LMFDB - Embedded newform 16.42.a.b.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [16,42,Mod(1,16)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(16, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 42, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("16.1");

S:= CuspForms(chi, 42);

N := Newforms(S);

Level:	$ N $	$=$	$ 16 = 2^{4} $
Weight:	$ k $	$=$	$ 42 $
Character orbit:	$[\chi]$	$=$	16.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:	$170.354672730$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\mathbb{Q}[x]/(x^{2} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1139917559892 $ x^2 - x - 1139917559892
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{7}\cdot 3^{2}\cdot 5 $
Twist minimal:	no (minimal twist has level 2)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.06767e6$ of defining polynomial
Character	$\chi$	$=$	16.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.71810e9 q^{3} +3.51885e14 q^{5} +7.89090e16 q^{7} -3.35211e19 q^{9} +O(q^{10})$	$q+1.71810e9 q^{3} +3.51885e14 q^{5} +7.89090e16 q^{7} -3.35211e19 q^{9} -1.20679e21 q^{11} +9.23598e22 q^{13} +6.04574e23 q^{15} -4.95668e24 q^{17} -2.41331e26 q^{19} +1.35574e26 q^{21} -8.23610e27 q^{23} +7.83484e28 q^{25} -1.20257e29 q^{27} +2.60718e29 q^{29} +6.40610e30 q^{31} -2.07339e30 q^{33} +2.77669e31 q^{35} +7.38149e31 q^{37} +1.58683e32 q^{39} +2.15386e33 q^{41} +5.67618e33 q^{43} -1.17956e34 q^{45} +6.45458e33 q^{47} -3.83410e34 q^{49} -8.51608e33 q^{51} -9.99163e34 q^{53} -4.24652e35 q^{55} -4.14631e35 q^{57} +3.33277e36 q^{59} +2.08511e36 q^{61} -2.64512e36 q^{63} +3.25000e37 q^{65} -2.21433e36 q^{67} -1.41505e37 q^{69} -5.59105e37 q^{71} -2.02164e38 q^{73} +1.34610e38 q^{75} -9.52266e37 q^{77} +2.46315e38 q^{79} +1.01600e39 q^{81} +3.59771e37 q^{83} -1.74418e39 q^{85} +4.47940e38 q^{87} +5.24223e38 q^{89} +7.28801e39 q^{91} +1.10063e40 q^{93} -8.49208e40 q^{95} -2.35889e40 q^{97} +4.04530e40 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 8863347528 q^{3} + 97599184325580 q^{5} - 21\!\cdots\!56 q^{7}+ \cdots + 41\!\cdots\!86 q^{9}+O(q^{10}) $ 2 * q - 8863347528 * q^3 + 97599184325580 * q^5 - 217106802373088656 * q^7 + 41972929616872509786 * q^9	$ 2 q - 8863347528 q^{3} + 97599184325580 q^{5} - 21\!\cdots\!56 q^{7}+ \cdots + 12\!\cdots\!88 q^{99}+O(q^{100}) $ 2 * q - 8863347528 * q^3 + 97599184325580 * q^5 - 217106802373088656 * q^7 + 41972929616872509786 * q^9 - 110594481484939565784 * q^11 + 173293106838448696895068 * q^13 + 3295287287252802285241680 * q^15 - 10287966260741677078429404 * q^17 - 206254882139859790701457000 * q^19 + 3267849131609686416057835584 * q^21 - 14925421378885128304361915568 * q^23 + 97534966241249753807444705150 * q^25 - 533156288863055843864697754320 * q^27 - 1375143011941884388614889894020 * q^29 + 3626496922711160469799126663616 * q^31 - 13672735988327258412044854187424 * q^33 + 103039524095905042910481769779360 * q^35 + 120356679589378915210866994599436 * q^37 - 697708668559917206762862430490352 * q^39 + 1270177139906840106861139346878164 * q^41 + 6045433421160616098770468790853352 * q^43 - 30992658299909156185379775337031460 * q^45 + 25416478966330542868882280891861664 * q^47 + 4716677344275473650585885334355954 * q^49 + 47896619495707425086522465903395056 * q^51 - 301376484284740318156836692232048852 * q^53 - 703399094180619772764269188861842960 * q^55 - 785787658029123941344672503009410400 * q^57 + 3282010250697938085995473868454788040 * q^59 - 4485157745840460017677520841591879236 * q^61 - 24992547810032633572829803615671997008 * q^63 + 11919812053150705903267451825832510120 * q^65 - 55055736076082381526441630307313037256 * q^67 + 56632222305135551516324020091548201152 * q^69 + 48857884738531480102553331099997138416 * q^71 - 485616958845162818165813572349849688812 * q^73 - 68411444542067712502035991381122300600 * q^75 - 419718086944271580508987308621981603648 * q^77 + 905766854341990828970937020702446890080 * q^79 + 2631580786121491108655810678136598411122 * q^81 - 1271755206567258248006583231290595969768 * q^83 - 388512546938141834318136915482004571560 * q^85 + 17757719150026706057484126704955838888080 * q^87 + 2692414558304473820098690391828757436980 * q^89 - 16669534332125295311104493248216990683104 * q^91 + 40418607330508619745600934617627243192576 * q^93 - 93840167647611953736551449995129877215600 * q^95 - 31989468315501178009691057062669320691004 * q^97 + 123209315561064920195916977740234432814088 * q^99

Display 100 coefficients 10 coefficients

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 $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	1.71810e9	0.284487	0.142244	−	0.989832i	$-0.454568\pi$
$3$	1.71810e9	0.284487	0.142244	+	0.989832i	$0.454568\pi$
$4$	0	0
$5$	3.51885e14	1.65012	0.825060	−	0.565044i	$-0.191141\pi$
$5$	3.51885e14	1.65012	0.825060	+	0.565044i	$0.191141\pi$
$6$	0	0
$7$	7.89090e16	0.373780	0.186890	−	0.982381i	$-0.440159\pi$
$7$	7.89090e16	0.373780	0.186890	+	0.982381i	$0.440159\pi$
$8$	0	0
$9$	−3.35211e19	−0.919067
$10$	0	0
$11$	−1.20679e21	−0.540856	−0.270428	−	0.962740i	$-0.587165\pi$
$11$	−1.20679e21	−0.540856	−0.270428	+	0.962740i	$0.587165\pi$
$12$	0	0
$13$	9.23598e22	1.34786	0.673929	−	0.738796i	$-0.264605\pi$
$13$	9.23598e22	1.34786	0.673929	+	0.738796i	$0.264605\pi$
$14$	0	0
$15$	6.04574e23	0.469439
$16$	0	0
$17$	−4.95668e24	−0.295793	−0.147897	−	0.989003i	$-0.547250\pi$
$17$	−4.95668e24	−0.295793	−0.147897	+	0.989003i	$0.547250\pi$
