LMFDB - Embedded newform 5292.2.a.p.1.1

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:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{7}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 7 $ x^2 - 7
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.64575$ 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)

$f(q)$	$=$	$q-2.64575 q^{5} +O(q^{10})$	$q-2.64575 q^{5} +2.64575 q^{11} -2.00000 q^{13} +1.00000 q^{19} +7.93725 q^{23} +2.00000 q^{25} -5.29150 q^{29} -7.00000 q^{31} -3.00000 q^{37} -7.93725 q^{41} -2.00000 q^{43} +5.29150 q^{47} +10.5830 q^{53} -7.00000 q^{55} +5.29150 q^{59} +8.00000 q^{61} +5.29150 q^{65} -2.00000 q^{67} +2.64575 q^{71} -10.0000 q^{73} +10.0000 q^{79} +15.8745 q^{83} +2.64575 q^{89} -2.64575 q^{95} -16.0000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q+O(q^{10}) $ 2 * q	$ 2 q - 4 q^{13} + 2 q^{19} + 4 q^{25} - 14 q^{31} - 6 q^{37} - 4 q^{43} - 14 q^{55} + 16 q^{61} - 4 q^{67} - 20 q^{73} + 20 q^{79} - 32 q^{97}+O(q^{100}) $ 2 * q - 4 * q^13 + 2 * q^19 + 4 * q^25 - 14 * q^31 - 6 * q^37 - 4 * q^43 - 14 * q^55 + 16 * q^61 - 4 * q^67 - 20 * q^73 + 20 * q^79 - 32 * 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
$5$	−2.64575	−1.18322	−0.591608	−	0.806226i	$-0.701507\pi$
$5$	−2.64575	−1.18322	−0.591608	+	0.806226i	$0.701507\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	0	0
$10$	0	0
$11$	2.64575	0.797724	0.398862	−	0.917011i	$-0.369405\pi$
$11$	2.64575	0.797724	0.398862	+	0.917011i	$0.369405\pi$
$12$	0	0
$13$	−2.00000	−0.554700	−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−2.00000	−0.554700	−0.277350	+	0.960769i	$0.589456\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	0	0
$19$	1.00000	0.229416	0.114708	−	0.993399i	$-0.463407\pi$
$19$	1.00000	0.229416	0.114708	+	0.993399i	$0.463407\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	7.93725	1.65503	0.827516	−	0.561442i	$-0.189753\pi$
$23$	7.93725	1.65503	0.827516	+	0.561442i	$0.189753\pi$
$24$	0	0
$25$	2.00000	0.400000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−5.29150	−0.982607	−0.491304	−	0.870988i	$-0.663479\pi$
$29$	−5.29150	−0.982607	−0.491304	+	0.870988i	$0.663479\pi$
$30$	0	0
$31$	−7.00000	−1.25724	−0.628619	−	0.777714i	$-0.716379\pi$
$31$	−7.00000	−1.25724	−0.628619	+	0.777714i	$0.716379\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−3.00000	−0.493197	−0.246598	−	0.969118i	$-0.579313\pi$
$37$	−3.00000	−0.493197	−0.246598	+	0.969118i	$0.579313\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−7.93725	−1.23959	−0.619795	−	0.784763i	$-0.712784\pi$
$41$	−7.93725	−1.23959	−0.619795	+	0.784763i	$0.712784\pi$
$42$	0	0
$43$	−2.00000	−0.304997	−0.152499	−	0.988304i	$-0.548732\pi$
$43$	−2.00000	−0.304997	−0.152499	+	0.988304i	$0.548732\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	5.29150	0.771845	0.385922	−	0.922531i	$-0.373883\pi$
$47$	5.29150	0.771845	0.385922	+	0.922531i	$0.373883\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	0	0
$52$	0	0
$53$	10.5830	1.45369	0.726844	−	0.686803i	$-0.240986\pi$
$53$	10.5830	1.45369	0.726844	+	0.686803i	$0.240986\pi$
$54$	0	0
$55$	−7.00000	−0.943880
$56$	0	0
$57$	0	0
$58$	0	0
$59$	5.29150	0.688895	0.344447	−	0.938806i	$-0.388066\pi$
$59$	5.29150	0.688895	0.344447	+	0.938806i	$0.388066\pi$
$60$	0	0
$61$	8.00000	1.02430	0.512148	−	0.858898i	$-0.328850\pi$
$61$	8.00000	1.02430	0.512148	+	0.858898i	$0.328850\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	5.29150	0.656330
$66$	0	0
$67$	−2.00000	−0.244339	−0.122169	−	0.992509i	$-0.538985\pi$
$67$	−2.00000	−0.244339	−0.122169	+	0.992509i	$0.538985\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	2.64575	0.313993	0.156996	−	0.987599i	$-0.449819\pi$
$71$	2.64575	0.313993	0.156996	+	0.987599i	$0.449819\pi$
$72$	0	0
$73$	−10.0000	−1.17041	−0.585206	−	0.810885i	$-0.698986\pi$
$73$	−10.0000	−1.17041	−0.585206	+	0.810885i	$0.698986\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	10.0000	1.12509	0.562544	−	0.826767i	$-0.309823\pi$
$79$	10.0000	1.12509	0.562544	+	0.826767i	$0.309823\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	15.8745	1.74245	0.871227	−	0.490881i	$-0.163325\pi$
