Properties

Label 2736.2.bm.b.559.1
Level $2736$
Weight $2$
Character 2736.559
Analytic conductor $21.847$
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] = [2736,2,Mod(559,2736)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2736, 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("2736.559");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2736 = 2^{4} \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2736.bm (of order \(6\), degree \(2\), 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: \(21.8470699930\)
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: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 912)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 559.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 2736.559
Dual form 2736.2.bm.b.1855.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.73205i q^{7} +O(q^{10})\) \(q+1.73205i q^{7} -3.46410i q^{11} +(-4.50000 + 2.59808i) q^{13} +(3.00000 - 5.19615i) q^{17} +(4.00000 + 1.73205i) q^{19} +(-3.00000 + 1.73205i) q^{23} +(2.50000 + 4.33013i) q^{25} +(3.00000 - 1.73205i) q^{29} -1.00000 q^{31} -8.66025i q^{37} +(6.00000 + 3.46410i) q^{41} +(4.50000 + 2.59808i) q^{43} +(-9.00000 + 5.19615i) q^{47} +4.00000 q^{49} +(9.00000 - 5.19615i) q^{53} +(6.00000 - 10.3923i) q^{59} +(2.50000 + 4.33013i) q^{61} +(6.50000 + 11.2583i) q^{67} +(2.50000 - 4.33013i) q^{73} +6.00000 q^{77} +(-0.500000 + 0.866025i) q^{79} +17.3205i q^{83} +(-4.50000 - 7.79423i) q^{91} +(12.0000 + 6.92820i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 9 q^{13} + 6 q^{17} + 8 q^{19} - 6 q^{23} + 5 q^{25} + 6 q^{29} - 2 q^{31} + 12 q^{41} + 9 q^{43} - 18 q^{47} + 8 q^{49} + 18 q^{53} + 12 q^{59} + 5 q^{61} + 13 q^{67} + 5 q^{73} + 12 q^{77} - q^{79} - 9 q^{91} + 24 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2736\mathbb{Z}\right)^\times\).

\(n\) \(1009\) \(1217\) \(1711\) \(2053\)
\(\chi(n)\) \(e\left(\frac{1}{6}\right)\) \(1\) \(-1\) \(1\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(6\) 0 0
\(7\) 1.73205i 0.654654i 0.944911 + 0.327327i \(0.106148\pi\)
−0.944911 + 0.327327i \(0.893852\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 3.46410i 1.04447i −0.852803 0.522233i \(-0.825099\pi\)
0.852803 0.522233i \(-0.174901\pi\)
\(12\) 0 0
\(13\) −4.50000 + 2.59808i −1.24808 + 0.720577i −0.970725 0.240192i \(-0.922790\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 3.00000 5.19615i 0.727607 1.26025i −0.230285 0.973123i \(-0.573966\pi\)
0.957892 0.287129i \(-0.0927008\pi\)
\(18\) 0 0
\(19\) 4.00000 + 1.73205i 0.917663 + 0.397360i
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −3.00000 + 1.73205i −0.625543 + 0.361158i −0.779024 0.626994i \(-0.784285\pi\)
0.153481 + 0.988152i \(0.450952\pi\)
\(24\) 0 0
\(25\) 2.50000 + 4.33013i 0.500000 + 0.866025i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 3.00000 1.73205i 0.557086 0.321634i −0.194889 0.980825i \(-0.562435\pi\)
0.751975 + 0.659192i \(0.229101\pi\)
\(30\) 0 0
\(31\) −1.00000 −0.179605 −0.0898027 0.995960i \(-0.528624\pi\)
−0.0898027 + 0.995960i \(0.528624\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 8.66025i 1.42374i −0.702313 0.711868i \(-0.747849\pi\)
0.702313 0.711868i \(-0.252151\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 6.00000 + 3.46410i 0.937043 + 0.541002i 0.889032 0.457845i \(-0.151379\pi\)
0.0480106 + 0.998847i \(0.484712\pi\)
\(42\) 0 0
\(43\) 4.50000 + 2.59808i 0.686244 + 0.396203i 0.802203 0.597051i \(-0.203661\pi\)
−0.115960 + 0.993254i \(0.536994\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −9.00000 + 5.19615i −1.31278 + 0.757937i −0.982556 0.185964i \(-0.940459\pi\)
−0.330228 + 0.943901i \(0.607126\pi\)
\(48\) 0 0
\(49\) 4.00000 0.571429
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 9.00000 5.19615i 1.23625 0.713746i 0.267920 0.963441i \(-0.413664\pi\)
0.968325 + 0.249695i \(0.0803302\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 6.00000 10.3923i 0.781133 1.35296i −0.150148 0.988663i \(-0.547975\pi\)
0.931282 0.364299i \(-0.118692\pi\)
\(60\) 0 0
\(61\) 2.50000 + 4.33013i 0.320092 + 0.554416i 0.980507 0.196485i \(-0.0629528\pi\)
