LMFDB - Embedded newform 2592.2.a.u.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2592,2,Mod(1,2592)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2592, 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("2592.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2592 = 2^{5} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2592.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:	$20.6972242039$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.13068.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 6x^{2} - x + 1 $ x^4 - x^3 - 6*x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2\cdot 3 $
Twist minimal:	no (minimal twist has level 288)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.82405$ of defining polynomial
Character	$\chi$	$=$	2592.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.37228 q^{5} -4.70285 q^{7} +O(q^{10})$	$q-3.37228 q^{5} -4.70285 q^{7} -0.875393 q^{11} -1.37228 q^{13} -2.37228 q^{17} -5.57825 q^{19} -4.70285 q^{23} +6.37228 q^{25} -5.37228 q^{29} +6.45364 q^{31} +15.8593 q^{35} +4.00000 q^{37} +1.00000 q^{41} +0.875393 q^{43} -4.70285 q^{47} +15.1168 q^{49} -4.00000 q^{53} +2.95207 q^{55} -8.53032 q^{59} -2.11684 q^{61} +4.62772 q^{65} +8.53032 q^{67} -9.40571 q^{71} +10.3723 q^{73} +4.11684 q^{77} -6.45364 q^{79} +2.95207 q^{83} +8.00000 q^{85} +12.7446 q^{89} +6.45364 q^{91} +18.8114 q^{95} +9.00000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{5}+O(q^{10}) $ 4 * q - 2 * q^5	$ 4 q - 2 q^{5} + 6 q^{13} + 2 q^{17} + 14 q^{25} - 10 q^{29} + 16 q^{37} + 4 q^{41} + 26 q^{49} - 16 q^{53} + 26 q^{61} + 30 q^{65} + 30 q^{73} - 18 q^{77} + 32 q^{85} + 28 q^{89} + 36 q^{97}+O(q^{100}) $ 4 * q - 2 * q^5 + 6 * q^13 + 2 * q^17 + 14 * q^25 - 10 * q^29 + 16 * q^37 + 4 * q^41 + 26 * q^49 - 16 * q^53 + 26 * q^61 + 30 * q^65 + 30 * q^73 - 18 * q^77 + 32 * q^85 + 28 * q^89 + 36 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	−3.37228	−1.50813	−0.754065	−	0.656800i	$-0.771910\pi$
$5$	−3.37228	−1.50813	−0.754065	+	0.656800i	$0.771910\pi$
$6$	0	0
$7$	−4.70285	−1.77751	−0.888756	−	0.458381i	$-0.848430\pi$
$7$	−4.70285	−1.77751	−0.888756	+	0.458381i	$0.848430\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−0.875393	−0.263941	−0.131970	−	0.991254i	$-0.542130\pi$
$11$	−0.875393	−0.263941	−0.131970	+	0.991254i	$0.542130\pi$
$12$	0	0
$13$	−1.37228	−0.380602	−0.190301	−	0.981726i	$-0.560946\pi$
$13$	−1.37228	−0.380602	−0.190301	+	0.981726i	$0.560946\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−2.37228	−0.575363	−0.287681	−	0.957726i	$-0.592884\pi$
$17$	−2.37228	−0.575363	−0.287681	+	0.957726i	$0.592884\pi$
$18$	0	0
$19$	−5.57825	−1.27974	−0.639869	−	0.768484i	$-0.721011\pi$
$19$	−5.57825	−1.27974	−0.639869	+	0.768484i	$0.721011\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−4.70285	−0.980613	−0.490307	−	0.871550i	$-0.663115\pi$
$23$	−4.70285	−0.980613	−0.490307	+	0.871550i	$0.663115\pi$
$24$	0	0
$25$	6.37228	1.27446
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−5.37228	−0.997608	−0.498804	−	0.866715i	$-0.666227\pi$
$29$	−5.37228	−0.997608	−0.498804	+	0.866715i	$0.666227\pi$
$30$	0	0
$31$	6.45364	1.15911	0.579554	−	0.814934i	$-0.303227\pi$
$31$	6.45364	1.15911	0.579554	+	0.814934i	$0.303227\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	15.8593	2.68072
$36$	0	0
$37$	4.00000	0.657596	0.328798	−	0.944400i	$-0.393356\pi$
$37$	4.00000	0.657596	0.328798	+	0.944400i	$0.393356\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	1.00000	0.156174	0.0780869	−	0.996947i	$-0.475119\pi$
$41$	1.00000	0.156174	0.0780869	+	0.996947i	$0.475119\pi$
$42$	0	0
$43$	0.875393	0.133496	0.0667481	−	0.997770i	$-0.478738\pi$
$43$	0.875393	0.133496	0.0667481	+	0.997770i	$0.478738\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−4.70285	−0.685982	−0.342991	−	0.939339i	$-0.611440\pi$
$47$	−4.70285	−0.685982	−0.342991	+	0.939339i	$0.611440\pi$
$48$	0	0
$49$	15.1168	2.15955
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−4.00000	−0.549442	−0.274721	−	0.961524i	$-0.588586\pi$
$53$	−4.00000	−0.549442	−0.274721	+	0.961524i	$0.588586\pi$
$54$	0	0
$55$	2.95207	0.398057
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−8.53032	−1.11055	−0.555276	−	0.831666i	$-0.687388\pi$
$59$	−8.53032	−1.11055	−0.555276	+	0.831666i	$0.687388\pi$
$60$	0	0
$61$	−2.11684	−0.271034	−0.135517	−	0.990775i	$-0.543270\pi$
$61$	−2.11684	−0.271034	−0.135517	+	0.990775i	$0.543270\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	4.62772	0.573998
$66$	0	0
$67$	8.53032	1.04214	0.521072	−	0.853513i	$-0.325532\pi$
