LMFDB - Embedded newform 5472.2.a.be.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5472,2,Mod(1,5472)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5472, 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("5472.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5472 = 2^{5} \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5472.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:	$43.6941399860$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{33}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 8 $ x^2 - x - 8
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1824)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$3.37228$ of defining polynomial
Character	$\chi$	$=$	5472.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.37228 q^{5} +3.37228 q^{7} +O(q^{10})$	$q+3.37228 q^{5} +3.37228 q^{7} +0.627719 q^{11} -4.00000 q^{13} +5.37228 q^{17} -1.00000 q^{19} -4.74456 q^{23} +6.37228 q^{25} +8.74456 q^{29} -6.00000 q^{31} +11.3723 q^{35} -4.00000 q^{37} -8.74456 q^{41} +7.37228 q^{43} +8.11684 q^{47} +4.37228 q^{49} +10.0000 q^{53} +2.11684 q^{55} +2.74456 q^{59} +9.37228 q^{61} -13.4891 q^{65} -6.74456 q^{67} +14.7446 q^{71} +2.62772 q^{73} +2.11684 q^{77} -2.00000 q^{79} +18.1168 q^{85} +7.48913 q^{89} -13.4891 q^{91} -3.37228 q^{95} +7.25544 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{5} + q^{7}+O(q^{10}) $ 2 * q + q^5 + q^7	$ 2 q + q^{5} + q^{7} + 7 q^{11} - 8 q^{13} + 5 q^{17} - 2 q^{19} + 2 q^{23} + 7 q^{25} + 6 q^{29} - 12 q^{31} + 17 q^{35} - 8 q^{37} - 6 q^{41} + 9 q^{43} - q^{47} + 3 q^{49} + 20 q^{53} - 13 q^{55} - 6 q^{59} + 13 q^{61} - 4 q^{65} - 2 q^{67} + 18 q^{71} + 11 q^{73} - 13 q^{77} - 4 q^{79} + 19 q^{85} - 8 q^{89} - 4 q^{91} - q^{95} + 26 q^{97}+O(q^{100}) $ 2 * q + q^5 + q^7 + 7 * q^11 - 8 * q^13 + 5 * q^17 - 2 * q^19 + 2 * q^23 + 7 * q^25 + 6 * q^29 - 12 * q^31 + 17 * q^35 - 8 * q^37 - 6 * q^41 + 9 * q^43 - q^47 + 3 * q^49 + 20 * q^53 - 13 * q^55 - 6 * q^59 + 13 * q^61 - 4 * q^65 - 2 * q^67 + 18 * q^71 + 11 * q^73 - 13 * q^77 - 4 * q^79 + 19 * q^85 - 8 * q^89 - 4 * q^91 - q^95 + 26 * 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
$5$	3.37228	1.50813	0.754065	−	0.656800i	$-0.228090\pi$
$5$	3.37228	1.50813	0.754065	+	0.656800i	$0.228090\pi$
$6$	0	0
$7$	3.37228	1.27460	0.637301	−	0.770615i	$-0.280051\pi$
$7$	3.37228	1.27460	0.637301	+	0.770615i	$0.280051\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	0.627719	0.189264	0.0946322	−	0.995512i	$-0.469833\pi$
$11$	0.627719	0.189264	0.0946322	+	0.995512i	$0.469833\pi$
$12$	0	0
$13$	−4.00000	−1.10940	−0.554700	−	0.832050i	$-0.687167\pi$
$13$	−4.00000	−1.10940	−0.554700	+	0.832050i	$0.687167\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	5.37228	1.30297	0.651485	−	0.758662i	$-0.274146\pi$
$17$	5.37228	1.30297	0.651485	+	0.758662i	$0.274146\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−4.74456	−0.989310	−0.494655	−	0.869090i	$-0.664706\pi$
$23$	−4.74456	−0.989310	−0.494655	+	0.869090i	$0.664706\pi$
$24$	0	0
$25$	6.37228	1.27446
$26$	0	0
$27$	0	0
$28$	0	0
$29$	8.74456	1.62382	0.811912	−	0.583779i	$-0.198427\pi$
$29$	8.74456	1.62382	0.811912	+	0.583779i	$0.198427\pi$
$30$	0	0
$31$	−6.00000	−1.07763	−0.538816	−	0.842424i	$-0.681128\pi$
$31$	−6.00000	−1.07763	−0.538816	+	0.842424i	$0.681128\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	11.3723	1.92227
$36$	0	0
$37$	−4.00000	−0.657596	−0.328798	−	0.944400i	$-0.606644\pi$
$37$	−4.00000	−0.657596	−0.328798	+	0.944400i	$0.606644\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−8.74456	−1.36567	−0.682836	−	0.730572i	$-0.739253\pi$
$41$	−8.74456	−1.36567	−0.682836	+	0.730572i	$0.739253\pi$
$42$	0	0
$43$	7.37228	1.12426	0.562131	−	0.827048i	$-0.309982\pi$
$43$	7.37228	1.12426	0.562131	+	0.827048i	$0.309982\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	8.11684	1.18396	0.591982	−	0.805951i	$-0.298346\pi$
$47$	8.11684	1.18396	0.591982	+	0.805951i	$0.298346\pi$
$48$	0	0
$49$	4.37228	0.624612
$50$	0	0
$51$	0	0
$52$	0	0
$53$	10.0000	1.37361	0.686803	−	0.726844i	$-0.259014\pi$
$53$	10.0000	1.37361	0.686803	+	0.726844i	$0.259014\pi$
$54$	0	0
$55$	2.11684	0.285435
$56$	0	0
$57$	0	0
$58$	0	0
$59$	2.74456	0.357312	0.178656	−	0.983912i	$-0.442825\pi$
$59$	2.74456	0.357312	0.178656	+	0.983912i	$0.442825\pi$
$60$	0	0
$61$	9.37228	1.20000	0.599999	−	0.800001i	$-0.295168\pi$
$61$	9.37228	1.20000	0.599999	+	0.800001i	$0.295168\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−13.4891	−1.67312
$66$	0	0
