LMFDB - Embedded newform 4536.2.a.y.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4536,2,Mod(1,4536)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4536, 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("4536.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4536 = 2^{3} \cdot 3^{4} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4536.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:	$36.2201423569$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.45729.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 11x^{2} + 12x - 3 $ x^4 - x^3 - 11x^2 + 12x - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 504)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.670984$ of defining polynomial
Character	$\chi$	$=$	4536.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.329016 q^{5} -1.00000 q^{7} +O(q^{10})$	$q-0.329016 q^{5} -1.00000 q^{7} +1.32902 q^{11} +3.07874 q^{13} -7.35741 q^{17} -2.93671 q^{19} +6.68643 q^{23} -4.89175 q^{25} +7.76516 q^{29} -3.27110 q^{31} +0.329016 q^{35} +0.329016 q^{37} -0.271104 q^{41} -10.9651 q^{43} -1.14203 q^{47} +1.00000 q^{49} -6.42828 q^{53} -0.437267 q^{55} -0.744341 q^{59} +8.84390 q^{61} -1.01295 q^{65} +8.57280 q^{67} -1.60769 q^{71} -13.4941 q^{73} -1.32902 q^{77} -1.25785 q^{79} +0.0632919 q^{83} +2.42070 q^{85} -11.3071 q^{89} -3.07874 q^{91} +0.966223 q^{95} -11.0284 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 3 q^{5} - 4 q^{7}+O(q^{10}) $ 4 * q - 3 * q^5 - 4 * q^7	$ 4 q - 3 q^{5} - 4 q^{7} + 7 q^{11} - 3 q^{13} - 3 q^{17} - 4 q^{19} + 2 q^{23} + 5 q^{25} - 9 q^{29} - 3 q^{31} + 3 q^{35} + 3 q^{37} + 9 q^{41} - 8 q^{43} + 3 q^{47} + 4 q^{49} - 6 q^{53} - 28 q^{55} + 10 q^{59} - 20 q^{61} + q^{65} - 11 q^{67} + 3 q^{71} - 24 q^{73} - 7 q^{77} - 21 q^{79} + 8 q^{83} - 9 q^{85} - 6 q^{89} + 3 q^{91} + 36 q^{95} - 16 q^{97}+O(q^{100}) $ 4 * q - 3 * q^5 - 4 * q^7 + 7 * q^11 - 3 * q^13 - 3 * q^17 - 4 * q^19 + 2 * q^23 + 5 * q^25 - 9 * q^29 - 3 * q^31 + 3 * q^35 + 3 * q^37 + 9 * q^41 - 8 * q^43 + 3 * q^47 + 4 * q^49 - 6 * q^53 - 28 * q^55 + 10 * q^59 - 20 * q^61 + q^65 - 11 * q^67 + 3 * q^71 - 24 * q^73 - 7 * q^77 - 21 * q^79 + 8 * q^83 - 9 * q^85 - 6 * q^89 + 3 * q^91 + 36 * q^95 - 16 * 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
$5$	−0.329016	−0.147140	−0.0735702	−	0.997290i	$-0.523439\pi$
$5$	−0.329016	−0.147140	−0.0735702	+	0.997290i	$0.523439\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	1.32902	0.400713	0.200357	−	0.979723i	$-0.435790\pi$
$11$	1.32902	0.400713	0.200357	+	0.979723i	$0.435790\pi$
$12$	0	0
$13$	3.07874	0.853888	0.426944	−	0.904278i	$-0.359590\pi$
$13$	3.07874	0.853888	0.426944	+	0.904278i	$0.359590\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−7.35741	−1.78443	−0.892217	−	0.451606i	$-0.850851\pi$
$17$	−7.35741	−1.78443	−0.892217	+	0.451606i	$0.850851\pi$
$18$	0	0
$19$	−2.93671	−0.673727	−0.336864	−	0.941553i	$-0.609366\pi$
$19$	−2.93671	−0.673727	−0.336864	+	0.941553i	$0.609366\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	6.68643	1.39422	0.697108	−	0.716966i	$-0.254470\pi$
$23$	6.68643	1.39422	0.697108	+	0.716966i	$0.254470\pi$
$24$	0	0
$25$	−4.89175	−0.978350
$26$	0	0
$27$	0	0
$28$	0	0
$29$	7.76516	1.44195	0.720977	−	0.692959i	$-0.243693\pi$
$29$	7.76516	1.44195	0.720977	+	0.692959i	$0.243693\pi$
$30$	0	0
$31$	−3.27110	−0.587508	−0.293754	−	0.955881i	$-0.594905\pi$
$31$	−3.27110	−0.587508	−0.293754	+	0.955881i	$0.594905\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0.329016	0.0556138
$36$	0	0
$37$	0.329016	0.0540899	0.0270449	−	0.999634i	$-0.491390\pi$
$37$	0.329016	0.0540899	0.0270449	+	0.999634i	$0.491390\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−0.271104	−0.0423393	−0.0211696	−	0.999776i	$-0.506739\pi$
$41$	−0.271104	−0.0423393	−0.0211696	+	0.999776i	$0.506739\pi$
$42$	0	0
$43$	−10.9651	−1.67216	−0.836081	−	0.548605i	$-0.815159\pi$
$43$	−10.9651	−1.67216	−0.836081	+	0.548605i	$0.815159\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−1.14203	−0.166582	−0.0832910	−	0.996525i	$-0.526543\pi$
$47$	−1.14203	−0.166582	−0.0832910	+	0.996525i	$0.526543\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−6.42828	−0.882992	−0.441496	−	0.897263i	$-0.645552\pi$
$53$	−6.42828	−0.882992	−0.441496	+	0.897263i	$0.645552\pi$
$54$	0	0
$55$	−0.437267	−0.0589611
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−0.744341	−0.0969049	−0.0484525	−	0.998825i	$-0.515429\pi$
$59$	−0.744341	−0.0969049	−0.0484525	+	0.998825i	$0.515429\pi$
$60$	0	0
$61$	8.84390	1.13235	0.566173	−	0.824287i	$-0.308424\pi$
$61$	8.84390	1.13235	0.566173	+	0.824287i	$0.308424\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−1.01295	−0.125641
