Properties

Label 1344.2.q.h.193.1
Level $1344$
Weight $2$
Character 1344.193
Analytic conductor $10.732$
Analytic rank $0$
Dimension $2$
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] = [1344,2,Mod(193,1344)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1344, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1344.193");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1344 = 2^{6} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1344.q (of order \(3\), degree \(2\), 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: \(10.7318940317\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 672)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 193.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 1344.193
Dual form 1344.2.q.h.961.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.500000 + 0.866025i) q^{3} +(0.500000 + 0.866025i) q^{5} +(2.50000 - 0.866025i) q^{7} +(-0.500000 - 0.866025i) q^{9} +O(q^{10})\) \(q+(-0.500000 + 0.866025i) q^{3} +(0.500000 + 0.866025i) q^{5} +(2.50000 - 0.866025i) q^{7} +(-0.500000 - 0.866025i) q^{9} +(0.500000 - 0.866025i) q^{11} -1.00000 q^{15} +(4.00000 - 6.92820i) q^{17} +(-2.00000 - 3.46410i) q^{19} +(-0.500000 + 2.59808i) q^{21} +(-2.00000 - 3.46410i) q^{23} +(2.00000 - 3.46410i) q^{25} +1.00000 q^{27} +5.00000 q^{29} +(-3.50000 + 6.06218i) q^{31} +(0.500000 + 0.866025i) q^{33} +(2.00000 + 1.73205i) q^{35} +(4.00000 + 6.92820i) q^{37} +4.00000 q^{41} -10.0000 q^{43} +(0.500000 - 0.866025i) q^{45} +(-3.00000 - 5.19615i) q^{47} +(5.50000 - 4.33013i) q^{49} +(4.00000 + 6.92820i) q^{51} +(-0.500000 + 0.866025i) q^{53} +1.00000 q^{55} +4.00000 q^{57} +(4.50000 - 7.79423i) q^{59} +(-1.00000 - 1.73205i) q^{61} +(-2.00000 - 1.73205i) q^{63} +(1.00000 - 1.73205i) q^{67} +4.00000 q^{69} +6.00000 q^{71} +(-1.00000 + 1.73205i) q^{73} +(2.00000 + 3.46410i) q^{75} +(0.500000 - 2.59808i) q^{77} +(4.50000 + 7.79423i) q^{79} +(-0.500000 + 0.866025i) q^{81} +3.00000 q^{83} +8.00000 q^{85} +(-2.50000 + 4.33013i) q^{87} +(3.00000 + 5.19615i) q^{89} +(-3.50000 - 6.06218i) q^{93} +(2.00000 - 3.46410i) q^{95} -1.00000 q^{97} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{3} + q^{5} + 5 q^{7} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{3} + q^{5} + 5 q^{7} - q^{9} + q^{11} - 2 q^{15} + 8 q^{17} - 4 q^{19} - q^{21} - 4 q^{23} + 4 q^{25} + 2 q^{27} + 10 q^{29} - 7 q^{31} + q^{33} + 4 q^{35} + 8 q^{37} + 8 q^{41} - 20 q^{43} + q^{45} - 6 q^{47} + 11 q^{49} + 8 q^{51} - q^{53} + 2 q^{55} + 8 q^{57} + 9 q^{59} - 2 q^{61} - 4 q^{63} + 2 q^{67} + 8 q^{69} + 12 q^{71} - 2 q^{73} + 4 q^{75} + q^{77} + 9 q^{79} - q^{81} + 6 q^{83} + 16 q^{85} - 5 q^{87} + 6 q^{89} - 7 q^{93} + 4 q^{95} - 2 q^{97} - 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1344\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(449\) \(577\) \(1093\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{2}{3}\right)\) \(1\)

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
\(3\) −0.500000 + 0.866025i −0.288675 + 0.500000i
\(4\) 0 0
\(5\) 0.500000 + 0.866025i 0.223607 + 0.387298i 0.955901 0.293691i \(-0.0948835\pi\)
−0.732294 + 0.680989i \(0.761550\pi\)
\(6\) 0 0
\(7\) 2.50000 0.866025i 0.944911 0.327327i
\(8\) 0 0
\(9\) −0.500000 0.866025i −0.166667 0.288675i
\(10\) 0 0
\(11\) 0.500000 0.866025i 0.150756 0.261116i −0.780750 0.624844i \(-0.785163\pi\)
0.931505 + 0.363727i \(0.118496\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 0 0
\(15\) −1.00000 −0.258199
\(16\) 0 0
\(17\) 4.00000 6.92820i 0.970143 1.68034i 0.275029 0.961436i \(-0.411312\pi\)
0.695113 0.718900i \(-0.255354\pi\)
\(18\) 0 0
\(19\) −2.00000 3.46410i −0.458831 0.794719i 0.540068 0.841621i \(-0.318398\pi\)
−0.998899 + 0.0469020i \(0.985065\pi\)
\(20\) 0 0
\(21\) −0.500000 + 2.59808i −0.109109 + 0.566947i
\(22\) 0 0
\(23\) −2.00000 3.46410i −0.417029 0.722315i 0.578610 0.815604i \(-0.303595\pi\)
−0.995639 + 0.0932891i \(0.970262\pi\)
\(24\) 0 0
\(25\) 2.00000 3.46410i 0.400000 0.692820i
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 5.00000 0.928477 0.464238 0.885710i \(-0.346328\pi\)
0.464238 + 0.885710i \(0.346328\pi\)
\(30\) 0 0
\(31\) −3.50000 + 6.06218i −0.628619 + 1.08880i 0.359211 + 0.933257i \(0.383046\pi\)
−0.987829 + 0.155543i \(0.950287\pi\)
\(32\) 0 0
\(33\) 0.500000 + 0.866025i 0.0870388 + 0.150756i
\(34\) 0 0
\(35\) 2.00000 + 1.73205i 0.338062 + 0.292770i
\(36\) 0 0
\(37\) 4.00000 + 6.92820i 0.657596 + 1.13899i 0.981236 + 0.192809i \(0.0617599\pi\)
−0.323640 + 0.946180i \(0.604907\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 4.00000 0.624695 0.312348 0.949968i \(-0.398885\pi\)
0.312348 + 0.949968i \(0.398885\pi\)
\(42\) 0 0
\(43\) −10.0000 −1.52499 −0.762493 0.646997i \(-0.776025\pi\)
−0.762493 + 0.646997i \(0.776025\pi\)
\(44\) 0 0
\(45\) 0.500000 0.866025i 0.0745356 0.129099i
\(46\) 0 0
\(47\) −3.00000 5.19615i −0.437595 0.757937i 0.559908 0.828554i \(-0.310836\pi\)
−0.997503 + 0.0706177i \(0.977503\pi\)
\(48\) 0 0
\(49\) 5.50000 4.33013i 0.785714 0.618590i
\(50\) 0 0
\(51\) 4.00000 + 6.92820i 0.560112 + 0.970143i
\(52\) 0 0
\(53\) −0.500000 + 0.866025i −0.0686803 + 0.118958i −0.898321 0.439340i \(-0.855212\pi\)
0.829640 + 0.558298i \(0.188546\pi\)
\(54\) 0 0
\(55\) 1.00000 0.134840
\(56\) 0 0
