LMFDB - Embedded newform 7803.2.a.cg.1.12

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)

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")

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);

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

Newform invariants

comment:select newform

sage:traces = [24,0,0,24,20,0,0,0,0,0,8,0,4,16,0,24,0,0,4,48,0,0,36,0,28,0, 0,0,64,0,0,0,0,0,0,0,0,0,0,0,36,0,4,16,0,0,0,0,24,0,0,16,0,0,20,80,0,0, 0,0,0,-16,0,-24,72,0,16,0,0,-48,72,0,0] 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:	$24$
Twist minimal:	no (minimal twist has level 459)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.12
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)

$f(q)$	$=$	$q-0.113371 q^{2} -1.98715 q^{4} +3.61829 q^{5} -4.87047 q^{7} +0.452028 q^{8} -0.410211 q^{10} +3.73056 q^{11} +2.58136 q^{13} +0.552172 q^{14} +3.92305 q^{16} -3.88598 q^{19} -7.19008 q^{20} -0.422939 q^{22} -1.43027 q^{23} +8.09204 q^{25} -0.292652 q^{26} +9.67835 q^{28} +7.92074 q^{29} -3.24883 q^{31} -1.34882 q^{32} -17.6228 q^{35} +6.58423 q^{37} +0.440560 q^{38} +1.63557 q^{40} -6.81800 q^{41} +1.38277 q^{43} -7.41318 q^{44} +0.162152 q^{46} -7.36992 q^{47} +16.7215 q^{49} -0.917406 q^{50} -5.12954 q^{52} +5.16969 q^{53} +13.4983 q^{55} -2.20159 q^{56} -0.897986 q^{58} -6.46653 q^{59} -4.65687 q^{61} +0.368324 q^{62} -7.69318 q^{64} +9.34010 q^{65} +7.37261 q^{67} +1.99792 q^{70} -1.35380 q^{71} +11.4130 q^{73} -0.746463 q^{74} +7.72202 q^{76} -18.1696 q^{77} -2.55583 q^{79} +14.1947 q^{80} +0.772966 q^{82} +8.86903 q^{83} -0.156766 q^{86} +1.68632 q^{88} +8.62930 q^{89} -12.5724 q^{91} +2.84217 q^{92} +0.835538 q^{94} -14.0606 q^{95} -3.63548 q^{97} -1.89574 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 24 q^{4} + 20 q^{5} + 8 q^{11} + 4 q^{13} + 16 q^{14} + 24 q^{16} + 4 q^{19} + 48 q^{20} + 36 q^{23} + 28 q^{25} + 64 q^{29} + 36 q^{41} + 4 q^{43} + 16 q^{44} + 24 q^{49} + 16 q^{52} + 20 q^{55}+ \cdots - 12 q^{95}+O(q^{100}) $ 24 * q + 24 * q^4 + 20 * q^5 + 8 * q^11 + 4 * q^13 + 16 * q^14 + 24 * q^16 + 4 * q^19 + 48 * q^20 + 36 * q^23 + 28 * q^25 + 64 * q^29 + 36 * q^41 + 4 * q^43 + 16 * q^44 + 24 * q^49 + 16 * q^52 + 20 * q^55 + 80 * q^56 - 16 * q^62 - 24 * q^64 + 72 * q^65 + 16 * q^67 - 48 * q^70 + 72 * q^71 + 80 * q^74 - 40 * q^76 + 104 * q^80 + 144 * q^92 - 72 * q^94 - 12 * q^95

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 $

$2$	−0.113371	−0.0801657	−0.0400828	−	0.999196i	$-0.512762\pi$
$2$	−0.113371	−0.0801657	−0.0400828	+	0.999196i	$0.512762\pi$
$3$	0	0
$3$	0	0
$4$	−1.98715	−0.993573
$5$	3.61829	1.61815	0.809075	−	0.587706i	$-0.199969\pi$
$5$	3.61829	1.61815	0.809075	+	0.587706i	$0.199969\pi$
$6$	0	0
$7$	−4.87047	−1.84087	−0.920433	−	0.390901i	$-0.872164\pi$
$7$	−4.87047	−1.84087	−0.920433	+	0.390901i	$0.872164\pi$
$8$	0.452028	0.159816
$9$	0	0
$10$	−0.410211	−0.129720
$11$	3.73056	1.12481	0.562404	−	0.826863i	$-0.309877\pi$
$11$	3.73056	1.12481	0.562404	+	0.826863i	$0.309877\pi$
$12$	0	0
$13$	2.58136	0.715940	0.357970	−	0.933733i	$-0.383469\pi$
$13$	2.58136	0.715940	0.357970	+	0.933733i	$0.383469\pi$
$14$	0.552172	0.147574
$15$	0	0
$16$	3.92305	0.980762
$17$	0	0
$17$	0	0
$18$	0	0
$19$	−3.88598	−0.891506	−0.445753	−	0.895156i	$-0.647064\pi$
$19$	−3.88598	−0.891506	−0.445753	+	0.895156i	$0.647064\pi$
$20$	−7.19008	−1.60775
$21$	0	0
$22$	−0.422939	−0.0901709
$23$	−1.43027	−0.298233	−0.149116	−	0.988820i	$-0.547643\pi$
$23$	−1.43027	−0.298233	−0.149116	+	0.988820i	$0.547643\pi$
$24$	0	0
$25$	8.09204	1.61841
$26$	−0.292652	−0.0573938
$27$	0	0
$28$	9.67835	1.82904
$29$	7.92074	1.47084	0.735422	−	0.677609i	$-0.236984\pi$
$29$	7.92074	1.47084	0.735422	+	0.677609i	$0.236984\pi$
$30$	0	0
$31$	−3.24883	−0.583507	−0.291754	−	0.956494i	$-0.594239\pi$
$31$	−3.24883	−0.583507	−0.291754	+	0.956494i	$0.594239\pi$
$32$	−1.34882	−0.238440
$33$	0	0
$34$	0	0
$35$	−17.6228	−2.97880
$36$	0	0
$37$	6.58423	1.08244	0.541220	−	0.840881i	$-0.317963\pi$
$37$	6.58423	1.08244	0.541220	+	0.840881i	$0.317963\pi$
$38$	0.440560	0.0714682
$39$	0	0
$40$	1.63557	0.258606
$41$	−6.81800	−1.06479	−0.532396	−	0.846495i	$-0.678708\pi$
$41$	−6.81800	−1.06479	−0.532396	+	0.846495i	$0.678708\pi$
$42$	0	0
$43$	1.38277	0.210870	0.105435	−	0.994426i	$-0.466376\pi$
$43$	1.38277	0.210870	0.105435	+	0.994426i	$0.466376\pi$
$44$	−7.41318	−1.11758
$45$	0	0
$46$	0.162152	0.0239080
$47$	−7.36992	−1.07501	−0.537506	−	0.843260i	$-0.680634\pi$
$47$	−7.36992	−1.07501	−0.537506	+	0.843260i	$0.680634\pi$
$48$	0	0
$49$	16.7215	2.38879
$50$	−0.917406	−0.129741
$51$	0	0
$52$	−5.12954	−0.711339
$53$	5.16969	0.710111	0.355056	−	0.934845i	$-0.384462\pi$
$53$	5.16969	0.710111	0.355056	+	0.934845i	$0.384462\pi$
$54$	0	0
$55$	13.4983	1.82011
$56$	−2.20159	−0.294200
$57$	0	0
$58$	−0.897986	−0.117911
$59$	−6.46653	−0.841870	−0.420935	−	0.907091i	$-0.638298\pi$
$59$	−6.46653	−0.841870	−0.420935	+	0.907091i	$0.638298\pi$
$60$	0	0
