Properties

Label 50.26.b.b.49.1
Level $50$
Weight $26$
Character 50.49
Analytic conductor $197.998$
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] = [50,26,Mod(49,50)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(50, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 26, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("50.49");
 
S:= CuspForms(chi, 26);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 50 = 2 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 26 \)
Character orbit: \([\chi]\) \(=\) 50.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: \(197.998389976\)
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: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 10)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 49.1
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 50.49
Dual form 50.26.b.b.49.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-4096.00i q^{2} +162864. i q^{3} -1.67772e7 q^{4} +6.67091e8 q^{6} +1.76009e10i q^{7} +6.87195e10i q^{8} +8.20764e11 q^{9} -1.12404e13 q^{11} -2.73240e12i q^{12} +4.24221e12i q^{13} +7.20933e13 q^{14} +2.81475e14 q^{16} +1.13786e15i q^{17} -3.36185e15i q^{18} -2.98985e15 q^{19} -2.86655e15 q^{21} +4.60406e16i q^{22} -1.22305e17i q^{23} -1.11919e16 q^{24} +1.73761e16 q^{26} +2.71666e17i q^{27} -2.95294e17i q^{28} +2.12048e18 q^{29} +4.22586e18 q^{31} -1.15292e18i q^{32} -1.83065e18i q^{33} +4.66066e18 q^{34} -1.37701e19 q^{36} -2.41679e19i q^{37} +1.22464e19i q^{38} -6.90903e17 q^{39} -1.61564e20 q^{41} +1.17414e19i q^{42} -3.11612e20i q^{43} +1.88582e20 q^{44} -5.00963e20 q^{46} +1.22586e21i q^{47} +4.58421e19i q^{48} +1.03128e21 q^{49} -1.85316e20 q^{51} -7.11725e19i q^{52} -4.78915e20i q^{53} +1.11274e21 q^{54} -1.20952e21 q^{56} -4.86938e20i q^{57} -8.68547e21i q^{58} -1.21434e22 q^{59} -2.03328e22 q^{61} -1.73091e22i q^{62} +1.44462e22i q^{63} -4.72237e21 q^{64} -7.49835e21 q^{66} -6.15653e22i q^{67} -1.90901e22i q^{68} +1.99192e22 q^{69} +1.34077e23 q^{71} +5.64025e22i q^{72} +2.23779e23i q^{73} -9.89916e22 q^{74} +5.01613e22 q^{76} -1.97841e23i q^{77} +2.82994e21i q^{78} -3.82350e23 q^{79} +6.51179e23 q^{81} +6.61767e23i q^{82} -7.71818e23i q^{83} +4.80928e22 q^{84} -1.27636e24 q^{86} +3.45349e23i q^{87} -7.72433e23i q^{88} +8.40581e23 q^{89} -7.46667e22 q^{91} +2.05195e24i q^{92} +6.88241e23i q^{93} +5.02111e24 q^{94} +1.87769e23 q^{96} +6.19164e24i q^{97} -4.22411e24i q^{98} -9.22569e24 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 33554432 q^{4} + 1334181888 q^{6} + 1641527853894 q^{9} - 22480747671096 q^{11} + 144186519486464 q^{14} + 562949953421312 q^{16} - 59\!\cdots\!40 q^{19} - 57\!\cdots\!76 q^{21} - 22\!\cdots\!08 q^{24}+ \cdots - 18\!\cdots\!12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/50\mathbb{Z}\right)^\times\).

\(n\) \(27\)
\(\chi(n)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 4096.00i − 0.707107i
\(3\) 162864.i 0.176933i 0.996079 + 0.0884666i \(0.0281967\pi\)
−0.996079 + 0.0884666i \(0.971803\pi\)
\(4\) −1.67772e7 −0.500000
\(5\) 0 0
\(6\) 6.67091e8 0.125111
\(7\) 1.76009e10i 0.480628i 0.970695 + 0.240314i \(0.0772504\pi\)
−0.970695 + 0.240314i \(0.922750\pi\)
\(8\) 6.87195e10i 0.353553i
\(9\) 8.20764e11 0.968695
\(10\) 0 0
\(11\) −1.12404e13 −1.07987 −0.539936 0.841706i \(-0.681552\pi\)
−0.539936 + 0.841706i \(0.681552\pi\)
\(12\) − 2.73240e12i − 0.0884666i
\(13\) 4.24221e12i 0.0505010i 0.999681 + 0.0252505i \(0.00803834\pi\)
−0.999681 + 0.0252505i \(0.991962\pi\)
\(14\) 7.20933e13 0.339855
\(15\) 0 0
\(16\) 2.81475e14 0.250000
\(17\) 1.13786e15i 0.473670i 0.971550 + 0.236835i \(0.0761101\pi\)
−0.971550 + 0.236835i \(0.923890\pi\)
\(18\) − 3.36185e15i − 0.684971i
\(19\) −2.98985e15 −0.309905 −0.154953 0.987922i \(-0.549522\pi\)
−0.154953 + 0.987922i \(0.549522\pi\)
\(20\) 0 0
\(21\) −2.86655e15 −0.0850391
\(22\) 4.60406e16i 0.763585i
\(23\) − 1.22305e17i − 1.16372i −0.813290 0.581859i \(-0.802325\pi\)
0.813290 0.581859i \(-0.197675\pi\)
\(24\) −1.11919e16 −0.0625553
\(25\) 0 0
\(26\) 1.73761e16 0.0357096
\(27\) 2.71666e17i 0.348328i
\(28\) − 2.95294e17i − 0.240314i
\(29\) 2.12048e18 1.11291 0.556453 0.830879i \(-0.312162\pi\)
0.556453 + 0.830879i \(0.312162\pi\)
\(30\) 0 0
\(31\) 4.22586e18 0.963594 0.481797 0.876283i \(-0.339984\pi\)
0.481797 + 0.876283i \(0.339984\pi\)
\(32\) − 1.15292e18i − 0.176777i
\(33\) − 1.83065e18i − 0.191065i
\(34\) 4.66066e18 0.334935
\(35\) 0 0
\(36\) −1.37701e19 −0.484347
\(37\) − 2.41679e19i − 0.603555i −0.953378 0.301778i \(-0.902420\pi\)
0.953378 0.301778i \(-0.0975800\pi\)
\(38\) 1.22464e19i 0.219136i
\(39\) −6.90903e17 −0.00893531
\(40\) 0 0
\(41\) −1.61564e20 −1.11827 −0.559136 0.829076i \(-0.688867\pi\)
−0.559136 + 0.829076i \(0.688867\pi\)
\(42\) 1.17414e19i 0.0601317i
\(43\) − 3.11612e20i − 1.18921i −0.804018 0.594605i \(-0.797308\pi\)
0.804018 0.594605i \(-0.202692\pi\)
\(44\) 1.88582e20 0.539936
\(45\) 0 0
\(46\) −5.00963e20 −0.822873
\(47\) 1.22586e21i 1.53892i 0.638694 + 0.769461i \(0.279475\pi\)
−0.638694 + 0.769461i \(0.720525\pi\)
\(48\) 4.58421e19i 0.0442333i
\(49\) 1.03128e21 0.768997
\(50\) 0 0
\(51\) −1.85316e20 −0.0838079
\(52\) − 7.11725e19i − 0.0252505i
\(53\) − 4.78915e20i − 0.133909i −0.997756 0.0669546i \(-0.978672\pi\)
0.997756 0.0669546i \(-0.0213283\pi\)
\(54\) 1.11274e21 0.246305
