Properties

Label 960.3.i.a.929.30
Level $960$
Weight $3$
Character 960.929
Analytic conductor $26.158$
Analytic rank $0$
Dimension $32$
CM no
Inner twists $16$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [960,3,Mod(929,960)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(960, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("960.929");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 960 = 2^{6} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 960.i (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: \(26.1581053786\)
Analytic rank: \(0\)
Dimension: \(32\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 929.30
Character \(\chi\) \(=\) 960.929
Dual form 960.3.i.a.929.25

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(2.81345 + 1.04140i) q^{3} +(-4.33946 + 2.48376i) q^{5} +10.8770i q^{7} +(6.83095 + 5.85987i) q^{9} +O(q^{10})\) \(q+(2.81345 + 1.04140i) q^{3} +(-4.33946 + 2.48376i) q^{5} +10.8770i q^{7} +(6.83095 + 5.85987i) q^{9} +15.2961 q^{11} -1.83874 q^{13} +(-14.7954 + 2.46878i) q^{15} -20.4760 q^{17} +13.6619i q^{19} +(-11.3274 + 30.6019i) q^{21} -19.4502 q^{23} +(12.6619 - 21.5563i) q^{25} +(13.1160 + 23.6002i) q^{27} -24.5696 q^{29} +50.8574 q^{31} +(43.0348 + 15.9295i) q^{33} +(-27.0159 - 47.2004i) q^{35} +16.2378 q^{37} +(-5.17319 - 1.91487i) q^{39} -42.3120i q^{41} -26.2320 q^{43} +(-44.1972 - 8.46228i) q^{45} -58.3505 q^{47} -69.3095 q^{49} +(-57.6081 - 21.3238i) q^{51} +63.7379i q^{53} +(-66.3770 + 37.9919i) q^{55} +(-14.2276 + 38.4370i) q^{57} +31.5828 q^{59} +81.8966i q^{61} +(-63.7379 + 74.3004i) q^{63} +(7.97914 - 4.56698i) q^{65} -50.6420 q^{67} +(-54.7220 - 20.2555i) q^{69} -135.929i q^{71} -33.7613i q^{73} +(58.0725 - 47.4614i) q^{75} +166.376i q^{77} -81.8966 q^{79} +(12.3238 + 80.0570i) q^{81} +75.6556i q^{83} +(88.8549 - 50.8574i) q^{85} +(-69.1253 - 25.5869i) q^{87} +124.350i q^{89} -20.0000i q^{91} +(143.085 + 52.9632i) q^{93} +(-33.9328 - 59.2853i) q^{95} +58.1713i q^{97} +(104.487 + 89.6333i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 32 q + 32 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 32 q + 32 q^{9} + 32 q^{25} - 352 q^{49} - 352 q^{81}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/960\mathbb{Z}\right)^\times\).

\(n\) \(511\) \(577\) \(641\) \(901\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(-1\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 2.81345 + 1.04140i 0.937815 + 0.347135i
\(4\) 0 0
\(5\) −4.33946 + 2.48376i −0.867893 + 0.496751i
\(6\) 0 0
\(7\) 10.8770i 1.55386i 0.629587 + 0.776930i \(0.283224\pi\)
−0.629587 + 0.776930i \(0.716776\pi\)
\(8\) 0 0
\(9\) 6.83095 + 5.85987i 0.758995 + 0.651097i
\(10\) 0 0
\(11\) 15.2961 1.39056 0.695278 0.718741i \(-0.255281\pi\)
0.695278 + 0.718741i \(0.255281\pi\)
\(12\) 0 0
\(13\) −1.83874 −0.141441 −0.0707207 0.997496i \(-0.522530\pi\)
−0.0707207 + 0.997496i \(0.522530\pi\)
\(14\) 0 0
\(15\) −14.7954 + 2.46878i −0.986363 + 0.164585i
\(16\) 0 0
\(17\) −20.4760 −1.20447 −0.602235 0.798319i \(-0.705723\pi\)
−0.602235 + 0.798319i \(0.705723\pi\)
\(18\) 0 0
\(19\) 13.6619i 0.719048i 0.933136 + 0.359524i \(0.117061\pi\)
−0.933136 + 0.359524i \(0.882939\pi\)
\(20\) 0 0
\(21\) −11.3274 + 30.6019i −0.539399 + 1.45723i
\(22\) 0 0
\(23\) −19.4502 −0.845660 −0.422830 0.906209i \(-0.638963\pi\)
−0.422830 + 0.906209i \(0.638963\pi\)
\(24\) 0 0
\(25\) 12.6619 21.5563i 0.506476 0.862254i
\(26\) 0 0
\(27\) 13.1160 + 23.6002i 0.485778 + 0.874082i
\(28\) 0 0
\(29\) −24.5696 −0.847229 −0.423614 0.905843i \(-0.639239\pi\)
−0.423614 + 0.905843i \(0.639239\pi\)
\(30\) 0 0
\(31\) 50.8574 1.64056 0.820281 0.571961i \(-0.193817\pi\)
0.820281 + 0.571961i \(0.193817\pi\)
\(32\) 0 0
\(33\) 43.0348 + 15.9295i 1.30409 + 0.482711i
\(34\) 0 0
\(35\) −27.0159 47.2004i −0.771882 1.34858i
\(36\) 0 0
\(37\) 16.2378 0.438860 0.219430 0.975628i \(-0.429580\pi\)
0.219430 + 0.975628i \(0.429580\pi\)
\(38\) 0 0
\(39\) −5.17319 1.91487i −0.132646 0.0490993i
\(40\) 0 0
\(41\) 42.3120i 1.03200i −0.856589 0.516000i \(-0.827420\pi\)
0.856589 0.516000i \(-0.172580\pi\)
\(42\) 0 0
\(43\) −26.2320 −0.610047 −0.305024 0.952345i \(-0.598664\pi\)
−0.305024 + 0.952345i \(0.598664\pi\)
\(44\) 0 0
\(45\) −44.1972 8.46228i −0.982159 0.188051i
\(46\) 0 0
\(47\) −58.3505 −1.24150 −0.620750 0.784008i \(-0.713172\pi\)
−0.620750 + 0.784008i \(0.713172\pi\)
\(48\) 0 0
\(49\) −69.3095 −1.41448
\(50\) 0 0
\(51\) −57.6081 21.3238i −1.12957 0.418114i
\(52\) 0 0
\(53\) 63.7379i 1.20260i 0.799022 + 0.601301i \(0.205351\pi\)
−0.799022 + 0.601301i \(0.794649\pi\)
\(54\) 0 0
\(55\) −66.3770 + 37.9919i −1.20685 + 0.690761i
\(56\) 0 0
\(57\) −14.2276 + 38.4370i −0.249607 + 0.674334i
\(58\) 0 0
\(59\) 31.5828 0.535303 0.267651 0.963516i \(-0.413752\pi\)
0.267651 + 0.963516i \(0.413752\pi\)
\(60\) 0 0
\(61\) 81.8966i 1.34257i 0.741201 + 0.671283i \(0.234257\pi\)
−0.741201 + 0.671283i \(0.765743\pi\)
\(62\) 0 0
\(63\) −63.7379 + 74.3004i −1.01171 + 1.17937i
\(64\) 0 0
\(65\) 7.97914 4.56698i 0.122756 0.0702613i
