LMFDB - Embedded newform 9248.2.a.bz.1.20

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [9248,2,Mod(1,9248)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9248, 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("9248.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 9248 = 2^{5} \cdot 17^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9248.a (trivial)

Newform invariants

comment:select newform

sage:traces = [20,0,0,0,0,0,0,0,28,0,0,0,-8,0,16,0,0,0,40,0,-32,0,0,0,28,0, 0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,40,0,0,0,32,0,36,0,0,0,-40,0,48,0,0,0, 8,0,0,0,0,0,0,0,72,0,-48,0,0,0,0,0,0,0,-48,0,0,0,36,0,24,0,0,0,96,0,64, 0,0,0,-32,0,0,0,0,0,0,0,-40,0,128,0,0,0,0,0,0,0,48,0,0,0,96,0,-72,0,0, 0,60,0,32,0,0,0,32,0,0,0,0,0,0,0,112,0,16,0,0,0,0,0,0,0,16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(145)] == traces)

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

Self dual:	yes
Analytic conductor:	$73.8456517893$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 40 x^{18} + 620 x^{16} - 4784 x^{14} + 19585 x^{12} - 41912 x^{10} + 43536 x^{8} - 20328 x^{6} + \cdots + 8 $ x^20 - 40x^18 + 620x^16 - 4784x^14 + 19585x^12 - 41912x^10 + 43536x^8 - 20328x^6 + 4208x^4 - 352*x^2 + 8
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{10} $
Twist minimal:	no (minimal twist has level 544)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.20
Root			$0.477107$ of defining polynomial
Character	$\chi$	$=$	9248.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+3.39376 q^{3} +2.27168 q^{5} +0.186067 q^{7} +8.51759 q^{9} +1.19207 q^{11} -2.84300 q^{13} +7.70955 q^{15} +4.60126 q^{19} +0.631465 q^{21} -5.60989 q^{23} +0.160551 q^{25} +18.7253 q^{27} +7.06390 q^{29} -8.72354 q^{31} +4.04561 q^{33} +0.422684 q^{35} -7.81018 q^{37} -9.64847 q^{39} +4.47854 q^{41} -0.493780 q^{43} +19.3493 q^{45} +8.40946 q^{47} -6.96538 q^{49} +8.95423 q^{53} +2.70801 q^{55} +15.6156 q^{57} +11.0983 q^{59} +4.88796 q^{61} +1.58484 q^{63} -6.45841 q^{65} -0.650160 q^{67} -19.0386 q^{69} +6.93455 q^{71} -4.09576 q^{73} +0.544870 q^{75} +0.221805 q^{77} +1.59878 q^{79} +37.9965 q^{81} +4.79814 q^{83} +23.9732 q^{87} +5.87757 q^{89} -0.528988 q^{91} -29.6056 q^{93} +10.4526 q^{95} +2.97084 q^{97} +10.1536 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 28 q^{9} - 8 q^{13} + 16 q^{15} + 40 q^{19} - 32 q^{21} + 28 q^{25} + 32 q^{35} + 40 q^{43} + 32 q^{47} + 36 q^{49} - 40 q^{53} + 48 q^{55} + 8 q^{59} + 72 q^{67} - 48 q^{69} - 48 q^{77} + 36 q^{81}+ \cdots - 32 q^{93}+O(q^{100}) $ 20 * q + 28 * q^9 - 8 * q^13 + 16 * q^15 + 40 * q^19 - 32 * q^21 + 28 * q^25 + 32 * q^35 + 40 * q^43 + 32 * q^47 + 36 * q^49 - 40 * q^53 + 48 * q^55 + 8 * q^59 + 72 * q^67 - 48 * q^69 - 48 * q^77 + 36 * q^81 + 24 * q^83 + 96 * q^87 + 64 * q^89 - 32 * q^93

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
$3$	3.39376	1.95939	0.979693	−	0.200502i	$-0.0642574\pi$
$3$	3.39376	1.95939	0.979693	+	0.200502i	$0.0642574\pi$
$4$	0	0
$5$	2.27168	1.01593	0.507964	−	0.861378i	$-0.330398\pi$
$5$	2.27168	1.01593	0.507964	+	0.861378i	$0.330398\pi$
$6$	0	0
$7$	0.186067	0.0703265	0.0351633	−	0.999382i	$-0.488805\pi$
$7$	0.186067	0.0703265	0.0351633	+	0.999382i	$0.488805\pi$
$8$	0	0
$9$	8.51759	2.83920
$10$	0	0
$11$	1.19207	0.359424	0.179712	−	0.983719i	$-0.442483\pi$
$11$	1.19207	0.359424	0.179712	+	0.983719i	$0.442483\pi$
$12$	0	0
$13$	−2.84300	−0.788507	−0.394254	−	0.919002i	$-0.628997\pi$
$13$	−2.84300	−0.788507	−0.394254	+	0.919002i	$0.628997\pi$
$14$	0	0
$15$	7.70955	1.99060
$16$	0	0
$17$	0	0
$17$	0	0
$18$	0	0
$19$	4.60126	1.05560	0.527801	−	0.849368i	$-0.323017\pi$
$19$	4.60126	1.05560	0.527801	+	0.849368i	$0.323017\pi$
$20$	0	0
$21$	0.631465	0.137797
$22$	0	0
$23$	−5.60989	−1.16974	−0.584872	−	0.811126i	$-0.698855\pi$
$23$	−5.60989	−1.16974	−0.584872	+	0.811126i	$0.698855\pi$
$24$	0	0
$25$	0.160551	0.0321101
$26$	0	0
$27$	18.7253	3.60369
$28$	0	0
$29$	7.06390	1.31173	0.655867	−	0.754876i	$-0.272303\pi$
$29$	7.06390	1.31173	0.655867	+	0.754876i	$0.272303\pi$
$30$	0	0
$31$	−8.72354	−1.56679	−0.783397	−	0.621521i	$-0.786515\pi$
$31$	−8.72354	−1.56679	−0.783397	+	0.621521i	$0.786515\pi$
$32$	0	0
$33$	4.04561	0.704250
$34$	0	0
$35$	0.422684	0.0714467
$36$	0	0
$37$	−7.81018	−1.28399	−0.641993	−	0.766711i	$-0.721892\pi$
$37$	−7.81018	−1.28399	−0.641993	+	0.766711i	$0.721892\pi$
$38$	0	0
$39$	−9.64847	−1.54499
$40$	0	0
$41$	4.47854	0.699430	0.349715	−	0.936856i	$-0.386278\pi$
$41$	4.47854	0.699430	0.349715	+	0.936856i	$0.386278\pi$
$42$	0	0
$43$	−0.493780	−0.0753007	−0.0376503	−	0.999291i	$-0.511987\pi$
$43$	−0.493780	−0.0753007	−0.0376503	+	0.999291i	$0.511987\pi$
$44$	0	0
$45$	19.3493	2.88442
$46$	0	0