$18$	0	0
$19$	−2.41331e26	−1.47287	−0.736435	−	0.676508i	$-0.763492\pi$
$19$	−2.41331e26	−1.47287	−0.736435	+	0.676508i	$0.763492\pi$
$20$	0	0
$21$	1.35574e26	0.106336
$22$	0	0
$23$	−8.23610e27	−1.00069	−0.500347	−	0.865825i	$-0.666794\pi$
$23$	−8.23610e27	−1.00069	−0.500347	+	0.865825i	$0.666794\pi$
$24$	0	0
$25$	7.83484e28	1.72290
$26$	0	0
$27$	−1.20257e29	−0.545950
$28$	0	0
$29$	2.60718e29	0.273534	0.136767	−	0.990603i	$-0.456329\pi$
$29$	2.60718e29	0.273534	0.136767	+	0.990603i	$0.456329\pi$
$30$	0	0
$31$	6.40610e30	1.71269	0.856346	−	0.516402i	$-0.172729\pi$
$31$	6.40610e30	1.71269	0.856346	+	0.516402i	$0.172729\pi$
$32$	0	0
$33$	−2.07339e30	−0.153867
$34$	0	0
$35$	2.77669e31	0.616783
$36$	0	0
$37$	7.38149e31	0.524818	0.262409	−	0.964957i	$-0.415483\pi$
$37$	7.38149e31	0.524818	0.262409	+	0.964957i	$0.415483\pi$
$38$	0	0
$39$	1.58683e32	0.383448
$40$	0	0
$41$	2.15386e33	1.86701	0.933507	−	0.358558i	$-0.116732\pi$
$41$	2.15386e33	1.86701	0.933507	+	0.358558i	$0.116732\pi$
$42$	0	0
$43$	5.67618e33	1.85333	0.926665	−	0.375889i	$-0.122663\pi$
$43$	5.67618e33	1.85333	0.926665	+	0.375889i	$0.122663\pi$
$44$	0	0
$45$	−1.17956e34	−1.51657
$46$	0	0
$47$	6.45458e33	0.340300	0.170150	−	0.985418i	$-0.445575\pi$
$47$	6.45458e33	0.340300	0.170150	+	0.985418i	$0.445575\pi$
$48$	0	0
$49$	−3.83410e34	−0.860288
$50$	0	0
$51$	−8.51608e33	−0.0841494
$52$	0	0
$53$	−9.99163e34	−0.448726	−0.224363	−	0.974506i	$-0.572030\pi$
$53$	−9.99163e34	−0.448726	−0.224363	+	0.974506i	$0.572030\pi$
$54$	0	0
$55$	−4.24652e35	−0.892479
$56$	0	0
$57$	−4.14631e35	−0.419013
$58$	0	0
$59$	3.33277e36	1.66088	0.830442	−	0.557105i	$-0.188088\pi$
$59$	3.33277e36	1.66088	0.830442	+	0.557105i	$0.188088\pi$
$60$	0	0
$61$	2.08511e36	0.524648	0.262324	−	0.964980i	$-0.415511\pi$
$61$	2.08511e36	0.524648	0.262324	+	0.964980i	$0.415511\pi$
$62$	0	0
$63$	−2.64512e36	−0.343529
$64$	0	0
$65$	3.25000e37	2.22413
$66$	0	0
$67$	−2.21433e36	−0.0814159	−0.0407080	−	0.999171i	$-0.512961\pi$
$67$	−2.21433e36	−0.0814159	−0.0407080	+	0.999171i	$0.512961\pi$
$68$	0	0
$69$	−1.41505e37	−0.284685
$70$	0	0
$71$	−5.59105e37	−0.626177	−0.313089	−	0.949724i	$-0.601364\pi$
$71$	−5.59105e37	−0.626177	−0.313089	+	0.949724i	$0.601364\pi$
$72$	0	0
$73$	−2.02164e38	−1.28110	−0.640552	−	0.767915i	$-0.721294\pi$
$73$	−2.02164e38	−1.28110	−0.640552	+	0.767915i	$0.721294\pi$
$74$	0	0
$75$	1.34610e38	0.490143
$76$	0	0
$77$	−9.52266e37	−0.202162
$78$	0	0
$79$	2.46315e38	0.309126	0.154563	−	0.987983i	$-0.450603\pi$
$79$	2.46315e38	0.309126	0.154563	+	0.987983i	$0.450603\pi$
$80$	0	0
$81$	1.01600e39	0.763751
$82$	0	0
$83$	3.59771e37	0.0164031	0.00820154	−	0.999966i	$-0.497389\pi$
$83$	3.59771e37	0.0164031	0.00820154	+	0.999966i	$0.497389\pi$
$84$	0	0
$85$	−1.74418e39	−0.488095
$86$	0	0
$87$	4.47940e38	0.0778171
$88$	0	0
$89$	5.24223e38	0.0571506	0.0285753	−	0.999592i	$-0.490903\pi$
$89$	5.24223e38	0.0571506	0.0285753	+	0.999592i	$0.490903\pi$
$90$	0	0
$91$	7.28801e39	0.503803
$92$	0	0
$93$	1.10063e40	0.487239
$94$	0	0
$95$	−8.49208e40	−2.43041
$96$	0	0
$97$	−2.35889e40	−0.440441	−0.220221	−	0.975450i	$-0.570678\pi$
$97$	−2.35889e40	−0.440441	−0.220221	+	0.975450i	$0.570678\pi$
$98$	0	0
$99$	4.04530e40	0.497083
$100$	0	0
$101$	−1.92595e40	−0.157057	−0.0785284	−	0.996912i	$-0.525022\pi$
$101$	−1.92595e40	−0.157057	−0.0785284	+	0.996912i	$0.525022\pi$
$102$	0	0
$103$	2.65701e41	1.44954	0.724770	−	0.688991i	$-0.241946\pi$
$103$	2.65701e41	1.44954	0.724770	+	0.688991i	$0.241946\pi$
$104$	0	0
$105$	4.77063e40	0.175467
$106$	0	0
$107$	2.47756e41	0.618951	0.309476	−	0.950907i	$-0.399846\pi$
$107$	2.47756e41	0.618951	0.309476	+	0.950907i	$0.399846\pi$
$108$	0	0
$109$	−5.73631e40	−0.0980369	−0.0490184	−	0.998798i	$-0.515609\pi$
$109$	−5.73631e40	−0.0980369	−0.0490184	+	0.998798i	$0.515609\pi$
$110$	0	0
$111$	1.26822e41	0.149304
$112$	0	0
$113$	1.51047e42	1.23312	0.616558	−	0.787310i	$-0.288527\pi$
$113$	1.51047e42	1.23312	0.616558	+	0.787310i	$0.288527\pi$
$114$	0	0
$115$	−2.89816e42	−1.65127
$116$	0	0
$117$	−3.09600e42	−1.23877
$118$	0	0
$119$	−3.91127e41	−0.110562
$120$	0	0
$121$	−3.52217e42	−0.707474
$122$	0	0
$123$	3.70054e42	0.531142
$124$	0	0
$125$	1.15677e43	1.19287
$126$	0	0
$127$	−7.49737e42	−0.558384	−0.279192	−	0.960235i	$-0.590067\pi$
$127$	−7.49737e42	−0.558384	−0.279192	+	0.960235i	$0.590067\pi$
$128$	0	0
$129$	9.75225e42	0.527249
$130$	0	0
$131$	4.11904e43	1.62456	0.812278	−	0.583271i	$-0.198227\pi$
$131$	4.11904e43	1.62456	0.812278	+	0.583271i	$0.198227\pi$
$132$	0	0
$133$	−1.90432e43	−0.550530
$134$	0	0
$135$	−4.23166e43	−0.900884
$136$	0	0
$137$	−3.66801e43	−0.577643	−0.288821	−	0.957383i	$-0.593263\pi$
$137$	−3.66801e43	−0.577643	−0.288821	+	0.957383i	$0.593263\pi$
$138$	0	0
$139$	8.88436e43	1.03950	0.519748	−	0.854320i	$-0.326026\pi$
$139$	8.88436e43	1.03950	0.519748	+	0.854320i	$0.326026\pi$
$140$	0	0
$141$	1.10896e43	0.0968110
$142$	0	0
$143$	−1.11459e44	−0.728998
$144$	0	0
$145$	9.17428e43	0.451365
$146$	0	0
$147$	−6.58737e43	−0.244741
$148$	0	0
$149$	3.54730e44	0.999033	0.499517	−	0.866304i	$-0.333511\pi$
$149$	3.54730e44	0.999033	0.499517	+	0.866304i	$0.333511\pi$
$150$	0	0
$151$	−4.26711e44	−0.914338	−0.457169	−	0.889380i	$-0.651137\pi$
$151$	−4.26711e44	−0.914338	−0.457169	+	0.889380i	$0.651137\pi$
$152$	0	0
$153$	1.66154e44	0.271854
$154$	0	0
$155$	2.25421e45	2.82615
$156$	0	0
$157$	−8.24494e43	−0.0794776	−0.0397388	−	0.999210i	$-0.512653\pi$
$157$	−8.24494e43	−0.0794776	−0.0397388	+	0.999210i	$0.512653\pi$
