Properties

Label 1764.2.x.b.293.1
Level $1764$
Weight $2$
Character 1764.293
Analytic conductor $14.086$
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] = [1764,2,Mod(293,1764)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1764, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 5, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1764.293");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1764.x (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: \(14.0856109166\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 2 x^{15} + 5 x^{14} - 17 x^{13} + 22 x^{12} - 31 x^{11} + 62 x^{10} - 52 x^{9} + 52 x^{8} - 156 x^{7} + 558 x^{6} - 837 x^{5} + 1782 x^{4} - 4131 x^{3} + 3645 x^{2} + \cdots + 6561 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 3^{4} \)
Twist minimal: no (minimal twist has level 252)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 293.1
Root \(-0.268067 + 1.71118i\) of defining polynomial
Character \(\chi\) \(=\) 1764.293
Dual form 1764.2.x.b.1469.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.56269 - 0.746985i) q^{3} +(0.842869 - 1.45989i) q^{5} +(1.88403 + 2.33462i) q^{9} +O(q^{10})\) \(q+(-1.56269 - 0.746985i) q^{3} +(0.842869 - 1.45989i) q^{5} +(1.88403 + 2.33462i) q^{9} +(-3.38216 + 1.95269i) q^{11} +(5.24391 + 3.02757i) q^{13} +(-2.40766 + 1.65175i) q^{15} -0.402488 q^{17} -0.168144i q^{19} +(-7.69373 - 4.44198i) q^{23} +(1.07914 + 1.86913i) q^{25} +(-1.20023 - 5.05563i) q^{27} +(-6.15380 + 3.55290i) q^{29} +(-5.44527 - 3.14383i) q^{31} +(6.74391 - 0.525036i) q^{33} -6.26515 q^{37} +(-5.93308 - 8.64829i) q^{39} +(-1.64707 + 2.85281i) q^{41} +(1.80474 + 3.12590i) q^{43} +(4.99628 - 0.782698i) q^{45} +(4.38482 + 7.59474i) q^{47} +(0.628965 + 0.300652i) q^{51} +5.71148i q^{53} +6.58345i q^{55} +(-0.125601 + 0.262757i) q^{57} +(2.25163 - 3.89994i) q^{59} +(-4.43678 + 2.56157i) q^{61} +(8.83986 - 5.10369i) q^{65} +(2.95521 - 5.11857i) q^{67} +(8.70486 + 12.6886i) q^{69} +11.4308i q^{71} +6.99239i q^{73} +(-0.290159 - 3.72699i) q^{75} +(-0.603968 - 1.04610i) q^{79} +(-1.90088 + 8.79697i) q^{81} +(-0.181350 - 0.314108i) q^{83} +(-0.339244 + 0.587588i) q^{85} +(12.2705 - 0.955298i) q^{87} -2.77052 q^{89} +(6.16090 + 8.98038i) q^{93} +(-0.245471 - 0.141723i) q^{95} +(0.508914 - 0.293821i) q^{97} +(-10.9309 - 4.21713i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 6 q^{9} + 6 q^{11} + 3 q^{13} - 3 q^{15} + 18 q^{17} - 21 q^{23} - 8 q^{25} - 9 q^{27} + 6 q^{29} - 6 q^{31} + 27 q^{33} - 2 q^{37} + 6 q^{39} + 6 q^{41} - 2 q^{43} - 15 q^{45} - 18 q^{47} + 18 q^{51} + 15 q^{57} - 15 q^{59} - 3 q^{61} + 39 q^{65} - 7 q^{67} - 21 q^{69} - 42 q^{75} - q^{79} - 18 q^{81} + 6 q^{85} + 51 q^{87} + 42 q^{89} + 48 q^{93} - 6 q^{95} + 3 q^{97} - 9 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1764\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(883\) \(1081\)
\(\chi(n)\) \(e\left(\frac{5}{6}\right)\) \(1\) \(-1\)

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
\(3\) −1.56269 0.746985i −0.902222 0.431272i
\(4\) 0 0
\(5\) 0.842869 1.45989i 0.376942 0.652883i −0.613673 0.789560i \(-0.710309\pi\)
0.990616 + 0.136677i \(0.0436422\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 1.88403 + 2.33462i 0.628009 + 0.778206i
\(10\) 0 0
\(11\) −3.38216 + 1.95269i −1.01976 + 0.588758i −0.914034 0.405637i \(-0.867050\pi\)
−0.105725 + 0.994395i \(0.533716\pi\)
\(12\) 0 0
\(13\) 5.24391 + 3.02757i 1.45440 + 0.839698i 0.998727 0.0504496i \(-0.0160654\pi\)
0.455673 + 0.890147i \(0.349399\pi\)
\(14\) 0 0
\(15\) −2.40766 + 1.65175i −0.621656 + 0.426481i
\(16\) 0 0
\(17\) −0.402488 −0.0976176 −0.0488088 0.998808i \(-0.515542\pi\)
−0.0488088 + 0.998808i \(0.515542\pi\)
\(18\) 0 0
\(19\) 0.168144i 0.0385748i −0.999814 0.0192874i \(-0.993860\pi\)
0.999814 0.0192874i \(-0.00613975\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −7.69373 4.44198i −1.60425 0.926216i −0.990623 0.136623i \(-0.956375\pi\)
−0.613630 0.789593i \(-0.710291\pi\)
\(24\) 0 0
\(25\) 1.07914 + 1.86913i 0.215829 + 0.373827i
\(26\) 0 0
\(27\) −1.20023 5.05563i −0.230985 0.972957i
\(28\) 0 0
\(29\) −6.15380 + 3.55290i −1.14273 + 0.659757i −0.947106 0.320921i \(-0.896007\pi\)
−0.195627 + 0.980678i \(0.562674\pi\)
\(30\) 0 0
\(31\) −5.44527 3.14383i −0.978000 0.564649i −0.0763342 0.997082i \(-0.524322\pi\)
−0.901666 + 0.432434i \(0.857655\pi\)
\(32\) 0 0
\(33\) 6.74391 0.525036i 1.17396 0.0913971i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −6.26515 −1.02998 −0.514992 0.857195i \(-0.672205\pi\)
−0.514992 + 0.857195i \(0.672205\pi\)
\(38\) 0 0
\(39\) −5.93308 8.64829i −0.950053 1.38484i
\(40\) 0 0
\(41\) −1.64707 + 2.85281i −0.257229 + 0.445534i −0.965499 0.260408i \(-0.916143\pi\)
0.708269 + 0.705942i \(0.249476\pi\)
\(42\) 0 0
\(43\) 1.80474 + 3.12590i 0.275220 + 0.476695i 0.970191 0.242343i \(-0.0779161\pi\)
−0.694971 + 0.719038i \(0.744583\pi\)
\(44\) 0 0
\(45\) 4.99628 0.782698i 0.744801 0.116678i
\(46\) 0 0
\(47\) 4.38482 + 7.59474i 0.639592 + 1.10781i 0.985522 + 0.169546i \(0.0542301\pi\)
−0.345930 + 0.938260i \(0.612437\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 0.628965 + 0.300652i 0.0880727 + 0.0420997i
\(52\) 0 0
\(53\) 5.71148i 0.784532i 0.919852 + 0.392266i \(0.128309\pi\)
−0.919852 + 0.392266i \(0.871691\pi\)
\(54\) 0 0
\(55\) 6.58345i 0.887712i
\(56\) 0 0
\(57\) −0.125601 + 0.262757i −0.0166362 + 0.0348030i
\(58\) 0 0
\(59\) 2.25163 3.89994i 0.293138 0.507729i −0.681412 0.731900i \(-0.738634\pi\)
