Properties

Label 1512.2.j.d.323.23
Level $1512$
Weight $2$
Character 1512.323
Analytic conductor $12.073$
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] = [1512,2,Mod(323,1512)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1512, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1512.323");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1512 = 2^{3} \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1512.j (of order \(2\), degree \(1\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(12.0733807856\)
Analytic rank: \(0\)
Dimension: \(48\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 323.23
Character \(\chi\) \(=\) 1512.323
Dual form 1512.2.j.d.323.24

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.0811990 - 1.41188i) q^{2} +(-1.98681 + 0.229287i) q^{4} -1.27819 q^{5} -1.00000i q^{7} +(0.485052 + 2.78653i) q^{8} +O(q^{10})\) \(q+(-0.0811990 - 1.41188i) q^{2} +(-1.98681 + 0.229287i) q^{4} -1.27819 q^{5} -1.00000i q^{7} +(0.485052 + 2.78653i) q^{8} +(0.103788 + 1.80466i) q^{10} +2.84853i q^{11} -0.223683i q^{13} +(-1.41188 + 0.0811990i) q^{14} +(3.89486 - 0.911099i) q^{16} +0.397844i q^{17} -4.64862 q^{19} +(2.53953 - 0.293073i) q^{20} +(4.02179 - 0.231298i) q^{22} +7.36177 q^{23} -3.36622 q^{25} +(-0.315814 + 0.0181628i) q^{26} +(0.229287 + 1.98681i) q^{28} +10.6317 q^{29} -7.67558i q^{31} +(-1.60262 - 5.42509i) q^{32} +(0.561708 - 0.0323045i) q^{34} +1.27819i q^{35} +4.74489i q^{37} +(0.377464 + 6.56330i) q^{38} +(-0.619991 - 3.56172i) q^{40} -1.97843i q^{41} +7.62888 q^{43} +(-0.653130 - 5.65950i) q^{44} +(-0.597768 - 10.3939i) q^{46} +11.0724 q^{47} -1.00000 q^{49} +(0.273334 + 4.75270i) q^{50} +(0.0512875 + 0.444416i) q^{52} -0.295366 q^{53} -3.64098i q^{55} +(2.78653 - 0.485052i) q^{56} +(-0.863279 - 15.0106i) q^{58} +7.25779i q^{59} -9.45592i q^{61} +(-10.8370 + 0.623249i) q^{62} +(-7.52945 + 2.70322i) q^{64} +0.285910i q^{65} -3.52321 q^{67} +(-0.0912202 - 0.790441i) q^{68} +(1.80466 - 0.103788i) q^{70} -8.37710 q^{71} +10.6704 q^{73} +(6.69921 - 0.385280i) q^{74} +(9.23595 - 1.06587i) q^{76} +2.84853 q^{77} -5.22135i q^{79} +(-4.97838 + 1.16456i) q^{80} +(-2.79331 + 0.160647i) q^{82} -9.11868i q^{83} -0.508521i q^{85} +(-0.619457 - 10.7711i) q^{86} +(-7.93751 + 1.38169i) q^{88} -8.94887i q^{89} -0.223683 q^{91} +(-14.6265 + 1.68795i) q^{92} +(-0.899068 - 15.6329i) q^{94} +5.94184 q^{95} +0.228078 q^{97} +(0.0811990 + 1.41188i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 48 q - 4 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 48 q - 4 q^{4} - 12 q^{10} - 12 q^{16} + 16 q^{19} - 24 q^{22} + 48 q^{25} + 8 q^{28} + 12 q^{34} + 32 q^{40} + 64 q^{43} + 60 q^{46} - 48 q^{49} + 16 q^{52} + 36 q^{58} - 4 q^{64} + 32 q^{67} + 12 q^{70} - 32 q^{76} + 36 q^{82} + 24 q^{88} - 60 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1512\mathbb{Z}\right)^\times\).

\(n\) \(757\) \(785\) \(1081\) \(1135\)
\(\chi(n)\) \(-1\) \(-1\) \(1\) \(-1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.0811990 1.41188i −0.0574164 0.998350i
\(3\) 0 0
\(4\) −1.98681 + 0.229287i −0.993407 + 0.114643i
\(5\) −1.27819 −0.571625 −0.285813 0.958285i \(-0.592264\pi\)
−0.285813 + 0.958285i \(0.592264\pi\)
\(6\) 0 0
\(7\) 1.00000i 0.377964i
\(8\) 0.485052 + 2.78653i 0.171492 + 0.985186i
\(9\) 0 0
\(10\) 0.103788 + 1.80466i 0.0328207 + 0.570682i
\(11\) 2.84853i 0.858865i 0.903099 + 0.429433i \(0.141286\pi\)
−0.903099 + 0.429433i \(0.858714\pi\)
\(12\) 0 0
\(13\) 0.223683i 0.0620385i −0.999519 0.0310192i \(-0.990125\pi\)
0.999519 0.0310192i \(-0.00987531\pi\)
\(14\) −1.41188 + 0.0811990i −0.377341 + 0.0217013i
\(15\) 0 0
\(16\) 3.89486 0.911099i 0.973714 0.227775i
\(17\) 0.397844i 0.0964913i 0.998836 + 0.0482456i \(0.0153630\pi\)
−0.998836 + 0.0482456i \(0.984637\pi\)
\(18\) 0 0
\(19\) −4.64862 −1.06647 −0.533234 0.845968i \(-0.679023\pi\)
−0.533234 + 0.845968i \(0.679023\pi\)
\(20\) 2.53953 0.293073i 0.567857 0.0655330i
\(21\) 0 0
\(22\) 4.02179 0.231298i 0.857448 0.0493129i
\(23\) 7.36177 1.53503 0.767517 0.641028i \(-0.221492\pi\)
0.767517 + 0.641028i \(0.221492\pi\)
\(24\) 0 0
\(25\) −3.36622 −0.673244
\(26\) −0.315814 + 0.0181628i −0.0619361 + 0.00356202i
\(27\) 0 0
\(28\) 0.229287 + 1.98681i 0.0433311 + 0.375472i
\(29\) 10.6317 1.97425 0.987124 0.159957i \(-0.0511356\pi\)
0.987124 + 0.159957i \(0.0511356\pi\)
\(30\) 0 0
\(31\) 7.67558i 1.37857i −0.724488 0.689287i \(-0.757924\pi\)
0.724488 0.689287i \(-0.242076\pi\)
\(32\) −1.60262 5.42509i −0.283306 0.959030i
\(33\) 0 0
\(34\) 0.561708 0.0323045i 0.0963321 0.00554018i
\(35\) 1.27819i 0.216054i
\(36\) 0 0
\(37\) 4.74489i 0.780055i 0.920803 + 0.390027i \(0.127534\pi\)
−0.920803 + 0.390027i \(0.872466\pi\)
\(38\) 0.377464 + 6.56330i 0.0612327 + 1.06471i
\(39\) 0 0
\(40\) −0.619991 3.56172i −0.0980292 0.563157i
\(41\) 1.97843i 0.308979i −0.987994 0.154490i \(-0.950627\pi\)
0.987994 0.154490i \(-0.0493733\pi\)
\(42\) 0 0
\(43\) 7.62888 1.16339 0.581697 0.813406i \(-0.302389\pi\)
0.581697 + 0.813406i \(0.302389\pi\)
\(44\) −0.653130 5.65950i −0.0984631 0.853202i
\(45\) 0 0
\(46\) −0.597768 10.3939i −0.0881361 1.53250i
\(47\) 11.0724 1.61508 0.807538 0.589815i \(-0.200799\pi\)
0.807538 + 0.589815i \(0.200799\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0.273334 + 4.75270i 0.0386552 + 0.672134i
\(51\) 0 0
\(52\) 0.0512875 + 0.444416i 0.00711229 + 0.0616294i
\(53\) −0.295366 −0.0405717 −0.0202858 0.999794i \(-0.506458\pi\)
−0.0202858 + 0.999794i \(0.506458\pi\)
\(54\) 0 0
\(55\) 3.64098i 0.490949i
\(56\) 2.78653 0.485052i 0.372365 0.0648179i
\(57\) 0 0
\(58\) −0.863279 15.0106i −0.113354 1.97099i
