Properties

Label 1148.2.r.a.737.12
Level $1148$
Weight $2$
Character 1148.737
Analytic conductor $9.167$
Analytic rank $0$
Dimension $56$
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] = [1148,2,Mod(81,1148)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1148, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 4, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1148.81");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1148 = 2^{2} \cdot 7 \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1148.r (of order \(6\), degree \(2\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(9.16682615204\)
Analytic rank: \(0\)
Dimension: \(56\)
Relative dimension: \(28\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 737.12
Character \(\chi\) \(=\) 1148.737
Dual form 1148.2.r.a.81.12

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.507835 + 0.293199i) q^{3} +(1.03735 - 1.79674i) q^{5} +(1.44101 - 2.21889i) q^{7} +(-1.32807 + 2.30028i) q^{9} +O(q^{10})\) \(q+(-0.507835 + 0.293199i) q^{3} +(1.03735 - 1.79674i) q^{5} +(1.44101 - 2.21889i) q^{7} +(-1.32807 + 2.30028i) q^{9} +(0.222860 - 0.128668i) q^{11} -3.45553i q^{13} +1.21660i q^{15} +(4.04528 - 2.33554i) q^{17} +(-1.80286 - 1.04088i) q^{19} +(-0.0812217 + 1.54933i) q^{21} +(-1.62764 + 2.81916i) q^{23} +(0.347816 + 0.602435i) q^{25} -3.31674i q^{27} -3.70628i q^{29} +(-2.78319 - 4.82063i) q^{31} +(-0.0754508 + 0.130685i) q^{33} +(-2.49194 - 4.89089i) q^{35} +(-0.268952 + 0.465838i) q^{37} +(1.01316 + 1.75484i) q^{39} +(-6.37105 + 0.640121i) q^{41} +8.37720 q^{43} +(2.75534 + 4.77239i) q^{45} +(-7.09770 - 4.09786i) q^{47} +(-2.84695 - 6.39491i) q^{49} +(-1.36955 + 2.37214i) q^{51} +(5.20381 - 3.00442i) q^{53} -0.533896i q^{55} +1.22074 q^{57} +(0.959511 + 1.66192i) q^{59} +(4.28053 - 7.41410i) q^{61} +(3.19031 + 6.26158i) q^{63} +(-6.20869 - 3.58459i) q^{65} +(-7.10708 + 4.10327i) q^{67} -1.90889i q^{69} -6.57120i q^{71} +(1.75029 + 3.03159i) q^{73} +(-0.353266 - 0.203958i) q^{75} +(0.0356436 - 0.679915i) q^{77} +(-7.43064 - 4.29008i) q^{79} +(-3.01174 - 5.21649i) q^{81} +5.65942 q^{83} -9.69108i q^{85} +(1.08667 + 1.88218i) q^{87} +(6.91750 + 3.99382i) q^{89} +(-7.66744 - 4.97947i) q^{91} +(2.82680 + 1.63206i) q^{93} +(-3.74039 + 2.15951i) q^{95} +2.39992i q^{97} +0.683522i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 56 q + 4 q^{5} + 32 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 56 q + 4 q^{5} + 32 q^{9} - 6 q^{21} + 2 q^{23} - 24 q^{25} - 4 q^{31} + 10 q^{33} + 10 q^{37} + 10 q^{39} + 20 q^{41} + 8 q^{43} - 22 q^{45} - 4 q^{49} + 18 q^{51} + 28 q^{57} - 16 q^{59} + 16 q^{61} + 2 q^{73} + 34 q^{77} - 28 q^{81} - 48 q^{83} - 2 q^{87} - 42 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1148\mathbb{Z}\right)^\times\).

\(n\) \(493\) \(575\) \(785\)
\(\chi(n)\) \(e\left(\frac{1}{3}\right)\) \(1\) \(-1\)

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
\(3\) −0.507835 + 0.293199i −0.293199 + 0.169278i −0.639383 0.768888i \(-0.720810\pi\)
0.346185 + 0.938166i \(0.387477\pi\)
\(4\) 0 0
\(5\) 1.03735 1.79674i 0.463916 0.803527i −0.535236 0.844703i \(-0.679777\pi\)
0.999152 + 0.0411762i \(0.0131105\pi\)
\(6\) 0 0
\(7\) 1.44101 2.21889i 0.544652 0.838662i
\(8\) 0 0
\(9\) −1.32807 + 2.30028i −0.442690 + 0.766761i
\(10\) 0 0
\(11\) 0.222860 0.128668i 0.0671949 0.0387950i −0.466026 0.884771i \(-0.654315\pi\)
0.533221 + 0.845976i \(0.320981\pi\)
\(12\) 0 0
\(13\) 3.45553i 0.958391i −0.877708 0.479196i \(-0.840928\pi\)
0.877708 0.479196i \(-0.159072\pi\)
\(14\) 0 0
\(15\) 1.21660i 0.314124i
\(16\) 0 0
\(17\) 4.04528 2.33554i 0.981123 0.566452i 0.0785143 0.996913i \(-0.474982\pi\)
0.902609 + 0.430461i \(0.141649\pi\)
\(18\) 0 0
\(19\) −1.80286 1.04088i −0.413604 0.238795i 0.278733 0.960369i \(-0.410086\pi\)
−0.692337 + 0.721574i \(0.743419\pi\)
\(20\) 0 0
\(21\) −0.0812217 + 1.54933i −0.0177240 + 0.338092i
\(22\) 0 0
\(23\) −1.62764 + 2.81916i −0.339387 + 0.587836i −0.984318 0.176406i \(-0.943553\pi\)
0.644931 + 0.764241i \(0.276886\pi\)
\(24\) 0 0
\(25\) 0.347816 + 0.602435i 0.0695633 + 0.120487i
\(26\) 0 0
\(27\) 3.31674i 0.638308i
\(28\) 0 0
\(29\) 3.70628i 0.688238i −0.938926 0.344119i \(-0.888178\pi\)
0.938926 0.344119i \(-0.111822\pi\)
\(30\) 0 0
\(31\) −2.78319 4.82063i −0.499876 0.865811i 0.500124 0.865954i \(-0.333288\pi\)
−1.00000 0.000142943i \(0.999954\pi\)
\(32\) 0 0
\(33\) −0.0754508 + 0.130685i −0.0131343 + 0.0227493i
\(34\) 0 0
\(35\) −2.49194 4.89089i −0.421214 0.826712i
\(36\) 0 0
\(37\) −0.268952 + 0.465838i −0.0442154 + 0.0765834i −0.887286 0.461219i \(-0.847412\pi\)
0.843071 + 0.537803i \(0.180745\pi\)
\(38\) 0 0
\(39\) 1.01316 + 1.75484i 0.162235 + 0.280999i
\(40\) 0 0
\(41\) −6.37105 + 0.640121i −0.994990 + 0.0999702i
\(42\) 0 0
\(43\) 8.37720 1.27751 0.638756 0.769410i \(-0.279450\pi\)
0.638756 + 0.769410i \(0.279450\pi\)
\(44\) 0 0
\(45\) 2.75534 + 4.77239i 0.410742 + 0.711426i
\(46\) 0 0
\(47\) −7.09770 4.09786i −1.03531 0.597734i −0.116805 0.993155i \(-0.537265\pi\)
−0.918500 + 0.395421i \(0.870599\pi\)
\(48\) 0 0
\(49\) −2.84695 6.39491i −0.406708 0.913558i
\(50\) 0 0
\(51\) −1.36955 + 2.37214i −0.191776 + 0.332166i
\(52\) 0 0
\(53\) 5.20381 3.00442i 0.714798 0.412689i −0.0980370 0.995183i \(-0.531256\pi\)
0.812835 + 0.582494i \(0.197923\pi\)
\(54\) 0 0
\(55\) 0.533896i 0.0719905i