\(57\) 4.00000 0.529813
\(58\) 0 0
\(59\) 4.50000 7.79423i 0.585850 1.01472i −0.408919 0.912571i \(-0.634094\pi\)
0.994769 0.102151i \(-0.0325726\pi\)
\(60\) 0 0
\(61\) −1.00000 1.73205i −0.128037 0.221766i 0.794879 0.606768i \(-0.207534\pi\)
−0.922916 + 0.385002i \(0.874201\pi\)
\(62\) 0 0
\(63\) −2.00000 1.73205i −0.251976 0.218218i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 1.00000 1.73205i 0.122169 0.211604i −0.798454 0.602056i \(-0.794348\pi\)
0.920623 + 0.390453i \(0.127682\pi\)
\(68\) 0 0
\(69\) 4.00000 0.481543
\(70\) 0 0
\(71\) 6.00000 0.712069 0.356034 0.934473i \(-0.384129\pi\)
0.356034 + 0.934473i \(0.384129\pi\)
\(72\) 0 0
\(73\) −1.00000 + 1.73205i −0.117041 + 0.202721i −0.918594 0.395203i \(-0.870674\pi\)
0.801553 + 0.597924i \(0.204008\pi\)
\(74\) 0 0
\(75\) 2.00000 + 3.46410i 0.230940 + 0.400000i
\(76\) 0 0
\(77\) 0.500000 2.59808i 0.0569803 0.296078i
\(78\) 0 0
\(79\) 4.50000 + 7.79423i 0.506290 + 0.876919i 0.999974 + 0.00727784i \(0.00231663\pi\)
−0.493684 + 0.869641i \(0.664350\pi\)
\(80\) 0 0
\(81\) −0.500000 + 0.866025i −0.0555556 + 0.0962250i
\(82\) 0 0
\(83\) 3.00000 0.329293 0.164646 0.986353i \(-0.447352\pi\)
0.164646 + 0.986353i \(0.447352\pi\)
\(84\) 0 0
\(85\) 8.00000 0.867722
\(86\) 0 0
\(87\) −2.50000 + 4.33013i −0.268028 + 0.464238i
\(88\) 0 0
\(89\) 3.00000 + 5.19615i 0.317999 + 0.550791i 0.980071 0.198650i \(-0.0636557\pi\)
−0.662071 + 0.749441i \(0.730322\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −3.50000 6.06218i −0.362933 0.628619i
\(94\) 0 0
\(95\) 2.00000 3.46410i 0.205196 0.355409i
\(96\) 0 0
\(97\) −1.00000 −0.101535 −0.0507673 0.998711i \(-0.516167\pi\)
−0.0507673 + 0.998711i \(0.516167\pi\)
\(98\) 0 0
\(99\) −1.00000 −0.100504
\(100\) 0 0
\(101\) 1.00000 1.73205i 0.0995037 0.172345i −0.811976 0.583691i \(-0.801608\pi\)
0.911479 + 0.411346i \(0.134941\pi\)
\(102\) 0 0
\(103\) 8.00000 + 13.8564i 0.788263 + 1.36531i 0.927030 + 0.374987i \(0.122353\pi\)
−0.138767 + 0.990325i \(0.544314\pi\)
\(104\) 0 0
\(105\) −2.50000 + 0.866025i −0.243975 + 0.0845154i
\(106\) 0 0
\(107\) 7.50000 + 12.9904i 0.725052 + 1.25583i 0.958952 + 0.283567i \(0.0915178\pi\)
−0.233900 + 0.972261i \(0.575149\pi\)
\(108\) 0 0
\(109\) 5.00000 8.66025i 0.478913 0.829502i −0.520794 0.853682i \(-0.674364\pi\)
0.999708 + 0.0241802i \(0.00769755\pi\)
\(110\) 0 0
\(111\) −8.00000 −0.759326
\(112\) 0 0
\(113\) 4.00000 0.376288 0.188144 0.982141i \(-0.439753\pi\)
0.188144 + 0.982141i \(0.439753\pi\)
\(114\) 0 0
\(115\) 2.00000 3.46410i 0.186501 0.323029i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 4.00000 20.7846i 0.366679 1.90532i
\(120\) 0 0
\(121\) 5.00000 + 8.66025i 0.454545 + 0.787296i
\(122\) 0 0
\(123\) −2.00000 + 3.46410i −0.180334 + 0.312348i
\(124\) 0 0
\(125\) 9.00000 0.804984
\(126\) 0 0
\(127\) 13.0000 1.15356 0.576782 0.816898i \(-0.304308\pi\)
0.576782 + 0.816898i \(0.304308\pi\)
\(128\) 0 0
\(129\) 5.00000 8.66025i 0.440225 0.762493i
\(130\) 0 0
\(131\) −9.50000 16.4545i −0.830019 1.43763i −0.898022 0.439950i \(-0.854996\pi\)
0.0680035 0.997685i \(-0.478337\pi\)
\(132\) 0 0
\(133\) −8.00000 6.92820i −0.693688 0.600751i
\(134\) 0 0
\(135\) 0.500000 + 0.866025i 0.0430331 + 0.0745356i
\(136\) 0 0
\(137\) −9.00000 + 15.5885i −0.768922 + 1.33181i 0.169226 + 0.985577i \(0.445873\pi\)
−0.938148 + 0.346235i \(0.887460\pi\)
\(138\) 0 0
\(139\) −10.0000 −0.848189 −0.424094 0.905618i \(-0.639408\pi\)
−0.424094 + 0.905618i \(0.639408\pi\)
\(140\) 0 0
\(141\) 6.00000 0.505291
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 2.50000 + 4.33013i 0.207614 + 0.359597i
\(146\) 0 0
\(147\) 1.00000 + 6.92820i 0.0824786 + 0.571429i
\(148\) 0 0
\(149\) 3.00000 + 5.19615i 0.245770 + 0.425685i 0.962348 0.271821i \(-0.0876260\pi\)
−0.716578 + 0.697507i \(0.754293\pi\)
\(150\) 0 0
\(151\) −3.50000 + 6.06218i −0.284826 + 0.493333i −0.972567 0.232623i \(-0.925269\pi\)
0.687741 + 0.725956i \(0.258602\pi\)
\(152\) 0 0
\(153\) −8.00000 −0.646762
\(154\) 0 0
\(155\) −7.00000 −0.562254
\(156\) 0 0
\(157\) 10.0000 17.3205i 0.798087 1.38233i −0.122774 0.992435i \(-0.539179\pi\)
0.920860 0.389892i \(-0.127488\pi\)
\(158\) 0 0
\(159\) −0.500000 0.866025i −0.0396526 0.0686803i
\(160\) 0 0
\(161\) −8.00000 6.92820i −0.630488 0.546019i
\(162\) 0 0
\(163\) 10.0000 + 17.3205i 0.783260 + 1.35665i 0.930033 + 0.367477i \(0.119778\pi\)
−0.146772 + 0.989170i \(0.546888\pi\)
\(164\) 0 0
\(165\) −0.500000 + 0.866025i −0.0389249 + 0.0674200i
\(166\) 0 0
\(167\) −18.0000 −1.39288 −0.696441 0.717614i \(-0.745234\pi\)
−0.696441 + 0.717614i \(0.745234\pi\)
\(168\) 0 0
\(169\) −13.0000 −1.00000
\(170\) 0 0
\(171\) −2.00000 + 3.46410i −0.152944 + 0.264906i
\(172\) 0 0
\(173\) −9.00000 15.5885i −0.684257 1.18517i −0.973670 0.227964i \(-0.926793\pi\)
0.289412 0.957205i \(-0.406540\pi\)
\(174\) 0 0
\(175\) 2.00000 10.3923i 0.151186 0.785584i
\(176\) 0 0
\(177\) 4.50000 + 7.79423i 0.338241 + 0.585850i
\(178\) 0 0
\(179\) 10.0000 17.3205i 0.747435 1.29460i −0.201613 0.979465i \(-0.564618\pi\)
0.949048 0.315130i \(-0.102048\pi\)
\(180\) 0 0
\(181\) −20.0000 −1.48659 −0.743294 0.668965i \(-0.766738\pi\)
−0.743294 + 0.668965i \(0.766738\pi\)
\(182\) 0 0
\(183\) 2.00000 0.147844
\(184\) 0 0
\(185\) −4.00000 + 6.92820i −0.294086 + 0.509372i
\(186\) 0 0
