Properties

Label 98.10.a.h.1.1
Level $98$
Weight $10$
Character 98.1
Self dual yes
Analytic conductor $50.474$
Analytic rank $0$
Dimension $3$
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] = [98,10,Mod(1,98)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(98, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("98.1");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 98 = 2 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 98.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: \(50.4735119441\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: \(\mathbb{Q}[x]/(x^{3} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 4037x + 70980 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{3}\cdot 7 \)
Twist minimal: no (minimal twist has level 14)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(19.2603\) of defining polynomial
Character \(\chi\) \(=\) 98.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-16.0000 q^{2} -179.327 q^{3} +256.000 q^{4} +168.288 q^{5} +2869.24 q^{6} -4096.00 q^{8} +12475.3 q^{9} +O(q^{10})\) \(q-16.0000 q^{2} -179.327 q^{3} +256.000 q^{4} +168.288 q^{5} +2869.24 q^{6} -4096.00 q^{8} +12475.3 q^{9} -2692.61 q^{10} +14493.8 q^{11} -45907.8 q^{12} +109153. q^{13} -30178.7 q^{15} +65536.0 q^{16} -300416. q^{17} -199605. q^{18} +491792. q^{19} +43081.8 q^{20} -231901. q^{22} -2.27046e6 q^{23} +734525. q^{24} -1.92480e6 q^{25} -1.74645e6 q^{26} +1.29253e6 q^{27} -3.80273e6 q^{29} +482859. q^{30} -7.90737e6 q^{31} -1.04858e6 q^{32} -2.59913e6 q^{33} +4.80666e6 q^{34} +3.19368e6 q^{36} +1.41551e7 q^{37} -7.86867e6 q^{38} -1.95742e7 q^{39} -689309. q^{40} -1.11075e7 q^{41} +8.50738e6 q^{43} +3.71041e6 q^{44} +2.09945e6 q^{45} +3.63274e7 q^{46} +4.37456e7 q^{47} -1.17524e7 q^{48} +3.07969e7 q^{50} +5.38729e7 q^{51} +2.79433e7 q^{52} -5.51883e7 q^{53} -2.06805e7 q^{54} +2.43914e6 q^{55} -8.81917e7 q^{57} +6.08436e7 q^{58} +4.90092e7 q^{59} -7.72575e6 q^{60} +1.41890e8 q^{61} +1.26518e8 q^{62} +1.67772e7 q^{64} +1.83692e7 q^{65} +4.15862e7 q^{66} -9.87576e7 q^{67} -7.69066e7 q^{68} +4.07156e8 q^{69} +2.96024e8 q^{71} -5.10989e7 q^{72} -4.60393e7 q^{73} -2.26482e8 q^{74} +3.45170e8 q^{75} +1.25899e8 q^{76} +3.13187e8 q^{78} -5.52570e8 q^{79} +1.10289e7 q^{80} -4.77339e8 q^{81} +1.77721e8 q^{82} +5.43217e8 q^{83} -5.05566e7 q^{85} -1.36118e8 q^{86} +6.81933e8 q^{87} -5.93666e7 q^{88} -5.08661e8 q^{89} -3.35912e7 q^{90} -5.81238e8 q^{92} +1.41801e9 q^{93} -6.99929e8 q^{94} +8.27628e7 q^{95} +1.88038e8 q^{96} +1.18179e9 q^{97} +1.80815e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 48 q^{2} + 71 q^{3} + 768 q^{4} - 1085 q^{5} - 1136 q^{6} - 12288 q^{8} + 9106 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 48 q^{2} + 71 q^{3} + 768 q^{4} - 1085 q^{5} - 1136 q^{6} - 12288 q^{8} + 9106 q^{9} + 17360 q^{10} - 2555 q^{11} + 18176 q^{12} - 18070 q^{13} - 353073 q^{15} + 196608 q^{16} + 20759 q^{17} - 145696 q^{18} + 1220649 q^{19} - 277760 q^{20} + 40880 q^{22} + 1960903 q^{23} - 290816 q^{24} + 863572 q^{25} + 289120 q^{26} - 2888659 q^{27} - 2606146 q^{29} + 5649168 q^{30} - 9377989 q^{31} - 3145728 q^{32} - 10778103 q^{33} - 332144 q^{34} + 2331136 q^{36} + 25814913 q^{37} - 19530384 q^{38} - 37990822 q^{39} + 4444160 q^{40} - 209250 q^{41} + 2944308 q^{43} - 654080 q^{44} - 37350558 q^{45} - 31374448 q^{46} + 48391269 q^{47} + 4653056 q^{48} - 13817152 q^{50} + 75441987 q^{51} - 4625920 q^{52} - 102186411 q^{53} + 46218544 q^{54} + 228278701 q^{55} + 43084589 q^{57} + 41698336 q^{58} + 144220135 q^{59} - 90386688 q^{60} + 280936871 q^{61} + 150047824 q^{62} + 50331648 q^{64} + 186819738 q^{65} + 172449648 q^{66} - 170710399 q^{67} + 5314304 q^{68} + 903654171 q^{69} + 469758688 q^{71} - 37298176 q^{72} + 613838539 q^{73} - 413038608 q^{74} + 902254676 q^{75} + 312486144 q^{76} + 607853152 q^{78} - 197445809 q^{79} - 71106560 q^{80} - 887872901 q^{81} + 3348000 q^{82} + 1074181436 q^{83} + 411272519 q^{85} - 47108928 q^{86} + 753428670 q^{87} + 10465280 q^{88} + 805730427 q^{89} + 597608928 q^{90} + 501991168 q^{92} + 1721516327 q^{93} - 774260304 q^{94} - 1799421743 q^{95} - 74448896 q^{96} + 2262918094 q^{97} - 1303712634 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −16.0000 −0.707107
\(3\) −179.327 −1.27821 −0.639103 0.769121i \(-0.720694\pi\)
−0.639103 + 0.769121i \(0.720694\pi\)
\(4\) 256.000 0.500000
\(5\) 168.288 0.120417 0.0602087 0.998186i \(-0.480823\pi\)
0.0602087 + 0.998186i \(0.480823\pi\)
\(6\) 2869.24 0.903829
\(7\) 0 0
\(8\) −4096.00 −0.353553
\(9\) 12475.3 0.633812
\(10\) −2692.61 −0.0851479
\(11\) 14493.8 0.298480 0.149240 0.988801i \(-0.452317\pi\)
0.149240 + 0.988801i \(0.452317\pi\)
\(12\) −45907.8 −0.639103
\(13\) 109153. 1.05997 0.529983 0.848008i \(-0.322198\pi\)
0.529983 + 0.848008i \(0.322198\pi\)
\(14\) 0 0
\(15\) −30178.7 −0.153918
\(16\) 65536.0 0.250000
\(17\) −300416. −0.872375 −0.436188 0.899856i \(-0.643672\pi\)
−0.436188 + 0.899856i \(0.643672\pi\)
\(18\) −199605. −0.448173
\(19\) 491792. 0.865745 0.432872 0.901455i \(-0.357500\pi\)
0.432872 + 0.901455i \(0.357500\pi\)
\(20\) 43081.8 0.0602087
\(21\) 0 0
\(22\) −231901. −0.211057
\(23\) −2.27046e6 −1.69176 −0.845880 0.533373i \(-0.820924\pi\)
−0.845880 + 0.533373i \(0.820924\pi\)
\(24\) 734525. 0.451914
\(25\) −1.92480e6 −0.985500
\(26\) −1.74645e6 −0.749509
\(27\) 1.29253e6 0.468064
\(28\) 0 0
\(29\) −3.80273e6 −0.998399 −0.499199 0.866487i \(-0.666372\pi\)