\(59\) 7.25779i 0.944884i 0.881362 + 0.472442i \(0.156627\pi\)
−0.881362 + 0.472442i \(0.843373\pi\)
\(60\) 0 0
\(61\) 9.45592i 1.21071i −0.795957 0.605354i \(-0.793032\pi\)
0.795957 0.605354i \(-0.206968\pi\)
\(62\) −10.8370 + 0.623249i −1.37630 + 0.0791527i
\(63\) 0 0
\(64\) −7.52945 + 2.70322i −0.941181 + 0.337903i
\(65\) 0.285910i 0.0354628i
\(66\) 0 0
\(67\) −3.52321 −0.430428 −0.215214 0.976567i \(-0.569045\pi\)
−0.215214 + 0.976567i \(0.569045\pi\)
\(68\) −0.0912202 0.790441i −0.0110621 0.0958551i
\(69\) 0 0
\(70\) 1.80466 0.103788i 0.215698 0.0124050i
\(71\) −8.37710 −0.994179 −0.497089 0.867699i \(-0.665598\pi\)
−0.497089 + 0.867699i \(0.665598\pi\)
\(72\) 0 0
\(73\) 10.6704 1.24887 0.624437 0.781076i \(-0.285329\pi\)
0.624437 + 0.781076i \(0.285329\pi\)
\(74\) 6.69921 0.385280i 0.778768 0.0447879i
\(75\) 0 0
\(76\) 9.23595 1.06587i 1.05944 0.122263i
\(77\) 2.84853 0.324621
\(78\) 0 0
\(79\) 5.22135i 0.587448i −0.955890 0.293724i \(-0.905105\pi\)
0.955890 0.293724i \(-0.0948947\pi\)
\(80\) −4.97838 + 1.16456i −0.556600 + 0.130202i
\(81\) 0 0
\(82\) −2.79331 + 0.160647i −0.308470 + 0.0177405i
\(83\) 9.11868i 1.00091i −0.865764 0.500453i \(-0.833167\pi\)
0.865764 0.500453i \(-0.166833\pi\)
\(84\) 0 0
\(85\) 0.508521i 0.0551569i
\(86\) −0.619457 10.7711i −0.0667978 1.16147i
\(87\) 0 0
\(88\) −7.93751 + 1.38169i −0.846142 + 0.147288i
\(89\) 8.94887i 0.948578i −0.880369 0.474289i \(-0.842705\pi\)
0.880369 0.474289i \(-0.157295\pi\)
\(90\) 0 0
\(91\) −0.223683 −0.0234483
\(92\) −14.6265 + 1.68795i −1.52491 + 0.175981i
\(93\) 0 0
\(94\) −0.899068 15.6329i −0.0927318 1.61241i
\(95\) 5.94184 0.609620
\(96\) 0 0
\(97\) 0.228078 0.0231578 0.0115789 0.999933i \(-0.496314\pi\)
0.0115789 + 0.999933i \(0.496314\pi\)
\(98\) 0.0811990 + 1.41188i 0.00820234 + 0.142621i
\(99\) 0 0
\(100\) 6.68805 0.771829i 0.668805 0.0771829i
\(101\) 6.36504 0.633346 0.316673 0.948535i \(-0.397434\pi\)
0.316673 + 0.948535i \(0.397434\pi\)
\(102\) 0 0
\(103\) 11.6557i 1.14847i −0.818690 0.574236i \(-0.805299\pi\)
0.818690 0.574236i \(-0.194701\pi\)
\(104\) 0.623298 0.108498i 0.0611194 0.0106391i
\(105\) 0 0
\(106\) 0.0239834 + 0.417022i 0.00232948 + 0.0405047i
\(107\) 9.01403i 0.871419i −0.900087 0.435709i \(-0.856498\pi\)
0.900087 0.435709i \(-0.143502\pi\)
\(108\) 0 0
\(109\) 5.45206i 0.522213i 0.965310 + 0.261106i \(0.0840873\pi\)
−0.965310 + 0.261106i \(0.915913\pi\)
\(110\) −5.14062 + 0.295644i −0.490139 + 0.0281885i
\(111\) 0 0
\(112\) −0.911099 3.89486i −0.0860908 0.368029i
\(113\) 7.00229i 0.658720i 0.944204 + 0.329360i \(0.106833\pi\)
−0.944204 + 0.329360i \(0.893167\pi\)
\(114\) 0 0
\(115\) −9.40976 −0.877465
\(116\) −21.1231 + 2.43769i −1.96123 + 0.226334i
\(117\) 0 0
\(118\) 10.2471 0.589326i 0.943326 0.0542518i
\(119\) 0.397844 0.0364703
\(120\) 0 0
\(121\) 2.88586 0.262351
\(122\) −13.3506 + 0.767812i −1.20871 + 0.0695144i
\(123\) 0 0
\(124\) 1.75991 + 15.2499i 0.158044 + 1.36948i
\(125\) 10.6936 0.956469
\(126\) 0 0
\(127\) 9.48046i 0.841255i −0.907234 0.420627i \(-0.861810\pi\)
0.907234 0.420627i \(-0.138190\pi\)
\(128\) 4.42801 + 10.4112i 0.391385 + 0.920227i
\(129\) 0 0
\(130\) 0.403671 0.0232156i 0.0354043 0.00203614i
\(131\) 6.34169i 0.554076i 0.960859 + 0.277038i \(0.0893528\pi\)
−0.960859 + 0.277038i \(0.910647\pi\)
\(132\) 0 0
\(133\) 4.64862i 0.403087i
\(134\) 0.286081 + 4.97435i 0.0247136 + 0.429718i
\(135\) 0 0
\(136\) −1.10860 + 0.192975i −0.0950618 + 0.0165475i
\(137\) 11.3108i 0.966349i 0.875524 + 0.483174i \(0.160516\pi\)
−0.875524 + 0.483174i \(0.839484\pi\)
\(138\) 0 0
\(139\) 21.7583 1.84551 0.922757 0.385382i \(-0.125930\pi\)
0.922757 + 0.385382i \(0.125930\pi\)
\(140\) −0.293073 2.53953i −0.0247692 0.214630i
\(141\) 0 0
\(142\) 0.680212 + 11.8275i 0.0570821 + 0.992539i
\(143\) 0.637168 0.0532827
\(144\) 0 0
\(145\) −13.5893 −1.12853
\(146\) −0.866424 15.0653i −0.0717057 1.24681i
\(147\) 0 0
\(148\) −1.08794 9.42721i −0.0894280 0.774912i
\(149\) 2.42505 0.198668 0.0993340 0.995054i \(-0.468329\pi\)
0.0993340 + 0.995054i \(0.468329\pi\)
\(150\) 0 0
\(151\) 20.0398i 1.63082i −0.578887 0.815408i \(-0.696513\pi\)
0.578887 0.815408i \(-0.303487\pi\)
\(152\) −2.25483 12.9535i −0.182891 1.05067i
\(153\) 0 0
\(154\) −0.231298 4.02179i −0.0186385 0.324085i
\(155\) 9.81087i 0.788028i
\(156\) 0 0
\(157\) 15.5990i 1.24494i 0.782644 + 0.622470i \(0.213871\pi\)
−0.782644 + 0.622470i \(0.786129\pi\)
\(158\) −7.37193 + 0.423969i −0.586479 + 0.0337291i
\(159\) 0 0
\(160\) 2.04846 + 6.93431i 0.161945 + 0.548206i
\(161\) 7.36177i 0.580189i
\(162\) 0 0
\(163\) 12.8755 1.00848 0.504242 0.863562i \(-0.331772\pi\)
0.504242 + 0.863562i \(0.331772\pi\)
\(164\) 0.453628 + 3.93078i 0.0354224 + 0.306942i
\(165\) 0 0
\(166\) −12.8745 + 0.740428i −0.999255 + 0.0574684i
\(167\) 1.55771 0.120539 0.0602697 0.998182i \(-0.480804\pi\)
0.0602697 + 0.998182i \(0.480804\pi\)
\(168\) 0 0
\(169\) 12.9500 0.996151
\(170\) −0.717971 + 0.0412914i −0.0550659 + 0.00316691i
\(171\) 0 0
\(172\) −15.1572 + 1.74920i −1.15572 + 0.133375i
\(173\) 2.60553 0.198095 0.0990474 0.995083i \(-0.468420\pi\)
0.0990474 + 0.995083i \(0.468420\pi\)
\(174\) 0 0
\(175\) 3.36622i 0.254462i
\(176\) 2.59530 + 11.0946i 0.195628 + 0.836289i
\(177\) 0 0
\(178\) −12.6347 + 0.726639i −0.947013 + 0.0544639i
\(179\) 18.9293i 1.41484i 0.706793 + 0.707420i \(0.250141\pi\)
−0.706793 + 0.707420i \(0.749859\pi\)
\(180\) 0 0
\(181\) 15.6813i 1.16558i 0.812621 + 0.582792i \(0.198040\pi\)
−0.812621 + 0.582792i \(0.801960\pi\)
\(182\) 0.0181628 + 0.315814i 0.00134632 + 0.0234097i
\(183\) 0 0
\(184\) 3.57084 + 20.5138i 0.263246 + 1.51229i
\(185\) 6.06488i 0.445899i
\(186\) 0 0
