Properties

Label 2352.2.bl.b.607.1
Level $2352$
Weight $2$
Character 2352.607
Analytic conductor $18.781$
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] = [2352,2,Mod(31,2352)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2352, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2352.31");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2352 = 2^{4} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2352.bl (of order \(6\), 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: \(18.7808145554\)
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 336)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 607.1
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 2352.607
Dual form 2352.2.bl.b.31.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.500000 - 0.866025i) q^{3} +(-1.50000 - 0.866025i) q^{5} +(-0.500000 + 0.866025i) q^{9} +O(q^{10})\) \(q+(-0.500000 - 0.866025i) q^{3} +(-1.50000 - 0.866025i) q^{5} +(-0.500000 + 0.866025i) q^{9} +(-1.50000 + 0.866025i) q^{11} +1.73205i q^{15} +(-3.00000 + 1.73205i) q^{17} +(1.00000 - 1.73205i) q^{19} +(-1.00000 - 1.73205i) q^{25} +1.00000 q^{27} +9.00000 q^{29} +(-2.50000 - 4.33013i) q^{31} +(1.50000 + 0.866025i) q^{33} +(-5.00000 + 8.66025i) q^{37} +10.3923i q^{41} -3.46410i q^{43} +(1.50000 - 0.866025i) q^{45} +(-6.00000 + 10.3923i) q^{47} +(3.00000 + 1.73205i) q^{51} +(4.50000 + 7.79423i) q^{53} +3.00000 q^{55} -2.00000 q^{57} +(4.50000 + 7.79423i) q^{59} +(12.0000 - 6.92820i) q^{67} -13.8564i q^{71} +(6.00000 - 3.46410i) q^{73} +(-1.00000 + 1.73205i) q^{75} +(-4.50000 - 2.59808i) q^{79} +(-0.500000 - 0.866025i) q^{81} +3.00000 q^{83} +6.00000 q^{85} +(-4.50000 - 7.79423i) q^{87} +(3.00000 + 1.73205i) q^{89} +(-2.50000 + 4.33013i) q^{93} +(-3.00000 + 1.73205i) q^{95} +19.0526i q^{97} -1.73205i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{3} - 3 q^{5} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{3} - 3 q^{5} - q^{9} - 3 q^{11} - 6 q^{17} + 2 q^{19} - 2 q^{25} + 2 q^{27} + 18 q^{29} - 5 q^{31} + 3 q^{33} - 10 q^{37} + 3 q^{45} - 12 q^{47} + 6 q^{51} + 9 q^{53} + 6 q^{55} - 4 q^{57} + 9 q^{59} + 24 q^{67} + 12 q^{73} - 2 q^{75} - 9 q^{79} - q^{81} + 6 q^{83} + 12 q^{85} - 9 q^{87} + 6 q^{89} - 5 q^{93} - 6 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2352\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(1471\) \(1765\) \(2257\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(e\left(\frac{5}{6}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −0.500000 0.866025i −0.288675 0.500000i
\(4\) 0 0
\(5\) −1.50000 0.866025i −0.670820 0.387298i 0.125567 0.992085i \(-0.459925\pi\)
−0.796387 + 0.604787i \(0.793258\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) −0.500000 + 0.866025i −0.166667 + 0.288675i
\(10\) 0 0
\(11\) −1.50000 + 0.866025i −0.452267 + 0.261116i −0.708787 0.705422i \(-0.750757\pi\)
0.256520 + 0.966539i \(0.417424\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) 1.73205i 0.447214i
\(16\) 0 0
\(17\) −3.00000 + 1.73205i −0.727607 + 0.420084i −0.817546 0.575863i \(-0.804666\pi\)
0.0899392 + 0.995947i \(0.471333\pi\)
\(18\) 0 0
\(19\) 1.00000 1.73205i 0.229416 0.397360i −0.728219 0.685344i \(-0.759652\pi\)
0.957635 + 0.287984i \(0.0929851\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(24\) 0 0
\(25\) −1.00000 1.73205i −0.200000 0.346410i
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 9.00000 1.67126 0.835629 0.549294i \(-0.185103\pi\)
0.835629 + 0.549294i \(0.185103\pi\)
\(30\) 0 0
\(31\) −2.50000 4.33013i −0.449013 0.777714i 0.549309 0.835619i \(-0.314891\pi\)
−0.998322 + 0.0579057i \(0.981558\pi\)
\(32\) 0 0
\(33\) 1.50000 + 0.866025i 0.261116 + 0.150756i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −5.00000 + 8.66025i −0.821995 + 1.42374i 0.0821995 + 0.996616i \(0.473806\pi\)
−0.904194 + 0.427121i \(0.859528\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 10.3923i 1.62301i 0.584349 + 0.811503i \(0.301350\pi\)
−0.584349 + 0.811503i \(0.698650\pi\)
\(42\) 0 0
\(43\) 3.46410i 0.528271i −0.964486 0.264135i \(-0.914913\pi\)
0.964486 0.264135i \(-0.0850865\pi\)
\(44\) 0 0
\(45\) 1.50000 0.866025i 0.223607 0.129099i
\(46\) 0 0
\(47\) −6.00000 + 10.3923i −0.875190 + 1.51587i −0.0186297 + 0.999826i \(0.505930\pi\)
−0.856560 + 0.516047i \(0.827403\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 3.00000 + 1.73205i 0.420084 + 0.242536i
\(52\) 0 0
\(53\) 4.50000 + 7.79423i 0.618123 + 1.07062i 0.989828 + 0.142269i \(0.0454398\pi\)
−0.371706 + 0.928351i \(0.621227\pi\)
\(54\) 0 0
\(55\) 3.00000 0.404520
\(56\) 0 0
\(57\) −2.00000 −0.264906
\(58\) 0 0
\(59\) 4.50000 + 7.79423i 0.585850 + 1.01472i 0.994769 + 0.102151i \(0.0325726\pi\)
−0.408919 + 0.912571i \(0.634094\pi\)
\(60\) 0 0
\(61\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 12.0000 6.92820i 1.46603 0.846415i 0.466755 0.884387i \(-0.345423\pi\)