$67$	8.53032	1.04214	0.521072	+	0.853513i	$0.325532\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−9.40571	−1.11625	−0.558126	−	0.829756i	$-0.688480\pi$
$71$	−9.40571	−1.11625	−0.558126	+	0.829756i	$0.688480\pi$
$72$	0	0
$73$	10.3723	1.21398	0.606992	−	0.794708i	$-0.292376\pi$
$73$	10.3723	1.21398	0.606992	+	0.794708i	$0.292376\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	4.11684	0.469158
$78$	0	0
$79$	−6.45364	−0.726091	−0.363046	−	0.931771i	$-0.618263\pi$
$79$	−6.45364	−0.726091	−0.363046	+	0.931771i	$0.618263\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	2.95207	0.324032	0.162016	−	0.986788i	$-0.448200\pi$
$83$	2.95207	0.324032	0.162016	+	0.986788i	$0.448200\pi$
$84$	0	0
$85$	8.00000	0.867722
$86$	0	0
$87$	0	0
$88$	0	0
$89$	12.7446	1.35092	0.675460	−	0.737396i	$-0.263945\pi$
$89$	12.7446	1.35092	0.675460	+	0.737396i	$0.263945\pi$
$90$	0	0
$91$	6.45364	0.676525
$92$	0	0
$93$	0	0
$94$	0	0
$95$	18.8114	1.93001
$96$	0	0
$97$	9.00000	0.913812	0.456906	−	0.889515i	$-0.348958\pi$
$97$	9.00000	0.913812	0.456906	+	0.889515i	$0.348958\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	2.11684	0.210634	0.105317	−	0.994439i	$-0.466414\pi$
$101$	2.11684	0.210634	0.105317	+	0.994439i	$0.466414\pi$
$102$	0	0
$103$	4.70285	0.463386	0.231693	−	0.972789i	$-0.425574\pi$
$103$	4.70285	0.463386	0.231693	+	0.972789i	$0.425574\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−5.57825	−0.539270	−0.269635	−	0.962963i	$-0.586903\pi$
$107$	−5.57825	−0.539270	−0.269635	+	0.962963i	$0.586903\pi$
$108$	0	0
$109$	−5.48913	−0.525763	−0.262881	−	0.964828i	$-0.584673\pi$
$109$	−5.48913	−0.525763	−0.262881	+	0.964828i	$0.584673\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	17.3723	1.63425	0.817123	−	0.576463i	$-0.195567\pi$
$113$	17.3723	1.63425	0.817123	+	0.576463i	$0.195567\pi$
$114$	0	0
$115$	15.8593	1.47889
$116$	0	0
$117$	0	0
$118$	0	0
$119$	11.1565	1.02271
$120$	0	0
$121$	−10.2337	−0.930335
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−4.62772	−0.413916
$126$	0	0
$127$	11.1565	0.989979	0.494989	−	0.868899i	$-0.335172\pi$
$127$	11.1565	0.989979	0.494989	+	0.868899i	$0.335172\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−6.45364	−0.563857	−0.281929	−	0.959435i	$-0.590974\pi$
$131$	−6.45364	−0.563857	−0.281929	+	0.959435i	$0.590974\pi$
$132$	0	0
$133$	26.2337	2.27475
$134$	0	0
$135$	0	0
$136$	0	0
$137$	10.4891	0.896146	0.448073	−	0.893997i	$-0.352110\pi$
$137$	10.4891	0.896146	0.448073	+	0.893997i	$0.352110\pi$
$138$	0	0
$139$	−19.6868	−1.66981	−0.834907	−	0.550391i	$-0.814479\pi$
$139$	−19.6868	−1.66981	−0.834907	+	0.550391i	$0.814479\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	1.20128	0.100456
$144$	0	0
$145$	18.1168	1.50452
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−17.3723	−1.42319	−0.711596	−	0.702589i	$-0.752027\pi$
$149$	−17.3723	−1.42319	−0.711596	+	0.702589i	$0.752027\pi$
$150$	0	0
$151$	4.70285	0.382713	0.191356	−	0.981521i	$-0.438711\pi$
$151$	4.70285	0.382713	0.191356	+	0.981521i	$0.438711\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−21.7635	−1.74809
$156$	0	0
$157$	14.1168	1.12665	0.563323	−	0.826236i	$-0.309523\pi$
$157$	14.1168	1.12665	0.563323	+	0.826236i	$0.309523\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	22.1168	1.74305
$162$	0	0
$163$	−18.8114	−1.47342	−0.736712	−	0.676207i	$-0.763623\pi$
$163$	−18.8114	−1.47342	−0.736712	+	0.676207i	$0.763623\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−6.45364	−0.499398	−0.249699	−	0.968324i	$-0.580332\pi$
$167$	−6.45364	−0.499398	−0.249699	+	0.968324i	$0.580332\pi$
$168$	0	0
$169$	−11.1168	−0.855142
$170$	0	0
$171$	0	0
$172$	0	0
$173$	5.37228	0.408447	0.204223	−	0.978924i	$-0.434533\pi$
$173$	5.37228	0.408447	0.204223	+	0.978924i	$0.434533\pi$
$174$	0	0
$175$	−29.9679	−2.26536
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−18.8114	−1.40603	−0.703016	−	0.711174i	$-0.748164\pi$
$179$	−18.8114	−1.40603	−0.703016	+	0.711174i	$0.748164\pi$
$180$	0	0
$181$	26.2337	1.94993	0.974967	−	0.222348i	$-0.0713722\pi$
$181$	26.2337	1.94993	0.974967	+	0.222348i	$0.0713722\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−13.4891	−0.991740
$186$	0	0
$187$	2.07668	0.151862
$188$	0	0
$189$	0	0
$190$	0	0
$191$	17.6101	1.27422	0.637112	−	0.770771i	$-0.280129\pi$
$191$	17.6101	1.27422	0.637112	+	0.770771i	$0.280129\pi$
$192$	0	0
$193$	1.00000	0.0719816	0.0359908	−	0.999352i	$-0.488541\pi$
$193$	1.00000	0.0719816	0.0359908	+	0.999352i	$0.488541\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−10.7446	−0.765518	−0.382759	−	0.923848i	$-0.625026\pi$
$197$	−10.7446	−0.765518	−0.382759	+	0.923848i	$0.625026\pi$
$198$	0	0
$199$	−17.0606	−1.20940	−0.604698	−	0.796455i	$-0.706706\pi$
