Properties

Label 1344.2.q.q.961.1
Level $1344$
Weight $2$
Character 1344.961
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 961.1
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 1344.961
Dual form 1344.2.q.q.193.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.500000 + 0.866025i) q^{3} +(2.50000 - 0.866025i) q^{7} +(-0.500000 + 0.866025i) q^{9} +O(q^{10})\) \(q+(0.500000 + 0.866025i) q^{3} +(2.50000 - 0.866025i) q^{7} +(-0.500000 + 0.866025i) q^{9} +(1.00000 + 1.73205i) q^{11} -1.00000 q^{13} +(1.00000 + 1.73205i) q^{17} +(-2.50000 + 4.33013i) q^{19} +(2.00000 + 1.73205i) q^{21} +(3.00000 - 5.19615i) q^{23} +(2.50000 + 4.33013i) q^{25} -1.00000 q^{27} +8.00000 q^{29} +(-1.50000 - 2.59808i) q^{31} +(-1.00000 + 1.73205i) q^{33} +(-4.50000 + 7.79423i) q^{37} +(-0.500000 - 0.866025i) q^{39} +2.00000 q^{41} +1.00000 q^{43} +(4.00000 - 6.92820i) q^{47} +(5.50000 - 4.33013i) q^{49} +(-1.00000 + 1.73205i) q^{51} +(3.00000 + 5.19615i) q^{53} -5.00000 q^{57} +(-3.00000 - 5.19615i) q^{59} +(-1.00000 + 1.73205i) q^{61} +(-0.500000 + 2.59808i) q^{63} +(2.50000 + 4.33013i) q^{67} +6.00000 q^{69} -4.00000 q^{71} +(5.50000 + 9.52628i) q^{73} +(-2.50000 + 4.33013i) q^{75} +(4.00000 + 3.46410i) q^{77} +(-2.50000 + 4.33013i) q^{79} +(-0.500000 - 0.866025i) q^{81} +(4.00000 + 6.92820i) q^{87} +(-6.00000 + 10.3923i) q^{89} +(-2.50000 + 0.866025i) q^{91} +(1.50000 - 2.59808i) q^{93} +18.0000 q^{97} -2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{3} + 5 q^{7} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{3} + 5 q^{7} - q^{9} + 2 q^{11} - 2 q^{13} + 2 q^{17} - 5 q^{19} + 4 q^{21} + 6 q^{23} + 5 q^{25} - 2 q^{27} + 16 q^{29} - 3 q^{31} - 2 q^{33} - 9 q^{37} - q^{39} + 4 q^{41} + 2 q^{43} + 8 q^{47} + 11 q^{49} - 2 q^{51} + 6 q^{53} - 10 q^{57} - 6 q^{59} - 2 q^{61} - q^{63} + 5 q^{67} + 12 q^{69} - 8 q^{71} + 11 q^{73} - 5 q^{75} + 8 q^{77} - 5 q^{79} - q^{81} + 8 q^{87} - 12 q^{89} - 5 q^{91} + 3 q^{93} + 36 q^{97} - 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(6\) 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
\(11\) 1.00000 + 1.73205i 0.301511 + 0.522233i 0.976478 0.215615i \(-0.0691756\pi\)
−0.674967 + 0.737848i \(0.735842\pi\)
\(12\) 0 0
\(13\) −1.00000 −0.277350 −0.138675 0.990338i \(-0.544284\pi\)
−0.138675 + 0.990338i \(0.544284\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 1.00000 + 1.73205i 0.242536 + 0.420084i 0.961436 0.275029i \(-0.0886875\pi\)
−0.718900 + 0.695113i \(0.755354\pi\)
\(18\) 0 0
\(19\) −2.50000 + 4.33013i −0.573539 + 0.993399i 0.422659 + 0.906289i \(0.361097\pi\)
−0.996199 + 0.0871106i \(0.972237\pi\)
\(20\) 0 0
\(21\) 2.00000 + 1.73205i 0.436436 + 0.377964i
\(22\) 0 0
\(23\) 3.00000 5.19615i 0.625543 1.08347i −0.362892 0.931831i \(-0.618211\pi\)
0.988436 0.151642i \(-0.0484560\pi\)
\(24\) 0 0
\(25\) 2.50000 + 4.33013i 0.500000 + 0.866025i
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 8.00000 1.48556 0.742781 0.669534i \(-0.233506\pi\)
0.742781 + 0.669534i \(0.233506\pi\)
\(30\) 0 0
\(31\) −1.50000 2.59808i −0.269408 0.466628i 0.699301 0.714827i \(-0.253495\pi\)
−0.968709 + 0.248199i \(0.920161\pi\)
\(32\) 0 0
\(33\) −1.00000 + 1.73205i −0.174078 + 0.301511i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −4.50000 + 7.79423i −0.739795 + 1.28136i 0.212792 + 0.977098i \(0.431744\pi\)
−0.952587 + 0.304266i \(0.901589\pi\)
\(38\) 0 0
\(39\) −0.500000 0.866025i −0.0800641 0.138675i
\(40\) 0 0
\(41\) 2.00000 0.312348 0.156174 0.987730i \(-0.450084\pi\)
0.156174 + 0.987730i \(0.450084\pi\)
\(42\) 0 0
\(43\) 1.00000 0.152499 0.0762493 0.997089i \(-0.475706\pi\)
0.0762493 + 0.997089i \(0.475706\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 4.00000 6.92820i 0.583460 1.01058i −0.411606 0.911362i \(-0.635032\pi\)
0.995066 0.0992202i \(-0.0316348\pi\)
\(48\) 0 0
\(49\) 5.50000 4.33013i 0.785714 0.618590i
\(50\) 0 0
\(51\) −1.00000 + 1.73205i −0.140028 + 0.242536i
\(52\) 0 0
\(53\) 3.00000 + 5.19615i 0.412082 + 0.713746i 0.995117 0.0987002i \(-0.0314685\pi\)
−0.583036 + 0.812447i \(0.698135\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −5.00000 −0.662266
\(58\) 0 0
\(59\) −3.00000 5.19615i −0.390567 0.676481i 0.601958 0.798528i \(-0.294388\pi\)
−0.992524 + 0.122047i \(0.961054\pi\)
\(60\) 0 0
\(61\) −1.00000 + 1.73205i −0.128037 + 0.221766i −0.922916 0.385002i \(-0.874201\pi\)
0.794879 + 0.606768i \(0.207534\pi\)
\(62\) 0 0
\(63\) −0.500000 + 2.59808i −0.0629941 + 0.327327i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 2.50000 + 4.33013i 0.305424 + 0.529009i 0.977356 0.211604i \(-0.0678686\pi\)
−0.671932 + 0.740613i \(0.734535\pi\)
\(68\) 0 0
\(69\) 6.00000 0.722315
\(70\) 0 0
\(71\) −4.00000 −0.474713 −0.237356 0.971423i \(-0.576281\pi\)
−0.237356 + 0.971423i \(0.576281\pi\)
\(72\) 0 0
\(73\) 5.50000 + 9.52628i 0.643726 + 1.11497i 0.984594 + 0.174855i \(0.0559458\pi\)
−0.340868 + 0.940111i \(0.610721\pi\)
\(74\) 0 0
\(75\) −2.50000 + 4.33013i −0.288675 + 0.500000i
\(76\) 0 0
