Properties

Label 1225.2.b.g.99.2
Level $1225$
Weight $2$
Character 1225.99
Analytic conductor $9.782$
Analytic rank $0$
Dimension $4$
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] = [1225,2,Mod(99,1225)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1225, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("1225.99");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1225 = 5^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1225.b (of order \(2\), degree \(1\), 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: \(9.78167424761\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 35)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 99.2
Root \(-0.707107 - 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 1225.99
Dual form 1225.2.b.g.99.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.414214i q^{2} -2.41421i q^{3} +1.82843 q^{4} -1.00000 q^{6} -1.58579i q^{8} -2.82843 q^{9} +O(q^{10})\) \(q-0.414214i q^{2} -2.41421i q^{3} +1.82843 q^{4} -1.00000 q^{6} -1.58579i q^{8} -2.82843 q^{9} +4.82843 q^{11} -4.41421i q^{12} -0.828427i q^{13} +3.00000 q^{16} -0.828427i q^{17} +1.17157i q^{18} +2.82843 q^{19} -2.00000i q^{22} +2.41421i q^{23} -3.82843 q^{24} -0.343146 q^{26} -0.414214i q^{27} +1.00000 q^{29} -6.00000 q^{31} -4.41421i q^{32} -11.6569i q^{33} -0.343146 q^{34} -5.17157 q^{36} -1.17157i q^{38} -2.00000 q^{39} -2.17157 q^{41} -6.41421i q^{43} +8.82843 q^{44} +1.00000 q^{46} +2.00000i q^{47} -7.24264i q^{48} -2.00000 q^{51} -1.51472i q^{52} +6.82843i q^{53} -0.171573 q^{54} -6.82843i q^{57} -0.414214i q^{58} +12.4853 q^{59} -11.4853 q^{61} +2.48528i q^{62} +4.17157 q^{64} -4.82843 q^{66} +12.4142i q^{67} -1.51472i q^{68} +5.82843 q^{69} -12.4853 q^{71} +4.48528i q^{72} -4.82843i q^{73} +5.17157 q^{76} +0.828427i q^{78} -9.17157 q^{79} -9.48528 q^{81} +0.899495i q^{82} +11.7279i q^{83} -2.65685 q^{86} -2.41421i q^{87} -7.65685i q^{88} -2.65685 q^{89} +4.41421i q^{92} +14.4853i q^{93} +0.828427 q^{94} -10.6569 q^{96} +0.343146i q^{97} -13.6569 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{4} - 4 q^{6}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{4} - 4 q^{6} + 8 q^{11} + 12 q^{16} - 4 q^{24} - 24 q^{26} + 4 q^{29} - 24 q^{31} - 24 q^{34} - 32 q^{36} - 8 q^{39} - 20 q^{41} + 24 q^{44} + 4 q^{46} - 8 q^{51} - 12 q^{54} + 16 q^{59} - 12 q^{61} + 28 q^{64} - 8 q^{66} + 12 q^{69} - 16 q^{71} + 32 q^{76} - 48 q^{79} - 4 q^{81} + 12 q^{86} + 12 q^{89} - 8 q^{94} - 20 q^{96} - 32 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1225\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(1177\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 0.414214i − 0.292893i −0.989219 0.146447i \(-0.953216\pi\)
0.989219 0.146447i \(-0.0467837\pi\)
\(3\) − 2.41421i − 1.39385i −0.717146 0.696923i \(-0.754552\pi\)
0.717146 0.696923i \(-0.245448\pi\)
\(4\) 1.82843 0.914214
\(5\) 0 0
\(6\) −1.00000 −0.408248
\(7\) 0 0
\(8\) − 1.58579i − 0.560660i
\(9\) −2.82843 −0.942809
\(10\) 0 0
\(11\) 4.82843 1.45583 0.727913 0.685670i \(-0.240491\pi\)
0.727913 + 0.685670i \(0.240491\pi\)
\(12\) − 4.41421i − 1.27427i
\(13\) − 0.828427i − 0.229764i −0.993379 0.114882i \(-0.963351\pi\)
0.993379 0.114882i \(-0.0366490\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 3.00000 0.750000
\(17\) − 0.828427i − 0.200923i −0.994941 0.100462i \(-0.967968\pi\)
0.994941 0.100462i \(-0.0320319\pi\)
\(18\) 1.17157i 0.276142i
\(19\) 2.82843 0.648886 0.324443 0.945905i \(-0.394823\pi\)
0.324443 + 0.945905i \(0.394823\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) − 2.00000i − 0.426401i
\(23\) 2.41421i 0.503398i 0.967806 + 0.251699i \(0.0809893\pi\)
−0.967806 + 0.251699i \(0.919011\pi\)
\(24\) −3.82843 −0.781474
\(25\) 0 0
\(26\) −0.343146 −0.0672964
\(27\) − 0.414214i − 0.0797154i
\(28\) 0 0
\(29\) 1.00000 0.185695 0.0928477 0.995680i \(-0.470403\pi\)
0.0928477 + 0.995680i \(0.470403\pi\)
\(30\) 0 0
\(31\) −6.00000 −1.07763 −0.538816 0.842424i \(-0.681128\pi\)
−0.538816 + 0.842424i \(0.681128\pi\)
\(32\) − 4.41421i − 0.780330i
\(33\) − 11.6569i − 2.02920i
\(34\) −0.343146 −0.0588490
\(35\) 0 0
\(36\) −5.17157 −0.861929
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) − 1.17157i − 0.190054i
\(39\) −2.00000 −0.320256
\(40\) 0 0
\(41\) −2.17157 −0.339143 −0.169571 0.985518i \(-0.554238\pi\)
−0.169571 + 0.985518i \(0.554238\pi\)
\(42\) 0 0
\(43\) − 6.41421i − 0.978158i −0.872239 0.489079i \(-0.837333\pi\)
0.872239 0.489079i \(-0.162667\pi\)
\(44\) 8.82843 1.33094
\(45\) 0 0
\(46\) 1.00000 0.147442
\(47\) 2.00000i 0.291730i 0.989305 + 0.145865i \(0.0465965\pi\)
−0.989305 + 0.145865i \(0.953403\pi\)
\(48\) − 7.24264i − 1.04539i
\(49\) 0 0
\(50\) 0 0
\(51\) −2.00000 −0.280056
\(52\) − 1.51472i − 0.210054i
\(53\) 6.82843i 0.937957i 0.883210 + 0.468978i \(0.155378\pi\)
−0.883210 + 0.468978i \(0.844622\pi\)
\(54\) −0.171573 −0.0233481
\(55\) 0 0
\(56\) 0 0
\(57\) − 6.82843i − 0.904447i
