Properties

Label 34.2.b.a.33.2
Level $34$
Weight $2$
Character 34.33
Analytic conductor $0.271$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

Related objects

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [34,2,Mod(33,34)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(34, 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("34.33");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 34 = 2 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 34.b (of order \(2\), degree \(1\), 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: \(0.271491366872\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-2}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 33.2
Root \(1.41421i\) of defining polynomial
Character \(\chi\) \(=\) 34.33
Dual form 34.2.b.a.33.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +2.82843i q^{3} +1.00000 q^{4} -2.82843i q^{5} -2.82843i q^{6} -1.00000 q^{8} -5.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +2.82843i q^{3} +1.00000 q^{4} -2.82843i q^{5} -2.82843i q^{6} -1.00000 q^{8} -5.00000 q^{9} +2.82843i q^{10} -2.82843i q^{11} +2.82843i q^{12} +2.00000 q^{13} +8.00000 q^{15} +1.00000 q^{16} +(-3.00000 + 2.82843i) q^{17} +5.00000 q^{18} -4.00000 q^{19} -2.82843i q^{20} +2.82843i q^{22} +5.65685i q^{23} -2.82843i q^{24} -3.00000 q^{25} -2.00000 q^{26} -5.65685i q^{27} -2.82843i q^{29} -8.00000 q^{30} -1.00000 q^{32} +8.00000 q^{33} +(3.00000 - 2.82843i) q^{34} -5.00000 q^{36} -8.48528i q^{37} +4.00000 q^{38} +5.65685i q^{39} +2.82843i q^{40} +5.65685i q^{41} -4.00000 q^{43} -2.82843i q^{44} +14.1421i q^{45} -5.65685i q^{46} +2.82843i q^{48} +7.00000 q^{49} +3.00000 q^{50} +(-8.00000 - 8.48528i) q^{51} +2.00000 q^{52} +6.00000 q^{53} +5.65685i q^{54} -8.00000 q^{55} -11.3137i q^{57} +2.82843i q^{58} +12.0000 q^{59} +8.00000 q^{60} +8.48528i q^{61} +1.00000 q^{64} -5.65685i q^{65} -8.00000 q^{66} -4.00000 q^{67} +(-3.00000 + 2.82843i) q^{68} -16.0000 q^{69} +5.65685i q^{71} +5.00000 q^{72} +8.48528i q^{74} -8.48528i q^{75} -4.00000 q^{76} -5.65685i q^{78} -16.9706i q^{79} -2.82843i q^{80} +1.00000 q^{81} -5.65685i q^{82} -12.0000 q^{83} +(8.00000 + 8.48528i) q^{85} +4.00000 q^{86} +8.00000 q^{87} +2.82843i q^{88} +6.00000 q^{89} -14.1421i q^{90} +5.65685i q^{92} +11.3137i q^{95} -2.82843i q^{96} +16.9706i q^{97} -7.00000 q^{98} +14.1421i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{8} - 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{8} - 10 q^{9} + 4 q^{13} + 16 q^{15} + 2 q^{16} - 6 q^{17} + 10 q^{18} - 8 q^{19} - 6 q^{25} - 4 q^{26} - 16 q^{30} - 2 q^{32} + 16 q^{33} + 6 q^{34} - 10 q^{36} + 8 q^{38} - 8 q^{43} + 14 q^{49} + 6 q^{50} - 16 q^{51} + 4 q^{52} + 12 q^{53} - 16 q^{55} + 24 q^{59} + 16 q^{60} + 2 q^{64} - 16 q^{66} - 8 q^{67} - 6 q^{68} - 32 q^{69} + 10 q^{72} - 8 q^{76} + 2 q^{81} - 24 q^{83} + 16 q^{85} + 8 q^{86} + 16 q^{87} + 12 q^{89} - 14 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/34\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 2.82843i 1.63299i 0.577350 + 0.816497i \(0.304087\pi\)
−0.577350 + 0.816497i \(0.695913\pi\)
\(4\) 1.00000 0.500000
\(5\) 2.82843i 1.26491i −0.774597 0.632456i \(-0.782047\pi\)
0.774597 0.632456i \(-0.217953\pi\)
\(6\) 2.82843i 1.15470i
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) −1.00000 −0.353553
\(9\) −5.00000 −1.66667
\(10\) 2.82843i 0.894427i
\(11\) 2.82843i 0.852803i −0.904534 0.426401i \(-0.859781\pi\)
0.904534 0.426401i \(-0.140219\pi\)
\(12\) 2.82843i 0.816497i
\(13\) 2.00000 0.554700 0.277350 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) 0 0
\(15\) 8.00000 2.06559
\(16\) 1.00000 0.250000
\(17\) −3.00000 + 2.82843i −0.727607 + 0.685994i
\(18\) 5.00000 1.17851
\(19\) −4.00000 −0.917663 −0.458831 0.888523i \(-0.651732\pi\)
−0.458831 + 0.888523i \(0.651732\pi\)
\(20\) 2.82843i 0.632456i
\(21\) 0 0
\(22\) 2.82843i 0.603023i
\(23\) 5.65685i 1.17954i 0.807573 + 0.589768i \(0.200781\pi\)
−0.807573 + 0.589768i \(0.799219\pi\)
\(24\) 2.82843i 0.577350i
\(25\) −3.00000 −0.600000
\(26\) −2.00000 −0.392232
\(27\) 5.65685i 1.08866i
\(28\) 0 0
\(29\) 2.82843i 0.525226i −0.964901 0.262613i \(-0.915416\pi\)
0.964901 0.262613i \(-0.0845842\pi\)
\(30\) −8.00000 −1.46059
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) −1.00000 −0.176777
\(33\) 8.00000 1.39262
\(34\) 3.00000 2.82843i 0.514496 0.485071i
\(35\) 0 0
\(36\) −5.00000 −0.833333
\(37\) 8.48528i 1.39497i −0.716599 0.697486i \(-0.754302\pi\)
0.716599 0.697486i \(-0.245698\pi\)
\(38\) 4.00000 0.648886
\(39\) 5.65685i 0.905822i
\(40\) 2.82843i 0.447214i
\(41\) 5.65685i 0.883452i 0.897150 + 0.441726i \(0.145634\pi\)
−0.897150 + 0.441726i \(0.854366\pi\)
\(42\) 0 0
\(43\) −4.00000 −0.609994 −0.304997 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) 2.82843i 0.426401i
\(45\) 14.1421i 2.10819i
\(46\) 5.65685i 0.834058i
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 2.82843i 0.408248i
\(49\) 7.00000 1.00000
\(50\) 3.00000 0.424264
\(51\) −8.00000 8.48528i −1.12022 1.18818i
\(52\) 2.00000 0.277350
\(53\) 6.00000 0.824163 0.412082 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) 5.65685i 0.769800i
\(55\) −8.00000 −1.07872
\(56\) 0 0
\(57\) 11.3137i 1.49854i
\(58\) 2.82843i 0.371391i
