LMFDB - Embedded newform 6552.2.a.br.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [6552,2,Mod(1,6552)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("6552.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6552, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 6552 = 2^{3} \cdot 3^{2} \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6552.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,-2,0,-4,0,0,0,1,0,-4,0,0,0,-16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$52.3179834043$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.64268.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 12x^{2} + 6x + 32 $ x^4 - x^3 - 12x^2 + 6x + 32
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 728)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.19455$ of defining polynomial
Character	$\chi$	$=$	6552.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.70715 q^{5} -1.00000 q^{7} -4.56246 q^{11} -1.00000 q^{13} -7.37850 q^{17} +6.46416 q^{19} -5.90170 q^{23} -2.08565 q^{25} -2.69655 q^{29} +0.523191 q^{31} -1.70715 q^{35} +2.17336 q^{37} +4.98735 q^{41} +2.32864 q^{43} +12.3266 q^{47} +1.00000 q^{49} +4.71774 q^{53} -7.78879 q^{55} +13.7822 q^{59} -0.137574 q^{61} -1.70715 q^{65} -7.96215 q^{67} +13.8392 q^{71} +2.52319 q^{73} +4.56246 q^{77} +4.85531 q^{79} +7.51054 q^{83} -12.5962 q^{85} +10.8321 q^{89} +1.00000 q^{91} +11.0353 q^{95} +17.6693 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{5} - 4 q^{7} + q^{11} - 4 q^{13} - 16 q^{17} - 10 q^{19} - 7 q^{23} + 14 q^{25} - 4 q^{29} - q^{31} + 2 q^{35} + 5 q^{37} - 19 q^{41} + 14 q^{43} + 13 q^{47} + 4 q^{49} + 8 q^{53} + 2 q^{55}+ \cdots + 25 q^{97}+O(q^{100}) $ 4 * q - 2 * q^5 - 4 * q^7 + q^11 - 4 * q^13 - 16 * q^17 - 10 * q^19 - 7 * q^23 + 14 * q^25 - 4 * q^29 - q^31 + 2 * q^35 + 5 * q^37 - 19 * q^41 + 14 * q^43 + 13 * q^47 + 4 * q^49 + 8 * q^53 + 2 * q^55 + 26 * q^59 - q^61 + 2 * q^65 + 5 * q^67 + 18 * q^71 + 7 * q^73 - q^77 + 9 * q^79 - 12 * q^83 + 14 * q^85 - 4 * q^89 + 4 * q^91 + 26 * q^95 + 25 * q^97

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
$5$	1.70715	0.763459	0.381730	−	0.924274i	$-0.375329\pi$
$5$	1.70715	0.763459	0.381730	+	0.924274i	$0.375329\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−4.56246	−1.37563	−0.687817	−	0.725884i	$-0.741431\pi$
$11$	−4.56246	−1.37563	−0.687817	+	0.725884i	$0.741431\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−7.37850	−1.78955	−0.894775	−	0.446517i	$-0.852664\pi$
$17$	−7.37850	−1.78955	−0.894775	+	0.446517i	$0.852664\pi$
$18$	0	0
$19$	6.46416	1.48298	0.741489	−	0.670964i	$-0.234120\pi$
$19$	6.46416	1.48298	0.741489	+	0.670964i	$0.234120\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−5.90170	−1.23059	−0.615294	−	0.788298i	$-0.710963\pi$
$23$	−5.90170	−1.23059	−0.615294	+	0.788298i	$0.710963\pi$
$24$	0	0
$25$	−2.08565	−0.417130
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−2.69655	−0.500737	−0.250369	−	0.968151i	$-0.580552\pi$
$29$	−2.69655	−0.500737	−0.250369	+	0.968151i	$0.580552\pi$
$30$	0	0
$31$	0.523191	0.0939678	0.0469839	−	0.998896i	$-0.485039\pi$
$31$	0.523191	0.0939678	0.0469839	+	0.998896i	$0.485039\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−1.70715	−0.288560
$36$	0	0
$37$	2.17336	0.357299	0.178649	−	0.983913i	$-0.442827\pi$
$37$	2.17336	0.357299	0.178649	+	0.983913i	$0.442827\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	4.98735	0.778893	0.389446	−	0.921049i	$-0.372666\pi$
$41$	4.98735	0.778893	0.389446	+	0.921049i	$0.372666\pi$
$42$	0	0
$43$	2.32864	0.355115	0.177557	−	0.984110i	$-0.443180\pi$
$43$	2.32864	0.355115	0.177557	+	0.984110i	$0.443180\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	12.3266	1.79802	0.899008	−	0.437932i	$-0.144289\pi$
$47$	12.3266	1.79802	0.899008	+	0.437932i	$0.144289\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	4.71774	0.648031	0.324016	−	0.946052i	$-0.394967\pi$
$53$	4.71774	0.648031	0.324016	+	0.946052i	$0.394967\pi$
$54$	0	0
$55$	−7.78879	−1.05024
$56$	0	0
$57$	0	0
$58$	0	0
$59$	13.7822	1.79429	0.897145	−	0.441736i	$-0.145637\pi$
$59$	13.7822	1.79429	0.897145	+	0.441736i	$0.145637\pi$
$60$	0	0
$61$	−0.137574	−0.0176146	−0.00880730	−	0.999961i	$-0.502803\pi$
$61$	−0.137574	−0.0176146	−0.00880730	+	0.999961i	$0.502803\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−1.70715	−0.211745
$66$	0	0
$67$	−7.96215	−0.972732	−0.486366	−	0.873755i	$-0.661678\pi$
$67$	−7.96215	−0.972732	−0.486366	+	0.873755i	$0.661678\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	13.8392	1.64241	0.821204	−	0.570634i	$-0.193303\pi$
