LMFDB - Embedded newform 3969.2.a.n.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3969,2,Mod(1,3969)] 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("3969.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3969, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 3969 = 3^{4} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3969.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,0,0,6,-3,0,0,-9,0,-3,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$31.6926245622$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.621.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 6x - 3 $ x^3 - 6*x - 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 567)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-2.14510$ of defining polynomial
Character	$\chi$	$=$	3969.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.14510 q^{2} +2.60147 q^{4} -3.74657 q^{5} +1.29021 q^{8} -8.03677 q^{10} -0.746568 q^{11} +6.03677 q^{13} -2.43531 q^{16} +0.543637 q^{17} +1.20293 q^{19} -9.74657 q^{20} -1.60147 q^{22} -7.49314 q^{23} +9.03677 q^{25} +12.9495 q^{26} +8.03677 q^{29} -2.00000 q^{31} -7.80440 q^{32} +1.16616 q^{34} +5.00000 q^{37} +2.58041 q^{38} -4.83384 q^{40} +2.79707 q^{41} +9.83384 q^{43} -1.94217 q^{44} -16.0735 q^{46} +4.29021 q^{47} +19.3848 q^{50} +15.7045 q^{52} +2.45636 q^{53} +2.79707 q^{55} +17.2397 q^{58} +14.5804 q^{59} +9.23970 q^{61} -4.29021 q^{62} -11.8706 q^{64} -22.6172 q^{65} +10.2397 q^{67} +1.41425 q^{68} -8.23970 q^{71} -10.0368 q^{73} +10.7255 q^{74} +3.12938 q^{76} +10.2397 q^{79} +9.12405 q^{80} +6.00000 q^{82} -2.79707 q^{83} -2.03677 q^{85} +21.0946 q^{86} -0.963226 q^{88} -6.54364 q^{89} -19.4931 q^{92} +9.20293 q^{94} -4.50686 q^{95} -5.20293 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 6 q^{4} - 3 q^{5} - 9 q^{8} - 3 q^{10} + 6 q^{11} - 3 q^{13} + 12 q^{16} - 3 q^{17} - 21 q^{20} - 3 q^{22} - 6 q^{23} + 6 q^{25} + 27 q^{26} + 3 q^{29} - 6 q^{31} - 18 q^{32} + 21 q^{34} + 15 q^{37}+ \cdots - 12 q^{97}+O(q^{100}) $ 3 * q + 6 * q^4 - 3 * q^5 - 9 * q^8 - 3 * q^10 + 6 * q^11 - 3 * q^13 + 12 * q^16 - 3 * q^17 - 21 * q^20 - 3 * q^22 - 6 * q^23 + 6 * q^25 + 27 * q^26 + 3 * q^29 - 6 * q^31 - 18 * q^32 + 21 * q^34 + 15 * q^37 - 18 * q^38 + 3 * q^40 + 12 * q^41 + 12 * q^43 - 3 * q^44 - 6 * q^46 + 27 * q^50 - 9 * q^52 + 12 * q^53 + 12 * q^55 + 27 * q^58 + 18 * q^59 + 3 * q^61 + 3 * q^64 - 21 * q^65 + 6 * q^67 - 39 * q^68 - 9 * q^73 + 48 * q^76 + 6 * q^79 - 3 * q^80 + 18 * q^82 - 12 * q^83 + 15 * q^85 + 45 * q^86 - 24 * q^88 - 15 * q^89 - 42 * q^92 + 24 * q^94 - 30 * q^95 - 12 * 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 $

$2$	2.14510	1.51682	0.758408	−	0.651780i	$-0.225977\pi$
$2$	2.14510	1.51682	0.758408	+	0.651780i	$0.225977\pi$
$3$	0	0
$3$	0	0
$4$	2.60147	1.30073
$5$	−3.74657	−1.67552	−0.837758	−	0.546041i	$-0.816134\pi$
$5$	−3.74657	−1.67552	−0.837758	+	0.546041i	$0.816134\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	1.29021	0.456156
$9$	0	0
$10$	−8.03677	−2.54145
$11$	−0.746568	−0.225099	−0.112549	−	0.993646i	$-0.535902\pi$
$11$	−0.746568	−0.225099	−0.112549	+	0.993646i	$0.535902\pi$
$12$	0	0
$13$	6.03677	1.67430	0.837150	−	0.546974i	$-0.184220\pi$
$13$	6.03677	1.67430	0.837150	+	0.546974i	$0.184220\pi$
$14$	0	0
$15$	0	0
$16$	−2.43531	−0.608827
$17$	0.543637	0.131851	0.0659257	−	0.997825i	$-0.479000\pi$
$17$	0.543637	0.131851	0.0659257	+	0.997825i	$0.479000\pi$
$18$	0	0
$19$	1.20293	0.275971	0.137986	−	0.990434i	$-0.455937\pi$
$19$	1.20293	0.275971	0.137986	+	0.990434i	$0.455937\pi$
$20$	−9.74657	−2.17940
$21$	0	0
$22$	−1.60147	−0.341434
$23$	−7.49314	−1.56243	−0.781213	−	0.624264i	$-0.785399\pi$
$23$	−7.49314	−1.56243	−0.781213	+	0.624264i	$0.785399\pi$
$24$	0	0
$25$	9.03677	1.80735
$26$	12.9495	2.53961
$27$	0	0
$28$	0	0
$29$	8.03677	1.49239	0.746196	−	0.665727i	$-0.231878\pi$
$29$	8.03677	1.49239	0.746196	+	0.665727i	$0.231878\pi$
$30$	0	0
$31$	−2.00000	−0.359211	−0.179605	−	0.983739i	$-0.557482\pi$
$31$	−2.00000	−0.359211	−0.179605	+	0.983739i	$0.557482\pi$
$32$	−7.80440	−1.37964
$33$	0	0
$34$	1.16616	0.199994
$35$	0	0
$36$	0	0
$37$	5.00000	0.821995	0.410997	−	0.911636i	$-0.365181\pi$
$37$	5.00000	0.821995	0.410997	+	0.911636i	$0.365181\pi$
$38$	2.58041	0.418598
$39$	0	0
$40$	−4.83384	−0.764298
$41$	2.79707	0.436829	0.218414	−	0.975856i	$-0.429912\pi$
$41$	2.79707	0.436829	0.218414	+	0.975856i	$0.429912\pi$
$42$	0	0
$43$	9.83384	1.49965	0.749823	−	0.661638i	$-0.230138\pi$
$43$	9.83384	1.49965	0.749823	+	0.661638i	$0.230138\pi$
$44$	−1.94217	−0.292793
$45$	0	0
$46$	−16.0735	−2.36992
$47$	4.29021	0.625791	0.312895	−	0.949788i	$-0.398701\pi$
$47$	4.29021	0.625791	0.312895	+	0.949788i	$0.398701\pi$
$48$	0	0
$49$	0	0
$50$	19.3848	2.74143
$51$	0	0
$52$	15.7045	2.17782
$53$	2.45636	0.337407	0.168704	−	0.985667i	$-0.446042\pi$
$53$	2.45636	0.337407	0.168704	+	0.985667i	$0.446042\pi$
$54$	0	0
$55$	2.79707	0.377157
$56$	0	0