\(59\) 12.0000 1.56227 0.781133 0.624364i \(-0.214642\pi\)
0.781133 + 0.624364i \(0.214642\pi\)
\(60\) 8.00000 1.03280
\(61\) 8.48528i 1.08643i 0.839594 + 0.543214i \(0.182793\pi\)
−0.839594 + 0.543214i \(0.817207\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 5.65685i 0.701646i
\(66\) −8.00000 −0.984732
\(67\) −4.00000 −0.488678 −0.244339 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) −3.00000 + 2.82843i −0.363803 + 0.342997i
\(69\) −16.0000 −1.92617
\(70\) 0 0
\(71\) 5.65685i 0.671345i 0.941979 + 0.335673i \(0.108964\pi\)
−0.941979 + 0.335673i \(0.891036\pi\)
\(72\) 5.00000 0.589256
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) 8.48528i 0.986394i
\(75\) 8.48528i 0.979796i
\(76\) −4.00000 −0.458831
\(77\) 0 0
\(78\) 5.65685i 0.640513i
\(79\) 16.9706i 1.90934i −0.297670 0.954669i \(-0.596210\pi\)
0.297670 0.954669i \(-0.403790\pi\)
\(80\) 2.82843i 0.316228i
\(81\) 1.00000 0.111111
\(82\) 5.65685i 0.624695i
\(83\) −12.0000 −1.31717 −0.658586 0.752506i \(-0.728845\pi\)
−0.658586 + 0.752506i \(0.728845\pi\)
\(84\) 0 0
\(85\) 8.00000 + 8.48528i 0.867722 + 0.920358i
\(86\) 4.00000 0.431331
\(87\) 8.00000 0.857690
\(88\) 2.82843i 0.301511i
\(89\) 6.00000 0.635999 0.317999 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 14.1421i 1.49071i
\(91\) 0 0
\(92\) 5.65685i 0.589768i
\(93\) 0 0
\(94\) 0 0
\(95\) 11.3137i 1.16076i
\(96\) 2.82843i 0.288675i
\(97\) 16.9706i 1.72310i 0.507673 + 0.861550i \(0.330506\pi\)
−0.507673 + 0.861550i \(0.669494\pi\)
\(98\) −7.00000 −0.707107
\(99\) 14.1421i 1.42134i
\(100\) −3.00000 −0.300000
\(101\) −6.00000 −0.597022 −0.298511 0.954406i \(-0.596490\pi\)
−0.298511 + 0.954406i \(0.596490\pi\)
\(102\) 8.00000 + 8.48528i 0.792118 + 0.840168i
\(103\) 8.00000 0.788263 0.394132 0.919054i \(-0.371045\pi\)
0.394132 + 0.919054i \(0.371045\pi\)
\(104\) −2.00000 −0.196116
\(105\) 0 0
\(106\) −6.00000 −0.582772
\(107\) 2.82843i 0.273434i −0.990610 0.136717i \(-0.956345\pi\)
0.990610 0.136717i \(-0.0436552\pi\)
\(108\) 5.65685i 0.544331i
\(109\) 8.48528i 0.812743i −0.913708 0.406371i \(-0.866794\pi\)
0.913708 0.406371i \(-0.133206\pi\)
\(110\) 8.00000 0.762770
\(111\) 24.0000 2.27798
\(112\) 0 0
\(113\) 11.3137i 1.06430i −0.846649 0.532152i \(-0.821383\pi\)
0.846649 0.532152i \(-0.178617\pi\)
\(114\) 11.3137i 1.05963i
\(115\) 16.0000 1.49201
\(116\) 2.82843i 0.262613i
\(117\) −10.0000 −0.924500
\(118\) −12.0000 −1.10469
\(119\) 0 0
\(120\) −8.00000 −0.730297
\(121\) 3.00000 0.272727
\(122\) 8.48528i 0.768221i
\(123\) −16.0000 −1.44267
\(124\) 0 0
\(125\) 5.65685i 0.505964i
\(126\) 0 0
\(127\) −16.0000 −1.41977 −0.709885 0.704317i \(-0.751253\pi\)
−0.709885 + 0.704317i \(0.751253\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 11.3137i 0.996116i
\(130\) 5.65685i 0.496139i
\(131\) 2.82843i 0.247121i −0.992337 0.123560i \(-0.960569\pi\)
0.992337 0.123560i \(-0.0394313\pi\)
\(132\) 8.00000 0.696311
\(133\) 0 0
\(134\) 4.00000 0.345547
\(135\) −16.0000 −1.37706
\(136\) 3.00000 2.82843i 0.257248 0.242536i
\(137\) −18.0000 −1.53784 −0.768922 0.639343i \(-0.779207\pi\)
−0.768922 + 0.639343i \(0.779207\pi\)
\(138\) 16.0000 1.36201
\(139\) 8.48528i 0.719712i 0.933008 + 0.359856i \(0.117174\pi\)
−0.933008 + 0.359856i \(0.882826\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 5.65685i 0.474713i
\(143\) 5.65685i 0.473050i
\(144\) −5.00000 −0.416667
\(145\) −8.00000 −0.664364
\(146\) 0 0
\(147\) 19.7990i 1.63299i
\(148\) 8.48528i 0.697486i
\(149\) 6.00000 0.491539 0.245770 0.969328i \(-0.420959\pi\)
0.245770 + 0.969328i \(0.420959\pi\)
\(150\) 8.48528i 0.692820i
\(151\) 8.00000 0.651031 0.325515 0.945537i \(-0.394462\pi\)
0.325515 + 0.945537i \(0.394462\pi\)
\(152\) 4.00000 0.324443
\(153\) 15.0000 14.1421i 1.21268 1.14332i
\(154\) 0 0
\(155\) 0 0
\(156\) 5.65685i 0.452911i
\(157\) 14.0000 1.11732 0.558661 0.829396i \(-0.311315\pi\)
0.558661 + 0.829396i \(0.311315\pi\)
\(158\) 16.9706i 1.35011i
\(159\) 16.9706i 1.34585i
\(160\) 2.82843i 0.223607i
\(161\) 0 0
\(162\) −1.00000 −0.0785674
\(163\) 8.48528i 0.664619i −0.943170 0.332309i \(-0.892172\pi\)
0.943170 0.332309i \(-0.107828\pi\)
\(164\) 5.65685i 0.441726i
\(165\) 22.6274i 1.76154i
\(166\) 12.0000 0.931381
\(167\) 11.3137i 0.875481i −0.899101 0.437741i \(-0.855779\pi\)
0.899101 0.437741i \(-0.144221\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) −8.00000 8.48528i −0.613572 0.650791i
\(171\) 20.0000 1.52944
\(172\) −4.00000 −0.304997
\(173\) 2.82843i 0.215041i −0.994203 0.107521i \(-0.965709\pi\)
0.994203 0.107521i \(-0.0342912\pi\)
\(174\) −8.00000 −0.606478
\(175\) 0 0
\(176\) 2.82843i 0.213201i
\(177\) 33.9411i 2.55117i
\(178\) −6.00000 −0.449719
\(179\) 12.0000 0.896922 0.448461 0.893802i \(-0.351972\pi\)
0.448461 + 0.893802i \(0.351972\pi\)
\(180\) 14.1421i 1.05409i
\(181\) 8.48528i 0.630706i 0.948974 + 0.315353i \(0.102123\pi\)
−0.948974 + 0.315353i \(0.897877\pi\)
\(182\) 0 0
\(183\) −24.0000 −1.77413
\(184\) 5.65685i 0.417029i
\(185\) −24.0000 −1.76452
\(186\) 0 0
\(187\) 8.00000 + 8.48528i 0.585018 + 0.620505i
\(188\) 0 0
\(189\) 0 0
\(190\) 11.3137i 0.820783i
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) 2.82843i 0.204124i
\(193\) 16.9706i 1.22157i 0.791797 + 0.610784i \(0.209146\pi\)