\(58\) − 0.414214i − 0.0543889i
\(59\) 12.4853 1.62545 0.812723 0.582651i \(-0.197984\pi\)
0.812723 + 0.582651i \(0.197984\pi\)
\(60\) 0 0
\(61\) −11.4853 −1.47054 −0.735270 0.677775i \(-0.762945\pi\)
−0.735270 + 0.677775i \(0.762945\pi\)
\(62\) 2.48528i 0.315631i
\(63\) 0 0
\(64\) 4.17157 0.521447
\(65\) 0 0
\(66\) −4.82843 −0.594338
\(67\) 12.4142i 1.51664i 0.651884 + 0.758319i \(0.273979\pi\)
−0.651884 + 0.758319i \(0.726021\pi\)
\(68\) − 1.51472i − 0.183687i
\(69\) 5.82843 0.701660
\(70\) 0 0
\(71\) −12.4853 −1.48173 −0.740865 0.671654i \(-0.765584\pi\)
−0.740865 + 0.671654i \(0.765584\pi\)
\(72\) 4.48528i 0.528595i
\(73\) − 4.82843i − 0.565125i −0.959249 0.282562i \(-0.908816\pi\)
0.959249 0.282562i \(-0.0911844\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 5.17157 0.593220
\(77\) 0 0
\(78\) 0.828427i 0.0938009i
\(79\) −9.17157 −1.03188 −0.515941 0.856624i \(-0.672558\pi\)
−0.515941 + 0.856624i \(0.672558\pi\)
\(80\) 0 0
\(81\) −9.48528 −1.05392
\(82\) 0.899495i 0.0993326i
\(83\) 11.7279i 1.28731i 0.765317 + 0.643653i \(0.222582\pi\)
−0.765317 + 0.643653i \(0.777418\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −2.65685 −0.286496
\(87\) − 2.41421i − 0.258831i
\(88\) − 7.65685i − 0.816223i
\(89\) −2.65685 −0.281626 −0.140813 0.990036i \(-0.544972\pi\)
−0.140813 + 0.990036i \(0.544972\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 4.41421i 0.460214i
\(93\) 14.4853i 1.50205i
\(94\) 0.828427 0.0854457
\(95\) 0 0
\(96\) −10.6569 −1.08766
\(97\) 0.343146i 0.0348412i 0.999848 + 0.0174206i \(0.00554543\pi\)
−0.999848 + 0.0174206i \(0.994455\pi\)
\(98\) 0 0
\(99\) −13.6569 −1.37257
\(100\) 0 0
\(101\) 12.3137 1.22526 0.612630 0.790370i \(-0.290112\pi\)
0.612630 + 0.790370i \(0.290112\pi\)
\(102\) 0.828427i 0.0820265i
\(103\) − 0.414214i − 0.0408137i −0.999792 0.0204068i \(-0.993504\pi\)
0.999792 0.0204068i \(-0.00649615\pi\)
\(104\) −1.31371 −0.128820
\(105\) 0 0
\(106\) 2.82843 0.274721
\(107\) 2.75736i 0.266564i 0.991078 + 0.133282i \(0.0425516\pi\)
−0.991078 + 0.133282i \(0.957448\pi\)
\(108\) − 0.757359i − 0.0728769i
\(109\) −3.48528 −0.333829 −0.166915 0.985971i \(-0.553380\pi\)
−0.166915 + 0.985971i \(0.553380\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) − 12.4853i − 1.17452i −0.809400 0.587258i \(-0.800207\pi\)
0.809400 0.587258i \(-0.199793\pi\)
\(114\) −2.82843 −0.264906
\(115\) 0 0
\(116\) 1.82843 0.169765
\(117\) 2.34315i 0.216624i
\(118\) − 5.17157i − 0.476082i
\(119\) 0 0
\(120\) 0 0
\(121\) 12.3137 1.11943
\(122\) 4.75736i 0.430711i
\(123\) 5.24264i 0.472713i
\(124\) −10.9706 −0.985186
\(125\) 0 0
\(126\) 0 0
\(127\) 13.3137i 1.18140i 0.806891 + 0.590700i \(0.201148\pi\)
−0.806891 + 0.590700i \(0.798852\pi\)
\(128\) − 10.5563i − 0.933058i
\(129\) −15.4853 −1.36340
\(130\) 0 0
\(131\) 3.31371 0.289520 0.144760 0.989467i \(-0.453759\pi\)
0.144760 + 0.989467i \(0.453759\pi\)
\(132\) − 21.3137i − 1.85512i
\(133\) 0 0
\(134\) 5.14214 0.444213
\(135\) 0 0
\(136\) −1.31371 −0.112650
\(137\) 1.65685i 0.141555i 0.997492 + 0.0707773i \(0.0225480\pi\)
−0.997492 + 0.0707773i \(0.977452\pi\)
\(138\) − 2.41421i − 0.205512i
\(139\) −12.1421 −1.02988 −0.514941 0.857225i \(-0.672186\pi\)
−0.514941 + 0.857225i \(0.672186\pi\)
\(140\) 0 0
\(141\) 4.82843 0.406627
\(142\) 5.17157i 0.433989i
\(143\) − 4.00000i − 0.334497i
\(144\) −8.48528 −0.707107
\(145\) 0 0
\(146\) −2.00000 −0.165521
\(147\) 0 0
\(148\) 0 0
\(149\) 7.82843 0.641330 0.320665 0.947193i \(-0.396094\pi\)
0.320665 + 0.947193i \(0.396094\pi\)
\(150\) 0 0
\(151\) 0.343146 0.0279248 0.0139624 0.999903i \(-0.495555\pi\)
0.0139624 + 0.999903i \(0.495555\pi\)
\(152\) − 4.48528i − 0.363804i
\(153\) 2.34315i 0.189432i
\(154\) 0 0
\(155\) 0 0
\(156\) −3.65685 −0.292783
\(157\) − 5.31371i − 0.424080i −0.977261 0.212040i \(-0.931989\pi\)
0.977261 0.212040i \(-0.0680107\pi\)
\(158\) 3.79899i 0.302231i
\(159\) 16.4853 1.30737
\(160\) 0 0
\(161\) 0 0
\(162\) 3.92893i 0.308686i
\(163\) − 23.6569i − 1.85295i −0.376359 0.926474i \(-0.622824\pi\)
0.376359 0.926474i \(-0.377176\pi\)
\(164\) −3.97056 −0.310049
\(165\) 0 0
\(166\) 4.85786 0.377043
\(167\) 19.5858i 1.51559i 0.652491 + 0.757797i \(0.273724\pi\)
−0.652491 + 0.757797i \(0.726276\pi\)
\(168\) 0 0
\(169\) 12.3137 0.947208
\(170\) 0 0
\(171\) −8.00000 −0.611775
\(172\) − 11.7279i − 0.894246i
\(173\) 19.3137i 1.46839i 0.678936 + 0.734197i \(0.262441\pi\)
−0.678936 + 0.734197i \(0.737559\pi\)
\(174\) −1.00000 −0.0758098
\(175\) 0 0
\(176\) 14.4853 1.09187
\(177\) − 30.1421i − 2.26562i
\(178\) 1.10051i 0.0824863i
\(179\) 10.0000 0.747435 0.373718 0.927543i \(-0.378083\pi\)
0.373718 + 0.927543i \(0.378083\pi\)
\(180\) 0 0
\(181\) −8.65685 −0.643459 −0.321729 0.946832i \(-0.604264\pi\)
−0.321729 + 0.946832i \(0.604264\pi\)
\(182\) 0 0
\(183\) 27.7279i 2.04971i
\(184\) 3.82843 0.282235
\(185\) 0 0
\(186\) 6.00000 0.439941
\(187\) − 4.00000i − 0.292509i
\(188\) 3.65685i 0.266704i
\(189\) 0 0
\(190\) 0 0