$47$	8.40946	1.22665	0.613323	−	0.789832i	$-0.289832\pi$
$47$	8.40946	1.22665	0.613323	+	0.789832i	$0.289832\pi$
$48$	0	0
$49$	−6.96538	−0.995054
$50$	0	0
$51$	0	0
$52$	0	0
$53$	8.95423	1.22996	0.614979	−	0.788544i	$-0.289165\pi$
$53$	8.95423	1.22996	0.614979	+	0.788544i	$0.289165\pi$
$54$	0	0
$55$	2.70801	0.365149
$56$	0	0
$57$	15.6156	2.06833
$58$	0	0
$59$	11.0983	1.44488	0.722440	−	0.691433i	$-0.243020\pi$
$59$	11.0983	1.44488	0.722440	+	0.691433i	$0.243020\pi$
$60$	0	0
$61$	4.88796	0.625839	0.312920	−	0.949780i	$-0.398693\pi$
$61$	4.88796	0.625839	0.312920	+	0.949780i	$0.398693\pi$
$62$	0	0
$63$	1.58484	0.199671
$64$	0	0
$65$	−6.45841	−0.801067
$66$	0	0
$67$	−0.650160	−0.0794297	−0.0397149	−	0.999211i	$-0.512645\pi$
$67$	−0.650160	−0.0794297	−0.0397149	+	0.999211i	$0.512645\pi$
$68$	0	0
$69$	−19.0386	−2.29198
$70$	0	0
$71$	6.93455	0.822979	0.411490	−	0.911414i	$-0.365009\pi$
$71$	6.93455	0.822979	0.411490	+	0.911414i	$0.365009\pi$
$72$	0	0
$73$	−4.09576	−0.479373	−0.239686	−	0.970850i	$-0.577045\pi$
$73$	−4.09576	−0.479373	−0.239686	+	0.970850i	$0.577045\pi$
$74$	0	0
$75$	0.544870	0.0629161
$76$	0	0
$77$	0.221805	0.0252770
$78$	0	0
$79$	1.59878	0.179876	0.0899382	−	0.995947i	$-0.471333\pi$
$79$	1.59878	0.179876	0.0899382	+	0.995947i	$0.471333\pi$
$80$	0	0
$81$	37.9965	4.22184
$82$	0	0
$83$	4.79814	0.526665	0.263332	−	0.964705i	$-0.415178\pi$
$83$	4.79814	0.526665	0.263332	+	0.964705i	$0.415178\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	23.9732	2.57019
$88$	0	0
$89$	5.87757	0.623021	0.311510	−	0.950243i	$-0.399165\pi$
$89$	5.87757	0.623021	0.311510	+	0.950243i	$0.399165\pi$
$90$	0	0
$91$	−0.528988	−0.0554530
$92$	0	0
$93$	−29.6056	−3.06996
$94$	0	0
$95$	10.4526	1.07242
$96$	0	0
$97$	2.97084	0.301643	0.150821	−	0.988561i	$-0.451808\pi$
$97$	2.97084	0.301643	0.150821	+	0.988561i	$0.451808\pi$
$98$	0	0
$99$	10.1536	1.02047
$100$	0	0
$101$	−8.27417	−0.823311	−0.411656	−	0.911340i	$-0.635049\pi$
$101$	−8.27417	−0.823311	−0.411656	+	0.911340i	$0.635049\pi$
$102$	0	0
$103$	0.744658	0.0733734	0.0366867	−	0.999327i	$-0.488320\pi$
$103$	0.744658	0.0733734	0.0366867	+	0.999327i	$0.488320\pi$
$104$	0	0
$105$	1.43449	0.139992
$106$	0	0
$107$	14.6373	1.41504	0.707521	−	0.706693i	$-0.249814\pi$
$107$	14.6373	1.41504	0.707521	+	0.706693i	$0.249814\pi$
$108$	0	0
$109$	12.0516	1.15434	0.577168	−	0.816625i	$-0.304158\pi$
$109$	12.0516	1.15434	0.577168	+	0.816625i	$0.304158\pi$
$110$	0	0
$111$	−26.5059	−2.51582
$112$	0	0
$113$	7.69564	0.723945	0.361972	−	0.932189i	$-0.382104\pi$
$113$	7.69564	0.723945	0.361972	+	0.932189i	$0.382104\pi$
$114$	0	0
$115$	−12.7439	−1.18838
$116$	0	0
$117$	−24.2155	−2.23873
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−9.57896	−0.870815
$122$	0	0
$123$	15.1991	1.37045
$124$	0	0
$125$	−10.9937	−0.983307
$126$	0	0
$127$	−4.77984	−0.424142	−0.212071	−	0.977254i	$-0.568021\pi$
$127$	−4.77984	−0.424142	−0.212071	+	0.977254i	$0.568021\pi$
$128$	0	0
$129$	−1.67577	−0.147543
$130$	0	0
$131$	5.03058	0.439524	0.219762	−	0.975554i	$-0.429472\pi$
$131$	5.03058	0.439524	0.219762	+	0.975554i	$0.429472\pi$
$132$	0	0
$133$	0.856141	0.0742368
$134$	0	0
$135$	42.5381	3.66110
$136$	0	0
$137$	−9.57731	−0.818244	−0.409122	−	0.912480i	$-0.634165\pi$
$137$	−9.57731	−0.818244	−0.409122	+	0.912480i	$0.634165\pi$
$138$	0	0
$139$	−11.1075	−0.942122	−0.471061	−	0.882101i	$-0.656129\pi$
$139$	−11.1075	−0.942122	−0.471061	+	0.882101i	$0.656129\pi$
$140$	0	0
$141$	28.5397	2.40347
$142$	0	0
$143$	−3.38907	−0.283408
$144$	0	0
$145$	16.0470	1.33263
$146$	0	0
$147$	−23.6388	−1.94970
$148$	0	0
$149$	−11.4334	−0.936657	−0.468329	−	0.883554i	$-0.655144\pi$
$149$	−11.4334	−0.936657	−0.468329	+	0.883554i	$0.655144\pi$
$150$	0	0
$151$	−20.1594	−1.64055	−0.820275	−	0.571970i	$-0.806179\pi$
$151$	−20.1594	−1.64055	−0.820275	+	0.571970i	$0.806179\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−19.8171	−1.59175
$156$	0	0
$157$	11.6143	0.926925	0.463463	−	0.886116i	$-0.346607\pi$
$157$	11.6143	0.926925	0.463463	+	0.886116i	$0.346607\pi$
$158$	0	0
$159$	30.3885	2.40996
$160$	0	0
$161$	−1.04381	−0.0822640
$162$	0	0
$163$	−13.0569	−1.02269	−0.511347	−	0.859375i	$-0.670853\pi$
$163$	−13.0569	−1.02269	−0.511347	+	0.859375i	$0.670853\pi$
$164$	0	0
$165$	9.19034	0.715467
$166$	0	0
$167$	12.1544	0.940538	0.470269	−	0.882523i	$-0.344157\pi$
$167$	12.1544	0.940538	0.470269	+	0.882523i	$0.344157\pi$
$168$	0	0
$169$	−4.91733	−0.378256
$170$	0	0
$171$	39.1917	2.99706
$172$	0	0
$173$	−10.0019	−0.760430	−0.380215	−	0.924898i	$-0.624150\pi$
$173$	−10.0019	−0.760430	−0.380215	+	0.924898i	$0.624150\pi$
$174$	0	0
$175$	0.0298731	0.00225819
$176$	0	0
$177$	37.6651	2.83108
$178$	0	0
$179$	10.0633	0.752167	0.376084	−	0.926586i	$-0.377271\pi$
