Properties

Label 25.34.b.a.24.3
Level $25$
Weight $34$
Character 25.24
Analytic conductor $172.457$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [25,34,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, 34, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("25.24");
 
S:= CuspForms(chi, 34);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 25 = 5^{2} \)
Weight: \( k \) \(=\) \( 34 \)
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: \(172.457072203\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 1178101x^{2} + 346979902500 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{10}\cdot 3^{4}\cdot 5^{2} \)
Twist minimal: no (minimal twist has level 1)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 24.3
Root \(766.996i\) of defining polynomial
Character \(\chi\) \(=\) 25.24
Dual form 25.34.b.a.24.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+49679.4i q^{2} +1.55221e7i q^{3} +6.12189e9 q^{4} -7.71130e11 q^{6} -1.22168e14i q^{7} +7.30875e14i q^{8} +5.31812e15 q^{9} +O(q^{10})\) \(q+49679.4i q^{2} +1.55221e7i q^{3} +6.12189e9 q^{4} -7.71130e11 q^{6} -1.22168e14i q^{7} +7.30875e14i q^{8} +5.31812e15 q^{9} +2.15763e17 q^{11} +9.50247e16i q^{12} +1.07762e18i q^{13} +6.06926e18 q^{14} +1.62772e19 q^{16} +2.54532e20i q^{17} +2.64201e20i q^{18} +1.22020e21 q^{19} +1.89631e21 q^{21} +1.07190e22i q^{22} +5.11280e21i q^{23} -1.13447e22 q^{24} -5.35354e22 q^{26} +1.68837e23i q^{27} -7.47901e23i q^{28} +1.64733e23 q^{29} -6.75706e24 q^{31} +7.08681e24i q^{32} +3.34910e24i q^{33} -1.26450e25 q^{34} +3.25570e25 q^{36} -7.46520e25i q^{37} +6.06190e25i q^{38} -1.67269e25 q^{39} +4.96453e26 q^{41} +9.42077e25i q^{42} +1.99347e26i q^{43} +1.32088e27 q^{44} -2.54001e26 q^{46} +2.16452e27i q^{47} +2.52656e26i q^{48} -7.19411e27 q^{49} -3.95088e27 q^{51} +6.59706e27i q^{52} +3.60439e28i q^{53} -8.38773e27 q^{54} +8.92898e28 q^{56} +1.89401e28i q^{57} +8.18385e27i q^{58} +1.87520e29 q^{59} +4.18340e27 q^{61} -3.35687e29i q^{62} -6.49706e29i q^{63} -2.12249e29 q^{64} -1.66381e29 q^{66} +4.85975e29i q^{67} +1.55822e30i q^{68} -7.93615e28 q^{69} -3.42819e30 q^{71} +3.88688e30i q^{72} -7.01467e30i q^{73} +3.70867e30 q^{74} +7.46995e30 q^{76} -2.63594e31i q^{77} -8.30984e29i q^{78} +2.95630e30 q^{79} +2.69431e31 q^{81} +2.46635e31i q^{82} +1.23020e31i q^{83} +1.16090e31 q^{84} -9.90343e30 q^{86} +2.55701e30i q^{87} +1.57696e32i q^{88} -7.05623e31 q^{89} +1.31651e32 q^{91} +3.13000e31i q^{92} -1.04884e32i q^{93} -1.07532e32 q^{94} -1.10002e32 q^{96} -7.71791e32i q^{97} -3.57399e32i q^{98} +1.14745e33 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 29304467968 q^{4} - 19857844386432 q^{6} + 16\!\cdots\!08 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 29304467968 q^{4} - 19857844386432 q^{6} + 16\!\cdots\!08 q^{9}+ \cdots + 18\!\cdots\!56 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 49679.4i 0.536021i 0.963416 + 0.268010i \(0.0863662\pi\)
−0.963416 + 0.268010i \(0.913634\pi\)
\(3\) 1.55221e7i 0.208185i 0.994568 + 0.104093i \(0.0331939\pi\)
−0.994568 + 0.104093i \(0.966806\pi\)
\(4\) 6.12189e9 0.712682
\(5\) 0 0
\(6\) −7.71130e11 −0.111592
\(7\) − 1.22168e14i − 1.38944i −0.719278 0.694722i \(-0.755527\pi\)
0.719278 0.694722i \(-0.244473\pi\)
\(8\) 7.30875e14i 0.918033i
\(9\) 5.31812e15 0.956659
\(10\) 0 0
\(11\) 2.15763e17 1.41578 0.707892 0.706321i \(-0.249646\pi\)
0.707892 + 0.706321i \(0.249646\pi\)
\(12\) 9.50247e16i 0.148370i
\(13\) 1.07762e18i 0.449158i 0.974456 + 0.224579i \(0.0721007\pi\)
−0.974456 + 0.224579i \(0.927899\pi\)
\(14\) 6.06926e18 0.744771
\(15\) 0 0
\(16\) 1.62772e19 0.220597
\(17\) 2.54532e20i 1.26863i 0.773074 + 0.634316i \(0.218718\pi\)
−0.773074 + 0.634316i \(0.781282\pi\)
\(18\) 2.64201e20i 0.512789i
\(19\) 1.22020e21 0.970505 0.485252 0.874374i \(-0.338728\pi\)
0.485252 + 0.874374i \(0.338728\pi\)
\(20\) 0 0
\(21\) 1.89631e21 0.289262
\(22\) 1.07190e22i 0.758890i
\(23\) 5.11280e21i 0.173840i 0.996215 + 0.0869199i \(0.0277024\pi\)
−0.996215 + 0.0869199i \(0.972298\pi\)
\(24\) −1.13447e22 −0.191121
\(25\) 0 0
\(26\) −5.35354e22 −0.240758
\(27\) 1.68837e23i 0.407348i
\(28\) − 7.47901e23i − 0.990231i
\(29\) 1.64733e23 0.122240 0.0611201 0.998130i \(-0.480533\pi\)
0.0611201 + 0.998130i \(0.480533\pi\)
\(30\) 0 0
\(31\) −6.75706e24 −1.66836 −0.834181 0.551491i \(-0.814059\pi\)
−0.834181 + 0.551491i \(0.814059\pi\)
\(32\) 7.08681e24i 1.03628i
\(33\) 3.34910e24i 0.294746i
\(34\) −1.26450e25 −0.680013
\(35\) 0 0
\(36\) 3.25570e25 0.681793
\(37\) − 7.46520e25i − 0.994748i −0.867536 0.497374i \(-0.834298\pi\)
0.867536 0.497374i \(-0.165702\pi\)
\(38\) 6.06190e25i 0.520211i
\(39\) −1.67269e25 −0.0935082
\(40\) 0 0
\(41\) 4.96453e26 1.21603 0.608016 0.793925i \(-0.291966\pi\)
0.608016 + 0.793925i \(0.291966\pi\)
\(42\) 9.42077e25i 0.155051i
\(43\) 1.99347e26i 0.222525i 0.993791 + 0.111263i \(0.0354895\pi\)
−0.993791 + 0.111263i \(0.964511\pi\)
\(44\) 1.32088e27 1.00900
\(45\) 0 0
\(46\) −2.54001e26 −0.0931818
\(47\) 2.16452e27i 0.556862i 0.960456 + 0.278431i \(0.0898143\pi\)
−0.960456 + 0.278431i \(0.910186\pi\)
\(48\) 2.52656e26i 0.0459250i
\(49\) −7.19411e27 −0.930555
\(50\) 0 0
\(51\) −3.95088e27 −0.264111
\(52\) 6.59706e27i 0.320107i
\(53\) 3.60439e28i 1.27726i 0.769513 + 0.638631i \(0.220499\pi\)
−0.769513 + 0.638631i \(0.779501\pi\)
