LMFDB - Embedded newform 8032.2.a.j.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8032,2,Mod(1,8032)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8032, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("8032.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8032 = 2^{5} \cdot 251 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8032.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$64.1358429035$
Analytic rank:	$0$
Dimension:	$30$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Character	$\chi$	$=$	8032.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.50495 q^{3} +0.621089 q^{5} +1.81379 q^{7} +3.27480 q^{9} +O(q^{10})$	$q-2.50495 q^{3} +0.621089 q^{5} +1.81379 q^{7} +3.27480 q^{9} -0.483706 q^{11} -2.34549 q^{13} -1.55580 q^{15} -4.38147 q^{17} +5.17518 q^{19} -4.54347 q^{21} -7.24965 q^{23} -4.61425 q^{25} -0.688350 q^{27} -5.05610 q^{29} -4.08995 q^{31} +1.21166 q^{33} +1.12653 q^{35} +1.40181 q^{37} +5.87535 q^{39} +7.17219 q^{41} -4.88547 q^{43} +2.03394 q^{45} +6.60949 q^{47} -3.71015 q^{49} +10.9754 q^{51} -2.78109 q^{53} -0.300424 q^{55} -12.9636 q^{57} +10.7127 q^{59} +6.80533 q^{61} +5.93980 q^{63} -1.45676 q^{65} +10.7694 q^{67} +18.1600 q^{69} -3.98861 q^{71} +4.34320 q^{73} +11.5585 q^{75} -0.877344 q^{77} -13.7650 q^{79} -8.10010 q^{81} +6.69438 q^{83} -2.72128 q^{85} +12.6653 q^{87} +7.88038 q^{89} -4.25424 q^{91} +10.2451 q^{93} +3.21424 q^{95} -12.5887 q^{97} -1.58404 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 30 q + 9 q^{3} + 5 q^{7} + 35 q^{9}+O(q^{10}) $ 30 * q + 9 * q^3 + 5 * q^7 + 35 * q^9	$ 30 q + 9 q^{3} + 5 q^{7} + 35 q^{9} + 31 q^{11} + 5 q^{13} + 8 q^{15} - q^{17} + 29 q^{19} + 6 q^{21} + 27 q^{23} + 34 q^{25} + 36 q^{27} + q^{29} - 9 q^{31} - 8 q^{33} + 51 q^{35} + 5 q^{37} - 8 q^{39} - 3 q^{41} + 43 q^{43} + 42 q^{47} + 41 q^{49} + 54 q^{51} - 11 q^{53} - 10 q^{55} + 2 q^{57} + 49 q^{59} + 21 q^{61} + 15 q^{63} - 14 q^{65} + 68 q^{67} - 10 q^{69} + 44 q^{71} + 25 q^{73} + 51 q^{75} - 20 q^{77} - 49 q^{79} + 6 q^{81} + 88 q^{83} - 5 q^{85} + 16 q^{87} - 16 q^{89} + 46 q^{91} - 4 q^{93} + 28 q^{95} - 4 q^{97} + 71 q^{99}+O(q^{100}) $ 30 * q + 9 * q^3 + 5 * q^7 + 35 * q^9 + 31 * q^11 + 5 * q^13 + 8 * q^15 - q^17 + 29 * q^19 + 6 * q^21 + 27 * q^23 + 34 * q^25 + 36 * q^27 + q^29 - 9 * q^31 - 8 * q^33 + 51 * q^35 + 5 * q^37 - 8 * q^39 - 3 * q^41 + 43 * q^43 + 42 * q^47 + 41 * q^49 + 54 * q^51 - 11 * q^53 - 10 * q^55 + 2 * q^57 + 49 * q^59 + 21 * q^61 + 15 * q^63 - 14 * q^65 + 68 * q^67 - 10 * q^69 + 44 * q^71 + 25 * q^73 + 51 * q^75 - 20 * q^77 - 49 * q^79 + 6 * q^81 + 88 * q^83 - 5 * q^85 + 16 * q^87 - 16 * q^89 + 46 * q^91 - 4 * q^93 + 28 * q^95 - 4 * q^97 + 71 * q^99

Display 100 coefficients 10 coefficients

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
$3$	−2.50495	−1.44624	−0.723118	−	0.690725i	$-0.757292\pi$
$3$	−2.50495	−1.44624	−0.723118	+	0.690725i	$0.757292\pi$
$4$	0	0
$5$	0.621089	0.277759	0.138880	−	0.990309i	$-0.455650\pi$
$5$	0.621089	0.277759	0.138880	+	0.990309i	$0.455650\pi$
$6$	0	0
$7$	1.81379	0.685550	0.342775	−	0.939418i	$-0.388633\pi$
$7$	1.81379	0.685550	0.342775	+	0.939418i	$0.388633\pi$
$8$	0	0
$9$	3.27480	1.09160
$10$	0	0
$11$	−0.483706	−0.145843	−0.0729215	−	0.997338i	$-0.523232\pi$
$11$	−0.483706	−0.145843	−0.0729215	+	0.997338i	$0.523232\pi$
$12$	0	0
$13$	−2.34549	−0.650523	−0.325261	−	0.945624i	$-0.605452\pi$
$13$	−2.34549	−0.650523	−0.325261	+	0.945624i	$0.605452\pi$
$14$	0	0
$15$	−1.55580	−0.401705
$16$	0	0
$17$	−4.38147	−1.06266	−0.531331	−	0.847165i	$-0.678308\pi$
$17$	−4.38147	−1.06266	−0.531331	+	0.847165i	$0.678308\pi$
$18$	0	0
$19$	5.17518	1.18727	0.593634	−	0.804735i	$-0.297693\pi$
$19$	5.17518	1.18727	0.593634	+	0.804735i	$0.297693\pi$
$20$	0	0
$21$	−4.54347	−0.991467
$22$	0	0
$23$	−7.24965	−1.51166	−0.755828	−	0.654770i	$-0.772765\pi$
$23$	−7.24965	−1.51166	−0.755828	+	0.654770i	$0.772765\pi$
$24$	0	0
$25$	−4.61425	−0.922850
$26$	0	0
$27$	−0.688350	−0.132473
$28$	0	0
$29$	−5.05610	−0.938893	−0.469447	−	0.882961i	$-0.655547\pi$
$29$	−5.05610	−0.938893	−0.469447	+	0.882961i	$0.655547\pi$
$30$	0	0
$31$	−4.08995	−0.734577	−0.367289	−	0.930107i	$-0.619714\pi$
$31$	−4.08995	−0.734577	−0.367289	+	0.930107i	$0.619714\pi$
$32$	0	0
$33$	1.21166	0.210923
$34$	0	0
$35$	1.12653	0.190418
$36$	0	0
$37$	1.40181	0.230457	0.115228	−	0.993339i	$-0.463240\pi$
$37$	1.40181	0.230457	0.115228	+	0.993339i	$0.463240\pi$
$38$	0	0
$39$	5.87535	0.940810
$40$	0	0
$41$	7.17219	1.12011	0.560054	−	0.828456i	$-0.310780\pi$
$41$	7.17219	1.12011	0.560054	+	0.828456i	$0.310780\pi$
$42$	0	0
$43$	−4.88547	−0.745027	−0.372514	−	0.928027i	$-0.621504\pi$
$43$	−4.88547	−0.745027	−0.372514	+	0.928027i	$0.621504\pi$
$44$	0	0
$45$	2.03394	0.303202
$46$	0	0
$47$	6.60949	0.964094	0.482047	−	0.876145i	$-0.339894\pi$