\(191\) −7.17157 −0.518917 −0.259458 0.965754i \(-0.583544\pi\)
−0.259458 + 0.965754i \(0.583544\pi\)
\(192\) − 10.0711i − 0.726817i
\(193\) 2.00000i 0.143963i 0.997406 + 0.0719816i \(0.0229323\pi\)
−0.997406 + 0.0719816i \(0.977068\pi\)
\(194\) 0.142136 0.0102047
\(195\) 0 0
\(196\) 0 0
\(197\) − 23.6569i − 1.68548i −0.538320 0.842741i \(-0.680941\pi\)
0.538320 0.842741i \(-0.319059\pi\)
\(198\) 5.65685i 0.402015i
\(199\) −1.65685 −0.117451 −0.0587256 0.998274i \(-0.518704\pi\)
−0.0587256 + 0.998274i \(0.518704\pi\)
\(200\) 0 0
\(201\) 29.9706 2.11396
\(202\) − 5.10051i − 0.358870i
\(203\) 0 0
\(204\) −3.65685 −0.256031
\(205\) 0 0
\(206\) −0.171573 −0.0119540
\(207\) − 6.82843i − 0.474608i
\(208\) − 2.48528i − 0.172323i
\(209\) 13.6569 0.944664
\(210\) 0 0
\(211\) 3.51472 0.241963 0.120982 0.992655i \(-0.461396\pi\)
0.120982 + 0.992655i \(0.461396\pi\)
\(212\) 12.4853i 0.857493i
\(213\) 30.1421i 2.06531i
\(214\) 1.14214 0.0780748
\(215\) 0 0
\(216\) −0.656854 −0.0446933
\(217\) 0 0
\(218\) 1.44365i 0.0977764i
\(219\) −11.6569 −0.787697
\(220\) 0 0
\(221\) −0.686292 −0.0461650
\(222\) 0 0
\(223\) − 11.6569i − 0.780601i −0.920688 0.390300i \(-0.872371\pi\)
0.920688 0.390300i \(-0.127629\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) −5.17157 −0.344008
\(227\) 26.9706i 1.79010i 0.445967 + 0.895050i \(0.352860\pi\)
−0.445967 + 0.895050i \(0.647140\pi\)
\(228\) − 12.4853i − 0.826858i
\(229\) 0.343146 0.0226757 0.0113379 0.999936i \(-0.496391\pi\)
0.0113379 + 0.999936i \(0.496391\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) − 1.58579i − 0.104112i
\(233\) 11.1716i 0.731874i 0.930640 + 0.365937i \(0.119251\pi\)
−0.930640 + 0.365937i \(0.880749\pi\)
\(234\) 0.970563 0.0634477
\(235\) 0 0
\(236\) 22.8284 1.48600
\(237\) 22.1421i 1.43829i
\(238\) 0 0
\(239\) −1.31371 −0.0849767 −0.0424884 0.999097i \(-0.513529\pi\)
−0.0424884 + 0.999097i \(0.513529\pi\)
\(240\) 0 0
\(241\) 16.3431 1.05275 0.526377 0.850251i \(-0.323550\pi\)
0.526377 + 0.850251i \(0.323550\pi\)
\(242\) − 5.10051i − 0.327873i
\(243\) 21.6569i 1.38929i
\(244\) −21.0000 −1.34439
\(245\) 0 0
\(246\) 2.17157 0.138454
\(247\) − 2.34315i − 0.149091i
\(248\) 9.51472i 0.604185i
\(249\) 28.3137 1.79431
\(250\) 0 0
\(251\) 13.3137 0.840354 0.420177 0.907442i \(-0.361968\pi\)
0.420177 + 0.907442i \(0.361968\pi\)
\(252\) 0 0
\(253\) 11.6569i 0.732860i
\(254\) 5.51472 0.346024
\(255\) 0 0
\(256\) 3.97056 0.248160
\(257\) 17.6569i 1.10140i 0.834702 + 0.550702i \(0.185640\pi\)
−0.834702 + 0.550702i \(0.814360\pi\)
\(258\) 6.41421i 0.399331i
\(259\) 0 0
\(260\) 0 0
\(261\) −2.82843 −0.175075
\(262\) − 1.37258i − 0.0847985i
\(263\) − 19.0416i − 1.17416i −0.809530 0.587079i \(-0.800278\pi\)
0.809530 0.587079i \(-0.199722\pi\)
\(264\) −18.4853 −1.13769
\(265\) 0 0
\(266\) 0 0
\(267\) 6.41421i 0.392543i
\(268\) 22.6985i 1.38653i
\(269\) 30.4558 1.85693 0.928463 0.371425i \(-0.121131\pi\)
0.928463 + 0.371425i \(0.121131\pi\)
\(270\) 0 0
\(271\) −0.485281 −0.0294787 −0.0147394 0.999891i \(-0.504692\pi\)
−0.0147394 + 0.999891i \(0.504692\pi\)
\(272\) − 2.48528i − 0.150692i
\(273\) 0 0
\(274\) 0.686292 0.0414604
\(275\) 0 0
\(276\) 10.6569 0.641467
\(277\) − 12.1421i − 0.729550i −0.931096 0.364775i \(-0.881146\pi\)
0.931096 0.364775i \(-0.118854\pi\)
\(278\) 5.02944i 0.301646i
\(279\) 16.9706 1.01600
\(280\) 0 0
\(281\) 26.2843 1.56799 0.783994 0.620768i \(-0.213179\pi\)
0.783994 + 0.620768i \(0.213179\pi\)
\(282\) − 2.00000i − 0.119098i
\(283\) − 14.0000i − 0.832214i −0.909316 0.416107i \(-0.863394\pi\)
0.909316 0.416107i \(-0.136606\pi\)
\(284\) −22.8284 −1.35462
\(285\) 0 0
\(286\) −1.65685 −0.0979718
\(287\) 0 0
\(288\) 12.4853i 0.735702i
\(289\) 16.3137 0.959630
\(290\) 0 0
\(291\) 0.828427 0.0485633
\(292\) − 8.82843i − 0.516645i
\(293\) 16.0000i 0.934730i 0.884064 + 0.467365i \(0.154797\pi\)
−0.884064 + 0.467365i \(0.845203\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) − 2.00000i − 0.116052i
\(298\) − 3.24264i − 0.187841i
\(299\) 2.00000 0.115663
\(300\) 0 0
\(301\) 0 0
\(302\) − 0.142136i − 0.00817899i
\(303\) − 29.7279i − 1.70782i
\(304\) 8.48528 0.486664
\(305\) 0 0
\(306\) 0.970563 0.0554834
\(307\) − 13.2426i − 0.755797i −0.925847 0.377899i \(-0.876647\pi\)
0.925847 0.377899i \(-0.123353\pi\)
\(308\) 0 0
\(309\) −1.00000 −0.0568880
\(310\) 0 0
\(311\) −18.8284 −1.06766 −0.533831 0.845591i \(-0.679248\pi\)
−0.533831 + 0.845591i \(0.679248\pi\)
\(312\) 3.17157i 0.179555i
\(313\) 17.6569i 0.998024i 0.866595 + 0.499012i \(0.166304\pi\)
−0.866595 + 0.499012i \(0.833696\pi\)
\(314\) −2.20101 −0.124210
\(315\) 0 0
\(316\) −16.7696 −0.943361
\(317\) 25.7990i 1.44902i 0.689267 + 0.724508i \(0.257933\pi\)
−0.689267 + 0.724508i \(0.742067\pi\)
\(318\) − 6.82843i − 0.382919i
\(319\) 4.82843 0.270340
\(320\) 0 0
\(321\) 6.65685 0.371549
\(322\) 0 0
\(323\) − 2.34315i − 0.130376i
\(324\) −17.3431 −0.963508
\(325\) 0 0
\(326\) −9.79899 −0.542716
