LMFDB - Embedded newform 4140.2.a.o.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4140,2,Mod(1,4140)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4140, 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("4140.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4140 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4140.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:	$33.0580664368$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1380)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.73205$ of defining polynomial
Character	$\chi$	$=$	4140.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{5} -2.46410 q^{7} +O(q^{10})$	$q-1.00000 q^{5} -2.46410 q^{7} -4.19615 q^{11} -3.26795 q^{13} +7.73205 q^{17} +0.732051 q^{19} +1.00000 q^{23} +1.00000 q^{25} -7.19615 q^{29} -1.00000 q^{31} +2.46410 q^{35} -11.3923 q^{37} +7.73205 q^{41} +3.46410 q^{43} -0.732051 q^{47} -0.928203 q^{49} -6.66025 q^{53} +4.19615 q^{55} +7.19615 q^{59} -10.7321 q^{61} +3.26795 q^{65} +5.00000 q^{67} +14.1244 q^{71} -11.2679 q^{73} +10.3397 q^{77} +4.00000 q^{79} +6.66025 q^{83} -7.73205 q^{85} +16.3923 q^{89} +8.05256 q^{91} -0.732051 q^{95} +4.00000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{5} + 2 q^{7}+O(q^{10}) $ 2 * q - 2 * q^5 + 2 * q^7	$ 2 q - 2 q^{5} + 2 q^{7} + 2 q^{11} - 10 q^{13} + 12 q^{17} - 2 q^{19} + 2 q^{23} + 2 q^{25} - 4 q^{29} - 2 q^{31} - 2 q^{35} - 2 q^{37} + 12 q^{41} + 2 q^{47} + 12 q^{49} + 4 q^{53} - 2 q^{55} + 4 q^{59} - 18 q^{61} + 10 q^{65} + 10 q^{67} + 4 q^{71} - 26 q^{73} + 38 q^{77} + 8 q^{79} - 4 q^{83} - 12 q^{85} + 12 q^{89} - 22 q^{91} + 2 q^{95} + 8 q^{97}+O(q^{100}) $ 2 * q - 2 * q^5 + 2 * q^7 + 2 * q^11 - 10 * q^13 + 12 * q^17 - 2 * q^19 + 2 * q^23 + 2 * q^25 - 4 * q^29 - 2 * q^31 - 2 * q^35 - 2 * q^37 + 12 * q^41 + 2 * q^47 + 12 * q^49 + 4 * q^53 - 2 * q^55 + 4 * q^59 - 18 * q^61 + 10 * q^65 + 10 * q^67 + 4 * q^71 - 26 * q^73 + 38 * q^77 + 8 * q^79 - 4 * q^83 - 12 * q^85 + 12 * q^89 - 22 * q^91 + 2 * q^95 + 8 * q^97

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−2.46410	−0.931343	−0.465671	−	0.884958i	$-0.654187\pi$
$7$	−2.46410	−0.931343	−0.465671	+	0.884958i	$0.654187\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−4.19615	−1.26519	−0.632594	−	0.774484i	$-0.718010\pi$
$11$	−4.19615	−1.26519	−0.632594	+	0.774484i	$0.718010\pi$
$12$	0	0
$13$	−3.26795	−0.906366	−0.453183	−	0.891417i	$-0.649712\pi$
$13$	−3.26795	−0.906366	−0.453183	+	0.891417i	$0.649712\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	7.73205	1.87530	0.937649	−	0.347584i	$-0.112998\pi$
$17$	7.73205	1.87530	0.937649	+	0.347584i	$0.112998\pi$
$18$	0	0
$19$	0.732051	0.167944	0.0839720	−	0.996468i	$-0.473239\pi$
$19$	0.732051	0.167944	0.0839720	+	0.996468i	$0.473239\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−7.19615	−1.33629	−0.668146	−	0.744030i	$-0.732912\pi$
$29$	−7.19615	−1.33629	−0.668146	+	0.744030i	$0.732912\pi$
$30$	0	0
$31$	−1.00000	−0.179605	−0.0898027	−	0.995960i	$-0.528624\pi$
$31$	−1.00000	−0.179605	−0.0898027	+	0.995960i	$0.528624\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	2.46410	0.416509
$36$	0	0
$37$	−11.3923	−1.87288	−0.936442	−	0.350823i	$-0.885902\pi$
$37$	−11.3923	−1.87288	−0.936442	+	0.350823i	$0.885902\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	7.73205	1.20754	0.603772	−	0.797157i	$-0.293664\pi$
$41$	7.73205	1.20754	0.603772	+	0.797157i	$0.293664\pi$
$42$	0	0
$43$	3.46410	0.528271	0.264135	−	0.964486i	$-0.414913\pi$
$43$	3.46410	0.528271	0.264135	+	0.964486i	$0.414913\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−0.732051	−0.106781	−0.0533903	−	0.998574i	$-0.517003\pi$
$47$	−0.732051	−0.106781	−0.0533903	+	0.998574i	$0.517003\pi$
$48$	0	0
$49$	−0.928203	−0.132600
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−6.66025	−0.914856	−0.457428	−	0.889247i	$-0.651229\pi$
$53$	−6.66025	−0.914856	−0.457428	+	0.889247i	$0.651229\pi$
$54$	0	0
$55$	4.19615	0.565809
$56$	0	0
$57$	0	0
$58$	0	0
$59$	7.19615	0.936859	0.468430	−	0.883501i	$-0.344820\pi$
$59$	7.19615	0.936859	0.468430	+	0.883501i	$0.344820\pi$
$60$	0	0
$61$	−10.7321	−1.37410	−0.687049	−	0.726611i	$-0.741094\pi$
$61$	−10.7321	−1.37410	−0.687049	+	0.726611i	$0.741094\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	3.26795	0.405339
$66$	0	0
$67$	5.00000	0.610847	0.305424	−	0.952217i	$-0.401202\pi$
$67$	5.00000	0.610847	0.305424	+	0.952217i	$0.401202\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	14.1244	1.67625	0.838126	−	0.545476i	$-0.183651\pi$