−0.499199 + 0.866487i \(0.666372\pi\)
\(30\) 482859. 0.108837
\(31\) −7.90737e6 −1.53782 −0.768908 0.639359i \(-0.779200\pi\)
−0.768908 + 0.639359i \(0.779200\pi\)
\(32\) −1.04858e6 −0.176777
\(33\) −2.59913e6 −0.381519
\(34\) 4.80666e6 0.616863
\(35\) 0 0
\(36\) 3.19368e6 0.316906
\(37\) 1.41551e7 1.24167 0.620835 0.783942i \(-0.286794\pi\)
0.620835 + 0.783942i \(0.286794\pi\)
\(38\) −7.86867e6 −0.612174
\(39\) −1.95742e7 −1.35486
\(40\) −689309. −0.0425740
\(41\) −1.11075e7 −0.613890 −0.306945 0.951727i \(-0.599307\pi\)
−0.306945 + 0.951727i \(0.599307\pi\)
\(42\) 0 0
\(43\) 8.50738e6 0.379479 0.189740 0.981834i \(-0.439236\pi\)
0.189740 + 0.981834i \(0.439236\pi\)
\(44\) 3.71041e6 0.149240
\(45\) 2.09945e6 0.0763220
\(46\) 3.63274e7 1.19625
\(47\) 4.37456e7 1.30766 0.653828 0.756643i \(-0.273162\pi\)
0.653828 + 0.756643i \(0.273162\pi\)
\(48\) −1.17524e7 −0.319552
\(49\) 0 0
\(50\) 3.07969e7 0.696853
\(51\) 5.38729e7 1.11508
\(52\) 2.79433e7 0.529983
\(53\) −5.51883e7 −0.960740 −0.480370 0.877066i \(-0.659498\pi\)
−0.480370 + 0.877066i \(0.659498\pi\)
\(54\) −2.06805e7 −0.330971
\(55\) 2.43914e6 0.0359421
\(56\) 0 0
\(57\) −8.81917e7 −1.10660
\(58\) 6.08436e7 0.705975
\(59\) 4.90092e7 0.526554 0.263277 0.964720i \(-0.415197\pi\)
0.263277 + 0.964720i \(0.415197\pi\)
\(60\) −7.72575e6 −0.0769591
\(61\) 1.41890e8 1.31210 0.656052 0.754716i \(-0.272225\pi\)
0.656052 + 0.754716i \(0.272225\pi\)
\(62\) 1.26518e8 1.08740
\(63\) 0 0
\(64\) 1.67772e7 0.125000
\(65\) 1.83692e7 0.127638
\(66\) 4.15862e7 0.269775
\(67\) −9.87576e7 −0.598734 −0.299367 0.954138i \(-0.596776\pi\)
−0.299367 + 0.954138i \(0.596776\pi\)
\(68\) −7.69066e7 −0.436188
\(69\) 4.07156e8 2.16242
\(70\) 0 0
\(71\) 2.96024e8 1.38250 0.691248 0.722617i \(-0.257061\pi\)
0.691248 + 0.722617i \(0.257061\pi\)
\(72\) −5.10989e7 −0.224086
\(73\) −4.60393e7 −0.189747 −0.0948737 0.995489i \(-0.530245\pi\)
−0.0948737 + 0.995489i \(0.530245\pi\)
\(74\) −2.26482e8 −0.877993
\(75\) 3.45170e8 1.25967
\(76\) 1.25899e8 0.432872
\(77\) 0 0
\(78\) 3.13187e8 0.958028
\(79\) −5.52570e8 −1.59612 −0.798060 0.602579i \(-0.794140\pi\)
−0.798060 + 0.602579i \(0.794140\pi\)
\(80\) 1.10289e7 0.0301043
\(81\) −4.77339e8 −1.23209
\(82\) 1.77721e8 0.434086
\(83\) 5.43217e8 1.25638 0.628191 0.778059i \(-0.283796\pi\)
0.628191 + 0.778059i \(0.283796\pi\)
\(84\) 0 0
\(85\) −5.05566e7 −0.105049
\(86\) −1.36118e8 −0.268332
\(87\) 6.81933e8 1.27616
\(88\) −5.93666e7 −0.105529
\(89\) −5.08661e8 −0.859356 −0.429678 0.902982i \(-0.641373\pi\)
−0.429678 + 0.902982i \(0.641373\pi\)
\(90\) −3.35912e7 −0.0539678
\(91\) 0 0
\(92\) −5.81238e8 −0.845880
\(93\) 1.41801e9 1.96565
\(94\) −6.99929e8 −0.924653
\(95\) 8.27628e7 0.104251
\(96\) 1.88038e8 0.225957
\(97\) 1.18179e9 1.35540 0.677701 0.735338i \(-0.262976\pi\)
0.677701 + 0.735338i \(0.262976\pi\)
\(98\) 0 0
\(99\) 1.80815e8 0.189180
\(100\) −4.92750e8 −0.492750
\(101\) 5.25710e8 0.502690 0.251345 0.967898i \(-0.419127\pi\)
0.251345 + 0.967898i \(0.419127\pi\)
\(102\) −8.61966e8 −0.788478
\(103\) 1.48474e9 1.29982 0.649909 0.760012i \(-0.274807\pi\)
0.649909 + 0.760012i \(0.274807\pi\)
\(104\) −4.47092e8 −0.374755
\(105\) 0 0
\(106\) 8.83014e8 0.679346
\(107\) 2.05930e9 1.51877 0.759385 0.650642i \(-0.225500\pi\)
0.759385 + 0.650642i \(0.225500\pi\)
\(108\) 3.30889e8 0.234032
\(109\) 1.62236e9 1.10085 0.550424 0.834886i \(-0.314466\pi\)
0.550424 + 0.834886i \(0.314466\pi\)
\(110\) −3.90262e7 −0.0254149
\(111\) −2.53840e9 −1.58711
\(112\) 0 0
\(113\) −2.79789e8 −0.161428 −0.0807139 0.996737i \(-0.525720\pi\)
−0.0807139 + 0.996737i \(0.525720\pi\)
\(114\) 1.41107e9 0.782485
\(115\) −3.82092e8 −0.203717
\(116\) −9.73498e8 −0.499199
\(117\) 1.36172e9 0.671819
\(118\) −7.84147e8 −0.372330
\(119\) 0 0
\(120\) 1.23612e8 0.0544183
\(121\) −2.14788e9 −0.910910
\(122\) −2.27024e9 −0.927797
\(123\) 1.99189e9 0.784679
\(124\) −2.02429e9 −0.768908
\(125\) −6.52610e8 −0.239089
\(126\) 0 0
\(127\) −2.04209e9 −0.696558 −0.348279 0.937391i \(-0.613234\pi\)
−0.348279 + 0.937391i \(0.613234\pi\)
\(128\) −2.68435e8 −0.0883883
\(129\) −1.52561e9 −0.485053
\(130\) −2.93908e8 −0.0902539
\(131\) −2.40237e9 −0.712719 −0.356359 0.934349i \(-0.615982\pi\)
−0.356359 + 0.934349i \(0.615982\pi\)
\(132\) −6.65378e8 −0.190759
\(133\) 0 0
\(134\) 1.58012e9 0.423369
\(135\) 2.17518e8 0.0563630
\(136\) 1.23051e9 0.308431
\(137\) −1.93337e9 −0.468891 −0.234446 0.972129i \(-0.575327\pi\)
−0.234446 + 0.972129i \(0.575327\pi\)
\(138\) −6.51449e9 −1.52906
\(139\) 8.99325e8 0.204338 0.102169 0.994767i \(-0.467422\pi\)
0.102169 + 0.994767i \(0.467422\pi\)
\(140\) 0 0
\(141\) −7.84478e9 −1.67146
\(142\) −4.73638e9 −0.977573
\(143\) 1.58205e9 0.316378
\(144\) 8.17583e8 0.158453
\(145\) −6.39954e8 −0.120225
\(146\) 7.36629e8 0.134172
\(147\) 0 0
\(148\) 3.62371e9 0.620835
\(149\) −9.43719e9 −1.56857 −0.784286 0.620399i \(-0.786971\pi\)
−0.784286 + 0.620399i \(0.786971\pi\)
\(150\) −5.52272e9 −0.890723
\(151\) 4.42810e9 0.693141 0.346571 0.938024i \(-0.387346\pi\)
0.346571 + 0.938024i \(0.387346\pi\)
\(152\) −2.01438e9 −0.306087
\(153\) −3.74779e9 −0.552922
\(154\) 0 0
\(155\) −1.33072e9 −0.185180
\(156\) −5.01099e9 −0.677428
\(157\) 5.24204e9 0.688576 0.344288 0.938864i \(-0.388120\pi\)
0.344288 + 0.938864i \(0.388120\pi\)
\(158\) 8.84112e9 1.12863
\(159\) 9.89678e9 1.22802
\(160\) −1.76463e8 −0.0212870
\(161\) 0 0
\(162\) 7.63742e9 0.871222
\(163\) 8.97873e9 0.996256 0.498128 0.867104i \(-0.334021\pi\)
