LMFDB - Embedded newform 7803.2.a.bs.1.8

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 = [10,0,0,12,2,0,0,0,0,0,-22,0,0,0,0,16,0,0,-10,-8,0,0,-28,0,8, 0,0,0,-6,0,0,0,0,0,0,0,0,0,0,0,-28,0,-8,-76,0,0,0,0,2,0,0,-4,0,0,42,64, 0,0,0,0,0,-52,0,-36,-32,0,18,0,0,-72,50,0,0] 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:	$1$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 16x^{8} + 86x^{6} - 170x^{4} + 73x^{2} - 8 $ x^10 - 16x^8 + 86x^6 - 170x^4 + 73x^2 - 8
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
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			1.8
Root			$2.06212$ 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)

$f(q)$	$=$	$q+2.06212 q^{2} +2.25232 q^{4} +1.44351 q^{5} -0.127581 q^{7} +0.520322 q^{8} +2.97669 q^{10} -5.72509 q^{11} +2.54466 q^{13} -0.263086 q^{14} -3.43168 q^{16} +1.47276 q^{19} +3.25126 q^{20} -11.8058 q^{22} -1.62839 q^{23} -2.91627 q^{25} +5.24738 q^{26} -0.287353 q^{28} -3.94816 q^{29} -0.413424 q^{31} -8.11718 q^{32} -0.184164 q^{35} -5.66603 q^{37} +3.03701 q^{38} +0.751092 q^{40} -1.89147 q^{41} +7.93369 q^{43} -12.8948 q^{44} -3.35792 q^{46} +0.484421 q^{47} -6.98372 q^{49} -6.01369 q^{50} +5.73140 q^{52} -9.05465 q^{53} -8.26423 q^{55} -0.0663830 q^{56} -8.14157 q^{58} -12.0518 q^{59} +4.31213 q^{61} -0.852529 q^{62} -9.87520 q^{64} +3.67325 q^{65} +9.00823 q^{67} -0.379768 q^{70} -10.9893 q^{71} -6.97856 q^{73} -11.6840 q^{74} +3.31714 q^{76} +0.730409 q^{77} +13.9312 q^{79} -4.95368 q^{80} -3.90044 q^{82} -4.09960 q^{83} +16.3602 q^{86} -2.97889 q^{88} +13.1764 q^{89} -0.324649 q^{91} -3.66765 q^{92} +0.998933 q^{94} +2.12595 q^{95} -3.59845 q^{97} -14.4013 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 12 q^{4} + 2 q^{5} - 22 q^{11} + 16 q^{16} - 10 q^{19} - 8 q^{20} - 28 q^{23} + 8 q^{25} - 6 q^{29} - 28 q^{41} - 8 q^{43} - 76 q^{44} + 2 q^{49} - 4 q^{52} + 42 q^{55} + 64 q^{56} - 52 q^{62} - 36 q^{64}+ \cdots - 38 q^{95}+O(q^{100}) $ 10 * q + 12 * q^4 + 2 * q^5 - 22 * q^11 + 16 * q^16 - 10 * q^19 - 8 * q^20 - 28 * q^23 + 8 * q^25 - 6 * q^29 - 28 * q^41 - 8 * q^43 - 76 * q^44 + 2 * q^49 - 4 * q^52 + 42 * q^55 + 64 * q^56 - 52 * q^62 - 36 * q^64 - 32 * q^65 + 18 * q^67 - 72 * q^70 + 50 * q^71 - 56 * q^74 - 28 * q^76 - 80 * q^80 - 56 * q^92 - 20 * q^94 - 38 * 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 $

$2$	2.06212	1.45814	0.729068	−	0.684441i	$-0.239954\pi$
$2$	2.06212	1.45814	0.729068	+	0.684441i	$0.239954\pi$
$3$	0	0
$3$	0	0
$4$	2.25232	1.12616
$5$	1.44351	0.645558	0.322779	−	0.946474i	$-0.395383\pi$
$5$	1.44351	0.645558	0.322779	+	0.946474i	$0.395383\pi$
$6$	0	0
$7$	−0.127581	−0.0482209	−0.0241105	−	0.999709i	$-0.507675\pi$
$7$	−0.127581	−0.0482209	−0.0241105	+	0.999709i	$0.507675\pi$
$8$	0.520322	0.183962
$9$	0	0
$10$	2.97669	0.941312
$11$	−5.72509	−1.72618	−0.863089	−	0.505052i	$-0.831473\pi$
$11$	−5.72509	−1.72618	−0.863089	+	0.505052i	$0.831473\pi$
$12$	0	0
$13$	2.54466	0.705762	0.352881	−	0.935668i	$-0.385202\pi$
$13$	2.54466	0.705762	0.352881	+	0.935668i	$0.385202\pi$
$14$	−0.263086	−0.0703127
$15$	0	0
$16$	−3.43168	−0.857921
$17$	0	0
$17$	0	0
$18$	0	0
$19$	1.47276	0.337875	0.168937	−	0.985627i	$-0.445966\pi$
$19$	1.47276	0.337875	0.168937	+	0.985627i	$0.445966\pi$
$20$	3.25126	0.727003
$21$	0	0
$22$	−11.8058	−2.51700
$23$	−1.62839	−0.339542	−0.169771	−	0.985484i	$-0.554303\pi$
$23$	−1.62839	−0.339542	−0.169771	+	0.985484i	$0.554303\pi$
$24$	0	0
$25$	−2.91627	−0.583255
$26$	5.24738	1.02910
$27$	0	0
$28$	−0.287353	−0.0543046
$29$	−3.94816	−0.733155	−0.366578	−	0.930388i	$-0.619471\pi$
$29$	−3.94816	−0.733155	−0.366578	+	0.930388i	$0.619471\pi$
$30$	0	0
$31$	−0.413424	−0.0742532	−0.0371266	−	0.999311i	$-0.511820\pi$
$31$	−0.413424	−0.0742532	−0.0371266	+	0.999311i	$0.511820\pi$
$32$	−8.11718	−1.43493
$33$	0	0
$34$	0	0
$35$	−0.184164	−0.0311294
$36$	0	0
$37$	−5.66603	−0.931489	−0.465745	−	0.884919i	$-0.654213\pi$
$37$	−5.66603	−0.931489	−0.465745	+	0.884919i	$0.654213\pi$
$38$	3.03701	0.492667
$39$	0	0
$40$	0.751092	0.118758
$41$	−1.89147	−0.295398	−0.147699	−	0.989032i	$-0.547187\pi$
$41$	−1.89147	−0.295398	−0.147699	+	0.989032i	$0.547187\pi$
$42$	0	0
$43$	7.93369	1.20988	0.604938	−	0.796272i	$-0.293198\pi$
$43$	7.93369	1.20988	0.604938	+	0.796272i	$0.293198\pi$
$44$	−12.8948	−1.94396
$45$	0	0
$46$	−3.35792	−0.495099
$47$	0.484421	0.0706601	0.0353300	−	0.999376i	$-0.488752\pi$
$47$	0.484421	0.0706601	0.0353300	+	0.999376i	$0.488752\pi$
$48$	0	0
$49$	−6.98372	−0.997675
$50$	−6.01369	−0.850465
$51$	0	0
$52$	5.73140	0.794802
$53$	−9.05465	−1.24375	−0.621876	−	0.783116i	$-0.713629\pi$
$53$	−9.05465	−1.24375	−0.621876	+	0.783116i	$0.713629\pi$
$54$	0	0
$55$	−8.26423	−1.11435
$56$	−0.0663830	−0.00887080
$57$	0	0
$58$	−8.14157	−1.06904
$59$	−12.0518	−1.56901	−0.784504	−	0.620124i	$-0.787082\pi$
$59$	−12.0518	−1.56901	−0.784504	+	0.620124i	$0.787082\pi$
$60$	0	0
$61$	4.31213	0.552111	0.276056	−	0.961142i	$-0.410973\pi$