$61$	−4.65687	−0.596251	−0.298125	−	0.954527i	$-0.596361\pi$
$61$	−4.65687	−0.596251	−0.298125	+	0.954527i	$0.596361\pi$
$62$	0.368324	0.0467772
$63$	0	0
$64$	−7.69318	−0.961647
$65$	9.34010	1.15850
$66$	0	0
$67$	7.37261	0.900708	0.450354	−	0.892850i	$-0.351298\pi$
$67$	7.37261	0.900708	0.450354	+	0.892850i	$0.351298\pi$
$68$	0	0
$69$	0	0
$70$	1.99792	0.238797
$71$	−1.35380	−0.160667	−0.0803334	−	0.996768i	$-0.525598\pi$
$71$	−1.35380	−0.160667	−0.0803334	+	0.996768i	$0.525598\pi$
$72$	0	0
$73$	11.4130	1.33579	0.667894	−	0.744257i	$-0.267196\pi$
$73$	11.4130	1.33579	0.667894	+	0.744257i	$0.267196\pi$
$74$	−0.746463	−0.0867745
$75$	0	0
$76$	7.72202	0.885777
$77$	−18.1696	−2.07062
$78$	0	0
$79$	−2.55583	−0.287553	−0.143777	−	0.989610i	$-0.545925\pi$
$79$	−2.55583	−0.287553	−0.143777	+	0.989610i	$0.545925\pi$
$80$	14.1947	1.58702
$81$	0	0
$82$	0.772966	0.0853598
$83$	8.86903	0.973503	0.486752	−	0.873540i	$-0.338182\pi$
$83$	8.86903	0.973503	0.486752	+	0.873540i	$0.338182\pi$
$84$	0	0
$85$	0	0
$86$	−0.156766	−0.0169046
$87$	0	0
$88$	1.68632	0.179762
$89$	8.62930	0.914704	0.457352	−	0.889286i	$-0.348798\pi$
$89$	8.62930	0.914704	0.457352	+	0.889286i	$0.348798\pi$
$90$	0	0
$91$	−12.5724	−1.31795
$92$	2.84217	0.296316
$93$	0	0
$94$	0.835538	0.0861791
$95$	−14.0606	−1.44259
$96$	0	0
$97$	−3.63548	−0.369127	−0.184563	−	0.982821i	$-0.559087\pi$
$97$	−3.63548	−0.369127	−0.184563	+	0.982821i	$0.559087\pi$
$98$	−1.89574	−0.191499
$99$	0	0
$100$	−16.0801	−1.60801
$101$	2.32609	0.231455	0.115728	−	0.993281i	$-0.463080\pi$
$101$	2.32609	0.231455	0.115728	+	0.993281i	$0.463080\pi$
$102$	0	0
$103$	4.01072	0.395188	0.197594	−	0.980284i	$-0.436687\pi$
$103$	4.01072	0.395188	0.197594	+	0.980284i	$0.436687\pi$
$104$	1.16685	0.114419
$105$	0	0
$106$	−0.586095	−0.0569265
$107$	−5.01007	−0.484342	−0.242171	−	0.970234i	$-0.577860\pi$
$107$	−5.01007	−0.484342	−0.242171	+	0.970234i	$0.577860\pi$
$108$	0	0
$109$	−8.74247	−0.837376	−0.418688	−	0.908130i	$-0.637510\pi$
$109$	−8.74247	−0.837376	−0.418688	+	0.908130i	$0.637510\pi$
$110$	−1.53032	−0.145910
$111$	0	0
$112$	−19.1071	−1.80545
$113$	−5.26185	−0.494993	−0.247497	−	0.968889i	$-0.579608\pi$
$113$	−5.26185	−0.494993	−0.247497	+	0.968889i	$0.579608\pi$
$114$	0	0
$115$	−5.17515	−0.482585
$116$	−15.7397	−1.46139
$117$	0	0
$118$	0.733119	0.0674891
$119$	0	0
$120$	0	0
$121$	2.91710	0.265191
$122$	0.527955	0.0477988
$123$	0	0
$124$	6.45590	0.579757
$125$	11.1879	1.00068
$126$	0	0
$127$	−3.13324	−0.278030	−0.139015	−	0.990290i	$-0.544394\pi$
$127$	−3.13324	−0.278030	−0.139015	+	0.990290i	$0.544394\pi$
$128$	3.56982	0.315531
$129$	0	0
$130$	−1.05890	−0.0928717
$131$	−2.43544	−0.212785	−0.106393	−	0.994324i	$-0.533930\pi$
$131$	−2.43544	−0.212785	−0.106393	+	0.994324i	$0.533930\pi$
$132$	0	0
$133$	18.9266	1.64114
$134$	−0.835844	−0.0722059
$135$	0	0
$136$	0	0
$137$	−21.8938	−1.87051	−0.935256	−	0.353971i	$-0.884831\pi$
$137$	−21.8938	−1.87051	−0.935256	+	0.353971i	$0.884831\pi$
$138$	0	0
$139$	5.05490	0.428751	0.214375	−	0.976751i	$-0.431228\pi$
$139$	5.05490	0.428751	0.214375	+	0.976751i	$0.431228\pi$
$140$	35.0191	2.95965
$141$	0	0
$142$	0.153482	0.0128800
$143$	9.62991	0.805294
$144$	0	0
$145$	28.6596	2.38005
$146$	−1.29390	−0.107084
$147$	0	0
$148$	−13.0838	−1.07548
$149$	21.1635	1.73378	0.866891	−	0.498497i	$-0.166115\pi$
$149$	21.1635	1.73378	0.866891	+	0.498497i	$0.166115\pi$
$150$	0	0
$151$	6.22312	0.506430	0.253215	−	0.967410i	$-0.418512\pi$
$151$	6.22312	0.506430	0.253215	+	0.967410i	$0.418512\pi$
$152$	−1.75658	−0.142477
$153$	0	0
$154$	2.05991	0.165993
$155$	−11.7552	−0.944202
$156$	0	0
$157$	−8.20400	−0.654751	−0.327375	−	0.944894i	$-0.606164\pi$
$157$	−8.20400	−0.654751	−0.327375	+	0.944894i	$0.606164\pi$
$158$	0.289758	0.0230519
$159$	0	0
$160$	−4.88042	−0.385831
$161$	6.96611	0.549007
$162$	0	0
$163$	15.9525	1.24950	0.624750	−	0.780825i	$-0.285201\pi$
$163$	15.9525	1.24950	0.624750	+	0.780825i	$0.285201\pi$
$164$	13.5484	1.05795
$165$	0	0
$166$	−1.00549	−0.0780415
$167$	−15.8269	−1.22472	−0.612361	−	0.790579i	$-0.709780\pi$
$167$	−15.8269	−1.22472	−0.612361	+	0.790579i	$0.709780\pi$
$168$	0	0
$169$	−6.33660	−0.487430
$170$	0	0
$171$	0	0
$172$	−2.74776	−0.209515
$173$	20.0361	1.52332	0.761658	−	0.647980i	$-0.224386\pi$
$173$	20.0361	1.52332	0.761658	+	0.647980i	$0.224386\pi$
$174$	0	0
$175$	−39.4121	−2.97927
$176$	14.6352	1.10317
$177$	0	0
$178$	−0.978316	−0.0733279
$179$	18.4967	1.38251	0.691254	−	0.722612i	$-0.257058\pi$
$179$	18.4967	1.38251	0.691254	+	0.722612i	$0.257058\pi$
$180$	0	0
$181$	4.85122	0.360588	0.180294	−	0.983613i	$-0.442295\pi$
$181$	4.85122	0.360588	0.180294	+	0.983613i	$0.442295\pi$
$182$	1.42535	0.105654
$183$	0	0
$184$	−0.646525	−0.0476624
$185$	23.8237	1.75155
$186$	0	0
$187$	0	0
$188$	14.6451	1.06810
$189$	0	0
$190$	1.59407	0.115646
$191$	−4.66086	−0.337248	−0.168624	−	0.985680i	$-0.553932\pi$
$191$	−4.66086	−0.337248	−0.168624	+	0.985680i	$0.553932\pi$
$192$	0	0
$193$	9.90846	0.713227	0.356613	−	0.934252i	$-0.383931\pi$