0.498128 + 0.867104i \(0.334021\pi\)
\(164\) −2.84353e9 −0.306945
\(165\) −4.37404e8 −0.0459415
\(166\) −8.69147e9 −0.888396
\(167\) −1.16267e10 −1.15673 −0.578367 0.815777i \(-0.696310\pi\)
−0.578367 + 0.815777i \(0.696310\pi\)
\(168\) 0 0
\(169\) 1.30995e9 0.123528
\(170\) 8.08905e8 0.0742809
\(171\) 6.13526e9 0.548720
\(172\) 2.17789e9 0.189740
\(173\) 6.13714e9 0.520905 0.260452 0.965487i \(-0.416128\pi\)
0.260452 + 0.965487i \(0.416128\pi\)
\(174\) −1.09109e10 −0.902381
\(175\) 0 0
\(176\) 9.49865e8 0.0746200
\(177\) −8.78869e9 −0.673045
\(178\) 8.13857e9 0.607656
\(179\) 2.33469e10 1.69977 0.849885 0.526969i \(-0.176672\pi\)
0.849885 + 0.526969i \(0.176672\pi\)
\(180\) 5.37460e8 0.0381610
\(181\) 5.45222e9 0.377590 0.188795 0.982017i \(-0.439542\pi\)
0.188795 + 0.982017i \(0.439542\pi\)
\(182\) 0 0
\(183\) −2.54448e10 −1.67714
\(184\) 9.29981e9 0.598127
\(185\) 2.38214e9 0.149518
\(186\) −2.26881e10 −1.38992
\(187\) −4.35417e9 −0.260386
\(188\) 1.11989e10 0.653828
\(189\) 0 0
\(190\) −1.32420e9 −0.0737164
\(191\) 2.34598e10 1.27548 0.637741 0.770250i \(-0.279869\pi\)
0.637741 + 0.770250i \(0.279869\pi\)
\(192\) −3.00861e9 −0.159776
\(193\) 8.76414e9 0.454675 0.227338 0.973816i \(-0.426998\pi\)
0.227338 + 0.973816i \(0.426998\pi\)
\(194\) −1.89087e10 −0.958414
\(195\) −3.29411e9 −0.163148
\(196\) 0 0
\(197\) 1.40897e10 0.666503 0.333252 0.942838i \(-0.391854\pi\)
0.333252 + 0.942838i \(0.391854\pi\)
\(198\) −2.89304e9 −0.133771
\(199\) 1.45952e10 0.659739 0.329870 0.944027i \(-0.392995\pi\)
0.329870 + 0.944027i \(0.392995\pi\)
\(200\) 7.88400e9 0.348427
\(201\) 1.77099e10 0.765306
\(202\) −8.41137e9 −0.355456
\(203\) 0 0
\(204\) 1.37915e10 0.557538
\(205\) −1.86927e9 −0.0739231
\(206\) −2.37558e10 −0.919110
\(207\) −2.83247e10 −1.07226
\(208\) 7.15347e9 0.264992
\(209\) 7.12793e9 0.258407
\(210\) 0 0
\(211\) 3.18799e10 1.10725 0.553625 0.832766i \(-0.313244\pi\)
0.553625 + 0.832766i \(0.313244\pi\)
\(212\) −1.41282e10 −0.480370
\(213\) −5.30852e10 −1.76712
\(214\) −3.29487e10 −1.07393
\(215\) 1.43169e9 0.0456959
\(216\) −5.29422e9 −0.165486
\(217\) 0 0
\(218\) −2.59577e10 −0.778417
\(219\) 8.25611e9 0.242536
\(220\) 6.24419e8 0.0179711
\(221\) −3.27914e10 −0.924688
\(222\) 4.06144e10 1.12226
\(223\) 5.49341e10 1.48755 0.743773 0.668433i \(-0.233035\pi\)
0.743773 + 0.668433i \(0.233035\pi\)
\(224\) 0 0
\(225\) −2.40126e10 −0.624622
\(226\) 4.47663e9 0.114147
\(227\) 2.54176e10 0.635358 0.317679 0.948198i \(-0.397097\pi\)
0.317679 + 0.948198i \(0.397097\pi\)
\(228\) −2.25771e10 −0.553300
\(229\) −3.95101e10 −0.949399 −0.474700 0.880148i \(-0.657443\pi\)
−0.474700 + 0.880148i \(0.657443\pi\)
\(230\) 6.11347e9 0.144050
\(231\) 0 0
\(232\) 1.55760e10 0.352987
\(233\) 1.03351e9 0.0229727 0.0114863 0.999934i \(-0.496344\pi\)
0.0114863 + 0.999934i \(0.496344\pi\)
\(234\) −2.17876e10 −0.475048
\(235\) 7.36187e9 0.157465
\(236\) 1.25463e10 0.263277
\(237\) 9.90909e10 2.04017
\(238\) 0 0
\(239\) 2.53952e10 0.503455 0.251727 0.967798i \(-0.419001\pi\)
0.251727 + 0.967798i \(0.419001\pi\)
\(240\) −1.97779e9 −0.0384796
\(241\) 4.44854e10 0.849455 0.424728 0.905321i \(-0.360370\pi\)
0.424728 + 0.905321i \(0.360370\pi\)
\(242\) 3.43660e10 0.644110
\(243\) 6.01590e10 1.10681
\(244\) 3.63239e10 0.656052
\(245\) 0 0
\(246\) −3.18702e10 −0.554852
\(247\) 5.36807e10 0.917660
\(248\) 3.23886e10 0.543700
\(249\) −9.74136e10 −1.60592
\(250\) 1.04418e10 0.169061
\(251\) −7.74824e10 −1.23217 −0.616086 0.787679i \(-0.711283\pi\)
−0.616086 + 0.787679i \(0.711283\pi\)
\(252\) 0 0
\(253\) −3.29076e10 −0.504956
\(254\) 3.26734e10 0.492541
\(255\) 9.06618e9 0.134274
\(256\) 4.29497e9 0.0625000
\(257\) −6.63813e10 −0.949177 −0.474588 0.880208i \(-0.657403\pi\)
−0.474588 + 0.880208i \(0.657403\pi\)
\(258\) 2.44097e10 0.342984
\(259\) 0 0
\(260\) 4.70252e9 0.0638191
\(261\) −4.74402e10 −0.632797
\(262\) 3.84378e10 0.503968
\(263\) 7.44561e10 0.959620 0.479810 0.877372i \(-0.340706\pi\)
0.479810 + 0.877372i \(0.340706\pi\)
\(264\) 1.06461e10 0.134887
\(265\) −9.28756e9 −0.115690
\(266\) 0 0
\(267\) 9.12168e10 1.09843
\(268\) −2.52820e10 −0.299367
\(269\) −2.00175e10 −0.233091 −0.116545 0.993185i \(-0.537182\pi\)
−0.116545 + 0.993185i \(0.537182\pi\)
\(270\) −3.48029e9 −0.0398547
\(271\) 1.45053e11 1.63367 0.816836 0.576869i \(-0.195726\pi\)
0.816836 + 0.576869i \(0.195726\pi\)
\(272\) −1.96881e10 −0.218094
\(273\) 0 0
\(274\) 3.09339e10 0.331556
\(275\) −2.78977e10 −0.294152
\(276\) 1.04232e11 1.08121
\(277\) 1.05211e11 1.07374 0.536871 0.843664i \(-0.319606\pi\)
0.536871 + 0.843664i \(0.319606\pi\)
\(278\) −1.43892e10 −0.144489
\(279\) −9.86470e10 −0.974687
\(280\) 0 0
\(281\) 1.15652e11 1.10656 0.553281 0.832995i \(-0.313376\pi\)
0.553281 + 0.832995i \(0.313376\pi\)
\(282\) 1.25516e11 1.18190
\(283\) −1.01339e11 −0.939155 −0.469578 0.882891i \(-0.655594\pi\)
−0.469578 + 0.882891i \(0.655594\pi\)
\(284\) 7.57821e10 0.691248
\(285\) −1.48416e10 −0.133254
\(286\) −2.53127e10 −0.223713
\(287\) 0 0
\(288\) −1.30813e10 −0.112043
\(289\) −2.83379e10 −0.238961
\(290\) 1.02393e10 0.0850116
\(291\) −2.11928e11 −1.73248
\(292\) −1.17861e10 −0.0948737
\(293\) 3.48160e10 0.275978 0.137989 0.990434i \(-0.455936\pi\)
0.137989 + 0.990434i \(0.455936\pi\)
\(294\) 0 0
\(295\) 8.24767e9 0.0634063
\(296\) −5.79794e10 −0.438996
\(297\) 1.87337e10 0.139708
\(298\) 1.50995e11 1.10915
\(299\) −2.47828e11 −1.79321
\(300\) 8.83636e10 0.629836
\(301\) 0 0