\(187\) −4.00000 6.92820i −0.292509 0.506640i
\(188\) 0 0
\(189\) 2.50000 0.866025i 0.181848 0.0629941i
\(190\) 0 0
\(191\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(192\) 0 0
\(193\) 5.50000 9.52628i 0.395899 0.685717i −0.597317 0.802005i \(-0.703766\pi\)
0.993215 + 0.116289i \(0.0370998\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −18.0000 −1.28245 −0.641223 0.767354i \(-0.721573\pi\)
−0.641223 + 0.767354i \(0.721573\pi\)
\(198\) 0 0
\(199\) −10.0000 + 17.3205i −0.708881 + 1.22782i 0.256391 + 0.966573i \(0.417466\pi\)
−0.965272 + 0.261245i \(0.915867\pi\)
\(200\) 0 0
\(201\) 1.00000 + 1.73205i 0.0705346 + 0.122169i
\(202\) 0 0
\(203\) 12.5000 4.33013i 0.877328 0.303915i
\(204\) 0 0
\(205\) 2.00000 + 3.46410i 0.139686 + 0.241943i
\(206\) 0 0
\(207\) −2.00000 + 3.46410i −0.139010 + 0.240772i
\(208\) 0 0
\(209\) −4.00000 −0.276686
\(210\) 0 0
\(211\) −22.0000 −1.51454 −0.757271 0.653101i \(-0.773468\pi\)
−0.757271 + 0.653101i \(0.773468\pi\)
\(212\) 0 0
\(213\) −3.00000 + 5.19615i −0.205557 + 0.356034i
\(214\) 0 0
\(215\) −5.00000 8.66025i −0.340997 0.590624i
\(216\) 0 0
\(217\) −3.50000 + 18.1865i −0.237595 + 1.23458i
\(218\) 0 0
\(219\) −1.00000 1.73205i −0.0675737 0.117041i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −19.0000 −1.27233 −0.636167 0.771551i \(-0.719481\pi\)
−0.636167 + 0.771551i \(0.719481\pi\)
\(224\) 0 0
\(225\) −4.00000 −0.266667
\(226\) 0 0
\(227\) 3.50000 6.06218i 0.232303 0.402361i −0.726182 0.687502i \(-0.758707\pi\)
0.958485 + 0.285141i \(0.0920405\pi\)
\(228\) 0 0
\(229\) 12.0000 + 20.7846i 0.792982 + 1.37349i 0.924113 + 0.382121i \(0.124806\pi\)
−0.131130 + 0.991365i \(0.541861\pi\)
\(230\) 0 0
\(231\) 2.00000 + 1.73205i 0.131590 + 0.113961i
\(232\) 0 0
\(233\) 4.00000 + 6.92820i 0.262049 + 0.453882i 0.966786 0.255586i \(-0.0822686\pi\)
−0.704737 + 0.709468i \(0.748935\pi\)
\(234\) 0 0
\(235\) 3.00000 5.19615i 0.195698 0.338960i
\(236\) 0 0
\(237\) −9.00000 −0.584613
\(238\) 0 0
\(239\) 12.0000 0.776215 0.388108 0.921614i \(-0.373129\pi\)
0.388108 + 0.921614i \(0.373129\pi\)
\(240\) 0 0
\(241\) −7.50000 + 12.9904i −0.483117 + 0.836784i −0.999812 0.0193858i \(-0.993829\pi\)
0.516695 + 0.856170i \(0.327162\pi\)
\(242\) 0 0
\(243\) −0.500000 0.866025i −0.0320750 0.0555556i
\(244\) 0 0
\(245\) 6.50000 + 2.59808i 0.415270 + 0.165985i
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) −1.50000 + 2.59808i −0.0950586 + 0.164646i
\(250\) 0 0
\(251\) −1.00000 −0.0631194 −0.0315597 0.999502i \(-0.510047\pi\)
−0.0315597 + 0.999502i \(0.510047\pi\)
\(252\) 0 0
\(253\) −4.00000 −0.251478
\(254\) 0 0
\(255\) −4.00000 + 6.92820i −0.250490 + 0.433861i
\(256\) 0 0
\(257\) −9.00000 15.5885i −0.561405 0.972381i −0.997374 0.0724199i \(-0.976928\pi\)
0.435970 0.899961i \(-0.356405\pi\)
\(258\) 0 0
\(259\) 16.0000 + 13.8564i 0.994192 + 0.860995i
\(260\) 0 0
\(261\) −2.50000 4.33013i −0.154746 0.268028i
\(262\) 0 0
\(263\) 15.0000 25.9808i 0.924940 1.60204i 0.133281 0.991078i \(-0.457449\pi\)
0.791658 0.610964i \(-0.209218\pi\)
\(264\) 0 0
\(265\) −1.00000 −0.0614295
\(266\) 0 0
\(267\) −6.00000 −0.367194
\(268\) 0 0
\(269\) −8.50000 + 14.7224i −0.518254 + 0.897643i 0.481521 + 0.876435i \(0.340085\pi\)
−0.999775 + 0.0212079i \(0.993249\pi\)
\(270\) 0 0
\(271\) −5.50000 9.52628i −0.334101 0.578680i 0.649211 0.760609i \(-0.275099\pi\)
−0.983312 + 0.181928i \(0.941766\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −2.00000 3.46410i −0.120605 0.208893i
\(276\) 0 0
\(277\) −12.0000 + 20.7846i −0.721010 + 1.24883i 0.239585 + 0.970875i \(0.422989\pi\)
−0.960595 + 0.277951i \(0.910345\pi\)
\(278\) 0 0
\(279\) 7.00000 0.419079
\(280\) 0 0
\(281\) 6.00000 0.357930 0.178965 0.983855i \(-0.442725\pi\)
0.178965 + 0.983855i \(0.442725\pi\)
\(282\) 0 0
\(283\) −3.00000 + 5.19615i −0.178331 + 0.308879i −0.941309 0.337546i \(-0.890403\pi\)
0.762978 + 0.646425i \(0.223737\pi\)
\(284\) 0 0
\(285\) 2.00000 + 3.46410i 0.118470 + 0.205196i
\(286\) 0 0
\(287\) 10.0000 3.46410i 0.590281 0.204479i
\(288\) 0 0
\(289\) −23.5000 40.7032i −1.38235 2.39431i
\(290\) 0 0
\(291\) 0.500000 0.866025i 0.0293105 0.0507673i
\(292\) 0 0
\(293\) −3.00000 −0.175262 −0.0876309 0.996153i \(-0.527930\pi\)
−0.0876309 + 0.996153i \(0.527930\pi\)
\(294\) 0 0
\(295\) 9.00000 0.524000
\(296\) 0 0
\(297\) 0.500000 0.866025i 0.0290129 0.0502519i
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) −25.0000 + 8.66025i −1.44098 + 0.499169i
\(302\) 0 0
\(303\) 1.00000 + 1.73205i 0.0574485 + 0.0995037i
\(304\) 0 0
\(305\) 1.00000 1.73205i 0.0572598 0.0991769i
\(306\) 0 0
\(307\) 4.00000 0.228292 0.114146 0.993464i \(-0.463587\pi\)
0.114146 + 0.993464i \(0.463587\pi\)
\(308\) 0 0
\(309\) −16.0000 −0.910208
\(310\) 0 0
\(311\) −14.0000 + 24.2487i −0.793867 + 1.37502i 0.129689 + 0.991555i \(0.458602\pi\)
−0.923556 + 0.383464i \(0.874731\pi\)
\(312\) 0 0
\(313\) −0.500000 0.866025i −0.0282617 0.0489506i 0.851549 0.524276i \(-0.175664\pi\)
−0.879810 + 0.475325i \(0.842331\pi\)
\(314\) 0 0
\(315\) 0.500000 2.59808i 0.0281718 0.146385i
\(316\) 0 0
\(317\) −2.50000 4.33013i −0.140414 0.243204i 0.787239 0.616649i \(-0.211510\pi\)
−0.927653 + 0.373444i \(0.878177\pi\)
\(318\) 0 0
\(319\) 2.50000 4.33013i 0.139973 0.242441i
\(320\) 0 0
\(321\) −15.0000 −0.837218
