Properties

Label 224.2.p.a.31.8
Level $224$
Weight $2$
Character 224.31
Analytic conductor $1.789$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [224,2,Mod(31,224)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(224, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("224.31");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 224 = 2^{5} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 224.p (of order \(6\), 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: \(1.78864900528\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: 16.0.2353561680715186176.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 4 x^{15} + 2 x^{14} + 41 x^{12} - 92 x^{11} + 66 x^{10} - 104 x^{9} + 291 x^{8} - 388 x^{7} + 366 x^{6} - 344 x^{5} + 286 x^{4} - 184 x^{3} + 84 x^{2} - 24 x + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{20} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 31.8
Root \(0.849168 - 0.0870829i\) of defining polynomial
Character \(\chi\) \(=\) 224.31
Dual form 224.2.p.a.159.8

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(1.60159 - 2.77404i) q^{3} +(-1.00367 + 0.579471i) q^{5} +(-1.44550 - 2.21597i) q^{7} +(-3.63019 - 6.28767i) q^{9} +O(q^{10})\) \(q+(1.60159 - 2.77404i) q^{3} +(-1.00367 + 0.579471i) q^{5} +(-1.44550 - 2.21597i) q^{7} +(-3.63019 - 6.28767i) q^{9} +(3.93239 + 2.27036i) q^{11} +2.08049i q^{13} +3.71230i q^{15} +(-0.301760 - 0.174221i) q^{17} +(-0.156095 - 0.270364i) q^{19} +(-8.46229 + 0.460776i) q^{21} +(4.08733 - 2.35982i) q^{23} +(-1.82843 + 3.16693i) q^{25} -13.6468 q^{27} +7.26038 q^{29} +(1.13445 - 1.96493i) q^{31} +(12.5961 - 7.27239i) q^{33} +(2.73490 + 1.38649i) q^{35} +(3.63386 + 6.29403i) q^{37} +(5.77137 + 3.33210i) q^{39} +2.08049i q^{41} +1.43929i q^{43} +(7.28704 + 4.20718i) q^{45} +(-4.02545 - 6.97228i) q^{47} +(-2.82108 + 6.40636i) q^{49} +(-0.966594 + 0.558063i) q^{51} +(0.805433 - 1.39505i) q^{53} -5.26244 q^{55} -1.00000 q^{57} +(-0.478903 + 0.829484i) q^{59} +(-9.88689 + 5.70820i) q^{61} +(-8.68589 + 17.1332i) q^{63} +(-1.20559 - 2.08814i) q^{65} +(0.592516 + 0.342089i) q^{67} -15.1179i q^{69} +4.57540i q^{71} +(7.58880 + 4.38140i) q^{73} +(5.85678 + 10.1442i) q^{75} +(-0.653181 - 11.9959i) q^{77} +(-13.6969 + 7.90790i) q^{79} +(-10.9660 + 18.9936i) q^{81} +10.0455 q^{83} +0.403825 q^{85} +(11.6282 - 20.1405i) q^{87} +(-7.11087 + 4.10546i) q^{89} +(4.61032 - 3.00735i) q^{91} +(-3.63386 - 6.29403i) q^{93} +(0.313336 + 0.180905i) q^{95} -13.0247i q^{97} -32.9674i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 8 q^{9} - 24 q^{21} + 16 q^{25} + 16 q^{29} + 24 q^{33} - 8 q^{37} - 24 q^{45} - 32 q^{49} - 8 q^{53} - 16 q^{57} - 24 q^{61} + 8 q^{65} - 24 q^{73} + 64 q^{77} - 48 q^{81} - 16 q^{85} - 72 q^{89} + 8 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/224\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(129\) \(197\)
\(\chi(n)\) \(-1\) \(e\left(\frac{1}{6}\right)\) \(1\)

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
\(3\) 1.60159 2.77404i 0.924679 1.60159i 0.132602 0.991169i \(-0.457667\pi\)
0.792077 0.610422i \(-0.209000\pi\)
\(4\) 0 0
\(5\) −1.00367 + 0.579471i −0.448856 + 0.259147i −0.707347 0.706866i \(-0.750108\pi\)
0.258491 + 0.966014i \(0.416775\pi\)
\(6\) 0 0
\(7\) −1.44550 2.21597i −0.546346 0.837559i
\(8\) 0 0
\(9\) −3.63019 6.28767i −1.21006 2.09589i
\(10\) 0 0
\(11\) 3.93239 + 2.27036i 1.18566 + 0.684541i 0.957317 0.289040i \(-0.0933362\pi\)
0.228342 + 0.973581i \(0.426670\pi\)
\(12\) 0 0
\(13\) 2.08049i 0.577025i 0.957476 + 0.288513i \(0.0931607\pi\)
−0.957476 + 0.288513i \(0.906839\pi\)
\(14\) 0 0
\(15\) 3.71230i 0.958512i
\(16\) 0 0
\(17\) −0.301760 0.174221i −0.0731876 0.0422549i 0.462960 0.886379i \(-0.346787\pi\)
−0.536147 + 0.844124i \(0.680121\pi\)
\(18\) 0 0
\(19\) −0.156095 0.270364i −0.0358106 0.0620258i 0.847565 0.530692i \(-0.178068\pi\)
−0.883375 + 0.468666i \(0.844735\pi\)
\(20\) 0 0
\(21\) −8.46229 + 0.460776i −1.84662 + 0.100550i
\(22\) 0 0
\(23\) 4.08733 2.35982i 0.852268 0.492057i −0.00914740 0.999958i \(-0.502912\pi\)
0.861415 + 0.507901i \(0.169578\pi\)
\(24\) 0 0
\(25\) −1.82843 + 3.16693i −0.365685 + 0.633386i
\(26\) 0 0
\(27\) −13.6468 −2.62632
\(28\) 0 0
\(29\) 7.26038 1.34822 0.674109 0.738632i \(-0.264528\pi\)
0.674109 + 0.738632i \(0.264528\pi\)
\(30\) 0 0
\(31\) 1.13445 1.96493i 0.203754 0.352912i −0.745981 0.665967i \(-0.768019\pi\)
0.949735 + 0.313055i \(0.101352\pi\)
\(32\) 0 0
\(33\) 12.5961 7.27239i 2.19271 1.26596i
\(34\) 0 0
\(35\) 2.73490 + 1.38649i 0.462282 + 0.234360i
\(36\) 0 0
\(37\) 3.63386 + 6.29403i 0.597403 + 1.03473i 0.993203 + 0.116396i \(0.0371340\pi\)
−0.395800 + 0.918337i \(0.629533\pi\)
\(38\) 0 0
\(39\) 5.77137 + 3.33210i 0.924158 + 0.533563i
\(40\) 0 0
\(41\) 2.08049i 0.324919i 0.986715 + 0.162459i \(0.0519426\pi\)
−0.986715 + 0.162459i \(0.948057\pi\)
\(42\) 0 0
\(43\) 1.43929i 0.219490i 0.993960 + 0.109745i \(0.0350035\pi\)
−0.993960 + 0.109745i \(0.964997\pi\)
\(44\) 0 0
\(45\) 7.28704 + 4.20718i 1.08629 + 0.627169i
\(46\) 0 0
\(47\) −4.02545 6.97228i −0.587172 1.01701i −0.994601 0.103774i \(-0.966908\pi\)
0.407429 0.913237i \(-0.366425\pi\)
\(48\) 0 0
\(49\) −2.82108 + 6.40636i −0.403012 + 0.915195i
\(50\) 0 0
\(51\) −0.966594 + 0.558063i −0.135350 + 0.0781445i
\(52\) 0 0
\(53\) 0.805433 1.39505i 0.110635 0.191625i −0.805392 0.592743i \(-0.798045\pi\)
0.916026 + 0.401118i \(0.131378\pi\)
\(54\) 0 0
\(55\) −5.26244 −0.709587
\(56\) 0 0
\(57\) −1.00000 −0.132453
