Properties

Label 2352.2.q.y.961.1
Level $2352$
Weight $2$
Character 2352.961
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(961,2352)]
 
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([0, 0, 0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2352.961");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2352 = 2^{4} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2352.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: \(18.7808145554\)
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 294)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 961.1
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 2352.961
Dual form 2352.2.q.y.1537.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.500000 + 0.866025i) q^{3} +(2.00000 - 3.46410i) q^{5} +(-0.500000 + 0.866025i) q^{9} +O(q^{10})\) \(q+(0.500000 + 0.866025i) q^{3} +(2.00000 - 3.46410i) q^{5} +(-0.500000 + 0.866025i) q^{9} +(-2.00000 - 3.46410i) q^{11} -4.00000 q^{13} +4.00000 q^{15} +(-2.00000 + 3.46410i) q^{19} +(-5.50000 - 9.52628i) q^{25} -1.00000 q^{27} +2.00000 q^{29} +(-4.00000 - 6.92820i) q^{31} +(2.00000 - 3.46410i) q^{33} +(3.00000 - 5.19615i) q^{37} +(-2.00000 - 3.46410i) q^{39} -4.00000 q^{43} +(2.00000 + 3.46410i) q^{45} +(4.00000 - 6.92820i) q^{47} +(5.00000 + 8.66025i) q^{53} -16.0000 q^{55} -4.00000 q^{57} +(-2.00000 - 3.46410i) q^{59} +(-2.00000 + 3.46410i) q^{61} +(-8.00000 + 13.8564i) q^{65} +(2.00000 + 3.46410i) q^{67} -8.00000 q^{71} +(-8.00000 - 13.8564i) q^{73} +(5.50000 - 9.52628i) q^{75} +(-4.00000 + 6.92820i) q^{79} +(-0.500000 - 0.866025i) q^{81} -12.0000 q^{83} +(1.00000 + 1.73205i) q^{87} +(4.00000 - 6.92820i) q^{89} +(4.00000 - 6.92820i) q^{93} +(8.00000 + 13.8564i) q^{95} -8.00000 q^{97} +4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{3} + 4 q^{5} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{3} + 4 q^{5} - q^{9} - 4 q^{11} - 8 q^{13} + 8 q^{15} - 4 q^{19} - 11 q^{25} - 2 q^{27} + 4 q^{29} - 8 q^{31} + 4 q^{33} + 6 q^{37} - 4 q^{39} - 8 q^{43} + 4 q^{45} + 8 q^{47} + 10 q^{53} - 32 q^{55} - 8 q^{57} - 4 q^{59} - 4 q^{61} - 16 q^{65} + 4 q^{67} - 16 q^{71} - 16 q^{73} + 11 q^{75} - 8 q^{79} - q^{81} - 24 q^{83} + 2 q^{87} + 8 q^{89} + 8 q^{93} + 16 q^{95} - 16 q^{97} + 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(785\) \(1471\) \(1765\) \(2257\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(e\left(\frac{1}{3}\right)\)

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
\(5\) 2.00000 3.46410i 0.894427 1.54919i 0.0599153 0.998203i \(-0.480917\pi\)
0.834512 0.550990i \(-0.185750\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) −0.500000 + 0.866025i −0.166667 + 0.288675i
\(10\) 0 0
\(11\) −2.00000 3.46410i −0.603023 1.04447i −0.992361 0.123371i \(-0.960630\pi\)
0.389338 0.921095i \(-0.372704\pi\)
\(12\) 0 0
\(13\) −4.00000 −1.10940 −0.554700 0.832050i \(-0.687167\pi\)
−0.554700 + 0.832050i \(0.687167\pi\)
\(14\) 0 0
\(15\) 4.00000 1.03280
\(16\) 0 0
\(17\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(18\) 0 0
\(19\) −2.00000 + 3.46410i −0.458831 + 0.794719i −0.998899 0.0469020i \(-0.985065\pi\)
0.540068 + 0.841621i \(0.318398\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(24\) 0 0
\(25\) −5.50000 9.52628i −1.10000 1.90526i
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 2.00000 0.371391 0.185695 0.982607i \(-0.440546\pi\)
0.185695 + 0.982607i \(0.440546\pi\)
\(30\) 0 0
\(31\) −4.00000 6.92820i −0.718421 1.24434i −0.961625 0.274367i \(-0.911532\pi\)
0.243204 0.969975i \(-0.421802\pi\)
\(32\) 0 0
\(33\) 2.00000 3.46410i 0.348155 0.603023i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 3.00000 5.19615i 0.493197 0.854242i −0.506772 0.862080i \(-0.669162\pi\)
0.999969 + 0.00783774i \(0.00249486\pi\)
\(38\) 0 0
\(39\) −2.00000 3.46410i −0.320256 0.554700i
\(40\) 0 0
\(41\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) −4.00000 −0.609994 −0.304997 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) 0 0
\(45\) 2.00000 + 3.46410i 0.298142 + 0.516398i
\(46\) 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
\(49\) 0 0
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 5.00000 + 8.66025i 0.686803 + 1.18958i 0.972867 + 0.231367i \(0.0743197\pi\)
−0.286064 + 0.958211i \(0.592347\pi\)
\(54\) 0 0
\(55\) −16.0000 −2.15744
\(56\) 0 0
\(57\) −4.00000 −0.529813
\(58\) 0 0
\(59\) −2.00000 3.46410i −0.260378 0.450988i 0.705965 0.708247i \(-0.250514\pi\)
−0.966342 + 0.257260i \(0.917180\pi\)
\(60\) 0 0
\(61\) −2.00000 + 3.46410i −0.256074 + 0.443533i −0.965187 0.261562i \(-0.915762\pi\)
0.709113 + 0.705095i \(0.249096\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −8.00000 + 13.8564i −0.992278 + 1.71868i
\(66\) 0 0