$199$	−17.0606	−1.20940	−0.604698	+	0.796455i	$0.706706\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	25.2651	1.77326
$204$	0	0
$205$	−3.37228	−0.235530
$206$	0	0
$207$	0	0
$208$	0	0
$209$	4.88316	0.337775
$210$	0	0
$211$	−15.8593	−1.09180	−0.545901	−	0.837849i	$-0.683813\pi$
$211$	−15.8593	−1.09180	−0.545901	+	0.837849i	$0.683813\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−2.95207	−0.201329
$216$	0	0
$217$	−30.3505	−2.06033
$218$	0	0
$219$	0	0
$220$	0	0
$221$	3.25544	0.218984
$222$	0	0
$223$	−4.70285	−0.314926	−0.157463	−	0.987525i	$-0.550332\pi$
$223$	−4.70285	−0.314926	−0.157463	+	0.987525i	$0.550332\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	21.4376	1.42286	0.711432	−	0.702755i	$-0.248047\pi$
$227$	21.4376	1.42286	0.711432	+	0.702755i	$0.248047\pi$
$228$	0	0
$229$	1.37228	0.0906829	0.0453415	−	0.998972i	$-0.485562\pi$
$229$	1.37228	0.0906829	0.0453415	+	0.998972i	$0.485562\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−25.8614	−1.69424	−0.847119	−	0.531404i	$-0.821665\pi$
$233$	−25.8614	−1.69424	−0.847119	+	0.531404i	$0.821665\pi$
$234$	0	0
$235$	15.8593	1.03455
$236$	0	0
$237$	0	0
$238$	0	0
$239$	17.6101	1.13910	0.569552	−	0.821955i	$-0.307117\pi$
$239$	17.6101	1.13910	0.569552	+	0.821955i	$0.307117\pi$
$240$	0	0
$241$	11.7446	0.756534	0.378267	−	0.925697i	$-0.376520\pi$
$241$	11.7446	0.756534	0.378267	+	0.925697i	$0.376520\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−50.9783	−3.25688
$246$	0	0
$247$	7.65492	0.487071
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−16.7347	−1.05629	−0.528144	−	0.849155i	$-0.677112\pi$
$251$	−16.7347	−1.05629	−0.528144	+	0.849155i	$0.677112\pi$
$252$	0	0
$253$	4.11684	0.258824
$254$	0	0
$255$	0	0
$256$	0	0
$257$	16.4891	1.02856	0.514282	−	0.857621i	$-0.328059\pi$
$257$	16.4891	1.02856	0.514282	+	0.857621i	$0.328059\pi$
$258$	0	0
$259$	−18.8114	−1.16888
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−17.6101	−1.08589	−0.542944	−	0.839769i	$-0.682690\pi$
$263$	−17.6101	−1.08589	−0.542944	+	0.839769i	$0.682690\pi$
$264$	0	0
$265$	13.4891	0.828630
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−13.4891	−0.822446	−0.411223	−	0.911535i	$-0.634898\pi$
$269$	−13.4891	−0.822446	−0.411223	+	0.911535i	$0.634898\pi$
$270$	0	0
$271$	22.3130	1.35542	0.677709	−	0.735330i	$-0.262973\pi$
$271$	22.3130	1.35542	0.677709	+	0.735330i	$0.262973\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−5.57825	−0.336381
$276$	0	0
$277$	−4.11684	−0.247357	−0.123679	−	0.992322i	$-0.539469\pi$
$277$	−4.11684	−0.247357	−0.123679	+	0.992322i	$0.539469\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−2.86141	−0.170697	−0.0853486	−	0.996351i	$-0.527200\pi$
$281$	−2.86141	−0.170697	−0.0853486	+	0.996351i	$0.527200\pi$
$282$	0	0
$283$	15.8593	0.942740	0.471370	−	0.881935i	$-0.343760\pi$
$283$	15.8593	0.942740	0.471370	+	0.881935i	$0.343760\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−4.70285	−0.277601
$288$	0	0
$289$	−11.3723	−0.668958
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−14.6277	−0.854560	−0.427280	−	0.904119i	$-0.640528\pi$
$293$	−14.6277	−0.854560	−0.427280	+	0.904119i	$0.640528\pi$
$294$	0	0
$295$	28.7666	1.67486
$296$	0	0
$297$	0	0
$298$	0	0
$299$	6.45364	0.373224
$300$	0	0
$301$	−4.11684	−0.237291
$302$	0	0
$303$	0	0
$304$	0	0
$305$	7.13859	0.408755
$306$	0	0
$307$	20.8881	1.19215	0.596073	−	0.802930i	$-0.296727\pi$
$307$	20.8881	1.19215	0.596073	+	0.802930i	$0.296727\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−27.0158	−1.53193	−0.765964	−	0.642883i	$-0.777738\pi$
$311$	−27.0158	−1.53193	−0.765964	+	0.642883i	$0.777738\pi$
$312$	0	0
$313$	−15.2337	−0.861059	−0.430529	−	0.902577i	$-0.641673\pi$
$313$	−15.2337	−0.861059	−0.430529	+	0.902577i	$0.641673\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	2.62772	0.147587	0.0737937	−	0.997274i	$-0.476489\pi$
$317$	2.62772	0.147587	0.0737937	+	0.997274i	$0.476489\pi$
$318$	0	0
$319$	4.70285	0.263309
$320$	0	0
$321$	0	0
$322$	0	0
$323$	13.2332	0.736313
$324$	0	0
$325$	−8.74456	−0.485061
$326$	0	0
$327$	0	0
$328$	0	0
$329$	22.1168	1.21934
$330$	0	0
$331$	−9.95521	−0.547188	−0.273594	−	0.961845i	$-0.588212\pi$
$331$	−9.95521	−0.547188	−0.273594	+	0.961845i	$0.588212\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−28.7666	−1.57169
$336$	0	0
$337$	−9.00000	−0.490261	−0.245131	−	0.969490i	$-0.578831\pi$
$337$	−9.00000	−0.490261	−0.245131	+	0.969490i	$0.578831\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−5.64947	−0.305936
$342$	0	0
$343$	−38.1723	−2.06111
$344$	0	0
$345$	0	0
$346$	0	0
$347$	8.53032	0.457931	0.228966	−	0.973434i	$-0.426466\pi$