\(58\) 0 0
\(59\) −0.478903 + 0.829484i −0.0623478 + 0.107990i −0.895514 0.445032i \(-0.853192\pi\)
0.833167 + 0.553022i \(0.186525\pi\)
\(60\) 0 0
\(61\) −9.88689 + 5.70820i −1.26589 + 0.730860i −0.974207 0.225656i \(-0.927547\pi\)
−0.291680 + 0.956516i \(0.594214\pi\)
\(62\) 0 0
\(63\) −8.68589 + 17.1332i −1.09432 + 2.15858i
\(64\) 0 0
\(65\) −1.20559 2.08814i −0.149534 0.259001i
\(66\) 0 0
\(67\) 0.592516 + 0.342089i 0.0723873 + 0.0417928i 0.535757 0.844372i \(-0.320026\pi\)
−0.463369 + 0.886165i \(0.653360\pi\)
\(68\) 0 0
\(69\) 15.1179i 1.81998i
\(70\) 0 0
\(71\) 4.57540i 0.543000i 0.962438 + 0.271500i \(0.0875197\pi\)
−0.962438 + 0.271500i \(0.912480\pi\)
\(72\) 0 0
\(73\) 7.58880 + 4.38140i 0.888202 + 0.512804i 0.873354 0.487086i \(-0.161940\pi\)
0.0148481 + 0.999890i \(0.495274\pi\)
\(74\) 0 0
\(75\) 5.85678 + 10.1442i 0.676283 + 1.17136i
\(76\) 0 0
\(77\) −0.653181 11.9959i −0.0744369 1.36706i
\(78\) 0 0
\(79\) −13.6969 + 7.90790i −1.54102 + 0.889708i −0.542245 + 0.840221i \(0.682425\pi\)
−0.998775 + 0.0494873i \(0.984241\pi\)
\(80\) 0 0
\(81\) −10.9660 + 18.9936i −1.21844 + 2.11040i
\(82\) 0 0
\(83\) 10.0455 1.10264 0.551319 0.834294i \(-0.314125\pi\)
0.551319 + 0.834294i \(0.314125\pi\)
\(84\) 0 0
\(85\) 0.403825 0.0438010
\(86\) 0 0
\(87\) 11.6282 20.1405i 1.24667 2.15929i
\(88\) 0 0
\(89\) −7.11087 + 4.10546i −0.753750 + 0.435178i −0.827047 0.562132i \(-0.809981\pi\)
0.0732970 + 0.997310i \(0.476648\pi\)
\(90\) 0 0
\(91\) 4.61032 3.00735i 0.483293 0.315256i
\(92\) 0 0
\(93\) −3.63386 6.29403i −0.376814 0.652661i
\(94\) 0 0
\(95\) 0.313336 + 0.180905i 0.0321476 + 0.0185604i
\(96\) 0 0
\(97\) 13.0247i 1.32245i −0.750185 0.661227i \(-0.770036\pi\)
0.750185 0.661227i \(-0.229964\pi\)
\(98\) 0 0
\(99\) 32.9674i 3.31335i
\(100\) 0 0
\(101\) 6.40015 + 3.69513i 0.636839 + 0.367679i 0.783396 0.621523i \(-0.213486\pi\)
−0.146557 + 0.989202i \(0.546819\pi\)
\(102\) 0 0
\(103\) −4.66044 8.07212i −0.459207 0.795370i 0.539712 0.841850i \(-0.318533\pi\)
−0.998919 + 0.0464796i \(0.985200\pi\)
\(104\) 0 0
\(105\) 8.22636 5.36612i 0.802811 0.523679i
\(106\) 0 0
\(107\) −5.96161 + 3.44194i −0.576331 + 0.332745i −0.759674 0.650304i \(-0.774641\pi\)
0.183343 + 0.983049i \(0.441308\pi\)
\(108\) 0 0
\(109\) −0.832100 + 1.44124i −0.0797007 + 0.138046i −0.903121 0.429387i \(-0.858730\pi\)
0.823420 + 0.567432i \(0.192063\pi\)
\(110\) 0 0
\(111\) 23.2798 2.20962
\(112\) 0 0
\(113\) −13.2751 −1.24881 −0.624407 0.781099i \(-0.714659\pi\)
−0.624407 + 0.781099i \(0.714659\pi\)
\(114\) 0 0
\(115\) −2.73490 + 4.73698i −0.255031 + 0.441726i
\(116\) 0 0
\(117\) 13.0815 7.55258i 1.20938 0.698236i
\(118\) 0 0
\(119\) 0.0501233 + 0.920530i 0.00459480 + 0.0843848i
\(120\) 0 0
\(121\) 4.80911 + 8.32962i 0.437191 + 0.757238i
\(122\) 0 0
\(123\) 5.77137 + 3.33210i 0.520387 + 0.300445i
\(124\) 0 0
\(125\) 10.0328i 0.897360i
\(126\) 0 0
\(127\) 8.75300i 0.776703i −0.921511 0.388352i \(-0.873045\pi\)
0.921511 0.388352i \(-0.126955\pi\)
\(128\) 0 0
\(129\) 3.99265 + 2.30516i 0.351534 + 0.202958i
\(130\) 0 0
\(131\) −3.73335 6.46636i −0.326185 0.564968i 0.655567 0.755137i \(-0.272430\pi\)
−0.981751 + 0.190169i \(0.939096\pi\)
\(132\) 0 0
\(133\) −0.373485 + 0.736712i −0.0323853 + 0.0638811i
\(134\) 0 0
\(135\) 13.6969 7.90790i 1.17884 0.680603i
\(136\) 0 0
\(137\) 3.70559 6.41826i 0.316590 0.548349i −0.663185 0.748456i \(-0.730796\pi\)
0.979774 + 0.200107i \(0.0641289\pi\)
\(138\) 0 0
\(139\) −21.3619 −1.81190 −0.905948 0.423390i \(-0.860840\pi\)
−0.905948 + 0.423390i \(0.860840\pi\)
\(140\) 0 0
\(141\) −25.7885 −2.17178
\(142\) 0 0
\(143\) −4.72348 + 8.18130i −0.394997 + 0.684155i
\(144\) 0 0
\(145\) −7.28704 + 4.20718i −0.605156 + 0.349387i
\(146\) 0 0
\(147\) 13.2533 + 18.0862i 1.09311 + 1.49172i
\(148\) 0 0
\(149\) 3.00367 + 5.20251i 0.246070 + 0.426207i 0.962432 0.271523i \(-0.0875272\pi\)
−0.716362 + 0.697729i \(0.754194\pi\)
\(150\) 0 0
\(151\) −4.95972 2.86350i −0.403616 0.233028i 0.284427 0.958698i \(-0.408197\pi\)
−0.688043 + 0.725670i \(0.741530\pi\)
\(152\) 0 0
\(153\) 2.52983i 0.204524i
\(154\) 0 0
\(155\) 2.62953i 0.211209i
\(156\) 0 0
\(157\) −17.2907 9.98280i −1.37995 0.796714i −0.387796 0.921745i \(-0.626764\pi\)
−0.992153 + 0.125031i \(0.960097\pi\)
\(158\) 0 0
\(159\) −2.57995 4.46860i −0.204603 0.354383i
\(160\) 0 0
\(161\) −11.1375 5.64631i −0.877761 0.444992i
\(162\) 0 0
\(163\) −12.1712 + 7.02707i −0.953325 + 0.550402i −0.894112 0.447843i \(-0.852192\pi\)
−0.0592126 + 0.998245i \(0.518859\pi\)
\(164\) 0 0
\(165\) −8.42828 + 14.5982i −0.656140 + 1.13647i
\(166\) 0 0
\(167\) 5.18343 0.401106 0.200553 0.979683i \(-0.435726\pi\)
0.200553 + 0.979683i \(0.435726\pi\)
\(168\) 0 0
\(169\) 8.67155 0.667042
\(170\) 0 0
\(171\) −1.13331 + 1.96294i −0.0866661 + 0.150110i
\(172\) 0 0
\(173\) 16.9904 9.80940i 1.29175 0.745795i 0.312790 0.949822i \(-0.398736\pi\)
0.978965 + 0.204027i \(0.0654031\pi\)
\(174\) 0 0
\(175\) 9.66082 0.526037i 0.730289 0.0397646i
\(176\) 0 0
\(177\) 1.53401 + 2.65699i 0.115303 + 0.199711i
\(178\) 0 0
\(179\) 10.1739 + 5.87388i 0.760431 + 0.439035i 0.829450 0.558581i \(-0.188654\pi\)
−0.0690198 + 0.997615i \(0.521987\pi\)
\(180\) 0 0
\(181\) 4.26353i 0.316905i 0.987367 + 0.158453i \(0.0506506\pi\)
−0.987367 + 0.158453i \(0.949349\pi\)
\(182\) 0 0
\(183\) 36.5688i 2.70324i
\(184\) 0 0
\(185\) −7.29441 4.21143i −0.536296 0.309631i
\(186\) 0 0
\(187\) −0.791092 1.37021i −0.0578504 0.100200i
\(188\) 0 0
\(189\) 19.7263 + 30.2409i 1.43488 + 2.19970i
