Properties

Label 64.4.e.a.49.1
Level $64$
Weight $4$
Character 64.49
Analytic conductor $3.776$
Analytic rank $0$
Dimension $10$
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] = [64,4,Mod(17,64)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(64, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 3]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("64.17");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 64 = 2^{6} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 64.e (of order \(4\), 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: \(3.77612224037\)
Analytic rank: \(0\)
Dimension: \(10\)
Relative dimension: \(5\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 2x^{9} - x^{8} + 6x^{7} + 14x^{6} - 80x^{5} + 56x^{4} + 96x^{3} - 64x^{2} - 512x + 1024 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{20} \)
Twist minimal: no (minimal twist has level 16)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 49.1
Root \(1.28199 - 1.53509i\) of defining polynomial
Character \(\chi\) \(=\) 64.49
Dual form 64.4.e.a.17.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-5.49618 + 5.49618i) q^{3} +(-4.66372 - 4.66372i) q^{5} -24.8965i q^{7} -33.4160i q^{9} +O(q^{10})\) \(q+(-5.49618 + 5.49618i) q^{3} +(-4.66372 - 4.66372i) q^{5} -24.8965i q^{7} -33.4160i q^{9} +(-22.3431 - 22.3431i) q^{11} +(-11.2714 + 11.2714i) q^{13} +51.2653 q^{15} -88.4846 q^{17} +(-37.8187 + 37.8187i) q^{19} +(136.836 + 136.836i) q^{21} +48.1224i q^{23} -81.4994i q^{25} +(35.2635 + 35.2635i) q^{27} +(10.4432 - 10.4432i) q^{29} +96.9578 q^{31} +245.604 q^{33} +(-116.110 + 116.110i) q^{35} +(-163.279 - 163.279i) q^{37} -123.899i q^{39} +360.519i q^{41} +(100.249 + 100.249i) q^{43} +(-155.843 + 155.843i) q^{45} -220.669 q^{47} -276.837 q^{49} +(486.327 - 486.327i) q^{51} +(-175.752 - 175.752i) q^{53} +208.404i q^{55} -415.717i q^{57} +(-405.008 - 405.008i) q^{59} +(664.576 - 664.576i) q^{61} -831.942 q^{63} +105.133 q^{65} +(107.377 - 107.377i) q^{67} +(-264.489 - 264.489i) q^{69} +215.050i q^{71} +668.587i q^{73} +(447.935 + 447.935i) q^{75} +(-556.266 + 556.266i) q^{77} +822.956 q^{79} +514.603 q^{81} +(-326.873 + 326.873i) q^{83} +(412.668 + 412.668i) q^{85} +114.795i q^{87} +262.733i q^{89} +(280.619 + 280.619i) q^{91} +(-532.898 + 532.898i) q^{93} +352.752 q^{95} -150.801 q^{97} +(-746.618 + 746.618i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 2 q^{3} - 2 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + 2 q^{3} - 2 q^{5} - 18 q^{11} - 2 q^{13} + 124 q^{15} - 4 q^{17} + 26 q^{19} + 52 q^{21} - 184 q^{27} - 202 q^{29} - 368 q^{31} - 4 q^{33} - 476 q^{35} - 10 q^{37} + 838 q^{43} + 194 q^{45} + 944 q^{47} + 94 q^{49} + 1500 q^{51} - 378 q^{53} - 1706 q^{59} + 910 q^{61} - 2628 q^{63} - 492 q^{65} - 1942 q^{67} + 580 q^{69} + 2954 q^{75} - 268 q^{77} + 4416 q^{79} + 482 q^{81} + 2562 q^{83} - 12 q^{85} - 3332 q^{91} - 2192 q^{93} - 6900 q^{95} - 4 q^{97} - 4958 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/64\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(63\)
\(\chi(n)\) \(e\left(\frac{1}{4}\right)\) \(1\)

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
\(3\) −5.49618 + 5.49618i −1.05774 + 1.05774i −0.0595129 + 0.998228i \(0.518955\pi\)
−0.998228 + 0.0595129i \(0.981045\pi\)
\(4\) 0 0
\(5\) −4.66372 4.66372i −0.417136 0.417136i 0.467079 0.884215i \(-0.345306\pi\)
−0.884215 + 0.467079i \(0.845306\pi\)
\(6\) 0 0
\(7\) 24.8965i 1.34429i −0.740422 0.672143i \(-0.765374\pi\)
0.740422 0.672143i \(-0.234626\pi\)
\(8\) 0 0
\(9\) 33.4160i 1.23763i
\(10\) 0 0
\(11\) −22.3431 22.3431i −0.612427 0.612427i 0.331151 0.943578i \(-0.392563\pi\)
−0.943578 + 0.331151i \(0.892563\pi\)
\(12\) 0 0
\(13\) −11.2714 + 11.2714i −0.240471 + 0.240471i −0.817045 0.576574i \(-0.804389\pi\)
0.576574 + 0.817045i \(0.304389\pi\)
\(14\) 0 0
\(15\) 51.2653 0.882443
\(16\) 0 0
\(17\) −88.4846 −1.26239 −0.631196 0.775623i \(-0.717436\pi\)
−0.631196 + 0.775623i \(0.717436\pi\)
\(18\) 0 0
\(19\) −37.8187 + 37.8187i −0.456643 + 0.456643i −0.897552 0.440909i \(-0.854656\pi\)
0.440909 + 0.897552i \(0.354656\pi\)
\(20\) 0 0
\(21\) 136.836 + 136.836i 1.42191 + 1.42191i
\(22\) 0 0
\(23\) 48.1224i 0.436270i 0.975919 + 0.218135i \(0.0699973\pi\)
−0.975919 + 0.218135i \(0.930003\pi\)
\(24\) 0 0
\(25\) 81.4994i 0.651995i
\(26\) 0 0
\(27\) 35.2635 + 35.2635i 0.251351 + 0.251351i
\(28\) 0 0
\(29\) 10.4432 10.4432i 0.0668705 0.0668705i −0.672881 0.739751i \(-0.734943\pi\)
0.739751 + 0.672881i \(0.234943\pi\)
\(30\) 0 0
\(31\) 96.9578 0.561746 0.280873 0.959745i \(-0.409376\pi\)
0.280873 + 0.959745i \(0.409376\pi\)
\(32\) 0 0
\(33\) 245.604 1.29558
\(34\) 0 0
\(35\) −116.110 + 116.110i −0.560750 + 0.560750i
\(36\) 0 0
\(37\) −163.279 163.279i −0.725484 0.725484i 0.244233 0.969717i \(-0.421464\pi\)
−0.969717 + 0.244233i \(0.921464\pi\)
\(38\) 0 0
\(39\) 123.899i 0.508713i
\(40\) 0 0
\(41\) 360.519i 1.37326i 0.727008 + 0.686629i \(0.240910\pi\)
−0.727008 + 0.686629i \(0.759090\pi\)
\(42\) 0 0
\(43\) 100.249 + 100.249i 0.355531 + 0.355531i 0.862163 0.506632i \(-0.169110\pi\)
−0.506632 + 0.862163i \(0.669110\pi\)
\(44\) 0 0
