Properties

Label 768.2.j.a.577.1
Level $768$
Weight $2$
Character 768.577
Analytic conductor $6.133$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [768,2,Mod(193,768)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(768, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("768.193");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 768 = 2^{8} \cdot 3 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 768.j (of order \(4\), degree \(2\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(6.13251087523\)
Analytic rank: \(1\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{8})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 577.1
Root \(0.707107 + 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 768.577
Dual form 768.2.j.a.193.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.707107 - 0.707107i) q^{3} +(-2.00000 + 2.00000i) q^{5} +4.24264i q^{7} +1.00000i q^{9} +O(q^{10})\) \(q+(-0.707107 - 0.707107i) q^{3} +(-2.00000 + 2.00000i) q^{5} +4.24264i q^{7} +1.00000i q^{9} +(2.82843 - 2.82843i) q^{11} +(-3.00000 - 3.00000i) q^{13} +2.82843 q^{15} -6.00000 q^{17} +(-1.41421 - 1.41421i) q^{19} +(3.00000 - 3.00000i) q^{21} -2.82843i q^{23} -3.00000i q^{25} +(0.707107 - 0.707107i) q^{27} +(-4.00000 - 4.00000i) q^{29} +4.24264 q^{31} -4.00000 q^{33} +(-8.48528 - 8.48528i) q^{35} +(-3.00000 + 3.00000i) q^{37} +4.24264i q^{39} -10.0000i q^{41} +(-4.24264 + 4.24264i) q^{43} +(-2.00000 - 2.00000i) q^{45} -2.82843 q^{47} -11.0000 q^{49} +(4.24264 + 4.24264i) q^{51} +(-4.00000 + 4.00000i) q^{53} +11.3137i q^{55} +2.00000i q^{57} +(-3.00000 - 3.00000i) q^{61} -4.24264 q^{63} +12.0000 q^{65} +(2.82843 + 2.82843i) q^{67} +(-2.00000 + 2.00000i) q^{69} +2.82843i q^{71} +16.0000i q^{73} +(-2.12132 + 2.12132i) q^{75} +(12.0000 + 12.0000i) q^{77} -4.24264 q^{79} -1.00000 q^{81} +(-11.3137 - 11.3137i) q^{83} +(12.0000 - 12.0000i) q^{85} +5.65685i q^{87} -14.0000i q^{89} +(12.7279 - 12.7279i) q^{91} +(-3.00000 - 3.00000i) q^{93} +5.65685 q^{95} -4.00000 q^{97} +(2.82843 + 2.82843i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 8 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 8 q^{5} - 12 q^{13} - 24 q^{17} + 12 q^{21} - 16 q^{29} - 16 q^{33} - 12 q^{37} - 8 q^{45} - 44 q^{49} - 16 q^{53} - 12 q^{61} + 48 q^{65} - 8 q^{69} + 48 q^{77} - 4 q^{81} + 48 q^{85} - 12 q^{93} - 16 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/768\mathbb{Z}\right)^\times\).

\(n\) \(257\) \(511\) \(517\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{3}{4}\right)\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −0.707107 0.707107i −0.408248 0.408248i
\(4\) 0 0
\(5\) −2.00000 + 2.00000i −0.894427 + 0.894427i −0.994936 0.100509i \(-0.967953\pi\)
0.100509 + 0.994936i \(0.467953\pi\)
\(6\) 0 0
\(7\) 4.24264i 1.60357i 0.597614 + 0.801784i \(0.296115\pi\)
−0.597614 + 0.801784i \(0.703885\pi\)
\(8\) 0 0
\(9\) 1.00000i 0.333333i
\(10\) 0 0
\(11\) 2.82843 2.82843i 0.852803 0.852803i −0.137675 0.990478i \(-0.543963\pi\)
0.990478 + 0.137675i \(0.0439628\pi\)
\(12\) 0 0
\(13\) −3.00000 3.00000i −0.832050 0.832050i 0.155747 0.987797i \(-0.450222\pi\)
−0.987797 + 0.155747i \(0.950222\pi\)
\(14\) 0 0
\(15\) 2.82843 0.730297
\(16\) 0 0
\(17\) −6.00000 −1.45521 −0.727607 0.685994i \(-0.759367\pi\)
−0.727607 + 0.685994i \(0.759367\pi\)
\(18\) 0 0
\(19\) −1.41421 1.41421i −0.324443 0.324443i 0.526026 0.850469i \(-0.323682\pi\)
−0.850469 + 0.526026i \(0.823682\pi\)
\(20\) 0 0
\(21\) 3.00000 3.00000i 0.654654 0.654654i
\(22\) 0 0
\(23\) 2.82843i 0.589768i −0.955533 0.294884i \(-0.904719\pi\)
0.955533 0.294884i \(-0.0952810\pi\)
\(24\) 0 0
\(25\) 3.00000i 0.600000i
\(26\) 0 0
\(27\) 0.707107 0.707107i 0.136083 0.136083i
\(28\) 0 0
\(29\) −4.00000 4.00000i −0.742781 0.742781i 0.230331 0.973112i \(-0.426019\pi\)
−0.973112 + 0.230331i \(0.926019\pi\)
\(30\) 0 0
\(31\) 4.24264 0.762001 0.381000 0.924575i \(-0.375580\pi\)
0.381000 + 0.924575i \(0.375580\pi\)
\(32\) 0 0
\(33\) −4.00000 −0.696311
\(34\) 0 0
\(35\) −8.48528 8.48528i −1.43427 1.43427i
\(36\) 0 0
\(37\) −3.00000 + 3.00000i −0.493197 + 0.493197i −0.909312 0.416115i \(-0.863391\pi\)
0.416115 + 0.909312i \(0.363391\pi\)
\(38\) 0 0
\(39\) 4.24264i 0.679366i
\(40\) 0 0
\(41\) 10.0000i 1.56174i −0.624695 0.780869i \(-0.714777\pi\)
0.624695 0.780869i \(-0.285223\pi\)
\(42\) 0 0
\(43\) −4.24264 + 4.24264i −0.646997 + 0.646997i −0.952266 0.305269i \(-0.901253\pi\)
0.305269 + 0.952266i \(0.401253\pi\)
\(44\) 0 0
\(45\) −2.00000 2.00000i −0.298142 0.298142i
\(46\) 0 0
\(47\) −2.82843 −0.412568 −0.206284 0.978492i \(-0.566137\pi\)
−0.206284 + 0.978492i \(0.566137\pi\)
\(48\) 0 0
\(49\) −11.0000 −1.57143
\(50\) 0 0
\(51\) 4.24264 + 4.24264i 0.594089 + 0.594089i
\(52\) 0 0
\(53\) −4.00000 + 4.00000i −0.549442 + 0.549442i −0.926279 0.376837i \(-0.877012\pi\)
0.376837 + 0.926279i \(0.377012\pi\)
\(54\) 0 0
\(55\) 11.3137i 1.52554i
\(56\) 0 0
\(57\) 2.00000i 0.264906i
\(58\) 0 0
\(59\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(60\) 0 0
\(61\) −3.00000 3.00000i −0.384111 0.384111i 0.488470 0.872581i \(-0.337555\pi\)
