Properties

Label 2352.2.h.p.2255.1
Level $2352$
Weight $2$
Character 2352.2255
Analytic conductor $18.781$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $8$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2352,2,Mod(2255,2352)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2352, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 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("2352.2255");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2352 = 2^{4} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2352.h (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: \(18.7808145554\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 12x^{14} + 97x^{12} - 432x^{10} + 1392x^{8} - 2502x^{6} + 3181x^{4} - 1650x^{2} + 625 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{16}\cdot 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 2255.1
Root \(2.01367 + 1.16260i\) of defining polynomial
Character \(\chi\) \(=\) 2352.2255
Dual form 2352.2.h.p.2255.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.70166 - 0.323042i) q^{3} -1.55481i q^{5} +(2.79129 + 1.09941i) q^{9} +O(q^{10})\) \(q+(-1.70166 - 0.323042i) q^{3} -1.55481i q^{5} +(2.79129 + 1.09941i) q^{9} -2.53326 q^{13} +(-0.502268 + 2.64575i) q^{15} -4.33991i q^{17} +6.83723i q^{19} +3.80848 q^{23} +2.58258 q^{25} +(-4.39466 - 2.77253i) q^{27} -8.33639i q^{29} -4.89898i q^{31} +1.58258 q^{37} +(4.31075 + 0.818350i) q^{39} -7.44953i q^{41} +6.20520i q^{43} +(1.70938 - 4.33991i) q^{45} +5.38601 q^{47} +(-1.40197 + 7.38505i) q^{51} +10.5352i q^{53} +(2.20871 - 11.6346i) q^{57} -10.2100 q^{59} +8.19012 q^{61} +3.93874i q^{65} -3.46410i q^{67} +(-6.48074 - 1.23030i) q^{69} -14.4391 q^{71} -11.5473 q^{73} +(-4.39466 - 0.834280i) q^{75} -6.92820i q^{79} +(6.58258 + 6.13756i) q^{81} -4.82395 q^{83} -6.74773 q^{85} +(-2.69300 + 14.1857i) q^{87} -4.33991i q^{89} +(-1.58258 + 8.33639i) q^{93} +10.6306 q^{95} +9.89949 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 8 q^{9} - 32 q^{25} - 48 q^{37} + 72 q^{57} + 32 q^{81} + 112 q^{85} + 48 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2352\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(1471\) \(1765\) \(2257\)
\(\chi(n)\) \(-1\) \(-1\) \(1\) \(1\)

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
\(3\) −1.70166 0.323042i −0.982453 0.186508i
\(4\) 0 0
\(5\) 1.55481i 0.695331i −0.937619 0.347665i \(-0.886975\pi\)
0.937619 0.347665i \(-0.113025\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 2.79129 + 1.09941i 0.930429 + 0.366471i
\(10\) 0 0
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) −2.53326 −0.702601 −0.351300 0.936263i \(-0.614260\pi\)
−0.351300 + 0.936263i \(0.614260\pi\)
\(14\) 0 0
\(15\) −0.502268 + 2.64575i −0.129685 + 0.683130i
\(16\) 0 0
\(17\) 4.33991i 1.05258i −0.850304 0.526292i \(-0.823582\pi\)
0.850304 0.526292i \(-0.176418\pi\)
\(18\) 0 0
\(19\) 6.83723i 1.56857i 0.620402 + 0.784284i \(0.286970\pi\)
−0.620402 + 0.784284i \(0.713030\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 3.80848 0.794124 0.397062 0.917792i \(-0.370030\pi\)
0.397062 + 0.917792i \(0.370030\pi\)
\(24\) 0 0
\(25\) 2.58258 0.516515
\(26\) 0 0
\(27\) −4.39466 2.77253i −0.845753 0.533574i
\(28\) 0 0
\(29\) 8.33639i 1.54803i −0.633168 0.774015i \(-0.718246\pi\)
0.633168 0.774015i \(-0.281754\pi\)
\(30\) 0 0
\(31\) 4.89898i 0.879883i −0.898027 0.439941i \(-0.854999\pi\)
0.898027 0.439941i \(-0.145001\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 1.58258 0.260174 0.130087 0.991503i \(-0.458474\pi\)
0.130087 + 0.991503i \(0.458474\pi\)
\(38\) 0 0
\(39\) 4.31075 + 0.818350i 0.690273 + 0.131041i
\(40\) 0 0
\(41\) 7.44953i 1.16342i −0.813396 0.581710i \(-0.802384\pi\)
0.813396 0.581710i \(-0.197616\pi\)
\(42\) 0 0
\(43\) 6.20520i 0.946285i 0.880986 + 0.473142i \(0.156880\pi\)
−0.880986 + 0.473142i \(0.843120\pi\)
\(44\) 0 0
\(45\) 1.70938 4.33991i 0.254819 0.646956i
\(46\) 0 0
\(47\) 5.38601 0.785630 0.392815 0.919617i \(-0.371501\pi\)
0.392815 + 0.919617i \(0.371501\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) −1.40197 + 7.38505i −0.196316 + 1.03411i
\(52\) 0 0
\(53\) 10.5352i 1.44712i 0.690259 + 0.723562i \(0.257496\pi\)
−0.690259 + 0.723562i \(0.742504\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 2.20871 11.6346i 0.292551 1.54105i
\(58\) 0 0
\(59\) −10.2100 −1.32922 −0.664611 0.747189i \(-0.731403\pi\)
−0.664611 + 0.747189i \(0.731403\pi\)
\(60\) 0 0
\(61\) 8.19012 1.04864 0.524319 0.851522i \(-0.324320\pi\)
