Properties

Label 37.8.a.b.1.1
Level $37$
Weight $8$
Character 37.1
Self dual yes
Analytic conductor $11.558$
Analytic rank $0$
Dimension $11$
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] = [37,8,Mod(1,37)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(37, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("37.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 37 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 37.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: \(11.5582459429\)
Analytic rank: \(0\)
Dimension: \(11\)
Coefficient field: \(\mathbb{Q}[x]/(x^{11} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{11} - 5 x^{10} - 1078 x^{9} + 4966 x^{8} + 379692 x^{7} - 1385588 x^{6} - 48765978 x^{5} + 87529978 x^{4} + 2159400643 x^{3} - 1763707223 x^{2} + \cdots + 6680404080 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{5} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-22.1395\) of defining polynomial
Character \(\chi\) \(=\) 37.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-21.1395 q^{2} +34.2904 q^{3} +318.879 q^{4} +463.709 q^{5} -724.881 q^{6} +1277.57 q^{7} -4035.09 q^{8} -1011.17 q^{9} +O(q^{10})\) \(q-21.1395 q^{2} +34.2904 q^{3} +318.879 q^{4} +463.709 q^{5} -724.881 q^{6} +1277.57 q^{7} -4035.09 q^{8} -1011.17 q^{9} -9802.57 q^{10} +5750.80 q^{11} +10934.5 q^{12} -6791.38 q^{13} -27007.2 q^{14} +15900.7 q^{15} +44483.3 q^{16} +20526.6 q^{17} +21375.7 q^{18} -14879.0 q^{19} +147867. q^{20} +43808.3 q^{21} -121569. q^{22} -45808.6 q^{23} -138365. q^{24} +136901. q^{25} +143566. q^{26} -109666. q^{27} +407390. q^{28} +90610.7 q^{29} -336134. q^{30} -254777. q^{31} -423864. q^{32} +197197. q^{33} -433923. q^{34} +592420. q^{35} -322441. q^{36} -50653.0 q^{37} +314534. q^{38} -232879. q^{39} -1.87110e6 q^{40} +297846. q^{41} -926087. q^{42} +588916. q^{43} +1.83381e6 q^{44} -468889. q^{45} +968371. q^{46} +510631. q^{47} +1.52535e6 q^{48} +808643. q^{49} -2.89401e6 q^{50} +703865. q^{51} -2.16563e6 q^{52} +271225. q^{53} +2.31829e6 q^{54} +2.66670e6 q^{55} -5.15511e6 q^{56} -510205. q^{57} -1.91547e6 q^{58} -1.25856e6 q^{59} +5.07041e6 q^{60} +536625. q^{61} +5.38587e6 q^{62} -1.29184e6 q^{63} +3.26641e6 q^{64} -3.14922e6 q^{65} -4.16865e6 q^{66} -2.35048e6 q^{67} +6.54551e6 q^{68} -1.57079e6 q^{69} -1.25235e7 q^{70} -217328. q^{71} +4.08017e6 q^{72} -5.42444e6 q^{73} +1.07078e6 q^{74} +4.69437e6 q^{75} -4.74459e6 q^{76} +7.34705e6 q^{77} +4.92294e6 q^{78} -2.42081e6 q^{79} +2.06273e7 q^{80} -1.54907e6 q^{81} -6.29632e6 q^{82} +1.00120e7 q^{83} +1.39696e7 q^{84} +9.51837e6 q^{85} -1.24494e7 q^{86} +3.10707e6 q^{87} -2.32050e7 q^{88} -2.99012e6 q^{89} +9.91208e6 q^{90} -8.67646e6 q^{91} -1.46074e7 q^{92} -8.73640e6 q^{93} -1.07945e7 q^{94} -6.89950e6 q^{95} -1.45344e7 q^{96} +8.35669e6 q^{97} -1.70943e7 q^{98} -5.81505e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 11 q + 16 q^{2} + 121 q^{3} + 794 q^{4} + 376 q^{5} + 519 q^{6} + 2243 q^{7} + 3870 q^{8} + 9826 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 11 q + 16 q^{2} + 121 q^{3} + 794 q^{4} + 376 q^{5} + 519 q^{6} + 2243 q^{7} + 3870 q^{8} + 9826 q^{9} - 12629 q^{10} + 9415 q^{11} + 27955 q^{12} + 12512 q^{13} + 18260 q^{14} + 25714 q^{15} + 167866 q^{16} + 54312 q^{17} + 163911 q^{18} + 97192 q^{19} + 85625 q^{20} + 97795 q^{21} - 12345 q^{22} + 107342 q^{23} + 163119 q^{24} + 165051 q^{25} + 61531 q^{26} + 446611 q^{27} + 215454 q^{28} + 41748 q^{29} - 1080964 q^{30} - 272248 q^{31} + 593306 q^{32} - 216525 q^{33} - 923600 q^{34} + 436814 q^{35} - 456119 q^{36} - 557183 q^{37} - 175872 q^{38} - 1587326 q^{39} - 3206863 q^{40} + 525465 q^{41} - 3814396 q^{42} - 1376086 q^{43} - 1337377 q^{44} - 2315492 q^{45} - 2037327 q^{46} + 2269179 q^{47} + 1779791 q^{48} + 2282536 q^{49} - 3881347 q^{50} - 103604 q^{51} - 4200495 q^{52} - 346415 q^{53} + 6349248 q^{54} + 4169374 q^{55} - 4307934 q^{56} + 6170792 q^{57} - 1334849 q^{58} + 4598828 q^{59} - 4448200 q^{60} + 6208418 q^{61} + 4732115 q^{62} + 6882994 q^{63} + 12483426 q^{64} + 9330160 q^{65} - 5715150 q^{66} + 2199016 q^{67} + 8095824 q^{68} + 13516268 q^{69} - 6471708 q^{70} + 4653285 q^{71} + 12839097 q^{72} - 1080699 q^{73} - 810448 q^{74} + 16194855 q^{75} + 1331888 q^{76} + 22058153 q^{77} - 23968103 q^{78} - 1336084 q^{79} - 89443 q^{80} + 9585355 q^{81} + 9689125 q^{82} + 28551309 q^{83} - 37602282 q^{84} + 13256012 q^{85} - 47733694 q^{86} - 5826578 q^{87} - 58704117 q^{88} - 8994788 q^{89} - 46526086 q^{90} - 696642 q^{91} - 41894465 q^{92} - 9859184 q^{93} - 26180048 q^{94} + 124152 q^{95} - 19485621 q^{96} - 3968264 q^{97} - 7312590 q^{98} - 14172918 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −21.1395 −1.86849 −0.934243 0.356636i \(-0.883923\pi\)
−0.934243 + 0.356636i \(0.883923\pi\)
\(3\) 34.2904 0.733242 0.366621 0.930370i \(-0.380514\pi\)
0.366621 + 0.930370i \(0.380514\pi\)
\(4\) 318.879 2.49124
\(5\) 463.709 1.65901 0.829507 0.558496i \(-0.188621\pi\)
0.829507 + 0.558496i \(0.188621\pi\)
\(6\) −724.881 −1.37005
\(7\) 1277.57 1.40780 0.703901 0.710298i \(-0.251440\pi\)
0.703901 + 0.710298i \(0.251440\pi\)
\(8\) −4035.09 −2.78637
\(9\) −1011.17 −0.462355
\(10\) −9802.57 −3.09985
\(11\) 5750.80 1.30273 0.651364 0.758765i \(-0.274197\pi\)
0.651364 + 0.758765i \(0.274197\pi\)
\(12\) 10934.5 1.82668
\(13\) −6791.38 −0.857346 −0.428673 0.903460i \(-0.641019\pi\)
−0.428673 + 0.903460i \(0.641019\pi\)
\(14\) −27007.2 −2.63046
\(15\) 15900.7 1.21646
\(16\) 44483.3 2.71504
\(17\) 20526.6 1.01332 0.506660 0.862146i \(-0.330880\pi\)
0.506660 + 0.862146i \(0.330880\pi\)
\(18\) 21375.7 0.863905