$61$	4.31213	0.552111	0.276056	+	0.961142i	$0.410973\pi$
$62$	−0.852529	−0.108271
$63$	0	0
$64$	−9.87520	−1.23440
$65$	3.67325	0.455610
$66$	0	0
$67$	9.00823	1.10053	0.550265	−	0.834990i	$-0.314527\pi$
$67$	9.00823	1.10053	0.550265	+	0.834990i	$0.314527\pi$
$68$	0	0
$69$	0	0
$70$	−0.379768	−0.0453909
$71$	−10.9893	−1.30419	−0.652096	−	0.758136i	$-0.726110\pi$
$71$	−10.9893	−1.30419	−0.652096	+	0.758136i	$0.726110\pi$
$72$	0	0
$73$	−6.97856	−0.816779	−0.408389	−	0.912808i	$-0.633909\pi$
$73$	−6.97856	−0.816779	−0.408389	+	0.912808i	$0.633909\pi$
$74$	−11.6840	−1.35824
$75$	0	0
$76$	3.31714	0.380502
$77$	0.730409	0.0832379
$78$	0	0
$79$	13.9312	1.56738	0.783692	−	0.621149i	$-0.213334\pi$
$79$	13.9312	1.56738	0.783692	+	0.621149i	$0.213334\pi$
$80$	−4.95368	−0.553838
$81$	0	0
$82$	−3.90044	−0.430731
$83$	−4.09960	−0.449989	−0.224995	−	0.974360i	$-0.572236\pi$
$83$	−4.09960	−0.449989	−0.224995	+	0.974360i	$0.572236\pi$
$84$	0	0
$85$	0	0
$86$	16.3602	1.76417
$87$	0	0
$88$	−2.97889	−0.317551
$89$	13.1764	1.39670	0.698349	−	0.715758i	$-0.253918\pi$
$89$	13.1764	1.39670	0.698349	+	0.715758i	$0.253918\pi$
$90$	0	0
$91$	−0.324649	−0.0340325
$92$	−3.66765	−0.382379
$93$	0	0
$94$	0.998933	0.103032
$95$	2.12595	0.218118
$96$	0	0
$97$	−3.59845	−0.365367	−0.182683	−	0.983172i	$-0.558478\pi$
$97$	−3.59845	−0.365367	−0.182683	+	0.983172i	$0.558478\pi$
$98$	−14.4013	−1.45475
$99$	0	0
$100$	−6.56839	−0.656839
$101$	3.12471	0.310920	0.155460	−	0.987842i	$-0.450314\pi$
$101$	3.12471	0.310920	0.155460	+	0.987842i	$0.450314\pi$
$102$	0	0
$103$	17.4113	1.71559	0.857793	−	0.513995i	$-0.171835\pi$
$103$	17.4113	1.71559	0.857793	+	0.513995i	$0.171835\pi$
$104$	1.32404	0.129833
$105$	0	0
$106$	−18.6718	−1.81356
$107$	1.25126	0.120964	0.0604818	−	0.998169i	$-0.480736\pi$
$107$	1.25126	0.120964	0.0604818	+	0.998169i	$0.480736\pi$
$108$	0	0
$109$	−17.9020	−1.71470	−0.857349	−	0.514736i	$-0.827890\pi$
$109$	−17.9020	−1.71470	−0.857349	+	0.514736i	$0.827890\pi$
$110$	−17.0418	−1.62487
$111$	0	0
$112$	0.437816	0.0413697
$113$	−12.9656	−1.21970	−0.609850	−	0.792517i	$-0.708770\pi$
$113$	−12.9656	−1.21970	−0.609850	+	0.792517i	$0.708770\pi$
$114$	0	0
$115$	−2.35060	−0.219194
$116$	−8.89254	−0.825652
$117$	0	0
$118$	−24.8522	−2.28783
$119$	0	0
$120$	0	0
$121$	21.7766	1.97969
$122$	8.89211	0.805054
$123$	0	0
$124$	−0.931165	−0.0836211
$125$	−11.4272	−1.02208
$126$	0	0
$127$	0.867311	0.0769614	0.0384807	−	0.999259i	$-0.487748\pi$
$127$	0.867311	0.0769614	0.0384807	+	0.999259i	$0.487748\pi$
$128$	−4.12945	−0.364995
$129$	0	0
$130$	7.57466	0.664342
$131$	−10.4046	−0.909058	−0.454529	−	0.890732i	$-0.650192\pi$
$131$	−10.4046	−0.909058	−0.454529	+	0.890732i	$0.650192\pi$
$132$	0	0
$133$	−0.187896	−0.0162926
$134$	18.5760	1.60472
$135$	0	0
$136$	0	0
$137$	13.1508	1.12355	0.561773	−	0.827292i	$-0.310120\pi$
$137$	13.1508	1.12355	0.561773	+	0.827292i	$0.310120\pi$
$138$	0	0
$139$	−22.3624	−1.89675	−0.948376	−	0.317147i	$-0.897275\pi$
$139$	−22.3624	−1.89675	−0.948376	+	0.317147i	$0.897275\pi$
$140$	−0.414797	−0.0350568
$141$	0	0
$142$	−22.6612	−1.90169
$143$	−14.5684	−1.21827
$144$	0	0
$145$	−5.69922	−0.473294
$146$	−14.3906	−1.19098
$147$	0	0
$148$	−12.7617	−1.04901
$149$	14.8896	1.21981	0.609903	−	0.792476i	$-0.291208\pi$
$149$	14.8896	1.21981	0.609903	+	0.792476i	$0.291208\pi$
$150$	0	0
$151$	−18.9574	−1.54273	−0.771367	−	0.636391i	$-0.780427\pi$
$151$	−18.9574	−1.54273	−0.771367	+	0.636391i	$0.780427\pi$
$152$	0.766311	0.0621560
$153$	0	0
$154$	1.50619	0.121372
$155$	−0.596783	−0.0479347
$156$	0	0
$157$	−17.0660	−1.36201	−0.681006	−	0.732278i	$-0.738457\pi$
$157$	−17.0660	−1.36201	−0.681006	+	0.732278i	$0.738457\pi$
$158$	28.7278	2.28546
$159$	0	0
$160$	−11.7172	−0.926329
$161$	0.207750	0.0163730
$162$	0	0
$163$	−9.55207	−0.748176	−0.374088	−	0.927393i	$-0.622044\pi$
$163$	−9.55207	−0.748176	−0.374088	+	0.927393i	$0.622044\pi$
$164$	−4.26021	−0.332666
$165$	0	0
$166$	−8.45385	−0.656146
$167$	−19.1189	−1.47946	−0.739731	−	0.672903i	$-0.765047\pi$
$167$	−19.1189	−1.47946	−0.739731	+	0.672903i	$0.765047\pi$
$168$	0	0
$169$	−6.52471	−0.501901
$170$	0	0
$171$	0	0
$172$	17.8693	1.36252
$173$	12.8270	0.975221	0.487611	−	0.873061i	$-0.337869\pi$
$173$	12.8270	0.975221	0.487611	+	0.873061i	$0.337869\pi$
$174$	0	0
$175$	0.372060	0.0281251
$176$	19.6467	1.48092
$177$	0	0
$178$	27.1713	2.03658
$179$	9.39230	0.702013	0.351007	−	0.936373i	$-0.385839\pi$
$179$	9.39230	0.702013	0.351007	+	0.936373i	$0.385839\pi$
$180$	0	0
$181$	7.19109	0.534510	0.267255	−	0.963626i	$-0.413883\pi$
$181$	7.19109	0.534510	0.267255	+	0.963626i	$0.413883\pi$
$182$	−0.669464	−0.0496240
$183$	0	0
$184$	−0.847286	−0.0624627
$185$	−8.17898	−0.601330
$186$	0	0
$187$	0	0
$188$	1.09107	0.0795747
$189$	0	0
$190$	4.38395	0.318045
$191$	9.81454	0.710155	0.355078	−	0.934837i	$-0.384454\pi$
$191$	9.81454	0.710155	0.355078	+	0.934837i	$0.384454\pi$
$192$	0	0
$193$	22.9729	1.65362	0.826812	−	0.562478i	$-0.190152\pi$