\(327\) 8.41421i 0.465307i
\(328\) 3.44365i 0.190144i
\(329\) 0 0
\(330\) 0 0
\(331\) −10.9706 −0.602997 −0.301498 0.953467i \(-0.597487\pi\)
−0.301498 + 0.953467i \(0.597487\pi\)
\(332\) 21.4437i 1.17687i
\(333\) 0 0
\(334\) 8.11270 0.443907
\(335\) 0 0
\(336\) 0 0
\(337\) 14.8284i 0.807756i 0.914813 + 0.403878i \(0.132338\pi\)
−0.914813 + 0.403878i \(0.867662\pi\)
\(338\) − 5.10051i − 0.277431i
\(339\) −30.1421 −1.63710
\(340\) 0 0
\(341\) −28.9706 −1.56884
\(342\) 3.31371i 0.179185i
\(343\) 0 0
\(344\) −10.1716 −0.548414
\(345\) 0 0
\(346\) 8.00000 0.430083
\(347\) 22.0711i 1.18484i 0.805630 + 0.592418i \(0.201827\pi\)
−0.805630 + 0.592418i \(0.798173\pi\)
\(348\) − 4.41421i − 0.236627i
\(349\) 26.6569 1.42691 0.713454 0.700702i \(-0.247130\pi\)
0.713454 + 0.700702i \(0.247130\pi\)
\(350\) 0 0
\(351\) −0.343146 −0.0183158
\(352\) − 21.3137i − 1.13602i
\(353\) 21.1716i 1.12685i 0.826168 + 0.563425i \(0.190516\pi\)
−0.826168 + 0.563425i \(0.809484\pi\)
\(354\) −12.4853 −0.663585
\(355\) 0 0
\(356\) −4.85786 −0.257466
\(357\) 0 0
\(358\) − 4.14214i − 0.218919i
\(359\) 10.0000 0.527780 0.263890 0.964553i \(-0.414994\pi\)
0.263890 + 0.964553i \(0.414994\pi\)
\(360\) 0 0
\(361\) −11.0000 −0.578947
\(362\) 3.58579i 0.188465i
\(363\) − 29.7279i − 1.56031i
\(364\) 0 0
\(365\) 0 0
\(366\) 11.4853 0.600345
\(367\) − 11.2426i − 0.586861i −0.955980 0.293431i \(-0.905203\pi\)
0.955980 0.293431i \(-0.0947969\pi\)
\(368\) 7.24264i 0.377549i
\(369\) 6.14214 0.319747
\(370\) 0 0
\(371\) 0 0
\(372\) 26.4853i 1.37320i
\(373\) − 12.9706i − 0.671590i −0.941935 0.335795i \(-0.890995\pi\)
0.941935 0.335795i \(-0.109005\pi\)
\(374\) −1.65685 −0.0856739
\(375\) 0 0
\(376\) 3.17157 0.163561
\(377\) − 0.828427i − 0.0426662i
\(378\) 0 0
\(379\) −21.1716 −1.08751 −0.543755 0.839244i \(-0.682998\pi\)
−0.543755 + 0.839244i \(0.682998\pi\)
\(380\) 0 0
\(381\) 32.1421 1.64669
\(382\) 2.97056i 0.151987i
\(383\) − 16.8995i − 0.863524i −0.901988 0.431762i \(-0.857892\pi\)
0.901988 0.431762i \(-0.142108\pi\)
\(384\) −25.4853 −1.30054
\(385\) 0 0
\(386\) 0.828427 0.0421658
\(387\) 18.1421i 0.922217i
\(388\) 0.627417i 0.0318523i
\(389\) −12.3431 −0.625822 −0.312911 0.949782i \(-0.601304\pi\)
−0.312911 + 0.949782i \(0.601304\pi\)
\(390\) 0 0
\(391\) 2.00000 0.101144
\(392\) 0 0
\(393\) − 8.00000i − 0.403547i
\(394\) −9.79899 −0.493666
\(395\) 0 0
\(396\) −24.9706 −1.25482
\(397\) 28.6274i 1.43677i 0.695646 + 0.718384i \(0.255118\pi\)
−0.695646 + 0.718384i \(0.744882\pi\)
\(398\) 0.686292i 0.0344007i
\(399\) 0 0
\(400\) 0 0
\(401\) 7.68629 0.383835 0.191918 0.981411i \(-0.438529\pi\)
0.191918 + 0.981411i \(0.438529\pi\)
\(402\) − 12.4142i − 0.619165i
\(403\) 4.97056i 0.247601i
\(404\) 22.5147 1.12015
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 3.17157i 0.157016i
\(409\) −24.7990 −1.22623 −0.613116 0.789993i \(-0.710084\pi\)
−0.613116 + 0.789993i \(0.710084\pi\)
\(410\) 0 0
\(411\) 4.00000 0.197305
\(412\) − 0.757359i − 0.0373124i
\(413\) 0 0
\(414\) −2.82843 −0.139010
\(415\) 0 0
\(416\) −3.65685 −0.179292
\(417\) 29.3137i 1.43550i
\(418\) − 5.65685i − 0.276686i
\(419\) 23.3137 1.13895 0.569475 0.822009i \(-0.307147\pi\)
0.569475 + 0.822009i \(0.307147\pi\)
\(420\) 0 0
\(421\) −3.48528 −0.169862 −0.0849311 0.996387i \(-0.527067\pi\)
−0.0849311 + 0.996387i \(0.527067\pi\)
\(422\) − 1.45584i − 0.0708694i
\(423\) − 5.65685i − 0.275046i
\(424\) 10.8284 0.525875
\(425\) 0 0
\(426\) 12.4853 0.604914
\(427\) 0 0
\(428\) 5.04163i 0.243696i
\(429\) −9.65685 −0.466237
\(430\) 0 0
\(431\) −21.7990 −1.05002 −0.525010 0.851096i \(-0.675938\pi\)
−0.525010 + 0.851096i \(0.675938\pi\)
\(432\) − 1.24264i − 0.0597866i
\(433\) 31.7990i 1.52816i 0.645120 + 0.764081i \(0.276807\pi\)
−0.645120 + 0.764081i \(0.723193\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −6.37258 −0.305191
\(437\) 6.82843i 0.326648i
\(438\) 4.82843i 0.230711i
\(439\) −33.9411 −1.61992 −0.809961 0.586484i \(-0.800512\pi\)
−0.809961 + 0.586484i \(0.800512\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0.284271i 0.0135214i
\(443\) 12.2132i 0.580267i 0.956986 + 0.290133i \(0.0936997\pi\)
−0.956986 + 0.290133i \(0.906300\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −4.82843 −0.228633
\(447\) − 18.8995i − 0.893915i
\(448\) 0 0
\(449\) 1.82843 0.0862888 0.0431444 0.999069i \(-0.486262\pi\)
0.0431444 + 0.999069i \(0.486262\pi\)
\(450\) 0 0
\(451\) −10.4853 −0.493733
\(452\) − 22.8284i − 1.07376i
\(453\) − 0.828427i − 0.0389229i
\(454\) 11.1716 0.524308
\(455\) 0 0
\(456\) −10.8284 −0.507088
\(457\) − 32.2843i − 1.51019i −0.655613 0.755097i \(-0.727590\pi\)
0.655613 0.755097i \(-0.272410\pi\)
\(458\) − 0.142136i − 0.00664156i
\(459\) −0.343146 −0.0160167
\(460\) 0 0
\(461\) 18.6863 0.870307 0.435154 0.900356i \(-0.356694\pi\)
0.435154 + 0.900356i \(0.356694\pi\)
\(462\) 0 0
\(463\) − 11.0416i − 0.513148i −0.966525 0.256574i \(-0.917406\pi\)
0.966525 0.256574i \(-0.0825937\pi\)
