Properties

Label 1380.2.t.a.1057.10
Level $1380$
Weight $2$
Character 1380.1057
Analytic conductor $11.019$
Analytic rank $0$
Dimension $48$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1380,2,Mod(1057,1380)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1380, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 0, 1, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1380.1057");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1380 = 2^{2} \cdot 3 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1380.t (of order \(4\), degree \(2\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(11.0193554789\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(24\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 1057.10
Character \(\chi\) \(=\) 1380.1057
Dual form 1380.2.t.a.1333.10

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.707107 + 0.707107i) q^{3} +(2.09989 - 0.768407i) q^{5} +(0.0117285 - 0.0117285i) q^{7} -1.00000i q^{9} +O(q^{10})\) \(q+(-0.707107 + 0.707107i) q^{3} +(2.09989 - 0.768407i) q^{5} +(0.0117285 - 0.0117285i) q^{7} -1.00000i q^{9} -1.72770i q^{11} +(-2.37219 + 2.37219i) q^{13} +(-0.941503 + 2.02819i) q^{15} +(-1.66583 + 1.66583i) q^{17} +6.73367 q^{19} +0.0165866i q^{21} +(4.79553 + 0.0541507i) q^{23} +(3.81910 - 3.22714i) q^{25} +(0.707107 + 0.707107i) q^{27} -0.894482i q^{29} +6.90064 q^{31} +(1.22167 + 1.22167i) q^{33} +(0.0156163 - 0.0336408i) q^{35} +(0.0575199 - 0.0575199i) q^{37} -3.35478i q^{39} +0.278794 q^{41} +(-4.31887 - 4.31887i) q^{43} +(-0.768407 - 2.09989i) q^{45} +(-7.61450 - 7.61450i) q^{47} +6.99972i q^{49} -2.35584i q^{51} +(9.64615 + 9.64615i) q^{53} +(-1.32757 - 3.62798i) q^{55} +(-4.76142 + 4.76142i) q^{57} -15.1371i q^{59} +5.91692i q^{61} +(-0.0117285 - 0.0117285i) q^{63} +(-3.15853 + 6.80414i) q^{65} +(1.99060 - 1.99060i) q^{67} +(-3.42924 + 3.35266i) q^{69} +12.6488 q^{71} +(7.55474 - 7.55474i) q^{73} +(-0.418577 + 4.98245i) q^{75} +(-0.0202632 - 0.0202632i) q^{77} -3.91526 q^{79} -1.00000 q^{81} +(12.2961 + 12.2961i) q^{83} +(-2.21803 + 4.77810i) q^{85} +(0.632494 + 0.632494i) q^{87} +6.61134 q^{89} +0.0556442i q^{91} +(-4.87949 + 4.87949i) q^{93} +(14.1400 - 5.17420i) q^{95} +(-9.80227 + 9.80227i) q^{97} -1.72770 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 48 q+O(q^{10}) \) Copy content Toggle raw display \( 48 q - 8 q^{13} - 16 q^{23} - 8 q^{25} + 8 q^{31} + 8 q^{35} - 24 q^{41} + 8 q^{47} - 32 q^{55} - 24 q^{71} + 8 q^{73} + 32 q^{75} + 40 q^{77} - 48 q^{81} + 24 q^{85} - 40 q^{87}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1380\mathbb{Z}\right)^\times\).

\(n\) \(277\) \(461\) \(691\) \(1201\)
\(\chi(n)\) \(e\left(\frac{1}{4}\right)\) \(1\) \(1\) \(-1\)

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
\(3\) −0.707107 + 0.707107i −0.408248 + 0.408248i
\(4\) 0 0
\(5\) 2.09989 0.768407i 0.939101 0.343642i
\(6\) 0 0
\(7\) 0.0117285 0.0117285i 0.00443294 0.00443294i −0.704887 0.709320i \(-0.749002\pi\)
0.709320 + 0.704887i \(0.249002\pi\)
\(8\) 0 0
\(9\) 1.00000i 0.333333i
\(10\) 0 0
\(11\) 1.72770i 0.520920i −0.965485 0.260460i \(-0.916126\pi\)
0.965485 0.260460i \(-0.0838742\pi\)
\(12\) 0 0
\(13\) −2.37219 + 2.37219i −0.657926 + 0.657926i −0.954889 0.296963i \(-0.904026\pi\)
0.296963 + 0.954889i \(0.404026\pi\)
\(14\) 0 0
\(15\) −0.941503 + 2.02819i −0.243095 + 0.523678i
\(16\) 0 0
\(17\) −1.66583 + 1.66583i −0.404023 + 0.404023i −0.879648 0.475625i \(-0.842222\pi\)
0.475625 + 0.879648i \(0.342222\pi\)
\(18\) 0 0
\(19\) 6.73367 1.54481 0.772405 0.635131i \(-0.219054\pi\)
0.772405 + 0.635131i \(0.219054\pi\)
\(20\) 0 0
\(21\) 0.0165866i 0.00361948i
\(22\) 0 0
\(23\) 4.79553 + 0.0541507i 0.999936 + 0.0112912i
\(24\) 0 0
\(25\) 3.81910 3.22714i 0.763820 0.645429i
\(26\) 0 0
\(27\) 0.707107 + 0.707107i 0.136083 + 0.136083i
\(28\) 0 0
\(29\) 0.894482i 0.166101i −0.996545 0.0830506i \(-0.973534\pi\)
0.996545 0.0830506i \(-0.0264663\pi\)
\(30\) 0 0
\(31\) 6.90064 1.23939 0.619696 0.784842i \(-0.287256\pi\)
0.619696 + 0.784842i \(0.287256\pi\)
\(32\) 0 0
\(33\) 1.22167 + 1.22167i 0.212665 + 0.212665i
\(34\) 0 0
\(35\) 0.0156163 0.0336408i 0.00263963 0.00568633i
\(36\) 0 0
\(37\) 0.0575199 0.0575199i 0.00945621 0.00945621i −0.702363 0.711819i \(-0.747872\pi\)
0.711819 + 0.702363i \(0.247872\pi\)
\(38\) 0 0
\(39\) 3.35478i 0.537194i
\(40\) 0 0
\(41\) 0.278794 0.0435403 0.0217702 0.999763i \(-0.493070\pi\)
0.0217702 + 0.999763i \(0.493070\pi\)
\(42\) 0 0
\(43\) −4.31887 4.31887i −0.658622 0.658622i 0.296432 0.955054i \(-0.404203\pi\)
−0.955054 + 0.296432i \(0.904203\pi\)
\(44\) 0 0
\(45\) −0.768407 2.09989i −0.114547 0.313034i
\(46\) 0 0
\(47\) −7.61450 7.61450i −1.11069 1.11069i −0.993057 0.117632i \(-0.962470\pi\)
−0.117632 0.993057i \(-0.537530\pi\)
\(48\) 0 0
\(49\) 6.99972i 0.999961i
\(50\) 0 0
\(51\) 2.35584i 0.329883i
\(52\) 0 0
\(53\) 9.64615 + 9.64615i 1.32500 + 1.32500i 0.909671 + 0.415329i \(0.136334\pi\)
0.415329 + 0.909671i \(0.363666\pi\)
\(54\) 0 0
\(55\) −1.32757 3.62798i −0.179010 0.489196i
\(56\) 0 0
\(57\) −4.76142 + 4.76142i −0.630666 + 0.630666i
\(58\) 0 0
\(59\) 15.1371i 1.97068i −0.170607 0.985339i \(-0.554573\pi\)
0.170607 0.985339i \(-0.445427\pi\)
\(60\) 0 0
