LMFDB - Embedded newform 7803.2.a.bw.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [7803,2,Mod(1,7803)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("7803.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(7803, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 7803 = 3^{3} \cdot 17^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7803.a (trivial)

Newform invariants

comment:select newform

sage:traces = [12,0,0,12,0,0,8,0,0,24,0,0,-4,0,0,20,0,0,-4,0,0,-28,0,0,0,0, 0,48,0,0,8,0,0,0,0,0,12,0,0,44,0,0,-20,0,0,36,0,0,28,0,0,8,0,0,-4,0,0, 28,0,0,48,0,0,40,0,0,-8,0,0,44,0,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(73)] == traces)

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

Self dual:	yes
Analytic conductor:	$62.3072686972$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 18x^{10} + 115x^{8} - 318x^{6} + 395x^{4} - 208x^{2} + 34 $ x^12 - 18x^10 + 115x^8 - 318x^6 + 395x^4 - 208*x^2 + 34
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 459)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.61349$ of defining polynomial
Character	$\chi$	$=$	7803.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.61349 q^{2} +0.603350 q^{4} +0.815992 q^{5} -2.52607 q^{7} +2.25348 q^{8} -1.31659 q^{10} +3.58347 q^{11} -5.59275 q^{13} +4.07580 q^{14} -4.84267 q^{16} -2.56848 q^{19} +0.492329 q^{20} -5.78189 q^{22} -1.99832 q^{23} -4.33416 q^{25} +9.02385 q^{26} -1.52411 q^{28} -5.86529 q^{29} +1.18717 q^{31} +3.30664 q^{32} -2.06126 q^{35} -6.74227 q^{37} +4.14421 q^{38} +1.83882 q^{40} -9.52343 q^{41} +9.26834 q^{43} +2.16209 q^{44} +3.22426 q^{46} +9.21835 q^{47} -0.618947 q^{49} +6.99312 q^{50} -3.37439 q^{52} -1.17050 q^{53} +2.92408 q^{55} -5.69246 q^{56} +9.46358 q^{58} -7.67537 q^{59} +4.20080 q^{61} -1.91549 q^{62} +4.35011 q^{64} -4.56364 q^{65} -4.68034 q^{67} +3.32582 q^{70} -8.39819 q^{71} -0.552268 q^{73} +10.8786 q^{74} -1.54969 q^{76} -9.05211 q^{77} +13.1827 q^{79} -3.95158 q^{80} +15.3660 q^{82} +8.60839 q^{83} -14.9544 q^{86} +8.07528 q^{88} +10.5202 q^{89} +14.1277 q^{91} -1.20568 q^{92} -14.8737 q^{94} -2.09586 q^{95} -3.32331 q^{97} +0.998664 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 12 q^{4} + 8 q^{7} + 24 q^{10} - 4 q^{13} + 20 q^{16} - 4 q^{19} - 28 q^{22} + 48 q^{28} + 8 q^{31} + 12 q^{37} + 44 q^{40} - 20 q^{43} + 36 q^{46} + 28 q^{49} + 8 q^{52} - 4 q^{55} + 28 q^{58} + 48 q^{61}+ \cdots + 24 q^{97}+O(q^{100}) $ 12 * q + 12 * q^4 + 8 * q^7 + 24 * q^10 - 4 * q^13 + 20 * q^16 - 4 * q^19 - 28 * q^22 + 48 * q^28 + 8 * q^31 + 12 * q^37 + 44 * q^40 - 20 * q^43 + 36 * q^46 + 28 * q^49 + 8 * q^52 - 4 * q^55 + 28 * q^58 + 48 * q^61 + 40 * q^64 - 8 * q^67 + 44 * q^70 + 4 * q^73 - 28 * q^76 + 40 * q^79 + 76 * q^82 - 40 * q^88 + 52 * q^91 + 28 * q^94 + 24 * q^97

Coefficient data

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

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

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

$2$	−1.61349	−1.14091	−0.570455	−	0.821329i	$-0.693233\pi$
$2$	−1.61349	−1.14091	−0.570455	+	0.821329i	$0.693233\pi$
$3$	0	0
$3$	0	0
$4$	0.603350	0.301675
$5$	0.815992	0.364922	0.182461	−	0.983213i	$-0.441594\pi$
$5$	0.815992	0.364922	0.182461	+	0.983213i	$0.441594\pi$
$6$	0	0
$7$	−2.52607	−0.954766	−0.477383	−	0.878695i	$-0.658415\pi$
$7$	−2.52607	−0.954766	−0.477383	+	0.878695i	$0.658415\pi$
$8$	2.25348	0.796726
$9$	0	0
$10$	−1.31659	−0.416344
$11$	3.58347	1.08046	0.540228	−	0.841519i	$-0.318338\pi$
$11$	3.58347	1.08046	0.540228	+	0.841519i	$0.318338\pi$
$12$	0	0
$13$	−5.59275	−1.55115	−0.775575	−	0.631255i	$-0.782540\pi$
$13$	−5.59275	−1.55115	−0.775575	+	0.631255i	$0.782540\pi$
$14$	4.07580	1.08930
$15$	0	0
$16$	−4.84267	−1.21067
$17$	0	0
$17$	0	0
$18$	0	0
$19$	−2.56848	−0.589249	−0.294625	−	0.955613i	$-0.595195\pi$
$19$	−2.56848	−0.589249	−0.294625	+	0.955613i	$0.595195\pi$
$20$	0.492329	0.110088
$21$	0	0
$22$	−5.78189	−1.23270
$23$	−1.99832	−0.416678	−0.208339	−	0.978057i	$-0.566806\pi$
$23$	−1.99832	−0.416678	−0.208339	+	0.978057i	$0.566806\pi$
$24$	0	0
$25$	−4.33416	−0.866832
$26$	9.02385	1.76972
$27$	0	0
$28$	−1.52411	−0.288029
$29$	−5.86529	−1.08916	−0.544578	−	0.838710i	$-0.683310\pi$
$29$	−5.86529	−1.08916	−0.544578	+	0.838710i	$0.683310\pi$
$30$	0	0
$31$	1.18717	0.213222	0.106611	−	0.994301i	$-0.466000\pi$
$31$	1.18717	0.213222	0.106611	+	0.994301i	$0.466000\pi$
$32$	3.30664	0.584536
$33$	0	0
$34$	0	0
$35$	−2.06126	−0.348416
$36$	0	0
$37$	−6.74227	−1.10842	−0.554211	−	0.832376i	$-0.686980\pi$
$37$	−6.74227	−1.10842	−0.554211	+	0.832376i	$0.686980\pi$
$38$	4.14421	0.672280
$39$	0	0
$40$	1.83882	0.290743
$41$	−9.52343	−1.48731	−0.743655	−	0.668563i	$-0.766910\pi$
$41$	−9.52343	−1.48731	−0.743655	+	0.668563i	$0.766910\pi$
$42$	0	0
$43$	9.26834	1.41341	0.706705	−	0.707509i	$-0.250181\pi$
$43$	9.26834	1.41341	0.706705	+	0.707509i	$0.250181\pi$
$44$	2.16209	0.325947
$45$	0	0
$46$	3.22426	0.475392
$47$	9.21835	1.34463	0.672317	−	0.740263i	$-0.265299\pi$
$47$	9.21835	1.34463	0.672317	+	0.740263i	$0.265299\pi$