\(56\) 0 0
\(57\) 1.22074 0.161691
\(58\) 0 0
\(59\) 0.959511 + 1.66192i 0.124918 + 0.216364i 0.921701 0.387902i \(-0.126800\pi\)
−0.796783 + 0.604265i \(0.793467\pi\)
\(60\) 0 0
\(61\) 4.28053 7.41410i 0.548066 0.949278i −0.450341 0.892857i \(-0.648698\pi\)
0.998407 0.0564216i \(-0.0179691\pi\)
\(62\) 0 0
\(63\) 3.19031 + 6.26158i 0.401941 + 0.788885i
\(64\) 0 0
\(65\) −6.20869 3.58459i −0.770093 0.444613i
\(66\) 0 0
\(67\) −7.10708 + 4.10327i −0.868268 + 0.501294i −0.866772 0.498704i \(-0.833809\pi\)
−0.00149539 + 0.999999i \(0.500476\pi\)
\(68\) 0 0
\(69\) 1.90889i 0.229803i
\(70\) 0 0
\(71\) 6.57120i 0.779858i −0.920845 0.389929i \(-0.872500\pi\)
0.920845 0.389929i \(-0.127500\pi\)
\(72\) 0 0
\(73\) 1.75029 + 3.03159i 0.204856 + 0.354820i 0.950087 0.311986i \(-0.100994\pi\)
−0.745231 + 0.666806i \(0.767661\pi\)
\(74\) 0 0
\(75\) −0.353266 0.203958i −0.0407917 0.0235511i
\(76\) 0 0
\(77\) 0.0356436 0.679915i 0.00406197 0.0774835i
\(78\) 0 0
\(79\) −7.43064 4.29008i −0.836013 0.482672i 0.0198941 0.999802i \(-0.493667\pi\)
−0.855907 + 0.517130i \(0.827000\pi\)
\(80\) 0 0
\(81\) −3.01174 5.21649i −0.334638 0.579610i
\(82\) 0 0
\(83\) 5.65942 0.621202 0.310601 0.950540i \(-0.399470\pi\)
0.310601 + 0.950540i \(0.399470\pi\)
\(84\) 0 0
\(85\) 9.69108i 1.05115i
\(86\) 0 0
\(87\) 1.08667 + 1.88218i 0.116504 + 0.201790i
\(88\) 0 0
\(89\) 6.91750 + 3.99382i 0.733254 + 0.423344i 0.819611 0.572920i \(-0.194189\pi\)
−0.0863576 + 0.996264i \(0.527523\pi\)
\(90\) 0 0
\(91\) −7.66744 4.97947i −0.803766 0.521990i
\(92\) 0 0
\(93\) 2.82680 + 1.63206i 0.293126 + 0.169236i
\(94\) 0 0
\(95\) −3.74039 + 2.15951i −0.383756 + 0.221561i
\(96\) 0 0
\(97\) 2.39992i 0.243675i 0.992550 + 0.121838i \(0.0388787\pi\)
−0.992550 + 0.121838i \(0.961121\pi\)
\(98\) 0 0
\(99\) 0.683522i 0.0686965i
\(100\) 0 0
\(101\) 16.9944 9.81172i 1.69101 0.976303i 0.737299 0.675567i \(-0.236101\pi\)
0.953707 0.300736i \(-0.0972322\pi\)
\(102\) 0 0
\(103\) 9.44694 16.3626i 0.930834 1.61225i 0.148936 0.988847i \(-0.452415\pi\)
0.781899 0.623406i \(-0.214251\pi\)
\(104\) 0 0
\(105\) 2.69949 + 1.75313i 0.263444 + 0.171088i
\(106\) 0 0
\(107\) −4.75048 + 8.22807i −0.459246 + 0.795437i −0.998921 0.0464359i \(-0.985214\pi\)
0.539675 + 0.841873i \(0.318547\pi\)
\(108\) 0 0
\(109\) 9.64417 5.56806i 0.923743 0.533324i 0.0389161 0.999242i \(-0.487610\pi\)
0.884827 + 0.465919i \(0.154276\pi\)
\(110\) 0 0
\(111\) 0.315425i 0.0299388i
\(112\) 0 0
\(113\) 15.9676 1.50211 0.751053 0.660242i \(-0.229546\pi\)
0.751053 + 0.660242i \(0.229546\pi\)
\(114\) 0 0
\(115\) 3.37687 + 5.84890i 0.314894 + 0.545413i
\(116\) 0 0
\(117\) 7.94869 + 4.58918i 0.734857 + 0.424270i
\(118\) 0 0
\(119\) 0.646990 12.3416i 0.0593095 1.13135i
\(120\) 0 0
\(121\) −5.46689 + 9.46893i −0.496990 + 0.860812i
\(122\) 0 0
\(123\) 3.04776 2.19306i 0.274807 0.197741i
\(124\) 0 0
\(125\) 11.8167 1.05692
\(126\) 0 0
\(127\) −20.9493 −1.85895 −0.929476 0.368883i \(-0.879740\pi\)
−0.929476 + 0.368883i \(0.879740\pi\)
\(128\) 0 0
\(129\) −4.25423 + 2.45618i −0.374564 + 0.216255i
\(130\) 0 0
\(131\) −5.69535 + 9.86464i −0.497605 + 0.861877i −0.999996 0.00276326i \(-0.999120\pi\)
0.502391 + 0.864640i \(0.332454\pi\)
\(132\) 0 0
\(133\) −4.90755 + 2.50042i −0.425539 + 0.216814i
\(134\) 0 0
\(135\) −5.95933 3.44062i −0.512897 0.296121i
\(136\) 0 0
\(137\) 10.0678 5.81267i 0.860154 0.496610i −0.00391012 0.999992i \(-0.501245\pi\)
0.864064 + 0.503382i \(0.167911\pi\)
\(138\) 0 0
\(139\) 0.426721 0.0361940 0.0180970 0.999836i \(-0.494239\pi\)
0.0180970 + 0.999836i \(0.494239\pi\)
\(140\) 0 0
\(141\) 4.80594 0.404733
\(142\) 0 0
\(143\) −0.444617 0.770100i −0.0371808 0.0643990i
\(144\) 0 0
\(145\) −6.65922 3.84470i −0.553018 0.319285i
\(146\) 0 0
\(147\) 3.32076 + 2.41283i 0.273892 + 0.199007i
\(148\) 0 0
\(149\) −5.86303 3.38502i −0.480318 0.277312i 0.240231 0.970716i \(-0.422777\pi\)
−0.720549 + 0.693404i \(0.756110\pi\)
\(150\) 0 0
\(151\) −0.667476 + 0.385367i −0.0543184 + 0.0313607i −0.526913 0.849919i \(-0.676651\pi\)
0.472595 + 0.881280i \(0.343317\pi\)
\(152\) 0 0
\(153\) 12.4070i 1.00305i
\(154\) 0 0
\(155\) −11.5486 −0.927603
\(156\) 0 0
\(157\) −12.6821 + 7.32204i −1.01215 + 0.584363i −0.911819 0.410592i \(-0.865322\pi\)
−0.100326 + 0.994955i \(0.531989\pi\)
\(158\) 0 0
\(159\) −1.76178 + 3.05150i −0.139718 + 0.242000i
\(160\) 0 0
\(161\) 3.90995 + 7.67401i 0.308147 + 0.604797i
\(162\) 0 0
\(163\) −2.29855 + 3.98120i −0.180036 + 0.311832i −0.941893 0.335914i \(-0.890955\pi\)
0.761856 + 0.647746i \(0.224288\pi\)
\(164\) 0 0
\(165\) 0.156537 + 0.271131i 0.0121864 + 0.0211075i
\(166\) 0 0
\(167\) 19.0661i 1.47538i 0.675141 + 0.737689i \(0.264083\pi\)
−0.675141 + 0.737689i \(0.735917\pi\)
\(168\) 0 0
\(169\) 1.05932 0.0814864
\(170\) 0 0
\(171\) 4.78864 2.76473i 0.366197 0.211424i
\(172\) 0 0
\(173\) −7.80164 + 13.5128i −0.593147 + 1.02736i 0.400658 + 0.916228i \(0.368782\pi\)
−0.993805 + 0.111134i \(0.964552\pi\)
\(174\) 0 0
\(175\) 1.83795 + 0.0963518i 0.138936 + 0.00728351i
\(176\) 0 0
\(177\) −0.974547 0.562655i −0.0732514 0.0422917i
\(178\) 0 0
\(179\) −15.8247 + 9.13642i −1.18280 + 0.682888i −0.956660 0.291207i \(-0.905943\pi\)
−0.226137 + 0.974095i \(0.572610\pi\)
\(180\) 0 0
\(181\) 7.36252i 0.547252i −0.961836 0.273626i \(-0.911777\pi\)
0.961836 0.273626i \(-0.0882230\pi\)