−0.791797 + 0.610784i \(0.790854\pi\)
\(194\) 16.9706i 1.21842i
\(195\) 16.0000 1.14578
\(196\) 7.00000 0.500000
\(197\) 2.82843i 0.201517i −0.994911 0.100759i \(-0.967873\pi\)
0.994911 0.100759i \(-0.0321270\pi\)
\(198\) 14.1421i 1.00504i
\(199\) 16.9706i 1.20301i −0.798869 0.601506i \(-0.794568\pi\)
0.798869 0.601506i \(-0.205432\pi\)
\(200\) 3.00000 0.212132
\(201\) 11.3137i 0.798007i
\(202\) 6.00000 0.422159
\(203\) 0 0
\(204\) −8.00000 8.48528i −0.560112 0.594089i
\(205\) 16.0000 1.11749
\(206\) −8.00000 −0.557386
\(207\) 28.2843i 1.96589i
\(208\) 2.00000 0.138675
\(209\) 11.3137i 0.782586i
\(210\) 0 0
\(211\) 8.48528i 0.584151i 0.956395 + 0.292075i \(0.0943458\pi\)
−0.956395 + 0.292075i \(0.905654\pi\)
\(212\) 6.00000 0.412082
\(213\) −16.0000 −1.09630
\(214\) 2.82843i 0.193347i
\(215\) 11.3137i 0.771589i
\(216\) 5.65685i 0.384900i
\(217\) 0 0
\(218\) 8.48528i 0.574696i
\(219\) 0 0
\(220\) −8.00000 −0.539360
\(221\) −6.00000 + 5.65685i −0.403604 + 0.380521i
\(222\) −24.0000 −1.61077
\(223\) −16.0000 −1.07144 −0.535720 0.844396i \(-0.679960\pi\)
−0.535720 + 0.844396i \(0.679960\pi\)
\(224\) 0 0
\(225\) 15.0000 1.00000
\(226\) 11.3137i 0.752577i
\(227\) 19.7990i 1.31411i −0.753845 0.657053i \(-0.771803\pi\)
0.753845 0.657053i \(-0.228197\pi\)
\(228\) 11.3137i 0.749269i
\(229\) −22.0000 −1.45380 −0.726900 0.686743i \(-0.759040\pi\)
−0.726900 + 0.686743i \(0.759040\pi\)
\(230\) −16.0000 −1.05501
\(231\) 0 0
\(232\) 2.82843i 0.185695i
\(233\) 22.6274i 1.48237i 0.671300 + 0.741186i \(0.265736\pi\)
−0.671300 + 0.741186i \(0.734264\pi\)
\(234\) 10.0000 0.653720
\(235\) 0 0
\(236\) 12.0000 0.781133
\(237\) 48.0000 3.11794
\(238\) 0 0
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 8.00000 0.516398
\(241\) 16.9706i 1.09317i 0.837404 + 0.546585i \(0.184072\pi\)
−0.837404 + 0.546585i \(0.815928\pi\)
\(242\) −3.00000 −0.192847
\(243\) 14.1421i 0.907218i
\(244\) 8.48528i 0.543214i
\(245\) 19.7990i 1.26491i
\(246\) 16.0000 1.02012
\(247\) −8.00000 −0.509028
\(248\) 0 0
\(249\) 33.9411i 2.15093i
\(250\) 5.65685i 0.357771i
\(251\) −12.0000 −0.757433 −0.378717 0.925513i \(-0.623635\pi\)
−0.378717 + 0.925513i \(0.623635\pi\)
\(252\) 0 0
\(253\) 16.0000 1.00591
\(254\) 16.0000 1.00393
\(255\) −24.0000 + 22.6274i −1.50294 + 1.41698i
\(256\) 1.00000 0.0625000
\(257\) 6.00000 0.374270 0.187135 0.982334i \(-0.440080\pi\)
0.187135 + 0.982334i \(0.440080\pi\)
\(258\) 11.3137i 0.704361i
\(259\) 0 0
\(260\) 5.65685i 0.350823i
\(261\) 14.1421i 0.875376i
\(262\) 2.82843i 0.174741i
\(263\) 24.0000 1.47990 0.739952 0.672660i \(-0.234848\pi\)
0.739952 + 0.672660i \(0.234848\pi\)
\(264\) −8.00000 −0.492366
\(265\) 16.9706i 1.04249i
\(266\) 0 0
\(267\) 16.9706i 1.03858i
\(268\) −4.00000 −0.244339
\(269\) 31.1127i 1.89697i 0.316815 + 0.948487i \(0.397387\pi\)
−0.316815 + 0.948487i \(0.602613\pi\)
\(270\) 16.0000 0.973729
\(271\) −16.0000 −0.971931 −0.485965 0.873978i \(-0.661532\pi\)
−0.485965 + 0.873978i \(0.661532\pi\)
\(272\) −3.00000 + 2.82843i −0.181902 + 0.171499i
\(273\) 0 0
\(274\) 18.0000 1.08742
\(275\) 8.48528i 0.511682i
\(276\) −16.0000 −0.963087
\(277\) 25.4558i 1.52949i −0.644331 0.764747i \(-0.722864\pi\)
0.644331 0.764747i \(-0.277136\pi\)
\(278\) 8.48528i 0.508913i
\(279\) 0 0
\(280\) 0 0
\(281\) 18.0000 1.07379 0.536895 0.843649i \(-0.319597\pi\)
0.536895 + 0.843649i \(0.319597\pi\)
\(282\) 0 0
\(283\) 8.48528i 0.504398i 0.967675 + 0.252199i \(0.0811537\pi\)
−0.967675 + 0.252199i \(0.918846\pi\)
\(284\) 5.65685i 0.335673i
\(285\) −32.0000 −1.89552
\(286\) 5.65685i 0.334497i
\(287\) 0 0
\(288\) 5.00000 0.294628
\(289\) 1.00000 16.9706i 0.0588235 0.998268i
\(290\) 8.00000 0.469776
\(291\) −48.0000 −2.81381
\(292\) 0 0
\(293\) 6.00000 0.350524 0.175262 0.984522i \(-0.443923\pi\)
0.175262 + 0.984522i \(0.443923\pi\)
\(294\) 19.7990i 1.15470i
\(295\) 33.9411i 1.97613i
\(296\) 8.48528i 0.493197i
\(297\) −16.0000 −0.928414
\(298\) −6.00000 −0.347571
\(299\) 11.3137i 0.654289i
\(300\) 8.48528i 0.489898i
\(301\) 0 0
\(302\) −8.00000 −0.460348
\(303\) 16.9706i 0.974933i
\(304\) −4.00000 −0.229416
\(305\) 24.0000 1.37424
\(306\) −15.0000 + 14.1421i −0.857493 + 0.808452i
\(307\) 20.0000 1.14146 0.570730 0.821138i \(-0.306660\pi\)
0.570730 + 0.821138i \(0.306660\pi\)
\(308\) 0 0
\(309\) 22.6274i 1.28723i
\(310\) 0 0
\(311\) 11.3137i 0.641542i −0.947157 0.320771i \(-0.896058\pi\)
0.947157 0.320771i \(-0.103942\pi\)
\(312\) 5.65685i 0.320256i
\(313\) 16.9706i 0.959233i −0.877478 0.479616i \(-0.840776\pi\)
0.877478 0.479616i \(-0.159224\pi\)
\(314\) −14.0000 −0.790066
\(315\) 0 0
\(316\) 16.9706i 0.954669i
\(317\) 2.82843i 0.158860i −0.996840 0.0794301i \(-0.974690\pi\)
0.996840 0.0794301i \(-0.0253101\pi\)
\(318\) 16.9706i 0.951662i
\(319\) −8.00000 −0.447914
\(320\) 2.82843i 0.158114i
\(321\) 8.00000 0.446516
\(322\) 0 0
\(323\) 12.0000 11.3137i 0.667698 0.629512i
\(324\) 1.00000 0.0555556
\(325\) −6.00000 −0.332820
\(326\) 8.48528i 0.469956i
\(327\) 24.0000 1.32720
\(328\) 5.65685i 0.312348i
\(329\) 0 0
\(330\) 22.6274i 1.24560i
\(331\) −4.00000 −0.219860 −0.109930 0.993939i \(-0.535063\pi\)
−0.109930 + 0.993939i \(0.535063\pi\)
\(332\) −12.0000 −0.658586
\(333\) 42.4264i 2.32495i
\(334\) 11.3137i 0.619059i
\(335\) 11.3137i 0.618134i
\(336\) 0 0