$71$	14.1244	1.67625	0.838126	+	0.545476i	$0.183651\pi$
$72$	0	0
$73$	−11.2679	−1.31881	−0.659407	−	0.751786i	$-0.729193\pi$
$73$	−11.2679	−1.31881	−0.659407	+	0.751786i	$0.729193\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	10.3397	1.17832
$78$	0	0
$79$	4.00000	0.450035	0.225018	−	0.974355i	$-0.427756\pi$
$79$	4.00000	0.450035	0.225018	+	0.974355i	$0.427756\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	6.66025	0.731058	0.365529	−	0.930800i	$-0.380888\pi$
$83$	6.66025	0.731058	0.365529	+	0.930800i	$0.380888\pi$
$84$	0	0
$85$	−7.73205	−0.838659
$86$	0	0
$87$	0	0
$88$	0	0
$89$	16.3923	1.73758	0.868790	−	0.495180i	$-0.164898\pi$
$89$	16.3923	1.73758	0.868790	+	0.495180i	$0.164898\pi$
$90$	0	0
$91$	8.05256	0.844138
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−0.732051	−0.0751068
$96$	0	0
$97$	4.00000	0.406138	0.203069	−	0.979164i	$-0.434908\pi$
$97$	4.00000	0.406138	0.203069	+	0.979164i	$0.434908\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	2.80385	0.278993	0.139497	−	0.990223i	$-0.455452\pi$
$101$	2.80385	0.278993	0.139497	+	0.990223i	$0.455452\pi$
$102$	0	0
$103$	16.9282	1.66799	0.833993	−	0.551775i	$-0.186049\pi$
$103$	16.9282	1.66799	0.833993	+	0.551775i	$0.186049\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−1.19615	−0.115636	−0.0578182	−	0.998327i	$-0.518414\pi$
$107$	−1.19615	−0.115636	−0.0578182	+	0.998327i	$0.518414\pi$
$108$	0	0
$109$	3.66025	0.350589	0.175294	−	0.984516i	$-0.443912\pi$
$109$	3.66025	0.350589	0.175294	+	0.984516i	$0.443912\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	15.1962	1.42953	0.714767	−	0.699363i	$-0.246533\pi$
$113$	15.1962	1.42953	0.714767	+	0.699363i	$0.246533\pi$
$114$	0	0
$115$	−1.00000	−0.0932505
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−19.0526	−1.74655
$120$	0	0
$121$	6.60770	0.600700
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	11.1244	0.987127	0.493563	−	0.869710i	$-0.335694\pi$
$127$	11.1244	0.987127	0.493563	+	0.869710i	$0.335694\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	21.8564	1.90960	0.954802	−	0.297244i	$-0.0960675\pi$
$131$	21.8564	1.90960	0.954802	+	0.297244i	$0.0960675\pi$
$132$	0	0
$133$	−1.80385	−0.156413
$134$	0	0
$135$	0	0
$136$	0	0
$137$	12.0000	1.02523	0.512615	−	0.858619i	$-0.328677\pi$
$137$	12.0000	1.02523	0.512615	+	0.858619i	$0.328677\pi$
$138$	0	0
$139$	7.00000	0.593732	0.296866	−	0.954919i	$-0.404058\pi$
$139$	7.00000	0.593732	0.296866	+	0.954919i	$0.404058\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	13.7128	1.14672
$144$	0	0
$145$	7.19615	0.597608
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−5.80385	−0.475470	−0.237735	−	0.971330i	$-0.576405\pi$
$149$	−5.80385	−0.475470	−0.237735	+	0.971330i	$0.576405\pi$
$150$	0	0
$151$	−2.39230	−0.194683	−0.0973415	−	0.995251i	$-0.531034\pi$
$151$	−2.39230	−0.194683	−0.0973415	+	0.995251i	$0.531034\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	1.00000	0.0803219
$156$	0	0
$157$	12.3205	0.983284	0.491642	−	0.870798i	$-0.336397\pi$
$157$	12.3205	0.983284	0.491642	+	0.870798i	$0.336397\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−2.46410	−0.194198
$162$	0	0
$163$	−3.85641	−0.302057	−0.151029	−	0.988529i	$-0.548259\pi$
$163$	−3.85641	−0.302057	−0.151029	+	0.988529i	$0.548259\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	5.80385	0.449115	0.224558	−	0.974461i	$-0.427906\pi$
$167$	5.80385	0.449115	0.224558	+	0.974461i	$0.427906\pi$
$168$	0	0
$169$	−2.32051	−0.178501
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−17.8564	−1.35760	−0.678799	−	0.734324i	$-0.737499\pi$
$173$	−17.8564	−1.35760	−0.678799	+	0.734324i	$0.737499\pi$
$174$	0	0
$175$	−2.46410	−0.186269
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−11.4641	−0.856867	−0.428434	−	0.903573i	$-0.640934\pi$
$179$	−11.4641	−0.856867	−0.428434	+	0.903573i	$0.640934\pi$
$180$	0	0
$181$	−12.3923	−0.921113	−0.460556	−	0.887630i	$-0.652350\pi$
$181$	−12.3923	−0.921113	−0.460556	+	0.887630i	$0.652350\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	11.3923	0.837579
$186$	0	0
$187$	−32.4449	−2.37260
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−5.80385	−0.419952	−0.209976	−	0.977707i	$-0.567339\pi$
$191$	−5.80385	−0.419952	−0.209976	+	0.977707i	$0.567339\pi$
$192$	0	0
$193$	24.3923	1.75580	0.877898	−	0.478847i	$-0.158945\pi$
$193$	24.3923	1.75580	0.877898	+	0.478847i	$0.158945\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	7.46410	0.531795	0.265898	−	0.964001i	$-0.414332\pi$
$197$	7.46410	0.531795	0.265898	+	0.964001i	$0.414332\pi$