−0.872581 + 0.488470i \(0.837555\pi\)
\(62\) 0 0
\(63\) −4.24264 −0.534522
\(64\) 0 0
\(65\) 12.0000 1.48842
\(66\) 0 0
\(67\) 2.82843 + 2.82843i 0.345547 + 0.345547i 0.858448 0.512901i \(-0.171429\pi\)
−0.512901 + 0.858448i \(0.671429\pi\)
\(68\) 0 0
\(69\) −2.00000 + 2.00000i −0.240772 + 0.240772i
\(70\) 0 0
\(71\) 2.82843i 0.335673i 0.985815 + 0.167836i \(0.0536780\pi\)
−0.985815 + 0.167836i \(0.946322\pi\)
\(72\) 0 0
\(73\) 16.0000i 1.87266i 0.351123 + 0.936329i \(0.385800\pi\)
−0.351123 + 0.936329i \(0.614200\pi\)
\(74\) 0 0
\(75\) −2.12132 + 2.12132i −0.244949 + 0.244949i
\(76\) 0 0
\(77\) 12.0000 + 12.0000i 1.36753 + 1.36753i
\(78\) 0 0
\(79\) −4.24264 −0.477334 −0.238667 0.971101i \(-0.576710\pi\)
−0.238667 + 0.971101i \(0.576710\pi\)
\(80\) 0 0
\(81\) −1.00000 −0.111111
\(82\) 0 0
\(83\) −11.3137 11.3137i −1.24184 1.24184i −0.959237 0.282604i \(-0.908802\pi\)
−0.282604 0.959237i \(-0.591198\pi\)
\(84\) 0 0
\(85\) 12.0000 12.0000i 1.30158 1.30158i
\(86\) 0 0
\(87\) 5.65685i 0.606478i
\(88\) 0 0
\(89\) 14.0000i 1.48400i −0.670402 0.741999i \(-0.733878\pi\)
0.670402 0.741999i \(-0.266122\pi\)
\(90\) 0 0
\(91\) 12.7279 12.7279i 1.33425 1.33425i
\(92\) 0 0
\(93\) −3.00000 3.00000i −0.311086 0.311086i
\(94\) 0 0
\(95\) 5.65685 0.580381
\(96\) 0 0
\(97\) −4.00000 −0.406138 −0.203069 0.979164i \(-0.565092\pi\)
−0.203069 + 0.979164i \(0.565092\pi\)
\(98\) 0 0
\(99\) 2.82843 + 2.82843i 0.284268 + 0.284268i
\(100\) 0 0
\(101\) −8.00000 + 8.00000i −0.796030 + 0.796030i −0.982467 0.186437i \(-0.940306\pi\)
0.186437 + 0.982467i \(0.440306\pi\)
\(102\) 0 0
\(103\) 18.3848i 1.81151i −0.423806 0.905753i \(-0.639306\pi\)
0.423806 0.905753i \(-0.360694\pi\)
\(104\) 0 0
\(105\) 12.0000i 1.17108i
\(106\) 0 0
\(107\) 2.82843 2.82843i 0.273434 0.273434i −0.557047 0.830481i \(-0.688066\pi\)
0.830481 + 0.557047i \(0.188066\pi\)
\(108\) 0 0
\(109\) 7.00000 + 7.00000i 0.670478 + 0.670478i 0.957826 0.287348i \(-0.0927736\pi\)
−0.287348 + 0.957826i \(0.592774\pi\)
\(110\) 0 0
\(111\) 4.24264 0.402694
\(112\) 0 0
\(113\) 6.00000 0.564433 0.282216 0.959351i \(-0.408930\pi\)
0.282216 + 0.959351i \(0.408930\pi\)
\(114\) 0 0
\(115\) 5.65685 + 5.65685i 0.527504 + 0.527504i
\(116\) 0 0
\(117\) 3.00000 3.00000i 0.277350 0.277350i
\(118\) 0 0
\(119\) 25.4558i 2.33353i
\(120\) 0 0
\(121\) 5.00000i 0.454545i
\(122\) 0 0
\(123\) −7.07107 + 7.07107i −0.637577 + 0.637577i
\(124\) 0 0
\(125\) −4.00000 4.00000i −0.357771 0.357771i
\(126\) 0 0
\(127\) 4.24264 0.376473 0.188237 0.982124i \(-0.439723\pi\)
0.188237 + 0.982124i \(0.439723\pi\)
\(128\) 0 0
\(129\) 6.00000 0.528271
\(130\) 0 0
\(131\) 14.1421 + 14.1421i 1.23560 + 1.23560i 0.961780 + 0.273824i \(0.0882887\pi\)
0.273824 + 0.961780i \(0.411711\pi\)
\(132\) 0 0
\(133\) 6.00000 6.00000i 0.520266 0.520266i
\(134\) 0 0
\(135\) 2.82843i 0.243432i
\(136\) 0 0
\(137\) 10.0000i 0.854358i 0.904167 + 0.427179i \(0.140493\pi\)
−0.904167 + 0.427179i \(0.859507\pi\)
\(138\) 0 0
\(139\) −8.48528 + 8.48528i −0.719712 + 0.719712i −0.968546 0.248834i \(-0.919953\pi\)
0.248834 + 0.968546i \(0.419953\pi\)
\(140\) 0 0
\(141\) 2.00000 + 2.00000i 0.168430 + 0.168430i
\(142\) 0 0
\(143\) −16.9706 −1.41915
\(144\) 0 0
\(145\) 16.0000 1.32873
\(146\) 0 0
\(147\) 7.77817 + 7.77817i 0.641533 + 0.641533i
\(148\) 0 0
\(149\) −6.00000 + 6.00000i −0.491539 + 0.491539i −0.908791 0.417252i \(-0.862993\pi\)
0.417252 + 0.908791i \(0.362993\pi\)
\(150\) 0 0
\(151\) 1.41421i 0.115087i 0.998343 + 0.0575435i \(0.0183268\pi\)
−0.998343 + 0.0575435i \(0.981673\pi\)
\(152\) 0 0
\(153\) 6.00000i 0.485071i
\(154\) 0 0
\(155\) −8.48528 + 8.48528i −0.681554 + 0.681554i
\(156\) 0 0
\(157\) −5.00000 5.00000i −0.399043 0.399043i 0.478852 0.877896i \(-0.341053\pi\)
−0.877896 + 0.478852i \(0.841053\pi\)
\(158\) 0 0
\(159\) 5.65685 0.448618
\(160\) 0 0
\(161\) 12.0000 0.945732
\(162\) 0 0
\(163\) 7.07107 + 7.07107i 0.553849 + 0.553849i 0.927549 0.373701i \(-0.121911\pi\)
−0.373701 + 0.927549i \(0.621911\pi\)
\(164\) 0 0
\(165\) 8.00000 8.00000i 0.622799 0.622799i
\(166\) 0 0
\(167\) 11.3137i 0.875481i −0.899101 0.437741i \(-0.855779\pi\)
0.899101 0.437741i \(-0.144221\pi\)
\(168\) 0 0
\(169\) 5.00000i 0.384615i
\(170\) 0 0
\(171\) 1.41421 1.41421i 0.108148 0.108148i
\(172\) 0 0
\(173\) 2.00000 + 2.00000i 0.152057 + 0.152057i 0.779036 0.626979i \(-0.215709\pi\)
−0.626979 + 0.779036i \(0.715709\pi\)
\(174\) 0 0
\(175\) 12.7279 0.962140
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(180\) 0 0
\(181\) 1.00000 1.00000i 0.0743294 0.0743294i −0.668965 0.743294i \(-0.733262\pi\)
0.743294 + 0.668965i \(0.233262\pi\)
\(182\) 0 0
\(183\) 4.24264i 0.313625i
\(184\) 0 0
\(185\) 12.0000i 0.882258i
\(186\) 0 0
\(187\) −16.9706 + 16.9706i −1.24101 + 1.24101i
\(188\) 0 0
\(189\) 3.00000 + 3.00000i 0.218218 + 0.218218i
\(190\) 0 0
\(191\) −22.6274 −1.63726 −0.818631 0.574320i \(-0.805267\pi\)
−0.818631 + 0.574320i \(0.805267\pi\)
\(192\) 0 0
\(193\) 18.0000 1.29567 0.647834 0.761781i \(-0.275675\pi\)
0.647834 + 0.761781i \(0.275675\pi\)
\(194\) 0 0
\(195\) −8.48528 8.48528i −0.607644 0.607644i