\(66\) 0 0
\(67\) −50.6420 −0.755851 −0.377926 0.925836i \(-0.623362\pi\)
−0.377926 + 0.925836i \(0.623362\pi\)
\(68\) 0 0
\(69\) −54.7220 20.2555i −0.793073 0.293558i
\(70\) 0 0
\(71\) 135.929i 1.91449i −0.289284 0.957243i \(-0.593417\pi\)
0.289284 0.957243i \(-0.406583\pi\)
\(72\) 0 0
\(73\) 33.7613i 0.462484i −0.972896 0.231242i \(-0.925721\pi\)
0.972896 0.231242i \(-0.0742789\pi\)
\(74\) 0 0
\(75\) 58.0725 47.4614i 0.774300 0.632819i
\(76\) 0 0
\(77\) 166.376i 2.16073i
\(78\) 0 0
\(79\) −81.8966 −1.03667 −0.518333 0.855179i \(-0.673447\pi\)
−0.518333 + 0.855179i \(0.673447\pi\)
\(80\) 0 0
\(81\) 12.3238 + 80.0570i 0.152146 + 0.988358i
\(82\) 0 0
\(83\) 75.6556i 0.911513i 0.890104 + 0.455756i \(0.150631\pi\)
−0.890104 + 0.455756i \(0.849369\pi\)
\(84\) 0 0
\(85\) 88.8549 50.8574i 1.04535 0.598322i
\(86\) 0 0
\(87\) −69.1253 25.5869i −0.794544 0.294103i
\(88\) 0 0
\(89\) 124.350i 1.39719i 0.715516 + 0.698597i \(0.246192\pi\)
−0.715516 + 0.698597i \(0.753808\pi\)
\(90\) 0 0
\(91\) 20.0000i 0.219780i
\(92\) 0 0
\(93\) 143.085 + 52.9632i 1.53854 + 0.569496i
\(94\) 0 0
\(95\) −33.9328 59.2853i −0.357188 0.624056i
\(96\) 0 0
\(97\) 58.1713i 0.599705i 0.953986 + 0.299852i \(0.0969374\pi\)
−0.953986 + 0.299852i \(0.903063\pi\)
\(98\) 0 0
\(99\) 104.487 + 89.6333i 1.05543 + 0.905387i
\(100\) 0 0
\(101\) 62.2197 0.616036 0.308018 0.951380i \(-0.400334\pi\)
0.308018 + 0.951380i \(0.400334\pi\)
\(102\) 0 0
\(103\) 94.5265i 0.917733i −0.888505 0.458867i \(-0.848256\pi\)
0.888505 0.458867i \(-0.151744\pi\)
\(104\) 0 0
\(105\) −26.8529 160.930i −0.255742 1.53267i
\(106\) 0 0
\(107\) 2.78700i 0.0260467i 0.999915 + 0.0130234i \(0.00414558\pi\)
−0.999915 + 0.0130234i \(0.995854\pi\)
\(108\) 0 0
\(109\) 19.8183i 0.181819i −0.995859 0.0909094i \(-0.971023\pi\)
0.995859 0.0909094i \(-0.0289774\pi\)
\(110\) 0 0
\(111\) 45.6842 + 16.9101i 0.411570 + 0.152344i
\(112\) 0 0
\(113\) −73.9249 −0.654202 −0.327101 0.944989i \(-0.606072\pi\)
−0.327101 + 0.944989i \(0.606072\pi\)
\(114\) 0 0
\(115\) 84.4034 48.3095i 0.733942 0.420083i
\(116\) 0 0
\(117\) −12.5603 10.7748i −0.107353 0.0920921i
\(118\) 0 0
\(119\) 222.718i 1.87158i
\(120\) 0 0
\(121\) 112.971 0.933648
\(122\) 0 0
\(123\) 44.0639 119.042i 0.358243 0.967825i
\(124\) 0 0
\(125\) −1.40515 + 124.992i −0.0112412 + 0.999937i
\(126\) 0 0
\(127\) 119.647i 0.942104i 0.882105 + 0.471052i \(0.156126\pi\)
−0.882105 + 0.471052i \(0.843874\pi\)
\(128\) 0 0
\(129\) −73.8024 27.3182i −0.572111 0.211769i
\(130\) 0 0
\(131\) 13.3149 0.101641 0.0508203 0.998708i \(-0.483816\pi\)
0.0508203 + 0.998708i \(0.483816\pi\)
\(132\) 0 0
\(133\) −148.601 −1.11730
\(134\) 0 0
\(135\) −115.534 69.8353i −0.855805 0.517299i
\(136\) 0 0
\(137\) 271.762 1.98366 0.991832 0.127550i \(-0.0407112\pi\)
0.991832 + 0.127550i \(0.0407112\pi\)
\(138\) 0 0
\(139\) 170.281i 1.22504i 0.790454 + 0.612521i \(0.209845\pi\)
−0.790454 + 0.612521i \(0.790155\pi\)
\(140\) 0 0
\(141\) −164.166 60.7665i −1.16430 0.430968i
\(142\) 0 0
\(143\) −28.1256 −0.196682
\(144\) 0 0
\(145\) 106.619 61.0250i 0.735304 0.420862i
\(146\) 0 0
\(147\) −194.999 72.1793i −1.32652 0.491015i
\(148\) 0 0
\(149\) 179.323 1.20351 0.601756 0.798680i \(-0.294468\pi\)
0.601756 + 0.798680i \(0.294468\pi\)
\(150\) 0 0
\(151\) −183.611 −1.21597 −0.607985 0.793949i \(-0.708022\pi\)
−0.607985 + 0.793949i \(0.708022\pi\)
\(152\) 0 0
\(153\) −139.871 119.987i −0.914187 0.784227i
\(154\) 0 0
\(155\) −220.694 + 126.317i −1.42383 + 0.814951i
\(156\) 0 0
\(157\) 175.560 1.11822 0.559109 0.829094i \(-0.311143\pi\)
0.559109 + 0.829094i \(0.311143\pi\)
\(158\) 0 0
\(159\) −66.3770 + 179.323i −0.417465 + 1.12782i
\(160\) 0 0
\(161\) 211.560i 1.31404i
\(162\) 0 0
\(163\) 266.206 1.63316 0.816581 0.577230i \(-0.195867\pi\)
0.816581 + 0.577230i \(0.195867\pi\)
\(164\) 0 0
\(165\) −226.313 + 37.7627i −1.37159 + 0.228865i
\(166\) 0 0
\(167\) 118.800 0.711380 0.355690 0.934604i \(-0.384246\pi\)
0.355690 + 0.934604i \(0.384246\pi\)
\(168\) 0 0
\(169\) −165.619 −0.979994
\(170\) 0 0
\(171\) −80.0570 + 93.3238i −0.468170 + 0.545753i
\(172\) 0 0
\(173\) 201.989i 1.16756i 0.811910 + 0.583782i \(0.198428\pi\)
−0.811910 + 0.583782i \(0.801572\pi\)
\(174\) 0 0
\(175\) 234.469 + 137.724i 1.33982 + 0.786993i
\(176\) 0 0
\(177\) 88.8566 + 32.8905i 0.502015 + 0.185822i
\(178\) 0 0
\(179\) 87.5958 0.489362 0.244681 0.969604i \(-0.421317\pi\)
0.244681 + 0.969604i \(0.421317\pi\)
\(180\) 0 0
\(181\) 163.793i 0.904934i −0.891781 0.452467i \(-0.850544\pi\)
0.891781 0.452467i \(-0.149456\pi\)
\(182\) 0 0
\(183\) −85.2875 + 230.412i −0.466052 + 1.25908i
\(184\) 0 0
\(185\) −70.4634 + 40.3308i −0.380883 + 0.218004i
\(186\) 0 0
\(187\) −313.203 −1.67488
\(188\) 0 0
\(189\) −256.700 + 142.663i −1.35820 + 0.754831i
\(190\) 0 0
\(191\) 11.4892i 0.0601530i 0.999548 + 0.0300765i \(0.00957510\pi\)
−0.999548 + 0.0300765i \(0.990425\pi\)
\(192\) 0 0
\(193\) 33.7613i 0.174929i −0.996168 0.0874646i \(-0.972124\pi\)
0.996168 0.0874646i \(-0.0278765\pi\)
\(194\) 0 0