\(464\) 3.00000 0.139272
\(465\) 0 0
\(466\) 4.62742 0.214361
\(467\) − 22.8995i − 1.05966i −0.848103 0.529831i \(-0.822255\pi\)
0.848103 0.529831i \(-0.177745\pi\)
\(468\) 4.28427i 0.198041i
\(469\) 0 0
\(470\) 0 0
\(471\) −12.8284 −0.591103
\(472\) − 19.7990i − 0.911322i
\(473\) − 30.9706i − 1.42403i
\(474\) 9.17157 0.421264
\(475\) 0 0
\(476\) 0 0
\(477\) − 19.3137i − 0.884314i
\(478\) 0.544156i 0.0248891i
\(479\) 24.3431 1.11227 0.556133 0.831093i \(-0.312284\pi\)
0.556133 + 0.831093i \(0.312284\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) − 6.76955i − 0.308345i
\(483\) 0 0
\(484\) 22.5147 1.02340
\(485\) 0 0
\(486\) 8.97056 0.406913
\(487\) 15.6569i 0.709480i 0.934965 + 0.354740i \(0.115431\pi\)
−0.934965 + 0.354740i \(0.884569\pi\)
\(488\) 18.2132i 0.824473i
\(489\) −57.1127 −2.58273
\(490\) 0 0
\(491\) −13.3137 −0.600839 −0.300420 0.953807i \(-0.597127\pi\)
−0.300420 + 0.953807i \(0.597127\pi\)
\(492\) 9.58579i 0.432161i
\(493\) − 0.828427i − 0.0373105i
\(494\) −0.970563 −0.0436677
\(495\) 0 0
\(496\) −18.0000 −0.808224
\(497\) 0 0
\(498\) − 11.7279i − 0.525541i
\(499\) −4.82843 −0.216150 −0.108075 0.994143i \(-0.534469\pi\)
−0.108075 + 0.994143i \(0.534469\pi\)
\(500\) 0 0
\(501\) 47.2843 2.11251
\(502\) − 5.51472i − 0.246134i
\(503\) − 37.8701i − 1.68854i −0.535916 0.844271i \(-0.680034\pi\)
0.535916 0.844271i \(-0.319966\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 4.82843 0.214650
\(507\) − 29.7279i − 1.32026i
\(508\) 24.3431i 1.08005i
\(509\) −24.6569 −1.09290 −0.546448 0.837493i \(-0.684020\pi\)
−0.546448 + 0.837493i \(0.684020\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) − 22.7574i − 1.00574i
\(513\) − 1.17157i − 0.0517262i
\(514\) 7.31371 0.322594
\(515\) 0 0
\(516\) −28.3137 −1.24644
\(517\) 9.65685i 0.424708i
\(518\) 0 0
\(519\) 46.6274 2.04672
\(520\) 0 0
\(521\) −18.9706 −0.831115 −0.415558 0.909567i \(-0.636414\pi\)
−0.415558 + 0.909567i \(0.636414\pi\)
\(522\) 1.17157i 0.0512784i
\(523\) − 24.3431i − 1.06445i −0.846602 0.532226i \(-0.821356\pi\)
0.846602 0.532226i \(-0.178644\pi\)
\(524\) 6.05887 0.264683
\(525\) 0 0
\(526\) −7.88730 −0.343903
\(527\) 4.97056i 0.216521i
\(528\) − 34.9706i − 1.52190i
\(529\) 17.1716 0.746590
\(530\) 0 0
\(531\) −35.3137 −1.53248
\(532\) 0 0
\(533\) 1.79899i 0.0779229i
\(534\) 2.65685 0.114973
\(535\) 0 0
\(536\) 19.6863 0.850318
\(537\) − 24.1421i − 1.04181i
\(538\) − 12.6152i − 0.543881i
\(539\) 0 0
\(540\) 0 0
\(541\) 18.6569 0.802121 0.401060 0.916052i \(-0.368642\pi\)
0.401060 + 0.916052i \(0.368642\pi\)
\(542\) 0.201010i 0.00863412i
\(543\) 20.8995i 0.896883i
\(544\) −3.65685 −0.156786
\(545\) 0 0
\(546\) 0 0
\(547\) 5.10051i 0.218082i 0.994037 + 0.109041i \(0.0347780\pi\)
−0.994037 + 0.109041i \(0.965222\pi\)
\(548\) 3.02944i 0.129411i
\(549\) 32.4853 1.38644
\(550\) 0 0
\(551\) 2.82843 0.120495
\(552\) − 9.24264i − 0.393393i
\(553\) 0 0
\(554\) −5.02944 −0.213680
\(555\) 0 0
\(556\) −22.2010 −0.941533
\(557\) − 34.2843i − 1.45267i −0.687340 0.726336i \(-0.741222\pi\)
0.687340 0.726336i \(-0.258778\pi\)
\(558\) − 7.02944i − 0.297580i
\(559\) −5.31371 −0.224746
\(560\) 0 0
\(561\) −9.65685 −0.407713
\(562\) − 10.8873i − 0.459253i
\(563\) − 16.2721i − 0.685786i −0.939374 0.342893i \(-0.888593\pi\)
0.939374 0.342893i \(-0.111407\pi\)
\(564\) 8.82843 0.371744
\(565\) 0 0
\(566\) −5.79899 −0.243750
\(567\) 0 0
\(568\) 19.7990i 0.830747i
\(569\) 3.65685 0.153303 0.0766517 0.997058i \(-0.475577\pi\)
0.0766517 + 0.997058i \(0.475577\pi\)
\(570\) 0 0
\(571\) 14.8284 0.620550 0.310275 0.950647i \(-0.399579\pi\)
0.310275 + 0.950647i \(0.399579\pi\)
\(572\) − 7.31371i − 0.305802i
\(573\) 17.3137i 0.723291i
\(574\) 0 0
\(575\) 0 0
\(576\) −11.7990 −0.491625
\(577\) 23.9411i 0.996682i 0.866981 + 0.498341i \(0.166057\pi\)
−0.866981 + 0.498341i \(0.833943\pi\)
\(578\) − 6.75736i − 0.281069i
\(579\) 4.82843 0.200663
\(580\) 0 0
\(581\) 0 0
\(582\) − 0.343146i − 0.0142238i
\(583\) 32.9706i 1.36550i
\(584\) −7.65685 −0.316843
\(585\) 0 0
\(586\) 6.62742 0.273776
\(587\) 22.2843i 0.919770i 0.887978 + 0.459885i \(0.152109\pi\)
−0.887978 + 0.459885i \(0.847891\pi\)
\(588\) 0 0
\(589\) −16.9706 −0.699260
\(590\) 0 0
\(591\) −57.1127 −2.34930
\(592\) 0 0
\(593\) − 43.7990i − 1.79861i −0.437323 0.899304i \(-0.644073\pi\)
0.437323 0.899304i \(-0.355927\pi\)
\(594\) −0.828427 −0.0339908
\(595\) 0 0
\(596\) 14.3137 0.586312
\(597\) 4.00000i 0.163709i
\(598\) − 0.828427i − 0.0338769i
\(599\) −17.6569 −0.721440 −0.360720 0.932674i \(-0.617469\pi\)
−0.360720 + 0.932674i \(0.617469\pi\)
\(600\) 0 0
\(601\) 8.34315 0.340324 0.170162 0.985416i \(-0.445571\pi\)
0.170162 + 0.985416i \(0.445571\pi\)
\(602\) 0 0
\(603\) − 35.1127i − 1.42990i
\(604\) 0.627417 0.0255292
\(605\) 0 0
\(606\) −12.3137 −0.500210
\(607\) − 4.21320i − 0.171009i −0.996338 0.0855043i \(-0.972750\pi\)
0.996338 0.0855043i \(-0.0272501\pi\)
\(608\) − 12.4853i − 0.506345i
\(609\) 0 0