$179$	10.0633	0.752167	0.376084	+	0.926586i	$0.377271\pi$
$180$	0	0
$181$	18.7651	1.39480	0.697399	−	0.716683i	$-0.254340\pi$
$181$	18.7651	1.39480	0.697399	+	0.716683i	$0.254340\pi$
$182$	0	0
$183$	16.5886	1.22626
$184$	0	0
$185$	−17.7423	−1.30444
$186$	0	0
$187$	0	0
$188$	0	0
$189$	3.48416	0.253435
$190$	0	0
$191$	−13.6260	−0.985940	−0.492970	−	0.870046i	$-0.664089\pi$
$191$	−13.6260	−0.985940	−0.492970	+	0.870046i	$0.664089\pi$
$192$	0	0
$193$	−20.7673	−1.49486	−0.747431	−	0.664340i	$-0.768713\pi$
$193$	−20.7673	−1.49486	−0.747431	+	0.664340i	$0.768713\pi$
$194$	0	0
$195$	−21.9183	−1.56960
$196$	0	0
$197$	13.4565	0.958738	0.479369	−	0.877614i	$-0.340866\pi$
$197$	13.4565	0.958738	0.479369	+	0.877614i	$0.340866\pi$
$198$	0	0
$199$	−18.1793	−1.28870	−0.644348	−	0.764733i	$-0.722871\pi$
$199$	−18.1793	−1.28870	−0.644348	+	0.764733i	$0.722871\pi$
$200$	0	0
$201$	−2.20649	−0.155633
$202$	0	0
$203$	1.31436	0.0922497
$204$	0	0
$205$	10.1738	0.710571
$206$	0	0
$207$	−47.7827	−3.32113
$208$	0	0
$209$	5.48504	0.379408
$210$	0	0
$211$	−5.24428	−0.361031	−0.180515	−	0.983572i	$-0.557777\pi$
$211$	−5.24428	−0.361031	−0.180515	+	0.983572i	$0.557777\pi$
$212$	0	0
$213$	23.5342	1.61253
$214$	0	0
$215$	−1.12171	−0.0765001
$216$	0	0
$217$	−1.62316	−0.110187
$218$	0	0
$219$	−13.9000	−0.939276
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−1.13949	−0.0763057	−0.0381529	−	0.999272i	$-0.512147\pi$
$223$	−1.13949	−0.0763057	−0.0381529	+	0.999272i	$0.512147\pi$
$224$	0	0
$225$	1.36750	0.0911669
$226$	0	0
$227$	−2.71599	−0.180266	−0.0901332	−	0.995930i	$-0.528729\pi$
$227$	−2.71599	−0.180266	−0.0901332	+	0.995930i	$0.528729\pi$
$228$	0	0
$229$	−3.97062	−0.262386	−0.131193	−	0.991357i	$-0.541881\pi$
$229$	−3.97062	−0.262386	−0.131193	+	0.991357i	$0.541881\pi$
$230$	0	0
$231$	0.752752	0.0495275
$232$	0	0
$233$	−7.09286	−0.464669	−0.232334	−	0.972636i	$-0.574636\pi$
$233$	−7.09286	−0.464669	−0.232334	+	0.972636i	$0.574636\pi$
$234$	0	0
$235$	19.1036	1.24618
$236$	0	0
$237$	5.42586	0.352447
$238$	0	0
$239$	−12.9067	−0.834866	−0.417433	−	0.908708i	$-0.637070\pi$
$239$	−12.9067	−0.834866	−0.417433	+	0.908708i	$0.637070\pi$
$240$	0	0
$241$	−12.7571	−0.821755	−0.410877	−	0.911691i	$-0.634778\pi$
$241$	−12.7571	−0.821755	−0.410877	+	0.911691i	$0.634778\pi$
$242$	0	0
$243$	72.7749	4.66851
$244$	0	0
$245$	−15.8231	−1.01090
$246$	0	0
$247$	−13.0814	−0.832350
$248$	0	0
$249$	16.2837	1.03194
$250$	0	0
$251$	−10.0282	−0.632974	−0.316487	−	0.948597i	$-0.602503\pi$
$251$	−10.0282	−0.632974	−0.316487	+	0.948597i	$0.602503\pi$
$252$	0	0
$253$	−6.68740	−0.420433
$254$	0	0
$255$	0	0
$256$	0	0
$257$	10.6402	0.663718	0.331859	−	0.943329i	$-0.392324\pi$
$257$	10.6402	0.663718	0.331859	+	0.943329i	$0.392324\pi$
$258$	0	0
$259$	−1.45321	−0.0902983
$260$	0	0
$261$	60.1674	3.72427
$262$	0	0
$263$	−29.8304	−1.83942	−0.919710	−	0.392599i	$-0.871576\pi$
$263$	−29.8304	−1.83942	−0.919710	+	0.392599i	$0.871576\pi$
$264$	0	0
$265$	20.3412	1.24955
$266$	0	0
$267$	19.9470	1.22074
$268$	0	0
$269$	−17.3642	−1.05871	−0.529357	−	0.848399i	$-0.677567\pi$
$269$	−17.3642	−1.05871	−0.529357	+	0.848399i	$0.677567\pi$
$270$	0	0
$271$	16.7401	1.01689	0.508445	−	0.861095i	$-0.330221\pi$
$271$	16.7401	1.01689	0.508445	+	0.861095i	$0.330221\pi$
$272$	0	0
$273$	−1.79526	−0.108654
$274$	0	0
$275$	0.191388	0.0115411
$276$	0	0
$277$	−10.9479	−0.657793	−0.328896	−	0.944366i	$-0.606677\pi$
$277$	−10.9479	−0.657793	−0.328896	+	0.944366i	$0.606677\pi$
$278$	0	0
$279$	−74.3035	−4.44844
$280$	0	0
$281$	2.14920	0.128211	0.0641053	−	0.997943i	$-0.479581\pi$
$281$	2.14920	0.128211	0.0641053	+	0.997943i	$0.479581\pi$
$282$	0	0
$283$	−6.02720	−0.358280	−0.179140	−	0.983824i	$-0.557331\pi$
$283$	−6.02720	−0.358280	−0.179140	+	0.983824i	$0.557331\pi$
$284$	0	0
$285$	35.4736	2.10128
$286$	0	0
$287$	0.833306	0.0491885
$288$	0	0
$289$	0	0
$290$	0	0
$291$	10.0823	0.591034
$292$	0	0
$293$	−16.0762	−0.939181	−0.469591	−	0.882884i	$-0.655599\pi$
$293$	−16.0762	−0.939181	−0.469591	+	0.882884i	$0.655599\pi$
$294$	0	0
$295$	25.2119	1.46790
$296$	0	0
$297$	22.3220	1.29525
$298$	0	0
$299$	15.9489	0.922351
$300$	0	0
$301$	−0.0918759	−0.00529564
$302$	0	0
$303$	−28.0805	−1.61318
$304$	0	0
$305$	11.1039	0.635808
$306$	0	0
$307$	13.6643	0.779861	0.389930	−	0.920844i	$-0.372499\pi$
$307$	13.6643	0.779861	0.389930	+	0.920844i	$0.372499\pi$
$308$	0	0
$309$	2.52719	0.143767
$310$	0	0
$311$	13.0617	0.740662	0.370331	−	0.928900i	$-0.379244\pi$
$311$	13.0617	0.740662	0.370331	+	0.928900i	$0.379244\pi$
$312$	0	0
$313$	−27.0908	−1.53126	−0.765631	−	0.643280i	$-0.777573\pi$
$313$	−27.0908	−1.53126	−0.765631	+	0.643280i	$0.777573\pi$
$314$	0	0
$315$	3.60025	0.202851
$316$	0	0
$317$	2.39770	0.134669	0.0673343	−	0.997730i	$-0.478551\pi$
