Properties

Label 25.36.b.a.24.6
Level $25$
Weight $36$
Character 25.24
Analytic conductor $193.988$
Analytic rank $0$
Dimension $6$
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] = [25,36,Mod(24,25)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(25, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 36, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("25.24");
 
S:= CuspForms(chi, 36);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 25 = 5^{2} \)
Weight: \( k \) \(=\) \( 36 \)
Character orbit: \([\chi]\) \(=\) 25.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: \(193.987826584\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 24844388x^{4} + 154310903773636x^{2} + 6999547445919666225 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{26}\cdot 3^{6}\cdot 5^{8}\cdot 7^{2} \)
Twist minimal: no (minimal twist has level 1)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 24.6
Root \(-3626.53i\) of defining polynomial
Character \(\chi\) \(=\) 25.24
Dual form 25.36.b.a.24.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+331573. i q^{2} +1.54691e8i q^{3} -7.55810e10 q^{4} -5.12913e13 q^{6} +3.96640e14i q^{7} -1.36679e16i q^{8} +2.61023e16 q^{9} +O(q^{10})\) \(q+331573. i q^{2} +1.54691e8i q^{3} -7.55810e10 q^{4} -5.12913e13 q^{6} +3.96640e14i q^{7} -1.36679e16i q^{8} +2.61023e16 q^{9} -6.82681e17 q^{11} -1.16917e19i q^{12} +1.17949e19i q^{13} -1.31515e20 q^{14} +1.93496e21 q^{16} -1.33083e21i q^{17} +8.65483e21i q^{18} +3.58945e22 q^{19} -6.13565e22 q^{21} -2.26359e23i q^{22} -6.92195e23i q^{23} +2.11429e24 q^{24} -3.91088e24 q^{26} +1.17772e25i q^{27} -2.99785e25i q^{28} +5.67568e25 q^{29} +1.02960e26 q^{31} +1.71955e26i q^{32} -1.05604e26i q^{33} +4.41267e26 q^{34} -1.97284e27 q^{36} +4.50412e27i q^{37} +1.19017e28i q^{38} -1.82456e27 q^{39} -6.23104e27 q^{41} -2.03442e28i q^{42} +2.75822e27i q^{43} +5.15977e28 q^{44} +2.29513e29 q^{46} +1.41103e29i q^{47} +2.99320e29i q^{48} +2.21495e29 q^{49} +2.05867e29 q^{51} -8.91472e29i q^{52} +2.48503e30i q^{53} -3.90500e30 q^{54} +5.42123e30 q^{56} +5.55255e30i q^{57} +1.88190e31i q^{58} -5.47495e30 q^{59} +2.30979e31 q^{61} +3.41386e31i q^{62} +1.03532e31i q^{63} +9.46893e30 q^{64} +3.50156e31 q^{66} +1.55814e31i q^{67} +1.00585e32i q^{68} +1.07076e32 q^{69} -1.13262e32 q^{71} -3.56763e32i q^{72} -3.23524e32i q^{73} -1.49345e33 q^{74} -2.71294e33 q^{76} -2.70778e32i q^{77} -6.04976e32i q^{78} +1.44166e32 q^{79} -5.15885e32 q^{81} -2.06605e33i q^{82} -3.50421e33i q^{83} +4.63739e33 q^{84} -9.14551e32 q^{86} +8.77975e33i q^{87} +9.33079e33i q^{88} -9.37543e33 q^{89} -4.67833e33 q^{91} +5.23168e34i q^{92} +1.59269e34i q^{93} -4.67858e34 q^{94} -2.65998e34 q^{96} +3.59603e33i q^{97} +7.34419e34i q^{98} -1.78195e34 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 69682524288 q^{4} - 9573061128768 q^{6} - 30\!\cdots\!02 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 69682524288 q^{4} - 9573061128768 q^{6} - 30\!\cdots\!02 q^{9}+ \cdots - 60\!\cdots\!04 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/25\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 331573.i 1.78877i 0.447298 + 0.894385i \(0.352386\pi\)
−0.447298 + 0.894385i \(0.647614\pi\)
\(3\) 1.54691e8i 0.691580i 0.938312 + 0.345790i \(0.112389\pi\)
−0.938312 + 0.345790i \(0.887611\pi\)
\(4\) −7.55810e10 −2.19970
\(5\) 0 0
\(6\) −5.12913e13 −1.23708
\(7\) 3.96640e14i 0.644438i 0.946665 + 0.322219i \(0.104429\pi\)
−0.946665 + 0.322219i \(0.895571\pi\)
\(8\) − 1.36679e16i − 2.14598i
\(9\) 2.61023e16 0.521717
\(10\) 0 0
\(11\) −6.82681e17 −0.407236 −0.203618 0.979050i \(-0.565270\pi\)
−0.203618 + 0.979050i \(0.565270\pi\)
\(12\) − 1.16917e19i − 1.52127i
\(13\) 1.17949e19i 0.378169i 0.981961 + 0.189085i \(0.0605520\pi\)
−0.981961 + 0.189085i \(0.939448\pi\)
\(14\) −1.31515e20 −1.15275
\(15\) 0 0
\(16\) 1.93496e21 1.63897
\(17\) − 1.33083e21i − 0.390181i −0.980785 0.195090i \(-0.937500\pi\)
0.980785 0.195090i \(-0.0625000\pi\)
\(18\) 8.65483e21i 0.933232i
\(19\) 3.58945e22 1.50259 0.751294 0.659967i \(-0.229430\pi\)
0.751294 + 0.659967i \(0.229430\pi\)
\(20\) 0 0
\(21\) −6.13565e22 −0.445680
\(22\) − 2.26359e23i − 0.728451i
\(23\) − 6.92195e23i − 1.02327i −0.859202 0.511636i \(-0.829039\pi\)
0.859202 0.511636i \(-0.170961\pi\)
\(24\) 2.11429e24 1.48412
\(25\) 0 0
\(26\) −3.91088e24 −0.676457
\(27\) 1.17772e25i 1.05239i
\(28\) − 2.99785e25i − 1.41757i
\(29\) 5.67568e25 1.45229 0.726144 0.687542i \(-0.241310\pi\)
0.726144 + 0.687542i \(0.241310\pi\)
\(30\) 0 0
\(31\) 1.02960e26 0.820043 0.410022 0.912076i \(-0.365521\pi\)
0.410022 + 0.912076i \(0.365521\pi\)
\(32\) 1.71955e26i 0.785760i
\(33\) − 1.05604e26i − 0.281636i
\(34\) 4.41267e26 0.697943
\(35\) 0 0
\(36\) −1.97284e27 −1.14762
\(37\) 4.50412e27i 1.62211i 0.584971 + 0.811054i \(0.301106\pi\)
−0.584971 + 0.811054i \(0.698894\pi\)
\(38\) 1.19017e28i 2.68779i
\(39\) −1.82456e27 −0.261534
\(40\) 0 0
\(41\) −6.23104e27 −0.372257 −0.186129 0.982525i \(-0.559594\pi\)
−0.186129 + 0.982525i \(0.559594\pi\)
\(42\) − 2.03442e28i − 0.797219i
\(43\) 2.75822e27i 0.0716029i 0.999359 + 0.0358014i \(0.0113984\pi\)
−0.999359 + 0.0358014i \(0.988602\pi\)
\(44\) 5.15977e28 0.895795
\(45\) 0 0
\(46\) 2.29513e29 1.83040
\(47\) 1.41103e29i 0.772365i 0.922422 + 0.386182i \(0.126207\pi\)
−0.922422 + 0.386182i \(0.873793\pi\)
