LMFDB - Embedded newform 6004.2.a.h.1.11

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6004,2,Mod(1,6004)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6004, 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("6004.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6004 = 2^{2} \cdot 19 \cdot 79 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6004.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:	$47.9421813736$
Analytic rank:	$0$
Dimension:	$31$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.11
Character	$\chi$	$=$	6004.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.68597 q^{3} +0.297828 q^{5} -1.50278 q^{7} -0.157509 q^{9} +O(q^{10})$	$q-1.68597 q^{3} +0.297828 q^{5} -1.50278 q^{7} -0.157509 q^{9} -5.06855 q^{11} -6.53241 q^{13} -0.502128 q^{15} +1.79434 q^{17} -1.00000 q^{19} +2.53365 q^{21} -4.87245 q^{23} -4.91130 q^{25} +5.32346 q^{27} +0.777394 q^{29} -8.49414 q^{31} +8.54542 q^{33} -0.447571 q^{35} -8.90835 q^{37} +11.0134 q^{39} -3.95229 q^{41} +3.75497 q^{43} -0.0469105 q^{45} +0.00550550 q^{47} -4.74164 q^{49} -3.02521 q^{51} +0.339792 q^{53} -1.50956 q^{55} +1.68597 q^{57} -12.1885 q^{59} -1.44351 q^{61} +0.236702 q^{63} -1.94553 q^{65} +4.39626 q^{67} +8.21479 q^{69} -3.38390 q^{71} +14.5164 q^{73} +8.28030 q^{75} +7.61694 q^{77} -1.00000 q^{79} -8.50266 q^{81} -12.7235 q^{83} +0.534406 q^{85} -1.31066 q^{87} -1.74243 q^{89} +9.81680 q^{91} +14.3209 q^{93} -0.297828 q^{95} +12.8208 q^{97} +0.798342 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 31 q - 4 q^{3} + 10 q^{5} + 11 q^{7} + 47 q^{9}+O(q^{10}) $ 31 * q - 4 * q^3 + 10 * q^5 + 11 * q^7 + 47 * q^9	$ 31 q - 4 q^{3} + 10 q^{5} + 11 q^{7} + 47 q^{9} - 4 q^{11} + 11 q^{13} + 5 q^{15} + 14 q^{17} - 31 q^{19} + 22 q^{21} + 15 q^{23} + 59 q^{25} + 5 q^{27} + 34 q^{29} - 12 q^{31} + 10 q^{33} + 8 q^{35} + 16 q^{37} + 18 q^{39} + 27 q^{41} + 2 q^{43} + 22 q^{45} + 30 q^{47} + 62 q^{49} - 14 q^{51} + 35 q^{53} + 8 q^{55} + 4 q^{57} - 16 q^{59} + 37 q^{61} + 31 q^{63} + 80 q^{65} + 16 q^{67} + q^{69} + 19 q^{71} + 38 q^{73} + 21 q^{75} + 44 q^{77} - 31 q^{79} + 55 q^{81} - 12 q^{83} + 66 q^{85} + 58 q^{87} + 16 q^{89} - 42 q^{91} + 10 q^{93} - 10 q^{95} - 12 q^{97} + 12 q^{99}+O(q^{100}) $ 31 * q - 4 * q^3 + 10 * q^5 + 11 * q^7 + 47 * q^9 - 4 * q^11 + 11 * q^13 + 5 * q^15 + 14 * q^17 - 31 * q^19 + 