$47$	6.60949	0.964094	0.482047	+	0.876145i	$0.339894\pi$
$48$	0	0
$49$	−3.71015	−0.530022
$50$	0	0
$51$	10.9754	1.53686
$52$	0	0
$53$	−2.78109	−0.382012	−0.191006	−	0.981589i	$-0.561175\pi$
$53$	−2.78109	−0.382012	−0.191006	+	0.981589i	$0.561175\pi$
$54$	0	0
$55$	−0.300424	−0.0405092
$56$	0	0
$57$	−12.9636	−1.71707
$58$	0	0
$59$	10.7127	1.39468	0.697338	−	0.716742i	$-0.254367\pi$
$59$	10.7127	1.39468	0.697338	+	0.716742i	$0.254367\pi$
$60$	0	0
$61$	6.80533	0.871333	0.435666	−	0.900108i	$-0.356513\pi$
$61$	6.80533	0.871333	0.435666	+	0.900108i	$0.356513\pi$
$62$	0	0
$63$	5.93980	0.748345
$64$	0	0
$65$	−1.45676	−0.180689
$66$	0	0
$67$	10.7694	1.31569	0.657844	−	0.753154i	$-0.271468\pi$
$67$	10.7694	1.31569	0.657844	+	0.753154i	$0.271468\pi$
$68$	0	0
$69$	18.1600	2.18621
$70$	0	0
$71$	−3.98861	−0.473361	−0.236680	−	0.971588i	$-0.576059\pi$
$71$	−3.98861	−0.473361	−0.236680	+	0.971588i	$0.576059\pi$
$72$	0	0
$73$	4.34320	0.508334	0.254167	−	0.967160i	$-0.418199\pi$
$73$	4.34320	0.508334	0.254167	+	0.967160i	$0.418199\pi$
$74$	0	0
$75$	11.5585	1.33466
$76$	0	0
$77$	−0.877344	−0.0999826
$78$	0	0
$79$	−13.7650	−1.54869	−0.774344	−	0.632765i	$-0.781920\pi$
$79$	−13.7650	−1.54869	−0.774344	+	0.632765i	$0.781920\pi$
$80$	0	0
$81$	−8.10010	−0.900011
$82$	0	0
$83$	6.69438	0.734803	0.367402	−	0.930062i	$-0.380247\pi$
$83$	6.69438	0.734803	0.367402	+	0.930062i	$0.380247\pi$
$84$	0	0
$85$	−2.72128	−0.295164
$86$	0	0
$87$	12.6653	1.35786
$88$	0	0
$89$	7.88038	0.835318	0.417659	−	0.908604i	$-0.362851\pi$
$89$	7.88038	0.835318	0.417659	+	0.908604i	$0.362851\pi$
$90$	0	0
$91$	−4.25424	−0.445966
$92$	0	0
$93$	10.2451	1.06237
$94$	0	0
$95$	3.21424	0.329775
$96$	0	0
$97$	−12.5887	−1.27819	−0.639096	−	0.769127i	$-0.720691\pi$
$97$	−12.5887	−1.27819	−0.639096	+	0.769127i	$0.720691\pi$
$98$	0	0
$99$	−1.58404	−0.159202
$100$	0	0
$101$	−12.2462	−1.21854	−0.609270	−	0.792963i	$-0.708537\pi$
$101$	−12.2462	−1.21854	−0.609270	+	0.792963i	$0.708537\pi$
$102$	0	0
$103$	17.0895	1.68388	0.841938	−	0.539575i	$-0.181415\pi$
$103$	17.0895	1.68388	0.841938	+	0.539575i	$0.181415\pi$
$104$	0	0
$105$	−2.82190	−0.275389
$106$	0	0
$107$	16.9018	1.63396	0.816980	−	0.576666i	$-0.195646\pi$
$107$	16.9018	1.63396	0.816980	+	0.576666i	$0.195646\pi$
$108$	0	0
$109$	−10.3339	−0.989805	−0.494902	−	0.868949i	$-0.664796\pi$
$109$	−10.3339	−0.989805	−0.494902	+	0.868949i	$0.664796\pi$
$110$	0	0
$111$	−3.51148	−0.333295
$112$	0	0
$113$	−6.41410	−0.603388	−0.301694	−	0.953405i	$-0.597552\pi$
$113$	−6.41410	−0.603388	−0.301694	+	0.953405i	$0.597552\pi$
$114$	0	0
$115$	−4.50267	−0.419876
$116$	0	0
$117$	−7.68101	−0.710110
$118$	0	0
$119$	−7.94707	−0.728507
$120$	0	0
$121$	−10.7660	−0.978730
$122$	0	0
$123$	−17.9660	−1.61994
$124$	0	0
$125$	−5.97130	−0.534089
$126$	0	0
$127$	18.1939	1.61444	0.807222	−	0.590247i	$-0.200970\pi$
$127$	18.1939	1.61444	0.807222	+	0.590247i	$0.200970\pi$
$128$	0	0
$129$	12.2379	1.07749
$130$	0	0
$131$	−9.50050	−0.830062	−0.415031	−	0.909807i	$-0.636229\pi$
$131$	−9.50050	−0.830062	−0.415031	+	0.909807i	$0.636229\pi$
$132$	0	0
$133$	9.38671	0.813931
$134$	0	0
$135$	−0.427527	−0.0367956
$136$	0	0
$137$	−9.92580	−0.848019	−0.424009	−	0.905658i	$-0.639378\pi$
$137$	−9.92580	−0.848019	−0.424009	+	0.905658i	$0.639378\pi$
$138$	0	0
$139$	0.655861	0.0556294	0.0278147	−	0.999613i	$-0.491145\pi$
$139$	0.655861	0.0556294	0.0278147	+	0.999613i	$0.491145\pi$
$140$	0	0
$141$	−16.5565	−1.39431
$142$	0	0
$143$	1.13453	0.0948742
$144$	0	0
$145$	−3.14028	−0.260786
$146$	0	0
$147$	9.29376	0.766536
$148$	0	0
$149$	20.9139	1.71333	0.856666	−	0.515871i	$-0.172532\pi$
$149$	20.9139	1.71333	0.856666	+	0.515871i	$0.172532\pi$
$150$	0	0
$151$	7.66675	0.623911	0.311956	−	0.950097i	$-0.399016\pi$
$151$	7.66675	0.623911	0.311956	+	0.950097i	$0.399016\pi$
$152$	0	0
$153$	−14.3484	−1.16000
$154$	0	0
$155$	−2.54022	−0.204036
$156$	0	0
$157$	−2.77007	−0.221076	−0.110538	−	0.993872i	$-0.535257\pi$
$157$	−2.77007	−0.221076	−0.110538	+	0.993872i	$0.535257\pi$
$158$	0	0
$159$	6.96650	0.552480
$160$	0	0
$161$	−13.1494	−1.03632
$162$	0	0
$163$	1.92833	0.151038	0.0755191	−	0.997144i	$-0.475939\pi$
$163$	1.92833	0.151038	0.0755191	+	0.997144i	$0.475939\pi$
$164$	0	0
$165$	0.752550	0.0585859
$166$	0	0
$167$	0.527552	0.0408232	0.0204116	−	0.999792i	$-0.493502\pi$
$167$	0.527552	0.0408232	0.0204116	+	0.999792i	$0.493502\pi$
$168$	0	0
$169$	−7.49866	−0.576820
$170$	0	0
$171$	16.9477	1.29602
$172$	0	0
$173$	18.4436	1.40224	0.701120	−	0.713043i	$-0.252684\pi$
$173$	18.4436	1.40224	0.701120	+	0.713043i	$0.252684\pi$
$174$	0	0
$175$	−8.36930	−0.632659
$176$	0	0
$177$	−26.8349	−2.01703
$178$	0	0
$179$	12.0608	0.901468	0.450734	−	0.892658i	$-0.351162\pi$
$179$	12.0608	0.901468	0.450734	+	0.892658i	$0.351162\pi$
$180$	0	0