$193$	9.90846	0.713227	0.356613	+	0.934252i	$0.383931\pi$
$194$	0.412159	0.0295913
$195$	0	0
$196$	−33.2281	−2.37344
$197$	12.9539	0.922928	0.461464	−	0.887159i	$-0.347324\pi$
$197$	12.9539	0.922928	0.461464	+	0.887159i	$0.347324\pi$
$198$	0	0
$199$	27.7148	1.96465	0.982325	−	0.187181i	$-0.0599352\pi$
$199$	27.7148	1.96465	0.982325	+	0.187181i	$0.0599352\pi$
$200$	3.65783	0.258648
$201$	0	0
$202$	−0.263713	−0.0185548
$203$	−38.5778	−2.70763
$204$	0	0
$205$	−24.6695	−1.72299
$206$	−0.454701	−0.0316805
$207$	0	0
$208$	10.1268	0.702166
$209$	−14.4969	−1.00277
$210$	0	0
$211$	22.7296	1.56477	0.782385	−	0.622795i	$-0.214003\pi$
$211$	22.7296	1.56477	0.782385	+	0.622795i	$0.214003\pi$
$212$	−10.2729	−0.705548
$213$	0	0
$214$	0.567999	0.0388276
$215$	5.00326	0.341220
$216$	0	0
$217$	15.8233	1.07416
$218$	0.991146	0.0671289
$219$	0	0
$220$	−26.8230	−1.80841
$221$	0	0
$222$	0	0
$223$	−12.0261	−0.805328	−0.402664	−	0.915348i	$-0.631916\pi$
$223$	−12.0261	−0.805328	−0.402664	+	0.915348i	$0.631916\pi$
$224$	6.56938	0.438935
$225$	0	0
$226$	0.596543	0.0396815
$227$	14.7781	0.980856	0.490428	−	0.871482i	$-0.336841\pi$
$227$	14.7781	0.980856	0.490428	+	0.871482i	$0.336841\pi$
$228$	0	0
$229$	−14.2110	−0.939093	−0.469546	−	0.882908i	$-0.655583\pi$
$229$	−14.2110	−0.939093	−0.469546	+	0.882908i	$0.655583\pi$
$230$	0.586714	0.0386868
$231$	0	0
$232$	3.58040	0.235065
$233$	−8.62121	−0.564794	−0.282397	−	0.959298i	$-0.591130\pi$
$233$	−8.62121	−0.564794	−0.282397	+	0.959298i	$0.591130\pi$
$234$	0	0
$235$	−26.6665	−1.73953
$236$	12.8499	0.836460
$237$	0	0
$238$	0	0
$239$	−16.0848	−1.04044	−0.520221	−	0.854032i	$-0.674150\pi$
$239$	−16.0848	−1.04044	−0.520221	+	0.854032i	$0.674150\pi$
$240$	0	0
$241$	−3.26675	−0.210430	−0.105215	−	0.994450i	$-0.533553\pi$
$241$	−3.26675	−0.210430	−0.105215	+	0.994450i	$0.533553\pi$
$242$	−0.330716	−0.0212592
$243$	0	0
$244$	9.25388	0.592419
$245$	60.5033	3.86541
$246$	0	0
$247$	−10.0311	−0.638265
$248$	−1.46856	−0.0932539
$249$	0	0
$250$	−1.26839	−0.0802199
$251$	−0.393311	−0.0248256	−0.0124128	−	0.999923i	$-0.503951\pi$
$251$	−0.393311	−0.0248256	−0.0124128	+	0.999923i	$0.503951\pi$
$252$	0	0
$253$	−5.33573	−0.335454
$254$	0.355220	0.0222885
$255$	0	0
$256$	14.9816	0.936352
$257$	−1.52997	−0.0954370	−0.0477185	−	0.998861i	$-0.515195\pi$
$257$	−1.52997	−0.0954370	−0.0477185	+	0.998861i	$0.515195\pi$
$258$	0	0
$259$	−32.0683	−1.99263
$260$	−18.5602	−1.15105
$261$	0	0
$262$	0.276109	0.0170581
$263$	10.3747	0.639729	0.319865	−	0.947463i	$-0.396363\pi$
$263$	10.3747	0.639729	0.319865	+	0.947463i	$0.396363\pi$
$264$	0	0
$265$	18.7054	1.14907
$266$	−2.14573	−0.131563
$267$	0	0
$268$	−14.6505	−0.894920
$269$	28.6320	1.74572	0.872861	−	0.487969i	$-0.162262\pi$
$269$	28.6320	1.74572	0.872861	+	0.487969i	$0.162262\pi$
$270$	0	0
$271$	3.90958	0.237490	0.118745	−	0.992925i	$-0.462113\pi$
$271$	3.90958	0.237490	0.118745	+	0.992925i	$0.462113\pi$
$272$	0	0
$273$	0	0
$274$	2.48213	0.149951
$275$	30.1879	1.82040
$276$	0	0
$277$	17.8515	1.07259	0.536295	−	0.844031i	$-0.319823\pi$
$277$	17.8515	1.07259	0.536295	+	0.844031i	$0.319823\pi$
$278$	−0.573081	−0.0343711
$279$	0	0
$280$	−7.96600	−0.476060
$281$	26.6936	1.59241	0.796204	−	0.605028i	$-0.206838\pi$
$281$	26.6936	1.59241	0.796204	+	0.605028i	$0.206838\pi$
$282$	0	0
$283$	−18.3255	−1.08934	−0.544670	−	0.838651i	$-0.683345\pi$
$283$	−18.3255	−1.08934	−0.544670	+	0.838651i	$0.683345\pi$
$284$	2.69020	0.159634
$285$	0	0
$286$	−1.09176	−0.0645569
$287$	33.2069	1.96014
$288$	0	0
$289$	0	0
$290$	−3.24917	−0.190798
$291$	0	0
$292$	−22.6792	−1.32720
$293$	16.6305	0.971564	0.485782	−	0.874080i	$-0.338535\pi$
$293$	16.6305	0.971564	0.485782	+	0.874080i	$0.338535\pi$
$294$	0	0
$295$	−23.3978	−1.36227
$296$	2.97626	0.172991
$297$	0	0
$298$	−2.39934	−0.138990
$299$	−3.69205	−0.213517
$300$	0	0
$301$	−6.73474	−0.388184
$302$	−0.705524	−0.0405983
$303$	0	0
$304$	−15.2449	−0.874355
$305$	−16.8499	−0.964823
$306$	0	0
$307$	27.9347	1.59432	0.797158	−	0.603771i	$-0.206336\pi$
$307$	27.9347	1.59432	0.797158	+	0.603771i	$0.206336\pi$
$308$	36.1057	2.05731
$309$	0	0
$310$	1.33271	0.0756926
$311$	−12.8189	−0.726893	−0.363446	−	0.931615i	$-0.618400\pi$
$311$	−12.8189	−0.726893	−0.363446	+	0.931615i	$0.618400\pi$
$312$	0	0
$313$	−0.228726	−0.0129284	−0.00646419	−	0.999979i	$-0.502058\pi$
$313$	−0.228726	−0.0129284	−0.00646419	+	0.999979i	$0.502058\pi$
$314$	0.930099	0.0524885
$315$	0	0
$316$	5.07881	0.285705
$317$	12.7565	0.716478	0.358239	−	0.933630i	$-0.383377\pi$
$317$	12.7565	0.716478	0.358239	+	0.933630i	$0.383377\pi$
$318$	0	0
$319$	29.5488	1.65442
$320$	−27.8362	−1.55609
$321$	0	0
$322$	−0.789758	−0.0440115
$323$	0	0
$324$	0	0
$325$	20.8884	1.15868
$326$	−1.80856	−0.100167
$327$	0	0
$328$	−3.08193	−0.170171
$329$	35.8950	1.97895
$330$	0	0
$331$	−4.03099	−0.221563	−0.110782	−	0.993845i	$-0.535335\pi$
$331$	−4.03099	−0.221563	−0.110782	+	0.993845i	$0.535335\pi$
$332$	−17.6241	−0.967247
$333$	0	0