$71$	13.8392	1.64241	0.821204	+	0.570634i	$0.193303\pi$
$72$	0	0
$73$	2.52319	0.295317	0.147659	−	0.989038i	$-0.452826\pi$
$73$	2.52319	0.295317	0.147659	+	0.989038i	$0.452826\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	4.56246	0.519941
$78$	0	0
$79$	4.85531	0.546265	0.273133	−	0.961976i	$-0.411940\pi$
$79$	4.85531	0.546265	0.273133	+	0.961976i	$0.411940\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	7.51054	0.824389	0.412194	−	0.911096i	$-0.364763\pi$
$83$	7.51054	0.824389	0.412194	+	0.911096i	$0.364763\pi$
$84$	0	0
$85$	−12.5962	−1.36625
$86$	0	0
$87$	0	0
$88$	0	0
$89$	10.8321	1.14820	0.574098	−	0.818786i	$-0.305353\pi$
$89$	10.8321	1.14820	0.574098	+	0.818786i	$0.305353\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	0	0
$93$	0	0
$94$	0	0
$95$	11.0353	1.13219
$96$	0	0
$97$	17.6693	1.79405	0.897023	−	0.441985i	$-0.145725\pi$
$97$	17.6693	1.79405	0.897023	+	0.441985i	$0.145725\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−18.1692	−1.80791	−0.903954	−	0.427631i	$-0.859348\pi$
$101$	−18.1692	−1.80791	−0.903954	+	0.427631i	$0.859348\pi$
$102$	0	0
$103$	15.1461	1.49239	0.746195	−	0.665727i	$-0.231879\pi$
$103$	15.1461	1.49239	0.746195	+	0.665727i	$0.231879\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−0.657284	−0.0635420	−0.0317710	−	0.999495i	$-0.510115\pi$
$107$	−0.657284	−0.0635420	−0.0317710	+	0.999495i	$0.510115\pi$
$108$	0	0
$109$	7.76760	0.744001	0.372001	−	0.928232i	$-0.378672\pi$
$109$	7.76760	0.744001	0.372001	+	0.928232i	$0.378672\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−4.87650	−0.458743	−0.229371	−	0.973339i	$-0.573667\pi$
$113$	−4.87650	−0.458743	−0.229371	+	0.973339i	$0.573667\pi$
$114$	0	0
$115$	−10.0751	−0.939504
$116$	0	0
$117$	0	0
$118$	0	0
$119$	7.37850	0.676386
$120$	0	0
$121$	9.81604	0.892368
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−12.0962	−1.08192
$126$	0	0
$127$	11.6446	1.03329	0.516647	−	0.856199i	$-0.327180\pi$
$127$	11.6446	1.03329	0.516647	+	0.856199i	$0.327180\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	15.4928	1.35362	0.676808	−	0.736160i	$-0.263363\pi$
$131$	15.4928	1.35362	0.676808	+	0.736160i	$0.263363\pi$
$132$	0	0
$133$	−6.46416	−0.560513
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−5.16071	−0.440909	−0.220455	−	0.975397i	$-0.570754\pi$
$137$	−5.16071	−0.440909	−0.220455	+	0.975397i	$0.570754\pi$
$138$	0	0
$139$	−9.76760	−0.828477	−0.414239	−	0.910168i	$-0.635952\pi$
$139$	−9.76760	−0.828477	−0.414239	+	0.910168i	$0.635952\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	4.56246	0.381532
$144$	0	0
$145$	−4.60341	−0.382293
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−8.38704	−0.687093	−0.343546	−	0.939136i	$-0.611628\pi$
$149$	−8.38704	−0.687093	−0.343546	+	0.939136i	$0.611628\pi$
$150$	0	0
$151$	−12.5246	−1.01924	−0.509619	−	0.860400i	$-0.670214\pi$
$151$	−12.5246	−1.01924	−0.509619	+	0.860400i	$0.670214\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0.893163	0.0717406
$156$	0	0
$157$	−5.19856	−0.414890	−0.207445	−	0.978247i	$-0.566515\pi$
$157$	−5.19856	−0.414890	−0.207445	+	0.978247i	$0.566515\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	5.90170	0.465119
$162$	0	0
$163$	16.8498	1.31978	0.659888	−	0.751364i	$-0.270604\pi$
$163$	16.8498	1.31978	0.659888	+	0.751364i	$0.270604\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	22.1960	1.71758	0.858788	−	0.512331i	$-0.171218\pi$
$167$	22.1960	1.71758	0.858788	+	0.512331i	$0.171218\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	0	0
$172$	0	0
$173$	6.98941	0.531395	0.265697	−	0.964056i	$-0.414398\pi$
$173$	6.98941	0.531395	0.265697	+	0.964056i	$0.414398\pi$
$174$	0	0
$175$	2.08565	0.157660
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−14.2106	−1.06215	−0.531074	−	0.847325i	$-0.678211\pi$
$179$	−14.2106	−1.06215	−0.531074	+	0.847325i	$0.678211\pi$
$180$	0	0
$181$	−18.9516	−1.40866	−0.704329	−	0.709874i	$-0.748752\pi$
$181$	−18.9516	−1.40866	−0.704329	+	0.709874i	$0.748752\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	3.71025	0.272783
$186$	0	0
$187$	33.6641	2.46177
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−3.34272	−0.241870	−0.120935	−	0.992660i	$-0.538589\pi$
$191$	−3.34272	−0.241870	−0.120935	+	0.992660i	$0.538589\pi$
$192$	0	0
$193$	3.12492	0.224937	0.112468	−	0.993655i	$-0.464124\pi$
$193$	3.12492	0.224937	0.112468	+	0.993655i	$0.464124\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	23.7867	1.69473	0.847367	−	0.531008i	$-0.178186\pi$