$193$	22.9729	1.65362	0.826812	+	0.562478i	$0.190152\pi$
$194$	−7.42042	−0.532755
$195$	0	0
$196$	−15.7296	−1.12354
$197$	−15.1901	−1.08225	−0.541126	−	0.840942i	$-0.682002\pi$
$197$	−15.1901	−1.08225	−0.541126	+	0.840942i	$0.682002\pi$
$198$	0	0
$199$	−5.15117	−0.365157	−0.182578	−	0.983191i	$-0.558444\pi$
$199$	−5.15117	−0.365157	−0.182578	+	0.983191i	$0.558444\pi$
$200$	−1.51740	−0.107297
$201$	0	0
$202$	6.44351	0.453364
$203$	0.503708	0.0353534
$204$	0	0
$205$	−2.73036	−0.190697
$206$	35.9041	2.50156
$207$	0	0
$208$	−8.73247	−0.605488
$209$	−8.43168	−0.583232
$210$	0	0
$211$	2.48185	0.170858	0.0854289	−	0.996344i	$-0.472774\pi$
$211$	2.48185	0.170858	0.0854289	+	0.996344i	$0.472774\pi$
$212$	−20.3940	−1.40067
$213$	0	0
$214$	2.58024	0.176381
$215$	11.4524	0.781046
$216$	0	0
$217$	0.0527449	0.00358055
$218$	−36.9160	−2.50026
$219$	0	0
$220$	−18.6137	−1.25494
$221$	0	0
$222$	0	0
$223$	−0.793309	−0.0531239	−0.0265620	−	0.999647i	$-0.508456\pi$
$223$	−0.793309	−0.0531239	−0.0265620	+	0.999647i	$0.508456\pi$
$224$	1.03559	0.0691935
$225$	0	0
$226$	−26.7365	−1.77849
$227$	4.44279	0.294878	0.147439	−	0.989071i	$-0.452897\pi$
$227$	4.44279	0.294878	0.147439	+	0.989071i	$0.452897\pi$
$228$	0	0
$229$	−28.4102	−1.87740	−0.938701	−	0.344733i	$-0.887969\pi$
$229$	−28.4102	−1.87740	−0.938701	+	0.344733i	$0.887969\pi$
$230$	−4.84720	−0.319615
$231$	0	0
$232$	−2.05432	−0.134872
$233$	23.1945	1.51952	0.759762	−	0.650201i	$-0.225315\pi$
$233$	23.1945	1.51952	0.759762	+	0.650201i	$0.225315\pi$
$234$	0	0
$235$	0.699268	0.0456152
$236$	−27.1445	−1.76696
$237$	0	0
$238$	0	0
$239$	−6.48768	−0.419653	−0.209827	−	0.977739i	$-0.567290\pi$
$239$	−6.48768	−0.419653	−0.209827	+	0.977739i	$0.567290\pi$
$240$	0	0
$241$	27.2402	1.75469	0.877346	−	0.479858i	$-0.159312\pi$
$241$	27.2402	1.75469	0.877346	+	0.479858i	$0.159312\pi$
$242$	44.9059	2.88666
$243$	0	0
$244$	9.71231	0.621767
$245$	−10.0811	−0.644057
$246$	0	0
$247$	3.74768	0.238459
$248$	−0.215114	−0.0136597
$249$	0	0
$250$	−23.5643	−1.49034
$251$	−10.8217	−0.683062	−0.341531	−	0.939871i	$-0.610945\pi$
$251$	−10.8217	−0.683062	−0.341531	+	0.939871i	$0.610945\pi$
$252$	0	0
$253$	9.32265	0.586110
$254$	1.78850	0.112220
$255$	0	0
$256$	11.2350	0.702186
$257$	0.736456	0.0459388	0.0229694	−	0.999736i	$-0.492688\pi$
$257$	0.736456	0.0459388	0.0229694	+	0.999736i	$0.492688\pi$
$258$	0	0
$259$	0.722875	0.0449172
$260$	8.27334	0.513091
$261$	0	0
$262$	−21.4556	−1.32553
$263$	21.9569	1.35392	0.676959	−	0.736020i	$-0.263297\pi$
$263$	21.9569	1.35392	0.676959	+	0.736020i	$0.263297\pi$
$264$	0	0
$265$	−13.0705	−0.802915
$266$	−0.387463	−0.0237569
$267$	0	0
$268$	20.2895	1.23938
$269$	9.15898	0.558433	0.279216	−	0.960228i	$-0.409925\pi$
$269$	9.15898	0.558433	0.279216	+	0.960228i	$0.409925\pi$
$270$	0	0
$271$	17.4194	1.05815	0.529077	−	0.848574i	$-0.322538\pi$
$271$	17.4194	1.05815	0.529077	+	0.848574i	$0.322538\pi$
$272$	0	0
$273$	0	0
$274$	27.1184	1.63828
$275$	16.6959	1.00680
$276$	0	0
$277$	−8.76381	−0.526567	−0.263283	−	0.964719i	$-0.584805\pi$
$277$	−8.76381	−0.526567	−0.263283	+	0.964719i	$0.584805\pi$
$278$	−46.1138	−2.76572
$279$	0	0
$280$	−0.0958247	−0.00572662
$281$	−3.29450	−0.196534	−0.0982668	−	0.995160i	$-0.531330\pi$
$281$	−3.29450	−0.196534	−0.0982668	+	0.995160i	$0.531330\pi$
$282$	0	0
$283$	14.7801	0.878589	0.439294	−	0.898343i	$-0.355229\pi$
$283$	14.7801	0.878589	0.439294	+	0.898343i	$0.355229\pi$
$284$	−24.7515	−1.46873
$285$	0	0
$286$	−30.0417	−1.77640
$287$	0.241315	0.0142444
$288$	0	0
$289$	0	0
$290$	−11.7525	−0.690128
$291$	0	0
$292$	−15.7180	−0.919825
$293$	−20.3034	−1.18614	−0.593068	−	0.805152i	$-0.702083\pi$
$293$	−20.3034	−1.18614	−0.593068	+	0.805152i	$0.702083\pi$
$294$	0	0
$295$	−17.3969	−1.01289
$296$	−2.94816	−0.171358
$297$	0	0
$298$	30.7041	1.77864
$299$	−4.14369	−0.239636
$300$	0	0
$301$	−1.01218	−0.0583414
$302$	−39.0924	−2.24952
$303$	0	0
$304$	−5.05405	−0.289870
$305$	6.22461	0.356420
$306$	0	0
$307$	1.64581	0.0939312	0.0469656	−	0.998897i	$-0.485045\pi$
$307$	1.64581	0.0939312	0.0469656	+	0.998897i	$0.485045\pi$
$308$	1.64512	0.0937394
$309$	0	0
$310$	−1.23064	−0.0698954
$311$	−31.4084	−1.78101	−0.890505	−	0.454974i	$-0.849649\pi$
$311$	−31.4084	−1.78101	−0.890505	+	0.454974i	$0.849649\pi$
$312$	0	0
$313$	−12.1758	−0.688216	−0.344108	−	0.938930i	$-0.611819\pi$
$313$	−12.1758	−0.688216	−0.344108	+	0.938930i	$0.611819\pi$
$314$	−35.1920	−1.98600
$315$	0	0
$316$	31.3776	1.76513
$317$	−3.05027	−0.171320	−0.0856600	−	0.996324i	$-0.527300\pi$
$317$	−3.05027	−0.171320	−0.0856600	+	0.996324i	$0.527300\pi$
$318$	0	0
$319$	22.6036	1.26556
$320$	−14.2550	−0.796877
$321$	0	0
$322$	0.428405	0.0238741
$323$	0	0
$324$	0	0
$325$	−7.42092	−0.411639
$326$	−19.6975	−1.09094
$327$	0	0
$328$	−0.984175	−0.0543420
$329$	−0.0618027	−0.00340729
$330$	0	0
$331$	11.9151	0.654915	0.327457	−	0.944866i	$-0.393808\pi$
$331$	11.9151	0.654915	0.327457	+	0.944866i	$0.393808\pi$
$332$	−9.23362	−0.506761
$333$	0	0