$57$	0	0
$58$	17.2397	2.26368
$59$	14.5804	1.89821	0.949104	−	0.314963i	$-0.101992\pi$
$59$	14.5804	1.89821	0.949104	+	0.314963i	$0.101992\pi$
$60$	0	0
$61$	9.23970	1.18302	0.591511	−	0.806297i	$-0.298531\pi$
$61$	9.23970	1.18302	0.591511	+	0.806297i	$0.298531\pi$
$62$	−4.29021	−0.544857
$63$	0	0
$64$	−11.8706	−1.48383
$65$	−22.6172	−2.80532
$66$	0	0
$67$	10.2397	1.25098	0.625490	−	0.780233i	$-0.284899\pi$
$67$	10.2397	1.25098	0.625490	+	0.780233i	$0.284899\pi$
$68$	1.41425	0.171503
$69$	0	0
$70$	0	0
$71$	−8.23970	−0.977873	−0.488937	−	0.872319i	$-0.662615\pi$
$71$	−8.23970	−0.977873	−0.488937	+	0.872319i	$0.662615\pi$
$72$	0	0
$73$	−10.0368	−1.17472	−0.587358	−	0.809327i	$-0.699832\pi$
$73$	−10.0368	−1.17472	−0.587358	+	0.809327i	$0.699832\pi$
$74$	10.7255	1.24682
$75$	0	0
$76$	3.12938	0.358965
$77$	0	0
$78$	0	0
$79$	10.2397	1.15206	0.576028	−	0.817430i	$-0.304602\pi$
$79$	10.2397	1.15206	0.576028	+	0.817430i	$0.304602\pi$
$80$	9.12405	1.02010
$81$	0	0
$82$	6.00000	0.662589
$83$	−2.79707	−0.307018	−0.153509	−	0.988147i	$-0.549057\pi$
$83$	−2.79707	−0.307018	−0.153509	+	0.988147i	$0.549057\pi$
$84$	0	0
$85$	−2.03677	−0.220919
$86$	21.0946	2.27469
$87$	0	0
$88$	−0.963226	−0.102680
$89$	−6.54364	−0.693624	−0.346812	−	0.937935i	$-0.612736\pi$
$89$	−6.54364	−0.693624	−0.346812	+	0.937935i	$0.612736\pi$
$90$	0	0
$91$	0	0
$92$	−19.4931	−2.03230
$93$	0	0
$94$	9.20293	0.949210
$95$	−4.50686	−0.462394
$96$	0	0
$97$	−5.20293	−0.528278	−0.264139	−	0.964485i	$-0.585088\pi$
$97$	−5.20293	−0.528278	−0.264139	+	0.964485i	$0.585088\pi$
$98$	0	0
$99$	0	0
$100$	23.5089	2.35089
$101$	4.50686	0.448450	0.224225	−	0.974537i	$-0.428015\pi$
$101$	4.50686	0.448450	0.224225	+	0.974537i	$0.428015\pi$
$102$	0	0
$103$	10.8706	1.07111	0.535557	−	0.844499i	$-0.320102\pi$
$103$	10.8706	1.07111	0.535557	+	0.844499i	$0.320102\pi$
$104$	7.78868	0.763743
$105$	0	0
$106$	5.26915	0.511785
$107$	−3.32698	−0.321631	−0.160816	−	0.986984i	$-0.551412\pi$
$107$	−3.32698	−0.321631	−0.160816	+	0.986984i	$0.551412\pi$
$108$	0	0
$109$	−5.63091	−0.539343	−0.269672	−	0.962952i	$-0.586915\pi$
$109$	−5.63091	−0.539343	−0.269672	+	0.962952i	$0.586915\pi$
$110$	6.00000	0.572078
$111$	0	0
$112$	0	0
$113$	2.59414	0.244036	0.122018	−	0.992528i	$-0.461063\pi$
$113$	2.59414	0.244036	0.122018	+	0.992528i	$0.461063\pi$
$114$	0	0
$115$	28.0735	2.61787
$116$	20.9074	1.94120
$117$	0	0
$118$	31.2765	2.87923
$119$	0	0
$120$	0	0
$121$	−10.4426	−0.949331
$122$	19.8201	1.79443
$123$	0	0
$124$	−5.20293	−0.467237
$125$	−15.1240	−1.35274
$126$	0	0
$127$	9.83384	0.872612	0.436306	−	0.899798i	$-0.356286\pi$
$127$	9.83384	0.872612	0.436306	+	0.899798i	$0.356286\pi$
$128$	−9.85490	−0.871058
$129$	0	0
$130$	−48.5162	−4.25515
$131$	−5.37748	−0.469833	−0.234916	−	0.972016i	$-0.575482\pi$
$131$	−5.37748	−0.469833	−0.234916	+	0.972016i	$0.575482\pi$
$132$	0	0
$133$	0	0
$134$	21.9652	1.89751
$135$	0	0
$136$	0.701404	0.0601449
$137$	10.4931	0.896489	0.448245	−	0.893911i	$-0.352049\pi$
$137$	10.4931	0.896489	0.448245	+	0.893911i	$0.352049\pi$
$138$	0	0
$139$	6.79707	0.576520	0.288260	−	0.957552i	$-0.406923\pi$
$139$	6.79707	0.576520	0.288260	+	0.957552i	$0.406923\pi$
$140$	0	0
$141$	0	0
$142$	−17.6750	−1.48325
$143$	−4.50686	−0.376883
$144$	0	0
$145$	−30.1103	−2.50053
$146$	−21.5299	−1.78183
$147$	0	0
$148$	13.0073	1.06920
$149$	1.50686	0.123447	0.0617235	−	0.998093i	$-0.480340\pi$
$149$	1.50686	0.123447	0.0617235	+	0.998093i	$0.480340\pi$
$150$	0	0
$151$	−6.23970	−0.507780	−0.253890	−	0.967233i	$-0.581710\pi$
$151$	−6.23970	−0.507780	−0.253890	+	0.967233i	$0.581710\pi$
$152$	1.55203	0.125886
$153$	0	0
$154$	0	0
$155$	7.49314	0.601863
$156$	0	0
$157$	−20.5162	−1.63737	−0.818685	−	0.574243i	$-0.805296\pi$
$157$	−20.5162	−1.63737	−0.818685	+	0.574243i	$0.805296\pi$
$158$	21.9652	1.74746
$159$	0	0
$160$	29.2397	2.31160
$161$	0	0
$162$	0	0
$163$	−12.2397	−0.958688	−0.479344	−	0.877627i	$-0.659125\pi$
$163$	−12.2397	−0.958688	−0.479344	+	0.877627i	$0.659125\pi$
$164$	7.27648	0.568198
$165$	0	0
$166$	−6.00000	−0.465690
$167$	14.7971	1.14503	0.572516	−	0.819894i	$-0.305968\pi$
$167$	14.7971	1.14503	0.572516	+	0.819894i	$0.305968\pi$
$168$	0	0
$169$	23.4426	1.80328
$170$	−4.36909	−0.335094
$171$	0	0
$172$	25.5824	1.95064
$173$	−1.82012	−0.138381	−0.0691904	−	0.997603i	$-0.522042\pi$
$173$	−1.82012	−0.138381	−0.0691904	+	0.997603i	$0.522042\pi$
$174$	0	0
$175$	0	0
$176$	1.81812	0.137046
$177$	0	0
$178$	−14.0368	−1.05210
$179$	1.08727	0.0812667	0.0406333	−	0.999174i	$-0.487062\pi$
$179$	1.08727	0.0812667	0.0406333	+	0.999174i	$0.487062\pi$
$180$	0	0
$181$	−2.00000	−0.148659	−0.0743294	−	0.997234i	$-0.523682\pi$
$181$	−2.00000	−0.148659	−0.0743294	+	0.997234i	$0.523682\pi$
$182$	0	0
$183$	0	0
$184$	−9.66769	−0.712711
$185$	−18.7328	−1.37727
$186$	0	0
$187$	−0.405862	−0.0296796
$188$	11.1608	0.813987
$189$	0	0
$190$	−9.66769	−0.701368
$191$	10.8201	0.782916	0.391458	−	0.920196i	$-0.371971\pi$