\(196\) 0 0
\(197\) −4.00000 + 4.00000i −0.284988 + 0.284988i −0.835095 0.550106i \(-0.814587\pi\)
0.550106 + 0.835095i \(0.314587\pi\)
\(198\) 0 0
\(199\) 7.07107i 0.501255i 0.968084 + 0.250627i \(0.0806369\pi\)
−0.968084 + 0.250627i \(0.919363\pi\)
\(200\) 0 0
\(201\) 4.00000i 0.282138i
\(202\) 0 0
\(203\) 16.9706 16.9706i 1.19110 1.19110i
\(204\) 0 0
\(205\) 20.0000 + 20.0000i 1.39686 + 1.39686i
\(206\) 0 0
\(207\) 2.82843 0.196589
\(208\) 0 0
\(209\) −8.00000 −0.553372
\(210\) 0 0
\(211\) 8.48528 + 8.48528i 0.584151 + 0.584151i 0.936041 0.351890i \(-0.114461\pi\)
−0.351890 + 0.936041i \(0.614461\pi\)
\(212\) 0 0
\(213\) 2.00000 2.00000i 0.137038 0.137038i
\(214\) 0 0
\(215\) 16.9706i 1.15738i
\(216\) 0 0
\(217\) 18.0000i 1.22192i
\(218\) 0 0
\(219\) 11.3137 11.3137i 0.764510 0.764510i
\(220\) 0 0
\(221\) 18.0000 + 18.0000i 1.21081 + 1.21081i
\(222\) 0 0
\(223\) −21.2132 −1.42054 −0.710271 0.703929i \(-0.751427\pi\)
−0.710271 + 0.703929i \(0.751427\pi\)
\(224\) 0 0
\(225\) 3.00000 0.200000
\(226\) 0 0
\(227\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(228\) 0 0
\(229\) −11.0000 + 11.0000i −0.726900 + 0.726900i −0.970001 0.243101i \(-0.921835\pi\)
0.243101 + 0.970001i \(0.421835\pi\)
\(230\) 0 0
\(231\) 16.9706i 1.11658i
\(232\) 0 0
\(233\) 18.0000i 1.17922i 0.807688 + 0.589610i \(0.200718\pi\)
−0.807688 + 0.589610i \(0.799282\pi\)
\(234\) 0 0
\(235\) 5.65685 5.65685i 0.369012 0.369012i
\(236\) 0 0
\(237\) 3.00000 + 3.00000i 0.194871 + 0.194871i
\(238\) 0 0
\(239\) 11.3137 0.731823 0.365911 0.930650i \(-0.380757\pi\)
0.365911 + 0.930650i \(0.380757\pi\)
\(240\) 0 0
\(241\) −18.0000 −1.15948 −0.579741 0.814801i \(-0.696846\pi\)
−0.579741 + 0.814801i \(0.696846\pi\)
\(242\) 0 0
\(243\) 0.707107 + 0.707107i 0.0453609 + 0.0453609i
\(244\) 0 0
\(245\) 22.0000 22.0000i 1.40553 1.40553i
\(246\) 0 0
\(247\) 8.48528i 0.539906i
\(248\) 0 0
\(249\) 16.0000i 1.01396i
\(250\) 0 0
\(251\) −2.82843 + 2.82843i −0.178529 + 0.178529i −0.790714 0.612185i \(-0.790291\pi\)
0.612185 + 0.790714i \(0.290291\pi\)
\(252\) 0 0
\(253\) −8.00000 8.00000i −0.502956 0.502956i
\(254\) 0 0
\(255\) −16.9706 −1.06274
\(256\) 0 0
\(257\) −14.0000 −0.873296 −0.436648 0.899632i \(-0.643834\pi\)
−0.436648 + 0.899632i \(0.643834\pi\)
\(258\) 0 0
\(259\) −12.7279 12.7279i −0.790875 0.790875i
\(260\) 0 0
\(261\) 4.00000 4.00000i 0.247594 0.247594i
\(262\) 0 0
\(263\) 22.6274i 1.39527i 0.716455 + 0.697633i \(0.245763\pi\)
−0.716455 + 0.697633i \(0.754237\pi\)
\(264\) 0 0
\(265\) 16.0000i 0.982872i
\(266\) 0 0
\(267\) −9.89949 + 9.89949i −0.605839 + 0.605839i
\(268\) 0 0
\(269\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(270\) 0 0
\(271\) −1.41421 −0.0859074 −0.0429537 0.999077i \(-0.513677\pi\)
−0.0429537 + 0.999077i \(0.513677\pi\)
\(272\) 0 0
\(273\) −18.0000 −1.08941
\(274\) 0 0
\(275\) −8.48528 8.48528i −0.511682 0.511682i
\(276\) 0 0
\(277\) −1.00000 + 1.00000i −0.0600842 + 0.0600842i −0.736510 0.676426i \(-0.763528\pi\)
0.676426 + 0.736510i \(0.263528\pi\)
\(278\) 0 0
\(279\) 4.24264i 0.254000i
\(280\) 0 0
\(281\) 6.00000i 0.357930i −0.983855 0.178965i \(-0.942725\pi\)
0.983855 0.178965i \(-0.0572749\pi\)
\(282\) 0 0
\(283\) −2.82843 + 2.82843i −0.168133 + 0.168133i −0.786158 0.618026i \(-0.787933\pi\)
0.618026 + 0.786158i \(0.287933\pi\)
\(284\) 0 0
\(285\) −4.00000 4.00000i −0.236940 0.236940i
\(286\) 0 0
\(287\) 42.4264 2.50435
\(288\) 0 0
\(289\) 19.0000 1.11765
\(290\) 0 0
\(291\) 2.82843 + 2.82843i 0.165805 + 0.165805i
\(292\) 0 0
\(293\) −4.00000 + 4.00000i −0.233682 + 0.233682i −0.814228 0.580545i \(-0.802839\pi\)
0.580545 + 0.814228i \(0.302839\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 4.00000i 0.232104i
\(298\) 0 0
\(299\) −8.48528 + 8.48528i −0.490716 + 0.490716i
\(300\) 0 0
\(301\) −18.0000 18.0000i −1.03750 1.03750i
\(302\) 0 0
\(303\) 11.3137 0.649956
\(304\) 0 0
\(305\) 12.0000 0.687118
\(306\) 0 0
\(307\) 2.82843 + 2.82843i 0.161427 + 0.161427i 0.783199 0.621772i \(-0.213587\pi\)
−0.621772 + 0.783199i \(0.713587\pi\)
\(308\) 0 0
\(309\) −13.0000 + 13.0000i −0.739544 + 0.739544i
\(310\) 0 0
\(311\) 28.2843i 1.60385i 0.597422 + 0.801927i \(0.296192\pi\)
−0.597422 + 0.801927i \(0.703808\pi\)
\(312\) 0 0
\(313\) 26.0000i 1.46961i −0.678280 0.734803i \(-0.737274\pi\)
0.678280 0.734803i \(-0.262726\pi\)
\(314\) 0 0
\(315\) 8.48528 8.48528i 0.478091 0.478091i
\(316\) 0 0
\(317\) −12.0000 12.0000i −0.673987 0.673987i 0.284646 0.958633i \(-0.408124\pi\)
−0.958633 + 0.284646i \(0.908124\pi\)
\(318\) 0 0
\(319\) −22.6274 −1.26689
\(320\) 0 0
\(321\) −4.00000 −0.223258
\(322\) 0 0
\(323\) 8.48528 + 8.48528i 0.472134 + 0.472134i
\(324\) 0 0
\(325\) −9.00000 + 9.00000i −0.499230 + 0.499230i
\(326\) 0 0
\(327\) 9.89949i 0.547443i
\(328\) 0 0
\(329\) 12.0000i 0.661581i
\(330\) 0 0
\(331\) 2.82843 2.82843i 0.155464 0.155464i −0.625089 0.780553i \(-0.714937\pi\)
0.780553 + 0.625089i \(0.214937\pi\)
\(332\) 0 0
\(333\) −3.00000 3.00000i −0.164399 0.164399i
\(334\) 0 0
\(335\) −11.3137 −0.618134
\(336\) 0 0
\(337\) −8.00000 −0.435788 −0.217894 0.975972i \(-0.569919\pi\)