$334$	1.79432	0.0981806
$335$	26.6763	1.45748
$336$	0	0
$337$	3.29186	0.179319	0.0896594	−	0.995972i	$-0.471422\pi$
$337$	3.29186	0.179319	0.0896594	+	0.995972i	$0.471422\pi$
$338$	0.718389	0.0390752
$339$	0	0
$340$	0	0
$341$	−12.1200	−0.656333
$342$	0	0
$343$	−47.3483	−2.55657
$344$	0.625051	0.0337005
$345$	0	0
$346$	−2.27152	−0.122118
$347$	10.4519	0.561089	0.280545	−	0.959841i	$-0.409485\pi$
$347$	10.4519	0.561089	0.280545	+	0.959841i	$0.409485\pi$
$348$	0	0
$349$	−0.501788	−0.0268601	−0.0134300	−	0.999910i	$-0.504275\pi$
$349$	−0.501788	−0.0268601	−0.0134300	+	0.999910i	$0.504275\pi$
$350$	4.46820	0.238835
$351$	0	0
$352$	−5.03185	−0.268199
$353$	−29.9020	−1.59152	−0.795761	−	0.605611i	$-0.792929\pi$
$353$	−29.9020	−1.59152	−0.795761	+	0.605611i	$0.792929\pi$
$354$	0	0
$355$	−4.89845	−0.259983
$356$	−17.1477	−0.908826
$357$	0	0
$358$	−2.09700	−0.110830
$359$	−13.6313	−0.719431	−0.359715	−	0.933062i	$-0.617126\pi$
$359$	−13.6313	−0.719431	−0.359715	+	0.933062i	$0.617126\pi$
$360$	0	0
$361$	−3.89912	−0.205217
$362$	−0.549989	−0.0289068
$363$	0	0
$364$	24.9833	1.30948
$365$	41.2955	2.16150
$366$	0	0
$367$	13.3648	0.697634	0.348817	−	0.937191i	$-0.386583\pi$
$367$	13.3648	0.697634	0.348817	+	0.937191i	$0.386583\pi$
$368$	−5.61103	−0.292495
$369$	0	0
$370$	−2.70092	−0.140414
$371$	−25.1788	−1.30722
$372$	0	0
$373$	16.3768	0.847961	0.423980	−	0.905671i	$-0.360633\pi$
$373$	16.3768	0.847961	0.423980	+	0.905671i	$0.360633\pi$
$374$	0	0
$375$	0	0
$376$	−3.33141	−0.171804
$377$	20.4463	1.05304
$378$	0	0
$379$	21.2846	1.09332	0.546659	−	0.837355i	$-0.315899\pi$
$379$	21.2846	1.09332	0.546659	+	0.837355i	$0.315899\pi$
$380$	27.9405	1.43332
$381$	0	0
$382$	0.528408	0.0270357
$383$	−9.76478	−0.498957	−0.249478	−	0.968380i	$-0.580259\pi$
$383$	−9.76478	−0.498957	−0.249478	+	0.968380i	$0.580259\pi$
$384$	0	0
$385$	−65.7429	−3.35057
$386$	−1.12334	−0.0571763
$387$	0	0
$388$	7.22422	0.366754
$389$	−0.183217	−0.00928949	−0.00464475	−	0.999989i	$-0.501478\pi$
$389$	−0.183217	−0.00928949	−0.00464475	+	0.999989i	$0.501478\pi$
$390$	0	0
$391$	0	0
$392$	7.55860	0.381767
$393$	0	0
$394$	−1.46860	−0.0739872
$395$	−9.24774	−0.465304
$396$	0	0
$397$	27.1242	1.36133	0.680663	−	0.732596i	$-0.261692\pi$
$397$	27.1242	1.36133	0.680663	+	0.732596i	$0.261692\pi$
$398$	−3.14207	−0.157498
$399$	0	0
$400$	31.7454	1.58727
$401$	25.0268	1.24978	0.624890	−	0.780713i	$-0.285144\pi$
$401$	25.0268	1.24978	0.624890	+	0.780713i	$0.285144\pi$
$402$	0	0
$403$	−8.38639	−0.417756
$404$	−4.62229	−0.229968
$405$	0	0
$406$	4.37361	0.217059
$407$	24.5629	1.21754
$408$	0	0
$409$	2.02920	0.100337	0.0501687	−	0.998741i	$-0.484024\pi$
$409$	2.02920	0.100337	0.0501687	+	0.998741i	$0.484024\pi$
$410$	2.79682	0.138125
$411$	0	0
$412$	−7.96990	−0.392649
$413$	31.4950	1.54977
$414$	0	0
$415$	32.0908	1.57527
$416$	−3.48178	−0.170708
$417$	0	0
$418$	1.64354	0.0803879
$419$	−8.58390	−0.419351	−0.209675	−	0.977771i	$-0.567241\pi$
$419$	−8.58390	−0.419351	−0.209675	+	0.977771i	$0.567241\pi$
$420$	0	0
$421$	15.1592	0.738815	0.369408	−	0.929267i	$-0.379561\pi$
$421$	15.1592	0.738815	0.369408	+	0.929267i	$0.379561\pi$
$422$	−2.57689	−0.125441
$423$	0	0
$424$	2.33685	0.113487
$425$	0	0
$426$	0	0
$427$	22.6811	1.09762
$428$	9.95575	0.481229
$429$	0	0
$430$	−0.567227	−0.0273541
$431$	29.6934	1.43028	0.715140	−	0.698981i	$-0.246363\pi$
$431$	29.6934	1.43028	0.715140	+	0.698981i	$0.246363\pi$
$432$	0	0
$433$	−15.8921	−0.763728	−0.381864	−	0.924219i	$-0.624718\pi$
$433$	−15.8921	−0.763728	−0.381864	+	0.924219i	$0.624718\pi$
$434$	−1.79391	−0.0861106
$435$	0	0
$436$	17.3726	0.831995
$437$	5.55802	0.265876
$438$	0	0
$439$	−10.2755	−0.490422	−0.245211	−	0.969470i	$-0.578857\pi$
$439$	−10.2755	−0.490422	−0.245211	+	0.969470i	$0.578857\pi$
$440$	6.10160	0.290882
$441$	0	0
$442$	0	0
$443$	4.79437	0.227787	0.113894	−	0.993493i	$-0.463668\pi$
$443$	4.79437	0.227787	0.113894	+	0.993493i	$0.463668\pi$
$444$	0	0
$445$	31.2233	1.48013
$446$	1.36342	0.0645597
$447$	0	0
$448$	37.4694	1.77026
$449$	19.1500	0.903742	0.451871	−	0.892083i	$-0.350757\pi$
$449$	19.1500	0.903742	0.451871	+	0.892083i	$0.350757\pi$
$450$	0	0
$451$	−25.4350	−1.19769
$452$	10.4561	0.491812
$453$	0	0
$454$	−1.67541	−0.0786310
$455$	−45.4907	−2.13264
$456$	0	0
$457$	−5.21495	−0.243945	−0.121973	−	0.992533i	$-0.538922\pi$
$457$	−5.21495	−0.243945	−0.121973	+	0.992533i	$0.538922\pi$
$458$	1.61113	0.0752830
$459$	0	0
$460$	10.2838	0.479484
$461$	−18.5337	−0.863199	−0.431600	−	0.902065i	$-0.642051\pi$
$461$	−18.5337	−0.863199	−0.431600	+	0.902065i	$0.642051\pi$
$462$	0	0
$463$	36.3578	1.68969	0.844846	−	0.535009i	$-0.179692\pi$
$463$	36.3578	1.68969	0.844846	+	0.535009i	$0.179692\pi$
$464$	31.0734	1.44255
$465$	0	0
$466$	0.977399	0.0452771
$467$	−15.3960	−0.712442	−0.356221	−	0.934402i	$-0.615935\pi$
$467$	−15.3960	−0.712442	−0.356221	+	0.934402i	$0.615935\pi$
$468$	0	0
$469$	−35.9081	−1.65808
$470$	3.02322	0.139451
$471$	0	0
$472$	−2.92305	−0.134544