$191$	10.8201	0.782916	0.391458	+	0.920196i	$0.371971\pi$
$192$	0	0
$193$	9.63091	0.693248	0.346624	−	0.938004i	$-0.387328\pi$
$193$	9.63091	0.693248	0.346624	+	0.938004i	$0.387328\pi$
$194$	−11.1608	−0.801300
$195$	0	0
$196$	0	0
$197$	19.4794	1.38785	0.693925	−	0.720047i	$-0.255880\pi$
$197$	19.4794	1.38785	0.693925	+	0.720047i	$0.255880\pi$
$198$	0	0
$199$	16.8706	1.19593	0.597963	−	0.801524i	$-0.295977\pi$
$199$	16.8706	1.19593	0.597963	+	0.801524i	$0.295977\pi$
$200$	11.6593	0.824437
$201$	0	0
$202$	9.66769	0.680216
$203$	0	0
$204$	0	0
$205$	−10.4794	−0.731914
$206$	23.3186	1.62468
$207$	0	0
$208$	−14.7014	−1.01936
$209$	−0.898070	−0.0621208
$210$	0	0
$211$	3.83384	0.263933	0.131966	−	0.991254i	$-0.457871\pi$
$211$	3.83384	0.263933	0.131966	+	0.991254i	$0.457871\pi$
$212$	6.39014	0.438877
$213$	0	0
$214$	−7.13671	−0.487856
$215$	−36.8432	−2.51268
$216$	0	0
$217$	0	0
$218$	−12.0789	−0.818085
$219$	0	0
$220$	7.27648	0.490580
$221$	3.28181	0.220759
$222$	0	0
$223$	−12.4794	−0.835683	−0.417842	−	0.908520i	$-0.637213\pi$
$223$	−12.4794	−0.835683	−0.417842	+	0.908520i	$0.637213\pi$
$224$	0	0
$225$	0	0
$226$	5.56469	0.370158
$227$	21.8569	1.45069	0.725346	−	0.688384i	$-0.241680\pi$
$227$	21.8569	1.45069	0.725346	+	0.688384i	$0.241680\pi$
$228$	0	0
$229$	−19.2397	−1.27140	−0.635698	−	0.771938i	$-0.719288\pi$
$229$	−19.2397	−1.27140	−0.635698	+	0.771938i	$0.719288\pi$
$230$	60.2206	3.97083
$231$	0	0
$232$	10.3691	0.680764
$233$	−11.4564	−0.750531	−0.375266	−	0.926917i	$-0.622449\pi$
$233$	−11.4564	−0.750531	−0.375266	+	0.926917i	$0.622449\pi$
$234$	0	0
$235$	−16.0735	−1.04852
$236$	37.9304	2.46906
$237$	0	0
$238$	0	0
$239$	−4.16616	−0.269486	−0.134743	−	0.990881i	$-0.543021\pi$
$239$	−4.16616	−0.269486	−0.134743	+	0.990881i	$0.543021\pi$
$240$	0	0
$241$	−13.2397	−0.852844	−0.426422	−	0.904524i	$-0.640226\pi$
$241$	−13.2397	−0.852844	−0.426422	+	0.904524i	$0.640226\pi$
$242$	−22.4005	−1.43996
$243$	0	0
$244$	24.0368	1.53880
$245$	0	0
$246$	0	0
$247$	7.26182	0.462059
$248$	−2.58041	−0.163856
$249$	0	0
$250$	−32.4426	−2.05185
$251$	12.8706	0.812386	0.406193	−	0.913787i	$-0.366856\pi$
$251$	12.8706	0.812386	0.406193	+	0.913787i	$0.366856\pi$
$252$	0	0
$253$	5.59414	0.351700
$254$	21.0946	1.32359
$255$	0	0
$256$	2.60147	0.162592
$257$	−0.137775	−0.00859416	−0.00429708	−	0.999991i	$-0.501368\pi$
$257$	−0.137775	−0.00859416	−0.00429708	+	0.999991i	$0.501368\pi$
$258$	0	0
$259$	0	0
$260$	−58.8378	−3.64897
$261$	0	0
$262$	−11.5352	−0.712650
$263$	11.2534	0.693916	0.346958	−	0.937881i	$-0.387215\pi$
$263$	11.2534	0.693916	0.346958	+	0.937881i	$0.387215\pi$
$264$	0	0
$265$	−9.20293	−0.565332
$266$	0	0
$267$	0	0
$268$	26.6382	1.62719
$269$	−5.45636	−0.332680	−0.166340	−	0.986068i	$-0.553195\pi$
$269$	−5.45636	−0.332680	−0.166340	+	0.986068i	$0.553195\pi$
$270$	0	0
$271$	3.12938	0.190097	0.0950483	−	0.995473i	$-0.469699\pi$
$271$	3.12938	0.190097	0.0950483	+	0.995473i	$0.469699\pi$
$272$	−1.32392	−0.0802747
$273$	0	0
$274$	22.5089	1.35981
$275$	−6.74657	−0.406833
$276$	0	0
$277$	−8.63091	−0.518581	−0.259291	−	0.965799i	$-0.583489\pi$
$277$	−8.63091	−0.518581	−0.259291	+	0.965799i	$0.583489\pi$
$278$	14.5804	0.874475
$279$	0	0
$280$	0	0
$281$	−13.0735	−0.779902	−0.389951	−	0.920836i	$-0.627508\pi$
$281$	−13.0735	−0.779902	−0.389951	+	0.920836i	$0.627508\pi$
$282$	0	0
$283$	13.6088	0.808959	0.404479	−	0.914547i	$-0.367453\pi$
$283$	13.6088	0.808959	0.404479	+	0.914547i	$0.367453\pi$
$284$	−21.4353	−1.27195
$285$	0	0
$286$	−9.66769	−0.571662
$287$	0	0
$288$	0	0
$289$	−16.7045	−0.982615
$290$	−64.5897	−3.79284
$291$	0	0
$292$	−26.1103	−1.52799
$293$	−10.8338	−0.632920	−0.316460	−	0.948606i	$-0.602494\pi$
$293$	−10.8338	−0.632920	−0.316460	+	0.948606i	$0.602494\pi$
$294$	0	0
$295$	−54.6265	−3.18048
$296$	6.45103	0.374958
$297$	0	0
$298$	3.23238	0.187247
$299$	−45.2344	−2.61597
$300$	0	0
$301$	0	0
$302$	−13.3848	−0.770209
$303$	0	0
$304$	−2.92951	−0.168019
$305$	−34.6172	−1.98217
$306$	0	0
$307$	−18.0735	−1.03151	−0.515756	−	0.856736i	$-0.672489\pi$
$307$	−18.0735	−1.03151	−0.515756	+	0.856736i	$0.672489\pi$
$308$	0	0
$309$	0	0
$310$	16.0735	0.912916
$311$	−4.29021	−0.243275	−0.121638	−	0.992575i	$-0.538815\pi$
$311$	−4.29021	−0.243275	−0.121638	+	0.992575i	$0.538815\pi$
$312$	0	0
$313$	−5.55736	−0.314121	−0.157060	−	0.987589i	$-0.550202\pi$
$313$	−5.55736	−0.314121	−0.157060	+	0.987589i	$0.550202\pi$
$314$	−44.0093	−2.48359
$315$	0	0
$316$	26.6382	1.49852
$317$	−27.5299	−1.54623	−0.773117	−	0.634264i	$-0.781303\pi$
$317$	−27.5299	−1.54623	−0.773117	+	0.634264i	$0.781303\pi$
$318$	0	0
$319$	−6.00000	−0.335936
$320$	44.4741	2.48618
$321$	0	0
$322$	0	0
$323$	0.653958	0.0363872
$324$	0	0
$325$	54.5530	3.02605
$326$	−26.2554	−1.45415
$327$	0	0
$328$	3.60879	0.199262
$329$	0	0
$330$	0	0
$331$	16.1471	0.887525	0.443762	−	0.896145i	$-0.353643\pi$
$331$	16.1471	0.887525	0.443762	+	0.896145i	$0.353643\pi$
