Properties

Label 144.3.q.e.65.3
Level $144$
Weight $3$
Character 144.65
Analytic conductor $3.924$
Analytic rank $0$
Dimension $8$
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] = [144,3,Mod(65,144)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(144, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("144.65");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 144 = 2^{4} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 144.q (of order \(6\), degree \(2\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(3.92371580679\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.19269881856.9
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} + 15x^{6} - 2x^{5} + 133x^{4} - 84x^{3} + 276x^{2} + 144x + 144 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2}\cdot 3 \)
Twist minimal: no (minimal twist has level 72)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 65.3
Root \(0.831167 + 1.43962i\) of defining polynomial
Character \(\chi\) \(=\) 144.65
Dual form 144.3.q.e.113.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.668833 + 2.92449i) q^{3} +(-0.0440114 + 0.0254100i) q^{5} +(-4.52944 + 7.84521i) q^{7} +(-8.10532 - 3.91200i) q^{9} +O(q^{10})\) \(q+(-0.668833 + 2.92449i) q^{3} +(-0.0440114 + 0.0254100i) q^{5} +(-4.52944 + 7.84521i) q^{7} +(-8.10532 - 3.91200i) q^{9} +(-3.29117 - 1.90016i) q^{11} +(0.216902 + 0.375686i) q^{13} +(-0.0448751 - 0.145706i) q^{15} +26.2355i q^{17} -34.2225 q^{19} +(-19.9138 - 18.4934i) q^{21} +(29.9930 - 17.3164i) q^{23} +(-12.4987 + 21.6484i) q^{25} +(16.8617 - 21.0875i) q^{27} +(14.0316 + 8.10114i) q^{29} +(17.1675 + 29.7350i) q^{31} +(7.75825 - 8.35413i) q^{33} -0.460372i q^{35} +29.2761 q^{37} +(-1.24376 + 0.383058i) q^{39} +(48.7026 - 28.1185i) q^{41} +(3.94539 - 6.83362i) q^{43} +(0.456130 - 0.0337838i) q^{45} +(33.4489 + 19.3117i) q^{47} +(-16.5316 - 28.6335i) q^{49} +(-76.7256 - 17.5472i) q^{51} -50.5273i q^{53} +0.193132 q^{55} +(22.8891 - 100.083i) q^{57} +(8.54743 - 4.93486i) q^{59} +(-36.5718 + 63.3442i) q^{61} +(67.4030 - 45.8689i) q^{63} +(-0.0190923 - 0.0110230i) q^{65} +(12.6797 + 21.9618i) q^{67} +(30.5815 + 99.2960i) q^{69} +97.8262i q^{71} -77.0599 q^{73} +(-54.9511 - 51.0316i) q^{75} +(29.8143 - 17.2133i) q^{77} +(42.1389 - 72.9868i) q^{79} +(50.3926 + 63.4160i) q^{81} +(-40.6763 - 23.4845i) q^{83} +(-0.666644 - 1.15466i) q^{85} +(-33.0765 + 35.6170i) q^{87} +108.587i q^{89} -3.92978 q^{91} +(-98.4420 + 30.3185i) q^{93} +(1.50618 - 0.869593i) q^{95} +(-32.4021 + 56.1221i) q^{97} +(19.2426 + 28.2765i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 10 q^{3} - 6 q^{5} - 6 q^{7} - 22 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 10 q^{3} - 6 q^{5} - 6 q^{7} - 22 q^{9} - 36 q^{11} + 14 q^{13} - 10 q^{15} - 4 q^{19} - 54 q^{21} + 102 q^{23} + 10 q^{25} + 20 q^{27} - 114 q^{29} + 50 q^{31} - 104 q^{33} + 120 q^{37} - 82 q^{39} + 264 q^{41} + 28 q^{43} + 206 q^{45} - 150 q^{47} + 94 q^{49} - 170 q^{51} + 244 q^{55} - 178 q^{57} + 108 q^{59} + 14 q^{61} + 210 q^{63} - 198 q^{65} + 20 q^{67} - 14 q^{69} - 76 q^{73} - 326 q^{75} + 66 q^{77} - 26 q^{79} + 194 q^{81} - 246 q^{83} - 224 q^{85} + 18 q^{87} - 108 q^{91} - 130 q^{93} + 456 q^{95} - 236 q^{97} + 634 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/144\mathbb{Z}\right)^\times\).

\(n\) \(37\) \(65\) \(127\)
\(\chi(n)\) \(1\) \(e\left(\frac{1}{6}\right)\) \(1\)

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
\(3\) −0.668833 + 2.92449i −0.222944 + 0.974831i
\(4\) 0 0
\(5\) −0.0440114 + 0.0254100i −0.00880228 + 0.00508200i −0.504395 0.863473i \(-0.668284\pi\)
0.495592 + 0.868555i \(0.334951\pi\)
\(6\) 0 0
\(7\) −4.52944 + 7.84521i −0.647062 + 1.12074i 0.336759 + 0.941591i \(0.390669\pi\)
−0.983821 + 0.179154i \(0.942664\pi\)
\(8\) 0 0
\(9\) −8.10532 3.91200i −0.900592 0.434666i
\(10\) 0 0
\(11\) −3.29117 1.90016i −0.299198 0.172742i 0.342885 0.939377i \(-0.388596\pi\)
−0.642082 + 0.766636i \(0.721929\pi\)
\(12\) 0 0
\(13\) 0.216902 + 0.375686i 0.0166848 + 0.0288989i 0.874247 0.485481i \(-0.161355\pi\)
−0.857562 + 0.514380i \(0.828022\pi\)
\(14\) 0 0
\(15\) −0.0448751 0.145706i −0.00299167 0.00971373i
\(16\) 0 0
\(17\) 26.2355i 1.54327i 0.636068 + 0.771633i \(0.280560\pi\)
−0.636068 + 0.771633i \(0.719440\pi\)
\(18\) 0 0
\(19\) −34.2225 −1.80118 −0.900592 0.434666i \(-0.856867\pi\)
−0.900592 + 0.434666i \(0.856867\pi\)
\(20\) 0 0
\(21\) −19.9138 18.4934i −0.948278 0.880640i
\(22\) 0 0
\(23\) 29.9930 17.3164i 1.30404 0.752889i 0.322947 0.946417i \(-0.395326\pi\)
0.981095 + 0.193528i \(0.0619931\pi\)
\(24\) 0 0
\(25\) −12.4987 + 21.6484i −0.499948 + 0.865936i
\(26\) 0 0
\(27\) 16.8617 21.0875i 0.624508 0.781018i
\(28\) 0 0
\(29\) 14.0316 + 8.10114i 0.483848 + 0.279350i 0.722019 0.691874i \(-0.243214\pi\)
−0.238171 + 0.971223i \(0.576548\pi\)
\(30\) 0 0
\(31\) 17.1675 + 29.7350i 0.553790 + 0.959193i 0.997997 + 0.0632685i \(0.0201524\pi\)
−0.444206 + 0.895925i \(0.646514\pi\)
\(32\) 0 0
\(33\) 7.75825 8.35413i 0.235099 0.253155i
\(34\) 0 0
\(35\) 0.460372i 0.0131535i
\(36\) 0 0
\(37\) 29.2761 0.791247 0.395623 0.918413i \(-0.370529\pi\)
0.395623 + 0.918413i \(0.370529\pi\)
\(38\) 0 0
\(39\) −1.24376 + 0.383058i −0.0318913 + 0.00982200i
\(40\) 0 0
\(41\) 48.7026 28.1185i 1.18787 0.685816i 0.230047 0.973180i \(-0.426112\pi\)
0.957822 + 0.287363i \(0.0927787\pi\)
\(42\) 0 0
\(43\) 3.94539 6.83362i 0.0917533 0.158921i −0.816496 0.577352i \(-0.804086\pi\)
0.908249 + 0.418430i \(0.137420\pi\)
\(44\) 0 0
\(45\) 0.456130 0.0337838i 0.0101362 0.000750752i
\(46\) 0 0
\(47\) 33.4489 + 19.3117i 0.711678 + 0.410887i 0.811682 0.584100i \(-0.198552\pi\)