\(182\) 0 0
\(183\) 5.02018i 0.371103i
\(184\) 0 0
\(185\) 0.557994 + 0.966473i 0.0410245 + 0.0710565i
\(186\) 0 0
\(187\) 0.601020 1.04100i 0.0439510 0.0761253i
\(188\) 0 0
\(189\) −7.35949 4.77948i −0.535324 0.347656i
\(190\) 0 0
\(191\) 11.0071 + 6.35493i 0.796443 + 0.459826i 0.842226 0.539125i \(-0.181245\pi\)
−0.0457831 + 0.998951i \(0.514578\pi\)
\(192\) 0 0
\(193\) −5.38830 + 3.11094i −0.387859 + 0.223930i −0.681232 0.732068i \(-0.738555\pi\)
0.293373 + 0.955998i \(0.405222\pi\)
\(194\) 0 0
\(195\) 4.20398 0.301053
\(196\) 0 0
\(197\) 8.53148 0.607843 0.303922 0.952697i \(-0.401704\pi\)
0.303922 + 0.952697i \(0.401704\pi\)
\(198\) 0 0
\(199\) −7.96960 + 4.60125i −0.564950 + 0.326174i −0.755130 0.655575i \(-0.772426\pi\)
0.190180 + 0.981749i \(0.439093\pi\)
\(200\) 0 0
\(201\) 2.40615 4.16757i 0.169717 0.293958i
\(202\) 0 0
\(203\) −8.22382 5.34080i −0.577199 0.374851i
\(204\) 0 0
\(205\) −5.45886 + 12.1111i −0.381264 + 0.845879i
\(206\) 0 0
\(207\) −4.32324 7.48808i −0.300486 0.520457i
\(208\) 0 0
\(209\) −0.535714 −0.0370561
\(210\) 0 0
\(211\) 25.0849i 1.72692i 0.504420 + 0.863459i \(0.331706\pi\)
−0.504420 + 0.863459i \(0.668294\pi\)
\(212\) 0 0
\(213\) 1.92667 + 3.33708i 0.132013 + 0.228653i
\(214\) 0 0
\(215\) 8.69008 15.0517i 0.592658 1.02651i
\(216\) 0 0
\(217\) −14.7071 0.770998i −0.998381 0.0523388i
\(218\) 0 0
\(219\) −1.77771 1.02636i −0.120127 0.0693552i
\(220\) 0 0
\(221\) −8.07053 13.9786i −0.542882 0.940300i
\(222\) 0 0
\(223\) 10.2367 0.685499 0.342750 0.939427i \(-0.388642\pi\)
0.342750 + 0.939427i \(0.388642\pi\)
\(224\) 0 0
\(225\) −1.84770 −0.123180
\(226\) 0 0
\(227\) −17.3358 + 10.0088i −1.15062 + 0.664309i −0.949038 0.315163i \(-0.897941\pi\)
−0.201580 + 0.979472i \(0.564608\pi\)
\(228\) 0 0
\(229\) 5.29971 + 3.05979i 0.350215 + 0.202197i 0.664780 0.747039i \(-0.268525\pi\)
−0.314565 + 0.949236i \(0.601859\pi\)
\(230\) 0 0
\(231\) 0.181249 + 0.355735i 0.0119253 + 0.0234057i
\(232\) 0 0
\(233\) 9.26585 + 5.34964i 0.607026 + 0.350467i 0.771801 0.635865i \(-0.219356\pi\)
−0.164775 + 0.986331i \(0.552690\pi\)
\(234\) 0 0
\(235\) −14.7256 + 8.50181i −0.960590 + 0.554597i
\(236\) 0 0
\(237\) 5.03139 0.326824
\(238\) 0 0
\(239\) 9.02006i 0.583459i 0.956501 + 0.291729i \(0.0942307\pi\)
−0.956501 + 0.291729i \(0.905769\pi\)
\(240\) 0 0
\(241\) 9.46522 + 16.3942i 0.609708 + 1.05605i 0.991288 + 0.131710i \(0.0420468\pi\)
−0.381580 + 0.924336i \(0.624620\pi\)
\(242\) 0 0
\(243\) 11.6761 + 6.74119i 0.749021 + 0.432448i
\(244\) 0 0
\(245\) −14.4433 1.51851i −0.922747 0.0970141i
\(246\) 0 0
\(247\) −3.59680 + 6.22983i −0.228859 + 0.396395i
\(248\) 0 0
\(249\) −2.87405 + 1.65933i −0.182136 + 0.105156i
\(250\) 0 0
\(251\) 15.9573 1.00721 0.503607 0.863933i \(-0.332006\pi\)
0.503607 + 0.863933i \(0.332006\pi\)
\(252\) 0 0
\(253\) 0.837705i 0.0526660i
\(254\) 0 0
\(255\) 2.84141 + 4.92147i 0.177936 + 0.308194i
\(256\) 0 0
\(257\) 2.82462 + 1.63080i 0.176195 + 0.101726i 0.585504 0.810670i \(-0.300897\pi\)
−0.409309 + 0.912396i \(0.634230\pi\)
\(258\) 0 0
\(259\) 0.646081 + 1.26805i 0.0401455 + 0.0787931i
\(260\) 0 0
\(261\) 8.52549 + 4.92219i 0.527714 + 0.304676i
\(262\) 0 0
\(263\) 5.38300 3.10788i 0.331930 0.191640i −0.324768 0.945794i \(-0.605286\pi\)
0.656698 + 0.754154i \(0.271953\pi\)
\(264\) 0 0
\(265\) 12.4665i 0.765812i
\(266\) 0 0
\(267\) −4.68393 −0.286652
\(268\) 0 0
\(269\) 1.71843 + 2.97641i 0.104775 + 0.181475i 0.913646 0.406510i \(-0.133255\pi\)
−0.808871 + 0.587986i \(0.799921\pi\)
\(270\) 0 0
\(271\) −3.75999 + 6.51250i −0.228403 + 0.395606i −0.957335 0.288980i \(-0.906684\pi\)
0.728932 + 0.684586i \(0.240017\pi\)
\(272\) 0 0
\(273\) 5.35377 + 0.280664i 0.324025 + 0.0169865i
\(274\) 0 0
\(275\) 0.155029 + 0.0895059i 0.00934859 + 0.00539741i
\(276\) 0 0
\(277\) −12.4158 21.5047i −0.745990 1.29209i −0.949731 0.313068i \(-0.898643\pi\)
0.203740 0.979025i \(-0.434690\pi\)
\(278\) 0 0
\(279\) 14.7851 0.885160
\(280\) 0 0
\(281\) 0.569323i 0.0339629i −0.999856 0.0169815i \(-0.994594\pi\)
0.999856 0.0169815i \(-0.00540563\pi\)
\(282\) 0 0
\(283\) 4.51558 + 7.82121i 0.268423 + 0.464922i 0.968455 0.249189i \(-0.0801641\pi\)
−0.700032 + 0.714112i \(0.746831\pi\)
\(284\) 0 0
\(285\) 1.26633 2.19335i 0.0750111 0.129923i
\(286\) 0 0
\(287\) −7.76041 + 15.0591i −0.458083 + 0.888910i
\(288\) 0 0
\(289\) 2.40950 4.17338i 0.141735 0.245493i
\(290\) 0 0
\(291\) −0.703654 1.21877i −0.0412490 0.0714453i
\(292\) 0 0
\(293\) 23.0084i 1.34417i 0.740475 + 0.672083i \(0.234600\pi\)
−0.740475 + 0.672083i \(0.765400\pi\)
\(294\) 0 0
\(295\) 3.98139 0.231806
\(296\) 0 0
\(297\) −0.426760 0.739170i −0.0247631 0.0428910i
\(298\) 0 0
\(299\) 9.74169 + 5.62437i 0.563376 + 0.325265i
\(300\) 0 0
\(301\) 12.0717 18.5881i 0.695799 1.07140i
\(302\) 0 0
\(303\) −5.75357 + 9.96547i −0.330534 + 0.572501i
\(304\) 0 0
\(305\) −8.88081 15.3820i −0.508514 0.880771i
\(306\) 0 0
\(307\) 30.7547 1.75526 0.877631 0.479336i \(-0.159123\pi\)
0.877631 + 0.479336i \(0.159123\pi\)
\(308\) 0 0
\(309\) 11.0793i 0.630280i
\(310\) 0 0
\(311\) 13.8312 7.98547i 0.784298 0.452815i −0.0536535 0.998560i \(-0.517087\pi\)
0.837951 + 0.545745i \(0.183753\pi\)
\(312\) 0 0
\(313\) −18.8425 10.8787i −1.06504 0.614902i −0.138219 0.990402i \(-0.544138\pi\)
−0.926823 + 0.375500i \(0.877471\pi\)
\(314\) 0 0
\(315\) 14.5599 + 0.763283i 0.820358 + 0.0430061i