\(77\) 4.00000 + 3.46410i 0.455842 + 0.394771i
\(78\) 0 0
\(79\) −2.50000 + 4.33013i −0.281272 + 0.487177i −0.971698 0.236225i \(-0.924090\pi\)
0.690426 + 0.723403i \(0.257423\pi\)
\(80\) 0 0
\(81\) −0.500000 0.866025i −0.0555556 0.0962250i
\(82\) 0 0
\(83\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 4.00000 + 6.92820i 0.428845 + 0.742781i
\(88\) 0 0
\(89\) −6.00000 + 10.3923i −0.635999 + 1.10158i 0.350304 + 0.936636i \(0.386078\pi\)
−0.986303 + 0.164946i \(0.947255\pi\)
\(90\) 0 0
\(91\) −2.50000 + 0.866025i −0.262071 + 0.0907841i
\(92\) 0 0
\(93\) 1.50000 2.59808i 0.155543 0.269408i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 18.0000 1.82762 0.913812 0.406138i \(-0.133125\pi\)
0.913812 + 0.406138i \(0.133125\pi\)
\(98\) 0 0
\(99\) −2.00000 −0.201008
\(100\) 0 0
\(101\) −3.00000 5.19615i −0.298511 0.517036i 0.677284 0.735721i \(-0.263157\pi\)
−0.975796 + 0.218685i \(0.929823\pi\)
\(102\) 0 0
\(103\) −5.50000 + 9.52628i −0.541931 + 0.938652i 0.456862 + 0.889538i \(0.348973\pi\)
−0.998793 + 0.0491146i \(0.984360\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 9.00000 15.5885i 0.870063 1.50699i 0.00813215 0.999967i \(-0.497411\pi\)
0.861931 0.507026i \(-0.169255\pi\)
\(108\) 0 0
\(109\) −1.50000 2.59808i −0.143674 0.248851i 0.785203 0.619238i \(-0.212558\pi\)
−0.928877 + 0.370387i \(0.879225\pi\)
\(110\) 0 0
\(111\) −9.00000 −0.854242
\(112\) 0 0
\(113\) −12.0000 −1.12887 −0.564433 0.825479i \(-0.690905\pi\)
−0.564433 + 0.825479i \(0.690905\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0.500000 0.866025i 0.0462250 0.0800641i
\(118\) 0 0
\(119\) 4.00000 + 3.46410i 0.366679 + 0.317554i
\(120\) 0 0
\(121\) 3.50000 6.06218i 0.318182 0.551107i
\(122\) 0 0
\(123\) 1.00000 + 1.73205i 0.0901670 + 0.156174i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −9.00000 −0.798621 −0.399310 0.916816i \(-0.630750\pi\)
−0.399310 + 0.916816i \(0.630750\pi\)
\(128\) 0 0
\(129\) 0.500000 + 0.866025i 0.0440225 + 0.0762493i
\(130\) 0 0
\(131\) 5.00000 8.66025i 0.436852 0.756650i −0.560593 0.828092i \(-0.689427\pi\)
0.997445 + 0.0714417i \(0.0227600\pi\)
\(132\) 0 0
\(133\) −2.50000 + 12.9904i −0.216777 + 1.12641i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 6.00000 + 10.3923i 0.512615 + 0.887875i 0.999893 + 0.0146279i \(0.00465636\pi\)
−0.487278 + 0.873247i \(0.662010\pi\)
\(138\) 0 0
\(139\) −15.0000 −1.27228 −0.636142 0.771572i \(-0.719471\pi\)
−0.636142 + 0.771572i \(0.719471\pi\)
\(140\) 0 0
\(141\) 8.00000 0.673722
\(142\) 0 0
\(143\) −1.00000 1.73205i −0.0836242 0.144841i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 6.50000 + 2.59808i 0.536111 + 0.214286i
\(148\) 0 0
\(149\) −10.0000 + 17.3205i −0.819232 + 1.41895i 0.0870170 + 0.996207i \(0.472267\pi\)
−0.906249 + 0.422744i \(0.861067\pi\)
\(150\) 0 0
\(151\) 8.00000 + 13.8564i 0.651031 + 1.12762i 0.982873 + 0.184284i \(0.0589965\pi\)
−0.331842 + 0.943335i \(0.607670\pi\)
\(152\) 0 0
\(153\) −2.00000 −0.161690
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −11.0000 19.0526i −0.877896 1.52056i −0.853646 0.520854i \(-0.825614\pi\)
−0.0242497 0.999706i \(-0.507720\pi\)
\(158\) 0 0
\(159\) −3.00000 + 5.19615i −0.237915 + 0.412082i
\(160\) 0 0
\(161\) 3.00000 15.5885i 0.236433 1.22854i
\(162\) 0 0
\(163\) 10.0000 17.3205i 0.783260 1.35665i −0.146772 0.989170i \(-0.546888\pi\)
0.930033 0.367477i \(-0.119778\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −12.0000 −0.928588 −0.464294 0.885681i \(-0.653692\pi\)
−0.464294 + 0.885681i \(0.653692\pi\)
\(168\) 0 0
\(169\) −12.0000 −0.923077
\(170\) 0 0
\(171\) −2.50000 4.33013i −0.191180 0.331133i
\(172\) 0 0
\(173\) 9.00000 15.5885i 0.684257 1.18517i −0.289412 0.957205i \(-0.593460\pi\)
0.973670 0.227964i \(-0.0732068\pi\)
\(174\) 0 0
\(175\) 10.0000 + 8.66025i 0.755929 + 0.654654i
\(176\) 0 0
\(177\) 3.00000 5.19615i 0.225494 0.390567i
\(178\) 0 0
\(179\) −2.00000 3.46410i −0.149487 0.258919i 0.781551 0.623841i \(-0.214429\pi\)
−0.931038 + 0.364922i \(0.881096\pi\)
\(180\) 0 0
\(181\) −5.00000 −0.371647 −0.185824 0.982583i \(-0.559495\pi\)
−0.185824 + 0.982583i \(0.559495\pi\)
\(182\) 0 0
\(183\) −2.00000 −0.147844
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −2.00000 + 3.46410i −0.146254 + 0.253320i
\(188\) 0 0
\(189\) −2.50000 + 0.866025i −0.181848 + 0.0629941i
\(190\) 0 0
\(191\) 10.0000 17.3205i 0.723575 1.25327i −0.235983 0.971757i \(-0.575831\pi\)
0.959558 0.281511i \(-0.0908356\pi\)
\(192\) 0 0
\(193\) −3.50000 6.06218i −0.251936 0.436365i 0.712123 0.702055i \(-0.247734\pi\)
−0.964059 + 0.265689i \(0.914400\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 10.0000 0.712470 0.356235 0.934396i \(-0.384060\pi\)
0.356235 + 0.934396i \(0.384060\pi\)
\(198\) 0 0
\(199\) −6.00000 10.3923i −0.425329 0.736691i 0.571122 0.820865i \(-0.306508\pi\)
−0.996451 + 0.0841740i \(0.973175\pi\)
\(200\) 0 0
\(201\) −2.50000 + 4.33013i −0.176336 + 0.305424i
\(202\) 0 0
