Properties

Label 1369.2.b.b.1368.1
Level $1369$
Weight $2$
Character 1369.1368
Analytic conductor $10.932$
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] = [1369,2,Mod(1368,1369)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1369, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1369.1368");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1369 = 37^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1369.b (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: \(10.9315200367\)
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 37)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1368.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 1369.1368
Dual form 1369.2.b.b.1368.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000i q^{2} +1.00000 q^{4} +1.00000i q^{5} +2.00000 q^{7} -3.00000i q^{8} -3.00000 q^{9} +O(q^{10})\) \(q-1.00000i q^{2} +1.00000 q^{4} +1.00000i q^{5} +2.00000 q^{7} -3.00000i q^{8} -3.00000 q^{9} +1.00000 q^{10} +2.00000 q^{11} -2.00000i q^{13} -2.00000i q^{14} -1.00000 q^{16} +3.00000i q^{17} +3.00000i q^{18} -6.00000i q^{19} +1.00000i q^{20} -2.00000i q^{22} -4.00000i q^{23} +4.00000 q^{25} -2.00000 q^{26} +2.00000 q^{28} -9.00000i q^{29} +10.0000i q^{31} -5.00000i q^{32} +3.00000 q^{34} +2.00000i q^{35} -3.00000 q^{36} -6.00000 q^{38} +3.00000 q^{40} +9.00000 q^{41} +2.00000i q^{43} +2.00000 q^{44} -3.00000i q^{45} -4.00000 q^{46} +6.00000 q^{47} -3.00000 q^{49} -4.00000i q^{50} -2.00000i q^{52} -2.00000 q^{53} +2.00000i q^{55} -6.00000i q^{56} -9.00000 q^{58} -4.00000i q^{59} -1.00000i q^{61} +10.0000 q^{62} -6.00000 q^{63} -7.00000 q^{64} +2.00000 q^{65} +10.0000 q^{67} +3.00000i q^{68} +2.00000 q^{70} +6.00000 q^{71} +9.00000i q^{72} +10.0000 q^{73} -6.00000i q^{76} +4.00000 q^{77} +10.0000i q^{79} -1.00000i q^{80} +9.00000 q^{81} -9.00000i q^{82} -12.0000 q^{83} -3.00000 q^{85} +2.00000 q^{86} -6.00000i q^{88} -7.00000i q^{89} -3.00000 q^{90} -4.00000i q^{91} -4.00000i q^{92} -6.00000i q^{94} +6.00000 q^{95} +7.00000i q^{97} +3.00000i q^{98} -6.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{4} + 4 q^{7} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{4} + 4 q^{7} - 6 q^{9} + 2 q^{10} + 4 q^{11} - 2 q^{16} + 8 q^{25} - 4 q^{26} + 4 q^{28} + 6 q^{34} - 6 q^{36} - 12 q^{38} + 6 q^{40} + 18 q^{41} + 4 q^{44} - 8 q^{46} + 12 q^{47} - 6 q^{49} - 4 q^{53} - 18 q^{58} + 20 q^{62} - 12 q^{63} - 14 q^{64} + 4 q^{65} + 20 q^{67} + 4 q^{70} + 12 q^{71} + 20 q^{73} + 8 q^{77} + 18 q^{81} - 24 q^{83} - 6 q^{85} + 4 q^{86} - 6 q^{90} + 12 q^{95} - 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1369\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-1\)

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 −0.935414 0.353553i \(-0.884973\pi\)
0.935414 0.353553i \(-0.115027\pi\)
\(3\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(4\) 1.00000 0.500000
\(5\) 1.00000i 0.447214i 0.974679 + 0.223607i \(0.0717831\pi\)
−0.974679 + 0.223607i \(0.928217\pi\)
\(6\) 0 0
\(7\) 2.00000 0.755929 0.377964 0.925820i \(-0.376624\pi\)
0.377964 + 0.925820i \(0.376624\pi\)
\(8\) − 3.00000i − 1.06066i
\(9\) −3.00000 −1.00000
\(10\) 1.00000 0.316228
\(11\) 2.00000 0.603023 0.301511 0.953463i \(-0.402509\pi\)
0.301511 + 0.953463i \(0.402509\pi\)
\(12\) 0 0
\(13\) − 2.00000i − 0.554700i −0.960769 0.277350i \(-0.910544\pi\)
0.960769 0.277350i \(-0.0894562\pi\)
\(14\) − 2.00000i − 0.534522i
\(15\) 0 0
\(16\) −1.00000 −0.250000
\(17\) 3.00000i 0.727607i 0.931476 + 0.363803i \(0.118522\pi\)
−0.931476 + 0.363803i \(0.881478\pi\)
\(18\) 3.00000i 0.707107i
\(19\) − 6.00000i − 1.37649i −0.725476 0.688247i \(-0.758380\pi\)
0.725476 0.688247i \(-0.241620\pi\)
\(20\) 1.00000i 0.223607i
\(21\) 0 0
\(22\) − 2.00000i − 0.426401i
\(23\) − 4.00000i − 0.834058i −0.908893 0.417029i \(-0.863071\pi\)
0.908893 0.417029i \(-0.136929\pi\)
\(24\) 0 0
\(25\) 4.00000 0.800000
\(26\) −2.00000 −0.392232
\(27\) 0 0
\(28\) 2.00000 0.377964
\(29\) − 9.00000i − 1.67126i −0.549294 0.835629i \(-0.685103\pi\)
0.549294 0.835629i \(-0.314897\pi\)
\(30\) 0 0
\(31\) 10.0000i 1.79605i 0.439941 + 0.898027i \(0.354999\pi\)
−0.439941 + 0.898027i \(0.645001\pi\)
\(32\) − 5.00000i − 0.883883i
\(33\) 0 0
\(34\) 3.00000 0.514496
\(35\) 2.00000i 0.338062i
\(36\) −3.00000 −0.500000
\(37\) 0 0
\(38\) −6.00000 −0.973329
\(39\) 0 0
\(40\) 3.00000 0.474342
\(41\) 9.00000 1.40556 0.702782 0.711405i \(-0.251941\pi\)
0.702782 + 0.711405i \(0.251941\pi\)
\(42\) 0 0
\(43\) 2.00000i 0.304997i 0.988304 + 0.152499i \(0.0487319\pi\)
−0.988304 + 0.152499i \(0.951268\pi\)
\(44\) 2.00000 0.301511
\(45\) − 3.00000i − 0.447214i
\(46\) −4.00000 −0.589768
\(47\) 6.00000 0.875190 0.437595 0.899172i \(-0.355830\pi\)
0.437595 + 0.899172i \(0.355830\pi\)
\(48\) 0 0
\(49\) −3.00000 −0.428571
\(50\) − 4.00000i − 0.565685i
\(51\) 0 0
\(52\) − 2.00000i − 0.277350i
\(53\) −2.00000 −0.274721 −0.137361 0.990521i \(-0.543862\pi\)
−0.137361 + 0.990521i \(0.543862\pi\)