$48$	0	0
$49$	−0.618947	−0.0884209
$50$	6.99312	0.988977
$51$	0	0
$52$	−3.37439	−0.467943
$53$	−1.17050	−0.160781	−0.0803905	−	0.996763i	$-0.525617\pi$
$53$	−1.17050	−0.160781	−0.0803905	+	0.996763i	$0.525617\pi$
$54$	0	0
$55$	2.92408	0.394283
$56$	−5.69246	−0.760687
$57$	0	0
$58$	9.46358	1.24263
$59$	−7.67537	−0.999248	−0.499624	−	0.866242i	$-0.666529\pi$
$59$	−7.67537	−0.999248	−0.499624	+	0.866242i	$0.666529\pi$
$60$	0	0
$61$	4.20080	0.537857	0.268929	−	0.963160i	$-0.413330\pi$
$61$	4.20080	0.537857	0.268929	+	0.963160i	$0.413330\pi$
$62$	−1.91549	−0.243267
$63$	0	0
$64$	4.35011	0.543764
$65$	−4.56364	−0.566050
$66$	0	0
$67$	−4.68034	−0.571794	−0.285897	−	0.958260i	$-0.592292\pi$
$67$	−4.68034	−0.571794	−0.285897	+	0.958260i	$0.592292\pi$
$68$	0	0
$69$	0	0
$70$	3.32582	0.397511
$71$	−8.39819	−0.996682	−0.498341	−	0.866981i	$-0.666057\pi$
$71$	−8.39819	−0.996682	−0.498341	+	0.866981i	$0.666057\pi$
$72$	0	0
$73$	−0.552268	−0.0646381	−0.0323191	−	0.999478i	$-0.510289\pi$
$73$	−0.552268	−0.0646381	−0.0323191	+	0.999478i	$0.510289\pi$
$74$	10.8786	1.26461
$75$	0	0
$76$	−1.54969	−0.177762
$77$	−9.05211	−1.03158
$78$	0	0
$79$	13.1827	1.48317	0.741586	−	0.670858i	$-0.234074\pi$
$79$	13.1827	1.48317	0.741586	+	0.670858i	$0.234074\pi$
$80$	−3.95158	−0.441800
$81$	0	0
$82$	15.3660	1.69689
$83$	8.60839	0.944893	0.472447	−	0.881359i	$-0.343371\pi$
$83$	8.60839	0.944893	0.472447	+	0.881359i	$0.343371\pi$
$84$	0	0
$85$	0	0
$86$	−14.9544	−1.61257
$87$	0	0
$88$	8.07528	0.860827
$89$	10.5202	1.11514	0.557570	−	0.830130i	$-0.311734\pi$
$89$	10.5202	1.11514	0.557570	+	0.830130i	$0.311734\pi$
$90$	0	0
$91$	14.1277	1.48099
$92$	−1.20568	−0.125701
$93$	0	0
$94$	−14.8737	−1.53411
$95$	−2.09586	−0.215030
$96$	0	0
$97$	−3.32331	−0.337431	−0.168715	−	0.985665i	$-0.553962\pi$
$97$	−3.32331	−0.337431	−0.168715	+	0.985665i	$0.553962\pi$
$98$	0.998664	0.100880
$99$	0	0
$100$	−2.61501	−0.261501
$101$	2.16031	0.214959	0.107479	−	0.994207i	$-0.465722\pi$
$101$	2.16031	0.214959	0.107479	+	0.994207i	$0.465722\pi$
$102$	0	0
$103$	5.73949	0.565529	0.282764	−	0.959189i	$-0.408749\pi$
$103$	5.73949	0.565529	0.282764	+	0.959189i	$0.408749\pi$
$104$	−12.6032	−1.23584
$105$	0	0
$106$	1.88860	0.183437
$107$	−11.0597	−1.06918	−0.534588	−	0.845113i	$-0.679533\pi$
$107$	−11.0597	−1.06918	−0.534588	+	0.845113i	$0.679533\pi$
$108$	0	0
$109$	10.2148	0.978404	0.489202	−	0.872170i	$-0.337288\pi$
$109$	10.2148	0.978404	0.489202	+	0.872170i	$0.337288\pi$
$110$	−4.71797	−0.449841
$111$	0	0
$112$	12.2329	1.15590
$113$	15.2226	1.43202	0.716009	−	0.698091i	$-0.245967\pi$
$113$	15.2226	1.43202	0.716009	+	0.698091i	$0.245967\pi$
$114$	0	0
$115$	−1.63061	−0.152055
$116$	−3.53882	−0.328571
$117$	0	0
$118$	12.3841	1.14005
$119$	0	0
$120$	0	0
$121$	1.84124	0.167386
$122$	−6.77795	−0.613647
$123$	0	0
$124$	0.716279	0.0643237
$125$	−7.61659	−0.681249
$126$	0	0
$127$	−4.45909	−0.395680	−0.197840	−	0.980234i	$-0.563393\pi$
$127$	−4.45909	−0.395680	−0.197840	+	0.980234i	$0.563393\pi$
$128$	−13.6321	−1.20492
$129$	0	0
$130$	7.36339	0.645812
$131$	21.7414	1.89955	0.949775	−	0.312933i	$-0.101311\pi$
$131$	21.7414	1.89955	0.949775	+	0.312933i	$0.101311\pi$
$132$	0	0
$133$	6.48817	0.562595
$134$	7.55168	0.652366
$135$	0	0
$136$	0	0
$137$	−12.8727	−1.09979	−0.549896	−	0.835233i	$-0.685333\pi$
$137$	−12.8727	−1.09979	−0.549896	+	0.835233i	$0.685333\pi$
$138$	0	0
$139$	−13.4259	−1.13877	−0.569386	−	0.822070i	$-0.692819\pi$
$139$	−13.4259	−1.13877	−0.569386	+	0.822070i	$0.692819\pi$
$140$	−1.24366	−0.105108
$141$	0	0
$142$	13.5504	1.13712
$143$	−20.0415	−1.67595
$144$	0	0
$145$	−4.78602	−0.397458
$146$	0.891079	0.0737462
$147$	0	0
$148$	−4.06795	−0.334383
$149$	−10.0770	−0.825541	−0.412770	−	0.910835i	$-0.635439\pi$
$149$	−10.0770	−0.825541	−0.412770	+	0.910835i	$0.635439\pi$
$150$	0	0
$151$	−17.4518	−1.42021	−0.710104	−	0.704097i	$-0.751352\pi$
$151$	−17.4518	−1.42021	−0.710104	+	0.704097i	$0.751352\pi$
$152$	−5.78801	−0.469470
$153$	0	0
$154$	14.6055	1.17694
$155$	0.968720	0.0778095
$156$	0	0
$157$	16.5280	1.31908	0.659538	−	0.751671i	$-0.270752\pi$
$157$	16.5280	1.31908	0.659538	+	0.751671i	$0.270752\pi$
$158$	−21.2702	−1.69217
$159$	0	0
$160$	2.69819	0.213310
$161$	5.04790	0.397830
$162$	0	0
$163$	11.1138	0.870502	0.435251	−	0.900309i	$-0.356660\pi$
$163$	11.1138	0.870502	0.435251	+	0.900309i	$0.356660\pi$
$164$	−5.74596	−0.448684
$165$	0	0
$166$	−13.8895	−1.07804
$167$	−2.19680	−0.169994	−0.0849969	−	0.996381i	$-0.527088\pi$
$167$	−2.19680	−0.169994	−0.0849969	+	0.996381i	$0.527088\pi$
$168$	0	0
$169$	18.2789	1.40607
$170$	0	0
$171$	0	0
$172$	5.59206	0.426390
$173$	7.63122	0.580191	0.290095	−	0.956998i	$-0.406313\pi$
$173$	7.63122	0.580191	0.290095	+	0.956998i	$0.406313\pi$
$174$	0	0
$175$	10.9484	0.827622
$176$	−17.3535	−1.30807
$177$	0	0
$178$	−16.9742	−1.27227
$179$	−16.9787	−1.26905	−0.634525	−	0.772902i	$-0.718804\pi$
$179$	−16.9787	−1.26905	−0.634525	+	0.772902i	$0.718804\pi$
$180$	0	0
$181$	−9.74474	−0.724321	−0.362160	−	0.932116i	$-0.617961\pi$
