LMFDB - Embedded newform 7803.2.a.bv.1.10

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:	$1$
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			1.10
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	$-0.306767\pi$
$2$	1.61349	1.14091	0.570455	+	0.821329i	$0.306767\pi$
$3$	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	$-0.341585\pi$
$7$	2.52607	0.954766	0.477383	+	0.878695i	$0.341585\pi$
$8$	−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	$-0.534000\pi$
$31$	−1.18717	−0.213222	−0.106611	+	0.994301i	$0.534000\pi$
$32$	−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	$-0.313020\pi$
$37$	6.74227	1.10842	0.554211	+	0.832376i	$0.313020\pi$
$38$	−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	$-0.734701\pi$
$47$	−9.21835	−1.34463	−0.672317	+	0.740263i	$0.734701\pi$
$48$	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	$-0.474383\pi$
$53$	1.17050	0.160781	0.0803905	+	0.996763i	$0.474383\pi$
$54$	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	$-0.333471\pi$
$59$	7.67537	0.999248	0.499624	+	0.866242i	$0.333471\pi$
$60$	0	0
$61$	−4.20080	−0.537857	−0.268929	−	0.963160i	$-0.586670\pi$
$61$	−4.20080	−0.537857	−0.268929	+	0.963160i	$0.586670\pi$
$62$	−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	$-0.489711\pi$
$73$	0.552268	0.0646381	0.0323191	+	0.999478i	$0.489711\pi$
$74$	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	$-0.765926\pi$
$79$	−13.1827	−1.48317	−0.741586	+	0.670858i	$0.765926\pi$
$80$	−3.95158	−0.441800
$81$	0	0
$82$	−15.3660	−1.69689
$83$	−8.60839	−0.944893	−0.472447	−	0.881359i	$-0.656629\pi$
$83$	−8.60839	−0.944893	−0.472447	+	0.881359i	$0.656629\pi$
$84$	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	$-0.688266\pi$
$89$	−10.5202	−1.11514	−0.557570	+	0.830130i	$0.688266\pi$
$90$	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	$-0.446038\pi$
$97$	3.32331	0.337431	0.168715	+	0.985665i	$0.446038\pi$
$98$	−0.998664	−0.100880
$99$	0	0
$100$	−2.61501	−0.261501
$101$	−2.16031	−0.214959	−0.107479	−	0.994207i	$-0.534278\pi$
$101$	−2.16031	−0.214959	−0.107479	+	0.994207i	$0.534278\pi$
$102$	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	$-0.662712\pi$
$109$	−10.2148	−0.978404	−0.489202	+	0.872170i	$0.662712\pi$
$110$	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	$-0.314667\pi$
$137$	12.8727	1.09979	0.549896	+	0.835233i	$0.314667\pi$
$138$	0	0
$139$	13.4259	1.13877	0.569386	−	0.822070i	$-0.307181\pi$
$139$	13.4259	1.13877	0.569386	+	0.822070i	$0.307181\pi$
$140$	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	$-0.364561\pi$
$149$	10.0770	0.825541	0.412770	+	0.910835i	$0.364561\pi$
$150$	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	$-0.643340\pi$
$163$	−11.1138	−0.870502	−0.435251	+	0.900309i	$0.643340\pi$
$164$	−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	$-0.281196\pi$
$179$	16.9787	1.26905	0.634525	+	0.772902i	$0.281196\pi$
$180$	0	0
$181$	9.74474	0.724321	0.362160	−	0.932116i	$-0.382039\pi$
$181$	9.74474	0.724321	0.362160	+	0.932116i	$0.382039\pi$
$182$	−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	$-0.893764\pi$
$191$	−26.1154	−1.88964	−0.944821	+	0.327588i	$0.893764\pi$
$192$	0	0
$193$	17.8318	1.28356	0.641781	−	0.766888i	$-0.278196\pi$
$193$	17.8318	1.28356	0.641781	+	0.766888i	$0.278196\pi$
$194$	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	$-0.620422\pi$
$199$	−10.4208	−0.738712	−0.369356	+	0.929288i	$0.620422\pi$
$200$	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	$-0.659302\pi$
$211$	−13.9399	−0.959664	−0.479832	+	0.877360i	$0.659302\pi$
$212$	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	$-0.553655\pi$
$239$	−5.18719	−0.335532	−0.167766	+	0.985827i	$0.553655\pi$
$240$	0	0
$241$	−1.27223	−0.0819515	−0.0409757	−	0.999160i	$-0.513047\pi$
$241$	−1.27223	−0.0819515	−0.0409757	+	0.999160i	$0.513047\pi$
$242$	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	$-0.461253\pi$
$251$	3.84757	0.242857	0.121428	+	0.992600i	$0.461253\pi$
$252$	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	$-0.104731\pi$
$257$	30.3426	1.89272	0.946359	+	0.323118i	$0.104731\pi$
$258$	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	$-0.816539\pi$
$263$	−27.1948	−1.67690	−0.838451	+	0.544977i	$0.816539\pi$