\(54\) 0 0
\(55\) 2.00000i 0.269680i
\(56\) − 6.00000i − 0.801784i
\(57\) 0 0
\(58\) −9.00000 −1.18176
\(59\) − 4.00000i − 0.520756i −0.965507 0.260378i \(-0.916153\pi\)
0.965507 0.260378i \(-0.0838471\pi\)
\(60\) 0 0
\(61\) − 1.00000i − 0.128037i −0.997949 0.0640184i \(-0.979608\pi\)
0.997949 0.0640184i \(-0.0203916\pi\)
\(62\) 10.0000 1.27000
\(63\) −6.00000 −0.755929
\(64\) −7.00000 −0.875000
\(65\) 2.00000 0.248069
\(66\) 0 0
\(67\) 10.0000 1.22169 0.610847 0.791748i \(-0.290829\pi\)
0.610847 + 0.791748i \(0.290829\pi\)
\(68\) 3.00000i 0.363803i
\(69\) 0 0
\(70\) 2.00000 0.239046
\(71\) 6.00000 0.712069 0.356034 0.934473i \(-0.384129\pi\)
0.356034 + 0.934473i \(0.384129\pi\)
\(72\) 9.00000i 1.06066i
\(73\) 10.0000 1.17041 0.585206 0.810885i \(-0.301014\pi\)
0.585206 + 0.810885i \(0.301014\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) − 6.00000i − 0.688247i
\(77\) 4.00000 0.455842
\(78\) 0 0
\(79\) 10.0000i 1.12509i 0.826767 + 0.562544i \(0.190177\pi\)
−0.826767 + 0.562544i \(0.809823\pi\)
\(80\) − 1.00000i − 0.111803i
\(81\) 9.00000 1.00000
\(82\) − 9.00000i − 0.993884i
\(83\) −12.0000 −1.31717 −0.658586 0.752506i \(-0.728845\pi\)
−0.658586 + 0.752506i \(0.728845\pi\)
\(84\) 0 0
\(85\) −3.00000 −0.325396
\(86\) 2.00000 0.215666
\(87\) 0 0
\(88\) − 6.00000i − 0.639602i
\(89\) − 7.00000i − 0.741999i −0.928633 0.370999i \(-0.879015\pi\)
0.928633 0.370999i \(-0.120985\pi\)
\(90\) −3.00000 −0.316228
\(91\) − 4.00000i − 0.419314i
\(92\) − 4.00000i − 0.417029i
\(93\) 0 0
\(94\) − 6.00000i − 0.618853i
\(95\) 6.00000 0.615587
\(96\) 0 0
\(97\) 7.00000i 0.710742i 0.934725 + 0.355371i \(0.115646\pi\)
−0.934725 + 0.355371i \(0.884354\pi\)
\(98\) 3.00000i 0.303046i
\(99\) −6.00000 −0.603023
\(100\) 4.00000 0.400000
\(101\) 3.00000 0.298511 0.149256 0.988799i \(-0.452312\pi\)
0.149256 + 0.988799i \(0.452312\pi\)
\(102\) 0 0
\(103\) − 6.00000i − 0.591198i −0.955312 0.295599i \(-0.904481\pi\)
0.955312 0.295599i \(-0.0955191\pi\)
\(104\) −6.00000 −0.588348
\(105\) 0 0
\(106\) 2.00000i 0.194257i
\(107\) −18.0000 −1.74013 −0.870063 0.492941i \(-0.835922\pi\)
−0.870063 + 0.492941i \(0.835922\pi\)
\(108\) 0 0
\(109\) 5.00000i 0.478913i 0.970907 + 0.239457i \(0.0769693\pi\)
−0.970907 + 0.239457i \(0.923031\pi\)
\(110\) 2.00000 0.190693
\(111\) 0 0
\(112\) −2.00000 −0.188982
\(113\) − 6.00000i − 0.564433i −0.959351 0.282216i \(-0.908930\pi\)
0.959351 0.282216i \(-0.0910696\pi\)
\(114\) 0 0
\(115\) 4.00000 0.373002
\(116\) − 9.00000i − 0.835629i
\(117\) 6.00000i 0.554700i
\(118\) −4.00000 −0.368230
\(119\) 6.00000i 0.550019i
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) −1.00000 −0.0905357
\(123\) 0 0
\(124\) 10.0000i 0.898027i
\(125\) 9.00000i 0.804984i
\(126\) 6.00000i 0.534522i
\(127\) −8.00000 −0.709885 −0.354943 0.934888i \(-0.615500\pi\)
−0.354943 + 0.934888i \(0.615500\pi\)
\(128\) − 3.00000i − 0.265165i
\(129\) 0 0
\(130\) − 2.00000i − 0.175412i
\(131\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(132\) 0 0
\(133\) − 12.0000i − 1.04053i
\(134\) − 10.0000i − 0.863868i
\(135\) 0 0
\(136\) 9.00000 0.771744
\(137\) −9.00000 −0.768922 −0.384461 0.923141i \(-0.625613\pi\)
−0.384461 + 0.923141i \(0.625613\pi\)
\(138\) 0 0
\(139\) −16.0000 −1.35710 −0.678551 0.734553i \(-0.737392\pi\)
−0.678551 + 0.734553i \(0.737392\pi\)
\(140\) 2.00000i 0.169031i
\(141\) 0 0
\(142\) − 6.00000i − 0.503509i
\(143\) − 4.00000i − 0.334497i
\(144\) 3.00000 0.250000
\(145\) 9.00000 0.747409
\(146\) − 10.0000i − 0.827606i
\(147\) 0 0
\(148\) 0 0
\(149\) −11.0000 −0.901155 −0.450578 0.892737i \(-0.648782\pi\)
−0.450578 + 0.892737i \(0.648782\pi\)
\(150\) 0 0
\(151\) −4.00000 −0.325515 −0.162758 0.986666i \(-0.552039\pi\)
−0.162758 + 0.986666i \(0.552039\pi\)
\(152\) −18.0000 −1.45999
\(153\) − 9.00000i − 0.727607i
\(154\) − 4.00000i − 0.322329i
\(155\) −10.0000 −0.803219
\(156\) 0 0
\(157\) 17.0000 1.35675 0.678374 0.734717i \(-0.262685\pi\)
0.678374 + 0.734717i \(0.262685\pi\)
\(158\) 10.0000 0.795557
\(159\) 0 0
\(160\) 5.00000 0.395285
\(161\) − 8.00000i − 0.630488i
\(162\) − 9.00000i − 0.707107i
\(163\) 6.00000i 0.469956i 0.972001 + 0.234978i \(0.0755019\pi\)
−0.972001 + 0.234978i \(0.924498\pi\)
\(164\) 9.00000 0.702782
\(165\) 0 0
\(166\) 12.0000i 0.931381i
\(167\) 12.0000i 0.928588i 0.885681 + 0.464294i \(0.153692\pi\)
−0.885681 + 0.464294i \(0.846308\pi\)
\(168\) 0 0
\(169\) 9.00000 0.692308
\(170\) 3.00000i 0.230089i
\(171\) 18.0000i 1.37649i
\(172\) 2.00000i 0.152499i
\(173\) −21.0000 −1.59660 −0.798300 0.602260i \(-0.794267\pi\)
−0.798300 + 0.602260i \(0.794267\pi\)
\(174\) 0 0
\(175\) 8.00000 0.604743
\(176\) −2.00000 −0.150756
\(177\) 0 0
\(178\) −7.00000 −0.524672
\(179\) − 12.0000i − 0.896922i −0.893802 0.448461i \(-0.851972\pi\)
0.893802 0.448461i \(-0.148028\pi\)
\(180\) − 3.00000i − 0.223607i
\(181\) 5.00000 0.371647 0.185824 0.982583i \(-0.440505\pi\)
0.185824 + 0.982583i \(0.440505\pi\)
\(182\) −4.00000 −0.296500