\(203\) 20.0000 6.92820i 1.40372 0.486265i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 3.00000 + 5.19615i 0.208514 + 0.361158i
\(208\) 0 0
\(209\) −10.0000 −0.691714
\(210\) 0 0
\(211\) 12.0000 0.826114 0.413057 0.910705i \(-0.364461\pi\)
0.413057 + 0.910705i \(0.364461\pi\)
\(212\) 0 0
\(213\) −2.00000 3.46410i −0.137038 0.237356i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −6.00000 5.19615i −0.407307 0.352738i
\(218\) 0 0
\(219\) −5.50000 + 9.52628i −0.371656 + 0.643726i
\(220\) 0 0
\(221\) −1.00000 1.73205i −0.0672673 0.116510i
\(222\) 0 0
\(223\) −8.00000 −0.535720 −0.267860 0.963458i \(-0.586316\pi\)
−0.267860 + 0.963458i \(0.586316\pi\)
\(224\) 0 0
\(225\) −5.00000 −0.333333
\(226\) 0 0
\(227\) −11.0000 19.0526i −0.730096 1.26456i −0.956842 0.290609i \(-0.906142\pi\)
0.226746 0.973954i \(-0.427191\pi\)
\(228\) 0 0
\(229\) 0.500000 0.866025i 0.0330409 0.0572286i −0.849032 0.528341i \(-0.822814\pi\)
0.882073 + 0.471113i \(0.156147\pi\)
\(230\) 0 0
\(231\) −1.00000 + 5.19615i −0.0657952 + 0.341882i
\(232\) 0 0
\(233\) −2.00000 + 3.46410i −0.131024 + 0.226941i −0.924072 0.382219i \(-0.875160\pi\)
0.793047 + 0.609160i \(0.208493\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −5.00000 −0.324785
\(238\) 0 0
\(239\) 22.0000 1.42306 0.711531 0.702655i \(-0.248002\pi\)
0.711531 + 0.702655i \(0.248002\pi\)
\(240\) 0 0
\(241\) −5.00000 8.66025i −0.322078 0.557856i 0.658838 0.752285i \(-0.271048\pi\)
−0.980917 + 0.194429i \(0.937715\pi\)
\(242\) 0 0
\(243\) 0.500000 0.866025i 0.0320750 0.0555556i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 2.50000 4.33013i 0.159071 0.275519i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 6.00000 0.378717 0.189358 0.981908i \(-0.439359\pi\)
0.189358 + 0.981908i \(0.439359\pi\)
\(252\) 0 0
\(253\) 12.0000 0.754434
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 16.0000 27.7128i 0.998053 1.72868i 0.445005 0.895528i \(-0.353202\pi\)
0.553047 0.833150i \(-0.313465\pi\)
\(258\) 0 0
\(259\) −4.50000 + 23.3827i −0.279616 + 1.45293i
\(260\) 0 0
\(261\) −4.00000 + 6.92820i −0.247594 + 0.428845i
\(262\) 0 0
\(263\) −4.00000 6.92820i −0.246651 0.427211i 0.715944 0.698158i \(-0.245997\pi\)
−0.962594 + 0.270947i \(0.912663\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) −12.0000 −0.734388
\(268\) 0 0
\(269\) −13.0000 22.5167i −0.792624 1.37287i −0.924337 0.381577i \(-0.875381\pi\)
0.131713 0.991288i \(-0.457952\pi\)
\(270\) 0 0
\(271\) −4.00000 + 6.92820i −0.242983 + 0.420858i −0.961563 0.274586i \(-0.911459\pi\)
0.718580 + 0.695444i \(0.244792\pi\)
\(272\) 0 0
\(273\) −2.00000 1.73205i −0.121046 0.104828i
\(274\) 0 0
\(275\) −5.00000 + 8.66025i −0.301511 + 0.522233i
\(276\) 0 0
\(277\) −5.50000 9.52628i −0.330463 0.572379i 0.652140 0.758099i \(-0.273872\pi\)
−0.982603 + 0.185720i \(0.940538\pi\)
\(278\) 0 0
\(279\) 3.00000 0.179605
\(280\) 0 0
\(281\) −14.0000 −0.835170 −0.417585 0.908638i \(-0.637123\pi\)
−0.417585 + 0.908638i \(0.637123\pi\)
\(282\) 0 0
\(283\) 15.5000 + 26.8468i 0.921379 + 1.59588i 0.797283 + 0.603606i \(0.206270\pi\)
0.124096 + 0.992270i \(0.460397\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 5.00000 1.73205i 0.295141 0.102240i
\(288\) 0 0
\(289\) 6.50000 11.2583i 0.382353 0.662255i
\(290\) 0 0
\(291\) 9.00000 + 15.5885i 0.527589 + 0.913812i
\(292\) 0 0
\(293\) −28.0000 −1.63578 −0.817889 0.575376i \(-0.804856\pi\)
−0.817889 + 0.575376i \(0.804856\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) −1.00000 1.73205i −0.0580259 0.100504i
\(298\) 0 0
\(299\) −3.00000 + 5.19615i −0.173494 + 0.300501i
\(300\) 0 0
\(301\) 2.50000 0.866025i 0.144098 0.0499169i
\(302\) 0 0
\(303\) 3.00000 5.19615i 0.172345 0.298511i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −25.0000 −1.42683 −0.713413 0.700744i \(-0.752851\pi\)
−0.713413 + 0.700744i \(0.752851\pi\)
\(308\) 0 0
\(309\) −11.0000 −0.625768
\(310\) 0 0
\(311\) −5.00000 8.66025i −0.283524 0.491078i 0.688726 0.725022i \(-0.258170\pi\)
−0.972250 + 0.233944i \(0.924837\pi\)
\(312\) 0 0
\(313\) −15.5000 + 26.8468i −0.876112 + 1.51747i −0.0205381 + 0.999789i \(0.506538\pi\)
−0.855574 + 0.517681i \(0.826795\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 2.00000 3.46410i 0.112331 0.194563i −0.804379 0.594117i \(-0.797502\pi\)
0.916710 + 0.399554i \(0.130835\pi\)
\(318\) 0 0
\(319\) 8.00000 + 13.8564i 0.447914 + 0.775810i
\(320\) 0 0
\(321\) 18.0000 1.00466
\(322\) 0 0
\(323\) −10.0000 −0.556415
\(324\) 0 0
\(325\) −2.50000 4.33013i −0.138675 0.240192i
\(326\) 0 0
\(327\) 1.50000 2.59808i 0.0829502 0.143674i
\(328\) 0 0
\(329\) 4.00000 20.7846i 0.220527 1.14589i
\(330\) 0 0
\(331\) 8.50000 14.7224i 0.467202 0.809218i −0.532096 0.846684i \(-0.678595\pi\)
0.999298 + 0.0374662i \(0.0119287\pi\)
\(332\) 0 0
\(333\) −4.50000 7.79423i −0.246598 0.427121i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −31.0000 −1.68868 −0.844339 0.535810i \(-0.820006\pi\)
−0.844339 + 0.535810i \(0.820006\pi\)
\(338\) 0 0
\(339\) −6.00000 10.3923i −0.325875 0.564433i
\(340\) 0 0
