LMFDB - Embedded newform 1600.4.a.cs.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1600,4,Mod(1,1600)] 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("1600.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1600, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 1600 = 2^{6} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1600.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,0,5,0,0,0,2,0,18,0,-31,0,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	yes
Analytic conductor:	$94.4030560092$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.5685.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 11x + 6 $ x^3 - x^2 - 11*x + 6
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 800)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$0.533386$ of defining polynomial
Character	$\chi$	$=$	1600.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.06677 q^{3} -28.8620 q^{7} -22.7285 q^{9} +18.7952 q^{11} +86.7195 q^{13} -64.7968 q^{17} +27.2566 q^{19} -59.6512 q^{21} +102.125 q^{23} -102.777 q^{27} -8.87408 q^{29} +272.771 q^{31} +38.8455 q^{33} +82.4677 q^{37} +179.230 q^{39} -249.119 q^{41} +137.227 q^{43} -439.114 q^{47} +490.015 q^{49} -133.920 q^{51} -490.724 q^{53} +56.3332 q^{57} -530.319 q^{59} +407.580 q^{61} +655.989 q^{63} +595.664 q^{67} +211.068 q^{69} +569.274 q^{71} -435.927 q^{73} -542.468 q^{77} -678.589 q^{79} +401.251 q^{81} -1277.18 q^{83} -18.3407 q^{87} -711.431 q^{89} -2502.90 q^{91} +563.756 q^{93} -1741.08 q^{97} -427.186 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 5 q^{3} + 2 q^{7} + 18 q^{9} - 31 q^{11} - 8 q^{13} - 89 q^{17} - 87 q^{19} + 70 q^{21} + 122 q^{23} + 143 q^{27} - 84 q^{29} + 294 q^{31} - 209 q^{33} + 94 q^{37} - 32 q^{39} - 345 q^{41} - 412 q^{43}+ \cdots - 1954 q^{99}+O(q^{100}) $ 3 * q + 5 * q^3 + 2 * q^7 + 18 * q^9 - 31 * q^11 - 8 * q^13 - 89 * q^17 - 87 * q^19 + 70 * q^21 + 122 * q^23 + 143 * q^27 - 84 * q^29 + 294 * q^31 - 209 * q^33 + 94 * q^37 - 32 * q^39 - 345 * q^41 - 412 * q^43 - 824 * q^47 + 359 * q^49 - 1351 * q^51 - 74 * q^53 - 913 * q^57 - 1040 * q^59 - 482 * q^61 + 1532 * q^63 + 735 * q^67 + 614 * q^69 + 2048 * q^71 - 15 * q^73 - 1474 * q^77 + 998 * q^79 - 549 * q^81 - 1221 * q^83 - 2564 * q^87 + 1189 * q^89 - 4032 * q^91 - 454 * q^93 - 1010 * q^97 - 1954 * 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
$3$	2.06677	0.397751	0.198875	−	0.980025i	$-0.436271\pi$
$3$	2.06677	0.397751	0.198875	+	0.980025i	$0.436271\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−28.8620	−1.55840	−0.779201	−	0.626775i	$-0.784375\pi$
$7$	−28.8620	−1.55840	−0.779201	+	0.626775i	$0.784375\pi$
$8$	0	0
$9$	−22.7285	−0.841795
$10$	0	0
$11$	18.7952	0.515179	0.257590	−	0.966254i	$-0.417072\pi$
$11$	18.7952	0.515179	0.257590	+	0.966254i	$0.417072\pi$
$12$	0	0
$13$	86.7195	1.85013	0.925064	−	0.379811i	$-0.124011\pi$
$13$	86.7195	1.85013	0.925064	+	0.379811i	$0.124011\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−64.7968	−0.924443	−0.462222	−	0.886764i	$-0.652948\pi$
$17$	−64.7968	−0.924443	−0.462222	+	0.886764i	$0.652948\pi$
$18$	0	0
$19$	27.2566	0.329110	0.164555	−	0.986368i	$-0.447381\pi$
$19$	27.2566	0.329110	0.164555	+	0.986368i	$0.447381\pi$
$20$	0	0
$21$	−59.6512	−0.619855
$22$	0	0
$23$	102.125	0.925846	0.462923	−	0.886399i	$-0.346801\pi$
$23$	102.125	0.925846	0.462923	+	0.886399i	$0.346801\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−102.777	−0.732575
$28$	0	0
$29$	−8.87408	−0.0568233	−0.0284117	−	0.999596i	$-0.509045\pi$
$29$	−8.87408	−0.0568233	−0.0284117	+	0.999596i	$0.509045\pi$
$30$	0	0
$31$	272.771	1.58036	0.790180	−	0.612874i	$-0.209987\pi$
$31$	272.771	1.58036	0.790180	+	0.612874i	$0.209987\pi$
$32$	0	0
$33$	38.8455	0.204913
$34$	0	0
$35$	0	0
$36$	0	0
$37$	82.4677	0.366422	0.183211	−	0.983074i	$-0.441351\pi$
$37$	82.4677	0.366422	0.183211	+	0.983074i	$0.441351\pi$
$38$	0	0
$39$	179.230	0.735890
$40$	0	0
$41$	−249.119	−0.948923	−0.474461	−	0.880276i	$-0.657357\pi$
$41$	−249.119	−0.948923	−0.474461	+	0.880276i	$0.657357\pi$
$42$	0	0
$43$	137.227	0.486672	0.243336	−	0.969942i	$-0.421758\pi$
$43$	137.227	0.486672	0.243336	+	0.969942i	$0.421758\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−439.114	−1.36280	−0.681398	−	0.731913i	$-0.738628\pi$
$47$	−439.114	−1.36280	−0.681398	+	0.731913i	$0.738628\pi$
$48$	0	0
$49$	490.015	1.42861
$50$	0	0
$51$	−133.920	−0.367698
$52$	0	0
$53$	−490.724	−1.27181	−0.635906	−	0.771766i	$-0.719374\pi$
$53$	−490.724	−1.27181	−0.635906	+	0.771766i	$0.719374\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	56.3332	0.130904
$58$	0	0
$59$	−530.319	−1.17020	−0.585099	−	0.810962i	$-0.698944\pi$
