Properties

Label 169.10.a.b.1.4
Level $169$
Weight $10$
Character 169.1
Self dual yes
Analytic conductor $87.041$
Analytic rank $1$
Dimension $5$
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] = [169,10,Mod(1,169)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(169, 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("169.1");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 169 = 13^{2} \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 169.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: \(87.0410563117\)
Analytic rank: \(1\)
Dimension: \(5\)
Coefficient field: \(\mathbb{Q}[x]/(x^{5} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - 1438x^{3} - 4164x^{2} + 396957x - 59580 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2}\cdot 3 \)
Twist minimal: no (minimal twist has level 13)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(-24.3176\) of defining polynomial
Character \(\chi\) \(=\) 169.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+21.3176 q^{2} -195.094 q^{3} -57.5590 q^{4} +1277.14 q^{5} -4158.93 q^{6} +2277.98 q^{7} -12141.6 q^{8} +18378.5 q^{9} +O(q^{10})\) \(q+21.3176 q^{2} -195.094 q^{3} -57.5590 q^{4} +1277.14 q^{5} -4158.93 q^{6} +2277.98 q^{7} -12141.6 q^{8} +18378.5 q^{9} +27225.6 q^{10} +7177.94 q^{11} +11229.4 q^{12} +48561.1 q^{14} -249162. q^{15} -229361. q^{16} -447890. q^{17} +391786. q^{18} +528333. q^{19} -73510.8 q^{20} -444419. q^{21} +153017. q^{22} +2.24354e6 q^{23} +2.36876e6 q^{24} -322041. q^{25} +254496. q^{27} -131118. q^{28} +5.98542e6 q^{29} -5.31153e6 q^{30} -169630. q^{31} +1.32709e6 q^{32} -1.40037e6 q^{33} -9.54795e6 q^{34} +2.90930e6 q^{35} -1.05785e6 q^{36} -1.26336e7 q^{37} +1.12628e7 q^{38} -1.55066e7 q^{40} +2.76549e7 q^{41} -9.47396e6 q^{42} -2.27606e7 q^{43} -413155. q^{44} +2.34719e7 q^{45} +4.78270e7 q^{46} -5.32103e7 q^{47} +4.47468e7 q^{48} -3.51644e7 q^{49} -6.86516e6 q^{50} +8.73804e7 q^{51} +3.18756e7 q^{53} +5.42525e6 q^{54} +9.16722e6 q^{55} -2.76584e7 q^{56} -1.03074e8 q^{57} +1.27595e8 q^{58} -1.14800e8 q^{59} +1.43415e7 q^{60} -7.80352e7 q^{61} -3.61610e6 q^{62} +4.18659e7 q^{63} +1.45723e8 q^{64} -2.98526e7 q^{66} -8.40538e7 q^{67} +2.57801e7 q^{68} -4.37700e8 q^{69} +6.20193e7 q^{70} -1.25752e8 q^{71} -2.23145e8 q^{72} +1.88250e8 q^{73} -2.69318e8 q^{74} +6.28282e7 q^{75} -3.04103e7 q^{76} +1.63512e7 q^{77} -4.28673e8 q^{79} -2.92926e8 q^{80} -4.11395e8 q^{81} +5.89536e8 q^{82} -2.43067e8 q^{83} +2.55803e7 q^{84} -5.72018e8 q^{85} -4.85202e8 q^{86} -1.16772e9 q^{87} -8.71520e7 q^{88} -2.92716e8 q^{89} +5.00366e8 q^{90} -1.29136e8 q^{92} +3.30937e7 q^{93} -1.13432e9 q^{94} +6.74754e8 q^{95} -2.58908e8 q^{96} -1.14275e9 q^{97} -7.49622e8 q^{98} +1.31920e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 15 q^{2} + 161 q^{3} + 361 q^{4} - 1803 q^{5} - 5693 q^{6} - 10099 q^{7} - 23151 q^{8} + 61060 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q - 15 q^{2} + 161 q^{3} + 361 q^{4} - 1803 q^{5} - 5693 q^{6} - 10099 q^{7} - 23151 q^{8} + 61060 q^{9} + 84505 q^{10} - 121746 q^{11} + 113389 q^{12} + 8475 q^{14} - 105973 q^{15} - 322463 q^{16} - 495669 q^{17} + 656228 q^{18} + 840738 q^{19} + 1595607 q^{20} + 1599467 q^{21} - 2023594 q^{22} - 592152 q^{23} + 2295657 q^{24} + 1670362 q^{25} + 6847883 q^{27} - 2587955 q^{28} + 10678182 q^{29} + 5491201 q^{30} - 12885296 q^{31} - 3282927 q^{32} - 17278298 q^{33} + 9934079 q^{34} + 8380731 q^{35} - 20483302 q^{36} - 7171823 q^{37} - 25568814 q^{38} - 54359445 q^{40} - 9294012 q^{41} - 69520457 q^{42} + 12831975 q^{43} + 41479074 q^{44} - 26135198 q^{45} + 59319696 q^{46} - 43354215 q^{47} - 86874671 q^{48} + 25249488 q^{49} + 16270770 q^{50} + 16905901 q^{51} + 93231780 q^{53} - 58983719 q^{54} + 99448846 q^{55} + 199599225 q^{56} - 90173382 q^{57} - 151020970 q^{58} - 246496182 q^{59} - 90097913 q^{60} - 132232612 q^{61} + 158135724 q^{62} + 416955202 q^{63} + 91019105 q^{64} - 323733130 q^{66} + 369388534 q^{67} + 238172073 q^{68} - 579986760 q^{69} + 144857425 q^{70} - 212150457 q^{71} + 415774278 q^{72} + 252729806 q^{73} + 192105957 q^{74} - 752457788 q^{75} + 953775990 q^{76} + 449666118 q^{77} - 1247271728 q^{79} - 900649725 q^{80} - 317713115 q^{81} + 169559388 q^{82} - 1696894296 q^{83} - 1247983739 q^{84} + 775363765 q^{85} - 3291621459 q^{86} - 614530466 q^{87} - 220227222 q^{88} + 753854382 q^{89} + 2296265882 q^{90} + 13876128 q^{92} + 892784668 q^{93} + 272071215 q^{94} + 1442632962 q^{95} - 930612847 q^{96} - 3824606 q^{97} - 1570614816 q^{98} - 3016199848 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 21.3176 0.942115 0.471057 0.882103i \(-0.343872\pi\)
0.471057 + 0.882103i \(0.343872\pi\)
\(3\) −195.094 −1.39058 −0.695292 0.718727i \(-0.744725\pi\)
−0.695292 + 0.718727i \(0.744725\pi\)
\(4\) −57.5590 −0.112420
\(5\) 1277.14 0.913846 0.456923 0.889506i \(-0.348951\pi\)
0.456923 + 0.889506i \(0.348951\pi\)
\(6\) −4158.93 −1.31009
\(7\) 2277.98 0.358599 0.179299 0.983795i \(-0.442617\pi\)
0.179299 + 0.983795i \(0.442617\pi\)
\(8\) −12141.6 −1.04803
\(9\) 18378.5 0.933725
\(10\) 27225.6 0.860948
\(11\) 7177.94 0.147820 0.0739099 0.997265i \(-0.476452\pi\)
0.0739099 + 0.997265i \(0.476452\pi\)
\(12\) 11229.4 0.156329
\(13\) 0 0
\(14\) 48561.1 0.337841
\(15\) −249162. −1.27078
\(16\) −229361. −0.874942
\(17\) −447890. −1.30062 −0.650311 0.759668i \(-0.725361\pi\)
−0.650311 + 0.759668i \(0.725361\pi\)
\(18\) 391786. 0.879677
\(19\) 528333. 0.930071 0.465036 0.885292i \(-0.346041\pi\)
0.465036 + 0.885292i \(0.346041\pi\)
\(20\) −73510.8 −0.102734
\(21\) −444419. −0.498662
\(22\) 153017. 0.139263
