LMFDB - Embedded newform 4056.2.a.bi.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4056,2,Mod(1,4056)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4056, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("4056.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4056 = 2^{3} \cdot 3 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4056.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:	$32.3873230598$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.27700337.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 19x^{4} + 17x^{3} + 103x^{2} - 71x - 127 $ x^6 - x^5 - 19x^4 + 17x^3 + 103x^2 - 71x - 127
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$3.16419$ of defining polynomial
Character	$\chi$	$=$	4056.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -2.71914 q^{5} +0.735542 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -2.71914 q^{5} +0.735542 q^{7} +1.00000 q^{9} +1.14374 q^{11} -2.71914 q^{15} +5.96612 q^{17} -3.07604 q^{19} +0.735542 q^{21} +5.31230 q^{23} +2.39374 q^{25} +1.00000 q^{27} -8.12923 q^{29} -1.67023 q^{31} +1.14374 q^{33} -2.00004 q^{35} -1.90945 q^{37} +8.69257 q^{41} +11.7282 q^{43} -2.71914 q^{45} -7.53981 q^{47} -6.45898 q^{49} +5.96612 q^{51} +1.86294 q^{53} -3.10999 q^{55} -3.07604 q^{57} -4.28323 q^{59} -2.91045 q^{61} +0.735542 q^{63} -0.596077 q^{67} +5.31230 q^{69} +2.30363 q^{71} +13.1830 q^{73} +2.39374 q^{75} +0.841267 q^{77} +12.6413 q^{79} +1.00000 q^{81} +9.10421 q^{83} -16.2227 q^{85} -8.12923 q^{87} +8.77241 q^{89} -1.67023 q^{93} +8.36419 q^{95} -6.86954 q^{97} +1.14374 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 6 q^{3} + q^{5} - 5 q^{7} + 6 q^{9}+O(q^{10}) $ 6 * q + 6 * q^3 + q^5 - 5 * q^7 + 6 * q^9	$ 6 q + 6 q^{3} + q^{5} - 5 q^{7} + 6 q^{9} - 6 q^{11} + q^{15} + 9 q^{17} + 7 q^{19} - 5 q^{21} + 12 q^{23} + 9 q^{25} + 6 q^{27} + 7 q^{29} - 11 q^{31} - 6 q^{33} + 6 q^{35} + 6 q^{37} + 13 q^{41} + 15 q^{43} + q^{45} - 9 q^{47} + 13 q^{49} + 9 q^{51} + 22 q^{53} + 3 q^{55} + 7 q^{57} - 7 q^{59} + 25 q^{61} - 5 q^{63} + 5 q^{67} + 12 q^{69} + 8 q^{71} + 15 q^{73} + 9 q^{75} + 45 q^{77} + 14 q^{79} + 6 q^{81} - 13 q^{83} - 35 q^{85} + 7 q^{87} + 33 q^{89} - 11 q^{93} + 47 q^{95} + 50 q^{97} - 6 q^{99}+O(q^{100}) $ 6 * q + 6 * q^3 + q^5 - 5 * q^7 + 6 * q^9 - 6 * q^11 + q^15 + 9 * q^17 + 7 * q^19 - 5 * q^21 + 12 * q^23 + 9 * q^25 + 6 * q^27 + 7 * q^29 - 11 * q^31 - 6 * q^33 + 6 * q^35 + 6 * q^37 + 13 * q^41 + 15 * q^43 + q^45 - 9 * q^47 + 13 * q^49 + 9 * q^51 + 22 * q^53 + 3 * q^55 + 7 * q^57 - 7 * q^59 + 25 * q^61 - 5 * q^63 + 5 * q^67 + 12 * q^69 + 8 * q^71 + 15 * q^73 + 9 * q^75 + 45 * q^77 + 14 * q^79 + 6 * q^81 - 13 * q^83 - 35 * q^85 + 7 * q^87 + 33 * q^89 - 11 * q^93 + 47 * q^95 + 50 * q^97 - 6 * 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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	−2.71914	−1.21604	−0.608019	−	0.793922i	$-0.708036\pi$
$5$	−2.71914	−1.21604	−0.608019	+	0.793922i	$0.708036\pi$
$6$	0	0
$7$	0.735542	0.278009	0.139004	−	0.990292i	$-0.455610\pi$
$7$	0.735542	0.278009	0.139004	+	0.990292i	$0.455610\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	1.14374	0.344850	0.172425	−	0.985023i	$-0.444840\pi$
$11$	1.14374	0.344850	0.172425	+	0.985023i	$0.444840\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	−2.71914	−0.702080
$16$	0	0
$17$	5.96612	1.44700	0.723499	−	0.690326i	$-0.242533\pi$
$17$	5.96612	1.44700	0.723499	+	0.690326i	$0.242533\pi$
$18$	0	0
$19$	−3.07604	−0.705692	−0.352846	−	0.935681i	$-0.614786\pi$
$19$	−3.07604	−0.705692	−0.352846	+	0.935681i	$0.614786\pi$
$20$	0	0
$21$	0.735542	0.160508
$22$	0	0
$23$	5.31230	1.10769	0.553845	−	0.832620i	$-0.313160\pi$
$23$	5.31230	1.10769	0.553845	+	0.832620i	$0.313160\pi$
$24$	0	0
$25$	2.39374	0.478749
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−8.12923	−1.50956	−0.754780	−	0.655978i	$-0.772256\pi$
$29$	−8.12923	−1.50956	−0.754780	+	0.655978i	$0.772256\pi$
$30$	0	0
$31$	−1.67023	−0.299982	−0.149991	−	0.988687i	$-0.547924\pi$
$31$	−1.67023	−0.299982	−0.149991	+	0.988687i	$0.547924\pi$
$32$	0	0
$33$	1.14374	0.199099
$34$	0	0
$35$	−2.00004	−0.338069
$36$	0	0
$37$	−1.90945	−0.313912	−0.156956	−	0.987606i	$-0.550168\pi$
$37$	−1.90945	−0.313912	−0.156956	+	0.987606i	$0.550168\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	8.69257	1.35755	0.678775	−	0.734346i	$-0.262511\pi$
$41$	8.69257	1.35755	0.678775	+	0.734346i	$0.262511\pi$
$42$	0	0
$43$	11.7282	1.78853	0.894266	−	0.447536i	$-0.147698\pi$
$43$	11.7282	1.78853	0.894266	+	0.447536i	$0.147698\pi$
$44$	0	0
$45$	−2.71914	−0.405346
$46$	0	0
$47$	−7.53981	−1.09979	−0.549897	−	0.835233i	$-0.685333\pi$
$47$	−7.53981	−1.09979	−0.549897	+	0.835233i	$0.685333\pi$
$48$	0	0
$49$	−6.45898	−0.922711
$50$	0	0
$51$	5.96612	0.835424