−0.660415 + 0.750901i \(0.729619\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 6.50000 + 11.2583i 0.794101 + 1.37542i 0.923408 + 0.383819i \(0.125391\pi\)
−0.129307 + 0.991605i \(0.541275\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(72\) 0 0
\(73\) 2.50000 4.33013i 0.292603 0.506803i −0.681822 0.731519i \(-0.738812\pi\)
0.974424 + 0.224716i \(0.0721453\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 6.00000 0.683763
\(78\) 0 0
\(79\) −0.500000 + 0.866025i −0.0562544 + 0.0974355i −0.892781 0.450490i \(-0.851249\pi\)
0.836527 + 0.547926i \(0.184582\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 17.3205i 1.90117i 0.310460 + 0.950586i \(0.399517\pi\)
−0.310460 + 0.950586i \(0.600483\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(90\) 0 0
\(91\) −4.50000 7.79423i −0.471728 0.817057i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 12.0000 + 6.92820i 1.21842 + 0.703452i 0.964579 0.263795i \(-0.0849741\pi\)
0.253837 + 0.967247i \(0.418307\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(102\) 0 0
\(103\) 1.00000 0.0985329 0.0492665 0.998786i \(-0.484312\pi\)
0.0492665 + 0.998786i \(0.484312\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(108\) 0 0
\(109\) −12.0000 6.92820i −1.14939 0.663602i −0.200653 0.979662i \(-0.564306\pi\)
−0.948739 + 0.316061i \(0.897640\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 6.92820i 0.651751i 0.945413 + 0.325875i \(0.105659\pi\)
−0.945413 + 0.325875i \(0.894341\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 9.00000 + 5.19615i 0.825029 + 0.476331i
\(120\) 0 0
\(121\) −1.00000 −0.0909091
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 8.00000 + 13.8564i 0.709885 + 1.22956i 0.964899 + 0.262620i \(0.0845865\pi\)
−0.255014 + 0.966937i \(0.582080\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(132\) 0 0
\(133\) −3.00000 + 6.92820i −0.260133 + 0.600751i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −6.00000 10.3923i −0.512615 0.887875i −0.999893 0.0146279i \(-0.995344\pi\)
0.487278 0.873247i \(-0.337990\pi\)
\(138\) 0 0
\(139\) −7.50000 + 4.33013i −0.636142 + 0.367277i −0.783127 0.621862i \(-0.786376\pi\)
0.146985 + 0.989139i \(0.453043\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 9.00000 + 15.5885i 0.752618 + 1.30357i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 9.00000 15.5885i 0.737309 1.27706i −0.216394 0.976306i \(-0.569430\pi\)
0.953703 0.300750i \(-0.0972370\pi\)
\(150\) 0 0
\(151\) −8.00000 −0.651031 −0.325515 0.945537i \(-0.605538\pi\)
−0.325515 + 0.945537i \(0.605538\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 8.50000 14.7224i 0.678374 1.17498i −0.297097 0.954847i \(-0.596018\pi\)
0.975470 0.220131i \(-0.0706483\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −3.00000 5.19615i −0.236433 0.409514i
\(162\) 0 0
\(163\) 22.5167i 1.76364i −0.471585 0.881820i \(-0.656318\pi\)
0.471585 0.881820i \(-0.343682\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 6.00000 + 10.3923i 0.464294 + 0.804181i 0.999169 0.0407502i \(-0.0129748\pi\)
−0.534875 + 0.844931i \(0.679641\pi\)
\(168\) 0 0
\(169\) 7.00000 12.1244i 0.538462 0.932643i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 21.0000 + 12.1244i 1.59660 + 0.921798i 0.992136 + 0.125166i \(0.0399462\pi\)
0.604465 + 0.796632i \(0.293387\pi\)
\(174\) 0 0
\(175\) −7.50000 + 4.33013i −0.566947 + 0.327327i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 6.00000 0.448461 0.224231 0.974536i \(-0.428013\pi\)
0.224231 + 0.974536i \(0.428013\pi\)
\(180\) 0 0
\(181\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −18.0000 10.3923i −1.31629 0.759961i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 6.92820i 0.501307i −0.968077 0.250654i \(-0.919354\pi\)
0.968077 0.250654i \(-0.0806455\pi\)
\(192\) 0 0
\(193\) 13.5000 + 7.79423i 0.971751 + 0.561041i 0.899770 0.436365i \(-0.143734\pi\)
0.0719816 + 0.997406i \(0.477068\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 12.0000 0.854965 0.427482 0.904024i \(-0.359401\pi\)
0.427482 + 0.904024i \(0.359401\pi\)
\(198\) 0 0
\(199\) 10.5000 6.06218i 0.744325 0.429736i −0.0793146 0.996850i \(-0.525273\pi\)