$67$	−6.74456	−0.823979	−0.411990	−	0.911188i	$-0.635166\pi$
$67$	−6.74456	−0.823979	−0.411990	+	0.911188i	$0.635166\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	14.7446	1.74986	0.874929	−	0.484252i	$-0.160908\pi$
$71$	14.7446	1.74986	0.874929	+	0.484252i	$0.160908\pi$
$72$	0	0
$73$	2.62772	0.307551	0.153776	−	0.988106i	$-0.450857\pi$
$73$	2.62772	0.307551	0.153776	+	0.988106i	$0.450857\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	2.11684	0.241237
$78$	0	0
$79$	−2.00000	−0.225018	−0.112509	−	0.993651i	$-0.535889\pi$
$79$	−2.00000	−0.225018	−0.112509	+	0.993651i	$0.535889\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	0	0
$85$	18.1168	1.96505
$86$	0	0
$87$	0	0
$88$	0	0
$89$	7.48913	0.793846	0.396923	−	0.917852i	$-0.370078\pi$
$89$	7.48913	0.793846	0.396923	+	0.917852i	$0.370078\pi$
$90$	0	0
$91$	−13.4891	−1.41404
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−3.37228	−0.345989
$96$	0	0
$97$	7.25544	0.736678	0.368339	−	0.929692i	$-0.379927\pi$
$97$	7.25544	0.736678	0.368339	+	0.929692i	$0.379927\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	14.7446	1.46714	0.733569	−	0.679615i	$-0.237853\pi$
$101$	14.7446	1.46714	0.733569	+	0.679615i	$0.237853\pi$
$102$	0	0
$103$	−10.0000	−0.985329	−0.492665	−	0.870219i	$-0.663977\pi$
$103$	−10.0000	−0.985329	−0.492665	+	0.870219i	$0.663977\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−2.74456	−0.265327	−0.132663	−	0.991161i	$-0.542353\pi$
$107$	−2.74456	−0.265327	−0.132663	+	0.991161i	$0.542353\pi$
$108$	0	0
$109$	6.74456	0.646012	0.323006	−	0.946397i	$-0.395307\pi$
$109$	6.74456	0.646012	0.323006	+	0.946397i	$0.395307\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	4.74456	0.446331	0.223165	−	0.974781i	$-0.428361\pi$
$113$	4.74456	0.446331	0.223165	+	0.974781i	$0.428361\pi$
$114$	0	0
$115$	−16.0000	−1.49201
$116$	0	0
$117$	0	0
$118$	0	0
$119$	18.1168	1.66077
$120$	0	0
$121$	−10.6060	−0.964179
$122$	0	0
$123$	0	0
$124$	0	0
$125$	4.62772	0.413916
$126$	0	0
$127$	−18.2337	−1.61798	−0.808989	−	0.587824i	$-0.799985\pi$
$127$	−18.2337	−1.61798	−0.808989	+	0.587824i	$0.799985\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−2.11684	−0.184950	−0.0924748	−	0.995715i	$-0.529478\pi$
$131$	−2.11684	−0.184950	−0.0924748	+	0.995715i	$0.529478\pi$
$132$	0	0
$133$	−3.37228	−0.292414
$134$	0	0
$135$	0	0
$136$	0	0
$137$	21.3723	1.82596	0.912979	−	0.408007i	$-0.133776\pi$
$137$	21.3723	1.82596	0.912979	+	0.408007i	$0.133776\pi$
$138$	0	0
$139$	−8.62772	−0.731794	−0.365897	−	0.930655i	$-0.619238\pi$
$139$	−8.62772	−0.731794	−0.365897	+	0.930655i	$0.619238\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−2.51087	−0.209970
$144$	0	0
$145$	29.4891	2.44894
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−12.8614	−1.05365	−0.526824	−	0.849975i	$-0.676617\pi$
$149$	−12.8614	−1.05365	−0.526824	+	0.849975i	$0.676617\pi$
$150$	0	0
$151$	11.4891	0.934972	0.467486	−	0.884001i	$-0.345160\pi$
$151$	11.4891	0.934972	0.467486	+	0.884001i	$0.345160\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−20.2337	−1.62521
$156$	0	0
$157$	2.00000	0.159617	0.0798087	−	0.996810i	$-0.474569\pi$
$157$	2.00000	0.159617	0.0798087	+	0.996810i	$0.474569\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−16.0000	−1.26098
$162$	0	0
$163$	12.0000	0.939913	0.469956	−	0.882690i	$-0.344270\pi$
$163$	12.0000	0.939913	0.469956	+	0.882690i	$0.344270\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−25.4891	−1.97241	−0.986204	−	0.165535i	$-0.947065\pi$
$167$	−25.4891	−1.97241	−0.986204	+	0.165535i	$0.947065\pi$
$168$	0	0
$169$	3.00000	0.230769
$170$	0	0
$171$	0	0
$172$	0	0
$173$	6.00000	0.456172	0.228086	−	0.973641i	$-0.426753\pi$
$173$	6.00000	0.456172	0.228086	+	0.973641i	$0.426753\pi$
$174$	0	0
$175$	21.4891	1.62443
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−4.00000	−0.298974	−0.149487	−	0.988764i	$-0.547762\pi$
$179$	−4.00000	−0.298974	−0.149487	+	0.988764i	$0.547762\pi$
$180$	0	0
$181$	−6.74456	−0.501319	−0.250660	−	0.968075i	$-0.580648\pi$
$181$	−6.74456	−0.501319	−0.250660	+	0.968075i	$0.580648\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−13.4891	−0.991740
$186$	0	0
$187$	3.37228	0.246606
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−21.6060	−1.56335	−0.781677	−	0.623684i	$-0.785635\pi$
$191$	−21.6060	−1.56335	−0.781677	+	0.623684i	$0.785635\pi$
$192$	0	0
$193$	24.9783	1.79797	0.898987	−	0.437975i	$-0.144304\pi$
$193$	24.9783	1.79797	0.898987	+	0.437975i	$0.144304\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−8.23369	−0.586626	−0.293313	−	0.956016i	$-0.594758\pi$