\(322\) 0 0
\(323\) −32.0000 −1.78053
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 5.00000 + 8.66025i 0.276501 + 0.478913i
\(328\) 0 0
\(329\) −12.0000 10.3923i −0.661581 0.572946i
\(330\) 0 0
\(331\) 10.0000 + 17.3205i 0.549650 + 0.952021i 0.998298 + 0.0583130i \(0.0185721\pi\)
−0.448649 + 0.893708i \(0.648095\pi\)
\(332\) 0 0
\(333\) 4.00000 6.92820i 0.219199 0.379663i
\(334\) 0 0
\(335\) 2.00000 0.109272
\(336\) 0 0
\(337\) 1.00000 0.0544735 0.0272367 0.999629i \(-0.491329\pi\)
0.0272367 + 0.999629i \(0.491329\pi\)
\(338\) 0 0
\(339\) −2.00000 + 3.46410i −0.108625 + 0.188144i
\(340\) 0 0
\(341\) 3.50000 + 6.06218i 0.189536 + 0.328285i
\(342\) 0 0
\(343\) 10.0000 15.5885i 0.539949 0.841698i
\(344\) 0 0
\(345\) 2.00000 + 3.46410i 0.107676 + 0.186501i
\(346\) 0 0
\(347\) 14.0000 24.2487i 0.751559 1.30174i −0.195507 0.980702i \(-0.562635\pi\)
0.947067 0.321037i \(-0.104031\pi\)
\(348\) 0 0
\(349\) −6.00000 −0.321173 −0.160586 0.987022i \(-0.551338\pi\)
−0.160586 + 0.987022i \(0.551338\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 12.0000 20.7846i 0.638696 1.10625i −0.347024 0.937856i \(-0.612808\pi\)
0.985719 0.168397i \(-0.0538590\pi\)
\(354\) 0 0
\(355\) 3.00000 + 5.19615i 0.159223 + 0.275783i
\(356\) 0 0
\(357\) 16.0000 + 13.8564i 0.846810 + 0.733359i
\(358\) 0 0
\(359\) −3.00000 5.19615i −0.158334 0.274242i 0.775934 0.630814i \(-0.217279\pi\)
−0.934268 + 0.356572i \(0.883946\pi\)
\(360\) 0 0
\(361\) 1.50000 2.59808i 0.0789474 0.136741i
\(362\) 0 0
\(363\) −10.0000 −0.524864
\(364\) 0 0
\(365\) −2.00000 −0.104685
\(366\) 0 0
\(367\) 1.50000 2.59808i 0.0782994 0.135618i −0.824217 0.566274i \(-0.808384\pi\)
0.902516 + 0.430656i \(0.141718\pi\)
\(368\) 0 0
\(369\) −2.00000 3.46410i −0.104116 0.180334i
\(370\) 0 0
\(371\) −0.500000 + 2.59808i −0.0259587 + 0.134885i
\(372\) 0 0
\(373\) −2.00000 3.46410i −0.103556 0.179364i 0.809591 0.586994i \(-0.199689\pi\)
−0.913147 + 0.407630i \(0.866355\pi\)
\(374\) 0 0
\(375\) −4.50000 + 7.79423i −0.232379 + 0.402492i
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 28.0000 1.43826 0.719132 0.694874i \(-0.244540\pi\)
0.719132 + 0.694874i \(0.244540\pi\)
\(380\) 0 0
\(381\) −6.50000 + 11.2583i −0.333005 + 0.576782i
\(382\) 0 0
\(383\) 17.0000 + 29.4449i 0.868659 + 1.50456i 0.863367 + 0.504576i \(0.168351\pi\)
0.00529229 + 0.999986i \(0.498315\pi\)
\(384\) 0 0
\(385\) 2.50000 0.866025i 0.127412 0.0441367i
\(386\) 0 0
\(387\) 5.00000 + 8.66025i 0.254164 + 0.440225i
\(388\) 0 0
\(389\) −17.0000 + 29.4449i −0.861934 + 1.49291i 0.00812520 + 0.999967i \(0.497414\pi\)
−0.870059 + 0.492947i \(0.835920\pi\)
\(390\) 0 0
\(391\) −32.0000 −1.61831
\(392\) 0 0
\(393\) 19.0000 0.958423
\(394\) 0 0
\(395\) −4.50000 + 7.79423i −0.226420 + 0.392170i
\(396\) 0 0
\(397\) 2.00000 + 3.46410i 0.100377 + 0.173858i 0.911840 0.410546i \(-0.134662\pi\)
−0.811463 + 0.584404i \(0.801328\pi\)
\(398\) 0 0
\(399\) 10.0000 3.46410i 0.500626 0.173422i
\(400\) 0 0
\(401\) 16.0000 + 27.7128i 0.799002 + 1.38391i 0.920267 + 0.391292i \(0.127972\pi\)
−0.121265 + 0.992620i \(0.538695\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) −1.00000 −0.0496904
\(406\) 0 0
\(407\) 8.00000 0.396545
\(408\) 0 0
\(409\) −11.5000 + 19.9186i −0.568638 + 0.984911i 0.428063 + 0.903749i \(0.359196\pi\)
−0.996701 + 0.0811615i \(0.974137\pi\)
\(410\) 0 0
\(411\) −9.00000 15.5885i −0.443937 0.768922i
\(412\) 0 0
\(413\) 4.50000 23.3827i 0.221431 1.15059i
\(414\) 0 0
\(415\) 1.50000 + 2.59808i 0.0736321 + 0.127535i
\(416\) 0 0
\(417\) 5.00000 8.66025i 0.244851 0.424094i
\(418\) 0 0
\(419\) −24.0000 −1.17248 −0.586238 0.810139i \(-0.699392\pi\)
−0.586238 + 0.810139i \(0.699392\pi\)
\(420\) 0 0
\(421\) −2.00000 −0.0974740 −0.0487370 0.998812i \(-0.515520\pi\)
−0.0487370 + 0.998812i \(0.515520\pi\)
\(422\) 0 0
\(423\) −3.00000 + 5.19615i −0.145865 + 0.252646i
\(424\) 0 0
\(425\) −16.0000 27.7128i −0.776114 1.34427i
\(426\) 0 0
\(427\) −4.00000 3.46410i −0.193574 0.167640i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(432\) 0 0
\(433\) 30.0000 1.44171 0.720854 0.693087i \(-0.243750\pi\)
0.720854 + 0.693087i \(0.243750\pi\)
\(434\) 0 0
\(435\) −5.00000 −0.239732
\(436\) 0 0
\(437\) −8.00000 + 13.8564i −0.382692 + 0.662842i
\(438\) 0 0
\(439\) −9.50000 16.4545i −0.453410 0.785330i 0.545185 0.838316i \(-0.316459\pi\)
−0.998595 + 0.0529862i \(0.983126\pi\)
\(440\) 0 0
\(441\) −6.50000 2.59808i −0.309524 0.123718i
\(442\) 0 0
\(443\) 14.5000 + 25.1147i 0.688916 + 1.19324i 0.972189 + 0.234198i \(0.0752464\pi\)
−0.283273 + 0.959039i \(0.591420\pi\)
\(444\) 0 0
\(445\) −3.00000 + 5.19615i −0.142214 + 0.246321i
\(446\) 0 0
\(447\) −6.00000 −0.283790
\(448\) 0 0
\(449\) −24.0000 −1.13263 −0.566315 0.824189i \(-0.691631\pi\)
−0.566315 + 0.824189i \(0.691631\pi\)
\(450\) 0 0
\(451\) 2.00000 3.46410i 0.0941763 0.163118i
\(452\) 0 0
\(453\) −3.50000 6.06218i −0.164444 0.284826i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −7.50000 12.9904i −0.350835 0.607664i 0.635561 0.772051i \(-0.280769\pi\)
−0.986396 + 0.164386i \(0.947436\pi\)
\(458\) 0 0
\(459\) 4.00000 6.92820i 0.186704 0.323381i
\(460\) 0 0
\(461\) 14.0000 0.652045 0.326023 0.945362i \(-0.394291\pi\)
0.326023 + 0.945362i \(0.394291\pi\)
\(462\) 0 0
\(463\) −40.0000 −1.85896 −0.929479 0.368875i \(-0.879743\pi\)