$198$	0	0
$199$	−4.53590	−0.321541	−0.160771	−	0.986992i	$-0.551398\pi$
$199$	−4.53590	−0.321541	−0.160771	+	0.986992i	$0.551398\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	17.7321	1.24455
$204$	0	0
$205$	−7.73205	−0.540030
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−3.07180	−0.212481
$210$	0	0
$211$	−17.9282	−1.23423	−0.617114	−	0.786874i	$-0.711698\pi$
$211$	−17.9282	−1.23423	−0.617114	+	0.786874i	$0.711698\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−3.46410	−0.236250
$216$	0	0
$217$	2.46410	0.167274
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−25.2679	−1.69971
$222$	0	0
$223$	14.0000	0.937509	0.468755	−	0.883328i	$-0.344703\pi$
$223$	14.0000	0.937509	0.468755	+	0.883328i	$0.344703\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	22.9282	1.52180	0.760899	−	0.648870i	$-0.224758\pi$
$227$	22.9282	1.52180	0.760899	+	0.648870i	$0.224758\pi$
$228$	0	0
$229$	17.4641	1.15406	0.577030	−	0.816723i	$-0.304212\pi$
$229$	17.4641	1.15406	0.577030	+	0.816723i	$0.304212\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−20.7846	−1.36165	−0.680823	−	0.732448i	$-0.738378\pi$
$233$	−20.7846	−1.36165	−0.680823	+	0.732448i	$0.738378\pi$
$234$	0	0
$235$	0.732051	0.0477537
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−2.12436	−0.137413	−0.0687066	−	0.997637i	$-0.521887\pi$
$239$	−2.12436	−0.137413	−0.0687066	+	0.997637i	$0.521887\pi$
$240$	0	0
$241$	−14.5885	−0.939725	−0.469863	−	0.882740i	$-0.655697\pi$
$241$	−14.5885	−0.939725	−0.469863	+	0.882740i	$0.655697\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0.928203	0.0593007
$246$	0	0
$247$	−2.39230	−0.152219
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−23.8564	−1.50580	−0.752902	−	0.658133i	$-0.771346\pi$
$251$	−23.8564	−1.50580	−0.752902	+	0.658133i	$0.771346\pi$
$252$	0	0
$253$	−4.19615	−0.263810
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−9.12436	−0.569162	−0.284581	−	0.958652i	$-0.591854\pi$
$257$	−9.12436	−0.569162	−0.284581	+	0.958652i	$0.591854\pi$
$258$	0	0
$259$	28.0718	1.74430
$260$	0	0
$261$	0	0
$262$	0	0
$263$	11.3397	0.699239	0.349619	−	0.936892i	$-0.386311\pi$
$263$	11.3397	0.699239	0.349619	+	0.936892i	$0.386311\pi$
$264$	0	0
$265$	6.66025	0.409136
$266$	0	0
$267$	0	0
$268$	0	0
$269$	1.73205	0.105605	0.0528025	−	0.998605i	$-0.483185\pi$
$269$	1.73205	0.105605	0.0528025	+	0.998605i	$0.483185\pi$
$270$	0	0
$271$	23.3923	1.42098	0.710491	−	0.703707i	$-0.248473\pi$
$271$	23.3923	1.42098	0.710491	+	0.703707i	$0.248473\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−4.19615	−0.253038
$276$	0	0
$277$	−4.14359	−0.248964	−0.124482	−	0.992222i	$-0.539727\pi$
$277$	−4.14359	−0.248964	−0.124482	+	0.992222i	$0.539727\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−15.5167	−0.925646	−0.462823	−	0.886451i	$-0.653164\pi$
$281$	−15.5167	−0.925646	−0.462823	+	0.886451i	$0.653164\pi$
$282$	0	0
$283$	30.8564	1.83422	0.917111	−	0.398631i	$-0.130515\pi$
$283$	30.8564	1.83422	0.917111	+	0.398631i	$0.130515\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−19.0526	−1.12464
$288$	0	0
$289$	42.7846	2.51674
$290$	0	0
$291$	0	0
$292$	0	0
$293$	24.6603	1.44067	0.720334	−	0.693628i	$-0.243989\pi$
$293$	24.6603	1.44067	0.720334	+	0.693628i	$0.243989\pi$
$294$	0	0
$295$	−7.19615	−0.418976
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−3.26795	−0.188990
$300$	0	0
$301$	−8.53590	−0.492001
$302$	0	0
$303$	0	0
$304$	0	0
$305$	10.7321	0.614515
$306$	0	0
$307$	3.66025	0.208902	0.104451	−	0.994530i	$-0.466692\pi$
$307$	3.66025	0.208902	0.104451	+	0.994530i	$0.466692\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−22.9282	−1.30014	−0.650070	−	0.759875i	$-0.725260\pi$
$311$	−22.9282	−1.30014	−0.650070	+	0.759875i	$0.725260\pi$
$312$	0	0
$313$	22.3205	1.26163	0.630815	−	0.775933i	$-0.282721\pi$
$313$	22.3205	1.26163	0.630815	+	0.775933i	$0.282721\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	13.5167	0.759171	0.379586	−	0.925157i	$-0.376067\pi$
$317$	13.5167	0.759171	0.379586	+	0.925157i	$0.376067\pi$
$318$	0	0
$319$	30.1962	1.69066
$320$	0	0
$321$	0	0
$322$	0	0
$323$	5.66025	0.314945
$324$	0	0
$325$	−3.26795	−0.181273
$326$	0	0
$327$	0	0
$328$	0	0
$329$	1.80385	0.0994493
$330$	0	0
$331$	−27.2487	−1.49772	−0.748862	−	0.662726i	$-0.769400\pi$
$331$	−27.2487	−1.49772	−0.748862	+	0.662726i	$0.769400\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−5.00000	−0.273179
$336$	0	0
$337$	15.0718	0.821013	0.410507	−	0.911858i	$-0.365352\pi$
$337$	15.0718	0.821013	0.410507	+	0.911858i	$0.365352\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	4.19615	0.227234
