LMFDB - Embedded newform 1984.2.a.n.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1984,2,Mod(1,1984)] 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("1984.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1984, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1984 = 2^{6} \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1984.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,-2,0,-2,0,4,0,6,0,4,0,2,0,2,0,6,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)

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

Self dual:	yes
Analytic conductor:	$15.8423197610$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 31)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.618034$ of defining polynomial
Character	$\chi$	$=$	1984.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.23607 q^{3} -1.00000 q^{5} +4.23607 q^{7} -1.47214 q^{9} +2.00000 q^{11} -1.23607 q^{13} -1.23607 q^{15} +5.23607 q^{17} +2.23607 q^{19} +5.23607 q^{21} +7.70820 q^{23} -4.00000 q^{25} -5.52786 q^{27} -7.23607 q^{29} -1.00000 q^{31} +2.47214 q^{33} -4.23607 q^{35} +2.00000 q^{37} -1.52786 q^{39} +7.00000 q^{41} -3.23607 q^{43} +1.47214 q^{45} +6.47214 q^{47} +10.9443 q^{49} +6.47214 q^{51} +1.52786 q^{53} -2.00000 q^{55} +2.76393 q^{57} -2.23607 q^{59} +14.1803 q^{61} -6.23607 q^{63} +1.23607 q^{65} +8.00000 q^{67} +9.52786 q^{69} -13.1803 q^{71} -0.472136 q^{73} -4.94427 q^{75} +8.47214 q^{77} -1.70820 q^{79} -2.41641 q^{81} +2.94427 q^{83} -5.23607 q^{85} -8.94427 q^{87} -1.70820 q^{89} -5.23607 q^{91} -1.23607 q^{93} -2.23607 q^{95} +1.94427 q^{97} -2.94427 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} - 2 q^{5} + 4 q^{7} + 6 q^{9} + 4 q^{11} + 2 q^{13} + 2 q^{15} + 6 q^{17} + 6 q^{21} + 2 q^{23} - 8 q^{25} - 20 q^{27} - 10 q^{29} - 2 q^{31} - 4 q^{33} - 4 q^{35} + 4 q^{37} - 12 q^{39} + 14 q^{41}+ \cdots + 12 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 - 2 * q^5 + 4 * q^7 + 6 * q^9 + 4 * q^11 + 2 * q^13 + 2 * q^15 + 6 * q^17 + 6 * q^21 + 2 * q^23 - 8 * q^25 - 20 * q^27 - 10 * q^29 - 2 * q^31 - 4 * q^33 - 4 * q^35 + 4 * q^37 - 12 * q^39 + 14 * q^41 - 2 * q^43 - 6 * q^45 + 4 * q^47 + 4 * q^49 + 4 * q^51 + 12 * q^53 - 4 * q^55 + 10 * q^57 + 6 * q^61 - 8 * q^63 - 2 * q^65 + 16 * q^67 + 28 * q^69 - 4 * q^71 + 8 * q^73 + 8 * q^75 + 8 * q^77 + 10 * q^79 + 22 * q^81 - 12 * q^83 - 6 * q^85 + 10 * q^89 - 6 * q^91 + 2 * q^93 - 14 * q^97 + 12 * q^99

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	0
$2$	0	0
$3$	1.23607	0.713644	0.356822	−	0.934172i	$-0.383860\pi$
$3$	1.23607	0.713644	0.356822	+	0.934172i	$0.383860\pi$
$4$	0	0
$5$	−1.00000	−0.447214	−0.223607	−	0.974679i	$-0.571783\pi$
$5$	−1.00000	−0.447214	−0.223607	+	0.974679i	$0.571783\pi$
$6$	0	0
$7$	4.23607	1.60108	0.800542	−	0.599277i	$-0.204545\pi$
$7$	4.23607	1.60108	0.800542	+	0.599277i	$0.204545\pi$
$8$	0	0
$9$	−1.47214	−0.490712
$10$	0	0
$11$	2.00000	0.603023	0.301511	−	0.953463i	$-0.402509\pi$
$11$	2.00000	0.603023	0.301511	+	0.953463i	$0.402509\pi$
$12$	0	0
$13$	−1.23607	−0.342824	−0.171412	−	0.985199i	$-0.554833\pi$
$13$	−1.23607	−0.342824	−0.171412	+	0.985199i	$0.554833\pi$
$14$	0	0
$15$	−1.23607	−0.319151
$16$	0	0
$17$	5.23607	1.26993	0.634967	−	0.772540i	$-0.281014\pi$
$17$	5.23607	1.26993	0.634967	+	0.772540i	$0.281014\pi$
$18$	0	0
$19$	2.23607	0.512989	0.256495	−	0.966546i	$-0.417432\pi$
$19$	2.23607	0.512989	0.256495	+	0.966546i	$0.417432\pi$
$20$	0	0
$21$	5.23607	1.14260
$22$	0	0
$23$	7.70820	1.60727	0.803636	−	0.595121i	$-0.202896\pi$
$23$	7.70820	1.60727	0.803636	+	0.595121i	$0.202896\pi$
$24$	0	0
$25$	−4.00000	−0.800000
$26$	0	0
$27$	−5.52786	−1.06384
$28$	0	0
$29$	−7.23607	−1.34370	−0.671852	−	0.740685i	$-0.734501\pi$
$29$	−7.23607	−1.34370	−0.671852	+	0.740685i	$0.734501\pi$
$30$	0	0
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	0	0
$33$	2.47214	0.430344
$34$	0	0
$35$	−4.23607	−0.716026
$36$	0	0
$37$	2.00000	0.328798	0.164399	−	0.986394i	$-0.447432\pi$
$37$	2.00000	0.328798	0.164399	+	0.986394i	$0.447432\pi$
$38$	0	0
$39$	−1.52786	−0.244654
$40$	0	0
$41$	7.00000	1.09322	0.546608	−	0.837389i	$-0.315919\pi$
$41$	7.00000	1.09322	0.546608	+	0.837389i	$0.315919\pi$
$42$	0	0
$43$	−3.23607	−0.493496	−0.246748	−	0.969080i	$-0.579362\pi$
$43$	−3.23607	−0.493496	−0.246748	+	0.969080i	$0.579362\pi$
$44$	0	0
$45$	1.47214	0.219453
$46$	0	0
$47$	6.47214	0.944058	0.472029	−	0.881583i	$-0.343522\pi$
$47$	6.47214	0.944058	0.472029	+	0.881583i	$0.343522\pi$
$48$	0	0
$49$	10.9443	1.56347
$50$	0	0
$51$	6.47214	0.906280
$52$	0	0
$53$	1.52786	0.209868	0.104934	−	0.994479i	$-0.466537\pi$
$53$	1.52786	0.209868	0.104934	+	0.994479i	$0.466537\pi$
$54$	0	0
$55$	−2.00000	−0.269680
$56$	0	0
$57$	2.76393	0.366092
$58$	0	0
$59$	−2.23607	−0.291111	−0.145556	−	0.989350i	$-0.546497\pi$
$59$	−2.23607	−0.291111	−0.145556	+	0.989350i	$0.546497\pi$
$60$	0	0
$61$	14.1803	1.81561	0.907803	−	0.419396i	$-0.137758\pi$
$61$	14.1803	1.81561	0.907803	+	0.419396i	$0.137758\pi$
$62$	0	0
$63$	−6.23607	−0.785671
$64$	0	0
$65$	1.23607	0.153315
$66$	0	0
