LMFDB - Embedded newform 16.46.a.c.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [16,46,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, 46, names="a")

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

chi := DirichletCharacter("16.1");

S:= CuspForms(chi, 46);

N := Newforms(S);

Level:	$ N $	$=$	$ 16 = 2^{4} $
Weight:	$ k $	$=$	$ 46 $
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:	$205.209161719$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	$\mathbb{Q}[x]/(x^{3} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 148878150x + 389915850150 $ x^3 - x^2 - 148878150*x + 389915850150
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{24}\cdot 3^{6}\cdot 5 $
Twist minimal:	no (minimal twist has level 1)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-13344.8$ 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)

$f(q)$	$=$	$q+1.72036e10 q^{3} +2.46761e15 q^{5} -1.29065e19 q^{7} -2.65835e21 q^{9} +O(q^{10})$	$q+1.72036e10 q^{3} +2.46761e15 q^{5} -1.29065e19 q^{7} -2.65835e21 q^{9} -7.78755e22 q^{11} -1.84585e25 q^{13} +4.24517e25 q^{15} -2.82858e27 q^{17} -8.52388e28 q^{19} -2.22037e29 q^{21} +2.09625e30 q^{23} -2.23326e31 q^{25} -9.65579e31 q^{27} -1.33372e33 q^{29} +9.42299e32 q^{31} -1.33974e33 q^{33} -3.18481e34 q^{35} -8.78606e34 q^{37} -3.17552e35 q^{39} +2.03901e36 q^{41} +6.02099e36 q^{43} -6.55977e36 q^{45} +4.75078e37 q^{47} +5.95698e37 q^{49} -4.86617e37 q^{51} -7.36820e38 q^{53} -1.92166e38 q^{55} -1.46641e39 q^{57} +9.51309e39 q^{59} -1.95631e39 q^{61} +3.43099e40 q^{63} -4.55483e40 q^{65} -2.10749e41 q^{67} +3.60631e40 q^{69} +5.43331e41 q^{71} +8.99506e41 q^{73} -3.84201e41 q^{75} +1.00510e42 q^{77} +7.61415e41 q^{79} +6.19245e42 q^{81} -6.62221e42 q^{83} -6.97983e42 q^{85} -2.29447e43 q^{87} -6.63932e43 q^{89} +2.38234e44 q^{91} +1.62109e43 q^{93} -2.10336e44 q^{95} +1.95163e44 q^{97} +2.07020e44 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 5359866876 q^{3} - 912448458460350 q^{5} + 76\!\cdots\!08 q^{7}+ \cdots + 11\!\cdots\!19 q^{9}+O(q^{10}) $ 3 * q - 5359866876 * q^3 - 912448458460350 * q^5 + 7619710926638056008 * q^7 + 1158701600689591796919 * q^9	$ 3 q - 5359866876 q^{3} - 912448458460350 q^{5} + 76\!\cdots\!08 q^{7}+ \cdots + 52\!\cdots\!12 q^{99}+O(q^{100}) $ 3 * q - 5359866876 * q^3 - 912448458460350 * q^5 + 7619710926638056008 * q^7 + 1158701600689591796919 * q^9 + 292602117773160656797644 * q^11 - 24359353574857845965202054 * q^13 + 1135189552611303064861248600 * q^15 + 1167342337360426664689201782 * q^17 - 139928385765722236845372310380 * q^19 + 2269674947269963115109120096 * q^21 + 10673923189590823281275419815384 * q^23 + 43966129876780298643382415638125 * q^25 - 286662461859179128178856521495640 * q^27 - 670860386246579691273176792023830 * q^29 + 4362750588615761992088215111558944 * q^31 + 32665703367813314965816678845383952 * q^33 - 15776204147695493575553971787662800 * q^35 - 389811033762613666259144433874550238 * q^37 + 409236468450733544586776272530045432 * q^39 + 2715552056267791317916826925062933406 * q^41 - 1244478438653295151469201253707220756 * q^43 - 36806221258052298400850661208655540550 * q^45 + 5368746178058366301880916464880208048 * q^47 + 78160841959900415140960388420190127371 * q^49 + 373116254852470180408674865132312910664 * q^51 - 1852254617049027724838330153197946097774 * q^53 + 3433604438201040460840734686798586808200 * q^55 + 103199557648847905550120265832946125680 * q^57 + 3587009869064273690797416085301514120060 * q^59 - 51762384161673790814764907347052617635894 * q^61 + 63198441434102160493715367231351517178664 * q^63 + 37939089079008549181090608539522846859900 * q^65 - 14148137855196379836825399018252188940732 * q^67 - 91367025965579022204230939913277492454112 * q^69 + 125388248998914140052658001286881482385544 * q^71 + 545662435246745842274304841909717562787966 * q^73 - 4696791604551172366648127047375667988322500 * q^75 + 6645319565466313145724427222526529624900384 * q^77 - 906179897072215837642402032445458679683120 * q^79 - 10433460971272354761639583481407161767228277 * q^81 - 1486742880234689875145309838364765216168236 * q^83 + 38251451691354483777201487197950382555147300 * q^85 + 19149930875978892871878601194088020234221880 * q^87 - 157452314112877728949303270670086721758716690 * q^89 + 209450940191589584493053736413801309667114864 * q^91 - 198656173601061645852049647068499329296299648 * q^93 - 11835463674693195649110972576862856536029000 * q^95 + 371229894708856356492851438771649466869172902 * q^97 + 52499767038004223859624250804879775541363612 * 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
$3$	1.72036e10	0.316512	0.158256	−	0.987398i	$-0.449413\pi$
$3$	1.72036e10	0.316512	0.158256	+	0.987398i	$0.449413\pi$
$4$	0	0
$5$	2.46761e15	0.462862	0.231431	−	0.972851i	$-0.425659\pi$
$5$	2.46761e15	0.462862	0.231431	+	0.972851i	$0.425659\pi$
$6$	0	0
$7$	−1.29065e19	−1.24767	−0.623837	−	0.781554i	$-0.714427\pi$
$7$	−1.29065e19	−1.24767	−0.623837	+	0.781554i	$0.714427\pi$
$8$	0	0
$9$	−2.65835e21	−0.899820
$10$	0	0
$11$	−7.78755e22	−0.288447	−0.144223	−	0.989545i	$-0.546068\pi$
$11$	−7.78755e22	−0.288447	−0.144223	+	0.989545i	$0.546068\pi$
$12$	0	0
$13$	−1.84585e25	−1.59393	−0.796967	−	0.604022i	$-0.793564\pi$
$13$	−1.84585e25	−1.59393	−0.796967	+	0.604022i	$0.793564\pi$
$14$	0	0
$15$	4.24517e25	0.146502
$16$	0	0
$17$	−2.82858e27	−0.584073	−0.292037	−	0.956407i	$-0.594333\pi$
$17$	−2.82858e27	−0.584073	−0.292037	+	0.956407i	$0.594333\pi$