$83$	15.8745	1.74245	0.871227	+	0.490881i	$0.163325\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	2.64575	0.280449	0.140225	−	0.990120i	$-0.455218\pi$
$89$	2.64575	0.280449	0.140225	+	0.990120i	$0.455218\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−2.64575	−0.271448
$96$	0	0
$97$	−16.0000	−1.62455	−0.812277	−	0.583272i	$-0.801772\pi$
$97$	−16.0000	−1.62455	−0.812277	+	0.583272i	$0.801772\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−10.5830	−1.05305	−0.526524	−	0.850160i	$-0.676505\pi$
$101$	−10.5830	−1.05305	−0.526524	+	0.850160i	$0.676505\pi$
$102$	0	0
$103$	1.00000	0.0985329	0.0492665	−	0.998786i	$-0.484312\pi$
$103$	1.00000	0.0985329	0.0492665	+	0.998786i	$0.484312\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	5.29150	0.511549	0.255774	−	0.966736i	$-0.417670\pi$
$107$	5.29150	0.511549	0.255774	+	0.966736i	$0.417670\pi$
$108$	0	0
$109$	−11.0000	−1.05361	−0.526804	−	0.849987i	$-0.676610\pi$
$109$	−11.0000	−1.05361	−0.526804	+	0.849987i	$0.676610\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−10.5830	−0.995565	−0.497783	−	0.867302i	$-0.665852\pi$
$113$	−10.5830	−0.995565	−0.497783	+	0.867302i	$0.665852\pi$
$114$	0	0
$115$	−21.0000	−1.95826
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−4.00000	−0.363636
$122$	0	0
$123$	0	0
$124$	0	0
$125$	7.93725	0.709930
$126$	0	0
$127$	2.00000	0.177471	0.0887357	−	0.996055i	$-0.471717\pi$
$127$	2.00000	0.177471	0.0887357	+	0.996055i	$0.471717\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−15.8745	−1.38696	−0.693481	−	0.720475i	$-0.743924\pi$
$131$	−15.8745	−1.38696	−0.693481	+	0.720475i	$0.743924\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−5.29150	−0.452084	−0.226042	−	0.974118i	$-0.572579\pi$
$137$	−5.29150	−0.452084	−0.226042	+	0.974118i	$0.572579\pi$
$138$	0	0
$139$	−12.0000	−1.01783	−0.508913	−	0.860818i	$-0.669953\pi$
$139$	−12.0000	−1.01783	−0.508913	+	0.860818i	$0.669953\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−5.29150	−0.442498
$144$	0	0
$145$	14.0000	1.16264
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−21.1660	−1.73399	−0.866994	−	0.498319i	$-0.833951\pi$
$149$	−21.1660	−1.73399	−0.866994	+	0.498319i	$0.833951\pi$
$150$	0	0
$151$	16.0000	1.30206	0.651031	−	0.759051i	$-0.274337\pi$
$151$	16.0000	1.30206	0.651031	+	0.759051i	$0.274337\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	18.5203	1.48758
$156$	0	0
$157$	−14.0000	−1.11732	−0.558661	−	0.829396i	$-0.688685\pi$
$157$	−14.0000	−1.11732	−0.558661	+	0.829396i	$0.688685\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−18.0000	−1.40987	−0.704934	−	0.709273i	$-0.749024\pi$
$163$	−18.0000	−1.40987	−0.704934	+	0.709273i	$0.749024\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−21.1660	−1.63788	−0.818938	−	0.573883i	$-0.805437\pi$
$167$	−21.1660	−1.63788	−0.818938	+	0.573883i	$0.805437\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	0	0
$171$	0	0
$172$	0	0
$173$	18.5203	1.40807	0.704035	−	0.710166i	$-0.251380\pi$
$173$	18.5203	1.40807	0.704035	+	0.710166i	$0.251380\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−15.8745	−1.18652	−0.593258	−	0.805012i	$-0.702159\pi$
$179$	−15.8745	−1.18652	−0.593258	+	0.805012i	$0.702159\pi$
$180$	0	0
$181$	−16.0000	−1.18927	−0.594635	−	0.803996i	$-0.702704\pi$
$181$	−16.0000	−1.18927	−0.594635	+	0.803996i	$0.702704\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	7.93725	0.583559
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−7.93725	−0.574320	−0.287160	−	0.957883i	$-0.592711\pi$
$191$	−7.93725	−0.574320	−0.287160	+	0.957883i	$0.592711\pi$
$192$	0	0
$193$	2.00000	0.143963	0.0719816	−	0.997406i	$-0.477068\pi$
$193$	2.00000	0.143963	0.0719816	+	0.997406i	$0.477068\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	0	0		−	1.00000i	$-0.5\pi$
$197$	0	0			1.00000i	$0.5\pi$
$198$	0	0
$199$	−25.0000	−1.77220	−0.886102	−	0.463491i	$-0.846597\pi$
$199$	−25.0000	−1.77220	−0.886102	+	0.463491i	$0.846597\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	21.0000	1.46670
$206$	0	0
$207$	0	0
$208$	0	0