−0.217894 + 0.975972i \(0.569919\pi\)
\(338\) 0 0
\(339\) −4.24264 4.24264i −0.230429 0.230429i
\(340\) 0 0
\(341\) 12.0000 12.0000i 0.649836 0.649836i
\(342\) 0 0
\(343\) 16.9706i 0.916324i
\(344\) 0 0
\(345\) 8.00000i 0.430706i
\(346\) 0 0
\(347\) −16.9706 + 16.9706i −0.911028 + 0.911028i −0.996353 0.0853256i \(-0.972807\pi\)
0.0853256 + 0.996353i \(0.472807\pi\)
\(348\) 0 0
\(349\) 3.00000 + 3.00000i 0.160586 + 0.160586i 0.782826 0.622240i \(-0.213777\pi\)
−0.622240 + 0.782826i \(0.713777\pi\)
\(350\) 0 0
\(351\) −4.24264 −0.226455
\(352\) 0 0
\(353\) −6.00000 −0.319348 −0.159674 0.987170i \(-0.551044\pi\)
−0.159674 + 0.987170i \(0.551044\pi\)
\(354\) 0 0
\(355\) −5.65685 5.65685i −0.300235 0.300235i
\(356\) 0 0
\(357\) −18.0000 + 18.0000i −0.952661 + 0.952661i
\(358\) 0 0
\(359\) 2.82843i 0.149279i 0.997211 + 0.0746393i \(0.0237806\pi\)
−0.997211 + 0.0746393i \(0.976219\pi\)
\(360\) 0 0
\(361\) 15.0000i 0.789474i
\(362\) 0 0
\(363\) −3.53553 + 3.53553i −0.185567 + 0.185567i
\(364\) 0 0
\(365\) −32.0000 32.0000i −1.67496 1.67496i
\(366\) 0 0
\(367\) 18.3848 0.959678 0.479839 0.877357i \(-0.340695\pi\)
0.479839 + 0.877357i \(0.340695\pi\)
\(368\) 0 0
\(369\) 10.0000 0.520579
\(370\) 0 0
\(371\) −16.9706 16.9706i −0.881068 0.881068i
\(372\) 0 0
\(373\) −13.0000 + 13.0000i −0.673114 + 0.673114i −0.958433 0.285318i \(-0.907901\pi\)
0.285318 + 0.958433i \(0.407901\pi\)
\(374\) 0 0
\(375\) 5.65685i 0.292119i
\(376\) 0 0
\(377\) 24.0000i 1.23606i
\(378\) 0 0
\(379\) 26.8701 26.8701i 1.38022 1.38022i 0.536011 0.844211i \(-0.319930\pi\)
0.844211 0.536011i \(-0.180070\pi\)
\(380\) 0 0
\(381\) −3.00000 3.00000i −0.153695 0.153695i
\(382\) 0 0
\(383\) 28.2843 1.44526 0.722629 0.691236i \(-0.242933\pi\)
0.722629 + 0.691236i \(0.242933\pi\)
\(384\) 0 0
\(385\) −48.0000 −2.44631
\(386\) 0 0
\(387\) −4.24264 4.24264i −0.215666 0.215666i
\(388\) 0 0
\(389\) 2.00000 2.00000i 0.101404 0.101404i −0.654585 0.755989i \(-0.727156\pi\)
0.755989 + 0.654585i \(0.227156\pi\)
\(390\) 0 0
\(391\) 16.9706i 0.858238i
\(392\) 0 0
\(393\) 20.0000i 1.00887i
\(394\) 0 0
\(395\) 8.48528 8.48528i 0.426941 0.426941i
\(396\) 0 0
\(397\) −23.0000 23.0000i −1.15434 1.15434i −0.985674 0.168663i \(-0.946055\pi\)
−0.168663 0.985674i \(-0.553945\pi\)
\(398\) 0 0
\(399\) −8.48528 −0.424795
\(400\) 0 0
\(401\) −10.0000 −0.499376 −0.249688 0.968326i \(-0.580328\pi\)
−0.249688 + 0.968326i \(0.580328\pi\)
\(402\) 0 0
\(403\) −12.7279 12.7279i −0.634023 0.634023i
\(404\) 0 0
\(405\) 2.00000 2.00000i 0.0993808 0.0993808i
\(406\) 0 0
\(407\) 16.9706i 0.841200i
\(408\) 0 0
\(409\) 28.0000i 1.38451i 0.721653 + 0.692255i \(0.243383\pi\)
−0.721653 + 0.692255i \(0.756617\pi\)
\(410\) 0 0
\(411\) 7.07107 7.07107i 0.348790 0.348790i
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 45.2548 2.22147
\(416\) 0 0
\(417\) 12.0000 0.587643
\(418\) 0 0
\(419\) 11.3137 + 11.3137i 0.552711 + 0.552711i 0.927222 0.374511i \(-0.122190\pi\)
−0.374511 + 0.927222i \(0.622190\pi\)
\(420\) 0 0
\(421\) 25.0000 25.0000i 1.21843 1.21843i 0.250242 0.968183i \(-0.419490\pi\)
0.968183 0.250242i \(-0.0805102\pi\)
\(422\) 0 0
\(423\) 2.82843i 0.137523i
\(424\) 0 0
\(425\) 18.0000i 0.873128i
\(426\) 0 0
\(427\) 12.7279 12.7279i 0.615947 0.615947i
\(428\) 0 0
\(429\) 12.0000 + 12.0000i 0.579365 + 0.579365i
\(430\) 0 0
\(431\) −36.7696 −1.77113 −0.885564 0.464518i \(-0.846227\pi\)
−0.885564 + 0.464518i \(0.846227\pi\)
\(432\) 0 0
\(433\) −8.00000 −0.384455 −0.192228 0.981350i \(-0.561571\pi\)
−0.192228 + 0.981350i \(0.561571\pi\)
\(434\) 0 0
\(435\) −11.3137 11.3137i −0.542451 0.542451i
\(436\) 0 0
\(437\) −4.00000 + 4.00000i −0.191346 + 0.191346i
\(438\) 0 0
\(439\) 1.41421i 0.0674967i −0.999430 0.0337484i \(-0.989256\pi\)
0.999430 0.0337484i \(-0.0107445\pi\)
\(440\) 0 0
\(441\) 11.0000i 0.523810i
\(442\) 0 0
\(443\) 16.9706 16.9706i 0.806296 0.806296i −0.177775 0.984071i \(-0.556890\pi\)
0.984071 + 0.177775i \(0.0568900\pi\)
\(444\) 0 0
\(445\) 28.0000 + 28.0000i 1.32733 + 1.32733i
\(446\) 0 0
\(447\) 8.48528 0.401340
\(448\) 0 0
\(449\) −26.0000 −1.22702 −0.613508 0.789689i \(-0.710242\pi\)
−0.613508 + 0.789689i \(0.710242\pi\)
\(450\) 0 0
\(451\) −28.2843 28.2843i −1.33185 1.33185i
\(452\) 0 0
\(453\) 1.00000 1.00000i 0.0469841 0.0469841i
\(454\) 0 0
\(455\) 50.9117i 2.38678i
\(456\) 0 0
\(457\) 12.0000i 0.561336i 0.959805 + 0.280668i \(0.0905560\pi\)
−0.959805 + 0.280668i \(0.909444\pi\)
\(458\) 0 0
\(459\) −4.24264 + 4.24264i −0.198030 + 0.198030i
\(460\) 0 0
\(461\) −6.00000 6.00000i −0.279448 0.279448i 0.553441 0.832889i \(-0.313315\pi\)
−0.832889 + 0.553441i \(0.813315\pi\)
\(462\) 0 0
\(463\) 12.7279 0.591517 0.295758 0.955263i \(-0.404428\pi\)
0.295758 + 0.955263i \(0.404428\pi\)
\(464\) 0 0
\(465\) 12.0000 0.556487
\(466\) 0 0
\(467\) 16.9706 + 16.9706i 0.785304 + 0.785304i 0.980720 0.195416i \(-0.0626058\pi\)
−0.195416 + 0.980720i \(0.562606\pi\)
\(468\) 0 0
\(469\) −12.0000 + 12.0000i −0.554109 + 0.554109i
\(470\) 0 0
\(471\) 7.07107i 0.325818i
\(472\) 0 0
\(473\) 24.0000i 1.10352i
\(474\) 0 0