\(195\) 27.2050 4.53943i 0.139513 0.0232791i
\(196\) 0 0
\(197\) 197.790i 1.00401i −0.864865 0.502005i \(-0.832596\pi\)
0.864865 0.502005i \(-0.167404\pi\)
\(198\) 0 0
\(199\) 316.365 1.58978 0.794888 0.606756i \(-0.207530\pi\)
0.794888 + 0.606756i \(0.207530\pi\)
\(200\) 0 0
\(201\) −142.479 52.7388i −0.708849 0.262382i
\(202\) 0 0
\(203\) 267.244i 1.31647i
\(204\) 0 0
\(205\) 105.093 + 183.611i 0.512647 + 0.895665i
\(206\) 0 0
\(207\) −132.863 113.976i −0.641851 0.550607i
\(208\) 0 0
\(209\) 208.974i 0.999876i
\(210\) 0 0
\(211\) 366.900i 1.73886i −0.494054 0.869431i \(-0.664485\pi\)
0.494054 0.869431i \(-0.335515\pi\)
\(212\) 0 0
\(213\) 141.557 382.428i 0.664585 1.79543i
\(214\) 0 0
\(215\) 113.833 65.1540i 0.529456 0.303042i
\(216\) 0 0
\(217\) 553.177i 2.54920i
\(218\) 0 0
\(219\) 35.1592 94.9857i 0.160544 0.433725i
\(220\) 0 0
\(221\) 37.6500 0.170362
\(222\) 0 0
\(223\) 256.905i 1.15204i −0.817436 0.576020i \(-0.804605\pi\)
0.817436 0.576020i \(-0.195395\pi\)
\(224\) 0 0
\(225\) 212.810 73.0533i 0.945824 0.324681i
\(226\) 0 0
\(227\) 266.570i 1.17432i 0.809472 + 0.587158i \(0.199753\pi\)
−0.809472 + 0.587158i \(0.800247\pi\)
\(228\) 0 0
\(229\) 141.351i 0.617255i 0.951183 + 0.308627i \(0.0998696\pi\)
−0.951183 + 0.308627i \(0.900130\pi\)
\(230\) 0 0
\(231\) −173.265 + 468.090i −0.750065 + 2.02637i
\(232\) 0 0
\(233\) −38.5469 −0.165437 −0.0827186 0.996573i \(-0.526360\pi\)
−0.0827186 + 0.996573i \(0.526360\pi\)
\(234\) 0 0
\(235\) 253.210 144.929i 1.07749 0.616717i
\(236\) 0 0
\(237\) −230.412 85.2875i −0.972201 0.359863i
\(238\) 0 0
\(239\) 49.1393i 0.205604i 0.994702 + 0.102802i \(0.0327807\pi\)
−0.994702 + 0.102802i \(0.967219\pi\)
\(240\) 0 0
\(241\) 106.338 0.441237 0.220618 0.975360i \(-0.429192\pi\)
0.220618 + 0.975360i \(0.429192\pi\)
\(242\) 0 0
\(243\) −48.6994 + 238.070i −0.200409 + 0.979712i
\(244\) 0 0
\(245\) 300.766 172.148i 1.22762 0.702645i
\(246\) 0 0
\(247\) 25.1207i 0.101703i
\(248\) 0 0
\(249\) −78.7881 + 212.853i −0.316418 + 0.854831i
\(250\) 0 0
\(251\) 155.933 0.621247 0.310624 0.950533i \(-0.399462\pi\)
0.310624 + 0.950533i \(0.399462\pi\)
\(252\) 0 0
\(253\) −297.512 −1.17594
\(254\) 0 0
\(255\) 302.952 50.5506i 1.18805 0.198238i
\(256\) 0 0
\(257\) −276.572 −1.07616 −0.538078 0.842895i \(-0.680850\pi\)
−0.538078 + 0.842895i \(0.680850\pi\)
\(258\) 0 0
\(259\) 176.619i 0.681927i
\(260\) 0 0
\(261\) −167.834 143.975i −0.643042 0.551628i
\(262\) 0 0
\(263\) 423.427 1.60999 0.804995 0.593282i \(-0.202168\pi\)
0.804995 + 0.593282i \(0.202168\pi\)
\(264\) 0 0
\(265\) −158.310 276.588i −0.597394 1.04373i
\(266\) 0 0
\(267\) −129.499 + 349.853i −0.485015 + 1.31031i
\(268\) 0 0
\(269\) 254.623 0.946555 0.473277 0.880913i \(-0.343071\pi\)
0.473277 + 0.880913i \(0.343071\pi\)
\(270\) 0 0
\(271\) −19.8183 −0.0731301 −0.0365650 0.999331i \(-0.511642\pi\)
−0.0365650 + 0.999331i \(0.511642\pi\)
\(272\) 0 0
\(273\) 20.8281 56.2689i 0.0762934 0.206113i
\(274\) 0 0
\(275\) 193.678 329.729i 0.704284 1.19901i
\(276\) 0 0
\(277\) −388.801 −1.40362 −0.701808 0.712367i \(-0.747623\pi\)
−0.701808 + 0.712367i \(0.747623\pi\)
\(278\) 0 0
\(279\) 347.405 + 298.018i 1.24518 + 1.06816i
\(280\) 0 0
\(281\) 256.458i 0.912661i 0.889810 + 0.456330i \(0.150837\pi\)
−0.889810 + 0.456330i \(0.849163\pi\)
\(282\) 0 0
\(283\) 3.88526 0.0137288 0.00686442 0.999976i \(-0.497815\pi\)
0.00686442 + 0.999976i \(0.497815\pi\)
\(284\) 0 0
\(285\) −33.7282 202.134i −0.118344 0.709242i
\(286\) 0 0
\(287\) 460.228 1.60358
\(288\) 0 0
\(289\) 130.267 0.450750
\(290\) 0 0
\(291\) −60.5799 + 163.662i −0.208178 + 0.562412i
\(292\) 0 0
\(293\) 491.642i 1.67796i −0.544163 0.838979i \(-0.683153\pi\)
0.544163 0.838979i \(-0.316847\pi\)
\(294\) 0 0
\(295\) −137.053 + 78.4441i −0.464585 + 0.265912i
\(296\) 0 0
\(297\) 200.624 + 360.992i 0.675502 + 1.21546i
\(298\) 0 0
\(299\) 35.7638 0.119611
\(300\) 0 0
\(301\) 285.326i 0.947928i
\(302\) 0 0
\(303\) 175.052 + 64.7959i 0.577728 + 0.213848i
\(304\) 0 0
\(305\) −203.411 355.387i −0.666922 1.16520i
\(306\) 0 0
\(307\) −341.499 −1.11237 −0.556187 0.831057i \(-0.687736\pi\)
−0.556187 + 0.831057i \(0.687736\pi\)
\(308\) 0 0
\(309\) 98.4404 265.945i 0.318577 0.860664i
\(310\) 0 0
\(311\) 75.3001i 0.242122i −0.992645 0.121061i \(-0.961370\pi\)
0.992645 0.121061i \(-0.0386297\pi\)
\(312\) 0 0
\(313\) 448.249i 1.43210i −0.698046 0.716052i \(-0.745947\pi\)
0.698046 0.716052i \(-0.254053\pi\)
\(314\) 0 0
\(315\) 92.0444 480.733i 0.292204 1.52614i
\(316\) 0 0
\(317\) 163.088i 0.514474i 0.966348 + 0.257237i \(0.0828121\pi\)
−0.966348 + 0.257237i \(0.917188\pi\)
\(318\) 0 0
\(319\) −375.820 −1.17812
\(320\) 0 0
\(321\) −2.90240 + 7.84107i −0.00904173 + 0.0244270i
\(322\) 0 0
\(323\) 279.741i 0.866072i
\(324\) 0 0
\(325\) −23.2819 + 39.6365i −0.0716367 + 0.121958i
\(326\) 0 0
\(327\) 20.6388 55.7576i 0.0631157 0.170512i
\(328\) 0 0
\(329\) 634.680i 1.92912i
\(330\) 0 0
\(331\) 320.138i 0.967184i 0.875293 + 0.483592i \(0.160668\pi\)
−0.875293 + 0.483592i \(0.839332\pi\)
\(332\) 0 0