$197$	23.7867	1.69473	0.847367	+	0.531008i	$0.178186\pi$
$198$	0	0
$199$	1.57511	0.111657	0.0558284	−	0.998440i	$-0.482220\pi$
$199$	1.57511	0.111657	0.0558284	+	0.998440i	$0.482220\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	2.69655	0.189261
$204$	0	0
$205$	8.51413	0.594653
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−29.4925	−2.04004
$210$	0	0
$211$	−13.5463	−0.932567	−0.466284	−	0.884635i	$-0.654407\pi$
$211$	−13.5463	−0.932567	−0.466284	+	0.884635i	$0.654407\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	3.97533	0.271115
$216$	0	0
$217$	−0.523191	−0.0355165
$218$	0	0
$219$	0	0
$220$	0	0
$221$	7.37850	0.496332
$222$	0	0
$223$	9.56957	0.640826	0.320413	−	0.947278i	$-0.396178\pi$
$223$	9.56957	0.640826	0.320413	+	0.947278i	$0.396178\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	12.7066	0.843368	0.421684	−	0.906743i	$-0.361439\pi$
$227$	12.7066	0.843368	0.421684	+	0.906743i	$0.361439\pi$
$228$	0	0
$229$	−2.97480	−0.196581	−0.0982903	−	0.995158i	$-0.531337\pi$
$229$	−2.97480	−0.196581	−0.0982903	+	0.995158i	$0.531337\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	9.06899	0.594129	0.297065	−	0.954857i	$-0.403992\pi$
$233$	9.06899	0.594129	0.297065	+	0.954857i	$0.403992\pi$
$234$	0	0
$235$	21.0433	1.37271
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−12.3891	−0.801384	−0.400692	−	0.916213i	$-0.631230\pi$
$239$	−12.3891	−0.801384	−0.400692	+	0.916213i	$0.631230\pi$
$240$	0	0
$241$	−22.9530	−1.47853	−0.739266	−	0.673414i	$-0.764827\pi$
$241$	−22.9530	−1.47853	−0.739266	+	0.673414i	$0.764827\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	1.70715	0.109066
$246$	0	0
$247$	−6.46416	−0.411304
$248$	0	0
$249$	0	0
$250$	0	0
$251$	7.16935	0.452526	0.226263	−	0.974066i	$-0.427349\pi$
$251$	7.16935	0.452526	0.226263	+	0.974066i	$0.427349\pi$
$252$	0	0
$253$	26.9263	1.69284
$254$	0	0
$255$	0	0
$256$	0	0
$257$	0.813983	0.0507749	0.0253874	−	0.999678i	$-0.491918\pi$
$257$	0.813983	0.0507749	0.0253874	+	0.999678i	$0.491918\pi$
$258$	0	0
$259$	−2.17336	−0.135046
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−22.0140	−1.35744	−0.678720	−	0.734398i	$-0.737465\pi$
$263$	−22.0140	−1.35744	−0.678720	+	0.734398i	$0.737465\pi$
$264$	0	0
$265$	8.05387	0.494745
$266$	0	0
$267$	0	0
$268$	0	0
$269$	6.77367	0.412998	0.206499	−	0.978447i	$-0.433793\pi$
$269$	6.77367	0.412998	0.206499	+	0.978447i	$0.433793\pi$
$270$	0	0
$271$	−15.3341	−0.931479	−0.465739	−	0.884922i	$-0.654212\pi$
$271$	−15.3341	−0.931479	−0.465739	+	0.884922i	$0.654212\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	9.51570	0.573818
$276$	0	0
$277$	−20.6712	−1.24201	−0.621007	−	0.783805i	$-0.713276\pi$
$277$	−20.6712	−1.24201	−0.621007	+	0.783805i	$0.713276\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−28.8925	−1.72358	−0.861792	−	0.507262i	$-0.830658\pi$
$281$	−28.8925	−1.72358	−0.861792	+	0.507262i	$0.830658\pi$
$282$	0	0
$283$	−6.98735	−0.415355	−0.207677	−	0.978197i	$-0.566590\pi$
$283$	−6.98735	−0.415355	−0.207677	+	0.978197i	$0.566590\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−4.98735	−0.294394
$288$	0	0
$289$	37.4423	2.20249
$290$	0	0
$291$	0	0
$292$	0	0
$293$	1.94613	0.113694	0.0568470	−	0.998383i	$-0.481895\pi$
$293$	1.94613	0.113694	0.0568470	+	0.998383i	$0.481895\pi$
$294$	0	0
$295$	23.5282	1.36987
$296$	0	0
$297$	0	0
$298$	0	0
$299$	5.90170	0.341304
$300$	0	0
$301$	−2.32864	−0.134221
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−0.234860	−0.0134480
$306$	0	0
$307$	−4.24636	−0.242353	−0.121176	−	0.992631i	$-0.538667\pi$
$307$	−4.24636	−0.242353	−0.121176	+	0.992631i	$0.538667\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−4.20709	−0.238562	−0.119281	−	0.992861i	$-0.538059\pi$
$311$	−4.20709	−0.238562	−0.119281	+	0.992861i	$0.538059\pi$
$312$	0	0
$313$	−5.08913	−0.287655	−0.143827	−	0.989603i	$-0.545941\pi$
$313$	−5.08913	−0.287655	−0.143827	+	0.989603i	$0.545941\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−9.29790	−0.522222	−0.261111	−	0.965309i	$-0.584089\pi$
$317$	−9.29790	−0.522222	−0.261111	+	0.965309i	$0.584089\pi$
$318$	0	0
$319$	12.3029	0.688831
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−47.6958	−2.65387
$324$	0	0
$325$	2.08565	0.115691
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−12.3266	−0.679586
$330$	0	0
$331$	−23.5981	−1.29707	−0.648535	−	0.761184i	$-0.724618\pi$
$331$	−23.5981	−1.29707	−0.648535	+	0.761184i	$0.724618\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−13.5926	−0.742641
