Properties

Label 10.22.a.c.1.1
Level $10$
Weight $22$
Character 10.1
Self dual yes
Analytic conductor $27.948$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [10,22,Mod(1,10)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(10, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 22, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("10.1");
 
S:= CuspForms(chi, 22);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 10 = 2 \cdot 5 \)
Weight: \( k \) \(=\) \( 22 \)
Character orbit: \([\chi]\) \(=\) 10.a (trivial)

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: yes
Analytic conductor: \(27.9477344287\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{474529}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 118632 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2}\cdot 3\cdot 5\cdot 7 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(344.930\) of defining polynomial
Character \(\chi\) \(=\) 10.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1024.00 q^{2} -94506.7 q^{3} +1.04858e6 q^{4} +9.76562e6 q^{5} +9.67749e7 q^{6} +3.65434e8 q^{7} -1.07374e9 q^{8} -1.52883e9 q^{9} -1.00000e10 q^{10} -3.92190e10 q^{11} -9.90975e10 q^{12} -4.64598e11 q^{13} -3.74204e11 q^{14} -9.22917e11 q^{15} +1.09951e12 q^{16} +2.43124e11 q^{17} +1.56552e12 q^{18} -6.74889e12 q^{19} +1.02400e13 q^{20} -3.45360e13 q^{21} +4.01602e13 q^{22} -1.66102e14 q^{23} +1.01476e14 q^{24} +9.53674e13 q^{25} +4.75749e14 q^{26} +1.13306e15 q^{27} +3.83185e14 q^{28} +2.85150e15 q^{29} +9.45067e14 q^{30} +3.48202e15 q^{31} -1.12590e15 q^{32} +3.70646e15 q^{33} -2.48959e14 q^{34} +3.56869e15 q^{35} -1.60309e15 q^{36} +3.87028e15 q^{37} +6.91086e15 q^{38} +4.39077e16 q^{39} -1.04858e16 q^{40} +1.53913e17 q^{41} +3.53648e16 q^{42} -6.75910e15 q^{43} -4.11241e16 q^{44} -1.49300e16 q^{45} +1.70089e17 q^{46} -1.11128e17 q^{47} -1.03911e17 q^{48} -4.25004e17 q^{49} -9.76562e16 q^{50} -2.29769e16 q^{51} -4.87167e17 q^{52} -2.06153e18 q^{53} -1.16025e18 q^{54} -3.82998e17 q^{55} -3.92382e17 q^{56} +6.37815e17 q^{57} -2.91994e18 q^{58} +5.84178e18 q^{59} -9.67749e17 q^{60} +4.90762e18 q^{61} -3.56559e18 q^{62} -5.58686e17 q^{63} +1.15292e18 q^{64} -4.53709e18 q^{65} -3.79541e18 q^{66} +1.35893e19 q^{67} +2.54934e17 q^{68} +1.56978e19 q^{69} -3.65434e18 q^{70} +3.28855e19 q^{71} +1.64157e18 q^{72} +3.25004e19 q^{73} -3.96317e18 q^{74} -9.01286e18 q^{75} -7.07672e18 q^{76} -1.43319e19 q^{77} -4.49615e19 q^{78} +5.11527e19 q^{79} +1.07374e19 q^{80} -9.10896e19 q^{81} -1.57607e20 q^{82} +2.02190e20 q^{83} -3.62136e19 q^{84} +2.37426e18 q^{85} +6.92131e18 q^{86} -2.69486e20 q^{87} +4.21111e19 q^{88} +2.17189e20 q^{89} +1.52883e19 q^{90} -1.69780e20 q^{91} -1.74171e20 q^{92} -3.29075e20 q^{93} +1.13795e20 q^{94} -6.59071e19 q^{95} +1.06405e20 q^{96} -1.03286e21 q^{97} +4.35204e20 q^{98} +5.99592e19 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2048 q^{2} + 100308 q^{3} + 2097152 q^{4} + 19531250 q^{5} - 102715392 q^{6} + 1328895316 q^{7} - 2147483648 q^{8} + 25963598826 q^{9} - 20000000000 q^{10} + 21869194224 q^{11} + 105180561408 q^{12}+ \cdots + 17\!\cdots\!12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1024.00 −0.707107
\(3\) −94506.7 −0.924037 −0.462019 0.886870i \(-0.652875\pi\)
−0.462019 + 0.886870i \(0.652875\pi\)
\(4\) 1.04858e6 0.500000
\(5\) 9.76562e6 0.447214
\(6\) 9.67749e7 0.653393
\(7\) 3.65434e8 0.488967 0.244483 0.969653i \(-0.421382\pi\)
0.244483 + 0.969653i \(0.421382\pi\)
\(8\) −1.07374e9 −0.353553
\(9\) −1.52883e9 −0.146155
\(10\) −1.00000e10 −0.316228
\(11\) −3.92190e10 −0.455904 −0.227952 0.973672i \(-0.573203\pi\)
−0.227952 + 0.973672i \(0.573203\pi\)
\(12\) −9.90975e10 −0.462019
\(13\) −4.64598e11 −0.934701 −0.467350 0.884072i \(-0.654791\pi\)
−0.467350 + 0.884072i \(0.654791\pi\)
\(14\) −3.74204e11 −0.345752
\(15\) −9.22917e11 −0.413242
\(16\) 1.09951e12 0.250000
\(17\) 2.43124e11 0.0292492 0.0146246 0.999893i \(-0.495345\pi\)
0.0146246 + 0.999893i \(0.495345\pi\)
\(18\) 1.56552e12 0.103347
\(19\) −6.74889e12 −0.252534 −0.126267 0.991996i \(-0.540300\pi\)
−0.126267 + 0.991996i \(0.540300\pi\)
\(20\) 1.02400e13 0.223607
\(21\) −3.45360e13 −0.451824
\(22\) 4.01602e13 0.322373
\(23\) −1.66102e14 −0.836052 −0.418026 0.908435i \(-0.637278\pi\)
−0.418026 + 0.908435i \(0.637278\pi\)
\(24\) 1.01476e14 0.326697
\(25\) 9.53674e13 0.200000
\(26\) 4.75749e14 0.660933
\(27\) 1.13306e15 1.05909
\(28\) 3.83185e14 0.244483
\(29\) 2.85150e15 1.25862 0.629310 0.777155i \(-0.283338\pi\)
0.629310 + 0.777155i \(0.283338\pi\)
\(30\) 9.45067e14 0.292206
\(31\) 3.48202e15 0.763017 0.381508 0.924365i \(-0.375405\pi\)
0.381508 + 0.924365i \(0.375405\pi\)
\(32\) −1.12590e15 −0.176777
\(33\) 3.70646e15 0.421272
\(34\) −2.48959e14 −0.0206823
\(35\) 3.56869e15 0.218673
\(36\) −1.60309e15 −0.0730774
\(37\) 3.87028e15 0.132320 0.0661598 0.997809i \(-0.478925\pi\)
0.0661598 + 0.997809i \(0.478925\pi\)
\(38\) 6.91086e15 0.178568
\(39\) 4.39077e16 0.863698
\(40\) −1.04858e16 −0.158114
\(41\) 1.53913e17 1.79079 0.895396 0.445272i \(-0.146893\pi\)
0.895396 + 0.445272i \(0.146893\pi\)
\(42\) 3.53648e16 0.319488
\(43\) −6.75910e15 −0.0476947 −0.0238473 0.999716i \(-0.507592\pi\)
−0.0238473 + 0.999716i \(0.507592\pi\)
\(44\) −4.11241e16 −0.227952
\(45\) −1.49300e16 −0.0653624
\(46\) 1.70089e17 0.591178
\(47\) −1.11128e17 −0.308174 −0.154087 0.988057i \(-0.549244\pi\)
−0.154087 + 0.988057i \(0.549244\pi\)
\(48\) −1.03911e17 −0.231009
\(49\) −4.25004e17 −0.760911
\(50\) −9.76562e16 −0.141421
\(51\) −2.29769e16 −0.0270274
\(52\) −4.87167e17 −0.467350
\(53\) −2.06153e18 −1.61917 −0.809584 0.587003i \(-0.800307\pi\)