$342$	0	0
$343$	19.5359	1.05484
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−7.60770	−0.408402	−0.204201	−	0.978929i	$-0.565460\pi$
$347$	−7.60770	−0.408402	−0.204201	+	0.978929i	$0.565460\pi$
$348$	0	0
$349$	0.464102	0.0248428	0.0124214	−	0.999923i	$-0.496046\pi$
$349$	0.464102	0.0248428	0.0124214	+	0.999923i	$0.496046\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−1.41154	−0.0751288	−0.0375644	−	0.999294i	$-0.511960\pi$
$353$	−1.41154	−0.0751288	−0.0375644	+	0.999294i	$0.511960\pi$
$354$	0	0
$355$	−14.1244	−0.749643
$356$	0	0
$357$	0	0
$358$	0	0
$359$	8.73205	0.460860	0.230430	−	0.973089i	$-0.425987\pi$
$359$	8.73205	0.460860	0.230430	+	0.973089i	$0.425987\pi$
$360$	0	0
$361$	−18.4641	−0.971795
$362$	0	0
$363$	0	0
$364$	0	0
$365$	11.2679	0.589791
$366$	0	0
$367$	−30.4641	−1.59021	−0.795107	−	0.606470i	$-0.792585\pi$
$367$	−30.4641	−1.59021	−0.795107	+	0.606470i	$0.792585\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	16.4115	0.852045
$372$	0	0
$373$	−9.46410	−0.490033	−0.245016	−	0.969519i	$-0.578793\pi$
$373$	−9.46410	−0.490033	−0.245016	+	0.969519i	$0.578793\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	23.5167	1.21117
$378$	0	0
$379$	9.85641	0.506290	0.253145	−	0.967428i	$-0.418535\pi$
$379$	9.85641	0.506290	0.253145	+	0.967428i	$0.418535\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−31.0526	−1.58671	−0.793356	−	0.608758i	$-0.791668\pi$
$383$	−31.0526	−1.58671	−0.793356	+	0.608758i	$0.791668\pi$
$384$	0	0
$385$	−10.3397	−0.526962
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−12.5359	−0.635595	−0.317798	−	0.948159i	$-0.602943\pi$
$389$	−12.5359	−0.635595	−0.317798	+	0.948159i	$0.602943\pi$
$390$	0	0
$391$	7.73205	0.391027
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−4.00000	−0.201262
$396$	0	0
$397$	32.0000	1.60603	0.803017	−	0.595956i	$-0.203227\pi$
$397$	32.0000	1.60603	0.803017	+	0.595956i	$0.203227\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−30.3923	−1.51772	−0.758860	−	0.651254i	$-0.774243\pi$
$401$	−30.3923	−1.51772	−0.758860	+	0.651254i	$0.774243\pi$
$402$	0	0
$403$	3.26795	0.162788
$404$	0	0
$405$	0	0
$406$	0	0
$407$	47.8038	2.36955
$408$	0	0
$409$	−32.1769	−1.59105	−0.795523	−	0.605923i	$-0.792804\pi$
$409$	−32.1769	−1.59105	−0.795523	+	0.605923i	$0.792804\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−17.7321	−0.872537
$414$	0	0
$415$	−6.66025	−0.326939
$416$	0	0
$417$	0	0
$418$	0	0
$419$	3.12436	0.152635	0.0763174	−	0.997084i	$-0.475684\pi$
$419$	3.12436	0.152635	0.0763174	+	0.997084i	$0.475684\pi$
$420$	0	0
$421$	13.5167	0.658762	0.329381	−	0.944197i	$-0.393160\pi$
$421$	13.5167	0.658762	0.329381	+	0.944197i	$0.393160\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	7.73205	0.375060
$426$	0	0
$427$	26.4449	1.27976
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−3.60770	−0.173777	−0.0868883	−	0.996218i	$-0.527692\pi$
$431$	−3.60770	−0.173777	−0.0868883	+	0.996218i	$0.527692\pi$
$432$	0	0
$433$	−29.3923	−1.41250	−0.706252	−	0.707961i	$-0.749615\pi$
$433$	−29.3923	−1.41250	−0.706252	+	0.707961i	$0.749615\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0.732051	0.0350187
$438$	0	0
$439$	18.7846	0.896541	0.448270	−	0.893898i	$-0.352040\pi$
$439$	18.7846	0.896541	0.448270	+	0.893898i	$0.352040\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	1.80385	0.0857034	0.0428517	−	0.999081i	$-0.486356\pi$
$443$	1.80385	0.0857034	0.0428517	+	0.999081i	$0.486356\pi$
$444$	0	0
$445$	−16.3923	−0.777070
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−4.26795	−0.201417	−0.100708	−	0.994916i	$-0.532111\pi$
$449$	−4.26795	−0.201417	−0.100708	+	0.994916i	$0.532111\pi$
$450$	0	0
$451$	−32.4449	−1.52777
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−8.05256	−0.377510
$456$	0	0
$457$	−26.4641	−1.23794	−0.618969	−	0.785415i	$-0.712449\pi$
$457$	−26.4641	−1.23794	−0.618969	+	0.785415i	$0.712449\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	29.3205	1.36559	0.682796	−	0.730609i	$-0.260764\pi$
$461$	29.3205	1.36559	0.682796	+	0.730609i	$0.260764\pi$
$462$	0	0
$463$	−10.5885	−0.492087	−0.246044	−	0.969259i	$-0.579131\pi$
$463$	−10.5885	−0.492087	−0.246044	+	0.969259i	$0.579131\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	29.4449	1.36255	0.681273	−	0.732030i	$-0.261427\pi$
$467$	29.4449	1.36255	0.681273	+	0.732030i	$0.261427\pi$
$468$	0	0
$469$	−12.3205	−0.568908
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−14.5359	−0.668361
$474$	0	0
$475$	0.732051	0.0335888