$197$	−8.23369	−0.586626	−0.293313	+	0.956016i	$0.594758\pi$
$198$	0	0
$199$	19.3723	1.37326	0.686632	−	0.727005i	$-0.259088\pi$
$199$	19.3723	1.37326	0.686632	+	0.727005i	$0.259088\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	29.4891	2.06973
$204$	0	0
$205$	−29.4891	−2.05961
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−0.627719	−0.0434202
$210$	0	0
$211$	−16.2337	−1.11757	−0.558787	−	0.829311i	$-0.688733\pi$
$211$	−16.2337	−1.11757	−0.558787	+	0.829311i	$0.688733\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	24.8614	1.69553
$216$	0	0
$217$	−20.2337	−1.37355
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−21.4891	−1.44551
$222$	0	0
$223$	−20.7446	−1.38916	−0.694579	−	0.719416i	$-0.744409\pi$
$223$	−20.7446	−1.38916	−0.694579	+	0.719416i	$0.744409\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−1.48913	−0.0988367	−0.0494184	−	0.998778i	$-0.515737\pi$
$227$	−1.48913	−0.0988367	−0.0494184	+	0.998778i	$0.515737\pi$
$228$	0	0
$229$	2.62772	0.173645	0.0868223	−	0.996224i	$-0.472329\pi$
$229$	2.62772	0.173645	0.0868223	+	0.996224i	$0.472329\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−13.3723	−0.876047	−0.438024	−	0.898963i	$-0.644321\pi$
$233$	−13.3723	−0.876047	−0.438024	+	0.898963i	$0.644321\pi$
$234$	0	0
$235$	27.3723	1.78557
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−20.1168	−1.30125	−0.650625	−	0.759399i	$-0.725493\pi$
$239$	−20.1168	−1.30125	−0.650625	+	0.759399i	$0.725493\pi$
$240$	0	0
$241$	−22.2337	−1.43220	−0.716099	−	0.697999i	$-0.754074\pi$
$241$	−22.2337	−1.43220	−0.716099	+	0.697999i	$0.754074\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	14.7446	0.941996
$246$	0	0
$247$	4.00000	0.254514
$248$	0	0
$249$	0	0
$250$	0	0
$251$	16.8614	1.06428	0.532141	−	0.846656i	$-0.321388\pi$
$251$	16.8614	1.06428	0.532141	+	0.846656i	$0.321388\pi$
$252$	0	0
$253$	−2.97825	−0.187241
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−15.2554	−0.951608	−0.475804	−	0.879551i	$-0.657843\pi$
$257$	−15.2554	−0.951608	−0.475804	+	0.879551i	$0.657843\pi$
$258$	0	0
$259$	−13.4891	−0.838173
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−13.3723	−0.824570	−0.412285	−	0.911055i	$-0.635269\pi$
$263$	−13.3723	−0.824570	−0.412285	+	0.911055i	$0.635269\pi$
$264$	0	0
$265$	33.7228	2.07158
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−20.9783	−1.27907	−0.639533	−	0.768763i	$-0.720872\pi$
$269$	−20.9783	−1.27907	−0.639533	+	0.768763i	$0.720872\pi$
$270$	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
$275$	4.00000	0.241209
$276$	0	0
$277$	32.1168	1.92971	0.964857	−	0.262775i	$-0.0846378\pi$
$277$	32.1168	1.92971	0.964857	+	0.262775i	$0.0846378\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	8.51087	0.507716	0.253858	−	0.967241i	$-0.418300\pi$
$281$	8.51087	0.507716	0.253858	+	0.967241i	$0.418300\pi$
$282$	0	0
$283$	4.86141	0.288981	0.144490	−	0.989506i	$-0.453846\pi$
$283$	4.86141	0.288981	0.144490	+	0.989506i	$0.453846\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−29.4891	−1.74069
$288$	0	0
$289$	11.8614	0.697730
$290$	0	0
$291$	0	0
$292$	0	0
$293$	28.7446	1.67928	0.839638	−	0.543147i	$-0.182767\pi$
$293$	28.7446	1.67928	0.839638	+	0.543147i	$0.182767\pi$
$294$	0	0
$295$	9.25544	0.538872
$296$	0	0
$297$	0	0
$298$	0	0
$299$	18.9783	1.09754
$300$	0	0
$301$	24.8614	1.43299
$302$	0	0
$303$	0	0
$304$	0	0
$305$	31.6060	1.80975
$306$	0	0
$307$	−20.2337	−1.15480	−0.577399	−	0.816462i	$-0.695932\pi$
$307$	−20.2337	−1.15480	−0.577399	+	0.816462i	$0.695932\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−2.86141	−0.162255	−0.0811277	−	0.996704i	$-0.525852\pi$
$311$	−2.86141	−0.162255	−0.0811277	+	0.996704i	$0.525852\pi$
$312$	0	0
$313$	−20.9783	−1.18576	−0.592880	−	0.805291i	$-0.702009\pi$
$313$	−20.9783	−1.18576	−0.592880	+	0.805291i	$0.702009\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	0.510875	0.0286936	0.0143468	−	0.999897i	$-0.495433\pi$
$317$	0.510875	0.0286936	0.0143468	+	0.999897i	$0.495433\pi$
$318$	0	0
$319$	5.48913	0.307332
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−5.37228	−0.298922
$324$	0	0
$325$	−25.4891	−1.41388
$326$	0	0
$327$	0	0
$328$	0	0
$329$	27.3723	1.50908
$330$	0	0
$331$	28.2337	1.55186	0.775932	−	0.630817i	$-0.217280\pi$
$331$	28.2337	1.55186	0.775932	+	0.630817i	$0.217280\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−22.7446	−1.24267
$336$	0	0
$337$	22.2337	1.21115	0.605573	−	0.795790i	$-0.292944\pi$