22 * q^21 + 15 * q^23 + 59 * q^25 + 5 * q^27 + 34 * q^29 - 12 * q^31 + 10 * q^33 + 8 * q^35 + 16 * q^37 + 18 * q^39 + 27 * q^41 + 2 * q^43 + 22 * q^45 + 30 * q^47 + 62 * q^49 - 14 * q^51 + 35 * q^53 + 8 * q^55 + 4 * q^57 - 16 * q^59 + 37 * q^61 + 31 * q^63 + 80 * q^65 + 16 * q^67 + q^69 + 19 * q^71 + 38 * q^73 + 21 * q^75 + 44 * q^77 - 31 * q^79 + 55 * q^81 - 12 * q^83 + 66 * q^85 + 58 * q^87 + 16 * q^89 - 42 * q^91 + 10 * q^93 - 10 * q^95 - 12 * q^97 + 12 * 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
$3$	−1.68597	−0.973395	−0.486697	−	0.873571i	$-0.661799\pi$
$3$	−1.68597	−0.973395	−0.486697	+	0.873571i	$0.661799\pi$
$4$	0	0
$5$	0.297828	0.133193	0.0665963	−	0.997780i	$-0.478786\pi$
$5$	0.297828	0.133193	0.0665963	+	0.997780i	$0.478786\pi$
$6$	0	0
$7$	−1.50278	−0.567999	−0.283999	−	0.958824i	$-0.591661\pi$
$7$	−1.50278	−0.567999	−0.283999	+	0.958824i	$0.591661\pi$
$8$	0	0
$9$	−0.157509	−0.0525030
$10$	0	0
$11$	−5.06855	−1.52823	−0.764113	−	0.645083i	$-0.776823\pi$
$11$	−5.06855	−1.52823	−0.764113	+	0.645083i	$0.776823\pi$
$12$	0	0
$13$	−6.53241	−1.81176	−0.905882	−	0.423531i	$-0.860791\pi$
$13$	−6.53241	−1.81176	−0.905882	+	0.423531i	$0.860791\pi$
$14$	0	0
$15$	−0.502128	−0.129649
$16$	0	0
$17$	1.79434	0.435192	0.217596	−	0.976039i	$-0.430178\pi$
$17$	1.79434	0.435192	0.217596	+	0.976039i	$0.430178\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	2.53365	0.552887
$22$	0	0
$23$	−4.87245	−1.01598	−0.507988	−	0.861364i	$-0.669610\pi$
$23$	−4.87245	−1.01598	−0.507988	+	0.861364i	$0.669610\pi$
$24$	0	0
$25$	−4.91130	−0.982260
$26$	0	0
$27$	5.32346	1.02450
$28$	0	0
$29$	0.777394	0.144358	0.0721792	−	0.997392i	$-0.477005\pi$
$29$	0.777394	0.144358	0.0721792	+	0.997392i	$0.477005\pi$
$30$	0	0
$31$	−8.49414	−1.52559	−0.762796	−	0.646639i	$-0.776174\pi$
$31$	−8.49414	−1.52559	−0.762796	+	0.646639i	$0.776174\pi$
$32$	0	0
$33$	8.54542	1.48757
$34$	0	0
$35$	−0.447571	−0.0756533
$36$	0	0
$37$	−8.90835	−1.46452	−0.732262	−	0.681023i	$-0.761535\pi$
$37$	−8.90835	−1.46452	−0.732262	+	0.681023i	$0.761535\pi$
$38$	0	0
$39$	11.0134	1.76356
$40$	0	0
$41$	−3.95229	−0.617244	−0.308622	−	0.951185i	$-0.599868\pi$
$41$	−3.95229	−0.617244	−0.308622	+	0.951185i	$0.599868\pi$
$42$	0	0
$43$	3.75497	0.572628	0.286314	−	0.958136i	$-0.407570\pi$
$43$	3.75497	0.572628	0.286314	+	0.958136i	$0.407570\pi$
$44$	0	0