$52$	0	0
$53$	1.86294	0.255894	0.127947	−	0.991781i	$-0.459161\pi$
$53$	1.86294	0.255894	0.127947	+	0.991781i	$0.459161\pi$
$54$	0	0
$55$	−3.10999	−0.419350
$56$	0	0
$57$	−3.07604	−0.407431
$58$	0	0
$59$	−4.28323	−0.557629	−0.278814	−	0.960345i	$-0.589941\pi$
$59$	−4.28323	−0.557629	−0.278814	+	0.960345i	$0.589941\pi$
$60$	0	0
$61$	−2.91045	−0.372645	−0.186323	−	0.982489i	$-0.559657\pi$
$61$	−2.91045	−0.372645	−0.186323	+	0.982489i	$0.559657\pi$
$62$	0	0
$63$	0.735542	0.0926696
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−0.596077	−0.0728224	−0.0364112	−	0.999337i	$-0.511593\pi$
$67$	−0.596077	−0.0728224	−0.0364112	+	0.999337i	$0.511593\pi$
$68$	0	0
$69$	5.31230	0.639525
$70$	0	0
$71$	2.30363	0.273391	0.136695	−	0.990613i	$-0.456352\pi$
$71$	2.30363	0.273391	0.136695	+	0.990613i	$0.456352\pi$
$72$	0	0
$73$	13.1830	1.54295	0.771477	−	0.636258i	$-0.219518\pi$
$73$	13.1830	1.54295	0.771477	+	0.636258i	$0.219518\pi$
$74$	0	0
$75$	2.39374	0.276406
$76$	0	0
$77$	0.841267	0.0958713
$78$	0	0
$79$	12.6413	1.42226	0.711129	−	0.703061i	$-0.248184\pi$
$79$	12.6413	1.42226	0.711129	+	0.703061i	$0.248184\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	9.10421	0.999317	0.499658	−	0.866222i	$-0.333459\pi$
$83$	9.10421	0.999317	0.499658	+	0.866222i	$0.333459\pi$
$84$	0	0
$85$	−16.2227	−1.75960
$86$	0	0
$87$	−8.12923	−0.871545
$88$	0	0
$89$	8.77241	0.929873	0.464937	−	0.885344i	$-0.346077\pi$
$89$	8.77241	0.929873	0.464937	+	0.885344i	$0.346077\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−1.67023	−0.173194
$94$	0	0
$95$	8.36419	0.858148
$96$	0	0
$97$	−6.86954	−0.697496	−0.348748	−	0.937216i	$-0.613393\pi$
$97$	−6.86954	−0.697496	−0.348748	+	0.937216i	$0.613393\pi$
$98$	0	0
$99$	1.14374	0.114950
$100$	0	0
$101$	7.11227	0.707697	0.353849	−	0.935303i	$-0.384873\pi$
$101$	7.11227	0.707697	0.353849	+	0.935303i	$0.384873\pi$
$102$	0	0
$103$	9.16550	0.903103	0.451552	−	0.892245i	$-0.350871\pi$
$103$	9.16550	0.903103	0.451552	+	0.892245i	$0.350871\pi$
$104$	0	0
$105$	−2.00004	−0.195184
$106$	0	0
$107$	4.98925	0.482329	0.241165	−	0.970484i	$-0.422471\pi$
$107$	4.98925	0.482329	0.241165	+	0.970484i	$0.422471\pi$
$108$	0	0
$109$	−15.6880	−1.50264	−0.751318	−	0.659940i	$-0.770582\pi$
$109$	−15.6880	−1.50264	−0.751318	+	0.659940i	$0.770582\pi$
$110$	0	0
$111$	−1.90945	−0.181237
$112$	0	0
$113$	14.7801	1.39039	0.695197	−	0.718819i	$-0.255317\pi$
$113$	14.7801	1.39039	0.695197	+	0.718819i	$0.255317\pi$
$114$	0	0
$115$	−14.4449	−1.34699
$116$	0	0
$117$	0	0
$118$	0	0
$119$	4.38833	0.402278
$120$	0	0
$121$	−9.69187	−0.881079
$122$	0	0
$123$	8.69257	0.783782
$124$	0	0
$125$	7.08679	0.633862
$126$	0	0
$127$	18.9211	1.67898	0.839490	−	0.543375i	$-0.182854\pi$
$127$	18.9211	1.67898	0.839490	+	0.543375i	$0.182854\pi$
$128$	0	0
$129$	11.7282	1.03261
$130$	0	0
$131$	16.4920	1.44091	0.720456	−	0.693500i	$-0.243932\pi$
$131$	16.4920	1.44091	0.720456	+	0.693500i	$0.243932\pi$
$132$	0	0
$133$	−2.26256	−0.196189
$134$	0	0
$135$	−2.71914	−0.234027
$136$	0	0
$137$	−10.0062	−0.854888	−0.427444	−	0.904042i	$-0.640586\pi$
$137$	−10.0062	−0.854888	−0.427444	+	0.904042i	$0.640586\pi$
$138$	0	0
$139$	−11.1940	−0.949464	−0.474732	−	0.880130i	$-0.657455\pi$
$139$	−11.1940	−0.949464	−0.474732	+	0.880130i	$0.657455\pi$
$140$	0	0
$141$	−7.53981	−0.634966
$142$	0	0
$143$	0	0
$144$	0	0
$145$	22.1045	1.83568
$146$	0	0
$147$	−6.45898	−0.532728
$148$	0	0
$149$	15.0579	1.23359	0.616795	−	0.787124i	$-0.288431\pi$
$149$	15.0579	1.23359	0.616795	+	0.787124i	$0.288431\pi$
$150$	0	0
$151$	18.1086	1.47365	0.736827	−	0.676081i	$-0.236323\pi$
$151$	18.1086	1.47365	0.736827	+	0.676081i	$0.236323\pi$
$152$	0	0
$153$	5.96612	0.482332
$154$	0	0
$155$	4.54159	0.364789
$156$	0	0
$157$	−3.72080	−0.296952	−0.148476	−	0.988916i	$-0.547437\pi$
$157$	−3.72080	−0.296952	−0.148476	+	0.988916i	$0.547437\pi$
$158$	0	0
$159$	1.86294	0.147740
$160$	0	0
$161$	3.90742	0.307948
$162$	0	0
$163$	−16.0780	−1.25932	−0.629661	−	0.776870i	$-0.716806\pi$
$163$	−16.0780	−1.25932	−0.629661	+	0.776870i	$0.716806\pi$
$164$	0	0
$165$	−3.10999	−0.242112
$166$	0	0
$167$	13.0746	1.01174	0.505871	−	0.862609i	$-0.331171\pi$
$167$	13.0746	1.01174	0.505871	+	0.862609i	$0.331171\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	−3.07604	−0.235231
$172$	0	0
$173$	17.0060	1.29294	0.646470	−	0.762940i	$-0.276245\pi$
$173$	17.0060	1.29294	0.646470	+	0.762940i	$0.276245\pi$
$174$	0	0
$175$	1.76070	0.133096
$176$	0	0
$177$	−4.28323	−0.321947
$178$	0	0
$179$	11.9269	0.891458	0.445729	−	0.895168i	$-0.352945\pi$
$179$	11.9269	0.891458	0.445729	+	0.895168i	$0.352945\pi$
$180$	0	0
$181$	0.248257	0.0184528	0.00922639	−	0.999957i	$-0.497063\pi$
$181$	0.248257	0.0184528	0.00922639	+	0.999957i	$0.497063\pi$
$182$	0	0
$183$	−2.91045	−0.215147
$184$	0	0
