LMFDB - Embedded newform 2088.2.a.s.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2088,2,Mod(1,2088)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2088, 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("2088.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2088 = 2^{3} \cdot 3^{2} \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2088.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:	$16.6727639420$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.568.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 6x - 2 $ x^3 - x^2 - 6*x - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 232)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-0.363328$ of defining polynomial
Character	$\chi$	$=$	2088.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.14134 q^{5} +O(q^{10})$	$q-4.14134 q^{5} +5.64600 q^{11} -2.86799 q^{13} -2.00000 q^{17} +4.28267 q^{19} +2.72666 q^{23} +12.1507 q^{25} -1.00000 q^{29} -5.36333 q^{31} -6.28267 q^{37} -11.7360 q^{41} +2.91934 q^{43} +4.19269 q^{47} -7.00000 q^{49} -1.41468 q^{53} -23.3820 q^{55} -1.27334 q^{59} +3.45331 q^{61} +11.8773 q^{65} +9.45331 q^{67} -13.8387 q^{71} -7.73599 q^{73} -14.9193 q^{79} -9.27334 q^{83} +8.28267 q^{85} +16.3013 q^{89} -17.7360 q^{95} -10.2827 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 4 q^{5}+O(q^{10}) $ 3 * q - 4 * q^5	$ 3 q - 4 q^{5} - 2 q^{11} + 4 q^{13} - 6 q^{17} - 4 q^{19} + 4 q^{23} + 7 q^{25} - 3 q^{29} - 14 q^{31} - 2 q^{37} - 10 q^{41} - 6 q^{43} + 2 q^{47} - 21 q^{49} - 26 q^{55} - 8 q^{59} + 2 q^{61} + 2 q^{65} + 20 q^{67} - 12 q^{71} + 2 q^{73} - 30 q^{79} - 32 q^{83} + 8 q^{85} - 10 q^{89} - 28 q^{95} - 14 q^{97}+O(q^{100}) $ 3 * q - 4 * q^5 - 2 * q^11 + 4 * q^13 - 6 * q^17 - 4 * q^19 + 4 * q^23 + 7 * q^25 - 3 * q^29 - 14 * q^31 - 2 * q^37 - 10 * q^41 - 6 * q^43 + 2 * q^47 - 21 * q^49 - 26 * q^55 - 8 * q^59 + 2 * q^61 + 2 * q^65 + 20 * q^67 - 12 * q^71 + 2 * q^73 - 30 * q^79 - 32 * q^83 + 8 * q^85 - 10 * q^89 - 28 * q^95 - 14 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	−4.14134	−1.85206	−0.926031	−	0.377448i	$-0.876802\pi$
$5$	−4.14134	−1.85206	−0.926031	+	0.377448i	$0.876802\pi$
$6$	0	0
$7$	0	0		−	1.00000i	$-0.5\pi$
$7$	0	0			1.00000i	$0.5\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	5.64600	1.70233	0.851167	−	0.524896i	$-0.175896\pi$
$11$	5.64600	1.70233	0.851167	+	0.524896i	$0.175896\pi$
$12$	0	0
$13$	−2.86799	−0.795438	−0.397719	−	0.917507i	$-0.630198\pi$
$13$	−2.86799	−0.795438	−0.397719	+	0.917507i	$0.630198\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−2.00000	−0.485071	−0.242536	−	0.970143i	$-0.577979\pi$
$17$	−2.00000	−0.485071	−0.242536	+	0.970143i	$0.577979\pi$
$18$	0	0
$19$	4.28267	0.982512	0.491256	−	0.871015i	$-0.336538\pi$
$19$	4.28267	0.982512	0.491256	+	0.871015i	$0.336538\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	2.72666	0.568547	0.284274	−	0.958743i	$-0.408248\pi$
$23$	2.72666	0.568547	0.284274	+	0.958743i	$0.408248\pi$
$24$	0	0
$25$	12.1507	2.43013
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	−5.36333	−0.963282	−0.481641	−	0.876369i	$-0.659959\pi$
$31$	−5.36333	−0.963282	−0.481641	+	0.876369i	$0.659959\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−6.28267	−1.03286	−0.516432	−	0.856328i	$-0.672740\pi$
$37$	−6.28267	−1.03286	−0.516432	+	0.856328i	$0.672740\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−11.7360	−1.83285	−0.916426	−	0.400203i	$-0.868940\pi$
$41$	−11.7360	−1.83285	−0.916426	+	0.400203i	$0.868940\pi$
$42$	0	0
$43$	2.91934	0.445196	0.222598	−	0.974910i	$-0.428546\pi$
$43$	2.91934	0.445196	0.222598	+	0.974910i	$0.428546\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	4.19269	0.611566	0.305783	−	0.952101i	$-0.401082\pi$
$47$	4.19269	0.611566	0.305783	+	0.952101i	$0.401082\pi$
$48$	0	0
$49$	−7.00000	−1.00000
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−1.41468	−0.194321	−0.0971606	−	0.995269i	$-0.530976\pi$
$53$	−1.41468	−0.194321	−0.0971606	+	0.995269i	$0.530976\pi$
$54$	0	0
$55$	−23.3820	−3.15283
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−1.27334	−0.165775	−0.0828876	−	0.996559i	$-0.526414\pi$
$59$	−1.27334	−0.165775	−0.0828876	+	0.996559i	$0.526414\pi$
$60$	0	0
$61$	3.45331	0.442151	0.221076	−	0.975257i	$-0.429043\pi$
$61$	3.45331	0.442151	0.221076	+	0.975257i	$0.429043\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	11.8773	1.47320
$66$	0	0
