LMFDB - Embedded newform 4732.2.a.s.1.8

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4732,2,Mod(1,4732)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4732, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("4732.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 4732 = 2^{2} \cdot 7 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4732.a (trivial)

Newform invariants

comment:select newform

sage:traces = [8,0,0,0,-6,0,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$37.7852102365$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 19x^{6} - 2x^{5} + 113x^{4} + 40x^{3} - 232x^{2} - 136x + 52 $ x^8 - 19x^6 - 2x^5 + 113x^4 + 40x^3 - 232x^2 - 136x + 52
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 364)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$-3.23100$ of defining polynomial
Character	$\chi$	$=$	4732.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+3.23100 q^{3} -3.14769 q^{5} -1.00000 q^{7} +7.43937 q^{9} -3.84820 q^{11} -10.1702 q^{15} +0.194369 q^{17} +6.15618 q^{19} -3.23100 q^{21} -8.02163 q^{23} +4.90796 q^{25} +14.3436 q^{27} -2.71904 q^{29} -8.51671 q^{31} -12.4335 q^{33} +3.14769 q^{35} +3.59029 q^{37} +3.98295 q^{41} -11.4951 q^{43} -23.4168 q^{45} -4.35126 q^{47} +1.00000 q^{49} +0.628005 q^{51} -0.576803 q^{53} +12.1129 q^{55} +19.8906 q^{57} -2.08672 q^{59} +3.40510 q^{61} -7.43937 q^{63} -5.42026 q^{67} -25.9179 q^{69} -9.15185 q^{71} -3.31209 q^{73} +15.8576 q^{75} +3.84820 q^{77} -8.98752 q^{79} +24.0261 q^{81} -7.66353 q^{83} -0.611813 q^{85} -8.78521 q^{87} +5.11527 q^{89} -27.5175 q^{93} -19.3778 q^{95} +9.62669 q^{97} -28.6282 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 6 q^{5} - 8 q^{7} + 14 q^{9} - 12 q^{11} - 12 q^{15} + 2 q^{17} - 6 q^{19} + 22 q^{25} - 6 q^{27} + 22 q^{29} - 14 q^{31} + 28 q^{33} + 6 q^{35} - 12 q^{37} - 4 q^{41} + 6 q^{43} - 20 q^{45} - 42 q^{47}+ \cdots - 80 q^{99}+O(q^{100}) $ 8 * q - 6 * q^5 - 8 * q^7 + 14 * q^9 - 12 * q^11 - 12 * q^15 + 2 * q^17 - 6 * q^19 + 22 * q^25 - 6 * q^27 + 22 * q^29 - 14 * q^31 + 28 * q^33 + 6 * q^35 - 12 * q^37 - 4 * q^41 + 6 * q^43 - 20 * q^45 - 42 * q^47 + 8 * q^49 + 2 * q^51 + 4 * q^53 - 2 * q^55 + 8 * q^57 - 2 * q^59 - 4 * q^61 - 14 * q^63 - 24 * q^67 - 52 * q^69 - 28 * q^71 - 28 * q^73 - 10 * q^75 + 12 * q^77 + 4 * q^79 - 38 * q^83 + 36 * q^85 - 26 * q^87 + 22 * q^89 - 56 * q^93 - 30 * q^95 + 4 * q^97 - 80 * q^99

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
$3$	3.23100	1.86542	0.932710	−	0.360628i	$-0.117438\pi$
$3$	3.23100	1.86542	0.932710	+	0.360628i	$0.117438\pi$
$4$	0	0
$5$	−3.14769	−1.40769	−0.703845	−	0.710353i	$-0.748535\pi$
$5$	−3.14769	−1.40769	−0.703845	+	0.710353i	$0.748535\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	7.43937	2.47979
$10$	0	0
$11$	−3.84820	−1.16028	−0.580138	−	0.814518i	$-0.697001\pi$
$11$	−3.84820	−1.16028	−0.580138	+	0.814518i	$0.697001\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	−10.1702	−2.62593
$16$	0	0
$17$	0.194369	0.0471413	0.0235707	−	0.999722i	$-0.492497\pi$
$17$	0.194369	0.0471413	0.0235707	+	0.999722i	$0.492497\pi$
$18$	0	0
$19$	6.15618	1.41232	0.706162	−	0.708050i	$-0.250425\pi$
$19$	6.15618	1.41232	0.706162	+	0.708050i	$0.250425\pi$
$20$	0	0
$21$	−3.23100	−0.705062
$22$	0	0
$23$	−8.02163	−1.67263	−0.836313	−	0.548253i	$-0.815293\pi$
$23$	−8.02163	−1.67263	−0.836313	+	0.548253i	$0.815293\pi$
$24$	0	0
$25$	4.90796	0.981593
$26$	0	0
$27$	14.3436	2.76043
$28$	0	0
$29$	−2.71904	−0.504912	−0.252456	−	0.967608i	$-0.581238\pi$
$29$	−2.71904	−0.504912	−0.252456	+	0.967608i	$0.581238\pi$
$30$	0	0
$31$	−8.51671	−1.52965	−0.764823	−	0.644240i	$-0.777174\pi$
$31$	−8.51671	−1.52965	−0.764823	+	0.644240i	$0.777174\pi$
$32$	0	0
$33$	−12.4335	−2.16440
$34$	0	0
$35$	3.14769	0.532057
$36$	0	0
$37$	3.59029	0.590240	0.295120	−	0.955460i	$-0.404640\pi$
$37$	3.59029	0.590240	0.295120	+	0.955460i	$0.404640\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	3.98295	0.622033	0.311016	−	0.950405i	$-0.399331\pi$
$41$	3.98295	0.622033	0.311016	+	0.950405i	$0.399331\pi$
$42$	0	0
$43$	−11.4951	−1.75299	−0.876494	−	0.481413i	$-0.840124\pi$
$43$	−11.4951	−1.75299	−0.876494	+	0.481413i	$0.840124\pi$
$44$	0	0
$45$	−23.4168	−3.49078
$46$	0	0
$47$	−4.35126	−0.634696	−0.317348	−	0.948309i	$-0.602792\pi$
$47$	−4.35126	−0.634696	−0.317348	+	0.948309i	$0.602792\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0.628005	0.0879383
$52$	0	0
$53$	−0.576803	−0.0792300	−0.0396150	−	0.999215i	$-0.512613\pi$
$53$	−0.576803	−0.0792300	−0.0396150	+	0.999215i	$0.512613\pi$
$54$	0	0
$55$	12.1129	1.63331
$56$	0	0
$57$	19.8906	2.63458
$58$	0	0
$59$	−2.08672	−0.271668	−0.135834	−	0.990732i	$-0.543371\pi$
$59$	−2.08672	−0.271668	−0.135834	+	0.990732i	$0.543371\pi$
$60$	0	0
$61$	3.40510	0.435979	0.217989	−	0.975951i	$-0.430050\pi$