$59$	−530.319	−1.17020	−0.585099	+	0.810962i	$0.698944\pi$
$60$	0	0
$61$	407.580	0.855496	0.427748	−	0.903898i	$-0.359307\pi$
$61$	407.580	0.855496	0.427748	+	0.903898i	$0.359307\pi$
$62$	0	0
$63$	655.989	1.31185
$64$	0	0
$65$	0	0
$66$	0	0
$67$	595.664	1.08615	0.543074	−	0.839685i	$-0.317260\pi$
$67$	595.664	1.08615	0.543074	+	0.839685i	$0.317260\pi$
$68$	0	0
$69$	211.068	0.368256
$70$	0	0
$71$	569.274	0.951555	0.475778	−	0.879566i	$-0.342167\pi$
$71$	569.274	0.951555	0.475778	+	0.879566i	$0.342167\pi$
$72$	0	0
$73$	−435.927	−0.698923	−0.349461	−	0.936951i	$-0.613635\pi$
$73$	−435.927	−0.698923	−0.349461	+	0.936951i	$0.613635\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−542.468	−0.802856
$78$	0	0
$79$	−678.589	−0.966421	−0.483210	−	0.875504i	$-0.660529\pi$
$79$	−678.589	−0.966421	−0.483210	+	0.875504i	$0.660529\pi$
$80$	0	0
$81$	401.251	0.550413
$82$	0	0
$83$	−1277.18	−1.68902	−0.844508	−	0.535543i	$-0.820107\pi$
$83$	−1277.18	−1.68902	−0.844508	+	0.535543i	$0.820107\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−18.3407	−0.0226015
$88$	0	0
$89$	−711.431	−0.847321	−0.423660	−	0.905821i	$-0.639255\pi$
$89$	−711.431	−0.847321	−0.423660	+	0.905821i	$0.639255\pi$
$90$	0	0
$91$	−2502.90	−2.88324
$92$	0	0
$93$	563.756	0.628589
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−1741.08	−1.82247	−0.911237	−	0.411882i	$-0.864871\pi$
$97$	−1741.08	−1.82247	−0.911237	+	0.411882i	$0.864871\pi$
$98$	0	0
$99$	−427.186	−0.433675
$100$	0	0
$101$	−1768.54	−1.74234	−0.871170	−	0.490981i	$-0.836639\pi$
$101$	−1768.54	−1.74234	−0.871170	+	0.490981i	$0.836639\pi$
$102$	0	0
$103$	−24.9858	−0.0239022	−0.0119511	−	0.999929i	$-0.503804\pi$
$103$	−24.9858	−0.0239022	−0.0119511	+	0.999929i	$0.503804\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1027.18	0.928052	0.464026	−	0.885821i	$-0.346404\pi$
$107$	1027.18	0.928052	0.464026	+	0.885821i	$0.346404\pi$
$108$	0	0
$109$	418.035	0.367344	0.183672	−	0.982988i	$-0.441202\pi$
$109$	418.035	0.367344	0.183672	+	0.982988i	$0.441202\pi$
$110$	0	0
$111$	170.442	0.145744
$112$	0	0
$113$	−527.349	−0.439016	−0.219508	−	0.975611i	$-0.570445\pi$
$113$	−527.349	−0.439016	−0.219508	+	0.975611i	$0.570445\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−1971.00	−1.55743
$118$	0	0
$119$	1870.17	1.44065
$120$	0	0
$121$	−977.740	−0.734590
$122$	0	0
$123$	−514.872	−0.377435
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−1958.66	−1.36853	−0.684264	−	0.729234i	$-0.739877\pi$
$127$	−1958.66	−1.36853	−0.684264	+	0.729234i	$0.739877\pi$
$128$	0	0
$129$	283.617	0.193574
$130$	0	0
$131$	−1566.40	−1.04471	−0.522354	−	0.852729i	$-0.674946\pi$
$131$	−1566.40	−1.04471	−0.522354	+	0.852729i	$0.674946\pi$
$132$	0	0
$133$	−786.680	−0.512885
$134$	0	0
$135$	0	0
$136$	0	0
$137$	496.635	0.309711	0.154856	−	0.987937i	$-0.450509\pi$
$137$	496.635	0.309711	0.154856	+	0.987937i	$0.450509\pi$
$138$	0	0
$139$	−2675.60	−1.63267	−0.816335	−	0.577579i	$-0.803997\pi$
$139$	−2675.60	−1.63267	−0.816335	+	0.577579i	$0.803997\pi$
$140$	0	0
$141$	−907.550	−0.542053
$142$	0	0
$143$	1629.91	0.953148
$144$	0	0
$145$	0	0
$146$	0	0
$147$	1012.75	0.568232
$148$	0	0
$149$	1454.05	0.799465	0.399732	−	0.916632i	$-0.369103\pi$
$149$	1454.05	0.799465	0.399732	+	0.916632i	$0.369103\pi$
$150$	0	0
$151$	2359.49	1.27160	0.635802	−	0.771852i	$-0.280669\pi$
$151$	2359.49	1.27160	0.635802	+	0.771852i	$0.280669\pi$
$152$	0	0
$153$	1472.73	0.778191
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−533.577	−0.271236	−0.135618	−	0.990761i	$-0.543302\pi$
$157$	−533.577	−0.271236	−0.135618	+	0.990761i	$0.543302\pi$
$158$	0	0
$159$	−1014.21	−0.505864
$160$	0	0
$161$	−2947.52	−1.44284
$162$	0	0
$163$	−1190.37	−0.572004	−0.286002	−	0.958229i	$-0.592326\pi$
$163$	−1190.37	−0.572004	−0.286002	+	0.958229i	$0.592326\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	294.777	0.136590	0.0682950	−	0.997665i	$-0.478244\pi$
$167$	294.777	0.136590	0.0682950	+	0.997665i	$0.478244\pi$
$168$	0	0
$169$	5323.28	2.42298
$170$	0	0
$171$	−619.500	−0.277043
$172$	0	0
$173$	−2806.61	−1.23343	−0.616713	−	0.787188i	$-0.711536\pi$
$173$	−2806.61	−1.23343	−0.616713	+	0.787188i	$0.711536\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−1096.05	−0.465447
$178$	0	0
$179$	4593.72	1.91816	0.959079	−	0.283137i	$-0.0913751\pi$
$179$	4593.72	1.91816	0.959079	+	0.283137i	$0.0913751\pi$
$180$	0	0
$181$	1411.60	0.579689	0.289844	−	0.957074i	$-0.406396\pi$
$181$	1411.60	0.579689	0.289844	+	0.957074i	$0.406396\pi$
$182$	0	0
$183$	842.375	0.340274
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−1217.87	−0.476254
$188$	0	0
$189$	2966.36	1.14165
$190$	0	0
$191$	3918.19	1.48435	0.742174	−	0.670207i	$-0.233795\pi$