\(475\) −4.24264 + 4.24264i −0.194666 + 0.194666i
\(476\) 0 0
\(477\) −4.00000 4.00000i −0.183147 0.183147i
\(478\) 0 0
\(479\) 25.4558 1.16311 0.581554 0.813508i \(-0.302445\pi\)
0.581554 + 0.813508i \(0.302445\pi\)
\(480\) 0 0
\(481\) 18.0000 0.820729
\(482\) 0 0
\(483\) −8.48528 8.48528i −0.386094 0.386094i
\(484\) 0 0
\(485\) 8.00000 8.00000i 0.363261 0.363261i
\(486\) 0 0
\(487\) 43.8406i 1.98661i −0.115529 0.993304i \(-0.536856\pi\)
0.115529 0.993304i \(-0.463144\pi\)
\(488\) 0 0
\(489\) 10.0000i 0.452216i
\(490\) 0 0
\(491\) 5.65685 5.65685i 0.255290 0.255290i −0.567845 0.823135i \(-0.692223\pi\)
0.823135 + 0.567845i \(0.192223\pi\)
\(492\) 0 0
\(493\) 24.0000 + 24.0000i 1.08091 + 1.08091i
\(494\) 0 0
\(495\) −11.3137 −0.508513
\(496\) 0 0
\(497\) −12.0000 −0.538274
\(498\) 0 0
\(499\) −14.1421 14.1421i −0.633089 0.633089i 0.315753 0.948842i \(-0.397743\pi\)
−0.948842 + 0.315753i \(0.897743\pi\)
\(500\) 0 0
\(501\) −8.00000 + 8.00000i −0.357414 + 0.357414i
\(502\) 0 0
\(503\) 14.1421i 0.630567i 0.948998 + 0.315283i \(0.102100\pi\)
−0.948998 + 0.315283i \(0.897900\pi\)
\(504\) 0 0
\(505\) 32.0000i 1.42398i
\(506\) 0 0
\(507\) 3.53553 3.53553i 0.157019 0.157019i
\(508\) 0 0
\(509\) −16.0000 16.0000i −0.709188 0.709188i 0.257177 0.966364i \(-0.417208\pi\)
−0.966364 + 0.257177i \(0.917208\pi\)
\(510\) 0 0
\(511\) −67.8823 −3.00293
\(512\) 0 0
\(513\) −2.00000 −0.0883022
\(514\) 0 0
\(515\) 36.7696 + 36.7696i 1.62026 + 1.62026i
\(516\) 0 0
\(517\) −8.00000 + 8.00000i −0.351840 + 0.351840i
\(518\) 0 0
\(519\) 2.82843i 0.124154i
\(520\) 0 0
\(521\) 22.0000i 0.963837i −0.876216 0.481919i \(-0.839940\pi\)
0.876216 0.481919i \(-0.160060\pi\)
\(522\) 0 0
\(523\) −15.5563 + 15.5563i −0.680232 + 0.680232i −0.960052 0.279821i \(-0.909725\pi\)
0.279821 + 0.960052i \(0.409725\pi\)
\(524\) 0 0
\(525\) −9.00000 9.00000i −0.392792 0.392792i
\(526\) 0 0
\(527\) −25.4558 −1.10887
\(528\) 0 0
\(529\) 15.0000 0.652174
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −30.0000 + 30.0000i −1.29944 + 1.29944i
\(534\) 0 0
\(535\) 11.3137i 0.489134i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −31.1127 + 31.1127i −1.34012 + 1.34012i
\(540\) 0 0
\(541\) −29.0000 29.0000i −1.24681 1.24681i −0.957122 0.289685i \(-0.906449\pi\)
−0.289685 0.957122i \(-0.593551\pi\)
\(542\) 0 0
\(543\) −1.41421 −0.0606897
\(544\) 0 0
\(545\) −28.0000 −1.19939
\(546\) 0 0
\(547\) −24.0416 24.0416i −1.02795 1.02795i −0.999598 0.0283478i \(-0.990975\pi\)
−0.0283478 0.999598i \(-0.509025\pi\)
\(548\) 0 0
\(549\) 3.00000 3.00000i 0.128037 0.128037i
\(550\) 0 0
\(551\) 11.3137i 0.481980i
\(552\) 0 0
\(553\) 18.0000i 0.765438i
\(554\) 0 0
\(555\) −8.48528 + 8.48528i −0.360180 + 0.360180i
\(556\) 0 0
\(557\) 14.0000 + 14.0000i 0.593199 + 0.593199i 0.938494 0.345295i \(-0.112221\pi\)
−0.345295 + 0.938494i \(0.612221\pi\)
\(558\) 0 0
\(559\) 25.4558 1.07667
\(560\) 0 0
\(561\) 24.0000 1.01328
\(562\) 0 0
\(563\) 2.82843 + 2.82843i 0.119204 + 0.119204i 0.764192 0.644988i \(-0.223138\pi\)
−0.644988 + 0.764192i \(0.723138\pi\)
\(564\) 0 0
\(565\) −12.0000 + 12.0000i −0.504844 + 0.504844i
\(566\) 0 0
\(567\) 4.24264i 0.178174i
\(568\) 0 0
\(569\) 18.0000i 0.754599i 0.926091 + 0.377300i \(0.123147\pi\)
−0.926091 + 0.377300i \(0.876853\pi\)
\(570\) 0 0
\(571\) 14.1421 14.1421i 0.591830 0.591830i −0.346296 0.938125i \(-0.612561\pi\)
0.938125 + 0.346296i \(0.112561\pi\)
\(572\) 0 0
\(573\) 16.0000 + 16.0000i 0.668410 + 0.668410i
\(574\) 0 0
\(575\) −8.48528 −0.353861
\(576\) 0 0
\(577\) 4.00000 0.166522 0.0832611 0.996528i \(-0.473466\pi\)
0.0832611 + 0.996528i \(0.473466\pi\)
\(578\) 0 0
\(579\) −12.7279 12.7279i −0.528954 0.528954i
\(580\) 0 0
\(581\) 48.0000 48.0000i 1.99138 1.99138i
\(582\) 0 0
\(583\) 22.6274i 0.937132i
\(584\) 0 0
\(585\) 12.0000i 0.496139i
\(586\) 0 0
\(587\) −16.9706 + 16.9706i −0.700450 + 0.700450i −0.964507 0.264057i \(-0.914939\pi\)
0.264057 + 0.964507i \(0.414939\pi\)
\(588\) 0 0
\(589\) −6.00000 6.00000i −0.247226 0.247226i
\(590\) 0 0
\(591\) 5.65685 0.232692
\(592\) 0 0
\(593\) 14.0000 0.574911 0.287456 0.957794i \(-0.407191\pi\)
0.287456 + 0.957794i \(0.407191\pi\)
\(594\) 0 0
\(595\) 50.9117 + 50.9117i 2.08718 + 2.08718i
\(596\) 0 0
\(597\) 5.00000 5.00000i 0.204636 0.204636i
\(598\) 0 0
\(599\) 48.0833i 1.96463i −0.187239 0.982314i \(-0.559954\pi\)
0.187239 0.982314i \(-0.440046\pi\)
\(600\) 0 0
\(601\) 28.0000i 1.14214i −0.820900 0.571072i \(-0.806528\pi\)
0.820900 0.571072i \(-0.193472\pi\)
\(602\) 0 0
\(603\) −2.82843 + 2.82843i −0.115182 + 0.115182i
\(604\) 0 0
\(605\) 10.0000 + 10.0000i 0.406558 + 0.406558i
\(606\) 0 0
\(607\) 4.24264 0.172203 0.0861017 0.996286i \(-0.472559\pi\)
0.0861017 + 0.996286i \(0.472559\pi\)
\(608\) 0 0
\(609\) −24.0000 −0.972529
\(610\) 0 0
\(611\) 8.48528 + 8.48528i 0.343278 + 0.343278i
\(612\) 0 0
\(613\) 13.0000 13.0000i 0.525065 0.525065i −0.394032 0.919097i \(-0.628920\pi\)
0.919097 + 0.394032i \(0.128920\pi\)
\(614\) 0 0
\(615\) 28.2843i 1.14053i
\(616\) 0 0
\(617\) 22.0000i 0.885687i 0.896599 + 0.442843i \(0.146030\pi\)