$334$	−39.4253	−2.15726
$335$	13.0035	0.710457
$336$	0	0
$337$	28.2176	1.53711	0.768555	−	0.639783i	$-0.220976\pi$
$337$	28.2176	1.53711	0.768555	+	0.639783i	$0.220976\pi$
$338$	−13.4547	−0.731840
$339$	0	0
$340$	0	0
$341$	2.36689	0.128174
$342$	0	0
$343$	1.78405	0.0963297
$344$	4.12808	0.222571
$345$	0	0
$346$	26.4508	1.42201
$347$	33.5125	1.79905	0.899523	−	0.436873i	$-0.143914\pi$
$347$	33.5125	1.79905	0.899523	+	0.436873i	$0.143914\pi$
$348$	0	0
$349$	15.8690	0.849446	0.424723	−	0.905323i	$-0.360372\pi$
$349$	15.8690	0.849446	0.424723	+	0.905323i	$0.360372\pi$
$350$	0.767230	0.0410102
$351$	0	0
$352$	46.4715	2.47694
$353$	−22.6893	−1.20763	−0.603816	−	0.797124i	$-0.706354\pi$
$353$	−22.6893	−1.20763	−0.603816	+	0.797124i	$0.706354\pi$
$354$	0	0
$355$	−15.8632	−0.841932
$356$	29.6776	1.57291
$357$	0	0
$358$	19.3680	1.02363
$359$	32.5288	1.71680	0.858402	−	0.512978i	$-0.171458\pi$
$359$	32.5288	1.71680	0.858402	+	0.512978i	$0.171458\pi$
$360$	0	0
$361$	−16.8310	−0.885841
$362$	14.8289	0.779388
$363$	0	0
$364$	−0.731215	−0.0383261
$365$	−10.0736	−0.527278
$366$	0	0
$367$	25.0942	1.30990	0.654952	−	0.755670i	$-0.272689\pi$
$367$	25.0942	1.30990	0.654952	+	0.755670i	$0.272689\pi$
$368$	5.58811	0.291300
$369$	0	0
$370$	−16.8660	−0.876822
$371$	1.15520	0.0599749
$372$	0	0
$373$	8.90157	0.460906	0.230453	−	0.973083i	$-0.425979\pi$
$373$	8.90157	0.460906	0.230453	+	0.973083i	$0.425979\pi$
$374$	0	0
$375$	0	0
$376$	0.252055	0.0129988
$377$	−10.0467	−0.517433
$378$	0	0
$379$	8.57311	0.440371	0.220186	−	0.975458i	$-0.429334\pi$
$379$	8.57311	0.440371	0.220186	+	0.975458i	$0.429334\pi$
$380$	4.78833	0.245636
$381$	0	0
$382$	20.2387	1.03550
$383$	−26.5050	−1.35434	−0.677170	−	0.735826i	$-0.736794\pi$
$383$	−26.5050	−1.35434	−0.677170	+	0.735826i	$0.736794\pi$
$384$	0	0
$385$	1.05435	0.0537349
$386$	47.3728	2.41121
$387$	0	0
$388$	−8.10487	−0.411462
$389$	−15.4897	−0.785361	−0.392681	−	0.919675i	$-0.628452\pi$
$389$	−15.4897	−0.785361	−0.392681	+	0.919675i	$0.628452\pi$
$390$	0	0
$391$	0	0
$392$	−3.63379	−0.183534
$393$	0	0
$394$	−31.3238	−1.57807
$395$	20.1099	1.01184
$396$	0	0
$397$	1.14485	0.0574585	0.0287293	−	0.999587i	$-0.490854\pi$
$397$	1.14485	0.0574585	0.0287293	+	0.999587i	$0.490854\pi$
$398$	−10.6223	−0.532448
$399$	0	0
$400$	10.0077	0.500386
$401$	−15.2415	−0.761126	−0.380563	−	0.924755i	$-0.624270\pi$
$401$	−15.2415	−0.761126	−0.380563	+	0.924755i	$0.624270\pi$
$402$	0	0
$403$	−1.05202	−0.0524050
$404$	7.03786	0.350146
$405$	0	0
$406$	1.03871	0.0515501
$407$	32.4385	1.60792
$408$	0	0
$409$	−20.8505	−1.03099	−0.515495	−	0.856893i	$-0.672392\pi$
$409$	−20.8505	−1.03099	−0.515495	+	0.856893i	$0.672392\pi$
$410$	−5.63033	−0.278062
$411$	0	0
$412$	39.2159	1.93203
$413$	1.53757	0.0756590
$414$	0	0
$415$	−5.91782	−0.290494
$416$	−20.6554	−1.01272
$417$	0	0
$418$	−17.3871	−0.850432
$419$	−1.58643	−0.0775020	−0.0387510	−	0.999249i	$-0.512338\pi$
$419$	−1.58643	−0.0775020	−0.0387510	+	0.999249i	$0.512338\pi$
$420$	0	0
$421$	6.69178	0.326138	0.163069	−	0.986615i	$-0.447861\pi$
$421$	6.69178	0.326138	0.163069	+	0.986615i	$0.447861\pi$
$422$	5.11787	0.249134
$423$	0	0
$424$	−4.71134	−0.228803
$425$	0	0
$426$	0	0
$427$	−0.550144	−0.0266233
$428$	2.81824	0.136225
$429$	0	0
$430$	23.6161	1.13887
$431$	−20.4283	−0.983995	−0.491998	−	0.870597i	$-0.663733\pi$
$431$	−20.4283	−0.983995	−0.491998	+	0.870597i	$0.663733\pi$
$432$	0	0
$433$	−37.9714	−1.82479	−0.912395	−	0.409310i	$-0.865769\pi$
$433$	−37.9714	−1.82479	−0.912395	+	0.409310i	$0.865769\pi$
$434$	0.108766	0.00522094
$435$	0	0
$436$	−40.3211	−1.93103
$437$	−2.39822	−0.114723
$438$	0	0
$439$	−18.8231	−0.898377	−0.449188	−	0.893437i	$-0.648287\pi$
$439$	−18.8231	−0.898377	−0.449188	+	0.893437i	$0.648287\pi$
$440$	−4.30006	−0.204998
$441$	0	0
$442$	0	0
$443$	−29.2336	−1.38893	−0.694465	−	0.719526i	$-0.744359\pi$
$443$	−29.2336	−1.38893	−0.694465	+	0.719526i	$0.744359\pi$
$444$	0	0
$445$	19.0203	0.901650
$446$	−1.63590	−0.0774619
$447$	0	0
$448$	1.25988	0.0595239
$449$	−8.55507	−0.403739	−0.201869	−	0.979412i	$-0.564702\pi$
$449$	−8.55507	−0.403739	−0.201869	+	0.979412i	$0.564702\pi$
$450$	0	0
$451$	10.8288	0.509910
$452$	−29.2027	−1.37358
$453$	0	0
$454$	9.16155	0.429973
$455$	−0.468635	−0.0219699
$456$	0	0
$457$	17.8952	0.837101	0.418550	−	0.908194i	$-0.362538\pi$
$457$	17.8952	0.837101	0.418550	+	0.908194i	$0.362538\pi$
$458$	−58.5852	−2.73751
$459$	0	0
$460$	−5.29430	−0.246848
$461$	30.3494	1.41351	0.706756	−	0.707457i	$-0.250158\pi$
$461$	30.3494	1.41351	0.706756	+	0.707457i	$0.250158\pi$
$462$	0	0
$463$	10.8211	0.502901	0.251451	−	0.967870i	$-0.419092\pi$
$463$	10.8211	0.502901	0.251451	+	0.967870i	$0.419092\pi$
$464$	13.5488	0.628989
$465$	0	0
$466$	47.8298	2.21567
$467$	33.7498	1.56175	0.780876	−	0.624686i	$-0.214773\pi$
$467$	33.7498	1.56175	0.780876	+	0.624686i	$0.214773\pi$
$468$	0	0
$469$	−1.14927	−0.0530686
$470$	1.44197	0.0665132
$471$	0	0
$472$	−6.27081	−0.288638
$473$	−45.4211	−2.08846