\(48\) 2.99320e29i 1.13348i
\(49\) 2.21495e29 0.584700
\(50\) 0 0
\(51\) 2.05867e29 0.269841
\(52\) − 8.91472e29i − 0.831858i
\(53\) 2.48503e30i 1.66151i 0.556638 + 0.830755i \(0.312091\pi\)
−0.556638 + 0.830755i \(0.687909\pi\)
\(54\) −3.90500e30 −1.88248
\(55\) 0 0
\(56\) 5.42123e30 1.38295
\(57\) 5.55255e30i 1.03916i
\(58\) 1.88190e31i 2.59781i
\(59\) −5.47495e30 −0.560364 −0.280182 0.959947i \(-0.590395\pi\)
−0.280182 + 0.959947i \(0.590395\pi\)
\(60\) 0 0
\(61\) 2.30979e31 1.31917 0.659585 0.751630i \(-0.270732\pi\)
0.659585 + 0.751630i \(0.270732\pi\)
\(62\) 3.41386e31i 1.46687i
\(63\) 1.03532e31i 0.336214i
\(64\) 9.46893e30 0.233427
\(65\) 0 0
\(66\) 3.50156e31 0.503782
\(67\) 1.55814e31i 0.172304i 0.996282 + 0.0861522i \(0.0274571\pi\)
−0.996282 + 0.0861522i \(0.972543\pi\)
\(68\) 1.00585e32i 0.858279i
\(69\) 1.07076e32 0.707675
\(70\) 0 0
\(71\) −1.13262e32 −0.454008 −0.227004 0.973894i \(-0.572893\pi\)
−0.227004 + 0.973894i \(0.572893\pi\)
\(72\) − 3.56763e32i − 1.11960i
\(73\) − 3.23524e32i − 0.797548i −0.917049 0.398774i \(-0.869436\pi\)
0.917049 0.398774i \(-0.130564\pi\)
\(74\) −1.49345e33 −2.90158
\(75\) 0 0
\(76\) −2.71294e33 −3.30524
\(77\) − 2.70778e32i − 0.262438i
\(78\) − 6.04976e32i − 0.467824i
\(79\) 1.44166e32 0.0892050 0.0446025 0.999005i \(-0.485798\pi\)
0.0446025 + 0.999005i \(0.485798\pi\)
\(80\) 0 0
\(81\) −5.15885e32 −0.206094
\(82\) − 2.06605e33i − 0.665882i
\(83\) − 3.50421e33i − 0.913532i −0.889587 0.456766i \(-0.849008\pi\)
0.889587 0.456766i \(-0.150992\pi\)
\(84\) 4.63739e33 0.980361
\(85\) 0 0
\(86\) −9.14551e32 −0.128081
\(87\) 8.77975e33i 1.00437i
\(88\) 9.33079e33i 0.873920i
\(89\) −9.37543e33 −0.720553 −0.360277 0.932846i \(-0.617318\pi\)
−0.360277 + 0.932846i \(0.617318\pi\)
\(90\) 0 0
\(91\) −4.67833e33 −0.243706
\(92\) 5.23168e34i 2.25089i
\(93\) 1.59269e34i 0.567126i
\(94\) −4.67858e34 −1.38158
\(95\) 0 0
\(96\) −2.65998e34 −0.543416
\(97\) 3.59603e33i 0.0612799i 0.999530 + 0.0306399i \(0.00975453\pi\)
−0.999530 + 0.0306399i \(0.990245\pi\)
\(98\) 7.34419e34i 1.04589i
\(99\) −1.78195e34 −0.212462
\(100\) 0 0
\(101\) 7.27571e34 0.611296 0.305648 0.952145i \(-0.401127\pi\)
0.305648 + 0.952145i \(0.401127\pi\)
\(102\) 6.82599e34i 0.482683i
\(103\) − 9.48038e34i − 0.565164i −0.959243 0.282582i \(-0.908809\pi\)
0.959243 0.282582i \(-0.0911909\pi\)
\(104\) 1.61211e35 0.811544
\(105\) 0 0
\(106\) −8.23968e35 −2.97206
\(107\) 5.43957e35i 1.66475i 0.554214 + 0.832374i \(0.313019\pi\)
−0.554214 + 0.832374i \(0.686981\pi\)
\(108\) − 8.90133e35i − 2.31494i
\(109\) 2.19201e35 0.485153 0.242577 0.970132i \(-0.422007\pi\)
0.242577 + 0.970132i \(0.422007\pi\)
\(110\) 0 0
\(111\) −6.96746e35 −1.12182
\(112\) 7.67481e35i 1.05621i
\(113\) 3.57083e35i 0.420627i 0.977634 + 0.210313i \(0.0674484\pi\)
−0.977634 + 0.210313i \(0.932552\pi\)
\(114\) −1.84108e36 −1.85882
\(115\) 0 0
\(116\) −4.28974e36 −3.19460
\(117\) 3.07875e35i 0.197297i
\(118\) − 1.81535e36i − 1.00236i
\(119\) 5.27859e35 0.251447
\(120\) 0 0
\(121\) −2.34419e36 −0.834159
\(122\) 7.65865e36i 2.35969i
\(123\) − 9.63884e35i − 0.257446i
\(124\) −7.78179e36 −1.80385
\(125\) 0 0
\(126\) −3.43285e36 −0.601410
\(127\) − 2.98205e36i − 0.454935i −0.973786 0.227467i \(-0.926955\pi\)
0.973786 0.227467i \(-0.0730445\pi\)
\(128\) 9.04797e36i 1.20331i
\(129\) −4.26671e35 −0.0495191
\(130\) 0 0
\(131\) 1.67513e37 1.48525 0.742627 0.669705i \(-0.233579\pi\)
0.742627 + 0.669705i \(0.233579\pi\)
\(132\) 7.98169e36i 0.619514i
\(133\) 1.42372e37i 0.968325i
\(134\) −5.16636e36 −0.308213
\(135\) 0 0
\(136\) −1.81896e37 −0.837321
\(137\) 7.05739e36i 0.285782i 0.989738 + 0.142891i \(0.0456398\pi\)
−0.989738 + 0.142891i \(0.954360\pi\)
\(138\) 3.55036e37i 1.26587i
\(139\) −5.54070e36 −0.174103 −0.0870514 0.996204i \(-0.527744\pi\)
−0.0870514 + 0.996204i \(0.527744\pi\)
\(140\) 0 0
\(141\) −2.18273e37 −0.534152
\(142\) − 3.75546e37i − 0.812115i
\(143\) − 8.05216e36i − 0.154004i
\(144\) 5.05068e37 0.855080
\(145\) 0 0
\(146\) 1.07272e38 1.42663
\(147\) 3.42633e37i 0.404367i
\(148\) − 3.40426e38i − 3.56815i
\(149\) 1.26594e38 1.17938 0.589689 0.807631i \(-0.299250\pi\)
0.589689 + 0.807631i \(0.299250\pi\)
\(150\) 0 0
\(151\) 1.46030e38 1.07732 0.538661 0.842523i \(-0.318930\pi\)
0.538661 + 0.842523i \(0.318930\pi\)
\(152\) − 4.90602e38i − 3.22453i
\(153\) − 3.47377e37i − 0.203564i
\(154\) 8.97829e37 0.469441
\(155\) 0 0
\(156\) 1.37902e38 0.575296
\(157\) − 2.57702e38i − 0.961332i −0.876904 0.480666i \(-0.840395\pi\)
0.876904 0.480666i \(-0.159605\pi\)
\(158\) 4.78016e37i 0.159567i
\(159\) −3.84410e38 −1.14907
\(160\) 0 0
\(161\) 2.74552e38 0.659436
\(162\) − 1.71054e38i − 0.368655i
\(163\) 1.00523e39i 1.94529i 0.232306 + 0.972643i \(0.425373\pi\)
−0.232306 + 0.972643i \(0.574627\pi\)
\(164\) 4.70948e38 0.818853
\(165\) 0 0
\(166\) 1.16190e39 1.63410
\(167\) 4.87583e38i 0.617320i 0.951172 + 0.308660i \(0.0998805\pi\)
−0.951172 + 0.308660i \(0.900119\pi\)
\(168\) 8.38614e38i 0.956422i
\(169\) 8.33666e38 0.856988
\(170\) 0 0
\(171\) 9.36930e38 0.783926
\(172\) − 2.08469e38i − 0.157505i
\(173\) 3.15688e37i 0.0215502i 0.999942 + 0.0107751i \(0.00342989\pi\)
−0.999942 + 0.0107751i \(0.996570\pi\)
\(174\) −2.91113e39 −1.79659
\(175\) 0 0
\(176\) −1.32096e39 −0.667447
\(177\) − 8.46924e38i − 0.387536i
\(178\) − 3.10864e39i − 1.28890i