−0.929479 + 0.368875i \(0.879743\pi\)
\(464\) 0 0
\(465\) 3.50000 6.06218i 0.162309 0.281127i
\(466\) 0 0
\(467\) −6.00000 10.3923i −0.277647 0.480899i 0.693153 0.720791i \(-0.256221\pi\)
−0.970799 + 0.239892i \(0.922888\pi\)
\(468\) 0 0
\(469\) 1.00000 5.19615i 0.0461757 0.239936i
\(470\) 0 0
\(471\) 10.0000 + 17.3205i 0.460776 + 0.798087i
\(472\) 0 0
\(473\) −5.00000 + 8.66025i −0.229900 + 0.398199i
\(474\) 0 0
\(475\) −16.0000 −0.734130
\(476\) 0 0
\(477\) 1.00000 0.0457869
\(478\) 0 0
\(479\) 9.00000 15.5885i 0.411220 0.712255i −0.583803 0.811895i \(-0.698436\pi\)
0.995023 + 0.0996406i \(0.0317693\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 10.0000 3.46410i 0.455016 0.157622i
\(484\) 0 0
\(485\) −0.500000 0.866025i −0.0227038 0.0393242i
\(486\) 0 0
\(487\) 15.5000 26.8468i 0.702372 1.21654i −0.265260 0.964177i \(-0.585458\pi\)
0.967632 0.252367i \(-0.0812090\pi\)
\(488\) 0 0
\(489\) −20.0000 −0.904431
\(490\) 0 0
\(491\) 27.0000 1.21849 0.609246 0.792981i \(-0.291472\pi\)
0.609246 + 0.792981i \(0.291472\pi\)
\(492\) 0 0
\(493\) 20.0000 34.6410i 0.900755 1.56015i
\(494\) 0 0
\(495\) −0.500000 0.866025i −0.0224733 0.0389249i
\(496\) 0 0
\(497\) 15.0000 5.19615i 0.672842 0.233079i
\(498\) 0 0
\(499\) 17.0000 + 29.4449i 0.761025 + 1.31813i 0.942323 + 0.334705i \(0.108637\pi\)
−0.181298 + 0.983428i \(0.558030\pi\)
\(500\) 0 0
\(501\) 9.00000 15.5885i 0.402090 0.696441i
\(502\) 0 0
\(503\) −12.0000 −0.535054 −0.267527 0.963550i \(-0.586206\pi\)
−0.267527 + 0.963550i \(0.586206\pi\)
\(504\) 0 0
\(505\) 2.00000 0.0889988
\(506\) 0 0
\(507\) 6.50000 11.2583i 0.288675 0.500000i
\(508\) 0 0
\(509\) −4.50000 7.79423i −0.199459 0.345473i 0.748894 0.662690i \(-0.230585\pi\)
−0.948353 + 0.317217i \(0.897252\pi\)
\(510\) 0 0
\(511\) −1.00000 + 5.19615i −0.0442374 + 0.229864i
\(512\) 0 0
\(513\) −2.00000 3.46410i −0.0883022 0.152944i
\(514\) 0 0
\(515\) −8.00000 + 13.8564i −0.352522 + 0.610586i
\(516\) 0 0
\(517\) −6.00000 −0.263880
\(518\) 0 0
\(519\) 18.0000 0.790112
\(520\) 0 0
\(521\) −7.00000 + 12.1244i −0.306676 + 0.531178i −0.977633 0.210318i \(-0.932550\pi\)
0.670957 + 0.741496i \(0.265883\pi\)
\(522\) 0 0
\(523\) 4.00000 + 6.92820i 0.174908 + 0.302949i 0.940129 0.340818i \(-0.110704\pi\)
−0.765222 + 0.643767i \(0.777371\pi\)
\(524\) 0 0
\(525\) 8.00000 + 6.92820i 0.349149 + 0.302372i
\(526\) 0 0
\(527\) 28.0000 + 48.4974i 1.21970 + 2.11258i
\(528\) 0 0
\(529\) 3.50000 6.06218i 0.152174 0.263573i
\(530\) 0 0
\(531\) −9.00000 −0.390567
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) −7.50000 + 12.9904i −0.324253 + 0.561623i
\(536\) 0 0
\(537\) 10.0000 + 17.3205i 0.431532 + 0.747435i
\(538\) 0 0
\(539\) −1.00000 6.92820i −0.0430730 0.298419i
\(540\) 0 0
\(541\) 11.0000 + 19.0526i 0.472927 + 0.819133i 0.999520 0.0309841i \(-0.00986412\pi\)
−0.526593 + 0.850118i \(0.676531\pi\)
\(542\) 0 0
\(543\) 10.0000 17.3205i 0.429141 0.743294i
\(544\) 0 0
\(545\) 10.0000 0.428353
\(546\) 0 0
\(547\) −8.00000 −0.342055 −0.171028 0.985266i \(-0.554709\pi\)
−0.171028 + 0.985266i \(0.554709\pi\)
\(548\) 0 0
\(549\) −1.00000 + 1.73205i −0.0426790 + 0.0739221i
\(550\) 0 0
\(551\) −10.0000 17.3205i −0.426014 0.737878i
\(552\) 0 0
\(553\) 18.0000 + 15.5885i 0.765438 + 0.662889i
\(554\) 0 0
\(555\) −4.00000 6.92820i −0.169791 0.294086i
\(556\) 0 0
\(557\) 4.50000 7.79423i 0.190671 0.330252i −0.754802 0.655953i \(-0.772267\pi\)
0.945473 + 0.325701i \(0.105600\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 8.00000 0.337760
\(562\) 0 0
\(563\) 14.5000 25.1147i 0.611102 1.05846i −0.379953 0.925006i \(-0.624060\pi\)
0.991055 0.133454i \(-0.0426069\pi\)
\(564\) 0 0
\(565\) 2.00000 + 3.46410i 0.0841406 + 0.145736i
\(566\) 0 0
\(567\) −0.500000 + 2.59808i −0.0209980 + 0.109109i
\(568\) 0 0
\(569\) 6.00000 + 10.3923i 0.251533 + 0.435668i 0.963948 0.266090i \(-0.0857319\pi\)
−0.712415 + 0.701758i \(0.752399\pi\)
\(570\) 0 0
\(571\) 7.00000 12.1244i 0.292941 0.507388i −0.681563 0.731760i \(-0.738699\pi\)
0.974504 + 0.224371i \(0.0720328\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −16.0000 −0.667246
\(576\) 0 0
\(577\) −11.5000 + 19.9186i −0.478751 + 0.829222i −0.999703 0.0243645i \(-0.992244\pi\)
0.520952 + 0.853586i \(0.325577\pi\)
\(578\) 0 0
\(579\) 5.50000 + 9.52628i 0.228572 + 0.395899i
\(580\) 0 0
\(581\) 7.50000 2.59808i 0.311152 0.107786i
\(582\) 0 0
\(583\) 0.500000 + 0.866025i 0.0207079 + 0.0358671i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −15.0000 −0.619116 −0.309558 0.950881i \(-0.600181\pi\)
−0.309558 + 0.950881i \(0.600181\pi\)
\(588\) 0 0
\(589\) 28.0000 1.15372
\(590\) 0 0
\(591\) 9.00000 15.5885i 0.370211 0.641223i
\(592\) 0 0
\(593\) 18.0000 + 31.1769i 0.739171 + 1.28028i 0.952869 + 0.303383i \(0.0981160\pi\)
−0.213697 + 0.976900i \(0.568551\pi\)
\(594\) 0 0
\(595\) 20.0000 6.92820i 0.819920 0.284029i
\(596\) 0 0
\(597\) −10.0000 17.3205i −0.409273 0.708881i
\(598\) 0 0
\(599\) −15.0000 + 25.9808i −0.612883 + 1.06155i 0.377869 + 0.925859i \(0.376657\pi\)
−0.990752 + 0.135686i \(0.956676\pi\)
\(600\) 0 0
\(601\) 19.0000 0.775026 0.387513 0.921864i \(-0.373334\pi\)
0.387513 + 0.921864i \(0.373334\pi\)
\(602\) 0 0
\(603\) −2.00000 −0.0814463
\(604\) 0 0
\(605\) −5.00000 + 8.66025i −0.203279 + 0.352089i