\(187\) −1.13327 −0.0828730
\(188\) −21.9988 + 2.53875i −1.60443 + 0.185158i
\(189\) 0 0
\(190\) −0.482472 8.38917i −0.0350022 0.608614i
\(191\) 3.86888 0.279943 0.139971 0.990156i \(-0.455299\pi\)
0.139971 + 0.990156i \(0.455299\pi\)
\(192\) 0 0
\(193\) −12.1061 −0.871419 −0.435710 0.900087i \(-0.643503\pi\)
−0.435710 + 0.900087i \(0.643503\pi\)
\(194\) −0.0185197 0.322019i −0.00132964 0.0231196i
\(195\) 0 0
\(196\) 1.98681 0.229287i 0.141915 0.0163776i
\(197\) −5.20058 −0.370526 −0.185263 0.982689i \(-0.559314\pi\)
−0.185263 + 0.982689i \(0.559314\pi\)
\(198\) 0 0
\(199\) 15.5030i 1.09898i 0.835501 + 0.549489i \(0.185177\pi\)
−0.835501 + 0.549489i \(0.814823\pi\)
\(200\) −1.63279 9.38006i −0.115456 0.663271i
\(201\) 0 0
\(202\) −0.516835 8.98668i −0.0363644 0.632301i
\(203\) 10.6317i 0.746196i
\(204\) 0 0
\(205\) 2.52882i 0.176620i
\(206\) −16.4565 + 0.946433i −1.14658 + 0.0659411i
\(207\) 0 0
\(208\) −0.203797 0.871213i −0.0141308 0.0604077i
\(209\) 13.2418i 0.915952i
\(210\) 0 0
\(211\) −9.04096 −0.622406 −0.311203 0.950344i \(-0.600732\pi\)
−0.311203 + 0.950344i \(0.600732\pi\)
\(212\) 0.586837 0.0677235i 0.0403042 0.00465127i
\(213\) 0 0
\(214\) −12.7267 + 0.731930i −0.869981 + 0.0500337i
\(215\) −9.75118 −0.665025
\(216\) 0 0
\(217\) −7.67558 −0.521052
\(218\) 7.69766 0.442702i 0.521351 0.0299835i
\(219\) 0 0
\(220\) 0.834827 + 7.23394i 0.0562840 + 0.487712i
\(221\) 0.0889909 0.00598617
\(222\) 0 0
\(223\) 10.2327i 0.685231i −0.939476 0.342616i \(-0.888687\pi\)
0.939476 0.342616i \(-0.111313\pi\)
\(224\) −5.42509 + 1.60262i −0.362479 + 0.107080i
\(225\) 0 0
\(226\) 9.88640 0.568579i 0.657633 0.0378213i
\(227\) 10.9987i 0.730007i −0.931006 0.365003i \(-0.881068\pi\)
0.931006 0.365003i \(-0.118932\pi\)
\(228\) 0 0
\(229\) 24.1988i 1.59910i 0.600600 + 0.799550i \(0.294928\pi\)
−0.600600 + 0.799550i \(0.705072\pi\)
\(230\) 0.764063 + 13.2855i 0.0503808 + 0.876017i
\(231\) 0 0
\(232\) 5.15691 + 29.6254i 0.338568 + 1.94500i
\(233\) 21.5448i 1.41144i 0.708489 + 0.705722i \(0.249377\pi\)
−0.708489 + 0.705722i \(0.750623\pi\)
\(234\) 0 0
\(235\) −14.1527 −0.923219
\(236\) −1.66411 14.4199i −0.108325 0.938655i
\(237\) 0 0
\(238\) −0.0323045 0.561708i −0.00209399 0.0364101i
\(239\) 20.5708 1.33061 0.665307 0.746570i \(-0.268301\pi\)
0.665307 + 0.746570i \(0.268301\pi\)
\(240\) 0 0
\(241\) −11.0840 −0.713984 −0.356992 0.934107i \(-0.616198\pi\)
−0.356992 + 0.934107i \(0.616198\pi\)
\(242\) −0.234329 4.07449i −0.0150632 0.261918i
\(243\) 0 0
\(244\) 2.16812 + 18.7872i 0.138799 + 1.20272i
\(245\) 1.27819 0.0816608
\(246\) 0 0
\(247\) 1.03982i 0.0661620i
\(248\) 21.3882 3.72306i 1.35815 0.236414i
\(249\) 0 0
\(250\) −0.868314 15.0982i −0.0549170 0.954891i
\(251\) 20.7016i 1.30667i −0.757068 0.653336i \(-0.773369\pi\)
0.757068 0.653336i \(-0.226631\pi\)
\(252\) 0 0
\(253\) 20.9702i 1.31839i
\(254\) −13.3853 + 0.769804i −0.839867 + 0.0483018i
\(255\) 0 0
\(256\) 14.3398 7.09720i 0.896237 0.443575i
\(257\) 10.6516i 0.664427i 0.943204 + 0.332213i \(0.107795\pi\)
−0.943204 + 0.332213i \(0.892205\pi\)
\(258\) 0 0
\(259\) 4.74489 0.294833
\(260\) −0.0655553 0.568050i −0.00406557 0.0352290i
\(261\) 0 0
\(262\) 8.95371 0.514939i 0.553162 0.0318130i
\(263\) −6.36313 −0.392367 −0.196184 0.980567i \(-0.562855\pi\)
−0.196184 + 0.980567i \(0.562855\pi\)
\(264\) 0 0
\(265\) 0.377535 0.0231918
\(266\) 6.56330 0.377464i 0.402422 0.0231438i
\(267\) 0 0
\(268\) 6.99996 0.807824i 0.427590 0.0493457i
\(269\) −14.3840 −0.877007 −0.438503 0.898730i \(-0.644491\pi\)
−0.438503 + 0.898730i \(0.644491\pi\)
\(270\) 0 0
\(271\) 12.9317i 0.785548i −0.919635 0.392774i \(-0.871515\pi\)
0.919635 0.392774i \(-0.128485\pi\)
\(272\) 0.362475 + 1.54954i 0.0219783 + 0.0939549i
\(273\) 0 0
\(274\) 15.9695 0.918427i 0.964755 0.0554842i
\(275\) 9.58880i 0.578226i
\(276\) 0 0
\(277\) 13.5242i 0.812593i 0.913741 + 0.406297i \(0.133180\pi\)
−0.913741 + 0.406297i \(0.866820\pi\)
\(278\) −1.76675 30.7201i −0.105963 1.84247i
\(279\) 0 0
\(280\) −3.56172 + 0.619991i −0.212853 + 0.0370515i
\(281\) 20.7882i 1.24012i 0.784554 + 0.620061i \(0.212892\pi\)
−0.784554 + 0.620061i \(0.787108\pi\)
\(282\) 0 0
\(283\) −17.6119 −1.04692 −0.523459 0.852051i \(-0.675359\pi\)
−0.523459 + 0.852051i \(0.675359\pi\)
\(284\) 16.6437 1.92076i 0.987624 0.113976i
\(285\) 0 0
\(286\) −0.0517374 0.899605i −0.00305930 0.0531948i
\(287\) −1.97843 −0.116783
\(288\) 0 0
\(289\) 16.8417 0.990689
\(290\) 1.10344 + 19.1865i 0.0647961 + 1.12667i
\(291\) 0 0
\(292\) −21.2000 + 2.44657i −1.24064 + 0.143175i
\(293\) −2.42581 −0.141718 −0.0708588 0.997486i \(-0.522574\pi\)
−0.0708588 + 0.997486i \(0.522574\pi\)
\(294\) 0 0
\(295\) 9.27687i 0.540120i
\(296\) −13.2218 + 2.30152i −0.768499 + 0.133773i
\(297\) 0 0
\(298\) −0.196912 3.42388i −0.0114068 0.198340i
\(299\) 1.64670i 0.0952312i
\(300\) 0 0
\(301\) 7.62888i 0.439721i
\(302\) −28.2938 + 1.62721i −1.62813 + 0.0936355i
\(303\) 0 0
\(304\) −18.1057 + 4.23536i −1.03843 + 0.242914i
\(305\) 12.0865i 0.692071i
\(306\) 0 0
\(307\) −5.15871 −0.294423 −0.147212 0.989105i \(-0.547030\pi\)
−0.147212 + 0.989105i \(0.547030\pi\)
\(308\) −5.65950 + 0.653130i −0.322480 + 0.0372156i
\(309\) 0 0
\(310\) 13.8518 0.796633i 0.786728 0.0452457i
\(311\) 31.1304 1.76524 0.882620 0.470086i \(-0.155777\pi\)
0.882620 + 0.470086i \(0.155777\pi\)
\(312\) 0 0
\(313\) −21.8080 −1.23266 −0.616330 0.787488i \(-0.711381\pi\)
−0.616330 + 0.787488i \(0.711381\pi\)
\(314\) 22.0240 1.26663i 1.24289 0.0714799i
\(315\) 0 0
\(316\) 1.19719 + 10.3739i 0.0673470 + 0.583575i
\(317\) −0.309869 −0.0174040 −0.00870199 0.999962i \(-0.502770\pi\)
−0.00870199 + 0.999962i \(0.502770\pi\)
