LMFDB - Embedded newform 5292.2.a.v.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5292,2,Mod(1,5292)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5292, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("5292.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5292 = 2^{2} \cdot 3^{3} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5292.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:	$42.2568327497$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.321.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 4x + 1 $ x^3 - x^2 - 4*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 3 $
Twist minimal:	no (minimal twist has level 756)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.46050$ of defining polynomial
Character	$\chi$	$=$	5292.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.866926 q^{5} +O(q^{10})$	$q+0.866926 q^{5} -3.51459 q^{11} +1.86693 q^{13} +6.51459 q^{17} +5.38151 q^{19} +8.64766 q^{23} -4.24844 q^{25} +3.51459 q^{29} +1.86693 q^{31} -2.78074 q^{37} -10.3815 q^{41} -5.78074 q^{43} -6.16225 q^{47} +5.60078 q^{53} -3.04689 q^{55} +5.64766 q^{59} -10.2953 q^{61} +1.61849 q^{65} -1.35234 q^{67} +2.08619 q^{71} -7.24844 q^{73} +11.6768 q^{79} -6.86693 q^{83} +5.64766 q^{85} +6.56148 q^{89} +4.66537 q^{95} +3.29533 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - q^{5}+O(q^{10}) $ 3 * q - q^5	$ 3 q - q^{5} + 5 q^{11} + 2 q^{13} + 4 q^{17} - 3 q^{19} + 14 q^{23} + 10 q^{25} - 5 q^{29} + 2 q^{31} - 12 q^{41} - 9 q^{43} + 9 q^{47} + 6 q^{53} - 8 q^{55} + 5 q^{59} - 7 q^{61} + 24 q^{65} - 16 q^{67} + 11 q^{71} + q^{73} - 8 q^{79} - 17 q^{83} + 5 q^{85} + 3 q^{89} + 32 q^{95} - 14 q^{97}+O(q^{100}) $ 3 * q - q^5 + 5 * q^11 + 2 * q^13 + 4 * q^17 - 3 * q^19 + 14 * q^23 + 10 * q^25 - 5 * q^29 + 2 * q^31 - 12 * q^41 - 9 * q^43 + 9 * q^47 + 6 * q^53 - 8 * q^55 + 5 * q^59 - 7 * q^61 + 24 * q^65 - 16 * q^67 + 11 * q^71 + q^73 - 8 * q^79 - 17 * q^83 + 5 * q^85 + 3 * q^89 + 32 * q^95 - 14 * 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
$5$	0.866926	0.387701	0.193850	−	0.981031i	$-0.437902\pi$
$5$	0.866926	0.387701	0.193850	+	0.981031i	$0.437902\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−3.51459	−1.05969	−0.529844	−	0.848095i	$-0.677750\pi$
$11$	−3.51459	−1.05969	−0.529844	+	0.848095i	$0.677750\pi$
$12$	0	0
$13$	1.86693	0.517792	0.258896	−	0.965905i	$-0.416641\pi$
$13$	1.86693	0.517792	0.258896	+	0.965905i	$0.416641\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	6.51459	1.58002	0.790010	−	0.613094i	$-0.210075\pi$
$17$	6.51459	1.58002	0.790010	+	0.613094i	$0.210075\pi$
$18$	0	0
$19$	5.38151	1.23460	0.617302	−	0.786726i	$-0.288226\pi$
$19$	5.38151	1.23460	0.617302	+	0.786726i	$0.288226\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	8.64766	1.80316	0.901581	−	0.432610i	$-0.142407\pi$
$23$	8.64766	1.80316	0.901581	+	0.432610i	$0.142407\pi$
$24$	0	0
$25$	−4.24844	−0.849688
$26$	0	0
$27$	0	0
$28$	0	0
$29$	3.51459	0.652643	0.326321	−	0.945259i	$-0.394191\pi$
$29$	3.51459	0.652643	0.326321	+	0.945259i	$0.394191\pi$
$30$	0	0
$31$	1.86693	0.335310	0.167655	−	0.985846i	$-0.446381\pi$
$31$	1.86693	0.335310	0.167655	+	0.985846i	$0.446381\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−2.78074	−0.457151	−0.228575	−	0.973526i	$-0.573407\pi$
$37$	−2.78074	−0.457151	−0.228575	+	0.973526i	$0.573407\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−10.3815	−1.62132	−0.810660	−	0.585517i	$-0.800892\pi$
$41$	−10.3815	−1.62132	−0.810660	+	0.585517i	$0.800892\pi$
$42$	0	0
$43$	−5.78074	−0.881554	−0.440777	−	0.897617i	$-0.645297\pi$
$43$	−5.78074	−0.881554	−0.440777	+	0.897617i	$0.645297\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−6.16225	−0.898857	−0.449428	−	0.893316i	$-0.648372\pi$
$47$	−6.16225	−0.898857	−0.449428	+	0.893316i	$0.648372\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	0	0
$52$	0	0
$53$	5.60078	0.769326	0.384663	−	0.923057i	$-0.374318\pi$
$53$	5.60078	0.769326	0.384663	+	0.923057i	$0.374318\pi$
$54$	0	0
$55$	−3.04689	−0.410842
$56$	0	0
$57$	0	0
$58$	0	0
$59$	5.64766	0.735263	0.367632	−	0.929972i	$-0.380169\pi$
$59$	5.64766	0.735263	0.367632	+	0.929972i	$0.380169\pi$
$60$	0	0
$61$	−10.2953	−1.31818	−0.659091	−	0.752063i	$-0.729059\pi$
$61$	−10.2953	−1.31818	−0.659091	+	0.752063i	$0.729059\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	1.61849	0.200748
$66$	0	0
$67$	−1.35234	−0.165214	−0.0826071	−	0.996582i	$-0.526325\pi$