$337$	22.2337	1.21115	0.605573	+	0.795790i	$0.292944\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−3.76631	−0.203957
$342$	0	0
$343$	−8.86141	−0.478471
$344$	0	0
$345$	0	0
$346$	0	0
$347$	1.88316	0.101093	0.0505466	−	0.998722i	$-0.483904\pi$
$347$	1.88316	0.101093	0.0505466	+	0.998722i	$0.483904\pi$
$348$	0	0
$349$	−1.37228	−0.0734565	−0.0367283	−	0.999325i	$-0.511694\pi$
$349$	−1.37228	−0.0734565	−0.0367283	+	0.999325i	$0.511694\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−20.9783	−1.11656	−0.558280	−	0.829653i	$-0.688538\pi$
$353$	−20.9783	−1.11656	−0.558280	+	0.829653i	$0.688538\pi$
$354$	0	0
$355$	49.7228	2.63901
$356$	0	0
$357$	0	0
$358$	0	0
$359$	16.3505	0.862948	0.431474	−	0.902125i	$-0.357994\pi$
$359$	16.3505	0.862948	0.431474	+	0.902125i	$0.357994\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	8.86141	0.463827
$366$	0	0
$367$	−32.4674	−1.69478	−0.847392	−	0.530968i	$-0.821828\pi$
$367$	−32.4674	−1.69478	−0.847392	+	0.530968i	$0.821828\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	33.7228	1.75080
$372$	0	0
$373$	9.48913	0.491328	0.245664	−	0.969355i	$-0.420994\pi$
$373$	9.48913	0.491328	0.245664	+	0.969355i	$0.420994\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−34.9783	−1.80147
$378$	0	0
$379$	30.7446	1.57924	0.789621	−	0.613595i	$-0.210277\pi$
$379$	30.7446	1.57924	0.789621	+	0.613595i	$0.210277\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−17.2554	−0.881712	−0.440856	−	0.897578i	$-0.645325\pi$
$383$	−17.2554	−0.881712	−0.440856	+	0.897578i	$0.645325\pi$
$384$	0	0
$385$	7.13859	0.363816
$386$	0	0
$387$	0	0
$388$	0	0
$389$	13.8832	0.703904	0.351952	−	0.936018i	$-0.385518\pi$
$389$	13.8832	0.703904	0.351952	+	0.936018i	$0.385518\pi$
$390$	0	0
$391$	−25.4891	−1.28904
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−6.74456	−0.339356
$396$	0	0
$397$	−36.3505	−1.82438	−0.912190	−	0.409766i	$-0.865610\pi$
$397$	−36.3505	−1.82438	−0.912190	+	0.409766i	$0.865610\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	3.48913	0.174239	0.0871193	−	0.996198i	$-0.472234\pi$
$401$	3.48913	0.174239	0.0871193	+	0.996198i	$0.472234\pi$
$402$	0	0
$403$	24.0000	1.19553
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−2.51087	−0.124459
$408$	0	0
$409$	32.9783	1.63067	0.815335	−	0.578990i	$-0.196553\pi$
$409$	32.9783	1.63067	0.815335	+	0.578990i	$0.196553\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	9.25544	0.455430
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	12.0000	0.586238	0.293119	−	0.956076i	$-0.405307\pi$
$419$	12.0000	0.586238	0.293119	+	0.956076i	$0.405307\pi$
$420$	0	0
$421$	−18.7446	−0.913554	−0.456777	−	0.889581i	$-0.650996\pi$
$421$	−18.7446	−0.913554	−0.456777	+	0.889581i	$0.650996\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	34.2337	1.66058
$426$	0	0
$427$	31.6060	1.52952
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−4.00000	−0.192673	−0.0963366	−	0.995349i	$-0.530713\pi$
$431$	−4.00000	−0.192673	−0.0963366	+	0.995349i	$0.530713\pi$
$432$	0	0
$433$	1.76631	0.0848835	0.0424418	−	0.999099i	$-0.486486\pi$
$433$	1.76631	0.0848835	0.0424418	+	0.999099i	$0.486486\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	4.74456	0.226963
$438$	0	0
$439$	6.23369	0.297518	0.148759	−	0.988874i	$-0.452472\pi$
$439$	6.23369	0.297518	0.148759	+	0.988874i	$0.452472\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−30.1168	−1.43089	−0.715447	−	0.698667i	$-0.753777\pi$
$443$	−30.1168	−1.43089	−0.715447	+	0.698667i	$0.753777\pi$
$444$	0	0
$445$	25.2554	1.19722
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−7.25544	−0.342405	−0.171203	−	0.985236i	$-0.554765\pi$
$449$	−7.25544	−0.342405	−0.171203	+	0.985236i	$0.554765\pi$
$450$	0	0
$451$	−5.48913	−0.258473
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−45.4891	−2.13256
$456$	0	0
$457$	−32.1168	−1.50236	−0.751181	−	0.660096i	$-0.770516\pi$
$457$	−32.1168	−1.50236	−0.751181	+	0.660096i	$0.770516\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−17.8832	−0.832902	−0.416451	−	0.909158i	$-0.636726\pi$
$461$	−17.8832	−0.832902	−0.416451	+	0.909158i	$0.636726\pi$
$462$	0	0
$463$	20.6277	0.958651	0.479326	−	0.877637i	$-0.340881\pi$
$463$	20.6277	0.958651	0.479326	+	0.877637i	$0.340881\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	19.3723	0.896442	0.448221	−	0.893923i	$-0.352058\pi$
$467$	19.3723	0.896442	0.448221	+	0.893923i	$0.352058\pi$
$468$	0	0
$469$	−22.7446	−1.05025
$470$	0	0
$471$	0	0
$472$	0	0
$473$	4.62772	0.212783
$474$	0	0
$475$	−6.37228	−0.292380