$67$	9.45331	1.15491	0.577453	−	0.816424i	$-0.304047\pi$
$67$	9.45331	1.15491	0.577453	+	0.816424i	$0.304047\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−13.8387	−1.64235	−0.821175	−	0.570676i	$-0.806681\pi$
$71$	−13.8387	−1.64235	−0.821175	+	0.570676i	$0.806681\pi$
$72$	0	0
$73$	−7.73599	−0.905429	−0.452714	−	0.891656i	$-0.649544\pi$
$73$	−7.73599	−0.905429	−0.452714	+	0.891656i	$0.649544\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−14.9193	−1.67856	−0.839279	−	0.543701i	$-0.817022\pi$
$79$	−14.9193	−1.67856	−0.839279	+	0.543701i	$0.817022\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−9.27334	−1.01788	−0.508941	−	0.860801i	$-0.669963\pi$
$83$	−9.27334	−1.01788	−0.508941	+	0.860801i	$0.669963\pi$
$84$	0	0
$85$	8.28267	0.898382
$86$	0	0
$87$	0	0
$88$	0	0
$89$	16.3013	1.72794	0.863969	−	0.503545i	$-0.167971\pi$
$89$	16.3013	1.72794	0.863969	+	0.503545i	$0.167971\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−17.7360	−1.81967
$96$	0	0
$97$	−10.2827	−1.04405	−0.522024	−	0.852931i	$-0.674823\pi$
$97$	−10.2827	−1.04405	−0.522024	+	0.852931i	$0.674823\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	0.829359	0.0825243	0.0412622	−	0.999148i	$-0.486862\pi$
$101$	0.829359	0.0825243	0.0412622	+	0.999148i	$0.486862\pi$
$102$	0	0
$103$	−2.72666	−0.268665	−0.134333	−	0.990936i	$-0.542889\pi$
$103$	−2.72666	−0.268665	−0.134333	+	0.990936i	$0.542889\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−18.0187	−1.74193	−0.870965	−	0.491346i	$-0.836505\pi$
$107$	−18.0187	−1.74193	−0.870965	+	0.491346i	$0.836505\pi$
$108$	0	0
$109$	−6.97070	−0.667672	−0.333836	−	0.942631i	$-0.608343\pi$
$109$	−6.97070	−0.667672	−0.333836	+	0.942631i	$0.608343\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−10.5653	−0.993904	−0.496952	−	0.867778i	$-0.665548\pi$
$113$	−10.5653	−0.993904	−0.496952	+	0.867778i	$0.665548\pi$
$114$	0	0
$115$	−11.2920	−1.05298
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	20.8773	1.89794
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−29.6133	−2.64869
$126$	0	0
$127$	5.73599	0.508986	0.254493	−	0.967075i	$-0.418091\pi$
$127$	5.73599	0.508986	0.254493	+	0.967075i	$0.418091\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−3.71733	−0.324784	−0.162392	−	0.986726i	$-0.551921\pi$
$131$	−3.71733	−0.324784	−0.162392	+	0.986726i	$0.551921\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−6.28267	−0.536765	−0.268382	−	0.963312i	$-0.586489\pi$
$137$	−6.28267	−0.536765	−0.268382	+	0.963312i	$0.586489\pi$
$138$	0	0
$139$	−4.17997	−0.354540	−0.177270	−	0.984162i	$-0.556727\pi$
$139$	−4.17997	−0.354540	−0.177270	+	0.984162i	$0.556727\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−16.1927	−1.35410
$144$	0	0
$145$	4.14134	0.343919
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−8.60398	−0.704865	−0.352433	−	0.935837i	$-0.614645\pi$
$149$	−8.60398	−0.704865	−0.352433	+	0.935837i	$0.614645\pi$
$150$	0	0
$151$	6.01866	0.489791	0.244896	−	0.969549i	$-0.421246\pi$
$151$	6.01866	0.489791	0.244896	+	0.969549i	$0.421246\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	22.2113	1.78406
$156$	0	0
$157$	1.71733	0.137058	0.0685288	−	0.997649i	$-0.478169\pi$
$157$	1.71733	0.137058	0.0685288	+	0.997649i	$0.478169\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	10.1086	0.791770	0.395885	−	0.918300i	$-0.370438\pi$
$163$	10.1086	0.791770	0.395885	+	0.918300i	$0.370438\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−13.8387	−1.07087	−0.535435	−	0.844576i	$-0.679852\pi$
$167$	−13.8387	−1.07087	−0.535435	+	0.844576i	$0.679852\pi$
$168$	0	0
$169$	−4.77462	−0.367278
$170$	0	0
$171$	0	0
$172$	0	0
$173$	10.5653	0.803268	0.401634	−	0.915800i	$-0.368442\pi$
$173$	10.5653	0.803268	0.401634	+	0.915800i	$0.368442\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	17.8387	1.33333	0.666663	−	0.745359i	$-0.267722\pi$
$179$	17.8387	1.33333	0.666663	+	0.745359i	$0.267722\pi$
$180$	0	0
$181$	10.9707	0.815445	0.407723	−	0.913106i	$-0.366323\pi$
$181$	10.9707	0.815445	0.407723	+	0.913106i	$0.366323\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	26.0187	1.91293
$186$	0	0
$187$	−11.2920	−0.825753
$188$	0	0
$189$	0	0
$190$	0	0
$191$	22.3013	1.61367	0.806834	−	0.590778i	$-0.201179\pi$
$191$	22.3013	1.61367	0.806834	+	0.590778i	$0.201179\pi$
$192$	0	0
$193$	−4.82936	−0.347625	−0.173812	−	0.984779i	$-0.555609\pi$