$18$	0	0
$19$	−8.52388e28	−1.44106	−0.720529	−	0.693425i	$-0.756101\pi$
$19$	−8.52388e28	−1.44106	−0.720529	+	0.693425i	$0.756101\pi$
$20$	0	0
$21$	−2.22037e29	−0.394904
$22$	0	0
$23$	2.09625e30	0.481468	0.240734	−	0.970591i	$-0.422612\pi$
$23$	2.09625e30	0.481468	0.240734	+	0.970591i	$0.422612\pi$
$24$	0	0
$25$	−2.23326e31	−0.785759
$26$	0	0
$27$	−9.65579e31	−0.601316
$28$	0	0
$29$	−1.33372e33	−1.66383	−0.831915	−	0.554903i	$-0.812755\pi$
$29$	−1.33372e33	−1.66383	−0.831915	+	0.554903i	$0.812755\pi$
$30$	0	0
$31$	9.42299e32	0.262150	0.131075	−	0.991372i	$-0.458157\pi$
$31$	9.42299e32	0.262150	0.131075	+	0.991372i	$0.458157\pi$
$32$	0	0
$33$	−1.33974e33	−0.0912969
$34$	0	0
$35$	−3.18481e34	−0.577501
$36$	0	0
$37$	−8.78606e34	−0.456305	−0.228153	−	0.973625i	$-0.573269\pi$
$37$	−8.78606e34	−0.456305	−0.228153	+	0.973625i	$0.573269\pi$
$38$	0	0
$39$	−3.17552e35	−0.504500
$40$	0	0
$41$	2.03901e36	1.05144	0.525718	−	0.850659i	$-0.323797\pi$
$41$	2.03901e36	1.05144	0.525718	+	0.850659i	$0.323797\pi$
$42$	0	0
$43$	6.02099e36	1.06323	0.531615	−	0.846986i	$-0.321585\pi$
$43$	6.02099e36	1.06323	0.531615	+	0.846986i	$0.321585\pi$
$44$	0	0
$45$	−6.55977e36	−0.416492
$46$	0	0
$47$	4.75078e37	1.13387	0.566935	−	0.823763i	$-0.308129\pi$
$47$	4.75078e37	1.13387	0.566935	+	0.823763i	$0.308129\pi$
$48$	0	0
$49$	5.95698e37	0.556691
$50$	0	0
$51$	−4.86617e37	−0.184866
$52$	0	0
$53$	−7.36820e38	−1.17802	−0.589012	−	0.808124i	$-0.700483\pi$
$53$	−7.36820e38	−1.17802	−0.589012	+	0.808124i	$0.700483\pi$
$54$	0	0
$55$	−1.92166e38	−0.133511
$56$	0	0
$57$	−1.46641e39	−0.456112
$58$	0	0
$59$	9.51309e39	1.36192	0.680960	−	0.732321i	$-0.261563\pi$
$59$	9.51309e39	1.36192	0.680960	+	0.732321i	$0.261563\pi$
$60$	0	0
$61$	−1.95631e39	−0.132287	−0.0661433	−	0.997810i	$-0.521069\pi$
$61$	−1.95631e39	−0.132287	−0.0661433	+	0.997810i	$0.521069\pi$
$62$	0	0
$63$	3.43099e40	1.12268
$64$	0	0
$65$	−4.55483e40	−0.737772
$66$	0	0
$67$	−2.10749e41	−1.72617	−0.863083	−	0.505062i	$-0.831470\pi$
$67$	−2.10749e41	−1.72617	−0.863083	+	0.505062i	$0.831470\pi$
$68$	0	0
$69$	3.60631e40	0.152391
$70$	0	0
$71$	5.43331e41	1.20712	0.603561	−	0.797317i	$-0.293748\pi$
$71$	5.43331e41	1.20712	0.603561	+	0.797317i	$0.293748\pi$
$72$	0	0
$73$	8.99506e41	1.06965	0.534823	−	0.844964i	$-0.320378\pi$
$73$	8.99506e41	1.06965	0.534823	+	0.844964i	$0.320378\pi$
$74$	0	0
$75$	−3.84201e41	−0.248702
$76$	0	0
$77$	1.00510e42	0.359888
$78$	0	0
$79$	7.61415e41	0.153113	0.0765565	−	0.997065i	$-0.475607\pi$
$79$	7.61415e41	0.153113	0.0765565	+	0.997065i	$0.475607\pi$
$80$	0	0
$81$	6.19245e42	0.709496
$82$	0	0
$83$	−6.62221e42	−0.438274	−0.219137	−	0.975694i	$-0.570324\pi$
$83$	−6.62221e42	−0.438274	−0.219137	+	0.975694i	$0.570324\pi$
$84$	0	0
$85$	−6.97983e42	−0.270345
$86$	0	0
$87$	−2.29447e43	−0.526623
$88$	0	0
$89$	−6.63932e43	−0.913794	−0.456897	−	0.889520i	$-0.651039\pi$
$89$	−6.63932e43	−0.913794	−0.456897	+	0.889520i	$0.651039\pi$
$90$	0	0
$91$	2.38234e44	1.98871
$92$	0	0
$93$	1.62109e43	0.0829738
$94$	0	0
$95$	−2.10336e44	−0.667011
$96$	0	0
$97$	1.95163e44	0.387288	0.193644	−	0.981072i	$-0.437969\pi$
$97$	1.95163e44	0.387288	0.193644	+	0.981072i	$0.437969\pi$
$98$	0	0
$99$	2.07020e44	0.259550
$100$	0	0
$101$	−3.07178e44	−0.245561	−0.122780	−	0.992434i	$-0.539181\pi$
$101$	−3.07178e44	−0.245561	−0.122780	+	0.992434i	$0.539181\pi$
$102$	0	0
$103$	−1.90315e45	−0.978666	−0.489333	−	0.872097i	$-0.662760\pi$
$103$	−1.90315e45	−0.978666	−0.489333	+	0.872097i	$0.662760\pi$
$104$	0	0
$105$	−5.47902e44	−0.182786
$106$	0	0
$107$	−1.11782e45	−0.243915	−0.121958	−	0.992535i	$-0.538917\pi$
$107$	−1.11782e45	−0.243915	−0.121958	+	0.992535i	$0.538917\pi$
$108$	0	0
$109$	3.21871e45	0.463005	0.231502	−	0.972834i	$-0.425636\pi$
$109$	3.21871e45	0.463005	0.231502	+	0.972834i	$0.425636\pi$
$110$	0	0
$111$	−1.51152e45	−0.144426
$112$	0	0
$113$	−1.15806e46	−0.740397	−0.370198	−	0.928953i	$-0.620710\pi$
$113$	−1.15806e46	−0.740397	−0.370198	+	0.928953i	$0.620710\pi$
$114$	0	0
$115$	5.17274e45	0.222853
$116$	0	0
$117$	4.90691e46	1.43425
$118$	0	0
$119$	3.65069e46	0.728733
$120$	0	0
$121$	−6.68259e46	−0.916798
$122$	0	0
$123$	3.50783e46	0.332792
$124$	0	0
$125$	−1.25242e47	−0.826560
$126$	0	0
$127$	−6.85398e46	−0.316489	−0.158245	−	0.987400i	$-0.550583\pi$
$127$	−6.85398e46	−0.316489	−0.158245	+	0.987400i	$0.550583\pi$
$128$	0	0
$129$	1.03583e47	0.336525
$130$	0	0
$131$	3.50099e46	0.0804613	0.0402306	−	0.999190i	$-0.487191\pi$
$131$	3.50099e46	0.0804613	0.0402306	+	0.999190i	$0.487191\pi$
$132$	0	0
$133$	1.10013e48	1.79797
$134$	0	0
$135$	−2.38267e47	−0.278326
$136$	0	0
$137$	−1.53504e48	−1.28798	−0.643989	−	0.765035i	$-0.722722\pi$
$137$	−1.53504e48	−1.28798	−0.643989	+	0.765035i	$0.722722\pi$
$138$	0	0
$139$	−1.45287e48	−0.879818	−0.439909	−	0.898042i	$-0.644989\pi$
$139$	−1.45287e48	−0.879818	−0.439909	+	0.898042i	$0.644989\pi$
$140$	0	0
$141$	8.17304e47	0.358884
$142$	0	0
$143$	1.43746e48	0.459765
$144$	0	0
$145$	−3.29110e48	−0.770123
$146$	0	0
$147$	1.02481e48	0.176200
$148$	0	0
$149$	6.21806e47	0.0788794	0.0394397	−	0.999222i	$-0.487443\pi$
$149$	6.21806e47	0.0788794	0.0394397	+	0.999222i	$0.487443\pi$
$150$	0	0
$151$	5.84550e48	0.549339	0.274670	−	0.961539i	$-0.411432\pi$
$151$	5.84550e48	0.549339	0.274670	+	0.961539i	$0.411432\pi$
$152$	0	0
$153$	7.51935e48	0.525561
$154$	0	0
$155$	2.32523e48	0.121339
$156$	0	0
$157$	2.07022e48	0.0809607	0.0404804	−	0.999180i	$-0.487111\pi$
$157$	2.07022e48	0.0809607	0.0404804	+	0.999180i	$0.487111\pi$