\(341\) 3.00000 5.19615i 0.162459 0.281387i
\(342\) 0 0
\(343\) 10.0000 15.5885i 0.539949 0.841698i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 9.00000 + 15.5885i 0.483145 + 0.836832i 0.999813 0.0193540i \(-0.00616095\pi\)
−0.516667 + 0.856186i \(0.672828\pi\)
\(348\) 0 0
\(349\) 14.0000 0.749403 0.374701 0.927146i \(-0.377745\pi\)
0.374701 + 0.927146i \(0.377745\pi\)
\(350\) 0 0
\(351\) 1.00000 0.0533761
\(352\) 0 0
\(353\) 12.0000 + 20.7846i 0.638696 + 1.10625i 0.985719 + 0.168397i \(0.0538590\pi\)
−0.347024 + 0.937856i \(0.612808\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) −1.00000 + 5.19615i −0.0529256 + 0.275010i
\(358\) 0 0
\(359\) 16.0000 27.7128i 0.844448 1.46263i −0.0416523 0.999132i \(-0.513262\pi\)
0.886100 0.463494i \(-0.153404\pi\)
\(360\) 0 0
\(361\) −3.00000 5.19615i −0.157895 0.273482i
\(362\) 0 0
\(363\) 7.00000 0.367405
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −18.5000 32.0429i −0.965692 1.67263i −0.707744 0.706469i \(-0.750287\pi\)
−0.257948 0.966159i \(-0.583046\pi\)
\(368\) 0 0
\(369\) −1.00000 + 1.73205i −0.0520579 + 0.0901670i
\(370\) 0 0
\(371\) 12.0000 + 10.3923i 0.623009 + 0.539542i
\(372\) 0 0
\(373\) −0.500000 + 0.866025i −0.0258890 + 0.0448411i −0.878680 0.477412i \(-0.841575\pi\)
0.852791 + 0.522253i \(0.174908\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −8.00000 −0.412021
\(378\) 0 0
\(379\) −25.0000 −1.28416 −0.642082 0.766636i \(-0.721929\pi\)
−0.642082 + 0.766636i \(0.721929\pi\)
\(380\) 0 0
\(381\) −4.50000 7.79423i −0.230542 0.399310i
\(382\) 0 0
\(383\) −1.00000 + 1.73205i −0.0510976 + 0.0885037i −0.890443 0.455095i \(-0.849605\pi\)
0.839345 + 0.543599i \(0.182939\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −0.500000 + 0.866025i −0.0254164 + 0.0440225i
\(388\) 0 0
\(389\) −4.00000 6.92820i −0.202808 0.351274i 0.746624 0.665246i \(-0.231673\pi\)
−0.949432 + 0.313972i \(0.898340\pi\)
\(390\) 0 0
\(391\) 12.0000 0.606866
\(392\) 0 0
\(393\) 10.0000 0.504433
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 7.50000 12.9904i 0.376414 0.651969i −0.614123 0.789210i \(-0.710490\pi\)
0.990538 + 0.137241i \(0.0438236\pi\)
\(398\) 0 0
\(399\) −12.5000 + 4.33013i −0.625783 + 0.216777i
\(400\) 0 0
\(401\) 11.0000 19.0526i 0.549314 0.951439i −0.449008 0.893528i \(-0.648223\pi\)
0.998322 0.0579116i \(-0.0184442\pi\)
\(402\) 0 0
\(403\) 1.50000 + 2.59808i 0.0747203 + 0.129419i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −18.0000 −0.892227
\(408\) 0 0
\(409\) −2.50000 4.33013i −0.123617 0.214111i 0.797574 0.603220i \(-0.206116\pi\)
−0.921192 + 0.389109i \(0.872783\pi\)
\(410\) 0 0
\(411\) −6.00000 + 10.3923i −0.295958 + 0.512615i
\(412\) 0 0
\(413\) −12.0000 10.3923i −0.590481 0.511372i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −7.50000 12.9904i −0.367277 0.636142i
\(418\) 0 0
\(419\) −34.0000 −1.66101 −0.830504 0.557012i \(-0.811948\pi\)
−0.830504 + 0.557012i \(0.811948\pi\)
\(420\) 0 0
\(421\) 11.0000 0.536107 0.268054 0.963404i \(-0.413620\pi\)
0.268054 + 0.963404i \(0.413620\pi\)
\(422\) 0 0
\(423\) 4.00000 + 6.92820i 0.194487 + 0.336861i
\(424\) 0 0
\(425\) −5.00000 + 8.66025i −0.242536 + 0.420084i
\(426\) 0 0
\(427\) −1.00000 + 5.19615i −0.0483934 + 0.251459i
\(428\) 0 0
\(429\) 1.00000 1.73205i 0.0482805 0.0836242i
\(430\) 0 0
\(431\) 6.00000 + 10.3923i 0.289010 + 0.500580i 0.973574 0.228373i \(-0.0733406\pi\)
−0.684564 + 0.728953i \(0.740007\pi\)
\(432\) 0 0
\(433\) −25.0000 −1.20142 −0.600712 0.799466i \(-0.705116\pi\)
−0.600712 + 0.799466i \(0.705116\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 15.0000 + 25.9808i 0.717547 + 1.24283i
\(438\) 0 0
\(439\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(440\) 0 0
\(441\) 1.00000 + 6.92820i 0.0476190 + 0.329914i
\(442\) 0 0
\(443\) 11.0000 19.0526i 0.522626 0.905214i −0.477028 0.878888i \(-0.658286\pi\)
0.999653 0.0263261i \(-0.00838082\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) −20.0000 −0.945968
\(448\) 0 0
\(449\) 24.0000 1.13263 0.566315 0.824189i \(-0.308369\pi\)
0.566315 + 0.824189i \(0.308369\pi\)
\(450\) 0 0
\(451\) 2.00000 + 3.46410i 0.0941763 + 0.163118i
\(452\) 0 0
\(453\) −8.00000 + 13.8564i −0.375873 + 0.651031i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −0.500000 + 0.866025i −0.0233890 + 0.0405110i −0.877483 0.479608i \(-0.840779\pi\)
0.854094 + 0.520119i \(0.174112\pi\)
\(458\) 0 0
\(459\) −1.00000 1.73205i −0.0466760 0.0808452i
\(460\) 0 0
\(461\) −12.0000 −0.558896 −0.279448 0.960161i \(-0.590151\pi\)
−0.279448 + 0.960161i \(0.590151\pi\)
\(462\) 0 0
\(463\) −11.0000 −0.511213 −0.255607 0.966781i \(-0.582275\pi\)
−0.255607 + 0.966781i \(0.582275\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −10.0000 + 17.3205i −0.462745 + 0.801498i −0.999097 0.0424970i \(-0.986469\pi\)
0.536352 + 0.843995i \(0.319802\pi\)
\(468\) 0 0
\(469\) 10.0000 + 8.66025i 0.461757 + 0.399893i
\(470\) 0 0
\(471\) 11.0000 19.0526i 0.506853 0.877896i
\(472\) 0 0
\(473\) 1.00000 + 1.73205i 0.0459800 + 0.0796398i
