LMFDB - Embedded newform 3969.2.a.u.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3969,2,Mod(1,3969)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3969, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("3969.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3969 = 3^{4} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3969.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:	$31.6926245622$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{3}, \sqrt{7})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 5x^{2} + 1 $ x^4 - 5*x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 567)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-2.18890$ of defining polynomial
Character	$\chi$	$=$	3969.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.456850 q^{2} -1.79129 q^{4} -4.37780 q^{5} -1.73205 q^{8} +O(q^{10})$	$q+0.456850 q^{2} -1.79129 q^{4} -4.37780 q^{5} -1.73205 q^{8} -2.00000 q^{10} +2.64575 q^{11} -4.00000 q^{13} +2.79129 q^{16} +3.46410 q^{17} +3.58258 q^{19} +7.84190 q^{20} +1.20871 q^{22} +3.46410 q^{23} +14.1652 q^{25} -1.82740 q^{26} -1.82740 q^{29} -9.16515 q^{31} +4.73930 q^{32} +1.58258 q^{34} +3.00000 q^{37} +1.63670 q^{38} +7.58258 q^{40} +4.37780 q^{41} -8.58258 q^{43} -4.73930 q^{44} +1.58258 q^{46} +2.74110 q^{47} +6.47135 q^{50} +7.16515 q^{52} +8.66025 q^{53} -11.5826 q^{55} -0.834849 q^{58} +3.46410 q^{59} -2.41742 q^{61} -4.18710 q^{62} -3.41742 q^{64} +17.5112 q^{65} -0.582576 q^{67} -6.20520 q^{68} -11.4014 q^{71} +3.16515 q^{73} +1.37055 q^{74} -6.41742 q^{76} -8.58258 q^{79} -12.2197 q^{80} +2.00000 q^{82} +6.20520 q^{83} -15.1652 q^{85} -3.92095 q^{86} -4.58258 q^{88} +8.75560 q^{89} -6.20520 q^{92} +1.25227 q^{94} -15.6838 q^{95} -7.58258 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{4}+O(q^{10}) $ 4 * q + 2 * q^4	$ 4 q + 2 q^{4} - 8 q^{10} - 16 q^{13} + 2 q^{16} - 4 q^{19} + 14 q^{22} + 20 q^{25} - 12 q^{34} + 12 q^{37} + 12 q^{40} - 16 q^{43} - 12 q^{46} - 8 q^{52} - 28 q^{55} - 40 q^{58} - 28 q^{61} - 32 q^{64} + 16 q^{67} - 24 q^{73} - 44 q^{76} - 16 q^{79} + 8 q^{82} - 24 q^{85} + 60 q^{94} - 12 q^{97}+O(q^{100}) $ 4 * q + 2 * q^4 - 8 * q^10 - 16 * q^13 + 2 * q^16 - 4 * q^19 + 14 * q^22 + 20 * q^25 - 12 * q^34 + 12 * q^37 + 12 * q^40 - 16 * q^43 - 12 * q^46 - 8 * q^52 - 28 * q^55 - 40 * q^58 - 28 * q^61 - 32 * q^64 + 16 * q^67 - 24 * q^73 - 44 * q^76 - 16 * q^79 + 8 * q^82 - 24 * q^85 + 60 * q^94 - 12 * 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 $

$2$	0.456850	0.323042	0.161521	−	0.986869i	$-0.448360\pi$
$2$	0.456850	0.323042	0.161521	+	0.986869i	$0.448360\pi$
$3$	0	0
$3$	0	0
$4$	−1.79129	−0.895644
$5$	−4.37780	−1.95781	−0.978906	−	0.204310i	$-0.934505\pi$
$5$	−4.37780	−1.95781	−0.978906	+	0.204310i	$0.934505\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−1.73205	−0.612372
$9$	0	0
$10$	−2.00000	−0.632456
$11$	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
$13$	−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
$16$	2.79129	0.697822
$17$	3.46410	0.840168	0.420084	−	0.907485i	$-0.362001\pi$
$17$	3.46410	0.840168	0.420084	+	0.907485i	$0.362001\pi$
$18$	0	0
$19$	3.58258	0.821899	0.410950	−	0.911658i	$-0.365197\pi$
$19$	3.58258	0.821899	0.410950	+	0.911658i	$0.365197\pi$
$20$	7.84190	1.75350
$21$	0	0
$22$	1.20871	0.257698
$23$	3.46410	0.722315	0.361158	−	0.932505i	$-0.382382\pi$
$23$	3.46410	0.722315	0.361158	+	0.932505i	$0.382382\pi$
$24$	0	0
$25$	14.1652	2.83303
$26$	−1.82740	−0.358383
$27$	0	0
$28$	0	0
$29$	−1.82740	−0.339340	−0.169670	−	0.985501i	$-0.554270\pi$
$29$	−1.82740	−0.339340	−0.169670	+	0.985501i	$0.554270\pi$
$30$	0	0
$31$	−9.16515	−1.64611	−0.823055	−	0.567962i	$-0.807732\pi$
$31$	−9.16515	−1.64611	−0.823055	+	0.567962i	$0.807732\pi$
$32$	4.73930	0.837798
$33$	0	0
$34$	1.58258	0.271409
$35$	0	0
$36$	0	0
$37$	3.00000	0.493197	0.246598	−	0.969118i	$-0.420687\pi$
$37$	3.00000	0.493197	0.246598	+	0.969118i	$0.420687\pi$
$38$	1.63670	0.265508
$39$	0	0
$40$	7.58258	1.19891
$41$	4.37780	0.683698	0.341849	−	0.939755i	$-0.388947\pi$
$41$	4.37780	0.683698	0.341849	+	0.939755i	$0.388947\pi$
$42$	0	0
$43$	−8.58258	−1.30883	−0.654415	−	0.756135i	$-0.727085\pi$
$43$	−8.58258	−1.30883	−0.654415	+	0.756135i	$0.727085\pi$
$44$	−4.73930	−0.714477
$45$	0	0
$46$	1.58258	0.233338
$47$	2.74110	0.399831	0.199915	−	0.979813i	$-0.435933\pi$
$47$	2.74110	0.399831	0.199915	+	0.979813i	$0.435933\pi$
$48$	0	0
$49$	0	0
$50$	6.47135	0.915188
$51$	0	0
$52$	7.16515	0.993628
$53$	8.66025	1.18958	0.594789	−	0.803882i	$-0.297236\pi$
$53$	8.66025	1.18958	0.594789	+	0.803882i	$0.297236\pi$
$54$	0	0
$55$	−11.5826	−1.56179
$56$	0	0
$57$	0	0
$58$	−0.834849	−0.109621
$59$	3.46410	0.450988	0.225494	−	0.974245i	$-0.427600\pi$
$59$	3.46410	0.450988	0.225494	+	0.974245i	$0.427600\pi$
$60$	0	0
$61$	−2.41742	−0.309519	−0.154760	−	0.987952i	$-0.549460\pi$