$181$	−8.18045	−0.608048	−0.304024	−	0.952664i	$-0.598330\pi$
$181$	−8.18045	−0.608048	−0.304024	+	0.952664i	$0.598330\pi$
$182$	0	0
$183$	−17.0470	−1.26015
$184$	0	0
$185$	0.870650	0.0640114
$186$	0	0
$187$	2.11934	0.154982
$188$	0	0
$189$	−1.24853	−0.0908169
$190$	0	0
$191$	−11.1426	−0.806252	−0.403126	−	0.915145i	$-0.632076\pi$
$191$	−11.1426	−0.806252	−0.403126	+	0.915145i	$0.632076\pi$
$192$	0	0
$193$	−6.72446	−0.484037	−0.242018	−	0.970272i	$-0.577809\pi$
$193$	−6.72446	−0.484037	−0.242018	+	0.970272i	$0.577809\pi$
$194$	0	0
$195$	3.64912	0.261319
$196$	0	0
$197$	21.8729	1.55838	0.779192	−	0.626786i	$-0.215630\pi$
$197$	21.8729	1.55838	0.779192	+	0.626786i	$0.215630\pi$
$198$	0	0
$199$	−21.9420	−1.55543	−0.777714	−	0.628618i	$-0.783621\pi$
$199$	−21.9420	−1.55543	−0.777714	+	0.628618i	$0.783621\pi$
$200$	0	0
$201$	−26.9768	−1.90280
$202$	0	0
$203$	−9.17071	−0.643658
$204$	0	0
$205$	4.45456	0.311120
$206$	0	0
$207$	−23.7411	−1.65012
$208$	0	0
$209$	−2.50327	−0.173155
$210$	0	0
$211$	−17.0857	−1.17623	−0.588116	−	0.808777i	$-0.700130\pi$
$211$	−17.0857	−1.17623	−0.588116	+	0.808777i	$0.700130\pi$
$212$	0	0
$213$	9.99128	0.684591
$214$	0	0
$215$	−3.03431	−0.206938
$216$	0	0
$217$	−7.41833	−0.503589
$218$	0	0
$219$	−10.8795	−0.735170
$220$	0	0
$221$	10.2767	0.691286
$222$	0	0
$223$	23.9559	1.60420	0.802102	−	0.597188i	$-0.203715\pi$
$223$	23.9559	1.60420	0.802102	+	0.597188i	$0.203715\pi$
$224$	0	0
$225$	−15.1107	−1.00738
$226$	0	0
$227$	7.20530	0.478232	0.239116	−	0.970991i	$-0.423142\pi$
$227$	7.20530	0.478232	0.239116	+	0.970991i	$0.423142\pi$
$228$	0	0
$229$	−6.16298	−0.407261	−0.203630	−	0.979048i	$-0.565274\pi$
$229$	−6.16298	−0.407261	−0.203630	+	0.979048i	$0.565274\pi$
$230$	0	0
$231$	2.19771	0.144598
$232$	0	0
$233$	13.7208	0.898876	0.449438	−	0.893311i	$-0.351624\pi$
$233$	13.7208	0.898876	0.449438	+	0.893311i	$0.351624\pi$
$234$	0	0
$235$	4.10508	0.267786
$236$	0	0
$237$	34.4808	2.23977
$238$	0	0
$239$	−13.9889	−0.904870	−0.452435	−	0.891797i	$-0.649445\pi$
$239$	−13.9889	−0.904870	−0.452435	+	0.891797i	$0.649445\pi$
$240$	0	0
$241$	24.3787	1.57037	0.785186	−	0.619260i	$-0.212567\pi$
$241$	24.3787	1.57037	0.785186	+	0.619260i	$0.212567\pi$
$242$	0	0
$243$	22.3554	1.43410
$244$	0	0
$245$	−2.30433	−0.147218
$246$	0	0
$247$	−12.1383	−0.772345
$248$	0	0
$249$	−16.7691	−1.06270
$250$	0	0
$251$	−1.00000	−0.0631194
$251$	−1.00000	−0.0631194
$252$	0	0
$253$	3.50670	0.220464
$254$	0	0
$255$	6.81668	0.426877
$256$	0	0
$257$	4.57991	0.285687	0.142844	−	0.989745i	$-0.454375\pi$
$257$	4.57991	0.285687	0.142844	+	0.989745i	$0.454375\pi$
$258$	0	0
$259$	2.54260	0.157989
$260$	0	0
$261$	−16.5577	−1.02489
$262$	0	0
$263$	11.4749	0.707570	0.353785	−	0.935327i	$-0.384895\pi$
$263$	11.4749	0.707570	0.353785	+	0.935327i	$0.384895\pi$
$264$	0	0
$265$	−1.72730	−0.106107
$266$	0	0
$267$	−19.7400	−1.20807
$268$	0	0
$269$	22.7037	1.38427	0.692136	−	0.721768i	$-0.256670\pi$
$269$	22.7037	1.38427	0.692136	+	0.721768i	$0.256670\pi$
$270$	0	0
$271$	−9.05963	−0.550333	−0.275167	−	0.961397i	$-0.588733\pi$
$271$	−9.05963	−0.550333	−0.275167	+	0.961397i	$0.588733\pi$
$272$	0	0
$273$	10.6567	0.644972
$274$	0	0
$275$	2.23194	0.134591
$276$	0	0
$277$	−27.1411	−1.63075	−0.815376	−	0.578931i	$-0.803470\pi$
$277$	−27.1411	−1.63075	−0.815376	+	0.578931i	$0.803470\pi$
$278$	0	0
$279$	−13.3938	−0.801863
$280$	0	0
$281$	21.1505	1.26173	0.630865	−	0.775893i	$-0.282700\pi$
$281$	21.1505	1.26173	0.630865	+	0.775893i	$0.282700\pi$
$282$	0	0
$283$	13.7849	0.819427	0.409714	−	0.912214i	$-0.365629\pi$
$283$	13.7849	0.819427	0.409714	+	0.912214i	$0.365629\pi$
$284$	0	0
$285$	−8.05154	−0.476932
$286$	0	0
$287$	13.0089	0.767889
$288$	0	0
$289$	2.19724	0.129249
$290$	0	0
$291$	31.5342	1.84857
$292$	0	0
$293$	−2.22689	−0.130096	−0.0650482	−	0.997882i	$-0.520720\pi$
$293$	−2.22689	−0.130096	−0.0650482	+	0.997882i	$0.520720\pi$
$294$	0	0
$295$	6.65354	0.387384
$296$	0	0
$297$	0.332960	0.0193203
$298$	0	0
$299$	17.0040	0.983367
$300$	0	0
$301$	−8.86123	−0.510753
$302$	0	0
$303$	30.6761	1.76230
$304$	0	0
$305$	4.22671	0.242021
$306$	0	0
$307$	15.8119	0.902436	0.451218	−	0.892414i	$-0.350990\pi$
$307$	15.8119	0.902436	0.451218	+	0.892414i	$0.350990\pi$
$308$	0	0
$309$	−42.8083	−2.43528
$310$	0	0
$311$	10.8409	0.614732	0.307366	−	0.951591i	$-0.400552\pi$
$311$	10.8409	0.614732	0.307366	+	0.951591i	$0.400552\pi$
$312$	0	0
$313$	−29.3039	−1.65636	−0.828178	−	0.560465i	$-0.810623\pi$
$313$	−29.3039	−1.65636	−0.828178	+	0.560465i	$0.810623\pi$
$314$	0	0
$315$	3.68914	0.207860
$316$	0	0
$317$	4.60716	0.258764	0.129382	−	0.991595i	$-0.458701\pi$
$317$	4.60716	0.258764	0.129382	+	0.991595i	$0.458701\pi$
$318$	0	0
$319$	2.44567	0.136931
$320$	0	0
$321$	−42.3383	−2.36309
$322$	0	0
$323$	−22.6749	−1.26166
$324$	0	0
