Properties

Label 2366.2.d.i.337.1
Level $2366$
Weight $2$
Character 2366.337
Analytic conductor $18.893$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2366,2,Mod(337,2366)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2366, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("2366.337");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2366 = 2 \cdot 7 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2366.d (of order \(2\), degree \(1\), 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: \(18.8926051182\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 182)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 337.1
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 2366.337
Dual form 2366.2.d.i.337.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000i q^{2} +3.00000 q^{3} -1.00000 q^{4} -3.00000i q^{6} -1.00000i q^{7} +1.00000i q^{8} +6.00000 q^{9} +O(q^{10})\) \(q-1.00000i q^{2} +3.00000 q^{3} -1.00000 q^{4} -3.00000i q^{6} -1.00000i q^{7} +1.00000i q^{8} +6.00000 q^{9} +5.00000i q^{11} -3.00000 q^{12} -1.00000 q^{14} +1.00000 q^{16} +4.00000 q^{17} -6.00000i q^{18} +2.00000i q^{19} -3.00000i q^{21} +5.00000 q^{22} -5.00000 q^{23} +3.00000i q^{24} +5.00000 q^{25} +9.00000 q^{27} +1.00000i q^{28} +4.00000 q^{29} +1.00000i q^{31} -1.00000i q^{32} +15.0000i q^{33} -4.00000i q^{34} -6.00000 q^{36} -7.00000i q^{37} +2.00000 q^{38} -9.00000i q^{41} -3.00000 q^{42} +12.0000 q^{43} -5.00000i q^{44} +5.00000i q^{46} +7.00000i q^{47} +3.00000 q^{48} -1.00000 q^{49} -5.00000i q^{50} +12.0000 q^{51} -4.00000 q^{53} -9.00000i q^{54} +1.00000 q^{56} +6.00000i q^{57} -4.00000i q^{58} +6.00000i q^{59} +13.0000 q^{61} +1.00000 q^{62} -6.00000i q^{63} -1.00000 q^{64} +15.0000 q^{66} +11.0000i q^{67} -4.00000 q^{68} -15.0000 q^{69} +6.00000i q^{72} -7.00000i q^{73} -7.00000 q^{74} +15.0000 q^{75} -2.00000i q^{76} +5.00000 q^{77} -17.0000 q^{79} +9.00000 q^{81} -9.00000 q^{82} +4.00000i q^{83} +3.00000i q^{84} -12.0000i q^{86} +12.0000 q^{87} -5.00000 q^{88} -14.0000i q^{89} +5.00000 q^{92} +3.00000i q^{93} +7.00000 q^{94} -3.00000i q^{96} +5.00000i q^{97} +1.00000i q^{98} +30.0000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 6 q^{3} - 2 q^{4} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 6 q^{3} - 2 q^{4} + 12 q^{9} - 6 q^{12} - 2 q^{14} + 2 q^{16} + 8 q^{17} + 10 q^{22} - 10 q^{23} + 10 q^{25} + 18 q^{27} + 8 q^{29} - 12 q^{36} + 4 q^{38} - 6 q^{42} + 24 q^{43} + 6 q^{48} - 2 q^{49} + 24 q^{51} - 8 q^{53} + 2 q^{56} + 26 q^{61} + 2 q^{62} - 2 q^{64} + 30 q^{66} - 8 q^{68} - 30 q^{69} - 14 q^{74} + 30 q^{75} + 10 q^{77} - 34 q^{79} + 18 q^{81} - 18 q^{82} + 24 q^{87} - 10 q^{88} + 10 q^{92} + 14 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2366\mathbb{Z}\right)^\times\).

\(n\) \(339\) \(2199\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 1.00000i − 0.707107i
\(3\) 3.00000 1.73205 0.866025 0.500000i \(-0.166667\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(4\) −1.00000 −0.500000
\(5\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(6\) − 3.00000i − 1.22474i
\(7\) − 1.00000i − 0.377964i
\(8\) 1.00000i 0.353553i
\(9\) 6.00000 2.00000
\(10\) 0 0
\(11\) 5.00000i 1.50756i 0.657129 + 0.753778i \(0.271771\pi\)
−0.657129 + 0.753778i \(0.728229\pi\)
\(12\) −3.00000 −0.866025
\(13\) 0 0
\(14\) −1.00000 −0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 4.00000 0.970143 0.485071 0.874475i \(-0.338794\pi\)
0.485071 + 0.874475i \(0.338794\pi\)
\(18\) − 6.00000i − 1.41421i
\(19\) 2.00000i 0.458831i 0.973329 + 0.229416i \(0.0736815\pi\)
−0.973329 + 0.229416i \(0.926318\pi\)
\(20\) 0 0
\(21\) − 3.00000i − 0.654654i
\(22\) 5.00000 1.06600
\(23\) −5.00000 −1.04257 −0.521286 0.853382i \(-0.674548\pi\)
−0.521286 + 0.853382i \(0.674548\pi\)
\(24\) 3.00000i 0.612372i
\(25\) 5.00000 1.00000
\(26\) 0 0
\(27\) 9.00000 1.73205
\(28\) 1.00000i 0.188982i
\(29\) 4.00000 0.742781 0.371391 0.928477i \(-0.378881\pi\)
0.371391 + 0.928477i \(0.378881\pi\)
\(30\) 0 0
\(31\) 1.00000i 0.179605i 0.995960 + 0.0898027i \(0.0286236\pi\)
−0.995960 + 0.0898027i \(0.971376\pi\)
\(32\) − 1.00000i − 0.176777i
\(33\) 15.0000i 2.61116i
\(34\) − 4.00000i − 0.685994i
\(35\) 0 0
\(36\) −6.00000 −1.00000
\(37\) − 7.00000i − 1.15079i −0.817875 0.575396i \(-0.804848\pi\)
0.817875 0.575396i \(-0.195152\pi\)
\(38\) 2.00000 0.324443
\(39\) 0 0
\(40\) 0 0
\(41\) − 9.00000i − 1.40556i −0.711405 0.702782i \(-0.751941\pi\)
0.711405 0.702782i \(-0.248059\pi\)
\(42\) −3.00000 −0.462910
\(43\) 12.0000 1.82998 0.914991 0.403473i \(-0.132197\pi\)
0.914991 + 0.403473i \(0.132197\pi\)
\(44\) − 5.00000i − 0.753778i
\(45\) 0 0
\(46\) 5.00000i 0.737210i
\(47\) 7.00000i 1.02105i 0.859861 + 0.510527i \(0.170550\pi\)
−0.859861 + 0.510527i \(0.829450\pi\)
\(48\) 3.00000 0.433013
\(49\) −1.00000 −0.142857
\(50\) − 5.00000i − 0.707107i
\(51\) 12.0000 1.68034
\(52\) 0 0
\(53\) −4.00000 −0.549442 −0.274721 0.961524i \(-0.588586\pi\)