\(55\) 0 0
\(56\) −1.20952e21 −0.169928
\(57\) − 4.86938e20i − 0.0548325i
\(58\) − 8.68547e21i − 0.786943i
\(59\) −1.21434e22 −0.888566 −0.444283 0.895887i \(-0.646541\pi\)
−0.444283 + 0.895887i \(0.646541\pi\)
\(60\) 0 0
\(61\) −2.03328e22 −0.980786 −0.490393 0.871501i \(-0.663147\pi\)
−0.490393 + 0.871501i \(0.663147\pi\)
\(62\) − 1.73091e22i − 0.681364i
\(63\) 1.44462e22i 0.465582i
\(64\) −4.72237e21 −0.125000
\(65\) 0 0
\(66\) −7.49835e21 −0.135104
\(67\) − 6.15653e22i − 0.919180i −0.888131 0.459590i \(-0.847996\pi\)
0.888131 0.459590i \(-0.152004\pi\)
\(68\) − 1.90901e22i − 0.236835i
\(69\) 1.99192e22 0.205900
\(70\) 0 0
\(71\) 1.34077e23 0.969669 0.484835 0.874606i \(-0.338880\pi\)
0.484835 + 0.874606i \(0.338880\pi\)
\(72\) 5.64025e22i 0.342485i
\(73\) 2.23779e23i 1.14362i 0.820385 + 0.571811i \(0.193759\pi\)
−0.820385 + 0.571811i \(0.806241\pi\)
\(74\) −9.89916e22 −0.426778
\(75\) 0 0
\(76\) 5.01613e22 0.154953
\(77\) − 1.97841e23i − 0.519017i
\(78\) 2.82994e21i 0.00631822i
\(79\) −3.82350e23 −0.727985 −0.363992 0.931402i \(-0.618587\pi\)
−0.363992 + 0.931402i \(0.618587\pi\)
\(80\) 0 0
\(81\) 6.51179e23 0.907064
\(82\) 6.61767e23i 0.790737i
\(83\) − 7.71818e23i − 0.792572i −0.918127 0.396286i \(-0.870299\pi\)
0.918127 0.396286i \(-0.129701\pi\)
\(84\) 4.80928e22 0.0425195
\(85\) 0 0
\(86\) −1.27636e24 −0.840899
\(87\) 3.45349e23i 0.196910i
\(88\) − 7.72433e23i − 0.381792i
\(89\) 8.40581e23 0.360748 0.180374 0.983598i \(-0.442269\pi\)
0.180374 + 0.983598i \(0.442269\pi\)
\(90\) 0 0
\(91\) −7.46667e22 −0.0242722
\(92\) 2.05195e24i 0.581859i
\(93\) 6.88241e23i 0.170492i
\(94\) 5.02111e24 1.08818
\(95\) 0 0
\(96\) 1.87769e23 0.0312777
\(97\) 6.19164e24i 0.906064i 0.891494 + 0.453032i \(0.149658\pi\)
−0.891494 + 0.453032i \(0.850342\pi\)
\(98\) − 4.22411e24i − 0.543763i
\(99\) −9.22569e24 −1.04607
\(100\) 0 0
\(101\) 6.64342e24 0.586644 0.293322 0.956014i \(-0.405239\pi\)
0.293322 + 0.956014i \(0.405239\pi\)
\(102\) 7.59053e23i 0.0592612i
\(103\) 1.36741e25i 0.945004i 0.881330 + 0.472502i \(0.156649\pi\)
−0.881330 + 0.472502i \(0.843351\pi\)
\(104\) −2.91522e23 −0.0178548
\(105\) 0 0
\(106\) −1.96164e24 −0.0946881
\(107\) − 1.97392e24i − 0.0847290i −0.999102 0.0423645i \(-0.986511\pi\)
0.999102 0.0423645i \(-0.0134891\pi\)
\(108\) − 4.55779e24i − 0.174164i
\(109\) −4.86252e25 −1.65588 −0.827942 0.560813i \(-0.810489\pi\)
−0.827942 + 0.560813i \(0.810489\pi\)
\(110\) 0 0
\(111\) 3.93608e24 0.106789
\(112\) 4.95421e24i 0.120157i
\(113\) 4.58410e25i 0.994886i 0.867497 + 0.497443i \(0.165728\pi\)
−0.867497 + 0.497443i \(0.834272\pi\)
\(114\) −1.99450e24 −0.0387724
\(115\) 0 0
\(116\) −3.55757e25 −0.556453
\(117\) 3.48185e24i 0.0489201i
\(118\) 4.97393e25i 0.628311i
\(119\) −2.00273e25 −0.227659
\(120\) 0 0
\(121\) 1.79989e25 0.166123
\(122\) 8.32832e25i 0.693521i
\(123\) − 2.63130e25i − 0.197859i
\(124\) −7.08982e25 −0.481797
\(125\) 0 0
\(126\) 5.91715e25 0.329216
\(127\) 1.31721e26i 0.663911i 0.943295 + 0.331955i \(0.107708\pi\)
−0.943295 + 0.331955i \(0.892292\pi\)
\(128\) 1.93428e25i 0.0883883i
\(129\) 5.07504e25 0.210411
\(130\) 0 0
\(131\) −7.36281e25 −0.251856 −0.125928 0.992039i \(-0.540191\pi\)
−0.125928 + 0.992039i \(0.540191\pi\)
\(132\) 3.07132e25i 0.0955326i
\(133\) − 5.26240e25i − 0.148949i
\(134\) −2.52171e26 −0.649959
\(135\) 0 0
\(136\) −7.81929e25 −0.167468
\(137\) 9.58320e26i 1.87285i 0.350866 + 0.936426i \(0.385887\pi\)
−0.350866 + 0.936426i \(0.614113\pi\)
\(138\) − 8.15889e25i − 0.145594i
\(139\) 4.90566e26 0.799858 0.399929 0.916546i \(-0.369035\pi\)
0.399929 + 0.916546i \(0.369035\pi\)
\(140\) 0 0
\(141\) −1.99648e26 −0.272286
\(142\) − 5.49178e26i − 0.685660i
\(143\) − 4.76840e25i − 0.0545346i
\(144\) 2.31025e26 0.242174
\(145\) 0 0
\(146\) 9.16597e26 0.808664
\(147\) 1.67958e26i 0.136061i
\(148\) 4.05470e26i 0.301778i
\(149\) −9.09615e26 −0.622342 −0.311171 0.950354i \(-0.600721\pi\)
−0.311171 + 0.950354i \(0.600721\pi\)
\(150\) 0 0
\(151\) 8.24560e26 0.477541 0.238770 0.971076i \(-0.423256\pi\)
0.238770 + 0.971076i \(0.423256\pi\)
\(152\) − 2.05461e26i − 0.109568i
\(153\) 9.33911e26i 0.458841i
\(154\) −8.10355e26 −0.367000
\(155\) 0 0
\(156\) 1.15914e25 0.00446765
\(157\) − 9.72907e26i − 0.346199i −0.984904 0.173099i \(-0.944622\pi\)
0.984904 0.173099i \(-0.0553782\pi\)
\(158\) 1.56610e27i 0.514763i
\(159\) 7.79981e25 0.0236930
\(160\) 0 0
\(161\) 2.15269e27 0.559316
\(162\) − 2.66723e27i − 0.641391i
\(163\) − 2.54973e27i − 0.567740i −0.958863 0.283870i \(-0.908382\pi\)
0.958863 0.283870i \(-0.0916184\pi\)
\(164\) 2.71060e27 0.559136
\(165\) 0 0
\(166\) −3.16137e27 −0.560433
\(167\) − 8.09067e27i − 1.33054i −0.746602 0.665271i \(-0.768316\pi\)
0.746602 0.665271i \(-0.231684\pi\)
\(168\) − 1.96988e26i − 0.0300659i
\(169\) 7.03841e27 0.997450
\(170\) 0 0
\(171\) −2.45396e27 −0.300203
\(172\) 5.22799e27i 0.594605i
\(173\) 4.51555e27i 0.477677i 0.971059 + 0.238838i \(0.0767667\pi\)
−0.971059 + 0.238838i \(0.923233\pi\)
\(174\) 1.41455e27 0.139236
\(175\) 0 0
\(176\) −3.16388e27 −0.269968
\(177\) − 1.97772e27i − 0.157217i
\(178\) − 3.44302e27i − 0.255088i
\(179\) 1.94694e28 1.34490 0.672450 0.740143i \(-0.265242\pi\)
0.672450 + 0.740143i \(0.265242\pi\)
\(180\) 0 0
\(181\) −1.21952e28 −0.733176 −0.366588 0.930383i \(-0.619474\pi\)
−0.366588 + 0.930383i \(0.619474\pi\)