\(67\) 2.00000 + 3.46410i 0.244339 + 0.423207i 0.961946 0.273241i \(-0.0880957\pi\)
−0.717607 + 0.696449i \(0.754762\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −8.00000 −0.949425 −0.474713 0.880141i \(-0.657448\pi\)
−0.474713 + 0.880141i \(0.657448\pi\)
\(72\) 0 0
\(73\) −8.00000 13.8564i −0.936329 1.62177i −0.772246 0.635323i \(-0.780867\pi\)
−0.164083 0.986447i \(-0.552466\pi\)
\(74\) 0 0
\(75\) 5.50000 9.52628i 0.635085 1.10000i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −4.00000 + 6.92820i −0.450035 + 0.779484i −0.998388 0.0567635i \(-0.981922\pi\)
0.548352 + 0.836247i \(0.315255\pi\)
\(80\) 0 0
\(81\) −0.500000 0.866025i −0.0555556 0.0962250i
\(82\) 0 0
\(83\) −12.0000 −1.31717 −0.658586 0.752506i \(-0.728845\pi\)
−0.658586 + 0.752506i \(0.728845\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 1.00000 + 1.73205i 0.107211 + 0.185695i
\(88\) 0 0
\(89\) 4.00000 6.92820i 0.423999 0.734388i −0.572327 0.820025i \(-0.693959\pi\)
0.996326 + 0.0856373i \(0.0272926\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 4.00000 6.92820i 0.414781 0.718421i
\(94\) 0 0
\(95\) 8.00000 + 13.8564i 0.820783 + 1.42164i
\(96\) 0 0
\(97\) −8.00000 −0.812277 −0.406138 0.913812i \(-0.633125\pi\)
−0.406138 + 0.913812i \(0.633125\pi\)
\(98\) 0 0
\(99\) 4.00000 0.402015
\(100\) 0 0
\(101\) 2.00000 + 3.46410i 0.199007 + 0.344691i 0.948207 0.317653i \(-0.102895\pi\)
−0.749199 + 0.662344i \(0.769562\pi\)
\(102\) 0 0
\(103\) 4.00000 6.92820i 0.394132 0.682656i −0.598858 0.800855i \(-0.704379\pi\)
0.992990 + 0.118199i \(0.0377120\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 2.00000 3.46410i 0.193347 0.334887i −0.753010 0.658009i \(-0.771399\pi\)
0.946357 + 0.323122i \(0.104732\pi\)
\(108\) 0 0
\(109\) 7.00000 + 12.1244i 0.670478 + 1.16130i 0.977769 + 0.209687i \(0.0672444\pi\)
−0.307290 + 0.951616i \(0.599422\pi\)
\(110\) 0 0
\(111\) 6.00000 0.569495
\(112\) 0 0
\(113\) −14.0000 −1.31701 −0.658505 0.752577i \(-0.728811\pi\)
−0.658505 + 0.752577i \(0.728811\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 2.00000 3.46410i 0.184900 0.320256i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −2.50000 + 4.33013i −0.227273 + 0.393648i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −24.0000 −2.14663
\(126\) 0 0
\(127\) 16.0000 1.41977 0.709885 0.704317i \(-0.248747\pi\)
0.709885 + 0.704317i \(0.248747\pi\)
\(128\) 0 0
\(129\) −2.00000 3.46410i −0.176090 0.304997i
\(130\) 0 0
\(131\) 6.00000 10.3923i 0.524222 0.907980i −0.475380 0.879781i \(-0.657689\pi\)
0.999602 0.0281993i \(-0.00897729\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) −2.00000 + 3.46410i −0.172133 + 0.298142i
\(136\) 0 0
\(137\) 5.00000 + 8.66025i 0.427179 + 0.739895i 0.996621 0.0821359i \(-0.0261741\pi\)
−0.569442 + 0.822031i \(0.692841\pi\)
\(138\) 0 0
\(139\) 12.0000 1.01783 0.508913 0.860818i \(-0.330047\pi\)
0.508913 + 0.860818i \(0.330047\pi\)
\(140\) 0 0
\(141\) 8.00000 0.673722
\(142\) 0 0
\(143\) 8.00000 + 13.8564i 0.668994 + 1.15873i
\(144\) 0 0
\(145\) 4.00000 6.92820i 0.332182 0.575356i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 5.00000 8.66025i 0.409616 0.709476i −0.585231 0.810867i \(-0.698996\pi\)
0.994847 + 0.101391i \(0.0323294\pi\)
\(150\) 0 0
\(151\) −4.00000 6.92820i −0.325515 0.563809i 0.656101 0.754673i \(-0.272204\pi\)
−0.981617 + 0.190864i \(0.938871\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −32.0000 −2.57030
\(156\) 0 0
\(157\) 2.00000 + 3.46410i 0.159617 + 0.276465i 0.934731 0.355357i \(-0.115641\pi\)
−0.775113 + 0.631822i \(0.782307\pi\)
\(158\) 0 0
\(159\) −5.00000 + 8.66025i −0.396526 + 0.686803i
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 6.00000 10.3923i 0.469956 0.813988i −0.529454 0.848339i \(-0.677603\pi\)
0.999410 + 0.0343508i \(0.0109363\pi\)
\(164\) 0 0
\(165\) −8.00000 13.8564i −0.622799 1.07872i
\(166\) 0 0
\(167\) 8.00000 0.619059 0.309529 0.950890i \(-0.399829\pi\)
0.309529 + 0.950890i \(0.399829\pi\)
\(168\) 0 0
\(169\) 3.00000 0.230769
\(170\) 0 0
\(171\) −2.00000 3.46410i −0.152944 0.264906i
\(172\) 0 0
\(173\) −2.00000 + 3.46410i −0.152057 + 0.263371i −0.931984 0.362500i \(-0.881923\pi\)
0.779926 + 0.625871i \(0.215256\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 2.00000 3.46410i 0.150329 0.260378i
\(178\) 0 0
\(179\) 6.00000 + 10.3923i 0.448461 + 0.776757i 0.998286 0.0585225i \(-0.0186389\pi\)
−0.549825 + 0.835280i \(0.685306\pi\)
\(180\) 0 0
\(181\) −20.0000 −1.48659 −0.743294 0.668965i \(-0.766738\pi\)
−0.743294 + 0.668965i \(0.766738\pi\)
\(182\) 0 0
\(183\) −4.00000 −0.295689
\(184\) 0 0
\(185\) −12.0000 20.7846i −0.882258 1.52811i
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(192\) 0 0
\(193\) −1.00000 1.73205i −0.0719816 0.124676i 0.827788 0.561041i \(-0.189599\pi\)