−0.274721 + 0.961524i \(0.588586\pi\)
\(54\) − 9.00000i − 1.22474i
\(55\) 0 0
\(56\) 1.00000 0.133631
\(57\) 6.00000i 0.794719i
\(58\) − 4.00000i − 0.525226i
\(59\) 6.00000i 0.781133i 0.920575 + 0.390567i \(0.127721\pi\)
−0.920575 + 0.390567i \(0.872279\pi\)
\(60\) 0 0
\(61\) 13.0000 1.66448 0.832240 0.554416i \(-0.187058\pi\)
0.832240 + 0.554416i \(0.187058\pi\)
\(62\) 1.00000 0.127000
\(63\) − 6.00000i − 0.755929i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 15.0000 1.84637
\(67\) 11.0000i 1.34386i 0.740613 + 0.671932i \(0.234535\pi\)
−0.740613 + 0.671932i \(0.765465\pi\)
\(68\) −4.00000 −0.485071
\(69\) −15.0000 −1.80579
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 6.00000i 0.707107i
\(73\) − 7.00000i − 0.819288i −0.912245 0.409644i \(-0.865653\pi\)
0.912245 0.409644i \(-0.134347\pi\)
\(74\) −7.00000 −0.813733
\(75\) 15.0000 1.73205
\(76\) − 2.00000i − 0.229416i
\(77\) 5.00000 0.569803
\(78\) 0 0
\(79\) −17.0000 −1.91265 −0.956325 0.292306i \(-0.905577\pi\)
−0.956325 + 0.292306i \(0.905577\pi\)
\(80\) 0 0
\(81\) 9.00000 1.00000
\(82\) −9.00000 −0.993884
\(83\) 4.00000i 0.439057i 0.975606 + 0.219529i \(0.0704519\pi\)
−0.975606 + 0.219529i \(0.929548\pi\)
\(84\) 3.00000i 0.327327i
\(85\) 0 0
\(86\) − 12.0000i − 1.29399i
\(87\) 12.0000 1.28654
\(88\) −5.00000 −0.533002
\(89\) − 14.0000i − 1.48400i −0.670402 0.741999i \(-0.733878\pi\)
0.670402 0.741999i \(-0.266122\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 5.00000 0.521286
\(93\) 3.00000i 0.311086i
\(94\) 7.00000 0.721995
\(95\) 0 0
\(96\) − 3.00000i − 0.306186i
\(97\) 5.00000i 0.507673i 0.967247 + 0.253837i \(0.0816925\pi\)
−0.967247 + 0.253837i \(0.918307\pi\)
\(98\) 1.00000i 0.101015i
\(99\) 30.0000i 3.01511i
\(100\) −5.00000 −0.500000
\(101\) −15.0000 −1.49256 −0.746278 0.665635i \(-0.768161\pi\)
−0.746278 + 0.665635i \(0.768161\pi\)
\(102\) − 12.0000i − 1.18818i
\(103\) −6.00000 −0.591198 −0.295599 0.955312i \(-0.595519\pi\)
−0.295599 + 0.955312i \(0.595519\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 4.00000i 0.388514i
\(107\) −8.00000 −0.773389 −0.386695 0.922208i \(-0.626383\pi\)
−0.386695 + 0.922208i \(0.626383\pi\)
\(108\) −9.00000 −0.866025
\(109\) − 18.0000i − 1.72409i −0.506834 0.862044i \(-0.669184\pi\)
0.506834 0.862044i \(-0.330816\pi\)
\(110\) 0 0
\(111\) − 21.0000i − 1.99323i
\(112\) − 1.00000i − 0.0944911i
\(113\) 1.00000 0.0940721 0.0470360 0.998893i \(-0.485022\pi\)
0.0470360 + 0.998893i \(0.485022\pi\)
\(114\) 6.00000 0.561951
\(115\) 0 0
\(116\) −4.00000 −0.371391
\(117\) 0 0
\(118\) 6.00000 0.552345
\(119\) − 4.00000i − 0.366679i
\(120\) 0 0
\(121\) −14.0000 −1.27273
\(122\) − 13.0000i − 1.17696i
\(123\) − 27.0000i − 2.43451i
\(124\) − 1.00000i − 0.0898027i
\(125\) 0 0
\(126\) −6.00000 −0.534522
\(127\) −9.00000 −0.798621 −0.399310 0.916816i \(-0.630750\pi\)
−0.399310 + 0.916816i \(0.630750\pi\)
\(128\) 1.00000i 0.0883883i
\(129\) 36.0000 3.16962
\(130\) 0 0
\(131\) 8.00000 0.698963 0.349482 0.936943i \(-0.386358\pi\)
0.349482 + 0.936943i \(0.386358\pi\)
\(132\) − 15.0000i − 1.30558i
\(133\) 2.00000 0.173422
\(134\) 11.0000 0.950255
\(135\) 0 0
\(136\) 4.00000i 0.342997i
\(137\) − 18.0000i − 1.53784i −0.639343 0.768922i \(-0.720793\pi\)
0.639343 0.768922i \(-0.279207\pi\)
\(138\) 15.0000i 1.27688i
\(139\) 4.00000 0.339276 0.169638 0.985506i \(-0.445740\pi\)
0.169638 + 0.985506i \(0.445740\pi\)
\(140\) 0 0
\(141\) 21.0000i 1.76852i
\(142\) 0 0
\(143\) 0 0
\(144\) 6.00000 0.500000
\(145\) 0 0
\(146\) −7.00000 −0.579324
\(147\) −3.00000 −0.247436
\(148\) 7.00000i 0.575396i
\(149\) 7.00000i 0.573462i 0.958011 + 0.286731i \(0.0925686\pi\)
−0.958011 + 0.286731i \(0.907431\pi\)
\(150\) − 15.0000i − 1.22474i
\(151\) 12.0000i 0.976546i 0.872691 + 0.488273i \(0.162373\pi\)
−0.872691 + 0.488273i \(0.837627\pi\)
\(152\) −2.00000 −0.162221
\(153\) 24.0000 1.94029
\(154\) − 5.00000i − 0.402911i
\(155\) 0 0
\(156\) 0 0
\(157\) 1.00000 0.0798087 0.0399043 0.999204i \(-0.487295\pi\)
0.0399043 + 0.999204i \(0.487295\pi\)
\(158\) 17.0000i 1.35245i
\(159\) −12.0000 −0.951662
\(160\) 0 0
\(161\) 5.00000i 0.394055i
\(162\) − 9.00000i − 0.707107i
\(163\) − 4.00000i − 0.313304i −0.987654 0.156652i \(-0.949930\pi\)
0.987654 0.156652i \(-0.0500701\pi\)
\(164\) 9.00000i 0.702782i
\(165\) 0 0
\(166\) 4.00000 0.310460
\(167\) − 8.00000i − 0.619059i −0.950890 0.309529i \(-0.899829\pi\)
0.950890 0.309529i \(-0.100171\pi\)
\(168\) 3.00000 0.231455
\(169\) 0 0
\(170\) 0 0
\(171\) 12.0000i 0.917663i
\(172\) −12.0000 −0.914991
\(173\) −18.0000 −1.36851 −0.684257 0.729241i \(-0.739873\pi\)
−0.684257 + 0.729241i \(0.739873\pi\)
\(174\) − 12.0000i − 0.909718i
\(175\) − 5.00000i − 0.377964i
\(176\) 5.00000i 0.376889i
\(177\) 18.0000i 1.35296i
\(178\) −14.0000 −1.04934
\(179\) 2.00000 0.149487 0.0747435 0.997203i \(-0.476186\pi\)
0.0747435 + 0.997203i \(0.476186\pi\)
\(180\) 0 0
\(181\) 5.00000 0.371647 0.185824 0.982583i \(-0.440505\pi\)
0.185824 + 0.982583i \(0.440505\pi\)