$45$	−0.0469105	−0.00699301
$46$	0	0
$47$	0.00550550	0.000803060 0	0.000401530	−	1.00000i	$-0.499872\pi$
$47$	0.00550550	0.000803060 0	0.000401530		1.00000i	$0.499872\pi$
$48$	0	0
$49$	−4.74164	−0.677377
$50$	0	0
$51$	−3.02521	−0.423614
$52$	0	0
$53$	0.339792	0.0466740	0.0233370	−	0.999728i	$-0.492571\pi$
$53$	0.339792	0.0466740	0.0233370	+	0.999728i	$0.492571\pi$
$54$	0	0
$55$	−1.50956	−0.203548
$56$	0	0
$57$	1.68597	0.223312
$58$	0	0
$59$	−12.1885	−1.58680	−0.793402	−	0.608698i	$-0.791692\pi$
$59$	−12.1885	−1.58680	−0.793402	+	0.608698i	$0.791692\pi$
$60$	0	0
$61$	−1.44351	−0.184823	−0.0924116	−	0.995721i	$-0.529458\pi$
$61$	−1.44351	−0.184823	−0.0924116	+	0.995721i	$0.529458\pi$
$62$	0	0
$63$	0.236702	0.0298216
$64$	0	0
$65$	−1.94553	−0.241314
$66$	0	0
$67$	4.39626	0.537089	0.268544	−	0.963267i	$-0.413457\pi$
$67$	4.39626	0.537089	0.268544	+	0.963267i	$0.413457\pi$
$68$	0	0
$69$	8.21479	0.988945
$70$	0	0
$71$	−3.38390	−0.401595	−0.200798	−	0.979633i	$-0.564353\pi$
$71$	−3.38390	−0.401595	−0.200798	+	0.979633i	$0.564353\pi$
$72$	0	0
$73$	14.5164	1.69902	0.849509	−	0.527575i	$-0.176899\pi$
$73$	14.5164	1.69902	0.849509	+	0.527575i	$0.176899\pi$
$74$	0	0
$75$	8.28030	0.956126
$76$	0	0
$77$	7.61694	0.868030
$78$	0	0
$79$	−1.00000	−0.112509
$79$	−1.00000	−0.112509
$80$	0	0
$81$	−8.50266	−0.944740
$82$	0	0
$83$	−12.7235	−1.39658	−0.698291	−	0.715814i	$-0.746056\pi$
$83$	−12.7235	−1.39658	−0.698291	+	0.715814i	$0.746056\pi$
$84$	0	0
$85$	0.534406	0.0579644
$86$	0	0
$87$	−1.31066	−0.140518
$88$	0	0
$89$	−1.74243	−0.184697	−0.0923487	−	0.995727i	$-0.529437\pi$
$89$	−1.74243	−0.184697	−0.0923487	+	0.995727i	$0.529437\pi$
$90$	0	0
$91$	9.81680	1.02908
$92$	0	0
$93$	14.3209	1.48500
$94$	0	0
$95$	−0.297828	−0.0305565
$96$	0	0
$97$	12.8208	1.30175	0.650876	−	0.759184i	$-0.274402\pi$
$97$	12.8208	1.30175	0.650876	+	0.759184i	$0.274402\pi$
$98$	0	0
$99$	0.798342	0.0802364
$100$	0	0
$101$	4.85586	0.483176	0.241588	−	0.970379i	$-0.422332\pi$
$101$	4.85586	0.483176	0.241588	+	0.970379i	$0.422332\pi$
$102$	0	0
$103$	2.19163	0.215947	0.107974	−	0.994154i	$-0.465564\pi$
$103$	2.19163	0.215947	0.107974	+	0.994154i	$0.465564\pi$
$104$	0	0
$105$	0.754590	0.0736405
$106$	0	0
$107$	−2.15053	−0.207900	−0.103950	−	0.994583i	$-0.533148\pi$
$107$	−2.15053	−0.207900	−0.103950	+	0.994583i	$0.533148\pi$
$108$	0	0
$109$	17.7618	1.70127	0.850637	−	0.525753i	$-0.176216\pi$