$67$	8.00000	0.977356	0.488678	−	0.872464i	$-0.337479\pi$
$67$	8.00000	0.977356	0.488678	+	0.872464i	$0.337479\pi$
$68$	0	0
$69$	9.52786	1.14702
$70$	0	0
$71$	−13.1803	−1.56422	−0.782109	−	0.623141i	$-0.785856\pi$
$71$	−13.1803	−1.56422	−0.782109	+	0.623141i	$0.785856\pi$
$72$	0	0
$73$	−0.472136	−0.0552593	−0.0276297	−	0.999618i	$-0.508796\pi$
$73$	−0.472136	−0.0552593	−0.0276297	+	0.999618i	$0.508796\pi$
$74$	0	0
$75$	−4.94427	−0.570915
$76$	0	0
$77$	8.47214	0.965489
$78$	0	0
$79$	−1.70820	−0.192188	−0.0960940	−	0.995372i	$-0.530635\pi$
$79$	−1.70820	−0.192188	−0.0960940	+	0.995372i	$0.530635\pi$
$80$	0	0
$81$	−2.41641	−0.268490
$82$	0	0
$83$	2.94427	0.323176	0.161588	−	0.986858i	$-0.448338\pi$
$83$	2.94427	0.323176	0.161588	+	0.986858i	$0.448338\pi$
$84$	0	0
$85$	−5.23607	−0.567931
$86$	0	0
$87$	−8.94427	−0.958927
$88$	0	0
$89$	−1.70820	−0.181069	−0.0905346	−	0.995893i	$-0.528858\pi$
$89$	−1.70820	−0.181069	−0.0905346	+	0.995893i	$0.528858\pi$
$90$	0	0
$91$	−5.23607	−0.548889
$92$	0	0
$93$	−1.23607	−0.128174
$94$	0	0
$95$	−2.23607	−0.229416
$96$	0	0
$97$	1.94427	0.197411	0.0987055	−	0.995117i	$-0.468530\pi$
$97$	1.94427	0.197411	0.0987055	+	0.995117i	$0.468530\pi$
$98$	0	0
$99$	−2.94427	−0.295910
$100$	0	0
$101$	3.00000	0.298511	0.149256	−	0.988799i	$-0.452312\pi$
$101$	3.00000	0.298511	0.149256	+	0.988799i	$0.452312\pi$
$102$	0	0
$103$	−1.76393	−0.173805	−0.0869027	−	0.996217i	$-0.527697\pi$
$103$	−1.76393	−0.173805	−0.0869027	+	0.996217i	$0.527697\pi$
$104$	0	0
$105$	−5.23607	−0.510988
$106$	0	0
$107$	10.2361	0.989558	0.494779	−	0.869019i	$-0.335249\pi$
$107$	10.2361	0.989558	0.494779	+	0.869019i	$0.335249\pi$
$108$	0	0
$109$	−3.94427	−0.377793	−0.188896	−	0.981997i	$-0.560491\pi$
$109$	−3.94427	−0.377793	−0.188896	+	0.981997i	$0.560491\pi$
$110$	0	0
$111$	2.47214	0.234645
$112$	0	0
$113$	−5.47214	−0.514775	−0.257388	−	0.966308i	$-0.582862\pi$
$113$	−5.47214	−0.514775	−0.257388	+	0.966308i	$0.582862\pi$
$114$	0	0
$115$	−7.70820	−0.718794
$116$	0	0
$117$	1.81966	0.168228
$118$	0	0
$119$	22.1803	2.03327
$120$	0	0
$121$	−7.00000	−0.636364
$122$	0	0
$123$	8.65248	0.780167
$124$	0	0
$125$	9.00000	0.804984
$126$	0	0
$127$	−3.52786	−0.313047	−0.156524	−	0.987674i	$-0.550029\pi$
$127$	−3.52786	−0.313047	−0.156524	+	0.987674i	$0.550029\pi$
$128$	0	0
$129$	−4.00000	−0.352180
$130$	0	0
$131$	12.0000	1.04844	0.524222	−	0.851581i	$-0.324356\pi$
$131$	12.0000	1.04844	0.524222	+	0.851581i	$0.324356\pi$
$132$	0	0
$133$	9.47214	0.821338
$134$	0	0
$135$	5.52786	0.475763
$136$	0	0
$137$	19.7082	1.68379	0.841893	−	0.539645i	$-0.181441\pi$
$137$	19.7082	1.68379	0.841893	+	0.539645i	$0.181441\pi$
$138$	0	0
$139$	−13.4164	−1.13796	−0.568982	−	0.822350i	$-0.692663\pi$
$139$	−13.4164	−1.13796	−0.568982	+	0.822350i	$0.692663\pi$
$140$	0	0
$141$	8.00000	0.673722
$142$	0	0
$143$	−2.47214	−0.206730
$144$	0	0
$145$	7.23607	0.600923
$146$	0	0
$147$	13.5279	1.11576
$148$	0	0
$149$	−10.0000	−0.819232	−0.409616	−	0.912258i	$-0.634337\pi$
$149$	−10.0000	−0.819232	−0.409616	+	0.912258i	$0.634337\pi$
$150$	0	0
$151$	−8.18034	−0.665707	−0.332853	−	0.942979i	$-0.608011\pi$
$151$	−8.18034	−0.665707	−0.332853	+	0.942979i	$0.608011\pi$
$152$	0	0
$153$	−7.70820	−0.623171
$154$	0	0
$155$	1.00000	0.0803219
$156$	0	0
$157$	14.8885	1.18824	0.594118	−	0.804378i	$-0.297501\pi$
$157$	14.8885	1.18824	0.594118	+	0.804378i	$0.297501\pi$
$158$	0	0
$159$	1.88854	0.149771
$160$	0	0
$161$	32.6525	2.57338
$162$	0	0
$163$	−2.70820	−0.212123	−0.106061	−	0.994360i	$-0.533824\pi$
$163$	−2.70820	−0.212123	−0.106061	+	0.994360i	$0.533824\pi$
$164$	0	0
$165$	−2.47214	−0.192456
$166$	0	0
$167$	−2.47214	−0.191300	−0.0956498	−	0.995415i	$-0.530493\pi$
$167$	−2.47214	−0.191300	−0.0956498	+	0.995415i	$0.530493\pi$
$168$	0	0
$169$	−11.4721	−0.882472
$170$	0	0
$171$	−3.29180	−0.251730
$172$	0	0
$173$	14.9443	1.13619	0.568096	−	0.822962i	$-0.307680\pi$
$173$	14.9443	1.13619	0.568096	+	0.822962i	$0.307680\pi$
$174$	0	0
$175$	−16.9443	−1.28087
$176$	0	0
$177$	−2.76393	−0.207750
$178$	0	0
$179$	−11.7082	−0.875112	−0.437556	−	0.899191i	$-0.644156\pi$
$179$	−11.7082	−0.875112	−0.437556	+	0.899191i	$0.644156\pi$
$180$	0	0
$181$	−18.1803	−1.35133	−0.675667	−	0.737207i	$-0.736144\pi$
$181$	−18.1803	−1.35133	−0.675667	+	0.737207i	$0.736144\pi$
$182$	0	0
$183$	17.5279	1.29570
$184$	0	0
$185$	−2.00000	−0.147043
$186$	0	0
$187$	10.4721	0.765798
$188$	0	0
$189$	−23.4164	−1.70329
$190$	0	0
$191$	−3.18034	−0.230121	−0.115061	−	0.993358i	$-0.536706\pi$
$191$	−3.18034	−0.230121	−0.115061	+	0.993358i	$0.536706\pi$
$192$	0	0
$193$	−5.47214	−0.393893	−0.196946	−	0.980414i	$-0.563103\pi$
$193$	−5.47214	−0.393893	−0.196946	+	0.980414i	$0.563103\pi$
$194$	0	0
$195$	1.52786	0.109413
$196$	0	0
$197$	15.4164	1.09837	0.549187	−	0.835700i	$-0.314938\pi$
$197$	15.4164	1.09837	0.549187	+	0.835700i	$0.314938\pi$
$198$	0	0
$199$	1.05573	0.0748386	0.0374193	−	0.999300i	$-0.488086\pi$
$199$	1.05573	0.0748386	0.0374193	+	0.999300i	$0.488086\pi$
$200$	0	0
$201$	9.88854	0.697484
$202$	0	0
$203$	−30.6525	−2.15138
$204$	0	0
$205$	−7.00000	−0.488901
$206$	0	0