$325$	10.8227	0.600335
$326$	0	0
$327$	25.8859	1.43149
$328$	0	0
$329$	11.9883	0.660934
$330$	0	0
$331$	−2.19899	−0.120868	−0.0604338	−	0.998172i	$-0.519248\pi$
$331$	−2.19899	−0.120868	−0.0604338	+	0.998172i	$0.519248\pi$
$332$	0	0
$333$	4.59065	0.251566
$334$	0	0
$335$	6.68874	0.365445
$336$	0	0
$337$	32.0537	1.74608	0.873038	−	0.487652i	$-0.162147\pi$
$337$	32.0537	1.74608	0.873038	+	0.487652i	$0.162147\pi$
$338$	0	0
$339$	16.0670	0.872641
$340$	0	0
$341$	1.97834	0.107133
$342$	0	0
$343$	−19.4260	−1.04891
$344$	0	0
$345$	11.2790	0.607240
$346$	0	0
$347$	−7.51822	−0.403599	−0.201800	−	0.979427i	$-0.564679\pi$
$347$	−7.51822	−0.403599	−0.201800	+	0.979427i	$0.564679\pi$
$348$	0	0
$349$	−10.5136	−0.562781	−0.281391	−	0.959593i	$-0.590796\pi$
$349$	−10.5136	−0.562781	−0.281391	+	0.959593i	$0.590796\pi$
$350$	0	0
$351$	1.61452	0.0861768
$352$	0	0
$353$	10.0061	0.532573	0.266287	−	0.963894i	$-0.414203\pi$
$353$	10.0061	0.532573	0.266287	+	0.963894i	$0.414203\pi$
$354$	0	0
$355$	−2.47728	−0.131480
$356$	0	0
$357$	19.9071	1.05359
$358$	0	0
$359$	29.7635	1.57086	0.785428	−	0.618953i	$-0.212443\pi$
$359$	29.7635	1.57086	0.785428	+	0.618953i	$0.212443\pi$
$360$	0	0
$361$	7.78248	0.409604
$362$	0	0
$363$	26.9684	1.41547
$364$	0	0
$365$	2.69751	0.141194
$366$	0	0
$367$	−20.0043	−1.04422	−0.522108	−	0.852880i	$-0.674854\pi$
$367$	−20.0043	−1.04422	−0.522108	+	0.852880i	$0.674854\pi$
$368$	0	0
$369$	23.4874	1.22271
$370$	0	0
$371$	−5.04432	−0.261888
$372$	0	0
$373$	21.9935	1.13878	0.569391	−	0.822067i	$-0.307179\pi$
$373$	21.9935	1.13878	0.569391	+	0.822067i	$0.307179\pi$
$374$	0	0
$375$	14.9578	0.772419
$376$	0	0
$377$	11.8590	0.610772
$378$	0	0
$379$	−0.536024	−0.0275337	−0.0137669	−	0.999905i	$-0.504382\pi$
$379$	−0.536024	−0.0275337	−0.0137669	+	0.999905i	$0.504382\pi$
$380$	0	0
$381$	−45.5748	−2.33487
$382$	0	0
$383$	−6.16948	−0.315246	−0.157623	−	0.987499i	$-0.550383\pi$
$383$	−6.16948	−0.315246	−0.157623	+	0.987499i	$0.550383\pi$
$384$	0	0
$385$	−0.544908	−0.0277711
$386$	0	0
$387$	−15.9989	−0.813271
$388$	0	0
$389$	−9.61873	−0.487689	−0.243844	−	0.969814i	$-0.578409\pi$
$389$	−9.61873	−0.487689	−0.243844	+	0.969814i	$0.578409\pi$
$390$	0	0
$391$	31.7641	1.60638
$392$	0	0
$393$	23.7983	1.20047
$394$	0	0
$395$	−8.54931	−0.430163
$396$	0	0
$397$	34.2608	1.71950	0.859749	−	0.510717i	$-0.170620\pi$
$397$	34.2608	1.71950	0.859749	+	0.510717i	$0.170620\pi$
$398$	0	0
$399$	−23.5133	−1.17714
$400$	0	0
$401$	−0.873610	−0.0436260	−0.0218130	−	0.999762i	$-0.506944\pi$
$401$	−0.873610	−0.0436260	−0.0218130	+	0.999762i	$0.506944\pi$
$402$	0	0
$403$	9.59296	0.477859
$404$	0	0
$405$	−5.03088	−0.249986
$406$	0	0
$407$	−0.678066	−0.0336105
$408$	0	0
$409$	−28.4608	−1.40729	−0.703647	−	0.710550i	$-0.748446\pi$
$409$	−28.4608	−1.40729	−0.703647	+	0.710550i	$0.748446\pi$
$410$	0	0
$411$	24.8637	1.22644
$412$	0	0
$413$	19.4307	0.956120
$414$	0	0
$415$	4.15780	0.204098
$416$	0	0
$417$	−1.64290	−0.0804532
$418$	0	0
$419$	−32.4800	−1.58675	−0.793377	−	0.608731i	$-0.791679\pi$
$419$	−32.4800	−1.58675	−0.793377	+	0.608731i	$0.791679\pi$
$420$	0	0
$421$	−31.3588	−1.52834	−0.764168	−	0.645018i	$-0.776850\pi$
$421$	−31.3588	−1.52834	−0.764168	+	0.645018i	$0.776850\pi$
$422$	0	0
$423$	21.6447	1.05240
$424$	0	0
$425$	20.2172	0.980677
$426$	0	0
$427$	12.3435	0.597342
$428$	0	0
$429$	−2.84195	−0.137210
$430$	0	0
$431$	31.7362	1.52868	0.764340	−	0.644814i	$-0.223065\pi$
$431$	31.7362	1.52868	0.764340	+	0.644814i	$0.223065\pi$
$432$	0	0
$433$	11.3700	0.546406	0.273203	−	0.961956i	$-0.411917\pi$
$433$	11.3700	0.546406	0.273203	+	0.961956i	$0.411917\pi$
$434$	0	0
$435$	7.86627	0.377159
$436$	0	0
$437$	−37.5182	−1.79474
$438$	0	0
$439$	−27.8131	−1.32745	−0.663723	−	0.747978i	$-0.731025\pi$
$439$	−27.8131	−1.32745	−0.663723	+	0.747978i	$0.731025\pi$
$440$	0	0
$441$	−12.1500	−0.578571
$442$	0	0
$443$	17.2294	0.818594	0.409297	−	0.912401i	$-0.365774\pi$
$443$	17.2294	0.818594	0.409297	+	0.912401i	$0.365774\pi$
$444$	0	0
$445$	4.89441	0.232017
$446$	0	0
$447$	−52.3883	−2.47788
$448$	0	0
$449$	7.24294	0.341815	0.170908	−	0.985287i	$-0.445330\pi$
$449$	7.24294	0.341815	0.170908	+	0.985287i	$0.445330\pi$
$450$	0	0
$451$	−3.46923	−0.163360
$452$	0	0
$453$	−19.2048	−0.902323
$454$	0	0
$455$	−2.64226	−0.123871
$456$	0	0
$457$	30.4163	1.42282	0.711408	−	0.702779i	$-0.248058\pi$
$457$	30.4163	1.42282	0.711408	+	0.702779i	$0.248058\pi$
$458$	0	0
$459$	3.01598	0.140774
$460$	0	0
$461$	34.9290	1.62681	0.813403	−	0.581700i	$-0.197612\pi$
$461$	34.9290	1.62681	0.813403	+	0.581700i	$0.197612\pi$
$462$	0	0
$463$	−10.0103	−0.465218	−0.232609	−	0.972570i	$-0.574726\pi$
$463$	−10.0103	−0.465218	−0.232609	+	0.972570i	$0.574726\pi$
$464$	0	0
$465$	6.36314	0.295084
$466$	0	0
$467$	2.45362	0.113540	0.0567700	−	0.998387i	$-0.481920\pi$