0.823640 + 0.567113i \(0.191940\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 3.00000 + 5.19615i 0.210559 + 0.364698i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 6.00000 13.8564i 0.415029 0.958468i
\(210\) 0 0
\(211\) −2.50000 + 4.33013i −0.172107 + 0.298098i −0.939156 0.343490i \(-0.888391\pi\)
0.767049 + 0.641588i \(0.221724\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 1.73205i 0.117579i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 31.1769i 2.09719i
\(222\) 0 0
\(223\) 5.50000 9.52628i 0.368307 0.637927i −0.620994 0.783815i \(-0.713271\pi\)
0.989301 + 0.145889i \(0.0466041\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 12.0000 0.796468 0.398234 0.917284i \(-0.369623\pi\)
0.398234 + 0.917284i \(0.369623\pi\)
\(228\) 0 0
\(229\) 23.0000 1.51988 0.759941 0.649992i \(-0.225228\pi\)
0.759941 + 0.649992i \(0.225228\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −12.0000 + 20.7846i −0.786146 + 1.36165i 0.142166 + 0.989843i \(0.454593\pi\)
−0.928312 + 0.371802i \(0.878740\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 6.92820i 0.448148i 0.974572 + 0.224074i \(0.0719358\pi\)
−0.974572 + 0.224074i \(0.928064\pi\)
\(240\) 0 0
\(241\) 7.50000 4.33013i 0.483117 0.278928i −0.238597 0.971119i \(-0.576688\pi\)
0.721715 + 0.692191i \(0.243354\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −22.5000 + 2.59808i −1.43164 + 0.165312i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −15.0000 + 8.66025i −0.946792 + 0.546630i −0.892083 0.451872i \(-0.850756\pi\)
−0.0547088 + 0.998502i \(0.517423\pi\)
\(252\) 0 0
\(253\) 6.00000 + 10.3923i 0.377217 + 0.653359i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 3.00000 1.73205i 0.187135 0.108042i −0.403506 0.914977i \(-0.632208\pi\)
0.590641 + 0.806935i \(0.298875\pi\)
\(258\) 0 0
\(259\) 15.0000 0.932055
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −18.0000 10.3923i −1.10993 0.640817i −0.171117 0.985251i \(-0.554738\pi\)
−0.938811 + 0.344434i \(0.888071\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −6.00000 3.46410i −0.365826 0.211210i 0.305807 0.952093i \(-0.401074\pi\)
−0.671634 + 0.740883i \(0.734407\pi\)
\(270\) 0 0
\(271\) −3.00000 1.73205i −0.182237 0.105215i 0.406106 0.913826i \(-0.366886\pi\)
−0.588343 + 0.808611i \(0.700220\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 15.0000 8.66025i 0.904534 0.522233i
\(276\) 0 0
\(277\) −10.0000 −0.600842 −0.300421 0.953807i \(-0.597127\pi\)
−0.300421 + 0.953807i \(0.597127\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 6.00000 3.46410i 0.357930 0.206651i −0.310242 0.950657i \(-0.600410\pi\)
0.668172 + 0.744007i \(0.267077\pi\)
\(282\) 0 0
\(283\) −3.00000 1.73205i −0.178331 0.102960i 0.408177 0.912903i \(-0.366165\pi\)
−0.586509 + 0.809943i \(0.699498\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −6.00000 + 10.3923i −0.354169 + 0.613438i
\(288\) 0 0
\(289\) −9.50000 16.4545i −0.558824 0.967911i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 24.2487i 1.41662i −0.705899 0.708312i \(-0.749457\pi\)
0.705899 0.708312i \(-0.250543\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 9.00000 15.5885i 0.520483 0.901504i
\(300\) 0 0
\(301\) −4.50000 + 7.79423i −0.259376 + 0.449252i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −2.00000 + 3.46410i −0.114146 + 0.197707i −0.917438 0.397879i \(-0.869747\pi\)
0.803292 + 0.595585i \(0.203080\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 20.7846i 1.17859i 0.807919 + 0.589294i \(0.200594\pi\)
−0.807919 + 0.589294i \(0.799406\pi\)
\(312\) 0 0
\(313\) −5.00000 8.66025i −0.282617 0.489506i 0.689412 0.724370i \(-0.257869\pi\)
−0.972028 + 0.234863i \(0.924536\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −24.0000 + 13.8564i −1.34797 + 0.778253i −0.987962 0.154694i \(-0.950561\pi\)
−0.360012 + 0.932948i \(0.617227\pi\)
\(318\) 0 0
\(319\) −6.00000 10.3923i −0.335936 0.581857i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 21.0000 15.5885i 1.16847 0.867365i
\(324\) 0 0
\(325\) −22.5000 12.9904i −1.24808 0.720577i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −9.00000 15.5885i −0.496186 0.859419i