\(61\) 5.91692i 0.757584i 0.925482 + 0.378792i \(0.123660\pi\)
−0.925482 + 0.378792i \(0.876340\pi\)
\(62\) 0 0
\(63\) −0.0117285 0.0117285i −0.00147765 0.00147765i
\(64\) 0 0
\(65\) −3.15853 + 6.80414i −0.391768 + 0.843950i
\(66\) 0 0
\(67\) 1.99060 1.99060i 0.243191 0.243191i −0.574978 0.818169i \(-0.694989\pi\)
0.818169 + 0.574978i \(0.194989\pi\)
\(68\) 0 0
\(69\) −3.42924 + 3.35266i −0.412832 + 0.403613i
\(70\) 0 0
\(71\) 12.6488 1.50113 0.750567 0.660795i \(-0.229781\pi\)
0.750567 + 0.660795i \(0.229781\pi\)
\(72\) 0 0
\(73\) 7.55474 7.55474i 0.884215 0.884215i −0.109745 0.993960i \(-0.535003\pi\)
0.993960 + 0.109745i \(0.0350034\pi\)
\(74\) 0 0
\(75\) −0.418577 + 4.98245i −0.0483331 + 0.575324i
\(76\) 0 0
\(77\) −0.0202632 0.0202632i −0.00230921 0.00230921i
\(78\) 0 0
\(79\) −3.91526 −0.440502 −0.220251 0.975443i \(-0.570688\pi\)
−0.220251 + 0.975443i \(0.570688\pi\)
\(80\) 0 0
\(81\) −1.00000 −0.111111
\(82\) 0 0
\(83\) 12.2961 + 12.2961i 1.34967 + 1.34967i 0.886014 + 0.463659i \(0.153464\pi\)
0.463659 + 0.886014i \(0.346536\pi\)
\(84\) 0 0
\(85\) −2.21803 + 4.77810i −0.240579 + 0.518257i
\(86\) 0 0
\(87\) 0.632494 + 0.632494i 0.0678105 + 0.0678105i
\(88\) 0 0
\(89\) 6.61134 0.700800 0.350400 0.936600i \(-0.386046\pi\)
0.350400 + 0.936600i \(0.386046\pi\)
\(90\) 0 0
\(91\) 0.0556442i 0.00583310i
\(92\) 0 0
\(93\) −4.87949 + 4.87949i −0.505979 + 0.505979i
\(94\) 0 0
\(95\) 14.1400 5.17420i 1.45073 0.530861i
\(96\) 0 0
\(97\) −9.80227 + 9.80227i −0.995269 + 0.995269i −0.999989 0.00471962i \(-0.998498\pi\)
0.00471962 + 0.999989i \(0.498498\pi\)
\(98\) 0 0
\(99\) −1.72770 −0.173640
\(100\) 0 0
\(101\) 8.42053 0.837874 0.418937 0.908015i \(-0.362403\pi\)
0.418937 + 0.908015i \(0.362403\pi\)
\(102\) 0 0
\(103\) 11.3577 + 11.3577i 1.11910 + 1.11910i 0.991873 + 0.127232i \(0.0406092\pi\)
0.127232 + 0.991873i \(0.459391\pi\)
\(104\) 0 0
\(105\) 0.0127452 + 0.0348300i 0.00124381 + 0.00339906i
\(106\) 0 0
\(107\) 7.94663 7.94663i 0.768230 0.768230i −0.209565 0.977795i \(-0.567205\pi\)
0.977795 + 0.209565i \(0.0672047\pi\)
\(108\) 0 0
\(109\) −10.3003 −0.986589 −0.493295 0.869862i \(-0.664208\pi\)
−0.493295 + 0.869862i \(0.664208\pi\)
\(110\) 0 0
\(111\) 0.0813454i 0.00772096i
\(112\) 0 0
\(113\) −6.53608 6.53608i −0.614863 0.614863i 0.329346 0.944209i \(-0.393172\pi\)
−0.944209 + 0.329346i \(0.893172\pi\)
\(114\) 0 0
\(115\) 10.1117 3.57120i 0.942921 0.333016i
\(116\) 0 0
\(117\) 2.37219 + 2.37219i 0.219309 + 0.219309i
\(118\) 0 0
\(119\) 0.0390752i 0.00358202i
\(120\) 0 0
\(121\) 8.01507 0.728642
\(122\) 0 0
\(123\) −0.197137 + 0.197137i −0.0177753 + 0.0177753i
\(124\) 0 0
\(125\) 5.53995 9.71128i 0.495508 0.868603i
\(126\) 0 0
\(127\) −2.73971 2.73971i −0.243110 0.243110i 0.575026 0.818135i \(-0.304992\pi\)
−0.818135 + 0.575026i \(0.804992\pi\)
\(128\) 0 0
\(129\) 6.10781 0.537762
\(130\) 0 0
\(131\) −11.6163 −1.01492 −0.507458 0.861676i \(-0.669415\pi\)
−0.507458 + 0.861676i \(0.669415\pi\)
\(132\) 0 0
\(133\) 0.0789756 0.0789756i 0.00684805 0.00684805i
\(134\) 0 0
\(135\) 2.02819 + 0.941503i 0.174559 + 0.0810317i
\(136\) 0 0
\(137\) −1.80967 + 1.80967i −0.154611 + 0.154611i −0.780174 0.625563i \(-0.784869\pi\)
0.625563 + 0.780174i \(0.284869\pi\)
\(138\) 0 0
\(139\) 5.50003i 0.466507i −0.972416 0.233253i \(-0.925063\pi\)
0.972416 0.233253i \(-0.0749371\pi\)
\(140\) 0 0
\(141\) 10.7685 0.906874
\(142\) 0 0
\(143\) 4.09842 + 4.09842i 0.342727 + 0.342727i
\(144\) 0 0
\(145\) −0.687326 1.87832i −0.0570793 0.155986i
\(146\) 0 0
\(147\) −4.94955 4.94955i −0.408232 0.408232i
\(148\) 0 0
\(149\) −1.57524 −0.129049 −0.0645243 0.997916i \(-0.520553\pi\)
−0.0645243 + 0.997916i \(0.520553\pi\)
\(150\) 0 0
\(151\) −17.0184 −1.38494 −0.692469 0.721448i \(-0.743477\pi\)
−0.692469 + 0.721448i \(0.743477\pi\)
\(152\) 0 0
\(153\) 1.66583 + 1.66583i 0.134674 + 0.134674i
\(154\) 0 0
\(155\) 14.4906 5.30250i 1.16391 0.425907i
\(156\) 0 0
\(157\) −8.29982 + 8.29982i −0.662398 + 0.662398i −0.955945 0.293547i \(-0.905164\pi\)
0.293547 + 0.955945i \(0.405164\pi\)
\(158\) 0 0
\(159\) −13.6417 −1.08186
\(160\) 0 0
\(161\) 0.0568793 0.0556091i 0.00448271 0.00438261i
\(162\) 0 0
\(163\) −16.4524 + 16.4524i −1.28865 + 1.28865i −0.353044 + 0.935607i \(0.614853\pi\)
−0.935607 + 0.353044i \(0.885147\pi\)
\(164\) 0 0
\(165\) 3.50410 + 1.62663i 0.272794 + 0.126633i
\(166\) 0 0
\(167\) −3.26933 3.26933i −0.252989 0.252989i 0.569206 0.822195i \(-0.307251\pi\)
−0.822195 + 0.569206i \(0.807251\pi\)
\(168\) 0 0
\(169\) 1.74546i 0.134266i
\(170\) 0 0
\(171\) 6.73367i 0.514936i
\(172\) 0 0
\(173\) 3.37850 3.37850i 0.256862 0.256862i −0.566914 0.823777i \(-0.691863\pi\)
0.823777 + 0.566914i \(0.191863\pi\)
\(174\) 0 0
\(175\) 0.00694275 0.0826417i 0.000524823 0.00624712i
\(176\) 0 0
\(177\) 10.7035 + 10.7035i 0.804526 + 0.804526i
\(178\) 0 0
\(179\) 2.90634i 0.217230i 0.994084 + 0.108615i \(0.0346416\pi\)
−0.994084 + 0.108615i \(0.965358\pi\)
\(180\) 0 0
\(181\) 4.39491i 0.326671i −0.986571 0.163336i \(-0.947775\pi\)
0.986571 0.163336i \(-0.0522254\pi\)
\(182\) 0 0
\(183\) −4.18390 4.18390i −0.309283 0.309283i
\(184\) 0 0
\(185\) 0.0765869 0.164984i 0.00563078 0.0121299i
\(186\) 0 0