0.974550 + 0.224171i \(0.0719673\pi\)
\(60\) 0 0
\(61\) −4.43678 + 2.56157i −0.568071 + 0.327976i −0.756379 0.654134i \(-0.773033\pi\)
0.188308 + 0.982110i \(0.439700\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 8.83986 5.10369i 1.09645 0.633035i
\(66\) 0 0
\(67\) 2.95521 5.11857i 0.361036 0.625332i −0.627096 0.778942i \(-0.715757\pi\)
0.988132 + 0.153610i \(0.0490899\pi\)
\(68\) 0 0
\(69\) 8.70486 + 12.6886i 1.04794 + 1.52752i
\(70\) 0 0
\(71\) 11.4308i 1.35658i 0.734792 + 0.678292i \(0.237280\pi\)
−0.734792 + 0.678292i \(0.762720\pi\)
\(72\) 0 0
\(73\) 6.99239i 0.818397i 0.912445 + 0.409199i \(0.134192\pi\)
−0.912445 + 0.409199i \(0.865808\pi\)
\(74\) 0 0
\(75\) −0.290159 3.72699i −0.0335046 0.430356i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −0.603968 1.04610i −0.0679517 0.117696i 0.830048 0.557692i \(-0.188313\pi\)
−0.898000 + 0.439996i \(0.854980\pi\)
\(80\) 0 0
\(81\) −1.90088 + 8.79697i −0.211209 + 0.977441i
\(82\) 0 0
\(83\) −0.181350 0.314108i −0.0199058 0.0344779i 0.855901 0.517140i \(-0.173003\pi\)
−0.875807 + 0.482662i \(0.839670\pi\)
\(84\) 0 0
\(85\) −0.339244 + 0.587588i −0.0367962 + 0.0637329i
\(86\) 0 0
\(87\) 12.2705 0.955298i 1.31553 0.102419i
\(88\) 0 0
\(89\) −2.77052 −0.293674 −0.146837 0.989161i \(-0.546909\pi\)
−0.146837 + 0.989161i \(0.546909\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 6.16090 + 8.98038i 0.638856 + 0.931222i
\(94\) 0 0
\(95\) −0.245471 0.141723i −0.0251848 0.0145405i
\(96\) 0 0
\(97\) 0.508914 0.293821i 0.0516723 0.0298330i −0.473941 0.880556i \(-0.657169\pi\)
0.525614 + 0.850723i \(0.323836\pi\)
\(98\) 0 0
\(99\) −10.9309 4.21713i −1.09859 0.423837i
\(100\) 0 0
\(101\) −6.92329 11.9915i −0.688893 1.19320i −0.972196 0.234167i \(-0.924764\pi\)
0.283303 0.959030i \(-0.408570\pi\)
\(102\) 0 0
\(103\) 10.4610 + 6.03967i 1.03075 + 0.595106i 0.917201 0.398425i \(-0.130443\pi\)
0.113554 + 0.993532i \(0.463777\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 18.3942i 1.77824i 0.457678 + 0.889118i \(0.348681\pi\)
−0.457678 + 0.889118i \(0.651319\pi\)
\(108\) 0 0
\(109\) 11.0207 1.05559 0.527796 0.849371i \(-0.323018\pi\)
0.527796 + 0.849371i \(0.323018\pi\)
\(110\) 0 0
\(111\) 9.79051 + 4.67997i 0.929274 + 0.444203i
\(112\) 0 0
\(113\) −7.36811 4.25398i −0.693133 0.400181i 0.111652 0.993747i \(-0.464386\pi\)
−0.804785 + 0.593567i \(0.797719\pi\)
\(114\) 0 0
\(115\) −12.9696 + 7.48801i −1.20942 + 0.698260i
\(116\) 0 0
\(117\) 2.81144 + 17.9466i 0.259918 + 1.65916i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 2.12600 3.68234i 0.193273 0.334758i
\(122\) 0 0
\(123\) 4.70487 3.22773i 0.424224 0.291035i
\(124\) 0 0
\(125\) 12.0670 1.07930
\(126\) 0 0
\(127\) −10.6312 −0.943365 −0.471682 0.881769i \(-0.656353\pi\)
−0.471682 + 0.881769i \(0.656353\pi\)
\(128\) 0 0
\(129\) −0.485255 6.23293i −0.0427243 0.548779i
\(130\) 0 0
\(131\) −3.16740 + 5.48610i −0.276737 + 0.479322i −0.970572 0.240812i \(-0.922586\pi\)
0.693835 + 0.720134i \(0.255920\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) −8.39232 2.50903i −0.722296 0.215942i
\(136\) 0 0
\(137\) −14.4158 + 8.32296i −1.23162 + 0.711078i −0.967368 0.253375i \(-0.918459\pi\)
−0.264255 + 0.964453i \(0.585126\pi\)
\(138\) 0 0
\(139\) −4.24007 2.44800i −0.359638 0.207637i 0.309284 0.950970i \(-0.399911\pi\)
−0.668922 + 0.743333i \(0.733244\pi\)
\(140\) 0 0
\(141\) −1.17898 15.1437i −0.0992884 1.27533i
\(142\) 0 0
\(143\) −23.6477 −1.97752
\(144\) 0 0
\(145\) 11.9785i 0.994761i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 4.57864 + 2.64348i 0.375097 + 0.216562i 0.675683 0.737192i \(-0.263849\pi\)
−0.300586 + 0.953755i \(0.597182\pi\)
\(150\) 0 0
\(151\) 7.29163 + 12.6295i 0.593385 + 1.02777i 0.993773 + 0.111427i \(0.0355421\pi\)
−0.400388 + 0.916346i \(0.631125\pi\)
\(152\) 0 0
\(153\) −0.758298 0.939655i −0.0613047 0.0759666i
\(154\) 0 0
\(155\) −9.17930 + 5.29967i −0.737299 + 0.425680i
\(156\) 0 0
\(157\) 15.4160 + 8.90044i 1.23033 + 0.710332i 0.967099 0.254400i \(-0.0818781\pi\)
0.263232 + 0.964732i \(0.415211\pi\)
\(158\) 0 0
\(159\) 4.26639 8.92529i 0.338346 0.707822i
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −0.0964456 −0.00755420 −0.00377710 0.999993i \(-0.501202\pi\)
−0.00377710 + 0.999993i \(0.501202\pi\)
\(164\) 0 0
\(165\) 4.91774 10.2879i 0.382845 0.800913i
\(166\) 0 0
\(167\) −2.47872 + 4.29327i −0.191809 + 0.332224i −0.945850 0.324604i \(-0.894769\pi\)
0.754041 + 0.656828i \(0.228102\pi\)
\(168\) 0 0
\(169\) 11.8324 + 20.4943i 0.910185 + 1.57649i
\(170\) 0 0
\(171\) 0.392551 0.316787i 0.0300191 0.0242253i
\(172\) 0 0
\(173\) −7.40033 12.8177i −0.562637 0.974515i −0.997265 0.0739055i \(-0.976454\pi\)
0.434629 0.900610i \(-0.356880\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −6.43181 + 4.41248i −0.483444 + 0.331662i
\(178\) 0 0
\(179\) 0.684450i 0.0511582i −0.999673 0.0255791i \(-0.991857\pi\)
0.999673 0.0255791i \(-0.00814297\pi\)
\(180\) 0 0
\(181\) 7.84745i 0.583297i −0.956526 0.291648i \(-0.905796\pi\)
0.956526 0.291648i \(-0.0942037\pi\)
\(182\) 0 0
\(183\) 8.84678 0.688752i 0.653973 0.0509140i
\(184\) 0 0
\(185\) −5.28070 + 9.14644i −0.388245 + 0.672459i
\(186\) 0 0
\(187\) 1.36128 0.785934i 0.0995464 0.0574732i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 16.9694 9.79729i 1.22786 0.708907i 0.261281 0.965263i \(-0.415855\pi\)
0.966582 + 0.256356i \(0.0825219\pi\)
\(192\) 0 0
\(193\) −9.18116 + 15.9022i −0.660875 + 1.14467i 0.319512 + 0.947582i \(0.396481\pi\)