$67$	−1.35234	−0.165214	−0.0826071	+	0.996582i	$0.526325\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	2.08619	0.247585	0.123792	−	0.992308i	$-0.460494\pi$
$71$	2.08619	0.247585	0.123792	+	0.992308i	$0.460494\pi$
$72$	0	0
$73$	−7.24844	−0.848366	−0.424183	−	0.905577i	$-0.639439\pi$
$73$	−7.24844	−0.848366	−0.424183	+	0.905577i	$0.639439\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	11.6768	1.31375	0.656874	−	0.754001i	$-0.271878\pi$
$79$	11.6768	1.31375	0.656874	+	0.754001i	$0.271878\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−6.86693	−0.753743	−0.376871	−	0.926266i	$-0.623000\pi$
$83$	−6.86693	−0.753743	−0.376871	+	0.926266i	$0.623000\pi$
$84$	0	0
$85$	5.64766	0.612575
$86$	0	0
$87$	0	0
$88$	0	0
$89$	6.56148	0.695515	0.347758	−	0.937585i	$-0.386943\pi$
$89$	6.56148	0.695515	0.347758	+	0.937585i	$0.386943\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	4.66537	0.478657
$96$	0	0
$97$	3.29533	0.334590	0.167295	−	0.985907i	$-0.446497\pi$
$97$	3.29533	0.334590	0.167295	+	0.985907i	$0.446497\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−1.89610	−0.188669	−0.0943347	−	0.995541i	$-0.530072\pi$
$101$	−1.89610	−0.188669	−0.0943347	+	0.995541i	$0.530072\pi$
$102$	0	0
$103$	7.00000	0.689730	0.344865	−	0.938652i	$-0.387925\pi$
$103$	7.00000	0.689730	0.344865	+	0.938652i	$0.387925\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	19.3815	1.87368	0.936841	−	0.349756i	$-0.113735\pi$
$107$	19.3815	1.87368	0.936841	+	0.349756i	$0.113735\pi$
$108$	0	0
$109$	9.02918	0.864838	0.432419	−	0.901673i	$-0.357660\pi$
$109$	9.02918	0.864838	0.432419	+	0.901673i	$0.357660\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−0.467702	−0.0439977	−0.0219989	−	0.999758i	$-0.507003\pi$
$113$	−0.467702	−0.0439977	−0.0219989	+	0.999758i	$0.507003\pi$
$114$	0	0
$115$	7.49688	0.699088
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	1.35234	0.122940
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−8.01771	−0.717126
$126$	0	0
$127$	14.6768	1.30236	0.651180	−	0.758924i	$-0.274274\pi$
$127$	14.6768	1.30236	0.651180	+	0.758924i	$0.274274\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	14.2484	1.24489	0.622446	−	0.782663i	$-0.286139\pi$
$131$	14.2484	1.24489	0.622446	+	0.782663i	$0.286139\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	17.2953	1.47764	0.738820	−	0.673903i	$-0.235383\pi$
$137$	17.2953	1.47764	0.738820	+	0.673903i	$0.235383\pi$
$138$	0	0
$139$	12.1331	1.02911	0.514557	−	0.857456i	$-0.327956\pi$
$139$	12.1331	1.02911	0.514557	+	0.857456i	$0.327956\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−6.56148	−0.548698
$144$	0	0
$145$	3.04689	0.253030
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−4.94299	−0.404946	−0.202473	−	0.979288i	$-0.564898\pi$
$149$	−4.94299	−0.404946	−0.202473	+	0.979288i	$0.564898\pi$
$150$	0	0
$151$	16.2484	1.32228	0.661140	−	0.750263i	$-0.270073\pi$
$151$	16.2484	1.32228	0.661140	+	0.750263i	$0.270073\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	1.61849	0.130000
$156$	0	0
$157$	1.39922	0.111670	0.0558351	−	0.998440i	$-0.482218\pi$
$157$	1.39922	0.111670	0.0558351	+	0.998440i	$0.482218\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−20.1445	−1.57784	−0.788921	−	0.614494i	$-0.789360\pi$
$163$	−20.1445	−1.57784	−0.788921	+	0.614494i	$0.789360\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	10.1914	0.788637	0.394318	−	0.918974i	$-0.370981\pi$
$167$	10.1914	0.788637	0.394318	+	0.918974i	$0.370981\pi$
$168$	0	0
$169$	−9.51459	−0.731891
$170$	0	0
$171$	0	0
$172$	0	0
$173$	20.8099	1.58215	0.791074	−	0.611720i	$-0.209522\pi$
$173$	20.8099	1.58215	0.791074	+	0.611720i	$0.209522\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	23.3638	1.74629	0.873146	−	0.487458i	$-0.162076\pi$
$179$	23.3638	1.74629	0.873146	+	0.487458i	$0.162076\pi$
$180$	0	0
$181$	−9.33463	−0.693837	−0.346919	−	0.937895i	$-0.612772\pi$
$181$	−9.33463	−0.693837	−0.346919	+	0.937895i	$0.612772\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−2.41069	−0.177238
$186$	0	0
$187$	−22.8961	−1.67433
$188$	0	0
$189$	0	0
$190$	0	0
$191$	24.9823	1.80766	0.903828	−	0.427897i	$-0.140745\pi$
$191$	24.9823	1.80766	0.903828	+	0.427897i	$0.140745\pi$
$192$	0	0
$193$	−23.9430	−1.72345	−0.861727	−	0.507372i	$-0.830617\pi$
$193$	−23.9430	−1.72345	−0.861727	+	0.507372i	$0.830617\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−5.13307	−0.365716	−0.182858	−	0.983139i	$-0.558535\pi$