\(474\) 0 0
\(475\) −25.0000 −1.14708
\(476\) 0 0
\(477\) −6.00000 −0.274721
\(478\) 0 0
\(479\) 3.00000 + 5.19615i 0.137073 + 0.237418i 0.926388 0.376571i \(-0.122897\pi\)
−0.789314 + 0.613990i \(0.789564\pi\)
\(480\) 0 0
\(481\) 4.50000 7.79423i 0.205182 0.355386i
\(482\) 0 0
\(483\) 15.0000 5.19615i 0.682524 0.236433i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 3.50000 + 6.06218i 0.158600 + 0.274703i 0.934364 0.356320i \(-0.115969\pi\)
−0.775764 + 0.631023i \(0.782635\pi\)
\(488\) 0 0
\(489\) 20.0000 0.904431
\(490\) 0 0
\(491\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(492\) 0 0
\(493\) 8.00000 + 13.8564i 0.360302 + 0.624061i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −10.0000 + 3.46410i −0.448561 + 0.155386i
\(498\) 0 0
\(499\) 11.5000 19.9186i 0.514811 0.891678i −0.485042 0.874491i \(-0.661196\pi\)
0.999852 0.0171872i \(-0.00547113\pi\)
\(500\) 0 0
\(501\) −6.00000 10.3923i −0.268060 0.464294i
\(502\) 0 0
\(503\) −42.0000 −1.87269 −0.936344 0.351085i \(-0.885813\pi\)
−0.936344 + 0.351085i \(0.885813\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −6.00000 10.3923i −0.266469 0.461538i
\(508\) 0 0
\(509\) −10.0000 + 17.3205i −0.443242 + 0.767718i −0.997928 0.0643419i \(-0.979505\pi\)
0.554686 + 0.832060i \(0.312839\pi\)
\(510\) 0 0
\(511\) 22.0000 + 19.0526i 0.973223 + 0.842836i
\(512\) 0 0
\(513\) 2.50000 4.33013i 0.110378 0.191180i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 16.0000 0.703679
\(518\) 0 0
\(519\) 18.0000 0.790112
\(520\) 0 0
\(521\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(522\) 0 0
\(523\) 6.50000 11.2583i 0.284225 0.492292i −0.688196 0.725525i \(-0.741597\pi\)
0.972421 + 0.233233i \(0.0749303\pi\)
\(524\) 0 0
\(525\) −2.50000 + 12.9904i −0.109109 + 0.566947i
\(526\) 0 0
\(527\) 3.00000 5.19615i 0.130682 0.226348i
\(528\) 0 0
\(529\) −6.50000 11.2583i −0.282609 0.489493i
\(530\) 0 0
\(531\) 6.00000 0.260378
\(532\) 0 0
\(533\) −2.00000 −0.0866296
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 2.00000 3.46410i 0.0863064 0.149487i
\(538\) 0 0
\(539\) 13.0000 + 5.19615i 0.559950 + 0.223814i
\(540\) 0 0
\(541\) 12.5000 21.6506i 0.537417 0.930834i −0.461625 0.887075i \(-0.652733\pi\)
0.999042 0.0437584i \(-0.0139332\pi\)
\(542\) 0 0
\(543\) −2.50000 4.33013i −0.107285 0.185824i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 28.0000 1.19719 0.598597 0.801050i \(-0.295725\pi\)
0.598597 + 0.801050i \(0.295725\pi\)
\(548\) 0 0
\(549\) −1.00000 1.73205i −0.0426790 0.0739221i
\(550\) 0 0
\(551\) −20.0000 + 34.6410i −0.852029 + 1.47576i
\(552\) 0 0
\(553\) −2.50000 + 12.9904i −0.106311 + 0.552407i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 13.0000 + 22.5167i 0.550828 + 0.954062i 0.998215 + 0.0597213i \(0.0190212\pi\)
−0.447387 + 0.894340i \(0.647645\pi\)
\(558\) 0 0
\(559\) −1.00000 −0.0422955
\(560\) 0 0
\(561\) −4.00000 −0.168880
\(562\) 0 0
\(563\) −10.0000 17.3205i −0.421450 0.729972i 0.574632 0.818412i \(-0.305145\pi\)
−0.996082 + 0.0884397i \(0.971812\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −2.00000 1.73205i −0.0839921 0.0727393i
\(568\) 0 0
\(569\) −13.0000 + 22.5167i −0.544988 + 0.943948i 0.453619 + 0.891196i \(0.350133\pi\)
−0.998608 + 0.0527519i \(0.983201\pi\)
\(570\) 0 0
\(571\) 3.50000 + 6.06218i 0.146470 + 0.253694i 0.929921 0.367760i \(-0.119875\pi\)
−0.783450 + 0.621455i \(0.786542\pi\)
\(572\) 0 0
\(573\) 20.0000 0.835512
\(574\) 0 0
\(575\) 30.0000 1.25109
\(576\) 0 0
\(577\) 8.50000 + 14.7224i 0.353860 + 0.612903i 0.986922 0.161198i \(-0.0515357\pi\)
−0.633062 + 0.774101i \(0.718202\pi\)
\(578\) 0 0
\(579\) 3.50000 6.06218i 0.145455 0.251936i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −6.00000 + 10.3923i −0.248495 + 0.430405i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 24.0000 0.990586 0.495293 0.868726i \(-0.335061\pi\)
0.495293 + 0.868726i \(0.335061\pi\)
\(588\) 0 0
\(589\) 15.0000 0.618064
\(590\) 0 0
\(591\) 5.00000 + 8.66025i 0.205673 + 0.356235i
\(592\) 0 0
\(593\) −7.00000 + 12.1244i −0.287456 + 0.497888i −0.973202 0.229953i \(-0.926143\pi\)
0.685746 + 0.727841i \(0.259476\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 6.00000 10.3923i 0.245564 0.425329i
\(598\) 0 0
\(599\) 10.0000 + 17.3205i 0.408589 + 0.707697i 0.994732 0.102511i \(-0.0326876\pi\)
−0.586143 + 0.810208i \(0.699354\pi\)
\(600\) 0 0
\(601\) 3.00000 0.122373 0.0611863 0.998126i \(-0.480512\pi\)
0.0611863 + 0.998126i \(0.480512\pi\)
\(602\) 0 0
\(603\) −5.00000 −0.203616
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −6.50000 + 11.2583i −0.263827 + 0.456962i −0.967256 0.253804i \(-0.918318\pi\)
0.703429 + 0.710766i \(0.251651\pi\)
\(608\) 0 0
\(609\) 16.0000 + 13.8564i 0.648353 + 0.561490i
\(610\) 0 0
\(611\) −4.00000 + 6.92820i −0.161823 + 0.280285i
\(612\) 0 0
\(613\) 5.00000 + 8.66025i 0.201948 + 0.349784i 0.949156 0.314806i \(-0.101939\pi\)
−0.747208 + 0.664590i \(0.768606\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 34.0000 1.36879 0.684394 0.729112i \(-0.260067\pi\)