\(330\) 0 0
\(331\) 25.0000 1.37412 0.687062 0.726599i \(-0.258900\pi\)
0.687062 + 0.726599i \(0.258900\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −7.50000 4.33013i −0.408551 0.235877i 0.281616 0.959527i \(-0.409130\pi\)
−0.690167 + 0.723650i \(0.742463\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 3.46410i 0.187592i
\(342\) 0 0
\(343\) 19.0526i 1.02874i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −12.0000 6.92820i −0.644194 0.371925i 0.142034 0.989862i \(-0.454636\pi\)
−0.786228 + 0.617936i \(0.787969\pi\)
\(348\) 0 0
\(349\) −19.0000 −1.01705 −0.508523 0.861048i \(-0.669808\pi\)
−0.508523 + 0.861048i \(0.669808\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −24.0000 −1.27739 −0.638696 0.769460i \(-0.720526\pi\)
−0.638696 + 0.769460i \(0.720526\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −18.0000 10.3923i −0.950004 0.548485i −0.0569216 0.998379i \(-0.518129\pi\)
−0.893082 + 0.449894i \(0.851462\pi\)
\(360\) 0 0
\(361\) 13.0000 + 13.8564i 0.684211 + 0.729285i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −1.50000 + 0.866025i −0.0782994 + 0.0452062i −0.538639 0.842537i \(-0.681061\pi\)
0.460339 + 0.887743i \(0.347728\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 9.00000 + 15.5885i 0.467257 + 0.809312i
\(372\) 0 0
\(373\) 13.8564i 0.717458i 0.933442 + 0.358729i \(0.116790\pi\)
−0.933442 + 0.358729i \(0.883210\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −9.00000 + 15.5885i −0.463524 + 0.802846i
\(378\) 0 0
\(379\) 25.0000 1.28416 0.642082 0.766636i \(-0.278071\pi\)
0.642082 + 0.766636i \(0.278071\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 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
\(387\) 0 0
\(388\) 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
\(391\) 20.7846i 1.05112i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 3.50000 6.06218i 0.175660 0.304252i −0.764730 0.644351i \(-0.777127\pi\)
0.940389 + 0.340099i \(0.110461\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −24.0000 13.8564i −1.19850 0.691956i −0.238282 0.971196i \(-0.576584\pi\)
−0.960221 + 0.279240i \(0.909917\pi\)
\(402\) 0 0
\(403\) 4.50000 2.59808i 0.224161 0.129419i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −30.0000 −1.48704
\(408\) 0 0
\(409\) −18.0000 + 10.3923i −0.890043 + 0.513866i −0.873956 0.486004i \(-0.838454\pi\)
−0.0160862 + 0.999871i \(0.505121\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 18.0000 + 10.3923i 0.885722 + 0.511372i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 31.1769i 1.52309i −0.648111 0.761546i \(-0.724441\pi\)
0.648111 0.761546i \(-0.275559\pi\)
\(420\) 0 0
\(421\) 18.0000 + 10.3923i 0.877266 + 0.506490i 0.869756 0.493482i \(-0.164276\pi\)
0.00751023 + 0.999972i \(0.497609\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 30.0000 1.45521
\(426\) 0 0
\(427\) −7.50000 + 4.33013i −0.362950 + 0.209550i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −15.0000 25.9808i −0.722525 1.25145i −0.959985 0.280052i \(-0.909648\pi\)
0.237460 0.971397i \(-0.423685\pi\)
\(432\) 0 0
\(433\) −22.5000 + 12.9904i −1.08128 + 0.624278i −0.931242 0.364402i \(-0.881273\pi\)
−0.150039 + 0.988680i \(0.547940\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −15.0000 + 1.73205i −0.717547 + 0.0828552i
\(438\) 0 0
\(439\) −9.50000 + 16.4545i −0.453410 + 0.785330i −0.998595 0.0529862i \(-0.983126\pi\)
0.545185 + 0.838316i \(0.316459\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 21.0000 12.1244i 0.997740 0.576046i 0.0901612 0.995927i \(-0.471262\pi\)
0.907579 + 0.419882i \(0.137928\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 10.3923i 0.490443i 0.969467 + 0.245222i \(0.0788607\pi\)
−0.969467 + 0.245222i \(0.921139\pi\)
\(450\) 0 0
\(451\) 12.0000 20.7846i 0.565058 0.978709i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −5.00000 −0.233890 −0.116945 0.993138i \(-0.537310\pi\)
−0.116945 + 0.993138i \(0.537310\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 6.00000 10.3923i 0.279448 0.484018i −0.691800 0.722089i \(-0.743182\pi\)
0.971248 + 0.238071i \(0.0765153\pi\)
\(462\) 0 0