$467$	2.45362	0.113540	0.0567700	+	0.998387i	$0.481920\pi$
$468$	0	0
$469$	19.5334	0.901970
$470$	0	0
$471$	6.93890	0.319728
$472$	0	0
$473$	2.36313	0.108657
$474$	0	0
$475$	−23.8796	−1.09567
$476$	0	0
$477$	−9.10750	−0.417004
$478$	0	0
$479$	41.2574	1.88510	0.942550	−	0.334064i	$-0.108420\pi$
$479$	41.2574	1.88510	0.942550	+	0.334064i	$0.108420\pi$
$480$	0	0
$481$	−3.28794	−0.149917
$482$	0	0
$483$	32.9386	1.49876
$484$	0	0
$485$	−7.81872	−0.355030
$486$	0	0
$487$	17.9091	0.811541	0.405770	−	0.913975i	$-0.367003\pi$
$487$	17.9091	0.811541	0.405770	+	0.913975i	$0.367003\pi$
$488$	0	0
$489$	−4.83037	−0.218437
$490$	0	0
$491$	−40.6674	−1.83529	−0.917646	−	0.397399i	$-0.869913\pi$
$491$	−40.6674	−1.83529	−0.917646	+	0.397399i	$0.869913\pi$
$492$	0	0
$493$	22.1531	0.997726
$494$	0	0
$495$	−0.983829	−0.0442198
$496$	0	0
$497$	−7.23451	−0.324512
$498$	0	0
$499$	37.0884	1.66030	0.830152	−	0.557537i	$-0.188254\pi$
$499$	37.0884	1.66030	0.830152	+	0.557537i	$0.188254\pi$
$500$	0	0
$501$	−1.32149	−0.0590400
$502$	0	0
$503$	−8.12814	−0.362416	−0.181208	−	0.983445i	$-0.558001\pi$
$503$	−8.12814	−0.362416	−0.181208	+	0.983445i	$0.558001\pi$
$504$	0	0
$505$	−7.60596	−0.338461
$506$	0	0
$507$	18.7838	0.834218
$508$	0	0
$509$	22.1394	0.981312	0.490656	−	0.871353i	$-0.336757\pi$
$509$	22.1394	0.981312	0.490656	+	0.871353i	$0.336757\pi$
$510$	0	0
$511$	7.87768	0.348488
$512$	0	0
$513$	−3.56234	−0.157281
$514$	0	0
$515$	10.6141	0.467712
$516$	0	0
$517$	−3.19705	−0.140606
$518$	0	0
$519$	−46.2003	−2.02797
$520$	0	0
$521$	−19.8659	−0.870342	−0.435171	−	0.900348i	$-0.643312\pi$
$521$	−19.8659	−0.870342	−0.435171	+	0.900348i	$0.643312\pi$
$522$	0	0
$523$	−27.4682	−1.20110	−0.600551	−	0.799587i	$-0.705052\pi$
$523$	−27.4682	−1.20110	−0.600551	+	0.799587i	$0.705052\pi$
$524$	0	0
$525$	20.9647	0.914975
$526$	0	0
$527$	17.9200	0.780607
$528$	0	0
$529$	29.5574	1.28510
$530$	0	0
$531$	35.0820	1.52243
$532$	0	0
$533$	−16.8223	−0.728656
$534$	0	0
$535$	10.4975	0.453848
$536$	0	0
$537$	−30.2118	−1.30374
$538$	0	0
$539$	1.79462	0.0772999
$540$	0	0
$541$	29.5152	1.26896	0.634479	−	0.772940i	$-0.281215\pi$
$541$	29.5152	1.26896	0.634479	+	0.772940i	$0.281215\pi$
$542$	0	0
$543$	20.4917	0.879381
$544$	0	0
$545$	−6.41824	−0.274927
$546$	0	0
$547$	1.03854	0.0444048	0.0222024	−	0.999753i	$-0.492932\pi$
$547$	1.03854	0.0444048	0.0222024	+	0.999753i	$0.492932\pi$
$548$	0	0
$549$	22.2861	0.951146
$550$	0	0
$551$	−26.1662	−1.11472
$552$	0	0
$553$	−24.9670	−1.06170
$554$	0	0
$555$	−2.18094	−0.0925756
$556$	0	0
$557$	−11.8847	−0.503573	−0.251786	−	0.967783i	$-0.581018\pi$
$557$	−11.8847	−0.503573	−0.251786	+	0.967783i	$0.581018\pi$
$558$	0	0
$559$	11.4588	0.484657
$560$	0	0
$561$	−5.30886	−0.224140
$562$	0	0
$563$	9.16899	0.386427	0.193213	−	0.981157i	$-0.438109\pi$
$563$	9.16899	0.386427	0.193213	+	0.981157i	$0.438109\pi$
$564$	0	0
$565$	−3.98372	−0.167596
$566$	0	0
$567$	−14.6919	−0.617002
$568$	0	0
$569$	41.8312	1.75365	0.876827	−	0.480805i	$-0.159656\pi$
$569$	41.8312	1.75365	0.876827	+	0.480805i	$0.159656\pi$
$570$	0	0
$571$	11.2916	0.472538	0.236269	−	0.971688i	$-0.424075\pi$
$571$	11.2916	0.472538	0.236269	+	0.971688i	$0.424075\pi$
$572$	0	0
$573$	27.9117	1.16603
$574$	0	0
$575$	33.4517	1.39503
$576$	0	0
$577$	45.1136	1.87811	0.939053	−	0.343773i	$-0.111705\pi$
$577$	45.1136	1.87811	0.939053	+	0.343773i	$0.111705\pi$
$578$	0	0
$579$	16.8445	0.700032
$580$	0	0
$581$	12.1422	0.503744
$582$	0	0
$583$	1.34523	0.0557138
$584$	0	0
$585$	−4.77059	−0.197240
$586$	0	0
$587$	45.1580	1.86387	0.931934	−	0.362627i	$-0.118120\pi$
$587$	45.1580	1.86387	0.931934	+	0.362627i	$0.118120\pi$
$588$	0	0
$589$	−21.1662	−0.872139
$590$	0	0
$591$	−54.7907	−2.25379
$592$	0	0
$593$	−6.11520	−0.251121	−0.125561	−	0.992086i	$-0.540073\pi$
$593$	−6.11520	−0.251121	−0.125561	+	0.992086i	$0.540073\pi$
$594$	0	0
$595$	−4.93584	−0.202350
$596$	0	0
$597$	54.9638	2.24952
$598$	0	0
$599$	22.1795	0.906230	0.453115	−	0.891452i	$-0.350313\pi$
$599$	22.1795	0.906230	0.453115	+	0.891452i	$0.350313\pi$
$600$	0	0
$601$	−10.0322	−0.409223	−0.204612	−	0.978843i	$-0.565593\pi$
$601$	−10.0322	−0.409223	−0.204612	+	0.978843i	$0.565593\pi$
$602$	0	0
$603$	35.2675	1.43620
$604$	0	0
$605$	−6.68666	−0.271851
$606$	0	0
$607$	12.8727	0.522485	0.261242	−	0.965273i	$-0.415868\pi$
$607$	12.8727	0.522485	0.261242	+	0.965273i	$0.415868\pi$
$608$	0	0
$609$	22.9722	0.930881
$610$	0	0
$611$	−15.5025	−0.627165
$612$	0	0
$613$	36.5880	1.47778	0.738888	−	0.673829i	$-0.235351\pi$
$613$	36.5880	1.47778	0.738888	+	0.673829i	$0.235351\pi$
$614$	0	0
$615$	−11.1585	−0.449953
$616$	0	0
$617$	32.1489	1.29427	0.647133	−	0.762377i	$-0.275968\pi$
$617$	32.1489	1.29427	0.647133	+	0.762377i	$0.275968\pi$
$618$	0	0
$619$	35.8596	1.44132	0.720659	−	0.693289i	$-0.243839\pi$