\(302\) −7.08497e10 −0.490125
\(303\) −9.42743e10 −0.642542
\(304\) 3.22301e10 0.216436
\(305\) 2.38785e10 0.158000
\(306\) 5.99647e10 0.390975
\(307\) 3.68120e10 0.236520 0.118260 0.992983i \(-0.462268\pi\)
0.118260 + 0.992983i \(0.462268\pi\)
\(308\) 0 0
\(309\) −2.66254e11 −1.66144
\(310\) 2.12915e10 0.130942
\(311\) −6.01195e10 −0.364413 −0.182206 0.983260i \(-0.558324\pi\)
−0.182206 + 0.983260i \(0.558324\pi\)
\(312\) 8.01759e10 0.479014
\(313\) −9.53249e10 −0.561380 −0.280690 0.959798i \(-0.590563\pi\)
−0.280690 + 0.959798i \(0.590563\pi\)
\(314\) −8.38727e10 −0.486897
\(315\) 0 0
\(316\) −1.41458e11 −0.798060
\(317\) −1.10625e11 −0.615298 −0.307649 0.951500i \(-0.599542\pi\)
−0.307649 + 0.951500i \(0.599542\pi\)
\(318\) −1.58349e11 −0.868344
\(319\) −5.51159e10 −0.298002
\(320\) 2.82341e9 0.0150522
\(321\) −3.69288e11 −1.94130
\(322\) 0 0
\(323\) −1.47742e11 −0.755254
\(324\) −1.22199e11 −0.616047
\(325\) −2.10099e11 −1.04460
\(326\) −1.43660e11 −0.704459
\(327\) −2.90933e11 −1.40711
\(328\) 4.54965e10 0.217043
\(329\) 0 0
\(330\) 6.99846e9 0.0324855
\(331\) −4.41494e10 −0.202162 −0.101081 0.994878i \(-0.532230\pi\)
−0.101081 + 0.994878i \(0.532230\pi\)
\(332\) 1.39063e11 0.628191
\(333\) 1.76590e11 0.786985
\(334\) 1.86028e11 0.817934
\(335\) −1.66198e10 −0.0720980
\(336\) 0 0
\(337\) 3.39384e11 1.43336 0.716682 0.697400i \(-0.245660\pi\)
0.716682 + 0.697400i \(0.245660\pi\)
\(338\) −2.09592e10 −0.0873475
\(339\) 5.01739e10 0.206338
\(340\) −1.29425e10 −0.0525246
\(341\) −1.14608e11 −0.459007
\(342\) −9.81642e10 −0.388003
\(343\) 0 0
\(344\) −3.48462e10 −0.134166
\(345\) 6.85196e10 0.260393
\(346\) −9.81942e10 −0.368335
\(347\) 2.36437e11 0.875453 0.437726 0.899108i \(-0.355784\pi\)
0.437726 + 0.899108i \(0.355784\pi\)
\(348\) 1.74575e11 0.638080
\(349\) −2.70871e11 −0.977346 −0.488673 0.872467i \(-0.662519\pi\)
−0.488673 + 0.872467i \(0.662519\pi\)
\(350\) 0 0
\(351\) 1.41084e11 0.496132
\(352\) −1.51978e10 −0.0527643
\(353\) −1.88618e11 −0.646544 −0.323272 0.946306i \(-0.604783\pi\)
−0.323272 + 0.946306i \(0.604783\pi\)
\(354\) 1.40619e11 0.475915
\(355\) 4.98174e10 0.166477
\(356\) −1.30217e11 −0.429678
\(357\) 0 0
\(358\) −3.73550e11 −1.20192
\(359\) −5.65261e10 −0.179607 −0.0898037 0.995959i \(-0.528624\pi\)
−0.0898037 + 0.995959i \(0.528624\pi\)
\(360\) −8.59935e9 −0.0269839
\(361\) −8.08287e10 −0.250486
\(362\) −8.72356e10 −0.266996
\(363\) 3.85173e11 1.16433
\(364\) 0 0
\(365\) −7.74788e9 −0.0228489
\(366\) 4.07117e11 1.18592
\(367\) −5.33289e10 −0.153450 −0.0767248 0.997052i \(-0.524446\pi\)
−0.0767248 + 0.997052i \(0.524446\pi\)
\(368\) −1.48797e11 −0.422940
\(369\) −1.38570e11 −0.389091
\(370\) −3.81143e10 −0.105726
\(371\) 0 0
\(372\) 3.63010e11 0.982823
\(373\) 3.62011e11 0.968349 0.484174 0.874972i \(-0.339120\pi\)
0.484174 + 0.874972i \(0.339120\pi\)
\(374\) 6.96668e10 0.184121
\(375\) 1.17031e11 0.305605
\(376\) −1.79182e11 −0.462326
\(377\) −4.15080e11 −1.05827
\(378\) 0 0
\(379\) −1.82450e11 −0.454222 −0.227111 0.973869i \(-0.572928\pi\)
−0.227111 + 0.973869i \(0.572928\pi\)
\(380\) 2.11873e10 0.0521253
\(381\) 3.66202e11 0.890345
\(382\) −3.75357e11 −0.901903
\(383\) 4.67701e10 0.111064 0.0555320 0.998457i \(-0.482315\pi\)
0.0555320 + 0.998457i \(0.482315\pi\)
\(384\) 4.81378e10 0.112979
\(385\) 0 0
\(386\) −1.40226e11 −0.321504
\(387\) 1.06132e11 0.240518
\(388\) 3.02539e11 0.677701
\(389\) 7.47331e11 1.65478 0.827389 0.561630i \(-0.189825\pi\)
0.827389 + 0.561630i \(0.189825\pi\)
\(390\) 5.27057e10 0.115363
\(391\) 6.82083e11 1.47585
\(392\) 0 0
\(393\) 4.30810e11 0.911002
\(394\) −2.25434e11 −0.471289
\(395\) −9.29911e10 −0.192200
\(396\) 4.62886e10 0.0945901
\(397\) 5.20994e11 1.05263 0.526315 0.850290i \(-0.323573\pi\)
0.526315 + 0.850290i \(0.323573\pi\)
\(398\) −2.33524e11 −0.466506
\(399\) 0 0
\(400\) −1.26144e11 −0.246375
\(401\) −2.80021e11 −0.540805 −0.270402 0.962747i \(-0.587157\pi\)
−0.270402 + 0.962747i \(0.587157\pi\)
\(402\) −2.83359e11 −0.541153
\(403\) −8.63116e11 −1.63003
\(404\) 1.34582e11 0.251345
\(405\) −8.03305e10 −0.148366
\(406\) 0 0
\(407\) 2.05161e11 0.370613
\(408\) −2.20663e11 −0.394239
\(409\) 1.02969e12 1.81950 0.909749 0.415159i \(-0.136274\pi\)
0.909749 + 0.415159i \(0.136274\pi\)
\(410\) 2.99083e10 0.0522715
\(411\) 3.46706e11 0.599340
\(412\) 3.80093e11 0.649909
\(413\) 0 0
\(414\) 4.53196e11 0.758201
\(415\) 9.14170e10 0.151290
\(416\) −1.14456e11 −0.187377
\(417\) −1.61274e11 −0.261187
\(418\) −1.14047e11 −0.182722
\(419\) 7.23331e11 1.14650 0.573250 0.819381i \(-0.305682\pi\)
0.573250 + 0.819381i \(0.305682\pi\)
\(420\) 0 0
\(421\) −6.04783e11 −0.938275 −0.469137 0.883125i \(-0.655435\pi\)
−0.469137 + 0.883125i \(0.655435\pi\)
\(422\) −5.10078e11 −0.782944
\(423\) 5.45740e11 0.828809
\(424\) 2.26051e11 0.339673
\(425\) 5.78243e11 0.859726
\(426\) 8.49363e11 1.24954
\(427\) 0 0
\(428\) 5.27180e11 0.759385
\(429\) −2.83704e11 −0.404397
\(430\) −2.29071e10 −0.0323118
\(431\) −1.27204e12 −1.77563 −0.887813 0.460204i \(-0.847776\pi\)
−0.887813 + 0.460204i \(0.847776\pi\)
\(432\) 8.47075e10 0.117016
\(433\) 3.81300e11 0.521280 0.260640 0.965436i \(-0.416066\pi\)
0.260640 + 0.965436i \(0.416066\pi\)
\(434\) 0 0
\(435\) 1.14761e11 0.153672
\(436\) 4.15323e11 0.550424
\(437\) −1.11659e12 −1.46463
\(438\) −1.32098e11 −0.171499
\(439\) 7.49056e11 0.962551 0.481276 0.876569i \(-0.340174\pi\)
0.481276 + 0.876569i \(0.340174\pi\)
\(440\) −9.99070e9 −0.0127075
\(441\) 0 0