\(610\) 0 0
\(611\) 1.65685 0.0670291
\(612\) 4.28427i 0.173181i
\(613\) 15.4558i 0.624256i 0.950040 + 0.312128i \(0.101042\pi\)
−0.950040 + 0.312128i \(0.898958\pi\)
\(614\) −5.48528 −0.221368
\(615\) 0 0
\(616\) 0 0
\(617\) − 11.3137i − 0.455473i −0.973723 0.227736i \(-0.926868\pi\)
0.973723 0.227736i \(-0.0731324\pi\)
\(618\) 0.414214i 0.0166621i
\(619\) −42.4853 −1.70763 −0.853814 0.520578i \(-0.825716\pi\)
−0.853814 + 0.520578i \(0.825716\pi\)
\(620\) 0 0
\(621\) 1.00000 0.0401286
\(622\) 7.79899i 0.312711i
\(623\) 0 0
\(624\) −6.00000 −0.240192
\(625\) 0 0
\(626\) 7.31371 0.292315
\(627\) − 32.9706i − 1.31672i
\(628\) − 9.71573i − 0.387700i
\(629\) 0 0
\(630\) 0 0
\(631\) 8.14214 0.324133 0.162067 0.986780i \(-0.448184\pi\)
0.162067 + 0.986780i \(0.448184\pi\)
\(632\) 14.5442i 0.578535i
\(633\) − 8.48528i − 0.337260i
\(634\) 10.6863 0.424407
\(635\) 0 0
\(636\) 30.1421 1.19521
\(637\) 0 0
\(638\) − 2.00000i − 0.0791808i
\(639\) 35.3137 1.39699
\(640\) 0 0
\(641\) 14.5147 0.573297 0.286648 0.958036i \(-0.407459\pi\)
0.286648 + 0.958036i \(0.407459\pi\)
\(642\) − 2.75736i − 0.108824i
\(643\) 30.2843i 1.19430i 0.802131 + 0.597148i \(0.203699\pi\)
−0.802131 + 0.597148i \(0.796301\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −0.970563 −0.0381863
\(647\) 17.0416i 0.669976i 0.942222 + 0.334988i \(0.108732\pi\)
−0.942222 + 0.334988i \(0.891268\pi\)
\(648\) 15.0416i 0.590891i
\(649\) 60.2843 2.36636
\(650\) 0 0
\(651\) 0 0
\(652\) − 43.2548i − 1.69399i
\(653\) 24.8284i 0.971611i 0.874067 + 0.485806i \(0.161474\pi\)
−0.874067 + 0.485806i \(0.838526\pi\)
\(654\) 3.48528 0.136285
\(655\) 0 0
\(656\) −6.51472 −0.254357
\(657\) 13.6569i 0.532805i
\(658\) 0 0
\(659\) −26.8284 −1.04509 −0.522544 0.852613i \(-0.675017\pi\)
−0.522544 + 0.852613i \(0.675017\pi\)
\(660\) 0 0
\(661\) 26.1716 1.01796 0.508978 0.860779i \(-0.330023\pi\)
0.508978 + 0.860779i \(0.330023\pi\)
\(662\) 4.54416i 0.176614i
\(663\) 1.65685i 0.0643469i
\(664\) 18.5980 0.721742
\(665\) 0 0
\(666\) 0 0
\(667\) 2.41421i 0.0934787i
\(668\) 35.8112i 1.38558i
\(669\) −28.1421 −1.08804
\(670\) 0 0
\(671\) −55.4558 −2.14085
\(672\) 0 0
\(673\) − 18.3431i − 0.707076i −0.935420 0.353538i \(-0.884978\pi\)
0.935420 0.353538i \(-0.115022\pi\)
\(674\) 6.14214 0.236586
\(675\) 0 0
\(676\) 22.5147 0.865951
\(677\) 0.142136i 0.00546272i 0.999996 + 0.00273136i \(0.000869419\pi\)
−0.999996 + 0.00273136i \(0.999131\pi\)
\(678\) 12.4853i 0.479494i
\(679\) 0 0
\(680\) 0 0
\(681\) 65.1127 2.49512
\(682\) 12.0000i 0.459504i
\(683\) 43.2426i 1.65463i 0.561736 + 0.827317i \(0.310134\pi\)
−0.561736 + 0.827317i \(0.689866\pi\)
\(684\) −14.6274 −0.559293
\(685\) 0 0
\(686\) 0 0
\(687\) − 0.828427i − 0.0316065i
\(688\) − 19.2426i − 0.733619i
\(689\) 5.65685 0.215509
\(690\) 0 0
\(691\) −4.82843 −0.183682 −0.0918410 0.995774i \(-0.529275\pi\)
−0.0918410 + 0.995774i \(0.529275\pi\)
\(692\) 35.3137i 1.34243i
\(693\) 0 0
\(694\) 9.14214 0.347031
\(695\) 0 0
\(696\) −3.82843 −0.145116
\(697\) 1.79899i 0.0681416i
\(698\) − 11.0416i − 0.417932i
\(699\) 26.9706 1.02012
\(700\) 0 0
\(701\) −42.7990 −1.61650 −0.808248 0.588843i \(-0.799584\pi\)
−0.808248 + 0.588843i \(0.799584\pi\)
\(702\) 0.142136i 0.00536456i
\(703\) 0 0
\(704\) 20.1421 0.759135
\(705\) 0 0
\(706\) 8.76955 0.330046
\(707\) 0 0
\(708\) − 55.1127i − 2.07126i
\(709\) −38.3137 −1.43890 −0.719451 0.694543i \(-0.755606\pi\)
−0.719451 + 0.694543i \(0.755606\pi\)
\(710\) 0 0
\(711\) 25.9411 0.972868
\(712\) 4.21320i 0.157896i
\(713\) − 14.4853i − 0.542478i
\(714\) 0 0
\(715\) 0 0
\(716\) 18.2843 0.683315
\(717\) 3.17157i 0.118445i
\(718\) − 4.14214i − 0.154583i
\(719\) 41.1127 1.53324 0.766622 0.642098i \(-0.221936\pi\)
0.766622 + 0.642098i \(0.221936\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 4.55635i 0.169570i
\(723\) − 39.4558i − 1.46738i
\(724\) −15.8284 −0.588259
\(725\) 0 0
\(726\) −12.3137 −0.457005
\(727\) − 40.4142i − 1.49888i −0.662072 0.749440i \(-0.730323\pi\)
0.662072 0.749440i \(-0.269677\pi\)
\(728\) 0 0
\(729\) 23.8284 0.882534
\(730\) 0 0
\(731\) −5.31371 −0.196535
\(732\) 50.6985i 1.87387i
\(733\) 22.0000i 0.812589i 0.913742 + 0.406294i \(0.133179\pi\)
−0.913742 + 0.406294i \(0.866821\pi\)
\(734\) −4.65685 −0.171888
\(735\) 0 0
\(736\) 10.6569 0.392817
\(737\) 59.9411i 2.20796i
\(738\) − 2.54416i − 0.0936517i
\(739\) −41.1127 −1.51236 −0.756178 0.654367i \(-0.772935\pi\)
−0.756178 + 0.654367i \(0.772935\pi\)
\(740\) 0 0
\(741\) −5.65685 −0.207810
\(742\) 0 0
\(743\) 1.92893i 0.0707657i 0.999374 + 0.0353828i \(0.0112651\pi\)
−0.999374 + 0.0353828i \(0.988735\pi\)
\(744\) 22.9706 0.842142
\(745\) 0 0
\(746\) −5.37258 −0.196704
\(747\) − 33.1716i − 1.21368i
\(748\) − 7.31371i − 0.267416i
\(749\) 0 0
\(750\) 0 0
\(751\) −41.6569 −1.52008 −0.760040 0.649876i \(-0.774821\pi\)
−0.760040 + 0.649876i \(0.774821\pi\)
\(752\) 6.00000i 0.218797i
\(753\) − 32.1421i − 1.17132i
\(754\) −0.343146 −0.0124966