−0.899770 + 0.436365i \(0.856266\pi\)
\(194\) 0 0
\(195\) −16.0000 −1.14578
\(196\) 0 0
\(197\) 6.00000 0.427482 0.213741 0.976890i \(-0.431435\pi\)
0.213741 + 0.976890i \(0.431435\pi\)
\(198\) 0 0
\(199\) −4.00000 6.92820i −0.283552 0.491127i 0.688705 0.725042i \(-0.258180\pi\)
−0.972257 + 0.233915i \(0.924846\pi\)
\(200\) 0 0
\(201\) −2.00000 + 3.46410i −0.141069 + 0.244339i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 16.0000 1.10674
\(210\) 0 0
\(211\) 28.0000 1.92760 0.963800 0.266627i \(-0.0859092\pi\)
0.963800 + 0.266627i \(0.0859092\pi\)
\(212\) 0 0
\(213\) −4.00000 6.92820i −0.274075 0.474713i
\(214\) 0 0
\(215\) −8.00000 + 13.8564i −0.545595 + 0.944999i
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 8.00000 13.8564i 0.540590 0.936329i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −16.0000 −1.07144 −0.535720 0.844396i \(-0.679960\pi\)
−0.535720 + 0.844396i \(0.679960\pi\)
\(224\) 0 0
\(225\) 11.0000 0.733333
\(226\) 0 0
\(227\) −10.0000 17.3205i −0.663723 1.14960i −0.979630 0.200812i \(-0.935642\pi\)
0.315906 0.948790i \(-0.397691\pi\)
\(228\) 0 0
\(229\) 2.00000 3.46410i 0.132164 0.228914i −0.792347 0.610071i \(-0.791141\pi\)
0.924510 + 0.381157i \(0.124474\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 5.00000 8.66025i 0.327561 0.567352i −0.654466 0.756091i \(-0.727107\pi\)
0.982027 + 0.188739i \(0.0604400\pi\)
\(234\) 0 0
\(235\) −16.0000 27.7128i −1.04372 1.80778i
\(236\) 0 0
\(237\) −8.00000 −0.519656
\(238\) 0 0
\(239\) 24.0000 1.55243 0.776215 0.630468i \(-0.217137\pi\)
0.776215 + 0.630468i \(0.217137\pi\)
\(240\) 0 0
\(241\) 4.00000 + 6.92820i 0.257663 + 0.446285i 0.965615 0.259975i \(-0.0837143\pi\)
−0.707953 + 0.706260i \(0.750381\pi\)
\(242\) 0 0
\(243\) 0.500000 0.866025i 0.0320750 0.0555556i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 8.00000 13.8564i 0.509028 0.881662i
\(248\) 0 0
\(249\) −6.00000 10.3923i −0.380235 0.658586i
\(250\) 0 0
\(251\) −20.0000 −1.26239 −0.631194 0.775625i \(-0.717435\pi\)
−0.631194 + 0.775625i \(0.717435\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 4.00000 6.92820i 0.249513 0.432169i −0.713878 0.700270i \(-0.753063\pi\)
0.963391 + 0.268101i \(0.0863961\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −1.00000 + 1.73205i −0.0618984 + 0.107211i
\(262\) 0 0
\(263\) 4.00000 + 6.92820i 0.246651 + 0.427211i 0.962594 0.270947i \(-0.0873367\pi\)
−0.715944 + 0.698158i \(0.754003\pi\)
\(264\) 0 0
\(265\) 40.0000 2.45718
\(266\) 0 0
\(267\) 8.00000 0.489592
\(268\) 0 0
\(269\) 14.0000 + 24.2487i 0.853595 + 1.47847i 0.877942 + 0.478766i \(0.158916\pi\)
−0.0243472 + 0.999704i \(0.507751\pi\)
\(270\) 0 0
\(271\) 16.0000 27.7128i 0.971931 1.68343i 0.282218 0.959350i \(-0.408930\pi\)
0.689713 0.724083i \(-0.257737\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −22.0000 + 38.1051i −1.32665 + 2.29783i
\(276\) 0 0
\(277\) −11.0000 19.0526i −0.660926 1.14476i −0.980373 0.197153i \(-0.936830\pi\)
0.319447 0.947604i \(-0.396503\pi\)
\(278\) 0 0
\(279\) 8.00000 0.478947
\(280\) 0 0
\(281\) 6.00000 0.357930 0.178965 0.983855i \(-0.442725\pi\)
0.178965 + 0.983855i \(0.442725\pi\)
\(282\) 0 0
\(283\) 14.0000 + 24.2487i 0.832214 + 1.44144i 0.896279 + 0.443491i \(0.146260\pi\)
−0.0640654 + 0.997946i \(0.520407\pi\)
\(284\) 0 0
\(285\) −8.00000 + 13.8564i −0.473879 + 0.820783i
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 8.50000 14.7224i 0.500000 0.866025i
\(290\) 0 0
\(291\) −4.00000 6.92820i −0.234484 0.406138i
\(292\) 0 0
\(293\) 12.0000 0.701047 0.350524 0.936554i \(-0.386004\pi\)
0.350524 + 0.936554i \(0.386004\pi\)
\(294\) 0 0
\(295\) −16.0000 −0.931556
\(296\) 0 0
\(297\) 2.00000 + 3.46410i 0.116052 + 0.201008i
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) −2.00000 + 3.46410i −0.114897 + 0.199007i
\(304\) 0 0
\(305\) 8.00000 + 13.8564i 0.458079 + 0.793416i
\(306\) 0 0
\(307\) 20.0000 1.14146 0.570730 0.821138i \(-0.306660\pi\)
0.570730 + 0.821138i \(0.306660\pi\)
\(308\) 0 0
\(309\) 8.00000 0.455104
\(310\) 0 0
\(311\) −16.0000 27.7128i −0.907277 1.57145i −0.817832 0.575458i \(-0.804824\pi\)
−0.0894452 0.995992i \(-0.528509\pi\)
\(312\) 0 0
\(313\) 12.0000 20.7846i 0.678280 1.17482i −0.297218 0.954810i \(-0.596059\pi\)
0.975499 0.220006i \(-0.0706077\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −15.0000 + 25.9808i −0.842484 + 1.45922i 0.0453045 + 0.998973i \(0.485574\pi\)
−0.887788 + 0.460252i \(0.847759\pi\)
\(318\) 0 0
\(319\) −4.00000 6.92820i −0.223957 0.387905i
\(320\) 0 0
\(321\) 4.00000 0.223258
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 22.0000 + 38.1051i 1.22034 + 2.11369i
\(326\) 0 0
\(327\) −7.00000 + 12.1244i −0.387101 + 0.670478i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 10.0000 17.3205i 0.549650 0.952021i −0.448649 0.893708i \(-0.648095\pi\)