$193$	−4.82936	−0.347625	−0.173812	+	0.984779i	$0.555609\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	18.5653	1.32273	0.661363	−	0.750066i	$-0.269978\pi$
$197$	18.5653	1.32273	0.661363	+	0.750066i	$0.269978\pi$
$198$	0	0
$199$	−1.98134	−0.140454	−0.0702268	−	0.997531i	$-0.522372\pi$
$199$	−1.98134	−0.140454	−0.0702268	+	0.997531i	$0.522372\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	48.6027	3.39456
$206$	0	0
$207$	0	0
$208$	0	0
$209$	24.1800	1.67256
$210$	0	0
$211$	20.6553	1.42197	0.710986	−	0.703206i	$-0.248249\pi$
$211$	20.6553	1.42197	0.710986	+	0.703206i	$0.248249\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−12.0900	−0.824530
$216$	0	0
$217$	0	0
$218$	0	0
$219$	0	0
$220$	0	0
$221$	5.73599	0.385844
$222$	0	0
$223$	−22.4040	−1.50028	−0.750142	−	0.661276i	$-0.770015\pi$
$223$	−22.4040	−1.50028	−0.750142	+	0.661276i	$0.770015\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−26.7640	−1.77639	−0.888194	−	0.459469i	$-0.848040\pi$
$227$	−26.7640	−1.77639	−0.888194	+	0.459469i	$0.848040\pi$
$228$	0	0
$229$	−0.829359	−0.0548056	−0.0274028	−	0.999624i	$-0.508724\pi$
$229$	−0.829359	−0.0548056	−0.0274028	+	0.999624i	$0.508724\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−23.1507	−1.51665	−0.758325	−	0.651876i	$-0.773982\pi$
$233$	−23.1507	−1.51665	−0.758325	+	0.651876i	$0.773982\pi$
$234$	0	0
$235$	−17.3633	−1.13266
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−10.7267	−0.693850	−0.346925	−	0.937893i	$-0.612774\pi$
$239$	−10.7267	−0.693850	−0.346925	+	0.937893i	$0.612774\pi$
$240$	0	0
$241$	−0.603978	−0.0389056	−0.0194528	−	0.999811i	$-0.506192\pi$
$241$	−0.603978	−0.0389056	−0.0194528	+	0.999811i	$0.506192\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	28.9894	1.85206
$246$	0	0
$247$	−12.2827	−0.781528
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−1.92867	−0.121737	−0.0608684	−	0.998146i	$-0.519387\pi$
$251$	−1.92867	−0.121737	−0.0608684	+	0.998146i	$0.519387\pi$
$252$	0	0
$253$	15.3947	0.967857
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−21.5946	−1.34704	−0.673519	−	0.739170i	$-0.735218\pi$
$257$	−21.5946	−1.34704	−0.673519	+	0.739170i	$0.735218\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−8.08998	−0.498850	−0.249425	−	0.968394i	$-0.580242\pi$
$263$	−8.08998	−0.498850	−0.249425	+	0.968394i	$0.580242\pi$
$264$	0	0
$265$	5.85866	0.359895
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−11.6587	−0.710845	−0.355422	−	0.934706i	$-0.615663\pi$
$269$	−11.6587	−0.710845	−0.355422	+	0.934706i	$0.615663\pi$
$270$	0	0
$271$	−2.07133	−0.125824	−0.0629121	−	0.998019i	$-0.520039\pi$
$271$	−2.07133	−0.125824	−0.0629121	+	0.998019i	$0.520039\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	68.6027	4.13690
$276$	0	0
$277$	−13.4720	−0.809452	−0.404726	−	0.914438i	$-0.632633\pi$
$277$	−13.4720	−0.809452	−0.404726	+	0.914438i	$0.632633\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	22.8680	1.36419	0.682095	−	0.731264i	$-0.261069\pi$
$281$	22.8680	1.36419	0.682095	+	0.731264i	$0.261069\pi$
$282$	0	0
$283$	0.707999	0.0420862	0.0210431	−	0.999779i	$-0.493301\pi$
$283$	0.707999	0.0420862	0.0210431	+	0.999779i	$0.493301\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−13.0000	−0.764706
$290$	0	0
$291$	0	0
$292$	0	0
$293$	18.5653	1.08460	0.542300	−	0.840185i	$-0.317554\pi$
$293$	18.5653	1.08460	0.542300	+	0.840185i	$0.317554\pi$
$294$	0	0
$295$	5.27334	0.307026
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−7.82003	−0.452244
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−14.3013	−0.818892
$306$	0	0
$307$	−8.55263	−0.488124	−0.244062	−	0.969760i	$-0.578480\pi$
$307$	−8.55263	−0.488124	−0.244062	+	0.969760i	$0.578480\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	13.1706	0.746839	0.373419	−	0.927663i	$-0.378185\pi$
$311$	13.1706	0.746839	0.373419	+	0.927663i	$0.378185\pi$
$312$	0	0
$313$	8.14134	0.460176	0.230088	−	0.973170i	$-0.426099\pi$
$313$	8.14134	0.460176	0.230088	+	0.973170i	$0.426099\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−17.1120	−0.961107	−0.480554	−	0.876965i	$-0.659564\pi$
$317$	−17.1120	−0.961107	−0.480554	+	0.876965i	$0.659564\pi$
$318$	0	0
$319$	−5.64600	−0.316115
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−8.56534	−0.476589
$324$	0	0
$325$	−34.8480	−1.93302
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	17.7487	0.975557	0.487778	−	0.872967i	$-0.337807\pi$