$185$	5.19208	0.381729
$186$	0	0
$187$	6.82368	0.498997
$188$	0	0
$189$	0.735542	0.0535028
$190$	0	0
$191$	−17.1792	−1.24304	−0.621522	−	0.783397i	$-0.713485\pi$
$191$	−17.1792	−1.24304	−0.621522	+	0.783397i	$0.713485\pi$
$192$	0	0
$193$	22.6570	1.63088	0.815442	−	0.578839i	$-0.196494\pi$
$193$	22.6570	1.63088	0.815442	+	0.578839i	$0.196494\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	4.47161	0.318589	0.159294	−	0.987231i	$-0.449078\pi$
$197$	4.47161	0.318589	0.159294	+	0.987231i	$0.449078\pi$
$198$	0	0
$199$	−21.1993	−1.50278	−0.751389	−	0.659859i	$-0.770616\pi$
$199$	−21.1993	−1.50278	−0.751389	+	0.659859i	$0.770616\pi$
$200$	0	0
$201$	−0.596077	−0.0420440
$202$	0	0
$203$	−5.97939	−0.419671
$204$	0	0
$205$	−23.6363	−1.65083
$206$	0	0
$207$	5.31230	0.369230
$208$	0	0
$209$	−3.51818	−0.243358
$210$	0	0
$211$	−23.4474	−1.61419	−0.807093	−	0.590424i	$-0.798960\pi$
$211$	−23.4474	−1.61419	−0.807093	+	0.590424i	$0.798960\pi$
$212$	0	0
$213$	2.30363	0.157842
$214$	0	0
$215$	−31.8906	−2.17492
$216$	0	0
$217$	−1.22852	−0.0833975
$218$	0	0
$219$	13.1830	0.890825
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−4.25181	−0.284722	−0.142361	−	0.989815i	$-0.545469\pi$
$223$	−4.25181	−0.284722	−0.142361	+	0.989815i	$0.545469\pi$
$224$	0	0
$225$	2.39374	0.159583
$226$	0	0
$227$	13.3153	0.883768	0.441884	−	0.897072i	$-0.354310\pi$
$227$	13.3153	0.883768	0.441884	+	0.897072i	$0.354310\pi$
$228$	0	0
$229$	11.6238	0.768123	0.384062	−	0.923307i	$-0.374525\pi$
$229$	11.6238	0.768123	0.384062	+	0.923307i	$0.374525\pi$
$230$	0	0
$231$	0.841267	0.0553513
$232$	0	0
$233$	29.4566	1.92977	0.964883	−	0.262682i	$-0.0846069\pi$
$233$	29.4566	1.92977	0.964883	+	0.262682i	$0.0846069\pi$
$234$	0	0
$235$	20.5018	1.33739
$236$	0	0
$237$	12.6413	0.821141
$238$	0	0
$239$	6.36845	0.411941	0.205971	−	0.978558i	$-0.433965\pi$
$239$	6.36845	0.411941	0.205971	+	0.978558i	$0.433965\pi$
$240$	0	0
$241$	−15.2290	−0.980984	−0.490492	−	0.871446i	$-0.663183\pi$
$241$	−15.2290	−0.980984	−0.490492	+	0.871446i	$0.663183\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	17.5629	1.12205
$246$	0	0
$247$	0	0
$248$	0	0
$249$	9.10421	0.576956
$250$	0	0
$251$	−1.57425	−0.0993655	−0.0496828	−	0.998765i	$-0.515821\pi$
$251$	−1.57425	−0.0993655	−0.0496828	+	0.998765i	$0.515821\pi$
$252$	0	0
$253$	6.07587	0.381987
$254$	0	0
$255$	−16.2227	−1.01591
$256$	0	0
$257$	13.1261	0.818782	0.409391	−	0.912359i	$-0.365741\pi$
$257$	13.1261	0.818782	0.409391	+	0.912359i	$0.365741\pi$
$258$	0	0
$259$	−1.40448	−0.0872703
$260$	0	0
$261$	−8.12923	−0.503187
$262$	0	0
$263$	10.1263	0.624412	0.312206	−	0.950014i	$-0.398932\pi$
$263$	10.1263	0.624412	0.312206	+	0.950014i	$0.398932\pi$
$264$	0	0
$265$	−5.06559	−0.311177
$266$	0	0
$267$	8.77241	0.536863
$268$	0	0
$269$	−17.1811	−1.04755	−0.523774	−	0.851857i	$-0.675476\pi$
$269$	−17.1811	−1.04755	−0.523774	+	0.851857i	$0.675476\pi$
$270$	0	0
$271$	−16.5460	−1.00510	−0.502550	−	0.864548i	$-0.667605\pi$
$271$	−16.5460	−1.00510	−0.502550	+	0.864548i	$0.667605\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	2.73781	0.165096
$276$	0	0
$277$	28.6891	1.72376	0.861882	−	0.507110i	$-0.169286\pi$
$277$	28.6891	1.72376	0.861882	+	0.507110i	$0.169286\pi$
$278$	0	0
$279$	−1.67023	−0.0999938
$280$	0	0
$281$	9.08857	0.542179	0.271089	−	0.962554i	$-0.412616\pi$
$281$	9.08857	0.542179	0.271089	+	0.962554i	$0.412616\pi$
$282$	0	0
$283$	−22.3186	−1.32671	−0.663353	−	0.748307i	$-0.730867\pi$
$283$	−22.3186	−1.32671	−0.663353	+	0.748307i	$0.730867\pi$
$284$	0	0
$285$	8.36419	0.495452
$286$	0	0
$287$	6.39375	0.377411
$288$	0	0
$289$	18.5946	1.09380
$290$	0	0
$291$	−6.86954	−0.402700
$292$	0	0
$293$	22.5171	1.31546	0.657731	−	0.753253i	$-0.271516\pi$
$293$	22.5171	1.31546	0.657731	+	0.753253i	$0.271516\pi$
$294$	0	0
$295$	11.6467	0.678098
$296$	0	0
$297$	1.14374	0.0663664
$298$	0	0
$299$	0	0
$300$	0	0
$301$	8.62658	0.497228
$302$	0	0
$303$	7.11227	0.408589
$304$	0	0
$305$	7.91394	0.453151
$306$	0	0
$307$	20.1371	1.14928	0.574641	−	0.818405i	$-0.305142\pi$
$307$	20.1371	1.14928	0.574641	+	0.818405i	$0.305142\pi$
$308$	0	0
$309$	9.16550	0.521407
$310$	0	0
$311$	−12.9556	−0.734645	−0.367322	−	0.930094i	$-0.619725\pi$
$311$	−12.9556	−0.734645	−0.367322	+	0.930094i	$0.619725\pi$
$312$	0	0
$313$	12.8936	0.728789	0.364394	−	0.931245i	$-0.381276\pi$
$313$	12.8936	0.728789	0.364394	+	0.931245i	$0.381276\pi$
$314$	0	0
$315$	−2.00004	−0.112690
$316$	0	0
$317$	−27.2814	−1.53228	−0.766138	−	0.642676i	$-0.777824\pi$
$317$	−27.2814	−1.53228	−0.766138	+	0.642676i	$0.777824\pi$
$318$	0	0
$319$	−9.29770	−0.520571
$320$	0	0
$321$	4.98925	0.278473
$322$	0	0
$323$	−18.3520	−1.02113
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−15.6880	−0.867547
$328$	0	0
$329$	−5.54584	−0.305752
$330$	0	0