\(54\) −8.38773e27 −0.218347
\(55\) 0 0
\(56\) 8.92898e28 1.27556
\(57\) 1.89401e28i 0.202045i
\(58\) 8.18385e27i 0.0655233i
\(59\) 1.87520e29 1.13237 0.566186 0.824278i \(-0.308418\pi\)
0.566186 + 0.824278i \(0.308418\pi\)
\(60\) 0 0
\(61\) 4.18340e27 0.0145743 0.00728715 0.999973i \(-0.497680\pi\)
0.00728715 + 0.999973i \(0.497680\pi\)
\(62\) − 3.35687e29i − 0.894276i
\(63\) − 6.49706e29i − 1.32922i
\(64\) −2.12249e29 −0.334870
\(65\) 0 0
\(66\) −1.66381e29 −0.157990
\(67\) 4.85975e29i 0.360064i 0.983661 + 0.180032i \(0.0576201\pi\)
−0.983661 + 0.180032i \(0.942380\pi\)
\(68\) 1.55822e30i 0.904130i
\(69\) −7.93615e28 −0.0361909
\(70\) 0 0
\(71\) −3.42819e30 −0.975667 −0.487834 0.872937i \(-0.662213\pi\)
−0.487834 + 0.872937i \(0.662213\pi\)
\(72\) 3.88688e30i 0.878244i
\(73\) − 7.01467e30i − 1.26235i −0.775640 0.631175i \(-0.782573\pi\)
0.775640 0.631175i \(-0.217427\pi\)
\(74\) 3.70867e30 0.533206
\(75\) 0 0
\(76\) 7.46995e30 0.691661
\(77\) − 2.63594e31i − 1.96715i
\(78\) − 8.30984e29i − 0.0501224i
\(79\) 2.95630e30 0.144511 0.0722555 0.997386i \(-0.476980\pi\)
0.0722555 + 0.997386i \(0.476980\pi\)
\(80\) 0 0
\(81\) 2.69431e31 0.871855
\(82\) 2.46635e31i 0.651818i
\(83\) 1.23020e31i 0.266187i 0.991103 + 0.133094i \(0.0424911\pi\)
−0.991103 + 0.133094i \(0.957509\pi\)
\(84\) 1.16090e31 0.206152
\(85\) 0 0
\(86\) −9.90343e30 −0.119278
\(87\) 2.55701e30i 0.0254486i
\(88\) 1.57696e32i 1.29974i
\(89\) −7.05623e31 −0.482656 −0.241328 0.970444i \(-0.577583\pi\)
−0.241328 + 0.970444i \(0.577583\pi\)
\(90\) 0 0
\(91\) 1.31651e32 0.624080
\(92\) 3.13000e31i 0.123892i
\(93\) − 1.04884e32i − 0.347329i
\(94\) −1.07532e32 −0.298490
\(95\) 0 0
\(96\) −1.10002e32 −0.215738
\(97\) − 7.71791e32i − 1.27575i −0.770140 0.637875i \(-0.779813\pi\)
0.770140 0.637875i \(-0.220187\pi\)
\(98\) − 3.57399e32i − 0.498797i
\(99\) 1.14745e33 1.35442
\(100\) 0 0
\(101\) 1.88601e33 1.60045 0.800224 0.599701i \(-0.204714\pi\)
0.800224 + 0.599701i \(0.204714\pi\)
\(102\) − 1.96278e32i − 0.141569i
\(103\) 4.05317e32i 0.248874i 0.992227 + 0.124437i \(0.0397125\pi\)
−0.992227 + 0.124437i \(0.960287\pi\)
\(104\) −7.87604e32 −0.412342
\(105\) 0 0
\(106\) −1.79064e33 −0.684640
\(107\) − 1.13905e33i − 0.373003i −0.982455 0.186501i \(-0.940285\pi\)
0.982455 0.186501i \(-0.0597149\pi\)
\(108\) 1.03360e33i 0.290309i
\(109\) −5.47789e32 −0.132153 −0.0660764 0.997815i \(-0.521048\pi\)
−0.0660764 + 0.997815i \(0.521048\pi\)
\(110\) 0 0
\(111\) 1.15876e33 0.207092
\(112\) − 1.98855e33i − 0.306507i
\(113\) − 7.05689e33i − 0.939332i −0.882844 0.469666i \(-0.844374\pi\)
0.882844 0.469666i \(-0.155626\pi\)
\(114\) −9.40936e32 −0.108300
\(115\) 0 0
\(116\) 1.00848e33 0.0871183
\(117\) 5.73090e33i 0.429691i
\(118\) 9.31588e33i 0.606975i
\(119\) 3.10958e34 1.76269
\(120\) 0 0
\(121\) 2.33284e34 1.00444
\(122\) 2.07829e32i 0.00781213i
\(123\) 7.70601e33i 0.253160i
\(124\) −4.13660e34 −1.18901
\(125\) 0 0
\(126\) 3.22771e34 0.712492
\(127\) − 1.88228e34i − 0.364689i −0.983235 0.182344i \(-0.941631\pi\)
0.983235 0.182344i \(-0.0583685\pi\)
\(128\) 5.03308e34i 0.856780i
\(129\) −3.09428e33 −0.0463265
\(130\) 0 0
\(131\) −1.26378e35 −1.46790 −0.733949 0.679204i \(-0.762325\pi\)
−0.733949 + 0.679204i \(0.762325\pi\)
\(132\) 2.05028e34i 0.210060i
\(133\) − 1.49070e35i − 1.34846i
\(134\) −2.41430e34 −0.193002
\(135\) 0 0
\(136\) −1.86031e35 −1.16465
\(137\) 2.95759e35i 1.64077i 0.571809 + 0.820387i \(0.306242\pi\)
−0.571809 + 0.820387i \(0.693758\pi\)
\(138\) − 3.94263e33i − 0.0193991i
\(139\) 8.22498e34 0.359245 0.179622 0.983736i \(-0.442512\pi\)
0.179622 + 0.983736i \(0.442512\pi\)
\(140\) 0 0
\(141\) −3.35980e34 −0.115931
\(142\) − 1.70310e35i − 0.522978i
\(143\) 2.32510e35i 0.635911i
\(144\) 8.65639e34 0.211036
\(145\) 0 0
\(146\) 3.48485e35 0.676646
\(147\) − 1.11668e35i − 0.193728i
\(148\) − 4.57011e35i − 0.708939i
\(149\) −5.29045e35 −0.734378 −0.367189 0.930146i \(-0.619680\pi\)
−0.367189 + 0.930146i \(0.619680\pi\)
\(150\) 0 0
\(151\) −5.18728e35 −0.577856 −0.288928 0.957351i \(-0.593299\pi\)
−0.288928 + 0.957351i \(0.593299\pi\)
\(152\) 8.91816e35i 0.890955i
\(153\) 1.35363e36i 1.21365i
\(154\) 1.30952e36 1.05444
\(155\) 0 0
\(156\) −1.02400e35 −0.0666416
\(157\) 1.31075e36i 0.767669i 0.923402 + 0.383835i \(0.125397\pi\)
−0.923402 + 0.383835i \(0.874603\pi\)
\(158\) 1.46867e35i 0.0774609i
\(159\) −5.59478e35 −0.265908
\(160\) 0 0
\(161\) 6.24622e35 0.241541
\(162\) 1.33852e36i 0.467332i
\(163\) − 1.61518e36i − 0.509478i −0.967010 0.254739i \(-0.918010\pi\)
0.967010 0.254739i \(-0.0819896\pi\)
\(164\) 3.03923e36 0.866643
\(165\) 0 0
\(166\) −6.11157e35 −0.142682
\(167\) − 6.45682e35i − 0.136520i −0.997668 0.0682601i \(-0.978255\pi\)
0.997668 0.0682601i \(-0.0217448\pi\)
\(168\) 1.38597e36i 0.265552i
\(169\) 4.59487e36 0.798257
\(170\) 0 0
\(171\) 6.48919e36 0.928442
\(172\) 1.22038e36i 0.158590i
\(173\) 6.58049e36i 0.777137i 0.921420 + 0.388569i \(0.127030\pi\)
−0.921420 + 0.388569i \(0.872970\pi\)
\(174\) −1.27031e35 −0.0136410
\(175\) 0 0
\(176\) 3.51200e36 0.312317
\(177\) 2.91071e36i 0.235743i
\(178\) − 3.50549e36i − 0.258714i
\(179\) 2.08823e37 1.40508 0.702542 0.711642i \(-0.252048\pi\)
0.702542 + 0.711642i \(0.252048\pi\)
\(180\) 0 0
\(181\) −2.42658e37 −1.35925 −0.679624 0.733561i \(-0.737857\pi\)
−0.679624 + 0.733561i \(0.737857\pi\)
