Properties

Label 1875.2.b.g.1249.14
Level $1875$
Weight $2$
Character 1875.1249
Analytic conductor $14.972$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1875,2,Mod(1249,1875)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1875, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("1875.1249");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1875 = 3 \cdot 5^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1875.b (of order \(2\), degree \(1\), not 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: \(14.9719503790\)
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: \( x^{16} + 25x^{14} + 246x^{12} + 1220x^{10} + 3281x^{8} + 4880x^{6} + 3936x^{4} + 1600x^{2} + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1249.14
Root \(0.895394i\) of defining polynomial
Character \(\chi\) \(=\) 1875.1249
Dual form 1875.2.b.g.1249.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.23365i q^{2} -1.00000i q^{3} -2.98921 q^{4} +2.23365 q^{6} -1.03143i q^{7} -2.20956i q^{8} -1.00000 q^{9} +O(q^{10})\) \(q+2.23365i q^{2} -1.00000i q^{3} -2.98921 q^{4} +2.23365 q^{6} -1.03143i q^{7} -2.20956i q^{8} -1.00000 q^{9} +6.17643 q^{11} +2.98921i q^{12} -0.937763i q^{13} +2.30385 q^{14} -1.04303 q^{16} -6.56329i q^{17} -2.23365i q^{18} -5.67453 q^{19} -1.03143 q^{21} +13.7960i q^{22} -1.64660i q^{23} -2.20956 q^{24} +2.09464 q^{26} +1.00000i q^{27} +3.08316i q^{28} -8.35819 q^{29} +5.53371 q^{31} -6.74889i q^{32} -6.17643i q^{33} +14.6601 q^{34} +2.98921 q^{36} -1.29548i q^{37} -12.6749i q^{38} -0.937763 q^{39} -4.98106 q^{41} -2.30385i q^{42} -7.75619i q^{43} -18.4627 q^{44} +3.67793 q^{46} -7.67288i q^{47} +1.04303i q^{48} +5.93616 q^{49} -6.56329 q^{51} +2.80317i q^{52} -0.500546i q^{53} -2.23365 q^{54} -2.27900 q^{56} +5.67453i q^{57} -18.6693i q^{58} +1.19340 q^{59} -12.3637 q^{61} +12.3604i q^{62} +1.03143i q^{63} +12.9886 q^{64} +13.7960 q^{66} -7.58851i q^{67} +19.6191i q^{68} -1.64660 q^{69} +10.6125 q^{71} +2.20956i q^{72} +7.98638i q^{73} +2.89365 q^{74} +16.9624 q^{76} -6.37054i q^{77} -2.09464i q^{78} +13.9213 q^{79} +1.00000 q^{81} -11.1260i q^{82} -1.46223i q^{83} +3.08316 q^{84} +17.3246 q^{86} +8.35819i q^{87} -13.6472i q^{88} +8.51161 q^{89} -0.967234 q^{91} +4.92203i q^{92} -5.53371i q^{93} +17.1386 q^{94} -6.74889 q^{96} +3.75623i q^{97} +13.2593i q^{98} -6.17643 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 18 q^{4} + 2 q^{6} - 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 18 q^{4} + 2 q^{6} - 16 q^{9} + 24 q^{11} - 32 q^{14} + 30 q^{16} - 32 q^{19} + 24 q^{21} - 6 q^{24} - 68 q^{26} - 4 q^{29} + 26 q^{31} + 74 q^{34} + 18 q^{36} - 28 q^{39} - 24 q^{41} - 94 q^{44} + 66 q^{46} - 60 q^{49} - 2 q^{51} - 2 q^{54} + 120 q^{56} - 28 q^{59} + 20 q^{61} - 82 q^{64} + 36 q^{66} + 8 q^{69} + 42 q^{71} + 18 q^{74} - 2 q^{76} - 20 q^{79} + 16 q^{81} + 42 q^{84} + 84 q^{86} + 18 q^{89} - 24 q^{91} - 28 q^{94} - 36 q^{96} - 24 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1875\mathbb{Z}\right)^\times\).

\(n\) \(626\) \(1252\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.23365i 1.57943i 0.613472 + 0.789716i \(0.289772\pi\)
−0.613472 + 0.789716i \(0.710228\pi\)
\(3\) − 1.00000i − 0.577350i
\(4\) −2.98921 −1.49461
\(5\) 0 0
\(6\) 2.23365 0.911886
\(7\) − 1.03143i − 0.389843i −0.980819 0.194921i \(-0.937555\pi\)
0.980819 0.194921i \(-0.0624452\pi\)
\(8\) − 2.20956i − 0.781198i
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 6.17643 1.86227 0.931133 0.364681i \(-0.118822\pi\)
0.931133 + 0.364681i \(0.118822\pi\)
\(12\) 2.98921i 0.862912i
\(13\) − 0.937763i − 0.260089i −0.991508 0.130044i \(-0.958488\pi\)
0.991508 0.130044i \(-0.0415120\pi\)
\(14\) 2.30385 0.615730
\(15\) 0 0
\(16\) −1.04303 −0.260757
\(17\) − 6.56329i − 1.59183i −0.605407 0.795916i \(-0.706990\pi\)
0.605407 0.795916i \(-0.293010\pi\)
\(18\) − 2.23365i − 0.526477i
\(19\) −5.67453 −1.30183 −0.650913 0.759152i \(-0.725614\pi\)
−0.650913 + 0.759152i \(0.725614\pi\)
\(20\) 0 0
\(21\) −1.03143 −0.225076
\(22\) 13.7960i 2.94132i
\(23\) − 1.64660i − 0.343339i −0.985155 0.171669i \(-0.945084\pi\)
0.985155 0.171669i \(-0.0549161\pi\)
\(24\) −2.20956 −0.451025
\(25\) 0 0
\(26\) 2.09464 0.410792
\(27\) 1.00000i 0.192450i
\(28\) 3.08316i 0.582662i
\(29\) −8.35819 −1.55208 −0.776039 0.630685i \(-0.782774\pi\)
−0.776039 + 0.630685i \(0.782774\pi\)
\(30\) 0 0
\(31\) 5.53371 0.993883 0.496942 0.867784i \(-0.334456\pi\)
0.496942 + 0.867784i \(0.334456\pi\)
\(32\) − 6.74889i − 1.19305i
\(33\) − 6.17643i − 1.07518i
\(34\) 14.6601 2.51419
\(35\) 0 0
\(36\) 2.98921 0.498202
\(37\) − 1.29548i − 0.212976i −0.994314 0.106488i \(-0.966040\pi\)
0.994314 0.106488i \(-0.0339605\pi\)
\(38\) − 12.6749i − 2.05615i
\(39\) −0.937763 −0.150162
\(40\) 0 0
\(41\) −4.98106 −0.777910 −0.388955 0.921257i \(-0.627164\pi\)
−0.388955 + 0.921257i \(0.627164\pi\)
\(42\) − 2.30385i − 0.355492i
\(43\) − 7.75619i − 1.18281i −0.806376 0.591404i \(-0.798574\pi\)
0.806376 0.591404i \(-0.201426\pi\)
\(44\) −18.4627 −2.78335
\(45\) 0 0
\(46\) 3.67793 0.542281