$158$	0	0
$159$	−1.26759e49	−0.372859
$160$	0	0
$161$	−2.70552e49	−0.600715
$162$	0	0
$163$	4.16253e49	0.700062	0.350031	−	0.936738i	$-0.386171\pi$
$163$	4.16253e49	0.700062	0.350031	+	0.936738i	$0.386171\pi$
$164$	0	0
$165$	−3.30595e48	−0.0422579
$166$	0	0
$167$	−1.93069e50	−1.88188	−0.940941	−	0.338570i	$-0.890057\pi$
$167$	−1.93069e50	−1.88188	−0.940941	+	0.338570i	$0.890057\pi$
$168$	0	0
$169$	2.06609e50	1.54063
$170$	0	0
$171$	2.26595e50	1.29669
$172$	0	0
$173$	−8.78028e48	−0.0386786	−0.0193393	−	0.999813i	$-0.506156\pi$
$173$	−8.78028e48	−0.0386786	−0.0193393	+	0.999813i	$0.506156\pi$
$174$	0	0
$175$	2.88235e50	0.980371
$176$	0	0
$177$	1.63659e50	0.431064
$178$	0	0
$179$	−3.03604e50	−0.621032	−0.310516	−	0.950568i	$-0.600502\pi$
$179$	−3.03604e50	−0.621032	−0.310516	+	0.950568i	$0.600502\pi$
$180$	0	0
$181$	−1.05625e51	−1.68268	−0.841338	−	0.540509i	$-0.818232\pi$
$181$	−1.05625e51	−1.68268	−0.841338	+	0.540509i	$0.818232\pi$
$182$	0	0
$183$	−3.36555e49	−0.0418703
$184$	0	0
$185$	−2.16806e50	−0.211206
$186$	0	0
$187$	2.20277e50	0.168474
$188$	0	0
$189$	1.24622e51	0.750247
$190$	0	0
$191$	1.95146e51	0.927060	0.463530	−	0.886081i	$-0.346583\pi$
$191$	1.95146e51	0.927060	0.463530	+	0.886081i	$0.346583\pi$
$192$	0	0
$193$	2.53211e51	0.951574	0.475787	−	0.879561i	$-0.342163\pi$
$193$	2.53211e51	0.951574	0.475787	+	0.879561i	$0.342163\pi$
$194$	0	0
$195$	−7.83594e50	−0.233514
$196$	0	0
$197$	5.70705e51	1.35183	0.675914	−	0.736981i	$-0.263749\pi$
$197$	5.70705e51	1.35183	0.675914	+	0.736981i	$0.263749\pi$
$198$	0	0
$199$	−2.85140e51	−0.538102	−0.269051	−	0.963126i	$-0.586710\pi$
$199$	−2.85140e51	−0.538102	−0.269051	+	0.963126i	$0.586710\pi$
$200$	0	0
$201$	−3.62564e51	−0.546353
$202$	0	0
$203$	1.72136e52	2.07592
$204$	0	0
$205$	5.03149e51	0.486670
$206$	0	0
$207$	−5.57257e51	−0.433235
$208$	0	0
$209$	6.63802e51	0.415668
$210$	0	0
$211$	1.00404e52	0.507455	0.253728	−	0.967276i	$-0.418343\pi$
$211$	1.00404e52	0.507455	0.253728	+	0.967276i	$0.418343\pi$
$212$	0	0
$213$	9.34723e51	0.382069
$214$	0	0
$215$	1.48574e52	0.492129
$216$	0	0
$217$	−1.21617e52	−0.327078
$218$	0	0
$219$	1.54747e52	0.338556
$220$	0	0
$221$	5.22112e52	0.930975
$222$	0	0
$223$	1.08955e53	1.58630	0.793151	−	0.609025i	$-0.208439\pi$
$223$	1.08955e53	1.58630	0.793151	+	0.609025i	$0.208439\pi$
$224$	0	0
$225$	5.93679e52	0.707041
$226$	0	0
$227$	−8.59683e51	−0.0838990	−0.0419495	−	0.999120i	$-0.513357\pi$
$227$	−8.59683e51	−0.0838990	−0.0419495	+	0.999120i	$0.513357\pi$
$228$	0	0
$229$	8.54768e52	0.684779	0.342389	−	0.939558i	$-0.388764\pi$
$229$	8.54768e52	0.684779	0.342389	+	0.939558i	$0.388764\pi$
$230$	0	0
$231$	1.72913e52	0.113909
$232$	0	0
$233$	1.94501e53	1.05539	0.527695	−	0.849434i	$-0.323057\pi$
$233$	1.94501e53	1.05539	0.527695	+	0.849434i	$0.323057\pi$
$234$	0	0
$235$	1.17231e53	0.524825
$236$	0	0
$237$	1.30991e52	0.0484621
$238$	0	0
$239$	4.67858e52	0.143272	0.0716360	−	0.997431i	$-0.477178\pi$
$239$	4.67858e52	0.143272	0.0716360	+	0.997431i	$0.477178\pi$
$240$	0	0
$241$	5.41561e53	1.37488	0.687438	−	0.726243i	$-0.258735\pi$
$241$	5.41561e53	1.37488	0.687438	+	0.726243i	$0.258735\pi$
$242$	0	0
$243$	3.91795e53	0.825881
$244$	0	0
$245$	1.46995e53	0.257671
$246$	0	0
$247$	1.57338e54	2.29695
$248$	0	0
$249$	−1.13926e53	−0.138719
$250$	0	0
$251$	−2.39905e53	−0.243995	−0.121997	−	0.992530i	$-0.538930\pi$
$251$	−2.39905e53	−0.243995	−0.121997	+	0.992530i	$0.538930\pi$
$252$	0	0
$253$	−1.63247e53	−0.138878
$254$	0	0
$255$	−1.20078e53	−0.0855676
$256$	0	0
$257$	1.10513e54	0.660569	0.330285	−	0.943881i	$-0.392855\pi$
$257$	1.10513e54	0.660569	0.330285	+	0.943881i	$0.392855\pi$
$258$	0	0
$259$	1.13397e54	0.569321
$260$	0	0
$261$	3.54549e54	1.49715
$262$	0	0
$263$	−9.95471e53	−0.354017	−0.177009	−	0.984209i	$-0.556642\pi$
$263$	−9.95471e53	−0.354017	−0.177009	+	0.984209i	$0.556642\pi$
$264$	0	0
$265$	−1.81818e54	−0.545263
$266$	0	0
$267$	−1.14220e54	−0.289227
$268$	0	0
$269$	−1.90365e54	−0.407532	−0.203766	−	0.979020i	$-0.565318\pi$
$269$	−1.90365e54	−0.407532	−0.203766	+	0.979020i	$0.565318\pi$
$270$	0	0
$271$	2.11803e54	0.383815	0.191908	−	0.981413i	$-0.438533\pi$
$271$	2.11803e54	0.383815	0.191908	+	0.981413i	$0.438533\pi$
$272$	0	0
$273$	4.09847e54	0.629452
$274$	0	0
$275$	1.73916e54	0.226650
$276$	0	0
$277$	7.86883e54	0.871195	0.435598	−	0.900141i	$-0.356537\pi$
$277$	7.86883e54	0.871195	0.435598	+	0.900141i	$0.356537\pi$
$278$	0	0
$279$	−2.50496e54	−0.235888
$280$	0	0
$281$	−2.25442e55	−1.80777	−0.903883	−	0.427781i	$-0.859296\pi$
$281$	−2.25442e55	−1.80777	−0.903883	+	0.427781i	$0.859296\pi$
$282$	0	0
$283$	5.22384e54	0.357104	0.178552	−	0.983930i	$-0.442859\pi$
$283$	5.22384e54	0.357104	0.178552	+	0.983930i	$0.442859\pi$
$284$	0	0
$285$	−3.61854e54	−0.211117
$286$	0	0
$287$	−2.63164e55	−1.31185
$288$	0	0
$289$	−1.54523e55	−0.658858
$290$	0	0
$291$	3.35751e54	0.122582
$292$	0	0
$293$	1.63631e55	0.512088	0.256044	−	0.966665i	$-0.417581\pi$
$293$	1.63631e55	0.512088	0.256044	+	0.966665i	$0.417581\pi$
$294$	0	0
$295$	2.34746e55	0.630381
$296$	0	0
$297$	7.51950e54	0.173448
$298$	0	0
$299$	−3.86936e55	−0.767428
$300$	0	0
$301$	−7.77096e55	−1.32657
$302$	0	0
$303$	−5.28456e54	−0.0777230
$304$	0	0
$305$	−4.82741e54	−0.0612305
$306$	0	0
$307$	−1.67269e56	−1.83148	−0.915741	−	0.401768i	$-0.868396\pi$
$307$	−1.67269e56	−1.83148	−0.915741	+	0.401768i	$0.868396\pi$
$308$	0	0
$309$	−3.27409e55	−0.309760
$310$	0	0
$311$	−2.38430e56	−1.95097	−0.975487	−	0.220058i	$-0.929375\pi$