0.998298 0.0583130i \(-0.0185721\pi\)
\(332\) 0 0
\(333\) 3.00000 + 5.19615i 0.164399 + 0.284747i
\(334\) 0 0
\(335\) 16.0000 0.874173
\(336\) 0 0
\(337\) 14.0000 0.762629 0.381314 0.924445i \(-0.375472\pi\)
0.381314 + 0.924445i \(0.375472\pi\)
\(338\) 0 0
\(339\) −7.00000 12.1244i −0.380188 0.658505i
\(340\) 0 0
\(341\) −16.0000 + 27.7128i −0.866449 + 1.50073i
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 6.00000 + 10.3923i 0.322097 + 0.557888i 0.980921 0.194409i \(-0.0622790\pi\)
−0.658824 + 0.752297i \(0.728946\pi\)
\(348\) 0 0
\(349\) 12.0000 0.642345 0.321173 0.947021i \(-0.395923\pi\)
0.321173 + 0.947021i \(0.395923\pi\)
\(350\) 0 0
\(351\) 4.00000 0.213504
\(352\) 0 0
\(353\) −12.0000 20.7846i −0.638696 1.10625i −0.985719 0.168397i \(-0.946141\pi\)
0.347024 0.937856i \(-0.387192\pi\)
\(354\) 0 0
\(355\) −16.0000 + 27.7128i −0.849192 + 1.47084i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 8.00000 13.8564i 0.422224 0.731313i −0.573933 0.818902i \(-0.694583\pi\)
0.996157 + 0.0875892i \(0.0279163\pi\)
\(360\) 0 0
\(361\) 1.50000 + 2.59808i 0.0789474 + 0.136741i
\(362\) 0 0
\(363\) −5.00000 −0.262432
\(364\) 0 0
\(365\) −64.0000 −3.34991
\(366\) 0 0
\(367\) 8.00000 + 13.8564i 0.417597 + 0.723299i 0.995697 0.0926670i \(-0.0295392\pi\)
−0.578101 + 0.815966i \(0.696206\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −11.0000 + 19.0526i −0.569558 + 0.986504i 0.427051 + 0.904227i \(0.359552\pi\)
−0.996610 + 0.0822766i \(0.973781\pi\)
\(374\) 0 0
\(375\) −12.0000 20.7846i −0.619677 1.07331i
\(376\) 0 0
\(377\) −8.00000 −0.412021
\(378\) 0 0
\(379\) −36.0000 −1.84920 −0.924598 0.380945i \(-0.875599\pi\)
−0.924598 + 0.380945i \(0.875599\pi\)
\(380\) 0 0
\(381\) 8.00000 + 13.8564i 0.409852 + 0.709885i
\(382\) 0 0
\(383\) −12.0000 + 20.7846i −0.613171 + 1.06204i 0.377531 + 0.925997i \(0.376773\pi\)
−0.990702 + 0.136047i \(0.956560\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 2.00000 3.46410i 0.101666 0.176090i
\(388\) 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
\(393\) 12.0000 0.605320
\(394\) 0 0
\(395\) 16.0000 + 27.7128i 0.805047 + 1.39438i
\(396\) 0 0
\(397\) 2.00000 3.46410i 0.100377 0.173858i −0.811463 0.584404i \(-0.801328\pi\)
0.911840 + 0.410546i \(0.134662\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 9.00000 15.5885i 0.449439 0.778450i −0.548911 0.835881i \(-0.684957\pi\)
0.998350 + 0.0574304i \(0.0182907\pi\)
\(402\) 0 0
\(403\) 16.0000 + 27.7128i 0.797017 + 1.38047i
\(404\) 0 0
\(405\) −4.00000 −0.198762
\(406\) 0 0
\(407\) −24.0000 −1.18964
\(408\) 0 0
\(409\) −12.0000 20.7846i −0.593362 1.02773i −0.993776 0.111398i \(-0.964467\pi\)
0.400414 0.916334i \(-0.368866\pi\)
\(410\) 0 0
\(411\) −5.00000 + 8.66025i −0.246632 + 0.427179i
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −24.0000 + 41.5692i −1.17811 + 2.04055i
\(416\) 0 0
\(417\) 6.00000 + 10.3923i 0.293821 + 0.508913i
\(418\) 0 0
\(419\) 28.0000 1.36789 0.683945 0.729534i \(-0.260263\pi\)
0.683945 + 0.729534i \(0.260263\pi\)
\(420\) 0 0
\(421\) 6.00000 0.292422 0.146211 0.989253i \(-0.453292\pi\)
0.146211 + 0.989253i \(0.453292\pi\)
\(422\) 0 0
\(423\) 4.00000 + 6.92820i 0.194487 + 0.336861i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) −8.00000 + 13.8564i −0.386244 + 0.668994i
\(430\) 0 0
\(431\) 20.0000 + 34.6410i 0.963366 + 1.66860i 0.713942 + 0.700205i \(0.246908\pi\)
0.249424 + 0.968394i \(0.419759\pi\)
\(432\) 0 0
\(433\) 8.00000 0.384455 0.192228 0.981350i \(-0.438429\pi\)
0.192228 + 0.981350i \(0.438429\pi\)
\(434\) 0 0
\(435\) 8.00000 0.383571
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −6.00000 + 10.3923i −0.285069 + 0.493753i −0.972626 0.232377i \(-0.925350\pi\)
0.687557 + 0.726130i \(0.258683\pi\)
\(444\) 0 0
\(445\) −16.0000 27.7128i −0.758473 1.31371i
\(446\) 0 0
\(447\) 10.0000 0.472984
\(448\) 0 0
\(449\) −30.0000 −1.41579 −0.707894 0.706319i \(-0.750354\pi\)
−0.707894 + 0.706319i \(0.750354\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 4.00000 6.92820i 0.187936 0.325515i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 11.0000 19.0526i 0.514558 0.891241i −0.485299 0.874348i \(-0.661289\pi\)
0.999857 0.0168929i \(-0.00537742\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −12.0000 −0.558896 −0.279448 0.960161i \(-0.590151\pi\)
−0.279448 + 0.960161i \(0.590151\pi\)
\(462\) 0 0
\(463\) −8.00000 −0.371792 −0.185896 0.982569i \(-0.559519\pi\)
−0.185896 + 0.982569i \(0.559519\pi\)
\(464\) 0 0
\(465\) −16.0000 27.7128i −0.741982 1.28515i
\(466\) 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
\(469\) 0 0
\(470\) 0 0
\(471\) −2.00000 + 3.46410i −0.0921551 + 0.159617i