0.684394 + 0.729112i \(0.260067\pi\)
\(618\) 0 0
\(619\) 2.50000 + 4.33013i 0.100483 + 0.174042i 0.911884 0.410448i \(-0.134628\pi\)
−0.811400 + 0.584491i \(0.801294\pi\)
\(620\) 0 0
\(621\) −3.00000 + 5.19615i −0.120386 + 0.208514i
\(622\) 0 0
\(623\) −6.00000 + 31.1769i −0.240385 + 1.24908i
\(624\) 0 0
\(625\) −12.5000 + 21.6506i −0.500000 + 0.866025i
\(626\) 0 0
\(627\) −5.00000 8.66025i −0.199681 0.345857i
\(628\) 0 0
\(629\) −18.0000 −0.717707
\(630\) 0 0
\(631\) −44.0000 −1.75161 −0.875806 0.482663i \(-0.839670\pi\)
−0.875806 + 0.482663i \(0.839670\pi\)
\(632\) 0 0
\(633\) 6.00000 + 10.3923i 0.238479 + 0.413057i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −5.50000 + 4.33013i −0.217918 + 0.171566i
\(638\) 0 0
\(639\) 2.00000 3.46410i 0.0791188 0.137038i
\(640\) 0 0
\(641\) 19.0000 + 32.9090i 0.750455 + 1.29983i 0.947602 + 0.319452i \(0.103499\pi\)
−0.197148 + 0.980374i \(0.563168\pi\)
\(642\) 0 0
\(643\) 13.0000 0.512670 0.256335 0.966588i \(-0.417485\pi\)
0.256335 + 0.966588i \(0.417485\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 21.0000 + 36.3731i 0.825595 + 1.42997i 0.901464 + 0.432855i \(0.142494\pi\)
−0.0758684 + 0.997118i \(0.524173\pi\)
\(648\) 0 0
\(649\) 6.00000 10.3923i 0.235521 0.407934i
\(650\) 0 0
\(651\) 1.50000 7.79423i 0.0587896 0.305480i
\(652\) 0 0
\(653\) −19.0000 + 32.9090i −0.743527 + 1.28783i 0.207352 + 0.978266i \(0.433515\pi\)
−0.950880 + 0.309561i \(0.899818\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −11.0000 −0.429151
\(658\) 0 0
\(659\) −6.00000 −0.233727 −0.116863 0.993148i \(-0.537284\pi\)
−0.116863 + 0.993148i \(0.537284\pi\)
\(660\) 0 0
\(661\) 7.50000 + 12.9904i 0.291716 + 0.505267i 0.974216 0.225619i \(-0.0724404\pi\)
−0.682499 + 0.730886i \(0.739107\pi\)
\(662\) 0 0
\(663\) 1.00000 1.73205i 0.0388368 0.0672673i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 24.0000 41.5692i 0.929284 1.60957i
\(668\) 0 0
\(669\) −4.00000 6.92820i −0.154649 0.267860i
\(670\) 0 0
\(671\) −4.00000 −0.154418
\(672\) 0 0
\(673\) 35.0000 1.34915 0.674575 0.738206i \(-0.264327\pi\)
0.674575 + 0.738206i \(0.264327\pi\)
\(674\) 0 0
\(675\) −2.50000 4.33013i −0.0962250 0.166667i
\(676\) 0 0
\(677\) 18.0000 31.1769i 0.691796 1.19823i −0.279453 0.960159i \(-0.590153\pi\)
0.971249 0.238067i \(-0.0765137\pi\)
\(678\) 0 0
\(679\) 45.0000 15.5885i 1.72694 0.598230i
\(680\) 0 0
\(681\) 11.0000 19.0526i 0.421521 0.730096i
\(682\) 0 0
\(683\) 9.00000 + 15.5885i 0.344375 + 0.596476i 0.985240 0.171178i \(-0.0547574\pi\)
−0.640865 + 0.767654i \(0.721424\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 1.00000 0.0381524
\(688\) 0 0
\(689\) −3.00000 5.19615i −0.114291 0.197958i
\(690\) 0 0
\(691\) 14.5000 25.1147i 0.551606 0.955410i −0.446553 0.894757i \(-0.647349\pi\)
0.998159 0.0606524i \(-0.0193181\pi\)
\(692\) 0 0
\(693\) −5.00000 + 1.73205i −0.189934 + 0.0657952i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 2.00000 + 3.46410i 0.0757554 + 0.131212i
\(698\) 0 0
\(699\) −4.00000 −0.151294
\(700\) 0 0
\(701\) 14.0000 0.528773 0.264386 0.964417i \(-0.414831\pi\)
0.264386 + 0.964417i \(0.414831\pi\)
\(702\) 0 0
\(703\) −22.5000 38.9711i −0.848604 1.46982i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −12.0000 10.3923i −0.451306 0.390843i
\(708\) 0 0
\(709\) 5.00000 8.66025i 0.187779 0.325243i −0.756730 0.653727i \(-0.773204\pi\)
0.944509 + 0.328484i \(0.106538\pi\)
\(710\) 0 0
\(711\) −2.50000 4.33013i −0.0937573 0.162392i
\(712\) 0 0
\(713\) −18.0000 −0.674105
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 11.0000 + 19.0526i 0.410803 + 0.711531i
\(718\) 0 0
\(719\) 15.0000 25.9808i 0.559406 0.968919i −0.438141 0.898906i \(-0.644363\pi\)
0.997546 0.0700124i \(-0.0223039\pi\)
\(720\) 0 0
\(721\) −5.50000 + 28.5788i −0.204831 + 1.06433i
\(722\) 0 0
\(723\) 5.00000 8.66025i 0.185952 0.322078i
\(724\) 0 0
\(725\) 20.0000 + 34.6410i 0.742781 + 1.28654i
\(726\) 0 0
\(727\) 17.0000 0.630495 0.315248 0.949009i \(-0.397912\pi\)
0.315248 + 0.949009i \(0.397912\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 1.00000 + 1.73205i 0.0369863 + 0.0640622i
\(732\) 0 0
\(733\) −19.5000 + 33.7750i −0.720249 + 1.24751i 0.240651 + 0.970612i \(0.422639\pi\)
−0.960900 + 0.276896i \(0.910694\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −5.00000 + 8.66025i −0.184177 + 0.319005i
\(738\) 0 0
\(739\) −16.5000 28.5788i −0.606962 1.05129i −0.991738 0.128279i \(-0.959055\pi\)
0.384776 0.923010i \(-0.374279\pi\)
\(740\) 0 0
\(741\) 5.00000 0.183680
\(742\) 0 0
\(743\) −54.0000 −1.98107 −0.990534 0.137268i \(-0.956168\pi\)
−0.990534 + 0.137268i \(0.956168\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 9.00000 46.7654i 0.328853 1.70877i
\(750\) 0 0
\(751\) −15.5000 + 26.8468i −0.565603 + 0.979653i 0.431390 + 0.902165i \(0.358023\pi\)
−0.996993 + 0.0774878i \(0.975310\pi\)
\(752\) 0 0