$311$	−2.38430e56	−1.95097	−0.975487	+	0.220058i	$0.929375\pi$
$312$	0	0
$313$	−7.01155e55	−0.496668	−0.248334	−	0.968674i	$-0.579883\pi$
$313$	−7.01155e55	−0.496668	−0.248334	+	0.968674i	$0.579883\pi$
$314$	0	0
$315$	8.46634e55	0.519647
$316$	0	0
$317$	−2.68335e56	−1.42838	−0.714191	−	0.699951i	$-0.753205\pi$
$317$	−2.68335e56	−1.42838	−0.714191	+	0.699951i	$0.753205\pi$
$318$	0	0
$319$	1.03864e56	0.479926
$320$	0	0
$321$	−1.92306e55	−0.0772021
$322$	0	0
$323$	2.41105e56	0.841683
$324$	0	0
$325$	4.12226e56	1.25245
$326$	0	0
$327$	5.53733e55	0.146547
$328$	0	0
$329$	−6.13157e56	−1.41470
$330$	0	0
$331$	−1.90982e56	−0.384470	−0.192235	−	0.981349i	$-0.561574\pi$
$331$	−1.90982e56	−0.384470	−0.192235	+	0.981349i	$0.561574\pi$
$332$	0	0
$333$	2.33564e56	0.410593
$334$	0	0
$335$	−5.20047e56	−0.798977
$336$	0	0
$337$	2.57253e56	0.345691	0.172845	−	0.984949i	$-0.444704\pi$
$337$	2.57253e56	0.345691	0.172845	+	0.984949i	$0.444704\pi$
$338$	0	0
$339$	−1.99227e56	−0.234345
$340$	0	0
$341$	−7.33820e55	−0.0756164
$342$	0	0
$343$	6.12245e56	0.553105
$344$	0	0
$345$	8.89896e55	0.0705358
$346$	0	0
$347$	−1.10908e57	−0.771878	−0.385939	−	0.922524i	$-0.626123\pi$
$347$	−1.10908e57	−0.771878	−0.385939	+	0.922524i	$0.626123\pi$
$348$	0	0
$349$	1.91424e57	1.17064	0.585321	−	0.810802i	$-0.300968\pi$
$349$	1.91424e57	1.17064	0.585321	+	0.810802i	$0.300968\pi$
$350$	0	0
$351$	1.78231e57	0.958459
$352$	0	0
$353$	−1.00890e57	−0.477437	−0.238719	−	0.971089i	$-0.576727\pi$
$353$	−1.00890e57	−0.477437	−0.238719	+	0.971089i	$0.576727\pi$
$354$	0	0
$355$	1.34073e57	0.558731
$356$	0	0
$357$	6.28050e56	0.230653
$358$	0	0
$359$	5.19166e57	1.68144	0.840720	−	0.541470i	$-0.182132\pi$
$359$	5.19166e57	1.68144	0.840720	+	0.541470i	$0.182132\pi$
$360$	0	0
$361$	3.76691e57	1.07665
$362$	0	0
$363$	−1.14964e57	−0.290178
$364$	0	0
$365$	2.21963e57	0.495098
$366$	0	0
$367$	−5.07031e57	−1.00011	−0.500056	−	0.865993i	$-0.666687\pi$
$367$	−5.07031e57	−1.00011	−0.500056	+	0.865993i	$0.666687\pi$
$368$	0	0
$369$	−5.42041e57	−0.946103
$370$	0	0
$371$	9.50974e57	1.46979
$372$	0	0
$373$	−5.35043e57	−0.732723	−0.366362	−	0.930473i	$-0.619397\pi$
$373$	−5.35043e57	−0.732723	−0.366362	+	0.930473i	$0.619397\pi$
$374$	0	0
$375$	−2.15461e57	−0.261616
$376$	0	0
$377$	2.46184e58	2.65204
$378$	0	0
$379$	1.65597e58	1.58369	0.791846	−	0.610720i	$-0.209120\pi$
$379$	1.65597e58	1.58369	0.791846	+	0.610720i	$0.209120\pi$
$380$	0	0
$381$	−1.17913e57	−0.100173
$382$	0	0
$383$	−4.08813e56	−0.0308711	−0.0154356	−	0.999881i	$-0.504913\pi$
$383$	−4.08813e56	−0.0308711	−0.0154356	+	0.999881i	$0.504913\pi$
$384$	0	0
$385$	2.48019e57	0.166578
$386$	0	0
$387$	−1.60059e58	−0.956716
$388$	0	0
$389$	8.62061e57	0.458851	0.229426	−	0.973326i	$-0.426315\pi$
$389$	8.62061e57	0.458851	0.229426	+	0.973326i	$0.426315\pi$
$390$	0	0
$391$	−5.92941e57	−0.281213
$392$	0	0
$393$	6.02296e56	0.0254670
$394$	0	0
$395$	1.87888e57	0.0708702
$396$	0	0
$397$	−9.58101e57	−0.322571	−0.161285	−	0.986908i	$-0.551564\pi$
$397$	−9.58101e57	−0.322571	−0.161285	+	0.986908i	$0.551564\pi$
$398$	0	0
$399$	1.89262e58	0.569080
$400$	0	0
$401$	−3.87566e58	−1.04135	−0.520676	−	0.853755i	$-0.674320\pi$
$401$	−3.87566e58	−1.04135	−0.520676	+	0.853755i	$0.674320\pi$
$402$	0	0
$403$	−1.73934e58	−0.417850
$404$	0	0
$405$	1.52806e58	0.328399
$406$	0	0
$407$	6.84219e57	0.131620
$408$	0	0
$409$	−2.12182e58	−0.365539	−0.182769	−	0.983156i	$-0.558506\pi$
$409$	−2.12182e58	−0.365539	−0.182769	+	0.983156i	$0.558506\pi$
$410$	0	0
$411$	−2.64083e58	−0.407661
$412$	0	0
$413$	−1.22780e59	−1.69923
$414$	0	0
$415$	−1.63410e58	−0.202860
$416$	0	0
$417$	−2.49945e58	−0.278473
$418$	0	0
$419$	−7.28603e58	−0.728913	−0.364456	−	0.931220i	$-0.618745\pi$
$419$	−7.28603e58	−0.728913	−0.364456	+	0.931220i	$0.618745\pi$
$420$	0	0
$421$	6.69387e58	0.601631	0.300815	−	0.953682i	$-0.402741\pi$
$421$	6.69387e58	0.601631	0.300815	+	0.953682i	$0.402741\pi$
$422$	0	0
$423$	−1.26292e59	−1.02028
$424$	0	0
$425$	6.31695e58	0.458941
$426$	0	0
$427$	2.52490e58	0.165051
$428$	0	0
$429$	2.47295e58	0.145521
$430$	0	0
$431$	−1.33907e59	−0.709684	−0.354842	−	0.934926i	$-0.615465\pi$
$431$	−1.33907e59	−0.709684	−0.354842	+	0.934926i	$0.615465\pi$
$432$	0	0
$433$	−1.66678e59	−0.795976	−0.397988	−	0.917391i	$-0.630292\pi$
$433$	−1.66678e59	−0.795976	−0.397988	+	0.917391i	$0.630292\pi$
$434$	0	0
$435$	−5.66186e58	−0.243754
$436$	0	0
$437$	−1.78682e59	−0.693823
$438$	0	0
$439$	1.80307e59	0.631774	0.315887	−	0.948797i	$-0.397698\pi$
$439$	1.80307e59	0.631774	0.315887	+	0.948797i	$0.397698\pi$
$440$	0	0
$441$	−1.58357e59	−0.500922
$442$	0	0
$443$	−1.21940e59	−0.348388	−0.174194	−	0.984711i	$-0.555732\pi$
$443$	−1.21940e59	−0.348388	−0.174194	+	0.984711i	$0.555732\pi$
$444$	0	0
$445$	−1.63833e59	−0.422961
$446$	0	0
$447$	1.06973e58	0.0249663
$448$	0	0
$449$	4.77895e59	1.00877	0.504383	−	0.863480i	$-0.331720\pi$
$449$	4.77895e59	1.00877	0.504383	+	0.863480i	$0.331720\pi$
$450$	0	0
$451$	−1.58789e59	−0.303283
$452$	0	0
$453$	1.00564e59	0.173873
$454$	0	0
$455$	5.87868e59	0.920499
$456$	0	0
$457$	2.48102e59	0.351978	0.175989	−	0.984392i	$-0.443688\pi$
$457$	2.48102e59	0.351978	0.175989	+	0.984392i	$0.443688\pi$
$458$	0	0
$459$	2.73121e59	0.351213
$460$	0	0
$461$	−1.59001e60	−1.85408	−0.927040	−	0.374962i	$-0.877656\pi$
$461$	−1.59001e60	−1.85408	−0.927040	+	0.374962i	$0.877656\pi$
$462$	0	0
$463$	−9.67043e59	−1.02300	−0.511498	−	0.859285i	$-0.670909\pi$
$463$	−9.67043e59	−1.02300	−0.511498	+	0.859285i	$0.670909\pi$