\(472\) 0 0
\(473\) 8.00000 + 13.8564i 0.367840 + 0.637118i
\(474\) 0 0
\(475\) 44.0000 2.01886
\(476\) 0 0
\(477\) −10.0000 −0.457869
\(478\) 0 0
\(479\) 12.0000 + 20.7846i 0.548294 + 0.949673i 0.998392 + 0.0566937i \(0.0180558\pi\)
−0.450098 + 0.892979i \(0.648611\pi\)
\(480\) 0 0
\(481\) −12.0000 + 20.7846i −0.547153 + 0.947697i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −16.0000 + 27.7128i −0.726523 + 1.25837i
\(486\) 0 0
\(487\) −4.00000 6.92820i −0.181257 0.313947i 0.761052 0.648691i \(-0.224683\pi\)
−0.942309 + 0.334744i \(0.891350\pi\)
\(488\) 0 0
\(489\) 12.0000 0.542659
\(490\) 0 0
\(491\) 12.0000 0.541552 0.270776 0.962642i \(-0.412720\pi\)
0.270776 + 0.962642i \(0.412720\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 8.00000 13.8564i 0.359573 0.622799i
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −6.00000 + 10.3923i −0.268597 + 0.465223i −0.968500 0.249015i \(-0.919893\pi\)
0.699903 + 0.714238i \(0.253227\pi\)
\(500\) 0 0
\(501\) 4.00000 + 6.92820i 0.178707 + 0.309529i
\(502\) 0 0
\(503\) −16.0000 −0.713405 −0.356702 0.934218i \(-0.616099\pi\)
−0.356702 + 0.934218i \(0.616099\pi\)
\(504\) 0 0
\(505\) 16.0000 0.711991
\(506\) 0 0
\(507\) 1.50000 + 2.59808i 0.0666173 + 0.115385i
\(508\) 0 0
\(509\) −2.00000 + 3.46410i −0.0886484 + 0.153544i −0.906940 0.421260i \(-0.861588\pi\)
0.818292 + 0.574803i \(0.194921\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 2.00000 3.46410i 0.0883022 0.152944i
\(514\) 0 0
\(515\) −16.0000 27.7128i −0.705044 1.22117i
\(516\) 0 0
\(517\) −32.0000 −1.40736
\(518\) 0 0
\(519\) −4.00000 −0.175581
\(520\) 0 0
\(521\) 16.0000 + 27.7128i 0.700973 + 1.21412i 0.968125 + 0.250466i \(0.0805839\pi\)
−0.267153 + 0.963654i \(0.586083\pi\)
\(522\) 0 0
\(523\) −2.00000 + 3.46410i −0.0874539 + 0.151475i −0.906434 0.422347i \(-0.861206\pi\)
0.818980 + 0.573822i \(0.194540\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 11.5000 + 19.9186i 0.500000 + 0.866025i
\(530\) 0 0
\(531\) 4.00000 0.173585
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) −8.00000 13.8564i −0.345870 0.599065i
\(536\) 0 0
\(537\) −6.00000 + 10.3923i −0.258919 + 0.448461i
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 1.00000 1.73205i 0.0429934 0.0744667i −0.843728 0.536771i \(-0.819644\pi\)
0.886721 + 0.462304i \(0.152977\pi\)
\(542\) 0 0
\(543\) −10.0000 17.3205i −0.429141 0.743294i
\(544\) 0 0
\(545\) 56.0000 2.39878
\(546\) 0 0
\(547\) 20.0000 0.855138 0.427569 0.903983i \(-0.359370\pi\)
0.427569 + 0.903983i \(0.359370\pi\)
\(548\) 0 0
\(549\) −2.00000 3.46410i −0.0853579 0.147844i
\(550\) 0 0
\(551\) −4.00000 + 6.92820i −0.170406 + 0.295151i
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 12.0000 20.7846i 0.509372 0.882258i
\(556\) 0 0
\(557\) 9.00000 + 15.5885i 0.381342 + 0.660504i 0.991254 0.131965i \(-0.0421286\pi\)
−0.609912 + 0.792469i \(0.708795\pi\)
\(558\) 0 0
\(559\) 16.0000 0.676728
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 2.00000 + 3.46410i 0.0842900 + 0.145994i 0.905088 0.425223i \(-0.139804\pi\)
−0.820798 + 0.571218i \(0.806471\pi\)
\(564\) 0 0
\(565\) −28.0000 + 48.4974i −1.17797 + 2.04030i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −3.00000 + 5.19615i −0.125767 + 0.217834i −0.922032 0.387113i \(-0.873472\pi\)
0.796266 + 0.604947i \(0.206806\pi\)
\(570\) 0 0
\(571\) 18.0000 + 31.1769i 0.753277 + 1.30471i 0.946227 + 0.323505i \(0.104861\pi\)
−0.192950 + 0.981209i \(0.561806\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −8.00000 13.8564i −0.333044 0.576850i 0.650063 0.759880i \(-0.274743\pi\)
−0.983107 + 0.183031i \(0.941409\pi\)
\(578\) 0 0
\(579\) 1.00000 1.73205i 0.0415586 0.0719816i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 20.0000 34.6410i 0.828315 1.43468i
\(584\) 0 0
\(585\) −8.00000 13.8564i −0.330759 0.572892i
\(586\) 0 0
\(587\) 28.0000 1.15568 0.577842 0.816149i \(-0.303895\pi\)
0.577842 + 0.816149i \(0.303895\pi\)
\(588\) 0 0
\(589\) 32.0000 1.31854
\(590\) 0 0
\(591\) 3.00000 + 5.19615i 0.123404 + 0.213741i
\(592\) 0 0
\(593\) −12.0000 + 20.7846i −0.492781 + 0.853522i −0.999965 0.00831589i \(-0.997353\pi\)
0.507184 + 0.861838i \(0.330686\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 4.00000 6.92820i 0.163709 0.283552i
\(598\) 0 0
\(599\) 12.0000 + 20.7846i 0.490307 + 0.849236i 0.999938 0.0111569i \(-0.00355143\pi\)
−0.509631 + 0.860393i \(0.670218\pi\)
\(600\) 0 0
\(601\) 32.0000 1.30531 0.652654 0.757656i \(-0.273656\pi\)
0.652654 + 0.757656i \(0.273656\pi\)
\(602\) 0 0
\(603\) −4.00000 −0.162893
\(604\) 0 0
\(605\) 10.0000 + 17.3205i 0.406558 + 0.704179i
\(606\) 0 0
\(607\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −16.0000 + 27.7128i −0.647291 + 1.12114i
\(612\) 0 0