\(333\) 110.920 + 95.1515i 0.333092 + 0.285740i
\(334\) 0 0
\(335\) 219.759 125.782i 0.655998 0.375470i
\(336\) 0 0
\(337\) 324.618i 0.963258i 0.876375 + 0.481629i \(0.159955\pi\)
−0.876375 + 0.481629i \(0.840045\pi\)
\(338\) 0 0
\(339\) −207.984 76.9857i −0.613521 0.227096i
\(340\) 0 0
\(341\) 777.921 2.28129
\(342\) 0 0
\(343\) 220.907i 0.644044i
\(344\) 0 0
\(345\) 287.774 48.0181i 0.834128 0.139183i
\(346\) 0 0
\(347\) 152.749i 0.440200i 0.975477 + 0.220100i \(0.0706383\pi\)
−0.975477 + 0.220100i \(0.929362\pi\)
\(348\) 0 0
\(349\) 101.715i 0.291446i −0.989325 0.145723i \(-0.953449\pi\)
0.989325 0.145723i \(-0.0465509\pi\)
\(350\) 0 0
\(351\) −24.1169 43.3946i −0.0687092 0.123631i
\(352\) 0 0
\(353\) 385.583 1.09230 0.546151 0.837687i \(-0.316092\pi\)
0.546151 + 0.837687i \(0.316092\pi\)
\(354\) 0 0
\(355\) 337.613 + 589.857i 0.951024 + 1.66157i
\(356\) 0 0
\(357\) 231.939 626.605i 0.649690 1.75519i
\(358\) 0 0
\(359\) 237.389i 0.661252i −0.943762 0.330626i \(-0.892740\pi\)
0.943762 0.330626i \(-0.107260\pi\)
\(360\) 0 0
\(361\) 174.352 0.482971
\(362\) 0 0
\(363\) 317.839 + 117.649i 0.875589 + 0.324102i
\(364\) 0 0
\(365\) 83.8550 + 146.506i 0.229740 + 0.401387i
\(366\) 0 0
\(367\) 286.946i 0.781870i −0.920418 0.390935i \(-0.872152\pi\)
0.920418 0.390935i \(-0.127848\pi\)
\(368\) 0 0
\(369\) 247.943 289.031i 0.671932 0.783282i
\(370\) 0 0
\(371\) −693.279 −1.86868
\(372\) 0 0
\(373\) 2.77125 0.00742961 0.00371481 0.999993i \(-0.498818\pi\)
0.00371481 + 0.999993i \(0.498818\pi\)
\(374\) 0 0
\(375\) −134.121 + 350.195i −0.357655 + 0.933854i
\(376\) 0 0
\(377\) 45.1771 0.119833
\(378\) 0 0
\(379\) 283.519i 0.748071i −0.927414 0.374036i \(-0.877974\pi\)
0.927414 0.374036i \(-0.122026\pi\)
\(380\) 0 0
\(381\) −124.601 + 336.621i −0.327037 + 0.883519i
\(382\) 0 0
\(383\) 397.679 1.03833 0.519163 0.854675i \(-0.326244\pi\)
0.519163 + 0.854675i \(0.326244\pi\)
\(384\) 0 0
\(385\) −413.238 721.984i −1.07335 1.87528i
\(386\) 0 0
\(387\) −179.190 153.716i −0.463022 0.397200i
\(388\) 0 0
\(389\) 500.320 1.28617 0.643084 0.765795i \(-0.277654\pi\)
0.643084 + 0.765795i \(0.277654\pi\)
\(390\) 0 0
\(391\) 398.262 1.01857
\(392\) 0 0
\(393\) 37.4608 + 13.8662i 0.0953201 + 0.0352830i
\(394\) 0 0
\(395\) 355.387 203.411i 0.899715 0.514965i
\(396\) 0 0
\(397\) 672.847 1.69483 0.847415 0.530932i \(-0.178158\pi\)
0.847415 + 0.530932i \(0.178158\pi\)
\(398\) 0 0
\(399\) −418.080 154.754i −1.04782 0.387854i
\(400\) 0 0
\(401\) 307.904i 0.767840i 0.923366 + 0.383920i \(0.125426\pi\)
−0.923366 + 0.383920i \(0.874574\pi\)
\(402\) 0 0
\(403\) −93.5135 −0.232043
\(404\) 0 0
\(405\) −252.321 316.795i −0.623014 0.782210i
\(406\) 0 0
\(407\) 248.376 0.610260
\(408\) 0 0
\(409\) 42.9571 0.105030 0.0525148 0.998620i \(-0.483276\pi\)
0.0525148 + 0.998620i \(0.483276\pi\)
\(410\) 0 0
\(411\) 764.588 + 283.014i 1.86031 + 0.688599i
\(412\) 0 0
\(413\) 343.527i 0.831785i
\(414\) 0 0
\(415\) −187.910 328.305i −0.452795 0.791096i
\(416\) 0 0
\(417\) −177.331 + 479.076i −0.425255 + 1.14886i
\(418\) 0 0
\(419\) 11.3337 0.0270495 0.0135247 0.999909i \(-0.495695\pi\)
0.0135247 + 0.999909i \(0.495695\pi\)
\(420\) 0 0
\(421\) 324.963i 0.771883i 0.922523 + 0.385941i \(0.126123\pi\)
−0.922523 + 0.385941i \(0.873877\pi\)
\(422\) 0 0
\(423\) −398.590 341.927i −0.942293 0.808337i
\(424\) 0 0
\(425\) −259.265 + 441.388i −0.610036 + 1.03856i
\(426\) 0 0
\(427\) −890.790 −2.08616
\(428\) 0 0
\(429\) −79.1298 29.2901i −0.184452 0.0682753i
\(430\) 0 0
\(431\) 332.486i 0.771428i −0.922618 0.385714i \(-0.873955\pi\)
0.922618 0.385714i \(-0.126045\pi\)
\(432\) 0 0
\(433\) 249.325i 0.575808i 0.957659 + 0.287904i \(0.0929584\pi\)
−0.957659 + 0.287904i \(0.907042\pi\)
\(434\) 0 0
\(435\) 363.519 60.6569i 0.835675 0.139441i
\(436\) 0 0
\(437\) 265.726i 0.608070i
\(438\) 0 0
\(439\) 152.572 0.347545 0.173772 0.984786i \(-0.444404\pi\)
0.173772 + 0.984786i \(0.444404\pi\)
\(440\) 0 0
\(441\) −473.450 406.145i −1.07358 0.920963i
\(442\) 0 0
\(443\) 759.343i 1.71409i 0.515240 + 0.857046i \(0.327703\pi\)
−0.515240 + 0.857046i \(0.672297\pi\)
\(444\) 0 0
\(445\) −308.856 539.613i −0.694058 1.21261i
\(446\) 0 0
\(447\) 504.516 + 186.748i 1.12867 + 0.417781i
\(448\) 0 0
\(449\) 356.596i 0.794201i 0.917775 + 0.397101i \(0.129984\pi\)
−0.917775 + 0.397101i \(0.870016\pi\)
\(450\) 0 0
\(451\) 647.209i 1.43505i
\(452\) 0 0
\(453\) −516.581 191.214i −1.14035 0.422105i
\(454\) 0 0
\(455\) 49.6751 + 86.7893i 0.109176 + 0.190746i
\(456\) 0 0
\(457\) 379.145i 0.829640i 0.909904 + 0.414820i \(0.136155\pi\)
−0.909904 + 0.414820i \(0.863845\pi\)
\(458\) 0 0
\(459\) −268.563 483.238i −0.585106 1.05281i
\(460\) 0 0
\(461\) −47.5481 −0.103141 −0.0515706 0.998669i \(-0.516423\pi\)
−0.0515706 + 0.998669i \(0.516423\pi\)
\(462\) 0 0
\(463\) 478.123i 1.03266i 0.856389 + 0.516331i \(0.172703\pi\)
−0.856389 + 0.516331i \(0.827297\pi\)
\(464\) 0 0
\(465\) −752.458 + 125.556i −1.61819 + 0.270012i
\(466\) 0 0
\(467\) 313.096i 0.670441i 0.942140 + 0.335220i \(0.108811\pi\)
−0.942140 + 0.335220i \(0.891189\pi\)