−0.809584 + 0.587003i \(0.800307\pi\)
\(54\) −1.16025e18 −0.748890
\(55\) −3.82998e17 −0.203886
\(56\) −3.92382e17 −0.172876
\(57\) 6.37815e17 0.233351
\(58\) −2.91994e18 −0.889978
\(59\) 5.84178e18 1.48799 0.743993 0.668188i \(-0.232930\pi\)
0.743993 + 0.668188i \(0.232930\pi\)
\(60\) −9.67749e17 −0.206621
\(61\) 4.90762e18 0.880861 0.440431 0.897787i \(-0.354826\pi\)
0.440431 + 0.897787i \(0.354826\pi\)
\(62\) −3.56559e18 −0.539534
\(63\) −5.58686e17 −0.0714648
\(64\) 1.15292e18 0.125000
\(65\) −4.53709e18 −0.418011
\(66\) −3.79541e18 −0.297884
\(67\) 1.35893e19 0.910776 0.455388 0.890293i \(-0.349500\pi\)
0.455388 + 0.890293i \(0.349500\pi\)
\(68\) 2.54934e17 0.0146246
\(69\) 1.56978e19 0.772543
\(70\) −3.65434e18 −0.154625
\(71\) 3.28855e19 1.19893 0.599463 0.800403i \(-0.295381\pi\)
0.599463 + 0.800403i \(0.295381\pi\)
\(72\) 1.64157e18 0.0516735
\(73\) 3.25004e19 0.885112 0.442556 0.896741i \(-0.354072\pi\)
0.442556 + 0.896741i \(0.354072\pi\)
\(74\) −3.96317e18 −0.0935641
\(75\) −9.01286e18 −0.184807
\(76\) −7.07672e18 −0.126267
\(77\) −1.43319e19 −0.222922
\(78\) −4.49615e19 −0.610727
\(79\) 5.11527e19 0.607832 0.303916 0.952699i \(-0.401706\pi\)
0.303916 + 0.952699i \(0.401706\pi\)
\(80\) 1.07374e19 0.111803
\(81\) −9.10896e19 −0.832484
\(82\) −1.57607e20 −1.26628
\(83\) 2.02190e20 1.43034 0.715171 0.698949i \(-0.246349\pi\)
0.715171 + 0.698949i \(0.246349\pi\)
\(84\) −3.62136e19 −0.225912
\(85\) 2.37426e18 0.0130807
\(86\) 6.92131e18 0.0337252
\(87\) −2.69486e20 −1.16301
\(88\) 4.21111e19 0.161186
\(89\) 2.17189e20 0.738316 0.369158 0.929367i \(-0.379646\pi\)
0.369158 + 0.929367i \(0.379646\pi\)
\(90\) 1.52883e19 0.0462182
\(91\) −1.69780e20 −0.457038
\(92\) −1.74171e20 −0.418026
\(93\) −3.29075e20 −0.705056
\(94\) 1.13795e20 0.217912
\(95\) −6.59071e19 −0.112937
\(96\) 1.06405e20 0.163348
\(97\) −1.03286e21 −1.42213 −0.711063 0.703129i \(-0.751786\pi\)
−0.711063 + 0.703129i \(0.751786\pi\)
\(98\) 4.35204e20 0.538046
\(99\) 5.99592e19 0.0666325
\(100\) 1.00000e20 0.100000
\(101\) −9.83683e20 −0.886096 −0.443048 0.896498i \(-0.646103\pi\)
−0.443048 + 0.896498i \(0.646103\pi\)
\(102\) 2.35283e19 0.0191113
\(103\) 1.00540e21 0.737140 0.368570 0.929600i \(-0.379848\pi\)
0.368570 + 0.929600i \(0.379848\pi\)
\(104\) 4.98859e20 0.330467
\(105\) −3.37265e20 −0.202062
\(106\) 2.11100e21 1.14493
\(107\) 3.28755e21 1.61563 0.807817 0.589433i \(-0.200649\pi\)
0.807817 + 0.589433i \(0.200649\pi\)
\(108\) 1.18810e21 0.529545
\(109\) 2.52264e21 1.02065 0.510324 0.859982i \(-0.329525\pi\)
0.510324 + 0.859982i \(0.329525\pi\)
\(110\) 3.92190e20 0.144169
\(111\) −3.65767e20 −0.122268
\(112\) 4.01799e20 0.122242
\(113\) −2.85082e21 −0.790034 −0.395017 0.918674i \(-0.629261\pi\)
−0.395017 + 0.918674i \(0.629261\pi\)
\(114\) −6.53123e20 −0.165004
\(115\) −1.62209e21 −0.373894
\(116\) 2.99001e21 0.629310
\(117\) 7.10292e20 0.136611
\(118\) −5.98198e21 −1.05216
\(119\) 8.88458e19 0.0143019
\(120\) 9.90975e20 0.146103
\(121\) −5.86212e21 −0.792152
\(122\) −5.02540e21 −0.622863
\(123\) −1.45458e22 −1.65476
\(124\) 3.65117e21 0.381508
\(125\) 9.31323e20 0.0894427
\(126\) 5.72095e20 0.0505332
\(127\) 1.38451e21 0.112553 0.0562764 0.998415i \(-0.482077\pi\)
0.0562764 + 0.998415i \(0.482077\pi\)
\(128\) −1.18059e21 −0.0883883
\(129\) 6.38780e20 0.0440717
\(130\) 4.64598e21 0.295578
\(131\) −6.11455e21 −0.358935 −0.179468 0.983764i \(-0.557438\pi\)
−0.179468 + 0.983764i \(0.557438\pi\)
\(132\) 3.88650e21 0.210636
\(133\) −2.46627e21 −0.123481
\(134\) −1.39154e22 −0.644016
\(135\) 1.10650e22 0.473639
\(136\) −2.61053e20 −0.0103412
\(137\) 3.06697e22 1.12498 0.562488 0.826806i \(-0.309844\pi\)
0.562488 + 0.826806i \(0.309844\pi\)
\(138\) −1.60745e22 −0.546271
\(139\) 2.77909e21 0.0875483 0.0437742 0.999041i \(-0.486062\pi\)
0.0437742 + 0.999041i \(0.486062\pi\)
\(140\) 3.74204e21 0.109336
\(141\) 1.05024e22 0.284765
\(142\) −3.36748e22 −0.847768
\(143\) 1.82211e22 0.426134
\(144\) −1.68097e21 −0.0365387
\(145\) 2.78467e22 0.562872
\(146\) −3.32804e22 −0.625868
\(147\) 4.01657e22 0.703111
\(148\) 4.05828e21 0.0661598
\(149\) 1.15060e23 1.74771 0.873854 0.486188i \(-0.161613\pi\)
0.873854 + 0.486188i \(0.161613\pi\)
\(150\) 9.22917e21 0.130679
\(151\) −9.22398e22 −1.21804 −0.609019 0.793156i \(-0.708437\pi\)
−0.609019 + 0.793156i \(0.708437\pi\)
\(152\) 7.24656e21 0.0892842
\(153\) −3.71696e20 −0.00427491
\(154\) 1.46759e22 0.157630
\(155\) 3.40041e22 0.341231
\(156\) 4.60405e22 0.431849
\(157\) −9.83630e22 −0.862751 −0.431375 0.902173i \(-0.641971\pi\)
−0.431375 + 0.902173i \(0.641971\pi\)
\(158\) −5.23803e22 −0.429802
\(159\) 1.94828e23 1.49617
\(160\) −1.09951e22 −0.0790569
\(161\) −6.06994e22 −0.408802
\(162\) 9.32757e22 0.588655
\(163\) −2.88242e23 −1.70525 −0.852623 0.522526i \(-0.824990\pi\)
−0.852623 + 0.522526i \(0.824990\pi\)
\(164\) 1.61390e23 0.895396
\(165\) 3.61959e22 0.188399
\(166\) −2.07043e23 −1.01140
\(167\) −1.80201e23 −0.826484 −0.413242 0.910621i \(-0.635604\pi\)
−0.413242 + 0.910621i \(0.635604\pi\)
\(168\) 3.70827e22 0.159744
\(169\) −3.12128e22 −0.126335
\(170\) −2.43124e21 −0.00924942
\(171\) 1.03179e22 0.0369090
\(172\) −7.08743e21 −0.0238473
\(173\) −4.53364e23 −1.43537 −0.717683 0.696370i \(-0.754797\pi\)
−0.717683 + 0.696370i \(0.754797\pi\)
\(174\) 2.75954e23 0.822373
\(175\) 3.48505e22 0.0977934
\(176\) −4.31217e22 −0.113976
\(177\) −5.52087e23 −1.37495
\(178\) −2.22401e23 −0.522068
\(179\) 6.58352e23 1.45714 0.728570 0.684971i \(-0.240185\pi\)
0.728570 + 0.684971i \(0.240185\pi\)
\(180\) −1.56552e22 −0.0326812
\(181\) 1.55219e23 0.305717 0.152858 0.988248i \(-0.451152\pi\)
0.152858 + 0.988248i \(0.451152\pi\)