$464$	0	0
$465$	4.00022e58	0.0384054
$466$	0	0
$467$	1.81440e59	0.158162	0.0790812	−	0.996868i	$-0.474801\pi$
$467$	1.81440e59	0.158162	0.0790812	+	0.996868i	$0.474801\pi$
$468$	0	0
$469$	2.72002e60	2.15369
$470$	0	0
$471$	3.56152e58	0.0256251
$472$	0	0
$473$	−4.68887e59	−0.306685
$474$	0	0
$475$	1.90360e60	1.13232
$476$	0	0
$477$	1.95872e60	1.06001
$478$	0	0
$479$	1.15748e60	0.570117	0.285059	−	0.958510i	$-0.407987\pi$
$479$	1.15748e60	0.570117	0.285059	+	0.958510i	$0.407987\pi$
$480$	0	0
$481$	1.62177e60	0.727321
$482$	0	0
$483$	−4.65446e59	−0.190134
$484$	0	0
$485$	4.81587e59	0.179261
$486$	0	0
$487$	1.82685e60	0.619871	0.309935	−	0.950758i	$-0.399693\pi$
$487$	1.82685e60	0.619871	0.309935	+	0.950758i	$0.399693\pi$
$488$	0	0
$489$	7.16104e59	0.221578
$490$	0	0
$491$	2.65564e60	0.749609	0.374805	−	0.927104i	$-0.377710\pi$
$491$	2.65564e60	0.749609	0.374805	+	0.927104i	$0.377710\pi$
$492$	0	0
$493$	3.77252e60	0.971799
$494$	0	0
$495$	5.10846e59	0.120136
$496$	0	0
$497$	−7.01248e60	−1.50610
$498$	0	0
$499$	−2.18380e60	−0.428500	−0.214250	−	0.976779i	$-0.568731\pi$
$499$	−2.18380e60	−0.428500	−0.214250	+	0.976779i	$0.568731\pi$
$500$	0	0
$501$	−3.32148e60	−0.595639
$502$	0	0
$503$	4.41130e60	0.723247	0.361624	−	0.932324i	$-0.382223\pi$
$503$	4.41130e60	0.723247	0.361624	+	0.932324i	$0.382223\pi$
$504$	0	0
$505$	−7.57995e59	−0.113661
$506$	0	0
$507$	3.55441e60	0.487628
$508$	0	0
$509$	3.40004e60	0.426908	0.213454	−	0.976953i	$-0.431529\pi$
$509$	3.40004e60	0.426908	0.213454	+	0.976953i	$0.431529\pi$
$510$	0	0
$511$	−1.16094e61	−1.33457
$512$	0	0
$513$	8.23048e60	0.866532
$514$	0	0
$515$	−4.69622e60	−0.452987
$516$	0	0
$517$	−3.69969e60	−0.327061
$518$	0	0
$519$	−1.51052e59	−0.0122423
$520$	0	0
$521$	7.11610e60	0.528924	0.264462	−	0.964396i	$-0.414806\pi$
$521$	7.11610e60	0.528924	0.264462	+	0.964396i	$0.414806\pi$
$522$	0	0
$523$	6.62508e60	0.451755	0.225878	−	0.974156i	$-0.427475\pi$
$523$	6.62508e60	0.451755	0.225878	+	0.974156i	$0.427475\pi$
$524$	0	0
$525$	4.95867e60	0.310300
$526$	0	0
$527$	−2.66536e60	−0.153115
$528$	0	0
$529$	−1.45620e61	−0.768189
$530$	0	0
$531$	−2.52891e61	−1.22548
$532$	0	0
$533$	−3.76371e61	−1.67592
$534$	0	0
$535$	−2.75836e60	−0.112899
$536$	0	0
$537$	−5.22307e60	−0.196564
$538$	0	0
$539$	−4.63903e60	−0.160576
$540$	0	0
$541$	−3.34138e61	−1.06411	−0.532056	−	0.846709i	$-0.678581\pi$
$541$	−3.34138e61	−1.06411	−0.532056	+	0.846709i	$0.678581\pi$
$542$	0	0
$543$	−1.81713e61	−0.532588
$544$	0	0
$545$	7.94252e60	0.214307
$546$	0	0
$547$	−3.86064e61	−0.959277	−0.479638	−	0.877466i	$-0.659232\pi$
$547$	−3.86064e61	−0.959277	−0.479638	+	0.877466i	$0.659232\pi$
$548$	0	0
$549$	5.20055e60	0.119034
$550$	0	0
$551$	1.13684e62	2.39767
$552$	0	0
$553$	−9.82717e60	−0.191035
$554$	0	0
$555$	−3.72984e60	−0.0668494
$556$	0	0
$557$	5.55735e61	0.918598	0.459299	−	0.888282i	$-0.348101\pi$
$557$	5.55735e61	0.918598	0.459299	+	0.888282i	$0.348101\pi$
$558$	0	0
$559$	−1.11138e62	−1.69472
$560$	0	0
$561$	3.78955e60	0.0533241
$562$	0	0
$563$	1.22198e61	0.158718	0.0793589	−	0.996846i	$-0.474713\pi$
$563$	1.22198e61	0.158718	0.0793589	+	0.996846i	$0.474713\pi$
$564$	0	0
$565$	−2.85763e61	−0.342701
$566$	0	0
$567$	−7.99227e61	−0.885220
$568$	0	0
$569$	−1.79144e62	−1.83306	−0.916529	−	0.399967i	$-0.869022\pi$
$569$	−1.79144e62	−1.83306	−0.916529	+	0.399967i	$0.869022\pi$
$570$	0	0
$571$	4.92064e61	0.465274	0.232637	−	0.972564i	$-0.425265\pi$
$571$	4.92064e61	0.465274	0.232637	+	0.972564i	$0.425265\pi$
$572$	0	0
$573$	3.35720e61	0.293426
$574$	0	0
$575$	−4.68148e61	−0.378318
$576$	0	0
$577$	1.61247e62	1.20513	0.602566	−	0.798069i	$-0.294145\pi$
$577$	1.61247e62	1.20513	0.602566	+	0.798069i	$0.294145\pi$
$578$	0	0
$579$	4.35613e61	0.301185
$580$	0	0
$581$	8.54693e61	0.546823
$582$	0	0
$583$	5.73802e61	0.339797
$584$	0	0
$585$	1.21083e62	0.663862
$586$	0	0
$587$	1.02006e62	0.517925	0.258962	−	0.965887i	$-0.416619\pi$
$587$	1.02006e62	0.517925	0.258962	+	0.965887i	$0.416619\pi$
$588$	0	0
$589$	−8.03204e61	−0.377774
$590$	0	0
$591$	9.81816e61	0.427870
$592$	0	0
$593$	9.42772e61	0.380782	0.190391	−	0.981708i	$-0.439024\pi$
$593$	9.42772e61	0.380782	0.190391	+	0.981708i	$0.439024\pi$
$594$	0	0
$595$	9.00849e61	0.337303
$596$	0	0
$597$	−4.90543e61	−0.170316
$598$	0	0
$599$	2.30170e62	0.741217	0.370609	−	0.928789i	$-0.379149\pi$
$599$	2.30170e62	0.741217	0.370609	+	0.928789i	$0.379149\pi$
$600$	0	0
$601$	1.32397e62	0.395552	0.197776	−	0.980247i	$-0.436628\pi$
$601$	1.32397e62	0.395552	0.197776	+	0.980247i	$0.436628\pi$
$602$	0	0
$603$	5.60245e62	1.55324
$604$	0	0
$605$	−1.64900e62	−0.424351
$606$	0	0
$607$	1.86733e62	0.446145	0.223073	−	0.974802i	$-0.428391\pi$
$607$	1.86733e62	0.446145	0.223073	+	0.974802i	$0.428391\pi$
$608$	0	0
$609$	2.96135e62	0.657053
$610$	0	0
$611$	−8.76921e62	−1.80731
$612$	0	0
$613$	7.96288e62	1.52479	0.762394	−	0.647113i	$-0.224024\pi$
$613$	7.96288e62	1.52479	0.762394	+	0.647113i	$0.224024\pi$
$614$	0	0
$615$	8.65596e61	0.154037
$616$	0	0
$617$	−1.66014e62	−0.274617	−0.137309	−	0.990528i	$-0.543845\pi$
$617$	−1.66014e62	−0.274617	−0.137309	+	0.990528i	$0.543845\pi$
$618$	0	0
$619$	−4.17933e62	−0.642786	−0.321393	−	0.946946i	$-0.604151\pi$
$619$	−4.17933e62	−0.642786	−0.321393	+	0.946946i	$0.604151\pi$
$620$	0	0
$621$	−2.02410e62	−0.289515
$622$	0	0
$623$	8.56901e62	1.14012
$624$	0	0
$625$	3.25683e62	0.403176
$626$	0	0
$627$	1.14198e62	0.131564
$628$	0	0
$629$	2.48521e62	0.266516
$630$	0	0