$474$	0	0
$475$	−4.29497	−0.197067
$476$	0	0
$477$	0	0
$478$	−13.3783	−0.611911
$479$	28.9330	1.32198	0.660990	−	0.750395i	$-0.270136\pi$
$479$	28.9330	1.32198	0.660990	+	0.750395i	$0.270136\pi$
$480$	0	0
$481$	−14.4181	−0.657409
$482$	56.1724	2.55858
$483$	0	0
$484$	49.0480	2.22945
$485$	−5.19440	−0.235866
$486$	0	0
$487$	28.0554	1.27131	0.635655	−	0.771973i	$-0.280730\pi$
$487$	28.0554	1.27131	0.635655	+	0.771973i	$0.280730\pi$
$488$	2.24370	0.101567
$489$	0	0
$490$	−20.7884	−0.939123
$491$	−20.1762	−0.910541	−0.455271	−	0.890353i	$-0.650457\pi$
$491$	−20.1762	−0.910541	−0.455271	+	0.890353i	$0.650457\pi$
$492$	0	0
$493$	0	0
$494$	7.72814	0.347706
$495$	0	0
$496$	1.41874	0.0637033
$497$	1.40202	0.0628893
$498$	0	0
$499$	−29.9876	−1.34243	−0.671214	−	0.741264i	$-0.734227\pi$
$499$	−29.9876	−1.34243	−0.671214	+	0.741264i	$0.734227\pi$
$500$	−25.7378	−1.15103
$501$	0	0
$502$	−22.3157	−0.995997
$503$	−1.08769	−0.0484979	−0.0242490	−	0.999706i	$-0.507719\pi$
$503$	−1.08769	−0.0484979	−0.0242490	+	0.999706i	$0.507719\pi$
$504$	0	0
$505$	4.51055	0.200717
$506$	19.2244	0.854629
$507$	0	0
$508$	1.95347	0.0866710
$509$	14.0982	0.624892	0.312446	−	0.949935i	$-0.398852\pi$
$509$	14.0982	0.624892	0.312446	+	0.949935i	$0.398852\pi$
$510$	0	0
$511$	0.890328	0.0393858
$512$	31.4267	1.38888
$513$	0	0
$514$	1.51866	0.0669851
$515$	25.1334	1.10751
$516$	0	0
$517$	−2.77335	−0.121972
$518$	1.49065	0.0654955
$519$	0	0
$520$	1.91127	0.0838148
$521$	0.519431	0.0227567	0.0113783	−	0.999935i	$-0.496378\pi$
$521$	0.519431	0.0227567	0.0113783	+	0.999935i	$0.496378\pi$
$522$	0	0
$523$	1.99821	0.0873757	0.0436879	−	0.999045i	$-0.486089\pi$
$523$	1.99821	0.0873757	0.0436879	+	0.999045i	$0.486089\pi$
$524$	−23.4346	−1.02375
$525$	0	0
$526$	45.2776	1.97420
$527$	0	0
$528$	0	0
$529$	−20.3484	−0.884711
$530$	−26.9529	−1.17076
$531$	0	0
$532$	−0.423202	−0.0183481
$533$	−4.81315	−0.208481
$534$	0	0
$535$	1.80621	0.0780891
$536$	4.68718	0.202456
$537$	0	0
$538$	18.8869	0.814271
$539$	39.9824	1.72216
$540$	0	0
$541$	−4.03513	−0.173484	−0.0867420	−	0.996231i	$-0.527646\pi$
$541$	−4.03513	−0.173484	−0.0867420	+	0.996231i	$0.527646\pi$
$542$	35.9209	1.54293
$543$	0	0
$544$	0	0
$545$	−25.8417	−1.10694
$546$	0	0
$547$	−2.46695	−0.105479	−0.0527396	−	0.998608i	$-0.516795\pi$
$547$	−2.46695	−0.105479	−0.0527396	+	0.998608i	$0.516795\pi$
$548$	29.6198	1.26529
$549$	0	0
$550$	34.4289	1.46805
$551$	−5.81470	−0.247714
$552$	0	0
$553$	−1.77735	−0.0755807
$554$	−18.0720	−0.767806
$555$	0	0
$556$	−50.3673	−2.13605
$557$	13.9487	0.591024	0.295512	−	0.955339i	$-0.404510\pi$
$557$	13.9487	0.591024	0.295512	+	0.955339i	$0.404510\pi$
$558$	0	0
$559$	20.1885	0.853885
$560$	0.631993	0.0267066
$561$	0	0
$562$	−6.79365	−0.286573
$563$	2.16638	0.0913019	0.0456510	−	0.998957i	$-0.485464\pi$
$563$	2.16638	0.0913019	0.0456510	+	0.998957i	$0.485464\pi$
$564$	0	0
$565$	−18.7160	−0.787387
$566$	30.4784	1.28110
$567$	0	0
$568$	−5.71799	−0.239921
$569$	−13.9324	−0.584078	−0.292039	−	0.956406i	$-0.594334\pi$
$569$	−13.9324	−0.584078	−0.292039	+	0.956406i	$0.594334\pi$
$570$	0	0
$571$	−5.20600	−0.217864	−0.108932	−	0.994049i	$-0.534743\pi$
$571$	−5.20600	−0.217864	−0.108932	+	0.994049i	$0.534743\pi$
$572$	−32.8127	−1.37197
$573$	0	0
$574$	0.497620	0.0207702
$575$	4.74882	0.198039
$576$	0	0
$577$	1.33032	0.0553821	0.0276911	−	0.999617i	$-0.491185\pi$
$577$	1.33032	0.0553821	0.0276911	+	0.999617i	$0.491185\pi$
$578$	0	0
$579$	0	0
$580$	−12.8365	−0.533006
$581$	0.523029	0.0216989
$582$	0	0
$583$	51.8387	2.14694
$584$	−3.63110	−0.150256
$585$	0	0
$586$	−41.8679	−1.72955
$587$	−10.1422	−0.418615	−0.209308	−	0.977850i	$-0.567121\pi$
$587$	−10.1422	−0.418615	−0.209308	+	0.977850i	$0.567121\pi$
$588$	0	0
$589$	−0.608875	−0.0250883
$590$	−35.8744	−1.47693
$591$	0	0
$592$	19.4440	0.799144
$593$	23.8630	0.979937	0.489968	−	0.871740i	$-0.337008\pi$
$593$	23.8630	0.979937	0.489968	+	0.871740i	$0.337008\pi$
$594$	0	0
$595$	0	0
$596$	33.5363	1.37370
$597$	0	0
$598$	−8.54477	−0.349422
$599$	−23.4361	−0.957574	−0.478787	−	0.877931i	$-0.658923\pi$
$599$	−23.4361	−0.957574	−0.478787	+	0.877931i	$0.658923\pi$
$600$	0	0
$601$	−13.1802	−0.537633	−0.268817	−	0.963191i	$-0.586633\pi$
$601$	−13.1802	−0.537633	−0.268817	+	0.963191i	$0.586633\pi$
$602$	−2.08724	−0.0850697
$603$	0	0
$604$	−42.6983	−1.73737
$605$	31.4348	1.27801
$606$	0	0
$607$	25.8950	1.05105	0.525523	−	0.850779i	$-0.323869\pi$
$607$	25.8950	1.05105	0.525523	+	0.850779i	$0.323869\pi$
$608$	−11.9547	−0.484826
$609$	0	0
$610$	12.8359	0.519709
$611$	1.23269	0.0498692
$612$	0	0
$613$	−32.9459	−1.33067	−0.665337	−	0.746543i	$-0.731712\pi$
$613$	−32.9459	−1.33067	−0.665337	+	0.746543i	$0.731712\pi$
$614$	3.39385	0.136964
$615$	0	0
$616$	0.380048	0.0153126
$617$	−3.62573	−0.145966	−0.0729831	−	0.997333i	$-0.523252\pi$
$617$	−3.62573	−0.145966	−0.0729831	+	0.997333i	$0.523252\pi$
$618$	0	0
$619$	38.7024	1.55558	0.777791	−	0.628523i	$-0.216340\pi$