$336$	0	0
$337$	4.87650	0.265640	0.132820	−	0.991140i	$-0.457597\pi$
$337$	4.87650	0.265640	0.132820	+	0.991140i	$0.457597\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−2.38704	−0.129265
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	33.1208	1.77802	0.889009	−	0.457890i	$-0.151395\pi$
$347$	33.1208	1.77802	0.889009	+	0.457890i	$0.151395\pi$
$348$	0	0
$349$	0.589188	0.0315385	0.0157693	−	0.999876i	$-0.494980\pi$
$349$	0.589188	0.0315385	0.0157693	+	0.999876i	$0.494980\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	3.08707	0.164308	0.0821541	−	0.996620i	$-0.473820\pi$
$353$	3.08707	0.164308	0.0821541	+	0.996620i	$0.473820\pi$
$354$	0	0
$355$	23.6255	1.25391
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−18.5750	−0.980351	−0.490176	−	0.871624i	$-0.663067\pi$
$359$	−18.5750	−0.980351	−0.490176	+	0.871624i	$0.663067\pi$
$360$	0	0
$361$	22.7853	1.19923
$362$	0	0
$363$	0	0
$364$	0	0
$365$	4.30746	0.225463
$366$	0	0
$367$	−25.3744	−1.32453	−0.662266	−	0.749269i	$-0.730405\pi$
$367$	−25.3744	−1.32453	−0.662266	+	0.749269i	$0.730405\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−4.71774	−0.244933
$372$	0	0
$373$	−18.5392	−0.959924	−0.479962	−	0.877289i	$-0.659350\pi$
$373$	−18.5392	−0.959924	−0.479962	+	0.877289i	$0.659350\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	2.69655	0.138880
$378$	0	0
$379$	1.76064	0.0904380	0.0452190	−	0.998977i	$-0.485601\pi$
$379$	1.76064	0.0904380	0.0452190	+	0.998977i	$0.485601\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−4.50136	−0.230009	−0.115004	−	0.993365i	$-0.536688\pi$
$383$	−4.50136	−0.230009	−0.115004	+	0.993365i	$0.536688\pi$
$384$	0	0
$385$	7.78879	0.396953
$386$	0	0
$387$	0	0
$388$	0	0
$389$	27.9817	1.41873	0.709363	−	0.704843i	$-0.248983\pi$
$389$	27.9817	1.41873	0.709363	+	0.704843i	$0.248983\pi$
$390$	0	0
$391$	43.5457	2.20220
$392$	0	0
$393$	0	0
$394$	0	0
$395$	8.28873	0.417051
$396$	0	0
$397$	3.64979	0.183178	0.0915889	−	0.995797i	$-0.470805\pi$
$397$	3.64979	0.183178	0.0915889	+	0.995797i	$0.470805\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−19.3174	−0.964665	−0.482333	−	0.875988i	$-0.660210\pi$
$401$	−19.3174	−0.964665	−0.482333	+	0.875988i	$0.660210\pi$
$402$	0	0
$403$	−0.523191	−0.0260620
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−9.91588	−0.491512
$408$	0	0
$409$	8.73234	0.431787	0.215893	−	0.976417i	$-0.430734\pi$
$409$	8.73234	0.431787	0.215893	+	0.976417i	$0.430734\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−13.7822	−0.678178
$414$	0	0
$415$	12.8216	0.629387
$416$	0	0
$417$	0	0
$418$	0	0
$419$	9.92636	0.484935	0.242467	−	0.970160i	$-0.422043\pi$
$419$	9.92636	0.484935	0.242467	+	0.970160i	$0.422043\pi$
$420$	0	0
$421$	12.4697	0.607736	0.303868	−	0.952714i	$-0.401722\pi$
$421$	12.4697	0.607736	0.303868	+	0.952714i	$0.401722\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	15.3890	0.746475
$426$	0	0
$427$	0.137574	0.00665769
$428$	0	0
$429$	0	0
$430$	0	0
$431$	0.154238	0.00742937	0.00371469	−	0.999993i	$-0.498818\pi$
$431$	0.154238	0.00742937	0.00371469	+	0.999993i	$0.498818\pi$
$432$	0	0
$433$	33.6853	1.61881	0.809407	−	0.587249i	$-0.199789\pi$
$433$	33.6853	1.61881	0.809407	+	0.587249i	$0.199789\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−38.1495	−1.82494
$438$	0	0
$439$	−14.5604	−0.694930	−0.347465	−	0.937693i	$-0.612957\pi$
$439$	−14.5604	−0.694930	−0.347465	+	0.937693i	$0.612957\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	9.82147	0.466632	0.233316	−	0.972401i	$-0.425042\pi$
$443$	9.82147	0.466632	0.233316	+	0.972401i	$0.425042\pi$
$444$	0	0
$445$	18.4919	0.876601
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−37.6137	−1.77510	−0.887551	−	0.460709i	$-0.847595\pi$
$449$	−37.6137	−1.77510	−0.887551	+	0.460709i	$0.847595\pi$
$450$	0	0
$451$	−22.7546	−1.07147
$452$	0	0
$453$	0	0
$454$	0	0
$455$	1.70715	0.0800323
$456$	0	0
$457$	22.0821	1.03296	0.516478	−	0.856301i	$-0.327243\pi$
$457$	22.0821	1.03296	0.516478	+	0.856301i	$0.327243\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	40.2881	1.87640	0.938202	−	0.346089i	$-0.112490\pi$
$461$	40.2881	1.87640	0.938202	+	0.346089i	$0.112490\pi$
$462$	0	0
$463$	−9.60432	−0.446351	−0.223175	−	0.974778i	$-0.571642\pi$
$463$	−9.60432	−0.446351	−0.223175	+	0.974778i	$0.571642\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−19.0822	−0.883018	−0.441509	−	0.897257i	$-0.645557\pi$
$467$	−19.0822	−0.883018	−0.441509	+	0.897257i	$0.645557\pi$
$468$	0	0