\(316\) 0 0
\(317\) 20.0409 + 11.5706i 1.12561 + 0.649872i 0.942827 0.333281i \(-0.108156\pi\)
0.182784 + 0.983153i \(0.441489\pi\)
\(318\) 0 0
\(319\) −0.476881 0.825981i −0.0267002 0.0462461i
\(320\) 0 0
\(321\) 5.57133i 0.310961i
\(322\) 0 0
\(323\) −9.72409 −0.541063
\(324\) 0 0
\(325\) 2.08173 1.20189i 0.115474 0.0666688i
\(326\) 0 0
\(327\) −3.26510 + 5.65531i −0.180560 + 0.312739i
\(328\) 0 0
\(329\) −19.3206 + 9.84394i −1.06518 + 0.542714i
\(330\) 0 0
\(331\) −8.36878 4.83172i −0.459990 0.265575i 0.252050 0.967714i \(-0.418895\pi\)
−0.712040 + 0.702139i \(0.752229\pi\)
\(332\) 0 0
\(333\) −0.714373 1.23733i −0.0391474 0.0678053i
\(334\) 0 0
\(335\) 17.0261i 0.930235i
\(336\) 0 0
\(337\) −3.60859 −0.196572 −0.0982862 0.995158i \(-0.531336\pi\)
−0.0982862 + 0.995158i \(0.531336\pi\)
\(338\) 0 0
\(339\) −8.10891 + 4.68168i −0.440416 + 0.254274i
\(340\) 0 0
\(341\) −1.24053 0.716218i −0.0671782 0.0387854i
\(342\) 0 0
\(343\) −18.2921 2.89807i −0.987681 0.156481i
\(344\) 0 0
\(345\) −3.42978 1.98018i −0.184653 0.106610i
\(346\) 0 0
\(347\) 27.1457 15.6726i 1.45726 0.841347i 0.458380 0.888756i \(-0.348430\pi\)
0.998876 + 0.0474092i \(0.0150965\pi\)
\(348\) 0 0
\(349\) 10.9809 0.587792 0.293896 0.955837i \(-0.405048\pi\)
0.293896 + 0.955837i \(0.405048\pi\)
\(350\) 0 0
\(351\) −11.4611 −0.611748
\(352\) 0 0
\(353\) −3.88754 6.73342i −0.206913 0.358384i 0.743827 0.668372i \(-0.233008\pi\)
−0.950741 + 0.309988i \(0.899675\pi\)
\(354\) 0 0
\(355\) −11.8067 6.81662i −0.626637 0.361789i
\(356\) 0 0
\(357\) 3.28997 + 6.45718i 0.174124 + 0.341750i
\(358\) 0 0
\(359\) −2.99511 + 5.18768i −0.158076 + 0.273795i −0.934175 0.356816i \(-0.883862\pi\)
0.776099 + 0.630611i \(0.217196\pi\)
\(360\) 0 0
\(361\) −7.33313 12.7014i −0.385954 0.668492i
\(362\) 0 0
\(363\) 6.41154i 0.336518i
\(364\) 0 0
\(365\) 7.26263 0.380143
\(366\) 0 0
\(367\) −5.45073 9.44094i −0.284526 0.492813i 0.687968 0.725741i \(-0.258503\pi\)
−0.972494 + 0.232928i \(0.925169\pi\)
\(368\) 0 0
\(369\) 6.98873 15.5053i 0.363819 0.807176i
\(370\) 0 0
\(371\) 0.832282 15.8761i 0.0432100 0.824246i
\(372\) 0 0
\(373\) −14.4469 + 25.0228i −0.748033 + 1.29563i 0.200731 + 0.979646i \(0.435668\pi\)
−0.948764 + 0.315985i \(0.897665\pi\)
\(374\) 0 0
\(375\) −6.00094 + 3.46464i −0.309887 + 0.178913i
\(376\) 0 0
\(377\) −12.8071 −0.659601
\(378\) 0 0
\(379\) 4.41610 0.226840 0.113420 0.993547i \(-0.463819\pi\)
0.113420 + 0.993547i \(0.463819\pi\)
\(380\) 0 0
\(381\) 10.6388 6.14231i 0.545042 0.314680i
\(382\) 0 0
\(383\) −7.49960 4.32990i −0.383212 0.221247i 0.296003 0.955187i \(-0.404346\pi\)
−0.679215 + 0.733940i \(0.737679\pi\)
\(384\) 0 0
\(385\) −1.18466 0.769351i −0.0603757 0.0392098i
\(386\) 0 0
\(387\) −11.1255 + 19.2699i −0.565541 + 0.979546i
\(388\) 0 0
\(389\) 1.95320 + 3.38304i 0.0990311 + 0.171527i 0.911284 0.411779i \(-0.135092\pi\)
−0.812253 + 0.583306i \(0.801759\pi\)
\(390\) 0 0
\(391\) 15.2057i 0.768986i
\(392\) 0 0
\(393\) 6.67947i 0.336935i
\(394\) 0 0
\(395\) −15.4163 + 8.90063i −0.775680 + 0.447839i
\(396\) 0 0
\(397\) 17.9795 + 10.3805i 0.902366 + 0.520981i 0.877967 0.478721i \(-0.158899\pi\)
0.0243990 + 0.999702i \(0.492233\pi\)
\(398\) 0 0
\(399\) 1.75910 2.70869i 0.0880653 0.135604i
\(400\) 0 0
\(401\) −2.74938 + 4.76207i −0.137298 + 0.237807i −0.926473 0.376362i \(-0.877175\pi\)
0.789175 + 0.614168i \(0.210508\pi\)
\(402\) 0 0
\(403\) −16.6578 + 9.61740i −0.829786 + 0.479077i
\(404\) 0 0
\(405\) −12.4969 −0.620976
\(406\) 0 0
\(407\) 0.138422i 0.00686134i
\(408\) 0 0
\(409\) −2.94311 5.09762i −0.145528 0.252061i 0.784042 0.620708i \(-0.213155\pi\)
−0.929570 + 0.368646i \(0.879821\pi\)
\(410\) 0 0
\(411\) −3.40853 + 5.90375i −0.168131 + 0.291211i
\(412\) 0 0
\(413\) 5.07029 + 0.265803i 0.249493 + 0.0130793i
\(414\) 0 0
\(415\) 5.87079 10.1685i 0.288186 0.499152i
\(416\) 0 0
\(417\) −0.216704 + 0.125114i −0.0106120 + 0.00612685i
\(418\) 0 0
\(419\) 17.1932 0.839941 0.419971 0.907538i \(-0.362040\pi\)
0.419971 + 0.907538i \(0.362040\pi\)
\(420\) 0 0
\(421\) 8.38887i 0.408849i 0.978882 + 0.204424i \(0.0655322\pi\)
−0.978882 + 0.204424i \(0.934468\pi\)
\(422\) 0 0
\(423\) 18.8525 10.8845i 0.916638 0.529221i
\(424\) 0 0
\(425\) 2.81403 + 1.62468i 0.136500 + 0.0788085i
\(426\) 0 0
\(427\) −10.2828 20.1819i −0.497618 0.976669i
\(428\) 0 0
\(429\) 0.451584 + 0.260722i 0.0218027 + 0.0125878i
\(430\) 0 0
\(431\) 7.02665 + 12.1705i 0.338462 + 0.586233i 0.984144 0.177374i \(-0.0567601\pi\)
−0.645682 + 0.763607i \(0.723427\pi\)
\(432\) 0 0
\(433\) 30.3289 1.45751 0.728756 0.684773i \(-0.240099\pi\)
0.728756 + 0.684773i \(0.240099\pi\)
\(434\) 0 0
\(435\) 4.50904 0.216192
\(436\) 0 0
\(437\) 5.86882 3.38837i 0.280744 0.162088i
\(438\) 0 0
\(439\) −24.3364 14.0506i −1.16151 0.670600i −0.209847 0.977734i \(-0.567297\pi\)
−0.951666 + 0.307134i \(0.900630\pi\)
\(440\) 0 0
\(441\) 18.4911 + 1.94408i 0.880526 + 0.0925752i
\(442\) 0 0
\(443\) 1.70957 2.96106i 0.0812239 0.140684i −0.822552 0.568690i \(-0.807450\pi\)
0.903776 + 0.428006i \(0.140784\pi\)
\(444\) 0 0
\(445\) 14.3517 8.28597i 0.680337 0.392793i
\(446\) 0 0
\(447\) 3.96993 0.187771
\(448\) 0 0
\(449\) 31.3383 1.47895 0.739473 0.673186i \(-0.235075\pi\)
0.739473 + 0.673186i \(0.235075\pi\)
\(450\) 0 0
\(451\) −1.33749 + 0.962410i −0.0629799 + 0.0453181i
\(452\) 0 0