$61$	−2.41742	−0.309519	−0.154760	+	0.987952i	$0.549460\pi$
$62$	−4.18710	−0.531762
$63$	0	0
$64$	−3.41742	−0.427178
$65$	17.5112	2.17200
$66$	0	0
$67$	−0.582576	−0.0711729	−0.0355865	−	0.999367i	$-0.511330\pi$
$67$	−0.582576	−0.0711729	−0.0355865	+	0.999367i	$0.511330\pi$
$68$	−6.20520	−0.752491
$69$	0	0
$70$	0	0
$71$	−11.4014	−1.35309	−0.676546	−	0.736400i	$-0.736524\pi$
$71$	−11.4014	−1.35309	−0.676546	+	0.736400i	$0.736524\pi$
$72$	0	0
$73$	3.16515	0.370453	0.185226	−	0.982696i	$-0.440698\pi$
$73$	3.16515	0.370453	0.185226	+	0.982696i	$0.440698\pi$
$74$	1.37055	0.159323
$75$	0	0
$76$	−6.41742	−0.736129
$77$	0	0
$78$	0	0
$79$	−8.58258	−0.965615	−0.482808	−	0.875726i	$-0.660383\pi$
$79$	−8.58258	−0.965615	−0.482808	+	0.875726i	$0.660383\pi$
$80$	−12.2197	−1.36620
$81$	0	0
$82$	2.00000	0.220863
$83$	6.20520	0.681110	0.340555	−	0.940225i	$-0.389385\pi$
$83$	6.20520	0.681110	0.340555	+	0.940225i	$0.389385\pi$
$84$	0	0
$85$	−15.1652	−1.64489
$86$	−3.92095	−0.422807
$87$	0	0
$88$	−4.58258	−0.488504
$89$	8.75560	0.928092	0.464046	−	0.885811i	$-0.346397\pi$
$89$	8.75560	0.928092	0.464046	+	0.885811i	$0.346397\pi$
$90$	0	0
$91$	0	0
$92$	−6.20520	−0.646937
$93$	0	0
$94$	1.25227	0.129162
$95$	−15.6838	−1.60912
$96$	0	0
$97$	−7.58258	−0.769894	−0.384947	−	0.922939i	$-0.625780\pi$
$97$	−7.58258	−0.769894	−0.384947	+	0.922939i	$0.625780\pi$
$98$	0	0
$99$	0	0
$100$	−25.3739	−2.53739
$101$	−1.82740	−0.181833	−0.0909166	−	0.995859i	$-0.528980\pi$
$101$	−1.82740	−0.181833	−0.0909166	+	0.995859i	$0.528980\pi$
$102$	0	0
$103$	−3.58258	−0.353002	−0.176501	−	0.984300i	$-0.556478\pi$
$103$	−3.58258	−0.353002	−0.176501	+	0.984300i	$0.556478\pi$
$104$	6.92820	0.679366
$105$	0	0
$106$	3.95644	0.384283
$107$	−11.2107	−1.08377	−0.541887	−	0.840451i	$-0.682290\pi$
$107$	−11.2107	−1.08377	−0.541887	+	0.840451i	$0.682290\pi$
$108$	0	0
$109$	6.00000	0.574696	0.287348	−	0.957826i	$-0.407226\pi$
$109$	6.00000	0.574696	0.287348	+	0.957826i	$0.407226\pi$
$110$	−5.29150	−0.504525
$111$	0	0
$112$	0	0
$113$	−13.9518	−1.31247	−0.656235	−	0.754556i	$-0.727852\pi$
$113$	−13.9518	−1.31247	−0.656235	+	0.754556i	$0.727852\pi$
$114$	0	0
$115$	−15.1652	−1.41416
$116$	3.27340	0.303928
$117$	0	0
$118$	1.58258	0.145688
$119$	0	0
$120$	0	0
$121$	−4.00000	−0.363636
$122$	−1.10440	−0.0999878
$123$	0	0
$124$	16.4174	1.47433
$125$	−40.1232	−3.58873
$126$	0	0
$127$	−7.41742	−0.658190	−0.329095	−	0.944297i	$-0.606744\pi$
$127$	−7.41742	−0.658190	−0.329095	+	0.944297i	$0.606744\pi$
$128$	−11.0399	−0.975795
$129$	0	0
$130$	8.00000	0.701646
$131$	18.4249	1.60979	0.804896	−	0.593416i	$-0.202221\pi$
$131$	18.4249	1.60979	0.804896	+	0.593416i	$0.202221\pi$
$132$	0	0
$133$	0	0
$134$	−0.266150	−0.0229918
$135$	0	0
$136$	−6.00000	−0.514496
$137$	−17.6066	−1.50423	−0.752115	−	0.659032i	$-0.770966\pi$
$137$	−17.6066	−1.50423	−0.752115	+	0.659032i	$0.770966\pi$
$138$	0	0
$139$	7.58258	0.643146	0.321573	−	0.946885i	$-0.395788\pi$
$139$	7.58258	0.643146	0.321573	+	0.946885i	$0.395788\pi$
$140$	0	0
$141$	0	0
$142$	−5.20871	−0.437105
$143$	−10.5830	−0.884995
$144$	0	0
$145$	8.00000	0.664364
$146$	1.44600	0.119672
$147$	0	0
$148$	−5.37386	−0.441729
$149$	10.6784	0.874805	0.437402	−	0.899266i	$-0.355899\pi$
$149$	10.6784	0.874805	0.437402	+	0.899266i	$0.355899\pi$
$150$	0	0
$151$	−20.5826	−1.67499	−0.837493	−	0.546448i	$-0.815980\pi$
$151$	−20.5826	−1.67499	−0.837493	+	0.546448i	$0.815980\pi$
$152$	−6.20520	−0.503308
$153$	0	0
$154$	0	0
$155$	40.1232	3.22277
$156$	0	0
$157$	−0.834849	−0.0666282	−0.0333141	−	0.999445i	$-0.510606\pi$
$157$	−0.834849	−0.0666282	−0.0333141	+	0.999445i	$0.510606\pi$
$158$	−3.92095	−0.311934
$159$	0	0
$160$	−20.7477	−1.64025
$161$	0	0
$162$	0	0
$163$	14.5826	1.14220	0.571098	−	0.820882i	$-0.306518\pi$
$163$	14.5826	1.14220	0.571098	+	0.820882i	$0.306518\pi$
$164$	−7.84190	−0.612350
$165$	0	0
$166$	2.83485	0.220027
$167$	2.55040	0.197356	0.0986780	−	0.995119i	$-0.468539\pi$
$167$	2.55040	0.197356	0.0986780	+	0.995119i	$0.468539\pi$
$168$	0	0
$169$	3.00000	0.230769
$170$	−6.92820	−0.531369
$171$	0	0
$172$	15.3739	1.17225
$173$	−11.4967	−0.874078	−0.437039	−	0.899442i	$-0.643973\pi$
$173$	−11.4967	−0.874078	−0.437039	+	0.899442i	$0.643973\pi$
$174$	0	0
$175$	0	0
$176$	7.38505	0.556669
$177$	0	0
$178$	4.00000	0.299813
$179$	−1.63670	−0.122333	−0.0611664	−	0.998128i	$-0.519482\pi$
$179$	−1.63670	−0.122333	−0.0611664	+	0.998128i	$0.519482\pi$
$180$	0	0
$181$	−13.1652	−0.978558	−0.489279	−	0.872127i	$-0.662740\pi$
$181$	−13.1652	−0.978558	−0.489279	+	0.872127i	$0.662740\pi$
$182$	0	0
$183$	0	0
$184$	−6.00000	−0.442326
$185$	−13.1334	−0.965587
$186$	0	0
$187$	9.16515	0.670222
$188$	−4.91010	−0.358106
$189$	0	0
$190$	−7.16515	−0.519815
$191$	6.10985	0.442093	0.221047	−	0.975263i	$-0.429053\pi$
$191$	6.10985	0.442093	0.221047	+	0.975263i	$0.429053\pi$
$192$	0	0