$181$	−9.74474	−0.724321	−0.362160	+	0.932116i	$0.617961\pi$
$182$	−22.7949	−1.68967
$183$	0	0
$184$	−4.50317	−0.331978
$185$	−5.50163	−0.404488
$186$	0	0
$187$	0	0
$188$	5.56189	0.405643
$189$	0	0
$190$	3.38164	0.245330
$191$	26.1154	1.88964	0.944821	−	0.327588i	$-0.106236\pi$
$191$	26.1154	1.88964	0.944821	+	0.327588i	$0.106236\pi$
$192$	0	0
$193$	−17.8318	−1.28356	−0.641781	−	0.766888i	$-0.721804\pi$
$193$	−17.8318	−1.28356	−0.641781	+	0.766888i	$0.721804\pi$
$194$	5.36212	0.384978
$195$	0	0
$196$	−0.373442	−0.0266744
$197$	14.7297	1.04945	0.524725	−	0.851272i	$-0.324168\pi$
$197$	14.7297	1.04945	0.524725	+	0.851272i	$0.324168\pi$
$198$	0	0
$199$	10.4208	0.738712	0.369356	−	0.929288i	$-0.379578\pi$
$199$	10.4208	0.738712	0.369356	+	0.929288i	$0.379578\pi$
$200$	−9.76694	−0.690627
$201$	0	0
$202$	−3.48564	−0.245249
$203$	14.8162	1.03989
$204$	0	0
$205$	−7.77104	−0.542753
$206$	−9.26061	−0.645217
$207$	0	0
$208$	27.0839	1.87793
$209$	−9.20406	−0.636658
$210$	0	0
$211$	13.9399	0.959664	0.479832	−	0.877360i	$-0.340698\pi$
$211$	13.9399	0.959664	0.479832	+	0.877360i	$0.340698\pi$
$212$	−0.706224	−0.0485036
$213$	0	0
$214$	17.8446	1.21983
$215$	7.56289	0.515785
$216$	0	0
$217$	−2.99888	−0.203577
$218$	−16.4815	−1.11627
$219$	0	0
$220$	1.76424	0.118945
$221$	0	0
$222$	0	0
$223$	18.9731	1.27054	0.635268	−	0.772292i	$-0.280890\pi$
$223$	18.9731	1.27054	0.635268	+	0.772292i	$0.280890\pi$
$224$	−8.35281	−0.558096
$225$	0	0
$226$	−24.5615	−1.63380
$227$	9.42310	0.625433	0.312716	−	0.949847i	$-0.398761\pi$
$227$	9.42310	0.625433	0.312716	+	0.949847i	$0.398761\pi$
$228$	0	0
$229$	−20.5779	−1.35982	−0.679912	−	0.733294i	$-0.737982\pi$
$229$	−20.5779	−1.35982	−0.679912	+	0.733294i	$0.737982\pi$
$230$	2.63097	0.173481
$231$	0	0
$232$	−13.2173	−0.867759
$233$	−25.0272	−1.63959	−0.819794	−	0.572658i	$-0.805912\pi$
$233$	−25.0272	−1.63959	−0.819794	+	0.572658i	$0.805912\pi$
$234$	0	0
$235$	7.52209	0.490687
$236$	−4.63094	−0.301448
$237$	0	0
$238$	0	0
$239$	5.18719	0.335532	0.167766	−	0.985827i	$-0.446345\pi$
$239$	5.18719	0.335532	0.167766	+	0.985827i	$0.446345\pi$
$240$	0	0
$241$	1.27223	0.0819515	0.0409757	−	0.999160i	$-0.486953\pi$
$241$	1.27223	0.0819515	0.0409757	+	0.999160i	$0.486953\pi$
$242$	−2.97082	−0.190972
$243$	0	0
$244$	2.53455	0.162258
$245$	−0.505055	−0.0322668
$246$	0	0
$247$	14.3649	0.914014
$248$	2.67526	0.169879
$249$	0	0
$250$	12.2893	0.777243
$251$	−3.84757	−0.242857	−0.121428	−	0.992600i	$-0.538747\pi$
$251$	−3.84757	−0.242857	−0.121428	+	0.992600i	$0.538747\pi$
$252$	0	0
$253$	−7.16090	−0.450202
$254$	7.19469	0.451435
$255$	0	0
$256$	13.2951	0.830943
$257$	−30.3426	−1.89272	−0.946359	−	0.323118i	$-0.895269\pi$
$257$	−30.3426	−1.89272	−0.946359	+	0.323118i	$0.895269\pi$
$258$	0	0
$259$	17.0315	1.05828
$260$	−2.75347	−0.170763
$261$	0	0
$262$	−35.0795	−2.16722
$263$	27.1948	1.67690	0.838451	−	0.544977i	$-0.183461\pi$
$263$	27.1948	1.67690	0.838451	+	0.544977i	$0.183461\pi$
$264$	0	0
$265$	−0.955121	−0.0586726
$266$	−10.4686	−0.641870
$267$	0	0
$268$	−2.82388	−0.172496
$269$	0.447683	0.0272957	0.0136478	−	0.999907i	$-0.495656\pi$
$269$	0.447683	0.0272957	0.0136478	+	0.999907i	$0.495656\pi$
$270$	0	0
$271$	−15.0696	−0.915411	−0.457705	−	0.889104i	$-0.651329\pi$
$271$	−15.0696	−0.915411	−0.457705	+	0.889104i	$0.651329\pi$
$272$	0	0
$273$	0	0
$274$	20.7700	1.25476
$275$	−15.5313	−0.936574
$276$	0	0
$277$	1.97884	0.118897	0.0594486	−	0.998231i	$-0.481066\pi$
$277$	1.97884	0.118897	0.0594486	+	0.998231i	$0.481066\pi$
$278$	21.6626	1.29924
$279$	0	0
$280$	−4.64500	−0.277592
$281$	29.1820	1.74085	0.870427	−	0.492297i	$-0.163843\pi$
$281$	29.1820	1.74085	0.870427	+	0.492297i	$0.163843\pi$
$282$	0	0
$283$	20.7155	1.23141	0.615705	−	0.787977i	$-0.288871\pi$
$283$	20.7155	1.23141	0.615705	+	0.787977i	$0.288871\pi$
$284$	−5.06705	−0.300674
$285$	0	0
$286$	32.3367	1.91211
$287$	24.0569	1.42003
$288$	0	0
$289$	0	0
$290$	7.72220	0.453463
$291$	0	0
$292$	−0.333211	−0.0194997
$293$	−5.05564	−0.295354	−0.147677	−	0.989036i	$-0.547180\pi$
$293$	−5.05564	−0.295354	−0.147677	+	0.989036i	$0.547180\pi$
$294$	0	0
$295$	−6.26304	−0.364648
$296$	−15.1936	−0.883109
$297$	0	0
$298$	16.2592	0.941867
$299$	11.1761	0.646330
$300$	0	0
$301$	−23.4125	−1.34948
$302$	28.1583	1.62033
$303$	0	0
$304$	12.4383	0.713385
$305$	3.42782	0.196276
$306$	0	0
$307$	−6.69816	−0.382284	−0.191142	−	0.981562i	$-0.561219\pi$
$307$	−6.69816	−0.382284	−0.191142	+	0.981562i	$0.561219\pi$
$308$	−5.46159	−0.311203
$309$	0	0
$310$	−1.56302	−0.0887736
$311$	5.27254	0.298978	0.149489	−	0.988763i	$-0.452237\pi$
$311$	5.27254	0.298978	0.149489	+	0.988763i	$0.452237\pi$
$312$	0	0
$313$	8.10801	0.458292	0.229146	−	0.973392i	$-0.426407\pi$
$313$	8.10801	0.458292	0.229146	+	0.973392i	$0.426407\pi$
$314$	−26.6677	−1.50495
$315$	0	0
$316$	7.95380	0.447436
$317$	−3.93073	−0.220772	−0.110386	−	0.993889i	$-0.535209\pi$
$317$	−3.93073	−0.220772	−0.110386	+	0.993889i	$0.535209\pi$
$318$	0	0
$319$	−21.0181	−1.17679
$320$	3.54965	0.198432