−0.100004 + 0.994987i \(0.531886\pi\)
\(48\) 0 0
\(49\) −16.5316 28.6335i −0.337379 0.584358i
\(50\) 0 0
\(51\) −76.7256 17.5472i −1.50442 0.344062i
\(52\) 0 0
\(53\) 50.5273i 0.953344i −0.879081 0.476672i \(-0.841843\pi\)
0.879081 0.476672i \(-0.158157\pi\)
\(54\) 0 0
\(55\) 0.193132 0.00351149
\(56\) 0 0
\(57\) 22.8891 100.083i 0.401564 1.75585i
\(58\) 0 0
\(59\) 8.54743 4.93486i 0.144872 0.0836417i −0.425812 0.904812i \(-0.640012\pi\)
0.570684 + 0.821170i \(0.306678\pi\)
\(60\) 0 0
\(61\) −36.5718 + 63.3442i −0.599537 + 1.03843i 0.393352 + 0.919388i \(0.371315\pi\)
−0.992889 + 0.119041i \(0.962018\pi\)
\(62\) 0 0
\(63\) 67.4030 45.8689i 1.06989 0.728077i
\(64\) 0 0
\(65\) −0.0190923 0.0110230i −0.000293728 0.000169584i
\(66\) 0 0
\(67\) 12.6797 + 21.9618i 0.189249 + 0.327789i 0.945000 0.327070i \(-0.106061\pi\)
−0.755751 + 0.654859i \(0.772728\pi\)
\(68\) 0 0
\(69\) 30.5815 + 99.2960i 0.443211 + 1.43907i
\(70\) 0 0
\(71\) 97.8262i 1.37783i 0.724840 + 0.688917i \(0.241914\pi\)
−0.724840 + 0.688917i \(0.758086\pi\)
\(72\) 0 0
\(73\) −77.0599 −1.05561 −0.527807 0.849364i \(-0.676986\pi\)
−0.527807 + 0.849364i \(0.676986\pi\)
\(74\) 0 0
\(75\) −54.9511 51.0316i −0.732681 0.680421i
\(76\) 0 0
\(77\) 29.8143 17.2133i 0.387199 0.223549i
\(78\) 0 0
\(79\) 42.1389 72.9868i 0.533404 0.923883i −0.465835 0.884872i \(-0.654246\pi\)
0.999239 0.0390112i \(-0.0124208\pi\)
\(80\) 0 0
\(81\) 50.3926 + 63.4160i 0.622131 + 0.782913i
\(82\) 0 0
\(83\) −40.6763 23.4845i −0.490076 0.282946i 0.234530 0.972109i \(-0.424645\pi\)
−0.724606 + 0.689163i \(0.757978\pi\)
\(84\) 0 0
\(85\) −0.666644 1.15466i −0.00784287 0.0135843i
\(86\) 0 0
\(87\) −33.0765 + 35.6170i −0.380190 + 0.409390i
\(88\) 0 0
\(89\) 108.587i 1.22008i 0.792371 + 0.610039i \(0.208846\pi\)
−0.792371 + 0.610039i \(0.791154\pi\)
\(90\) 0 0
\(91\) −3.92978 −0.0431844
\(92\) 0 0
\(93\) −98.4420 + 30.3185i −1.05852 + 0.326005i
\(94\) 0 0
\(95\) 1.50618 0.869593i 0.0158545 0.00915361i
\(96\) 0 0
\(97\) −32.4021 + 56.1221i −0.334043 + 0.578579i −0.983300 0.181990i \(-0.941746\pi\)
0.649258 + 0.760568i \(0.275080\pi\)
\(98\) 0 0
\(99\) 19.2426 + 28.2765i 0.194370 + 0.285621i
\(100\) 0 0
\(101\) −168.478 97.2705i −1.66809 0.963075i −0.968664 0.248373i \(-0.920104\pi\)
−0.699430 0.714701i \(-0.746563\pi\)
\(102\) 0 0
\(103\) 12.4420 + 21.5502i 0.120796 + 0.209225i 0.920082 0.391726i \(-0.128122\pi\)
−0.799286 + 0.600951i \(0.794789\pi\)
\(104\) 0 0
\(105\) 1.34635 + 0.307912i 0.0128224 + 0.00293249i
\(106\) 0 0
\(107\) 23.2306i 0.217108i 0.994091 + 0.108554i \(0.0346221\pi\)
−0.994091 + 0.108554i \(0.965378\pi\)
\(108\) 0 0
\(109\) 157.077 1.44108 0.720538 0.693416i \(-0.243895\pi\)
0.720538 + 0.693416i \(0.243895\pi\)
\(110\) 0 0
\(111\) −19.5808 + 85.6179i −0.176404 + 0.771332i
\(112\) 0 0
\(113\) −32.5614 + 18.7994i −0.288154 + 0.166366i −0.637109 0.770774i \(-0.719870\pi\)
0.348955 + 0.937140i \(0.386537\pi\)
\(114\) 0 0
\(115\) −0.880021 + 1.52424i −0.00765236 + 0.0132543i
\(116\) 0 0
\(117\) −0.288382 3.89357i −0.00246481 0.0332784i
\(118\) 0 0
\(119\) −205.823 118.832i −1.72961 0.998589i
\(120\) 0 0
\(121\) −53.2788 92.2816i −0.440321 0.762658i
\(122\) 0 0
\(123\) 49.6584 + 161.237i 0.403727 + 1.31087i
\(124\) 0 0
\(125\) 2.54087i 0.0203269i
\(126\) 0 0
\(127\) −48.4364 −0.381389 −0.190694 0.981649i \(-0.561074\pi\)
−0.190694 + 0.981649i \(0.561074\pi\)
\(128\) 0 0
\(129\) 17.3461 + 16.1088i 0.134466 + 0.124875i
\(130\) 0 0
\(131\) −0.0274376 + 0.0158411i −0.000209447 + 0.000120924i −0.500105 0.865965i \(-0.666705\pi\)
0.499895 + 0.866086i \(0.333372\pi\)
\(132\) 0 0
\(133\) 155.009 268.483i 1.16548 2.01867i
\(134\) 0 0
\(135\) −0.206274 + 1.35655i −0.00152796 + 0.0100485i
\(136\) 0 0
\(137\) 0.913705 + 0.527528i 0.00666938 + 0.00385057i 0.503331 0.864094i \(-0.332108\pi\)
−0.496662 + 0.867944i \(0.665441\pi\)
\(138\) 0 0
\(139\) −45.3655 78.5754i −0.326371 0.565290i 0.655418 0.755266i \(-0.272492\pi\)
−0.981789 + 0.189976i \(0.939159\pi\)
\(140\) 0 0
\(141\) −78.8487 + 84.9047i −0.559211 + 0.602161i
\(142\) 0 0
\(143\) 1.64860i 0.0115286i
\(144\) 0 0
\(145\) −0.823399 −0.00567862
\(146\) 0 0
\(147\) 94.7955 29.1955i 0.644867 0.198609i
\(148\) 0 0
\(149\) 15.1086 8.72295i 0.101400 0.0585433i −0.448442 0.893812i \(-0.648021\pi\)
0.549842 + 0.835268i \(0.314688\pi\)
\(150\) 0 0
\(151\) −40.8713 + 70.7912i −0.270671 + 0.468816i −0.969034 0.246928i \(-0.920579\pi\)
0.698363 + 0.715744i \(0.253912\pi\)
\(152\) 0 0
\(153\) 102.633 212.647i 0.670806 1.38985i
\(154\) 0 0
\(155\) −1.51113 0.872452i −0.00974923 0.00562872i
\(156\) 0 0
\(157\) 96.4835 + 167.114i 0.614544 + 1.06442i 0.990464 + 0.137770i \(0.0439934\pi\)
−0.375920 + 0.926652i \(0.622673\pi\)
\(158\) 0 0
\(159\) 147.767 + 33.7943i 0.929350 + 0.212543i
\(160\) 0 0
\(161\) 313.735i 1.94866i
\(162\) 0 0
\(163\) 165.401 1.01473 0.507364 0.861732i \(-0.330620\pi\)
0.507364 + 0.861732i \(0.330620\pi\)
\(164\) 0 0
\(165\) −0.129173 + 0.564814i −0.000782868 + 0.00342311i
\(166\) 0 0
\(167\) 215.643 124.502i 1.29128 0.745520i 0.312398 0.949951i \(-0.398868\pi\)
0.978881 + 0.204432i \(0.0655346\pi\)
\(168\) 0 0
\(169\) 84.4059 146.195i 0.499443 0.865061i
\(170\) 0 0
\(171\) 277.384 + 133.878i 1.62213 + 0.782914i
\(172\) 0 0
\(173\) −117.476 67.8248i −0.679052 0.392051i 0.120446 0.992720i \(-0.461568\pi\)
−0.799498 + 0.600669i \(0.794901\pi\)
\(174\) 0 0
\(175\) −113.224 196.110i −0.646995 1.12063i
\(176\) 0 0
\(177\) 8.71516 + 28.2975i 0.0492382 + 0.159873i
\(178\) 0 0
\(179\) 95.4526i 0.533255i −0.963800 0.266627i \(-0.914091\pi\)