\(23\) 2.24354e6 1.67170 0.835851 0.548957i \(-0.184975\pi\)
0.835851 + 0.548957i \(0.184975\pi\)
\(24\) 2.36876e6 1.45737
\(25\) −322041. −0.164885
\(26\) 0 0
\(27\) 254496. 0.0921603
\(28\) −131118. −0.0403136
\(29\) 5.98542e6 1.57146 0.785730 0.618569i \(-0.212287\pi\)
0.785730 + 0.618569i \(0.212287\pi\)
\(30\) −5.31153e6 −1.19722
\(31\) −169630. −0.0329894 −0.0164947 0.999864i \(-0.505251\pi\)
−0.0164947 + 0.999864i \(0.505251\pi\)
\(32\) 1.32709e6 0.223731
\(33\) −1.40037e6 −0.205556
\(34\) −9.54795e6 −1.22534
\(35\) 2.90930e6 0.327704
\(36\) −1.05785e6 −0.104969
\(37\) −1.26336e7 −1.10820 −0.554100 0.832450i \(-0.686938\pi\)
−0.554100 + 0.832450i \(0.686938\pi\)
\(38\) 1.12628e7 0.876234
\(39\) 0 0
\(40\) −1.55066e7 −0.957736
\(41\) 2.76549e7 1.52843 0.764213 0.644964i \(-0.223128\pi\)
0.764213 + 0.644964i \(0.223128\pi\)
\(42\) −9.47396e6 −0.469796
\(43\) −2.27606e7 −1.01526 −0.507628 0.861576i \(-0.669478\pi\)
−0.507628 + 0.861576i \(0.669478\pi\)
\(44\) −413155. −0.0166179
\(45\) 2.34719e7 0.853281
\(46\) 4.78270e7 1.57493
\(47\) −5.32103e7 −1.59058 −0.795289 0.606230i \(-0.792681\pi\)
−0.795289 + 0.606230i \(0.792681\pi\)
\(48\) 4.47468e7 1.21668
\(49\) −3.51644e7 −0.871407
\(50\) −6.86516e6 −0.155341
\(51\) 8.73804e7 1.80862
\(52\) 0 0
\(53\) 3.18756e7 0.554904 0.277452 0.960740i \(-0.410510\pi\)
0.277452 + 0.960740i \(0.410510\pi\)
\(54\) 5.42525e6 0.0868256
\(55\) 9.16722e6 0.135085
\(56\) −2.76584e7 −0.375821
\(57\) −1.03074e8 −1.29334
\(58\) 1.27595e8 1.48050
\(59\) −1.14800e8 −1.23341 −0.616705 0.787194i \(-0.711533\pi\)
−0.616705 + 0.787194i \(0.711533\pi\)
\(60\) 1.43415e7 0.142861
\(61\) −7.80352e7 −0.721616 −0.360808 0.932640i \(-0.617499\pi\)
−0.360808 + 0.932640i \(0.617499\pi\)
\(62\) −3.61610e6 −0.0310798
\(63\) 4.18659e7 0.334833
\(64\) 1.45723e8 1.08572
\(65\) 0 0
\(66\) −2.98526e7 −0.193657
\(67\) −8.40538e7 −0.509590 −0.254795 0.966995i \(-0.582008\pi\)
−0.254795 + 0.966995i \(0.582008\pi\)
\(68\) 2.57801e7 0.146216
\(69\) −4.37700e8 −2.32464
\(70\) 6.20193e7 0.308735
\(71\) −1.25752e8 −0.587288 −0.293644 0.955915i \(-0.594868\pi\)
−0.293644 + 0.955915i \(0.594868\pi\)
\(72\) −2.23145e8 −0.978570
\(73\) 1.88250e8 0.775859 0.387929 0.921689i \(-0.373190\pi\)
0.387929 + 0.921689i \(0.373190\pi\)
\(74\) −2.69318e8 −1.04405
\(75\) 6.28282e7 0.229287
\(76\) −3.04103e7 −0.104558
\(77\) 1.63512e7 0.0530080
\(78\) 0 0
\(79\) −4.28673e8 −1.23824 −0.619120 0.785297i \(-0.712510\pi\)
−0.619120 + 0.785297i \(0.712510\pi\)
\(80\) −2.92926e8 −0.799562
\(81\) −4.11395e8 −1.06188
\(82\) 5.89536e8 1.43995
\(83\) −2.43067e8 −0.562180 −0.281090 0.959681i \(-0.590696\pi\)
−0.281090 + 0.959681i \(0.590696\pi\)
\(84\) 2.55803e7 0.0560594
\(85\) −5.72018e8 −1.18857
\(86\) −4.85202e8 −0.956488
\(87\) −1.16772e9 −2.18525
\(88\) −8.71520e7 −0.154919
\(89\) −2.92716e8 −0.494529 −0.247265 0.968948i \(-0.579532\pi\)
−0.247265 + 0.968948i \(0.579532\pi\)
\(90\) 5.00366e8 0.803889
\(91\) 0 0
\(92\) −1.29136e8 −0.187932
\(93\) 3.30937e7 0.0458746
\(94\) −1.13432e9 −1.49851
\(95\) 6.74754e8 0.849942
\(96\) −2.58908e8 −0.311117
\(97\) −1.14275e9 −1.31063 −0.655313 0.755358i \(-0.727463\pi\)
−0.655313 + 0.755358i \(0.727463\pi\)
\(98\) −7.49622e8 −0.820965
\(99\) 1.31920e8 0.138023
\(100\) 1.85364e7 0.0185364
\(101\) −8.98629e8 −0.859279 −0.429640 0.903000i \(-0.641359\pi\)
−0.429640 + 0.903000i \(0.641359\pi\)
\(102\) 1.86274e9 1.70393
\(103\) 6.13518e8 0.537106 0.268553 0.963265i \(-0.413455\pi\)
0.268553 + 0.963265i \(0.413455\pi\)
\(104\) 0 0
\(105\) −5.67585e8 −0.455700
\(106\) 6.79513e8 0.522783
\(107\) 1.46257e9 1.07868 0.539338 0.842090i \(-0.318675\pi\)
0.539338 + 0.842090i \(0.318675\pi\)
\(108\) −1.46485e7 −0.0103607
\(109\) −7.17715e8 −0.487005 −0.243502 0.969900i \(-0.578296\pi\)
−0.243502 + 0.969900i \(0.578296\pi\)
\(110\) 1.95423e8 0.127265
\(111\) 2.46473e9 1.54105
\(112\) −5.22479e8 −0.313753
\(113\) 9.27444e7 0.0535099 0.0267550 0.999642i \(-0.491483\pi\)
0.0267550 + 0.999642i \(0.491483\pi\)
\(114\) −2.19730e9 −1.21848
\(115\) 2.86531e9 1.52768
\(116\) −3.44514e8 −0.176663
\(117\) 0 0
\(118\) −2.44726e9 −1.16201
\(119\) −1.02028e9 −0.466401
\(120\) 3.02523e9 1.33181
\(121\) −2.30642e9 −0.978149
\(122\) −1.66352e9 −0.679845
\(123\) −5.39529e9 −2.12540
\(124\) 9.76371e6 0.00370866
\(125\) −2.90570e9 −1.06453
\(126\) 8.92481e8 0.315451
\(127\) 4.40060e9 1.50105 0.750524 0.660843i \(-0.229801\pi\)
0.750524 + 0.660843i \(0.229801\pi\)
\(128\) 2.42700e9 0.799144
\(129\) 4.44045e9 1.41180
\(130\) 0 0
\(131\) −1.21951e9 −0.361798 −0.180899 0.983502i \(-0.557901\pi\)
−0.180899 + 0.983502i \(0.557901\pi\)
\(132\) 8.06038e7 0.0231086
\(133\) 1.20353e9 0.333522
\(134\) −1.79183e9 −0.480092
\(135\) 3.25027e8 0.0842204
\(136\) 5.43812e9 1.36309
\(137\) 6.73258e9 1.63282 0.816411 0.577472i \(-0.195961\pi\)
0.816411 + 0.577472i \(0.195961\pi\)
\(138\) −9.33073e9 −2.19008
\(139\) 4.12694e8 0.0937695 0.0468847 0.998900i \(-0.485071\pi\)
0.0468847 + 0.998900i \(0.485071\pi\)
\(140\) −1.67456e8 −0.0368404
\(141\) 1.03810e10 2.21183
\(142\) −2.68073e9 −0.553293
\(143\) 0 0
\(144\) −4.21531e9 −0.816956
\(145\) 7.64421e9 1.43607
\(146\) 4.01305e9 0.730948
\(147\) 6.86035e9 1.21177
\(148\) 7.27175e8 0.124584
\(149\) −7.17339e9 −1.19230 −0.596151 0.802872i \(-0.703304\pi\)
−0.596151 + 0.802872i \(0.703304\pi\)
\(150\) 1.33935e9 0.216015
\(151\) −9.30445e9 −1.45645 −0.728223 0.685340i \(-0.759654\pi\)
−0.728223 + 0.685340i \(0.759654\pi\)
\(152\) −6.41483e9 −0.974740
\(153\) −8.23155e9 −1.21442
\(154\) 3.48569e8 0.0499396
\(155\) −2.16641e8 −0.0301472
\(156\) 0 0