\(190\) 0 0
\(191\) −5.89629 + 3.40422i −0.426641 + 0.246321i −0.697914 0.716181i \(-0.745888\pi\)
0.271274 + 0.962502i \(0.412555\pi\)
\(192\) 0 0
\(193\) 9.39791 16.2777i 0.676476 1.17169i −0.299559 0.954078i \(-0.596839\pi\)
0.976035 0.217613i \(-0.0698272\pi\)
\(194\) 0 0
\(195\) −7.72342 −0.553086
\(196\) 0 0
\(197\) 14.4674 1.03076 0.515380 0.856962i \(-0.327651\pi\)
0.515380 + 0.856962i \(0.327651\pi\)
\(198\) 0 0
\(199\) −7.27990 + 12.6092i −0.516058 + 0.893839i 0.483768 + 0.875196i \(0.339268\pi\)
−0.999826 + 0.0186428i \(0.994065\pi\)
\(200\) 0 0
\(201\) 1.89794 1.09577i 0.133870 0.0772899i
\(202\) 0 0
\(203\) −10.4948 16.0888i −0.736594 1.12921i
\(204\) 0 0
\(205\) −1.20559 2.08814i −0.0842017 0.145842i
\(206\) 0 0
\(207\) −29.6756 17.1332i −2.06260 1.19084i
\(208\) 0 0
\(209\) 1.41757i 0.0980552i
\(210\) 0 0
\(211\) 5.87441i 0.404411i −0.979343 0.202206i \(-0.935189\pi\)
0.979343 0.202206i \(-0.0648110\pi\)
\(212\) 0 0
\(213\) 12.6923 + 7.32792i 0.869664 + 0.502101i
\(214\) 0 0
\(215\) −0.834029 1.44458i −0.0568803 0.0985195i
\(216\) 0 0
\(217\) −5.99408 + 0.326381i −0.406905 + 0.0221562i
\(218\) 0 0
\(219\) 24.3083 14.0344i 1.64260 0.948358i
\(220\) 0 0
\(221\) 0.362467 0.627811i 0.0243821 0.0422311i
\(222\) 0 0
\(223\) 10.4178 0.697624 0.348812 0.937193i \(-0.386585\pi\)
0.348812 + 0.937193i \(0.386585\pi\)
\(224\) 0 0
\(225\) 26.5501 1.77001
\(226\) 0 0
\(227\) 1.61221 2.79243i 0.107006 0.185340i −0.807550 0.589799i \(-0.799207\pi\)
0.914556 + 0.404459i \(0.132540\pi\)
\(228\) 0 0
\(229\) −14.0851 + 8.13205i −0.930772 + 0.537381i −0.887056 0.461663i \(-0.847253\pi\)
−0.0437160 + 0.999044i \(0.513920\pi\)
\(230\) 0 0
\(231\) −34.3231 17.4005i −2.25829 1.14487i
\(232\) 0 0
\(233\) 8.78704 + 15.2196i 0.575658 + 0.997069i 0.995970 + 0.0896895i \(0.0285875\pi\)
−0.420312 + 0.907380i \(0.638079\pi\)
\(234\) 0 0
\(235\) 8.08046 + 4.66526i 0.527111 + 0.304328i
\(236\) 0 0
\(237\) 50.6609i 3.29078i
\(238\) 0 0
\(239\) 3.56805i 0.230798i 0.993319 + 0.115399i \(0.0368147\pi\)
−0.993319 + 0.115399i \(0.963185\pi\)
\(240\) 0 0
\(241\) 15.1182 + 8.72850i 0.973850 + 0.562252i 0.900408 0.435047i \(-0.143268\pi\)
0.0734420 + 0.997299i \(0.476602\pi\)
\(242\) 0 0
\(243\) 14.6558 + 25.3846i 0.940172 + 1.62843i
\(244\) 0 0
\(245\) −0.880858 8.06463i −0.0562760 0.515230i
\(246\) 0 0
\(247\) 0.562491 0.324754i 0.0357904 0.0206636i
\(248\) 0 0
\(249\) 16.0888 27.8666i 1.01959 1.76598i
\(250\) 0 0
\(251\) −4.28476 −0.270452 −0.135226 0.990815i \(-0.543176\pi\)
−0.135226 + 0.990815i \(0.543176\pi\)
\(252\) 0 0
\(253\) 21.4306 1.34733
\(254\) 0 0
\(255\) 0.646762 1.12023i 0.0405018 0.0701512i
\(256\) 0 0
\(257\) −16.0829 + 9.28546i −1.00322 + 0.579211i −0.909200 0.416359i \(-0.863306\pi\)
−0.0940226 + 0.995570i \(0.529973\pi\)
\(258\) 0 0
\(259\) 8.69468 17.1505i 0.540261 1.06568i
\(260\) 0 0
\(261\) −26.3565 45.6508i −1.63143 2.82572i
\(262\) 0 0
\(263\) 18.2190 + 10.5188i 1.12343 + 0.648615i 0.942275 0.334839i \(-0.108682\pi\)
0.181158 + 0.983454i \(0.442015\pi\)
\(264\) 0 0
\(265\) 1.86690i 0.114683i
\(266\) 0 0
\(267\) 26.3011i 1.60960i
\(268\) 0 0
\(269\) −13.6872 7.90231i −0.834523 0.481812i 0.0208758 0.999782i \(-0.493355\pi\)
−0.855399 + 0.517970i \(0.826688\pi\)
\(270\) 0 0
\(271\) 8.36423 + 14.4873i 0.508091 + 0.880039i 0.999956 + 0.00936761i \(0.00298185\pi\)
−0.491865 + 0.870671i \(0.663685\pi\)
\(272\) 0 0
\(273\) −0.958642 17.6057i −0.0580196 1.06555i
\(274\) 0 0
\(275\) −14.3802 + 8.30239i −0.867156 + 0.500653i
\(276\) 0 0
\(277\) −8.81278 + 15.2642i −0.529509 + 0.917136i 0.469899 + 0.882720i \(0.344290\pi\)
−0.999408 + 0.0344156i \(0.989043\pi\)
\(278\) 0 0
\(279\) −16.4731 −0.986219
\(280\) 0 0
\(281\) −6.06803 −0.361988 −0.180994 0.983484i \(-0.557931\pi\)
−0.180994 + 0.983484i \(0.557931\pi\)
\(282\) 0 0
\(283\) −0.917617 + 1.58936i −0.0545467 + 0.0944776i −0.892009 0.452017i \(-0.850705\pi\)
0.837463 + 0.546494i \(0.184038\pi\)
\(284\) 0 0
\(285\) 1.00367 0.579471i 0.0594525 0.0343249i
\(286\) 0 0
\(287\) 4.61032 3.00735i 0.272139 0.177518i
\(288\) 0 0
\(289\) −8.43929 14.6173i −0.496429 0.859840i
\(290\) 0 0
\(291\) −36.1309 20.8602i −2.11803 1.22285i
\(292\) 0 0
\(293\) 22.6107i 1.32093i 0.750857 + 0.660465i \(0.229641\pi\)
−0.750857 + 0.660465i \(0.770359\pi\)
\(294\) 0 0
\(295\) 1.11004i 0.0646291i
\(296\) 0 0
\(297\) −53.6643 30.9831i −3.11392 1.79782i
\(298\) 0 0
\(299\) 4.90960 + 8.50367i 0.283929 + 0.491780i
\(300\) 0 0
\(301\) 3.18944 2.08049i 0.183836 0.119918i
\(302\) 0 0
\(303\) 20.5009 11.8362i 1.17774 0.679970i
\(304\) 0 0
\(305\) 6.61547 11.4583i 0.378801 0.656102i
\(306\) 0 0
\(307\) −0.372239 −0.0212448 −0.0106224 0.999944i \(-0.503381\pi\)
−0.0106224 + 0.999944i \(0.503381\pi\)
\(308\) 0 0
\(309\) −29.8565 −1.69848
\(310\) 0 0
\(311\) 8.22708 14.2497i 0.466515 0.808028i −0.532753 0.846271i \(-0.678843\pi\)
0.999268 + 0.0382426i \(0.0121760\pi\)
\(312\) 0 0
\(313\) −12.9853 + 7.49706i −0.733971 + 0.423759i −0.819873 0.572545i \(-0.805956\pi\)
0.0859018 + 0.996304i \(0.472623\pi\)
\(314\) 0 0
\(315\) −1.21040 22.2294i −0.0681983 1.25248i
\(316\) 0 0
\(317\) 12.4623 + 21.5853i 0.699952 + 1.21235i 0.968483 + 0.249081i \(0.0801285\pi\)
−0.268531 + 0.963271i \(0.586538\pi\)
\(318\) 0 0
\(319\) 28.5506 + 16.4837i 1.59853 + 0.922910i
\(320\) 0 0
\(321\) 22.0503i 1.23073i
\(322\) 0 0
\(323\) 0.108780i 0.00605269i
\(324\) 0 0
\(325\) −6.58878 3.80403i −0.365479 0.211010i
\(326\) 0 0
