Properties

Label 75.22.b.c.49.2
Level $75$
Weight $22$
Character 75.49
Analytic conductor $209.608$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [75,22,Mod(49,75)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(75, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 22, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("75.49");
 
S:= CuspForms(chi, 22);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 22 \)
Character orbit: \([\chi]\) \(=\) 75.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: \(209.608008215\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 15)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 49.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 75.49
Dual form 75.22.b.c.49.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+544.000i q^{2} -59049.0i q^{3} +1.80122e6 q^{4} +3.21227e7 q^{6} +1.27770e9i q^{7} +2.12071e9i q^{8} -3.48678e9 q^{9} +O(q^{10})\) \(q+544.000i q^{2} -59049.0i q^{3} +1.80122e6 q^{4} +3.21227e7 q^{6} +1.27770e9i q^{7} +2.12071e9i q^{8} -3.48678e9 q^{9} -7.75859e10 q^{11} -1.06360e11i q^{12} +4.34111e11i q^{13} -6.95068e11 q^{14} +2.62376e12 q^{16} +1.28489e13i q^{17} -1.89681e12i q^{18} +2.86053e13 q^{19} +7.54468e13 q^{21} -4.22067e13i q^{22} -2.24022e14i q^{23} +1.25226e14 q^{24} -2.36156e14 q^{26} +2.05891e14i q^{27} +2.30141e15i q^{28} +5.16760e13 q^{29} +8.92111e15 q^{31} +5.87478e15i q^{32} +4.58137e15i q^{33} -6.98981e15 q^{34} -6.28045e15 q^{36} +4.39772e16i q^{37} +1.55613e16i q^{38} +2.56338e16 q^{39} +5.81681e16 q^{41} +4.10431e16i q^{42} +1.61438e17i q^{43} -1.39749e17 q^{44} +1.21868e17 q^{46} -1.60065e17i q^{47} -1.54930e17i q^{48} -1.07397e18 q^{49} +7.58716e17 q^{51} +7.81927e17i q^{52} -2.29953e18i q^{53} -1.12005e17 q^{54} -2.70963e18 q^{56} -1.68911e18i q^{57} +2.81118e16i q^{58} -5.15426e18 q^{59} +1.25169e18 q^{61} +4.85308e18i q^{62} -4.45506e18i q^{63} +2.30654e18 q^{64} -2.49227e18 q^{66} -5.40779e18i q^{67} +2.31437e19i q^{68} -1.32283e19 q^{69} -1.10432e19 q^{71} -7.39447e18i q^{72} +3.77012e19i q^{73} -2.39236e19 q^{74} +5.15242e19 q^{76} -9.91314e19i q^{77} +1.39448e19i q^{78} -6.31554e19 q^{79} +1.21577e19 q^{81} +3.16434e19i q^{82} +1.45158e20i q^{83} +1.35896e20 q^{84} -8.78222e19 q^{86} -3.05142e18i q^{87} -1.64537e20i q^{88} -1.37255e20 q^{89} -5.54663e20 q^{91} -4.03512e20i q^{92} -5.26783e20i q^{93} +8.70752e19 q^{94} +3.46900e20 q^{96} -3.24306e20i q^{97} -5.84238e20i q^{98} +2.70525e20 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 3602432 q^{4} + 64245312 q^{6} - 6973568802 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 3602432 q^{4} + 64245312 q^{6} - 6973568802 q^{9} - 155171843488 q^{11} - 1390135837440 q^{14} + 5247512608768 q^{16} + 57210512319592 q^{19} + 150893623281240 q^{21} + 250451868450816 q^{24} - 472312657787776 q^{26} + 103352061666284 q^{29} + 17\!\cdots\!00 q^{31}+ \cdots + 54\!\cdots\!88 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/75\mathbb{Z}\right)^\times\).

\(n\) \(26\) \(52\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 544.000i 0.375650i 0.982202 + 0.187825i \(0.0601439\pi\)
−0.982202 + 0.187825i \(0.939856\pi\)
\(3\) − 59049.0i − 0.577350i
\(4\) 1.80122e6 0.858887
\(5\) 0 0
\(6\) 3.21227e7 0.216882
\(7\) 1.27770e9i 1.70962i 0.518943 + 0.854809i \(0.326326\pi\)
−0.518943 + 0.854809i \(0.673674\pi\)
\(8\) 2.12071e9i 0.698292i
\(9\) −3.48678e9 −0.333333
\(10\) 0 0
\(11\) −7.75859e10 −0.901903 −0.450951 0.892549i \(-0.648915\pi\)
−0.450951 + 0.892549i \(0.648915\pi\)
\(12\) − 1.06360e11i − 0.495878i
\(13\) 4.34111e11i 0.873364i 0.899616 + 0.436682i \(0.143847\pi\)
−0.899616 + 0.436682i \(0.856153\pi\)
\(14\) −6.95068e11 −0.642219
\(15\) 0 0
\(16\) 2.62376e12 0.596573
\(17\) 1.28489e13i 1.54580i 0.634529 + 0.772899i \(0.281194\pi\)
−0.634529 + 0.772899i \(0.718806\pi\)
\(18\) − 1.89681e12i − 0.125217i
\(19\) 2.86053e13 1.07037 0.535184 0.844736i \(-0.320242\pi\)
0.535184 + 0.844736i \(0.320242\pi\)
\(20\) 0 0
\(21\) 7.54468e13 0.987048
\(22\) − 4.22067e13i − 0.338800i
\(23\) − 2.24022e14i − 1.12758i −0.825917 0.563792i \(-0.809342\pi\)
0.825917 0.563792i \(-0.190658\pi\)
\(24\) 1.25226e14 0.403159
\(25\) 0 0
\(26\) −2.36156e14 −0.328080
\(27\) 2.05891e14i 0.192450i
\(28\) 2.30141e15i 1.46837i
\(29\) 5.16760e13 0.0228092 0.0114046 0.999935i \(-0.496370\pi\)
0.0114046 + 0.999935i \(0.496370\pi\)
\(30\) 0 0
\(31\) 8.92111e15 1.95488 0.977442 0.211203i \(-0.0677382\pi\)
0.977442 + 0.211203i \(0.0677382\pi\)
\(32\) 5.87478e15i 0.922395i
\(33\) 4.58137e15i 0.520714i
\(34\) −6.98981e15 −0.580680
\(35\) 0 0
\(36\) −6.28045e15 −0.286296
\(37\) 4.39772e16i 1.50352i 0.659437 + 0.751760i \(0.270795\pi\)
−0.659437 + 0.751760i \(0.729205\pi\)
\(38\) 1.55613e16i 0.402084i
\(39\) 2.56338e16 0.504237
\(40\) 0 0
\(41\) 5.81681e16 0.676791 0.338395 0.941004i \(-0.390116\pi\)
0.338395 + 0.941004i \(0.390116\pi\)
\(42\) 4.10431e16i 0.370785i
\(43\) 1.61438e17i 1.13916i 0.821934 + 0.569582i \(0.192895\pi\)
−0.821934 + 0.569582i \(0.807105\pi\)
\(44\) −1.39749e17 −0.774632
\(45\) 0 0
\(46\) 1.21868e17 0.423577
\(47\) − 1.60065e17i − 0.443883i −0.975060 0.221941i \(-0.928761\pi\)
0.975060 0.221941i \(-0.0712394\pi\)
\(48\) − 1.54930e17i − 0.344432i
\(49\) −1.07397e18 −1.92279
\(50\) 0 0
\(51\) 7.58716e17 0.892467
\(52\) 7.81927e17i 0.750121i