$631$	−4.78834e62	−0.478106	−0.239053	−	0.971007i	$-0.576837\pi$
$631$	−4.78834e62	−0.478106	−0.239053	+	0.971007i	$0.576837\pi$
$632$	0	0
$633$	1.72732e62	0.160616
$634$	0	0
$635$	−1.69129e62	−0.146491
$636$	0	0
$637$	−1.09957e63	−0.887330
$638$	0	0
$639$	−1.44436e63	−1.08619
$640$	0	0
$641$	−2.18288e63	−1.53011	−0.765056	−	0.643964i	$-0.777289\pi$
$641$	−2.18288e63	−1.53011	−0.765056	+	0.643964i	$0.777289\pi$
$642$	0	0
$643$	1.90128e63	1.24251	0.621255	−	0.783609i	$-0.286623\pi$
$643$	1.90128e63	1.24251	0.621255	+	0.783609i	$0.286623\pi$
$644$	0	0
$645$	2.55601e62	0.155765
$646$	0	0
$647$	−5.73318e62	−0.325874	−0.162937	−	0.986637i	$-0.552097\pi$
$647$	−5.73318e62	−0.325874	−0.162937	+	0.986637i	$0.552097\pi$
$648$	0	0
$649$	−7.40837e62	−0.392841
$650$	0	0
$651$	−2.09225e62	−0.103524
$652$	0	0
$653$	1.09427e63	0.505335	0.252667	−	0.967553i	$-0.418692\pi$
$653$	1.09427e63	0.505335	0.252667	+	0.967553i	$0.418692\pi$
$654$	0	0
$655$	8.63908e61	0.0372425
$656$	0	0
$657$	−2.39120e63	−0.962488
$658$	0	0
$659$	−3.42421e63	−1.28718	−0.643589	−	0.765372i	$-0.722555\pi$
$659$	−3.42421e63	−1.28718	−0.643589	+	0.765372i	$0.722555\pi$
$660$	0	0
$661$	2.99634e61	0.0105210	0.00526051	−	0.999986i	$-0.498326\pi$
$661$	2.99634e61	0.0105210	0.00526051	+	0.999986i	$0.498326\pi$
$662$	0	0
$663$	8.98220e62	0.294665
$664$	0	0
$665$	2.71470e63	0.832212
$666$	0	0
$667$	−2.79581e63	−0.801081
$668$	0	0
$669$	1.87441e63	0.502084
$670$	0	0
$671$	1.52349e62	0.0381576
$672$	0	0
$673$	−5.41615e63	−1.26868	−0.634340	−	0.773054i	$-0.718728\pi$
$673$	−5.41615e63	−1.26868	−0.634340	+	0.773054i	$0.718728\pi$
$674$	0	0
$675$	2.15639e63	0.472490
$676$	0	0
$677$	−4.55896e63	−0.934588	−0.467294	−	0.884102i	$-0.654771\pi$
$677$	−4.55896e63	−0.934588	−0.467294	+	0.884102i	$0.654771\pi$
$678$	0	0
$679$	−2.51887e63	−0.483210
$680$	0	0
$681$	−1.47896e62	−0.0265551
$682$	0	0
$683$	−1.04266e64	−1.75257	−0.876287	−	0.481790i	$-0.839987\pi$
$683$	−1.04266e64	−1.75257	−0.876287	+	0.481790i	$0.839987\pi$
$684$	0	0
$685$	−3.78789e63	−0.596156
$686$	0	0
$687$	1.47051e63	0.216741
$688$	0	0
$689$	1.36006e64	1.87769
$690$	0	0
$691$	9.60653e63	1.24254	0.621270	−	0.783597i	$-0.286617\pi$
$691$	9.60653e63	1.24254	0.621270	+	0.783597i	$0.286617\pi$
$692$	0	0
$693$	−2.67190e63	−0.323834
$694$	0	0
$695$	−3.58511e63	−0.407234
$696$	0	0
$697$	−5.76750e63	−0.614116
$698$	0	0
$699$	3.34611e63	0.334044
$700$	0	0
$701$	−4.79975e63	−0.449327	−0.224664	−	0.974436i	$-0.572128\pi$
$701$	−4.79975e63	−0.449327	−0.224664	+	0.974436i	$0.572128\pi$
$702$	0	0
$703$	7.48914e63	0.657562
$704$	0	0
$705$	2.01679e63	0.166114
$706$	0	0
$707$	3.96458e63	0.306380
$708$	0	0
$709$	9.62702e63	0.698155	0.349077	−	0.937094i	$-0.386495\pi$
$709$	9.62702e63	0.698155	0.349077	+	0.937094i	$0.386495\pi$
$710$	0	0
$711$	−2.02411e63	−0.137774
$712$	0	0
$713$	1.97530e63	0.126217
$714$	0	0
$715$	3.54710e63	0.212808
$716$	0	0
$717$	8.04883e62	0.0453474
$718$	0	0
$719$	2.96105e64	1.56692	0.783459	−	0.621443i	$-0.213453\pi$
$719$	2.96105e64	1.56692	0.783459	+	0.621443i	$0.213453\pi$
$720$	0	0
$721$	2.45629e64	1.22106
$722$	0	0
$723$	9.31678e63	0.435165
$724$	0	0
$725$	2.97854e64	1.30737
$726$	0	0
$727$	−2.01758e64	−0.832350	−0.416175	−	0.909284i	$-0.636630\pi$
$727$	−2.01758e64	−0.832350	−0.416175	+	0.909284i	$0.636630\pi$
$728$	0	0
$729$	−1.15542e64	−0.448095
$730$	0	0
$731$	−1.70308e64	−0.621005
$732$	0	0
$733$	−2.30546e64	−0.790530	−0.395265	−	0.918567i	$-0.629347\pi$
$733$	−2.30546e64	−0.790530	−0.395265	+	0.918567i	$0.629347\pi$
$734$	0	0
$735$	2.52884e63	0.0815561
$736$	0	0
$737$	1.64122e64	0.497907
$738$	0	0
$739$	−4.27723e64	−1.22085	−0.610425	−	0.792074i	$-0.709001\pi$
$739$	−4.27723e64	−1.22085	−0.610425	+	0.792074i	$0.709001\pi$
$740$	0	0
$741$	2.70677e64	0.727013
$742$	0	0
$743$	6.07984e64	1.53689	0.768447	−	0.639913i	$-0.221030\pi$
$743$	6.07984e64	1.53689	0.768447	+	0.639913i	$0.221030\pi$
$744$	0	0
$745$	1.53437e63	0.0365103
$746$	0	0
$747$	1.76041e64	0.394368
$748$	0	0
$749$	1.44272e64	0.304327
$750$	0	0
$751$	8.42750e64	1.67417	0.837086	−	0.547071i	$-0.184257\pi$
$751$	8.42750e64	1.67417	0.837086	+	0.547071i	$0.184257\pi$
$752$	0	0
$753$	−4.12723e63	−0.0772274
$754$	0	0
$755$	1.44244e64	0.254268
$756$	0	0
$757$	−1.62790e64	−0.270377	−0.135189	−	0.990820i	$-0.543164\pi$
$757$	−1.62790e64	−0.270377	−0.135189	+	0.990820i	$0.543164\pi$
$758$	0	0
$759$	−2.80843e63	−0.0439565
$760$	0	0
$761$	−1.06343e65	−1.56875	−0.784373	−	0.620289i	$-0.787015\pi$
$761$	−1.06343e65	−1.56875	−0.784373	+	0.620289i	$0.787015\pi$
$762$	0	0
$763$	−4.15421e64	−0.577679
$764$	0	0
$765$	1.85548e64	0.243262
$766$	0	0
$767$	−1.75597e65	−2.17081
$768$	0	0
$769$	3.91583e64	0.456542	0.228271	−	0.973598i	$-0.426693\pi$
$769$	3.91583e64	0.456542	0.228271	+	0.973598i	$0.426693\pi$
$770$	0	0
$771$	1.90122e64	0.209078
$772$	0	0
$773$	−1.00996e65	−1.04777	−0.523885	−	0.851789i	$-0.675518\pi$
$773$	−1.00996e65	−1.04777	−0.523885	+	0.851789i	$0.675518\pi$
$774$	0	0
$775$	−2.10440e64	−0.205987
$776$	0	0
$777$	1.95083e64	0.180197
$778$	0	0
$779$	−1.73803e65	−1.51518
$780$	0	0
$781$	−4.23122e64	−0.348190
$782$	0	0
$783$	1.28781e65	1.00049
$784$	0	0
$785$	5.10850e63	0.0374736
$786$	0	0
$787$	−1.02621e65	−0.710896	−0.355448	−	0.934696i	$-0.615672\pi$
$787$	−1.02621e65	−0.710896	−0.355448	+	0.934696i	$0.615672\pi$
$788$	0	0
$789$	−1.71257e64	−0.112051
$790$	0	0
$791$	1.49464e65	0.923774
$792$	0	0
$793$	3.61105e64	0.210856
$794$	0	0
$795$	−3.12793e64	−0.172582
$796$	0	0