\(47\) − 7.67288i − 1.11920i −0.828761 0.559602i \(-0.810954\pi\)
0.828761 0.559602i \(-0.189046\pi\)
\(48\) 1.04303i 0.150548i
\(49\) 5.93616 0.848023
\(50\) 0 0
\(51\) −6.56329 −0.919044
\(52\) 2.80317i 0.388730i
\(53\) − 0.500546i − 0.0687553i −0.999409 0.0343777i \(-0.989055\pi\)
0.999409 0.0343777i \(-0.0109449\pi\)
\(54\) −2.23365 −0.303962
\(55\) 0 0
\(56\) −2.27900 −0.304544
\(57\) 5.67453i 0.751609i
\(58\) − 18.6693i − 2.45140i
\(59\) 1.19340 0.155367 0.0776837 0.996978i \(-0.475248\pi\)
0.0776837 + 0.996978i \(0.475248\pi\)
\(60\) 0 0
\(61\) −12.3637 −1.58301 −0.791504 0.611164i \(-0.790701\pi\)
−0.791504 + 0.611164i \(0.790701\pi\)
\(62\) 12.3604i 1.56977i
\(63\) 1.03143i 0.129948i
\(64\) 12.9886 1.62358
\(65\) 0 0
\(66\) 13.7960 1.69817
\(67\) − 7.58851i − 0.927084i −0.886075 0.463542i \(-0.846578\pi\)
0.886075 0.463542i \(-0.153422\pi\)
\(68\) 19.6191i 2.37916i
\(69\) −1.64660 −0.198227
\(70\) 0 0
\(71\) 10.6125 1.25948 0.629739 0.776807i \(-0.283162\pi\)
0.629739 + 0.776807i \(0.283162\pi\)
\(72\) 2.20956i 0.260399i
\(73\) 7.98638i 0.934735i 0.884063 + 0.467367i \(0.154797\pi\)
−0.884063 + 0.467367i \(0.845203\pi\)
\(74\) 2.89365 0.336380
\(75\) 0 0
\(76\) 16.9624 1.94572
\(77\) − 6.37054i − 0.725990i
\(78\) − 2.09464i − 0.237171i
\(79\) 13.9213 1.56627 0.783134 0.621854i \(-0.213620\pi\)
0.783134 + 0.621854i \(0.213620\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) − 11.1260i − 1.22866i
\(83\) − 1.46223i − 0.160500i −0.996775 0.0802500i \(-0.974428\pi\)
0.996775 0.0802500i \(-0.0255719\pi\)
\(84\) 3.08316 0.336400
\(85\) 0 0
\(86\) 17.3246 1.86816
\(87\) 8.35819i 0.896092i
\(88\) − 13.6472i − 1.45480i
\(89\) 8.51161 0.902229 0.451115 0.892466i \(-0.351027\pi\)
0.451115 + 0.892466i \(0.351027\pi\)
\(90\) 0 0
\(91\) −0.967234 −0.101394
\(92\) 4.92203i 0.513157i
\(93\) − 5.53371i − 0.573819i
\(94\) 17.1386 1.76771
\(95\) 0 0
\(96\) −6.74889 −0.688806
\(97\) 3.75623i 0.381387i 0.981650 + 0.190693i \(0.0610737\pi\)
−0.981650 + 0.190693i \(0.938926\pi\)
\(98\) 13.2593i 1.33939i
\(99\) −6.17643 −0.620755
\(100\) 0 0
\(101\) −2.60566 −0.259273 −0.129636 0.991562i \(-0.541381\pi\)
−0.129636 + 0.991562i \(0.541381\pi\)
\(102\) − 14.6601i − 1.45157i
\(103\) − 17.2182i − 1.69656i −0.529550 0.848279i \(-0.677639\pi\)
0.529550 0.848279i \(-0.322361\pi\)
\(104\) −2.07204 −0.203181
\(105\) 0 0
\(106\) 1.11805 0.108594
\(107\) 5.54296i 0.535858i 0.963439 + 0.267929i \(0.0863393\pi\)
−0.963439 + 0.267929i \(0.913661\pi\)
\(108\) − 2.98921i − 0.287637i
\(109\) 5.98458 0.573219 0.286610 0.958047i \(-0.407472\pi\)
0.286610 + 0.958047i \(0.407472\pi\)
\(110\) 0 0
\(111\) −1.29548 −0.122961
\(112\) 1.07581i 0.101654i
\(113\) − 0.323636i − 0.0304451i −0.999884 0.0152226i \(-0.995154\pi\)
0.999884 0.0152226i \(-0.00484568\pi\)
\(114\) −12.6749 −1.18712
\(115\) 0 0
\(116\) 24.9844 2.31975
\(117\) 0.937763i 0.0866962i
\(118\) 2.66564i 0.245392i
\(119\) −6.76955 −0.620564
\(120\) 0 0
\(121\) 27.1483 2.46803
\(122\) − 27.6162i − 2.50025i
\(123\) 4.98106i 0.449127i
\(124\) −16.5414 −1.48546
\(125\) 0 0
\(126\) −2.30385 −0.205243
\(127\) − 0.0904114i − 0.00802271i −0.999992 0.00401136i \(-0.998723\pi\)
0.999992 0.00401136i \(-0.00127686\pi\)
\(128\) 15.5143i 1.37129i
\(129\) −7.75619 −0.682894
\(130\) 0 0
\(131\) −13.5700 −1.18562 −0.592809 0.805343i \(-0.701981\pi\)
−0.592809 + 0.805343i \(0.701981\pi\)
\(132\) 18.4627i 1.60697i
\(133\) 5.85286i 0.507507i
\(134\) 16.9501 1.46427
\(135\) 0 0
\(136\) −14.5020 −1.24354
\(137\) − 13.9795i − 1.19435i −0.802111 0.597174i \(-0.796290\pi\)
0.802111 0.597174i \(-0.203710\pi\)
\(138\) − 3.67793i − 0.313086i
\(139\) 2.21762 0.188096 0.0940481 0.995568i \(-0.470019\pi\)
0.0940481 + 0.995568i \(0.470019\pi\)
\(140\) 0 0
\(141\) −7.67288 −0.646173
\(142\) 23.7048i 1.98926i
\(143\) − 5.79203i − 0.484354i
\(144\) 1.04303 0.0869191
\(145\) 0 0
\(146\) −17.8388 −1.47635
\(147\) − 5.93616i − 0.489606i
\(148\) 3.87247i 0.318315i
\(149\) −3.59402 −0.294433 −0.147217 0.989104i \(-0.547031\pi\)
−0.147217 + 0.989104i \(0.547031\pi\)
\(150\) 0 0
\(151\) 16.8624 1.37224 0.686121 0.727487i \(-0.259312\pi\)
0.686121 + 0.727487i \(0.259312\pi\)
\(152\) 12.5382i 1.01698i
\(153\) 6.56329i 0.530611i
\(154\) 14.2296 1.14665
\(155\) 0 0
\(156\) 2.80317 0.224433
\(157\) − 8.42895i − 0.672703i −0.941736 0.336352i \(-0.890807\pi\)
0.941736 0.336352i \(-0.109193\pi\)
\(158\) 31.0953i 2.47381i
\(159\) −0.500546 −0.0396959
\(160\) 0 0
\(161\) −1.69834 −0.133848
\(162\) 2.23365i 0.175492i
\(163\) 11.3806i 0.891394i 0.895184 + 0.445697i \(0.147044\pi\)
−0.895184 + 0.445697i \(0.852956\pi\)
\(164\) 14.8894 1.16267
\(165\) 0 0
\(166\) 3.26611 0.253499
\(167\) 0.886875i 0.0686285i 0.999411 + 0.0343142i \(0.0109247\pi\)
−0.999411 + 0.0343142i \(0.989075\pi\)
\(168\) 2.27900i 0.175829i
\(169\) 12.1206 0.932354
\(170\) 0 0
\(171\) 5.67453 0.433942
\(172\) 23.1849i 1.76783i
\(173\) 20.0184i 1.52197i 0.648769 + 0.760986i \(0.275284\pi\)
−0.648769 + 0.760986i \(0.724716\pi\)
\(174\) −18.6693 −1.41532
\(175\) 0 0
\(176\) −6.44220 −0.485599