\(182\) 1.73855e23 0.323174
\(183\) −4.63803e23 −0.813949
\(184\) 1.78351e23 0.295589
\(185\) 3.77957e22 0.0591751
\(186\) 3.36973e23 0.498550
\(187\) −9.53509e21 −0.0133348
\(188\) −1.16526e23 −0.154087
\(189\) 4.14058e23 0.517860
\(190\) 6.74889e22 0.0798582
\(191\) 6.36717e23 0.713010 0.356505 0.934293i \(-0.383968\pi\)
0.356505 + 0.934293i \(0.383968\pi\)
\(192\) −1.08959e23 −0.115505
\(193\) −4.28896e23 −0.430527 −0.215263 0.976556i \(-0.569061\pi\)
−0.215263 + 0.976556i \(0.569061\pi\)
\(194\) 1.05765e24 1.00559
\(195\) 4.28786e23 0.386258
\(196\) −4.45649e23 −0.380456
\(197\) −2.72735e23 −0.220722 −0.110361 0.993892i \(-0.535201\pi\)
−0.110361 + 0.993892i \(0.535201\pi\)
\(198\) −6.13982e22 −0.0471163
\(199\) 2.24441e24 1.63360 0.816799 0.576923i \(-0.195747\pi\)
0.816799 + 0.576923i \(0.195747\pi\)
\(200\) −1.02400e23 −0.0707107
\(201\) −1.28428e24 −0.841591
\(202\) 1.00729e24 0.626565
\(203\) 1.04203e24 0.615423
\(204\) −2.40930e22 −0.0135137
\(205\) 1.50306e24 0.800866
\(206\) −1.02953e24 −0.521236
\(207\) 2.53942e23 0.122193
\(208\) −5.10831e23 −0.233675
\(209\) 2.64685e23 0.115131
\(210\) 3.45360e23 0.142879
\(211\) −3.61254e24 −1.42183 −0.710913 0.703280i \(-0.751718\pi\)
−0.710913 + 0.703280i \(0.751718\pi\)
\(212\) −2.16167e24 −0.809584
\(213\) −3.10790e24 −1.10785
\(214\) −3.36646e24 −1.14243
\(215\) −6.60068e22 −0.0213297
\(216\) −1.21661e24 −0.374445
\(217\) 1.27245e24 0.373090
\(218\) −2.58318e24 −0.721708
\(219\) −3.07150e24 −0.817876
\(220\) −4.01602e23 −0.101943
\(221\) −1.12955e23 −0.0273393
\(222\) 3.74546e23 0.0864567
\(223\) 4.19001e24 0.922600 0.461300 0.887244i \(-0.347383\pi\)
0.461300 + 0.887244i \(0.347383\pi\)
\(224\) −4.11442e23 −0.0864379
\(225\) −1.45801e23 −0.0292309
\(226\) 2.91924e24 0.558638
\(227\) 7.86051e24 1.43608 0.718041 0.696001i \(-0.245039\pi\)
0.718041 + 0.696001i \(0.245039\pi\)
\(228\) 6.68798e23 0.116675
\(229\) 3.29155e23 0.0548439 0.0274220 0.999624i \(-0.491270\pi\)
0.0274220 + 0.999624i \(0.491270\pi\)
\(230\) 1.66102e24 0.264383
\(231\) 1.35447e24 0.205988
\(232\) −3.06177e24 −0.444989
\(233\) 3.62417e24 0.503468 0.251734 0.967796i \(-0.418999\pi\)
0.251734 + 0.967796i \(0.418999\pi\)
\(234\) −7.27339e23 −0.0965985
\(235\) −1.08524e24 −0.137820
\(236\) 6.12555e24 0.743993
\(237\) −4.83427e24 −0.561660
\(238\) −9.09781e22 −0.0101130
\(239\) −5.27026e23 −0.0560601 −0.0280301 0.999607i \(-0.508923\pi\)
−0.0280301 + 0.999607i \(0.508923\pi\)
\(240\) −1.01476e24 −0.103311
\(241\) −3.17826e24 −0.309749 −0.154874 0.987934i \(-0.549497\pi\)
−0.154874 + 0.987934i \(0.549497\pi\)
\(242\) 6.00281e24 0.560136
\(243\) −3.24361e24 −0.289843
\(244\) 5.14601e24 0.440431
\(245\) −4.15043e24 −0.340290
\(246\) 1.48949e25 1.17009
\(247\) 3.13552e24 0.236043
\(248\) −3.73880e24 −0.269767
\(249\) −1.91083e25 −1.32169
\(250\) −9.53674e23 −0.0632456
\(251\) 1.26683e25 0.805649 0.402824 0.915277i \(-0.368029\pi\)
0.402824 + 0.915277i \(0.368029\pi\)
\(252\) −5.85825e23 −0.0357324
\(253\) 6.51436e24 0.381159
\(254\) −1.41774e24 −0.0795868
\(255\) −2.24384e23 −0.0120870
\(256\) 1.20893e24 0.0625000
\(257\) −1.86982e24 −0.0927902 −0.0463951 0.998923i \(-0.514773\pi\)
−0.0463951 + 0.998923i \(0.514773\pi\)
\(258\) −6.54111e23 −0.0311634
\(259\) 1.41433e24 0.0646999
\(260\) −4.75749e24 −0.209005
\(261\) −4.35946e24 −0.183953
\(262\) 6.26130e24 0.253806
\(263\) 4.09180e25 1.59360 0.796800 0.604243i \(-0.206524\pi\)
0.796800 + 0.604243i \(0.206524\pi\)
\(264\) −3.97978e24 −0.148942
\(265\) −2.01321e25 −0.724114
\(266\) 2.52546e24 0.0873140
\(267\) −2.05258e25 −0.682232
\(268\) 1.42494e25 0.455388
\(269\) −3.93628e25 −1.20973 −0.604864 0.796329i \(-0.706772\pi\)
−0.604864 + 0.796329i \(0.706772\pi\)
\(270\) −1.13306e25 −0.334914
\(271\) −2.31984e25 −0.659600 −0.329800 0.944051i \(-0.606981\pi\)
−0.329800 + 0.944051i \(0.606981\pi\)
\(272\) 2.67318e23 0.00731231
\(273\) 1.60454e25 0.422320
\(274\) −3.14057e25 −0.795478
\(275\) −3.74021e24 −0.0911808
\(276\) 1.64603e25 0.386272
\(277\) −6.39590e25 −1.44499 −0.722494 0.691378i \(-0.757004\pi\)
−0.722494 + 0.691378i \(0.757004\pi\)
\(278\) −2.84579e24 −0.0619060
\(279\) −5.32342e24 −0.111518
\(280\) −3.83185e24 −0.0773124
\(281\) −4.18321e25 −0.813005 −0.406503 0.913650i \(-0.633252\pi\)
−0.406503 + 0.913650i \(0.633252\pi\)
\(282\) −1.07544e25 −0.201359
\(283\) 7.89427e25 1.42415 0.712073 0.702106i \(-0.247757\pi\)
0.712073 + 0.702106i \(0.247757\pi\)
\(284\) 3.44830e25 0.599463
\(285\) 6.22867e24 0.104358
\(286\) −1.86584e25 −0.301322
\(287\) 5.62450e25 0.875638
\(288\) 1.72131e24 0.0258367
\(289\) −6.90328e25 −0.999144
\(290\) −2.85150e25 −0.398010
\(291\) 9.76121e25 1.31410
\(292\) 3.40791e25 0.442556
\(293\) 1.30571e26 1.63583 0.817913 0.575343i \(-0.195131\pi\)
0.817913 + 0.575343i \(0.195131\pi\)
\(294\) −4.11297e25 −0.497174
\(295\) 5.70486e25 0.665447
\(296\) −4.15568e24 −0.0467820
\(297\) −4.44374e25 −0.482843
\(298\) −1.17822e26 −1.23582
\(299\) 7.71709e25 0.781458
\(300\) −9.45067e24 −0.0924037
\(301\) −2.47000e24 −0.0233211
\(302\) 9.44536e25 0.861282
\(303\) 9.29647e25 0.818786
\(304\) −7.42048e24 −0.0631334
\(305\) 4.79259e25 0.393933
\(306\) 3.80616e23 0.00302282
\(307\) 3.70803e25 0.284571 0.142285 0.989826i \(-0.454555\pi\)
0.142285 + 0.989826i \(0.454555\pi\)
\(308\) −1.50281e25 −0.111461
\(309\) −9.50175e25 −0.681145
\(310\) −3.48202e25 −0.241287
\(311\) 1.83082e26 1.22648 0.613242 0.789895i \(-0.289865\pi\)
0.613242 + 0.789895i \(0.289865\pi\)
\(312\) −4.71455e25 −0.305363
\(313\) −2.37945e26 −1.49026 −0.745129 0.666921i \(-0.767612\pi\)
−0.745129 + 0.666921i \(0.767612\pi\)
\(314\) 1.00724e26 0.610057