$331$	−14.4726	−0.795484	−0.397742	−	0.917497i	$-0.630206\pi$
$331$	−14.4726	−0.795484	−0.397742	+	0.917497i	$0.630206\pi$
$332$	0	0
$333$	−1.90945	−0.104637
$334$	0	0
$335$	1.62082	0.0885548
$336$	0	0
$337$	−9.27680	−0.505339	−0.252670	−	0.967553i	$-0.581309\pi$
$337$	−9.27680	−0.505339	−0.252670	+	0.967553i	$0.581309\pi$
$338$	0	0
$339$	14.7801	0.802744
$340$	0	0
$341$	−1.91030	−0.103449
$342$	0	0
$343$	−9.89964	−0.534531
$344$	0	0
$345$	−14.4449	−0.777687
$346$	0	0
$347$	−24.9968	−1.34190	−0.670951	−	0.741502i	$-0.734114\pi$
$347$	−24.9968	−1.34190	−0.670951	+	0.741502i	$0.734114\pi$
$348$	0	0
$349$	18.8625	1.00969	0.504844	−	0.863211i	$-0.331550\pi$
$349$	18.8625	1.00969	0.504844	+	0.863211i	$0.331550\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−6.99213	−0.372154	−0.186077	−	0.982535i	$-0.559577\pi$
$353$	−6.99213	−0.372154	−0.186077	+	0.982535i	$0.559577\pi$
$354$	0	0
$355$	−6.26390	−0.332453
$356$	0	0
$357$	4.38833	0.232255
$358$	0	0
$359$	−26.5257	−1.39997	−0.699987	−	0.714156i	$-0.746811\pi$
$359$	−26.5257	−1.39997	−0.699987	+	0.714156i	$0.746811\pi$
$360$	0	0
$361$	−9.53798	−0.501999
$362$	0	0
$363$	−9.69187	−0.508691
$364$	0	0
$365$	−35.8465	−1.87629
$366$	0	0
$367$	−32.6137	−1.70242	−0.851211	−	0.524824i	$-0.824131\pi$
$367$	−32.6137	−1.70242	−0.851211	+	0.524824i	$0.824131\pi$
$368$	0	0
$369$	8.69257	0.452517
$370$	0	0
$371$	1.37027	0.0711408
$372$	0	0
$373$	−10.4172	−0.539384	−0.269692	−	0.962947i	$-0.586922\pi$
$373$	−10.4172	−0.539384	−0.269692	+	0.962947i	$0.586922\pi$
$374$	0	0
$375$	7.08679	0.365960
$376$	0	0
$377$	0	0
$378$	0	0
$379$	20.4662	1.05128	0.525640	−	0.850707i	$-0.323826\pi$
$379$	20.4662	1.05128	0.525640	+	0.850707i	$0.323826\pi$
$380$	0	0
$381$	18.9211	0.969360
$382$	0	0
$383$	−35.4182	−1.80979	−0.904894	−	0.425637i	$-0.860050\pi$
$383$	−35.4182	−1.80979	−0.904894	+	0.425637i	$0.860050\pi$
$384$	0	0
$385$	−2.28753	−0.116583
$386$	0	0
$387$	11.7282	0.596177
$388$	0	0
$389$	6.20097	0.314402	0.157201	−	0.987567i	$-0.449753\pi$
$389$	6.20097	0.314402	0.157201	+	0.987567i	$0.449753\pi$
$390$	0	0
$391$	31.6938	1.60282
$392$	0	0
$393$	16.4920	0.831911
$394$	0	0
$395$	−34.3735	−1.72952
$396$	0	0
$397$	3.93338	0.197410	0.0987052	−	0.995117i	$-0.468530\pi$
$397$	3.93338	0.197410	0.0987052	+	0.995117i	$0.468530\pi$
$398$	0	0
$399$	−2.26256	−0.113270
$400$	0	0
$401$	19.6651	0.982027	0.491013	−	0.871152i	$-0.336627\pi$
$401$	19.6651	0.982027	0.491013	+	0.871152i	$0.336627\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	−2.71914	−0.135115
$406$	0	0
$407$	−2.18391	−0.108252
$408$	0	0
$409$	−14.2584	−0.705033	−0.352516	−	0.935806i	$-0.614674\pi$
$409$	−14.2584	−0.705033	−0.352516	+	0.935806i	$0.614674\pi$
$410$	0	0
$411$	−10.0062	−0.493570
$412$	0	0
$413$	−3.15049	−0.155026
$414$	0	0
$415$	−24.7557	−1.21521
$416$	0	0
$417$	−11.1940	−0.548174
$418$	0	0
$419$	10.3494	0.505603	0.252802	−	0.967518i	$-0.418648\pi$
$419$	10.3494	0.505603	0.252802	+	0.967518i	$0.418648\pi$
$420$	0	0
$421$	−35.5629	−1.73323	−0.866614	−	0.498979i	$-0.833709\pi$
$421$	−35.5629	−1.73323	−0.866614	+	0.498979i	$0.833709\pi$
$422$	0	0
$423$	−7.53981	−0.366598
$424$	0	0
$425$	14.2814	0.692748
$426$	0	0
$427$	−2.14076	−0.103599
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−16.9667	−0.817257	−0.408629	−	0.912701i	$-0.633993\pi$
$431$	−16.9667	−0.817257	−0.408629	+	0.912701i	$0.633993\pi$
$432$	0	0
$433$	−18.2275	−0.875958	−0.437979	−	0.898985i	$-0.644306\pi$
$433$	−18.2275	−0.875958	−0.437979	+	0.898985i	$0.644306\pi$
$434$	0	0
$435$	22.1045	1.05983
$436$	0	0
$437$	−16.3408	−0.781688
$438$	0	0
$439$	−9.98284	−0.476455	−0.238228	−	0.971209i	$-0.576566\pi$
$439$	−9.98284	−0.476455	−0.238228	+	0.971209i	$0.576566\pi$
$440$	0	0
$441$	−6.45898	−0.307570
$442$	0	0
$443$	11.8110	0.561156	0.280578	−	0.959831i	$-0.409474\pi$
$443$	11.8110	0.561156	0.280578	+	0.959831i	$0.409474\pi$
$444$	0	0
$445$	−23.8534	−1.13076
$446$	0	0
$447$	15.0579	0.712213
$448$	0	0
$449$	31.3334	1.47872	0.739358	−	0.673313i	$-0.235129\pi$
$449$	31.3334	1.47872	0.739358	+	0.673313i	$0.235129\pi$
$450$	0	0
$451$	9.94201	0.468151
$452$	0	0
$453$	18.1086	0.850815
$454$	0	0
$455$	0	0
$456$	0	0
$457$	0.275381	0.0128818	0.00644089	−	0.999979i	$-0.497950\pi$
$457$	0.275381	0.0128818	0.00644089	+	0.999979i	$0.497950\pi$
$458$	0	0
$459$	5.96612	0.278475
$460$	0	0
$461$	−13.7955	−0.642519	−0.321260	−	0.946991i	$-0.604106\pi$
$461$	−13.7955	−0.642519	−0.321260	+	0.946991i	$0.604106\pi$
$462$	0	0
$463$	18.6326	0.865929	0.432965	−	0.901411i	$-0.357467\pi$
$463$	18.6326	0.865929	0.432965	+	0.901411i	$0.357467\pi$
$464$	0	0
$465$	4.54159	0.210611
$466$	0	0
$467$	20.1171	0.930909	0.465454	−	0.885072i	$-0.345891\pi$
$467$	20.1171	0.930909	0.465454	+	0.885072i	$0.345891\pi$
$468$	0	0
$469$	−0.438440	−0.0202453