$207$	−11.3475	−0.788707
$208$	0	0
$209$	4.47214	0.309344
$210$	0	0
$211$	0.819660	0.0564277	0.0282139	−	0.999602i	$-0.491018\pi$
$211$	0.819660	0.0564277	0.0282139	+	0.999602i	$0.491018\pi$
$212$	0	0
$213$	−16.2918	−1.11630
$214$	0	0
$215$	3.23607	0.220698
$216$	0	0
$217$	−4.23607	−0.287563
$218$	0	0
$219$	−0.583592	−0.0394355
$220$	0	0
$221$	−6.47214	−0.435363
$222$	0	0
$223$	−4.00000	−0.267860	−0.133930	−	0.990991i	$-0.542760\pi$
$223$	−4.00000	−0.267860	−0.133930	+	0.990991i	$0.542760\pi$
$224$	0	0
$225$	5.88854	0.392570
$226$	0	0
$227$	2.47214	0.164081	0.0820407	−	0.996629i	$-0.473856\pi$
$227$	2.47214	0.164081	0.0820407	+	0.996629i	$0.473856\pi$
$228$	0	0
$229$	−13.4164	−0.886581	−0.443291	−	0.896378i	$-0.646189\pi$
$229$	−13.4164	−0.886581	−0.443291	+	0.896378i	$0.646189\pi$
$230$	0	0
$231$	10.4721	0.689016
$232$	0	0
$233$	0.0557281	0.00365087	0.00182543	−	0.999998i	$-0.499419\pi$
$233$	0.0557281	0.00365087	0.00182543	+	0.999998i	$0.499419\pi$
$234$	0	0
$235$	−6.47214	−0.422196
$236$	0	0
$237$	−2.11146	−0.137154
$238$	0	0
$239$	−1.70820	−0.110495	−0.0552473	−	0.998473i	$-0.517595\pi$
$239$	−1.70820	−0.110495	−0.0552473	+	0.998473i	$0.517595\pi$
$240$	0	0
$241$	−30.3607	−1.95570	−0.977852	−	0.209299i	$-0.932882\pi$
$241$	−30.3607	−1.95570	−0.977852	+	0.209299i	$0.932882\pi$
$242$	0	0
$243$	13.5967	0.872232
$244$	0	0
$245$	−10.9443	−0.699204
$246$	0	0
$247$	−2.76393	−0.175865
$248$	0	0
$249$	3.63932	0.230633
$250$	0	0
$251$	−24.1803	−1.52625	−0.763125	−	0.646251i	$-0.776336\pi$
$251$	−24.1803	−1.52625	−0.763125	+	0.646251i	$0.776336\pi$
$252$	0	0
$253$	15.4164	0.969221
$254$	0	0
$255$	−6.47214	−0.405301
$256$	0	0
$257$	−15.9443	−0.994576	−0.497288	−	0.867585i	$-0.665671\pi$
$257$	−15.9443	−0.994576	−0.497288	+	0.867585i	$0.665671\pi$
$258$	0	0
$259$	8.47214	0.526433
$260$	0	0
$261$	10.6525	0.659372
$262$	0	0
$263$	18.7639	1.15703	0.578517	−	0.815670i	$-0.303632\pi$
$263$	18.7639	1.15703	0.578517	+	0.815670i	$0.303632\pi$
$264$	0	0
$265$	−1.52786	−0.0938559
$266$	0	0
$267$	−2.11146	−0.129219
$268$	0	0
$269$	28.9443	1.76476	0.882382	−	0.470534i	$-0.155939\pi$
$269$	28.9443	1.76476	0.882382	+	0.470534i	$0.155939\pi$
$270$	0	0
$271$	−8.18034	−0.496920	−0.248460	−	0.968642i	$-0.579924\pi$
$271$	−8.18034	−0.496920	−0.248460	+	0.968642i	$0.579924\pi$
$272$	0	0
$273$	−6.47214	−0.391711
$274$	0	0
$275$	−8.00000	−0.482418
$276$	0	0
$277$	−18.6525	−1.12072	−0.560359	−	0.828250i	$-0.689337\pi$
$277$	−18.6525	−1.12072	−0.560359	+	0.828250i	$0.689337\pi$
$278$	0	0
$279$	1.47214	0.0881345
$280$	0	0
$281$	17.0000	1.01413	0.507067	−	0.861906i	$-0.330729\pi$
$281$	17.0000	1.01413	0.507067	+	0.861906i	$0.330729\pi$
$282$	0	0
$283$	21.8885	1.30114	0.650569	−	0.759447i	$-0.274530\pi$
$283$	21.8885	1.30114	0.650569	+	0.759447i	$0.274530\pi$
$284$	0	0
$285$	−2.76393	−0.163721
$286$	0	0
$287$	29.6525	1.75033
$288$	0	0
$289$	10.4164	0.612730
$290$	0	0
$291$	2.40325	0.140881
$292$	0	0
$293$	−8.47214	−0.494947	−0.247474	−	0.968895i	$-0.579600\pi$
$293$	−8.47214	−0.494947	−0.247474	+	0.968895i	$0.579600\pi$
$294$	0	0
$295$	2.23607	0.130189
$296$	0	0
$297$	−11.0557	−0.641518
$298$	0	0
$299$	−9.52786	−0.551011
$300$	0	0
$301$	−13.7082	−0.790128
$302$	0	0
$303$	3.70820	0.213031
$304$	0	0
$305$	−14.1803	−0.811964
$306$	0	0
$307$	−15.2918	−0.872749	−0.436374	−	0.899765i	$-0.643738\pi$
$307$	−15.2918	−0.872749	−0.436374	+	0.899765i	$0.643738\pi$
$308$	0	0
$309$	−2.18034	−0.124035
$310$	0	0
$311$	6.81966	0.386707	0.193354	−	0.981129i	$-0.438064\pi$
$311$	6.81966	0.386707	0.193354	+	0.981129i	$0.438064\pi$
$312$	0	0
$313$	21.2361	1.20033	0.600167	−	0.799875i	$-0.295101\pi$
$313$	21.2361	1.20033	0.600167	+	0.799875i	$0.295101\pi$
$314$	0	0
$315$	6.23607	0.351363
$316$	0	0
$317$	−21.9443	−1.23251	−0.616257	−	0.787545i	$-0.711352\pi$
$317$	−21.9443	−1.23251	−0.616257	+	0.787545i	$0.711352\pi$
$318$	0	0
$319$	−14.4721	−0.810284
$320$	0	0
$321$	12.6525	0.706192
$322$	0	0
$323$	11.7082	0.651462
$324$	0	0
$325$	4.94427	0.274259
$326$	0	0
$327$	−4.87539	−0.269610
$328$	0	0
$329$	27.4164	1.51152
$330$	0	0
$331$	2.00000	0.109930	0.0549650	−	0.998488i	$-0.482495\pi$
$331$	2.00000	0.109930	0.0549650	+	0.998488i	$0.482495\pi$
$332$	0	0
$333$	−2.94427	−0.161345
$334$	0	0
$335$	−8.00000	−0.437087
$336$	0	0
$337$	−19.2361	−1.04786	−0.523928	−	0.851763i	$-0.675534\pi$
$337$	−19.2361	−1.04786	−0.523928	+	0.851763i	$0.675534\pi$
$338$	0	0
$339$	−6.76393	−0.367366
$340$	0	0
$341$	−2.00000	−0.108306
$342$	0	0
$343$	16.7082	0.902158
$344$	0	0
$345$	−9.52786	−0.512963
$346$	0	0
$347$	1.81966	0.0976845	0.0488422	−	0.998807i	$-0.484447\pi$
$347$	1.81966	0.0976845	0.0488422	+	0.998807i	$0.484447\pi$
$348$	0	0
$349$	−27.8885	−1.49284	−0.746420	−	0.665475i	$-0.768229\pi$
$349$	−27.8885	−1.49284	−0.746420	+	0.665475i	$0.768229\pi$
$350$	0	0
$351$	6.83282	0.364709
$352$	0	0
$353$	−19.4164	−1.03343	−0.516716	−	0.856157i	$-0.672846\pi$
$353$	−19.4164	−1.03343	−0.516716	+	0.856157i	$0.672846\pi$
$354$	0	0
$355$	13.1803	0.699540
$356$	0	0
$357$	27.4164	1.45103
$358$	0	0