$332$	−7.27648	−0.399349
$333$	0	0
$334$	31.7412	1.73680
$335$	−38.3638	−2.09604
$336$	0	0
$337$	−1.11032	−0.0604830	−0.0302415	−	0.999543i	$-0.509628\pi$
$337$	−1.11032	−0.0604830	−0.0302415	+	0.999543i	$0.509628\pi$
$338$	50.2869	2.73524
$339$	0	0
$340$	−5.29860	−0.287357
$341$	1.49314	0.0808579
$342$	0	0
$343$	0	0
$344$	12.6877	0.684074
$345$	0	0
$346$	−3.90433	−0.209898
$347$	−28.8201	−1.54714	−0.773572	−	0.633708i	$-0.781532\pi$
$347$	−28.8201	−1.54714	−0.773572	+	0.633708i	$0.781532\pi$
$348$	0	0
$349$	23.2765	1.24596	0.622981	−	0.782237i	$-0.285922\pi$
$349$	23.2765	1.24596	0.622981	+	0.782237i	$0.285922\pi$
$350$	0	0
$351$	0	0
$352$	5.82651	0.310554
$353$	−2.36375	−0.125810	−0.0629049	−	0.998020i	$-0.520036\pi$
$353$	−2.36375	−0.125810	−0.0629049	+	0.998020i	$0.520036\pi$
$354$	0	0
$355$	30.8706	1.63844
$356$	−17.0230	−0.902220
$357$	0	0
$358$	2.33231	0.123267
$359$	−3.32698	−0.175591	−0.0877956	−	0.996139i	$-0.527982\pi$
$359$	−3.32698	−0.175591	−0.0877956	+	0.996139i	$0.527982\pi$
$360$	0	0
$361$	−17.5530	−0.923840
$362$	−4.29021	−0.225488
$363$	0	0
$364$	0	0
$365$	37.6035	1.96825
$366$	0	0
$367$	−5.66769	−0.295851	−0.147925	−	0.988999i	$-0.547260\pi$
$367$	−5.66769	−0.295851	−0.147925	+	0.988999i	$0.547260\pi$
$368$	18.2481	0.951248
$369$	0	0
$370$	−40.1839	−2.08906
$371$	0	0
$372$	0	0
$373$	23.5162	1.21762	0.608811	−	0.793315i	$-0.291647\pi$
$373$	23.5162	1.21762	0.608811	+	0.793315i	$0.291647\pi$
$374$	−0.870616	−0.0450185
$375$	0	0
$376$	5.53525	0.285459
$377$	48.5162	2.49871
$378$	0	0
$379$	36.7927	1.88991	0.944956	−	0.327197i	$-0.106104\pi$
$379$	36.7927	1.88991	0.944956	+	0.327197i	$0.106104\pi$
$380$	−11.7245	−0.601452
$381$	0	0
$382$	23.2103	1.18754
$383$	−26.9863	−1.37893	−0.689467	−	0.724317i	$-0.742155\pi$
$383$	−26.9863	−1.37893	−0.689467	+	0.724317i	$0.742155\pi$
$384$	0	0
$385$	0	0
$386$	20.6593	1.05153
$387$	0	0
$388$	−13.5352	−0.687148
$389$	−8.98627	−0.455622	−0.227811	−	0.973705i	$-0.573157\pi$
$389$	−8.98627	−0.455622	−0.227811	+	0.973705i	$0.573157\pi$
$390$	0	0
$391$	−4.07355	−0.206008
$392$	0	0
$393$	0	0
$394$	41.7853	2.10511
$395$	−38.3638	−1.93029
$396$	0	0
$397$	−28.5015	−1.43045	−0.715225	−	0.698894i	$-0.753676\pi$
$397$	−28.5015	−1.43045	−0.715225	+	0.698894i	$0.753676\pi$
$398$	36.1892	1.81400
$399$	0	0
$400$	−22.0073	−1.10037
$401$	7.50686	0.374875	0.187437	−	0.982277i	$-0.439982\pi$
$401$	7.50686	0.374875	0.187437	+	0.982277i	$0.439982\pi$
$402$	0	0
$403$	−12.0735	−0.601426
$404$	11.7245	0.583313
$405$	0	0
$406$	0	0
$407$	−3.73284	−0.185030
$408$	0	0
$409$	27.2397	1.34692	0.673458	−	0.739225i	$-0.264808\pi$
$409$	27.2397	1.34692	0.673458	+	0.739225i	$0.264808\pi$
$410$	−22.4794	−1.11018
$411$	0	0
$412$	28.2795	1.39323
$413$	0	0
$414$	0	0
$415$	10.4794	0.514414
$416$	−47.1134	−2.30992
$417$	0	0
$418$	−1.92645	−0.0942259
$419$	−19.6823	−0.961545	−0.480773	−	0.876845i	$-0.659644\pi$
$419$	−19.6823	−0.961545	−0.480773	+	0.876845i	$0.659644\pi$
$420$	0	0
$421$	21.4794	1.04684	0.523421	−	0.852074i	$-0.324655\pi$
$421$	21.4794	1.04684	0.523421	+	0.852074i	$0.324655\pi$
$422$	8.22399	0.400337
$423$	0	0
$424$	3.16921	0.153911
$425$	4.91273	0.238302
$426$	0	0
$427$	0	0
$428$	−8.65502	−0.418356
$429$	0	0
$430$	−79.0324	−3.81128
$431$	−2.17455	−0.104744	−0.0523722	−	0.998628i	$-0.516678\pi$
$431$	−2.17455	−0.104744	−0.0523722	+	0.998628i	$0.516678\pi$
$432$	0	0
$433$	−35.3133	−1.69705	−0.848523	−	0.529158i	$-0.822508\pi$
$433$	−35.3133	−1.69705	−0.848523	+	0.529158i	$0.822508\pi$
$434$	0	0
$435$	0	0
$436$	−14.6486	−0.701542
$437$	−9.01373	−0.431185
$438$	0	0
$439$	19.2029	0.916506	0.458253	−	0.888822i	$-0.348475\pi$
$439$	19.2029	0.916506	0.458253	+	0.888822i	$0.348475\pi$
$440$	3.60879	0.172042
$441$	0	0
$442$	7.03983	0.334851
$443$	5.31859	0.252694	0.126347	−	0.991986i	$-0.459675\pi$
$443$	5.31859	0.252694	0.126347	+	0.991986i	$0.459675\pi$
$444$	0	0
$445$	24.5162	1.16218
$446$	−26.7696	−1.26758
$447$	0	0
$448$	0	0
$449$	−7.36909	−0.347769	−0.173884	−	0.984766i	$-0.555632\pi$
$449$	−7.36909	−0.347769	−0.173884	+	0.984766i	$0.555632\pi$
$450$	0	0
$451$	−2.08820	−0.0983296
$452$	6.74856	0.317426
$453$	0	0
$454$	46.8853	2.20043
$455$	0	0
$456$	0	0
$457$	−1.00000	−0.0467780	−0.0233890	−	0.999726i	$-0.507446\pi$
$457$	−1.00000	−0.0467780	−0.0233890	+	0.999726i	$0.507446\pi$
$458$	−41.2711	−1.92847
$459$	0	0
$460$	73.0324	3.40515
$461$	30.4648	1.41889	0.709443	−	0.704763i	$-0.248947\pi$
$461$	30.4648	1.41889	0.709443	+	0.704763i	$0.248947\pi$
$462$	0	0
$463$	−27.5015	−1.27810	−0.639052	−	0.769163i	$-0.720673\pi$
$463$	−27.5015	−1.27810	−0.639052	+	0.769163i	$0.720673\pi$
$464$	−19.5720	−0.908608
$465$	0	0
$466$	−24.5751	−1.13842
$467$	39.1755	1.81282	0.906412	−	0.422394i	$-0.138810\pi$
$467$	39.1755	1.81282	0.906412	+	0.422394i	$0.138810\pi$
$468$	0	0
$469$	0	0
$470$	−34.4794	−1.59042
$471$	0	0
$472$	18.8117	0.865880
$473$	−7.34163	−0.337569
$474$	0	0
$475$	10.8706	0.498778