$347$	8.53032	0.457931	0.228966	+	0.973434i	$0.426466\pi$
$348$	0	0
$349$	−5.88316	−0.314918	−0.157459	−	0.987526i	$-0.550330\pi$
$349$	−5.88316	−0.314918	−0.157459	+	0.987526i	$0.550330\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	15.2337	0.810807	0.405404	−	0.914138i	$-0.367131\pi$
$353$	15.2337	0.810807	0.405404	+	0.914138i	$0.367131\pi$
$354$	0	0
$355$	31.7187	1.68345
$356$	0	0
$357$	0	0
$358$	0	0
$359$	33.4695	1.76645	0.883226	−	0.468948i	$-0.155367\pi$
$359$	33.4695	1.76645	0.883226	+	0.468948i	$0.155367\pi$
$360$	0	0
$361$	12.1168	0.637729
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−34.9783	−1.83085
$366$	0	0
$367$	31.1692	1.62702	0.813509	−	0.581552i	$-0.197554\pi$
$367$	31.1692	1.62702	0.813509	+	0.581552i	$0.197554\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	18.8114	0.976640
$372$	0	0
$373$	0.116844	0.00604995	0.00302498	−	0.999995i	$-0.499037\pi$
$373$	0.116844	0.00604995	0.00302498	+	0.999995i	$0.499037\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	7.37228	0.379692
$378$	0	0
$379$	3.82746	0.196604	0.0983018	−	0.995157i	$-0.468659\pi$
$379$	3.82746	0.196604	0.0983018	+	0.995157i	$0.468659\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−21.7635	−1.11206	−0.556031	−	0.831161i	$-0.687676\pi$
$383$	−21.7635	−1.11206	−0.556031	+	0.831161i	$0.687676\pi$
$384$	0	0
$385$	−13.8832	−0.707551
$386$	0	0
$387$	0	0
$388$	0	0
$389$	32.3505	1.64024	0.820119	−	0.572194i	$-0.193907\pi$
$389$	32.3505	1.64024	0.820119	+	0.572194i	$0.193907\pi$
$390$	0	0
$391$	11.1565	0.564208
$392$	0	0
$393$	0	0
$394$	0	0
$395$	21.7635	1.09504
$396$	0	0
$397$	−4.00000	−0.200754	−0.100377	−	0.994949i	$-0.532005\pi$
$397$	−4.00000	−0.200754	−0.100377	+	0.994949i	$0.532005\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−17.9783	−0.897791	−0.448895	−	0.893584i	$-0.648182\pi$
$401$	−17.9783	−0.897791	−0.448895	+	0.893584i	$0.648182\pi$
$402$	0	0
$403$	−8.85621	−0.441159
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−3.50157	−0.173566
$408$	0	0
$409$	−29.7446	−1.47077	−0.735387	−	0.677647i	$-0.763000\pi$
$409$	−29.7446	−1.47077	−0.735387	+	0.677647i	$0.763000\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	40.1168	1.97402
$414$	0	0
$415$	−9.95521	−0.488682
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−15.8593	−0.774780	−0.387390	−	0.921916i	$-0.626623\pi$
$419$	−15.8593	−0.774780	−0.387390	+	0.921916i	$0.626623\pi$
$420$	0	0
$421$	−16.6277	−0.810385	−0.405193	−	0.914231i	$-0.632796\pi$
$421$	−16.6277	−0.810385	−0.405193	+	0.914231i	$0.632796\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−15.1168	−0.733275
$426$	0	0
$427$	9.95521	0.481766
$428$	0	0
$429$	0	0
$430$	0	0
$431$	11.1565	0.537389	0.268695	−	0.963225i	$-0.413408\pi$
$431$	11.1565	0.537389	0.268695	+	0.963225i	$0.413408\pi$
$432$	0	0
$433$	−0.883156	−0.0424418	−0.0212209	−	0.999775i	$-0.506755\pi$
$433$	−0.883156	−0.0424418	−0.0212209	+	0.999775i	$0.506755\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	26.2337	1.25493
$438$	0	0
$439$	−21.7635	−1.03871	−0.519357	−	0.854557i	$-0.673829\pi$
$439$	−21.7635	−1.03871	−0.519357	+	0.854557i	$0.673829\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−21.4376	−1.01853	−0.509265	−	0.860609i	$-0.670083\pi$
$443$	−21.4376	−1.01853	−0.509265	+	0.860609i	$0.670083\pi$
$444$	0	0
$445$	−42.9783	−2.03736
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−0.883156	−0.0416787	−0.0208394	−	0.999783i	$-0.506634\pi$
$449$	−0.883156	−0.0416787	−0.0208394	+	0.999783i	$0.506634\pi$
$450$	0	0
$451$	−0.875393	−0.0412206
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−21.7635	−1.02029
$456$	0	0
$457$	19.9783	0.934543	0.467272	−	0.884114i	$-0.345237\pi$
$457$	19.9783	0.934543	0.467272	+	0.884114i	$0.345237\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−11.8832	−0.553454	−0.276727	−	0.960949i	$-0.589250\pi$
$461$	−11.8832	−0.553454	−0.276727	+	0.960949i	$0.589250\pi$
$462$	0	0
$463$	−0.549500	−0.0255374	−0.0127687	−	0.999918i	$-0.504065\pi$
$463$	−0.549500	−0.0255374	−0.0127687	+	0.999918i	$0.504065\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−9.73160	−0.450325	−0.225162	−	0.974321i	$-0.572291\pi$
$467$	−9.73160	−0.450325	−0.225162	+	0.974321i	$0.572291\pi$
$468$	0	0
$469$	−40.1168	−1.85242
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−0.766312	−0.0352351
$474$	0	0
$475$	−35.5462	−1.63097
$476$	0	0
$477$	0	0
$478$	0	0
$479$	4.70285	0.214879	0.107439	−	0.994212i	$-0.465735\pi$
$479$	4.70285	0.214879	0.107439	+	0.994212i	$0.465735\pi$
$480$	0	0
$481$	−5.48913	−0.250283
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−30.3505	−1.37815
$486$	0	0