$473$	5.15851	0.237188
$474$	0	0
$475$	−31.4455	−1.44282
$476$	0	0
$477$	0	0
$478$	1.82356	0.0834077
$479$	13.8811	0.634243	0.317121	−	0.948385i	$-0.397284\pi$
$479$	13.8811	0.634243	0.317121	+	0.948385i	$0.397284\pi$
$480$	0	0
$481$	16.9962	0.774962
$482$	0.370356	0.0168693
$483$	0	0
$484$	−5.79670	−0.263487
$485$	−13.1542	−0.597302
$486$	0	0
$487$	23.7739	1.07730	0.538650	−	0.842530i	$-0.318935\pi$
$487$	23.7739	1.07730	0.538650	+	0.842530i	$0.318935\pi$
$488$	−2.10504	−0.0952905
$489$	0	0
$490$	−6.85934	−0.309874
$491$	−11.6093	−0.523921	−0.261961	−	0.965079i	$-0.584369\pi$
$491$	−11.6093	−0.523921	−0.261961	+	0.965079i	$0.584369\pi$
$492$	0	0
$493$	0	0
$494$	1.13724	0.0511669
$495$	0	0
$496$	−12.7453	−0.572281
$497$	6.59366	0.295766
$498$	0	0
$499$	−0.392624	−0.0175762	−0.00878812	−	0.999961i	$-0.502797\pi$
$499$	−0.392624	−0.0175762	−0.00878812	+	0.999961i	$0.502797\pi$
$500$	−22.2320	−0.994245
$501$	0	0
$502$	0.0445902	0.00199016
$503$	−24.7781	−1.10480	−0.552401	−	0.833579i	$-0.686288\pi$
$503$	−24.7781	−1.10480	−0.552401	+	0.833579i	$0.686288\pi$
$504$	0	0
$505$	8.41649	0.374529
$506$	0.604919	0.0268919
$507$	0	0
$508$	6.22621	0.276244
$509$	17.1712	0.761100	0.380550	−	0.924760i	$-0.375735\pi$
$509$	17.1712	0.761100	0.380550	+	0.924760i	$0.375735\pi$
$510$	0	0
$511$	−55.5866	−2.45900
$512$	−8.83813	−0.390594
$513$	0	0
$514$	0.173455	0.00765077
$515$	14.5120	0.639474
$516$	0	0
$517$	−27.4939	−1.20918
$518$	3.63563	0.159740
$519$	0	0
$520$	4.22199	0.185147
$521$	30.2285	1.32433	0.662167	−	0.749356i	$-0.269637\pi$
$521$	30.2285	1.32433	0.662167	+	0.749356i	$0.269637\pi$
$522$	0	0
$523$	−32.1131	−1.40421	−0.702104	−	0.712074i	$-0.747756\pi$
$523$	−32.1131	−1.40421	−0.702104	+	0.712074i	$0.747756\pi$
$524$	4.83958	0.211418
$525$	0	0
$526$	−1.17619	−0.0512843
$527$	0	0
$528$	0	0
$529$	−20.9543	−0.911057
$530$	−2.12066	−0.0921157
$531$	0	0
$532$	−37.6099	−1.63060
$533$	−17.5997	−0.762327
$534$	0	0
$535$	−18.1279	−0.783738
$536$	3.33263	0.143948
$537$	0	0
$538$	−3.24605	−0.139947
$539$	62.3806	2.68692
$540$	0	0
$541$	−11.7509	−0.505209	−0.252604	−	0.967570i	$-0.581287\pi$
$541$	−11.7509	−0.505209	−0.252604	+	0.967570i	$0.581287\pi$
$542$	−0.443235	−0.0190386
$543$	0	0
$544$	0	0
$545$	−31.6328	−1.35500
$546$	0	0
$547$	24.4542	1.04559	0.522793	−	0.852460i	$-0.324890\pi$
$547$	24.4542	1.04559	0.522793	+	0.852460i	$0.324890\pi$
$548$	43.5062	1.85849
$549$	0	0
$550$	−3.42244	−0.145933
$551$	−30.7799	−1.31127
$552$	0	0
$553$	12.4481	0.529347
$554$	−2.02384	−0.0859849
$555$	0	0
$556$	−10.0448	−0.425996
$557$	−31.7081	−1.34351	−0.671757	−	0.740772i	$-0.734460\pi$
$557$	−31.7081	−1.34351	−0.671757	+	0.740772i	$0.734460\pi$
$558$	0	0
$559$	3.56942	0.150970
$560$	−69.1350	−2.92149
$561$	0	0
$562$	−3.02629	−0.127657
$563$	35.8638	1.51148	0.755740	−	0.654872i	$-0.227277\pi$
$563$	35.8638	1.51148	0.755740	+	0.654872i	$0.227277\pi$
$564$	0	0
$565$	−19.0389	−0.800973
$566$	2.07759	0.0873277
$567$	0	0
$568$	−0.611957	−0.0256771
$569$	29.4017	1.23258	0.616291	−	0.787518i	$-0.288634\pi$
$569$	29.4017	1.23258	0.616291	+	0.787518i	$0.288634\pi$
$570$	0	0
$571$	40.7825	1.70670	0.853348	−	0.521342i	$-0.174568\pi$
$571$	40.7825	1.70670	0.853348	+	0.521342i	$0.174568\pi$
$572$	−19.1361	−0.800119
$573$	0	0
$574$	−3.76471	−0.157136
$575$	−11.5738	−0.482662
$576$	0	0
$577$	8.38687	0.349150	0.174575	−	0.984644i	$-0.444145\pi$
$577$	8.38687	0.349150	0.174575	+	0.984644i	$0.444145\pi$
$578$	0	0
$579$	0	0
$580$	−56.9507	−2.36475
$581$	−43.1964	−1.79209
$582$	0	0
$583$	19.2858	0.798738
$584$	5.15899	0.213480
$585$	0	0
$586$	−1.88542	−0.0778861
$587$	−24.7388	−1.02108	−0.510539	−	0.859855i	$-0.670554\pi$
$587$	−24.7388	−1.02108	−0.510539	+	0.859855i	$0.670554\pi$
$588$	0	0
$589$	12.6249	0.520200
$590$	2.65264	0.109207
$591$	0	0
$592$	25.8302	1.06162
$593$	0.713305	0.0292919	0.0146460	−	0.999893i	$-0.495338\pi$
$593$	0.713305	0.0292919	0.0146460	+	0.999893i	$0.495338\pi$
$594$	0	0
$595$	0	0
$596$	−42.0550	−1.72264
$597$	0	0
$598$	0.418573	0.0171167
$599$	−26.1042	−1.06659	−0.533294	−	0.845930i	$-0.679046\pi$
$599$	−26.1042	−1.06659	−0.533294	+	0.845930i	$0.679046\pi$
$600$	0	0
$601$	−36.0612	−1.47097	−0.735484	−	0.677542i	$-0.763045\pi$
$601$	−36.0612	−1.47097	−0.735484	+	0.677542i	$0.763045\pi$
$602$	0.763527	0.0311190
$603$	0	0
$604$	−12.3663	−0.503176
$605$	10.5549	0.429118
$606$	0	0
$607$	32.9308	1.33662	0.668310	−	0.743883i	$-0.267018\pi$
$607$	32.9308	1.33662	0.668310	+	0.743883i	$0.267018\pi$
$608$	5.24149	0.212570
$609$	0	0
$610$	1.91030	0.0773457
$611$	−19.0244	−0.769644
$612$	0	0
$613$	16.1615	0.652756	0.326378	−	0.945239i	$-0.394172\pi$
$613$	16.1615	0.652756	0.326378	+	0.945239i	$0.394172\pi$
$614$	−3.16699	−0.127809
$615$	0	0
$616$	−8.21318	−0.330918
$617$	−29.8804	−1.20294	−0.601470	−	0.798895i	$-0.705418\pi$
$617$	−29.8804	−1.20294	−0.601470	+	0.798895i	$0.705418\pi$
$618$	0	0
$619$	−37.6734	−1.51422	−0.757110	−	0.653287i	$-0.773389\pi$