\(157\) −5.86131e9 −0.769921 −0.384961 0.922933i \(-0.625785\pi\)
−0.384961 + 0.922933i \(0.625785\pi\)
\(158\) −9.13830e9 −1.16656
\(159\) −6.21873e9 −0.771640
\(160\) 1.69488e9 0.204456
\(161\) 5.11074e9 0.599470
\(162\) −8.76996e9 −1.00041
\(163\) −1.60676e10 −1.78282 −0.891409 0.453199i \(-0.850283\pi\)
−0.891409 + 0.453199i \(0.850283\pi\)
\(164\) −1.59179e9 −0.171825
\(165\) −1.78847e9 −0.187847
\(166\) −5.18162e9 −0.529638
\(167\) 1.49042e9 0.148281 0.0741404 0.997248i \(-0.476379\pi\)
0.0741404 + 0.997248i \(0.476379\pi\)
\(168\) 5.39598e9 0.522611
\(169\) 0 0
\(170\) −1.21941e10 −1.11977
\(171\) 9.70997e9 0.868431
\(172\) 1.31008e9 0.114135
\(173\) −6.62250e9 −0.562101 −0.281050 0.959693i \(-0.590683\pi\)
−0.281050 + 0.959693i \(0.590683\pi\)
\(174\) −2.48930e10 −2.05876
\(175\) −7.33604e8 −0.0591276
\(176\) −1.64634e9 −0.129334
\(177\) 2.23967e10 1.71516
\(178\) −6.24002e9 −0.465903
\(179\) 1.41019e10 1.02669 0.513344 0.858183i \(-0.328406\pi\)
0.513344 + 0.858183i \(0.328406\pi\)
\(180\) −1.35102e9 −0.0959257
\(181\) 2.38898e10 1.65447 0.827234 0.561857i \(-0.189913\pi\)
0.827234 + 0.561857i \(0.189913\pi\)
\(182\) 0 0
\(183\) 1.52242e10 1.00347
\(184\) −2.72403e10 −1.75199
\(185\) −1.61348e10 −1.01272
\(186\) 7.05479e8 0.0432191
\(187\) −3.21493e9 −0.192258
\(188\) 3.06273e9 0.178813
\(189\) 5.79737e8 0.0330486
\(190\) 1.43842e10 0.800743
\(191\) −2.32674e10 −1.26502 −0.632509 0.774553i \(-0.717975\pi\)
−0.632509 + 0.774553i \(0.717975\pi\)
\(192\) −2.84297e10 −1.50979
\(193\) −8.87539e9 −0.460447 −0.230224 0.973138i \(-0.573946\pi\)
−0.230224 + 0.973138i \(0.573946\pi\)
\(194\) −2.43607e10 −1.23476
\(195\) 0 0
\(196\) 2.02403e9 0.0979634
\(197\) 9.96936e9 0.471595 0.235798 0.971802i \(-0.424230\pi\)
0.235798 + 0.971802i \(0.424230\pi\)
\(198\) 2.81222e9 0.130034
\(199\) −5.92044e9 −0.267618 −0.133809 0.991007i \(-0.542721\pi\)
−0.133809 + 0.991007i \(0.542721\pi\)
\(200\) 3.91011e9 0.172804
\(201\) 1.63984e10 0.708628
\(202\) −1.91566e10 −0.809540
\(203\) 1.36347e10 0.563524
\(204\) −5.02953e9 −0.203325
\(205\) 3.53191e10 1.39675
\(206\) 1.30787e10 0.506015
\(207\) 4.12330e10 1.56091
\(208\) 0 0
\(209\) 3.79234e9 0.137483
\(210\) −1.20996e10 −0.429322
\(211\) −3.18290e10 −1.10548 −0.552741 0.833353i \(-0.686418\pi\)
−0.552741 + 0.833353i \(0.686418\pi\)
\(212\) −1.83473e9 −0.0623822
\(213\) 2.45334e10 0.816674
\(214\) 3.11786e10 1.01624
\(215\) −2.90685e10 −0.927788
\(216\) −3.09000e9 −0.0965865
\(217\) −3.86413e8 −0.0118300
\(218\) −1.53000e10 −0.458814
\(219\) −3.67264e10 −1.07890
\(220\) −5.27656e8 −0.0151862
\(221\) 0 0
\(222\) 5.25422e10 1.45184
\(223\) 2.33567e10 0.632470 0.316235 0.948681i \(-0.397581\pi\)
0.316235 + 0.948681i \(0.397581\pi\)
\(224\) 3.02309e9 0.0802298
\(225\) −5.91865e9 −0.153958
\(226\) 1.97709e9 0.0504125
\(227\) −2.96928e10 −0.742223 −0.371112 0.928588i \(-0.621023\pi\)
−0.371112 + 0.928588i \(0.621023\pi\)
\(228\) 5.93285e9 0.145397
\(229\) −4.24937e10 −1.02109 −0.510546 0.859850i \(-0.670557\pi\)
−0.510546 + 0.859850i \(0.670557\pi\)
\(230\) 6.10817e10 1.43925
\(231\) −3.19001e9 −0.0737121
\(232\) −7.26728e10 −1.64693
\(233\) −3.84693e10 −0.855092 −0.427546 0.903994i \(-0.640622\pi\)
−0.427546 + 0.903994i \(0.640622\pi\)
\(234\) 0 0
\(235\) −6.79569e10 −1.45354
\(236\) 6.60776e9 0.138660
\(237\) 8.36314e10 1.72188
\(238\) −2.17500e10 −0.439403
\(239\) 1.98946e10 0.394407 0.197203 0.980363i \(-0.436814\pi\)
0.197203 + 0.980363i \(0.436814\pi\)
\(240\) 5.71479e10 1.11186
\(241\) 5.31240e10 1.01441 0.507206 0.861825i \(-0.330678\pi\)
0.507206 + 0.861825i \(0.330678\pi\)
\(242\) −4.91675e10 −0.921529
\(243\) 7.52513e10 1.38448
\(244\) 4.49162e9 0.0811240
\(245\) −4.49098e10 −0.796332
\(246\) −1.15015e11 −2.00238
\(247\) 0 0
\(248\) 2.05958e9 0.0345738
\(249\) 4.74209e10 0.781758
\(250\) −6.19427e10 −1.00291
\(251\) 1.26269e10 0.200801 0.100400 0.994947i \(-0.467988\pi\)
0.100400 + 0.994947i \(0.467988\pi\)
\(252\) −2.40976e9 −0.0376418
\(253\) 1.61040e10 0.247111
\(254\) 9.38102e10 1.41416
\(255\) 1.11597e11 1.65280
\(256\) −2.28724e10 −0.332837
\(257\) 1.11000e11 1.58717 0.793583 0.608461i \(-0.208213\pi\)
0.793583 + 0.608461i \(0.208213\pi\)
\(258\) 9.46598e10 1.33008
\(259\) −2.87790e10 −0.397399
\(260\) 0 0
\(261\) 1.10003e11 1.46731
\(262\) −2.59971e10 −0.340855
\(263\) 7.01020e10 0.903503 0.451752 0.892144i \(-0.350799\pi\)
0.451752 + 0.892144i \(0.350799\pi\)
\(264\) 1.70028e10 0.215428
\(265\) 4.07096e10 0.507097
\(266\) 2.56564e10 0.314216
\(267\) 5.71071e10 0.687685
\(268\) 4.83805e9 0.0572880
\(269\) 7.41364e10 0.863269 0.431635 0.902049i \(-0.357937\pi\)
0.431635 + 0.902049i \(0.357937\pi\)
\(270\) 6.92880e9 0.0793453
\(271\) −8.92477e10 −1.00516 −0.502580 0.864531i \(-0.667616\pi\)
−0.502580 + 0.864531i \(0.667616\pi\)
\(272\) 1.02728e11 1.13797
\(273\) 0 0
\(274\) 1.43523e11 1.53831
\(275\) −2.31159e9 −0.0243733
\(276\) 2.51936e10 0.261336
\(277\) −6.68313e10 −0.682057 −0.341028 0.940053i \(-0.610775\pi\)
−0.341028 + 0.940053i \(0.610775\pi\)
\(278\) 8.79765e9 0.0883416
\(279\) −3.11754e9 −0.0308031
\(280\) −3.53236e10 −0.343443
\(281\) 8.13670e10 0.778520 0.389260 0.921128i \(-0.372731\pi\)
0.389260 + 0.921128i \(0.372731\pi\)
\(282\) 2.21298e11 2.08380
\(283\) −1.19236e11 −1.10502 −0.552508 0.833507i \(-0.686329\pi\)
−0.552508 + 0.833507i \(0.686329\pi\)
\(284\) 7.23814e9 0.0660229
\(285\) −1.31640e11 −1.18192
\(286\) 0 0
\(287\) 6.29972e10 0.548091
\(288\) 2.43900e10 0.208904
\(289\) 8.20174e10 0.691617
\(290\) 1.62956e11 1.35295
\(291\) 2.22943e11 1.82254
\(292\) −1.08355e10 −0.0872219