$264$	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	$-0.518934\pi$
$277$	−1.97884	−0.118897	−0.0594486	+	0.998231i	$0.518934\pi$
$278$	21.6626	1.29924
$279$	0	0
$280$	−4.64500	−0.277592
$281$	−29.1820	−1.74085	−0.870427	−	0.492297i	$-0.836157\pi$
$281$	−29.1820	−1.74085	−0.870427	+	0.492297i	$0.836157\pi$
$282$	0	0
$283$	−20.7155	−1.23141	−0.615705	−	0.787977i	$-0.711129\pi$
$283$	−20.7155	−1.23141	−0.615705	+	0.787977i	$0.711129\pi$
$284$	−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	$-0.452820\pi$
$293$	5.05564	0.295354	0.147677	+	0.989036i	$0.452820\pi$
$294$	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	$-0.573593\pi$
$313$	−8.10801	−0.458292	−0.229146	+	0.973392i	$0.573593\pi$
$314$	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	$-0.817419\pi$
$337$	−30.8390	−1.67991	−0.839955	+	0.542656i	$0.817419\pi$
$338$	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	$-0.295002\pi$
$353$	22.5616	1.20083	0.600416	+	0.799688i	$0.295002\pi$
$354$	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	$-0.522060\pi$
$359$	−2.62410	−0.138495	−0.0692474	+	0.997600i	$0.522060\pi$
$360$	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	$-0.643207\pi$
$367$	−16.6620	−0.869747	−0.434874	+	0.900492i	$0.643207\pi$
$368$	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	$-0.371331\pi$
$379$	15.3138	0.786615	0.393307	+	0.919407i	$0.371331\pi$
$380$	−1.26453	−0.0648693
$381$	0	0
$382$	−42.1369	−2.15591
$383$	21.9992	1.12411	0.562053	−	0.827101i	$-0.310012\pi$
$383$	21.9992	1.12411	0.562053	+	0.827101i	$0.310012\pi$
$384$	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	$-0.763794\pi$
$389$	−29.0749	−1.47415	−0.737077	+	0.675809i	$0.763794\pi$
$390$	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	$-0.815342\pi$
$397$	−33.3302	−1.67279	−0.836397	+	0.548125i	$0.815342\pi$
$398$	−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	$-0.356975\pi$
$439$	18.2017	0.868718	0.434359	+	0.900740i	$0.356975\pi$
$440$	−6.58936	−0.314135
$441$	0	0
$442$	0	0
$443$	14.4462	0.686361	0.343181	−	0.939269i	$-0.388496\pi$
$443$	14.4462	0.686361	0.343181	+	0.939269i	$0.388496\pi$
$444$	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	$-0.605634\pi$
$461$	−13.9905	−0.651604	−0.325802	+	0.945438i	$0.605634\pi$
$462$	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	$-0.595379\pi$
$467$	−12.7577	−0.590354	−0.295177	+	0.955443i	$0.595379\pi$
$468$	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	$-0.630229\pi$
$487$	−17.5577	−0.795617	−0.397809	+	0.917468i	$0.630229\pi$
$488$	9.46642	0.428525
$489$	0	0
$490$	−0.814901	−0.0368135
$491$	35.8504	1.61790	0.808952	−	0.587874i	$-0.200035\pi$
$491$	35.8504	1.61790	0.808952	+	0.587874i	$0.200035\pi$
$492$	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	$-0.715252\pi$
$499$	−27.9614	−1.25172	−0.625861	+	0.779934i	$0.715252\pi$
$500$	−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	$-0.153554\pi$
$509$	39.9728	1.77176	0.885882	+	0.463910i	$0.153554\pi$
$510$	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	$-0.712760\pi$
$541$	−28.8294	−1.23947	−0.619736	+	0.784810i	$0.712760\pi$
$542$	−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	$-0.457213\pi$
$547$	6.26865	0.268028	0.134014	+	0.990979i	$0.457213\pi$
$548$	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	$-0.464403\pi$
$557$	5.26768	0.223199	0.111599	+	0.993753i	$0.464403\pi$
$558$	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	$-0.586778\pi$
$563$	−12.7777	−0.538515	−0.269257	+	0.963068i	$0.586778\pi$
$564$	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	$-0.546299\pi$
$569$	−6.91475	−0.289881	−0.144941	+	0.989440i	$0.546299\pi$
$570$	0	0
$571$	−9.01348	−0.377202	−0.188601	−	0.982054i	$-0.560395\pi$
$571$	−9.01348	−0.377202	−0.188601	+	0.982054i	$0.560395\pi$
$572$	−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	$-0.641830\pi$
$587$	−20.8834	−0.861950	−0.430975	+	0.902364i	$0.641830\pi$
$588$	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	$-0.306400\pi$
$593$	27.8291	1.14280	0.571402	+	0.820670i	$0.306400\pi$
$594$	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	$-0.206947\pi$
$599$	38.9632	1.59199	0.795997	+	0.605300i	$0.206947\pi$
$600$	0	0
$601$	−24.5350	−1.00080	−0.500402	−	0.865793i	$-0.666814\pi$