\(315\) −5.45592e24 −0.0319600
\(316\) 5.36375e25 0.303916
\(317\) 1.60193e26 0.878055 0.439027 0.898474i \(-0.355323\pi\)
0.439027 + 0.898474i \(0.355323\pi\)
\(318\) −1.99504e26 −1.05795
\(319\) −1.11833e26 −0.573809
\(320\) 1.12590e25 0.0559017
\(321\) −3.10696e26 −1.49291
\(322\) 6.21562e25 0.289066
\(323\) −1.64082e24 −0.00738642
\(324\) −9.55143e25 −0.416242
\(325\) −4.43076e25 −0.186940
\(326\) 2.95160e26 1.20579
\(327\) −2.38406e26 −0.943118
\(328\) −1.65263e26 −0.633140
\(329\) −4.06100e25 −0.150687
\(330\) −3.70646e25 −0.133218
\(331\) −5.15992e26 −1.79659 −0.898295 0.439393i \(-0.855193\pi\)
−0.898295 + 0.439393i \(0.855193\pi\)
\(332\) 2.12012e26 0.715171
\(333\) −5.91700e24 −0.0193391
\(334\) 1.84526e26 0.584413
\(335\) 1.32708e26 0.407312
\(336\) −3.79727e25 −0.112956
\(337\) −2.13485e26 −0.615535 −0.307768 0.951462i \(-0.599582\pi\)
−0.307768 + 0.951462i \(0.599582\pi\)
\(338\) 3.19620e25 0.0893322
\(339\) 2.69421e26 0.730021
\(340\) 2.48959e24 0.00654033
\(341\) −1.36561e26 −0.347862
\(342\) −1.05655e25 −0.0260986
\(343\) −3.59422e26 −0.861027
\(344\) 7.25752e24 0.0168626
\(345\) 1.53299e26 0.345492
\(346\) 4.64244e26 1.01496
\(347\) 2.02825e26 0.430191 0.215095 0.976593i \(-0.430994\pi\)
0.215095 + 0.976593i \(0.430994\pi\)
\(348\) −2.82577e26 −0.581506
\(349\) 7.65093e26 1.52773 0.763866 0.645375i \(-0.223299\pi\)
0.763866 + 0.645375i \(0.223299\pi\)
\(350\) −3.56869e25 −0.0691504
\(351\) −5.26417e26 −0.989932
\(352\) 4.41567e25 0.0805932
\(353\) 5.36101e26 0.949756 0.474878 0.880052i \(-0.342492\pi\)
0.474878 + 0.880052i \(0.342492\pi\)
\(354\) 5.65337e26 0.972240
\(355\) 3.21148e26 0.536176
\(356\) 2.27739e26 0.369158
\(357\) −8.39653e24 −0.0132155
\(358\) −6.74153e26 −1.03035
\(359\) 9.97471e26 1.48050 0.740250 0.672332i \(-0.234707\pi\)
0.740250 + 0.672332i \(0.234707\pi\)
\(360\) 1.60309e25 0.0231091
\(361\) −6.68662e26 −0.936227
\(362\) −1.58944e26 −0.216175
\(363\) 5.54010e26 0.731978
\(364\) −1.78027e26 −0.228519
\(365\) 3.17386e26 0.395834
\(366\) 4.74934e26 0.575549
\(367\) −7.27262e26 −0.856441 −0.428220 0.903674i \(-0.640859\pi\)
−0.428220 + 0.903674i \(0.640859\pi\)
\(368\) −1.82631e26 −0.209013
\(369\) −2.35307e26 −0.261733
\(370\) −3.87028e25 −0.0418431
\(371\) −7.53351e26 −0.791720
\(372\) −3.45060e26 −0.352528
\(373\) 8.47344e26 0.841622 0.420811 0.907148i \(-0.361746\pi\)
0.420811 + 0.907148i \(0.361746\pi\)
\(374\) 9.76393e24 0.00942915
\(375\) −8.80163e25 −0.0826484
\(376\) 1.19323e26 0.108956
\(377\) −1.32480e27 −1.17643
\(378\) −4.23995e26 −0.366182
\(379\) 9.08041e26 0.762771 0.381385 0.924416i \(-0.375447\pi\)
0.381385 + 0.924416i \(0.375447\pi\)
\(380\) −6.91086e25 −0.0564683
\(381\) −1.30845e26 −0.104003
\(382\) −6.51998e26 −0.504175
\(383\) −3.75246e25 −0.0282312 −0.0141156 0.999900i \(-0.504493\pi\)
−0.0141156 + 0.999900i \(0.504493\pi\)
\(384\) 1.11574e26 0.0816741
\(385\) −1.39960e26 −0.0996937
\(386\) 4.39189e26 0.304428
\(387\) 1.03335e25 0.00697080
\(388\) −1.08303e27 −0.711063
\(389\) −5.30842e26 −0.339230 −0.169615 0.985510i \(-0.554252\pi\)
−0.169615 + 0.985510i \(0.554252\pi\)
\(390\) −4.39077e26 −0.273125
\(391\) −4.03835e25 −0.0244539
\(392\) 4.56344e26 0.269023
\(393\) 5.77867e26 0.331670
\(394\) 2.79281e26 0.156074
\(395\) 4.99538e26 0.271831
\(396\) 6.28717e25 0.0333162
\(397\) 1.80269e27 0.930297 0.465149 0.885233i \(-0.346001\pi\)
0.465149 + 0.885233i \(0.346001\pi\)
\(398\) −2.29828e27 −1.15513
\(399\) 2.33079e26 0.114101
\(400\) 1.04858e26 0.0500000
\(401\) 1.75068e27 0.813187 0.406594 0.913609i \(-0.366717\pi\)
0.406594 + 0.913609i \(0.366717\pi\)
\(402\) 1.31510e27 0.595095
\(403\) −1.61774e27 −0.713192
\(404\) −1.03147e27 −0.443048
\(405\) −8.89547e26 −0.372298
\(406\) −1.06704e27 −0.435170
\(407\) −1.51788e26 −0.0603250
\(408\) 2.46712e25 0.00955563
\(409\) −1.99836e27 −0.754362 −0.377181 0.926140i \(-0.623107\pi\)
−0.377181 + 0.926140i \(0.623107\pi\)
\(410\) −1.53913e27 −0.566298
\(411\) −2.89849e27 −1.03952
\(412\) 1.05424e27 0.368570
\(413\) 2.13478e27 0.727576
\(414\) −2.60037e26 −0.0864035
\(415\) 1.97451e27 0.639669
\(416\) 5.23091e26 0.165233
\(417\) −2.62643e26 −0.0808979
\(418\) −2.71037e26 −0.0814100
\(419\) −2.81050e27 −0.823259 −0.411630 0.911351i \(-0.635040\pi\)
−0.411630 + 0.911351i \(0.635040\pi\)
\(420\) −3.53648e26 −0.101031
\(421\) 4.66949e27 1.30109 0.650546 0.759467i \(-0.274540\pi\)
0.650546 + 0.759467i \(0.274540\pi\)
\(422\) 3.69924e27 1.00538
\(423\) 1.69896e26 0.0450411
\(424\) 2.21355e27 0.572463
\(425\) 2.31861e25 0.00584985
\(426\) 3.18249e27 0.783370
\(427\) 1.79341e27 0.430712
\(428\) 3.44725e27 0.807817
\(429\) −1.72201e27 −0.393763
\(430\) 6.75910e25 0.0150824
\(431\) −6.66126e26 −0.145059 −0.0725296 0.997366i \(-0.523107\pi\)
−0.0725296 + 0.997366i \(0.523107\pi\)
\(432\) 1.24581e27 0.264772
\(433\) 8.02347e27 1.66433 0.832165 0.554528i \(-0.187101\pi\)
0.832165 + 0.554528i \(0.187101\pi\)
\(434\) −1.30299e27 −0.263814
\(435\) −2.63170e27 −0.520115
\(436\) 2.64518e27 0.510324
\(437\) 1.12101e27 0.211131
\(438\) 3.14522e27 0.578326
\(439\) −6.00429e27 −1.07791 −0.538957 0.842333i \(-0.681182\pi\)
−0.538957 + 0.842333i \(0.681182\pi\)
\(440\) 4.11241e26 0.0720847
\(441\) 6.49759e26 0.111211
\(442\) 1.15666e26 0.0193318
\(443\) −2.47404e27 −0.403802 −0.201901 0.979406i \(-0.564712\pi\)
−0.201901 + 0.979406i \(0.564712\pi\)
\(444\) −3.83535e26 −0.0611341
\(445\) 2.12098e27 0.330185
\(446\) −4.29057e27 −0.652377
\(447\) −1.08740e28 −1.61495
\(448\) 4.21317e26 0.0611209
\(449\) −1.39880e28 −1.98230 −0.991151 0.132738i \(-0.957623\pi\)
−0.991151 + 0.132738i \(0.957623\pi\)
\(450\) 1.49300e26 0.0206694