$158$	0	0
$159$	−1.71666e44	−0.127657
$160$	0	0
$161$	−6.49902e44	−0.374040
$162$	0	0
$163$	−1.78991e44	−0.0799806	−0.0399903	−	0.999200i	$-0.512733\pi$
$163$	−1.78991e44	−0.0799806	−0.0399903	+	0.999200i	$0.512733\pi$
$164$	0	0
$165$	−7.29595e44	−0.253899
$166$	0	0
$167$	−3.96773e45	−1.07859	−0.539293	−	0.842118i	$-0.681308\pi$
$167$	−3.96773e45	−1.07859	−0.539293	+	0.842118i	$0.681308\pi$
$168$	0	0
$169$	3.83487e45	0.816721
$170$	0	0
$171$	8.08969e45	1.35367
$172$	0	0
$173$	6.93399e45	0.914193	0.457097	−	0.889417i	$-0.348889\pi$
$173$	6.93399e45	0.914193	0.457097	+	0.889417i	$0.348889\pi$
$174$	0	0
$175$	6.18239e45	0.643986
$176$	0	0
$177$	5.72604e45	0.472500
$178$	0	0
$179$	1.62403e46	1.06441	0.532203	−	0.846617i	$-0.321364\pi$
$179$	1.62403e46	1.06441	0.532203	+	0.846617i	$0.321364\pi$
$180$	0	0
$181$	2.21915e46	1.15818	0.579090	−	0.815264i	$-0.303408\pi$
$181$	2.21915e46	1.15818	0.579090	+	0.815264i	$0.303408\pi$
$182$	0	0
$183$	3.58243e45	0.149256
$184$	0	0
$185$	2.59744e46	0.866014
$186$	0	0
$187$	5.98168e45	0.159982
$188$	0	0
$189$	−9.48935e45	−0.204066
$190$	0	0
$191$	4.97685e46	0.862523	0.431262	−	0.902227i	$-0.358069\pi$
$191$	4.97685e46	0.862523	0.431262	+	0.902227i	$0.358069\pi$
$192$	0	0
$193$	4.02520e46	0.563459	0.281730	−	0.959494i	$-0.409092\pi$
$193$	4.02520e46	0.563459	0.281730	+	0.959494i	$0.409092\pi$
$194$	0	0
$195$	5.58383e46	0.632736
$196$	0	0
$197$	−7.00300e46	−0.643764	−0.321882	−	0.946780i	$-0.604315\pi$
$197$	−7.00300e46	−0.643764	−0.321882	+	0.946780i	$0.604315\pi$
$198$	0	0
$199$	1.17193e47	0.875818	0.437909	−	0.899019i	$-0.355719\pi$
$199$	1.17193e47	0.875818	0.437909	+	0.899019i	$0.355719\pi$
$200$	0	0
$201$	−3.80445e45	−0.0231618
$202$	0	0
$203$	2.05730e46	0.102242
$204$	0	0
$205$	7.57910e47	3.08080
$206$	0	0
$207$	2.76083e47	0.919704
$208$	0	0
$209$	2.91236e47	0.796611
$210$	0	0
$211$	−5.52367e47	−1.24290	−0.621452	−	0.783452i	$-0.713457\pi$
$211$	−5.52367e47	−1.24290	−0.621452	+	0.783452i	$0.713457\pi$
$212$	0	0
$213$	−9.60599e46	−0.178139
$214$	0	0
$215$	1.99736e48	3.05822
$216$	0	0
$217$	5.05499e47	0.640171
$218$	0	0
$219$	−3.47337e47	−0.364458
$220$	0	0
$221$	−4.57798e47	−0.398687
$222$	0	0
$223$	−1.20926e48	−0.875531	−0.437765	−	0.899089i	$-0.644230\pi$
$223$	−1.20926e48	−0.875531	−0.437765	+	0.899089i	$0.644230\pi$
$224$	0	0
$225$	−2.62633e48	−1.58346
$226$	0	0
$227$	1.02944e48	0.517690	0.258845	−	0.965919i	$-0.416658\pi$
$227$	1.02944e48	0.517690	0.258845	+	0.965919i	$0.416658\pi$
$228$	0	0
$229$	2.04406e48	0.858750	0.429375	−	0.903126i	$-0.358734\pi$
$229$	2.04406e48	0.858750	0.429375	+	0.903126i	$0.358734\pi$
$230$	0	0
$231$	−1.63609e47	−0.0575124
$232$	0	0
$233$	4.38442e48	1.29156	0.645781	−	0.763523i	$-0.276532\pi$
$233$	4.38442e48	1.29156	0.645781	+	0.763523i	$0.276532\pi$
$234$	0	0
$235$	2.27127e48	0.561536
$236$	0	0
$237$	4.23194e47	0.0879423
$238$	0	0
$239$	−4.33337e48	−0.757999	−0.379000	−	0.925397i	$-0.623732\pi$
$239$	−4.33337e48	−0.757999	−0.379000	+	0.925397i	$0.623732\pi$
$240$	0	0
$241$	−3.97635e48	−0.586320	−0.293160	−	0.956063i	$-0.594707\pi$
$241$	−3.97635e48	−0.586320	−0.293160	+	0.956063i	$0.594707\pi$
$242$	0	0
$243$	6.13173e48	0.763228
$244$	0	0
$245$	−1.34916e49	−1.41958
$246$	0	0
$247$	−2.22893e49	−1.98522
$248$	0	0
$249$	6.18123e46	0.00466647
$250$	0	0
$251$	−1.35701e49	−0.869507	−0.434754	−	0.900549i	$-0.643165\pi$
$251$	−1.35701e49	−0.869507	−0.434754	+	0.900549i	$0.643165\pi$
$252$	0	0
$253$	9.93926e48	0.541232
$254$	0	0
$255$	−2.99668e48	−0.138857
$256$	0	0
$257$	−4.30252e49	−1.69861	−0.849304	−	0.527903i	$-0.822978\pi$
$257$	−4.30252e49	−1.69861	−0.849304	+	0.527903i	$0.822978\pi$
$258$	0	0
$259$	5.82466e48	0.196167
$260$	0	0
$261$	−8.73956e48	−0.251397
$262$	0	0
$263$	1.54140e49	0.379160	0.189580	−	0.981865i	$-0.439287\pi$
$263$	1.54140e49	0.379160	0.189580	+	0.981865i	$0.439287\pi$
$264$	0	0
$265$	−3.51591e49	−0.740452
$266$	0	0
$267$	9.00667e47	0.0162586
$268$	0	0
$269$	2.17495e49	0.336920	0.168460	−	0.985708i	$-0.446121\pi$
$269$	2.17495e49	0.336920	0.168460	+	0.985708i	$0.446121\pi$
$270$	0	0
$271$	−5.12539e49	−0.682113	−0.341056	−	0.940043i	$-0.610785\pi$
$271$	−5.12539e49	−0.682113	−0.341056	+	0.940043i	$0.610785\pi$
$272$	0	0
$273$	1.25215e49	0.143326
$274$	0	0
$275$	−9.45501e49	−0.931841
$276$	0	0
$277$	−5.73007e49	−0.486771	−0.243385	−	0.969930i	$-0.578258\pi$
$277$	−5.73007e49	−0.486771	−0.243385	+	0.969930i	$0.578258\pi$
$278$	0	0
$279$	−2.14740e50	−1.57408
$280$	0	0
$281$	1.97230e49	0.124880	0.0624400	−	0.998049i	$-0.480112\pi$
$281$	1.97230e49	0.124880	0.0624400	+	0.998049i	$0.480112\pi$
$282$	0	0
$283$	1.39119e49	0.0761663	0.0380832	−	0.999275i	$-0.487875\pi$
$283$	1.39119e49	0.0761663	0.0380832	+	0.999275i	$0.487875\pi$
$284$	0	0
$285$	−1.45902e50	−0.691422
$286$	0	0
$287$	1.69959e50	0.697853
$288$	0	0
$289$	−2.56237e50	−0.912506
$290$	0	0
$291$	−4.05282e49	−0.125300
$292$	0	0
$293$	3.84683e50	1.03351	0.516757	−	0.856132i	$-0.327139\pi$
$293$	3.84683e50	1.03351	0.516757	+	0.856132i	$0.327139\pi$
$294$	0	0
$295$	1.17275e51	2.74066
$296$	0	0
$297$	1.45125e50	0.295281
$298$	0	0
$299$	−7.60684e50	−1.34879
$300$	0	0
$301$	4.47901e50	0.692738
$302$	0	0
$303$	−3.30897e49	−0.0446807
$304$	0	0
$305$	7.33720e50	0.865732
$306$	0	0
$307$	1.33309e51	1.37569	0.687846	−	0.725856i	$-0.258556\pi$
$307$	1.33309e51	1.37569	0.687846	+	0.725856i	$0.258556\pi$
$308$	0	0
$309$	4.56501e50	0.412376
$310$	0	0