$359$	−17.7639	−0.937544	−0.468772	−	0.883319i	$-0.655303\pi$
$359$	−17.7639	−0.937544	−0.468772	+	0.883319i	$0.655303\pi$
$360$	0	0
$361$	−14.0000	−0.736842
$362$	0	0
$363$	−8.65248	−0.454137
$364$	0	0
$365$	0.472136	0.0247127
$366$	0	0
$367$	−18.0000	−0.939592	−0.469796	−	0.882775i	$-0.655673\pi$
$367$	−18.0000	−0.939592	−0.469796	+	0.882775i	$0.655673\pi$
$368$	0	0
$369$	−10.3050	−0.536454
$370$	0	0
$371$	6.47214	0.336017
$372$	0	0
$373$	−19.0000	−0.983783	−0.491891	−	0.870657i	$-0.663694\pi$
$373$	−19.0000	−0.983783	−0.491891	+	0.870657i	$0.663694\pi$
$374$	0	0
$375$	11.1246	0.574472
$376$	0	0
$377$	8.94427	0.460653
$378$	0	0
$379$	−37.8885	−1.94620	−0.973102	−	0.230375i	$-0.926005\pi$
$379$	−37.8885	−1.94620	−0.973102	+	0.230375i	$0.926005\pi$
$380$	0	0
$381$	−4.36068	−0.223404
$382$	0	0
$383$	−11.8885	−0.607476	−0.303738	−	0.952756i	$-0.598235\pi$
$383$	−11.8885	−0.607476	−0.303738	+	0.952756i	$0.598235\pi$
$384$	0	0
$385$	−8.47214	−0.431780
$386$	0	0
$387$	4.76393	0.242164
$388$	0	0
$389$	−17.8885	−0.906985	−0.453493	−	0.891260i	$-0.649822\pi$
$389$	−17.8885	−0.906985	−0.453493	+	0.891260i	$0.649822\pi$
$390$	0	0
$391$	40.3607	2.04113
$392$	0	0
$393$	14.8328	0.748217
$394$	0	0
$395$	1.70820	0.0859491
$396$	0	0
$397$	7.00000	0.351320	0.175660	−	0.984451i	$-0.443794\pi$
$397$	7.00000	0.351320	0.175660	+	0.984451i	$0.443794\pi$
$398$	0	0
$399$	11.7082	0.586143
$400$	0	0
$401$	15.8197	0.789996	0.394998	−	0.918682i	$-0.370745\pi$
$401$	15.8197	0.789996	0.394998	+	0.918682i	$0.370745\pi$
$402$	0	0
$403$	1.23607	0.0615729
$404$	0	0
$405$	2.41641	0.120072
$406$	0	0
$407$	4.00000	0.198273
$408$	0	0
$409$	−26.1803	−1.29453	−0.647267	−	0.762263i	$-0.724088\pi$
$409$	−26.1803	−1.29453	−0.647267	+	0.762263i	$0.724088\pi$
$410$	0	0
$411$	24.3607	1.20162
$412$	0	0
$413$	−9.47214	−0.466093
$414$	0	0
$415$	−2.94427	−0.144529
$416$	0	0
$417$	−16.5836	−0.812102
$418$	0	0
$419$	30.1246	1.47168	0.735842	−	0.677153i	$-0.236787\pi$
$419$	30.1246	1.47168	0.735842	+	0.677153i	$0.236787\pi$
$420$	0	0
$421$	15.3607	0.748634	0.374317	−	0.927301i	$-0.377877\pi$
$421$	15.3607	0.748634	0.374317	+	0.927301i	$0.377877\pi$
$422$	0	0
$423$	−9.52786	−0.463261
$424$	0	0
$425$	−20.9443	−1.01595
$426$	0	0
$427$	60.0689	2.90694
$428$	0	0
$429$	−3.05573	−0.147532
$430$	0	0
$431$	−12.0000	−0.578020	−0.289010	−	0.957326i	$-0.593326\pi$
$431$	−12.0000	−0.578020	−0.289010	+	0.957326i	$0.593326\pi$
$432$	0	0
$433$	−12.1803	−0.585350	−0.292675	−	0.956212i	$-0.594545\pi$
$433$	−12.1803	−0.585350	−0.292675	+	0.956212i	$0.594545\pi$
$434$	0	0
$435$	8.94427	0.428845
$436$	0	0
$437$	17.2361	0.824513
$438$	0	0
$439$	−21.1803	−1.01088	−0.505441	−	0.862861i	$-0.668670\pi$
$439$	−21.1803	−1.01088	−0.505441	+	0.862861i	$0.668670\pi$
$440$	0	0
$441$	−16.1115	−0.767212
$442$	0	0
$443$	17.2918	0.821558	0.410779	−	0.911735i	$-0.365257\pi$
$443$	17.2918	0.821558	0.410779	+	0.911735i	$0.365257\pi$
$444$	0	0
$445$	1.70820	0.0809766
$446$	0	0
$447$	−12.3607	−0.584640
$448$	0	0
$449$	31.3050	1.47737	0.738686	−	0.674050i	$-0.235447\pi$
$449$	31.3050	1.47737	0.738686	+	0.674050i	$0.235447\pi$
$450$	0	0
$451$	14.0000	0.659234
$452$	0	0
$453$	−10.1115	−0.475078
$454$	0	0
$455$	5.23607	0.245471
$456$	0	0
$457$	−20.9443	−0.979732	−0.489866	−	0.871798i	$-0.662954\pi$
$457$	−20.9443	−0.979732	−0.489866	+	0.871798i	$0.662954\pi$
$458$	0	0
$459$	−28.9443	−1.35100
$460$	0	0
$461$	10.3607	0.482545	0.241272	−	0.970457i	$-0.422435\pi$
$461$	10.3607	0.482545	0.241272	+	0.970457i	$0.422435\pi$
$462$	0	0
$463$	29.4164	1.36710	0.683548	−	0.729905i	$-0.260436\pi$
$463$	29.4164	1.36710	0.683548	+	0.729905i	$0.260436\pi$
$464$	0	0
$465$	1.23607	0.0573213
$466$	0	0
$467$	−8.70820	−0.402968	−0.201484	−	0.979492i	$-0.564576\pi$
$467$	−8.70820	−0.402968	−0.201484	+	0.979492i	$0.564576\pi$
$468$	0	0
$469$	33.8885	1.56483
$470$	0	0
$471$	18.4033	0.847977
$472$	0	0
$473$	−6.47214	−0.297589
$474$	0	0
$475$	−8.94427	−0.410391
$476$	0	0
$477$	−2.24922	−0.102985
$478$	0	0
$479$	−36.7082	−1.67724	−0.838620	−	0.544716i	$-0.816637\pi$
$479$	−36.7082	−1.67724	−0.838620	+	0.544716i	$0.816637\pi$
$480$	0	0
$481$	−2.47214	−0.112720
$482$	0	0
$483$	40.3607	1.83647
$484$	0	0
$485$	−1.94427	−0.0882848
$486$	0	0
$487$	14.7639	0.669018	0.334509	−	0.942393i	$-0.391430\pi$
$487$	14.7639	0.669018	0.334509	+	0.942393i	$0.391430\pi$
$488$	0	0
$489$	−3.34752	−0.151380
$490$	0	0
$491$	−40.3607	−1.82145	−0.910726	−	0.413011i	$-0.864477\pi$
$491$	−40.3607	−1.82145	−0.910726	+	0.413011i	$0.864477\pi$
$492$	0	0
$493$	−37.8885	−1.70641
$494$	0	0
$495$	2.94427	0.132335
$496$	0	0
$497$	−55.8328	−2.50444
$498$	0	0
$499$	−33.4164	−1.49592	−0.747962	−	0.663742i	$-0.768967\pi$
$499$	−33.4164	−1.49592	−0.747962	+	0.663742i	$0.768967\pi$
$500$	0	0
$501$	−3.05573	−0.136520
$502$	0	0
$503$	1.65248	0.0736803	0.0368401	−	0.999321i	$-0.488271\pi$
$503$	1.65248	0.0736803	0.0368401	+	0.999321i	$0.488271\pi$
$504$	0	0
$505$	−3.00000	−0.133498
$506$	0	0
$507$	−14.1803	−0.629771
$508$	0	0
$509$	19.5967	0.868611	0.434305	−	0.900766i	$-0.356994\pi$
$509$	19.5967	0.868611	0.434305	+	0.900766i	$0.356994\pi$