−0.980386 + 0.197086i \(0.936852\pi\)
\(194\) 0 0
\(195\) −17.6264 + 1.37227i −1.26225 + 0.0982705i
\(196\) 0 0
\(197\) 5.92313i 0.422006i 0.977485 + 0.211003i \(0.0676730\pi\)
−0.977485 + 0.211003i \(0.932327\pi\)
\(198\) 0 0
\(199\) 15.7348i 1.11541i 0.830039 + 0.557706i \(0.188318\pi\)
−0.830039 + 0.557706i \(0.811682\pi\)
\(200\) 0 0
\(201\) −8.44157 + 5.79126i −0.595423 + 0.408484i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 2.77653 + 4.80909i 0.193921 + 0.335881i
\(206\) 0 0
\(207\) −4.12487 26.3307i −0.286699 1.83011i
\(208\) 0 0
\(209\) 0.328332 + 0.568688i 0.0227112 + 0.0393370i
\(210\) 0 0
\(211\) 5.06619 8.77489i 0.348771 0.604088i −0.637261 0.770648i \(-0.719933\pi\)
0.986031 + 0.166560i \(0.0532659\pi\)
\(212\) 0 0
\(213\) 8.53862 17.8628i 0.585057 1.22394i
\(214\) 0 0
\(215\) 6.08462 0.414968
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 5.22321 10.9270i 0.352952 0.738376i
\(220\) 0 0
\(221\) −2.11061 1.21856i −0.141975 0.0819693i
\(222\) 0 0
\(223\) −13.3944 + 7.73325i −0.896955 + 0.517857i −0.876211 0.481928i \(-0.839937\pi\)
−0.0207437 + 0.999785i \(0.506603\pi\)
\(224\) 0 0
\(225\) −2.33058 + 6.04089i −0.155372 + 0.402726i
\(226\) 0 0
\(227\) −14.0360 24.3110i −0.931600 1.61358i −0.780588 0.625046i \(-0.785080\pi\)
−0.151011 0.988532i \(-0.548253\pi\)
\(228\) 0 0
\(229\) −14.7453 8.51319i −0.974396 0.562568i −0.0738222 0.997271i \(-0.523520\pi\)
−0.900573 + 0.434704i \(0.856853\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 18.4769i 1.21047i −0.796049 0.605233i \(-0.793080\pi\)
0.796049 0.605233i \(-0.206920\pi\)
\(234\) 0 0
\(235\) 14.7833 0.964358
\(236\) 0 0
\(237\) 0.162394 + 2.08590i 0.0105486 + 0.135493i
\(238\) 0 0
\(239\) −6.06656 3.50253i −0.392413 0.226560i 0.290792 0.956786i \(-0.406081\pi\)
−0.683205 + 0.730226i \(0.739415\pi\)
\(240\) 0 0
\(241\) 5.38459 3.10879i 0.346852 0.200255i −0.316446 0.948611i \(-0.602490\pi\)
0.663298 + 0.748355i \(0.269156\pi\)
\(242\) 0 0
\(243\) 9.54170 12.3270i 0.612101 0.790780i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 0.509067 0.881730i 0.0323912 0.0561031i
\(248\) 0 0
\(249\) 0.0487612 + 0.626321i 0.00309012 + 0.0396915i
\(250\) 0 0
\(251\) −9.81844 −0.619734 −0.309867 0.950780i \(-0.600285\pi\)
−0.309867 + 0.950780i \(0.600285\pi\)
\(252\) 0 0
\(253\) 34.6952 2.18127
\(254\) 0 0
\(255\) 0.969055 0.664810i 0.0606846 0.0416320i
\(256\) 0 0
\(257\) 0.667904 1.15684i 0.0416627 0.0721619i −0.844442 0.535647i \(-0.820068\pi\)
0.886105 + 0.463485i \(0.153401\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −19.8886 7.67302i −1.23107 0.474948i
\(262\) 0 0
\(263\) −17.6238 + 10.1751i −1.08673 + 0.627424i −0.932704 0.360643i \(-0.882557\pi\)
−0.154026 + 0.988067i \(0.549224\pi\)
\(264\) 0 0
\(265\) 8.33814 + 4.81402i 0.512208 + 0.295723i
\(266\) 0 0
\(267\) 4.32947 + 2.06954i 0.264959 + 0.126654i
\(268\) 0 0
\(269\) 26.7228 1.62932 0.814659 0.579940i \(-0.196924\pi\)
0.814659 + 0.579940i \(0.196924\pi\)
\(270\) 0 0
\(271\) 4.34764i 0.264100i 0.991243 + 0.132050i \(0.0421560\pi\)
−0.991243 + 0.132050i \(0.957844\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −7.29968 4.21447i −0.440187 0.254142i
\(276\) 0 0
\(277\) 2.19901 + 3.80880i 0.132126 + 0.228849i 0.924496 0.381192i \(-0.124486\pi\)
−0.792370 + 0.610041i \(0.791153\pi\)
\(278\) 0 0
\(279\) −2.91940 18.6357i −0.174780 1.11569i
\(280\) 0 0
\(281\) 4.62273 2.66893i 0.275769 0.159215i −0.355738 0.934586i \(-0.615770\pi\)
0.631506 + 0.775371i \(0.282437\pi\)
\(282\) 0 0
\(283\) 15.5431 + 8.97381i 0.923941 + 0.533437i 0.884890 0.465800i \(-0.154233\pi\)
0.0390505 + 0.999237i \(0.487567\pi\)
\(284\) 0 0
\(285\) 0.277732 + 0.404833i 0.0164514 + 0.0239802i
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −16.8380 −0.990471
\(290\) 0 0
\(291\) −1.01476 + 0.0790022i −0.0594861 + 0.00463119i
\(292\) 0 0
\(293\) 13.1126 22.7117i 0.766048 1.32683i −0.173642 0.984809i \(-0.555554\pi\)
0.939691 0.342026i \(-0.111113\pi\)
\(294\) 0 0
\(295\) −3.79566 6.57428i −0.220992 0.382769i
\(296\) 0 0
\(297\) 13.9315 + 14.7553i 0.808386 + 0.856188i
\(298\) 0 0
\(299\) −26.8968 46.5867i −1.55548 2.69418i
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 1.86152 + 23.9106i 0.106942 + 1.37363i
\(304\) 0 0
\(305\) 8.63628i 0.494512i
\(306\) 0 0
\(307\) 7.19520i 0.410652i −0.978694 0.205326i \(-0.934175\pi\)
0.978694 0.205326i \(-0.0658254\pi\)
\(308\) 0 0
\(309\) −11.8358 17.2524i −0.673317 0.981454i
\(310\) 0 0
\(311\) 1.08721 1.88311i 0.0616503 0.106781i −0.833553 0.552440i \(-0.813697\pi\)
0.895203 + 0.445658i \(0.147030\pi\)
\(312\) 0 0
\(313\) 10.2870 5.93922i 0.581457 0.335704i −0.180255 0.983620i \(-0.557692\pi\)
0.761712 + 0.647916i \(0.224359\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −7.09969 + 4.09901i −0.398758 + 0.230223i −0.685948 0.727651i \(-0.740612\pi\)
0.287190 + 0.957874i \(0.407279\pi\)
\(318\) 0 0
\(319\) 13.8754 24.0329i 0.776875 1.34559i
\(320\) 0 0
\(321\) 13.7402 28.7445i 0.766903 1.60436i
\(322\) 0 0
\(323\) 0.0676757i 0.00376558i
\(324\) 0 0
\(325\) 13.0688i 0.724924i
\(326\) 0 0
\(327\) −17.2220 8.23231i −0.952379 0.455248i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −8.58540 14.8704i −0.471897 0.817349i 0.527586 0.849501i \(-0.323097\pi\)
−0.999483 + 0.0321526i \(0.989764\pi\)
\(332\) 0 0
\(333\) −11.8037 14.6267i −0.646839 0.801540i
\(334\) 0 0
\(335\) −4.98170 8.62856i −0.272179 0.471428i
\(336\) 0 0