$476$	0	0
$477$	0	0
$478$	−8.93684	−0.408761
$479$	20.9863	0.958887	0.479444	−	0.877573i	$-0.340838\pi$
$479$	20.9863	0.958887	0.479444	+	0.877573i	$0.340838\pi$
$480$	0	0
$481$	30.1839	1.37627
$482$	−28.4005	−1.29361
$483$	0	0
$484$	−27.1662	−1.23483
$485$	19.4931	0.885138
$486$	0	0
$487$	28.6456	1.29805	0.649027	−	0.760765i	$-0.275176\pi$
$487$	28.6456	1.29805	0.649027	+	0.760765i	$0.275176\pi$
$488$	11.9211	0.539644
$489$	0	0
$490$	0	0
$491$	−32.1471	−1.45078	−0.725389	−	0.688339i	$-0.758340\pi$
$491$	−32.1471	−1.45078	−0.725389	+	0.688339i	$0.758340\pi$
$492$	0	0
$493$	4.36909	0.196774
$494$	15.5774	0.700858
$495$	0	0
$496$	4.87062	0.218697
$497$	0	0
$498$	0	0
$499$	−19.2618	−0.862278	−0.431139	−	0.902286i	$-0.641888\pi$
$499$	−19.2618	−0.862278	−0.431139	+	0.902286i	$0.641888\pi$
$500$	−39.3447	−1.75955
$501$	0	0
$502$	27.6088	1.23224
$503$	−36.2206	−1.61500	−0.807499	−	0.589869i	$-0.799180\pi$
$503$	−36.2206	−1.61500	−0.807499	+	0.589869i	$0.799180\pi$
$504$	0	0
$505$	−16.8853	−0.751385
$506$	12.0000	0.533465
$507$	0	0
$508$	25.5824	1.13504
$509$	−8.98627	−0.398310	−0.199155	−	0.979968i	$-0.563820\pi$
$509$	−8.98627	−0.398310	−0.199155	+	0.979968i	$0.563820\pi$
$510$	0	0
$511$	0	0
$512$	25.2902	1.11768
$513$	0	0
$514$	−0.295541	−0.0130358
$515$	−40.7275	−1.79467
$516$	0	0
$517$	−3.20293	−0.140865
$518$	0	0
$519$	0	0
$520$	−29.1808	−1.27966
$521$	−17.5667	−0.769610	−0.384805	−	0.922998i	$-0.625731\pi$
$521$	−17.5667	−0.769610	−0.384805	+	0.922998i	$0.625731\pi$
$522$	0	0
$523$	25.6088	1.11979	0.559897	−	0.828562i	$-0.310841\pi$
$523$	25.6088	1.11979	0.559897	+	0.828562i	$0.310841\pi$
$524$	−13.9893	−0.611127
$525$	0	0
$526$	24.1398	1.05254
$527$	−1.08727	−0.0473624
$528$	0	0
$529$	33.1471	1.44118
$530$	−19.7412	−0.857504
$531$	0	0
$532$	0	0
$533$	16.8853	0.731382
$534$	0	0
$535$	12.4648	0.538898
$536$	13.2113	0.570642
$537$	0	0
$538$	−11.7045	−0.504615
$539$	0	0
$540$	0	0
$541$	−41.0735	−1.76589	−0.882945	−	0.469477i	$-0.844443\pi$
$541$	−41.0735	−1.76589	−0.882945	+	0.469477i	$0.844443\pi$
$542$	6.71285	0.288342
$543$	0	0
$544$	−4.24276	−0.181907
$545$	21.0966	0.903679
$546$	0	0
$547$	−36.6456	−1.56685	−0.783426	−	0.621486i	$-0.786529\pi$
$547$	−36.6456	−1.56685	−0.783426	+	0.621486i	$0.786529\pi$
$548$	27.2975	1.16609
$549$	0	0
$550$	−14.4721	−0.617092
$551$	9.66769	0.411857
$552$	0	0
$553$	0	0
$554$	−18.5142	−0.786593
$555$	0	0
$556$	17.6823	0.749898
$557$	−13.9127	−0.589501	−0.294751	−	0.955574i	$-0.595237\pi$
$557$	−13.9127	−0.589501	−0.294751	+	0.955574i	$0.595237\pi$
$558$	0	0
$559$	59.3647	2.51086
$560$	0	0
$561$	0	0
$562$	−28.0441	−1.18297
$563$	−0.681412	−0.0287181	−0.0143590	−	0.999897i	$-0.504571\pi$
$563$	−0.681412	−0.0287181	−0.0143590	+	0.999897i	$0.504571\pi$
$564$	0	0
$565$	−9.71911	−0.408886
$566$	29.1923	1.22704
$567$	0	0
$568$	−10.6309	−0.446063
$569$	34.0598	1.42786	0.713931	−	0.700216i	$-0.246913\pi$
$569$	34.0598	1.42786	0.713931	+	0.700216i	$0.246913\pi$
$570$	0	0
$571$	17.6677	0.739370	0.369685	−	0.929157i	$-0.379466\pi$
$571$	17.6677	0.739370	0.369685	+	0.929157i	$0.379466\pi$
$572$	−11.7245	−0.490224
$573$	0	0
$574$	0	0
$575$	−67.7138	−2.82386
$576$	0	0
$577$	12.5015	0.520445	0.260223	−	0.965549i	$-0.416204\pi$
$577$	12.5015	0.520445	0.260223	+	0.965549i	$0.416204\pi$
$578$	−35.8328	−1.49045
$579$	0	0
$580$	−78.3310	−3.25252
$581$	0	0
$582$	0	0
$583$	−1.83384	−0.0759500
$584$	−12.9495	−0.535854
$585$	0	0
$586$	−23.2397	−0.960023
$587$	−45.3647	−1.87240	−0.936200	−	0.351467i	$-0.885683\pi$
$587$	−45.3647	−1.87240	−0.936200	+	0.351467i	$0.885683\pi$
$588$	0	0
$589$	−2.40586	−0.0991318
$590$	−117.179	−4.82420
$591$	0	0
$592$	−12.1765	−0.500453
$593$	9.93577	0.408013	0.204007	−	0.978970i	$-0.434604\pi$
$593$	9.93577	0.408013	0.204007	+	0.978970i	$0.434604\pi$
$594$	0	0
$595$	0	0
$596$	3.92005	0.160572
$597$	0	0
$598$	−97.0324	−3.96795
$599$	−18.3407	−0.749381	−0.374690	−	0.927150i	$-0.622251\pi$
$599$	−18.3407	−0.749381	−0.374690	+	0.927150i	$0.622251\pi$
$600$	0	0
$601$	−2.35443	−0.0960393	−0.0480197	−	0.998846i	$-0.515291\pi$
$601$	−2.35443	−0.0960393	−0.0480197	+	0.998846i	$0.515291\pi$
$602$	0	0
$603$	0	0
$604$	−16.2324	−0.660486
$605$	39.1240	1.59062
$606$	0	0
$607$	41.6823	1.69183	0.845917	−	0.533315i	$-0.179054\pi$
$607$	41.6823	1.69183	0.845917	+	0.533315i	$0.179054\pi$
$608$	−9.38815	−0.380740
$609$	0	0
$610$	−74.2574	−3.00659
$611$	25.8990	1.04776
$612$	0	0
$613$	21.5897	0.872001	0.436000	−	0.899946i	$-0.356395\pi$
$613$	21.5897	0.872001	0.436000	+	0.899946i	$0.356395\pi$
$614$	−38.7696	−1.56461
$615$	0	0
$616$	0	0
$617$	−0.819187	−0.0329792	−0.0164896	−	0.999864i	$-0.505249\pi$
$617$	−0.819187	−0.0329792	−0.0164896	+	0.999864i	$0.505249\pi$
$618$	0	0
$619$	−31.3500	−1.26006	−0.630032	−	0.776569i	$-0.716958\pi$
$619$	−31.3500	−1.26006	−0.630032	+	0.776569i	$0.716958\pi$
$620$	19.4931	0.782863
$621$	0	0
$622$	−9.20293	−0.369004
$623$	0	0
$624$	0	0