$510$	0	0
$511$	−2.00000	−0.0884748
$512$	0	0
$513$	−12.3607	−0.545737
$514$	0	0
$515$	1.76393	0.0777281
$516$	0	0
$517$	12.9443	0.569288
$518$	0	0
$519$	18.4721	0.810837
$520$	0	0
$521$	2.00000	0.0876216	0.0438108	−	0.999040i	$-0.486050\pi$
$521$	2.00000	0.0876216	0.0438108	+	0.999040i	$0.486050\pi$
$522$	0	0
$523$	−4.29180	−0.187667	−0.0938336	−	0.995588i	$-0.529912\pi$
$523$	−4.29180	−0.187667	−0.0938336	+	0.995588i	$0.529912\pi$
$524$	0	0
$525$	−20.9443	−0.914083
$526$	0	0
$527$	−5.23607	−0.228087
$528$	0	0
$529$	36.4164	1.58332
$530$	0	0
$531$	3.29180	0.142852
$532$	0	0
$533$	−8.65248	−0.374780
$534$	0	0
$535$	−10.2361	−0.442544
$536$	0	0
$537$	−14.4721	−0.624519
$538$	0	0
$539$	21.8885	0.942806
$540$	0	0
$541$	−19.3607	−0.832381	−0.416190	−	0.909278i	$-0.636635\pi$
$541$	−19.3607	−0.832381	−0.416190	+	0.909278i	$0.636635\pi$
$542$	0	0
$543$	−22.4721	−0.964372
$544$	0	0
$545$	3.94427	0.168954
$546$	0	0
$547$	28.1246	1.20252	0.601261	−	0.799053i	$-0.294665\pi$
$547$	28.1246	1.20252	0.601261	+	0.799053i	$0.294665\pi$
$548$	0	0
$549$	−20.8754	−0.890940
$550$	0	0
$551$	−16.1803	−0.689306
$552$	0	0
$553$	−7.23607	−0.307709
$554$	0	0
$555$	−2.47214	−0.104936
$556$	0	0
$557$	12.0000	0.508456	0.254228	−	0.967144i	$-0.418179\pi$
$557$	12.0000	0.508456	0.254228	+	0.967144i	$0.418179\pi$
$558$	0	0
$559$	4.00000	0.169182
$560$	0	0
$561$	12.9443	0.546508
$562$	0	0
$563$	−39.5410	−1.66646	−0.833228	−	0.552930i	$-0.813510\pi$
$563$	−39.5410	−1.66646	−0.833228	+	0.552930i	$0.813510\pi$
$564$	0	0
$565$	5.47214	0.230214
$566$	0	0
$567$	−10.2361	−0.429874
$568$	0	0
$569$	14.4721	0.606704	0.303352	−	0.952879i	$-0.401894\pi$
$569$	14.4721	0.606704	0.303352	+	0.952879i	$0.401894\pi$
$570$	0	0
$571$	5.81966	0.243545	0.121773	−	0.992558i	$-0.461142\pi$
$571$	5.81966	0.243545	0.121773	+	0.992558i	$0.461142\pi$
$572$	0	0
$573$	−3.93112	−0.164225
$574$	0	0
$575$	−30.8328	−1.28582
$576$	0	0
$577$	24.8328	1.03380	0.516902	−	0.856045i	$-0.327085\pi$
$577$	24.8328	1.03380	0.516902	+	0.856045i	$0.327085\pi$
$578$	0	0
$579$	−6.76393	−0.281099
$580$	0	0
$581$	12.4721	0.517431
$582$	0	0
$583$	3.05573	0.126555
$584$	0	0
$585$	−1.81966	−0.0752337
$586$	0	0
$587$	2.47214	0.102036	0.0510180	−	0.998698i	$-0.483753\pi$
$587$	2.47214	0.102036	0.0510180	+	0.998698i	$0.483753\pi$
$588$	0	0
$589$	−2.23607	−0.0921356
$590$	0	0
$591$	19.0557	0.783848
$592$	0	0
$593$	−15.4721	−0.635364	−0.317682	−	0.948197i	$-0.602905\pi$
$593$	−15.4721	−0.635364	−0.317682	+	0.948197i	$0.602905\pi$
$594$	0	0
$595$	−22.1803	−0.909305
$596$	0	0
$597$	1.30495	0.0534081
$598$	0	0
$599$	−34.5967	−1.41358	−0.706792	−	0.707421i	$-0.749859\pi$
$599$	−34.5967	−1.41358	−0.706792	+	0.707421i	$0.749859\pi$
$600$	0	0
$601$	−36.5410	−1.49054	−0.745270	−	0.666763i	$-0.767679\pi$
$601$	−36.5410	−1.49054	−0.745270	+	0.666763i	$0.767679\pi$
$602$	0	0
$603$	−11.7771	−0.479600
$604$	0	0
$605$	7.00000	0.284590
$606$	0	0
$607$	−13.5279	−0.549079	−0.274540	−	0.961576i	$-0.588525\pi$
$607$	−13.5279	−0.549079	−0.274540	+	0.961576i	$0.588525\pi$
$608$	0	0
$609$	−37.8885	−1.53532
$610$	0	0
$611$	−8.00000	−0.323645
$612$	0	0
$613$	8.11146	0.327619	0.163809	−	0.986492i	$-0.447622\pi$
$613$	8.11146	0.327619	0.163809	+	0.986492i	$0.447622\pi$
$614$	0	0
$615$	−8.65248	−0.348902
$616$	0	0
$617$	23.5279	0.947196	0.473598	−	0.880741i	$-0.342955\pi$
$617$	23.5279	0.947196	0.473598	+	0.880741i	$0.342955\pi$
$618$	0	0
$619$	−16.1803	−0.650343	−0.325171	−	0.945655i	$-0.605422\pi$
$619$	−16.1803	−0.650343	−0.325171	+	0.945655i	$0.605422\pi$
$620$	0	0
$621$	−42.6099	−1.70988
$622$	0	0
$623$	−7.23607	−0.289907
$624$	0	0
$625$	11.0000	0.440000
$626$	0	0
$627$	5.52786	0.220762
$628$	0	0
$629$	10.4721	0.417551
$630$	0	0
$631$	10.3607	0.412452	0.206226	−	0.978504i	$-0.433882\pi$
$631$	10.3607	0.412452	0.206226	+	0.978504i	$0.433882\pi$
$632$	0	0
$633$	1.01316	0.0402693
$634$	0	0
$635$	3.52786	0.139999
$636$	0	0
$637$	−13.5279	−0.535993
$638$	0	0
$639$	19.4033	0.767581
$640$	0	0
$641$	12.0000	0.473972	0.236986	−	0.971513i	$-0.423841\pi$
$641$	12.0000	0.473972	0.236986	+	0.971513i	$0.423841\pi$
$642$	0	0
$643$	28.4721	1.12283	0.561416	−	0.827534i	$-0.310257\pi$
$643$	28.4721	1.12283	0.561416	+	0.827534i	$0.310257\pi$
$644$	0	0
$645$	4.00000	0.157500
$646$	0	0
$647$	−16.9443	−0.666148	−0.333074	−	0.942901i	$-0.608086\pi$
$647$	−16.9443	−0.666148	−0.333074	+	0.942901i	$0.608086\pi$
$648$	0	0
$649$	−4.47214	−0.175547
$650$	0	0
$651$	−5.23607	−0.205218
$652$	0	0
$653$	−15.3050	−0.598929	−0.299465	−	0.954107i	$-0.596808\pi$
$653$	−15.3050	−0.598929	−0.299465	+	0.954107i	$0.596808\pi$
$654$	0	0
$655$	−12.0000	−0.468879
$656$	0	0
$657$	0.695048	0.0271164
$658$	0	0
$659$	−5.65248	−0.220189	−0.110095	−	0.993921i	$-0.535115\pi$
$659$	−5.65248	−0.220189	−0.110095	+	0.993921i	$0.535115\pi$
$660$	0	0
$661$	45.3607	1.76433	0.882163	−	0.470944i	$-0.156087\pi$
$661$	45.3607	1.76433	0.882163	+	0.470944i	$0.156087\pi$
$662$	0	0
$663$	−8.00000	−0.310694
$664$	0	0
$665$	−9.47214	−0.367314
$666$	0	0
$667$	−55.7771	−2.15970
$668$	0	0