\(179\) 4.86041e38 0.182702 0.0913512 0.995819i \(-0.470881\pi\)
0.0913512 + 0.995819i \(0.470881\pi\)
\(180\) 0 0
\(181\) −4.89892e39 −1.51609 −0.758045 0.652202i \(-0.773845\pi\)
−0.758045 + 0.652202i \(0.773845\pi\)
\(182\) − 1.55121e39i − 0.435935i
\(183\) 3.57303e39i 0.912311i
\(184\) −9.46084e39 −2.19593
\(185\) 0 0
\(186\) −5.28093e39 −1.01446
\(187\) 9.08530e38i 0.158895i
\(188\) − 1.06647e40i − 1.69897i
\(189\) −4.67131e39 −0.678199
\(190\) 0 0
\(191\) −8.79853e39 −1.06249 −0.531246 0.847217i \(-0.678276\pi\)
−0.531246 + 0.847217i \(0.678276\pi\)
\(192\) 1.46476e39i 0.161434i
\(193\) − 1.65403e40i − 1.66452i −0.554383 0.832262i \(-0.687046\pi\)
0.554383 0.832262i \(-0.312954\pi\)
\(194\) −1.19235e39 −0.109616
\(195\) 0 0
\(196\) −1.67408e40 −1.28616
\(197\) 3.96581e36i 0 0.000278723i 1.00000 0.000139362i \(4.43602e-5\pi\)
−1.00000 0.000139362i \(0.999956\pi\)
\(198\) − 5.90848e39i − 0.380045i
\(199\) 1.61546e40 0.951409 0.475705 0.879605i \(-0.342193\pi\)
0.475705 + 0.879605i \(0.342193\pi\)
\(200\) 0 0
\(201\) −2.41029e39 −0.119162
\(202\) 2.41243e40i 1.09347i
\(203\) 2.25120e40i 0.935909i
\(204\) −1.55596e40 −0.593569
\(205\) 0 0
\(206\) 3.14344e40 1.01095
\(207\) − 1.80679e40i − 0.533859i
\(208\) 2.28226e40i 0.619808i
\(209\) −2.45045e40 −0.611908
\(210\) 0 0
\(211\) 2.70344e40 0.571444 0.285722 0.958313i \(-0.407767\pi\)
0.285722 + 0.958313i \(0.407767\pi\)
\(212\) − 1.87821e41i − 3.65482i
\(213\) − 1.75206e40i − 0.313983i
\(214\) −1.80362e41 −2.97785
\(215\) 0 0
\(216\) 1.60969e41 2.25841
\(217\) 4.08379e40i 0.528467i
\(218\) 7.26812e40i 0.867827i
\(219\) 5.00462e40 0.551569
\(220\) 0 0
\(221\) 1.56970e40 0.147554
\(222\) − 2.31022e41i − 2.00667i
\(223\) − 1.60448e41i − 1.28825i −0.764921 0.644124i \(-0.777222\pi\)
0.764921 0.644124i \(-0.222778\pi\)
\(224\) −6.82042e40 −0.506374
\(225\) 0 0
\(226\) −1.18399e41 −0.752405
\(227\) − 1.32023e41i − 0.776597i −0.921534 0.388299i \(-0.873063\pi\)
0.921534 0.388299i \(-0.126937\pi\)
\(228\) − 4.19667e41i − 2.28584i
\(229\) 1.27675e41 0.644147 0.322073 0.946715i \(-0.395620\pi\)
0.322073 + 0.946715i \(0.395620\pi\)
\(230\) 0 0
\(231\) 4.18869e40 0.181497
\(232\) − 7.75745e41i − 3.11659i
\(233\) 1.15249e41i 0.429445i 0.976675 + 0.214722i \(0.0688847\pi\)
−0.976675 + 0.214722i \(0.931115\pi\)
\(234\) −1.02083e41 −0.352919
\(235\) 0 0
\(236\) 4.13802e41 1.23263
\(237\) 2.23012e40i 0.0616924i
\(238\) 1.75024e41i 0.449781i
\(239\) −1.42881e41 −0.341201 −0.170601 0.985340i \(-0.554571\pi\)
−0.170601 + 0.985340i \(0.554571\pi\)
\(240\) 0 0
\(241\) −1.47367e41 −0.304160 −0.152080 0.988368i \(-0.548597\pi\)
−0.152080 + 0.988368i \(0.548597\pi\)
\(242\) − 7.77271e41i − 1.49212i
\(243\) 5.09429e41i 0.909859i
\(244\) −1.74576e42 −2.90177
\(245\) 0 0
\(246\) 3.19598e41 0.460511
\(247\) 4.23373e41i 0.568233i
\(248\) − 1.40724e42i − 1.75980i
\(249\) 5.42069e41 0.631781
\(250\) 0 0
\(251\) 1.16925e42 1.18473 0.592363 0.805671i \(-0.298195\pi\)
0.592363 + 0.805671i \(0.298195\pi\)
\(252\) − 7.82507e41i − 0.739569i
\(253\) 4.72548e41i 0.416713i
\(254\) 9.88766e41 0.813773
\(255\) 0 0
\(256\) −2.67471e42 −1.91901
\(257\) 6.14627e41i 0.411890i 0.978564 + 0.205945i \(0.0660269\pi\)
−0.978564 + 0.205945i \(0.933973\pi\)
\(258\) − 1.41473e41i − 0.0885783i
\(259\) −1.78651e42 −1.04535
\(260\) 0 0
\(261\) 1.48148e42 0.757684
\(262\) 5.55428e42i 2.65678i
\(263\) − 5.29428e41i − 0.236909i −0.992959 0.118454i \(-0.962206\pi\)
0.992959 0.118454i \(-0.0377940\pi\)
\(264\) −1.44339e42 −0.604386
\(265\) 0 0
\(266\) −4.72067e42 −1.73211
\(267\) − 1.45029e42i − 0.498320i
\(268\) − 1.17766e42i − 0.379018i
\(269\) −2.61222e42 −0.787670 −0.393835 0.919181i \(-0.628852\pi\)
−0.393835 + 0.919181i \(0.628852\pi\)
\(270\) 0 0
\(271\) −6.35646e42 −1.68365 −0.841827 0.539748i \(-0.818520\pi\)
−0.841827 + 0.539748i \(0.818520\pi\)
\(272\) − 2.57509e42i − 0.639495i
\(273\) − 7.23695e41i − 0.168542i
\(274\) −2.34004e42 −0.511197
\(275\) 0 0
\(276\) −8.09293e42 −1.55667
\(277\) 1.32088e41i 0.0238488i 0.999929 + 0.0119244i \(0.00379574\pi\)
−0.999929 + 0.0119244i \(0.996204\pi\)
\(278\) − 1.83715e42i − 0.311430i
\(279\) 2.68748e42 0.427831
\(280\) 0 0
\(281\) 1.21053e43 1.70065 0.850324 0.526260i \(-0.176406\pi\)
0.850324 + 0.526260i \(0.176406\pi\)
\(282\) − 7.23734e42i − 0.955475i
\(283\) − 1.08291e41i − 0.0134378i −0.999977 0.00671891i \(-0.997861\pi\)
0.999977 0.00671891i \(-0.00213871\pi\)
\(284\) 8.56046e42 0.998680
\(285\) 0 0
\(286\) 2.66988e42 0.275477
\(287\) − 2.47148e42i − 0.239896i
\(288\) 4.48842e42i 0.409945i
\(289\) 9.86245e42 0.847759
\(290\) 0 0
\(291\) −5.56273e41 −0.0423799
\(292\) 2.44523e43i 1.75437i
\(293\) 2.38381e42i 0.161097i 0.996751 + 0.0805486i \(0.0256672\pi\)
−0.996751 + 0.0805486i \(0.974333\pi\)
\(294\) −1.13608e43 −0.723319
\(295\) 0 0
\(296\) 6.15617e43 3.48102
\(297\) − 8.04007e42i − 0.428570i
\(298\) 4.19751e43i 2.10963i
\(299\) 8.16438e42 0.386970
\(300\) 0 0
\(301\) −1.09402e42 −0.0461436
\(302\) 4.84196e43i 1.92708i
\(303\) 1.12548e43i 0.422760i
\(304\) 6.94543e43 2.46270
\(305\) 0 0
\(306\) 1.15181e43 0.364129
\(307\) − 4.84424e43i − 1.44645i −0.690611 0.723226i \(-0.742658\pi\)
0.690611 0.723226i \(-0.257342\pi\)
\(308\) 2.04657e43i 0.577284i
\(309\) 1.46653e43 0.390856
\(310\) 0 0
\(311\) 1.85715e43 0.442119 0.221060 0.975260i \(-0.429048\pi\)
0.221060 + 0.975260i \(0.429048\pi\)