\(19\) −14879.0 −0.497662 −0.248831 0.968547i \(-0.580046\pi\)
−0.248831 + 0.968547i \(0.580046\pi\)
\(20\) 147867. 4.13301
\(21\) 43808.3 1.03226
\(22\) −121569. −2.43413
\(23\) −45808.6 −0.785054 −0.392527 0.919741i \(-0.628399\pi\)
−0.392527 + 0.919741i \(0.628399\pi\)
\(24\) −138365. −2.04308
\(25\) 136901. 1.75233
\(26\) 143566. 1.60194
\(27\) −109666. −1.07226
\(28\) 407390. 3.50718
\(29\) 90610.7 0.689900 0.344950 0.938621i \(-0.387896\pi\)
0.344950 + 0.938621i \(0.387896\pi\)
\(30\) −336134. −2.27294
\(31\) −254777. −1.53601 −0.768006 0.640442i \(-0.778751\pi\)
−0.768006 + 0.640442i \(0.778751\pi\)
\(32\) −423864. −2.28666
\(33\) 197197. 0.955216
\(34\) −433923. −1.89337
\(35\) 592420. 2.33556
\(36\) −322441. −1.15184
\(37\) −50653.0 −0.164399
\(38\) 314534. 0.929876
\(39\) −232879. −0.628643
\(40\) −1.87110e6 −4.62262
\(41\) 297846. 0.674913 0.337457 0.941341i \(-0.390433\pi\)
0.337457 + 0.941341i \(0.390433\pi\)
\(42\) −926087. −1.92876
\(43\) 588916. 1.12957 0.564785 0.825238i \(-0.308959\pi\)
0.564785 + 0.825238i \(0.308959\pi\)
\(44\) 1.83381e6 3.24541
\(45\) −468889. −0.767054
\(46\) 968371. 1.46686
\(47\) 510631. 0.717405 0.358702 0.933452i \(-0.383219\pi\)
0.358702 + 0.933452i \(0.383219\pi\)
\(48\) 1.52535e6 1.99079
\(49\) 808643. 0.981907
\(50\) −2.89401e6 −3.27420
\(51\) 703865. 0.743009
\(52\) −2.16563e6 −2.13586
\(53\) 271225. 0.250245 0.125122 0.992141i \(-0.460068\pi\)
0.125122 + 0.992141i \(0.460068\pi\)
\(54\) 2.31829e6 2.00351
\(55\) 2.66670e6 2.16125
\(56\) −5.15511e6 −3.92265
\(57\) −510205. −0.364907
\(58\) −1.91547e6 −1.28907
\(59\) −1.25856e6 −0.797796 −0.398898 0.916995i \(-0.630607\pi\)
−0.398898 + 0.916995i \(0.630607\pi\)
\(60\) 5.07041e6 3.03050
\(61\) 536625. 0.302703 0.151351 0.988480i \(-0.451637\pi\)
0.151351 + 0.988480i \(0.451637\pi\)
\(62\) 5.38587e6 2.87002
\(63\) −1.29184e6 −0.650905
\(64\) 3.26641e6 1.55755
\(65\) −3.14922e6 −1.42235
\(66\) −4.16865e6 −1.78481
\(67\) −2.35048e6 −0.954762 −0.477381 0.878696i \(-0.658414\pi\)
−0.477381 + 0.878696i \(0.658414\pi\)
\(68\) 6.54551e6 2.52442
\(69\) −1.57079e6 −0.575635
\(70\) −1.25235e7 −4.36397
\(71\) −217328. −0.0720630 −0.0360315 0.999351i \(-0.511472\pi\)
−0.0360315 + 0.999351i \(0.511472\pi\)
\(72\) 4.08017e6 1.28829
\(73\) −5.42444e6 −1.63202 −0.816009 0.578039i \(-0.803818\pi\)
−0.816009 + 0.578039i \(0.803818\pi\)
\(74\) 1.07078e6 0.307177
\(75\) 4.69437e6 1.28488
\(76\) −4.74459e6 −1.23980
\(77\) 7.34705e6 1.83398
\(78\) 4.92294e6 1.17461
\(79\) −2.42081e6 −0.552416 −0.276208 0.961098i \(-0.589078\pi\)
−0.276208 + 0.961098i \(0.589078\pi\)
\(80\) 2.06273e7 4.50430
\(81\) −1.54907e6 −0.323872
\(82\) −6.29632e6 −1.26107
\(83\) 1.00120e7 1.92198 0.960991 0.276581i \(-0.0892015\pi\)
0.960991 + 0.276581i \(0.0892015\pi\)
\(84\) 1.39696e7 2.57161
\(85\) 9.51837e6 1.68111
\(86\) −1.24494e7 −2.11059
\(87\) 3.10707e6 0.505864
\(88\) −2.32050e7 −3.62988
\(89\) −2.99012e6 −0.449598 −0.224799 0.974405i \(-0.572172\pi\)
−0.224799 + 0.974405i \(0.572172\pi\)
\(90\) 9.91208e6 1.43323
\(91\) −8.67646e6 −1.20697
\(92\) −1.46074e7 −1.95576
\(93\) −8.73640e6 −1.12627
\(94\) −1.07945e7 −1.34046
\(95\) −6.89950e6 −0.825629
\(96\) −1.45344e7 −1.67667
\(97\) 8.35669e6 0.929680 0.464840 0.885395i \(-0.346112\pi\)
0.464840 + 0.885395i \(0.346112\pi\)
\(98\) −1.70943e7 −1.83468
\(99\) −5.81505e6 −0.602324
\(100\) 4.36547e7 4.36547
\(101\) −2.81077e6 −0.271457 −0.135729 0.990746i \(-0.543338\pi\)
−0.135729 + 0.990746i \(0.543338\pi\)
\(102\) −1.48794e7 −1.38830
\(103\) 1.63311e6 0.147260 0.0736300 0.997286i \(-0.476542\pi\)
0.0736300 + 0.997286i \(0.476542\pi\)
\(104\) 2.74038e7 2.38888
\(105\) 2.03143e7 1.71253
\(106\) −5.73357e6 −0.467579
\(107\) 1.44703e7 1.14192 0.570959 0.820979i \(-0.306572\pi\)
0.570959 + 0.820979i \(0.306572\pi\)
\(108\) −3.49703e7 −2.67126
\(109\) −1.76263e7 −1.30367 −0.651835 0.758361i \(-0.726000\pi\)
−0.651835 + 0.758361i \(0.726000\pi\)
\(110\) −5.63727e7 −4.03826
\(111\) −1.73691e6 −0.120544
\(112\) 5.68305e7 3.82224
\(113\) −8.66686e6 −0.565050 −0.282525 0.959260i \(-0.591172\pi\)
−0.282525 + 0.959260i \(0.591172\pi\)
\(114\) 1.07855e7 0.681824
\(115\) −2.12418e7 −1.30242
\(116\) 2.88938e7 1.71871
\(117\) 6.86725e6 0.396399
\(118\) 2.66053e7 1.49067
\(119\) 2.62242e7 1.42655
\(120\) −6.41609e7 −3.38950
\(121\) 1.35845e7 0.697102
\(122\) −1.13440e7 −0.565596
\(123\) 1.02132e7 0.494875
\(124\) −8.12431e7 −3.82658
\(125\) 2.72548e7 1.24812
\(126\) 2.73089e7 1.21621
\(127\) 2.07630e7 0.899449 0.449724 0.893167i \(-0.351522\pi\)
0.449724 + 0.893167i \(0.351522\pi\)
\(128\) −1.47958e7 −0.623595
\(129\) 2.01941e7 0.828249
\(130\) 6.65730e7 2.65764
\(131\) −3.89604e7 −1.51417 −0.757084 0.653318i \(-0.773376\pi\)
−0.757084 + 0.653318i \(0.773376\pi\)
\(132\) 6.28820e7 2.37967
\(133\) −1.90089e7 −0.700610
\(134\) 4.96880e7 1.78396
\(135\) −5.08533e7 −1.77890
\(136\) −8.28267e7 −2.82348
\(137\) −1.86250e7 −0.618833 −0.309416 0.950927i \(-0.600134\pi\)
−0.309416 + 0.950927i \(0.600134\pi\)
\(138\) 3.32058e7 1.07557
\(139\) 5.31015e6 0.167708 0.0838542 0.996478i \(-0.473277\pi\)
0.0838542 + 0.996478i \(0.473277\pi\)
\(140\) 1.88910e8 5.81845
\(141\) 1.75097e7 0.526032
\(142\) 4.59421e6 0.134649
\(143\) −3.90559e7 −1.11689
\(144\) −4.49802e7 −1.25532
\(145\) 4.20169e7 1.14455
\(146\) 1.14670e8 3.04940
\(147\) 2.77286e7 0.719976
\(148\) −1.61522e7 −0.409558
\(149\) −2.48865e7 −0.616329 −0.308164 0.951333i \(-0.599715\pi\)
−0.308164 + 0.951333i \(0.599715\pi\)
\(150\) −9.92367e7 −2.40078
\(151\) 3.67328e7 0.868230 0.434115 0.900858i \(-0.357061\pi\)
0.434115 + 0.900858i \(0.357061\pi\)
\(152\) 6.00379e7 1.38667
\(153\) −2.07559e7 −0.468514