$66$	0	0
$67$	8.57280	1.04733	0.523667	−	0.851923i	$-0.324564\pi$
$67$	8.57280	1.04733	0.523667	+	0.851923i	$0.324564\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−1.60769	−0.190798	−0.0953990	−	0.995439i	$-0.530413\pi$
$71$	−1.60769	−0.190798	−0.0953990	+	0.995439i	$0.530413\pi$
$72$	0	0
$73$	−13.4941	−1.57936	−0.789680	−	0.613519i	$-0.789754\pi$
$73$	−13.4941	−1.57936	−0.789680	+	0.613519i	$0.789754\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−1.32902	−0.151455
$78$	0	0
$79$	−1.25785	−0.141519	−0.0707597	−	0.997493i	$-0.522542\pi$
$79$	−1.25785	−0.141519	−0.0707597	+	0.997493i	$0.522542\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	0.0632919	0.00694718	0.00347359	−	0.999994i	$-0.498894\pi$
$83$	0.0632919	0.00694718	0.00347359	+	0.999994i	$0.498894\pi$
$84$	0	0
$85$	2.42070	0.262562
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−11.3071	−1.19855	−0.599274	−	0.800544i	$-0.704544\pi$
$89$	−11.3071	−1.19855	−0.599274	+	0.800544i	$0.704544\pi$
$90$	0	0
$91$	−3.07874	−0.322739
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0.966223	0.0991324
$96$	0	0
$97$	−11.0284	−1.11976	−0.559882	−	0.828572i	$-0.689154\pi$
$97$	−11.0284	−1.11976	−0.559882	+	0.828572i	$0.689154\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−14.2416	−1.41709	−0.708546	−	0.705665i	$-0.750648\pi$
$101$	−14.2416	−1.41709	−0.708546	+	0.705665i	$0.750648\pi$
$102$	0	0
$103$	10.5858	1.04304	0.521522	−	0.853238i	$-0.325364\pi$
$103$	10.5858	1.04304	0.521522	+	0.853238i	$0.325364\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	15.5574	1.50399	0.751993	−	0.659171i	$-0.229093\pi$
$107$	15.5574	1.50399	0.751993	+	0.659171i	$0.229093\pi$
$108$	0	0
$109$	−14.3574	−1.37519	−0.687595	−	0.726094i	$-0.741334\pi$
$109$	−14.3574	−1.37519	−0.687595	+	0.726094i	$0.741334\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	3.89932	0.366817	0.183409	−	0.983037i	$-0.441287\pi$
$113$	3.89932	0.366817	0.183409	+	0.983037i	$0.441287\pi$
$114$	0	0
$115$	−2.19994	−0.205145
$116$	0	0
$117$	0	0
$118$	0	0
$119$	7.35741	0.674453
$120$	0	0
$121$	−9.23372	−0.839429
$122$	0	0
$123$	0	0
$124$	0	0
$125$	3.25454	0.291095
$126$	0	0
$127$	−7.00787	−0.621848	−0.310924	−	0.950435i	$-0.600638\pi$
$127$	−7.00787	−0.621848	−0.310924	+	0.950435i	$0.600638\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−6.66560	−0.582377	−0.291188	−	0.956666i	$-0.594051\pi$
$131$	−6.66560	−0.582377	−0.291188	+	0.956666i	$0.594051\pi$
$132$	0	0
$133$	2.93671	0.254645
$134$	0	0
$135$	0	0
$136$	0	0
$137$	16.8385	1.43861	0.719306	−	0.694693i	$-0.244460\pi$
$137$	16.8385	1.43861	0.719306	+	0.694693i	$0.244460\pi$
$138$	0	0
$139$	−15.6206	−1.32493	−0.662463	−	0.749095i	$-0.730489\pi$
$139$	−15.6206	−1.32493	−0.662463	+	0.749095i	$0.730489\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	4.09169	0.342164
$144$	0	0
$145$	−2.55486	−0.212170
$146$	0	0
$147$	0	0
$148$	0	0
$149$	15.0647	1.23415	0.617073	−	0.786906i	$-0.288318\pi$
$149$	15.0647	1.23415	0.617073	+	0.786906i	$0.288318\pi$
$150$	0	0
$151$	−7.73428	−0.629406	−0.314703	−	0.949190i	$-0.601905\pi$
$151$	−7.73428	−0.629406	−0.314703	+	0.949190i	$0.601905\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	1.07624	0.0864460
$156$	0	0
$157$	−8.71624	−0.695632	−0.347816	−	0.937563i	$-0.613077\pi$
$157$	−8.71624	−0.695632	−0.347816	+	0.937563i	$0.613077\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−6.68643	−0.526964
$162$	0	0
$163$	13.4513	1.05359	0.526793	−	0.849993i	$-0.323394\pi$
$163$	13.4513	1.05359	0.526793	+	0.849993i	$0.323394\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	11.6361	0.900428	0.450214	−	0.892921i	$-0.351348\pi$
$167$	11.6361	0.900428	0.450214	+	0.892921i	$0.351348\pi$
$168$	0	0
$169$	−3.52138	−0.270876
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−1.77166	−0.134697	−0.0673485	−	0.997730i	$-0.521454\pi$
$173$	−1.77166	−0.134697	−0.0673485	+	0.997730i	$0.521454\pi$
$174$	0	0
$175$	4.89175	0.369781
$176$	0	0
$177$	0	0
$178$	0	0
$179$	6.73139	0.503128	0.251564	−	0.967841i	$-0.419055\pi$
$179$	6.73139	0.503128	0.251564	+	0.967841i	$0.419055\pi$
$180$	0	0
$181$	−13.2711	−0.986433	−0.493217	−	0.869906i	$-0.664179\pi$
$181$	−13.2711	−0.986433	−0.493217	+	0.869906i	$0.664179\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−0.108251	−0.00795880
$186$	0	0
$187$	−9.77812	−0.715047
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−20.8166	−1.50623	−0.753117	−	0.657886i	$-0.771451\pi$
$191$	−20.8166	−1.50623	−0.753117	+	0.657886i	$0.771451\pi$
$192$	0	0
$193$	−20.5171	−1.47685	−0.738426	−	0.674335i	$-0.764431\pi$