$625$	11.4794	0.459176
$626$	−11.9211	−0.476464
$627$	0	0
$628$	−53.3721	−2.12978
$629$	2.71819	0.108381
$630$	0	0
$631$	−1.66769	−0.0663895	−0.0331947	−	0.999449i	$-0.510568\pi$
$631$	−1.66769	−0.0663895	−0.0331947	+	0.999449i	$0.510568\pi$
$632$	13.2113	0.525518
$633$	0	0
$634$	−59.0545	−2.34535
$635$	−36.8432	−1.46208
$636$	0	0
$637$	0	0
$638$	−12.8706	−0.509553
$639$	0	0
$640$	36.9220	1.45947
$641$	−15.2481	−0.602264	−0.301132	−	0.953583i	$-0.597364\pi$
$641$	−15.2481	−0.602264	−0.301132	+	0.953583i	$0.597364\pi$
$642$	0	0
$643$	22.6265	0.892302	0.446151	−	0.894958i	$-0.352794\pi$
$643$	22.6265	0.892302	0.446151	+	0.894958i	$0.352794\pi$
$644$	0	0
$645$	0	0
$646$	1.40281	0.0551927
$647$	42.0040	1.65135	0.825673	−	0.564148i	$-0.190795\pi$
$647$	42.0040	1.65135	0.825673	+	0.564148i	$0.190795\pi$
$648$	0	0
$649$	−10.8853	−0.427284
$650$	117.022	4.58997
$651$	0	0
$652$	−31.8412	−1.24700
$653$	22.3554	0.874833	0.437416	−	0.899259i	$-0.355894\pi$
$653$	22.3554	0.874833	0.437416	+	0.899259i	$0.355894\pi$
$654$	0	0
$655$	20.1471	0.787212
$656$	−6.81172	−0.265953
$657$	0	0
$658$	0	0
$659$	31.4280	1.22426	0.612130	−	0.790757i	$-0.290313\pi$
$659$	31.4280	1.22426	0.612130	+	0.790757i	$0.290313\pi$
$660$	0	0
$661$	24.4426	0.950708	0.475354	−	0.879795i	$-0.342320\pi$
$661$	24.4426	0.950708	0.475354	+	0.879795i	$0.342320\pi$
$662$	34.6372	1.34621
$663$	0	0
$664$	−3.60879	−0.140048
$665$	0	0
$666$	0	0
$667$	−60.2206	−2.33175
$668$	38.4941	1.48938
$669$	0	0
$670$	−82.2942	−3.17930
$671$	−6.89807	−0.266297
$672$	0	0
$673$	39.4794	1.52182	0.760910	−	0.648858i	$-0.224753\pi$
$673$	39.4794	1.52182	0.760910	+	0.648858i	$0.224753\pi$
$674$	−2.38175	−0.0917417
$675$	0	0
$676$	60.9852	2.34558
$677$	42.3784	1.62873	0.814367	−	0.580350i	$-0.197084\pi$
$677$	42.3784	1.62873	0.814367	+	0.580350i	$0.197084\pi$
$678$	0	0
$679$	0	0
$680$	−2.62786	−0.100774
$681$	0	0
$682$	3.20293	0.122647
$683$	−22.4143	−0.857658	−0.428829	−	0.903386i	$-0.641074\pi$
$683$	−22.4143	−0.857658	−0.428829	+	0.903386i	$0.641074\pi$
$684$	0	0
$685$	−39.3133	−1.50208
$686$	0	0
$687$	0	0
$688$	−23.9484	−0.913026
$689$	14.8285	0.564921
$690$	0	0
$691$	−12.4794	−0.474739	−0.237370	−	0.971419i	$-0.576285\pi$
$691$	−12.4794	−0.474739	−0.237370	+	0.971419i	$0.576285\pi$
$692$	−4.73497	−0.179996
$693$	0	0
$694$	−61.8221	−2.34674
$695$	−25.4657	−0.965968
$696$	0	0
$697$	1.52059	0.0575965
$698$	49.9304	1.88989
$699$	0	0
$700$	0	0
$701$	15.0000	0.566542	0.283271	−	0.959040i	$-0.408580\pi$
$701$	15.0000	0.566542	0.283271	+	0.959040i	$0.408580\pi$
$702$	0	0
$703$	6.01466	0.226847
$704$	8.86223	0.334008
$705$	0	0
$706$	−5.07049	−0.190830
$707$	0	0
$708$	0	0
$709$	−31.0000	−1.16423	−0.582115	−	0.813107i	$-0.697775\pi$
$709$	−31.0000	−1.16423	−0.582115	+	0.813107i	$0.697775\pi$
$710$	66.2206	2.48522
$711$	0	0
$712$	−8.44264	−0.316401
$713$	14.9863	0.561240
$714$	0	0
$715$	16.8853	0.631473
$716$	2.82851	0.105706
$717$	0	0
$718$	−7.13671	−0.266340
$719$	−6.65396	−0.248151	−0.124075	−	0.992273i	$-0.539596\pi$
$719$	−6.65396	−0.248151	−0.124075	+	0.992273i	$0.539596\pi$
$720$	0	0
$721$	0	0
$722$	−37.6529	−1.40130
$723$	0	0
$724$	−5.20293	−0.193365
$725$	72.6265	2.69728
$726$	0	0
$727$	−36.9442	−1.37018	−0.685092	−	0.728457i	$-0.740238\pi$
$727$	−36.9442	−1.37018	−0.685092	+	0.728457i	$0.740238\pi$
$728$	0	0
$729$	0	0
$730$	80.6633	2.98548
$731$	5.34604	0.197731
$732$	0	0
$733$	38.1324	1.40845	0.704227	−	0.709975i	$-0.251294\pi$
$733$	38.1324	1.40845	0.704227	+	0.709975i	$0.251294\pi$
$734$	−12.1578	−0.448751
$735$	0	0
$736$	58.4794	2.15558
$737$	−7.64464	−0.281594
$738$	0	0
$739$	−6.23970	−0.229531	−0.114766	−	0.993393i	$-0.536612\pi$
$739$	−6.23970	−0.229531	−0.114766	+	0.993393i	$0.536612\pi$
$740$	−48.7328	−1.79145
$741$	0	0
$742$	0	0
$743$	11.0957	0.407060	0.203530	−	0.979069i	$-0.434758\pi$
$743$	11.0957	0.407060	0.203530	+	0.979069i	$0.434758\pi$
$744$	0	0
$745$	−5.64557	−0.206838
$746$	50.4446	1.84691
$747$	0	0
$748$	−1.05584	−0.0386052
$749$	0	0
$750$	0	0
$751$	36.3868	1.32777	0.663887	−	0.747833i	$-0.268906\pi$
$751$	36.3868	1.32777	0.663887	+	0.747833i	$0.268906\pi$
$752$	−10.4480	−0.380998
$753$	0	0
$754$	104.072	3.79009
$755$	23.3775	0.850794
$756$	0	0
$757$	−4.40586	−0.160134	−0.0800669	−	0.996789i	$-0.525513\pi$
$757$	−4.40586	−0.160134	−0.0800669	+	0.996789i	$0.525513\pi$
$758$	78.9240	2.86665
$759$	0	0
$760$	−5.81478	−0.210924
$761$	47.2123	1.71144	0.855721	−	0.517437i	$-0.173114\pi$
$761$	47.2123	1.71144	0.855721	+	0.517437i	$0.173114\pi$
$762$	0	0
$763$	0	0
$764$	28.1482	1.01836
$765$	0	0
$766$	−57.8883	−2.09159
$767$	88.0186	3.17817
$768$	0	0
$769$	−20.7603	−0.748635	−0.374318	−	0.927301i	$-0.622123\pi$
$769$	−20.7603	−0.748635	−0.374318	+	0.927301i	$0.622123\pi$
$770$	0	0
$771$	0	0
$772$	25.0545	0.901731
$773$	−0.976953	−0.0351386	−0.0175693	−	0.999846i	$-0.505593\pi$
$773$	−0.976953	−0.0351386	−0.0175693	+	0.999846i	$0.505593\pi$
$774$	0	0