$331$	17.7487	0.975557	0.487778	+	0.872967i	$0.337807\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−39.1493	−2.13896
$336$	0	0
$337$	12.5467	0.683462	0.341731	−	0.939798i	$-0.388987\pi$
$337$	12.5467	0.683462	0.341731	+	0.939798i	$0.388987\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−30.2814	−1.63983
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−21.3107	−1.14402	−0.572008	−	0.820248i	$-0.693835\pi$
$347$	−21.3107	−1.14402	−0.572008	+	0.820248i	$0.693835\pi$
$348$	0	0
$349$	20.3213	1.08777	0.543887	−	0.839158i	$-0.316952\pi$
$349$	20.3213	1.08777	0.543887	+	0.839158i	$0.316952\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	3.09337	0.164644	0.0823218	−	0.996606i	$-0.473766\pi$
$353$	3.09337	0.164644	0.0823218	+	0.996606i	$0.473766\pi$
$354$	0	0
$355$	57.3107	3.04173
$356$	0	0
$357$	0	0
$358$	0	0
$359$	15.4847	0.817251	0.408625	−	0.912702i	$-0.366008\pi$
$359$	15.4847	0.817251	0.408625	+	0.912702i	$0.366008\pi$
$360$	0	0
$361$	−0.658719	−0.0346694
$362$	0	0
$363$	0	0
$364$	0	0
$365$	32.0373	1.67691
$366$	0	0
$367$	−32.8480	−1.71465	−0.857326	−	0.514773i	$-0.827876\pi$
$367$	−32.8480	−1.71465	−0.857326	+	0.514773i	$0.827876\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−0.706681	−0.0365905	−0.0182953	−	0.999833i	$-0.505824\pi$
$373$	−0.706681	−0.0365905	−0.0182953	+	0.999833i	$0.505824\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	2.86799	0.147709
$378$	0	0
$379$	−15.3947	−0.790773	−0.395386	−	0.918515i	$-0.629389\pi$
$379$	−15.3947	−0.790773	−0.395386	+	0.918515i	$0.629389\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−7.43466	−0.379893	−0.189947	−	0.981794i	$-0.560831\pi$
$383$	−7.43466	−0.379893	−0.189947	+	0.981794i	$0.560831\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	33.1893	1.68276	0.841382	−	0.540441i	$-0.181742\pi$
$389$	33.1893	1.68276	0.841382	+	0.540441i	$0.181742\pi$
$390$	0	0
$391$	−5.45331	−0.275786
$392$	0	0
$393$	0	0
$394$	0	0
$395$	61.7860	3.10879
$396$	0	0
$397$	6.44267	0.323348	0.161674	−	0.986844i	$-0.448311\pi$
$397$	6.44267	0.323348	0.161674	+	0.986844i	$0.448311\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	4.32131	0.215796	0.107898	−	0.994162i	$-0.465588\pi$
$401$	4.32131	0.215796	0.107898	+	0.994162i	$0.465588\pi$
$402$	0	0
$403$	15.3820	0.766231
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−35.4720	−1.75828
$408$	0	0
$409$	−22.3599	−1.10563	−0.552814	−	0.833305i	$-0.686446\pi$
$409$	−22.3599	−1.10563	−0.552814	+	0.833305i	$0.686446\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	38.4040	1.88518
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−1.63328	−0.0797911	−0.0398955	−	0.999204i	$-0.512703\pi$
$419$	−1.63328	−0.0797911	−0.0398955	+	0.999204i	$0.512703\pi$
$420$	0	0
$421$	14.5653	0.709871	0.354936	−	0.934891i	$-0.384503\pi$
$421$	14.5653	0.709871	0.354936	+	0.934891i	$0.384503\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−24.3013	−1.17879
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	38.7640	1.86719	0.933597	−	0.358324i	$-0.116652\pi$
$431$	38.7640	1.86719	0.933597	+	0.358324i	$0.116652\pi$
$432$	0	0
$433$	3.73599	0.179540	0.0897700	−	0.995963i	$-0.471387\pi$
$433$	3.73599	0.179540	0.0897700	+	0.995963i	$0.471387\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	11.6774	0.558605
$438$	0	0
$439$	−33.3107	−1.58983	−0.794915	−	0.606720i	$-0.792485\pi$
$439$	−33.3107	−1.58983	−0.794915	+	0.606720i	$0.792485\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−12.8480	−0.610428	−0.305214	−	0.952284i	$-0.598728\pi$
$443$	−12.8480	−0.610428	−0.305214	+	0.952284i	$0.598728\pi$
$444$	0	0
$445$	−67.5093	−3.20025
$446$	0	0
$447$	0	0
$448$	0	0
$449$	0.341281	0.0161061	0.00805303	−	0.999968i	$-0.497437\pi$
$449$	0.341281	0.0161061	0.00805303	+	0.999968i	$0.497437\pi$
$450$	0	0
$451$	−66.2614	−3.12013
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−16.9066	−0.790859	−0.395429	−	0.918496i	$-0.629404\pi$
$457$	−16.9066	−0.790859	−0.395429	+	0.918496i	$0.629404\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−26.0373	−1.21268	−0.606339	−	0.795206i	$-0.707363\pi$
$461$	−26.0373	−1.21268	−0.606339	+	0.795206i	$0.707363\pi$
$462$	0	0
$463$	41.3107	1.91987	0.959935	−	0.280224i	$-0.0904088\pi$
$463$	41.3107	1.91987	0.959935	+	0.280224i	$0.0904088\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	33.1180	1.53252	0.766258	−	0.642532i	$-0.222116\pi$
