LMFDB - Embedded newform 6272.2.a.bu.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6272,2,Mod(1,6272)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6272, 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("6272.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6272 = 2^{7} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6272.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:	$50.0821721477$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.11344.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 4x^{2} + 4x + 3 $ x^4 - 2x^3 - 4x^2 + 4*x + 3
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 896)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-0.552409$ of defining polynomial
Character	$\chi$	$=$	6272.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.14243 q^{3} +3.87834 q^{5} +1.59002 q^{9} +O(q^{10})$	$q+2.14243 q^{3} +3.87834 q^{5} +1.59002 q^{9} -3.84073 q^{11} +4.69484 q^{13} +8.30910 q^{15} -0.514794 q^{17} +3.91596 q^{19} +1.95893 q^{23} +10.0416 q^{25} -3.02078 q^{27} +9.34671 q^{29} -1.44759 q^{31} -8.22851 q^{33} -4.64842 q^{37} +10.0584 q^{39} +7.06185 q^{41} +5.75669 q^{43} +6.16666 q^{45} -3.37236 q^{47} -1.10291 q^{51} -0.449497 q^{53} -14.8957 q^{55} +8.38969 q^{57} -3.61426 q^{59} -14.6147 q^{61} +18.2082 q^{65} -8.50944 q^{67} +4.19689 q^{69} -6.00000 q^{71} +11.4945 q^{73} +21.5134 q^{75} +6.89567 q^{79} -11.2419 q^{81} -14.9031 q^{83} -1.99655 q^{85} +20.0247 q^{87} +7.57664 q^{89} -3.10137 q^{93} +15.1874 q^{95} +3.59693 q^{97} -6.10686 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{3} + 2 q^{5} + 4 q^{9}+O(q^{10}) $ 4 * q + 2 * q^3 + 2 * q^5 + 4 * q^9	$ 4 q + 2 q^{3} + 2 q^{5} + 4 q^{9} + 8 q^{13} + 2 q^{15} + 4 q^{17} + 4 q^{19} + 4 q^{23} + 8 q^{27} + 8 q^{29} - 10 q^{31} + 8 q^{33} + 2 q^{37} + 22 q^{39} + 12 q^{41} - 4 q^{43} - 14 q^{47} + 8 q^{51} - 10 q^{53} - 24 q^{55} + 12 q^{57} + 6 q^{59} + 6 q^{61} + 8 q^{65} - 22 q^{67} + 34 q^{69} - 24 q^{71} + 16 q^{73} + 32 q^{75} - 8 q^{79} - 24 q^{81} + 16 q^{83} - 6 q^{85} + 18 q^{87} + 8 q^{89} - 2 q^{93} + 16 q^{95} + 16 q^{97} + 30 q^{99}+O(q^{100}) $ 4 * q + 2 * q^3 + 2 * q^5 + 4 * q^9 + 8 * q^13 + 2 * q^15 + 4 * q^17 + 4 * q^19 + 4 * q^23 + 8 * q^27 + 8 * q^29 - 10 * q^31 + 8 * q^33 + 2 * q^37 + 22 * q^39 + 12 * q^41 - 4 * q^43 - 14 * q^47 + 8 * q^51 - 10 * q^53 - 24 * q^55 + 12 * q^57 + 6 * q^59 + 6 * q^61 + 8 * q^65 - 22 * q^67 + 34 * q^69 - 24 * q^71 + 16 * q^73 + 32 * q^75 - 8 * q^79 - 24 * q^81 + 16 * q^83 - 6 * q^85 + 18 * q^87 + 8 * q^89 - 2 * q^93 + 16 * q^95 + 16 * q^97 + 30 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	2.14243	1.23694	0.618468	−	0.785810i	$-0.287754\pi$
$3$	2.14243	1.23694	0.618468	+	0.785810i	$0.287754\pi$
$4$	0	0
$5$	3.87834	1.73445	0.867224	−	0.497918i	$-0.165902\pi$
$5$	3.87834	1.73445	0.867224	+	0.497918i	$0.165902\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	1.59002	0.530008
$10$	0	0
$11$	−3.84073	−1.15802	−0.579012	−	0.815319i	$-0.696561\pi$
$11$	−3.84073	−1.15802	−0.579012	+	0.815319i	$0.696561\pi$
$12$	0	0
$13$	4.69484	1.30212	0.651058	−	0.759028i	$-0.274326\pi$
$13$	4.69484	1.30212	0.651058	+	0.759028i	$0.274326\pi$
$14$	0	0
$15$	8.30910	2.14540
$16$	0	0
$17$	−0.514794	−0.124856	−0.0624279	−	0.998049i	$-0.519884\pi$
$17$	−0.514794	−0.124856	−0.0624279	+	0.998049i	$0.519884\pi$
$18$	0	0
$19$	3.91596	0.898383	0.449191	−	0.893436i	$-0.351712\pi$
$19$	3.91596	0.898383	0.449191	+	0.893436i	$0.351712\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	1.95893	0.408466	0.204233	−	0.978922i	$-0.434530\pi$
$23$	1.95893	0.408466	0.204233	+	0.978922i	$0.434530\pi$
$24$	0	0
$25$	10.0416	2.00831
$26$	0	0
$27$	−3.02078	−0.581349
$28$	0	0
$29$	9.34671	1.73564	0.867821	−	0.496878i	$-0.165520\pi$
$29$	9.34671	1.73564	0.867821	+	0.496878i	$0.165520\pi$
$30$	0	0
$31$	−1.44759	−0.259995	−0.129997	−	0.991514i	$-0.541497\pi$
$31$	−1.44759	−0.259995	−0.129997	+	0.991514i	$0.541497\pi$
$32$	0	0
$33$	−8.22851	−1.43240
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−4.64842	−0.764195	−0.382098	−	0.924122i	$-0.624798\pi$
$37$	−4.64842	−0.764195	−0.382098	+	0.924122i	$0.624798\pi$
$38$	0	0
$39$	10.0584	1.61063
$40$	0	0
$41$	7.06185	1.10288	0.551438	−	0.834216i	$-0.314080\pi$
$41$	7.06185	1.10288	0.551438	+	0.834216i	$0.314080\pi$
$42$	0	0
$43$	5.75669	0.877887	0.438943	−	0.898515i	$-0.355353\pi$
$43$	5.75669	0.877887	0.438943	+	0.898515i	$0.355353\pi$
$44$	0	0
$45$	6.16666	0.919272
$46$	0	0
$47$	−3.37236	−0.491909	−0.245955	−	0.969281i	$-0.579101\pi$
$47$	−3.37236	−0.491909	−0.245955	+	0.969281i	$0.579101\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−1.10291	−0.154439
$52$	0	0
$53$	−0.449497	−0.0617432	−0.0308716	−	0.999523i	$-0.509828\pi$
$53$	−0.449497	−0.0617432	−0.0308716	+	0.999523i	$0.509828\pi$