$476$	0	0
$477$	0	0
$478$	0	0
$479$	34.9808	1.59831	0.799156	−	0.601124i	$-0.205280\pi$
$479$	34.9808	1.59831	0.799156	+	0.601124i	$0.205280\pi$
$480$	0	0
$481$	37.2295	1.69752
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−4.00000	−0.181631
$486$	0	0
$487$	−2.73205	−0.123801	−0.0619005	−	0.998082i	$-0.519716\pi$
$487$	−2.73205	−0.123801	−0.0619005	+	0.998082i	$0.519716\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−19.0526	−0.859830	−0.429915	−	0.902869i	$-0.641456\pi$
$491$	−19.0526	−0.859830	−0.429915	+	0.902869i	$0.641456\pi$
$492$	0	0
$493$	−55.6410	−2.50595
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−34.8038	−1.56117
$498$	0	0
$499$	−28.4641	−1.27423	−0.637114	−	0.770770i	$-0.719872\pi$
$499$	−28.4641	−1.27423	−0.637114	+	0.770770i	$0.719872\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−17.1962	−0.766739	−0.383369	−	0.923595i	$-0.625236\pi$
$503$	−17.1962	−0.766739	−0.383369	+	0.923595i	$0.625236\pi$
$504$	0	0
$505$	−2.80385	−0.124770
$506$	0	0
$507$	0	0
$508$	0	0
$509$	32.7846	1.45315	0.726576	−	0.687086i	$-0.241110\pi$
$509$	32.7846	1.45315	0.726576	+	0.687086i	$0.241110\pi$
$510$	0	0
$511$	27.7654	1.22827
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−16.9282	−0.745946
$516$	0	0
$517$	3.07180	0.135097
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−22.0526	−0.966140	−0.483070	−	0.875582i	$-0.660478\pi$
$521$	−22.0526	−0.966140	−0.483070	+	0.875582i	$0.660478\pi$
$522$	0	0
$523$	36.7846	1.60848	0.804239	−	0.594306i	$-0.202573\pi$
$523$	36.7846	1.60848	0.804239	+	0.594306i	$0.202573\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−7.73205	−0.336813
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−25.2679	−1.09448
$534$	0	0
$535$	1.19615	0.0517142
$536$	0	0
$537$	0	0
$538$	0	0
$539$	3.89488	0.167764
$540$	0	0
$541$	−14.9282	−0.641814	−0.320907	−	0.947111i	$-0.603988\pi$
$541$	−14.9282	−0.641814	−0.320907	+	0.947111i	$0.603988\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−3.66025	−0.156788
$546$	0	0
$547$	−2.92820	−0.125201	−0.0626005	−	0.998039i	$-0.519939\pi$
$547$	−2.92820	−0.125201	−0.0626005	+	0.998039i	$0.519939\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−5.26795	−0.224422
$552$	0	0
$553$	−9.85641	−0.419137
$554$	0	0
$555$	0	0
$556$	0	0
$557$	15.9808	0.677127	0.338563	−	0.940944i	$-0.390059\pi$
$557$	15.9808	0.677127	0.338563	+	0.940944i	$0.390059\pi$
$558$	0	0
$559$	−11.3205	−0.478806
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−15.5885	−0.656975	−0.328488	−	0.944508i	$-0.606539\pi$
$563$	−15.5885	−0.656975	−0.328488	+	0.944508i	$0.606539\pi$
$564$	0	0
$565$	−15.1962	−0.639307
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−0.535898	−0.0224660	−0.0112330	−	0.999937i	$-0.503576\pi$
$569$	−0.535898	−0.0224660	−0.0112330	+	0.999937i	$0.503576\pi$
$570$	0	0
$571$	−24.5885	−1.02899	−0.514497	−	0.857492i	$-0.672022\pi$
$571$	−24.5885	−1.02899	−0.514497	+	0.857492i	$0.672022\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	1.00000	0.0417029
$576$	0	0
$577$	16.7846	0.698752	0.349376	−	0.936983i	$-0.386394\pi$
$577$	16.7846	0.698752	0.349376	+	0.936983i	$0.386394\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−16.4115	−0.680866
$582$	0	0
$583$	27.9474	1.15746
$584$	0	0
$585$	0	0
$586$	0	0
$587$	24.9282	1.02890	0.514449	−	0.857521i	$-0.327997\pi$
$587$	24.9282	1.02890	0.514449	+	0.857521i	$0.327997\pi$
$588$	0	0
$589$	−0.732051	−0.0301636
$590$	0	0
$591$	0	0
$592$	0	0
$593$	31.2679	1.28402	0.642010	−	0.766696i	$-0.278101\pi$
$593$	31.2679	1.28402	0.642010	+	0.766696i	$0.278101\pi$
$594$	0	0
$595$	19.0526	0.781079
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−3.60770	−0.147406	−0.0737032	−	0.997280i	$-0.523482\pi$
$599$	−3.60770	−0.147406	−0.0737032	+	0.997280i	$0.523482\pi$
$600$	0	0
$601$	−6.60770	−0.269534	−0.134767	−	0.990877i	$-0.543029\pi$
$601$	−6.60770	−0.269534	−0.134767	+	0.990877i	$0.543029\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−6.60770	−0.268641
$606$	0	0
$607$	−28.9808	−1.17629	−0.588146	−	0.808754i	$-0.700142\pi$
$607$	−28.9808	−1.17629	−0.588146	+	0.808754i	$0.700142\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	2.39230	0.0967823
$612$	0	0
$613$	−41.7128	−1.68476	−0.842382	−	0.538880i	$-0.818847\pi$
$613$	−41.7128	−1.68476	−0.842382	+	0.538880i	$0.818847\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	11.8756	0.478095	0.239048	−	0.971008i	$-0.423165\pi$
$617$	11.8756	0.478095	0.239048	+	0.971008i	$0.423165\pi$
$618$	0	0
$619$	10.9282	0.439242	0.219621	−	0.975585i	$-0.429518\pi$