\(183\) 0 0
\(184\) −12.0000 −0.884652
\(185\) 0 0
\(186\) 0 0
\(187\) 6.00000i 0.438763i
\(188\) 6.00000 0.437595
\(189\) 0 0
\(190\) − 6.00000i − 0.435286i
\(191\) − 4.00000i − 0.289430i −0.989473 0.144715i \(-0.953773\pi\)
0.989473 0.144715i \(-0.0462265\pi\)
\(192\) 0 0
\(193\) 19.0000i 1.36765i 0.729646 + 0.683825i \(0.239685\pi\)
−0.729646 + 0.683825i \(0.760315\pi\)
\(194\) 7.00000 0.502571
\(195\) 0 0
\(196\) −3.00000 −0.214286
\(197\) −15.0000 −1.06871 −0.534353 0.845262i \(-0.679445\pi\)
−0.534353 + 0.845262i \(0.679445\pi\)
\(198\) 6.00000i 0.426401i
\(199\) 10.0000i 0.708881i 0.935079 + 0.354441i \(0.115329\pi\)
−0.935079 + 0.354441i \(0.884671\pi\)
\(200\) − 12.0000i − 0.848528i
\(201\) 0 0
\(202\) − 3.00000i − 0.211079i
\(203\) − 18.0000i − 1.26335i
\(204\) 0 0
\(205\) 9.00000i 0.628587i
\(206\) −6.00000 −0.418040
\(207\) 12.0000i 0.834058i
\(208\) 2.00000i 0.138675i
\(209\) − 12.0000i − 0.830057i
\(210\) 0 0
\(211\) 2.00000 0.137686 0.0688428 0.997628i \(-0.478069\pi\)
0.0688428 + 0.997628i \(0.478069\pi\)
\(212\) −2.00000 −0.137361
\(213\) 0 0
\(214\) 18.0000i 1.23045i
\(215\) −2.00000 −0.136399
\(216\) 0 0
\(217\) 20.0000i 1.35769i
\(218\) 5.00000 0.338643
\(219\) 0 0
\(220\) 2.00000i 0.134840i
\(221\) 6.00000 0.403604
\(222\) 0 0
\(223\) 28.0000 1.87502 0.937509 0.347960i \(-0.113126\pi\)
0.937509 + 0.347960i \(0.113126\pi\)
\(224\) − 10.0000i − 0.668153i
\(225\) −12.0000 −0.800000
\(226\) −6.00000 −0.399114
\(227\) 14.0000i 0.929213i 0.885517 + 0.464606i \(0.153804\pi\)
−0.885517 + 0.464606i \(0.846196\pi\)
\(228\) 0 0
\(229\) 1.00000 0.0660819 0.0330409 0.999454i \(-0.489481\pi\)
0.0330409 + 0.999454i \(0.489481\pi\)
\(230\) − 4.00000i − 0.263752i
\(231\) 0 0
\(232\) −27.0000 −1.77264
\(233\) −3.00000 −0.196537 −0.0982683 0.995160i \(-0.531330\pi\)
−0.0982683 + 0.995160i \(0.531330\pi\)
\(234\) 6.00000 0.392232
\(235\) 6.00000i 0.391397i
\(236\) − 4.00000i − 0.260378i
\(237\) 0 0
\(238\) 6.00000 0.388922
\(239\) 6.00000i 0.388108i 0.980991 + 0.194054i \(0.0621637\pi\)
−0.980991 + 0.194054i \(0.937836\pi\)
\(240\) 0 0
\(241\) 14.0000i 0.901819i 0.892570 + 0.450910i \(0.148900\pi\)
−0.892570 + 0.450910i \(0.851100\pi\)
\(242\) 7.00000i 0.449977i
\(243\) 0 0
\(244\) − 1.00000i − 0.0640184i
\(245\) − 3.00000i − 0.191663i
\(246\) 0 0
\(247\) −12.0000 −0.763542
\(248\) 30.0000 1.90500
\(249\) 0 0
\(250\) 9.00000 0.569210
\(251\) 14.0000i 0.883672i 0.897096 + 0.441836i \(0.145673\pi\)
−0.897096 + 0.441836i \(0.854327\pi\)
\(252\) −6.00000 −0.377964
\(253\) − 8.00000i − 0.502956i
\(254\) 8.00000i 0.501965i
\(255\) 0 0
\(256\) −17.0000 −1.06250
\(257\) − 21.0000i − 1.30994i −0.755653 0.654972i \(-0.772680\pi\)
0.755653 0.654972i \(-0.227320\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 2.00000 0.124035
\(261\) 27.0000i 1.67126i
\(262\) 0 0
\(263\) −4.00000 −0.246651 −0.123325 0.992366i \(-0.539356\pi\)
−0.123325 + 0.992366i \(0.539356\pi\)
\(264\) 0 0
\(265\) − 2.00000i − 0.122859i
\(266\) −12.0000 −0.735767
\(267\) 0 0
\(268\) 10.0000 0.610847
\(269\) 18.0000 1.09748 0.548740 0.835993i \(-0.315108\pi\)
0.548740 + 0.835993i \(0.315108\pi\)
\(270\) 0 0
\(271\) 14.0000 0.850439 0.425220 0.905090i \(-0.360197\pi\)
0.425220 + 0.905090i \(0.360197\pi\)
\(272\) − 3.00000i − 0.181902i
\(273\) 0 0
\(274\) 9.00000i 0.543710i
\(275\) 8.00000 0.482418
\(276\) 0 0
\(277\) 3.00000i 0.180253i 0.995930 + 0.0901263i \(0.0287271\pi\)
−0.995930 + 0.0901263i \(0.971273\pi\)
\(278\) 16.0000i 0.959616i
\(279\) − 30.0000i − 1.79605i
\(280\) 6.00000 0.358569
\(281\) 27.0000i 1.61068i 0.592810 + 0.805342i \(0.298019\pi\)
−0.592810 + 0.805342i \(0.701981\pi\)
\(282\) 0 0
\(283\) 14.0000i 0.832214i 0.909316 + 0.416107i \(0.136606\pi\)
−0.909316 + 0.416107i \(0.863394\pi\)
\(284\) 6.00000 0.356034
\(285\) 0 0
\(286\) −4.00000 −0.236525
\(287\) 18.0000 1.06251
\(288\) 15.0000i 0.883883i
\(289\) 8.00000 0.470588
\(290\) − 9.00000i − 0.528498i
\(291\) 0 0
\(292\) 10.0000 0.585206
\(293\) −23.0000 −1.34367 −0.671837 0.740699i \(-0.734495\pi\)
−0.671837 + 0.740699i \(0.734495\pi\)
\(294\) 0 0
\(295\) 4.00000 0.232889
\(296\) 0 0
\(297\) 0 0
\(298\) 11.0000i 0.637213i
\(299\) −8.00000 −0.462652
\(300\) 0 0
\(301\) 4.00000i 0.230556i
\(302\) 4.00000i 0.230174i
\(303\) 0 0
\(304\) 6.00000i 0.344124i
\(305\) 1.00000 0.0572598
\(306\) −9.00000 −0.514496
\(307\) 14.0000 0.799022 0.399511 0.916728i \(-0.369180\pi\)
0.399511 + 0.916728i \(0.369180\pi\)
\(308\) 4.00000 0.227921
\(309\) 0 0
\(310\) 10.0000i 0.567962i
\(311\) 18.0000i 1.02069i 0.859971 + 0.510343i \(0.170482\pi\)
−0.859971 + 0.510343i \(0.829518\pi\)
\(312\) 0 0
\(313\) 7.00000i 0.395663i 0.980236 + 0.197832i \(0.0633900\pi\)
−0.980236 + 0.197832i \(0.936610\pi\)
\(314\) − 17.0000i − 0.959366i
\(315\) − 6.00000i − 0.338062i
\(316\) 10.0000i 0.562544i
\(317\) −1.00000 −0.0561656 −0.0280828 0.999606i \(-0.508940\pi\)
−0.0280828 + 0.999606i \(0.508940\pi\)
\(318\) 0 0