$54$	0	0
$55$	−14.8957	−2.00853
$56$	0	0
$57$	8.38969	1.11124
$58$	0	0
$59$	−3.61426	−0.470536	−0.235268	−	0.971931i	$-0.575597\pi$
$59$	−3.61426	−0.470536	−0.235268	+	0.971931i	$0.575597\pi$
$60$	0	0
$61$	−14.6147	−1.87123	−0.935613	−	0.353027i	$-0.885153\pi$
$61$	−14.6147	−1.87123	−0.935613	+	0.353027i	$0.885153\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	18.2082	2.25845
$66$	0	0
$67$	−8.50944	−1.03959	−0.519797	−	0.854290i	$-0.673992\pi$
$67$	−8.50944	−1.03959	−0.519797	+	0.854290i	$0.673992\pi$
$68$	0	0
$69$	4.19689	0.505246
$70$	0	0
$71$	−6.00000	−0.712069	−0.356034	−	0.934473i	$-0.615871\pi$
$71$	−6.00000	−0.712069	−0.356034	+	0.934473i	$0.615871\pi$
$72$	0	0
$73$	11.4945	1.34533	0.672665	−	0.739947i	$-0.265149\pi$
$73$	11.4945	1.34533	0.672665	+	0.739947i	$0.265149\pi$
$74$	0	0
$75$	21.5134	2.48415
$76$	0	0
$77$	0	0
$78$	0	0
$79$	6.89567	0.775824	0.387912	−	0.921696i	$-0.373196\pi$
$79$	6.89567	0.775824	0.387912	+	0.921696i	$0.373196\pi$
$80$	0	0
$81$	−11.2419	−1.24910
$82$	0	0
$83$	−14.9031	−1.63582	−0.817912	−	0.575343i	$-0.804868\pi$
$83$	−14.9031	−1.63582	−0.817912	+	0.575343i	$0.804868\pi$
$84$	0	0
$85$	−1.99655	−0.216556
$86$	0	0
$87$	20.0247	2.14688
$88$	0	0
$89$	7.57664	0.803122	0.401561	−	0.915832i	$-0.368468\pi$
$89$	7.57664	0.803122	0.401561	+	0.915832i	$0.368468\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−3.10137	−0.321597
$94$	0	0
$95$	15.1874	1.55820
$96$	0	0
$97$	3.59693	0.365213	0.182606	−	0.983186i	$-0.441547\pi$
$97$	3.59693	0.365213	0.182606	+	0.983186i	$0.441547\pi$
$98$	0	0
$99$	−6.10686	−0.613762
$100$	0	0
$101$	−10.8151	−1.07614	−0.538071	−	0.842900i	$-0.680847\pi$
$101$	−10.8151	−1.07614	−0.538071	+	0.842900i	$0.680847\pi$
$102$	0	0
$103$	−10.0826	−0.993471	−0.496735	−	0.867902i	$-0.665468\pi$
$103$	−10.0826	−0.993471	−0.496735	+	0.867902i	$0.665468\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	18.4105	1.77981	0.889903	−	0.456149i	$-0.150772\pi$
$107$	18.4105	1.77981	0.889903	+	0.456149i	$0.150772\pi$
$108$	0	0
$109$	−1.38384	−0.132548	−0.0662738	−	0.997801i	$-0.521111\pi$
$109$	−1.38384	−0.132548	−0.0662738	+	0.997801i	$0.521111\pi$
$110$	0	0
$111$	−9.95893	−0.945260
$112$	0	0
$113$	−9.18935	−0.864461	−0.432231	−	0.901763i	$-0.642273\pi$
$113$	−9.18935	−0.864461	−0.432231	+	0.901763i	$0.642273\pi$
$114$	0	0
$115$	7.59742	0.708463
$116$	0	0
$117$	7.46492	0.690132
$118$	0	0
$119$	0	0
$120$	0	0
$121$	3.75120	0.341018
$122$	0	0
$123$	15.1295	1.36418
$124$	0	0
$125$	19.5529	1.74886
$126$	0	0
$127$	2.97041	0.263581	0.131791	−	0.991278i	$-0.457927\pi$
$127$	2.97041	0.263581	0.131791	+	0.991278i	$0.457927\pi$
$128$	0	0
$129$	12.3333	1.08589
$130$	0	0
$131$	−8.05649	−0.703899	−0.351949	−	0.936019i	$-0.614481\pi$
$131$	−8.05649	−0.703899	−0.351949	+	0.936019i	$0.614481\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−11.7156	−1.00832
$136$	0	0
$137$	7.81163	0.667393	0.333696	−	0.942681i	$-0.391704\pi$
$137$	7.81163	0.667393	0.333696	+	0.942681i	$0.391704\pi$
$138$	0	0
$139$	19.1533	1.62456	0.812280	−	0.583268i	$-0.198226\pi$
$139$	19.1533	1.62456	0.812280	+	0.583268i	$0.198226\pi$
$140$	0	0
$141$	−7.22506	−0.608460
$142$	0	0
$143$	−18.0316	−1.50788
$144$	0	0
$145$	36.2498	3.01038
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−8.29762	−0.679768	−0.339884	−	0.940467i	$-0.610388\pi$
$149$	−8.29762	−0.679768	−0.339884	+	0.940467i	$0.610388\pi$
$150$	0	0
$151$	11.2902	0.918786	0.459393	−	0.888233i	$-0.348067\pi$
$151$	11.2902	0.918786	0.459393	+	0.888233i	$0.348067\pi$
$152$	0	0
$153$	−0.818535	−0.0661747
$154$	0	0
$155$	−5.61426	−0.450948
$156$	0	0
$157$	6.43878	0.513871	0.256935	−	0.966429i	$-0.417287\pi$
$157$	6.43878	0.513871	0.256935	+	0.966429i	$0.417287\pi$
$158$	0	0
$159$	−0.963018	−0.0763723
$160$	0	0
$161$	0	0
$162$	0	0
$163$	14.9932	1.17436	0.587180	−	0.809456i	$-0.300238\pi$
$163$	14.9932	1.17436	0.587180	+	0.809456i	$0.300238\pi$
$164$	0	0
$165$	−31.9130	−2.48442
$166$	0	0
$167$	2.34326	0.181327	0.0906636	−	0.995882i	$-0.471101\pi$
$167$	2.34326	0.181327	0.0906636	+	0.995882i	$0.471101\pi$
$168$	0	0
$169$	9.04156	0.695504
$170$	0	0
$171$	6.22647	0.476150
$172$	0	0
$173$	2.82340	0.214659	0.107330	−	0.994223i	$-0.465770\pi$
$173$	2.82340	0.214659	0.107330	+	0.994223i	$0.465770\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−7.74330	−0.582022
$178$	0	0
$179$	−12.2937	−0.918873	−0.459436	−	0.888211i	$-0.651949\pi$
$179$	−12.2937	−0.918873	−0.459436	+	0.888211i	$0.651949\pi$
$180$	0	0
$181$	6.50141	0.483246	0.241623	−	0.970370i	$-0.422320\pi$
$181$	6.50141	0.483246	0.241623	+	0.970370i	$0.422320\pi$
$182$	0	0
$183$	−31.3111	−2.31459
$184$	0	0
$185$	−18.0282	−1.32546
$186$	0	0
$187$	1.97718	0.144586
$188$	0	0
$189$	0	0
$190$	0	0
$191$	6.88574	0.498235	0.249117	−	0.968473i	$-0.419860\pi$