0.999279 + 0.0379722i \(0.0120898\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 13.8564i 1.64445i −0.569160 0.822226i \(-0.692732\pi\)
0.569160 0.822226i \(-0.307268\pi\)
\(72\) 0 0
\(73\) 6.00000 3.46410i 0.702247 0.405442i −0.105937 0.994373i \(-0.533784\pi\)
0.808184 + 0.588930i \(0.200451\pi\)
\(74\) 0 0
\(75\) −1.00000 + 1.73205i −0.115470 + 0.200000i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −4.50000 2.59808i −0.506290 0.292306i 0.225018 0.974355i \(-0.427756\pi\)
−0.731307 + 0.682048i \(0.761089\pi\)
\(80\) 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
\(85\) 6.00000 0.650791
\(86\) 0 0
\(87\) −4.50000 7.79423i −0.482451 0.835629i
\(88\) 0 0
\(89\) 3.00000 + 1.73205i 0.317999 + 0.183597i 0.650500 0.759506i \(-0.274559\pi\)
−0.332501 + 0.943103i \(0.607893\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −2.50000 + 4.33013i −0.259238 + 0.449013i
\(94\) 0 0
\(95\) −3.00000 + 1.73205i −0.307794 + 0.177705i
\(96\) 0 0
\(97\) 19.0526i 1.93449i 0.253837 + 0.967247i \(0.418307\pi\)
−0.253837 + 0.967247i \(0.581693\pi\)
\(98\) 0 0
\(99\) 1.73205i 0.174078i
\(100\) 0 0
\(101\) 12.0000 6.92820i 1.19404 0.689382i 0.234823 0.972038i \(-0.424549\pi\)
0.959221 + 0.282656i \(0.0912155\pi\)
\(102\) 0 0
\(103\) 2.00000 3.46410i 0.197066 0.341328i −0.750510 0.660859i \(-0.770192\pi\)
0.947576 + 0.319531i \(0.103525\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 10.5000 + 6.06218i 1.01507 + 0.586053i 0.912673 0.408690i \(-0.134014\pi\)
0.102400 + 0.994743i \(0.467348\pi\)
\(108\) 0 0
\(109\) 2.00000 + 3.46410i 0.191565 + 0.331801i 0.945769 0.324840i \(-0.105310\pi\)
−0.754204 + 0.656640i \(0.771977\pi\)
\(110\) 0 0
\(111\) 10.0000 0.949158
\(112\) 0 0
\(113\) 6.00000 0.564433 0.282216 0.959351i \(-0.408930\pi\)
0.282216 + 0.959351i \(0.408930\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −4.00000 + 6.92820i −0.363636 + 0.629837i
\(122\) 0 0
\(123\) 9.00000 5.19615i 0.811503 0.468521i
\(124\) 0 0
\(125\) 12.1244i 1.08444i
\(126\) 0 0
\(127\) 5.19615i 0.461084i −0.973062 0.230542i \(-0.925950\pi\)
0.973062 0.230542i \(-0.0740499\pi\)
\(128\) 0 0
\(129\) −3.00000 + 1.73205i −0.264135 + 0.152499i
\(130\) 0 0
\(131\) 4.50000 7.79423i 0.393167 0.680985i −0.599699 0.800226i \(-0.704713\pi\)
0.992865 + 0.119241i \(0.0380462\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) −1.50000 0.866025i −0.129099 0.0745356i
\(136\) 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
\(139\) 14.0000 1.18746 0.593732 0.804663i \(-0.297654\pi\)
0.593732 + 0.804663i \(0.297654\pi\)
\(140\) 0 0
\(141\) 12.0000 1.01058
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) −13.5000 7.79423i −1.12111 0.647275i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −3.00000 + 5.19615i −0.245770 + 0.425685i −0.962348 0.271821i \(-0.912374\pi\)
0.716578 + 0.697507i \(0.245707\pi\)
\(150\) 0 0
\(151\) −4.50000 + 2.59808i −0.366205 + 0.211428i −0.671799 0.740733i \(-0.734478\pi\)
0.305594 + 0.952162i \(0.401145\pi\)
\(152\) 0 0
\(153\) 3.46410i 0.280056i
\(154\) 0 0
\(155\) 8.66025i 0.695608i
\(156\) 0 0
\(157\) −6.00000 + 3.46410i −0.478852 + 0.276465i −0.719938 0.694038i \(-0.755830\pi\)
0.241086 + 0.970504i \(0.422496\pi\)
\(158\) 0 0
\(159\) 4.50000 7.79423i 0.356873 0.618123i
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 12.0000 + 6.92820i 0.939913 + 0.542659i 0.889933 0.456091i \(-0.150751\pi\)
0.0499796 + 0.998750i \(0.484084\pi\)
\(164\) 0 0
\(165\) −1.50000 2.59808i −0.116775 0.202260i
\(166\) 0 0
\(167\) −12.0000 −0.928588 −0.464294 0.885681i \(-0.653692\pi\)
−0.464294 + 0.885681i \(0.653692\pi\)
\(168\) 0 0
\(169\) 13.0000 1.00000
\(170\) 0 0
\(171\) 1.00000 + 1.73205i 0.0764719 + 0.132453i
\(172\) 0 0
\(173\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 4.50000 7.79423i 0.338241 0.585850i
\(178\) 0 0
\(179\) −21.0000 + 12.1244i −1.56961 + 0.906217i −0.573400 + 0.819275i \(0.694376\pi\)
−0.996213 + 0.0869415i \(0.972291\pi\)
\(180\) 0 0
\(181\) 10.3923i 0.772454i −0.922404 0.386227i \(-0.873778\pi\)
0.922404 0.386227i \(-0.126222\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 15.0000 8.66025i 1.10282 0.636715i
\(186\) 0 0
\(187\) 3.00000 5.19615i 0.219382 0.379980i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 12.0000 + 6.92820i 0.868290 + 0.501307i 0.866779 0.498692i \(-0.166186\pi\)
0.00151007 + 0.999999i \(0.499519\pi\)
\(192\) 0 0
\(193\) 2.50000 + 4.33013i 0.179954 + 0.311689i 0.941865 0.335993i \(-0.109072\pi\)
−0.761911 + 0.647682i \(0.775738\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −6.00000 −0.427482 −0.213741 0.976890i \(-0.568565\pi\)
−0.213741 + 0.976890i \(0.568565\pi\)
\(198\) 0 0
\(199\) −8.00000 13.8564i −0.567105 0.982255i −0.996850 0.0793045i \(-0.974730\pi\)
0.429745 0.902950i \(-0.358603\pi\)