$193$	−20.5171	−1.47685	−0.738426	+	0.674335i	$0.764431\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−7.20781	−0.513535	−0.256768	−	0.966473i	$-0.582658\pi$
$197$	−7.20781	−0.513535	−0.256768	+	0.966473i	$0.582658\pi$
$198$	0	0
$199$	−14.3276	−1.01566	−0.507828	−	0.861458i	$-0.669552\pi$
$199$	−14.3276	−1.01566	−0.507828	+	0.861458i	$0.669552\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−7.76516	−0.545008
$204$	0	0
$205$	0.0891973	0.00622981
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−3.90293	−0.269971
$210$	0	0
$211$	−7.37794	−0.507918	−0.253959	−	0.967215i	$-0.581733\pi$
$211$	−7.37794	−0.507918	−0.253959	+	0.967215i	$0.581733\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	3.60769	0.246043
$216$	0	0
$217$	3.27110	0.222057
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−22.6515	−1.52371
$222$	0	0
$223$	−26.7097	−1.78862	−0.894308	−	0.447451i	$-0.852332\pi$
$223$	−26.7097	−1.78862	−0.894308	+	0.447451i	$0.852332\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	16.9277	1.12353	0.561766	−	0.827296i	$-0.310122\pi$
$227$	16.9277	1.12353	0.561766	+	0.827296i	$0.310122\pi$
$228$	0	0
$229$	−16.4516	−1.08715	−0.543576	−	0.839360i	$-0.682930\pi$
$229$	−16.4516	−1.08715	−0.543576	+	0.839360i	$0.682930\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−4.10855	−0.269160	−0.134580	−	0.990903i	$-0.542969\pi$
$233$	−4.10855	−0.269160	−0.134580	+	0.990903i	$0.542969\pi$
$234$	0	0
$235$	0.375745	0.0245109
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−20.8363	−1.34779	−0.673895	−	0.738827i	$-0.735380\pi$
$239$	−20.8363	−1.34779	−0.673895	+	0.738827i	$0.735380\pi$
$240$	0	0
$241$	22.6954	1.46194	0.730969	−	0.682411i	$-0.239068\pi$
$241$	22.6954	1.46194	0.730969	+	0.682411i	$0.239068\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−0.329016	−0.0210200
$246$	0	0
$247$	−9.04135	−0.575287
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−4.00030	−0.252497	−0.126248	−	0.991999i	$-0.540294\pi$
$251$	−4.00030	−0.252497	−0.126248	+	0.991999i	$0.540294\pi$
$252$	0	0
$253$	8.88637	0.558681
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−0.509506	−0.0317821	−0.0158911	−	0.999874i	$-0.505058\pi$
$257$	−0.509506	−0.0317821	−0.0158911	+	0.999874i	$0.505058\pi$
$258$	0	0
$259$	−0.329016	−0.0204440
$260$	0	0
$261$	0	0
$262$	0	0
$263$	3.12688	0.192812	0.0964059	−	0.995342i	$-0.469265\pi$
$263$	3.12688	0.192812	0.0964059	+	0.995342i	$0.469265\pi$
$264$	0	0
$265$	2.11500	0.129924
$266$	0	0
$267$	0	0
$268$	0	0
$269$	4.16505	0.253947	0.126974	−	0.991906i	$-0.459474\pi$
$269$	4.16505	0.253947	0.126974	+	0.991906i	$0.459474\pi$
$270$	0	0
$271$	−13.0230	−0.791092	−0.395546	−	0.918446i	$-0.629445\pi$
$271$	−13.0230	−0.791092	−0.395546	+	0.918446i	$0.629445\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−6.50121	−0.392038
$276$	0	0
$277$	−0.695418	−0.0417836	−0.0208918	−	0.999782i	$-0.506651\pi$
$277$	−0.695418	−0.0417836	−0.0208918	+	0.999782i	$0.506651\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−32.2977	−1.92672	−0.963359	−	0.268214i	$-0.913566\pi$
$281$	−32.2977	−1.92672	−0.963359	+	0.268214i	$0.913566\pi$
$282$	0	0
$283$	−14.1277	−0.839802	−0.419901	−	0.907570i	$-0.637935\pi$
$283$	−14.1277	−0.839802	−0.419901	+	0.907570i	$0.637935\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0.271104	0.0160027
$288$	0	0
$289$	37.1315	2.18421
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−16.6764	−0.974244	−0.487122	−	0.873334i	$-0.661953\pi$
$293$	−16.6764	−0.974244	−0.487122	+	0.873334i	$0.661953\pi$
$294$	0	0
$295$	0.244900	0.0142586
$296$	0	0
$297$	0	0
$298$	0	0
$299$	20.5858	1.19050
$300$	0	0
$301$	10.9651	0.632018
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−2.90978	−0.166614
$306$	0	0
$307$	27.0345	1.54294	0.771469	−	0.636267i	$-0.219522\pi$
$307$	27.0345	1.54294	0.771469	+	0.636267i	$0.219522\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	23.4753	1.33116	0.665581	−	0.746325i	$-0.268184\pi$
$311$	23.4753	1.33116	0.665581	+	0.746325i	$0.268184\pi$
$312$	0	0
$313$	−0.729195	−0.0412165	−0.0206083	−	0.999788i	$-0.506560\pi$
$313$	−0.729195	−0.0412165	−0.0206083	+	0.999788i	$0.506560\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−33.4797	−1.88041	−0.940203	−	0.340613i	$-0.889365\pi$
$317$	−33.4797	−1.88041	−0.940203	+	0.340613i	$0.889365\pi$
$318$	0	0
$319$	10.3200	0.577811
$320$	0	0
$321$	0	0
$322$	0	0
$323$	21.6066	1.20222
$324$	0	0
$325$	−15.0604	−0.835401
$326$	0	0
$327$	0	0
$328$	0	0
$329$	1.14203	0.0629620
$330$	0	0
$331$	−9.84101	−0.540911	−0.270456	−	0.962732i	$-0.587174\pi$
$331$	−9.84101	−0.540911	−0.270456	+	0.962732i	$0.587174\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−2.82059	−0.154105