\(293\) 6.95594e10 0.551381 0.275690 0.961246i \(-0.411094\pi\)
0.275690 + 0.961246i \(0.411094\pi\)
\(294\) 1.46246e11 1.14162
\(295\) −1.46615e11 −1.12715
\(296\) 1.53392e11 1.16142
\(297\) 1.82676e9 0.0136231
\(298\) −1.52920e11 −1.12328
\(299\) 0 0
\(300\) −3.61633e9 −0.0257764
\(301\) −5.18482e10 −0.364069
\(302\) −1.98349e11 −1.37214
\(303\) 1.75317e11 1.19490
\(304\) −1.21179e11 −0.813759
\(305\) −9.96618e10 −0.659446
\(306\) −1.75477e11 −1.14413
\(307\) 4.02660e10 0.258711 0.129356 0.991598i \(-0.458709\pi\)
0.129356 + 0.991598i \(0.458709\pi\)
\(308\) −9.41157e8 −0.00595915
\(309\) −1.19693e11 −0.746891
\(310\) −4.61827e9 −0.0284022
\(311\) −1.78556e11 −1.08231 −0.541157 0.840921i \(-0.682014\pi\)
−0.541157 + 0.840921i \(0.682014\pi\)
\(312\) 0 0
\(313\) 9.44550e9 0.0556257 0.0278128 0.999613i \(-0.491146\pi\)
0.0278128 + 0.999613i \(0.491146\pi\)
\(314\) −1.24949e11 −0.725354
\(315\) 5.34685e10 0.305985
\(316\) 2.46740e10 0.139203
\(317\) −5.42593e10 −0.301792 −0.150896 0.988550i \(-0.548216\pi\)
−0.150896 + 0.988550i \(0.548216\pi\)
\(318\) −1.32569e11 −0.726974
\(319\) 4.29630e10 0.232293
\(320\) 1.86109e11 0.992183
\(321\) −2.85339e11 −1.49999
\(322\) 1.08949e11 0.564769
\(323\) −2.36635e11 −1.20967
\(324\) 2.36795e10 0.119377
\(325\) 0 0
\(326\) −3.42523e11 −1.67962
\(327\) 1.40022e11 0.677221
\(328\) −3.35776e11 −1.60183
\(329\) −1.21212e11 −0.570379
\(330\) −3.81259e10 −0.176973
\(331\) −6.66790e10 −0.305325 −0.152663 0.988278i \(-0.548785\pi\)
−0.152663 + 0.988278i \(0.548785\pi\)
\(332\) 1.39907e10 0.0632001
\(333\) −2.32186e11 −1.03476
\(334\) 3.17722e10 0.139697
\(335\) −1.07348e11 −0.465687
\(336\) 1.01932e11 0.436300
\(337\) 2.13457e11 0.901522 0.450761 0.892645i \(-0.351153\pi\)
0.450761 + 0.892645i \(0.351153\pi\)
\(338\) 0 0
\(339\) −1.80938e10 −0.0744101
\(340\) 3.29247e10 0.133619
\(341\) −1.21759e9 −0.00487649
\(342\) 2.06994e11 0.818162
\(343\) −1.72028e11 −0.671084
\(344\) 2.76351e11 1.06402
\(345\) −5.59004e11 −2.12437
\(346\) −1.41176e11 −0.529564
\(347\) −9.49978e10 −0.351748 −0.175874 0.984413i \(-0.556275\pi\)
−0.175874 + 0.984413i \(0.556275\pi\)
\(348\) 6.72126e10 0.245665
\(349\) −2.52647e11 −0.911590 −0.455795 0.890085i \(-0.650645\pi\)
−0.455795 + 0.890085i \(0.650645\pi\)
\(350\) −1.56387e10 −0.0557050
\(351\) 0 0
\(352\) 9.52580e9 0.0330719
\(353\) −4.50376e11 −1.54379 −0.771896 0.635749i \(-0.780691\pi\)
−0.771896 + 0.635749i \(0.780691\pi\)
\(354\) 4.77445e11 1.61588
\(355\) −1.60602e11 −0.536691
\(356\) 1.68485e10 0.0555949
\(357\) 1.99051e11 0.648570
\(358\) 3.00619e11 0.967258
\(359\) −3.93740e11 −1.25108 −0.625539 0.780193i \(-0.715121\pi\)
−0.625539 + 0.780193i \(0.715121\pi\)
\(360\) −2.84988e11 −0.894262
\(361\) −4.35522e10 −0.134967
\(362\) 5.09273e11 1.55870
\(363\) 4.49969e11 1.36020
\(364\) 0 0
\(365\) 2.40422e11 0.709015
\(366\) 3.24543e11 0.945382
\(367\) −2.26305e11 −0.651173 −0.325586 0.945512i \(-0.605562\pi\)
−0.325586 + 0.945512i \(0.605562\pi\)
\(368\) −5.14580e11 −1.46264
\(369\) 5.08255e11 1.42713
\(370\) −3.43956e11 −0.954103
\(371\) 7.26120e10 0.198988
\(372\) −1.90484e9 −0.00515721
\(373\) 4.75023e11 1.27065 0.635323 0.772246i \(-0.280867\pi\)
0.635323 + 0.772246i \(0.280867\pi\)
\(374\) −6.85346e10 −0.181129
\(375\) 5.66884e11 1.48031
\(376\) 6.46060e11 1.66697
\(377\) 0 0
\(378\) 1.23586e10 0.0311355
\(379\) 1.67293e11 0.416486 0.208243 0.978077i \(-0.433225\pi\)
0.208243 + 0.978077i \(0.433225\pi\)
\(380\) −3.88381e10 −0.0955504
\(381\) −8.58528e11 −2.08733
\(382\) −4.96005e11 −1.19179
\(383\) 5.36589e11 1.27423 0.637114 0.770769i \(-0.280128\pi\)
0.637114 + 0.770769i \(0.280128\pi\)
\(384\) −4.73492e11 −1.11128
\(385\) 2.08827e10 0.0484411
\(386\) −1.89202e11 −0.433794
\(387\) −4.18306e11 −0.947971
\(388\) 6.57755e10 0.147340
\(389\) 5.95613e11 1.31884 0.659418 0.751776i \(-0.270803\pi\)
0.659418 + 0.751776i \(0.270803\pi\)
\(390\) 0 0
\(391\) −1.00486e12 −2.17425
\(392\) 4.26954e11 0.913258
\(393\) 2.37919e11 0.503110
\(394\) 2.12523e11 0.444297
\(395\) −5.47475e11 −1.13156
\(396\) −7.59317e9 −0.0155165
\(397\) 6.68797e11 1.35125 0.675627 0.737243i \(-0.263873\pi\)
0.675627 + 0.737243i \(0.263873\pi\)
\(398\) −1.26210e11 −0.252127
\(399\) −2.34801e11 −0.463791
\(400\) 7.38637e10 0.144265
\(401\) −4.27579e11 −0.825785 −0.412892 0.910780i \(-0.635481\pi\)
−0.412892 + 0.910780i \(0.635481\pi\)
\(402\) 3.49574e11 0.667609
\(403\) 0 0
\(404\) 5.17241e10 0.0966000
\(405\) −5.25408e11 −0.970397
\(406\) 2.90658e11 0.530904
\(407\) −9.06830e10 −0.163814
\(408\) −1.06094e12 −1.89549
\(409\) 4.37893e10 0.0773772 0.0386886 0.999251i \(-0.487682\pi\)
0.0386886 + 0.999251i \(0.487682\pi\)
\(410\) 7.52919e11 1.31589
\(411\) −1.31348e12 −2.27058
\(412\) −3.53135e10 −0.0603813
\(413\) −2.61512e11 −0.442299
\(414\) 8.78989e11 1.47056
\(415\) −3.10431e11 −0.513746
\(416\) 0 0
\(417\) −8.05139e10 −0.130394
\(418\) 8.08437e10 0.129525
\(419\) −1.14243e11 −0.181079 −0.0905395 0.995893i \(-0.528859\pi\)
−0.0905395 + 0.995893i \(0.528859\pi\)
\(420\) 3.26696e10 0.0512297
\(421\) 5.06520e11 0.785828 0.392914 0.919575i \(-0.371467\pi\)
0.392914 + 0.919575i \(0.371467\pi\)
\(422\) −6.78518e11 −1.04149
\(423\) −9.77926e11 −1.48516
\(424\) −3.87023e11 −0.581554
\(425\) 1.44239e11 0.214453
\(426\) 5.22993e11 0.769401
\(427\) −1.77763e11 −0.258770
\(428\) −8.41842e10 −0.121264
\(429\) 0 0
\(430\) −6.19670e11 −0.874083
\(431\) 1.32605e11 0.185103 0.0925513 0.995708i \(-0.470498\pi\)
0.0925513 + 0.995708i \(0.470498\pi\)
\(432\) −5.83714e10 −0.0806350
\(433\) 1.21346e12 1.65894 0.829469 0.558553i \(-0.188643\pi\)
0.829469 + 0.558553i \(0.188643\pi\)