$191$	3918.19	1.48435	0.742174	+	0.670207i	$0.233795\pi$
$192$	0	0
$193$	−2227.06	−0.830606	−0.415303	−	0.909683i	$-0.636325\pi$
$193$	−2227.06	−0.830606	−0.415303	+	0.909683i	$0.636325\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−3321.98	−1.20143	−0.600714	−	0.799464i	$-0.705117\pi$
$197$	−3321.98	−1.20143	−0.600714	+	0.799464i	$0.705117\pi$
$198$	0	0
$199$	1939.22	0.690792	0.345396	−	0.938457i	$-0.387745\pi$
$199$	1939.22	0.690792	0.345396	+	0.938457i	$0.387745\pi$
$200$	0	0
$201$	1231.10	0.432016
$202$	0	0
$203$	256.124	0.0885535
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−2321.13	−0.779372
$208$	0	0
$209$	512.294	0.169551
$210$	0	0
$211$	4774.66	1.55783	0.778913	−	0.627132i	$-0.215771\pi$
$211$	4774.66	1.55783	0.778913	+	0.627132i	$0.215771\pi$
$212$	0	0
$213$	1176.56	0.378482
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−7872.73	−2.46284
$218$	0	0
$219$	−900.961	−0.277997
$220$	0	0
$221$	−5619.15	−1.71034
$222$	0	0
$223$	1874.06	0.562764	0.281382	−	0.959596i	$-0.409207\pi$
$223$	1874.06	0.562764	0.281382	+	0.959596i	$0.409207\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	1933.07	0.565209	0.282604	−	0.959237i	$-0.408802\pi$
$227$	1933.07	0.565209	0.282604	+	0.959237i	$0.408802\pi$
$228$	0	0
$229$	162.819	0.0469843	0.0234921	−	0.999724i	$-0.492522\pi$
$229$	162.819	0.0469843	0.0234921	+	0.999724i	$0.492522\pi$
$230$	0	0
$231$	−1121.16	−0.319337
$232$	0	0
$233$	−3496.60	−0.983133	−0.491566	−	0.870840i	$-0.663576\pi$
$233$	−3496.60	−0.983133	−0.491566	+	0.870840i	$0.663576\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−1402.49	−0.384394
$238$	0	0
$239$	3571.35	0.966576	0.483288	−	0.875462i	$-0.339442\pi$
$239$	3571.35	0.966576	0.483288	+	0.875462i	$0.339442\pi$
$240$	0	0
$241$	−2751.24	−0.735365	−0.367682	−	0.929951i	$-0.619849\pi$
$241$	−2751.24	−0.735365	−0.367682	+	0.929951i	$0.619849\pi$
$242$	0	0
$243$	3604.28	0.951502
$244$	0	0
$245$	0	0
$246$	0	0
$247$	2363.68	0.608896
$248$	0	0
$249$	−2639.63	−0.671807
$250$	0	0
$251$	2136.31	0.537222	0.268611	−	0.963249i	$-0.413435\pi$
$251$	2136.31	0.537222	0.268611	+	0.963249i	$0.413435\pi$
$252$	0	0
$253$	1919.46	0.476977
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−4446.43	−1.07922	−0.539612	−	0.841914i	$-0.681429\pi$
$257$	−4446.43	−1.07922	−0.539612	+	0.841914i	$0.681429\pi$
$258$	0	0
$259$	−2380.18	−0.571032
$260$	0	0
$261$	201.694	0.0478336
$262$	0	0
$263$	−6431.62	−1.50795	−0.753975	−	0.656903i	$-0.771866\pi$
$263$	−6431.62	−1.50795	−0.753975	+	0.656903i	$0.771866\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−1470.37	−0.337022
$268$	0	0
$269$	−1026.67	−0.232703	−0.116352	−	0.993208i	$-0.537120\pi$
$269$	−1026.67	−0.232703	−0.116352	+	0.993208i	$0.537120\pi$
$270$	0	0
$271$	−3555.80	−0.797046	−0.398523	−	0.917158i	$-0.630477\pi$
$271$	−3555.80	−0.797046	−0.398523	+	0.917158i	$0.630477\pi$
$272$	0	0
$273$	−5172.92	−1.14681
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−4641.61	−1.00681	−0.503406	−	0.864050i	$-0.667920\pi$
$277$	−4641.61	−1.00681	−0.503406	+	0.864050i	$0.667920\pi$
$278$	0	0
$279$	−6199.67	−1.33034
$280$	0	0
$281$	−5560.83	−1.18054	−0.590269	−	0.807206i	$-0.700978\pi$
$281$	−5560.83	−1.18054	−0.590269	+	0.807206i	$0.700978\pi$
$282$	0	0
$283$	−960.808	−0.201817	−0.100908	−	0.994896i	$-0.532175\pi$
$283$	−960.808	−0.201817	−0.100908	+	0.994896i	$0.532175\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	7190.07	1.47880
$288$	0	0
$289$	−714.373	−0.145405
$290$	0	0
$291$	−3598.42	−0.724890
$292$	0	0
$293$	−6120.18	−1.22029	−0.610144	−	0.792290i	$-0.708889\pi$
$293$	−6120.18	−1.22029	−0.610144	+	0.792290i	$0.708889\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−1931.72	−0.377407
$298$	0	0
$299$	8856.20	1.71293
$300$	0	0
$301$	−3960.64	−0.758431
$302$	0	0
$303$	−3655.17	−0.693017
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−8419.04	−1.56515	−0.782574	−	0.622558i	$-0.786093\pi$
$307$	−8419.04	−1.56515	−0.782574	+	0.622558i	$0.786093\pi$
$308$	0	0
$309$	−51.6400	−0.00950711
$310$	0	0
$311$	−10558.8	−1.92519	−0.962594	−	0.270947i	$-0.912663\pi$
$311$	−10558.8	−1.92519	−0.962594	+	0.270947i	$0.912663\pi$
$312$	0	0
$313$	7401.02	1.33652	0.668259	−	0.743928i	$-0.267040\pi$
$313$	7401.02	1.33652	0.668259	+	0.743928i	$0.267040\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−5845.54	−1.03570	−0.517852	−	0.855470i	$-0.673268\pi$
$317$	−5845.54	−1.03570	−0.517852	+	0.855470i	$0.673268\pi$
$318$	0	0
$319$	−166.790	−0.0292742
$320$	0	0
$321$	2122.96	0.369133
$322$	0	0
$323$	−1766.14	−0.304244
$324$	0	0
$325$	0	0
$326$	0	0
$327$	863.984	0.146111
$328$	0	0
$329$	12673.7	2.12378
$330$	0	0
$331$	2795.86	0.464273	0.232137	−	0.972683i	$-0.425428\pi$