\(453\) 0.225978 0.391406i 0.0106174 0.0183899i
\(454\) 0 0
\(455\) −16.9006 + 8.61096i −0.792313 + 0.403688i
\(456\) 0 0
\(457\) −36.2715 20.9414i −1.69671 0.979595i −0.948845 0.315743i \(-0.897746\pi\)
−0.747864 0.663852i \(-0.768921\pi\)
\(458\) 0 0
\(459\) −7.74639 13.4171i −0.361571 0.626259i
\(460\) 0 0
\(461\) 16.9008 0.787147 0.393573 0.919293i \(-0.371239\pi\)
0.393573 + 0.919293i \(0.371239\pi\)
\(462\) 0 0
\(463\) 20.2856i 0.942750i −0.881933 0.471375i \(-0.843758\pi\)
0.881933 0.471375i \(-0.156242\pi\)
\(464\) 0 0
\(465\) 5.86476 3.38602i 0.271972 0.157023i
\(466\) 0 0
\(467\) −1.31869 + 2.28403i −0.0610215 + 0.105692i −0.894922 0.446222i \(-0.852769\pi\)
0.833901 + 0.551914i \(0.186102\pi\)
\(468\) 0 0
\(469\) −1.13669 + 21.6827i −0.0524873 + 1.00121i
\(470\) 0 0
\(471\) 4.29362 7.43678i 0.197840 0.342669i
\(472\) 0 0
\(473\) 1.86694 1.07788i 0.0858422 0.0495610i
\(474\) 0 0
\(475\) 1.44814i 0.0664453i
\(476\) 0 0
\(477\) 15.9603i 0.730772i
\(478\) 0 0
\(479\) −0.408256 + 0.235707i −0.0186537 + 0.0107697i −0.509298 0.860590i \(-0.670095\pi\)
0.490644 + 0.871360i \(0.336762\pi\)
\(480\) 0 0
\(481\) 1.60972 + 0.929371i 0.0733968 + 0.0423757i
\(482\) 0 0
\(483\) −4.23562 2.75074i −0.192727 0.125163i
\(484\) 0 0
\(485\) 4.31204 + 2.48956i 0.195800 + 0.113045i
\(486\) 0 0
\(487\) −10.5814 18.3275i −0.479487 0.830496i 0.520236 0.854022i \(-0.325844\pi\)
−0.999723 + 0.0235264i \(0.992511\pi\)
\(488\) 0 0
\(489\) 2.69573i 0.121905i
\(490\) 0 0
\(491\) 28.8212 1.30068 0.650342 0.759641i \(-0.274626\pi\)
0.650342 + 0.759641i \(0.274626\pi\)
\(492\) 0 0
\(493\) −8.65616 14.9929i −0.389854 0.675247i
\(494\) 0 0
\(495\) 1.22811 + 0.709050i 0.0551995 + 0.0318694i
\(496\) 0 0
\(497\) −14.5808 9.46919i −0.654037 0.424751i
\(498\) 0 0
\(499\) 4.77029 + 2.75413i 0.213547 + 0.123292i 0.602959 0.797772i \(-0.293988\pi\)
−0.389412 + 0.921064i \(0.627322\pi\)
\(500\) 0 0
\(501\) −5.59015 9.68242i −0.249749 0.432579i
\(502\) 0 0
\(503\) 4.46515i 0.199091i 0.995033 + 0.0995455i \(0.0317389\pi\)
−0.995033 + 0.0995455i \(0.968261\pi\)
\(504\) 0 0
\(505\) 40.7127i 1.81169i
\(506\) 0 0
\(507\) −0.537961 + 0.310592i −0.0238917 + 0.0137939i
\(508\) 0 0
\(509\) 23.8708 + 13.7818i 1.05805 + 0.610868i 0.924893 0.380227i \(-0.124154\pi\)
0.133161 + 0.991094i \(0.457487\pi\)
\(510\) 0 0
\(511\) 9.24895 + 0.484863i 0.409149 + 0.0214491i
\(512\) 0 0
\(513\) −3.45234 + 5.97962i −0.152424 + 0.264007i
\(514\) 0 0
\(515\) −19.5995 33.9474i −0.863659 1.49590i
\(516\) 0 0
\(517\) −2.10906 −0.0927563
\(518\) 0 0
\(519\) 9.14971i 0.401628i
\(520\) 0 0
\(521\) −3.68697 + 2.12868i −0.161529 + 0.0932590i −0.578586 0.815622i \(-0.696395\pi\)
0.417056 + 0.908881i \(0.363062\pi\)
\(522\) 0 0
\(523\) 9.22671 15.9811i 0.403456 0.698806i −0.590684 0.806903i \(-0.701142\pi\)
0.994140 + 0.108096i \(0.0344755\pi\)
\(524\) 0 0
\(525\) −0.961624 + 0.489953i −0.0419687 + 0.0213833i
\(526\) 0 0
\(527\) −22.5176 13.0005i −0.980881 0.566312i
\(528\) 0 0
\(529\) 6.20156 + 10.7414i 0.269633 + 0.467018i
\(530\) 0 0
\(531\) −5.09719 −0.221199
\(532\) 0 0
\(533\) 2.21196 + 22.0153i 0.0958105 + 0.953590i
\(534\) 0 0
\(535\) 9.85580 + 17.0707i 0.426103 + 0.738033i
\(536\) 0 0
\(537\) 5.35757 9.27959i 0.231196 0.400444i
\(538\) 0 0
\(539\) −1.45729 1.05886i −0.0627701 0.0456082i
\(540\) 0 0
\(541\) −15.0566 + 26.0789i −0.647335 + 1.12122i 0.336421 + 0.941712i \(0.390783\pi\)
−0.983757 + 0.179506i \(0.942550\pi\)
\(542\) 0 0
\(543\) 2.15868 + 3.73894i 0.0926378 + 0.160453i
\(544\) 0 0
\(545\) 23.1041i 0.989670i
\(546\) 0 0
\(547\) 20.8394i 0.891028i −0.895275 0.445514i \(-0.853021\pi\)
0.895275 0.445514i \(-0.146979\pi\)
\(548\) 0 0
\(549\) 11.3697 + 19.6929i 0.485246 + 0.840471i
\(550\) 0 0
\(551\) −3.85779 + 6.68190i −0.164348 + 0.284658i
\(552\) 0 0
\(553\) −20.2269 + 10.3057i −0.860135 + 0.438244i
\(554\) 0 0
\(555\) −0.566737 0.327206i −0.0240567 0.0138891i
\(556\) 0 0
\(557\) −27.1658 + 15.6842i −1.15105 + 0.664561i −0.949144 0.314843i \(-0.898048\pi\)
−0.201910 + 0.979404i \(0.564715\pi\)
\(558\) 0 0
\(559\) 28.9477i 1.22436i
\(560\) 0 0
\(561\) 0.704873i 0.0297598i
\(562\) 0 0
\(563\) −2.00473 + 1.15743i −0.0844894 + 0.0487800i −0.541649 0.840604i \(-0.682200\pi\)
0.457160 + 0.889384i \(0.348867\pi\)
\(564\) 0 0
\(565\) 16.5640 28.6897i 0.696852 1.20698i
\(566\) 0 0
\(567\) −15.9148 0.834311i −0.668359 0.0350378i
\(568\) 0 0
\(569\) −19.9287 + 34.5175i −0.835454 + 1.44705i 0.0582056 + 0.998305i \(0.481462\pi\)
−0.893660 + 0.448745i \(0.851871\pi\)
\(570\) 0 0
\(571\) 33.3747 19.2689i 1.39669 0.806378i 0.402643 0.915357i \(-0.368092\pi\)
0.994044 + 0.108980i \(0.0347583\pi\)
\(572\) 0 0
\(573\) −7.45302 −0.311354
\(574\) 0 0
\(575\) −2.26448 −0.0944355
\(576\) 0 0
\(577\) −33.5277 + 19.3572i −1.39578 + 0.805851i −0.993947 0.109864i \(-0.964959\pi\)
−0.401829 + 0.915715i \(0.631625\pi\)
\(578\) 0 0
\(579\) 1.82425 3.15969i 0.0758131 0.131312i
\(580\) 0 0
\(581\) 8.15531 12.5576i 0.338339 0.520978i
\(582\) 0 0
\(583\) 0.773148 1.33913i 0.0320205 0.0554611i
\(584\) 0 0
\(585\) 16.4911 9.52116i 0.681824 0.393651i
\(586\) 0 0
\(587\) 25.2840i 1.04358i −0.853074 0.521790i \(-0.825264\pi\)
0.853074 0.521790i \(-0.174736\pi\)
\(588\) 0 0
\(589\) 11.5879i 0.477471i
\(590\) 0 0
\(591\) −4.33258 + 2.50142i −0.178219 + 0.102895i
\(592\) 0 0