\(337\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(338\) 9.00000 0.489535
\(339\) 32.0000 1.73800
\(340\) 8.00000 + 8.48528i 0.433861 + 0.460179i
\(341\) 0 0
\(342\) −20.0000 −1.08148
\(343\) 0 0
\(344\) 4.00000 0.215666
\(345\) 45.2548i 2.43644i
\(346\) 2.82843i 0.152057i
\(347\) 14.1421i 0.759190i 0.925153 + 0.379595i \(0.123937\pi\)
−0.925153 + 0.379595i \(0.876063\pi\)
\(348\) 8.00000 0.428845
\(349\) −34.0000 −1.81998 −0.909989 0.414632i \(-0.863910\pi\)
−0.909989 + 0.414632i \(0.863910\pi\)
\(350\) 0 0
\(351\) 11.3137i 0.603881i
\(352\) 2.82843i 0.150756i
\(353\) −6.00000 −0.319348 −0.159674 0.987170i \(-0.551044\pi\)
−0.159674 + 0.987170i \(0.551044\pi\)
\(354\) 33.9411i 1.80395i
\(355\) 16.0000 0.849192
\(356\) 6.00000 0.317999
\(357\) 0 0
\(358\) −12.0000 −0.634220
\(359\) 24.0000 1.26667 0.633336 0.773877i \(-0.281685\pi\)
0.633336 + 0.773877i \(0.281685\pi\)
\(360\) 14.1421i 0.745356i
\(361\) −3.00000 −0.157895
\(362\) 8.48528i 0.445976i
\(363\) 8.48528i 0.445362i
\(364\) 0 0
\(365\) 0 0
\(366\) 24.0000 1.25450
\(367\) 16.9706i 0.885856i 0.896557 + 0.442928i \(0.146060\pi\)
−0.896557 + 0.442928i \(0.853940\pi\)
\(368\) 5.65685i 0.294884i
\(369\) 28.2843i 1.47242i
\(370\) 24.0000 1.24770
\(371\) 0 0
\(372\) 0 0
\(373\) −22.0000 −1.13912 −0.569558 0.821951i \(-0.692886\pi\)
−0.569558 + 0.821951i \(0.692886\pi\)
\(374\) −8.00000 8.48528i −0.413670 0.438763i
\(375\) 16.0000 0.826236
\(376\) 0 0
\(377\) 5.65685i 0.291343i
\(378\) 0 0
\(379\) 25.4558i 1.30758i 0.756677 + 0.653789i \(0.226822\pi\)
−0.756677 + 0.653789i \(0.773178\pi\)
\(380\) 11.3137i 0.580381i
\(381\) 45.2548i 2.31848i
\(382\) 0 0
\(383\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(384\) 2.82843i 0.144338i
\(385\) 0 0
\(386\) 16.9706i 0.863779i
\(387\) 20.0000 1.01666
\(388\) 16.9706i 0.861550i
\(389\) −6.00000 −0.304212 −0.152106 0.988364i \(-0.548606\pi\)
−0.152106 + 0.988364i \(0.548606\pi\)
\(390\) −16.0000 −0.810191
\(391\) −16.0000 16.9706i −0.809155 0.858238i
\(392\) −7.00000 −0.353553
\(393\) 8.00000 0.403547
\(394\) 2.82843i 0.142494i
\(395\) −48.0000 −2.41514
\(396\) 14.1421i 0.710669i
\(397\) 8.48528i 0.425864i 0.977067 + 0.212932i \(0.0683013\pi\)
−0.977067 + 0.212932i \(0.931699\pi\)
\(398\) 16.9706i 0.850657i
\(399\) 0 0
\(400\) −3.00000 −0.150000
\(401\) 28.2843i 1.41245i −0.707988 0.706225i \(-0.750397\pi\)
0.707988 0.706225i \(-0.249603\pi\)
\(402\) 11.3137i 0.564276i
\(403\) 0 0
\(404\) −6.00000 −0.298511
\(405\) 2.82843i 0.140546i
\(406\) 0 0
\(407\) −24.0000 −1.18964
\(408\) 8.00000 + 8.48528i 0.396059 + 0.420084i
\(409\) 2.00000 0.0988936 0.0494468 0.998777i \(-0.484254\pi\)
0.0494468 + 0.998777i \(0.484254\pi\)
\(410\) −16.0000 −0.790184
\(411\) 50.9117i 2.51129i
\(412\) 8.00000 0.394132
\(413\) 0 0
\(414\) 28.2843i 1.39010i
\(415\) 33.9411i 1.66610i
\(416\) −2.00000 −0.0980581
\(417\) −24.0000 −1.17529
\(418\) 11.3137i 0.553372i
\(419\) 14.1421i 0.690889i 0.938439 + 0.345444i \(0.112272\pi\)
−0.938439 + 0.345444i \(0.887728\pi\)
\(420\) 0 0
\(421\) 26.0000 1.26716 0.633581 0.773676i \(-0.281584\pi\)
0.633581 + 0.773676i \(0.281584\pi\)
\(422\) 8.48528i 0.413057i
\(423\) 0 0
\(424\) −6.00000 −0.291386
\(425\) 9.00000 8.48528i 0.436564 0.411597i
\(426\) 16.0000 0.775203
\(427\) 0 0
\(428\) 2.82843i 0.136717i
\(429\) 16.0000 0.772487
\(430\) 11.3137i 0.545595i
\(431\) 5.65685i 0.272481i 0.990676 + 0.136241i \(0.0435020\pi\)
−0.990676 + 0.136241i \(0.956498\pi\)
\(432\) 5.65685i 0.272166i
\(433\) 14.0000 0.672797 0.336399 0.941720i \(-0.390791\pi\)
0.336399 + 0.941720i \(0.390791\pi\)
\(434\) 0 0
\(435\) 22.6274i 1.08490i
\(436\) 8.48528i 0.406371i
\(437\) 22.6274i 1.08242i
\(438\) 0 0
\(439\) 16.9706i 0.809961i 0.914325 + 0.404980i \(0.132722\pi\)
−0.914325 + 0.404980i \(0.867278\pi\)
\(440\) 8.00000 0.381385
\(441\) −35.0000 −1.66667
\(442\) 6.00000 5.65685i 0.285391 0.269069i
\(443\) −12.0000 −0.570137 −0.285069 0.958507i \(-0.592016\pi\)
−0.285069 + 0.958507i \(0.592016\pi\)
\(444\) 24.0000 1.13899
\(445\) 16.9706i 0.804482i
\(446\) 16.0000 0.757622
\(447\) 16.9706i 0.802680i
\(448\) 0 0
\(449\) 5.65685i 0.266963i 0.991051 + 0.133482i \(0.0426157\pi\)
−0.991051 + 0.133482i \(0.957384\pi\)
\(450\) −15.0000 −0.707107
\(451\) 16.0000 0.753411
\(452\) 11.3137i 0.532152i
\(453\) 22.6274i 1.06313i
\(454\) 19.7990i 0.929213i
\(455\) 0 0
\(456\) 11.3137i 0.529813i
\(457\) −10.0000 −0.467780 −0.233890 0.972263i \(-0.575146\pi\)
−0.233890 + 0.972263i \(0.575146\pi\)
\(458\) 22.0000 1.02799
\(459\) 16.0000 + 16.9706i 0.746816 + 0.792118i
\(460\) 16.0000 0.746004
\(461\) −18.0000 −0.838344 −0.419172 0.907907i \(-0.637680\pi\)
−0.419172 + 0.907907i \(0.637680\pi\)
\(462\) 0 0
\(463\) 32.0000 1.48717 0.743583 0.668644i \(-0.233125\pi\)
0.743583 + 0.668644i \(0.233125\pi\)
\(464\) 2.82843i 0.131306i
\(465\) 0 0
\(466\) 22.6274i 1.04819i
\(467\) −36.0000 −1.66588 −0.832941 0.553362i \(-0.813345\pi\)
−0.832941 + 0.553362i \(0.813345\pi\)
\(468\) −10.0000 −0.462250
\(469\) 0 0
\(470\) 0 0
\(471\) 39.5980i 1.82458i
\(472\) −12.0000 −0.552345
\(473\) 11.3137i 0.520205i
\(474\) −48.0000 −2.20471
\(475\) 12.0000 0.550598
\(476\) 0 0
\(477\) −30.0000 −1.37361
\(478\) 0 0
\(479\) 5.65685i 0.258468i 0.991614 + 0.129234i \(0.0412519\pi\)