$61$	3.40510	0.435979	0.217989	+	0.975951i	$0.430050\pi$
$62$	0	0
$63$	−7.43937	−0.937272
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−5.42026	−0.662190	−0.331095	−	0.943597i	$-0.607418\pi$
$67$	−5.42026	−0.662190	−0.331095	+	0.943597i	$0.607418\pi$
$68$	0	0
$69$	−25.9179	−3.12015
$70$	0	0
$71$	−9.15185	−1.08613	−0.543063	−	0.839692i	$-0.682735\pi$
$71$	−9.15185	−1.08613	−0.543063	+	0.839692i	$0.682735\pi$
$72$	0	0
$73$	−3.31209	−0.387650	−0.193825	−	0.981036i	$-0.562089\pi$
$73$	−3.31209	−0.387650	−0.193825	+	0.981036i	$0.562089\pi$
$74$	0	0
$75$	15.8576	1.83108
$76$	0	0
$77$	3.84820	0.438543
$78$	0	0
$79$	−8.98752	−1.01117	−0.505587	−	0.862775i	$-0.668724\pi$
$79$	−8.98752	−1.01117	−0.505587	+	0.862775i	$0.668724\pi$
$80$	0	0
$81$	24.0261	2.66957
$82$	0	0
$83$	−7.66353	−0.841181	−0.420591	−	0.907251i	$-0.638177\pi$
$83$	−7.66353	−0.841181	−0.420591	+	0.907251i	$0.638177\pi$
$84$	0	0
$85$	−0.611813	−0.0663604
$86$	0	0
$87$	−8.78521	−0.941873
$88$	0	0
$89$	5.11527	0.542218	0.271109	−	0.962549i	$-0.412610\pi$
$89$	5.11527	0.542218	0.271109	+	0.962549i	$0.412610\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−27.5175	−2.85343
$94$	0	0
$95$	−19.3778	−1.98812
$96$	0	0
$97$	9.62669	0.977442	0.488721	−	0.872440i	$-0.337464\pi$
$97$	9.62669	0.977442	0.488721	+	0.872440i	$0.337464\pi$
$98$	0	0
$99$	−28.6282	−2.87724
$100$	0	0
$101$	−2.54158	−0.252896	−0.126448	−	0.991973i	$-0.540358\pi$
$101$	−2.54158	−0.252896	−0.126448	+	0.991973i	$0.540358\pi$
$102$	0	0
$103$	−3.35392	−0.330471	−0.165236	−	0.986254i	$-0.552838\pi$
$103$	−3.35392	−0.330471	−0.165236	+	0.986254i	$0.552838\pi$
$104$	0	0
$105$	10.1702	0.992510
$106$	0	0
$107$	−12.9130	−1.24835	−0.624174	−	0.781286i	$-0.714564\pi$
$107$	−12.9130	−1.24835	−0.624174	+	0.781286i	$0.714564\pi$
$108$	0	0
$109$	8.04268	0.770349	0.385174	−	0.922844i	$-0.374141\pi$
$109$	8.04268	0.770349	0.385174	+	0.922844i	$0.374141\pi$
$110$	0	0
$111$	11.6002	1.10105
$112$	0	0
$113$	−8.00469	−0.753018	−0.376509	−	0.926413i	$-0.622876\pi$
$113$	−8.00469	−0.753018	−0.376509	+	0.926413i	$0.622876\pi$
$114$	0	0
$115$	25.2496	2.35454
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−0.194369	−0.0178177
$120$	0	0
$121$	3.80865	0.346241
$122$	0	0
$123$	12.8689	1.16035
$124$	0	0
$125$	0.289701	0.0259116
$126$	0	0
$127$	1.32877	0.117909	0.0589545	−	0.998261i	$-0.481223\pi$
$127$	1.32877	0.117909	0.0589545	+	0.998261i	$0.481223\pi$
$128$	0	0
$129$	−37.1407	−3.27006
$130$	0	0
$131$	−0.592761	−0.0517897	−0.0258949	−	0.999665i	$-0.508244\pi$
$131$	−0.592761	−0.0517897	−0.0258949	+	0.999665i	$0.508244\pi$
$132$	0	0
$133$	−6.15618	−0.533808
$134$	0	0
$135$	−45.1492	−3.88583
$136$	0	0
$137$	2.56034	0.218745	0.109372	−	0.994001i	$-0.465116\pi$
$137$	2.56034	0.218745	0.109372	+	0.994001i	$0.465116\pi$
$138$	0	0
$139$	−18.0230	−1.52869	−0.764345	−	0.644807i	$-0.776938\pi$
$139$	−18.0230	−1.52869	−0.764345	+	0.644807i	$0.776938\pi$
$140$	0	0
$141$	−14.0589	−1.18397
$142$	0	0
$143$	0	0
$144$	0	0
$145$	8.55869	0.710760
$146$	0	0
$147$	3.23100	0.266488
$148$	0	0
$149$	8.10789	0.664224	0.332112	−	0.943240i	$-0.392239\pi$
$149$	8.10789	0.664224	0.332112	+	0.943240i	$0.392239\pi$
$150$	0	0
$151$	−3.58804	−0.291991	−0.145995	−	0.989285i	$-0.546638\pi$
$151$	−3.58804	−0.291991	−0.145995	+	0.989285i	$0.546638\pi$
$152$	0	0
$153$	1.44598	0.116901
$154$	0	0
$155$	26.8080	2.15327
$156$	0	0
$157$	14.3447	1.14483	0.572417	−	0.819963i	$-0.306006\pi$
$157$	14.3447	1.14483	0.572417	+	0.819963i	$0.306006\pi$
$158$	0	0
$159$	−1.86365	−0.147797
$160$	0	0
$161$	8.02163	0.632193
$162$	0	0
$163$	−0.246141	−0.0192792	−0.00963961	−	0.999954i	$-0.503068\pi$
$163$	−0.246141	−0.0192792	−0.00963961	+	0.999954i	$0.503068\pi$
$164$	0	0
$165$	39.1370	3.04681
$166$	0	0
$167$	1.80831	0.139931	0.0699657	−	0.997549i	$-0.477711\pi$
$167$	1.80831	0.139931	0.0699657	+	0.997549i	$0.477711\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	45.7981	3.50227
$172$	0	0
$173$	−10.7996	−0.821080	−0.410540	−	0.911843i	$-0.634660\pi$
$173$	−10.7996	−0.821080	−0.410540	+	0.911843i	$0.634660\pi$
$174$	0	0
$175$	−4.90796	−0.371007
$176$	0	0
$177$	−6.74220	−0.506775
$178$	0	0
$179$	8.67647	0.648510	0.324255	−	0.945970i	$-0.394886\pi$
$179$	8.67647	0.648510	0.324255	+	0.945970i	$0.394886\pi$
$180$	0	0
$181$	10.0087	0.743940	0.371970	−	0.928245i	$-0.378682\pi$
$181$	10.0087	0.743940	0.371970	+	0.928245i	$0.378682\pi$
$182$	0	0
$183$	11.0019	0.813283
$184$	0	0
$185$	−11.3011	−0.830876
$186$	0	0
$187$	−0.747970	−0.0546970
$188$	0	0
$189$	−14.3436	−1.04334
$190$	0	0
$191$	−24.0312	−1.73884	−0.869420	−	0.494074i	$-0.835507\pi$
$191$	−24.0312	−1.73884	−0.869420	+	0.494074i	$0.835507\pi$
$192$	0	0
$193$	−3.27308	−0.235601	−0.117801	−	0.993037i	$-0.537584\pi$