\(318\) 0 0
\(319\) 30.2846i 1.69561i
\(320\) 9.62409 3.45524i 0.538003 0.193154i
\(321\) 0 0
\(322\) −10.3939 + 0.597768i −0.579231 + 0.0333123i
\(323\) 1.84943i 0.102905i
\(324\) 0 0
\(325\) 0.752966i 0.0417670i
\(326\) −1.04547 18.1786i −0.0579035 1.00682i
\(327\) 0 0
\(328\) 5.51295 0.959644i 0.304402 0.0529875i
\(329\) 11.0724i 0.610441i
\(330\) 0 0
\(331\) 28.5893 1.57141 0.785706 0.618600i \(-0.212300\pi\)
0.785706 + 0.618600i \(0.212300\pi\)
\(332\) 2.09079 + 18.1171i 0.114747 + 0.994306i
\(333\) 0 0
\(334\) −0.126485 2.19930i −0.00692093 0.120341i
\(335\) 4.50334 0.246044
\(336\) 0 0
\(337\) 26.7103 1.45500 0.727502 0.686106i \(-0.240681\pi\)
0.727502 + 0.686106i \(0.240681\pi\)
\(338\) −1.05152 18.2838i −0.0571954 0.994508i
\(339\) 0 0
\(340\) 0.116597 + 1.01034i 0.00632337 + 0.0547932i
\(341\) 21.8641 1.18401
\(342\) 0 0
\(343\) 1.00000i 0.0539949i
\(344\) 3.70041 + 21.2581i 0.199513 + 1.14616i
\(345\) 0 0
\(346\) −0.211566 3.67870i −0.0113739 0.197768i
\(347\) 11.0302i 0.592132i 0.955168 + 0.296066i \(0.0956748\pi\)
−0.955168 + 0.296066i \(0.904325\pi\)
\(348\) 0 0
\(349\) 9.30132i 0.497888i −0.968518 0.248944i \(-0.919916\pi\)
0.968518 0.248944i \(-0.0800836\pi\)
\(350\) 4.75270 0.273334i 0.254043 0.0146103i
\(351\) 0 0
\(352\) 15.4536 4.56512i 0.823677 0.243322i
\(353\) 18.4585i 0.982446i −0.871034 0.491223i \(-0.836550\pi\)
0.871034 0.491223i \(-0.163450\pi\)
\(354\) 0 0
\(355\) 10.7076 0.568298
\(356\) 2.05186 + 17.7797i 0.108748 + 0.942324i
\(357\) 0 0
\(358\) 26.7259 1.53704i 1.41251 0.0812350i
\(359\) −7.70098 −0.406442 −0.203221 0.979133i \(-0.565141\pi\)
−0.203221 + 0.979133i \(0.565141\pi\)
\(360\) 0 0
\(361\) 2.60971 0.137353
\(362\) 22.1402 1.27331i 1.16366 0.0669236i
\(363\) 0 0
\(364\) 0.444416 0.0512875i 0.0232937 0.00268819i
\(365\) −13.6388 −0.713888
\(366\) 0 0
\(367\) 36.0127i 1.87985i −0.341386 0.939923i \(-0.610896\pi\)
0.341386 0.939923i \(-0.389104\pi\)
\(368\) 28.6730 6.70730i 1.49468 0.349642i
\(369\) 0 0
\(370\) −8.56289 + 0.492462i −0.445164 + 0.0256019i
\(371\) 0.295366i 0.0153346i
\(372\) 0 0
\(373\) 32.8248i 1.69960i 0.527103 + 0.849801i \(0.323278\pi\)
−0.527103 + 0.849801i \(0.676722\pi\)
\(374\) 0.0920205 + 1.60004i 0.00475827 + 0.0827363i
\(375\) 0 0
\(376\) 5.37070 + 30.8535i 0.276973 + 1.59115i
\(377\) 2.37812i 0.122479i
\(378\) 0 0
\(379\) −27.8200 −1.42902 −0.714509 0.699626i \(-0.753350\pi\)
−0.714509 + 0.699626i \(0.753350\pi\)
\(380\) −11.8053 + 1.36238i −0.605601 + 0.0698888i
\(381\) 0 0
\(382\) −0.314150 5.46240i −0.0160733 0.279481i
\(383\) 11.9047 0.608301 0.304151 0.952624i \(-0.401627\pi\)
0.304151 + 0.952624i \(0.401627\pi\)
\(384\) 0 0
\(385\) −3.64098 −0.185561
\(386\) 0.983007 + 17.0924i 0.0500337 + 0.869982i
\(387\) 0 0
\(388\) −0.453148 + 0.0522952i −0.0230051 + 0.00265489i
\(389\) 24.8388 1.25938 0.629688 0.776848i \(-0.283183\pi\)
0.629688 + 0.776848i \(0.283183\pi\)
\(390\) 0 0
\(391\) 2.92883i 0.148117i
\(392\) −0.485052 2.78653i −0.0244988 0.140741i
\(393\) 0 0
\(394\) 0.422282 + 7.34260i 0.0212743 + 0.369915i
\(395\) 6.67390i 0.335800i
\(396\) 0 0
\(397\) 4.49543i 0.225619i 0.993617 + 0.112810i \(0.0359850\pi\)
−0.993617 + 0.112810i \(0.964015\pi\)
\(398\) 21.8884 1.25883i 1.09716 0.0630993i
\(399\) 0 0
\(400\) −13.1109 + 3.06696i −0.655547 + 0.153348i
\(401\) 14.8094i 0.739545i −0.929122 0.369772i \(-0.879436\pi\)
0.929122 0.369772i \(-0.120564\pi\)
\(402\) 0 0
\(403\) −1.71689 −0.0855246
\(404\) −12.6462 + 1.45942i −0.629170 + 0.0726088i
\(405\) 0 0
\(406\) −15.0106 + 0.863279i −0.744965 + 0.0428438i
\(407\) −13.5160 −0.669962
\(408\) 0 0
\(409\) −33.4088 −1.65196 −0.825980 0.563699i \(-0.809378\pi\)
−0.825980 + 0.563699i \(0.809378\pi\)
\(410\) 3.57039 0.205338i 0.176329 0.0101409i
\(411\) 0 0
\(412\) 2.67250 + 23.1577i 0.131665 + 1.14090i
\(413\) 7.25779 0.357133
\(414\) 0 0
\(415\) 11.6554i 0.572143i
\(416\) −1.21350 + 0.358479i −0.0594967 + 0.0175759i
\(417\) 0 0
\(418\) −18.6958 + 1.07522i −0.914441 + 0.0525906i
\(419\) 6.31558i 0.308536i 0.988029 + 0.154268i \(0.0493020\pi\)
−0.988029 + 0.154268i \(0.950698\pi\)
\(420\) 0 0
\(421\) 10.6716i 0.520104i −0.965595 0.260052i \(-0.916260\pi\)
0.965595 0.260052i \(-0.0837397\pi\)
\(422\) 0.734117 + 12.7648i 0.0357363 + 0.621379i
\(423\) 0 0
\(424\) −0.143268 0.823045i −0.00695771 0.0399706i
\(425\) 1.33923i 0.0649622i
\(426\) 0 0
\(427\) −9.45592 −0.457604
\(428\) 2.06680 + 17.9092i 0.0999023 + 0.865673i
\(429\) 0 0
\(430\) 0.791786 + 13.7675i 0.0381833 + 0.663928i
\(431\) −35.8686 −1.72773 −0.863865 0.503723i \(-0.831963\pi\)
−0.863865 + 0.503723i \(0.831963\pi\)
\(432\) 0 0
\(433\) 4.43908 0.213328 0.106664 0.994295i \(-0.465983\pi\)
0.106664 + 0.994295i \(0.465983\pi\)
\(434\) 0.623249 + 10.8370i 0.0299169 + 0.520192i
\(435\) 0 0
\(436\) −1.25008 10.8322i −0.0598682 0.518770i
\(437\) −34.2221 −1.63706
\(438\) 0 0
\(439\) 18.7939i 0.896985i −0.893787 0.448493i \(-0.851961\pi\)
0.893787 0.448493i \(-0.148039\pi\)
\(440\) 10.1457 1.76606i 0.483676 0.0841938i
\(441\) 0 0
\(442\) −0.00722597 0.125644i −0.000343704 0.00597630i
\(443\) 30.9630i 1.47109i 0.677473 + 0.735547i \(0.263075\pi\)
−0.677473 + 0.735547i \(0.736925\pi\)
\(444\) 0 0
\(445\) 11.4384i 0.542232i
\(446\) −14.4473 + 0.830883i −0.684101 + 0.0393435i
\(447\) 0 0
\(448\) 2.70322 + 7.52945i 0.127715 + 0.355733i
\(449\) 24.6744i 1.16446i −0.813025 0.582229i \(-0.802181\pi\)
0.813025 0.582229i \(-0.197819\pi\)
\(450\) 0 0
\(451\) 5.63563 0.265372
\(452\) −1.60553 13.9122i −0.0755178 0.654377i
\(453\) 0 0
\(454\) −15.5288 + 0.893080i −0.728803 + 0.0419143i
\(455\) 0.285910 0.0134037
\(456\) 0 0