$669$	−4.94427	−0.191157
$670$	0	0
$671$	28.3607	1.09485
$672$	0	0
$673$	47.0132	1.81222	0.906112	−	0.423038i	$-0.139036\pi$
$673$	47.0132	1.81222	0.906112	+	0.423038i	$0.139036\pi$
$674$	0	0
$675$	22.1115	0.851070
$676$	0	0
$677$	−42.7214	−1.64192	−0.820958	−	0.570989i	$-0.806560\pi$
$677$	−42.7214	−1.64192	−0.820958	+	0.570989i	$0.806560\pi$
$678$	0	0
$679$	8.23607	0.316071
$680$	0	0
$681$	3.05573	0.117096
$682$	0	0
$683$	−17.1803	−0.657387	−0.328694	−	0.944437i	$-0.606608\pi$
$683$	−17.1803	−0.657387	−0.328694	+	0.944437i	$0.606608\pi$
$684$	0	0
$685$	−19.7082	−0.753012
$686$	0	0
$687$	−16.5836	−0.632704
$688$	0	0
$689$	−1.88854	−0.0719478
$690$	0	0
$691$	−19.1803	−0.729655	−0.364827	−	0.931075i	$-0.618872\pi$
$691$	−19.1803	−0.729655	−0.364827	+	0.931075i	$0.618872\pi$
$692$	0	0
$693$	−12.4721	−0.473777
$694$	0	0
$695$	13.4164	0.508913
$696$	0	0
$697$	36.6525	1.38831
$698$	0	0
$699$	0.0688837	0.00260542
$700$	0	0
$701$	−7.00000	−0.264386	−0.132193	−	0.991224i	$-0.542202\pi$
$701$	−7.00000	−0.264386	−0.132193	+	0.991224i	$0.542202\pi$
$702$	0	0
$703$	4.47214	0.168670
$704$	0	0
$705$	−8.00000	−0.301297
$706$	0	0
$707$	12.7082	0.477941
$708$	0	0
$709$	−34.4721	−1.29463	−0.647314	−	0.762223i	$-0.724108\pi$
$709$	−34.4721	−1.29463	−0.647314	+	0.762223i	$0.724108\pi$
$710$	0	0
$711$	2.51471	0.0943089
$712$	0	0
$713$	−7.70820	−0.288675
$714$	0	0
$715$	2.47214	0.0924526
$716$	0	0
$717$	−2.11146	−0.0788538
$718$	0	0
$719$	36.1803	1.34930	0.674649	−	0.738138i	$-0.264295\pi$
$719$	36.1803	1.34930	0.674649	+	0.738138i	$0.264295\pi$
$720$	0	0
$721$	−7.47214	−0.278277
$722$	0	0
$723$	−37.5279	−1.39568
$724$	0	0
$725$	28.9443	1.07496
$726$	0	0
$727$	39.7639	1.47476	0.737381	−	0.675477i	$-0.236062\pi$
$727$	39.7639	1.47476	0.737381	+	0.675477i	$0.236062\pi$
$728$	0	0
$729$	24.0557	0.890953
$730$	0	0
$731$	−16.9443	−0.626707
$732$	0	0
$733$	5.47214	0.202118	0.101059	−	0.994880i	$-0.467777\pi$
$733$	5.47214	0.202118	0.101059	+	0.994880i	$0.467777\pi$
$734$	0	0
$735$	−13.5279	−0.498983
$736$	0	0
$737$	16.0000	0.589368
$738$	0	0
$739$	−16.1803	−0.595203	−0.297602	−	0.954690i	$-0.596187\pi$
$739$	−16.1803	−0.595203	−0.297602	+	0.954690i	$0.596187\pi$
$740$	0	0
$741$	−3.41641	−0.125505
$742$	0	0
$743$	−27.8197	−1.02060	−0.510302	−	0.859995i	$-0.670466\pi$
$743$	−27.8197	−1.02060	−0.510302	+	0.859995i	$0.670466\pi$
$744$	0	0
$745$	10.0000	0.366372
$746$	0	0
$747$	−4.33437	−0.158586
$748$	0	0
$749$	43.3607	1.58436
$750$	0	0
$751$	−45.5410	−1.66182	−0.830908	−	0.556410i	$-0.812178\pi$
$751$	−45.5410	−1.66182	−0.830908	+	0.556410i	$0.812178\pi$
$752$	0	0
$753$	−29.8885	−1.08920
$754$	0	0
$755$	8.18034	0.297713
$756$	0	0
$757$	22.6525	0.823318	0.411659	−	0.911338i	$-0.364949\pi$
$757$	22.6525	0.823318	0.411659	+	0.911338i	$0.364949\pi$
$758$	0	0
$759$	19.0557	0.691679
$760$	0	0
$761$	2.00000	0.0724999	0.0362500	−	0.999343i	$-0.488459\pi$
$761$	2.00000	0.0724999	0.0362500	+	0.999343i	$0.488459\pi$
$762$	0	0
$763$	−16.7082	−0.604878
$764$	0	0
$765$	7.70820	0.278691
$766$	0	0
$767$	2.76393	0.0997998
$768$	0	0
$769$	−2.63932	−0.0951763	−0.0475882	−	0.998867i	$-0.515154\pi$
$769$	−2.63932	−0.0951763	−0.0475882	+	0.998867i	$0.515154\pi$
$770$	0	0
$771$	−19.7082	−0.709774
$772$	0	0
$773$	−29.1246	−1.04754	−0.523770	−	0.851860i	$-0.675475\pi$
$773$	−29.1246	−1.04754	−0.523770	+	0.851860i	$0.675475\pi$
$774$	0	0
$775$	4.00000	0.143684
$776$	0	0
$777$	10.4721	0.375686
$778$	0	0
$779$	15.6525	0.560808
$780$	0	0
$781$	−26.3607	−0.943259
$782$	0	0
$783$	40.0000	1.42948
$784$	0	0
$785$	−14.8885	−0.531395
$786$	0	0
$787$	38.6525	1.37781	0.688906	−	0.724851i	$-0.258091\pi$
$787$	38.6525	1.37781	0.688906	+	0.724851i	$0.258091\pi$
$788$	0	0
$789$	23.1935	0.825710
$790$	0	0
$791$	−23.1803	−0.824198
$792$	0	0
$793$	−17.5279	−0.622433
$794$	0	0
$795$	−1.88854	−0.0669797
$796$	0	0
$797$	28.5836	1.01248	0.506241	−	0.862392i	$-0.331034\pi$
$797$	28.5836	1.01248	0.506241	+	0.862392i	$0.331034\pi$
$798$	0	0
$799$	33.8885	1.19889
$800$	0	0
$801$	2.51471	0.0888529
$802$	0	0
$803$	−0.944272	−0.0333226
$804$	0	0
$805$	−32.6525	−1.15085
$806$	0	0
$807$	35.7771	1.25941
$808$	0	0
$809$	−3.41641	−0.120115	−0.0600573	−	0.998195i	$-0.519128\pi$
$809$	−3.41641	−0.120115	−0.0600573	+	0.998195i	$0.519128\pi$
$810$	0	0
$811$	−28.0000	−0.983213	−0.491606	−	0.870817i	$-0.663590\pi$
$811$	−28.0000	−0.983213	−0.491606	+	0.870817i	$0.663590\pi$
$812$	0	0
$813$	−10.1115	−0.354624
$814$	0	0
$815$	2.70820	0.0948642
$816$	0	0
$817$	−7.23607	−0.253158
$818$	0	0
$819$	7.70820	0.269346
$820$	0	0
$821$	36.5410	1.27529	0.637645	−	0.770330i	$-0.279909\pi$
$821$	36.5410	1.27529	0.637645	+	0.770330i	$0.279909\pi$
$822$	0	0
$823$	27.7082	0.965847	0.482924	−	0.875662i	$-0.339575\pi$
$823$	27.7082	0.965847	0.482924	+	0.875662i	$0.339575\pi$
$824$	0	0
$825$	−9.88854	−0.344275
$826$	0	0
$827$	48.6525	1.69181	0.845906	−	0.533332i	$-0.179060\pi$
$827$	48.6525	1.69181	0.845906	+	0.533332i	$0.179060\pi$
$828$	0	0
$829$	36.8328	1.27926	0.639628	−	0.768684i	$-0.279088\pi$
$829$	36.8328	1.27926	0.639628	+	0.768684i	$0.279088\pi$