\(606\) 0 0
\(607\) −12.5000 21.6506i −0.507359 0.878772i −0.999964 0.00851879i \(-0.997288\pi\)
0.492604 0.870253i \(-0.336045\pi\)
\(608\) 0 0
\(609\) −2.50000 + 12.9904i −0.101305 + 0.526397i
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 22.0000 38.1051i 0.888572 1.53905i 0.0470071 0.998895i \(-0.485032\pi\)
0.841564 0.540157i \(-0.181635\pi\)
\(614\) 0 0
\(615\) −4.00000 −0.161296
\(616\) 0 0
\(617\) 6.00000 0.241551 0.120775 0.992680i \(-0.461462\pi\)
0.120775 + 0.992680i \(0.461462\pi\)
\(618\) 0 0
\(619\) −5.00000 + 8.66025i −0.200967 + 0.348085i −0.948840 0.315757i \(-0.897742\pi\)
0.747873 + 0.663842i \(0.231075\pi\)
\(620\) 0 0
\(621\) −2.00000 3.46410i −0.0802572 0.139010i
\(622\) 0 0
\(623\) 12.0000 + 10.3923i 0.480770 + 0.416359i
\(624\) 0 0
\(625\) −5.50000 9.52628i −0.220000 0.381051i
\(626\) 0 0
\(627\) 2.00000 3.46410i 0.0798723 0.138343i
\(628\) 0 0
\(629\) 64.0000 2.55185
\(630\) 0 0
\(631\) 33.0000 1.31371 0.656855 0.754017i \(-0.271887\pi\)
0.656855 + 0.754017i \(0.271887\pi\)
\(632\) 0 0
\(633\) 11.0000 19.0526i 0.437211 0.757271i
\(634\) 0 0
\(635\) 6.50000 + 11.2583i 0.257945 + 0.446773i
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) −3.00000 5.19615i −0.118678 0.205557i
\(640\) 0 0
\(641\) −19.0000 + 32.9090i −0.750455 + 1.29983i 0.197148 + 0.980374i \(0.436832\pi\)
−0.947602 + 0.319452i \(0.896501\pi\)
\(642\) 0 0
\(643\) −2.00000 −0.0788723 −0.0394362 0.999222i \(-0.512556\pi\)
−0.0394362 + 0.999222i \(0.512556\pi\)
\(644\) 0 0
\(645\) 10.0000 0.393750
\(646\) 0 0
\(647\) −5.00000 + 8.66025i −0.196570 + 0.340470i −0.947414 0.320010i \(-0.896314\pi\)
0.750844 + 0.660480i \(0.229647\pi\)
\(648\) 0 0
\(649\) −4.50000 7.79423i −0.176640 0.305950i
\(650\) 0 0
\(651\) −14.0000 12.1244i −0.548703 0.475191i
\(652\) 0 0
\(653\) −15.5000 26.8468i −0.606562 1.05060i −0.991803 0.127780i \(-0.959215\pi\)
0.385241 0.922816i \(-0.374118\pi\)
\(654\) 0 0
\(655\) 9.50000 16.4545i 0.371196 0.642930i
\(656\) 0 0
\(657\) 2.00000 0.0780274
\(658\) 0 0
\(659\) −16.0000 −0.623272 −0.311636 0.950202i \(-0.600877\pi\)
−0.311636 + 0.950202i \(0.600877\pi\)
\(660\) 0 0
\(661\) 9.00000 15.5885i 0.350059 0.606321i −0.636200 0.771524i \(-0.719495\pi\)
0.986260 + 0.165203i \(0.0528281\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 2.00000 10.3923i 0.0775567 0.402996i
\(666\) 0 0
\(667\) −10.0000 17.3205i −0.387202 0.670653i
\(668\) 0 0
\(669\) 9.50000 16.4545i 0.367291 0.636167i
\(670\) 0 0
\(671\) −2.00000 −0.0772091
\(672\) 0 0
\(673\) −19.0000 −0.732396 −0.366198 0.930537i \(-0.619341\pi\)
−0.366198 + 0.930537i \(0.619341\pi\)
\(674\) 0 0
\(675\) 2.00000 3.46410i 0.0769800 0.133333i
\(676\) 0 0
\(677\) −13.5000 23.3827i −0.518847 0.898670i −0.999760 0.0219013i \(-0.993028\pi\)
0.480913 0.876768i \(-0.340305\pi\)
\(678\) 0 0
\(679\) −2.50000 + 0.866025i −0.0959412 + 0.0332350i
\(680\) 0 0
\(681\) 3.50000 + 6.06218i 0.134120 + 0.232303i
\(682\) 0 0
\(683\) −18.5000 + 32.0429i −0.707883 + 1.22609i 0.257758 + 0.966209i \(0.417016\pi\)
−0.965641 + 0.259880i \(0.916317\pi\)
\(684\) 0 0
\(685\) −18.0000 −0.687745
\(686\) 0 0
\(687\) −24.0000 −0.915657
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) −6.00000 10.3923i −0.228251 0.395342i 0.729039 0.684472i \(-0.239967\pi\)
−0.957290 + 0.289130i \(0.906634\pi\)
\(692\) 0 0
\(693\) −2.50000 + 0.866025i −0.0949671 + 0.0328976i
\(694\) 0 0
\(695\) −5.00000 8.66025i −0.189661 0.328502i
\(696\) 0 0
\(697\) 16.0000 27.7128i 0.606043 1.04970i
\(698\) 0 0
\(699\) −8.00000 −0.302588
\(700\) 0 0
\(701\) −27.0000 −1.01978 −0.509888 0.860241i \(-0.670313\pi\)
−0.509888 + 0.860241i \(0.670313\pi\)
\(702\) 0 0
\(703\) 16.0000 27.7128i 0.603451 1.04521i
\(704\) 0 0
\(705\) 3.00000 + 5.19615i 0.112987 + 0.195698i
\(706\) 0 0
\(707\) 1.00000 5.19615i 0.0376089 0.195421i
\(708\) 0 0
\(709\) 7.00000 + 12.1244i 0.262891 + 0.455340i 0.967009 0.254743i \(-0.0819909\pi\)
−0.704118 + 0.710083i \(0.748658\pi\)
\(710\) 0 0
\(711\) 4.50000 7.79423i 0.168763 0.292306i
\(712\) 0 0
\(713\) 28.0000 1.04861
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −6.00000 + 10.3923i −0.224074 + 0.388108i
\(718\) 0 0
\(719\) −5.00000 8.66025i −0.186469 0.322973i 0.757602 0.652717i \(-0.226371\pi\)
−0.944070 + 0.329744i \(0.893038\pi\)
\(720\) 0 0
\(721\) 32.0000 + 27.7128i 1.19174 + 1.03208i
\(722\) 0 0
\(723\) −7.50000 12.9904i −0.278928 0.483117i
\(724\) 0 0
\(725\) 10.0000 17.3205i 0.371391 0.643268i
\(726\) 0 0
\(727\) 19.0000 0.704671 0.352335 0.935874i \(-0.385388\pi\)
0.352335 + 0.935874i \(0.385388\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −40.0000 + 69.2820i −1.47945 + 2.56249i
\(732\) 0 0
\(733\) −7.00000 12.1244i −0.258551 0.447823i 0.707303 0.706910i \(-0.249912\pi\)
−0.965854 + 0.259087i \(0.916578\pi\)
\(734\) 0 0
\(735\) −5.50000 + 4.33013i −0.202871 + 0.159719i
\(736\) 0 0
\(737\) −1.00000 1.73205i −0.0368355 0.0638009i
\(738\) 0 0
\(739\) 17.0000 29.4449i 0.625355 1.08315i −0.363117 0.931744i \(-0.618287\pi\)
0.988472 0.151403i \(-0.0483792\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −38.0000 −1.39408 −0.697042 0.717030i \(-0.745501\pi\)
−0.697042 + 0.717030i \(0.745501\pi\)
\(744\) 0 0
\(745\) −3.00000 + 5.19615i −0.109911 + 0.190372i
\(746\) 0 0