\(177\) − 1.19340i − 0.0897014i
\(178\) 19.0120i 1.42501i
\(179\) 3.38799 0.253231 0.126615 0.991952i \(-0.459589\pi\)
0.126615 + 0.991952i \(0.459589\pi\)
\(180\) 0 0
\(181\) 5.41991 0.402859 0.201429 0.979503i \(-0.435441\pi\)
0.201429 + 0.979503i \(0.435441\pi\)
\(182\) − 2.16047i − 0.160144i
\(183\) 12.3637i 0.913950i
\(184\) −3.63825 −0.268216
\(185\) 0 0
\(186\) 12.3604 0.906308
\(187\) − 40.5377i − 2.96441i
\(188\) 22.9359i 1.67277i
\(189\) 1.03143 0.0750253
\(190\) 0 0
\(191\) −11.2760 −0.815901 −0.407950 0.913004i \(-0.633756\pi\)
−0.407950 + 0.913004i \(0.633756\pi\)
\(192\) − 12.9886i − 0.937374i
\(193\) 6.22383i 0.448001i 0.974589 + 0.224000i \(0.0719117\pi\)
−0.974589 + 0.224000i \(0.928088\pi\)
\(194\) −8.39011 −0.602375
\(195\) 0 0
\(196\) −17.7444 −1.26746
\(197\) − 19.7296i − 1.40568i −0.711349 0.702839i \(-0.751916\pi\)
0.711349 0.702839i \(-0.248084\pi\)
\(198\) − 13.7960i − 0.980441i
\(199\) −18.5731 −1.31661 −0.658307 0.752750i \(-0.728727\pi\)
−0.658307 + 0.752750i \(0.728727\pi\)
\(200\) 0 0
\(201\) −7.58851 −0.535252
\(202\) − 5.82014i − 0.409504i
\(203\) 8.62087i 0.605066i
\(204\) 19.6191 1.37361
\(205\) 0 0
\(206\) 38.4595 2.67960
\(207\) 1.64660i 0.114446i
\(208\) 0.978114i 0.0678200i
\(209\) −35.0483 −2.42434
\(210\) 0 0
\(211\) 2.71998 0.187251 0.0936255 0.995607i \(-0.470154\pi\)
0.0936255 + 0.995607i \(0.470154\pi\)
\(212\) 1.49624i 0.102762i
\(213\) − 10.6125i − 0.727160i
\(214\) −12.3811 −0.846352
\(215\) 0 0
\(216\) 2.20956 0.150342
\(217\) − 5.70762i − 0.387458i
\(218\) 13.3675i 0.905361i
\(219\) 7.98638 0.539669
\(220\) 0 0
\(221\) −6.15481 −0.414017
\(222\) − 2.89365i − 0.194209i
\(223\) − 7.04569i − 0.471814i −0.971776 0.235907i \(-0.924194\pi\)
0.971776 0.235907i \(-0.0758061\pi\)
\(224\) −6.96099 −0.465101
\(225\) 0 0
\(226\) 0.722892 0.0480860
\(227\) 16.9671i 1.12615i 0.826407 + 0.563073i \(0.190381\pi\)
−0.826407 + 0.563073i \(0.809619\pi\)
\(228\) − 16.9624i − 1.12336i
\(229\) −5.14606 −0.340061 −0.170030 0.985439i \(-0.554387\pi\)
−0.170030 + 0.985439i \(0.554387\pi\)
\(230\) 0 0
\(231\) −6.37054 −0.419151
\(232\) 18.4679i 1.21248i
\(233\) − 27.3476i − 1.79160i −0.444460 0.895799i \(-0.646604\pi\)
0.444460 0.895799i \(-0.353396\pi\)
\(234\) −2.09464 −0.136931
\(235\) 0 0
\(236\) −3.56733 −0.232213
\(237\) − 13.9213i − 0.904285i
\(238\) − 15.1208i − 0.980139i
\(239\) −19.6994 −1.27425 −0.637123 0.770762i \(-0.719876\pi\)
−0.637123 + 0.770762i \(0.719876\pi\)
\(240\) 0 0
\(241\) −9.57114 −0.616532 −0.308266 0.951300i \(-0.599749\pi\)
−0.308266 + 0.951300i \(0.599749\pi\)
\(242\) 60.6400i 3.89809i
\(243\) − 1.00000i − 0.0641500i
\(244\) 36.9577 2.36597
\(245\) 0 0
\(246\) −11.1260 −0.709365
\(247\) 5.32136i 0.338590i
\(248\) − 12.2271i − 0.776420i
\(249\) −1.46223 −0.0926648
\(250\) 0 0
\(251\) −5.82514 −0.367680 −0.183840 0.982956i \(-0.558853\pi\)
−0.183840 + 0.982956i \(0.558853\pi\)
\(252\) − 3.08316i − 0.194221i
\(253\) − 10.1701i − 0.639388i
\(254\) 0.201948 0.0126713
\(255\) 0 0
\(256\) −8.67641 −0.542276
\(257\) − 3.47558i − 0.216801i −0.994107 0.108400i \(-0.965427\pi\)
0.994107 0.108400i \(-0.0345729\pi\)
\(258\) − 17.3246i − 1.07859i
\(259\) −1.33619 −0.0830270
\(260\) 0 0
\(261\) 8.35819 0.517359
\(262\) − 30.3107i − 1.87260i
\(263\) 22.7227i 1.40114i 0.713584 + 0.700570i \(0.247071\pi\)
−0.713584 + 0.700570i \(0.752929\pi\)
\(264\) −13.6472 −0.839928
\(265\) 0 0
\(266\) −13.0733 −0.801573
\(267\) − 8.51161i − 0.520902i
\(268\) 22.6837i 1.38563i
\(269\) −1.63546 −0.0997160 −0.0498580 0.998756i \(-0.515877\pi\)
−0.0498580 + 0.998756i \(0.515877\pi\)
\(270\) 0 0
\(271\) 12.4498 0.756275 0.378137 0.925750i \(-0.376565\pi\)
0.378137 + 0.925750i \(0.376565\pi\)
\(272\) 6.84571i 0.415082i
\(273\) 0.967234i 0.0585397i
\(274\) 31.2254 1.88639
\(275\) 0 0
\(276\) 4.92203 0.296271
\(277\) 2.16353i 0.129994i 0.997885 + 0.0649971i \(0.0207038\pi\)
−0.997885 + 0.0649971i \(0.979296\pi\)
\(278\) 4.95340i 0.297085i
\(279\) −5.53371 −0.331294
\(280\) 0 0
\(281\) 4.97817 0.296973 0.148486 0.988914i \(-0.452560\pi\)
0.148486 + 0.988914i \(0.452560\pi\)
\(282\) − 17.1386i − 1.02059i
\(283\) 17.3357i 1.03050i 0.857041 + 0.515249i \(0.172301\pi\)
−0.857041 + 0.515249i \(0.827699\pi\)
\(284\) −31.7232 −1.88242
\(285\) 0 0
\(286\) 12.9374 0.765004
\(287\) 5.13759i 0.303263i
\(288\) 6.74889i 0.397682i
\(289\) −26.0768 −1.53393
\(290\) 0 0
\(291\) 3.75623 0.220194
\(292\) − 23.8730i − 1.39706i
\(293\) − 20.1863i − 1.17930i −0.807661 0.589648i \(-0.799267\pi\)
0.807661 0.589648i \(-0.200733\pi\)
\(294\) 13.2593 0.773300
\(295\) 0 0
\(296\) −2.86244 −0.166376
\(297\) 6.17643i 0.358393i
\(298\) − 8.02779i − 0.465038i
\(299\) −1.54412 −0.0892986
\(300\) 0 0
\(301\) −7.99994 −0.461109
\(302\) 37.6648i 2.16736i
\(303\) 2.60566i 0.149691i
\(304\) 5.91870 0.339461
\(305\) 0 0
\(306\) −14.6601 −0.838064
\(307\) 24.7882i 1.41474i 0.706844 + 0.707369i \(0.250118\pi\)
−0.706844 + 0.707369i \(0.749882\pi\)
\(308\) 19.0429i 1.08507i
\(309\) −17.2182 −0.979508
\(310\) 0 0