\(53\) − 2.29953e18i − 1.80610i −0.429535 0.903050i \(-0.641322\pi\)
0.429535 0.903050i \(-0.358678\pi\)
\(54\) −1.12005e17 −0.0722940
\(55\) 0 0
\(56\) −2.70963e18 −1.19381
\(57\) − 1.68911e18i − 0.617977i
\(58\) 2.81118e16i 0.00856829i
\(59\) −5.15426e18 −1.31286 −0.656432 0.754385i \(-0.727935\pi\)
−0.656432 + 0.754385i \(0.727935\pi\)
\(60\) 0 0
\(61\) 1.25169e18 0.224663 0.112332 0.993671i \(-0.464168\pi\)
0.112332 + 0.993671i \(0.464168\pi\)
\(62\) 4.85308e18i 0.734353i
\(63\) − 4.45506e18i − 0.569872i
\(64\) 2.30654e18 0.250075
\(65\) 0 0
\(66\) −2.49227e18 −0.195606
\(67\) − 5.40779e18i − 0.362438i −0.983443 0.181219i \(-0.941996\pi\)
0.983443 0.181219i \(-0.0580043\pi\)
\(68\) 2.31437e19i 1.32767i
\(69\) −1.32283e19 −0.651011
\(70\) 0 0
\(71\) −1.10432e19 −0.402609 −0.201305 0.979529i \(-0.564518\pi\)
−0.201305 + 0.979529i \(0.564518\pi\)
\(72\) − 7.39447e18i − 0.232764i
\(73\) 3.77012e19i 1.02675i 0.858164 + 0.513375i \(0.171605\pi\)
−0.858164 + 0.513375i \(0.828395\pi\)
\(74\) −2.39236e19 −0.564798
\(75\) 0 0
\(76\) 5.15242e19 0.919325
\(77\) − 9.91314e19i − 1.54191i
\(78\) 1.39448e19i 0.189417i
\(79\) −6.31554e19 −0.750457 −0.375229 0.926932i \(-0.622436\pi\)
−0.375229 + 0.926932i \(0.622436\pi\)
\(80\) 0 0
\(81\) 1.21577e19 0.111111
\(82\) 3.16434e19i 0.254237i
\(83\) 1.45158e20i 1.02688i 0.858124 + 0.513442i \(0.171630\pi\)
−0.858124 + 0.513442i \(0.828370\pi\)
\(84\) 1.35896e20 0.847762
\(85\) 0 0
\(86\) −8.78222e19 −0.427928
\(87\) − 3.05142e18i − 0.0131689i
\(88\) − 1.64537e20i − 0.629791i
\(89\) −1.37255e20 −0.466588 −0.233294 0.972406i \(-0.574950\pi\)
−0.233294 + 0.972406i \(0.574950\pi\)
\(90\) 0 0
\(91\) −5.54663e20 −1.49312
\(92\) − 4.03512e20i − 0.968467i
\(93\) − 5.26783e20i − 1.12865i
\(94\) 8.70752e19 0.166745
\(95\) 0 0
\(96\) 3.46900e20 0.532545
\(97\) − 3.24306e20i − 0.446532i −0.974758 0.223266i \(-0.928328\pi\)
0.974758 0.223266i \(-0.0716718\pi\)
\(98\) − 5.84238e20i − 0.722298i
\(99\) 2.70525e20 0.300634
\(100\) 0 0
\(101\) 1.14557e21 1.03192 0.515962 0.856611i \(-0.327435\pi\)
0.515962 + 0.856611i \(0.327435\pi\)
\(102\) 4.12741e20i 0.335256i
\(103\) − 8.95787e20i − 0.656771i −0.944544 0.328385i \(-0.893496\pi\)
0.944544 0.328385i \(-0.106504\pi\)
\(104\) −9.20624e20 −0.609863
\(105\) 0 0
\(106\) 1.25094e21 0.678463
\(107\) 1.25783e21i 0.618146i 0.951038 + 0.309073i \(0.100019\pi\)
−0.951038 + 0.309073i \(0.899981\pi\)
\(108\) 3.70854e20i 0.165293i
\(109\) 1.66725e21 0.674562 0.337281 0.941404i \(-0.390493\pi\)
0.337281 + 0.941404i \(0.390493\pi\)
\(110\) 0 0
\(111\) 2.59681e21 0.868057
\(112\) 3.35237e21i 1.01991i
\(113\) 6.51198e21i 1.80463i 0.431073 + 0.902317i \(0.358135\pi\)
−0.431073 + 0.902317i \(0.641865\pi\)
\(114\) 9.18877e20 0.232143
\(115\) 0 0
\(116\) 9.30797e19 0.0195905
\(117\) − 1.51365e21i − 0.291121i
\(118\) − 2.80392e21i − 0.493178i
\(119\) −1.64170e22 −2.64272
\(120\) 0 0
\(121\) −1.38067e21 −0.186571
\(122\) 6.80917e20i 0.0843949i
\(123\) − 3.43477e21i − 0.390745i
\(124\) 1.60688e22 1.67902
\(125\) 0 0
\(126\) 2.42355e21 0.214073
\(127\) − 7.99589e21i − 0.650022i −0.945710 0.325011i \(-0.894632\pi\)
0.945710 0.325011i \(-0.105368\pi\)
\(128\) 1.35751e22i 1.01634i
\(129\) 9.53274e21 0.657697
\(130\) 0 0
\(131\) −5.53406e21 −0.324859 −0.162430 0.986720i \(-0.551933\pi\)
−0.162430 + 0.986720i \(0.551933\pi\)
\(132\) 8.25204e21i 0.447234i
\(133\) 3.65489e22i 1.82992i
\(134\) 2.94184e21 0.136150
\(135\) 0 0
\(136\) −2.72489e22 −1.07942
\(137\) 5.02831e21i 0.184440i 0.995739 + 0.0922201i \(0.0293963\pi\)
−0.995739 + 0.0922201i \(0.970604\pi\)
\(138\) − 7.19619e21i − 0.244552i
\(139\) −3.06514e21 −0.0965594 −0.0482797 0.998834i \(-0.515374\pi\)
−0.0482797 + 0.998834i \(0.515374\pi\)
\(140\) 0 0
\(141\) −9.45166e21 −0.256276
\(142\) − 6.00752e21i − 0.151240i
\(143\) − 3.36809e22i − 0.787690i
\(144\) −9.14847e21 −0.198858
\(145\) 0 0
\(146\) −2.05094e22 −0.385699
\(147\) 6.34167e22i 1.11012i
\(148\) 7.92124e22i 1.29135i
\(149\) −3.18365e22 −0.483581 −0.241791 0.970328i \(-0.577735\pi\)
−0.241791 + 0.970328i \(0.577735\pi\)
\(150\) 0 0
\(151\) 6.56248e22 0.866583 0.433292 0.901254i \(-0.357352\pi\)
0.433292 + 0.901254i \(0.357352\pi\)
\(152\) 6.06635e22i 0.747429i
\(153\) − 4.48014e22i − 0.515266i
\(154\) 5.39275e22 0.579219
\(155\) 0 0
\(156\) 4.61720e22 0.433083
\(157\) − 6.59284e22i − 0.578264i −0.957289 0.289132i \(-0.906633\pi\)
0.957289 0.289132i \(-0.0933667\pi\)
\(158\) − 3.43565e22i − 0.281910i
\(159\) −1.35785e23 −1.04275
\(160\) 0 0
\(161\) 2.86233e23 1.92774
\(162\) 6.61377e21i 0.0417389i
\(163\) 8.79055e21i 0.0520052i 0.999662 + 0.0260026i \(0.00827781\pi\)
−0.999662 + 0.0260026i \(0.991722\pi\)
\(164\) 1.04773e23 0.581286
\(165\) 0 0
\(166\) −7.89661e22 −0.385750
\(167\) − 1.51028e23i − 0.692683i −0.938109 0.346341i \(-0.887424\pi\)
0.938109 0.346341i \(-0.112576\pi\)
\(168\) 1.60001e23i 0.689247i
\(169\) 5.86123e22 0.237235
\(170\) 0 0
\(171\) −9.97404e22 −0.356789
\(172\) 2.90784e23i 0.978414i
\(173\) − 6.14868e23i − 1.94669i −0.229338 0.973347i \(-0.573656\pi\)
0.229338 0.973347i \(-0.426344\pi\)
\(174\) 1.65997e21 0.00494690
\(175\) 0 0
\(176\) −2.03567e23 −0.538051
\(177\) 3.04354e23i 0.757983i
\(178\) − 7.46667e22i − 0.175274i
\(179\) −4.83508e23 −1.07016 −0.535078 0.844803i \(-0.679718\pi\)
−0.535078 + 0.844803i \(0.679718\pi\)