$193$	−3.27308	−0.235601	−0.117801	+	0.993037i	$0.537584\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−12.7524	−0.908570	−0.454285	−	0.890857i	$-0.650105\pi$
$197$	−12.7524	−0.908570	−0.454285	+	0.890857i	$0.650105\pi$
$198$	0	0
$199$	27.3231	1.93688	0.968442	−	0.249239i	$-0.0801806\pi$
$199$	27.3231	1.93688	0.968442	+	0.249239i	$0.0801806\pi$
$200$	0	0
$201$	−17.5129	−1.23526
$202$	0	0
$203$	2.71904	0.190839
$204$	0	0
$205$	−12.5371	−0.875629
$206$	0	0
$207$	−59.6758	−4.14776
$208$	0	0
$209$	−23.6902	−1.63869
$210$	0	0
$211$	26.5857	1.83024	0.915118	−	0.403185i	$-0.132097\pi$
$211$	26.5857	1.83024	0.915118	+	0.403185i	$0.132097\pi$
$212$	0	0
$213$	−29.5696	−2.02608
$214$	0	0
$215$	36.1831	2.46766
$216$	0	0
$217$	8.51671	0.578152
$218$	0	0
$219$	−10.7014	−0.723130
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−15.1338	−1.01344	−0.506718	−	0.862112i	$-0.669141\pi$
$223$	−15.1338	−1.01344	−0.506718	+	0.862112i	$0.669141\pi$
$224$	0	0
$225$	36.5122	2.43414
$226$	0	0
$227$	−21.4054	−1.42072	−0.710362	−	0.703836i	$-0.751469\pi$
$227$	−21.4054	−1.42072	−0.710362	+	0.703836i	$0.751469\pi$
$228$	0	0
$229$	−8.36123	−0.552526	−0.276263	−	0.961082i	$-0.589096\pi$
$229$	−8.36123	−0.552526	−0.276263	+	0.961082i	$0.589096\pi$
$230$	0	0
$231$	12.4335	0.818067
$232$	0	0
$233$	−26.1663	−1.71421	−0.857107	−	0.515138i	$-0.827740\pi$
$233$	−26.1663	−1.71421	−0.857107	+	0.515138i	$0.827740\pi$
$234$	0	0
$235$	13.6964	0.893456
$236$	0	0
$237$	−29.0387	−1.88626
$238$	0	0
$239$	−18.3562	−1.18736	−0.593682	−	0.804700i	$-0.702326\pi$
$239$	−18.3562	−1.18736	−0.593682	+	0.804700i	$0.702326\pi$
$240$	0	0
$241$	3.33482	0.214814	0.107407	−	0.994215i	$-0.465745\pi$
$241$	3.33482	0.214814	0.107407	+	0.994215i	$0.465745\pi$
$242$	0	0
$243$	34.5975	2.21943
$244$	0	0
$245$	−3.14769	−0.201099
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−24.7609	−1.56916
$250$	0	0
$251$	−12.1977	−0.769911	−0.384956	−	0.922935i	$-0.625783\pi$
$251$	−12.1977	−0.769911	−0.384956	+	0.922935i	$0.625783\pi$
$252$	0	0
$253$	30.8688	1.94071
$254$	0	0
$255$	−1.97677	−0.123790
$256$	0	0
$257$	15.2730	0.952701	0.476350	−	0.879256i	$-0.341959\pi$
$257$	15.2730	0.952701	0.476350	+	0.879256i	$0.341959\pi$
$258$	0	0
$259$	−3.59029	−0.223090
$260$	0	0
$261$	−20.2279	−1.25208
$262$	0	0
$263$	24.8989	1.53533	0.767666	−	0.640850i	$-0.221418\pi$
$263$	24.8989	1.53533	0.767666	+	0.640850i	$0.221418\pi$
$264$	0	0
$265$	1.81560	0.111531
$266$	0	0
$267$	16.5275	1.01146
$268$	0	0
$269$	−7.85673	−0.479033	−0.239517	−	0.970892i	$-0.576989\pi$
$269$	−7.85673	−0.479033	−0.239517	+	0.970892i	$0.576989\pi$
$270$	0	0
$271$	2.63865	0.160287	0.0801433	−	0.996783i	$-0.474462\pi$
$271$	2.63865	0.160287	0.0801433	+	0.996783i	$0.474462\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−18.8868	−1.13892
$276$	0	0
$277$	11.6549	0.700273	0.350136	−	0.936699i	$-0.386135\pi$
$277$	11.6549	0.700273	0.350136	+	0.936699i	$0.386135\pi$
$278$	0	0
$279$	−63.3589	−3.79320
$280$	0	0
$281$	3.86501	0.230567	0.115283	−	0.993333i	$-0.463222\pi$
$281$	3.86501	0.230567	0.115283	+	0.993333i	$0.463222\pi$
$282$	0	0
$283$	6.84863	0.407109	0.203554	−	0.979064i	$-0.434751\pi$
$283$	6.84863	0.407109	0.203554	+	0.979064i	$0.434751\pi$
$284$	0	0
$285$	−62.6095	−3.70867
$286$	0	0
$287$	−3.98295	−0.235106
$288$	0	0
$289$	−16.9622	−0.997778
$290$	0	0
$291$	31.1038	1.82334
$292$	0	0
$293$	−1.48894	−0.0869846	−0.0434923	−	0.999054i	$-0.513848\pi$
$293$	−1.48894	−0.0869846	−0.0434923	+	0.999054i	$0.513848\pi$
$294$	0	0
$295$	6.56836	0.382425
$296$	0	0
$297$	−55.1971	−3.20286
$298$	0	0
$299$	0	0
$300$	0	0
$301$	11.4951	0.662567
$302$	0	0
$303$	−8.21183	−0.471757
$304$	0	0
$305$	−10.7182	−0.613723
$306$	0	0
$307$	2.33128	0.133053	0.0665265	−	0.997785i	$-0.478808\pi$
$307$	2.33128	0.133053	0.0665265	+	0.997785i	$0.478808\pi$
$308$	0	0
$309$	−10.8365	−0.616468
$310$	0	0
$311$	−11.8244	−0.670502	−0.335251	−	0.942129i	$-0.608821\pi$
$311$	−11.8244	−0.670502	−0.335251	+	0.942129i	$0.608821\pi$
$312$	0	0
$313$	−18.4087	−1.04052	−0.520260	−	0.854008i	$-0.674165\pi$
$313$	−18.4087	−1.04052	−0.520260	+	0.854008i	$0.674165\pi$
$314$	0	0
$315$	23.4168	1.31939
$316$	0	0
$317$	8.42274	0.473068	0.236534	−	0.971623i	$-0.423988\pi$
$317$	8.42274	0.473068	0.236534	+	0.971623i	$0.423988\pi$
$318$	0	0
$319$	10.4634	0.585838
$320$	0	0
$321$	−41.7219	−2.32869
$322$	0	0
$323$	1.19657	0.0665788
$324$	0	0
$325$	0	0
$326$	0	0
$327$	25.9859	1.43702
$328$	0	0
$329$	4.35126	0.239893
$330$	0	0
$331$	18.0566	0.992481	0.496240	−	0.868185i	$-0.334713\pi$
$331$	18.0566	0.992481	0.496240	+	0.868185i	$0.334713\pi$
$332$	0	0
$333$	26.7095	1.46367
$334$	0	0
$335$	17.0613	0.932159
$336$	0	0
$337$	3.64765	0.198700	0.0993500	−	0.995053i	$-0.468324\pi$
$337$	3.64765	0.198700	0.0993500	+	0.995053i	$0.468324\pi$
$338$	0	0