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−1.76631	−0.0807049	−0.0403524	−	0.999186i	$-0.512848\pi$
$479$	−1.76631	−0.0807049	−0.0403524	+	0.999186i	$0.512848\pi$
$480$	0	0
$481$	16.0000	0.729537
$482$	0	0
$483$	0	0
$484$	0	0
$485$	24.4674	1.11101
$486$	0	0
$487$	30.4674	1.38061	0.690304	−	0.723519i	$-0.257477\pi$
$487$	30.4674	1.38061	0.690304	+	0.723519i	$0.257477\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	5.48913	0.247721	0.123860	−	0.992300i	$-0.460473\pi$
$491$	5.48913	0.247721	0.123860	+	0.992300i	$0.460473\pi$
$492$	0	0
$493$	46.9783	2.11579
$494$	0	0
$495$	0	0
$496$	0	0
$497$	49.7228	2.23037
$498$	0	0
$499$	11.6060	0.519555	0.259777	−	0.965669i	$-0.416351\pi$
$499$	11.6060	0.519555	0.259777	+	0.965669i	$0.416351\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	31.7228	1.41445	0.707225	−	0.706988i	$-0.249947\pi$
$503$	31.7228	1.41445	0.707225	+	0.706988i	$0.249947\pi$
$504$	0	0
$505$	49.7228	2.21264
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−8.74456	−0.387596	−0.193798	−	0.981041i	$-0.562081\pi$
$509$	−8.74456	−0.387596	−0.193798	+	0.981041i	$0.562081\pi$
$510$	0	0
$511$	8.86141	0.392006
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−33.7228	−1.48600
$516$	0	0
$517$	5.09509	0.224082
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−6.00000	−0.262865	−0.131432	−	0.991325i	$-0.541958\pi$
$521$	−6.00000	−0.262865	−0.131432	+	0.991325i	$0.541958\pi$
$522$	0	0
$523$	17.7228	0.774965	0.387482	−	0.921877i	$-0.373345\pi$
$523$	17.7228	0.774965	0.387482	+	0.921877i	$0.373345\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−32.2337	−1.40412
$528$	0	0
$529$	−0.489125	−0.0212663
$530$	0	0
$531$	0	0
$532$	0	0
$533$	34.9783	1.51508
$534$	0	0
$535$	−9.25544	−0.400147
$536$	0	0
$537$	0	0
$538$	0	0
$539$	2.74456	0.118217
$540$	0	0
$541$	32.3505	1.39086	0.695429	−	0.718595i	$-0.255214\pi$
$541$	32.3505	1.39086	0.695429	+	0.718595i	$0.255214\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	22.7446	0.974270
$546$	0	0
$547$	−2.51087	−0.107357	−0.0536786	−	0.998558i	$-0.517095\pi$
$547$	−2.51087	−0.107357	−0.0536786	+	0.998558i	$0.517095\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−8.74456	−0.372531
$552$	0	0
$553$	−6.74456	−0.286808
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−44.8614	−1.90084	−0.950419	−	0.310971i	$-0.899346\pi$
$557$	−44.8614	−1.90084	−0.950419	+	0.310971i	$0.899346\pi$
$558$	0	0
$559$	−29.4891	−1.24726
$560$	0	0
$561$	0	0
$562$	0	0
$563$	33.4891	1.41140	0.705699	−	0.708512i	$-0.250633\pi$
$563$	33.4891	1.41140	0.705699	+	0.708512i	$0.250633\pi$
$564$	0	0
$565$	16.0000	0.673125
$566$	0	0
$567$	0	0
$568$	0	0
$569$	19.2554	0.807230	0.403615	−	0.914929i	$-0.367754\pi$
$569$	19.2554	0.807230	0.403615	+	0.914929i	$0.367754\pi$
$570$	0	0
$571$	−17.4891	−0.731897	−0.365949	−	0.930635i	$-0.619255\pi$
$571$	−17.4891	−0.731897	−0.365949	+	0.930635i	$0.619255\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−30.2337	−1.26083
$576$	0	0
$577$	−5.13859	−0.213922	−0.106961	−	0.994263i	$-0.534112\pi$
$577$	−5.13859	−0.213922	−0.106961	+	0.994263i	$0.534112\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	6.27719	0.259975
$584$	0	0
$585$	0	0
$586$	0	0
$587$	42.3505	1.74799	0.873997	−	0.485932i	$-0.161520\pi$
$587$	42.3505	1.74799	0.873997	+	0.485932i	$0.161520\pi$
$588$	0	0
$589$	6.00000	0.247226
$590$	0	0
$591$	0	0
$592$	0	0
$593$	11.4891	0.471802	0.235901	−	0.971777i	$-0.424196\pi$
$593$	11.4891	0.471802	0.235901	+	0.971777i	$0.424196\pi$
$594$	0	0
$595$	61.0951	2.50465
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−8.23369	−0.336419	−0.168210	−	0.985751i	$-0.553799\pi$
$599$	−8.23369	−0.336419	−0.168210	+	0.985751i	$0.553799\pi$
$600$	0	0
$601$	−40.9783	−1.67154	−0.835769	−	0.549081i	$-0.814978\pi$
$601$	−40.9783	−1.67154	−0.835769	+	0.549081i	$0.814978\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−35.7663	−1.45411
$606$	0	0
$607$	−41.2119	−1.67274	−0.836370	−	0.548165i	$-0.815327\pi$
$607$	−41.2119	−1.67274	−0.836370	+	0.548165i	$0.815327\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−32.4674	−1.31349
$612$	0	0
$613$	22.6277	0.913925	0.456962	−	0.889486i	$-0.348937\pi$
$613$	22.6277	0.913925	0.456962	+	0.889486i	$0.348937\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−12.1168	−0.487806	−0.243903	−	0.969800i	$-0.578428\pi$
$617$	−12.1168	−0.487806	−0.243903	+	0.969800i	$0.578428\pi$
$618$	0	0
$619$	−14.9783	−0.602027	−0.301013	−	0.953620i	$-0.597325\pi$