0.963800 0.266627i \(-0.0859093\pi\)
\(180\) 0 0
\(181\) 58.9249 0.325552 0.162776 0.986663i \(-0.447955\pi\)
0.162776 + 0.986663i \(0.447955\pi\)
\(182\) 0 0
\(183\) −160.789 149.321i −0.878630 0.815960i
\(184\) 0 0
\(185\) −1.28848 + 0.743906i −0.00696477 + 0.00402111i
\(186\) 0 0
\(187\) 49.8517 86.3457i 0.266587 0.461742i
\(188\) 0 0
\(189\) 89.0619 + 227.798i 0.471227 + 1.20528i
\(190\) 0 0
\(191\) −164.852 95.1775i −0.863101 0.498311i 0.00194880 0.999998i \(-0.499380\pi\)
−0.865049 + 0.501687i \(0.832713\pi\)
\(192\) 0 0
\(193\) 5.29645 + 9.17373i 0.0274428 + 0.0475323i 0.879421 0.476046i \(-0.157930\pi\)
−0.851978 + 0.523578i \(0.824597\pi\)
\(194\) 0 0
\(195\) 0.0450062 0.0484629i 0.000230801 0.000248528i
\(196\) 0 0
\(197\) 215.874i 1.09581i −0.836541 0.547904i \(-0.815426\pi\)
0.836541 0.547904i \(-0.184574\pi\)
\(198\) 0 0
\(199\) 146.668 0.737026 0.368513 0.929623i \(-0.379867\pi\)
0.368513 + 0.929623i \(0.379867\pi\)
\(200\) 0 0
\(201\) −72.7079 + 22.3928i −0.361731 + 0.111407i
\(202\) 0 0
\(203\) −127.110 + 73.3872i −0.626159 + 0.361513i
\(204\) 0 0
\(205\) −1.42898 + 2.47506i −0.00697063 + 0.0120735i
\(206\) 0 0
\(207\) −310.845 + 23.0231i −1.50166 + 0.111223i
\(208\) 0 0
\(209\) 112.632 + 65.0282i 0.538910 + 0.311140i
\(210\) 0 0
\(211\) 54.8335 + 94.9744i 0.259874 + 0.450116i 0.966208 0.257763i \(-0.0829855\pi\)
−0.706334 + 0.707879i \(0.749652\pi\)
\(212\) 0 0
\(213\) −286.092 65.4294i −1.34316 0.307180i
\(214\) 0 0
\(215\) 0.401009i 0.00186516i
\(216\) 0 0
\(217\) −311.036 −1.43335
\(218\) 0 0
\(219\) 51.5402 225.361i 0.235343 1.02905i
\(220\) 0 0
\(221\) −9.85631 + 5.69054i −0.0445987 + 0.0257491i
\(222\) 0 0
\(223\) −73.8403 + 127.895i −0.331123 + 0.573521i −0.982732 0.185033i \(-0.940761\pi\)
0.651610 + 0.758554i \(0.274094\pi\)
\(224\) 0 0
\(225\) 185.995 126.572i 0.826642 0.562544i
\(226\) 0 0
\(227\) 346.255 + 199.911i 1.52535 + 0.880664i 0.999548 + 0.0300589i \(0.00956949\pi\)
0.525806 + 0.850605i \(0.323764\pi\)
\(228\) 0 0
\(229\) 39.1692 + 67.8430i 0.171044 + 0.296258i 0.938785 0.344503i \(-0.111953\pi\)
−0.767741 + 0.640760i \(0.778619\pi\)
\(230\) 0 0
\(231\) 30.3994 + 98.7046i 0.131599 + 0.427293i
\(232\) 0 0
\(233\) 352.995i 1.51500i −0.652835 0.757500i \(-0.726420\pi\)
0.652835 0.757500i \(-0.273580\pi\)
\(234\) 0 0
\(235\) −1.96284 −0.00835251
\(236\) 0 0
\(237\) 185.265 + 172.051i 0.781711 + 0.725953i
\(238\) 0 0
\(239\) −279.549 + 161.397i −1.16966 + 0.675303i −0.953600 0.301077i \(-0.902654\pi\)
−0.216060 + 0.976380i \(0.569321\pi\)
\(240\) 0 0
\(241\) −105.601 + 182.907i −0.438180 + 0.758949i −0.997549 0.0699691i \(-0.977710\pi\)
0.559370 + 0.828918i \(0.311043\pi\)
\(242\) 0 0
\(243\) −219.164 + 104.958i −0.901909 + 0.431926i
\(244\) 0 0
\(245\) 1.45516 + 0.840135i 0.00593941 + 0.00342912i
\(246\) 0 0
\(247\) −7.42293 12.8569i −0.0300524 0.0520522i
\(248\) 0 0
\(249\) 95.8859 103.250i 0.385084 0.414661i
\(250\) 0 0
\(251\) 206.824i 0.824001i 0.911184 + 0.412001i \(0.135170\pi\)
−0.911184 + 0.412001i \(0.864830\pi\)
\(252\) 0 0
\(253\) −131.616 −0.520222
\(254\) 0 0
\(255\) 3.82267 1.17732i 0.0149909 0.00461694i
\(256\) 0 0
\(257\) 20.7432 11.9761i 0.0807127 0.0465995i −0.459100 0.888384i \(-0.651828\pi\)
0.539813 + 0.841785i \(0.318495\pi\)
\(258\) 0 0
\(259\) −132.604 + 229.678i −0.511986 + 0.886786i
\(260\) 0 0
\(261\) −82.0389 120.554i −0.314325 0.461892i
\(262\) 0 0
\(263\) 119.369 + 68.9175i 0.453873 + 0.262044i 0.709464 0.704741i \(-0.248937\pi\)
−0.255592 + 0.966785i \(0.582270\pi\)
\(264\) 0 0
\(265\) 1.28390 + 2.22377i 0.00484489 + 0.00839160i
\(266\) 0 0
\(267\) −317.562 72.6265i −1.18937 0.272009i
\(268\) 0 0
\(269\) 249.461i 0.927366i 0.886001 + 0.463683i \(0.153472\pi\)
−0.886001 + 0.463683i \(0.846528\pi\)
\(270\) 0 0
\(271\) −72.6700 −0.268155 −0.134078 0.990971i \(-0.542807\pi\)
−0.134078 + 0.990971i \(0.542807\pi\)
\(272\) 0 0
\(273\) 2.62837 11.4926i 0.00962771 0.0420975i
\(274\) 0 0
\(275\) 82.2709 47.4991i 0.299167 0.172724i
\(276\) 0 0
\(277\) −182.021 + 315.270i −0.657117 + 1.13816i 0.324242 + 0.945974i \(0.394891\pi\)
−0.981359 + 0.192185i \(0.938443\pi\)
\(278\) 0 0
\(279\) −22.8251 308.171i −0.0818102 1.10456i
\(280\) 0 0
\(281\) 241.120 + 139.211i 0.858077 + 0.495411i 0.863368 0.504575i \(-0.168351\pi\)
−0.00529060 + 0.999986i \(0.501684\pi\)
\(282\) 0 0
\(283\) 13.7745 + 23.8581i 0.0486732 + 0.0843044i 0.889336 0.457255i \(-0.151167\pi\)
−0.840662 + 0.541560i \(0.817834\pi\)
\(284\) 0 0
\(285\) 1.53574 + 4.98642i 0.00538855 + 0.0174962i
\(286\) 0 0
\(287\) 509.443i 1.77506i
\(288\) 0 0
\(289\) −399.303 −1.38167
\(290\) 0 0
\(291\) −142.457 132.296i −0.489544 0.454626i
\(292\) 0 0
\(293\) 342.145 197.537i 1.16773 0.674189i 0.214585 0.976705i \(-0.431160\pi\)
0.953144 + 0.302517i \(0.0978267\pi\)
\(294\) 0 0
\(295\) −0.250789 + 0.434380i −0.000850133 + 0.00147247i
\(296\) 0 0
\(297\) −95.5645 + 37.3627i −0.321766 + 0.125800i
\(298\) 0 0
\(299\) 13.0111 + 7.51195i 0.0435153 + 0.0251236i
\(300\) 0 0
\(301\) 35.7408 + 61.9049i 0.118740 + 0.205664i
\(302\) 0 0
\(303\) 397.150 427.654i 1.31073 1.41140i
\(304\) 0 0
\(305\) 3.71715i 0.0121874i
\(306\) 0 0
\(307\) 122.443 0.398836 0.199418 0.979915i \(-0.436095\pi\)
0.199418 + 0.979915i \(0.436095\pi\)
\(308\) 0 0
\(309\) −71.3449 + 21.9731i −0.230890 + 0.0711102i
\(310\) 0 0
\(311\) 420.591 242.829i 1.35238 0.780799i 0.363801 0.931477i \(-0.381479\pi\)
0.988583 + 0.150677i \(0.0481455\pi\)
\(312\) 0 0
\(313\) 5.15434 8.92759i 0.0164676 0.0285226i −0.857674 0.514194i \(-0.828091\pi\)
0.874142 + 0.485671i \(0.161425\pi\)
\(314\) 0 0
\(315\) −1.80097 + 3.73146i −0.00571737 + 0.0118459i