$109$	17.7618	1.70127	0.850637	+	0.525753i	$0.176216\pi$
$110$	0	0
$111$	15.0192	1.42556
$112$	0	0
$113$	−5.63940	−0.530510	−0.265255	−	0.964178i	$-0.585456\pi$
$113$	−5.63940	−0.530510	−0.265255	+	0.964178i	$0.585456\pi$
$114$	0	0
$115$	−1.45115	−0.135320
$116$	0	0
$117$	1.02891	0.0951230
$118$	0	0
$119$	−2.69651	−0.247189
$120$	0	0
$121$	14.6902	1.33547
$122$	0	0
$123$	6.66344	0.600822
$124$	0	0
$125$	−2.95186	−0.264022
$126$	0	0
$127$	−12.7675	−1.13294	−0.566468	−	0.824084i	$-0.691690\pi$
$127$	−12.7675	−1.13294	−0.566468	+	0.824084i	$0.691690\pi$
$128$	0	0
$129$	−6.33077	−0.557393
$130$	0	0
$131$	−8.54323	−0.746425	−0.373213	−	0.927746i	$-0.621744\pi$
$131$	−8.54323	−0.746425	−0.373213	+	0.927746i	$0.621744\pi$
$132$	0	0
$133$	1.50278	0.130308
$134$	0	0
$135$	1.58547	0.136456
$136$	0	0
$137$	−13.3371	−1.13947	−0.569735	−	0.821829i	$-0.692954\pi$
$137$	−13.3371	−1.13947	−0.569735	+	0.821829i	$0.692954\pi$
$138$	0	0
$139$	−17.4757	−1.48227	−0.741136	−	0.671355i	$-0.765713\pi$
$139$	−17.4757	−1.48227	−0.741136	+	0.671355i	$0.765713\pi$
$140$	0	0
$141$	−0.00928210	−0.000781694 0
$142$	0	0
$143$	33.1098	2.76878
$144$	0	0
$145$	0.231529	0.0192275
$146$	0	0
$147$	7.99426	0.659355
$148$	0	0
$149$	−1.88925	−0.154773	−0.0773866	−	0.997001i	$-0.524658\pi$
$149$	−1.88925	−0.154773	−0.0773866	+	0.997001i	$0.524658\pi$
$150$	0	0
$151$	−2.42258	−0.197146	−0.0985732	−	0.995130i	$-0.531428\pi$
$151$	−2.42258	−0.197146	−0.0985732	+	0.995130i	$0.531428\pi$
$152$	0	0
$153$	−0.282625	−0.0228489
$154$	0	0
$155$	−2.52979	−0.203198
$156$	0	0
$157$	−5.53535	−0.441769	−0.220885	−	0.975300i	$-0.570894\pi$
$157$	−5.53535	−0.441769	−0.220885	+	0.975300i	$0.570894\pi$
$158$	0	0
$159$	−0.572878	−0.0454322
$160$	0	0
$161$	7.32223	0.577073
$162$	0	0
$163$	−7.51696	−0.588774	−0.294387	−	0.955686i	$-0.595115\pi$
$163$	−7.51696	−0.588774	−0.294387	+	0.955686i	$0.595115\pi$
$164$	0	0
$165$	2.54506	0.198133
$166$	0	0
$167$	−22.9549	−1.77630	−0.888152	−	0.459549i	$-0.848011\pi$
$167$	−22.9549	−1.77630	−0.888152	+	0.459549i	$0.848011\pi$
$168$	0	0
$169$	29.6723	2.28249
$170$	0	0
$171$	0.157509	0.0120450
$172$	0	0
$173$	13.9783	1.06275	0.531375	−	0.847137i	$-0.321676\pi$
$173$	13.9783	1.06275	0.531375	+	0.847137i	$0.321676\pi$
$174$	0	0
$175$	7.38062	0.557922
$176$	0	0
$177$	20.5494	1.54459
$178$	0	0