$331$	2795.86	0.464273	0.232137	+	0.972683i	$0.425428\pi$
$332$	0	0
$333$	−1874.36	−0.308452
$334$	0	0
$335$	0	0
$336$	0	0
$337$	4087.70	0.660745	0.330372	−	0.943851i	$-0.392826\pi$
$337$	4087.70	0.660745	0.330372	+	0.943851i	$0.392826\pi$
$338$	0	0
$339$	−1089.91	−0.174619
$340$	0	0
$341$	5126.80	0.814169
$342$	0	0
$343$	−4243.14	−0.667954
$344$	0	0
$345$	0	0
$346$	0	0
$347$	8513.81	1.31713	0.658566	−	0.752523i	$-0.271163\pi$
$347$	8513.81	1.31713	0.658566	+	0.752523i	$0.271163\pi$
$348$	0	0
$349$	10514.7	1.61271	0.806356	−	0.591430i	$-0.201437\pi$
$349$	10514.7	1.61271	0.806356	+	0.591430i	$0.201437\pi$
$350$	0	0
$351$	−8912.81	−1.35536
$352$	0	0
$353$	1812.42	0.273273	0.136636	−	0.990621i	$-0.456371\pi$
$353$	1812.42	0.273273	0.136636	+	0.990621i	$0.456371\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	3865.21	0.573021
$358$	0	0
$359$	−5373.97	−0.790048	−0.395024	−	0.918671i	$-0.629264\pi$
$359$	−5373.97	−0.790048	−0.395024	+	0.918671i	$0.629264\pi$
$360$	0	0
$361$	−6116.08	−0.891687
$362$	0	0
$363$	−2020.77	−0.292184
$364$	0	0
$365$	0	0
$366$	0	0
$367$	4596.67	0.653799	0.326899	−	0.945059i	$-0.393996\pi$
$367$	4596.67	0.653799	0.326899	+	0.945059i	$0.393996\pi$
$368$	0	0
$369$	5662.09	0.798798
$370$	0	0
$371$	14163.3	1.98199
$372$	0	0
$373$	−7835.69	−1.08771	−0.543856	−	0.839179i	$-0.683036\pi$
$373$	−7835.69	−1.08771	−0.543856	+	0.839179i	$0.683036\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−769.556	−0.105130
$378$	0	0
$379$	−2494.50	−0.338084	−0.169042	−	0.985609i	$-0.554067\pi$
$379$	−2494.50	−0.338084	−0.169042	+	0.985609i	$0.554067\pi$
$380$	0	0
$381$	−4048.11	−0.544333
$382$	0	0
$383$	−7274.12	−0.970470	−0.485235	−	0.874384i	$-0.661266\pi$
$383$	−7274.12	−0.970470	−0.485235	+	0.874384i	$0.661266\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−3118.96	−0.409678
$388$	0	0
$389$	3306.03	0.430906	0.215453	−	0.976514i	$-0.430877\pi$
$389$	3306.03	0.430906	0.215453	+	0.976514i	$0.430877\pi$
$390$	0	0
$391$	−6617.35	−0.855892
$392$	0	0
$393$	−3237.39	−0.415533
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−8077.86	−1.02120	−0.510600	−	0.859819i	$-0.670577\pi$
$397$	−8077.86	−1.02120	−0.510600	+	0.859819i	$0.670577\pi$
$398$	0	0
$399$	−1625.89	−0.204000
$400$	0	0
$401$	12015.5	1.49632	0.748160	−	0.663518i	$-0.230937\pi$
$401$	12015.5	1.49632	0.748160	+	0.663518i	$0.230937\pi$
$402$	0	0
$403$	23654.6	2.92387
$404$	0	0
$405$	0	0
$406$	0	0
$407$	1550.00	0.188773
$408$	0	0
$409$	−7199.69	−0.870420	−0.435210	−	0.900329i	$-0.643326\pi$
$409$	−7199.69	−0.870420	−0.435210	+	0.900329i	$0.643326\pi$
$410$	0	0
$411$	1026.43	0.123188
$412$	0	0
$413$	15306.1	1.82364
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−5529.85	−0.649395
$418$	0	0
$419$	−1429.38	−0.166658	−0.0833292	−	0.996522i	$-0.526555\pi$
$419$	−1429.38	−0.166658	−0.0833292	+	0.996522i	$0.526555\pi$
$420$	0	0
$421$	−13142.9	−1.52148	−0.760741	−	0.649056i	$-0.775164\pi$
$421$	−13142.9	−1.52148	−0.760741	+	0.649056i	$0.775164\pi$
$422$	0	0
$423$	9980.39	1.14719
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−11763.6	−1.33321
$428$	0	0
$429$	3368.66	0.379115
$430$	0	0
$431$	7241.89	0.809349	0.404675	−	0.914461i	$-0.367385\pi$
$431$	7241.89	0.809349	0.404675	+	0.914461i	$0.367385\pi$
$432$	0	0
$433$	−5596.76	−0.621162	−0.310581	−	0.950547i	$-0.600524\pi$
$433$	−5596.76	−0.621162	−0.310581	+	0.950547i	$0.600524\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	2783.57	0.304705
$438$	0	0
$439$	−5214.92	−0.566958	−0.283479	−	0.958978i	$-0.591489\pi$
$439$	−5214.92	−0.566958	−0.283479	+	0.958978i	$0.591489\pi$
$440$	0	0
$441$	−11137.3	−1.20260
$442$	0	0
$443$	−2732.92	−0.293104	−0.146552	−	0.989203i	$-0.546818\pi$
$443$	−2732.92	−0.293104	−0.146552	+	0.989203i	$0.546818\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	3005.19	0.317988
$448$	0	0
$449$	−15217.8	−1.59949	−0.799747	−	0.600338i	$-0.795033\pi$
$449$	−15217.8	−1.59949	−0.799747	+	0.600338i	$0.795033\pi$
$450$	0	0
$451$	−4682.25	−0.488865
$452$	0	0
$453$	4876.52	0.505781
$454$	0	0
$455$	0	0
$456$	0	0
$457$	12772.1	1.30734	0.653670	−	0.756780i	$-0.273229\pi$
$457$	12772.1	1.30734	0.653670	+	0.756780i	$0.273229\pi$
$458$	0	0
$459$	6659.65	0.677224
$460$	0	0
$461$	10121.5	1.02257	0.511285	−	0.859411i	$-0.329170\pi$
$461$	10121.5	1.02257	0.511285	+	0.859411i	$0.329170\pi$
$462$	0	0
$463$	−5422.15	−0.544252	−0.272126	−	0.962262i	$-0.587727\pi$
$463$	−5422.15	−0.544252	−0.272126	+	0.962262i	$0.587727\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−16663.3	−1.65115	−0.825575	−	0.564292i	$-0.809149\pi$
$467$	−16663.3	−1.65115	−0.825575	+	0.564292i	$0.809149\pi$
$468$	0	0