$619$	38.7024	1.55558	0.777791	+	0.628523i	$0.216340\pi$
$620$	−1.34415	−0.0539823
$621$	0	0
$622$	−64.7679	−2.59696
$623$	−1.68105	−0.0673500
$624$	0	0
$625$	−1.91399	−0.0765594
$626$	−25.1079	−1.00351
$627$	0	0
$628$	−38.4381	−1.53385
$629$	0	0
$630$	0	0
$631$	−20.0655	−0.798793	−0.399396	−	0.916778i	$-0.630780\pi$
$631$	−20.0655	−0.798793	−0.399396	+	0.916778i	$0.630780\pi$
$632$	7.24873	0.288339
$633$	0	0
$634$	−6.29001	−0.249808
$635$	1.25197	0.0496830
$636$	0	0
$637$	−17.7712	−0.704120
$638$	46.6112	1.84535
$639$	0	0
$640$	−5.96091	−0.235626
$641$	−40.6812	−1.60681	−0.803405	−	0.595432i	$-0.796981\pi$
$641$	−40.6812	−1.60681	−0.803405	+	0.595432i	$0.796981\pi$
$642$	0	0
$643$	26.7094	1.05332	0.526659	−	0.850077i	$-0.323445\pi$
$643$	26.7094	1.05332	0.526659	+	0.850077i	$0.323445\pi$
$644$	0.467921	0.0184387
$645$	0	0
$646$	0	0
$647$	−1.47428	−0.0579601	−0.0289800	−	0.999580i	$-0.509226\pi$
$647$	−1.47428	−0.0579601	−0.0289800	+	0.999580i	$0.509226\pi$
$648$	0	0
$649$	68.9975	2.70839
$650$	−15.3028	−0.600225
$651$	0	0
$652$	−21.5144	−0.842568
$653$	−1.95400	−0.0764658	−0.0382329	−	0.999269i	$-0.512173\pi$
$653$	−1.95400	−0.0764658	−0.0382329	+	0.999269i	$0.512173\pi$
$654$	0	0
$655$	−15.0192	−0.586850
$656$	6.49093	0.253428
$657$	0	0
$658$	−0.127444	−0.00496830
$659$	−33.4094	−1.30145	−0.650723	−	0.759315i	$-0.725534\pi$
$659$	−33.4094	−1.30145	−0.650723	+	0.759315i	$0.725534\pi$
$660$	0	0
$661$	−23.3283	−0.907366	−0.453683	−	0.891163i	$-0.649890\pi$
$661$	−23.3283	−0.907366	−0.453683	+	0.891163i	$0.649890\pi$
$662$	24.5704	0.954955
$663$	0	0
$664$	−2.13311	−0.0827808
$665$	−0.271230	−0.0105178
$666$	0	0
$667$	6.42913	0.248937
$668$	−43.0619	−1.66611
$669$	0	0
$670$	26.8147	1.03594
$671$	−24.6873	−0.953043
$672$	0	0
$673$	−8.54944	−0.329557	−0.164778	−	0.986331i	$-0.552691\pi$
$673$	−8.54944	−0.329557	−0.164778	+	0.986331i	$0.552691\pi$
$674$	58.1880	2.24132
$675$	0	0
$676$	−14.6958	−0.565222
$677$	19.1127	0.734562	0.367281	−	0.930110i	$-0.380289\pi$
$677$	19.1127	0.734562	0.367281	+	0.930110i	$0.380289\pi$
$678$	0	0
$679$	0.459092	0.0176183
$680$	0	0
$681$	0	0
$682$	4.88080	0.186895
$683$	−0.0672121	−0.00257180	−0.00128590	−	0.999999i	$-0.500409\pi$
$683$	−0.0672121	−0.00257180	−0.00128590	+	0.999999i	$0.500409\pi$
$684$	0	0
$685$	18.9833	0.725314
$686$	3.67892	0.140462
$687$	0	0
$688$	−27.2259	−1.03798
$689$	−23.0410	−0.877793
$690$	0	0
$691$	−0.152462	−0.00579991	−0.00289996	−	0.999996i	$-0.500923\pi$
$691$	−0.152462	−0.00579991	−0.00289996	+	0.999996i	$0.500923\pi$
$692$	28.8906	1.09826
$693$	0	0
$694$	69.1067	2.62326
$695$	−32.2804	−1.22446
$696$	0	0
$697$	0	0
$698$	32.7236	1.23861
$699$	0	0
$700$	0.837999	0.0316734
$701$	28.8361	1.08912	0.544562	−	0.838721i	$-0.316696\pi$
$701$	28.8361	1.08912	0.544562	+	0.838721i	$0.316696\pi$
$702$	0	0
$703$	−8.34470	−0.314726
$704$	56.5363	2.13079
$705$	0	0
$706$	−46.7881	−1.76089
$707$	−0.398652	−0.0149928
$708$	0	0
$709$	24.7973	0.931281	0.465641	−	0.884974i	$-0.345824\pi$
$709$	24.7973	0.931281	0.465641	+	0.884974i	$0.345824\pi$
$710$	−32.7118	−1.22765
$711$	0	0
$712$	6.85599	0.256939
$713$	0.673214	0.0252121
$714$	0	0
$715$	−21.0297	−0.786464
$716$	21.1545	0.790581
$717$	0	0
$718$	67.0781	2.50333
$719$	−9.66469	−0.360432	−0.180216	−	0.983627i	$-0.557680\pi$
$719$	−9.66469	−0.360432	−0.180216	+	0.983627i	$0.557680\pi$
$720$	0	0
$721$	−2.22134	−0.0827271
$722$	−34.7074	−1.29168
$723$	0	0
$724$	16.1967	0.601945
$725$	11.5139	0.427616
$726$	0	0
$727$	32.5242	1.20626	0.603129	−	0.797644i	$-0.293920\pi$
$727$	32.5242	1.20626	0.603129	+	0.797644i	$0.293920\pi$
$728$	−0.168922	−0.00626067
$729$	0	0
$730$	−20.7730	−0.768844
$731$	0	0
$732$	0	0
$733$	−33.0498	−1.22072	−0.610361	−	0.792123i	$-0.708976\pi$
$733$	−33.0498	−1.22072	−0.610361	+	0.792123i	$0.708976\pi$
$734$	51.7471	1.91002
$735$	0	0
$736$	13.2179	0.487218
$737$	−51.5729	−1.89971
$738$	0	0
$739$	−3.87288	−0.142466	−0.0712330	−	0.997460i	$-0.522693\pi$
$739$	−3.87288	−0.142466	−0.0712330	+	0.997460i	$0.522693\pi$
$740$	−18.4217	−0.677196
$741$	0	0
$742$	2.38215	0.0874515
$743$	−22.8764	−0.839253	−0.419626	−	0.907697i	$-0.637839\pi$
$743$	−22.8764	−0.839253	−0.419626	+	0.907697i	$0.637839\pi$
$744$	0	0
$745$	21.4934	0.787456
$746$	18.3561	0.672063
$747$	0	0
$748$	0	0
$749$	−0.159636	−0.00583297
$750$	0	0
$751$	47.9834	1.75094	0.875469	−	0.483274i	$-0.160552\pi$
$751$	47.9834	1.75094	0.875469	+	0.483274i	$0.160552\pi$
$752$	−1.66238	−0.0606208
$753$	0	0
$754$	−20.7175	−0.754487
$755$	−27.3653	−0.995924
$756$	0	0
$757$	7.24009	0.263146	0.131573	−	0.991307i	$-0.457997\pi$
$757$	7.24009	0.263146	0.131573	+	0.991307i	$0.457997\pi$
$758$	17.6788	0.642122
$759$	0	0
$760$	1.10618	0.0401253
$761$	23.8187	0.863426	0.431713	−	0.902011i	$-0.357909\pi$
$761$	23.8187	0.863426	0.431713	+	0.902011i	$0.357909\pi$
$762$	0	0
$763$	2.28394	0.0826843
$764$	22.1055	0.799750
$765$	0	0
$766$	−54.6563	−1.97481
$767$	−30.6677	−1.10735
$768$	0	0