$317$	2.39770	0.134669	0.0673343	+	0.997730i	$0.478551\pi$
$318$	0	0
$319$	8.42069	0.471468
$320$	0	0
$321$	49.6754	2.77261
$322$	0	0
$323$	0	0
$324$	0	0
$325$	−0.456446	−0.0253191
$326$	0	0
$327$	40.9003	2.26179
$328$	0	0
$329$	1.56472	0.0862658
$330$	0	0
$331$	1.77188	0.0973911	0.0486955	−	0.998814i	$-0.484494\pi$
$331$	1.77188	0.0973911	0.0486955	+	0.998814i	$0.484494\pi$
$332$	0	0
$333$	−66.5239	−3.64549
$334$	0	0
$335$	−1.47696	−0.0806949
$336$	0	0
$337$	−29.1295	−1.58678	−0.793392	−	0.608711i	$-0.791687\pi$
$337$	−29.1295	−1.58678	−0.793392	+	0.608711i	$0.791687\pi$
$338$	0	0
$339$	26.1171	1.41849
$340$	0	0
$341$	−10.3991	−0.563143
$342$	0	0
$343$	−2.59849	−0.140305
$344$	0	0
$345$	−43.2497	−2.32849
$346$	0	0
$347$	6.69745	0.359538	0.179769	−	0.983709i	$-0.442465\pi$
$347$	6.69745	0.359538	0.179769	+	0.983709i	$0.442465\pi$
$348$	0	0
$349$	−5.37824	−0.287891	−0.143945	−	0.989586i	$-0.545979\pi$
$349$	−5.37824	−0.287891	−0.143945	+	0.989586i	$0.545979\pi$
$350$	0	0
$351$	−53.2362	−2.84154
$352$	0	0
$353$	−15.2597	−0.812192	−0.406096	−	0.913831i	$-0.633110\pi$
$353$	−15.2597	−0.812192	−0.406096	+	0.913831i	$0.633110\pi$
$354$	0	0
$355$	15.7531	0.836088
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−4.25634	−0.224641	−0.112321	−	0.993672i	$-0.535828\pi$
$359$	−4.25634	−0.224641	−0.112321	+	0.993672i	$0.535828\pi$
$360$	0	0
$361$	2.17162	0.114296
$362$	0	0
$363$	−32.5087	−1.70626
$364$	0	0
$365$	−9.30428	−0.487008
$366$	0	0
$367$	−13.9583	−0.728618	−0.364309	−	0.931278i	$-0.618695\pi$
$367$	−13.9583	−0.728618	−0.364309	+	0.931278i	$0.618695\pi$
$368$	0	0
$369$	38.1463	1.98582
$370$	0	0
$371$	1.66608	0.0864987
$372$	0	0
$373$	31.1872	1.61481	0.807407	−	0.589995i	$-0.200870\pi$
$373$	31.1872	1.61481	0.807407	+	0.589995i	$0.200870\pi$
$374$	0	0
$375$	−37.3100	−1.92668
$376$	0	0
$377$	−20.0827	−1.03431
$378$	0	0
$379$	−2.13418	−0.109626	−0.0548128	−	0.998497i	$-0.517456\pi$
$379$	−2.13418	−0.109626	−0.0548128	+	0.998497i	$0.517456\pi$
$380$	0	0
$381$	−16.2216	−0.831059
$382$	0	0
$383$	3.09120	0.157953	0.0789765	−	0.996876i	$-0.474835\pi$
$383$	3.09120	0.157953	0.0789765	+	0.996876i	$0.474835\pi$
$384$	0	0
$385$	0.503871	0.0256796
$386$	0	0
$387$	−4.20581	−0.213793
$388$	0	0
$389$	−8.78360	−0.445346	−0.222673	−	0.974893i	$-0.571478\pi$
$389$	−8.78360	−0.445346	−0.222673	+	0.974893i	$0.571478\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	17.0726	0.861197
$394$	0	0
$395$	3.63191	0.182741
$396$	0	0
$397$	−1.15420	−0.0579276	−0.0289638	−	0.999580i	$-0.509221\pi$
$397$	−1.15420	−0.0579276	−0.0289638	+	0.999580i	$0.509221\pi$
$398$	0	0
$399$	2.90553	0.145459
$400$	0	0
$401$	27.0893	1.35278	0.676388	−	0.736546i	$-0.263544\pi$
$401$	27.0893	1.35278	0.676388	+	0.736546i	$0.263544\pi$
$402$	0	0
$403$	24.8011	1.23543
$404$	0	0
$405$	86.3161	4.28908
$406$	0	0
$407$	−9.31031	−0.461495
$408$	0	0
$409$	−24.1618	−1.19473	−0.597363	−	0.801971i	$-0.703785\pi$
$409$	−24.1618	−1.19473	−0.597363	+	0.801971i	$0.703785\pi$
$410$	0	0
$411$	−32.5030	−1.60326
$412$	0	0
$413$	2.06503	0.101613
$414$	0	0
$415$	10.8999	0.535054
$416$	0	0
$417$	−37.6960	−1.84598
$418$	0	0
$419$	−9.17329	−0.448145	−0.224072	−	0.974573i	$-0.571935\pi$
$419$	−9.17329	−0.448145	−0.224072	+	0.974573i	$0.571935\pi$
$420$	0	0
$421$	−22.0797	−1.07610	−0.538049	−	0.842914i	$-0.680838\pi$
$421$	−22.0797	−1.07610	−0.538049	+	0.842914i	$0.680838\pi$
$422$	0	0
$423$	71.6283	3.48269
$424$	0	0
$425$	0	0
$426$	0	0
$427$	0.909486	0.0440131
$428$	0	0
$429$	−11.5017	−0.555306
$430$	0	0
$431$	−32.3906	−1.56020	−0.780100	−	0.625655i	$-0.784832\pi$
$431$	−32.3906	−1.56020	−0.780100	+	0.625655i	$0.784832\pi$
$432$	0	0
$433$	−15.2827	−0.734438	−0.367219	−	0.930135i	$-0.619690\pi$
$433$	−15.2827	−0.734438	−0.367219	+	0.930135i	$0.619690\pi$
$434$	0	0
$435$	54.4595	2.61113
$436$	0	0
$437$	−25.8126	−1.23478
$438$	0	0
$439$	−27.6550	−1.31990	−0.659950	−	0.751310i	$-0.729423\pi$
$439$	−27.6550	−1.31990	−0.659950	+	0.751310i	$0.729423\pi$
$440$	0	0
$441$	−59.3282	−2.82515
$442$	0	0
$443$	27.8405	1.32274	0.661371	−	0.750059i	$-0.269975\pi$
$443$	27.8405	1.32274	0.661371	+	0.750059i	$0.269975\pi$
$444$	0	0
$445$	13.3520	0.632944
$446$	0	0
$447$	−38.8020	−1.83527
$448$	0	0
$449$	9.74950	0.460107	0.230054	−	0.973178i	$-0.426110\pi$
$449$	9.74950	0.460107	0.230054	+	0.973178i	$0.426110\pi$
$450$	0	0
$451$	5.33875	0.251392
$452$	0	0
$453$	−68.4161	−3.21447
$454$	0	0
$455$	−1.20169	−0.0563363
$456$	0	0
$457$	23.4839	1.09853	0.549266	−	0.835647i	$-0.314907\pi$
$457$	23.4839	1.09853	0.549266	+	0.835647i	$0.314907\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−29.6741	−1.38206	−0.691030	−	0.722826i	$-0.742843\pi$
$461$	−29.6741	−1.38206	−0.691030	+	0.722826i	$0.742843\pi$