$619$	10.9282	0.439242	0.219621	+	0.975585i	$0.429518\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−40.3923	−1.61828
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−88.0859	−3.51221
$630$	0	0
$631$	34.8372	1.38685	0.693423	−	0.720531i	$-0.256102\pi$
$631$	34.8372	1.38685	0.693423	+	0.720531i	$0.256102\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−11.1244	−0.441457
$636$	0	0
$637$	3.03332	0.120185
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−10.0526	−0.397052	−0.198526	−	0.980096i	$-0.563615\pi$
$641$	−10.0526	−0.397052	−0.198526	+	0.980096i	$0.563615\pi$
$642$	0	0
$643$	−1.53590	−0.0605699	−0.0302850	−	0.999541i	$-0.509641\pi$
$643$	−1.53590	−0.0605699	−0.0302850	+	0.999541i	$0.509641\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	26.7321	1.05095	0.525473	−	0.850810i	$-0.323889\pi$
$647$	26.7321	1.05095	0.525473	+	0.850810i	$0.323889\pi$
$648$	0	0
$649$	−30.1962	−1.18530
$650$	0	0
$651$	0	0
$652$	0	0
$653$	33.3731	1.30599	0.652995	−	0.757363i	$-0.273512\pi$
$653$	33.3731	1.30599	0.652995	+	0.757363i	$0.273512\pi$
$654$	0	0
$655$	−21.8564	−0.854000
$656$	0	0
$657$	0	0
$658$	0	0
$659$	50.4449	1.96505	0.982526	−	0.186123i	$-0.0595923\pi$
$659$	50.4449	1.96505	0.982526	+	0.186123i	$0.0595923\pi$
$660$	0	0
$661$	28.5359	1.10992	0.554959	−	0.831878i	$-0.312734\pi$
$661$	28.5359	1.10992	0.554959	+	0.831878i	$0.312734\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	1.80385	0.0699502
$666$	0	0
$667$	−7.19615	−0.278636
$668$	0	0
$669$	0	0
$670$	0	0
$671$	45.0333	1.73849
$672$	0	0
$673$	−27.9090	−1.07581	−0.537906	−	0.843005i	$-0.680784\pi$
$673$	−27.9090	−1.07581	−0.537906	+	0.843005i	$0.680784\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	20.8038	0.799557	0.399778	−	0.916612i	$-0.369087\pi$
$677$	20.8038	0.799557	0.399778	+	0.916612i	$0.369087\pi$
$678$	0	0
$679$	−9.85641	−0.378254
$680$	0	0
$681$	0	0
$682$	0	0
$683$	7.26795	0.278100	0.139050	−	0.990285i	$-0.455595\pi$
$683$	7.26795	0.278100	0.139050	+	0.990285i	$0.455595\pi$
$684$	0	0
$685$	−12.0000	−0.458496
$686$	0	0
$687$	0	0
$688$	0	0
$689$	21.7654	0.829195
$690$	0	0
$691$	−0.535898	−0.0203865	−0.0101933	−	0.999948i	$-0.503245\pi$
$691$	−0.535898	−0.0203865	−0.0101933	+	0.999948i	$0.503245\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−7.00000	−0.265525
$696$	0	0
$697$	59.7846	2.26450
$698$	0	0
$699$	0	0
$700$	0	0
$701$	18.5885	0.702076	0.351038	−	0.936361i	$-0.385829\pi$
$701$	18.5885	0.702076	0.351038	+	0.936361i	$0.385829\pi$
$702$	0	0
$703$	−8.33975	−0.314539
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−6.90897	−0.259838
$708$	0	0
$709$	31.1244	1.16890	0.584450	−	0.811430i	$-0.301310\pi$
$709$	31.1244	1.16890	0.584450	+	0.811430i	$0.301310\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−1.00000	−0.0374503
$714$	0	0
$715$	−13.7128	−0.512830
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−14.2679	−0.532105	−0.266052	−	0.963959i	$-0.585719\pi$
$719$	−14.2679	−0.532105	−0.266052	+	0.963959i	$0.585719\pi$
$720$	0	0
$721$	−41.7128	−1.55347
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−7.19615	−0.267258
$726$	0	0
$727$	29.0000	1.07555	0.537775	−	0.843088i	$-0.319265\pi$
$727$	29.0000	1.07555	0.537775	+	0.843088i	$0.319265\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	26.7846	0.990665
$732$	0	0
$733$	15.6410	0.577714	0.288857	−	0.957372i	$-0.406725\pi$
$733$	15.6410	0.577714	0.288857	+	0.957372i	$0.406725\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−20.9808	−0.772836
$738$	0	0
$739$	−1.67949	−0.0617811	−0.0308906	−	0.999523i	$-0.509834\pi$
$739$	−1.67949	−0.0617811	−0.0308906	+	0.999523i	$0.509834\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	17.8564	0.655088	0.327544	−	0.944836i	$-0.393779\pi$
$743$	17.8564	0.655088	0.327544	+	0.944836i	$0.393779\pi$
$744$	0	0
$745$	5.80385	0.212637
$746$	0	0
$747$	0	0
$748$	0	0
$749$	2.94744	0.107697
$750$	0	0
$751$	−49.1244	−1.79257	−0.896287	−	0.443475i	$-0.853745\pi$
$751$	−49.1244	−1.79257	−0.896287	+	0.443475i	$0.853745\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	2.39230	0.0870649
$756$	0	0
$757$	23.1436	0.841168	0.420584	−	0.907254i	$-0.361825\pi$
$757$	23.1436	0.841168	0.420584	+	0.907254i	$0.361825\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−5.19615	−0.188360	−0.0941802	−	0.995555i	$-0.530023\pi$
$761$	−5.19615	−0.188360	−0.0941802	+	0.995555i	$0.530023\pi$
$762$	0	0
$763$	−9.01924	−0.326518
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−23.5167	−0.849137
$768$	0	0