\(311\) −28.1046 −1.59367 −0.796834 0.604198i \(-0.793493\pi\)
−0.796834 + 0.604198i \(0.793493\pi\)
\(312\) 2.07204i 0.117306i
\(313\) 27.1858i 1.53663i 0.640070 + 0.768317i \(0.278905\pi\)
−0.640070 + 0.768317i \(0.721095\pi\)
\(314\) 18.8274 1.06249
\(315\) 0 0
\(316\) −41.6137 −2.34095
\(317\) − 14.9749i − 0.841073i −0.907276 0.420536i \(-0.861842\pi\)
0.907276 0.420536i \(-0.138158\pi\)
\(318\) − 1.11805i − 0.0626970i
\(319\) −51.6238 −2.89038
\(320\) 0 0
\(321\) 5.54296 0.309378
\(322\) − 3.79351i − 0.211404i
\(323\) 37.2436i 2.07229i
\(324\) −2.98921 −0.166067
\(325\) 0 0
\(326\) −25.4202 −1.40790
\(327\) − 5.98458i − 0.330948i
\(328\) 11.0059i 0.607702i
\(329\) −7.91401 −0.436314
\(330\) 0 0
\(331\) −13.9393 −0.766173 −0.383086 0.923713i \(-0.625139\pi\)
−0.383086 + 0.923713i \(0.625139\pi\)
\(332\) 4.37090i 0.239885i
\(333\) 1.29548i 0.0709918i
\(334\) −1.98097 −0.108394
\(335\) 0 0
\(336\) 1.07581 0.0586902
\(337\) 35.3670i 1.92656i 0.268492 + 0.963282i \(0.413475\pi\)
−0.268492 + 0.963282i \(0.586525\pi\)
\(338\) 27.0732i 1.47259i
\(339\) −0.323636 −0.0175775
\(340\) 0 0
\(341\) 34.1786 1.85087
\(342\) 12.6749i 0.685382i
\(343\) − 13.3427i − 0.720438i
\(344\) −17.1378 −0.924007
\(345\) 0 0
\(346\) −44.7142 −2.40385
\(347\) 9.34346i 0.501583i 0.968041 + 0.250792i \(0.0806909\pi\)
−0.968041 + 0.250792i \(0.919309\pi\)
\(348\) − 24.9844i − 1.33931i
\(349\) −6.92379 −0.370622 −0.185311 0.982680i \(-0.559329\pi\)
−0.185311 + 0.982680i \(0.559329\pi\)
\(350\) 0 0
\(351\) 0.937763 0.0500541
\(352\) − 41.6841i − 2.22177i
\(353\) − 5.13902i − 0.273522i −0.990604 0.136761i \(-0.956331\pi\)
0.990604 0.136761i \(-0.0436693\pi\)
\(354\) 2.66564 0.141677
\(355\) 0 0
\(356\) −25.4430 −1.34848
\(357\) 6.76955i 0.358283i
\(358\) 7.56761i 0.399961i
\(359\) 19.0429 1.00504 0.502522 0.864564i \(-0.332406\pi\)
0.502522 + 0.864564i \(0.332406\pi\)
\(360\) 0 0
\(361\) 13.2003 0.694750
\(362\) 12.1062i 0.636288i
\(363\) − 27.1483i − 1.42492i
\(364\) 2.89127 0.151544
\(365\) 0 0
\(366\) −27.6162 −1.44352
\(367\) 14.1916i 0.740796i 0.928873 + 0.370398i \(0.120779\pi\)
−0.928873 + 0.370398i \(0.879221\pi\)
\(368\) 1.71745i 0.0895282i
\(369\) 4.98106 0.259303
\(370\) 0 0
\(371\) −0.516277 −0.0268038
\(372\) 16.5414i 0.857633i
\(373\) − 35.4663i − 1.83637i −0.396148 0.918187i \(-0.629653\pi\)
0.396148 0.918187i \(-0.370347\pi\)
\(374\) 90.5473 4.68209
\(375\) 0 0
\(376\) −16.9537 −0.874320
\(377\) 7.83800i 0.403678i
\(378\) 2.30385i 0.118497i
\(379\) 1.99692 0.102575 0.0512874 0.998684i \(-0.483668\pi\)
0.0512874 + 0.998684i \(0.483668\pi\)
\(380\) 0 0
\(381\) −0.0904114 −0.00463191
\(382\) − 25.1866i − 1.28866i
\(383\) 9.43941i 0.482331i 0.970484 + 0.241166i \(0.0775297\pi\)
−0.970484 + 0.241166i \(0.922470\pi\)
\(384\) 15.5143 0.791713
\(385\) 0 0
\(386\) −13.9019 −0.707587
\(387\) 7.75619i 0.394269i
\(388\) − 11.2282i − 0.570024i
\(389\) 1.31535 0.0666908 0.0333454 0.999444i \(-0.489384\pi\)
0.0333454 + 0.999444i \(0.489384\pi\)
\(390\) 0 0
\(391\) −10.8071 −0.546538
\(392\) − 13.1163i − 0.662474i
\(393\) 13.5700i 0.684517i
\(394\) 44.0692 2.22017
\(395\) 0 0
\(396\) 18.4627 0.927785
\(397\) − 25.3086i − 1.27020i −0.772430 0.635100i \(-0.780959\pi\)
0.772430 0.635100i \(-0.219041\pi\)
\(398\) − 41.4860i − 2.07950i
\(399\) 5.85286 0.293009
\(400\) 0 0
\(401\) 13.4276 0.670543 0.335271 0.942122i \(-0.391172\pi\)
0.335271 + 0.942122i \(0.391172\pi\)
\(402\) − 16.9501i − 0.845395i
\(403\) − 5.18931i − 0.258498i
\(404\) 7.78887 0.387511
\(405\) 0 0
\(406\) −19.2560 −0.955661
\(407\) − 8.00144i − 0.396617i
\(408\) 14.5020i 0.717956i
\(409\) 18.8088 0.930035 0.465018 0.885301i \(-0.346048\pi\)
0.465018 + 0.885301i \(0.346048\pi\)
\(410\) 0 0
\(411\) −13.9795 −0.689558
\(412\) 51.4688i 2.53569i
\(413\) − 1.23090i − 0.0605688i
\(414\) −3.67793 −0.180760
\(415\) 0 0
\(416\) −6.32886 −0.310298
\(417\) − 2.21762i − 0.108597i
\(418\) − 78.2859i − 3.82909i
\(419\) 21.7812 1.06408 0.532042 0.846718i \(-0.321425\pi\)
0.532042 + 0.846718i \(0.321425\pi\)
\(420\) 0 0
\(421\) 28.3679 1.38257 0.691283 0.722584i \(-0.257046\pi\)
0.691283 + 0.722584i \(0.257046\pi\)
\(422\) 6.07549i 0.295750i
\(423\) 7.67288i 0.373068i
\(424\) −1.10599 −0.0537115
\(425\) 0 0
\(426\) 23.7048 1.14850
\(427\) 12.7522i 0.617124i
\(428\) − 16.5691i − 0.800898i
\(429\) −5.79203 −0.279642
\(430\) 0 0
\(431\) −23.7069 −1.14192 −0.570960 0.820978i \(-0.693429\pi\)
−0.570960 + 0.820978i \(0.693429\pi\)
\(432\) − 1.04303i − 0.0501828i
\(433\) − 7.11846i − 0.342091i −0.985263 0.171046i \(-0.945285\pi\)
0.985263 0.171046i \(-0.0547146\pi\)
\(434\) 12.7488 0.611964
\(435\) 0 0
\(436\) −17.8892 −0.856737
\(437\) 9.34365i 0.446967i
\(438\) 17.8388i 0.852371i
\(439\) −6.68142 −0.318887 −0.159443 0.987207i \(-0.550970\pi\)
−0.159443 + 0.987207i \(0.550970\pi\)
\(440\) 0 0
\(441\) −5.93616 −0.282674
\(442\) − 13.7477i − 0.653912i
\(443\) − 20.7065i − 0.983797i −0.870653 0.491898i \(-0.836303\pi\)
0.870653 0.491898i \(-0.163697\pi\)
\(444\) 3.87247 0.183779
\(445\) 0 0
\(446\) 15.7376 0.745199
\(447\) 3.59402i 0.169991i