\(182\) 3.05835e26i 0.0171630i
\(183\) − 3.31148e27i − 0.173534i
\(184\) 8.40477e27 0.411437
\(185\) 0 0
\(186\) 2.81903e27 0.120556
\(187\) − 1.27899e28i − 0.511503i
\(188\) − 2.05665e28i − 0.769461i
\(189\) −4.78156e27 −0.167416
\(190\) 0 0
\(191\) 6.11119e28 1.87590 0.937948 0.346775i \(-0.112723\pi\)
0.937948 + 0.346775i \(0.112723\pi\)
\(192\) − 7.69103e26i − 0.0221167i
\(193\) 6.48935e28i 1.74878i 0.485226 + 0.874389i \(0.338737\pi\)
−0.485226 + 0.874389i \(0.661263\pi\)
\(194\) 2.53610e28 0.640684
\(195\) 0 0
\(196\) −1.73020e28 −0.384498
\(197\) 3.17525e28i 0.662140i 0.943606 + 0.331070i \(0.107410\pi\)
−0.943606 + 0.331070i \(0.892590\pi\)
\(198\) 3.77884e28i 0.739680i
\(199\) 7.39236e28 1.35869 0.679344 0.733820i \(-0.262265\pi\)
0.679344 + 0.733820i \(0.262265\pi\)
\(200\) 0 0
\(201\) 1.00268e28 0.162634
\(202\) − 2.72115e28i − 0.414820i
\(203\) 3.73223e28i 0.534894i
\(204\) 3.10908e27 0.0419040
\(205\) 0 0
\(206\) 5.60091e28 0.668218
\(207\) − 1.00384e29i − 1.12729i
\(208\) 1.19408e27i 0.0126253i
\(209\) 3.36070e28 0.334658
\(210\) 0 0
\(211\) 1.18016e28 0.104330 0.0521652 0.998638i \(-0.483388\pi\)
0.0521652 + 0.998638i \(0.483388\pi\)
\(212\) 8.03487e27i 0.0669546i
\(213\) 2.18363e28i 0.171567i
\(214\) −8.08517e27 −0.0599124
\(215\) 0 0
\(216\) −1.86687e28 −0.123152
\(217\) 7.43790e28i 0.463131i
\(218\) 1.99169e29i 1.17089i
\(219\) −3.64455e28 −0.202345
\(220\) 0 0
\(221\) −4.82702e27 −0.0239208
\(222\) − 1.61222e28i − 0.0755112i
\(223\) 1.55992e28i 0.0690704i 0.999403 + 0.0345352i \(0.0109951\pi\)
−0.999403 + 0.0345352i \(0.989005\pi\)
\(224\) 2.02924e28 0.0849639
\(225\) 0 0
\(226\) 1.87765e29 0.703491
\(227\) 3.57748e29i 1.26839i 0.773171 + 0.634197i \(0.218669\pi\)
−0.773171 + 0.634197i \(0.781331\pi\)
\(228\) 8.16947e27i 0.0274163i
\(229\) 6.16468e29 1.95870 0.979348 0.202179i \(-0.0648024\pi\)
0.979348 + 0.202179i \(0.0648024\pi\)
\(230\) 0 0
\(231\) 3.22211e28 0.0918313
\(232\) 1.45718e29i 0.393472i
\(233\) 5.68344e29i 1.45433i 0.686464 + 0.727163i \(0.259162\pi\)
−0.686464 + 0.727163i \(0.740838\pi\)
\(234\) 1.42617e28 0.0345917
\(235\) 0 0
\(236\) 2.03732e29 0.444283
\(237\) − 6.22710e28i − 0.128805i
\(238\) 8.20317e28i 0.160979i
\(239\) 5.96388e29 1.11059 0.555296 0.831653i \(-0.312605\pi\)
0.555296 + 0.831653i \(0.312605\pi\)
\(240\) 0 0
\(241\) −4.97938e29 −0.835530 −0.417765 0.908555i \(-0.637187\pi\)
−0.417765 + 0.908555i \(0.637187\pi\)
\(242\) − 7.37237e28i − 0.117467i
\(243\) 3.36233e29i 0.508817i
\(244\) 3.41128e29 0.490393
\(245\) 0 0
\(246\) −1.07778e29 −0.139908
\(247\) − 1.26835e28i − 0.0156505i
\(248\) 2.90399e29i 0.340682i
\(249\) 1.25701e29 0.140232
\(250\) 0 0
\(251\) 1.75610e30 1.77267 0.886335 0.463044i \(-0.153243\pi\)
0.886335 + 0.463044i \(0.153243\pi\)
\(252\) − 2.42367e29i − 0.232791i
\(253\) 1.37476e30i 1.25667i
\(254\) 5.39531e29 0.469456
\(255\) 0 0
\(256\) 7.92282e28 0.0625000
\(257\) − 1.22065e30i − 0.917123i −0.888663 0.458562i \(-0.848365\pi\)
0.888663 0.458562i \(-0.151635\pi\)
\(258\) − 2.07874e29i − 0.148783i
\(259\) 4.25376e29 0.290086
\(260\) 0 0
\(261\) 1.74041e30 1.07807
\(262\) 3.01581e29i 0.178089i
\(263\) 2.27827e30i 1.28280i 0.767207 + 0.641399i \(0.221646\pi\)
−0.767207 + 0.641399i \(0.778354\pi\)
\(264\) 1.25801e29 0.0675518
\(265\) 0 0
\(266\) −2.15548e29 −0.105323
\(267\) 1.36900e29i 0.0638284i
\(268\) 1.03289e30i 0.459590i
\(269\) 3.12815e30 1.32857 0.664284 0.747480i \(-0.268736\pi\)
0.664284 + 0.747480i \(0.268736\pi\)
\(270\) 0 0
\(271\) 1.31818e30 0.510339 0.255170 0.966896i \(-0.417869\pi\)
0.255170 + 0.966896i \(0.417869\pi\)
\(272\) 3.20278e29i 0.118417i
\(273\) − 1.21605e28i − 0.00429456i
\(274\) 3.92528e30 1.32431
\(275\) 0 0
\(276\) −3.34188e29 −0.102950
\(277\) − 2.11862e30i − 0.623815i −0.950112 0.311908i \(-0.899032\pi\)
0.950112 0.311908i \(-0.100968\pi\)
\(278\) − 2.00936e30i − 0.565585i
\(279\) 3.46844e30 0.933429
\(280\) 0 0
\(281\) −6.37141e30 −1.56822 −0.784109 0.620623i \(-0.786880\pi\)
−0.784109 + 0.620623i \(0.786880\pi\)
\(282\) 8.17758e29i 0.192536i
\(283\) 3.00725e30i 0.677391i 0.940896 + 0.338695i \(0.109986\pi\)
−0.940896 + 0.338695i \(0.890014\pi\)
\(284\) −2.24943e30 −0.484835
\(285\) 0 0
\(286\) −1.95314e29 −0.0385618
\(287\) − 2.84368e30i − 0.537473i
\(288\) − 9.46276e29i − 0.171243i
\(289\) 4.47591e30 0.775637
\(290\) 0 0
\(291\) −1.00840e30 −0.160313
\(292\) − 3.75438e30i − 0.571811i
\(293\) − 7.19334e30i − 1.04975i −0.851180 0.524875i \(-0.824112\pi\)
0.851180 0.524875i \(-0.175888\pi\)
\(294\) 6.87956e29 0.0962097
\(295\) 0 0
\(296\) 1.66080e30 0.213389
\(297\) − 3.05362e30i − 0.376149i
\(298\) 3.72578e30i 0.440063i
\(299\) 5.18845e29 0.0587690
\(300\) 0 0
\(301\) 5.48465e30 0.571568
\(302\) − 3.37740e30i − 0.337672i
\(303\) 1.08197e30i 0.103797i
\(304\) −8.41567e29 −0.0774763
\(305\) 0 0
\(306\) 3.82530e30 0.324450
\(307\) 6.08638e30i 0.495597i 0.968812 + 0.247799i \(0.0797072\pi\)
−0.968812 + 0.247799i \(0.920293\pi\)
\(308\) 3.31921e30i 0.259508i
\(309\) −2.22702e30 −0.167203
\(310\) 0 0
\(311\) 3.60168e30 0.249460 0.124730 0.992191i \(-0.460194\pi\)
0.124730 + 0.992191i \(0.460194\pi\)
\(312\) − 4.74785e28i − 0.00315911i
\(313\) 3.07166e30i 0.196367i 0.995168 + 0.0981835i \(0.0313032\pi\)
−0.995168 + 0.0981835i \(0.968697\pi\)
\(314\) −3.98503e30 −0.244800
\(315\) 0 0
\(316\) 6.41476e30 0.363992