\(187\) 2.87805 + 2.87805i 0.210464 + 0.210464i
\(188\) 0 0
\(189\) 0.0165866 0.00120649
\(190\) 0 0
\(191\) 7.01334i 0.507467i −0.967274 0.253734i \(-0.918341\pi\)
0.967274 0.253734i \(-0.0816587\pi\)
\(192\) 0 0
\(193\) −10.7410 + 10.7410i −0.773151 + 0.773151i −0.978656 0.205505i \(-0.934116\pi\)
0.205505 + 0.978656i \(0.434116\pi\)
\(194\) 0 0
\(195\) −2.57783 7.04468i −0.184603 0.504480i
\(196\) 0 0
\(197\) −1.15071 1.15071i −0.0819848 0.0819848i 0.664925 0.746910i \(-0.268463\pi\)
−0.746910 + 0.664925i \(0.768463\pi\)
\(198\) 0 0
\(199\) 17.6728 1.25279 0.626397 0.779504i \(-0.284529\pi\)
0.626397 + 0.779504i \(0.284529\pi\)
\(200\) 0 0
\(201\) 2.81513i 0.198564i
\(202\) 0 0
\(203\) −0.0104909 0.0104909i −0.000736317 0.000736317i
\(204\) 0 0
\(205\) 0.585438 0.214227i 0.0408888 0.0149623i
\(206\) 0 0
\(207\) 0.0541507 4.79553i 0.00376374 0.333312i
\(208\) 0 0
\(209\) 11.6337i 0.804722i
\(210\) 0 0
\(211\) 6.29187 0.433150 0.216575 0.976266i \(-0.430511\pi\)
0.216575 + 0.976266i \(0.430511\pi\)
\(212\) 0 0
\(213\) −8.94403 + 8.94403i −0.612835 + 0.612835i
\(214\) 0 0
\(215\) −12.3878 5.75052i −0.844842 0.392182i
\(216\) 0 0
\(217\) 0.0809339 0.0809339i 0.00549415 0.00549415i
\(218\) 0 0
\(219\) 10.6840i 0.721958i
\(220\) 0 0
\(221\) 7.90331i 0.531634i
\(222\) 0 0
\(223\) −6.61400 + 6.61400i −0.442906 + 0.442906i −0.892988 0.450081i \(-0.851395\pi\)
0.450081 + 0.892988i \(0.351395\pi\)
\(224\) 0 0
\(225\) −3.22714 3.81910i −0.215143 0.254607i
\(226\) 0 0
\(227\) 0.842807 0.842807i 0.0559391 0.0559391i −0.678584 0.734523i \(-0.737406\pi\)
0.734523 + 0.678584i \(0.237406\pi\)
\(228\) 0 0
\(229\) −20.3773 −1.34657 −0.673285 0.739383i \(-0.735117\pi\)
−0.673285 + 0.739383i \(0.735117\pi\)
\(230\) 0 0
\(231\) 0.0286565 0.00188546
\(232\) 0 0
\(233\) −15.0006 + 15.0006i −0.982722 + 0.982722i −0.999853 0.0171312i \(-0.994547\pi\)
0.0171312 + 0.999853i \(0.494547\pi\)
\(234\) 0 0
\(235\) −21.8407 10.1386i −1.42473 0.661370i
\(236\) 0 0
\(237\) 2.76851 2.76851i 0.179834 0.179834i
\(238\) 0 0
\(239\) 17.0614i 1.10361i −0.833974 0.551804i \(-0.813940\pi\)
0.833974 0.551804i \(-0.186060\pi\)
\(240\) 0 0
\(241\) 23.3681i 1.50527i −0.658438 0.752635i \(-0.728783\pi\)
0.658438 0.752635i \(-0.271217\pi\)
\(242\) 0 0
\(243\) 0.707107 0.707107i 0.0453609 0.0453609i
\(244\) 0 0
\(245\) 5.37864 + 14.6987i 0.343628 + 0.939064i
\(246\) 0 0
\(247\) −15.9735 + 15.9735i −1.01637 + 1.01637i
\(248\) 0 0
\(249\) −17.3893 −1.10200
\(250\) 0 0
\(251\) 11.0277i 0.696063i 0.937483 + 0.348032i \(0.113150\pi\)
−0.937483 + 0.348032i \(0.886850\pi\)
\(252\) 0 0
\(253\) 0.0935560 8.28521i 0.00588181 0.520887i
\(254\) 0 0
\(255\) −1.81024 4.94701i −0.113362 0.309794i
\(256\) 0 0
\(257\) −1.64897 1.64897i −0.102860 0.102860i 0.653804 0.756664i \(-0.273172\pi\)
−0.756664 + 0.653804i \(0.773172\pi\)
\(258\) 0 0
\(259\) 0.00134924i 8.38377e-5i
\(260\) 0 0
\(261\) −0.894482 −0.0553670
\(262\) 0 0
\(263\) −1.85419 1.85419i −0.114334 0.114334i 0.647625 0.761959i \(-0.275762\pi\)
−0.761959 + 0.647625i \(0.775762\pi\)
\(264\) 0 0
\(265\) 27.6680 + 12.8437i 1.69963 + 0.788983i
\(266\) 0 0
\(267\) −4.67492 + 4.67492i −0.286101 + 0.286101i
\(268\) 0 0
\(269\) 18.0253i 1.09902i −0.835486 0.549511i \(-0.814814\pi\)
0.835486 0.549511i \(-0.185186\pi\)
\(270\) 0 0
\(271\) −31.0616 −1.88686 −0.943429 0.331574i \(-0.892420\pi\)
−0.943429 + 0.331574i \(0.892420\pi\)
\(272\) 0 0
\(273\) −0.0393464 0.0393464i −0.00238135 0.00238135i
\(274\) 0 0
\(275\) −5.57552 6.59825i −0.336217 0.397889i
\(276\) 0 0
\(277\) −17.2388 17.2388i −1.03578 1.03578i −0.999336 0.0364415i \(-0.988398\pi\)
−0.0364415 0.999336i \(-0.511602\pi\)
\(278\) 0 0
\(279\) 6.90064i 0.413130i
\(280\) 0 0
\(281\) 28.7503i 1.71510i −0.514401 0.857550i \(-0.671986\pi\)
0.514401 0.857550i \(-0.328014\pi\)
\(282\) 0 0
\(283\) 1.63943 + 1.63943i 0.0974541 + 0.0974541i 0.754153 0.656699i \(-0.228048\pi\)
−0.656699 + 0.754153i \(0.728048\pi\)
\(284\) 0 0
\(285\) −6.33977 + 13.6572i −0.375535 + 0.808982i
\(286\) 0 0
\(287\) 0.00326983 0.00326983i 0.000193012 0.000193012i
\(288\) 0 0
\(289\) 11.4500i 0.673531i
\(290\) 0 0
\(291\) 13.8625i 0.812634i
\(292\) 0 0
\(293\) 2.90209 + 2.90209i 0.169542 + 0.169542i 0.786778 0.617236i \(-0.211748\pi\)
−0.617236 + 0.786778i \(0.711748\pi\)
\(294\) 0 0
\(295\) −11.6314 31.7862i −0.677208 1.85067i
\(296\) 0 0
\(297\) 1.22167 1.22167i 0.0708882 0.0708882i
\(298\) 0 0
\(299\) −11.5043 + 11.2474i −0.665313 + 0.650455i
\(300\) 0 0
\(301\) −0.101307 −0.00583926
\(302\) 0 0
\(303\) −5.95421 + 5.95421i −0.342060 + 0.342060i
\(304\) 0 0
\(305\) 4.54660 + 12.4249i 0.260338 + 0.711448i
\(306\) 0 0
\(307\) −20.9933 20.9933i −1.19815 1.19815i −0.974719 0.223434i \(-0.928273\pi\)
−0.223434 0.974719i \(-0.571727\pi\)
\(308\) 0 0
\(309\) −16.0622 −0.913745
\(310\) 0 0
\(311\) 26.0182 1.47536 0.737678 0.675153i \(-0.235922\pi\)
0.737678 + 0.675153i \(0.235922\pi\)
\(312\) 0 0
\(313\) 13.7233 + 13.7233i 0.775690 + 0.775690i 0.979095 0.203405i \(-0.0652008\pi\)
−0.203405 + 0.979095i \(0.565201\pi\)
\(314\) 0 0
\(315\) −0.0336408 0.0156163i −0.00189544 0.000879878i
\(316\) 0 0
\(317\) −18.5746 18.5746i −1.04325 1.04325i −0.999021 0.0442316i \(-0.985916\pi\)