$487$	11.1565	0.505549	0.252775	−	0.967525i	$-0.418657\pi$
$487$	11.1565	0.505549	0.252775	+	0.967525i	$0.418657\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	40.2490	1.81641	0.908206	−	0.418523i	$-0.137452\pi$
$491$	40.2490	1.81641	0.908206	+	0.418523i	$0.137452\pi$
$492$	0	0
$493$	12.7446	0.573986
$494$	0	0
$495$	0	0
$496$	0	0
$497$	44.2337	1.98415
$498$	0	0
$499$	13.7827	0.616997	0.308499	−	0.951225i	$-0.400173\pi$
$499$	13.7827	0.616997	0.308499	+	0.951225i	$0.400173\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	31.7187	1.41427	0.707133	−	0.707080i	$-0.249988\pi$
$503$	31.7187	1.41427	0.707133	+	0.707080i	$0.249988\pi$
$504$	0	0
$505$	−7.13859	−0.317663
$506$	0	0
$507$	0	0
$508$	0	0
$509$	38.3505	1.69986	0.849929	−	0.526898i	$-0.176645\pi$
$509$	38.3505	1.69986	0.849929	+	0.526898i	$0.176645\pi$
$510$	0	0
$511$	−48.7793	−2.15787
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−15.8593	−0.698846
$516$	0	0
$517$	4.11684	0.181059
$518$	0	0
$519$	0	0
$520$	0	0
$521$	35.3505	1.54873	0.774367	−	0.632736i	$-0.218068\pi$
$521$	35.3505	1.54873	0.774367	+	0.632736i	$0.218068\pi$
$522$	0	0
$523$	−12.9073	−0.564396	−0.282198	−	0.959356i	$-0.591064\pi$
$523$	−12.9073	−0.564396	−0.282198	+	0.959356i	$0.591064\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−15.3098	−0.666908
$528$	0	0
$529$	−0.883156	−0.0383981
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−1.37228	−0.0594401
$534$	0	0
$535$	18.8114	0.813289
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−13.2332	−0.569993
$540$	0	0
$541$	−2.74456	−0.117998	−0.0589990	−	0.998258i	$-0.518791\pi$
$541$	−2.74456	−0.117998	−0.0589990	+	0.998258i	$0.518791\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	18.5109	0.792919
$546$	0	0
$547$	10.2811	0.439588	0.219794	−	0.975546i	$-0.429461\pi$
$547$	10.2811	0.439588	0.219794	+	0.975546i	$0.429461\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	29.9679	1.27668
$552$	0	0
$553$	30.3505	1.29064
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−7.25544	−0.307423	−0.153711	−	0.988116i	$-0.549123\pi$
$557$	−7.25544	−0.307423	−0.153711	+	0.988116i	$0.549123\pi$
$558$	0	0
$559$	−1.20128	−0.0508089
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−6.77953	−0.285723	−0.142862	−	0.989743i	$-0.545630\pi$
$563$	−6.77953	−0.285723	−0.142862	+	0.989743i	$0.545630\pi$
$564$	0	0
$565$	−58.5842	−2.46466
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−28.2554	−1.18453	−0.592265	−	0.805743i	$-0.701766\pi$
$569$	−28.2554	−1.18453	−0.592265	+	0.805743i	$0.701766\pi$
$570$	0	0
$571$	−40.2490	−1.68437	−0.842184	−	0.539190i	$-0.818731\pi$
$571$	−40.2490	−1.68437	−0.842184	+	0.539190i	$0.818731\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−29.9679	−1.24975
$576$	0	0
$577$	0.883156	0.0367663	0.0183831	−	0.999831i	$-0.494148\pi$
$577$	0.883156	0.0367663	0.0183831	+	0.999831i	$0.494148\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−13.8832	−0.575970
$582$	0	0
$583$	3.50157	0.145020
$584$	0	0
$585$	0	0
$586$	0	0
$587$	38.4982	1.58899	0.794496	−	0.607269i	$-0.207735\pi$
$587$	38.4982	1.58899	0.794496	+	0.607269i	$0.207735\pi$
$588$	0	0
$589$	−36.0000	−1.48335
$590$	0	0
$591$	0	0
$592$	0	0
$593$	39.7228	1.63122	0.815610	−	0.578602i	$-0.196401\pi$
$593$	39.7228	1.63122	0.815610	+	0.578602i	$0.196401\pi$
$594$	0	0
$595$	−37.6228	−1.54239
$596$	0	0
$597$	0	0
$598$	0	0
$599$	31.1692	1.27354	0.636769	−	0.771054i	$-0.280270\pi$
$599$	31.1692	1.27354	0.636769	+	0.771054i	$0.280270\pi$
$600$	0	0
$601$	−37.2337	−1.51879	−0.759397	−	0.650628i	$-0.774506\pi$
$601$	−37.2337	−1.51879	−0.759397	+	0.650628i	$0.774506\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	34.5109	1.40307
$606$	0	0
$607$	0.549500	0.0223035	0.0111518	−	0.999938i	$-0.496450\pi$
$607$	0.549500	0.0223035	0.0111518	+	0.999938i	$0.496450\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	6.45364	0.261086
$612$	0	0
$613$	−32.4674	−1.31134	−0.655672	−	0.755045i	$-0.727615\pi$
$613$	−32.4674	−1.31134	−0.655672	+	0.755045i	$0.727615\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−26.2554	−1.05700	−0.528502	−	0.848932i	$-0.677246\pi$
$617$	−26.2554	−1.05700	−0.528502	+	0.848932i	$0.677246\pi$
$618$	0	0
$619$	10.2811	0.413232	0.206616	−	0.978422i	$-0.433755\pi$
$619$	10.2811	0.413232	0.206616	+	0.978422i	$0.433755\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−59.9358	−2.40128
$624$	0	0
$625$	−16.2554	−0.650217
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−9.48913	−0.378356
$630$	0	0
$631$	0	0		−	1.00000i	$-0.5\pi$
$631$	0	0			1.00000i	$0.5\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−37.6228	−1.49302
$636$	0	0