$321$	0	0
$322$	−8.14473	−0.453888
$323$	0	0
$324$	0	0
$325$	24.2399	1.34459
$326$	−17.9320	−0.993164
$327$	0	0
$328$	−21.4609	−1.18498
$329$	−23.2862	−1.28381
$330$	0	0
$331$	−19.9729	−1.09781	−0.548906	−	0.835884i	$-0.684955\pi$
$331$	−19.9729	−1.09781	−0.548906	+	0.835884i	$0.684955\pi$
$332$	5.19387	0.285051
$333$	0	0
$334$	3.54452	0.193947
$335$	−3.81912	−0.208661
$336$	0	0
$337$	30.8390	1.67991	0.839955	−	0.542656i	$-0.182581\pi$
$337$	30.8390	1.67991	0.839955	+	0.542656i	$0.182581\pi$
$338$	−29.4928	−1.60420
$339$	0	0
$340$	0	0
$341$	4.25418	0.230377
$342$	0	0
$343$	19.2460	1.03919
$344$	20.8860	1.12610
$345$	0	0
$346$	−12.3129	−0.661945
$347$	−16.8330	−0.903641	−0.451820	−	0.892109i	$-0.649225\pi$
$347$	−16.8330	−0.903641	−0.451820	+	0.892109i	$0.649225\pi$
$348$	0	0
$349$	−27.9391	−1.49555	−0.747774	−	0.663954i	$-0.768877\pi$
$349$	−27.9391	−1.49555	−0.747774	+	0.663954i	$0.768877\pi$
$350$	−17.6651	−0.944242
$351$	0	0
$352$	11.8492	0.631566
$353$	−22.5616	−1.20083	−0.600416	−	0.799688i	$-0.704998\pi$
$353$	−22.5616	−1.20083	−0.600416	+	0.799688i	$0.704998\pi$
$354$	0	0
$355$	−6.85285	−0.363712
$356$	6.34737	0.336410
$357$	0	0
$358$	27.3950	1.44787
$359$	2.62410	0.138495	0.0692474	−	0.997600i	$-0.477940\pi$
$359$	2.62410	0.138495	0.0692474	+	0.997600i	$0.477940\pi$
$360$	0	0
$361$	−12.4029	−0.652786
$362$	15.7230	0.826385
$363$	0	0
$364$	8.52396	0.446777
$365$	−0.450646	−0.0235879
$366$	0	0
$367$	16.6620	0.869747	0.434874	−	0.900492i	$-0.356793\pi$
$367$	16.6620	0.869747	0.434874	+	0.900492i	$0.356793\pi$
$368$	9.67719	0.504458
$369$	0	0
$370$	8.87683	0.461485
$371$	2.95678	0.153508
$372$	0	0
$373$	26.4354	1.36877	0.684386	−	0.729120i	$-0.260070\pi$
$373$	26.4354	1.36877	0.684386	+	0.729120i	$0.260070\pi$
$374$	0	0
$375$	0	0
$376$	20.7734	1.07130
$377$	32.8031	1.68945
$378$	0	0
$379$	−15.3138	−0.786615	−0.393307	−	0.919407i	$-0.628669\pi$
$379$	−15.3138	−0.786615	−0.393307	+	0.919407i	$0.628669\pi$
$380$	−1.26453	−0.0648693
$381$	0	0
$382$	−42.1369	−2.15591
$383$	−21.9992	−1.12411	−0.562053	−	0.827101i	$-0.689988\pi$
$383$	−21.9992	−1.12411	−0.562053	+	0.827101i	$0.689988\pi$
$384$	0	0
$385$	−7.38644	−0.376448
$386$	28.7715	1.46443
$387$	0	0
$388$	−2.00512	−0.101794
$389$	29.0749	1.47415	0.737077	−	0.675809i	$-0.236206\pi$
$389$	29.0749	1.47415	0.737077	+	0.675809i	$0.236206\pi$
$390$	0	0
$391$	0	0
$392$	−1.39478	−0.0704472
$393$	0	0
$394$	−23.7663	−1.19733
$395$	10.7570	0.541243
$396$	0	0
$397$	33.3302	1.67279	0.836397	−	0.548125i	$-0.184658\pi$
$397$	33.3302	1.67279	0.836397	+	0.548125i	$0.184658\pi$
$398$	−16.8139	−0.842804
$399$	0	0
$400$	20.9889	1.04944
$401$	−21.1160	−1.05448	−0.527241	−	0.849716i	$-0.676773\pi$
$401$	−21.1160	−1.05448	−0.527241	+	0.849716i	$0.676773\pi$
$402$	0	0
$403$	−6.63955	−0.330739
$404$	1.30342	0.0648477
$405$	0	0
$406$	−23.9057	−1.18642
$407$	−24.1607	−1.19760
$408$	0	0
$409$	−16.0388	−0.793066	−0.396533	−	0.918020i	$-0.629787\pi$
$409$	−16.0388	−0.793066	−0.396533	+	0.918020i	$0.629787\pi$
$410$	12.5385	0.619232
$411$	0	0
$412$	3.46292	0.170606
$413$	19.3886	0.954049
$414$	0	0
$415$	7.02437	0.344813
$416$	−18.4932	−0.906704
$417$	0	0
$418$	14.8507	0.726369
$419$	30.3271	1.48157	0.740787	−	0.671739i	$-0.234453\pi$
$419$	30.3271	1.48157	0.740787	+	0.671739i	$0.234453\pi$
$420$	0	0
$421$	−3.28147	−0.159929	−0.0799645	−	0.996798i	$-0.525481\pi$
$421$	−3.28147	−0.159929	−0.0799645	+	0.996798i	$0.525481\pi$
$422$	−22.4919	−1.09489
$423$	0	0
$424$	−2.63771	−0.128098
$425$	0	0
$426$	0	0
$427$	−10.6115	−0.513528
$428$	−6.67284	−0.322544
$429$	0	0
$430$	−12.2026	−0.588464
$431$	−0.636903	−0.0306786	−0.0153393	−	0.999882i	$-0.504883\pi$
$431$	−0.636903	−0.0306786	−0.0153393	+	0.999882i	$0.504883\pi$
$432$	0	0
$433$	38.8945	1.86915	0.934576	−	0.355764i	$-0.115779\pi$
$433$	38.8945	1.86915	0.934576	+	0.355764i	$0.115779\pi$
$434$	4.83866	0.232263
$435$	0	0
$436$	6.16312	0.295160
$437$	5.13263	0.245527
$438$	0	0
$439$	−18.2017	−0.868718	−0.434359	−	0.900740i	$-0.643025\pi$
$439$	−18.2017	−0.868718	−0.434359	+	0.900740i	$0.643025\pi$
$440$	6.58936	0.314135
$441$	0	0
$442$	0	0
$443$	−14.4462	−0.686361	−0.343181	−	0.939269i	$-0.611504\pi$
$443$	−14.4462	−0.686361	−0.343181	+	0.939269i	$0.611504\pi$
$444$	0	0
$445$	8.58440	0.406939
$446$	−30.6130	−1.44957
$447$	0	0
$448$	−10.9887	−0.519168
$449$	37.0123	1.74672	0.873360	−	0.487076i	$-0.161937\pi$
$449$	37.0123	1.74672	0.873360	+	0.487076i	$0.161937\pi$
$450$	0	0
$451$	−34.1269	−1.60697
$452$	9.18454	0.432004
$453$	0	0
$454$	−15.2041	−0.713562
$455$	11.5281	0.540445
$456$	0	0
$457$	15.0383	0.703461	0.351731	−	0.936101i	$-0.385593\pi$
$457$	15.0383	0.703461	0.351731	+	0.936101i	$0.385593\pi$
$458$	33.2022	1.55144
$459$	0	0
$460$	−0.983828	−0.0458712
$461$	13.9905	0.651604	0.325802	−	0.945438i	$-0.394366\pi$
$461$	13.9905	0.651604	0.325802	+	0.945438i	$0.394366\pi$
$462$	0	0
$463$	2.40768	0.111895	0.0559473	−	0.998434i	$-0.482182\pi$
$463$	2.40768	0.111895	0.0559473	+	0.998434i	$0.482182\pi$
$464$	28.4036	1.31861