$209$	2.64575	0.183010
$210$	0	0
$211$	8.00000	0.550743	0.275371	−	0.961338i	$-0.411199\pi$
$211$	8.00000	0.550743	0.275371	+	0.961338i	$0.411199\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	5.29150	0.360877
$216$	0	0
$217$	0	0
$218$	0	0
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−7.00000	−0.468755	−0.234377	−	0.972146i	$-0.575305\pi$
$223$	−7.00000	−0.468755	−0.234377	+	0.972146i	$0.575305\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	10.5830	0.702419	0.351209	−	0.936297i	$-0.385771\pi$
$227$	10.5830	0.702419	0.351209	+	0.936297i	$0.385771\pi$
$228$	0	0
$229$	28.0000	1.85029	0.925146	−	0.379611i	$-0.123942\pi$
$229$	28.0000	1.85029	0.925146	+	0.379611i	$0.123942\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−21.1660	−1.38663	−0.693316	−	0.720634i	$-0.743851\pi$
$233$	−21.1660	−1.38663	−0.693316	+	0.720634i	$0.743851\pi$
$234$	0	0
$235$	−14.0000	−0.913259
$236$	0	0
$237$	0	0
$238$	0	0
$239$	15.8745	1.02684	0.513418	−	0.858138i	$-0.328379\pi$
$239$	15.8745	1.02684	0.513418	+	0.858138i	$0.328379\pi$
$240$	0	0
$241$	−10.0000	−0.644157	−0.322078	−	0.946713i	$-0.604381\pi$
$241$	−10.0000	−0.644157	−0.322078	+	0.946713i	$0.604381\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−2.00000	−0.127257
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−10.5830	−0.667993	−0.333997	−	0.942574i	$-0.608397\pi$
$251$	−10.5830	−0.667993	−0.333997	+	0.942574i	$0.608397\pi$
$252$	0	0
$253$	21.0000	1.32026
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−13.2288	−0.825187	−0.412594	−	0.910915i	$-0.635377\pi$
$257$	−13.2288	−0.825187	−0.412594	+	0.910915i	$0.635377\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−13.2288	−0.815720	−0.407860	−	0.913044i	$-0.633725\pi$
$263$	−13.2288	−0.815720	−0.407860	+	0.913044i	$0.633725\pi$
$264$	0	0
$265$	−28.0000	−1.72003
$266$	0	0
$267$	0	0
$268$	0	0
$269$	13.2288	0.806572	0.403286	−	0.915074i	$-0.367868\pi$
$269$	13.2288	0.806572	0.403286	+	0.915074i	$0.367868\pi$
$270$	0	0
$271$	−8.00000	−0.485965	−0.242983	−	0.970031i	$-0.578126\pi$
$271$	−8.00000	−0.485965	−0.242983	+	0.970031i	$0.578126\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	5.29150	0.319090
$276$	0	0
$277$	25.0000	1.50210	0.751052	−	0.660243i	$-0.229547\pi$
$277$	25.0000	1.50210	0.751052	+	0.660243i	$0.229547\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−15.8745	−0.946994	−0.473497	−	0.880795i	$-0.657008\pi$
$281$	−15.8745	−0.946994	−0.473497	+	0.880795i	$0.657008\pi$
$282$	0	0
$283$	−28.0000	−1.66443	−0.832214	−	0.554455i	$-0.812927\pi$
$283$	−28.0000	−1.66443	−0.832214	+	0.554455i	$0.812927\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−17.0000	−1.00000
$290$	0	0
$291$	0	0
$292$	0	0
$293$	0	0		−	1.00000i	$-0.5\pi$
$293$	0	0			1.00000i	$0.5\pi$
$294$	0	0
$295$	−14.0000	−0.815112
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−15.8745	−0.918046
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−21.1660	−1.21196
$306$	0	0
$307$	−7.00000	−0.399511	−0.199756	−	0.979846i	$-0.564015\pi$
$307$	−7.00000	−0.399511	−0.199756	+	0.979846i	$0.564015\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−5.29150	−0.300054	−0.150027	−	0.988682i	$-0.547936\pi$
$311$	−5.29150	−0.300054	−0.150027	+	0.988682i	$0.547936\pi$
$312$	0	0
$313$	6.00000	0.339140	0.169570	−	0.985518i	$-0.445762\pi$
$313$	6.00000	0.339140	0.169570	+	0.985518i	$0.445762\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−15.8745	−0.891601	−0.445801	−	0.895132i	$-0.647081\pi$
$317$	−15.8745	−0.891601	−0.445801	+	0.895132i	$0.647081\pi$
$318$	0	0
$319$	−14.0000	−0.783850
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0	0
$324$	0	0
$325$	−4.00000	−0.221880
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	32.0000	1.75888	0.879440	−	0.476011i	$-0.157918\pi$
$331$	32.0000	1.75888	0.879440	+	0.476011i	$0.157918\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	5.29150	0.289106
$336$	0	0
$337$	−13.0000	−0.708155	−0.354078	−	0.935216i	$-0.615205\pi$
$337$	−13.0000	−0.708155	−0.354078	+	0.935216i	$0.615205\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−18.5203	−1.00293