$470$	0	0
$471$	−3.72080	−0.171446
$472$	0	0
$473$	13.4140	0.616775
$474$	0	0
$475$	−7.36325	−0.337849
$476$	0	0
$477$	1.86294	0.0852980
$478$	0	0
$479$	−32.7364	−1.49576	−0.747882	−	0.663832i	$-0.768929\pi$
$479$	−32.7364	−1.49576	−0.747882	+	0.663832i	$0.768929\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	3.90742	0.177794
$484$	0	0
$485$	18.6793	0.848182
$486$	0	0
$487$	18.2763	0.828177	0.414089	−	0.910237i	$-0.364100\pi$
$487$	18.2763	0.828177	0.414089	+	0.910237i	$0.364100\pi$
$488$	0	0
$489$	−16.0780	−0.727070
$490$	0	0
$491$	−25.2495	−1.13949	−0.569747	−	0.821820i	$-0.692959\pi$
$491$	−25.2495	−1.13949	−0.569747	+	0.821820i	$0.692959\pi$
$492$	0	0
$493$	−48.5000	−2.18433
$494$	0	0
$495$	−3.10999	−0.139783
$496$	0	0
$497$	1.69442	0.0760050
$498$	0	0
$499$	38.3791	1.71808	0.859042	−	0.511906i	$-0.171060\pi$
$499$	38.3791	1.71808	0.859042	+	0.511906i	$0.171060\pi$
$500$	0	0
$501$	13.0746	0.584129
$502$	0	0
$503$	−0.196327	−0.00875380	−0.00437690	−	0.999990i	$-0.501393\pi$
$503$	−0.196327	−0.00875380	−0.00437690	+	0.999990i	$0.501393\pi$
$504$	0	0
$505$	−19.3393	−0.860587
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−25.7356	−1.14071	−0.570355	−	0.821399i	$-0.693194\pi$
$509$	−25.7356	−1.14071	−0.570355	+	0.821399i	$0.693194\pi$
$510$	0	0
$511$	9.69665	0.428955
$512$	0	0
$513$	−3.07604	−0.135810
$514$	0	0
$515$	−24.9223	−1.09821
$516$	0	0
$517$	−8.62356	−0.379264
$518$	0	0
$519$	17.0060	0.746479
$520$	0	0
$521$	11.0712	0.485040	0.242520	−	0.970146i	$-0.422026\pi$
$521$	11.0712	0.485040	0.242520	+	0.970146i	$0.422026\pi$
$522$	0	0
$523$	−24.2974	−1.06245	−0.531227	−	0.847230i	$-0.678269\pi$
$523$	−24.2974	−1.06245	−0.531227	+	0.847230i	$0.678269\pi$
$524$	0	0
$525$	1.76070	0.0768432
$526$	0	0
$527$	−9.96478	−0.434072
$528$	0	0
$529$	5.22048	0.226977
$530$	0	0
$531$	−4.28323	−0.185876
$532$	0	0
$533$	0	0
$534$	0	0
$535$	−13.5665	−0.586531
$536$	0	0
$537$	11.9269	0.514683
$538$	0	0
$539$	−7.38737	−0.318197
$540$	0	0
$541$	31.5652	1.35710	0.678548	−	0.734556i	$-0.262610\pi$
$541$	31.5652	1.35710	0.678548	+	0.734556i	$0.262610\pi$
$542$	0	0
$543$	0.248257	0.0106537
$544$	0	0
$545$	42.6579	1.82726
$546$	0	0
$547$	−1.79582	−0.0767838	−0.0383919	−	0.999263i	$-0.512224\pi$
$547$	−1.79582	−0.0767838	−0.0383919	+	0.999263i	$0.512224\pi$
$548$	0	0
$549$	−2.91045	−0.124215
$550$	0	0
$551$	25.0058	1.06528
$552$	0	0
$553$	9.29822	0.395400
$554$	0	0
$555$	5.19208	0.220391
$556$	0	0
$557$	13.1019	0.555146	0.277573	−	0.960705i	$-0.410470\pi$
$557$	13.1019	0.555146	0.277573	+	0.960705i	$0.410470\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	6.82368	0.288096
$562$	0	0
$563$	−5.44099	−0.229310	−0.114655	−	0.993405i	$-0.536576\pi$
$563$	−5.44099	−0.229310	−0.114655	+	0.993405i	$0.536576\pi$
$564$	0	0
$565$	−40.1892	−1.69077
$566$	0	0
$567$	0.735542	0.0308899
$568$	0	0
$569$	8.65863	0.362989	0.181494	−	0.983392i	$-0.441907\pi$
$569$	8.65863	0.362989	0.181494	+	0.983392i	$0.441907\pi$
$570$	0	0
$571$	−16.0718	−0.672585	−0.336293	−	0.941758i	$-0.609173\pi$
$571$	−16.0718	−0.672585	−0.336293	+	0.941758i	$0.609173\pi$
$572$	0	0
$573$	−17.1792	−0.717672
$574$	0	0
$575$	12.7163	0.530305
$576$	0	0
$577$	35.8157	1.49103	0.745513	−	0.666491i	$-0.232204\pi$
$577$	35.8157	1.49103	0.745513	+	0.666491i	$0.232204\pi$
$578$	0	0
$579$	22.6570	0.941591
$580$	0	0
$581$	6.69653	0.277819
$582$	0	0
$583$	2.13071	0.0882450
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−24.0888	−0.994250	−0.497125	−	0.867679i	$-0.665611\pi$
$587$	−24.0888	−0.994250	−0.497125	+	0.867679i	$0.665611\pi$
$588$	0	0
$589$	5.13768	0.211695
$590$	0	0
$591$	4.47161	0.183937
$592$	0	0
$593$	32.7042	1.34300	0.671501	−	0.741004i	$-0.265650\pi$
$593$	32.7042	1.34300	0.671501	+	0.741004i	$0.265650\pi$
$594$	0	0
$595$	−11.9325	−0.489185
$596$	0	0
$597$	−21.1993	−0.867630
$598$	0	0
$599$	−30.2139	−1.23451	−0.617254	−	0.786764i	$-0.711755\pi$
$599$	−30.2139	−1.23451	−0.617254	+	0.786764i	$0.711755\pi$
$600$	0	0
$601$	1.78275	0.0727200	0.0363600	−	0.999339i	$-0.488424\pi$
$601$	1.78275	0.0727200	0.0363600	+	0.999339i	$0.488424\pi$
$602$	0	0
$603$	−0.596077	−0.0242741
$604$	0	0
$605$	26.3536	1.07143
$606$	0	0
$607$	23.5806	0.957109	0.478554	−	0.878058i	$-0.341161\pi$
$607$	23.5806	0.957109	0.478554	+	0.878058i	$0.341161\pi$
$608$	0	0
$609$	−5.97939	−0.242297
$610$	0	0
$611$	0	0
$612$	0	0
$613$	17.4093	0.703156	0.351578	−	0.936159i	$-0.385645\pi$
$613$	17.4093	0.703156	0.351578	+	0.936159i	$0.385645\pi$
$614$	0	0
$615$	−23.6363	−0.953109
$616$	0	0
$617$	30.9193	1.24476	0.622382	−	0.782714i	$-0.286165\pi$
$617$	30.9193	1.24476	0.622382	+	0.782714i	$0.286165\pi$
$618$	0	0
$619$	−32.6808	−1.31355	−0.656777	−	0.754085i	$-0.728081\pi$
$619$	−32.6808	−1.31355	−0.656777	+	0.754085i	$0.728081\pi$