$336$	0	0
$337$	−7.86981	−0.428696	−0.214348	−	0.976757i	$-0.568763\pi$
$337$	−7.86981	−0.428696	−0.214348	+	0.976757i	$0.568763\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−4.34735	−0.235422
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	16.2944	0.874730	0.437365	−	0.899284i	$-0.355912\pi$
$347$	16.2944	0.874730	0.437365	+	0.899284i	$0.355912\pi$
$348$	0	0
$349$	−3.58438	−0.191867	−0.0959336	−	0.995388i	$-0.530584\pi$
$349$	−3.58438	−0.191867	−0.0959336	+	0.995388i	$0.530584\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	19.6066	1.04355	0.521776	−	0.853082i	$-0.325270\pi$
$353$	19.6066	1.04355	0.521776	+	0.853082i	$0.325270\pi$
$354$	0	0
$355$	0.528956	0.0280741
$356$	0	0
$357$	0	0
$358$	0	0
$359$	16.8040	0.886882	0.443441	−	0.896303i	$-0.353757\pi$
$359$	16.8040	0.886882	0.443441	+	0.896303i	$0.353757\pi$
$360$	0	0
$361$	−10.3757	−0.546092
$362$	0	0
$363$	0	0
$364$	0	0
$365$	4.43976	0.232388
$366$	0	0
$367$	33.6321	1.75558	0.877790	−	0.479045i	$-0.159017\pi$
$367$	33.6321	1.75558	0.877790	+	0.479045i	$0.159017\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	6.42828	0.333740
$372$	0	0
$373$	−34.4318	−1.78281	−0.891407	−	0.453204i	$-0.850281\pi$
$373$	−34.4318	−1.78281	−0.891407	+	0.453204i	$0.850281\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	23.9069	1.23127
$378$	0	0
$379$	18.0913	0.929287	0.464644	−	0.885498i	$-0.346182\pi$
$379$	18.0913	0.929287	0.464644	+	0.885498i	$0.346182\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	10.0758	0.514852	0.257426	−	0.966298i	$-0.417126\pi$
$383$	10.0758	0.514852	0.257426	+	0.966298i	$0.417126\pi$
$384$	0	0
$385$	0.437267	0.0222852
$386$	0	0
$387$	0	0
$388$	0	0
$389$	1.43047	0.0725278	0.0362639	−	0.999342i	$-0.488454\pi$
$389$	1.43047	0.0725278	0.0362639	+	0.999342i	$0.488454\pi$
$390$	0	0
$391$	−49.1948	−2.48789
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0.413853	0.0208232
$396$	0	0
$397$	−10.5433	−0.529152	−0.264576	−	0.964365i	$-0.585232\pi$
$397$	−10.5433	−0.529152	−0.264576	+	0.964365i	$0.585232\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−1.60809	−0.0803041	−0.0401521	−	0.999194i	$-0.512784\pi$
$401$	−1.60809	−0.0803041	−0.0401521	+	0.999194i	$0.512784\pi$
$402$	0	0
$403$	−10.0709	−0.501666
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0.437267	0.0216745
$408$	0	0
$409$	26.0349	1.28734	0.643670	−	0.765303i	$-0.277411\pi$
$409$	26.0349	1.28734	0.643670	+	0.765303i	$0.277411\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0.744341	0.0366266
$414$	0	0
$415$	−0.0208240	−0.00102221
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−4.01764	−0.196274	−0.0981372	−	0.995173i	$-0.531288\pi$
$419$	−4.01764	−0.196274	−0.0981372	+	0.995173i	$0.531288\pi$
$420$	0	0
$421$	5.26861	0.256776	0.128388	−	0.991724i	$-0.459020\pi$
$421$	5.26861	0.256776	0.128388	+	0.991724i	$0.459020\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	35.9906	1.74580
$426$	0	0
$427$	−8.84390	−0.427986
$428$	0	0
$429$	0	0
$430$	0	0
$431$	17.7343	0.854230	0.427115	−	0.904197i	$-0.359530\pi$
$431$	17.7343	0.854230	0.427115	+	0.904197i	$0.359530\pi$
$432$	0	0
$433$	−4.76835	−0.229152	−0.114576	−	0.993414i	$-0.536551\pi$
$433$	−4.76835	−0.229152	−0.114576	+	0.993414i	$0.536551\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−19.6361	−0.939322
$438$	0	0
$439$	36.2268	1.72901	0.864506	−	0.502623i	$-0.167631\pi$
$439$	36.2268	1.72901	0.864506	+	0.502623i	$0.167631\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−15.6242	−0.742329	−0.371164	−	0.928567i	$-0.621041\pi$
$443$	−15.6242	−0.742329	−0.371164	+	0.928567i	$0.621041\pi$
$444$	0	0
$445$	3.72021	0.176355
$446$	0	0
$447$	0	0
$448$	0	0
$449$	15.4142	0.727441	0.363721	−	0.931508i	$-0.381506\pi$
$449$	15.4142	0.727441	0.363721	+	0.931508i	$0.381506\pi$
$450$	0	0
$451$	−0.360301	−0.0169659
$452$	0	0
$453$	0	0
$454$	0	0
$455$	1.01295	0.0474880
$456$	0	0
$457$	−9.42427	−0.440849	−0.220424	−	0.975404i	$-0.570744\pi$
$457$	−9.42427	−0.440849	−0.220424	+	0.975404i	$0.570744\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−40.0812	−1.86677	−0.933383	−	0.358881i	$-0.883159\pi$
$461$	−40.0812	−1.86677	−0.933383	+	0.358881i	$0.883159\pi$
$462$	0	0
$463$	21.9435	1.01980	0.509900	−	0.860234i	$-0.329683\pi$
$463$	21.9435	1.01980	0.509900	+	0.860234i	$0.329683\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	20.9808	0.970878	0.485439	−	0.874271i	$-0.338660\pi$
$467$	20.9808	0.970878	0.485439	+	0.874271i	$0.338660\pi$
$468$	0	0
$469$	−8.57280	−0.395855
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−14.5728	−0.670058
$474$	0	0