$191$	6.88574	0.498235	0.249117	+	0.968473i	$0.419860\pi$
$192$	0	0
$193$	−3.85602	−0.277562	−0.138781	−	0.990323i	$-0.544318\pi$
$193$	−3.85602	−0.277562	−0.138781	+	0.990323i	$0.544318\pi$
$194$	0	0
$195$	39.0099	2.79356
$196$	0	0
$197$	5.22993	0.372617	0.186308	−	0.982491i	$-0.440348\pi$
$197$	5.22993	0.372617	0.186308	+	0.982491i	$0.440348\pi$
$198$	0	0
$199$	2.70977	0.192091	0.0960454	−	0.995377i	$-0.469381\pi$
$199$	2.70977	0.192091	0.0960454	+	0.995377i	$0.469381\pi$
$200$	0	0
$201$	−18.2309	−1.28591
$202$	0	0
$203$	0	0
$204$	0	0
$205$	27.3883	1.91288
$206$	0	0
$207$	3.11475	0.216490
$208$	0	0
$209$	−15.0401	−1.04035
$210$	0	0
$211$	16.1237	1.11000	0.555000	−	0.831850i	$-0.312718\pi$
$211$	16.1237	1.11000	0.555000	+	0.831850i	$0.312718\pi$
$212$	0	0
$213$	−12.8546	−0.880783
$214$	0	0
$215$	22.3264	1.52265
$216$	0	0
$217$	0	0
$218$	0	0
$219$	24.6262	1.66409
$220$	0	0
$221$	−2.41688	−0.162577
$222$	0	0
$223$	−2.48662	−0.166516	−0.0832582	−	0.996528i	$-0.526533\pi$
$223$	−2.48662	−0.166516	−0.0832582	+	0.996528i	$0.526533\pi$
$224$	0	0
$225$	15.9663	1.06442
$226$	0	0
$227$	−3.39850	−0.225566	−0.112783	−	0.993620i	$-0.535977\pi$
$227$	−3.39850	−0.225566	−0.112783	+	0.993620i	$0.535977\pi$
$228$	0	0
$229$	14.8595	0.981942	0.490971	−	0.871176i	$-0.336642\pi$
$229$	14.8595	0.981942	0.490971	+	0.871176i	$0.336642\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	5.98661	0.392196	0.196098	−	0.980584i	$-0.437173\pi$
$233$	5.98661	0.392196	0.196098	+	0.980584i	$0.437173\pi$
$234$	0	0
$235$	−13.0792	−0.853191
$236$	0	0
$237$	14.7735	0.959644
$238$	0	0
$239$	15.0128	0.971094	0.485547	−	0.874211i	$-0.338620\pi$
$239$	15.0128	0.971094	0.485547	+	0.874211i	$0.338620\pi$
$240$	0	0
$241$	−10.6638	−0.686918	−0.343459	−	0.939168i	$-0.611599\pi$
$241$	−10.6638	−0.686918	−0.343459	+	0.939168i	$0.611599\pi$
$242$	0	0
$243$	−15.0227	−0.963706
$244$	0	0
$245$	0	0
$246$	0	0
$247$	18.3848	1.16980
$248$	0	0
$249$	−31.9288	−2.02341
$250$	0	0
$251$	9.84876	0.621648	0.310824	−	0.950467i	$-0.399395\pi$
$251$	9.84876	0.621648	0.310824	+	0.950467i	$0.399395\pi$
$252$	0	0
$253$	−7.52373	−0.473013
$254$	0	0
$255$	−4.27747	−0.267866
$256$	0	0
$257$	12.0549	0.751967	0.375983	−	0.926626i	$-0.377305\pi$
$257$	12.0549	0.751967	0.375983	+	0.926626i	$0.377305\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	14.8615	0.919904
$262$	0	0
$263$	−24.6118	−1.51763	−0.758814	−	0.651307i	$-0.774221\pi$
$263$	−24.6118	−1.51763	−0.758814	+	0.651307i	$0.774221\pi$
$264$	0	0
$265$	−1.74330	−0.107090
$266$	0	0
$267$	16.2325	0.993410
$268$	0	0
$269$	−16.7721	−1.02261	−0.511307	−	0.859398i	$-0.670838\pi$
$269$	−16.7721	−1.02261	−0.511307	+	0.859398i	$0.670838\pi$
$270$	0	0
$271$	−30.2992	−1.84054	−0.920272	−	0.391280i	$-0.872032\pi$
$271$	−30.2992	−1.84054	−0.920272	+	0.391280i	$0.872032\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−38.5669	−2.32567
$276$	0	0
$277$	−22.0006	−1.32189	−0.660945	−	0.750435i	$-0.729844\pi$
$277$	−22.0006	−1.32189	−0.660945	+	0.750435i	$0.729844\pi$
$278$	0	0
$279$	−2.30170	−0.137799
$280$	0	0
$281$	12.4031	0.739905	0.369953	−	0.929051i	$-0.379374\pi$
$281$	12.4031	0.739905	0.369953	+	0.929051i	$0.379374\pi$
$282$	0	0
$283$	11.4215	0.678935	0.339467	−	0.940618i	$-0.389753\pi$
$283$	11.4215	0.678935	0.339467	+	0.940618i	$0.389753\pi$
$284$	0	0
$285$	32.5381	1.92739
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−16.7350	−0.984411
$290$	0	0
$291$	7.70618	0.451744
$292$	0	0
$293$	24.9436	1.45722	0.728612	−	0.684927i	$-0.240166\pi$
$293$	24.9436	1.45722	0.728612	+	0.684927i	$0.240166\pi$
$294$	0	0
$295$	−14.0173	−0.816120
$296$	0	0
$297$	11.6020	0.673216
$298$	0	0
$299$	9.19689	0.531870
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−23.1706	−1.33112
$304$	0	0
$305$	−56.6810	−3.24555
$306$	0	0
$307$	−1.67074	−0.0953544	−0.0476772	−	0.998863i	$-0.515182\pi$
$307$	−1.67074	−0.0953544	−0.0476772	+	0.998863i	$0.515182\pi$
$308$	0	0
$309$	−21.6014	−1.22886
$310$	0	0
$311$	−30.4575	−1.72709	−0.863544	−	0.504274i	$-0.831760\pi$
$311$	−30.4575	−1.72709	−0.863544	+	0.504274i	$0.831760\pi$
$312$	0	0
$313$	20.8319	1.17749	0.588745	−	0.808319i	$-0.299622\pi$
$313$	20.8319	1.17749	0.588745	+	0.808319i	$0.299622\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	1.47909	0.0830737	0.0415368	−	0.999137i	$-0.486775\pi$
$317$	1.47909	0.0830737	0.0415368	+	0.999137i	$0.486775\pi$
$318$	0	0
$319$	−35.8982	−2.00991
$320$	0	0
$321$	39.4432	2.20151
$322$	0	0
$323$	−2.01591	−0.112168
$324$	0	0
$325$	47.1435	2.61505
$326$	0	0
$327$	−2.96478	−0.163953
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−12.5441	−0.689486	−0.344743	−	0.938697i	$-0.612034\pi$
$331$	−12.5441	−0.689486	−0.344743	+	0.938697i	$0.612034\pi$
$332$	0	0
$333$	−7.39110	−0.405030
$334$	0	0
$335$	−33.0025	−1.80312
$336$	0	0