$637$	−20.7446	−0.821929
$638$	0	0
$639$	0	0
$640$	0	0
$641$	2.25544	0.0890844	0.0445422	−	0.999008i	$-0.485817\pi$
$641$	2.25544	0.0890844	0.0445422	+	0.999008i	$0.485817\pi$
$642$	0	0
$643$	−2.62618	−0.103566	−0.0517832	−	0.998658i	$-0.516490\pi$
$643$	−2.62618	−0.103566	−0.0517832	+	0.998658i	$0.516490\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	4.15335	0.163285	0.0816426	−	0.996662i	$-0.473983\pi$
$647$	4.15335	0.163285	0.0816426	+	0.996662i	$0.473983\pi$
$648$	0	0
$649$	7.46738	0.293120
$650$	0	0
$651$	0	0
$652$	0	0
$653$	23.8832	0.934620	0.467310	−	0.884093i	$-0.345223\pi$
$653$	23.8832	0.934620	0.467310	+	0.884093i	$0.345223\pi$
$654$	0	0
$655$	21.7635	0.850370
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−40.5749	−1.58057	−0.790287	−	0.612737i	$-0.790069\pi$
$659$	−40.5749	−1.58057	−0.790287	+	0.612737i	$0.790069\pi$
$660$	0	0
$661$	14.1168	0.549082	0.274541	−	0.961575i	$-0.411474\pi$
$661$	14.1168	0.549082	0.274541	+	0.961575i	$0.411474\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−88.4674	−3.43062
$666$	0	0
$667$	25.2651	0.978267
$668$	0	0
$669$	0	0
$670$	0	0
$671$	1.85307	0.0715369
$672$	0	0
$673$	−28.3505	−1.09283	−0.546416	−	0.837514i	$-0.684008\pi$
$673$	−28.3505	−1.09283	−0.546416	+	0.837514i	$0.684008\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−50.3505	−1.93513	−0.967564	−	0.252626i	$-0.918706\pi$
$677$	−50.3505	−1.93513	−0.967564	+	0.252626i	$0.918706\pi$
$678$	0	0
$679$	−42.3257	−1.62431
$680$	0	0
$681$	0	0
$682$	0	0
$683$	9.73160	0.372369	0.186185	−	0.982515i	$-0.440388\pi$
$683$	9.73160	0.372369	0.186185	+	0.982515i	$0.440388\pi$
$684$	0	0
$685$	−35.3723	−1.35151
$686$	0	0
$687$	0	0
$688$	0	0
$689$	5.48913	0.209119
$690$	0	0
$691$	18.2619	0.694716	0.347358	−	0.937733i	$-0.387079\pi$
$691$	18.2619	0.694716	0.347358	+	0.937733i	$0.387079\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	66.3895	2.51830
$696$	0	0
$697$	−2.37228	−0.0898566
$698$	0	0
$699$	0	0
$700$	0	0
$701$	23.2554	0.878346	0.439173	−	0.898403i	$-0.355272\pi$
$701$	23.2554	0.878346	0.439173	+	0.898403i	$0.355272\pi$
$702$	0	0
$703$	−22.3130	−0.841550
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−9.95521	−0.374404
$708$	0	0
$709$	27.6060	1.03676	0.518382	−	0.855149i	$-0.326535\pi$
$709$	27.6060	1.03676	0.518382	+	0.855149i	$0.326535\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−30.3505	−1.13664
$714$	0	0
$715$	−4.05107	−0.151501
$716$	0	0
$717$	0	0
$718$	0	0
$719$	15.3098	0.570961	0.285481	−	0.958385i	$-0.407847\pi$
$719$	15.3098	0.570961	0.285481	+	0.958385i	$0.407847\pi$
$720$	0	0
$721$	−22.1168	−0.823674
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−34.2337	−1.27141
$726$	0	0
$727$	20.0127	0.742230	0.371115	−	0.928587i	$-0.378976\pi$
$727$	20.0127	0.742230	0.371115	+	0.928587i	$0.378976\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−2.07668	−0.0768087
$732$	0	0
$733$	40.1168	1.48175	0.740875	−	0.671643i	$-0.234411\pi$
$733$	40.1168	1.48175	0.740875	+	0.671643i	$0.234411\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−7.46738	−0.275064
$738$	0	0
$739$	7.32903	0.269603	0.134801	−	0.990873i	$-0.456960\pi$
$739$	7.32903	0.269603	0.134801	+	0.990873i	$0.456960\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	49.3288	1.80970	0.904850	−	0.425731i	$-0.139983\pi$
$743$	49.3288	1.80970	0.904850	+	0.425731i	$0.139983\pi$
$744$	0	0
$745$	58.5842	2.14636
$746$	0	0
$747$	0	0
$748$	0	0
$749$	26.2337	0.958558
$750$	0	0
$751$	−21.7635	−0.794161	−0.397081	−	0.917784i	$-0.629977\pi$
$751$	−21.7635	−0.794161	−0.397081	+	0.917784i	$0.629977\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−15.8593	−0.577181
$756$	0	0
$757$	34.4674	1.25274	0.626369	−	0.779527i	$-0.284540\pi$
$757$	34.4674	1.25274	0.626369	+	0.779527i	$0.284540\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	14.6277	0.530254	0.265127	−	0.964213i	$-0.414586\pi$
$761$	14.6277	0.530254	0.265127	+	0.964213i	$0.414586\pi$
$762$	0	0
$763$	25.8146	0.934550
$764$	0	0
$765$	0	0
$766$	0	0
$767$	11.7060	0.422679
$768$	0	0
$769$	−32.1168	−1.15816	−0.579082	−	0.815270i	$-0.696589\pi$
$769$	−32.1168	−1.15816	−0.579082	+	0.815270i	$0.696589\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−7.25544	−0.260960	−0.130480	−	0.991451i	$-0.541652\pi$
$773$	−7.25544	−0.260960	−0.130480	+	0.991451i	$0.541652\pi$
$774$	0	0
$775$	41.1244	1.47723
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−5.57825	−0.199861
$780$	0	0
$781$	8.23369	0.294625
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−47.6060	−1.69913
$786$	0	0
$787$	8.85621	0.315690	0.157845	−	0.987464i	$-0.449545\pi$