$179$	−8.43551	−0.630500	−0.315250	−	0.949009i	$-0.602088\pi$
$179$	−8.43551	−0.630500	−0.315250	+	0.949009i	$0.602088\pi$
$180$	0	0
$181$	24.8912	1.85015	0.925076	−	0.379783i	$-0.124001\pi$
$181$	24.8912	1.85015	0.925076	+	0.379783i	$0.124001\pi$
$182$	0	0
$183$	2.43372	0.179906
$184$	0	0
$185$	−2.65315	−0.195064
$186$	0	0
$187$	−9.09472	−0.665072
$188$	0	0
$189$	−8.00001	−0.581915
$190$	0	0
$191$	7.60482	0.550265	0.275133	−	0.961406i	$-0.411278\pi$
$191$	7.60482	0.550265	0.275133	+	0.961406i	$0.411278\pi$
$192$	0	0
$193$	−0.690100	−0.0496745	−0.0248372	−	0.999692i	$-0.507907\pi$
$193$	−0.690100	−0.0496745	−0.0248372	+	0.999692i	$0.507907\pi$
$194$	0	0
$195$	3.28011	0.234893
$196$	0	0
$197$	−5.78924	−0.412466	−0.206233	−	0.978503i	$-0.566121\pi$
$197$	−5.78924	−0.412466	−0.206233	+	0.978503i	$0.566121\pi$
$198$	0	0
$199$	−17.9566	−1.27291	−0.636455	−	0.771314i	$-0.719600\pi$
$199$	−17.9566	−1.27291	−0.636455	+	0.771314i	$0.719600\pi$
$200$	0	0
$201$	−7.41196	−0.522799
$202$	0	0
$203$	−1.16825	−0.0819954
$204$	0	0
$205$	−1.17710	−0.0822124
$206$	0	0
$207$	0.767454	0.0533417
$208$	0	0
$209$	5.06855	0.350599
$210$	0	0
$211$	−13.7725	−0.948141	−0.474070	−	0.880487i	$-0.657216\pi$
$211$	−13.7725	−0.948141	−0.474070	+	0.880487i	$0.657216\pi$
$212$	0	0
$213$	5.70516	0.390911
$214$	0	0
$215$	1.11833	0.0762698
$216$	0	0
$217$	12.7649	0.866535
$218$	0	0
$219$	−24.4742	−1.65381
$220$	0	0
$221$	−11.7214	−0.788466
$222$	0	0
$223$	0.222929	0.0149284	0.00746422	−	0.999972i	$-0.497624\pi$
$223$	0.222929	0.0149284	0.00746422	+	0.999972i	$0.497624\pi$
$224$	0	0
$225$	0.773574	0.0515716
$226$	0	0
$227$	8.46050	0.561543	0.280772	−	0.959775i	$-0.409410\pi$
$227$	8.46050	0.561543	0.280772	+	0.959775i	$0.409410\pi$
$228$	0	0
$229$	5.02736	0.332217	0.166109	−	0.986107i	$-0.446880\pi$
$229$	5.02736	0.332217	0.166109	+	0.986107i	$0.446880\pi$
$230$	0	0
$231$	−12.8419	−0.844936
$232$	0	0
$233$	−8.21855	−0.538415	−0.269208	−	0.963082i	$-0.586762\pi$
$233$	−8.21855	−0.538415	−0.269208	+	0.963082i	$0.586762\pi$
$234$	0	0
$235$	0.00163969	0.000106962 0
$236$	0	0
$237$	1.68597	0.109515
$238$	0	0
$239$	26.7260	1.72876	0.864382	−	0.502836i	$-0.167710\pi$
$239$	26.7260	1.72876	0.864382	+	0.502836i	$0.167710\pi$
$240$	0	0
$241$	−5.67390	−0.365488	−0.182744	−	0.983161i	$-0.558498\pi$
$241$	−5.67390	−0.365488	−0.182744	+	0.983161i	$0.558498\pi$
$242$	0	0