$337$	−34.8425	−1.89799	−0.948995	−	0.315290i	$-0.897898\pi$
$337$	−34.8425	−1.89799	−0.948995	+	0.315290i	$0.897898\pi$
$338$	0	0
$339$	−19.6876	−1.06928
$340$	0	0
$341$	5.55980	0.301080
$342$	0	0
$343$	0	0
$344$	0	0
$345$	16.2770	0.876323
$346$	0	0
$347$	−1.77543	−0.0953102	−0.0476551	−	0.998864i	$-0.515175\pi$
$347$	−1.77543	−0.0953102	−0.0476551	+	0.998864i	$0.515175\pi$
$348$	0	0
$349$	−22.9193	−1.22684	−0.613420	−	0.789757i	$-0.710207\pi$
$349$	−22.9193	−1.22684	−0.613420	+	0.789757i	$0.710207\pi$
$350$	0	0
$351$	−14.1821	−0.756984
$352$	0	0
$353$	−11.3763	−0.605499	−0.302750	−	0.953070i	$-0.597905\pi$
$353$	−11.3763	−0.605499	−0.302750	+	0.953070i	$0.597905\pi$
$354$	0	0
$355$	−23.2701	−1.23505
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−29.1231	−1.53706	−0.768530	−	0.639814i	$-0.779011\pi$
$359$	−29.1231	−1.53706	−0.768530	+	0.639814i	$0.779011\pi$
$360$	0	0
$361$	−3.66526	−0.192908
$362$	0	0
$363$	8.03670	0.421818
$364$	0	0
$365$	44.5797	2.33341
$366$	0	0
$367$	−3.34074	−0.174385	−0.0871925	−	0.996191i	$-0.527790\pi$
$367$	−3.34074	−0.174385	−0.0871925	+	0.996191i	$0.527790\pi$
$368$	0	0
$369$	11.2285	0.584533
$370$	0	0
$371$	0	0
$372$	0	0
$373$	17.9292	0.928339	0.464169	−	0.885746i	$-0.346353\pi$
$373$	17.9292	0.928339	0.464169	+	0.885746i	$0.346353\pi$
$374$	0	0
$375$	41.8908	2.16323
$376$	0	0
$377$	43.8814	2.26001
$378$	0	0
$379$	−28.0247	−1.43953	−0.719767	−	0.694216i	$-0.755751\pi$
$379$	−28.0247	−1.43953	−0.719767	+	0.694216i	$0.755751\pi$
$380$	0	0
$381$	6.36391	0.326033
$382$	0	0
$383$	−15.8184	−0.808283	−0.404141	−	0.914697i	$-0.632430\pi$
$383$	−15.8184	−0.808283	−0.404141	+	0.914697i	$0.632430\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	9.15328	0.465287
$388$	0	0
$389$	15.3511	0.778334	0.389167	−	0.921167i	$-0.372763\pi$
$389$	15.3511	0.778334	0.389167	+	0.921167i	$0.372763\pi$
$390$	0	0
$391$	−1.00845	−0.0509994
$392$	0	0
$393$	−17.2605	−0.870677
$394$	0	0
$395$	26.7438	1.34563
$396$	0	0
$397$	2.80720	0.140889	0.0704446	−	0.997516i	$-0.477558\pi$
$397$	2.80720	0.140889	0.0704446	+	0.997516i	$0.477558\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−10.8680	−0.542722	−0.271361	−	0.962478i	$-0.587474\pi$
$401$	−10.8680	−0.542722	−0.271361	+	0.962478i	$0.587474\pi$
$402$	0	0
$403$	−6.79621	−0.338543
$404$	0	0
$405$	−43.5999	−2.16650
$406$	0	0
$407$	17.8533	0.884956
$408$	0	0
$409$	17.0055	0.840867	0.420434	−	0.907323i	$-0.361878\pi$
$409$	17.0055	0.840867	0.420434	+	0.907323i	$0.361878\pi$
$410$	0	0
$411$	16.7359	0.825522
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−57.7992	−2.83725
$416$	0	0
$417$	41.0346	2.00948
$418$	0	0
$419$	3.54705	0.173285	0.0866424	−	0.996239i	$-0.472386\pi$
$419$	3.54705	0.173285	0.0866424	+	0.996239i	$0.472386\pi$
$420$	0	0
$421$	−9.11412	−0.444195	−0.222098	−	0.975024i	$-0.571290\pi$
$421$	−9.11412	−0.444195	−0.222098	+	0.975024i	$0.571290\pi$
$422$	0	0
$423$	−5.36214	−0.260716
$424$	0	0
$425$	−5.16933	−0.250750
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−38.6316	−1.86515
$430$	0	0
$431$	9.57128	0.461032	0.230516	−	0.973068i	$-0.425959\pi$
$431$	9.57128	0.461032	0.230516	+	0.973068i	$0.425959\pi$
$432$	0	0
$433$	3.74739	0.180088	0.0900440	−	0.995938i	$-0.471299\pi$
$433$	3.74739	0.180088	0.0900440	+	0.995938i	$0.471299\pi$
$434$	0	0
$435$	77.6628	3.72365
$436$	0	0
$437$	7.67111	0.366959
$438$	0	0
$439$	17.6236	0.841126	0.420563	−	0.907263i	$-0.361832\pi$
$439$	17.6236	0.841126	0.420563	+	0.907263i	$0.361832\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−26.5688	−1.26232	−0.631161	−	0.775652i	$-0.717421\pi$
$443$	−26.5688	−1.26232	−0.631161	+	0.775652i	$0.717421\pi$
$444$	0	0
$445$	29.3848	1.39297
$446$	0	0
$447$	−17.7771	−0.840828
$448$	0	0
$449$	36.2460	1.71055	0.855276	−	0.518172i	$-0.173387\pi$
$449$	36.2460	1.71055	0.855276	+	0.518172i	$0.173387\pi$
$450$	0	0
$451$	−27.1226	−1.27716
$452$	0	0
$453$	24.1886	1.13648
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−9.70414	−0.453941	−0.226970	−	0.973902i	$-0.572882\pi$
$457$	−9.70414	−0.453941	−0.226970	+	0.973902i	$0.572882\pi$
$458$	0	0
$459$	1.55508	0.0725849
$460$	0	0
$461$	−14.0332	−0.653593	−0.326797	−	0.945095i	$-0.605969\pi$
$461$	−14.0332	−0.653593	−0.326797	+	0.945095i	$0.605969\pi$
$462$	0	0
$463$	−23.6824	−1.10062	−0.550308	−	0.834962i	$-0.685490\pi$
$463$	−23.6824	−1.10062	−0.550308	+	0.834962i	$0.685490\pi$
$464$	0	0
$465$	−12.0282	−0.557793
$466$	0	0
$467$	22.7915	1.05466	0.527332	−	0.849659i	$-0.323192\pi$
$467$	22.7915	1.05466	0.527332	+	0.849659i	$0.323192\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	13.7947	0.635625
$472$	0	0
$473$	−22.1099	−1.01661
$474$	0	0
$475$	39.3223	1.80423
$476$	0	0
$477$	−0.714711	−0.0327244
$478$	0	0
$479$	−14.7276	−0.672921	−0.336460	−	0.941698i	$-0.609230\pi$