$619$	−14.9783	−0.602027	−0.301013	+	0.953620i	$0.597325\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	25.2554	1.01184
$624$	0	0
$625$	−16.2554	−0.650217
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−21.4891	−0.856828
$630$	0	0
$631$	−14.1168	−0.561983	−0.280991	−	0.959710i	$-0.590663\pi$
$631$	−14.1168	−0.561983	−0.280991	+	0.959710i	$0.590663\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−61.4891	−2.44012
$636$	0	0
$637$	−17.4891	−0.692944
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−2.23369	−0.0882254	−0.0441127	−	0.999027i	$-0.514046\pi$
$641$	−2.23369	−0.0882254	−0.0441127	+	0.999027i	$0.514046\pi$
$642$	0	0
$643$	4.39403	0.173284	0.0866418	−	0.996240i	$-0.472386\pi$
$643$	4.39403	0.173284	0.0866418	+	0.996240i	$0.472386\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	40.3505	1.58634	0.793172	−	0.608998i	$-0.208428\pi$
$647$	40.3505	1.58634	0.793172	+	0.608998i	$0.208428\pi$
$648$	0	0
$649$	1.72281	0.0676263
$650$	0	0
$651$	0	0
$652$	0	0
$653$	11.1386	0.435887	0.217943	−	0.975961i	$-0.430065\pi$
$653$	11.1386	0.435887	0.217943	+	0.975961i	$0.430065\pi$
$654$	0	0
$655$	−7.13859	−0.278928
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−8.23369	−0.320739	−0.160369	−	0.987057i	$-0.551269\pi$
$659$	−8.23369	−0.320739	−0.160369	+	0.987057i	$0.551269\pi$
$660$	0	0
$661$	−6.51087	−0.253244	−0.126622	−	0.991951i	$-0.540413\pi$
$661$	−6.51087	−0.253244	−0.126622	+	0.991951i	$0.540413\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−11.3723	−0.440998
$666$	0	0
$667$	−41.4891	−1.60647
$668$	0	0
$669$	0	0
$670$	0	0
$671$	5.88316	0.227117
$672$	0	0
$673$	6.00000	0.231283	0.115642	−	0.993291i	$-0.463108\pi$
$673$	6.00000	0.231283	0.115642	+	0.993291i	$0.463108\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−39.7228	−1.52667	−0.763336	−	0.646002i	$-0.776440\pi$
$677$	−39.7228	−1.52667	−0.763336	+	0.646002i	$0.776440\pi$
$678$	0	0
$679$	24.4674	0.938972
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−17.4891	−0.669203	−0.334601	−	0.942360i	$-0.608602\pi$
$683$	−17.4891	−0.669203	−0.334601	+	0.942360i	$0.608602\pi$
$684$	0	0
$685$	72.0733	2.75378
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−40.0000	−1.52388
$690$	0	0
$691$	8.62772	0.328214	0.164107	−	0.986443i	$-0.447526\pi$
$691$	8.62772	0.328214	0.164107	+	0.986443i	$0.447526\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−29.0951	−1.10364
$696$	0	0
$697$	−46.9783	−1.77943
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−1.25544	−0.0474172	−0.0237086	−	0.999719i	$-0.507547\pi$
$701$	−1.25544	−0.0474172	−0.0237086	+	0.999719i	$0.507547\pi$
$702$	0	0
$703$	4.00000	0.150863
$704$	0	0
$705$	0	0
$706$	0	0
$707$	49.7228	1.87002
$708$	0	0
$709$	−42.0000	−1.57734	−0.788672	−	0.614815i	$-0.789231\pi$
$709$	−42.0000	−1.57734	−0.788672	+	0.614815i	$0.789231\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	28.4674	1.06611
$714$	0	0
$715$	−8.46738	−0.316662
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−22.6277	−0.843872	−0.421936	−	0.906626i	$-0.638649\pi$
$719$	−22.6277	−0.843872	−0.421936	+	0.906626i	$0.638649\pi$
$720$	0	0
$721$	−33.7228	−1.25590
$722$	0	0
$723$	0	0
$724$	0	0
$725$	55.7228	2.06949
$726$	0	0
$727$	−39.8397	−1.47757	−0.738786	−	0.673941i	$-0.764600\pi$
$727$	−39.8397	−1.47757	−0.738786	+	0.673941i	$0.764600\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	39.6060	1.46488
$732$	0	0
$733$	34.0000	1.25582	0.627909	−	0.778287i	$-0.283911\pi$
$733$	34.0000	1.25582	0.627909	+	0.778287i	$0.283911\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−4.23369	−0.155950
$738$	0	0
$739$	−30.1168	−1.10787	−0.553933	−	0.832561i	$-0.686874\pi$
$739$	−30.1168	−1.10787	−0.553933	+	0.832561i	$0.686874\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−13.7228	−0.503441	−0.251721	−	0.967800i	$-0.580996\pi$
$743$	−13.7228	−0.503441	−0.251721	+	0.967800i	$0.580996\pi$
$744$	0	0
$745$	−43.3723	−1.58904
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−9.25544	−0.338186
$750$	0	0
$751$	−8.74456	−0.319094	−0.159547	−	0.987190i	$-0.551003\pi$
$751$	−8.74456	−0.319094	−0.159547	+	0.987190i	$0.551003\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	38.7446	1.41006
$756$	0	0
$757$	−35.0951	−1.27555	−0.637776	−	0.770222i	$-0.720146\pi$
$757$	−35.0951	−1.27555	−0.637776	+	0.770222i	$0.720146\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	13.3723	0.484745	0.242372	−	0.970183i	$-0.422074\pi$
$761$	13.3723	0.484745	0.242372	+	0.970183i	$0.422074\pi$
$762$	0	0
$763$	22.7446	0.823408