$243$	−1.63516	−0.104895
$244$	0	0
$245$	−1.41219	−0.0902216
$246$	0	0
$247$	6.53241	0.415647
$248$	0	0
$249$	21.4514	1.35943
$250$	0	0
$251$	6.17045	0.389476	0.194738	−	0.980855i	$-0.437614\pi$
$251$	6.17045	0.389476	0.194738	+	0.980855i	$0.437614\pi$
$252$	0	0
$253$	24.6962	1.55264
$254$	0	0
$255$	−0.900991	−0.0564222
$256$	0	0
$257$	−6.69541	−0.417648	−0.208824	−	0.977953i	$-0.566964\pi$
$257$	−6.69541	−0.417648	−0.208824	+	0.977953i	$0.566964\pi$
$258$	0	0
$259$	13.3873	0.831848
$260$	0	0
$261$	−0.122447	−0.00757925
$262$	0	0
$263$	28.9891	1.78754	0.893771	−	0.448524i	$-0.148050\pi$
$263$	28.9891	1.78754	0.893771	+	0.448524i	$0.148050\pi$
$264$	0	0
$265$	0.101199	0.00621663
$266$	0	0
$267$	2.93769	0.179783
$268$	0	0
$269$	−18.1102	−1.10420	−0.552098	−	0.833779i	$-0.686173\pi$
$269$	−18.1102	−1.10420	−0.552098	+	0.833779i	$0.686173\pi$
$270$	0	0
$271$	19.8829	1.20780	0.603901	−	0.797059i	$-0.293612\pi$
$271$	19.8829	1.20780	0.603901	+	0.797059i	$0.293612\pi$
$272$	0	0
$273$	−16.5508	−1.00170
$274$	0	0
$275$	24.8932	1.50111
$276$	0	0
$277$	−24.6095	−1.47864	−0.739321	−	0.673353i	$-0.764853\pi$
$277$	−24.6095	−1.47864	−0.739321	+	0.673353i	$0.764853\pi$
$278$	0	0
$279$	1.33790	0.0800982
$280$	0	0
$281$	3.42088	0.204073	0.102036	−	0.994781i	$-0.467464\pi$
$281$	3.42088	0.204073	0.102036	+	0.994781i	$0.467464\pi$
$282$	0	0
$283$	−8.27689	−0.492010	−0.246005	−	0.969269i	$-0.579118\pi$
$283$	−8.27689	−0.492010	−0.246005	+	0.969269i	$0.579118\pi$
$284$	0	0
$285$	0.502128	0.0297435
$286$	0	0
$287$	5.93944	0.350594
$288$	0	0
$289$	−13.7803	−0.810608
$290$	0	0
$291$	−21.6154	−1.26712
$292$	0	0
$293$	−15.1737	−0.886457	−0.443229	−	0.896409i	$-0.646167\pi$
$293$	−15.1737	−0.886457	−0.443229	+	0.896409i	$0.646167\pi$
$294$	0	0
$295$	−3.63007	−0.211351
$296$	0	0
$297$	−26.9822	−1.56567
$298$	0	0
$299$	31.8288	1.84071
$300$	0	0
$301$	−5.64291	−0.325252
$302$	0	0
$303$	−8.18682	−0.470321
$304$	0	0
$305$	−0.429919	−0.0246171
$306$	0	0
$307$	−27.4398	−1.56607	−0.783037	−	0.621975i	$-0.786331\pi$
$307$	−27.4398	−1.56607	−0.783037	+	0.621975i	$0.786331\pi$
$308$	0	0
$309$	−3.69501	−0.210202
$310$	0	0
$311$	24.5181	1.39029	0.695146	−	0.718868i	$-0.255340\pi$
$311$	24.5181	1.39029	0.695146	+	0.718868i	$0.255340\pi$
$312$	0	0
$313$	18.5172	1.04666	0.523328	−	0.852131i	$-0.324690\pi$
$313$	18.5172	1.04666	0.523328	+	0.852131i	$0.324690\pi$