$479$	−14.7276	−0.672921	−0.336460	+	0.941698i	$0.609230\pi$
$480$	0	0
$481$	−21.8236	−0.995071
$482$	0	0
$483$	0	0
$484$	0	0
$485$	13.9501	0.633443
$486$	0	0
$487$	−6.06882	−0.275004	−0.137502	−	0.990501i	$-0.543907\pi$
$487$	−6.06882	−0.275004	−0.137502	+	0.990501i	$0.543907\pi$
$488$	0	0
$489$	32.1220	1.45261
$490$	0	0
$491$	−32.6083	−1.47159	−0.735795	−	0.677204i	$-0.763191\pi$
$491$	−32.6083	−1.47159	−0.735795	+	0.677204i	$0.763191\pi$
$492$	0	0
$493$	−4.81163	−0.216705
$494$	0	0
$495$	−23.6845	−1.06454
$496$	0	0
$497$	0	0
$498$	0	0
$499$	34.4421	1.54184	0.770920	−	0.636932i	$-0.219797\pi$
$499$	34.4421	1.54184	0.770920	+	0.636932i	$0.219797\pi$
$500$	0	0
$501$	5.02029	0.224290
$502$	0	0
$503$	−18.2550	−0.813951	−0.406975	−	0.913439i	$-0.633416\pi$
$503$	−18.2550	−0.813951	−0.406975	+	0.913439i	$0.633416\pi$
$504$	0	0
$505$	−41.9446	−1.86651
$506$	0	0
$507$	19.3709	0.860294
$508$	0	0
$509$	−7.60419	−0.337050	−0.168525	−	0.985697i	$-0.553900\pi$
$509$	−7.60419	−0.337050	−0.168525	+	0.985697i	$0.553900\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−11.8293	−0.522274
$514$	0	0
$515$	−39.1039	−1.72312
$516$	0	0
$517$	12.9523	0.569642
$518$	0	0
$519$	6.04895	0.265520
$520$	0	0
$521$	−9.56283	−0.418955	−0.209478	−	0.977813i	$-0.567176\pi$
$521$	−9.56283	−0.418955	−0.209478	+	0.977813i	$0.567176\pi$
$522$	0	0
$523$	22.7428	0.994474	0.497237	−	0.867615i	$-0.334348\pi$
$523$	22.7428	0.994474	0.497237	+	0.867615i	$0.334348\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0.745211	0.0324619
$528$	0	0
$529$	−19.1626	−0.833156
$530$	0	0
$531$	−5.74676	−0.249388
$532$	0	0
$533$	33.1543	1.43607
$534$	0	0
$535$	71.4021	3.08698
$536$	0	0
$537$	−26.3384	−1.13659
$538$	0	0
$539$	0	0
$540$	0	0
$541$	1.66180	0.0714466	0.0357233	−	0.999362i	$-0.488627\pi$
$541$	1.66180	0.0714466	0.0357233	+	0.999362i	$0.488627\pi$
$542$	0	0
$543$	13.9288	0.597744
$544$	0	0
$545$	−5.36700	−0.229897
$546$	0	0
$547$	7.03571	0.300825	0.150413	−	0.988623i	$-0.451940\pi$
$547$	7.03571	0.300825	0.150413	+	0.988623i	$0.451940\pi$
$548$	0	0
$549$	−23.2378	−0.991766
$550$	0	0
$551$	36.6014	1.55927
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−38.6242	−1.63951
$556$	0	0
$557$	−21.3165	−0.903209	−0.451604	−	0.892218i	$-0.649148\pi$
$557$	−21.3165	−0.903209	−0.451604	+	0.892218i	$0.649148\pi$
$558$	0	0
$559$	27.0268	1.14311
$560$	0	0
$561$	4.23599	0.178844
$562$	0	0
$563$	−9.76472	−0.411534	−0.205767	−	0.978601i	$-0.565969\pi$
$563$	−9.76472	−0.411534	−0.205767	+	0.978601i	$0.565969\pi$
$564$	0	0
$565$	−35.6395	−1.49936
$566$	0	0
$567$	0	0
$568$	0	0
$569$	5.80923	0.243536	0.121768	−	0.992559i	$-0.461144\pi$
$569$	5.80923	0.243536	0.121768	+	0.992559i	$0.461144\pi$
$570$	0	0
$571$	−40.9774	−1.71485	−0.857425	−	0.514609i	$-0.827937\pi$
$571$	−40.9774	−1.71485	−0.857425	+	0.514609i	$0.827937\pi$
$572$	0	0
$573$	14.7522	0.616284
$574$	0	0
$575$	19.6707	0.820327
$576$	0	0
$577$	1.48662	0.0618888	0.0309444	−	0.999521i	$-0.490149\pi$
$577$	1.48662	0.0618888	0.0309444	+	0.999521i	$0.490149\pi$
$578$	0	0
$579$	−8.26127	−0.343327
$580$	0	0
$581$	0	0
$582$	0	0
$583$	1.72640	0.0715000
$584$	0	0
$585$	28.9515	1.19700
$586$	0	0
$587$	−37.9043	−1.56448	−0.782239	−	0.622978i	$-0.785923\pi$
$587$	−37.9043	−1.56448	−0.782239	+	0.622978i	$0.785923\pi$
$588$	0	0
$589$	−5.66871	−0.233575
$590$	0	0
$591$	11.2048	0.460903
$592$	0	0
$593$	−19.3426	−0.794307	−0.397153	−	0.917752i	$-0.630002\pi$
$593$	−19.3426	−0.794307	−0.397153	+	0.917752i	$0.630002\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	5.80551	0.237604
$598$	0	0
$599$	20.1510	0.823346	0.411673	−	0.911332i	$-0.364945\pi$
$599$	20.1510	0.823346	0.411673	+	0.911332i	$0.364945\pi$
$600$	0	0
$601$	−17.6731	−0.720902	−0.360451	−	0.932778i	$-0.617377\pi$
$601$	−17.6731	−0.720902	−0.360451	+	0.932778i	$0.617377\pi$
$602$	0	0
$603$	−13.5302	−0.550993
$604$	0	0
$605$	14.5485	0.591479
$606$	0	0
$607$	−43.0094	−1.74570	−0.872849	−	0.487990i	$-0.837730\pi$
$607$	−43.0094	−1.74570	−0.872849	+	0.487990i	$0.837730\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−15.8327	−0.640523
$612$	0	0
$613$	−27.1729	−1.09750	−0.548752	−	0.835985i	$-0.684897\pi$
$613$	−27.1729	−1.09750	−0.548752	+	0.835985i	$0.684897\pi$
$614$	0	0
$615$	58.6776	2.36611
$616$	0	0
$617$	34.4770	1.38799	0.693997	−	0.719978i	$-0.255848\pi$
$617$	34.4770	1.38799	0.693997	+	0.719978i	$0.255848\pi$
$618$	0	0
$619$	−16.5104	−0.663610	−0.331805	−	0.943348i	$-0.607658\pi$
$619$	−16.5104	−0.663610	−0.331805	+	0.943348i	$0.607658\pi$
$620$	0	0
$621$	−5.91750	−0.237461
$622$	0	0
$623$	0	0
$624$	0	0
$625$	25.6251	1.02500
$626$	0	0
$627$	−32.2225	−1.28684
$628$	0	0
$629$	2.39298	0.0954143
$630$	0	0
$631$	22.9890	0.915178	0.457589	−	0.889164i	$-0.348713\pi$
$631$	22.9890	0.915178	0.457589	+	0.889164i	$0.348713\pi$
$632$	0	0
$633$	34.5440	1.37300