$619$	35.8596	1.44132	0.720659	+	0.693289i	$0.243839\pi$
$620$	0	0
$621$	4.99030	0.200254
$622$	0	0
$623$	14.2934	0.572652
$624$	0	0
$625$	19.3625	0.774502
$626$	0	0
$627$	6.27057	0.250422
$628$	0	0
$629$	−6.14199	−0.244897
$630$	0	0
$631$	−6.23671	−0.248280	−0.124140	−	0.992265i	$-0.539617\pi$
$631$	−6.23671	−0.248280	−0.124140	+	0.992265i	$0.539617\pi$
$632$	0	0
$633$	42.7990	1.70111
$634$	0	0
$635$	11.3000	0.448427
$636$	0	0
$637$	8.70214	0.344791
$638$	0	0
$639$	−13.0619	−0.516720
$640$	0	0
$641$	−31.3451	−1.23806	−0.619029	−	0.785368i	$-0.712474\pi$
$641$	−31.3451	−1.23806	−0.619029	+	0.785368i	$0.712474\pi$
$642$	0	0
$643$	12.8041	0.504943	0.252472	−	0.967604i	$-0.418757\pi$
$643$	12.8041	0.504943	0.252472	+	0.967604i	$0.418757\pi$
$644$	0	0
$645$	7.60081	0.299281
$646$	0	0
$647$	1.25878	0.0494879	0.0247439	−	0.999694i	$-0.492123\pi$
$647$	1.25878	0.0494879	0.0247439	+	0.999694i	$0.492123\pi$
$648$	0	0
$649$	−5.18181	−0.203404
$650$	0	0
$651$	18.5826	0.728309
$652$	0	0
$653$	26.9061	1.05292	0.526459	−	0.850200i	$-0.323519\pi$
$653$	26.9061	1.05292	0.526459	+	0.850200i	$0.323519\pi$
$654$	0	0
$655$	−5.90065	−0.230557
$656$	0	0
$657$	14.2231	0.554896
$658$	0	0
$659$	29.6019	1.15313	0.576564	−	0.817052i	$-0.304393\pi$
$659$	29.6019	1.15313	0.576564	+	0.817052i	$0.304393\pi$
$660$	0	0
$661$	−1.68103	−0.0653846	−0.0326923	−	0.999465i	$-0.510408\pi$
$661$	−1.68103	−0.0653846	−0.0326923	+	0.999465i	$0.510408\pi$
$662$	0	0
$663$	−25.7427	−0.999762
$664$	0	0
$665$	5.82998	0.226077
$666$	0	0
$667$	36.6549	1.41928
$668$	0	0
$669$	−60.0083	−2.32006
$670$	0	0
$671$	−3.29178	−0.127078
$672$	0	0
$673$	45.3536	1.74825	0.874126	−	0.485699i	$-0.161435\pi$
$673$	45.3536	1.74825	0.874126	+	0.485699i	$0.161435\pi$
$674$	0	0
$675$	3.17622	0.122253
$676$	0	0
$677$	−28.9128	−1.11121	−0.555604	−	0.831447i	$-0.687513\pi$
$677$	−28.9128	−1.11121	−0.555604	+	0.831447i	$0.687513\pi$
$678$	0	0
$679$	−22.8334	−0.876265
$680$	0	0
$681$	−18.0489	−0.691637
$682$	0	0
$683$	−44.1499	−1.68935	−0.844674	−	0.535281i	$-0.820206\pi$
$683$	−44.1499	−1.68935	−0.844674	+	0.535281i	$0.820206\pi$
$684$	0	0
$685$	−6.16480	−0.235545
$686$	0	0
$687$	15.4380	0.588995
$688$	0	0
$689$	6.52303	0.248508
$690$	0	0
$691$	−35.4255	−1.34765	−0.673825	−	0.738891i	$-0.735350\pi$
$691$	−35.4255	−1.34765	−0.673825	+	0.738891i	$0.735350\pi$
$692$	0	0
$693$	−2.87312	−0.109141
$694$	0	0
$695$	0.407348	0.0154516
$696$	0	0
$697$	−31.4247	−1.19030
$698$	0	0
$699$	−34.3699	−1.29999
$700$	0	0
$701$	−40.4330	−1.52713	−0.763567	−	0.645728i	$-0.776554\pi$
$701$	−40.4330	−1.52713	−0.763567	+	0.645728i	$0.776554\pi$
$702$	0	0
$703$	7.25463	0.273614
$704$	0	0
$705$	−10.2830	−0.387282
$706$	0	0
$707$	−22.2120	−0.835370
$708$	0	0
$709$	13.4983	0.506938	0.253469	−	0.967343i	$-0.418428\pi$
$709$	13.4983	0.506938	0.253469	+	0.967343i	$0.418428\pi$
$710$	0	0
$711$	−45.0777	−1.69055
$712$	0	0
$713$	29.6507	1.11043
$714$	0	0
$715$	0.704644	0.0263522
$716$	0	0
$717$	35.0417	1.30866
$718$	0	0
$719$	−3.56277	−0.132869	−0.0664345	−	0.997791i	$-0.521162\pi$
$719$	−3.56277	−0.132869	−0.0664345	+	0.997791i	$0.521162\pi$
$720$	0	0
$721$	30.9968	1.15438
$722$	0	0
$723$	−61.0676	−2.27113
$724$	0	0
$725$	23.3301	0.866458
$726$	0	0
$727$	−10.8832	−0.403637	−0.201819	−	0.979423i	$-0.564685\pi$
$727$	−10.8832	−0.403637	−0.201819	+	0.979423i	$0.564685\pi$
$728$	0	0
$729$	−31.6990	−1.17404
$730$	0	0
$731$	21.4055	0.791712
$732$	0	0
$733$	48.9800	1.80912	0.904558	−	0.426350i	$-0.140201\pi$
$733$	48.9800	1.80912	0.904558	+	0.426350i	$0.140201\pi$
$734$	0	0
$735$	5.77225	0.212913
$736$	0	0
$737$	−5.20922	−0.191884
$738$	0	0
$739$	37.1912	1.36810	0.684050	−	0.729435i	$-0.260217\pi$
$739$	37.1912	1.36810	0.684050	+	0.729435i	$0.260217\pi$
$740$	0	0
$741$	30.4060	1.11699
$742$	0	0
$743$	3.76311	0.138055	0.0690276	−	0.997615i	$-0.478010\pi$
$743$	3.76311	0.138055	0.0690276	+	0.997615i	$0.478010\pi$
$744$	0	0
$745$	12.9894	0.475894
$746$	0	0
$747$	21.9227	0.802110
$748$	0	0
$749$	30.6564	1.12016
$750$	0	0
$751$	−27.6703	−1.00970	−0.504852	−	0.863206i	$-0.668453\pi$
$751$	−27.6703	−1.00970	−0.504852	+	0.863206i	$0.668453\pi$
$752$	0	0
$753$	2.50495	0.0912856
$754$	0	0
$755$	4.76173	0.173297
$756$	0	0
$757$	−47.1437	−1.71347	−0.856733	−	0.515760i	$-0.827510\pi$
$757$	−47.1437	−1.71347	−0.856733	+	0.515760i	$0.827510\pi$
$758$	0	0
$759$	−8.78412	−0.318844
$760$	0	0
$761$	16.0482	0.581748	0.290874	−	0.956761i	$-0.406054\pi$
$761$	16.0482	0.581748	0.290874	+	0.956761i	$0.406054\pi$
$762$	0	0
$763$	−18.7435	−0.678560
$764$	0	0
$765$	−8.91163	−0.322201
$766$	0	0
$767$	−25.1266	−0.907269
$768$	0	0
$769$	15.6630	0.564822	0.282411	−	0.959294i	$-0.408866\pi$
$769$	15.6630	0.564822	0.282411	+	0.959294i	$0.408866\pi$
$770$	0	0
$771$	−11.4725	−0.413171
$772$	0	0