\(182\) 0 0
\(183\) 39.0000 2.88296
\(184\) − 5.00000i − 0.368605i
\(185\) 0 0
\(186\) 3.00000 0.219971
\(187\) 20.0000i 1.46254i
\(188\) − 7.00000i − 0.510527i
\(189\) − 9.00000i − 0.654654i
\(190\) 0 0
\(191\) −16.0000 −1.15772 −0.578860 0.815427i \(-0.696502\pi\)
−0.578860 + 0.815427i \(0.696502\pi\)
\(192\) −3.00000 −0.216506
\(193\) − 4.00000i − 0.287926i −0.989583 0.143963i \(-0.954015\pi\)
0.989583 0.143963i \(-0.0459847\pi\)
\(194\) 5.00000 0.358979
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) 27.0000i 1.92367i 0.273629 + 0.961835i \(0.411776\pi\)
−0.273629 + 0.961835i \(0.588224\pi\)
\(198\) 30.0000 2.13201
\(199\) 20.0000 1.41776 0.708881 0.705328i \(-0.249200\pi\)
0.708881 + 0.705328i \(0.249200\pi\)
\(200\) 5.00000i 0.353553i
\(201\) 33.0000i 2.32764i
\(202\) 15.0000i 1.05540i
\(203\) − 4.00000i − 0.280745i
\(204\) −12.0000 −0.840168
\(205\) 0 0
\(206\) 6.00000i 0.418040i
\(207\) −30.0000 −2.08514
\(208\) 0 0
\(209\) −10.0000 −0.691714
\(210\) 0 0
\(211\) −14.0000 −0.963800 −0.481900 0.876226i \(-0.660053\pi\)
−0.481900 + 0.876226i \(0.660053\pi\)
\(212\) 4.00000 0.274721
\(213\) 0 0
\(214\) 8.00000i 0.546869i
\(215\) 0 0
\(216\) 9.00000i 0.612372i
\(217\) 1.00000 0.0678844
\(218\) −18.0000 −1.21911
\(219\) − 21.0000i − 1.41905i
\(220\) 0 0
\(221\) 0 0
\(222\) −21.0000 −1.40943
\(223\) 3.00000i 0.200895i 0.994942 + 0.100447i \(0.0320274\pi\)
−0.994942 + 0.100447i \(0.967973\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 30.0000 2.00000
\(226\) − 1.00000i − 0.0665190i
\(227\) − 12.0000i − 0.796468i −0.917284 0.398234i \(-0.869623\pi\)
0.917284 0.398234i \(-0.130377\pi\)
\(228\) − 6.00000i − 0.397360i
\(229\) − 16.0000i − 1.05731i −0.848837 0.528655i \(-0.822697\pi\)
0.848837 0.528655i \(-0.177303\pi\)
\(230\) 0 0
\(231\) 15.0000 0.986928
\(232\) 4.00000i 0.262613i
\(233\) 21.0000 1.37576 0.687878 0.725826i \(-0.258542\pi\)
0.687878 + 0.725826i \(0.258542\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) − 6.00000i − 0.390567i
\(237\) −51.0000 −3.31281
\(238\) −4.00000 −0.259281
\(239\) − 6.00000i − 0.388108i −0.980991 0.194054i \(-0.937836\pi\)
0.980991 0.194054i \(-0.0621637\pi\)
\(240\) 0 0
\(241\) 30.0000i 1.93247i 0.257663 + 0.966235i \(0.417048\pi\)
−0.257663 + 0.966235i \(0.582952\pi\)
\(242\) 14.0000i 0.899954i
\(243\) 0 0
\(244\) −13.0000 −0.832240
\(245\) 0 0
\(246\) −27.0000 −1.72146
\(247\) 0 0
\(248\) −1.00000 −0.0635001
\(249\) 12.0000i 0.760469i
\(250\) 0 0
\(251\) 13.0000 0.820553 0.410276 0.911961i \(-0.365432\pi\)
0.410276 + 0.911961i \(0.365432\pi\)
\(252\) 6.00000i 0.377964i
\(253\) − 25.0000i − 1.57174i
\(254\) 9.00000i 0.564710i
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −28.0000 −1.74659 −0.873296 0.487190i \(-0.838022\pi\)
−0.873296 + 0.487190i \(0.838022\pi\)
\(258\) − 36.0000i − 2.24126i
\(259\) −7.00000 −0.434959
\(260\) 0 0
\(261\) 24.0000 1.48556
\(262\) − 8.00000i − 0.494242i
\(263\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(264\) −15.0000 −0.923186
\(265\) 0 0
\(266\) − 2.00000i − 0.122628i
\(267\) − 42.0000i − 2.57036i
\(268\) − 11.0000i − 0.671932i
\(269\) 13.0000 0.792624 0.396312 0.918116i \(-0.370290\pi\)
0.396312 + 0.918116i \(0.370290\pi\)
\(270\) 0 0
\(271\) 1.00000i 0.0607457i 0.999539 + 0.0303728i \(0.00966946\pi\)
−0.999539 + 0.0303728i \(0.990331\pi\)
\(272\) 4.00000 0.242536
\(273\) 0 0
\(274\) −18.0000 −1.08742
\(275\) 25.0000i 1.50756i
\(276\) 15.0000 0.902894
\(277\) 10.0000 0.600842 0.300421 0.953807i \(-0.402873\pi\)
0.300421 + 0.953807i \(0.402873\pi\)
\(278\) − 4.00000i − 0.239904i
\(279\) 6.00000i 0.359211i
\(280\) 0 0
\(281\) 16.0000i 0.954480i 0.878773 + 0.477240i \(0.158363\pi\)
−0.878773 + 0.477240i \(0.841637\pi\)
\(282\) 21.0000 1.25053
\(283\) −3.00000 −0.178331 −0.0891657 0.996017i \(-0.528420\pi\)
−0.0891657 + 0.996017i \(0.528420\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −9.00000 −0.531253
\(288\) − 6.00000i − 0.353553i
\(289\) −1.00000 −0.0588235
\(290\) 0 0
\(291\) 15.0000i 0.879316i
\(292\) 7.00000i 0.409644i
\(293\) 6.00000i 0.350524i 0.984522 + 0.175262i \(0.0560772\pi\)
−0.984522 + 0.175262i \(0.943923\pi\)
\(294\) 3.00000i 0.174964i
\(295\) 0 0
\(296\) 7.00000 0.406867
\(297\) 45.0000i 2.61116i
\(298\) 7.00000 0.405499
\(299\) 0 0
\(300\) −15.0000 −0.866025
\(301\) − 12.0000i − 0.691669i
\(302\) 12.0000 0.690522
\(303\) −45.0000 −2.58518
\(304\) 2.00000i 0.114708i
\(305\) 0 0
\(306\) − 24.0000i − 1.37199i
\(307\) − 32.0000i − 1.82634i −0.407583 0.913168i \(-0.633628\pi\)
0.407583 0.913168i \(-0.366372\pi\)
\(308\) −5.00000 −0.284901
\(309\) −18.0000 −1.02398
\(310\) 0 0
\(311\) −26.0000 −1.47432 −0.737162 0.675716i \(-0.763835\pi\)
−0.737162 + 0.675716i \(0.763835\pi\)
\(312\) 0 0
\(313\) −30.0000 −1.69570 −0.847850 0.530236i \(-0.822103\pi\)
−0.847850 + 0.530236i \(0.822103\pi\)
\(314\) − 1.00000i − 0.0564333i
\(315\) 0 0
\(316\) 17.0000 0.956325
\(317\) 11.0000i 0.617822i 0.951091 + 0.308911i \(0.0999645\pi\)