\(200\) 0 0
\(201\) −12.0000 6.92820i −0.846415 0.488678i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 9.00000 15.5885i 0.628587 1.08875i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 3.46410i 0.239617i
\(210\) 0 0
\(211\) 6.92820i 0.476957i 0.971148 + 0.238479i \(0.0766487\pi\)
−0.971148 + 0.238479i \(0.923351\pi\)
\(212\) 0 0
\(213\) −12.0000 + 6.92820i −0.822226 + 0.474713i
\(214\) 0 0
\(215\) −3.00000 + 5.19615i −0.204598 + 0.354375i
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −6.00000 3.46410i −0.405442 0.234082i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 19.0000 1.27233 0.636167 0.771551i \(-0.280519\pi\)
0.636167 + 0.771551i \(0.280519\pi\)
\(224\) 0 0
\(225\) 2.00000 0.133333
\(226\) 0 0
\(227\) 10.5000 + 18.1865i 0.696909 + 1.20708i 0.969533 + 0.244962i \(0.0787754\pi\)
−0.272623 + 0.962121i \(0.587891\pi\)
\(228\) 0 0
\(229\) 3.00000 + 1.73205i 0.198246 + 0.114457i 0.595837 0.803105i \(-0.296820\pi\)
−0.397591 + 0.917563i \(0.630154\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −9.00000 + 15.5885i −0.589610 + 1.02123i 0.404674 + 0.914461i \(0.367385\pi\)
−0.994283 + 0.106773i \(0.965948\pi\)
\(234\) 0 0
\(235\) 18.0000 10.3923i 1.17419 0.677919i
\(236\) 0 0
\(237\) 5.19615i 0.337526i
\(238\) 0 0
\(239\) 6.92820i 0.448148i −0.974572 0.224074i \(-0.928064\pi\)
0.974572 0.224074i \(-0.0719358\pi\)
\(240\) 0 0
\(241\) −19.5000 + 11.2583i −1.25611 + 0.725213i −0.972315 0.233674i \(-0.924925\pi\)
−0.283790 + 0.958886i \(0.591592\pi\)
\(242\) 0 0
\(243\) −0.500000 + 0.866025i −0.0320750 + 0.0555556i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) −1.50000 2.59808i −0.0950586 0.164646i
\(250\) 0 0
\(251\) −15.0000 −0.946792 −0.473396 0.880850i \(-0.656972\pi\)
−0.473396 + 0.880850i \(0.656972\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) −3.00000 5.19615i −0.187867 0.325396i
\(256\) 0 0
\(257\) 9.00000 + 5.19615i 0.561405 + 0.324127i 0.753709 0.657208i \(-0.228263\pi\)
−0.192304 + 0.981335i \(0.561596\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −4.50000 + 7.79423i −0.278543 + 0.482451i
\(262\) 0 0
\(263\) −15.0000 + 8.66025i −0.924940 + 0.534014i −0.885208 0.465196i \(-0.845984\pi\)
−0.0397320 + 0.999210i \(0.512650\pi\)
\(264\) 0 0
\(265\) 15.5885i 0.957591i
\(266\) 0 0
\(267\) 3.46410i 0.212000i
\(268\) 0 0
\(269\) 13.5000 7.79423i 0.823110 0.475223i −0.0283781 0.999597i \(-0.509034\pi\)
0.851488 + 0.524375i \(0.175701\pi\)
\(270\) 0 0
\(271\) −5.50000 + 9.52628i −0.334101 + 0.578680i −0.983312 0.181928i \(-0.941766\pi\)
0.649211 + 0.760609i \(0.275099\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 3.00000 + 1.73205i 0.180907 + 0.104447i
\(276\) 0 0
\(277\) −14.0000 24.2487i −0.841178 1.45696i −0.888899 0.458103i \(-0.848529\pi\)
0.0477206 0.998861i \(-0.484804\pi\)
\(278\) 0 0
\(279\) 5.00000 0.299342
\(280\) 0 0
\(281\) −24.0000 −1.43172 −0.715860 0.698244i \(-0.753965\pi\)
−0.715860 + 0.698244i \(0.753965\pi\)
\(282\) 0 0
\(283\) 7.00000 + 12.1244i 0.416107 + 0.720718i 0.995544 0.0942988i \(-0.0300609\pi\)
−0.579437 + 0.815017i \(0.696728\pi\)
\(284\) 0 0
\(285\) 3.00000 + 1.73205i 0.177705 + 0.102598i
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −2.50000 + 4.33013i −0.147059 + 0.254713i
\(290\) 0 0
\(291\) 16.5000 9.52628i 0.967247 0.558440i
\(292\) 0 0
\(293\) 5.19615i 0.303562i −0.988414 0.151781i \(-0.951499\pi\)
0.988414 0.151781i \(-0.0485009\pi\)
\(294\) 0 0
\(295\) 15.5885i 0.907595i
\(296\) 0 0
\(297\) −1.50000 + 0.866025i −0.0870388 + 0.0502519i
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) −12.0000 6.92820i −0.689382 0.398015i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −20.0000 −1.14146 −0.570730 0.821138i \(-0.693340\pi\)
−0.570730 + 0.821138i \(0.693340\pi\)
\(308\) 0 0
\(309\) −4.00000 −0.227552
\(310\) 0 0
\(311\) 9.00000 + 15.5885i 0.510343 + 0.883940i 0.999928 + 0.0119847i \(0.00381495\pi\)
−0.489585 + 0.871956i \(0.662852\pi\)
\(312\) 0 0
\(313\) 4.50000 + 2.59808i 0.254355 + 0.146852i 0.621757 0.783210i \(-0.286419\pi\)
−0.367402 + 0.930062i \(0.619753\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 10.5000 18.1865i 0.589739 1.02146i −0.404528 0.914526i \(-0.632564\pi\)
0.994266 0.106932i \(-0.0341026\pi\)
\(318\) 0 0
\(319\) −13.5000 + 7.79423i −0.755855 + 0.436393i
\(320\) 0 0
\(321\) 12.1244i 0.676716i
\(322\) 0 0
\(323\) 6.92820i 0.385496i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 2.00000 3.46410i 0.110600 0.191565i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −12.0000 6.92820i −0.659580 0.380808i 0.132537 0.991178i \(-0.457688\pi\)
−0.792117 + 0.610370i \(0.791021\pi\)
\(332\) 0 0
\(333\) −5.00000 8.66025i −0.273998 0.474579i
\(334\) 0 0
\(335\) −24.0000 −1.31126
\(336\) 0 0
\(337\) −1.00000 −0.0544735 −0.0272367 0.999629i \(-0.508671\pi\)