\(319\) − 18.0000i − 1.00781i
\(320\) − 7.00000i − 0.391312i
\(321\) 0 0
\(322\) −8.00000 −0.445823
\(323\) 18.0000 1.00155
\(324\) 9.00000 0.500000
\(325\) − 8.00000i − 0.443760i
\(326\) 6.00000 0.332309
\(327\) 0 0
\(328\) − 27.0000i − 1.49083i
\(329\) 12.0000 0.661581
\(330\) 0 0
\(331\) − 20.0000i − 1.09930i −0.835395 0.549650i \(-0.814761\pi\)
0.835395 0.549650i \(-0.185239\pi\)
\(332\) −12.0000 −0.658586
\(333\) 0 0
\(334\) 12.0000 0.656611
\(335\) 10.0000i 0.546358i
\(336\) 0 0
\(337\) 1.00000 0.0544735 0.0272367 0.999629i \(-0.491329\pi\)
0.0272367 + 0.999629i \(0.491329\pi\)
\(338\) − 9.00000i − 0.489535i
\(339\) 0 0
\(340\) −3.00000 −0.162698
\(341\) 20.0000i 1.08306i
\(342\) 18.0000 0.973329
\(343\) −20.0000 −1.07990
\(344\) 6.00000 0.323498
\(345\) 0 0
\(346\) 21.0000i 1.12897i
\(347\) 16.0000i 0.858925i 0.903085 + 0.429463i \(0.141297\pi\)
−0.903085 + 0.429463i \(0.858703\pi\)
\(348\) 0 0
\(349\) 9.00000 0.481759 0.240879 0.970555i \(-0.422564\pi\)
0.240879 + 0.970555i \(0.422564\pi\)
\(350\) − 8.00000i − 0.427618i
\(351\) 0 0
\(352\) − 10.0000i − 0.533002i
\(353\) 25.0000i 1.33062i 0.746569 + 0.665308i \(0.231700\pi\)
−0.746569 + 0.665308i \(0.768300\pi\)
\(354\) 0 0
\(355\) 6.00000i 0.318447i
\(356\) − 7.00000i − 0.370999i
\(357\) 0 0
\(358\) −12.0000 −0.634220
\(359\) 24.0000 1.26667 0.633336 0.773877i \(-0.281685\pi\)
0.633336 + 0.773877i \(0.281685\pi\)
\(360\) −9.00000 −0.474342
\(361\) −17.0000 −0.894737
\(362\) − 5.00000i − 0.262794i
\(363\) 0 0
\(364\) − 4.00000i − 0.209657i
\(365\) 10.0000i 0.523424i
\(366\) 0 0
\(367\) −28.0000 −1.46159 −0.730794 0.682598i \(-0.760850\pi\)
−0.730794 + 0.682598i \(0.760850\pi\)
\(368\) 4.00000i 0.208514i
\(369\) −27.0000 −1.40556
\(370\) 0 0
\(371\) −4.00000 −0.207670
\(372\) 0 0
\(373\) −17.0000 −0.880227 −0.440113 0.897942i \(-0.645062\pi\)
−0.440113 + 0.897942i \(0.645062\pi\)
\(374\) 6.00000 0.310253
\(375\) 0 0
\(376\) − 18.0000i − 0.928279i
\(377\) −18.0000 −0.927047
\(378\) 0 0
\(379\) 6.00000 0.308199 0.154100 0.988055i \(-0.450752\pi\)
0.154100 + 0.988055i \(0.450752\pi\)
\(380\) 6.00000 0.307794
\(381\) 0 0
\(382\) −4.00000 −0.204658
\(383\) − 34.0000i − 1.73732i −0.495410 0.868659i \(-0.664982\pi\)
0.495410 0.868659i \(-0.335018\pi\)
\(384\) 0 0
\(385\) 4.00000i 0.203859i
\(386\) 19.0000 0.967075
\(387\) − 6.00000i − 0.304997i
\(388\) 7.00000i 0.355371i
\(389\) 25.0000i 1.26755i 0.773517 + 0.633775i \(0.218496\pi\)
−0.773517 + 0.633775i \(0.781504\pi\)
\(390\) 0 0
\(391\) 12.0000 0.606866
\(392\) 9.00000i 0.454569i
\(393\) 0 0
\(394\) 15.0000i 0.755689i
\(395\) −10.0000 −0.503155
\(396\) −6.00000 −0.301511
\(397\) −13.0000 −0.652451 −0.326226 0.945292i \(-0.605777\pi\)
−0.326226 + 0.945292i \(0.605777\pi\)
\(398\) 10.0000 0.501255
\(399\) 0 0
\(400\) −4.00000 −0.200000
\(401\) 18.0000i 0.898877i 0.893311 + 0.449439i \(0.148376\pi\)
−0.893311 + 0.449439i \(0.851624\pi\)
\(402\) 0 0
\(403\) 20.0000 0.996271
\(404\) 3.00000 0.149256
\(405\) 9.00000i 0.447214i
\(406\) −18.0000 −0.893325
\(407\) 0 0
\(408\) 0 0
\(409\) 25.0000i 1.23617i 0.786111 + 0.618085i \(0.212091\pi\)
−0.786111 + 0.618085i \(0.787909\pi\)
\(410\) 9.00000 0.444478
\(411\) 0 0
\(412\) − 6.00000i − 0.295599i
\(413\) − 8.00000i − 0.393654i
\(414\) 12.0000 0.589768
\(415\) − 12.0000i − 0.589057i
\(416\) −10.0000 −0.490290
\(417\) 0 0
\(418\) −12.0000 −0.586939
\(419\) −26.0000 −1.27018 −0.635092 0.772437i \(-0.719038\pi\)
−0.635092 + 0.772437i \(0.719038\pi\)
\(420\) 0 0
\(421\) − 9.00000i − 0.438633i −0.975654 0.219317i \(-0.929617\pi\)
0.975654 0.219317i \(-0.0703828\pi\)
\(422\) − 2.00000i − 0.0973585i
\(423\) −18.0000 −0.875190
\(424\) 6.00000i 0.291386i
\(425\) 12.0000i 0.582086i
\(426\) 0 0
\(427\) − 2.00000i − 0.0967868i
\(428\) −18.0000 −0.870063
\(429\) 0 0
\(430\) 2.00000i 0.0964486i
\(431\) − 18.0000i − 0.867029i −0.901146 0.433515i \(-0.857273\pi\)
0.901146 0.433515i \(-0.142727\pi\)
\(432\) 0 0
\(433\) −21.0000 −1.00920 −0.504598 0.863355i \(-0.668359\pi\)
−0.504598 + 0.863355i \(0.668359\pi\)
\(434\) 20.0000 0.960031
\(435\) 0 0
\(436\) 5.00000i 0.239457i
\(437\) −24.0000 −1.14808
\(438\) 0 0
\(439\) 20.0000i 0.954548i 0.878755 + 0.477274i \(0.158375\pi\)
−0.878755 + 0.477274i \(0.841625\pi\)
\(440\) 6.00000 0.286039
\(441\) 9.00000 0.428571
\(442\) − 6.00000i − 0.285391i
\(443\) 32.0000 1.52037 0.760183 0.649709i \(-0.225109\pi\)
0.760183 + 0.649709i \(0.225109\pi\)
\(444\) 0 0
\(445\) 7.00000 0.331832
\(446\) − 28.0000i − 1.32584i
\(447\) 0 0
\(448\) −14.0000 −0.661438
\(449\) 6.00000i 0.283158i 0.989927 + 0.141579i \(0.0452178\pi\)
−0.989927 + 0.141579i \(0.954782\pi\)
\(450\) 12.0000i 0.565685i
\(451\) 18.0000 0.847587
\(452\) − 6.00000i − 0.282216i
\(453\) 0 0
\(454\) 14.0000 0.657053
\(455\) 4.00000 0.187523
\(456\) 0 0
\(457\) 3.00000i 0.140334i 0.997535 + 0.0701670i \(0.0223532\pi\)
−0.997535 + 0.0701670i \(0.977647\pi\)
\(458\) − 1.00000i − 0.0467269i
\(459\) 0 0