−0.0442316 0.999021i \(-0.514084\pi\)
\(318\) 0 0
\(319\) −1.54539 −0.0865254
\(320\) 0 0
\(321\) 11.2382i 0.627257i
\(322\) 0 0
\(323\) −11.2171 + 11.2171i −0.624138 + 0.624138i
\(324\) 0 0
\(325\) −1.40423 + 16.7150i −0.0778929 + 0.927182i
\(326\) 0 0
\(327\) 7.28341 7.28341i 0.402773 0.402773i
\(328\) 0 0
\(329\) −0.178613 −0.00984725
\(330\) 0 0
\(331\) −8.54938 −0.469916 −0.234958 0.972006i \(-0.575495\pi\)
−0.234958 + 0.972006i \(0.575495\pi\)
\(332\) 0 0
\(333\) −0.0575199 0.0575199i −0.00315207 0.00315207i
\(334\) 0 0
\(335\) 2.65046 5.70964i 0.144810 0.311951i
\(336\) 0 0
\(337\) −13.7074 + 13.7074i −0.746687 + 0.746687i −0.973855 0.227168i \(-0.927053\pi\)
0.227168 + 0.973855i \(0.427053\pi\)
\(338\) 0 0
\(339\) 9.24341 0.502033
\(340\) 0 0
\(341\) 11.9222i 0.645624i
\(342\) 0 0
\(343\) 0.164195 + 0.164195i 0.00886571 + 0.00886571i
\(344\) 0 0
\(345\) −4.62483 + 9.67528i −0.248992 + 0.520899i
\(346\) 0 0
\(347\) 7.02177 + 7.02177i 0.376949 + 0.376949i 0.870000 0.493052i \(-0.164119\pi\)
−0.493052 + 0.870000i \(0.664119\pi\)
\(348\) 0 0
\(349\) 21.7920i 1.16650i 0.812294 + 0.583249i \(0.198219\pi\)
−0.812294 + 0.583249i \(0.801781\pi\)
\(350\) 0 0
\(351\) −3.35478 −0.179065
\(352\) 0 0
\(353\) −14.0973 + 14.0973i −0.750326 + 0.750326i −0.974540 0.224214i \(-0.928019\pi\)
0.224214 + 0.974540i \(0.428019\pi\)
\(354\) 0 0
\(355\) 26.5611 9.71940i 1.40972 0.515852i
\(356\) 0 0
\(357\) −0.0276304 0.0276304i −0.00146235 0.00146235i
\(358\) 0 0
\(359\) 21.4103 1.12999 0.564997 0.825093i \(-0.308877\pi\)
0.564997 + 0.825093i \(0.308877\pi\)
\(360\) 0 0
\(361\) 26.3423 1.38644
\(362\) 0 0
\(363\) −5.66751 + 5.66751i −0.297467 + 0.297467i
\(364\) 0 0
\(365\) 10.0590 21.6692i 0.526514 1.13422i
\(366\) 0 0
\(367\) 13.3282 13.3282i 0.695726 0.695726i −0.267760 0.963486i \(-0.586283\pi\)
0.963486 + 0.267760i \(0.0862834\pi\)
\(368\) 0 0
\(369\) 0.278794i 0.0145134i
\(370\) 0 0
\(371\) 0.226269 0.0117473
\(372\) 0 0
\(373\) −24.6632 24.6632i −1.27701 1.27701i −0.942331 0.334681i \(-0.891371\pi\)
−0.334681 0.942331i \(-0.608629\pi\)
\(374\) 0 0
\(375\) 2.94958 + 10.7842i 0.152316 + 0.556896i
\(376\) 0 0
\(377\) 2.12188 + 2.12188i 0.109282 + 0.109282i
\(378\) 0 0
\(379\) −11.8558 −0.608994 −0.304497 0.952513i \(-0.598488\pi\)
−0.304497 + 0.952513i \(0.598488\pi\)
\(380\) 0 0
\(381\) 3.87453 0.198498
\(382\) 0 0
\(383\) 11.5598 + 11.5598i 0.590679 + 0.590679i 0.937815 0.347136i \(-0.112846\pi\)
−0.347136 + 0.937815i \(0.612846\pi\)
\(384\) 0 0
\(385\) −0.0581210 0.0269802i −0.00296212 0.00137504i
\(386\) 0 0
\(387\) −4.31887 + 4.31887i −0.219541 + 0.219541i
\(388\) 0 0
\(389\) −11.8359 −0.600106 −0.300053 0.953923i \(-0.597004\pi\)
−0.300053 + 0.953923i \(0.597004\pi\)
\(390\) 0 0
\(391\) −8.07873 + 7.89832i −0.408559 + 0.399435i
\(392\) 0 0
\(393\) 8.21393 8.21393i 0.414338 0.414338i
\(394\) 0 0
\(395\) −8.22164 + 3.00852i −0.413675 + 0.151375i
\(396\) 0 0
\(397\) 3.53582 + 3.53582i 0.177458 + 0.177458i 0.790247 0.612789i \(-0.209952\pi\)
−0.612789 + 0.790247i \(0.709952\pi\)
\(398\) 0 0
\(399\) 0.111688i 0.00559141i
\(400\) 0 0
\(401\) 22.6536i 1.13127i 0.824657 + 0.565633i \(0.191368\pi\)
−0.824657 + 0.565633i \(0.808632\pi\)
\(402\) 0 0
\(403\) −16.3696 + 16.3696i −0.815428 + 0.815428i
\(404\) 0 0
\(405\) −2.09989 + 0.768407i −0.104345 + 0.0381824i
\(406\) 0 0
\(407\) −0.0993769 0.0993769i −0.00492593 0.00492593i
\(408\) 0 0
\(409\) 2.72111i 0.134550i −0.997734 0.0672750i \(-0.978570\pi\)
0.997734 0.0672750i \(-0.0214305\pi\)
\(410\) 0 0
\(411\) 2.55927i 0.126239i
\(412\) 0 0
\(413\) −0.177535 0.177535i −0.00873590 0.00873590i
\(414\) 0 0
\(415\) 35.2689 + 16.3721i 1.73128 + 0.803675i
\(416\) 0 0
\(417\) 3.88911 + 3.88911i 0.190451 + 0.190451i
\(418\) 0 0
\(419\) −29.6820 −1.45006 −0.725030 0.688717i \(-0.758174\pi\)
−0.725030 + 0.688717i \(0.758174\pi\)
\(420\) 0 0
\(421\) 12.9116i 0.629271i 0.949213 + 0.314636i \(0.101882\pi\)
−0.949213 + 0.314636i \(0.898118\pi\)
\(422\) 0 0
\(423\) −7.61450 + 7.61450i −0.370230 + 0.370230i
\(424\) 0 0
\(425\) −0.986100 + 11.7378i −0.0478329 + 0.569369i
\(426\) 0 0
\(427\) 0.0693964 + 0.0693964i 0.00335833 + 0.00335833i
\(428\) 0 0
\(429\) −5.79604 −0.279835
\(430\) 0 0
\(431\) 17.6210i 0.848772i 0.905481 + 0.424386i \(0.139510\pi\)
−0.905481 + 0.424386i \(0.860490\pi\)
\(432\) 0 0
\(433\) 9.30573 + 9.30573i 0.447205 + 0.447205i 0.894424 0.447220i \(-0.147586\pi\)
−0.447220 + 0.894424i \(0.647586\pi\)
\(434\) 0 0
\(435\) 1.81418 + 0.842157i 0.0869834 + 0.0403784i
\(436\) 0 0
\(437\) 32.2915 + 0.364633i 1.54471 + 0.0174428i
\(438\) 0 0
\(439\) 2.67753i 0.127791i −0.997957 0.0638957i \(-0.979648\pi\)
0.997957 0.0638957i \(-0.0203525\pi\)
\(440\) 0 0
\(441\) 6.99972 0.333320
\(442\) 0 0
\(443\) 14.8957 14.8957i 0.707717 0.707717i −0.258338 0.966055i \(-0.583175\pi\)
0.966055 + 0.258338i \(0.0831748\pi\)
\(444\) 0 0
\(445\) 13.8831 5.08020i 0.658122 0.240824i
\(446\) 0 0
\(447\) 1.11386 1.11386i 0.0526839 0.0526839i
\(448\) 0 0
\(449\) 24.4085i 1.15191i −0.817482 0.575954i \(-0.804631\pi\)
0.817482 0.575954i \(-0.195369\pi\)
\(450\) 0 0
\(451\) 0.481672i 0.0226810i
\(452\) 0 0
\(453\) 12.0338 12.0338i 0.565398 0.565398i
\(454\) 0 0