\(182\) 6.54034e36i 0.334520i
\(183\) 6.49353e34i 0.00303416i
\(184\) −3.73682e36 −0.159591
\(185\) 0 0
\(186\) 5.21058e36 0.186175
\(187\) 5.49185e37i 1.79611i
\(188\) 1.32510e37i 0.396865i
\(189\) 2.06265e37 0.565987
\(190\) 0 0
\(191\) 4.06968e37 0.938664 0.469332 0.883022i \(-0.344495\pi\)
0.469332 + 0.883022i \(0.344495\pi\)
\(192\) − 3.29455e36i − 0.0697150i
\(193\) 5.23242e37i 1.01627i 0.861279 + 0.508133i \(0.169664\pi\)
−0.861279 + 0.508133i \(0.830336\pi\)
\(194\) 3.83421e37 0.683829
\(195\) 0 0
\(196\) −4.40416e37 −0.663189
\(197\) 1.35006e38i 1.86922i 0.355672 + 0.934611i \(0.384252\pi\)
−0.355672 + 0.934611i \(0.615748\pi\)
\(198\) 5.70048e37i 0.725999i
\(199\) 9.91562e37 1.16210 0.581051 0.813867i \(-0.302642\pi\)
0.581051 + 0.813867i \(0.302642\pi\)
\(200\) 0 0
\(201\) −7.54336e36 −0.0749601
\(202\) 9.36960e37i 0.857874i
\(203\) − 2.01252e37i − 0.169846i
\(204\) −2.41868e37 −0.188227
\(205\) 0 0
\(206\) −2.01359e37 −0.133402
\(207\) 2.71905e37i 0.166305i
\(208\) 1.75406e37i 0.0990828i
\(209\) 2.63274e38 1.37403
\(210\) 0 0
\(211\) −2.07128e38 −0.923802 −0.461901 0.886931i \(-0.652832\pi\)
−0.461901 + 0.886931i \(0.652832\pi\)
\(212\) 2.20657e38i 0.910282i
\(213\) − 5.32127e37i − 0.203120i
\(214\) 5.65876e37 0.199937
\(215\) 0 0
\(216\) −1.23399e38 −0.373959
\(217\) 8.25499e38i 2.31809i
\(218\) − 2.72139e37i − 0.0708367i
\(219\) 1.08883e38 0.262803
\(220\) 0 0
\(221\) −2.74288e38 −0.569816
\(222\) 5.75665e37i 0.111006i
\(223\) 4.51334e38i 0.808106i 0.914736 + 0.404053i \(0.132399\pi\)
−0.914736 + 0.404053i \(0.867601\pi\)
\(224\) 8.65784e38 1.43985
\(225\) 0 0
\(226\) 3.50582e38 0.503501
\(227\) − 2.43217e38i − 0.324764i −0.986728 0.162382i \(-0.948082\pi\)
0.986728 0.162382i \(-0.0519176\pi\)
\(228\) 1.15949e38i 0.143994i
\(229\) −3.41418e38 −0.394458 −0.197229 0.980357i \(-0.563194\pi\)
−0.197229 + 0.980357i \(0.563194\pi\)
\(230\) 0 0
\(231\) 4.09153e38 0.409533
\(232\) 1.20399e38i 0.112221i
\(233\) 6.99608e37i 0.0607410i 0.999539 + 0.0303705i \(0.00966871\pi\)
−0.999539 + 0.0303705i \(0.990331\pi\)
\(234\) −2.84708e38 −0.230323
\(235\) 0 0
\(236\) 1.14798e39 0.807020
\(237\) 4.58880e37i 0.0300851i
\(238\) 1.54482e39i 0.944840i
\(239\) 1.63523e39 0.933282 0.466641 0.884447i \(-0.345464\pi\)
0.466641 + 0.884447i \(0.345464\pi\)
\(240\) 0 0
\(241\) −3.64560e38 −0.181338 −0.0906689 0.995881i \(-0.528900\pi\)
−0.0906689 + 0.995881i \(0.528900\pi\)
\(242\) 1.15894e39i 0.538403i
\(243\) 1.35679e39i 0.588856i
\(244\) 2.56103e37 0.0103868
\(245\) 0 0
\(246\) −3.82830e38 −0.135699
\(247\) 1.31491e39i 0.435910i
\(248\) − 4.93857e39i − 1.53161i
\(249\) −1.90953e38 −0.0554164
\(250\) 0 0
\(251\) 1.94693e39 0.495147 0.247573 0.968869i \(-0.420367\pi\)
0.247573 + 0.968869i \(0.420367\pi\)
\(252\) − 3.97743e39i − 0.947313i
\(253\) 1.10315e39i 0.246120i
\(254\) 9.35105e38 0.195481
\(255\) 0 0
\(256\) −4.32361e39 −0.794122
\(257\) − 5.64464e39i − 0.972164i −0.873913 0.486082i \(-0.838426\pi\)
0.873913 0.486082i \(-0.161574\pi\)
\(258\) − 1.53722e38i − 0.0248320i
\(259\) −9.12012e39 −1.38215
\(260\) 0 0
\(261\) 8.76071e38 0.116942
\(262\) − 6.27840e39i − 0.786824i
\(263\) − 9.75930e39i − 1.14855i −0.818663 0.574274i \(-0.805285\pi\)
0.818663 0.574274i \(-0.194715\pi\)
\(264\) −2.44777e39 −0.270586
\(265\) 0 0
\(266\) 7.40572e39 0.722804
\(267\) − 1.09528e39i − 0.100482i
\(268\) 2.97508e39i 0.256611i
\(269\) 1.08142e40 0.877168 0.438584 0.898690i \(-0.355480\pi\)
0.438584 + 0.898690i \(0.355480\pi\)
\(270\) 0 0
\(271\) −6.19121e39 −0.444408 −0.222204 0.975000i \(-0.571325\pi\)
−0.222204 + 0.975000i \(0.571325\pi\)
\(272\) 4.14306e39i 0.279856i
\(273\) 2.04350e39i 0.129924i
\(274\) −1.46931e40 −0.879489
\(275\) 0 0
\(276\) −4.85842e38 −0.0257926
\(277\) − 5.15662e39i − 0.257898i −0.991651 0.128949i \(-0.958840\pi\)
0.991651 0.128949i \(-0.0411604\pi\)
\(278\) 4.08612e39i 0.192563i
\(279\) −3.59349e40 −1.59605
\(280\) 0 0
\(281\) −1.28386e40 −0.506832 −0.253416 0.967357i \(-0.581554\pi\)
−0.253416 + 0.967357i \(0.581554\pi\)
\(282\) − 1.66913e39i − 0.0621412i
\(283\) − 3.83806e40i − 1.34783i −0.738809 0.673915i \(-0.764611\pi\)
0.738809 0.673915i \(-0.235389\pi\)
\(284\) −2.09870e40 −0.695340
\(285\) 0 0
\(286\) −1.15510e40 −0.340862
\(287\) − 6.06509e40i − 1.68961i
\(288\) 3.76885e40i 0.991364i
\(289\) −2.45321e40 −0.609426
\(290\) 0 0
\(291\) 1.19798e40 0.265593
\(292\) − 4.29430e40i − 0.899654i
\(293\) − 1.90344e40i − 0.376897i −0.982083 0.188449i \(-0.939654\pi\)
0.982083 0.188449i \(-0.0603459\pi\)
\(294\) 5.54760e39 0.103842
\(295\) 0 0
\(296\) 5.45613e40 0.913212
\(297\) 3.64287e40i 0.576717i
\(298\) − 2.62827e40i − 0.393642i
\(299\) −5.50964e39 −0.0780816
\(300\) 0 0
\(301\) 2.43538e40 0.309186
\(302\) − 2.57701e40i − 0.309743i
\(303\) 2.92749e40i 0.333190i
\(304\) 1.98614e40 0.214090
\(305\) 0 0
\(306\) −6.72478e40 −0.650540
\(307\) − 1.48955e41i − 1.36544i −0.730682 0.682718i \(-0.760798\pi\)
0.730682 0.682718i \(-0.239202\pi\)
\(308\) − 1.61369e41i − 1.40195i
\(309\) −6.29138e39 −0.0518121
\(310\) 0 0
\(311\) −7.37928e39 −0.0546345 −0.0273173 0.999627i \(-0.508696\pi\)
−0.0273173 + 0.999627i \(0.508696\pi\)
\(312\) − 1.22253e40i − 0.0858437i
\(313\) 2.63324e40i 0.175391i 0.996147 + 0.0876957i \(0.0279503\pi\)
−0.996147 + 0.0876957i \(0.972050\pi\)
\(314\) −6.51171e40 −0.411487