\(45\) −155.843 + 155.843i −0.516260 + 0.516260i
\(46\) 0 0
\(47\) −220.669 −0.684849 −0.342425 0.939545i \(-0.611248\pi\)
−0.342425 + 0.939545i \(0.611248\pi\)
\(48\) 0 0
\(49\) −276.837 −0.807104
\(50\) 0 0
\(51\) 486.327 486.327i 1.33528 1.33528i
\(52\) 0 0
\(53\) −175.752 175.752i −0.455498 0.455498i 0.441676 0.897174i \(-0.354384\pi\)
−0.897174 + 0.441676i \(0.854384\pi\)
\(54\) 0 0
\(55\) 208.404i 0.510931i
\(56\) 0 0
\(57\) 415.717i 0.966019i
\(58\) 0 0
\(59\) −405.008 405.008i −0.893687 0.893687i 0.101181 0.994868i \(-0.467738\pi\)
−0.994868 + 0.101181i \(0.967738\pi\)
\(60\) 0 0
\(61\) 664.576 664.576i 1.39492 1.39492i 0.581061 0.813860i \(-0.302638\pi\)
0.813860 0.581061i \(-0.197362\pi\)
\(62\) 0 0
\(63\) −831.942 −1.66373
\(64\) 0 0
\(65\) 105.133 0.200619
\(66\) 0 0
\(67\) 107.377 107.377i 0.195794 0.195794i −0.602400 0.798194i \(-0.705789\pi\)
0.798194 + 0.602400i \(0.205789\pi\)
\(68\) 0 0
\(69\) −264.489 264.489i −0.461461 0.461461i
\(70\) 0 0
\(71\) 215.050i 0.359461i 0.983716 + 0.179731i \(0.0575226\pi\)
−0.983716 + 0.179731i \(0.942477\pi\)
\(72\) 0 0
\(73\) 668.587i 1.07195i 0.844235 + 0.535974i \(0.180055\pi\)
−0.844235 + 0.535974i \(0.819945\pi\)
\(74\) 0 0
\(75\) 447.935 + 447.935i 0.689642 + 0.689642i
\(76\) 0 0
\(77\) −556.266 + 556.266i −0.823277 + 0.823277i
\(78\) 0 0
\(79\) 822.956 1.17202 0.586012 0.810303i \(-0.300697\pi\)
0.586012 + 0.810303i \(0.300697\pi\)
\(80\) 0 0
\(81\) 514.603 0.705902
\(82\) 0 0
\(83\) −326.873 + 326.873i −0.432277 + 0.432277i −0.889402 0.457125i \(-0.848879\pi\)
0.457125 + 0.889402i \(0.348879\pi\)
\(84\) 0 0
\(85\) 412.668 + 412.668i 0.526589 + 0.526589i
\(86\) 0 0
\(87\) 114.795i 0.141463i
\(88\) 0 0
\(89\) 262.733i 0.312918i 0.987684 + 0.156459i \(0.0500079\pi\)
−0.987684 + 0.156459i \(0.949992\pi\)
\(90\) 0 0
\(91\) 280.619 + 280.619i 0.323262 + 0.323262i
\(92\) 0 0
\(93\) −532.898 + 532.898i −0.594182 + 0.594182i
\(94\) 0 0
\(95\) 352.752 0.380964
\(96\) 0 0
\(97\) −150.801 −0.157850 −0.0789251 0.996881i \(-0.525149\pi\)
−0.0789251 + 0.996881i \(0.525149\pi\)
\(98\) 0 0
\(99\) −746.618 + 746.618i −0.757958 + 0.757958i
\(100\) 0 0
\(101\) 487.985 + 487.985i 0.480755 + 0.480755i 0.905373 0.424617i \(-0.139591\pi\)
−0.424617 + 0.905373i \(0.639591\pi\)
\(102\) 0 0
\(103\) 1840.58i 1.76075i −0.474275 0.880377i \(-0.657290\pi\)
0.474275 0.880377i \(-0.342710\pi\)
\(104\) 0 0
\(105\) 1276.33i 1.18626i
\(106\) 0 0
\(107\) 79.4098 + 79.4098i 0.0717461 + 0.0717461i 0.742069 0.670323i \(-0.233845\pi\)
−0.670323 + 0.742069i \(0.733845\pi\)
\(108\) 0 0
\(109\) 952.979 952.979i 0.837421 0.837421i −0.151098 0.988519i \(-0.548281\pi\)
0.988519 + 0.151098i \(0.0482809\pi\)
\(110\) 0 0
\(111\) 1794.82 1.53475
\(112\) 0 0
\(113\) −720.469 −0.599788 −0.299894 0.953973i \(-0.596951\pi\)
−0.299894 + 0.953973i \(0.596951\pi\)
\(114\) 0 0
\(115\) 224.430 224.430i 0.181984 0.181984i
\(116\) 0 0
\(117\) 376.646 + 376.646i 0.297615 + 0.297615i
\(118\) 0 0
\(119\) 2202.96i 1.69702i
\(120\) 0 0
\(121\) 332.571i 0.249865i
\(122\) 0 0
\(123\) −1981.48 1981.48i −1.45255 1.45255i
\(124\) 0 0
\(125\) −963.056 + 963.056i −0.689107 + 0.689107i
\(126\) 0 0
\(127\) −2622.35 −1.83225 −0.916124 0.400895i \(-0.868699\pi\)
−0.916124 + 0.400895i \(0.868699\pi\)
\(128\) 0 0
\(129\) −1101.97 −0.752119
\(130\) 0 0
\(131\) −657.574 + 657.574i −0.438569 + 0.438569i −0.891530 0.452961i \(-0.850368\pi\)
0.452961 + 0.891530i \(0.350368\pi\)
\(132\) 0 0
\(133\) 941.555 + 941.555i 0.613858 + 0.613858i
\(134\) 0 0
\(135\) 328.919i 0.209695i
\(136\) 0 0
\(137\) 2511.52i 1.56623i −0.621874 0.783117i \(-0.713629\pi\)
0.621874 0.783117i \(-0.286371\pi\)
\(138\) 0 0
\(139\) 1086.02 + 1086.02i 0.662697 + 0.662697i 0.956015 0.293318i \(-0.0947595\pi\)
−0.293318 + 0.956015i \(0.594759\pi\)
\(140\) 0 0
\(141\) 1212.84 1212.84i 0.724393 0.724393i
\(142\) 0 0
\(143\) 503.677 0.294543
\(144\) 0 0
\(145\) −97.4080 −0.0557882
\(146\) 0 0
\(147\) 1521.54 1521.54i 0.853706 0.853706i
\(148\) 0 0
\(149\) −2284.63 2284.63i −1.25614 1.25614i −0.952922 0.303214i \(-0.901940\pi\)
−0.303214 0.952922i \(-0.598060\pi\)
\(150\) 0 0
\(151\) 2814.39i 1.51677i −0.651809 0.758383i \(-0.725990\pi\)
0.651809 0.758383i \(-0.274010\pi\)
\(152\) 0 0
\(153\) 2956.80i 1.56237i
\(154\) 0 0
\(155\) −452.184 452.184i −0.234325 0.234325i
\(156\) 0 0
\(157\) −906.308 + 906.308i −0.460709 + 0.460709i −0.898888 0.438179i \(-0.855624\pi\)
0.438179 + 0.898888i \(0.355624\pi\)
\(158\) 0 0
\(159\) 1931.93 0.963598
\(160\) 0 0
\(161\) 1198.08 0.586472
\(162\) 0 0
\(163\) −1392.36 + 1392.36i −0.669067 + 0.669067i −0.957500 0.288433i \(-0.906866\pi\)
0.288433 + 0.957500i \(0.406866\pi\)
\(164\) 0 0
\(165\) −1145.43 1145.43i −0.540433 0.540433i
\(166\) 0 0
\(167\) 1221.66i 0.566075i −0.959109 0.283038i \(-0.908658\pi\)
0.959109 0.283038i \(-0.0913421\pi\)
\(168\) 0 0
\(169\) 1942.91i 0.884347i
\(170\) 0 0
\(171\) 1263.75 + 1263.75i 0.565155 + 0.565155i
\(172\) 0 0
\(173\) −563.418 + 563.418i −0.247606 + 0.247606i −0.819988 0.572381i \(-0.806020\pi\)
0.572381 + 0.819988i \(0.306020\pi\)
\(174\) 0 0
\(175\) −2029.05 −0.876468
\(176\) 0 0
\(177\) 4451.99 1.89058
\(178\) 0 0