−0.896599 + 0.442843i \(0.853970\pi\)
\(618\) 0 0
\(619\) −25.4558 + 25.4558i −1.02316 + 1.02316i −0.0234313 + 0.999725i \(0.507459\pi\)
−0.999725 + 0.0234313i \(0.992541\pi\)
\(620\) 0 0
\(621\) −2.00000 2.00000i −0.0802572 0.0802572i
\(622\) 0 0
\(623\) 59.3970 2.37969
\(624\) 0 0
\(625\) 31.0000 1.24000
\(626\) 0 0
\(627\) 5.65685 + 5.65685i 0.225913 + 0.225913i
\(628\) 0 0
\(629\) 18.0000 18.0000i 0.717707 0.717707i
\(630\) 0 0
\(631\) 35.3553i 1.40747i −0.710461 0.703737i \(-0.751513\pi\)
0.710461 0.703737i \(-0.248487\pi\)
\(632\) 0 0
\(633\) 12.0000i 0.476957i
\(634\) 0 0
\(635\) −8.48528 + 8.48528i −0.336728 + 0.336728i
\(636\) 0 0
\(637\) 33.0000 + 33.0000i 1.30751 + 1.30751i
\(638\) 0 0
\(639\) −2.82843 −0.111891
\(640\) 0 0
\(641\) 18.0000 0.710957 0.355479 0.934684i \(-0.384318\pi\)
0.355479 + 0.934684i \(0.384318\pi\)
\(642\) 0 0
\(643\) 9.89949 + 9.89949i 0.390398 + 0.390398i 0.874829 0.484431i \(-0.160973\pi\)
−0.484431 + 0.874829i \(0.660973\pi\)
\(644\) 0 0
\(645\) −12.0000 + 12.0000i −0.472500 + 0.472500i
\(646\) 0 0
\(647\) 14.1421i 0.555985i 0.960583 + 0.277992i \(0.0896690\pi\)
−0.960583 + 0.277992i \(0.910331\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 12.7279 12.7279i 0.498847 0.498847i
\(652\) 0 0
\(653\) 6.00000 + 6.00000i 0.234798 + 0.234798i 0.814692 0.579894i \(-0.196906\pi\)
−0.579894 + 0.814692i \(0.696906\pi\)
\(654\) 0 0
\(655\) −56.5685 −2.21032
\(656\) 0 0
\(657\) −16.0000 −0.624219
\(658\) 0 0
\(659\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(660\) 0 0
\(661\) 9.00000 9.00000i 0.350059 0.350059i −0.510072 0.860132i \(-0.670381\pi\)
0.860132 + 0.510072i \(0.170381\pi\)
\(662\) 0 0
\(663\) 25.4558i 0.988623i
\(664\) 0 0
\(665\) 24.0000i 0.930680i
\(666\) 0 0
\(667\) −11.3137 + 11.3137i −0.438069 + 0.438069i
\(668\) 0 0
\(669\) 15.0000 + 15.0000i 0.579934 + 0.579934i
\(670\) 0 0
\(671\) −16.9706 −0.655141
\(672\) 0 0
\(673\) −40.0000 −1.54189 −0.770943 0.636904i \(-0.780215\pi\)
−0.770943 + 0.636904i \(0.780215\pi\)
\(674\) 0 0
\(675\) −2.12132 2.12132i −0.0816497 0.0816497i
\(676\) 0 0
\(677\) 26.0000 26.0000i 0.999261 0.999261i −0.000738553 1.00000i \(-0.500235\pi\)
1.00000 0.000738553i \(0.000235089\pi\)
\(678\) 0 0
\(679\) 16.9706i 0.651270i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 16.9706 16.9706i 0.649361 0.649361i −0.303478 0.952838i \(-0.598148\pi\)
0.952838 + 0.303478i \(0.0981479\pi\)
\(684\) 0 0
\(685\) −20.0000 20.0000i −0.764161 0.764161i
\(686\) 0 0
\(687\) 15.5563 0.593512
\(688\) 0 0
\(689\) 24.0000 0.914327
\(690\) 0 0
\(691\) 15.5563 + 15.5563i 0.591791 + 0.591791i 0.938115 0.346324i \(-0.112570\pi\)
−0.346324 + 0.938115i \(0.612570\pi\)
\(692\) 0 0
\(693\) −12.0000 + 12.0000i −0.455842 + 0.455842i
\(694\) 0 0
\(695\) 33.9411i 1.28746i
\(696\) 0 0
\(697\) 60.0000i 2.27266i
\(698\) 0 0
\(699\) 12.7279 12.7279i 0.481414 0.481414i
\(700\) 0 0
\(701\) 4.00000 + 4.00000i 0.151078 + 0.151078i 0.778599 0.627521i \(-0.215931\pi\)
−0.627521 + 0.778599i \(0.715931\pi\)
\(702\) 0 0
\(703\) 8.48528 0.320028
\(704\) 0 0
\(705\) −8.00000 −0.301297
\(706\) 0 0
\(707\) −33.9411 33.9411i −1.27649 1.27649i
\(708\) 0 0
\(709\) −13.0000 + 13.0000i −0.488225 + 0.488225i −0.907746 0.419521i \(-0.862198\pi\)
0.419521 + 0.907746i \(0.362198\pi\)
\(710\) 0 0
\(711\) 4.24264i 0.159111i
\(712\) 0 0
\(713\) 12.0000i 0.449404i
\(714\) 0 0
\(715\) 33.9411 33.9411i 1.26933 1.26933i
\(716\) 0 0
\(717\) −8.00000 8.00000i −0.298765 0.298765i
\(718\) 0 0
\(719\) 8.48528 0.316448 0.158224 0.987403i \(-0.449423\pi\)
0.158224 + 0.987403i \(0.449423\pi\)
\(720\) 0 0
\(721\) 78.0000 2.90487
\(722\) 0 0
\(723\) 12.7279 + 12.7279i 0.473357 + 0.473357i
\(724\) 0 0
\(725\) −12.0000 + 12.0000i −0.445669 + 0.445669i
\(726\) 0 0
\(727\) 26.8701i 0.996555i 0.867018 + 0.498278i \(0.166034\pi\)
−0.867018 + 0.498278i \(0.833966\pi\)
\(728\) 0 0
\(729\) 1.00000i 0.0370370i
\(730\) 0 0
\(731\) 25.4558 25.4558i 0.941518 0.941518i
\(732\) 0 0
\(733\) 25.0000 + 25.0000i 0.923396 + 0.923396i 0.997268 0.0738717i \(-0.0235355\pi\)
−0.0738717 + 0.997268i \(0.523536\pi\)
\(734\) 0 0
\(735\) −31.1127 −1.14761
\(736\) 0 0
\(737\) 16.0000 0.589368
\(738\) 0 0
\(739\) 25.4558 + 25.4558i 0.936408 + 0.936408i 0.998096 0.0616872i \(-0.0196481\pi\)
−0.0616872 + 0.998096i \(0.519648\pi\)
\(740\) 0 0
\(741\) 6.00000 6.00000i 0.220416 0.220416i
\(742\) 0 0
\(743\) 22.6274i 0.830119i −0.909794 0.415060i \(-0.863761\pi\)
0.909794 0.415060i \(-0.136239\pi\)
\(744\) 0 0
\(745\) 24.0000i 0.879292i
\(746\) 0 0
\(747\) 11.3137 11.3137i 0.413947 0.413947i
\(748\) 0 0
\(749\) 12.0000 + 12.0000i 0.438470 + 0.438470i
\(750\) 0 0
\(751\) −12.7279 −0.464448 −0.232224 0.972662i \(-0.574600\pi\)
−0.232224 + 0.972662i \(0.574600\pi\)
\(752\) 0 0
\(753\) 4.00000 0.145768
\(754\) 0 0
\(755\) −2.82843 2.82843i −0.102937 0.102937i
\(756\) 0 0
\(757\) −27.0000 + 27.0000i −0.981332 + 0.981332i −0.999829 0.0184972i \(-0.994112\pi\)
0.0184972 + 0.999829i \(0.494112\pi\)
\(758\) 0 0
\(759\) 11.3137i 0.410662i
\(760\) 0 0
\(761\) 34.0000i 1.23250i 0.787551 + 0.616250i \(0.211349\pi\)