\(312\) 2.49379e43i 0.561248i
\(313\) − 2.67734e43i − 0.569740i −0.958566 0.284870i \(-0.908049\pi\)
0.958566 0.284870i \(-0.0919505\pi\)
\(314\) 8.54470e43 1.71960
\(315\) 0 0
\(316\) −1.08962e43 −0.196224
\(317\) − 2.42612e43i − 0.413405i −0.978404 0.206703i \(-0.933727\pi\)
0.978404 0.206703i \(-0.0662733\pi\)
\(318\) − 1.27460e44i − 2.05542i
\(319\) −3.87468e43 −0.591423
\(320\) 0 0
\(321\) −8.41452e43 −1.15131
\(322\) 9.10342e43i 1.17958i
\(323\) − 4.77694e43i − 0.586281i
\(324\) 3.89911e43 0.453344
\(325\) 0 0
\(326\) −3.33308e44 −3.47967
\(327\) 3.39084e43i 0.335522i
\(328\) 8.51651e43i 0.798857i
\(329\) −5.59669e43 −0.497741
\(330\) 0 0
\(331\) 1.83689e44 1.46924 0.734620 0.678479i \(-0.237361\pi\)
0.734620 + 0.678479i \(0.237361\pi\)
\(332\) 2.64852e44i 2.00950i
\(333\) 1.17568e44i 0.846282i
\(334\) −1.61669e44 −1.10424
\(335\) 0 0
\(336\) −1.18722e44 −0.730457
\(337\) 9.63203e42i 0.0562593i 0.999604 + 0.0281297i \(0.00895513\pi\)
−0.999604 + 0.0281297i \(0.991045\pi\)
\(338\) 2.76421e44i 1.53295i
\(339\) −5.52375e43 −0.290897
\(340\) 0 0
\(341\) −7.02885e43 −0.333951
\(342\) 3.10661e44i 1.40226i
\(343\) 2.38109e44i 1.02124i
\(344\) 3.76990e43 0.153659
\(345\) 0 0
\(346\) −1.04674e43 −0.0385483
\(347\) 2.05198e44i 0.718468i 0.933247 + 0.359234i \(0.116962\pi\)
−0.933247 + 0.359234i \(0.883038\pi\)
\(348\) − 6.63583e44i − 2.20932i
\(349\) 1.00117e44 0.317003 0.158501 0.987359i \(-0.449334\pi\)
0.158501 + 0.987359i \(0.449334\pi\)
\(350\) 0 0
\(351\) −1.38911e44 −0.397981
\(352\) − 1.17390e44i − 0.319990i
\(353\) 2.71539e44i 0.704328i 0.935938 + 0.352164i \(0.114554\pi\)
−0.935938 + 0.352164i \(0.885446\pi\)
\(354\) 2.80817e44 0.693213
\(355\) 0 0
\(356\) 7.08605e44 1.58500
\(357\) 8.16550e43i 0.173896i
\(358\) 1.61158e44i 0.326813i
\(359\) −8.81356e44 −1.70215 −0.851076 0.525043i \(-0.824049\pi\)
−0.851076 + 0.525043i \(0.824049\pi\)
\(360\) 0 0
\(361\) 7.17758e44 1.25777
\(362\) − 1.62435e45i − 2.71194i
\(363\) − 3.62625e44i − 0.576888i
\(364\) 3.53593e44 0.536080
\(365\) 0 0
\(366\) −1.18472e45 −1.63191
\(367\) − 1.11298e45i − 1.46161i −0.682585 0.730806i \(-0.739144\pi\)
0.682585 0.730806i \(-0.260856\pi\)
\(368\) − 1.33937e45i − 1.67712i
\(369\) −1.62645e44 −0.194213
\(370\) 0 0
\(371\) −9.85661e44 −1.07074
\(372\) − 1.20377e45i − 1.24750i
\(373\) 5.50495e44i 0.544313i 0.962253 + 0.272157i \(0.0877368\pi\)
−0.962253 + 0.272157i \(0.912263\pi\)
\(374\) −3.01244e44 −0.284227
\(375\) 0 0
\(376\) 1.92857e45 1.65748
\(377\) 6.69441e44i 0.549211i
\(378\) − 1.54888e45i − 1.21314i
\(379\) 1.81534e45 1.35760 0.678799 0.734325i \(-0.262501\pi\)
0.678799 + 0.734325i \(0.262501\pi\)
\(380\) 0 0
\(381\) 4.61295e44 0.314624
\(382\) − 2.91736e45i − 1.90056i
\(383\) − 5.42819e44i − 0.337813i −0.985632 0.168907i \(-0.945976\pi\)
0.985632 0.168907i \(-0.0540236\pi\)
\(384\) −1.39964e45 −0.832184
\(385\) 0 0
\(386\) 5.48431e45 2.97745
\(387\) 7.19958e43i 0.0373564i
\(388\) − 2.71792e44i − 0.134797i
\(389\) −2.27596e45 −1.07906 −0.539532 0.841965i \(-0.681399\pi\)
−0.539532 + 0.841965i \(0.681399\pi\)
\(390\) 0 0
\(391\) −9.21192e44 −0.399261
\(392\) − 3.02737e45i − 1.25476i
\(393\) 2.59127e45i 1.02717i
\(394\) −1.31496e42 −0.000498572 0
\(395\) 0 0
\(396\) 1.34682e45 0.467352
\(397\) 2.97575e45i 0.988013i 0.869458 + 0.494006i \(0.164468\pi\)
−0.869458 + 0.494006i \(0.835532\pi\)
\(398\) 5.35643e45i 1.70185i
\(399\) −2.20236e45 −0.669674
\(400\) 0 0
\(401\) −3.35408e45 −0.934429 −0.467215 0.884144i \(-0.654742\pi\)
−0.467215 + 0.884144i \(0.654742\pi\)
\(402\) − 7.99189e44i − 0.213154i
\(403\) 1.21440e45i 0.310115i
\(404\) −5.49906e45 −1.34467
\(405\) 0 0
\(406\) −7.46438e45 −1.67413
\(407\) − 3.07488e45i − 0.660580i
\(408\) − 2.81376e45i − 0.579074i
\(409\) −8.75180e45 −1.72560 −0.862799 0.505547i \(-0.831291\pi\)
−0.862799 + 0.505547i \(0.831291\pi\)
\(410\) 0 0
\(411\) −1.09171e45 −0.197641
\(412\) 7.16537e45i 1.24319i
\(413\) − 2.17158e45i − 0.361119i
\(414\) 5.99083e45 0.954951
\(415\) 0 0
\(416\) −2.02819e45 −0.297150
\(417\) − 8.57095e44i − 0.120406i
\(418\) − 8.12503e45i − 1.09456i
\(419\) −1.99625e45 −0.257911 −0.128955 0.991650i \(-0.541162\pi\)
−0.128955 + 0.991650i \(0.541162\pi\)
\(420\) 0 0
\(421\) −1.23389e46 −1.46670 −0.733348 0.679854i \(-0.762043\pi\)
−0.733348 + 0.679854i \(0.762043\pi\)
\(422\) 8.96388e45i 1.02218i
\(423\) 3.68311e45i 0.402956i
\(424\) 3.39650e46 3.56557
\(425\) 0 0
\(426\) 5.80936e45 0.561643
\(427\) 9.16156e45i 0.850122i
\(428\) − 4.11129e46i − 3.66194i
\(429\) 1.24559e45 0.106506
\(430\) 0 0
\(431\) −1.20357e46 −0.948677 −0.474339 0.880343i \(-0.657313\pi\)
−0.474339 + 0.880343i \(0.657313\pi\)
\(432\) 2.27884e46i 1.72484i
\(433\) 2.06332e46i 1.49979i 0.661559 + 0.749893i \(0.269895\pi\)
−0.661559 + 0.749893i \(0.730105\pi\)
\(434\) −1.35407e46 −0.945305
\(435\) 0 0
\(436\) −1.65674e46 −1.06719
\(437\) − 2.48460e46i − 1.53756i
\(438\) 1.65940e46i 0.986629i
\(439\) −3.14998e46 −1.79961 −0.899805 0.436293i \(-0.856291\pi\)
−0.899805 + 0.436293i \(0.856291\pi\)
\(440\) 0 0
\(441\) 5.78154e45 0.305048
\(442\) 5.20470e45i 0.263941i
\(443\) 2.49595e46i 1.21667i 0.793682 + 0.608333i \(0.208161\pi\)
−0.793682 + 0.608333i \(0.791839\pi\)
\(444\) 5.26608e46 2.46766
\(445\) 0 0
\(446\) 5.32003e46 2.30438
\(447\) 1.95829e46i 0.815634i
\(448\) 3.75576e45i 0.150429i