\(179\) 2202.23 2202.23i 0.919565 0.919565i −0.0774329 0.996998i \(-0.524672\pi\)
0.996998 + 0.0774329i \(0.0246724\pi\)
\(180\) 0 0
\(181\) 121.294 + 121.294i 0.0498104 + 0.0498104i 0.731573 0.681763i \(-0.238786\pi\)
−0.681763 + 0.731573i \(0.738786\pi\)
\(182\) 0 0
\(183\) 7305.26i 2.95093i
\(184\) 0 0
\(185\) 1522.98i 0.605251i
\(186\) 0 0
\(187\) 1977.02 + 1977.02i 0.773124 + 0.773124i
\(188\) 0 0
\(189\) 877.939 877.939i 0.337887 0.337887i
\(190\) 0 0
\(191\) −3927.65 −1.48793 −0.743966 0.668218i \(-0.767058\pi\)
−0.743966 + 0.668218i \(0.767058\pi\)
\(192\) 0 0
\(193\) −3249.02 −1.21176 −0.605880 0.795556i \(-0.707179\pi\)
−0.605880 + 0.795556i \(0.707179\pi\)
\(194\) 0 0
\(195\) −577.833 + 577.833i −0.212202 + 0.212202i
\(196\) 0 0
\(197\) 2420.90 + 2420.90i 0.875545 + 0.875545i 0.993070 0.117525i \(-0.0374961\pi\)
−0.117525 + 0.993070i \(0.537496\pi\)
\(198\) 0 0
\(199\) 1371.30i 0.488488i 0.969714 + 0.244244i \(0.0785397\pi\)
−0.969714 + 0.244244i \(0.921460\pi\)
\(200\) 0 0
\(201\) 1180.33i 0.414198i
\(202\) 0 0
\(203\) −259.998 259.998i −0.0898931 0.0898931i
\(204\) 0 0
\(205\) 1681.36 1681.36i 0.572836 0.572836i
\(206\) 0 0
\(207\) 1608.06 0.539941
\(208\) 0 0
\(209\) 1689.98 0.559321
\(210\) 0 0
\(211\) 1620.50 1620.50i 0.528719 0.528719i −0.391471 0.920190i \(-0.628034\pi\)
0.920190 + 0.391471i \(0.128034\pi\)
\(212\) 0 0
\(213\) −1181.95 1181.95i −0.380217 0.380217i
\(214\) 0 0
\(215\) 935.068i 0.296610i
\(216\) 0 0
\(217\) 2413.91i 0.755148i
\(218\) 0 0
\(219\) −3674.67 3674.67i −1.13384 1.13384i
\(220\) 0 0
\(221\) 997.347 997.347i 0.303569 0.303569i
\(222\) 0 0
\(223\) 419.617 0.126007 0.0630036 0.998013i \(-0.479932\pi\)
0.0630036 + 0.998013i \(0.479932\pi\)
\(224\) 0 0
\(225\) −2723.38 −0.806929
\(226\) 0 0
\(227\) 2133.64 2133.64i 0.623853 0.623853i −0.322661 0.946515i \(-0.604577\pi\)
0.946515 + 0.322661i \(0.104577\pi\)
\(228\) 0 0
\(229\) −1574.42 1574.42i −0.454325 0.454325i 0.442462 0.896787i \(-0.354105\pi\)
−0.896787 + 0.442462i \(0.854105\pi\)
\(230\) 0 0
\(231\) 6114.67i 1.74163i
\(232\) 0 0
\(233\) 1194.86i 0.335957i −0.985791 0.167978i \(-0.946276\pi\)
0.985791 0.167978i \(-0.0537239\pi\)
\(234\) 0 0
\(235\) 1029.14 + 1029.14i 0.285675 + 0.285675i
\(236\) 0 0
\(237\) −4523.12 + 4523.12i −1.23970 + 1.23970i
\(238\) 0 0
\(239\) 4241.03 1.14782 0.573911 0.818917i \(-0.305425\pi\)
0.573911 + 0.818917i \(0.305425\pi\)
\(240\) 0 0
\(241\) −5571.19 −1.48910 −0.744548 0.667569i \(-0.767335\pi\)
−0.744548 + 0.667569i \(0.767335\pi\)
\(242\) 0 0
\(243\) −3780.46 + 3780.46i −0.998012 + 0.998012i
\(244\) 0 0
\(245\) 1291.09 + 1291.09i 0.336672 + 0.336672i
\(246\) 0 0
\(247\) 852.541i 0.219619i
\(248\) 0 0
\(249\) 3593.11i 0.914474i
\(250\) 0 0
\(251\) 482.728 + 482.728i 0.121393 + 0.121393i 0.765193 0.643801i \(-0.222643\pi\)
−0.643801 + 0.765193i \(0.722643\pi\)
\(252\) 0 0
\(253\) 1075.20 1075.20i 0.267184 0.267184i
\(254\) 0 0
\(255\) −4536.19 −1.11399
\(256\) 0 0
\(257\) 8093.12 1.96434 0.982169 0.188002i \(-0.0602013\pi\)
0.982169 + 0.188002i \(0.0602013\pi\)
\(258\) 0 0
\(259\) −4065.08 + 4065.08i −0.975257 + 0.975257i
\(260\) 0 0
\(261\) −348.969 348.969i −0.0827610 0.0827610i
\(262\) 0 0
\(263\) 410.300i 0.0961984i −0.998843 0.0480992i \(-0.984684\pi\)
0.998843 0.0480992i \(-0.0153164\pi\)
\(264\) 0 0
\(265\) 1639.32i 0.380009i
\(266\) 0 0
\(267\) −1444.03 1444.03i −0.330986 0.330986i
\(268\) 0 0
\(269\) −4.77962 + 4.77962i −0.00108334 + 0.00108334i −0.707648 0.706565i \(-0.750244\pi\)
0.706565 + 0.707648i \(0.250244\pi\)
\(270\) 0 0
\(271\) 2833.98 0.635247 0.317623 0.948217i \(-0.397115\pi\)
0.317623 + 0.948217i \(0.397115\pi\)
\(272\) 0 0
\(273\) −3084.67 −0.683855
\(274\) 0 0
\(275\) −1820.95 + 1820.95i −0.399300 + 0.399300i
\(276\) 0 0
\(277\) 1525.80 + 1525.80i 0.330962 + 0.330962i 0.852952 0.521989i \(-0.174810\pi\)
−0.521989 + 0.852952i \(0.674810\pi\)
\(278\) 0 0
\(279\) 3239.94i 0.695234i
\(280\) 0 0
\(281\) 4750.23i 1.00845i −0.863572 0.504226i \(-0.831778\pi\)
0.863572 0.504226i \(-0.168222\pi\)
\(282\) 0 0
\(283\) −644.104 644.104i −0.135293 0.135293i 0.636217 0.771510i \(-0.280498\pi\)
−0.771510 + 0.636217i \(0.780498\pi\)
\(284\) 0 0
\(285\) −1938.79 + 1938.79i −0.402961 + 0.402961i
\(286\) 0 0
\(287\) 8975.67 1.84605
\(288\) 0 0
\(289\) 2916.53 0.593635
\(290\) 0 0
\(291\) 828.827 828.827i 0.166965 0.166965i
\(292\) 0 0
\(293\) −1433.16 1433.16i −0.285755 0.285755i 0.549644 0.835399i \(-0.314763\pi\)
−0.835399 + 0.549644i \(0.814763\pi\)
\(294\) 0 0
\(295\) 3777.69i 0.745578i
\(296\) 0 0
\(297\) 1575.79i 0.307868i
\(298\) 0 0
\(299\) −542.407 542.407i −0.104910 0.104910i
\(300\) 0 0
\(301\) 2495.85 2495.85i 0.477935 0.477935i
\(302\) 0 0
\(303\) −5364.10 −1.01703
\(304\) 0 0
\(305\) −6198.79 −1.16374
\(306\) 0 0
\(307\) 231.211 231.211i 0.0429834 0.0429834i −0.685288 0.728272i \(-0.740324\pi\)
0.728272 + 0.685288i \(0.240324\pi\)
\(308\) 0 0
\(309\) 10116.2 + 10116.2i 1.86242 + 1.86242i
\(310\) 0 0
\(311\) 871.410i 0.158885i 0.996839 + 0.0794423i \(0.0253140\pi\)
−0.996839 + 0.0794423i \(0.974686\pi\)
\(312\) 0 0
\(313\) 3515.02i 0.634762i 0.948298 + 0.317381i \(0.102803\pi\)
−0.948298 + 0.317381i \(0.897197\pi\)