$475$	14.3656	0.659141
$476$	0	0
$477$	0	0
$478$	0	0
$479$	13.3039	0.607870	0.303935	−	0.952693i	$-0.401699\pi$
$479$	13.3039	0.607870	0.303935	+	0.952693i	$0.401699\pi$
$480$	0	0
$481$	1.01295	0.0461867
$482$	0	0
$483$	0	0
$484$	0	0
$485$	3.62852	0.164762
$486$	0	0
$487$	−38.5519	−1.74695	−0.873477	−	0.486865i	$-0.838140\pi$
$487$	−38.5519	−1.74695	−0.873477	+	0.486865i	$0.838140\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	6.51919	0.294207	0.147103	−	0.989121i	$-0.453005\pi$
$491$	6.51919	0.294207	0.147103	+	0.989121i	$0.453005\pi$
$492$	0	0
$493$	−57.1315	−2.57307
$494$	0	0
$495$	0	0
$496$	0	0
$497$	1.60769	0.0721149
$498$	0	0
$499$	28.6518	1.28263	0.641316	−	0.767277i	$-0.278389\pi$
$499$	28.6518	1.28263	0.641316	+	0.767277i	$0.278389\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−16.2801	−0.725892	−0.362946	−	0.931810i	$-0.618229\pi$
$503$	−16.2801	−0.725892	−0.362946	+	0.931810i	$0.618229\pi$
$504$	0	0
$505$	4.68571	0.208511
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−26.4592	−1.17278	−0.586391	−	0.810028i	$-0.699452\pi$
$509$	−26.4592	−1.17278	−0.586391	+	0.810028i	$0.699452\pi$
$510$	0	0
$511$	13.4941	0.596942
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−3.48288	−0.153474
$516$	0	0
$517$	−1.51777	−0.0667516
$518$	0	0
$519$	0	0
$520$	0	0
$521$	7.72559	0.338464	0.169232	−	0.985576i	$-0.445871\pi$
$521$	7.72559	0.338464	0.169232	+	0.985576i	$0.445871\pi$
$522$	0	0
$523$	−16.9546	−0.741375	−0.370687	−	0.928758i	$-0.620878\pi$
$523$	−16.9546	−0.741375	−0.370687	+	0.928758i	$0.620878\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	24.0669	1.04837
$528$	0	0
$529$	21.7083	0.943840
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−0.834656	−0.0361530
$534$	0	0
$535$	−5.11861	−0.221297
$536$	0	0
$537$	0	0
$538$	0	0
$539$	1.32902	0.0572448
$540$	0	0
$541$	21.9353	0.943072	0.471536	−	0.881847i	$-0.343700\pi$
$541$	21.9353	0.943072	0.471536	+	0.881847i	$0.343700\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	4.72382	0.202346
$546$	0	0
$547$	42.8068	1.83029	0.915144	−	0.403128i	$-0.132077\pi$
$547$	42.8068	1.83029	0.915144	+	0.403128i	$0.132077\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−22.8040	−0.971484
$552$	0	0
$553$	1.25785	0.0534893
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−36.5498	−1.54866	−0.774332	−	0.632780i	$-0.781914\pi$
$557$	−36.5498	−1.54866	−0.774332	+	0.632780i	$0.781914\pi$
$558$	0	0
$559$	−33.7587	−1.42784
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−39.7165	−1.67385	−0.836925	−	0.547317i	$-0.815649\pi$
$563$	−39.7165	−1.67385	−0.836925	+	0.547317i	$0.815649\pi$
$564$	0	0
$565$	−1.28294	−0.0539736
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−30.8345	−1.29265	−0.646325	−	0.763062i	$-0.723695\pi$
$569$	−30.8345	−1.29265	−0.646325	+	0.763062i	$0.723695\pi$
$570$	0	0
$571$	−38.8017	−1.62380	−0.811901	−	0.583795i	$-0.801567\pi$
$571$	−38.8017	−1.62380	−0.811901	+	0.583795i	$0.801567\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−32.7083	−1.36403
$576$	0	0
$577$	−2.10713	−0.0877211	−0.0438606	−	0.999038i	$-0.513966\pi$
$577$	−2.10713	−0.0877211	−0.0438606	+	0.999038i	$0.513966\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−0.0632919	−0.00262579
$582$	0	0
$583$	−8.54328	−0.353827
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−6.92914	−0.285996	−0.142998	−	0.989723i	$-0.545674\pi$
$587$	−6.92914	−0.285996	−0.142998	+	0.989723i	$0.545674\pi$
$588$	0	0
$589$	9.60628	0.395820
$590$	0	0
$591$	0	0
$592$	0	0
$593$	10.0323	0.411977	0.205989	−	0.978554i	$-0.433959\pi$
$593$	10.0323	0.411977	0.205989	+	0.978554i	$0.433959\pi$
$594$	0	0
$595$	−2.42070	−0.0992392
$596$	0	0
$597$	0	0
$598$	0	0
$599$	43.3290	1.77037	0.885187	−	0.465236i	$-0.154030\pi$
$599$	43.3290	1.77037	0.885187	+	0.465236i	$0.154030\pi$
$600$	0	0
$601$	−46.5157	−1.89741	−0.948707	−	0.316157i	$-0.897607\pi$
$601$	−46.5157	−1.89741	−0.948707	+	0.316157i	$0.897607\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	3.03804	0.123514
$606$	0	0
$607$	37.8454	1.53610	0.768048	−	0.640392i	$-0.221228\pi$
$607$	37.8454	1.53610	0.768048	+	0.640392i	$0.221228\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−3.51600	−0.142242
$612$	0	0
$613$	30.2811	1.22304	0.611522	−	0.791228i	$-0.290558\pi$
$613$	30.2811	1.22304	0.611522	+	0.791228i	$0.290558\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	16.1754	0.651197	0.325599	−	0.945508i	$-0.394434\pi$
$617$	16.1754	0.651197	0.325599	+	0.945508i	$0.394434\pi$
$618$	0	0
$619$	−25.8755	−1.04002	−0.520012	−	0.854159i	$-0.674072\pi$
$619$	−25.8755	−1.04002	−0.520012	+	0.854159i	$0.674072\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	11.3071	0.453008