$769$	6.92589	0.249754	0.124877	−	0.992172i	$-0.460146\pi$
$769$	6.92589	0.249754	0.124877	+	0.992172i	$0.460146\pi$
$770$	2.17420	0.0783528
$771$	0	0
$772$	51.7424	1.86225
$773$	8.04876	0.289494	0.144747	−	0.989469i	$-0.453763\pi$
$773$	8.04876	0.289494	0.144747	+	0.989469i	$0.453763\pi$
$774$	0	0
$775$	1.20566	0.0433085
$776$	−1.87235	−0.0672135
$777$	0	0
$778$	−31.9417	−1.14516
$779$	−2.78569	−0.0998076
$780$	0	0
$781$	62.9148	2.25127
$782$	0	0
$783$	0	0
$784$	23.9659	0.855926
$785$	−24.6349	−0.879258
$786$	0	0
$787$	38.6780	1.37872	0.689360	−	0.724419i	$-0.257892\pi$
$787$	38.6780	1.37872	0.689360	+	0.724419i	$0.257892\pi$
$788$	−34.2131	−1.21879
$789$	0	0
$790$	41.4689	1.47540
$791$	1.65416	0.0588150
$792$	0	0
$793$	10.9729	0.389659
$794$	2.36082	0.0837824
$795$	0	0
$796$	−11.6021	−0.411226
$797$	−50.6758	−1.79503	−0.897514	−	0.440985i	$-0.854629\pi$
$797$	−50.6758	−1.79503	−0.897514	+	0.440985i	$0.854629\pi$
$798$	0	0
$799$	0	0
$800$	23.6719	0.836928
$801$	0	0
$802$	−31.4298	−1.10983
$803$	39.9529	1.40991
$804$	0	0
$805$	0.299890	0.0105697
$806$	−2.16940	−0.0764137
$807$	0	0
$808$	1.62586	0.0571974
$809$	−34.7611	−1.22214	−0.611068	−	0.791578i	$-0.709260\pi$
$809$	−34.7611	−1.22214	−0.611068	+	0.791578i	$0.709260\pi$
$810$	0	0
$811$	23.2649	0.816942	0.408471	−	0.912771i	$-0.366062\pi$
$811$	23.2649	0.816942	0.408471	+	0.912771i	$0.366062\pi$
$812$	1.13451	0.0398137
$813$	0	0
$814$	66.8919	2.34456
$815$	−13.7885	−0.482991
$816$	0	0
$817$	11.6844	0.408787
$818$	−42.9961	−1.50332
$819$	0	0
$820$	−6.14966	−0.214756
$821$	−9.90703	−0.345758	−0.172879	−	0.984943i	$-0.555307\pi$
$821$	−9.90703	−0.345758	−0.172879	+	0.984943i	$0.555307\pi$
$822$	0	0
$823$	3.84078	0.133881	0.0669406	−	0.997757i	$-0.478676\pi$
$823$	3.84078	0.133881	0.0669406	+	0.997757i	$0.478676\pi$
$824$	9.05949	0.315602
$825$	0	0
$826$	3.17065	0.110321
$827$	12.1752	0.423373	0.211687	−	0.977338i	$-0.432104\pi$
$827$	12.1752	0.423373	0.211687	+	0.977338i	$0.432104\pi$
$828$	0	0
$829$	−40.9472	−1.42216	−0.711078	−	0.703113i	$-0.751793\pi$
$829$	−40.9472	−1.42216	−0.711078	+	0.703113i	$0.751793\pi$
$830$	−12.2032	−0.423580
$831$	0	0
$832$	−25.1290	−0.871192
$833$	0	0
$834$	0	0
$835$	−27.5983	−0.955079
$836$	−18.9909	−0.656814
$837$	0	0
$838$	−3.27140	−0.113009
$839$	9.30970	0.321407	0.160703	−	0.987003i	$-0.448624\pi$
$839$	9.30970	0.321407	0.160703	+	0.987003i	$0.448624\pi$
$840$	0	0
$841$	−13.4120	−0.462484
$842$	13.7992	0.475553
$843$	0	0
$844$	5.58994	0.192414
$845$	−9.41849	−0.324006
$846$	0	0
$847$	−2.77827	−0.0954625
$848$	31.0727	1.06704
$849$	0	0
$850$	0	0
$851$	9.22648	0.316280
$852$	0	0
$853$	−26.1377	−0.894939	−0.447469	−	0.894299i	$-0.647675\pi$
$853$	−26.1377	−0.894939	−0.447469	+	0.894299i	$0.647675\pi$
$854$	−1.13446	−0.0388204
$855$	0	0
$856$	0.651057	0.0222527
$857$	−45.2942	−1.54722	−0.773611	−	0.633661i	$-0.781552\pi$
$857$	−45.2942	−1.54722	−0.773611	+	0.633661i	$0.781552\pi$
$858$	0	0
$859$	26.0657	0.889351	0.444675	−	0.895692i	$-0.353319\pi$
$859$	26.0657	0.889351	0.444675	+	0.895692i	$0.353319\pi$
$860$	25.7945	0.879584
$861$	0	0
$862$	−42.1255	−1.43480
$863$	−7.52107	−0.256020	−0.128010	−	0.991773i	$-0.540859\pi$
$863$	−7.52107	−0.256020	−0.128010	+	0.991773i	$0.540859\pi$
$864$	0	0
$865$	18.5160	0.629562
$866$	−78.3015	−2.66079
$867$	0	0
$868$	0.118799	0.00403229
$869$	−79.7574	−2.70559
$870$	0	0
$871$	22.9229	0.776712
$872$	−9.31480	−0.315439
$873$	0	0
$874$	−4.94542	−0.167281
$875$	1.45789	0.0492858
$876$	0	0
$877$	10.3399	0.349155	0.174577	−	0.984643i	$-0.444144\pi$
$877$	10.3399	0.349155	0.174577	+	0.984643i	$0.444144\pi$
$878$	−38.8154	−1.30996
$879$	0	0
$880$	28.3602	0.956023
$881$	29.7263	1.00150	0.500752	−	0.865591i	$-0.333057\pi$
$881$	29.7263	1.00150	0.500752	+	0.865591i	$0.333057\pi$
$882$	0	0
$883$	33.0395	1.11187	0.555933	−	0.831227i	$-0.312361\pi$
$883$	33.0395	1.11187	0.555933	+	0.831227i	$0.312361\pi$
$884$	0	0
$885$	0	0
$886$	−60.2831	−2.02525
$887$	11.4794	0.385440	0.192720	−	0.981254i	$-0.438269\pi$
$887$	11.4794	0.385440	0.192720	+	0.981254i	$0.438269\pi$
$888$	0	0
$889$	−0.110652	−0.00371115
$890$	39.2221	1.31473
$891$	0	0
$892$	−1.78679	−0.0598262
$893$	0.713437	0.0238742
$894$	0	0
$895$	13.5579	0.453191
$896$	0.526838	0.0176004
$897$	0	0
$898$	−17.6416	−0.588707
$899$	1.63227	0.0544391
$900$	0	0
$901$	0	0
$902$	22.3303	0.743519
$903$	0	0
$904$	−6.74628	−0.224378
$905$	10.3804	0.345057
$906$	0	0
$907$	−16.4437	−0.546004	−0.273002	−	0.962014i	$-0.588016\pi$
$907$	−16.4437	−0.546004	−0.273002	+	0.962014i	$0.588016\pi$
$908$	10.0066	0.332081
$909$	0	0
$910$	−0.966379	−0.0320352
$911$	−11.9515	−0.395971	−0.197985	−	0.980205i	$-0.563440\pi$
$911$	−11.9515	−0.395971	−0.197985	+	0.980205i	$0.563440\pi$
$912$	0	0
$913$	23.4705	0.776762
$914$	36.9019	1.22061
$915$	0	0
$916$	−63.9891	−2.11426
$917$	1.32743	0.0438356
$918$	0	0
$919$	43.9032	1.44823	0.724116	−	0.689678i	$-0.242248\pi$
$919$	43.9032	1.44823	0.724116	+	0.689678i	$0.242248\pi$