\(337\) 3.95399 6.84850i 0.215387 0.373062i −0.738005 0.674795i \(-0.764232\pi\)
0.953392 + 0.301733i \(0.0975653\pi\)
\(338\) 0 0
\(339\) 8.33644 + 12.1515i 0.452773 + 0.659981i
\(340\) 0 0
\(341\) 24.5557 1.32977
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 25.8610 2.01336i 1.39231 0.108396i
\(346\) 0 0
\(347\) 0.443850 + 0.256257i 0.0238271 + 0.0137566i 0.511866 0.859065i \(-0.328954\pi\)
−0.488039 + 0.872822i \(0.662288\pi\)
\(348\) 0 0
\(349\) −5.74612 + 3.31752i −0.307583 + 0.177583i −0.645844 0.763469i \(-0.723494\pi\)
0.338262 + 0.941052i \(0.390161\pi\)
\(350\) 0 0
\(351\) 9.01239 30.1451i 0.481046 1.60903i
\(352\) 0 0
\(353\) −9.03437 15.6480i −0.480851 0.832858i 0.518908 0.854830i \(-0.326339\pi\)
−0.999759 + 0.0219721i \(0.993006\pi\)
\(354\) 0 0
\(355\) 16.6877 + 9.63465i 0.885691 + 0.511354i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 1.76296i 0.0930453i −0.998917 0.0465227i \(-0.985186\pi\)
0.998917 0.0465227i \(-0.0148140\pi\)
\(360\) 0 0
\(361\) 18.9717 0.998512
\(362\) 0 0
\(363\) −6.07294 + 4.16628i −0.318747 + 0.218673i
\(364\) 0 0
\(365\) 10.2081 + 5.89367i 0.534318 + 0.308489i
\(366\) 0 0
\(367\) −28.9614 + 16.7209i −1.51177 + 0.872822i −0.511867 + 0.859065i \(0.671046\pi\)
−0.999905 + 0.0137576i \(0.995621\pi\)
\(368\) 0 0
\(369\) −9.76335 + 1.52949i −0.508260 + 0.0796221i
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −12.7844 + 22.1433i −0.661952 + 1.14653i 0.318150 + 0.948040i \(0.396938\pi\)
−0.980102 + 0.198494i \(0.936395\pi\)
\(374\) 0 0
\(375\) −18.8570 9.01386i −0.973773 0.465474i
\(376\) 0 0
\(377\) −43.0267 −2.21599
\(378\) 0 0
\(379\) 25.7920 1.32485 0.662423 0.749130i \(-0.269528\pi\)
0.662423 + 0.749130i \(0.269528\pi\)
\(380\) 0 0
\(381\) 16.6133 + 7.94133i 0.851124 + 0.406847i
\(382\) 0 0
\(383\) −16.4158 + 28.4330i −0.838808 + 1.45286i 0.0520838 + 0.998643i \(0.483414\pi\)
−0.890892 + 0.454215i \(0.849920\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −3.89760 + 10.1026i −0.198126 + 0.513546i
\(388\) 0 0
\(389\) 17.4542 10.0772i 0.884965 0.510935i 0.0126730 0.999920i \(-0.495966\pi\)
0.872292 + 0.488985i \(0.162633\pi\)
\(390\) 0 0
\(391\) 3.09663 + 1.78784i 0.156603 + 0.0904150i
\(392\) 0 0
\(393\) 9.04771 6.20709i 0.456396 0.313106i
\(394\) 0 0
\(395\) −2.03626 −0.102456
\(396\) 0 0
\(397\) 34.8864i 1.75090i −0.483311 0.875449i \(-0.660566\pi\)
0.483311 0.875449i \(-0.339434\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −8.36793 4.83122i −0.417874 0.241260i 0.276293 0.961073i \(-0.410894\pi\)
−0.694167 + 0.719814i \(0.744227\pi\)
\(402\) 0 0
\(403\) −19.0364 32.9719i −0.948268 1.64245i
\(404\) 0 0
\(405\) 11.2404 + 10.1898i 0.558541 + 0.506334i
\(406\) 0 0
\(407\) 21.1897 12.2339i 1.05034 0.606412i
\(408\) 0 0
\(409\) −32.1202 18.5446i −1.58824 0.916973i −0.993597 0.112986i \(-0.963958\pi\)
−0.594647 0.803987i \(-0.702708\pi\)
\(410\) 0 0
\(411\) 28.7446 2.23786i 1.41787 0.110386i
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −0.611419 −0.0300134
\(416\) 0 0
\(417\) 4.79731 + 6.99275i 0.234925 + 0.342436i
\(418\) 0 0
\(419\) −1.84193 + 3.19031i −0.0899841 + 0.155857i −0.907504 0.420043i \(-0.862015\pi\)
0.817520 + 0.575900i \(0.195348\pi\)
\(420\) 0 0
\(421\) −8.55139 14.8114i −0.416769 0.721866i 0.578843 0.815439i \(-0.303504\pi\)
−0.995612 + 0.0935732i \(0.970171\pi\)
\(422\) 0 0
\(423\) −9.46969 + 24.5456i −0.460432 + 1.19345i
\(424\) 0 0
\(425\) −0.434342 0.752303i −0.0210687 0.0364921i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 36.9541 + 17.6644i 1.78416 + 0.852847i
\(430\) 0 0
\(431\) 31.5512i 1.51977i 0.650058 + 0.759885i \(0.274745\pi\)
−0.650058 + 0.759885i \(0.725255\pi\)
\(432\) 0 0
\(433\) 10.0692i 0.483893i −0.970290 0.241947i \(-0.922214\pi\)
0.970290 0.241947i \(-0.0777859\pi\)
\(434\) 0 0
\(435\) 8.94777 18.7188i 0.429013 0.897496i
\(436\) 0 0
\(437\) −0.746890 + 1.29365i −0.0357286 + 0.0618837i
\(438\) 0 0
\(439\) −24.1966 + 13.9699i −1.15484 + 0.666748i −0.950062 0.312060i \(-0.898981\pi\)
−0.204779 + 0.978808i \(0.565648\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 30.1930 17.4319i 1.43451 0.828215i 0.437050 0.899437i \(-0.356023\pi\)
0.997460 + 0.0712223i \(0.0226900\pi\)
\(444\) 0 0
\(445\) −2.33518 + 4.04466i −0.110698 + 0.191735i
\(446\) 0 0
\(447\) −5.18038 7.55113i −0.245023 0.357156i
\(448\) 0 0
\(449\) 23.2411i 1.09682i 0.836211 + 0.548408i \(0.184766\pi\)
−0.836211 + 0.548408i \(0.815234\pi\)
\(450\) 0 0
\(451\) 12.8649i 0.605783i
\(452\) 0 0
\(453\) −1.96056 25.1828i −0.0921153 1.18319i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 3.10938 + 5.38560i 0.145451 + 0.251928i 0.929541 0.368719i \(-0.120203\pi\)
−0.784090 + 0.620647i \(0.786870\pi\)
\(458\) 0 0
\(459\) 0.483079 + 2.03483i 0.0225482 + 0.0949777i
\(460\) 0 0
\(461\) 2.17165 + 3.76140i 0.101144 + 0.175186i 0.912156 0.409843i \(-0.134416\pi\)
−0.811012 + 0.585029i \(0.801083\pi\)
\(462\) 0 0
\(463\) 3.57451 6.19124i 0.166122 0.287731i −0.770931 0.636918i \(-0.780209\pi\)
0.937053 + 0.349187i \(0.113542\pi\)
\(464\) 0 0
\(465\) 18.3032 1.42497i 0.848791 0.0660813i
\(466\) 0 0
\(467\) 1.88890 0.0874080 0.0437040 0.999045i \(-0.486084\pi\)
0.0437040 + 0.999045i \(0.486084\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) −17.4420 25.4242i −0.803686 1.17148i
\(472\) 0 0
\(473\) −12.2078 7.04818i −0.561316 0.324076i
\(474\) 0 0
\(475\) 0.314283 0.181451i 0.0144203 0.00832555i
\(476\) 0 0
\(477\) −13.3341 + 10.7606i −0.610527 + 0.492693i