\(451\) −6.03631e27 −0.816429
\(452\) −2.98930e27 −0.395017
\(453\) 8.71729e27 1.12551
\(454\) −8.04916e27 −1.01546
\(455\) −1.65801e27 −0.204393
\(456\) −6.84849e26 −0.0825019
\(457\) 5.38517e27 0.633985 0.316993 0.948428i \(-0.397327\pi\)
0.316993 + 0.948428i \(0.397327\pi\)
\(458\) −3.37055e26 −0.0387805
\(459\) 2.75474e26 0.0309776
\(460\) −1.70089e27 −0.186947
\(461\) 4.10176e27 0.440667 0.220334 0.975425i \(-0.429285\pi\)
0.220334 + 0.975425i \(0.429285\pi\)
\(462\) −1.38697e27 −0.145656
\(463\) −1.59610e27 −0.163855 −0.0819276 0.996638i \(-0.526108\pi\)
−0.0819276 + 0.996638i \(0.526108\pi\)
\(464\) 3.13526e27 0.314655
\(465\) −3.21362e27 −0.315311
\(466\) −3.71115e27 −0.356006
\(467\) −5.00681e27 −0.469607 −0.234803 0.972043i \(-0.575445\pi\)
−0.234803 + 0.972043i \(0.575445\pi\)
\(468\) 7.44795e26 0.0683054
\(469\) 4.96599e27 0.445339
\(470\) 1.11128e27 0.0974532
\(471\) 9.29596e27 0.797214
\(472\) −6.27256e27 −0.526082
\(473\) 2.65085e26 0.0217442
\(474\) 4.95029e27 0.397153
\(475\) −6.43624e26 −0.0505067
\(476\) 9.31616e25 0.00715095
\(477\) 3.15172e27 0.236649
\(478\) 5.39674e26 0.0396405
\(479\) 4.44191e27 0.319188 0.159594 0.987183i \(-0.448981\pi\)
0.159594 + 0.987183i \(0.448981\pi\)
\(480\) 1.03911e27 0.0730516
\(481\) −1.79813e27 −0.123679
\(482\) 3.25453e27 0.219026
\(483\) 5.73650e27 0.377748
\(484\) −6.14688e27 −0.396076
\(485\) −1.00865e28 −0.635994
\(486\) 3.32146e27 0.204950
\(487\) 1.46905e28 0.887119 0.443559 0.896245i \(-0.353716\pi\)
0.443559 + 0.896245i \(0.353716\pi\)
\(488\) −5.26951e27 −0.311431
\(489\) 2.72408e28 1.57571
\(490\) 4.25004e27 0.240621
\(491\) −2.20895e27 −0.122414 −0.0612069 0.998125i \(-0.519495\pi\)
−0.0612069 + 0.998125i \(0.519495\pi\)
\(492\) −1.52524e28 −0.827379
\(493\) 6.93269e26 0.0368137
\(494\) −3.21078e27 −0.166908
\(495\) 5.85539e26 0.0297990
\(496\) 3.82853e27 0.190754
\(497\) 1.20175e28 0.586235
\(498\) 1.95669e28 0.934576
\(499\) 3.64836e28 1.70625 0.853125 0.521707i \(-0.174705\pi\)
0.853125 + 0.521707i \(0.174705\pi\)
\(500\) 9.76563e26 0.0447214
\(501\) 1.70302e28 0.763702
\(502\) −1.29724e28 −0.569680
\(503\) −4.00705e28 −1.72330 −0.861651 0.507502i \(-0.830569\pi\)
−0.861651 + 0.507502i \(0.830569\pi\)
\(504\) 5.99885e26 0.0252666
\(505\) −9.60628e27 −0.396274
\(506\) −6.67071e27 −0.269520
\(507\) 2.94982e27 0.116738
\(508\) 1.45176e27 0.0562764
\(509\) −4.63213e28 −1.75891 −0.879456 0.475979i \(-0.842094\pi\)
−0.879456 + 0.475979i \(0.842094\pi\)
\(510\) 2.29769e26 0.00854681
\(511\) 1.18767e28 0.432790
\(512\) −1.23794e27 −0.0441942
\(513\) −7.64689e27 −0.267456
\(514\) 1.91470e27 0.0656126
\(515\) 9.81840e27 0.329659
\(516\) 6.69810e26 0.0220358
\(517\) 4.35833e27 0.140498
\(518\) −1.44828e27 −0.0457497
\(519\) 4.28459e28 1.32633
\(520\) 4.87167e27 0.147789
\(521\) −1.41568e28 −0.420891 −0.210445 0.977606i \(-0.567491\pi\)
−0.210445 + 0.977606i \(0.567491\pi\)
\(522\) 4.46409e27 0.130075
\(523\) −5.46471e28 −1.56063 −0.780314 0.625388i \(-0.784941\pi\)
−0.780314 + 0.625388i \(0.784941\pi\)
\(524\) −6.41158e27 −0.179468
\(525\) −3.29361e27 −0.0903647
\(526\) −4.19001e28 −1.12685
\(527\) 8.46564e26 0.0223177
\(528\) 4.07529e27 0.105318
\(529\) −1.18816e28 −0.301017
\(530\) 2.06153e28 0.512026
\(531\) −8.93108e27 −0.217476
\(532\) −2.58607e27 −0.0617403
\(533\) −7.15078e28 −1.67385
\(534\) 2.10184e28 0.482411
\(535\) 3.21050e28 0.722534
\(536\) −1.45914e28 −0.322008
\(537\) −6.22187e28 −1.34645
\(538\) 4.03076e28 0.855406
\(539\) 1.66682e28 0.346902
\(540\) 1.16025e28 0.236820
\(541\) −1.97332e28 −0.395026 −0.197513 0.980300i \(-0.563286\pi\)
−0.197513 + 0.980300i \(0.563286\pi\)
\(542\) 2.37552e28 0.466408
\(543\) −1.46692e28 −0.282494
\(544\) −2.73734e26 −0.00517058
\(545\) 2.46351e28 0.456448
\(546\) −1.64304e28 −0.298625
\(547\) 4.31761e28 0.769797 0.384899 0.922959i \(-0.374236\pi\)
0.384899 + 0.922959i \(0.374236\pi\)
\(548\) 3.21595e28 0.562488
\(549\) −7.50291e27 −0.128742
\(550\) 3.82998e27 0.0644745
\(551\) −1.92445e28 −0.317844
\(552\) −1.68554e28 −0.273135
\(553\) 1.86929e28 0.297210
\(554\) 6.54940e28 1.02176
\(555\) −3.57195e27 −0.0546800
\(556\) 2.91409e27 0.0437742
\(557\) 3.91712e27 0.0577414 0.0288707 0.999583i \(-0.490809\pi\)
0.0288707 + 0.999583i \(0.490809\pi\)
\(558\) 5.45119e27 0.0788555
\(559\) 3.14027e27 0.0445802
\(560\) 3.92382e27 0.0546682
\(561\) 9.01130e26 0.0123219
\(562\) 4.28361e28 0.574881
\(563\) −2.70683e28 −0.356551 −0.178276 0.983981i \(-0.557052\pi\)
−0.178276 + 0.983981i \(0.557052\pi\)
\(564\) 1.10125e28 0.142382
\(565\) −2.78400e28 −0.353314
\(566\) −8.08373e28 −1.00702
\(567\) −3.32872e28 −0.407057
\(568\) −3.53106e28 −0.423884
\(569\) 4.56161e28 0.537575 0.268788 0.963199i \(-0.413377\pi\)
0.268788 + 0.963199i \(0.413377\pi\)
\(570\) −6.37815e27 −0.0737920
\(571\) 1.31174e29 1.48994 0.744968 0.667100i \(-0.232465\pi\)
0.744968 + 0.667100i \(0.232465\pi\)
\(572\) 1.91062e28 0.213067
\(573\) −6.01740e28 −0.658848
\(574\) −5.75949e28 −0.619169
\(575\) −1.58407e28 −0.167210
\(576\) −1.76262e27 −0.0182693
\(577\) 1.42017e29 1.44542 0.722710 0.691151i \(-0.242896\pi\)
0.722710 + 0.691151i \(0.242896\pi\)
\(578\) 7.06896e28 0.706502
\(579\) 4.05336e28 0.397823
\(580\) 2.91994e28 0.281436
\(581\) 7.38871e28 0.699390
\(582\) −9.99548e28 −0.929207
\(583\) 8.08510e28 0.738185
\(584\) −3.48970e28 −0.312934
\(585\) 6.93644e27 0.0610942
\(586\) −1.33705e29 −1.15670
\(587\) 9.99241e28 0.849122 0.424561 0.905399i \(-0.360428\pi\)
0.424561 + 0.905399i \(0.360428\pi\)
\(588\) 4.21168e28 0.351555
\(589\) −2.34998e28 −0.192687
\(590\) −5.84178e28 −0.470542
\(591\) 2.57753e28 0.203955