$314$	0	0
$315$	0.0704964	0.00397202
$316$	0	0
$317$	−4.58121	−0.257307	−0.128653	−	0.991690i	$-0.541065\pi$
$317$	−4.58121	−0.257307	−0.128653	+	0.991690i	$0.541065\pi$
$318$	0	0
$319$	−3.94026	−0.220612
$320$	0	0
$321$	3.62573	0.202369
$322$	0	0
$323$	−1.79434	−0.0998400
$324$	0	0
$325$	32.0826	1.77962
$326$	0	0
$327$	−29.9459	−1.65601
$328$	0	0
$329$	−0.00827358	−0.000456137 0
$330$	0	0
$331$	−27.0701	−1.48791	−0.743955	−	0.668230i	$-0.767052\pi$
$331$	−27.0701	−1.48791	−0.743955	+	0.668230i	$0.767052\pi$
$332$	0	0
$333$	1.40314	0.0768919
$334$	0	0
$335$	1.30933	0.0715362
$336$	0	0
$337$	14.3817	0.783418	0.391709	−	0.920089i	$-0.371884\pi$
$337$	14.3817	0.783418	0.391709	+	0.920089i	$0.371884\pi$
$338$	0	0
$339$	9.50785	0.516395
$340$	0	0
$341$	43.0530	2.33145
$342$	0	0
$343$	17.6451	0.952748
$344$	0	0
$345$	2.44659	0.131720
$346$	0	0
$347$	−29.1543	−1.56508	−0.782542	−	0.622598i	$-0.786077\pi$
$347$	−29.1543	−1.56508	−0.782542	+	0.622598i	$0.786077\pi$
$348$	0	0
$349$	−11.9332	−0.638768	−0.319384	−	0.947625i	$-0.603476\pi$
$349$	−11.9332	−0.638768	−0.319384	+	0.947625i	$0.603476\pi$
$350$	0	0
$351$	−34.7750	−1.85615
$352$	0	0
$353$	25.8377	1.37520	0.687601	−	0.726089i	$-0.258664\pi$
$353$	25.8377	1.37520	0.687601	+	0.726089i	$0.258664\pi$
$354$	0	0
$355$	−1.00782	−0.0534896
$356$	0	0
$357$	4.54623	0.240612
$358$	0	0
$359$	4.08127	0.215401	0.107701	−	0.994183i	$-0.465651\pi$
$359$	4.08127	0.215401	0.107701	+	0.994183i	$0.465651\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−24.7672	−1.29994
$364$	0	0
$365$	4.32339	0.226297
$366$	0	0
$367$	−16.8659	−0.880391	−0.440196	−	0.897902i	$-0.645091\pi$
$367$	−16.8659	−0.880391	−0.440196	+	0.897902i	$0.645091\pi$
$368$	0	0
$369$	0.622522	0.0324072
$370$	0	0
$371$	−0.510633	−0.0265108
$372$	0	0
$373$	−21.9376	−1.13589	−0.567943	−	0.823068i	$-0.692261\pi$
$373$	−21.9376	−1.13589	−0.567943	+	0.823068i	$0.692261\pi$
$374$	0	0
$375$	4.97674	0.256998
$376$	0	0
$377$	−5.07825	−0.261543
$378$	0	0
$379$	−18.4943	−0.949990	−0.474995	−	0.879988i	$-0.657550\pi$
$379$	−18.4943	−0.949990	−0.474995	+	0.879988i	$0.657550\pi$
$380$	0	0
$381$	21.5257	1.10279
$382$	0	0
$383$	17.8259	0.910861	0.455430	−	0.890271i	$-0.349485\pi$
$383$	17.8259	0.910861	0.455430	+	0.890271i	$0.349485\pi$
$384$	0	0
$385$	2.26854	0.115615
$386$	0	0
$387$	−0.591442	−0.0300647
$388$	0	0