$311$	3.42217e50	0.270841	0.135420	−	0.990788i	$-0.456761\pi$
$311$	3.42217e50	0.270841	0.135420	+	0.990788i	$0.456761\pi$
$312$	0	0
$313$	8.29482e50	0.575636	0.287818	−	0.957685i	$-0.407070\pi$
$313$	8.29482e50	0.575636	0.287818	+	0.957685i	$0.407070\pi$
$314$	0	0
$315$	−9.30777e50	−0.566865
$316$	0	0
$317$	−2.94058e51	−1.57296	−0.786481	−	0.617615i	$-0.788099\pi$
$317$	−2.94058e51	−1.57296	−0.786481	+	0.617615i	$0.788099\pi$
$318$	0	0
$319$	−3.14632e50	−0.147943
$320$	0	0
$321$	4.25670e50	0.176084
$322$	0	0
$323$	1.19620e51	0.435665
$324$	0	0
$325$	7.23624e51	2.32222
$326$	0	0
$327$	−9.85556e49	−0.0278902
$328$	0	0
$329$	5.09324e50	0.127197
$330$	0	0
$331$	8.47531e51	1.86931	0.934656	−	0.355554i	$-0.115708\pi$
$331$	8.47531e51	1.86931	0.934656	+	0.355554i	$0.115708\pi$
$332$	0	0
$333$	−2.47436e51	−0.482343
$334$	0	0
$335$	−7.79191e50	−0.134346
$336$	0	0
$337$	2.49054e51	0.380085	0.190042	−	0.981776i	$-0.439137\pi$
$337$	2.49054e51	0.380085	0.190042	+	0.981776i	$0.439137\pi$
$338$	0	0
$339$	2.59514e51	0.350806
$340$	0	0
$341$	−7.73083e51	−0.926321
$342$	0	0
$343$	−6.54224e51	−0.695339
$344$	0	0
$345$	−4.97933e51	−0.469764
$346$	0	0
$347$	1.02424e51	0.0858314	0.0429157	−	0.999079i	$-0.486335\pi$
$347$	1.02424e51	0.0858314	0.0429157	+	0.999079i	$0.486335\pi$
$348$	0	0
$349$	7.39758e51	0.551022	0.275511	−	0.961298i	$-0.411153\pi$
$349$	7.39758e51	0.551022	0.275511	+	0.961298i	$0.411153\pi$
$350$	0	0
$351$	−1.11069e52	−0.735863
$352$	0	0
$353$	−6.47846e51	−0.382024	−0.191012	−	0.981588i	$-0.561177\pi$
$353$	−6.47846e51	−0.382024	−0.191012	+	0.981588i	$0.561177\pi$
$354$	0	0
$355$	−1.96741e52	−1.03327
$356$	0	0
$357$	−6.71995e50	−0.0314534
$358$	0	0
$359$	−2.46025e52	−1.02693	−0.513467	−	0.858109i	$-0.671639\pi$
$359$	−2.46025e52	−1.02693	−0.513467	+	0.858109i	$0.671639\pi$
$360$	0	0
$361$	3.13936e52	1.16935
$362$	0	0
$363$	−6.05145e51	−0.201267
$364$	0	0
$365$	−7.11383e52	−2.11398
$366$	0	0
$367$	4.93112e52	1.31006	0.655030	−	0.755603i	$-0.272656\pi$
$367$	4.93112e52	1.31006	0.655030	+	0.755603i	$0.272656\pi$
$368$	0	0
$369$	−7.21997e52	−1.71591
$370$	0	0
$371$	−7.88429e51	−0.167725
$372$	0	0
$373$	3.70443e52	0.705815	0.352907	−	0.935658i	$-0.385193\pi$
$373$	3.70443e52	0.705815	0.352907	+	0.935658i	$0.385193\pi$
$374$	0	0
$375$	1.98746e52	0.339357
$376$	0	0
$377$	2.40799e52	0.368686
$378$	0	0
$379$	−1.03809e53	−1.42604	−0.713021	−	0.701143i	$-0.752673\pi$
$379$	−1.03809e53	−1.42604	−0.713021	+	0.701143i	$0.752673\pi$
$380$	0	0
$381$	−1.28812e52	−0.158853
$382$	0	0
$383$	1.27369e52	0.141088	0.0705440	−	0.997509i	$-0.477526\pi$
$383$	1.27369e52	0.141088	0.0705440	+	0.997509i	$0.477526\pi$
$384$	0	0
$385$	−3.35088e52	−0.333591
$386$	0	0
$387$	−1.90272e53	−1.70333
$388$	0	0
$389$	−1.61862e53	−1.30371	−0.651853	−	0.758346i	$-0.726008\pi$
$389$	−1.61862e53	−1.30371	−0.651853	+	0.758346i	$0.726008\pi$
$390$	0	0
$391$	4.08237e52	0.295998
$392$	0	0
$393$	7.07693e52	0.462166
$394$	0	0
$395$	8.66745e52	0.510095
$396$	0	0
$397$	2.20821e53	1.17175	0.585876	−	0.810401i	$-0.300751\pi$
$397$	2.20821e53	1.17175	0.585876	+	0.810401i	$0.300751\pi$
$398$	0	0
$399$	−3.27181e52	−0.156619
$400$	0	0
$401$	−1.18852e52	−0.0513509	−0.0256755	−	0.999670i	$-0.508174\pi$
$401$	−1.18852e52	−0.0513509	−0.0256755	+	0.999670i	$0.508174\pi$
$402$	0	0
$403$	5.91666e53	2.30847
$404$	0	0
$405$	3.57516e53	1.26028
$406$	0	0
$407$	−8.90792e52	−0.283851
$408$	0	0
$409$	−9.36392e52	−0.269855	−0.134927	−	0.990856i	$-0.543080\pi$
$409$	−9.36392e52	−0.269855	−0.134927	+	0.990856i	$0.543080\pi$
$410$	0	0
$411$	−6.30202e52	−0.164332
$412$	0	0
$413$	2.62985e53	0.620806
$414$	0	0
$415$	1.26598e52	0.0270671
$416$	0	0
$417$	1.52642e53	0.295723
$418$	0	0
$419$	4.45575e53	0.782588	0.391294	−	0.920266i	$-0.372028\pi$
$419$	4.45575e53	0.782588	0.391294	+	0.920266i	$0.372028\pi$
$420$	0	0
$421$	−5.72623e53	−0.912191	−0.456096	−	0.889931i	$-0.650752\pi$
$421$	−5.72623e53	−0.912191	−0.456096	+	0.889931i	$0.650752\pi$
$422$	0	0
$423$	−2.16365e53	−0.312758
$424$	0	0
$425$	−3.88348e53	−0.509622
$426$	0	0
$427$	1.64534e53	0.196103
$428$	0	0
$429$	−1.91498e53	−0.207391
$430$	0	0
$431$	7.99569e53	0.787176	0.393588	−	0.919287i	$-0.371234\pi$
$431$	7.99569e53	0.787176	0.393588	+	0.919287i	$0.371234\pi$
$432$	0	0
$433$	1.41016e54	1.26261	0.631303	−	0.775536i	$-0.282520\pi$
$433$	1.41016e54	1.26261	0.631303	+	0.775536i	$0.282520\pi$
$434$	0	0
$435$	1.57623e53	0.128408
$436$	0	0
$437$	1.98763e54	1.47389
$438$	0	0
$439$	−9.48293e53	−0.640353	−0.320177	−	0.947358i	$-0.603742\pi$
$439$	−9.48293e53	−0.640353	−0.320177	+	0.947358i	$0.603742\pi$
$440$	0	0
$441$	1.28523e54	0.790662
$442$	0	0
$443$	1.15125e54	0.645494	0.322747	−	0.946485i	$-0.395394\pi$
$443$	1.15125e54	0.645494	0.322747	+	0.946485i	$0.395394\pi$
$444$	0	0
$445$	1.84466e53	0.0943054
$446$	0	0
$447$	6.09463e53	0.284212
$448$	0	0
$449$	−1.29162e54	−0.549649	−0.274824	−	0.961494i	$-0.588620\pi$
$449$	−1.29162e54	−0.549649	−0.274824	+	0.961494i	$0.588620\pi$
$450$	0	0
$451$	−2.59925e54	−1.00979
$452$	0	0
$453$	−7.33133e53	−0.260118
$454$	0	0
$455$	2.56454e54	0.831336
$456$	0	0
$457$	−5.70144e54	−1.68928	−0.844641	−	0.535332i	$-0.820186\pi$
$457$	−5.70144e54	−1.68928	−0.844641	+	0.535332i	$0.820186\pi$
$458$	0	0
$459$	5.96076e53	0.161488
$460$	0	0
$461$	−5.36766e54	−1.33020	−0.665098	−	0.746756i	$-0.731610\pi$
$461$	−5.36766e54	−1.33020	−0.665098	+	0.746756i	$0.731610\pi$
$462$	0	0
$463$	−6.24453e54	−1.41609	−0.708043	−	0.706170i	$-0.750422\pi$