\(592\) 4.25542e27 0.0330799
\(593\) −1.01433e29 −0.774647 −0.387323 0.921944i \(-0.626600\pi\)
−0.387323 + 0.921944i \(0.626600\pi\)
\(594\) 4.55039e28 0.341422
\(595\) 8.67635e26 0.00639601
\(596\) 1.20649e29 0.873854
\(597\) −2.12112e29 −1.50950
\(598\) −7.90230e28 −0.552574
\(599\) 5.05093e28 0.347048 0.173524 0.984830i \(-0.444485\pi\)
0.173524 + 0.984830i \(0.444485\pi\)
\(600\) 9.67749e27 0.0653393
\(601\) 1.29341e29 0.858130 0.429065 0.903274i \(-0.358843\pi\)
0.429065 + 0.903274i \(0.358843\pi\)
\(602\) 2.52928e27 0.0164905
\(603\) −2.07757e28 −0.133114
\(604\) −9.67205e28 −0.609019
\(605\) −5.72473e28 −0.354261
\(606\) −9.51958e28 −0.578969
\(607\) −8.14047e28 −0.486596 −0.243298 0.969952i \(-0.578229\pi\)
−0.243298 + 0.969952i \(0.578229\pi\)
\(608\) 7.59857e27 0.0446421
\(609\) −9.84793e28 −0.568674
\(610\) −4.90762e28 −0.278553
\(611\) 5.16299e28 0.288051
\(612\) −3.89751e26 −0.00213746
\(613\) 2.20870e28 0.119070 0.0595349 0.998226i \(-0.481038\pi\)
0.0595349 + 0.998226i \(0.481038\pi\)
\(614\) −3.79702e28 −0.201222
\(615\) −1.42049e29 −0.740030
\(616\) 1.53888e28 0.0788148
\(617\) −1.09793e29 −0.552818 −0.276409 0.961040i \(-0.589144\pi\)
−0.276409 + 0.961040i \(0.589144\pi\)
\(618\) 9.72979e28 0.481642
\(619\) −1.54602e29 −0.752424 −0.376212 0.926534i \(-0.622774\pi\)
−0.376212 + 0.926534i \(0.622774\pi\)
\(620\) 3.56559e28 0.170616
\(621\) −1.88204e29 −0.885454
\(622\) −1.87476e29 −0.867255
\(623\) 7.93682e28 0.361012
\(624\) 4.82770e28 0.215925
\(625\) 9.09495e27 0.0400000
\(626\) 2.43656e29 1.05377
\(627\) −2.50145e28 −0.106385
\(628\) −1.03141e29 −0.431375
\(629\) 9.40958e26 0.00387025
\(630\) 5.58686e27 0.0225992
\(631\) −1.10811e29 −0.440832 −0.220416 0.975406i \(-0.570742\pi\)
−0.220416 + 0.975406i \(0.570742\pi\)
\(632\) −5.49248e28 −0.214901
\(633\) 3.41409e29 1.31382
\(634\) −1.64038e29 −0.620878
\(635\) 1.35206e28 0.0503351
\(636\) 2.04292e29 0.748086
\(637\) 1.97456e29 0.711224
\(638\) 1.14517e29 0.405744
\(639\) −5.02764e28 −0.175229
\(640\) −1.15292e28 −0.0395285
\(641\) 1.41424e29 0.476995 0.238497 0.971143i \(-0.423345\pi\)
0.238497 + 0.971143i \(0.423345\pi\)
\(642\) 3.18153e29 1.05564
\(643\) 5.97065e29 1.94897 0.974487 0.224442i \(-0.0720559\pi\)
0.974487 + 0.224442i \(0.0720559\pi\)
\(644\) −6.36479e28 −0.204401
\(645\) 6.23809e27 0.0197094
\(646\) 1.68020e27 0.00522299
\(647\) 4.89100e29 1.49590 0.747951 0.663754i \(-0.231038\pi\)
0.747951 + 0.663754i \(0.231038\pi\)
\(648\) 9.78067e28 0.294328
\(649\) −2.29109e29 −0.678378
\(650\) 4.53709e28 0.132187
\(651\) −1.20255e29 −0.344749
\(652\) −3.02244e29 −0.852623
\(653\) 2.34337e29 0.650508 0.325254 0.945627i \(-0.394550\pi\)
0.325254 + 0.945627i \(0.394550\pi\)
\(654\) 2.44128e29 0.666885
\(655\) −5.97124e28 −0.160521
\(656\) 1.69229e29 0.447698
\(657\) −4.96875e28 −0.129363
\(658\) 4.15846e28 0.106552
\(659\) −4.06286e29 −1.02455 −0.512277 0.858820i \(-0.671198\pi\)
−0.512277 + 0.858820i \(0.671198\pi\)
\(660\) 3.79541e28 0.0941993
\(661\) −3.10306e29 −0.758009 −0.379005 0.925395i \(-0.623734\pi\)
−0.379005 + 0.925395i \(0.623734\pi\)
\(662\) 5.28376e29 1.27038
\(663\) 1.06750e28 0.0252625
\(664\) −2.17100e29 −0.505702
\(665\) −2.40847e28 −0.0552222
\(666\) 6.05901e27 0.0136748
\(667\) −4.73641e29 −1.05227
\(668\) −1.88955e29 −0.413242
\(669\) −3.95984e29 −0.852517
\(670\) −1.35893e29 −0.288013
\(671\) −1.92472e29 −0.401588
\(672\) 3.88840e28 0.0798719
\(673\) −1.94077e29 −0.392478 −0.196239 0.980556i \(-0.562873\pi\)
−0.196239 + 0.980556i \(0.562873\pi\)
\(674\) 2.18608e29 0.435249
\(675\) 1.08057e29 0.211818
\(676\) −3.27290e28 −0.0631674
\(677\) 2.34383e29 0.445395 0.222697 0.974888i \(-0.428514\pi\)
0.222697 + 0.974888i \(0.428514\pi\)
\(678\) −2.75887e29 −0.516203
\(679\) −3.77442e29 −0.695372
\(680\) −2.54934e27 −0.00462471
\(681\) −7.42871e29 −1.32699
\(682\) 1.39839e29 0.245976
\(683\) −3.67410e29 −0.636405 −0.318202 0.948023i \(-0.603079\pi\)
−0.318202 + 0.948023i \(0.603079\pi\)
\(684\) 1.08191e28 0.0184545
\(685\) 2.99508e29 0.503104
\(686\) 3.68049e29 0.608838
\(687\) −3.11074e28 −0.0506778
\(688\) −7.43171e27 −0.0119237
\(689\) 9.57782e29 1.51344
\(690\) −1.56978e29 −0.244300
\(691\) 9.60338e27 0.0147199 0.00735994 0.999973i \(-0.497657\pi\)
0.00735994 + 0.999973i \(0.497657\pi\)
\(692\) −4.75386e29 −0.717683
\(693\) 2.19111e28 0.0325811
\(694\) −2.07692e29 −0.304191
\(695\) 2.71396e28 0.0391528
\(696\) 2.89358e29 0.411187
\(697\) 3.74200e28 0.0523793
\(698\) −7.83455e29 −1.08027
\(699\) −3.42509e29 −0.465223
\(700\) 3.65434e28 0.0488967
\(701\) 9.60216e29 1.26570 0.632849 0.774276i \(-0.281886\pi\)
0.632849 + 0.774276i \(0.281886\pi\)
\(702\) 5.39051e29 0.699988
\(703\) −2.61201e28 −0.0334152
\(704\) −4.52164e28 −0.0569880
\(705\) 1.02562e29 0.127351
\(706\) −5.48967e29 −0.671579
\(707\) −3.59471e29 −0.433272
\(708\) −5.78905e29 −0.687477
\(709\) 1.47754e30 1.72884 0.864418 0.502775i \(-0.167687\pi\)
0.864418 + 0.502775i \(0.167687\pi\)
\(710\) −3.28855e29 −0.379133
\(711\) −7.82037e28 −0.0888376
\(712\) −2.33205e29 −0.261034
\(713\) −5.78372e29 −0.637922
\(714\) 8.59805e27 0.00934477
\(715\) 1.77940e29 0.190573
\(716\) 6.90332e29 0.728570
\(717\) 4.98075e28 0.0518016
\(718\) −1.02141e30 −1.04687
\(719\) −5.91742e29 −0.597695 −0.298847 0.954301i \(-0.596602\pi\)
−0.298847 + 0.954301i \(0.596602\pi\)
\(720\) −1.64157e28 −0.0163406
\(721\) 3.67409e29 0.360437
\(722\) 6.84710e29 0.662012
\(723\) 3.00367e29 0.286220
\(724\) 1.62759e29 0.152858
\(725\) 2.71940e29 0.251724
\(726\) −5.67306e29 −0.517587
\(727\) −6.05957e29 −0.544917 −0.272459 0.962168i \(-0.587837\pi\)
−0.272459 + 0.962168i \(0.587837\pi\)