\(455\) 0.0427574 + 0.116847i 0.00200450 + 0.00547787i
\(456\) 0 0
\(457\) −3.49970 + 3.49970i −0.163709 + 0.163709i −0.784208 0.620498i \(-0.786930\pi\)
0.620498 + 0.784208i \(0.286930\pi\)
\(458\) 0 0
\(459\) −2.35584 −0.109961
\(460\) 0 0
\(461\) −15.6454 −0.728680 −0.364340 0.931266i \(-0.618705\pi\)
−0.364340 + 0.931266i \(0.618705\pi\)
\(462\) 0 0
\(463\) 19.9374 19.9374i 0.926569 0.926569i −0.0709131 0.997482i \(-0.522591\pi\)
0.997482 + 0.0709131i \(0.0225913\pi\)
\(464\) 0 0
\(465\) −6.49697 + 13.9958i −0.301290 + 0.649041i
\(466\) 0 0
\(467\) −4.29888 + 4.29888i −0.198929 + 0.198929i −0.799541 0.600612i \(-0.794924\pi\)
0.600612 + 0.799541i \(0.294924\pi\)
\(468\) 0 0
\(469\) 0.0466934i 0.00215610i
\(470\) 0 0
\(471\) 11.7377i 0.540845i
\(472\) 0 0
\(473\) −7.46169 + 7.46169i −0.343089 + 0.343089i
\(474\) 0 0
\(475\) 25.7166 21.7305i 1.17996 0.997065i
\(476\) 0 0
\(477\) 9.64615 9.64615i 0.441667 0.441667i
\(478\) 0 0
\(479\) −25.9392 −1.18519 −0.592596 0.805500i \(-0.701897\pi\)
−0.592596 + 0.805500i \(0.701897\pi\)
\(480\) 0 0
\(481\) 0.272896i 0.0124430i
\(482\) 0 0
\(483\) −0.000898174 0.0795412i −4.08683e−5 0.00361925i
\(484\) 0 0
\(485\) −13.0516 + 28.1158i −0.592642 + 1.27667i
\(486\) 0 0
\(487\) 11.3525 + 11.3525i 0.514430 + 0.514430i 0.915881 0.401451i \(-0.131494\pi\)
−0.401451 + 0.915881i \(0.631494\pi\)
\(488\) 0 0
\(489\) 23.2672i 1.05218i
\(490\) 0 0
\(491\) −33.9816 −1.53357 −0.766784 0.641906i \(-0.778144\pi\)
−0.766784 + 0.641906i \(0.778144\pi\)
\(492\) 0 0
\(493\) 1.49005 + 1.49005i 0.0671086 + 0.0671086i
\(494\) 0 0
\(495\) −3.62798 + 1.32757i −0.163065 + 0.0596700i
\(496\) 0 0
\(497\) 0.148351 0.148351i 0.00665444 0.00665444i
\(498\) 0 0
\(499\) 7.63815i 0.341930i 0.985277 + 0.170965i \(0.0546886\pi\)
−0.985277 + 0.170965i \(0.945311\pi\)
\(500\) 0 0
\(501\) 4.62353 0.206564
\(502\) 0 0
\(503\) 10.4184 + 10.4184i 0.464535 + 0.464535i 0.900139 0.435603i \(-0.143465\pi\)
−0.435603 + 0.900139i \(0.643465\pi\)
\(504\) 0 0
\(505\) 17.6822 6.47039i 0.786848 0.287929i
\(506\) 0 0
\(507\) −1.23423 1.23423i −0.0548140 0.0548140i
\(508\) 0 0
\(509\) 13.7018i 0.607320i −0.952780 0.303660i \(-0.901791\pi\)
0.952780 0.303660i \(-0.0982088\pi\)
\(510\) 0 0
\(511\) 0.177211i 0.00783935i
\(512\) 0 0
\(513\) 4.76142 + 4.76142i 0.210222 + 0.210222i
\(514\) 0 0
\(515\) 32.5772 + 15.1226i 1.43552 + 0.666381i
\(516\) 0 0
\(517\) −13.1555 + 13.1555i −0.578580 + 0.578580i
\(518\) 0 0
\(519\) 4.77792i 0.209727i
\(520\) 0 0
\(521\) 10.4040i 0.455806i −0.973684 0.227903i \(-0.926813\pi\)
0.973684 0.227903i \(-0.0731870\pi\)
\(522\) 0 0
\(523\) −19.8761 19.8761i −0.869120 0.869120i 0.123255 0.992375i \(-0.460667\pi\)
−0.992375 + 0.123255i \(0.960667\pi\)
\(524\) 0 0
\(525\) 0.0535272 + 0.0633457i 0.00233612 + 0.00276463i
\(526\) 0 0
\(527\) −11.4953 + 11.4953i −0.500742 + 0.500742i
\(528\) 0 0
\(529\) 22.9941 + 0.519362i 0.999745 + 0.0225810i
\(530\) 0 0
\(531\) −15.1371 −0.656893
\(532\) 0 0
\(533\) −0.661352 + 0.661352i −0.0286463 + 0.0286463i
\(534\) 0 0
\(535\) 10.5808 22.7933i 0.457449 0.985441i
\(536\) 0 0
\(537\) −2.05510 2.05510i −0.0886839 0.0886839i
\(538\) 0 0
\(539\) 12.0934 0.520899
\(540\) 0 0
\(541\) 3.70430 0.159260 0.0796302 0.996824i \(-0.474626\pi\)
0.0796302 + 0.996824i \(0.474626\pi\)
\(542\) 0 0
\(543\) 3.10767 + 3.10767i 0.133363 + 0.133363i
\(544\) 0 0
\(545\) −21.6295 + 7.91481i −0.926506 + 0.339033i
\(546\) 0 0
\(547\) 2.28076 + 2.28076i 0.0975182 + 0.0975182i 0.754183 0.656665i \(-0.228033\pi\)
−0.656665 + 0.754183i \(0.728033\pi\)
\(548\) 0 0
\(549\) 5.91692 0.252528
\(550\) 0 0
\(551\) 6.02314i 0.256595i
\(552\) 0 0
\(553\) −0.0459200 + 0.0459200i −0.00195272 + 0.00195272i
\(554\) 0 0
\(555\) 0.0625064 + 0.170817i 0.00265325 + 0.00725076i
\(556\) 0 0
\(557\) −26.0661 + 26.0661i −1.10445 + 1.10445i −0.110588 + 0.993866i \(0.535273\pi\)
−0.993866 + 0.110588i \(0.964727\pi\)
\(558\) 0 0
\(559\) 20.4903 0.866649
\(560\) 0 0
\(561\) −4.07017 −0.171843
\(562\) 0 0
\(563\) −13.3486 13.3486i −0.562578 0.562578i 0.367461 0.930039i \(-0.380227\pi\)
−0.930039 + 0.367461i \(0.880227\pi\)
\(564\) 0 0
\(565\) −18.7474 8.70270i −0.788711 0.366125i
\(566\) 0 0
\(567\) −0.0117285 + 0.0117285i −0.000492549 + 0.000492549i
\(568\) 0 0
\(569\) −24.4230 −1.02386 −0.511932 0.859026i \(-0.671070\pi\)
−0.511932 + 0.859026i \(0.671070\pi\)
\(570\) 0 0
\(571\) 6.36696i 0.266449i 0.991086 + 0.133225i \(0.0425332\pi\)
−0.991086 + 0.133225i \(0.957467\pi\)
\(572\) 0 0
\(573\) 4.95918 + 4.95918i 0.207173 + 0.207173i
\(574\) 0 0
\(575\) 18.4894 15.2690i 0.771059 0.636763i
\(576\) 0 0
\(577\) −5.01040 5.01040i −0.208586 0.208586i 0.595081 0.803666i \(-0.297120\pi\)
−0.803666 + 0.595081i \(0.797120\pi\)
\(578\) 0 0
\(579\) 15.1900i 0.631275i
\(580\) 0 0
\(581\) 0.288429 0.0119660
\(582\) 0 0
\(583\) 16.6656 16.6656i 0.690219 0.690219i
\(584\) 0 0
\(585\) 6.80414 + 3.15853i 0.281317 + 0.130589i
\(586\) 0 0
\(587\) 11.4787 + 11.4787i 0.473776 + 0.473776i 0.903134 0.429358i \(-0.141260\pi\)
−0.429358 + 0.903134i \(0.641260\pi\)
\(588\) 0 0
\(589\) 46.4666 1.91462
\(590\) 0 0
\(591\) 1.62735 0.0669403
\(592\) 0 0
\(593\) 10.5347 10.5347i 0.432610 0.432610i −0.456905 0.889515i \(-0.651042\pi\)