\(316\) 0 0
\(317\) −144.879 83.6462i −0.457033 0.263868i 0.253763 0.967266i \(-0.418332\pi\)
−0.710796 + 0.703398i \(0.751665\pi\)
\(318\) 0 0
\(319\) −30.7869 53.3245i −0.0965107 0.167162i
\(320\) 0 0
\(321\) −67.9378 15.5374i −0.211644 0.0484031i
\(322\) 0 0
\(323\) 897.845i 2.77971i
\(324\) 0 0
\(325\) −10.8440 −0.0333661
\(326\) 0 0
\(327\) −105.058 + 459.371i −0.321280 + 1.40481i
\(328\) 0 0
\(329\) −303.009 + 174.942i −0.921000 + 0.531740i
\(330\) 0 0
\(331\) −52.2422 + 90.4861i −0.157831 + 0.273372i −0.934086 0.357047i \(-0.883784\pi\)
0.776255 + 0.630419i \(0.217117\pi\)
\(332\) 0 0
\(333\) −237.293 114.528i −0.712590 0.343928i
\(334\) 0 0
\(335\) −1.11610 0.644381i −0.00333164 0.00192352i
\(336\) 0 0
\(337\) 196.086 + 339.631i 0.581858 + 1.00781i 0.995259 + 0.0972586i \(0.0310074\pi\)
−0.413401 + 0.910549i \(0.635659\pi\)
\(338\) 0 0
\(339\) −33.2004 107.799i −0.0979364 0.317992i
\(340\) 0 0
\(341\) 130.484i 0.382651i
\(342\) 0 0
\(343\) −144.370 −0.420903
\(344\) 0 0
\(345\) −3.86905 3.59308i −0.0112146 0.0104147i
\(346\) 0 0
\(347\) −247.220 + 142.732i −0.712449 + 0.411333i −0.811967 0.583703i \(-0.801603\pi\)
0.0995180 + 0.995036i \(0.468270\pi\)
\(348\) 0 0
\(349\) 151.562 262.514i 0.434276 0.752189i −0.562960 0.826484i \(-0.690337\pi\)
0.997236 + 0.0742956i \(0.0236708\pi\)
\(350\) 0 0
\(351\) 11.5796 + 1.76078i 0.0329903 + 0.00501646i
\(352\) 0 0
\(353\) 308.183 + 177.929i 0.873039 + 0.504049i 0.868357 0.495940i \(-0.165176\pi\)
0.00468222 + 0.999989i \(0.498510\pi\)
\(354\) 0 0
\(355\) −2.48576 4.30547i −0.00700215 0.0121281i
\(356\) 0 0
\(357\) 485.185 522.450i 1.35906 1.46345i
\(358\) 0 0
\(359\) 148.171i 0.412733i −0.978475 0.206367i \(-0.933836\pi\)
0.978475 0.206367i \(-0.0661639\pi\)
\(360\) 0 0
\(361\) 810.179 2.24426
\(362\) 0 0
\(363\) 305.511 94.0925i 0.841629 0.259208i
\(364\) 0 0
\(365\) 3.39151 1.95809i 0.00929181 0.00536463i
\(366\) 0 0
\(367\) 123.772 214.380i 0.337254 0.584141i −0.646661 0.762777i \(-0.723835\pi\)
0.983915 + 0.178636i \(0.0571686\pi\)
\(368\) 0 0
\(369\) −504.750 + 37.3849i −1.36789 + 0.101314i
\(370\) 0 0
\(371\) 396.397 + 228.860i 1.06846 + 0.616873i
\(372\) 0 0
\(373\) −224.520 388.881i −0.601931 1.04258i −0.992528 0.122014i \(-0.961065\pi\)
0.390597 0.920562i \(-0.372269\pi\)
\(374\) 0 0
\(375\) 7.43075 + 1.69942i 0.0198153 + 0.00453177i
\(376\) 0 0
\(377\) 7.02862i 0.0186436i
\(378\) 0 0
\(379\) −618.282 −1.63135 −0.815675 0.578510i \(-0.803634\pi\)
−0.815675 + 0.578510i \(0.803634\pi\)
\(380\) 0 0
\(381\) 32.3959 141.652i 0.0850285 0.371790i
\(382\) 0 0
\(383\) 370.803 214.083i 0.968155 0.558965i 0.0694818 0.997583i \(-0.477865\pi\)
0.898673 + 0.438619i \(0.144532\pi\)
\(384\) 0 0
\(385\) −0.874780 + 1.51516i −0.00227216 + 0.00393549i
\(386\) 0 0
\(387\) −58.7118 + 39.9544i −0.151710 + 0.103241i
\(388\) 0 0
\(389\) 585.313 + 337.930i 1.50466 + 0.868716i 0.999985 + 0.00540555i \(0.00172065\pi\)
0.504674 + 0.863310i \(0.331613\pi\)
\(390\) 0 0
\(391\) 454.306 + 786.881i 1.16191 + 2.01248i
\(392\) 0 0
\(393\) −0.0279760 0.0908360i −7.11857e−5 0.000231135i
\(394\) 0 0
\(395\) 4.28300i 0.0108430i
\(396\) 0 0
\(397\) 209.902 0.528721 0.264361 0.964424i \(-0.414839\pi\)
0.264361 + 0.964424i \(0.414839\pi\)
\(398\) 0 0
\(399\) 681.501 + 632.892i 1.70802 + 1.58619i
\(400\) 0 0
\(401\) 175.023 101.050i 0.436467 0.251994i −0.265631 0.964075i \(-0.585580\pi\)
0.702098 + 0.712080i \(0.252247\pi\)
\(402\) 0 0
\(403\) −7.44734 + 12.8992i −0.0184797 + 0.0320079i
\(404\) 0 0
\(405\) −3.82925 1.51055i −0.00945493 0.00372975i
\(406\) 0 0
\(407\) −96.3528 55.6293i −0.236739 0.136681i
\(408\) 0 0
\(409\) −291.252 504.464i −0.712108 1.23341i −0.964064 0.265669i \(-0.914407\pi\)
0.251957 0.967739i \(-0.418926\pi\)
\(410\) 0 0
\(411\) −2.15387 + 2.31930i −0.00524056 + 0.00564306i
\(412\) 0 0
\(413\) 89.4085i 0.216486i
\(414\) 0 0
\(415\) 2.38696 0.00575172
\(416\) 0 0
\(417\) 260.135 80.1173i 0.623825 0.192128i
\(418\) 0 0
\(419\) −675.460 + 389.977i −1.61208 + 0.930733i −0.623189 + 0.782071i \(0.714163\pi\)
−0.988888 + 0.148662i \(0.952503\pi\)
\(420\) 0 0
\(421\) −84.1068 + 145.677i −0.199779 + 0.346027i −0.948457 0.316907i \(-0.897356\pi\)
0.748678 + 0.662934i \(0.230689\pi\)
\(422\) 0 0
\(423\) −195.567 287.380i −0.462332 0.679384i
\(424\) 0 0
\(425\) −567.957 327.910i −1.33637 0.771553i
\(426\) 0 0
\(427\) −331.299 573.827i −0.775876 1.34386i
\(428\) 0 0
\(429\) 4.82131 + 1.10264i 0.0112385 + 0.00257024i
\(430\) 0 0
\(431\) 182.732i 0.423973i −0.977273 0.211986i \(-0.932007\pi\)
0.977273 0.211986i \(-0.0679933\pi\)
\(432\) 0 0
\(433\) 447.193 1.03278 0.516389 0.856354i \(-0.327276\pi\)
0.516389 + 0.856354i \(0.327276\pi\)
\(434\) 0 0
\(435\) 0.550717 2.40803i 0.00126602 0.00553569i
\(436\) 0 0
\(437\) −1026.43 + 592.612i −2.34882 + 1.35609i
\(438\) 0 0
\(439\) −98.6108 + 170.799i −0.224626 + 0.389063i −0.956207 0.292691i \(-0.905449\pi\)
0.731581 + 0.681754i \(0.238783\pi\)
\(440\) 0 0
\(441\) 21.9796 + 296.756i 0.0498403 + 0.672915i
\(442\) 0 0
\(443\) −244.803 141.337i −0.552603 0.319045i 0.197568 0.980289i \(-0.436696\pi\)
−0.750171 + 0.661244i \(0.770029\pi\)
\(444\) 0 0
\(445\) −2.75919 4.77906i −0.00620043 0.0107395i
\(446\) 0 0
\(447\) 15.4051 + 50.0192i 0.0344633 + 0.111900i
\(448\) 0 0
\(449\) 349.046i 0.777386i −0.921367 0.388693i \(-0.872927\pi\)
0.921367 0.388693i \(-0.127073\pi\)
\(450\) 0 0
\(451\) −213.718 −0.473877
\(452\) 0 0
\(453\) −179.692 166.875i −0.396672 0.368378i
\(454\) 0 0
\(455\) 0.172955 0.0998556i 0.000380121 0.000219463i
\(456\) 0 0