\(327\) 2.66537 + 4.61655i 0.147395 + 0.255296i
\(328\) 0 0
\(329\) −9.63162 + 18.9987i −0.531008 + 1.04743i
\(330\) 0 0
\(331\) 25.5548 14.7541i 1.40462 0.810956i 0.409756 0.912195i \(-0.365614\pi\)
0.994862 + 0.101239i \(0.0322807\pi\)
\(332\) 0 0
\(333\) 26.3832 45.6970i 1.44579 2.50418i
\(334\) 0 0
\(335\) −0.792923 −0.0433220
\(336\) 0 0
\(337\) 12.8878 0.702046 0.351023 0.936367i \(-0.385834\pi\)
0.351023 + 0.936367i \(0.385834\pi\)
\(338\) 0 0
\(339\) −21.2612 + 36.8255i −1.15475 + 2.00009i
\(340\) 0 0
\(341\) 8.92222 5.15124i 0.483165 0.278956i
\(342\) 0 0
\(343\) 18.2742 3.00893i 0.986714 0.162467i
\(344\) 0 0
\(345\) 8.76038 + 15.1734i 0.471643 + 0.816909i
\(346\) 0 0
\(347\) −23.5256 13.5825i −1.26292 0.729146i −0.289280 0.957245i \(-0.593416\pi\)
−0.973638 + 0.228098i \(0.926749\pi\)
\(348\) 0 0
\(349\) 28.9547i 1.54991i −0.632016 0.774955i \(-0.717772\pi\)
0.632016 0.774955i \(-0.282228\pi\)
\(350\) 0 0
\(351\) 28.3920i 1.51545i
\(352\) 0 0
\(353\) 19.9867 + 11.5393i 1.06379 + 0.614178i 0.926477 0.376351i \(-0.122821\pi\)
0.137309 + 0.990528i \(0.456155\pi\)
\(354\) 0 0
\(355\) −2.65131 4.59220i −0.140717 0.243729i
\(356\) 0 0
\(357\) 2.63386 + 1.33527i 0.139399 + 0.0706699i
\(358\) 0 0
\(359\) 18.2218 10.5204i 0.961709 0.555243i 0.0650103 0.997885i \(-0.479292\pi\)
0.896699 + 0.442642i \(0.145959\pi\)
\(360\) 0 0
\(361\) 9.45127 16.3701i 0.497435 0.861583i
\(362\) 0 0
\(363\) 30.8089 1.61705
\(364\) 0 0
\(365\) −10.1556 −0.531567
\(366\) 0 0
\(367\) 3.97372 6.88269i 0.207427 0.359273i −0.743477 0.668762i \(-0.766825\pi\)
0.950903 + 0.309489i \(0.100158\pi\)
\(368\) 0 0
\(369\) 13.0815 7.55258i 0.680993 0.393172i
\(370\) 0 0
\(371\) −4.25565 + 0.231722i −0.220942 + 0.0120304i
\(372\) 0 0
\(373\) 2.19457 + 3.80110i 0.113630 + 0.196814i 0.917231 0.398355i \(-0.130419\pi\)
−0.803601 + 0.595168i \(0.797085\pi\)
\(374\) 0 0
\(375\) −27.8313 16.0684i −1.43720 0.829770i
\(376\) 0 0
\(377\) 15.1052i 0.777956i
\(378\) 0 0
\(379\) 8.93801i 0.459115i −0.973295 0.229557i \(-0.926272\pi\)
0.973295 0.229557i \(-0.0737278\pi\)
\(380\) 0 0
\(381\) −24.2812 14.0187i −1.24396 0.718201i
\(382\) 0 0
\(383\) −18.6813 32.3569i −0.954569 1.65336i −0.735352 0.677685i \(-0.762983\pi\)
−0.219217 0.975676i \(-0.570350\pi\)
\(384\) 0 0
\(385\) 7.60684 + 11.6614i 0.387680 + 0.594321i
\(386\) 0 0
\(387\) 9.04980 5.22491i 0.460027 0.265597i
\(388\) 0 0
\(389\) 1.25670 2.17667i 0.0637173 0.110362i −0.832407 0.554165i \(-0.813038\pi\)
0.896124 + 0.443803i \(0.146371\pi\)
\(390\) 0 0
\(391\) −1.64453 −0.0831673
\(392\) 0 0
\(393\) −23.9172 −1.20646
\(394\) 0 0
\(395\) 9.16479 15.8739i 0.461131 0.798702i
\(396\) 0 0
\(397\) 2.27748 1.31490i 0.114303 0.0659931i −0.441758 0.897134i \(-0.645645\pi\)
0.556062 + 0.831141i \(0.312312\pi\)
\(398\) 0 0
\(399\) 1.44550 + 2.21597i 0.0723653 + 0.110937i
\(400\) 0 0
\(401\) −1.34040 2.32164i −0.0669365 0.115937i 0.830615 0.556847i \(-0.187989\pi\)
−0.897551 + 0.440910i \(0.854656\pi\)
\(402\) 0 0
\(403\) 4.08803 + 2.36022i 0.203639 + 0.117571i
\(404\) 0 0
\(405\) 25.4178i 1.26302i
\(406\) 0 0
\(407\) 33.0007i 1.63579i
\(408\) 0 0
\(409\) −6.49854 3.75194i −0.321332 0.185521i 0.330654 0.943752i \(-0.392731\pi\)
−0.651986 + 0.758231i \(0.726064\pi\)
\(410\) 0 0
\(411\) −11.8697 20.5589i −0.585487 1.01409i
\(412\) 0 0
\(413\) 2.53037 0.137780i 0.124511 0.00677970i
\(414\) 0 0
\(415\) −10.0824 + 5.82108i −0.494926 + 0.285746i
\(416\) 0 0
\(417\) −34.2131 + 59.2588i −1.67542 + 2.90191i
\(418\) 0 0
\(419\) 30.5826 1.49406 0.747028 0.664792i \(-0.231480\pi\)
0.747028 + 0.664792i \(0.231480\pi\)
\(420\) 0 0
\(421\) 31.0386 1.51273 0.756364 0.654151i \(-0.226974\pi\)
0.756364 + 0.654151i \(0.226974\pi\)
\(422\) 0 0
\(423\) −29.2262 + 50.6213i −1.42103 + 2.46129i
\(424\) 0 0
\(425\) 1.10349 0.637102i 0.0535273 0.0309040i
\(426\) 0 0
\(427\) 26.9407 + 13.6579i 1.30375 + 0.660953i
\(428\) 0 0
\(429\) 15.1302 + 26.2062i 0.730491 + 1.26525i
\(430\) 0 0
\(431\) −18.6600 10.7734i −0.898823 0.518935i −0.0220046 0.999758i \(-0.507005\pi\)
−0.876818 + 0.480822i \(0.840338\pi\)
\(432\) 0 0
\(433\) 29.6992i 1.42725i −0.700527 0.713626i \(-0.747052\pi\)
0.700527 0.713626i \(-0.252948\pi\)
\(434\) 0 0
\(435\) 26.9527i 1.29228i
\(436\) 0 0
\(437\) −1.27602 0.736712i −0.0610405 0.0352417i
\(438\) 0 0
\(439\) 14.2081 + 24.6092i 0.678116 + 1.17453i 0.975548 + 0.219788i \(0.0705365\pi\)
−0.297432 + 0.954743i \(0.596130\pi\)
\(440\) 0 0
\(441\) 50.5222 5.51828i 2.40582 0.262775i
\(442\) 0 0
\(443\) −25.9034 + 14.9553i −1.23071 + 0.710550i −0.967178 0.254101i \(-0.918220\pi\)
−0.263531 + 0.964651i \(0.584887\pi\)
\(444\) 0 0
\(445\) 4.75799 8.24108i 0.225550 0.390665i
\(446\) 0 0
\(447\) 19.2426 0.910145
\(448\) 0 0
\(449\) −27.4086 −1.29349 −0.646746 0.762706i \(-0.723871\pi\)
−0.646746 + 0.762706i \(0.723871\pi\)
\(450\) 0 0
\(451\) −4.72348 + 8.18130i −0.222420 + 0.385243i
\(452\) 0 0
\(453\) −15.8869 + 9.17230i −0.746431 + 0.430952i
\(454\) 0 0
\(455\) −2.88458 + 5.68994i −0.135231 + 0.266748i
\(456\) 0 0
\(457\) −2.34312 4.05840i −0.109606 0.189844i 0.806004 0.591910i \(-0.201626\pi\)
−0.915611 + 0.402066i \(0.868292\pi\)
\(458\) 0 0
\(459\) 4.11805 + 2.37756i 0.192214 + 0.110975i
\(460\) 0 0
\(461\) 4.70270i 0.219026i 0.993985 + 0.109513i \(0.0349292\pi\)
−0.993985 + 0.109513i \(0.965071\pi\)
\(462\) 0 0
\(463\) 23.4246i 1.08863i −0.838880 0.544317i \(-0.816789\pi\)
0.838880 0.544317i \(-0.183211\pi\)
\(464\) 0 0
\(465\) 7.29441 + 4.21143i 0.338270 + 0.195300i