$193$	14.0000	1.00774	0.503871	−	0.863779i	$-0.331909\pi$
$193$	14.0000	1.00774	0.503871	+	0.863779i	$0.331909\pi$
$194$	−3.46410	−0.248708
$195$	0	0
$196$	0	0
$197$	−0.0953502	−0.00679342	−0.00339671	−	0.999994i	$-0.501081\pi$
$197$	−0.0953502	−0.00679342	−0.00339671	+	0.999994i	$0.501081\pi$
$198$	0	0
$199$	16.7477	1.18721	0.593607	−	0.804755i	$-0.297703\pi$
$199$	16.7477	1.18721	0.593607	+	0.804755i	$0.297703\pi$
$200$	−24.5348	−1.73487
$201$	0	0
$202$	−0.834849	−0.0587397
$203$	0	0
$204$	0	0
$205$	−19.1652	−1.33855
$206$	−1.63670	−0.114034
$207$	0	0
$208$	−11.1652	−0.774164
$209$	9.47860	0.655649
$210$	0	0
$211$	11.7477	0.808747	0.404373	−	0.914594i	$-0.367490\pi$
$211$	11.7477	0.808747	0.404373	+	0.914594i	$0.367490\pi$
$212$	−15.5130	−1.06544
$213$	0	0
$214$	−5.12159	−0.350105
$215$	37.5728	2.56244
$216$	0	0
$217$	0	0
$218$	2.74110	0.185651
$219$	0	0
$220$	20.7477	1.39881
$221$	−13.8564	−0.932083
$222$	0	0
$223$	−22.0000	−1.47323	−0.736614	−	0.676313i	$-0.763577\pi$
$223$	−22.0000	−1.47323	−0.736614	+	0.676313i	$0.763577\pi$
$224$	0	0
$225$	0	0
$226$	−6.37386	−0.423983
$227$	13.1334	0.871695	0.435847	−	0.900021i	$-0.356449\pi$
$227$	13.1334	0.871695	0.435847	+	0.900021i	$0.356449\pi$
$228$	0	0
$229$	−21.9129	−1.44804	−0.724022	−	0.689777i	$-0.757709\pi$
$229$	−21.9129	−1.44804	−0.724022	+	0.689777i	$0.757709\pi$
$230$	−6.92820	−0.456832
$231$	0	0
$232$	3.16515	0.207802
$233$	0	0		−	1.00000i	$-0.5\pi$
$233$	0	0			1.00000i	$0.5\pi$
$234$	0	0
$235$	−12.0000	−0.782794
$236$	−6.20520	−0.403924
$237$	0	0
$238$	0	0
$239$	25.2578	1.63379	0.816894	−	0.576787i	$-0.195694\pi$
$239$	25.2578	1.63379	0.816894	+	0.576787i	$0.195694\pi$
$240$	0	0
$241$	−18.4174	−1.18637	−0.593185	−	0.805066i	$-0.702130\pi$
$241$	−18.4174	−1.18637	−0.593185	+	0.805066i	$0.702130\pi$
$242$	−1.82740	−0.117470
$243$	0	0
$244$	4.33030	0.277219
$245$	0	0
$246$	0	0
$247$	−14.3303	−0.911815
$248$	15.8745	1.00803
$249$	0	0
$250$	−18.3303	−1.15931
$251$	−7.84190	−0.494977	−0.247488	−	0.968891i	$-0.579605\pi$
$251$	−7.84190	−0.494977	−0.247488	+	0.968891i	$0.579605\pi$
$252$	0	0
$253$	9.16515	0.576208
$254$	−3.38865	−0.212623
$255$	0	0
$256$	1.79129	0.111955
$257$	−10.2016	−0.636359	−0.318179	−	0.948031i	$-0.603071\pi$
$257$	−10.2016	−0.636359	−0.318179	+	0.948031i	$0.603071\pi$
$258$	0	0
$259$	0	0
$260$	−31.3676	−1.94534
$261$	0	0
$262$	8.41742	0.520030
$263$	−4.28245	−0.264067	−0.132034	−	0.991245i	$-0.542151\pi$
$263$	−4.28245	−0.264067	−0.132034	+	0.991245i	$0.542151\pi$
$264$	0	0
$265$	−37.9129	−2.32897
$266$	0	0
$267$	0	0
$268$	1.04356	0.0637456
$269$	6.92820	0.422420	0.211210	−	0.977441i	$-0.432260\pi$
$269$	6.92820	0.422420	0.211210	+	0.977441i	$0.432260\pi$
$270$	0	0
$271$	−8.74773	−0.531387	−0.265693	−	0.964058i	$-0.585601\pi$
$271$	−8.74773	−0.531387	−0.265693	+	0.964058i	$0.585601\pi$
$272$	9.66930	0.586288
$273$	0	0
$274$	−8.04356	−0.485929
$275$	37.4775	2.25998
$276$	0	0
$277$	7.00000	0.420589	0.210295	−	0.977638i	$-0.432558\pi$
$277$	7.00000	0.420589	0.210295	+	0.977638i	$0.432558\pi$
$278$	3.46410	0.207763
$279$	0	0
$280$	0	0
$281$	−29.6356	−1.76791	−0.883955	−	0.467572i	$-0.845129\pi$
$281$	−29.6356	−1.76791	−0.883955	+	0.467572i	$0.845129\pi$
$282$	0	0
$283$	−17.5826	−1.04518	−0.522588	−	0.852585i	$-0.675033\pi$
$283$	−17.5826	−1.04518	−0.522588	+	0.852585i	$0.675033\pi$
$284$	20.4231	1.21189
$285$	0	0
$286$	−4.83485	−0.285891
$287$	0	0
$288$	0	0
$289$	−5.00000	−0.294118
$290$	3.65480	0.214617
$291$	0	0
$292$	−5.66970	−0.331794
$293$	21.6983	1.26763	0.633814	−	0.773485i	$-0.281488\pi$
$293$	21.6983	1.26763	0.633814	+	0.773485i	$0.281488\pi$
$294$	0	0
$295$	−15.1652	−0.882949
$296$	−5.19615	−0.302020
$297$	0	0
$298$	4.87841	0.282599
$299$	−13.8564	−0.801337
$300$	0	0
$301$	0	0
$302$	−9.40315	−0.541091
$303$	0	0
$304$	10.0000	0.573539
$305$	10.5830	0.605981
$306$	0	0
$307$	−16.3303	−0.932020	−0.466010	−	0.884780i	$-0.654309\pi$
$307$	−16.3303	−0.932020	−0.466010	+	0.884780i	$0.654309\pi$
$308$	0	0
$309$	0	0
$310$	18.3303	1.04109
$311$	−18.0435	−1.02315	−0.511577	−	0.859238i	$-0.670938\pi$
$311$	−18.0435	−1.02315	−0.511577	+	0.859238i	$0.670938\pi$
$312$	0	0
$313$	1.16515	0.0658583	0.0329291	−	0.999458i	$-0.489516\pi$
$313$	1.16515	0.0658583	0.0329291	+	0.999458i	$0.489516\pi$
$314$	−0.381401	−0.0215237
$315$	0	0
$316$	15.3739	0.864847
$317$	0	0		−	1.00000i	$-0.5\pi$
$317$	0	0			1.00000i	$0.5\pi$
$318$	0	0
$319$	−4.83485	−0.270700
$320$	14.9608	0.836335
$321$	0	0
$322$	0	0
$323$	12.4104	0.690533
$324$	0	0
$325$	−56.6606	−3.14296
$326$	6.66205	0.368977
$327$	0	0
$328$	−7.58258	−0.418678
$329$	0	0
$330$	0	0
$331$	14.3303	0.787665	0.393832	−	0.919182i	$-0.371149\pi$
$331$	14.3303	0.787665	0.393832	+	0.919182i	$0.371149\pi$
$332$	−11.1153	−0.610032
$333$	0	0
$334$	1.16515	0.0637542
$335$	2.55040	0.139343
$336$	0	0
$337$	−20.1652	−1.09847	−0.549233	−	0.835669i	$-0.685080\pi$