−0.0272367 + 0.999629i \(0.508671\pi\)
\(338\) 0 0
\(339\) −3.00000 5.19615i −0.162938 0.282216i
\(340\) 0 0
\(341\) 7.50000 + 4.33013i 0.406148 + 0.234490i
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 15.0000 8.66025i 0.805242 0.464907i −0.0400587 0.999197i \(-0.512754\pi\)
0.845301 + 0.534291i \(0.179421\pi\)
\(348\) 0 0
\(349\) 6.92820i 0.370858i 0.982658 + 0.185429i \(0.0593675\pi\)
−0.982658 + 0.185429i \(0.940632\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −6.00000 + 3.46410i −0.319348 + 0.184376i −0.651102 0.758990i \(-0.725693\pi\)
0.331754 + 0.943366i \(0.392360\pi\)
\(354\) 0 0
\(355\) −12.0000 + 20.7846i −0.636894 + 1.10313i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −24.0000 13.8564i −1.26667 0.731313i −0.292315 0.956322i \(-0.594426\pi\)
−0.974357 + 0.225009i \(0.927759\pi\)
\(360\) 0 0
\(361\) 7.50000 + 12.9904i 0.394737 + 0.683704i
\(362\) 0 0
\(363\) 8.00000 0.419891
\(364\) 0 0
\(365\) −12.0000 −0.628109
\(366\) 0 0
\(367\) −9.50000 16.4545i −0.495896 0.858917i 0.504093 0.863649i \(-0.331827\pi\)
−0.999989 + 0.00473247i \(0.998494\pi\)
\(368\) 0 0
\(369\) −9.00000 5.19615i −0.468521 0.270501i
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −1.00000 + 1.73205i −0.0517780 + 0.0896822i −0.890753 0.454488i \(-0.849822\pi\)
0.838975 + 0.544170i \(0.183156\pi\)
\(374\) 0 0
\(375\) 10.5000 6.06218i 0.542218 0.313050i
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 24.2487i 1.24557i 0.782392 + 0.622786i \(0.213999\pi\)
−0.782392 + 0.622786i \(0.786001\pi\)
\(380\) 0 0
\(381\) −4.50000 + 2.59808i −0.230542 + 0.133103i
\(382\) 0 0
\(383\) −9.00000 + 15.5885i −0.459879 + 0.796533i −0.998954 0.0457244i \(-0.985440\pi\)
0.539076 + 0.842257i \(0.318774\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 3.00000 + 1.73205i 0.152499 + 0.0880451i
\(388\) 0 0
\(389\) 3.00000 + 5.19615i 0.152106 + 0.263455i 0.932002 0.362454i \(-0.118061\pi\)
−0.779895 + 0.625910i \(0.784728\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) −9.00000 −0.453990
\(394\) 0 0
\(395\) 4.50000 + 7.79423i 0.226420 + 0.392170i
\(396\) 0 0
\(397\) 18.0000 + 10.3923i 0.903394 + 0.521575i 0.878300 0.478110i \(-0.158678\pi\)
0.0250943 + 0.999685i \(0.492011\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 6.00000 10.3923i 0.299626 0.518967i −0.676425 0.736512i \(-0.736472\pi\)
0.976050 + 0.217545i \(0.0698049\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 1.73205i 0.0860663i
\(406\) 0 0
\(407\) 17.3205i 0.858546i
\(408\) 0 0
\(409\) 19.5000 11.2583i 0.964213 0.556689i 0.0667458 0.997770i \(-0.478738\pi\)
0.897467 + 0.441081i \(0.145405\pi\)
\(410\) 0 0
\(411\) 6.00000 10.3923i 0.295958 0.512615i
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −4.50000 2.59808i −0.220896 0.127535i
\(416\) 0 0
\(417\) −7.00000 12.1244i −0.342791 0.593732i
\(418\) 0 0
\(419\) −12.0000 −0.586238 −0.293119 0.956076i \(-0.594693\pi\)
−0.293119 + 0.956076i \(0.594693\pi\)
\(420\) 0 0
\(421\) −20.0000 −0.974740 −0.487370 0.873195i \(-0.662044\pi\)
−0.487370 + 0.873195i \(0.662044\pi\)
\(422\) 0 0
\(423\) −6.00000 10.3923i −0.291730 0.505291i
\(424\) 0 0
\(425\) 6.00000 + 3.46410i 0.291043 + 0.168034i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −33.0000 + 19.0526i −1.58955 + 0.917729i −0.596174 + 0.802855i \(0.703313\pi\)
−0.993380 + 0.114874i \(0.963353\pi\)
\(432\) 0 0
\(433\) 34.6410i 1.66474i −0.554220 0.832370i \(-0.686983\pi\)
0.554220 0.832370i \(-0.313017\pi\)
\(434\) 0 0
\(435\) 15.5885i 0.747409i
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 14.5000 25.1147i 0.692047 1.19866i −0.279119 0.960257i \(-0.590042\pi\)
0.971166 0.238404i \(-0.0766244\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −10.5000 6.06218i −0.498870 0.288023i 0.229377 0.973338i \(-0.426331\pi\)
−0.728247 + 0.685315i \(0.759665\pi\)
\(444\) 0 0
\(445\) −3.00000 5.19615i −0.142214 0.246321i
\(446\) 0 0
\(447\) 6.00000 0.283790
\(448\) 0 0
\(449\) 12.0000 0.566315 0.283158 0.959073i \(-0.408618\pi\)
0.283158 + 0.959073i \(0.408618\pi\)
\(450\) 0 0
\(451\) −9.00000 15.5885i −0.423793 0.734032i
\(452\) 0 0
\(453\) 4.50000 + 2.59808i 0.211428 + 0.122068i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 15.5000 26.8468i 0.725059 1.25584i −0.233890 0.972263i \(-0.575146\pi\)
0.958950 0.283577i \(-0.0915211\pi\)
\(458\) 0 0
\(459\) −3.00000 + 1.73205i −0.140028 + 0.0808452i
\(460\) 0 0
\(461\) 6.92820i 0.322679i 0.986899 + 0.161339i \(0.0515813\pi\)
−0.986899 + 0.161339i \(0.948419\pi\)
\(462\) 0 0
\(463\) 38.1051i 1.77090i 0.464739 + 0.885448i \(0.346148\pi\)
−0.464739 + 0.885448i \(0.653852\pi\)
\(464\) 0 0
\(465\) 7.50000 4.33013i 0.347804 0.200805i
\(466\) 0 0
\(467\) −6.00000 + 10.3923i −0.277647 + 0.480899i −0.970799 0.239892i \(-0.922888\pi\)
0.693153 + 0.720791i \(0.256221\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 6.00000 + 3.46410i 0.276465 + 0.159617i