0.889515 + 0.456905i \(0.151042\pi\)
\(594\) 0 0
\(595\) 0.0300257 + 0.0820538i 0.00123093 + 0.00336388i
\(596\) 0 0
\(597\) −12.4966 + 12.4966i −0.511451 + 0.511451i
\(598\) 0 0
\(599\) 13.4401i 0.549146i −0.961566 0.274573i \(-0.911463\pi\)
0.961566 0.274573i \(-0.0885365\pi\)
\(600\) 0 0
\(601\) 8.20348 0.334627 0.167313 0.985904i \(-0.446491\pi\)
0.167313 + 0.985904i \(0.446491\pi\)
\(602\) 0 0
\(603\) −1.99060 1.99060i −0.0810635 0.0810635i
\(604\) 0 0
\(605\) 16.8308 6.15883i 0.684269 0.250392i
\(606\) 0 0
\(607\) −11.5736 11.5736i −0.469756 0.469756i 0.432080 0.901835i \(-0.357780\pi\)
−0.901835 + 0.432080i \(0.857780\pi\)
\(608\) 0 0
\(609\) 0.0148364 0.000601200
\(610\) 0 0
\(611\) 36.1260 1.46150
\(612\) 0 0
\(613\) 27.9916 + 27.9916i 1.13057 + 1.13057i 0.990083 + 0.140486i \(0.0448664\pi\)
0.140486 + 0.990083i \(0.455134\pi\)
\(614\) 0 0
\(615\) −0.262486 + 0.565449i −0.0105844 + 0.0228011i
\(616\) 0 0
\(617\) −6.87175 + 6.87175i −0.276646 + 0.276646i −0.831769 0.555122i \(-0.812671\pi\)
0.555122 + 0.831769i \(0.312671\pi\)
\(618\) 0 0
\(619\) −43.3388 −1.74193 −0.870967 0.491342i \(-0.836506\pi\)
−0.870967 + 0.491342i \(0.836506\pi\)
\(620\) 0 0
\(621\) 3.35266 + 3.42924i 0.134538 + 0.137611i
\(622\) 0 0
\(623\) 0.0775408 0.0775408i 0.00310661 0.00310661i
\(624\) 0 0
\(625\) 4.17108 24.6496i 0.166843 0.985983i
\(626\) 0 0
\(627\) 8.22629 + 8.22629i 0.328526 + 0.328526i
\(628\) 0 0
\(629\) 0.191637i 0.00764105i
\(630\) 0 0
\(631\) 29.6526i 1.18045i 0.807238 + 0.590225i \(0.200961\pi\)
−0.807238 + 0.590225i \(0.799039\pi\)
\(632\) 0 0
\(633\) −4.44902 + 4.44902i −0.176833 + 0.176833i
\(634\) 0 0
\(635\) −7.85830 3.64788i −0.311847 0.144762i
\(636\) 0 0
\(637\) −16.6047 16.6047i −0.657900 0.657900i
\(638\) 0 0
\(639\) 12.6488i 0.500378i
\(640\) 0 0
\(641\) 7.85702i 0.310333i 0.987888 + 0.155167i \(0.0495915\pi\)
−0.987888 + 0.155167i \(0.950409\pi\)
\(642\) 0 0
\(643\) −8.26989 8.26989i −0.326133 0.326133i 0.524981 0.851114i \(-0.324072\pi\)
−0.851114 + 0.524981i \(0.824072\pi\)
\(644\) 0 0
\(645\) 12.8257 4.69328i 0.505013 0.184798i
\(646\) 0 0
\(647\) −2.77391 2.77391i −0.109054 0.109054i 0.650475 0.759528i \(-0.274570\pi\)
−0.759528 + 0.650475i \(0.774570\pi\)
\(648\) 0 0
\(649\) −26.1522 −1.02657
\(650\) 0 0
\(651\) 0.114458i 0.00448596i
\(652\) 0 0
\(653\) 7.86077 7.86077i 0.307616 0.307616i −0.536368 0.843984i \(-0.680204\pi\)
0.843984 + 0.536368i \(0.180204\pi\)
\(654\) 0 0
\(655\) −24.3929 + 8.92601i −0.953109 + 0.348768i
\(656\) 0 0
\(657\) −7.55474 7.55474i −0.294738 0.294738i
\(658\) 0 0
\(659\) 26.7922 1.04368 0.521839 0.853044i \(-0.325246\pi\)
0.521839 + 0.853044i \(0.325246\pi\)
\(660\) 0 0
\(661\) 42.4339i 1.65049i 0.564777 + 0.825244i \(0.308962\pi\)
−0.564777 + 0.825244i \(0.691038\pi\)
\(662\) 0 0
\(663\) 5.58849 + 5.58849i 0.217039 + 0.217039i
\(664\) 0 0
\(665\) 0.105155 0.226526i 0.00407773 0.00878429i
\(666\) 0 0
\(667\) 0.0484368 4.28951i 0.00187548 0.166091i
\(668\) 0 0
\(669\) 9.35361i 0.361631i
\(670\) 0 0
\(671\) 10.2226 0.394641
\(672\) 0 0
\(673\) −11.5454 + 11.5454i −0.445043 + 0.445043i −0.893703 0.448659i \(-0.851902\pi\)
0.448659 + 0.893703i \(0.351902\pi\)
\(674\) 0 0
\(675\) 4.98245 + 0.418577i 0.191775 + 0.0161110i
\(676\) 0 0
\(677\) 19.8948 19.8948i 0.764619 0.764619i −0.212535 0.977153i \(-0.568172\pi\)
0.977153 + 0.212535i \(0.0681719\pi\)
\(678\) 0 0
\(679\) 0.229931i 0.00882394i
\(680\) 0 0
\(681\) 1.19191i 0.0456741i
\(682\) 0 0
\(683\) −19.3260 + 19.3260i −0.739487 + 0.739487i −0.972479 0.232991i \(-0.925149\pi\)
0.232991 + 0.972479i \(0.425149\pi\)
\(684\) 0 0
\(685\) −2.40956 + 5.19069i −0.0920644 + 0.198326i
\(686\) 0 0
\(687\) 14.4089 14.4089i 0.549735 0.549735i
\(688\) 0 0
\(689\) −45.7649 −1.74350
\(690\) 0 0
\(691\) −22.0154 −0.837505 −0.418752 0.908100i \(-0.637532\pi\)
−0.418752 + 0.908100i \(0.637532\pi\)
\(692\) 0 0
\(693\) −0.0202632 + 0.0202632i −0.000769736 + 0.000769736i
\(694\) 0 0
\(695\) −4.22626 11.5495i −0.160311 0.438097i
\(696\) 0 0
\(697\) −0.464424 + 0.464424i −0.0175913 + 0.0175913i
\(698\) 0 0
\(699\) 21.2141i 0.802389i
\(700\) 0 0
\(701\) 33.7843i 1.27602i −0.770030 0.638008i \(-0.779759\pi\)
0.770030 0.638008i \(-0.220241\pi\)
\(702\) 0 0
\(703\) 0.387320 0.387320i 0.0146080 0.0146080i
\(704\) 0 0
\(705\) 22.6128 8.27462i 0.851646 0.311640i
\(706\) 0 0
\(707\) 0.0987598 0.0987598i 0.00371425 0.00371425i
\(708\) 0 0
\(709\) 21.5113 0.807876 0.403938 0.914786i \(-0.367641\pi\)
0.403938 + 0.914786i \(0.367641\pi\)
\(710\) 0 0
\(711\) 3.91526i 0.146834i
\(712\) 0 0
\(713\) 33.0922 + 0.373675i 1.23931 + 0.0139942i
\(714\) 0 0
\(715\) 11.7555 + 5.45699i 0.439630 + 0.204080i
\(716\) 0 0
\(717\) 12.0642 + 12.0642i 0.450546 + 0.450546i
\(718\) 0 0
\(719\) 14.6800i 0.547472i 0.961805 + 0.273736i \(0.0882594\pi\)
−0.961805 + 0.273736i \(0.911741\pi\)
\(720\) 0 0
\(721\) 0.266416 0.00992186
\(722\) 0 0
\(723\) 16.5237 + 16.5237i 0.614524 + 0.614524i
\(724\) 0 0
\(725\) −2.88662 3.41612i −0.107206 0.126871i
\(726\) 0 0
\(727\) 31.3379 31.3379i 1.16226 1.16226i 0.178277 0.983980i \(-0.442948\pi\)
0.983980 0.178277i \(-0.0570522\pi\)
\(728\) 0 0
\(729\) 1.00000i 0.0370370i
\(730\) 0 0