\(317\) 2.80660e31i 1.53088i 0.643510 + 0.765438i \(0.277478\pi\)
−0.643510 + 0.765438i \(0.722522\pi\)
\(318\) − 3.19480e29i − 0.0167535i
\(319\) −2.38349e31 −1.20180
\(320\) 0 0
\(321\) 3.21480e29 0.0149914
\(322\) − 8.81740e30i − 0.395496i
\(323\) − 3.40201e30i − 0.146793i
\(324\) −1.09250e31 −0.453532
\(325\) 0 0
\(326\) −1.04437e31 −0.401453
\(327\) − 7.91929e30i − 0.292981i
\(328\) − 1.11026e31i − 0.395369i
\(329\) −2.15762e31 −0.739649
\(330\) 0 0
\(331\) −1.79893e31 −0.571694 −0.285847 0.958275i \(-0.592275\pi\)
−0.285847 + 0.958275i \(0.592275\pi\)
\(332\) 1.29490e31i 0.396286i
\(333\) − 1.98361e31i − 0.584661i
\(334\) −3.31394e31 −0.940835
\(335\) 0 0
\(336\) −8.06863e29 −0.0212598
\(337\) 3.37680e31i 0.857296i 0.903472 + 0.428648i \(0.141010\pi\)
−0.903472 + 0.428648i \(0.858990\pi\)
\(338\) − 2.88293e31i − 0.705303i
\(339\) −7.46585e30 −0.176028
\(340\) 0 0
\(341\) −4.75003e31 −1.04056
\(342\) 1.00514e31i 0.212276i
\(343\) 4.17554e31i 0.850230i
\(344\) 2.14138e31 0.420450
\(345\) 0 0
\(346\) 1.84957e31 0.337769
\(347\) 3.07070e30i 0.0540902i 0.999634 + 0.0270451i \(0.00860977\pi\)
−0.999634 + 0.0270451i \(0.991390\pi\)
\(348\) − 5.79400e30i − 0.0984550i
\(349\) −4.73553e31 −0.776338 −0.388169 0.921588i \(-0.626892\pi\)
−0.388169 + 0.921588i \(0.626892\pi\)
\(350\) 0 0
\(351\) −1.15246e30 −0.0175909
\(352\) 1.29593e31i 0.190896i
\(353\) − 1.13735e32i − 1.61700i −0.588496 0.808500i \(-0.700280\pi\)
0.588496 0.808500i \(-0.299720\pi\)
\(354\) −8.10074e30 −0.111169
\(355\) 0 0
\(356\) −1.41026e31 −0.180374
\(357\) − 3.26172e30i − 0.0402805i
\(358\) − 7.97467e31i − 0.950988i
\(359\) −7.58317e31 −0.873313 −0.436656 0.899628i \(-0.643837\pi\)
−0.436656 + 0.899628i \(0.643837\pi\)
\(360\) 0 0
\(361\) −8.41373e31 −0.903959
\(362\) 4.99517e31i 0.518434i
\(363\) 2.93138e30i 0.0293927i
\(364\) 1.25270e30 0.0121361
\(365\) 0 0
\(366\) −1.35638e31 −0.122707
\(367\) − 6.65888e31i − 0.582205i −0.956692 0.291102i \(-0.905978\pi\)
0.956692 0.291102i \(-0.0940220\pi\)
\(368\) − 3.44259e31i − 0.290930i
\(369\) −1.32606e32 −1.08326
\(370\) 0 0
\(371\) 8.42934e30 0.0643605
\(372\) − 1.15468e31i − 0.0852459i
\(373\) 4.03892e31i 0.288340i 0.989553 + 0.144170i \(0.0460512\pi\)
−0.989553 + 0.144170i \(0.953949\pi\)
\(374\) −5.23875e31 −0.361687
\(375\) 0 0
\(376\) −8.42403e31 −0.544091
\(377\) 8.99550e30i 0.0562029i
\(378\) 1.95853e31i 0.118381i
\(379\) −3.00036e32 −1.75462 −0.877309 0.479926i \(-0.840664\pi\)
−0.877309 + 0.479926i \(0.840664\pi\)
\(380\) 0 0
\(381\) −2.14527e31 −0.117468
\(382\) − 2.50314e32i − 1.32646i
\(383\) − 5.83515e31i − 0.299273i −0.988741 0.149636i \(-0.952190\pi\)
0.988741 0.149636i \(-0.0478103\pi\)
\(384\) −3.15025e30 −0.0156388
\(385\) 0 0
\(386\) 2.65804e32 1.23657
\(387\) − 2.55760e32i − 1.15198i
\(388\) − 1.03879e32i − 0.453032i
\(389\) 4.65526e32 1.96595 0.982977 0.183726i \(-0.0588161\pi\)
0.982977 + 0.183726i \(0.0588161\pi\)
\(390\) 0 0
\(391\) 1.39166e32 0.551218
\(392\) 7.08688e31i 0.271881i
\(393\) − 1.19914e31i − 0.0445617i
\(394\) 1.30058e32 0.468204
\(395\) 0 0
\(396\) 1.54781e32 0.523033
\(397\) 3.57088e32i 1.16922i 0.811316 + 0.584608i \(0.198752\pi\)
−0.811316 + 0.584608i \(0.801248\pi\)
\(398\) − 3.02791e32i − 0.960737i
\(399\) 8.57055e30 0.0263540
\(400\) 0 0
\(401\) −5.85410e32 −1.69105 −0.845523 0.533940i \(-0.820711\pi\)
−0.845523 + 0.533940i \(0.820711\pi\)
\(402\) − 4.10696e31i − 0.114999i
\(403\) 1.79270e31i 0.0486625i
\(404\) −1.11458e32 −0.293322
\(405\) 0 0
\(406\) 1.52872e32 0.378227
\(407\) 2.71656e32i 0.651762i
\(408\) − 1.27348e31i − 0.0296306i
\(409\) 5.85778e32 1.32188 0.660939 0.750439i \(-0.270158\pi\)
0.660939 + 0.750439i \(0.270158\pi\)
\(410\) 0 0
\(411\) −1.56076e32 −0.331370
\(412\) − 2.29413e32i − 0.472502i
\(413\) − 2.13734e32i − 0.427070i
\(414\) −4.11172e32 −0.797113
\(415\) 0 0
\(416\) 4.89093e30 0.00892740
\(417\) 7.98956e31i 0.141521i
\(418\) − 1.37654e32i − 0.236639i
\(419\) 2.15945e32 0.360303 0.180152 0.983639i \(-0.442341\pi\)
0.180152 + 0.983639i \(0.442341\pi\)
\(420\) 0 0
\(421\) −2.57301e32 −0.404497 −0.202249 0.979334i \(-0.564825\pi\)
−0.202249 + 0.979334i \(0.564825\pi\)
\(422\) − 4.83394e31i − 0.0737727i
\(423\) 1.00614e33i 1.49075i
\(424\) 3.29108e31 0.0473441
\(425\) 0 0
\(426\) 8.94413e31 0.121316
\(427\) − 3.57876e32i − 0.471394i
\(428\) 3.31169e31i 0.0423645i
\(429\) 7.76601e30 0.00964899
\(430\) 0 0
\(431\) −4.87420e32 −0.571396 −0.285698 0.958320i \(-0.592225\pi\)
−0.285698 + 0.958320i \(0.592225\pi\)
\(432\) 7.64671e31i 0.0870819i
\(433\) 1.34127e33i 1.48394i 0.670433 + 0.741970i \(0.266108\pi\)
−0.670433 + 0.741970i \(0.733892\pi\)
\(434\) 3.04656e32 0.327483
\(435\) 0 0
\(436\) 8.15795e32 0.827942
\(437\) 3.65674e32i 0.360642i
\(438\) 1.49281e32i 0.143079i
\(439\) −3.20033e32 −0.298118 −0.149059 0.988828i \(-0.547624\pi\)
−0.149059 + 0.988828i \(0.547624\pi\)
\(440\) 0 0
\(441\) 8.46435e32 0.744923
\(442\) 1.97715e31i 0.0169146i
\(443\) 1.50899e33i 1.25499i 0.778622 + 0.627494i \(0.215919\pi\)
−0.778622 + 0.627494i \(0.784081\pi\)
\(444\) −6.60364e31 −0.0533945
\(445\) 0 0
\(446\) 6.38944e31 0.0488402
\(447\) − 1.48144e32i − 0.110113i
\(448\) − 8.31179e31i − 0.0600785i
\(449\) 7.32432e32 0.514859 0.257429 0.966297i \(-0.417125\pi\)
0.257429 + 0.966297i \(0.417125\pi\)
\(450\) 0 0
\(451\) 1.81604e33 1.20759