$465$	0	0
$466$	40.3812	1.87062
$467$	12.7577	0.590354	0.295177	−	0.955443i	$-0.404621\pi$
$467$	12.7577	0.590354	0.295177	+	0.955443i	$0.404621\pi$
$468$	0	0
$469$	11.8229	0.545930
$470$	−12.1368	−0.559830
$471$	0	0
$472$	−17.2963	−0.796127
$473$	33.2128	1.52713
$474$	0	0
$475$	11.1322	0.510780
$476$	0	0
$477$	0	0
$478$	−8.36949	−0.382811
$479$	32.9435	1.50523	0.752613	−	0.658463i	$-0.228793\pi$
$479$	32.9435	1.50523	0.752613	+	0.658463i	$0.228793\pi$
$480$	0	0
$481$	37.7078	1.71933
$482$	−2.05273	−0.0934992
$483$	0	0
$484$	1.11091	0.0504961
$485$	−2.71179	−0.123136
$486$	0	0
$487$	17.5577	0.795617	0.397809	−	0.917468i	$-0.369771\pi$
$487$	17.5577	0.795617	0.397809	+	0.917468i	$0.369771\pi$
$488$	9.46642	0.428525
$489$	0	0
$490$	0.814901	0.0368135
$491$	−35.8504	−1.61790	−0.808952	−	0.587874i	$-0.799965\pi$
$491$	−35.8504	−1.61790	−0.808952	+	0.587874i	$0.799965\pi$
$492$	0	0
$493$	0	0
$494$	−23.1776	−1.04281
$495$	0	0
$496$	−5.74907	−0.258141
$497$	21.2145	0.951599
$498$	0	0
$499$	27.9614	1.25172	0.625861	−	0.779934i	$-0.284748\pi$
$499$	27.9614	1.25172	0.625861	+	0.779934i	$0.284748\pi$
$500$	−4.59547	−0.205516
$501$	0	0
$502$	6.20802	0.277077
$503$	−25.7397	−1.14768	−0.573838	−	0.818969i	$-0.694546\pi$
$503$	−25.7397	−1.14768	−0.573838	+	0.818969i	$0.694546\pi$
$504$	0	0
$505$	1.76279	0.0784433
$506$	11.5540	0.513640
$507$	0	0
$508$	−2.69039	−0.119367
$509$	−39.9728	−1.77176	−0.885882	−	0.463910i	$-0.846446\pi$
$509$	−39.9728	−1.77176	−0.885882	+	0.463910i	$0.846446\pi$
$510$	0	0
$511$	1.39507	0.0617143
$512$	5.81277	0.256891
$513$	0	0
$514$	48.9574	2.15942
$515$	4.68337	0.206374
$516$	0	0
$517$	33.0336	1.45282
$518$	−27.4801	−1.20741
$519$	0	0
$520$	−10.2841	−0.450986
$521$	−12.9910	−0.569144	−0.284572	−	0.958655i	$-0.591851\pi$
$521$	−12.9910	−0.569144	−0.284572	+	0.958655i	$0.591851\pi$
$522$	0	0
$523$	1.53869	0.0672822	0.0336411	−	0.999434i	$-0.489290\pi$
$523$	1.53869	0.0672822	0.0336411	+	0.999434i	$0.489290\pi$
$524$	13.1176	0.573047
$525$	0	0
$526$	−43.8785	−1.91319
$527$	0	0
$528$	0	0
$529$	−19.0067	−0.826380
$530$	1.54108	0.0669402
$531$	0	0
$532$	3.91464	0.169721
$533$	53.2622	2.30704
$534$	0	0
$535$	−9.02458	−0.390167
$536$	−10.5471	−0.455563
$537$	0	0
$538$	−0.722331	−0.0311419
$539$	−2.21798	−0.0955350
$540$	0	0
$541$	28.8294	1.23947	0.619736	−	0.784810i	$-0.287240\pi$
$541$	28.8294	1.23947	0.619736	+	0.784810i	$0.287240\pi$
$542$	24.3146	1.04440
$543$	0	0
$544$	0	0
$545$	8.33522	0.357042
$546$	0	0
$547$	−6.26865	−0.268028	−0.134014	−	0.990979i	$-0.542787\pi$
$547$	−6.26865	−0.268028	−0.134014	+	0.990979i	$0.542787\pi$
$548$	−7.76677	−0.331780
$549$	0	0
$550$	25.0596	1.06855
$551$	15.0649	0.641784
$552$	0	0
$553$	−33.3006	−1.41608
$554$	−3.19284	−0.135651
$555$	0	0
$556$	−8.10054	−0.343539
$557$	−5.26768	−0.223199	−0.111599	−	0.993753i	$-0.535597\pi$
$557$	−5.26768	−0.223199	−0.111599	+	0.993753i	$0.535597\pi$
$558$	0	0
$559$	−51.8356	−2.19241
$560$	9.98198	0.421816
$561$	0	0
$562$	−47.0849	−1.98616
$563$	12.7777	0.538515	0.269257	−	0.963068i	$-0.413222\pi$
$563$	12.7777	0.538515	0.269257	+	0.963068i	$0.413222\pi$
$564$	0	0
$565$	12.4215	0.522576
$566$	−33.4243	−1.40493
$567$	0	0
$568$	−18.9252	−0.794082
$569$	6.91475	0.289881	0.144941	−	0.989440i	$-0.453701\pi$
$569$	6.91475	0.289881	0.144941	+	0.989440i	$0.453701\pi$
$570$	0	0
$571$	9.01348	0.377202	0.188601	−	0.982054i	$-0.439605\pi$
$571$	9.01348	0.377202	0.188601	+	0.982054i	$0.439605\pi$
$572$	−12.0920	−0.505592
$573$	0	0
$574$	−38.8156	−1.62013
$575$	8.66102	0.361189
$576$	0	0
$577$	37.5084	1.56149	0.780747	−	0.624848i	$-0.214839\pi$
$577$	37.5084	1.56149	0.780747	+	0.624848i	$0.214839\pi$
$578$	0	0
$579$	0	0
$580$	−2.88765	−0.119903
$581$	−21.7454	−0.902152
$582$	0	0
$583$	−4.19446	−0.173717
$584$	−1.24453	−0.0514988
$585$	0	0
$586$	8.15723	0.336972
$587$	20.8834	0.861950	0.430975	−	0.902364i	$-0.358170\pi$
$587$	20.8834	0.861950	0.430975	+	0.902364i	$0.358170\pi$
$588$	0	0
$589$	−3.04922	−0.125641
$590$	10.1053	0.416031
$591$	0	0
$592$	32.6506	1.34193
$593$	−27.8291	−1.14280	−0.571402	−	0.820670i	$-0.693600\pi$
$593$	−27.8291	−1.14280	−0.571402	+	0.820670i	$0.693600\pi$
$594$	0	0
$595$	0	0
$596$	−6.07996	−0.249045
$597$	0	0
$598$	−18.0325	−0.737404
$599$	−38.9632	−1.59199	−0.795997	−	0.605300i	$-0.793053\pi$
$599$	−38.9632	−1.59199	−0.795997	+	0.605300i	$0.793053\pi$
$600$	0	0
$601$	24.5350	1.00080	0.500402	−	0.865793i	$-0.333186\pi$
$601$	24.5350	1.00080	0.500402	+	0.865793i	$0.333186\pi$
$602$	37.7759	1.53963
$603$	0	0
$604$	−10.5296	−0.428441
$605$	1.50244	0.0610828
$606$	0	0
$607$	−15.7988	−0.641253	−0.320626	−	0.947206i	$-0.603893\pi$
$607$	−15.7988	−0.641253	−0.320626	+	0.947206i	$0.603893\pi$
$608$	−8.49302	−0.344437
$609$	0	0
$610$	−5.53075	−0.223934
$611$	−51.5559	−2.08573
$612$	0	0
$613$	13.1247	0.530103	0.265052	−	0.964234i	$-0.414611\pi$
$613$	13.1247	0.530103	0.265052	+	0.964234i	$0.414611\pi$
$614$	10.8074	0.436152
$615$	0	0
$616$	−20.3987	−0.821889
$617$	−14.3978	−0.579635	−0.289817	−	0.957082i	$-0.593595\pi$