$620$	0	0
$621$	5.31230	0.213175
$622$	0	0
$623$	6.45248	0.258513
$624$	0	0
$625$	−31.2387	−1.24955
$626$	0	0
$627$	−3.51818	−0.140503
$628$	0	0
$629$	−11.3920	−0.454230
$630$	0	0
$631$	11.7322	0.467053	0.233527	−	0.972350i	$-0.424973\pi$
$631$	11.7322	0.467053	0.233527	+	0.972350i	$0.424973\pi$
$632$	0	0
$633$	−23.4474	−0.931951
$634$	0	0
$635$	−51.4493	−2.04170
$636$	0	0
$637$	0	0
$638$	0	0
$639$	2.30363	0.0911302
$640$	0	0
$641$	−38.0086	−1.50125	−0.750624	−	0.660729i	$-0.770247\pi$
$641$	−38.0086	−1.50125	−0.750624	+	0.660729i	$0.770247\pi$
$642$	0	0
$643$	3.53115	0.139255	0.0696275	−	0.997573i	$-0.477819\pi$
$643$	3.53115	0.139255	0.0696275	+	0.997573i	$0.477819\pi$
$644$	0	0
$645$	−31.8906	−1.25569
$646$	0	0
$647$	26.2407	1.03163	0.515814	−	0.856701i	$-0.327490\pi$
$647$	26.2407	1.03163	0.515814	+	0.856701i	$0.327490\pi$
$648$	0	0
$649$	−4.89889	−0.192298
$650$	0	0
$651$	−1.22852	−0.0481496
$652$	0	0
$653$	2.79758	0.109478	0.0547389	−	0.998501i	$-0.482567\pi$
$653$	2.79758	0.109478	0.0547389	+	0.998501i	$0.482567\pi$
$654$	0	0
$655$	−44.8441	−1.75220
$656$	0	0
$657$	13.1830	0.514318
$658$	0	0
$659$	10.5249	0.409991	0.204996	−	0.978763i	$-0.434282\pi$
$659$	10.5249	0.409991	0.204996	+	0.978763i	$0.434282\pi$
$660$	0	0
$661$	−34.8706	−1.35631	−0.678155	−	0.734919i	$-0.737220\pi$
$661$	−34.8706	−1.35631	−0.678155	+	0.734919i	$0.737220\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	6.15222	0.238573
$666$	0	0
$667$	−43.1849	−1.67212
$668$	0	0
$669$	−4.25181	−0.164385
$670$	0	0
$671$	−3.32879	−0.128507
$672$	0	0
$673$	−10.3696	−0.399718	−0.199859	−	0.979825i	$-0.564048\pi$
$673$	−10.3696	−0.399718	−0.199859	+	0.979825i	$0.564048\pi$
$674$	0	0
$675$	2.39374	0.0921352
$676$	0	0
$677$	12.0671	0.463778	0.231889	−	0.972742i	$-0.425509\pi$
$677$	12.0671	0.463778	0.231889	+	0.972742i	$0.425509\pi$
$678$	0	0
$679$	−5.05284	−0.193910
$680$	0	0
$681$	13.3153	0.510244
$682$	0	0
$683$	−9.35234	−0.357857	−0.178929	−	0.983862i	$-0.557263\pi$
$683$	−9.35234	−0.357857	−0.178929	+	0.983862i	$0.557263\pi$
$684$	0	0
$685$	27.2083	1.03958
$686$	0	0
$687$	11.6238	0.443476
$688$	0	0
$689$	0	0
$690$	0	0
$691$	−10.8063	−0.411092	−0.205546	−	0.978647i	$-0.565897\pi$
$691$	−10.8063	−0.411092	−0.205546	+	0.978647i	$0.565897\pi$
$692$	0	0
$693$	0.841267	0.0319571
$694$	0	0
$695$	30.4382	1.15458
$696$	0	0
$697$	51.8609	1.96437
$698$	0	0
$699$	29.4566	1.11415
$700$	0	0
$701$	5.25030	0.198301	0.0991505	−	0.995072i	$-0.468387\pi$
$701$	5.25030	0.198301	0.0991505	+	0.995072i	$0.468387\pi$
$702$	0	0
$703$	5.87355	0.221525
$704$	0	0
$705$	20.5018	0.772143
$706$	0	0
$707$	5.23137	0.196746
$708$	0	0
$709$	16.6495	0.625284	0.312642	−	0.949871i	$-0.398786\pi$
$709$	16.6495	0.625284	0.312642	+	0.949871i	$0.398786\pi$
$710$	0	0
$711$	12.6413	0.474086
$712$	0	0
$713$	−8.87274	−0.332287
$714$	0	0
$715$	0	0
$716$	0	0
$717$	6.36845	0.237834
$718$	0	0
$719$	11.0141	0.410756	0.205378	−	0.978683i	$-0.434158\pi$
$719$	11.0141	0.410756	0.205378	+	0.978683i	$0.434158\pi$
$720$	0	0
$721$	6.74161	0.251071
$722$	0	0
$723$	−15.2290	−0.566371
$724$	0	0
$725$	−19.4593	−0.722700
$726$	0	0
$727$	−33.3243	−1.23593	−0.617965	−	0.786205i	$-0.712043\pi$
$727$	−33.3243	−1.23593	−0.617965	+	0.786205i	$0.712043\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	69.9718	2.58800
$732$	0	0
$733$	31.4717	1.16243	0.581216	−	0.813749i	$-0.302577\pi$
$733$	31.4717	1.16243	0.581216	+	0.813749i	$0.302577\pi$
$734$	0	0
$735$	17.5629	0.647817
$736$	0	0
$737$	−0.681755	−0.0251128
$738$	0	0
$739$	7.78683	0.286443	0.143222	−	0.989691i	$-0.454254\pi$
$739$	7.78683	0.286443	0.143222	+	0.989691i	$0.454254\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−12.4688	−0.457437	−0.228718	−	0.973493i	$-0.573454\pi$
$743$	−12.4688	−0.457437	−0.228718	+	0.973493i	$0.573454\pi$
$744$	0	0
$745$	−40.9445	−1.50009
$746$	0	0
$747$	9.10421	0.333106
$748$	0	0
$749$	3.66980	0.134092
$750$	0	0
$751$	19.0652	0.695699	0.347849	−	0.937550i	$-0.386912\pi$
$751$	19.0652	0.695699	0.347849	+	0.937550i	$0.386912\pi$
$752$	0	0
$753$	−1.57425	−0.0573687
$754$	0	0
$755$	−49.2398	−1.79202
$756$	0	0
$757$	11.1165	0.404037	0.202018	−	0.979382i	$-0.435250\pi$
$757$	11.1165	0.404037	0.202018	+	0.979382i	$0.435250\pi$
$758$	0	0
$759$	6.07587	0.220540
$760$	0	0
$761$	25.9184	0.939542	0.469771	−	0.882788i	$-0.344336\pi$
$761$	25.9184	0.939542	0.469771	+	0.882788i	$0.344336\pi$
$762$	0	0
$763$	−11.5392	−0.417746
$764$	0	0
$765$	−16.2227	−0.586535
$766$	0	0
$767$	0	0
$768$	0	0
$769$	51.8255	1.86888	0.934438	−	0.356125i	$-0.115902\pi$
$769$	51.8255	1.86888	0.934438	+	0.356125i	$0.115902\pi$
$770$	0	0
$771$	13.1261	0.472724
$772$	0	0
$773$	0.0857906	0.00308567	0.00154284	−	0.999999i	$-0.499509\pi$
$773$	0.0857906	0.00308567	0.00154284	+	0.999999i	$0.499509\pi$