\(731\) 14.3890 0.532196
\(732\) 0 0
\(733\) 13.5985 + 13.5985i 0.502273 + 0.502273i 0.912144 0.409870i \(-0.134426\pi\)
−0.409870 + 0.912144i \(0.634426\pi\)
\(734\) 0 0
\(735\) −14.1968 6.59026i −0.523657 0.243085i
\(736\) 0 0
\(737\) −3.43915 3.43915i −0.126683 0.126683i
\(738\) 0 0
\(739\) 31.5137i 1.15925i −0.814883 0.579625i \(-0.803199\pi\)
0.814883 0.579625i \(-0.196801\pi\)
\(740\) 0 0
\(741\) 22.5900i 0.829863i
\(742\) 0 0
\(743\) 33.9355 + 33.9355i 1.24497 + 1.24497i 0.957912 + 0.287061i \(0.0926781\pi\)
0.287061 + 0.957912i \(0.407322\pi\)
\(744\) 0 0
\(745\) −3.30783 + 1.21042i −0.121190 + 0.0443465i
\(746\) 0 0
\(747\) 12.2961 12.2961i 0.449891 0.449891i
\(748\) 0 0
\(749\) 0.186404i 0.00681104i
\(750\) 0 0
\(751\) 13.9950i 0.510686i 0.966851 + 0.255343i \(0.0821883\pi\)
−0.966851 + 0.255343i \(0.917812\pi\)
\(752\) 0 0
\(753\) −7.79778 7.79778i −0.284167 0.284167i
\(754\) 0 0
\(755\) −35.7368 + 13.0771i −1.30060 + 0.475923i
\(756\) 0 0
\(757\) −23.8848 + 23.8848i −0.868109 + 0.868109i −0.992263 0.124154i \(-0.960378\pi\)
0.124154 + 0.992263i \(0.460378\pi\)
\(758\) 0 0
\(759\) 5.79237 + 5.92468i 0.210250 + 0.215052i
\(760\) 0 0
\(761\) 27.0267 0.979718 0.489859 0.871802i \(-0.337048\pi\)
0.489859 + 0.871802i \(0.337048\pi\)
\(762\) 0 0
\(763\) −0.120807 + 0.120807i −0.00437349 + 0.00437349i
\(764\) 0 0
\(765\) 4.77810 + 2.21803i 0.172752 + 0.0801930i
\(766\) 0 0
\(767\) 35.9079 + 35.9079i 1.29656 + 1.29656i
\(768\) 0 0
\(769\) 52.5610 1.89540 0.947699 0.319165i \(-0.103402\pi\)
0.947699 + 0.319165i \(0.103402\pi\)
\(770\) 0 0
\(771\) 2.33200 0.0839849
\(772\) 0 0
\(773\) 11.7819 + 11.7819i 0.423764 + 0.423764i 0.886497 0.462734i \(-0.153131\pi\)
−0.462734 + 0.886497i \(0.653131\pi\)
\(774\) 0 0
\(775\) 26.3542 22.2694i 0.946672 0.799939i
\(776\) 0 0
\(777\) 0.000954057 0 0.000954057i 3.42266e−5 0 3.42266e-5i
\(778\) 0 0
\(779\) 1.87731 0.0672615
\(780\) 0 0
\(781\) 21.8532i 0.781970i
\(782\) 0 0
\(783\) 0.632494 0.632494i 0.0226035 0.0226035i
\(784\) 0 0
\(785\) −11.0511 + 23.8064i −0.394430 + 0.849686i
\(786\) 0 0
\(787\) −9.86015 + 9.86015i −0.351476 + 0.351476i −0.860659 0.509182i \(-0.829948\pi\)
0.509182 + 0.860659i \(0.329948\pi\)
\(788\) 0 0
\(789\) 2.62222 0.0933535
\(790\) 0 0
\(791\) −0.153316 −0.00545130
\(792\) 0 0
\(793\) −14.0360 14.0360i −0.498435 0.498435i
\(794\) 0 0
\(795\) −28.6461 + 10.4824i −1.01597 + 0.371772i
\(796\) 0 0
\(797\) 23.1046 23.1046i 0.818406 0.818406i −0.167471 0.985877i \(-0.553560\pi\)
0.985877 + 0.167471i \(0.0535601\pi\)
\(798\) 0 0
\(799\) 25.3689 0.897488
\(800\) 0 0
\(801\) 6.61134i 0.233600i
\(802\) 0 0
\(803\) −13.0523 13.0523i −0.460605 0.460605i
\(804\) 0 0
\(805\) 0.0767100 0.160479i 0.00270367 0.00565616i
\(806\) 0 0
\(807\) 12.7458 + 12.7458i 0.448674 + 0.448674i
\(808\) 0 0
\(809\) 54.7553i 1.92509i 0.271116 + 0.962547i \(0.412607\pi\)
−0.271116 + 0.962547i \(0.587393\pi\)
\(810\) 0 0
\(811\) −38.5346 −1.35313 −0.676567 0.736381i \(-0.736533\pi\)
−0.676567 + 0.736381i \(0.736533\pi\)
\(812\) 0 0
\(813\) 21.9639 21.9639i 0.770307 0.770307i
\(814\) 0 0
\(815\) −21.9061 + 47.1904i −0.767338 + 1.65301i
\(816\) 0 0
\(817\) −29.0818 29.0818i −1.01744 1.01744i
\(818\) 0 0
\(819\) 0.0556442 0.00194437
\(820\) 0 0
\(821\) 1.82678 0.0637551 0.0318776 0.999492i \(-0.489851\pi\)
0.0318776 + 0.999492i \(0.489851\pi\)
\(822\) 0 0
\(823\) 30.5211 30.5211i 1.06390 1.06390i 0.0660840 0.997814i \(-0.478949\pi\)
0.997814 0.0660840i \(-0.0210505\pi\)
\(824\) 0 0
\(825\) 8.60816 + 0.723174i 0.299698 + 0.0251777i
\(826\) 0 0
\(827\) −3.34366 + 3.34366i −0.116270 + 0.116270i −0.762848 0.646578i \(-0.776200\pi\)
0.646578 + 0.762848i \(0.276200\pi\)
\(828\) 0 0
\(829\) 4.28008i 0.148653i 0.997234 + 0.0743266i \(0.0236807\pi\)
−0.997234 + 0.0743266i \(0.976319\pi\)
\(830\) 0 0
\(831\) 24.3793 0.845709
\(832\) 0 0
\(833\) −11.6603 11.6603i −0.404007 0.404007i
\(834\) 0 0
\(835\) −9.37743 4.35307i −0.324519 0.150644i
\(836\) 0 0
\(837\) 4.87949 + 4.87949i 0.168660 + 0.168660i
\(838\) 0 0
\(839\) 16.0883 0.555429 0.277714 0.960664i \(-0.410423\pi\)
0.277714 + 0.960664i \(0.410423\pi\)
\(840\) 0 0
\(841\) 28.1999 0.972410
\(842\) 0 0
\(843\) 20.3295 + 20.3295i 0.700187 + 0.700187i
\(844\) 0 0
\(845\) 1.34122 + 3.66528i 0.0461395 + 0.126090i
\(846\) 0 0
\(847\) 0.0940044 0.0940044i 0.00323003 0.00323003i
\(848\) 0 0
\(849\) −2.31851 −0.0795709
\(850\) 0 0
\(851\) 0.278953 0.272723i 0.00956238 0.00934884i
\(852\) 0 0
\(853\) 29.6273 29.6273i 1.01442 1.01442i 0.0145238 0.999895i \(-0.495377\pi\)
0.999895 0.0145238i \(-0.00462323\pi\)
\(854\) 0 0
\(855\) −5.17420 14.1400i −0.176954 0.483577i
\(856\) 0 0
\(857\) −22.9180 22.9180i −0.782865 0.782865i 0.197449 0.980313i \(-0.436734\pi\)
−0.980313 + 0.197449i \(0.936734\pi\)
\(858\) 0 0
\(859\) 44.9995i 1.53536i 0.640831 + 0.767682i \(0.278590\pi\)
−0.640831 + 0.767682i \(0.721410\pi\)
\(860\) 0 0
\(861\) 0.00462424i 0.000157594i
\(862\) 0 0
\(863\) −22.5165 + 22.5165i −0.766469 + 0.766469i −0.977483 0.211014i \(-0.932324\pi\)
0.211014 + 0.977483i \(0.432324\pi\)
\(864\) 0 0
\(865\) 4.49842 9.69054i 0.152951 0.329488i
\(866\) 0 0
\(867\) −8.09639 8.09639i −0.274968 0.274968i
\(868\) 0 0