$830$	0	0
$831$	−23.0557	−0.799794
$832$	0	0
$833$	57.3050	1.98550
$834$	0	0
$835$	2.47214	0.0855518
$836$	0	0
$837$	5.52786	0.191071
$838$	0	0
$839$	−11.0557	−0.381686	−0.190843	−	0.981621i	$-0.561122\pi$
$839$	−11.0557	−0.381686	−0.190843	+	0.981621i	$0.561122\pi$
$840$	0	0
$841$	23.3607	0.805541
$842$	0	0
$843$	21.0132	0.723732
$844$	0	0
$845$	11.4721	0.394653
$846$	0	0
$847$	−29.6525	−1.01887
$848$	0	0
$849$	27.0557	0.928550
$850$	0	0
$851$	15.4164	0.528468
$852$	0	0
$853$	−37.4164	−1.28111	−0.640557	−	0.767911i	$-0.721296\pi$
$853$	−37.4164	−1.28111	−0.640557	+	0.767911i	$0.721296\pi$
$854$	0	0
$855$	3.29180	0.112577
$856$	0	0
$857$	51.6656	1.76486	0.882432	−	0.470440i	$-0.155905\pi$
$857$	51.6656	1.76486	0.882432	+	0.470440i	$0.155905\pi$
$858$	0	0
$859$	37.8885	1.29274	0.646370	−	0.763024i	$-0.276286\pi$
$859$	37.8885	1.29274	0.646370	+	0.763024i	$0.276286\pi$
$860$	0	0
$861$	36.6525	1.24911
$862$	0	0
$863$	32.1803	1.09543	0.547716	−	0.836664i	$-0.315498\pi$
$863$	32.1803	1.09543	0.547716	+	0.836664i	$0.315498\pi$
$864$	0	0
$865$	−14.9443	−0.508120
$866$	0	0
$867$	12.8754	0.437271
$868$	0	0
$869$	−3.41641	−0.115894
$870$	0	0
$871$	−9.88854	−0.335061
$872$	0	0
$873$	−2.86223	−0.0968719
$874$	0	0
$875$	38.1246	1.28885
$876$	0	0
$877$	35.9443	1.21375	0.606876	−	0.794797i	$-0.292422\pi$
$877$	35.9443	1.21375	0.606876	+	0.794797i	$0.292422\pi$
$878$	0	0
$879$	−10.4721	−0.353216
$880$	0	0
$881$	24.3607	0.820732	0.410366	−	0.911921i	$-0.365401\pi$
$881$	24.3607	0.820732	0.410366	+	0.911921i	$0.365401\pi$
$882$	0	0
$883$	39.7771	1.33861	0.669303	−	0.742990i	$-0.266593\pi$
$883$	39.7771	1.33861	0.669303	+	0.742990i	$0.266593\pi$
$884$	0	0
$885$	2.76393	0.0929086
$886$	0	0
$887$	31.0689	1.04319	0.521596	−	0.853193i	$-0.325337\pi$
$887$	31.0689	1.04319	0.521596	+	0.853193i	$0.325337\pi$
$888$	0	0
$889$	−14.9443	−0.501215
$890$	0	0
$891$	−4.83282	−0.161905
$892$	0	0
$893$	14.4721	0.484292
$894$	0	0
$895$	11.7082	0.391362
$896$	0	0
$897$	−11.7771	−0.393226
$898$	0	0
$899$	7.23607	0.241336
$900$	0	0
$901$	8.00000	0.266519
$902$	0	0
$903$	−16.9443	−0.563870
$904$	0	0
$905$	18.1803	0.604335
$906$	0	0
$907$	−19.7639	−0.656251	−0.328125	−	0.944634i	$-0.606417\pi$
$907$	−19.7639	−0.656251	−0.328125	+	0.944634i	$0.606417\pi$
$908$	0	0
$909$	−4.41641	−0.146483
$910$	0	0
$911$	4.18034	0.138501	0.0692504	−	0.997599i	$-0.477939\pi$
$911$	4.18034	0.138501	0.0692504	+	0.997599i	$0.477939\pi$
$912$	0	0
$913$	5.88854	0.194882
$914$	0	0
$915$	−17.5279	−0.579453
$916$	0	0
$917$	50.8328	1.67865
$918$	0	0
$919$	5.52786	0.182347	0.0911737	−	0.995835i	$-0.470938\pi$
$919$	5.52786	0.182347	0.0911737	+	0.995835i	$0.470938\pi$
$920$	0	0
$921$	−18.9017	−0.622832
$922$	0	0
$923$	16.2918	0.536251
$924$	0	0
$925$	−8.00000	−0.263038
$926$	0	0
$927$	2.59675	0.0852884
$928$	0	0
$929$	−20.0000	−0.656179	−0.328089	−	0.944647i	$-0.606405\pi$
$929$	−20.0000	−0.656179	−0.328089	+	0.944647i	$0.606405\pi$
$930$	0	0
$931$	24.4721	0.802042
$932$	0	0
$933$	8.42956	0.275972
$934$	0	0
$935$	−10.4721	−0.342475
$936$	0	0
$937$	26.9443	0.880231	0.440115	−	0.897941i	$-0.354937\pi$
$937$	26.9443	0.880231	0.440115	+	0.897941i	$0.354937\pi$
$938$	0	0
$939$	26.2492	0.856611
$940$	0	0
$941$	38.0000	1.23876	0.619382	−	0.785090i	$-0.287383\pi$
$941$	38.0000	1.23876	0.619382	+	0.785090i	$0.287383\pi$
$942$	0	0
$943$	53.9574	1.75710
$944$	0	0
$945$	23.4164	0.761736
$946$	0	0
$947$	−30.9443	−1.00555	−0.502777	−	0.864416i	$-0.667688\pi$
$947$	−30.9443	−1.00555	−0.502777	+	0.864416i	$0.667688\pi$
$948$	0	0
$949$	0.583592	0.0189442
$950$	0	0
$951$	−27.1246	−0.879576
$952$	0	0
$953$	32.2918	1.04603	0.523017	−	0.852322i	$-0.324806\pi$
$953$	32.2918	1.04603	0.523017	+	0.852322i	$0.324806\pi$
$954$	0	0
$955$	3.18034	0.102913
$956$	0	0
$957$	−17.8885	−0.578254
$958$	0	0
$959$	83.4853	2.69588
$960$	0	0
$961$	1.00000	0.0322581
$962$	0	0
$963$	−15.0689	−0.485588
$964$	0	0
$965$	5.47214	0.176154
$966$	0	0
$967$	−15.6393	−0.502927	−0.251463	−	0.967867i	$-0.580912\pi$
$967$	−15.6393	−0.502927	−0.251463	+	0.967867i	$0.580912\pi$
$968$	0	0
$969$	14.4721	0.464912
$970$	0	0
$971$	−28.0000	−0.898563	−0.449281	−	0.893390i	$-0.648320\pi$
$971$	−28.0000	−0.898563	−0.449281	+	0.893390i	$0.648320\pi$
$972$	0	0
$973$	−56.8328	−1.82198
$974$	0	0
$975$	6.11146	0.195723
$976$	0	0
$977$	33.2492	1.06374	0.531868	−	0.846827i	$-0.321490\pi$
$977$	33.2492	1.06374	0.531868	+	0.846827i	$0.321490\pi$
$978$	0	0
$979$	−3.41641	−0.109189
$980$	0	0
$981$	5.80650	0.185387
$982$	0	0
$983$	−48.4721	−1.54602	−0.773011	−	0.634393i	$-0.781250\pi$
$983$	−48.4721	−1.54602	−0.773011	+	0.634393i	$0.781250\pi$
$984$	0	0
$985$	−15.4164	−0.491208
$986$	0	0
$987$	33.8885	1.07868
$988$	0	0
$989$	−24.9443	−0.793182
$990$	0	0
$991$	−50.5410	−1.60549	−0.802744	−	0.596324i	$-0.796628\pi$
$991$	−50.5410	−1.60549	−0.802744	+	0.596324i	$0.796628\pi$
$992$	0	0
$993$	2.47214	0.0784509
$994$	0	0
$995$	−1.05573	−0.0334688
$996$	0	0
$997$	−15.3607	−0.486478	−0.243239	−	0.969966i	$-0.578210\pi$
$997$	−15.3607	−0.486478	−0.243239	+	0.969966i	$0.578210\pi$
$998$	0	0