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−10.9783	−0.396402
$768$	0	0
$769$	−31.0951	−1.12132	−0.560659	−	0.828047i	$-0.689452\pi$
$769$	−31.0951	−1.12132	−0.560659	+	0.828047i	$0.689452\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−2.00000	−0.0719350	−0.0359675	−	0.999353i	$-0.511451\pi$
$773$	−2.00000	−0.0719350	−0.0359675	+	0.999353i	$0.511451\pi$
$774$	0	0
$775$	−38.2337	−1.37339
$776$	0	0
$777$	0	0
$778$	0	0
$779$	8.74456	0.313306
$780$	0	0
$781$	9.25544	0.331186
$782$	0	0
$783$	0	0
$784$	0	0
$785$	6.74456	0.240724
$786$	0	0
$787$	−41.9565	−1.49559	−0.747794	−	0.663931i	$-0.768887\pi$
$787$	−41.9565	−1.49559	−0.747794	+	0.663931i	$0.768887\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	16.0000	0.568895
$792$	0	0
$793$	−37.4891	−1.33128
$794$	0	0
$795$	0	0
$796$	0	0
$797$	11.4891	0.406966	0.203483	−	0.979079i	$-0.434774\pi$
$797$	11.4891	0.406966	0.203483	+	0.979079i	$0.434774\pi$
$798$	0	0
$799$	43.6060	1.54267
$800$	0	0
$801$	0	0
$802$	0	0
$803$	1.64947	0.0582085
$804$	0	0
$805$	−53.9565	−1.90172
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−27.8832	−0.980320	−0.490160	−	0.871633i	$-0.663062\pi$
$809$	−27.8832	−0.980320	−0.490160	+	0.871633i	$0.663062\pi$
$810$	0	0
$811$	−32.0000	−1.12367	−0.561836	−	0.827249i	$-0.689905\pi$
$811$	−32.0000	−1.12367	−0.561836	+	0.827249i	$0.689905\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	40.4674	1.41751
$816$	0	0
$817$	−7.37228	−0.257923
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−38.3505	−1.33844	−0.669221	−	0.743063i	$-0.733372\pi$
$821$	−38.3505	−1.33844	−0.669221	+	0.743063i	$0.733372\pi$
$822$	0	0
$823$	−30.1168	−1.04981	−0.524904	−	0.851162i	$-0.675899\pi$
$823$	−30.1168	−1.04981	−0.524904	+	0.851162i	$0.675899\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	26.7446	0.930000	0.465000	−	0.885311i	$-0.346054\pi$
$827$	26.7446	0.930000	0.465000	+	0.885311i	$0.346054\pi$
$828$	0	0
$829$	18.5109	0.642909	0.321455	−	0.946925i	$-0.395828\pi$
$829$	18.5109	0.642909	0.321455	+	0.946925i	$0.395828\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	23.4891	0.813850
$834$	0	0
$835$	−85.9565	−2.97465
$836$	0	0
$837$	0	0
$838$	0	0
$839$	2.74456	0.0947528	0.0473764	−	0.998877i	$-0.484914\pi$
$839$	2.74456	0.0947528	0.0473764	+	0.998877i	$0.484914\pi$
$840$	0	0
$841$	47.4674	1.63681
$842$	0	0
$843$	0	0
$844$	0	0
$845$	10.1168	0.348030
$846$	0	0
$847$	−35.7663	−1.22895
$848$	0	0
$849$	0	0
$850$	0	0
$851$	18.9783	0.650566
$852$	0	0
$853$	−15.4891	−0.530338	−0.265169	−	0.964202i	$-0.585428\pi$
$853$	−15.4891	−0.530338	−0.265169	+	0.964202i	$0.585428\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−0.978251	−0.0334164	−0.0167082	−	0.999860i	$-0.505319\pi$
$857$	−0.978251	−0.0334164	−0.0167082	+	0.999860i	$0.505319\pi$
$858$	0	0
$859$	34.3505	1.17203	0.586013	−	0.810302i	$-0.300697\pi$
$859$	34.3505	1.17203	0.586013	+	0.810302i	$0.300697\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	52.4674	1.78601	0.893005	−	0.450046i	$-0.148593\pi$
$863$	52.4674	1.78601	0.893005	+	0.450046i	$0.148593\pi$
$864$	0	0
$865$	20.2337	0.687966
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−1.25544	−0.0425878
$870$	0	0
$871$	26.9783	0.914123
$872$	0	0
$873$	0	0
$874$	0	0
$875$	15.6060	0.527578
$876$	0	0
$877$	−15.7663	−0.532391	−0.266195	−	0.963919i	$-0.585767\pi$
$877$	−15.7663	−0.532391	−0.266195	+	0.963919i	$0.585767\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	25.3723	0.854814	0.427407	−	0.904059i	$-0.359427\pi$
$881$	25.3723	0.854814	0.427407	+	0.904059i	$0.359427\pi$
$882$	0	0
$883$	52.8614	1.77893	0.889464	−	0.457005i	$-0.151078\pi$
$883$	52.8614	1.77893	0.889464	+	0.457005i	$0.151078\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−37.7228	−1.26661	−0.633304	−	0.773903i	$-0.718302\pi$
$887$	−37.7228	−1.26661	−0.633304	+	0.773903i	$0.718302\pi$
$888$	0	0
$889$	−61.4891	−2.06228
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−8.11684	−0.271620
$894$	0	0
$895$	−13.4891	−0.450892
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−52.4674	−1.74988
$900$	0	0
$901$	53.7228	1.78977
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−22.7446	−0.756055
$906$	0	0
$907$	−25.2554	−0.838593	−0.419297	−	0.907849i	$-0.637723\pi$
$907$	−25.2554	−0.838593	−0.419297	+	0.907849i	$0.637723\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	5.25544	0.174120	0.0870602	−	0.996203i	$-0.472253\pi$
$911$	5.25544	0.174120	0.0870602	+	0.996203i	$0.472253\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−7.13859	−0.235737