\(468\) 0 0
\(469\) 550.834i 1.17449i
\(470\) 0 0
\(471\) 493.929 + 182.829i 1.04868 + 0.388172i
\(472\) 0 0
\(473\) −401.248 −0.848305
\(474\) 0 0
\(475\) 294.501 + 172.986i 0.620002 + 0.364180i
\(476\) 0 0
\(477\) −373.496 + 435.391i −0.783011 + 0.912769i
\(478\) 0 0
\(479\) 260.368i 0.543565i 0.962359 + 0.271783i \(0.0876132\pi\)
−0.962359 + 0.271783i \(0.912387\pi\)
\(480\) 0 0
\(481\) −29.8571 −0.0620730
\(482\) 0 0
\(483\) 220.320 595.212i 0.456148 1.23232i
\(484\) 0 0
\(485\) −144.483 252.432i −0.297904 0.520479i
\(486\) 0 0
\(487\) 286.325i 0.587935i 0.955815 + 0.293968i \(0.0949758\pi\)
−0.955815 + 0.293968i \(0.905024\pi\)
\(488\) 0 0
\(489\) 748.955 + 277.228i 1.53160 + 0.566928i
\(490\) 0 0
\(491\) 302.514 0.616117 0.308059 0.951367i \(-0.400321\pi\)
0.308059 + 0.951367i \(0.400321\pi\)
\(492\) 0 0
\(493\) 503.088 1.02046
\(494\) 0 0
\(495\) −676.045 129.440i −1.36575 0.261495i
\(496\) 0 0
\(497\) 1478.50 2.97484
\(498\) 0 0
\(499\) 192.929i 0.386630i −0.981137 0.193315i \(-0.938076\pi\)
0.981137 0.193315i \(-0.0619240\pi\)
\(500\) 0 0
\(501\) 334.239 + 123.719i 0.667143 + 0.246945i
\(502\) 0 0
\(503\) −248.654 −0.494341 −0.247171 0.968972i \(-0.579501\pi\)
−0.247171 + 0.968972i \(0.579501\pi\)
\(504\) 0 0
\(505\) −270.000 + 154.539i −0.534653 + 0.306017i
\(506\) 0 0
\(507\) −465.960 172.476i −0.919054 0.340190i
\(508\) 0 0
\(509\) −730.373 −1.43492 −0.717459 0.696601i \(-0.754695\pi\)
−0.717459 + 0.696601i \(0.754695\pi\)
\(510\) 0 0
\(511\) 367.223 0.718636
\(512\) 0 0
\(513\) −322.424 + 179.190i −0.628507 + 0.349298i
\(514\) 0 0
\(515\) 234.781 + 410.194i 0.455885 + 0.796494i
\(516\) 0 0
\(517\) −892.537 −1.72638
\(518\) 0 0
\(519\) −210.352 + 568.284i −0.405302 + 1.09496i
\(520\) 0 0
\(521\) 263.006i 0.504810i 0.967622 + 0.252405i \(0.0812215\pi\)
−0.967622 + 0.252405i \(0.918779\pi\)
\(522\) 0 0
\(523\) 764.855 1.46244 0.731219 0.682143i \(-0.238952\pi\)
0.731219 + 0.682143i \(0.238952\pi\)
\(524\) 0 0
\(525\) 516.239 + 631.655i 0.983312 + 1.20315i
\(526\) 0 0
\(527\) −1041.36 −1.97601
\(528\) 0 0
\(529\) −150.690 −0.284859
\(530\) 0 0
\(531\) 215.741 + 185.071i 0.406292 + 0.348534i
\(532\) 0 0
\(533\) 77.8007i 0.145968i
\(534\) 0 0
\(535\) −6.92223 12.0941i −0.0129387 0.0226058i
\(536\) 0 0
\(537\) 246.446 + 91.2227i 0.458931 + 0.169875i
\(538\) 0 0
\(539\) −1060.17 −1.96691
\(540\) 0 0
\(541\) 1062.03i 1.96309i 0.191231 + 0.981545i \(0.438752\pi\)
−0.191231 + 0.981545i \(0.561248\pi\)
\(542\) 0 0
\(543\) 170.575 460.823i 0.314134 0.848661i
\(544\) 0 0
\(545\) 49.2237 + 86.0006i 0.0903188 + 0.157799i
\(546\) 0 0
\(547\) −886.905 −1.62140 −0.810699 0.585463i \(-0.800913\pi\)
−0.810699 + 0.585463i \(0.800913\pi\)
\(548\) 0 0
\(549\) −479.903 + 559.432i −0.874141 + 1.01900i
\(550\) 0 0
\(551\) 335.668i 0.609198i
\(552\) 0 0
\(553\) 890.790i 1.61083i
\(554\) 0 0
\(555\) −240.246 + 40.0875i −0.432875 + 0.0722298i
\(556\) 0 0
\(557\) 690.342i 1.23939i 0.784841 + 0.619697i \(0.212745\pi\)
−0.784841 + 0.619697i \(0.787255\pi\)
\(558\) 0 0
\(559\) 48.2339 0.0862860
\(560\) 0 0
\(561\) −881.181 326.172i −1.57073 0.581411i
\(562\) 0 0
\(563\) 282.528i 0.501826i −0.968010 0.250913i \(-0.919269\pi\)
0.968010 0.250913i \(-0.0807308\pi\)
\(564\) 0 0
\(565\) 320.794 183.611i 0.567778 0.324976i
\(566\) 0 0
\(567\) −870.781 + 134.046i −1.53577 + 0.236413i
\(568\) 0 0
\(569\) 987.481i 1.73547i −0.497029 0.867734i \(-0.665576\pi\)
0.497029 0.867734i \(-0.334424\pi\)
\(570\) 0 0
\(571\) 13.8334i 0.0242266i −0.999927 0.0121133i \(-0.996144\pi\)
0.999927 0.0121133i \(-0.00385587\pi\)
\(572\) 0 0
\(573\) −11.9649 + 32.3243i −0.0208812 + 0.0564124i
\(574\) 0 0
\(575\) −246.276 + 419.275i −0.428307 + 0.729174i
\(576\) 0 0
\(577\) 849.259i 1.47185i 0.677062 + 0.735926i \(0.263253\pi\)
−0.677062 + 0.735926i \(0.736747\pi\)
\(578\) 0 0
\(579\) 35.1592 94.9857i 0.0607241 0.164051i
\(580\) 0 0
\(581\) −822.907 −1.41636
\(582\) 0 0
\(583\) 974.943i 1.67229i
\(584\) 0 0
\(585\) 81.2671 + 15.5599i 0.138918 + 0.0265982i
\(586\) 0 0
\(587\) 493.447i 0.840626i −0.907379 0.420313i \(-0.861920\pi\)
0.907379 0.420313i \(-0.138080\pi\)
\(588\) 0 0
\(589\) 694.809i 1.17964i
\(590\) 0 0
\(591\) 205.979 556.471i 0.348527 0.941575i
\(592\) 0 0
\(593\) −604.188 −1.01887 −0.509434 0.860510i \(-0.670145\pi\)
−0.509434 + 0.860510i \(0.670145\pi\)
\(594\) 0 0
\(595\) 553.177 + 966.476i 0.929709 + 1.62433i
\(596\) 0 0
\(597\) 890.077 + 329.464i 1.49092 + 0.551867i
\(598\) 0 0
\(599\) 268.675i 0.448539i −0.974527 0.224269i \(-0.928000\pi\)
0.974527 0.224269i \(-0.0719996\pi\)
\(600\) 0 0
\(601\) 799.576 1.33041 0.665205 0.746661i \(-0.268344\pi\)
0.665205 + 0.746661i \(0.268344\pi\)
\(602\) 0 0
\(603\) −345.933 296.756i −0.573687 0.492132i
\(604\) 0 0
\(605\) −490.235 + 280.594i −0.810307 + 0.463791i
\(606\) 0 0
\(607\) 253.227i 0.417178i −0.978003 0.208589i \(-0.933113\pi\)
0.978003 0.208589i \(-0.0668871\pi\)
\(608\) 0 0
\(609\) 278.310 751.877i 0.456994 1.23461i
\(610\) 0 0
\(611\) 107.291 0.175600
\(612\) 0 0