\(448\) − 13.3968i − 0.632940i
\(449\) 0.218310 0.0103027 0.00515134 0.999987i \(-0.498360\pi\)
0.00515134 + 0.999987i \(0.498360\pi\)
\(450\) 0 0
\(451\) −30.7652 −1.44867
\(452\) 0.967418i 0.0455035i
\(453\) − 16.8624i − 0.792264i
\(454\) −37.8987 −1.77867
\(455\) 0 0
\(456\) 12.5382 0.587156
\(457\) 5.96061i 0.278826i 0.990234 + 0.139413i \(0.0445215\pi\)
−0.990234 + 0.139413i \(0.955479\pi\)
\(458\) − 11.4945i − 0.537103i
\(459\) 6.56329 0.306348
\(460\) 0 0
\(461\) −12.8249 −0.597314 −0.298657 0.954360i \(-0.596539\pi\)
−0.298657 + 0.954360i \(0.596539\pi\)
\(462\) − 14.2296i − 0.662020i
\(463\) − 26.9181i − 1.25099i −0.780227 0.625496i \(-0.784897\pi\)
0.780227 0.625496i \(-0.215103\pi\)
\(464\) 8.71784 0.404716
\(465\) 0 0
\(466\) 61.0850 2.82971
\(467\) − 26.8600i − 1.24293i −0.783441 0.621466i \(-0.786537\pi\)
0.783441 0.621466i \(-0.213463\pi\)
\(468\) − 2.80317i − 0.129577i
\(469\) −7.82699 −0.361417
\(470\) 0 0
\(471\) −8.42895 −0.388386
\(472\) − 2.63689i − 0.121373i
\(473\) − 47.9056i − 2.20270i
\(474\) 31.0953 1.42826
\(475\) 0 0
\(476\) 20.2356 0.927499
\(477\) 0.500546i 0.0229184i
\(478\) − 44.0016i − 2.01258i
\(479\) −39.2952 −1.79544 −0.897722 0.440562i \(-0.854779\pi\)
−0.897722 + 0.440562i \(0.854779\pi\)
\(480\) 0 0
\(481\) −1.21485 −0.0553925
\(482\) − 21.3786i − 0.973770i
\(483\) 1.69834i 0.0772773i
\(484\) −81.1522 −3.68874
\(485\) 0 0
\(486\) 2.23365 0.101321
\(487\) 37.3269i 1.69144i 0.533626 + 0.845721i \(0.320829\pi\)
−0.533626 + 0.845721i \(0.679171\pi\)
\(488\) 27.3183i 1.23664i
\(489\) 11.3806 0.514647
\(490\) 0 0
\(491\) −16.8123 −0.758729 −0.379364 0.925247i \(-0.623857\pi\)
−0.379364 + 0.925247i \(0.623857\pi\)
\(492\) − 14.8894i − 0.671268i
\(493\) 54.8573i 2.47065i
\(494\) −11.8861 −0.534780
\(495\) 0 0
\(496\) −5.77182 −0.259162
\(497\) − 10.9461i − 0.490998i
\(498\) − 3.26611i − 0.146358i
\(499\) 20.6970 0.926526 0.463263 0.886221i \(-0.346678\pi\)
0.463263 + 0.886221i \(0.346678\pi\)
\(500\) 0 0
\(501\) 0.886875 0.0396227
\(502\) − 13.0114i − 0.580725i
\(503\) 38.8631i 1.73282i 0.499333 + 0.866410i \(0.333579\pi\)
−0.499333 + 0.866410i \(0.666421\pi\)
\(504\) 2.27900 0.101515
\(505\) 0 0
\(506\) 22.7165 1.00987
\(507\) − 12.1206i − 0.538295i
\(508\) 0.270259i 0.0119908i
\(509\) −20.8260 −0.923096 −0.461548 0.887115i \(-0.652706\pi\)
−0.461548 + 0.887115i \(0.652706\pi\)
\(510\) 0 0
\(511\) 8.23736 0.364400
\(512\) 11.6486i 0.514799i
\(513\) − 5.67453i − 0.250536i
\(514\) 7.76325 0.342422
\(515\) 0 0
\(516\) 23.1849 1.02066
\(517\) − 47.3910i − 2.08425i
\(518\) − 2.98459i − 0.131135i
\(519\) 20.0184 0.878710
\(520\) 0 0
\(521\) 10.0302 0.439430 0.219715 0.975564i \(-0.429487\pi\)
0.219715 + 0.975564i \(0.429487\pi\)
\(522\) 18.6693i 0.817134i
\(523\) − 21.8640i − 0.956046i −0.878347 0.478023i \(-0.841354\pi\)
0.878347 0.478023i \(-0.158646\pi\)
\(524\) 40.5637 1.77203
\(525\) 0 0
\(526\) −50.7546 −2.21300
\(527\) − 36.3193i − 1.58210i
\(528\) 6.44220i 0.280361i
\(529\) 20.2887 0.882118
\(530\) 0 0
\(531\) −1.19340 −0.0517891
\(532\) − 17.4954i − 0.758524i
\(533\) 4.67105i 0.202326i
\(534\) 19.0120 0.822730
\(535\) 0 0
\(536\) −16.7673 −0.724236
\(537\) − 3.38799i − 0.146203i
\(538\) − 3.65306i − 0.157495i
\(539\) 36.6643 1.57924
\(540\) 0 0
\(541\) 39.2383 1.68699 0.843493 0.537140i \(-0.180495\pi\)
0.843493 + 0.537140i \(0.180495\pi\)
\(542\) 27.8087i 1.19448i
\(543\) − 5.41991i − 0.232591i
\(544\) −44.2949 −1.89913
\(545\) 0 0
\(546\) −2.16047 −0.0924594
\(547\) 2.82535i 0.120803i 0.998174 + 0.0604016i \(0.0192381\pi\)
−0.998174 + 0.0604016i \(0.980762\pi\)
\(548\) 41.7877i 1.78508i
\(549\) 12.3637 0.527669
\(550\) 0 0
\(551\) 47.4288 2.02053
\(552\) 3.63825i 0.154854i
\(553\) − 14.3588i − 0.610598i
\(554\) −4.83259 −0.205317
\(555\) 0 0
\(556\) −6.62895 −0.281130
\(557\) − 10.4495i − 0.442759i −0.975188 0.221379i \(-0.928944\pi\)
0.975188 0.221379i \(-0.0710559\pi\)
\(558\) − 12.3604i − 0.523257i
\(559\) −7.27347 −0.307635
\(560\) 0 0
\(561\) −40.5377 −1.71150
\(562\) 11.1195i 0.469048i
\(563\) − 5.74113i − 0.241960i −0.992655 0.120980i \(-0.961396\pi\)
0.992655 0.120980i \(-0.0386037\pi\)
\(564\) 22.9359 0.965774
\(565\) 0 0
\(566\) −38.7219 −1.62760
\(567\) − 1.03143i − 0.0433159i
\(568\) − 23.4491i − 0.983901i
\(569\) −5.50079 −0.230605 −0.115302 0.993330i \(-0.536784\pi\)
−0.115302 + 0.993330i \(0.536784\pi\)
\(570\) 0 0
\(571\) −9.36436 −0.391886 −0.195943 0.980615i \(-0.562777\pi\)
−0.195943 + 0.980615i \(0.562777\pi\)
\(572\) 17.3136i 0.723919i
\(573\) 11.2760i 0.471061i
\(574\) −11.4756 −0.478983
\(575\) 0 0
\(576\) −12.9886 −0.541193
\(577\) 23.9595i 0.997448i 0.866761 + 0.498724i \(0.166198\pi\)
−0.866761 + 0.498724i \(0.833802\pi\)
\(578\) − 58.2465i − 2.42274i
\(579\) 6.22383 0.258653
\(580\) 0 0
\(581\) −1.50818 −0.0625698
\(582\) 8.39011i 0.347781i
\(583\) − 3.09159i − 0.128041i
\(584\) 17.6464 0.730213
\(585\) 0 0
\(586\) 45.0892 1.86262
\(587\) − 22.4831i − 0.927978i −0.885841 0.463989i \(-0.846418\pi\)
0.885841 0.463989i \(-0.153582\pi\)
\(588\) 17.7444i 0.731769i