\(478\) 0 0
\(479\) 5.22491 + 9.04981i 0.238732 + 0.413497i 0.960351 0.278794i \(-0.0899348\pi\)
−0.721618 + 0.692291i \(0.756601\pi\)
\(480\) 0 0
\(481\) −32.8539 18.9682i −1.49801 0.864875i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0.990611i 0.0449813i
\(486\) 0 0
\(487\) 23.6596 1.07212 0.536060 0.844180i \(-0.319912\pi\)
0.536060 + 0.844180i \(0.319912\pi\)
\(488\) 0 0
\(489\) 0.150715 + 0.0720434i 0.00681557 + 0.00325792i
\(490\) 0 0
\(491\) −11.6767 6.74152i −0.526960 0.304241i 0.212817 0.977092i \(-0.431736\pi\)
−0.739778 + 0.672851i \(0.765069\pi\)
\(492\) 0 0
\(493\) 2.47683 1.43000i 0.111551 0.0644039i
\(494\) 0 0
\(495\) −15.3698 + 12.4034i −0.690823 + 0.557491i
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 6.04035 10.4622i 0.270403 0.468352i −0.698562 0.715550i \(-0.746176\pi\)
0.968965 + 0.247197i \(0.0795096\pi\)
\(500\) 0 0
\(501\) 7.08050 4.85751i 0.316333 0.217017i
\(502\) 0 0
\(503\) −20.5283 −0.915310 −0.457655 0.889130i \(-0.651310\pi\)
−0.457655 + 0.889130i \(0.651310\pi\)
\(504\) 0 0
\(505\) −23.3417 −1.03869
\(506\) 0 0
\(507\) −3.18148 40.8650i −0.141294 1.81488i
\(508\) 0 0
\(509\) −4.09043 + 7.08483i −0.181305 + 0.314029i −0.942325 0.334699i \(-0.891365\pi\)
0.761020 + 0.648728i \(0.224699\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) −0.850073 + 0.201812i −0.0375316 + 0.00891020i
\(514\) 0 0
\(515\) 17.6345 10.1813i 0.777070 0.448642i
\(516\) 0 0
\(517\) −29.6603 17.1244i −1.30446 0.753131i
\(518\) 0 0
\(519\) 1.98979 + 25.5582i 0.0873421 + 1.12188i
\(520\) 0 0
\(521\) 27.7491 1.21571 0.607856 0.794048i \(-0.292030\pi\)
0.607856 + 0.794048i \(0.292030\pi\)
\(522\) 0 0
\(523\) 22.9604i 1.00399i 0.864872 + 0.501993i \(0.167400\pi\)
−0.864872 + 0.501993i \(0.832600\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 2.19166 + 1.26535i 0.0954700 + 0.0551196i
\(528\) 0 0
\(529\) 27.9623 + 48.4322i 1.21575 + 2.10575i
\(530\) 0 0
\(531\) 13.3470 2.09089i 0.579211 0.0907370i
\(532\) 0 0
\(533\) −17.2742 + 9.97325i −0.748228 + 0.431990i
\(534\) 0 0
\(535\) 26.8536 + 15.5039i 1.16098 + 0.670292i
\(536\) 0 0
\(537\) −0.511274 + 1.06959i −0.0220631 + 0.0461561i
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 5.20810 0.223914 0.111957 0.993713i \(-0.464288\pi\)
0.111957 + 0.993713i \(0.464288\pi\)
\(542\) 0 0
\(543\) −5.86193 + 12.2632i −0.251559 + 0.526263i
\(544\) 0 0
\(545\) 9.28902 16.0890i 0.397898 0.689179i
\(546\) 0 0
\(547\) 10.6224 + 18.3985i 0.454181 + 0.786664i 0.998641 0.0521229i \(-0.0165988\pi\)
−0.544460 + 0.838787i \(0.683265\pi\)
\(548\) 0 0
\(549\) −14.3393 5.53210i −0.611987 0.236104i
\(550\) 0 0
\(551\) 0.597397 + 1.03472i 0.0254500 + 0.0440807i
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 15.0844 10.3485i 0.640296 0.439269i
\(556\) 0 0
\(557\) 12.8109i 0.542813i 0.962465 + 0.271407i \(0.0874888\pi\)
−0.962465 + 0.271407i \(0.912511\pi\)
\(558\) 0 0
\(559\) 21.8559i 0.924406i
\(560\) 0 0
\(561\) −2.71434 + 0.211321i −0.114600 + 0.00892197i
\(562\) 0 0
\(563\) −18.7396 + 32.4580i −0.789781 + 1.36794i 0.136319 + 0.990665i \(0.456473\pi\)
−0.926101 + 0.377277i \(0.876861\pi\)
\(564\) 0 0
\(565\) −12.4207 + 7.17109i −0.522542 + 0.301690i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −5.94906 + 3.43469i −0.249397 + 0.143990i −0.619488 0.785006i \(-0.712660\pi\)
0.370091 + 0.928996i \(0.379327\pi\)
\(570\) 0 0
\(571\) −0.0847909 + 0.146862i −0.00354839 + 0.00614599i −0.867794 0.496924i \(-0.834463\pi\)
0.864246 + 0.503070i \(0.167796\pi\)
\(572\) 0 0
\(573\) −33.8364 + 2.63428i −1.41354 + 0.110049i
\(574\) 0 0
\(575\) 19.1741i 0.799617i
\(576\) 0 0
\(577\) 6.24916i 0.260156i 0.991504 + 0.130078i \(0.0415228\pi\)
−0.991504 + 0.130078i \(0.958477\pi\)
\(578\) 0 0
\(579\) 26.2261 17.9921i 1.08992 0.747728i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −11.1527 19.3171i −0.461900 0.800033i
\(584\) 0 0
\(585\) 28.5697 + 11.0222i 1.18121 + 0.455712i
\(586\) 0 0
\(587\) 10.7881 + 18.6855i 0.445273 + 0.771235i 0.998071 0.0620801i \(-0.0197734\pi\)
−0.552799 + 0.833315i \(0.686440\pi\)
\(588\) 0 0
\(589\) −0.528615 + 0.915588i −0.0217812 + 0.0377261i
\(590\) 0 0
\(591\) 4.42449 9.25605i 0.181999 0.380743i
\(592\) 0 0
\(593\) −8.26071 −0.339227 −0.169613 0.985511i \(-0.554252\pi\)
−0.169613 + 0.985511i \(0.554252\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 11.7537 24.5887i 0.481046 1.00635i
\(598\) 0 0
\(599\) −30.7618 17.7603i −1.25689 0.725667i −0.284422 0.958699i \(-0.591802\pi\)
−0.972469 + 0.233033i \(0.925135\pi\)
\(600\) 0 0
\(601\) 35.8981 20.7258i 1.46432 0.845423i 0.465109 0.885254i \(-0.346015\pi\)
0.999206 + 0.0398308i \(0.0126819\pi\)
\(602\) 0 0
\(603\) 17.5176 2.74424i 0.713371 0.111754i
\(604\) 0 0
\(605\) −3.58388 6.20746i −0.145705 0.252369i
\(606\) 0 0
\(607\) −2.09569 1.20995i −0.0850616 0.0491103i 0.456866 0.889536i \(-0.348972\pi\)
−0.541927 + 0.840425i \(0.682305\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 53.1015i 2.14826i
\(612\) 0 0
\(613\) −42.6456 −1.72244 −0.861219 0.508234i \(-0.830298\pi\)
−0.861219 + 0.508234i \(0.830298\pi\)
\(614\) 0 0
\(615\) −0.746549 9.58916i −0.0301038 0.386672i
\(616\) 0 0
\(617\) 13.2535 + 7.65193i 0.533567 + 0.308055i 0.742468 0.669882i \(-0.233655\pi\)
−0.208901 + 0.977937i \(0.566989\pi\)
\(618\) 0 0
\(619\) 23.9177 13.8089i 0.961334 0.555026i 0.0647505 0.997901i \(-0.479375\pi\)
0.896583 + 0.442875i \(0.146042\pi\)
\(620\) 0 0
\(621\) −13.2227 + 44.2281i −0.530610 + 1.77481i