−0.951091 + 0.308911i \(0.900036\pi\)
\(318\) 12.0000i 0.672927i
\(319\) 20.0000i 1.11979i
\(320\) 0 0
\(321\) −24.0000 −1.33955
\(322\) 5.00000 0.278639
\(323\) 8.00000i 0.445132i
\(324\) −9.00000 −0.500000
\(325\) 0 0
\(326\) −4.00000 −0.221540
\(327\) − 54.0000i − 2.98621i
\(328\) 9.00000 0.496942
\(329\) 7.00000 0.385922
\(330\) 0 0
\(331\) 3.00000i 0.164895i 0.996595 + 0.0824475i \(0.0262737\pi\)
−0.996595 + 0.0824475i \(0.973726\pi\)
\(332\) − 4.00000i − 0.219529i
\(333\) − 42.0000i − 2.30159i
\(334\) −8.00000 −0.437741
\(335\) 0 0
\(336\) − 3.00000i − 0.163663i
\(337\) −9.00000 −0.490261 −0.245131 0.969490i \(-0.578831\pi\)
−0.245131 + 0.969490i \(0.578831\pi\)
\(338\) 0 0
\(339\) 3.00000 0.162938
\(340\) 0 0
\(341\) −5.00000 −0.270765
\(342\) 12.0000 0.648886
\(343\) 1.00000i 0.0539949i
\(344\) 12.0000i 0.646997i
\(345\) 0 0
\(346\) 18.0000i 0.967686i
\(347\) 16.0000 0.858925 0.429463 0.903085i \(-0.358703\pi\)
0.429463 + 0.903085i \(0.358703\pi\)
\(348\) −12.0000 −0.643268
\(349\) − 2.00000i − 0.107058i −0.998566 0.0535288i \(-0.982953\pi\)
0.998566 0.0535288i \(-0.0170469\pi\)
\(350\) −5.00000 −0.267261
\(351\) 0 0
\(352\) 5.00000 0.266501
\(353\) 3.00000i 0.159674i 0.996808 + 0.0798369i \(0.0254400\pi\)
−0.996808 + 0.0798369i \(0.974560\pi\)
\(354\) 18.0000 0.956689
\(355\) 0 0
\(356\) 14.0000i 0.741999i
\(357\) − 12.0000i − 0.635107i
\(358\) − 2.00000i − 0.105703i
\(359\) 4.00000i 0.211112i 0.994413 + 0.105556i \(0.0336622\pi\)
−0.994413 + 0.105556i \(0.966338\pi\)
\(360\) 0 0
\(361\) 15.0000 0.789474
\(362\) − 5.00000i − 0.262794i
\(363\) −42.0000 −2.20443
\(364\) 0 0
\(365\) 0 0
\(366\) − 39.0000i − 2.03856i
\(367\) 8.00000 0.417597 0.208798 0.977959i \(-0.433045\pi\)
0.208798 + 0.977959i \(0.433045\pi\)
\(368\) −5.00000 −0.260643
\(369\) − 54.0000i − 2.81113i
\(370\) 0 0
\(371\) 4.00000i 0.207670i
\(372\) − 3.00000i − 0.155543i
\(373\) 4.00000 0.207112 0.103556 0.994624i \(-0.466978\pi\)
0.103556 + 0.994624i \(0.466978\pi\)
\(374\) 20.0000 1.03418
\(375\) 0 0
\(376\) −7.00000 −0.360997
\(377\) 0 0
\(378\) −9.00000 −0.462910
\(379\) − 16.0000i − 0.821865i −0.911666 0.410932i \(-0.865203\pi\)
0.911666 0.410932i \(-0.134797\pi\)
\(380\) 0 0
\(381\) −27.0000 −1.38325
\(382\) 16.0000i 0.818631i
\(383\) − 9.00000i − 0.459879i −0.973205 0.229939i \(-0.926147\pi\)
0.973205 0.229939i \(-0.0738528\pi\)
\(384\) 3.00000i 0.153093i
\(385\) 0 0
\(386\) −4.00000 −0.203595
\(387\) 72.0000 3.65997
\(388\) − 5.00000i − 0.253837i
\(389\) −8.00000 −0.405616 −0.202808 0.979219i \(-0.565007\pi\)
−0.202808 + 0.979219i \(0.565007\pi\)
\(390\) 0 0
\(391\) −20.0000 −1.01144
\(392\) − 1.00000i − 0.0505076i
\(393\) 24.0000 1.21064
\(394\) 27.0000 1.36024
\(395\) 0 0
\(396\) − 30.0000i − 1.50756i
\(397\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(398\) − 20.0000i − 1.00251i
\(399\) 6.00000 0.300376
\(400\) 5.00000 0.250000
\(401\) 28.0000i 1.39825i 0.714998 + 0.699127i \(0.246428\pi\)
−0.714998 + 0.699127i \(0.753572\pi\)
\(402\) 33.0000 1.64589
\(403\) 0 0
\(404\) 15.0000 0.746278
\(405\) 0 0
\(406\) −4.00000 −0.198517
\(407\) 35.0000 1.73489
\(408\) 12.0000i 0.594089i
\(409\) − 6.00000i − 0.296681i −0.988936 0.148340i \(-0.952607\pi\)
0.988936 0.148340i \(-0.0473931\pi\)
\(410\) 0 0
\(411\) − 54.0000i − 2.66362i
\(412\) 6.00000 0.295599
\(413\) 6.00000 0.295241
\(414\) 30.0000i 1.47442i
\(415\) 0 0
\(416\) 0 0
\(417\) 12.0000 0.587643
\(418\) 10.0000i 0.489116i
\(419\) −9.00000 −0.439679 −0.219839 0.975536i \(-0.570553\pi\)
−0.219839 + 0.975536i \(0.570553\pi\)
\(420\) 0 0
\(421\) − 35.0000i − 1.70580i −0.522078 0.852898i \(-0.674843\pi\)
0.522078 0.852898i \(-0.325157\pi\)
\(422\) 14.0000i 0.681509i
\(423\) 42.0000i 2.04211i
\(424\) − 4.00000i − 0.194257i
\(425\) 20.0000 0.970143
\(426\) 0 0
\(427\) − 13.0000i − 0.629114i
\(428\) 8.00000 0.386695
\(429\) 0 0
\(430\) 0 0
\(431\) − 6.00000i − 0.289010i −0.989504 0.144505i \(-0.953841\pi\)
0.989504 0.144505i \(-0.0461589\pi\)
\(432\) 9.00000 0.433013
\(433\) −26.0000 −1.24948 −0.624740 0.780833i \(-0.714795\pi\)
−0.624740 + 0.780833i \(0.714795\pi\)
\(434\) − 1.00000i − 0.0480015i
\(435\) 0 0
\(436\) 18.0000i 0.862044i
\(437\) − 10.0000i − 0.478365i
\(438\) −21.0000 −1.00342
\(439\) 2.00000 0.0954548 0.0477274 0.998860i \(-0.484802\pi\)
0.0477274 + 0.998860i \(0.484802\pi\)
\(440\) 0 0
\(441\) −6.00000 −0.285714
\(442\) 0 0
\(443\) −12.0000 −0.570137 −0.285069 0.958507i \(-0.592016\pi\)
−0.285069 + 0.958507i \(0.592016\pi\)
\(444\) 21.0000i 0.996616i
\(445\) 0 0
\(446\) 3.00000 0.142054
\(447\) 21.0000i 0.993266i
\(448\) 1.00000i 0.0472456i
\(449\) − 18.0000i − 0.849473i −0.905317 0.424736i \(-0.860367\pi\)
0.905317 0.424736i \(-0.139633\pi\)
\(450\) − 30.0000i − 1.41421i
\(451\) 45.0000 2.11897
\(452\) −1.00000 −0.0470360
\(453\) 36.0000i 1.69143i
\(454\) −12.0000 −0.563188
\(455\) 0 0
\(456\) −6.00000 −0.280976
\(457\) − 10.0000i − 0.467780i −0.972263 0.233890i \(-0.924854\pi\)