$339$	−25.8632	−1.40469
$340$	0	0
$341$	32.7740	1.77481
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	81.5815	4.39220
$346$	0	0
$347$	14.5945	0.783476	0.391738	−	0.920077i	$-0.371874\pi$
$347$	14.5945	0.783476	0.391738	+	0.920077i	$0.371874\pi$
$348$	0	0
$349$	−17.8836	−0.957289	−0.478645	−	0.878009i	$-0.658872\pi$
$349$	−17.8836	−0.957289	−0.478645	+	0.878009i	$0.658872\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−6.38272	−0.339718	−0.169859	−	0.985468i	$-0.554331\pi$
$353$	−6.38272	−0.339718	−0.169859	+	0.985468i	$0.554331\pi$
$354$	0	0
$355$	28.8072	1.52893
$356$	0	0
$357$	−0.628005	−0.0332376
$358$	0	0
$359$	30.4944	1.60943	0.804717	−	0.593659i	$-0.202317\pi$
$359$	30.4944	1.60943	0.804717	+	0.593659i	$0.202317\pi$
$360$	0	0
$361$	18.8985	0.994660
$362$	0	0
$363$	12.3057	0.645884
$364$	0	0
$365$	10.4254	0.545692
$366$	0	0
$367$	−7.16337	−0.373925	−0.186962	−	0.982367i	$-0.559864\pi$
$367$	−7.16337	−0.373925	−0.186962	+	0.982367i	$0.559864\pi$
$368$	0	0
$369$	29.6306	1.54251
$370$	0	0
$371$	0.576803	0.0299461
$372$	0	0
$373$	−1.16467	−0.0603045	−0.0301523	−	0.999545i	$-0.509599\pi$
$373$	−1.16467	−0.0603045	−0.0301523	+	0.999545i	$0.509599\pi$
$374$	0	0
$375$	0.936023	0.0483360
$376$	0	0
$377$	0	0
$378$	0	0
$379$	29.0197	1.49064	0.745322	−	0.666705i	$-0.232296\pi$
$379$	29.0197	1.49064	0.745322	+	0.666705i	$0.232296\pi$
$380$	0	0
$381$	4.29325	0.219950
$382$	0	0
$383$	21.9571	1.12196	0.560978	−	0.827831i	$-0.310425\pi$
$383$	21.9571	1.12196	0.560978	+	0.827831i	$0.310425\pi$
$384$	0	0
$385$	−12.1129	−0.617333
$386$	0	0
$387$	−85.5163	−4.34704
$388$	0	0
$389$	29.1973	1.48036	0.740180	−	0.672409i	$-0.234740\pi$
$389$	29.1973	1.48036	0.740180	+	0.672409i	$0.234740\pi$
$390$	0	0
$391$	−1.55915	−0.0788498
$392$	0	0
$393$	−1.91521	−0.0966096
$394$	0	0
$395$	28.2899	1.42342
$396$	0	0
$397$	−31.9503	−1.60354	−0.801771	−	0.597632i	$-0.796108\pi$
$397$	−31.9503	−1.60354	−0.801771	+	0.597632i	$0.796108\pi$
$398$	0	0
$399$	−19.8906	−0.995776
$400$	0	0
$401$	36.8552	1.84046	0.920232	−	0.391374i	$-0.128000\pi$
$401$	36.8552	1.84046	0.920232	+	0.391374i	$0.128000\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	−75.6267	−3.75792
$406$	0	0
$407$	−13.8162	−0.684842
$408$	0	0
$409$	5.65493	0.279618	0.139809	−	0.990178i	$-0.455351\pi$
$409$	5.65493	0.279618	0.139809	+	0.990178i	$0.455351\pi$
$410$	0	0
$411$	8.27247	0.408051
$412$	0	0
$413$	2.08672	0.102681
$414$	0	0
$415$	24.1224	1.18412
$416$	0	0
$417$	−58.2323	−2.85165
$418$	0	0
$419$	−25.8062	−1.26072	−0.630358	−	0.776304i	$-0.717092\pi$
$419$	−25.8062	−1.26072	−0.630358	+	0.776304i	$0.717092\pi$
$420$	0	0
$421$	−28.8606	−1.40658	−0.703290	−	0.710903i	$-0.748286\pi$
$421$	−28.8606	−1.40658	−0.703290	+	0.710903i	$0.748286\pi$
$422$	0	0
$423$	−32.3706	−1.57391
$424$	0	0
$425$	0.953954	0.0462736
$426$	0	0
$427$	−3.40510	−0.164785
$428$	0	0
$429$	0	0
$430$	0	0
$431$	31.6712	1.52555	0.762774	−	0.646665i	$-0.223837\pi$
$431$	31.6712	1.52555	0.762774	+	0.646665i	$0.223837\pi$
$432$	0	0
$433$	31.3034	1.50434	0.752172	−	0.658967i	$-0.229006\pi$
$433$	31.3034	1.50434	0.752172	+	0.658967i	$0.229006\pi$
$434$	0	0
$435$	27.6531	1.32587
$436$	0	0
$437$	−49.3826	−2.36229
$438$	0	0
$439$	36.9572	1.76387	0.881935	−	0.471371i	$-0.156241\pi$
$439$	36.9572	1.76387	0.881935	+	0.471371i	$0.156241\pi$
$440$	0	0
$441$	7.43937	0.354256
$442$	0	0
$443$	28.2871	1.34396	0.671981	−	0.740568i	$-0.265444\pi$
$443$	28.2871	1.34396	0.671981	+	0.740568i	$0.265444\pi$
$444$	0	0
$445$	−16.1013	−0.763275
$446$	0	0
$447$	26.1966	1.23906
$448$	0	0
$449$	−24.3284	−1.14813	−0.574064	−	0.818811i	$-0.694634\pi$
$449$	−24.3284	−1.14813	−0.574064	+	0.818811i	$0.694634\pi$
$450$	0	0
$451$	−15.3272	−0.721730
$452$	0	0
$453$	−11.5930	−0.544686
$454$	0	0
$455$	0	0
$456$	0	0
$457$	1.54960	0.0724870	0.0362435	−	0.999343i	$-0.488461\pi$
$457$	1.54960	0.0724870	0.0362435	+	0.999343i	$0.488461\pi$
$458$	0	0
$459$	2.78795	0.130130
$460$	0	0
$461$	12.9552	0.603383	0.301692	−	0.953406i	$-0.402449\pi$
$461$	12.9552	0.603383	0.301692	+	0.953406i	$0.402449\pi$
$462$	0	0
$463$	−28.1690	−1.30912	−0.654562	−	0.756008i	$-0.727147\pi$
$463$	−28.1690	−1.30912	−0.654562	+	0.756008i	$0.727147\pi$
$464$	0	0
$465$	86.6166	4.01675
$466$	0	0
$467$	29.2908	1.35542	0.677709	−	0.735330i	$-0.262973\pi$
$467$	29.2908	1.35542	0.677709	+	0.735330i	$0.262973\pi$
$468$	0	0
$469$	5.42026	0.250284
$470$	0	0
$471$	46.3478	2.13559
$472$	0	0
$473$	44.2355	2.03395
$474$	0	0
$475$	30.2143	1.38633
$476$	0	0
$477$	−4.29105	−0.196474
$478$	0	0
$479$	−19.4711	−0.889658	−0.444829	−	0.895616i	$-0.646735\pi$
$479$	−19.4711	−0.889658	−0.444829	+	0.895616i	$0.646735\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	25.9179	1.17930
$484$	0	0
$485$	−30.3018	−1.37594
$486$	0	0
$487$	4.53546	0.205521	0.102761	−	0.994706i	$-0.467232\pi$