−0.991614 + 0.129234i \(0.958748\pi\)
\(480\) −8.00000 −0.365148
\(481\) 16.9706i 0.773791i
\(482\) 16.9706i 0.772988i
\(483\) 0 0
\(484\) 3.00000 0.136364
\(485\) 48.0000 2.17957
\(486\) 14.1421i 0.641500i
\(487\) 16.9706i 0.769010i 0.923123 + 0.384505i \(0.125628\pi\)
−0.923123 + 0.384505i \(0.874372\pi\)
\(488\) 8.48528i 0.384111i
\(489\) 24.0000 1.08532
\(490\) 19.7990i 0.894427i
\(491\) −12.0000 −0.541552 −0.270776 0.962642i \(-0.587280\pi\)
−0.270776 + 0.962642i \(0.587280\pi\)
\(492\) −16.0000 −0.721336
\(493\) 8.00000 + 8.48528i 0.360302 + 0.382158i
\(494\) 8.00000 0.359937
\(495\) 40.0000 1.79787
\(496\) 0 0
\(497\) 0 0
\(498\) 33.9411i 1.52094i
\(499\) 25.4558i 1.13956i −0.821797 0.569780i \(-0.807028\pi\)
0.821797 0.569780i \(-0.192972\pi\)
\(500\) 5.65685i 0.252982i
\(501\) 32.0000 1.42965
\(502\) 12.0000 0.535586
\(503\) 28.2843i 1.26113i −0.776135 0.630567i \(-0.782823\pi\)
0.776135 0.630567i \(-0.217177\pi\)
\(504\) 0 0
\(505\) 16.9706i 0.755180i
\(506\) −16.0000 −0.711287
\(507\) 25.4558i 1.13053i
\(508\) −16.0000 −0.709885
\(509\) −18.0000 −0.797836 −0.398918 0.916987i \(-0.630614\pi\)
−0.398918 + 0.916987i \(0.630614\pi\)
\(510\) 24.0000 22.6274i 1.06274 1.00196i
\(511\) 0 0
\(512\) −1.00000 −0.0441942
\(513\) 22.6274i 0.999025i
\(514\) −6.00000 −0.264649
\(515\) 22.6274i 0.997083i
\(516\) 11.3137i 0.498058i
\(517\) 0 0
\(518\) 0 0
\(519\) 8.00000 0.351161
\(520\) 5.65685i 0.248069i
\(521\) 5.65685i 0.247831i 0.992293 + 0.123916i \(0.0395452\pi\)
−0.992293 + 0.123916i \(0.960455\pi\)
\(522\) 14.1421i 0.618984i
\(523\) 20.0000 0.874539 0.437269 0.899331i \(-0.355946\pi\)
0.437269 + 0.899331i \(0.355946\pi\)
\(524\) 2.82843i 0.123560i
\(525\) 0 0
\(526\) −24.0000 −1.04645
\(527\) 0 0
\(528\) 8.00000 0.348155
\(529\) −9.00000 −0.391304
\(530\) 16.9706i 0.737154i
\(531\) −60.0000 −2.60378
\(532\) 0 0
\(533\) 11.3137i 0.490051i
\(534\) 16.9706i 0.734388i
\(535\) −8.00000 −0.345870
\(536\) 4.00000 0.172774
\(537\) 33.9411i 1.46467i
\(538\) 31.1127i 1.34136i
\(539\) 19.7990i 0.852803i
\(540\) −16.0000 −0.688530
\(541\) 8.48528i 0.364811i −0.983223 0.182405i \(-0.941612\pi\)
0.983223 0.182405i \(-0.0583883\pi\)
\(542\) 16.0000 0.687259
\(543\) −24.0000 −1.02994
\(544\) 3.00000 2.82843i 0.128624 0.121268i
\(545\) −24.0000 −1.02805
\(546\) 0 0
\(547\) 42.4264i 1.81402i 0.421107 + 0.907011i \(0.361642\pi\)
−0.421107 + 0.907011i \(0.638358\pi\)
\(548\) −18.0000 −0.768922
\(549\) 42.4264i 1.81071i
\(550\) 8.48528i 0.361814i
\(551\) 11.3137i 0.481980i
\(552\) 16.0000 0.681005
\(553\) 0 0
\(554\) 25.4558i 1.08152i
\(555\) 67.8823i 2.88144i
\(556\) 8.48528i 0.359856i
\(557\) 18.0000 0.762684 0.381342 0.924434i \(-0.375462\pi\)
0.381342 + 0.924434i \(0.375462\pi\)
\(558\) 0 0
\(559\) −8.00000 −0.338364
\(560\) 0 0
\(561\) −24.0000 + 22.6274i −1.01328 + 0.955330i
\(562\) −18.0000 −0.759284
\(563\) 36.0000 1.51722 0.758610 0.651546i \(-0.225879\pi\)
0.758610 + 0.651546i \(0.225879\pi\)
\(564\) 0 0
\(565\) −32.0000 −1.34625
\(566\) 8.48528i 0.356663i
\(567\) 0 0
\(568\) 5.65685i 0.237356i
\(569\) −6.00000 −0.251533 −0.125767 0.992060i \(-0.540139\pi\)
−0.125767 + 0.992060i \(0.540139\pi\)
\(570\) 32.0000 1.34033
\(571\) 25.4558i 1.06529i 0.846338 + 0.532647i \(0.178803\pi\)
−0.846338 + 0.532647i \(0.821197\pi\)
\(572\) 5.65685i 0.236525i
\(573\) 0 0
\(574\) 0 0
\(575\) 16.9706i 0.707721i
\(576\) −5.00000 −0.208333
\(577\) 38.0000 1.58196 0.790980 0.611842i \(-0.209571\pi\)
0.790980 + 0.611842i \(0.209571\pi\)
\(578\) −1.00000 + 16.9706i −0.0415945 + 0.705882i
\(579\) −48.0000 −1.99481
\(580\) −8.00000 −0.332182
\(581\) 0 0
\(582\) 48.0000 1.98966
\(583\) 16.9706i 0.702849i
\(584\) 0 0
\(585\) 28.2843i 1.16941i
\(586\) −6.00000 −0.247858
\(587\) −12.0000 −0.495293 −0.247647 0.968850i \(-0.579657\pi\)
−0.247647 + 0.968850i \(0.579657\pi\)
\(588\) 19.7990i 0.816497i
\(589\) 0 0
\(590\) 33.9411i 1.39733i
\(591\) 8.00000 0.329076
\(592\) 8.48528i 0.348743i
\(593\) 18.0000 0.739171 0.369586 0.929197i \(-0.379500\pi\)
0.369586 + 0.929197i \(0.379500\pi\)
\(594\) 16.0000 0.656488
\(595\) 0 0
\(596\) 6.00000 0.245770
\(597\) 48.0000 1.96451
\(598\) 11.3137i 0.462652i
\(599\) 24.0000 0.980613 0.490307 0.871550i \(-0.336885\pi\)
0.490307 + 0.871550i \(0.336885\pi\)
\(600\) 8.48528i 0.346410i
\(601\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(602\) 0 0
\(603\) 20.0000 0.814463
\(604\) 8.00000 0.325515
\(605\) 8.48528i 0.344976i
\(606\) 16.9706i 0.689382i
\(607\) 33.9411i 1.37763i −0.724938 0.688814i \(-0.758132\pi\)
0.724938 0.688814i \(-0.241868\pi\)
\(608\) 4.00000 0.162221
\(609\) 0 0
\(610\) −24.0000 −0.971732
\(611\) 0 0
\(612\) 15.0000 14.1421i 0.606339 0.571662i
\(613\) −10.0000 −0.403896 −0.201948 0.979396i \(-0.564727\pi\)
−0.201948 + 0.979396i \(0.564727\pi\)
\(614\) −20.0000 −0.807134
\(615\) 45.2548i 1.82485i
\(616\) 0 0
\(617\) 11.3137i 0.455473i −0.973723 0.227736i \(-0.926868\pi\)
0.973723 0.227736i \(-0.0731324\pi\)
\(618\) 22.6274i 0.910208i
\(619\) 8.48528i 0.341052i −0.985353 0.170526i \(-0.945453\pi\)
0.985353 0.170526i \(-0.0545467\pi\)
\(620\) 0 0
\(621\) 32.0000 1.28412
\(622\) 11.3137i 0.453638i
\(623\) 0 0
\(624\) 5.65685i 0.226455i
\(625\) −31.0000 −1.24000