\(180\) 0 0
\(181\) −9.28497e23 −1.82876 −0.914378 0.404862i \(-0.867320\pi\)
−0.914378 + 0.404862i \(0.867320\pi\)
\(182\) − 3.01737e23i − 0.560891i
\(183\) − 7.39108e22i − 0.129709i
\(184\) 4.75087e23 0.787382
\(185\) 0 0
\(186\) 2.86570e23 0.423979
\(187\) − 9.96895e23i − 1.39416i
\(188\) − 2.88311e23i − 0.381245i
\(189\) −2.63067e23 −0.329016
\(190\) 0 0
\(191\) −5.96634e23 −0.668125 −0.334062 0.942551i \(-0.608420\pi\)
−0.334062 + 0.942551i \(0.608420\pi\)
\(192\) − 1.36199e23i − 0.144381i
\(193\) 8.56373e23i 0.859629i 0.902917 + 0.429815i \(0.141421\pi\)
−0.902917 + 0.429815i \(0.858579\pi\)
\(194\) 1.76423e23 0.167740
\(195\) 0 0
\(196\) −1.93445e24 −1.65146
\(197\) 1.10520e24i 0.894430i 0.894427 + 0.447215i \(0.147584\pi\)
−0.894427 + 0.447215i \(0.852416\pi\)
\(198\) 1.47166e23i 0.112933i
\(199\) −1.53187e24 −1.11497 −0.557487 0.830186i \(-0.688234\pi\)
−0.557487 + 0.830186i \(0.688234\pi\)
\(200\) 0 0
\(201\) −3.19324e23 −0.209254
\(202\) 6.23191e23i 0.387643i
\(203\) 6.60264e22i 0.0389950i
\(204\) 1.36661e24 0.766528
\(205\) 0 0
\(206\) 4.87308e23 0.246716
\(207\) 7.81117e23i 0.375861i
\(208\) 1.13900e24i 0.521026i
\(209\) −2.21937e24 −0.965368
\(210\) 0 0
\(211\) 2.15918e24 0.849811 0.424906 0.905238i \(-0.360307\pi\)
0.424906 + 0.905238i \(0.360307\pi\)
\(212\) − 4.14195e24i − 1.55124i
\(213\) 6.52092e23i 0.232446i
\(214\) −6.84258e23 −0.232207
\(215\) 0 0
\(216\) −4.36636e23 −0.134386
\(217\) 1.13985e25i 3.34210i
\(218\) 9.06983e23i 0.253400i
\(219\) 2.22622e24 0.592795
\(220\) 0 0
\(221\) −5.57785e24 −1.35005
\(222\) 1.41266e24i 0.326086i
\(223\) − 7.86309e24i − 1.73138i −0.500582 0.865689i \(-0.666881\pi\)
0.500582 0.865689i \(-0.333119\pi\)
\(224\) −7.50620e24 −1.57694
\(225\) 0 0
\(226\) −3.54252e24 −0.677912
\(227\) 5.06979e24i 0.926229i 0.886298 + 0.463115i \(0.153268\pi\)
−0.886298 + 0.463115i \(0.846732\pi\)
\(228\) − 3.04246e24i − 0.530772i
\(229\) 6.10932e24 1.01793 0.508967 0.860786i \(-0.330027\pi\)
0.508967 + 0.860786i \(0.330027\pi\)
\(230\) 0 0
\(231\) −5.85361e24 −0.890221
\(232\) 1.09590e23i 0.0159275i
\(233\) − 2.40865e24i − 0.334609i −0.985905 0.167304i \(-0.946494\pi\)
0.985905 0.167304i \(-0.0535063\pi\)
\(234\) 8.23426e23 0.109360
\(235\) 0 0
\(236\) −9.28393e24 −1.12760
\(237\) 3.72926e24i 0.433277i
\(238\) − 8.93087e24i − 0.992740i
\(239\) 8.12956e23 0.0864748 0.0432374 0.999065i \(-0.486233\pi\)
0.0432374 + 0.999065i \(0.486233\pi\)
\(240\) 0 0
\(241\) 1.98089e24 0.193055 0.0965277 0.995330i \(-0.469226\pi\)
0.0965277 + 0.995330i \(0.469226\pi\)
\(242\) − 7.51087e23i − 0.0700856i
\(243\) − 7.17898e23i − 0.0641500i
\(244\) 2.25456e24 0.192960
\(245\) 0 0
\(246\) 1.86851e24 0.146784
\(247\) 1.24179e25i 0.934821i
\(248\) 1.89191e25i 1.36508i
\(249\) 8.57145e24 0.592872
\(250\) 0 0
\(251\) −1.71966e25 −1.09363 −0.546813 0.837255i \(-0.684159\pi\)
−0.546813 + 0.837255i \(0.684159\pi\)
\(252\) − 8.02452e24i − 0.489456i
\(253\) 1.73810e25i 1.01697i
\(254\) 4.34976e24 0.244181
\(255\) 0 0
\(256\) −2.54768e24 −0.131712
\(257\) 1.42771e25i 0.708503i 0.935150 + 0.354251i \(0.115264\pi\)
−0.935150 + 0.354251i \(0.884736\pi\)
\(258\) 5.18581e24i 0.247064i
\(259\) −5.61895e25 −2.57044
\(260\) 0 0
\(261\) −1.80183e23 −0.00760307
\(262\) − 3.01053e24i − 0.122034i
\(263\) − 1.23443e25i − 0.480765i −0.970678 0.240382i \(-0.922727\pi\)
0.970678 0.240382i \(-0.0772729\pi\)
\(264\) −9.71577e24 −0.363610
\(265\) 0 0
\(266\) −1.98826e25 −0.687410
\(267\) 8.10477e24i 0.269384i
\(268\) − 9.74059e24i − 0.311293i
\(269\) 4.23127e24 0.130039 0.0650193 0.997884i \(-0.479289\pi\)
0.0650193 + 0.997884i \(0.479289\pi\)
\(270\) 0 0
\(271\) −5.48006e25 −1.55815 −0.779073 0.626933i \(-0.784310\pi\)
−0.779073 + 0.626933i \(0.784310\pi\)
\(272\) 3.37124e25i 0.922182i
\(273\) 3.27523e25i 0.862053i
\(274\) −2.73540e24 −0.0692851
\(275\) 0 0
\(276\) −2.38270e25 −0.559144
\(277\) 1.74992e25i 0.395349i 0.980268 + 0.197675i \(0.0633390\pi\)
−0.980268 + 0.197675i \(0.936661\pi\)
\(278\) − 1.66743e24i − 0.0362726i
\(279\) −3.11060e25 −0.651628
\(280\) 0 0
\(281\) 2.52311e25 0.490365 0.245183 0.969477i \(-0.421152\pi\)
0.245183 + 0.969477i \(0.421152\pi\)
\(282\) − 5.14171e24i − 0.0962701i
\(283\) 1.19713e25i 0.215966i 0.994153 + 0.107983i \(0.0344391\pi\)
−0.994153 + 0.107983i \(0.965561\pi\)
\(284\) −1.98912e25 −0.345796
\(285\) 0 0
\(286\) 1.83224e25 0.295896
\(287\) 7.43213e25i 1.15705i
\(288\) − 2.04841e25i − 0.307465i
\(289\) −9.60027e25 −1.38949
\(290\) 0 0
\(291\) −1.91500e25 −0.257805
\(292\) 6.79080e25i 0.881863i
\(293\) − 6.89938e25i − 0.864371i −0.901785 0.432186i \(-0.857743\pi\)
0.901785 0.432186i \(-0.142257\pi\)
\(294\) −3.44987e25 −0.417019
\(295\) 0 0
\(296\) −9.32629e25 −1.04990
\(297\) − 1.59743e25i − 0.173571i
\(298\) − 1.73191e25i − 0.181657i
\(299\) 9.72505e25 0.984791
\(300\) 0 0
\(301\) −2.06269e26 −1.94754
\(302\) 3.56999e25i 0.325532i
\(303\) − 6.76449e25i − 0.595782i
\(304\) 7.50532e25 0.638553
\(305\) 0 0
\(306\) 2.43720e25 0.193560
\(307\) − 6.78268e25i − 0.520533i −0.965537 0.260267i \(-0.916190\pi\)
0.965537 0.260267i \(-0.0838104\pi\)
\(308\) − 1.78557e26i − 1.32432i
\(309\) −5.28953e25 −0.379187
\(310\) 0 0
\(311\) 1.00215e26 0.671347 0.335673 0.941978i \(-0.391036\pi\)
0.335673 + 0.941978i \(0.391036\pi\)
\(312\) 5.43619e25i 0.352105i