\(589\) −31.4012 −1.29386
\(590\) 0 0
\(591\) −19.7296 −0.811568
\(592\) 1.35122i 0.0555349i
\(593\) 31.0854i 1.27652i 0.769819 + 0.638262i \(0.220346\pi\)
−0.769819 + 0.638262i \(0.779654\pi\)
\(594\) −13.7960 −0.566058
\(595\) 0 0
\(596\) 10.7433 0.440062
\(597\) 18.5731i 0.760148i
\(598\) − 3.44902i − 0.141041i
\(599\) 9.30580 0.380225 0.190112 0.981762i \(-0.439115\pi\)
0.190112 + 0.981762i \(0.439115\pi\)
\(600\) 0 0
\(601\) 39.0074 1.59114 0.795572 0.605859i \(-0.207170\pi\)
0.795572 + 0.605859i \(0.207170\pi\)
\(602\) − 17.8691i − 0.728290i
\(603\) 7.58851i 0.309028i
\(604\) −50.4053 −2.05096
\(605\) 0 0
\(606\) −5.82014 −0.236427
\(607\) 18.6229i 0.755880i 0.925830 + 0.377940i \(0.123368\pi\)
−0.925830 + 0.377940i \(0.876632\pi\)
\(608\) 38.2968i 1.55314i
\(609\) 8.62087 0.349335
\(610\) 0 0
\(611\) −7.19534 −0.291092
\(612\) − 19.6191i − 0.793054i
\(613\) 36.3935i 1.46992i 0.678111 + 0.734960i \(0.262799\pi\)
−0.678111 + 0.734960i \(0.737201\pi\)
\(614\) −55.3683 −2.23448
\(615\) 0 0
\(616\) −14.0761 −0.567142
\(617\) 0.139173i 0.00560289i 0.999996 + 0.00280145i \(0.000891729\pi\)
−0.999996 + 0.00280145i \(0.999108\pi\)
\(618\) − 38.4595i − 1.54707i
\(619\) −1.04686 −0.0420769 −0.0210384 0.999779i \(-0.506697\pi\)
−0.0210384 + 0.999779i \(0.506697\pi\)
\(620\) 0 0
\(621\) 1.64660 0.0660756
\(622\) − 62.7761i − 2.51709i
\(623\) − 8.77910i − 0.351727i
\(624\) 0.978114 0.0391559
\(625\) 0 0
\(626\) −60.7237 −2.42701
\(627\) 35.0483i 1.39970i
\(628\) 25.1959i 1.00543i
\(629\) −8.50261 −0.339021
\(630\) 0 0
\(631\) 25.6813 1.02236 0.511178 0.859475i \(-0.329209\pi\)
0.511178 + 0.859475i \(0.329209\pi\)
\(632\) − 30.7599i − 1.22356i
\(633\) − 2.71998i − 0.108109i
\(634\) 33.4487 1.32842
\(635\) 0 0
\(636\) 1.49624 0.0593298
\(637\) − 5.56671i − 0.220561i
\(638\) − 115.310i − 4.56516i
\(639\) −10.6125 −0.419826
\(640\) 0 0
\(641\) −12.7784 −0.504718 −0.252359 0.967634i \(-0.581206\pi\)
−0.252359 + 0.967634i \(0.581206\pi\)
\(642\) 12.3811i 0.488642i
\(643\) − 14.8374i − 0.585131i −0.956246 0.292565i \(-0.905491\pi\)
0.956246 0.292565i \(-0.0945089\pi\)
\(644\) 5.07671 0.200050
\(645\) 0 0
\(646\) −83.1893 −3.27304
\(647\) 17.9733i 0.706602i 0.935510 + 0.353301i \(0.114941\pi\)
−0.935510 + 0.353301i \(0.885059\pi\)
\(648\) − 2.20956i − 0.0867998i
\(649\) 7.37095 0.289335
\(650\) 0 0
\(651\) −5.70762 −0.223699
\(652\) − 34.0189i − 1.33228i
\(653\) − 3.36997i − 0.131877i −0.997824 0.0659385i \(-0.978996\pi\)
0.997824 0.0659385i \(-0.0210041\pi\)
\(654\) 13.3675 0.522710
\(655\) 0 0
\(656\) 5.19539 0.202846
\(657\) − 7.98638i − 0.311578i
\(658\) − 17.6772i − 0.689128i
\(659\) 28.9767 1.12877 0.564386 0.825511i \(-0.309113\pi\)
0.564386 + 0.825511i \(0.309113\pi\)
\(660\) 0 0
\(661\) 18.3608 0.714153 0.357076 0.934075i \(-0.383774\pi\)
0.357076 + 0.934075i \(0.383774\pi\)
\(662\) − 31.1356i − 1.21012i
\(663\) 6.15481i 0.239033i
\(664\) −3.23088 −0.125382
\(665\) 0 0
\(666\) −2.89365 −0.112127
\(667\) 13.7626i 0.532889i
\(668\) − 2.65106i − 0.102573i
\(669\) −7.04569 −0.272402
\(670\) 0 0
\(671\) −76.3635 −2.94798
\(672\) 6.96099i 0.268526i
\(673\) 15.6585i 0.603591i 0.953373 + 0.301796i \(0.0975860\pi\)
−0.953373 + 0.301796i \(0.902414\pi\)
\(674\) −78.9977 −3.04288
\(675\) 0 0
\(676\) −36.2311 −1.39350
\(677\) 32.5070i 1.24934i 0.780887 + 0.624672i \(0.214767\pi\)
−0.780887 + 0.624672i \(0.785233\pi\)
\(678\) − 0.722892i − 0.0277625i
\(679\) 3.87427 0.148681
\(680\) 0 0
\(681\) 16.9671 0.650181
\(682\) 76.3432i 2.92333i
\(683\) 0.00991954i 0 0.000379561i 1.00000 0.000189780i \(6.04090e-5\pi\)
−1.00000 0.000189780i \(0.999940\pi\)
\(684\) −16.9624 −0.648572
\(685\) 0 0
\(686\) 29.8030 1.13788
\(687\) 5.14606i 0.196334i
\(688\) 8.08993i 0.308426i
\(689\) −0.469394 −0.0178825
\(690\) 0 0
\(691\) 15.1856 0.577688 0.288844 0.957376i \(-0.406729\pi\)
0.288844 + 0.957376i \(0.406729\pi\)
\(692\) − 59.8393i − 2.27475i
\(693\) 6.37054i 0.241997i
\(694\) −20.8701 −0.792217
\(695\) 0 0
\(696\) 18.4679 0.700026
\(697\) 32.6921i 1.23830i
\(698\) − 15.4654i − 0.585373i
\(699\) −27.3476 −1.03438
\(700\) 0 0
\(701\) 38.0017 1.43530 0.717652 0.696402i \(-0.245217\pi\)
0.717652 + 0.696402i \(0.245217\pi\)
\(702\) 2.09464i 0.0790570i
\(703\) 7.35123i 0.277257i
\(704\) 80.2234 3.02353
\(705\) 0 0
\(706\) 11.4788 0.432010
\(707\) 2.68754i 0.101076i
\(708\) 3.56733i 0.134068i
\(709\) −23.2204 −0.872062 −0.436031 0.899932i \(-0.643616\pi\)
−0.436031 + 0.899932i \(0.643616\pi\)
\(710\) 0 0
\(711\) −13.9213 −0.522089
\(712\) − 18.8069i − 0.704819i
\(713\) − 9.11178i − 0.341239i
\(714\) −15.1208 −0.565883
\(715\) 0 0
\(716\) −10.1274 −0.378480
\(717\) 19.6994i 0.735686i
\(718\) 42.5352i 1.58740i
\(719\) −23.3002 −0.868951 −0.434476 0.900684i \(-0.643066\pi\)
−0.434476 + 0.900684i \(0.643066\pi\)
\(720\) 0 0
\(721\) −17.7593 −0.661390
\(722\) 29.4848i 1.09731i
\(723\) 9.57114i 0.355955i
\(724\) −16.2013 −0.602116
\(725\) 0 0
\(726\) 60.6400 2.25056
\(727\) 23.3640i 0.866523i 0.901268 + 0.433261i \(0.142637\pi\)
−0.901268 + 0.433261i \(0.857363\pi\)