\(755\) 0 0
\(756\) 0 0
\(757\) 19.4558i 0.707135i 0.935409 + 0.353567i \(0.115031\pi\)
−0.935409 + 0.353567i \(0.884969\pi\)
\(758\) 8.76955i 0.318524i
\(759\) 28.1421 1.02149
\(760\) 0 0
\(761\) −13.3137 −0.482622 −0.241311 0.970448i \(-0.577577\pi\)
−0.241311 + 0.970448i \(0.577577\pi\)
\(762\) − 13.3137i − 0.482305i
\(763\) 0 0
\(764\) −13.1127 −0.474401
\(765\) 0 0
\(766\) −7.00000 −0.252920
\(767\) − 10.3431i − 0.373469i
\(768\) − 9.58579i − 0.345897i
\(769\) −44.6274 −1.60931 −0.804653 0.593745i \(-0.797649\pi\)
−0.804653 + 0.593745i \(0.797649\pi\)
\(770\) 0 0
\(771\) 42.6274 1.53519
\(772\) 3.65685i 0.131613i
\(773\) − 25.1127i − 0.903241i −0.892210 0.451620i \(-0.850846\pi\)
0.892210 0.451620i \(-0.149154\pi\)
\(774\) 7.51472 0.270111
\(775\) 0 0
\(776\) 0.544156 0.0195341
\(777\) 0 0
\(778\) 5.11270i 0.183299i
\(779\) −6.14214 −0.220065
\(780\) 0 0
\(781\) −60.2843 −2.15714
\(782\) − 0.828427i − 0.0296245i
\(783\) − 0.414214i − 0.0148028i
\(784\) 0 0
\(785\) 0 0
\(786\) −3.31371 −0.118196
\(787\) − 28.5563i − 1.01792i −0.860789 0.508962i \(-0.830029\pi\)
0.860789 0.508962i \(-0.169971\pi\)
\(788\) − 43.2548i − 1.54089i
\(789\) −45.9706 −1.63660
\(790\) 0 0
\(791\) 0 0
\(792\) 21.6569i 0.769543i
\(793\) 9.51472i 0.337878i
\(794\) 11.8579 0.420820
\(795\) 0 0
\(796\) −3.02944 −0.107376
\(797\) − 8.00000i − 0.283375i −0.989911 0.141687i \(-0.954747\pi\)
0.989911 0.141687i \(-0.0452527\pi\)
\(798\) 0 0
\(799\) 1.65685 0.0586153
\(800\) 0 0
\(801\) 7.51472 0.265520
\(802\) − 3.18377i − 0.112423i
\(803\) − 23.3137i − 0.822723i
\(804\) 54.7990 1.93261
\(805\) 0 0
\(806\) 2.05887 0.0725208
\(807\) − 73.5269i − 2.58827i
\(808\) − 19.5269i − 0.686954i
\(809\) 9.62742 0.338482 0.169241 0.985575i \(-0.445868\pi\)
0.169241 + 0.985575i \(0.445868\pi\)
\(810\) 0 0
\(811\) 24.6274 0.864786 0.432393 0.901685i \(-0.357669\pi\)
0.432393 + 0.901685i \(0.357669\pi\)
\(812\) 0 0
\(813\) 1.17157i 0.0410889i
\(814\) 0 0
\(815\) 0 0
\(816\) −6.00000 −0.210042
\(817\) − 18.1421i − 0.634713i
\(818\) 10.2721i 0.359155i
\(819\) 0 0
\(820\) 0 0
\(821\) −19.9411 −0.695950 −0.347975 0.937504i \(-0.613131\pi\)
−0.347975 + 0.937504i \(0.613131\pi\)
\(822\) − 1.65685i − 0.0577894i
\(823\) 12.0711i 0.420771i 0.977619 + 0.210385i \(0.0674719\pi\)
−0.977619 + 0.210385i \(0.932528\pi\)
\(824\) −0.656854 −0.0228826
\(825\) 0 0
\(826\) 0 0
\(827\) 16.2132i 0.563788i 0.959446 + 0.281894i \(0.0909627\pi\)
−0.959446 + 0.281894i \(0.909037\pi\)
\(828\) − 12.4853i − 0.433894i
\(829\) −6.68629 −0.232225 −0.116112 0.993236i \(-0.537043\pi\)
−0.116112 + 0.993236i \(0.537043\pi\)
\(830\) 0 0
\(831\) −29.3137 −1.01688
\(832\) − 3.45584i − 0.119810i
\(833\) 0 0
\(834\) 12.1421 0.420448
\(835\) 0 0
\(836\) 24.9706 0.863625
\(837\) 2.48528i 0.0859039i
\(838\) − 9.65685i − 0.333590i
\(839\) −20.8284 −0.719077 −0.359539 0.933130i \(-0.617066\pi\)
−0.359539 + 0.933130i \(0.617066\pi\)
\(840\) 0 0
\(841\) −28.0000 −0.965517
\(842\) 1.44365i 0.0497515i
\(843\) − 63.4558i − 2.18554i
\(844\) 6.42641 0.221206
\(845\) 0 0
\(846\) −2.34315 −0.0805590
\(847\) 0 0
\(848\) 20.4853i 0.703467i
\(849\) −33.7990 −1.15998
\(850\) 0 0
\(851\) 0 0
\(852\) 55.1127i 1.88813i
\(853\) − 53.4558i − 1.83029i −0.403121 0.915147i \(-0.632075\pi\)
0.403121 0.915147i \(-0.367925\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 4.37258 0.149452
\(857\) − 22.2843i − 0.761216i −0.924736 0.380608i \(-0.875715\pi\)
0.924736 0.380608i \(-0.124285\pi\)
\(858\) 4.00000i 0.136558i
\(859\) −46.6274 −1.59091 −0.795453 0.606015i \(-0.792767\pi\)
−0.795453 + 0.606015i \(0.792767\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 9.02944i 0.307544i
\(863\) 16.5563i 0.563585i 0.959475 + 0.281792i \(0.0909289\pi\)
−0.959475 + 0.281792i \(0.909071\pi\)
\(864\) −1.82843 −0.0622044
\(865\) 0 0
\(866\) 13.1716 0.447588
\(867\) − 39.3848i − 1.33758i
\(868\) 0 0
\(869\) −44.2843 −1.50224
\(870\) 0 0
\(871\) 10.2843 0.348469
\(872\) 5.52691i 0.187165i
\(873\) − 0.970563i − 0.0328486i
\(874\) 2.82843 0.0956730
\(875\) 0 0
\(876\) −21.3137 −0.720123
\(877\) 30.8284i 1.04100i 0.853861 + 0.520501i \(0.174255\pi\)
−0.853861 + 0.520501i \(0.825745\pi\)
\(878\) 14.0589i 0.474464i
\(879\) 38.6274 1.30287
\(880\) 0 0
\(881\) −3.82843 −0.128983 −0.0644915 0.997918i \(-0.520543\pi\)
−0.0644915 + 0.997918i \(0.520543\pi\)
\(882\) 0 0
\(883\) 38.2843i 1.28837i 0.764870 + 0.644184i \(0.222803\pi\)
−0.764870 + 0.644184i \(0.777197\pi\)
\(884\) −1.25483 −0.0422046
\(885\) 0 0
\(886\) 5.05887 0.169956
\(887\) 44.0711i 1.47976i 0.672738 + 0.739881i \(0.265118\pi\)
−0.672738 + 0.739881i \(0.734882\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −45.7990 −1.53432
\(892\) − 21.3137i − 0.713636i
\(893\) 5.65685i 0.189299i
\(894\) −7.82843 −0.261822
\(895\) 0 0
\(896\) 0 0
\(897\) − 4.82843i − 0.161216i
\(898\) − 0.757359i − 0.0252734i
\(899\) −6.00000 −0.200111
\(900\) 0 0
\(901\) 5.65685 0.188457
\(902\) 4.34315i 0.144611i
