.0590i 1.59585i
\(834\) 0 0
\(835\) 5.70288i 0.197356i
\(836\) 0 0
\(837\) −22.9506 30.0800i −0.793287 1.03972i
\(838\) 0 0
\(839\) 34.0659 1.17609 0.588043 0.808830i \(-0.299899\pi\)
0.588043 + 0.808830i \(0.299899\pi\)
\(840\) 0 0
\(841\) 0.620465 0.0213954
\(842\) 0 0
\(843\) 16.0938 3.55248i 0.554298 0.122354i
\(844\) 0 0
\(845\) 4.69659i 0.161568i
\(846\) 0 0
\(847\) 8.56861 0.294421
\(848\) 0 0
\(849\) 13.8029 3.04680i 0.473714 0.104566i
\(850\) 0 0
\(851\) −38.3261 −1.31380
\(852\) 0 0
\(853\) 15.6050 0.534304 0.267152 0.963654i \(-0.413917\pi\)
0.267152 + 0.963654i \(0.413917\pi\)
\(854\) 0 0
\(855\) −6.35191 3.87275i −0.217231 0.132445i
\(856\) 0 0
\(857\) −25.5605 −0.873128 −0.436564 0.899673i \(-0.643805\pi\)
−0.436564 + 0.899673i \(0.643805\pi\)
\(858\) 0 0
\(859\) 38.6287 1.31800 0.658998 0.752145i \(-0.270981\pi\)
0.658998 + 0.752145i \(0.270981\pi\)
\(860\) 0 0
\(861\) −10.7240 + 2.36718i −0.365473 + 0.0806734i
\(862\) 0 0
\(863\) 16.2452 0.552994 0.276497 0.961015i \(-0.410826\pi\)
0.276497 + 0.961015i \(0.410826\pi\)
\(864\) 0 0
\(865\) 8.63114i 0.293467i
\(866\) 0 0
\(867\) −56.5963 + 12.4929i −1.92211 + 0.424281i
\(868\) 0 0
\(869\) −8.61318 −0.292182
\(870\) 0 0
\(871\) 17.8597 0.605151
\(872\) 0 0
\(873\) −11.6837 25.1758i −0.395434 0.852071i
\(874\) 0 0
\(875\) 3.95510i 0.133707i
\(876\) 0 0
\(877\) 19.8852i 0.671475i −0.941956 0.335737i \(-0.891015\pi\)
0.941956 0.335737i \(-0.108985\pi\)
\(878\) 0 0
\(879\) 5.47762 1.20911i 0.184756 0.0407824i
\(880\) 0 0
\(881\) 32.9959i 1.11166i −0.831296 0.555829i \(-0.812401\pi\)
0.831296 0.555829i \(-0.187599\pi\)
\(882\) 0 0
\(883\) −18.7441 −0.630789 −0.315394 0.948961i \(-0.602137\pi\)
−0.315394 + 0.948961i \(0.602137\pi\)
\(884\) 0 0
\(885\) 1.79572 + 8.13513i 0.0603625 + 0.273459i
\(886\) 0 0
\(887\) −5.71898 −0.192024 −0.0960122 0.995380i \(-0.530609\pi\)
−0.0960122 + 0.995380i \(0.530609\pi\)
\(888\) 0 0
\(889\) 8.24907i 0.276665i
\(890\) 0 0
\(891\) 32.9099 + 27.8202i 1.10252 + 0.932012i
\(892\) 0 0
\(893\) −1.88600 2.82430i −0.0631125 0.0945115i
\(894\) 0 0
\(895\) 11.2238i 0.375171i
\(896\) 0 0
\(897\) 49.8229 10.9978i 1.66354 0.367204i
\(898\) 0 0
\(899\) 39.6292i 1.32171i
\(900\) 0 0
\(901\) 41.6792i 1.38854i
\(902\) 0 0
\(903\) 3.75364 0.828567i 0.124913 0.0275730i
\(904\) 0 0
\(905\) −11.0477 −0.367237
\(906\) 0 0
\(907\) 27.4189i 0.910431i −0.890381 0.455216i \(-0.849562\pi\)
0.890381 0.455216i \(-0.150438\pi\)
\(908\) 0 0
\(909\) 13.7774 + 29.6872i 0.456969 + 0.984663i
\(910\) 0 0
\(911\) −36.7522 −1.21766 −0.608828 0.793302i \(-0.708360\pi\)
−0.608828 + 0.793302i \(0.708360\pi\)
\(912\) 0 0
\(913\) −6.14387 −0.203332
\(914\) 0 0
\(915\) −2.44029 11.0552i −0.0806734 0.365473i
\(916\) 0 0
\(917\) 4.25759i 0.140598i
\(918\) 0 0
\(919\) 47.7497 1.57512 0.787558 0.616240i \(-0.211345\pi\)
0.787558 + 0.616240i \(0.211345\pi\)
\(920\) 0 0
\(921\) −0.0709270 0.321319i −0.00233712 0.0105878i
\(922\) 0 0
\(923\) 33.7005i 1.10927i
\(924\) 0 0
\(925\) 28.0502i 0.922286i
\(926\) 0 0
\(927\) 23.8017 + 51.2873i 0.781751 + 1.68449i
\(928\) 0 0