\(466\) 0 0
\(467\) −0.0216998 0.0375851i −0.00100415 0.00173923i 0.865523 0.500869i \(-0.166986\pi\)
−0.866527 + 0.499130i \(0.833653\pi\)
\(468\) 0 0
\(469\) −0.0984187 1.80749i −0.00454455 0.0834620i
\(470\) 0 0
\(471\) −55.3853 + 31.9767i −2.55202 + 1.47341i
\(472\) 0 0
\(473\) −3.26772 + 5.65986i −0.150250 + 0.260241i
\(474\) 0 0
\(475\) 1.14163 0.0523817
\(476\) 0 0
\(477\) −11.6955 −0.535500
\(478\) 0 0
\(479\) −14.8431 + 25.7090i −0.678198 + 1.17467i 0.297324 + 0.954777i \(0.403906\pi\)
−0.975523 + 0.219898i \(0.929428\pi\)
\(480\) 0 0
\(481\) −13.0947 + 7.56022i −0.597067 + 0.344717i
\(482\) 0 0
\(483\) −33.5008 + 21.8529i −1.52434 + 0.994339i
\(484\) 0 0
\(485\) 7.54742 + 13.0725i 0.342711 + 0.593592i
\(486\) 0 0
\(487\) −4.68168 2.70297i −0.212147 0.122483i 0.390162 0.920746i \(-0.372419\pi\)
−0.602309 + 0.798263i \(0.705752\pi\)
\(488\) 0 0
\(489\) 45.0179i 2.03578i
\(490\) 0 0
\(491\) 20.9674i 0.946245i −0.880997 0.473123i \(-0.843127\pi\)
0.880997 0.473123i \(-0.156873\pi\)
\(492\) 0 0
\(493\) −2.19089 1.26491i −0.0986729 0.0569688i
\(494\) 0 0
\(495\) 19.1036 + 33.0885i 0.858645 + 1.48722i
\(496\) 0 0
\(497\) 10.1390 6.61372i 0.454795 0.296666i
\(498\) 0 0
\(499\) −12.1712 + 7.02707i −0.544859 + 0.314575i −0.747046 0.664772i \(-0.768529\pi\)
0.202187 + 0.979347i \(0.435195\pi\)
\(500\) 0 0
\(501\) 8.30173 14.3790i 0.370894 0.642407i
\(502\) 0 0
\(503\) 9.67327 0.431310 0.215655 0.976470i \(-0.430811\pi\)
0.215655 + 0.976470i \(0.430811\pi\)
\(504\) 0 0
\(505\) −8.56488 −0.381132
\(506\) 0 0
\(507\) 13.8883 24.0552i 0.616800 1.06833i
\(508\) 0 0
\(509\) −31.8068 + 18.3637i −1.40981 + 0.813956i −0.995370 0.0961191i \(-0.969357\pi\)
−0.414443 + 0.910075i \(0.636024\pi\)
\(510\) 0 0
\(511\) −1.26052 23.1499i −0.0557623 1.02409i
\(512\) 0 0
\(513\) 2.13019 + 3.68959i 0.0940501 + 0.162900i
\(514\) 0 0
\(515\) 9.35512 + 5.40118i 0.412236 + 0.238005i
\(516\) 0 0
\(517\) 36.5569i 1.60777i
\(518\) 0 0
\(519\) 62.8426i 2.75848i
\(520\) 0 0
\(521\) 34.4661 + 19.8990i 1.50999 + 0.871791i 0.999932 + 0.0116486i \(0.00370794\pi\)
0.510054 + 0.860142i \(0.329625\pi\)
\(522\) 0 0
\(523\) 3.64794 + 6.31841i 0.159513 + 0.276285i 0.934693 0.355455i \(-0.115674\pi\)
−0.775180 + 0.631740i \(0.782341\pi\)
\(524\) 0 0
\(525\) 14.0134 27.6420i 0.611596 1.20639i
\(526\) 0 0
\(527\) −0.684666 + 0.395292i −0.0298245 + 0.0172192i
\(528\) 0 0
\(529\) −0.362467 + 0.627811i −0.0157594 + 0.0272961i
\(530\) 0 0
\(531\) 6.95403 0.301779
\(532\) 0 0
\(533\) −4.32845 −0.187486
\(534\) 0 0
\(535\) 3.98900 6.90916i 0.172460 0.298709i
\(536\) 0 0
\(537\) 32.5887 18.8151i 1.40631 0.811932i
\(538\) 0 0
\(539\) −25.6384 + 18.7874i −1.10432 + 0.809231i
\(540\) 0 0
\(541\) 2.35416 + 4.07753i 0.101213 + 0.175307i 0.912185 0.409779i \(-0.134394\pi\)
−0.810971 + 0.585086i \(0.801061\pi\)
\(542\) 0 0
\(543\) 11.8272 + 6.82843i 0.507553 + 0.293036i
\(544\) 0 0
\(545\) 1.92871i 0.0826169i
\(546\) 0 0
\(547\) 4.76040i 0.203540i 0.994808 + 0.101770i \(0.0324506\pi\)
−0.994808 + 0.101770i \(0.967549\pi\)
\(548\) 0 0
\(549\) 71.7825 + 41.4437i 3.06360 + 1.76877i
\(550\) 0 0
\(551\) −1.13331 1.96294i −0.0482805 0.0836243i
\(552\) 0 0
\(553\) 37.3225 + 18.9211i 1.58711 + 0.804607i
\(554\) 0 0
\(555\) −23.3653 + 13.4900i −0.991803 + 0.572618i
\(556\) 0 0
\(557\) −4.43562 + 7.68272i −0.187943 + 0.325527i −0.944564 0.328326i \(-0.893515\pi\)
0.756621 + 0.653854i \(0.226849\pi\)
\(558\) 0 0
\(559\) −2.99444 −0.126651
\(560\) 0 0
\(561\) −5.06803 −0.213972
\(562\) 0 0
\(563\) 11.4979 19.9149i 0.484578 0.839313i −0.515265 0.857031i \(-0.672307\pi\)
0.999843 + 0.0177173i \(0.00563989\pi\)
\(564\) 0 0
\(565\) 13.3238 7.69251i 0.560538 0.323626i
\(566\) 0 0
\(567\) 57.9406 3.15490i 2.43328 0.132493i
\(568\) 0 0
\(569\) −20.4674 35.4506i −0.858038 1.48616i −0.873798 0.486289i \(-0.838350\pi\)
0.0157605 0.999876i \(-0.494983\pi\)
\(570\) 0 0
\(571\) 15.4751 + 8.93456i 0.647614 + 0.373900i 0.787541 0.616262i \(-0.211354\pi\)
−0.139928 + 0.990162i \(0.544687\pi\)
\(572\) 0 0
\(573\) 21.8087i 0.911072i
\(574\) 0 0
\(575\) 17.2591i 0.719753i
\(576\) 0 0
\(577\) −35.6511 20.5832i −1.48417 0.856888i −0.484336 0.874882i \(-0.660939\pi\)
−0.999838 + 0.0179940i \(0.994272\pi\)
\(578\) 0 0
\(579\) −30.1032 52.1403i −1.25105 2.16688i
\(580\) 0 0
\(581\) −14.5208 22.2606i −0.602422 0.923525i
\(582\) 0 0
\(583\) 6.33455 3.65725i 0.262350 0.151468i
\(584\) 0 0
\(585\) −8.75300 + 15.1606i −0.361892 + 0.626816i
\(586\) 0 0
\(587\) 0.372239 0.0153640 0.00768198 0.999970i \(-0.497555\pi\)
0.00768198 + 0.999970i \(0.497555\pi\)
\(588\) 0 0
\(589\) −0.708329 −0.0291862
\(590\) 0 0
\(591\) 23.1709 40.1331i 0.953123 1.65086i
\(592\) 0 0
\(593\) 36.9573 21.3373i 1.51765 0.876218i 0.517870 0.855459i \(-0.326725\pi\)
0.999785 0.0207588i \(-0.00660821\pi\)
\(594\) 0 0
\(595\) −0.583728 0.894866i −0.0239305 0.0366859i
\(596\) 0 0
\(597\) 23.3188 + 40.3894i 0.954376 + 1.65303i
\(598\) 0 0
\(599\) −28.6691 16.5521i −1.17139 0.676301i −0.217382 0.976087i \(-0.569752\pi\)
−0.954007 + 0.299785i \(0.903085\pi\)
\(600\) 0 0
\(601\) 24.7937i 1.01136i −0.862722 0.505678i \(-0.831242\pi\)
0.862722 0.505678i \(-0.168758\pi\)
\(602\) 0 0
\(603\) 4.96739i 0.202288i
\(604\) 0 0
\(605\) −9.65354 5.57347i −0.392472 0.226594i
\(606\) 0 0
\(607\) 10.4041 + 18.0204i 0.422288 + 0.731425i 0.996163 0.0875185i \(-0.0278937\pi\)
−0.573875 + 0.818943i \(0.694560\pi\)
\(608\) 0 0
\(609\) −61.4394 + 3.34541i −2.48965 + 0.135563i
\(610\) 0 0