\(472\) 0 0
\(473\) 3.00000 + 5.19615i 0.137940 + 0.238919i
\(474\) 0 0
\(475\) −4.00000 −0.183533
\(476\) 0 0
\(477\) −9.00000 −0.412082
\(478\) 0 0
\(479\) −3.00000 5.19615i −0.137073 0.237418i 0.789314 0.613990i \(-0.210436\pi\)
−0.926388 + 0.376571i \(0.877103\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 16.5000 28.5788i 0.749226 1.29770i
\(486\) 0 0
\(487\) 7.50000 4.33013i 0.339857 0.196217i −0.320352 0.947299i \(-0.603801\pi\)
0.660209 + 0.751082i \(0.270468\pi\)
\(488\) 0 0
\(489\) 13.8564i 0.626608i
\(490\) 0 0
\(491\) 25.9808i 1.17250i 0.810132 + 0.586248i \(0.199395\pi\)
−0.810132 + 0.586248i \(0.800605\pi\)
\(492\) 0 0
\(493\) −27.0000 + 15.5885i −1.21602 + 0.702069i
\(494\) 0 0
\(495\) −1.50000 + 2.59808i −0.0674200 + 0.116775i
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −3.00000 1.73205i −0.134298 0.0775372i 0.431346 0.902187i \(-0.358039\pi\)
−0.565644 + 0.824650i \(0.691372\pi\)
\(500\) 0 0
\(501\) 6.00000 + 10.3923i 0.268060 + 0.464294i
\(502\) 0 0
\(503\) 6.00000 0.267527 0.133763 0.991013i \(-0.457294\pi\)
0.133763 + 0.991013i \(0.457294\pi\)
\(504\) 0 0
\(505\) −24.0000 −1.06799
\(506\) 0 0
\(507\) −6.50000 11.2583i −0.288675 0.500000i
\(508\) 0 0
\(509\) −28.5000 16.4545i −1.26324 0.729332i −0.289540 0.957166i \(-0.593502\pi\)
−0.973700 + 0.227834i \(0.926836\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 1.00000 1.73205i 0.0441511 0.0764719i
\(514\) 0 0
\(515\) −6.00000 + 3.46410i −0.264392 + 0.152647i
\(516\) 0 0
\(517\) 20.7846i 0.914106i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −33.0000 + 19.0526i −1.44576 + 0.834708i −0.998225 0.0595604i \(-0.981030\pi\)
−0.447532 + 0.894268i \(0.647697\pi\)
\(522\) 0 0
\(523\) 14.0000 24.2487i 0.612177 1.06032i −0.378695 0.925521i \(-0.623627\pi\)
0.990873 0.134801i \(-0.0430394\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 15.0000 + 8.66025i 0.653410 + 0.377247i
\(528\) 0 0
\(529\) −11.5000 19.9186i −0.500000 0.866025i
\(530\) 0 0
\(531\) −9.00000 −0.390567
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) −10.5000 18.1865i −0.453955 0.786272i
\(536\) 0 0
\(537\) 21.0000 + 12.1244i 0.906217 + 0.523205i
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −11.0000 + 19.0526i −0.472927 + 0.819133i −0.999520 0.0309841i \(-0.990136\pi\)
0.526593 + 0.850118i \(0.323469\pi\)
\(542\) 0 0
\(543\) −9.00000 + 5.19615i −0.386227 + 0.222988i
\(544\) 0 0
\(545\) 6.92820i 0.296772i
\(546\) 0 0
\(547\) 10.3923i 0.444343i 0.975008 + 0.222171i \(0.0713145\pi\)
−0.975008 + 0.222171i \(0.928686\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 9.00000 15.5885i 0.383413 0.664091i
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) −15.0000 8.66025i −0.636715 0.367607i
\(556\) 0 0
\(557\) 13.5000 + 23.3827i 0.572013 + 0.990756i 0.996359 + 0.0852559i \(0.0271708\pi\)
−0.424346 + 0.905500i \(0.639496\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) −6.00000 −0.253320
\(562\) 0 0
\(563\) 22.5000 + 38.9711i 0.948262 + 1.64244i 0.749085 + 0.662474i \(0.230494\pi\)
0.199177 + 0.979963i \(0.436173\pi\)
\(564\) 0 0
\(565\) −9.00000 5.19615i −0.378633 0.218604i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 3.00000 5.19615i 0.125767 0.217834i −0.796266 0.604947i \(-0.793194\pi\)
0.922032 + 0.387113i \(0.126528\pi\)
\(570\) 0 0
\(571\) 18.0000 10.3923i 0.753277 0.434904i −0.0736000 0.997288i \(-0.523449\pi\)
0.826877 + 0.562383i \(0.190115\pi\)
\(572\) 0 0
\(573\) 13.8564i 0.578860i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 1.50000 0.866025i 0.0624458 0.0360531i −0.468452 0.883489i \(-0.655188\pi\)
0.530898 + 0.847436i \(0.321855\pi\)
\(578\) 0 0
\(579\) 2.50000 4.33013i 0.103896 0.179954i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −13.5000 7.79423i −0.559113 0.322804i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −33.0000 −1.36206 −0.681028 0.732257i \(-0.738467\pi\)
−0.681028 + 0.732257i \(0.738467\pi\)
\(588\) 0 0
\(589\) −10.0000 −0.412043
\(590\) 0 0
\(591\) 3.00000 + 5.19615i 0.123404 + 0.213741i
\(592\) 0 0
\(593\) 24.0000 + 13.8564i 0.985562 + 0.569014i 0.903945 0.427649i \(-0.140658\pi\)
0.0816172 + 0.996664i \(0.473992\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −8.00000 + 13.8564i −0.327418 + 0.567105i
\(598\) 0 0
\(599\) −24.0000 + 13.8564i −0.980613 + 0.566157i −0.902455 0.430784i \(-0.858237\pi\)
−0.0781581 + 0.996941i \(0.524904\pi\)
\(600\) 0 0
\(601\) 15.5885i 0.635866i 0.948113 + 0.317933i \(0.102989\pi\)
−0.948113 + 0.317933i \(0.897011\pi\)
\(602\) 0 0
\(603\) 13.8564i 0.564276i
\(604\) 0 0
\(605\) 12.0000 6.92820i 0.487869 0.281672i
\(606\) 0 0
\(607\) −9.50000 + 16.4545i −0.385593 + 0.667867i −0.991851 0.127401i \(-0.959336\pi\)