\(929\) 14.9076i 0.489104i 0.969636 + 0.244552i \(0.0786409\pi\)
−0.969636 + 0.244552i \(0.921359\pi\)
\(930\) 0 0
\(931\) 23.5036 15.6951i 0.770299 0.514387i
\(932\) 0 0
\(933\) 2.08343 + 9.43854i 0.0682085 + 0.309004i
\(934\) 0 0
\(935\) 19.3505i 0.632829i
\(936\) 0 0
\(937\) −26.4925 −0.865473 −0.432737 0.901520i \(-0.642452\pi\)
−0.432737 + 0.901520i \(0.642452\pi\)
\(938\) 0 0
\(939\) −30.2770 + 6.68324i −0.988052 + 0.218099i
\(940\) 0 0
\(941\) 8.03967 0.262086 0.131043 0.991377i \(-0.458167\pi\)
0.131043 + 0.991377i \(0.458167\pi\)
\(942\) 0 0
\(943\) 56.3879i 1.83624i
\(944\) 0 0
\(945\) −1.28831 1.68852i −0.0419087 0.0549274i
\(946\) 0 0
\(947\) 41.6266i 1.35268i −0.736589 0.676341i \(-0.763565\pi\)
0.736589 0.676341i \(-0.236435\pi\)
\(948\) 0 0
\(949\) 14.7672i 0.479365i
\(950\) 0 0
\(951\) −37.4856 + 8.27445i −1.21555 + 0.268317i
\(952\) 0 0
\(953\) 33.5570 1.08702 0.543510 0.839403i \(-0.317095\pi\)
0.543510 + 0.839403i \(0.317095\pi\)
\(954\) 0 0
\(955\) 9.01584 0.291746
\(956\) 0 0
\(957\) 9.72898 + 44.0750i 0.314493 + 1.42474i
\(958\) 0 0
\(959\) 10.8790i 0.351300i
\(960\) 0 0
\(961\) −22.0198 −0.710316
\(962\) 0 0
\(963\) −12.9827 + 6.02509i −0.418361 + 0.194156i
\(964\) 0 0
\(965\) 4.84119 0.155843
\(966\) 0 0
\(967\) 27.3041 0.878042 0.439021 0.898477i \(-0.355325\pi\)
0.439021 + 0.898477i \(0.355325\pi\)
\(968\) 0 0
\(969\) −38.6974 37.1332i −1.24314 1.19289i
\(970\) 0 0
\(971\) −12.4588 −0.399820 −0.199910 0.979814i \(-0.564065\pi\)
−0.199910 + 0.979814i \(0.564065\pi\)
\(972\) 0 0
\(973\) 3.85798 0.123681
\(974\) 0 0
\(975\) 8.04909 + 36.4646i 0.257777 + 1.16780i
\(976\) 0 0
\(977\) −38.7035 −1.23824 −0.619118 0.785298i \(-0.712510\pi\)
−0.619118 + 0.785298i \(0.712510\pi\)
\(978\) 0 0
\(979\) 0.662100i 0.0211608i
\(980\) 0 0
\(981\) −13.5179 29.1280i −0.431594 0.929986i
\(982\) 0 0
\(983\) 24.8496 0.792579 0.396289 0.918126i \(-0.370298\pi\)
0.396289 + 0.918126i \(0.370298\pi\)
\(984\) 0 0
\(985\) 7.96795 0.253880
\(986\) 0 0
\(987\) −0.208986 0.946764i −0.00665208 0.0301358i
\(988\) 0 0
\(989\) 19.7370i 0.627601i
\(990\) 0 0
\(991\) 54.3454i 1.72634i 0.504915 + 0.863169i \(0.331524\pi\)
−0.504915 + 0.863169i \(0.668476\pi\)
\(992\) 0 0
\(993\) −9.68941 43.8958i −0.307484 1.39299i
\(994\) 0 0
\(995\) 11.2058i 0.355249i
\(996\) 0 0
\(997\) −46.5392 −1.47391 −0.736955 0.675941i \(-0.763737\pi\)
−0.736955 + 0.675941i \(0.763737\pi\)
\(998\) 0 0
\(999\) −18.9062 24.7793i −0.598166 0.783982i
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 912.2.f.h.113.9 10
3.2 odd 2 912.2.f.i.113.1 10
4.3 odd 2 456.2.f.b.113.2 yes 10
12.11 even 2 456.2.f.a.113.10 yes 10
19.18 odd 2 912.2.f.i.113.2 10
57.56 even 2 inner 912.2.f.h.113.10 10
76.75 even 2 456.2.f.a.113.9 10
228.227 odd 2 456.2.f.b.113.1 yes 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
456.2.f.a.113.9 10 76.75 even 2
456.2.f.a.113.10 yes 10 12.11 even 2
456.2.f.b.113.1 yes 10 228.227 odd 2
456.2.f.b.113.2 yes 10 4.3 odd 2
912.2.f.h.113.9 10 1.1 even 1 trivial
912.2.f.h.113.10 10 57.56 even 2 inner
912.2.f.i.113.1 10 3.2 odd 2
912.2.f.i.113.2 10 19.18 odd 2