\(611\) 14.5058 8.37491i 0.586841 0.338813i
\(612\) 0 0
\(613\) −14.2359 + 24.6573i −0.574984 + 0.995901i 0.421060 + 0.907033i \(0.361658\pi\)
−0.996043 + 0.0888679i \(0.971675\pi\)
\(614\) 0 0
\(615\) −7.72342 −0.311438
\(616\) 0 0
\(617\) 3.08277 0.124108 0.0620538 0.998073i \(-0.480235\pi\)
0.0620538 + 0.998073i \(0.480235\pi\)
\(618\) 0 0
\(619\) −20.7929 + 36.0144i −0.835739 + 1.44754i 0.0576884 + 0.998335i \(0.481627\pi\)
−0.893427 + 0.449208i \(0.851706\pi\)
\(620\) 0 0
\(621\) −55.7789 + 32.2039i −2.23833 + 1.29230i
\(622\) 0 0
\(623\) 19.3763 + 9.82307i 0.776296 + 0.393553i
\(624\) 0 0
\(625\) −3.32843 5.76500i −0.133137 0.230600i
\(626\) 0 0
\(627\) −3.93239 2.27036i −0.157044 0.0906696i
\(628\) 0 0
\(629\) 2.53239i 0.100973i
\(630\) 0 0
\(631\) 32.0373i 1.27539i −0.770291 0.637693i \(-0.779889\pi\)
0.770291 0.637693i \(-0.220111\pi\)
\(632\) 0 0
\(633\) −16.2958 9.40841i −0.647702 0.373951i
\(634\) 0 0
\(635\) 5.07211 + 8.78515i 0.201281 + 0.348628i
\(636\) 0 0
\(637\) −13.3284 5.86924i −0.528090 0.232548i
\(638\) 0 0
\(639\) 28.7686 16.6096i 1.13807 0.657064i
\(640\) 0 0
\(641\) 4.96742 8.60382i 0.196201 0.339830i −0.751092 0.660197i \(-0.770473\pi\)
0.947294 + 0.320367i \(0.103806\pi\)
\(642\) 0 0
\(643\) −22.7132 −0.895722 −0.447861 0.894103i \(-0.647814\pi\)
−0.447861 + 0.894103i \(0.647814\pi\)
\(644\) 0 0
\(645\) −5.34309 −0.210384
\(646\) 0 0
\(647\) −13.1348 + 22.7502i −0.516384 + 0.894403i 0.483435 + 0.875380i \(0.339389\pi\)
−0.999819 + 0.0190231i \(0.993944\pi\)
\(648\) 0 0
\(649\) −3.76646 + 2.17457i −0.147847 + 0.0853592i
\(650\) 0 0
\(651\) −8.69468 + 17.1505i −0.340771 + 0.672183i
\(652\) 0 0
\(653\) −16.1118 27.9064i −0.630503 1.09206i −0.987449 0.157939i \(-0.949515\pi\)
0.356946 0.934125i \(-0.383818\pi\)
\(654\) 0 0
\(655\) 7.49413 + 4.32674i 0.292820 + 0.169060i
\(656\) 0 0
\(657\) 63.6212i 2.48210i
\(658\) 0 0
\(659\) 44.0667i 1.71660i 0.513152 + 0.858298i \(0.328478\pi\)
−0.513152 + 0.858298i \(0.671522\pi\)
\(660\) 0 0
\(661\) −29.1380 16.8228i −1.13334 0.654332i −0.188565 0.982061i \(-0.560384\pi\)
−0.944772 + 0.327728i \(0.893717\pi\)
\(662\) 0 0
\(663\) −1.16105 2.01099i −0.0450913 0.0781004i
\(664\) 0 0
\(665\) −0.0520461 0.955842i −0.00201826 0.0370660i
\(666\) 0 0
\(667\) 29.6756 17.1332i 1.14904 0.663400i
\(668\) 0 0
\(669\) 16.6850 28.8992i 0.645078 1.11731i
\(670\) 0 0
\(671\) −51.8388 −2.00121
\(672\) 0 0
\(673\) −4.12427 −0.158979 −0.0794895 0.996836i \(-0.525329\pi\)
−0.0794895 + 0.996836i \(0.525329\pi\)
\(674\) 0 0
\(675\) 24.9521 43.2183i 0.960407 1.66347i
\(676\) 0 0
\(677\) 15.0630 8.69665i 0.578920 0.334239i −0.181784 0.983338i \(-0.558187\pi\)
0.760704 + 0.649099i \(0.224854\pi\)
\(678\) 0 0
\(679\) −28.8623 + 18.8271i −1.10763 + 0.722518i
\(680\) 0 0
\(681\) −5.16420 8.94466i −0.197893 0.342760i
\(682\) 0 0
\(683\) −14.8844 8.59353i −0.569537 0.328822i 0.187427 0.982278i \(-0.439985\pi\)
−0.756964 + 0.653456i \(0.773318\pi\)
\(684\) 0 0
\(685\) 8.58911i 0.328173i
\(686\) 0 0
\(687\) 52.0969i 1.98762i
\(688\) 0 0
\(689\) 2.90240 + 1.67570i 0.110572 + 0.0638390i
\(690\) 0 0
\(691\) 7.43714 + 12.8815i 0.282922 + 0.490035i 0.972103 0.234554i \(-0.0753629\pi\)
−0.689181 + 0.724589i \(0.742030\pi\)
\(692\) 0 0
\(693\) −73.0549 + 47.6542i −2.77513 + 1.81023i
\(694\) 0 0
\(695\) 21.4404 12.3786i 0.813280 0.469548i
\(696\) 0 0
\(697\) 0.362467 0.627811i 0.0137294 0.0237800i
\(698\) 0 0
\(699\) 56.2930 2.12920
\(700\) 0 0
\(701\) 9.31371 0.351774 0.175887 0.984410i \(-0.443721\pi\)
0.175887 + 0.984410i \(0.443721\pi\)
\(702\) 0 0
\(703\) 1.13445 1.96493i 0.0427867 0.0741088i
\(704\) 0 0
\(705\) 25.8832 14.9437i 0.974817 0.562811i
\(706\) 0 0
\(707\) −1.06308 19.5239i −0.0399814 0.734271i
\(708\) 0 0
\(709\) −3.10576 5.37934i −0.116639 0.202025i 0.801794 0.597600i \(-0.203879\pi\)
−0.918434 + 0.395574i \(0.870546\pi\)
\(710\) 0 0
\(711\) 99.4445 + 57.4143i 3.72946 + 2.15320i
\(712\) 0 0
\(713\) 10.7084i 0.401034i
\(714\) 0 0
\(715\) 10.9485i 0.409450i
\(716\) 0 0
\(717\) 9.89791 + 5.71456i 0.369644 + 0.213414i
\(718\) 0 0
\(719\) −23.9419 41.4685i −0.892881 1.54652i −0.836405 0.548112i \(-0.815347\pi\)
−0.0564765 0.998404i \(-0.517987\pi\)
\(720\) 0 0
\(721\) −11.1510 + 21.9956i −0.415284 + 0.819161i
\(722\) 0 0
\(723\) 48.4264 27.9590i 1.80100 1.03981i
\(724\) 0 0
\(725\) −13.2751 + 22.9931i −0.493024 + 0.853942i
\(726\) 0 0
\(727\) −51.2000 −1.89890 −0.949452 0.313913i \(-0.898360\pi\)
−0.949452 + 0.313913i \(0.898360\pi\)
\(728\) 0 0
\(729\) 28.0948 1.04055
\(730\) 0 0
\(731\) 0.250756 0.434322i 0.00927454 0.0160640i
\(732\) 0 0
\(733\) 6.79809 3.92488i 0.251093 0.144969i −0.369172 0.929361i \(-0.620359\pi\)
0.620265 + 0.784393i \(0.287025\pi\)
\(734\) 0 0
\(735\) −23.7824 10.4727i −0.877225 0.386291i
\(736\) 0 0
\(737\) 1.55333 + 2.69045i 0.0572178 + 0.0991041i
\(738\) 0 0
\(739\) 4.02476 + 2.32370i 0.148053 + 0.0854786i 0.572197 0.820116i \(-0.306091\pi\)
−0.424143 + 0.905595i \(0.639425\pi\)
\(740\) 0 0
\(741\) 2.08049i 0.0764288i
\(742\) 0 0
\(743\) 49.8239i 1.82786i −0.405871 0.913931i \(-0.633032\pi\)
0.405871 0.913931i \(-0.366968\pi\)
\(744\) 0 0
\(745\) −6.02941 3.48108i −0.220901 0.127537i
\(746\) 0 0
\(747\) −36.4671 63.1629i −1.33426 2.31101i
\(748\) 0 0
\(749\) 16.2447 + 8.23547i 0.593569 + 0.300917i
\(750\) 0 0
\(751\) −23.9959 + 13.8540i −0.875623 + 0.505541i −0.869213 0.494438i \(-0.835374\pi\)
−0.00641005 + 0.999979i \(0.502040\pi\)
\(752\) 0 0