$487$	4.53546	0.205521	0.102761	+	0.994706i	$0.467232\pi$
$488$	0	0
$489$	−0.795281	−0.0359638
$490$	0	0
$491$	18.1487	0.819037	0.409519	−	0.912302i	$-0.365697\pi$
$491$	18.1487	0.819037	0.409519	+	0.912302i	$0.365697\pi$
$492$	0	0
$493$	−0.528495	−0.0238022
$494$	0	0
$495$	90.1127	4.05026
$496$	0	0
$497$	9.15185	0.410517
$498$	0	0
$499$	6.65532	0.297933	0.148967	−	0.988842i	$-0.452405\pi$
$499$	6.65532	0.297933	0.148967	+	0.988842i	$0.452405\pi$
$500$	0	0
$501$	5.84266	0.261031
$502$	0	0
$503$	14.9196	0.665230	0.332615	−	0.943063i	$-0.392069\pi$
$503$	14.9196	0.665230	0.332615	+	0.943063i	$0.392069\pi$
$504$	0	0
$505$	8.00010	0.356000
$506$	0	0
$507$	0	0
$508$	0	0
$509$	17.7673	0.787522	0.393761	−	0.919213i	$-0.371174\pi$
$509$	17.7673	0.787522	0.393761	+	0.919213i	$0.371174\pi$
$510$	0	0
$511$	3.31209	0.146518
$512$	0	0
$513$	88.3018	3.89862
$514$	0	0
$515$	10.5571	0.465201
$516$	0	0
$517$	16.7445	0.736423
$518$	0	0
$519$	−34.8936	−1.53166
$520$	0	0
$521$	38.0579	1.66735	0.833673	−	0.552258i	$-0.186234\pi$
$521$	38.0579	1.66735	0.833673	+	0.552258i	$0.186234\pi$
$522$	0	0
$523$	13.8905	0.607390	0.303695	−	0.952769i	$-0.401780\pi$
$523$	13.8905	0.607390	0.303695	+	0.952769i	$0.401780\pi$
$524$	0	0
$525$	−15.8576	−0.692084
$526$	0	0
$527$	−1.65538	−0.0721096
$528$	0	0
$529$	41.3465	1.79767
$530$	0	0
$531$	−15.5239	−0.673680
$532$	0	0
$533$	0	0
$534$	0	0
$535$	40.6462	1.75729
$536$	0	0
$537$	28.0337	1.20974
$538$	0	0
$539$	−3.84820	−0.165754
$540$	0	0
$541$	−41.8453	−1.79907	−0.899535	−	0.436850i	$-0.856094\pi$
$541$	−41.8453	−1.79907	−0.899535	+	0.436850i	$0.856094\pi$
$542$	0	0
$543$	32.3381	1.38776
$544$	0	0
$545$	−25.3159	−1.08441
$546$	0	0
$547$	−22.3475	−0.955509	−0.477754	−	0.878493i	$-0.658549\pi$
$547$	−22.3475	−0.955509	−0.477754	+	0.878493i	$0.658549\pi$
$548$	0	0
$549$	25.3318	1.08114
$550$	0	0
$551$	−16.7389	−0.713100
$552$	0	0
$553$	8.98752	0.382188
$554$	0	0
$555$	−36.5140	−1.54993
$556$	0	0
$557$	30.6572	1.29899	0.649493	−	0.760368i	$-0.274981\pi$
$557$	30.6572	1.29899	0.649493	+	0.760368i	$0.274981\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	−2.41669	−0.102033
$562$	0	0
$563$	−33.2473	−1.40121	−0.700604	−	0.713550i	$-0.747086\pi$
$563$	−33.2473	−1.40121	−0.700604	+	0.713550i	$0.747086\pi$
$564$	0	0
$565$	25.1963	1.06002
$566$	0	0
$567$	−24.0261	−1.00900
$568$	0	0
$569$	11.6102	0.486725	0.243363	−	0.969935i	$-0.421749\pi$
$569$	11.6102	0.486725	0.243363	+	0.969935i	$0.421749\pi$
$570$	0	0
$571$	−16.6994	−0.698847	−0.349423	−	0.936965i	$-0.613623\pi$
$571$	−16.6994	−0.698847	−0.349423	+	0.936965i	$0.613623\pi$
$572$	0	0
$573$	−77.6450	−3.24366
$574$	0	0
$575$	−39.3699	−1.64184
$576$	0	0
$577$	37.2583	1.55108	0.775541	−	0.631297i	$-0.217477\pi$
$577$	37.2583	1.55108	0.775541	+	0.631297i	$0.217477\pi$
$578$	0	0
$579$	−10.5753	−0.439495
$580$	0	0
$581$	7.66353	0.317937
$582$	0	0
$583$	2.21965	0.0919287
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−41.7900	−1.72486	−0.862429	−	0.506178i	$-0.831058\pi$
$587$	−41.7900	−1.72486	−0.862429	+	0.506178i	$0.831058\pi$
$588$	0	0
$589$	−52.4304	−2.16036
$590$	0	0
$591$	−41.2030	−1.69486
$592$	0	0
$593$	−4.33672	−0.178088	−0.0890438	−	0.996028i	$-0.528381\pi$
$593$	−4.33672	−0.178088	−0.0890438	+	0.996028i	$0.528381\pi$
$594$	0	0
$595$	0.611813	0.0250819
$596$	0	0
$597$	88.2810	3.61310
$598$	0	0
$599$	22.7578	0.929858	0.464929	−	0.885348i	$-0.346080\pi$
$599$	22.7578	0.929858	0.464929	+	0.885348i	$0.346080\pi$
$600$	0	0
$601$	−27.5268	−1.12284	−0.561421	−	0.827530i	$-0.689745\pi$
$601$	−27.5268	−1.12284	−0.561421	+	0.827530i	$0.689745\pi$
$602$	0	0
$603$	−40.3233	−1.64209
$604$	0	0
$605$	−11.9885	−0.487400
$606$	0	0
$607$	−9.06967	−0.368126	−0.184063	−	0.982914i	$-0.558925\pi$
$607$	−9.06967	−0.368126	−0.184063	+	0.982914i	$0.558925\pi$
$608$	0	0
$609$	8.78521	0.355995
$610$	0	0
$611$	0	0
$612$	0	0
$613$	9.37895	0.378812	0.189406	−	0.981899i	$-0.439344\pi$
$613$	9.37895	0.378812	0.189406	+	0.981899i	$0.439344\pi$
$614$	0	0
$615$	−40.5074	−1.63342
$616$	0	0
$617$	−30.1129	−1.21230	−0.606151	−	0.795350i	$-0.707287\pi$
$617$	−30.1129	−1.21230	−0.606151	+	0.795350i	$0.707287\pi$
$618$	0	0
$619$	39.8942	1.60348	0.801742	−	0.597671i	$-0.203907\pi$
$619$	39.8942	1.60348	0.801742	+	0.597671i	$0.203907\pi$
$620$	0	0
$621$	−115.059	−4.61716
$622$	0	0
$623$	−5.11527	−0.204939
$624$	0	0
$625$	−25.4517	−1.01807
$626$	0	0
$627$	−76.5431	−3.05684
$628$	0	0
$629$	0.697840	0.0278247
$630$	0	0
$631$	24.6880	0.982812	0.491406	−	0.870931i	$-0.336483\pi$
$631$	24.6880	0.982812	0.491406	+	0.870931i	$0.336483\pi$
$632$	0	0
$633$	85.8985	3.41416
$634$	0	0
$635$	−4.18255	−0.165979
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−68.0840	−2.69336
$640$	0	0
$641$	−18.6177	−0.735353	−0.367677	−	0.929954i	$-0.619847\pi$