$617$	−14.3978	−0.579635	−0.289817	+	0.957082i	$0.593595\pi$
$618$	0	0
$619$	−20.2851	−0.815325	−0.407663	−	0.913133i	$-0.633656\pi$
$619$	−20.2851	−0.815325	−0.407663	+	0.913133i	$0.633656\pi$
$620$	0.584477	0.0234732
$621$	0	0
$622$	−8.50718	−0.341107
$623$	−26.5748	−1.06470
$624$	0	0
$625$	15.4557	0.618229
$626$	−13.0822	−0.522870
$627$	0	0
$628$	9.97216	0.397933
$629$	0	0
$630$	0	0
$631$	−0.933597	−0.0371659	−0.0185830	−	0.999827i	$-0.505915\pi$
$631$	−0.933597	−0.0371659	−0.0185830	+	0.999827i	$0.505915\pi$
$632$	29.7070	1.18168
$633$	0	0
$634$	6.34219	0.251881
$635$	−3.63858	−0.144393
$636$	0	0
$637$	3.46162	0.137154
$638$	33.9124	1.34261
$639$	0	0
$640$	−11.1237	−0.439703
$641$	−5.69896	−0.225095	−0.112548	−	0.993646i	$-0.535901\pi$
$641$	−5.69896	−0.225095	−0.112548	+	0.993646i	$0.535901\pi$
$642$	0	0
$643$	−2.74575	−0.108282	−0.0541408	−	0.998533i	$-0.517242\pi$
$643$	−2.74575	−0.108282	−0.0541408	+	0.998533i	$0.517242\pi$
$644$	3.04565	0.120015
$645$	0	0
$646$	0	0
$647$	8.48539	0.333595	0.166798	−	0.985991i	$-0.446657\pi$
$647$	8.48539	0.333595	0.166798	+	0.985991i	$0.446657\pi$
$648$	0	0
$649$	−27.5044	−1.07964
$650$	−39.1108	−1.53405
$651$	0	0
$652$	6.70553	0.262609
$653$	9.61573	0.376293	0.188146	−	0.982141i	$-0.439752\pi$
$653$	9.61573	0.376293	0.188146	+	0.982141i	$0.439752\pi$
$654$	0	0
$655$	17.7408	0.693189
$656$	46.1188	1.80064
$657$	0	0
$658$	37.5721	1.46471
$659$	−16.9543	−0.660446	−0.330223	−	0.943903i	$-0.607124\pi$
$659$	−16.9543	−0.660446	−0.330223	+	0.943903i	$0.607124\pi$
$660$	0	0
$661$	22.8810	0.889968	0.444984	−	0.895539i	$-0.353209\pi$
$661$	22.8810	0.889968	0.444984	+	0.895539i	$0.353209\pi$
$662$	32.2261	1.25250
$663$	0	0
$664$	19.3988	0.752821
$665$	5.29429	0.205304
$666$	0	0
$667$	11.7207	0.453827
$668$	−1.32544	−0.0512829
$669$	0	0
$670$	6.16211	0.238063
$671$	15.0534	0.581131
$672$	0	0
$673$	0.972535	0.0374885	0.0187442	−	0.999824i	$-0.494033\pi$
$673$	0.972535	0.0374885	0.0187442	+	0.999824i	$0.494033\pi$
$674$	−49.7585	−1.91663
$675$	0	0
$676$	11.0286	0.424176
$677$	−20.6935	−0.795315	−0.397657	−	0.917534i	$-0.630177\pi$
$677$	−20.6935	−0.795315	−0.397657	+	0.917534i	$0.630177\pi$
$678$	0	0
$679$	8.39492	0.322167
$680$	0	0
$681$	0	0
$682$	−6.86408	−0.262839
$683$	21.9329	0.839237	0.419619	−	0.907701i	$-0.362164\pi$
$683$	21.9329	0.839237	0.419619	+	0.907701i	$0.362164\pi$
$684$	0	0
$685$	−10.5040	−0.401339
$686$	−31.0533	−1.18562
$687$	0	0
$688$	−44.8835	−1.71117
$689$	6.54634	0.249396
$690$	0	0
$691$	30.0105	1.14165	0.570826	−	0.821071i	$-0.306623\pi$
$691$	30.0105	1.14165	0.570826	+	0.821071i	$0.306623\pi$
$692$	4.60430	0.175029
$693$	0	0
$694$	27.1598	1.03097
$695$	−10.9554	−0.415564
$696$	0	0
$697$	0	0
$698$	45.0795	1.70628
$699$	0	0
$700$	6.60572	0.249673
$701$	−13.0830	−0.494140	−0.247070	−	0.968998i	$-0.579468\pi$
$701$	−13.0830	−0.494140	−0.247070	+	0.968998i	$0.579468\pi$
$702$	0	0
$703$	17.3174	0.653137
$704$	15.5885	0.587513
$705$	0	0
$706$	36.4029	1.37004
$707$	−5.45711	−0.205236
$708$	0	0
$709$	−32.8467	−1.23358	−0.616792	−	0.787126i	$-0.711568\pi$
$709$	−32.8467	−1.23358	−0.616792	+	0.787126i	$0.711568\pi$
$710$	11.0570	0.414962
$711$	0	0
$712$	23.7071	0.888460
$713$	−2.37234	−0.0888448
$714$	0	0
$715$	−16.3537	−0.611592
$716$	−10.2441	−0.382841
$717$	0	0
$718$	−4.23396	−0.158010
$719$	13.6307	0.508340	0.254170	−	0.967160i	$-0.418198\pi$
$719$	13.6307	0.508340	0.254170	+	0.967160i	$0.418198\pi$
$720$	0	0
$721$	−14.4984	−0.539948
$722$	20.0120	0.744769
$723$	0	0
$724$	−5.87949	−0.218510
$725$	25.4211	0.944115
$726$	0	0
$727$	8.12446	0.301320	0.150660	−	0.988586i	$-0.451860\pi$
$727$	8.12446	0.301320	0.150660	+	0.988586i	$0.451860\pi$
$728$	31.8365	1.17994
$729$	0	0
$730$	0.727113	0.0269117
$731$	0	0
$732$	0	0
$733$	31.1522	1.15063	0.575316	−	0.817931i	$-0.304879\pi$
$733$	31.1522	1.15063	0.575316	+	0.817931i	$0.304879\pi$
$734$	−26.8839	−0.992303
$735$	0	0
$736$	−6.60771	−0.243563
$737$	−16.7718	−0.617799
$738$	0	0
$739$	34.6895	1.27607	0.638037	−	0.770005i	$-0.279747\pi$
$739$	34.6895	1.27607	0.638037	+	0.770005i	$0.279747\pi$
$740$	−3.31941	−0.122024
$741$	0	0
$742$	−4.77074	−0.175139
$743$	48.6272	1.78396	0.891979	−	0.452076i	$-0.149317\pi$
$743$	48.6272	1.78396	0.891979	+	0.452076i	$0.149317\pi$
$744$	0	0
$745$	−8.22275	−0.301258
$746$	−42.6532	−1.56164
$747$	0	0
$748$	0	0
$749$	27.9375	1.02081
$750$	0	0
$751$	−41.9289	−1.53001	−0.765004	−	0.644025i	$-0.777263\pi$
$751$	−41.9289	−1.53001	−0.765004	+	0.644025i	$0.777263\pi$
$752$	−44.6414	−1.62790
$753$	0	0
$754$	−52.9275	−1.92751
$755$	−14.2405	−0.518266
$756$	0	0
$757$	10.7455	0.390553	0.195277	−	0.980748i	$-0.437439\pi$
$757$	10.7455	0.390553	0.195277	+	0.980748i	$0.437439\pi$
$758$	24.7086	0.897456
$759$	0	0
$760$	−4.72297	−0.171320
$761$	7.39335	0.268009	0.134004	−	0.990981i	$-0.457216\pi$
$761$	7.39335	0.268009	0.134004	+	0.990981i	$0.457216\pi$
$762$	0	0
$763$	−25.8034	−0.934148
$764$	15.7567	0.570058
$765$	0	0
$766$	35.4955	1.28250
$767$	42.9265	1.54998