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0	0
$347$	7.93725	0.426094	0.213047	−	0.977042i	$-0.431661\pi$
$347$	7.93725	0.426094	0.213047	+	0.977042i	$0.431661\pi$
$348$	0	0
$349$	−12.0000	−0.642345	−0.321173	−	0.947021i	$-0.604077\pi$
$349$	−12.0000	−0.642345	−0.321173	+	0.947021i	$0.604077\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−23.8118	−1.26737	−0.633686	−	0.773590i	$-0.718459\pi$
$353$	−23.8118	−1.26737	−0.633686	+	0.773590i	$0.718459\pi$
$354$	0	0
$355$	−7.00000	−0.371521
$356$	0	0
$357$	0	0
$358$	0	0
$359$	5.29150	0.279275	0.139637	−	0.990203i	$-0.455406\pi$
$359$	5.29150	0.279275	0.139637	+	0.990203i	$0.455406\pi$
$360$	0	0
$361$	−18.0000	−0.947368
$362$	0	0
$363$	0	0
$364$	0	0
$365$	26.4575	1.38485
$366$	0	0
$367$	−25.0000	−1.30499	−0.652495	−	0.757793i	$-0.726278\pi$
$367$	−25.0000	−1.30499	−0.652495	+	0.757793i	$0.726278\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−11.0000	−0.569558	−0.284779	−	0.958593i	$-0.591920\pi$
$373$	−11.0000	−0.569558	−0.284779	+	0.958593i	$0.591920\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	10.5830	0.545053
$378$	0	0
$379$	−6.00000	−0.308199	−0.154100	−	0.988055i	$-0.549248\pi$
$379$	−6.00000	−0.308199	−0.154100	+	0.988055i	$0.549248\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	31.7490	1.62230	0.811149	−	0.584839i	$-0.198842\pi$
$383$	31.7490	1.62230	0.811149	+	0.584839i	$0.198842\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	31.7490	1.60974	0.804869	−	0.593452i	$-0.202235\pi$
$389$	31.7490	1.60974	0.804869	+	0.593452i	$0.202235\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−26.4575	−1.33122
$396$	0	0
$397$	28.0000	1.40528	0.702640	−	0.711546i	$-0.252005\pi$
$397$	28.0000	1.40528	0.702640	+	0.711546i	$0.252005\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	15.8745	0.792735	0.396368	−	0.918092i	$-0.370271\pi$
$401$	15.8745	0.792735	0.396368	+	0.918092i	$0.370271\pi$
$402$	0	0
$403$	14.0000	0.697390
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−7.93725	−0.393435
$408$	0	0
$409$	18.0000	0.890043	0.445021	−	0.895520i	$-0.353196\pi$
$409$	18.0000	0.890043	0.445021	+	0.895520i	$0.353196\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−42.0000	−2.06170
$416$	0	0
$417$	0	0
$418$	0	0
$419$	21.1660	1.03403	0.517014	−	0.855977i	$-0.327044\pi$
$419$	21.1660	1.03403	0.517014	+	0.855977i	$0.327044\pi$
$420$	0	0
$421$	−15.0000	−0.731055	−0.365528	−	0.930800i	$-0.619111\pi$
$421$	−15.0000	−0.731055	−0.365528	+	0.930800i	$0.619111\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	13.2288	0.637207	0.318603	−	0.947888i	$-0.396786\pi$
$431$	13.2288	0.637207	0.318603	+	0.947888i	$0.396786\pi$
$432$	0	0
$433$	−14.0000	−0.672797	−0.336399	−	0.941720i	$-0.609209\pi$
$433$	−14.0000	−0.672797	−0.336399	+	0.941720i	$0.609209\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	7.93725	0.379690
$438$	0	0
$439$	0	0		−	1.00000i	$-0.5\pi$
$439$	0	0			1.00000i	$0.5\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	7.93725	0.377110	0.188555	−	0.982063i	$-0.439620\pi$
$443$	7.93725	0.377110	0.188555	+	0.982063i	$0.439620\pi$
$444$	0	0
$445$	−7.00000	−0.331832
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−31.7490	−1.49833	−0.749164	−	0.662384i	$-0.769545\pi$
$449$	−31.7490	−1.49833	−0.749164	+	0.662384i	$0.769545\pi$
$450$	0	0
$451$	−21.0000	−0.988851
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	41.0000	1.91790	0.958950	−	0.283577i	$-0.0915211\pi$
$457$	41.0000	1.91790	0.958950	+	0.283577i	$0.0915211\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−23.8118	−1.10902	−0.554512	−	0.832176i	$-0.687095\pi$
$461$	−23.8118	−1.10902	−0.554512	+	0.832176i	$0.687095\pi$
$462$	0	0
$463$	20.0000	0.929479	0.464739	−	0.885448i	$-0.346148\pi$
$463$	20.0000	0.929479	0.464739	+	0.885448i	$0.346148\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	26.4575	1.22431	0.612154	−	0.790739i	$-0.290303\pi$
$467$	26.4575	1.22431	0.612154	+	0.790739i	$0.290303\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−5.29150	−0.243304
$474$	0	0
$475$	2.00000	0.0917663
$476$	0	0