\(457\) −40.2987 + 69.7995i −0.0881811 + 0.152734i −0.906742 0.421685i \(-0.861439\pi\)
0.818561 + 0.574419i \(0.194772\pi\)
\(458\) 0 0
\(459\) 553.242 + 442.376i 1.20532 + 0.963782i
\(460\) 0 0
\(461\) 495.135 + 285.866i 1.07405 + 0.620100i 0.929284 0.369366i \(-0.120425\pi\)
0.144761 + 0.989467i \(0.453759\pi\)
\(462\) 0 0
\(463\) −139.837 242.205i −0.302024 0.523120i 0.674571 0.738210i \(-0.264329\pi\)
−0.976594 + 0.215090i \(0.930996\pi\)
\(464\) 0 0
\(465\) 3.56217 3.83577i 0.00766059 0.00824896i
\(466\) 0 0
\(467\) 506.702i 1.08501i −0.840051 0.542507i \(-0.817475\pi\)
0.840051 0.542507i \(-0.182525\pi\)
\(468\) 0 0
\(469\) −229.727 −0.489823
\(470\) 0 0
\(471\) −553.256 + 170.394i −1.17464 + 0.361770i
\(472\) 0 0
\(473\) −25.9699 + 14.9938i −0.0549048 + 0.0316993i
\(474\) 0 0
\(475\) 427.737 740.862i 0.900499 1.55971i
\(476\) 0 0
\(477\) −197.662 + 409.540i −0.414387 + 0.858574i
\(478\) 0 0
\(479\) 217.596 + 125.629i 0.454271 + 0.262274i 0.709632 0.704572i \(-0.248861\pi\)
−0.255361 + 0.966846i \(0.582194\pi\)
\(480\) 0 0
\(481\) 6.35006 + 10.9986i 0.0132018 + 0.0228662i
\(482\) 0 0
\(483\) −917.516 209.836i −1.89962 0.434444i
\(484\) 0 0
\(485\) 3.29335i 0.00679041i
\(486\) 0 0
\(487\) 313.224 0.643170 0.321585 0.946881i \(-0.395784\pi\)
0.321585 + 0.946881i \(0.395784\pi\)
\(488\) 0 0
\(489\) −110.625 + 483.713i −0.226228 + 0.989188i
\(490\) 0 0
\(491\) −533.397 + 307.957i −1.08635 + 0.627204i −0.932602 0.360906i \(-0.882467\pi\)
−0.153747 + 0.988110i \(0.549134\pi\)
\(492\) 0 0
\(493\) −212.538 + 368.126i −0.431111 + 0.746706i
\(494\) 0 0
\(495\) −1.56540 0.755532i −0.00316242 0.00152633i
\(496\) 0 0
\(497\) −767.467 443.098i −1.54420 0.891544i
\(498\) 0 0
\(499\) 412.029 + 713.654i 0.825709 + 1.43017i 0.901377 + 0.433036i \(0.142558\pi\)
−0.0756679 + 0.997133i \(0.524109\pi\)
\(500\) 0 0
\(501\) 219.875 + 713.919i 0.438873 + 1.42499i
\(502\) 0 0
\(503\) 175.718i 0.349340i 0.984627 + 0.174670i \(0.0558858\pi\)
−0.984627 + 0.174670i \(0.944114\pi\)
\(504\) 0 0
\(505\) 9.88657 0.0195774
\(506\) 0 0
\(507\) 371.094 + 344.625i 0.731940 + 0.679733i
\(508\) 0 0
\(509\) −314.934 + 181.827i −0.618730 + 0.357224i −0.776374 0.630272i \(-0.782943\pi\)
0.157644 + 0.987496i \(0.449610\pi\)
\(510\) 0 0
\(511\) 349.038 604.551i 0.683049 1.18307i
\(512\) 0 0
\(513\) −577.050 + 721.667i −1.12485 + 1.40676i
\(514\) 0 0
\(515\) −1.09518 0.632301i −0.00212656 0.00122777i
\(516\) 0 0
\(517\) −73.3907 127.116i −0.141955 0.245873i
\(518\) 0 0
\(519\) 276.925 298.194i 0.533574 0.574556i
\(520\) 0 0
\(521\) 458.709i 0.880440i −0.897890 0.440220i \(-0.854901\pi\)
0.897890 0.440220i \(-0.145099\pi\)
\(522\) 0 0
\(523\) 458.289 0.876270 0.438135 0.898909i \(-0.355639\pi\)
0.438135 + 0.898909i \(0.355639\pi\)
\(524\) 0 0
\(525\) 649.251 199.959i 1.23667 0.380873i
\(526\) 0 0
\(527\) −780.113 + 450.398i −1.48029 + 0.854646i
\(528\) 0 0
\(529\) 335.219 580.616i 0.633683 1.09757i
\(530\) 0 0
\(531\) −88.5848 + 6.56114i −0.166826 + 0.0123562i
\(532\) 0 0
\(533\) 21.1274 + 12.1979i 0.0396387 + 0.0228854i
\(534\) 0 0
\(535\) −0.590289 1.02241i −0.00110334 0.00191105i
\(536\) 0 0
\(537\) 279.150 + 63.8418i 0.519833 + 0.118886i
\(538\) 0 0
\(539\) 125.651i 0.233118i
\(540\) 0 0
\(541\) −824.876 −1.52472 −0.762362 0.647150i \(-0.775961\pi\)
−0.762362 + 0.647150i \(0.775961\pi\)
\(542\) 0 0
\(543\) −39.4109 + 172.325i −0.0725800 + 0.317358i
\(544\) 0 0
\(545\) −6.91319 + 3.99133i −0.0126847 + 0.00732354i
\(546\) 0 0
\(547\) 16.8719 29.2230i 0.0308444 0.0534241i −0.850191 0.526474i \(-0.823514\pi\)
0.881036 + 0.473050i \(0.156847\pi\)
\(548\) 0 0
\(549\) 544.228 370.357i 0.991309 0.674602i
\(550\) 0 0
\(551\) −480.196 277.241i −0.871499 0.503160i
\(552\) 0 0
\(553\) 381.731 + 661.178i 0.690291 + 1.19562i
\(554\) 0 0
\(555\) −1.31377 4.26571i −0.00236715 0.00768596i
\(556\) 0 0
\(557\) 541.032i 0.971332i −0.874145 0.485666i \(-0.838577\pi\)
0.874145 0.485666i \(-0.161423\pi\)
\(558\) 0 0
\(559\) 3.42306 0.00612354
\(560\) 0 0
\(561\) 219.175 + 203.542i 0.390686 + 0.362820i
\(562\) 0 0
\(563\) 97.8909 56.5173i 0.173874 0.100386i −0.410537 0.911844i \(-0.634659\pi\)
0.584411 + 0.811458i \(0.301326\pi\)
\(564\) 0 0
\(565\) 0.955383 1.65477i 0.00169094 0.00292880i
\(566\) 0 0
\(567\) −725.762 + 108.102i −1.28000 + 0.190656i
\(568\) 0 0
\(569\) 236.524 + 136.557i 0.415684 + 0.239995i 0.693229 0.720717i \(-0.256187\pi\)
−0.277545 + 0.960713i \(0.589521\pi\)
\(570\) 0 0
\(571\) −122.654 212.443i −0.214806 0.372054i 0.738407 0.674356i \(-0.235579\pi\)
−0.953212 + 0.302301i \(0.902245\pi\)
\(572\) 0 0
\(573\) 388.604 418.451i 0.678193 0.730282i
\(574\) 0 0
\(575\) 865.733i 1.50562i
\(576\) 0 0
\(577\) 632.666 1.09648 0.548238 0.836323i \(-0.315299\pi\)
0.548238 + 0.836323i \(0.315299\pi\)
\(578\) 0 0
\(579\) −30.3709 + 9.35375i −0.0524541 + 0.0161550i
\(580\) 0 0
\(581\) 368.482 212.743i 0.634220 0.366167i
\(582\) 0 0
\(583\) −96.0099 + 166.294i −0.164682 + 0.285238i
\(584\) 0 0
\(585\) 0.111628 + 0.164034i 0.000190817 + 0.000280400i
\(586\) 0 0
\(587\) 980.129 + 565.878i 1.66973 + 0.964017i 0.967784 + 0.251782i \(0.0810167\pi\)
0.701942 + 0.712234i \(0.252317\pi\)
\(588\) 0 0
\(589\) −587.515 1017.61i −0.997478 1.72768i
\(590\) 0 0
\(591\) 631.322 + 144.384i 1.06823 + 0.244304i
\(592\) 0 0
\(593\) 180.213i 0.303900i −0.988388 0.151950i \(-0.951445\pi\)
0.988388 0.151950i \(-0.0485553\pi\)
\(594\) 0 0
\(595\) 12.0781 0.0202993
\(596\) 0 0
\(597\) −98.0965 + 428.930i −0.164316 + 0.718476i
\(598\) 0 0
\(599\) 566.086 326.830i 0.945052 0.545626i 0.0535119 0.998567i \(-0.482959\pi\)
0.891541 + 0.452941i \(0.149625\pi\)
\(600\) 0 0