0.972263 0.233890i \(-0.0751456\pi\)
\(458\) −16.0000 −0.747631
\(459\) 36.0000 1.68034
\(460\) 0 0
\(461\) 12.0000i 0.558896i 0.960161 + 0.279448i \(0.0901514\pi\)
−0.960161 + 0.279448i \(0.909849\pi\)
\(462\) − 15.0000i − 0.697863i
\(463\) − 12.0000i − 0.557687i −0.960337 0.278844i \(-0.910049\pi\)
0.960337 0.278844i \(-0.0899511\pi\)
\(464\) 4.00000 0.185695
\(465\) 0 0
\(466\) − 21.0000i − 0.972806i
\(467\) 36.0000 1.66588 0.832941 0.553362i \(-0.186655\pi\)
0.832941 + 0.553362i \(0.186655\pi\)
\(468\) 0 0
\(469\) 11.0000 0.507933
\(470\) 0 0
\(471\) 3.00000 0.138233
\(472\) −6.00000 −0.276172
\(473\) 60.0000i 2.75880i
\(474\) 51.0000i 2.34251i
\(475\) 10.0000i 0.458831i
\(476\) 4.00000i 0.183340i
\(477\) −24.0000 −1.09888
\(478\) −6.00000 −0.274434
\(479\) 16.0000i 0.731059i 0.930800 + 0.365529i \(0.119112\pi\)
−0.930800 + 0.365529i \(0.880888\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 30.0000 1.36646
\(483\) 15.0000i 0.682524i
\(484\) 14.0000 0.636364
\(485\) 0 0
\(486\) 0 0
\(487\) − 26.0000i − 1.17817i −0.808070 0.589086i \(-0.799488\pi\)
0.808070 0.589086i \(-0.200512\pi\)
\(488\) 13.0000i 0.588482i
\(489\) − 12.0000i − 0.542659i
\(490\) 0 0
\(491\) −28.0000 −1.26362 −0.631811 0.775122i \(-0.717688\pi\)
−0.631811 + 0.775122i \(0.717688\pi\)
\(492\) 27.0000i 1.21725i
\(493\) 16.0000 0.720604
\(494\) 0 0
\(495\) 0 0
\(496\) 1.00000i 0.0449013i
\(497\) 0 0
\(498\) 12.0000 0.537733
\(499\) 1.00000i 0.0447661i 0.999749 + 0.0223831i \(0.00712535\pi\)
−0.999749 + 0.0223831i \(0.992875\pi\)
\(500\) 0 0
\(501\) − 24.0000i − 1.07224i
\(502\) − 13.0000i − 0.580218i
\(503\) −40.0000 −1.78351 −0.891756 0.452517i \(-0.850526\pi\)
−0.891756 + 0.452517i \(0.850526\pi\)
\(504\) 6.00000 0.267261
\(505\) 0 0
\(506\) −25.0000 −1.11139
\(507\) 0 0
\(508\) 9.00000 0.399310
\(509\) 6.00000i 0.265945i 0.991120 + 0.132973i \(0.0424523\pi\)
−0.991120 + 0.132973i \(0.957548\pi\)
\(510\) 0 0
\(511\) −7.00000 −0.309662
\(512\) − 1.00000i − 0.0441942i
\(513\) 18.0000i 0.794719i
\(514\) 28.0000i 1.23503i
\(515\) 0 0
\(516\) −36.0000 −1.58481
\(517\) −35.0000 −1.53930
\(518\) 7.00000i 0.307562i
\(519\) −54.0000 −2.37034
\(520\) 0 0
\(521\) −14.0000 −0.613351 −0.306676 0.951814i \(-0.599217\pi\)
−0.306676 + 0.951814i \(0.599217\pi\)
\(522\) − 24.0000i − 1.05045i
\(523\) −13.0000 −0.568450 −0.284225 0.958758i \(-0.591736\pi\)
−0.284225 + 0.958758i \(0.591736\pi\)
\(524\) −8.00000 −0.349482
\(525\) − 15.0000i − 0.654654i
\(526\) 0 0
\(527\) 4.00000i 0.174243i
\(528\) 15.0000i 0.652791i
\(529\) 2.00000 0.0869565
\(530\) 0 0
\(531\) 36.0000i 1.56227i
\(532\) −2.00000 −0.0867110
\(533\) 0 0
\(534\) −42.0000 −1.81752
\(535\) 0 0
\(536\) −11.0000 −0.475128
\(537\) 6.00000 0.258919
\(538\) − 13.0000i − 0.560470i
\(539\) − 5.00000i − 0.215365i
\(540\) 0 0
\(541\) − 18.0000i − 0.773880i −0.922105 0.386940i \(-0.873532\pi\)
0.922105 0.386940i \(-0.126468\pi\)
\(542\) 1.00000 0.0429537
\(543\) 15.0000 0.643712
\(544\) − 4.00000i − 0.171499i
\(545\) 0 0
\(546\) 0 0
\(547\) −32.0000 −1.36822 −0.684111 0.729378i \(-0.739809\pi\)
−0.684111 + 0.729378i \(0.739809\pi\)
\(548\) 18.0000i 0.768922i
\(549\) 78.0000 3.32896
\(550\) 25.0000 1.06600
\(551\) 8.00000i 0.340811i
\(552\) − 15.0000i − 0.638442i
\(553\) 17.0000i 0.722914i
\(554\) − 10.0000i − 0.424859i
\(555\) 0 0
\(556\) −4.00000 −0.169638
\(557\) 9.00000i 0.381342i 0.981654 + 0.190671i \(0.0610664\pi\)
−0.981654 + 0.190671i \(0.938934\pi\)
\(558\) 6.00000 0.254000
\(559\) 0 0
\(560\) 0 0
\(561\) 60.0000i 2.53320i
\(562\) 16.0000 0.674919
\(563\) −35.0000 −1.47507 −0.737537 0.675307i \(-0.764011\pi\)
−0.737537 + 0.675307i \(0.764011\pi\)
\(564\) − 21.0000i − 0.884260i
\(565\) 0 0
\(566\) 3.00000i 0.126099i
\(567\) − 9.00000i − 0.377964i
\(568\) 0 0
\(569\) −21.0000 −0.880366 −0.440183 0.897908i \(-0.645086\pi\)
−0.440183 + 0.897908i \(0.645086\pi\)
\(570\) 0 0
\(571\) −22.0000 −0.920671 −0.460336 0.887745i \(-0.652271\pi\)
−0.460336 + 0.887745i \(0.652271\pi\)
\(572\) 0 0
\(573\) −48.0000 −2.00523
\(574\) 9.00000i 0.375653i
\(575\) −25.0000 −1.04257
\(576\) −6.00000 −0.250000
\(577\) 2.00000i 0.0832611i 0.999133 + 0.0416305i \(0.0132552\pi\)
−0.999133 + 0.0416305i \(0.986745\pi\)
\(578\) 1.00000i 0.0415945i
\(579\) − 12.0000i − 0.498703i
\(580\) 0 0
\(581\) 4.00000 0.165948
\(582\) 15.0000 0.621770
\(583\) − 20.0000i − 0.828315i
\(584\) 7.00000 0.289662
\(585\) 0 0
\(586\) 6.00000 0.247858
\(587\) 18.0000i 0.742940i 0.928445 + 0.371470i \(0.121146\pi\)
−0.928445 + 0.371470i \(0.878854\pi\)
\(588\) 3.00000 0.123718
\(589\) −2.00000 −0.0824086
\(590\) 0 0
\(591\) 81.0000i 3.33189i
\(592\) − 7.00000i − 0.287698i
\(593\) 46.0000i 1.88899i 0.328521 + 0.944497i \(0.393450\pi\)
−0.328521 + 0.944497i \(0.606550\pi\)
\(594\) 45.0000 1.84637
\(595\) 0 0
\(596\) − 7.00000i − 0.286731i
\(597\) 60.0000 2.45564
\(598\) 0 0
\(599\) −9.00000 −0.367730 −0.183865 0.982952i \(-0.558861\pi\)
−0.183865 + 0.982952i \(0.558861\pi\)