\(434\) −8.23741e9 −0.0111452
\(435\) −1.49134e12 −1.99698
\(436\) 4.13110e10 0.0547490
\(437\) 1.18534e12 1.55480
\(438\) −7.82920e11 −1.01644
\(439\) −7.26961e11 −0.934159 −0.467079 0.884215i \(-0.654694\pi\)
−0.467079 + 0.884215i \(0.654694\pi\)
\(440\) −1.11305e11 −0.141572
\(441\) −6.46270e11 −0.813655
\(442\) 0 0
\(443\) −1.02161e12 −1.26028 −0.630142 0.776480i \(-0.717003\pi\)
−0.630142 + 0.776480i \(0.717003\pi\)
\(444\) −1.41867e11 −0.173244
\(445\) −3.73840e11 −0.451924
\(446\) 4.97910e11 0.595859
\(447\) 1.39948e12 1.65800
\(448\) 3.31954e11 0.389339
\(449\) −5.91722e11 −0.687083 −0.343542 0.939137i \(-0.611627\pi\)
−0.343542 + 0.939137i \(0.611627\pi\)
\(450\) −1.26171e11 −0.145046
\(451\) 1.98505e11 0.225932
\(452\) −5.33827e9 −0.00601558
\(453\) 1.81524e12 2.02531
\(454\) −6.32979e11 −0.699259
\(455\) 0 0
\(456\) 1.25149e12 1.35546
\(457\) 8.30664e11 0.890845 0.445423 0.895320i \(-0.353053\pi\)
0.445423 + 0.895320i \(0.353053\pi\)
\(458\) −9.05865e11 −0.961986
\(459\) −1.13986e11 −0.119866
\(460\) −1.64924e11 −0.171741
\(461\) −8.73246e11 −0.900497 −0.450249 0.892903i \(-0.648665\pi\)
−0.450249 + 0.892903i \(0.648665\pi\)
\(462\) −6.80035e10 −0.0694452
\(463\) −7.70415e11 −0.779130 −0.389565 0.920999i \(-0.627375\pi\)
−0.389565 + 0.920999i \(0.627375\pi\)
\(464\) −1.37282e12 −1.37494
\(465\) 4.22652e10 0.0419223
\(466\) −8.20074e11 −0.805595
\(467\) 4.50995e11 0.438779 0.219390 0.975637i \(-0.429593\pi\)
0.219390 + 0.975637i \(0.429593\pi\)
\(468\) 0 0
\(469\) −1.91473e11 −0.182738
\(470\) −1.44868e12 −1.36940
\(471\) 1.14350e12 1.07064
\(472\) 1.39386e12 1.29265
\(473\) −1.63374e11 −0.150075
\(474\) 1.78282e12 1.62221
\(475\) −1.70145e11 −0.153355
\(476\) 5.87265e10 0.0524327
\(477\) 5.85827e11 0.518128
\(478\) 4.24105e11 0.371576
\(479\) 5.26966e11 0.457376 0.228688 0.973500i \(-0.426556\pi\)
0.228688 + 0.973500i \(0.426556\pi\)
\(480\) −3.30661e11 −0.284313
\(481\) 0 0
\(482\) 1.13248e12 0.955692
\(483\) −9.97072e11 −0.833613
\(484\) 1.32755e11 0.109963
\(485\) −1.45945e12 −1.19771
\(486\) 1.60418e12 1.30434
\(487\) −1.88076e12 −1.51514 −0.757570 0.652754i \(-0.773614\pi\)
−0.757570 + 0.652754i \(0.773614\pi\)
\(488\) 9.47475e11 0.756273
\(489\) 3.13469e12 2.47916
\(490\) −9.57371e11 −0.750236
\(491\) 1.68889e12 1.31140 0.655699 0.755022i \(-0.272374\pi\)
0.655699 + 0.755022i \(0.272374\pi\)
\(492\) 3.10547e11 0.238938
\(493\) −2.68081e12 −2.04388
\(494\) 0 0
\(495\) 1.68480e11 0.126132
\(496\) 3.89064e10 0.0288638
\(497\) −2.86460e11 −0.210601
\(498\) 1.01090e12 0.736506
\(499\) −1.60699e12 −1.16027 −0.580137 0.814519i \(-0.697001\pi\)
−0.580137 + 0.814519i \(0.697001\pi\)
\(500\) 1.67249e11 0.119674
\(501\) −2.90772e11 −0.206197
\(502\) 2.69176e11 0.189177
\(503\) 2.73565e12 1.90548 0.952740 0.303786i \(-0.0982507\pi\)
0.952740 + 0.303786i \(0.0982507\pi\)
\(504\) −5.08321e11 −0.350914
\(505\) −1.14767e12 −0.785249
\(506\) 3.43299e11 0.232807
\(507\) 0 0
\(508\) −2.53294e11 −0.168748
\(509\) −7.49256e11 −0.494766 −0.247383 0.968918i \(-0.579571\pi\)
−0.247383 + 0.968918i \(0.579571\pi\)
\(510\) 2.37898e12 1.55713
\(511\) 4.28830e11 0.278222
\(512\) −1.73021e12 −1.11271
\(513\) 1.34459e11 0.0857157
\(514\) 2.36625e12 1.49529
\(515\) 7.83548e11 0.490832
\(516\) −2.55588e11 −0.158714
\(517\) −3.81940e11 −0.235119
\(518\) −6.13500e11 −0.374396
\(519\) 1.29201e12 0.781649
\(520\) 0 0
\(521\) 1.07862e12 0.641355 0.320678 0.947188i \(-0.396089\pi\)
0.320678 + 0.947188i \(0.396089\pi\)
\(522\) 2.34501e12 1.38238
\(523\) −2.69236e12 −1.57353 −0.786765 0.617253i \(-0.788246\pi\)
−0.786765 + 0.617253i \(0.788246\pi\)
\(524\) 7.01939e10 0.0406732
\(525\) 1.43121e11 0.0822219
\(526\) 1.49441e12 0.851204
\(527\) 7.59755e10 0.0429067
\(528\) 3.21190e11 0.179850
\(529\) 3.23232e12 1.79459
\(530\) 8.67832e11 0.477743
\(531\) −2.10985e12 −1.15167
\(532\) −6.92740e10 −0.0374945
\(533\) 0 0
\(534\) 1.21739e12 0.647878
\(535\) 1.86791e12 0.985743
\(536\) 1.02055e12 0.534064
\(537\) −2.75119e12 −1.42770
\(538\) 1.58041e12 0.813299
\(539\) −2.52408e11 −0.128811
\(540\) −1.87082e10 −0.00946804
\(541\) −1.94584e12 −0.976604 −0.488302 0.872675i \(-0.662384\pi\)
−0.488302 + 0.872675i \(0.662384\pi\)
\(542\) −1.90255e12 −0.946976
\(543\) −4.66074e12 −2.30068
\(544\) −5.94392e11 −0.290990
\(545\) −9.16622e11 −0.445047
\(546\) 0 0
\(547\) −7.49274e11 −0.357847 −0.178924 0.983863i \(-0.557261\pi\)
−0.178924 + 0.983863i \(0.557261\pi\)
\(548\) −3.87520e11 −0.183562
\(549\) −1.43417e12 −0.673791
\(550\) −4.92777e10 −0.0229625
\(551\) 3.16229e12 1.46157
\(552\) 5.31440e12 2.43629
\(553\) −9.76509e11 −0.444031
\(554\) −1.42468e12 −0.642576
\(555\) 3.14780e12 1.40828
\(556\) −2.37542e10 −0.0105415
\(557\) −2.43252e12 −1.07080 −0.535399 0.844599i \(-0.679839\pi\)
−0.535399 + 0.844599i \(0.679839\pi\)
\(558\) −6.64586e10 −0.0290200
\(559\) 0 0
\(560\) −6.67278e11 −0.286722
\(561\) 6.27212e11 0.267351
\(562\) 1.73455e12 0.733455
\(563\) −2.26566e12 −0.950401 −0.475200 0.879878i \(-0.657624\pi\)
−0.475200 + 0.879878i \(0.657624\pi\)
\(564\) −5.97518e11 −0.248654
\(565\) 1.18447e11 0.0488999
\(566\) −2.54183e12 −1.04105
\(567\) −9.37149e11 −0.380789
\(568\) 1.52683e12 0.615494
\(569\) 4.39567e12 1.75800 0.879002 0.476817i \(-0.158210\pi\)
0.879002 + 0.476817i \(0.158210\pi\)
\(570\) −2.80626e12 −1.11350
\(571\) 2.81010e12 1.10626 0.553132 0.833094i \(-0.313433\pi\)
0.553132 + 0.833094i \(0.313433\pi\)
\(572\) 0 0
\(573\) 4.53931e12 1.75912
\(574\) 1.34295e12 0.516365
\(575\) −7.22513e11 −0.275639
\(576\) 2.67818e12 1.01377
\(577\) 3.06576e12 1.15146 0.575728 0.817642i \(-0.304719\pi\)