\(613\) 620.741 1.01263 0.506314 0.862349i \(-0.331008\pi\)
0.506314 + 0.862349i \(0.331008\pi\)
\(614\) 0 0
\(615\) 104.459 + 626.025i 0.169852 + 1.01793i
\(616\) 0 0
\(617\) −230.046 −0.372847 −0.186423 0.982470i \(-0.559690\pi\)
−0.186423 + 0.982470i \(0.559690\pi\)
\(618\) 0 0
\(619\) 303.405i 0.490153i −0.969504 0.245077i \(-0.921187\pi\)
0.969504 0.245077i \(-0.0788131\pi\)
\(620\) 0 0
\(621\) −255.109 459.028i −0.410803 0.739176i
\(622\) 0 0
\(623\) −1352.56 −2.17104
\(624\) 0 0
\(625\) −304.352 545.889i −0.486964 0.873422i
\(626\) 0 0
\(627\) −217.627 + 587.937i −0.347092 + 0.937699i
\(628\) 0 0
\(629\) −332.486 −0.528594
\(630\) 0 0
\(631\) −440.522 −0.698133 −0.349067 0.937098i \(-0.613501\pi\)
−0.349067 + 0.937098i \(0.613501\pi\)
\(632\) 0 0
\(633\) 382.091 1032.25i 0.603620 1.63073i
\(634\) 0 0
\(635\) −297.175 519.205i −0.467991 0.817645i
\(636\) 0 0
\(637\) 127.442 0.200066
\(638\) 0 0
\(639\) 796.524 928.521i 1.24652 1.45309i
\(640\) 0 0
\(641\) 283.860i 0.442839i 0.975179 + 0.221419i \(0.0710690\pi\)
−0.975179 + 0.221419i \(0.928931\pi\)
\(642\) 0 0
\(643\) 241.796 0.376043 0.188021 0.982165i \(-0.439793\pi\)
0.188021 + 0.982165i \(0.439793\pi\)
\(644\) 0 0
\(645\) 388.114 64.7610i 0.601728 0.100405i
\(646\) 0 0
\(647\) −142.727 −0.220599 −0.110299 0.993898i \(-0.535181\pi\)
−0.110299 + 0.993898i \(0.535181\pi\)
\(648\) 0 0
\(649\) 483.095 0.744369
\(650\) 0 0
\(651\) −576.081 + 1556.33i −0.884917 + 2.39068i
\(652\) 0 0
\(653\) 282.167i 0.432108i 0.976381 + 0.216054i \(0.0693187\pi\)
−0.976381 + 0.216054i \(0.930681\pi\)
\(654\) 0 0
\(655\) −57.7796 + 33.0710i −0.0882132 + 0.0504901i
\(656\) 0 0
\(657\) 197.837 230.622i 0.301122 0.351023i
\(658\) 0 0
\(659\) 497.619 0.755113 0.377556 0.925987i \(-0.376764\pi\)
0.377556 + 0.925987i \(0.376764\pi\)
\(660\) 0 0
\(661\) 59.4548i 0.0899467i −0.998988 0.0449733i \(-0.985680\pi\)
0.998988 0.0449733i \(-0.0143203\pi\)
\(662\) 0 0
\(663\) 105.926 + 39.2089i 0.159768 + 0.0591386i
\(664\) 0 0
\(665\) 644.848 369.088i 0.969696 0.555020i
\(666\) 0 0
\(667\) 477.884 0.716467
\(668\) 0 0
\(669\) 267.542 722.787i 0.399913 1.08040i
\(670\) 0 0
\(671\) 1252.70i 1.86692i
\(672\) 0 0
\(673\) 302.271i 0.449140i 0.974458 + 0.224570i \(0.0720978\pi\)
−0.974458 + 0.224570i \(0.927902\pi\)
\(674\) 0 0
\(675\) 674.808 + 16.0903i 0.999716 + 0.0238375i
\(676\) 0 0
\(677\) 219.339i 0.323987i 0.986792 + 0.161994i \(0.0517924\pi\)
−0.986792 + 0.161994i \(0.948208\pi\)
\(678\) 0 0
\(679\) −632.731 −0.931857
\(680\) 0 0
\(681\) −277.607 + 749.980i −0.407646 + 1.10129i
\(682\) 0 0
\(683\) 442.875i 0.648426i −0.945984 0.324213i \(-0.894901\pi\)
0.945984 0.324213i \(-0.105099\pi\)
\(684\) 0 0
\(685\) −1179.30 + 674.991i −1.72161 + 0.985388i
\(686\) 0 0
\(687\) −147.204 + 397.684i −0.214271 + 0.578871i
\(688\) 0 0
\(689\) 117.197i 0.170098i
\(690\) 0 0
\(691\) 586.338i 0.848536i −0.905537 0.424268i \(-0.860532\pi\)
0.905537 0.424268i \(-0.139468\pi\)
\(692\) 0 0
\(693\) −974.943 + 1136.51i −1.40684 + 1.63998i
\(694\) 0 0
\(695\) −422.936 738.928i −0.608542 1.06321i
\(696\) 0 0
\(697\) 866.380i 1.24301i
\(698\) 0 0
\(699\) −108.449 40.1429i −0.155149 0.0574290i
\(700\) 0 0
\(701\) −488.830 −0.697333 −0.348666 0.937247i \(-0.613365\pi\)
−0.348666 + 0.937247i \(0.613365\pi\)
\(702\) 0 0
\(703\) 221.840i 0.315561i
\(704\) 0 0
\(705\) 863.322 144.054i 1.22457 0.204332i
\(706\) 0 0
\(707\) 676.764i 0.957234i
\(708\) 0 0
\(709\) 22.4418i 0.0316528i 0.999875 + 0.0158264i \(0.00503790\pi\)
−0.999875 + 0.0158264i \(0.994962\pi\)
\(710\) 0 0
\(711\) −559.432 479.903i −0.786824 0.674970i
\(712\) 0 0
\(713\) −989.186 −1.38736
\(714\) 0 0
\(715\) 122.050 69.8571i 0.170699 0.0977023i
\(716\) 0 0
\(717\) −51.1739 + 138.251i −0.0713722 + 0.192818i
\(718\) 0 0
\(719\) 83.6070i 0.116282i −0.998308 0.0581411i \(-0.981483\pi\)
0.998308 0.0581411i \(-0.0185173\pi\)
\(720\) 0 0
\(721\) 1028.17 1.42603
\(722\) 0 0
\(723\) 299.176 + 110.741i 0.413799 + 0.153169i
\(724\) 0 0
\(725\) −311.098 + 529.632i −0.429101 + 0.730526i
\(726\) 0 0
\(727\) 359.563i 0.494585i −0.968941 0.247292i \(-0.920459\pi\)
0.968941 0.247292i \(-0.0795408\pi\)
\(728\) 0 0
\(729\) −384.940 + 619.081i −0.528039 + 0.849220i
\(730\) 0 0
\(731\) 537.127 0.734784
\(732\) 0 0
\(733\) 626.594 0.854835 0.427418 0.904054i \(-0.359423\pi\)
0.427418 + 0.904054i \(0.359423\pi\)
\(734\) 0 0
\(735\) 1025.47 171.110i 1.39519 0.232802i
\(736\) 0 0
\(737\) −774.627 −1.05105
\(738\) 0 0
\(739\) 80.3095i 0.108673i 0.998523 + 0.0543366i \(0.0173044\pi\)
−0.998523 + 0.0543366i \(0.982696\pi\)
\(740\) 0 0
\(741\) 26.1608 70.6757i 0.0353047 0.0953788i
\(742\) 0 0
\(743\) −133.774 −0.180046 −0.0900229 0.995940i \(-0.528694\pi\)
−0.0900229 + 0.995940i \(0.528694\pi\)
\(744\) 0 0
\(745\) −778.167 + 445.395i −1.04452 + 0.597846i
\(746\) 0 0
\(747\) −443.332 + 516.800i −0.593483 + 0.691833i
\(748\) 0 0
\(749\) −30.3142 −0.0404730
\(750\) 0 0
\(751\) −1237.05 −1.64720 −0.823599 0.567172i \(-0.808037\pi\)
−0.823599 + 0.567172i \(0.808037\pi\)