$469$	−17192.0	−1.69265
$470$	0	0
$471$	−1102.78	−0.107884
$472$	0	0
$473$	2579.21	0.250724
$474$	0	0
$475$	0	0
$476$	0	0
$477$	11153.4	1.07060
$478$	0	0
$479$	6745.65	0.643458	0.321729	−	0.946832i	$-0.395736\pi$
$479$	6745.65	0.643458	0.321729	+	0.946832i	$0.395736\pi$
$480$	0	0
$481$	7151.56	0.677927
$482$	0	0
$483$	−6091.85	−0.573890
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−3148.75	−0.292985	−0.146492	−	0.989212i	$-0.546798\pi$
$487$	−3148.75	−0.292985	−0.146492	+	0.989212i	$0.546798\pi$
$488$	0	0
$489$	−2460.22	−0.227515
$490$	0	0
$491$	16871.1	1.55067	0.775336	−	0.631548i	$-0.217580\pi$
$491$	16871.1	1.55067	0.775336	+	0.631548i	$0.217580\pi$
$492$	0	0
$493$	575.012	0.0525299
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−16430.4	−1.48290
$498$	0	0
$499$	−6873.07	−0.616595	−0.308297	−	0.951290i	$-0.599759\pi$
$499$	−6873.07	−0.616595	−0.308297	+	0.951290i	$0.599759\pi$
$500$	0	0
$501$	609.237	0.0543288
$502$	0	0
$503$	12534.1	1.11107	0.555536	−	0.831492i	$-0.312513\pi$
$503$	12534.1	1.11107	0.555536	+	0.831492i	$0.312513\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	11002.0	0.963740
$508$	0	0
$509$	−6046.18	−0.526507	−0.263254	−	0.964727i	$-0.584796\pi$
$509$	−6046.18	−0.526507	−0.263254	+	0.964727i	$0.584796\pi$
$510$	0	0
$511$	12581.7	1.08920
$512$	0	0
$513$	−2801.36	−0.241098
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−8253.25	−0.702085
$518$	0	0
$519$	−5800.63	−0.490596
$520$	0	0
$521$	3643.76	0.306403	0.153202	−	0.988195i	$-0.451042\pi$
$521$	3643.76	0.306403	0.153202	+	0.988195i	$0.451042\pi$
$522$	0	0
$523$	14510.8	1.21322	0.606611	−	0.794999i	$-0.292529\pi$
$523$	14510.8	1.21322	0.606611	+	0.794999i	$0.292529\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−17674.7	−1.46095
$528$	0	0
$529$	−1737.56	−0.142809
$530$	0	0
$531$	12053.3	0.985067
$532$	0	0
$533$	−21603.5	−1.75563
$534$	0	0
$535$	0	0
$536$	0	0
$537$	9494.17	0.762949
$538$	0	0
$539$	9209.94	0.735993
$540$	0	0
$541$	5064.47	0.402475	0.201237	−	0.979543i	$-0.435504\pi$
$541$	5064.47	0.402475	0.201237	+	0.979543i	$0.435504\pi$
$542$	0	0
$543$	2917.46	0.230571
$544$	0	0
$545$	0	0
$546$	0	0
$547$	11557.1	0.903376	0.451688	−	0.892176i	$-0.350822\pi$
$547$	11557.1	0.903376	0.451688	+	0.892176i	$0.350822\pi$
$548$	0	0
$549$	−9263.66	−0.720152
$550$	0	0
$551$	−241.877	−0.0187011
$552$	0	0
$553$	19585.4	1.50607
$554$	0	0
$555$	0	0
$556$	0	0
$557$	13511.4	1.02782	0.513911	−	0.857843i	$-0.328196\pi$
$557$	13511.4	1.02782	0.513911	+	0.857843i	$0.328196\pi$
$558$	0	0
$559$	11900.3	0.900407
$560$	0	0
$561$	−2517.06	−0.189430
$562$	0	0
$563$	−9383.50	−0.702429	−0.351214	−	0.936295i	$-0.614231\pi$
$563$	−9383.50	−0.702429	−0.351214	+	0.936295i	$0.614231\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−11580.9	−0.857764
$568$	0	0
$569$	20395.3	1.50266	0.751332	−	0.659924i	$-0.229412\pi$
$569$	20395.3	1.50266	0.751332	+	0.659924i	$0.229412\pi$
$570$	0	0
$571$	−24939.7	−1.82783	−0.913917	−	0.405902i	$-0.866957\pi$
$571$	−24939.7	−1.82783	−0.913917	+	0.405902i	$0.866957\pi$
$572$	0	0
$573$	8098.02	0.590400
$574$	0	0
$575$	0	0
$576$	0	0
$577$	13633.6	0.983665	0.491832	−	0.870690i	$-0.336327\pi$
$577$	13633.6	0.983665	0.491832	+	0.870690i	$0.336327\pi$
$578$	0	0
$579$	−4602.82	−0.330374
$580$	0	0
$581$	36861.9	2.63217
$582$	0	0
$583$	−9223.26	−0.655212
$584$	0	0
$585$	0	0
$586$	0	0
$587$	1261.57	0.0887065	0.0443532	−	0.999016i	$-0.485877\pi$
$587$	1261.57	0.0887065	0.0443532	+	0.999016i	$0.485877\pi$
$588$	0	0
$589$	7434.82	0.520112
$590$	0	0
$591$	−6865.78	−0.477869
$592$	0	0
$593$	−9137.25	−0.632752	−0.316376	−	0.948634i	$-0.602466\pi$
$593$	−9137.25	−0.632752	−0.316376	+	0.948634i	$0.602466\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	4007.92	0.274763
$598$	0	0
$599$	−5793.88	−0.395211	−0.197606	−	0.980282i	$-0.563317\pi$
$599$	−5793.88	−0.395211	−0.197606	+	0.980282i	$0.563317\pi$
$600$	0	0
$601$	10763.6	0.730545	0.365272	−	0.930901i	$-0.380976\pi$
$601$	10763.6	0.730545	0.365272	+	0.930901i	$0.380976\pi$
$602$	0	0
$603$	−13538.5	−0.914314
$604$	0	0
$605$	0	0
$606$	0	0
$607$	10668.7	0.713393	0.356697	−	0.934220i	$-0.383903\pi$
$607$	10668.7	0.713393	0.356697	+	0.934220i	$0.383903\pi$
$608$	0	0
$609$	529.350	0.0352222
$610$	0	0
$611$	−38079.8	−2.52135
$612$	0	0
$613$	−1699.04	−0.111947	−0.0559735	−	0.998432i	$-0.517826\pi$
$613$	−1699.04	−0.111947	−0.0559735	+	0.998432i	$0.517826\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	13951.6	0.910324	0.455162	−	0.890409i	$-0.349581\pi$
$617$	13951.6	0.910324	0.455162	+	0.890409i	$0.349581\pi$
$618$	0	0
$619$	−20560.1	−1.33502	−0.667512	−	0.744599i	$-0.732641\pi$
$619$	−20560.1	−1.33502	−0.667512	+	0.744599i	$0.732641\pi$