$768$	0	0
$769$	−17.4382	−0.628838	−0.314419	−	0.949284i	$-0.601810\pi$
$769$	−17.4382	−0.628838	−0.314419	+	0.949284i	$0.601810\pi$
$770$	11.9180	0.429493
$771$	0	0
$772$	−10.7588	−0.387219
$773$	31.1238	1.11945	0.559723	−	0.828680i	$-0.310907\pi$
$773$	31.1238	1.11945	0.559723	+	0.828680i	$0.310907\pi$
$774$	0	0
$775$	−5.14538	−0.184827
$776$	−7.48901	−0.268840
$777$	0	0
$778$	−46.9120	−1.68188
$779$	24.4607	0.876396
$780$	0	0
$781$	−30.0947	−1.07687
$782$	0	0
$783$	0	0
$784$	2.99735	0.107048
$785$	13.4867	0.481361
$786$	0	0
$787$	−7.21080	−0.257037	−0.128519	−	0.991707i	$-0.541022\pi$
$787$	−7.21080	−0.257037	−0.128519	+	0.991707i	$0.541022\pi$
$788$	8.88719	0.316593
$789$	0	0
$790$	−17.3563	−0.617509
$791$	−38.4533	−1.36724
$792$	0	0
$793$	−23.4940	−0.834298
$794$	−53.7779	−1.90851
$795$	0	0
$796$	6.28740	0.222851
$797$	7.58702	0.268746	0.134373	−	0.990931i	$-0.457098\pi$
$797$	7.58702	0.268746	0.134373	+	0.990931i	$0.457098\pi$
$798$	0	0
$799$	0	0
$800$	−14.3315	−0.506695
$801$	0	0
$802$	34.0704	1.20307
$803$	−1.97904	−0.0698386
$804$	0	0
$805$	4.11904	0.145177
$806$	10.7128	0.377344
$807$	0	0
$808$	4.86822	0.171263
$809$	5.33119	0.187435	0.0937174	−	0.995599i	$-0.470125\pi$
$809$	5.33119	0.187435	0.0937174	+	0.995599i	$0.470125\pi$
$810$	0	0
$811$	46.9291	1.64790	0.823951	−	0.566661i	$-0.191765\pi$
$811$	46.9291	1.64790	0.823951	+	0.566661i	$0.191765\pi$
$812$	8.93933	0.313709
$813$	0	0
$814$	38.9831	1.36636
$815$	9.06879	0.317666
$816$	0	0
$817$	−23.8055	−0.832850
$818$	25.8784	0.904817
$819$	0	0
$820$	−4.68866	−0.163735
$821$	20.4129	0.712416	0.356208	−	0.934407i	$-0.384069\pi$
$821$	20.4129	0.712416	0.356208	+	0.934407i	$0.384069\pi$
$822$	0	0
$823$	33.6464	1.17284	0.586419	−	0.810008i	$-0.300537\pi$
$823$	33.6464	1.17284	0.586419	+	0.810008i	$0.300537\pi$
$824$	12.9338	0.450571
$825$	0	0
$826$	−31.2833	−1.08848
$827$	13.9070	0.483592	0.241796	−	0.970327i	$-0.422264\pi$
$827$	13.9070	0.483592	0.241796	+	0.970327i	$0.422264\pi$
$828$	0	0
$829$	41.0730	1.42652	0.713262	−	0.700897i	$-0.247217\pi$
$829$	41.0730	1.42652	0.713262	+	0.700897i	$0.247217\pi$
$830$	−11.3338	−0.393400
$831$	0	0
$832$	−24.3291	−0.843460
$833$	0	0
$834$	0	0
$835$	−1.79257	−0.0620345
$836$	−5.55327	−0.192064
$837$	0	0
$838$	−48.9324	−1.69034
$839$	14.9174	0.515006	0.257503	−	0.966278i	$-0.417100\pi$
$839$	14.9174	0.515006	0.257503	+	0.966278i	$0.417100\pi$
$840$	0	0
$841$	5.40160	0.186262
$842$	5.29462	0.182465
$843$	0	0
$844$	8.41066	0.289507
$845$	14.9154	0.513106
$846$	0	0
$847$	−4.65111	−0.159814
$848$	5.66836	0.194652
$849$	0	0
$850$	0	0
$851$	13.4732	0.461855
$852$	0	0
$853$	−12.1679	−0.416621	−0.208311	−	0.978063i	$-0.566796\pi$
$853$	−12.1679	−0.416621	−0.208311	+	0.978063i	$0.566796\pi$
$854$	17.1216	0.585889
$855$	0	0
$856$	−24.9227	−0.851841
$857$	45.4324	1.55194	0.775970	−	0.630770i	$-0.217261\pi$
$857$	45.4324	1.55194	0.775970	+	0.630770i	$0.217261\pi$
$858$	0	0
$859$	24.5741	0.838458	0.419229	−	0.907880i	$-0.362300\pi$
$859$	24.5741	0.838458	0.419229	+	0.907880i	$0.362300\pi$
$860$	4.56307	0.155599
$861$	0	0
$862$	1.02764	0.0350015
$863$	−2.35798	−0.0802667	−0.0401334	−	0.999194i	$-0.512778\pi$
$863$	−2.35798	−0.0802667	−0.0401334	+	0.999194i	$0.512778\pi$
$864$	0	0
$865$	6.22701	0.211725
$866$	−62.7559	−2.13253
$867$	0	0
$868$	−1.80937	−0.0614141
$869$	47.2399	1.60250
$870$	0	0
$871$	26.1760	0.886939
$872$	23.0189	0.779520
$873$	0	0
$874$	−8.28145	−0.280124
$875$	19.2401	0.650434
$876$	0	0
$877$	37.0944	1.25259	0.626294	−	0.779587i	$-0.284571\pi$
$877$	37.0944	1.25259	0.626294	+	0.779587i	$0.284571\pi$
$878$	29.3682	0.991129
$879$	0	0
$880$	−14.1603	−0.477345
$881$	35.0035	1.17930	0.589648	−	0.807660i	$-0.299266\pi$
$881$	35.0035	1.17930	0.589648	+	0.807660i	$0.299266\pi$
$882$	0	0
$883$	−4.09966	−0.137964	−0.0689822	−	0.997618i	$-0.521975\pi$
$883$	−4.09966	−0.137964	−0.0689822	+	0.997618i	$0.521975\pi$
$884$	0	0
$885$	0	0
$886$	23.3089	0.783076
$887$	−4.54407	−0.152575	−0.0762874	−	0.997086i	$-0.524307\pi$
$887$	−4.54407	−0.152575	−0.0762874	+	0.997086i	$0.524307\pi$
$888$	0	0
$889$	11.2640	0.377782
$890$	−13.8508	−0.464281
$891$	0	0
$892$	11.4474	0.383289
$893$	−23.6771	−0.792324
$894$	0	0
$895$	−13.8545	−0.463105
$896$	34.4358	1.15042
$897$	0	0
$898$	−59.7190	−1.99285
$899$	−6.96309	−0.232232
$900$	0	0
$901$	0	0
$902$	55.0634	1.83341
$903$	0	0
$904$	34.3038	1.14093
$905$	−7.95163	−0.264321
$906$	0	0
$907$	−8.69309	−0.288649	−0.144325	−	0.989530i	$-0.546101\pi$
$907$	−8.69309	−0.288649	−0.144325	+	0.989530i	$0.546101\pi$
$908$	5.68543	0.188677
$909$	0	0
$910$	−18.6005	−0.616599
$911$	−45.8572	−1.51932	−0.759658	−	0.650322i	$-0.774634\pi$
$911$	−45.8572	−1.51932	−0.759658	+	0.650322i	$0.774634\pi$
$912$	0	0
$913$	30.8479	1.02092
$914$	−24.2641	−0.802586
$915$	0	0
$916$	−12.4157	−0.410225
$917$	−54.9203	−1.81363
$918$	0	0
$919$	36.9580	1.21913	0.609566	−	0.792735i	$-0.291344\pi$
$919$	36.9580	1.21913	0.609566	+	0.792735i	$0.291344\pi$
$920$	−3.67455	−0.121146
$921$	0	0
$922$	−22.5736	−0.743421