0.524319 + 0.851522i \(0.324320\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 3.93874i 0.488540i
\(66\) 0 0
\(67\) 3.46410i 0.423207i −0.977356 0.211604i \(-0.932131\pi\)
0.977356 0.211604i \(-0.0678686\pi\)
\(68\) 0 0
\(69\) −6.48074 1.23030i −0.780189 0.148111i
\(70\) 0 0
\(71\) −14.4391 −1.71360 −0.856800 0.515648i \(-0.827551\pi\)
−0.856800 + 0.515648i \(0.827551\pi\)
\(72\) 0 0
\(73\) −11.5473 −1.35151 −0.675753 0.737128i \(-0.736181\pi\)
−0.675753 + 0.737128i \(0.736181\pi\)
\(74\) 0 0
\(75\) −4.39466 0.834280i −0.507452 0.0963344i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 6.92820i 0.779484i −0.920924 0.389742i \(-0.872564\pi\)
0.920924 0.389742i \(-0.127436\pi\)
\(80\) 0 0
\(81\) 6.58258 + 6.13756i 0.731397 + 0.681952i
\(82\) 0 0
\(83\) −4.82395 −0.529497 −0.264748 0.964318i \(-0.585289\pi\)
−0.264748 + 0.964318i \(0.585289\pi\)
\(84\) 0 0
\(85\) −6.74773 −0.731894
\(86\) 0 0
\(87\) −2.69300 + 14.1857i −0.288720 + 1.52087i
\(88\) 0 0
\(89\) 4.33991i 0.460030i −0.973187 0.230015i \(-0.926122\pi\)
0.973187 0.230015i \(-0.0738775\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −1.58258 + 8.33639i −0.164105 + 0.864444i
\(94\) 0 0
\(95\) 10.6306 1.09067
\(96\) 0 0
\(97\) 9.89949 1.00514 0.502571 0.864536i \(-0.332388\pi\)
0.502571 + 0.864536i \(0.332388\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 19.5635i 1.94664i −0.229455 0.973319i \(-0.573694\pi\)
0.229455 0.973319i \(-0.426306\pi\)
\(102\) 0 0
\(103\) 3.87650i 0.381963i −0.981594 0.190982i \(-0.938833\pi\)
0.981594 0.190982i \(-0.0611671\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 14.4391 1.39588 0.697938 0.716158i \(-0.254101\pi\)
0.697938 + 0.716158i \(0.254101\pi\)
\(108\) 0 0
\(109\) −13.5826 −1.30097 −0.650487 0.759517i \(-0.725435\pi\)
−0.650487 + 0.759517i \(0.725435\pi\)
\(110\) 0 0
\(111\) −2.69300 0.511238i −0.255609 0.0485246i
\(112\) 0 0
\(113\) 14.4740i 1.36160i −0.732471 0.680798i \(-0.761633\pi\)
0.732471 0.680798i \(-0.238367\pi\)
\(114\) 0 0
\(115\) 5.92146i 0.552179i
\(116\) 0 0
\(117\) −7.07107 2.78511i −0.653720 0.257483i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −11.0000 −1.00000
\(122\) 0 0
\(123\) −2.40651 + 12.6766i −0.216988 + 1.14301i
\(124\) 0 0
\(125\) 11.7894i 1.05448i
\(126\) 0 0
\(127\) 10.3923i 0.922168i −0.887357 0.461084i \(-0.847461\pi\)
0.887357 0.461084i \(-0.152539\pi\)
\(128\) 0 0
\(129\) 2.00454 10.5591i 0.176490 0.929681i
\(130\) 0 0
\(131\) −15.5960 −1.36263 −0.681313 0.731992i \(-0.738591\pi\)
−0.681313 + 0.731992i \(0.738591\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) −4.31075 + 6.83285i −0.371010 + 0.588078i
\(136\) 0 0
\(137\) 1.73991i 0.148650i 0.997234 + 0.0743251i \(0.0236803\pi\)
−0.997234 + 0.0743251i \(0.976320\pi\)
\(138\) 0 0
\(139\) 11.7362i 0.995452i 0.867334 + 0.497726i \(0.165832\pi\)
−0.867334 + 0.497726i \(0.834168\pi\)
\(140\) 0 0
\(141\) −9.16515 1.73991i −0.771845 0.146527i
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) −12.9615 −1.07639
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 10.5352i 0.863079i 0.902094 + 0.431540i \(0.142029\pi\)
−0.902094 + 0.431540i \(0.857971\pi\)
\(150\) 0 0
\(151\) 15.8745i 1.29185i −0.763401 0.645925i \(-0.776472\pi\)
0.763401 0.645925i \(-0.223528\pi\)
\(152\) 0 0
\(153\) 4.77136 12.1139i 0.385742 0.979355i
\(154\) 0 0
\(155\) −7.61697 −0.611809
\(156\) 0 0
\(157\) −8.19012 −0.653643 −0.326821 0.945086i \(-0.605978\pi\)
−0.326821 + 0.945086i \(0.605978\pi\)
\(158\) 0 0
\(159\) 3.40332 17.9274i 0.269901 1.42173i
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 20.0616i 1.57135i −0.618642 0.785673i \(-0.712317\pi\)
0.618642 0.785673i \(-0.287683\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 0 0
\(169\) −6.58258 −0.506352
\(170\) 0 0
\(171\) −7.51695 + 19.0847i −0.574836 + 1.45944i
\(172\) 0 0
\(173\) 1.55481i 0.118210i −0.998252 0.0591049i \(-0.981175\pi\)
0.998252 0.0591049i \(-0.0188246\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 17.3739 + 3.29824i 1.30590 + 0.247911i
\(178\) 0 0
\(179\) −14.4391 −1.07923 −0.539613 0.841913i \(-0.681429\pi\)
−0.539613 + 0.841913i \(0.681429\pi\)
\(180\) 0 0
\(181\) −10.4282 −0.775123 −0.387562 0.921844i \(-0.626683\pi\)
−0.387562 + 0.921844i \(0.626683\pi\)
\(182\) 0 0
\(183\) −13.9368 2.64575i −1.03024 0.195580i
\(184\) 0 0
\(185\) 2.46060i 0.180907i
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −22.0560 −1.59592 −0.797959 0.602712i \(-0.794087\pi\)
−0.797959 + 0.602712i \(0.794087\pi\)