\(315\) 0 0
\(316\) 1.80981e40 0.102990
\(317\) 1.79226e41i 0.968109i 0.875038 + 0.484054i \(0.160836\pi\)
−0.875038 + 0.484054i \(0.839164\pi\)
\(318\) − 2.77946e40i − 0.142532i
\(319\) 3.55433e40 0.173066
\(320\) 0 0
\(321\) 1.76805e40 0.0776538
\(322\) 3.10309e40i 0.129471i
\(323\) 3.10581e41i 1.23121i
\(324\) 1.64942e41 0.621355
\(325\) 0 0
\(326\) 8.02413e40 0.273091
\(327\) − 8.50285e39i − 0.0275123i
\(328\) 3.62845e41i 1.11636i
\(329\) 2.64436e41 0.773728
\(330\) 0 0
\(331\) 6.16181e41 1.63135 0.815674 0.578512i \(-0.196366\pi\)
0.815674 + 0.578512i \(0.196366\pi\)
\(332\) 7.53115e40i 0.189707i
\(333\) − 3.97009e41i − 0.951635i
\(334\) 3.20771e40 0.0731777
\(335\) 0 0
\(336\) 3.08666e40 0.0638102
\(337\) 4.51357e41i 0.888437i 0.895919 + 0.444218i \(0.146519\pi\)
−0.895919 + 0.444218i \(0.853481\pi\)
\(338\) 2.28271e41i 0.427882i
\(339\) 1.09538e41 0.195555
\(340\) 0 0
\(341\) −1.45792e42 −2.36204
\(342\) 3.22379e41i 0.497664i
\(343\) − 6.55899e40i − 0.0964903i
\(344\) −1.45697e41 −0.204286
\(345\) 0 0
\(346\) −3.26915e41 −0.416562
\(347\) 8.17900e41i 0.993720i 0.867831 + 0.496860i \(0.165514\pi\)
−0.867831 + 0.496860i \(0.834486\pi\)
\(348\) 1.56537e40i 0.0181368i
\(349\) 1.22969e42 1.35886 0.679432 0.733739i \(-0.262226\pi\)
0.679432 + 0.733739i \(0.262226\pi\)
\(350\) 0 0
\(351\) −1.81942e41 −0.182964
\(352\) 1.52907e42i 1.46715i
\(353\) − 1.62570e41i − 0.148853i −0.997226 0.0744266i \(-0.976287\pi\)
0.997226 0.0744266i \(-0.0237126\pi\)
\(354\) −1.44602e41 −0.126363
\(355\) 0 0
\(356\) −4.31974e41 −0.343980
\(357\) 4.82673e41i 0.366967i
\(358\) 1.03742e42i 0.753155i
\(359\) −3.64036e41 −0.252398 −0.126199 0.992005i \(-0.540278\pi\)
−0.126199 + 0.992005i \(0.540278\pi\)
\(360\) 0 0
\(361\) −9.18755e40 −0.0581207
\(362\) − 1.20551e42i − 0.728585i
\(363\) 3.62106e41i 0.209111i
\(364\) 8.05951e41 0.444771
\(365\) 0 0
\(366\) −3.22595e39 −0.00162637
\(367\) 1.35581e42i 0.653442i 0.945121 + 0.326721i \(0.105944\pi\)
−0.945121 + 0.326721i \(0.894056\pi\)
\(368\) 8.32218e40i 0.0383485i
\(369\) 2.64020e42 1.16333
\(370\) 0 0
\(371\) 4.40343e42 1.77469
\(372\) − 6.42088e41i − 0.247535i
\(373\) − 1.93457e42i − 0.713493i −0.934201 0.356746i \(-0.883886\pi\)
0.934201 0.356746i \(-0.116114\pi\)
\(374\) −2.72832e42 −0.962752
\(375\) 0 0
\(376\) −1.58200e42 −0.511218
\(377\) 1.77519e41i 0.0549052i
\(378\) 1.02471e42i 0.303381i
\(379\) 5.45789e42 1.54696 0.773478 0.633823i \(-0.218515\pi\)
0.773478 + 0.633823i \(0.218515\pi\)
\(380\) 0 0
\(381\) 2.92169e41 0.0759229
\(382\) 2.02180e42i 0.503143i
\(383\) 5.06942e42i 1.20831i 0.796867 + 0.604155i \(0.206489\pi\)
−0.796867 + 0.604155i \(0.793511\pi\)
\(384\) −7.81241e41 −0.178369
\(385\) 0 0
\(386\) −2.59944e42 −0.544740
\(387\) 1.06015e42i 0.212881i
\(388\) − 4.72482e42i − 0.909204i
\(389\) −7.51028e42 −1.38512 −0.692560 0.721360i \(-0.743517\pi\)
−0.692560 + 0.721360i \(0.743517\pi\)
\(390\) 0 0
\(391\) −1.30137e42 −0.220539
\(392\) − 5.25800e42i − 0.854280i
\(393\) − 1.96166e42i − 0.305595i
\(394\) −6.70703e42 −1.00194
\(395\) 0 0
\(396\) 7.02458e42 0.965272
\(397\) − 1.26926e43i − 1.67304i −0.547938 0.836519i \(-0.684587\pi\)
0.547938 0.836519i \(-0.315413\pi\)
\(398\) 4.92603e42i 0.622911i
\(399\) 2.31389e42 0.280730
\(400\) 0 0
\(401\) 5.29553e42 0.591598 0.295799 0.955250i \(-0.404414\pi\)
0.295799 + 0.955250i \(0.404414\pi\)
\(402\) − 3.74750e41i − 0.0401802i
\(403\) − 7.28153e42i − 0.749358i
\(404\) 1.15459e43 1.14061
\(405\) 0 0
\(406\) 9.99807e41 0.0910410
\(407\) − 1.61071e43i − 1.40835i
\(408\) − 2.88760e42i − 0.242462i
\(409\) −1.80907e43 −1.45888 −0.729442 0.684042i \(-0.760220\pi\)
−0.729442 + 0.684042i \(0.760220\pi\)
\(410\) 0 0
\(411\) −4.59081e42 −0.341585
\(412\) 2.48130e42i 0.177368i
\(413\) − 2.29090e43i − 1.57337i
\(414\) −1.35081e42 −0.0891432
\(415\) 0 0
\(416\) −7.63687e42 −0.465453
\(417\) 1.27669e42i 0.0747896i
\(418\) 1.30793e43i 0.736506i
\(419\) −9.59306e42 −0.519309 −0.259654 0.965702i \(-0.583609\pi\)
−0.259654 + 0.965702i \(0.583609\pi\)
\(420\) 0 0
\(421\) −1.95406e43 −0.977875 −0.488938 0.872319i \(-0.662615\pi\)
−0.488938 + 0.872319i \(0.662615\pi\)
\(422\) − 1.02900e43i − 0.495177i
\(423\) 1.15112e43i 0.532727i
\(424\) −2.63436e43 −1.17257
\(425\) 0 0
\(426\) 2.64358e42 0.108876
\(427\) − 5.11080e41i − 0.0202502i
\(428\) − 6.97316e42i − 0.265832i
\(429\) −3.60904e42 −0.132387
\(430\) 0 0
\(431\) 4.01382e43 1.36359 0.681794 0.731544i \(-0.261200\pi\)
0.681794 + 0.731544i \(0.261200\pi\)
\(432\) 2.74819e42i 0.0898596i
\(433\) 1.24146e43i 0.390735i 0.980730 + 0.195367i \(0.0625899\pi\)
−0.980730 + 0.195367i \(0.937410\pi\)
\(434\) −4.10103e43 −1.24255
\(435\) 0 0
\(436\) −3.35350e42 −0.0941829
\(437\) 6.23865e42i 0.168712i
\(438\) 5.40922e42i 0.140868i
\(439\) 6.07044e43 1.52249 0.761247 0.648462i \(-0.224587\pi\)
0.761247 + 0.648462i \(0.224587\pi\)
\(440\) 0 0
\(441\) −3.82592e43 −0.890223
\(442\) − 1.36265e43i − 0.305433i
\(443\) − 6.61294e43i − 1.42802i −0.700137 0.714009i \(-0.746878\pi\)
0.700137 0.714009i \(-0.253122\pi\)
\(444\) 7.09379e42 0.147591
\(445\) 0 0
\(446\) −2.24220e43 −0.433162
\(447\) − 8.21191e42i − 0.152887i
\(448\) 2.59301e43i 0.465283i
\(449\) 1.54504e43 0.267225 0.133612 0.991034i \(-0.457342\pi\)
0.133612 + 0.991034i \(0.457342\pi\)
\(450\) 0 0
\(451\) 1.07116e44 1.72164