\(154\) −1.55313e8 −3.42677
\(155\) −1.18142e8 −2.54827
\(156\) −7.42601e7 −1.56610
\(157\) 4.28398e7 0.883483 0.441741 0.897142i \(-0.354361\pi\)
0.441741 + 0.897142i \(0.354361\pi\)
\(158\) 5.11748e7 1.03218
\(159\) 9.30041e6 0.183490
\(160\) −1.96549e8 −3.79360
\(161\) −5.85237e7 −1.10520
\(162\) 3.27466e7 0.605150
\(163\) −3.01638e7 −0.545543 −0.272772 0.962079i \(-0.587940\pi\)
−0.272772 + 0.962079i \(0.587940\pi\)
\(164\) 9.49768e7 1.68137
\(165\) 9.14420e7 1.58472
\(166\) −2.11650e8 −3.59120
\(167\) 4.97323e7 0.826288 0.413144 0.910666i \(-0.364431\pi\)
0.413144 + 0.910666i \(0.364431\pi\)
\(168\) −1.76770e8 −2.87625
\(169\) −1.66257e7 −0.264958
\(170\) −2.01214e8 −3.14113
\(171\) 1.50452e7 0.230097
\(172\) 1.87793e8 2.81403
\(173\) 3.40863e7 0.500516 0.250258 0.968179i \(-0.419485\pi\)
0.250258 + 0.968179i \(0.419485\pi\)
\(174\) −6.56820e7 −0.945200
\(175\) 1.74900e8 2.46693
\(176\) 2.55815e8 3.53697
\(177\) −4.31564e7 −0.584978
\(178\) 6.32097e7 0.840067
\(179\) −7.71597e7 −1.00555 −0.502776 0.864416i \(-0.667688\pi\)
−0.502776 + 0.864416i \(0.667688\pi\)
\(180\) −1.49519e8 −1.91092
\(181\) −1.42751e8 −1.78939 −0.894695 0.446678i \(-0.852607\pi\)
−0.894695 + 0.446678i \(0.852607\pi\)
\(182\) 1.83416e8 2.25521
\(183\) 1.84011e7 0.221955
\(184\) 1.84842e8 2.18745
\(185\) −2.34882e7 −0.272740
\(186\) 1.84683e8 2.10442
\(187\) 1.18045e8 1.32008
\(188\) 1.62829e8 1.78723
\(189\) −1.40107e8 −1.50953
\(190\) 1.45852e8 1.54268
\(191\) −1.36996e8 −1.42262 −0.711312 0.702876i \(-0.751899\pi\)
−0.711312 + 0.702876i \(0.751899\pi\)
\(192\) 1.12006e8 1.14206
\(193\) −3.22984e7 −0.323393 −0.161696 0.986841i \(-0.551697\pi\)
−0.161696 + 0.986841i \(0.551697\pi\)
\(194\) −1.76656e8 −1.73709
\(195\) −1.07988e8 −1.04293
\(196\) 2.57859e8 2.44617
\(197\) −1.43900e8 −1.34100 −0.670498 0.741911i \(-0.733920\pi\)
−0.670498 + 0.741911i \(0.733920\pi\)
\(198\) 1.22927e8 1.12543
\(199\) 5.36192e6 0.0482319 0.0241160 0.999709i \(-0.492323\pi\)
0.0241160 + 0.999709i \(0.492323\pi\)
\(200\) −5.52406e8 −4.88263
\(201\) −8.05989e7 −0.700072
\(202\) 5.94184e7 0.507214
\(203\) 1.15761e8 0.971243
\(204\) 2.24448e8 1.85101
\(205\) 1.38114e8 1.11969
\(206\) −3.45231e7 −0.275153
\(207\) 4.63203e7 0.362974
\(208\) −3.02103e8 −2.32773
\(209\) −8.55659e7 −0.648319
\(210\) −4.29434e8 −3.19985
\(211\) 753318. 0.00552064 0.00276032 0.999996i \(-0.499121\pi\)
0.00276032 + 0.999996i \(0.499121\pi\)
\(212\) 8.64881e7 0.623420
\(213\) −7.45226e6 −0.0528396
\(214\) −3.05895e8 −2.13366
\(215\) 2.73085e8 1.87397
\(216\) 4.42514e8 2.98771
\(217\) −3.25496e8 −2.16240
\(218\) 3.72611e8 2.43589
\(219\) −1.86006e8 −1.19666
\(220\) 8.50353e8 5.38419
\(221\) −1.39404e8 −0.868765
\(222\) 3.67174e7 0.225235
\(223\) 2.36905e8 1.43057 0.715283 0.698835i \(-0.246298\pi\)
0.715283 + 0.698835i \(0.246298\pi\)
\(224\) −5.41515e8 −3.21916
\(225\) −1.38430e8 −0.810199
\(226\) 1.83213e8 1.05579
\(227\) −9.12788e7 −0.517940 −0.258970 0.965885i \(-0.583383\pi\)
−0.258970 + 0.965885i \(0.583383\pi\)
\(228\) −1.62694e8 −0.909072
\(229\) −7.06190e7 −0.388595 −0.194297 0.980943i \(-0.562243\pi\)
−0.194297 + 0.980943i \(0.562243\pi\)
\(230\) 4.49042e8 2.43355
\(231\) 2.51933e8 1.34476
\(232\) −3.65622e8 −1.92231
\(233\) −1.45360e8 −0.752836 −0.376418 0.926450i \(-0.622844\pi\)
−0.376418 + 0.926450i \(0.622844\pi\)
\(234\) −1.45170e8 −0.740665
\(235\) 2.36784e8 1.19019
\(236\) −4.01328e8 −1.98750
\(237\) −8.30106e7 −0.405055
\(238\) −5.54367e8 −2.66549
\(239\) 2.12567e8 1.00717 0.503585 0.863946i \(-0.332014\pi\)
0.503585 + 0.863946i \(0.332014\pi\)
\(240\) 7.07317e8 3.30274
\(241\) 1.12071e7 0.0515743 0.0257872 0.999667i \(-0.491791\pi\)
0.0257872 + 0.999667i \(0.491791\pi\)
\(242\) −2.87171e8 −1.30253
\(243\) 1.86722e8 0.834784
\(244\) 1.71118e8 0.754106
\(245\) 3.74975e8 1.62900
\(246\) −2.15903e8 −0.924668
\(247\) 1.01049e8 0.426669
\(248\) 1.02805e9 4.27989
\(249\) 3.43316e8 1.40928
\(250\) −5.76153e8 −2.33210
\(251\) −8.14863e7 −0.325257 −0.162629 0.986687i \(-0.551997\pi\)
−0.162629 + 0.986687i \(0.551997\pi\)
\(252\) −4.11941e8 −1.62156
\(253\) −2.63436e8 −1.02271
\(254\) −4.38919e8 −1.68061
\(255\) 3.26388e8 1.23266
\(256\) −1.05325e8 −0.392367
\(257\) 1.82069e8 0.669066 0.334533 0.942384i \(-0.391421\pi\)
0.334533 + 0.942384i \(0.391421\pi\)
\(258\) −4.26894e8 −1.54757
\(259\) −6.47128e7 −0.231441
\(260\) −1.00422e9 −3.54342
\(261\) −9.16229e7 −0.318979
\(262\) 8.23604e8 2.82920
\(263\) −5.00302e8 −1.69585 −0.847924 0.530117i \(-0.822148\pi\)
−0.847924 + 0.530117i \(0.822148\pi\)
\(264\) −7.95707e8 −2.66158
\(265\) 1.25770e8 0.415159
\(266\) 4.01839e8 1.30908
\(267\) −1.02532e8 −0.329664
\(268\) −7.49519e8 −2.37854
\(269\) 3.67711e8 1.15179 0.575895 0.817523i \(-0.304654\pi\)
0.575895 + 0.817523i \(0.304654\pi\)
\(270\) 1.07501e9 3.32384
\(271\) 4.33982e8 1.32458 0.662291 0.749247i \(-0.269584\pi\)
0.662291 + 0.749247i \(0.269584\pi\)
\(272\) 9.13091e8 2.75121
\(273\) −2.97519e8 −0.885004
\(274\) 3.93722e8 1.15628
\(275\) 7.87289e8 2.28281
\(276\) −5.00893e8 −1.43405
\(277\) 5.04242e8 1.42548 0.712738 0.701430i \(-0.247455\pi\)
0.712738 + 0.701430i \(0.247455\pi\)
\(278\) −1.12254e8 −0.313361
\(279\) 2.57623e8 0.710184
\(280\) −2.39047e9 −6.50773
\(281\) 3.34393e7 0.0899052 0.0449526 0.998989i \(-0.485686\pi\)
0.0449526 + 0.998989i \(0.485686\pi\)
\(282\) −3.70147e8 −0.982883
\(283\) −4.04643e8 −1.06125 −0.530627 0.847605i \(-0.678044\pi\)
−0.530627 + 0.847605i \(0.678044\pi\)
\(284\) −6.93014e7 −0.179526
\(285\) −2.36586e8 −0.605386
\(286\) 8.25622e8 2.08689
\(287\) 3.80519e8 0.950145
\(288\) 4.28599e8 1.05725
\(289\) 1.10034e7 0.0268154