$197$	−5.13307	−0.365716	−0.182858	+	0.983139i	$0.558535\pi$
$198$	0	0
$199$	7.00000	0.496217	0.248108	−	0.968732i	$-0.420191\pi$
$199$	7.00000	0.496217	0.248108	+	0.968732i	$0.420191\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−9.00000	−0.628587
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−18.9138	−1.30830
$210$	0	0
$211$	−5.54377	−0.381649	−0.190824	−	0.981624i	$-0.561116\pi$
$211$	−5.54377	−0.381649	−0.190824	+	0.981624i	$0.561116\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−5.01147	−0.341779
$216$	0	0
$217$	0	0
$218$	0	0
$219$	0	0
$220$	0	0
$221$	12.1623	0.818122
$222$	0	0
$223$	−19.6008	−1.31257	−0.656283	−	0.754515i	$-0.727872\pi$
$223$	−19.6008	−1.31257	−0.656283	+	0.754515i	$0.727872\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	16.1445	1.07155	0.535776	−	0.844360i	$-0.320019\pi$
$227$	16.1445	1.07155	0.535776	+	0.844360i	$0.320019\pi$
$228$	0	0
$229$	22.6768	1.49853	0.749264	−	0.662272i	$-0.230407\pi$
$229$	22.6768	1.49853	0.749264	+	0.662272i	$0.230407\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	3.70467	0.242701	0.121351	−	0.992610i	$-0.461277\pi$
$233$	3.70467	0.242701	0.121351	+	0.992610i	$0.461277\pi$
$234$	0	0
$235$	−5.34221	−0.348488
$236$	0	0
$237$	0	0
$238$	0	0
$239$	1.61849	0.104691	0.0523456	−	0.998629i	$-0.483330\pi$
$239$	1.61849	0.104691	0.0523456	+	0.998629i	$0.483330\pi$
$240$	0	0
$241$	−4.20155	−0.270646	−0.135323	−	0.990802i	$-0.543207\pi$
$241$	−4.20155	−0.270646	−0.135323	+	0.990802i	$0.543207\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	10.0469	0.639268
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−19.5438	−1.23359	−0.616796	−	0.787123i	$-0.711570\pi$
$251$	−19.5438	−1.23359	−0.616796	+	0.787123i	$0.711570\pi$
$252$	0	0
$253$	−30.3930	−1.91079
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−6.56148	−0.409294	−0.204647	−	0.978836i	$-0.565605\pi$
$257$	−6.56148	−0.409294	−0.204647	+	0.978836i	$0.565605\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	0	0
$263$	19.3815	1.19512	0.597558	−	0.801826i	$-0.296138\pi$
$263$	19.3815	1.19512	0.597558	+	0.801826i	$0.296138\pi$
$264$	0	0
$265$	4.85546	0.298268
$266$	0	0
$267$	0	0
$268$	0	0
$269$	14.4854	0.883191	0.441596	−	0.897214i	$-0.354413\pi$
$269$	14.4854	0.883191	0.441596	+	0.897214i	$0.354413\pi$
$270$	0	0
$271$	−5.35234	−0.325131	−0.162566	−	0.986698i	$-0.551977\pi$
$271$	−5.35234	−0.325131	−0.162566	+	0.986698i	$0.551977\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	14.9315	0.900405
$276$	0	0
$277$	12.2193	0.734184	0.367092	−	0.930185i	$-0.380353\pi$
$277$	12.2193	0.734184	0.367092	+	0.930185i	$0.380353\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−9.11537	−0.543777	−0.271889	−	0.962329i	$-0.587648\pi$
$281$	−9.11537	−0.543777	−0.271889	+	0.962329i	$0.587648\pi$
$282$	0	0
$283$	−13.8099	−0.820914	−0.410457	−	0.911880i	$-0.634631\pi$
$283$	−13.8099	−0.820914	−0.410457	+	0.911880i	$0.634631\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	25.4399	1.49646
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−28.1445	−1.64422	−0.822111	−	0.569327i	$-0.807204\pi$
$293$	−28.1445	−1.64422	−0.822111	+	0.569327i	$0.807204\pi$
$294$	0	0
$295$	4.89610	0.285062
$296$	0	0
$297$	0	0
$298$	0	0
$299$	16.1445	0.933663
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−8.92528	−0.511060
$306$	0	0
$307$	−7.24844	−0.413690	−0.206845	−	0.978374i	$-0.566320\pi$
$307$	−7.24844	−0.413690	−0.206845	+	0.978374i	$0.566320\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−7.73385	−0.438546	−0.219273	−	0.975663i	$-0.570369\pi$
$311$	−7.73385	−0.438546	−0.219273	+	0.975663i	$0.570369\pi$
$312$	0	0
$313$	9.85680	0.557139	0.278570	−	0.960416i	$-0.410140\pi$
$313$	9.85680	0.557139	0.278570	+	0.960416i	$0.410140\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	18.7237	1.05163	0.525815	−	0.850599i	$-0.323761\pi$
$317$	18.7237	1.05163	0.525815	+	0.850599i	$0.323761\pi$
$318$	0	0
$319$	−12.3523	−0.691598
$320$	0	0
$321$	0	0
$322$	0	0
$323$	35.0584	1.95070
$324$	0	0
$325$	−7.93152	−0.439962
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−18.9253	−1.04023	−0.520114	−	0.854097i	$-0.674110\pi$
$331$	−18.9253	−1.04023	−0.520114	+	0.854097i	$0.674110\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−1.17237	−0.0640537
$336$	0	0
$337$	29.2776	1.59485	0.797427	−	0.603416i	$-0.206194\pi$
$337$	29.2776	1.59485	0.797427	+	0.603416i	$0.206194\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−6.56148	−0.355324