$463$	−6.24453e54	−1.41609	−0.708043	+	0.706170i	$0.750422\pi$
$464$	0	0
$465$	3.87296e54	0.804004
$466$	0	0
$467$	8.52716e53	0.162109	0.0810547	−	0.996710i	$-0.474171\pi$
$467$	8.52716e53	0.162109	0.0810547	+	0.996710i	$0.474171\pi$
$468$	0	0
$469$	−1.74731e53	−0.0304317
$470$	0	0
$471$	−1.41656e53	−0.0226104
$472$	0	0
$473$	−6.84996e54	−1.00239
$474$	0	0
$475$	−1.89079e55	−2.53761
$476$	0	0
$477$	3.34931e54	0.412409
$478$	0	0
$479$	6.54754e54	0.739947	0.369973	−	0.929042i	$-0.379367\pi$
$479$	6.54754e54	0.739947	0.369973	+	0.929042i	$0.379367\pi$
$480$	0	0
$481$	6.81753e54	0.707380
$482$	0	0
$483$	−1.11660e54	−0.106410
$484$	0	0
$485$	−8.30060e54	−0.726781
$486$	0	0
$487$	4.90744e54	0.394922	0.197461	−	0.980311i	$-0.436730\pi$
$487$	4.90744e54	0.394922	0.197461	+	0.980311i	$0.436730\pi$
$488$	0	0
$489$	−3.07524e53	−0.0227535
$490$	0	0
$491$	−2.65373e54	−0.180587	−0.0902935	−	0.995915i	$-0.528781\pi$
$491$	−2.65373e54	−0.180587	−0.0902935	+	0.995915i	$0.528781\pi$
$492$	0	0
$493$	−1.29230e54	−0.0809097
$494$	0	0
$495$	1.42348e55	0.820248
$496$	0	0
$497$	−4.41184e54	−0.234053
$498$	0	0
$499$	−6.02973e53	−0.0294603	−0.0147301	−	0.999892i	$-0.504689\pi$
$499$	−6.02973e53	−0.0294603	−0.0147301	+	0.999892i	$0.504689\pi$
$500$	0	0
$501$	−6.81696e54	−0.306844
$502$	0	0
$503$	−4.08674e55	−1.69525	−0.847625	−	0.530596i	$-0.821968\pi$
$503$	−4.08674e55	−1.69525	−0.847625	+	0.530596i	$0.821968\pi$
$504$	0	0
$505$	−6.77713e54	−0.259163
$506$	0	0
$507$	6.58870e54	0.232347
$508$	0	0
$509$	4.17871e54	0.135934	0.0679670	−	0.997688i	$-0.478349\pi$
$509$	4.17871e54	0.135934	0.0679670	+	0.997688i	$0.478349\pi$
$510$	0	0
$511$	−1.59525e55	−0.478851
$512$	0	0
$513$	2.90217e55	0.804114
$514$	0	0
$515$	9.34962e55	2.39192
$516$	0	0
$517$	−7.78933e54	−0.184053
$518$	0	0
$519$	1.19133e55	0.260076
$520$	0	0
$521$	2.55519e55	0.515525	0.257763	−	0.966208i	$-0.417015\pi$
$521$	2.55519e55	0.515525	0.257763	+	0.966208i	$0.417015\pi$
$522$	0	0
$523$	6.45058e55	1.20314	0.601568	−	0.798822i	$-0.294543\pi$
$523$	6.45058e55	1.20314	0.601568	+	0.798822i	$0.294543\pi$
$524$	0	0
$525$	1.06220e55	0.183206
$526$	0	0
$527$	−3.17530e55	−0.506603
$528$	0	0
$529$	9.40022e52	0.00138770
$530$	0	0
$531$	−1.11718e56	−1.52646
$532$	0	0
$533$	1.98930e56	2.51647
$534$	0	0
$535$	8.71816e55	1.02134
$536$	0	0
$537$	2.79025e55	0.302810
$538$	0	0
$539$	4.62696e55	0.465292
$540$	0	0
$541$	−1.14581e56	−1.06799	−0.533995	−	0.845488i	$-0.679310\pi$
$541$	−1.14581e56	−1.06799	−0.533995	+	0.845488i	$0.679310\pi$
$542$	0	0
$543$	3.81273e55	0.329488
$544$	0	0
$545$	−2.01852e55	−0.161773
$546$	0	0
$547$	−6.76717e55	−0.503114	−0.251557	−	0.967842i	$-0.580943\pi$
$547$	−6.76717e55	−0.503114	−0.251557	+	0.967842i	$0.580943\pi$
$548$	0	0
$549$	−6.98953e55	−0.482186
$550$	0	0
$551$	−6.29194e55	−0.402881
$552$	0	0
$553$	1.94364e55	0.115545
$554$	0	0
$555$	4.46266e55	0.246370
$556$	0	0
$557$	−1.91017e56	−0.979584	−0.489792	−	0.871839i	$-0.662927\pi$
$557$	−1.91017e56	−0.979584	−0.489792	+	0.871839i	$0.662927\pi$
$558$	0	0
$559$	5.24250e56	2.49802
$560$	0	0
$561$	1.02771e55	0.0455128
$562$	0	0
$563$	−2.46904e56	−1.01650	−0.508249	−	0.861210i	$-0.669707\pi$
$563$	−2.46904e56	−1.01650	−0.508249	+	0.861210i	$0.669707\pi$
$564$	0	0
$565$	5.31511e56	2.03479
$566$	0	0
$567$	8.01717e55	0.285475
$568$	0	0
$569$	−4.36145e56	−1.44487	−0.722437	−	0.691437i	$-0.756978\pi$
$569$	−4.36145e56	−1.44487	−0.722437	+	0.691437i	$0.756978\pi$
$570$	0	0
$571$	−3.87246e56	−1.19384	−0.596920	−	0.802300i	$-0.703609\pi$
$571$	−3.87246e56	−1.19384	−0.596920	+	0.802300i	$0.703609\pi$
$572$	0	0
$573$	8.55074e55	0.245377
$574$	0	0
$575$	−6.45285e56	−1.72409
$576$	0	0
$577$	3.88180e56	0.965893	0.482947	−	0.875650i	$-0.339567\pi$
$577$	3.88180e56	0.965893	0.482947	+	0.875650i	$0.339567\pi$
$578$	0	0
$579$	6.91570e55	0.160297
$580$	0	0
$581$	2.83892e54	0.00613115
$582$	0	0
$583$	1.20578e56	0.242696
$584$	0	0
$585$	−1.08944e57	−2.04412
$586$	0	0
$587$	−5.00890e56	−0.876317	−0.438158	−	0.898898i	$-0.644369\pi$
$587$	−5.00890e56	−0.876317	−0.438158	+	0.898898i	$0.644369\pi$
$588$	0	0
$589$	−1.54599e57	−2.52257
$590$	0	0
$591$	−1.20319e56	−0.183143
$592$	0	0
$593$	9.63914e56	1.36904	0.684521	−	0.728993i	$-0.260011\pi$
$593$	9.63914e56	1.36904	0.684521	+	0.728993i	$0.260011\pi$
$594$	0	0
$595$	−1.37632e56	−0.182440
$596$	0	0
$597$	2.01350e56	0.249159
$598$	0	0
$599$	1.39170e56	0.160803	0.0804016	−	0.996763i	$-0.474380\pi$
$599$	1.39170e56	0.160803	0.0804016	+	0.996763i	$0.474380\pi$
$600$	0	0
$601$	−1.72237e56	−0.185866	−0.0929330	−	0.995672i	$-0.529624\pi$
$601$	−1.72237e56	−0.185866	−0.0929330	+	0.995672i	$0.529624\pi$
$602$	0	0
$603$	7.42269e55	0.0748267
$604$	0	0
$605$	−1.23940e57	−1.16742
$606$	0	0
$607$	−1.22974e57	−1.08254	−0.541270	−	0.840849i	$-0.682056\pi$
$607$	−1.22974e57	−1.08254	−0.541270	+	0.840849i	$0.682056\pi$
$608$	0	0
$609$	3.53465e55	0.0290865
$610$	0	0
$611$	5.96143e56	0.458676
$612$	0	0
$613$	2.53270e57	1.82240	0.911199	−	0.411966i	$-0.135158\pi$
$613$	2.53270e57	1.82240	0.911199	+	0.411966i	$0.135158\pi$
$614$	0	0
$615$	1.30217e57	0.876449
$616$	0	0
$617$	−2.74089e56	−0.172602	−0.0863011	−	0.996269i	$-0.527505\pi$
$617$	−2.74089e56	−0.172602	−0.0863011	+	0.996269i	$0.527505\pi$
$618$	0	0
$619$	1.90431e57	1.12222	0.561112	−	0.827740i	$-0.310374\pi$
$619$	1.90431e57	1.12222	0.561112	+	0.827740i	$0.310374\pi$
$620$	0	0
$621$	9.90449e56	0.546329
$622$	0	0
$623$	4.13659e55	0.0213618
$624$	0	0
$625$	5.07645e56	0.245482
$626$	0	0