$787$	8.85621	0.315690	0.157845	+	0.987464i	$0.449545\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−81.6993	−2.90489
$792$	0	0
$793$	2.90491	0.103156
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−17.0951	−0.605539	−0.302770	−	0.953064i	$-0.597911\pi$
$797$	−17.0951	−0.605539	−0.302770	+	0.953064i	$0.597911\pi$
$798$	0	0
$799$	11.1565	0.394688
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−9.07982	−0.320420
$804$	0	0
$805$	−74.5842	−2.62875
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−18.3723	−0.645935	−0.322968	−	0.946410i	$-0.604680\pi$
$809$	−18.3723	−0.645935	−0.322968	+	0.946410i	$0.604680\pi$
$810$	0	0
$811$	50.2042	1.76291	0.881454	−	0.472269i	$-0.156565\pi$
$811$	50.2042	1.76291	0.881454	+	0.472269i	$0.156565\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	63.4374	2.22212
$816$	0	0
$817$	−4.88316	−0.170840
$818$	0	0
$819$	0	0
$820$	0	0
$821$	2.62772	0.0917080	0.0458540	−	0.998948i	$-0.485399\pi$
$821$	2.62772	0.0917080	0.0458540	+	0.998948i	$0.485399\pi$
$822$	0	0
$823$	21.7635	0.758628	0.379314	−	0.925268i	$-0.376160\pi$
$823$	21.7635	0.758628	0.379314	+	0.925268i	$0.376160\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−18.8114	−0.654137	−0.327069	−	0.945001i	$-0.606061\pi$
$827$	−18.8114	−0.654137	−0.327069	+	0.945001i	$0.606061\pi$
$828$	0	0
$829$	26.2337	0.911134	0.455567	−	0.890202i	$-0.349437\pi$
$829$	26.2337	0.911134	0.455567	+	0.890202i	$0.349437\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−35.8614	−1.24252
$834$	0	0
$835$	21.7635	0.753157
$836$	0	0
$837$	0	0
$838$	0	0
$839$	6.45364	0.222804	0.111402	−	0.993775i	$-0.464466\pi$
$839$	6.45364	0.222804	0.111402	+	0.993775i	$0.464466\pi$
$840$	0	0
$841$	−0.138593	−0.00477908
$842$	0	0
$843$	0	0
$844$	0	0
$845$	37.4891	1.28967
$846$	0	0
$847$	48.1275	1.65368
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−18.8114	−0.644847
$852$	0	0
$853$	−3.88316	−0.132957	−0.0664784	−	0.997788i	$-0.521176\pi$
$853$	−3.88316	−0.132957	−0.0664784	+	0.997788i	$0.521176\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	19.0951	0.652276	0.326138	−	0.945322i	$-0.394253\pi$
$857$	19.0951	0.652276	0.326138	+	0.945322i	$0.394253\pi$
$858$	0	0
$859$	−2.62618	−0.0896040	−0.0448020	−	0.998996i	$-0.514266\pi$
$859$	−2.62618	−0.0896040	−0.0448020	+	0.998996i	$0.514266\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−37.6228	−1.28070	−0.640348	−	0.768085i	$-0.721210\pi$
$863$	−37.6228	−1.28070	−0.640348	+	0.768085i	$0.721210\pi$
$864$	0	0
$865$	−18.1168	−0.615991
$866$	0	0
$867$	0	0
$868$	0	0
$869$	5.64947	0.191645
$870$	0	0
$871$	−11.7060	−0.396643
$872$	0	0
$873$	0	0
$874$	0	0
$875$	21.7635	0.735740
$876$	0	0
$877$	38.3505	1.29501	0.647503	−	0.762063i	$-0.275813\pi$
$877$	38.3505	1.29501	0.647503	+	0.762063i	$0.275813\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−30.2337	−1.01860	−0.509299	−	0.860589i	$-0.670095\pi$
$881$	−30.2337	−1.01860	−0.509299	+	0.860589i	$0.670095\pi$
$882$	0	0
$883$	−32.0446	−1.07839	−0.539193	−	0.842182i	$-0.681271\pi$
$883$	−32.0446	−1.07839	−0.539193	+	0.842182i	$0.681271\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	21.7635	0.730746	0.365373	−	0.930861i	$-0.380941\pi$
$887$	21.7635	0.730746	0.365373	+	0.930861i	$0.380941\pi$
$888$	0	0
$889$	−52.4674	−1.75970
$890$	0	0
$891$	0	0
$892$	0	0
$893$	26.2337	0.877877
$894$	0	0
$895$	63.4374	2.12048
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−34.6708	−1.15633
$900$	0	0
$901$	9.48913	0.316129
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−88.4674	−2.94076
$906$	0	0
$907$	34.3449	1.14040	0.570201	−	0.821505i	$-0.306865\pi$
$907$	34.3449	1.14040	0.570201	+	0.821505i	$0.306865\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	22.8625	0.757468	0.378734	−	0.925506i	$-0.376360\pi$
$911$	22.8625	0.757468	0.378734	+	0.925506i	$0.376360\pi$
$912$	0	0
$913$	−2.58422	−0.0855252
$914$	0	0
$915$	0	0
$916$	0	0
$917$	30.3505	1.00226
$918$	0	0
$919$	26.4663	0.873044	0.436522	−	0.899694i	$-0.356210\pi$
$919$	26.4663	0.873044	0.436522	+	0.899694i	$0.356210\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	12.9073	0.424848
$924$	0	0
$925$	25.4891	0.838077
$926$	0	0
$927$	0	0
$928$	0	0
$929$	30.6277	1.00486	0.502431	−	0.864617i	$-0.332439\pi$
$929$	30.6277	1.00486	0.502431	+	0.864617i	$0.332439\pi$
$930$	0	0
$931$	−84.3255	−2.76366
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−7.00314	−0.229027
$936$	0	0
$937$	−22.2337	−0.726343	−0.363171	−	0.931722i	$-0.618306\pi$
$937$	−22.2337	−0.726343	−0.363171	+	0.931722i	$0.618306\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	17.0951	0.557284	0.278642	−	0.960395i	$-0.410116\pi$
$941$	17.0951	0.557284	0.278642	+	0.960395i	$0.410116\pi$