0.606258 + 0.795268i \(0.292670\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 2.00000 + 3.46410i 0.0807792 + 0.139914i 0.903585 0.428409i \(-0.140926\pi\)
−0.822806 + 0.568323i \(0.807592\pi\)
\(614\) 0 0
\(615\) −18.0000 −0.725830
\(616\) 0 0
\(617\) 12.0000 0.483102 0.241551 0.970388i \(-0.422344\pi\)
0.241551 + 0.970388i \(0.422344\pi\)
\(618\) 0 0
\(619\) 14.0000 + 24.2487i 0.562708 + 0.974638i 0.997259 + 0.0739910i \(0.0235736\pi\)
−0.434551 + 0.900647i \(0.643093\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 5.50000 9.52628i 0.220000 0.381051i
\(626\) 0 0
\(627\) 3.00000 1.73205i 0.119808 0.0691714i
\(628\) 0 0
\(629\) 34.6410i 1.38123i
\(630\) 0 0
\(631\) 32.9090i 1.31009i −0.755592 0.655043i \(-0.772651\pi\)
0.755592 0.655043i \(-0.227349\pi\)
\(632\) 0 0
\(633\) 6.00000 3.46410i 0.238479 0.137686i
\(634\) 0 0
\(635\) −4.50000 + 7.79423i −0.178577 + 0.309305i
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 12.0000 + 6.92820i 0.474713 + 0.274075i
\(640\) 0 0
\(641\) −18.0000 31.1769i −0.710957 1.23141i −0.964498 0.264089i \(-0.914929\pi\)
0.253541 0.967325i \(-0.418405\pi\)
\(642\) 0 0
\(643\) 16.0000 0.630978 0.315489 0.948929i \(-0.397831\pi\)
0.315489 + 0.948929i \(0.397831\pi\)
\(644\) 0 0
\(645\) 6.00000 0.236250
\(646\) 0 0
\(647\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(648\) 0 0
\(649\) −13.5000 7.79423i −0.529921 0.305950i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 13.5000 23.3827i 0.528296 0.915035i −0.471160 0.882048i \(-0.656165\pi\)
0.999456 0.0329874i \(-0.0105021\pi\)
\(654\) 0 0
\(655\) −13.5000 + 7.79423i −0.527489 + 0.304546i
\(656\) 0 0
\(657\) 6.92820i 0.270295i
\(658\) 0 0
\(659\) 10.3923i 0.404827i −0.979300 0.202413i \(-0.935122\pi\)
0.979300 0.202413i \(-0.0648785\pi\)
\(660\) 0 0
\(661\) −3.00000 + 1.73205i −0.116686 + 0.0673690i −0.557207 0.830373i \(-0.688127\pi\)
0.440521 + 0.897742i \(0.354794\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) −9.50000 16.4545i −0.367291 0.636167i
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −35.0000 −1.34915 −0.674575 0.738206i \(-0.735673\pi\)
−0.674575 + 0.738206i \(0.735673\pi\)
\(674\) 0 0
\(675\) −1.00000 1.73205i −0.0384900 0.0666667i
\(676\) 0 0
\(677\) −25.5000 14.7224i −0.980045 0.565829i −0.0777610 0.996972i \(-0.524777\pi\)
−0.902284 + 0.431143i \(0.858110\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 10.5000 18.1865i 0.402361 0.696909i
\(682\) 0 0
\(683\) 22.5000 12.9904i 0.860939 0.497063i −0.00338791 0.999994i \(-0.501078\pi\)
0.864326 + 0.502931i \(0.167745\pi\)
\(684\) 0 0
\(685\) 20.7846i 0.794139i
\(686\) 0 0
\(687\) 3.46410i 0.132164i
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 11.0000 19.0526i 0.418460 0.724793i −0.577325 0.816514i \(-0.695903\pi\)
0.995785 + 0.0917209i \(0.0292368\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −21.0000 12.1244i −0.796575 0.459903i
\(696\) 0 0
\(697\) −18.0000 31.1769i −0.681799 1.18091i
\(698\) 0 0
\(699\) 18.0000 0.680823
\(700\) 0 0
\(701\) −45.0000 −1.69963 −0.849813 0.527084i \(-0.823285\pi\)
−0.849813 + 0.527084i \(0.823285\pi\)
\(702\) 0 0
\(703\) 10.0000 + 17.3205i 0.377157 + 0.653255i
\(704\) 0 0
\(705\) −18.0000 10.3923i −0.677919 0.391397i
\(706\) 0 0
\(707\) 0 0
\(708\) 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
\(711\) 4.50000 2.59808i 0.168763 0.0974355i
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −6.00000 + 3.46410i −0.224074 + 0.129369i
\(718\) 0 0
\(719\) −9.00000 + 15.5885i −0.335643 + 0.581351i −0.983608 0.180319i \(-0.942287\pi\)
0.647965 + 0.761670i \(0.275620\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 19.5000 + 11.2583i 0.725213 + 0.418702i
\(724\) 0 0
\(725\) −9.00000 15.5885i −0.334252 0.578941i
\(726\) 0 0
\(727\) −37.0000 −1.37225 −0.686127 0.727482i \(-0.740691\pi\)
−0.686127 + 0.727482i \(0.740691\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 6.00000 + 10.3923i 0.221918 + 0.384373i
\(732\) 0 0
\(733\) 9.00000 + 5.19615i 0.332423 + 0.191924i 0.656916 0.753964i \(-0.271861\pi\)
−0.324494 + 0.945888i \(0.605194\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −12.0000 + 20.7846i −0.442026 + 0.765611i
\(738\) 0 0
\(739\) −9.00000 + 5.19615i −0.331070 + 0.191144i −0.656316 0.754486i \(-0.727886\pi\)
0.325246 + 0.945629i \(0.394553\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 13.8564i 0.508342i 0.967159 + 0.254171i \(0.0818026\pi\)
−0.967159 + 0.254171i \(0.918197\pi\)
\(744\) 0 0
\(745\) 9.00000 5.19615i 0.329734 0.190372i
\(746\) 0 0
\(747\) −1.50000 + 2.59808i −0.0548821 + 0.0950586i
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −4.50000 2.59808i −0.164207 0.0948051i 0.415644 0.909527i \(-0.363556\pi\)
−0.579852 + 0.814722i \(0.696889\pi\)
\(752\) 0 0
\(753\) 7.50000 + 12.9904i 0.273315 + 0.473396i
\(754\) 0 0
\(755\) 9.00000 0.327544
\(756\) 0 0
\(757\) −4.00000 −0.145382 −0.0726912 0.997354i \(-0.523159\pi\)