\(452\) − 4.32015e43i − 0.669444i
\(453\) − 8.05175e42i − 0.120301i
\(454\) 1.20829e43 0.174080
\(455\) 0 0
\(456\) −1.38429e43 −0.185484
\(457\) 7.12331e43i 0.920586i 0.887767 + 0.460293i \(0.152256\pi\)
−0.887767 + 0.460293i \(0.847744\pi\)
\(458\) − 1.69615e43i − 0.211438i
\(459\) −4.29745e43 −0.516774
\(460\) 0 0
\(461\) −1.34004e44 −1.49986 −0.749932 0.661515i \(-0.769914\pi\)
−0.749932 + 0.661515i \(0.769914\pi\)
\(462\) 2.03265e43i 0.219518i
\(463\) 5.53073e43i 0.576363i 0.957576 + 0.288182i \(0.0930507\pi\)
−0.957576 + 0.288182i \(0.906949\pi\)
\(464\) 2.68139e42 0.0269658
\(465\) 0 0
\(466\) −3.47561e42 −0.0325584
\(467\) − 3.51798e43i − 0.318101i −0.987270 0.159050i \(-0.949157\pi\)
0.987270 0.159050i \(-0.0508432\pi\)
\(468\) 3.50840e43i 0.306233i
\(469\) 5.93707e43 0.500288
\(470\) 0 0
\(471\) −2.03456e43 −0.159818
\(472\) 1.37054e44i 1.03955i
\(473\) 4.30116e43i 0.315048i
\(474\) −2.27969e42 −0.0161262
\(475\) 0 0
\(476\) 1.90365e44 1.25624
\(477\) 1.91686e44i 1.22190i
\(478\) 8.12372e43i 0.500258i
\(479\) 2.90520e44 1.72838 0.864192 0.503162i \(-0.167830\pi\)
0.864192 + 0.503162i \(0.167830\pi\)
\(480\) 0 0
\(481\) 8.04464e43 0.446799
\(482\) − 1.81112e43i − 0.0972008i
\(483\) 9.69546e42i 0.0502853i
\(484\) 1.42814e44 0.715849
\(485\) 0 0
\(486\) −6.74045e43 −0.315639
\(487\) − 1.45706e44i − 0.659551i −0.944059 0.329776i \(-0.893027\pi\)
0.944059 0.329776i \(-0.106973\pi\)
\(488\) 3.05755e42i 0.0133797i
\(489\) 2.50710e43 0.106066
\(490\) 0 0
\(491\) 1.43984e44 0.569469 0.284735 0.958606i \(-0.408095\pi\)
0.284735 + 0.958606i \(0.408095\pi\)
\(492\) 4.71753e43i 0.180423i
\(493\) 4.19299e43i 0.155078i
\(494\) −6.53241e43 −0.233657
\(495\) 0 0
\(496\) −1.09986e44 −0.368035
\(497\) 4.18816e44i 1.35564i
\(498\) − 9.48646e42i − 0.0297043i
\(499\) −1.76176e44 −0.533687 −0.266843 0.963740i \(-0.585981\pi\)
−0.266843 + 0.963740i \(0.585981\pi\)
\(500\) 0 0
\(501\) 1.00224e43 0.0284215
\(502\) 9.67224e43i 0.265409i
\(503\) 2.43829e44i 0.647460i 0.946149 + 0.323730i \(0.104937\pi\)
−0.946149 + 0.323730i \(0.895063\pi\)
\(504\) 4.74854e44 1.22027
\(505\) 0 0
\(506\) −5.48039e43 −0.131925
\(507\) 7.13222e43i 0.166185i
\(508\) − 1.15231e44i − 0.259907i
\(509\) −3.49786e44 −0.763763 −0.381882 0.924211i \(-0.624724\pi\)
−0.381882 + 0.924211i \(0.624724\pi\)
\(510\) 0 0
\(511\) −8.56970e44 −1.75396
\(512\) 2.17544e44i 0.431114i
\(513\) 2.06015e44i 0.395333i
\(514\) 2.80422e44 0.521100
\(515\) 0 0
\(516\) −1.89429e43 −0.0330161
\(517\) 4.67023e44i 0.788396i
\(518\) − 4.53082e44i − 0.740860i
\(519\) −1.02143e44 −0.161789
\(520\) 0 0
\(521\) −3.31981e44 −0.493505 −0.246752 0.969079i \(-0.579363\pi\)
−0.246752 + 0.969079i \(0.579363\pi\)
\(522\) 4.35227e43i 0.0626834i
\(523\) − 1.02645e45i − 1.43238i −0.697905 0.716190i \(-0.745884\pi\)
0.697905 0.716190i \(-0.254116\pi\)
\(524\) −7.73674e44 −1.04614
\(525\) 0 0
\(526\) 4.84837e44 0.615645
\(527\) − 1.71989e45i − 2.11654i
\(528\) 5.45137e43i 0.0650199i
\(529\) 8.38864e44 0.969780
\(530\) 0 0
\(531\) 9.97254e44 1.08329
\(532\) − 9.12591e44i − 0.961024i
\(533\) 5.34987e44i 0.546191i
\(534\) 5.44127e43 0.0538604
\(535\) 0 0
\(536\) −3.55187e44 −0.330550
\(537\) 3.24137e44i 0.292518i
\(538\) 5.37245e44i 0.470180i
\(539\) −1.55222e45 −1.31746
\(540\) 0 0
\(541\) 4.08746e44 0.326361 0.163180 0.986596i \(-0.447825\pi\)
0.163180 + 0.986596i \(0.447825\pi\)
\(542\) − 3.07576e44i − 0.238212i
\(543\) − 3.76657e44i − 0.282976i
\(544\) −1.80382e45 −1.31465
\(545\) 0 0
\(546\) −1.01520e44 −0.0696422
\(547\) 7.54748e44i 0.502355i 0.967941 + 0.251178i \(0.0808178\pi\)
−0.967941 + 0.251178i \(0.919182\pi\)
\(548\) 1.81060e45i 1.16935i
\(549\) 2.22479e43 0.0139426
\(550\) 0 0
\(551\) 2.01008e44 0.118635
\(552\) − 5.80033e43i − 0.0332245i
\(553\) − 3.61166e44i − 0.200790i
\(554\) 2.56178e44 0.138239
\(555\) 0 0
\(556\) 5.03524e44 0.256027
\(557\) − 3.77046e45i − 1.86116i −0.366087 0.930581i \(-0.619303\pi\)
0.366087 0.930581i \(-0.380697\pi\)
\(558\) − 1.78523e45i − 0.855517i
\(559\) −2.14819e44 −0.0999491
\(560\) 0 0
\(561\) −8.52453e44 −0.373924
\(562\) − 6.37816e44i − 0.271673i
\(563\) − 1.99523e45i − 0.825284i −0.910893 0.412642i \(-0.864606\pi\)
0.910893 0.412642i \(-0.135394\pi\)
\(564\) −2.05683e44 −0.0826216
\(565\) 0 0
\(566\) 1.90672e45 0.722465
\(567\) − 3.29159e45i − 1.21139i
\(568\) − 2.50558e45i − 0.895695i
\(569\) −2.38895e45 −0.829574 −0.414787 0.909919i \(-0.636144\pi\)
−0.414787 + 0.909919i \(0.636144\pi\)
\(570\) 0 0
\(571\) 9.25875e44 0.303429 0.151714 0.988424i \(-0.451521\pi\)
0.151714 + 0.988424i \(0.451521\pi\)
\(572\) 1.42340e45i 0.453202i
\(573\) 6.31701e44i 0.195416i
\(574\) 3.01310e45 0.905665
\(575\) 0 0
\(576\) −1.12877e45 −0.320356
\(577\) − 3.73598e45i − 1.03040i −0.857071 0.515198i \(-0.827719\pi\)
0.857071 0.515198i \(-0.172281\pi\)
\(578\) − 1.21874e45i − 0.326665i
\(579\) −8.12183e44 −0.211572
\(580\) 0 0
\(581\) 1.50292e45 0.369853
\(582\) 5.95151e44i 0.142363i
\(583\) 7.77694e45i 1.80833i
\(584\) 5.12684e45 1.15888
\(585\) 0 0
\(586\) 9.45617e44 0.202025
\(587\) 6.35230e45i 1.31948i 0.751495 + 0.659739i \(0.229333\pi\)
−0.751495 + 0.659739i \(0.770667\pi\)
\(588\) − 6.83618e44i − 0.138066i
\(589\) −8.24499e45 −1.61915
\(590\) 0 0
\(591\) −2.09558e45 −0.389145
\(592\) − 1.21512e45i − 0.219438i