$601$	−24.5350	−1.00080	−0.500402	+	0.865793i	$0.666814\pi$
$602$	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	$-0.396107\pi$
$607$	15.7988	0.641253	0.320626	+	0.947206i	$0.396107\pi$
$608$	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	$-0.366344\pi$
$619$	20.2851	0.815325	0.407663	+	0.913133i	$0.366344\pi$
$620$	−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	$-0.482758\pi$
$643$	2.74575	0.108282	0.0541408	+	0.998533i	$0.482758\pi$
$644$	−3.04565	−0.120015
$645$	0	0
$646$	0	0
$647$	−8.48539	−0.333595	−0.166798	−	0.985991i	$-0.553343\pi$
$647$	−8.48539	−0.333595	−0.166798	+	0.985991i	$0.553343\pi$
$648$	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	$-0.392876\pi$
$659$	16.9543	0.660446	0.330223	+	0.943903i	$0.392876\pi$
$660$	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	$-0.505967\pi$
$673$	−0.972535	−0.0374885	−0.0187442	+	0.999824i	$0.505967\pi$
$674$	−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	$-0.693377\pi$
$691$	−30.0105	−1.14165	−0.570826	+	0.821071i	$0.693377\pi$
$692$	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	$-0.420532\pi$
$701$	13.0830	0.494140	0.247070	+	0.968998i	$0.420532\pi$
$702$	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	$-0.288432\pi$
$709$	32.8467	1.23358	0.616792	+	0.787126i	$0.288432\pi$
$710$	−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	$-0.222737\pi$
$751$	41.9289	1.53001	0.765004	+	0.644025i	$0.222737\pi$
$752$	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	$-0.542784\pi$
$761$	−7.39335	−0.268009	−0.134004	+	0.990981i	$0.542784\pi$
$762$	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	$-0.689093\pi$
$773$	−31.1238	−1.11945	−0.559723	+	0.828680i	$0.689093\pi$
$774$	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	$-0.458978\pi$
$787$	7.21080	0.257037	0.128519	+	0.991707i	$0.458978\pi$
$788$	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	$-0.542902\pi$
$797$	−7.58702	−0.268746	−0.134373	+	0.990931i	$0.542902\pi$
$798$	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	$-0.808235\pi$
$811$	−46.9291	−1.64790	−0.823951	+	0.566661i	$0.808235\pi$
$812$	−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	$-0.699463\pi$
$823$	−33.6464	−1.17284	−0.586419	+	0.810008i	$0.699463\pi$
$824$	−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	$-0.433204\pi$
$853$	12.1679	0.416621	0.208311	+	0.978063i	$0.433204\pi$
$854$	−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	$-0.487222\pi$
$863$	2.35798	0.0802667	0.0401334	+	0.999194i	$0.487222\pi$
$864$	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	$-0.715429\pi$
$877$	−37.0944	−1.25259	−0.626294	+	0.779587i	$0.715429\pi$
$878$	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	$-0.453899\pi$
$907$	8.69309	0.288649	0.144325	+	0.989530i	$0.453899\pi$
$908$	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	$-0.500291\pi$
$953$	−0.0564747	−0.00182940	−0.000914698		1.00000i	$0.500291\pi$
$954$	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	$-0.445981\pi$
$971$	10.5257	0.337785	0.168892	+	0.985634i	$0.445981\pi$
$972$	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	$-0.571465\pi$
$977$	−13.9176	−0.445264	−0.222632	+	0.974903i	$0.571465\pi$
$978$	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	$-0.678129\pi$
$991$	−33.4227	−1.06171	−0.530854	+	0.847463i	$0.678129\pi$
$992$	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	$-0.730349\pi$
$997$	−41.8141	−1.32427	−0.662133	+	0.749386i	$0.730349\pi$
$998$	−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.bv.1.10		12
3.2	odd	2	inner	7803.2.a.bv.1.3		12
17.8	even	8		459.2.f.c.217.3	yes	24
17.15	even	8		459.2.f.c.55.10	yes	24
17.16	even	2		7803.2.a.bw.1.10		12
51.8	odd	8		459.2.f.c.217.10	yes	24
51.32	odd	8		459.2.f.c.55.3	✓	24
51.50	odd	2		7803.2.a.bw.1.3		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
459.2.f.c.55.3	✓	24	51.32	odd	8
459.2.f.c.55.10	yes	24	17.15	even	8
459.2.f.c.217.3	yes	24	17.8	even	8
459.2.f.c.217.10	yes	24	51.8	odd	8
7803.2.a.bv.1.3		12	3.2	odd	2	inner
7803.2.a.bv.1.10		12	1.1	even	1	trivial
7803.2.a.bw.1.3		12	51.50	odd	2
7803.2.a.bw.1.10		12	17.16	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

L-functions

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