$469$	7.96215	0.367658
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−10.6243	−0.488508
$474$	0	0
$475$	−13.4820	−0.618595
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−0.825105	−0.0377000	−0.0188500	−	0.999822i	$-0.506000\pi$
$479$	−0.825105	−0.0377000	−0.0188500	+	0.999822i	$0.506000\pi$
$480$	0	0
$481$	−2.17336	−0.0990968
$482$	0	0
$483$	0	0
$484$	0	0
$485$	30.1641	1.36968
$486$	0	0
$487$	41.3466	1.87359	0.936797	−	0.349874i	$-0.113775\pi$
$487$	41.3466	1.87359	0.936797	+	0.349874i	$0.113775\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	40.9283	1.84707	0.923534	−	0.383516i	$-0.125287\pi$
$491$	40.9283	1.84707	0.923534	+	0.383516i	$0.125287\pi$
$492$	0	0
$493$	19.8965	0.896095
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−13.8392	−0.620772
$498$	0	0
$499$	12.8306	0.574379	0.287189	−	0.957874i	$-0.407279\pi$
$499$	12.8306	0.574379	0.287189	+	0.957874i	$0.407279\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	17.3931	0.775520	0.387760	−	0.921760i	$-0.373249\pi$
$503$	17.3931	0.775520	0.387760	+	0.921760i	$0.373249\pi$
$504$	0	0
$505$	−31.0176	−1.38026
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−7.19608	−0.318961	−0.159480	−	0.987201i	$-0.550982\pi$
$509$	−7.19608	−0.318961	−0.159480	+	0.987201i	$0.550982\pi$
$510$	0	0
$511$	−2.52319	−0.111619
$512$	0	0
$513$	0	0
$514$	0	0
$515$	25.8566	1.13938
$516$	0	0
$517$	−56.2395	−2.47341
$518$	0	0
$519$	0	0
$520$	0	0
$521$	24.9495	1.09306	0.546529	−	0.837440i	$-0.315949\pi$
$521$	24.9495	1.09306	0.546529	+	0.837440i	$0.315949\pi$
$522$	0	0
$523$	25.5901	1.11898	0.559489	−	0.828838i	$-0.310997\pi$
$523$	25.5901	1.11898	0.559489	+	0.828838i	$0.310997\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−3.86036	−0.168160
$528$	0	0
$529$	11.8300	0.514348
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−4.98735	−0.216026
$534$	0	0
$535$	−1.12208	−0.0485117
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−4.56246	−0.196519
$540$	0	0
$541$	25.5457	1.09829	0.549147	−	0.835726i	$-0.314953\pi$
$541$	25.5457	1.09829	0.549147	+	0.835726i	$0.314953\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	13.2604	0.568015
$546$	0	0
$547$	21.3567	0.913146	0.456573	−	0.889686i	$-0.349077\pi$
$547$	21.3567	0.913146	0.456573	+	0.889686i	$0.349077\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−17.4309	−0.742583
$552$	0	0
$553$	−4.85531	−0.206469
$554$	0	0
$555$	0	0
$556$	0	0
$557$	0.651705	0.0276136	0.0138068	−	0.999905i	$-0.495605\pi$
$557$	0.651705	0.0276136	0.0138068	+	0.999905i	$0.495605\pi$
$558$	0	0
$559$	−2.32864	−0.0984911
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−12.5479	−0.528829	−0.264415	−	0.964409i	$-0.585179\pi$
$563$	−12.5479	−0.528829	−0.264415	+	0.964409i	$0.585179\pi$
$564$	0	0
$565$	−8.32490	−0.350231
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−20.0983	−0.842565	−0.421282	−	0.906930i	$-0.638420\pi$
$569$	−20.0983	−0.842565	−0.421282	+	0.906930i	$0.638420\pi$
$570$	0	0
$571$	−14.2953	−0.598240	−0.299120	−	0.954215i	$-0.596693\pi$
$571$	−14.2953	−0.598240	−0.299120	+	0.954215i	$0.596693\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	12.3089	0.513316
$576$	0	0
$577$	−1.26418	−0.0526284	−0.0263142	−	0.999654i	$-0.508377\pi$
$577$	−1.26418	−0.0526284	−0.0263142	+	0.999654i	$0.508377\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−7.51054	−0.311590
$582$	0	0
$583$	−21.5245	−0.891454
$584$	0	0
$585$	0	0
$586$	0	0
$587$	28.8210	1.18957	0.594785	−	0.803885i	$-0.297237\pi$
$587$	28.8210	1.18957	0.594785	+	0.803885i	$0.297237\pi$
$588$	0	0
$589$	3.38199	0.139352
$590$	0	0
$591$	0	0
$592$	0	0
$593$	19.7534	0.811176	0.405588	−	0.914056i	$-0.367067\pi$
$593$	19.7534	0.811176	0.405588	+	0.914056i	$0.367067\pi$
$594$	0	0
$595$	12.5962	0.516393
$596$	0	0
$597$	0	0
$598$	0	0
$599$	14.1727	0.579082	0.289541	−	0.957166i	$-0.406497\pi$
$599$	14.1727	0.579082	0.289541	+	0.957166i	$0.406497\pi$
$600$	0	0
$601$	−36.2922	−1.48039	−0.740195	−	0.672392i	$-0.765267\pi$
$601$	−36.2922	−1.48039	−0.740195	+	0.672392i	$0.765267\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	16.7574	0.681286
$606$	0	0
$607$	41.5780	1.68760	0.843799	−	0.536660i	$-0.180314\pi$
$607$	41.5780	1.68760	0.843799	+	0.536660i	$0.180314\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−12.3266	−0.498680
$612$	0	0
$613$	−3.98334	−0.160885	−0.0804427	−	0.996759i	$-0.525633\pi$
$613$	−3.98334	−0.160885	−0.0804427	+	0.996759i	$0.525633\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	33.5644	1.35125	0.675626	−	0.737244i	$-0.263873\pi$