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−21.0875	−1.13204	−0.566019	−	0.824392i	$-0.691517\pi$
$347$	−21.0875	−1.13204	−0.566019	+	0.824392i	$0.691517\pi$
$348$	0	0
$349$	−33.1914	−1.77670	−0.888348	−	0.459170i	$-0.848147\pi$
$349$	−33.1914	−1.77670	−0.888348	+	0.459170i	$0.848147\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	21.7237	1.15624	0.578119	−	0.815953i	$-0.303787\pi$
$353$	21.7237	1.15624	0.578119	+	0.815953i	$0.303787\pi$
$354$	0	0
$355$	1.80857	0.0959889
$356$	0	0
$357$	0	0
$358$	0	0
$359$	6.75156	0.356334	0.178167	−	0.984000i	$-0.442983\pi$
$359$	6.75156	0.356334	0.178167	+	0.984000i	$0.442983\pi$
$360$	0	0
$361$	9.96070	0.524247
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−6.28386	−0.328912
$366$	0	0
$367$	−35.0875	−1.83155	−0.915777	−	0.401687i	$-0.868424\pi$
$367$	−35.0875	−1.83155	−0.915777	+	0.401687i	$0.868424\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	17.4677	0.904443	0.452222	−	0.891906i	$-0.350632\pi$
$373$	17.4677	0.904443	0.452222	+	0.891906i	$0.350632\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	6.56148	0.337933
$378$	0	0
$379$	2.86693	0.147264	0.0736320	−	0.997285i	$-0.476541\pi$
$379$	2.86693	0.147264	0.0736320	+	0.997285i	$0.476541\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−4.84922	−0.247783	−0.123892	−	0.992296i	$-0.539538\pi$
$383$	−4.84922	−0.247783	−0.123892	+	0.992296i	$0.539538\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−26.4107	−1.33908	−0.669538	−	0.742778i	$-0.733508\pi$
$389$	−26.4107	−1.33908	−0.669538	+	0.742778i	$0.733508\pi$
$390$	0	0
$391$	56.3360	2.84903
$392$	0	0
$393$	0	0
$394$	0	0
$395$	10.1230	0.509341
$396$	0	0
$397$	−29.0191	−1.45642	−0.728212	−	0.685352i	$-0.759649\pi$
$397$	−29.0191	−1.45642	−0.728212	+	0.685352i	$0.759649\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	33.9076	1.69326	0.846632	−	0.532179i	$-0.178627\pi$
$401$	33.9076	1.69326	0.846632	+	0.532179i	$0.178627\pi$
$402$	0	0
$403$	3.48541	0.173621
$404$	0	0
$405$	0	0
$406$	0	0
$407$	9.77315	0.484437
$408$	0	0
$409$	36.2675	1.79331	0.896656	−	0.442728i	$-0.145989\pi$
$409$	36.2675	1.79331	0.896656	+	0.442728i	$0.145989\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−5.95311	−0.292227
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−22.1230	−1.08078	−0.540388	−	0.841416i	$-0.681723\pi$
$419$	−22.1230	−1.08078	−0.540388	+	0.841416i	$0.681723\pi$
$420$	0	0
$421$	6.47529	0.315586	0.157793	−	0.987472i	$-0.449562\pi$
$421$	6.47529	0.315586	0.157793	+	0.987472i	$0.449562\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−27.6768	−1.34252
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−3.98229	−0.191820	−0.0959101	−	0.995390i	$-0.530576\pi$
$431$	−3.98229	−0.191820	−0.0959101	+	0.995390i	$0.530576\pi$
$432$	0	0
$433$	26.4690	1.27202	0.636011	−	0.771680i	$-0.280583\pi$
$433$	26.4690	1.27202	0.636011	+	0.771680i	$0.280583\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	46.5375	2.22619
$438$	0	0
$439$	10.7922	0.515084	0.257542	−	0.966267i	$-0.417087\pi$
$439$	10.7922	0.515084	0.257542	+	0.966267i	$0.417087\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−0.277618	−0.0131900	−0.00659502	−	0.999978i	$-0.502099\pi$
$443$	−0.277618	−0.0131900	−0.00659502	+	0.999978i	$0.502099\pi$
$444$	0	0
$445$	5.68831	0.269652
$446$	0	0
$447$	0	0
$448$	0	0
$449$	0.277618	0.0131016	0.00655081	−	0.999979i	$-0.497915\pi$
$449$	0.277618	0.0131016	0.00655081	+	0.999979i	$0.497915\pi$
$450$	0	0
$451$	36.4868	1.71809
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−37.6768	−1.76245	−0.881224	−	0.472699i	$-0.843280\pi$
$457$	−37.6768	−1.76245	−0.881224	+	0.472699i	$0.843280\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	7.84922	0.365574	0.182787	−	0.983152i	$-0.441488\pi$
$461$	7.84922	0.365574	0.182787	+	0.983152i	$0.441488\pi$
$462$	0	0
$463$	−0.532298	−0.0247380	−0.0123690	−	0.999924i	$-0.503937\pi$
$463$	−0.532298	−0.0247380	−0.0123690	+	0.999924i	$0.503937\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	7.68696	0.355710	0.177855	−	0.984057i	$-0.443084\pi$
$467$	7.68696	0.355710	0.177855	+	0.984057i	$0.443084\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	20.3169	0.934173
$474$	0	0
$475$	−22.8630	−1.04903
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−22.1914	−1.01395	−0.506976	−	0.861960i	$-0.669237\pi$
$479$	−22.1914	−1.01395	−0.506976	+	0.861960i	$0.669237\pi$
$480$	0	0
$481$	−5.19143	−0.236709