$462$	0	0
$463$	−1.76853	−0.0821906	−0.0410953	−	0.999155i	$-0.513085\pi$
$463$	−1.76853	−0.0821906	−0.0410953	+	0.999155i	$0.513085\pi$
$464$	0	0
$465$	−67.2546	−3.11886
$466$	0	0
$467$	13.3902	0.619626	0.309813	−	0.950797i	$-0.399734\pi$
$467$	13.3902	0.619626	0.309813	+	0.950797i	$0.399734\pi$
$468$	0	0
$469$	−0.120973	−0.00558602
$470$	0	0
$471$	39.4163	1.81620
$472$	0	0
$473$	−0.588622	−0.0270648
$474$	0	0
$475$	0.738735	0.0338955
$476$	0	0
$477$	76.2684	3.49209
$478$	0	0
$479$	0.908139	0.0414940	0.0207470	−	0.999785i	$-0.493396\pi$
$479$	0.908139	0.0414940	0.0207470	+	0.999785i	$0.493396\pi$
$480$	0	0
$481$	22.2044	1.01243
$482$	0	0
$483$	−3.54245	−0.161187
$484$	0	0
$485$	6.74880	0.306447
$486$	0	0
$487$	23.8055	1.07873	0.539365	−	0.842072i	$-0.318664\pi$
$487$	23.8055	1.07873	0.539365	+	0.842072i	$0.318664\pi$
$488$	0	0
$489$	−44.3118	−2.00385
$490$	0	0
$491$	40.1358	1.81130	0.905652	−	0.424023i	$-0.139382\pi$
$491$	40.1358	1.81130	0.905652	+	0.424023i	$0.139382\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	23.0657	1.03673
$496$	0	0
$497$	1.29029	0.0578773
$498$	0	0
$499$	22.3207	0.999210	0.499605	−	0.866253i	$-0.333478\pi$
$499$	22.3207	0.999210	0.499605	+	0.866253i	$0.333478\pi$
$500$	0	0
$501$	41.2492	1.84288
$502$	0	0
$503$	5.49623	0.245065	0.122532	−	0.992465i	$-0.460898\pi$
$503$	5.49623	0.245065	0.122532	+	0.992465i	$0.460898\pi$
$504$	0	0
$505$	−18.7963	−0.836425
$506$	0	0
$507$	−16.6882	−0.741150
$508$	0	0
$509$	8.82625	0.391217	0.195608	−	0.980682i	$-0.437332\pi$
$509$	8.82625	0.391217	0.195608	+	0.980682i	$0.437332\pi$
$510$	0	0
$511$	−0.762084	−0.0337126
$512$	0	0
$513$	86.1602	3.80407
$514$	0	0
$515$	1.69163	0.0745421
$516$	0	0
$517$	10.0247	0.440886
$518$	0	0
$519$	−33.9440	−1.48998
$520$	0	0
$521$	−23.9271	−1.04827	−0.524133	−	0.851637i	$-0.675610\pi$
$521$	−23.9271	−1.04827	−0.524133	+	0.851637i	$0.675610\pi$
$522$	0	0
$523$	16.4633	0.719891	0.359946	−	0.932973i	$-0.382795\pi$
$523$	16.4633	0.719891	0.359946	+	0.932973i	$0.382795\pi$
$524$	0	0
$525$	0.101382	0.00442467
$526$	0	0
$527$	0	0
$528$	0	0
$529$	8.47089	0.368300
$530$	0	0
$531$	94.5311	4.10230
$532$	0	0
$533$	−12.7325	−0.551506
$534$	0	0
$535$	33.2513	1.43758
$536$	0	0
$537$	34.1524	1.47379
$538$	0	0
$539$	−8.30324	−0.357646
$540$	0	0
$541$	−11.9309	−0.512950	−0.256475	−	0.966551i	$-0.582561\pi$
$541$	−11.9309	−0.512950	−0.256475	+	0.966551i	$0.582561\pi$
$542$	0	0
$543$	63.6842	2.73295
$544$	0	0
$545$	27.3775	1.17272
$546$	0	0
$547$	37.9117	1.62099	0.810494	−	0.585747i	$-0.199199\pi$
$547$	37.9117	1.62099	0.810494	+	0.585747i	$0.199199\pi$
$548$	0	0
$549$	41.6336	1.77688
$550$	0	0
$551$	32.5029	1.38467
$552$	0	0
$553$	0.297479	0.0126501
$554$	0	0
$555$	−60.2129	−2.55590
$556$	0	0
$557$	−9.57355	−0.405644	−0.202822	−	0.979216i	$-0.565011\pi$
$557$	−9.57355	−0.405644	−0.202822	+	0.979216i	$0.565011\pi$
$558$	0	0
$559$	1.40382	0.0593752
$560$	0	0
$561$	0	0
$562$	0	0
$563$	6.05324	0.255114	0.127557	−	0.991831i	$-0.459286\pi$
$563$	6.05324	0.255114	0.127557	+	0.991831i	$0.459286\pi$
$564$	0	0
$565$	17.4821	0.735476
$566$	0	0
$567$	7.06988	0.296907
$568$	0	0
$569$	−0.439066	−0.0184066	−0.00920329	−	0.999958i	$-0.502930\pi$
$569$	−0.439066	−0.0184066	−0.00920329	+	0.999958i	$0.502930\pi$
$570$	0	0
$571$	−10.7246	−0.448813	−0.224406	−	0.974496i	$-0.572044\pi$
$571$	−10.7246	−0.448813	−0.224406	+	0.974496i	$0.572044\pi$
$572$	0	0
$573$	−46.2432	−1.93184
$574$	0	0
$575$	−0.900671	−0.0375606
$576$	0	0
$577$	−10.2170	−0.425341	−0.212671	−	0.977124i	$-0.568216\pi$
$577$	−10.2170	−0.425341	−0.212671	+	0.977124i	$0.568216\pi$
$578$	0	0
$579$	−70.4791	−2.92901
$580$	0	0
$581$	0.892774	0.0370385
$582$	0	0
$583$	10.6741	0.442076
$584$	0	0
$585$	−55.0101	−2.27439
$586$	0	0
$587$	32.1892	1.32859	0.664296	−	0.747470i	$-0.268731\pi$
$587$	32.1892	1.32859	0.664296	+	0.747470i	$0.268731\pi$
$588$	0	0
$589$	−40.1393	−1.65391
$590$	0	0
$591$	45.6682	1.87854
$592$	0	0
$593$	6.18495	0.253985	0.126993	−	0.991904i	$-0.459468\pi$
$593$	6.18495	0.253985	0.126993	+	0.991904i	$0.459468\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−61.6961	−2.52505
$598$	0	0
$599$	41.9523	1.71412	0.857062	−	0.515214i	$-0.172287\pi$
$599$	41.9523	1.71412	0.857062	+	0.515214i	$0.172287\pi$
$600$	0	0
$601$	−1.95380	−0.0796972	−0.0398486	−	0.999206i	$-0.512688\pi$
$601$	−1.95380	−0.0796972	−0.0398486	+	0.999206i	$0.512688\pi$
$602$	0	0
$603$	−5.53780	−0.225516
$604$	0	0
$605$	−21.7604	−0.884685
$606$	0	0
$607$	−9.30960	−0.377865	−0.188933	−	0.981990i	$-0.560503\pi$
$607$	−9.30960	−0.377865	−0.188933	+	0.981990i	$0.560503\pi$
$608$	0	0
$609$	4.46061	0.180753
$610$	0	0
$611$	−23.9081	−0.967220
$612$	0	0
$613$	−20.2988	−0.819861	−0.409931	−	0.912117i	$-0.634447\pi$