0.575728 + 0.817642i \(0.304719\pi\)
\(578\) 1.74842e12 0.651583
\(579\) 1.73153e12 0.640291
\(580\) −4.39993e11 −0.161443
\(581\) −5.53702e11 −0.201597
\(582\) 4.75262e12 1.71704
\(583\) 2.28801e11 0.0820257
\(584\) −2.28567e12 −0.813121
\(585\) 0 0
\(586\) 1.48284e12 0.519464
\(587\) −1.53677e12 −0.534240 −0.267120 0.963663i \(-0.586072\pi\)
−0.267120 + 0.963663i \(0.586072\pi\)
\(588\) −3.94875e11 −0.136226
\(589\) −8.96210e10 −0.0306825
\(590\) −3.12549e12 −1.06190
\(591\) −1.94496e12 −0.655793
\(592\) 2.89765e12 0.969611
\(593\) 3.34336e12 1.11029 0.555146 0.831753i \(-0.312662\pi\)
0.555146 + 0.831753i \(0.312662\pi\)
\(594\) 3.89421e10 0.0128345
\(595\) −1.30304e12 −0.426219
\(596\) 4.12893e11 0.134038
\(597\) 1.15504e12 0.372145
\(598\) 0 0
\(599\) −9.97101e11 −0.316460 −0.158230 0.987402i \(-0.550579\pi\)
−0.158230 + 0.987402i \(0.550579\pi\)
\(600\) −7.62838e11 −0.240299
\(601\) −2.24332e12 −0.701386 −0.350693 0.936491i \(-0.614054\pi\)
−0.350693 + 0.936491i \(0.614054\pi\)
\(602\) −1.10528e12 −0.342995
\(603\) −1.54478e12 −0.475817
\(604\) 5.35555e11 0.163733
\(605\) −2.94562e12 −0.893878
\(606\) 3.73734e12 1.12573
\(607\) −5.75690e12 −1.72123 −0.860617 0.509253i \(-0.829922\pi\)
−0.860617 + 0.509253i \(0.829922\pi\)
\(608\) 7.01147e11 0.208086
\(609\) −2.66003e12 −0.783627
\(610\) −2.12455e12 −0.621274
\(611\) 0 0
\(612\) 4.73799e11 0.136525
\(613\) −3.44873e12 −0.986479 −0.493239 0.869894i \(-0.664187\pi\)
−0.493239 + 0.869894i \(0.664187\pi\)
\(614\) 8.58375e11 0.243736
\(615\) −6.89053e12 −1.94229
\(616\) −1.98530e11 −0.0555538
\(617\) 5.13402e12 1.42618 0.713089 0.701073i \(-0.247295\pi\)
0.713089 + 0.701073i \(0.247295\pi\)
\(618\) −2.55158e12 −0.703657
\(619\) −1.83851e12 −0.503336 −0.251668 0.967814i \(-0.580979\pi\)
−0.251668 + 0.967814i \(0.580979\pi\)
\(620\) 1.24696e10 0.00338915
\(621\) 5.70972e11 0.154065
\(622\) −3.80640e12 −1.01966
\(623\) −6.66802e11 −0.177338
\(624\) 0 0
\(625\) −3.08200e12 −0.807928
\(626\) 2.01356e11 0.0524058
\(627\) −7.39861e11 −0.191182
\(628\) 3.37371e11 0.0865544
\(629\) 5.65845e12 1.44135
\(630\) 1.13982e12 0.288273
\(631\) −3.01780e12 −0.757805 −0.378903 0.925437i \(-0.623699\pi\)
−0.378903 + 0.925437i \(0.623699\pi\)
\(632\) 5.20480e12 1.29771
\(633\) 6.20963e12 1.53727
\(634\) −1.15668e12 −0.284322
\(635\) 5.62017e12 1.37173
\(636\) 3.57944e11 0.0867477
\(637\) 0 0
\(638\) 9.15868e11 0.218847
\(639\) −2.31113e12 −0.548366
\(640\) 3.09962e12 0.730295
\(641\) 2.47728e12 0.579580 0.289790 0.957090i \(-0.406415\pi\)
0.289790 + 0.957090i \(0.406415\pi\)
\(642\) −6.08274e12 −1.41316
\(643\) −5.96725e12 −1.37665 −0.688327 0.725401i \(-0.741654\pi\)
−0.688327 + 0.725401i \(0.741654\pi\)
\(644\) −2.94169e11 −0.0673923
\(645\) 5.67107e12 1.29017
\(646\) −5.04449e12 −1.13965
\(647\) −4.85153e12 −1.08845 −0.544227 0.838938i \(-0.683177\pi\)
−0.544227 + 0.838938i \(0.683177\pi\)
\(648\) 4.99501e12 1.11288
\(649\) −8.24027e11 −0.182322
\(650\) 0 0
\(651\) 7.53867e10 0.0164506
\(652\) 9.24835e11 0.200424
\(653\) −1.50523e12 −0.323961 −0.161980 0.986794i \(-0.551788\pi\)
−0.161980 + 0.986794i \(0.551788\pi\)
\(654\) 2.98493e12 0.638020
\(655\) −1.55749e12 −0.330628
\(656\) −6.34294e12 −1.33728
\(657\) 3.45976e12 0.724439
\(658\) −2.58395e12 −0.537362
\(659\) −3.29009e12 −0.679552 −0.339776 0.940506i \(-0.610351\pi\)
−0.339776 + 0.940506i \(0.610351\pi\)
\(660\) 1.02942e11 0.0211177
\(661\) −7.79224e12 −1.58765 −0.793827 0.608144i \(-0.791914\pi\)
−0.793827 + 0.608144i \(0.791914\pi\)
\(662\) −1.42144e12 −0.287652
\(663\) 0 0
\(664\) 2.95124e12 0.589179
\(665\) 1.53708e12 0.304788
\(666\) −4.94966e12 −0.974858
\(667\) 1.34285e13 2.62701
\(668\) −8.57871e10 −0.0166697
\(669\) −4.55675e12 −0.879503
\(670\) −2.28841e12 −0.438730
\(671\) −5.60132e11 −0.106669
\(672\) −5.89786e11 −0.111566
\(673\) −6.76161e12 −1.27052 −0.635261 0.772297i \(-0.719108\pi\)
−0.635261 + 0.772297i \(0.719108\pi\)
\(674\) 4.55040e12 0.849337
\(675\) −8.19583e10 −0.0151959
\(676\) 0 0
\(677\) 6.04459e12 1.10590 0.552952 0.833213i \(-0.313501\pi\)
0.552952 + 0.833213i \(0.313501\pi\)
\(678\) −3.85718e11 −0.0701029
\(679\) −2.60316e12 −0.469988
\(680\) 6.94523e12 1.24565
\(681\) 5.79287e12 1.03212
\(682\) −2.59562e10 −0.00459421
\(683\) −3.24587e12 −0.570740 −0.285370 0.958417i \(-0.592116\pi\)
−0.285370 + 0.958417i \(0.592116\pi\)
\(684\) −5.58896e11 −0.0976289
\(685\) 8.59843e12 1.49215
\(686\) −3.66724e12 −0.632238
\(687\) 8.29025e12 1.41992
\(688\) 5.22039e12 0.888290
\(689\) 0 0
\(690\) −1.19166e13 −2.00140
\(691\) 4.14584e12 0.691769 0.345885 0.938277i \(-0.387579\pi\)
0.345885 + 0.938277i \(0.387579\pi\)
\(692\) 3.81184e11 0.0631913
\(693\) 3.00511e11 0.0494949
\(694\) −2.02513e12 −0.331387
\(695\) 5.27067e11 0.0856908
\(696\) 1.41780e13 2.29020
\(697\) −1.23863e13 −1.98790
\(698\) −5.38583e12 −0.858822
\(699\) 7.50512e12 1.18908
\(700\) 4.22255e10 0.00664711
\(701\) −5.47240e12 −0.855947 −0.427974 0.903791i \(-0.640772\pi\)
−0.427974 + 0.903791i \(0.640772\pi\)
\(702\) 0 0
\(703\) −6.67473e12 −1.03071
\(704\) 1.04599e12 0.160491
\(705\) 1.32580e13 2.02128
\(706\) −9.60094e12 −1.45443
\(707\) −2.04706e12 −0.308136
\(708\) −1.28913e12 −0.192818
\(709\) −1.68810e12 −0.250895 −0.125447 0.992100i \(-0.540037\pi\)
−0.125447 + 0.992100i \(0.540037\pi\)
\(710\) −3.42366e12 −0.505625
\(711\) −7.87838e12 −1.15618
\(712\) 3.55406e12 0.518280
\(713\) −3.80571e11 −0.0551484
\(714\) 4.24329e12 0.611028
\(715\) 0 0
\(716\) −8.11690e11 −0.115420
\(717\) −3.88131e12 −0.548456
\(718\) −8.39360e12 −1.17866
\(719\) −3.99409e12 −0.557363 −0.278681 0.960384i \(-0.589897\pi\)