−0.0726912 + 0.997354i \(0.523159\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 42.0000 + 24.2487i 1.52250 + 0.879015i 0.999646 + 0.0265919i \(0.00846546\pi\)
0.522852 + 0.852423i \(0.324868\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) −3.00000 + 5.19615i −0.108465 + 0.187867i
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 19.0526i 0.687053i −0.939143 0.343526i \(-0.888379\pi\)
0.939143 0.343526i \(-0.111621\pi\)
\(770\) 0 0
\(771\) 10.3923i 0.374270i
\(772\) 0 0
\(773\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(774\) 0 0
\(775\) −5.00000 + 8.66025i −0.179605 + 0.311086i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 18.0000 + 10.3923i 0.644917 + 0.372343i
\(780\) 0 0
\(781\) 12.0000 + 20.7846i 0.429394 + 0.743732i
\(782\) 0 0
\(783\) 9.00000 0.321634
\(784\) 0 0
\(785\) 12.0000 0.428298
\(786\) 0 0
\(787\) 14.0000 + 24.2487i 0.499046 + 0.864373i 0.999999 0.00110111i \(-0.000350496\pi\)
−0.500953 + 0.865474i \(0.667017\pi\)
\(788\) 0 0
\(789\) 15.0000 + 8.66025i 0.534014 + 0.308313i
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) −13.5000 + 7.79423i −0.478796 + 0.276433i
\(796\) 0 0
\(797\) 5.19615i 0.184057i −0.995756 0.0920286i \(-0.970665\pi\)
0.995756 0.0920286i \(-0.0293351\pi\)
\(798\) 0 0
\(799\) 41.5692i 1.47061i
\(800\) 0 0
\(801\) −3.00000 + 1.73205i −0.106000 + 0.0611990i
\(802\) 0 0
\(803\) −6.00000 + 10.3923i −0.211735 + 0.366736i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −13.5000 7.79423i −0.475223 0.274370i
\(808\) 0 0
\(809\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(810\) 0 0
\(811\) −16.0000 −0.561836 −0.280918 0.959732i \(-0.590639\pi\)
−0.280918 + 0.959732i \(0.590639\pi\)
\(812\) 0 0
\(813\) 11.0000 0.385787
\(814\) 0 0
\(815\) −12.0000 20.7846i −0.420342 0.728053i
\(816\) 0 0
\(817\) −6.00000 3.46410i −0.209913 0.121194i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −1.50000 + 2.59808i −0.0523504 + 0.0906735i −0.891013 0.453978i \(-0.850005\pi\)
0.838663 + 0.544651i \(0.183338\pi\)
\(822\) 0 0
\(823\) 33.0000 19.0526i 1.15031 0.664130i 0.201345 0.979520i \(-0.435469\pi\)
0.948962 + 0.315390i \(0.102135\pi\)
\(824\) 0 0
\(825\) 3.46410i 0.120605i
\(826\) 0 0
\(827\) 39.8372i 1.38527i −0.721286 0.692637i \(-0.756449\pi\)
0.721286 0.692637i \(-0.243551\pi\)
\(828\) 0 0
\(829\) 15.0000 8.66025i 0.520972 0.300783i −0.216361 0.976314i \(-0.569419\pi\)
0.737332 + 0.675530i \(0.236085\pi\)
\(830\) 0 0
\(831\) −14.0000 + 24.2487i −0.485655 + 0.841178i
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 18.0000 + 10.3923i 0.622916 + 0.359641i
\(836\) 0 0
\(837\) −2.50000 4.33013i −0.0864126 0.149671i
\(838\) 0 0
\(839\) 30.0000 1.03572 0.517858 0.855467i \(-0.326730\pi\)
0.517858 + 0.855467i \(0.326730\pi\)
\(840\) 0 0
\(841\) 52.0000 1.79310
\(842\) 0 0
\(843\) 12.0000 + 20.7846i 0.413302 + 0.715860i
\(844\) 0 0
\(845\) −19.5000 11.2583i −0.670820 0.387298i
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 7.00000 12.1244i 0.240239 0.416107i
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 48.4974i 1.66052i 0.557376 + 0.830260i \(0.311808\pi\)
−0.557376 + 0.830260i \(0.688192\pi\)
\(854\) 0 0
\(855\) 3.46410i 0.118470i
\(856\) 0 0
\(857\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(858\) 0 0
\(859\) 7.00000 12.1244i 0.238837 0.413678i −0.721544 0.692369i \(-0.756567\pi\)
0.960381 + 0.278691i \(0.0899005\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 27.0000 + 15.5885i 0.919091 + 0.530637i 0.883345 0.468724i \(-0.155286\pi\)
0.0357458 + 0.999361i \(0.488619\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 5.00000 0.169809
\(868\) 0 0
\(869\) 9.00000 0.305304
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) −16.5000 9.52628i −0.558440 0.322416i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −16.0000 + 27.7128i −0.540282 + 0.935795i 0.458606 + 0.888640i \(0.348349\pi\)
−0.998888 + 0.0471555i \(0.984984\pi\)
\(878\) 0 0
\(879\) −4.50000 + 2.59808i −0.151781 + 0.0876309i
\(880\) 0 0
\(881\) 13.8564i 0.466834i 0.972377 + 0.233417i \(0.0749907\pi\)
−0.972377 + 0.233417i \(0.925009\pi\)
\(882\) 0 0
\(883\) 24.2487i 0.816034i 0.912974 + 0.408017i \(0.133780\pi\)
−0.912974 + 0.408017i \(0.866220\pi\)
\(884\) 0 0
\(885\) −13.5000 + 7.79423i −0.453798 + 0.262000i
\(886\) 0 0
\(887\) 24.0000 41.5692i 0.805841 1.39576i −0.109881 0.993945i \(-0.535047\pi\)
0.915722 0.401813i \(-0.131620\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 1.50000 + 0.866025i 0.0502519 + 0.0290129i
\(892\) 0 0
\(893\) 12.0000 + 20.7846i 0.401565 + 0.695530i
\(894\) 0 0
\(895\) 42.0000 1.40391
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −22.5000 38.9711i −0.750417 1.29976i
\(900\) 0 0
\(901\) −27.0000 15.5885i −0.899500 0.519327i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −9.00000 + 15.5885i −0.299170 + 0.518178i