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 4.77517 8.27084i 0.191007 0.330834i
\(626\) 0 0
\(627\) −0.0882815 1.13395i −0.00352562 0.0452854i
\(628\) 0 0
\(629\) 2.52164 0.100545
\(630\) 0 0
\(631\) 8.28775 0.329930 0.164965 0.986299i \(-0.447249\pi\)
0.164965 + 0.986299i \(0.447249\pi\)
\(632\) 0 0
\(633\) −14.4716 + 9.92811i −0.575195 + 0.394607i
\(634\) 0 0
\(635\) −8.96069 + 15.5204i −0.355594 + 0.615907i
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) −26.6865 + 21.5359i −1.05570 + 0.851947i
\(640\) 0 0
\(641\) 8.58307 4.95544i 0.339011 0.195728i −0.320824 0.947139i \(-0.603960\pi\)
0.659835 + 0.751411i \(0.270626\pi\)
\(642\) 0 0
\(643\) 6.83668 + 3.94716i 0.269612 + 0.155661i 0.628711 0.777639i \(-0.283583\pi\)
−0.359099 + 0.933299i \(0.616916\pi\)
\(644\) 0 0
\(645\) −9.50841 4.54512i −0.374393 0.178964i
\(646\) 0 0
\(647\) −4.31931 −0.169810 −0.0849049 0.996389i \(-0.527059\pi\)
−0.0849049 + 0.996389i \(0.527059\pi\)
\(648\) 0 0
\(649\) 17.5870i 0.690349i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −37.5853 21.6999i −1.47082 0.849181i −0.471361 0.881940i \(-0.656237\pi\)
−0.999463 + 0.0327591i \(0.989571\pi\)
\(654\) 0 0
\(655\) 5.33940 + 9.24812i 0.208628 + 0.361354i
\(656\) 0 0
\(657\) −16.3246 + 13.1739i −0.636882 + 0.513961i
\(658\) 0 0
\(659\) −9.34894 + 5.39761i −0.364183 + 0.210261i −0.670914 0.741535i \(-0.734098\pi\)
0.306731 + 0.951796i \(0.400765\pi\)
\(660\) 0 0
\(661\) −3.39495 1.96008i −0.132048 0.0762381i 0.432521 0.901624i \(-0.357624\pi\)
−0.564569 + 0.825386i \(0.690958\pi\)
\(662\) 0 0
\(663\) 2.38799 + 3.48083i 0.0927419 + 0.135184i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 63.1276 2.44431
\(668\) 0 0
\(669\) 26.7080 2.07931i 1.03259 0.0803906i
\(670\) 0 0
\(671\) 10.0039 17.3273i 0.386197 0.668913i
\(672\) 0 0
\(673\) −12.3404 21.3742i −0.475687 0.823915i 0.523925 0.851765i \(-0.324467\pi\)
−0.999612 + 0.0278497i \(0.991134\pi\)
\(674\) 0 0
\(675\) 8.15443 7.69916i 0.313864 0.296341i
\(676\) 0 0
\(677\) 7.36327 + 12.7536i 0.282994 + 0.490159i 0.972121 0.234481i \(-0.0753392\pi\)
−0.689127 + 0.724641i \(0.742006\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 3.77397 + 48.4753i 0.144619 + 1.85758i
\(682\) 0 0
\(683\) 1.84900i 0.0707499i 0.999374 + 0.0353750i \(0.0112625\pi\)
−0.999374 + 0.0353750i \(0.988737\pi\)
\(684\) 0 0
\(685\) 28.0606i 1.07214i
\(686\) 0 0
\(687\) 16.6831 + 24.3180i 0.636502 + 0.927790i
\(688\) 0 0
\(689\) −17.2919 + 29.9505i −0.658769 + 1.14102i
\(690\) 0 0
\(691\) 33.7613 19.4921i 1.28434 0.741514i 0.306701 0.951806i \(-0.400775\pi\)
0.977639 + 0.210292i \(0.0674415\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −7.14764 + 4.12669i −0.271126 + 0.156534i
\(696\) 0 0
\(697\) 0.662926 1.14822i 0.0251101 0.0434920i
\(698\) 0 0
\(699\) −13.8020 + 28.8738i −0.522040 + 1.09211i
\(700\) 0 0
\(701\) 25.4389i 0.960813i −0.877046 0.480406i \(-0.840489\pi\)
0.877046 0.480406i \(-0.159511\pi\)
\(702\) 0 0
\(703\) 1.05344i 0.0397314i
\(704\) 0 0
\(705\) −23.1018 11.0429i −0.870065 0.415900i
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 7.14517 + 12.3758i 0.268342 + 0.464783i 0.968434 0.249270i \(-0.0801908\pi\)
−0.700092 + 0.714053i \(0.746857\pi\)
\(710\) 0 0
\(711\) 1.30436 3.38092i 0.0489173 0.126794i
\(712\) 0 0
\(713\) 27.9296 + 48.3756i 1.04597 + 1.81168i
\(714\) 0 0
\(715\) −19.9319 + 34.5230i −0.745410 + 1.29109i
\(716\) 0 0
\(717\) 6.86384 + 10.0050i 0.256335 + 0.373644i
\(718\) 0 0
\(719\) 33.4688 1.24818 0.624088 0.781354i \(-0.285471\pi\)
0.624088 + 0.781354i \(0.285471\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) −10.7367 + 0.835888i −0.399302 + 0.0310870i
\(724\) 0 0
\(725\) −13.2817 7.66819i −0.493269 0.284789i
\(726\) 0 0
\(727\) 12.1354 7.00636i 0.450076 0.259851i −0.257786 0.966202i \(-0.582993\pi\)
0.707862 + 0.706350i \(0.249660\pi\)
\(728\) 0 0
\(729\) −24.1189 + 12.1359i −0.893292 + 0.449477i
\(730\) 0 0
\(731\) −0.726384 1.25813i −0.0268663 0.0465338i
\(732\) 0 0
\(733\) 23.6491 + 13.6538i 0.873501 + 0.504316i 0.868510 0.495672i \(-0.165078\pi\)
0.00499085 + 0.999988i \(0.498411\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 23.0824i 0.850251i
\(738\) 0 0
\(739\) −52.6314 −1.93608 −0.968039 0.250801i \(-0.919306\pi\)
−0.968039 + 0.250801i \(0.919306\pi\)
\(740\) 0 0
\(741\) −1.45416 + 0.997609i −0.0534197 + 0.0366481i
\(742\) 0 0
\(743\) −30.9523 17.8703i −1.13553 0.655599i −0.190211 0.981743i \(-0.560917\pi\)
−0.945320 + 0.326144i \(0.894250\pi\)
\(744\) 0 0
\(745\) 7.71839 4.45621i 0.282780 0.163263i
\(746\) 0 0
\(747\) 0.391654 1.01517i 0.0143299 0.0371432i
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 16.5641 28.6899i 0.604433 1.04691i −0.387708 0.921782i \(-0.626733\pi\)
0.992141 0.125126i \(-0.0399336\pi\)
\(752\) 0 0
\(753\) 15.3432 + 7.33423i 0.559138 + 0.267274i
\(754\) 0 0
\(755\) 24.5836 0.894687
\(756\) 0 0
\(757\) −13.6903 −0.497584 −0.248792 0.968557i \(-0.580034\pi\)
−0.248792 + 0.968557i \(0.580034\pi\)
\(758\) 0 0
\(759\) −54.2180 25.9168i −1.96799 0.940721i
\(760\) 0 0
\(761\) 6.51737 11.2884i 0.236255 0.409205i −0.723382 0.690448i \(-0.757413\pi\)
0.959637 + 0.281243i \(0.0907467\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) −2.01094 + 0.315026i −0.0727057 + 0.0113898i
\(766\) 0 0
\(767\) 23.6147 13.6340i 0.852678 0.492294i
\(768\) 0 0
\(769\) −18.4866 10.6732i −0.666642 0.384886i 0.128161 0.991753i \(-0.459093\pi\)