\(192\) 0 0
\(193\) 8.74773 0.629675 0.314838 0.949146i \(-0.398050\pi\)
0.314838 + 0.949146i \(0.398050\pi\)
\(194\) 0 0
\(195\) 1.27238 6.70239i 0.0911168 0.479968i
\(196\) 0 0
\(197\) 10.5352i 0.750604i −0.926903 0.375302i \(-0.877539\pi\)
0.926903 0.375302i \(-0.122461\pi\)
\(198\) 0 0
\(199\) 9.79796i 0.694559i −0.937762 0.347279i \(-0.887106\pi\)
0.937762 0.347279i \(-0.112894\pi\)
\(200\) 0 0
\(201\) −1.11905 + 5.89472i −0.0789317 + 0.415782i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −11.5826 −0.808962
\(206\) 0 0
\(207\) 10.6306 + 4.18710i 0.738876 + 0.291024i
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 15.8745i 1.09285i −0.837509 0.546423i \(-0.815989\pi\)
0.837509 0.546423i \(-0.184011\pi\)
\(212\) 0 0
\(213\) 24.5704 + 4.66442i 1.68353 + 0.319601i
\(214\) 0 0
\(215\) 9.64789 0.657981
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 19.6495 + 3.73025i 1.32779 + 0.252067i
\(220\) 0 0
\(221\) 10.9941i 0.739546i
\(222\) 0 0
\(223\) 23.4724i 1.57183i 0.618335 + 0.785915i \(0.287808\pi\)
−0.618335 + 0.785915i \(0.712192\pi\)
\(224\) 0 0
\(225\) 7.20871 + 2.83932i 0.480581 + 0.189288i
\(226\) 0 0
\(227\) −4.82395 −0.320177 −0.160088 0.987103i \(-0.551178\pi\)
−0.160088 + 0.987103i \(0.551178\pi\)
\(228\) 0 0
\(229\) 8.78044 0.580228 0.290114 0.956992i \(-0.406307\pi\)
0.290114 + 0.956992i \(0.406307\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 1.73991i 0.113985i −0.998375 0.0569925i \(-0.981849\pi\)
0.998375 0.0569925i \(-0.0181511\pi\)
\(234\) 0 0
\(235\) 8.37420i 0.546273i
\(236\) 0 0
\(237\) −2.23810 + 11.7894i −0.145380 + 0.765806i
\(238\) 0 0
\(239\) 3.80848 0.246350 0.123175 0.992385i \(-0.460692\pi\)
0.123175 + 0.992385i \(0.460692\pi\)
\(240\) 0 0
\(241\) 3.06199 0.197240 0.0986199 0.995125i \(-0.468557\pi\)
0.0986199 + 0.995125i \(0.468557\pi\)
\(242\) 0 0
\(243\) −9.21861 12.5705i −0.591374 0.806397i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 17.3205i 1.10208i
\(248\) 0 0
\(249\) 8.20871 + 1.55834i 0.520206 + 0.0987556i
\(250\) 0 0
\(251\) −0.562063 −0.0354771 −0.0177385 0.999843i \(-0.505647\pi\)
−0.0177385 + 0.999843i \(0.505647\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 11.4823 + 2.17980i 0.719052 + 0.136504i
\(256\) 0 0
\(257\) 13.6688i 0.852633i 0.904574 + 0.426317i \(0.140189\pi\)
−0.904574 + 0.426317i \(0.859811\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 9.16515 23.2693i 0.567309 1.44033i
\(262\) 0 0
\(263\) 29.6730 1.82971 0.914857 0.403777i \(-0.132303\pi\)
0.914857 + 0.403777i \(0.132303\pi\)
\(264\) 0 0
\(265\) 16.3802 1.00623
\(266\) 0 0
\(267\) −1.40197 + 7.38505i −0.0857994 + 0.451958i
\(268\) 0 0
\(269\) 1.55481i 0.0947982i 0.998876 + 0.0473991i \(0.0150933\pi\)
−0.998876 + 0.0473991i \(0.984907\pi\)
\(270\) 0 0
\(271\) 19.5959i 1.19037i 0.803590 + 0.595184i \(0.202921\pi\)
−0.803590 + 0.595184i \(0.797079\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 2.00000 0.120168 0.0600842 0.998193i \(-0.480863\pi\)
0.0600842 + 0.998193i \(0.480863\pi\)
\(278\) 0 0
\(279\) 5.38601 13.6745i 0.322452 0.818669i
\(280\) 0 0
\(281\) 14.9329i 0.890821i 0.895326 + 0.445410i \(0.146942\pi\)
−0.895326 + 0.445410i \(0.853058\pi\)
\(282\) 0 0
\(283\) 26.4331i 1.57129i −0.618679 0.785644i \(-0.712332\pi\)
0.618679 0.785644i \(-0.287668\pi\)
\(284\) 0 0
\(285\) −18.0896 3.43412i −1.07154 0.203420i
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −1.83485 −0.107932
\(290\) 0 0
\(291\) −16.8456 3.19795i −0.987505 0.187467i
\(292\) 0 0
\(293\) 13.3442i 0.779579i 0.920904 + 0.389790i \(0.127452\pi\)
−0.920904 + 0.389790i \(0.872548\pi\)
\(294\) 0 0
\(295\) 15.8745i 0.924250i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −9.64789 −0.557952
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) −6.31982 + 33.2904i −0.363064 + 1.91248i
\(304\) 0 0
\(305\) 12.7341i 0.729150i
\(306\) 0 0
\(307\) 11.7362i 0.669821i −0.942250 0.334910i \(-0.891294\pi\)
0.942250 0.334910i \(-0.108706\pi\)
\(308\) 0 0
\(309\) −1.25227 + 6.59649i −0.0712393 + 0.375261i
\(310\) 0 0
\(311\) 15.0339 0.852494 0.426247 0.904607i \(-0.359835\pi\)
0.426247 + 0.904607i \(0.359835\pi\)
\(312\) 0 0
\(313\) 5.89041 0.332946 0.166473 0.986046i \(-0.446762\pi\)
0.166473 + 0.986046i \(0.446762\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 2.65775i 0.149274i 0.997211 + 0.0746371i \(0.0237798\pi\)
−0.997211 + 0.0746371i \(0.976220\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) −24.5704 4.66442i −1.37138 0.260343i
\(322\) 0 0