\(442\) 5.24663e11 0.653853
\(443\) 6.69286e10 0.0825648 0.0412824 0.999148i \(-0.486856\pi\)
0.0412824 + 0.999148i \(0.486856\pi\)
\(444\) −6.49831e11 −0.793555
\(445\) −8.56017e10 −0.103481
\(446\) −8.78946e11 −1.05185
\(447\) 1.69235e12 2.00496
\(448\) 0 0
\(449\) −7.18015e11 −0.833729 −0.416865 0.908969i \(-0.636871\pi\)
−0.416865 + 0.908969i \(0.636871\pi\)
\(450\) 3.84201e11 0.441674
\(451\) −1.60991e11 −0.183234
\(452\) −7.16261e10 −0.0807139
\(453\) −7.94081e11 −0.885977
\(454\) −4.06682e11 −0.449266
\(455\) 0 0
\(456\) 3.61233e11 0.391242
\(457\) 1.66054e12 1.78085 0.890426 0.455128i \(-0.150407\pi\)
0.890426 + 0.455128i \(0.150407\pi\)
\(458\) 6.32162e11 0.671327
\(459\) −3.88298e11 −0.408327
\(460\) −9.78156e10 −0.101859
\(461\) −5.97375e11 −0.616018 −0.308009 0.951384i \(-0.599663\pi\)
−0.308009 + 0.951384i \(0.599663\pi\)
\(462\) 0 0
\(463\) −6.50336e11 −0.657693 −0.328847 0.944383i \(-0.606660\pi\)
−0.328847 + 0.944383i \(0.606660\pi\)
\(464\) −2.49215e11 −0.249600
\(465\) 2.38634e11 0.236698
\(466\) −1.65361e10 −0.0162441
\(467\) 1.14333e12 1.11236 0.556180 0.831062i \(-0.312267\pi\)
0.556180 + 0.831062i \(0.312267\pi\)
\(468\) 3.48601e11 0.335910
\(469\) 0 0
\(470\) −1.17790e11 −0.111344
\(471\) −9.40042e11 −0.880143
\(472\) −2.00742e11 −0.186165
\(473\) 1.23304e11 0.113267
\(474\) −1.58545e12 −1.44262
\(475\) −9.46602e11 −0.853191
\(476\) 0 0
\(477\) −6.88493e11 −0.608929
\(478\) −4.06322e11 −0.355996
\(479\) −1.93813e11 −0.168218 −0.0841090 0.996457i \(-0.526804\pi\)
−0.0841090 + 0.996457i \(0.526804\pi\)
\(480\) 3.16447e10 0.0272092
\(481\) 1.54508e12 1.31613
\(482\) −7.11766e11 −0.600656
\(483\) 0 0
\(484\) −5.49857e11 −0.455455
\(485\) 1.98882e11 0.163214
\(486\) −9.62543e11 −0.782631
\(487\) 4.62927e11 0.372934 0.186467 0.982461i \(-0.440296\pi\)
0.186467 + 0.982461i \(0.440296\pi\)
\(488\) −5.81182e11 −0.463899
\(489\) −1.61013e12 −1.27342
\(490\) 0 0
\(491\) 1.20314e12 0.934222 0.467111 0.884199i \(-0.345295\pi\)
0.467111 + 0.884199i \(0.345295\pi\)
\(492\) 5.09923e11 0.392339
\(493\) 1.14240e12 0.870978
\(494\) −8.58891e11 −0.648884
\(495\) 3.04290e10 0.0227806
\(496\) −5.18217e11 −0.384454
\(497\) 0 0
\(498\) 1.55862e12 1.13555
\(499\) 3.45574e11 0.249510 0.124755 0.992188i \(-0.460185\pi\)
0.124755 + 0.992188i \(0.460185\pi\)
\(500\) −1.67068e11 −0.119544
\(501\) 2.08499e12 1.47854
\(502\) 1.23972e12 0.871277
\(503\) −1.55705e12 −1.08454 −0.542270 0.840204i \(-0.682435\pi\)
−0.542270 + 0.840204i \(0.682435\pi\)
\(504\) 0 0
\(505\) 8.84709e10 0.0605326
\(506\) 5.26521e11 0.357058
\(507\) −2.34910e11 −0.157894
\(508\) −5.22774e11 −0.348279
\(509\) −1.97093e12 −1.30149 −0.650746 0.759295i \(-0.725544\pi\)
−0.650746 + 0.759295i \(0.725544\pi\)
\(510\) −1.45059e11 −0.0949464
\(511\) 0 0
\(512\) −6.87195e10 −0.0441942
\(513\) 6.35657e11 0.405224
\(514\) 1.06210e12 0.671169
\(515\) 2.49864e11 0.156521
\(516\) −3.90555e11 −0.242526
\(517\) 6.34039e11 0.390309
\(518\) 0 0
\(519\) −1.10056e12 −0.665824
\(520\) −7.52404e10 −0.0451270
\(521\) −1.48877e12 −0.885235 −0.442617 0.896711i \(-0.645950\pi\)
−0.442617 + 0.896711i \(0.645950\pi\)
\(522\) 7.59044e11 0.447455
\(523\) 2.06845e12 1.20889 0.604446 0.796646i \(-0.293394\pi\)
0.604446 + 0.796646i \(0.293394\pi\)
\(524\) −6.15005e11 −0.356359
\(525\) 0 0
\(526\) −1.19130e12 −0.678554
\(527\) 2.37550e12 1.34155
\(528\) −1.70337e11 −0.0953797
\(529\) 3.35384e12 1.86205
\(530\) 1.48601e11 0.0818050
\(531\) 6.11405e11 0.333737
\(532\) 0 0
\(533\) −1.21243e12 −0.650703
\(534\) −1.45947e12 −0.776710
\(535\) 3.46556e11 0.182886
\(536\) 4.04511e11 0.211684
\(537\) −4.18673e12 −2.17266
\(538\) 3.20281e11 0.164820
\(539\) 0 0
\(540\) 5.56847e10 0.0281815
\(541\) −3.40957e12 −1.71124 −0.855622 0.517601i \(-0.826825\pi\)
−0.855622 + 0.517601i \(0.826825\pi\)
\(542\) −2.32085e12 −1.15518
\(543\) −9.77733e11 −0.482637
\(544\) 3.15009e11 0.154216
\(545\) 2.73024e11 0.132561
\(546\) 0 0
\(547\) 9.65758e10 0.0461238 0.0230619 0.999734i \(-0.492659\pi\)
0.0230619 + 0.999734i \(0.492659\pi\)
\(548\) −4.94942e11 −0.234446
\(549\) 1.77013e12 0.831627
\(550\) 4.46363e11 0.207997
\(551\) −1.87015e12 −0.864359
\(552\) −1.66771e12 −0.764530
\(553\) 0 0
\(554\) −1.68337e12 −0.759251
\(555\) −4.27183e11 −0.191116
\(556\) 2.30227e11 0.102169
\(557\) −8.96612e11 −0.394690 −0.197345 0.980334i \(-0.563232\pi\)
−0.197345 + 0.980334i \(0.563232\pi\)
\(558\) 1.57835e12 0.689207
\(559\) 9.28609e11 0.402235
\(560\) 0 0
\(561\) 7.80822e11 0.332828
\(562\) −1.85044e12 −0.782457
\(563\) 4.25576e12 1.78521 0.892605 0.450839i \(-0.148875\pi\)
0.892605 + 0.450839i \(0.148875\pi\)
\(564\) −2.00826e12 −0.835728
\(565\) −4.70853e10 −0.0194387
\(566\) 1.62142e12 0.664083
\(567\) 0 0
\(568\) −1.21251e12 −0.488786
\(569\) −4.79023e12 −1.91581 −0.957903 0.287092i \(-0.907311\pi\)
−0.957903 + 0.287092i \(0.907311\pi\)
\(570\) 2.37466e11 0.0942247
\(571\) 2.37883e12 0.936483 0.468242 0.883600i \(-0.344888\pi\)
0.468242 + 0.883600i \(0.344888\pi\)
\(572\) 4.05004e11 0.158189
\(573\) −4.20699e12 −1.63033
\(574\) 0 0
\(575\) 4.37019e12 1.66723
\(576\) 2.09301e11 0.0792265
\(577\) −4.13934e12 −1.55468 −0.777339 0.629082i \(-0.783431\pi\)
−0.777339 + 0.629082i \(0.783431\pi\)
\(578\) 4.53407e11 0.168971
\(579\) −1.57165e12 −0.581169
\(580\) −1.63828e11 −0.0601123
\(581\) 0 0
\(582\) 3.39084e12 1.22505
\(583\) −7.99889e11 −0.286762
\(584\) 1.88577e11 0.0670858
\(585\) 2.29162e11 0.0808987
\(586\) −5.57056e11 −0.195146