$624$	0	0
$625$	23.3879	0.935518
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−2.42070	−0.0965198
$630$	0	0
$631$	29.2969	1.16629	0.583146	−	0.812368i	$-0.301822\pi$
$631$	29.2969	1.16629	0.583146	+	0.812368i	$0.301822\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	2.30570	0.0914989
$636$	0	0
$637$	3.07874	0.121984
$638$	0	0
$639$	0	0
$640$	0	0
$641$	17.6526	0.697236	0.348618	−	0.937265i	$-0.386651\pi$
$641$	17.6526	0.697236	0.348618	+	0.937265i	$0.386651\pi$
$642$	0	0
$643$	−29.0852	−1.14701	−0.573504	−	0.819203i	$-0.694416\pi$
$643$	−29.0852	−1.14701	−0.573504	+	0.819203i	$0.694416\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−39.8747	−1.56764	−0.783819	−	0.620989i	$-0.786731\pi$
$647$	−39.8747	−1.56764	−0.783819	+	0.620989i	$0.786731\pi$
$648$	0	0
$649$	−0.989241	−0.0388311
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−14.1797	−0.554895	−0.277448	−	0.960741i	$-0.589488\pi$
$653$	−14.1797	−0.554895	−0.277448	+	0.960741i	$0.589488\pi$
$654$	0	0
$655$	2.19309	0.0856911
$656$	0	0
$657$	0	0
$658$	0	0
$659$	22.7184	0.884983	0.442491	−	0.896773i	$-0.354095\pi$
$659$	22.7184	0.884983	0.442491	+	0.896773i	$0.354095\pi$
$660$	0	0
$661$	40.0103	1.55622	0.778111	−	0.628127i	$-0.216178\pi$
$661$	40.0103	1.55622	0.778111	+	0.628127i	$0.216178\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−0.966223	−0.0374685
$666$	0	0
$667$	51.9212	2.01040
$668$	0	0
$669$	0	0
$670$	0	0
$671$	11.7537	0.453746
$672$	0	0
$673$	−1.82776	−0.0704551	−0.0352275	−	0.999379i	$-0.511216\pi$
$673$	−1.82776	−0.0704551	−0.0352275	+	0.999379i	$0.511216\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	28.4491	1.09339	0.546694	−	0.837333i	$-0.315886\pi$
$677$	28.4491	1.09339	0.546694	+	0.837333i	$0.315886\pi$
$678$	0	0
$679$	11.0284	0.423231
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−0.868335	−0.0332259	−0.0166130	−	0.999862i	$-0.505288\pi$
$683$	−0.868335	−0.0332259	−0.0166130	+	0.999862i	$0.505288\pi$
$684$	0	0
$685$	−5.54014	−0.211678
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−19.7910	−0.753976
$690$	0	0
$691$	−17.4386	−0.663397	−0.331699	−	0.943385i	$-0.607622\pi$
$691$	−17.4386	−0.663397	−0.331699	+	0.943385i	$0.607622\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	5.13944	0.194950
$696$	0	0
$697$	1.99462	0.0755516
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−9.23113	−0.348655	−0.174327	−	0.984688i	$-0.555775\pi$
$701$	−9.23113	−0.348655	−0.174327	+	0.984688i	$0.555775\pi$
$702$	0	0
$703$	−0.966223	−0.0364418
$704$	0	0
$705$	0	0
$706$	0	0
$707$	14.2416	0.535610
$708$	0	0
$709$	21.6462	0.812938	0.406469	−	0.913664i	$-0.366760\pi$
$709$	21.6462	0.812938	0.406469	+	0.913664i	$0.366760\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−21.8720	−0.819113
$714$	0	0
$715$	−1.34623	−0.0503462
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−48.6526	−1.81444	−0.907218	−	0.420661i	$-0.861798\pi$
$719$	−48.6526	−1.81444	−0.907218	+	0.420661i	$0.861798\pi$
$720$	0	0
$721$	−10.5858	−0.394234
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−37.9852	−1.41074
$726$	0	0
$727$	2.22834	0.0826445	0.0413222	−	0.999146i	$-0.486843\pi$
$727$	2.22834	0.0826445	0.0413222	+	0.999146i	$0.486843\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	80.6748	2.98387
$732$	0	0
$733$	38.0258	1.40452	0.702258	−	0.711923i	$-0.252175\pi$
$733$	38.0258	1.40452	0.702258	+	0.711923i	$0.252175\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	11.3934	0.419681
$738$	0	0
$739$	−48.6048	−1.78796	−0.893978	−	0.448112i	$-0.852097\pi$
$739$	−48.6048	−1.78796	−0.893978	+	0.448112i	$0.852097\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	42.5401	1.56064	0.780322	−	0.625377i	$-0.215055\pi$
$743$	42.5401	1.56064	0.780322	+	0.625377i	$0.215055\pi$
$744$	0	0
$745$	−4.95651	−0.181593
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−15.5574	−0.568453
$750$	0	0
$751$	9.25785	0.337824	0.168912	−	0.985631i	$-0.445975\pi$
$751$	9.25785	0.337824	0.168912	+	0.985631i	$0.445975\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	2.54470	0.0926111
$756$	0	0
$757$	41.0363	1.49149	0.745744	−	0.666232i	$-0.232094\pi$
$757$	41.0363	1.49149	0.745744	+	0.666232i	$0.232094\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	38.3717	1.39097	0.695487	−	0.718538i	$-0.255189\pi$
$761$	38.3717	1.39097	0.695487	+	0.718538i	$0.255189\pi$
$762$	0	0
$763$	14.3574	0.519773
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−2.29163	−0.0827459
$768$	0	0
$769$	−0.299543	−0.0108018	−0.00540090	−	0.999985i	$-0.501719\pi$