\(626\) 16.9706i 0.678280i
\(627\) −32.0000 −1.27796
\(628\) 14.0000 0.558661
\(629\) 24.0000 + 25.4558i 0.956943 + 1.01499i
\(630\) 0 0
\(631\) 8.00000 0.318475 0.159237 0.987240i \(-0.449096\pi\)
0.159237 + 0.987240i \(0.449096\pi\)
\(632\) 16.9706i 0.675053i
\(633\) −24.0000 −0.953914
\(634\) 2.82843i 0.112331i
\(635\) 45.2548i 1.79588i
\(636\) 16.9706i 0.672927i
\(637\) 14.0000 0.554700
\(638\) 8.00000 0.316723
\(639\) 28.2843i 1.11891i
\(640\) 2.82843i 0.111803i
\(641\) 5.65685i 0.223432i 0.993740 + 0.111716i \(0.0356347\pi\)
−0.993740 + 0.111716i \(0.964365\pi\)
\(642\) −8.00000 −0.315735
\(643\) 8.48528i 0.334627i −0.985904 0.167313i \(-0.946491\pi\)
0.985904 0.167313i \(-0.0535092\pi\)
\(644\) 0 0
\(645\) −32.0000 −1.26000
\(646\) −12.0000 + 11.3137i −0.472134 + 0.445132i
\(647\) −24.0000 −0.943537 −0.471769 0.881722i \(-0.656384\pi\)
−0.471769 + 0.881722i \(0.656384\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 33.9411i 1.33231i
\(650\) 6.00000 0.235339
\(651\) 0 0
\(652\) 8.48528i 0.332309i
\(653\) 14.1421i 0.553425i 0.960953 + 0.276712i \(0.0892449\pi\)
−0.960953 + 0.276712i \(0.910755\pi\)
\(654\) −24.0000 −0.938474
\(655\) −8.00000 −0.312586
\(656\) 5.65685i 0.220863i
\(657\) 0 0
\(658\) 0 0
\(659\) −12.0000 −0.467454 −0.233727 0.972302i \(-0.575092\pi\)
−0.233727 + 0.972302i \(0.575092\pi\)
\(660\) 22.6274i 0.880771i
\(661\) −10.0000 −0.388955 −0.194477 0.980907i \(-0.562301\pi\)
−0.194477 + 0.980907i \(0.562301\pi\)
\(662\) 4.00000 0.155464
\(663\) −16.0000 16.9706i −0.621389 0.659082i
\(664\) 12.0000 0.465690
\(665\) 0 0
\(666\) 42.4264i 1.64399i
\(667\) 16.0000 0.619522
\(668\) 11.3137i 0.437741i
\(669\) 45.2548i 1.74965i
\(670\) 11.3137i 0.437087i
\(671\) 24.0000 0.926510
\(672\) 0 0
\(673\) 16.9706i 0.654167i −0.944995 0.327084i \(-0.893934\pi\)
0.944995 0.327084i \(-0.106066\pi\)
\(674\) 0 0
\(675\) 16.9706i 0.653197i
\(676\) −9.00000 −0.346154
\(677\) 14.1421i 0.543526i 0.962364 + 0.271763i \(0.0876068\pi\)
−0.962364 + 0.271763i \(0.912393\pi\)
\(678\) −32.0000 −1.22895
\(679\) 0 0
\(680\) −8.00000 8.48528i −0.306786 0.325396i
\(681\) 56.0000 2.14592
\(682\) 0 0
\(683\) 19.7990i 0.757587i −0.925481 0.378794i \(-0.876339\pi\)
0.925481 0.378794i \(-0.123661\pi\)
\(684\) 20.0000 0.764719
\(685\) 50.9117i 1.94524i
\(686\) 0 0
\(687\) 62.2254i 2.37405i
\(688\) −4.00000 −0.152499
\(689\) 12.0000 0.457164
\(690\) 45.2548i 1.72282i
\(691\) 8.48528i 0.322795i 0.986889 + 0.161398i \(0.0516002\pi\)
−0.986889 + 0.161398i \(0.948400\pi\)
\(692\) 2.82843i 0.107521i
\(693\) 0 0
\(694\) 14.1421i 0.536828i
\(695\) 24.0000 0.910372
\(696\) −8.00000 −0.303239
\(697\) −16.0000 16.9706i −0.606043 0.642806i
\(698\) 34.0000 1.28692
\(699\) −64.0000 −2.42070
\(700\) 0 0
\(701\) −30.0000 −1.13308 −0.566542 0.824033i \(-0.691719\pi\)
−0.566542 + 0.824033i \(0.691719\pi\)
\(702\) 11.3137i 0.427008i
\(703\) 33.9411i 1.28011i
\(704\) 2.82843i 0.106600i
\(705\) 0 0
\(706\) 6.00000 0.225813
\(707\) 0 0
\(708\) 33.9411i 1.27559i
\(709\) 8.48528i 0.318671i 0.987224 + 0.159336i \(0.0509352\pi\)
−0.987224 + 0.159336i \(0.949065\pi\)
\(710\) −16.0000 −0.600469
\(711\) 84.8528i 3.18223i
\(712\) −6.00000 −0.224860
\(713\) 0 0
\(714\) 0 0
\(715\) −16.0000 −0.598366
\(716\) 12.0000 0.448461
\(717\) 0 0
\(718\) −24.0000 −0.895672
\(719\) 39.5980i 1.47676i 0.674387 + 0.738378i \(0.264408\pi\)
−0.674387 + 0.738378i \(0.735592\pi\)
\(720\) 14.1421i 0.527046i
\(721\) 0 0
\(722\) 3.00000 0.111648
\(723\) −48.0000 −1.78514
\(724\) 8.48528i 0.315353i
\(725\) 8.48528i 0.315135i
\(726\) 8.48528i 0.314918i
\(727\) 8.00000 0.296704 0.148352 0.988935i \(-0.452603\pi\)
0.148352 + 0.988935i \(0.452603\pi\)
\(728\) 0 0
\(729\) 43.0000 1.59259
\(730\) 0 0
\(731\) 12.0000 11.3137i 0.443836 0.418453i
\(732\) −24.0000 −0.887066
\(733\) 14.0000 0.517102 0.258551 0.965998i \(-0.416755\pi\)
0.258551 + 0.965998i \(0.416755\pi\)
\(734\) 16.9706i 0.626395i
\(735\) 56.0000 2.06559
\(736\) 5.65685i 0.208514i
\(737\) 11.3137i 0.416746i
\(738\) 28.2843i 1.04116i
\(739\) 44.0000 1.61857 0.809283 0.587419i \(-0.199856\pi\)
0.809283 + 0.587419i \(0.199856\pi\)
\(740\) −24.0000 −0.882258
\(741\) 22.6274i 0.831239i
\(742\) 0 0
\(743\) 11.3137i 0.415060i −0.978229 0.207530i \(-0.933458\pi\)
0.978229 0.207530i \(-0.0665424\pi\)
\(744\) 0 0
\(745\) 16.9706i 0.621753i
\(746\) 22.0000 0.805477
\(747\) 60.0000 2.19529
\(748\) 8.00000 + 8.48528i 0.292509 + 0.310253i
\(749\) 0 0
\(750\) −16.0000 −0.584237
\(751\) 16.9706i 0.619265i −0.950856 0.309632i \(-0.899794\pi\)
0.950856 0.309632i \(-0.100206\pi\)
\(752\) 0 0
\(753\) 33.9411i 1.23688i
\(754\) 5.65685i 0.206010i
\(755\) 22.6274i 0.823496i
\(756\) 0 0
\(757\) −22.0000 −0.799604 −0.399802 0.916602i \(-0.630921\pi\)
−0.399802 + 0.916602i \(0.630921\pi\)
\(758\) 25.4558i 0.924598i
\(759\) 45.2548i 1.64265i
\(760\) 11.3137i 0.410391i
\(761\) 30.0000 1.08750 0.543750 0.839248i \(-0.317004\pi\)
0.543750 + 0.839248i \(0.317004\pi\)
\(762\) 45.2548i 1.63941i
\(763\) 0 0
\(764\) 0 0
\(765\) −40.0000 42.4264i −1.44620 1.53393i
\(766\) 0 0
\(767\) 24.0000 0.866590
\(768\) 2.82843i 0.102062i
\(769\) −10.0000 −0.360609 −0.180305 0.983611i \(-0.557708\pi\)
−0.180305 + 0.983611i \(0.557708\pi\)
\(770\) 0 0
\(771\) 16.9706i 0.611180i