$613$	−20.2988	−0.819861	−0.409931	+	0.912117i	$0.634447\pi$
$614$	0	0
$615$	34.5275	1.39228
$616$	0	0
$617$	−32.6776	−1.31555	−0.657776	−	0.753213i	$-0.728503\pi$
$617$	−32.6776	−1.31555	−0.657776	+	0.753213i	$0.728503\pi$
$618$	0	0
$619$	−34.1665	−1.37327	−0.686633	−	0.727004i	$-0.740912\pi$
$619$	−34.1665	−1.37327	−0.686633	+	0.727004i	$0.740912\pi$
$620$	0	0
$621$	−105.047	−4.21540
$622$	0	0
$623$	1.09362	0.0438149
$624$	0	0
$625$	−25.7770	−1.03108
$626$	0	0
$627$	18.6149	0.743408
$628$	0	0
$629$	0	0
$630$	0	0
$631$	49.7912	1.98216	0.991078	−	0.133285i	$-0.0425526\pi$
$631$	49.7912	1.98216	0.991078	+	0.133285i	$0.0425526\pi$
$632$	0	0
$633$	−17.7978	−0.707399
$634$	0	0
$635$	−10.8583	−0.430898
$636$	0	0
$637$	19.8026	0.784608
$638$	0	0
$639$	59.0656	2.33660
$640$	0	0
$641$	−5.67460	−0.224133	−0.112067	−	0.993701i	$-0.535747\pi$
$641$	−5.67460	−0.224133	−0.112067	+	0.993701i	$0.535747\pi$
$642$	0	0
$643$	−12.6399	−0.498469	−0.249234	−	0.968443i	$-0.580179\pi$
$643$	−12.6399	−0.498469	−0.249234	+	0.968443i	$0.580179\pi$
$644$	0	0
$645$	−3.80682	−0.149893
$646$	0	0
$647$	12.8430	0.504909	0.252454	−	0.967609i	$-0.418762\pi$
$647$	12.8430	0.504909	0.252454	+	0.967609i	$0.418762\pi$
$648$	0	0
$649$	13.2300	0.519324
$650$	0	0
$651$	−5.50861	−0.215899
$652$	0	0
$653$	30.0133	1.17451	0.587255	−	0.809402i	$-0.300209\pi$
$653$	30.0133	1.17451	0.587255	+	0.809402i	$0.300209\pi$
$654$	0	0
$655$	11.4279	0.446525
$656$	0	0
$657$	−34.8860	−1.36103
$658$	0	0
$659$	−41.3664	−1.61141	−0.805703	−	0.592319i	$-0.798212\pi$
$659$	−41.3664	−1.61141	−0.805703	+	0.592319i	$0.798212\pi$
$660$	0	0
$661$	−32.4280	−1.26130	−0.630652	−	0.776066i	$-0.717213\pi$
$661$	−32.4280	−1.26130	−0.630652	+	0.776066i	$0.717213\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	1.94488	0.0754193
$666$	0	0
$667$	−39.6277	−1.53439
$668$	0	0
$669$	−3.86714	−0.149512
$670$	0	0
$671$	5.82681	0.224942
$672$	0	0
$673$	−23.0871	−0.889941	−0.444970	−	0.895545i	$-0.646786\pi$
$673$	−23.0871	−0.889941	−0.444970	+	0.895545i	$0.646786\pi$
$674$	0	0
$675$	3.00636	0.115715
$676$	0	0
$677$	−36.1481	−1.38928	−0.694642	−	0.719355i	$-0.744437\pi$
$677$	−36.1481	−1.38928	−0.694642	+	0.719355i	$0.744437\pi$
$678$	0	0
$679$	0.552773	0.0212135
$680$	0	0
$681$	−9.21740	−0.353211
$682$	0	0
$683$	−16.6737	−0.638003	−0.319002	−	0.947754i	$-0.603348\pi$
$683$	−16.6737	−0.638003	−0.319002	+	0.947754i	$0.603348\pi$
$684$	0	0
$685$	−21.7566	−0.831278
$686$	0	0
$687$	−13.4753	−0.514115
$688$	0	0
$689$	−25.4569	−0.969831
$690$	0	0
$691$	17.6213	0.670345	0.335172	−	0.942157i	$-0.391205\pi$
$691$	17.6213	0.670345	0.335172	+	0.942157i	$0.391205\pi$
$692$	0	0
$693$	1.88924	0.0717664
$694$	0	0
$695$	−25.2326	−0.957128
$696$	0	0
$697$	0	0
$698$	0	0
$699$	−24.0714	−0.910466
$700$	0	0
$701$	17.8045	0.672468	0.336234	−	0.941778i	$-0.390847\pi$
$701$	17.8045	0.672468	0.336234	+	0.941778i	$0.390847\pi$
$702$	0	0
$703$	−35.9367	−1.35538
$704$	0	0
$705$	64.8331	2.44176
$706$	0	0
$707$	−1.53955	−0.0579006
$708$	0	0
$709$	−20.5394	−0.771373	−0.385686	−	0.922630i	$-0.626035\pi$
$709$	−20.5394	−0.771373	−0.385686	+	0.922630i	$0.626035\pi$
$710$	0	0
$711$	13.6177	0.510704
$712$	0	0
$713$	48.9381	1.83275
$714$	0	0
$715$	−7.69890	−0.287922
$716$	0	0
$717$	−43.8022	−1.63582
$718$	0	0
$719$	−3.20609	−0.119567	−0.0597835	−	0.998211i	$-0.519041\pi$
$719$	−3.20609	−0.119567	−0.0597835	+	0.998211i	$0.519041\pi$
$720$	0	0
$721$	0.138556	0.00516010
$722$	0	0
$723$	−43.2944	−1.61013
$724$	0	0
$725$	1.13411	0.0421199
$726$	0	0
$727$	3.21632	0.119287	0.0596433	−	0.998220i	$-0.481004\pi$
$727$	3.21632	0.119287	0.0596433	+	0.998220i	$0.481004\pi$
$728$	0	0
$729$	132.991	4.92559
$730$	0	0
$731$	0	0
$732$	0	0
$733$	23.0730	0.852221	0.426111	−	0.904671i	$-0.359883\pi$
$733$	23.0730	0.852221	0.426111	+	0.904671i	$0.359883\pi$
$734$	0	0
$735$	−53.6999	−1.98075
$736$	0	0
$737$	−0.775039	−0.0285489
$738$	0	0
$739$	22.4355	0.825303	0.412652	−	0.910889i	$-0.364603\pi$
$739$	22.4355	0.825303	0.412652	+	0.910889i	$0.364603\pi$
$740$	0	0
$741$	−44.3951	−1.63090
$742$	0	0
$743$	24.5950	0.902304	0.451152	−	0.892447i	$-0.351013\pi$
$743$	24.5950	0.902304	0.451152	+	0.892447i	$0.351013\pi$
$744$	0	0
$745$	−25.9730	−0.951577
$746$	0	0
$747$	40.8686	1.49530
$748$	0	0
$749$	2.72351	0.0995150
$750$	0	0
$751$	−30.2007	−1.10204	−0.551019	−	0.834493i	$-0.685761\pi$
$751$	−30.2007	−1.10204	−0.551019	+	0.834493i	$0.685761\pi$
$752$	0	0
$753$	−34.0333	−1.24024
$754$	0	0
$755$	−45.7958	−1.66668
$756$	0	0
$757$	−10.7066	−0.389138	−0.194569	−	0.980889i	$-0.562331\pi$
$757$	−10.7066	−0.389138	−0.194569	+	0.980889i	$0.562331\pi$
$758$	0	0
$759$	−22.6954	−0.823792
$760$	0	0
$761$	−5.58389	−0.202416	−0.101208	−	0.994865i	$-0.532271\pi$