\(314\) 0 0
\(315\) 3879.95 + 3879.95i 0.694001 + 0.694001i
\(316\) 0 0
\(317\) 4723.77 4723.77i 0.836951 0.836951i −0.151506 0.988456i \(-0.548412\pi\)
0.988456 + 0.151506i \(0.0484121\pi\)
\(318\) 0 0
\(319\) −466.665 −0.0819067
\(320\) 0 0
\(321\) −872.901 −0.151778
\(322\) 0 0
\(323\) 3346.38 3346.38i 0.576462 0.576462i
\(324\) 0 0
\(325\) 918.613 + 918.613i 0.156786 + 0.156786i
\(326\) 0 0
\(327\) 10475.5i 1.77155i
\(328\) 0 0
\(329\) 5493.90i 0.920633i
\(330\) 0 0
\(331\) 1820.26 + 1820.26i 0.302268 + 0.302268i 0.841901 0.539633i \(-0.181437\pi\)
−0.539633 + 0.841901i \(0.681437\pi\)
\(332\) 0 0
\(333\) −5456.13 + 5456.13i −0.897880 + 0.897880i
\(334\) 0 0
\(335\) −1001.55 −0.163345
\(336\) 0 0
\(337\) 74.0970 0.0119772 0.00598861 0.999982i \(-0.498094\pi\)
0.00598861 + 0.999982i \(0.498094\pi\)
\(338\) 0 0
\(339\) 3959.83 3959.83i 0.634420 0.634420i
\(340\) 0 0
\(341\) −2166.34 2166.34i −0.344029 0.344029i
\(342\) 0 0
\(343\) 1647.24i 0.259307i
\(344\) 0 0
\(345\) 2467.01i 0.384984i
\(346\) 0 0
\(347\) −2102.73 2102.73i −0.325305 0.325305i 0.525493 0.850798i \(-0.323881\pi\)
−0.850798 + 0.525493i \(0.823881\pi\)
\(348\) 0 0
\(349\) −6612.85 + 6612.85i −1.01426 + 1.01426i −0.0143661 + 0.999897i \(0.504573\pi\)
−0.999897 + 0.0143661i \(0.995427\pi\)
\(350\) 0 0
\(351\) −794.940 −0.120885
\(352\) 0 0
\(353\) 2216.90 0.334259 0.167130 0.985935i \(-0.446550\pi\)
0.167130 + 0.985935i \(0.446550\pi\)
\(354\) 0 0
\(355\) 1002.93 1002.93i 0.149944 0.149944i
\(356\) 0 0
\(357\) −12107.9 12107.9i −1.79500 1.79500i
\(358\) 0 0
\(359\) 2082.23i 0.306117i −0.988217 0.153059i \(-0.951088\pi\)
0.988217 0.153059i \(-0.0489123\pi\)
\(360\) 0 0
\(361\) 3998.49i 0.582955i
\(362\) 0 0
\(363\) 1827.87 + 1827.87i 0.264293 + 0.264293i
\(364\) 0 0
\(365\) 3118.10 3118.10i 0.447148 0.447148i
\(366\) 0 0
\(367\) −4509.22 −0.641360 −0.320680 0.947188i \(-0.603911\pi\)
−0.320680 + 0.947188i \(0.603911\pi\)
\(368\) 0 0
\(369\) 12047.1 1.69959
\(370\) 0 0
\(371\) −4375.61 + 4375.61i −0.612320 + 0.612320i
\(372\) 0 0
\(373\) −8661.56 8661.56i −1.20236 1.20236i −0.973448 0.228908i \(-0.926485\pi\)
−0.228908 0.973448i \(-0.573515\pi\)
\(374\) 0 0
\(375\) 10586.3i 1.45779i
\(376\) 0 0
\(377\) 235.418i 0.0321609i
\(378\) 0 0
\(379\) −3522.46 3522.46i −0.477405 0.477405i 0.426896 0.904301i \(-0.359607\pi\)
−0.904301 + 0.426896i \(0.859607\pi\)
\(380\) 0 0
\(381\) 14412.9 14412.9i 1.93804 1.93804i
\(382\) 0 0
\(383\) −3044.88 −0.406229 −0.203115 0.979155i \(-0.565106\pi\)
−0.203115 + 0.979155i \(0.565106\pi\)
\(384\) 0 0
\(385\) 5188.54 0.686837
\(386\) 0 0
\(387\) 3349.92 3349.92i 0.440016 0.440016i
\(388\) 0 0
\(389\) 1932.73 + 1932.73i 0.251911 + 0.251911i 0.821754 0.569843i \(-0.192996\pi\)
−0.569843 + 0.821754i \(0.692996\pi\)
\(390\) 0 0
\(391\) 4258.09i 0.550744i
\(392\) 0 0
\(393\) 7228.29i 0.927784i
\(394\) 0 0
\(395\) −3838.04 3838.04i −0.488893 0.488893i
\(396\) 0 0
\(397\) −672.457 + 672.457i −0.0850117 + 0.0850117i −0.748334 0.663322i \(-0.769146\pi\)
0.663322 + 0.748334i \(0.269146\pi\)
\(398\) 0 0
\(399\) −10349.9 −1.29861
\(400\) 0 0
\(401\) −7606.74 −0.947288 −0.473644 0.880716i \(-0.657062\pi\)
−0.473644 + 0.880716i \(0.657062\pi\)
\(402\) 0 0
\(403\) −1092.85 + 1092.85i −0.135084 + 0.135084i
\(404\) 0 0
\(405\) −2399.96 2399.96i −0.294457 0.294457i
\(406\) 0 0
\(407\) 7296.32i 0.888612i
\(408\) 0 0
\(409\) 4981.58i 0.602257i −0.953584 0.301129i \(-0.902637\pi\)
0.953584 0.301129i \(-0.0973634\pi\)
\(410\) 0 0
\(411\) 13803.8 + 13803.8i 1.65667 + 1.65667i
\(412\) 0 0
\(413\) −10083.3 + 10083.3i −1.20137 + 1.20137i
\(414\) 0 0
\(415\) 3048.89 0.360637
\(416\) 0 0
\(417\) −11937.9 −1.40192
\(418\) 0 0
\(419\) 3433.38 3433.38i 0.400314 0.400314i −0.478030 0.878344i \(-0.658649\pi\)
0.878344 + 0.478030i \(0.158649\pi\)
\(420\) 0 0
\(421\) 7973.72 + 7973.72i 0.923077 + 0.923077i 0.997246 0.0741686i \(-0.0236303\pi\)
−0.0741686 + 0.997246i \(0.523630\pi\)
\(422\) 0 0
\(423\) 7373.88i 0.847590i
\(424\) 0 0
\(425\) 7211.44i 0.823074i
\(426\) 0 0
\(427\) −16545.6 16545.6i −1.87517 1.87517i
\(428\) 0 0
\(429\) −2768.30 + 2768.30i −0.311550 + 0.311550i
\(430\) 0 0
\(431\) −4800.16 −0.536463 −0.268232 0.963354i \(-0.586439\pi\)
−0.268232 + 0.963354i \(0.586439\pi\)
\(432\) 0 0
\(433\) 6242.32 0.692810 0.346405 0.938085i \(-0.387402\pi\)
0.346405 + 0.938085i \(0.387402\pi\)
\(434\) 0 0
\(435\) 535.372 535.372i 0.0590095 0.0590095i
\(436\) 0 0
\(437\) −1819.93 1819.93i −0.199220 0.199220i
\(438\) 0 0
\(439\) 4929.27i 0.535903i 0.963432 + 0.267951i \(0.0863466\pi\)
−0.963432 + 0.267951i \(0.913653\pi\)
\(440\) 0 0
\(441\) 9250.77i 0.998896i
\(442\) 0 0
\(443\) 7670.67 + 7670.67i 0.822674 + 0.822674i 0.986491 0.163817i \(-0.0523807\pi\)
−0.163817 + 0.986491i \(0.552381\pi\)
\(444\) 0 0
\(445\) 1225.32 1225.32i 0.130529 0.130529i
\(446\) 0 0
\(447\) 25113.5 2.65733
\(448\) 0 0
\(449\) −11515.2 −1.21032 −0.605162 0.796102i \(-0.706892\pi\)
−0.605162 + 0.796102i \(0.706892\pi\)
\(450\) 0 0
\(451\) 8055.12 8055.12i 0.841021 0.841021i
\(452\) 0 0
\(453\) 15468.4 + 15468.4i 1.60434 + 1.60434i
\(454\) 0 0
\(455\) 2617.46i 0.269689i
\(456\) 0 0