$482$	0	0
$483$	0	0
$484$	0	0
$485$	2.85680	0.129721
$486$	0	0
$487$	−7.98229	−0.361712	−0.180856	−	0.983510i	$-0.557887\pi$
$487$	−7.98229	−0.361712	−0.180856	+	0.983510i	$0.557887\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	38.1052	1.71967	0.859833	−	0.510576i	$-0.170568\pi$
$491$	38.1052	1.71967	0.859833	+	0.510576i	$0.170568\pi$
$492$	0	0
$493$	22.8961	1.03119
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	15.4500	0.691637	0.345818	−	0.938301i	$-0.387601\pi$
$499$	15.4500	0.691637	0.345818	+	0.938301i	$0.387601\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	1.78074	0.0793992	0.0396996	−	0.999212i	$-0.487360\pi$
$503$	1.78074	0.0793992	0.0396996	+	0.999212i	$0.487360\pi$
$504$	0	0
$505$	−1.64378	−0.0731473
$506$	0	0
$507$	0	0
$508$	0	0
$509$	7.78074	0.344875	0.172438	−	0.985020i	$-0.444836\pi$
$509$	7.78074	0.344875	0.172438	+	0.985020i	$0.444836\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	0	0
$514$	0	0
$515$	6.06848	0.267409
$516$	0	0
$517$	21.6578	0.952508
$518$	0	0
$519$	0	0
$520$	0	0
$521$	4.21926	0.184849	0.0924246	−	0.995720i	$-0.470538\pi$
$521$	4.21926	0.184849	0.0924246	+	0.995720i	$0.470538\pi$
$522$	0	0
$523$	2.05701	0.0899467	0.0449734	−	0.998988i	$-0.485680\pi$
$523$	2.05701	0.0899467	0.0449734	+	0.998988i	$0.485680\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	12.1623	0.529796
$528$	0	0
$529$	51.7821	2.25140
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−19.3815	−0.839507
$534$	0	0
$535$	16.8023	0.726428
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	17.2776	0.742823	0.371411	−	0.928468i	$-0.378874\pi$
$541$	17.2776	0.742823	0.371411	+	0.928468i	$0.378874\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	7.82763	0.335299
$546$	0	0
$547$	−20.8492	−0.891448	−0.445724	−	0.895170i	$-0.647054\pi$
$547$	−20.8492	−0.891448	−0.445724	+	0.895170i	$0.647054\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	18.9138	0.805756
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−13.2877	−0.563020	−0.281510	−	0.959558i	$-0.590835\pi$
$557$	−13.2877	−0.563020	−0.281510	+	0.959558i	$0.590835\pi$
$558$	0	0
$559$	−10.7922	−0.456462
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−30.3930	−1.28091	−0.640456	−	0.767995i	$-0.721255\pi$
$563$	−30.3930	−1.28091	−0.640456	+	0.767995i	$0.721255\pi$
$564$	0	0
$565$	−0.405463	−0.0170580
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−43.7060	−1.83225	−0.916126	−	0.400891i	$-0.868701\pi$
$569$	−43.7060	−1.83225	−0.916126	+	0.400891i	$0.868701\pi$
$570$	0	0
$571$	−38.2130	−1.59917	−0.799583	−	0.600556i	$-0.794946\pi$
$571$	−38.2130	−1.59917	−0.799583	+	0.600556i	$0.794946\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−36.7391	−1.53213
$576$	0	0
$577$	−33.1914	−1.38178	−0.690889	−	0.722961i	$-0.742781\pi$
$577$	−33.1914	−1.38178	−0.690889	+	0.722961i	$0.742781\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−19.6844	−0.815246
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−3.79221	−0.156521	−0.0782606	−	0.996933i	$-0.524937\pi$
$587$	−3.79221	−0.156521	−0.0782606	+	0.996933i	$0.524937\pi$
$588$	0	0
$589$	10.0469	0.413975
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−3.63619	−0.149321	−0.0746603	−	0.997209i	$-0.523787\pi$
$593$	−3.63619	−0.149321	−0.0746603	+	0.997209i	$0.523787\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	38.8506	1.58739	0.793696	−	0.608315i	$-0.208154\pi$
$599$	38.8506	1.58739	0.793696	+	0.608315i	$0.208154\pi$
$600$	0	0
$601$	−32.7237	−1.33483	−0.667414	−	0.744687i	$-0.732599\pi$
$601$	−32.7237	−1.33483	−0.667414	+	0.744687i	$0.732599\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	1.17237	0.0476638
$606$	0	0
$607$	27.8099	1.12877	0.564385	−	0.825512i	$-0.309113\pi$
$607$	27.8099	1.12877	0.564385	+	0.825512i	$0.309113\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−11.5045	−0.465421
$612$	0	0
$613$	−29.9368	−1.20913	−0.604567	−	0.796554i	$-0.706654\pi$
$613$	−29.9368	−1.20913	−0.604567	+	0.796554i	$0.706654\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−30.4183	−1.22459	−0.612297	−	0.790628i	$-0.709754\pi$
$617$	−30.4183	−1.22459	−0.612297	+	0.790628i	$0.709754\pi$
$618$	0	0
$619$	−23.3054	−0.936725	−0.468363	−	0.883536i	$-0.655156\pi$
$619$	−23.3054	−0.936725	−0.468363	+	0.883536i	$0.655156\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	14.2914	0.571658
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−18.1154	−0.722307