$641$	−18.6177	−0.735353	−0.367677	+	0.929954i	$0.619847\pi$
$642$	0	0
$643$	−31.6831	−1.24946	−0.624729	−	0.780841i	$-0.714791\pi$
$643$	−31.6831	−1.24946	−0.624729	+	0.780841i	$0.714791\pi$
$644$	0	0
$645$	116.908	4.60323
$646$	0	0
$647$	29.6583	1.16599	0.582994	−	0.812477i	$-0.301881\pi$
$647$	29.6583	1.16599	0.582994	+	0.812477i	$0.301881\pi$
$648$	0	0
$649$	8.03013	0.315210
$650$	0	0
$651$	27.5175	1.07850
$652$	0	0
$653$	12.6897	0.496585	0.248292	−	0.968685i	$-0.420131\pi$
$653$	12.6897	0.496585	0.248292	+	0.968685i	$0.420131\pi$
$654$	0	0
$655$	1.86583	0.0729039
$656$	0	0
$657$	−24.6398	−0.961291
$658$	0	0
$659$	−24.5526	−0.956434	−0.478217	−	0.878242i	$-0.658717\pi$
$659$	−24.5526	−0.956434	−0.478217	+	0.878242i	$0.658717\pi$
$660$	0	0
$661$	25.5574	0.994067	0.497033	−	0.867731i	$-0.334423\pi$
$661$	25.5574	0.994067	0.497033	+	0.867731i	$0.334423\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	19.3778	0.751437
$666$	0	0
$667$	21.8111	0.844529
$668$	0	0
$669$	−48.8974	−1.89048
$670$	0	0
$671$	−13.1035	−0.505856
$672$	0	0
$673$	−6.79658	−0.261989	−0.130994	−	0.991383i	$-0.541817\pi$
$673$	−6.79658	−0.261989	−0.130994	+	0.991383i	$0.541817\pi$
$674$	0	0
$675$	70.3979	2.70962
$676$	0	0
$677$	−26.0956	−1.00294	−0.501468	−	0.865176i	$-0.667207\pi$
$677$	−26.0956	−1.00294	−0.501468	+	0.865176i	$0.667207\pi$
$678$	0	0
$679$	−9.62669	−0.369438
$680$	0	0
$681$	−69.1608	−2.65025
$682$	0	0
$683$	−48.6898	−1.86306	−0.931532	−	0.363660i	$-0.881527\pi$
$683$	−48.6898	−1.86306	−0.931532	+	0.363660i	$0.881527\pi$
$684$	0	0
$685$	−8.05917	−0.307925
$686$	0	0
$687$	−27.0152	−1.03069
$688$	0	0
$689$	0	0
$690$	0	0
$691$	3.59045	0.136587	0.0682935	−	0.997665i	$-0.478245\pi$
$691$	3.59045	0.136587	0.0682935	+	0.997665i	$0.478245\pi$
$692$	0	0
$693$	28.6282	1.08749
$694$	0	0
$695$	56.7308	2.15192
$696$	0	0
$697$	0.774161	0.0293234
$698$	0	0
$699$	−84.5435	−3.19773
$700$	0	0
$701$	15.8746	0.599575	0.299787	−	0.954006i	$-0.403084\pi$
$701$	15.8746	0.599575	0.299787	+	0.954006i	$0.403084\pi$
$702$	0	0
$703$	22.1025	0.833611
$704$	0	0
$705$	44.2532	1.66667
$706$	0	0
$707$	2.54158	0.0955858
$708$	0	0
$709$	22.6500	0.850637	0.425318	−	0.905044i	$-0.360162\pi$
$709$	22.6500	0.850637	0.425318	+	0.905044i	$0.360162\pi$
$710$	0	0
$711$	−66.8614	−2.50750
$712$	0	0
$713$	68.3179	2.55852
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−59.3089	−2.21493
$718$	0	0
$719$	10.2526	0.382358	0.191179	−	0.981555i	$-0.438769\pi$
$719$	10.2526	0.382358	0.191179	+	0.981555i	$0.438769\pi$
$720$	0	0
$721$	3.35392	0.124906
$722$	0	0
$723$	10.7748	0.400719
$724$	0	0
$725$	−13.3449	−0.495618
$726$	0	0
$727$	5.60059	0.207715	0.103857	−	0.994592i	$-0.466881\pi$
$727$	5.60059	0.207715	0.103857	+	0.994592i	$0.466881\pi$
$728$	0	0
$729$	39.7064	1.47061
$730$	0	0
$731$	−2.23429	−0.0826382
$732$	0	0
$733$	41.1556	1.52012	0.760058	−	0.649855i	$-0.225170\pi$
$733$	41.1556	1.52012	0.760058	+	0.649855i	$0.225170\pi$
$734$	0	0
$735$	−10.1702	−0.375133
$736$	0	0
$737$	20.8582	0.768323
$738$	0	0
$739$	12.1469	0.446833	0.223416	−	0.974723i	$-0.428279\pi$
$739$	12.1469	0.446833	0.223416	+	0.974723i	$0.428279\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−0.562043	−0.0206194	−0.0103097	−	0.999947i	$-0.503282\pi$
$743$	−0.562043	−0.0206194	−0.0103097	+	0.999947i	$0.503282\pi$
$744$	0	0
$745$	−25.5211	−0.935022
$746$	0	0
$747$	−57.0118	−2.08595
$748$	0	0
$749$	12.9130	0.471831
$750$	0	0
$751$	−22.7138	−0.828837	−0.414418	−	0.910086i	$-0.636015\pi$
$751$	−22.7138	−0.828837	−0.414418	+	0.910086i	$0.636015\pi$
$752$	0	0
$753$	−39.4107	−1.43621
$754$	0	0
$755$	11.2941	0.411033
$756$	0	0
$757$	19.4820	0.708085	0.354042	−	0.935229i	$-0.384807\pi$
$757$	19.4820	0.708085	0.354042	+	0.935229i	$0.384807\pi$
$758$	0	0
$759$	99.7372	3.62023
$760$	0	0
$761$	−42.7334	−1.54909	−0.774543	−	0.632522i	$-0.782020\pi$
$761$	−42.7334	−1.54909	−0.774543	+	0.632522i	$0.782020\pi$
$762$	0	0
$763$	−8.04268	−0.291164
$764$	0	0
$765$	−4.55150	−0.164560
$766$	0	0
$767$	0	0
$768$	0	0
$769$	34.1187	1.23035	0.615176	−	0.788390i	$-0.289085\pi$
$769$	34.1187	1.23035	0.615176	+	0.788390i	$0.289085\pi$
$770$	0	0
$771$	49.3469	1.77719
$772$	0	0
$773$	−28.8705	−1.03840	−0.519200	−	0.854653i	$-0.673770\pi$
$773$	−28.8705	−1.03840	−0.519200	+	0.854653i	$0.673770\pi$
$774$	0	0
$775$	−41.7997	−1.50149
$776$	0	0
$777$	−11.6002	−0.416156
$778$	0	0
$779$	24.5198	0.878512
$780$	0	0
$781$	35.2182	1.26021
$782$	0	0
$783$	−39.0008	−1.39377
$784$	0	0
$785$	−45.1528	−1.61157
$786$	0	0
$787$	−44.9696	−1.60299	−0.801497	−	0.597999i	$-0.795963\pi$
$787$	−44.9696	−1.60299	−0.801497	+	0.597999i	$0.795963\pi$
$788$	0	0
$789$	80.4484	2.86404
$790$	0	0
$791$	8.00469	0.284614
$792$	0	0
$793$	0	0
$794$	0	0
$795$	5.86620	0.208053
$796$	0	0