$619$	−37.6734	−1.51422	−0.757110	+	0.653287i	$0.773389\pi$
$620$	23.3593	0.938134
$621$	0	0
$622$	1.45330	0.0582719
$623$	−42.0288	−1.68385
$624$	0	0
$625$	0.0208904	0.000835616 0
$626$	0.0259310	0.00103641
$627$	0	0
$628$	16.3026	0.650543
$629$	0	0
$630$	0	0
$631$	20.2110	0.804589	0.402294	−	0.915510i	$-0.368213\pi$
$631$	20.2110	0.804589	0.402294	+	0.915510i	$0.368213\pi$
$632$	−1.15531	−0.0459557
$633$	0	0
$634$	−1.44623	−0.0574370
$635$	−11.3370	−0.449895
$636$	0	0
$637$	43.1642	1.71023
$638$	−3.34999	−0.132627
$639$	0	0
$640$	12.9167	0.510576
$641$	−10.9697	−0.433279	−0.216639	−	0.976252i	$-0.569510\pi$
$641$	−10.9697	−0.433279	−0.216639	+	0.976252i	$0.569510\pi$
$642$	0	0
$643$	4.07380	0.160655	0.0803274	−	0.996769i	$-0.474403\pi$
$643$	4.07380	0.160655	0.0803274	+	0.996769i	$0.474403\pi$
$644$	−13.8427	−0.545478
$645$	0	0
$646$	0	0
$647$	−17.4383	−0.685570	−0.342785	−	0.939414i	$-0.611370\pi$
$647$	−17.4383	−0.685570	−0.342785	+	0.939414i	$0.611370\pi$
$648$	0	0
$649$	−24.1238	−0.946941
$650$	−2.36815	−0.0928866
$651$	0	0
$652$	−31.7001	−1.24147
$653$	21.3395	0.835081	0.417540	−	0.908658i	$-0.362892\pi$
$653$	21.3395	0.835081	0.417540	+	0.908658i	$0.362892\pi$
$654$	0	0
$655$	−8.81213	−0.344319
$656$	−26.7473	−1.04431
$657$	0	0
$658$	−4.06946	−0.158644
$659$	−31.9158	−1.24326	−0.621631	−	0.783310i	$-0.713530\pi$
$659$	−31.9158	−1.24326	−0.621631	+	0.783310i	$0.713530\pi$
$660$	0	0
$661$	−39.5393	−1.53790	−0.768949	−	0.639310i	$-0.779220\pi$
$661$	−39.5393	−1.53790	−0.768949	+	0.639310i	$0.779220\pi$
$662$	0.456999	0.0177618
$663$	0	0
$664$	4.00906	0.155582
$665$	68.4819	2.65561
$666$	0	0
$667$	−11.3288	−0.438654
$668$	31.4504	1.21685
$669$	0	0
$670$	−3.02433	−0.116840
$671$	−17.3727	−0.670667
$672$	0	0
$673$	−27.2101	−1.04887	−0.524435	−	0.851450i	$-0.675724\pi$
$673$	−27.2101	−1.04887	−0.524435	+	0.851450i	$0.675724\pi$
$674$	−0.373202	−0.0143752
$675$	0	0
$676$	12.5917	0.484298
$677$	24.0531	0.924436	0.462218	−	0.886766i	$-0.347054\pi$
$677$	24.0531	0.924436	0.462218	+	0.886766i	$0.347054\pi$
$678$	0	0
$679$	17.7065	0.679513
$680$	0	0
$681$	0	0
$682$	1.37406	0.0526154
$683$	−12.8949	−0.493410	−0.246705	−	0.969091i	$-0.579348\pi$
$683$	−12.8949	−0.493410	−0.246705	+	0.969091i	$0.579348\pi$
$684$	0	0
$685$	−79.2181	−3.02677
$686$	5.36795	0.204949
$687$	0	0
$688$	5.42467	0.206813
$689$	13.3448	0.508397
$690$	0	0
$691$	48.7884	1.85600	0.927999	−	0.372582i	$-0.121528\pi$
$691$	48.7884	1.85600	0.927999	+	0.372582i	$0.121528\pi$
$692$	−39.8146	−1.51353
$693$	0	0
$694$	−1.18495	−0.0449801
$695$	18.2901	0.693783
$696$	0	0
$697$	0	0
$698$	0.0568884	0.00215326
$699$	0	0
$700$	78.3175	2.96012
$701$	−13.4261	−0.507097	−0.253548	−	0.967323i	$-0.581598\pi$
$701$	−13.4261	−0.507097	−0.253548	+	0.967323i	$0.581598\pi$
$702$	0	0
$703$	−25.5862	−0.965002
$704$	−28.6999	−1.08167
$705$	0	0
$706$	3.39003	0.127585
$707$	−11.3292	−0.426078
$708$	0	0
$709$	−28.7368	−1.07923	−0.539617	−	0.841910i	$-0.681431\pi$
$709$	−28.7368	−1.07923	−0.539617	+	0.841910i	$0.681431\pi$
$710$	0.555344	0.0208417
$711$	0	0
$712$	3.90069	0.146185
$713$	4.64672	0.174021
$714$	0	0
$715$	34.8438	1.30309
$716$	−36.7557	−1.37362
$717$	0	0
$718$	1.54540	0.0576737
$719$	11.0096	0.410589	0.205295	−	0.978700i	$-0.434185\pi$
$719$	11.0096	0.410589	0.205295	+	0.978700i	$0.434185\pi$
$720$	0	0
$721$	−19.5341	−0.727489
$722$	0.442049	0.0164514
$723$	0	0
$724$	−9.64008	−0.358271
$725$	64.0949	2.38043
$726$	0	0
$727$	−5.45319	−0.202248	−0.101124	−	0.994874i	$-0.532244\pi$
$727$	−5.45319	−0.202248	−0.101124	+	0.994874i	$0.532244\pi$
$728$	−5.68310	−0.210630
$729$	0	0
$730$	−4.68172	−0.173278
$731$	0	0
$732$	0	0
$733$	15.8460	0.585284	0.292642	−	0.956222i	$-0.405466\pi$
$733$	15.8460	0.585284	0.292642	+	0.956222i	$0.405466\pi$
$734$	−1.51518	−0.0559263
$735$	0	0
$736$	1.92918	0.0711105
$737$	27.5040	1.01312
$738$	0	0
$739$	32.9777	1.21311	0.606553	−	0.795043i	$-0.292552\pi$
$739$	32.9777	1.21311	0.606553	+	0.795043i	$0.292552\pi$
$740$	−47.3411	−1.74029
$741$	0	0
$742$	2.85456	0.104794
$743$	41.9905	1.54048	0.770241	−	0.637752i	$-0.220136\pi$
$743$	41.9905	1.54048	0.770241	+	0.637752i	$0.220136\pi$
$744$	0	0
$745$	76.5758	2.80552
$746$	−1.85667	−0.0679774
$747$	0	0
$748$	0	0
$749$	24.4014	0.891609
$750$	0	0
$751$	14.6066	0.533002	0.266501	−	0.963835i	$-0.414132\pi$
$751$	14.6066	0.533002	0.266501	+	0.963835i	$0.414132\pi$
$752$	−28.9125	−1.05433
$753$	0	0
$754$	−2.31802	−0.0844174
$755$	22.5171	0.819480
$756$	0	0
$757$	34.0891	1.23899	0.619495	−	0.785001i	$-0.287338\pi$
$757$	34.0891	1.23899	0.619495	+	0.785001i	$0.287338\pi$
$758$	−2.41307	−0.0876466
$759$	0	0
$760$	−6.35580	−0.230549
$761$	−0.721195	−0.0261433	−0.0130716	−	0.999915i	$-0.504161\pi$
$761$	−0.721195	−0.0261433	−0.0130716	+	0.999915i	$0.504161\pi$
$762$	0	0
$763$	42.5799	1.54150
$764$	9.26180	0.335080
$765$	0	0
$766$	1.10705	0.0399992
$767$	−16.6924	−0.602728
$768$	0	0
$769$	34.6936	1.25108	0.625542	−	0.780191i	$-0.284878\pi$