\(313\) 2.29792e26i 1.43920i 0.694390 + 0.719599i \(0.255674\pi\)
−0.694390 + 0.719599i \(0.744326\pi\)
\(314\) 3.58651e25 0.217225
\(315\) 0 0
\(316\) −1.13756e26 −0.644558
\(317\) 2.06457e26i 1.13164i 0.824530 + 0.565818i \(0.191439\pi\)
−0.824530 + 0.565818i \(0.808561\pi\)
\(318\) − 7.38669e25i − 0.391711i
\(319\) −4.00933e24 −0.0205717
\(320\) 0 0
\(321\) 7.42734e25 0.356887
\(322\) 1.55711e26i 0.724155i
\(323\) 3.67547e26i 1.65457i
\(324\) 2.18986e25 0.0954319
\(325\) 0 0
\(326\) −4.78206e24 −0.0195358
\(327\) − 9.84493e25i − 0.389459i
\(328\) 1.23358e26i 0.472597i
\(329\) 2.04515e26 0.758870
\(330\) 0 0
\(331\) −1.09215e26 −0.380268 −0.190134 0.981758i \(-0.560892\pi\)
−0.190134 + 0.981758i \(0.560892\pi\)
\(332\) 2.61461e26i 0.881978i
\(333\) − 1.53339e26i − 0.501173i
\(334\) 8.21592e25 0.260207
\(335\) 0 0
\(336\) 1.97954e26 0.588846
\(337\) − 4.76493e26i − 1.37386i −0.726724 0.686930i \(-0.758958\pi\)
0.726724 0.686930i \(-0.241042\pi\)
\(338\) 3.18851e25i 0.0891173i
\(339\) 3.84526e26 1.04191
\(340\) 0 0
\(341\) −6.92152e26 −1.76312
\(342\) − 5.42588e25i − 0.134028i
\(343\) − 6.58553e26i − 1.57762i
\(344\) −3.42363e26 −0.795469
\(345\) 0 0
\(346\) 3.34488e26 0.731276
\(347\) − 5.67729e26i − 1.20415i −0.798438 0.602077i \(-0.794340\pi\)
0.798438 0.602077i \(-0.205660\pi\)
\(348\) − 5.49626e24i − 0.0113106i
\(349\) 4.85816e26 0.970075 0.485037 0.874493i \(-0.338806\pi\)
0.485037 + 0.874493i \(0.338806\pi\)
\(350\) 0 0
\(351\) −8.93796e25 −0.168079
\(352\) − 4.55800e26i − 0.831910i
\(353\) − 1.71788e26i − 0.304340i −0.988354 0.152170i \(-0.951374\pi\)
0.988354 0.152170i \(-0.0486261\pi\)
\(354\) −1.65568e26 −0.284737
\(355\) 0 0
\(356\) −2.47226e26 −0.400746
\(357\) 9.69410e26i 1.52578i
\(358\) − 2.63028e26i − 0.402004i
\(359\) −1.33028e27 −1.97448 −0.987239 0.159247i \(-0.949093\pi\)
−0.987239 + 0.159247i \(0.949093\pi\)
\(360\) 0 0
\(361\) 1.04051e26 0.145687
\(362\) − 5.05103e26i − 0.686973i
\(363\) 8.15275e25i 0.107717i
\(364\) −9.99067e26 −1.28242
\(365\) 0 0
\(366\) 4.02075e25 0.0487254
\(367\) 3.40629e26i 0.401132i 0.979680 + 0.200566i \(0.0642781\pi\)
−0.979680 + 0.200566i \(0.935722\pi\)
\(368\) − 5.87780e26i − 0.672686i
\(369\) −2.02820e26 −0.225597
\(370\) 0 0
\(371\) 2.93810e27 3.08774
\(372\) − 9.48849e26i − 0.969385i
\(373\) 1.62912e27i 1.61812i 0.587729 + 0.809058i \(0.300022\pi\)
−0.587729 + 0.809058i \(0.699978\pi\)
\(374\) 5.42311e26 0.523717
\(375\) 0 0
\(376\) 3.39451e26 0.309960
\(377\) 2.24331e25i 0.0199207i
\(378\) − 1.43108e26i − 0.123595i
\(379\) −3.25550e26 −0.273467 −0.136734 0.990608i \(-0.543660\pi\)
−0.136734 + 0.990608i \(0.543660\pi\)
\(380\) 0 0
\(381\) −4.72149e26 −0.375290
\(382\) − 3.24569e26i − 0.250981i
\(383\) 1.61833e27i 1.21753i 0.793350 + 0.608765i \(0.208335\pi\)
−0.793350 + 0.608765i \(0.791665\pi\)
\(384\) 8.01594e26 0.586782
\(385\) 0 0
\(386\) −4.65867e26 −0.322920
\(387\) − 5.62899e26i − 0.379722i
\(388\) − 5.84146e26i − 0.383520i
\(389\) 2.02736e27 1.29557 0.647784 0.761824i \(-0.275696\pi\)
0.647784 + 0.761824i \(0.275696\pi\)
\(390\) 0 0
\(391\) 2.87844e27 1.74302
\(392\) − 2.27758e27i − 1.34267i
\(393\) 3.26781e26i 0.187558i
\(394\) −6.01230e26 −0.335993
\(395\) 0 0
\(396\) 4.87275e26 0.258211
\(397\) 1.57647e27i 0.813551i 0.913528 + 0.406775i \(0.133347\pi\)
−0.913528 + 0.406775i \(0.866653\pi\)
\(398\) − 8.33337e26i − 0.418840i
\(399\) 2.15818e27 1.05650
\(400\) 0 0
\(401\) 2.38019e27 1.10559 0.552797 0.833316i \(-0.313561\pi\)
0.552797 + 0.833316i \(0.313561\pi\)
\(402\) − 1.73712e26i − 0.0786063i
\(403\) 3.87275e27i 1.70733i
\(404\) 2.06342e27 0.886306
\(405\) 0 0
\(406\) −3.59184e25 −0.0146485
\(407\) − 3.41201e27i − 1.35603i
\(408\) 1.60902e27i 0.623202i
\(409\) −1.28701e27 −0.485832 −0.242916 0.970047i \(-0.578104\pi\)
−0.242916 + 0.970047i \(0.578104\pi\)
\(410\) 0 0
\(411\) 2.96916e26 0.106487
\(412\) − 1.61351e27i − 0.564092i
\(413\) − 6.58558e27i − 2.24450i
\(414\) −4.24928e26 −0.141192
\(415\) 0 0
\(416\) −2.55031e27 −0.805587
\(417\) 1.80993e26i 0.0557486i
\(418\) − 1.20733e27i − 0.362641i
\(419\) 6.36510e27 1.86448 0.932241 0.361839i \(-0.117851\pi\)
0.932241 + 0.361839i \(0.117851\pi\)
\(420\) 0 0
\(421\) 1.16839e27 0.325556 0.162778 0.986663i \(-0.447955\pi\)
0.162778 + 0.986663i \(0.447955\pi\)
\(422\) 1.17459e27i 0.319232i
\(423\) 5.58111e26i 0.147961i
\(424\) 4.87664e27 1.26119
\(425\) 0 0
\(426\) −3.54738e26 −0.0873186
\(427\) 1.59928e27i 0.384088i
\(428\) 2.26562e27i 0.530917i
\(429\) −1.98882e27 −0.454773
\(430\) 0 0
\(431\) 4.87470e27 1.06154 0.530770 0.847516i \(-0.321903\pi\)
0.530770 + 0.847516i \(0.321903\pi\)
\(432\) 5.40208e26i 0.114811i
\(433\) − 3.42952e27i − 0.711395i −0.934601 0.355698i \(-0.884243\pi\)
0.934601 0.355698i \(-0.115757\pi\)
\(434\) −6.20078e27 −1.25546
\(435\) 0 0
\(436\) 3.00307e27 0.579372
\(437\) − 6.40821e27i − 1.20693i
\(438\) 1.21106e27i 0.222684i
\(439\) −6.10587e27 −1.09615 −0.548075 0.836429i \(-0.684639\pi\)
−0.548075 + 0.836429i \(0.684639\pi\)
\(440\) 0 0
\(441\) 3.74469e27 0.640931
\(442\) − 3.03435e27i − 0.507145i
\(443\) 6.52956e27i 1.06572i 0.846202 + 0.532862i \(0.178884\pi\)
−0.846202 + 0.532862i \(0.821116\pi\)
\(444\) 4.67741e27 0.745563
\(445\) 0 0
\(446\) 4.27752e27 0.650393
\(447\) 1.87991e27i 0.279196i
\(448\) 2.94706e27i 0.427533i
\(449\) −3.56364e27 −0.505018 −0.252509 0.967595i \(-0.581256\pi\)