$774$	0	0
$775$	−3.99809	−0.143616
$776$	0	0
$777$	−1.40448	−0.0503855
$778$	0	0
$779$	−26.7387	−0.958012
$780$	0	0
$781$	2.63475	0.0942787
$782$	0	0
$783$	−8.12923	−0.290515
$784$	0	0
$785$	10.1174	0.361105
$786$	0	0
$787$	8.76170	0.312321	0.156160	−	0.987732i	$-0.450088\pi$
$787$	8.76170	0.312321	0.156160	+	0.987732i	$0.450088\pi$
$788$	0	0
$789$	10.1263	0.360504
$790$	0	0
$791$	10.8714	0.386542
$792$	0	0
$793$	0	0
$794$	0	0
$795$	−5.06559	−0.179658
$796$	0	0
$797$	0.288970	0.0102358	0.00511792	−	0.999987i	$-0.498371\pi$
$797$	0.288970	0.0102358	0.00511792	+	0.999987i	$0.498371\pi$
$798$	0	0
$799$	−44.9834	−1.59140
$800$	0	0
$801$	8.77241	0.309958
$802$	0	0
$803$	15.0779	0.532087
$804$	0	0
$805$	−10.6248	−0.374476
$806$	0	0
$807$	−17.1811	−0.604802
$808$	0	0
$809$	−10.0782	−0.354330	−0.177165	−	0.984181i	$-0.556693\pi$
$809$	−10.0782	−0.354330	−0.177165	+	0.984181i	$0.556693\pi$
$810$	0	0
$811$	48.3498	1.69779	0.848896	−	0.528559i	$-0.177268\pi$
$811$	48.3498	1.69779	0.848896	+	0.528559i	$0.177268\pi$
$812$	0	0
$813$	−16.5460	−0.580295
$814$	0	0
$815$	43.7183	1.53138
$816$	0	0
$817$	−36.0764	−1.26215
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−40.6203	−1.41766	−0.708829	−	0.705381i	$-0.750776\pi$
$821$	−40.6203	−1.41766	−0.708829	+	0.705381i	$0.750776\pi$
$822$	0	0
$823$	−34.6609	−1.20820	−0.604101	−	0.796908i	$-0.706468\pi$
$823$	−34.6609	−1.20820	−0.604101	+	0.796908i	$0.706468\pi$
$824$	0	0
$825$	2.73781	0.0953184
$826$	0	0
$827$	1.13065	0.0393164	0.0196582	−	0.999807i	$-0.493742\pi$
$827$	1.13065	0.0393164	0.0196582	+	0.999807i	$0.493742\pi$
$828$	0	0
$829$	−10.5479	−0.366342	−0.183171	−	0.983081i	$-0.558636\pi$
$829$	−10.5479	−0.366342	−0.183171	+	0.983081i	$0.558636\pi$
$830$	0	0
$831$	28.6891	0.995215
$832$	0	0
$833$	−38.5351	−1.33516
$834$	0	0
$835$	−35.5517	−1.23032
$836$	0	0
$837$	−1.67023	−0.0577315
$838$	0	0
$839$	23.0198	0.794732	0.397366	−	0.917660i	$-0.369924\pi$
$839$	23.0198	0.794732	0.397366	+	0.917660i	$0.369924\pi$
$840$	0	0
$841$	37.0844	1.27877
$842$	0	0
$843$	9.08857	0.313027
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−7.12877	−0.244948
$848$	0	0
$849$	−22.3186	−0.765974
$850$	0	0
$851$	−10.1436	−0.347717
$852$	0	0
$853$	−15.9208	−0.545119	−0.272560	−	0.962139i	$-0.587870\pi$
$853$	−15.9208	−0.545119	−0.272560	+	0.962139i	$0.587870\pi$
$854$	0	0
$855$	8.36419	0.286049
$856$	0	0
$857$	−42.0212	−1.43542	−0.717708	−	0.696344i	$-0.754809\pi$
$857$	−42.0212	−1.43542	−0.717708	+	0.696344i	$0.754809\pi$
$858$	0	0
$859$	−12.6037	−0.430033	−0.215017	−	0.976610i	$-0.568981\pi$
$859$	−12.6037	−0.430033	−0.215017	+	0.976610i	$0.568981\pi$
$860$	0	0
$861$	6.39375	0.217898
$862$	0	0
$863$	53.7003	1.82798	0.913990	−	0.405737i	$-0.132985\pi$
$863$	53.7003	1.82798	0.913990	+	0.405737i	$0.132985\pi$
$864$	0	0
$865$	−46.2417	−1.57226
$866$	0	0
$867$	18.5946	0.631507
$868$	0	0
$869$	14.4583	0.490465
$870$	0	0
$871$	0	0
$872$	0	0
$873$	−6.86954	−0.232499
$874$	0	0
$875$	5.21263	0.176219
$876$	0	0
$877$	−24.3597	−0.822569	−0.411284	−	0.911507i	$-0.634920\pi$
$877$	−24.3597	−0.822569	−0.411284	+	0.911507i	$0.634920\pi$
$878$	0	0
$879$	22.5171	0.759482
$880$	0	0
$881$	3.00559	0.101261	0.0506305	−	0.998717i	$-0.483877\pi$
$881$	3.00559	0.101261	0.0506305	+	0.998717i	$0.483877\pi$
$882$	0	0
$883$	19.8422	0.667745	0.333872	−	0.942618i	$-0.391645\pi$
$883$	19.8422	0.667745	0.333872	+	0.942618i	$0.391645\pi$
$884$	0	0
$885$	11.6467	0.391500
$886$	0	0
$887$	−9.19535	−0.308750	−0.154375	−	0.988012i	$-0.549336\pi$
$887$	−9.19535	−0.308750	−0.154375	+	0.988012i	$0.549336\pi$
$888$	0	0
$889$	13.9173	0.466771
$890$	0	0
$891$	1.14374	0.0383166
$892$	0	0
$893$	23.1927	0.776116
$894$	0	0
$895$	−32.4309	−1.08405
$896$	0	0
$897$	0	0
$898$	0	0
$899$	13.5777	0.452840
$900$	0	0
$901$	11.1145	0.370278
$902$	0	0
$903$	8.62658	0.287074
$904$	0	0
$905$	−0.675046	−0.0224393
$906$	0	0
$907$	−55.1117	−1.82995	−0.914977	−	0.403507i	$-0.867791\pi$
$907$	−55.1117	−1.82995	−0.914977	+	0.403507i	$0.867791\pi$
$908$	0	0
$909$	7.11227	0.235899
$910$	0	0
$911$	−43.1678	−1.43021	−0.715107	−	0.699015i	$-0.753622\pi$
$911$	−43.1678	−1.43021	−0.715107	+	0.699015i	$0.753622\pi$
$912$	0	0
$913$	10.4128	0.344614
$914$	0	0
$915$	7.91394	0.261627
$916$	0	0
$917$	12.1306	0.400586
$918$	0	0
$919$	−7.83265	−0.258375	−0.129188	−	0.991620i	$-0.541237\pi$
$919$	−7.83265	−0.258375	−0.129188	+	0.991620i	$0.541237\pi$
$920$	0	0
$921$	20.1371	0.663539
$922$	0	0
$923$	0	0
$924$	0	0
$925$	−4.57074	−0.150285
$926$	0	0
$927$	9.16550	0.301034
$928$	0	0
$929$	−10.9070	−0.357847	−0.178923	−	0.983863i	$-0.557261\pi$
$929$	−10.9070	−0.357847	−0.178923	+	0.983863i	$0.557261\pi$
$930$	0	0
$931$	19.8681	0.651150
$932$	0	0
$933$	−12.9556	−0.424147
$934$	0	0
$935$	−18.5546	−0.606799
$936$	0	0