\(593\) −17.1142 9.88092i −0.702798 0.405761i 0.105591 0.994410i \(-0.466327\pi\)
−0.808389 + 0.588649i \(0.799660\pi\)
\(594\) 0 0
\(595\) −21.5034 13.9650i −0.881555 0.572509i
\(596\) 0 0
\(597\) 2.69816 4.67335i 0.110428 0.191268i
\(598\) 0 0
\(599\) 5.67447 + 9.82846i 0.231852 + 0.401580i 0.958353 0.285585i \(-0.0921880\pi\)
−0.726501 + 0.687166i \(0.758855\pi\)
\(600\) 0 0
\(601\) 32.4655i 1.32429i −0.749374 0.662147i \(-0.769645\pi\)
0.749374 0.662147i \(-0.230355\pi\)
\(602\) 0 0
\(603\) 21.7977i 0.887672i
\(604\) 0 0
\(605\) 11.3421 + 19.6452i 0.461123 + 0.798689i
\(606\) 0 0
\(607\) 3.98589 6.90377i 0.161782 0.280215i −0.773726 0.633521i \(-0.781609\pi\)
0.935508 + 0.353305i \(0.114942\pi\)
\(608\) 0 0
\(609\) 5.74226 + 0.301030i 0.232688 + 0.0121984i
\(610\) 0 0
\(611\) −14.1603 + 24.5263i −0.572863 + 0.992227i
\(612\) 0 0
\(613\) 6.40119 + 11.0872i 0.258542 + 0.447807i 0.965851 0.259096i \(-0.0834247\pi\)
−0.707310 + 0.706904i \(0.750091\pi\)
\(614\) 0 0
\(615\) −0.778769 7.75099i −0.0314030 0.312550i
\(616\) 0 0
\(617\) −7.07093 −0.284665 −0.142333 0.989819i \(-0.545460\pi\)
−0.142333 + 0.989819i \(0.545460\pi\)
\(618\) 0 0
\(619\) −1.72125 2.98130i −0.0691830 0.119829i 0.829359 0.558716i \(-0.188706\pi\)
−0.898542 + 0.438888i \(0.855373\pi\)
\(620\) 0 0
\(621\) 9.35043 + 5.39847i 0.375220 + 0.216633i
\(622\) 0 0
\(623\) 18.8301 9.59403i 0.754411 0.384377i
\(624\) 0 0
\(625\) 10.5190 18.2194i 0.420759 0.728775i
\(626\) 0 0
\(627\) 0.272054 0.157071i 0.0108648 0.00627280i
\(628\) 0 0
\(629\) 2.51259i 0.100184i
\(630\) 0 0
\(631\) −28.8212 −1.14735 −0.573677 0.819082i \(-0.694483\pi\)
−0.573677 + 0.819082i \(0.694483\pi\)
\(632\) 0 0
\(633\) −7.35486 12.7390i −0.292330 0.506330i
\(634\) 0 0
\(635\) −21.7317 + 37.6405i −0.862398 + 1.49372i
\(636\) 0 0
\(637\) −22.0978 + 9.83773i −0.875546 + 0.389785i
\(638\) 0 0
\(639\) 15.1156 + 8.72701i 0.597965 + 0.345235i
\(640\) 0 0
\(641\) 36.8207 21.2585i 1.45433 0.839658i 0.455608 0.890180i \(-0.349422\pi\)
0.998723 + 0.0505221i \(0.0160885\pi\)
\(642\) 0 0
\(643\) 18.0505i 0.711840i 0.934516 + 0.355920i \(0.115833\pi\)
−0.934516 + 0.355920i \(0.884167\pi\)
\(644\) 0 0
\(645\) 10.1917i 0.401297i
\(646\) 0 0
\(647\) 12.4612 + 21.5834i 0.489900 + 0.848531i 0.999932 0.0116236i \(-0.00370000\pi\)
−0.510033 + 0.860155i \(0.670367\pi\)
\(648\) 0 0
\(649\) 0.427674 + 0.246918i 0.0167877 + 0.00969236i
\(650\) 0 0
\(651\) 7.69482 3.92055i 0.301584 0.153659i
\(652\) 0 0
\(653\) 0.739738 + 0.427088i 0.0289482 + 0.0167133i 0.514404 0.857548i \(-0.328013\pi\)
−0.485456 + 0.874261i \(0.661346\pi\)
\(654\) 0 0
\(655\) 11.8161 + 20.4661i 0.461694 + 0.799678i
\(656\) 0 0
\(657\) −9.29801 −0.362750
\(658\) 0 0
\(659\) 8.38863i 0.326775i −0.986562 0.163387i \(-0.947758\pi\)
0.986562 0.163387i \(-0.0522420\pi\)
\(660\) 0 0
\(661\) −3.94238 6.82841i −0.153341 0.265594i 0.779113 0.626884i \(-0.215670\pi\)
−0.932454 + 0.361289i \(0.882337\pi\)
\(662\) 0 0
\(663\) 8.19699 + 4.73253i 0.318345 + 0.183796i
\(664\) 0 0
\(665\) −0.598227 + 11.4114i −0.0231983 + 0.442515i
\(666\) 0 0
\(667\) 10.4486 + 6.03249i 0.404571 + 0.233579i
\(668\) 0 0
\(669\) −5.19855 + 3.00138i −0.200987 + 0.116040i
\(670\) 0 0
\(671\) 2.20308i 0.0850488i
\(672\) 0 0
\(673\) 36.9646i 1.42488i −0.701733 0.712440i \(-0.747590\pi\)
0.701733 0.712440i \(-0.252410\pi\)
\(674\) 0 0
\(675\) 1.99812 1.15362i 0.0769078 0.0444028i
\(676\) 0 0
\(677\) 1.13106 1.95905i 0.0434702 0.0752925i −0.843472 0.537174i \(-0.819492\pi\)
0.886942 + 0.461881i \(0.152825\pi\)
\(678\) 0 0
\(679\) 5.32517 + 3.45833i 0.204361 + 0.132718i
\(680\) 0 0
\(681\) 5.86915 10.1657i 0.224906 0.389549i
\(682\) 0 0
\(683\) 9.13602 5.27469i 0.349580 0.201830i −0.314920 0.949118i \(-0.601978\pi\)
0.664500 + 0.747288i \(0.268644\pi\)
\(684\) 0 0
\(685\) 24.1191i 0.921542i
\(686\) 0 0
\(687\) −3.58850 −0.136910
\(688\) 0 0
\(689\) −10.3819 17.9819i −0.395517 0.685056i
\(690\) 0 0
\(691\) −18.9468 10.9390i −0.720772 0.416138i 0.0942647 0.995547i \(-0.469950\pi\)
−0.815037 + 0.579409i \(0.803283\pi\)
\(692\) 0 0
\(693\) 1.51666 + 0.984965i 0.0576132 + 0.0374157i
\(694\) 0 0
\(695\) 0.442658 0.766706i 0.0167910 0.0290828i
\(696\) 0 0
\(697\) −24.2776 + 17.4693i −0.919580 + 0.661697i
\(698\) 0 0
\(699\) −6.27403 −0.237306
\(700\) 0 0
\(701\) 42.8380 1.61797 0.808985 0.587830i \(-0.200017\pi\)
0.808985 + 0.587830i \(0.200017\pi\)
\(702\) 0 0
\(703\) 0.969765 0.559894i 0.0365754 0.0211168i
\(704\) 0 0
\(705\) 4.98544 8.63503i 0.187762 0.325214i
\(706\) 0 0
\(707\) 2.71804 51.8476i 0.102222 1.94993i
\(708\) 0 0
\(709\) −10.4160 6.01368i −0.391181 0.225848i 0.291491 0.956574i \(-0.405849\pi\)
−0.682672 + 0.730725i \(0.739182\pi\)
\(710\) 0 0
\(711\) 19.7368 11.3951i 0.740189 0.427348i
\(712\) 0 0
\(713\) 18.1202 0.678606
\(714\) 0 0
\(715\) −1.84489 −0.0689950
\(716\) 0 0
\(717\) −2.64467 4.58070i −0.0987669 0.171069i
\(718\) 0 0
\(719\) 29.3267 + 16.9318i 1.09370 + 0.631449i 0.934560 0.355807i \(-0.115794\pi\)
0.159142 + 0.987256i \(0.449127\pi\)
\(720\) 0 0
\(721\) −22.6936 44.5404i −0.845154 1.65877i
\(722\) 0 0
\(723\) −9.61354 5.55038i −0.357531 0.206421i
\(724\) 0 0
\(725\) 2.23279 1.28910i 0.0829238 0.0478761i
\(726\) 0 0
\(727\) 7.88181i 0.292320i 0.989261 + 0.146160i \(0.0466914\pi\)
−0.989261 + 0.146160i \(0.953309\pi\)
\(728\) 0 0
\(729\) 10.1644 0.376460