$775$	−18.0735	−0.649221
$776$	−6.71285	−0.240977
$777$	0	0
$778$	−19.2765	−0.691095
$779$	3.36468	0.120552
$780$	0	0
$781$	6.15150	0.220118
$782$	−8.73818	−0.312477
$783$	0	0
$784$	0	0
$785$	76.8653	2.74344
$786$	0	0
$787$	−2.87062	−0.102326	−0.0511632	−	0.998690i	$-0.516293\pi$
$787$	−2.87062	−0.102326	−0.0511632	+	0.998690i	$0.516293\pi$
$788$	50.6750	1.80522
$789$	0	0
$790$	−82.2942	−2.92790
$791$	0	0
$792$	0	0
$793$	55.7780	1.98074
$794$	−61.1387	−2.16973
$795$	0	0
$796$	43.8883	1.55558
$797$	−1.22505	−0.0433935	−0.0216967	−	0.999765i	$-0.506907\pi$
$797$	−1.22505	−0.0433935	−0.0216967	+	0.999765i	$0.506907\pi$
$798$	0	0
$799$	2.33231	0.0825114
$800$	−70.5266	−2.49349
$801$	0	0
$802$	16.1030	0.568616
$803$	7.49314	0.264427
$804$	0	0
$805$	0	0
$806$	−25.8990	−0.912253
$807$	0	0
$808$	5.81478	0.204563
$809$	−15.4333	−0.542607	−0.271303	−	0.962494i	$-0.587455\pi$
$809$	−15.4333	−0.542607	−0.271303	+	0.962494i	$0.587455\pi$
$810$	0	0
$811$	2.47941	0.0870638	0.0435319	−	0.999052i	$-0.486139\pi$
$811$	2.47941	0.0870638	0.0435319	+	0.999052i	$0.486139\pi$
$812$	0	0
$813$	0	0
$814$	−8.00733	−0.280657
$815$	45.8569	1.60630
$816$	0	0
$817$	11.8294	0.413860
$818$	58.4320	2.04303
$819$	0	0
$820$	−27.2618	−0.952024
$821$	−28.8990	−1.00858	−0.504291	−	0.863534i	$-0.668246\pi$
$821$	−28.8990	−1.00858	−0.504291	+	0.863534i	$0.668246\pi$
$822$	0	0
$823$	−24.9588	−0.870010	−0.435005	−	0.900428i	$-0.643253\pi$
$823$	−24.9588	−0.870010	−0.435005	+	0.900428i	$0.643253\pi$
$824$	14.0253	0.488595
$825$	0	0
$826$	0	0
$827$	12.7191	0.442287	0.221143	−	0.975241i	$-0.429021\pi$
$827$	12.7191	0.442287	0.221143	+	0.975241i	$0.429021\pi$
$828$	0	0
$829$	6.33231	0.219930	0.109965	−	0.993935i	$-0.464926\pi$
$829$	6.33231	0.219930	0.109965	+	0.993935i	$0.464926\pi$
$830$	22.4794	0.780272
$831$	0	0
$832$	−71.6602	−2.48437
$833$	0	0
$834$	0	0
$835$	−55.4382	−1.91852
$836$	−2.33630	−0.0808026
$837$	0	0
$838$	−42.2206	−1.45849
$839$	45.4236	1.56820	0.784098	−	0.620637i	$-0.213126\pi$
$839$	45.4236	1.56820	0.784098	+	0.620637i	$0.213126\pi$
$840$	0	0
$841$	35.5897	1.22723
$842$	46.0755	1.58787
$843$	0	0
$844$	9.97361	0.343306
$845$	−87.8294	−3.02142
$846$	0	0
$847$	0	0
$848$	−5.98200	−0.205423
$849$	0	0
$850$	10.5383	0.361461
$851$	−37.4657	−1.28431
$852$	0	0
$853$	24.0882	0.824764	0.412382	−	0.911011i	$-0.364697\pi$
$853$	24.0882	0.824764	0.412382	+	0.911011i	$0.364697\pi$
$854$	0	0
$855$	0	0
$856$	−4.29249	−0.146714
$857$	−17.8622	−0.610162	−0.305081	−	0.952326i	$-0.598684\pi$
$857$	−17.8622	−0.610162	−0.305081	+	0.952326i	$0.598684\pi$
$858$	0	0
$859$	−33.7412	−1.15124	−0.575618	−	0.817719i	$-0.695238\pi$
$859$	−33.7412	−1.15124	−0.575618	+	0.817719i	$0.695238\pi$
$860$	−95.8462	−3.26833
$861$	0	0
$862$	−4.66463	−0.158878
$863$	54.1555	1.84347	0.921737	−	0.387815i	$-0.126770\pi$
$863$	54.1555	1.84347	0.921737	+	0.387815i	$0.126770\pi$
$864$	0	0
$865$	6.81919	0.231859
$866$	−75.7506	−2.57411
$867$	0	0
$868$	0	0
$869$	−7.64464	−0.259327
$870$	0	0
$871$	61.8148	2.09451
$872$	−7.26503	−0.246025
$873$	0	0
$874$	−19.3354	−0.654029
$875$	0	0
$876$	0	0
$877$	−19.0000	−0.641584	−0.320792	−	0.947150i	$-0.603949\pi$
$877$	−19.0000	−0.641584	−0.320792	+	0.947150i	$0.603949\pi$
$878$	41.1923	1.39017
$879$	0	0
$880$	−6.81172	−0.229623
$881$	−41.3500	−1.39312	−0.696559	−	0.717500i	$-0.745287\pi$
$881$	−41.3500	−1.39312	−0.696559	+	0.717500i	$0.745287\pi$
$882$	0	0
$883$	52.4603	1.76543	0.882716	−	0.469908i	$-0.155713\pi$
$883$	52.4603	1.76543	0.882716	+	0.469908i	$0.155713\pi$
$884$	8.53753	0.287148
$885$	0	0
$886$	11.4089	0.383290
$887$	21.2344	0.712980	0.356490	−	0.934299i	$-0.383973\pi$
$887$	21.2344	0.712980	0.356490	+	0.934299i	$0.383973\pi$
$888$	0	0
$889$	0	0
$890$	52.5897	1.76281
$891$	0	0
$892$	−32.4648	−1.08700
$893$	5.16082	0.172700
$894$	0	0
$895$	−4.07355	−0.136164
$896$	0	0
$897$	0	0
$898$	−15.8075	−0.527501
$899$	−16.0735	−0.536083
$900$	0	0
$901$	1.33537	0.0444876
$902$	−4.47941	−0.149148
$903$	0	0
$904$	3.34697	0.111319
$905$	7.49314	0.249080
$906$	0	0
$907$	40.2397	1.33614	0.668069	−	0.744100i	$-0.267121\pi$
$907$	40.2397	1.33614	0.668069	+	0.744100i	$0.267121\pi$
$908$	56.8599	1.88696
$909$	0	0
$910$	0	0
$911$	−0.811724	−0.0268936	−0.0134468	−	0.999910i	$-0.504280\pi$
$911$	−0.811724	−0.0268936	−0.0134468	+	0.999910i	$0.504280\pi$
$912$	0	0
$913$	2.08820	0.0691094
$914$	−2.14510	−0.0709537
$915$	0	0
$916$	−50.0514	−1.65375
$917$	0	0
$918$	0	0
$919$	17.7603	0.585858	0.292929	−	0.956134i	$-0.405370\pi$
$919$	17.7603	0.585858	0.292929	+	0.956134i	$0.405370\pi$
$920$	36.2206	1.19416
$921$	0	0
$922$	65.3500	2.15219
$923$	−49.7412	−1.63725
$924$	0	0
$925$	45.1839	1.48564
$926$	−58.9936	−1.93865
$927$	0	0
$928$	−62.7222	−2.05896
$929$	−1.59947	−0.0524770	−0.0262385	−	0.999656i	$-0.508353\pi$
$929$	−1.59947	−0.0524770	−0.0262385	+	0.999656i	$0.508353\pi$
$930$	0	0
$931$	0	0
$932$	−29.8033	−0.976241
$933$	0	0
$934$	84.0354	2.74972