\(449\) −3.14792e46 −1.21259 −0.606294 0.795241i \(-0.707344\pi\)
−0.606294 + 0.795241i \(0.707344\pi\)
\(450\) 0 0
\(451\) 4.25381e45 0.151596
\(452\) − 2.69887e46i − 0.925252i
\(453\) 2.25895e46i 0.745054i
\(454\) 4.37752e46 1.38915
\(455\) 0 0
\(456\) 7.58916e46 2.23002
\(457\) 3.21994e46i 0.910572i 0.890345 + 0.455286i \(0.150463\pi\)
−0.890345 + 0.455286i \(0.849537\pi\)
\(458\) 4.23336e46i 1.15223i
\(459\) 1.56734e46 0.410622
\(460\) 0 0
\(461\) 8.96567e45 0.217679 0.108840 0.994059i \(-0.465287\pi\)
0.108840 + 0.994059i \(0.465287\pi\)
\(462\) 1.38886e46i 0.324656i
\(463\) 2.74379e46i 0.617566i 0.951133 + 0.308783i \(0.0999216\pi\)
−0.951133 + 0.308783i \(0.900078\pi\)
\(464\) 1.09822e47 2.38026
\(465\) 0 0
\(466\) −3.82134e46 −0.768178
\(467\) − 8.40024e46i − 1.62647i −0.581935 0.813235i \(-0.697704\pi\)
0.581935 0.813235i \(-0.302296\pi\)
\(468\) − 2.32695e46i − 0.433994i
\(469\) −6.18019e45 −0.111039
\(470\) 0 0
\(471\) 3.98641e46 0.664838
\(472\) 7.48309e46i 1.20253i
\(473\) − 1.88298e45i − 0.0291592i
\(474\) −7.39447e45 −0.110353
\(475\) 0 0
\(476\) −3.98962e46 −0.553107
\(477\) 6.48649e46i 0.866839i
\(478\) − 4.73755e46i − 0.610331i
\(479\) −1.51913e47 −1.88679 −0.943395 0.331671i \(-0.892388\pi\)
−0.943395 + 0.331671i \(0.892388\pi\)
\(480\) 0 0
\(481\) −5.31257e46 −0.613431
\(482\) − 4.88629e46i − 0.544072i
\(483\) 4.24707e46i 0.456052i
\(484\) 1.77176e47 1.83490
\(485\) 0 0
\(486\) −1.68913e47 −1.62753
\(487\) − 9.57009e46i − 0.889527i −0.895648 0.444764i \(-0.853288\pi\)
0.895648 0.444764i \(-0.146712\pi\)
\(488\) − 3.15699e47i − 2.83091i
\(489\) −1.55500e47 −1.34532
\(490\) 0 0
\(491\) −9.00218e46 −0.725139 −0.362570 0.931957i \(-0.618100\pi\)
−0.362570 + 0.931957i \(0.618100\pi\)
\(492\) 7.28514e46i 0.566302i
\(493\) − 7.55335e46i − 0.566655i
\(494\) −1.40379e47 −1.01644
\(495\) 0 0
\(496\) 1.99222e47 1.34403
\(497\) − 4.49242e46i − 0.292580i
\(498\) 1.79736e47i 1.13011i
\(499\) 1.31845e47 0.800394 0.400197 0.916429i \(-0.368942\pi\)
0.400197 + 0.916429i \(0.368942\pi\)
\(500\) 0 0
\(501\) −7.54246e46 −0.426926
\(502\) 3.87693e47i 2.11920i
\(503\) − 3.29261e46i − 0.173820i −0.996216 0.0869101i \(-0.972301\pi\)
0.996216 0.0869101i \(-0.0276993\pi\)
\(504\) 1.41507e47 0.721510
\(505\) 0 0
\(506\) −1.56684e47 −0.745404
\(507\) 1.28960e47i 0.592676i
\(508\) 2.25386e47i 1.00072i
\(509\) −1.43449e47 −0.615370 −0.307685 0.951488i \(-0.599554\pi\)
−0.307685 + 0.951488i \(0.599554\pi\)
\(510\) 0 0
\(511\) 1.28323e47 0.513970
\(512\) − 5.75978e47i − 2.22937i
\(513\) 4.22737e47i 1.58131i
\(514\) −2.03794e47 −0.736777
\(515\) 0 0
\(516\) 3.22482e46 0.108927
\(517\) − 9.63280e46i − 0.314534i
\(518\) − 5.92360e47i − 1.86989i
\(519\) −4.88340e45 −0.0149037
\(520\) 0 0
\(521\) −2.33055e47 −0.664966 −0.332483 0.943109i \(-0.607886\pi\)
−0.332483 + 0.943109i \(0.607886\pi\)
\(522\) 4.91220e47i 1.35532i
\(523\) 4.26380e47i 1.13767i 0.822451 + 0.568836i \(0.192606\pi\)
−0.822451 + 0.568836i \(0.807394\pi\)
\(524\) −1.26608e48 −3.26711
\(525\) 0 0
\(526\) 1.75544e47 0.423776
\(527\) − 1.37021e47i − 0.319965i
\(528\) − 2.04340e47i − 0.461593i
\(529\) −2.15467e46 −0.0470875
\(530\) 0 0
\(531\) −1.42909e47 −0.292351
\(532\) − 1.07606e48i − 2.13002i
\(533\) − 7.34945e46i − 0.140776i
\(534\) 4.80878e47 0.891380
\(535\) 0 0
\(536\) 2.12964e47 0.369762
\(537\) 7.51860e46i 0.126353i
\(538\) − 8.66140e47i − 1.40896i
\(539\) −1.51211e47 −0.238111
\(540\) 0 0
\(541\) −5.43835e45 −0.00802629 −0.00401314 0.999992i \(-0.501277\pi\)
−0.00401314 + 0.999992i \(0.501277\pi\)
\(542\) − 2.10763e48i − 3.01167i
\(543\) − 7.57817e47i − 1.04850i
\(544\) 2.28842e47 0.306588
\(545\) 0 0
\(546\) 2.39958e47 0.301484
\(547\) − 4.15694e47i − 0.505820i −0.967490 0.252910i \(-0.918612\pi\)
0.967490 0.252910i \(-0.0813876\pi\)
\(548\) − 5.33405e47i − 0.628633i
\(549\) 6.02909e47 0.688233
\(550\) 0 0
\(551\) 2.03726e48 2.18219
\(552\) − 1.46350e48i − 1.51866i
\(553\) 5.71821e46i 0.0574871i
\(554\) −4.37967e46 −0.0426600
\(555\) 0 0
\(556\) 4.18772e47 0.382973
\(557\) 1.81613e45i 0.00160947i 1.00000 0.000804733i \(0.000256154\pi\)
−1.00000 0.000804733i \(0.999744\pi\)
\(558\) 8.91097e47i 0.765291i
\(559\) −3.25329e46 −0.0270780
\(560\) 0 0
\(561\) −1.40541e47 −0.109889
\(562\) 4.01379e48i 3.04207i
\(563\) 2.19502e48i 1.61266i 0.591468 + 0.806329i \(0.298549\pi\)
−0.591468 + 0.806329i \(0.701451\pi\)
\(564\) 1.64973e48 1.17497
\(565\) 0 0
\(566\) 3.59063e46 0.0240372
\(567\) − 2.04621e47i − 0.132815i
\(568\) 1.54805e48i 0.974293i
\(569\) −1.19614e48 −0.729993 −0.364996 0.931009i \(-0.618930\pi\)
−0.364996 + 0.931009i \(0.618930\pi\)
\(570\) 0 0
\(571\) −1.26426e48 −0.725610 −0.362805 0.931865i \(-0.618181\pi\)
−0.362805 + 0.931865i \(0.618181\pi\)
\(572\) 6.08590e47i 0.338762i
\(573\) − 1.36105e48i − 0.734799i
\(574\) 8.19476e47 0.429120
\(575\) 0 0
\(576\) 2.47161e47 0.121783
\(577\) 2.42022e48i 1.15686i 0.815734 + 0.578428i \(0.196333\pi\)
−0.815734 + 0.578428i \(0.803667\pi\)
\(578\) 3.27012e48i 1.51645i
\(579\) 2.55863e48 1.15115
\(580\) 0 0
\(581\) 1.38991e48 0.588715
\(582\) − 1.84445e47i − 0.0758080i
\(583\) − 1.69648e48i − 0.676626i
\(584\) −4.42189e48 −1.71153
\(585\) 0 0
\(586\) −7.90407e47 −0.288166
\(587\) − 1.04500e48i − 0.369787i −0.982759 0.184893i \(-0.940806\pi\)
0.982759 0.184893i \(-0.0591940\pi\)
\(588\) − 2.58965e48i − 0.889485i