\(728\) 2.13716i 0.0792085i
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) −50.9061 −1.88283
\(732\) − 36.9577i − 1.36600i
\(733\) − 10.8277i − 0.399931i −0.979803 0.199966i \(-0.935917\pi\)
0.979803 0.199966i \(-0.0640831\pi\)
\(734\) −31.6992 −1.17004
\(735\) 0 0
\(736\) −11.1127 −0.409619
\(737\) − 46.8699i − 1.72648i
\(738\) 11.1260i 0.409552i
\(739\) −24.5505 −0.903105 −0.451553 0.892244i \(-0.649130\pi\)
−0.451553 + 0.892244i \(0.649130\pi\)
\(740\) 0 0
\(741\) 5.32136 0.195485
\(742\) − 1.15318i − 0.0423347i
\(743\) − 0.745387i − 0.0273456i −0.999907 0.0136728i \(-0.995648\pi\)
0.999907 0.0136728i \(-0.00435232\pi\)
\(744\) −12.2271 −0.448266
\(745\) 0 0
\(746\) 79.2194 2.90043
\(747\) 1.46223i 0.0535000i
\(748\) 121.176i 4.43063i
\(749\) 5.71716 0.208900
\(750\) 0 0
\(751\) 1.84004 0.0671440 0.0335720 0.999436i \(-0.489312\pi\)
0.0335720 + 0.999436i \(0.489312\pi\)
\(752\) 8.00304i 0.291841i
\(753\) 5.82514i 0.212280i
\(754\) −17.5074 −0.637582
\(755\) 0 0
\(756\) −3.08316 −0.112133
\(757\) 24.4525i 0.888742i 0.895843 + 0.444371i \(0.146573\pi\)
−0.895843 + 0.444371i \(0.853427\pi\)
\(758\) 4.46043i 0.162010i
\(759\) −10.1701 −0.369151
\(760\) 0 0
\(761\) 15.6021 0.565574 0.282787 0.959183i \(-0.408741\pi\)
0.282787 + 0.959183i \(0.408741\pi\)
\(762\) − 0.201948i − 0.00731579i
\(763\) − 6.17266i − 0.223465i
\(764\) 33.7063 1.21945
\(765\) 0 0
\(766\) −21.0844 −0.761810
\(767\) − 1.11913i − 0.0404093i
\(768\) 8.67641i 0.313083i
\(769\) 9.63438 0.347425 0.173712 0.984796i \(-0.444424\pi\)
0.173712 + 0.984796i \(0.444424\pi\)
\(770\) 0 0
\(771\) −3.47558 −0.125170
\(772\) − 18.6043i − 0.669585i
\(773\) 5.00666i 0.180077i 0.995938 + 0.0900386i \(0.0286990\pi\)
−0.995938 + 0.0900386i \(0.971301\pi\)
\(774\) −17.3246 −0.622722
\(775\) 0 0
\(776\) 8.29961 0.297939
\(777\) 1.33619i 0.0479356i
\(778\) 2.93803i 0.105334i
\(779\) 28.2651 1.01270
\(780\) 0 0
\(781\) 65.5477 2.34548
\(782\) − 24.1393i − 0.863220i
\(783\) − 8.35819i − 0.298697i
\(784\) −6.19159 −0.221128
\(785\) 0 0
\(786\) −30.3107 −1.08115
\(787\) 45.0282i 1.60508i 0.596597 + 0.802541i \(0.296519\pi\)
−0.596597 + 0.802541i \(0.703481\pi\)
\(788\) 58.9761i 2.10093i
\(789\) 22.7227 0.808948
\(790\) 0 0
\(791\) −0.333807 −0.0118688
\(792\) 13.6472i 0.484933i
\(793\) 11.5942i 0.411722i
\(794\) 56.5306 2.00620
\(795\) 0 0
\(796\) 55.5190 1.96782
\(797\) − 3.26600i − 0.115688i −0.998326 0.0578438i \(-0.981577\pi\)
0.998326 0.0578438i \(-0.0184225\pi\)
\(798\) 13.0733i 0.462789i
\(799\) −50.3593 −1.78158
\(800\) 0 0
\(801\) −8.51161 −0.300743
\(802\) 29.9926i 1.05908i
\(803\) 49.3273i 1.74072i
\(804\) 22.6837 0.799992
\(805\) 0 0
\(806\) 11.5911 0.408280
\(807\) 1.63546i 0.0575710i
\(808\) 5.75736i 0.202543i
\(809\) 24.5785 0.864135 0.432067 0.901841i \(-0.357784\pi\)
0.432067 + 0.901841i \(0.357784\pi\)
\(810\) 0 0
\(811\) −17.0396 −0.598340 −0.299170 0.954200i \(-0.596710\pi\)
−0.299170 + 0.954200i \(0.596710\pi\)
\(812\) − 25.7696i − 0.904336i
\(813\) − 12.4498i − 0.436635i
\(814\) 17.8725 0.626430
\(815\) 0 0
\(816\) 6.84571 0.239648
\(817\) 44.0127i 1.53981i
\(818\) 42.0124i 1.46893i
\(819\) 0.967234 0.0337979
\(820\) 0 0
\(821\) −3.30808 −0.115453 −0.0577264 0.998332i \(-0.518385\pi\)
−0.0577264 + 0.998332i \(0.518385\pi\)
\(822\) − 31.2254i − 1.08911i
\(823\) − 14.5583i − 0.507469i −0.967274 0.253735i \(-0.918341\pi\)
0.967274 0.253735i \(-0.0816590\pi\)
\(824\) −38.0446 −1.32535
\(825\) 0 0
\(826\) 2.74941 0.0956644
\(827\) 8.34632i 0.290230i 0.989415 + 0.145115i \(0.0463552\pi\)
−0.989415 + 0.145115i \(0.953645\pi\)
\(828\) − 4.92203i − 0.171052i
\(829\) −11.1514 −0.387303 −0.193652 0.981070i \(-0.562033\pi\)
−0.193652 + 0.981070i \(0.562033\pi\)
\(830\) 0 0
\(831\) 2.16353 0.0750522
\(832\) − 12.1803i − 0.422274i
\(833\) − 38.9607i − 1.34991i
\(834\) 4.95340 0.171522
\(835\) 0 0
\(836\) 104.767 3.62344
\(837\) 5.53371i 0.191273i
\(838\) 48.6518i 1.68065i
\(839\) −10.3600 −0.357665 −0.178833 0.983879i \(-0.557232\pi\)
−0.178833 + 0.983879i \(0.557232\pi\)
\(840\) 0 0
\(841\) 40.8594 1.40894
\(842\) 63.3641i 2.18367i
\(843\) − 4.97817i − 0.171457i
\(844\) −8.13060 −0.279867
\(845\) 0 0
\(846\) −17.1386 −0.589236
\(847\) − 28.0015i − 0.962144i
\(848\) 0.522085i 0.0179285i
\(849\) 17.3357 0.594958
\(850\) 0 0
\(851\) −2.13313 −0.0731228
\(852\) 31.7232i 1.08682i
\(853\) 16.3865i 0.561064i 0.959845 + 0.280532i \(0.0905108\pi\)
−0.959845 + 0.280532i \(0.909489\pi\)
\(854\) −28.4841 −0.974706
\(855\) 0 0
\(856\) 12.2475 0.418611
\(857\) 0.794707i 0.0271467i 0.999908 + 0.0135733i \(0.00432066\pi\)
−0.999908 + 0.0135733i \(0.995679\pi\)
\(858\) − 12.9374i − 0.441675i
\(859\) 1.06774 0.0364307 0.0182154 0.999834i \(-0.494202\pi\)
0.0182154 + 0.999834i \(0.494202\pi\)
\(860\) 0 0
\(861\) 5.13759 0.175089
\(862\) − 52.9530i − 1.80359i
\(863\) − 49.3832i − 1.68102i −0.541794 0.840511i \(-0.682255\pi\)
0.541794 0.840511i \(-0.317745\pi\)
\(864\) 6.74889 0.229602
\(865\) 0 0
\(866\) 15.9002 0.540310
\(867\) 26.0768i 0.885614i
\(868\) 17.0613i 0.579098i