\(290\) −8.88218e8 −2.13858
\(291\) 2.86554e8 0.681681
\(292\) −1.72974e9 −4.06575
\(293\) −7.89372e8 −1.83335 −0.916674 0.399635i \(-0.869137\pi\)
−0.916674 + 0.399635i \(0.869137\pi\)
\(294\) −5.86170e8 −1.34527
\(295\) −5.83605e8 −1.32355
\(296\) 2.04389e8 0.458076
\(297\) −6.30670e8 −1.39687
\(298\) 5.26089e8 1.15160
\(299\) 3.11103e8 0.673063
\(300\) 1.49694e9 3.20095
\(301\) 7.52381e8 1.59021
\(302\) −7.76513e8 −1.62228
\(303\) −9.63825e7 −0.199044
\(304\) −6.61865e8 −1.35118
\(305\) 2.48838e8 0.502189
\(306\) 4.38770e8 0.875411
\(307\) −1.98514e7 −0.0391567 −0.0195784 0.999808i \(-0.506232\pi\)
−0.0195784 + 0.999808i \(0.506232\pi\)
\(308\) 2.34282e9 4.56890
\(309\) 5.59999e7 0.107977
\(310\) 2.49747e9 4.76140
\(311\) −6.48559e8 −1.22261 −0.611305 0.791395i \(-0.709355\pi\)
−0.611305 + 0.791395i \(0.709355\pi\)
\(312\) 9.39686e8 1.75163
\(313\) −3.98800e8 −0.735106 −0.367553 0.930003i \(-0.619804\pi\)
−0.367553 + 0.930003i \(0.619804\pi\)
\(314\) −9.05611e8 −1.65078
\(315\) −5.99038e8 −1.07986
\(316\) −7.71947e8 −1.37620
\(317\) 1.08007e9 1.90434 0.952170 0.305569i \(-0.0988466\pi\)
0.952170 + 0.305569i \(0.0988466\pi\)
\(318\) −1.96606e8 −0.342849
\(319\) 5.21084e8 0.898753
\(320\) 1.51466e9 2.58399
\(321\) 4.96192e8 0.837303
\(322\) 1.23716e9 2.06505
\(323\) −3.05415e8 −0.504291
\(324\) −4.93966e8 −0.806843
\(325\) −9.29744e8 −1.50235
\(326\) 6.37648e8 1.01934
\(327\) −6.04411e8 −0.955906
\(328\) −1.20183e9 −1.88056
\(329\) 6.52366e8 1.00996
\(330\) −1.93304e9 −2.96102
\(331\) −4.44536e8 −0.673766 −0.336883 0.941547i \(-0.609373\pi\)
−0.336883 + 0.941547i \(0.609373\pi\)
\(332\) 3.19263e9 4.78812
\(333\) 5.12189e7 0.0760108
\(334\) −1.05132e9 −1.54391
\(335\) −1.08994e9 −1.58396
\(336\) 1.94874e9 2.80263
\(337\) 1.13274e9 1.61223 0.806113 0.591762i \(-0.201567\pi\)
0.806113 + 0.591762i \(0.201567\pi\)
\(338\) 3.51459e8 0.495070
\(339\) −2.97190e8 −0.414319
\(340\) 3.03521e9 4.18805
\(341\) −1.46517e9 −2.00101
\(342\) −3.18048e8 −0.429933
\(343\) −1.90364e7 −0.0254715
\(344\) −2.37633e9 −3.14740
\(345\) −7.28390e8 −0.954986
\(346\) −7.20567e8 −0.935208
\(347\) −9.61527e7 −0.123540 −0.0617701 0.998090i \(-0.519675\pi\)
−0.0617701 + 0.998090i \(0.519675\pi\)
\(348\) 9.90780e8 1.26023
\(349\) −1.44389e9 −1.81821 −0.909105 0.416568i \(-0.863233\pi\)
−0.909105 + 0.416568i \(0.863233\pi\)
\(350\) −3.69731e9 −4.60943
\(351\) 7.44786e8 0.919299
\(352\) −2.43756e9 −2.97889
\(353\) 9.51558e8 1.15139 0.575697 0.817663i \(-0.304731\pi\)
0.575697 + 0.817663i \(0.304731\pi\)
\(354\) 9.12306e8 1.09302
\(355\) −1.00777e8 −0.119553
\(356\) −9.53487e8 −1.12006
\(357\) 8.99237e8 1.04601
\(358\) 1.63112e9 1.87886
\(359\) 7.14118e8 0.814591 0.407296 0.913296i \(-0.366472\pi\)
0.407296 + 0.913296i \(0.366472\pi\)
\(360\) 1.89201e9 2.13729
\(361\) −6.72488e8 −0.752332
\(362\) 3.01769e9 3.34345
\(363\) 4.65819e8 0.511145
\(364\) −2.76674e9 −3.00686
\(365\) −2.51536e9 −2.70754
\(366\) −3.88990e8 −0.414719
\(367\) 4.02540e8 0.425087 0.212543 0.977152i \(-0.431825\pi\)
0.212543 + 0.977152i \(0.431825\pi\)
\(368\) −2.03772e9 −2.13146
\(369\) −3.01173e8 −0.312050
\(370\) 4.96530e8 0.509612
\(371\) 3.46509e8 0.352295
\(372\) −2.78586e9 −2.80581
\(373\) −1.63963e9 −1.63593 −0.817966 0.575266i \(-0.804898\pi\)
−0.817966 + 0.575266i \(0.804898\pi\)
\(374\) −2.49540e9 −2.46655
\(375\) 9.34576e8 0.915177
\(376\) −2.06044e9 −1.99895
\(377\) −6.15371e8 −0.591483
\(378\) 2.96178e9 2.82054
\(379\) 1.85624e9 1.75145 0.875724 0.482812i \(-0.160384\pi\)
0.875724 + 0.482812i \(0.160384\pi\)
\(380\) −2.20011e9 −2.05684
\(381\) 7.11970e8 0.659514
\(382\) 2.89602e9 2.65815
\(383\) −3.89277e8 −0.354049 −0.177024 0.984206i \(-0.556647\pi\)
−0.177024 + 0.984206i \(0.556647\pi\)
\(384\) −5.07352e8 −0.457246
\(385\) 3.40689e9 3.04261
\(386\) 6.82772e8 0.604255
\(387\) −5.95495e8 −0.522263
\(388\) 2.66477e9 2.31606
\(389\) 1.57210e8 0.135412 0.0677059 0.997705i \(-0.478432\pi\)
0.0677059 + 0.997705i \(0.478432\pi\)
\(390\) 2.28281e9 1.94869
\(391\) −9.40295e8 −0.795510
\(392\) −3.26294e9 −2.73595
\(393\) −1.33597e9 −1.11025
\(394\) 3.04197e9 2.50563
\(395\) −1.12255e9 −0.916467
\(396\) −1.85430e9 −1.50053
\(397\) −2.38343e9 −1.91177 −0.955883 0.293746i \(-0.905098\pi\)
−0.955883 + 0.293746i \(0.905098\pi\)
\(398\) −1.13348e8 −0.0901207
\(399\) −6.51822e8 −0.513717
\(400\) 6.08979e9 4.75765
\(401\) −5.12906e8 −0.397222 −0.198611 0.980078i \(-0.563643\pi\)
−0.198611 + 0.980078i \(0.563643\pi\)
\(402\) 1.70382e9 1.30808
\(403\) 1.73029e9 1.31689
\(404\) −8.96297e8 −0.676265
\(405\) −7.18317e8 −0.537308
\(406\) −2.44714e9 −1.81475
\(407\) −2.91295e8 −0.214167
\(408\) −2.84016e9 −2.07029
\(409\) 5.86360e8 0.423772 0.211886 0.977294i \(-0.432039\pi\)
0.211886 + 0.977294i \(0.432039\pi\)
\(410\) −2.91966e9 −2.09213
\(411\) −6.38656e8 −0.453754
\(412\) 5.20764e8 0.366860
\(413\) −1.60790e9 −1.12314
\(414\) −9.79189e8 −0.678212
\(415\) 4.64267e9 3.18859
\(416\) 2.87862e9 1.96046
\(417\) 1.82087e8 0.122971
\(418\) 1.80882e9 1.21138
\(419\) −6.22382e8 −0.413341 −0.206670 0.978411i \(-0.566263\pi\)
−0.206670 + 0.978411i \(0.566263\pi\)
\(420\) 6.47780e9 4.26634
\(421\) −2.20235e9 −1.43847 −0.719233 0.694769i \(-0.755507\pi\)
−0.719233 + 0.694769i \(0.755507\pi\)
\(422\) −1.59248e7 −0.0103152
\(423\) −5.16335e8 −0.331696
\(424\) −1.09442e9 −0.697273
\(425\) 2.81011e9 1.77567
\(426\) 1.57537e8 0.0987301
\(427\) 6.85576e8 0.426146
\(428\) 4.61428e9 2.84479
\(429\) −1.33924e9 −0.818951
\(430\) −5.77289e9 −3.50150
\(431\) 1.93493e9 1.16412 0.582058 0.813148i \(-0.302248\pi\)
0.582058 + 0.813148i \(0.302248\pi\)
\(432\) −4.87832e9 −2.91124