$617$	33.5644	1.35125	0.675626	+	0.737244i	$0.263873\pi$
$618$	0	0
$619$	6.75289	0.271421	0.135711	−	0.990749i	$-0.456668\pi$
$619$	6.75289	0.271421	0.135711	+	0.990749i	$0.456668\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−10.8321	−0.433978
$624$	0	0
$625$	−10.2218	−0.408872
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−16.0362	−0.639404
$630$	0	0
$631$	6.05323	0.240975	0.120488	−	0.992715i	$-0.461554\pi$
$631$	6.05323	0.240975	0.120488	+	0.992715i	$0.461554\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	19.8791	0.788878
$636$	0	0
$637$	−1.00000	−0.0396214
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−34.1807	−1.35006	−0.675029	−	0.737791i	$-0.735869\pi$
$641$	−34.1807	−1.35006	−0.675029	+	0.737791i	$0.735869\pi$
$642$	0	0
$643$	−37.1500	−1.46505	−0.732527	−	0.680738i	$-0.761659\pi$
$643$	−37.1500	−1.46505	−0.732527	+	0.680738i	$0.761659\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	11.7172	0.460651	0.230326	−	0.973114i	$-0.426021\pi$
$647$	11.7172	0.460651	0.230326	+	0.973114i	$0.426021\pi$
$648$	0	0
$649$	−62.8808	−2.46829
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−28.7529	−1.12519	−0.562594	−	0.826734i	$-0.690196\pi$
$653$	−28.7529	−1.12519	−0.562594	+	0.826734i	$0.690196\pi$
$654$	0	0
$655$	26.4485	1.03343
$656$	0	0
$657$	0	0
$658$	0	0
$659$	25.2740	0.984536	0.492268	−	0.870444i	$-0.336168\pi$
$659$	25.2740	0.984536	0.492268	+	0.870444i	$0.336168\pi$
$660$	0	0
$661$	32.8491	1.27768	0.638842	−	0.769338i	$-0.279414\pi$
$661$	32.8491	1.27768	0.638842	+	0.769338i	$0.279414\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−11.0353	−0.427929
$666$	0	0
$667$	15.9142	0.616202
$668$	0	0
$669$	0	0
$670$	0	0
$671$	0.627678	0.0242312
$672$	0	0
$673$	47.5870	1.83434	0.917172	−	0.398491i	$-0.130466\pi$
$673$	47.5870	1.83434	0.917172	+	0.398491i	$0.130466\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−33.5481	−1.28936	−0.644680	−	0.764453i	$-0.723009\pi$
$677$	−33.5481	−1.28936	−0.644680	+	0.764453i	$0.723009\pi$
$678$	0	0
$679$	−17.6693	−0.678085
$680$	0	0
$681$	0	0
$682$	0	0
$683$	29.7908	1.13992	0.569958	−	0.821674i	$-0.306960\pi$
$683$	29.7908	1.13992	0.569958	+	0.821674i	$0.306960\pi$
$684$	0	0
$685$	−8.81009	−0.336616
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−4.71774	−0.179732
$690$	0	0
$691$	−15.1959	−0.578078	−0.289039	−	0.957317i	$-0.593336\pi$
$691$	−15.1959	−0.578078	−0.289039	+	0.957317i	$0.593336\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−16.6747	−0.632508
$696$	0	0
$697$	−36.7992	−1.39387
$698$	0	0
$699$	0	0
$700$	0	0
$701$	28.2106	1.06550	0.532749	−	0.846273i	$-0.321159\pi$
$701$	28.2106	1.06550	0.532749	+	0.846273i	$0.321159\pi$
$702$	0	0
$703$	14.0490	0.529866
$704$	0	0
$705$	0	0
$706$	0	0
$707$	18.1692	0.683325
$708$	0	0
$709$	21.8546	0.820765	0.410383	−	0.911913i	$-0.365395\pi$
$709$	21.8546	0.820765	0.410383	+	0.911913i	$0.365395\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−3.08771	−0.115636
$714$	0	0
$715$	7.78879	0.291284
$716$	0	0
$717$	0	0
$718$	0	0
$719$	45.2563	1.68778	0.843888	−	0.536520i	$-0.180261\pi$
$719$	45.2563	1.68778	0.843888	+	0.536520i	$0.180261\pi$
$720$	0	0
$721$	−15.1461	−0.564070
$722$	0	0
$723$	0	0
$724$	0	0
$725$	5.62407	0.208873
$726$	0	0
$727$	44.8818	1.66457	0.832287	−	0.554345i	$-0.187031\pi$
$727$	44.8818	1.66457	0.832287	+	0.554345i	$0.187031\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−17.1819	−0.635495
$732$	0	0
$733$	42.0205	1.55206	0.776032	−	0.630693i	$-0.217229\pi$
$733$	42.0205	1.55206	0.776032	+	0.630693i	$0.217229\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	36.3270	1.33812
$738$	0	0
$739$	29.2958	1.07767	0.538833	−	0.842413i	$-0.318866\pi$
$739$	29.2958	1.07767	0.538833	+	0.842413i	$0.318866\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−35.1128	−1.28816	−0.644081	−	0.764957i	$-0.722760\pi$
$743$	−35.1128	−1.28816	−0.644081	+	0.764957i	$0.722760\pi$
$744$	0	0
$745$	−14.3179	−0.524567
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0.657284	0.0240166
$750$	0	0
$751$	−17.8128	−0.649999	−0.325000	−	0.945714i	$-0.605364\pi$
$751$	−17.8128	−0.649999	−0.325000	+	0.945714i	$0.605364\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−21.3813	−0.778147
$756$	0	0
$757$	−25.8850	−0.940807	−0.470404	−	0.882451i	$-0.655892\pi$
$757$	−25.8850	−0.940807	−0.470404	+	0.882451i	$0.655892\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−27.3306	−0.990733	−0.495367	−	0.868684i	$-0.664966\pi$