\(747\) −1.50000 2.59808i −0.0548821 0.0950586i
\(748\) 0 0
\(749\) 30.0000 + 25.9808i 1.09618 + 0.949316i
\(750\) 0 0
\(751\) 3.50000 + 6.06218i 0.127717 + 0.221212i 0.922792 0.385299i \(-0.125902\pi\)
−0.795075 + 0.606511i \(0.792568\pi\)
\(752\) 0 0
\(753\) 0.500000 0.866025i 0.0182210 0.0315597i
\(754\) 0 0
\(755\) −7.00000 −0.254756
\(756\) 0 0
\(757\) 26.0000 0.944986 0.472493 0.881334i \(-0.343354\pi\)
0.472493 + 0.881334i \(0.343354\pi\)
\(758\) 0 0
\(759\) 2.00000 3.46410i 0.0725954 0.125739i
\(760\) 0 0
\(761\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(762\) 0 0
\(763\) 5.00000 25.9808i 0.181012 0.940567i
\(764\) 0 0
\(765\) −4.00000 6.92820i −0.144620 0.250490i
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −19.0000 −0.685158 −0.342579 0.939489i \(-0.611300\pi\)
−0.342579 + 0.939489i \(0.611300\pi\)
\(770\) 0 0
\(771\) 18.0000 0.648254
\(772\) 0 0
\(773\) 9.00000 15.5885i 0.323708 0.560678i −0.657542 0.753418i \(-0.728404\pi\)
0.981250 + 0.192740i \(0.0617373\pi\)
\(774\) 0 0
\(775\) 14.0000 + 24.2487i 0.502895 + 0.871039i
\(776\) 0 0
\(777\) −20.0000 + 6.92820i −0.717496 + 0.248548i
\(778\) 0 0
\(779\) −8.00000 13.8564i −0.286630 0.496457i
\(780\) 0 0
\(781\) 3.00000 5.19615i 0.107348 0.185933i
\(782\) 0 0
\(783\) 5.00000 0.178685
\(784\) 0 0
\(785\) 20.0000 0.713831
\(786\) 0 0
\(787\) −19.0000 + 32.9090i −0.677277 + 1.17308i 0.298521 + 0.954403i \(0.403507\pi\)
−0.975798 + 0.218675i \(0.929827\pi\)
\(788\) 0 0
\(789\) 15.0000 + 25.9808i 0.534014 + 0.924940i
\(790\) 0 0
\(791\) 10.0000 3.46410i 0.355559 0.123169i
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0.500000 0.866025i 0.0177332 0.0307148i
\(796\) 0 0
\(797\) 51.0000 1.80651 0.903256 0.429101i \(-0.141170\pi\)
0.903256 + 0.429101i \(0.141170\pi\)
\(798\) 0 0
\(799\) −48.0000 −1.69812
\(800\) 0 0
\(801\) 3.00000 5.19615i 0.106000 0.183597i
\(802\) 0 0
\(803\) 1.00000 + 1.73205i 0.0352892 + 0.0611227i
\(804\) 0 0
\(805\) 2.00000 10.3923i 0.0704907 0.366281i
\(806\) 0 0
\(807\) −8.50000 14.7224i −0.299214 0.518254i
\(808\) 0 0
\(809\) 16.0000 27.7128i 0.562530 0.974331i −0.434745 0.900554i \(-0.643161\pi\)
0.997275 0.0737769i \(-0.0235053\pi\)
\(810\) 0 0
\(811\) 2.00000 0.0702295 0.0351147 0.999383i \(-0.488820\pi\)
0.0351147 + 0.999383i \(0.488820\pi\)
\(812\) 0 0
\(813\) 11.0000 0.385787
\(814\) 0 0
\(815\) −10.0000 + 17.3205i −0.350285 + 0.606711i
\(816\) 0 0
\(817\) 20.0000 + 34.6410i 0.699711 + 1.21194i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −0.500000 0.866025i −0.0174501 0.0302245i 0.857168 0.515036i \(-0.172221\pi\)
−0.874619 + 0.484812i \(0.838888\pi\)
\(822\) 0 0
\(823\) −8.00000 + 13.8564i −0.278862 + 0.483004i −0.971102 0.238664i \(-0.923291\pi\)
0.692240 + 0.721668i \(0.256624\pi\)
\(824\) 0 0
\(825\) 4.00000 0.139262
\(826\) 0 0
\(827\) 43.0000 1.49526 0.747628 0.664117i \(-0.231193\pi\)
0.747628 + 0.664117i \(0.231193\pi\)
\(828\) 0 0
\(829\) −10.0000 + 17.3205i −0.347314 + 0.601566i −0.985771 0.168091i \(-0.946240\pi\)
0.638457 + 0.769657i \(0.279573\pi\)
\(830\) 0 0
\(831\) −12.0000 20.7846i −0.416275 0.721010i
\(832\) 0 0
\(833\) −8.00000 55.4256i −0.277184 1.92038i
\(834\) 0 0
\(835\) −9.00000 15.5885i −0.311458 0.539461i
\(836\) 0 0
\(837\) −3.50000 + 6.06218i −0.120978 + 0.209540i
\(838\) 0 0
\(839\) −8.00000 −0.276191 −0.138095 0.990419i \(-0.544098\pi\)
−0.138095 + 0.990419i \(0.544098\pi\)
\(840\) 0 0
\(841\) −4.00000 −0.137931
\(842\) 0 0
\(843\) −3.00000 + 5.19615i −0.103325 + 0.178965i
\(844\) 0 0
\(845\) −6.50000 11.2583i −0.223607 0.387298i
\(846\) 0 0
\(847\) 20.0000 + 17.3205i 0.687208 + 0.595140i
\(848\) 0 0
\(849\) −3.00000 5.19615i −0.102960 0.178331i
\(850\) 0 0
\(851\) 16.0000 27.7128i 0.548473 0.949983i
\(852\) 0 0
\(853\) −42.0000 −1.43805 −0.719026 0.694983i \(-0.755412\pi\)
−0.719026 + 0.694983i \(0.755412\pi\)
\(854\) 0 0
\(855\) −4.00000 −0.136797
\(856\) 0 0
\(857\) 3.00000 5.19615i 0.102478 0.177497i −0.810227 0.586116i \(-0.800656\pi\)
0.912705 + 0.408619i \(0.133990\pi\)
\(858\) 0 0
\(859\) −9.00000 15.5885i −0.307076 0.531871i 0.670645 0.741778i \(-0.266017\pi\)
−0.977721 + 0.209907i \(0.932684\pi\)
\(860\) 0 0
\(861\) −2.00000 + 10.3923i −0.0681598 + 0.354169i
\(862\) 0 0
\(863\) 23.0000 + 39.8372i 0.782929 + 1.35607i 0.930228 + 0.366981i \(0.119609\pi\)
−0.147299 + 0.989092i \(0.547058\pi\)
\(864\) 0 0
\(865\) 9.00000 15.5885i 0.306009 0.530023i
\(866\) 0 0
\(867\) 47.0000 1.59620
\(868\) 0 0
\(869\) 9.00000 0.305304
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 0.500000 + 0.866025i 0.0169224 + 0.0293105i
\(874\) 0 0
\(875\) 22.5000 7.79423i 0.760639 0.263493i
\(876\) 0 0
\(877\) 16.0000 + 27.7128i 0.540282 + 0.935795i 0.998888 + 0.0471555i \(0.0150156\pi\)
−0.458606 + 0.888640i \(0.651651\pi\)
\(878\) 0 0
\(879\) 1.50000 2.59808i 0.0505937 0.0876309i
\(880\) 0 0
\(881\) 18.0000 0.606435 0.303218 0.952921i \(-0.401939\pi\)
0.303218 + 0.952921i \(0.401939\pi\)
\(882\) 0 0
\(883\) 28.0000 0.942275 0.471138 0.882060i \(-0.343844\pi\)
0.471138 + 0.882060i \(0.343844\pi\)
\(884\) 0 0
\(885\) −4.50000 + 7.79423i −0.151266 + 0.262000i
\(886\) 0 0
\(887\) −2.00000 3.46410i −0.0671534 0.116313i 0.830494 0.557028i \(-0.188058\pi\)
−0.897647 + 0.440715i \(0.854725\pi\)
\(888\) 0 0