\(601\) −178.947 + 309.945i −0.297749 + 0.515716i −0.975621 0.219464i \(-0.929569\pi\)
0.677872 + 0.735180i \(0.262902\pi\)
\(602\) 0 0
\(603\) −16.8583 227.611i −0.0279573 0.377464i
\(604\) 0 0
\(605\) 4.68975 + 2.70763i 0.00775164 + 0.00447541i
\(606\) 0 0
\(607\) −320.064 554.367i −0.527288 0.913290i −0.999494 0.0318015i \(-0.989876\pi\)
0.472206 0.881488i \(-0.343458\pi\)
\(608\) 0 0
\(609\) −129.605 420.817i −0.212816 0.690997i
\(610\) 0 0
\(611\) 16.7550i 0.0274223i
\(612\) 0 0
\(613\) 246.093 0.401457 0.200729 0.979647i \(-0.435669\pi\)
0.200729 + 0.979647i \(0.435669\pi\)
\(614\) 0 0
\(615\) −6.28256 5.83445i −0.0102155 0.00948690i
\(616\) 0 0
\(617\) 309.912 178.928i 0.502289 0.289997i −0.227369 0.973809i \(-0.573013\pi\)
0.729658 + 0.683812i \(0.239679\pi\)
\(618\) 0 0
\(619\) 40.5053 70.1573i 0.0654368 0.113340i −0.831451 0.555598i \(-0.812489\pi\)
0.896888 + 0.442258i \(0.145823\pi\)
\(620\) 0 0
\(621\) 140.572 924.462i 0.226364 1.48867i
\(622\) 0 0
\(623\) −851.888 491.838i −1.36740 0.789467i
\(624\) 0 0
\(625\) −312.403 541.098i −0.499845 0.865757i
\(626\) 0 0
\(627\) −265.507 + 285.899i −0.423456 + 0.455979i
\(628\) 0 0
\(629\) 768.075i 1.22110i
\(630\) 0 0
\(631\) 252.241 0.399748 0.199874 0.979822i \(-0.435947\pi\)
0.199874 + 0.979822i \(0.435947\pi\)
\(632\) 0 0
\(633\) −314.426 + 96.8382i −0.496724 + 0.152983i
\(634\) 0 0
\(635\) 2.13175 1.23077i 0.00335709 0.00193822i
\(636\) 0 0
\(637\) 7.17147 12.4214i 0.0112582 0.0194998i
\(638\) 0 0
\(639\) 382.696 792.913i 0.598898 1.24087i
\(640\) 0 0
\(641\) −843.278 486.867i −1.31557 0.759542i −0.332554 0.943084i \(-0.607910\pi\)
−0.983012 + 0.183542i \(0.941244\pi\)
\(642\) 0 0
\(643\) 341.530 + 591.547i 0.531151 + 0.919980i 0.999339 + 0.0363512i \(0.0115735\pi\)
−0.468188 + 0.883629i \(0.655093\pi\)
\(644\) 0 0
\(645\) −1.17275 0.268208i −0.00181822 0.000415827i
\(646\) 0 0
\(647\) 390.640i 0.603771i −0.953344 0.301885i \(-0.902384\pi\)
0.953344 0.301885i \(-0.0976160\pi\)
\(648\) 0 0
\(649\) −37.5081 −0.0577937
\(650\) 0 0
\(651\) 208.031 909.624i 0.319557 1.39727i
\(652\) 0 0
\(653\) 542.529 313.230i 0.830826 0.479678i −0.0233093 0.999728i \(-0.507420\pi\)
0.854135 + 0.520051i \(0.174087\pi\)
\(654\) 0 0
\(655\) 0.000805043 0.00139438i 1.22907e−6 2.12882e-6i
\(656\) 0 0
\(657\) 624.595 + 301.458i 0.950678 + 0.458840i
\(658\) 0 0
\(659\) −51.4518 29.7057i −0.0780755 0.0450769i 0.460454 0.887684i \(-0.347687\pi\)
−0.538530 + 0.842607i \(0.681020\pi\)
\(660\) 0 0
\(661\) 425.950 + 737.767i 0.644402 + 1.11614i 0.984439 + 0.175725i \(0.0562270\pi\)
−0.340037 + 0.940412i \(0.610440\pi\)
\(662\) 0 0
\(663\) −10.0497 32.6307i −0.0151580 0.0492168i
\(664\) 0 0
\(665\) 15.7551i 0.0236918i
\(666\) 0 0
\(667\) 561.132 0.841277
\(668\) 0 0
\(669\) −324.642 301.486i −0.485264 0.450652i
\(670\) 0 0
\(671\) 240.728 138.985i 0.358760 0.207130i
\(672\) 0 0
\(673\) 210.489 364.577i 0.312762 0.541720i −0.666197 0.745776i \(-0.732079\pi\)
0.978959 + 0.204056i \(0.0654124\pi\)
\(674\) 0 0
\(675\) 245.761 + 628.596i 0.364090 + 0.931253i
\(676\) 0 0
\(677\) 313.326 + 180.899i 0.462816 + 0.267207i 0.713227 0.700933i \(-0.247233\pi\)
−0.250412 + 0.968139i \(0.580566\pi\)
\(678\) 0 0
\(679\) −293.527 508.403i −0.432293 0.748753i
\(680\) 0 0
\(681\) −816.224 + 878.915i −1.19857 + 1.29062i
\(682\) 0 0
\(683\) 705.769i 1.03334i −0.856186 0.516668i \(-0.827172\pi\)
0.856186 0.516668i \(-0.172828\pi\)
\(684\) 0 0
\(685\) −0.0536179 −7.82743e−5
\(686\) 0 0
\(687\) −224.604 + 69.1743i −0.326934 + 0.100690i
\(688\) 0 0
\(689\) 18.9824 10.9595i 0.0275506 0.0159063i
\(690\) 0 0
\(691\) −345.384 + 598.222i −0.499832 + 0.865734i −1.00000 0.000194148i \(-0.999938\pi\)
0.500168 + 0.865928i \(0.333272\pi\)
\(692\) 0 0
\(693\) −308.993 + 22.8860i −0.445878 + 0.0330245i
\(694\) 0 0
\(695\) 3.99320 + 2.30547i 0.00574561 + 0.00331723i
\(696\) 0 0
\(697\) 737.703 + 1277.74i 1.05840 + 1.83320i
\(698\) 0 0
\(699\) 1032.33 + 236.095i 1.47687 + 0.337761i
\(700\) 0 0
\(701\) 1213.04i 1.73045i 0.501384 + 0.865225i \(0.332824\pi\)
−0.501384 + 0.865225i \(0.667176\pi\)
\(702\) 0 0
\(703\) −1001.90 −1.42518
\(704\) 0 0
\(705\) 1.31281 5.74032i 0.00186215 0.00814229i
\(706\) 0 0
\(707\) 1526.22 881.161i 2.15872 1.24634i
\(708\) 0 0
\(709\) 226.667 392.598i 0.319699 0.553735i −0.660726 0.750627i \(-0.729751\pi\)
0.980425 + 0.196892i \(0.0630848\pi\)
\(710\) 0 0
\(711\) −627.074 + 426.734i −0.881960 + 0.600189i
\(712\) 0 0
\(713\) 1029.81 + 594.560i 1.44433 + 0.833885i
\(714\) 0 0
\(715\) 0.0418908 + 0.0725570i 5.85885e−5 + 0.000101478i
\(716\) 0 0
\(717\) −285.035 925.486i −0.397538 1.29078i
\(718\) 0 0
\(719\) 418.833i 0.582522i 0.956644 + 0.291261i \(0.0940748\pi\)
−0.956644 + 0.291261i \(0.905925\pi\)
\(720\) 0 0
\(721\) −225.421 −0.312650
\(722\) 0 0
\(723\) −464.280 431.164i −0.642158 0.596355i
\(724\) 0 0
\(725\) −350.753 + 202.508i −0.483798 + 0.279321i
\(726\) 0 0
\(727\) −376.166 + 651.539i −0.517423 + 0.896203i 0.482372 + 0.875966i \(0.339775\pi\)
−0.999795 + 0.0202365i \(0.993558\pi\)
\(728\) 0 0
\(729\) −160.365 711.143i −0.219980 0.975504i
\(730\) 0 0
\(731\) 179.284 + 103.509i 0.245258 + 0.141600i
\(732\) 0 0
\(733\) 255.863 + 443.168i 0.349063 + 0.604595i 0.986083 0.166252i \(-0.0531665\pi\)
−0.637020 + 0.770847i \(0.719833\pi\)
\(734\) 0 0
\(735\) −3.43022 + 3.69368i −0.00466697 + 0.00502542i
\(736\) 0 0
\(737\) 96.3737i 0.130765i
\(738\) 0 0
\(739\) 466.830 0.631705 0.315853 0.948808i \(-0.397710\pi\)
0.315853 + 0.948808i \(0.397710\pi\)
\(740\) 0 0
\(741\) 42.5646 13.1092i 0.0574421 0.0176912i
\(742\) 0 0