\(728\) 1.82300e29 0.161587
\(729\) 1.25937e30 1.10031
\(730\) −3.25004e29 −0.279897
\(731\) −1.64330e27 −0.00139503
\(732\) −4.86333e29 −0.406974
\(733\) 2.10206e30 1.73402 0.867010 0.498291i \(-0.166039\pi\)
0.867010 + 0.498291i \(0.166039\pi\)
\(734\) 7.44717e29 0.605595
\(735\) 3.92243e29 0.314441
\(736\) 1.87015e29 0.147795
\(737\) −5.32959e29 −0.415226
\(738\) 2.40954e29 0.185073
\(739\) 9.22542e29 0.698585 0.349293 0.937014i \(-0.386422\pi\)
0.349293 + 0.937014i \(0.386422\pi\)
\(740\) 3.96317e28 0.0295876
\(741\) −2.96328e29 −0.218113
\(742\) 7.71432e29 0.559831
\(743\) −6.11017e29 −0.437191 −0.218595 0.975816i \(-0.570147\pi\)
−0.218595 + 0.975816i \(0.570147\pi\)
\(744\) 3.53341e29 0.249275
\(745\) 1.12363e30 0.781599
\(746\) −8.67680e29 −0.595116
\(747\) −3.09114e29 −0.209051
\(748\) −9.99826e27 −0.00666742
\(749\) 1.20138e30 0.789992
\(750\) 9.01286e28 0.0584413
\(751\) −1.36857e30 −0.875078 −0.437539 0.899199i \(-0.644150\pi\)
−0.437539 + 0.899199i \(0.644150\pi\)
\(752\) −1.22187e29 −0.0770436
\(753\) −1.19724e30 −0.744450
\(754\) 1.35660e30 0.831863
\(755\) −9.00780e29 −0.544723
\(756\) 4.34171e29 0.258930
\(757\) −5.97591e29 −0.351477 −0.175739 0.984437i \(-0.556231\pi\)
−0.175739 + 0.984437i \(0.556231\pi\)
\(758\) −9.29834e29 −0.539360
\(759\) −6.15651e29 −0.352205
\(760\) 7.07672e28 0.0399291
\(761\) −2.76098e30 −1.53647 −0.768236 0.640166i \(-0.778865\pi\)
−0.768236 + 0.640166i \(0.778865\pi\)
\(762\) 1.33986e29 0.0735412
\(763\) 9.21857e29 0.499064
\(764\) 6.67646e29 0.356505
\(765\) −3.62984e27 −0.00191180
\(766\) 3.84252e28 0.0199625
\(767\) −2.71408e30 −1.39082
\(768\) −1.14252e29 −0.0577523
\(769\) 2.46811e30 1.23066 0.615330 0.788269i \(-0.289023\pi\)
0.615330 + 0.788269i \(0.289023\pi\)
\(770\) 1.43319e29 0.0704941
\(771\) 1.76711e29 0.0857416
\(772\) −4.49730e29 −0.215263
\(773\) −2.37027e30 −1.11921 −0.559606 0.828759i \(-0.689048\pi\)
−0.559606 + 0.828759i \(0.689048\pi\)
\(774\) −1.05815e28 −0.00492910
\(775\) 3.32072e29 0.152603
\(776\) 1.10902e30 0.502797
\(777\) −1.33664e29 −0.0597851
\(778\) 5.43582e29 0.239872
\(779\) −1.03874e30 −0.452235
\(780\) 4.49615e29 0.193129
\(781\) −1.28974e30 −0.546595
\(782\) 4.13527e28 0.0172915
\(783\) 3.23092e30 1.33299
\(784\) −4.67297e29 −0.190228
\(785\) −9.60576e29 −0.385834
\(786\) −5.91735e29 −0.234526
\(787\) 2.43834e30 0.953585 0.476792 0.879016i \(-0.341799\pi\)
0.476792 + 0.879016i \(0.341799\pi\)
\(788\) −2.85983e29 −0.110361
\(789\) −3.86703e30 −1.47255
\(790\) −5.11527e29 −0.192213
\(791\) −1.04178e30 −0.386300
\(792\) −6.43807e28 −0.0235581
\(793\) −2.28007e30 −0.823341
\(794\) −1.84596e30 −0.657820
\(795\) 1.90262e30 0.669109
\(796\) 2.35343e30 0.816799
\(797\) −1.54506e29 −0.0529215 −0.0264608 0.999650i \(-0.508424\pi\)
−0.0264608 + 0.999650i \(0.508424\pi\)
\(798\) −2.38673e29 −0.0806814
\(799\) −2.70179e28 −0.00901386
\(800\) −1.07374e29 −0.0353553
\(801\) −3.32045e29 −0.107908
\(802\) −1.79269e30 −0.575010
\(803\) −1.27463e30 −0.403526
\(804\) −1.34667e30 −0.420796
\(805\) −5.92768e29 −0.182822
\(806\) 1.65657e30 0.504303
\(807\) 3.72005e30 1.11783
\(808\) 1.05622e30 0.313282
\(809\) −2.18140e30 −0.638668 −0.319334 0.947642i \(-0.603459\pi\)
−0.319334 + 0.947642i \(0.603459\pi\)
\(810\) 9.10896e29 0.263255
\(811\) 2.64431e30 0.754387 0.377193 0.926134i \(-0.376889\pi\)
0.377193 + 0.926134i \(0.376889\pi\)
\(812\) 1.09265e30 0.307712
\(813\) 2.19241e30 0.609495
\(814\) 1.55431e29 0.0426562
\(815\) −2.81486e30 −0.762610
\(816\) −2.52633e28 −0.00675685
\(817\) 4.56164e28 0.0120445
\(818\) 2.04632e30 0.533414
\(819\) 2.59565e29 0.0667982
\(820\) 1.57607e30 0.400433
\(821\) 4.66260e30 1.16957 0.584784 0.811189i \(-0.301179\pi\)
0.584784 + 0.811189i \(0.301179\pi\)
\(822\) 2.96805e30 0.735051
\(823\) −2.92987e30 −0.716392 −0.358196 0.933646i \(-0.616608\pi\)
−0.358196 + 0.933646i \(0.616608\pi\)
\(824\) −1.07954e30 −0.260618
\(825\) 3.53475e29 0.0842544
\(826\) −2.18602e30 −0.514474
\(827\) 5.72515e30 1.33039 0.665195 0.746669i \(-0.268348\pi\)
0.665195 + 0.746669i \(0.268348\pi\)
\(828\) 2.66278e29 0.0610965
\(829\) −7.48372e29 −0.169549 −0.0847745 0.996400i \(-0.527017\pi\)
−0.0847745 + 0.996400i \(0.527017\pi\)
\(830\) −2.02190e30 −0.452314
\(831\) 6.04456e30 1.33522
\(832\) −5.35645e29 −0.116838
\(833\) −1.03329e29 −0.0222561
\(834\) 2.68947e29 0.0572035
\(835\) −1.75978e30 −0.369615
\(836\) 2.77542e29 0.0575655
\(837\) 3.94534e30 0.808103
\(838\) 2.87795e30 0.582132
\(839\) 2.16889e30 0.433249 0.216625 0.976255i \(-0.430495\pi\)
0.216625 + 0.976255i \(0.430495\pi\)
\(840\) 3.62136e29 0.0714396
\(841\) 2.99821e30 0.584123
\(842\) −4.78156e30 −0.920010
\(843\) 3.95342e30 0.751247
\(844\) −3.78802e30 −0.710913
\(845\) −3.04813e29 −0.0564986
\(846\) −1.73973e29 −0.0318489
\(847\) −2.14222e30 −0.387336
\(848\) −2.26667e30 −0.404792
\(849\) −7.46062e30 −1.31596
\(850\) −2.37426e28 −0.00413647
\(851\) −6.42862e29 −0.110626
\(852\) −3.25887e30 −0.553926
\(853\) 4.43176e30 0.744065 0.372033 0.928220i \(-0.378661\pi\)
0.372033 + 0.928220i \(0.378661\pi\)
\(854\) −1.83645e30 −0.304559
\(855\) 1.00761e29 0.0165062
\(856\) −3.52998e30 −0.571213
\(857\) −1.50790e30 −0.241031 −0.120515 0.992711i \(-0.538455\pi\)
−0.120515 + 0.992711i \(0.538455\pi\)
\(858\) 1.76334e30 0.278433
\(859\) 1.77910e30 0.277506 0.138753 0.990327i \(-0.455691\pi\)
0.138753 + 0.990327i \(0.455691\pi\)
\(860\) −6.92131e28 −0.0106649
\(861\) −5.31554e30 −0.809122
\(862\) 6.82113e29 0.102572
\(863\) −5.78269e30 −0.859047 −0.429523 0.903056i \(-0.641318\pi\)
−0.429523 + 0.903056i \(0.641318\pi\)
\(864\) −1.27571e30 −0.187222
\(865\) −4.42738e30 −0.641915
\(866\) −8.21604e30 −1.17686