\(753\) −6.86244 + 11.8861i −0.250081 + 0.433153i
\(754\) 0 0
\(755\) 6.63725 0.241554
\(756\) 0 0
\(757\) 43.5089 1.58136 0.790679 0.612231i \(-0.209728\pi\)
0.790679 + 0.612231i \(0.209728\pi\)
\(758\) 0 0
\(759\) 34.3231 59.4494i 1.24585 2.15788i
\(760\) 0 0
\(761\) −15.0726 + 8.70219i −0.546382 + 0.315454i −0.747662 0.664080i \(-0.768823\pi\)
0.201279 + 0.979534i \(0.435490\pi\)
\(762\) 0 0
\(763\) 4.39655 0.239394i 0.159166 0.00866666i
\(764\) 0 0
\(765\) −1.46596 2.53912i −0.0530019 0.0918020i
\(766\) 0 0
\(767\) −1.72574 0.996354i −0.0623127 0.0359763i
\(768\) 0 0
\(769\) 30.4935i 1.09962i 0.835289 + 0.549812i \(0.185301\pi\)
−0.835289 + 0.549812i \(0.814699\pi\)
\(770\) 0 0
\(771\) 59.4860i 2.14234i
\(772\) 0 0
\(773\) −7.39566 4.26989i −0.266004 0.153577i 0.361067 0.932540i \(-0.382413\pi\)
−0.627070 + 0.778963i \(0.715746\pi\)
\(774\) 0 0
\(775\) 4.14853 + 7.18547i 0.149020 + 0.258110i
\(776\) 0 0
\(777\) −33.6509 51.5875i −1.20722 1.85069i
\(778\) 0 0
\(779\) 0.562491 0.324754i 0.0201533 0.0116355i
\(780\) 0 0
\(781\) −10.3878 + 17.9922i −0.371705 + 0.643813i
\(782\) 0 0
\(783\) −99.0806 −3.54085
\(784\) 0 0
\(785\) 23.1390 0.825865
\(786\) 0 0
\(787\) 15.5818 26.9884i 0.555430 0.962033i −0.442440 0.896798i \(-0.645887\pi\)
0.997870 0.0652352i \(-0.0207798\pi\)
\(788\) 0 0
\(789\) 58.3589 33.6935i 2.07763 1.19952i
\(790\) 0 0
\(791\) 19.1891 + 29.4172i 0.682284 + 1.04596i
\(792\) 0 0
\(793\) −11.8759 20.5696i −0.421725 0.730448i
\(794\) 0 0
\(795\) 5.17885 + 2.99001i 0.183675 + 0.106045i
\(796\) 0 0
\(797\) 37.2767i 1.32041i 0.751086 + 0.660204i \(0.229530\pi\)
−0.751086 + 0.660204i \(0.770470\pi\)
\(798\) 0 0
\(799\) 2.80528i 0.0992435i
\(800\) 0 0
\(801\) 51.6276 + 29.8072i 1.82417 + 1.05319i
\(802\) 0 0
\(803\) 19.8947 + 34.4587i 0.702070 + 1.21602i
\(804\) 0 0
\(805\) 14.4503 0.786827i 0.509307 0.0277320i
\(806\) 0 0
\(807\) −43.8426 + 25.3125i −1.54333 + 0.891043i
\(808\) 0 0
\(809\) 19.8105 34.3129i 0.696501 1.20638i −0.273171 0.961966i \(-0.588072\pi\)
0.969672 0.244410i \(-0.0785943\pi\)
\(810\) 0 0
\(811\) −10.9442 −0.384302 −0.192151 0.981365i \(-0.561546\pi\)
−0.192151 + 0.981365i \(0.561546\pi\)
\(812\) 0 0
\(813\) 53.5843 1.87928
\(814\) 0 0
\(815\) 8.14396 14.1058i 0.285270 0.494103i
\(816\) 0 0
\(817\) 0.389133 0.224666i 0.0136141 0.00786008i
\(818\) 0 0
\(819\) −35.6455 18.0709i −1.24556 0.631450i
\(820\) 0 0
\(821\) −15.3675 26.6174i −0.536331 0.928952i −0.999098 0.0424723i \(-0.986477\pi\)
0.462767 0.886480i \(-0.346857\pi\)
\(822\) 0 0
\(823\) −18.2514 10.5374i −0.636202 0.367312i 0.146948 0.989144i \(-0.453055\pi\)
−0.783150 + 0.621833i \(0.786388\pi\)
\(824\) 0 0
\(825\) 53.1881i 1.85177i
\(826\) 0 0
\(827\) 17.1524i 0.596447i 0.954496 + 0.298224i \(0.0963941\pi\)
−0.954496 + 0.298224i \(0.903606\pi\)
\(828\) 0 0
\(829\) 41.5212 + 23.9723i 1.44209 + 0.832591i 0.997989 0.0633850i \(-0.0201896\pi\)
0.444102 + 0.895976i \(0.353523\pi\)
\(830\) 0 0
\(831\) 28.2289 + 48.8940i 0.979251 + 1.69611i
\(832\) 0 0
\(833\) 1.96742 1.44169i 0.0681669 0.0499517i
\(834\) 0 0
\(835\) −5.20247 + 3.00365i −0.180039 + 0.103945i
\(836\) 0 0
\(837\) −15.4816 + 26.8149i −0.535123 + 0.926860i
\(838\) 0 0
\(839\) 21.2881 0.734948 0.367474 0.930034i \(-0.380223\pi\)
0.367474 + 0.930034i \(0.380223\pi\)
\(840\) 0 0
\(841\) 23.7130 0.817691
\(842\) 0 0
\(843\) −9.71849 + 16.8329i −0.334723 + 0.579757i
\(844\) 0 0
\(845\) −8.70340 + 5.02491i −0.299406 + 0.172862i
\(846\) 0 0
\(847\) 11.5067 22.6973i 0.395374 0.779888i
\(848\) 0 0
\(849\) 2.93929 + 5.09101i 0.100876 + 0.174723i
\(850\) 0 0
\(851\) 29.7056 + 17.1505i 1.01829 + 0.587913i
\(852\) 0 0
\(853\) 16.3608i 0.560183i −0.959973 0.280091i \(-0.909635\pi\)
0.959973 0.280091i \(-0.0903648\pi\)
\(854\) 0 0
\(855\) 2.62687i 0.0898371i
\(856\) 0 0
\(857\) 14.1776 + 8.18543i 0.484297 + 0.279609i 0.722205 0.691679i \(-0.243129\pi\)
−0.237909 + 0.971288i \(0.576462\pi\)
\(858\) 0 0
\(859\) 17.1788 + 29.7545i 0.586133 + 1.01521i 0.994733 + 0.102498i \(0.0326836\pi\)
−0.408601 + 0.912713i \(0.633983\pi\)
\(860\) 0 0
\(861\) −0.958642 17.6057i −0.0326704 0.600002i
\(862\) 0 0
\(863\) 28.4284 16.4132i 0.967715 0.558710i 0.0691759 0.997604i \(-0.477963\pi\)
0.898539 + 0.438894i \(0.144630\pi\)
\(864\) 0 0
\(865\) −11.3685 + 19.6909i −0.386541 + 0.669509i
\(866\) 0 0
\(867\) −54.0652 −1.83615
\(868\) 0 0
\(869\) −71.8152 −2.43616
\(870\) 0 0
\(871\) −0.711714 + 1.23273i −0.0241155 + 0.0417693i
\(872\) 0 0
\(873\) −81.8948 + 47.2820i −2.77172 + 1.60025i
\(874\) 0 0
\(875\) −22.2324 + 14.5024i −0.751592 + 0.490269i
\(876\) 0 0
\(877\) −14.4711 25.0647i −0.488654 0.846374i 0.511261 0.859426i \(-0.329179\pi\)
−0.999915 + 0.0130521i \(0.995845\pi\)
\(878\) 0 0
\(879\) 62.7229 + 36.2131i 2.11559 + 1.22144i
\(880\) 0 0
\(881\) 28.9616i 0.975740i 0.872916 + 0.487870i \(0.162226\pi\)
−0.872916 + 0.487870i \(0.837774\pi\)
\(882\) 0 0
\(883\) 21.5875i 0.726478i −0.931696 0.363239i \(-0.881671\pi\)
0.931696 0.363239i \(-0.118329\pi\)
\(884\) 0 0
\(885\) −3.07929 1.77783i −0.103509 0.0597611i
\(886\) 0 0
\(887\) 8.47280 + 14.6753i 0.284489 + 0.492749i 0.972485 0.232965i \(-0.0748428\pi\)
−0.687996 + 0.725714i \(0.741509\pi\)
\(888\) 0 0
\(889\) −19.3964 + 12.6524i −0.650535 + 0.424349i
\(890\) 0 0
\(891\) −86.2448 + 49.7934i −2.88931 + 1.66814i
\(892\) 0 0
\(893\) −1.25670 + 2.17667i −0.0420539 + 0.0728396i
\(894\) 0 0
\(895\) −13.6150 −0.455099
\(896\) 0 0
\(897\) 31.4527 1.05017
\(898\) 0 0
\(899\) 8.23656 14.2661i 0.274705 0.475802i