\(730\) 0 0
\(731\) 33.8881 19.5653i 1.25340 0.723649i
\(732\) 0 0
\(733\) 5.99235 10.3790i 0.221332 0.383359i −0.733880 0.679279i \(-0.762293\pi\)
0.955213 + 0.295920i \(0.0956261\pi\)
\(734\) 0 0
\(735\) 7.78002 3.46359i 0.286970 0.127757i
\(736\) 0 0
\(737\) −1.05592 + 1.82891i −0.0388954 + 0.0673688i
\(738\) 0 0
\(739\) 3.88348 + 6.72638i 0.142856 + 0.247434i 0.928571 0.371155i \(-0.121038\pi\)
−0.785715 + 0.618589i \(0.787705\pi\)
\(740\) 0 0
\(741\) 4.21830i 0.154963i
\(742\) 0 0
\(743\) −21.1316 −0.775245 −0.387622 0.921818i \(-0.626704\pi\)
−0.387622 + 0.921818i \(0.626704\pi\)
\(744\) 0 0
\(745\) −12.1640 + 7.02289i −0.445655 + 0.257299i
\(746\) 0 0
\(747\) −7.51610 + 13.0183i −0.275000 + 0.476314i
\(748\) 0 0
\(749\) 11.4117 + 22.3976i 0.416974 + 0.818389i
\(750\) 0 0
\(751\) 16.1002 + 9.29546i 0.587505 + 0.339196i 0.764110 0.645086i \(-0.223178\pi\)
−0.176605 + 0.984282i \(0.556512\pi\)
\(752\) 0 0
\(753\) −8.10365 + 4.67865i −0.295313 + 0.170499i
\(754\) 0 0
\(755\) 1.59904i 0.0581950i
\(756\) 0 0
\(757\) 18.9035i 0.687061i −0.939142 0.343530i \(-0.888377\pi\)
0.939142 0.343530i \(-0.111623\pi\)
\(758\) 0 0
\(759\) −0.245614 0.425416i −0.00891521 0.0154416i
\(760\) 0 0
\(761\) −1.95049 + 3.37835i −0.0707052 + 0.122465i −0.899211 0.437516i \(-0.855858\pi\)
0.828505 + 0.559981i \(0.189192\pi\)
\(762\) 0 0
\(763\) 1.54246 29.4230i 0.0558408 1.06518i
\(764\) 0 0
\(765\) 22.2922 + 12.8704i 0.805977 + 0.465331i
\(766\) 0 0
\(767\) 5.74282 3.31562i 0.207361 0.119720i
\(768\) 0 0
\(769\) 6.88478 0.248272 0.124136 0.992265i \(-0.460384\pi\)
0.124136 + 0.992265i \(0.460384\pi\)
\(770\) 0 0
\(771\) −1.91259 −0.0688802
\(772\) 0 0
\(773\) −3.73918 + 2.15882i −0.134489 + 0.0776473i −0.565735 0.824587i \(-0.691407\pi\)
0.431246 + 0.902234i \(0.358074\pi\)
\(774\) 0 0
\(775\) 1.93608 3.35339i 0.0695460 0.120457i
\(776\) 0 0
\(777\) −0.699894 0.454532i −0.0251086 0.0163063i
\(778\) 0 0
\(779\) 12.1524 + 5.47746i 0.435405 + 0.196250i
\(780\) 0 0
\(781\) −0.845506 1.46446i −0.0302546 0.0524024i
\(782\) 0 0
\(783\) −12.2928 −0.439308
\(784\) 0 0
\(785\) 30.3820i 1.08438i
\(786\) 0 0
\(787\) 18.2446 + 31.6005i 0.650349 + 1.12644i 0.983038 + 0.183400i \(0.0587105\pi\)
−0.332690 + 0.943036i \(0.607956\pi\)
\(788\) 0 0
\(789\) −1.82245 + 3.15658i −0.0648810 + 0.112377i
\(790\) 0 0
\(791\) 23.0096 35.4304i 0.818126 1.25976i
\(792\) 0 0
\(793\) −25.6196 14.7915i −0.909780 0.525262i
\(794\) 0 0
\(795\) 3.65517 + 6.33093i 0.129635 + 0.224535i
\(796\) 0 0
\(797\) −12.4618 −0.441419 −0.220709 0.975340i \(-0.570837\pi\)
−0.220709 + 0.975340i \(0.570837\pi\)
\(798\) 0 0
\(799\) −38.2828 −1.35435
\(800\) 0 0
\(801\) −18.3738 + 10.6081i −0.649208 + 0.374820i
\(802\) 0 0
\(803\) 0.780138 + 0.450413i 0.0275305 + 0.0158947i
\(804\) 0 0
\(805\) 17.8442 + 0.935457i 0.628925 + 0.0329705i
\(806\) 0 0
\(807\) −1.74536 1.00768i −0.0614396 0.0354722i
\(808\) 0 0
\(809\) −32.3793 + 18.6942i −1.13840 + 0.657253i −0.946033 0.324071i \(-0.894948\pi\)
−0.192363 + 0.981324i \(0.561615\pi\)
\(810\) 0 0
\(811\) 9.01558 0.316580 0.158290 0.987393i \(-0.449402\pi\)
0.158290 + 0.987393i \(0.449402\pi\)
\(812\) 0 0
\(813\) 4.40970i 0.154655i
\(814\) 0 0
\(815\) 4.76879 + 8.25979i 0.167044 + 0.289328i
\(816\) 0 0
\(817\) −15.1029 8.71967i −0.528384 0.305063i
\(818\) 0 0
\(819\) 21.6371 11.0242i 0.756061 0.385217i
\(820\) 0 0
\(821\) −4.55132 + 7.88312i −0.158842 + 0.275123i −0.934451 0.356091i \(-0.884109\pi\)
0.775609 + 0.631213i \(0.217443\pi\)
\(822\) 0 0
\(823\) 11.1005 6.40890i 0.386941 0.223400i −0.293893 0.955838i \(-0.594951\pi\)
0.680834 + 0.732438i \(0.261618\pi\)
\(824\) 0 0
\(825\) −0.104972 −0.00365466
\(826\) 0 0
\(827\) 52.4173i 1.82273i 0.411602 + 0.911364i \(0.364970\pi\)
−0.411602 + 0.911364i \(0.635030\pi\)
\(828\) 0 0
\(829\) −19.7389 34.1888i −0.685562 1.18743i −0.973260 0.229707i \(-0.926223\pi\)
0.287698 0.957721i \(-0.407110\pi\)
\(830\) 0 0
\(831\) 12.6103 + 7.28056i 0.437447 + 0.252560i
\(832\) 0 0
\(833\) −26.4523 19.2200i −0.916517 0.665933i
\(834\) 0 0
\(835\) 34.2568 + 19.7782i 1.18551 + 0.684452i
\(836\) 0 0
\(837\) −15.9888 + 9.23114i −0.552654 + 0.319075i
\(838\) 0 0
\(839\) 37.7810i 1.30435i −0.758071 0.652173i \(-0.773858\pi\)
0.758071 0.652173i \(-0.226142\pi\)
\(840\) 0 0
\(841\) 15.2635 0.526328
\(842\) 0 0
\(843\) 0.166925 + 0.289122i 0.00574919 + 0.00995789i
\(844\) 0 0
\(845\) 1.09889 1.90333i 0.0378029 0.0654765i
\(846\) 0 0
\(847\) 13.1327 + 25.7753i 0.451243 + 0.885650i
\(848\) 0 0
\(849\) −4.58633 2.64792i −0.157403 0.0908764i
\(850\) 0 0
\(851\) −0.875515 1.51644i −0.0300123 0.0519828i
\(852\) 0 0
\(853\) −14.0326 −0.480468 −0.240234 0.970715i \(-0.577224\pi\)
−0.240234 + 0.970715i \(0.577224\pi\)
\(854\) 0 0
\(855\) 11.4719i 0.392332i
\(856\) 0 0
\(857\) −4.29615 7.44115i −0.146754 0.254185i 0.783272 0.621679i \(-0.213549\pi\)
−0.930026 + 0.367494i \(0.880216\pi\)
\(858\) 0 0
\(859\) 3.19621 5.53600i 0.109053 0.188886i −0.806334 0.591461i \(-0.798551\pi\)
0.915387 + 0.402575i \(0.131885\pi\)
\(860\) 0 0
\(861\) −0.474294 9.92287i −0.0161639 0.338170i
\(862\) 0 0
\(863\) 13.0029 22.5217i 0.442623 0.766646i −0.555260 0.831677i \(-0.687381\pi\)
0.997883 + 0.0650308i \(0.0207146\pi\)
\(864\) 0 0
\(865\) 16.1860 + 28.0350i 0.550342 + 0.953220i
\(866\) 0 0
\(867\) 2.82585i 0.0959710i