$797$	−21.4184	−0.758679	−0.379340	−	0.925257i	$-0.623849\pi$
$797$	−21.4184	−0.758679	−0.379340	+	0.925257i	$0.623849\pi$
$798$	0	0
$799$	−0.845748	−0.0299204
$800$	0	0
$801$	38.0544	1.34459
$802$	0	0
$803$	12.7456	0.449781
$804$	0	0
$805$	−25.2496	−0.889932
$806$	0	0
$807$	−25.3851	−0.893598
$808$	0	0
$809$	−8.25498	−0.290230	−0.145115	−	0.989415i	$-0.546355\pi$
$809$	−8.25498	−0.290230	−0.145115	+	0.989415i	$0.546355\pi$
$810$	0	0
$811$	−43.7679	−1.53690	−0.768450	−	0.639910i	$-0.778971\pi$
$811$	−43.7679	−1.53690	−0.768450	+	0.639910i	$0.778971\pi$
$812$	0	0
$813$	8.52548	0.299002
$814$	0	0
$815$	0.774775	0.0271392
$816$	0	0
$817$	−70.7659	−2.47579
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−22.1084	−0.771589	−0.385795	−	0.922585i	$-0.626073\pi$
$821$	−22.1084	−0.771589	−0.385795	+	0.922585i	$0.626073\pi$
$822$	0	0
$823$	−42.0711	−1.46650	−0.733252	−	0.679957i	$-0.761999\pi$
$823$	−42.0711	−1.46650	−0.733252	+	0.679957i	$0.761999\pi$
$824$	0	0
$825$	−61.0234	−2.12456
$826$	0	0
$827$	−44.4242	−1.54478	−0.772390	−	0.635148i	$-0.780939\pi$
$827$	−44.4242	−1.54478	−0.772390	+	0.635148i	$0.780939\pi$
$828$	0	0
$829$	−17.9479	−0.623355	−0.311678	−	0.950188i	$-0.600891\pi$
$829$	−17.9479	−0.623355	−0.311678	+	0.950188i	$0.600891\pi$
$830$	0	0
$831$	37.6569	1.30630
$832$	0	0
$833$	0.194369	0.00673448
$834$	0	0
$835$	−5.69201	−0.196980
$836$	0	0
$837$	−122.160	−4.22248
$838$	0	0
$839$	−3.57419	−0.123395	−0.0616973	−	0.998095i	$-0.519651\pi$
$839$	−3.57419	−0.123395	−0.0616973	+	0.998095i	$0.519651\pi$
$840$	0	0
$841$	−21.6068	−0.745064
$842$	0	0
$843$	12.4878	0.430104
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−3.80865	−0.130867
$848$	0	0
$849$	22.1279	0.759429
$850$	0	0
$851$	−28.8000	−0.987251
$852$	0	0
$853$	24.4780	0.838111	0.419055	−	0.907961i	$-0.362361\pi$
$853$	24.4780	0.838111	0.419055	+	0.907961i	$0.362361\pi$
$854$	0	0
$855$	−144.158	−4.93011
$856$	0	0
$857$	26.9438	0.920384	0.460192	−	0.887820i	$-0.347781\pi$
$857$	26.9438	0.920384	0.460192	+	0.887820i	$0.347781\pi$
$858$	0	0
$859$	−17.9537	−0.612573	−0.306287	−	0.951939i	$-0.599087\pi$
$859$	−17.9537	−0.612573	−0.306287	+	0.951939i	$0.599087\pi$
$860$	0	0
$861$	−12.8689	−0.438572
$862$	0	0
$863$	−11.0841	−0.377308	−0.188654	−	0.982044i	$-0.560412\pi$
$863$	−11.0841	−0.377308	−0.188654	+	0.982044i	$0.560412\pi$
$864$	0	0
$865$	33.9939	1.15583
$866$	0	0
$867$	−54.8050	−1.86127
$868$	0	0
$869$	34.5858	1.17324
$870$	0	0
$871$	0	0
$872$	0	0
$873$	71.6165	2.42385
$874$	0	0
$875$	−0.289701	−0.00979367
$876$	0	0
$877$	−4.60737	−0.155580	−0.0777899	−	0.996970i	$-0.524786\pi$
$877$	−4.60737	−0.155580	−0.0777899	+	0.996970i	$0.524786\pi$
$878$	0	0
$879$	−4.81075	−0.162263
$880$	0	0
$881$	−34.3107	−1.15596	−0.577979	−	0.816052i	$-0.696158\pi$
$881$	−34.3107	−1.15596	−0.577979	+	0.816052i	$0.696158\pi$
$882$	0	0
$883$	−42.5926	−1.43336	−0.716678	−	0.697404i	$-0.754338\pi$
$883$	−42.5926	−1.43336	−0.716678	+	0.697404i	$0.754338\pi$
$884$	0	0
$885$	21.2224	0.713382
$886$	0	0
$887$	41.6052	1.39697	0.698483	−	0.715627i	$-0.253859\pi$
$887$	41.6052	1.39697	0.698483	+	0.715627i	$0.253859\pi$
$888$	0	0
$889$	−1.32877	−0.0445654
$890$	0	0
$891$	−92.4572	−3.09743
$892$	0	0
$893$	−26.7871	−0.896397
$894$	0	0
$895$	−27.3108	−0.912901
$896$	0	0
$897$	0	0
$898$	0	0
$899$	23.1572	0.772337
$900$	0	0
$901$	−0.112112	−0.00373501
$902$	0	0
$903$	37.1407	1.23597
$904$	0	0
$905$	−31.5043	−1.04724
$906$	0	0
$907$	15.6716	0.520367	0.260183	−	0.965559i	$-0.416217\pi$
$907$	15.6716	0.520367	0.260183	+	0.965559i	$0.416217\pi$
$908$	0	0
$909$	−18.9077	−0.627129
$910$	0	0
$911$	−33.5478	−1.11149	−0.555743	−	0.831354i	$-0.687566\pi$
$911$	−33.5478	−1.11149	−0.555743	+	0.831354i	$0.687566\pi$
$912$	0	0
$913$	29.4908	0.976002
$914$	0	0
$915$	−34.6306	−1.14485
$916$	0	0
$917$	0.592761	0.0195747
$918$	0	0
$919$	3.33761	0.110098	0.0550488	−	0.998484i	$-0.482469\pi$
$919$	3.33761	0.110098	0.0550488	+	0.998484i	$0.482469\pi$
$920$	0	0
$921$	7.53236	0.248200
$922$	0	0
$923$	0	0
$924$	0	0
$925$	17.6210	0.579376
$926$	0	0
$927$	−24.9510	−0.819499
$928$	0	0
$929$	9.09688	0.298459	0.149230	−	0.988803i	$-0.452321\pi$
$929$	9.09688	0.298459	0.149230	+	0.988803i	$0.452321\pi$
$930$	0	0
$931$	6.15618	0.201761
$932$	0	0
$933$	−38.2047	−1.25077
$934$	0	0
$935$	2.35438	0.0769964
$936$	0	0
$937$	9.08442	0.296775	0.148388	−	0.988929i	$-0.452592\pi$
$937$	9.08442	0.296775	0.148388	+	0.988929i	$0.452592\pi$
$938$	0	0
$939$	−59.4785	−1.94101
$940$	0	0
$941$	−8.23345	−0.268403	−0.134201	−	0.990954i	$-0.542847\pi$
$941$	−8.23345	−0.268403	−0.134201	+	0.990954i	$0.542847\pi$
$942$	0	0
$943$	−31.9498	−1.04043
$944$	0	0
$945$	45.1492	1.46871
$946$	0	0
$947$	5.05532	0.164276	0.0821380	−	0.996621i	$-0.473825\pi$