\(906\) 0 0
\(907\) −18.0000 + 10.3923i −0.597680 + 0.345071i −0.768128 0.640296i \(-0.778812\pi\)
0.170448 + 0.985367i \(0.445478\pi\)
\(908\) 0 0
\(909\) 13.8564i 0.459588i
\(910\) 0 0
\(911\) 41.5692i 1.37725i 0.725118 + 0.688625i \(0.241785\pi\)
−0.725118 + 0.688625i \(0.758215\pi\)
\(912\) 0 0
\(913\) −4.50000 + 2.59808i −0.148928 + 0.0859838i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −15.0000 8.66025i −0.494804 0.285675i 0.231761 0.972773i \(-0.425551\pi\)
−0.726565 + 0.687097i \(0.758885\pi\)
\(920\) 0 0
\(921\) 10.0000 + 17.3205i 0.329511 + 0.570730i
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 20.0000 0.657596
\(926\) 0 0
\(927\) 2.00000 + 3.46410i 0.0656886 + 0.113776i
\(928\) 0 0
\(929\) 15.0000 + 8.66025i 0.492134 + 0.284134i 0.725459 0.688265i \(-0.241627\pi\)
−0.233325 + 0.972399i \(0.574961\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 9.00000 15.5885i 0.294647 0.510343i
\(934\) 0 0
\(935\) −9.00000 + 5.19615i −0.294331 + 0.169932i
\(936\) 0 0
\(937\) 19.0526i 0.622420i 0.950341 + 0.311210i \(0.100734\pi\)
−0.950341 + 0.311210i \(0.899266\pi\)
\(938\) 0 0
\(939\) 5.19615i 0.169570i
\(940\) 0 0
\(941\) 34.5000 19.9186i 1.12467 0.649327i 0.182079 0.983284i \(-0.441717\pi\)
0.942588 + 0.333957i \(0.108384\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 33.0000 + 19.0526i 1.07236 + 0.619125i 0.928824 0.370521i \(-0.120821\pi\)
0.143532 + 0.989646i \(0.454154\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) −21.0000 −0.680972
\(952\) 0 0
\(953\) −18.0000 −0.583077 −0.291539 0.956559i \(-0.594167\pi\)
−0.291539 + 0.956559i \(0.594167\pi\)
\(954\) 0 0
\(955\) −12.0000 20.7846i −0.388311 0.672574i
\(956\) 0 0
\(957\) 13.5000 + 7.79423i 0.436393 + 0.251952i
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 3.00000 5.19615i 0.0967742 0.167618i
\(962\) 0 0
\(963\) −10.5000 + 6.06218i −0.338358 + 0.195351i
\(964\) 0 0
\(965\) 8.66025i 0.278783i
\(966\) 0 0
\(967\) 39.8372i 1.28108i 0.767926 + 0.640538i \(0.221289\pi\)
−0.767926 + 0.640538i \(0.778711\pi\)
\(968\) 0 0
\(969\) 6.00000 3.46410i 0.192748 0.111283i
\(970\) 0 0
\(971\) 16.5000 28.5788i 0.529510 0.917139i −0.469897 0.882721i \(-0.655709\pi\)
0.999408 0.0344175i \(-0.0109576\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 9.00000 + 15.5885i 0.287936 + 0.498719i 0.973317 0.229465i \(-0.0736978\pi\)
−0.685381 + 0.728184i \(0.740364\pi\)
\(978\) 0 0
\(979\) −6.00000 −0.191761
\(980\) 0 0
\(981\) −4.00000 −0.127710
\(982\) 0 0
\(983\) −9.00000 15.5885i −0.287055 0.497195i 0.686050 0.727554i \(-0.259343\pi\)
−0.973106 + 0.230360i \(0.926010\pi\)
\(984\) 0 0
\(985\) 9.00000 + 5.19615i 0.286764 + 0.165563i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) −10.5000 + 6.06218i −0.333543 + 0.192571i −0.657413 0.753530i \(-0.728349\pi\)
0.323870 + 0.946102i \(0.395016\pi\)
\(992\) 0 0
\(993\) 13.8564i 0.439720i
\(994\) 0 0
\(995\) 27.7128i 0.878555i
\(996\) 0 0
\(997\) −24.0000 + 13.8564i −0.760088 + 0.438837i −0.829327 0.558763i \(-0.811276\pi\)
0.0692396 + 0.997600i \(0.477943\pi\)
\(998\) 0 0
\(999\) −5.00000 + 8.66025i −0.158193 + 0.273998i
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 2352.2.bl.b.607.1 2
4.3 odd 2 2352.2.bl.h.607.1 2
7.2 even 3 2352.2.b.g.1567.2 2
7.3 odd 6 2352.2.bl.h.31.1 2
7.4 even 3 336.2.bl.c.31.1 2
7.5 odd 6 2352.2.b.c.1567.1 2
7.6 odd 2 336.2.bl.g.271.1 yes 2
21.2 odd 6 7056.2.b.e.1567.1 2
21.5 even 6 7056.2.b.i.1567.2 2
21.11 odd 6 1008.2.cs.e.703.1 2
21.20 even 2 1008.2.cs.d.271.1 2
28.3 even 6 inner 2352.2.bl.b.31.1 2
28.11 odd 6 336.2.bl.g.31.1 yes 2
28.19 even 6 2352.2.b.g.1567.1 2
28.23 odd 6 2352.2.b.c.1567.2 2
28.27 even 2 336.2.bl.c.271.1 yes 2
56.11 odd 6 1344.2.bl.b.703.1 2
56.13 odd 2 1344.2.bl.b.1279.1 2
56.27 even 2 1344.2.bl.f.1279.1 2
56.53 even 6 1344.2.bl.f.703.1 2
84.11 even 6 1008.2.cs.d.703.1 2
84.23 even 6 7056.2.b.i.1567.1 2
84.47 odd 6 7056.2.b.e.1567.2 2
84.83 odd 2 1008.2.cs.e.271.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
336.2.bl.c.31.1 2 7.4 even 3
336.2.bl.c.271.1 yes 2 28.27 even 2
336.2.bl.g.31.1 yes 2 28.11 odd 6
336.2.bl.g.271.1 yes 2 7.6 odd 2
1008.2.cs.d.271.1 2 21.20 even 2
1008.2.cs.d.703.1 2 84.11 even 6
1008.2.cs.e.271.1 2 84.83 odd 2
1008.2.cs.e.703.1 2 21.11 odd 6
1344.2.bl.b.703.1 2 56.11 odd 6
1344.2.bl.b.1279.1 2 56.13 odd 2
1344.2.bl.f.703.1 2 56.53 even 6
1344.2.bl.f.1279.1 2 56.27 even 2
2352.2.b.c.1567.1 2 7.5 odd 6
2352.2.b.c.1567.2 2 28.23 odd 6
2352.2.b.g.1567.1 2 28.19 even 6
2352.2.b.g.1567.2 2 7.2 even 3
2352.2.bl.b.31.1 2 28.3 even 6 inner
2352.2.bl.b.607.1 2 1.1 even 1 trivial
2352.2.bl.h.31.1 2 7.3 odd 6
2352.2.bl.h.607.1 2 4.3 odd 2
7056.2.b.e.1567.1 2 21.2 odd 6
7056.2.b.e.1567.2 2 84.47 odd 6
7056.2.b.i.1567.1 2 84.23 even 6
7056.2.b.i.1567.2 2 21.5 even 6