$773$	−16.8227	−0.605071	−0.302535	−	0.953138i	$-0.597833\pi$
$773$	−16.8227	−0.605071	−0.302535	+	0.953138i	$0.597833\pi$
$774$	0	0
$775$	18.8721	0.677904
$776$	0	0
$777$	−6.36909	−0.228490
$778$	0	0
$779$	37.1174	1.32987
$780$	0	0
$781$	1.92931	0.0690363
$782$	0	0
$783$	3.48037	0.124378
$784$	0	0
$785$	−1.72046	−0.0614059
$786$	0	0
$787$	−21.8777	−0.779856	−0.389928	−	0.920845i	$-0.627500\pi$
$787$	−21.8777	−0.779856	−0.389928	+	0.920845i	$0.627500\pi$
$788$	0	0
$789$	−28.7440	−1.02331
$790$	0	0
$791$	−11.6339	−0.413652
$792$	0	0
$793$	−15.9619	−0.566822
$794$	0	0
$795$	4.32682	0.153456
$796$	0	0
$797$	−2.94083	−0.104169	−0.0520847	−	0.998643i	$-0.516587\pi$
$797$	−2.94083	−0.104169	−0.0520847	+	0.998643i	$0.516587\pi$
$798$	0	0
$799$	−28.9593	−1.02451
$800$	0	0
$801$	25.8066	0.911832
$802$	0	0
$803$	−2.10084	−0.0741369
$804$	0	0
$805$	−8.16692	−0.287846
$806$	0	0
$807$	−56.8718	−2.00198
$808$	0	0
$809$	21.9712	0.772466	0.386233	−	0.922401i	$-0.373776\pi$
$809$	21.9712	0.772466	0.386233	+	0.922401i	$0.373776\pi$
$810$	0	0
$811$	26.8297	0.942119	0.471059	−	0.882101i	$-0.343872\pi$
$811$	26.8297	0.942119	0.471059	+	0.882101i	$0.343872\pi$
$812$	0	0
$813$	22.6940	0.795912
$814$	0	0
$815$	1.19766	0.0419523
$816$	0	0
$817$	−25.2832	−0.884547
$818$	0	0
$819$	−13.9318	−0.486816
$820$	0	0
$821$	−40.8802	−1.42673	−0.713365	−	0.700793i	$-0.752830\pi$
$821$	−40.8802	−1.42673	−0.713365	+	0.700793i	$0.752830\pi$
$822$	0	0
$823$	−28.5531	−0.995299	−0.497650	−	0.867378i	$-0.665803\pi$
$823$	−28.5531	−0.995299	−0.497650	+	0.867378i	$0.665803\pi$
$824$	0	0
$825$	−5.59091	−0.194651
$826$	0	0
$827$	14.9847	0.521068	0.260534	−	0.965465i	$-0.416101\pi$
$827$	14.9847	0.521068	0.260534	+	0.965465i	$0.416101\pi$
$828$	0	0
$829$	48.6095	1.68828	0.844139	−	0.536124i	$-0.180112\pi$
$829$	48.6095	1.68828	0.844139	+	0.536124i	$0.180112\pi$
$830$	0	0
$831$	67.9873	2.35845
$832$	0	0
$833$	16.2559	0.563234
$834$	0	0
$835$	0.327656	0.0113390
$836$	0	0
$837$	2.81532	0.0973117
$838$	0	0
$839$	−7.42401	−0.256305	−0.128153	−	0.991754i	$-0.540905\pi$
$839$	−7.42401	−0.256305	−0.128153	+	0.991754i	$0.540905\pi$
$840$	0	0
$841$	−3.43590	−0.118479
$842$	0	0
$843$	−52.9809	−1.82476
$844$	0	0
$845$	−4.65733	−0.160217
$846$	0	0
$847$	−19.5274	−0.670968
$848$	0	0
$849$	−34.5305	−1.18509
$850$	0	0
$851$	−10.1626	−0.348371
$852$	0	0
$853$	−10.1871	−0.348800	−0.174400	−	0.984675i	$-0.555799\pi$
$853$	−10.1871	−0.348800	−0.174400	+	0.984675i	$0.555799\pi$
$854$	0	0
$855$	10.5260	0.359981
$856$	0	0
$857$	−8.60159	−0.293825	−0.146912	−	0.989150i	$-0.546934\pi$
$857$	−8.60159	−0.293825	−0.146912	+	0.989150i	$0.546934\pi$
$858$	0	0
$859$	22.1434	0.755523	0.377761	−	0.925903i	$-0.376694\pi$
$859$	22.1434	0.755523	0.377761	+	0.925903i	$0.376694\pi$
$860$	0	0
$861$	−32.5866	−1.11055
$862$	0	0
$863$	11.6665	0.397133	0.198566	−	0.980087i	$-0.436371\pi$
$863$	11.6665	0.397133	0.198566	+	0.980087i	$0.436371\pi$
$864$	0	0
$865$	11.4551	0.389485
$866$	0	0
$867$	−5.50397	−0.186925
$868$	0	0
$869$	6.65824	0.225865
$870$	0	0
$871$	−25.2595	−0.855886
$872$	0	0
$873$	−41.2255	−1.39527
$874$	0	0
$875$	−10.8307	−0.366145
$876$	0	0
$877$	−12.2217	−0.412698	−0.206349	−	0.978478i	$-0.566158\pi$
$877$	−12.2217	−0.412698	−0.206349	+	0.978478i	$0.566158\pi$
$878$	0	0
$879$	5.57826	0.188150
$880$	0	0
$881$	38.1594	1.28562	0.642811	−	0.766025i	$-0.277768\pi$
$881$	38.1594	1.28562	0.642811	+	0.766025i	$0.277768\pi$
$882$	0	0
$883$	50.6040	1.70296	0.851481	−	0.524386i	$-0.175705\pi$
$883$	50.6040	1.70296	0.851481	+	0.524386i	$0.175705\pi$
$884$	0	0
$885$	−16.6668	−0.560249
$886$	0	0
$887$	15.6838	0.526610	0.263305	−	0.964713i	$-0.415187\pi$
$887$	15.6838	0.526610	0.263305	+	0.964713i	$0.415187\pi$
$888$	0	0
$889$	32.9999	1.10678
$890$	0	0
$891$	3.91807	0.131260
$892$	0	0
$893$	34.2053	1.14464
$894$	0	0
$895$	7.49084	0.250391
$896$	0	0
$897$	−42.5942	−1.42218
$898$	0	0
$899$	20.6792	0.689690
$900$	0	0
$901$	12.1853	0.405950
$902$	0	0
$903$	22.1970	0.738669
$904$	0	0
$905$	−5.08078	−0.168891
$906$	0	0
$907$	34.8671	1.15774	0.578872	−	0.815418i	$-0.303493\pi$
$907$	34.8671	1.15774	0.578872	+	0.815418i	$0.303493\pi$
$908$	0	0
$909$	−40.1037	−1.33016
$910$	0	0
$911$	−24.6095	−0.815350	−0.407675	−	0.913127i	$-0.633660\pi$
$911$	−24.6095	−0.815350	−0.407675	+	0.913127i	$0.633660\pi$
$912$	0	0
$913$	−3.23811	−0.107166
$914$	0	0
$915$	−10.5877	−0.350019
$916$	0	0
$917$	−17.2319	−0.569049
$918$	0	0
$919$	−1.20539	−0.0397621	−0.0198811	−	0.999802i	$-0.506329\pi$
$919$	−1.20539	−0.0397621	−0.0198811	+	0.999802i	$0.506329\pi$
$920$	0	0
$921$	−39.6082	−1.30513
$922$	0	0
$923$	9.35525	0.307932
$924$	0	0
$925$	−6.46831	−0.212677
$926$	0	0
$927$	55.9645	1.83812
$928$	0	0
$929$	−14.8578	−0.487470	−0.243735	−	0.969842i	$-0.578373\pi$
$929$	−14.8578	−0.487470	−0.243735	+	0.969842i	$0.578373\pi$