\(452\) − 7.69084e32i − 0.497443i
\(453\) 1.34291e32i 0.0844928i
\(454\) 1.46534e33 0.896890
\(455\) 0 0
\(456\) 3.34621e31 0.0193862
\(457\) 1.80848e33i 1.01944i 0.860341 + 0.509718i \(0.170250\pi\)
−0.860341 + 0.509718i \(0.829750\pi\)
\(458\) − 2.52505e33i − 1.38501i
\(459\) −3.09116e32 −0.164992
\(460\) 0 0
\(461\) −2.02348e33 −1.02291 −0.511454 0.859311i \(-0.670893\pi\)
−0.511454 + 0.859311i \(0.670893\pi\)
\(462\) − 1.31978e32i − 0.0649345i
\(463\) 4.00896e33i 1.91986i 0.280239 + 0.959930i \(0.409586\pi\)
−0.280239 + 0.959930i \(0.590414\pi\)
\(464\) 5.96861e32 0.278226
\(465\) 0 0
\(466\) 2.32794e33 1.02836
\(467\) 1.11951e33i 0.481466i 0.970591 + 0.240733i \(0.0773878\pi\)
−0.970591 + 0.240733i \(0.922612\pi\)
\(468\) − 5.84158e31i − 0.0244600i
\(469\) 1.08360e33 0.441784
\(470\) 0 0
\(471\) 1.58452e32 0.0612541
\(472\) − 8.34487e32i − 0.314155i
\(473\) 3.50264e33i 1.28420i
\(474\) −2.55062e32 −0.0910786
\(475\) 0 0
\(476\) 3.36002e32 0.113830
\(477\) − 3.93077e32i − 0.129717i
\(478\) − 2.44280e33i − 0.785308i
\(479\) 5.66968e33 1.77568 0.887841 0.460151i \(-0.152205\pi\)
0.887841 + 0.460151i \(0.152205\pi\)
\(480\) 0 0
\(481\) 1.02525e32 0.0304802
\(482\) 2.03955e33i 0.590809i
\(483\) 3.50595e32i 0.0989616i
\(484\) −3.01972e32 −0.0830615
\(485\) 0 0
\(486\) 1.37721e33 0.359788
\(487\) − 1.76093e33i − 0.448363i −0.974547 0.224182i \(-0.928029\pi\)
0.974547 0.224182i \(-0.0719709\pi\)
\(488\) − 1.39726e33i − 0.346760i
\(489\) 4.15260e32 0.100452
\(490\) 0 0
\(491\) −5.58354e33 −1.28348 −0.641742 0.766920i \(-0.721788\pi\)
−0.641742 + 0.766920i \(0.721788\pi\)
\(492\) 4.41459e32i 0.0989297i
\(493\) 2.41280e33i 0.527150i
\(494\) −5.19518e31 −0.0110666
\(495\) 0 0
\(496\) 1.18947e33 0.240899
\(497\) 2.35987e33i 0.466050i
\(498\) − 5.14873e32i − 0.0991593i
\(499\) 4.37368e33 0.821468 0.410734 0.911755i \(-0.365273\pi\)
0.410734 + 0.911755i \(0.365273\pi\)
\(500\) 0 0
\(501\) 1.31768e33 0.235417
\(502\) − 7.19299e33i − 1.25347i
\(503\) − 4.32379e33i − 0.734962i −0.930031 0.367481i \(-0.880220\pi\)
0.930031 0.367481i \(-0.119780\pi\)
\(504\) −9.92734e32 −0.164608
\(505\) 0 0
\(506\) 5.63101e33 0.888597
\(507\) 1.14630e33i 0.176482i
\(508\) − 2.20992e33i − 0.331955i
\(509\) −1.81736e33 −0.266360 −0.133180 0.991092i \(-0.542519\pi\)
−0.133180 + 0.991092i \(0.542519\pi\)
\(510\) 0 0
\(511\) −3.93870e33 −0.549657
\(512\) − 3.24519e32i − 0.0441942i
\(513\) − 8.12238e32i − 0.107948i
\(514\) −4.99979e33 −0.648504
\(515\) 0 0
\(516\) −8.51451e32 −0.105205
\(517\) − 1.37791e34i − 1.66184i
\(518\) − 1.74234e33i − 0.205121i
\(519\) −7.35420e32 −0.0845169
\(520\) 0 0
\(521\) −7.48131e33 −0.819419 −0.409710 0.912216i \(-0.634370\pi\)
−0.409710 + 0.912216i \(0.634370\pi\)
\(522\) − 7.12872e33i − 0.762308i
\(523\) 1.34405e34i 1.40328i 0.712531 + 0.701641i \(0.247549\pi\)
−0.712531 + 0.701641i \(0.752451\pi\)
\(524\) 1.23527e33 0.125928
\(525\) 0 0
\(526\) 9.33180e33 0.907076
\(527\) 4.80842e33i 0.456426i
\(528\) − 5.15283e32i − 0.0477663i
\(529\) −3.91286e33 −0.354240
\(530\) 0 0
\(531\) −9.96685e33 −0.860749
\(532\) 8.82883e32i 0.0744746i
\(533\) − 6.85390e32i − 0.0564739i
\(534\) 5.60744e32 0.0451335
\(535\) 0 0
\(536\) 4.23073e33 0.324979
\(537\) 3.17087e33i 0.237958i
\(538\) − 1.28129e34i − 0.939440i
\(539\) −1.15919e34 −0.830418
\(540\) 0 0
\(541\) −9.16620e33 −0.626937 −0.313468 0.949599i \(-0.601491\pi\)
−0.313468 + 0.949599i \(0.601491\pi\)
\(542\) − 5.39927e33i − 0.360864i
\(543\) − 1.98617e33i − 0.129723i
\(544\) 1.31186e33 0.0837338
\(545\) 0 0
\(546\) −4.98095e31 −0.00303671
\(547\) 2.47403e34i 1.47423i 0.675770 + 0.737113i \(0.263811\pi\)
−0.675770 + 0.737113i \(0.736189\pi\)
\(548\) − 1.60779e34i − 0.936426i
\(549\) −1.66884e34 −0.950082
\(550\) 0 0
\(551\) −6.33989e33 −0.344895
\(552\) 1.36883e33i 0.0727968i
\(553\) − 6.72970e33i − 0.349890i
\(554\) −8.67787e33 −0.441104
\(555\) 0 0
\(556\) −8.23034e33 −0.399929
\(557\) 2.03083e34i 0.964905i 0.875922 + 0.482452i \(0.160254\pi\)
−0.875922 + 0.482452i \(0.839746\pi\)
\(558\) − 1.42067e34i − 0.660034i
\(559\) 1.32192e33 0.0600564
\(560\) 0 0
\(561\) 2.08302e33 0.0905018
\(562\) 2.60973e34i 1.10890i
\(563\) − 3.26297e34i − 1.35599i −0.735064 0.677997i \(-0.762848\pi\)
0.735064 0.677997i \(-0.237152\pi\)
\(564\) 3.34954e33 0.136143
\(565\) 0 0
\(566\) 1.23177e34 0.478988
\(567\) 1.14613e34i 0.435960i
\(568\) 9.21368e33i 0.342830i
\(569\) −3.41443e34 −1.24284 −0.621419 0.783478i \(-0.713444\pi\)
−0.621419 + 0.783478i \(0.713444\pi\)
\(570\) 0 0
\(571\) 1.36450e33 0.0475359 0.0237679 0.999718i \(-0.492434\pi\)
0.0237679 + 0.999718i \(0.492434\pi\)
\(572\) 8.00005e32i 0.0272673i
\(573\) 9.95293e33i 0.331908i
\(574\) −1.16477e34 −0.380051
\(575\) 0 0
\(576\) −3.87595e33 −0.121087
\(577\) − 3.92987e34i − 1.20138i −0.799481 0.600691i \(-0.794892\pi\)
0.799481 0.600691i \(-0.205108\pi\)
\(578\) − 1.83333e34i − 0.548458i
\(579\) −1.05688e34 −0.309417
\(580\) 0 0
\(581\) 1.35847e34 0.380933
\(582\) 4.13039e33i 0.113358i
\(583\) 5.38319e33i 0.144605i
\(584\) −1.53780e34 −0.404332
\(585\) 0 0
\(586\) −2.94639e34 −0.742285
\(587\) − 3.72718e34i − 0.919187i −0.888129 0.459594i \(-0.847995\pi\)
0.888129 0.459594i \(-0.152005\pi\)
\(588\) − 2.81787e33i − 0.0680305i
\(589\) −1.26347e34 −0.298623
\(590\) 0 0
\(591\) −5.17134e33 −0.117155