\(457\) 4829.89i 0.494383i −0.968967 0.247191i \(-0.920492\pi\)
0.968967 0.247191i \(-0.0795076\pi\)
\(458\) 0 0
\(459\) −3120.28 3120.28i −0.317303 0.317303i
\(460\) 0 0
\(461\) 8265.79 8265.79i 0.835090 0.835090i −0.153118 0.988208i \(-0.548931\pi\)
0.988208 + 0.153118i \(0.0489315\pi\)
\(462\) 0 0
\(463\) 5043.86 0.506281 0.253141 0.967430i \(-0.418536\pi\)
0.253141 + 0.967430i \(0.418536\pi\)
\(464\) 0 0
\(465\) 4970.57 0.495709
\(466\) 0 0
\(467\) −12438.8 + 12438.8i −1.23255 + 1.23255i −0.269564 + 0.962982i \(0.586880\pi\)
−0.962982 + 0.269564i \(0.913120\pi\)
\(468\) 0 0
\(469\) −2673.31 2673.31i −0.263203 0.263203i
\(470\) 0 0
\(471\) 9962.47i 0.974621i
\(472\) 0 0
\(473\) 4479.75i 0.435474i
\(474\) 0 0
\(475\) 3082.20 + 3082.20i 0.297729 + 0.297729i
\(476\) 0 0
\(477\) −5872.93 + 5872.93i −0.563738 + 0.563738i
\(478\) 0 0
\(479\) −13059.7 −1.24575 −0.622875 0.782321i \(-0.714036\pi\)
−0.622875 + 0.782321i \(0.714036\pi\)
\(480\) 0 0
\(481\) 3680.77 0.348916
\(482\) 0 0
\(483\) −6584.87 + 6584.87i −0.620335 + 0.620335i
\(484\) 0 0
\(485\) 703.292 + 703.292i 0.0658450 + 0.0658450i
\(486\) 0 0
\(487\) 15549.3i 1.44683i 0.690414 + 0.723414i \(0.257428\pi\)
−0.690414 + 0.723414i \(0.742572\pi\)
\(488\) 0 0
\(489\) 15305.3i 1.41540i
\(490\) 0 0
\(491\) 8628.34 + 8628.34i 0.793058 + 0.793058i 0.981990 0.188932i \(-0.0605027\pi\)
−0.188932 + 0.981990i \(0.560503\pi\)
\(492\) 0 0
\(493\) −924.059 + 924.059i −0.0844169 + 0.0844169i
\(494\) 0 0
\(495\) 6964.03 0.632344
\(496\) 0 0
\(497\) 5354.00 0.483219
\(498\) 0 0
\(499\) −1732.42 + 1732.42i −0.155418 + 0.155418i −0.780533 0.625115i \(-0.785052\pi\)
0.625115 + 0.780533i \(0.285052\pi\)
\(500\) 0 0
\(501\) 6714.44 + 6714.44i 0.598761 + 0.598761i
\(502\) 0 0
\(503\) 16579.6i 1.46968i −0.678241 0.734839i \(-0.737258\pi\)
0.678241 0.734839i \(-0.262742\pi\)
\(504\) 0 0
\(505\) 4551.65i 0.401081i
\(506\) 0 0
\(507\) −10678.6 10678.6i −0.935410 0.935410i
\(508\) 0 0
\(509\) −7830.92 + 7830.92i −0.681924 + 0.681924i −0.960434 0.278509i \(-0.910160\pi\)
0.278509 + 0.960434i \(0.410160\pi\)
\(510\) 0 0
\(511\) 16645.5 1.44100
\(512\) 0 0
\(513\) −2667.24 −0.229555
\(514\) 0 0
\(515\) −8583.95 + 8583.95i −0.734474 + 0.734474i
\(516\) 0 0
\(517\) 4930.44 + 4930.44i 0.419420 + 0.419420i
\(518\) 0 0
\(519\) 6193.30i 0.523807i
\(520\) 0 0
\(521\) 3400.02i 0.285907i 0.989729 + 0.142953i \(0.0456599\pi\)
−0.989729 + 0.142953i \(0.954340\pi\)
\(522\) 0 0
\(523\) 2019.50 + 2019.50i 0.168847 + 0.168847i 0.786472 0.617626i \(-0.211905\pi\)
−0.617626 + 0.786472i \(0.711905\pi\)
\(524\) 0 0
\(525\) 11152.0 11152.0i 0.927075 0.927075i
\(526\) 0 0
\(527\) −8579.28 −0.709144
\(528\) 0 0
\(529\) 9851.23 0.809668
\(530\) 0 0
\(531\) −13533.7 + 13533.7i −1.10605 + 1.10605i
\(532\) 0 0
\(533\) −4063.56 4063.56i −0.330229 0.330229i
\(534\) 0 0
\(535\) 740.691i 0.0598558i
\(536\) 0 0
\(537\) 24207.7i 1.94532i
\(538\) 0 0
\(539\) 6185.39 + 6185.39i 0.494293 + 0.494293i
\(540\) 0 0
\(541\) −13432.6 + 13432.6i −1.06749 + 1.06749i −0.0699388 + 0.997551i \(0.522280\pi\)
−0.997551 + 0.0699388i \(0.977720\pi\)
\(542\) 0 0
\(543\) −1333.30 −0.105373
\(544\) 0 0
\(545\) −8888.86 −0.698637
\(546\) 0 0
\(547\) 5376.46 5376.46i 0.420257 0.420257i −0.465035 0.885292i \(-0.653958\pi\)
0.885292 + 0.465035i \(0.153958\pi\)
\(548\) 0 0
\(549\) −22207.5 22207.5i −1.72640 1.72640i
\(550\) 0 0
\(551\) 789.894i 0.0610719i
\(552\) 0 0
\(553\) 20488.7i 1.57553i
\(554\) 0 0
\(555\) −8370.55 8370.55i −0.640198 0.640198i
\(556\) 0 0
\(557\) −2076.28 + 2076.28i −0.157944 + 0.157944i −0.781655 0.623711i \(-0.785624\pi\)
0.623711 + 0.781655i \(0.285624\pi\)
\(558\) 0 0
\(559\) −2259.90 −0.170990
\(560\) 0 0
\(561\) −21732.1 −1.63553
\(562\) 0 0
\(563\) 16643.1 16643.1i 1.24587 1.24587i 0.288339 0.957528i \(-0.406897\pi\)
0.957528 0.288339i \(-0.0931031\pi\)
\(564\) 0 0
\(565\) 3360.07 + 3360.07i 0.250193 + 0.250193i
\(566\) 0 0
\(567\) 12811.8i 0.948934i
\(568\) 0 0
\(569\) 5659.60i 0.416982i −0.978024 0.208491i \(-0.933145\pi\)
0.978024 0.208491i \(-0.0668551\pi\)
\(570\) 0 0
\(571\) −4872.67 4872.67i −0.357119 0.357119i 0.505631 0.862750i \(-0.331260\pi\)
−0.862750 + 0.505631i \(0.831260\pi\)
\(572\) 0 0
\(573\) 21587.1 21587.1i 1.57385 1.57385i
\(574\) 0 0
\(575\) 3921.95 0.284446
\(576\) 0 0
\(577\) 10652.2 0.768556 0.384278 0.923217i \(-0.374450\pi\)
0.384278 + 0.923217i \(0.374450\pi\)
\(578\) 0 0
\(579\) 17857.2 17857.2i 1.28173 1.28173i
\(580\) 0 0
\(581\) 8138.01 + 8138.01i 0.581104 + 0.581104i
\(582\) 0 0
\(583\) 7853.69i 0.557919i
\(584\) 0 0
\(585\) 3513.14i 0.248291i
\(586\) 0 0
\(587\) −9903.95 9903.95i −0.696388 0.696388i 0.267242 0.963630i \(-0.413888\pi\)
−0.963630 + 0.267242i \(0.913888\pi\)
\(588\) 0 0
\(589\) −3666.82 + 3666.82i −0.256517 + 0.256517i
\(590\) 0 0
\(591\) −26611.5 −1.85220
\(592\) 0 0
\(593\) −3528.04 −0.244316 −0.122158 0.992511i \(-0.538981\pi\)
−0.122158 + 0.992511i \(0.538981\pi\)
\(594\) 0 0
\(595\) 10274.0 10274.0i 0.707887 0.707887i
\(596\) 0 0
\(597\) −7536.93 7536.93i −0.516693 0.516693i
\(598\) 0 0
\(599\) 19024.9i 1.29772i 0.760907 + 0.648861i \(0.224754\pi\)
−0.760907 + 0.648861i \(0.775246\pi\)
\(600\) 0 0