$761$	−5.58389	−0.202416	−0.101208	+	0.994865i	$0.532271\pi$
$762$	0	0
$763$	2.24240	0.0811805
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−31.5526	−1.13930
$768$	0	0
$769$	19.5700	0.705714	0.352857	−	0.935677i	$-0.385210\pi$
$769$	19.5700	0.705714	0.352857	+	0.935677i	$0.385210\pi$
$770$	0	0
$771$	36.1103	1.30048
$772$	0	0
$773$	9.61450	0.345810	0.172905	−	0.984939i	$-0.444685\pi$
$773$	9.61450	0.345810	0.172905	+	0.984939i	$0.444685\pi$
$774$	0	0
$775$	−1.40057	−0.0503099
$776$	0	0
$777$	−4.93185	−0.176929
$778$	0	0
$779$	20.6069	0.738320
$780$	0	0
$781$	8.26649	0.295798
$782$	0	0
$783$	132.274	4.72709
$784$	0	0
$785$	26.3841	0.941690
$786$	0	0
$787$	14.2050	0.506354	0.253177	−	0.967420i	$-0.418525\pi$
$787$	14.2050	0.506354	0.253177	+	0.967420i	$0.418525\pi$
$788$	0	0
$789$	−101.237	−3.60413
$790$	0	0
$791$	1.43190	0.0509125
$792$	0	0
$793$	−13.8965	−0.493479
$794$	0	0
$795$	69.0330	2.44835
$796$	0	0
$797$	−4.63015	−0.164008	−0.0820042	−	0.996632i	$-0.526132\pi$
$797$	−4.63015	−0.164008	−0.0820042	+	0.996632i	$0.526132\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	50.0627	1.76888
$802$	0	0
$803$	−4.88245	−0.172298
$804$	0	0
$805$	−2.37121	−0.0835743
$806$	0	0
$807$	−58.9298	−2.07443
$808$	0	0
$809$	−5.51571	−0.193922	−0.0969610	−	0.995288i	$-0.530912\pi$
$809$	−5.51571	−0.193922	−0.0969610	+	0.995288i	$0.530912\pi$
$810$	0	0
$811$	−51.4712	−1.80740	−0.903699	−	0.428168i	$-0.859159\pi$
$811$	−51.4712	−1.80740	−0.903699	+	0.428168i	$0.859159\pi$
$812$	0	0
$813$	56.8119	1.99248
$814$	0	0
$815$	−29.6611	−1.03898
$816$	0	0
$817$	−2.27201	−0.0794876
$818$	0	0
$819$	−4.50570	−0.157442
$820$	0	0
$821$	−42.6171	−1.48735	−0.743673	−	0.668544i	$-0.766918\pi$
$821$	−42.6171	−1.48735	−0.743673	+	0.668544i	$0.766918\pi$
$822$	0	0
$823$	5.47648	0.190898	0.0954491	−	0.995434i	$-0.469571\pi$
$823$	5.47648	0.190898	0.0954491	+	0.995434i	$0.469571\pi$
$824$	0	0
$825$	0.649524	0.0226135
$826$	0	0
$827$	−29.5473	−1.02746	−0.513730	−	0.857952i	$-0.671737\pi$
$827$	−29.5473	−1.02746	−0.513730	+	0.857952i	$0.671737\pi$
$828$	0	0
$829$	−38.7372	−1.34540	−0.672700	−	0.739916i	$-0.734865\pi$
$829$	−38.7372	−1.34540	−0.672700	+	0.739916i	$0.734865\pi$
$830$	0	0
$831$	−37.1544	−1.28887
$832$	0	0
$833$	0	0
$834$	0	0
$835$	27.6110	0.955519
$836$	0	0
$837$	−163.351	−5.64625
$838$	0	0
$839$	48.0586	1.65917	0.829583	−	0.558383i	$-0.188578\pi$
$839$	48.0586	1.65917	0.829583	+	0.558383i	$0.188578\pi$
$840$	0	0
$841$	20.8987	0.720646
$842$	0	0
$843$	7.29387	0.251214
$844$	0	0
$845$	−11.1706	−0.384281
$846$	0	0
$847$	−1.78232	−0.0612414
$848$	0	0
$849$	−20.4549	−0.702009
$850$	0	0
$851$	43.8143	1.50193
$852$	0	0
$853$	−29.6676	−1.01580	−0.507899	−	0.861416i	$-0.669578\pi$
$853$	−29.6676	−1.01580	−0.507899	+	0.861416i	$0.669578\pi$
$854$	0	0
$855$	89.0311	3.04480
$856$	0	0
$857$	−1.74727	−0.0596855	−0.0298428	−	0.999555i	$-0.509501\pi$
$857$	−1.74727	−0.0596855	−0.0298428	+	0.999555i	$0.509501\pi$
$858$	0	0
$859$	−21.5736	−0.736082	−0.368041	−	0.929810i	$-0.619971\pi$
$859$	−21.5736	−0.736082	−0.368041	+	0.929810i	$0.619971\pi$
$860$	0	0
$861$	2.82804	0.0963793
$862$	0	0
$863$	9.73076	0.331239	0.165619	−	0.986190i	$-0.447038\pi$
$863$	9.73076	0.331239	0.165619	+	0.986190i	$0.447038\pi$
$864$	0	0
$865$	−22.7212	−0.772543
$866$	0	0
$867$	0	0
$868$	0	0
$869$	1.90586	0.0646518
$870$	0	0
$871$	1.84841	0.0626309
$872$	0	0
$873$	25.3043	0.856422
$874$	0	0
$875$	−2.04556	−0.0691526
$876$	0	0
$877$	29.6861	1.00243	0.501214	−	0.865323i	$-0.332887\pi$
$877$	29.6861	1.00243	0.501214	+	0.865323i	$0.332887\pi$
$878$	0	0
$879$	−54.5587	−1.84022
$880$	0	0
$881$	9.94921	0.335197	0.167599	−	0.985855i	$-0.446399\pi$
$881$	9.94921	0.335197	0.167599	+	0.985855i	$0.446399\pi$
$882$	0	0
$883$	49.2844	1.65855	0.829277	−	0.558838i	$-0.188753\pi$
$883$	49.2844	1.65855	0.829277	+	0.558838i	$0.188753\pi$
$884$	0	0
$885$	85.5632	2.87617
$886$	0	0
$887$	−9.46670	−0.317861	−0.158930	−	0.987290i	$-0.550805\pi$
$887$	−9.46670	−0.317861	−0.158930	+	0.987290i	$0.550805\pi$
$888$	0	0
$889$	−0.889368	−0.0298285
$890$	0	0
$891$	45.2946	1.51743
$892$	0	0
$893$	38.6941	1.29485
$894$	0	0
$895$	22.8607	0.764148
$896$	0	0
$897$	54.1269	1.80724
$898$	0	0
$899$	−61.6223	−2.05522
$900$	0	0
$901$	0	0
$902$	0	0
$903$	−0.311804	−0.0103762
$904$	0	0
$905$	42.6284	1.41702
$906$	0	0
$907$	28.3490	0.941314	0.470657	−	0.882316i	$-0.344017\pi$
$907$	28.3490	0.941314	0.470657	+	0.882316i	$0.344017\pi$
$908$	0	0
$909$	−70.4760	−2.33754
$910$	0	0
$911$	−14.4019	−0.477157	−0.238579	−	0.971123i	$-0.576681\pi$
$911$	−14.4019	−0.477157	−0.238579	+	0.971123i	$0.576681\pi$
$912$	0	0
$913$	5.71974	0.189296
$914$	0	0
$915$	37.6840	1.24579
$916$	0	0
$917$	0.936023	0.0309102
$918$	0	0