$999$	−11.0557	−0.349788

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1984.2.a.n.1.2		2
4.3	odd	2		1984.2.a.r.1.1		2
8.3	odd	2		31.2.a.a.1.1	✓	2
8.5	even	2		496.2.a.i.1.1		2
24.5	odd	2		4464.2.a.bf.1.2		2
24.11	even	2		279.2.a.a.1.2		2
40.3	even	4		775.2.b.d.249.3		4
40.19	odd	2		775.2.a.d.1.2		2
40.27	even	4		775.2.b.d.249.2		4
56.27	even	2		1519.2.a.a.1.1		2
88.43	even	2		3751.2.a.b.1.2		2
104.51	odd	2		5239.2.a.f.1.2		2
120.59	even	2		6975.2.a.y.1.1		2
136.67	odd	2		8959.2.a.b.1.1		2
248.3	even	30		961.2.g.d.846.1		8
248.11	even	30		961.2.g.e.338.1		8
248.19	odd	30		961.2.g.a.547.1		8
248.27	even	10		961.2.d.g.388.1		4
248.35	odd	10		961.2.d.d.388.1		4
248.43	even	30		961.2.g.d.547.1		8
248.51	odd	30		961.2.g.h.338.1		8
248.59	odd	30		961.2.g.a.846.1		8
248.67	odd	6		961.2.c.e.521.1		4
248.75	even	30		961.2.g.d.448.1		8
248.83	even	30		961.2.g.d.844.1		8
248.91	even	10		961.2.d.a.531.1		4
248.99	even	6		961.2.c.c.439.1		4
248.107	odd	30		961.2.g.h.816.1		8
248.115	even	30		961.2.g.e.732.1		8
248.123	even	2		961.2.a.f.1.1		2
248.131	odd	30		961.2.g.h.235.1		8
248.139	even	10		961.2.d.a.628.1		4
248.147	even	10		961.2.d.g.374.1		4
248.163	odd	10		961.2.d.d.374.1		4
248.171	odd	10		961.2.d.c.628.1		4
248.179	even	30		961.2.g.e.235.1		8
248.195	odd	30		961.2.g.h.732.1		8
248.203	even	30		961.2.g.e.816.1		8
248.211	odd	6		961.2.c.e.439.1		4
248.219	odd	10		961.2.d.c.531.1		4
248.227	odd	30		961.2.g.a.844.1		8
248.235	odd	30		961.2.g.a.448.1		8
248.243	even	6		961.2.c.c.521.1		4
744.371	odd	2		8649.2.a.c.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
31.2.a.a.1.1	✓	2	8.3	odd	2
279.2.a.a.1.2		2	24.11	even	2
496.2.a.i.1.1		2	8.5	even	2
775.2.a.d.1.2		2	40.19	odd	2
775.2.b.d.249.2		4	40.27	even	4
775.2.b.d.249.3		4	40.3	even	4
961.2.a.f.1.1		2	248.123	even	2
961.2.c.c.439.1		4	248.99	even	6
961.2.c.c.521.1		4	248.243	even	6
961.2.c.e.439.1		4	248.211	odd	6
961.2.c.e.521.1		4	248.67	odd	6
961.2.d.a.531.1		4	248.91	even	10
961.2.d.a.628.1		4	248.139	even	10
961.2.d.c.531.1		4	248.219	odd	10
961.2.d.c.628.1		4	248.171	odd	10
961.2.d.d.374.1		4	248.163	odd	10
961.2.d.d.388.1		4	248.35	odd	10
961.2.d.g.374.1		4	248.147	even	10
961.2.d.g.388.1		4	248.27	even	10
961.2.g.a.448.1		8	248.235	odd	30
961.2.g.a.547.1		8	248.19	odd	30
961.2.g.a.844.1		8	248.227	odd	30
961.2.g.a.846.1		8	248.59	odd	30
961.2.g.d.448.1		8	248.75	even	30
961.2.g.d.547.1		8	248.43	even	30
961.2.g.d.844.1		8	248.83	even	30
961.2.g.d.846.1		8	248.3	even	30
961.2.g.e.235.1		8	248.179	even	30
961.2.g.e.338.1		8	248.11	even	30
961.2.g.e.732.1		8	248.115	even	30
961.2.g.e.816.1		8	248.203	even	30
961.2.g.h.235.1		8	248.131	odd	30
961.2.g.h.338.1		8	248.51	odd	30
961.2.g.h.732.1		8	248.195	odd	30
961.2.g.h.816.1		8	248.107	odd	30
1519.2.a.a.1.1		2	56.27	even	2
1984.2.a.n.1.2		2	1.1	even	1	trivial
1984.2.a.r.1.1		2	4.3	odd	2
3751.2.a.b.1.2		2	88.43	even	2
4464.2.a.bf.1.2		2	24.5	odd	2
5239.2.a.f.1.2		2	104.51	odd	2
6975.2.a.y.1.1		2	120.59	even	2
8649.2.a.c.1.2		2	744.371	odd	2
8959.2.a.b.1.1		2	136.67	odd	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:	\( N \)	\(=\)	\( 1984 = 2^{6} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1984.a (trivial)

\(f(q)\)	\(=\)	\(q+1.23607 q^{3} -1.00000 q^{5} +4.23607 q^{7} -1.47214 q^{9} +2.00000 q^{11} -1.23607 q^{13} -1.23607 q^{15} +5.23607 q^{17} +2.23607 q^{19} +5.23607 q^{21} +7.70820 q^{23} -4.00000 q^{25} -5.52786 q^{27} -7.23607 q^{29} -1.00000 q^{31} +2.47214 q^{33} -4.23607 q^{35} +2.00000 q^{37} -1.52786 q^{39} +7.00000 q^{41} -3.23607 q^{43} +1.47214 q^{45} +6.47214 q^{47} +10.9443 q^{49} +6.47214 q^{51} +1.52786 q^{53} -2.00000 q^{55} +2.76393 q^{57} -2.23607 q^{59} +14.1803 q^{61} -6.23607 q^{63} +1.23607 q^{65} +8.00000 q^{67} +9.52786 q^{69} -13.1803 q^{71} -0.472136 q^{73} -4.94427 q^{75} +8.47214 q^{77} -1.70820 q^{79} -2.41641 q^{81} +2.94427 q^{83} -5.23607 q^{85} -8.94427 q^{87} -1.70820 q^{89} -5.23607 q^{91} -1.23607 q^{93} -2.23607 q^{95} +1.94427 q^{97} -2.94427 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} - 2 q^{5} + 4 q^{7} + 6 q^{9} + 4 q^{11} + 2 q^{13} + 2 q^{15} + 6 q^{17} + 6 q^{21} + 2 q^{23} - 8 q^{25} - 20 q^{27} - 10 q^{29} - 2 q^{31} - 4 q^{33} - 4 q^{35} + 4 q^{37} - 12 q^{39} + 14 q^{41}+ \cdots + 12 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 - 2 * q^5 + 4 * q^7 + 6 * q^9 + 4 * q^11 + 2 * q^13 + 2 * q^15 + 6 * q^17 + 6 * q^21 + 2 * q^23 - 8 * q^25 - 20 * q^27 - 10 * q^29 - 2 * q^31 - 4 * q^33 - 4 * q^35 + 4 * q^37 - 12 * q^39 + 14 * q^41 - 2 * q^43 - 6 * q^45 + 4 * q^47 + 4 * q^49 + 4 * q^51 + 12 * q^53 - 4 * q^55 + 10 * q^57 + 6 * q^61 - 8 * q^63 - 2 * q^65 + 16 * q^67 + 28 * q^69 - 4 * q^71 + 8 * q^73 + 8 * q^75 + 8 * q^77 + 10 * q^79 + 22 * q^81 - 12 * q^83 - 6 * q^85 + 10 * q^89 - 6 * q^91 + 2 * q^93 - 14 * q^97 + 12 * q^99

Introduction

L-functions

Modular forms

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Representations

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Database

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