\(589\) 3.69568e48 1.23219
\(590\) 0 0
\(591\) −6.13475e44 −0.000192760 0
\(592\) 8.71527e48i 2.65859i
\(593\) − 4.87793e48i − 1.44470i −0.691526 0.722351i \(-0.743061\pi\)
0.691526 0.722351i \(-0.256939\pi\)
\(594\) 2.66587e48 0.766613
\(595\) 0 0
\(596\) −9.56809e48 −2.59427
\(597\) 2.49897e48i 0.657976i
\(598\) 2.70709e48i 0.692201i
\(599\) −8.48111e47 −0.210612 −0.105306 0.994440i \(-0.533582\pi\)
−0.105306 + 0.994440i \(0.533582\pi\)
\(600\) 0 0
\(601\) −6.23099e48 −1.45967 −0.729834 0.683624i \(-0.760403\pi\)
−0.729834 + 0.683624i \(0.760403\pi\)
\(602\) − 3.62747e47i − 0.0825402i
\(603\) 4.06710e47i 0.0898942i
\(604\) −1.10371e49 −2.36978
\(605\) 0 0
\(606\) −3.73181e48 −0.756220
\(607\) 2.65746e48i 0.523197i 0.965177 + 0.261598i \(0.0842496\pi\)
−0.965177 + 0.261598i \(0.915750\pi\)
\(608\) 6.17224e48i 1.18067i
\(609\) −3.48240e48 −0.647256
\(610\) 0 0
\(611\) −1.66429e48 −0.292085
\(612\) 2.62551e48i 0.447779i
\(613\) − 5.89105e48i − 0.976415i −0.872727 0.488208i \(-0.837651\pi\)
0.872727 0.488208i \(-0.162349\pi\)
\(614\) 1.60622e49 2.58737
\(615\) 0 0
\(616\) −3.70097e48 −0.563187
\(617\) − 2.86204e48i − 0.423337i −0.977342 0.211668i \(-0.932110\pi\)
0.977342 0.211668i \(-0.0678897\pi\)
\(618\) 4.86261e48i 0.699151i
\(619\) 9.42149e48 1.31684 0.658420 0.752651i \(-0.271225\pi\)
0.658420 + 0.752651i \(0.271225\pi\)
\(620\) 0 0
\(621\) 8.15212e48 1.07688
\(622\) 6.15780e48i 0.790850i
\(623\) − 3.71867e48i − 0.464352i
\(624\) −3.53045e48 −0.428647
\(625\) 0 0
\(626\) 8.87733e48 1.01913
\(627\) − 3.79062e48i − 0.423183i
\(628\) 1.94774e49i 2.11464i
\(629\) 5.99421e48 0.632915
\(630\) 0 0
\(631\) 8.74950e48 0.873915 0.436958 0.899482i \(-0.356056\pi\)
0.436958 + 0.899482i \(0.356056\pi\)
\(632\) − 1.97045e48i − 0.191432i
\(633\) 4.18197e48i 0.395199i
\(634\) 8.04438e48 0.739487
\(635\) 0 0
\(636\) 2.90541e49 2.52760
\(637\) 2.61252e48i 0.221116i
\(638\) − 1.28474e49i − 1.05792i
\(639\) −2.95640e48 −0.236864
\(640\) 0 0
\(641\) 5.35983e48 0.406571 0.203285 0.979120i \(-0.434838\pi\)
0.203285 + 0.979120i \(0.434838\pi\)
\(642\) − 2.79003e49i − 2.05942i
\(643\) − 2.04799e49i − 1.47108i −0.677483 0.735538i \(-0.736929\pi\)
0.677483 0.735538i \(-0.263071\pi\)
\(644\) −2.07509e49 −1.45056
\(645\) 0 0
\(646\) 1.58391e49 1.04872
\(647\) 2.13324e49i 1.37472i 0.726317 + 0.687360i \(0.241231\pi\)
−0.726317 + 0.687360i \(0.758769\pi\)
\(648\) 7.05105e48i 0.442274i
\(649\) 3.73764e48 0.228200
\(650\) 0 0
\(651\) −6.31724e48 −0.365477
\(652\) − 7.59765e49i − 4.27904i
\(653\) 2.78018e49i 1.52438i 0.647354 + 0.762189i \(0.275876\pi\)
−0.647354 + 0.762189i \(0.724124\pi\)
\(654\) −1.12431e49 −0.600172
\(655\) 0 0
\(656\) −1.20568e49 −0.610119
\(657\) − 8.44474e48i − 0.416095i
\(658\) − 1.85571e49i − 0.890344i
\(659\) 2.64746e49 1.23690 0.618450 0.785824i \(-0.287761\pi\)
0.618450 + 0.785824i \(0.287761\pi\)
\(660\) 0 0
\(661\) 2.31405e49 1.02529 0.512646 0.858600i \(-0.328665\pi\)
0.512646 + 0.858600i \(0.328665\pi\)
\(662\) 6.09062e49i 2.62813i
\(663\) 2.42818e48i 0.102046i
\(664\) −4.78952e49 −1.96042
\(665\) 0 0
\(666\) −3.89824e49 −1.51380
\(667\) − 3.92868e49i − 1.48609i
\(668\) − 3.68520e49i − 1.35792i
\(669\) 2.48198e49 0.890927
\(670\) 0 0
\(671\) −1.57685e49 −0.537213
\(672\) − 1.05506e49i − 0.350198i
\(673\) − 3.41502e49i − 1.10441i −0.833709 0.552204i \(-0.813787\pi\)
0.833709 0.552204i \(-0.186213\pi\)
\(674\) −3.19372e48 −0.100635
\(675\) 0 0
\(676\) −6.30093e49 −1.88511
\(677\) 1.81760e49i 0.529903i 0.964262 + 0.264951i \(0.0853559\pi\)
−0.964262 + 0.264951i \(0.914644\pi\)
\(678\) − 1.83153e49i − 0.520348i
\(679\) −1.42633e48 −0.0394911
\(680\) 0 0
\(681\) 2.04227e49 0.537079
\(682\) − 2.33058e49i − 0.597361i
\(683\) 1.52341e49i 0.380587i 0.981727 + 0.190294i \(0.0609440\pi\)
−0.981727 + 0.190294i \(0.939056\pi\)
\(684\) −7.08142e49 −1.72440
\(685\) 0 0
\(686\) −7.89504e49 −1.82676
\(687\) 1.97501e49i 0.445479i
\(688\) 5.33703e48i 0.117355i
\(689\) −2.93107e49 −0.628332
\(690\) 0 0
\(691\) −1.25243e49 −0.255203 −0.127602 0.991825i \(-0.540728\pi\)
−0.127602 + 0.991825i \(0.540728\pi\)
\(692\) − 2.38600e48i − 0.0474039i
\(693\) − 7.06795e48i − 0.136918i
\(694\) −6.80382e49 −1.28517
\(695\) 0 0
\(696\) 1.20001e50 2.15537
\(697\) 8.29244e48i 0.145247i
\(698\) 3.31961e49i 0.567045i
\(699\) −1.78279e49 −0.296995
\(700\) 0 0
\(701\) 4.20207e49 0.665883 0.332942 0.942947i \(-0.391959\pi\)
0.332942 + 0.942947i \(0.391959\pi\)
\(702\) − 4.60592e49i − 0.711896i
\(703\) 1.61673e50i 2.43736i
\(704\) −6.46426e48 −0.0950598
\(705\) 0 0
\(706\) −9.00351e49 −1.25988
\(707\) 2.88584e49i 0.393942i
\(708\) 6.40114e49i 0.852463i
\(709\) −9.39998e48 −0.122129 −0.0610644 0.998134i \(-0.519449\pi\)
−0.0610644 + 0.998134i \(0.519449\pi\)
\(710\) 0 0
\(711\) 3.76307e48 0.0465398
\(712\) 1.28142e50i 1.54629i
\(713\) − 7.12681e49i − 0.839128i
\(714\) −2.70746e49 −0.311059
\(715\) 0 0
\(716\) −3.67354e49 −0.401890
\(717\) − 2.21023e49i − 0.235968i
\(718\) − 2.92234e50i − 3.04476i
\(719\) 7.29315e49 0.741583 0.370792 0.928716i \(-0.379086\pi\)
0.370792 + 0.928716i \(0.379086\pi\)
\(720\) 0 0
\(721\) 3.76030e49 0.364213
\(722\) 2.37989e50i 2.24987i
\(723\) − 2.27963e49i − 0.210351i
\(724\) 3.70265e50 3.33494
\(725\) 0 0
\(726\) 1.20237e50 1.03192
\(727\) − 4.92620e49i − 0.412724i −0.978476 0.206362i \(-0.933838\pi\)