$919$	−55.5023	−1.83085	−0.915427	−	0.402485i	$-0.868147\pi$
$919$	−55.5023	−1.83085	−0.915427	+	0.402485i	$0.868147\pi$
$920$	0	0
$921$	46.3732	1.52805
$922$	0	0
$923$	−19.7149	−0.648925
$924$	0	0
$925$	−1.25393	−0.0412289
$926$	0	0
$927$	6.34269	0.208321
$928$	0	0
$929$	−32.3658	−1.06189	−0.530944	−	0.847407i	$-0.678163\pi$
$929$	−32.3658	−1.06189	−0.530944	+	0.847407i	$0.678163\pi$
$930$	0	0
$931$	−32.0495	−1.05038
$932$	0	0
$933$	44.3283	1.45124
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−43.4263	−1.41868	−0.709338	−	0.704868i	$-0.751006\pi$
$937$	−43.4263	−1.41868	−0.709338	+	0.704868i	$0.751006\pi$
$938$	0	0
$939$	−91.9396	−3.00033
$940$	0	0
$941$	−21.2959	−0.694227	−0.347114	−	0.937823i	$-0.612838\pi$
$941$	−21.2959	−0.694227	−0.347114	+	0.937823i	$0.612838\pi$
$942$	0	0
$943$	−25.1241	−0.818154
$944$	0	0
$945$	7.91491	0.257472
$946$	0	0
$947$	11.7935	0.383238	0.191619	−	0.981469i	$-0.438626\pi$
$947$	11.7935	0.383238	0.191619	+	0.981469i	$0.438626\pi$
$948$	0	0
$949$	11.6443	0.377989
$950$	0	0
$951$	8.13723	0.263868
$952$	0	0
$953$	29.3924	0.952114	0.476057	−	0.879414i	$-0.342066\pi$
$953$	29.3924	0.952114	0.476057	+	0.879414i	$0.342066\pi$
$954$	0	0
$955$	−30.9539	−1.00164
$956$	0	0
$957$	28.5778	0.923788
$958$	0	0
$959$	−1.78202	−0.0575443
$960$	0	0
$961$	45.1002	1.45485
$962$	0	0
$963$	124.674	4.01758
$964$	0	0
$965$	−47.1767	−1.51867
$966$	0	0
$967$	6.19462	0.199205	0.0996027	−	0.995027i	$-0.468243\pi$
$967$	6.19462	0.199205	0.0996027	+	0.995027i	$0.468243\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	1.28848	0.0413494	0.0206747	−	0.999786i	$-0.493419\pi$
$971$	1.28848	0.0413494	0.0206747	+	0.999786i	$0.493419\pi$
$972$	0	0
$973$	−2.06673	−0.0662562
$974$	0	0
$975$	−1.54907	−0.0496098
$976$	0	0
$977$	43.5640	1.39373	0.696867	−	0.717200i	$-0.254577\pi$
$977$	43.5640	1.39373	0.696867	+	0.717200i	$0.254577\pi$
$978$	0	0
$979$	7.00649	0.223928
$980$	0	0
$981$	102.651	3.27739
$982$	0	0
$983$	52.2558	1.66670	0.833351	−	0.552744i	$-0.186419\pi$
$983$	52.2558	1.66670	0.833351	+	0.552744i	$0.186419\pi$
$984$	0	0
$985$	30.5690	0.974009
$986$	0	0
$987$	5.31028	0.169028
$988$	0	0
$989$	2.77005	0.0880825
$990$	0	0
$991$	38.6122	1.22656	0.613278	−	0.789867i	$-0.289851\pi$
$991$	38.6122	1.22656	0.613278	+	0.789867i	$0.289851\pi$
$992$	0	0
$993$	6.01331	0.190827
$994$	0	0
$995$	−41.2976	−1.30922
$996$	0	0
$997$	−21.4135	−0.678173	−0.339086	−	0.940755i	$-0.610118\pi$
$997$	−21.4135	−0.678173	−0.339086	+	0.940755i	$0.610118\pi$
$998$	0	0
$999$	−146.248	−4.62709

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9248.2.a.bz.1.20		20
4.3	odd	2		9248.2.a.by.1.1		20
17.3	odd	16		544.2.bb.f.417.5	yes	20
17.6	odd	16		544.2.bb.f.257.5	yes	20
17.16	even	2	inner	9248.2.a.bz.1.1		20
68.3	even	16		544.2.bb.e.417.1	yes	20
68.23	even	16		544.2.bb.e.257.1	✓	20
68.67	odd	2		9248.2.a.by.1.20		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
544.2.bb.e.257.1	✓	20	68.23	even	16
544.2.bb.e.417.1	yes	20	68.3	even	16
544.2.bb.f.257.5	yes	20	17.6	odd	16
544.2.bb.f.417.5	yes	20	17.3	odd	16
9248.2.a.by.1.1		20	4.3	odd	2
9248.2.a.by.1.20		20	68.67	odd	2
9248.2.a.bz.1.1		20	17.16	even	2	inner
9248.2.a.bz.1.20		20	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 9248 = 2^{5} \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9248.a (trivial)

\(f(q)\)	\(=\)	\(q+3.39376 q^{3} +2.27168 q^{5} +0.186067 q^{7} +8.51759 q^{9} +1.19207 q^{11} -2.84300 q^{13} +7.70955 q^{15} +4.60126 q^{19} +0.631465 q^{21} -5.60989 q^{23} +0.160551 q^{25} +18.7253 q^{27} +7.06390 q^{29} -8.72354 q^{31} +4.04561 q^{33} +0.422684 q^{35} -7.81018 q^{37} -9.64847 q^{39} +4.47854 q^{41} -0.493780 q^{43} +19.3493 q^{45} +8.40946 q^{47} -6.96538 q^{49} +8.95423 q^{53} +2.70801 q^{55} +15.6156 q^{57} +11.0983 q^{59} +4.88796 q^{61} +1.58484 q^{63} -6.45841 q^{65} -0.650160 q^{67} -19.0386 q^{69} +6.93455 q^{71} -4.09576 q^{73} +0.544870 q^{75} +0.221805 q^{77} +1.59878 q^{79} +37.9965 q^{81} +4.79814 q^{83} +23.9732 q^{87} +5.87757 q^{89} -0.528988 q^{91} -29.6056 q^{93} +10.4526 q^{95} +2.97084 q^{97} +10.1536 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 28 q^{9} - 8 q^{13} + 16 q^{15} + 40 q^{19} - 32 q^{21} + 28 q^{25} + 32 q^{35} + 40 q^{43} + 32 q^{47} + 36 q^{49} - 40 q^{53} + 48 q^{55} + 8 q^{59} + 72 q^{67} - 48 q^{69} - 48 q^{77} + 36 q^{81}+ \cdots - 32 q^{93}+O(q^{100}) \) 20 * q + 28 * q^9 - 8 * q^13 + 16 * q^15 + 40 * q^19 - 32 * q^21 + 28 * q^25 + 32 * q^35 + 40 * q^43 + 32 * q^47 + 36 * q^49 - 40 * q^53 + 48 * q^55 + 8 * q^59 + 72 * q^67 - 48 * q^69 - 48 * q^77 + 36 * q^81 + 24 * q^83 + 96 * q^87 + 64 * q^89 - 32 * q^93

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

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