\(752\) 0 0
\(753\) 438.709 + 162.389i 0.582615 + 0.215657i
\(754\) 0 0
\(755\) 796.775 456.046i 1.05533 0.604034i
\(756\) 0 0
\(757\) 436.609 0.576762 0.288381 0.957516i \(-0.406883\pi\)
0.288381 + 0.957516i \(0.406883\pi\)
\(758\) 0 0
\(759\) −837.035 309.831i −1.10281 0.408209i
\(760\) 0 0
\(761\) 610.636i 0.802412i −0.915988 0.401206i \(-0.868591\pi\)
0.915988 0.401206i \(-0.131409\pi\)
\(762\) 0 0
\(763\) 215.563 0.282521
\(764\) 0 0
\(765\) 904.981 + 173.274i 1.18298 + 0.226502i
\(766\) 0 0
\(767\) −58.0726 −0.0757140
\(768\) 0 0
\(769\) 41.2667 0.0536627 0.0268314 0.999640i \(-0.491458\pi\)
0.0268314 + 0.999640i \(0.491458\pi\)
\(770\) 0 0
\(771\) −778.121 288.024i −1.00924 0.373572i
\(772\) 0 0
\(773\) 502.417i 0.649957i 0.945721 + 0.324978i \(0.105357\pi\)
−0.945721 + 0.324978i \(0.894643\pi\)
\(774\) 0 0
\(775\) 643.952 1096.30i 0.830905 1.41458i
\(776\) 0 0
\(777\) −183.932 + 496.908i −0.236721 + 0.639521i
\(778\) 0 0
\(779\) 578.062 0.742057
\(780\) 0 0
\(781\) 2079.18i 2.66220i
\(782\) 0 0
\(783\) −322.256 579.849i −0.411565 0.740547i
\(784\) 0 0
\(785\) −761.837 + 436.049i −0.970493 + 0.555476i
\(786\) 0 0
\(787\) −95.8180 −0.121751 −0.0608754 0.998145i \(-0.519389\pi\)
−0.0608754 + 0.998145i \(0.519389\pi\)
\(788\) 0 0
\(789\) 1191.29 + 440.959i 1.50987 + 0.558884i
\(790\) 0 0
\(791\) 804.082i 1.01654i
\(792\) 0 0
\(793\) 150.586i 0.189895i
\(794\) 0 0
\(795\) −157.355 943.031i −0.197930 1.18620i
\(796\) 0 0
\(797\) 808.865i 1.01489i −0.861685 0.507444i \(-0.830591\pi\)
0.861685 0.507444i \(-0.169409\pi\)
\(798\) 0 0
\(799\) 1194.79 1.49535
\(800\) 0 0
\(801\) −728.676 + 849.430i −0.909708 + 1.06046i
\(802\) 0 0
\(803\) 516.418i 0.643111i
\(804\) 0 0
\(805\) 525.464 + 918.057i 0.652750 + 1.14044i
\(806\) 0 0
\(807\) 716.369 + 265.166i 0.887694 + 0.328582i
\(808\) 0 0
\(809\) 1115.63i 1.37902i 0.724277 + 0.689510i \(0.242174\pi\)
−0.724277 + 0.689510i \(0.757826\pi\)
\(810\) 0 0
\(811\) 1266.17i 1.56124i −0.625005 0.780621i \(-0.714903\pi\)
0.625005 0.780621i \(-0.285097\pi\)
\(812\) 0 0
\(813\) −55.7576 20.6388i −0.0685825 0.0253860i
\(814\) 0 0
\(815\) −1155.19 + 661.190i −1.41741 + 0.811276i
\(816\) 0 0
\(817\) 358.379i 0.438653i
\(818\) 0 0
\(819\) 117.197 136.619i 0.143098 0.166812i
\(820\) 0 0
\(821\) −537.970 −0.655261 −0.327631 0.944806i \(-0.606250\pi\)
−0.327631 + 0.944806i \(0.606250\pi\)
\(822\) 0 0
\(823\) 74.9483i 0.0910672i −0.998963 0.0455336i \(-0.985501\pi\)
0.998963 0.0455336i \(-0.0144988\pi\)
\(824\) 0 0
\(825\) 888.284 725.976i 1.07671 0.879971i
\(826\) 0 0
\(827\) 848.170i 1.02560i 0.858509 + 0.512799i \(0.171391\pi\)
−0.858509 + 0.512799i \(0.828609\pi\)
\(828\) 0 0
\(829\) 223.248i 0.269298i 0.990893 + 0.134649i \(0.0429907\pi\)
−0.990893 + 0.134649i \(0.957009\pi\)
\(830\) 0 0
\(831\) −1093.87 404.900i −1.31633 0.487244i
\(832\) 0 0
\(833\) 1419.18 1.70370
\(834\) 0 0
\(835\) −515.530 + 295.071i −0.617402 + 0.353379i
\(836\) 0 0
\(837\) 667.046 + 1200.25i 0.796949 + 1.43399i
\(838\) 0 0
\(839\) 139.111i 0.165806i −0.996558 0.0829028i \(-0.973581\pi\)
0.996558 0.0829028i \(-0.0264191\pi\)
\(840\) 0 0
\(841\) −237.333 −0.282204
\(842\) 0 0
\(843\) −267.076 + 721.530i −0.316817 + 0.855907i
\(844\) 0 0
\(845\) 718.698 411.357i 0.850530 0.486814i
\(846\) 0 0
\(847\) 1228.79i 1.45076i
\(848\) 0 0
\(849\) 10.9310 + 4.04613i 0.0128751 + 0.00476576i
\(850\) 0 0
\(851\) −315.828 −0.371126
\(852\) 0 0
\(853\) 1056.44 1.23850 0.619252 0.785193i \(-0.287436\pi\)
0.619252 + 0.785193i \(0.287436\pi\)
\(854\) 0 0
\(855\) 115.611 603.817i 0.135217 0.706219i
\(856\) 0 0
\(857\) −508.439 −0.593277 −0.296639 0.954990i \(-0.595866\pi\)
−0.296639 + 0.954990i \(0.595866\pi\)
\(858\) 0 0
\(859\) 972.814i 1.13250i 0.824235 + 0.566248i \(0.191606\pi\)
−0.824235 + 0.566248i \(0.808394\pi\)
\(860\) 0 0
\(861\) 1294.83 + 479.284i 1.50386 + 0.556660i
\(862\) 0 0
\(863\) 287.554 0.333203 0.166601 0.986024i \(-0.446721\pi\)
0.166601 + 0.986024i \(0.446721\pi\)
\(864\) 0 0
\(865\) −501.690 876.522i −0.579989 1.01332i
\(866\) 0 0
\(867\) 366.498 + 135.660i 0.422720 + 0.156471i
\(868\) 0 0
\(869\) −1252.70 −1.44154
\(870\) 0 0
\(871\) 93.1175 0.106909
\(872\) 0 0
\(873\) −340.877 + 397.366i −0.390466 + 0.455173i
\(874\) 0 0
\(875\) −1359.54 15.2839i −1.55376 0.0174673i
\(876\) 0 0
\(877\) −34.0035 −0.0387726 −0.0193863 0.999812i \(-0.506171\pi\)
−0.0193863 + 0.999812i \(0.506171\pi\)
\(878\) 0 0
\(879\) 511.998 1383.21i 0.582478 1.57362i
\(880\) 0 0
\(881\) 1453.52i 1.64985i −0.565242 0.824925i \(-0.691217\pi\)
0.565242 0.824925i \(-0.308783\pi\)
\(882\) 0 0
\(883\) −689.562 −0.780931 −0.390465 0.920618i \(-0.627686\pi\)
−0.390465 + 0.920618i \(0.627686\pi\)
\(884\) 0 0
\(885\) −467.282 + 77.9710i −0.528003 + 0.0881028i
\(886\) 0 0
\(887\) −116.423 −0.131255 −0.0656275 0.997844i \(-0.520905\pi\)
−0.0656275 + 0.997844i \(0.520905\pi\)
\(888\) 0 0
\(889\) −1301.40 −1.46390
\(890\) 0 0
\(891\) 188.506 + 1224.56i 0.211567 + 1.37437i
\(892\) 0 0
\(893\) 797.179i 0.892698i
\(894\) 0 0