\(463\) 5.19615i 0.241486i 0.992684 + 0.120743i \(0.0385276\pi\)
−0.992684 + 0.120743i \(0.961472\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(468\) 0 0
\(469\) −19.5000 + 11.2583i −0.900426 + 0.519861i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 9.00000 15.5885i 0.413820 0.716758i
\(474\) 0 0
\(475\) 2.50000 + 21.6506i 0.114708 + 0.993399i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 12.0000 6.92820i 0.548294 0.316558i −0.200140 0.979767i \(-0.564140\pi\)
0.748434 + 0.663210i \(0.230806\pi\)
\(480\) 0 0
\(481\) 22.5000 + 38.9711i 1.02591 + 1.77693i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 8.00000 0.362515 0.181257 0.983436i \(-0.441983\pi\)
0.181257 + 0.983436i \(0.441983\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −12.0000 6.92820i −0.541552 0.312665i 0.204155 0.978938i \(-0.434555\pi\)
−0.745708 + 0.666273i \(0.767889\pi\)
\(492\) 0 0
\(493\) 20.7846i 0.936092i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 7.50000 + 4.33013i 0.335746 + 0.193843i 0.658389 0.752678i \(-0.271238\pi\)
−0.322643 + 0.946521i \(0.604571\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −15.0000 + 8.66025i −0.668817 + 0.386142i −0.795628 0.605785i \(-0.792859\pi\)
0.126811 + 0.991927i \(0.459526\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 21.0000 12.1244i 0.930809 0.537403i 0.0437414 0.999043i \(-0.486072\pi\)
0.887067 + 0.461640i \(0.152739\pi\)
\(510\) 0 0
\(511\) 7.50000 + 4.33013i 0.331780 + 0.191554i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 18.0000 + 31.1769i 0.791639 + 1.37116i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 3.46410i 0.151765i 0.997117 + 0.0758825i \(0.0241774\pi\)
−0.997117 + 0.0758825i \(0.975823\pi\)
\(522\) 0 0
\(523\) −14.5000 25.1147i −0.634041 1.09819i −0.986718 0.162446i \(-0.948062\pi\)
0.352677 0.935745i \(-0.385272\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −3.00000 + 5.19615i −0.130682 + 0.226348i
\(528\) 0 0
\(529\) −5.50000 + 9.52628i −0.239130 + 0.414186i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −36.0000 −1.55933
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 13.8564i 0.596838i
\(540\) 0 0
\(541\) −12.5000 21.6506i −0.537417 0.930834i −0.999042 0.0437584i \(-0.986067\pi\)
0.461625 0.887075i \(-0.347267\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −6.50000 11.2583i −0.277920 0.481371i 0.692948 0.720988i \(-0.256312\pi\)
−0.970868 + 0.239616i \(0.922978\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 15.0000 1.73205i 0.639021 0.0737878i
\(552\) 0 0
\(553\) −1.50000 0.866025i −0.0637865 0.0368271i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −12.0000 20.7846i −0.508456 0.880672i −0.999952 0.00979220i \(-0.996883\pi\)
0.491496 0.870880i \(-0.336450\pi\)
\(558\) 0 0
\(559\) −27.0000 −1.14198
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −24.0000 −1.01148 −0.505740 0.862686i \(-0.668780\pi\)
−0.505740 + 0.862686i \(0.668780\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 34.6410i 1.45223i 0.687575 + 0.726113i \(0.258675\pi\)
−0.687575 + 0.726113i \(0.741325\pi\)
\(570\) 0 0
\(571\) 32.9090i 1.37720i −0.725143 0.688599i \(-0.758226\pi\)
0.725143 0.688599i \(-0.241774\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −15.0000 8.66025i −0.625543 0.361158i
\(576\) 0 0
\(577\) −38.0000 −1.58196 −0.790980 0.611842i \(-0.790429\pi\)
−0.790980 + 0.611842i \(0.790429\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −30.0000 −1.24461
\(582\) 0 0
\(583\) −18.0000 31.1769i −0.745484 1.29122i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −36.0000 20.7846i −1.48588 0.857873i −0.486008 0.873954i \(-0.661548\pi\)
−0.999871 + 0.0160815i \(0.994881\pi\)
\(588\) 0 0
\(589\) −4.00000 1.73205i −0.164817 0.0713679i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −12.0000 20.7846i −0.492781 0.853522i 0.507184 0.861838i \(-0.330686\pi\)
−0.999965 + 0.00831589i \(0.997353\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −21.0000 36.3731i −0.858037 1.48616i −0.873799 0.486287i \(-0.838351\pi\)
0.0157622 0.999876i \(-0.494983\pi\)
\(600\) 0 0
\(601\) 46.7654i 1.90760i −0.300443 0.953800i \(-0.597135\pi\)