\(753\) 3.00000 + 5.19615i 0.109326 + 0.189358i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −38.0000 −1.38113 −0.690567 0.723269i \(-0.742639\pi\)
−0.690567 + 0.723269i \(0.742639\pi\)
\(758\) 0 0
\(759\) 6.00000 + 10.3923i 0.217786 + 0.377217i
\(760\) 0 0
\(761\) 25.0000 43.3013i 0.906249 1.56967i 0.0870179 0.996207i \(-0.472266\pi\)
0.819231 0.573463i \(-0.194400\pi\)
\(762\) 0 0
\(763\) −6.00000 5.19615i −0.217215 0.188113i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 3.00000 + 5.19615i 0.108324 + 0.187622i
\(768\) 0 0
\(769\) 23.0000 0.829401 0.414701 0.909958i \(-0.363886\pi\)
0.414701 + 0.909958i \(0.363886\pi\)
\(770\) 0 0
\(771\) 32.0000 1.15245
\(772\) 0 0
\(773\) 21.0000 + 36.3731i 0.755318 + 1.30825i 0.945216 + 0.326445i \(0.105851\pi\)
−0.189899 + 0.981804i \(0.560816\pi\)
\(774\) 0 0
\(775\) 7.50000 12.9904i 0.269408 0.466628i
\(776\) 0 0
\(777\) −22.5000 + 7.79423i −0.807183 + 0.279616i
\(778\) 0 0
\(779\) −5.00000 + 8.66025i −0.179144 + 0.310286i
\(780\) 0 0
\(781\) −4.00000 6.92820i −0.143131 0.247911i
\(782\) 0 0
\(783\) −8.00000 −0.285897
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 4.00000 + 6.92820i 0.142585 + 0.246964i 0.928469 0.371409i \(-0.121125\pi\)
−0.785885 + 0.618373i \(0.787792\pi\)
\(788\) 0 0
\(789\) 4.00000 6.92820i 0.142404 0.246651i
\(790\) 0 0
\(791\) −30.0000 + 10.3923i −1.06668 + 0.369508i
\(792\) 0 0
\(793\) 1.00000 1.73205i 0.0355110 0.0615069i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −6.00000 −0.212531 −0.106265 0.994338i \(-0.533889\pi\)
−0.106265 + 0.994338i \(0.533889\pi\)
\(798\) 0 0
\(799\) 16.0000 0.566039
\(800\) 0 0
\(801\) −6.00000 10.3923i −0.212000 0.367194i
\(802\) 0 0
\(803\) −11.0000 + 19.0526i −0.388182 + 0.672350i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 13.0000 22.5167i 0.457622 0.792624i
\(808\) 0 0
\(809\) 14.0000 + 24.2487i 0.492214 + 0.852539i 0.999960 0.00896753i \(-0.00285449\pi\)
−0.507746 + 0.861507i \(0.669521\pi\)
\(810\) 0 0
\(811\) −28.0000 −0.983213 −0.491606 0.870817i \(-0.663590\pi\)
−0.491606 + 0.870817i \(0.663590\pi\)
\(812\) 0 0
\(813\) −8.00000 −0.280572
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −2.50000 + 4.33013i −0.0874639 + 0.151492i
\(818\) 0 0
\(819\) 0.500000 2.59808i 0.0174714 0.0907841i
\(820\) 0 0
\(821\) 24.0000 41.5692i 0.837606 1.45078i −0.0542853 0.998525i \(-0.517288\pi\)
0.891891 0.452250i \(-0.149379\pi\)
\(822\) 0 0
\(823\) 16.0000 + 27.7128i 0.557725 + 0.966008i 0.997686 + 0.0679910i \(0.0216589\pi\)
−0.439961 + 0.898017i \(0.645008\pi\)
\(824\) 0 0
\(825\) −10.0000 −0.348155
\(826\) 0 0
\(827\) −24.0000 −0.834562 −0.417281 0.908778i \(-0.637017\pi\)
−0.417281 + 0.908778i \(0.637017\pi\)
\(828\) 0 0
\(829\) 24.5000 + 42.4352i 0.850920 + 1.47384i 0.880379 + 0.474271i \(0.157288\pi\)
−0.0294587 + 0.999566i \(0.509378\pi\)
\(830\) 0 0
\(831\) 5.50000 9.52628i 0.190793 0.330463i
\(832\) 0 0
\(833\) 13.0000 + 5.19615i 0.450423 + 0.180036i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 1.50000 + 2.59808i 0.0518476 + 0.0898027i
\(838\) 0 0
\(839\) 4.00000 0.138095 0.0690477 0.997613i \(-0.478004\pi\)
0.0690477 + 0.997613i \(0.478004\pi\)
\(840\) 0 0
\(841\) 35.0000 1.20690
\(842\) 0 0
\(843\) −7.00000 12.1244i −0.241093 0.417585i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 3.50000 18.1865i 0.120261 0.624897i
\(848\) 0 0
\(849\) −15.5000 + 26.8468i −0.531959 + 0.921379i
\(850\) 0 0
\(851\) 27.0000 + 46.7654i 0.925548 + 1.60310i
\(852\) 0 0
\(853\) 21.0000 0.719026 0.359513 0.933140i \(-0.382943\pi\)
0.359513 + 0.933140i \(0.382943\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −4.00000 6.92820i −0.136637 0.236663i 0.789584 0.613642i \(-0.210296\pi\)
−0.926222 + 0.376979i \(0.876963\pi\)
\(858\) 0 0
\(859\) −22.0000 + 38.1051i −0.750630 + 1.30013i 0.196887 + 0.980426i \(0.436917\pi\)
−0.947518 + 0.319704i \(0.896417\pi\)
\(860\) 0 0
\(861\) 4.00000 + 3.46410i 0.136320 + 0.118056i
\(862\) 0 0
\(863\) −9.00000 + 15.5885i −0.306364 + 0.530637i −0.977564 0.210639i \(-0.932446\pi\)
0.671200 + 0.741276i \(0.265779\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 13.0000 0.441503
\(868\) 0 0
\(869\) −10.0000 −0.339227
\(870\) 0 0
\(871\) −2.50000 4.33013i −0.0847093 0.146721i
\(872\) 0 0
\(873\) −9.00000 + 15.5885i −0.304604 + 0.527589i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 5.00000 8.66025i 0.168838 0.292436i −0.769174 0.639040i \(-0.779332\pi\)
0.938012 + 0.346604i \(0.112665\pi\)
\(878\) 0 0
\(879\) −14.0000 24.2487i −0.472208 0.817889i
\(880\) 0 0
\(881\) −54.0000 −1.81931 −0.909653 0.415369i \(-0.863653\pi\)
−0.909653 + 0.415369i \(0.863653\pi\)
\(882\) 0 0
\(883\) 19.0000 0.639401 0.319700 0.947519i \(-0.396418\pi\)
0.319700 + 0.947519i \(0.396418\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −3.00000 + 5.19615i −0.100730 + 0.174470i −0.911986 0.410222i \(-0.865451\pi\)
0.811256 + 0.584692i \(0.198785\pi\)
\(888\) 0 0
\(889\) −22.5000 + 7.79423i −0.754626 + 0.261410i
\(890\) 0 0
\(891\) 1.00000 1.73205i 0.0335013 0.0580259i
\(892\) 0 0