$477$	0	0
$478$	0	0
$479$	31.7490	1.45065	0.725325	−	0.688407i	$-0.241690\pi$
$479$	31.7490	1.45065	0.725325	+	0.688407i	$0.241690\pi$
$480$	0	0
$481$	6.00000	0.273576
$482$	0	0
$483$	0	0
$484$	0	0
$485$	42.3320	1.92220
$486$	0	0
$487$	−26.0000	−1.17817	−0.589086	−	0.808070i	$-0.700512\pi$
$487$	−26.0000	−1.17817	−0.589086	+	0.808070i	$0.700512\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−23.8118	−1.07461	−0.537305	−	0.843388i	$-0.680558\pi$
$491$	−23.8118	−1.07461	−0.537305	+	0.843388i	$0.680558\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	6.00000	0.268597	0.134298	−	0.990941i	$-0.457122\pi$
$499$	6.00000	0.268597	0.134298	+	0.990941i	$0.457122\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−5.29150	−0.235936	−0.117968	−	0.993017i	$-0.537638\pi$
$503$	−5.29150	−0.235936	−0.117968	+	0.993017i	$0.537638\pi$
$504$	0	0
$505$	28.0000	1.24598
$506$	0	0
$507$	0	0
$508$	0	0
$509$	10.5830	0.469083	0.234542	−	0.972106i	$-0.424641\pi$
$509$	10.5830	0.469083	0.234542	+	0.972106i	$0.424641\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−2.64575	−0.116586
$516$	0	0
$517$	14.0000	0.615719
$518$	0	0
$519$	0	0
$520$	0	0
$521$	34.3948	1.50686	0.753431	−	0.657527i	$-0.228397\pi$
$521$	34.3948	1.50686	0.753431	+	0.657527i	$0.228397\pi$
$522$	0	0
$523$	−29.0000	−1.26808	−0.634041	−	0.773300i	$-0.718605\pi$
$523$	−29.0000	−1.26808	−0.634041	+	0.773300i	$0.718605\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	40.0000	1.73913
$530$	0	0
$531$	0	0
$532$	0	0
$533$	15.8745	0.687601
$534$	0	0
$535$	−14.0000	−0.605273
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−11.0000	−0.472927	−0.236463	−	0.971640i	$-0.575988\pi$
$541$	−11.0000	−0.472927	−0.236463	+	0.971640i	$0.575988\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	29.1033	1.24665
$546$	0	0
$547$	−8.00000	−0.342055	−0.171028	−	0.985266i	$-0.554709\pi$
$547$	−8.00000	−0.342055	−0.171028	+	0.985266i	$0.554709\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−5.29150	−0.225426
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	31.7490	1.34525	0.672624	−	0.739984i	$-0.265167\pi$
$557$	31.7490	1.34525	0.672624	+	0.739984i	$0.265167\pi$
$558$	0	0
$559$	4.00000	0.169182
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−37.0405	−1.56107	−0.780536	−	0.625111i	$-0.785054\pi$
$563$	−37.0405	−1.56107	−0.780536	+	0.625111i	$0.785054\pi$
$564$	0	0
$565$	28.0000	1.17797
$566$	0	0
$567$	0	0
$568$	0	0
$569$	31.7490	1.33099	0.665494	−	0.746403i	$-0.268221\pi$
$569$	31.7490	1.33099	0.665494	+	0.746403i	$0.268221\pi$
$570$	0	0
$571$	12.0000	0.502184	0.251092	−	0.967963i	$-0.419210\pi$
$571$	12.0000	0.502184	0.251092	+	0.967963i	$0.419210\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	15.8745	0.662013
$576$	0	0
$577$	32.0000	1.33218	0.666089	−	0.745873i	$-0.267967\pi$
$577$	32.0000	1.33218	0.666089	+	0.745873i	$0.267967\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	28.0000	1.15964
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−10.5830	−0.436807	−0.218404	−	0.975859i	$-0.570085\pi$
$587$	−10.5830	−0.436807	−0.218404	+	0.975859i	$0.570085\pi$
$588$	0	0
$589$	−7.00000	−0.288430
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−7.93725	−0.325944	−0.162972	−	0.986631i	$-0.552108\pi$
$593$	−7.93725	−0.325944	−0.162972	+	0.986631i	$0.552108\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	23.8118	0.972922	0.486461	−	0.873702i	$-0.338288\pi$
$599$	23.8118	0.972922	0.486461	+	0.873702i	$0.338288\pi$
$600$	0	0
$601$	−14.0000	−0.571072	−0.285536	−	0.958368i	$-0.592172\pi$
$601$	−14.0000	−0.571072	−0.285536	+	0.958368i	$0.592172\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	10.5830	0.430260
$606$	0	0
$607$	−36.0000	−1.46119	−0.730597	−	0.682808i	$-0.760758\pi$
$607$	−36.0000	−1.46119	−0.730597	+	0.682808i	$0.760758\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−10.5830	−0.428143
$612$	0	0
$613$	−37.0000	−1.49442	−0.747208	−	0.664590i	$-0.768606\pi$
$613$	−37.0000	−1.49442	−0.747208	+	0.664590i	$0.768606\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−26.4575	−1.06514	−0.532570	−	0.846386i	$-0.678774\pi$