$627$	5.00373e56	0.226626
$628$	0	0
$629$	−3.65877e56	−0.155238
$630$	0	0
$631$	−1.91059e57	−0.759565	−0.379782	−	0.925076i	$-0.624001\pi$
$631$	−1.91059e57	−0.759565	−0.379782	+	0.925076i	$0.624001\pi$
$632$	0	0
$633$	−9.49023e56	−0.353590
$634$	0	0
$635$	−2.63821e57	−0.921400
$636$	0	0
$637$	−3.54117e57	−1.15955
$638$	0	0
$639$	1.87418e57	0.575499
$640$	0	0
$641$	6.02067e57	1.73403	0.867013	−	0.498286i	$-0.166037\pi$
$641$	6.02067e57	1.73403	0.867013	+	0.498286i	$0.166037\pi$
$642$	0	0
$643$	−5.41099e57	−1.46201	−0.731007	−	0.682370i	$-0.760949\pi$
$643$	−5.41099e57	−1.46201	−0.731007	+	0.682370i	$0.760949\pi$
$644$	0	0
$645$	3.43167e57	0.870024
$646$	0	0
$647$	6.44976e57	1.53464	0.767319	−	0.641266i	$-0.221590\pi$
$647$	6.44976e57	1.53464	0.767319	+	0.641266i	$0.221590\pi$
$648$	0	0
$649$	−4.02196e57	−0.898300
$650$	0	0
$651$	8.68498e56	0.182120
$652$	0	0
$653$	5.84519e57	1.15101	0.575504	−	0.817799i	$-0.304806\pi$
$653$	5.84519e57	1.15101	0.575504	+	0.817799i	$0.304806\pi$
$654$	0	0
$655$	1.44943e58	2.68071
$656$	0	0
$657$	6.77675e57	1.17742
$658$	0	0
$659$	3.68490e57	0.601554	0.300777	−	0.953694i	$-0.402754\pi$
$659$	3.68490e57	0.601554	0.300777	+	0.953694i	$0.402754\pi$
$660$	0	0
$661$	−6.04679e57	−0.927672	−0.463836	−	0.885921i	$-0.653527\pi$
$661$	−6.04679e57	−0.927672	−0.463836	+	0.885921i	$0.653527\pi$
$662$	0	0
$663$	−7.86543e56	−0.113421
$664$	0	0
$665$	−6.70101e57	−0.908441
$666$	0	0
$667$	−2.14730e57	−0.273724
$668$	0	0
$669$	−2.07764e57	−0.249077
$670$	0	0
$671$	−2.51629e57	−0.283759
$672$	0	0
$673$	6.56510e57	0.696518	0.348259	−	0.937398i	$-0.386773\pi$
$673$	6.56510e57	0.696518	0.348259	+	0.937398i	$0.386773\pi$
$674$	0	0
$675$	−9.42194e57	−0.940617
$676$	0	0
$677$	1.10721e58	1.04031	0.520155	−	0.854072i	$-0.325874\pi$
$677$	1.10721e58	1.04031	0.520155	+	0.854072i	$0.325874\pi$
$678$	0	0
$679$	−1.86138e57	−0.164628
$680$	0	0
$681$	1.76868e57	0.147276
$682$	0	0
$683$	−1.97103e58	−1.54550	−0.772749	−	0.634711i	$-0.781119\pi$
$683$	−1.97103e58	−1.54550	−0.772749	+	0.634711i	$0.781119\pi$
$684$	0	0
$685$	−1.29072e58	−0.953180
$686$	0	0
$687$	3.51190e57	0.244303
$688$	0	0
$689$	−9.22825e57	−0.604819
$690$	0	0
$691$	−1.78821e58	−1.10438	−0.552191	−	0.833718i	$-0.686208\pi$
$691$	−1.78821e58	−1.10438	−0.552191	+	0.833718i	$0.686208\pi$
$692$	0	0
$693$	3.19210e57	0.185800
$694$	0	0
$695$	3.12627e58	1.71529
$696$	0	0
$697$	−1.06760e58	−0.552250
$698$	0	0
$699$	7.53287e57	0.367433
$700$	0	0
$701$	−2.83192e58	−1.30275	−0.651375	−	0.758756i	$-0.725807\pi$
$701$	−2.83192e58	−1.30275	−0.651375	+	0.758756i	$0.725807\pi$
$702$	0	0
$703$	−1.78138e58	−0.772989
$704$	0	0
$705$	3.90227e57	0.159750
$706$	0	0
$707$	−1.51975e57	−0.0587047
$708$	0	0
$709$	3.01107e58	1.09767	0.548836	−	0.835930i	$-0.315071\pi$
$709$	3.01107e58	1.09767	0.548836	+	0.835930i	$0.315071\pi$
$710$	0	0
$711$	−8.25675e57	−0.284107
$712$	0	0
$713$	−5.27613e58	−1.71388
$714$	0	0
$715$	−3.92207e58	−1.20293
$716$	0	0
$717$	−7.44516e57	−0.215641
$718$	0	0
$719$	3.81485e58	1.04360	0.521802	−	0.853067i	$-0.325260\pi$
$719$	3.81485e58	1.04360	0.521802	+	0.853067i	$0.325260\pi$
$720$	0	0
$721$	2.09662e58	0.541809
$722$	0	0
$723$	−6.83176e57	−0.166801
$724$	0	0
$725$	2.04268e58	0.471272
$726$	0	0
$727$	−3.32784e58	−0.725617	−0.362808	−	0.931864i	$-0.618182\pi$
$727$	−3.32784e58	−0.725617	−0.362808	+	0.931864i	$0.618182\pi$
$728$	0	0
$729$	−2.65217e58	−0.546622
$730$	0	0
$731$	−2.81350e58	−0.548202
$732$	0	0
$733$	−2.73207e58	−0.503338	−0.251669	−	0.967813i	$-0.580979\pi$
$733$	−2.73207e58	−0.503338	−0.251669	+	0.967813i	$0.580979\pi$
$734$	0	0
$735$	−2.31800e58	−0.403852
$736$	0	0
$737$	2.67224e57	0.0440343
$738$	0	0
$739$	2.78923e58	0.434784	0.217392	−	0.976084i	$-0.430245\pi$
$739$	2.78923e58	0.434784	0.217392	+	0.976084i	$0.430245\pi$
$740$	0	0
$741$	−3.82952e58	−0.564770
$742$	0	0
$743$	5.29149e58	0.738427	0.369214	−	0.929345i	$-0.379627\pi$
$743$	5.29149e58	0.738427	0.369214	+	0.929345i	$0.379627\pi$
$744$	0	0
$745$	1.24824e59	1.64853
$746$	0	0
$747$	−1.20599e57	−0.0150755
$748$	0	0
$749$	1.95502e58	0.231352
$750$	0	0
$751$	1.65688e59	1.85641	0.928203	−	0.372075i	$-0.121353\pi$
$751$	1.65688e59	1.85641	0.928203	+	0.372075i	$0.121353\pi$
$752$	0	0
$753$	−2.33149e58	−0.247364
$754$	0	0
$755$	−1.50153e59	−1.50877
$756$	0	0
$757$	1.30792e59	1.24485	0.622424	−	0.782680i	$-0.286148\pi$
$757$	1.30792e59	1.24485	0.622424	+	0.782680i	$0.286148\pi$
$758$	0	0
$759$	1.70766e58	0.153974
$760$	0	0
$761$	8.10267e58	0.692218	0.346109	−	0.938194i	$-0.387503\pi$
$761$	8.10267e58	0.692218	0.346109	+	0.938194i	$0.387503\pi$
$762$	0	0
$763$	−4.52646e57	−0.0366443
$764$	0	0
$765$	5.84670e58	0.448592
$766$	0	0
$767$	3.07814e59	2.23863
$768$	0	0
$769$	−5.87182e58	−0.404839	−0.202420	−	0.979299i	$-0.564881\pi$
$769$	−5.87182e58	−0.404839	−0.202420	+	0.979299i	$0.564881\pi$
$770$	0	0
$771$	−7.39216e58	−0.483233
$772$	0	0
$773$	−2.32645e59	−1.44216	−0.721081	−	0.692850i	$-0.756355\pi$
$773$	−2.32645e59	−1.44216	−0.721081	+	0.692850i	$0.756355\pi$
$774$	0	0
$775$	5.01908e59	2.95080
$776$	0	0
$777$	1.00074e58	0.0558070
$778$	0	0
$779$	−5.19792e59	−2.74987
$780$	0	0
$781$	6.74723e58	0.338672
$782$	0	0
$783$	−3.13532e58	−0.149336
$784$	0	0
$785$	−2.90127e58	−0.131148
$786$	0	0
$787$	−2.24048e59	−0.961299	−0.480649	−	0.876913i	$-0.659599\pi$
$787$	−2.24048e59	−0.961299	−0.480649	+	0.876913i	$0.659599\pi$
$788$	0	0
$789$	2.64828e58	0.107866
$790$	0	0
$791$	1.19190e59	0.460914
$792$	0	0