\(457\) 13.3915 0.626429 0.313214 0.949682i \(-0.398594\pi\)
0.313214 + 0.949682i \(0.398594\pi\)
\(458\) 34.1658 1.96492i 1.59646 0.0918145i
\(459\) 0 0
\(460\) 18.6954 2.15753i 0.871679 0.100595i
\(461\) 12.3835 0.576759 0.288379 0.957516i \(-0.406884\pi\)
0.288379 + 0.957516i \(0.406884\pi\)
\(462\) 0 0
\(463\) 3.34707i 0.155551i −0.996971 0.0777757i \(-0.975218\pi\)
0.996971 0.0777757i \(-0.0247818\pi\)
\(464\) 41.4087 9.68649i 1.92235 0.449684i
\(465\) 0 0
\(466\) 30.4186 1.74941i 1.40912 0.0810400i
\(467\) 3.05078i 0.141173i −0.997506 0.0705866i \(-0.977513\pi\)
0.997506 0.0705866i \(-0.0224871\pi\)
\(468\) 0 0
\(469\) 3.52321i 0.162687i
\(470\) 1.14918 + 19.9819i 0.0530079 + 0.921696i
\(471\) 0 0
\(472\) −20.2240 + 3.52041i −0.930886 + 0.162040i
\(473\) 21.7311i 0.999198i
\(474\) 0 0
\(475\) 15.6483 0.717993
\(476\) −0.790441 + 0.0912202i −0.0362298 + 0.00418107i
\(477\) 0 0
\(478\) −1.67033 29.0435i −0.0763990 1.32842i
\(479\) 20.4934 0.936368 0.468184 0.883631i \(-0.344908\pi\)
0.468184 + 0.883631i \(0.344908\pi\)
\(480\) 0 0
\(481\) 1.06135 0.0483934
\(482\) 0.900011 + 15.6493i 0.0409944 + 0.712807i
\(483\) 0 0
\(484\) −5.73366 + 0.661688i −0.260621 + 0.0300767i
\(485\) −0.291528 −0.0132376
\(486\) 0 0
\(487\) 21.9186i 0.993226i −0.867972 0.496613i \(-0.834577\pi\)
0.867972 0.496613i \(-0.165423\pi\)
\(488\) 26.3492 4.58662i 1.19277 0.207627i
\(489\) 0 0
\(490\) −0.103788 1.80466i −0.00468866 0.0815261i
\(491\) 10.2880i 0.464291i −0.972681 0.232146i \(-0.925425\pi\)
0.972681 0.232146i \(-0.0745746\pi\)
\(492\) 0 0
\(493\) 4.22974i 0.190498i
\(494\) 1.46810 0.0844322i 0.0660529 0.00379878i
\(495\) 0 0
\(496\) −6.99321 29.8953i −0.314004 1.34234i
\(497\) 8.37710i 0.375764i
\(498\) 0 0
\(499\) −1.87931 −0.0841294 −0.0420647 0.999115i \(-0.513394\pi\)
−0.0420647 + 0.999115i \(0.513394\pi\)
\(500\) −21.2463 + 2.45191i −0.950163 + 0.109653i
\(501\) 0 0
\(502\) −29.2281 + 1.68095i −1.30452 + 0.0750243i
\(503\) −14.5489 −0.648702 −0.324351 0.945937i \(-0.605146\pi\)
−0.324351 + 0.945937i \(0.605146\pi\)
\(504\) 0 0
\(505\) −8.13576 −0.362036
\(506\) 29.6075 1.70276i 1.31621 0.0756970i
\(507\) 0 0
\(508\) 2.17374 + 18.8359i 0.0964442 + 0.835708i
\(509\) −25.7122 −1.13967 −0.569837 0.821758i \(-0.692994\pi\)
−0.569837 + 0.821758i \(0.692994\pi\)
\(510\) 0 0
\(511\) 10.6704i 0.472030i
\(512\) −11.1848 19.6698i −0.494302 0.869290i
\(513\) 0 0
\(514\) 15.0387 0.864897i 0.663331 0.0381490i
\(515\) 14.8983i 0.656496i
\(516\) 0 0
\(517\) 31.5401i 1.38713i
\(518\) −0.385280 6.69921i −0.0169282 0.294347i
\(519\) 0 0
\(520\) −0.796695 + 0.138681i −0.0349374 + 0.00608158i
\(521\) 8.03293i 0.351929i 0.984397 + 0.175965i \(0.0563044\pi\)
−0.984397 + 0.175965i \(0.943696\pi\)
\(522\) 0 0
\(523\) 17.2938 0.756206 0.378103 0.925764i \(-0.376577\pi\)
0.378103 + 0.925764i \(0.376577\pi\)
\(524\) −1.45406 12.5998i −0.0635211 0.550423i
\(525\) 0 0
\(526\) 0.516680 + 8.98398i 0.0225283 + 0.391720i
\(527\) 3.05368 0.133020
\(528\) 0 0
\(529\) 31.1956 1.35633
\(530\) −0.0306555 0.533034i −0.00133159 0.0231535i
\(531\) 0 0
\(532\) −1.06587 9.23595i −0.0462112 0.400429i
\(533\) −0.442542 −0.0191686
\(534\) 0 0
\(535\) 11.5217i 0.498125i
\(536\) −1.70894 9.81751i −0.0738150 0.424052i
\(537\) 0 0
\(538\) 1.16796 + 20.3085i 0.0503545 + 0.875560i
\(539\) 2.84853i 0.122695i
\(540\) 0 0
\(541\) 6.58504i 0.283113i 0.989930 + 0.141557i \(0.0452107\pi\)
−0.989930 + 0.141557i \(0.954789\pi\)
\(542\) −18.2581 + 1.05004i −0.784252 + 0.0451033i
\(543\) 0 0
\(544\) 2.15834 0.637593i 0.0925380 0.0273366i
\(545\) 6.96879i 0.298510i
\(546\) 0 0
\(547\) −33.4478 −1.43012 −0.715062 0.699061i \(-0.753602\pi\)
−0.715062 + 0.699061i \(0.753602\pi\)
\(548\) −2.59342 22.4725i −0.110785 0.959978i
\(549\) 0 0
\(550\) −13.5382 + 0.778600i −0.577272 + 0.0331996i
\(551\) −49.4226 −2.10547
\(552\) 0 0
\(553\) −5.22135 −0.222035
\(554\) 19.0946 1.09816i 0.811253 0.0466561i
\(555\) 0 0
\(556\) −43.2297 + 4.98888i −1.83335 + 0.211576i
\(557\) 5.92223 0.250933 0.125467 0.992098i \(-0.459957\pi\)
0.125467 + 0.992098i \(0.459957\pi\)
\(558\) 0 0
\(559\) 1.70645i 0.0721751i
\(560\) 1.16456 + 4.97838i 0.0492117 + 0.210375i
\(561\) 0 0
\(562\) 29.3505 1.68798i 1.23808 0.0712033i
\(563\) 23.1453i 0.975459i −0.872995 0.487730i \(-0.837825\pi\)
0.872995 0.487730i \(-0.162175\pi\)
\(564\) 0 0
\(565\) 8.95028i 0.376541i
\(566\) 1.43007 + 24.8659i 0.0601102 + 1.04519i
\(567\) 0 0
\(568\) −4.06333 23.3430i −0.170494 0.979451i
\(569\) 1.41139i 0.0591686i −0.999562 0.0295843i \(-0.990582\pi\)
0.999562 0.0295843i \(-0.00941835\pi\)
\(570\) 0 0
\(571\) −6.82194 −0.285489 −0.142745 0.989760i \(-0.545593\pi\)
−0.142745 + 0.989760i \(0.545593\pi\)
\(572\) −1.26593 + 0.146094i −0.0529314 + 0.00610850i
\(573\) 0 0
\(574\) 0.160647 + 2.79331i 0.00670527 + 0.116591i
\(575\) −24.7813 −1.03345
\(576\) 0 0
\(577\) 6.97587 0.290409 0.145205 0.989402i \(-0.453616\pi\)
0.145205 + 0.989402i \(0.453616\pi\)
\(578\) −1.36753 23.7785i −0.0568818 0.989055i
\(579\) 0 0
\(580\) 26.9994 3.11584i 1.12109 0.129378i
\(581\) −9.11868 −0.378307
\(582\) 0 0
\(583\) 0.841360i 0.0348456i
\(584\) 5.17569 + 29.7333i 0.214172 + 1.23037i
\(585\) 0 0
\(586\) 0.196974 + 3.42496i 0.00813691 + 0.141484i
\(587\) 40.3481i 1.66534i −0.553766 0.832672i \(-0.686810\pi\)
0.553766 0.832672i \(-0.313190\pi\)
\(588\) 0 0
\(589\) 35.6809i 1.47020i
\(590\) −13.0978 + 0.753272i −0.539229 + 0.0310117i
\(591\) 0 0
\(592\) 4.32306 + 18.4807i 0.177677 + 0.759550i
\(593\) 6.40544i 0.263040i 0.991314 + 0.131520i \(0.0419858\pi\)
−0.991314 + 0.131520i \(0.958014\pi\)
\(594\) 0 0
\(595\) −0.508521 −0.0208473
\(596\) −4.81813 + 0.556032i −0.197358 + 0.0227760i