$769$	−0.299543	−0.0108018	−0.00540090	+	0.999985i	$0.501719\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−40.2941	−1.44928	−0.724639	−	0.689128i	$-0.757994\pi$
$773$	−40.2941	−1.44928	−0.724639	+	0.689128i	$0.757994\pi$
$774$	0	0
$775$	16.0014	0.574788
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0.796152	0.0285251
$780$	0	0
$781$	−2.13665	−0.0764553
$782$	0	0
$783$	0	0
$784$	0	0
$785$	2.86778	0.102355
$786$	0	0
$787$	24.5130	0.873794	0.436897	−	0.899512i	$-0.356077\pi$
$787$	24.5130	0.873794	0.436897	+	0.899512i	$0.356077\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−3.89932	−0.138644
$792$	0	0
$793$	27.2280	0.966896
$794$	0	0
$795$	0	0
$796$	0	0
$797$	53.8280	1.90669	0.953343	−	0.301889i	$-0.0976172\pi$
$797$	53.8280	1.90669	0.953343	+	0.301889i	$0.0976172\pi$
$798$	0	0
$799$	8.40237	0.297255
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−17.9338	−0.632871
$804$	0	0
$805$	2.19994	0.0775377
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−7.44011	−0.261580	−0.130790	−	0.991410i	$-0.541751\pi$
$809$	−7.44011	−0.261580	−0.130790	+	0.991410i	$0.541751\pi$
$810$	0	0
$811$	2.94854	0.103537	0.0517687	−	0.998659i	$-0.483514\pi$
$811$	2.94854	0.103537	0.0517687	+	0.998659i	$0.483514\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−4.42569	−0.155025
$816$	0	0
$817$	32.2013	1.12658
$818$	0	0
$819$	0	0
$820$	0	0
$821$	16.8866	0.589347	0.294674	−	0.955598i	$-0.404789\pi$
$821$	16.8866	0.589347	0.294674	+	0.955598i	$0.404789\pi$
$822$	0	0
$823$	48.7658	1.69987	0.849935	−	0.526887i	$-0.176641\pi$
$823$	48.7658	1.69987	0.849935	+	0.526887i	$0.176641\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	14.4541	0.502618	0.251309	−	0.967907i	$-0.419139\pi$
$827$	14.4541	0.502618	0.251309	+	0.967907i	$0.419139\pi$
$828$	0	0
$829$	−32.5659	−1.13106	−0.565530	−	0.824728i	$-0.691328\pi$
$829$	−32.5659	−1.13106	−0.565530	+	0.824728i	$0.691328\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−7.35741	−0.254919
$834$	0	0
$835$	−3.82846	−0.132489
$836$	0	0
$837$	0	0
$838$	0	0
$839$	48.8824	1.68761	0.843803	−	0.536653i	$-0.180311\pi$
$839$	48.8824	1.68761	0.843803	+	0.536653i	$0.180311\pi$
$840$	0	0
$841$	31.2978	1.07923
$842$	0	0
$843$	0	0
$844$	0	0
$845$	1.15859	0.0398567
$846$	0	0
$847$	9.23372	0.317274
$848$	0	0
$849$	0	0
$850$	0	0
$851$	2.19994	0.0754130
$852$	0	0
$853$	−21.0571	−0.720981	−0.360491	−	0.932763i	$-0.617391\pi$
$853$	−21.0571	−0.720981	−0.360491	+	0.932763i	$0.617391\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−37.4580	−1.27954	−0.639771	−	0.768565i	$-0.720971\pi$
$857$	−37.4580	−1.27954	−0.639771	+	0.768565i	$0.720971\pi$
$858$	0	0
$859$	26.4247	0.901598	0.450799	−	0.892625i	$-0.351139\pi$
$859$	26.4247	0.901598	0.450799	+	0.892625i	$0.351139\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	29.1888	0.993597	0.496798	−	0.867866i	$-0.334509\pi$
$863$	29.1888	0.993597	0.496798	+	0.867866i	$0.334509\pi$
$864$	0	0
$865$	0.582905	0.0198194
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−1.67171	−0.0567087
$870$	0	0
$871$	26.3934	0.894306
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−3.25454	−0.110024
$876$	0	0
$877$	11.4053	0.385128	0.192564	−	0.981284i	$-0.438320\pi$
$877$	11.4053	0.385128	0.192564	+	0.981284i	$0.438320\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	16.7109	0.563005	0.281503	−	0.959560i	$-0.409167\pi$
$881$	16.7109	0.563005	0.281503	+	0.959560i	$0.409167\pi$
$882$	0	0
$883$	11.8347	0.398268	0.199134	−	0.979972i	$-0.436187\pi$
$883$	11.8347	0.398268	0.199134	+	0.979972i	$0.436187\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−35.7601	−1.20071	−0.600353	−	0.799735i	$-0.704973\pi$
$887$	−35.7601	−1.20071	−0.600353	+	0.799735i	$0.704973\pi$
$888$	0	0
$889$	7.00787	0.235036
$890$	0	0
$891$	0	0
$892$	0	0
$893$	3.35380	0.112231
$894$	0	0
$895$	−2.21473	−0.0740304
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−25.4007	−0.847159
$900$	0	0
$901$	47.2955	1.57564
$902$	0	0
$903$	0	0
$904$	0	0
$905$	4.36640	0.145144
$906$	0	0
$907$	−16.6462	−0.552726	−0.276363	−	0.961053i	$-0.589129\pi$
$907$	−16.6462	−0.552726	−0.276363	+	0.961053i	$0.589129\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−19.7652	−0.654849	−0.327425	−	0.944877i	$-0.606181\pi$
$911$	−19.7652	−0.654849	−0.327425	+	0.944877i	$0.606181\pi$
$912$	0	0
$913$	0.0841159	0.00278383
$914$	0	0
$915$	0	0
$916$	0	0
$917$	6.66560	0.220118
$918$	0	0
$919$	−10.2384	−0.337734	−0.168867	−	0.985639i	$-0.554011\pi$
$919$	−10.2384	−0.337734	−0.168867	+	0.985639i	$0.554011\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−4.94966	−0.162920
$924$	0	0
$925$	−1.60946	−0.0529188
$926$	0	0
$927$	0	0
$928$	0	0