\(323\) 29.6730 1.65105
\(324\) 0 0
\(325\) −6.54234 −0.362904
\(326\) 0 0
\(327\) 23.1129 + 4.38774i 1.27815 + 0.242643i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 13.1334i 0.721877i −0.932590 0.360939i \(-0.882456\pi\)
0.932590 0.360939i \(-0.117544\pi\)
\(332\) 0 0
\(333\) 4.41742 + 1.73991i 0.242073 + 0.0953463i
\(334\) 0 0
\(335\) −5.38601 −0.294269
\(336\) 0 0
\(337\) −1.25227 −0.0682157 −0.0341078 0.999418i \(-0.510859\pi\)
−0.0341078 + 0.999418i \(0.510859\pi\)
\(338\) 0 0
\(339\) −4.67569 + 24.6297i −0.253949 + 1.33770i
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) −1.91288 + 10.0763i −0.102986 + 0.542490i
\(346\) 0 0
\(347\) −36.4951 −1.95916 −0.979579 0.201058i \(-0.935562\pi\)
−0.979579 + 0.201058i \(0.935562\pi\)
\(348\) 0 0
\(349\) 4.77136 0.255405 0.127703 0.991813i \(-0.459240\pi\)
0.127703 + 0.991813i \(0.459240\pi\)
\(350\) 0 0
\(351\) 11.1328 + 7.02355i 0.594227 + 0.374890i
\(352\) 0 0
\(353\) 27.9188i 1.48597i 0.669309 + 0.742984i \(0.266590\pi\)
−0.669309 + 0.742984i \(0.733410\pi\)
\(354\) 0 0
\(355\) 22.4499i 1.19152i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −3.80848 −0.201004 −0.100502 0.994937i \(-0.532045\pi\)
−0.100502 + 0.994937i \(0.532045\pi\)
\(360\) 0 0
\(361\) −27.7477 −1.46041
\(362\) 0 0
\(363\) 18.7183 + 3.55346i 0.982453 + 0.186508i
\(364\) 0 0
\(365\) 17.9538i 0.939743i
\(366\) 0 0
\(367\) 23.4724i 1.22525i −0.790374 0.612625i \(-0.790114\pi\)
0.790374 0.612625i \(-0.209886\pi\)
\(368\) 0 0
\(369\) 8.19012 20.7938i 0.426361 1.08248i
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 10.0000 0.517780 0.258890 0.965907i \(-0.416643\pi\)
0.258890 + 0.965907i \(0.416643\pi\)
\(374\) 0 0
\(375\) −3.80848 + 20.0616i −0.196669 + 1.03598i
\(376\) 0 0
\(377\) 21.1183i 1.08765i
\(378\) 0 0
\(379\) 6.20520i 0.318740i −0.987219 0.159370i \(-0.949054\pi\)
0.987219 0.159370i \(-0.0509463\pi\)
\(380\) 0 0
\(381\) −3.35715 + 17.6842i −0.171992 + 0.905987i
\(382\) 0 0
\(383\) 15.0339 0.768196 0.384098 0.923292i \(-0.374512\pi\)
0.384098 + 0.923292i \(0.374512\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −6.82209 + 17.3205i −0.346786 + 0.880451i
\(388\) 0 0
\(389\) 38.2022i 1.93693i −0.249157 0.968463i \(-0.580154\pi\)
0.249157 0.968463i \(-0.419846\pi\)
\(390\) 0 0
\(391\) 16.5285i 0.835882i
\(392\) 0 0
\(393\) 26.5390 + 5.03815i 1.33872 + 0.254141i
\(394\) 0 0
\(395\) −10.7720 −0.541999
\(396\) 0 0
\(397\) 38.1222 1.91330 0.956648 0.291246i \(-0.0940698\pi\)
0.956648 + 0.291246i \(0.0940698\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 21.5294i 1.07513i −0.843224 0.537563i \(-0.819345\pi\)
0.843224 0.537563i \(-0.180655\pi\)
\(402\) 0 0
\(403\) 12.4104i 0.618206i
\(404\) 0 0
\(405\) 9.54273 10.2346i 0.474182 0.508563i
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −17.7944 −0.879879 −0.439939 0.898027i \(-0.645000\pi\)
−0.439939 + 0.898027i \(0.645000\pi\)
\(410\) 0 0
\(411\) 0.562063 2.96073i 0.0277245 0.146042i
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 7.50030i 0.368175i
\(416\) 0 0
\(417\) 3.79129 19.9710i 0.185660 0.977986i
\(418\) 0 0
\(419\) −15.5960 −0.761913 −0.380956 0.924593i \(-0.624405\pi\)
−0.380956 + 0.924593i \(0.624405\pi\)
\(420\) 0 0
\(421\) 35.4955 1.72994 0.864971 0.501821i \(-0.167337\pi\)
0.864971 + 0.501821i \(0.167337\pi\)
\(422\) 0 0
\(423\) 15.0339 + 5.92146i 0.730973 + 0.287911i
\(424\) 0 0
\(425\) 11.2082i 0.543675i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 17.4527 0.840665 0.420333 0.907370i \(-0.361913\pi\)
0.420333 + 0.907370i \(0.361913\pi\)
\(432\) 0 0
\(433\) 0.823886 0.0395935 0.0197967 0.999804i \(-0.493698\pi\)
0.0197967 + 0.999804i \(0.493698\pi\)
\(434\) 0 0
\(435\) 22.0560 + 4.18710i 1.05751 + 0.200756i
\(436\) 0 0
\(437\) 26.0395i 1.24564i
\(438\) 0 0
\(439\) 14.6969i 0.701447i 0.936479 + 0.350723i \(0.114064\pi\)
−0.936479 + 0.350723i \(0.885936\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 6.82209 0.324127 0.162064 0.986780i \(-0.448185\pi\)
0.162064 + 0.986780i \(0.448185\pi\)
\(444\) 0 0
\(445\) −6.74773 −0.319873
\(446\) 0 0
\(447\) 3.40332 17.9274i 0.160971 0.847935i
\(448\) 0 0
\(449\) 28.9479i 1.36614i 0.730355 + 0.683068i \(0.239355\pi\)
−0.730355 + 0.683068i \(0.760645\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) −5.12813 + 27.0130i −0.240941 + 1.26918i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −24.7477 −1.15765 −0.578825 0.815452i \(-0.696489\pi\)
−0.578825 + 0.815452i \(0.696489\pi\)
\(458\) 0 0