\(867\) 6.52407e30 0.923247
\(868\) 1.33426e30 0.186545
\(869\) −2.00616e30 −0.277113
\(870\) 2.69486e30 0.367777
\(871\) −6.31357e30 −0.851303
\(872\) −2.70866e30 −0.360854
\(873\) 1.57907e30 0.207850
\(874\) −1.14791e30 −0.149292
\(875\) 3.40337e29 0.0437345
\(876\) −3.22070e30 −0.408938
\(877\) −1.01800e31 −1.27718 −0.638589 0.769548i \(-0.720482\pi\)
−0.638589 + 0.769548i \(0.720482\pi\)
\(878\) 6.14839e30 0.762201
\(879\) −1.23398e31 −1.51156
\(880\) −4.21111e29 −0.0509716
\(881\) 1.30842e31 1.56495 0.782473 0.622685i \(-0.213958\pi\)
0.782473 + 0.622685i \(0.213958\pi\)
\(882\) −6.65353e29 −0.0786379
\(883\) 6.52955e30 0.762598 0.381299 0.924452i \(-0.375477\pi\)
0.381299 + 0.924452i \(0.375477\pi\)
\(884\) −1.18442e29 −0.0136696
\(885\) −5.39148e30 −0.614898
\(886\) 2.53342e30 0.285531
\(887\) −1.68400e30 −0.187562 −0.0937810 0.995593i \(-0.529895\pi\)
−0.0937810 + 0.995593i \(0.529895\pi\)
\(888\) 3.92740e29 0.0432284
\(889\) 5.05946e29 0.0550346
\(890\) −2.17189e30 −0.233476
\(891\) 3.57244e30 0.379533
\(892\) 4.39354e30 0.461300
\(893\) 7.49991e29 0.0778244
\(894\) 1.11349e31 1.14194
\(895\) 6.42922e30 0.651653
\(896\) −4.31428e29 −0.0432190
\(897\) −7.29317e30 −0.722097
\(898\) 1.43237e31 1.40170
\(899\) 9.92899e30 0.960348
\(900\) −1.52883e29 −0.0146155
\(901\) −5.01207e29 −0.0473594
\(902\) 6.18119e30 0.577302
\(903\) 2.33432e29 0.0215496
\(904\) 3.06104e30 0.279319
\(905\) 1.51581e30 0.136721
\(906\) −8.92650e30 −0.795857
\(907\) −6.67368e30 −0.588151 −0.294076 0.955782i \(-0.595012\pi\)
−0.294076 + 0.955782i \(0.595012\pi\)
\(908\) 8.24234e30 0.718041
\(909\) 1.50388e30 0.129507
\(910\) 1.69780e30 0.144528
\(911\) 1.76568e31 1.48583 0.742915 0.669385i \(-0.233442\pi\)
0.742915 + 0.669385i \(0.233442\pi\)
\(912\) 7.01285e29 0.0583377
\(913\) −7.92969e30 −0.652098
\(914\) −5.51441e30 −0.448295
\(915\) −4.52932e30 −0.364009
\(916\) 3.45144e29 0.0274220
\(917\) −2.23447e30 −0.175507
\(918\) −2.82085e29 −0.0219044
\(919\) −2.65243e30 −0.203625 −0.101812 0.994804i \(-0.532464\pi\)
−0.101812 + 0.994804i \(0.532464\pi\)
\(920\) 1.74171e30 0.132191
\(921\) −3.50434e30 −0.262954
\(922\) −4.20020e30 −0.311599
\(923\) −1.52786e31 −1.12064
\(924\) 1.42026e30 0.102994
\(925\) 3.69099e29 0.0264639
\(926\) 1.63441e30 0.115863
\(927\) −1.53709e30 −0.107736
\(928\) −3.21050e30 −0.222495
\(929\) 1.93439e31 1.32550 0.662748 0.748842i \(-0.269390\pi\)
0.662748 + 0.748842i \(0.269390\pi\)
\(930\) 3.29075e30 0.222958
\(931\) 2.86830e30 0.192156
\(932\) 3.80022e30 0.251734
\(933\) −1.73025e31 −1.13332
\(934\) 5.12698e30 0.332062
\(935\) −9.31161e28 −0.00596352
\(936\) −7.62670e29 −0.0482992
\(937\) 5.66629e30 0.354841 0.177420 0.984135i \(-0.443225\pi\)
0.177420 + 0.984135i \(0.443225\pi\)
\(938\) −5.08518e30 −0.314903
\(939\) 2.24874e31 1.37705
\(940\) −1.13795e30 −0.0689098
\(941\) 5.09478e30 0.305095 0.152547 0.988296i \(-0.451252\pi\)
0.152547 + 0.988296i \(0.451252\pi\)
\(942\) −9.51907e30 −0.563715
\(943\) −2.55653e31 −1.49719
\(944\) 6.42310e30 0.371996
\(945\) 4.04354e30 0.231594
\(946\) −2.71447e29 −0.0153755
\(947\) 1.88265e31 1.05462 0.527310 0.849673i \(-0.323201\pi\)
0.527310 + 0.849673i \(0.323201\pi\)
\(948\) −5.06910e30 −0.280830
\(949\) −1.50996e31 −0.827314
\(950\) 6.59071e29 0.0357137
\(951\) −1.51393e31 −0.811355
\(952\) −9.53975e28 −0.00505649
\(953\) −1.70882e29 −0.00895823 −0.00447911 0.999990i \(-0.501426\pi\)
−0.00447911 + 0.999990i \(0.501426\pi\)
\(954\) −3.22736e30 −0.167336
\(955\) 6.21794e30 0.318868
\(956\) −5.52627e29 −0.0280301
\(957\) 1.05690e31 0.530221
\(958\) −4.54852e30 −0.225700
\(959\) 1.12077e31 0.550076
\(960\) −1.06405e30 −0.0516553
\(961\) −8.70101e30 −0.417806
\(962\) 1.84128e30 0.0874544
\(963\) −5.02611e30 −0.236133
\(964\) −3.33264e30 −0.154874
\(965\) −4.18844e30 −0.192537
\(966\) −5.87418e30 −0.267108
\(967\) −4.87103e30 −0.219100 −0.109550 0.993981i \(-0.534941\pi\)
−0.109550 + 0.993981i \(0.534941\pi\)
\(968\) 6.29440e30 0.280068
\(969\) 1.55068e29 0.00682533
\(970\) 1.03286e31 0.449715
\(971\) −1.12704e31 −0.485443 −0.242722 0.970096i \(-0.578040\pi\)
−0.242722 + 0.970096i \(0.578040\pi\)
\(972\) −3.40118e30 −0.144922
\(973\) 1.01558e30 0.0428082
\(974\) −1.50430e31 −0.627288
\(975\) 4.18736e30 0.172740
\(976\) 5.39598e30 0.220215
\(977\) 1.66556e31 0.672462 0.336231 0.941780i \(-0.390848\pi\)
0.336231 + 0.941780i \(0.390848\pi\)
\(978\) −2.78946e31 −1.11420
\(979\) −8.51793e30 −0.336601
\(980\) −4.35204e30 −0.170145
\(981\) −3.85668e30 −0.149173
\(982\) 2.26196e30 0.0865596
\(983\) −3.01316e31 −1.14080 −0.570402 0.821366i \(-0.693213\pi\)
−0.570402 + 0.821366i \(0.693213\pi\)
\(984\) 1.56185e31 0.585045
\(985\) −2.66343e30 −0.0987098
\(986\) −7.09907e29 −0.0260312
\(987\) 3.83792e30 0.139240
\(988\) 3.28783e30 0.118022
\(989\) 1.12270e30 0.0398752
\(990\) −5.99592e29 −0.0210710
\(991\) −3.83614e31 −1.33389 −0.666946 0.745106i \(-0.732399\pi\)
−0.666946 + 0.745106i \(0.732399\pi\)
\(992\) −3.92041e30 −0.134884
\(993\) 4.87647e31 1.66012
\(994\) −1.23059e31 −0.414531
\(995\) 2.19181e31 0.730567
\(996\) −2.00365e31 −0.660845
\(997\) −9.24504e30 −0.301724 −0.150862 0.988555i \(-0.548205\pi\)
−0.150862 + 0.988555i \(0.548205\pi\)
\(998\) −3.73592e31 −1.20650
\(999\) 4.38525e30 0.140138
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 10.22.a.c.1.1 2
4.3 odd 2 80.22.a.b.1.2 2
5.2 odd 4 50.22.b.d.49.2 4
5.3 odd 4 50.22.b.d.49.3 4
5.4 even 2 50.22.a.e.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
10.22.a.c.1.1 2 1.1 even 1 trivial
50.22.a.e.1.2 2 5.4 even 2
50.22.b.d.49.2 4 5.2 odd 4
50.22.b.d.49.3 4 5.3 odd 4
80.22.a.b.1.2 2 4.3 odd 2