$942$	0	0
$943$	−4.70285	−0.153146
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−5.02875	−0.163412	−0.0817062	−	0.996656i	$-0.526037\pi$
$947$	−5.02875	−0.163412	−0.0817062	+	0.996656i	$0.526037\pi$
$948$	0	0
$949$	−14.2337	−0.462045
$950$	0	0
$951$	0	0
$952$	0	0
$953$	8.60597	0.278775	0.139387	−	0.990238i	$-0.455487\pi$
$953$	8.60597	0.278775	0.139387	+	0.990238i	$0.455487\pi$
$954$	0	0
$955$	−59.3863	−1.92170
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−49.3288	−1.59291
$960$	0	0
$961$	10.6495	0.343531
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−3.37228	−0.108558
$966$	0	0
$967$	−49.3288	−1.58631	−0.793154	−	0.609021i	$-0.791563\pi$
$967$	−49.3288	−1.58631	−0.793154	+	0.609021i	$0.791563\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	57.5333	1.84633	0.923165	−	0.384404i	$-0.125593\pi$
$971$	57.5333	1.84633	0.923165	+	0.384404i	$0.125593\pi$
$972$	0	0
$973$	92.5842	2.96811
$974$	0	0
$975$	0	0
$976$	0	0
$977$	45.2337	1.44715	0.723577	−	0.690244i	$-0.242497\pi$
$977$	45.2337	1.44715	0.723577	+	0.690244i	$0.242497\pi$
$978$	0	0
$979$	−11.1565	−0.356563
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−59.3863	−1.89413	−0.947065	−	0.321042i	$-0.895967\pi$
$983$	−59.3863	−1.89413	−0.947065	+	0.321042i	$0.895967\pi$
$984$	0	0
$985$	36.2337	1.15450
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−4.11684	−0.130908
$990$	0	0
$991$	−7.00314	−0.222462	−0.111231	−	0.993795i	$-0.535479\pi$
$991$	−7.00314	−0.222462	−0.111231	+	0.993795i	$0.535479\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	57.5333	1.82393
$996$	0	0
$997$	−10.3505	−0.327805	−0.163902	−	0.986477i	$-0.552408\pi$
$997$	−10.3505	−0.327805	−0.163902	+	0.986477i	$0.552408\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2592.2.a.u.1.1		4
3.2	odd	2		2592.2.a.x.1.3		4
4.3	odd	2	inner	2592.2.a.u.1.2		4
8.3	odd	2		5184.2.a.cf.1.4		4
8.5	even	2		5184.2.a.cf.1.3		4
9.2	odd	6		864.2.i.f.577.2		8
9.4	even	3		288.2.i.f.97.3	yes	8
9.5	odd	6		864.2.i.f.289.2		8
9.7	even	3		288.2.i.f.193.3	yes	8
12.11	even	2		2592.2.a.x.1.4		4
24.5	odd	2		5184.2.a.cc.1.1		4
24.11	even	2		5184.2.a.cc.1.2		4
36.7	odd	6		288.2.i.f.193.2	yes	8
36.11	even	6		864.2.i.f.577.1		8
36.23	even	6		864.2.i.f.289.1		8
36.31	odd	6		288.2.i.f.97.2	✓	8
72.5	odd	6		1728.2.i.n.1153.4		8
72.11	even	6		1728.2.i.n.577.3		8
72.13	even	6		576.2.i.n.385.2		8
72.29	odd	6		1728.2.i.n.577.4		8
72.43	odd	6		576.2.i.n.193.3		8
72.59	even	6		1728.2.i.n.1153.3		8
72.61	even	6		576.2.i.n.193.2		8
72.67	odd	6		576.2.i.n.385.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
288.2.i.f.97.2	✓	8	36.31	odd	6
288.2.i.f.97.3	yes	8	9.4	even	3
288.2.i.f.193.2	yes	8	36.7	odd	6
288.2.i.f.193.3	yes	8	9.7	even	3
576.2.i.n.193.2		8	72.61	even	6
576.2.i.n.193.3		8	72.43	odd	6
576.2.i.n.385.2		8	72.13	even	6
576.2.i.n.385.3		8	72.67	odd	6
864.2.i.f.289.1		8	36.23	even	6
864.2.i.f.289.2		8	9.5	odd	6
864.2.i.f.577.1		8	36.11	even	6
864.2.i.f.577.2		8	9.2	odd	6
1728.2.i.n.577.3		8	72.11	even	6
1728.2.i.n.577.4		8	72.29	odd	6
1728.2.i.n.1153.3		8	72.59	even	6
1728.2.i.n.1153.4		8	72.5	odd	6
2592.2.a.u.1.1		4	1.1	even	1	trivial
2592.2.a.u.1.2		4	4.3	odd	2	inner
2592.2.a.x.1.3		4	3.2	odd	2
2592.2.a.x.1.4		4	12.11	even	2
5184.2.a.cc.1.1		4	24.5	odd	2
5184.2.a.cc.1.2		4	24.11	even	2
5184.2.a.cf.1.3		4	8.5	even	2
5184.2.a.cf.1.4		4	8.3	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:	\( N \)	\(=\)	\( 2592 = 2^{5} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2592.a (trivial)

\(f(q)\)	\(=\)	\(q-3.37228 q^{5} -4.70285 q^{7} +O(q^{10})\)	\(q-3.37228 q^{5} -4.70285 q^{7} -0.875393 q^{11} -1.37228 q^{13} -2.37228 q^{17} -5.57825 q^{19} -4.70285 q^{23} +6.37228 q^{25} -5.37228 q^{29} +6.45364 q^{31} +15.8593 q^{35} +4.00000 q^{37} +1.00000 q^{41} +0.875393 q^{43} -4.70285 q^{47} +15.1168 q^{49} -4.00000 q^{53} +2.95207 q^{55} -8.53032 q^{59} -2.11684 q^{61} +4.62772 q^{65} +8.53032 q^{67} -9.40571 q^{71} +10.3723 q^{73} +4.11684 q^{77} -6.45364 q^{79} +2.95207 q^{83} +8.00000 q^{85} +12.7446 q^{89} +6.45364 q^{91} +18.8114 q^{95} +9.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{5}+O(q^{10}) \) 4 * q - 2 * q^5	\( 4 q - 2 q^{5} + 6 q^{13} + 2 q^{17} + 14 q^{25} - 10 q^{29} + 16 q^{37} + 4 q^{41} + 26 q^{49} - 16 q^{53} + 26 q^{61} + 30 q^{65} + 30 q^{73} - 18 q^{77} + 32 q^{85} + 28 q^{89} + 36 q^{97}+O(q^{100}) \) 4 * q - 2 * q^5 + 6 * q^13 + 2 * q^17 + 14 * q^25 - 10 * q^29 + 16 * q^37 + 4 * q^41 + 26 * q^49 - 16 * q^53 + 26 * q^61 + 30 * q^65 + 30 * q^73 - 18 * q^77 + 32 * q^85 + 28 * q^89 + 36 * q^97

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