\(459\) −12.0325 + 19.0725i −0.561631 + 0.890226i
\(460\) 0 0
\(461\) 10.2346i 0.476674i 0.971182 + 0.238337i \(0.0766023\pi\)
−0.971182 + 0.238337i \(0.923398\pi\)
\(462\) 0 0
\(463\) 26.2668i 1.22072i 0.792123 + 0.610361i \(0.208976\pi\)
−0.792123 + 0.610361i \(0.791024\pi\)
\(464\) 0 0
\(465\) 12.9615 + 2.46060i 0.601074 + 0.114108i
\(466\) 0 0
\(467\) −25.2439 −1.16815 −0.584073 0.811701i \(-0.698542\pi\)
−0.584073 + 0.811701i \(0.698542\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 13.9368 + 2.64575i 0.642173 + 0.121910i
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 17.6577i 0.810189i
\(476\) 0 0
\(477\) −11.5826 + 29.4068i −0.530330 + 1.34645i
\(478\) 0 0
\(479\) 31.1919 1.42520 0.712598 0.701573i \(-0.247518\pi\)
0.712598 + 0.701573i \(0.247518\pi\)
\(480\) 0 0
\(481\) −4.00908 −0.182798
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 15.3918i 0.698906i
\(486\) 0 0
\(487\) 36.6591i 1.66118i −0.556882 0.830592i \(-0.688002\pi\)
0.556882 0.830592i \(-0.311998\pi\)
\(488\) 0 0
\(489\) −6.48074 + 34.1380i −0.293069 + 1.54377i
\(490\) 0 0
\(491\) 22.0560 0.995374 0.497687 0.867357i \(-0.334183\pi\)
0.497687 + 0.867357i \(0.334183\pi\)
\(492\) 0 0
\(493\) −36.1792 −1.62943
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 8.94630i 0.400492i 0.979746 + 0.200246i \(0.0641741\pi\)
−0.979746 + 0.200246i \(0.935826\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 31.1919 1.39078 0.695390 0.718633i \(-0.255232\pi\)
0.695390 + 0.718633i \(0.255232\pi\)
\(504\) 0 0
\(505\) −30.4174 −1.35356
\(506\) 0 0
\(507\) 11.2013 + 2.12645i 0.497467 + 0.0944389i
\(508\) 0 0
\(509\) 0.905793i 0.0401486i −0.999798 0.0200743i \(-0.993610\pi\)
0.999798 0.0200743i \(-0.00639027\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 18.9564 30.0473i 0.836947 1.32662i
\(514\) 0 0
\(515\) −6.02721 −0.265591
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) −0.502268 + 2.64575i −0.0220471 + 0.116136i
\(520\) 0 0
\(521\) 27.9188i 1.22314i 0.791189 + 0.611572i \(0.209463\pi\)
−0.791189 + 0.611572i \(0.790537\pi\)
\(522\) 0 0
\(523\) 0.915775i 0.0400440i −0.999800 0.0200220i \(-0.993626\pi\)
0.999800 0.0200220i \(-0.00637363\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −21.2611 −0.926150
\(528\) 0 0
\(529\) −8.49545 −0.369368
\(530\) 0 0
\(531\) −28.4989 11.2250i −1.23675 0.487122i
\(532\) 0 0
\(533\) 18.8716i 0.817420i
\(534\) 0 0
\(535\) 22.4499i 0.970596i
\(536\) 0 0
\(537\) 24.5704 + 4.66442i 1.06029 + 0.201285i
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −23.4955 −1.01015 −0.505074 0.863076i \(-0.668535\pi\)
−0.505074 + 0.863076i \(0.668535\pi\)
\(542\) 0 0
\(543\) 17.7453 + 3.36875i 0.761523 + 0.144567i
\(544\) 0 0
\(545\) 21.1183i 0.904608i
\(546\) 0 0
\(547\) 11.6874i 0.499717i −0.968282 0.249859i \(-0.919616\pi\)
0.968282 0.249859i \(-0.0803842\pi\)
\(548\) 0 0
\(549\) 22.8610 + 9.00433i 0.975683 + 0.384296i
\(550\) 0 0
\(551\) 56.9978 2.42819
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) −0.794877 + 4.18710i −0.0337406 + 0.177733i
\(556\) 0 0
\(557\) 2.65775i 0.112613i 0.998414 + 0.0563063i \(0.0179323\pi\)
−0.998414 + 0.0563063i \(0.982068\pi\)
\(558\) 0 0
\(559\) 15.7194i 0.664860i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −0.562063 −0.0236881 −0.0118441 0.999930i \(-0.503770\pi\)
−0.0118441 + 0.999930i \(0.503770\pi\)
\(564\) 0 0
\(565\) −22.5042 −0.946759
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 4.85658i 0.203598i 0.994805 + 0.101799i \(0.0324599\pi\)
−0.994805 + 0.101799i \(0.967540\pi\)
\(570\) 0 0
\(571\) 0.723000i 0.0302566i −0.999886 0.0151283i \(-0.995184\pi\)
0.999886 0.0151283i \(-0.00481567\pi\)
\(572\) 0 0
\(573\) 37.5318 + 7.12502i 1.56791 + 0.297652i
\(574\) 0 0
\(575\) 9.83570 0.410177
\(576\) 0 0
\(577\) 43.7174 1.81998 0.909990 0.414631i \(-0.136089\pi\)
0.909990 + 0.414631i \(0.136089\pi\)
\(578\) 0 0
\(579\) −14.8856 2.82588i −0.618627 0.117440i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) −4.33030 + 10.9941i −0.179036 + 0.454552i
\(586\) 0 0
\(587\) 36.0159 1.48653 0.743267 0.668995i \(-0.233275\pi\)
0.743267 + 0.668995i \(0.233275\pi\)
\(588\) 0 0
\(589\) 33.4955 1.38016
\(590\) 0 0
\(591\) −3.40332 + 17.9274i −0.139994 + 0.737433i
\(592\) 0 0
\(593\) 28.5678i 1.17314i 0.809899 + 0.586570i \(0.199522\pi\)
−0.809899 + 0.586570i \(0.800478\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −3.16515 + 16.6728i −0.129541 + 0.682372i
\(598\) 0 0
\(599\) −29.6730 −1.21241 −0.606203 0.795310i \(-0.707308\pi\)