\(593\) − 7.20521e45i − 1.26545i −0.774378 0.632724i \(-0.781937\pi\)
0.774378 0.632724i \(-0.218063\pi\)
\(594\) −1.80976e45 −0.309132
\(595\) 0 0
\(596\) −3.23876e45 −0.523377
\(597\) 1.53912e45i 0.241933i
\(598\) − 2.73716e44i − 0.0418534i
\(599\) 6.37291e45 0.947971 0.473985 0.880533i \(-0.342815\pi\)
0.473985 + 0.880533i \(0.342815\pi\)
\(600\) 0 0
\(601\) −8.13523e45 −1.14536 −0.572679 0.819780i \(-0.694096\pi\)
−0.572679 + 0.819780i \(0.694096\pi\)
\(602\) 1.20989e45i 0.165730i
\(603\) 2.58447e45i 0.344458i
\(604\) −3.17559e45 −0.411827
\(605\) 0 0
\(606\) −1.45436e45 −0.178597
\(607\) 8.31211e45i 0.993339i 0.867940 + 0.496670i \(0.165444\pi\)
−0.867940 + 0.496670i \(0.834556\pi\)
\(608\) 8.64734e45i 1.00571i
\(609\) 3.12386e44 0.0353595
\(610\) 0 0
\(611\) −2.33253e45 −0.250119
\(612\) 8.28679e45i 0.864944i
\(613\) 2.99559e44i 0.0304358i 0.999884 + 0.0152179i \(0.00484420\pi\)
−0.999884 + 0.0152179i \(0.995156\pi\)
\(614\) 7.40000e45 0.731903
\(615\) 0 0
\(616\) 1.92654e46 1.80591
\(617\) 9.03503e45i 0.824564i 0.911056 + 0.412282i \(0.135268\pi\)
−0.911056 + 0.412282i \(0.864732\pi\)
\(618\) − 3.12552e44i − 0.0277723i
\(619\) 1.15706e46 1.00106 0.500528 0.865720i \(-0.333139\pi\)
0.500528 + 0.865720i \(0.333139\pi\)
\(620\) 0 0
\(621\) −8.63230e44 −0.0708133
\(622\) − 3.66598e44i − 0.0292853i
\(623\) 8.62048e45i 0.670623i
\(624\) −2.72267e44 −0.0206276
\(625\) 0 0
\(626\) −1.30818e45 −0.0940135
\(627\) 4.08658e45i 0.286052i
\(628\) 8.02424e45i 0.547104i
\(629\) 1.90013e46 1.26197
\(630\) 0 0
\(631\) −1.43742e46 −0.905939 −0.452970 0.891526i \(-0.649635\pi\)
−0.452970 + 0.891526i \(0.649635\pi\)
\(632\) 2.16068e45i 0.132666i
\(633\) − 3.21507e45i − 0.192322i
\(634\) −8.90387e45 −0.518927
\(635\) 0 0
\(636\) −3.42506e45 −0.189507
\(637\) − 7.75250e45i − 0.417966i
\(638\) 1.76577e45i 0.0927668i
\(639\) −1.82315e46 −0.933381
\(640\) 0 0
\(641\) 1.34213e46 0.652586 0.326293 0.945269i \(-0.394200\pi\)
0.326293 + 0.945269i \(0.394200\pi\)
\(642\) 8.78360e44i 0.0416241i
\(643\) − 3.48572e46i − 1.60995i −0.593312 0.804973i \(-0.702180\pi\)
0.593312 0.804973i \(-0.297820\pi\)
\(644\) 3.82387e45 0.172142
\(645\) 0 0
\(646\) −1.54295e46 −0.659956
\(647\) 3.47978e46i 1.45088i 0.688285 + 0.725440i \(0.258364\pi\)
−0.688285 + 0.725440i \(0.741636\pi\)
\(648\) 1.96920e46i 0.800392i
\(649\) 4.04598e46 1.60319
\(650\) 0 0
\(651\) −1.28135e46 −0.482594
\(652\) − 9.88796e45i − 0.363096i
\(653\) − 1.19815e46i − 0.428986i −0.976726 0.214493i \(-0.931190\pi\)
0.976726 0.214493i \(-0.0688099\pi\)
\(654\) 4.22417e44 0.0147472
\(655\) 0 0
\(656\) 8.08085e45 0.268252
\(657\) − 3.73049e46i − 1.20764i
\(658\) 1.31370e46i 0.414735i
\(659\) 2.00047e46 0.615918 0.307959 0.951400i \(-0.400354\pi\)
0.307959 + 0.951400i \(0.400354\pi\)
\(660\) 0 0
\(661\) 3.80588e46 1.11463 0.557316 0.830301i \(-0.311831\pi\)
0.557316 + 0.830301i \(0.311831\pi\)
\(662\) 3.06115e46i 0.874437i
\(663\) − 4.25754e45i − 0.118628i
\(664\) −8.99123e45 −0.244369
\(665\) 0 0
\(666\) 1.97232e46 0.510096
\(667\) 8.42247e44i 0.0212502i
\(668\) − 3.95279e45i − 0.0972954i
\(669\) −7.00567e45 −0.168236
\(670\) 0 0
\(671\) 9.02622e44 0.0206341
\(672\) 1.34388e46i 0.299756i
\(673\) 5.38328e46i 1.17165i 0.810437 + 0.585825i \(0.199230\pi\)
−0.810437 + 0.585825i \(0.800770\pi\)
\(674\) −2.24231e46 −0.476221
\(675\) 0 0
\(676\) 2.81293e46 0.568903
\(677\) − 9.38585e46i − 1.85251i −0.376898 0.926255i \(-0.623009\pi\)
0.376898 0.926255i \(-0.376991\pi\)
\(678\) 5.44178e45i 0.104822i
\(679\) −9.42884e46 −1.77258
\(680\) 0 0
\(681\) 3.77525e45 0.0676111
\(682\) − 7.24288e46i − 1.26610i
\(683\) − 1.22152e46i − 0.208430i −0.994555 0.104215i \(-0.966767\pi\)
0.994555 0.104215i \(-0.0332329\pi\)
\(684\) 3.97261e46 0.661683
\(685\) 0 0
\(686\) 3.25847e45 0.0517208
\(687\) − 5.29953e45i − 0.0821205i
\(688\) 3.24480e45i 0.0490883i
\(689\) −3.88416e46 −0.573693
\(690\) 0 0
\(691\) −1.01396e47 −1.42768 −0.713841 0.700308i \(-0.753046\pi\)
−0.713841 + 0.700308i \(0.753046\pi\)
\(692\) 4.02850e46i 0.553851i
\(693\) − 1.40182e47i − 1.88189i
\(694\) −4.06328e46 −0.532655
\(695\) 0 0
\(696\) −1.86885e45 −0.0233627
\(697\) 1.26363e47i 1.54270i
\(698\) 6.10903e46i 0.728379i
\(699\) −1.08594e45 −0.0126454
\(700\) 0 0
\(701\) −1.07227e47 −1.19112 −0.595560 0.803311i \(-0.703070\pi\)
−0.595560 + 0.803311i \(0.703070\pi\)
\(702\) − 9.03876e45i − 0.0980724i
\(703\) − 9.10906e46i − 0.965408i
\(704\) −4.57954e46 −0.474103
\(705\) 0 0
\(706\) 8.07639e45 0.0797884
\(707\) − 2.30411e47i − 2.22373i
\(708\) 1.78190e46i 0.168010i
\(709\) 3.54939e45 0.0326957 0.0163478 0.999866i \(-0.494796\pi\)
0.0163478 + 0.999866i \(0.494796\pi\)
\(710\) 0 0
\(711\) 1.57219e46 0.138248
\(712\) − 5.15722e46i − 0.443094i
\(713\) − 3.45475e46i − 0.290028i
\(714\) −2.39789e46 −0.196702
\(715\) 0 0
\(716\) 1.27839e47 1.00138
\(717\) 2.53822e46i 0.194296i
\(718\) − 1.80851e46i − 0.135290i
\(719\) −6.94896e46 −0.508035 −0.254017 0.967200i \(-0.581752\pi\)
−0.254017 + 0.967200i \(0.581752\pi\)
\(720\) 0 0
\(721\) 4.95169e46 0.345797
\(722\) − 4.56432e45i − 0.0311539i
\(723\) − 5.65875e45i − 0.0377519i
\(724\) −1.48553e47 −0.968711
\(725\) 0 0
\(726\) −1.79892e46 −0.112088
\(727\) − 1.73359e47i − 1.05592i −0.849270 0.527958i \(-0.822958\pi\)
0.849270 0.527958i \(-0.177042\pi\)
\(728\) 9.62203e46i 0.572926i
\(729\) 1.28718e47 0.749264