$918$	0	0
$919$	−25.4891	−0.840809	−0.420404	−	0.907337i	$-0.638112\pi$
$919$	−25.4891	−0.840809	−0.420404	+	0.907337i	$0.638112\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−58.9783	−1.94129
$924$	0	0
$925$	−25.4891	−0.838077
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−16.9783	−0.557038	−0.278519	−	0.960431i	$-0.589844\pi$
$929$	−16.9783	−0.557038	−0.278519	+	0.960431i	$0.589844\pi$
$930$	0	0
$931$	−4.37228	−0.143296
$932$	0	0
$933$	0	0
$934$	0	0
$935$	11.3723	0.371913
$936$	0	0
$937$	48.5842	1.58718	0.793589	−	0.608455i	$-0.208210\pi$
$937$	48.5842	1.58718	0.793589	+	0.608455i	$0.208210\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−28.9783	−0.944664	−0.472332	−	0.881421i	$-0.656588\pi$
$941$	−28.9783	−0.944664	−0.472332	+	0.881421i	$0.656588\pi$
$942$	0	0
$943$	41.4891	1.35107
$944$	0	0
$945$	0	0
$946$	0	0
$947$	37.4891	1.21823	0.609116	−	0.793081i	$-0.291524\pi$
$947$	37.4891	1.21823	0.609116	+	0.793081i	$0.291524\pi$
$948$	0	0
$949$	−10.5109	−0.341197
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−23.4891	−0.760887	−0.380444	−	0.924804i	$-0.624229\pi$
$953$	−23.4891	−0.760887	−0.380444	+	0.924804i	$0.624229\pi$
$954$	0	0
$955$	−72.8614	−2.35774
$956$	0	0
$957$	0	0
$958$	0	0
$959$	72.0733	2.32737
$960$	0	0
$961$	5.00000	0.161290
$962$	0	0
$963$	0	0
$964$	0	0
$965$	84.2337	2.71158
$966$	0	0
$967$	−50.9783	−1.63935	−0.819675	−	0.572829i	$-0.805846\pi$
$967$	−50.9783	−1.63935	−0.819675	+	0.572829i	$0.805846\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−22.9783	−0.737407	−0.368704	−	0.929547i	$-0.620198\pi$
$971$	−22.9783	−0.737407	−0.368704	+	0.929547i	$0.620198\pi$
$972$	0	0
$973$	−29.0951	−0.932746
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−20.9783	−0.671154	−0.335577	−	0.942013i	$-0.608931\pi$
$977$	−20.9783	−0.671154	−0.335577	+	0.942013i	$0.608931\pi$
$978$	0	0
$979$	4.70106	0.150247
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−2.51087	−0.0800845	−0.0400422	−	0.999198i	$-0.512749\pi$
$983$	−2.51087	−0.0800845	−0.0400422	+	0.999198i	$0.512749\pi$
$984$	0	0
$985$	−27.7663	−0.884708
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−34.9783	−1.11224
$990$	0	0
$991$	−18.2337	−0.579212	−0.289606	−	0.957146i	$-0.593524\pi$
$991$	−18.2337	−0.579212	−0.289606	+	0.957146i	$0.593524\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	65.3288	2.07106
$996$	0	0
$997$	12.1168	0.383744	0.191872	−	0.981420i	$-0.438544\pi$
$997$	12.1168	0.383744	0.191872	+	0.981420i	$0.438544\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5472.2.a.be.1.2		2
3.2	odd	2		1824.2.a.r.1.1	yes	2
4.3	odd	2		5472.2.a.bd.1.2		2
12.11	even	2		1824.2.a.o.1.1	✓	2
24.5	odd	2		3648.2.a.bm.1.2		2
24.11	even	2		3648.2.a.bt.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1824.2.a.o.1.1	✓	2	12.11	even	2
1824.2.a.r.1.1	yes	2	3.2	odd	2
3648.2.a.bm.1.2		2	24.5	odd	2
3648.2.a.bt.1.2		2	24.11	even	2
5472.2.a.bd.1.2		2	4.3	odd	2
5472.2.a.be.1.2		2	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 5472 = 2^{5} \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5472.a (trivial)

\(f(q)\)	\(=\)	\(q+3.37228 q^{5} +3.37228 q^{7} +O(q^{10})\)	\(q+3.37228 q^{5} +3.37228 q^{7} +0.627719 q^{11} -4.00000 q^{13} +5.37228 q^{17} -1.00000 q^{19} -4.74456 q^{23} +6.37228 q^{25} +8.74456 q^{29} -6.00000 q^{31} +11.3723 q^{35} -4.00000 q^{37} -8.74456 q^{41} +7.37228 q^{43} +8.11684 q^{47} +4.37228 q^{49} +10.0000 q^{53} +2.11684 q^{55} +2.74456 q^{59} +9.37228 q^{61} -13.4891 q^{65} -6.74456 q^{67} +14.7446 q^{71} +2.62772 q^{73} +2.11684 q^{77} -2.00000 q^{79} +18.1168 q^{85} +7.48913 q^{89} -13.4891 q^{91} -3.37228 q^{95} +7.25544 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{5} + q^{7}+O(q^{10}) \) 2 * q + q^5 + q^7	\( 2 q + q^{5} + q^{7} + 7 q^{11} - 8 q^{13} + 5 q^{17} - 2 q^{19} + 2 q^{23} + 7 q^{25} + 6 q^{29} - 12 q^{31} + 17 q^{35} - 8 q^{37} - 6 q^{41} + 9 q^{43} - q^{47} + 3 q^{49} + 20 q^{53} - 13 q^{55} - 6 q^{59} + 13 q^{61} - 4 q^{65} - 2 q^{67} + 18 q^{71} + 11 q^{73} - 13 q^{77} - 4 q^{79} + 19 q^{85} - 8 q^{89} - 4 q^{91} - q^{95} + 26 q^{97}+O(q^{100}) \) 2 * q + q^5 + q^7 + 7 * q^11 - 8 * q^13 + 5 * q^17 - 2 * q^19 + 2 * q^23 + 7 * q^25 + 6 * q^29 - 12 * q^31 + 17 * q^35 - 8 * q^37 - 6 * q^41 + 9 * q^43 - q^47 + 3 * q^49 + 20 * q^53 - 13 * q^55 - 6 * q^59 + 13 * q^61 - 4 * q^65 - 2 * q^67 + 18 * q^71 + 11 * q^73 - 13 * q^77 - 4 * q^79 + 19 * q^85 - 8 * q^89 - 4 * q^91 - q^95 + 26 * q^97

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