\(587\) 3.51788e12 1.22295 0.611476 0.791263i \(-0.290576\pi\)
0.611476 + 0.791263i \(0.290576\pi\)
\(588\) 0 0
\(589\) −3.88878e12 −1.33136
\(590\) −1.31963e11 −0.0448350
\(591\) −2.52666e12 −0.851929
\(592\) 9.27670e11 0.310417
\(593\) −1.46555e10 −0.00486691 −0.00243346 0.999997i \(-0.500775\pi\)
−0.00243346 + 0.999997i \(0.500775\pi\)
\(594\) −2.99739e11 −0.0987882
\(595\) 0 0
\(596\) −2.41592e12 −0.784286
\(597\) −2.61733e12 −0.843283
\(598\) 3.96525e12 1.26799
\(599\) −3.03476e12 −0.963170 −0.481585 0.876399i \(-0.659939\pi\)
−0.481585 + 0.876399i \(0.659939\pi\)
\(600\) −1.41382e12 −0.445361
\(601\) −9.37778e11 −0.293201 −0.146600 0.989196i \(-0.546833\pi\)
−0.146600 + 0.989196i \(0.546833\pi\)
\(602\) 0 0
\(603\) −1.23203e12 −0.379485
\(604\) 1.13359e12 0.346571
\(605\) −3.61463e11 −0.109689
\(606\) 1.50839e12 0.454346
\(607\) 4.36820e12 1.30603 0.653015 0.757345i \(-0.273504\pi\)
0.653015 + 0.757345i \(0.273504\pi\)
\(608\) −5.15681e11 −0.153044
\(609\) 0 0
\(610\) −3.82056e11 −0.111723
\(611\) 4.77498e12 1.38607
\(612\) −9.59435e11 −0.276461
\(613\) −1.25555e12 −0.359138 −0.179569 0.983745i \(-0.557470\pi\)
−0.179569 + 0.983745i \(0.557470\pi\)
\(614\) −5.88992e11 −0.167245
\(615\) 3.35212e11 0.0944889
\(616\) 0 0
\(617\) 5.91635e12 1.64350 0.821751 0.569846i \(-0.192997\pi\)
0.821751 + 0.569846i \(0.192997\pi\)
\(618\) 4.26007e12 1.17481
\(619\) 8.44127e11 0.231100 0.115550 0.993302i \(-0.463137\pi\)
0.115550 + 0.993302i \(0.463137\pi\)
\(620\) −3.40664e11 −0.0925899
\(621\) −2.93465e12 −0.791851
\(622\) 9.61912e11 0.257679
\(623\) 0 0
\(624\) −1.28281e12 −0.338714
\(625\) 3.64956e12 0.956709
\(626\) 1.52520e12 0.396956
\(627\) −1.27823e12 −0.330298
\(628\) 1.34196e12 0.344288
\(629\) −4.25243e12 −1.08320
\(630\) 0 0
\(631\) −4.70645e11 −0.118185 −0.0590924 0.998253i \(-0.518821\pi\)
−0.0590924 + 0.998253i \(0.518821\pi\)
\(632\) 2.26333e12 0.564313
\(633\) −5.71694e12 −1.41529
\(634\) 1.77000e12 0.435081
\(635\) −3.43659e11 −0.0838777
\(636\) 2.53358e12 0.614012
\(637\) 0 0
\(638\) 8.81855e11 0.210719
\(639\) 3.69299e12 0.876243
\(640\) −4.51746e10 −0.0106435
\(641\) 7.35770e11 0.172140 0.0860698 0.996289i \(-0.472569\pi\)
0.0860698 + 0.996289i \(0.472569\pi\)
\(642\) 5.90861e12 1.37271
\(643\) 2.51018e12 0.579103 0.289551 0.957162i \(-0.406494\pi\)
0.289551 + 0.957162i \(0.406494\pi\)
\(644\) 0 0
\(645\) −2.56742e11 −0.0584087
\(646\) 2.36388e12 0.534046
\(647\) −1.18989e12 −0.266955 −0.133477 0.991052i \(-0.542614\pi\)
−0.133477 + 0.991052i \(0.542614\pi\)
\(648\) 1.95518e12 0.435611
\(649\) 7.10329e11 0.157166
\(650\) 3.36158e12 0.738641
\(651\) 0 0
\(652\) 2.29856e12 0.498128
\(653\) −4.19215e12 −0.902250 −0.451125 0.892461i \(-0.648977\pi\)
−0.451125 + 0.892461i \(0.648977\pi\)
\(654\) 4.65493e12 0.994977
\(655\) −4.04290e11 −0.0858237
\(656\) −7.27944e11 −0.153473
\(657\) −5.74355e11 −0.120264
\(658\) 0 0
\(659\) 3.83674e11 0.0792461 0.0396230 0.999215i \(-0.487384\pi\)
0.0396230 + 0.999215i \(0.487384\pi\)
\(660\) −1.11975e11 −0.0229707
\(661\) 7.20723e12 1.46846 0.734230 0.678901i \(-0.237543\pi\)
0.734230 + 0.678901i \(0.237543\pi\)
\(662\) 7.06390e11 0.142950
\(663\) 5.88041e12 1.18194
\(664\) −2.22502e12 −0.444198
\(665\) 0 0
\(666\) −2.82544e12 −0.556482
\(667\) 8.63394e12 1.68905
\(668\) −2.97644e12 −0.578367
\(669\) −9.85119e12 −1.90139
\(670\) 2.65916e11 0.0509810
\(671\) 2.05653e12 0.391636
\(672\) 0 0
\(673\) −5.29279e12 −0.994529 −0.497264 0.867599i \(-0.665662\pi\)
−0.497264 + 0.867599i \(0.665662\pi\)
\(674\) −5.43014e12 −1.01354
\(675\) −2.48787e12 −0.461277
\(676\) 3.35348e11 0.0617640
\(677\) −8.77680e11 −0.160578 −0.0802892 0.996772i \(-0.525584\pi\)
−0.0802892 + 0.996772i \(0.525584\pi\)
\(678\) −8.02782e11 −0.145903
\(679\) 0 0
\(680\) 2.07080e11 0.0371405
\(681\) −4.55808e12 −0.812119
\(682\) 1.83372e12 0.324567
\(683\) −1.02795e13 −1.80749 −0.903747 0.428068i \(-0.859195\pi\)
−0.903747 + 0.428068i \(0.859195\pi\)
\(684\) 1.57063e12 0.274360
\(685\) −3.25363e11 −0.0564626
\(686\) 0 0
\(687\) 7.08525e12 1.21353
\(688\) 5.57540e11 0.0948698
\(689\) −6.02399e12 −1.01835
\(690\) −1.09631e12 −0.184125
\(691\) −1.34749e12 −0.224841 −0.112420 0.993661i \(-0.535860\pi\)
−0.112420 + 0.993661i \(0.535860\pi\)
\(692\) 1.57111e12 0.260452
\(693\) 0 0
\(694\) −3.78299e12 −0.619039
\(695\) 1.51346e11 0.0246059
\(696\) −2.79320e12 −0.451191
\(697\) 3.33689e12 0.535543
\(698\) 4.33394e12 0.691088
\(699\) −1.85336e11 −0.0293639
\(700\) 0 0
\(701\) 6.86975e12 1.07451 0.537254 0.843420i \(-0.319462\pi\)
0.537254 + 0.843420i \(0.319462\pi\)
\(702\) −2.25735e12 −0.350818
\(703\) 6.96137e12 1.07497
\(704\) 2.43165e11 0.0373100
\(705\) −1.32019e12 −0.201272
\(706\) 3.01789e12 0.457175
\(707\) 0 0
\(708\) −2.24990e12 −0.336523
\(709\) −1.40987e12 −0.209542 −0.104771 0.994496i \(-0.533411\pi\)
−0.104771 + 0.994496i \(0.533411\pi\)
\(710\) −7.97078e11 −0.117717
\(711\) −6.89349e12 −1.01164
\(712\) 2.08347e12 0.303828
\(713\) 1.79534e13 2.60162
\(714\) 0 0
\(715\) 2.66240e11 0.0380975
\(716\) 5.97680e12 0.849885
\(717\) −4.55405e12 −0.643519
\(718\) 9.04418e11 0.127002
\(719\) −6.92395e12 −0.966216 −0.483108 0.875561i \(-0.660492\pi\)
−0.483108 + 0.875561i \(0.660492\pi\)
\(720\) 1.37590e11 0.0190805
\(721\) 0 0
\(722\) 1.29326e12 0.177120
\(723\) −7.97745e12 −1.08578
\(724\) 1.39577e12 0.188795
\(725\) 7.31950e12 0.983922
\(726\) −6.16277e12 −0.823306
\(727\) 9.30979e12 1.23605 0.618024 0.786160i \(-0.287934\pi\)