0.978476 0.206362i \(-0.0661624\pi\)
\(728\) 6.39429e49i 0.522990i
\(729\) −1.04614e50 −0.835334
\(730\) 0 0
\(731\) 3.67071e48 0.0279380
\(732\) − 2.70054e50i − 2.00681i
\(733\) 1.28080e50i 0.929310i 0.885492 + 0.464655i \(0.153822\pi\)
−0.885492 + 0.464655i \(0.846178\pi\)
\(734\) 3.69035e50 2.61449
\(735\) 0 0
\(736\) 1.19026e50 0.804047
\(737\) − 1.06371e49i − 0.0701685i
\(738\) − 5.39286e49i − 0.347402i
\(739\) −1.99479e50 −1.25493 −0.627466 0.778644i \(-0.715908\pi\)
−0.627466 + 0.778644i \(0.715908\pi\)
\(740\) 0 0
\(741\) −6.54918e49 −0.392978
\(742\) − 3.26819e50i − 1.91531i
\(743\) − 1.05855e50i − 0.605907i −0.953005 0.302954i \(-0.902027\pi\)
0.953005 0.302954i \(-0.0979728\pi\)
\(744\) 2.17687e50 1.21704
\(745\) 0 0
\(746\) −1.82529e50 −0.973651
\(747\) − 9.14681e49i − 0.476606i
\(748\) − 6.86676e49i − 0.349522i
\(749\) −2.15755e50 −1.07283
\(750\) 0 0
\(751\) −1.20549e50 −0.572090 −0.286045 0.958216i \(-0.592341\pi\)
−0.286045 + 0.958216i \(0.592341\pi\)
\(752\) 2.73027e50i 1.26588i
\(753\) 1.80872e50i 0.819333i
\(754\) −2.21969e50 −0.982411
\(755\) 0 0
\(756\) 3.53062e50 1.49183
\(757\) 1.18495e50i 0.489240i 0.969619 + 0.244620i \(0.0786633\pi\)
−0.969619 + 0.244620i \(0.921337\pi\)
\(758\) 6.01917e50i 2.42843i
\(759\) −7.30989e49 −0.288190
\(760\) 0 0
\(761\) 2.89900e50 1.09148 0.545741 0.837954i \(-0.316248\pi\)
0.545741 + 0.837954i \(0.316248\pi\)
\(762\) 1.52953e50i 0.562789i
\(763\) 8.69439e49i 0.312651i
\(764\) 6.65002e50 2.33716
\(765\) 0 0
\(766\) 1.79984e50 0.604270
\(767\) − 6.45766e49i − 0.211912i
\(768\) − 4.13754e50i − 1.32715i
\(769\) −6.18664e49 −0.193974 −0.0969870 0.995286i \(-0.530921\pi\)
−0.0969870 + 0.995286i \(0.530921\pi\)
\(770\) 0 0
\(771\) −9.50771e49 −0.284855
\(772\) 1.25013e51i 3.66145i
\(773\) 6.37753e50i 1.82604i 0.407911 + 0.913022i \(0.366257\pi\)
−0.407911 + 0.913022i \(0.633743\pi\)
\(774\) −2.38719e49 −0.0668221
\(775\) 0 0
\(776\) 4.91501e49 0.131506
\(777\) − 2.76357e50i − 0.722941i
\(778\) − 7.54649e50i − 1.93020i
\(779\) −2.23660e50 −0.559349
\(780\) 0 0
\(781\) 7.73218e49 0.184888
\(782\) − 3.05443e50i − 0.714186i
\(783\) 6.68436e50i 1.52837i
\(784\) 4.28584e50 0.958307
\(785\) 0 0
\(786\) −8.59196e50 −1.83737
\(787\) 6.23653e50i 1.30432i 0.758080 + 0.652161i \(0.226138\pi\)
−0.758080 + 0.652161i \(0.773862\pi\)
\(788\) − 2.99740e47i 0 0.000613107i
\(789\) 8.18976e49 0.163841
\(790\) 0 0
\(791\) −1.41634e50 −0.271068
\(792\) 2.43555e50i 0.455939i
\(793\) 2.72438e50i 0.498869i
\(794\) −9.86679e50 −1.76733
\(795\) 0 0
\(796\) −1.22098e51 −2.09281
\(797\) 9.07097e50i 1.52101i 0.649329 + 0.760507i \(0.275050\pi\)
−0.649329 + 0.760507i \(0.724950\pi\)
\(798\) − 7.30245e50i − 1.19789i
\(799\) 1.87783e50 0.301362
\(800\) 0 0
\(801\) −2.44720e50 −0.375925
\(802\) − 1.11212e51i − 1.67148i
\(803\) 2.20864e50i 0.324790i
\(804\) 1.82172e50 0.262121
\(805\) 0 0
\(806\) −4.02662e50 −0.554724
\(807\) − 4.04086e50i − 0.544736i
\(808\) − 9.94435e50i − 1.31183i
\(809\) −1.43609e51 −1.85389 −0.926943 0.375202i \(-0.877573\pi\)
−0.926943 + 0.375202i \(0.877573\pi\)
\(810\) 0 0
\(811\) 6.18627e50 0.764828 0.382414 0.923991i \(-0.375093\pi\)
0.382414 + 0.923991i \(0.375093\pi\)
\(812\) − 1.70148e51i − 2.05872i
\(813\) − 9.83286e50i − 1.16438i
\(814\) 1.01955e51 1.18163
\(815\) 0 0
\(816\) 3.98343e50 0.442262
\(817\) 9.90049e49i 0.107590i
\(818\) − 2.90186e51i − 3.08670i
\(819\) −1.22115e50 −0.127146
\(820\) 0 0
\(821\) −9.87204e50 −0.984922 −0.492461 0.870335i \(-0.663903\pi\)
−0.492461 + 0.870335i \(0.663903\pi\)
\(822\) − 3.61983e50i − 0.353534i
\(823\) 1.56871e51i 1.49985i 0.661525 + 0.749923i \(0.269909\pi\)
−0.661525 + 0.749923i \(0.730091\pi\)
\(824\) −1.29577e51 −1.21283
\(825\) 0 0
\(826\) 7.20039e50 0.645960
\(827\) − 3.34712e49i − 0.0293985i −0.999892 0.0146992i \(-0.995321\pi\)
0.999892 0.0146992i \(-0.00467908\pi\)
\(828\) 1.36559e51i 1.17433i
\(829\) 1.24277e51 1.04638 0.523188 0.852217i \(-0.324743\pi\)
0.523188 + 0.852217i \(0.324743\pi\)
\(830\) 0 0
\(831\) −2.04327e49 −0.0164933
\(832\) 1.11685e50i 0.0882749i
\(833\) − 2.94772e50i − 0.228139i
\(834\) 2.84190e50 0.215379
\(835\) 0 0
\(836\) 1.85208e51 1.34601
\(837\) 1.21258e51i 0.863005i
\(838\) − 6.61903e50i − 0.461343i
\(839\) −7.06560e50 −0.482297 −0.241149 0.970488i \(-0.577524\pi\)
−0.241149 + 0.970488i \(0.577524\pi\)
\(840\) 0 0
\(841\) 1.69401e51 1.10914
\(842\) − 4.09125e51i − 2.62358i
\(843\) 1.87258e51i 1.17613i
\(844\) −2.04329e51 −1.25700
\(845\) 0 0
\(846\) −1.22122e51 −0.720796
\(847\) − 9.29800e50i − 0.537564i
\(848\) 4.80842e51i 2.72317i
\(849\) 1.67516e49 0.00929333
\(850\) 0 0
\(851\) 3.11773e51 1.65986
\(852\) 1.32422e51i 0.690667i
\(853\) 4.40952e50i 0.225311i 0.993634 + 0.112656i \(0.0359357\pi\)
−0.993634 + 0.112656i \(0.964064\pi\)
\(854\) −3.03773e51 −1.52067
\(855\) 0 0
\(856\) 7.43474e51 3.57252
\(857\) − 2.01998e51i − 0.951004i −0.879715 0.475502i \(-0.842267\pi\)
0.879715 0.475502i \(-0.157733\pi\)
\(858\) 4.13006e50i 0.190515i
\(859\) 2.47558e51 1.11891 0.559456 0.828860i \(-0.311010\pi\)
0.559456 + 0.828860i \(0.311010\pi\)
\(860\) 0 0
\(861\) 3.82315e50 0.165908
\(862\) − 3.99071e51i − 1.69696i
\(863\) − 1.15182e51i − 0.479948i −0.970779 0.239974i \(-0.922861\pi\)
0.970779 0.239974i \(-0.0771389\pi\)
\(864\) −2.02515e51 −0.826926
\(865\) 0 0
\(866\) −6.84143e51 −2.68277