\(601\) 1065.92i 0.0723460i −0.999346 0.0361730i \(-0.988483\pi\)
0.999346 0.0361730i \(-0.0115167\pi\)
\(602\) 0 0
\(603\) −3588.11 3588.11i −0.242320 0.242320i
\(604\) 0 0
\(605\) −1551.02 + 1551.02i −0.104228 + 0.104228i
\(606\) 0 0
\(607\) −12909.7 −0.863242 −0.431621 0.902055i \(-0.642058\pi\)
−0.431621 + 0.902055i \(0.642058\pi\)
\(608\) 0 0
\(609\) 2858.00 0.190167
\(610\) 0 0
\(611\) 2487.25 2487.25i 0.164687 0.164687i
\(612\) 0 0
\(613\) −8763.41 8763.41i −0.577408 0.577408i 0.356781 0.934188i \(-0.383874\pi\)
−0.934188 + 0.356781i \(0.883874\pi\)
\(614\) 0 0
\(615\) 18482.1i 1.21182i
\(616\) 0 0
\(617\) 18921.2i 1.23459i 0.786733 + 0.617293i \(0.211771\pi\)
−0.786733 + 0.617293i \(0.788229\pi\)
\(618\) 0 0
\(619\) 15116.6 + 15116.6i 0.981565 + 0.981565i 0.999833 0.0182677i \(-0.00581513\pi\)
−0.0182677 + 0.999833i \(0.505815\pi\)
\(620\) 0 0
\(621\) −1696.97 + 1696.97i −0.109657 + 0.109657i
\(622\) 0 0
\(623\) 6541.15 0.420651
\(624\) 0 0
\(625\) −1204.57 −0.0770926
\(626\) 0 0
\(627\) −9288.42 + 9288.42i −0.591617 + 0.591617i
\(628\) 0 0
\(629\) 14447.7 + 14447.7i 0.915845 + 0.915845i
\(630\) 0 0
\(631\) 9602.80i 0.605834i −0.953017 0.302917i \(-0.902039\pi\)
0.953017 0.302917i \(-0.0979605\pi\)
\(632\) 0 0
\(633\) 17813.1i 1.11849i
\(634\) 0 0
\(635\) 12229.9 + 12229.9i 0.764297 + 0.764297i
\(636\) 0 0
\(637\) 3120.34 3120.34i 0.194085 0.194085i
\(638\) 0 0
\(639\) 7186.12 0.444880
\(640\) 0 0
\(641\) 4450.84 0.274256 0.137128 0.990553i \(-0.456213\pi\)
0.137128 + 0.990553i \(0.456213\pi\)
\(642\) 0 0
\(643\) −6491.27 + 6491.27i −0.398119 + 0.398119i −0.877569 0.479450i \(-0.840836\pi\)
0.479450 + 0.877569i \(0.340836\pi\)
\(644\) 0 0
\(645\) 5139.30 + 5139.30i 0.313736 + 0.313736i
\(646\) 0 0
\(647\) 5546.17i 0.337005i −0.985701 0.168503i \(-0.946107\pi\)
0.985701 0.168503i \(-0.0538932\pi\)
\(648\) 0 0
\(649\) 18098.3i 1.09464i
\(650\) 0 0
\(651\) 13267.3 + 13267.3i 0.798750 + 0.798750i
\(652\) 0 0
\(653\) 6327.58 6327.58i 0.379200 0.379200i −0.491614 0.870813i \(-0.663593\pi\)
0.870813 + 0.491614i \(0.163593\pi\)
\(654\) 0 0
\(655\) 6133.49 0.365886
\(656\) 0 0
\(657\) 22341.5 1.32667
\(658\) 0 0
\(659\) −15135.2 + 15135.2i −0.894663 + 0.894663i −0.994958 0.100294i \(-0.968022\pi\)
0.100294 + 0.994958i \(0.468022\pi\)
\(660\) 0 0
\(661\) −23460.1 23460.1i −1.38047 1.38047i −0.843777 0.536694i \(-0.819673\pi\)
−0.536694 0.843777i \(-0.680327\pi\)
\(662\) 0 0
\(663\) 10963.2i 0.642195i
\(664\) 0 0
\(665\) 8782.30i 0.512125i
\(666\) 0 0
\(667\) 502.550 + 502.550i 0.0291736 + 0.0291736i
\(668\) 0 0
\(669\) −2306.29 + 2306.29i −0.133283 + 0.133283i
\(670\) 0 0
\(671\) −29697.4 −1.70858
\(672\) 0 0
\(673\) 30638.5 1.75487 0.877436 0.479694i \(-0.159252\pi\)
0.877436 + 0.479694i \(0.159252\pi\)
\(674\) 0 0
\(675\) 2873.96 2873.96i 0.163879 0.163879i
\(676\) 0 0
\(677\) −12468.9 12468.9i −0.707855 0.707855i 0.258229 0.966084i \(-0.416861\pi\)
−0.966084 + 0.258229i \(0.916861\pi\)
\(678\) 0 0
\(679\) 3754.41i 0.212196i
\(680\) 0 0
\(681\) 23453.8i 1.31975i
\(682\) 0 0
\(683\) −13838.5 13838.5i −0.775280 0.775280i 0.203744 0.979024i \(-0.434689\pi\)
−0.979024 + 0.203744i \(0.934689\pi\)
\(684\) 0 0
\(685\) −11713.1 + 11713.1i −0.653333 + 0.653333i
\(686\) 0 0
\(687\) 17306.6 0.961116
\(688\) 0 0
\(689\) 3961.95 0.219068
\(690\) 0 0
\(691\) 106.012 106.012i 0.00583628 0.00583628i −0.704183 0.710019i \(-0.748686\pi\)
0.710019 + 0.704183i \(0.248686\pi\)
\(692\) 0 0
\(693\) 18588.2 + 18588.2i 1.01891 + 1.01891i
\(694\) 0 0
\(695\) 10129.8i 0.552870i
\(696\) 0 0
\(697\) 31900.4i 1.73359i
\(698\) 0 0
\(699\) 6567.17 + 6567.17i 0.355355 + 0.355355i
\(700\) 0 0
\(701\) 7839.35 7839.35i 0.422380 0.422380i −0.463643 0.886022i \(-0.653458\pi\)
0.886022 + 0.463643i \(0.153458\pi\)
\(702\) 0 0
\(703\) 12350.0 0.662574
\(704\) 0 0
\(705\) −11312.7 −0.604341
\(706\) 0 0
\(707\) 12149.1 12149.1i 0.646273 0.646273i
\(708\) 0 0
\(709\) 2728.30 + 2728.30i 0.144518 + 0.144518i 0.775664 0.631146i \(-0.217415\pi\)
−0.631146 + 0.775664i \(0.717415\pi\)
\(710\) 0 0
\(711\) 27499.9i 1.45053i
\(712\) 0 0
\(713\) 4665.84i 0.245073i
\(714\) 0 0
\(715\) −2349.01 2349.01i −0.122864 0.122864i
\(716\) 0 0
\(717\) −23309.5 + 23309.5i −1.21410 + 1.21410i
\(718\) 0 0
\(719\) 32717.8 1.69704 0.848519 0.529166i \(-0.177495\pi\)
0.848519 + 0.529166i \(0.177495\pi\)
\(720\) 0 0
\(721\) −45824.0 −2.36696
\(722\) 0 0
\(723\) 30620.3 30620.3i 1.57508 1.57508i
\(724\) 0 0
\(725\) −851.111 851.111i −0.0435993 0.0435993i
\(726\) 0 0
\(727\) 25847.2i 1.31859i 0.751883 + 0.659297i \(0.229146\pi\)
−0.751883 + 0.659297i \(0.770854\pi\)
\(728\) 0 0
\(729\) 27662.0i 1.40537i
\(730\) 0 0
\(731\) −8870.50 8870.50i −0.448820 0.448820i
\(732\) 0 0
\(733\) 10131.9 10131.9i 0.510547 0.510547i −0.404147 0.914694i \(-0.632432\pi\)
0.914694 + 0.404147i \(0.132432\pi\)
\(734\) 0 0
\(735\) −14192.1 −0.712223
\(736\) 0 0
\(737\) −4798.27 −0.239819
\(738\) 0 0
\(739\) −19163.7 + 19163.7i −0.953921 + 0.953921i −0.998984 0.0450633i \(-0.985651\pi\)
0.0450633 + 0.998984i \(0.485651\pi\)
\(740\) 0 0
\(741\) 4685.72 + 4685.72i 0.232300 + 0.232300i
\(742\) 0 0