$793$	1.92580e59	0.707150
$794$	0	0
$795$	−6.04068e58	−0.210649
$796$	0	0
$797$	3.32067e59	1.09984	0.549921	−	0.835217i	$-0.314658\pi$
$797$	3.32067e59	1.09984	0.549921	+	0.835217i	$0.314658\pi$
$798$	0	0
$799$	−3.19933e58	−0.100658
$800$	0	0
$801$	−1.75725e58	−0.0525252
$802$	0	0
$803$	2.43969e59	0.692893
$804$	0	0
$805$	−2.28691e59	−0.617211
$806$	0	0
$807$	3.73678e58	0.0958495
$808$	0	0
$809$	7.35141e59	1.79236	0.896182	−	0.443687i	$-0.146330\pi$
$809$	7.35141e59	1.79236	0.896182	+	0.443687i	$0.146330\pi$
$810$	0	0
$811$	7.34195e58	0.170170	0.0850852	−	0.996374i	$-0.472884\pi$
$811$	7.34195e58	0.170170	0.0850852	+	0.996374i	$0.472884\pi$
$812$	0	0
$813$	−8.80593e58	−0.194053
$814$	0	0
$815$	−6.29842e58	−0.131978
$816$	0	0
$817$	−1.36984e60	−2.72971
$818$	0	0
$819$	−2.44302e59	−0.463029
$820$	0	0
$821$	−7.00875e59	−1.26359	−0.631794	−	0.775136i	$-0.717681\pi$
$821$	−7.00875e59	−1.26359	−0.631794	+	0.775136i	$0.717681\pi$
$822$	0	0
$823$	4.10799e59	0.704584	0.352292	−	0.935890i	$-0.385402\pi$
$823$	4.10799e59	0.704584	0.352292	+	0.935890i	$0.385402\pi$
$824$	0	0
$825$	−1.62447e59	−0.265097
$826$	0	0
$827$	−2.98123e59	−0.462949	−0.231474	−	0.972841i	$-0.574355\pi$
$827$	−2.98123e59	−0.462949	−0.231474	+	0.972841i	$0.574355\pi$
$828$	0	0
$829$	3.83752e59	0.567130	0.283565	−	0.958953i	$-0.408483\pi$
$829$	3.83752e59	0.567130	0.283565	+	0.958953i	$0.408483\pi$
$830$	0	0
$831$	−9.84484e58	−0.138480
$832$	0	0
$833$	1.90044e59	0.254467
$834$	0	0
$835$	−1.39618e60	−1.77980
$836$	0	0
$837$	−7.70379e59	−0.935045
$838$	0	0
$839$	−9.98054e59	−1.15354	−0.576772	−	0.816905i	$-0.695688\pi$
$839$	−9.98054e59	−1.15354	−0.576772	+	0.816905i	$0.695688\pi$
$840$	0	0
$841$	−8.40511e59	−0.925179
$842$	0	0
$843$	3.38861e58	0.0355268
$844$	0	0
$845$	1.34943e60	1.34769
$846$	0	0
$847$	−2.77931e59	−0.264440
$848$	0	0
$849$	2.39020e58	0.0216684
$850$	0	0
$851$	−6.07948e59	−0.525182
$852$	0	0
$853$	−1.04133e60	−0.857302	−0.428651	−	0.903470i	$-0.641011\pi$
$853$	−1.04133e60	−0.857302	−0.428651	+	0.903470i	$0.641011\pi$
$854$	0	0
$855$	2.84664e60	2.23371
$856$	0	0
$857$	−2.22118e59	−0.166141	−0.0830705	−	0.996544i	$-0.526473\pi$
$857$	−2.22118e59	−0.166141	−0.0830705	+	0.996544i	$0.526473\pi$
$858$	0	0
$859$	−1.83508e60	−1.30856	−0.654282	−	0.756250i	$-0.727029\pi$
$859$	−1.83508e60	−1.30856	−0.654282	+	0.756250i	$0.727029\pi$
$860$	0	0
$861$	2.92006e59	0.198530
$862$	0	0
$863$	2.51134e60	1.62811	0.814055	−	0.580788i	$-0.197256\pi$
$863$	2.51134e60	1.62811	0.814055	+	0.580788i	$0.197256\pi$
$864$	0	0
$865$	2.43997e60	1.50853
$866$	0	0
$867$	−4.40241e59	−0.259597
$868$	0	0
$869$	−2.97250e59	−0.167193
$870$	0	0
$871$	−2.04515e59	−0.109737
$872$	0	0
$873$	7.90728e59	0.404795
$874$	0	0
$875$	9.12799e59	0.445872
$876$	0	0
$877$	1.15614e60	0.538913	0.269457	−	0.963013i	$-0.413156\pi$
$877$	1.15614e60	0.538913	0.269457	+	0.963013i	$0.413156\pi$
$878$	0	0
$879$	6.60924e59	0.294022
$880$	0	0
$881$	−2.91398e60	−1.23732	−0.618658	−	0.785661i	$-0.712323\pi$
$881$	−2.91398e60	−1.23732	−0.618658	+	0.785661i	$0.712323\pi$
$882$	0	0
$883$	−1.84407e60	−0.747452	−0.373726	−	0.927539i	$-0.621920\pi$
$883$	−1.84407e60	−0.747452	−0.373726	+	0.927539i	$0.621920\pi$
$884$	0	0
$885$	2.01491e60	0.779683
$886$	0	0
$887$	4.08939e59	0.151086	0.0755431	−	0.997143i	$-0.475931\pi$
$887$	4.08939e59	0.151086	0.0755431	+	0.997143i	$0.475931\pi$
$888$	0	0
$889$	−5.91610e59	−0.208713
$890$	0	0
$891$	−1.22610e60	−0.413080
$892$	0	0
$893$	−1.55769e60	−0.501217
$894$	0	0
$895$	5.71472e60	1.75640
$896$	0	0
$897$	−1.30693e60	−0.383714
$898$	0	0
$899$	1.67019e60	0.468480
$900$	0	0
$901$	4.95254e59	0.132730
$902$	0	0
$903$	7.69540e59	0.197075
$904$	0	0
$905$	7.80886e60	1.91114
$906$	0	0
$907$	−1.20769e60	−0.282492	−0.141246	−	0.989975i	$-0.545111\pi$
$907$	−1.20769e60	−0.282492	−0.141246	+	0.989975i	$0.545111\pi$
$908$	0	0
$909$	6.45600e59	0.144346
$910$	0	0
$911$	−5.09758e59	−0.108953	−0.0544763	−	0.998515i	$-0.517349\pi$
$911$	−5.09758e59	−0.108953	−0.0544763	+	0.998515i	$0.517349\pi$
$912$	0	0
$913$	−4.34169e58	−0.00887172
$914$	0	0
$915$	1.26060e60	0.246290
$916$	0	0
$917$	3.25029e60	0.607227
$918$	0	0
$919$	5.01795e60	0.896517	0.448259	−	0.893904i	$-0.352044\pi$
$919$	5.01795e60	0.896517	0.448259	+	0.893904i	$0.352044\pi$
$920$	0	0
$921$	2.29038e60	0.391367
$922$	0	0
$923$	−5.16388e60	−0.843998
$924$	0	0
$925$	5.78328e60	0.904209
$926$	0	0
$927$	−8.90660e60	−1.33222
$928$	0	0
$929$	−3.15656e60	−0.451744	−0.225872	−	0.974157i	$-0.572523\pi$
$929$	−3.15656e60	−0.451744	−0.225872	+	0.974157i	$0.572523\pi$
$930$	0	0
$931$	9.25288e60	1.26709
$932$	0	0
$933$	5.87964e59	0.0770508
$934$	0	0
$935$	2.10486e60	0.263989
$936$	0	0
$937$	−5.64418e60	−0.677547	−0.338773	−	0.940868i	$-0.610012\pi$
$937$	−5.64418e60	−0.677547	−0.338773	+	0.940868i	$0.610012\pi$
$938$	0	0
$939$	1.42513e60	0.163761
$940$	0	0
$941$	−1.57154e61	−1.72877	−0.864385	−	0.502831i	$-0.832292\pi$
$941$	−1.57154e61	−1.72877	−0.864385	+	0.502831i	$0.832292\pi$
$942$	0	0
$943$	−1.77394e61	−1.86831
$944$	0	0
$945$	−3.33916e60	−0.336733
$946$	0	0
$947$	3.54084e60	0.341926	0.170963	−	0.985277i	$-0.445312\pi$
$947$	3.54084e60	0.341926	0.170963	+	0.985277i	$0.445312\pi$
$948$	0	0
$949$	−1.86718e61	−1.72675
$950$	0	0
$951$	−5.05221e60	−0.447488
$952$	0	0
$953$	−3.82394e60	−0.324420	−0.162210	−	0.986756i	$-0.551862\pi$
$953$	−3.82394e60	−0.324420	−0.162210	+	0.986756i	$0.551862\pi$
$954$	0	0
$955$	1.75128e61	1.42327
$956$	0	0
$957$	−5.40570e59	−0.0420879