0.300443 0.953800i \(-0.402865\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 47.0000 1.90767 0.953836 0.300329i \(-0.0970966\pi\)
0.953836 + 0.300329i \(0.0970966\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 27.0000 46.7654i 1.09230 1.89192i
\(612\) 0 0
\(613\) 19.0000 32.9090i 0.767403 1.32918i −0.171564 0.985173i \(-0.554882\pi\)
0.938967 0.344008i \(-0.111785\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 21.0000 + 36.3731i 0.845428 + 1.46432i 0.885249 + 0.465118i \(0.153988\pi\)
−0.0398207 + 0.999207i \(0.512679\pi\)
\(618\) 0 0
\(619\) 39.8372i 1.60119i 0.599205 + 0.800595i \(0.295483\pi\)
−0.599205 + 0.800595i \(0.704517\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −12.5000 + 21.6506i −0.500000 + 0.866025i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −45.0000 25.9808i −1.79427 1.03592i
\(630\) 0 0
\(631\) 19.5000 11.2583i 0.776283 0.448187i −0.0588285 0.998268i \(-0.518737\pi\)
0.835111 + 0.550081i \(0.185403\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −18.0000 + 10.3923i −0.713186 + 0.411758i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −3.00000 1.73205i −0.118493 0.0684119i 0.439582 0.898202i \(-0.355127\pi\)
−0.558075 + 0.829790i \(0.688460\pi\)
\(642\) 0 0
\(643\) 10.5000 + 6.06218i 0.414080 + 0.239069i 0.692541 0.721378i \(-0.256491\pi\)
−0.278462 + 0.960447i \(0.589824\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 24.2487i 0.953315i 0.879089 + 0.476658i \(0.158152\pi\)
−0.879089 + 0.476658i \(0.841848\pi\)
\(648\) 0 0
\(649\) −36.0000 20.7846i −1.41312 0.815867i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −48.0000 −1.87839 −0.939193 0.343391i \(-0.888424\pi\)
−0.939193 + 0.343391i \(0.888424\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −9.00000 15.5885i −0.350590 0.607240i 0.635763 0.771885i \(-0.280686\pi\)
−0.986353 + 0.164644i \(0.947352\pi\)
\(660\) 0 0
\(661\) −6.00000 + 3.46410i −0.233373 + 0.134738i −0.612127 0.790759i \(-0.709686\pi\)
0.378754 + 0.925497i \(0.376353\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −6.00000 + 10.3923i −0.232321 + 0.402392i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 15.0000 8.66025i 0.579069 0.334325i
\(672\) 0 0
\(673\) 8.66025i 0.333828i 0.985971 + 0.166914i \(0.0533803\pi\)
−0.985971 + 0.166914i \(0.946620\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 17.3205i 0.665681i 0.942983 + 0.332841i \(0.108007\pi\)
−0.942983 + 0.332841i \(0.891993\pi\)
\(678\) 0 0
\(679\) −12.0000 + 20.7846i −0.460518 + 0.797640i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 12.0000 0.459167 0.229584 0.973289i \(-0.426264\pi\)
0.229584 + 0.973289i \(0.426264\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −27.0000 + 46.7654i −1.02862 + 1.78162i
\(690\) 0 0
\(691\) 3.46410i 0.131781i −0.997827 0.0658903i \(-0.979011\pi\)
0.997827 0.0658903i \(-0.0209887\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 36.0000 20.7846i 1.36360 0.787273i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −12.0000 + 20.7846i −0.453234 + 0.785024i −0.998585 0.0531839i \(-0.983063\pi\)
0.545351 + 0.838208i \(0.316396\pi\)
\(702\) 0 0
\(703\) 15.0000 34.6410i 0.565736 1.30651i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 17.5000 + 30.3109i 0.657226 + 1.13835i 0.981331 + 0.192328i \(0.0616038\pi\)
−0.324104 + 0.946021i \(0.605063\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 3.00000 1.73205i 0.112351 0.0648658i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 39.0000 + 22.5167i 1.45445 + 0.839730i 0.998730 0.0503909i \(-0.0160467\pi\)
0.455725 + 0.890121i \(0.349380\pi\)
\(720\) 0 0
\(721\) 1.73205i 0.0645049i
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 15.0000 + 8.66025i 0.557086 + 0.321634i
\(726\) 0 0
\(727\) −16.5000 9.52628i −0.611951 0.353310i 0.161778 0.986827i \(-0.448277\pi\)
−0.773729 + 0.633517i \(0.781611\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 27.0000 15.5885i 0.998631 0.576560i
\(732\) 0 0
\(733\) 22.0000 0.812589 0.406294 0.913742i \(-0.366821\pi\)
0.406294 + 0.913742i \(0.366821\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 39.0000 22.5167i 1.43658 0.829412i