\(772\) 16.9706i 0.610784i
\(773\) −54.0000 −1.94225 −0.971123 0.238581i \(-0.923318\pi\)
−0.971123 + 0.238581i \(0.923318\pi\)
\(774\) −20.0000 −0.718885
\(775\) 0 0
\(776\) 16.9706i 0.609208i
\(777\) 0 0
\(778\) 6.00000 0.215110
\(779\) 22.6274i 0.810711i
\(780\) 16.0000 0.572892
\(781\) 16.0000 0.572525
\(782\) 16.0000 + 16.9706i 0.572159 + 0.606866i
\(783\) −16.0000 −0.571793
\(784\) 7.00000 0.250000
\(785\) 39.5980i 1.41331i
\(786\) −8.00000 −0.285351
\(787\) 42.4264i 1.51234i 0.654376 + 0.756169i \(0.272931\pi\)
−0.654376 + 0.756169i \(0.727069\pi\)
\(788\) 2.82843i 0.100759i
\(789\) 67.8823i 2.41667i
\(790\) 48.0000 1.70776
\(791\) 0 0
\(792\) 14.1421i 0.502519i
\(793\) 16.9706i 0.602642i
\(794\) 8.48528i 0.301131i
\(795\) 48.0000 1.70238
\(796\) 16.9706i 0.601506i
\(797\) −18.0000 −0.637593 −0.318796 0.947823i \(-0.603279\pi\)
−0.318796 + 0.947823i \(0.603279\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 3.00000 0.106066
\(801\) −30.0000 −1.06000
\(802\) 28.2843i 0.998752i
\(803\) 0 0
\(804\) 11.3137i 0.399004i
\(805\) 0 0
\(806\) 0 0
\(807\) −88.0000 −3.09775
\(808\) 6.00000 0.211079
\(809\) 39.5980i 1.39219i 0.717949 + 0.696095i \(0.245081\pi\)
−0.717949 + 0.696095i \(0.754919\pi\)
\(810\) 2.82843i 0.0993808i
\(811\) 42.4264i 1.48979i −0.667180 0.744896i \(-0.732499\pi\)
0.667180 0.744896i \(-0.267501\pi\)
\(812\) 0 0
\(813\) 45.2548i 1.58716i
\(814\) 24.0000 0.841200
\(815\) −24.0000 −0.840683
\(816\) −8.00000 8.48528i −0.280056 0.297044i
\(817\) 16.0000 0.559769
\(818\) −2.00000 −0.0699284
\(819\) 0 0
\(820\) 16.0000 0.558744
\(821\) 48.0833i 1.67812i 0.544041 + 0.839059i \(0.316894\pi\)
−0.544041 + 0.839059i \(0.683106\pi\)
\(822\) 50.9117i 1.77575i
\(823\) 16.9706i 0.591557i −0.955257 0.295778i \(-0.904421\pi\)
0.955257 0.295778i \(-0.0955790\pi\)
\(824\) −8.00000 −0.278693
\(825\) −24.0000 −0.835573
\(826\) 0 0
\(827\) 36.7696i 1.27860i −0.768956 0.639301i \(-0.779224\pi\)
0.768956 0.639301i \(-0.220776\pi\)
\(828\) 28.2843i 0.982946i
\(829\) −34.0000 −1.18087 −0.590434 0.807086i \(-0.701044\pi\)
−0.590434 + 0.807086i \(0.701044\pi\)
\(830\) 33.9411i 1.17811i
\(831\) 72.0000 2.49765
\(832\) 2.00000 0.0693375
\(833\) −21.0000 + 19.7990i −0.727607 + 0.685994i
\(834\) 24.0000 0.831052
\(835\) −32.0000 −1.10741
\(836\) 11.3137i 0.391293i
\(837\) 0 0
\(838\) 14.1421i 0.488532i
\(839\) 45.2548i 1.56237i −0.624299 0.781185i \(-0.714615\pi\)
0.624299 0.781185i \(-0.285385\pi\)
\(840\) 0 0
\(841\) 21.0000 0.724138
\(842\) −26.0000 −0.896019
\(843\) 50.9117i 1.75349i
\(844\) 8.48528i 0.292075i
\(845\) 25.4558i 0.875708i
\(846\) 0 0
\(847\) 0 0
\(848\) 6.00000 0.206041
\(849\) −24.0000 −0.823678
\(850\) −9.00000 + 8.48528i −0.308697 + 0.291043i
\(851\) 48.0000 1.64542
\(852\) −16.0000 −0.548151
\(853\) 25.4558i 0.871592i −0.900046 0.435796i \(-0.856467\pi\)
0.900046 0.435796i \(-0.143533\pi\)
\(854\) 0 0
\(855\) 56.5685i 1.93460i
\(856\) 2.82843i 0.0966736i
\(857\) 28.2843i 0.966172i −0.875573 0.483086i \(-0.839516\pi\)
0.875573 0.483086i \(-0.160484\pi\)
\(858\) −16.0000 −0.546231
\(859\) −4.00000 −0.136478 −0.0682391 0.997669i \(-0.521738\pi\)
−0.0682391 + 0.997669i \(0.521738\pi\)
\(860\) 11.3137i 0.385794i
\(861\) 0 0
\(862\) 5.65685i 0.192673i
\(863\) −48.0000 −1.63394 −0.816970 0.576681i \(-0.804348\pi\)
−0.816970 + 0.576681i \(0.804348\pi\)
\(864\) 5.65685i 0.192450i
\(865\) −8.00000 −0.272008
\(866\) −14.0000 −0.475739
\(867\) 48.0000 + 2.82843i 1.63017 + 0.0960584i
\(868\) 0 0
\(869\) −48.0000 −1.62829
\(870\) 22.6274i 0.767141i
\(871\) −8.00000 −0.271070
\(872\) 8.48528i 0.287348i
\(873\) 84.8528i 2.87183i
\(874\) 22.6274i 0.765384i
\(875\) 0 0
\(876\) 0 0
\(877\) 8.48528i 0.286528i −0.989685 0.143264i \(-0.954240\pi\)
0.989685 0.143264i \(-0.0457597\pi\)
\(878\) 16.9706i 0.572729i
\(879\) 16.9706i 0.572403i
\(880\) −8.00000 −0.269680
\(881\) 45.2548i 1.52467i −0.647180 0.762337i \(-0.724052\pi\)
0.647180 0.762337i \(-0.275948\pi\)
\(882\) 35.0000 1.17851
\(883\) 44.0000 1.48072 0.740359 0.672212i \(-0.234656\pi\)
0.740359 + 0.672212i \(0.234656\pi\)
\(884\) −6.00000 + 5.65685i −0.201802 + 0.190261i
\(885\) 96.0000 3.22700
\(886\) 12.0000 0.403148
\(887\) 39.5980i 1.32957i 0.747035 + 0.664785i \(0.231477\pi\)
−0.747035 + 0.664785i \(0.768523\pi\)
\(888\) −24.0000 −0.805387
\(889\) 0 0
\(890\) 16.9706i 0.568855i
\(891\) 2.82843i 0.0947559i
\(892\) −16.0000 −0.535720
\(893\) 0 0
\(894\) 16.9706i 0.567581i
\(895\) 33.9411i 1.13453i
\(896\) 0 0
\(897\) −32.0000 −1.06845
\(898\) 5.65685i 0.188772i
\(899\) 0 0
\(900\) 15.0000 0.500000
\(901\) −18.0000 + 16.9706i −0.599667 + 0.565371i
\(902\) −16.0000 −0.532742
\(903\) 0 0
\(904\) 11.3137i 0.376288i
\(905\) 24.0000 0.797787
\(906\) 22.6274i 0.751746i
\(907\) 42.4264i 1.40875i 0.709830 + 0.704373i \(0.248772\pi\)
−0.709830 + 0.704373i \(0.751228\pi\)
\(908\) 19.7990i 0.657053i
\(909\) 30.0000 0.995037
\(910\) 0 0
\(911\) 56.5685i 1.87420i 0.349062 + 0.937100i \(0.386500\pi\)
−0.349062 + 0.937100i \(0.613500\pi\)
\(912\) 11.3137i 0.374634i
\(913\) 33.9411i 1.12329i
\(914\) 10.0000 0.330771
\(915\) 67.8823i 2.24412i
\(916\) −22.0000 −0.726900
\(917\) 0 0
\(918\) −16.0000 16.9706i −0.528079 0.560112i
\(919\) −40.0000 −1.31948 −0.659739 0.751495i \(-0.729333\pi\)