$630$	0	0
$631$	4.60078	0.183154	0.0915770	−	0.995798i	$-0.470809\pi$
$631$	4.60078	0.183154	0.0915770	+	0.995798i	$0.470809\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	12.7237	0.504926
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	4.66537	0.184271	0.0921356	−	0.995746i	$-0.470631\pi$
$641$	4.66537	0.184271	0.0921356	+	0.995746i	$0.470631\pi$
$642$	0	0
$643$	22.3992	0.883339	0.441670	−	0.897178i	$-0.354386\pi$
$643$	22.3992	0.883339	0.441670	+	0.897178i	$0.354386\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−39.7237	−1.56170	−0.780850	−	0.624718i	$-0.785214\pi$
$647$	−39.7237	−1.56170	−0.780850	+	0.624718i	$0.785214\pi$
$648$	0	0
$649$	−19.8492	−0.779150
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−38.1052	−1.49117	−0.745587	−	0.666409i	$-0.767831\pi$
$653$	−38.1052	−1.49117	−0.745587	+	0.666409i	$0.767831\pi$
$654$	0	0
$655$	12.3523	0.482646
$656$	0	0
$657$	0	0
$658$	0	0
$659$	28.3321	1.10366	0.551831	−	0.833956i	$-0.313929\pi$
$659$	28.3321	1.10366	0.551831	+	0.833956i	$0.313929\pi$
$660$	0	0
$661$	−43.0774	−1.67552	−0.837759	−	0.546041i	$-0.816134\pi$
$661$	−43.0774	−1.67552	−0.837759	+	0.546041i	$0.816134\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	30.3930	1.17682
$668$	0	0
$669$	0	0
$670$	0	0
$671$	36.1838	1.39686
$672$	0	0
$673$	22.7630	0.877450	0.438725	−	0.898621i	$-0.355430\pi$
$673$	22.7630	0.877450	0.438725	+	0.898621i	$0.355430\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−38.8506	−1.49315	−0.746574	−	0.665302i	$-0.768303\pi$
$677$	−38.8506	−1.49315	−0.746574	+	0.665302i	$0.768303\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−0.657786	−0.0251695	−0.0125847	−	0.999921i	$-0.504006\pi$
$683$	−0.657786	−0.0251695	−0.0125847	+	0.999921i	$0.504006\pi$
$684$	0	0
$685$	14.9938	0.572882
$686$	0	0
$687$	0	0
$688$	0	0
$689$	10.4562	0.398351
$690$	0	0
$691$	4.72373	0.179699	0.0898496	−	0.995955i	$-0.471361\pi$
$691$	4.72373	0.179699	0.0898496	+	0.995955i	$0.471361\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	10.5185	0.398988
$696$	0	0
$697$	−67.6313	−2.56172
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−48.1560	−1.81883	−0.909414	−	0.415893i	$-0.863469\pi$
$701$	−48.1560	−1.81883	−0.909414	+	0.415893i	$0.863469\pi$
$702$	0	0
$703$	−14.9646	−0.564400
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	0.543767	0.0204216	0.0102108	−	0.999948i	$-0.496750\pi$
$709$	0.543767	0.0204216	0.0102108	+	0.999948i	$0.496750\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	16.1445	0.604618
$714$	0	0
$715$	−5.68831	−0.212731
$716$	0	0
$717$	0	0
$718$	0	0
$719$	50.6136	1.88757	0.943784	−	0.330562i	$-0.107238\pi$
$719$	50.6136	1.88757	0.943784	+	0.330562i	$0.107238\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−14.9315	−0.554543
$726$	0	0
$727$	−23.3054	−0.864351	−0.432176	−	0.901789i	$-0.642254\pi$
$727$	−23.3054	−0.864351	−0.432176	+	0.901789i	$0.642254\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−37.6591	−1.39287
$732$	0	0
$733$	44.6122	1.64779	0.823895	−	0.566742i	$-0.191796\pi$
$733$	44.6122	1.64779	0.823895	+	0.566742i	$0.191796\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	4.75291	0.175076
$738$	0	0
$739$	37.8784	1.39338	0.696690	−	0.717373i	$-0.254655\pi$
$739$	37.8784	1.39338	0.696690	+	0.717373i	$0.254655\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	16.0570	0.589075	0.294537	−	0.955640i	$-0.404834\pi$
$743$	16.0570	0.589075	0.294537	+	0.955640i	$0.404834\pi$
$744$	0	0
$745$	−4.28520	−0.156998
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	9.73385	0.355193	0.177597	−	0.984103i	$-0.443168\pi$
$751$	9.73385	0.355193	0.177597	+	0.984103i	$0.443168\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	14.0862	0.512649
$756$	0	0
$757$	−16.7922	−0.610323	−0.305162	−	0.952301i	$-0.598710\pi$
$757$	−16.7922	−0.610323	−0.305162	+	0.952301i	$0.598710\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−19.2877	−0.699180	−0.349590	−	0.936903i	$-0.613679\pi$
$761$	−19.2877	−0.699180	−0.349590	+	0.936903i	$0.613679\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	10.5438	0.380713
$768$	0	0
$769$	1.58931	0.0573119	0.0286559	−	0.999589i	$-0.490877\pi$
$769$	1.58931	0.0573119	0.0286559	+	0.999589i	$0.490877\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−20.5539	−0.739272	−0.369636	−	0.929177i	$-0.620518\pi$
$773$	−20.5539	−0.739272	−0.369636	+	0.929177i	$0.620518\pi$
$774$	0	0
$775$	−7.93152	−0.284909