\(597\) 0 0
\(598\) −2.32495 + 0.133710i −0.0950741 + 0.00546783i
\(599\) −2.80603 −0.114651 −0.0573257 0.998356i \(-0.518257\pi\)
−0.0573257 + 0.998356i \(0.518257\pi\)
\(600\) 0 0
\(601\) 11.6165 0.473847 0.236924 0.971528i \(-0.423861\pi\)
0.236924 + 0.971528i \(0.423861\pi\)
\(602\) −10.7711 + 0.619457i −0.438996 + 0.0252472i
\(603\) 0 0
\(604\) 4.59486 + 39.8153i 0.186962 + 1.62006i
\(605\) −3.68868 −0.149966
\(606\) 0 0
\(607\) 38.5388i 1.56424i 0.623126 + 0.782122i \(0.285862\pi\)
−0.623126 + 0.782122i \(0.714138\pi\)
\(608\) 7.44999 + 25.2192i 0.302137 + 1.02277i
\(609\) 0 0
\(610\) 17.0647 0.981412i 0.690929 0.0397362i
\(611\) 2.47671i 0.100197i
\(612\) 0 0
\(613\) 15.8274i 0.639261i −0.947542 0.319631i \(-0.896441\pi\)
0.947542 0.319631i \(-0.103559\pi\)
\(614\) 0.418882 + 7.28348i 0.0169047 + 0.293937i
\(615\) 0 0
\(616\) 1.38169 + 7.93751i 0.0556698 + 0.319811i
\(617\) 23.9040i 0.962339i −0.876628 0.481170i \(-0.840212\pi\)
0.876628 0.481170i \(-0.159788\pi\)
\(618\) 0 0
\(619\) −31.4193 −1.26285 −0.631424 0.775437i \(-0.717529\pi\)
−0.631424 + 0.775437i \(0.717529\pi\)
\(620\) −2.24950 19.4924i −0.0903421 0.782832i
\(621\) 0 0
\(622\) −2.52775 43.9523i −0.101354 1.76233i
\(623\) −8.94887 −0.358529
\(624\) 0 0
\(625\) 3.16256 0.126502
\(626\) 1.77079 + 30.7902i 0.0707748 + 1.23063i
\(627\) 0 0
\(628\) −3.57665 30.9924i −0.142724 1.23673i
\(629\) −1.88772 −0.0752685
\(630\) 0 0
\(631\) 6.94219i 0.276364i 0.990407 + 0.138182i \(0.0441260\pi\)
−0.990407 + 0.138182i \(0.955874\pi\)
\(632\) 14.5494 2.53263i 0.578746 0.100743i
\(633\) 0 0
\(634\) 0.0251610 + 0.437498i 0.000999273 + 0.0173753i
\(635\) 12.1179i 0.480883i
\(636\) 0 0
\(637\) 0.223683i 0.00886264i
\(638\) 42.7583 2.45908i 1.69282 0.0973559i
\(639\) 0 0
\(640\) −5.65985 13.3075i −0.223725 0.526025i
\(641\) 31.1823i 1.23163i 0.787892 + 0.615814i \(0.211173\pi\)
−0.787892 + 0.615814i \(0.788827\pi\)
\(642\) 0 0
\(643\) 22.6881 0.894733 0.447366 0.894351i \(-0.352362\pi\)
0.447366 + 0.894351i \(0.352362\pi\)
\(644\) 1.68795 + 14.6265i 0.0665147 + 0.576363i
\(645\) 0 0
\(646\) −2.61117 + 0.150172i −0.102735 + 0.00590842i
\(647\) −37.7608 −1.48453 −0.742266 0.670106i \(-0.766249\pi\)
−0.742266 + 0.670106i \(0.766249\pi\)
\(648\) 0 0
\(649\) −20.6741 −0.811528
\(650\) 1.06310 0.0611401i 0.0416981 0.00239811i
\(651\) 0 0
\(652\) −25.5811 + 2.95217i −1.00183 + 0.115616i
\(653\) −47.8848 −1.87388 −0.936939 0.349494i \(-0.886353\pi\)
−0.936939 + 0.349494i \(0.886353\pi\)
\(654\) 0 0
\(655\) 8.10591i 0.316724i
\(656\) −1.80255 7.70571i −0.0703777 0.300857i
\(657\) 0 0
\(658\) −15.6329 + 0.899068i −0.609434 + 0.0350493i
\(659\) 7.00120i 0.272728i −0.990659 0.136364i \(-0.956458\pi\)
0.990659 0.136364i \(-0.0435417\pi\)
\(660\) 0 0
\(661\) 41.8042i 1.62599i −0.582267 0.812997i \(-0.697834\pi\)
0.582267 0.812997i \(-0.302166\pi\)
\(662\) −2.32142 40.3647i −0.0902247 1.56882i
\(663\) 0 0
\(664\) 25.4094 4.42304i 0.986078 0.171647i
\(665\) 5.94184i 0.230415i
\(666\) 0 0
\(667\) 78.2677 3.03054
\(668\) −3.09488 + 0.357163i −0.119745 + 0.0138190i
\(669\) 0 0
\(670\) −0.365667 6.35818i −0.0141269 0.245638i
\(671\) 26.9355 1.03983
\(672\) 0 0
\(673\) 22.2519 0.857747 0.428874 0.903364i \(-0.358911\pi\)
0.428874 + 0.903364i \(0.358911\pi\)
\(674\) −2.16885 37.7118i −0.0835410 1.45260i
\(675\) 0 0
\(676\) −25.7292 + 2.96925i −0.989583 + 0.114202i
\(677\) −29.2219 −1.12309 −0.561544 0.827447i \(-0.689793\pi\)
−0.561544 + 0.827447i \(0.689793\pi\)
\(678\) 0 0
\(679\) 0.228078i 0.00875283i
\(680\) 1.41701 0.246660i 0.0543398 0.00945896i
\(681\) 0 0
\(682\) −1.77535 30.8695i −0.0679815 1.18206i
\(683\) 35.0080i 1.33954i 0.742567 + 0.669772i \(0.233608\pi\)
−0.742567 + 0.669772i \(0.766392\pi\)
\(684\) 0 0
\(685\) 14.4574i 0.552390i
\(686\) 1.41188 0.0811990i 0.0539059 0.00310019i
\(687\) 0 0
\(688\) 29.7134 6.95067i 1.13281 0.264992i
\(689\) 0.0660684i 0.00251700i
\(690\) 0 0
\(691\) 1.33849 0.0509185 0.0254593 0.999676i \(-0.491895\pi\)
0.0254593 + 0.999676i \(0.491895\pi\)
\(692\) −5.17670 + 0.597413i −0.196789 + 0.0227102i
\(693\) 0 0
\(694\) 15.5733 0.895640i 0.591155 0.0339980i
\(695\) −27.8113 −1.05494
\(696\) 0 0
\(697\) 0.787107 0.0298138
\(698\) −13.1324 + 0.755258i −0.497067 + 0.0285869i
\(699\) 0 0
\(700\) −0.771829 6.68805i −0.0291724 0.252785i
\(701\) 23.0621 0.871044 0.435522 0.900178i \(-0.356564\pi\)
0.435522 + 0.900178i \(0.356564\pi\)
\(702\) 0 0
\(703\) 22.0572i 0.831903i
\(704\) −7.70022 21.4479i −0.290213 0.808348i
\(705\) 0 0
\(706\) −26.0612 + 1.49881i −0.980825 + 0.0564085i
\(707\) 6.36504i 0.239382i
\(708\) 0 0
\(709\) 46.8150i 1.75818i −0.476660 0.879088i \(-0.658153\pi\)
0.476660 0.879088i \(-0.341847\pi\)
\(710\) −0.869443 15.1178i −0.0326296 0.567360i
\(711\) 0 0
\(712\) 24.9363 4.34067i 0.934526 0.162674i
\(713\) 56.5058i 2.11616i
\(714\) 0 0
\(715\) −0.814424 −0.0304577
\(716\) −4.34023 37.6090i −0.162202 1.40551i
\(717\) 0 0
\(718\) 0.625312 + 10.8729i 0.0233364 + 0.405772i
\(719\) 34.8849 1.30099 0.650493 0.759512i \(-0.274562\pi\)
0.650493 + 0.759512i \(0.274562\pi\)
\(720\) 0 0
\(721\) −11.6557 −0.434082
\(722\) −0.211906 3.68460i −0.00788633 0.137127i
\(723\) 0 0
\(724\) −3.59552 31.1559i −0.133626 1.15790i
\(725\) −35.7885 −1.32915
\(726\) 0 0
\(727\) 16.4491i 0.610063i 0.952342 + 0.305032i \(0.0986671\pi\)
−0.952342 + 0.305032i \(0.901333\pi\)
\(728\) −0.108498 0.623298i −0.00402120 0.0231010i
\(729\) 0 0
\(730\) 1.10746 + 19.2564i 0.0409888 + 0.712710i
\(731\) 3.03510i 0.112257i
\(732\) 0 0
\(733\) 35.9855i 1.32915i −0.747220 0.664577i \(-0.768612\pi\)
0.747220 0.664577i \(-0.231388\pi\)