$634$	0	0
$635$	11.5203	0.457168
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−9.54015	−0.377402
$640$	0	0
$641$	40.6893	1.60713	0.803566	−	0.595215i	$-0.202933\pi$
$641$	40.6893	1.60713	0.803566	+	0.595215i	$0.202933\pi$
$642$	0	0
$643$	−7.63300	−0.301016	−0.150508	−	0.988609i	$-0.548091\pi$
$643$	−7.63300	−0.301016	−0.150508	+	0.988609i	$0.548091\pi$
$644$	0	0
$645$	47.8329	1.88342
$646$	0	0
$647$	−30.1104	−1.18376	−0.591881	−	0.806026i	$-0.701614\pi$
$647$	−30.1104	−1.18376	−0.591881	+	0.806026i	$0.701614\pi$
$648$	0	0
$649$	13.8814	0.544892
$650$	0	0
$651$	0	0
$652$	0	0
$653$	49.9264	1.95377	0.976885	−	0.213766i	$-0.0685729\pi$
$653$	49.9264	1.95377	0.976885	+	0.213766i	$0.0685729\pi$
$654$	0	0
$655$	−31.2458	−1.22088
$656$	0	0
$657$	18.2766	0.713036
$658$	0	0
$659$	−25.8329	−1.00631	−0.503154	−	0.864197i	$-0.667827\pi$
$659$	−25.8329	−1.00631	−0.503154	+	0.864197i	$0.667827\pi$
$660$	0	0
$661$	17.4347	0.678131	0.339066	−	0.940763i	$-0.389889\pi$
$661$	17.4347	0.678131	0.339066	+	0.940763i	$0.389889\pi$
$662$	0	0
$663$	−5.17800	−0.201097
$664$	0	0
$665$	0	0
$666$	0	0
$667$	18.3096	0.708950
$668$	0	0
$669$	−5.32742	−0.205970
$670$	0	0
$671$	56.1313	2.16692
$672$	0	0
$673$	−16.6879	−0.643273	−0.321636	−	0.946863i	$-0.604233\pi$
$673$	−16.6879	−0.643273	−0.321636	+	0.946863i	$0.604233\pi$
$674$	0	0
$675$	−30.3333	−1.16753
$676$	0	0
$677$	−10.7330	−0.412501	−0.206250	−	0.978499i	$-0.566126\pi$
$677$	−10.7330	−0.412501	−0.206250	+	0.978499i	$0.566126\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−7.28106	−0.279011
$682$	0	0
$683$	28.5181	1.09121	0.545607	−	0.838041i	$-0.316299\pi$
$683$	28.5181	1.09121	0.545607	+	0.838041i	$0.316299\pi$
$684$	0	0
$685$	30.2962	1.15756
$686$	0	0
$687$	31.8354	1.21460
$688$	0	0
$689$	−2.11032	−0.0803967
$690$	0	0
$691$	50.2334	1.91097	0.955485	−	0.295040i	$-0.0953329\pi$
$691$	50.2334	1.91097	0.955485	+	0.295040i	$0.0953329\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	74.2830	2.81772
$696$	0	0
$697$	−3.63540	−0.137700
$698$	0	0
$699$	12.8259	0.485121
$700$	0	0
$701$	15.7567	0.595122	0.297561	−	0.954703i	$-0.403827\pi$
$701$	15.7567	0.595122	0.297561	+	0.954703i	$0.403827\pi$
$702$	0	0
$703$	−18.2030	−0.686540
$704$	0	0
$705$	−28.0213	−1.05534
$706$	0	0
$707$	0	0
$708$	0	0
$709$	40.3192	1.51422	0.757109	−	0.653289i	$-0.226611\pi$
$709$	40.3192	1.51422	0.757109	+	0.653289i	$0.226611\pi$
$710$	0	0
$711$	10.9643	0.411193
$712$	0	0
$713$	−2.83573	−0.106199
$714$	0	0
$715$	−69.9329	−2.61534
$716$	0	0
$717$	32.1638	1.20118
$718$	0	0
$719$	22.0528	0.822429	0.411215	−	0.911539i	$-0.365105\pi$
$719$	22.0528	0.822429	0.411215	+	0.911539i	$0.365105\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−22.8466	−0.849673
$724$	0	0
$725$	93.8556	3.48571
$726$	0	0
$727$	−26.9538	−0.999660	−0.499830	−	0.866123i	$-0.666604\pi$
$727$	−26.9538	−0.999660	−0.499830	+	0.866123i	$0.666604\pi$
$728$	0	0
$729$	1.54057	0.0570581
$730$	0	0
$731$	−2.96351	−0.109609
$732$	0	0
$733$	18.3444	0.677566	0.338783	−	0.940865i	$-0.389985\pi$
$733$	18.3444	0.677566	0.338783	+	0.940865i	$0.389985\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	32.6824	1.20387
$738$	0	0
$739$	25.0356	0.920948	0.460474	−	0.887673i	$-0.347679\pi$
$739$	25.0356	0.920948	0.460474	+	0.887673i	$0.347679\pi$
$740$	0	0
$741$	39.3883	1.44696
$742$	0	0
$743$	−43.6233	−1.60038	−0.800191	−	0.599745i	$-0.795269\pi$
$743$	−43.6233	−1.60038	−0.800191	+	0.599745i	$0.795269\pi$
$744$	0	0
$745$	−32.1810	−1.17902
$746$	0	0
$747$	−23.6962	−0.867000
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−1.06072	−0.0387063	−0.0193531	−	0.999813i	$-0.506161\pi$
$751$	−1.06072	−0.0387063	−0.0193531	+	0.999813i	$0.506161\pi$
$752$	0	0
$753$	21.1003	0.768938
$754$	0	0
$755$	43.7874	1.59359
$756$	0	0
$757$	−14.6533	−0.532583	−0.266291	−	0.963893i	$-0.585798\pi$
$757$	−14.6533	−0.532583	−0.266291	+	0.963893i	$0.585798\pi$
$758$	0	0
$759$	−16.1191	−0.585086
$760$	0	0
$761$	17.6782	0.640835	0.320417	−	0.947276i	$-0.396177\pi$
$761$	17.6782	0.640835	0.320417	+	0.947276i	$0.396177\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−3.17456	−0.114777
$766$	0	0
$767$	−16.9684	−0.612692
$768$	0	0
$769$	5.72443	0.206428	0.103214	−	0.994659i	$-0.467087\pi$
$769$	5.72443	0.206428	0.103214	+	0.994659i	$0.467087\pi$
$770$	0	0
$771$	25.8269	0.930134
$772$	0	0
$773$	−31.0288	−1.11603	−0.558014	−	0.829831i	$-0.688436\pi$
$773$	−31.0288	−1.11603	−0.558014	+	0.829831i	$0.688436\pi$
$774$	0	0
$775$	−14.5361	−0.522151
$776$	0	0
$777$	0	0
$778$	0	0
$779$	27.6539	0.990804
$780$	0	0
$781$	23.0444	0.824593
$782$	0	0
$783$	−28.2344	−1.00901
$784$	0	0
$785$	24.9718	0.891282
$786$	0	0
$787$	28.7361	1.02433	0.512166	−	0.858887i	$-0.328843\pi$
$787$	28.7361	1.02433	0.512166	+	0.858887i	$0.328843\pi$
$788$	0	0