$929$	35.1683	1.15383	0.576917	−	0.816802i	$-0.304255\pi$
$929$	35.1683	1.15383	0.576917	+	0.816802i	$0.304255\pi$
$930$	0	0
$931$	−2.93671	−0.0962467
$932$	0	0
$933$	0	0
$934$	0	0
$935$	3.21715	0.105212
$936$	0	0
$937$	−31.5829	−1.03177	−0.515884	−	0.856659i	$-0.672536\pi$
$937$	−31.5829	−1.03177	−0.515884	+	0.856659i	$0.672536\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−12.9928	−0.423554	−0.211777	−	0.977318i	$-0.567925\pi$
$941$	−12.9928	−0.423554	−0.211777	+	0.977318i	$0.567925\pi$
$942$	0	0
$943$	−1.81271	−0.0590301
$944$	0	0
$945$	0	0
$946$	0	0
$947$	35.4491	1.15194	0.575971	−	0.817470i	$-0.304624\pi$
$947$	35.4491	1.15194	0.575971	+	0.817470i	$0.304624\pi$
$948$	0	0
$949$	−41.5447	−1.34860
$950$	0	0
$951$	0	0
$952$	0	0
$953$	40.0040	1.29586	0.647928	−	0.761702i	$-0.275636\pi$
$953$	40.0040	1.29586	0.647928	+	0.761702i	$0.275636\pi$
$954$	0	0
$955$	6.84898	0.221628
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−16.8385	−0.543744
$960$	0	0
$961$	−20.2999	−0.654835
$962$	0	0
$963$	0	0
$964$	0	0
$965$	6.75044	0.217304
$966$	0	0
$967$	6.07629	0.195400	0.0977001	−	0.995216i	$-0.468851\pi$
$967$	6.07629	0.195400	0.0977001	+	0.995216i	$0.468851\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−57.1397	−1.83370	−0.916850	−	0.399232i	$-0.869277\pi$
$971$	−57.1397	−1.83370	−0.916850	+	0.399232i	$0.869277\pi$
$972$	0	0
$973$	15.6206	0.500775
$974$	0	0
$975$	0	0
$976$	0	0
$977$	11.8213	0.378196	0.189098	−	0.981958i	$-0.439444\pi$
$977$	11.8213	0.378196	0.189098	+	0.981958i	$0.439444\pi$
$978$	0	0
$979$	−15.0273	−0.480274
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−18.5267	−0.590911	−0.295455	−	0.955357i	$-0.595471\pi$
$983$	−18.5267	−0.590911	−0.295455	+	0.955357i	$0.595471\pi$
$984$	0	0
$985$	2.37148	0.0755618
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−73.3174	−2.33136
$990$	0	0
$991$	12.5408	0.398371	0.199186	−	0.979962i	$-0.436170\pi$
$991$	12.5408	0.398371	0.199186	+	0.979962i	$0.436170\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	4.71401	0.149444
$996$	0	0
$997$	15.8842	0.503057	0.251528	−	0.967850i	$-0.419067\pi$
$997$	15.8842	0.503057	0.251528	+	0.967850i	$0.419067\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4536.2.a.y.1.3		4
3.2	odd	2		4536.2.a.z.1.2		4
4.3	odd	2		9072.2.a.cg.1.3		4
9.2	odd	6		504.2.r.e.337.1	yes	8
9.4	even	3		1512.2.r.e.505.2		8
9.5	odd	6		504.2.r.e.169.1	✓	8
9.7	even	3		1512.2.r.e.1009.2		8
12.11	even	2		9072.2.a.cj.1.2		4
36.7	odd	6		3024.2.r.m.1009.2		8
36.11	even	6		1008.2.r.l.337.4		8
36.23	even	6		1008.2.r.l.673.4		8
36.31	odd	6		3024.2.r.m.2017.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
504.2.r.e.169.1	✓	8	9.5	odd	6
504.2.r.e.337.1	yes	8	9.2	odd	6
1008.2.r.l.337.4		8	36.11	even	6
1008.2.r.l.673.4		8	36.23	even	6
1512.2.r.e.505.2		8	9.4	even	3
1512.2.r.e.1009.2		8	9.7	even	3
3024.2.r.m.1009.2		8	36.7	odd	6
3024.2.r.m.2017.2		8	36.31	odd	6
4536.2.a.y.1.3		4	1.1	even	1	trivial
4536.2.a.z.1.2		4	3.2	odd	2
9072.2.a.cg.1.3		4	4.3	odd	2
9072.2.a.cj.1.2		4	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 4536 = 2^{3} \cdot 3^{4} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4536.a (trivial)

\(f(q)\)	\(=\)	\(q-0.329016 q^{5} -1.00000 q^{7} +O(q^{10})\)	\(q-0.329016 q^{5} -1.00000 q^{7} +1.32902 q^{11} +3.07874 q^{13} -7.35741 q^{17} -2.93671 q^{19} +6.68643 q^{23} -4.89175 q^{25} +7.76516 q^{29} -3.27110 q^{31} +0.329016 q^{35} +0.329016 q^{37} -0.271104 q^{41} -10.9651 q^{43} -1.14203 q^{47} +1.00000 q^{49} -6.42828 q^{53} -0.437267 q^{55} -0.744341 q^{59} +8.84390 q^{61} -1.01295 q^{65} +8.57280 q^{67} -1.60769 q^{71} -13.4941 q^{73} -1.32902 q^{77} -1.25785 q^{79} +0.0632919 q^{83} +2.42070 q^{85} -11.3071 q^{89} -3.07874 q^{91} +0.966223 q^{95} -11.0284 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 3 q^{5} - 4 q^{7}+O(q^{10}) \) 4 * q - 3 * q^5 - 4 * q^7	\( 4 q - 3 q^{5} - 4 q^{7} + 7 q^{11} - 3 q^{13} - 3 q^{17} - 4 q^{19} + 2 q^{23} + 5 q^{25} - 9 q^{29} - 3 q^{31} + 3 q^{35} + 3 q^{37} + 9 q^{41} - 8 q^{43} + 3 q^{47} + 4 q^{49} - 6 q^{53} - 28 q^{55} + 10 q^{59} - 20 q^{61} + q^{65} - 11 q^{67} + 3 q^{71} - 24 q^{73} - 7 q^{77} - 21 q^{79} + 8 q^{83} - 9 q^{85} - 6 q^{89} + 3 q^{91} + 36 q^{95} - 16 q^{97}+O(q^{100}) \) 4 * q - 3 * q^5 - 4 * q^7 + 7 * q^11 - 3 * q^13 - 3 * q^17 - 4 * q^19 + 2 * q^23 + 5 * q^25 - 9 * q^29 - 3 * q^31 + 3 * q^35 + 3 * q^37 + 9 * q^41 - 8 * q^43 + 3 * q^47 + 4 * q^49 - 6 * q^53 - 28 * q^55 + 10 * q^59 - 20 * q^61 + q^65 - 11 * q^67 + 3 * q^71 - 24 * q^73 - 7 * q^77 - 21 * q^79 + 8 * q^83 - 9 * q^85 - 6 * q^89 + 3 * q^91 + 36 * q^95 - 16 * q^97

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