\(734\) −50.8456 + 2.92419i −1.87675 + 0.107934i
\(735\) 0 0
\(736\) −11.7981 39.9382i −0.434885 1.47214i
\(737\) 10.0360i 0.369680i
\(738\) 0 0
\(739\) −5.44866 −0.200432 −0.100216 0.994966i \(-0.531953\pi\)
−0.100216 + 0.994966i \(0.531953\pi\)
\(740\) 1.39060 + 12.0498i 0.0511193 + 0.442959i
\(741\) 0 0
\(742\) 0.417022 0.0239834i 0.0153093 0.000880459i
\(743\) −31.2115 −1.14504 −0.572520 0.819891i \(-0.694034\pi\)
−0.572520 + 0.819891i \(0.694034\pi\)
\(744\) 0 0
\(745\) −3.09969 −0.113564
\(746\) 46.3447 2.66534i 1.69680 0.0975850i
\(747\) 0 0
\(748\) 2.25160 0.259844i 0.0823266 0.00950083i
\(749\) −9.01403 −0.329365
\(750\) 0 0
\(751\) 23.1812i 0.845894i −0.906154 0.422947i \(-0.860996\pi\)
0.906154 0.422947i \(-0.139004\pi\)
\(752\) 43.1254 10.0881i 1.57262 0.367874i
\(753\) 0 0
\(754\) −3.35762 + 0.193101i −0.122277 + 0.00703232i
\(755\) 25.6147i 0.932216i
\(756\) 0 0
\(757\) 39.3962i 1.43188i −0.698162 0.715940i \(-0.745998\pi\)
0.698162 0.715940i \(-0.254002\pi\)
\(758\) 2.25896 + 39.2785i 0.0820490 + 1.42666i
\(759\) 0 0
\(760\) 2.88210 + 16.5571i 0.104545 + 0.600589i
\(761\) 17.1356i 0.621166i 0.950546 + 0.310583i \(0.100524\pi\)
−0.950546 + 0.310583i \(0.899476\pi\)
\(762\) 0 0
\(763\) 5.45206 0.197378
\(764\) −7.68675 + 0.887083i −0.278097 + 0.0320935i
\(765\) 0 0
\(766\) −0.966649 16.8080i −0.0349265 0.607298i
\(767\) 1.62344 0.0586192
\(768\) 0 0
\(769\) −44.9084 −1.61944 −0.809718 0.586819i \(-0.800380\pi\)
−0.809718 + 0.586819i \(0.800380\pi\)
\(770\) 0.295644 + 5.14062i 0.0106543 + 0.185255i
\(771\) 0 0
\(772\) 24.0526 2.77578i 0.865674 0.0999024i
\(773\) 27.0273 0.972106 0.486053 0.873929i \(-0.338436\pi\)
0.486053 + 0.873929i \(0.338436\pi\)
\(774\) 0 0
\(775\) 25.8377i 0.928117i
\(776\) 0.110630 + 0.635545i 0.00397138 + 0.0228147i
\(777\) 0 0
\(778\) −2.01688 35.0694i −0.0723088 1.25730i
\(779\) 9.19699i 0.329516i
\(780\) 0 0
\(781\) 23.8624i 0.853866i
\(782\) 4.13516 0.237818i 0.147873 0.00850437i
\(783\) 0 0
\(784\) −3.89486 + 0.911099i −0.139102 + 0.0325393i
\(785\) 19.9386i 0.711639i
\(786\) 0 0
\(787\) −34.1562 −1.21754 −0.608769 0.793348i \(-0.708336\pi\)
−0.608769 + 0.793348i \(0.708336\pi\)
\(788\) 10.3326 1.19242i 0.368083 0.0424783i
\(789\) 0 0
\(790\) 9.42275 0.541914i 0.335246 0.0192804i
\(791\) 7.00229 0.248973
\(792\) 0 0
\(793\) −2.11513 −0.0751104
\(794\) 6.34702 0.365025i 0.225247 0.0129542i
\(795\) 0 0
\(796\) −3.55463 30.8015i −0.125990 1.09173i
\(797\) −46.2712 −1.63901 −0.819505 0.573072i \(-0.805751\pi\)
−0.819505 + 0.573072i \(0.805751\pi\)
\(798\) 0 0
\(799\) 4.40509i 0.155841i
\(800\) 5.39478 + 18.2621i 0.190734 + 0.645661i
\(801\) 0 0
\(802\) −20.9091 + 1.20251i −0.738325 + 0.0424620i
\(803\) 30.3949i 1.07261i
\(804\) 0 0
\(805\) 9.40976i 0.331651i
\(806\) 0.139410 + 2.42405i 0.00491051 + 0.0853835i
\(807\) 0 0
\(808\) 3.08738 + 17.7364i 0.108614 + 0.623963i
\(809\) 26.3429i 0.926167i 0.886315 + 0.463084i \(0.153257\pi\)
−0.886315 + 0.463084i \(0.846743\pi\)
\(810\) 0 0
\(811\) 2.96401 0.104080 0.0520402 0.998645i \(-0.483428\pi\)
0.0520402 + 0.998645i \(0.483428\pi\)
\(812\) 2.43769 + 21.1231i 0.0855463 + 0.741276i
\(813\) 0 0
\(814\) 1.09748 + 19.0829i 0.0384668 + 0.668857i
\(815\) −16.4573 −0.576475
\(816\) 0 0
\(817\) −35.4638 −1.24072
\(818\) 2.71276 + 47.1693i 0.0948496 + 1.64924i
\(819\) 0 0
\(820\) −0.579824 5.02429i −0.0202483 0.175456i
\(821\) −10.3996 −0.362949 −0.181475 0.983396i \(-0.558087\pi\)
−0.181475 + 0.983396i \(0.558087\pi\)
\(822\) 0 0
\(823\) 39.8598i 1.38942i −0.719288 0.694712i \(-0.755532\pi\)
0.719288 0.694712i \(-0.244468\pi\)
\(824\) 32.4790 5.65364i 1.13146 0.196954i
\(825\) 0 0
\(826\) −0.589326 10.2471i −0.0205053 0.356544i
\(827\) 27.5060i 0.956477i −0.878230 0.478239i \(-0.841275\pi\)
0.878230 0.478239i \(-0.158725\pi\)
\(828\) 0 0
\(829\) 38.0086i 1.32009i 0.751225 + 0.660047i \(0.229464\pi\)
−0.751225 + 0.660047i \(0.770536\pi\)
\(830\) 16.4561 0.946410i 0.571199 0.0328504i
\(831\) 0 0
\(832\) 0.604665 + 1.68421i 0.0209630 + 0.0583894i
\(833\) 0.397844i 0.0137845i
\(834\) 0 0
\(835\) −1.99106 −0.0689034
\(836\) 3.03616 + 26.3089i 0.105008 + 0.909913i
\(837\) 0 0
\(838\) 8.91684 0.512819i 0.308027 0.0177150i
\(839\) −30.9037 −1.06691 −0.533457 0.845827i \(-0.679107\pi\)
−0.533457 + 0.845827i \(0.679107\pi\)
\(840\) 0 0
\(841\) 84.0320 2.89765
\(842\) −15.0671 + 0.866526i −0.519246 + 0.0298625i
\(843\) 0 0
\(844\) 17.9627 2.07297i 0.618302 0.0713546i
\(845\) −16.5526 −0.569425
\(846\) 0 0
\(847\) 2.88586i 0.0991592i
\(848\) −1.15041 + 0.269108i −0.0395052 + 0.00924120i
\(849\) 0 0
\(850\) −1.89083 + 0.108744i −0.0648551 + 0.00372989i
\(851\) 34.9308i 1.19741i
\(852\) 0 0
\(853\) 25.3302i 0.867290i −0.901084 0.433645i \(-0.857227\pi\)
0.901084 0.433645i \(-0.142773\pi\)
\(854\) 0.767812 + 13.3506i 0.0262740 + 0.456849i
\(855\) 0 0
\(856\) 25.1178 4.37228i 0.858509 0.149441i
\(857\) 39.1574i 1.33759i −0.743447 0.668795i \(-0.766810\pi\)
0.743447 0.668795i \(-0.233190\pi\)
\(858\) 0 0
\(859\) 11.2783 0.384810 0.192405 0.981316i \(-0.438371\pi\)
0.192405 + 0.981316i \(0.438371\pi\)
\(860\) 19.3738 2.23582i 0.660641 0.0762407i
\(861\) 0 0
\(862\) 2.91249 + 50.6422i 0.0992000 + 1.72488i
\(863\) 15.6543 0.532878 0.266439 0.963852i \(-0.414153\pi\)
0.266439 + 0.963852i \(0.414153\pi\)
\(864\) 0 0
\(865\) −3.33037 −0.113236
\(866\) −0.360449 6.26745i −0.0122485 0.212976i
\(867\) 0 0
\(868\) 15.2499 1.75991i 0.517617 0.0597351i
\(869\) 14.8732 0.504539
\(870\) 0 0
\(871\) 0.788081i 0.0267031i
\(872\) −15.1923 + 2.64454i −0.514476 + 0.0895553i
\(873\) 0 0
\(874\) 2.77880 + 48.3175i 0.0939943 + 1.63436i
\(875\) 10.6936i 0.361511i