\(460\) 4.00000 0.186501
\(461\) 30.0000i 1.39724i 0.715493 + 0.698620i \(0.246202\pi\)
−0.715493 + 0.698620i \(0.753798\pi\)
\(462\) 0 0
\(463\) − 22.0000i − 1.02243i −0.859454 0.511213i \(-0.829196\pi\)
0.859454 0.511213i \(-0.170804\pi\)
\(464\) 9.00000i 0.417815i
\(465\) 0 0
\(466\) 3.00000i 0.138972i
\(467\) − 8.00000i − 0.370196i −0.982720 0.185098i \(-0.940740\pi\)
0.982720 0.185098i \(-0.0592602\pi\)
\(468\) 6.00000i 0.277350i
\(469\) 20.0000 0.923514
\(470\) 6.00000 0.276759
\(471\) 0 0
\(472\) −12.0000 −0.552345
\(473\) 4.00000i 0.183920i
\(474\) 0 0
\(475\) − 24.0000i − 1.10120i
\(476\) 6.00000i 0.275010i
\(477\) 6.00000 0.274721
\(478\) 6.00000 0.274434
\(479\) − 4.00000i − 0.182765i −0.995816 0.0913823i \(-0.970871\pi\)
0.995816 0.0913823i \(-0.0291285\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 14.0000 0.637683
\(483\) 0 0
\(484\) −7.00000 −0.318182
\(485\) −7.00000 −0.317854
\(486\) 0 0
\(487\) 12.0000i 0.543772i 0.962329 + 0.271886i \(0.0876473\pi\)
−0.962329 + 0.271886i \(0.912353\pi\)
\(488\) −3.00000 −0.135804
\(489\) 0 0
\(490\) −3.00000 −0.135526
\(491\) 2.00000 0.0902587 0.0451294 0.998981i \(-0.485630\pi\)
0.0451294 + 0.998981i \(0.485630\pi\)
\(492\) 0 0
\(493\) 27.0000 1.21602
\(494\) 12.0000i 0.539906i
\(495\) − 6.00000i − 0.269680i
\(496\) − 10.0000i − 0.449013i
\(497\) 12.0000 0.538274
\(498\) 0 0
\(499\) 18.0000i 0.805791i 0.915246 + 0.402895i \(0.131996\pi\)
−0.915246 + 0.402895i \(0.868004\pi\)
\(500\) 9.00000i 0.402492i
\(501\) 0 0
\(502\) 14.0000 0.624851
\(503\) − 26.0000i − 1.15928i −0.814872 0.579641i \(-0.803193\pi\)
0.814872 0.579641i \(-0.196807\pi\)
\(504\) 18.0000i 0.801784i
\(505\) 3.00000i 0.133498i
\(506\) −8.00000 −0.355643
\(507\) 0 0
\(508\) −8.00000 −0.354943
\(509\) 7.00000 0.310270 0.155135 0.987893i \(-0.450419\pi\)
0.155135 + 0.987893i \(0.450419\pi\)
\(510\) 0 0
\(511\) 20.0000 0.884748
\(512\) 11.0000i 0.486136i
\(513\) 0 0
\(514\) −21.0000 −0.926270
\(515\) 6.00000 0.264392
\(516\) 0 0
\(517\) 12.0000 0.527759
\(518\) 0 0
\(519\) 0 0
\(520\) − 6.00000i − 0.263117i
\(521\) 42.0000 1.84005 0.920027 0.391856i \(-0.128167\pi\)
0.920027 + 0.391856i \(0.128167\pi\)
\(522\) 27.0000 1.18176
\(523\) 8.00000i 0.349816i 0.984585 + 0.174908i \(0.0559627\pi\)
−0.984585 + 0.174908i \(0.944037\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 4.00000i 0.174408i
\(527\) −30.0000 −1.30682
\(528\) 0 0
\(529\) 7.00000 0.304348
\(530\) −2.00000 −0.0868744
\(531\) 12.0000i 0.520756i
\(532\) − 12.0000i − 0.520266i
\(533\) − 18.0000i − 0.779667i
\(534\) 0 0
\(535\) − 18.0000i − 0.778208i
\(536\) − 30.0000i − 1.29580i
\(537\) 0 0
\(538\) − 18.0000i − 0.776035i
\(539\) −6.00000 −0.258438
\(540\) 0 0
\(541\) 17.0000i 0.730887i 0.930834 + 0.365444i \(0.119083\pi\)
−0.930834 + 0.365444i \(0.880917\pi\)
\(542\) − 14.0000i − 0.601351i
\(543\) 0 0
\(544\) 15.0000 0.643120
\(545\) −5.00000 −0.214176
\(546\) 0 0
\(547\) − 2.00000i − 0.0855138i −0.999086 0.0427569i \(-0.986386\pi\)
0.999086 0.0427569i \(-0.0136141\pi\)
\(548\) −9.00000 −0.384461
\(549\) 3.00000i 0.128037i
\(550\) − 8.00000i − 0.341121i
\(551\) −54.0000 −2.30048
\(552\) 0 0
\(553\) 20.0000i 0.850487i
\(554\) 3.00000 0.127458
\(555\) 0 0
\(556\) −16.0000 −0.678551
\(557\) − 33.0000i − 1.39825i −0.714997 0.699127i \(-0.753572\pi\)
0.714997 0.699127i \(-0.246428\pi\)
\(558\) −30.0000 −1.27000
\(559\) 4.00000 0.169182
\(560\) − 2.00000i − 0.0845154i
\(561\) 0 0
\(562\) 27.0000 1.13893
\(563\) − 24.0000i − 1.01148i −0.862686 0.505740i \(-0.831220\pi\)
0.862686 0.505740i \(-0.168780\pi\)
\(564\) 0 0
\(565\) 6.00000 0.252422
\(566\) 14.0000 0.588464
\(567\) 18.0000 0.755929
\(568\) − 18.0000i − 0.755263i
\(569\) − 3.00000i − 0.125767i −0.998021 0.0628833i \(-0.979970\pi\)
0.998021 0.0628833i \(-0.0200296\pi\)
\(570\) 0 0
\(571\) −44.0000 −1.84134 −0.920671 0.390339i \(-0.872358\pi\)
−0.920671 + 0.390339i \(0.872358\pi\)
\(572\) − 4.00000i − 0.167248i
\(573\) 0 0
\(574\) − 18.0000i − 0.751305i
\(575\) − 16.0000i − 0.667246i
\(576\) 21.0000 0.875000
\(577\) − 18.0000i − 0.749350i −0.927156 0.374675i \(-0.877754\pi\)
0.927156 0.374675i \(-0.122246\pi\)
\(578\) − 8.00000i − 0.332756i
\(579\) 0 0
\(580\) 9.00000 0.373705
\(581\) −24.0000 −0.995688
\(582\) 0 0
\(583\) −4.00000 −0.165663
\(584\) − 30.0000i − 1.24141i
\(585\) −6.00000 −0.248069
\(586\) 23.0000i 0.950121i
\(587\) − 40.0000i − 1.65098i −0.564419 0.825488i \(-0.690900\pi\)
0.564419 0.825488i \(-0.309100\pi\)
\(588\) 0 0
\(589\) 60.0000 2.47226
\(590\) − 4.00000i − 0.164677i
\(591\) 0 0
\(592\) 0 0
\(593\) −29.0000 −1.19089 −0.595444 0.803397i \(-0.703024\pi\)
−0.595444 + 0.803397i \(0.703024\pi\)
\(594\) 0 0
\(595\) −6.00000 −0.245976
\(596\) −11.0000 −0.450578
\(597\) 0 0
\(598\) 8.00000i 0.327144i
\(599\) −38.0000 −1.55264 −0.776319 0.630340i \(-0.782915\pi\)
−0.776319 + 0.630340i \(0.782915\pi\)
\(600\) 0 0
\(601\) 23.0000 0.938190 0.469095 0.883148i \(-0.344580\pi\)