$947$	5.05532	0.164276	0.0821380	+	0.996621i	$0.473825\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	27.2139	0.882471
$952$	0	0
$953$	39.3158	1.27356	0.636782	−	0.771044i	$-0.280265\pi$
$953$	39.3158	1.27356	0.636782	+	0.771044i	$0.280265\pi$
$954$	0	0
$955$	75.6429	2.44775
$956$	0	0
$957$	33.8072	1.09283
$958$	0	0
$959$	−2.56034	−0.0826778
$960$	0	0
$961$	41.5344	1.33982
$962$	0	0
$963$	−96.0646	−3.09564
$964$	0	0
$965$	10.3026	0.331653
$966$	0	0
$967$	−21.2101	−0.682071	−0.341036	−	0.940050i	$-0.610778\pi$
$967$	−21.2101	−0.682071	−0.341036	+	0.940050i	$0.610778\pi$
$968$	0	0
$969$	3.86611	0.124197
$970$	0	0
$971$	−23.5410	−0.755466	−0.377733	−	0.925915i	$-0.623296\pi$
$971$	−23.5410	−0.755466	−0.377733	+	0.925915i	$0.623296\pi$
$972$	0	0
$973$	18.0230	0.577791
$974$	0	0
$975$	0	0
$976$	0	0
$977$	12.3992	0.396684	0.198342	−	0.980133i	$-0.436444\pi$
$977$	12.3992	0.396684	0.198342	+	0.980133i	$0.436444\pi$
$978$	0	0
$979$	−19.6846	−0.629123
$980$	0	0
$981$	59.8324	1.91030
$982$	0	0
$983$	−36.0862	−1.15097	−0.575486	−	0.817812i	$-0.695187\pi$
$983$	−36.0862	−1.15097	−0.575486	+	0.817812i	$0.695187\pi$
$984$	0	0
$985$	40.1406	1.27898
$986$	0	0
$987$	14.0589	0.447500
$988$	0	0
$989$	92.2095	2.93209
$990$	0	0
$991$	52.8491	1.67881	0.839403	−	0.543510i	$-0.182905\pi$
$991$	52.8491	1.67881	0.839403	+	0.543510i	$0.182905\pi$
$992$	0	0
$993$	58.3409	1.85139
$994$	0	0
$995$	−86.0047	−2.72653
$996$	0	0
$997$	1.40828	0.0446006	0.0223003	−	0.999751i	$-0.492901\pi$
$997$	1.40828	0.0446006	0.0223003	+	0.999751i	$0.492901\pi$
$998$	0	0
$999$	51.4977	1.62932

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4732.2.a.s.1.8		8
13.5	odd	4		4732.2.g.k.337.16		16
13.6	odd	12		364.2.u.a.309.1	yes	16
13.8	odd	4		4732.2.g.k.337.15		16
13.11	odd	12		364.2.u.a.225.1	✓	16
13.12	even	2		4732.2.a.t.1.8		8
39.11	even	12		3276.2.cf.c.2773.7		16
39.32	even	12		3276.2.cf.c.1765.2		16
52.11	even	12		1456.2.cc.f.225.8		16
52.19	even	12		1456.2.cc.f.673.8		16
91.6	even	12		2548.2.u.c.1765.8		16
91.11	odd	12		2548.2.bq.e.1941.8		16
91.19	even	12		2548.2.bq.c.361.1		16
91.24	even	12		2548.2.bq.c.1941.1		16
91.32	odd	12		2548.2.bb.d.569.1		16
91.37	odd	12		2548.2.bb.d.1733.1		16
91.45	even	12		2548.2.bb.c.569.8		16
91.58	odd	12		2548.2.bq.e.361.8		16
91.76	even	12		2548.2.u.c.589.8		16
91.89	even	12		2548.2.bb.c.1733.8		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
364.2.u.a.225.1	✓	16	13.11	odd	12
364.2.u.a.309.1	yes	16	13.6	odd	12
1456.2.cc.f.225.8		16	52.11	even	12
1456.2.cc.f.673.8		16	52.19	even	12
2548.2.u.c.589.8		16	91.76	even	12
2548.2.u.c.1765.8		16	91.6	even	12
2548.2.bb.c.569.8		16	91.45	even	12
2548.2.bb.c.1733.8		16	91.89	even	12
2548.2.bb.d.569.1		16	91.32	odd	12
2548.2.bb.d.1733.1		16	91.37	odd	12
2548.2.bq.c.361.1		16	91.19	even	12
2548.2.bq.c.1941.1		16	91.24	even	12
2548.2.bq.e.361.8		16	91.58	odd	12
2548.2.bq.e.1941.8		16	91.11	odd	12
3276.2.cf.c.1765.2		16	39.32	even	12
3276.2.cf.c.2773.7		16	39.11	even	12
4732.2.a.s.1.8		8	1.1	even	1	trivial
4732.2.a.t.1.8		8	13.12	even	2
4732.2.g.k.337.15		16	13.8	odd	4
4732.2.g.k.337.16		16	13.5	odd	4

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:	\( N \)	\(=\)	\( 4732 = 2^{2} \cdot 7 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4732.a (trivial)

\(f(q)\)	\(=\)	\(q+3.23100 q^{3} -3.14769 q^{5} -1.00000 q^{7} +7.43937 q^{9} -3.84820 q^{11} -10.1702 q^{15} +0.194369 q^{17} +6.15618 q^{19} -3.23100 q^{21} -8.02163 q^{23} +4.90796 q^{25} +14.3436 q^{27} -2.71904 q^{29} -8.51671 q^{31} -12.4335 q^{33} +3.14769 q^{35} +3.59029 q^{37} +3.98295 q^{41} -11.4951 q^{43} -23.4168 q^{45} -4.35126 q^{47} +1.00000 q^{49} +0.628005 q^{51} -0.576803 q^{53} +12.1129 q^{55} +19.8906 q^{57} -2.08672 q^{59} +3.40510 q^{61} -7.43937 q^{63} -5.42026 q^{67} -25.9179 q^{69} -9.15185 q^{71} -3.31209 q^{73} +15.8576 q^{75} +3.84820 q^{77} -8.98752 q^{79} +24.0261 q^{81} -7.66353 q^{83} -0.611813 q^{85} -8.78521 q^{87} +5.11527 q^{89} -27.5175 q^{93} -19.3778 q^{95} +9.62669 q^{97} -28.6282 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 6 q^{5} - 8 q^{7} + 14 q^{9} - 12 q^{11} - 12 q^{15} + 2 q^{17} - 6 q^{19} + 22 q^{25} - 6 q^{27} + 22 q^{29} - 14 q^{31} + 28 q^{33} + 6 q^{35} - 12 q^{37} - 4 q^{41} + 6 q^{43} - 20 q^{45} - 42 q^{47}+ \cdots - 80 q^{99}+O(q^{100}) \) 8 * q - 6 * q^5 - 8 * q^7 + 14 * q^9 - 12 * q^11 - 12 * q^15 + 2 * q^17 - 6 * q^19 + 22 * q^25 - 6 * q^27 + 22 * q^29 - 14 * q^31 + 28 * q^33 + 6 * q^35 - 12 * q^37 - 4 * q^41 + 6 * q^43 - 20 * q^45 - 42 * q^47 + 8 * q^49 + 2 * q^51 + 4 * q^53 - 2 * q^55 + 8 * q^57 - 2 * q^59 - 4 * q^61 - 14 * q^63 - 24 * q^67 - 52 * q^69 - 28 * q^71 - 28 * q^73 - 10 * q^75 + 12 * q^77 + 4 * q^79 - 38 * q^83 + 36 * q^85 - 26 * q^87 + 22 * q^89 - 56 * q^93 - 30 * q^95 + 4 * q^97 - 80 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more