−0.252509 + 0.967595i \(0.581256\pi\)
\(450\) 0 0
\(451\) −4.51302e27 −0.610399
\(452\) 1.17295e28i 1.54998i
\(453\) − 3.87508e27i − 0.500322i
\(454\) −2.75797e27 −0.347939
\(455\) 0 0
\(456\) 3.58212e27 0.431528
\(457\) − 5.64234e25i − 0.00664262i −0.999994 0.00332131i \(-0.998943\pi\)
0.999994 0.00332131i \(-0.00105721\pi\)
\(458\) 3.32347e27i 0.382388i
\(459\) −2.64548e27 −0.297489
\(460\) 0 0
\(461\) 3.68168e27 0.395537 0.197768 0.980249i \(-0.436631\pi\)
0.197768 + 0.980249i \(0.436631\pi\)
\(462\) − 3.18436e27i − 0.334412i
\(463\) 6.25388e27i 0.642020i 0.947076 + 0.321010i \(0.104022\pi\)
−0.947076 + 0.321010i \(0.895978\pi\)
\(464\) 1.35585e26 0.0136074
\(465\) 0 0
\(466\) 1.31031e27 0.125696
\(467\) − 1.81487e26i − 0.0170223i −0.999964 0.00851114i \(-0.997291\pi\)
0.999964 0.00851114i \(-0.00270921\pi\)
\(468\) − 2.72641e27i − 0.250040i
\(469\) 6.90952e27 0.619631
\(470\) 0 0
\(471\) −3.89301e27 −0.333861
\(472\) − 1.09307e28i − 0.916762i
\(473\) − 1.25253e28i − 1.02742i
\(474\) −2.02872e27 −0.162761
\(475\) 0 0
\(476\) −2.95706e28 −2.26980
\(477\) 8.01796e27i 0.602034i
\(478\) 4.42248e26i 0.0324843i
\(479\) −4.83395e27 −0.347360 −0.173680 0.984802i \(-0.555566\pi\)
−0.173680 + 0.984802i \(0.555566\pi\)
\(480\) 0 0
\(481\) −1.90910e28 −1.31312
\(482\) 1.07761e27i 0.0725214i
\(483\) − 1.69018e28i − 1.11298i
\(484\) −2.48689e27 −0.160244
\(485\) 0 0
\(486\) 3.90537e26 0.0240980
\(487\) − 6.30377e27i − 0.380668i −0.981719 0.190334i \(-0.939043\pi\)
0.981719 0.190334i \(-0.0609571\pi\)
\(488\) 2.65447e27i 0.156881i
\(489\) 5.19073e26 0.0300252
\(490\) 0 0
\(491\) 3.67372e27 0.203587 0.101794 0.994806i \(-0.467542\pi\)
0.101794 + 0.994806i \(0.467542\pi\)
\(492\) − 6.18676e27i − 0.335606i
\(493\) 6.63981e26i 0.0352584i
\(494\) −6.75531e27 −0.351166
\(495\) 0 0
\(496\) 2.34068e28 1.16623
\(497\) − 1.41099e28i − 0.688307i
\(498\) 4.66287e27i 0.222713i
\(499\) 2.00088e28 0.935761 0.467881 0.883792i \(-0.345018\pi\)
0.467881 + 0.883792i \(0.345018\pi\)
\(500\) 0 0
\(501\) −8.91805e27 −0.399921
\(502\) − 9.35496e27i − 0.410821i
\(503\) − 1.40332e28i − 0.603521i −0.953384 0.301761i \(-0.902426\pi\)
0.953384 0.301761i \(-0.0975743\pi\)
\(504\) 9.44790e27 0.397937
\(505\) 0 0
\(506\) −9.45525e27 −0.382026
\(507\) − 3.46100e27i − 0.136967i
\(508\) − 1.44023e28i − 0.558295i
\(509\) 6.58158e27 0.249916 0.124958 0.992162i \(-0.460120\pi\)
0.124958 + 0.992162i \(0.460120\pi\)
\(510\) 0 0
\(511\) −4.81708e28 −1.75535
\(512\) 2.70830e28i 0.966858i
\(513\) 5.88957e27i 0.205992i
\(514\) −7.76673e27 −0.266149
\(515\) 0 0
\(516\) 1.71705e28 0.564887
\(517\) 1.24188e28i 0.400339i
\(518\) − 3.05671e28i − 0.965588i
\(519\) −3.63073e28 −1.12392
\(520\) 0 0
\(521\) −4.65750e28 −1.38470 −0.692351 0.721561i \(-0.743425\pi\)
−0.692351 + 0.721561i \(0.743425\pi\)
\(522\) − 9.80196e25i − 0.00285610i
\(523\) 3.89578e28i 1.11257i 0.830992 + 0.556284i \(0.187773\pi\)
−0.830992 + 0.556284i \(0.812227\pi\)
\(524\) −9.96804e27 −0.279017
\(525\) 0 0
\(526\) 6.71532e27 0.180600
\(527\) 1.14627e29i 3.02186i
\(528\) 1.20204e28i 0.310644i
\(529\) −1.07144e28 −0.271445
\(530\) 0 0
\(531\) 1.79718e28 0.437621
\(532\) 6.58324e28i 1.57169i
\(533\) 2.52514e28i 0.591085i
\(534\) −4.40900e27 −0.101194
\(535\) 0 0
\(536\) 1.14684e28 0.253088
\(537\) 2.85507e28i 0.617854i
\(538\) 2.30181e27i 0.0488491i
\(539\) 8.33247e28 1.73417
\(540\) 0 0
\(541\) 7.65629e27 0.153267 0.0766333 0.997059i \(-0.475583\pi\)
0.0766333 + 0.997059i \(0.475583\pi\)
\(542\) − 2.98116e28i − 0.585318i
\(543\) 5.48268e28i 1.05583i
\(544\) −7.54846e28 −1.42584
\(545\) 0 0
\(546\) −1.78172e28 −0.323830
\(547\) 6.77366e28i 1.20769i 0.797100 + 0.603847i \(0.206366\pi\)
−0.797100 + 0.603847i \(0.793634\pi\)
\(548\) 9.05707e27i 0.158413i
\(549\) −4.36436e27 −0.0748878
\(550\) 0 0
\(551\) 1.47821e27 0.0244142
\(552\) − 2.80534e28i − 0.454595i
\(553\) − 8.06935e28i − 1.28299i
\(554\) −9.51957e27 −0.148513
\(555\) 0 0
\(556\) −5.52097e27 −0.0829336
\(557\) − 5.64828e28i − 0.832600i −0.909227 0.416300i \(-0.863327\pi\)
0.909227 0.416300i \(-0.136673\pi\)
\(558\) − 1.69217e28i − 0.244784i
\(559\) −7.00819e28 −0.994906
\(560\) 0 0
\(561\) −5.88657e28 −0.804919
\(562\) 1.37257e28i 0.184206i
\(563\) − 1.32120e28i − 0.174033i −0.996207 0.0870166i \(-0.972267\pi\)
0.996207 0.0870166i \(-0.0277333\pi\)
\(564\) −1.70245e28 −0.220112
\(565\) 0 0
\(566\) −6.51240e27 −0.0811275
\(567\) 1.55338e28i 0.189957i
\(568\) − 2.34195e28i − 0.281139i
\(569\) −3.77888e27 −0.0445332 −0.0222666 0.999752i \(-0.507088\pi\)
−0.0222666 + 0.999752i \(0.507088\pi\)
\(570\) 0 0
\(571\) −9.70592e28 −1.10245 −0.551223 0.834358i \(-0.685839\pi\)
−0.551223 + 0.834358i \(0.685839\pi\)
\(572\) − 6.06666e28i − 0.676536i
\(573\) 3.52306e28i 0.385742i
\(574\) −4.04308e28 −0.434647
\(575\) 0 0
\(576\) −8.04239e27 −0.0833584
\(577\) − 1.40057e27i − 0.0142547i −0.999975 0.00712737i \(-0.997731\pi\)
0.999975 0.00712737i \(-0.00226873\pi\)
\(578\) − 5.22255e28i − 0.521964i
\(579\) 5.05680e28 0.496307
\(580\) 0 0
\(581\) −1.85468e29 −1.75558
\(582\) − 1.04176e28i − 0.0968447i
\(583\) 1.78411e29i 1.62893i
\(584\) −7.99534e28 −0.716972
\(585\) 0 0
\(586\) 3.75326e28 0.324701
\(587\) − 9.16040e28i − 0.778421i −0.921149 0.389211i \(-0.872748\pi\)
0.921149 0.389211i \(-0.127252\pi\)
\(588\) 1.14227e29i 0.953471i