0.469095 + 0.883148i \(0.344580\pi\)
\(602\) 4.00000 0.163028
\(603\) −30.0000 −1.22169
\(604\) −4.00000 −0.162758
\(605\) − 7.00000i − 0.284590i
\(606\) 0 0
\(607\) 14.0000i 0.568242i 0.958788 + 0.284121i \(0.0917018\pi\)
−0.958788 + 0.284121i \(0.908298\pi\)
\(608\) −30.0000 −1.21666
\(609\) 0 0
\(610\) − 1.00000i − 0.0404888i
\(611\) − 12.0000i − 0.485468i
\(612\) − 9.00000i − 0.363803i
\(613\) −33.0000 −1.33286 −0.666429 0.745569i \(-0.732178\pi\)
−0.666429 + 0.745569i \(0.732178\pi\)
\(614\) − 14.0000i − 0.564994i
\(615\) 0 0
\(616\) − 12.0000i − 0.483494i
\(617\) 22.0000 0.885687 0.442843 0.896599i \(-0.353970\pi\)
0.442843 + 0.896599i \(0.353970\pi\)
\(618\) 0 0
\(619\) 10.0000 0.401934 0.200967 0.979598i \(-0.435592\pi\)
0.200967 + 0.979598i \(0.435592\pi\)
\(620\) −10.0000 −0.401610
\(621\) 0 0
\(622\) 18.0000 0.721734
\(623\) − 14.0000i − 0.560898i
\(624\) 0 0
\(625\) 11.0000 0.440000
\(626\) 7.00000 0.279776
\(627\) 0 0
\(628\) 17.0000 0.678374
\(629\) 0 0
\(630\) −6.00000 −0.239046
\(631\) − 20.0000i − 0.796187i −0.917345 0.398094i \(-0.869672\pi\)
0.917345 0.398094i \(-0.130328\pi\)
\(632\) 30.0000 1.19334
\(633\) 0 0
\(634\) 1.00000i 0.0397151i
\(635\) − 8.00000i − 0.317470i
\(636\) 0 0
\(637\) 6.00000i 0.237729i
\(638\) −18.0000 −0.712627
\(639\) −18.0000 −0.712069
\(640\) 3.00000 0.118585
\(641\) −37.0000 −1.46141 −0.730706 0.682692i \(-0.760809\pi\)
−0.730706 + 0.682692i \(0.760809\pi\)
\(642\) 0 0
\(643\) − 14.0000i − 0.552106i −0.961142 0.276053i \(-0.910973\pi\)
0.961142 0.276053i \(-0.0890266\pi\)
\(644\) − 8.00000i − 0.315244i
\(645\) 0 0
\(646\) − 18.0000i − 0.708201i
\(647\) 14.0000i 0.550397i 0.961387 + 0.275198i \(0.0887435\pi\)
−0.961387 + 0.275198i \(0.911256\pi\)
\(648\) − 27.0000i − 1.06066i
\(649\) − 8.00000i − 0.314027i
\(650\) −8.00000 −0.313786
\(651\) 0 0
\(652\) 6.00000i 0.234978i
\(653\) 39.0000i 1.52619i 0.646288 + 0.763094i \(0.276321\pi\)
−0.646288 + 0.763094i \(0.723679\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −9.00000 −0.351391
\(657\) −30.0000 −1.17041
\(658\) − 12.0000i − 0.467809i
\(659\) 36.0000 1.40236 0.701180 0.712984i \(-0.252657\pi\)
0.701180 + 0.712984i \(0.252657\pi\)
\(660\) 0 0
\(661\) 7.00000i 0.272268i 0.990690 + 0.136134i \(0.0434678\pi\)
−0.990690 + 0.136134i \(0.956532\pi\)
\(662\) −20.0000 −0.777322
\(663\) 0 0
\(664\) 36.0000i 1.39707i
\(665\) 12.0000 0.465340
\(666\) 0 0
\(667\) −36.0000 −1.39393
\(668\) 12.0000i 0.464294i
\(669\) 0 0
\(670\) 10.0000 0.386334
\(671\) − 2.00000i − 0.0772091i
\(672\) 0 0
\(673\) 6.00000 0.231283 0.115642 0.993291i \(-0.463108\pi\)
0.115642 + 0.993291i \(0.463108\pi\)
\(674\) − 1.00000i − 0.0385186i
\(675\) 0 0
\(676\) 9.00000 0.346154
\(677\) −13.0000 −0.499631 −0.249815 0.968294i \(-0.580370\pi\)
−0.249815 + 0.968294i \(0.580370\pi\)
\(678\) 0 0
\(679\) 14.0000i 0.537271i
\(680\) 9.00000i 0.345134i
\(681\) 0 0
\(682\) 20.0000 0.765840
\(683\) − 30.0000i − 1.14792i −0.818884 0.573959i \(-0.805407\pi\)
0.818884 0.573959i \(-0.194593\pi\)
\(684\) 18.0000i 0.688247i
\(685\) − 9.00000i − 0.343872i
\(686\) 20.0000i 0.763604i
\(687\) 0 0
\(688\) − 2.00000i − 0.0762493i
\(689\) 4.00000i 0.152388i
\(690\) 0 0
\(691\) 14.0000 0.532585 0.266293 0.963892i \(-0.414201\pi\)
0.266293 + 0.963892i \(0.414201\pi\)
\(692\) −21.0000 −0.798300
\(693\) −12.0000 −0.455842
\(694\) 16.0000 0.607352
\(695\) − 16.0000i − 0.606915i
\(696\) 0 0
\(697\) 27.0000i 1.02270i
\(698\) − 9.00000i − 0.340655i
\(699\) 0 0
\(700\) 8.00000 0.302372
\(701\) 6.00000i 0.226617i 0.993560 + 0.113308i \(0.0361448\pi\)
−0.993560 + 0.113308i \(0.963855\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) −14.0000 −0.527645
\(705\) 0 0
\(706\) 25.0000 0.940887
\(707\) 6.00000 0.225653
\(708\) 0 0
\(709\) 10.0000i 0.375558i 0.982211 + 0.187779i \(0.0601289\pi\)
−0.982211 + 0.187779i \(0.939871\pi\)
\(710\) 6.00000 0.225176
\(711\) − 30.0000i − 1.12509i
\(712\) −21.0000 −0.787008
\(713\) 40.0000 1.49801
\(714\) 0 0
\(715\) 4.00000 0.149592
\(716\) − 12.0000i − 0.448461i
\(717\) 0 0
\(718\) − 24.0000i − 0.895672i
\(719\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 3.00000i 0.111803i
\(721\) − 12.0000i − 0.446903i
\(722\) 17.0000i 0.632674i
\(723\) 0 0
\(724\) 5.00000 0.185824
\(725\) − 36.0000i − 1.33701i
\(726\) 0 0
\(727\) − 40.0000i − 1.48352i −0.670667 0.741759i \(-0.733992\pi\)
0.670667 0.741759i \(-0.266008\pi\)
\(728\) −12.0000 −0.444750
\(729\) −27.0000 −1.00000
\(730\) 10.0000 0.370117
\(731\) −6.00000 −0.221918
\(732\) 0 0
\(733\) −22.0000 −0.812589 −0.406294 0.913742i \(-0.633179\pi\)
−0.406294 + 0.913742i \(0.633179\pi\)
\(734\) 28.0000i 1.03350i
\(735\) 0 0
\(736\) −20.0000 −0.737210
\(737\) 20.0000 0.736709
\(738\) 27.0000i 0.993884i
\(739\) −12.0000 −0.441427 −0.220714 0.975339i \(-0.570839\pi\)
−0.220714 + 0.975339i \(0.570839\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 4.00000i 0.146845i
\(743\) −6.00000 −0.220119 −0.110059 0.993925i \(-0.535104\pi\)