$797$	−7.78531e64	−0.405942	−0.202971	−	0.979185i	$-0.565060\pi$
$797$	−7.78531e64	−0.405942	−0.202971	+	0.979185i	$0.565060\pi$
$798$	0	0
$799$	−1.34379e65	−0.662263
$800$	0	0
$801$	1.76496e65	0.822250
$802$	0	0
$803$	−7.00495e64	−0.308536
$804$	0	0
$805$	−6.67617e64	−0.278048
$806$	0	0
$807$	−3.27496e64	−0.128989
$808$	0	0
$809$	−3.71145e65	−1.38262	−0.691308	−	0.722560i	$-0.742965\pi$
$809$	−3.71145e65	−1.38262	−0.691308	+	0.722560i	$0.742965\pi$
$810$	0	0
$811$	1.97077e65	0.694490	0.347245	−	0.937774i	$-0.387117\pi$
$811$	1.97077e65	0.694490	0.347245	+	0.937774i	$0.387117\pi$
$812$	0	0
$813$	3.64376e64	0.121482
$814$	0	0
$815$	1.02715e65	0.324032
$816$	0	0
$817$	−5.13222e65	−1.53218
$818$	0	0
$819$	−6.33308e65	−1.78948
$820$	0	0
$821$	4.18847e65	1.12030	0.560150	−	0.828391i	$-0.310743\pi$
$821$	4.18847e65	1.12030	0.560150	+	0.828391i	$0.310743\pi$
$822$	0	0
$823$	5.89925e64	0.149383	0.0746914	−	0.997207i	$-0.476203\pi$
$823$	5.89925e64	0.149383	0.0746914	+	0.997207i	$0.476203\pi$
$824$	0	0
$825$	2.99198e64	0.0717374
$826$	0	0
$827$	4.27471e65	0.970583	0.485292	−	0.874352i	$-0.338713\pi$
$827$	4.27471e65	0.970583	0.485292	+	0.874352i	$0.338713\pi$
$828$	0	0
$829$	6.40554e64	0.137746	0.0688730	−	0.997625i	$-0.478060\pi$
$829$	6.40554e64	0.137746	0.0688730	+	0.997625i	$0.478060\pi$
$830$	0	0
$831$	1.35372e65	0.275744
$832$	0	0
$833$	−1.68498e65	−0.325149
$834$	0	0
$835$	−4.76420e65	−0.871052
$836$	0	0
$837$	−9.09863e64	−0.157635
$838$	0	0
$839$	9.87581e65	1.62154	0.810772	−	0.585363i	$-0.199048\pi$
$839$	9.87581e65	1.62154	0.810772	+	0.585363i	$0.199048\pi$
$840$	0	0
$841$	1.13625e66	1.76833
$842$	0	0
$843$	−3.87840e65	−0.572180
$844$	0	0
$845$	5.09830e65	0.713098
$846$	0	0
$847$	8.62486e65	1.14387
$848$	0	0
$849$	8.98687e64	0.113028
$850$	0	0
$851$	−1.84178e65	−0.219696
$852$	0	0
$853$	5.47101e65	0.619034	0.309517	−	0.950894i	$-0.399833\pi$
$853$	5.47101e65	0.619034	0.309517	+	0.950894i	$0.399833\pi$
$854$	0	0
$855$	5.59147e65	0.600190
$856$	0	0
$857$	3.91129e65	0.398338	0.199169	−	0.979965i	$-0.436176\pi$
$857$	3.91129e65	0.398338	0.199169	+	0.979965i	$0.436176\pi$
$858$	0	0
$859$	−1.59577e66	−1.54214	−0.771072	−	0.636748i	$-0.780279\pi$
$859$	−1.59577e66	−1.54214	−0.771072	+	0.636748i	$0.780279\pi$
$860$	0	0
$861$	−4.52737e65	−0.415217
$862$	0	0
$863$	1.43745e65	0.125127	0.0625633	−	0.998041i	$-0.480072\pi$
$863$	1.43745e65	0.125127	0.0625633	+	0.998041i	$0.480072\pi$
$864$	0	0
$865$	−2.16663e64	−0.0179029
$866$	0	0
$867$	−2.65835e65	−0.208537
$868$	0	0
$869$	−5.92956e64	−0.0441649
$870$	0	0
$871$	3.89011e66	2.75140
$872$	0	0
$873$	−5.18813e65	−0.348490
$874$	0	0
$875$	1.61643e66	1.03128
$876$	0	0
$877$	−9.07110e65	−0.549755	−0.274878	−	0.961479i	$-0.588637\pi$
$877$	−9.07110e65	−0.549755	−0.274878	+	0.961479i	$0.588637\pi$
$878$	0	0
$879$	2.81504e65	0.162082
$880$	0	0
$881$	2.58911e66	1.41642	0.708212	−	0.706000i	$-0.249502\pi$
$881$	2.58911e66	1.41642	0.708212	+	0.706000i	$0.249502\pi$
$882$	0	0
$883$	−1.38977e66	−0.722480	−0.361240	−	0.932473i	$-0.617647\pi$
$883$	−1.38977e66	−0.722480	−0.361240	+	0.932473i	$0.617647\pi$
$884$	0	0
$885$	4.03847e65	0.199523
$886$	0	0
$887$	−1.62550e66	−0.763319	−0.381660	−	0.924303i	$-0.624647\pi$
$887$	−1.62550e66	−0.763319	−0.381660	+	0.924303i	$0.624647\pi$
$888$	0	0
$889$	8.84606e65	0.394875
$890$	0	0
$891$	−4.82241e65	−0.204652
$892$	0	0
$893$	−4.04951e66	−1.63397
$894$	0	0
$895$	−7.49176e65	−0.287452
$896$	0	0
$897$	−6.65669e65	−0.242901
$898$	0	0
$899$	−1.25676e66	−0.436173
$900$	0	0
$901$	2.08415e66	0.688052
$902$	0	0
$903$	−1.33688e66	−0.419874
$904$	0	0
$905$	−2.60642e66	−0.778847
$906$	0	0
$907$	−6.17539e66	−1.75590	−0.877951	−	0.478751i	$-0.841090\pi$
$907$	−6.17539e66	−1.75590	−0.877951	+	0.478751i	$0.841090\pi$
$908$	0	0
$909$	8.16586e65	0.220960
$910$	0	0
$911$	2.64162e66	0.680311	0.340155	−	0.940369i	$-0.389520\pi$
$911$	2.64162e66	0.680311	0.340155	+	0.940369i	$0.389520\pi$
$912$	0	0
$913$	5.15708e65	0.126419
$914$	0	0
$915$	−8.30487e64	−0.0193802
$916$	0	0
$917$	−4.51854e65	−0.100389
$918$	0	0
$919$	8.03936e66	1.70068	0.850340	−	0.526234i	$-0.176396\pi$
$919$	8.03936e66	1.70068	0.850340	+	0.526234i	$0.176396\pi$
$920$	0	0
$921$	−2.87763e66	−0.579687
$922$	0	0
$923$	−1.00291e67	−1.92407
$924$	0	0
$925$	1.96216e66	0.358546
$926$	0	0
$927$	5.05923e66	0.880623
$928$	0	0
$929$	−6.85607e66	−1.13690	−0.568449	−	0.822718i	$-0.692457\pi$
$929$	−6.85607e66	−1.13690	−0.568449	+	0.822718i	$0.692457\pi$
$930$	0	0
$931$	−5.07766e66	−0.802225
$932$	0	0
$933$	−4.10184e66	−0.617507
$934$	0	0
$935$	5.43558e65	0.0779802
$936$	0	0
$937$	−5.58369e66	−0.763450	−0.381725	−	0.924276i	$-0.624670\pi$
$937$	−5.58369e66	−0.763450	−0.381725	+	0.924276i	$0.624670\pi$
$938$	0	0
$939$	−1.20624e66	−0.157202
$940$	0	0
$941$	8.59076e66	1.06725	0.533624	−	0.845722i	$-0.320830\pi$
$941$	8.59076e66	1.06725	0.533624	+	0.845722i	$0.320830\pi$
$942$	0	0
$943$	4.27428e66	0.506233
$944$	0	0
$945$	3.07519e66	0.347261
$946$	0	0
$947$	−1.38497e67	−1.49131	−0.745654	−	0.666334i	$-0.767863\pi$
$947$	−1.38497e67	−1.49131	−0.745654	+	0.666334i	$0.767863\pi$
$948$	0	0
$949$	−1.66035e67	−1.70495
$950$	0	0
$951$	−4.61633e66	−0.452101
$952$	0	0
$953$	1.29710e67	1.21166	0.605831	−	0.795593i	$-0.292841\pi$
$953$	1.29710e67	1.21166	0.605831	+	0.795593i	$0.292841\pi$
$954$	0	0
$955$	4.81543e66	0.429101
$956$	0	0
$957$	1.78683e66	0.151903
$958$	0	0
$959$	1.98120e67	1.60698
$960$	0	0
$961$	−1.20325e67	−0.931277
$962$	0	0
$963$	2.97157e66	0.219480