−0.787551 + 0.616250i \(0.788651\pi\)
\(762\) 0 0
\(763\) −29.6985 + 29.6985i −1.07516 + 1.07516i
\(764\) 0 0
\(765\) 12.0000 + 12.0000i 0.433861 + 0.433861i
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 14.0000 0.504853 0.252426 0.967616i \(-0.418771\pi\)
0.252426 + 0.967616i \(0.418771\pi\)
\(770\) 0 0
\(771\) 9.89949 + 9.89949i 0.356522 + 0.356522i
\(772\) 0 0
\(773\) 4.00000 4.00000i 0.143870 0.143870i −0.631503 0.775373i \(-0.717562\pi\)
0.775373 + 0.631503i \(0.217562\pi\)
\(774\) 0 0
\(775\) 12.7279i 0.457200i
\(776\) 0 0
\(777\) 18.0000i 0.645746i
\(778\) 0 0
\(779\) −14.1421 + 14.1421i −0.506695 + 0.506695i
\(780\) 0 0
\(781\) 8.00000 + 8.00000i 0.286263 + 0.286263i
\(782\) 0 0
\(783\) −5.65685 −0.202159
\(784\) 0 0
\(785\) 20.0000 0.713831
\(786\) 0 0
\(787\) −26.8701 26.8701i −0.957814 0.957814i 0.0413314 0.999145i \(-0.486840\pi\)
−0.999145 + 0.0413314i \(0.986840\pi\)
\(788\) 0 0
\(789\) 16.0000 16.0000i 0.569615 0.569615i
\(790\) 0 0
\(791\) 25.4558i 0.905106i
\(792\) 0 0
\(793\) 18.0000i 0.639199i
\(794\) 0 0
\(795\) −11.3137 + 11.3137i −0.401256 + 0.401256i
\(796\) 0 0
\(797\) 22.0000 + 22.0000i 0.779280 + 0.779280i 0.979708 0.200428i \(-0.0642334\pi\)
−0.200428 + 0.979708i \(0.564233\pi\)
\(798\) 0 0
\(799\) 16.9706 0.600375
\(800\) 0 0
\(801\) 14.0000 0.494666
\(802\) 0 0
\(803\) 45.2548 + 45.2548i 1.59701 + 1.59701i
\(804\) 0 0
\(805\) −24.0000 + 24.0000i −0.845889 + 0.845889i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 30.0000i 1.05474i −0.849635 0.527372i \(-0.823177\pi\)
0.849635 0.527372i \(-0.176823\pi\)
\(810\) 0 0
\(811\) −1.41421 + 1.41421i −0.0496598 + 0.0496598i −0.731501 0.681841i \(-0.761180\pi\)
0.681841 + 0.731501i \(0.261180\pi\)
\(812\) 0 0
\(813\) 1.00000 + 1.00000i 0.0350715 + 0.0350715i
\(814\) 0 0
\(815\) −28.2843 −0.990755
\(816\) 0 0
\(817\) 12.0000 0.419827
\(818\) 0 0
\(819\) 12.7279 + 12.7279i 0.444750 + 0.444750i
\(820\) 0 0
\(821\) 20.0000 20.0000i 0.698005 0.698005i −0.265975 0.963980i \(-0.585694\pi\)
0.963980 + 0.265975i \(0.0856939\pi\)
\(822\) 0 0
\(823\) 4.24264i 0.147889i 0.997262 + 0.0739446i \(0.0235588\pi\)
−0.997262 + 0.0739446i \(0.976441\pi\)
\(824\) 0 0
\(825\) 12.0000i 0.417786i
\(826\) 0 0
\(827\) −25.4558 + 25.4558i −0.885186 + 0.885186i −0.994056 0.108870i \(-0.965277\pi\)
0.108870 + 0.994056i \(0.465277\pi\)
\(828\) 0 0
\(829\) −15.0000 15.0000i −0.520972 0.520972i 0.396893 0.917865i \(-0.370088\pi\)
−0.917865 + 0.396893i \(0.870088\pi\)
\(830\) 0 0
\(831\) 1.41421 0.0490585
\(832\) 0 0
\(833\) 66.0000 2.28676
\(834\) 0 0
\(835\) 22.6274 + 22.6274i 0.783054 + 0.783054i
\(836\) 0 0
\(837\) 3.00000 3.00000i 0.103695 0.103695i
\(838\) 0 0
\(839\) 31.1127i 1.07413i 0.843541 + 0.537065i \(0.180467\pi\)
−0.843541 + 0.537065i \(0.819533\pi\)
\(840\) 0 0
\(841\) 3.00000i 0.103448i
\(842\) 0 0
\(843\) −4.24264 + 4.24264i −0.146124 + 0.146124i
\(844\) 0 0
\(845\) −10.0000 10.0000i −0.344010 0.344010i
\(846\) 0 0
\(847\) 21.2132 0.728894
\(848\) 0 0
\(849\) 4.00000 0.137280
\(850\) 0 0
\(851\) 8.48528 + 8.48528i 0.290872 + 0.290872i
\(852\) 0 0
\(853\) 15.0000 15.0000i 0.513590 0.513590i −0.402034 0.915625i \(-0.631697\pi\)
0.915625 + 0.402034i \(0.131697\pi\)
\(854\) 0 0
\(855\) 5.65685i 0.193460i
\(856\) 0 0
\(857\) 30.0000i 1.02478i −0.858753 0.512390i \(-0.828760\pi\)
0.858753 0.512390i \(-0.171240\pi\)
\(858\) 0 0
\(859\) 7.07107 7.07107i 0.241262 0.241262i −0.576110 0.817372i \(-0.695430\pi\)
0.817372 + 0.576110i \(0.195430\pi\)
\(860\) 0 0
\(861\) −30.0000 30.0000i −1.02240 1.02240i
\(862\) 0 0
\(863\) 28.2843 0.962808 0.481404 0.876499i \(-0.340127\pi\)
0.481404 + 0.876499i \(0.340127\pi\)
\(864\) 0 0
\(865\) −8.00000 −0.272008
\(866\) 0 0
\(867\) −13.4350 13.4350i −0.456278 0.456278i
\(868\) 0 0
\(869\) −12.0000 + 12.0000i −0.407072 + 0.407072i
\(870\) 0 0
\(871\) 16.9706i 0.575026i
\(872\) 0 0
\(873\) 4.00000i 0.135379i
\(874\) 0 0
\(875\) 16.9706 16.9706i 0.573710 0.573710i
\(876\) 0 0
\(877\) 11.0000 + 11.0000i 0.371444 + 0.371444i 0.868003 0.496559i \(-0.165403\pi\)
−0.496559 + 0.868003i \(0.665403\pi\)
\(878\) 0 0
\(879\) 5.65685 0.190801
\(880\) 0 0
\(881\) −58.0000 −1.95407 −0.977035 0.213080i \(-0.931651\pi\)
−0.977035 + 0.213080i \(0.931651\pi\)
\(882\) 0 0
\(883\) −24.0416 24.0416i −0.809065 0.809065i 0.175427 0.984492i \(-0.443869\pi\)
−0.984492 + 0.175427i \(0.943869\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 16.9706i 0.569816i −0.958555 0.284908i \(-0.908037\pi\)
0.958555 0.284908i \(-0.0919630\pi\)
\(888\) 0 0
\(889\) 18.0000i 0.603701i
\(890\) 0 0
\(891\) −2.82843 + 2.82843i −0.0947559 + 0.0947559i
\(892\) 0 0
\(893\) 4.00000 + 4.00000i 0.133855 + 0.133855i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 12.0000 0.400668
\(898\) 0 0
\(899\) −16.9706 16.9706i −0.566000 0.566000i
\(900\) 0 0
\(901\) 24.0000 24.0000i 0.799556 0.799556i
\(902\) 0 0
\(903\) 25.4558i 0.847117i
\(904\) 0 0
\(905\) 4.00000i 0.132964i
\(906\) 0 0
\(907\) −41.0122 + 41.0122i −1.36179 + 1.36179i −0.490149 + 0.871639i \(0.663058\pi\)