\(867\) 1.52563e51i 0.586293i
\(868\) − 3.08657e51i − 1.16247i
\(869\) −9.84195e49 −0.0363274
\(870\) 0 0
\(871\) −1.83781e50 −0.0651602
\(872\) − 2.99601e51i − 1.04113i
\(873\) 9.38647e49i 0.0319708i
\(874\) 8.23827e51 2.75034
\(875\) 0 0
\(876\) −3.78255e51 −1.21328
\(877\) 1.21793e50i 0.0382938i 0.999817 + 0.0191469i \(0.00609502\pi\)
−0.999817 + 0.0191469i \(0.993905\pi\)
\(878\) − 1.04445e52i − 3.21909i
\(879\) −3.68753e50 −0.111412
\(880\) 0 0
\(881\) 1.43159e51 0.415663 0.207832 0.978165i \(-0.433359\pi\)
0.207832 + 0.978165i \(0.433359\pi\)
\(882\) 1.91700e51i 0.545661i
\(883\) − 4.21625e51i − 1.17656i −0.808658 0.588280i \(-0.799805\pi\)
0.808658 0.588280i \(-0.200195\pi\)
\(884\) −1.18639e51 −0.324575
\(885\) 0 0
\(886\) −8.27591e51 −2.17634
\(887\) 1.06484e51i 0.274549i 0.990533 + 0.137274i \(0.0438342\pi\)
−0.990533 + 0.137274i \(0.956166\pi\)
\(888\) 9.52303e51i 2.40740i
\(889\) 1.18280e51 0.293177
\(890\) 0 0
\(891\) 3.52185e50 0.0839288
\(892\) 1.21268e52i 2.83376i
\(893\) 5.06481e51i 1.16055i
\(894\) −6.49316e51 −1.45898
\(895\) 0 0
\(896\) −3.58879e51 −0.775457
\(897\) 1.26295e51i 0.267621i
\(898\) − 1.04377e52i − 2.16904i
\(899\) 5.84365e51 1.19094
\(900\) 0 0
\(901\) 3.30714e51 0.648289
\(902\) 1.41045e51i 0.271171i
\(903\) − 1.69235e50i − 0.0319120i
\(904\) 4.88057e51 0.902658
\(905\) 0 0
\(906\) −7.49007e51 −1.33273
\(907\) 5.18332e51i 0.904649i 0.891853 + 0.452325i \(0.149405\pi\)
−0.891853 + 0.452325i \(0.850595\pi\)
\(908\) 9.97841e51i 1.70828i
\(909\) 1.89913e51 0.318923
\(910\) 0 0
\(911\) 9.24676e51 1.49423 0.747116 0.664694i \(-0.231438\pi\)
0.747116 + 0.664694i \(0.231438\pi\)
\(912\) 1.07439e52i 1.70315i
\(913\) 2.39226e51i 0.372023i
\(914\) −1.06765e52 −1.62880
\(915\) 0 0
\(916\) −9.64981e51 −1.41693
\(917\) 6.64424e51i 0.957154i
\(918\) 5.19689e51i 0.734508i
\(919\) 4.29279e51 0.595275 0.297638 0.954679i \(-0.403801\pi\)
0.297638 + 0.954679i \(0.403801\pi\)
\(920\) 0 0
\(921\) 7.49359e51 1.00034
\(922\) 2.97278e51i 0.389378i
\(923\) − 1.33592e51i − 0.171692i
\(924\) −3.16586e51 −0.399238
\(925\) 0 0
\(926\) −9.09766e51 −1.10468
\(927\) − 2.47460e51i − 0.294856i
\(928\) 9.75961e51i 1.14115i
\(929\) −4.47733e51 −0.513740 −0.256870 0.966446i \(-0.582691\pi\)
−0.256870 + 0.966446i \(0.582691\pi\)
\(930\) 0 0
\(931\) 7.95047e51 0.878564
\(932\) − 8.71061e51i − 0.944649i
\(933\) 2.87284e51i 0.305761i
\(934\) 2.78529e52 2.90938
\(935\) 0 0
\(936\) 4.20799e51 0.423397
\(937\) − 1.28073e50i − 0.0126478i −0.999980 0.00632388i \(-0.997987\pi\)
0.999980 0.00632388i \(-0.00201297\pi\)
\(938\) − 2.04919e51i − 0.198624i
\(939\) 4.14159e51 0.394021
\(940\) 0 0
\(941\) 3.26378e50 0.0299160 0.0149580 0.999888i \(-0.495239\pi\)
0.0149580 + 0.999888i \(0.495239\pi\)
\(942\) 1.32179e52i 1.18924i
\(943\) 4.31310e51i 0.380921i
\(944\) −1.05938e52 −0.918420
\(945\) 0 0
\(946\) 6.24346e50 0.0521591
\(947\) − 1.57573e52i − 1.29228i −0.763219 0.646140i \(-0.776382\pi\)
0.763219 0.646140i \(-0.223618\pi\)
\(948\) − 1.68555e51i − 0.135705i
\(949\) 3.81594e51 0.301608
\(950\) 0 0
\(951\) 3.75299e51 0.285903
\(952\) − 7.21472e51i − 0.539601i
\(953\) 9.47096e51i 0.695454i 0.937596 + 0.347727i \(0.113046\pi\)
−0.937596 + 0.347727i \(0.886954\pi\)
\(954\) −2.15075e52 −1.55057
\(955\) 0 0
\(956\) 1.07991e52 0.750540
\(957\) − 5.99377e51i − 0.409017i
\(958\) − 5.03703e52i − 3.37503i
\(959\) −2.79924e51 −0.184168
\(960\) 0 0
\(961\) −5.16308e51 −0.327529
\(962\) − 1.76151e52i − 1.09729i
\(963\) 1.41985e52i 0.868528i
\(964\) 1.11381e52 0.669059
\(965\) 0 0
\(966\) −1.40821e52 −0.815773
\(967\) 1.42806e52i 0.812424i 0.913779 + 0.406212i \(0.133151\pi\)
−0.913779 + 0.406212i \(0.866849\pi\)
\(968\) 3.20401e52i 1.79009i
\(969\) 7.38949e51 0.405460
\(970\) 0 0
\(971\) −4.08920e51 −0.216422 −0.108211 0.994128i \(-0.534512\pi\)
−0.108211 + 0.994128i \(0.534512\pi\)
\(972\) − 3.85032e52i − 2.00141i
\(973\) − 2.19766e51i − 0.112198i
\(974\) 3.17319e52 1.59116
\(975\) 0 0
\(976\) 4.46934e52 2.16208
\(977\) 2.55940e52i 1.21614i 0.793883 + 0.608071i \(0.208056\pi\)
−0.793883 + 0.608071i \(0.791944\pi\)
\(978\) − 5.15597e52i − 2.40647i
\(979\) 6.40042e51 0.293435
\(980\) 0 0
\(981\) 5.72166e51 0.253113
\(982\) − 2.98488e52i − 1.29711i
\(983\) − 1.82270e52i − 0.778087i −0.921219 0.389044i \(-0.872806\pi\)
0.921219 0.389044i \(-0.127194\pi\)
\(984\) −1.31742e52 −0.552474
\(985\) 0 0
\(986\) 2.50449e52 1.01361
\(987\) − 8.65757e51i − 0.344228i
\(988\) − 3.19989e52i − 1.24994i
\(989\) 1.90922e51 0.0732693
\(990\) 0 0
\(991\) 8.47738e51 0.314031 0.157015 0.987596i \(-0.449813\pi\)
0.157015 + 0.987596i \(0.449813\pi\)
\(992\) 1.77044e52i 0.644358i
\(993\) 2.84149e52i 1.01610i
\(994\) 1.48957e52 0.523358
\(995\) 0 0
\(996\) −4.09702e52 −1.38973
\(997\) − 4.09490e52i − 1.36483i −0.730966 0.682414i \(-0.760930\pi\)
0.730966 0.682414i \(-0.239070\pi\)
\(998\) 4.37162e52i 1.43172i
\(999\) −5.30459e52 −1.70709
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 25.36.b.a.24.6 6
5.2 odd 4 25.36.a.a.1.1 3
5.3 odd 4 1.36.a.a.1.3 3
5.4 even 2 inner 25.36.b.a.24.1 6
15.8 even 4 9.36.a.b.1.1 3
20.3 even 4 16.36.a.d.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1.36.a.a.1.3 3 5.3 odd 4
9.36.a.b.1.1 3 15.8 even 4
16.36.a.d.1.2 3 20.3 even 4
25.36.a.a.1.1 3 5.2 odd 4
25.36.b.a.24.1 6 5.4 even 2 inner
25.36.b.a.24.6 6 1.1 even 1 trivial