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−55.8683	−2.00169
$780$	0	0
$781$	−7.33209	−0.262363
$782$	0	0
$783$	0	0
$784$	0	0
$785$	1.21302	0.0432946
$786$	0	0
$787$	30.3891	1.08325	0.541627	−	0.840619i	$-0.317808\pi$
$787$	30.3891	1.08325	0.541627	+	0.840619i	$0.317808\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−19.2206	−0.682544
$794$	0	0
$795$	0	0
$796$	0	0
$797$	45.4120	1.60858	0.804288	−	0.594239i	$-0.202547\pi$
$797$	45.4120	1.60858	0.804288	+	0.594239i	$0.202547\pi$
$798$	0	0
$799$	−40.1445	−1.42021
$800$	0	0
$801$	0	0
$802$	0	0
$803$	25.4753	0.899003
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	15.0191	0.528042	0.264021	−	0.964517i	$-0.414951\pi$
$809$	15.0191	0.528042	0.264021	+	0.964517i	$0.414951\pi$
$810$	0	0
$811$	−23.3930	−0.821439	−0.410719	−	0.911762i	$-0.634722\pi$
$811$	−23.3930	−0.821439	−0.410719	+	0.911762i	$0.634722\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−17.4638	−0.611731
$816$	0	0
$817$	−31.1091	−1.08837
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−29.1547	−1.01750	−0.508752	−	0.860913i	$-0.669893\pi$
$821$	−29.1547	−1.01750	−0.508752	+	0.860913i	$0.669893\pi$
$822$	0	0
$823$	3.19143	0.111246	0.0556231	−	0.998452i	$-0.482285\pi$
$823$	3.19143	0.111246	0.0556231	+	0.998452i	$0.482285\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	34.5654	1.20196	0.600978	−	0.799266i	$-0.294778\pi$
$827$	34.5654	1.20196	0.600978	+	0.799266i	$0.294778\pi$
$828$	0	0
$829$	23.4222	0.813485	0.406743	−	0.913543i	$-0.366665\pi$
$829$	23.4222	0.813485	0.406743	+	0.913543i	$0.366665\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0	0
$834$	0	0
$835$	8.83521	0.305755
$836$	0	0
$837$	0	0
$838$	0	0
$839$	25.0977	0.866467	0.433234	−	0.901282i	$-0.357372\pi$
$839$	25.0977	0.866467	0.433234	+	0.901282i	$0.357372\pi$
$840$	0	0
$841$	−16.6477	−0.574057
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−8.24844	−0.283755
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−24.0469	−0.824317
$852$	0	0
$853$	11.9430	0.408920	0.204460	−	0.978875i	$-0.434456\pi$
$853$	11.9430	0.408920	0.204460	+	0.978875i	$0.434456\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	10.2193	0.349083	0.174542	−	0.984650i	$-0.444156\pi$
$857$	10.2193	0.349083	0.174542	+	0.984650i	$0.444156\pi$
$858$	0	0
$859$	11.9430	0.407490	0.203745	−	0.979024i	$-0.434689\pi$
$859$	11.9430	0.407490	0.203745	+	0.979024i	$0.434689\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−15.8669	−0.540116	−0.270058	−	0.962844i	$-0.587043\pi$
$863$	−15.8669	−0.540116	−0.270058	+	0.962844i	$0.587043\pi$
$864$	0	0
$865$	18.0406	0.613400
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−41.0393	−1.39216
$870$	0	0
$871$	−2.52471	−0.0855466
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−31.1813	−1.05292	−0.526459	−	0.850201i	$-0.676481\pi$
$877$	−31.1813	−1.05292	−0.526459	+	0.850201i	$0.676481\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	33.1623	1.11726	0.558632	−	0.829415i	$-0.311326\pi$
$881$	33.1623	1.11726	0.558632	+	0.829415i	$0.311326\pi$
$882$	0	0
$883$	19.5045	0.656378	0.328189	−	0.944612i	$-0.393562\pi$
$883$	19.5045	0.656378	0.328189	+	0.944612i	$0.393562\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−29.5261	−0.991388	−0.495694	−	0.868497i	$-0.665086\pi$
$887$	−29.5261	−0.991388	−0.495694	+	0.868497i	$0.665086\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−33.1623	−1.10973
$894$	0	0
$895$	20.2547	0.677039
$896$	0	0
$897$	0	0
$898$	0	0
$899$	6.56148	0.218837
$900$	0	0
$901$	36.4868	1.21555
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−8.09243	−0.269001
$906$	0	0
$907$	32.4183	1.07643	0.538216	−	0.842807i	$-0.319099\pi$
$907$	32.4183	1.07643	0.538216	+	0.842807i	$0.319099\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	30.5831	1.01326	0.506631	−	0.862163i	$-0.330891\pi$
$911$	30.5831	1.01326	0.506631	+	0.862163i	$0.330891\pi$
$912$	0	0
$913$	24.1344	0.798733
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−48.2953	−1.59312	−0.796558	−	0.604562i	$-0.793348\pi$
$919$	−48.2953	−1.59312	−0.796558	+	0.604562i	$0.793348\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	3.89476	0.128197
$924$	0	0
$925$	11.8138	0.388435
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−43.2193	−1.41798	−0.708989	−	0.705220i	$-0.750848\pi$
$929$	−43.2193	−1.41798	−0.708989	+	0.705220i	$0.750848\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−19.8492	−0.649139
$936$	0	0