\(743\) 23322.1i 1.15155i −0.817607 0.575777i \(-0.804699\pi\)
0.817607 0.575777i \(-0.195301\pi\)
\(744\) 0 0
\(745\) 21309.8i 1.04796i
\(746\) 0 0
\(747\) 10922.8 + 10922.8i 0.534999 + 0.534999i
\(748\) 0 0
\(749\) 1977.03 1977.03i 0.0964473 0.0964473i
\(750\) 0 0
\(751\) −25994.0 −1.26303 −0.631515 0.775364i \(-0.717566\pi\)
−0.631515 + 0.775364i \(0.717566\pi\)
\(752\) 0 0
\(753\) −5306.32 −0.256804
\(754\) 0 0
\(755\) −13125.5 + 13125.5i −0.632698 + 0.632698i
\(756\) 0 0
\(757\) 22145.0 + 22145.0i 1.06324 + 1.06324i 0.997860 + 0.0653808i \(0.0208262\pi\)
0.0653808 + 0.997860i \(0.479174\pi\)
\(758\) 0 0
\(759\) 11819.0i 0.565222i
\(760\) 0 0
\(761\) 16497.5i 0.785853i −0.919570 0.392926i \(-0.871463\pi\)
0.919570 0.392926i \(-0.128537\pi\)
\(762\) 0 0
\(763\) −23725.9 23725.9i −1.12573 1.12573i
\(764\) 0 0
\(765\) 13789.7 13789.7i 0.651723 0.651723i
\(766\) 0 0
\(767\) 9130.02 0.429812
\(768\) 0 0
\(769\) −24867.3 −1.16611 −0.583055 0.812433i \(-0.698143\pi\)
−0.583055 + 0.812433i \(0.698143\pi\)
\(770\) 0 0
\(771\) −44481.2 + 44481.2i −2.07776 + 2.07776i
\(772\) 0 0
\(773\) 1881.72 + 1881.72i 0.0875559 + 0.0875559i 0.749528 0.661972i \(-0.230280\pi\)
−0.661972 + 0.749528i \(0.730280\pi\)
\(774\) 0 0
\(775\) 7902.00i 0.366256i
\(776\) 0 0
\(777\) 44684.8i 2.06314i
\(778\) 0 0
\(779\) −13634.4 13634.4i −0.627089 0.627089i
\(780\) 0 0
\(781\) 4804.89 4804.89i 0.220144 0.220144i
\(782\) 0 0
\(783\) 736.525 0.0336159
\(784\) 0 0
\(785\) 8453.54 0.384356
\(786\) 0 0
\(787\) 7790.94 7790.94i 0.352881 0.352881i −0.508300 0.861180i \(-0.669726\pi\)
0.861180 + 0.508300i \(0.169726\pi\)
\(788\) 0 0
\(789\) 2255.08 + 2255.08i 0.101753 + 0.101753i
\(790\) 0 0
\(791\) 17937.2i 0.806286i
\(792\) 0 0
\(793\) 14981.4i 0.670877i
\(794\) 0 0
\(795\) −9009.98 9009.98i −0.401951 0.401951i
\(796\) 0 0
\(797\) −4209.84 + 4209.84i −0.187102 + 0.187102i −0.794442 0.607340i \(-0.792237\pi\)
0.607340 + 0.794442i \(0.292237\pi\)
\(798\) 0 0
\(799\) 19525.8 0.864549
\(800\) 0 0
\(801\) 8779.50 0.387276
\(802\) 0 0
\(803\) 14938.3 14938.3i 0.656490 0.656490i
\(804\) 0 0
\(805\) −5587.51 5587.51i −0.244639 0.244639i
\(806\) 0 0
\(807\) 52.5393i 0.00229178i
\(808\) 0 0
\(809\) 27554.3i 1.19747i −0.800946 0.598737i \(-0.795670\pi\)
0.800946 0.598737i \(-0.204330\pi\)
\(810\) 0 0
\(811\) −3406.54 3406.54i −0.147497 0.147497i 0.629502 0.776999i \(-0.283259\pi\)
−0.776999 + 0.629502i \(0.783259\pi\)
\(812\) 0 0
\(813\) −15576.1 + 15576.1i −0.671926 + 0.671926i
\(814\) 0 0
\(815\) 12987.1 0.558184
\(816\) 0 0
\(817\) −7582.59 −0.324701
\(818\) 0 0
\(819\) 9377.16 9377.16i 0.400079 0.400079i
\(820\) 0 0
\(821\) −9839.14 9839.14i −0.418256 0.418256i 0.466346 0.884602i \(-0.345570\pi\)
−0.884602 + 0.466346i \(0.845570\pi\)
\(822\) 0 0
\(823\) 36653.5i 1.55244i −0.630461 0.776221i \(-0.717134\pi\)
0.630461 0.776221i \(-0.282866\pi\)
\(824\) 0 0
\(825\) 20016.5i 0.844711i
\(826\) 0 0
\(827\) 22223.2 + 22223.2i 0.934431 + 0.934431i 0.997979 0.0635475i \(-0.0202414\pi\)
−0.0635475 + 0.997979i \(0.520241\pi\)
\(828\) 0 0
\(829\) 14715.5 14715.5i 0.616516 0.616516i −0.328120 0.944636i \(-0.606415\pi\)
0.944636 + 0.328120i \(0.106415\pi\)
\(830\) 0 0
\(831\) −16772.2 −0.700145
\(832\) 0 0
\(833\) 24495.8 1.01888
\(834\) 0 0
\(835\) −5697.46 + 5697.46i −0.236130 + 0.236130i
\(836\) 0 0
\(837\) 3419.07 + 3419.07i 0.141195 + 0.141195i
\(838\) 0 0
\(839\) 11010.0i 0.453050i −0.974005 0.226525i \(-0.927263\pi\)
0.974005 0.226525i \(-0.0727365\pi\)
\(840\) 0 0
\(841\) 24170.9i 0.991057i
\(842\) 0 0
\(843\) 26108.1 + 26108.1i 1.06668 + 1.06668i
\(844\) 0 0
\(845\) 9061.20 9061.20i 0.368893 0.368893i
\(846\) 0 0
\(847\) −8279.85 −0.335890
\(848\) 0 0
\(849\) 7080.22 0.286210
\(850\) 0 0
\(851\) 7857.37 7857.37i 0.316507 0.316507i
\(852\) 0 0
\(853\) 12809.9 + 12809.9i 0.514190 + 0.514190i 0.915808 0.401617i \(-0.131552\pi\)
−0.401617 + 0.915808i \(0.631552\pi\)
\(854\) 0 0
\(855\) 11787.6i 0.471493i
\(856\) 0 0
\(857\) 38510.8i 1.53501i 0.641043 + 0.767505i \(0.278502\pi\)
−0.641043 + 0.767505i \(0.721498\pi\)
\(858\) 0 0
\(859\) −23234.6 23234.6i −0.922882 0.922882i 0.0743503 0.997232i \(-0.476312\pi\)
−0.997232 + 0.0743503i \(0.976312\pi\)
\(860\) 0 0
\(861\) −49331.9 + 49331.9i −1.95264 + 1.95264i
\(862\) 0 0
\(863\) 22079.5 0.870911 0.435456 0.900210i \(-0.356587\pi\)
0.435456 + 0.900210i \(0.356587\pi\)
\(864\) 0 0
\(865\) 5255.26 0.206571
\(866\) 0 0
\(867\) −16029.8 + 16029.8i −0.627912 + 0.627912i
\(868\) 0 0
\(869\) −18387.4 18387.4i −0.717779 0.717779i
\(870\) 0 0
\(871\) 2420.58i 0.0941656i
\(872\) 0 0
\(873\) 5039.15i 0.195360i
\(874\) 0 0
\(875\) 23976.7 + 23976.7i 0.926356 + 0.926356i
\(876\) 0 0
\(877\) 4082.13 4082.13i 0.157176 0.157176i −0.624138 0.781314i \(-0.714550\pi\)
0.781314 + 0.624138i \(0.214550\pi\)
\(878\) 0 0
\(879\) 15753.9 0.604510
\(880\) 0 0
\(881\) 7132.59 0.272762 0.136381 0.990656i \(-0.456453\pi\)
0.136381 + 0.990656i \(0.456453\pi\)
\(882\) 0 0
\(883\) −19170.0 + 19170.0i −0.730601 + 0.730601i −0.970739 0.240138i \(-0.922807\pi\)
0.240138 + 0.970739i \(0.422807\pi\)
\(884\) 0 0
\(885\) −20762.9 20762.9i −0.788628 0.788628i
\(886\) 0 0