$789$	−52.7291	−1.87721
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−68.6140	−2.43655
$794$	0	0
$795$	−3.73492	−0.132464
$796$	0	0
$797$	−31.8711	−1.12893	−0.564466	−	0.825456i	$-0.690918\pi$
$797$	−31.8711	−1.12893	−0.564466	+	0.825456i	$0.690918\pi$
$798$	0	0
$799$	1.73607	0.0614178
$800$	0	0
$801$	12.0470	0.425661
$802$	0	0
$803$	−44.1473	−1.55792
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−35.9331	−1.26491
$808$	0	0
$809$	−10.3076	−0.362394	−0.181197	−	0.983447i	$-0.557997\pi$
$809$	−10.3076	−0.362394	−0.181197	+	0.983447i	$0.557997\pi$
$810$	0	0
$811$	−43.1650	−1.51573	−0.757864	−	0.652413i	$-0.773757\pi$
$811$	−43.1650	−1.51573	−0.757864	+	0.652413i	$0.773757\pi$
$812$	0	0
$813$	−64.9140	−2.27663
$814$	0	0
$815$	58.1489	2.03687
$816$	0	0
$817$	22.5430	0.788679
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−43.6628	−1.52384	−0.761921	−	0.647670i	$-0.775744\pi$
$821$	−43.6628	−1.52384	−0.761921	+	0.647670i	$0.775744\pi$
$822$	0	0
$823$	23.0272	0.802679	0.401340	−	0.915929i	$-0.368545\pi$
$823$	23.0272	0.802679	0.401340	+	0.915929i	$0.368545\pi$
$824$	0	0
$825$	−82.6271	−2.87671
$826$	0	0
$827$	−5.57382	−0.193821	−0.0969104	−	0.995293i	$-0.530896\pi$
$827$	−5.57382	−0.193821	−0.0969104	+	0.995293i	$0.530896\pi$
$828$	0	0
$829$	−23.3027	−0.809336	−0.404668	−	0.914464i	$-0.632613\pi$
$829$	−23.3027	−0.809336	−0.404668	+	0.914464i	$0.632613\pi$
$830$	0	0
$831$	−47.1349	−1.63509
$832$	0	0
$833$	0	0
$834$	0	0
$835$	9.08798	0.314503
$836$	0	0
$837$	4.37285	0.151148
$838$	0	0
$839$	34.1423	1.17872	0.589361	−	0.807869i	$-0.299379\pi$
$839$	34.1423	1.17872	0.589361	+	0.807869i	$0.299379\pi$
$840$	0	0
$841$	58.3611	2.01245
$842$	0	0
$843$	26.5728	0.915215
$844$	0	0
$845$	35.0663	1.20632
$846$	0	0
$847$	0	0
$848$	0	0
$849$	24.4697	0.839798
$850$	0	0
$851$	−9.10594	−0.312148
$852$	0	0
$853$	−2.61863	−0.0896602	−0.0448301	−	0.998995i	$-0.514275\pi$
$853$	−2.61863	−0.0896602	−0.0448301	+	0.998995i	$0.514275\pi$
$854$	0	0
$855$	24.1484	0.825858
$856$	0	0
$857$	−56.4325	−1.92770	−0.963849	−	0.266451i	$-0.914149\pi$
$857$	−56.4325	−1.92770	−0.963849	+	0.266451i	$0.914149\pi$
$858$	0	0
$859$	37.3510	1.27440	0.637200	−	0.770698i	$-0.280092\pi$
$859$	37.3510	1.27440	0.637200	+	0.770698i	$0.280092\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	3.02753	0.103058	0.0515292	−	0.998671i	$-0.483590\pi$
$863$	3.02753	0.103058	0.0515292	+	0.998671i	$0.483590\pi$
$864$	0	0
$865$	10.9501	0.372315
$866$	0	0
$867$	−35.8536	−1.21765
$868$	0	0
$869$	−26.4844	−0.898422
$870$	0	0
$871$	−39.9505	−1.35367
$872$	0	0
$873$	5.71920	0.193566
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−24.6508	−0.832399	−0.416199	−	0.909273i	$-0.636638\pi$
$877$	−24.6508	−0.832399	−0.416199	+	0.909273i	$0.636638\pi$
$878$	0	0
$879$	53.4401	1.80249
$880$	0	0
$881$	−16.3563	−0.551057	−0.275529	−	0.961293i	$-0.588853\pi$
$881$	−16.3563	−0.551057	−0.275529	+	0.961293i	$0.588853\pi$
$882$	0	0
$883$	−24.4570	−0.823044	−0.411522	−	0.911400i	$-0.635003\pi$
$883$	−24.4570	−0.823044	−0.411522	+	0.911400i	$0.635003\pi$
$884$	0	0
$885$	−30.0312	−1.00949
$886$	0	0
$887$	2.95690	0.0992829	0.0496414	−	0.998767i	$-0.484192\pi$
$887$	2.95690	0.0992829	0.0496414	+	0.998767i	$0.484192\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	43.1771	1.44649
$892$	0	0
$893$	−13.2060	−0.441923
$894$	0	0
$895$	−47.6791	−1.59374
$896$	0	0
$897$	19.7037	0.657888
$898$	0	0
$899$	−13.5302	−0.451258
$900$	0	0
$901$	0.231398	0.00770900
$902$	0	0
$903$	0	0
$904$	0	0
$905$	25.2147	0.838165
$906$	0	0
$907$	27.0307	0.897540	0.448770	−	0.893647i	$-0.351862\pi$
$907$	27.0307	0.897540	0.448770	+	0.893647i	$0.351862\pi$
$908$	0	0
$909$	−17.1963	−0.570364
$910$	0	0
$911$	7.18005	0.237886	0.118943	−	0.992901i	$-0.462049\pi$
$911$	7.18005	0.237886	0.118943	+	0.992901i	$0.462049\pi$
$912$	0	0
$913$	57.2386	1.89432
$914$	0	0
$915$	−121.435	−4.01453
$916$	0	0
$917$	0	0
$918$	0	0
$919$	0.655394	0.0216194	0.0108097	−	0.999942i	$-0.496559\pi$
$919$	0.655394	0.0216194	0.0108097	+	0.999942i	$0.496559\pi$
$920$	0	0
$921$	−3.57946	−0.117947
$922$	0	0
$923$	−28.1691	−0.927196
$924$	0	0
$925$	−46.6774	−1.53474
$926$	0	0
$927$	−16.0316	−0.526548
$928$	0	0
$929$	−29.0938	−0.954538	−0.477269	−	0.878757i	$-0.658373\pi$
$929$	−29.0938	−0.954538	−0.477269	+	0.878757i	$0.658373\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−65.2532	−2.13629
$934$	0	0
$935$	7.66820	0.250777
$936$	0	0
$937$	−13.8159	−0.451345	−0.225672	−	0.974203i	$-0.572458\pi$
$937$	−13.8159	−0.451345	−0.225672	+	0.974203i	$0.572458\pi$
$938$	0	0
$939$	44.6310	1.45648
$940$	0	0
$941$	−5.09222	−0.166001	−0.0830007	−	0.996549i	$-0.526450\pi$
$941$	−5.09222	−0.166001	−0.0830007	+	0.996549i	$0.526450\pi$
$942$	0	0
$943$	13.8337	0.450487