0.618024 + 0.786160i \(0.287934\pi\)
\(728\) 0 0
\(729\) −1.39269e12 −0.182634
\(730\) 1.23966e11 0.0161566
\(731\) −2.55576e12 −0.331048
\(732\) −6.51387e12 −0.838570
\(733\) −8.26111e12 −1.05699 −0.528494 0.848937i \(-0.677243\pi\)
−0.528494 + 0.848937i \(0.677243\pi\)
\(734\) 8.53263e11 0.108505
\(735\) 0 0
\(736\) 2.38075e12 0.299064
\(737\) −1.43137e12 −0.178710
\(738\) 2.21712e12 0.275129
\(739\) 5.12673e12 0.632325 0.316163 0.948705i \(-0.397606\pi\)
0.316163 + 0.948705i \(0.397606\pi\)
\(740\) 6.09828e11 0.0747592
\(741\) −9.62642e12 −1.17296
\(742\) 0 0
\(743\) 1.04980e12 0.126374 0.0631870 0.998002i \(-0.479874\pi\)
0.0631870 + 0.998002i \(0.479874\pi\)
\(744\) −5.80816e12 −0.694961
\(745\) −1.58817e12 −0.188883
\(746\) −5.79217e12 −0.684726
\(747\) 6.77680e12 0.796310
\(748\) −1.11467e12 −0.130193
\(749\) 0 0
\(750\) −1.87249e12 −0.216095
\(751\) −1.15116e12 −0.132055 −0.0660277 0.997818i \(-0.521033\pi\)
−0.0660277 + 0.997818i \(0.521033\pi\)
\(752\) 2.86691e12 0.326914
\(753\) 1.38947e13 1.57497
\(754\) 6.64128e12 0.748309
\(755\) 7.45198e11 0.0834662
\(756\) 0 0
\(757\) −4.77630e12 −0.528641 −0.264320 0.964435i \(-0.585148\pi\)
−0.264320 + 0.964435i \(0.585148\pi\)
\(758\) 2.91920e12 0.321183
\(759\) 5.90123e12 0.645438
\(760\) −3.38996e11 −0.0368582
\(761\) 1.16815e12 0.126260 0.0631301 0.998005i \(-0.479892\pi\)
0.0631301 + 0.998005i \(0.479892\pi\)
\(762\) −5.85923e12 −0.629569
\(763\) 0 0
\(764\) 6.00572e12 0.637741
\(765\) −6.30710e11 −0.0665814
\(766\) −7.48321e11 −0.0785342
\(767\) 5.34951e12 0.558130
\(768\) −7.70205e11 −0.0798879
\(769\) −1.52183e13 −1.56927 −0.784634 0.619959i \(-0.787149\pi\)
−0.784634 + 0.619959i \(0.787149\pi\)
\(770\) 0 0
\(771\) 1.19040e13 1.21324
\(772\) 2.24362e12 0.227338
\(773\) 1.75227e13 1.76520 0.882600 0.470124i \(-0.155791\pi\)
0.882600 + 0.470124i \(0.155791\pi\)
\(774\) −1.69812e12 −0.170072
\(775\) 1.52201e13 1.51552
\(776\) −4.84062e12 −0.479207
\(777\) 0 0
\(778\) −1.19573e13 −1.17010
\(779\) −5.46260e12 −0.531472
\(780\) −8.43292e11 −0.0815741
\(781\) 4.29051e12 0.412647
\(782\) −1.09133e13 −1.04358
\(783\) −4.91515e12 −0.467314
\(784\) 0 0
\(785\) 8.82175e11 0.0829165
\(786\) −6.89296e12 −0.644176
\(787\) −1.12156e12 −0.104216 −0.0521081 0.998641i \(-0.516594\pi\)
−0.0521081 + 0.998641i \(0.516594\pi\)
\(788\) 3.60695e12 0.333252
\(789\) −1.33520e13 −1.22659
\(790\) 1.48786e12 0.135906
\(791\) 0 0
\(792\) −7.40617e11 −0.0668853
\(793\) 1.54878e13 1.39079
\(794\) −8.33591e12 −0.744322
\(795\) 1.66551e12 0.147875
\(796\) 3.73638e12 0.329870
\(797\) −1.56799e13 −1.37651 −0.688257 0.725467i \(-0.741624\pi\)
−0.688257 + 0.725467i \(0.741624\pi\)
\(798\) 0 0
\(799\) −1.31419e13 −1.14077
\(800\) 2.01830e12 0.174213
\(801\) −6.34571e12 −0.544670
\(802\) 4.48033e12 0.382407
\(803\) −6.67284e11 −0.0566358
\(804\) 4.53375e12 0.382653
\(805\) 0 0
\(806\) 1.38099e13 1.15261
\(807\) 3.58969e12 0.297938
\(808\) −2.15331e12 −0.177728
\(809\) 1.31404e13 1.07855 0.539275 0.842130i \(-0.318698\pi\)
0.539275 + 0.842130i \(0.318698\pi\)
\(810\) 1.28529e12 0.104910
\(811\) −6.99946e12 −0.568160 −0.284080 0.958801i \(-0.591688\pi\)
−0.284080 + 0.958801i \(0.591688\pi\)
\(812\) 0 0
\(813\) −2.60120e13 −2.08817
\(814\) −3.28258e12 −0.262063
\(815\) 1.51102e12 0.119966
\(816\) 3.53061e12 0.278769
\(817\) 4.18386e12 0.328532
\(818\) −1.64750e13 −1.28658
\(819\) 0 0
\(820\) −4.78533e11 −0.0369615
\(821\) 6.26383e11 0.0481167 0.0240583 0.999711i \(-0.492341\pi\)
0.0240583 + 0.999711i \(0.492341\pi\)
\(822\) −5.54730e12 −0.423797
\(823\) 1.29677e13 0.985290 0.492645 0.870230i \(-0.336030\pi\)
0.492645 + 0.870230i \(0.336030\pi\)
\(824\) −6.08149e12 −0.459555
\(825\) 5.00282e12 0.375987
\(826\) 0 0
\(827\) 9.65832e11 0.0718004 0.0359002 0.999355i \(-0.488570\pi\)
0.0359002 + 0.999355i \(0.488570\pi\)
\(828\) −7.25113e12 −0.536129
\(829\) 1.92526e13 1.41578 0.707888 0.706325i \(-0.249648\pi\)
0.707888 + 0.706325i \(0.249648\pi\)
\(830\) −1.46267e12 −0.106978
\(831\) −1.88671e13 −1.37246
\(832\) 1.83129e12 0.132496
\(833\) 0 0
\(834\) 2.58038e12 0.184687
\(835\) −1.95664e12 −0.139291
\(836\) 1.82475e12 0.129204
\(837\) −1.02205e13 −0.719796
\(838\) −1.15733e13 −0.810698
\(839\) 1.29497e13 0.902256 0.451128 0.892459i \(-0.351022\pi\)
0.451128 + 0.892459i \(0.351022\pi\)
\(840\) 0 0
\(841\) −4.64218e10 −0.00319993
\(842\) 9.67653e12 0.663460
\(843\) −2.07396e13 −1.41441
\(844\) 8.16125e12 0.553625
\(845\) 2.20450e11 0.0148749
\(846\) −8.73184e12 −0.586056
\(847\) 0 0
\(848\) −3.61682e12 −0.240185
\(849\) 1.81728e13 1.20043
\(850\) −9.25188e12 −0.607918
\(851\) −3.21387e13 −2.10061
\(852\) −1.35898e13 −0.883558
\(853\) −1.06726e13 −0.690241 −0.345120 0.938558i \(-0.612162\pi\)
−0.345120 + 0.938558i \(0.612162\pi\)
\(854\) 0 0
\(855\) 1.03249e12 0.0660754
\(856\) −8.43488e12 −0.536966
\(857\) −2.42456e13 −1.53539 −0.767697 0.640813i \(-0.778598\pi\)
−0.767697 + 0.640813i \(0.778598\pi\)
\(858\) 4.53927e12 0.285952
\(859\) 2.36434e13 1.48163 0.740817 0.671707i \(-0.234439\pi\)
0.740817 + 0.671707i \(0.234439\pi\)
\(860\) 3.66513e11 0.0228479
\(861\) 0 0
\(862\) 2.03526e13 1.25556
\(863\) −1.34443e13 −0.825069 −0.412534 0.910942i \(-0.635356\pi\)
−0.412534 + 0.910942i \(0.635356\pi\)
\(864\) −1.35532e12 −0.0827428
\(865\) 1.03281e12 0.0627260
\(866\) −6.10080e12 −0.368601
\(867\) 5.08176e12 0.305442
\(868\) 0 0
\(869\) −8.00883e12 −0.476409
\(870\) −1.83618e12 −0.108662