\(869\) 85.9839 2.91680
\(870\) 0 0
\(871\) −7.11622 −0.241124
\(872\) − 13.2233i − 0.447798i
\(873\) − 3.75623i − 0.127129i
\(874\) −20.8705 −0.705955
\(875\) 0 0
\(876\) −23.8730 −0.806594
\(877\) − 46.1096i − 1.55701i −0.627638 0.778506i \(-0.715978\pi\)
0.627638 0.778506i \(-0.284022\pi\)
\(878\) − 14.9240i − 0.503660i
\(879\) −20.1863 −0.680866
\(880\) 0 0
\(881\) −14.0984 −0.474988 −0.237494 0.971389i \(-0.576326\pi\)
−0.237494 + 0.971389i \(0.576326\pi\)
\(882\) − 13.2593i − 0.446465i
\(883\) 54.9493i 1.84919i 0.380951 + 0.924595i \(0.375597\pi\)
−0.380951 + 0.924595i \(0.624403\pi\)
\(884\) 18.3980 0.618793
\(885\) 0 0
\(886\) 46.2512 1.55384
\(887\) 0.338258i 0.0113576i 0.999984 + 0.00567879i \(0.00180763\pi\)
−0.999984 + 0.00567879i \(0.998192\pi\)
\(888\) 2.86244i 0.0960573i
\(889\) −0.0932527 −0.00312759
\(890\) 0 0
\(891\) 6.17643 0.206918
\(892\) 21.0611i 0.705177i
\(893\) 43.5399i 1.45701i
\(894\) −8.02779 −0.268490
\(895\) 0 0
\(896\) 16.0019 0.534586
\(897\) 1.54412i 0.0515565i
\(898\) 0.487629i 0.0162724i
\(899\) −46.2518 −1.54258
\(900\) 0 0
\(901\) −3.28523 −0.109447
\(902\) − 68.7188i − 2.28808i
\(903\) 7.99994i 0.266221i
\(904\) −0.715094 −0.0237837
\(905\) 0 0
\(906\) 37.6648 1.25133
\(907\) 25.1534i 0.835204i 0.908630 + 0.417602i \(0.137129\pi\)
−0.908630 + 0.417602i \(0.862871\pi\)
\(908\) − 50.7183i − 1.68315i
\(909\) 2.60566 0.0864242
\(910\) 0 0
\(911\) −16.2238 −0.537519 −0.268759 0.963207i \(-0.586614\pi\)
−0.268759 + 0.963207i \(0.586614\pi\)
\(912\) − 5.91870i − 0.195988i
\(913\) − 9.03134i − 0.298894i
\(914\) −13.3139 −0.440386
\(915\) 0 0
\(916\) 15.3827 0.508257
\(917\) 13.9965i 0.462205i
\(918\) 14.6601i 0.483856i
\(919\) −21.5696 −0.711515 −0.355758 0.934578i \(-0.615777\pi\)
−0.355758 + 0.934578i \(0.615777\pi\)
\(920\) 0 0
\(921\) 24.7882 0.816799
\(922\) − 28.6464i − 0.943417i
\(923\) − 9.95205i − 0.327576i
\(924\) 19.0429 0.626466
\(925\) 0 0
\(926\) 60.1258 1.97586
\(927\) 17.2182i 0.565519i
\(928\) 56.4085i 1.85170i
\(929\) 37.6727 1.23600 0.618001 0.786177i \(-0.287943\pi\)
0.618001 + 0.786177i \(0.287943\pi\)
\(930\) 0 0
\(931\) −33.6849 −1.10398
\(932\) 81.7477i 2.67773i
\(933\) 28.1046i 0.920105i
\(934\) 59.9960 1.96313
\(935\) 0 0
\(936\) 2.07204 0.0677269
\(937\) − 42.7073i − 1.39519i −0.716495 0.697593i \(-0.754255\pi\)
0.716495 0.697593i \(-0.245745\pi\)
\(938\) − 17.4828i − 0.570834i
\(939\) 27.1858 0.887176
\(940\) 0 0
\(941\) −4.27023 −0.139206 −0.0696028 0.997575i \(-0.522173\pi\)
−0.0696028 + 0.997575i \(0.522173\pi\)
\(942\) − 18.8274i − 0.613429i
\(943\) 8.20179i 0.267087i
\(944\) −1.24475 −0.0405132
\(945\) 0 0
\(946\) 107.005 3.47902
\(947\) − 26.4988i − 0.861097i −0.902567 0.430548i \(-0.858320\pi\)
0.902567 0.430548i \(-0.141680\pi\)
\(948\) 41.6137i 1.35155i
\(949\) 7.48933 0.243114
\(950\) 0 0
\(951\) −14.9749 −0.485594
\(952\) 14.9577i 0.484783i
\(953\) 16.9907i 0.550383i 0.961389 + 0.275191i \(0.0887412\pi\)
−0.961389 + 0.275191i \(0.911259\pi\)
\(954\) −1.11805 −0.0361981
\(955\) 0 0
\(956\) 58.8856 1.90450
\(957\) 51.6238i 1.66876i
\(958\) − 87.7719i − 2.83578i
\(959\) −14.4188 −0.465608
\(960\) 0 0
\(961\) −0.378072 −0.0121959
\(962\) − 2.71356i − 0.0874887i
\(963\) − 5.54296i − 0.178619i
\(964\) 28.6102 0.921472
\(965\) 0 0
\(966\) −3.79351 −0.122054
\(967\) 2.79897i 0.0900089i 0.998987 + 0.0450045i \(0.0143302\pi\)
−0.998987 + 0.0450045i \(0.985670\pi\)
\(968\) − 59.9859i − 1.92802i
\(969\) 37.2436 1.19644
\(970\) 0 0
\(971\) −5.63046 −0.180690 −0.0903450 0.995911i \(-0.528797\pi\)
−0.0903450 + 0.995911i \(0.528797\pi\)
\(972\) 2.98921i 0.0958791i
\(973\) − 2.28731i − 0.0733279i
\(974\) −83.3753 −2.67152
\(975\) 0 0
\(976\) 12.8957 0.412781
\(977\) − 40.5445i − 1.29713i −0.761157 0.648567i \(-0.775368\pi\)
0.761157 0.648567i \(-0.224632\pi\)
\(978\) 25.4202i 0.812850i
\(979\) 52.5714 1.68019
\(980\) 0 0
\(981\) −5.98458 −0.191073
\(982\) − 37.5529i − 1.19836i
\(983\) 16.6381i 0.530674i 0.964156 + 0.265337i \(0.0854832\pi\)
−0.964156 + 0.265337i \(0.914517\pi\)
\(984\) 11.0059 0.350857
\(985\) 0 0
\(986\) −122.532 −3.90222
\(987\) 7.91401i 0.251906i
\(988\) − 15.9067i − 0.506059i
\(989\) −12.7713 −0.406104
\(990\) 0 0
\(991\) 1.05069 0.0333764 0.0166882 0.999861i \(-0.494688\pi\)
0.0166882 + 0.999861i \(0.494688\pi\)
\(992\) − 37.3464i − 1.18575i
\(993\) 13.9393i 0.442350i
\(994\) 24.4497 0.775498
\(995\) 0 0
\(996\) 4.37090 0.138497
\(997\) − 0.318717i − 0.0100939i −0.999987 0.00504693i \(-0.998394\pi\)
0.999987 0.00504693i \(-0.00160650\pi\)
\(998\) 46.2300i 1.46339i
\(999\) 1.29548 0.0409872
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 1875.2.b.g.1249.14 16
5.2 odd 4 1875.2.a.n.1.2 8
5.3 odd 4 1875.2.a.o.1.7 yes 8
5.4 even 2 inner 1875.2.b.g.1249.3 16
15.2 even 4 5625.2.a.bc.1.7 8
15.8 even 4 5625.2.a.u.1.2 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1875.2.a.n.1.2 8 5.2 odd 4
1875.2.a.o.1.7 yes 8 5.3 odd 4
1875.2.b.g.1249.3 16 5.4 even 2 inner
1875.2.b.g.1249.14 16 1.1 even 1 trivial
5625.2.a.u.1.2 8 15.8 even 4
5625.2.a.bc.1.7 8 15.2 even 4