\(600\) 15.0000i 0.612372i
\(601\) 26.0000 1.06056 0.530281 0.847822i \(-0.322086\pi\)
0.530281 + 0.847822i \(0.322086\pi\)
\(602\) −12.0000 −0.489083
\(603\) 66.0000i 2.68773i
\(604\) − 12.0000i − 0.488273i
\(605\) 0 0
\(606\) 45.0000i 1.82800i
\(607\) 22.0000 0.892952 0.446476 0.894795i \(-0.352679\pi\)
0.446476 + 0.894795i \(0.352679\pi\)
\(608\) 2.00000 0.0811107
\(609\) − 12.0000i − 0.486265i
\(610\) 0 0
\(611\) 0 0
\(612\) −24.0000 −0.970143
\(613\) − 23.0000i − 0.928961i −0.885583 0.464481i \(-0.846241\pi\)
0.885583 0.464481i \(-0.153759\pi\)
\(614\) −32.0000 −1.29141
\(615\) 0 0
\(616\) 5.00000i 0.201456i
\(617\) 6.00000i 0.241551i 0.992680 + 0.120775i \(0.0385381\pi\)
−0.992680 + 0.120775i \(0.961462\pi\)
\(618\) 18.0000i 0.724066i
\(619\) − 6.00000i − 0.241160i −0.992704 0.120580i \(-0.961525\pi\)
0.992704 0.120580i \(-0.0384755\pi\)
\(620\) 0 0
\(621\) −45.0000 −1.80579
\(622\) 26.0000i 1.04251i
\(623\) −14.0000 −0.560898
\(624\) 0 0
\(625\) 25.0000 1.00000
\(626\) 30.0000i 1.19904i
\(627\) −30.0000 −1.19808
\(628\) −1.00000 −0.0399043
\(629\) − 28.0000i − 1.11643i
\(630\) 0 0
\(631\) 26.0000i 1.03504i 0.855670 + 0.517522i \(0.173145\pi\)
−0.855670 + 0.517522i \(0.826855\pi\)
\(632\) − 17.0000i − 0.676224i
\(633\) −42.0000 −1.66935
\(634\) 11.0000 0.436866
\(635\) 0 0
\(636\) 12.0000 0.475831
\(637\) 0 0
\(638\) 20.0000 0.791808
\(639\) 0 0
\(640\) 0 0
\(641\) 33.0000 1.30342 0.651711 0.758468i \(-0.274052\pi\)
0.651711 + 0.758468i \(0.274052\pi\)
\(642\) 24.0000i 0.947204i
\(643\) − 36.0000i − 1.41970i −0.704352 0.709851i \(-0.748762\pi\)
0.704352 0.709851i \(-0.251238\pi\)
\(644\) − 5.00000i − 0.197028i
\(645\) 0 0
\(646\) 8.00000 0.314756
\(647\) 12.0000 0.471769 0.235884 0.971781i \(-0.424201\pi\)
0.235884 + 0.971781i \(0.424201\pi\)
\(648\) 9.00000i 0.353553i
\(649\) −30.0000 −1.17760
\(650\) 0 0
\(651\) 3.00000 0.117579
\(652\) 4.00000i 0.156652i
\(653\) −12.0000 −0.469596 −0.234798 0.972044i \(-0.575443\pi\)
−0.234798 + 0.972044i \(0.575443\pi\)
\(654\) −54.0000 −2.11157
\(655\) 0 0
\(656\) − 9.00000i − 0.351391i
\(657\) − 42.0000i − 1.63858i
\(658\) − 7.00000i − 0.272888i
\(659\) 26.0000 1.01282 0.506408 0.862294i \(-0.330973\pi\)
0.506408 + 0.862294i \(0.330973\pi\)
\(660\) 0 0
\(661\) 20.0000i 0.777910i 0.921257 + 0.388955i \(0.127164\pi\)
−0.921257 + 0.388955i \(0.872836\pi\)
\(662\) 3.00000 0.116598
\(663\) 0 0
\(664\) −4.00000 −0.155230
\(665\) 0 0
\(666\) −42.0000 −1.62747
\(667\) −20.0000 −0.774403
\(668\) 8.00000i 0.309529i
\(669\) 9.00000i 0.347960i
\(670\) 0 0
\(671\) 65.0000i 2.50930i
\(672\) −3.00000 −0.115728
\(673\) 9.00000 0.346925 0.173462 0.984841i \(-0.444505\pi\)
0.173462 + 0.984841i \(0.444505\pi\)
\(674\) 9.00000i 0.346667i
\(675\) 45.0000 1.73205
\(676\) 0 0
\(677\) 3.00000 0.115299 0.0576497 0.998337i \(-0.481639\pi\)
0.0576497 + 0.998337i \(0.481639\pi\)
\(678\) − 3.00000i − 0.115214i
\(679\) 5.00000 0.191882
\(680\) 0 0
\(681\) − 36.0000i − 1.37952i
\(682\) 5.00000i 0.191460i
\(683\) − 3.00000i − 0.114792i −0.998351 0.0573959i \(-0.981720\pi\)
0.998351 0.0573959i \(-0.0182797\pi\)
\(684\) − 12.0000i − 0.458831i
\(685\) 0 0
\(686\) 1.00000 0.0381802
\(687\) − 48.0000i − 1.83131i
\(688\) 12.0000 0.457496
\(689\) 0 0
\(690\) 0 0
\(691\) − 36.0000i − 1.36950i −0.728776 0.684752i \(-0.759910\pi\)
0.728776 0.684752i \(-0.240090\pi\)
\(692\) 18.0000 0.684257
\(693\) 30.0000 1.13961
\(694\) − 16.0000i − 0.607352i
\(695\) 0 0
\(696\) 12.0000i 0.454859i
\(697\) − 36.0000i − 1.36360i
\(698\) −2.00000 −0.0757011
\(699\) 63.0000 2.38288
\(700\) 5.00000i 0.188982i
\(701\) 12.0000 0.453234 0.226617 0.973984i \(-0.427233\pi\)
0.226617 + 0.973984i \(0.427233\pi\)
\(702\) 0 0
\(703\) 14.0000 0.528020
\(704\) − 5.00000i − 0.188445i
\(705\) 0 0
\(706\) 3.00000 0.112906
\(707\) 15.0000i 0.564133i
\(708\) − 18.0000i − 0.676481i
\(709\) − 29.0000i − 1.08912i −0.838723 0.544559i \(-0.816697\pi\)
0.838723 0.544559i \(-0.183303\pi\)
\(710\) 0 0
\(711\) −102.000 −3.82530
\(712\) 14.0000 0.524672
\(713\) − 5.00000i − 0.187251i
\(714\) −12.0000 −0.449089
\(715\) 0 0
\(716\) −2.00000 −0.0747435
\(717\) − 18.0000i − 0.672222i
\(718\) 4.00000 0.149279
\(719\) 18.0000 0.671287 0.335643 0.941989i \(-0.391046\pi\)
0.335643 + 0.941989i \(0.391046\pi\)
\(720\) 0 0
\(721\) 6.00000i 0.223452i
\(722\) − 15.0000i − 0.558242i
\(723\) 90.0000i 3.34714i
\(724\) −5.00000 −0.185824
\(725\) 20.0000 0.742781
\(726\) 42.0000i 1.55877i
\(727\) −14.0000 −0.519231 −0.259616 0.965712i \(-0.583596\pi\)
−0.259616 + 0.965712i \(0.583596\pi\)
\(728\) 0 0
\(729\) −27.0000 −1.00000
\(730\) 0 0
\(731\) 48.0000 1.77534
\(732\) −39.0000 −1.44148
\(733\) 20.0000i 0.738717i 0.929287 + 0.369358i \(0.120423\pi\)
−0.929287 + 0.369358i \(0.879577\pi\)
\(734\) − 8.00000i − 0.295285i
\(735\) 0 0
\(736\) 5.00000i 0.184302i
\(737\) −55.0000 −2.02595
\(738\) −54.0000 −1.98777
\(739\) 16.0000i 0.588570i 0.955718 + 0.294285i \(0.0950814\pi\)