$920$	−1.22307	−0.0403233
$921$	0	0
$922$	62.5840	2.06109
$923$	−27.9641	−0.920448
$924$	0	0
$925$	16.5237	0.543295
$926$	22.3145	0.733299
$927$	0	0
$928$	32.0479	1.05202
$929$	−15.2019	−0.498760	−0.249380	−	0.968406i	$-0.580227\pi$
$929$	−15.2019	−0.498760	−0.249380	+	0.968406i	$0.580227\pi$
$930$	0	0
$931$	−10.2854	−0.337089
$932$	52.2416	1.71123
$933$	0	0
$934$	69.5959	2.27725
$935$	0	0
$936$	0	0
$937$	−44.2687	−1.44620	−0.723098	−	0.690745i	$-0.757283\pi$
$937$	−44.2687	−1.44620	−0.723098	+	0.690745i	$0.757283\pi$
$938$	−2.36994	−0.0773812
$939$	0	0
$940$	1.57498	0.0513701
$941$	9.10999	0.296977	0.148489	−	0.988914i	$-0.452559\pi$
$941$	9.10999	0.296977	0.148489	+	0.988914i	$0.452559\pi$
$942$	0	0
$943$	3.08005	0.100300
$944$	41.3579	1.34608
$945$	0	0
$946$	−93.6635	−3.04526
$947$	−28.2018	−0.916435	−0.458217	−	0.888840i	$-0.651512\pi$
$947$	−28.2018	−0.916435	−0.458217	+	0.888840i	$0.651512\pi$
$948$	0	0
$949$	−17.7581	−0.576451
$950$	−8.85674	−0.287350
$951$	0	0
$952$	0	0
$953$	−29.0372	−0.940607	−0.470304	−	0.882505i	$-0.655856\pi$
$953$	−29.0372	−0.940607	−0.470304	+	0.882505i	$0.655856\pi$
$954$	0	0
$955$	14.1674	0.458447
$956$	−14.6124	−0.472597
$957$	0	0
$958$	59.6631	1.92763
$959$	−1.67778	−0.0541784
$960$	0	0
$961$	−30.8291	−0.994486
$962$	−29.7318	−0.958592
$963$	0	0
$964$	61.3537	1.97607
$965$	33.1616	1.06751
$966$	0	0
$967$	40.3570	1.29779	0.648897	−	0.760876i	$-0.275231\pi$
$967$	40.3570	1.29779	0.648897	+	0.760876i	$0.275231\pi$
$968$	11.3309	0.364187
$969$	0	0
$970$	−10.7115	−0.343924
$971$	18.1373	0.582053	0.291026	−	0.956715i	$-0.406003\pi$
$971$	18.1373	0.582053	0.291026	+	0.956715i	$0.406003\pi$
$972$	0	0
$973$	2.85300	0.0914631
$974$	57.8534	1.85374
$975$	0	0
$976$	−14.7979	−0.473668
$977$	41.5209	1.32837	0.664185	−	0.747568i	$-0.268779\pi$
$977$	41.5209	1.32837	0.664185	+	0.747568i	$0.268779\pi$
$978$	0	0
$979$	−75.4361	−2.41095
$980$	−22.7059	−0.725313
$981$	0	0
$982$	−41.6058	−1.32769
$983$	6.41860	0.204722	0.102361	−	0.994747i	$-0.467360\pi$
$983$	6.41860	0.204722	0.102361	+	0.994747i	$0.467360\pi$
$984$	0	0
$985$	−21.9271	−0.698656
$986$	0	0
$987$	0	0
$988$	8.44098	0.268543
$989$	−12.9191	−0.410804
$990$	0	0
$991$	−13.5445	−0.430255	−0.215128	−	0.976586i	$-0.569017\pi$
$991$	−13.5445	−0.430255	−0.215128	+	0.976586i	$0.569017\pi$
$992$	3.35584	0.106548
$993$	0	0
$994$	2.89113	0.0917012
$995$	−7.43577	−0.235730
$996$	0	0
$997$	−9.01169	−0.285403	−0.142702	−	0.989766i	$-0.545579\pi$
$997$	−9.01169	−0.285403	−0.142702	+	0.989766i	$0.545579\pi$
$998$	−61.8378	−1.95744
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7803.2.a.bs.1.8		10
3.2	odd	2		7803.2.a.br.1.3		10
17.8	even	8		459.2.f.b.217.3	yes	20
17.15	even	8		459.2.f.b.55.8	yes	20
17.16	even	2		7803.2.a.br.1.8		10
51.8	odd	8		459.2.f.b.217.8	yes	20
51.32	odd	8		459.2.f.b.55.3	✓	20
51.50	odd	2	inner	7803.2.a.bs.1.3		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
459.2.f.b.55.3	✓	20	51.32	odd	8
459.2.f.b.55.8	yes	20	17.15	even	8
459.2.f.b.217.3	yes	20	17.8	even	8
459.2.f.b.217.8	yes	20	51.8	odd	8
7803.2.a.br.1.3		10	3.2	odd	2
7803.2.a.br.1.8		10	17.16	even	2
7803.2.a.bs.1.3		10	51.50	odd	2	inner
7803.2.a.bs.1.8		10	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+2.06212 q^{2} +2.25232 q^{4} +1.44351 q^{5} -0.127581 q^{7} +0.520322 q^{8} +2.97669 q^{10} -5.72509 q^{11} +2.54466 q^{13} -0.263086 q^{14} -3.43168 q^{16} +1.47276 q^{19} +3.25126 q^{20} -11.8058 q^{22} -1.62839 q^{23} -2.91627 q^{25} +5.24738 q^{26} -0.287353 q^{28} -3.94816 q^{29} -0.413424 q^{31} -8.11718 q^{32} -0.184164 q^{35} -5.66603 q^{37} +3.03701 q^{38} +0.751092 q^{40} -1.89147 q^{41} +7.93369 q^{43} -12.8948 q^{44} -3.35792 q^{46} +0.484421 q^{47} -6.98372 q^{49} -6.01369 q^{50} +5.73140 q^{52} -9.05465 q^{53} -8.26423 q^{55} -0.0663830 q^{56} -8.14157 q^{58} -12.0518 q^{59} +4.31213 q^{61} -0.852529 q^{62} -9.87520 q^{64} +3.67325 q^{65} +9.00823 q^{67} -0.379768 q^{70} -10.9893 q^{71} -6.97856 q^{73} -11.6840 q^{74} +3.31714 q^{76} +0.730409 q^{77} +13.9312 q^{79} -4.95368 q^{80} -3.90044 q^{82} -4.09960 q^{83} +16.3602 q^{86} -2.97889 q^{88} +13.1764 q^{89} -0.324649 q^{91} -3.66765 q^{92} +0.998933 q^{94} +2.12595 q^{95} -3.59845 q^{97} -14.4013 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 12 q^{4} + 2 q^{5} - 22 q^{11} + 16 q^{16} - 10 q^{19} - 8 q^{20} - 28 q^{23} + 8 q^{25} - 6 q^{29} - 28 q^{41} - 8 q^{43} - 76 q^{44} + 2 q^{49} - 4 q^{52} + 42 q^{55} + 64 q^{56} - 52 q^{62} - 36 q^{64}+ \cdots - 38 q^{95}+O(q^{100}) \) 10 * q + 12 * q^4 + 2 * q^5 - 22 * q^11 + 16 * q^16 - 10 * q^19 - 8 * q^20 - 28 * q^23 + 8 * q^25 - 6 * q^29 - 28 * q^41 - 8 * q^43 - 76 * q^44 + 2 * q^49 - 4 * q^52 + 42 * q^55 + 64 * q^56 - 52 * q^62 - 36 * q^64 - 32 * q^65 + 18 * q^67 - 72 * q^70 + 50 * q^71 - 56 * q^74 - 28 * q^76 - 80 * q^80 - 56 * q^92 - 20 * q^94 - 38 * q^95

Introduction

L-functions

Modular forms

Varieties

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