−0.110059 + 0.993925i \(0.535104\pi\)
\(744\) 0 0
\(745\) − 11.0000i − 0.403009i
\(746\) 17.0000i 0.622414i
\(747\) 36.0000 1.31717
\(748\) 6.00000i 0.219382i
\(749\) −36.0000 −1.31541
\(750\) 0 0
\(751\) −10.0000 −0.364905 −0.182453 0.983215i \(-0.558404\pi\)
−0.182453 + 0.983215i \(0.558404\pi\)
\(752\) −6.00000 −0.218797
\(753\) 0 0
\(754\) 18.0000i 0.655521i
\(755\) − 4.00000i − 0.145575i
\(756\) 0 0
\(757\) 13.0000i 0.472493i 0.971693 + 0.236247i \(0.0759173\pi\)
−0.971693 + 0.236247i \(0.924083\pi\)
\(758\) − 6.00000i − 0.217930i
\(759\) 0 0
\(760\) − 18.0000i − 0.652929i
\(761\) −7.00000 −0.253750 −0.126875 0.991919i \(-0.540495\pi\)
−0.126875 + 0.991919i \(0.540495\pi\)
\(762\) 0 0
\(763\) 10.0000i 0.362024i
\(764\) − 4.00000i − 0.144715i
\(765\) 9.00000 0.325396
\(766\) −34.0000 −1.22847
\(767\) −8.00000 −0.288863
\(768\) 0 0
\(769\) − 14.0000i − 0.504853i −0.967616 0.252426i \(-0.918771\pi\)
0.967616 0.252426i \(-0.0812286\pi\)
\(770\) 4.00000 0.144150
\(771\) 0 0
\(772\) 19.0000i 0.683825i
\(773\) 21.0000 0.755318 0.377659 0.925945i \(-0.376729\pi\)
0.377659 + 0.925945i \(0.376729\pi\)
\(774\) −6.00000 −0.215666
\(775\) 40.0000i 1.43684i
\(776\) 21.0000 0.753856
\(777\) 0 0
\(778\) 25.0000 0.896293
\(779\) − 54.0000i − 1.93475i
\(780\) 0 0
\(781\) 12.0000 0.429394
\(782\) − 12.0000i − 0.429119i
\(783\) 0 0
\(784\) 3.00000 0.107143
\(785\) 17.0000i 0.606756i
\(786\) 0 0
\(787\) 4.00000 0.142585 0.0712923 0.997455i \(-0.477288\pi\)
0.0712923 + 0.997455i \(0.477288\pi\)
\(788\) −15.0000 −0.534353
\(789\) 0 0
\(790\) 10.0000i 0.355784i
\(791\) − 12.0000i − 0.426671i
\(792\) 18.0000i 0.639602i
\(793\) −2.00000 −0.0710221
\(794\) 13.0000i 0.461353i
\(795\) 0 0
\(796\) 10.0000i 0.354441i
\(797\) 26.0000i 0.920967i 0.887668 + 0.460484i \(0.152324\pi\)
−0.887668 + 0.460484i \(0.847676\pi\)
\(798\) 0 0
\(799\) 18.0000i 0.636794i
\(800\) − 20.0000i − 0.707107i
\(801\) 21.0000i 0.741999i
\(802\) 18.0000 0.635602
\(803\) 20.0000 0.705785
\(804\) 0 0
\(805\) 8.00000 0.281963
\(806\) − 20.0000i − 0.704470i
\(807\) 0 0
\(808\) − 9.00000i − 0.316619i
\(809\) 10.0000i 0.351581i 0.984428 + 0.175791i \(0.0562482\pi\)
−0.984428 + 0.175791i \(0.943752\pi\)
\(810\) 9.00000 0.316228
\(811\) −16.0000 −0.561836 −0.280918 0.959732i \(-0.590639\pi\)
−0.280918 + 0.959732i \(0.590639\pi\)
\(812\) − 18.0000i − 0.631676i
\(813\) 0 0
\(814\) 0 0
\(815\) −6.00000 −0.210171
\(816\) 0 0
\(817\) 12.0000 0.419827
\(818\) 25.0000 0.874105
\(819\) 12.0000i 0.419314i
\(820\) 9.00000i 0.314294i
\(821\) 10.0000 0.349002 0.174501 0.984657i \(-0.444169\pi\)
0.174501 + 0.984657i \(0.444169\pi\)
\(822\) 0 0
\(823\) −16.0000 −0.557725 −0.278862 0.960331i \(-0.589957\pi\)
−0.278862 + 0.960331i \(0.589957\pi\)
\(824\) −18.0000 −0.627060
\(825\) 0 0
\(826\) −8.00000 −0.278356
\(827\) − 8.00000i − 0.278187i −0.990279 0.139094i \(-0.955581\pi\)
0.990279 0.139094i \(-0.0444189\pi\)
\(828\) 12.0000i 0.417029i
\(829\) − 38.0000i − 1.31979i −0.751356 0.659897i \(-0.770600\pi\)
0.751356 0.659897i \(-0.229400\pi\)
\(830\) −12.0000 −0.416526
\(831\) 0 0
\(832\) 14.0000i 0.485363i
\(833\) − 9.00000i − 0.311832i
\(834\) 0 0
\(835\) −12.0000 −0.415277
\(836\) − 12.0000i − 0.415029i
\(837\) 0 0
\(838\) 26.0000i 0.898155i
\(839\) 40.0000 1.38095 0.690477 0.723355i \(-0.257401\pi\)
0.690477 + 0.723355i \(0.257401\pi\)
\(840\) 0 0
\(841\) −52.0000 −1.79310
\(842\) −9.00000 −0.310160
\(843\) 0 0
\(844\) 2.00000 0.0688428
\(845\) 9.00000i 0.309609i
\(846\) 18.0000i 0.618853i
\(847\) −14.0000 −0.481046
\(848\) 2.00000 0.0686803
\(849\) 0 0
\(850\) 12.0000 0.411597
\(851\) 0 0
\(852\) 0 0
\(853\) 19.0000i 0.650548i 0.945620 + 0.325274i \(0.105456\pi\)
−0.945620 + 0.325274i \(0.894544\pi\)
\(854\) −2.00000 −0.0684386
\(855\) −18.0000 −0.615587
\(856\) 54.0000i 1.84568i
\(857\) − 45.0000i − 1.53717i −0.639747 0.768585i \(-0.720961\pi\)
0.639747 0.768585i \(-0.279039\pi\)
\(858\) 0 0
\(859\) − 50.0000i − 1.70598i −0.521929 0.852989i \(-0.674787\pi\)
0.521929 0.852989i \(-0.325213\pi\)
\(860\) −2.00000 −0.0681994
\(861\) 0 0
\(862\) −18.0000 −0.613082
\(863\) 36.0000 1.22545 0.612727 0.790295i \(-0.290072\pi\)
0.612727 + 0.790295i \(0.290072\pi\)
\(864\) 0 0
\(865\) − 21.0000i − 0.714021i
\(866\) 21.0000i 0.713609i
\(867\) 0 0
\(868\) 20.0000i 0.678844i
\(869\) 20.0000i 0.678454i
\(870\) 0 0
\(871\) − 20.0000i − 0.677674i
\(872\) 15.0000 0.507964
\(873\) − 21.0000i − 0.710742i
\(874\) 24.0000i 0.811812i
\(875\) 18.0000i 0.608511i
\(876\) 0 0
\(877\) 17.0000 0.574049 0.287025 0.957923i \(-0.407334\pi\)
0.287025 + 0.957923i \(0.407334\pi\)
\(878\) 20.0000 0.674967
\(879\) 0 0
\(880\) − 2.00000i − 0.0674200i
\(881\) −19.0000 −0.640126 −0.320063 0.947396i \(-0.603704\pi\)
−0.320063 + 0.947396i \(0.603704\pi\)
\(882\) − 9.00000i − 0.303046i
\(883\) − 48.0000i − 1.61533i −0.589643 0.807664i \(-0.700731\pi\)
0.589643 0.807664i \(-0.299269\pi\)
\(884\) 6.00000 0.201802