\(900\) 0 0
\(901\) −0.486096 + 0.280648i −0.0161942 + 0.00934972i
\(902\) 0 0
\(903\) −0.663192 12.1797i −0.0220697 0.405316i
\(904\) 0 0
\(905\) −2.47059 4.27919i −0.0821252 0.142245i
\(906\) 0 0
\(907\) −32.8183 18.9477i −1.08971 0.629147i −0.156214 0.987723i \(-0.549929\pi\)
−0.933500 + 0.358576i \(0.883262\pi\)
\(908\) 0 0
\(909\) 53.6561i 1.77966i
\(910\) 0 0
\(911\) 21.3300i 0.706694i 0.935492 + 0.353347i \(0.114957\pi\)
−0.935492 + 0.353347i \(0.885043\pi\)
\(912\) 0 0
\(913\) 39.5028 + 22.8070i 1.30735 + 0.754801i
\(914\) 0 0
\(915\) −21.1906 36.7031i −0.700538 1.21337i
\(916\) 0 0
\(917\) −8.93274 + 17.6201i −0.294985 + 0.581867i
\(918\) 0 0
\(919\) −6.92890 + 4.00040i −0.228563 + 0.131961i −0.609909 0.792471i \(-0.708794\pi\)
0.381346 + 0.924432i \(0.375461\pi\)
\(920\) 0 0
\(921\) −0.596175 + 1.03261i −0.0196446 + 0.0340255i
\(922\) 0 0
\(923\) −9.51909 −0.313325
\(924\) 0 0
\(925\) −26.5770 −0.873846
\(926\) 0 0
\(927\) −33.8366 + 58.6067i −1.11134 + 1.92489i
\(928\) 0 0
\(929\) 40.3773 23.3118i 1.32474 0.764836i 0.340255 0.940333i \(-0.389486\pi\)
0.984480 + 0.175497i \(0.0561532\pi\)
\(930\) 0 0
\(931\) 2.17241 0.237281i 0.0711978 0.00777657i
\(932\) 0 0
\(933\) −26.3529 45.6445i −0.862754 1.49433i
\(934\) 0 0
\(935\) 1.58800 + 0.916830i 0.0519330 + 0.0299835i
\(936\) 0 0
\(937\) 2.78754i 0.0910651i 0.998963 + 0.0455325i \(0.0144985\pi\)
−0.998963 + 0.0455325i \(0.985502\pi\)
\(938\) 0 0
\(939\) 48.0289i 1.56736i
\(940\) 0 0
\(941\) 7.53957 + 4.35297i 0.245783 + 0.141903i 0.617832 0.786310i \(-0.288011\pi\)
−0.372049 + 0.928213i \(0.621345\pi\)
\(942\) 0 0
\(943\) 4.90960 + 8.50367i 0.159879 + 0.276918i
\(944\) 0 0
\(945\) −37.3225 18.9211i −1.21410 0.615503i
\(946\) 0 0
\(947\) −0.851519 + 0.491624i −0.0276706 + 0.0159756i −0.513772 0.857927i \(-0.671752\pi\)
0.486101 + 0.873903i \(0.338419\pi\)
\(948\) 0 0
\(949\) −9.11547 + 15.7885i −0.295901 + 0.512515i
\(950\) 0 0
\(951\) 79.8380 2.58892
\(952\) 0 0
\(953\) −27.9025 −0.903850 −0.451925 0.892056i \(-0.649263\pi\)
−0.451925 + 0.892056i \(0.649263\pi\)
\(954\) 0 0
\(955\) 3.94530 6.83346i 0.127667 0.221125i
\(956\) 0 0
\(957\) 91.4528 52.8003i 2.95625 1.70679i
\(958\) 0 0
\(959\) −19.5791 + 1.06609i −0.632242 + 0.0344259i
\(960\) 0 0
\(961\) 12.9260 + 22.3885i 0.416969 + 0.722211i
\(962\) 0 0
\(963\) 43.2835 + 24.9898i 1.39479 + 0.805284i
\(964\) 0 0
\(965\) 21.7833i 0.701228i
\(966\) 0 0
\(967\) 36.4955i 1.17362i −0.809726 0.586808i \(-0.800384\pi\)
0.809726 0.586808i \(-0.199616\pi\)
\(968\) 0 0
\(969\) 0.301760 + 0.174221i 0.00969394 + 0.00559680i
\(970\) 0 0
\(971\) 9.88987 + 17.1298i 0.317381 + 0.549720i 0.979941 0.199289i \(-0.0638632\pi\)
−0.662560 + 0.749009i \(0.730530\pi\)
\(972\) 0 0
\(973\) 30.8786 + 47.3375i 0.989922 + 1.51757i
\(974\) 0 0
\(975\) −21.1050 + 12.1850i −0.675902 + 0.390232i
\(976\) 0 0
\(977\) −11.1375 + 19.2908i −0.356321 + 0.617167i −0.987343 0.158598i \(-0.949302\pi\)
0.631022 + 0.775765i \(0.282636\pi\)
\(978\) 0 0
\(979\) −37.2836 −1.19159
\(980\) 0 0
\(981\) 12.0827 0.385771
\(982\) 0 0
\(983\) 9.09947 15.7607i 0.290228 0.502690i −0.683635 0.729824i \(-0.739602\pi\)
0.973864 + 0.227134i \(0.0729355\pi\)
\(984\) 0 0
\(985\) −14.5206 + 8.38345i −0.462663 + 0.267119i
\(986\) 0 0
\(987\) 37.2771 + 57.1466i 1.18654 + 1.81900i
\(988\) 0 0
\(989\) 3.39648 + 5.88287i 0.108002 + 0.187065i
\(990\) 0 0
\(991\) −7.96425 4.59816i −0.252993 0.146065i 0.368141 0.929770i \(-0.379994\pi\)
−0.621134 + 0.783705i \(0.713328\pi\)
\(992\) 0 0
\(993\) 94.5199i 2.99950i
\(994\) 0 0
\(995\) 16.8740i 0.534940i
\(996\) 0 0
\(997\) −10.4757 6.04813i −0.331768 0.191546i 0.324858 0.945763i \(-0.394683\pi\)
−0.656626 + 0.754217i \(0.728017\pi\)
\(998\) 0 0
\(999\) −49.5904 85.8931i −1.56897 2.71754i
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 224.2.p.a.31.8 yes 16
3.2 odd 2 2016.2.cs.b.703.5 16
4.3 odd 2 inner 224.2.p.a.31.1 16
7.2 even 3 1568.2.p.b.607.8 16
7.3 odd 6 1568.2.f.b.1567.16 16
7.4 even 3 1568.2.f.b.1567.1 16
7.5 odd 6 inner 224.2.p.a.159.1 yes 16
7.6 odd 2 1568.2.p.b.31.1 16
8.3 odd 2 448.2.p.e.255.8 16
8.5 even 2 448.2.p.e.255.1 16
12.11 even 2 2016.2.cs.b.703.6 16
21.5 even 6 2016.2.cs.b.1279.6 16
28.3 even 6 1568.2.f.b.1567.2 16
28.11 odd 6 1568.2.f.b.1567.15 16
28.19 even 6 inner 224.2.p.a.159.8 yes 16
28.23 odd 6 1568.2.p.b.607.1 16
28.27 even 2 1568.2.p.b.31.8 16
56.3 even 6 3136.2.f.j.3135.15 16
56.5 odd 6 448.2.p.e.383.8 16
56.11 odd 6 3136.2.f.j.3135.2 16
56.19 even 6 448.2.p.e.383.1 16
56.45 odd 6 3136.2.f.j.3135.1 16
56.53 even 6 3136.2.f.j.3135.16 16
84.47 odd 6 2016.2.cs.b.1279.5 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
224.2.p.a.31.1 16 4.3 odd 2 inner
224.2.p.a.31.8 yes 16 1.1 even 1 trivial
224.2.p.a.159.1 yes 16 7.5 odd 6 inner
224.2.p.a.159.8 yes 16 28.19 even 6 inner
448.2.p.e.255.1 16 8.5 even 2
448.2.p.e.255.8 16 8.3 odd 2
448.2.p.e.383.1 16 56.19 even 6
448.2.p.e.383.8 16 56.5 odd 6
1568.2.f.b.1567.1 16 7.4 even 3
1568.2.f.b.1567.2 16 28.3 even 6
1568.2.f.b.1567.15 16 28.11 odd 6
1568.2.f.b.1567.16 16 7.3 odd 6
1568.2.p.b.31.1 16 7.6 odd 2
1568.2.p.b.31.8 16 28.27 even 2
1568.2.p.b.607.1 16 28.23 odd 6
1568.2.p.b.607.8 16 7.2 even 3
2016.2.cs.b.703.5 16 3.2 odd 2
2016.2.cs.b.703.6 16 12.11 even 2
2016.2.cs.b.1279.5 16 84.47 odd 6
2016.2.cs.b.1279.6 16 21.5 even 6
3136.2.f.j.3135.1 16 56.45 odd 6
3136.2.f.j.3135.2 16 56.11 odd 6
3136.2.f.j.3135.15 16 56.3 even 6
3136.2.f.j.3135.16 16 56.53 even 6