−0.606203 + 0.795310i \(0.707308\pi\)
\(600\) 0 0
\(601\) −0.823886 −0.0336070 −0.0168035 0.999859i \(-0.505349\pi\)
−0.0168035 + 0.999859i \(0.505349\pi\)
\(602\) 0 0
\(603\) 3.80848 9.66930i 0.155093 0.393765i
\(604\) 0 0
\(605\) 17.1029i 0.695331i
\(606\) 0 0
\(607\) 18.5734i 0.753873i −0.926239 0.376936i \(-0.876978\pi\)
0.926239 0.376936i \(-0.123022\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −13.6442 −0.551984
\(612\) 0 0
\(613\) 38.2432 1.54463 0.772314 0.635241i \(-0.219099\pi\)
0.772314 + 0.635241i \(0.219099\pi\)
\(614\) 0 0
\(615\) 19.7096 + 3.74166i 0.794768 + 0.150878i
\(616\) 0 0
\(617\) 21.5294i 0.866740i 0.901216 + 0.433370i \(0.142676\pi\)
−0.901216 + 0.433370i \(0.857324\pi\)
\(618\) 0 0
\(619\) 7.85971i 0.315908i −0.987446 0.157954i \(-0.949510\pi\)
0.987446 0.157954i \(-0.0504898\pi\)
\(620\) 0 0
\(621\) −16.7370 10.5591i −0.671633 0.423724i
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −5.41742 −0.216697
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 6.86824i 0.273855i
\(630\) 0 0
\(631\) 40.1232i 1.59728i −0.601809 0.798640i \(-0.705553\pi\)
0.601809 0.798640i \(-0.294447\pi\)
\(632\) 0 0
\(633\) −5.12813 + 27.0130i −0.203825 + 1.07367i
\(634\) 0 0
\(635\) −16.1580 −0.641212
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) −40.3036 15.8745i −1.59438 0.627986i
\(640\) 0 0
\(641\) 4.85658i 0.191823i 0.995390 + 0.0959117i \(0.0305766\pi\)
−0.995390 + 0.0959117i \(0.969423\pi\)
\(642\) 0 0
\(643\) 39.0851i 1.54137i 0.637218 + 0.770684i \(0.280085\pi\)
−0.637218 + 0.770684i \(0.719915\pi\)
\(644\) 0 0
\(645\) −16.4174 3.11667i −0.646435 0.122719i
\(646\) 0 0
\(647\) 25.8059 1.01454 0.507268 0.861789i \(-0.330656\pi\)
0.507268 + 0.861789i \(0.330656\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 25.0092i 0.978685i −0.872092 0.489342i \(-0.837237\pi\)
0.872092 0.489342i \(-0.162763\pi\)
\(654\) 0 0
\(655\) 24.2487i 0.947476i
\(656\) 0 0
\(657\) −32.2317 12.6952i −1.25748 0.495288i
\(658\) 0 0
\(659\) 13.6442 0.531502 0.265751 0.964042i \(-0.414380\pi\)
0.265751 + 0.964042i \(0.414380\pi\)
\(660\) 0 0
\(661\) 9.24756 0.359689 0.179844 0.983695i \(-0.442441\pi\)
0.179844 + 0.983695i \(0.442441\pi\)
\(662\) 0 0
\(663\) 3.55157 18.7083i 0.137932 0.726570i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 31.7490i 1.22933i
\(668\) 0 0
\(669\) 7.58258 39.9421i 0.293159 1.54425i
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −15.1652 −0.584574 −0.292287 0.956331i \(-0.594416\pi\)
−0.292287 + 0.956331i \(0.594416\pi\)
\(674\) 0 0
\(675\) −11.3496 7.16027i −0.436844 0.275599i
\(676\) 0 0
\(677\) 0.905793i 0.0348124i 0.999849 + 0.0174062i \(0.00554085\pi\)
−0.999849 + 0.0174062i \(0.994459\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 8.20871 + 1.55834i 0.314559 + 0.0597156i
\(682\) 0 0
\(683\) −22.0560 −0.843950 −0.421975 0.906607i \(-0.638663\pi\)
−0.421975 + 0.906607i \(0.638663\pi\)
\(684\) 0 0
\(685\) 2.70522 0.103361
\(686\) 0 0
\(687\) −14.9413 2.83645i −0.570047 0.108217i
\(688\) 0 0
\(689\) 26.6885i 1.01675i
\(690\) 0 0
\(691\) 12.7587i 0.485363i −0.970106 0.242682i \(-0.921973\pi\)
0.970106 0.242682i \(-0.0780270\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 18.2475 0.692169
\(696\) 0 0
\(697\) −32.3303 −1.22460
\(698\) 0 0
\(699\) −0.562063 + 2.96073i −0.0212592 + 0.111985i
\(700\) 0 0
\(701\) 4.85658i 0.183431i 0.995785 + 0.0917153i \(0.0292350\pi\)
−0.995785 + 0.0917153i \(0.970765\pi\)
\(702\) 0 0
\(703\) 10.8204i 0.408100i
\(704\) 0 0
\(705\) −2.70522 + 14.2500i −0.101884 + 0.536688i
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 47.0780 1.76805 0.884026 0.467437i \(-0.154823\pi\)
0.884026 + 0.467437i \(0.154823\pi\)
\(710\) 0 0
\(711\) 7.61697 19.3386i 0.285659 0.725255i
\(712\) 0 0
\(713\) 18.6577i 0.698736i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −6.48074 1.23030i −0.242028 0.0459464i
\(718\) 0 0
\(719\) −25.8059 −0.962398 −0.481199 0.876611i \(-0.659799\pi\)
−0.481199 + 0.876611i \(0.659799\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) −5.21046 0.989150i −0.193779 0.0367869i
\(724\) 0 0
\(725\) 21.5294i 0.799581i
\(726\) 0 0
\(727\) 20.6184i 0.764694i 0.924019 + 0.382347i \(0.124884\pi\)
−0.924019 + 0.382347i \(0.875116\pi\)
\(728\) 0 0
\(729\) 11.6261 + 24.3687i 0.430598 + 0.902544i
\(730\) 0 0
\(731\) 26.9300 0.996044
\(732\) 0 0
\(733\) −1.94294 −0.0717640 −0.0358820 0.999356i \(-0.511424\pi\)