\(889\) 32.5000 11.2583i 1.09002 0.377592i
\(890\) 0 0
\(891\) 0.500000 + 0.866025i 0.0167506 + 0.0290129i
\(892\) 0 0
\(893\) −12.0000 + 20.7846i −0.401565 + 0.695530i
\(894\) 0 0
\(895\) 20.0000 0.668526
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −17.5000 + 30.3109i −0.583658 + 1.01092i
\(900\) 0 0
\(901\) 4.00000 + 6.92820i 0.133259 + 0.230812i
\(902\) 0 0
\(903\) 5.00000 25.9808i 0.166390 0.864586i
\(904\) 0 0
\(905\) −10.0000 17.3205i −0.332411 0.575753i
\(906\) 0 0
\(907\) −6.00000 + 10.3923i −0.199227 + 0.345071i −0.948278 0.317441i \(-0.897176\pi\)
0.749051 + 0.662512i \(0.230510\pi\)
\(908\) 0 0
\(909\) −2.00000 −0.0663358
\(910\) 0 0
\(911\) −6.00000 −0.198789 −0.0993944 0.995048i \(-0.531691\pi\)
−0.0993944 + 0.995048i \(0.531691\pi\)
\(912\) 0 0
\(913\) 1.50000 2.59808i 0.0496428 0.0859838i
\(914\) 0 0
\(915\) 1.00000 + 1.73205i 0.0330590 + 0.0572598i
\(916\) 0 0
\(917\) −38.0000 32.9090i −1.25487 1.08675i
\(918\) 0 0
\(919\) 16.0000 + 27.7128i 0.527791 + 0.914161i 0.999475 + 0.0323936i \(0.0103130\pi\)
−0.471684 + 0.881768i \(0.656354\pi\)
\(920\) 0 0
\(921\) −2.00000 + 3.46410i −0.0659022 + 0.114146i
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 32.0000 1.05215
\(926\) 0 0
\(927\) 8.00000 13.8564i 0.262754 0.455104i
\(928\) 0 0
\(929\) −17.0000 29.4449i −0.557752 0.966055i −0.997684 0.0680235i \(-0.978331\pi\)
0.439932 0.898031i \(-0.355003\pi\)
\(930\) 0 0
\(931\) −26.0000 10.3923i −0.852116 0.340594i
\(932\) 0 0
\(933\) −14.0000 24.2487i −0.458339 0.793867i
\(934\) 0 0
\(935\) 4.00000 6.92820i 0.130814 0.226576i
\(936\) 0 0
\(937\) −29.0000 −0.947389 −0.473694 0.880689i \(-0.657080\pi\)
−0.473694 + 0.880689i \(0.657080\pi\)
\(938\) 0 0
\(939\) 1.00000 0.0326338
\(940\) 0 0
\(941\) −25.5000 + 44.1673i −0.831276 + 1.43981i 0.0657503 + 0.997836i \(0.479056\pi\)
−0.897027 + 0.441977i \(0.854277\pi\)
\(942\) 0 0
\(943\) −8.00000 13.8564i −0.260516 0.451227i
\(944\) 0 0
\(945\) 2.00000 + 1.73205i 0.0650600 + 0.0563436i
\(946\) 0 0
\(947\) 20.0000 + 34.6410i 0.649913 + 1.12568i 0.983143 + 0.182836i \(0.0585279\pi\)
−0.333231 + 0.942845i \(0.608139\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 5.00000 0.162136
\(952\) 0 0
\(953\) −6.00000 −0.194359 −0.0971795 0.995267i \(-0.530982\pi\)
−0.0971795 + 0.995267i \(0.530982\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 2.50000 + 4.33013i 0.0808135 + 0.139973i
\(958\) 0 0
\(959\) −9.00000 + 46.7654i −0.290625 + 1.51013i
\(960\) 0 0
\(961\) −9.00000 15.5885i −0.290323 0.502853i
\(962\) 0 0
\(963\) 7.50000 12.9904i 0.241684 0.418609i
\(964\) 0 0
\(965\) 11.0000 0.354103
\(966\) 0 0
\(967\) 7.00000 0.225105 0.112552 0.993646i \(-0.464097\pi\)
0.112552 + 0.993646i \(0.464097\pi\)
\(968\) 0 0
\(969\) 16.0000 27.7128i 0.513994 0.890264i
\(970\) 0 0
\(971\) 29.5000 + 51.0955i 0.946700 + 1.63973i 0.752311 + 0.658808i \(0.228939\pi\)
0.194389 + 0.980925i \(0.437728\pi\)
\(972\) 0 0
\(973\) −25.0000 + 8.66025i −0.801463 + 0.277635i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −25.0000 + 43.3013i −0.799821 + 1.38533i 0.119912 + 0.992785i \(0.461739\pi\)
−0.919732 + 0.392546i \(0.871594\pi\)
\(978\) 0 0
\(979\) 6.00000 0.191761
\(980\) 0 0
\(981\) −10.0000 −0.319275
\(982\) 0 0
\(983\) 12.0000 20.7846i 0.382741 0.662926i −0.608712 0.793391i \(-0.708314\pi\)
0.991453 + 0.130465i \(0.0416470\pi\)
\(984\) 0 0
\(985\) −9.00000 15.5885i −0.286764 0.496690i
\(986\) 0 0
\(987\) 15.0000 5.19615i 0.477455 0.165395i
\(988\) 0 0
\(989\) 20.0000 + 34.6410i 0.635963 + 1.10152i
\(990\) 0 0
\(991\) 2.50000 4.33013i 0.0794151 0.137551i −0.823583 0.567196i \(-0.808028\pi\)
0.902998 + 0.429645i \(0.141361\pi\)
\(992\) 0 0
\(993\) −20.0000 −0.634681
\(994\) 0 0
\(995\) −20.0000 −0.634043
\(996\) 0 0
\(997\) 5.00000 8.66025i 0.158352 0.274273i −0.775923 0.630828i \(-0.782715\pi\)
0.934274 + 0.356555i \(0.116049\pi\)
\(998\) 0 0
\(999\) 4.00000 + 6.92820i 0.126554 + 0.219199i
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 1344.2.q.h.193.1 2
4.3 odd 2 1344.2.q.r.193.1 2
7.2 even 3 inner 1344.2.q.h.961.1 2
7.3 odd 6 9408.2.a.bb.1.1 1
7.4 even 3 9408.2.a.cg.1.1 1
8.3 odd 2 672.2.q.b.193.1 2
8.5 even 2 672.2.q.g.193.1 yes 2
24.5 odd 2 2016.2.s.j.865.1 2
24.11 even 2 2016.2.s.i.865.1 2
28.3 even 6 9408.2.a.cp.1.1 1
28.11 odd 6 9408.2.a.o.1.1 1
28.23 odd 6 1344.2.q.r.961.1 2
56.3 even 6 4704.2.a.f.1.1 1
56.11 odd 6 4704.2.a.bc.1.1 1
56.37 even 6 672.2.q.g.289.1 yes 2
56.45 odd 6 4704.2.a.w.1.1 1
56.51 odd 6 672.2.q.b.289.1 yes 2
56.53 even 6 4704.2.a.l.1.1 1
168.107 even 6 2016.2.s.i.289.1 2
168.149 odd 6 2016.2.s.j.289.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
672.2.q.b.193.1 2 8.3 odd 2
672.2.q.b.289.1 yes 2 56.51 odd 6
672.2.q.g.193.1 yes 2 8.5 even 2
672.2.q.g.289.1 yes 2 56.37 even 6
1344.2.q.h.193.1 2 1.1 even 1 trivial
1344.2.q.h.961.1 2 7.2 even 3 inner
1344.2.q.r.193.1 2 4.3 odd 2
1344.2.q.r.961.1 2 28.23 odd 6
2016.2.s.i.289.1 2 168.107 even 6
2016.2.s.i.865.1 2 24.11 even 2
2016.2.s.j.289.1 2 168.149 odd 6
2016.2.s.j.865.1 2 24.5 odd 2
4704.2.a.f.1.1 1 56.3 even 6
4704.2.a.l.1.1 1 56.53 even 6
4704.2.a.w.1.1 1 56.45 odd 6
4704.2.a.bc.1.1 1 56.11 odd 6
9408.2.a.o.1.1 1 28.11 odd 6
9408.2.a.bb.1.1 1 7.3 odd 6
9408.2.a.cg.1.1 1 7.4 even 3
9408.2.a.cp.1.1 1 28.3 even 6