\(592\) − 6.80265e33i − 0.150889i
\(593\) 7.75169e34i 1.68350i 0.539870 + 0.841749i \(0.318474\pi\)
−0.539870 + 0.841749i \(0.681526\pi\)
\(594\) −1.25076e34 −0.265978
\(595\) 0 0
\(596\) 1.52608e34 0.311171
\(597\) 1.20395e34i 0.240397i
\(598\) − 2.12519e33i − 0.0415559i
\(599\) −6.18820e34 −1.18503 −0.592515 0.805559i \(-0.701865\pi\)
−0.592515 + 0.805559i \(0.701865\pi\)
\(600\) 0 0
\(601\) −2.95852e34 −0.543430 −0.271715 0.962378i \(-0.587591\pi\)
−0.271715 + 0.962378i \(0.587591\pi\)
\(602\) − 2.24651e34i − 0.404160i
\(603\) − 5.05305e34i − 0.890405i
\(604\) −1.38338e34 −0.238770
\(605\) 0 0
\(606\) 4.43177e33 0.0733954
\(607\) 1.00617e35i 1.63234i 0.577810 + 0.816171i \(0.303907\pi\)
−0.577810 + 0.816171i \(0.696093\pi\)
\(608\) 3.44706e33i 0.0547840i
\(609\) −6.07845e33 −0.0946405
\(610\) 0 0
\(611\) −5.20034e33 −0.0777171
\(612\) − 1.56684e34i − 0.229421i
\(613\) − 4.47446e34i − 0.641926i −0.947092 0.320963i \(-0.895993\pi\)
0.947092 0.320963i \(-0.104007\pi\)
\(614\) 2.49298e34 0.350440
\(615\) 0 0
\(616\) 1.35955e34 0.183500
\(617\) − 2.58799e34i − 0.342293i −0.985246 0.171147i \(-0.945253\pi\)
0.985246 0.171147i \(-0.0547472\pi\)
\(618\) 9.12187e33i 0.118230i
\(619\) 1.37687e35 1.74889 0.874443 0.485129i \(-0.161227\pi\)
0.874443 + 0.485129i \(0.161227\pi\)
\(620\) 0 0
\(621\) 3.32262e34 0.405355
\(622\) − 1.47525e34i − 0.176395i
\(623\) 1.47950e34i 0.173386i
\(624\) −1.94472e32 −0.00223383
\(625\) 0 0
\(626\) 1.25815e34 0.138852
\(627\) 5.47337e33i 0.0592121i
\(628\) 1.63227e34i 0.173099i
\(629\) 2.74996e34 0.285886
\(630\) 0 0
\(631\) −2.83484e34 −0.283244 −0.141622 0.989921i \(-0.545232\pi\)
−0.141622 + 0.989921i \(0.545232\pi\)
\(632\) − 2.62749e34i − 0.257381i
\(633\) 1.92206e33i 0.0184595i
\(634\) 1.14958e35 1.08249
\(635\) 0 0
\(636\) −1.30859e33 −0.0118465
\(637\) 4.37489e33i 0.0388351i
\(638\) 9.76279e34i 0.849798i
\(639\) 1.10045e35 0.939314
\(640\) 0 0
\(641\) −1.36368e34 −0.111940 −0.0559701 0.998432i \(-0.517825\pi\)
−0.0559701 + 0.998432i \(0.517825\pi\)
\(642\) − 1.31678e33i − 0.0106005i
\(643\) 8.47939e32i 0.00669464i 0.999994 + 0.00334732i \(0.00106549\pi\)
−0.999994 + 0.00334732i \(0.998935\pi\)
\(644\) −3.61161e34 −0.279658
\(645\) 0 0
\(646\) −1.39346e34 −0.103798
\(647\) − 2.50478e35i − 1.83007i −0.403379 0.915033i \(-0.632164\pi\)
0.403379 0.915033i \(-0.367836\pi\)
\(648\) 4.47487e34i 0.320696i
\(649\) 1.36496e35 0.959537
\(650\) 0 0
\(651\) −1.21137e34 −0.0819432
\(652\) 4.27774e34i 0.283870i
\(653\) − 1.55775e35i − 1.01410i −0.861916 0.507051i \(-0.830735\pi\)
0.861916 0.507051i \(-0.169265\pi\)
\(654\) −3.24374e34 −0.207169
\(655\) 0 0
\(656\) −4.54763e34 −0.279568
\(657\) 1.83669e35i 1.10782i
\(658\) 8.83760e34i 0.523011i
\(659\) 1.85485e35 1.07706 0.538531 0.842605i \(-0.318979\pi\)
0.538531 + 0.842605i \(0.318979\pi\)
\(660\) 0 0
\(661\) 2.92623e35 1.63602 0.818012 0.575202i \(-0.195076\pi\)
0.818012 + 0.575202i \(0.195076\pi\)
\(662\) 7.36840e34i 0.404248i
\(663\) − 7.86148e32i − 0.00423239i
\(664\) 5.30389e34 0.280217
\(665\) 0 0
\(666\) −8.12488e34 −0.413417
\(667\) − 2.59346e35i − 1.29511i
\(668\) 1.35739e35i 0.665271i
\(669\) −2.54055e33 −0.0122209
\(670\) 0 0
\(671\) 2.28548e35 1.05912
\(672\) 3.30491e33i 0.0150329i
\(673\) 2.13829e35i 0.954724i 0.878707 + 0.477362i \(0.158407\pi\)
−0.878707 + 0.477362i \(0.841593\pi\)
\(674\) 1.38314e35 0.606200
\(675\) 0 0
\(676\) −1.18085e35 −0.498725
\(677\) − 1.59208e35i − 0.660096i −0.943964 0.330048i \(-0.892935\pi\)
0.943964 0.330048i \(-0.107065\pi\)
\(678\) 3.05801e34i 0.124471i
\(679\) −1.08978e35 −0.435480
\(680\) 0 0
\(681\) −5.82643e34 −0.224421
\(682\) 1.94561e35i 0.735786i
\(683\) 4.23003e35i 1.57067i 0.619071 + 0.785335i \(0.287509\pi\)
−0.619071 + 0.785335i \(0.712491\pi\)
\(684\) 4.11706e34 0.150102
\(685\) 0 0
\(686\) 1.71030e35 0.601203
\(687\) 1.00401e35i 0.346559i
\(688\) − 8.77111e34i − 0.297303i
\(689\) 2.03166e33 0.00676255
\(690\) 0 0
\(691\) 3.02759e35 0.971899 0.485950 0.873987i \(-0.338474\pi\)
0.485950 + 0.873987i \(0.338474\pi\)
\(692\) − 7.57584e34i − 0.238838i
\(693\) − 1.62380e35i − 0.502769i
\(694\) 1.25776e34 0.0382476
\(695\) 0 0
\(696\) −2.37322e34 −0.0696182
\(697\) − 1.83837e35i − 0.529692i
\(698\) 1.93967e35i 0.548954i
\(699\) −9.25627e34 −0.257319
\(700\) 0 0
\(701\) 2.66145e35 0.713909 0.356954 0.934122i \(-0.383815\pi\)
0.356954 + 0.934122i \(0.383815\pi\)
\(702\) 4.72049e33i 0.0124386i
\(703\) 7.22582e34i 0.187045i
\(704\) 5.30812e34 0.134984
\(705\) 0 0
\(706\) −4.65859e35 −1.14339
\(707\) 1.16930e35i 0.281958i
\(708\) 3.31806e34i 0.0786084i
\(709\) −2.81745e34 −0.0655810 −0.0327905 0.999462i \(-0.510439\pi\)
−0.0327905 + 0.999462i \(0.510439\pi\)
\(710\) 0 0
\(711\) −3.13819e35 −0.705195
\(712\) 5.77643e34i 0.127544i
\(713\) − 5.16846e35i − 1.12135i
\(714\) −1.33600e34 −0.0284826
\(715\) 0 0
\(716\) −3.26643e35 −0.672450
\(717\) 9.71301e34i 0.196501i
\(718\) 3.10606e35i 0.617526i
\(719\) −2.87441e35 −0.561613 −0.280806 0.959764i \(-0.590602\pi\)
−0.280806 + 0.959764i \(0.590602\pi\)
\(720\) 0 0
\(721\) −2.40676e35 −0.454195
\(722\) 3.44626e35i 0.639195i
\(723\) − 8.10962e34i − 0.147833i
\(724\) 2.04602e35 0.366588
\(725\) 0 0
\(726\) 1.20069e34 0.0207838
\(727\) 4.98046e35i 0.847402i 0.905802 + 0.423701i \(0.139269\pi\)
−0.905802 + 0.423701i \(0.860731\pi\)