−0.794803 + 0.606867i \(0.792426\pi\)
\(770\) 0 0
\(771\) −1.90788 + 1.30888i −0.0687104 + 0.0471381i
\(772\) 0 0
\(773\) 11.4788 0.412864 0.206432 0.978461i \(-0.433815\pi\)
0.206432 + 0.978461i \(0.433815\pi\)
\(774\) 0 0
\(775\) 13.5706i 0.487470i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0.479682 + 0.276944i 0.0171864 + 0.00992256i
\(780\) 0 0
\(781\) −22.3208 38.6607i −0.798701 1.38339i
\(782\) 0 0
\(783\) 25.3482 + 26.8471i 0.905870 + 0.959436i
\(784\) 0 0
\(785\) 25.9873 15.0038i 0.927528 0.535509i
\(786\) 0 0
\(787\) −35.6808 20.6003i −1.27188 0.734322i −0.296541 0.955020i \(-0.595833\pi\)
−0.975342 + 0.220698i \(0.929166\pi\)
\(788\) 0 0
\(789\) 35.1413 2.73587i 1.25106 0.0973994i
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −31.0214 −1.10160
\(794\) 0 0
\(795\) −9.43395 13.7513i −0.334588 0.487709i
\(796\) 0 0
\(797\) 25.0066 43.3127i 0.885779 1.53421i 0.0409600 0.999161i \(-0.486958\pi\)
0.844819 0.535053i \(-0.179708\pi\)
\(798\) 0 0
\(799\) −1.76484 3.05679i −0.0624355 0.108141i
\(800\) 0 0
\(801\) −5.21973 6.46810i −0.184430 0.228539i
\(802\) 0 0
\(803\) −13.6540 23.6494i −0.481838 0.834568i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −41.7596 19.9615i −1.47001 0.702679i
\(808\) 0 0
\(809\) 50.6908i 1.78219i −0.453812 0.891097i \(-0.649936\pi\)
0.453812 0.891097i \(-0.350064\pi\)
\(810\) 0 0
\(811\) 8.96566i 0.314827i −0.987533 0.157413i \(-0.949684\pi\)
0.987533 0.157413i \(-0.0503155\pi\)
\(812\) 0 0
\(813\) 3.24762 6.79403i 0.113899 0.238277i
\(814\) 0 0
\(815\) −0.0812910 + 0.140800i −0.00284750 + 0.00493201i
\(816\) 0 0
\(817\) 0.525599 0.303455i 0.0183884 0.0106165i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 28.2190 16.2922i 0.984849 0.568603i 0.0811184 0.996704i \(-0.474151\pi\)
0.903731 + 0.428102i \(0.140817\pi\)
\(822\) 0 0
\(823\) 10.0877 17.4724i 0.351636 0.609051i −0.634901 0.772594i \(-0.718959\pi\)
0.986536 + 0.163543i \(0.0522923\pi\)
\(824\) 0 0
\(825\) 8.25902 + 12.0387i 0.287542 + 0.419133i
\(826\) 0 0
\(827\) 0.253288i 0.00880770i −0.999990 0.00440385i \(-0.998598\pi\)
0.999990 0.00440385i \(-0.00140179\pi\)
\(828\) 0 0
\(829\) 7.05416i 0.245001i 0.992468 + 0.122501i \(0.0390913\pi\)
−0.992468 + 0.122501i \(0.960909\pi\)
\(830\) 0 0
\(831\) −0.591268 7.59463i −0.0205108 0.263455i
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 4.17848 + 7.23733i 0.144602 + 0.250458i
\(836\) 0 0
\(837\) −9.35845 + 31.3026i −0.323475 + 1.08198i
\(838\) 0 0
\(839\) −17.0936 29.6069i −0.590136 1.02215i −0.994214 0.107420i \(-0.965741\pi\)
0.404078 0.914725i \(-0.367592\pi\)
\(840\) 0 0
\(841\) 10.7462 18.6130i 0.370558 0.641826i
\(842\) 0 0
\(843\) −9.21756 + 0.717619i −0.317470 + 0.0247161i
\(844\) 0 0
\(845\) 39.8926 1.37235
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −17.5858 25.6338i −0.603543 0.879749i
\(850\) 0 0
\(851\) 48.2024 + 27.8296i 1.65236 + 0.953988i
\(852\) 0 0
\(853\) −21.7586 + 12.5623i −0.745000 + 0.430126i −0.823884 0.566758i \(-0.808198\pi\)
0.0788844 + 0.996884i \(0.474864\pi\)
\(854\) 0 0
\(855\) −0.131606 0.840092i −0.00450082 0.0287305i
\(856\) 0 0
\(857\) −21.0954 36.5383i −0.720604 1.24812i −0.960758 0.277388i \(-0.910531\pi\)
0.240154 0.970735i \(-0.422802\pi\)
\(858\) 0 0
\(859\) 4.08139 + 2.35639i 0.139255 + 0.0803990i 0.568009 0.823022i \(-0.307714\pi\)
−0.428754 + 0.903421i \(0.641047\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 35.6120i 1.21225i 0.795371 + 0.606123i \(0.207276\pi\)
−0.795371 + 0.606123i \(0.792724\pi\)
\(864\) 0 0
\(865\) −24.9500 −0.848326
\(866\) 0 0
\(867\) 26.3127 + 12.5777i 0.893625 + 0.427162i
\(868\) 0 0
\(869\) 4.08543 + 2.35873i 0.138589 + 0.0800143i
\(870\) 0 0
\(871\) 30.9937 17.8942i 1.05018 0.606322i
\(872\) 0 0
\(873\) 1.64477 + 0.634551i 0.0556669 + 0.0214763i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −20.4532 + 35.4260i −0.690655 + 1.19625i 0.280969 + 0.959717i \(0.409344\pi\)
−0.971624 + 0.236532i \(0.923989\pi\)
\(878\) 0 0
\(879\) −37.4564 + 25.6966i −1.26337 + 0.866724i
\(880\) 0 0
\(881\) 37.4443 1.26153 0.630765 0.775974i \(-0.282741\pi\)
0.630765 + 0.775974i \(0.282741\pi\)
\(882\) 0 0
\(883\) −49.8357 −1.67711 −0.838553 0.544821i \(-0.816598\pi\)
−0.838553 + 0.544821i \(0.816598\pi\)
\(884\) 0 0
\(885\) 1.02057 + 13.1089i 0.0343061 + 0.440650i
\(886\) 0 0
\(887\) −14.4482 + 25.0251i −0.485124 + 0.840260i −0.999854 0.0170929i \(-0.994559\pi\)
0.514730 + 0.857352i \(0.327892\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −10.7487 33.4646i −0.360094 1.12111i
\(892\) 0 0
\(893\) 1.27701 0.737280i 0.0427334 0.0246721i
\(894\) 0 0
\(895\) −0.999223 0.576902i −0.0334003 0.0192837i
\(896\) 0 0
\(897\) 7.23198 + 92.8922i 0.241469 + 3.10158i
\(898\) 0 0
\(899\) 44.6789 1.49012
\(900\) 0 0
\(901\) 2.29880i 0.0765841i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −11.4564 6.61437i −0.380825 0.219869i
\(906\) 0 0
\(907\) 7.43498 + 12.8778i 0.246874 + 0.427599i 0.962657 0.270724i \(-0.0872632\pi\)
−0.715783 + 0.698323i \(0.753930\pi\)
\(908\) 0 0
\(909\) 14.9519 38.7555i 0.495923 1.28544i
\(910\) 0 0
\(911\) 7.81616 4.51266i 0.258961 0.149511i −0.364899 0.931047i \(-0.618897\pi\)
0.623861 + 0.781536i \(0.285563\pi\)
\(912\) 0 0
\(913\) 1.22671 + 0.708243i 0.0405982 + 0.0234394i
\(914\) 0 0
\(915\) 6.45117 13.4959i 0.213269 0.446160i
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −26.4166 −0.871403 −0.435702 0.900091i \(-0.643500\pi\)
−0.435702 + 0.900091i \(0.643500\pi\)