\(613\) −13.0000 22.5167i −0.525065 0.909439i −0.999574 0.0291886i \(-0.990708\pi\)
0.474509 0.880251i \(-0.342626\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 22.0000 0.885687 0.442843 0.896599i \(-0.353970\pi\)
0.442843 + 0.896599i \(0.353970\pi\)
\(618\) 0 0
\(619\) −10.0000 17.3205i −0.401934 0.696170i 0.592025 0.805919i \(-0.298329\pi\)
−0.993959 + 0.109749i \(0.964995\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −20.5000 + 35.5070i −0.820000 + 1.42028i
\(626\) 0 0
\(627\) 8.00000 + 13.8564i 0.319489 + 0.553372i
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(632\) 0 0
\(633\) 14.0000 + 24.2487i 0.556450 + 0.963800i
\(634\) 0 0
\(635\) 32.0000 55.4256i 1.26988 2.19950i
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 4.00000 6.92820i 0.158238 0.274075i
\(640\) 0 0
\(641\) 1.00000 + 1.73205i 0.0394976 + 0.0684119i 0.885098 0.465404i \(-0.154091\pi\)
−0.845601 + 0.533816i \(0.820758\pi\)
\(642\) 0 0
\(643\) −36.0000 −1.41970 −0.709851 0.704352i \(-0.751238\pi\)
−0.709851 + 0.704352i \(0.751238\pi\)
\(644\) 0 0
\(645\) −16.0000 −0.629999
\(646\) 0 0
\(647\) 12.0000 + 20.7846i 0.471769 + 0.817127i 0.999478 0.0322975i \(-0.0102824\pi\)
−0.527710 + 0.849425i \(0.676949\pi\)
\(648\) 0 0
\(649\) −8.00000 + 13.8564i −0.314027 + 0.543912i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 23.0000 39.8372i 0.900060 1.55895i 0.0726446 0.997358i \(-0.476856\pi\)
0.827415 0.561591i \(-0.189811\pi\)
\(654\) 0 0
\(655\) −24.0000 41.5692i −0.937758 1.62424i
\(656\) 0 0
\(657\) 16.0000 0.624219
\(658\) 0 0
\(659\) −4.00000 −0.155818 −0.0779089 0.996960i \(-0.524824\pi\)
−0.0779089 + 0.996960i \(0.524824\pi\)
\(660\) 0 0
\(661\) 14.0000 + 24.2487i 0.544537 + 0.943166i 0.998636 + 0.0522143i \(0.0166279\pi\)
−0.454099 + 0.890951i \(0.650039\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) −8.00000 13.8564i −0.309298 0.535720i
\(670\) 0 0
\(671\) 16.0000 0.617673
\(672\) 0 0
\(673\) −34.0000 −1.31060 −0.655302 0.755367i \(-0.727459\pi\)
−0.655302 + 0.755367i \(0.727459\pi\)
\(674\) 0 0
\(675\) 5.50000 + 9.52628i 0.211695 + 0.366667i
\(676\) 0 0
\(677\) 6.00000 10.3923i 0.230599 0.399409i −0.727386 0.686229i \(-0.759265\pi\)
0.957984 + 0.286820i \(0.0925982\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 10.0000 17.3205i 0.383201 0.663723i
\(682\) 0 0
\(683\) −22.0000 38.1051i −0.841807 1.45805i −0.888366 0.459136i \(-0.848159\pi\)
0.0465592 0.998916i \(-0.485174\pi\)
\(684\) 0 0
\(685\) 40.0000 1.52832
\(686\) 0 0
\(687\) 4.00000 0.152610
\(688\) 0 0
\(689\) −20.0000 34.6410i −0.761939 1.31972i
\(690\) 0 0
\(691\) −10.0000 + 17.3205i −0.380418 + 0.658903i −0.991122 0.132956i \(-0.957553\pi\)
0.610704 + 0.791859i \(0.290887\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 24.0000 41.5692i 0.910372 1.57681i
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 10.0000 0.378235
\(700\) 0 0
\(701\) 34.0000 1.28416 0.642081 0.766637i \(-0.278071\pi\)
0.642081 + 0.766637i \(0.278071\pi\)
\(702\) 0 0
\(703\) 12.0000 + 20.7846i 0.452589 + 0.783906i
\(704\) 0 0
\(705\) 16.0000 27.7128i 0.602595 1.04372i
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 19.0000 32.9090i 0.713560 1.23592i −0.249952 0.968258i \(-0.580415\pi\)
0.963512 0.267664i \(-0.0862517\pi\)
\(710\) 0 0
\(711\) −4.00000 6.92820i −0.150012 0.259828i
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 64.0000 2.39346
\(716\) 0 0
\(717\) 12.0000 + 20.7846i 0.448148 + 0.776215i
\(718\) 0 0
\(719\) 12.0000 20.7846i 0.447524 0.775135i −0.550700 0.834703i \(-0.685639\pi\)
0.998224 + 0.0595683i \(0.0189724\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) −4.00000 + 6.92820i −0.148762 + 0.257663i
\(724\) 0 0
\(725\) −11.0000 19.0526i −0.408530 0.707594i
\(726\) 0 0
\(727\) −8.00000 −0.296704 −0.148352 0.988935i \(-0.547397\pi\)
−0.148352 + 0.988935i \(0.547397\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) −2.00000 + 3.46410i −0.0738717 + 0.127950i −0.900595 0.434659i \(-0.856869\pi\)
0.826723 + 0.562609i \(0.190202\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 8.00000 13.8564i 0.294684 0.510407i
\(738\) 0 0
\(739\) −10.0000 17.3205i −0.367856 0.637145i 0.621374 0.783514i \(-0.286575\pi\)
−0.989230 + 0.146369i \(0.953241\pi\)
\(740\) 0 0
\(741\) 16.0000 0.587775
\(742\) 0 0
\(743\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(744\) 0 0
\(745\) −20.0000 34.6410i −0.732743 1.26915i
\(746\) 0 0
\(747\) 6.00000 10.3923i 0.219529 0.380235i
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(752\) 0 0
\(753\) −10.0000 17.3205i −0.364420 0.631194i
\(754\) 0 0
\(755\) −32.0000 −1.16460
\(756\) 0 0
\(757\) 10.0000 0.363456 0.181728 0.983349i \(-0.441831\pi\)