\(738\) 0 0
\(739\) −28.5000 16.4545i −1.04839 0.605288i −0.126191 0.992006i \(-0.540275\pi\)
−0.922198 + 0.386718i \(0.873609\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −21.0000 + 36.3731i −0.770415 + 1.33440i 0.166920 + 0.985970i \(0.446618\pi\)
−0.937336 + 0.348428i \(0.886716\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −6.50000 11.2583i −0.237188 0.410822i 0.722718 0.691143i \(-0.242893\pi\)
−0.959906 + 0.280321i \(0.909559\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −17.5000 + 30.3109i −0.636048 + 1.10167i 0.350244 + 0.936659i \(0.386099\pi\)
−0.986292 + 0.165009i \(0.947235\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −12.0000 −0.435000 −0.217500 0.976060i \(-0.569790\pi\)
−0.217500 + 0.976060i \(0.569790\pi\)
\(762\) 0 0
\(763\) 12.0000 20.7846i 0.434429 0.752453i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 62.3538i 2.25147i
\(768\) 0 0
\(769\) −0.500000 0.866025i −0.0180305 0.0312297i 0.856869 0.515534i \(-0.172406\pi\)
−0.874900 + 0.484304i \(0.839073\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 3.00000 1.73205i 0.107903 0.0622975i −0.445078 0.895492i \(-0.646824\pi\)
0.552980 + 0.833194i \(0.313491\pi\)
\(774\) 0 0
\(775\) −2.50000 4.33013i −0.0898027 0.155543i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 18.0000 + 24.2487i 0.644917 + 0.868800i
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −31.0000 −1.10503 −0.552515 0.833503i \(-0.686332\pi\)
−0.552515 + 0.833503i \(0.686332\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −12.0000 −0.426671
\(792\) 0 0
\(793\) −22.5000 12.9904i −0.798998 0.461302i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 48.4974i 1.71787i 0.512087 + 0.858933i \(0.328872\pi\)
−0.512087 + 0.858933i \(0.671128\pi\)
\(798\) 0 0
\(799\) 62.3538i 2.20592i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −15.0000 8.66025i −0.529339 0.305614i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −12.0000 −0.421898 −0.210949 0.977497i \(-0.567655\pi\)
−0.210949 + 0.977497i \(0.567655\pi\)
\(810\) 0 0
\(811\) 10.0000 + 17.3205i 0.351147 + 0.608205i 0.986451 0.164057i \(-0.0524582\pi\)
−0.635303 + 0.772263i \(0.719125\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 13.5000 + 18.1865i 0.472305 + 0.636266i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −9.00000 15.5885i −0.314102 0.544041i 0.665144 0.746715i \(-0.268370\pi\)
−0.979246 + 0.202674i \(0.935037\pi\)
\(822\) 0 0
\(823\) −15.0000 + 8.66025i −0.522867 + 0.301877i −0.738107 0.674684i \(-0.764280\pi\)
0.215240 + 0.976561i \(0.430947\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 9.00000 + 15.5885i 0.312961 + 0.542064i 0.979002 0.203851i \(-0.0653459\pi\)
−0.666041 + 0.745915i \(0.732013\pi\)
\(828\) 0 0
\(829\) 53.6936i 1.86485i −0.361357 0.932427i \(-0.617686\pi\)
0.361357 0.932427i \(-0.382314\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 12.0000 20.7846i 0.415775 0.720144i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 18.0000 31.1769i 0.621429 1.07635i −0.367791 0.929909i \(-0.619886\pi\)
0.989220 0.146438i \(-0.0467809\pi\)
\(840\) 0 0
\(841\) −8.50000 + 14.7224i −0.293103 + 0.507670i
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 1.73205i 0.0595140i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 15.0000 + 25.9808i 0.514193 + 0.890609i
\(852\) 0 0
\(853\) −14.5000 + 25.1147i −0.496471 + 0.859912i −0.999992 0.00407068i \(-0.998704\pi\)
0.503521 + 0.863983i \(0.332038\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −9.00000 5.19615i −0.307434 0.177497i 0.338344 0.941023i \(-0.390133\pi\)
−0.645778 + 0.763525i \(0.723467\pi\)
\(858\) 0 0
\(859\) −19.5000 + 11.2583i −0.665331 + 0.384129i −0.794305 0.607519i \(-0.792165\pi\)
0.128974 + 0.991648i \(0.458832\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 6.00000 0.204242 0.102121 0.994772i \(-0.467437\pi\)
0.102121 + 0.994772i \(0.467437\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 3.00000 + 1.73205i 0.101768 + 0.0587558i
\(870\) 0 0
\(871\) −58.5000 33.7750i −1.98220 1.14442i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 46.5000 + 26.8468i 1.57019 + 0.906552i 0.996144 + 0.0877308i \(0.0279615\pi\)