\(728\) − 5.13105e33i − 0.00858152i
\(729\) 4.96977e35 0.817037
\(730\) 0 0
\(731\) 3.54570e35 0.563293
\(732\) 5.55575e34i 0.0867668i
\(733\) − 8.47339e35i − 1.30094i −0.759531 0.650471i \(-0.774572\pi\)
0.759531 0.650471i \(-0.225428\pi\)
\(734\) −2.72748e35 −0.411681
\(735\) 0 0
\(736\) −1.41009e35 −0.205718
\(737\) 6.92016e35i 0.992597i
\(738\) 5.43155e35i 0.765983i
\(739\) −7.12136e35 −0.987433 −0.493716 0.869623i \(-0.664362\pi\)
−0.493716 + 0.869623i \(0.664362\pi\)
\(740\) 0 0
\(741\) 2.06569e33 0.00276910
\(742\) − 3.45266e34i − 0.0455098i
\(743\) − 6.18867e35i − 0.802116i −0.916053 0.401058i \(-0.868643\pi\)
0.916053 0.401058i \(-0.131357\pi\)
\(744\) −4.72956e34 −0.0602780
\(745\) 0 0
\(746\) 1.65434e35 0.203887
\(747\) − 6.33481e35i − 0.767761i
\(748\) 2.14579e35i 0.255751i
\(749\) 3.47427e34 0.0407231
\(750\) 0 0
\(751\) 1.34082e36 1.52009 0.760046 0.649869i \(-0.225176\pi\)
0.760046 + 0.649869i \(0.225176\pi\)
\(752\) 3.45048e35i 0.384731i
\(753\) 2.86006e35i 0.313644i
\(754\) 3.68456e34 0.0397414
\(755\) 0 0
\(756\) 8.02213e34 0.0837080
\(757\) − 8.81045e35i − 0.904274i −0.891949 0.452137i \(-0.850662\pi\)
0.891949 0.452137i \(-0.149338\pi\)
\(758\) 1.22895e36i 1.24070i
\(759\) −2.23899e35 −0.222346
\(760\) 0 0
\(761\) −9.08045e35 −0.872568 −0.436284 0.899809i \(-0.643706\pi\)
−0.436284 + 0.899809i \(0.643706\pi\)
\(762\) 8.78701e34i 0.0830623i
\(763\) − 8.55847e35i − 0.795865i
\(764\) −1.02529e36 −0.937948
\(765\) 0 0
\(766\) −2.39008e35 −0.211618
\(767\) − 5.15148e34i − 0.0448735i
\(768\) 1.29034e34i 0.0110583i
\(769\) −1.57427e36 −1.32739 −0.663696 0.748003i \(-0.731013\pi\)
−0.663696 + 0.748003i \(0.731013\pi\)
\(770\) 0 0
\(771\) 1.98800e35 0.162270
\(772\) − 1.08873e36i − 0.874389i
\(773\) − 1.97307e36i − 1.55919i −0.626284 0.779595i \(-0.715425\pi\)
0.626284 0.779595i \(-0.284575\pi\)
\(774\) −1.04759e36 −0.814574
\(775\) 0 0
\(776\) −4.25486e35 −0.320342
\(777\) 6.92785e34i 0.0513258i
\(778\) − 1.90679e36i − 1.39014i
\(779\) 4.83052e35 0.346558
\(780\) 0 0
\(781\) −1.50707e36 −1.04712
\(782\) − 5.70024e35i − 0.389770i
\(783\) 5.76061e35i 0.387656i
\(784\) 2.90279e35 0.192249
\(785\) 0 0
\(786\) −4.91166e34 −0.0315099
\(787\) − 9.52553e35i − 0.601458i −0.953710 0.300729i \(-0.902770\pi\)
0.953710 0.300729i \(-0.0972299\pi\)
\(788\) − 5.32719e35i − 0.331070i
\(789\) −3.71048e35 −0.226970
\(790\) 0 0
\(791\) −8.06842e35 −0.478170
\(792\) − 6.33985e35i − 0.369840i
\(793\) − 8.62560e34i − 0.0495307i
\(794\) 1.46263e36 0.826760
\(795\) 0 0
\(796\) −1.24023e36 −0.679344
\(797\) − 3.70863e35i − 0.199979i −0.994989 0.0999894i \(-0.968119\pi\)
0.994989 0.0999894i \(-0.0318809\pi\)
\(798\) − 3.51050e34i − 0.0186351i
\(799\) −1.39485e36 −0.728941
\(800\) 0 0
\(801\) 6.89918e35 0.349455
\(802\) 2.39784e36i 1.19575i
\(803\) − 2.51536e36i − 1.23497i
\(804\) −1.68221e35 −0.0813168
\(805\) 0 0
\(806\) 7.34290e34 0.0344096
\(807\) 5.09463e35i 0.235068i
\(808\) 4.56532e35i 0.207410i
\(809\) 2.08152e36 0.931159 0.465580 0.885006i \(-0.345846\pi\)
0.465580 + 0.885006i \(0.345846\pi\)
\(810\) 0 0
\(811\) −9.20046e34 −0.0399070 −0.0199535 0.999801i \(-0.506352\pi\)
−0.0199535 + 0.999801i \(0.506352\pi\)
\(812\) − 6.26164e35i − 0.267447i
\(813\) 2.14684e35i 0.0902960i
\(814\) 1.11270e36 0.460865
\(815\) 0 0
\(816\) −5.21618e34 −0.0209520
\(817\) 9.31673e35i 0.368543i
\(818\) − 2.39935e36i − 0.934709i
\(819\) −6.12837e34 −0.0235124
\(820\) 0 0
\(821\) 9.64905e35 0.359083 0.179542 0.983750i \(-0.442539\pi\)
0.179542 + 0.983750i \(0.442539\pi\)
\(822\) 6.39286e35i 0.234314i
\(823\) 4.33977e35i 0.156664i 0.996927 + 0.0783319i \(0.0249594\pi\)
−0.996927 + 0.0783319i \(0.975041\pi\)
\(824\) −9.39677e35 −0.334109
\(825\) 0 0
\(826\) −8.75456e35 −0.301984
\(827\) − 1.42223e36i − 0.483227i −0.970373 0.241614i \(-0.922323\pi\)
0.970373 0.241614i \(-0.0776766\pi\)
\(828\) 1.68416e36i 0.563644i
\(829\) −4.07234e36 −1.34249 −0.671247 0.741234i \(-0.734241\pi\)
−0.671247 + 0.741234i \(0.734241\pi\)
\(830\) 0 0
\(831\) 3.45047e35 0.110374
\(832\) − 2.00333e34i − 0.00631263i
\(833\) 1.17344e36i 0.364251i
\(834\) 3.27252e35 0.100071
\(835\) 0 0
\(836\) −5.63832e35 −0.167329
\(837\) 1.14802e36i 0.335646i
\(838\) − 8.84511e35i − 0.254773i
\(839\) −1.54259e36 −0.437750 −0.218875 0.975753i \(-0.570239\pi\)
−0.218875 + 0.975753i \(0.570239\pi\)
\(840\) 0 0
\(841\) 8.66055e35 0.238559
\(842\) 1.05391e36i 0.286023i
\(843\) − 1.03767e36i − 0.277470i
\(844\) −1.97998e35 −0.0521652
\(845\) 0 0
\(846\) 4.12115e36 1.05412
\(847\) 3.16798e35i 0.0798434i
\(848\) − 1.34803e35i − 0.0334773i
\(849\) −4.89773e35 −0.119853
\(850\) 0 0
\(851\) −2.95586e36 −0.702368
\(852\) − 3.66352e35i − 0.0857834i
\(853\) 7.94652e36i 1.83364i 0.399300 + 0.916820i \(0.369253\pi\)
−0.399300 + 0.916820i \(0.630747\pi\)
\(854\) −1.46586e36 −0.333326
\(855\) 0 0
\(856\) 1.35647e35 0.0299562
\(857\) 7.37212e35i 0.160447i 0.996777 + 0.0802236i \(0.0255634\pi\)
−0.996777 + 0.0802236i \(0.974437\pi\)
\(858\) − 3.18096e34i − 0.00682286i
\(859\) 6.08537e36 1.28639 0.643195 0.765703i \(-0.277609\pi\)
0.643195 + 0.765703i \(0.277609\pi\)
\(860\) 0 0
\(861\) 4.63133e35 0.0950968
\(862\) 1.99647e36i 0.404038i
\(863\) 3.30325e36i 0.658880i 0.944177 + 0.329440i \(0.106860\pi\)
−0.944177 + 0.329440i \(0.893140\pi\)
\(864\) 3.13209e35 0.0615762
\(865\) 0 0
\(866\) 5.49382e36 1.04930