−0.0358820 + 0.999356i \(0.511424\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 40.8462i 1.50255i −0.659988 0.751276i \(-0.729439\pi\)
0.659988 0.751276i \(-0.270561\pi\)
\(740\) 0 0
\(741\) −5.59525 + 29.4736i −0.205547 + 1.08274i
\(742\) 0 0
\(743\) −25.0696 −0.919716 −0.459858 0.887993i \(-0.652100\pi\)
−0.459858 + 0.887993i \(0.652100\pi\)
\(744\) 0 0
\(745\) 16.3802 0.600125
\(746\) 0 0
\(747\) −13.4650 5.30352i −0.492659 0.194046i
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 22.8027i 0.832083i −0.909346 0.416041i \(-0.863417\pi\)
0.909346 0.416041i \(-0.136583\pi\)
\(752\) 0 0
\(753\) 0.956439 + 0.181570i 0.0348546 + 0.00661677i
\(754\) 0 0
\(755\) −24.6818 −0.898262
\(756\) 0 0
\(757\) −4.74773 −0.172559 −0.0862795 0.996271i \(-0.527498\pi\)
−0.0862795 + 0.996271i \(0.527498\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 13.6688i 0.495492i −0.968825 0.247746i \(-0.920310\pi\)
0.968825 0.247746i \(-0.0796898\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) −18.8348 7.41855i −0.680975 0.268218i
\(766\) 0 0
\(767\) 25.8645 0.933913
\(768\) 0 0
\(769\) 11.0801 0.399560 0.199780 0.979841i \(-0.435977\pi\)
0.199780 + 0.979841i \(0.435977\pi\)
\(770\) 0 0
\(771\) 4.41558 23.2596i 0.159023 0.837673i
\(772\) 0 0
\(773\) 1.55481i 0.0559225i −0.999609 0.0279613i \(-0.991098\pi\)
0.999609 0.0279613i \(-0.00890150\pi\)
\(774\) 0 0
\(775\) 12.6520i 0.454473i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 50.9341 1.82490
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) −23.1129 + 36.6356i −0.825988 + 1.30925i
\(784\) 0 0
\(785\) 12.7341i 0.454498i
\(786\) 0 0
\(787\) 29.2872i 1.04398i 0.852953 + 0.521988i \(0.174809\pi\)
−0.852953 + 0.521988i \(0.825191\pi\)
\(788\) 0 0
\(789\) −50.4933 9.58562i −1.79761 0.341257i
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −20.7477 −0.736773
\(794\) 0 0
\(795\) −27.8736 5.29150i −0.988574 0.187670i
\(796\) 0 0
\(797\) 7.77403i 0.275370i −0.990476 0.137685i \(-0.956034\pi\)
0.990476 0.137685i \(-0.0439662\pi\)
\(798\) 0 0
\(799\) 23.3748i 0.826941i
\(800\) 0 0
\(801\) 4.77136 12.1139i 0.168588 0.428025i
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0.502268 2.64575i 0.0176807 0.0931349i
\(808\) 0 0
\(809\) 22.3514i 0.785834i −0.919574 0.392917i \(-0.871466\pi\)
0.919574 0.392917i \(-0.128534\pi\)
\(810\) 0 0
\(811\) 54.8045i 1.92445i −0.272259 0.962224i \(-0.587771\pi\)
0.272259 0.962224i \(-0.412229\pi\)
\(812\) 0 0
\(813\) 6.33030 33.3456i 0.222013 1.16948i
\(814\) 0 0
\(815\) −31.1919 −1.09261
\(816\) 0 0
\(817\) −42.4264 −1.48431
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 2.65775i 0.0927562i 0.998924 + 0.0463781i \(0.0147679\pi\)
−0.998924 + 0.0463781i \(0.985232\pi\)
\(822\) 0 0
\(823\) 40.1232i 1.39861i 0.714825 + 0.699304i \(0.246507\pi\)
−0.714825 + 0.699304i \(0.753493\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −22.0560 −0.766963 −0.383482 0.923549i \(-0.625275\pi\)
−0.383482 + 0.923549i \(0.625275\pi\)
\(828\) 0 0
\(829\) 3.71392 0.128990 0.0644948 0.997918i \(-0.479456\pi\)
0.0644948 + 0.997918i \(0.479456\pi\)
\(830\) 0 0
\(831\) −3.40332 0.646084i −0.118060 0.0224124i
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −13.5826 + 21.5294i −0.469482 + 0.744164i
\(838\) 0 0
\(839\) 25.8059 0.890919 0.445460 0.895302i \(-0.353040\pi\)
0.445460 + 0.895302i \(0.353040\pi\)
\(840\) 0 0
\(841\) −40.4955 −1.39639
\(842\) 0 0
\(843\) 4.82395 25.4107i 0.166146 0.875190i
\(844\) 0 0
\(845\) 10.2346i 0.352082i
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −8.53901 + 44.9802i −0.293058 + 1.54372i
\(850\) 0 0
\(851\) 6.02721 0.206610
\(852\) 0 0
\(853\) 30.2272 1.03496 0.517480 0.855695i \(-0.326870\pi\)
0.517480 + 0.855695i \(0.326870\pi\)
\(854\) 0 0
\(855\) 29.6730 + 11.6874i 1.01479 + 0.399701i
\(856\) 0 0
\(857\) 22.9976i 0.785583i −0.919628 0.392791i \(-0.871509\pi\)
0.919628 0.392791i \(-0.128491\pi\)
\(858\) 0 0
\(859\) 5.81475i 0.198397i 0.995068 + 0.0991984i \(0.0316279\pi\)
−0.995068 + 0.0991984i \(0.968372\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −35.7002 −1.21525 −0.607625 0.794224i \(-0.707877\pi\)
−0.607625 + 0.794224i \(0.707877\pi\)
\(864\) 0 0
\(865\) −2.41742 −0.0821949
\(866\) 0 0
\(867\) 3.12229 + 0.592733i 0.106038 + 0.0201303i
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 8.77548i 0.297346i
\(872\) 0 0
\(873\) 27.6323 + 10.8836i 0.935213 + 0.368356i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 10.4174 0.351771 0.175886 0.984411i \(-0.443721\pi\)
0.175886 + 0.984411i \(0.443721\pi\)