$337$	−20.1652	−1.09847	−0.549233	+	0.835669i	$0.685080\pi$
$338$	1.37055	0.0745481
$339$	0	0
$340$	27.1652	1.47324
$341$	−24.2487	−1.31314
$342$	0	0
$343$	0	0
$344$	14.8655	0.801492
$345$	0	0
$346$	−5.25227	−0.282364
$347$	36.0315	1.93427	0.967135	−	0.254262i	$-0.0818327\pi$
$347$	36.0315	1.93427	0.967135	+	0.254262i	$0.0818327\pi$
$348$	0	0
$349$	24.7477	1.32472	0.662358	−	0.749188i	$-0.269556\pi$
$349$	24.7477	1.32472	0.662358	+	0.749188i	$0.269556\pi$
$350$	0	0
$351$	0	0
$352$	12.5390	0.668332
$353$	35.7454	1.90254	0.951268	−	0.308364i	$-0.0997816\pi$
$353$	35.7454	1.90254	0.951268	+	0.308364i	$0.0997816\pi$
$354$	0	0
$355$	49.9129	2.64910
$356$	−15.6838	−0.831240
$357$	0	0
$358$	−0.747727	−0.0395186
$359$	−30.3586	−1.60226	−0.801132	−	0.598488i	$-0.795768\pi$
$359$	−30.3586	−1.60226	−0.801132	+	0.598488i	$0.795768\pi$
$360$	0	0
$361$	−6.16515	−0.324482
$362$	−6.01450	−0.316115
$363$	0	0
$364$	0	0
$365$	−13.8564	−0.725277
$366$	0	0
$367$	8.83485	0.461175	0.230588	−	0.973052i	$-0.425935\pi$
$367$	8.83485	0.461175	0.230588	+	0.973052i	$0.425935\pi$
$368$	9.66930	0.504047
$369$	0	0
$370$	−6.00000	−0.311925
$371$	0	0
$372$	0	0
$373$	−7.33030	−0.379549	−0.189774	−	0.981828i	$-0.560776\pi$
$373$	−7.33030	−0.379549	−0.189774	+	0.981828i	$0.560776\pi$
$374$	4.18710	0.216510
$375$	0	0
$376$	−4.74773	−0.244845
$377$	7.30960	0.376464
$378$	0	0
$379$	−28.9129	−1.48515	−0.742577	−	0.669760i	$-0.766397\pi$
$379$	−28.9129	−1.48515	−0.742577	+	0.669760i	$0.766397\pi$
$380$	28.0942	1.44120
$381$	0	0
$382$	2.79129	0.142815
$383$	−19.3386	−0.988157	−0.494078	−	0.869417i	$-0.664494\pi$
$383$	−19.3386	−0.988157	−0.494078	+	0.869417i	$0.664494\pi$
$384$	0	0
$385$	0	0
$386$	6.39590	0.325543
$387$	0	0
$388$	13.5826	0.689551
$389$	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
$391$	12.0000	0.606866
$392$	0	0
$393$	0	0
$394$	−0.0435608	−0.00219456
$395$	37.5728	1.89049
$396$	0	0
$397$	1.58258	0.0794272	0.0397136	−	0.999211i	$-0.487355\pi$
$397$	1.58258	0.0794272	0.0397136	+	0.999211i	$0.487355\pi$
$398$	7.65120	0.383520
$399$	0	0
$400$	39.5390	1.97695
$401$	−14.1425	−0.706241	−0.353120	−	0.935578i	$-0.614879\pi$
$401$	−14.1425	−0.706241	−0.353120	+	0.935578i	$0.614879\pi$
$402$	0	0
$403$	36.6606	1.82619
$404$	3.27340	0.162858
$405$	0	0
$406$	0	0
$407$	7.93725	0.393435
$408$	0	0
$409$	−9.25227	−0.457495	−0.228748	−	0.973486i	$-0.573463\pi$
$409$	−9.25227	−0.457495	−0.228748	+	0.973486i	$0.573463\pi$
$410$	−8.75560	−0.432408
$411$	0	0
$412$	6.41742	0.316164
$413$	0	0
$414$	0	0
$415$	−27.1652	−1.33348
$416$	−18.9572	−0.929454
$417$	0	0
$418$	4.33030	0.211802
$419$	18.6156	0.909432	0.454716	−	0.890636i	$-0.349741\pi$
$419$	18.6156	0.909432	0.454716	+	0.890636i	$0.349741\pi$
$420$	0	0
$421$	−16.1652	−0.787841	−0.393921	−	0.919144i	$-0.628882\pi$
$421$	−16.1652	−0.787841	−0.393921	+	0.919144i	$0.628882\pi$
$422$	5.36695	0.261259
$423$	0	0
$424$	−15.0000	−0.728464
$425$	49.0695	2.38022
$426$	0	0
$427$	0	0
$428$	20.0815	0.970676
$429$	0	0
$430$	17.1652	0.827777
$431$	36.6591	1.76581	0.882904	−	0.469554i	$-0.155585\pi$
$431$	36.6591	1.76581	0.882904	+	0.469554i	$0.155585\pi$
$432$	0	0
$433$	21.9129	1.05307	0.526533	−	0.850155i	$-0.323492\pi$
$433$	21.9129	1.05307	0.526533	+	0.850155i	$0.323492\pi$
$434$	0	0
$435$	0	0
$436$	−10.7477	−0.514723
$437$	12.4104	0.593670
$438$	0	0
$439$	−25.5826	−1.22099	−0.610495	−	0.792020i	$-0.709029\pi$
$439$	−25.5826	−1.22099	−0.610495	+	0.792020i	$0.709029\pi$
$440$	20.0616	0.956400
$441$	0	0
$442$	−6.33030	−0.301102
$443$	−17.3205	−0.822922	−0.411461	−	0.911427i	$-0.634981\pi$
$443$	−17.3205	−0.822922	−0.411461	+	0.911427i	$0.634981\pi$
$444$	0	0
$445$	−38.3303	−1.81703
$446$	−10.0507	−0.475915
$447$	0	0
$448$	0	0
$449$	−26.5529	−1.25311	−0.626554	−	0.779378i	$-0.715535\pi$
$449$	−26.5529	−1.25311	−0.626554	+	0.779378i	$0.715535\pi$
$450$	0	0
$451$	11.5826	0.545402
$452$	24.9916	1.17551
$453$	0	0
$454$	6.00000	0.281594
$455$	0	0
$456$	0	0
$457$	31.3303	1.46557	0.732785	−	0.680460i	$-0.238220\pi$
$457$	31.3303	1.46557	0.732785	+	0.680460i	$0.238220\pi$
$458$	−10.0109	−0.467779
$459$	0	0
$460$	27.1652	1.26658
$461$	13.3241	0.620566	0.310283	−	0.950644i	$-0.399576\pi$
$461$	13.3241	0.620566	0.310283	+	0.950644i	$0.399576\pi$
$462$	0	0
$463$	20.9129	0.971904	0.485952	−	0.873986i	$-0.338473\pi$
$463$	20.9129	0.971904	0.485952	+	0.873986i	$0.338473\pi$
$464$	−5.10080	−0.236799
$465$	0	0
$466$	0	0
$467$	−6.20520	−0.287143	−0.143571	−	0.989640i	$-0.545859\pi$
$467$	−6.20520	−0.287143	−0.143571	+	0.989640i	$0.545859\pi$
$468$	0	0
$469$	0	0
$470$	−5.48220	−0.252875
$471$	0	0
$472$	−6.00000	−0.276172
$473$	−22.7074	−1.04409
$474$	0	0
$475$	50.7477	2.32847
$476$	0	0
$477$	0	0
$478$	11.5390	0.527782
$479$	−24.6301	−1.12538	−0.562689	−	0.826668i	$-0.690233\pi$
$479$	−24.6301	−1.12538	−0.562689	+	0.826668i	$0.690233\pi$