\(893\) 20.0000 + 34.6410i 0.669274 + 1.15922i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) −6.00000 −0.200334
\(898\) 0 0
\(899\) −12.0000 20.7846i −0.400222 0.693206i
\(900\) 0 0
\(901\) −6.00000 + 10.3923i −0.199889 + 0.346218i
\(902\) 0 0
\(903\) 2.00000 + 1.73205i 0.0665558 + 0.0576390i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 0.500000 + 0.866025i 0.0166022 + 0.0287559i 0.874207 0.485553i \(-0.161382\pi\)
−0.857605 + 0.514309i \(0.828048\pi\)
\(908\) 0 0
\(909\) 6.00000 0.199007
\(910\) 0 0
\(911\) 22.0000 0.728893 0.364446 0.931224i \(-0.381258\pi\)
0.364446 + 0.931224i \(0.381258\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 5.00000 25.9808i 0.165115 0.857960i
\(918\) 0 0
\(919\) 17.5000 30.3109i 0.577272 0.999864i −0.418519 0.908208i \(-0.637451\pi\)
0.995791 0.0916559i \(-0.0292160\pi\)
\(920\) 0 0
\(921\) −12.5000 21.6506i −0.411889 0.713413i
\(922\) 0 0
\(923\) 4.00000 0.131662
\(924\) 0 0
\(925\) −45.0000 −1.47959
\(926\) 0 0
\(927\) −5.50000 9.52628i −0.180644 0.312884i
\(928\) 0 0
\(929\) 8.00000 13.8564i 0.262471 0.454614i −0.704427 0.709777i \(-0.748796\pi\)
0.966898 + 0.255163i \(0.0821291\pi\)
\(930\) 0 0
\(931\) 5.00000 + 34.6410i 0.163868 + 1.13531i
\(932\) 0 0
\(933\) 5.00000 8.66025i 0.163693 0.283524i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 51.0000 1.66610 0.833049 0.553200i \(-0.186593\pi\)
0.833049 + 0.553200i \(0.186593\pi\)
\(938\) 0 0
\(939\) −31.0000 −1.01165
\(940\) 0 0
\(941\) 13.0000 + 22.5167i 0.423788 + 0.734022i 0.996306 0.0858697i \(-0.0273669\pi\)
−0.572518 + 0.819892i \(0.694034\pi\)
\(942\) 0 0
\(943\) 6.00000 10.3923i 0.195387 0.338420i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −25.0000 + 43.3013i −0.812391 + 1.40710i 0.0987955 + 0.995108i \(0.468501\pi\)
−0.911186 + 0.411994i \(0.864832\pi\)
\(948\) 0 0
\(949\) −5.50000 9.52628i −0.178538 0.309236i
\(950\) 0 0
\(951\) 4.00000 0.129709
\(952\) 0 0
\(953\) 14.0000 0.453504 0.226752 0.973952i \(-0.427189\pi\)
0.226752 + 0.973952i \(0.427189\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) −8.00000 + 13.8564i −0.258603 + 0.447914i
\(958\) 0 0
\(959\) 24.0000 + 20.7846i 0.775000 + 0.671170i
\(960\) 0 0
\(961\) 11.0000 19.0526i 0.354839 0.614599i
\(962\) 0 0
\(963\) 9.00000 + 15.5885i 0.290021 + 0.502331i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 13.0000 0.418052 0.209026 0.977910i \(-0.432971\pi\)
0.209026 + 0.977910i \(0.432971\pi\)
\(968\) 0 0
\(969\) −5.00000 8.66025i −0.160623 0.278207i
\(970\) 0 0
\(971\) −13.0000 + 22.5167i −0.417190 + 0.722594i −0.995656 0.0931127i \(-0.970318\pi\)
0.578466 + 0.815707i \(0.303652\pi\)
\(972\) 0 0
\(973\) −37.5000 + 12.9904i −1.20219 + 0.416452i
\(974\) 0 0
\(975\) 2.50000 4.33013i 0.0800641 0.138675i
\(976\) 0 0
\(977\) −15.0000 25.9808i −0.479893 0.831198i 0.519841 0.854263i \(-0.325991\pi\)
−0.999734 + 0.0230645i \(0.992658\pi\)
\(978\) 0 0
\(979\) −24.0000 −0.767043
\(980\) 0 0
\(981\) 3.00000 0.0957826
\(982\) 0 0
\(983\) −1.00000 1.73205i −0.0318950 0.0552438i 0.849637 0.527368i \(-0.176821\pi\)
−0.881532 + 0.472124i \(0.843488\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 20.0000 6.92820i 0.636607 0.220527i
\(988\) 0 0
\(989\) 3.00000 5.19615i 0.0953945 0.165228i
\(990\) 0 0
\(991\) −27.5000 47.6314i −0.873566 1.51306i −0.858282 0.513178i \(-0.828468\pi\)
−0.0152841 0.999883i \(-0.504865\pi\)
\(992\) 0 0
\(993\) 17.0000 0.539479
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −20.5000 35.5070i −0.649242 1.12452i −0.983304 0.181968i \(-0.941753\pi\)
0.334063 0.942551i \(-0.391580\pi\)
\(998\) 0 0
\(999\) 4.50000 7.79423i 0.142374 0.246598i
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.q.961.1 2
4.3 odd 2 1344.2.q.e.961.1 2
7.2 even 3 9408.2.a.t.1.1 1
7.4 even 3 inner 1344.2.q.q.193.1 2
7.5 odd 6 9408.2.a.ci.1.1 1
8.3 odd 2 672.2.q.h.289.1 yes 2
8.5 even 2 672.2.q.d.289.1 yes 2
24.5 odd 2 2016.2.s.h.289.1 2
24.11 even 2 2016.2.s.e.289.1 2
28.11 odd 6 1344.2.q.e.193.1 2
28.19 even 6 9408.2.a.x.1.1 1
28.23 odd 6 9408.2.a.cn.1.1 1
56.5 odd 6 4704.2.a.j.1.1 1
56.11 odd 6 672.2.q.h.193.1 yes 2
56.19 even 6 4704.2.a.y.1.1 1
56.37 even 6 4704.2.a.ba.1.1 1
56.51 odd 6 4704.2.a.g.1.1 1
56.53 even 6 672.2.q.d.193.1 2
168.11 even 6 2016.2.s.e.865.1 2
168.53 odd 6 2016.2.s.h.865.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
672.2.q.d.193.1 2 56.53 even 6
672.2.q.d.289.1 yes 2 8.5 even 2
672.2.q.h.193.1 yes 2 56.11 odd 6
672.2.q.h.289.1 yes 2 8.3 odd 2
1344.2.q.e.193.1 2 28.11 odd 6
1344.2.q.e.961.1 2 4.3 odd 2
1344.2.q.q.193.1 2 7.4 even 3 inner
1344.2.q.q.961.1 2 1.1 even 1 trivial
2016.2.s.e.289.1 2 24.11 even 2
2016.2.s.e.865.1 2 168.11 even 6
2016.2.s.h.289.1 2 24.5 odd 2
2016.2.s.h.865.1 2 168.53 odd 6
4704.2.a.g.1.1 1 56.51 odd 6
4704.2.a.j.1.1 1 56.5 odd 6
4704.2.a.y.1.1 1 56.19 even 6
4704.2.a.ba.1.1 1 56.37 even 6
9408.2.a.t.1.1 1 7.2 even 3
9408.2.a.x.1.1 1 28.19 even 6
9408.2.a.ci.1.1 1 7.5 odd 6
9408.2.a.cn.1.1 1 28.23 odd 6