\(433\) 2.83343e9 1.67728 0.838640 0.544687i \(-0.183351\pi\)
0.838640 + 0.544687i \(0.183351\pi\)
\(434\) 6.88082e9 4.04042
\(435\) 1.44078e9 0.839236
\(436\) −5.62064e9 −3.24776
\(437\) 6.81584e8 0.390692
\(438\) 3.93208e9 2.23595
\(439\) 2.70042e8 0.152337 0.0761685 0.997095i \(-0.475731\pi\)
0.0761685 + 0.997095i \(0.475731\pi\)
\(440\) −1.07604e10 −6.02202
\(441\) −8.17676e8 −0.453990
\(442\) 2.94693e9 1.62328
\(443\) 1.71410e9 0.936750 0.468375 0.883530i \(-0.344840\pi\)
0.468375 + 0.883530i \(0.344840\pi\)
\(444\) −5.53864e8 −0.300305
\(445\) −1.38655e9 −0.745889
\(446\) −5.00807e9 −2.67299
\(447\) −8.53368e8 −0.451918
\(448\) 4.17307e9 2.19272
\(449\) 1.07309e9 0.559468 0.279734 0.960078i \(-0.409754\pi\)
0.279734 + 0.960078i \(0.409754\pi\)
\(450\) 2.92634e9 1.51385
\(451\) 1.71285e9 0.879229
\(452\) −2.76368e9 −1.40768
\(453\) 1.25958e9 0.636623
\(454\) 1.92959e9 0.967764
\(455\) −4.02335e9 −2.00239
\(456\) 2.05872e9 1.01676
\(457\) −1.32100e9 −0.647434 −0.323717 0.946154i \(-0.604933\pi\)
−0.323717 + 0.946154i \(0.604933\pi\)
\(458\) 1.49285e9 0.726084
\(459\) −2.25108e9 −1.08654
\(460\) −6.77357e9 −3.24463
\(461\) 2.81759e9 1.33944 0.669721 0.742613i \(-0.266414\pi\)
0.669721 + 0.742613i \(0.266414\pi\)
\(462\) −5.32574e9 −2.51266
\(463\) 1.21643e9 0.569580 0.284790 0.958590i \(-0.408076\pi\)
0.284790 + 0.958590i \(0.408076\pi\)
\(464\) 4.03066e9 1.87311
\(465\) −4.05115e9 −1.86850
\(466\) 3.07285e9 1.40666
\(467\) 1.33377e9 0.606000 0.303000 0.952991i \(-0.402012\pi\)
0.303000 + 0.952991i \(0.402012\pi\)
\(468\) 2.18982e9 0.987525
\(469\) −3.00291e9 −1.34412
\(470\) −5.00549e9 −2.22384
\(471\) 1.46899e9 0.647807
\(472\) 5.07840e9 2.22295
\(473\) 3.38674e9 1.47152
\(474\) 1.75480e9 0.756840
\(475\) −2.03694e9 −0.872068
\(476\) 8.36234e9 3.55389
\(477\) −2.74255e8 −0.115702
\(478\) −4.49356e9 −1.88188
\(479\) 5.14618e8 0.213949 0.106975 0.994262i \(-0.465884\pi\)
0.106975 + 0.994262i \(0.465884\pi\)
\(480\) −6.73974e9 −2.78163
\(481\) 3.44004e8 0.140947
\(482\) −2.36913e8 −0.0963660
\(483\) −2.00680e9 −0.810380
\(484\) 4.33183e9 1.73665
\(485\) 3.87507e9 1.54235
\(486\) −3.94722e9 −1.55978
\(487\) −2.97786e8 −0.116830 −0.0584149 0.998292i \(-0.518605\pi\)
−0.0584149 + 0.998292i \(0.518605\pi\)
\(488\) −2.16533e9 −0.843441
\(489\) −1.03433e9 −0.400015
\(490\) −7.92678e9 −3.04376
\(491\) −4.51780e8 −0.172243 −0.0861216 0.996285i \(-0.527447\pi\)
−0.0861216 + 0.996285i \(0.527447\pi\)
\(492\) 3.25679e9 1.23285
\(493\) 1.85993e9 0.699089
\(494\) −2.13612e9 −0.797225
\(495\) −2.69649e9 −0.999264
\(496\) −1.13333e10 −4.17034
\(497\) −2.77652e8 −0.101450
\(498\) −7.25754e9 −2.63322
\(499\) 4.48528e9 1.61598 0.807992 0.589193i \(-0.200554\pi\)
0.807992 + 0.589193i \(0.200554\pi\)
\(500\) 8.69098e9 3.10938
\(501\) 1.70534e9 0.605869
\(502\) 1.72258e9 0.607738
\(503\) −1.27132e9 −0.445419 −0.222709 0.974885i \(-0.571490\pi\)
−0.222709 + 0.974885i \(0.571490\pi\)
\(504\) 5.21270e9 1.81366
\(505\) −1.30338e9 −0.450351
\(506\) 5.56891e9 1.91092
\(507\) −5.70102e8 −0.194278
\(508\) 6.62087e9 2.24074
\(509\) −2.73217e8 −0.0918324 −0.0459162 0.998945i \(-0.514621\pi\)
−0.0459162 + 0.998945i \(0.514621\pi\)
\(510\) −6.89969e9 −2.30321
\(511\) −6.93010e9 −2.29756
\(512\) 4.12038e9 1.35673
\(513\) 1.63172e9 0.533624
\(514\) −3.84884e9 −1.25014
\(515\) 7.57287e8 0.244306
\(516\) 6.43948e9 2.06337
\(517\) 2.93654e9 0.934584
\(518\) 1.36800e9 0.432445
\(519\) 1.16883e9 0.367000
\(520\) 1.27074e10 3.96318
\(521\) −1.56251e9 −0.484050 −0.242025 0.970270i \(-0.577812\pi\)
−0.242025 + 0.970270i \(0.577812\pi\)
\(522\) 1.93686e9 0.596008
\(523\) −1.80565e9 −0.551923 −0.275961 0.961169i \(-0.588996\pi\)
−0.275961 + 0.961169i \(0.588996\pi\)
\(524\) −1.24237e10 −3.77216
\(525\) 5.99739e9 1.80886
\(526\) 1.05761e10 3.16867
\(527\) −5.22972e9 −1.55647
\(528\) 8.77197e9 2.59345
\(529\) −1.30640e9 −0.383691
\(530\) −2.65871e9 −0.775720
\(531\) 1.27262e9 0.368865
\(532\) −6.06154e9 −1.74539
\(533\) −2.02278e9 −0.578634
\(534\) 2.16748e9 0.615973
\(535\) 6.71001e9 1.89446
\(536\) 9.48440e9 2.66032
\(537\) −2.64583e9 −0.737314
\(538\) −7.77323e9 −2.15211
\(539\) 4.65034e9 1.27916
\(540\) −1.62160e10 −4.43166
\(541\) 3.11802e8 0.0846619 0.0423310 0.999104i \(-0.486522\pi\)
0.0423310 + 0.999104i \(0.486522\pi\)
\(542\) −9.17416e9 −2.47496
\(543\) −4.89499e9 −1.31206
\(544\) −8.70049e9 −2.31711
\(545\) −8.17345e9 −2.16281
\(546\) 6.28941e9 1.65362
\(547\) 5.63775e9 1.47282 0.736411 0.676535i \(-0.236519\pi\)
0.736411 + 0.676535i \(0.236519\pi\)
\(548\) −5.93911e9 −1.54166
\(549\) −5.42620e8 −0.139956
\(550\) −1.66429e10 −4.26540
\(551\) −1.34819e9 −0.343337
\(552\) 6.33829e9 1.60393
\(553\) −3.09276e9 −0.777693
\(554\) −1.06594e10 −2.66348
\(555\) −8.05420e8 −0.199985
\(556\) 1.69329e9 0.417802
\(557\) −1.42514e9 −0.349434 −0.174717 0.984619i \(-0.555901\pi\)
−0.174717 + 0.984619i \(0.555901\pi\)
\(558\) −5.44603e9 −1.32697
\(559\) −3.99955e9 −0.968433
\(560\) 2.63528e10 6.34116
\(561\) 4.04779e9 0.967939
\(562\) −7.06890e8 −0.167987
\(563\) −1.42064e8 −0.0335508 −0.0167754 0.999859i \(-0.505340\pi\)
−0.0167754 + 0.999859i \(0.505340\pi\)
\(564\) 5.58348e9 1.31047
\(565\) −4.01890e9 −0.937426
\(566\) 8.55395e9 1.98294
\(567\) −1.97905e9 −0.455948
\(568\) 8.76939e8 0.200794
\(569\) 6.41286e9 1.45935 0.729674 0.683795i \(-0.239672\pi\)
0.729674 + 0.683795i \(0.239672\pi\)
\(570\) 5.00132e9 1.13116
\(571\) 2.13891e9 0.480801 0.240401 0.970674i \(-0.422721\pi\)
0.240401 + 0.970674i \(0.422721\pi\)
\(572\) −1.24541e10 −2.78244
\(573\) −4.69763e9 −1.04313
\(574\) −8.04399e9 −1.77533
\(575\) −6.27123e9 −1.37567
\(576\) −3.30290e9 −0.720140