\(876\) 0 0
\(877\) 14.6357i 0.494211i −0.968989 0.247106i \(-0.920521\pi\)
0.968989 0.247106i \(-0.0794795\pi\)
\(878\) −26.5348 + 1.52605i −0.895506 + 0.0515016i
\(879\) 0 0
\(880\) −3.31729 14.1811i −0.111826 0.478044i
\(881\) 21.9054i 0.738013i −0.929427 0.369007i \(-0.879698\pi\)
0.929427 0.369007i \(-0.120302\pi\)
\(882\) 0 0
\(883\) −36.3852 −1.22446 −0.612230 0.790680i \(-0.709727\pi\)
−0.612230 + 0.790680i \(0.709727\pi\)
\(884\) −0.176808 + 0.0204044i −0.00594670 + 0.000686274i
\(885\) 0 0
\(886\) 43.7160 2.51416i 1.46867 0.0844649i
\(887\) 10.3554 0.347700 0.173850 0.984772i \(-0.444379\pi\)
0.173850 + 0.984772i \(0.444379\pi\)
\(888\) 0 0
\(889\) −9.48046 −0.317964
\(890\) 16.1496 0.928785i 0.541337 0.0311330i
\(891\) 0 0
\(892\) 2.34622 + 20.3304i 0.0785571 + 0.680713i
\(893\) −51.4715 −1.72243
\(894\) 0 0
\(895\) 24.1953i 0.808759i
\(896\) 10.4112 4.42801i 0.347813 0.147929i
\(897\) 0 0
\(898\) −34.8373 + 2.00354i −1.16254 + 0.0668589i
\(899\) 81.6040i 2.72165i
\(900\) 0 0
\(901\) 0.117510i 0.00391481i
\(902\) −0.457608 7.95684i −0.0152367 0.264934i
\(903\) 0 0
\(904\) −19.5121 + 3.39648i −0.648962 + 0.112965i
\(905\) 20.0438i 0.666278i
\(906\) 0 0
\(907\) 10.5615 0.350689 0.175344 0.984507i \(-0.443896\pi\)
0.175344 + 0.984507i \(0.443896\pi\)
\(908\) 2.52185 + 21.8523i 0.0836904 + 0.725194i
\(909\) 0 0
\(910\) −0.0232156 0.403671i −0.000769590 0.0133816i
\(911\) 18.4732 0.612044 0.306022 0.952024i \(-0.401002\pi\)
0.306022 + 0.952024i \(0.401002\pi\)
\(912\) 0 0
\(913\) 25.9749 0.859643
\(914\) −1.08738 18.9072i −0.0359673 0.625395i
\(915\) 0 0
\(916\) −5.54845 48.0784i −0.183326 1.58856i
\(917\) 6.34169 0.209421
\(918\) 0 0
\(919\) 27.7252i 0.914569i −0.889320 0.457285i \(-0.848822\pi\)
0.889320 0.457285i \(-0.151178\pi\)
\(920\) −4.56423 26.2205i −0.150478 0.864466i
\(921\) 0 0
\(922\) −1.00553 17.4841i −0.0331154 0.575807i
\(923\) 1.87381i 0.0616773i
\(924\) 0 0
\(925\) 15.9723i 0.525167i
\(926\) −4.72566 + 0.271779i −0.155295 + 0.00893120i
\(927\) 0 0
\(928\) −17.0385 57.6777i −0.559317 1.89336i
\(929\) 57.6318i 1.89084i 0.325858 + 0.945419i \(0.394347\pi\)
−0.325858 + 0.945419i \(0.605653\pi\)
\(930\) 0 0
\(931\) 4.64862 0.152353
\(932\) −4.93992 42.8054i −0.161813 1.40214i
\(933\) 0 0
\(934\) −4.30734 + 0.247720i −0.140940 + 0.00810565i
\(935\) 1.44854 0.0473723
\(936\) 0 0
\(937\) −44.9081 −1.46708 −0.733542 0.679644i \(-0.762134\pi\)
−0.733542 + 0.679644i \(0.762134\pi\)
\(938\) 4.97435 0.286081i 0.162418 0.00934087i
\(939\) 0 0
\(940\) 28.1187 3.24502i 0.917132 0.105841i
\(941\) 25.0167 0.815521 0.407761 0.913089i \(-0.366310\pi\)
0.407761 + 0.913089i \(0.366310\pi\)
\(942\) 0 0
\(943\) 14.5648i 0.474294i
\(944\) 6.61257 + 28.2681i 0.215221 + 0.920047i
\(945\) 0 0
\(946\) 30.6817 1.76455i 0.997550 0.0573703i
\(947\) 39.7909i 1.29303i 0.762901 + 0.646516i \(0.223775\pi\)
−0.762901 + 0.646516i \(0.776225\pi\)
\(948\) 0 0
\(949\) 2.38678i 0.0774782i
\(950\) −1.27063 22.0935i −0.0412246 0.716809i
\(951\) 0 0
\(952\) 0.192975 + 1.10860i 0.00625436 + 0.0359300i
\(953\) 12.4544i 0.403437i −0.979444 0.201718i \(-0.935347\pi\)
0.979444 0.201718i \(-0.0646526\pi\)
\(954\) 0 0
\(955\) −4.94518 −0.160022
\(956\) −40.8703 + 4.71661i −1.32184 + 0.152546i
\(957\) 0 0
\(958\) −1.66405 28.9343i −0.0537629 0.934824i
\(959\) 11.3108 0.365246
\(960\) 0 0
\(961\) −27.9145 −0.900466
\(962\) −0.0861806 1.49850i −0.00277857 0.0483136i
\(963\) 0 0
\(964\) 22.0219 2.54142i 0.709277 0.0818535i
\(965\) 15.4740 0.498125
\(966\) 0 0
\(967\) 6.97106i 0.224174i 0.993698 + 0.112087i \(0.0357536\pi\)
−0.993698 + 0.112087i \(0.964246\pi\)
\(968\) 1.39979 + 8.04151i 0.0449910 + 0.258464i
\(969\) 0 0
\(970\) 0.0236718 + 0.411602i 0.000760055 + 0.0132158i
\(971\) 9.18374i 0.294720i 0.989083 + 0.147360i \(0.0470776\pi\)
−0.989083 + 0.147360i \(0.952922\pi\)
\(972\) 0 0
\(973\) 21.7583i 0.697539i
\(974\) −30.9464 + 1.77977i −0.991587 + 0.0570274i
\(975\) 0 0
\(976\) −8.61529 36.8295i −0.275769 1.17888i
\(977\) 39.1316i 1.25193i 0.779851 + 0.625965i \(0.215295\pi\)
−0.779851 + 0.625965i \(0.784705\pi\)
\(978\) 0 0
\(979\) 25.4912 0.814701
\(980\) −2.53953 + 0.293073i −0.0811224 + 0.00936186i
\(981\) 0 0
\(982\) −14.5254 + 0.835376i −0.463525 + 0.0266579i
\(983\) −50.4121 −1.60790 −0.803948 0.594700i \(-0.797271\pi\)
−0.803948 + 0.594700i \(0.797271\pi\)
\(984\) 0 0
\(985\) 6.64735 0.211802
\(986\) 5.97188 0.343450i 0.190183 0.0109377i
\(987\) 0 0
\(988\) −0.238416 2.06592i −0.00758503 0.0657258i
\(989\) 56.1620 1.78585
\(990\) 0 0
\(991\) 55.9071i 1.77595i 0.459894 + 0.887974i \(0.347887\pi\)
−0.459894 + 0.887974i \(0.652113\pi\)
\(992\) −41.6407 + 12.3010i −1.32209 + 0.390558i
\(993\) 0 0
\(994\) 11.8275 0.680212i 0.375144 0.0215750i
\(995\) 19.8158i 0.628204i
\(996\) 0 0
\(997\) 11.6669i 0.369494i −0.982786 0.184747i \(-0.940853\pi\)
0.982786 0.184747i \(-0.0591466\pi\)
\(998\) 0.152598 + 2.65336i 0.00483040 + 0.0839906i
\(999\) 0 0
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 1512.2.j.d.323.23 48
3.2 odd 2 inner 1512.2.j.d.323.26 yes 48
4.3 odd 2 6048.2.j.d.5615.16 48
8.3 odd 2 inner 1512.2.j.d.323.25 yes 48
8.5 even 2 6048.2.j.d.5615.34 48
12.11 even 2 6048.2.j.d.5615.33 48
24.5 odd 2 6048.2.j.d.5615.15 48
24.11 even 2 inner 1512.2.j.d.323.24 yes 48
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1512.2.j.d.323.23 48 1.1 even 1 trivial
1512.2.j.d.323.24 yes 48 24.11 even 2 inner
1512.2.j.d.323.25 yes 48 8.3 odd 2 inner
1512.2.j.d.323.26 yes 48 3.2 odd 2 inner
6048.2.j.d.5615.15 48 24.5 odd 2
6048.2.j.d.5615.16 48 4.3 odd 2
6048.2.j.d.5615.33 48 12.11 even 2
6048.2.j.d.5615.34 48 8.5 even 2