\(887\) 45045.7i 1.70517i 0.522589 + 0.852585i \(0.324966\pi\)
−0.522589 + 0.852585i \(0.675034\pi\)
\(888\) 0 0
\(889\) 65287.3i 2.46306i
\(890\) 0 0
\(891\) −11497.8 11497.8i −0.432314 0.432314i
\(892\) 0 0
\(893\) 8345.43 8345.43i 0.312731 0.312731i
\(894\) 0 0
\(895\) −20541.1 −0.767167
\(896\) 0 0
\(897\) 5962.34 0.221936
\(898\) 0 0
\(899\) 1012.55 1012.55i 0.0375643 0.0375643i
\(900\) 0 0
\(901\) 15551.4 + 15551.4i 0.575017 + 0.575017i
\(902\) 0 0
\(903\) 27435.3i 1.01106i
\(904\) 0 0
\(905\) 1131.36i 0.0415554i
\(906\) 0 0
\(907\) 33658.2 + 33658.2i 1.23220 + 1.23220i 0.963118 + 0.269079i \(0.0867192\pi\)
0.269079 + 0.963118i \(0.413281\pi\)
\(908\) 0 0
\(909\) 16306.5 16306.5i 0.594997 0.594997i
\(910\) 0 0
\(911\) 42503.8 1.54579 0.772895 0.634534i \(-0.218808\pi\)
0.772895 + 0.634534i \(0.218808\pi\)
\(912\) 0 0
\(913\) 14606.7 0.529477
\(914\) 0 0
\(915\) 34069.7 34069.7i 1.23094 1.23094i
\(916\) 0 0
\(917\) 16371.3 + 16371.3i 0.589562 + 0.589562i
\(918\) 0 0
\(919\) 8819.41i 0.316568i 0.987394 + 0.158284i \(0.0505961\pi\)
−0.987394 + 0.158284i \(0.949404\pi\)
\(920\) 0 0
\(921\) 2541.55i 0.0909305i
\(922\) 0 0
\(923\) −2423.92 2423.92i −0.0864402 0.0864402i
\(924\) 0 0
\(925\) −13307.1 + 13307.1i −0.473012 + 0.473012i
\(926\) 0 0
\(927\) −61504.8 −2.17916
\(928\) 0 0
\(929\) −14155.6 −0.499925 −0.249963 0.968256i \(-0.580418\pi\)
−0.249963 + 0.968256i \(0.580418\pi\)
\(930\) 0 0
\(931\) 10469.6 10469.6i 0.368558 0.368558i
\(932\) 0 0
\(933\) −4789.43 4789.43i −0.168059 0.168059i
\(934\) 0 0
\(935\) 18440.6i 0.644996i
\(936\) 0 0
\(937\) 38518.7i 1.34296i −0.741023 0.671479i \(-0.765659\pi\)
0.741023 0.671479i \(-0.234341\pi\)
\(938\) 0 0
\(939\) −19319.2 19319.2i −0.671414 0.671414i
\(940\) 0 0
\(941\) 27998.2 27998.2i 0.969942 0.969942i −0.0296196 0.999561i \(-0.509430\pi\)
0.999561 + 0.0296196i \(0.00942960\pi\)
\(942\) 0 0
\(943\) −17349.0 −0.599112
\(944\) 0 0
\(945\) −8188.93 −0.281890
\(946\) 0 0
\(947\) 32839.5 32839.5i 1.12687 1.12687i 0.136181 0.990684i \(-0.456517\pi\)
0.990684 0.136181i \(-0.0434829\pi\)
\(948\) 0 0
\(949\) −7535.92 7535.92i −0.257773 0.257773i
\(950\) 0 0
\(951\) 51925.4i 1.77055i
\(952\) 0 0
\(953\) 20600.1i 0.700211i −0.936710 0.350106i \(-0.886146\pi\)
0.936710 0.350106i \(-0.113854\pi\)
\(954\) 0 0
\(955\) 18317.5 + 18317.5i 0.620670 + 0.620670i
\(956\) 0 0
\(957\) 2564.88 2564.88i 0.0866360 0.0866360i
\(958\) 0 0
\(959\) −62528.2 −2.10547
\(960\) 0 0
\(961\) −20390.2 −0.684441
\(962\) 0 0
\(963\) 2653.56 2653.56i 0.0887951 0.0887951i
\(964\) 0 0
\(965\) 15152.5 + 15152.5i 0.505469 + 0.505469i
\(966\) 0 0
\(967\) 38210.9i 1.27071i −0.772219 0.635356i \(-0.780853\pi\)
0.772219 0.635356i \(-0.219147\pi\)
\(968\) 0 0
\(969\) 36784.6i 1.21950i
\(970\) 0 0
\(971\) −37224.9 37224.9i −1.23028 1.23028i −0.963855 0.266427i \(-0.914157\pi\)
−0.266427 0.963855i \(-0.585843\pi\)
\(972\) 0 0
\(973\) 27038.1 27038.1i 0.890854 0.890854i
\(974\) 0 0
\(975\) −10097.7 −0.331678
\(976\) 0 0
\(977\) 7985.95 0.261508 0.130754 0.991415i \(-0.458260\pi\)
0.130754 + 0.991415i \(0.458260\pi\)
\(978\) 0 0
\(979\) 5870.28 5870.28i 0.191639 0.191639i
\(980\) 0 0
\(981\) −31844.8 31844.8i −1.03642 1.03642i
\(982\) 0 0
\(983\) 10703.1i 0.347279i 0.984809 + 0.173639i \(0.0555527\pi\)
−0.984809 + 0.173639i \(0.944447\pi\)
\(984\) 0 0
\(985\) 22580.9i 0.730442i
\(986\) 0 0
\(987\) −30195.4 30195.4i −0.973791 0.973791i
\(988\) 0 0
\(989\) −4824.23 + 4824.23i −0.155108 + 0.155108i
\(990\) 0 0
\(991\) 23945.4 0.767558 0.383779 0.923425i \(-0.374623\pi\)
0.383779 + 0.923425i \(0.374623\pi\)
\(992\) 0 0
\(993\) −20009.0 −0.639442
\(994\) 0 0
\(995\) 6395.37 6395.37i 0.203766 0.203766i
\(996\) 0 0
\(997\) −14292.5 14292.5i −0.454010 0.454010i 0.442673 0.896683i \(-0.354030\pi\)
−0.896683 + 0.442673i \(0.854030\pi\)
\(998\) 0 0
\(999\) 11515.6i 0.364702i
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 64.4.e.a.49.1 10
3.2 odd 2 576.4.k.a.433.4 10
4.3 odd 2 16.4.e.a.5.1 10
8.3 odd 2 128.4.e.b.97.1 10
8.5 even 2 128.4.e.a.97.5 10
12.11 even 2 144.4.k.a.37.5 10
16.3 odd 4 16.4.e.a.13.1 yes 10
16.5 even 4 128.4.e.a.33.5 10
16.11 odd 4 128.4.e.b.33.1 10
16.13 even 4 inner 64.4.e.a.17.1 10
32.3 odd 8 1024.4.a.n.1.9 10
32.5 even 8 1024.4.b.k.513.2 10
32.11 odd 8 1024.4.b.j.513.2 10
32.13 even 8 1024.4.a.m.1.9 10
32.19 odd 8 1024.4.a.n.1.2 10
32.21 even 8 1024.4.b.k.513.9 10
32.27 odd 8 1024.4.b.j.513.9 10
32.29 even 8 1024.4.a.m.1.2 10
48.29 odd 4 576.4.k.a.145.4 10
48.35 even 4 144.4.k.a.109.5 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
16.4.e.a.5.1 10 4.3 odd 2
16.4.e.a.13.1 yes 10 16.3 odd 4
64.4.e.a.17.1 10 16.13 even 4 inner
64.4.e.a.49.1 10 1.1 even 1 trivial
128.4.e.a.33.5 10 16.5 even 4
128.4.e.a.97.5 10 8.5 even 2
128.4.e.b.33.1 10 16.11 odd 4
128.4.e.b.97.1 10 8.3 odd 2
144.4.k.a.37.5 10 12.11 even 2
144.4.k.a.109.5 10 48.35 even 4
576.4.k.a.145.4 10 48.29 odd 4
576.4.k.a.433.4 10 3.2 odd 2
1024.4.a.m.1.2 10 32.29 even 8
1024.4.a.m.1.9 10 32.13 even 8
1024.4.a.n.1.2 10 32.19 odd 8
1024.4.a.n.1.9 10 32.3 odd 8
1024.4.b.j.513.2 10 32.11 odd 8
1024.4.b.j.513.9 10 32.27 odd 8
1024.4.b.k.513.2 10 32.5 even 8
1024.4.b.k.513.9 10 32.21 even 8