\(577\) 1.52123e9 0.329670 0.164835 0.986321i \(-0.447291\pi\)
0.164835 + 0.986321i \(0.447291\pi\)
\(578\) −2.32606e8 −0.0501042
\(579\) −1.10752e9 −0.237125
\(580\) 1.33983e10 2.85136
\(581\) 1.27911e10 2.70577
\(582\) −6.05761e9 −1.27371
\(583\) 1.55976e9 0.326001
\(584\) 2.18881e10 4.54740
\(585\) 3.18440e9 0.657631
\(586\) 1.66869e10 3.42559
\(587\) −1.70891e9 −0.348728 −0.174364 0.984681i \(-0.555787\pi\)
−0.174364 + 0.984681i \(0.555787\pi\)
\(588\) 8.84208e9 1.79363
\(589\) 3.79082e9 0.764416
\(590\) 1.23371e10 2.47304
\(591\) −4.93437e9 −0.983276
\(592\) −2.25321e9 −0.446350
\(593\) 5.64855e9 1.11236 0.556180 0.831062i \(-0.312267\pi\)
0.556180 + 0.831062i \(0.312267\pi\)
\(594\) 1.33321e10 2.61002
\(595\) 1.21604e10 2.36667
\(596\) −7.93579e9 −1.53542
\(597\) 1.83862e8 0.0353657
\(598\) −6.57657e9 −1.25761
\(599\) 2.09091e9 0.397504 0.198752 0.980050i \(-0.436311\pi\)
0.198752 + 0.980050i \(0.436311\pi\)
\(600\) −1.89422e10 −3.58015
\(601\) −2.90384e9 −0.545647 −0.272823 0.962064i \(-0.587957\pi\)
−0.272823 + 0.962064i \(0.587957\pi\)
\(602\) −1.59050e10 −2.97129
\(603\) 2.37674e9 0.441439
\(604\) 1.17133e10 2.16297
\(605\) 6.29927e9 1.15650
\(606\) 2.03748e9 0.371911
\(607\) 2.35727e9 0.427809 0.213904 0.976855i \(-0.431382\pi\)
0.213904 + 0.976855i \(0.431382\pi\)
\(608\) 6.30665e9 1.13798
\(609\) 3.96950e9 0.712157
\(610\) −5.26031e9 −0.938333
\(611\) −3.46789e9 −0.615064
\(612\) −6.61863e9 −1.16718
\(613\) −5.84011e9 −1.02402 −0.512011 0.858979i \(-0.671099\pi\)
−0.512011 + 0.858979i \(0.671099\pi\)
\(614\) 4.19648e8 0.0731638
\(615\) 4.73597e9 0.821005
\(616\) −2.96460e10 −5.11015
\(617\) 1.14662e9 0.196527 0.0982637 0.995160i \(-0.468671\pi\)
0.0982637 + 0.995160i \(0.468671\pi\)
\(618\) −1.18381e9 −0.201754
\(619\) −8.82033e9 −1.49475 −0.747373 0.664405i \(-0.768685\pi\)
−0.747373 + 0.664405i \(0.768685\pi\)
\(620\) −3.76731e10 −6.34835
\(621\) 5.02366e9 0.841783
\(622\) 1.37102e10 2.28443
\(623\) −3.82009e9 −0.632944
\(624\) −1.03592e10 −1.70679
\(625\) 1.94291e9 0.318327
\(626\) 8.43043e9 1.37354
\(627\) −2.93409e9 −0.475375
\(628\) 1.36607e10 2.20097
\(629\) −1.03973e9 −0.166589
\(630\) 1.26634e10 2.01771
\(631\) 1.55001e9 0.245602 0.122801 0.992431i \(-0.460812\pi\)
0.122801 + 0.992431i \(0.460812\pi\)
\(632\) 9.76820e9 1.53923
\(633\) 2.58315e7 0.00404797
\(634\) −2.28322e10 −3.55823
\(635\) 9.62797e9 1.49220
\(636\) 2.96571e9 0.457118
\(637\) −5.49180e9 −0.841834
\(638\) −1.10155e10 −1.67931
\(639\) 2.19756e8 0.0333187
\(640\) −6.86092e9 −1.03455
\(641\) −4.65696e9 −0.698392 −0.349196 0.937050i \(-0.613545\pi\)
−0.349196 + 0.937050i \(0.613545\pi\)
\(642\) −1.04893e10 −1.56449
\(643\) 3.05737e9 0.453533 0.226767 0.973949i \(-0.427185\pi\)
0.226767 + 0.973949i \(0.427185\pi\)
\(644\) −1.86620e10 −2.75332
\(645\) 9.36419e9 1.37408
\(646\) 6.45632e9 0.942261
\(647\) −7.51191e8 −0.109040 −0.0545199 0.998513i \(-0.517363\pi\)
−0.0545199 + 0.998513i \(0.517363\pi\)
\(648\) 6.25063e9 0.902426
\(649\) −7.23772e9 −1.03931
\(650\) 1.96543e10 2.80712
\(651\) −1.11614e10 −1.58556
\(652\) −9.61859e9 −1.35908
\(653\) 8.40509e9 1.18126 0.590631 0.806942i \(-0.298879\pi\)
0.590631 + 0.806942i \(0.298879\pi\)
\(654\) 1.27769e10 1.78610
\(655\) −1.80663e10 −2.51203
\(656\) 1.32492e10 1.83242
\(657\) 5.48504e9 0.754572
\(658\) −1.37907e10 −1.88710
\(659\) 9.63741e9 1.31178 0.655890 0.754856i \(-0.272293\pi\)
0.655890 + 0.754856i \(0.272293\pi\)
\(660\) 2.91589e10 3.94791
\(661\) −2.15965e9 −0.290856 −0.145428 0.989369i \(-0.546456\pi\)
−0.145428 + 0.989369i \(0.546456\pi\)
\(662\) 9.39727e9 1.25892
\(663\) −4.78021e9 −0.637015
\(664\) −4.03994e10 −5.35534
\(665\) −8.81460e9 −1.16232
\(666\) −1.08274e9 −0.142025
\(667\) −4.15075e9 −0.541609
\(668\) 1.58586e10 2.05848
\(669\) 8.12357e9 1.04895
\(670\) 2.30408e10 2.95962
\(671\) 3.08603e9 0.394340
\(672\) −1.85688e10 −2.36043
\(673\) −1.11530e10 −1.41038 −0.705192 0.709016i \(-0.749139\pi\)
−0.705192 + 0.709016i \(0.749139\pi\)
\(674\) −2.39456e10 −3.01242
\(675\) −1.50134e10 −1.87895
\(676\) −5.30159e9 −0.660074
\(677\) −1.43527e10 −1.77776 −0.888882 0.458137i \(-0.848517\pi\)
−0.888882 + 0.458137i \(0.848517\pi\)
\(678\) 6.28244e9 0.774149
\(679\) 1.06763e10 1.30881
\(680\) −3.84075e10 −4.68419
\(681\) −3.12998e9 −0.379776
\(682\) 3.09731e10 3.73886
\(683\) 7.89840e9 0.948564 0.474282 0.880373i \(-0.342708\pi\)
0.474282 + 0.880373i \(0.342708\pi\)
\(684\) 4.79759e9 0.573227
\(685\) −8.63655e9 −1.02665
\(686\) 4.02419e8 0.0475931
\(687\) −2.42155e9 −0.284934
\(688\) 2.61969e10 3.06683
\(689\) −1.84199e9 −0.214546
\(690\) 1.53978e10 1.78438
\(691\) −2.20339e9 −0.254050 −0.127025 0.991900i \(-0.540543\pi\)
−0.127025 + 0.991900i \(0.540543\pi\)
\(692\) 1.08694e10 1.24691
\(693\) −7.42913e9 −0.847953
\(694\) 2.03262e9 0.230833
\(695\) 2.46236e9 0.278231
\(696\) −1.25373e10 −1.40952
\(697\) 6.11377e9 0.683903
\(698\) 3.05230e10 3.39730
\(699\) −4.98446e9 −0.552011
\(700\) 5.57720e10 6.14572
\(701\) 1.36665e10 1.49846 0.749228 0.662312i \(-0.230425\pi\)
0.749228 + 0.662312i \(0.230425\pi\)
\(702\) −1.57444e10 −1.71770
\(703\) 7.53664e8 0.0818152
\(704\) 1.87845e10 2.02906
\(705\) 8.11940e9 0.872694
\(706\) −2.01155e10 −2.15136
\(707\) −3.59096e9 −0.382158
\(708\) −1.37617e10 −1.45732
\(709\) −1.02722e10 −1.08243 −0.541216 0.840884i \(-0.682036\pi\)
−0.541216 + 0.840884i \(0.682036\pi\)
\(710\) 2.13038e9 0.223384
\(711\) 2.44786e9 0.255413
\(712\) 1.20654e10 1.25274
\(713\) 1.16710e10 1.20585
\(714\) −1.90094e10 −1.95445
\(715\) −1.81105e10 −1.85294
\(716\) −2.46046e10 −2.50508
\(717\) 7.28899e9 0.738500
\(718\) −1.50961e10 −1.52205