−0.871639 + 0.490149i \(0.836942\pi\)
\(908\) 0 0
\(909\) −8.00000 8.00000i −0.265343 0.265343i
\(910\) 0 0
\(911\) −22.6274 −0.749680 −0.374840 0.927090i \(-0.622302\pi\)
−0.374840 + 0.927090i \(0.622302\pi\)
\(912\) 0 0
\(913\) −64.0000 −2.11809
\(914\) 0 0
\(915\) −8.48528 8.48528i −0.280515 0.280515i
\(916\) 0 0
\(917\) −60.0000 + 60.0000i −1.98137 + 1.98137i
\(918\) 0 0
\(919\) 18.3848i 0.606458i −0.952918 0.303229i \(-0.901935\pi\)
0.952918 0.303229i \(-0.0980647\pi\)
\(920\) 0 0
\(921\) 4.00000i 0.131804i
\(922\) 0 0
\(923\) 8.48528 8.48528i 0.279296 0.279296i
\(924\) 0 0
\(925\) 9.00000 + 9.00000i 0.295918 + 0.295918i
\(926\) 0 0
\(927\) 18.3848 0.603835
\(928\) 0 0
\(929\) −46.0000 −1.50921 −0.754606 0.656179i \(-0.772172\pi\)
−0.754606 + 0.656179i \(0.772172\pi\)
\(930\) 0 0
\(931\) 15.5563 + 15.5563i 0.509839 + 0.509839i
\(932\) 0 0
\(933\) 20.0000 20.0000i 0.654771 0.654771i
\(934\) 0 0
\(935\) 67.8823i 2.21999i
\(936\) 0 0
\(937\) 2.00000i 0.0653372i 0.999466 + 0.0326686i \(0.0104006\pi\)
−0.999466 + 0.0326686i \(0.989599\pi\)
\(938\) 0 0
\(939\) −18.3848 + 18.3848i −0.599965 + 0.599965i
\(940\) 0 0
\(941\) −26.0000 26.0000i −0.847576 0.847576i 0.142254 0.989830i \(-0.454565\pi\)
−0.989830 + 0.142254i \(0.954565\pi\)
\(942\) 0 0
\(943\) −28.2843 −0.921063
\(944\) 0 0
\(945\) −12.0000 −0.390360
\(946\) 0 0
\(947\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(948\) 0 0
\(949\) 48.0000 48.0000i 1.55815 1.55815i
\(950\) 0 0
\(951\) 16.9706i 0.550308i
\(952\) 0 0
\(953\) 18.0000i 0.583077i −0.956559 0.291539i \(-0.905833\pi\)
0.956559 0.291539i \(-0.0941672\pi\)
\(954\) 0 0
\(955\) 45.2548 45.2548i 1.46441 1.46441i
\(956\) 0 0
\(957\) 16.0000 + 16.0000i 0.517207 + 0.517207i
\(958\) 0 0
\(959\) −42.4264 −1.37002
\(960\) 0 0
\(961\) −13.0000 −0.419355
\(962\) 0 0
\(963\) 2.82843 + 2.82843i 0.0911448 + 0.0911448i
\(964\) 0 0
\(965\) −36.0000 + 36.0000i −1.15888 + 1.15888i
\(966\) 0 0
\(967\) 35.3553i 1.13695i −0.822700 0.568476i \(-0.807533\pi\)
0.822700 0.568476i \(-0.192467\pi\)
\(968\) 0 0
\(969\) 12.0000i 0.385496i
\(970\) 0 0
\(971\) 22.6274 22.6274i 0.726148 0.726148i −0.243702 0.969850i \(-0.578362\pi\)
0.969850 + 0.243702i \(0.0783618\pi\)
\(972\) 0 0
\(973\) −36.0000 36.0000i −1.15411 1.15411i
\(974\) 0 0
\(975\) 12.7279 0.407620
\(976\) 0 0
\(977\) −2.00000 −0.0639857 −0.0319928 0.999488i \(-0.510185\pi\)
−0.0319928 + 0.999488i \(0.510185\pi\)
\(978\) 0 0
\(979\) −39.5980 39.5980i −1.26556 1.26556i
\(980\) 0 0
\(981\) −7.00000 + 7.00000i −0.223493 + 0.223493i
\(982\) 0 0
\(983\) 22.6274i 0.721703i −0.932623 0.360851i \(-0.882486\pi\)
0.932623 0.360851i \(-0.117514\pi\)
\(984\) 0 0
\(985\) 16.0000i 0.509802i
\(986\) 0 0
\(987\) −8.48528 + 8.48528i −0.270089 + 0.270089i
\(988\) 0 0
\(989\) 12.0000 + 12.0000i 0.381578 + 0.381578i
\(990\) 0 0
\(991\) 46.6690 1.48249 0.741246 0.671234i \(-0.234235\pi\)
0.741246 + 0.671234i \(0.234235\pi\)
\(992\) 0 0
\(993\) −4.00000 −0.126936
\(994\) 0 0
\(995\) −14.1421 14.1421i −0.448336 0.448336i
\(996\) 0 0
\(997\) 33.0000 33.0000i 1.04512 1.04512i 0.0461877 0.998933i \(-0.485293\pi\)
0.998933 0.0461877i \(-0.0147072\pi\)
\(998\) 0 0
\(999\) 4.24264i 0.134231i
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 768.2.j.a.577.1 yes 4
3.2 odd 2 2304.2.k.d.577.2 4
4.3 odd 2 inner 768.2.j.a.577.2 yes 4
8.3 odd 2 768.2.j.d.577.1 yes 4
8.5 even 2 768.2.j.d.577.2 yes 4
12.11 even 2 2304.2.k.d.577.1 4
16.3 odd 4 768.2.j.d.193.1 yes 4
16.5 even 4 inner 768.2.j.a.193.1 4
16.11 odd 4 inner 768.2.j.a.193.2 yes 4
16.13 even 4 768.2.j.d.193.2 yes 4
24.5 odd 2 2304.2.k.a.577.2 4
24.11 even 2 2304.2.k.a.577.1 4
32.3 odd 8 3072.2.d.d.1537.3 4
32.5 even 8 3072.2.a.f.1.1 2
32.11 odd 8 3072.2.a.f.1.2 2
32.13 even 8 3072.2.d.d.1537.4 4
32.19 odd 8 3072.2.d.d.1537.2 4
32.21 even 8 3072.2.a.d.1.2 2
32.27 odd 8 3072.2.a.d.1.1 2
32.29 even 8 3072.2.d.d.1537.1 4
48.5 odd 4 2304.2.k.d.1729.1 4
48.11 even 4 2304.2.k.d.1729.2 4
48.29 odd 4 2304.2.k.a.1729.1 4
48.35 even 4 2304.2.k.a.1729.2 4
96.5 odd 8 9216.2.a.q.1.2 2
96.11 even 8 9216.2.a.q.1.1 2
96.53 odd 8 9216.2.a.e.1.1 2
96.59 even 8 9216.2.a.e.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
768.2.j.a.193.1 4 16.5 even 4 inner
768.2.j.a.193.2 yes 4 16.11 odd 4 inner
768.2.j.a.577.1 yes 4 1.1 even 1 trivial
768.2.j.a.577.2 yes 4 4.3 odd 2 inner
768.2.j.d.193.1 yes 4 16.3 odd 4
768.2.j.d.193.2 yes 4 16.13 even 4
768.2.j.d.577.1 yes 4 8.3 odd 2
768.2.j.d.577.2 yes 4 8.5 even 2
2304.2.k.a.577.1 4 24.11 even 2
2304.2.k.a.577.2 4 24.5 odd 2
2304.2.k.a.1729.1 4 48.29 odd 4
2304.2.k.a.1729.2 4 48.35 even 4
2304.2.k.d.577.1 4 12.11 even 2
2304.2.k.d.577.2 4 3.2 odd 2
2304.2.k.d.1729.1 4 48.5 odd 4
2304.2.k.d.1729.2 4 48.11 even 4
3072.2.a.d.1.1 2 32.27 odd 8
3072.2.a.d.1.2 2 32.21 even 8
3072.2.a.f.1.1 2 32.5 even 8
3072.2.a.f.1.2 2 32.11 odd 8
3072.2.d.d.1537.1 4 32.29 even 8
3072.2.d.d.1537.2 4 32.19 odd 8
3072.2.d.d.1537.3 4 32.3 odd 8
3072.2.d.d.1537.4 4 32.13 even 8
9216.2.a.e.1.1 2 96.53 odd 8
9216.2.a.e.1.2 2 96.59 even 8
9216.2.a.q.1.1 2 96.11 even 8
9216.2.a.q.1.2 2 96.5 odd 8