−0.955718 + 0.294285i \(0.904919\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 4.00000 0.146845
\(743\) − 36.0000i − 1.32071i −0.750953 0.660356i \(-0.770405\pi\)
0.750953 0.660356i \(-0.229595\pi\)
\(744\) −3.00000 −0.109985
\(745\) 0 0
\(746\) − 4.00000i − 0.146450i
\(747\) 24.0000i 0.878114i
\(748\) − 20.0000i − 0.731272i
\(749\) 8.00000i 0.292314i
\(750\) 0 0
\(751\) −25.0000 −0.912263 −0.456131 0.889912i \(-0.650765\pi\)
−0.456131 + 0.889912i \(0.650765\pi\)
\(752\) 7.00000i 0.255264i
\(753\) 39.0000 1.42124
\(754\) 0 0
\(755\) 0 0
\(756\) 9.00000i 0.327327i
\(757\) −18.0000 −0.654221 −0.327111 0.944986i \(-0.606075\pi\)
−0.327111 + 0.944986i \(0.606075\pi\)
\(758\) −16.0000 −0.581146
\(759\) − 75.0000i − 2.72233i
\(760\) 0 0
\(761\) 33.0000i 1.19625i 0.801403 + 0.598125i \(0.204087\pi\)
−0.801403 + 0.598125i \(0.795913\pi\)
\(762\) 27.0000i 0.978107i
\(763\) −18.0000 −0.651644
\(764\) 16.0000 0.578860
\(765\) 0 0
\(766\) −9.00000 −0.325183
\(767\) 0 0
\(768\) 3.00000 0.108253
\(769\) − 1.00000i − 0.0360609i −0.999837 0.0180305i \(-0.994260\pi\)
0.999837 0.0180305i \(-0.00573959\pi\)
\(770\) 0 0
\(771\) −84.0000 −3.02519
\(772\) 4.00000i 0.143963i
\(773\) 34.0000i 1.22290i 0.791285 + 0.611448i \(0.209412\pi\)
−0.791285 + 0.611448i \(0.790588\pi\)
\(774\) − 72.0000i − 2.58799i
\(775\) 5.00000i 0.179605i
\(776\) −5.00000 −0.179490
\(777\) −21.0000 −0.753371
\(778\) 8.00000i 0.286814i
\(779\) 18.0000 0.644917
\(780\) 0 0
\(781\) 0 0
\(782\) 20.0000i 0.715199i
\(783\) 36.0000 1.28654
\(784\) −1.00000 −0.0357143
\(785\) 0 0
\(786\) − 24.0000i − 0.856052i
\(787\) − 8.00000i − 0.285169i −0.989783 0.142585i \(-0.954459\pi\)
0.989783 0.142585i \(-0.0455413\pi\)
\(788\) − 27.0000i − 0.961835i
\(789\) 0 0
\(790\) 0 0
\(791\) − 1.00000i − 0.0355559i
\(792\) −30.0000 −1.06600
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) −20.0000 −0.708881
\(797\) −3.00000 −0.106265 −0.0531327 0.998587i \(-0.516921\pi\)
−0.0531327 + 0.998587i \(0.516921\pi\)
\(798\) − 6.00000i − 0.212398i
\(799\) 28.0000i 0.990569i
\(800\) − 5.00000i − 0.176777i
\(801\) − 84.0000i − 2.96799i
\(802\) 28.0000 0.988714
\(803\) 35.0000 1.23512
\(804\) − 33.0000i − 1.16382i
\(805\) 0 0
\(806\) 0 0
\(807\) 39.0000 1.37287
\(808\) − 15.0000i − 0.527698i
\(809\) 6.00000 0.210949 0.105474 0.994422i \(-0.466364\pi\)
0.105474 + 0.994422i \(0.466364\pi\)
\(810\) 0 0
\(811\) 52.0000i 1.82597i 0.407997 + 0.912983i \(0.366228\pi\)
−0.407997 + 0.912983i \(0.633772\pi\)
\(812\) 4.00000i 0.140372i
\(813\) 3.00000i 0.105215i
\(814\) − 35.0000i − 1.22675i
\(815\) 0 0
\(816\) 12.0000 0.420084
\(817\) 24.0000i 0.839654i
\(818\) −6.00000 −0.209785
\(819\) 0 0
\(820\) 0 0
\(821\) 6.00000i 0.209401i 0.994504 + 0.104701i \(0.0333885\pi\)
−0.994504 + 0.104701i \(0.966612\pi\)
\(822\) −54.0000 −1.88347
\(823\) 15.0000 0.522867 0.261434 0.965221i \(-0.415805\pi\)
0.261434 + 0.965221i \(0.415805\pi\)
\(824\) − 6.00000i − 0.209020i
\(825\) 75.0000i 2.61116i
\(826\) − 6.00000i − 0.208767i
\(827\) 36.0000i 1.25184i 0.779886 + 0.625921i \(0.215277\pi\)
−0.779886 + 0.625921i \(0.784723\pi\)
\(828\) 30.0000 1.04257
\(829\) 54.0000 1.87550 0.937749 0.347314i \(-0.112906\pi\)
0.937749 + 0.347314i \(0.112906\pi\)
\(830\) 0 0
\(831\) 30.0000 1.04069
\(832\) 0 0
\(833\) −4.00000 −0.138592
\(834\) − 12.0000i − 0.415526i
\(835\) 0 0
\(836\) 10.0000 0.345857
\(837\) 9.00000i 0.311086i
\(838\) 9.00000i 0.310900i
\(839\) − 47.0000i − 1.62262i −0.584616 0.811310i \(-0.698755\pi\)
0.584616 0.811310i \(-0.301245\pi\)
\(840\) 0 0
\(841\) −13.0000 −0.448276
\(842\) −35.0000 −1.20618
\(843\) 48.0000i 1.65321i
\(844\) 14.0000 0.481900
\(845\) 0 0
\(846\) 42.0000 1.44399
\(847\) 14.0000i 0.481046i
\(848\) −4.00000 −0.137361
\(849\) −9.00000 −0.308879
\(850\) − 20.0000i − 0.685994i
\(851\) 35.0000i 1.19978i
\(852\) 0 0
\(853\) 16.0000i 0.547830i 0.961754 + 0.273915i \(0.0883186\pi\)
−0.961754 + 0.273915i \(0.911681\pi\)
\(854\) −13.0000 −0.444851
\(855\) 0 0
\(856\) − 8.00000i − 0.273434i
\(857\) −2.00000 −0.0683187 −0.0341593 0.999416i \(-0.510875\pi\)
−0.0341593 + 0.999416i \(0.510875\pi\)
\(858\) 0 0
\(859\) 29.0000 0.989467 0.494734 0.869045i \(-0.335266\pi\)
0.494734 + 0.869045i \(0.335266\pi\)
\(860\) 0 0
\(861\) −27.0000 −0.920158
\(862\) −6.00000 −0.204361
\(863\) − 24.0000i − 0.816970i −0.912765 0.408485i \(-0.866057\pi\)
0.912765 0.408485i \(-0.133943\pi\)
\(864\) − 9.00000i − 0.306186i
\(865\) 0 0
\(866\) 26.0000i 0.883516i
\(867\) −3.00000 −0.101885
\(868\) −1.00000 −0.0339422
\(869\) − 85.0000i − 2.88343i
\(870\) 0 0
\(871\) 0 0
\(872\) 18.0000 0.609557
\(873\) 30.0000i 1.01535i
\(874\) −10.0000 −0.338255
\(875\) 0 0
\(876\) 21.0000i 0.709524i
\(877\) 13.0000i 0.438979i 0.975615 + 0.219489i \(0.0704391\pi\)
−0.975615 + 0.219489i \(0.929561\pi\)
\(878\) − 2.00000i − 0.0674967i
\(879\) 18.0000i 0.607125i
\(880\) 0 0
\(881\) −6.00000 −0.202145 −0.101073 0.994879i \(-0.532227\pi\)
−0.101073 + 0.994879i \(0.532227\pi\)