\(589\) 2.55191e29 2.09245
\(590\) 0 0
\(591\) 6.52610e28 0.516399
\(592\) 1.15385e29i 0.896959i
\(593\) − 1.47495e29i − 1.12643i −0.826311 0.563214i \(-0.809565\pi\)
0.826311 0.563214i \(-0.190435\pi\)
\(594\) 8.68999e27 0.0652021
\(595\) 0 0
\(596\) −5.73444e28 −0.415341
\(597\) 9.04554e28i 0.643730i
\(598\) 5.29043e28i 0.369937i
\(599\) −1.19182e29 −0.818899 −0.409450 0.912333i \(-0.634279\pi\)
−0.409450 + 0.912333i \(0.634279\pi\)
\(600\) 0 0
\(601\) 2.64125e29 1.75238 0.876190 0.481966i \(-0.160077\pi\)
0.876190 + 0.481966i \(0.160077\pi\)
\(602\) − 1.12210e29i − 0.731593i
\(603\) 1.88558e28i 0.120813i
\(604\) 1.18205e29 0.744297
\(605\) 0 0
\(606\) 3.67988e28 0.223806
\(607\) − 8.82263e28i − 0.527371i −0.964609 0.263686i \(-0.915062\pi\)
0.964609 0.263686i \(-0.0849382\pi\)
\(608\) 1.68050e29i 0.987302i
\(609\) 3.89879e27 0.0225138
\(610\) 0 0
\(611\) 6.94859e28 0.387671
\(612\) − 8.06970e28i − 0.442555i
\(613\) − 3.27173e29i − 1.76377i −0.471465 0.881885i \(-0.656275\pi\)
0.471465 0.881885i \(-0.343725\pi\)
\(614\) 3.68978e28 0.195538
\(615\) 0 0
\(616\) 2.10229e29 1.07670
\(617\) 7.46428e28i 0.375832i 0.982185 + 0.187916i \(0.0601733\pi\)
−0.982185 + 0.187916i \(0.939827\pi\)
\(618\) − 2.87751e28i − 0.142442i
\(619\) 8.32478e28 0.405155 0.202577 0.979266i \(-0.435068\pi\)
0.202577 + 0.979266i \(0.435068\pi\)
\(620\) 0 0
\(621\) 4.61242e28 0.217004
\(622\) 5.45167e28i 0.252192i
\(623\) − 1.75371e29i − 0.797686i
\(624\) 6.72569e28 0.300814
\(625\) 0 0
\(626\) −1.25007e29 −0.540635
\(627\) 1.31051e29i 0.557355i
\(628\) − 1.18751e29i − 0.496663i
\(629\) −5.65059e29 −2.32414
\(630\) 0 0
\(631\) 3.63173e29 1.44479 0.722396 0.691479i \(-0.243041\pi\)
0.722396 + 0.691479i \(0.243041\pi\)
\(632\) − 1.33934e29i − 0.524038i
\(633\) − 1.27497e29i − 0.490639i
\(634\) −1.12312e29 −0.425099
\(635\) 0 0
\(636\) −2.44578e29 −0.895606
\(637\) − 4.66221e29i − 1.67930i
\(638\) − 2.18108e27i − 0.00772776i
\(639\) 3.85054e28 0.134203
\(640\) 0 0
\(641\) −4.09788e28 −0.138213 −0.0691067 0.997609i \(-0.522015\pi\)
−0.0691067 + 0.997609i \(0.522015\pi\)
\(642\) 4.04047e28i 0.134065i
\(643\) 3.44481e29i 1.12448i 0.826976 + 0.562238i \(0.190059\pi\)
−0.826976 + 0.562238i \(0.809941\pi\)
\(644\) 5.15567e29 1.65571
\(645\) 0 0
\(646\) −1.99945e29 −0.621541
\(647\) − 2.99052e29i − 0.914645i −0.889301 0.457322i \(-0.848809\pi\)
0.889301 0.457322i \(-0.151191\pi\)
\(648\) 2.57829e28i 0.0775880i
\(649\) 3.99898e29 1.18408
\(650\) 0 0
\(651\) 6.73069e29 1.92956
\(652\) 1.58337e28i 0.0446665i
\(653\) − 6.43185e29i − 1.78545i −0.450603 0.892724i \(-0.648791\pi\)
0.450603 0.892724i \(-0.351209\pi\)
\(654\) 5.35564e28 0.146300
\(655\) 0 0
\(656\) 1.52619e29 0.403755
\(657\) − 1.31456e29i − 0.342250i
\(658\) 1.11256e29i 0.285070i
\(659\) −1.00447e29 −0.253303 −0.126652 0.991947i \(-0.540423\pi\)
−0.126652 + 0.991947i \(0.540423\pi\)
\(660\) 0 0
\(661\) −4.70181e29 −1.14855 −0.574274 0.818663i \(-0.694716\pi\)
−0.574274 + 0.818663i \(0.694716\pi\)
\(662\) − 5.94131e28i − 0.142848i
\(663\) 3.29367e29i 0.779449i
\(664\) −3.07839e29 −0.717065
\(665\) 0 0
\(666\) 8.34163e28 0.188266
\(667\) − 1.15766e28i − 0.0257193i
\(668\) − 2.72034e29i − 0.594936i
\(669\) −4.64307e29 −0.999612
\(670\) 0 0
\(671\) −9.71132e28 −0.202624
\(672\) 4.43233e29i 0.910448i
\(673\) 3.92345e29i 0.793432i 0.917941 + 0.396716i \(0.129850\pi\)
−0.917941 + 0.396716i \(0.870150\pi\)
\(674\) 2.59212e29 0.516091
\(675\) 0 0
\(676\) 1.05573e29 0.203758
\(677\) − 1.52718e29i − 0.290207i −0.989416 0.145104i \(-0.953648\pi\)
0.989416 0.145104i \(-0.0463516\pi\)
\(678\) 2.09182e29i 0.391392i
\(679\) 4.14366e29 0.763399
\(680\) 0 0
\(681\) 2.99366e29 0.534759
\(682\) − 3.76531e29i − 0.662315i
\(683\) 4.36102e29i 0.755389i 0.925930 + 0.377695i \(0.123283\pi\)
−0.925930 + 0.377695i \(0.876717\pi\)
\(684\) −1.79654e29 −0.306442
\(685\) 0 0
\(686\) 3.58253e29 0.592634
\(687\) − 3.60749e29i − 0.587705i
\(688\) 4.23574e29i 0.679595i
\(689\) 9.98250e29 1.57738
\(690\) 0 0
\(691\) 5.96732e29 0.914660 0.457330 0.889297i \(-0.348806\pi\)
0.457330 + 0.889297i \(0.348806\pi\)
\(692\) − 1.10751e30i − 1.67199i
\(693\) 3.45650e29i 0.513970i
\(694\) 3.08845e29 0.452341
\(695\) 0 0
\(696\) 6.47118e27 0.00919573
\(697\) 7.47397e29i 1.04618i
\(698\) 2.64284e29i 0.364409i
\(699\) −1.42229e29 −0.193186
\(700\) 0 0
\(701\) 5.37209e29 0.708116 0.354058 0.935223i \(-0.384802\pi\)
0.354058 + 0.935223i \(0.384802\pi\)
\(702\) − 4.86225e28i − 0.0631390i
\(703\) 1.25798e30i 1.60932i
\(704\) −1.78955e29 −0.225543
\(705\) 0 0
\(706\) 9.34527e28 0.114325
\(707\) 1.46370e30i 1.76420i
\(708\) 5.48207e29i 0.651021i
\(709\) 1.27317e29 0.148971 0.0744854 0.997222i \(-0.476269\pi\)
0.0744854 + 0.997222i \(0.476269\pi\)
\(710\) 0 0
\(711\) 2.20209e29 0.250152
\(712\) − 2.91078e29i − 0.325814i
\(713\) − 1.99853e30i − 2.20430i
\(714\) −5.27359e29 −0.573159
\(715\) 0 0
\(716\) −8.70902e29 −0.919142
\(717\) − 4.80043e28i − 0.0499262i
\(718\) − 7.23674e29i − 0.741713i
\(719\) 1.86676e30 1.88553 0.942767 0.333451i \(-0.108213\pi\)
0.942767 + 0.333451i \(0.108213\pi\)
\(720\) 0 0
\(721\) 1.14455e30 1.12283
\(722\) 5.66038e28i 0.0547275i
\(723\) − 1.16970e29i − 0.111461i
\(724\) −1.67242e30 −1.57069
\(725\) 0 0
\(726\) −4.43509e28 −0.0404640
\(727\) − 1.86570e30i − 1.67776i −0.544314 0.838882i \(-0.683210\pi\)