0.181728 + 0.983349i \(0.441831\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 8.00000 13.8564i 0.290000 0.502294i −0.683810 0.729661i \(-0.739678\pi\)
0.973809 + 0.227366i \(0.0730114\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 8.00000 + 13.8564i 0.288863 + 0.500326i
\(768\) 0 0
\(769\) −40.0000 −1.44244 −0.721218 0.692708i \(-0.756418\pi\)
−0.721218 + 0.692708i \(0.756418\pi\)
\(770\) 0 0
\(771\) 8.00000 0.288113
\(772\) 0 0
\(773\) 18.0000 + 31.1769i 0.647415 + 1.12136i 0.983738 + 0.179609i \(0.0574833\pi\)
−0.336323 + 0.941747i \(0.609183\pi\)
\(774\) 0 0
\(775\) −44.0000 + 76.2102i −1.58053 + 2.73755i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 16.0000 + 27.7128i 0.572525 + 0.991642i
\(782\) 0 0
\(783\) −2.00000 −0.0714742
\(784\) 0 0
\(785\) 16.0000 0.571064
\(786\) 0 0
\(787\) −10.0000 17.3205i −0.356462 0.617409i 0.630905 0.775860i \(-0.282684\pi\)
−0.987367 + 0.158450i \(0.949350\pi\)
\(788\) 0 0
\(789\) −4.00000 + 6.92820i −0.142404 + 0.246651i
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 8.00000 13.8564i 0.284088 0.492055i
\(794\) 0 0
\(795\) 20.0000 + 34.6410i 0.709327 + 1.22859i
\(796\) 0 0
\(797\) −12.0000 −0.425062 −0.212531 0.977154i \(-0.568171\pi\)
−0.212531 + 0.977154i \(0.568171\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 4.00000 + 6.92820i 0.141333 + 0.244796i
\(802\) 0 0
\(803\) −32.0000 + 55.4256i −1.12926 + 1.95593i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −14.0000 + 24.2487i −0.492823 + 0.853595i
\(808\) 0 0
\(809\) −21.0000 36.3731i −0.738321 1.27881i −0.953251 0.302180i \(-0.902286\pi\)
0.214930 0.976629i \(-0.431048\pi\)
\(810\) 0 0
\(811\) −44.0000 −1.54505 −0.772524 0.634985i \(-0.781006\pi\)
−0.772524 + 0.634985i \(0.781006\pi\)
\(812\) 0 0
\(813\) 32.0000 1.12229
\(814\) 0 0
\(815\) −24.0000 41.5692i −0.840683 1.45611i
\(816\) 0 0
\(817\) 8.00000 13.8564i 0.279885 0.484774i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 5.00000 8.66025i 0.174501 0.302245i −0.765487 0.643451i \(-0.777502\pi\)
0.939989 + 0.341206i \(0.110835\pi\)
\(822\) 0 0
\(823\) 8.00000 + 13.8564i 0.278862 + 0.483004i 0.971102 0.238664i \(-0.0767093\pi\)
−0.692240 + 0.721668i \(0.743376\pi\)
\(824\) 0 0
\(825\) −44.0000 −1.53188
\(826\) 0 0
\(827\) −36.0000 −1.25184 −0.625921 0.779886i \(-0.715277\pi\)
−0.625921 + 0.779886i \(0.715277\pi\)
\(828\) 0 0
\(829\) −26.0000 45.0333i −0.903017 1.56407i −0.823557 0.567234i \(-0.808014\pi\)
−0.0794606 0.996838i \(-0.525320\pi\)
\(830\) 0 0
\(831\) 11.0000 19.0526i 0.381586 0.660926i
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 16.0000 27.7128i 0.553703 0.959041i
\(836\) 0 0
\(837\) 4.00000 + 6.92820i 0.138260 + 0.239474i
\(838\) 0 0
\(839\) 24.0000 0.828572 0.414286 0.910147i \(-0.364031\pi\)
0.414286 + 0.910147i \(0.364031\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) 0 0
\(843\) 3.00000 + 5.19615i 0.103325 + 0.178965i
\(844\) 0 0
\(845\) 6.00000 10.3923i 0.206406 0.357506i
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −14.0000 + 24.2487i −0.480479 + 0.832214i
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) −52.0000 −1.78045 −0.890223 0.455525i \(-0.849452\pi\)
−0.890223 + 0.455525i \(0.849452\pi\)
\(854\) 0 0
\(855\) −16.0000 −0.547188
\(856\) 0 0
\(857\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(858\) 0 0
\(859\) 18.0000 31.1769i 0.614152 1.06374i −0.376381 0.926465i \(-0.622831\pi\)
0.990533 0.137277i \(-0.0438352\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 24.0000 41.5692i 0.816970 1.41503i −0.0909355 0.995857i \(-0.528986\pi\)
0.907905 0.419176i \(-0.137681\pi\)
\(864\) 0 0
\(865\) 8.00000 + 13.8564i 0.272008 + 0.471132i
\(866\) 0 0
\(867\) 17.0000 0.577350
\(868\) 0 0
\(869\) 32.0000 1.08553
\(870\) 0 0
\(871\) −8.00000 13.8564i −0.271070 0.469506i
\(872\) 0 0
\(873\) 4.00000 6.92820i 0.135379 0.234484i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −9.00000 + 15.5885i −0.303908 + 0.526385i −0.977018 0.213158i \(-0.931625\pi\)
0.673109 + 0.739543i \(0.264958\pi\)
\(878\) 0 0
\(879\) 6.00000 + 10.3923i 0.202375 + 0.350524i
\(880\) 0 0
\(881\) 8.00000 0.269527 0.134763 0.990878i \(-0.456973\pi\)
0.134763 + 0.990878i \(0.456973\pi\)
\(882\) 0 0
\(883\) −4.00000 −0.134611 −0.0673054 0.997732i \(-0.521440\pi\)
−0.0673054 + 0.997732i \(0.521440\pi\)
\(884\) 0 0
\(885\) −8.00000 13.8564i −0.268917 0.465778i
\(886\) 0 0
\(887\) −12.0000 + 20.7846i −0.402921 + 0.697879i −0.994077 0.108678i \(-0.965338\pi\)
0.591156 + 0.806557i \(0.298672\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −2.00000 + 3.46410i −0.0670025 + 0.116052i
\(892\) 0 0
\(893\) 16.0000 + 27.7128i 0.535420 + 0.927374i
\(894\) 0 0
\(895\) 48.0000 1.60446
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −8.00000 13.8564i −0.266815 0.462137i