$620$	0	0
$621$	−10496.1	−0.678251
$622$	0	0
$623$	20533.3	1.32047
$624$	0	0
$625$	0	0
$626$	0	0
$627$	1058.79	0.0674389
$628$	0	0
$629$	−5343.64	−0.338736
$630$	0	0
$631$	14838.5	0.936154	0.468077	−	0.883688i	$-0.344947\pi$
$631$	14838.5	0.936154	0.468077	+	0.883688i	$0.344947\pi$
$632$	0	0
$633$	9868.14	0.619626
$634$	0	0
$635$	0	0
$636$	0	0
$637$	42493.9	2.64312
$638$	0	0
$639$	−12938.7	−0.801014
$640$	0	0
$641$	7766.13	0.478539	0.239270	−	0.970953i	$-0.423092\pi$
$641$	7766.13	0.478539	0.239270	+	0.970953i	$0.423092\pi$
$642$	0	0
$643$	−22756.2	−1.39567	−0.697834	−	0.716259i	$-0.745853\pi$
$643$	−22756.2	−1.39567	−0.697834	+	0.716259i	$0.745853\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	9541.64	0.579785	0.289892	−	0.957059i	$-0.406381\pi$
$647$	9541.64	0.579785	0.289892	+	0.957059i	$0.406381\pi$
$648$	0	0
$649$	−9967.47	−0.602862
$650$	0	0
$651$	−16271.1	−0.979594
$652$	0	0
$653$	23220.2	1.39154	0.695770	−	0.718264i	$-0.255063\pi$
$653$	23220.2	1.39154	0.695770	+	0.718264i	$0.255063\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	9907.94	0.588349
$658$	0	0
$659$	3704.40	0.218973	0.109486	−	0.993988i	$-0.465079\pi$
$659$	3704.40	0.218973	0.109486	+	0.993988i	$0.465079\pi$
$660$	0	0
$661$	7879.78	0.463673	0.231837	−	0.972755i	$-0.425527\pi$
$661$	7879.78	0.463673	0.231837	+	0.972755i	$0.425527\pi$
$662$	0	0
$663$	−11613.5	−0.680288
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−906.263	−0.0526096
$668$	0	0
$669$	3873.26	0.223840
$670$	0	0
$671$	7660.55	0.440734
$672$	0	0
$673$	−22045.2	−1.26267	−0.631336	−	0.775509i	$-0.717493\pi$
$673$	−22045.2	−1.26267	−0.631336	+	0.775509i	$0.717493\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−4640.91	−0.263463	−0.131732	−	0.991285i	$-0.542054\pi$
$677$	−4640.91	−0.263463	−0.131732	+	0.991285i	$0.542054\pi$
$678$	0	0
$679$	50251.1	2.84015
$680$	0	0
$681$	3995.22	0.224812
$682$	0	0
$683$	−2734.15	−0.153176	−0.0765880	−	0.997063i	$-0.524403\pi$
$683$	−2734.15	−0.153176	−0.0765880	+	0.997063i	$0.524403\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	336.510	0.0186880
$688$	0	0
$689$	−42555.3	−2.35302
$690$	0	0
$691$	1039.34	0.0572189	0.0286095	−	0.999591i	$-0.490892\pi$
$691$	1039.34	0.0572189	0.0286095	+	0.999591i	$0.490892\pi$
$692$	0	0
$693$	12329.5	0.675840
$694$	0	0
$695$	0	0
$696$	0	0
$697$	16142.1	0.877225
$698$	0	0
$699$	−7226.68	−0.391042
$700$	0	0
$701$	−29579.2	−1.59371	−0.796856	−	0.604169i	$-0.793505\pi$
$701$	−29579.2	−1.59371	−0.796856	+	0.604169i	$0.793505\pi$
$702$	0	0
$703$	2247.79	0.120593
$704$	0	0
$705$	0	0
$706$	0	0
$707$	51043.6	2.71527
$708$	0	0
$709$	−2106.06	−0.111558	−0.0557790	−	0.998443i	$-0.517764\pi$
$709$	−2106.06	−0.111558	−0.0557790	+	0.998443i	$0.517764\pi$
$710$	0	0
$711$	15423.3	0.813528
$712$	0	0
$713$	27856.7	1.46317
$714$	0	0
$715$	0	0
$716$	0	0
$717$	7381.17	0.384456
$718$	0	0
$719$	−26617.9	−1.38064	−0.690321	−	0.723503i	$-0.742531\pi$
$719$	−26617.9	−1.38064	−0.690321	+	0.723503i	$0.742531\pi$
$720$	0	0
$721$	721.141	0.0372492
$722$	0	0
$723$	−5686.19	−0.292492
$724$	0	0
$725$	0	0
$726$	0	0
$727$	2710.31	0.138266	0.0691332	−	0.997607i	$-0.477977\pi$
$727$	2710.31	0.138266	0.0691332	+	0.997607i	$0.477977\pi$
$728$	0	0
$729$	−3384.54	−0.171952
$730$	0	0
$731$	−8891.87	−0.449901
$732$	0	0
$733$	31768.6	1.60082	0.800408	−	0.599455i	$-0.204616\pi$
$733$	31768.6	1.60082	0.800408	+	0.599455i	$0.204616\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	11195.6	0.559561
$738$	0	0
$739$	−7867.06	−0.391603	−0.195801	−	0.980644i	$-0.562731\pi$
$739$	−7867.06	−0.391603	−0.195801	+	0.980644i	$0.562731\pi$
$740$	0	0
$741$	4885.19	0.242189
$742$	0	0
$743$	6820.78	0.336783	0.168392	−	0.985720i	$-0.446143\pi$
$743$	6820.78	0.336783	0.168392	+	0.985720i	$0.446143\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	29028.2	1.42180
$748$	0	0
$749$	−29646.6	−1.44628
$750$	0	0
$751$	10534.9	0.511884	0.255942	−	0.966692i	$-0.417614\pi$
$751$	10534.9	0.511884	0.255942	+	0.966692i	$0.417614\pi$
$752$	0	0
$753$	4415.27	0.213680
$754$	0	0
$755$	0	0
$756$	0	0
$757$	12358.9	0.593384	0.296692	−	0.954973i	$-0.404116\pi$
$757$	12358.9	0.593384	0.296692	+	0.954973i	$0.404116\pi$
$758$	0	0
$759$	3967.08	0.189718
$760$	0	0
$761$	19823.6	0.944292	0.472146	−	0.881520i	$-0.343480\pi$
$761$	19823.6	0.944292	0.472146	+	0.881520i	$0.343480\pi$
$762$	0	0
$763$	−12065.3	−0.572470
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−45989.1	−2.16502
$768$	0	0
$769$	1342.30	0.0629449	0.0314725	−	0.999505i	$-0.489980\pi$
$769$	1342.30	0.0629449	0.0314725	+	0.999505i	$0.489980\pi$
$770$	0	0
$771$	−9189.76	−0.429262
$772$	0	0
$773$	−25002.9	−1.16338	−0.581689	−	0.813411i	$-0.697608\pi$