\(730\) 0 0
\(731\) −5.07401e46 −0.282303
\(732\) 3.97527e44i 0.00216239i
\(733\) − 4.59301e46i − 0.244277i −0.992513 0.122138i \(-0.961025\pi\)
0.992513 0.122138i \(-0.0389752\pi\)
\(734\) −6.73556e46 −0.350259
\(735\) 0 0
\(736\) −3.62334e46 −0.180146
\(737\) 1.04855e47i 0.509773i
\(738\) 1.31164e47i 0.623568i
\(739\) 3.24514e47 1.50869 0.754346 0.656477i \(-0.227954\pi\)
0.754346 + 0.656477i \(0.227954\pi\)
\(740\) 0 0
\(741\) −2.04102e46 −0.0907502
\(742\) 2.18760e47i 0.951268i
\(743\) − 3.63726e47i − 1.54689i −0.633866 0.773443i \(-0.718533\pi\)
0.633866 0.773443i \(-0.281467\pi\)
\(744\) 7.66571e46 0.318859
\(745\) 0 0
\(746\) 9.61085e46 0.382447
\(747\) 6.54236e46i 0.254651i
\(748\) 3.36205e47i 1.28005i
\(749\) −1.39156e47 −0.518267
\(750\) 0 0
\(751\) 2.28361e47 0.813885 0.406943 0.913454i \(-0.366595\pi\)
0.406943 + 0.913454i \(0.366595\pi\)
\(752\) 3.52323e46i 0.122842i
\(753\) 3.02205e46i 0.103082i
\(754\) −8.81906e45 −0.0294303
\(755\) 0 0
\(756\) 1.26273e47 0.403369
\(757\) − 1.23646e47i − 0.386453i −0.981154 0.193226i \(-0.938105\pi\)
0.981154 0.193226i \(-0.0618952\pi\)
\(758\) 2.71145e47i 0.829201i
\(759\) −1.71233e46 −0.0512386
\(760\) 0 0
\(761\) −1.65073e47 −0.472966 −0.236483 0.971636i \(-0.575995\pi\)
−0.236483 + 0.971636i \(0.575995\pi\)
\(762\) 1.45148e46i 0.0406962i
\(763\) 6.69225e46i 0.183619i
\(764\) 2.49141e47 0.668968
\(765\) 0 0
\(766\) −2.51846e47 −0.647679
\(767\) 2.02075e47i 0.508614i
\(768\) − 6.71116e46i − 0.165325i
\(769\) 2.15810e47 0.520339 0.260170 0.965563i \(-0.416222\pi\)
0.260170 + 0.965563i \(0.416222\pi\)
\(770\) 0 0
\(771\) 8.76167e46 0.202390
\(772\) 3.20323e47i 0.724274i
\(773\) − 5.97126e46i − 0.132161i −0.997814 0.0660806i \(-0.978951\pi\)
0.997814 0.0660806i \(-0.0210495\pi\)
\(774\) −5.26677e46 −0.114109
\(775\) 0 0
\(776\) 5.64083e47 1.17118
\(777\) − 1.41564e47i − 0.287743i
\(778\) − 3.73106e47i − 0.742453i
\(779\) 6.05774e47 1.18016
\(780\) 0 0
\(781\) −7.39675e47 −1.38133
\(782\) − 6.46514e46i − 0.118213i
\(783\) 2.78131e46i 0.0497943i
\(784\) −1.17100e47 −0.205277
\(785\) 0 0
\(786\) 9.74541e46 0.163805
\(787\) 2.40999e47i 0.396673i 0.980134 + 0.198336i \(0.0635539\pi\)
−0.980134 + 0.198336i \(0.936446\pi\)
\(788\) 8.26493e47i 1.33216i
\(789\) 1.51485e47 0.239111
\(790\) 0 0
\(791\) −8.62128e47 −1.30515
\(792\) 8.38645e47i 1.24340i
\(793\) 4.50811e45i 0.00654616i
\(794\) 6.30559e47 0.896783
\(795\) 0 0
\(796\) 6.07023e47 0.828209
\(797\) 5.30320e47i 0.708722i 0.935109 + 0.354361i \(0.115302\pi\)
−0.935109 + 0.354361i \(0.884698\pi\)
\(798\) 1.14953e47i 0.150477i
\(799\) −5.50941e47 −0.706452
\(800\) 0 0
\(801\) −3.75259e47 −0.461737
\(802\) 2.63079e47i 0.317109i
\(803\) − 1.51350e48i − 1.78722i
\(804\) −4.61796e46 −0.0534226
\(805\) 0 0
\(806\) 3.61742e47 0.401672
\(807\) 1.67860e47i 0.182614i
\(808\) 1.37844e48i 1.46926i
\(809\) −6.52623e47 −0.681572 −0.340786 0.940141i \(-0.610693\pi\)
−0.340786 + 0.940141i \(0.610693\pi\)
\(810\) 0 0
\(811\) −4.54256e47 −0.455466 −0.227733 0.973724i \(-0.573131\pi\)
−0.227733 + 0.973724i \(0.573131\pi\)
\(812\) − 1.23204e47i − 0.121046i
\(813\) − 9.61007e46i − 0.0925194i
\(814\) 8.00193e47 0.754904
\(815\) 0 0
\(816\) −6.43091e46 −0.0582619
\(817\) 2.43243e47i 0.215962i
\(818\) − 8.98736e47i − 0.781993i
\(819\) 7.00135e47 0.597032
\(820\) 0 0
\(821\) −3.88660e47 −0.318352 −0.159176 0.987250i \(-0.550884\pi\)
−0.159176 + 0.987250i \(0.550884\pi\)
\(822\) − 2.28069e47i − 0.183097i
\(823\) 2.16262e48i 1.70170i 0.525410 + 0.850849i \(0.323912\pi\)
−0.525410 + 0.850849i \(0.676088\pi\)
\(824\) −2.96236e47 −0.228475
\(825\) 0 0
\(826\) 1.13811e48 0.843358
\(827\) − 5.70292e47i − 0.414244i −0.978315 0.207122i \(-0.933590\pi\)
0.978315 0.207122i \(-0.0664097\pi\)
\(828\) 1.66457e47i 0.118523i
\(829\) −1.81254e48 −1.26514 −0.632569 0.774504i \(-0.717999\pi\)
−0.632569 + 0.774504i \(0.717999\pi\)
\(830\) 0 0
\(831\) 8.00417e46 0.0536907
\(832\) − 2.28723e47i − 0.150410i
\(833\) − 1.83113e48i − 1.18053i
\(834\) −6.34253e46 −0.0400888
\(835\) 0 0
\(836\) 1.61174e48 0.979242
\(837\) − 1.14084e48i − 0.679603i
\(838\) − 4.76578e47i − 0.278360i
\(839\) 1.88483e47 0.107945 0.0539723 0.998542i \(-0.482812\pi\)
0.0539723 + 0.998542i \(0.482812\pi\)
\(840\) 0 0
\(841\) −1.78894e48 −0.985057
\(842\) − 9.70766e47i − 0.524162i
\(843\) − 1.99283e47i − 0.105515i
\(844\) −1.26802e48 −0.658377
\(845\) 0 0
\(846\) −5.71870e47 −0.285553
\(847\) − 2.84999e48i − 1.39562i
\(848\) 5.86693e47i 0.281760i
\(849\) 5.95748e47 0.280599
\(850\) 0 0
\(851\) 3.81681e47 0.172927
\(852\) − 3.25762e47i − 0.144760i
\(853\) − 3.65000e48i − 1.59087i −0.606040 0.795434i \(-0.707243\pi\)
0.606040 0.795434i \(-0.292757\pi\)
\(854\) 2.53901e46 0.0108545
\(855\) 0 0
\(856\) 8.32507e47 0.342429
\(857\) − 1.58574e48i − 0.639807i −0.947450 0.319903i \(-0.896350\pi\)
0.947450 0.319903i \(-0.103650\pi\)
\(858\) − 1.79295e47i − 0.0709625i
\(859\) −2.88906e48 −1.12168 −0.560840 0.827924i \(-0.689522\pi\)
−0.560840 + 0.827924i \(0.689522\pi\)
\(860\) 0 0
\(861\) 9.41431e47 0.351752
\(862\) 1.99404e48i 0.730912i
\(863\) − 3.38580e46i − 0.0121754i −0.999981 0.00608770i \(-0.998062\pi\)
0.999981 0.00608770i \(-0.00193779\pi\)
\(864\) −1.19652e48 −0.422126
\(865\) 0 0
\(866\) −6.16750e47 −0.209442
\(867\) − 3.80791e47i − 0.126874i