\(719\) −9.86949e9 −0.990247 −0.495123 0.868823i \(-0.664877\pi\)
−0.495123 + 0.868823i \(0.664877\pi\)
\(720\) −2.08577e10 −2.08259
\(721\) 2.08641e9 0.207313
\(722\) 1.42161e10 1.40572
\(723\) 3.84296e8 0.0378165
\(724\) −4.55204e10 −4.45780
\(725\) 1.24047e10 1.20893
\(726\) −9.84718e9 −0.955067
\(727\) −1.04805e10 −1.01160 −0.505802 0.862650i \(-0.668803\pi\)
−0.505802 + 0.862650i \(0.668803\pi\)
\(728\) 3.50103e10 3.36307
\(729\) 9.79059e9 0.935971
\(730\) 5.31735e10 5.05900
\(731\) 1.20884e10 1.14462
\(732\) 5.86771e9 0.552943
\(733\) −1.01654e10 −0.953369 −0.476685 0.879074i \(-0.658162\pi\)
−0.476685 + 0.879074i \(0.658162\pi\)
\(734\) −8.50949e9 −0.794269
\(735\) 1.28580e10 1.19445
\(736\) 1.94166e10 1.79515
\(737\) −1.35172e10 −1.24380
\(738\) 6.36665e9 0.583061
\(739\) 1.10451e10 1.00673 0.503367 0.864073i \(-0.332095\pi\)
0.503367 + 0.864073i \(0.332095\pi\)
\(740\) −7.48990e9 −0.679462
\(741\) 3.46499e9 0.312852
\(742\) −7.32504e9 −0.658258
\(743\) −2.89185e9 −0.258652 −0.129326 0.991602i \(-0.541281\pi\)
−0.129326 + 0.991602i \(0.541281\pi\)
\(744\) 3.52522e10 3.13820
\(745\) −1.15401e10 −1.02250
\(746\) 3.46610e10 3.05672
\(747\) −1.01239e10 −0.888638
\(748\) 3.76419e10 3.28864
\(749\) 1.84868e10 1.60759
\(750\) −1.97565e10 −1.71000
\(751\) −7.39191e9 −0.636820 −0.318410 0.947953i \(-0.603149\pi\)
−0.318410 + 0.947953i \(0.603149\pi\)
\(752\) 2.27145e10 1.94779
\(753\) −2.79420e9 −0.238492
\(754\) 1.30086e10 1.10518
\(755\) 1.70333e10 1.44041
\(756\) −4.46770e10 −3.76061
\(757\) 1.37247e10 1.14992 0.574959 0.818183i \(-0.305018\pi\)
0.574959 + 0.818183i \(0.305018\pi\)
\(758\) −3.92401e10 −3.27256
\(759\) −9.03332e9 −0.749896
\(760\) 2.78401e10 2.30050
\(761\) 3.46687e9 0.285161 0.142581 0.989783i \(-0.454460\pi\)
0.142581 + 0.989783i \(0.454460\pi\)
\(762\) −1.50507e10 −1.23229
\(763\) −2.25188e10 −1.83531
\(764\) −4.36851e10 −3.54410
\(765\) −9.62470e9 −0.777271
\(766\) 8.22912e9 0.661535
\(767\) 8.54735e9 0.683987
\(768\) −3.61164e9 −0.287700
\(769\) 1.82468e10 1.44692 0.723461 0.690365i \(-0.242550\pi\)
0.723461 + 0.690365i \(0.242550\pi\)
\(770\) −7.20200e10 −5.68507
\(771\) 6.24319e9 0.490588
\(772\) −1.02993e10 −0.805649
\(773\) −1.29659e10 −1.00966 −0.504828 0.863220i \(-0.668444\pi\)
−0.504828 + 0.863220i \(0.668444\pi\)
\(774\) 1.25885e10 0.975842
\(775\) −3.48792e10 −2.69160
\(776\) −3.37200e10 −2.59043
\(777\) −2.21902e9 −0.169703
\(778\) −3.32334e9 −0.253015
\(779\) −4.43164e9 −0.335879
\(780\) −3.44351e10 −2.59818
\(781\) −1.24981e9 −0.0938785
\(782\) 1.98774e10 1.48640
\(783\) −9.93695e9 −0.739753
\(784\) 3.59711e10 2.66592
\(785\) 1.98652e10 1.46571
\(786\) 2.82417e10 2.07449
\(787\) 1.69567e9 0.124003 0.0620013 0.998076i \(-0.480252\pi\)
0.0620013 + 0.998076i \(0.480252\pi\)
\(788\) −4.58865e10 −3.34075
\(789\) −1.71555e10 −1.24347
\(790\) 2.37302e10 1.71241
\(791\) −1.10725e10 −0.795479
\(792\) 2.34642e10 1.67829
\(793\) −3.64442e9 −0.259521
\(794\) 5.03845e10 3.57211
\(795\) 4.31268e9 0.304413
\(796\) 1.70980e9 0.120157
\(797\) 1.49735e10 1.04766 0.523828 0.851824i \(-0.324503\pi\)
0.523828 + 0.851824i \(0.324503\pi\)
\(798\) 1.37792e10 0.959874
\(799\) 1.04815e10 0.726960
\(800\) −5.80272e10 −4.00698
\(801\) 3.02353e9 0.207874
\(802\) 1.08426e10 0.742203
\(803\) −3.11949e10 −2.12608
\(804\) −2.57013e10 −1.74405
\(805\) −2.71379e10 −1.83354
\(806\) −3.65775e10 −2.46060
\(807\) 1.26089e10 0.844542
\(808\) 1.13417e10 0.756379
\(809\) −9.82683e9 −0.652520 −0.326260 0.945280i \(-0.605788\pi\)
−0.326260 + 0.945280i \(0.605788\pi\)
\(810\) 1.51849e10 1.00395
\(811\) 2.21461e10 1.45789 0.728945 0.684572i \(-0.240011\pi\)
0.728945 + 0.684572i \(0.240011\pi\)
\(812\) 3.69139e10 2.41960
\(813\) 1.48814e10 0.971240
\(814\) 6.15784e9 0.400169
\(815\) −1.39872e10 −0.905064
\(816\) 3.13102e10 2.01730
\(817\) −8.76245e9 −0.562145
\(818\) −1.23954e10 −0.791813
\(819\) 8.77339e9 0.558051
\(820\) 4.40415e10 2.78942
\(821\) 5.54795e8 0.0349890 0.0174945 0.999847i \(-0.494431\pi\)
0.0174945 + 0.999847i \(0.494431\pi\)
\(822\) 1.35009e10 0.847834
\(823\) −2.49245e10 −1.55857 −0.779286 0.626668i \(-0.784418\pi\)
−0.779286 + 0.626668i \(0.784418\pi\)
\(824\) −6.58974e9 −0.410320
\(825\) 2.69964e10 1.67385
\(826\) 3.39902e10 2.09857
\(827\) 4.78357e9 0.294092 0.147046 0.989130i \(-0.453023\pi\)
0.147046 + 0.989130i \(0.453023\pi\)
\(828\) 1.47706e10 0.904256
\(829\) 7.08659e9 0.432012 0.216006 0.976392i \(-0.430697\pi\)
0.216006 + 0.976392i \(0.430697\pi\)
\(830\) −9.81437e10 −5.95784
\(831\) 1.72907e10 1.04522
\(832\) −2.21834e10 −1.33536
\(833\) 1.65987e10 0.994985
\(834\) −3.84923e9 −0.229769
\(835\) 2.30613e10 1.37082
\(836\) −2.72852e10 −1.61512
\(837\) 2.79405e10 1.64701
\(838\) 1.31569e10 0.772322
\(839\) 1.68635e10 0.985782 0.492891 0.870091i \(-0.335940\pi\)
0.492891 + 0.870091i \(0.335940\pi\)
\(840\) −8.19700e10 −4.77175
\(841\) −9.03959e9 −0.524038
\(842\) 4.65567e10 2.68776
\(843\) 1.14664e9 0.0659223
\(844\) 2.40217e8 0.0137533
\(845\) −7.70949e9 −0.439569
\(846\) 1.09151e10 0.619770
\(847\) 1.73552e10 0.981382
\(848\) 1.20650e10 0.679425
\(849\) −1.38754e10 −0.778157
\(850\) −5.94043e10 −3.31781
\(851\) 2.32034e9 0.129062
\(852\) −2.37637e9 −0.131636
\(853\) 1.23581e10 0.681759 0.340880 0.940107i \(-0.389275\pi\)
0.340880 + 0.940107i \(0.389275\pi\)
\(854\) −1.44928e10 −0.796248
\(855\) 6.97658e9 0.381734
\(856\) −5.83890e10 −3.18180
\(857\) 3.83883e9 0.208337 0.104168 0.994560i \(-0.466782\pi\)
0.104168 + 0.994560i \(0.466782\pi\)
\(858\) 2.83109e10 1.53020
\(859\) 6.19149e9 0.333288 0.166644 0.986017i \(-0.446707\pi\)
0.166644 + 0.986017i \(0.446707\pi\)
\(860\) 8.70811e10 4.66852
\(861\) 1.30481e10 0.696686