$769$	34.6936	1.25108	0.625542	+	0.780191i	$0.284878\pi$
$770$	7.45337	0.268601
$771$	0	0
$772$	−19.6896	−0.708643
$773$	−10.4552	−0.376047	−0.188024	−	0.982164i	$-0.560208\pi$
$773$	−10.4552	−0.376047	−0.188024	+	0.982164i	$0.560208\pi$
$774$	0	0
$775$	−26.2897	−0.944352
$776$	−1.64334	−0.0589924
$777$	0	0
$778$	0.0207716	0.000744698 0
$779$	26.4946	0.949269
$780$	0	0
$781$	−5.05044	−0.180719
$782$	0	0
$783$	0	0
$784$	65.5993	2.34283
$785$	−29.6845	−1.05948
$786$	0	0
$787$	52.5295	1.87248	0.936238	−	0.351368i	$-0.114283\pi$
$787$	52.5295	1.87248	0.936238	+	0.351368i	$0.114283\pi$
$788$	−25.7413	−0.916997
$789$	0	0
$790$	1.04843	0.0373014
$791$	25.6277	0.911216
$792$	0	0
$793$	−12.0210	−0.426879
$794$	−3.07511	−0.109132
$795$	0	0
$796$	−55.0734	−1.95202
$797$	19.1586	0.678632	0.339316	−	0.940672i	$-0.389804\pi$
$797$	19.1586	0.678632	0.339316	+	0.940672i	$0.389804\pi$
$798$	0	0
$799$	0	0
$800$	−10.9147	−0.385893
$801$	0	0
$802$	−2.83733	−0.100189
$803$	42.5768	1.50250
$804$	0	0
$805$	25.2054	0.888375
$806$	0.950777	0.0334897
$807$	0	0
$808$	1.05146	0.0369903
$809$	40.4768	1.42309	0.711544	−	0.702641i	$-0.247996\pi$
$809$	40.4768	1.42309	0.711544	+	0.702641i	$0.247996\pi$
$810$	0	0
$811$	52.2706	1.83547	0.917734	−	0.397195i	$-0.130016\pi$
$811$	52.2706	1.83547	0.917734	+	0.397195i	$0.130016\pi$
$812$	76.6597	2.69023
$813$	0	0
$814$	−2.78473	−0.0976046
$815$	57.7210	2.02188
$816$	0	0
$817$	−5.37342	−0.187992
$818$	−0.230053	−0.00804362
$819$	0	0
$820$	49.0219	1.71192
$821$	−45.2023	−1.57757	−0.788785	−	0.614670i	$-0.789289\pi$
$821$	−45.2023	−1.57757	−0.788785	+	0.614670i	$0.789289\pi$
$822$	0	0
$823$	−19.6813	−0.686047	−0.343023	−	0.939327i	$-0.611451\pi$
$823$	−19.6813	−0.686047	−0.343023	+	0.939327i	$0.611451\pi$
$824$	1.81296	0.0631575
$825$	0	0
$826$	−3.57064	−0.124238
$827$	34.1215	1.18652	0.593260	−	0.805011i	$-0.297841\pi$
$827$	34.1215	1.18652	0.593260	+	0.805011i	$0.297841\pi$
$828$	0	0
$829$	−43.1191	−1.49759	−0.748794	−	0.662802i	$-0.769367\pi$
$829$	−43.1191	−1.49759	−0.748794	+	0.662802i	$0.769367\pi$
$830$	−3.63817	−0.126283
$831$	0	0
$832$	−19.8588	−0.688481
$833$	0	0
$834$	0	0
$835$	−57.2663	−1.98178
$836$	28.8075	0.996328
$837$	0	0
$838$	0.973168	0.0336175
$839$	10.7739	0.371957	0.185978	−	0.982554i	$-0.440455\pi$
$839$	10.7739	0.371957	0.185978	+	0.982554i	$0.440455\pi$
$840$	0	0
$841$	33.7381	1.16338
$842$	−1.71862	−0.0592276
$843$	0	0
$844$	−45.1671	−1.55471
$845$	−22.9277	−0.788735
$846$	0	0
$847$	−14.2077	−0.488181
$848$	20.2809	0.696450
$849$	0	0
$850$	0	0
$851$	−9.41725	−0.322819
$852$	0	0
$853$	34.0697	1.16652	0.583262	−	0.812284i	$-0.301776\pi$
$853$	34.0697	1.16652	0.583262	+	0.812284i	$0.301776\pi$
$854$	−2.57139	−0.0879912
$855$	0	0
$856$	−2.26470	−0.0774057
$857$	−24.7691	−0.846097	−0.423049	−	0.906107i	$-0.639040\pi$
$857$	−24.7691	−0.846097	−0.423049	+	0.906107i	$0.639040\pi$
$858$	0	0
$859$	−0.964082	−0.0328941	−0.0164470	−	0.999865i	$-0.505235\pi$
$859$	−0.964082	−0.0328941	−0.0164470	+	0.999865i	$0.505235\pi$
$860$	−9.94222	−0.339027
$861$	0	0
$862$	−3.36638	−0.114659
$863$	42.1294	1.43410	0.717050	−	0.697022i	$-0.245492\pi$
$863$	42.1294	1.43410	0.717050	+	0.697022i	$0.245492\pi$
$864$	0	0
$865$	72.4964	2.46495
$866$	1.80172	0.0612248
$867$	0	0
$868$	−31.4433	−1.06725
$869$	−9.53468	−0.323442
$870$	0	0
$871$	19.0313	0.644853
$872$	−3.95184	−0.133826
$873$	0	0
$874$	−0.630121	−0.0213142
$875$	−54.4904	−1.84211
$876$	0	0
$877$	−52.1405	−1.76066	−0.880330	−	0.474362i	$-0.842679\pi$
$877$	−52.1405	−1.76066	−0.880330	+	0.474362i	$0.842679\pi$
$878$	1.16494	0.0393150
$879$	0	0
$880$	52.9543	1.78509
$881$	−17.2430	−0.580933	−0.290466	−	0.956885i	$-0.593810\pi$
$881$	−17.2430	−0.580933	−0.290466	+	0.956885i	$0.593810\pi$
$882$	0	0
$883$	19.4857	0.655747	0.327873	−	0.944722i	$-0.393668\pi$
$883$	19.4857	0.655747	0.327873	+	0.944722i	$0.393668\pi$
$884$	0	0
$885$	0	0
$886$	−0.543544	−0.0182607
$887$	5.55458	0.186505	0.0932523	−	0.995643i	$-0.470274\pi$
$887$	5.55458	0.186505	0.0932523	+	0.995643i	$0.470274\pi$
$888$	0	0
$889$	15.2604	0.511817
$890$	−3.53983	−0.118656
$891$	0	0
$892$	23.8977	0.800153
$893$	28.6394	0.958380
$894$	0	0
$895$	66.9265	2.23711
$896$	−17.3867	−0.580850
$897$	0	0
$898$	−2.17106	−0.0724491
$899$	−25.7331	−0.858248
$900$	0	0
$901$	0	0
$902$	2.88360	0.0960133
$903$	0	0
$904$	−2.37851	−0.0791079
$905$	17.5531	0.583485
$906$	0	0
$907$	11.1136	0.369020	0.184510	−	0.982831i	$-0.440930\pi$
$907$	11.1136	0.369020	0.184510	+	0.982831i	$0.440930\pi$
$908$	−29.3662	−0.974552
$909$	0	0
$910$	5.15735	0.170964
$911$	−30.8962	−1.02364	−0.511818	−	0.859094i	$-0.671028\pi$
$911$	−30.8962	−1.02364	−0.511818	+	0.859094i	$0.671028\pi$
$912$	0	0
$913$	33.0865	1.09500
$914$	0.591227	0.0195560
$915$	0	0
$916$	28.2394	0.933057
$917$	11.8617	0.391709
$918$	0	0
$919$	−26.7590	−0.882699	−0.441349	−	0.897335i	$-0.645500\pi$
$919$	−26.7590	−0.882699	−0.441349	+	0.897335i	$0.645500\pi$
$920$	−2.33932	−0.0771249
$921$	0	0
$922$	2.10119	0.0691989