$937$	−37.3638	−1.22062	−0.610311	−	0.792162i	$-0.708956\pi$
$937$	−37.3638	−1.22062	−0.610311	+	0.792162i	$0.708956\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−27.5146	−0.896950	−0.448475	−	0.893795i	$-0.648033\pi$
$941$	−27.5146	−0.896950	−0.448475	+	0.893795i	$0.648033\pi$
$942$	0	0
$943$	−89.7758	−2.92350
$944$	0	0
$945$	0	0
$946$	0	0
$947$	15.6768	0.509429	0.254714	−	0.967016i	$-0.418019\pi$
$947$	15.6768	0.509429	0.254714	+	0.967016i	$0.418019\pi$
$948$	0	0
$949$	−13.5323	−0.439277
$950$	0	0
$951$	0	0
$952$	0	0
$953$	11.2268	0.363673	0.181837	−	0.983329i	$-0.441796\pi$
$953$	11.2268	0.363673	0.181837	+	0.983329i	$0.441796\pi$
$954$	0	0
$955$	21.6578	0.700829
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−27.5146	−0.887567
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−20.7568	−0.668185
$966$	0	0
$967$	38.4868	1.23765	0.618825	−	0.785529i	$-0.287609\pi$
$967$	38.4868	1.23765	0.618825	+	0.785529i	$0.287609\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−14.1800	−0.455057	−0.227528	−	0.973771i	$-0.573064\pi$
$971$	−14.1800	−0.455057	−0.227528	+	0.973771i	$0.573064\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−49.3943	−1.58026	−0.790132	−	0.612936i	$-0.789988\pi$
$977$	−49.3943	−1.58026	−0.790132	+	0.612936i	$0.789988\pi$
$978$	0	0
$979$	−23.0609	−0.737029
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−13.4562	−0.429187	−0.214594	−	0.976703i	$-0.568843\pi$
$983$	−13.4562	−0.429187	−0.214594	+	0.976703i	$0.568843\pi$
$984$	0	0
$985$	−4.44999	−0.141789
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−49.9899	−1.58959
$990$	0	0
$991$	−2.19767	−0.0698113	−0.0349056	−	0.999391i	$-0.511113\pi$
$991$	−2.19767	−0.0698113	−0.0349056	+	0.999391i	$0.511113\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	6.06848	0.192384
$996$	0	0
$997$	49.8329	1.57822	0.789111	−	0.614250i	$-0.210542\pi$
$997$	49.8329	1.57822	0.789111	+	0.614250i	$0.210542\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5292.2.a.v.1.2		3
3.2	odd	2		5292.2.a.w.1.2		3
7.3	odd	6		756.2.k.e.541.2	yes	6
7.5	odd	6		756.2.k.e.109.2	✓	6
7.6	odd	2		5292.2.a.x.1.2		3
21.5	even	6		756.2.k.f.109.2	yes	6
21.17	even	6		756.2.k.f.541.2	yes	6
21.20	even	2		5292.2.a.u.1.2		3
63.5	even	6		2268.2.i.k.865.2		6
63.31	odd	6		2268.2.l.k.541.2		6
63.38	even	6		2268.2.i.k.2053.2		6
63.40	odd	6		2268.2.i.j.865.2		6
63.47	even	6		2268.2.l.j.109.2		6
63.52	odd	6		2268.2.i.j.2053.2		6
63.59	even	6		2268.2.l.j.541.2		6
63.61	odd	6		2268.2.l.k.109.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
756.2.k.e.109.2	✓	6	7.5	odd	6
756.2.k.e.541.2	yes	6	7.3	odd	6
756.2.k.f.109.2	yes	6	21.5	even	6
756.2.k.f.541.2	yes	6	21.17	even	6
2268.2.i.j.865.2		6	63.40	odd	6
2268.2.i.j.2053.2		6	63.52	odd	6
2268.2.i.k.865.2		6	63.5	even	6
2268.2.i.k.2053.2		6	63.38	even	6
2268.2.l.j.109.2		6	63.47	even	6
2268.2.l.j.541.2		6	63.59	even	6
2268.2.l.k.109.2		6	63.61	odd	6
2268.2.l.k.541.2		6	63.31	odd	6
5292.2.a.u.1.2		3	21.20	even	2
5292.2.a.v.1.2		3	1.1	even	1	trivial
5292.2.a.w.1.2		3	3.2	odd	2
5292.2.a.x.1.2		3	7.6	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:	\( N \)	\(=\)	\( 5292 = 2^{2} \cdot 3^{3} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5292.a (trivial)

\(f(q)\)	\(=\)	\(q+0.866926 q^{5} +O(q^{10})\)	\(q+0.866926 q^{5} -3.51459 q^{11} +1.86693 q^{13} +6.51459 q^{17} +5.38151 q^{19} +8.64766 q^{23} -4.24844 q^{25} +3.51459 q^{29} +1.86693 q^{31} -2.78074 q^{37} -10.3815 q^{41} -5.78074 q^{43} -6.16225 q^{47} +5.60078 q^{53} -3.04689 q^{55} +5.64766 q^{59} -10.2953 q^{61} +1.61849 q^{65} -1.35234 q^{67} +2.08619 q^{71} -7.24844 q^{73} +11.6768 q^{79} -6.86693 q^{83} +5.64766 q^{85} +6.56148 q^{89} +4.66537 q^{95} +3.29533 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - q^{5}+O(q^{10}) \) 3 * q - q^5	\( 3 q - q^{5} + 5 q^{11} + 2 q^{13} + 4 q^{17} - 3 q^{19} + 14 q^{23} + 10 q^{25} - 5 q^{29} + 2 q^{31} - 12 q^{41} - 9 q^{43} + 9 q^{47} + 6 q^{53} - 8 q^{55} + 5 q^{59} - 7 q^{61} + 24 q^{65} - 16 q^{67} + 11 q^{71} + q^{73} - 8 q^{79} - 17 q^{83} + 5 q^{85} + 3 q^{89} + 32 q^{95} - 14 q^{97}+O(q^{100}) \) 3 * q - q^5 + 5 * q^11 + 2 * q^13 + 4 * q^17 - 3 * q^19 + 14 * q^23 + 10 * q^25 - 5 * q^29 + 2 * q^31 - 12 * q^41 - 9 * q^43 + 9 * q^47 + 6 * q^53 - 8 * q^55 + 5 * q^59 - 7 * q^61 + 24 * q^65 - 16 * q^67 + 11 * q^71 + q^73 - 8 * q^79 - 17 * q^83 + 5 * q^85 + 3 * q^89 + 32 * q^95 - 14 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more