$480$	0	0
$481$	−12.0000	−0.547153
$482$	−8.41400	−0.383247
$483$	0	0
$484$	7.16515	0.325689
$485$	33.1950	1.50731
$486$	0	0
$487$	−6.58258	−0.298285	−0.149142	−	0.988816i	$-0.547651\pi$
$487$	−6.58258	−0.298285	−0.149142	+	0.988816i	$0.547651\pi$
$488$	4.18710	0.189541
$489$	0	0
$490$	0	0
$491$	−22.4213	−1.01186	−0.505930	−	0.862575i	$-0.668851\pi$
$491$	−22.4213	−1.01186	−0.505930	+	0.862575i	$0.668851\pi$
$492$	0	0
$493$	−6.33030	−0.285102
$494$	−6.54680	−0.294555
$495$	0	0
$496$	−25.5826	−1.14869
$497$	0	0
$498$	0	0
$499$	−20.0000	−0.895323	−0.447661	−	0.894203i	$-0.647743\pi$
$499$	−20.0000	−0.895323	−0.447661	+	0.894203i	$0.647743\pi$
$500$	71.8722	3.21422
$501$	0	0
$502$	−3.58258	−0.159898
$503$	−5.48220	−0.244439	−0.122220	−	0.992503i	$-0.539001\pi$
$503$	−5.48220	−0.244439	−0.122220	+	0.992503i	$0.539001\pi$
$504$	0	0
$505$	8.00000	0.355995
$506$	4.18710	0.186139
$507$	0	0
$508$	13.2867	0.589504
$509$	−19.1479	−0.848716	−0.424358	−	0.905494i	$-0.639500\pi$
$509$	−19.1479	−0.848716	−0.424358	+	0.905494i	$0.639500\pi$
$510$	0	0
$511$	0	0
$512$	22.8981	1.01196
$513$	0	0
$514$	−4.66061	−0.205570
$515$	15.6838	0.691111
$516$	0	0
$517$	7.25227	0.318955
$518$	0	0
$519$	0	0
$520$	−30.3303	−1.33007
$521$	3.46410	0.151765	0.0758825	−	0.997117i	$-0.475823\pi$
$521$	3.46410	0.151765	0.0758825	+	0.997117i	$0.475823\pi$
$522$	0	0
$523$	6.41742	0.280614	0.140307	−	0.990108i	$-0.455191\pi$
$523$	6.41742	0.280614	0.140307	+	0.990108i	$0.455191\pi$
$524$	−33.0043	−1.44180
$525$	0	0
$526$	−1.95644	−0.0853048
$527$	−31.7490	−1.38301
$528$	0	0
$529$	−11.0000	−0.478261
$530$	−17.3205	−0.752355
$531$	0	0
$532$	0	0
$533$	−17.5112	−0.758495
$534$	0	0
$535$	49.0780	2.12183
$536$	1.00905	0.0435844
$537$	0	0
$538$	3.16515	0.136459
$539$	0	0
$540$	0	0
$541$	11.0000	0.472927	0.236463	−	0.971640i	$-0.424012\pi$
$541$	11.0000	0.472927	0.236463	+	0.971640i	$0.424012\pi$
$542$	−3.99640	−0.171660
$543$	0	0
$544$	16.4174	0.703891
$545$	−26.2668	−1.12515
$546$	0	0
$547$	−12.9129	−0.552115	−0.276057	−	0.961141i	$-0.589028\pi$
$547$	−12.9129	−0.552115	−0.276057	+	0.961141i	$0.589028\pi$
$548$	31.5384	1.34725
$549$	0	0
$550$	17.1216	0.730067
$551$	−6.54680	−0.278903
$552$	0	0
$553$	0	0
$554$	3.19795	0.135868
$555$	0	0
$556$	−13.5826	−0.576030
$557$	−12.5058	−0.529886	−0.264943	−	0.964264i	$-0.585353\pi$
$557$	−12.5058	−0.529886	−0.264943	+	0.964264i	$0.585353\pi$
$558$	0	0
$559$	34.3303	1.45202
$560$	0	0
$561$	0	0
$562$	−13.5390	−0.571109
$563$	−41.7599	−1.75997	−0.879985	−	0.475001i	$-0.842448\pi$
$563$	−41.7599	−1.75997	−0.879985	+	0.475001i	$0.842448\pi$
$564$	0	0
$565$	61.0780	2.56957
$566$	−8.03260	−0.337636
$567$	0	0
$568$	19.7477	0.828596
$569$	−11.9337	−0.500285	−0.250142	−	0.968209i	$-0.580477\pi$
$569$	−11.9337	−0.500285	−0.250142	+	0.968209i	$0.580477\pi$
$570$	0	0
$571$	−0.834849	−0.0349373	−0.0174687	−	0.999847i	$-0.505561\pi$
$571$	−0.834849	−0.0349373	−0.0174687	+	0.999847i	$0.505561\pi$
$572$	18.9572	0.792641
$573$	0	0
$574$	0	0
$575$	49.0695	2.04634
$576$	0	0
$577$	−32.7477	−1.36331	−0.681653	−	0.731676i	$-0.738739\pi$
$577$	−32.7477	−1.36331	−0.681653	+	0.731676i	$0.738739\pi$
$578$	−2.28425	−0.0950123
$579$	0	0
$580$	−14.3303	−0.595033
$581$	0	0
$582$	0	0
$583$	22.9129	0.948954
$584$	−5.48220	−0.226855
$585$	0	0
$586$	9.91288	0.409497
$587$	−8.56490	−0.353511	−0.176756	−	0.984255i	$-0.556560\pi$
$587$	−8.56490	−0.353511	−0.176756	+	0.984255i	$0.556560\pi$
$588$	0	0
$589$	−32.8348	−1.35294
$590$	−6.92820	−0.285230
$591$	0	0
$592$	8.37386	0.344164
$593$	−43.5873	−1.78992	−0.894958	−	0.446150i	$-0.852795\pi$
$593$	−43.5873	−1.78992	−0.894958	+	0.446150i	$0.852795\pi$
$594$	0	0
$595$	0	0
$596$	−19.1280	−0.783514
$597$	0	0
$598$	−6.33030	−0.258865
$599$	−4.47315	−0.182768	−0.0913840	−	0.995816i	$-0.529129\pi$
$599$	−4.47315	−0.182768	−0.0913840	+	0.995816i	$0.529129\pi$
$600$	0	0
$601$	−4.41742	−0.180190	−0.0900952	−	0.995933i	$-0.528717\pi$
$601$	−4.41742	−0.180190	−0.0900952	+	0.995933i	$0.528717\pi$
$602$	0	0
$603$	0	0
$604$	36.8693	1.50019
$605$	17.5112	0.711932
$606$	0	0
$607$	−21.5826	−0.876010	−0.438005	−	0.898973i	$-0.644315\pi$
$607$	−21.5826	−0.876010	−0.438005	+	0.898973i	$0.644315\pi$
$608$	16.9789	0.688586
$609$	0	0
$610$	4.83485	0.195757
$611$	−10.9644	−0.443572
$612$	0	0
$613$	1.00000	0.0403896	0.0201948	−	0.999796i	$-0.493571\pi$
$613$	1.00000	0.0403896	0.0201948	+	0.999796i	$0.493571\pi$
$614$	−7.46050	−0.301081
$615$	0	0
$616$	0	0
$617$	−24.8208	−0.999248	−0.499624	−	0.866242i	$-0.666529\pi$
$617$	−24.8208	−0.999248	−0.499624	+	0.866242i	$0.666529\pi$
$618$	0	0
$619$	27.9129	1.12191	0.560957	−	0.827845i	$-0.310433\pi$
$619$	27.9129	1.12191	0.560957	+	0.827845i	$0.310433\pi$
$620$	−71.8722	−2.88646
$621$	0	0
$622$	−8.24318	−0.330521
$623$	0	0
$624$	0	0
$625$	104.826	4.19303
$626$	0.532300	0.0212750