\(895\) −380.119 + 217.567i −0.424714 + 0.243091i
\(896\) 0 0
\(897\) 100.620 + 37.2446i 0.112173 + 0.0415213i
\(898\) 0 0
\(899\) −1249.55 −1.38993
\(900\) 0 0
\(901\) 1305.10i 1.44850i
\(902\) 0 0
\(903\) 297.140 802.750i 0.329059 0.888981i
\(904\) 0 0
\(905\) 406.822 + 710.774i 0.449527 + 0.785386i
\(906\) 0 0
\(907\) 189.331 0.208745 0.104372 0.994538i \(-0.466717\pi\)
0.104372 + 0.994538i \(0.466717\pi\)
\(908\) 0 0
\(909\) 425.019 + 364.599i 0.467568 + 0.401099i
\(910\) 0 0
\(911\) 1079.12i 1.18455i 0.805737 + 0.592273i \(0.201769\pi\)
−0.805737 + 0.592273i \(0.798231\pi\)
\(912\) 0 0
\(913\) 1157.24i 1.26751i
\(914\) 0 0
\(915\) −202.184 1211.70i −0.220966 1.32426i
\(916\) 0 0
\(917\) 144.827i 0.157935i
\(918\) 0 0
\(919\) −1223.20 −1.33101 −0.665507 0.746392i \(-0.731785\pi\)
−0.665507 + 0.746392i \(0.731785\pi\)
\(920\) 0 0
\(921\) −960.788 355.638i −1.04320 0.386144i
\(922\) 0 0
\(923\) 249.937i 0.270788i
\(924\) 0 0
\(925\) 205.602 350.028i 0.222272 0.378409i
\(926\) 0 0
\(927\) 553.913 645.706i 0.597533 0.696555i
\(928\) 0 0
\(929\) 291.450i 0.313724i −0.987621 0.156862i \(-0.949862\pi\)
0.987621 0.156862i \(-0.0501378\pi\)
\(930\) 0 0
\(931\) 946.900i 1.01708i
\(932\) 0 0
\(933\) 78.4178 211.853i 0.0840491 0.227066i
\(934\) 0 0
\(935\) 1359.14 777.921i 1.45362 0.832001i
\(936\) 0 0
\(937\) 706.443i 0.753941i −0.926225 0.376970i \(-0.876966\pi\)
0.926225 0.376970i \(-0.123034\pi\)
\(938\) 0 0
\(939\) 466.809 1261.12i 0.497134 1.34305i
\(940\) 0 0
\(941\) −661.438 −0.702909 −0.351455 0.936205i \(-0.614313\pi\)
−0.351455 + 0.936205i \(0.614313\pi\)
\(942\) 0 0
\(943\) 822.976i 0.872721i
\(944\) 0 0
\(945\) 759.600 1256.66i 0.803809 1.32980i
\(946\) 0 0
\(947\) 1699.89i 1.79503i −0.440988 0.897513i \(-0.645372\pi\)
0.440988 0.897513i \(-0.354628\pi\)
\(948\) 0 0
\(949\) 62.0783i 0.0654145i
\(950\) 0 0
\(951\) −169.841 + 458.840i −0.178592 + 0.482481i
\(952\) 0 0
\(953\) 1691.82 1.77526 0.887629 0.460558i \(-0.152351\pi\)
0.887629 + 0.460558i \(0.152351\pi\)
\(954\) 0 0
\(955\) −28.5365 49.8571i −0.0298811 0.0522064i
\(956\) 0 0
\(957\) −1057.35 391.381i −1.10486 0.408966i
\(958\) 0 0
\(959\) 2955.96i 3.08234i
\(960\) 0 0
\(961\) 1625.48 1.69144
\(962\) 0 0
\(963\) −16.3315 + 19.0379i −0.0169589 + 0.0197693i
\(964\) 0 0
\(965\) 83.8550 + 146.506i 0.0868963 + 0.151820i
\(966\) 0 0
\(967\) 1252.98i 1.29574i −0.761751 0.647870i \(-0.775660\pi\)
0.761751 0.647870i \(-0.224340\pi\)
\(968\) 0 0
\(969\) 291.324 787.037i 0.300644 0.812215i
\(970\) 0 0
\(971\) 1013.73 1.04400 0.522001 0.852945i \(-0.325186\pi\)
0.522001 + 0.852945i \(0.325186\pi\)
\(972\) 0 0
\(973\) −1852.15 −1.90354
\(974\) 0 0
\(975\) −106.780 + 87.2692i −0.109518 + 0.0895069i
\(976\) 0 0
\(977\) 7.21543 0.00738529 0.00369265 0.999993i \(-0.498825\pi\)
0.00369265 + 0.999993i \(0.498825\pi\)
\(978\) 0 0
\(979\) 1902.08i 1.94288i
\(980\) 0 0
\(981\) 116.132 135.378i 0.118382 0.138000i
\(982\) 0 0
\(983\) 483.877 0.492245 0.246123 0.969239i \(-0.420843\pi\)
0.246123 + 0.969239i \(0.420843\pi\)
\(984\) 0 0
\(985\) 491.262 + 858.302i 0.498743 + 0.871372i
\(986\) 0 0
\(987\) 660.959 1785.64i 0.669664 1.80916i
\(988\) 0 0
\(989\) 510.218 0.515892
\(990\) 0 0
\(991\) 799.147 0.806405 0.403203 0.915111i \(-0.367897\pi\)
0.403203 + 0.915111i \(0.367897\pi\)
\(992\) 0 0
\(993\) −333.393 + 900.691i −0.335744 + 0.907040i
\(994\) 0 0
\(995\) −1372.86 + 785.775i −1.37976 + 0.789723i
\(996\) 0 0
\(997\) −1692.54 −1.69763 −0.848817 0.528686i \(-0.822685\pi\)
−0.848817 + 0.528686i \(0.822685\pi\)
\(998\) 0 0
\(999\) 212.975 + 383.216i 0.213189 + 0.383600i
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 960.3.i.a.929.30 yes 32
3.2 odd 2 inner 960.3.i.a.929.27 yes 32
4.3 odd 2 inner 960.3.i.a.929.2 yes 32
5.4 even 2 inner 960.3.i.a.929.1 32
8.3 odd 2 inner 960.3.i.a.929.31 yes 32
8.5 even 2 inner 960.3.i.a.929.3 yes 32
12.11 even 2 inner 960.3.i.a.929.7 yes 32
15.14 odd 2 inner 960.3.i.a.929.8 yes 32
20.19 odd 2 inner 960.3.i.a.929.29 yes 32
24.5 odd 2 inner 960.3.i.a.929.6 yes 32
24.11 even 2 inner 960.3.i.a.929.26 yes 32
40.19 odd 2 inner 960.3.i.a.929.4 yes 32
40.29 even 2 inner 960.3.i.a.929.32 yes 32
60.59 even 2 inner 960.3.i.a.929.28 yes 32
120.29 odd 2 inner 960.3.i.a.929.25 yes 32
120.59 even 2 inner 960.3.i.a.929.5 yes 32
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
960.3.i.a.929.1 32 5.4 even 2 inner
960.3.i.a.929.2 yes 32 4.3 odd 2 inner
960.3.i.a.929.3 yes 32 8.5 even 2 inner
960.3.i.a.929.4 yes 32 40.19 odd 2 inner
960.3.i.a.929.5 yes 32 120.59 even 2 inner
960.3.i.a.929.6 yes 32 24.5 odd 2 inner
960.3.i.a.929.7 yes 32 12.11 even 2 inner
960.3.i.a.929.8 yes 32 15.14 odd 2 inner
960.3.i.a.929.25 yes 32 120.29 odd 2 inner
960.3.i.a.929.26 yes 32 24.11 even 2 inner
960.3.i.a.929.27 yes 32 3.2 odd 2 inner
960.3.i.a.929.28 yes 32 60.59 even 2 inner
960.3.i.a.929.29 yes 32 20.19 odd 2 inner
960.3.i.a.929.30 yes 32 1.1 even 1 trivial
960.3.i.a.929.31 yes 32 8.3 odd 2 inner
960.3.i.a.929.32 yes 32 40.29 even 2 inner