\(903\) 0 0
\(904\) −19.7990 −0.658505
\(905\) 0 0
\(906\) −0.343146 −0.0114003
\(907\) 28.2132i 0.936804i 0.883515 + 0.468402i \(0.155170\pi\)
−0.883515 + 0.468402i \(0.844830\pi\)
\(908\) 49.3137i 1.63653i
\(909\) −34.8284 −1.15519
\(910\) 0 0
\(911\) −49.7990 −1.64991 −0.824957 0.565195i \(-0.808801\pi\)
−0.824957 + 0.565195i \(0.808801\pi\)
\(912\) − 20.4853i − 0.678335i
\(913\) 56.6274i 1.87409i
\(914\) −13.3726 −0.442326
\(915\) 0 0
\(916\) 0.627417 0.0207304
\(917\) 0 0
\(918\) 0.142136i 0.00469117i
\(919\) −19.1127 −0.630470 −0.315235 0.949014i \(-0.602083\pi\)
−0.315235 + 0.949014i \(0.602083\pi\)
\(920\) 0 0
\(921\) −31.9706 −1.05347
\(922\) − 7.74012i − 0.254907i
\(923\) 10.3431i 0.340449i
\(924\) 0 0
\(925\) 0 0
\(926\) −4.57359 −0.150298
\(927\) 1.17157i 0.0384795i
\(928\) − 4.41421i − 0.144904i
\(929\) 11.4853 0.376820 0.188410 0.982090i \(-0.439667\pi\)
0.188410 + 0.982090i \(0.439667\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 20.4264i 0.669089i
\(933\) 45.4558i 1.48816i
\(934\) −9.48528 −0.310368
\(935\) 0 0
\(936\) 3.71573 0.121452
\(937\) − 10.6274i − 0.347183i −0.984818 0.173591i \(-0.944463\pi\)
0.984818 0.173591i \(-0.0555372\pi\)
\(938\) 0 0
\(939\) 42.6274 1.39109
\(940\) 0 0
\(941\) 10.2843 0.335258 0.167629 0.985850i \(-0.446389\pi\)
0.167629 + 0.985850i \(0.446389\pi\)
\(942\) 5.31371i 0.173130i
\(943\) − 5.24264i − 0.170724i
\(944\) 37.4558 1.21908
\(945\) 0 0
\(946\) −12.8284 −0.417088
\(947\) − 43.1838i − 1.40328i −0.712530 0.701642i \(-0.752451\pi\)
0.712530 0.701642i \(-0.247549\pi\)
\(948\) 40.4853i 1.31490i
\(949\) −4.00000 −0.129845
\(950\) 0 0
\(951\) 62.2843 2.01971
\(952\) 0 0
\(953\) 2.34315i 0.0759019i 0.999280 + 0.0379510i \(0.0120831\pi\)
−0.999280 + 0.0379510i \(0.987917\pi\)
\(954\) −8.00000 −0.259010
\(955\) 0 0
\(956\) −2.40202 −0.0776869
\(957\) − 11.6569i − 0.376813i
\(958\) − 10.0833i − 0.325775i
\(959\) 0 0
\(960\) 0 0
\(961\) 5.00000 0.161290
\(962\) 0 0
\(963\) − 7.79899i − 0.251319i
\(964\) 29.8823 0.962442
\(965\) 0 0
\(966\) 0 0
\(967\) − 27.5269i − 0.885206i −0.896718 0.442603i \(-0.854055\pi\)
0.896718 0.442603i \(-0.145945\pi\)
\(968\) − 19.5269i − 0.627619i
\(969\) −5.65685 −0.181724
\(970\) 0 0
\(971\) −24.0000 −0.770197 −0.385098 0.922876i \(-0.625832\pi\)
−0.385098 + 0.922876i \(0.625832\pi\)
\(972\) 39.5980i 1.27011i
\(973\) 0 0
\(974\) 6.48528 0.207802
\(975\) 0 0
\(976\) −34.4558 −1.10290
\(977\) 21.3137i 0.681886i 0.940084 + 0.340943i \(0.110746\pi\)
−0.940084 + 0.340943i \(0.889254\pi\)
\(978\) 23.6569i 0.756463i
\(979\) −12.8284 −0.409998
\(980\) 0 0
\(981\) 9.85786 0.314737
\(982\) 5.51472i 0.175982i
\(983\) 14.2132i 0.453331i 0.973973 + 0.226665i \(0.0727824\pi\)
−0.973973 + 0.226665i \(0.927218\pi\)
\(984\) 8.31371 0.265031
\(985\) 0 0
\(986\) −0.343146 −0.0109280
\(987\) 0 0
\(988\) − 4.28427i − 0.136301i
\(989\) 15.4853 0.492403
\(990\) 0 0
\(991\) −15.6569 −0.497356 −0.248678 0.968586i \(-0.579996\pi\)
−0.248678 + 0.968586i \(0.579996\pi\)
\(992\) 26.4853i 0.840909i
\(993\) 26.4853i 0.840485i
\(994\) 0 0
\(995\) 0 0
\(996\) 51.7696 1.64038
\(997\) − 17.4558i − 0.552832i −0.961038 0.276416i \(-0.910853\pi\)
0.961038 0.276416i \(-0.0891468\pi\)
\(998\) 2.00000i 0.0633089i
\(999\) 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 1225.2.b.g.99.2 4
5.2 odd 4 1225.2.a.k.1.2 2
5.3 odd 4 245.2.a.h.1.1 2
5.4 even 2 inner 1225.2.b.g.99.3 4
7.2 even 3 175.2.k.a.74.2 8
7.4 even 3 175.2.k.a.149.3 8
7.6 odd 2 1225.2.b.h.99.2 4
15.8 even 4 2205.2.a.n.1.2 2
20.3 even 4 3920.2.a.bq.1.1 2
35.2 odd 12 175.2.e.c.151.1 4
35.3 even 12 245.2.e.e.226.2 4
35.4 even 6 175.2.k.a.149.2 8
35.9 even 6 175.2.k.a.74.3 8
35.13 even 4 245.2.a.g.1.1 2
35.18 odd 12 35.2.e.a.16.2 yes 4
35.23 odd 12 35.2.e.a.11.2 4
35.27 even 4 1225.2.a.m.1.2 2
35.32 odd 12 175.2.e.c.51.1 4
35.33 even 12 245.2.e.e.116.2 4
35.34 odd 2 1225.2.b.h.99.3 4
105.23 even 12 315.2.j.e.46.1 4
105.53 even 12 315.2.j.e.226.1 4
105.83 odd 4 2205.2.a.q.1.2 2
140.23 even 12 560.2.q.k.81.2 4
140.83 odd 4 3920.2.a.bv.1.2 2
140.123 even 12 560.2.q.k.401.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
35.2.e.a.11.2 4 35.23 odd 12
35.2.e.a.16.2 yes 4 35.18 odd 12
175.2.e.c.51.1 4 35.32 odd 12
175.2.e.c.151.1 4 35.2 odd 12
175.2.k.a.74.2 8 7.2 even 3
175.2.k.a.74.3 8 35.9 even 6
175.2.k.a.149.2 8 35.4 even 6
175.2.k.a.149.3 8 7.4 even 3
245.2.a.g.1.1 2 35.13 even 4
245.2.a.h.1.1 2 5.3 odd 4
245.2.e.e.116.2 4 35.33 even 12
245.2.e.e.226.2 4 35.3 even 12
315.2.j.e.46.1 4 105.23 even 12
315.2.j.e.226.1 4 105.53 even 12
560.2.q.k.81.2 4 140.23 even 12
560.2.q.k.401.2 4 140.123 even 12
1225.2.a.k.1.2 2 5.2 odd 4
1225.2.a.m.1.2 2 35.27 even 4
1225.2.b.g.99.2 4 1.1 even 1 trivial
1225.2.b.g.99.3 4 5.4 even 2 inner
1225.2.b.h.99.2 4 7.6 odd 2
1225.2.b.h.99.3 4 35.34 odd 2
2205.2.a.n.1.2 2 15.8 even 4
2205.2.a.q.1.2 2 105.83 odd 4
3920.2.a.bq.1.1 2 20.3 even 4
3920.2.a.bv.1.2 2 140.83 odd 4