$773$	−25002.9	−1.16338	−0.581689	+	0.813411i	$0.697608\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−4919.30	−0.227128
$778$	0	0
$779$	−6790.13	−0.312300
$780$	0	0
$781$	10699.6	0.490222
$782$	0	0
$783$	912.055	0.0416273
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−9270.25	−0.419884	−0.209942	−	0.977714i	$-0.567328\pi$
$787$	−9270.25	−0.419884	−0.209942	+	0.977714i	$0.567328\pi$
$788$	0	0
$789$	−13292.7	−0.599788
$790$	0	0
$791$	15220.3	0.684163
$792$	0	0
$793$	35345.1	1.58278
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−4918.58	−0.218601	−0.109301	−	0.994009i	$-0.534861\pi$
$797$	−4918.58	−0.218601	−0.109301	+	0.994009i	$0.534861\pi$
$798$	0	0
$799$	28453.2	1.25983
$800$	0	0
$801$	16169.7	0.713270
$802$	0	0
$803$	−8193.34	−0.360071
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−2121.89	−0.0925578
$808$	0	0
$809$	−23789.6	−1.03387	−0.516933	−	0.856026i	$-0.672926\pi$
$809$	−23789.6	−1.03387	−0.516933	+	0.856026i	$0.672926\pi$
$810$	0	0
$811$	1656.24	0.0717122	0.0358561	−	0.999357i	$-0.488584\pi$
$811$	1656.24	0.0717122	0.0358561	+	0.999357i	$0.488584\pi$
$812$	0	0
$813$	−7349.03	−0.317025
$814$	0	0
$815$	0	0
$816$	0	0
$817$	3740.34	0.160169
$818$	0	0
$819$	56887.0	2.42710
$820$	0	0
$821$	33677.4	1.43161	0.715804	−	0.698301i	$-0.246060\pi$
$821$	33677.4	1.43161	0.715804	+	0.698301i	$0.246060\pi$
$822$	0	0
$823$	20918.5	0.885995	0.442998	−	0.896523i	$-0.353915\pi$
$823$	20918.5	0.885995	0.442998	+	0.896523i	$0.353915\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−46941.5	−1.97378	−0.986891	−	0.161391i	$-0.948402\pi$
$827$	−46941.5	−1.97378	−0.986891	+	0.161391i	$0.948402\pi$
$828$	0	0
$829$	1512.56	0.0633695	0.0316848	−	0.999498i	$-0.489913\pi$
$829$	1512.56	0.0633695	0.0316848	+	0.999498i	$0.489913\pi$
$830$	0	0
$831$	−9593.14	−0.400460
$832$	0	0
$833$	−31751.4	−1.32067
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−28034.7	−1.15773
$838$	0	0
$839$	−8785.71	−0.361521	−0.180761	−	0.983527i	$-0.557856\pi$
$839$	−8785.71	−0.361521	−0.180761	+	0.983527i	$0.557856\pi$
$840$	0	0
$841$	−24310.3	−0.996771
$842$	0	0
$843$	−11493.0	−0.469560
$844$	0	0
$845$	0	0
$846$	0	0
$847$	28219.5	1.14479
$848$	0	0
$849$	−1985.77	−0.0802726
$850$	0	0
$851$	8421.98	0.339250
$852$	0	0
$853$	−17742.0	−0.712162	−0.356081	−	0.934455i	$-0.615887\pi$
$853$	−17742.0	−0.712162	−0.356081	+	0.934455i	$0.615887\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	48201.1	1.92126	0.960629	−	0.277835i	$-0.0896169\pi$
$857$	48201.1	1.92126	0.960629	+	0.277835i	$0.0896169\pi$
$858$	0	0
$859$	−27332.1	−1.08563	−0.542817	−	0.839851i	$-0.682642\pi$
$859$	−27332.1	−1.08563	−0.542817	+	0.839851i	$0.682642\pi$
$860$	0	0
$861$	14860.2	0.588194
$862$	0	0
$863$	21577.8	0.851119	0.425560	−	0.904930i	$-0.360077\pi$
$863$	21577.8	0.851119	0.425560	+	0.904930i	$0.360077\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−1476.45	−0.0578348
$868$	0	0
$869$	−12754.2	−0.497880
$870$	0	0
$871$	51655.7	2.00951
$872$	0	0
$873$	39572.1	1.53415
$874$	0	0
$875$	0	0
$876$	0	0
$877$	25212.9	0.970785	0.485392	−	0.874296i	$-0.338677\pi$
$877$	25212.9	0.970785	0.485392	+	0.874296i	$0.338677\pi$
$878$	0	0
$879$	−12649.0	−0.485370
$880$	0	0
$881$	33688.3	1.28829	0.644147	−	0.764902i	$-0.277213\pi$
$881$	33688.3	1.28829	0.644147	+	0.764902i	$0.277213\pi$
$882$	0	0
$883$	25749.9	0.981372	0.490686	−	0.871336i	$-0.336746\pi$
$883$	25749.9	0.981372	0.490686	+	0.871336i	$0.336746\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−25524.9	−0.966227	−0.483114	−	0.875558i	$-0.660494\pi$
$887$	−25524.9	−0.966227	−0.483114	+	0.875558i	$0.660494\pi$
$888$	0	0
$889$	56530.9	2.13272
$890$	0	0
$891$	7541.60	0.283561
$892$	0	0
$893$	−11968.8	−0.448510
$894$	0	0
$895$	0	0
$896$	0	0
$897$	18303.7	0.681320
$898$	0	0
$899$	−2420.60	−0.0898013
$900$	0	0
$901$	31797.3	1.17572
$902$	0	0
$903$	−8185.75	−0.301666
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−24092.9	−0.882018	−0.441009	−	0.897503i	$-0.645379\pi$
$907$	−24092.9	−0.882018	−0.441009	+	0.897503i	$0.645379\pi$
$908$	0	0
$909$	40196.2	1.46669
$910$	0	0
$911$	−9445.56	−0.343519	−0.171759	−	0.985139i	$-0.554945\pi$
$911$	−9445.56	−0.343519	−0.171759	+	0.985139i	$0.554945\pi$
$912$	0	0
$913$	−24004.8	−0.870146
$914$	0	0
$915$	0	0
$916$	0	0
$917$	45209.4	1.62808
$918$	0	0
$919$	47739.0	1.71356	0.856782	−	0.515679i	$-0.172460\pi$
$919$	47739.0	1.71356	0.856782	+	0.515679i	$0.172460\pi$
$920$	0	0
$921$	−17400.2	−0.622538
$922$	0	0
$923$	49367.2	1.76050
$924$	0	0
$925$	0	0
$926$	0	0
$927$	567.889	0.0201207
$928$	0	0
$929$	−30097.7	−1.06294	−0.531471	−	0.847076i	$-0.678361\pi$
$929$	−30097.7	−1.06294	−0.531471	+	0.847076i	$0.678361\pi$