0.574049 + 0.818821i \(0.305372\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 12.0000 0.404290 0.202145 0.979356i \(-0.435209\pi\)
0.202145 + 0.979356i \(0.435209\pi\)
\(882\) 0 0
\(883\) 22.5000 12.9904i 0.757185 0.437161i −0.0710990 0.997469i \(-0.522651\pi\)
0.828284 + 0.560308i \(0.189317\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(888\) 0 0
\(889\) −24.0000 + 13.8564i −0.804934 + 0.464729i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −45.0000 + 5.19615i −1.50587 + 0.173883i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −3.00000 + 1.73205i −0.100056 + 0.0577671i
\(900\) 0 0
\(901\) 62.3538i 2.07731i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 14.0000 24.2487i 0.464862 0.805165i −0.534333 0.845274i \(-0.679437\pi\)
0.999195 + 0.0401089i \(0.0127705\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 12.0000 0.397578 0.198789 0.980042i \(-0.436299\pi\)
0.198789 + 0.980042i \(0.436299\pi\)
\(912\) 0 0
\(913\) 60.0000 1.98571
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 32.9090i 1.08557i −0.839873 0.542783i \(-0.817370\pi\)
0.839873 0.542783i \(-0.182630\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 37.5000 21.6506i 1.23299 0.711868i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 12.0000 20.7846i 0.393707 0.681921i −0.599228 0.800578i \(-0.704526\pi\)
0.992935 + 0.118657i \(0.0378590\pi\)
\(930\) 0 0
\(931\) 16.0000 + 6.92820i 0.524379 + 0.227063i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −23.5000 40.7032i −0.767712 1.32972i −0.938801 0.344460i \(-0.888062\pi\)
0.171089 0.985255i \(-0.445271\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 12.0000 6.92820i 0.391189 0.225853i −0.291486 0.956575i \(-0.594150\pi\)
0.682675 + 0.730722i \(0.260816\pi\)
\(942\) 0 0
\(943\) −24.0000 −0.781548
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 6.00000 + 3.46410i 0.194974 + 0.112568i 0.594309 0.804237i \(-0.297426\pi\)
−0.399335 + 0.916805i \(0.630759\pi\)
\(948\) 0 0
\(949\) 25.9808i 0.843371i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 9.00000 + 5.19615i 0.291539 + 0.168320i 0.638636 0.769509i \(-0.279499\pi\)
−0.347097 + 0.937829i \(0.612833\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 18.0000 10.3923i 0.581250 0.335585i
\(960\) 0 0
\(961\) −30.0000 −0.967742
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 4.50000 + 2.59808i 0.144710 + 0.0835485i 0.570607 0.821223i \(-0.306708\pi\)
−0.425897 + 0.904772i \(0.640041\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −27.0000 + 46.7654i −0.866471 + 1.50077i −0.000892350 1.00000i \(0.500284\pi\)
−0.865579 + 0.500773i \(0.833049\pi\)
\(972\) 0 0
\(973\) −7.50000 12.9904i −0.240439 0.416452i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 17.3205i 0.554132i 0.960851 + 0.277066i \(0.0893621\pi\)
−0.960851 + 0.277066i \(0.910638\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −15.0000 + 25.9808i −0.478426 + 0.828658i −0.999694 0.0247352i \(-0.992126\pi\)
0.521268 + 0.853393i \(0.325459\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −18.0000 −0.572367
\(990\) 0 0
\(991\) 9.50000 16.4545i 0.301777 0.522694i −0.674761 0.738036i \(-0.735753\pi\)
0.976539 + 0.215342i \(0.0690867\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 26.5000 + 45.8993i 0.839263 + 1.45365i 0.890511 + 0.454961i \(0.150347\pi\)
−0.0512480 + 0.998686i \(0.516320\pi\)
\(998\) 0 0
\(999\) 0 0
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 2736.2.bm.b.559.1 2
3.2 odd 2 912.2.bb.b.559.1 yes 2
4.3 odd 2 2736.2.bm.a.559.1 2
12.11 even 2 912.2.bb.a.559.1 yes 2
19.12 odd 6 2736.2.bm.a.1855.1 2
57.50 even 6 912.2.bb.a.31.1 2
76.31 even 6 inner 2736.2.bm.b.1855.1 2
228.107 odd 6 912.2.bb.b.31.1 yes 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
912.2.bb.a.31.1 2 57.50 even 6
912.2.bb.a.559.1 yes 2 12.11 even 2
912.2.bb.b.31.1 yes 2 228.107 odd 6
912.2.bb.b.559.1 yes 2 3.2 odd 2
2736.2.bm.a.559.1 2 4.3 odd 2
2736.2.bm.a.1855.1 2 19.12 odd 6
2736.2.bm.b.559.1 2 1.1 even 1 trivial
2736.2.bm.b.1855.1 2 76.31 even 6 inner