$958$	0	0
$959$	−2.89439e60	−0.215912
$960$	0	0
$961$	2.70478e61	1.93331
$962$	0	0
$963$	−8.30505e60	−0.568858
$964$	0	0
$965$	1.41641e61	0.929776
$966$	0	0
$967$	−1.08706e61	−0.683930	−0.341965	−	0.939713i	$-0.611092\pi$
$967$	−1.08706e61	−0.683930	−0.341965	+	0.939713i	$0.611092\pi$
$968$	0	0
$969$	2.05519e60	0.123941
$970$	0	0
$971$	−1.86961e61	−1.08083	−0.540416	−	0.841398i	$-0.681733\pi$
$971$	−1.86961e61	−1.08083	−0.540416	+	0.841398i	$0.681733\pi$
$972$	0	0
$973$	7.01055e60	0.388543
$974$	0	0
$975$	1.24326e61	0.660643
$976$	0	0
$977$	−2.93813e61	−1.49704	−0.748518	−	0.663115i	$-0.769234\pi$
$977$	−2.93813e61	−1.49704	−0.748518	+	0.663115i	$0.769234\pi$
$978$	0	0
$979$	−6.32627e59	−0.0309103
$980$	0	0
$981$	1.92288e60	0.0901024
$982$	0	0
$983$	−2.55719e61	−1.14925	−0.574626	−	0.818416i	$-0.694852\pi$
$983$	−2.55719e61	−1.14925	−0.574626	+	0.818416i	$0.694852\pi$
$984$	0	0
$985$	−2.46425e61	−1.06229
$986$	0	0
$987$	8.75070e59	0.0361861
$988$	0	0
$989$	−4.67496e61	−1.85461
$990$	0	0
$991$	−1.53747e61	−0.585188	−0.292594	−	0.956237i	$-0.594518\pi$
$991$	−1.53747e61	−0.585188	−0.292594	+	0.956237i	$0.594518\pi$
$992$	0	0
$993$	1.45614e61	0.531796
$994$	0	0
$995$	4.12385e61	1.44521
$996$	0	0
$997$	4.42141e61	1.48700	0.743499	−	0.668737i	$-0.233165\pi$
$997$	4.42141e61	1.48700	0.743499	+	0.668737i	$0.233165\pi$
$998$	0	0
$999$	−8.87676e60	−0.286525

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	16.42.a.b.1.2		2
4.3	odd	2		2.42.a.b.1.1	✓	2
12.11	even	2		18.42.a.c.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2.42.a.b.1.1	✓	2	4.3	odd	2
16.42.a.b.1.2		2	1.1	even	1	trivial
18.42.a.c.1.1		2	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 16 = 2^{4} \)
Weight:	\( k \)	\(=\)	\( 42 \)
Character orbit:	\([\chi]\)	\(=\)	16.a (trivial)

\(f(q)\)	\(=\)	\(q+1.71810e9 q^{3} +3.51885e14 q^{5} +7.89090e16 q^{7} -3.35211e19 q^{9} +O(q^{10})\)	\(q+1.71810e9 q^{3} +3.51885e14 q^{5} +7.89090e16 q^{7} -3.35211e19 q^{9} -1.20679e21 q^{11} +9.23598e22 q^{13} +6.04574e23 q^{15} -4.95668e24 q^{17} -2.41331e26 q^{19} +1.35574e26 q^{21} -8.23610e27 q^{23} +7.83484e28 q^{25} -1.20257e29 q^{27} +2.60718e29 q^{29} +6.40610e30 q^{31} -2.07339e30 q^{33} +2.77669e31 q^{35} +7.38149e31 q^{37} +1.58683e32 q^{39} +2.15386e33 q^{41} +5.67618e33 q^{43} -1.17956e34 q^{45} +6.45458e33 q^{47} -3.83410e34 q^{49} -8.51608e33 q^{51} -9.99163e34 q^{53} -4.24652e35 q^{55} -4.14631e35 q^{57} +3.33277e36 q^{59} +2.08511e36 q^{61} -2.64512e36 q^{63} +3.25000e37 q^{65} -2.21433e36 q^{67} -1.41505e37 q^{69} -5.59105e37 q^{71} -2.02164e38 q^{73} +1.34610e38 q^{75} -9.52266e37 q^{77} +2.46315e38 q^{79} +1.01600e39 q^{81} +3.59771e37 q^{83} -1.74418e39 q^{85} +4.47940e38 q^{87} +5.24223e38 q^{89} +7.28801e39 q^{91} +1.10063e40 q^{93} -8.49208e40 q^{95} -2.35889e40 q^{97} +4.04530e40 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 8863347528 q^{3} + 97599184325580 q^{5} - 21\!\cdots\!56 q^{7}+ \cdots + 41\!\cdots\!86 q^{9}+O(q^{10}) \) 2 * q - 8863347528 * q^3 + 97599184325580 * q^5 - 217106802373088656 * q^7 + 41972929616872509786 * q^9	\( 2 q - 8863347528 q^{3} + 97599184325580 q^{5} - 21\!\cdots\!56 q^{7}+ \cdots + 12\!\cdots\!88 q^{99}+O(q^{100}) \) 2 * q - 8863347528 * q^3 + 97599184325580 * q^5 - 217106802373088656 * q^7 + 41972929616872509786 * q^9 - 110594481484939565784 * q^11 + 173293106838448696895068 * q^13 + 3295287287252802285241680 * q^15 - 10287966260741677078429404 * q^17 - 206254882139859790701457000 * q^19 + 3267849131609686416057835584 * q^21 - 14925421378885128304361915568 * q^23 + 97534966241249753807444705150 * q^25 - 533156288863055843864697754320 * q^27 - 1375143011941884388614889894020 * q^29 + 3626496922711160469799126663616 * q^31 - 13672735988327258412044854187424 * q^33 + 103039524095905042910481769779360 * q^35 + 120356679589378915210866994599436 * q^37 - 697708668559917206762862430490352 * q^39 + 1270177139906840106861139346878164 * q^41 + 6045433421160616098770468790853352 * q^43 - 30992658299909156185379775337031460 * q^45 + 25416478966330542868882280891861664 * q^47 + 4716677344275473650585885334355954 * q^49 + 47896619495707425086522465903395056 * q^51 - 301376484284740318156836692232048852 * q^53 - 703399094180619772764269188861842960 * q^55 - 785787658029123941344672503009410400 * q^57 + 3282010250697938085995473868454788040 * q^59 - 4485157745840460017677520841591879236 * q^61 - 24992547810032633572829803615671997008 * q^63 + 11919812053150705903267451825832510120 * q^65 - 55055736076082381526441630307313037256 * q^67 + 56632222305135551516324020091548201152 * q^69 + 48857884738531480102553331099997138416 * q^71 - 485616958845162818165813572349849688812 * q^73 - 68411444542067712502035991381122300600 * q^75 - 419718086944271580508987308621981603648 * q^77 + 905766854341990828970937020702446890080 * q^79 + 2631580786121491108655810678136598411122 * q^81 - 1271755206567258248006583231290595969768 * q^83 - 388512546938141834318136915482004571560 * q^85 + 17757719150026706057484126704955838888080 * q^87 + 2692414558304473820098690391828757436980 * q^89 - 16669534332125295311104493248216990683104 * q^91 + 40418607330508619745600934617627243192576 * q^93 - 93840167647611953736551449995129877215600 * q^95 - 31989468315501178009691057062669320691004 * q^97 + 123209315561064920195916977740234432814088 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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