\(869\) 6.76439i 0.229466i
\(870\) 0 0
\(871\) 9.44415i 0.320003i
\(872\) 0 0
\(873\) 9.80227 + 9.80227i 0.331756 + 0.331756i
\(874\) 0 0
\(875\) −0.0489234 0.178873i −0.00165391 0.00604703i
\(876\) 0 0
\(877\) −0.130809 0.130809i −0.00441710 0.00441710i 0.704895 0.709312i \(-0.250994\pi\)
−0.709312 + 0.704895i \(0.750994\pi\)
\(878\) 0 0
\(879\) −4.10417 −0.138430
\(880\) 0 0
\(881\) 8.93033i 0.300871i 0.988620 + 0.150435i \(0.0480675\pi\)
−0.988620 + 0.150435i \(0.951932\pi\)
\(882\) 0 0
\(883\) 15.0677 15.0677i 0.507070 0.507070i −0.406556 0.913626i \(-0.633270\pi\)
0.913626 + 0.406556i \(0.133270\pi\)
\(884\) 0 0
\(885\) 30.7009 + 14.2516i 1.03200 + 0.479062i
\(886\) 0 0
\(887\) 4.25851 + 4.25851i 0.142987 + 0.142987i 0.774977 0.631990i \(-0.217762\pi\)
−0.631990 + 0.774977i \(0.717762\pi\)
\(888\) 0 0
\(889\) −0.0642651 −0.00215538
\(890\) 0 0
\(891\) 1.72770i 0.0578800i
\(892\) 0 0
\(893\) −51.2735 51.2735i −1.71580 1.71580i
\(894\) 0 0
\(895\) 2.23325 + 6.10301i 0.0746495 + 0.204001i
\(896\) 0 0
\(897\) 0.181664 16.0879i 0.00606557 0.537160i
\(898\) 0 0
\(899\) 6.17250i 0.205864i
\(900\) 0 0
\(901\) −32.1377 −1.07066
\(902\) 0 0
\(903\) 0.0716352 0.0716352i 0.00238387 0.00238387i
\(904\) 0 0
\(905\) −3.37708 9.22885i −0.112258 0.306777i
\(906\) 0 0
\(907\) −6.23724 + 6.23724i −0.207104 + 0.207104i −0.803035 0.595931i \(-0.796783\pi\)
0.595931 + 0.803035i \(0.296783\pi\)
\(908\) 0 0
\(909\) 8.42053i 0.279291i
\(910\) 0 0
\(911\) 46.8771i 1.55311i 0.630050 + 0.776554i \(0.283034\pi\)
−0.630050 + 0.776554i \(0.716966\pi\)
\(912\) 0 0
\(913\) 21.2439 21.2439i 0.703071 0.703071i
\(914\) 0 0
\(915\) −12.0007 5.57080i −0.396730 0.184165i
\(916\) 0 0
\(917\) −0.136241 + 0.136241i −0.00449907 + 0.00449907i
\(918\) 0 0
\(919\) 35.5200 1.17170 0.585848 0.810421i \(-0.300762\pi\)
0.585848 + 0.810421i \(0.300762\pi\)
\(920\) 0 0
\(921\) 29.6891 0.978288
\(922\) 0 0
\(923\) −30.0053 + 30.0053i −0.987635 + 0.987635i
\(924\) 0 0
\(925\) 0.0340493 0.405299i 0.00111954 0.0133262i
\(926\) 0 0
\(927\) 11.3577 11.3577i 0.373035 0.373035i
\(928\) 0 0
\(929\) 15.2702i 0.501000i 0.968117 + 0.250500i \(0.0805949\pi\)
−0.968117 + 0.250500i \(0.919405\pi\)
\(930\) 0 0
\(931\) 47.1338i 1.54475i
\(932\) 0 0
\(933\) −18.3976 + 18.3976i −0.602311 + 0.602311i
\(934\) 0 0
\(935\) 8.25510 + 3.83208i 0.269971 + 0.125322i
\(936\) 0 0
\(937\) −26.0645 + 26.0645i −0.851491 + 0.851491i −0.990317 0.138826i \(-0.955667\pi\)
0.138826 + 0.990317i \(0.455667\pi\)
\(938\) 0 0
\(939\) −19.4077 −0.633348
\(940\) 0 0
\(941\) 7.43783i 0.242466i −0.992624 0.121233i \(-0.961315\pi\)
0.992624 0.121233i \(-0.0386849\pi\)
\(942\) 0 0
\(943\) 1.33697 + 0.0150969i 0.0435376 + 0.000491623i
\(944\) 0 0
\(945\) 0.0348300 0.0127452i 0.00113302 0.000414602i
\(946\) 0 0
\(947\) −19.8343 19.8343i −0.644528 0.644528i 0.307137 0.951665i \(-0.400629\pi\)
−0.951665 + 0.307137i \(0.900629\pi\)
\(948\) 0 0
\(949\) 35.8425i 1.16350i
\(950\) 0 0
\(951\) 26.2684 0.851812
\(952\) 0 0
\(953\) −31.6040 31.6040i −1.02375 1.02375i −0.999711 0.0240441i \(-0.992346\pi\)
−0.0240441 0.999711i \(-0.507654\pi\)
\(954\) 0 0
\(955\) −5.38910 14.7273i −0.174387 0.476563i
\(956\) 0 0
\(957\) 1.09276 1.09276i 0.0353238 0.0353238i
\(958\) 0 0
\(959\) 0.0424494i 0.00137076i
\(960\) 0 0
\(961\) 16.6188 0.536091
\(962\) 0 0
\(963\) −7.94663 7.94663i −0.256077 0.256077i
\(964\) 0 0
\(965\) −14.3014 + 30.8083i −0.460379 + 0.991753i
\(966\) 0 0
\(967\) 11.9956 + 11.9956i 0.385754 + 0.385754i 0.873170 0.487416i \(-0.162060\pi\)
−0.487416 + 0.873170i \(0.662060\pi\)
\(968\) 0 0
\(969\) 15.8634i 0.509607i
\(970\) 0 0
\(971\) 20.2973i 0.651373i −0.945478 0.325686i \(-0.894405\pi\)
0.945478 0.325686i \(-0.105595\pi\)
\(972\) 0 0
\(973\) −0.0645069 0.0645069i −0.00206800 0.00206800i
\(974\) 0 0
\(975\) −10.8264 12.8122i −0.346721 0.410320i
\(976\) 0 0
\(977\) 23.2680 23.2680i 0.744411 0.744411i −0.229013 0.973423i \(-0.573550\pi\)
0.973423 + 0.229013i \(0.0735497\pi\)
\(978\) 0 0
\(979\) 11.4224i 0.365061i
\(980\) 0 0
\(981\) 10.3003i 0.328863i
\(982\) 0 0
\(983\) 7.96631 + 7.96631i 0.254086 + 0.254086i 0.822643 0.568558i \(-0.192498\pi\)
−0.568558 + 0.822643i \(0.692498\pi\)
\(984\) 0 0
\(985\) −3.30059 1.53216i −0.105165 0.0488186i
\(986\) 0 0
\(987\) 0.126298 0.126298i 0.00402012 0.00402012i
\(988\) 0 0
\(989\) −20.4774 20.9451i −0.651143 0.666016i
\(990\) 0 0
\(991\) 12.4553 0.395655 0.197827 0.980237i \(-0.436611\pi\)
0.197827 + 0.980237i \(0.436611\pi\)
\(992\) 0 0
\(993\) 6.04532 6.04532i 0.191842 0.191842i
\(994\) 0 0
\(995\) 37.1111 13.5799i 1.17650 0.430513i
\(996\) 0 0
\(997\) 21.3456 + 21.3456i 0.676021 + 0.676021i 0.959097 0.283077i \(-0.0913551\pi\)
−0.283077 + 0.959097i \(0.591355\pi\)
\(998\) 0 0
\(999\) 0.0813454 0.00257365
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 1380.2.t.a.1057.10 yes 48
5.3 odd 4 inner 1380.2.t.a.1333.9 yes 48
23.22 odd 2 inner 1380.2.t.a.1057.9 48
115.68 even 4 inner 1380.2.t.a.1333.10 yes 48
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1380.2.t.a.1057.9 48 23.22 odd 2 inner
1380.2.t.a.1057.10 yes 48 1.1 even 1 trivial
1380.2.t.a.1333.9 yes 48 5.3 odd 4 inner
1380.2.t.a.1333.10 yes 48 115.68 even 4 inner