\(868\) 0 0
\(869\) −2.20799 −0.0749010
\(870\) 0 0
\(871\) 14.1790 + 24.5587i 0.480436 + 0.832140i
\(872\) 0 0
\(873\) −5.52051 3.18727i −0.186841 0.107873i
\(874\) 0 0
\(875\) 17.0281 26.2200i 0.575653 0.886398i
\(876\) 0 0
\(877\) 18.5426 32.1168i 0.626140 1.08451i −0.362179 0.932109i \(-0.617967\pi\)
0.988319 0.152398i \(-0.0486997\pi\)
\(878\) 0 0
\(879\) −6.74604 11.6845i −0.227538 0.394108i
\(880\) 0 0
\(881\) 12.5556 0.423008 0.211504 0.977377i \(-0.432164\pi\)
0.211504 + 0.977377i \(0.432164\pi\)
\(882\) 0 0
\(883\) 46.7251i 1.57242i 0.617957 + 0.786212i \(0.287960\pi\)
−0.617957 + 0.786212i \(0.712040\pi\)
\(884\) 0 0
\(885\) −2.02189 + 1.16734i −0.0679650 + 0.0392396i
\(886\) 0 0
\(887\) −43.2765 24.9857i −1.45308 0.838937i −0.454426 0.890784i \(-0.650156\pi\)
−0.998655 + 0.0518476i \(0.983489\pi\)
\(888\) 0 0
\(889\) −30.1883 + 46.4843i −1.01248 + 1.55903i
\(890\) 0 0
\(891\) −1.34240 0.775032i −0.0449719 0.0259646i
\(892\) 0 0
\(893\) 8.53077 + 14.7757i 0.285471 + 0.494451i
\(894\) 0 0
\(895\) 37.9106i 1.26721i
\(896\) 0 0
\(897\) −6.59622 −0.220242
\(898\) 0 0
\(899\) −17.8666 + 10.3153i −0.595884 + 0.344034i
\(900\) 0 0
\(901\) 14.0339 24.3074i 0.467537 0.809797i
\(902\) 0 0
\(903\) −0.680410 + 12.9791i −0.0226426 + 0.431917i
\(904\) 0 0
\(905\) −13.2285 7.63750i −0.439731 0.253879i
\(906\) 0 0
\(907\) 6.14774 + 10.6482i 0.204132 + 0.353568i 0.949856 0.312688i \(-0.101229\pi\)
−0.745724 + 0.666255i \(0.767896\pi\)
\(908\) 0 0
\(909\) 52.1226i 1.72880i
\(910\) 0 0
\(911\) 7.78973 0.258085 0.129043 0.991639i \(-0.458810\pi\)
0.129043 + 0.991639i \(0.458810\pi\)
\(912\) 0 0
\(913\) 1.26126 0.728188i 0.0417416 0.0240995i
\(914\) 0 0
\(915\) 9.01997 + 5.20768i 0.298191 + 0.172161i
\(916\) 0 0
\(917\) 13.6815 + 26.8524i 0.451802 + 0.886746i
\(918\) 0 0
\(919\) −26.8463 15.4997i −0.885576 0.511288i −0.0130835 0.999914i \(-0.504165\pi\)
−0.872493 + 0.488627i \(0.837498\pi\)
\(920\) 0 0
\(921\) −15.6183 + 9.01723i −0.514641 + 0.297128i
\(922\) 0 0
\(923\) −22.7070 −0.747409
\(924\) 0 0
\(925\) −0.374183 −0.0123031
\(926\) 0 0
\(927\) 25.0924 + 43.4613i 0.824142 + 1.42746i
\(928\) 0 0
\(929\) −14.7331 8.50615i −0.483377 0.279078i 0.238446 0.971156i \(-0.423362\pi\)
−0.721823 + 0.692078i \(0.756695\pi\)
\(930\) 0 0
\(931\) −1.52368 + 14.4925i −0.0499367 + 0.474971i
\(932\) 0 0
\(933\) −4.68266 + 8.11060i −0.153303 + 0.265529i
\(934\) 0 0
\(935\) −1.24694 2.15976i −0.0407791 0.0706316i
\(936\) 0 0
\(937\) 38.2023i 1.24802i 0.781418 + 0.624008i \(0.214497\pi\)
−0.781418 + 0.624008i \(0.785503\pi\)
\(938\) 0 0
\(939\) 12.7585 0.416358
\(940\) 0 0
\(941\) −7.07502 12.2543i −0.230639 0.399478i 0.727357 0.686259i \(-0.240748\pi\)
−0.957996 + 0.286781i \(0.907415\pi\)
\(942\) 0 0
\(943\) 8.56519 19.0029i 0.278921 0.618819i
\(944\) 0 0
\(945\) −16.2218 + 8.26511i −0.527696 + 0.268864i
\(946\) 0 0
\(947\) 11.1438 19.3015i 0.362123 0.627216i −0.626187 0.779673i \(-0.715385\pi\)
0.988310 + 0.152457i \(0.0487186\pi\)
\(948\) 0 0
\(949\) 10.4757 6.04817i 0.340057 0.196332i
\(950\) 0 0
\(951\) −13.5700 −0.440037
\(952\) 0 0
\(953\) −16.2735 −0.527151 −0.263576 0.964639i \(-0.584902\pi\)
−0.263576 + 0.964639i \(0.584902\pi\)
\(954\) 0 0
\(955\) 22.8363 13.1845i 0.738966 0.426642i
\(956\) 0 0
\(957\) 0.484353 + 0.279641i 0.0156569 + 0.00903952i
\(958\) 0 0
\(959\) 1.61022 30.7156i 0.0519968 0.991858i
\(960\) 0 0
\(961\) 0.00767448 0.0132926i 0.000247564 0.000428793i
\(962\) 0 0
\(963\) −12.6179 21.8549i −0.406607 0.704264i
\(964\) 0 0
\(965\) 12.9085i 0.415540i
\(966\) 0 0
\(967\) 14.3396i 0.461130i 0.973057 + 0.230565i \(0.0740574\pi\)
−0.973057 + 0.230565i \(0.925943\pi\)
\(968\) 0 0
\(969\) 4.93823 2.85109i 0.158639 0.0915901i
\(970\) 0 0
\(971\) −20.6839 11.9419i −0.663779 0.383233i 0.129937 0.991522i \(-0.458523\pi\)
−0.793715 + 0.608289i \(0.791856\pi\)
\(972\) 0 0
\(973\) 0.614911 0.946847i 0.0197131 0.0303545i
\(974\) 0 0
\(975\) −0.704784 + 1.22072i −0.0225712 + 0.0390944i
\(976\) 0 0
\(977\) −13.3035 + 7.68080i −0.425618 + 0.245730i −0.697478 0.716606i \(-0.745694\pi\)
0.271860 + 0.962337i \(0.412361\pi\)
\(978\) 0 0
\(979\) 2.05551 0.0656945
\(980\) 0 0
\(981\) 29.5791i 0.944387i
\(982\) 0 0
\(983\) 4.42677 + 7.66738i 0.141192 + 0.244552i 0.927946 0.372715i \(-0.121573\pi\)
−0.786754 + 0.617267i \(0.788240\pi\)
\(984\) 0 0
\(985\) 8.85012 15.3289i 0.281988 0.488418i
\(986\) 0 0
\(987\) 6.92543 10.6639i 0.220439 0.339435i
\(988\) 0 0
\(989\) −13.6351 + 23.6167i −0.433571 + 0.750966i
\(990\) 0 0
\(991\) −22.9929 + 13.2750i −0.730394 + 0.421693i −0.818566 0.574412i \(-0.805231\pi\)
0.0881722 + 0.996105i \(0.471897\pi\)
\(992\) 0 0
\(993\) 5.66661 0.179824
\(994\) 0 0
\(995\) 19.0924i 0.605270i
\(996\) 0 0
\(997\) 3.26235 1.88352i 0.103320 0.0596517i −0.447450 0.894309i \(-0.647668\pi\)
0.550770 + 0.834657i \(0.314334\pi\)
\(998\) 0 0
\(999\) 1.54507 + 0.892044i 0.0488837 + 0.0282230i
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 1148.2.r.a.737.12 yes 56
7.4 even 3 inner 1148.2.r.a.81.17 yes 56
41.40 even 2 inner 1148.2.r.a.737.17 yes 56
287.81 even 6 inner 1148.2.r.a.81.12 56
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1148.2.r.a.81.12 56 287.81 even 6 inner
1148.2.r.a.81.17 yes 56 7.4 even 3 inner
1148.2.r.a.737.12 yes 56 1.1 even 1 trivial
1148.2.r.a.737.17 yes 56 41.40 even 2 inner