−0.278681 + 0.960384i \(0.589897\pi\)
\(720\) −5.38354e12 −0.746572
\(721\) 1.39758e12 0.192605
\(722\) −9.28430e11 −0.127154
\(723\) −1.03642e13 −1.41062
\(724\) −1.37507e12 −0.185995
\(725\) −1.92755e12 −0.259111
\(726\) 9.59226e12 1.28146
\(727\) −1.96928e12 −0.261459 −0.130730 0.991418i \(-0.541732\pi\)
−0.130730 + 0.991418i \(0.541732\pi\)
\(728\) 0 0
\(729\) −6.58356e12 −0.863350
\(730\) 5.12522e12 0.667974
\(731\) 1.01942e13 1.32046
\(732\) −8.76287e11 −0.112810
\(733\) −2.48265e12 −0.317649 −0.158825 0.987307i \(-0.550770\pi\)
−0.158825 + 0.987307i \(0.550770\pi\)
\(734\) −4.82428e12 −0.613479
\(735\) 8.76162e12 1.10737
\(736\) 2.97739e12 0.374012
\(737\) −6.03333e11 −0.0753275
\(738\) 1.08348e13 1.34452
\(739\) 8.83198e12 1.08933 0.544664 0.838655i \(-0.316657\pi\)
0.544664 + 0.838655i \(0.316657\pi\)
\(740\) 9.28704e11 0.113850
\(741\) 0 0
\(742\) 1.54792e12 0.187469
\(743\) −5.88065e12 −0.707906 −0.353953 0.935263i \(-0.615163\pi\)
−0.353953 + 0.935263i \(0.615163\pi\)
\(744\) −4.01812e11 −0.0480778
\(745\) −9.16141e12 −1.08958
\(746\) 1.01264e13 1.19710
\(747\) −4.46722e12 −0.524921
\(748\) 1.85048e11 0.0216136
\(749\) 3.33171e12 0.386811
\(750\) 1.20846e13 1.39462
\(751\) −9.39265e12 −1.07748 −0.538739 0.842473i \(-0.681099\pi\)
−0.538739 + 0.842473i \(0.681099\pi\)
\(752\) 1.22043e13 1.39166
\(753\) −2.46343e12 −0.279230
\(754\) 0 0
\(755\) −1.18831e13 −1.33097
\(756\) −3.33690e10 −0.00371531
\(757\) 1.20690e13 1.33579 0.667897 0.744254i \(-0.267195\pi\)
0.667897 + 0.744254i \(0.267195\pi\)
\(758\) 3.56628e12 0.392378
\(759\) −3.14179e12 −0.343628
\(760\) −8.19263e12 −0.890762
\(761\) −1.73414e12 −0.187436 −0.0937182 0.995599i \(-0.529875\pi\)
−0.0937182 + 0.995599i \(0.529875\pi\)
\(762\) −1.83018e13 −1.96651
\(763\) −1.63494e12 −0.174639
\(764\) 1.33924e12 0.142213
\(765\) −1.05128e13 −1.10980
\(766\) 1.14388e13 1.20047
\(767\) 0 0
\(768\) 4.46226e12 0.462839
\(769\) 1.87490e13 1.93335 0.966675 0.256006i \(-0.0824068\pi\)
0.966675 + 0.256006i \(0.0824068\pi\)
\(770\) 4.45170e11 0.0456371
\(771\) −2.16553e13 −2.20709
\(772\) 5.10858e11 0.0517634
\(773\) 2.23308e12 0.224956 0.112478 0.993654i \(-0.464121\pi\)
0.112478 + 0.993654i \(0.464121\pi\)
\(774\) −8.91729e12 −0.893097
\(775\) 5.46278e10 0.00543947
\(776\) 1.38749e13 1.37357
\(777\) 5.61460e12 0.552617
\(778\) 1.26971e13 1.24250
\(779\) 1.46110e13 1.42154
\(780\) 0 0
\(781\) −9.02638e11 −0.0868129
\(782\) −2.14212e13 −2.04839
\(783\) 1.52327e12 0.144826
\(784\) 8.06534e12 0.762431
\(785\) −7.48571e12 −0.703590
\(786\) 5.07188e12 0.473988
\(787\) −7.05497e12 −0.655555 −0.327777 0.944755i \(-0.606300\pi\)
−0.327777 + 0.944755i \(0.606300\pi\)
\(788\) −5.73826e11 −0.0530166
\(789\) −1.36765e13 −1.25640
\(790\) −1.16709e13 −1.06606
\(791\) 2.11270e11 0.0191886
\(792\) −1.60172e12 −0.144652
\(793\) 0 0
\(794\) 1.42572e13 1.27304
\(795\) −7.94219e12 −0.705161
\(796\) 3.40774e11 0.0300855
\(797\) 1.16654e13 1.02409 0.512045 0.858959i \(-0.328888\pi\)
0.512045 + 0.858959i \(0.328888\pi\)
\(798\) −5.00540e12 −0.436944
\(799\) 2.38323e13 2.06874
\(800\) −4.27379e11 −0.0368900
\(801\) −5.37969e12 −0.461755
\(802\) −9.11497e12 −0.777984
\(803\) 1.35125e12 0.114687
\(804\) −9.43873e11 −0.0796638
\(805\) 6.52712e12 0.547823
\(806\) 0 0
\(807\) −1.44635e13 −1.20045
\(808\) 1.09108e13 0.900548
\(809\) −1.37043e13 −1.12483 −0.562415 0.826855i \(-0.690128\pi\)
−0.562415 + 0.826855i \(0.690128\pi\)
\(810\) −1.12005e13 −0.914225
\(811\) −2.07440e12 −0.168383 −0.0841917 0.996450i \(-0.526831\pi\)
−0.0841917 + 0.996450i \(0.526831\pi\)
\(812\) −7.84797e11 −0.0633512
\(813\) 1.74117e13 1.39776
\(814\) −1.93315e12 −0.154332
\(815\) −2.05206e13 −1.62922
\(816\) −2.00416e13 −1.58244
\(817\) −1.20252e13 −0.944261
\(818\) 9.33483e11 0.0728982
\(819\) 0 0
\(820\) −2.03293e12 −0.157022
\(821\) 1.51952e13 1.16724 0.583622 0.812026i \(-0.301635\pi\)
0.583622 + 0.812026i \(0.301635\pi\)
\(822\) −2.80003e13 −2.13914
\(823\) 8.32881e12 0.632825 0.316412 0.948622i \(-0.397522\pi\)
0.316412 + 0.948622i \(0.397522\pi\)
\(824\) −7.44912e12 −0.562901
\(825\) 4.50977e11 0.0338931
\(826\) −5.57481e12 −0.416697
\(827\) 1.50997e13 1.12252 0.561258 0.827641i \(-0.310317\pi\)
0.561258 + 0.827641i \(0.310317\pi\)
\(828\) −2.37333e12 −0.175477
\(829\) −9.65383e12 −0.709912 −0.354956 0.934883i \(-0.615504\pi\)
−0.354956 + 0.934883i \(0.615504\pi\)
\(830\) −6.61764e12 −0.484007
\(831\) 1.30384e13 0.948458
\(832\) 0 0
\(833\) 1.57498e13 1.13337
\(834\) −1.71637e12 −0.122846
\(835\) 1.90347e12 0.135506
\(836\) −2.18283e11 −0.0154558
\(837\) −4.31701e10 −0.00304032
\(838\) −2.43540e12 −0.170597
\(839\) −1.96235e13 −1.36725 −0.683623 0.729835i \(-0.739597\pi\)
−0.683623 + 0.729835i \(0.739597\pi\)
\(840\) 6.89141e12 0.477586
\(841\) 2.13181e13 1.46949
\(842\) 1.07978e13 0.740340
\(843\) −1.58742e13 −1.08260
\(844\) 1.83204e12 0.124278
\(845\) 0 0
\(846\) −2.08471e13 −1.39919
\(847\) −5.25399e12 −0.350763
\(848\) −7.31102e12 −0.485508
\(849\) 2.32622e13 1.53662
\(850\) 3.07483e12 0.202040
\(851\) −2.83439e13 −1.85258
\(852\) −1.41211e12 −0.0918104
\(853\) 1.66671e13 1.07793 0.538963 0.842329i \(-0.318816\pi\)
0.538963 + 0.842329i \(0.318816\pi\)
\(854\) −3.78947e12 −0.243791
\(855\) 1.24010e13 0.793613
\(856\) −1.77580e13 −1.13048
\(857\) −8.56434e12 −0.542351 −0.271175 0.962530i \(-0.587412\pi\)
−0.271175 + 0.962530i \(0.587412\pi\)
\(858\) 0 0
\(859\) 1.87649e12 0.117592 0.0587958 0.998270i \(-0.481274\pi\)
0.0587958 + 0.998270i \(0.481274\pi\)
\(860\) 1.67315e12 0.104302
\(861\) −1.22904e13 −0.762167
\(862\) 2.82683e12 0.174388