$769$	−9.66025	−0.348358	−0.174179	−	0.984714i	$-0.555727\pi$
$769$	−9.66025	−0.348358	−0.174179	+	0.984714i	$0.555727\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−45.0333	−1.61974	−0.809868	−	0.586612i	$-0.800461\pi$
$773$	−45.0333	−1.61974	−0.809868	+	0.586612i	$0.800461\pi$
$774$	0	0
$775$	−1.00000	−0.0359211
$776$	0	0
$777$	0	0
$778$	0	0
$779$	5.66025	0.202800
$780$	0	0
$781$	−59.2679	−2.12077
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−12.3205	−0.439738
$786$	0	0
$787$	45.3923	1.61806	0.809030	−	0.587767i	$-0.199993\pi$
$787$	45.3923	1.61806	0.809030	+	0.587767i	$0.199993\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−37.4449	−1.33139
$792$	0	0
$793$	35.0718	1.24544
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−4.26795	−0.151179	−0.0755893	−	0.997139i	$-0.524084\pi$
$797$	−4.26795	−0.151179	−0.0755893	+	0.997139i	$0.524084\pi$
$798$	0	0
$799$	−5.66025	−0.200245
$800$	0	0
$801$	0	0
$802$	0	0
$803$	47.2820	1.66855
$804$	0	0
$805$	2.46410	0.0868482
$806$	0	0
$807$	0	0
$808$	0	0
$809$	23.1962	0.815533	0.407767	−	0.913086i	$-0.366308\pi$
$809$	23.1962	0.815533	0.407767	+	0.913086i	$0.366308\pi$
$810$	0	0
$811$	−33.9282	−1.19138	−0.595690	−	0.803214i	$-0.703121\pi$
$811$	−33.9282	−1.19138	−0.595690	+	0.803214i	$0.703121\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	3.85641	0.135084
$816$	0	0
$817$	2.53590	0.0887199
$818$	0	0
$819$	0	0
$820$	0	0
$821$	1.85641	0.0647890	0.0323945	−	0.999475i	$-0.489687\pi$
$821$	1.85641	0.0647890	0.0323945	+	0.999475i	$0.489687\pi$
$822$	0	0
$823$	20.7846	0.724506	0.362253	−	0.932080i	$-0.382008\pi$
$823$	20.7846	0.724506	0.362253	+	0.932080i	$0.382008\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	40.5167	1.40890	0.704451	−	0.709752i	$-0.251193\pi$
$827$	40.5167	1.40890	0.704451	+	0.709752i	$0.251193\pi$
$828$	0	0
$829$	55.4974	1.92751	0.963753	−	0.266798i	$-0.0859655\pi$
$829$	55.4974	1.92751	0.963753	+	0.266798i	$0.0859655\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−7.17691	−0.248665
$834$	0	0
$835$	−5.80385	−0.200850
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−42.0000	−1.45000	−0.725001	−	0.688748i	$-0.758161\pi$
$839$	−42.0000	−1.45000	−0.725001	+	0.688748i	$0.758161\pi$
$840$	0	0
$841$	22.7846	0.785676
$842$	0	0
$843$	0	0
$844$	0	0
$845$	2.32051	0.0798279
$846$	0	0
$847$	−16.2820	−0.559457
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−11.3923	−0.390523
$852$	0	0
$853$	−31.4641	−1.07731	−0.538655	−	0.842526i	$-0.681067\pi$
$853$	−31.4641	−1.07731	−0.538655	+	0.842526i	$0.681067\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−33.7128	−1.15161	−0.575804	−	0.817588i	$-0.695311\pi$
$857$	−33.7128	−1.15161	−0.575804	+	0.817588i	$0.695311\pi$
$858$	0	0
$859$	−19.0000	−0.648272	−0.324136	−	0.946011i	$-0.605073\pi$
$859$	−19.0000	−0.648272	−0.324136	+	0.946011i	$0.605073\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−51.9615	−1.76879	−0.884395	−	0.466738i	$-0.845429\pi$
$863$	−51.9615	−1.76879	−0.884395	+	0.466738i	$0.845429\pi$
$864$	0	0
$865$	17.8564	0.607136
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−16.7846	−0.569379
$870$	0	0
$871$	−16.3397	−0.553651
$872$	0	0
$873$	0	0
$874$	0	0
$875$	2.46410	0.0833018
$876$	0	0
$877$	−7.46410	−0.252045	−0.126022	−	0.992027i	$-0.540221\pi$
$877$	−7.46410	−0.252045	−0.126022	+	0.992027i	$0.540221\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−7.94744	−0.267756	−0.133878	−	0.990998i	$-0.542743\pi$
$881$	−7.94744	−0.267756	−0.133878	+	0.990998i	$0.542743\pi$
$882$	0	0
$883$	−45.6603	−1.53659	−0.768295	−	0.640096i	$-0.778895\pi$
$883$	−45.6603	−1.53659	−0.768295	+	0.640096i	$0.778895\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	17.3205	0.581566	0.290783	−	0.956789i	$-0.406084\pi$
$887$	17.3205	0.581566	0.290783	+	0.956789i	$0.406084\pi$
$888$	0	0
$889$	−27.4115	−0.919354
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−0.535898	−0.0179332
$894$	0	0
$895$	11.4641	0.383203
$896$	0	0
$897$	0	0
$898$	0	0
$899$	7.19615	0.240005
$900$	0	0
$901$	−51.4974	−1.71563
$902$	0	0
$903$	0	0
$904$	0	0
$905$	12.3923	0.411934
$906$	0	0
$907$	19.9282	0.661705	0.330853	−	0.943682i	$-0.392664\pi$
$907$	19.9282	0.661705	0.330853	+	0.943682i	$0.392664\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	24.0000	0.795155	0.397578	−	0.917568i	$-0.369851\pi$
$911$	24.0000	0.795155	0.397578	+	0.917568i	$0.369851\pi$
$912$	0	0
$913$	−27.9474	−0.924925
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−53.8564	−1.77850
$918$	0	0
$919$	−22.9282	−0.756332	−0.378166	−	0.925738i	$-0.623445\pi$