$761$	−27.3306	−0.990733	−0.495367	+	0.868684i	$0.664966\pi$
$762$	0	0
$763$	−7.76760	−0.281206
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−13.7822	−0.497647
$768$	0	0
$769$	−25.5192	−0.920245	−0.460123	−	0.887855i	$-0.652195\pi$
$769$	−25.5192	−0.920245	−0.460123	+	0.887855i	$0.652195\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	3.62797	0.130489	0.0652444	−	0.997869i	$-0.479217\pi$
$773$	3.62797	0.130489	0.0652444	+	0.997869i	$0.479217\pi$
$774$	0	0
$775$	−1.09119	−0.0391968
$776$	0	0
$777$	0	0
$778$	0	0
$779$	32.2390	1.15508
$780$	0	0
$781$	−63.1407	−2.25935
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−8.87470	−0.316752
$786$	0	0
$787$	13.3886	0.477251	0.238625	−	0.971112i	$-0.423303\pi$
$787$	13.3886	0.477251	0.238625	+	0.971112i	$0.423303\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	4.87650	0.173388
$792$	0	0
$793$	0.137574	0.00488541
$794$	0	0
$795$	0	0
$796$	0	0
$797$	5.95519	0.210944	0.105472	−	0.994422i	$-0.466365\pi$
$797$	5.95519	0.210944	0.105472	+	0.994422i	$0.466365\pi$
$798$	0	0
$799$	−90.9517	−3.21764
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−11.5120	−0.406248
$804$	0	0
$805$	10.0751	0.355099
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−39.3205	−1.38244	−0.691218	−	0.722647i	$-0.742925\pi$
$809$	−39.3205	−1.38244	−0.691218	+	0.722647i	$0.742925\pi$
$810$	0	0
$811$	−40.9353	−1.43743	−0.718716	−	0.695304i	$-0.755270\pi$
$811$	−40.9353	−1.43743	−0.718716	+	0.695304i	$0.755270\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	28.7650	1.00759
$816$	0	0
$817$	15.0527	0.526627
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−12.7994	−0.446701	−0.223351	−	0.974738i	$-0.571700\pi$
$821$	−12.7994	−0.446701	−0.223351	+	0.974738i	$0.571700\pi$
$822$	0	0
$823$	−2.76153	−0.0962609	−0.0481305	−	0.998841i	$-0.515326\pi$
$823$	−2.76153	−0.0962609	−0.0481305	+	0.998841i	$0.515326\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−7.37813	−0.256563	−0.128281	−	0.991738i	$-0.540946\pi$
$827$	−7.37813	−0.256563	−0.128281	+	0.991738i	$0.540946\pi$
$828$	0	0
$829$	39.5245	1.37274	0.686371	−	0.727251i	$-0.259203\pi$
$829$	39.5245	1.37274	0.686371	+	0.727251i	$0.259203\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−7.37850	−0.255650
$834$	0	0
$835$	37.8918	1.31130
$836$	0	0
$837$	0	0
$838$	0	0
$839$	13.5123	0.466498	0.233249	−	0.972417i	$-0.425064\pi$
$839$	13.5123	0.466498	0.233249	+	0.972417i	$0.425064\pi$
$840$	0	0
$841$	−21.7286	−0.749262
$842$	0	0
$843$	0	0
$844$	0	0
$845$	1.70715	0.0587276
$846$	0	0
$847$	−9.81604	−0.337283
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−12.8265	−0.439688
$852$	0	0
$853$	6.40680	0.219365	0.109682	−	0.993967i	$-0.465017\pi$
$853$	6.40680	0.219365	0.109682	+	0.993967i	$0.465017\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−27.5856	−0.942306	−0.471153	−	0.882052i	$-0.656162\pi$
$857$	−27.5856	−0.942306	−0.471153	+	0.882052i	$0.656162\pi$
$858$	0	0
$859$	26.9943	0.921034	0.460517	−	0.887651i	$-0.347664\pi$
$859$	26.9943	0.921034	0.460517	+	0.887651i	$0.347664\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−29.1208	−0.991284	−0.495642	−	0.868527i	$-0.665067\pi$
$863$	−29.1208	−0.991284	−0.495642	+	0.868527i	$0.665067\pi$
$864$	0	0
$865$	11.9319	0.405698
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−22.1522	−0.751461
$870$	0	0
$871$	7.96215	0.269787
$872$	0	0
$873$	0	0
$874$	0	0
$875$	12.0962	0.408928
$876$	0	0
$877$	12.3126	0.415768	0.207884	−	0.978154i	$-0.433342\pi$
$877$	12.3126	0.415768	0.207884	+	0.978154i	$0.433342\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−7.93644	−0.267386	−0.133693	−	0.991023i	$-0.542684\pi$
$881$	−7.93644	−0.267386	−0.133693	+	0.991023i	$0.542684\pi$
$882$	0	0
$883$	−18.1672	−0.611374	−0.305687	−	0.952132i	$-0.598886\pi$
$883$	−18.1672	−0.611374	−0.305687	+	0.952132i	$0.598886\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	7.17531	0.240923	0.120462	−	0.992718i	$-0.461563\pi$
$887$	7.17531	0.240923	0.120462	+	0.992718i	$0.461563\pi$
$888$	0	0
$889$	−11.6446	−0.390548
$890$	0	0
$891$	0	0
$892$	0	0
$893$	79.6809	2.66642
$894$	0	0
$895$	−24.2595	−0.810907
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−1.41081	−0.0470532
$900$	0	0
$901$	−34.8099	−1.15968
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−32.3531	−1.07545
$906$	0	0
$907$	48.8890	1.62333	0.811667	−	0.584120i	$-0.198560\pi$
$907$	48.8890	1.62333	0.811667	+	0.584120i	$0.198560\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	42.2780	1.40073	0.700367	−	0.713783i	$-0.253020\pi$