\(863\) −2.54184e11 −0.0155991 −0.00779956 0.999970i \(-0.502483\pi\)
−0.00779956 + 0.999970i \(0.502483\pi\)
\(864\) 3.37740e11 0.0206192
\(865\) −8.45785e12 −0.513674
\(866\) 2.58681e13 1.56291
\(867\) −1.60011e13 −0.961753
\(868\) 2.22415e10 0.00132992
\(869\) −3.07699e12 −0.183036
\(870\) −3.17918e13 −1.88139
\(871\) 0 0
\(872\) 8.71425e12 0.510394
\(873\) −2.10021e13 −1.22376
\(874\) 2.52685e13 1.46480
\(875\) −6.61913e12 −0.381737
\(876\) 2.11393e12 0.121289
\(877\) 2.88730e13 1.64814 0.824069 0.566490i \(-0.191699\pi\)
0.824069 + 0.566490i \(0.191699\pi\)
\(878\) −1.54971e13 −0.880085
\(879\) −1.35706e13 −0.766742
\(880\) −2.10260e12 −0.118191
\(881\) −1.43284e13 −0.801317 −0.400659 0.916227i \(-0.631219\pi\)
−0.400659 + 0.916227i \(0.631219\pi\)
\(882\) −1.37769e13 −0.766556
\(883\) 3.54065e12 0.196001 0.0980007 0.995186i \(-0.468755\pi\)
0.0980007 + 0.995186i \(0.468755\pi\)
\(884\) 0 0
\(885\) 2.86037e13 1.56739
\(886\) −2.17783e13 −1.18733
\(887\) 1.07018e13 0.580497 0.290249 0.956951i \(-0.406262\pi\)
0.290249 + 0.956951i \(0.406262\pi\)
\(888\) −2.99259e13 −1.61506
\(889\) 1.00245e13 0.538274
\(890\) −7.96937e12 −0.425764
\(891\) −2.95297e12 −0.156967
\(892\) −1.34439e12 −0.0711022
\(893\) −2.81127e13 −1.47935
\(894\) 2.98336e13 1.56202
\(895\) 1.80101e13 0.938235
\(896\) 5.52866e12 0.286572
\(897\) 0 0
\(898\) −1.26141e13 −0.647311
\(899\) −1.01531e12 −0.0518416
\(900\) 3.40671e11 0.0173079
\(901\) −1.42768e13 −0.721720
\(902\) 4.23165e12 0.212853
\(903\) 1.01152e13 0.506269
\(904\) −1.12607e12 −0.0560799
\(905\) 3.05106e13 1.51193
\(906\) 3.86966e13 1.90808
\(907\) 2.74516e13 1.34690 0.673448 0.739234i \(-0.264812\pi\)
0.673448 + 0.739234i \(0.264812\pi\)
\(908\) 1.70909e12 0.0834406
\(909\) −1.65155e13 −0.802331
\(910\) 0 0
\(911\) 1.57903e13 0.759550 0.379775 0.925079i \(-0.376001\pi\)
0.379775 + 0.925079i \(0.376001\pi\)
\(912\) 2.36412e13 1.13160
\(913\) −1.74472e12 −0.0831013
\(914\) 1.77078e13 0.839279
\(915\) 1.94434e13 0.917016
\(916\) 2.44589e12 0.114791
\(917\) −2.77803e12 −0.129740
\(918\) −2.42991e12 −0.112927
\(919\) 3.33575e13 1.54267 0.771335 0.636429i \(-0.219589\pi\)
0.771335 + 0.636429i \(0.219589\pi\)
\(920\) −3.47896e13 −1.60105
\(921\) −7.85564e12 −0.359760
\(922\) −1.86155e13 −0.848372
\(923\) 0 0
\(924\) 1.83614e11 0.00828670
\(925\) 4.06853e12 0.182726
\(926\) −1.64234e13 −0.734030
\(927\) 1.12756e13 0.501509
\(928\) 7.94322e12 0.351585
\(929\) 5.90953e12 0.260305 0.130152 0.991494i \(-0.458453\pi\)
0.130152 + 0.991494i \(0.458453\pi\)
\(930\) 9.00994e11 0.0394956
\(931\) −1.85785e13 −0.810471
\(932\) 2.21425e12 0.0961293
\(933\) 3.48352e13 1.50505
\(934\) 9.61415e12 0.413380
\(935\) −4.10591e12 −0.175694
\(936\) 0 0
\(937\) 2.94135e13 1.24657 0.623287 0.781993i \(-0.285797\pi\)
0.623287 + 0.781993i \(0.285797\pi\)
\(938\) −4.08175e12 −0.172160
\(939\) −1.84276e12 −0.0773522
\(940\) 3.91153e12 0.163407
\(941\) 1.78277e13 0.741212 0.370606 0.928790i \(-0.379150\pi\)
0.370606 + 0.928790i \(0.379150\pi\)
\(942\) 2.43768e13 1.00867
\(943\) 6.20448e13 2.55507
\(944\) 2.63306e13 1.07916
\(945\) 7.40404e11 0.0302013
\(946\) −3.48275e12 −0.141388
\(947\) 2.20380e13 0.890427 0.445213 0.895425i \(-0.353128\pi\)
0.445213 + 0.895425i \(0.353128\pi\)
\(948\) −4.81374e12 −0.193573
\(949\) 0 0
\(950\) −3.62709e12 −0.144478
\(951\) 1.05856e13 0.419667
\(952\) 1.23879e13 0.488801
\(953\) 3.10318e13 1.21868 0.609340 0.792909i \(-0.291435\pi\)
0.609340 + 0.792909i \(0.291435\pi\)
\(954\) 1.24884e13 0.488136
\(955\) −2.97156e13 −1.15603
\(956\) −1.14511e12 −0.0443391
\(957\) −8.38180e12 −0.323023
\(958\) 1.12337e13 0.430900
\(959\) 1.53367e13 0.585527
\(960\) −3.63086e13 −1.37971
\(961\) −2.64108e13 −0.998912
\(962\) 0 0
\(963\) 2.68799e13 1.00719
\(964\) −3.05776e12 −0.114040
\(965\) −1.13351e13 −0.420778
\(966\) −2.12552e13 −0.785359
\(967\) −3.86641e13 −1.42196 −0.710982 0.703210i \(-0.751749\pi\)
−0.710982 + 0.703210i \(0.751749\pi\)
\(968\) 2.80038e13 1.02513
\(969\) 4.61660e13 1.68215
\(970\) −3.11120e13 −1.12838
\(971\) 2.91204e13 1.05126 0.525631 0.850713i \(-0.323829\pi\)
0.525631 + 0.850713i \(0.323829\pi\)
\(972\) −4.33138e12 −0.155643
\(973\) 9.40108e11 0.0336256
\(974\) −4.00933e13 −1.42744
\(975\) 0 0
\(976\) 1.78982e13 0.631372
\(977\) 2.43957e13 0.856618 0.428309 0.903632i \(-0.359109\pi\)
0.428309 + 0.903632i \(0.359109\pi\)
\(978\) 6.68241e13 2.33565
\(979\) −2.10110e12 −0.0731012
\(980\) 2.58496e12 0.0895235
\(981\) −1.31905e13 −0.454729
\(982\) 3.60031e13 1.23549
\(983\) −1.65480e13 −0.565268 −0.282634 0.959228i \(-0.591208\pi\)
−0.282634 + 0.959228i \(0.591208\pi\)
\(984\) 6.55077e13 2.22748
\(985\) 1.27323e13 0.430965
\(986\) −5.71485e13 −1.92557
\(987\) 2.36477e13 0.793160
\(988\) 0 0
\(989\) −5.10643e13 −1.69721
\(990\) 3.59159e12 0.118831
\(991\) −6.37548e12 −0.209982 −0.104991 0.994473i \(-0.533481\pi\)
−0.104991 + 0.994473i \(0.533481\pi\)
\(992\) −2.25115e11 −0.00738077
\(993\) 1.30086e13 0.424581
\(994\) −6.10664e12 −0.198410
\(995\) −7.56122e12 −0.244561
\(996\) −2.72950e12 −0.0878851
\(997\) 1.52049e13 0.487366 0.243683 0.969855i \(-0.421644\pi\)
0.243683 + 0.969855i \(0.421644\pi\)
\(998\) −3.42572e13 −1.09311
\(999\) −3.21519e12 −0.102132
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 169.10.a.b.1.4 5
13.12 even 2 13.10.a.b.1.2 5
39.38 odd 2 117.10.a.e.1.4 5
52.51 odd 2 208.10.a.h.1.5 5
65.64 even 2 325.10.a.b.1.4 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
13.10.a.b.1.2 5 13.12 even 2
117.10.a.e.1.4 5 39.38 odd 2
169.10.a.b.1.4 5 1.1 even 1 trivial
208.10.a.h.1.5 5 52.51 odd 2
325.10.a.b.1.4 5 65.64 even 2