$923$	−3.49465	−0.115028
$924$	0	0
$925$	53.2798	1.75183
$926$	−4.12194	−0.135455
$927$	0	0
$928$	−10.6836	−0.350708
$929$	11.8167	0.387694	0.193847	−	0.981032i	$-0.437903\pi$
$929$	11.8167	0.387694	0.193847	+	0.981032i	$0.437903\pi$
$930$	0	0
$931$	−64.9795	−2.12962
$932$	17.1316	0.561165
$933$	0	0
$934$	1.74547	0.0571134
$935$	0	0
$936$	0	0
$937$	14.2979	0.467092	0.233546	−	0.972346i	$-0.424967\pi$
$937$	14.2979	0.467092	0.233546	+	0.972346i	$0.424967\pi$
$938$	4.07095	0.132921
$939$	0	0
$940$	52.9903	1.72835
$941$	−17.2993	−0.563942	−0.281971	−	0.959423i	$-0.590988\pi$
$941$	−17.2993	−0.563942	−0.281971	+	0.959423i	$0.590988\pi$
$942$	0	0
$943$	9.75161	0.317556
$944$	−25.3685	−0.825674
$945$	0	0
$946$	−0.584827	−0.0190144
$947$	41.1695	1.33783	0.668914	−	0.743340i	$-0.266760\pi$
$947$	41.1695	1.33783	0.668914	+	0.743340i	$0.266760\pi$
$948$	0	0
$949$	29.4610	0.956343
$950$	3.56502	0.115665
$951$	0	0
$952$	0	0
$953$	11.2870	0.365623	0.182812	−	0.983148i	$-0.441480\pi$
$953$	11.2870	0.365623	0.182812	+	0.983148i	$0.441480\pi$
$954$	0	0
$955$	−16.8643	−0.545717
$956$	31.9629	1.03375
$957$	0	0
$958$	−1.57372	−0.0508445
$959$	106.633	3.44336
$960$	0	0
$961$	−20.4451	−0.659520
$962$	−1.92689	−0.0621253
$963$	0	0
$964$	6.49151	0.209078
$965$	35.8517	1.15411
$966$	0	0
$967$	−14.9595	−0.481065	−0.240532	−	0.970641i	$-0.577322\pi$
$967$	−14.9595	−0.481065	−0.240532	+	0.970641i	$0.577322\pi$
$968$	1.31861	0.0423818
$969$	0	0
$970$	1.49131	0.0478831
$971$	−51.2684	−1.64528	−0.822641	−	0.568561i	$-0.807500\pi$
$971$	−51.2684	−1.64528	−0.822641	+	0.568561i	$0.807500\pi$
$972$	0	0
$973$	−24.6198	−0.789273
$974$	−2.69528	−0.0863625
$975$	0	0
$976$	−18.2691	−0.584780
$977$	−35.5301	−1.13671	−0.568355	−	0.822784i	$-0.692420\pi$
$977$	−35.5301	−1.13671	−0.568355	+	0.822784i	$0.692420\pi$
$978$	0	0
$979$	32.1922	1.02887
$980$	−120.229	−3.84057
$981$	0	0
$982$	1.31617	0.0420005
$983$	−58.2701	−1.85853	−0.929263	−	0.369419i	$-0.879557\pi$
$983$	−58.2701	−1.85853	−0.929263	+	0.369419i	$0.879557\pi$
$984$	0	0
$985$	46.8710	1.49344
$986$	0	0
$987$	0	0
$988$	19.9333	0.634163
$989$	−1.97774	−0.0628884
$990$	0	0
$991$	−2.86644	−0.0910556	−0.0455278	−	0.998963i	$-0.514497\pi$
$991$	−2.86644	−0.0910556	−0.0455278	+	0.998963i	$0.514497\pi$
$992$	4.38208	0.139131
$993$	0	0
$994$	−0.747532	−0.0237103
$995$	100.280	3.17910
$996$	0	0
$997$	−38.1559	−1.20841	−0.604205	−	0.796829i	$-0.706509\pi$
$997$	−38.1559	−1.20841	−0.604205	+	0.796829i	$0.706509\pi$
$998$	0.0445123	0.00140901
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7803.2.a.cg.1.12		24
3.2	odd	2		7803.2.a.cd.1.13		24
17.5	odd	16		459.2.l.a.433.7	yes	48
17.7	odd	16		459.2.l.a.406.7	yes	48
17.16	even	2		7803.2.a.cd.1.12		24
51.5	even	16		459.2.l.a.433.6	yes	48
51.41	even	16		459.2.l.a.406.6	✓	48
51.50	odd	2	inner	7803.2.a.cg.1.13		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
459.2.l.a.406.6	✓	48	51.41	even	16
459.2.l.a.406.7	yes	48	17.7	odd	16
459.2.l.a.433.6	yes	48	51.5	even	16
459.2.l.a.433.7	yes	48	17.5	odd	16
7803.2.a.cd.1.12		24	17.16	even	2
7803.2.a.cd.1.13		24	3.2	odd	2
7803.2.a.cg.1.12		24	1.1	even	1	trivial
7803.2.a.cg.1.13		24	51.50	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)

\(f(q)\)	\(=\)	\(q-0.113371 q^{2} -1.98715 q^{4} +3.61829 q^{5} -4.87047 q^{7} +0.452028 q^{8} -0.410211 q^{10} +3.73056 q^{11} +2.58136 q^{13} +0.552172 q^{14} +3.92305 q^{16} -3.88598 q^{19} -7.19008 q^{20} -0.422939 q^{22} -1.43027 q^{23} +8.09204 q^{25} -0.292652 q^{26} +9.67835 q^{28} +7.92074 q^{29} -3.24883 q^{31} -1.34882 q^{32} -17.6228 q^{35} +6.58423 q^{37} +0.440560 q^{38} +1.63557 q^{40} -6.81800 q^{41} +1.38277 q^{43} -7.41318 q^{44} +0.162152 q^{46} -7.36992 q^{47} +16.7215 q^{49} -0.917406 q^{50} -5.12954 q^{52} +5.16969 q^{53} +13.4983 q^{55} -2.20159 q^{56} -0.897986 q^{58} -6.46653 q^{59} -4.65687 q^{61} +0.368324 q^{62} -7.69318 q^{64} +9.34010 q^{65} +7.37261 q^{67} +1.99792 q^{70} -1.35380 q^{71} +11.4130 q^{73} -0.746463 q^{74} +7.72202 q^{76} -18.1696 q^{77} -2.55583 q^{79} +14.1947 q^{80} +0.772966 q^{82} +8.86903 q^{83} -0.156766 q^{86} +1.68632 q^{88} +8.62930 q^{89} -12.5724 q^{91} +2.84217 q^{92} +0.835538 q^{94} -14.0606 q^{95} -3.63548 q^{97} -1.89574 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 24 q^{4} + 20 q^{5} + 8 q^{11} + 4 q^{13} + 16 q^{14} + 24 q^{16} + 4 q^{19} + 48 q^{20} + 36 q^{23} + 28 q^{25} + 64 q^{29} + 36 q^{41} + 4 q^{43} + 16 q^{44} + 24 q^{49} + 16 q^{52} + 20 q^{55}+ \cdots - 12 q^{95}+O(q^{100}) \) 24 * q + 24 * q^4 + 20 * q^5 + 8 * q^11 + 4 * q^13 + 16 * q^14 + 24 * q^16 + 4 * q^19 + 48 * q^20 + 36 * q^23 + 28 * q^25 + 64 * q^29 + 36 * q^41 + 4 * q^43 + 16 * q^44 + 24 * q^49 + 16 * q^52 + 20 * q^55 + 80 * q^56 - 16 * q^62 - 24 * q^64 + 72 * q^65 + 16 * q^67 - 48 * q^70 + 72 * q^71 + 80 * q^74 - 40 * q^76 + 104 * q^80 + 144 * q^92 - 72 * q^94 - 12 * q^95

Introduction

L-functions

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