$627$	0	0
$628$	1.49545	0.0596751
$629$	10.3923	0.414368
$630$	0	0
$631$	−31.1652	−1.24067	−0.620333	−	0.784339i	$-0.713002\pi$
$631$	−31.1652	−1.24067	−0.620333	+	0.784339i	$0.713002\pi$
$632$	14.8655	0.591316
$633$	0	0
$634$	0	0
$635$	32.4720	1.28861
$636$	0	0
$637$	0	0
$638$	−2.20880	−0.0874473
$639$	0	0
$640$	48.3303	1.91042
$641$	−12.1244	−0.478883	−0.239442	−	0.970911i	$-0.576964\pi$
$641$	−12.1244	−0.478883	−0.239442	+	0.970911i	$0.576964\pi$
$642$	0	0
$643$	34.3303	1.35385	0.676927	−	0.736050i	$-0.263311\pi$
$643$	34.3303	1.35385	0.676927	+	0.736050i	$0.263311\pi$
$644$	0	0
$645$	0	0
$646$	5.66970	0.223071
$647$	11.3060	0.444485	0.222242	−	0.974991i	$-0.428662\pi$
$647$	11.3060	0.444485	0.222242	+	0.974991i	$0.428662\pi$
$648$	0	0
$649$	9.16515	0.359764
$650$	−25.8854	−1.01531
$651$	0	0
$652$	−26.1216	−1.02300
$653$	−12.1244	−0.474463	−0.237231	−	0.971453i	$-0.576240\pi$
$653$	−12.1244	−0.474463	−0.237231	+	0.971453i	$0.576240\pi$
$654$	0	0
$655$	−80.6606	−3.15167
$656$	12.2197	0.477099
$657$	0	0
$658$	0	0
$659$	−6.49125	−0.252863	−0.126432	−	0.991975i	$-0.540352\pi$
$659$	−6.49125	−0.252863	−0.126432	+	0.991975i	$0.540352\pi$
$660$	0	0
$661$	8.00000	0.311164	0.155582	−	0.987823i	$-0.450275\pi$
$661$	8.00000	0.311164	0.155582	+	0.987823i	$0.450275\pi$
$662$	6.54680	0.254449
$663$	0	0
$664$	−10.7477	−0.417093
$665$	0	0
$666$	0	0
$667$	−6.33030	−0.245110
$668$	−4.56850	−0.176761
$669$	0	0
$670$	1.16515	0.0450137
$671$	−6.39590	−0.246911
$672$	0	0
$673$	28.4955	1.09842	0.549210	−	0.835685i	$-0.314929\pi$
$673$	28.4955	1.09842	0.549210	+	0.835685i	$0.314929\pi$
$674$	−9.21245	−0.354850
$675$	0	0
$676$	−5.37386	−0.206687
$677$	11.8383	0.454983	0.227492	−	0.973780i	$-0.426948\pi$
$677$	11.8383	0.454983	0.227492	+	0.973780i	$0.426948\pi$
$678$	0	0
$679$	0	0
$680$	26.2668	1.00729
$681$	0	0
$682$	−11.0780	−0.424200
$683$	−4.47315	−0.171160	−0.0855802	−	0.996331i	$-0.527274\pi$
$683$	−4.47315	−0.171160	−0.0855802	+	0.996331i	$0.527274\pi$
$684$	0	0
$685$	77.0780	2.94500
$686$	0	0
$687$	0	0
$688$	−23.9564	−0.913331
$689$	−34.6410	−1.31972
$690$	0	0
$691$	−10.8348	−0.412177	−0.206089	−	0.978533i	$-0.566073\pi$
$691$	−10.8348	−0.412177	−0.206089	+	0.978533i	$0.566073\pi$
$692$	20.5939	0.782863
$693$	0	0
$694$	16.4610	0.624850
$695$	−33.1950	−1.25916
$696$	0	0
$697$	15.1652	0.574421
$698$	11.3060	0.427939
$699$	0	0
$700$	0	0
$701$	29.4449	1.11212	0.556059	−	0.831143i	$-0.312313\pi$
$701$	29.4449	1.11212	0.556059	+	0.831143i	$0.312313\pi$
$702$	0	0
$703$	10.7477	0.405358
$704$	−9.04165	−0.340770
$705$	0	0
$706$	16.3303	0.614599
$707$	0	0
$708$	0	0
$709$	15.0000	0.563337	0.281668	−	0.959512i	$-0.409112\pi$
$709$	15.0000	0.563337	0.281668	+	0.959512i	$0.409112\pi$
$710$	22.8027	0.855770
$711$	0	0
$712$	−15.1652	−0.568338
$713$	−31.7490	−1.18901
$714$	0	0
$715$	46.3303	1.73266
$716$	2.93180	0.109567
$717$	0	0
$718$	−13.8693	−0.517598
$719$	49.0695	1.82998	0.914992	−	0.403471i	$-0.132197\pi$
$719$	49.0695	1.82998	0.914992	+	0.403471i	$0.132197\pi$
$720$	0	0
$721$	0	0
$722$	−2.81655	−0.104821
$723$	0	0
$724$	23.5826	0.876440
$725$	−25.8854	−0.961360
$726$	0	0
$727$	−46.2432	−1.71506	−0.857532	−	0.514430i	$-0.828004\pi$
$727$	−46.2432	−1.71506	−0.857532	+	0.514430i	$0.828004\pi$
$728$	0	0
$729$	0	0
$730$	−6.33030	−0.234295
$731$	−29.7309	−1.09964
$732$	0	0
$733$	−10.7477	−0.396976	−0.198488	−	0.980103i	$-0.563603\pi$
$733$	−10.7477	−0.396976	−0.198488	+	0.980103i	$0.563603\pi$
$734$	4.03620	0.148979
$735$	0	0
$736$	16.4174	0.605154
$737$	−1.54135	−0.0567764
$738$	0	0
$739$	−19.4174	−0.714281	−0.357141	−	0.934051i	$-0.616248\pi$
$739$	−19.4174	−0.714281	−0.357141	+	0.934051i	$0.616248\pi$
$740$	23.5257	0.864822
$741$	0	0
$742$	0	0
$743$	13.4195	0.492312	0.246156	−	0.969230i	$-0.420832\pi$
$743$	13.4195	0.492312	0.246156	+	0.969230i	$0.420832\pi$
$744$	0	0
$745$	−46.7477	−1.71270
$746$	−3.34885	−0.122610
$747$	0	0
$748$	−16.4174	−0.600280
$749$	0	0
$750$	0	0
$751$	−9.41742	−0.343647	−0.171823	−	0.985128i	$-0.554966\pi$
$751$	−9.41742	−0.343647	−0.171823	+	0.985128i	$0.554966\pi$
$752$	7.65120	0.279011
$753$	0	0
$754$	3.33939	0.121614
$755$	90.1064	3.27931
$756$	0	0
$757$	−6.83485	−0.248417	−0.124208	−	0.992256i	$-0.539639\pi$
$757$	−6.83485	−0.248417	−0.124208	+	0.992256i	$0.539639\pi$
$758$	−13.2089	−0.479767
$759$	0	0
$760$	27.1652	0.985384
$761$	−30.4539	−1.10395	−0.551977	−	0.833859i	$-0.686126\pi$
$761$	−30.4539	−1.10395	−0.551977	+	0.833859i	$0.686126\pi$
$762$	0	0
$763$	0	0
$764$	−10.9445	−0.395958
$765$	0	0
$766$	−8.83485	−0.319216
$767$	−13.8564	−0.500326
$768$	0	0
$769$	45.0780	1.62556	0.812778	−	0.582574i	$-0.197954\pi$
$769$	45.0780	1.62556	0.812778	+	0.582574i	$0.197954\pi$
$770$	0	0
$771$	0	0
$772$	−25.0780	−0.902578
$773$	−31.5583	−1.13507	−0.567537	−	0.823348i	$-0.692104\pi$
$773$	−31.5583	−1.13507	−0.567537	+	0.823348i	$0.692104\pi$