\(862\) −4.09036e10 −2.17513
\(863\) 1.63864e10 0.867851 0.433926 0.900949i \(-0.357128\pi\)
0.433926 + 0.900949i \(0.357128\pi\)
\(864\) 4.64836e10 2.45189
\(865\) 1.58061e10 0.830363
\(866\) −5.98974e10 −3.13397
\(867\) 3.77310e8 0.0196622
\(868\) −1.03794e11 −5.38707
\(869\) −1.39216e10 −0.719649
\(870\) −3.04573e10 −1.56810
\(871\) 1.59630e10 0.818561
\(872\) 7.11235e10 3.63250
\(873\) −8.45005e9 −0.429842
\(874\) −1.44083e10 −0.730002
\(875\) 3.48199e10 1.75711
\(876\) −5.93134e10 −2.98118
\(877\) −1.42291e10 −0.712326 −0.356163 0.934424i \(-0.615915\pi\)
−0.356163 + 0.934424i \(0.615915\pi\)
\(878\) −5.70855e9 −0.284639
\(879\) −2.70678e10 −1.34429
\(880\) 1.18623e11 5.86788
\(881\) −2.08797e9 −0.102875 −0.0514375 0.998676i \(-0.516380\pi\)
−0.0514375 + 0.998676i \(0.516380\pi\)
\(882\) 1.72853e10 0.848274
\(883\) −8.08685e9 −0.395291 −0.197645 0.980274i \(-0.563329\pi\)
−0.197645 + 0.980274i \(0.563329\pi\)
\(884\) −4.44530e10 −2.16430
\(885\) −2.00120e10 −0.970486
\(886\) −3.62353e10 −1.75031
\(887\) 3.27815e10 1.57723 0.788617 0.614884i \(-0.210797\pi\)
0.788617 + 0.614884i \(0.210797\pi\)
\(888\) 7.00858e9 0.335881
\(889\) 2.65261e10 1.26625
\(890\) 2.93109e10 1.39368
\(891\) −8.90839e9 −0.421917
\(892\) 7.55442e10 3.56389
\(893\) −7.59765e9 −0.357026
\(894\) 1.80398e10 0.844404
\(895\) −3.57796e10 −1.66823
\(896\) −1.89026e10 −0.877898
\(897\) 1.06678e10 0.493518
\(898\) −2.26847e10 −1.04536
\(899\) −2.30855e10 −1.05970
\(900\) −4.41424e10 −2.01840
\(901\) 5.56734e9 0.253578
\(902\) −3.62089e10 −1.64283
\(903\) 2.57994e10 1.16601
\(904\) 3.49715e10 1.57444
\(905\) −6.61950e10 −2.96862
\(906\) −2.66269e10 −1.18952
\(907\) 1.06210e10 0.472651 0.236325 0.971674i \(-0.424057\pi\)
0.236325 + 0.971674i \(0.424057\pi\)
\(908\) −2.91069e10 −1.29031
\(909\) 2.84218e9 0.125510
\(910\) 8.50516e10 3.74143
\(911\) 2.54777e10 1.11647 0.558233 0.829684i \(-0.311480\pi\)
0.558233 + 0.829684i \(0.311480\pi\)
\(912\) −2.26956e10 −0.990739
\(913\) 5.75772e10 2.50382
\(914\) 2.79253e10 1.20972
\(915\) 8.53274e9 0.368226
\(916\) −2.25189e10 −0.968084
\(917\) −4.97746e10 −2.13165
\(918\) 4.75868e10 2.03019
\(919\) −1.29356e10 −0.549772 −0.274886 0.961477i \(-0.588640\pi\)
−0.274886 + 0.961477i \(0.588640\pi\)
\(920\) 8.57127e10 3.62901
\(921\) −6.80711e8 −0.0287114
\(922\) −5.95624e10 −2.50273
\(923\) 1.47596e9 0.0617829
\(924\) 8.03361e10 3.35011
\(925\) −6.93443e9 −0.288081
\(926\) −2.57148e10 −1.06425
\(927\) −1.65135e9 −0.0680865
\(928\) −3.84066e10 −1.57757
\(929\) 1.63560e10 0.669303 0.334652 0.942342i \(-0.391381\pi\)
0.334652 + 0.942342i \(0.391381\pi\)
\(930\) 8.56392e10 3.49126
\(931\) −1.20318e10 −0.488658
\(932\) −4.63524e10 −1.87550
\(933\) −2.22393e10 −0.896470
\(934\) −2.81953e10 −1.13230
\(935\) 5.47383e10 2.19003
\(936\) −2.77099e10 −1.10451
\(937\) 1.25975e10 0.500259 0.250129 0.968212i \(-0.419527\pi\)
0.250129 + 0.968212i \(0.419527\pi\)
\(938\) 6.34800e10 2.51146
\(939\) −1.36750e10 −0.539011
\(940\) 7.55054e10 2.96504
\(941\) 4.27267e10 1.67161 0.835805 0.549026i \(-0.185001\pi\)
0.835805 + 0.549026i \(0.185001\pi\)
\(942\) −3.10537e10 −1.21042
\(943\) −1.36439e10 −0.529843
\(944\) −5.59848e10 −2.16605
\(945\) −6.49686e10 −2.50433
\(946\) −7.15940e10 −2.74952
\(947\) −2.14494e9 −0.0820713 −0.0410356 0.999158i \(-0.513066\pi\)
−0.0410356 + 0.999158i \(0.513066\pi\)
\(948\) −2.64703e10 −1.00909
\(949\) 3.68394e10 1.39920
\(950\) 4.30599e10 1.62945
\(951\) 3.70360e10 1.39634
\(952\) −1.05817e11 −3.97490
\(953\) −4.21227e10 −1.57649 −0.788246 0.615361i \(-0.789010\pi\)
−0.788246 + 0.615361i \(0.789010\pi\)
\(954\) 5.79762e9 0.216188
\(955\) −6.35261e10 −2.36015
\(956\) 6.77831e10 2.50910
\(957\) 1.78682e10 0.659004
\(958\) −1.08788e10 −0.399761
\(959\) −2.37947e10 −0.871194
\(960\) 5.19383e10 1.89469
\(961\) 3.73988e10 1.35933
\(962\) −7.27207e9 −0.263357
\(963\) −1.46320e10 −0.527972
\(964\) 3.57371e9 0.128484
\(965\) −1.49770e10 −0.536513
\(966\) 4.24227e10 1.51418
\(967\) 4.69149e10 1.66847 0.834235 0.551409i \(-0.185909\pi\)
0.834235 + 0.551409i \(0.185909\pi\)
\(968\) −5.48148e10 −1.94238
\(969\) −1.04728e10 −0.369767
\(970\) −8.19171e10 −2.88186
\(971\) 3.82889e10 1.34216 0.671082 0.741383i \(-0.265830\pi\)
0.671082 + 0.741383i \(0.265830\pi\)
\(972\) 5.95418e10 2.07965
\(973\) 6.78408e9 0.236100
\(974\) 6.29506e9 0.218295
\(975\) −3.18813e10 −1.10159
\(976\) 2.38709e10 0.821852
\(977\) −4.93896e10 −1.69436 −0.847178 0.531309i \(-0.821700\pi\)
−0.847178 + 0.531309i \(0.821700\pi\)
\(978\) 2.18652e10 0.747423
\(979\) −1.71956e10 −0.585704
\(980\) 1.19571e11 4.05823
\(981\) 1.78232e10 0.602759
\(982\) 9.55041e9 0.321834
\(983\) 2.20718e10 0.741141 0.370570 0.928804i \(-0.379162\pi\)
0.370570 + 0.928804i \(0.379162\pi\)
\(984\) −4.12113e10 −1.37890
\(985\) −6.67275e10 −2.22473
\(986\) −3.93180e10 −1.30624
\(987\) 2.23699e10 0.740549
\(988\) 3.22223e10 1.06294
\(989\) −2.69774e10 −0.886774
\(990\) 5.70024e10 1.86711
\(991\) −4.94638e10 −1.61447 −0.807235 0.590231i \(-0.799037\pi\)
−0.807235 + 0.590231i \(0.799037\pi\)
\(992\) 1.07991e11 3.51233
\(993\) −1.52433e10 −0.494034
\(994\) 5.86943e9 0.189559
\(995\) 2.48637e9 0.0800174
\(996\) 1.09476e11 3.51085
\(997\) −3.10203e9 −0.0991318 −0.0495659 0.998771i \(-0.515784\pi\)
−0.0495659 + 0.998771i \(0.515784\pi\)
\(998\) −9.48165e10 −3.01945
\(999\) 5.55493e9 0.176279
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 37.8.a.b.1.1 11
3.2 odd 2 333.8.a.d.1.11 11
4.3 odd 2 592.8.a.g.1.5 11
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
37.8.a.b.1.1 11 1.1 even 1 trivial
333.8.a.d.1.11 11 3.2 odd 2
592.8.a.g.1.5 11 4.3 odd 2