\(867\) 7.28965e35i 0.137236i
\(868\) − 1.24787e36i − 0.231565i
\(869\) 4.29775e36 0.786130
\(870\) 0 0
\(871\) 2.61173e35 0.0464195
\(872\) − 3.34150e36i − 0.585444i
\(873\) 5.08188e36i 0.877700i
\(874\) 1.49780e36 0.255013
\(875\) 0 0
\(876\) 6.11454e35 0.101172
\(877\) 2.53964e35i 0.0414264i 0.999785 + 0.0207132i \(0.00659369\pi\)
−0.999785 + 0.0207132i \(0.993406\pi\)
\(878\) 1.31085e36i 0.210801i
\(879\) 1.17154e36 0.185736
\(880\) 0 0
\(881\) 7.01270e35 0.108065 0.0540327 0.998539i \(-0.482792\pi\)
0.0540327 + 0.998539i \(0.482792\pi\)
\(882\) − 3.46700e36i − 0.526740i
\(883\) 7.67538e36i 1.14972i 0.818253 + 0.574858i \(0.194943\pi\)
−0.818253 + 0.574858i \(0.805057\pi\)
\(884\) 8.09840e34 0.0119604
\(885\) 0 0
\(886\) 6.18082e36 0.887410
\(887\) 1.19088e37i 1.68586i 0.538024 + 0.842929i \(0.319171\pi\)
−0.538024 + 0.842929i \(0.680829\pi\)
\(888\) 2.70485e35i 0.0377556i
\(889\) −2.31841e36 −0.319094
\(890\) 0 0
\(891\) −7.31950e36 −0.979513
\(892\) − 2.61712e35i − 0.0345352i
\(893\) − 3.66512e36i − 0.476920i
\(894\) −6.06796e35 −0.0778617
\(895\) 0 0
\(896\) −3.40451e35 −0.0424819
\(897\) 8.45012e34i 0.0103982i
\(898\) − 3.00004e36i − 0.364060i
\(899\) 8.96084e36 1.07239
\(900\) 0 0
\(901\) 5.44937e35 0.0634288
\(902\) − 7.43851e36i − 0.853895i
\(903\) 8.93253e35i 0.101129i
\(904\) −3.15017e36 −0.351745
\(905\) 0 0
\(906\) 5.50056e35 0.0597454
\(907\) − 6.98769e36i − 0.748587i −0.927310 0.374294i \(-0.877885\pi\)
0.927310 0.374294i \(-0.122115\pi\)
\(908\) − 6.00202e36i − 0.634197i
\(909\) 5.45268e36 0.568279
\(910\) 0 0
\(911\) −8.23719e36 −0.835217 −0.417608 0.908627i \(-0.637132\pi\)
−0.417608 + 0.908627i \(0.637132\pi\)
\(912\) − 1.37061e35i − 0.0137081i
\(913\) 8.67553e36i 0.855877i
\(914\) 7.40751e36 0.720850
\(915\) 0 0
\(916\) −1.03426e37 −0.979348
\(917\) − 1.29592e36i − 0.121049i
\(918\) 1.26614e36i 0.116667i
\(919\) 8.31177e36 0.755526 0.377763 0.925902i \(-0.376693\pi\)
0.377763 + 0.925902i \(0.376693\pi\)
\(920\) 0 0
\(921\) −9.91251e35 −0.0876877
\(922\) 8.28816e36i 0.723305i
\(923\) 5.68781e35i 0.0489693i
\(924\) −5.40581e35 −0.0459157
\(925\) 0 0
\(926\) 1.64207e37 1.35755
\(927\) 1.12232e37i 0.915420i
\(928\) − 2.44474e36i − 0.196736i
\(929\) 1.48672e37 1.18041 0.590205 0.807253i \(-0.299047\pi\)
0.590205 + 0.807253i \(0.299047\pi\)
\(930\) 0 0
\(931\) −3.08336e36 −0.238316
\(932\) − 9.53522e36i − 0.727163i
\(933\) 5.86585e35i 0.0441378i
\(934\) 4.58550e36 0.340448
\(935\) 0 0
\(936\) −2.39271e35 −0.0172959
\(937\) − 2.25559e37i − 1.60885i −0.594055 0.804424i \(-0.702474\pi\)
0.594055 0.804424i \(-0.297526\pi\)
\(938\) − 4.43844e36i − 0.312388i
\(939\) −5.00263e35 −0.0347439
\(940\) 0 0
\(941\) 2.40580e37 1.62701 0.813504 0.581559i \(-0.197557\pi\)
0.813504 + 0.581559i \(0.197557\pi\)
\(942\) − 6.49017e35i − 0.0433132i
\(943\) 1.97602e37i 1.30135i
\(944\) −3.41806e36 −0.222141
\(945\) 0 0
\(946\) 1.43468e37 0.908063
\(947\) 2.05868e35i 0.0128592i 0.999979 + 0.00642962i \(0.00204662\pi\)
−0.999979 + 0.00642962i \(0.997953\pi\)
\(948\) 1.04473e36i 0.0644023i
\(949\) −9.49316e35 −0.0577541
\(950\) 0 0
\(951\) −4.57094e36 −0.270863
\(952\) − 1.37626e36i − 0.0804896i
\(953\) 7.96109e36i 0.459528i 0.973246 + 0.229764i \(0.0737954\pi\)
−0.973246 + 0.229764i \(0.926205\pi\)
\(954\) −1.61004e36 −0.0917239
\(955\) 0 0
\(956\) −1.00057e37 −0.555296
\(957\) − 3.88185e36i − 0.212638i
\(958\) − 2.32230e37i − 1.25560i
\(959\) −1.68673e37 −0.900145
\(960\) 0 0
\(961\) −1.37487e36 −0.0714859
\(962\) − 4.19943e35i − 0.0215527i
\(963\) − 1.62012e36i − 0.0820765i
\(964\) 8.35401e36 0.417765
\(965\) 0 0
\(966\) 1.43604e36 0.0699764
\(967\) − 4.83856e36i − 0.232748i −0.993205 0.116374i \(-0.962873\pi\)
0.993205 0.116374i \(-0.0371271\pi\)
\(968\) 1.23688e36i 0.0587334i
\(969\) 5.54066e35 0.0259725
\(970\) 0 0
\(971\) 1.24060e37 0.566749 0.283375 0.959009i \(-0.408546\pi\)
0.283375 + 0.959009i \(0.408546\pi\)
\(972\) − 5.64105e36i − 0.254409i
\(973\) 8.63441e36i 0.384434i
\(974\) −7.21277e36 −0.317041
\(975\) 0 0
\(976\) −5.72318e36 −0.245197
\(977\) − 5.52484e36i − 0.233688i −0.993150 0.116844i \(-0.962722\pi\)
0.993150 0.116844i \(-0.0372778\pi\)
\(978\) − 1.70090e36i − 0.0710303i
\(979\) −9.44844e36 −0.389562
\(980\) 0 0
\(981\) −3.99098e37 −1.60405
\(982\) 2.28702e37i 0.907561i
\(983\) − 3.62327e37i − 1.41965i −0.704377 0.709826i \(-0.748774\pi\)
0.704377 0.709826i \(-0.251226\pi\)
\(984\) 1.80822e36 0.0699539
\(985\) 0 0
\(986\) 9.88281e36 0.372751
\(987\) − 3.51398e36i − 0.130869i
\(988\) 2.12795e35i 0.00782526i
\(989\) −3.81119e37 −1.38391
\(990\) 0 0
\(991\) −1.55999e37 −0.552334 −0.276167 0.961110i \(-0.589064\pi\)
−0.276167 + 0.961110i \(0.589064\pi\)
\(992\) − 4.87209e36i − 0.170341i
\(993\) − 2.92980e36i − 0.101152i
\(994\) 9.66602e36 0.329547
\(995\) 0 0
\(996\) −2.10892e36 −0.0701162
\(997\) 5.89404e37i 1.93519i 0.252505 + 0.967596i \(0.418746\pi\)
−0.252505 + 0.967596i \(0.581254\pi\)
\(998\) − 1.79146e37i − 0.580865i
\(999\) 6.56558e36 0.210235
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 50.26.b.b.49.1 2
5.2 odd 4 10.26.a.a.1.1 1
5.3 odd 4 50.26.a.a.1.1 1
5.4 even 2 inner 50.26.b.b.49.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
10.26.a.a.1.1 1 5.2 odd 4
50.26.a.a.1.1 1 5.3 odd 4
50.26.b.b.49.1 2 1.1 even 1 trivial
50.26.b.b.49.2 2 5.4 even 2 inner