0.544314 0.838882i \(-0.316790\pi\)
\(728\) − 1.17628e30i − 1.04263i
\(729\) −4.23912e28 −0.0370370
\(730\) 0 0
\(731\) −2.07430e30 −1.76092
\(732\) − 1.33129e29i − 0.111406i
\(733\) 4.88128e29i 0.402663i 0.979523 + 0.201331i \(0.0645268\pi\)
−0.979523 + 0.201331i \(0.935473\pi\)
\(734\) −1.85302e29 −0.150686
\(735\) 0 0
\(736\) 1.31608e30 1.04008
\(737\) 4.19568e29i 0.326884i
\(738\) − 1.10334e29i − 0.0847456i
\(739\) −1.67069e30 −1.26511 −0.632556 0.774515i \(-0.717994\pi\)
−0.632556 + 0.774515i \(0.717994\pi\)
\(740\) 0 0
\(741\) 7.33262e29 0.539719
\(742\) 1.59833e30i 1.15991i
\(743\) − 1.62834e30i − 1.16509i −0.812797 0.582547i \(-0.802056\pi\)
0.812797 0.582547i \(-0.197944\pi\)
\(744\) 1.11715e30 0.788129
\(745\) 0 0
\(746\) −8.86240e29 −0.607846
\(747\) − 5.06136e29i − 0.342295i
\(748\) − 1.79562e30i − 1.19743i
\(749\) −1.60712e30 −1.05679
\(750\) 0 0
\(751\) 5.83073e29 0.372824 0.186412 0.982472i \(-0.440314\pi\)
0.186412 + 0.982472i \(0.440314\pi\)
\(752\) − 4.19971e29i − 0.264809i
\(753\) 1.01544e30i 0.631406i
\(754\) −1.22036e28 −0.00748324
\(755\) 0 0
\(756\) −4.73840e29 −0.282587
\(757\) 1.39333e30i 0.819494i 0.912199 + 0.409747i \(0.134383\pi\)
−0.912199 + 0.409747i \(0.865617\pi\)
\(758\) − 1.77099e29i − 0.102728i
\(759\) 1.02633e30 0.587148
\(760\) 0 0
\(761\) −9.17379e29 −0.510517 −0.255258 0.966873i \(-0.582161\pi\)
−0.255258 + 0.966873i \(0.582161\pi\)
\(762\) − 2.56849e29i − 0.140978i
\(763\) 2.13024e30i 1.15324i
\(764\) −1.07467e30 −0.573843
\(765\) 0 0
\(766\) −8.80371e29 −0.457366
\(767\) − 2.23752e30i − 1.14661i
\(768\) 1.50438e29i 0.0760438i
\(769\) 1.13366e30 0.565273 0.282636 0.959227i \(-0.408791\pi\)
0.282636 + 0.959227i \(0.408791\pi\)
\(770\) 0 0
\(771\) 8.43047e29 0.409054
\(772\) 1.54251e30i 0.738324i
\(773\) 1.81526e29i 0.0857143i 0.999081 + 0.0428572i \(0.0136460\pi\)
−0.999081 + 0.0428572i \(0.986354\pi\)
\(774\) 3.06217e29 0.142643
\(775\) 0 0
\(776\) 6.87761e29 0.311809
\(777\) 3.31794e30i 1.48405i
\(778\) 1.10289e30i 0.486681i
\(779\) 1.66391e30 0.724415
\(780\) 0 0
\(781\) 8.56799e29 0.363114
\(782\) 1.56587e30i 0.654765i
\(783\) 1.06396e28i 0.00438963i
\(784\) −2.81783e30 −1.14709
\(785\) 0 0
\(786\) −1.77769e29 −0.0704561
\(787\) 7.88071e29i 0.308199i 0.988055 + 0.154099i \(0.0492476\pi\)
−0.988055 + 0.154099i \(0.950752\pi\)
\(788\) 1.99071e30i 0.768214i
\(789\) −7.28921e29 −0.277570
\(790\) 0 0
\(791\) −8.32034e30 −3.08523
\(792\) 5.73706e29i 0.209930i
\(793\) 5.43371e29i 0.196213i
\(794\) −8.57598e29 −0.305611
\(795\) 0 0
\(796\) −2.75923e30 −0.957636
\(797\) − 3.61162e30i − 1.23706i −0.785763 0.618528i \(-0.787729\pi\)
0.785763 0.618528i \(-0.212271\pi\)
\(798\) 1.17405e30i 0.396876i
\(799\) 2.05666e30 0.686153
\(800\) 0 0
\(801\) 4.78579e29 0.155529
\(802\) 1.29482e30i 0.415317i
\(803\) − 2.92508e30i − 0.926029i
\(804\) −5.75172e29 −0.179725
\(805\) 0 0
\(806\) −2.10678e30 −0.641358
\(807\) − 2.49853e29i − 0.0750778i
\(808\) 2.42943e30i 0.720584i
\(809\) −6.55296e30 −1.91857 −0.959286 0.282436i \(-0.908858\pi\)
−0.959286 + 0.282436i \(0.908858\pi\)
\(810\) 0 0
\(811\) −2.95834e30 −0.843975 −0.421988 0.906602i \(-0.638667\pi\)
−0.421988 + 0.906602i \(0.638667\pi\)
\(812\) 1.18928e29i 0.0334923i
\(813\) 3.23592e30i 0.899596i
\(814\) 1.85613e30 0.509393
\(815\) 0 0
\(816\) 1.99069e30 0.532422
\(817\) 4.61797e30i 1.21933i
\(818\) − 7.00132e29i − 0.182503i
\(819\) 1.93399e30 0.497706
\(820\) 0 0
\(821\) −6.60712e30 −1.65733 −0.828665 0.559744i \(-0.810899\pi\)
−0.828665 + 0.559744i \(0.810899\pi\)
\(822\) 1.61523e29i 0.0400018i
\(823\) − 4.27938e30i − 1.04636i −0.852221 0.523182i \(-0.824745\pi\)
0.852221 0.523182i \(-0.175255\pi\)
\(824\) 1.89971e30 0.458618
\(825\) 0 0
\(826\) 3.58256e30 0.843146
\(827\) 8.08422e29i 0.187858i 0.995579 + 0.0939291i \(0.0299427\pi\)
−0.995579 + 0.0939291i \(0.970057\pi\)
\(828\) 1.40696e30i 0.322822i
\(829\) −6.58152e30 −1.49109 −0.745544 0.666456i \(-0.767810\pi\)
−0.745544 + 0.666456i \(0.767810\pi\)
\(830\) 0 0
\(831\) 1.03331e30 0.228255
\(832\) 1.00129e30i 0.218407i
\(833\) − 1.37993e31i − 2.97225i
\(834\) −9.84603e28 −0.0209420
\(835\) 0 0
\(836\) −3.99756e30 −0.829141
\(837\) 1.83678e30i 0.376218i
\(838\) 3.46262e30i 0.700393i
\(839\) 8.11970e30 1.62196 0.810979 0.585075i \(-0.198935\pi\)
0.810979 + 0.585075i \(0.198935\pi\)
\(840\) 0 0
\(841\) −5.13017e30 −0.999480
\(842\) 6.35604e29i 0.122295i
\(843\) − 1.48987e30i − 0.283113i
\(844\) 3.88914e30 0.729892
\(845\) 0 0
\(846\) −3.03613e29 −0.0555816
\(847\) − 1.76409e30i − 0.318966i
\(848\) − 6.03340e30i − 1.07747i
\(849\) 7.06894e29 0.124688
\(850\) 0 0
\(851\) 9.85186e30 1.69534
\(852\) 1.17456e30i 0.199645i
\(853\) − 1.29820e30i − 0.217960i −0.994044 0.108980i \(-0.965241\pi\)
0.994044 0.108980i \(-0.0347585\pi\)
\(854\) −8.70007e29 −0.144283
\(855\) 0 0
\(856\) −2.66749e30 −0.431646
\(857\) − 1.05399e30i − 0.168476i −0.996446 0.0842380i \(-0.973154\pi\)
0.996446 0.0842380i \(-0.0268456\pi\)
\(858\) − 1.08192e30i − 0.170836i
\(859\) −6.98517e30 −1.08955 −0.544777 0.838581i \(-0.683386\pi\)
−0.544777 + 0.838581i \(0.683386\pi\)
\(860\) 0 0
\(861\) 4.38860e30 0.668025
\(862\) 2.65184e30i 0.398768i
\(863\) − 1.03987e31i − 1.54478i −0.635146 0.772392i \(-0.719060\pi\)
0.635146 0.772392i \(-0.280940\pi\)
\(864\) −1.20956e30 −0.177515
\(865\) 0 0
\(866\) 1.86566e30 0.267236