\(882\) 6.00000i 0.202031i
\(883\) −30.0000 −1.00958 −0.504790 0.863242i \(-0.668430\pi\)
−0.504790 + 0.863242i \(0.668430\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 12.0000i 0.403148i
\(887\) 54.0000 1.81314 0.906571 0.422053i \(-0.138690\pi\)
0.906571 + 0.422053i \(0.138690\pi\)
\(888\) 21.0000 0.704714
\(889\) 9.00000i 0.301850i
\(890\) 0 0
\(891\) 45.0000i 1.50756i
\(892\) − 3.00000i − 0.100447i
\(893\) −14.0000 −0.468492
\(894\) 21.0000 0.702345
\(895\) 0 0
\(896\) 1.00000 0.0334077
\(897\) 0 0
\(898\) −18.0000 −0.600668
\(899\) 4.00000i 0.133407i
\(900\) −30.0000 −1.00000
\(901\) −16.0000 −0.533037
\(902\) − 45.0000i − 1.49834i
\(903\) − 36.0000i − 1.19800i
\(904\) 1.00000i 0.0332595i
\(905\) 0 0
\(906\) 36.0000 1.19602
\(907\) −14.0000 −0.464862 −0.232431 0.972613i \(-0.574668\pi\)
−0.232431 + 0.972613i \(0.574668\pi\)
\(908\) 12.0000i 0.398234i
\(909\) −90.0000 −2.98511
\(910\) 0 0
\(911\) 28.0000 0.927681 0.463841 0.885919i \(-0.346471\pi\)
0.463841 + 0.885919i \(0.346471\pi\)
\(912\) 6.00000i 0.198680i
\(913\) −20.0000 −0.661903
\(914\) −10.0000 −0.330771
\(915\) 0 0
\(916\) 16.0000i 0.528655i
\(917\) − 8.00000i − 0.264183i
\(918\) − 36.0000i − 1.18818i
\(919\) 31.0000 1.02260 0.511298 0.859404i \(-0.329165\pi\)
0.511298 + 0.859404i \(0.329165\pi\)
\(920\) 0 0
\(921\) − 96.0000i − 3.16331i
\(922\) 12.0000 0.395199
\(923\) 0 0
\(924\) −15.0000 −0.493464
\(925\) − 35.0000i − 1.15079i
\(926\) −12.0000 −0.394344
\(927\) −36.0000 −1.18240
\(928\) − 4.00000i − 0.131306i
\(929\) − 29.0000i − 0.951459i −0.879592 0.475730i \(-0.842184\pi\)
0.879592 0.475730i \(-0.157816\pi\)
\(930\) 0 0
\(931\) − 2.00000i − 0.0655474i
\(932\) −21.0000 −0.687878
\(933\) −78.0000 −2.55361
\(934\) − 36.0000i − 1.17796i
\(935\) 0 0
\(936\) 0 0
\(937\) −8.00000 −0.261349 −0.130674 0.991425i \(-0.541714\pi\)
−0.130674 + 0.991425i \(0.541714\pi\)
\(938\) − 11.0000i − 0.359163i
\(939\) −90.0000 −2.93704
\(940\) 0 0
\(941\) 6.00000i 0.195594i 0.995206 + 0.0977972i \(0.0311797\pi\)
−0.995206 + 0.0977972i \(0.968820\pi\)
\(942\) − 3.00000i − 0.0977453i
\(943\) 45.0000i 1.46540i
\(944\) 6.00000i 0.195283i
\(945\) 0 0
\(946\) 60.0000 1.95077
\(947\) 16.0000i 0.519930i 0.965618 + 0.259965i \(0.0837111\pi\)
−0.965618 + 0.259965i \(0.916289\pi\)
\(948\) 51.0000 1.65640
\(949\) 0 0
\(950\) 10.0000 0.324443
\(951\) 33.0000i 1.07010i
\(952\) 4.00000 0.129641
\(953\) −6.00000 −0.194359 −0.0971795 0.995267i \(-0.530982\pi\)
−0.0971795 + 0.995267i \(0.530982\pi\)
\(954\) 24.0000i 0.777029i
\(955\) 0 0
\(956\) 6.00000i 0.194054i
\(957\) 60.0000i 1.93952i
\(958\) 16.0000 0.516937
\(959\) −18.0000 −0.581250
\(960\) 0 0
\(961\) 30.0000 0.967742
\(962\) 0 0
\(963\) −48.0000 −1.54678
\(964\) − 30.0000i − 0.966235i
\(965\) 0 0
\(966\) 15.0000 0.482617
\(967\) 2.00000i 0.0643157i 0.999483 + 0.0321578i \(0.0102379\pi\)
−0.999483 + 0.0321578i \(0.989762\pi\)
\(968\) − 14.0000i − 0.449977i
\(969\) 24.0000i 0.770991i
\(970\) 0 0
\(971\) 9.00000 0.288824 0.144412 0.989518i \(-0.453871\pi\)
0.144412 + 0.989518i \(0.453871\pi\)
\(972\) 0 0
\(973\) − 4.00000i − 0.128234i
\(974\) −26.0000 −0.833094
\(975\) 0 0
\(976\) 13.0000 0.416120
\(977\) − 26.0000i − 0.831814i −0.909407 0.415907i \(-0.863464\pi\)
0.909407 0.415907i \(-0.136536\pi\)
\(978\) −12.0000 −0.383718
\(979\) 70.0000 2.23721
\(980\) 0 0
\(981\) − 108.000i − 3.44817i
\(982\) 28.0000i 0.893516i
\(983\) 56.0000i 1.78612i 0.449935 + 0.893061i \(0.351447\pi\)
−0.449935 + 0.893061i \(0.648553\pi\)
\(984\) 27.0000 0.860729
\(985\) 0 0
\(986\) − 16.0000i − 0.509544i
\(987\) 21.0000 0.668437
\(988\) 0 0
\(989\) −60.0000 −1.90789
\(990\) 0 0
\(991\) 59.0000 1.87420 0.937098 0.349065i \(-0.113501\pi\)
0.937098 + 0.349065i \(0.113501\pi\)
\(992\) 1.00000 0.0317500
\(993\) 9.00000i 0.285606i
\(994\) 0 0
\(995\) 0 0
\(996\) − 12.0000i − 0.380235i
\(997\) 33.0000 1.04512 0.522560 0.852602i \(-0.324977\pi\)
0.522560 + 0.852602i \(0.324977\pi\)
\(998\) 1.00000 0.0316544
\(999\) − 63.0000i − 1.99323i
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 2366.2.d.i.337.1 2
13.5 odd 4 182.2.a.b.1.1 1
13.8 odd 4 2366.2.a.o.1.1 1
13.12 even 2 inner 2366.2.d.i.337.2 2
39.5 even 4 1638.2.a.q.1.1 1
52.31 even 4 1456.2.a.b.1.1 1
65.44 odd 4 4550.2.a.o.1.1 1
91.5 even 12 1274.2.f.u.1145.1 2
91.18 odd 12 1274.2.f.m.79.1 2
91.31 even 12 1274.2.f.u.79.1 2
91.44 odd 12 1274.2.f.m.1145.1 2
91.83 even 4 1274.2.a.a.1.1 1
104.5 odd 4 5824.2.a.a.1.1 1
104.83 even 4 5824.2.a.be.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
182.2.a.b.1.1 1 13.5 odd 4
1274.2.a.a.1.1 1 91.83 even 4
1274.2.f.m.79.1 2 91.18 odd 12
1274.2.f.m.1145.1 2 91.44 odd 12
1274.2.f.u.79.1 2 91.31 even 12
1274.2.f.u.1145.1 2 91.5 even 12
1456.2.a.b.1.1 1 52.31 even 4
1638.2.a.q.1.1 1 39.5 even 4
2366.2.a.o.1.1 1 13.8 odd 4
2366.2.d.i.337.1 2 1.1 even 1 trivial
2366.2.d.i.337.2 2 13.12 even 2 inner
4550.2.a.o.1.1 1 65.44 odd 4
5824.2.a.a.1.1 1 104.5 odd 4
5824.2.a.be.1.1 1 104.83 even 4