$617$	−26.4575	−1.06514	−0.532570	+	0.846386i	$0.678774\pi$
$618$	0	0
$619$	17.0000	0.683288	0.341644	−	0.939829i	$-0.389016\pi$
$619$	17.0000	0.683288	0.341644	+	0.939829i	$0.389016\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−31.0000	−1.24000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	6.00000	0.238856	0.119428	−	0.992843i	$-0.461894\pi$
$631$	6.00000	0.238856	0.119428	+	0.992843i	$0.461894\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−5.29150	−0.209987
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−10.5830	−0.418004	−0.209002	−	0.977915i	$-0.567021\pi$
$641$	−10.5830	−0.418004	−0.209002	+	0.977915i	$0.567021\pi$
$642$	0	0
$643$	21.0000	0.828159	0.414080	−	0.910241i	$-0.364104\pi$
$643$	21.0000	0.828159	0.414080	+	0.910241i	$0.364104\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−15.8745	−0.624091	−0.312046	−	0.950067i	$-0.601014\pi$
$647$	−15.8745	−0.624091	−0.312046	+	0.950067i	$0.601014\pi$
$648$	0	0
$649$	14.0000	0.549548
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−5.29150	−0.207072	−0.103536	−	0.994626i	$-0.533016\pi$
$653$	−5.29150	−0.207072	−0.103536	+	0.994626i	$0.533016\pi$
$654$	0	0
$655$	42.0000	1.64108
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−18.5203	−0.721447	−0.360723	−	0.932673i	$-0.617470\pi$
$659$	−18.5203	−0.721447	−0.360723	+	0.932673i	$0.617470\pi$
$660$	0	0
$661$	10.0000	0.388955	0.194477	−	0.980907i	$-0.437699\pi$
$661$	10.0000	0.388955	0.194477	+	0.980907i	$0.437699\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−42.0000	−1.62625
$668$	0	0
$669$	0	0
$670$	0	0
$671$	21.1660	0.817105
$672$	0	0
$673$	−34.0000	−1.31060	−0.655302	−	0.755367i	$-0.727459\pi$
$673$	−34.0000	−1.31060	−0.655302	+	0.755367i	$0.727459\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−39.6863	−1.52527	−0.762634	−	0.646831i	$-0.776094\pi$
$677$	−39.6863	−1.52527	−0.762634	+	0.646831i	$0.776094\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	29.1033	1.11361	0.556803	−	0.830645i	$-0.312028\pi$
$683$	29.1033	1.11361	0.556803	+	0.830645i	$0.312028\pi$
$684$	0	0
$685$	14.0000	0.534913
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−21.1660	−0.806361
$690$	0	0
$691$	−28.0000	−1.06517	−0.532585	−	0.846376i	$-0.678779\pi$
$691$	−28.0000	−1.06517	−0.532585	+	0.846376i	$0.678779\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	31.7490	1.20431
$696$	0	0
$697$	0	0
$698$	0	0
$699$	0	0
$700$	0	0
$701$	37.0405	1.39900	0.699501	−	0.714632i	$-0.253406\pi$
$701$	37.0405	1.39900	0.699501	+	0.714632i	$0.253406\pi$
$702$	0	0
$703$	−3.00000	−0.113147
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	25.0000	0.938895	0.469447	−	0.882960i	$-0.344453\pi$
$709$	25.0000	0.938895	0.469447	+	0.882960i	$0.344453\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−55.5608	−2.08077
$714$	0	0
$715$	14.0000	0.523570
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−47.6235	−1.77606	−0.888029	−	0.459788i	$-0.847926\pi$
$719$	−47.6235	−1.77606	−0.888029	+	0.459788i	$0.847926\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−10.5830	−0.393043
$726$	0	0
$727$	28.0000	1.03846	0.519231	−	0.854634i	$-0.326218\pi$
$727$	28.0000	1.03846	0.519231	+	0.854634i	$0.326218\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	0	0
$732$	0	0
$733$	14.0000	0.517102	0.258551	−	0.965998i	$-0.416755\pi$
$733$	14.0000	0.517102	0.258551	+	0.965998i	$0.416755\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−5.29150	−0.194915
$738$	0	0
$739$	4.00000	0.147142	0.0735712	−	0.997290i	$-0.476560\pi$
$739$	4.00000	0.147142	0.0735712	+	0.997290i	$0.476560\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−39.6863	−1.45595	−0.727974	−	0.685605i	$-0.759538\pi$
$743$	−39.6863	−1.45595	−0.727974	+	0.685605i	$0.759538\pi$
$744$	0	0
$745$	56.0000	2.05168
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	18.0000	0.656829	0.328415	−	0.944534i	$-0.393486\pi$
$751$	18.0000	0.656829	0.328415	+	0.944534i	$0.393486\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−42.3320	−1.54062
$756$	0	0
$757$	22.0000	0.799604	0.399802	−	0.916602i	$-0.369079\pi$