\(867\) 5.66887e30i 0.802224i
\(868\) 2.05311e31i 2.87049i
\(869\) 4.89997e30 0.676839
\(870\) 0 0
\(871\) 2.34758e30 0.316541
\(872\) 3.53575e30i 0.471041i
\(873\) 1.13079e30i 0.148844i
\(874\) 3.48607e30 0.453384
\(875\) 0 0
\(876\) 4.00990e30 0.509144
\(877\) 5.66163e30i 0.710306i 0.934808 + 0.355153i \(0.115571\pi\)
−0.934808 + 0.355153i \(0.884429\pi\)
\(878\) − 3.32159e30i − 0.411770i
\(879\) −4.07402e30 −0.499045
\(880\) 0 0
\(881\) 3.59833e30 0.430382 0.215191 0.976572i \(-0.430963\pi\)
0.215191 + 0.976572i \(0.430963\pi\)
\(882\) 2.03711e30i 0.240766i
\(883\) 1.24249e31i 1.45112i 0.688157 + 0.725562i \(0.258420\pi\)
−0.688157 + 0.725562i \(0.741580\pi\)
\(884\) −1.00469e31 −1.15954
\(885\) 0 0
\(886\) −3.55208e30 −0.400340
\(887\) − 9.60349e30i − 1.06962i −0.844971 0.534812i \(-0.820383\pi\)
0.844971 0.534812i \(-0.179617\pi\)
\(888\) 5.50708e30i 0.606157i
\(889\) 1.02163e31 1.11129
\(890\) 0 0
\(891\) −9.43264e29 −0.100211
\(892\) − 1.41631e31i − 1.48706i
\(893\) − 4.57869e30i − 0.475118i
\(894\) −1.02267e30 −0.104880
\(895\) 0 0
\(896\) −1.73448e31 −1.73754
\(897\) − 5.74254e30i − 0.568570i
\(898\) − 1.93862e30i − 0.189710i
\(899\) 4.61007e29 0.0445894
\(900\) 0 0
\(901\) 2.95464e31 2.79187
\(902\) − 2.45509e30i − 0.229297i
\(903\) 1.21800e31i 1.12441i
\(904\) −1.38100e31 −1.26016
\(905\) 0 0
\(906\) 2.10804e30 0.187946
\(907\) − 1.94141e31i − 1.71097i −0.517830 0.855483i \(-0.673260\pi\)
0.517830 0.855483i \(-0.326740\pi\)
\(908\) 9.13179e30i 0.795526i
\(909\) −3.99436e30 −0.343975
\(910\) 0 0
\(911\) −4.71847e30 −0.397062 −0.198531 0.980095i \(-0.563617\pi\)
−0.198531 + 0.980095i \(0.563617\pi\)
\(912\) − 4.43182e30i − 0.368669i
\(913\) − 1.12622e31i − 0.926150i
\(914\) 3.06943e28 0.00249530
\(915\) 0 0
\(916\) 1.10042e31 0.874291
\(917\) − 7.07086e30i − 0.555385i
\(918\) − 1.43914e30i − 0.111752i
\(919\) 1.97911e31 1.51935 0.759673 0.650305i \(-0.225359\pi\)
0.759673 + 0.650305i \(0.225359\pi\)
\(920\) 0 0
\(921\) −4.00510e30 −0.300530
\(922\) 2.00284e30i 0.148584i
\(923\) − 4.79399e30i − 0.351624i
\(924\) −1.05436e31 −0.764599
\(925\) 0 0
\(926\) −3.40211e30 −0.241175
\(927\) 3.12342e30i 0.218924i
\(928\) 3.03585e29i 0.0210391i
\(929\) 2.30269e31 1.57786 0.788932 0.614480i \(-0.210634\pi\)
0.788932 + 0.614480i \(0.210634\pi\)
\(930\) 0 0
\(931\) −3.07211e31 −2.05809
\(932\) − 4.33851e30i − 0.287391i
\(933\) − 5.91757e30i − 0.387602i
\(934\) 9.87288e28 0.00639443
\(935\) 0 0
\(936\) 3.21002e30 0.203288
\(937\) 2.58396e31i 1.61816i 0.587701 + 0.809078i \(0.300033\pi\)
−0.587701 + 0.809078i \(0.699967\pi\)
\(938\) 3.75878e30i 0.232765i
\(939\) 1.35690e31 0.830921
\(940\) 0 0
\(941\) 2.93678e31 1.75865 0.879326 0.476221i \(-0.157994\pi\)
0.879326 + 0.476221i \(0.157994\pi\)
\(942\) − 2.11780e30i − 0.125415i
\(943\) − 1.30309e31i − 0.763138i
\(944\) −1.35235e31 −0.783220
\(945\) 0 0
\(946\) 6.81377e30 0.385949
\(947\) − 8.23900e29i − 0.0461530i −0.999734 0.0230765i \(-0.992654\pi\)
0.999734 0.0230765i \(-0.00734613\pi\)
\(948\) 6.71721e30i 0.372135i
\(949\) −1.63665e31 −0.896728
\(950\) 0 0
\(951\) 1.21911e31 0.653350
\(952\) − 3.48158e31i − 1.84539i
\(953\) 3.34266e31i 1.75233i 0.482008 + 0.876167i \(0.339907\pi\)
−0.482008 + 0.876167i \(0.660093\pi\)
\(954\) −4.36177e30 −0.226154
\(955\) 0 0
\(956\) 1.46431e30 0.0742720
\(957\) 2.36747e29i 0.0118771i
\(958\) − 2.62967e30i − 0.130486i
\(959\) −6.42466e30 −0.315322
\(960\) 0 0
\(961\) 5.87607e31 2.82157
\(962\) − 1.03855e31i − 0.493274i
\(963\) − 4.38577e30i − 0.206049i
\(964\) 3.56802e30 0.165813
\(965\) 0 0
\(966\) 9.19456e30 0.418091
\(967\) 2.28891e31i 1.02956i 0.857323 + 0.514779i \(0.172126\pi\)
−0.857323 + 0.514779i \(0.827874\pi\)
\(968\) − 2.92801e30i − 0.130281i
\(969\) 2.17033e31 0.955268
\(970\) 0 0
\(971\) −4.60425e31 −1.98316 −0.991578 0.129508i \(-0.958660\pi\)
−0.991578 + 0.129508i \(0.958660\pi\)
\(972\) − 1.29309e30i − 0.0550976i
\(973\) − 3.91632e30i − 0.165080i
\(974\) 3.42925e30 0.142998
\(975\) 0 0
\(976\) 3.28412e30 0.134028
\(977\) 7.20727e30i 0.290990i 0.989359 + 0.145495i \(0.0464775\pi\)
−0.989359 + 0.145495i \(0.953523\pi\)
\(978\) 2.82376e29i 0.0112790i
\(979\) 1.06491e31 0.420817
\(980\) 0 0
\(981\) −5.81333e30 −0.224854
\(982\) 1.99850e30i 0.0764776i
\(983\) − 1.58951e31i − 0.601800i −0.953656 0.300900i \(-0.902713\pi\)
0.953656 0.300900i \(-0.0972870\pi\)
\(984\) 7.28415e30 0.272854
\(985\) 0 0
\(986\) −3.61206e29 −0.0132448
\(987\) − 1.20764e31i − 0.438134i
\(988\) 2.23672e31i 0.802905i
\(989\) 3.61657e31 1.28450
\(990\) 0 0
\(991\) 4.62622e31 1.60862 0.804309 0.594211i \(-0.202536\pi\)
0.804309 + 0.594211i \(0.202536\pi\)
\(992\) 5.24095e31i 1.80317i
\(993\) 6.44905e30i 0.219548i
\(994\) 7.67580e30 0.258563
\(995\) 0 0
\(996\) 1.54390e31 0.509210
\(997\) − 1.13237e31i − 0.369562i −0.982780 0.184781i \(-0.940842\pi\)
0.982780 0.184781i \(-0.0591576\pi\)
\(998\) 1.08848e31i 0.351519i
\(999\) −9.05451e30 −0.289352
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 75.22.b.c.49.2 2
5.2 odd 4 75.22.a.b.1.1 1
5.3 odd 4 15.22.a.a.1.1 1
5.4 even 2 inner 75.22.b.c.49.1 2
15.8 even 4 45.22.a.a.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
15.22.a.a.1.1 1 5.3 odd 4
45.22.a.a.1.1 1 15.8 even 4
75.22.a.b.1.1 1 5.2 odd 4
75.22.b.c.49.1 2 5.4 even 2 inner
75.22.b.c.49.2 2 1.1 even 1 trivial