$930$	0	0
$931$	13356.1	0.470171
$932$	0	0
$933$	−21822.6	−0.765745
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−22895.9	−0.798266	−0.399133	−	0.916893i	$-0.630689\pi$
$937$	−22895.9	−0.798266	−0.399133	+	0.916893i	$0.630689\pi$
$938$	0	0
$939$	15296.2	0.531601
$940$	0	0
$941$	−31370.1	−1.08676	−0.543378	−	0.839488i	$-0.682855\pi$
$941$	−31370.1	−1.08676	−0.543378	+	0.839488i	$0.682855\pi$
$942$	0	0
$943$	−25441.2	−0.878556
$944$	0	0
$945$	0	0
$946$	0	0
$947$	26224.3	0.899867	0.449934	−	0.893062i	$-0.351448\pi$
$947$	26224.3	0.899867	0.449934	+	0.893062i	$0.351448\pi$
$948$	0	0
$949$	−37803.4	−1.29310
$950$	0	0
$951$	−12081.4	−0.411952
$952$	0	0
$953$	−57163.4	−1.94303	−0.971513	−	0.236988i	$-0.923840\pi$
$953$	−57163.4	−1.94303	−0.971513	+	0.236988i	$0.923840\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−344.718	−0.0116438
$958$	0	0
$959$	−14333.9	−0.482654
$960$	0	0
$961$	44613.2	1.49754
$962$	0	0
$963$	−23346.3	−0.781229
$964$	0	0
$965$	0	0
$966$	0	0
$967$	12182.2	0.405122	0.202561	−	0.979270i	$-0.435074\pi$
$967$	12182.2	0.405122	0.202561	+	0.979270i	$0.435074\pi$
$968$	0	0
$969$	−3650.21	−0.121013
$970$	0	0
$971$	39736.0	1.31327	0.656637	−	0.754207i	$-0.271978\pi$
$971$	39736.0	1.31327	0.656637	+	0.754207i	$0.271978\pi$
$972$	0	0
$973$	77223.0	2.54435
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−15841.8	−0.518756	−0.259378	−	0.965776i	$-0.583518\pi$
$977$	−15841.8	−0.518756	−0.259378	+	0.965776i	$0.583518\pi$
$978$	0	0
$979$	−13371.5	−0.436522
$980$	0	0
$981$	−9501.30	−0.309228
$982$	0	0
$983$	−19382.2	−0.628887	−0.314444	−	0.949276i	$-0.601818\pi$
$983$	−19382.2	−0.628887	−0.314444	+	0.949276i	$0.601818\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	26193.7	0.844736
$988$	0	0
$989$	14014.3	0.450584
$990$	0	0
$991$	−18065.8	−0.579090	−0.289545	−	0.957164i	$-0.593504\pi$
$991$	−18065.8	−0.579090	−0.289545	+	0.957164i	$0.593504\pi$
$992$	0	0
$993$	5778.41	0.184665
$994$	0	0
$995$	0	0
$996$	0	0
$997$	23743.9	0.754241	0.377120	−	0.926164i	$-0.376914\pi$
$997$	23743.9	0.754241	0.377120	+	0.926164i	$0.376914\pi$
$998$	0	0
$999$	−8475.82	−0.268431

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1600.4.a.cs.1.2		3
4.3	odd	2		1600.4.a.cr.1.2		3
5.4	even	2		1600.4.a.cq.1.2		3
8.3	odd	2		800.4.a.w.1.2	yes	3
8.5	even	2		800.4.a.v.1.2	yes	3
20.19	odd	2		1600.4.a.ct.1.2		3
40.3	even	4		800.4.c.m.449.4		6
40.13	odd	4		800.4.c.n.449.3		6
40.19	odd	2		800.4.a.u.1.2	✓	3
40.27	even	4		800.4.c.m.449.3		6
40.29	even	2		800.4.a.x.1.2	yes	3
40.37	odd	4		800.4.c.n.449.4		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
800.4.a.u.1.2	✓	3	40.19	odd	2
800.4.a.v.1.2	yes	3	8.5	even	2
800.4.a.w.1.2	yes	3	8.3	odd	2
800.4.a.x.1.2	yes	3	40.29	even	2
800.4.c.m.449.3		6	40.27	even	4
800.4.c.m.449.4		6	40.3	even	4
800.4.c.n.449.3		6	40.13	odd	4
800.4.c.n.449.4		6	40.37	odd	4
1600.4.a.cq.1.2		3	5.4	even	2
1600.4.a.cr.1.2		3	4.3	odd	2
1600.4.a.cs.1.2		3	1.1	even	1	trivial
1600.4.a.ct.1.2		3	20.19	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:	\( N \)	\(=\)	\( 1600 = 2^{6} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1600.a (trivial)

\(f(q)\)	\(=\)	\(q+2.06677 q^{3} -28.8620 q^{7} -22.7285 q^{9} +18.7952 q^{11} +86.7195 q^{13} -64.7968 q^{17} +27.2566 q^{19} -59.6512 q^{21} +102.125 q^{23} -102.777 q^{27} -8.87408 q^{29} +272.771 q^{31} +38.8455 q^{33} +82.4677 q^{37} +179.230 q^{39} -249.119 q^{41} +137.227 q^{43} -439.114 q^{47} +490.015 q^{49} -133.920 q^{51} -490.724 q^{53} +56.3332 q^{57} -530.319 q^{59} +407.580 q^{61} +655.989 q^{63} +595.664 q^{67} +211.068 q^{69} +569.274 q^{71} -435.927 q^{73} -542.468 q^{77} -678.589 q^{79} +401.251 q^{81} -1277.18 q^{83} -18.3407 q^{87} -711.431 q^{89} -2502.90 q^{91} +563.756 q^{93} -1741.08 q^{97} -427.186 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 5 q^{3} + 2 q^{7} + 18 q^{9} - 31 q^{11} - 8 q^{13} - 89 q^{17} - 87 q^{19} + 70 q^{21} + 122 q^{23} + 143 q^{27} - 84 q^{29} + 294 q^{31} - 209 q^{33} + 94 q^{37} - 32 q^{39} - 345 q^{41} - 412 q^{43}+ \cdots - 1954 q^{99}+O(q^{100}) \) 3 * q + 5 * q^3 + 2 * q^7 + 18 * q^9 - 31 * q^11 - 8 * q^13 - 89 * q^17 - 87 * q^19 + 70 * q^21 + 122 * q^23 + 143 * q^27 - 84 * q^29 + 294 * q^31 - 209 * q^33 + 94 * q^37 - 32 * q^39 - 345 * q^41 - 412 * q^43 - 824 * q^47 + 359 * q^49 - 1351 * q^51 - 74 * q^53 - 913 * q^57 - 1040 * q^59 - 482 * q^61 + 1532 * q^63 + 735 * q^67 + 614 * q^69 + 2048 * q^71 - 15 * q^73 - 1474 * q^77 + 998 * q^79 - 549 * q^81 - 1221 * q^83 - 2564 * q^87 + 1189 * q^89 - 4032 * q^91 - 454 * q^93 - 1010 * q^97 - 1954 * q^99

Introduction

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