$757$	22.0000	0.799604	0.399802	+	0.916602i	$0.369079\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	52.9150	1.91817	0.959084	−	0.283121i	$-0.0913699\pi$
$761$	52.9150	1.91817	0.959084	+	0.283121i	$0.0913699\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−10.5830	−0.382130
$768$	0	0
$769$	−26.0000	−0.937584	−0.468792	−	0.883309i	$-0.655311\pi$
$769$	−26.0000	−0.937584	−0.468792	+	0.883309i	$0.655311\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−7.93725	−0.285483	−0.142742	−	0.989760i	$-0.545592\pi$
$773$	−7.93725	−0.285483	−0.142742	+	0.989760i	$0.545592\pi$
$774$	0	0
$775$	−14.0000	−0.502895
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−7.93725	−0.284382
$780$	0	0
$781$	7.00000	0.250480
$782$	0	0
$783$	0	0
$784$	0	0
$785$	37.0405	1.32203
$786$	0	0
$787$	−4.00000	−0.142585	−0.0712923	−	0.997455i	$-0.522712\pi$
$787$	−4.00000	−0.142585	−0.0712923	+	0.997455i	$0.522712\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−16.0000	−0.568177
$794$	0	0
$795$	0	0
$796$	0	0
$797$	23.8118	0.843456	0.421728	−	0.906722i	$-0.361424\pi$
$797$	23.8118	0.843456	0.421728	+	0.906722i	$0.361424\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−26.4575	−0.933665
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−47.6235	−1.67435	−0.837177	−	0.546932i	$-0.815796\pi$
$809$	−47.6235	−1.67435	−0.837177	+	0.546932i	$0.815796\pi$
$810$	0	0
$811$	49.0000	1.72062	0.860311	−	0.509769i	$-0.170269\pi$
$811$	49.0000	1.72062	0.860311	+	0.509769i	$0.170269\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	47.6235	1.66818
$816$	0	0
$817$	−2.00000	−0.0699711
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−37.0405	−1.29272	−0.646362	−	0.763031i	$-0.723710\pi$
$821$	−37.0405	−1.29272	−0.646362	+	0.763031i	$0.723710\pi$
$822$	0	0
$823$	−44.0000	−1.53374	−0.766872	−	0.641800i	$-0.778188\pi$
$823$	−44.0000	−1.53374	−0.766872	+	0.641800i	$0.778188\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−7.93725	−0.276005	−0.138003	−	0.990432i	$-0.544068\pi$
$827$	−7.93725	−0.276005	−0.138003	+	0.990432i	$0.544068\pi$
$828$	0	0
$829$	38.0000	1.31979	0.659897	−	0.751356i	$-0.270600\pi$
$829$	38.0000	1.31979	0.659897	+	0.751356i	$0.270600\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0	0
$834$	0	0
$835$	56.0000	1.93796
$836$	0	0
$837$	0	0
$838$	0	0
$839$	15.8745	0.548049	0.274024	−	0.961723i	$-0.411645\pi$
$839$	15.8745	0.548049	0.274024	+	0.961723i	$0.411645\pi$
$840$	0	0
$841$	−1.00000	−0.0344828
$842$	0	0
$843$	0	0
$844$	0	0
$845$	23.8118	0.819150
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−23.8118	−0.816257
$852$	0	0
$853$	40.0000	1.36957	0.684787	−	0.728743i	$-0.259895\pi$
$853$	40.0000	1.36957	0.684787	+	0.728743i	$0.259895\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−7.93725	−0.271131	−0.135566	−	0.990768i	$-0.543285\pi$
$857$	−7.93725	−0.271131	−0.135566	+	0.990768i	$0.543285\pi$
$858$	0	0
$859$	−15.0000	−0.511793	−0.255897	−	0.966704i	$-0.582371\pi$
$859$	−15.0000	−0.511793	−0.255897	+	0.966704i	$0.582371\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−47.6235	−1.62112	−0.810562	−	0.585653i	$-0.800838\pi$
$863$	−47.6235	−1.62112	−0.810562	+	0.585653i	$0.800838\pi$
$864$	0	0
$865$	−49.0000	−1.66605
$866$	0	0
$867$	0	0
$868$	0	0
$869$	26.4575	0.897510
$870$	0	0
$871$	4.00000	0.135535
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−18.0000	−0.607817	−0.303908	−	0.952701i	$-0.598292\pi$
$877$	−18.0000	−0.607817	−0.303908	+	0.952701i	$0.598292\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	29.1033	0.980514	0.490257	−	0.871578i	$-0.336903\pi$
$881$	29.1033	0.980514	0.490257	+	0.871578i	$0.336903\pi$
$882$	0	0
$883$	−20.0000	−0.673054	−0.336527	−	0.941674i	$-0.609252\pi$
$883$	−20.0000	−0.673054	−0.336527	+	0.941674i	$0.609252\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	15.8745	0.533014	0.266507	−	0.963833i	$-0.414130\pi$
$887$	15.8745	0.533014	0.266507	+	0.963833i	$0.414130\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	0	0
$893$	5.29150	0.177073
$894$	0	0
$895$	42.0000	1.40391
$896$	0	0
$897$	0	0
$898$	0	0