\(868\) 5.05361e48i 1.65206i
\(869\) 6.37858e47 0.204596
\(870\) 0 0
\(871\) −5.23695e47 −0.161726
\(872\) − 4.00366e47i − 0.121321i
\(873\) − 4.10448e48i − 1.22046i
\(874\) −3.09933e47 −0.0904334
\(875\) 0 0
\(876\) 6.66567e47 0.187295
\(877\) − 4.24252e48i − 1.16985i −0.811087 0.584925i \(-0.801124\pi\)
0.811087 0.584925i \(-0.198876\pi\)
\(878\) 3.01576e48i 0.816089i
\(879\) 2.95454e47 0.0784646
\(880\) 0 0
\(881\) −2.17405e48 −0.556119 −0.278059 0.960564i \(-0.589691\pi\)
−0.278059 + 0.960564i \(0.589691\pi\)
\(882\) − 1.90069e48i − 0.477178i
\(883\) − 4.77705e48i − 1.17708i −0.808466 0.588542i \(-0.799702\pi\)
0.808466 0.588542i \(-0.200298\pi\)
\(884\) −1.67916e48 −0.406098
\(885\) 0 0
\(886\) 3.28527e48 0.765447
\(887\) − 5.12275e48i − 1.17156i −0.810471 0.585779i \(-0.800789\pi\)
0.810471 0.585779i \(-0.199211\pi\)
\(888\) 8.46907e47i 0.190117i
\(889\) −2.29955e48 −0.506714
\(890\) 0 0
\(891\) 5.81331e48 1.23436
\(892\) 2.76302e48i 0.575922i
\(893\) 2.64116e48i 0.540437i
\(894\) 4.07963e47 0.0819505
\(895\) 0 0
\(896\) 6.14883e48 1.19045
\(897\) − 8.55214e46i − 0.0162555i
\(898\) 7.67570e47i 0.143238i
\(899\) −1.11311e48 −0.203941
\(900\) 0 0
\(901\) −9.17434e48 −1.62038
\(902\) 5.32147e48i 0.922834i
\(903\) 3.78023e47i 0.0643681i
\(904\) 5.15770e48 0.862338
\(905\) 0 0
\(906\) 4.00007e47 0.0644840
\(907\) 7.71888e48i 1.22190i 0.791671 + 0.610948i \(0.209211\pi\)
−0.791671 + 0.610948i \(0.790789\pi\)
\(908\) − 1.48895e48i − 0.231453i
\(909\) 1.00300e49 1.53108
\(910\) 0 0
\(911\) −1.06274e49 −1.56450 −0.782250 0.622964i \(-0.785928\pi\)
−0.782250 + 0.622964i \(0.785928\pi\)
\(912\) 3.08292e47i 0.0445704i
\(913\) 2.65432e48i 0.376864i
\(914\) −3.53882e48 −0.493454
\(915\) 0 0
\(916\) −2.09012e48 −0.281123
\(917\) 1.54394e49i 2.03956i
\(918\) − 2.13495e48i − 0.277002i
\(919\) −8.89051e48 −1.13298 −0.566488 0.824070i \(-0.691698\pi\)
−0.566488 + 0.824070i \(0.691698\pi\)
\(920\) 0 0
\(921\) 2.31210e48 0.284264
\(922\) − 6.65723e48i − 0.803959i
\(923\) − 3.69428e48i − 0.438229i
\(924\) 2.50479e48 0.291866
\(925\) 0 0
\(926\) −2.74764e48 −0.308943
\(927\) 2.15552e48i 0.238088i
\(928\) 1.16743e48i 0.126675i
\(929\) 7.35883e48 0.784422 0.392211 0.919875i \(-0.371710\pi\)
0.392211 + 0.919875i \(0.371710\pi\)
\(930\) 0 0
\(931\) −8.77828e48 −0.903108
\(932\) 4.28292e47i 0.0432890i
\(933\) − 1.14542e47i − 0.0113741i
\(934\) 1.74771e48 0.170509
\(935\) 0 0
\(936\) −4.18858e48 −0.394471
\(937\) − 7.92924e48i − 0.733717i −0.930277 0.366858i \(-0.880433\pi\)
0.930277 0.366858i \(-0.119567\pi\)
\(938\) 2.94950e48i 0.268165i
\(939\) −4.08734e47 −0.0365140
\(940\) 0 0
\(941\) 1.44230e49 1.24402 0.622008 0.783011i \(-0.286317\pi\)
0.622008 + 0.783011i \(0.286317\pi\)
\(942\) − 1.01076e48i − 0.0856656i
\(943\) 2.53827e48i 0.211395i
\(944\) 3.05229e48 0.249797
\(945\) 0 0
\(946\) −2.13679e48 −0.168872
\(947\) 9.83625e48i 0.763932i 0.924176 + 0.381966i \(0.124753\pi\)
−0.924176 + 0.381966i \(0.875247\pi\)
\(948\) 2.80921e47i 0.0214411i
\(949\) 7.55913e48 0.566995
\(950\) 0 0
\(951\) −2.78198e48 −0.201546
\(952\) 2.27271e49i 1.61821i
\(953\) 8.19274e48i 0.573319i 0.958033 + 0.286660i \(0.0925449\pi\)
−0.958033 + 0.286660i \(0.907455\pi\)
\(954\) −9.52286e48 −0.654967
\(955\) 0 0
\(956\) 1.00107e49 0.665133
\(957\) 5.51707e47i 0.0360298i
\(958\) 1.44329e49i 0.926450i
\(959\) 3.61324e49 2.27976
\(960\) 0 0
\(961\) 2.92544e49 1.78343
\(962\) 3.99653e48i 0.239494i
\(963\) − 6.05763e48i − 0.356837i
\(964\) −2.23180e48 −0.129236
\(965\) 0 0
\(966\) −4.81665e47 −0.0269540
\(967\) − 1.65015e49i − 0.907791i −0.891055 0.453895i \(-0.850034\pi\)
0.891055 0.453895i \(-0.149966\pi\)
\(968\) 1.70501e49i 0.922113i
\(969\) −4.82088e48 −0.256321
\(970\) 0 0
\(971\) −3.28602e49 −1.68870 −0.844352 0.535789i \(-0.820014\pi\)
−0.844352 + 0.535789i \(0.820014\pi\)
\(972\) 8.30611e48i 0.419666i
\(973\) − 1.00483e49i − 0.499151i
\(974\) 7.23857e48 0.353533
\(975\) 0 0
\(976\) 6.80939e46 0.00321504
\(977\) 1.17177e49i 0.543977i 0.962300 + 0.271989i \(0.0876813\pi\)
−0.962300 + 0.271989i \(0.912319\pi\)
\(978\) 1.24552e48i 0.0568536i
\(979\) −1.52247e49 −0.683336
\(980\) 0 0
\(981\) −2.91321e48 −0.126425
\(982\) 7.15306e48i 0.305247i
\(983\) − 3.61572e49i − 1.51726i −0.651519 0.758632i \(-0.725868\pi\)
0.651519 0.758632i \(-0.274132\pi\)
\(984\) −5.63213e48 −0.232409
\(985\) 0 0
\(986\) −2.08305e48 −0.0831249
\(987\) 4.10461e48i 0.161079i
\(988\) 8.04975e48i 0.310665i
\(989\) −1.01922e48 −0.0386838
\(990\) 0 0
\(991\) 2.27623e49 0.835603 0.417802 0.908538i \(-0.362801\pi\)
0.417802 + 0.908538i \(0.362801\pi\)
\(992\) − 4.78860e49i − 1.72889i
\(993\) 9.56445e48i 0.339623i
\(994\) −2.08065e49 −0.726649
\(995\) 0 0
\(996\) −1.16900e48 −0.0394942
\(997\) − 5.01160e49i − 1.66535i −0.553761 0.832676i \(-0.686808\pi\)
0.553761 0.832676i \(-0.313192\pi\)
\(998\) − 8.75230e48i − 0.286067i
\(999\) 1.26040e49 0.405209
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 25.34.b.a.24.3 4
5.2 odd 4 25.34.a.a.1.1 2
5.3 odd 4 1.34.a.a.1.2 2
5.4 even 2 inner 25.34.b.a.24.2 4
15.8 even 4 9.34.a.b.1.1 2
20.3 even 4 16.34.a.b.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1.34.a.a.1.2 2 5.3 odd 4
9.34.a.b.1.1 2 15.8 even 4
16.34.a.b.1.2 2 20.3 even 4
25.34.a.a.1.1 2 5.2 odd 4
25.34.b.a.24.2 4 5.4 even 2 inner
25.34.b.a.24.3 4 1.1 even 1 trivial