−0.659739 + 0.751495i \(0.729333\pi\)
\(920\) −16.0000 −0.527504
\(921\) 56.5685i 1.86400i
\(922\) 18.0000 0.592798
\(923\) 11.3137i 0.372395i
\(924\) 0 0
\(925\) 25.4558i 0.836983i
\(926\) −32.0000 −1.05159
\(927\) −40.0000 −1.31377
\(928\) 2.82843i 0.0928477i
\(929\) 5.65685i 0.185595i 0.995685 + 0.0927977i \(0.0295810\pi\)
−0.995685 + 0.0927977i \(0.970419\pi\)
\(930\) 0 0
\(931\) −28.0000 −0.917663
\(932\) 22.6274i 0.741186i
\(933\) 32.0000 1.04763
\(934\) 36.0000 1.17796
\(935\) 24.0000 22.6274i 0.784884 0.739996i
\(936\) 10.0000 0.326860
\(937\) 2.00000 0.0653372 0.0326686 0.999466i \(-0.489599\pi\)
0.0326686 + 0.999466i \(0.489599\pi\)
\(938\) 0 0
\(939\) 48.0000 1.56642
\(940\) 0 0
\(941\) 36.7696i 1.19865i −0.800505 0.599327i \(-0.795435\pi\)
0.800505 0.599327i \(-0.204565\pi\)
\(942\) 39.5980i 1.29017i
\(943\) −32.0000 −1.04206
\(944\) 12.0000 0.390567
\(945\) 0 0
\(946\) 11.3137i 0.367840i
\(947\) 2.82843i 0.0919115i −0.998943 0.0459558i \(-0.985367\pi\)
0.998943 0.0459558i \(-0.0146333\pi\)
\(948\) 48.0000 1.55897
\(949\) 0 0
\(950\) −12.0000 −0.389331
\(951\) 8.00000 0.259418
\(952\) 0 0
\(953\) 6.00000 0.194359 0.0971795 0.995267i \(-0.469018\pi\)
0.0971795 + 0.995267i \(0.469018\pi\)
\(954\) 30.0000 0.971286
\(955\) 0 0
\(956\) 0 0
\(957\) 22.6274i 0.731441i
\(958\) 5.65685i 0.182765i
\(959\) 0 0
\(960\) 8.00000 0.258199
\(961\) 31.0000 1.00000
\(962\) 16.9706i 0.547153i
\(963\) 14.1421i 0.455724i
\(964\) 16.9706i 0.546585i
\(965\) 48.0000 1.54517
\(966\) 0 0
\(967\) 8.00000 0.257263 0.128631 0.991692i \(-0.458942\pi\)
0.128631 + 0.991692i \(0.458942\pi\)
\(968\) −3.00000 −0.0964237
\(969\) 32.0000 + 33.9411i 1.02799 + 1.09035i
\(970\) −48.0000 −1.54119
\(971\) 12.0000 0.385098 0.192549 0.981287i \(-0.438325\pi\)
0.192549 + 0.981287i \(0.438325\pi\)
\(972\) 14.1421i 0.453609i
\(973\) 0 0
\(974\) 16.9706i 0.543772i
\(975\) 16.9706i 0.543493i
\(976\) 8.48528i 0.271607i
\(977\) 18.0000 0.575871 0.287936 0.957650i \(-0.407031\pi\)
0.287936 + 0.957650i \(0.407031\pi\)
\(978\) −24.0000 −0.767435
\(979\) 16.9706i 0.542382i
\(980\) 19.7990i 0.632456i
\(981\) 42.4264i 1.35457i
\(982\) 12.0000 0.382935
\(983\) 5.65685i 0.180426i 0.995923 + 0.0902128i \(0.0287547\pi\)
−0.995923 + 0.0902128i \(0.971245\pi\)
\(984\) 16.0000 0.510061
\(985\) −8.00000 −0.254901
\(986\) −8.00000 8.48528i −0.254772 0.270226i
\(987\) 0 0
\(988\) −8.00000 −0.254514
\(989\) 22.6274i 0.719510i
\(990\) −40.0000 −1.27128
\(991\) 50.9117i 1.61726i −0.588315 0.808632i \(-0.700209\pi\)
0.588315 0.808632i \(-0.299791\pi\)
\(992\) 0 0
\(993\) 11.3137i 0.359030i
\(994\) 0 0
\(995\) −48.0000 −1.52170
\(996\) 33.9411i 1.07547i
\(997\) 59.3970i 1.88112i −0.339626 0.940560i \(-0.610301\pi\)
0.339626 0.940560i \(-0.389699\pi\)
\(998\) 25.4558i 0.805791i
\(999\) −48.0000 −1.51865
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 34.2.b.a.33.2 yes 2
3.2 odd 2 306.2.b.d.271.2 2
4.3 odd 2 272.2.b.a.33.1 2
5.2 odd 4 850.2.d.i.849.2 4
5.3 odd 4 850.2.d.i.849.3 4
5.4 even 2 850.2.b.f.101.1 2
7.6 odd 2 1666.2.b.c.883.1 2
8.3 odd 2 1088.2.b.a.577.2 2
8.5 even 2 1088.2.b.b.577.1 2
12.11 even 2 2448.2.c.n.577.2 2
17.2 even 8 578.2.c.d.251.1 2
17.3 odd 16 578.2.d.g.399.1 8
17.4 even 4 578.2.a.d.1.2 2
17.5 odd 16 578.2.d.g.179.1 8
17.6 odd 16 578.2.d.g.423.2 8
17.7 odd 16 578.2.d.g.155.2 8
17.8 even 8 578.2.c.d.327.1 2
17.9 even 8 578.2.c.a.327.1 2
17.10 odd 16 578.2.d.g.155.1 8
17.11 odd 16 578.2.d.g.423.1 8
17.12 odd 16 578.2.d.g.179.2 8
17.13 even 4 578.2.a.d.1.1 2
17.14 odd 16 578.2.d.g.399.2 8
17.15 even 8 578.2.c.a.251.1 2
17.16 even 2 inner 34.2.b.a.33.1 2
51.38 odd 4 5202.2.a.u.1.2 2
51.47 odd 4 5202.2.a.u.1.1 2
51.50 odd 2 306.2.b.d.271.1 2
68.47 odd 4 4624.2.a.s.1.2 2
68.55 odd 4 4624.2.a.s.1.1 2
68.67 odd 2 272.2.b.a.33.2 2
85.33 odd 4 850.2.d.i.849.4 4
85.67 odd 4 850.2.d.i.849.1 4
85.84 even 2 850.2.b.f.101.2 2
119.118 odd 2 1666.2.b.c.883.2 2
136.67 odd 2 1088.2.b.a.577.1 2
136.101 even 2 1088.2.b.b.577.2 2
204.203 even 2 2448.2.c.n.577.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
34.2.b.a.33.1 2 17.16 even 2 inner
34.2.b.a.33.2 yes 2 1.1 even 1 trivial
272.2.b.a.33.1 2 4.3 odd 2
272.2.b.a.33.2 2 68.67 odd 2
306.2.b.d.271.1 2 51.50 odd 2
306.2.b.d.271.2 2 3.2 odd 2
578.2.a.d.1.1 2 17.13 even 4
578.2.a.d.1.2 2 17.4 even 4
578.2.c.a.251.1 2 17.15 even 8
578.2.c.a.327.1 2 17.9 even 8
578.2.c.d.251.1 2 17.2 even 8
578.2.c.d.327.1 2 17.8 even 8
578.2.d.g.155.1 8 17.10 odd 16
578.2.d.g.155.2 8 17.7 odd 16
578.2.d.g.179.1 8 17.5 odd 16
578.2.d.g.179.2 8 17.12 odd 16
578.2.d.g.399.1 8 17.3 odd 16
578.2.d.g.399.2 8 17.14 odd 16
578.2.d.g.423.1 8 17.11 odd 16
578.2.d.g.423.2 8 17.6 odd 16
850.2.b.f.101.1 2 5.4 even 2
850.2.b.f.101.2 2 85.84 even 2
850.2.d.i.849.1 4 85.67 odd 4
850.2.d.i.849.2 4 5.2 odd 4
850.2.d.i.849.3 4 5.3 odd 4
850.2.d.i.849.4 4 85.33 odd 4
1088.2.b.a.577.1 2 136.67 odd 2
1088.2.b.a.577.2 2 8.3 odd 2
1088.2.b.b.577.1 2 8.5 even 2
1088.2.b.b.577.2 2 136.101 even 2
1666.2.b.c.883.1 2 7.6 odd 2
1666.2.b.c.883.2 2 119.118 odd 2
2448.2.c.n.577.1 2 204.203 even 2
2448.2.c.n.577.2 2 12.11 even 2
4624.2.a.s.1.1 2 68.55 odd 4
4624.2.a.s.1.2 2 68.47 odd 4
5202.2.a.u.1.1 2 51.47 odd 4
5202.2.a.u.1.2 2 51.38 odd 4