$774$	0	0
$775$	−129.826	−4.66348
$776$	13.1334	0.471462
$777$	0	0
$778$	14.5045	0.520013
$779$	15.6838	0.561931
$780$	0	0
$781$	−30.1652	−1.07939
$782$	5.48220	0.196043
$783$	0	0
$784$	0	0
$785$	3.65480	0.130445
$786$	0	0
$787$	−53.9129	−1.92179	−0.960893	−	0.276919i	$-0.910687\pi$
$787$	−53.9129	−1.92179	−0.960893	+	0.276919i	$0.910687\pi$
$788$	0.170800	0.00608449
$789$	0	0
$790$	17.1652	0.610709
$791$	0	0
$792$	0	0
$793$	9.66970	0.343381
$794$	0.723000	0.0256583
$795$	0	0
$796$	−30.0000	−1.06332
$797$	−50.7062	−1.79611	−0.898053	−	0.439887i	$-0.855019\pi$
$797$	−50.7062	−1.79611	−0.898053	+	0.439887i	$0.855019\pi$
$798$	0	0
$799$	9.49545	0.335925
$800$	67.1329	2.37351
$801$	0	0
$802$	−6.46099	−0.228145
$803$	8.37420	0.295519
$804$	0	0
$805$	0	0
$806$	16.7484	0.589937
$807$	0	0
$808$	3.16515	0.111350
$809$	−10.1063	−0.355317	−0.177658	−	0.984092i	$-0.556852\pi$
$809$	−10.1063	−0.355317	−0.177658	+	0.984092i	$0.556852\pi$
$810$	0	0
$811$	16.0000	0.561836	0.280918	−	0.959732i	$-0.409361\pi$
$811$	16.0000	0.561836	0.280918	+	0.959732i	$0.409361\pi$
$812$	0	0
$813$	0	0
$814$	3.62614	0.127096
$815$	−63.8396	−2.23620
$816$	0	0
$817$	−30.7477	−1.07573
$818$	−4.22690	−0.147790
$819$	0	0
$820$	34.3303	1.19887
$821$	−33.6718	−1.17515	−0.587576	−	0.809169i	$-0.699918\pi$
$821$	−33.6718	−1.17515	−0.587576	+	0.809169i	$0.699918\pi$
$822$	0	0
$823$	12.0000	0.418294	0.209147	−	0.977884i	$-0.432931\pi$
$823$	12.0000	0.418294	0.209147	+	0.977884i	$0.432931\pi$
$824$	6.20520	0.216168
$825$	0	0
$826$	0	0
$827$	25.2578	0.878298	0.439149	−	0.898414i	$-0.355280\pi$
$827$	25.2578	0.878298	0.439149	+	0.898414i	$0.355280\pi$
$828$	0	0
$829$	10.3303	0.358786	0.179393	−	0.983777i	$-0.442587\pi$
$829$	10.3303	0.358786	0.179393	+	0.983777i	$0.442587\pi$
$830$	−12.4104	−0.430771
$831$	0	0
$832$	13.6697	0.473911
$833$	0	0
$834$	0	0
$835$	−11.1652	−0.386386
$836$	−16.9789	−0.587228
$837$	0	0
$838$	8.50455	0.293785
$839$	−39.0188	−1.34708	−0.673540	−	0.739151i	$-0.735227\pi$
$839$	−39.0188	−1.34708	−0.673540	+	0.739151i	$0.735227\pi$
$840$	0	0
$841$	−25.6606	−0.884848
$842$	−7.38505	−0.254506
$843$	0	0
$844$	−21.0436	−0.724349
$845$	−13.1334	−0.451803
$846$	0	0
$847$	0	0
$848$	24.1733	0.830113
$849$	0	0
$850$	22.4174	0.768911
$851$	10.3923	0.356244
$852$	0	0
$853$	−12.7477	−0.436474	−0.218237	−	0.975896i	$-0.570031\pi$
$853$	−12.7477	−0.436474	−0.218237	+	0.975896i	$0.570031\pi$
$854$	0	0
$855$	0	0
$856$	19.4174	0.663674
$857$	−40.3139	−1.37710	−0.688549	−	0.725190i	$-0.741752\pi$
$857$	−40.3139	−1.37710	−0.688549	+	0.725190i	$0.741752\pi$
$858$	0	0
$859$	−26.3303	−0.898378	−0.449189	−	0.893437i	$-0.648287\pi$
$859$	−26.3303	−0.898378	−0.449189	+	0.893437i	$0.648287\pi$
$860$	−67.3037	−2.29504
$861$	0	0
$862$	16.7477	0.570430
$863$	−2.45505	−0.0835709	−0.0417855	−	0.999127i	$-0.513305\pi$
$863$	−2.45505	−0.0835709	−0.0417855	+	0.999127i	$0.513305\pi$
$864$	0	0
$865$	50.3303	1.71128
$866$	10.0109	0.340184
$867$	0	0
$868$	0	0
$869$	−22.7074	−0.770294
$870$	0	0
$871$	2.33030	0.0789593
$872$	−10.3923	−0.351928
$873$	0	0
$874$	5.66970	0.191780
$875$	0	0
$876$	0	0
$877$	13.0000	0.438979	0.219489	−	0.975615i	$-0.429561\pi$
$877$	13.0000	0.438979	0.219489	+	0.975615i	$0.429561\pi$
$878$	−11.6874	−0.394431
$879$	0	0
$880$	−32.3303	−1.08985
$881$	21.5076	0.724610	0.362305	−	0.932060i	$-0.381990\pi$
$881$	21.5076	0.724610	0.362305	+	0.932060i	$0.381990\pi$
$882$	0	0
$883$	48.9129	1.64605	0.823025	−	0.568006i	$-0.192285\pi$
$883$	48.9129	1.64605	0.823025	+	0.568006i	$0.192285\pi$
$884$	24.8208	0.834814
$885$	0	0
$886$	−7.91288	−0.265838
$887$	22.9934	0.772043	0.386022	−	0.922490i	$-0.373849\pi$
$887$	22.9934	0.772043	0.386022	+	0.922490i	$0.373849\pi$
$888$	0	0
$889$	0	0
$890$	−17.5112	−0.586977
$891$	0	0
$892$	39.4083	1.31949
$893$	9.82020	0.328621
$894$	0	0
$895$	7.16515	0.239505
$896$	0	0
$897$	0	0
$898$	−12.1307	−0.404806
$899$	16.7484	0.558591
$900$	0	0
$901$	30.0000	0.999445
$902$	5.29150	0.176188
$903$	0	0
$904$	24.1652	0.803721
$905$	57.6344	1.91583
$906$	0	0
$907$	18.5826	0.617024	0.308512	−	0.951220i	$-0.400169\pi$
$907$	18.5826	0.617024	0.308512	+	0.951220i	$0.400169\pi$
$908$	−23.5257	−0.780728
$909$	0	0
$910$	0	0
$911$	−8.94630	−0.296404	−0.148202	−	0.988957i	$-0.547349\pi$
$911$	−8.94630	−0.296404	−0.148202	+	0.988957i	$0.547349\pi$
$912$	0	0
$913$	16.4174	0.543337
$914$	14.3133	0.473440
$915$	0	0
$916$	39.2523	1.29693
$917$	0	0
$918$	0	0
$919$	16.0780	0.530365	0.265183	−	0.964198i	$-0.414568\pi$
$919$	16.0780	0.530365	0.265183	+	0.964198i	$0.414568\pi$
$920$	26.2668	0.865991
$921$	0	0
$922$	6.08712	0.200469
$923$	45.6054	1.50112
$924$	0	0
$925$	42.4955	1.39724
$926$	9.55405	0.313966
$927$	0	0
$928$	−8.66061	−0.284298
$929$	2.35970	0.0774193	0.0387096	−	0.999251i	$-0.487675\pi$