\(878\) 0 0
\(879\) 4.31075 22.7074i 0.145398 0.765900i
\(880\) 0 0
\(881\) 16.7784i 0.565278i 0.959226 + 0.282639i \(0.0912098\pi\)
−0.959226 + 0.282639i \(0.908790\pi\)
\(882\) 0 0
\(883\) 21.3567i 0.718711i 0.933201 + 0.359355i \(0.117003\pi\)
−0.933201 + 0.359355i \(0.882997\pi\)
\(884\) 0 0
\(885\) 5.12813 27.0130i 0.172380 0.908032i
\(886\) 0 0
\(887\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 36.8254i 1.23231i
\(894\) 0 0
\(895\) 22.4499i 0.750419i
\(896\) 0 0
\(897\) 16.4174 + 3.11667i 0.548162 + 0.104063i
\(898\) 0 0
\(899\) −40.8398 −1.36208
\(900\) 0 0
\(901\) 45.7220 1.52322
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 16.2139i 0.538967i
\(906\) 0 0
\(907\) 43.5873i 1.44729i 0.690171 + 0.723647i \(0.257535\pi\)
−0.690171 + 0.723647i \(0.742465\pi\)
\(908\) 0 0
\(909\) 21.5084 54.6073i 0.713388 1.81121i
\(910\) 0 0
\(911\) −25.0696 −0.830594 −0.415297 0.909686i \(-0.636322\pi\)
−0.415297 + 0.909686i \(0.636322\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) −4.11363 + 21.6690i −0.135992 + 0.716356i
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 15.3024i 0.504780i 0.967626 + 0.252390i \(0.0812166\pi\)
−0.967626 + 0.252390i \(0.918783\pi\)
\(920\) 0 0
\(921\) −3.79129 + 19.9710i −0.124927 + 0.658068i
\(922\) 0 0
\(923\) 36.5779 1.20398
\(924\) 0 0
\(925\) 4.08712 0.134384
\(926\) 0 0
\(927\) 4.26188 10.8204i 0.139979 0.355390i
\(928\) 0 0
\(929\) 51.4977i 1.68958i −0.535095 0.844792i \(-0.679724\pi\)
0.535095 0.844792i \(-0.320276\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) −25.5826 4.85658i −0.837536 0.158997i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −37.0031 −1.20884 −0.604419 0.796666i \(-0.706595\pi\)
−0.604419 + 0.796666i \(0.706595\pi\)
\(938\) 0 0
\(939\) −10.0235 1.90285i −0.327104 0.0620972i
\(940\) 0 0
\(941\) 22.6731i 0.739122i 0.929207 + 0.369561i \(0.120492\pi\)
−0.929207 + 0.369561i \(0.879508\pi\)
\(942\) 0 0
\(943\) 28.3714i 0.923900i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −22.8509 −0.742554 −0.371277 0.928522i \(-0.621080\pi\)
−0.371277 + 0.928522i \(0.621080\pi\)
\(948\) 0 0
\(949\) 29.2523 0.949569
\(950\) 0 0
\(951\) 0.858565 4.52259i 0.0278409 0.146655i
\(952\) 0 0
\(953\) 55.3339i 1.79244i −0.443610 0.896220i \(-0.646303\pi\)
0.443610 0.896220i \(-0.353697\pi\)
\(954\) 0 0
\(955\) 34.2929i 1.10969i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 7.00000 0.225806
\(962\) 0 0
\(963\) 40.3036 + 15.8745i 1.29876 + 0.511549i
\(964\) 0 0
\(965\) 13.6010i 0.437833i
\(966\) 0 0
\(967\) 2.01810i 0.0648978i 0.999473 + 0.0324489i \(0.0103306\pi\)
−0.999473 + 0.0324489i \(0.989669\pi\)
\(968\) 0 0
\(969\) −50.4933 9.58562i −1.62208 0.307934i
\(970\) 0 0
\(971\) 36.0159 1.15580 0.577902 0.816106i \(-0.303872\pi\)
0.577902 + 0.816106i \(0.303872\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 11.1328 + 2.11345i 0.356536 + 0.0676846i
\(976\) 0 0
\(977\) 44.7986i 1.43324i 0.697466 + 0.716618i \(0.254311\pi\)
−0.697466 + 0.716618i \(0.745689\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) −37.9129 14.9329i −1.21047 0.476770i
\(982\) 0 0
\(983\) −5.38601 −0.171787 −0.0858935 0.996304i \(-0.527374\pi\)
−0.0858935 + 0.996304i \(0.527374\pi\)
\(984\) 0 0
\(985\) −16.3802 −0.521918
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 23.6324i 0.751467i
\(990\) 0 0
\(991\) 59.4618i 1.88887i −0.328702 0.944434i \(-0.606611\pi\)
0.328702 0.944434i \(-0.393389\pi\)
\(992\) 0 0
\(993\) −4.24264 + 22.3486i −0.134636 + 0.709211i
\(994\) 0 0
\(995\) −15.2339 −0.482948
\(996\) 0 0
\(997\) −38.5893 −1.22213 −0.611067 0.791579i \(-0.709260\pi\)
−0.611067 + 0.791579i \(0.709260\pi\)
\(998\) 0 0
\(999\) −6.95489 4.38774i −0.220043 0.138822i
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 2352.2.h.p.2255.1 16
3.2 odd 2 inner 2352.2.h.p.2255.14 yes 16
4.3 odd 2 inner 2352.2.h.p.2255.15 yes 16
7.6 odd 2 inner 2352.2.h.p.2255.16 yes 16
12.11 even 2 inner 2352.2.h.p.2255.4 yes 16
21.20 even 2 inner 2352.2.h.p.2255.3 yes 16
28.27 even 2 inner 2352.2.h.p.2255.2 yes 16
84.83 odd 2 inner 2352.2.h.p.2255.13 yes 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2352.2.h.p.2255.1 16 1.1 even 1 trivial
2352.2.h.p.2255.2 yes 16 28.27 even 2 inner
2352.2.h.p.2255.3 yes 16 21.20 even 2 inner
2352.2.h.p.2255.4 yes 16 12.11 even 2 inner
2352.2.h.p.2255.13 yes 16 84.83 odd 2 inner
2352.2.h.p.2255.14 yes 16 3.2 odd 2 inner
2352.2.h.p.2255.15 yes 16 4.3 odd 2 inner
2352.2.h.p.2255.16 yes 16 7.6 odd 2 inner