\(743\) 546.320 315.418i 0.735290 0.424520i −0.0850643 0.996375i \(-0.527110\pi\)
0.820354 + 0.571856i \(0.193776\pi\)
\(744\) 0 0
\(745\) −0.443300 + 0.767818i −0.000595034 + 0.00103063i
\(746\) 0 0
\(747\) 237.824 + 349.475i 0.318372 + 0.467838i
\(748\) 0 0
\(749\) −182.249 105.222i −0.243323 0.140483i
\(750\) 0 0
\(751\) −90.7172 157.127i −0.120795 0.209223i 0.799286 0.600950i \(-0.205211\pi\)
−0.920081 + 0.391727i \(0.871878\pi\)
\(752\) 0 0
\(753\) −604.856 138.331i −0.803262 0.183706i
\(754\) 0 0
\(755\) 4.15416i 0.00550220i
\(756\) 0 0
\(757\) −1381.82 −1.82539 −0.912696 0.408640i \(-0.866003\pi\)
−0.912696 + 0.408640i \(0.866003\pi\)
\(758\) 0 0
\(759\) 88.0292 384.910i 0.115980 0.507128i
\(760\) 0 0
\(761\) −590.093 + 340.690i −0.775418 + 0.447688i −0.834804 0.550548i \(-0.814419\pi\)
0.0593862 + 0.998235i \(0.481086\pi\)
\(762\) 0 0
\(763\) −711.471 + 1232.30i −0.932466 + 1.61508i
\(764\) 0 0
\(765\) 0.886337 + 11.9668i 0.00115861 + 0.0156429i
\(766\) 0 0
\(767\) 3.70791 + 2.14076i 0.00483430 + 0.00279109i
\(768\) 0 0
\(769\) −45.1754 78.2461i −0.0587457 0.101750i 0.835157 0.550012i \(-0.185377\pi\)
−0.893903 + 0.448261i \(0.852043\pi\)
\(770\) 0 0
\(771\) 21.1502 + 68.6732i 0.0274322 + 0.0890703i
\(772\) 0 0
\(773\) 87.4025i 0.113069i 0.998401 + 0.0565346i \(0.0180051\pi\)
−0.998401 + 0.0565346i \(0.981995\pi\)
\(774\) 0 0
\(775\) −858.286 −1.10747
\(776\) 0 0
\(777\) −583.000 541.417i −0.750322 0.696804i
\(778\) 0 0
\(779\) −1666.72 + 962.284i −2.13957 + 1.23528i
\(780\) 0 0
\(781\) 185.885 321.963i 0.238010 0.412245i
\(782\) 0 0
\(783\) 407.429 159.292i 0.520344 0.203438i
\(784\) 0 0
\(785\) −8.49274 4.90329i −0.0108188 0.00624622i
\(786\) 0 0
\(787\) −366.731 635.196i −0.465986 0.807111i 0.533260 0.845951i \(-0.320967\pi\)
−0.999245 + 0.0388408i \(0.987633\pi\)
\(788\) 0 0
\(789\) −281.386 + 302.998i −0.356637 + 0.384028i
\(790\) 0 0
\(791\) 340.602i 0.430597i
\(792\) 0 0
\(793\) −31.7300 −0.0400126
\(794\) 0 0
\(795\) −7.36213 + 2.26741i −0.00926054 + 0.00285209i
\(796\) 0 0
\(797\) 853.500 492.768i 1.07089 0.618279i 0.142466 0.989800i \(-0.454497\pi\)
0.928425 + 0.371521i \(0.121164\pi\)
\(798\) 0 0
\(799\) −506.653 + 877.549i −0.634109 + 1.09831i
\(800\) 0 0
\(801\) 424.792 880.133i 0.530327 1.09879i
\(802\) 0 0
\(803\) 253.617 + 146.426i 0.315837 + 0.182349i
\(804\) 0 0
\(805\) −7.97200 13.8079i −0.00990310 0.0171527i
\(806\) 0 0
\(807\) −729.548 166.848i −0.904025 0.206751i
\(808\) 0 0
\(809\) 459.662i 0.568185i 0.958797 + 0.284093i \(0.0916923\pi\)
−0.958797 + 0.284093i \(0.908308\pi\)
\(810\) 0 0
\(811\) −912.030 −1.12457 −0.562287 0.826942i \(-0.690078\pi\)
−0.562287 + 0.826942i \(0.690078\pi\)
\(812\) 0 0
\(813\) 48.6041 212.523i 0.0597836 0.261406i
\(814\) 0 0
\(815\) −7.27951 + 4.20283i −0.00893191 + 0.00515684i
\(816\) 0 0
\(817\) −135.021 + 233.864i −0.165265 + 0.286247i
\(818\) 0 0
\(819\) 31.8521 + 15.3733i 0.0388915 + 0.0187708i
\(820\) 0 0
\(821\) 224.455 + 129.589i 0.273393 + 0.157843i 0.630428 0.776247i \(-0.282879\pi\)
−0.357036 + 0.934091i \(0.616213\pi\)
\(822\) 0 0
\(823\) 721.885 + 1250.34i 0.877139 + 1.51925i 0.854467 + 0.519506i \(0.173884\pi\)
0.0226714 + 0.999743i \(0.492783\pi\)
\(824\) 0 0
\(825\) 83.8853 + 272.370i 0.101679 + 0.330145i
\(826\) 0 0
\(827\) 1148.68i 1.38897i 0.719505 + 0.694487i \(0.244369\pi\)
−0.719505 + 0.694487i \(0.755631\pi\)
\(828\) 0 0
\(829\) −424.655 −0.512250 −0.256125 0.966644i \(-0.582446\pi\)
−0.256125 + 0.966644i \(0.582446\pi\)
\(830\) 0 0
\(831\) −800.264 743.183i −0.963013 0.894324i
\(832\) 0 0
\(833\) 751.216 433.715i 0.901820 0.520666i
\(834\) 0 0
\(835\) −6.32718 + 10.9590i −0.00757746 + 0.0131245i
\(836\) 0 0
\(837\) 916.510 + 139.363i 1.09499 + 0.166503i
\(838\) 0 0
\(839\) 52.2029 + 30.1394i 0.0622204 + 0.0359230i 0.530787 0.847505i \(-0.321896\pi\)
−0.468567 + 0.883428i \(0.655230\pi\)
\(840\) 0 0
\(841\) −289.243 500.984i −0.343928 0.595700i
\(842\) 0 0
\(843\) −568.389 + 612.045i −0.674246 + 0.726031i
\(844\) 0 0
\(845\) 8.57901i 0.0101527i
\(846\) 0 0
\(847\) 965.291 1.13966
\(848\) 0 0
\(849\) −78.9858 + 24.3263i −0.0930340 + 0.0286529i
\(850\) 0 0
\(851\) 878.078 506.959i 1.03182 0.595721i
\(852\) 0 0
\(853\) −602.705 + 1043.92i −0.706571 + 1.22382i 0.259550 + 0.965730i \(0.416426\pi\)
−0.966122 + 0.258087i \(0.916908\pi\)
\(854\) 0 0
\(855\) −15.6099 + 1.15617i −0.0182572 + 0.00135224i
\(856\) 0 0
\(857\) 174.400 + 100.690i 0.203500 + 0.117491i 0.598287 0.801282i \(-0.295848\pi\)
−0.394787 + 0.918773i \(0.629182\pi\)
\(858\) 0 0
\(859\) −706.859 1224.31i −0.822885 1.42528i −0.903525 0.428536i \(-0.859030\pi\)
0.0806394 0.996743i \(-0.474304\pi\)
\(860\) 0 0
\(861\) −1489.86 340.732i −1.73039 0.395740i
\(862\) 0 0
\(863\) 299.383i 0.346909i 0.984842 + 0.173455i \(0.0554930\pi\)
−0.984842 + 0.173455i \(0.944507\pi\)
\(864\) 0 0
\(865\) 6.89371 0.00796961
\(866\) 0 0
\(867\) 267.067 1167.76i 0.308035 1.34690i
\(868\) 0 0
\(869\) −277.373 + 160.141i −0.319186 + 0.184282i
\(870\) 0 0
\(871\) −5.50050 + 9.52714i −0.00631515 + 0.0109382i
\(872\) 0 0
\(873\) 482.179 328.131i 0.552325 0.375866i
\(874\) 0 0
\(875\) 19.9336 + 11.5087i 0.0227813 + 0.0131528i
\(876\) 0 0
\(877\) 247.122 + 428.029i 0.281782 + 0.488060i 0.971824 0.235709i \(-0.0757413\pi\)
−0.690042 + 0.723769i \(0.742408\pi\)
\(878\) 0 0
\(879\) 348.859 + 1132.72i 0.396882 + 1.28865i
\(880\) 0 0
\(881\) 211.533i 0.240105i −0.992768 0.120053i \(-0.961694\pi\)
0.992768 0.120053i \(-0.0383063\pi\)
\(882\) 0 0
\(883\) 410.167 0.464515 0.232258 0.972654i \(-0.425389\pi\)
0.232258 + 0.972654i \(0.425389\pi\)
\(884\) 0 0
\(885\) −1.10261 1.02396i −0.00124588 0.00115702i