$467$	33.1180	1.53252	0.766258	+	0.642532i	$0.222116\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	16.4826	0.757872
$474$	0	0
$475$	52.0373	2.38764
$476$	0	0
$477$	0	0
$478$	0	0
$479$	24.6553	1.12653	0.563265	−	0.826276i	$-0.309545\pi$
$479$	24.6553	1.12653	0.563265	+	0.826276i	$0.309545\pi$
$480$	0	0
$481$	18.0187	0.821580
$482$	0	0
$483$	0	0
$484$	0	0
$485$	42.5840	1.93364
$486$	0	0
$487$	5.45331	0.247113	0.123557	−	0.992338i	$-0.460570\pi$
$487$	5.45331	0.247113	0.123557	+	0.992338i	$0.460570\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	16.3727	0.738888	0.369444	−	0.929253i	$-0.379548\pi$
$491$	16.3727	0.738888	0.369444	+	0.929253i	$0.379548\pi$
$492$	0	0
$493$	2.00000	0.0900755
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−25.6333	−1.14750	−0.573752	−	0.819029i	$-0.694513\pi$
$499$	−25.6333	−1.14750	−0.573752	+	0.819029i	$0.694513\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−0.475360	−0.0211952	−0.0105976	−	0.999944i	$-0.503373\pi$
$503$	−0.475360	−0.0211952	−0.0105976	+	0.999944i	$0.503373\pi$
$504$	0	0
$505$	−3.43466	−0.152840
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−31.9987	−1.41832	−0.709158	−	0.705049i	$-0.750925\pi$
$509$	−31.9987	−1.41832	−0.709158	+	0.705049i	$0.750925\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	0	0
$514$	0	0
$515$	11.2920	0.497585
$516$	0	0
$517$	23.6719	1.04109
$518$	0	0
$519$	0	0
$520$	0	0
$521$	26.7253	1.17086	0.585429	−	0.810724i	$-0.300926\pi$
$521$	26.7253	1.17086	0.585429	+	0.810724i	$0.300926\pi$
$522$	0	0
$523$	9.45331	0.413365	0.206682	−	0.978408i	$-0.433733\pi$
$523$	9.45331	0.413365	0.206682	+	0.978408i	$0.433733\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	10.7267	0.467260
$528$	0	0
$529$	−15.5653	−0.676754
$530$	0	0
$531$	0	0
$532$	0	0
$533$	33.6587	1.45792
$534$	0	0
$535$	74.6213	3.22616
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−39.5220	−1.70233
$540$	0	0
$541$	−20.8667	−0.897128	−0.448564	−	0.893751i	$-0.648064\pi$
$541$	−20.8667	−0.897128	−0.448564	+	0.893751i	$0.648064\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	28.8680	1.23657
$546$	0	0
$547$	−12.5653	−0.537255	−0.268628	−	0.963244i	$-0.586570\pi$
$547$	−12.5653	−0.537255	−0.268628	+	0.963244i	$0.586570\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−4.28267	−0.182448
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−28.3786	−1.20244	−0.601220	−	0.799084i	$-0.705318\pi$
$557$	−28.3786	−1.20244	−0.601220	+	0.799084i	$0.705318\pi$
$558$	0	0
$559$	−8.37266	−0.354126
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−3.16470	−0.133376	−0.0666881	−	0.997774i	$-0.521243\pi$
$563$	−3.16470	−0.133376	−0.0666881	+	0.997774i	$0.521243\pi$
$564$	0	0
$565$	43.7546	1.84077
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−10.3599	−0.434311	−0.217156	−	0.976137i	$-0.569678\pi$
$569$	−10.3599	−0.434311	−0.217156	+	0.976137i	$0.569678\pi$
$570$	0	0
$571$	−18.5840	−0.777716	−0.388858	−	0.921298i	$-0.627130\pi$
$571$	−18.5840	−0.777716	−0.388858	+	0.921298i	$0.627130\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	33.1307	1.38165
$576$	0	0
$577$	−28.0187	−1.16643	−0.583216	−	0.812317i	$-0.698206\pi$
$577$	−28.0187	−1.16643	−0.583216	+	0.812317i	$0.698206\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−7.98728	−0.330799
$584$	0	0
$585$	0	0
$586$	0	0
$587$	45.8760	1.89351	0.946753	−	0.321962i	$-0.104342\pi$
$587$	45.8760	1.89351	0.946753	+	0.321962i	$0.104342\pi$
$588$	0	0
$589$	−22.9694	−0.946437
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−42.8026	−1.75769	−0.878846	−	0.477105i	$-0.841686\pi$
$593$	−42.8026	−1.75769	−0.878846	+	0.477105i	$0.841686\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	8.90069	0.363672	0.181836	−	0.983329i	$-0.441796\pi$
$599$	8.90069	0.363672	0.181836	+	0.983329i	$0.441796\pi$
$600$	0	0
$601$	13.7173	0.559541	0.279771	−	0.960067i	$-0.409742\pi$
$601$	13.7173	0.559541	0.279771	+	0.960067i	$0.409742\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−86.4600	−3.51510
$606$	0	0
$607$	24.4754	0.993424	0.496712	−	0.867915i	$-0.334540\pi$
$607$	24.4754	0.993424	0.496712	+	0.867915i	$0.334540\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−12.0246	−0.486463
$612$	0	0
$613$	−8.74663	−0.353273	−0.176637	−	0.984276i	$-0.556522\pi$
$613$	−8.74663	−0.353273	−0.176637	+	0.984276i	$0.556522\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−22.2827	−0.897067	−0.448533	−	0.893766i	$-0.648053\pi$