$929$	2.35970	0.0774193	0.0387096	+	0.999251i	$0.487675\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	−2.83485	−0.0927591
$935$	−40.1232	−1.31217
$936$	0	0
$937$	−33.4955	−1.09425	−0.547124	−	0.837051i	$-0.684278\pi$
$937$	−33.4955	−1.09425	−0.547124	+	0.837051i	$0.684278\pi$
$938$	0	0
$939$	0	0
$940$	21.4955	0.701104
$941$	32.8136	1.06969	0.534847	−	0.844949i	$-0.320369\pi$
$941$	32.8136	1.06969	0.534847	+	0.844949i	$0.320369\pi$
$942$	0	0
$943$	15.1652	0.493845
$944$	9.66930	0.314709
$945$	0	0
$946$	−10.3739	−0.337283
$947$	−41.7599	−1.35702	−0.678508	−	0.734593i	$-0.737373\pi$
$947$	−41.7599	−1.35702	−0.678508	+	0.734593i	$0.737373\pi$
$948$	0	0
$949$	−12.6606	−0.410981
$950$	23.1841	0.752192
$951$	0	0
$952$	0	0
$953$	−3.65480	−0.118391	−0.0591953	−	0.998246i	$-0.518853\pi$
$953$	−3.65480	−0.118391	−0.0591953	+	0.998246i	$0.518853\pi$
$954$	0	0
$955$	−26.7477	−0.865536
$956$	−45.2439	−1.46329
$957$	0	0
$958$	−11.2523	−0.363544
$959$	0	0
$960$	0	0
$961$	53.0000	1.70968
$962$	−5.48220	−0.176753
$963$	0	0
$964$	32.9909	1.06257
$965$	−61.2892	−1.97297
$966$	0	0
$967$	4.00000	0.128631	0.0643157	−	0.997930i	$-0.479514\pi$
$967$	4.00000	0.128631	0.0643157	+	0.997930i	$0.479514\pi$
$968$	6.92820	0.222681
$969$	0	0
$970$	15.1652	0.486924
$971$	−35.2131	−1.13004	−0.565021	−	0.825076i	$-0.691132\pi$
$971$	−35.2131	−1.13004	−0.565021	+	0.825076i	$0.691132\pi$
$972$	0	0
$973$	0	0
$974$	−3.00725	−0.0963585
$975$	0	0
$976$	−6.74773	−0.215989
$977$	24.8208	0.794088	0.397044	−	0.917800i	$-0.370036\pi$
$977$	24.8208	0.794088	0.397044	+	0.917800i	$0.370036\pi$
$978$	0	0
$979$	23.1652	0.740361
$980$	0	0
$981$	0	0
$982$	−10.2432	−0.326873
$983$	44.3103	1.41328	0.706640	−	0.707573i	$-0.250210\pi$
$983$	44.3103	1.41328	0.706640	+	0.707573i	$0.250210\pi$
$984$	0	0
$985$	0.417424	0.0133002
$986$	−2.89200	−0.0921001
$987$	0	0
$988$	25.6697	0.816662
$989$	−29.7309	−0.945388
$990$	0	0
$991$	−35.7477	−1.13556	−0.567782	−	0.823179i	$-0.692198\pi$
$991$	−35.7477	−1.13556	−0.567782	+	0.823179i	$0.692198\pi$
$992$	−43.4364	−1.37911
$993$	0	0
$994$	0	0
$995$	−73.3182	−2.32434
$996$	0	0
$997$	−6.83485	−0.216462	−0.108231	−	0.994126i	$-0.534519\pi$
$997$	−6.83485	−0.216462	−0.108231	+	0.994126i	$0.534519\pi$
$998$	−9.13701	−0.289227
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3969.2.a.u.1.3		4
3.2	odd	2	inner	3969.2.a.u.1.2		4
7.6	odd	2		567.2.a.i.1.3	yes	4
21.20	even	2		567.2.a.i.1.2	✓	4
28.27	even	2		9072.2.a.ci.1.4		4
63.13	odd	6		567.2.f.n.379.2		8
63.20	even	6		567.2.f.n.190.3		8
63.34	odd	6		567.2.f.n.190.2		8
63.41	even	6		567.2.f.n.379.3		8
84.83	odd	2		9072.2.a.ci.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
567.2.a.i.1.2	✓	4	21.20	even	2
567.2.a.i.1.3	yes	4	7.6	odd	2
567.2.f.n.190.2		8	63.34	odd	6
567.2.f.n.190.3		8	63.20	even	6
567.2.f.n.379.2		8	63.13	odd	6
567.2.f.n.379.3		8	63.41	even	6
3969.2.a.u.1.2		4	3.2	odd	2	inner
3969.2.a.u.1.3		4	1.1	even	1	trivial
9072.2.a.ci.1.1		4	84.83	odd	2
9072.2.a.ci.1.4		4	28.27	even	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:	\( N \)	\(=\)	\( 3969 = 3^{4} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3969.a (trivial)

\(f(q)\)	\(=\)	\(q+0.456850 q^{2} -1.79129 q^{4} -4.37780 q^{5} -1.73205 q^{8} +O(q^{10})\)	\(q+0.456850 q^{2} -1.79129 q^{4} -4.37780 q^{5} -1.73205 q^{8} -2.00000 q^{10} +2.64575 q^{11} -4.00000 q^{13} +2.79129 q^{16} +3.46410 q^{17} +3.58258 q^{19} +7.84190 q^{20} +1.20871 q^{22} +3.46410 q^{23} +14.1652 q^{25} -1.82740 q^{26} -1.82740 q^{29} -9.16515 q^{31} +4.73930 q^{32} +1.58258 q^{34} +3.00000 q^{37} +1.63670 q^{38} +7.58258 q^{40} +4.37780 q^{41} -8.58258 q^{43} -4.73930 q^{44} +1.58258 q^{46} +2.74110 q^{47} +6.47135 q^{50} +7.16515 q^{52} +8.66025 q^{53} -11.5826 q^{55} -0.834849 q^{58} +3.46410 q^{59} -2.41742 q^{61} -4.18710 q^{62} -3.41742 q^{64} +17.5112 q^{65} -0.582576 q^{67} -6.20520 q^{68} -11.4014 q^{71} +3.16515 q^{73} +1.37055 q^{74} -6.41742 q^{76} -8.58258 q^{79} -12.2197 q^{80} +2.00000 q^{82} +6.20520 q^{83} -15.1652 q^{85} -3.92095 q^{86} -4.58258 q^{88} +8.75560 q^{89} -6.20520 q^{92} +1.25227 q^{94} -15.6838 q^{95} -7.58258 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{4}+O(q^{10}) \) 4 * q + 2 * q^4	\( 4 q + 2 q^{4} - 8 q^{10} - 16 q^{13} + 2 q^{16} - 4 q^{19} + 14 q^{22} + 20 q^{25} - 12 q^{34} + 12 q^{37} + 12 q^{40} - 16 q^{43} - 12 q^{46} - 8 q^{52} - 28 q^{55} - 40 q^{58} - 28 q^{61} - 32 q^{64} + 16 q^{67} - 24 q^{73} - 44 q^{76} - 16 q^{79} + 8 q^{82} - 24 q^{85} + 60 q^{94} - 12 q^{97}+O(q^{100}) \) 4 * q + 2 * q^4 - 8 * q^10 - 16 * q^13 + 2 * q^16 - 4 * q^19 + 14 * q^22 + 20 * q^25 - 12 * q^34 + 12 * q^37 + 12 * q^40 - 16 * q^43 - 12 * q^46 - 8 * q^52 - 28 * q^55 - 40 * q^58 - 28 * q^61 - 32 * q^64 + 16 * q^67 - 24 * q^73 - 44 * q^76 - 16 * q^79 + 8 * q^82 - 24 * q^85 + 60 * q^94 - 12 * q^97

Introduction

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