$923$	46.9690	1.54600
$924$	0	0
$925$	29.2221	0.960815
$926$	−3.88477	−0.127662
$927$	0	0
$928$	−19.3944	−0.636652
$929$	37.1359	1.21839	0.609195	−	0.793020i	$-0.291493\pi$
$929$	37.1359	1.21839	0.609195	+	0.793020i	$0.291493\pi$
$930$	0	0
$931$	1.58975	0.0521020
$932$	−15.1002	−0.494623
$933$	0	0
$934$	−20.5844	−0.673541
$935$	0	0
$936$	0	0
$937$	−20.2047	−0.660058	−0.330029	−	0.943971i	$-0.607059\pi$
$937$	−20.2047	−0.660058	−0.330029	+	0.943971i	$0.607059\pi$
$938$	−19.0761	−0.622857
$939$	0	0
$940$	4.53845	0.148028
$941$	−48.0881	−1.56763	−0.783814	−	0.620996i	$-0.786728\pi$
$941$	−48.0881	−1.56763	−0.783814	+	0.620996i	$0.786728\pi$
$942$	0	0
$943$	19.0308	0.619729
$944$	37.1693	1.20976
$945$	0	0
$946$	−53.5885	−1.74231
$947$	23.0130	0.747821	0.373911	−	0.927465i	$-0.378017\pi$
$947$	23.0130	0.747821	0.373911	+	0.927465i	$0.378017\pi$
$948$	0	0
$949$	3.08870	0.100263
$950$	−17.9617	−0.582754
$951$	0	0
$952$	0	0
$953$	0.0564747	0.00182940	0.000914698	−	1.00000i	$-0.499709\pi$
$953$	0.0564747	0.00182940	0.000914698		1.00000i	$0.499709\pi$
$954$	0	0
$955$	21.3099	0.689573
$956$	3.12969	0.101222
$957$	0	0
$958$	−53.1540	−1.71733
$959$	32.5175	1.05004
$960$	0	0
$961$	−29.5906	−0.954536
$962$	−60.8412	−1.96160
$963$	0	0
$964$	0.767599	0.0247227
$965$	−14.5506	−0.468401
$966$	0	0
$967$	0.706923	0.0227331	0.0113666	−	0.999935i	$-0.496382\pi$
$967$	0.706923	0.0227331	0.0113666	+	0.999935i	$0.496382\pi$
$968$	4.14920	0.133360
$969$	0	0
$970$	4.37545	0.140487
$971$	−10.5257	−0.337785	−0.168892	−	0.985634i	$-0.554019\pi$
$971$	−10.5257	−0.337785	−0.168892	+	0.985634i	$0.554019\pi$
$972$	0	0
$973$	33.9149	1.08726
$974$	−28.3293	−0.907728
$975$	0	0
$976$	−20.3431	−0.651166
$977$	13.9176	0.445264	0.222632	−	0.974903i	$-0.428535\pi$
$977$	13.9176	0.445264	0.222632	+	0.974903i	$0.428535\pi$
$978$	0	0
$979$	37.6988	1.20486
$980$	−0.304725	−0.00973409
$981$	0	0
$982$	57.8442	1.84588
$983$	−18.7022	−0.596509	−0.298254	−	0.954486i	$-0.596404\pi$
$983$	−18.7022	−0.596509	−0.298254	+	0.954486i	$0.596404\pi$
$984$	0	0
$985$	12.0193	0.382968
$986$	0	0
$987$	0	0
$988$	8.66704	0.275735
$989$	−18.5211	−0.588936
$990$	0	0
$991$	33.4227	1.06171	0.530854	−	0.847463i	$-0.321871\pi$
$991$	33.4227	1.06171	0.530854	+	0.847463i	$0.321871\pi$
$992$	3.92554	0.124636
$993$	0	0
$994$	−34.2293	−1.08569
$995$	8.50330	0.269573
$996$	0	0
$997$	41.8141	1.32427	0.662133	−	0.749386i	$-0.269651\pi$
$997$	41.8141	1.32427	0.662133	+	0.749386i	$0.269651\pi$
$998$	−45.1154	−1.42810
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7803.2.a.bw.1.3		12
3.2	odd	2	inner	7803.2.a.bw.1.10		12
17.2	even	8		459.2.f.c.55.3	✓	24
17.9	even	8		459.2.f.c.217.10	yes	24
17.16	even	2		7803.2.a.bv.1.3		12
51.2	odd	8		459.2.f.c.55.10	yes	24
51.26	odd	8		459.2.f.c.217.3	yes	24
51.50	odd	2		7803.2.a.bv.1.10		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
459.2.f.c.55.3	✓	24	17.2	even	8
459.2.f.c.55.10	yes	24	51.2	odd	8
459.2.f.c.217.3	yes	24	51.26	odd	8
459.2.f.c.217.10	yes	24	17.9	even	8
7803.2.a.bv.1.3		12	17.16	even	2
7803.2.a.bv.1.10		12	51.50	odd	2
7803.2.a.bw.1.3		12	1.1	even	1	trivial
7803.2.a.bw.1.10		12	3.2	odd	2	inner

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:	\( N \)	\(=\)	\( 7803 = 3^{3} \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7803.a (trivial)

\(f(q)\)	\(=\)	\(q-1.61349 q^{2} +0.603350 q^{4} +0.815992 q^{5} -2.52607 q^{7} +2.25348 q^{8} -1.31659 q^{10} +3.58347 q^{11} -5.59275 q^{13} +4.07580 q^{14} -4.84267 q^{16} -2.56848 q^{19} +0.492329 q^{20} -5.78189 q^{22} -1.99832 q^{23} -4.33416 q^{25} +9.02385 q^{26} -1.52411 q^{28} -5.86529 q^{29} +1.18717 q^{31} +3.30664 q^{32} -2.06126 q^{35} -6.74227 q^{37} +4.14421 q^{38} +1.83882 q^{40} -9.52343 q^{41} +9.26834 q^{43} +2.16209 q^{44} +3.22426 q^{46} +9.21835 q^{47} -0.618947 q^{49} +6.99312 q^{50} -3.37439 q^{52} -1.17050 q^{53} +2.92408 q^{55} -5.69246 q^{56} +9.46358 q^{58} -7.67537 q^{59} +4.20080 q^{61} -1.91549 q^{62} +4.35011 q^{64} -4.56364 q^{65} -4.68034 q^{67} +3.32582 q^{70} -8.39819 q^{71} -0.552268 q^{73} +10.8786 q^{74} -1.54969 q^{76} -9.05211 q^{77} +13.1827 q^{79} -3.95158 q^{80} +15.3660 q^{82} +8.60839 q^{83} -14.9544 q^{86} +8.07528 q^{88} +10.5202 q^{89} +14.1277 q^{91} -1.20568 q^{92} -14.8737 q^{94} -2.09586 q^{95} -3.32331 q^{97} +0.998664 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 12 q^{4} + 8 q^{7} + 24 q^{10} - 4 q^{13} + 20 q^{16} - 4 q^{19} - 28 q^{22} + 48 q^{28} + 8 q^{31} + 12 q^{37} + 44 q^{40} - 20 q^{43} + 36 q^{46} + 28 q^{49} + 8 q^{52} - 4 q^{55} + 28 q^{58} + 48 q^{61}+ \cdots + 24 q^{97}+O(q^{100}) \) 12 * q + 12 * q^4 + 8 * q^7 + 24 * q^10 - 4 * q^13 + 20 * q^16 - 4 * q^19 - 28 * q^22 + 48 * q^28 + 8 * q^31 + 12 * q^37 + 44 * q^40 - 20 * q^43 + 36 * q^46 + 28 * q^49 + 8 * q^52 - 4 * q^55 + 28 * q^58 + 48 * q^61 + 40 * q^64 - 8 * q^67 + 44 * q^70 + 4 * q^73 - 28 * q^76 + 40 * q^79 + 76 * q^82 - 40 * q^88 + 52 * q^91 + 28 * q^94 + 24 * q^97

Introduction

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