\(920\) 0 0
\(921\) −5.37471 + 11.2439i −0.177103 + 0.370499i
\(922\) 0 0
\(923\) −34.6075 + 59.9420i −1.13912 + 1.97302i
\(924\) 0 0
\(925\) −6.76100 11.7104i −0.222300 0.385036i
\(926\) 0 0
\(927\) 5.60851 + 35.8014i 0.184208 + 1.17587i
\(928\) 0 0
\(929\) 11.1259 + 19.2706i 0.365029 + 0.632249i 0.988781 0.149373i \(-0.0477257\pi\)
−0.623752 + 0.781623i \(0.714392\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) −3.10564 + 2.13059i −0.101674 + 0.0697525i
\(934\) 0 0
\(935\) 2.64976i 0.0866563i
\(936\) 0 0
\(937\) 14.6822i 0.479647i 0.970817 + 0.239823i \(0.0770896\pi\)
−0.970817 + 0.239823i \(0.922910\pi\)
\(938\) 0 0
\(939\) −20.5120 + 1.59693i −0.669383 + 0.0521137i
\(940\) 0 0
\(941\) 23.0396 39.9058i 0.751070 1.30089i −0.196235 0.980557i \(-0.562871\pi\)
0.947305 0.320334i \(-0.103795\pi\)
\(942\) 0 0
\(943\) 25.3442 14.6325i 0.825322 0.476500i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −6.96116 + 4.01903i −0.226207 + 0.130601i −0.608821 0.793308i \(-0.708357\pi\)
0.382614 + 0.923908i \(0.375024\pi\)
\(948\) 0 0
\(949\) −21.1700 + 36.6675i −0.687206 + 1.19028i
\(950\) 0 0
\(951\) 14.1565 1.10213i 0.459057 0.0357392i
\(952\) 0 0
\(953\) 54.9348i 1.77951i 0.456437 + 0.889756i \(0.349125\pi\)
−0.456437 + 0.889756i \(0.650875\pi\)
\(954\) 0 0
\(955\) 33.0313i 1.06887i
\(956\) 0 0
\(957\) −39.6353 + 27.1914i −1.28123 + 0.878974i
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 4.26733 + 7.39124i 0.137656 + 0.238427i
\(962\) 0 0
\(963\) −42.9435 + 34.6552i −1.38383 + 1.11675i
\(964\) 0 0
\(965\) 15.4770 + 26.8070i 0.498223 + 0.862948i
\(966\) 0 0
\(967\) −26.5917 + 46.0582i −0.855132 + 1.48113i 0.0213900 + 0.999771i \(0.493191\pi\)
−0.876522 + 0.481361i \(0.840143\pi\)
\(968\) 0 0
\(969\) 0.0505527 0.105756i 0.00162399 0.00339739i
\(970\) 0 0
\(971\) −15.2281 −0.488692 −0.244346 0.969688i \(-0.578573\pi\)
−0.244346 + 0.969688i \(0.578573\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 9.76217 20.4225i 0.312640 0.654043i
\(976\) 0 0
\(977\) −1.49418 0.862667i −0.0478031 0.0275992i 0.475908 0.879495i \(-0.342120\pi\)
−0.523711 + 0.851896i \(0.675453\pi\)
\(978\) 0 0
\(979\) 9.37033 5.40997i 0.299477 0.172903i
\(980\) 0 0
\(981\) 20.7633 + 25.7292i 0.662922 + 0.821469i
\(982\) 0 0
\(983\) 30.1191 + 52.1679i 0.960651 + 1.66390i 0.720871 + 0.693070i \(0.243742\pi\)
0.239780 + 0.970827i \(0.422925\pi\)
\(984\) 0 0
\(985\) 8.64713 + 4.99242i 0.275521 + 0.159072i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 32.0664i 1.01965i
\(990\) 0 0
\(991\) 5.74624 0.182535 0.0912676 0.995826i \(-0.470908\pi\)
0.0912676 + 0.995826i \(0.470908\pi\)
\(992\) 0 0
\(993\) 2.30843 + 29.6510i 0.0732558 + 0.940946i
\(994\) 0 0
\(995\) 22.9711 + 13.2624i 0.728234 + 0.420446i
\(996\) 0 0
\(997\) 0.0224508 0.0129620i 0.000711024 0.000410510i −0.499644 0.866231i \(-0.666536\pi\)
0.500355 + 0.865820i \(0.333203\pi\)
\(998\) 0 0
\(999\) 7.51964 + 31.6743i 0.237911 + 1.00213i
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 1764.2.x.b.293.1 16
3.2 odd 2 5292.2.x.b.881.3 16
7.2 even 3 252.2.bm.a.185.7 yes 16
7.3 odd 6 252.2.w.a.5.4 16
7.4 even 3 1764.2.w.b.509.5 16
7.5 odd 6 1764.2.bm.a.1697.2 16
7.6 odd 2 1764.2.x.a.293.8 16
9.2 odd 6 1764.2.x.a.1469.8 16
9.7 even 3 5292.2.x.a.4409.6 16
21.2 odd 6 756.2.bm.a.17.6 16
21.5 even 6 5292.2.bm.a.2285.3 16
21.11 odd 6 5292.2.w.b.1097.3 16
21.17 even 6 756.2.w.a.341.6 16
21.20 even 2 5292.2.x.a.881.6 16
28.3 even 6 1008.2.ca.d.257.5 16
28.23 odd 6 1008.2.df.d.689.2 16
63.2 odd 6 252.2.w.a.101.4 yes 16
63.11 odd 6 1764.2.bm.a.1685.2 16
63.16 even 3 756.2.w.a.521.6 16
63.20 even 6 inner 1764.2.x.b.1469.1 16
63.23 odd 6 2268.2.t.b.1781.3 16
63.25 even 3 5292.2.bm.a.4625.3 16
63.31 odd 6 2268.2.t.b.2105.3 16
63.34 odd 6 5292.2.x.b.4409.3 16
63.38 even 6 252.2.bm.a.173.7 yes 16
63.47 even 6 1764.2.w.b.1109.5 16
63.52 odd 6 756.2.bm.a.89.6 16
63.58 even 3 2268.2.t.a.1781.6 16
63.59 even 6 2268.2.t.a.2105.6 16
63.61 odd 6 5292.2.w.b.521.3 16
84.23 even 6 3024.2.df.d.17.6 16
84.59 odd 6 3024.2.ca.d.2609.6 16
252.79 odd 6 3024.2.ca.d.2033.6 16
252.115 even 6 3024.2.df.d.1601.6 16
252.191 even 6 1008.2.ca.d.353.5 16
252.227 odd 6 1008.2.df.d.929.2 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
252.2.w.a.5.4 16 7.3 odd 6
252.2.w.a.101.4 yes 16 63.2 odd 6
252.2.bm.a.173.7 yes 16 63.38 even 6
252.2.bm.a.185.7 yes 16 7.2 even 3
756.2.w.a.341.6 16 21.17 even 6
756.2.w.a.521.6 16 63.16 even 3
756.2.bm.a.17.6 16 21.2 odd 6
756.2.bm.a.89.6 16 63.52 odd 6
1008.2.ca.d.257.5 16 28.3 even 6
1008.2.ca.d.353.5 16 252.191 even 6
1008.2.df.d.689.2 16 28.23 odd 6
1008.2.df.d.929.2 16 252.227 odd 6
1764.2.w.b.509.5 16 7.4 even 3
1764.2.w.b.1109.5 16 63.47 even 6
1764.2.x.a.293.8 16 7.6 odd 2
1764.2.x.a.1469.8 16 9.2 odd 6
1764.2.x.b.293.1 16 1.1 even 1 trivial
1764.2.x.b.1469.1 16 63.20 even 6 inner
1764.2.bm.a.1685.2 16 63.11 odd 6
1764.2.bm.a.1697.2 16 7.5 odd 6
2268.2.t.a.1781.6 16 63.58 even 3
2268.2.t.a.2105.6 16 63.59 even 6
2268.2.t.b.1781.3 16 63.23 odd 6
2268.2.t.b.2105.3 16 63.31 odd 6
3024.2.ca.d.2033.6 16 252.79 odd 6
3024.2.ca.d.2609.6 16 84.59 odd 6
3024.2.df.d.17.6 16 84.23 even 6
3024.2.df.d.1601.6 16 252.115 even 6
5292.2.w.b.521.3 16 63.61 odd 6
5292.2.w.b.1097.3 16 21.11 odd 6
5292.2.x.a.881.6 16 21.20 even 2
5292.2.x.a.4409.6 16 9.7 even 3
5292.2.x.b.881.3 16 3.2 odd 2
5292.2.x.b.4409.3 16 63.34 odd 6
5292.2.bm.a.2285.3 16 21.5 even 6
5292.2.bm.a.4625.3 16 63.25 even 3