\(885\) 0 0
\(886\) − 32.0000i − 1.07506i
\(887\) 8.00000 0.268614 0.134307 0.990940i \(-0.457119\pi\)
0.134307 + 0.990940i \(0.457119\pi\)
\(888\) 0 0
\(889\) −16.0000 −0.536623
\(890\) − 7.00000i − 0.234641i
\(891\) 18.0000 0.603023
\(892\) 28.0000 0.937509
\(893\) − 36.0000i − 1.20469i
\(894\) 0 0
\(895\) 12.0000 0.401116
\(896\) − 6.00000i − 0.200446i
\(897\) 0 0
\(898\) 6.00000 0.200223
\(899\) 90.0000 3.00167
\(900\) −12.0000 −0.400000
\(901\) − 6.00000i − 0.199889i
\(902\) − 18.0000i − 0.599334i
\(903\) 0 0
\(904\) −18.0000 −0.598671
\(905\) 5.00000i 0.166206i
\(906\) 0 0
\(907\) − 38.0000i − 1.26177i −0.775877 0.630885i \(-0.782692\pi\)
0.775877 0.630885i \(-0.217308\pi\)
\(908\) 14.0000i 0.464606i
\(909\) −9.00000 −0.298511
\(910\) − 4.00000i − 0.132599i
\(911\) − 40.0000i − 1.32526i −0.748947 0.662630i \(-0.769440\pi\)
0.748947 0.662630i \(-0.230560\pi\)
\(912\) 0 0
\(913\) −24.0000 −0.794284
\(914\) 3.00000 0.0992312
\(915\) 0 0
\(916\) 1.00000 0.0330409
\(917\) 0 0
\(918\) 0 0
\(919\) 4.00000i 0.131948i 0.997821 + 0.0659739i \(0.0210154\pi\)
−0.997821 + 0.0659739i \(0.978985\pi\)
\(920\) − 12.0000i − 0.395628i
\(921\) 0 0
\(922\) 30.0000 0.987997
\(923\) − 12.0000i − 0.394985i
\(924\) 0 0
\(925\) 0 0
\(926\) −22.0000 −0.722965
\(927\) 18.0000i 0.591198i
\(928\) −45.0000 −1.47720
\(929\) −15.0000 −0.492134 −0.246067 0.969253i \(-0.579138\pi\)
−0.246067 + 0.969253i \(0.579138\pi\)
\(930\) 0 0
\(931\) 18.0000i 0.589926i
\(932\) −3.00000 −0.0982683
\(933\) 0 0
\(934\) −8.00000 −0.261768
\(935\) −6.00000 −0.196221
\(936\) 18.0000 0.588348
\(937\) −5.00000 −0.163343 −0.0816714 0.996659i \(-0.526026\pi\)
−0.0816714 + 0.996659i \(0.526026\pi\)
\(938\) − 20.0000i − 0.653023i
\(939\) 0 0
\(940\) 6.00000i 0.195698i
\(941\) 17.0000 0.554184 0.277092 0.960843i \(-0.410629\pi\)
0.277092 + 0.960843i \(0.410629\pi\)
\(942\) 0 0
\(943\) − 36.0000i − 1.17232i
\(944\) 4.00000i 0.130189i
\(945\) 0 0
\(946\) 4.00000 0.130051
\(947\) 36.0000i 1.16984i 0.811090 + 0.584921i \(0.198875\pi\)
−0.811090 + 0.584921i \(0.801125\pi\)
\(948\) 0 0
\(949\) − 20.0000i − 0.649227i
\(950\) −24.0000 −0.778663
\(951\) 0 0
\(952\) 18.0000 0.583383
\(953\) 50.0000 1.61966 0.809829 0.586665i \(-0.199560\pi\)
0.809829 + 0.586665i \(0.199560\pi\)
\(954\) − 6.00000i − 0.194257i
\(955\) 4.00000 0.129437
\(956\) 6.00000i 0.194054i
\(957\) 0 0
\(958\) −4.00000 −0.129234
\(959\) −18.0000 −0.581250
\(960\) 0 0
\(961\) −69.0000 −2.22581
\(962\) 0 0
\(963\) 54.0000 1.74013
\(964\) 14.0000i 0.450910i
\(965\) −19.0000 −0.611632
\(966\) 0 0
\(967\) − 8.00000i − 0.257263i −0.991692 0.128631i \(-0.958942\pi\)
0.991692 0.128631i \(-0.0410584\pi\)
\(968\) 21.0000i 0.674966i
\(969\) 0 0
\(970\) 7.00000i 0.224756i
\(971\) 28.0000 0.898563 0.449281 0.893390i \(-0.351680\pi\)
0.449281 + 0.893390i \(0.351680\pi\)
\(972\) 0 0
\(973\) −32.0000 −1.02587
\(974\) 12.0000 0.384505
\(975\) 0 0
\(976\) 1.00000i 0.0320092i
\(977\) 2.00000i 0.0639857i 0.999488 + 0.0319928i \(0.0101854\pi\)
−0.999488 + 0.0319928i \(0.989815\pi\)
\(978\) 0 0
\(979\) − 14.0000i − 0.447442i
\(980\) − 3.00000i − 0.0958315i
\(981\) − 15.0000i − 0.478913i
\(982\) − 2.00000i − 0.0638226i
\(983\) 18.0000 0.574111 0.287055 0.957914i \(-0.407324\pi\)
0.287055 + 0.957914i \(0.407324\pi\)
\(984\) 0 0
\(985\) − 15.0000i − 0.477940i
\(986\) − 27.0000i − 0.859855i
\(987\) 0 0
\(988\) −12.0000 −0.381771
\(989\) 8.00000 0.254385
\(990\) −6.00000 −0.190693
\(991\) 30.0000i 0.952981i 0.879180 + 0.476491i \(0.158091\pi\)
−0.879180 + 0.476491i \(0.841909\pi\)
\(992\) 50.0000 1.58750
\(993\) 0 0
\(994\) − 12.0000i − 0.380617i
\(995\) −10.0000 −0.317021
\(996\) 0 0
\(997\) − 18.0000i − 0.570066i −0.958518 0.285033i \(-0.907995\pi\)
0.958518 0.285033i \(-0.0920045\pi\)
\(998\) 18.0000 0.569780
\(999\) 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 1369.2.b.b.1368.1 2
37.6 odd 4 1369.2.a.b.1.1 1
37.14 odd 12 37.2.c.a.26.1 yes 2
37.29 odd 12 37.2.c.a.10.1 2
37.31 odd 4 1369.2.a.d.1.1 1
37.36 even 2 inner 1369.2.b.b.1368.2 2
111.14 even 12 333.2.f.a.100.1 2
111.29 even 12 333.2.f.a.10.1 2
148.51 even 12 592.2.i.c.433.1 2
148.103 even 12 592.2.i.c.417.1 2
185.14 odd 12 925.2.e.a.26.1 2
185.29 odd 12 925.2.e.a.676.1 2
185.88 even 12 925.2.o.a.174.1 4
185.103 even 12 925.2.o.a.824.2 4
185.162 even 12 925.2.o.a.174.2 4
185.177 even 12 925.2.o.a.824.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
37.2.c.a.10.1 2 37.29 odd 12
37.2.c.a.26.1 yes 2 37.14 odd 12
333.2.f.a.10.1 2 111.29 even 12
333.2.f.a.100.1 2 111.14 even 12
592.2.i.c.417.1 2 148.103 even 12
592.2.i.c.433.1 2 148.51 even 12
925.2.e.a.26.1 2 185.14 odd 12
925.2.e.a.676.1 2 185.29 odd 12
925.2.o.a.174.1 4 185.88 even 12
925.2.o.a.174.2 4 185.162 even 12
925.2.o.a.824.1 4 185.177 even 12
925.2.o.a.824.2 4 185.103 even 12
1369.2.a.b.1.1 1 37.6 odd 4
1369.2.a.d.1.1 1 37.31 odd 4
1369.2.b.b.1368.1 2 1.1 even 1 trivial
1369.2.b.b.1368.2 2 37.36 even 2 inner