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −40.0000 + 69.2820i −1.32964 + 2.30301i
\(906\) 0 0
\(907\) −10.0000 17.3205i −0.332045 0.575118i 0.650868 0.759191i \(-0.274405\pi\)
−0.982913 + 0.184073i \(0.941072\pi\)
\(908\) 0 0
\(909\) −4.00000 −0.132672
\(910\) 0 0
\(911\) 8.00000 0.265052 0.132526 0.991180i \(-0.457691\pi\)
0.132526 + 0.991180i \(0.457691\pi\)
\(912\) 0 0
\(913\) 24.0000 + 41.5692i 0.794284 + 1.37574i
\(914\) 0 0
\(915\) −8.00000 + 13.8564i −0.264472 + 0.458079i
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −24.0000 + 41.5692i −0.791687 + 1.37124i 0.133235 + 0.991084i \(0.457464\pi\)
−0.924922 + 0.380158i \(0.875870\pi\)
\(920\) 0 0
\(921\) 10.0000 + 17.3205i 0.329511 + 0.570730i
\(922\) 0 0
\(923\) 32.0000 1.05329
\(924\) 0 0
\(925\) −66.0000 −2.17007
\(926\) 0 0
\(927\) 4.00000 + 6.92820i 0.131377 + 0.227552i
\(928\) 0 0
\(929\) 24.0000 41.5692i 0.787414 1.36384i −0.140132 0.990133i \(-0.544753\pi\)
0.927546 0.373709i \(-0.121914\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 16.0000 27.7128i 0.523816 0.907277i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(938\) 0 0
\(939\) 24.0000 0.783210
\(940\) 0 0
\(941\) −6.00000 10.3923i −0.195594 0.338779i 0.751501 0.659732i \(-0.229330\pi\)
−0.947095 + 0.320953i \(0.895997\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −14.0000 + 24.2487i −0.454939 + 0.787977i −0.998685 0.0512727i \(-0.983672\pi\)
0.543746 + 0.839250i \(0.317006\pi\)
\(948\) 0 0
\(949\) 32.0000 + 55.4256i 1.03876 + 1.79919i
\(950\) 0 0
\(951\) −30.0000 −0.972817
\(952\) 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
\(957\) 4.00000 6.92820i 0.129302 0.223957i
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −16.5000 + 28.5788i −0.532258 + 0.921898i
\(962\) 0 0
\(963\) 2.00000 + 3.46410i 0.0644491 + 0.111629i
\(964\) 0 0
\(965\) −8.00000 −0.257529
\(966\) 0 0
\(967\) 8.00000 0.257263 0.128631 0.991692i \(-0.458942\pi\)
0.128631 + 0.991692i \(0.458942\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −2.00000 + 3.46410i −0.0641831 + 0.111168i −0.896331 0.443385i \(-0.853777\pi\)
0.832148 + 0.554553i \(0.187111\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) −22.0000 + 38.1051i −0.704564 + 1.22034i
\(976\) 0 0
\(977\) −7.00000 12.1244i −0.223950 0.387893i 0.732054 0.681247i \(-0.238562\pi\)
−0.956004 + 0.293354i \(0.905229\pi\)
\(978\) 0 0
\(979\) −32.0000 −1.02272
\(980\) 0 0
\(981\) −14.0000 −0.446986
\(982\) 0 0
\(983\) 4.00000 + 6.92820i 0.127580 + 0.220975i 0.922739 0.385426i \(-0.125946\pi\)
−0.795158 + 0.606402i \(0.792612\pi\)
\(984\) 0 0
\(985\) 12.0000 20.7846i 0.382352 0.662253i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 4.00000 + 6.92820i 0.127064 + 0.220082i 0.922538 0.385906i \(-0.126111\pi\)
−0.795474 + 0.605988i \(0.792778\pi\)
\(992\) 0 0
\(993\) 20.0000 0.634681
\(994\) 0 0
\(995\) −32.0000 −1.01447
\(996\) 0 0
\(997\) 14.0000 + 24.2487i 0.443384 + 0.767964i 0.997938 0.0641836i \(-0.0204443\pi\)
−0.554554 + 0.832148i \(0.687111\pi\)
\(998\) 0 0
\(999\) −3.00000 + 5.19615i −0.0949158 + 0.164399i
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.q.y.961.1 2
4.3 odd 2 294.2.e.d.79.1 2
7.2 even 3 2352.2.a.b.1.1 1
7.3 odd 6 2352.2.q.a.1537.1 2
7.4 even 3 inner 2352.2.q.y.1537.1 2
7.5 odd 6 2352.2.a.y.1.1 1
7.6 odd 2 2352.2.q.a.961.1 2
12.11 even 2 882.2.g.a.667.1 2
21.2 odd 6 7056.2.a.ca.1.1 1
21.5 even 6 7056.2.a.a.1.1 1
28.3 even 6 294.2.e.e.67.1 2
28.11 odd 6 294.2.e.d.67.1 2
28.19 even 6 294.2.a.b.1.1 1
28.23 odd 6 294.2.a.c.1.1 yes 1
28.27 even 2 294.2.e.e.79.1 2
56.5 odd 6 9408.2.a.b.1.1 1
56.19 even 6 9408.2.a.br.1.1 1
56.37 even 6 9408.2.a.de.1.1 1
56.51 odd 6 9408.2.a.bo.1.1 1
84.11 even 6 882.2.g.a.361.1 2
84.23 even 6 882.2.a.l.1.1 1
84.47 odd 6 882.2.a.f.1.1 1
84.59 odd 6 882.2.g.f.361.1 2
84.83 odd 2 882.2.g.f.667.1 2
140.19 even 6 7350.2.a.cj.1.1 1
140.79 odd 6 7350.2.a.br.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
294.2.a.b.1.1 1 28.19 even 6
294.2.a.c.1.1 yes 1 28.23 odd 6
294.2.e.d.67.1 2 28.11 odd 6
294.2.e.d.79.1 2 4.3 odd 2
294.2.e.e.67.1 2 28.3 even 6
294.2.e.e.79.1 2 28.27 even 2
882.2.a.f.1.1 1 84.47 odd 6
882.2.a.l.1.1 1 84.23 even 6
882.2.g.a.361.1 2 84.11 even 6
882.2.g.a.667.1 2 12.11 even 2
882.2.g.f.361.1 2 84.59 odd 6
882.2.g.f.667.1 2 84.83 odd 2
2352.2.a.b.1.1 1 7.2 even 3
2352.2.a.y.1.1 1 7.5 odd 6
2352.2.q.a.961.1 2 7.6 odd 2
2352.2.q.a.1537.1 2 7.3 odd 6
2352.2.q.y.961.1 2 1.1 even 1 trivial
2352.2.q.y.1537.1 2 7.4 even 3 inner
7056.2.a.a.1.1 1 21.5 even 6
7056.2.a.ca.1.1 1 21.2 odd 6
7350.2.a.br.1.1 1 140.79 odd 6
7350.2.a.cj.1.1 1 140.19 even 6
9408.2.a.b.1.1 1 56.5 odd 6
9408.2.a.bo.1.1 1 56.51 odd 6
9408.2.a.br.1.1 1 56.19 even 6
9408.2.a.de.1.1 1 56.37 even 6