$930$	0	0
$931$	−19.2007	−0.629278
$932$	0	0
$933$	−27.1560	−0.889048
$934$	0	0
$935$	1.31630	0.0430476
$936$	0	0
$937$	6.67858	0.218180	0.109090	−	0.994032i	$-0.465206\pi$
$937$	6.67858	0.218180	0.109090	+	0.994032i	$0.465206\pi$
$938$	0	0
$939$	73.4050	2.39548
$940$	0	0
$941$	−35.3735	−1.15314	−0.576572	−	0.817046i	$-0.695610\pi$
$941$	−35.3735	−1.15314	−0.576572	+	0.817046i	$0.695610\pi$
$942$	0	0
$943$	−51.9958	−1.69322
$944$	0	0
$945$	−0.775445	−0.0252252
$946$	0	0
$947$	7.03274	0.228534	0.114267	−	0.993450i	$-0.463548\pi$
$947$	7.03274	0.228534	0.114267	+	0.993450i	$0.463548\pi$
$948$	0	0
$949$	−10.1870	−0.330683
$950$	0	0
$951$	−11.5407	−0.374233
$952$	0	0
$953$	−25.8616	−0.837740	−0.418870	−	0.908046i	$-0.637574\pi$
$953$	−25.8616	−0.837740	−0.418870	+	0.908046i	$0.637574\pi$
$954$	0	0
$955$	−6.92055	−0.223944
$956$	0	0
$957$	−6.12628	−0.198035
$958$	0	0
$959$	−18.0034	−0.581359
$960$	0	0
$961$	−14.2723	−0.460397
$962$	0	0
$963$	55.3500	1.78363
$964$	0	0
$965$	−4.17648	−0.134446
$966$	0	0
$967$	−15.1823	−0.488231	−0.244116	−	0.969746i	$-0.578498\pi$
$967$	−15.1823	−0.488231	−0.244116	+	0.969746i	$0.578498\pi$
$968$	0	0
$969$	56.7995	1.82466
$970$	0	0
$971$	−7.04484	−0.226080	−0.113040	−	0.993590i	$-0.536059\pi$
$971$	−7.04484	−0.226080	−0.113040	+	0.993590i	$0.536059\pi$
$972$	0	0
$973$	1.18960	0.0381367
$974$	0	0
$975$	−27.1103	−0.868226
$976$	0	0
$977$	14.3944	0.460517	0.230258	−	0.973130i	$-0.426043\pi$
$977$	14.3944	0.460517	0.230258	+	0.973130i	$0.426043\pi$
$978$	0	0
$979$	−3.81179	−0.121825
$980$	0	0
$981$	−33.8413	−1.08047
$982$	0	0
$983$	−49.2859	−1.57198	−0.785988	−	0.618242i	$-0.787845\pi$
$983$	−49.2859	−1.57198	−0.785988	+	0.618242i	$0.787845\pi$
$984$	0	0
$985$	13.5850	0.432855
$986$	0	0
$987$	−30.0300	−0.955867
$988$	0	0
$989$	35.4179	1.12622
$990$	0	0
$991$	−34.6000	−1.09910	−0.549552	−	0.835459i	$-0.685202\pi$
$991$	−34.6000	−1.09910	−0.549552	+	0.835459i	$0.685202\pi$
$992$	0	0
$993$	5.50837	0.174803
$994$	0	0
$995$	−13.6279	−0.432035
$996$	0	0
$997$	−26.1713	−0.828853	−0.414427	−	0.910083i	$-0.636018\pi$
$997$	−26.1713	−0.828853	−0.414427	+	0.910083i	$0.636018\pi$
$998$	0	0
$999$	−0.964938	−0.0305293

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8032.2.a.j.1.4	yes	30
4.3	odd	2		8032.2.a.g.1.27	✓	30

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8032.2.a.g.1.27	✓	30	4.3	odd	2
8032.2.a.j.1.4	yes	30	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 8032 = 2^{5} \cdot 251 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8032.a (trivial)

\(f(q)\)	\(=\)	\(q-2.50495 q^{3} +0.621089 q^{5} +1.81379 q^{7} +3.27480 q^{9} +O(q^{10})\)	\(q-2.50495 q^{3} +0.621089 q^{5} +1.81379 q^{7} +3.27480 q^{9} -0.483706 q^{11} -2.34549 q^{13} -1.55580 q^{15} -4.38147 q^{17} +5.17518 q^{19} -4.54347 q^{21} -7.24965 q^{23} -4.61425 q^{25} -0.688350 q^{27} -5.05610 q^{29} -4.08995 q^{31} +1.21166 q^{33} +1.12653 q^{35} +1.40181 q^{37} +5.87535 q^{39} +7.17219 q^{41} -4.88547 q^{43} +2.03394 q^{45} +6.60949 q^{47} -3.71015 q^{49} +10.9754 q^{51} -2.78109 q^{53} -0.300424 q^{55} -12.9636 q^{57} +10.7127 q^{59} +6.80533 q^{61} +5.93980 q^{63} -1.45676 q^{65} +10.7694 q^{67} +18.1600 q^{69} -3.98861 q^{71} +4.34320 q^{73} +11.5585 q^{75} -0.877344 q^{77} -13.7650 q^{79} -8.10010 q^{81} +6.69438 q^{83} -2.72128 q^{85} +12.6653 q^{87} +7.88038 q^{89} -4.25424 q^{91} +10.2451 q^{93} +3.21424 q^{95} -12.5887 q^{97} -1.58404 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q + 9 q^{3} + 5 q^{7} + 35 q^{9}+O(q^{10}) \) 30 * q + 9 * q^3 + 5 * q^7 + 35 * q^9	\( 30 q + 9 q^{3} + 5 q^{7} + 35 q^{9} + 31 q^{11} + 5 q^{13} + 8 q^{15} - q^{17} + 29 q^{19} + 6 q^{21} + 27 q^{23} + 34 q^{25} + 36 q^{27} + q^{29} - 9 q^{31} - 8 q^{33} + 51 q^{35} + 5 q^{37} - 8 q^{39} - 3 q^{41} + 43 q^{43} + 42 q^{47} + 41 q^{49} + 54 q^{51} - 11 q^{53} - 10 q^{55} + 2 q^{57} + 49 q^{59} + 21 q^{61} + 15 q^{63} - 14 q^{65} + 68 q^{67} - 10 q^{69} + 44 q^{71} + 25 q^{73} + 51 q^{75} - 20 q^{77} - 49 q^{79} + 6 q^{81} + 88 q^{83} - 5 q^{85} + 16 q^{87} - 16 q^{89} + 46 q^{91} - 4 q^{93} + 28 q^{95} - 4 q^{97} + 71 q^{99}+O(q^{100}) \) 30 * q + 9 * q^3 + 5 * q^7 + 35 * q^9 + 31 * q^11 + 5 * q^13 + 8 * q^15 - q^17 + 29 * q^19 + 6 * q^21 + 27 * q^23 + 34 * q^25 + 36 * q^27 + q^29 - 9 * q^31 - 8 * q^33 + 51 * q^35 + 5 * q^37 - 8 * q^39 - 3 * q^41 + 43 * q^43 + 42 * q^47 + 41 * q^49 + 54 * q^51 - 11 * q^53 - 10 * q^55 + 2 * q^57 + 49 * q^59 + 21 * q^61 + 15 * q^63 - 14 * q^65 + 68 * q^67 - 10 * q^69 + 44 * q^71 + 25 * q^73 + 51 * q^75 - 20 * q^77 - 49 * q^79 + 6 * q^81 + 88 * q^83 - 5 * q^85 + 16 * q^87 - 16 * q^89 + 46 * q^91 - 4 * q^93 + 28 * q^95 - 4 * q^97 + 71 * q^99

Introduction

L-functions

Modular forms

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