$937$	−53.4124	−1.74491	−0.872453	−	0.488697i	$-0.837472\pi$
$937$	−53.4124	−1.74491	−0.872453	+	0.488697i	$0.837472\pi$
$938$	0	0
$939$	12.8936	0.420766
$940$	0	0
$941$	−2.56819	−0.0837207	−0.0418604	−	0.999123i	$-0.513328\pi$
$941$	−2.56819	−0.0837207	−0.0418604	+	0.999123i	$0.513328\pi$
$942$	0	0
$943$	46.1775	1.50375
$944$	0	0
$945$	−2.00004	−0.0650615
$946$	0	0
$947$	−22.6747	−0.736827	−0.368414	−	0.929662i	$-0.620099\pi$
$947$	−22.6747	−0.736827	−0.368414	+	0.929662i	$0.620099\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	−27.2814	−0.884660
$952$	0	0
$953$	−35.2443	−1.14168	−0.570838	−	0.821063i	$-0.693382\pi$
$953$	−35.2443	−1.14168	−0.570838	+	0.821063i	$0.693382\pi$
$954$	0	0
$955$	46.7128	1.51159
$956$	0	0
$957$	−9.29770	−0.300552
$958$	0	0
$959$	−7.35999	−0.237666
$960$	0	0
$961$	−28.2103	−0.910011
$962$	0	0
$963$	4.98925	0.160776
$964$	0	0
$965$	−61.6075	−1.98322
$966$	0	0
$967$	−48.6851	−1.56561	−0.782804	−	0.622268i	$-0.786211\pi$
$967$	−48.6851	−1.56561	−0.782804	+	0.622268i	$0.786211\pi$
$968$	0	0
$969$	−18.3520	−0.589552
$970$	0	0
$971$	−30.7782	−0.987718	−0.493859	−	0.869542i	$-0.664414\pi$
$971$	−30.7782	−0.987718	−0.493859	+	0.869542i	$0.664414\pi$
$972$	0	0
$973$	−8.23367	−0.263959
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−18.9491	−0.606234	−0.303117	−	0.952953i	$-0.598027\pi$
$977$	−18.9491	−0.606234	−0.303117	+	0.952953i	$0.598027\pi$
$978$	0	0
$979$	10.0333	0.320667
$980$	0	0
$981$	−15.6880	−0.500879
$982$	0	0
$983$	−53.9304	−1.72011	−0.860056	−	0.510200i	$-0.829571\pi$
$983$	−53.9304	−1.72011	−0.860056	+	0.510200i	$0.829571\pi$
$984$	0	0
$985$	−12.1589	−0.387416
$986$	0	0
$987$	−5.54584	−0.176526
$988$	0	0
$989$	62.3036	1.98114
$990$	0	0
$991$	58.1777	1.84807	0.924037	−	0.382303i	$-0.124869\pi$
$991$	58.1777	1.84807	0.924037	+	0.382303i	$0.124869\pi$
$992$	0	0
$993$	−14.4726	−0.459273
$994$	0	0
$995$	57.6440	1.82744
$996$	0	0
$997$	−37.3857	−1.18402	−0.592009	−	0.805931i	$-0.701665\pi$
$997$	−37.3857	−1.18402	−0.592009	+	0.805931i	$0.701665\pi$
$998$	0	0
$999$	−1.90945	−0.0604124

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4056.2.a.bi.1.2	yes	6
4.3	odd	2		8112.2.a.cu.1.2		6
13.5	odd	4		4056.2.c.r.337.9		12
13.8	odd	4		4056.2.c.r.337.4		12
13.12	even	2		4056.2.a.bh.1.5	✓	6
52.51	odd	2		8112.2.a.ct.1.5		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4056.2.a.bh.1.5	✓	6	13.12	even	2
4056.2.a.bi.1.2	yes	6	1.1	even	1	trivial
4056.2.c.r.337.4		12	13.8	odd	4
4056.2.c.r.337.9		12	13.5	odd	4
8112.2.a.ct.1.5		6	52.51	odd	2
8112.2.a.cu.1.2		6	4.3	odd	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:	\( N \)	\(=\)	\( 4056 = 2^{3} \cdot 3 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4056.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -2.71914 q^{5} +0.735542 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -2.71914 q^{5} +0.735542 q^{7} +1.00000 q^{9} +1.14374 q^{11} -2.71914 q^{15} +5.96612 q^{17} -3.07604 q^{19} +0.735542 q^{21} +5.31230 q^{23} +2.39374 q^{25} +1.00000 q^{27} -8.12923 q^{29} -1.67023 q^{31} +1.14374 q^{33} -2.00004 q^{35} -1.90945 q^{37} +8.69257 q^{41} +11.7282 q^{43} -2.71914 q^{45} -7.53981 q^{47} -6.45898 q^{49} +5.96612 q^{51} +1.86294 q^{53} -3.10999 q^{55} -3.07604 q^{57} -4.28323 q^{59} -2.91045 q^{61} +0.735542 q^{63} -0.596077 q^{67} +5.31230 q^{69} +2.30363 q^{71} +13.1830 q^{73} +2.39374 q^{75} +0.841267 q^{77} +12.6413 q^{79} +1.00000 q^{81} +9.10421 q^{83} -16.2227 q^{85} -8.12923 q^{87} +8.77241 q^{89} -1.67023 q^{93} +8.36419 q^{95} -6.86954 q^{97} +1.14374 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 6 q^{3} + q^{5} - 5 q^{7} + 6 q^{9}+O(q^{10}) \) 6 * q + 6 * q^3 + q^5 - 5 * q^7 + 6 * q^9	\( 6 q + 6 q^{3} + q^{5} - 5 q^{7} + 6 q^{9} - 6 q^{11} + q^{15} + 9 q^{17} + 7 q^{19} - 5 q^{21} + 12 q^{23} + 9 q^{25} + 6 q^{27} + 7 q^{29} - 11 q^{31} - 6 q^{33} + 6 q^{35} + 6 q^{37} + 13 q^{41} + 15 q^{43} + q^{45} - 9 q^{47} + 13 q^{49} + 9 q^{51} + 22 q^{53} + 3 q^{55} + 7 q^{57} - 7 q^{59} + 25 q^{61} - 5 q^{63} + 5 q^{67} + 12 q^{69} + 8 q^{71} + 15 q^{73} + 9 q^{75} + 45 q^{77} + 14 q^{79} + 6 q^{81} - 13 q^{83} - 35 q^{85} + 7 q^{87} + 33 q^{89} - 11 q^{93} + 47 q^{95} + 50 q^{97} - 6 q^{99}+O(q^{100}) \) 6 * q + 6 * q^3 + q^5 - 5 * q^7 + 6 * q^9 - 6 * q^11 + q^15 + 9 * q^17 + 7 * q^19 - 5 * q^21 + 12 * q^23 + 9 * q^25 + 6 * q^27 + 7 * q^29 - 11 * q^31 - 6 * q^33 + 6 * q^35 + 6 * q^37 + 13 * q^41 + 15 * q^43 + q^45 - 9 * q^47 + 13 * q^49 + 9 * q^51 + 22 * q^53 + 3 * q^55 + 7 * q^57 - 7 * q^59 + 25 * q^61 - 5 * q^63 + 5 * q^67 + 12 * q^69 + 8 * q^71 + 15 * q^73 + 9 * q^75 + 45 * q^77 + 14 * q^79 + 6 * q^81 - 13 * q^83 - 35 * q^85 + 7 * q^87 + 33 * q^89 - 11 * q^93 + 47 * q^95 + 50 * q^97 - 6 * q^99

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