$617$	−22.2827	−0.897067	−0.448533	+	0.893766i	$0.648053\pi$
$618$	0	0
$619$	14.9966	0.602765	0.301382	−	0.953503i	$-0.402552\pi$
$619$	14.9966	0.602765	0.301382	+	0.953503i	$0.402552\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	61.8853	2.47541
$626$	0	0
$627$	0	0
$628$	0	0
$629$	12.5653	0.501013
$630$	0	0
$631$	−10.7267	−0.427021	−0.213511	−	0.976941i	$-0.568490\pi$
$631$	−10.7267	−0.427021	−0.213511	+	0.976941i	$0.568490\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−23.7546	−0.942674
$636$	0	0
$637$	20.0759	0.795438
$638$	0	0
$639$	0	0
$640$	0	0
$641$	22.9253	0.905494	0.452747	−	0.891639i	$-0.350444\pi$
$641$	22.9253	0.905494	0.452747	+	0.891639i	$0.350444\pi$
$642$	0	0
$643$	30.5467	1.20464	0.602322	−	0.798253i	$-0.294242\pi$
$643$	30.5467	1.20464	0.602322	+	0.798253i	$0.294242\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−24.7453	−0.972839	−0.486419	−	0.873725i	$-0.661697\pi$
$647$	−24.7453	−0.972839	−0.486419	+	0.873725i	$0.661697\pi$
$648$	0	0
$649$	−7.18930	−0.282205
$650$	0	0
$651$	0	0
$652$	0	0
$653$	15.4134	0.603171	0.301586	−	0.953439i	$-0.402484\pi$
$653$	15.4134	0.603171	0.301586	+	0.953439i	$0.402484\pi$
$654$	0	0
$655$	15.3947	0.601521
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−18.1341	−0.706403	−0.353202	−	0.935547i	$-0.614907\pi$
$659$	−18.1341	−0.706403	−0.353202	+	0.935547i	$0.614907\pi$
$660$	0	0
$661$	11.0934	0.431482	0.215741	−	0.976451i	$-0.430783\pi$
$661$	11.0934	0.431482	0.215741	+	0.976451i	$0.430783\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−2.72666	−0.105577
$668$	0	0
$669$	0	0
$670$	0	0
$671$	19.4974	0.752689
$672$	0	0
$673$	−35.7160	−1.37675	−0.688375	−	0.725355i	$-0.741676\pi$
$673$	−35.7160	−1.37675	−0.688375	+	0.725355i	$0.741676\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−5.64006	−0.216765	−0.108383	−	0.994109i	$-0.534567\pi$
$677$	−5.64006	−0.216765	−0.108383	+	0.994109i	$0.534567\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	26.0187	0.995576	0.497788	−	0.867299i	$-0.334146\pi$
$683$	26.0187	0.995576	0.497788	+	0.867299i	$0.334146\pi$
$684$	0	0
$685$	26.0187	0.994122
$686$	0	0
$687$	0	0
$688$	0	0
$689$	4.05729	0.154570
$690$	0	0
$691$	9.65872	0.367435	0.183717	−	0.982979i	$-0.441187\pi$
$691$	9.65872	0.367435	0.183717	+	0.982979i	$0.441187\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	17.3107	0.656631
$696$	0	0
$697$	23.4720	0.889064
$698$	0	0
$699$	0	0
$700$	0	0
$701$	16.8867	0.637800	0.318900	−	0.947788i	$-0.396687\pi$
$701$	16.8867	0.637800	0.318900	+	0.947788i	$0.396687\pi$
$702$	0	0
$703$	−26.9066	−1.01480
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	36.2814	1.36257	0.681287	−	0.732016i	$-0.261420\pi$
$709$	36.2814	1.36257	0.681287	+	0.732016i	$0.261420\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−14.6240	−0.547671
$714$	0	0
$715$	67.0594	2.50788
$716$	0	0
$717$	0	0
$718$	0	0
$719$	47.3293	1.76509	0.882543	−	0.470232i	$-0.155830\pi$
$719$	47.3293	1.76509	0.882543	+	0.470232i	$0.155830\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−12.1507	−0.451264
$726$	0	0
$727$	−52.5254	−1.94806	−0.974029	−	0.226421i	$-0.927297\pi$
$727$	−52.5254	−1.94806	−0.974029	+	0.226421i	$0.927297\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−5.83869	−0.215952
$732$	0	0
$733$	−29.4320	−1.08710	−0.543548	−	0.839378i	$-0.682919\pi$
$733$	−29.4320	−1.08710	−0.543548	+	0.839378i	$0.682919\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	53.3734	1.96603
$738$	0	0
$739$	−4.08998	−0.150453	−0.0752263	−	0.997166i	$-0.523968\pi$
$739$	−4.08998	−0.150453	−0.0752263	+	0.997166i	$0.523968\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	18.8294	0.690782	0.345391	−	0.938459i	$-0.387746\pi$
$743$	18.8294	0.690782	0.345391	+	0.938459i	$0.387746\pi$
$744$	0	0
$745$	35.6320	1.30545
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	22.6613	0.826921	0.413461	−	0.910522i	$-0.364320\pi$
$751$	22.6613	0.826921	0.413461	+	0.910522i	$0.364320\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−24.9253	−0.907124
$756$	0	0
$757$	35.1680	1.27820	0.639101	−	0.769122i	$-0.279306\pi$
$757$	35.1680	1.27820	0.639101	+	0.769122i	$0.279306\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−13.4720	−0.488359	−0.244179	−	0.969730i	$-0.578519\pi$
$761$	−13.4720	−0.488359	−0.244179	+	0.969730i	$0.578519\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	3.65194	0.131864