\(886\) 0 0
\(887\) 734.748 424.207i 0.828352 0.478249i −0.0249363 0.999689i \(-0.507938\pi\)
0.853288 + 0.521440i \(0.174605\pi\)
\(888\) 0 0
\(889\) 219.390 379.994i 0.246782 0.427440i
\(890\) 0 0
\(891\) −45.3502 304.467i −0.0508981 0.341714i
\(892\) 0 0
\(893\) −1144.70 660.895i −1.28186 0.740084i
\(894\) 0 0
\(895\) 2.42545 + 4.20100i 0.00271000 + 0.00469385i
\(896\) 0 0
\(897\) −30.6709 + 33.0266i −0.0341927 + 0.0368189i
\(898\) 0 0
\(899\) 556.305i 0.618805i
\(900\) 0 0
\(901\) 1325.61 1.47126
\(902\) 0 0
\(903\) −204.945 + 63.1197i −0.226960 + 0.0699000i
\(904\) 0 0
\(905\) −2.59337 + 1.49728i −0.00286560 + 0.00165445i
\(906\) 0 0
\(907\) 577.488 1000.24i 0.636701 1.10280i −0.349451 0.936955i \(-0.613632\pi\)
0.986152 0.165844i \(-0.0530347\pi\)
\(908\) 0 0
\(909\) 985.043 + 1447.49i 1.08366 + 1.59240i
\(910\) 0 0
\(911\) −454.824 262.593i −0.499258 0.288247i 0.229149 0.973391i \(-0.426406\pi\)
−0.728407 + 0.685145i \(0.759739\pi\)
\(912\) 0 0
\(913\) 89.2486 + 154.583i 0.0977531 + 0.169313i
\(914\) 0 0
\(915\) 10.8708 + 2.48615i 0.0118806 + 0.00271711i
\(916\) 0 0
\(917\) 0.287005i 0.000312982i
\(918\) 0 0
\(919\) 1548.07 1.68452 0.842260 0.539072i \(-0.181225\pi\)
0.842260 + 0.539072i \(0.181225\pi\)
\(920\) 0 0
\(921\) −81.8937 + 358.083i −0.0889182 + 0.388798i
\(922\) 0 0
\(923\) −36.7519 + 21.2187i −0.0398179 + 0.0229889i
\(924\) 0 0
\(925\) −365.914 + 633.781i −0.395583 + 0.685169i
\(926\) 0 0
\(927\) −16.5422 223.344i −0.0178449 0.240932i
\(928\) 0 0
\(929\) −531.821 307.047i −0.572466 0.330513i 0.185668 0.982613i \(-0.440555\pi\)
−0.758134 + 0.652099i \(0.773889\pi\)
\(930\) 0 0
\(931\) 565.752 + 979.911i 0.607682 + 1.05254i
\(932\) 0 0
\(933\) 428.845 + 1392.43i 0.459641 + 1.49242i
\(934\) 0 0
\(935\) 5.06692i 0.00541917i
\(936\) 0 0
\(937\) −1763.19 −1.88174 −0.940868 0.338774i \(-0.889988\pi\)
−0.940868 + 0.338774i \(0.889988\pi\)
\(938\) 0 0
\(939\) 22.6613 + 21.0449i 0.0241334 + 0.0224120i
\(940\) 0 0
\(941\) 1201.52 693.697i 1.27685 0.737191i 0.300584 0.953755i \(-0.402818\pi\)
0.976268 + 0.216564i \(0.0694851\pi\)
\(942\) 0 0
\(943\) 973.824 1686.71i 1.03269 1.78867i
\(944\) 0 0
\(945\) −9.70808 7.76265i −0.0102731 0.00821445i
\(946\) 0 0
\(947\) −1073.21 619.615i −1.13327 0.654293i −0.188514 0.982071i \(-0.560367\pi\)
−0.944755 + 0.327778i \(0.893700\pi\)
\(948\) 0 0
\(949\) −16.7145 28.9503i −0.0176127 0.0305061i
\(950\) 0 0
\(951\) 341.523 367.754i 0.359120 0.386702i
\(952\) 0 0
\(953\) 460.952i 0.483686i −0.970315 0.241843i \(-0.922248\pi\)
0.970315 0.241843i \(-0.0777518\pi\)
\(954\) 0 0
\(955\) 9.67383 0.0101297
\(956\) 0 0
\(957\) 176.539 54.3710i 0.184471 0.0568140i
\(958\) 0 0
\(959\) −8.27714 + 4.77881i −0.00863101 + 0.00498312i
\(960\) 0 0
\(961\) −108.946 + 188.700i −0.113368 + 0.196358i
\(962\) 0 0
\(963\) 90.8780 188.292i 0.0943697 0.195526i
\(964\) 0 0
\(965\) −0.466208 0.269166i −0.000483117 0.000278928i
\(966\) 0 0
\(967\) 9.76447 + 16.9126i 0.0100977 + 0.0174897i 0.871030 0.491230i \(-0.163452\pi\)
−0.860932 + 0.508719i \(0.830119\pi\)
\(968\) 0 0
\(969\) 2625.74 + 600.508i 2.70974 + 0.619720i
\(970\) 0 0
\(971\) 509.199i 0.524406i −0.965013 0.262203i \(-0.915551\pi\)
0.965013 0.262203i \(-0.0844491\pi\)
\(972\) 0 0
\(973\) 821.921 0.844728
\(974\) 0 0
\(975\) 7.25282 31.7132i 0.00743879 0.0325263i
\(976\) 0 0
\(977\) 886.588 511.872i 0.907460 0.523922i 0.0278467 0.999612i \(-0.491135\pi\)
0.879613 + 0.475690i \(0.157802\pi\)
\(978\) 0 0
\(979\) 206.333 357.379i 0.210759 0.365045i
\(980\) 0 0
\(981\) −1273.16 614.485i −1.29782 0.626387i
\(982\) 0 0
\(983\) 1161.49 + 670.589i 1.18158 + 0.682187i 0.956380 0.292126i \(-0.0943625\pi\)
0.225202 + 0.974312i \(0.427696\pi\)
\(984\) 0 0
\(985\) 5.48536 + 9.50092i 0.00556889 + 0.00964560i
\(986\) 0 0
\(987\) −308.955 1003.16i −0.313025 1.01637i
\(988\) 0 0
\(989\) 273.281i 0.276320i
\(990\) 0 0
\(991\) −505.274 −0.509863 −0.254931 0.966959i \(-0.582053\pi\)
−0.254931 + 0.966959i \(0.582053\pi\)
\(992\) 0 0
\(993\) −229.685 213.302i −0.231304 0.214806i
\(994\) 0 0
\(995\) −6.45507 + 3.72683i −0.00648750 + 0.00374556i
\(996\) 0 0
\(997\) 878.908 1522.31i 0.881553 1.52689i 0.0319379 0.999490i \(-0.489832\pi\)
0.849615 0.527404i \(-0.176835\pi\)
\(998\) 0 0
\(999\) 493.646 617.360i 0.494140 0.617978i
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 144.3.q.e.65.3 8
3.2 odd 2 432.3.q.e.305.3 8
4.3 odd 2 72.3.m.b.65.2 yes 8
8.3 odd 2 576.3.q.i.65.3 8
8.5 even 2 576.3.q.j.65.2 8
9.2 odd 6 1296.3.e.i.161.5 8
9.4 even 3 432.3.q.e.17.3 8
9.5 odd 6 inner 144.3.q.e.113.3 8
9.7 even 3 1296.3.e.i.161.4 8
12.11 even 2 216.3.m.b.89.3 8
24.5 odd 2 1728.3.q.i.1601.2 8
24.11 even 2 1728.3.q.j.1601.2 8
36.7 odd 6 648.3.e.c.161.4 8
36.11 even 6 648.3.e.c.161.5 8
36.23 even 6 72.3.m.b.41.2 8
36.31 odd 6 216.3.m.b.17.3 8
72.5 odd 6 576.3.q.j.257.2 8
72.13 even 6 1728.3.q.i.449.2 8
72.59 even 6 576.3.q.i.257.3 8
72.67 odd 6 1728.3.q.j.449.2 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
72.3.m.b.41.2 8 36.23 even 6
72.3.m.b.65.2 yes 8 4.3 odd 2
144.3.q.e.65.3 8 1.1 even 1 trivial
144.3.q.e.113.3 8 9.5 odd 6 inner
216.3.m.b.17.3 8 36.31 odd 6
216.3.m.b.89.3 8 12.11 even 2
432.3.q.e.17.3 8 9.4 even 3
432.3.q.e.305.3 8 3.2 odd 2
576.3.q.i.65.3 8 8.3 odd 2
576.3.q.i.257.3 8 72.59 even 6
576.3.q.j.65.2 8 8.5 even 2
576.3.q.j.257.2 8 72.5 odd 6
648.3.e.c.161.4 8 36.7 odd 6
648.3.e.c.161.5 8 36.11 even 6
1296.3.e.i.161.4 8 9.7 even 3
1296.3.e.i.161.5 8 9.2 odd 6
1728.3.q.i.449.2 8 72.13 even 6
1728.3.q.i.1601.2 8 24.5 odd 2
1728.3.q.j.449.2 8 72.67 odd 6
1728.3.q.j.1601.2 8 24.11 even 2