$911$	42.2780	1.40073	0.700367	+	0.713783i	$0.253020\pi$
$912$	0	0
$913$	−34.2665	−1.13406
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−15.4928	−0.511618
$918$	0	0
$919$	24.7334	0.815879	0.407940	−	0.913009i	$-0.366247\pi$
$919$	24.7334	0.815879	0.407940	+	0.913009i	$0.366247\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−13.8392	−0.455522
$924$	0	0
$925$	−4.53288	−0.149040
$926$	0	0
$927$	0	0
$928$	0	0
$929$	17.1260	0.561885	0.280942	−	0.959725i	$-0.409353\pi$
$929$	17.1260	0.561885	0.280942	+	0.959725i	$0.409353\pi$
$930$	0	0
$931$	6.46416	0.211854
$932$	0	0
$933$	0	0
$934$	0	0
$935$	57.4696	1.87946
$936$	0	0
$937$	−17.8221	−0.582223	−0.291112	−	0.956689i	$-0.594025\pi$
$937$	−17.8221	−0.582223	−0.291112	+	0.956689i	$0.594025\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	32.2675	1.05189	0.525946	−	0.850518i	$-0.323711\pi$
$941$	32.2675	1.05189	0.525946	+	0.850518i	$0.323711\pi$
$942$	0	0
$943$	−29.4338	−0.958496
$944$	0	0
$945$	0	0
$946$	0	0
$947$	19.4143	0.630880	0.315440	−	0.948946i	$-0.397848\pi$
$947$	19.4143	0.630880	0.315440	+	0.948946i	$0.397848\pi$
$948$	0	0
$949$	−2.52319	−0.0819062
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−30.0644	−0.973880	−0.486940	−	0.873435i	$-0.661887\pi$
$953$	−30.0644	−0.973880	−0.486940	+	0.873435i	$0.661887\pi$
$954$	0	0
$955$	−5.70651	−0.184658
$956$	0	0
$957$	0	0
$958$	0	0
$959$	5.16071	0.166648
$960$	0	0
$961$	−30.7263	−0.991170
$962$	0	0
$963$	0	0
$964$	0	0
$965$	5.33470	0.171730
$966$	0	0
$967$	20.4288	0.656945	0.328473	−	0.944513i	$-0.393466\pi$
$967$	20.4288	0.656945	0.328473	+	0.944513i	$0.393466\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−47.3661	−1.52005	−0.760026	−	0.649893i	$-0.774814\pi$
$971$	−47.3661	−1.52005	−0.760026	+	0.649893i	$0.774814\pi$
$972$	0	0
$973$	9.76760	0.313135
$974$	0	0
$975$	0	0
$976$	0	0
$977$	31.8816	1.01998	0.509991	−	0.860180i	$-0.329649\pi$
$977$	31.8816	1.01998	0.509991	+	0.860180i	$0.329649\pi$
$978$	0	0
$979$	−49.4209	−1.57950
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−55.3529	−1.76548	−0.882742	−	0.469858i	$-0.844305\pi$
$983$	−55.3529	−1.76548	−0.882742	+	0.469858i	$0.844305\pi$
$984$	0	0
$985$	40.6074	1.29386
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−13.7429	−0.437000
$990$	0	0
$991$	19.9622	0.634119	0.317059	−	0.948406i	$-0.397305\pi$
$991$	19.9622	0.634119	0.317059	+	0.948406i	$0.397305\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	2.68895	0.0852455
$996$	0	0
$997$	54.7905	1.73523	0.867616	−	0.497234i	$-0.165651\pi$
$997$	54.7905	1.73523	0.867616	+	0.497234i	$0.165651\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6552.2.a.br.1.3		4
3.2	odd	2		728.2.a.i.1.3	✓	4
12.11	even	2		1456.2.a.v.1.2		4
21.20	even	2		5096.2.a.s.1.2		4
24.5	odd	2		5824.2.a.cb.1.2		4
24.11	even	2		5824.2.a.ce.1.3		4
39.38	odd	2		9464.2.a.z.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
728.2.a.i.1.3	✓	4	3.2	odd	2
1456.2.a.v.1.2		4	12.11	even	2
5096.2.a.s.1.2		4	21.20	even	2
5824.2.a.cb.1.2		4	24.5	odd	2
5824.2.a.ce.1.3		4	24.11	even	2
6552.2.a.br.1.3		4	1.1	even	1	trivial
9464.2.a.z.1.3		4	39.38	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 6552 = 2^{3} \cdot 3^{2} \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6552.a (trivial)

\(f(q)\)	\(=\)	\(q+1.70715 q^{5} -1.00000 q^{7} -4.56246 q^{11} -1.00000 q^{13} -7.37850 q^{17} +6.46416 q^{19} -5.90170 q^{23} -2.08565 q^{25} -2.69655 q^{29} +0.523191 q^{31} -1.70715 q^{35} +2.17336 q^{37} +4.98735 q^{41} +2.32864 q^{43} +12.3266 q^{47} +1.00000 q^{49} +4.71774 q^{53} -7.78879 q^{55} +13.7822 q^{59} -0.137574 q^{61} -1.70715 q^{65} -7.96215 q^{67} +13.8392 q^{71} +2.52319 q^{73} +4.56246 q^{77} +4.85531 q^{79} +7.51054 q^{83} -12.5962 q^{85} +10.8321 q^{89} +1.00000 q^{91} +11.0353 q^{95} +17.6693 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{5} - 4 q^{7} + q^{11} - 4 q^{13} - 16 q^{17} - 10 q^{19} - 7 q^{23} + 14 q^{25} - 4 q^{29} - q^{31} + 2 q^{35} + 5 q^{37} - 19 q^{41} + 14 q^{43} + 13 q^{47} + 4 q^{49} + 8 q^{53} + 2 q^{55}+ \cdots + 25 q^{97}+O(q^{100}) \) 4 * q - 2 * q^5 - 4 * q^7 + q^11 - 4 * q^13 - 16 * q^17 - 10 * q^19 - 7 * q^23 + 14 * q^25 - 4 * q^29 - q^31 + 2 * q^35 + 5 * q^37 - 19 * q^41 + 14 * q^43 + 13 * q^47 + 4 * q^49 + 8 * q^53 + 2 * q^55 + 26 * q^59 - q^61 + 2 * q^65 + 5 * q^67 + 18 * q^71 + 7 * q^73 - q^77 + 9 * q^79 - 12 * q^83 + 14 * q^85 - 4 * q^89 + 4 * q^91 + 26 * q^95 + 25 * q^97

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