$768$	0	0
$769$	44.0560	1.58870	0.794349	−	0.607461i	$-0.207812\pi$
$769$	44.0560	1.58870	0.794349	+	0.607461i	$0.207812\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−4.26401	−0.153366	−0.0766830	−	0.997056i	$-0.524433\pi$
$773$	−4.26401	−0.153366	−0.0766830	+	0.997056i	$0.524433\pi$
$774$	0	0
$775$	−65.1680	−2.34090
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−50.2614	−1.80080
$780$	0	0
$781$	−78.1332	−2.79583
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−7.11203	−0.253839
$786$	0	0
$787$	7.08660	0.252610	0.126305	−	0.991991i	$-0.459688\pi$
$787$	7.08660	0.252610	0.126305	+	0.991991i	$0.459688\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−9.90408	−0.351704
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−19.4533	−0.689072	−0.344536	−	0.938773i	$-0.611964\pi$
$797$	−19.4533	−0.689072	−0.344536	+	0.938773i	$0.611964\pi$
$798$	0	0
$799$	−8.38538	−0.296653
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−43.6774	−1.54134
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	14.3200	0.503464	0.251732	−	0.967797i	$-0.419000\pi$
$809$	14.3200	0.503464	0.251732	+	0.967797i	$0.419000\pi$
$810$	0	0
$811$	3.43466	0.120607	0.0603035	−	0.998180i	$-0.480793\pi$
$811$	3.43466	0.120607	0.0603035	+	0.998180i	$0.480793\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−41.8633	−1.46641
$816$	0	0
$817$	12.5026	0.437410
$818$	0	0
$819$	0	0
$820$	0	0
$821$	33.0921	1.15492	0.577460	−	0.816419i	$-0.304044\pi$
$821$	33.0921	1.15492	0.577460	+	0.816419i	$0.304044\pi$
$822$	0	0
$823$	−9.20796	−0.320969	−0.160485	−	0.987038i	$-0.551306\pi$
$823$	−9.20796	−0.320969	−0.160485	+	0.987038i	$0.551306\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−20.8607	−0.725399	−0.362699	−	0.931906i	$-0.618145\pi$
$827$	−20.8607	−0.725399	−0.362699	+	0.931906i	$0.618145\pi$
$828$	0	0
$829$	−11.4160	−0.396494	−0.198247	−	0.980152i	$-0.563525\pi$
$829$	−11.4160	−0.396494	−0.198247	+	0.980152i	$0.563525\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	14.0000	0.485071
$834$	0	0
$835$	57.3107	1.98332
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−33.8260	−1.16780	−0.583901	−	0.811825i	$-0.698474\pi$
$839$	−33.8260	−1.16780	−0.583901	+	0.811825i	$0.698474\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	0	0
$844$	0	0
$845$	19.7733	0.680222
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−17.1307	−0.587232
$852$	0	0
$853$	−17.4347	−0.596951	−0.298476	−	0.954417i	$-0.596478\pi$
$853$	−17.4347	−0.596951	−0.298476	+	0.954417i	$0.596478\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	8.98935	0.307070	0.153535	−	0.988143i	$-0.450934\pi$
$857$	8.98935	0.307070	0.153535	+	0.988143i	$0.450934\pi$
$858$	0	0
$859$	−3.48469	−0.118896	−0.0594480	−	0.998231i	$-0.518934\pi$
$859$	−3.48469	−0.118896	−0.0594480	+	0.998231i	$0.518934\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−31.8947	−1.08571	−0.542853	−	0.839827i	$-0.682656\pi$
$863$	−31.8947	−1.08571	−0.542853	+	0.839827i	$0.682656\pi$
$864$	0	0
$865$	−43.7546	−1.48770
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−84.2346	−2.85746
$870$	0	0
$871$	−27.1120	−0.918656
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−38.5853	−1.30293	−0.651467	−	0.758677i	$-0.725846\pi$
$877$	−38.5853	−1.30293	−0.651467	+	0.758677i	$0.725846\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	8.54669	0.287945	0.143973	−	0.989582i	$-0.454012\pi$
$881$	8.54669	0.287945	0.143973	+	0.989582i	$0.454012\pi$
$882$	0	0
$883$	−20.9253	−0.704192	−0.352096	−	0.935964i	$-0.614531\pi$
$883$	−20.9253	−0.704192	−0.352096	+	0.935964i	$0.614531\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	38.2886	1.28561	0.642803	−	0.766032i	$-0.277771\pi$
$887$	38.2886	1.28561	0.642803	+	0.766032i	$0.277771\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	0	0
$893$	17.9559	0.600871
$894$	0	0
$895$	−73.8760	−2.46940
$896$	0	0
$897$	0	0
$898$	0	0
$899$	5.36333	0.178877
$900$	0	0
$901$	2.82936	0.0942596
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−45.4333	−1.51026
$906$	0	0
$907$	−4.28267	−0.142204	−0.0711019	−	0.997469i	$-0.522652\pi$
$907$	−4.28267	−0.142204	−0.0711019	+	0.997469i	$0.522652\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−41.9660	−1.39040	−0.695198	−	0.718819i	$-0.744683\pi$
$911$	−41.9660	−1.39040	−0.695198	+	0.718819i	$0.744683\pi$
$912$	0	0
$913$	−52.3573	−1.73277