\(871\) −1.07797e13 −0.634638
\(872\) −6.64517e12 −0.389208
\(873\) 1.47432e13 0.859070
\(874\) 1.78655e13 1.03565
\(875\) 0 0
\(876\) 2.11356e12 0.121268
\(877\) −2.24398e13 −1.28092 −0.640458 0.767993i \(-0.721255\pi\)
−0.640458 + 0.767993i \(0.721255\pi\)
\(878\) −1.19849e13 −0.680626
\(879\) −6.24346e12 −0.352757
\(880\) 1.59851e11 0.00898554
\(881\) −2.34578e13 −1.31188 −0.655942 0.754811i \(-0.727728\pi\)
−0.655942 + 0.754811i \(0.727728\pi\)
\(882\) 0 0
\(883\) −2.46146e13 −1.36260 −0.681302 0.732002i \(-0.738586\pi\)
−0.681302 + 0.732002i \(0.738586\pi\)
\(884\) −8.39461e12 −0.462344
\(885\) −1.47903e12 −0.0810463
\(886\) −1.07086e12 −0.0583821
\(887\) −1.00413e13 −0.544669 −0.272334 0.962203i \(-0.587796\pi\)
−0.272334 + 0.962203i \(0.587796\pi\)
\(888\) 1.03973e13 0.561128
\(889\) 0 0
\(890\) 1.36963e12 0.0731724
\(891\) −6.91845e12 −0.367755
\(892\) 1.40631e13 0.743773
\(893\) 2.15137e13 1.13210
\(894\) −2.70776e13 −1.41772
\(895\) 3.92901e12 0.204682
\(896\) 0 0
\(897\) 4.44424e13 2.29209
\(898\) 1.14882e13 0.589535
\(899\) 3.00696e13 1.53535
\(900\) −6.14721e12 −0.312311
\(901\) 1.65795e13 0.838126
\(902\) 2.57585e12 0.129566
\(903\) 0 0
\(904\) 1.14602e12 0.0570733
\(905\) 9.17546e11 0.0454683
\(906\) 1.27053e13 0.626481
\(907\) −4.19206e12 −0.205681 −0.102841 0.994698i \(-0.532793\pi\)
−0.102841 + 0.994698i \(0.532793\pi\)
\(908\) 6.50691e12 0.317679
\(909\) 6.55841e12 0.318611
\(910\) 0 0
\(911\) −3.27191e13 −1.57387 −0.786935 0.617036i \(-0.788333\pi\)
−0.786935 + 0.617036i \(0.788333\pi\)
\(912\) −5.77973e12 −0.276650
\(913\) 7.87327e12 0.375005
\(914\) −2.65687e13 −1.25925
\(915\) −4.28206e12 −0.201957
\(916\) −1.01146e13 −0.474700
\(917\) 0 0
\(918\) 6.21277e12 0.288731
\(919\) 3.99412e11 0.0184715 0.00923573 0.999957i \(-0.497060\pi\)
0.00923573 + 0.999957i \(0.497060\pi\)
\(920\) 1.56505e12 0.0720249
\(921\) −6.60141e12 −0.302321
\(922\) 9.55800e12 0.435590
\(923\) 3.23120e13 1.46540
\(924\) 0 0
\(925\) −2.72458e13 −1.22366
\(926\) 1.04054e13 0.465059
\(927\) 1.85226e13 0.823840
\(928\) 3.98745e12 0.176494
\(929\) 2.25881e13 0.994968 0.497484 0.867473i \(-0.334257\pi\)
0.497484 + 0.867473i \(0.334257\pi\)
\(930\) −3.81815e12 −0.167371
\(931\) 0 0
\(932\) 2.64578e11 0.0114863
\(933\) 1.07811e13 0.465795
\(934\) −1.82933e13 −0.786557
\(935\) −7.32756e11 −0.0313550
\(936\) −5.57762e12 −0.237524
\(937\) 2.21481e13 0.938658 0.469329 0.883023i \(-0.344496\pi\)
0.469329 + 0.883023i \(0.344496\pi\)
\(938\) 0 0
\(939\) 1.70944e13 0.717559
\(940\) 1.88464e12 0.0787323
\(941\) −5.09190e12 −0.211703 −0.105851 0.994382i \(-0.533757\pi\)
−0.105851 + 0.994382i \(0.533757\pi\)
\(942\) 1.50407e13 0.622355
\(943\) 2.52193e13 1.03856
\(944\) 3.21187e12 0.131639
\(945\) 0 0
\(946\) −1.97287e12 −0.0800917
\(947\) 1.20342e13 0.486229 0.243114 0.969998i \(-0.421831\pi\)
0.243114 + 0.969998i \(0.421831\pi\)
\(948\) 2.53673e13 1.02008
\(949\) −5.02534e12 −0.201126
\(950\) 1.51456e13 0.603297
\(951\) 1.98380e13 0.786478
\(952\) 0 0
\(953\) 2.76503e13 1.08588 0.542940 0.839772i \(-0.317311\pi\)
0.542940 + 0.839772i \(0.317311\pi\)
\(954\) 1.10159e13 0.430578
\(955\) 3.94802e12 0.153590
\(956\) 6.50116e12 0.251727
\(957\) 9.88380e12 0.380908
\(958\) 3.10100e12 0.118948
\(959\) 0 0
\(960\) −5.06315e11 −0.0192398
\(961\) 3.60869e13 1.36488
\(962\) −2.47213e13 −0.930642
\(963\) 2.56904e13 0.962614
\(964\) 1.13883e13 0.424728
\(965\) 1.47490e12 0.0547508
\(966\) 0 0
\(967\) 2.75896e13 1.01467 0.507337 0.861748i \(-0.330630\pi\)
0.507337 + 0.861748i \(0.330630\pi\)
\(968\) 8.79771e12 0.322055
\(969\) 2.64942e13 0.965371
\(970\) −3.18211e12 −0.115410
\(971\) −3.98756e13 −1.43953 −0.719766 0.694217i \(-0.755751\pi\)
−0.719766 + 0.694217i \(0.755751\pi\)
\(972\) 1.54007e13 0.553404
\(973\) 0 0
\(974\) −7.40683e12 −0.263704
\(975\) 3.76765e13 1.33521
\(976\) 9.29892e12 0.328026
\(977\) −1.41342e13 −0.496301 −0.248151 0.968721i \(-0.579823\pi\)
−0.248151 + 0.968721i \(0.579823\pi\)
\(978\) 2.57621e13 0.900444
\(979\) −7.37242e12 −0.256500
\(980\) 0 0
\(981\) 2.02394e13 0.697730
\(982\) −1.92503e13 −0.660595
\(983\) 7.41532e11 0.0253302 0.0126651 0.999920i \(-0.495968\pi\)
0.0126651 + 0.999920i \(0.495968\pi\)
\(984\) −8.15877e12 −0.277426
\(985\) 2.37112e12 0.0802585
\(986\) −1.82784e13 −0.615875
\(987\) 0 0
\(988\) 1.37423e13 0.458830
\(989\) −1.93157e13 −0.641987
\(990\) −4.86864e11 −0.0161083
\(991\) −3.88208e13 −1.27859 −0.639297 0.768960i \(-0.720775\pi\)
−0.639297 + 0.768960i \(0.720775\pi\)
\(992\) 8.29148e12 0.271850
\(993\) 7.91719e12 0.258404
\(994\) 0 0
\(995\) 2.45621e12 0.0794440
\(996\) −2.49379e13 −0.802958
\(997\) 4.33172e13 1.38846 0.694229 0.719755i \(-0.255746\pi\)
0.694229 + 0.719755i \(0.255746\pi\)
\(998\) −5.52918e12 −0.176430
\(999\) 1.82960e13 0.581180
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 98.10.a.h.1.1 3
7.2 even 3 98.10.c.l.67.3 6
7.3 odd 6 14.10.c.b.9.1 6
7.4 even 3 98.10.c.l.79.3 6
7.5 odd 6 14.10.c.b.11.1 yes 6
7.6 odd 2 98.10.a.g.1.3 3
21.5 even 6 126.10.g.e.109.2 6
21.17 even 6 126.10.g.e.37.2 6
28.3 even 6 112.10.i.a.65.3 6
28.19 even 6 112.10.i.a.81.3 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
14.10.c.b.9.1 6 7.3 odd 6
14.10.c.b.11.1 yes 6 7.5 odd 6
98.10.a.g.1.3 3 7.6 odd 2
98.10.a.h.1.1 3 1.1 even 1 trivial
98.10.c.l.67.3 6 7.2 even 3
98.10.c.l.79.3 6 7.4 even 3
112.10.i.a.65.3 6 28.3 even 6
112.10.i.a.81.3 6 28.19 even 6
126.10.g.e.37.2 6 21.17 even 6
126.10.g.e.109.2 6 21.5 even 6