$964$	0	0
$965$	6.24825e66	0.440447
$966$	0	0
$967$	−2.84025e66	−0.191100	−0.0955502	−	0.995425i	$-0.530461\pi$
$967$	−2.84025e66	−0.191100	−0.0955502	+	0.995425i	$0.530461\pi$
$968$	0	0
$969$	4.14786e66	0.266403
$970$	0	0
$971$	−2.77039e67	−1.69866	−0.849332	−	0.527858i	$-0.822995\pi$
$971$	−2.77039e67	−1.69866	−0.849332	+	0.527858i	$0.822995\pi$
$972$	0	0
$973$	1.87514e67	1.09773
$974$	0	0
$975$	7.09176e66	0.396415
$976$	0	0
$977$	1.30344e67	0.695766	0.347883	−	0.937538i	$-0.386901\pi$
$977$	1.30344e67	0.695766	0.347883	+	0.937538i	$0.386901\pi$
$978$	0	0
$979$	5.17040e66	0.263581
$980$	0	0
$981$	−8.55645e66	−0.416621
$982$	0	0
$983$	2.69857e67	1.25510	0.627551	−	0.778576i	$-0.284057\pi$
$983$	2.69857e67	1.25510	0.627551	+	0.778576i	$0.284057\pi$
$984$	0	0
$985$	1.40828e67	0.625710
$986$	0	0
$987$	−1.05485e67	−0.447770
$988$	0	0
$989$	1.26215e67	0.511911
$990$	0	0
$991$	3.22029e67	1.24807	0.624034	−	0.781397i	$-0.285493\pi$
$991$	3.22029e67	1.24807	0.624034	+	0.781397i	$0.285493\pi$
$992$	0	0
$993$	−3.28557e66	−0.121689
$994$	0	0
$995$	−7.03615e66	−0.249067
$996$	0	0
$997$	−1.99799e67	−0.676009	−0.338005	−	0.941144i	$-0.609752\pi$
$997$	−1.99799e67	−0.676009	−0.338005	+	0.941144i	$0.609752\pi$
$998$	0	0
$999$	8.48364e66	0.274384

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	16.46.a.c.1.2		3
4.3	odd	2		1.46.a.a.1.1	✓	3
12.11	even	2		9.46.a.b.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1.46.a.a.1.1	✓	3	4.3	odd	2
9.46.a.b.1.3		3	12.11	even	2
16.46.a.c.1.2		3	1.1	even	1	trivial

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:	\( k \)	\(=\)	\( 46 \)
Character orbit:	\([\chi]\)	\(=\)	16.a (trivial)

\(f(q)\)	\(=\)	\(q+1.72036e10 q^{3} +2.46761e15 q^{5} -1.29065e19 q^{7} -2.65835e21 q^{9} +O(q^{10})\)	\(q+1.72036e10 q^{3} +2.46761e15 q^{5} -1.29065e19 q^{7} -2.65835e21 q^{9} -7.78755e22 q^{11} -1.84585e25 q^{13} +4.24517e25 q^{15} -2.82858e27 q^{17} -8.52388e28 q^{19} -2.22037e29 q^{21} +2.09625e30 q^{23} -2.23326e31 q^{25} -9.65579e31 q^{27} -1.33372e33 q^{29} +9.42299e32 q^{31} -1.33974e33 q^{33} -3.18481e34 q^{35} -8.78606e34 q^{37} -3.17552e35 q^{39} +2.03901e36 q^{41} +6.02099e36 q^{43} -6.55977e36 q^{45} +4.75078e37 q^{47} +5.95698e37 q^{49} -4.86617e37 q^{51} -7.36820e38 q^{53} -1.92166e38 q^{55} -1.46641e39 q^{57} +9.51309e39 q^{59} -1.95631e39 q^{61} +3.43099e40 q^{63} -4.55483e40 q^{65} -2.10749e41 q^{67} +3.60631e40 q^{69} +5.43331e41 q^{71} +8.99506e41 q^{73} -3.84201e41 q^{75} +1.00510e42 q^{77} +7.61415e41 q^{79} +6.19245e42 q^{81} -6.62221e42 q^{83} -6.97983e42 q^{85} -2.29447e43 q^{87} -6.63932e43 q^{89} +2.38234e44 q^{91} +1.62109e43 q^{93} -2.10336e44 q^{95} +1.95163e44 q^{97} +2.07020e44 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 5359866876 q^{3} - 912448458460350 q^{5} + 76\!\cdots\!08 q^{7}+ \cdots + 11\!\cdots\!19 q^{9}+O(q^{10}) \) 3 * q - 5359866876 * q^3 - 912448458460350 * q^5 + 7619710926638056008 * q^7 + 1158701600689591796919 * q^9	\( 3 q - 5359866876 q^{3} - 912448458460350 q^{5} + 76\!\cdots\!08 q^{7}+ \cdots + 52\!\cdots\!12 q^{99}+O(q^{100}) \) 3 * q - 5359866876 * q^3 - 912448458460350 * q^5 + 7619710926638056008 * q^7 + 1158701600689591796919 * q^9 + 292602117773160656797644 * q^11 - 24359353574857845965202054 * q^13 + 1135189552611303064861248600 * q^15 + 1167342337360426664689201782 * q^17 - 139928385765722236845372310380 * q^19 + 2269674947269963115109120096 * q^21 + 10673923189590823281275419815384 * q^23 + 43966129876780298643382415638125 * q^25 - 286662461859179128178856521495640 * q^27 - 670860386246579691273176792023830 * q^29 + 4362750588615761992088215111558944 * q^31 + 32665703367813314965816678845383952 * q^33 - 15776204147695493575553971787662800 * q^35 - 389811033762613666259144433874550238 * q^37 + 409236468450733544586776272530045432 * q^39 + 2715552056267791317916826925062933406 * q^41 - 1244478438653295151469201253707220756 * q^43 - 36806221258052298400850661208655540550 * q^45 + 5368746178058366301880916464880208048 * q^47 + 78160841959900415140960388420190127371 * q^49 + 373116254852470180408674865132312910664 * q^51 - 1852254617049027724838330153197946097774 * q^53 + 3433604438201040460840734686798586808200 * q^55 + 103199557648847905550120265832946125680 * q^57 + 3587009869064273690797416085301514120060 * q^59 - 51762384161673790814764907347052617635894 * q^61 + 63198441434102160493715367231351517178664 * q^63 + 37939089079008549181090608539522846859900 * q^65 - 14148137855196379836825399018252188940732 * q^67 - 91367025965579022204230939913277492454112 * q^69 + 125388248998914140052658001286881482385544 * q^71 + 545662435246745842274304841909717562787966 * q^73 - 4696791604551172366648127047375667988322500 * q^75 + 6645319565466313145724427222526529624900384 * q^77 - 906179897072215837642402032445458679683120 * q^79 - 10433460971272354761639583481407161767228277 * q^81 - 1486742880234689875145309838364765216168236 * q^83 + 38251451691354483777201487197950382555147300 * q^85 + 19149930875978892871878601194088020234221880 * q^87 - 157452314112877728949303270670086721758716690 * q^89 + 209450940191589584493053736413801309667114864 * q^91 - 198656173601061645852049647068499329296299648 * q^93 - 11835463674693195649110972576862856536029000 * q^95 + 371229894708856356492851438771649466869172902 * q^97 + 52499767038004223859624250804879775541363612 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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