$935$	1.52059	0.0497286
$936$	0	0
$937$	−2.70140	−0.0882510	−0.0441255	−	0.999026i	$-0.514050\pi$
$937$	−2.70140	−0.0882510	−0.0441255	+	0.999026i	$0.514050\pi$
$938$	0	0
$939$	0	0
$940$	−41.8148	−1.36385
$941$	−36.7917	−1.19938	−0.599688	−	0.800234i	$-0.704709\pi$
$941$	−36.7917	−1.19938	−0.599688	+	0.800234i	$0.704709\pi$
$942$	0	0
$943$	−20.9588	−0.682513
$944$	−35.5078	−1.15568
$945$	0	0
$946$	−15.7486	−0.512030
$947$	−24.3784	−0.792192	−0.396096	−	0.918209i	$-0.629635\pi$
$947$	−24.3784	−0.792192	−0.396096	+	0.918209i	$0.629635\pi$
$948$	0	0
$949$	−60.5897	−1.96683
$950$	23.3186	0.756555
$951$	0	0
$952$	0	0
$953$	−1.63091	−0.0528304	−0.0264152	−	0.999651i	$-0.508409\pi$
$953$	−1.63091	−0.0528304	−0.0264152	+	0.999651i	$0.508409\pi$
$954$	0	0
$955$	−40.5383	−1.31179
$956$	−10.8381	−0.350530
$957$	0	0
$958$	45.0177	1.45446
$959$	0	0
$960$	0	0
$961$	−27.0000	−0.870968
$962$	64.7475	2.08754
$963$	0	0
$964$	−34.4426	−1.10932
$965$	−36.0829	−1.16155
$966$	0	0
$967$	48.0735	1.54594	0.772971	−	0.634442i	$-0.218770\pi$
$967$	48.0735	1.54594	0.772971	+	0.634442i	$0.218770\pi$
$968$	−13.4731	−0.433043
$969$	0	0
$970$	41.8148	1.34259
$971$	10.3491	0.332118	0.166059	−	0.986116i	$-0.446896\pi$
$971$	10.3491	0.332118	0.166059	+	0.986116i	$0.446896\pi$
$972$	0	0
$973$	0	0
$974$	61.4477	1.96891
$975$	0	0
$976$	−22.5015	−0.720256
$977$	11.1608	0.357066	0.178533	−	0.983934i	$-0.442865\pi$
$977$	11.1608	0.357066	0.178533	+	0.983934i	$0.442865\pi$
$978$	0	0
$979$	4.88527	0.156134
$980$	0	0
$981$	0	0
$982$	−68.9588	−2.20056
$983$	41.3775	1.31974	0.659868	−	0.751381i	$-0.270612\pi$
$983$	41.3775	1.31974	0.659868	+	0.751381i	$0.270612\pi$
$984$	0	0
$985$	−72.9809	−2.32537
$986$	9.37214	0.298470
$987$	0	0
$988$	18.8914	0.601015
$989$	−73.6863	−2.34309
$990$	0	0
$991$	0.977882	0.0310634	0.0155317	−	0.999879i	$-0.495056\pi$
$991$	0.977882	0.0310634	0.0155317	+	0.999879i	$0.495056\pi$
$992$	15.6088	0.495580
$993$	0	0
$994$	0	0
$995$	−63.2069	−2.00379
$996$	0	0
$997$	28.1103	0.890263	0.445131	−	0.895465i	$-0.353157\pi$
$997$	28.1103	0.890263	0.445131	+	0.895465i	$0.353157\pi$
$998$	−41.3186	−1.30792
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3969.2.a.n.1.3		3
3.2	odd	2		3969.2.a.o.1.1		3
7.6	odd	2		567.2.a.f.1.3	yes	3
21.20	even	2		567.2.a.e.1.1	✓	3
28.27	even	2		9072.2.a.cb.1.3		3
63.13	odd	6		567.2.f.l.379.1		6
63.20	even	6		567.2.f.m.190.3		6
63.34	odd	6		567.2.f.l.190.1		6
63.41	even	6		567.2.f.m.379.3		6
84.83	odd	2		9072.2.a.bu.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
567.2.a.e.1.1	✓	3	21.20	even	2
567.2.a.f.1.3	yes	3	7.6	odd	2
567.2.f.l.190.1		6	63.34	odd	6
567.2.f.l.379.1		6	63.13	odd	6
567.2.f.m.190.3		6	63.20	even	6
567.2.f.m.379.3		6	63.41	even	6
3969.2.a.n.1.3		3	1.1	even	1	trivial
3969.2.a.o.1.1		3	3.2	odd	2
9072.2.a.bu.1.1		3	84.83	odd	2
9072.2.a.cb.1.3		3	28.27	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:	\( N \)	\(=\)	\( 3969 = 3^{4} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3969.a (trivial)

\(f(q)\)	\(=\)	\(q+2.14510 q^{2} +2.60147 q^{4} -3.74657 q^{5} +1.29021 q^{8} -8.03677 q^{10} -0.746568 q^{11} +6.03677 q^{13} -2.43531 q^{16} +0.543637 q^{17} +1.20293 q^{19} -9.74657 q^{20} -1.60147 q^{22} -7.49314 q^{23} +9.03677 q^{25} +12.9495 q^{26} +8.03677 q^{29} -2.00000 q^{31} -7.80440 q^{32} +1.16616 q^{34} +5.00000 q^{37} +2.58041 q^{38} -4.83384 q^{40} +2.79707 q^{41} +9.83384 q^{43} -1.94217 q^{44} -16.0735 q^{46} +4.29021 q^{47} +19.3848 q^{50} +15.7045 q^{52} +2.45636 q^{53} +2.79707 q^{55} +17.2397 q^{58} +14.5804 q^{59} +9.23970 q^{61} -4.29021 q^{62} -11.8706 q^{64} -22.6172 q^{65} +10.2397 q^{67} +1.41425 q^{68} -8.23970 q^{71} -10.0368 q^{73} +10.7255 q^{74} +3.12938 q^{76} +10.2397 q^{79} +9.12405 q^{80} +6.00000 q^{82} -2.79707 q^{83} -2.03677 q^{85} +21.0946 q^{86} -0.963226 q^{88} -6.54364 q^{89} -19.4931 q^{92} +9.20293 q^{94} -4.50686 q^{95} -5.20293 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 6 q^{4} - 3 q^{5} - 9 q^{8} - 3 q^{10} + 6 q^{11} - 3 q^{13} + 12 q^{16} - 3 q^{17} - 21 q^{20} - 3 q^{22} - 6 q^{23} + 6 q^{25} + 27 q^{26} + 3 q^{29} - 6 q^{31} - 18 q^{32} + 21 q^{34} + 15 q^{37}+ \cdots - 12 q^{97}+O(q^{100}) \) 3 * q + 6 * q^4 - 3 * q^5 - 9 * q^8 - 3 * q^10 + 6 * q^11 - 3 * q^13 + 12 * q^16 - 3 * q^17 - 21 * q^20 - 3 * q^22 - 6 * q^23 + 6 * q^25 + 27 * q^26 + 3 * q^29 - 6 * q^31 - 18 * q^32 + 21 * q^34 + 15 * q^37 - 18 * q^38 + 3 * q^40 + 12 * q^41 + 12 * q^43 - 3 * q^44 - 6 * q^46 + 27 * q^50 - 9 * q^52 + 12 * q^53 + 12 * q^55 + 27 * q^58 + 18 * q^59 + 3 * q^61 + 3 * q^64 - 21 * q^65 + 6 * q^67 - 39 * q^68 - 9 * q^73 + 48 * q^76 + 6 * q^79 - 3 * q^80 + 18 * q^82 - 12 * q^83 + 15 * q^85 + 45 * q^86 - 24 * q^88 - 15 * q^89 - 42 * q^92 + 24 * q^94 - 30 * q^95 - 12 * q^97

Introduction

L-functions

Modular forms

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Database

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