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−8.35799	−0.271598	−0.135799	−	0.990736i	$-0.543360\pi$
$947$	−8.35799	−0.271598	−0.135799	+	0.990736i	$0.543360\pi$
$948$	0	0
$949$	53.9649	1.75178
$950$	0	0
$951$	3.16884	0.102757
$952$	0	0
$953$	−24.6058	−0.797060	−0.398530	−	0.917155i	$-0.630480\pi$
$953$	−24.6058	−0.797060	−0.398530	+	0.917155i	$0.630480\pi$
$954$	0	0
$955$	26.7053	0.864162
$956$	0	0
$957$	−76.9095	−2.48613
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−28.9045	−0.932403
$962$	0	0
$963$	29.2731	0.943312
$964$	0	0
$965$	−14.9550	−0.481418
$966$	0	0
$967$	−25.3815	−0.816215	−0.408107	−	0.912934i	$-0.633811\pi$
$967$	−25.3815	−0.816215	−0.408107	+	0.912934i	$0.633811\pi$
$968$	0	0
$969$	−4.31896	−0.138745
$970$	0	0
$971$	−38.2970	−1.22901	−0.614504	−	0.788914i	$-0.710644\pi$
$971$	−38.2970	−1.22901	−0.614504	+	0.788914i	$0.710644\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	101.002	3.23465
$976$	0	0
$977$	5.94197	0.190100	0.0950502	−	0.995472i	$-0.469699\pi$
$977$	5.94197	0.190100	0.0950502	+	0.995472i	$0.469699\pi$
$978$	0	0
$979$	−29.0998	−0.930034
$980$	0	0
$981$	−2.20034	−0.0702514
$982$	0	0
$983$	33.5317	1.06949	0.534747	−	0.845012i	$-0.320407\pi$
$983$	33.5317	1.06949	0.534747	+	0.845012i	$0.320407\pi$
$984$	0	0
$985$	20.2835	0.646285
$986$	0	0
$987$	0	0
$988$	0	0
$989$	11.2770	0.358587
$990$	0	0
$991$	−31.9658	−1.01543	−0.507714	−	0.861526i	$-0.669509\pi$
$991$	−31.9658	−1.01543	−0.507714	+	0.861526i	$0.669509\pi$
$992$	0	0
$993$	−26.8749	−0.852849
$994$	0	0
$995$	10.5094	0.333172
$996$	0	0
$997$	40.2141	1.27359	0.636796	−	0.771032i	$-0.280259\pi$
$997$	40.2141	1.27359	0.636796	+	0.771032i	$0.280259\pi$
$998$	0	0
$999$	14.0418	0.444264

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6272.2.a.bu.1.3		4
4.3	odd	2		6272.2.a.be.1.2		4
7.3	odd	6		896.2.i.g.513.3	yes	8
7.5	odd	6		896.2.i.g.641.3	yes	8
7.6	odd	2		6272.2.a.z.1.2		4
8.3	odd	2		6272.2.a.bq.1.3		4
8.5	even	2		6272.2.a.ba.1.2		4
28.3	even	6		896.2.i.d.513.2	yes	8
28.19	even	6		896.2.i.d.641.2	yes	8
28.27	even	2		6272.2.a.bp.1.3		4
56.3	even	6		896.2.i.e.513.3	yes	8
56.5	odd	6		896.2.i.b.641.2	yes	8
56.13	odd	2		6272.2.a.bt.1.3		4
56.19	even	6		896.2.i.e.641.3	yes	8
56.27	even	2		6272.2.a.bd.1.2		4
56.45	odd	6		896.2.i.b.513.2	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
896.2.i.b.513.2	✓	8	56.45	odd	6
896.2.i.b.641.2	yes	8	56.5	odd	6
896.2.i.d.513.2	yes	8	28.3	even	6
896.2.i.d.641.2	yes	8	28.19	even	6
896.2.i.e.513.3	yes	8	56.3	even	6
896.2.i.e.641.3	yes	8	56.19	even	6
896.2.i.g.513.3	yes	8	7.3	odd	6
896.2.i.g.641.3	yes	8	7.5	odd	6
6272.2.a.z.1.2		4	7.6	odd	2
6272.2.a.ba.1.2		4	8.5	even	2
6272.2.a.bd.1.2		4	56.27	even	2
6272.2.a.be.1.2		4	4.3	odd	2
6272.2.a.bp.1.3		4	28.27	even	2
6272.2.a.bq.1.3		4	8.3	odd	2
6272.2.a.bt.1.3		4	56.13	odd	2
6272.2.a.bu.1.3		4	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:	\( N \)	\(=\)	\( 6272 = 2^{7} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6272.a (trivial)

\(f(q)\)	\(=\)	\(q+2.14243 q^{3} +3.87834 q^{5} +1.59002 q^{9} +O(q^{10})\)	\(q+2.14243 q^{3} +3.87834 q^{5} +1.59002 q^{9} -3.84073 q^{11} +4.69484 q^{13} +8.30910 q^{15} -0.514794 q^{17} +3.91596 q^{19} +1.95893 q^{23} +10.0416 q^{25} -3.02078 q^{27} +9.34671 q^{29} -1.44759 q^{31} -8.22851 q^{33} -4.64842 q^{37} +10.0584 q^{39} +7.06185 q^{41} +5.75669 q^{43} +6.16666 q^{45} -3.37236 q^{47} -1.10291 q^{51} -0.449497 q^{53} -14.8957 q^{55} +8.38969 q^{57} -3.61426 q^{59} -14.6147 q^{61} +18.2082 q^{65} -8.50944 q^{67} +4.19689 q^{69} -6.00000 q^{71} +11.4945 q^{73} +21.5134 q^{75} +6.89567 q^{79} -11.2419 q^{81} -14.9031 q^{83} -1.99655 q^{85} +20.0247 q^{87} +7.57664 q^{89} -3.10137 q^{93} +15.1874 q^{95} +3.59693 q^{97} -6.10686 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{3} + 2 q^{5} + 4 q^{9}+O(q^{10}) \) 4 * q + 2 * q^3 + 2 * q^5 + 4 * q^9	\( 4 q + 2 q^{3} + 2 q^{5} + 4 q^{9} + 8 q^{13} + 2 q^{15} + 4 q^{17} + 4 q^{19} + 4 q^{23} + 8 q^{27} + 8 q^{29} - 10 q^{31} + 8 q^{33} + 2 q^{37} + 22 q^{39} + 12 q^{41} - 4 q^{43} - 14 q^{47} + 8 q^{51} - 10 q^{53} - 24 q^{55} + 12 q^{57} + 6 q^{59} + 6 q^{61} + 8 q^{65} - 22 q^{67} + 34 q^{69} - 24 q^{71} + 16 q^{73} + 32 q^{75} - 8 q^{79} - 24 q^{81} + 16 q^{83} - 6 q^{85} + 18 q^{87} + 8 q^{89} - 2 q^{93} + 16 q^{95} + 16 q^{97} + 30 q^{99}+O(q^{100}) \) 4 * q + 2 * q^3 + 2 * q^5 + 4 * q^9 + 8 * q^13 + 2 * q^15 + 4 * q^17 + 4 * q^19 + 4 * q^23 + 8 * q^27 + 8 * q^29 - 10 * q^31 + 8 * q^33 + 2 * q^37 + 22 * q^39 + 12 * q^41 - 4 * q^43 - 14 * q^47 + 8 * q^51 - 10 * q^53 - 24 * q^55 + 12 * q^57 + 6 * q^59 + 6 * q^61 + 8 * q^65 - 22 * q^67 + 34 * q^69 - 24 * q^71 + 16 * q^73 + 32 * q^75 - 8 * q^79 - 24 * q^81 + 16 * q^83 - 6 * q^85 + 18 * q^87 + 8 * q^89 - 2 * q^93 + 16 * q^95 + 16 * q^97 + 30 * q^99

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