$899$	37.0405	1.23537
$900$	0	0
$901$	0	0
$902$	0	0
$903$	0	0
$904$	0	0
$905$	42.3320	1.40716
$906$	0	0
$907$	16.0000	0.531271	0.265636	−	0.964073i	$-0.414418\pi$
$907$	16.0000	0.531271	0.265636	+	0.964073i	$0.414418\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−26.4575	−0.876577	−0.438288	−	0.898834i	$-0.644415\pi$
$911$	−26.4575	−0.876577	−0.438288	+	0.898834i	$0.644415\pi$
$912$	0	0
$913$	42.0000	1.39000
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−36.0000	−1.18753	−0.593765	−	0.804638i	$-0.702359\pi$
$919$	−36.0000	−1.18753	−0.593765	+	0.804638i	$0.702359\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−5.29150	−0.174172
$924$	0	0
$925$	−6.00000	−0.197279
$926$	0	0
$927$	0	0
$928$	0	0
$929$	31.7490	1.04165	0.520826	−	0.853663i	$-0.325624\pi$
$929$	31.7490	1.04165	0.520826	+	0.853663i	$0.325624\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−12.0000	−0.392023	−0.196011	−	0.980602i	$-0.562799\pi$
$937$	−12.0000	−0.392023	−0.196011	+	0.980602i	$0.562799\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−23.8118	−0.776241	−0.388121	−	0.921609i	$-0.626876\pi$
$941$	−23.8118	−0.776241	−0.388121	+	0.921609i	$0.626876\pi$
$942$	0	0
$943$	−63.0000	−2.05156
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−60.8523	−1.97743	−0.988717	−	0.149797i	$-0.952138\pi$
$947$	−60.8523	−1.97743	−0.988717	+	0.149797i	$0.952138\pi$
$948$	0	0
$949$	20.0000	0.649227
$950$	0	0
$951$	0	0
$952$	0	0
$953$	15.8745	0.514226	0.257113	−	0.966381i	$-0.417229\pi$
$953$	15.8745	0.514226	0.257113	+	0.966381i	$0.417229\pi$
$954$	0	0
$955$	21.0000	0.679544
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	18.0000	0.580645
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−5.29150	−0.170339
$966$	0	0
$967$	−8.00000	−0.257263	−0.128631	−	0.991692i	$-0.541058\pi$
$967$	−8.00000	−0.257263	−0.128631	+	0.991692i	$0.541058\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	37.0405	1.18869	0.594343	−	0.804211i	$-0.297412\pi$
$971$	37.0405	1.18869	0.594343	+	0.804211i	$0.297412\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−21.1660	−0.677161	−0.338580	−	0.940937i	$-0.609947\pi$
$977$	−21.1660	−0.677161	−0.338580	+	0.940937i	$0.609947\pi$
$978$	0	0
$979$	7.00000	0.223721
$980$	0	0
$981$	0	0
$982$	0	0
$983$	37.0405	1.18141	0.590705	−	0.806888i	$-0.298850\pi$
$983$	37.0405	1.18141	0.590705	+	0.806888i	$0.298850\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−15.8745	−0.504780
$990$	0	0
$991$	2.00000	0.0635321	0.0317660	−	0.999495i	$-0.489887\pi$
$991$	2.00000	0.0635321	0.0317660	+	0.999495i	$0.489887\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	66.1438	2.09690
$996$	0	0
$997$	28.0000	0.886769	0.443384	−	0.896332i	$-0.353778\pi$
$997$	28.0000	0.886769	0.443384	+	0.896332i	$0.353778\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5292.2.a.p.1.1	✓	2
3.2	odd	2	inner	5292.2.a.p.1.2	yes	2
7.6	odd	2		5292.2.a.s.1.2	yes	2
21.20	even	2		5292.2.a.s.1.1	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5292.2.a.p.1.1	✓	2	1.1	even	1	trivial
5292.2.a.p.1.2	yes	2	3.2	odd	2	inner
5292.2.a.s.1.1	yes	2	21.20	even	2
5292.2.a.s.1.2	yes	2	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)

\(f(q)\)	\(=\)	\(q-2.64575 q^{5} +O(q^{10})\)	\(q-2.64575 q^{5} +2.64575 q^{11} -2.00000 q^{13} +1.00000 q^{19} +7.93725 q^{23} +2.00000 q^{25} -5.29150 q^{29} -7.00000 q^{31} -3.00000 q^{37} -7.93725 q^{41} -2.00000 q^{43} +5.29150 q^{47} +10.5830 q^{53} -7.00000 q^{55} +5.29150 q^{59} +8.00000 q^{61} +5.29150 q^{65} -2.00000 q^{67} +2.64575 q^{71} -10.0000 q^{73} +10.0000 q^{79} +15.8745 q^{83} +2.64575 q^{89} -2.64575 q^{95} -16.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q+O(q^{10}) \) 2 * q	\( 2 q - 4 q^{13} + 2 q^{19} + 4 q^{25} - 14 q^{31} - 6 q^{37} - 4 q^{43} - 14 q^{55} + 16 q^{61} - 4 q^{67} - 20 q^{73} + 20 q^{79} - 32 q^{97}+O(q^{100}) \) 2 * q - 4 * q^13 + 2 * q^19 + 4 * q^25 - 14 * q^31 - 6 * q^37 - 4 * q^43 - 14 * q^55 + 16 * q^61 - 4 * q^67 - 20 * q^73 + 20 * q^79 - 32 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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