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−5.06794	−0.167176	−0.0835879	−	0.996500i	$-0.526638\pi$
$919$	−5.06794	−0.167176	−0.0835879	+	0.996500i	$0.526638\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	39.6893	1.30639
$924$	0	0
$925$	−76.3386	−2.51000
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−27.6960	−0.908677	−0.454339	−	0.890829i	$-0.650124\pi$
$929$	−27.6960	−0.908677	−0.454339	+	0.890829i	$0.650124\pi$
$930$	0	0
$931$	−29.9787	−0.982512
$932$	0	0
$933$	0	0
$934$	0	0
$935$	46.7640	1.52935
$936$	0	0
$937$	14.6027	0.477048	0.238524	−	0.971137i	$-0.423336\pi$
$937$	14.6027	0.477048	0.238524	+	0.971137i	$0.423336\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−10.1600	−0.331206	−0.165603	−	0.986192i	$-0.552957\pi$
$941$	−10.1600	−0.331206	−0.165603	+	0.986192i	$0.552957\pi$
$942$	0	0
$943$	−32.0000	−1.04206
$944$	0	0
$945$	0	0
$946$	0	0
$947$	18.2487	0.593002	0.296501	−	0.955033i	$-0.404180\pi$
$947$	18.2487	0.593002	0.296501	+	0.955033i	$0.404180\pi$
$948$	0	0
$949$	22.1867	0.720212
$950$	0	0
$951$	0	0
$952$	0	0
$953$	32.5640	1.05485	0.527426	−	0.849601i	$-0.323157\pi$
$953$	32.5640	1.05485	0.527426	+	0.849601i	$0.323157\pi$
$954$	0	0
$955$	−92.3573	−2.98861
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−2.23471	−0.0720874
$962$	0	0
$963$	0	0
$964$	0	0
$965$	20.0000	0.643823
$966$	0	0
$967$	15.1247	0.486379	0.243190	−	0.969979i	$-0.421806\pi$
$967$	15.1247	0.486379	0.243190	+	0.969979i	$0.421806\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	8.81070	0.282749	0.141374	−	0.989956i	$-0.454848\pi$
$971$	8.81070	0.282749	0.141374	+	0.989956i	$0.454848\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	6.08273	0.194604	0.0973018	−	0.995255i	$-0.468979\pi$
$977$	6.08273	0.194604	0.0973018	+	0.995255i	$0.468979\pi$
$978$	0	0
$979$	92.0373	2.94153
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−0.335342	−0.0106958	−0.00534788	−	0.999986i	$-0.501702\pi$
$983$	−0.335342	−0.0106958	−0.00534788	+	0.999986i	$0.501702\pi$
$984$	0	0
$985$	−76.8853	−2.44977
$986$	0	0
$987$	0	0
$988$	0	0
$989$	7.96005	0.253115
$990$	0	0
$991$	22.1986	0.705163	0.352581	−	0.935781i	$-0.385304\pi$
$991$	22.1986	0.705163	0.352581	+	0.935781i	$0.385304\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	8.20541	0.260129
$996$	0	0
$997$	29.1893	0.924434	0.462217	−	0.886767i	$-0.347054\pi$
$997$	29.1893	0.924434	0.462217	+	0.886767i	$0.347054\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2088.2.a.s.1.1		3
3.2	odd	2		232.2.a.d.1.2	✓	3
4.3	odd	2		4176.2.a.bu.1.1		3
12.11	even	2		464.2.a.j.1.2		3
15.14	odd	2		5800.2.a.p.1.2		3
24.5	odd	2		1856.2.a.x.1.2		3
24.11	even	2		1856.2.a.y.1.2		3
87.86	odd	2		6728.2.a.j.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
232.2.a.d.1.2	✓	3	3.2	odd	2
464.2.a.j.1.2		3	12.11	even	2
1856.2.a.x.1.2		3	24.5	odd	2
1856.2.a.y.1.2		3	24.11	even	2
2088.2.a.s.1.1		3	1.1	even	1	trivial
4176.2.a.bu.1.1		3	4.3	odd	2
5800.2.a.p.1.2		3	15.14	odd	2
6728.2.a.j.1.2		3	87.86	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2088 = 2^{3} \cdot 3^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2088.a (trivial)

\(f(q)\)	\(=\)	\(q-4.14134 q^{5} +O(q^{10})\)	\(q-4.14134 q^{5} +5.64600 q^{11} -2.86799 q^{13} -2.00000 q^{17} +4.28267 q^{19} +2.72666 q^{23} +12.1507 q^{25} -1.00000 q^{29} -5.36333 q^{31} -6.28267 q^{37} -11.7360 q^{41} +2.91934 q^{43} +4.19269 q^{47} -7.00000 q^{49} -1.41468 q^{53} -23.3820 q^{55} -1.27334 q^{59} +3.45331 q^{61} +11.8773 q^{65} +9.45331 q^{67} -13.8387 q^{71} -7.73599 q^{73} -14.9193 q^{79} -9.27334 q^{83} +8.28267 q^{85} +16.3013 q^{89} -17.7360 q^{95} -10.2827 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 4 q^{5}+O(q^{10}) \) 3 * q - 4 * q^5	\( 3 q - 4 q^{5} - 2 q^{11} + 4 q^{13} - 6 q^{17} - 4 q^{19} + 4 q^{23} + 7 q^{25} - 3 q^{29} - 14 q^{31} - 2 q^{37} - 10 q^{41} - 6 q^{43} + 2 q^{47} - 21 q^{49} - 26 q^{55} - 8 q^{59} + 2 q^{61} + 2 q^{65} + 20 q^{67} - 12 q^{71} + 2 q^{73} - 30 q^{79} - 32 q^{83} + 8 q^{85} - 10 q^{89} - 28 q^{95} - 14 q^{97}+O(q^{100}) \) 3 * q - 4 * q^5 - 2 * q^11 + 4 * q^13 - 6 * q^17 - 4 * q^19 + 4 * q^23 + 7 * q^25 - 3 * q^29 - 14 * q^31 - 2 * q^37 - 10 * q^41 - 6 * q^43 + 2 * q^47 - 21 * q^49 - 26 * q^55 - 8 * q^59 + 2 * q^61 + 2 * q^65 + 20 * q^67 - 12 * q^71 + 2 * q^73 - 30 * q^79 - 32 * q^83 + 8 * q^85 - 10 * q^89 - 28 * q^95 - 14 * q^97

Introduction

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