$919$	−22.9282	−0.756332	−0.378166	+	0.925738i	$0.623445\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−46.1577	−1.51930
$924$	0	0
$925$	−11.3923	−0.374577
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−10.5167	−0.345040	−0.172520	−	0.985006i	$-0.555191\pi$
$929$	−10.5167	−0.345040	−0.172520	+	0.985006i	$0.555191\pi$
$930$	0	0
$931$	−0.679492	−0.0222694
$932$	0	0
$933$	0	0
$934$	0	0
$935$	32.4449	1.06106
$936$	0	0
$937$	5.07180	0.165688	0.0828442	−	0.996563i	$-0.473600\pi$
$937$	5.07180	0.165688	0.0828442	+	0.996563i	$0.473600\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−22.1962	−0.723574	−0.361787	−	0.932261i	$-0.617833\pi$
$941$	−22.1962	−0.723574	−0.361787	+	0.932261i	$0.617833\pi$
$942$	0	0
$943$	7.73205	0.251790
$944$	0	0
$945$	0	0
$946$	0	0
$947$	8.67949	0.282046	0.141023	−	0.990006i	$-0.454961\pi$
$947$	8.67949	0.282046	0.141023	+	0.990006i	$0.454961\pi$
$948$	0	0
$949$	36.8231	1.19533
$950$	0	0
$951$	0	0
$952$	0	0
$953$	25.7128	0.832920	0.416460	−	0.909154i	$-0.363271\pi$
$953$	25.7128	0.832920	0.416460	+	0.909154i	$0.363271\pi$
$954$	0	0
$955$	5.80385	0.187808
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−29.5692	−0.954840
$960$	0	0
$961$	−30.0000	−0.967742
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−24.3923	−0.785216
$966$	0	0
$967$	−16.8756	−0.542684	−0.271342	−	0.962483i	$-0.587467\pi$
$967$	−16.8756	−0.542684	−0.271342	+	0.962483i	$0.587467\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	47.3205	1.51859	0.759294	−	0.650748i	$-0.225545\pi$
$971$	47.3205	1.51859	0.759294	+	0.650748i	$0.225545\pi$
$972$	0	0
$973$	−17.2487	−0.552968
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−9.87564	−0.315950	−0.157975	−	0.987443i	$-0.550497\pi$
$977$	−9.87564	−0.315950	−0.157975	+	0.987443i	$0.550497\pi$
$978$	0	0
$979$	−68.7846	−2.19837
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−11.1962	−0.357102	−0.178551	−	0.983931i	$-0.557141\pi$
$983$	−11.1962	−0.357102	−0.178551	+	0.983931i	$0.557141\pi$
$984$	0	0
$985$	−7.46410	−0.237826
$986$	0	0
$987$	0	0
$988$	0	0
$989$	3.46410	0.110152
$990$	0	0
$991$	28.7128	0.912093	0.456046	−	0.889956i	$-0.349265\pi$
$991$	28.7128	0.912093	0.456046	+	0.889956i	$0.349265\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	4.53590	0.143798
$996$	0	0
$997$	−53.4641	−1.69323	−0.846613	−	0.532210i	$-0.821362\pi$
$997$	−53.4641	−1.69323	−0.846613	+	0.532210i	$0.821362\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4140.2.a.o.1.1		2
3.2	odd	2		1380.2.a.h.1.1	✓	2
12.11	even	2		5520.2.a.bp.1.2		2
15.2	even	4		6900.2.f.i.6349.3		4
15.8	even	4		6900.2.f.i.6349.2		4
15.14	odd	2		6900.2.a.s.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1380.2.a.h.1.1	✓	2	3.2	odd	2
4140.2.a.o.1.1		2	1.1	even	1	trivial
5520.2.a.bp.1.2		2	12.11	even	2
6900.2.a.s.1.2		2	15.14	odd	2
6900.2.f.i.6349.2		4	15.8	even	4
6900.2.f.i.6349.3		4	15.2	even	4

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:	\( N \)	\(=\)	\( 4140 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4140.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{5} -2.46410 q^{7} +O(q^{10})\)	\(q-1.00000 q^{5} -2.46410 q^{7} -4.19615 q^{11} -3.26795 q^{13} +7.73205 q^{17} +0.732051 q^{19} +1.00000 q^{23} +1.00000 q^{25} -7.19615 q^{29} -1.00000 q^{31} +2.46410 q^{35} -11.3923 q^{37} +7.73205 q^{41} +3.46410 q^{43} -0.732051 q^{47} -0.928203 q^{49} -6.66025 q^{53} +4.19615 q^{55} +7.19615 q^{59} -10.7321 q^{61} +3.26795 q^{65} +5.00000 q^{67} +14.1244 q^{71} -11.2679 q^{73} +10.3397 q^{77} +4.00000 q^{79} +6.66025 q^{83} -7.73205 q^{85} +16.3923 q^{89} +8.05256 q^{91} -0.732051 q^{95} +4.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{5} + 2 q^{7}+O(q^{10}) \) 2 * q - 2 * q^5 + 2 * q^7	\( 2 q - 2 q^{5} + 2 q^{7} + 2 q^{11} - 10 q^{13} + 12 q^{17} - 2 q^{19} + 2 q^{23} + 2 q^{25} - 4 q^{29} - 2 q^{31} - 2 q^{35} - 2 q^{37} + 12 q^{41} + 2 q^{47} + 12 q^{49} + 4 q^{53} - 2 q^{55} + 4 q^{59} - 18 q^{61} + 10 q^{65} + 10 q^{67} + 4 q^{71} - 26 q^{73} + 38 q^{77} + 8 q^{79} - 4 q^{83} - 12 q^{85} + 12 q^{89} - 22 q^{91} + 2 q^{95} + 8 q^{97}+O(q^{100}) \) 2 * q - 2 * q^5 + 2 * q^7 + 2 * q^11 - 10 * q^13 + 12 * q^17 - 2 * q^19 + 2 * q^23 + 2 * q^25 - 4 * q^29 - 2 * q^31 - 2 * q^35 - 2 * q^37 + 12 * q^41 + 2 * q^47 + 12 * q^49 + 4 * q^53 - 2 * q^55 + 4 * q^59 - 18 * q^61 + 10 * q^65 + 10 * q^67 + 4 * q^71 - 26 * q^73 + 38 * q^77 + 8 * q^79 - 4 * q^83 - 12 * q^85 + 12 * q^89 - 22 * q^91 + 2 * q^95 + 8 * q^97

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