$389$	8.16346	0.413904	0.206952	−	0.978351i	$-0.433646\pi$
$389$	8.16346	0.413904	0.206952	+	0.978351i	$0.433646\pi$
$390$	0	0
$391$	−8.74285	−0.442145
$392$	0	0
$393$	14.4036	0.726566
$394$	0	0
$395$	−0.297828	−0.0149853
$396$	0	0
$397$	5.40704	0.271371	0.135686	−	0.990752i	$-0.456676\pi$
$397$	5.40704	0.271371	0.135686	+	0.990752i	$0.456676\pi$
$398$	0	0
$399$	−2.53365	−0.126841
$400$	0	0
$401$	28.8033	1.43837	0.719185	−	0.694819i	$-0.244516\pi$
$401$	28.8033	1.43837	0.719185	+	0.694819i	$0.244516\pi$
$402$	0	0
$403$	55.4872	2.76401
$404$	0	0
$405$	−2.53233	−0.125832
$406$	0	0
$407$	45.1524	2.23812
$408$	0	0
$409$	4.11449	0.203449	0.101724	−	0.994813i	$-0.467564\pi$
$409$	4.11449	0.203449	0.101724	+	0.994813i	$0.467564\pi$
$410$	0	0
$411$	22.4860	1.10915
$412$	0	0
$413$	18.3166	0.901303
$414$	0	0
$415$	−3.78940	−0.186014
$416$	0	0
$417$	29.4635	1.44284
$418$	0	0
$419$	13.9069	0.679398	0.339699	−	0.940534i	$-0.389675\pi$
$419$	13.9069	0.679398	0.339699	+	0.940534i	$0.389675\pi$
$420$	0	0
$421$	1.91134	0.0931528	0.0465764	−	0.998915i	$-0.485169\pi$
$421$	1.91134	0.0931528	0.0465764	+	0.998915i	$0.485169\pi$
$422$	0	0
$423$	−0.000867166 0	−4.21630e−5 0
$424$	0	0
$425$	−8.81256	−0.427472
$426$	0	0
$427$	2.16929	0.104979
$428$	0	0
$429$	−55.8221	−2.69512
$430$	0	0
$431$	10.7911	0.519791	0.259895	−	0.965637i	$-0.416312\pi$
$431$	10.7911	0.519791	0.259895	+	0.965637i	$0.416312\pi$
$432$	0	0
$433$	16.9559	0.814848	0.407424	−	0.913239i	$-0.366427\pi$
$433$	16.9559	0.814848	0.407424	+	0.913239i	$0.366427\pi$
$434$	0	0
$435$	−0.390351	−0.0187159
$436$	0	0
$437$	4.87245	0.233081
$438$	0	0
$439$	27.6326	1.31883	0.659416	−	0.751778i	$-0.270804\pi$
$439$	27.6326	1.31883	0.659416	+	0.751778i	$0.270804\pi$
$440$	0	0
$441$	0.746851	0.0355643
$442$	0	0
$443$	−20.0539	−0.952790	−0.476395	−	0.879231i	$-0.658057\pi$
$443$	−20.0539	−0.952790	−0.476395	+	0.879231i	$0.658057\pi$
$444$	0	0
$445$	−0.518945	−0.0246003
$446$	0	0
$447$	3.18521	0.150655
$448$	0	0
$449$	24.4699	1.15480	0.577402	−	0.816460i	$-0.304066\pi$
$449$	24.4699	1.15480	0.577402	+	0.816460i	$0.304066\pi$
$450$	0	0
$451$	20.0324	0.943289
$452$	0	0
$453$	4.08439	0.191901
$454$	0	0
$455$	2.92371	0.137066
$456$	0	0
$457$	−4.45857	−0.208563	−0.104281	−	0.994548i	$-0.533254\pi$
$457$	−4.45857	−0.208563	−0.104281	+	0.994548i	$0.533254\pi$
$458$	0	0
$459$	9.55212	0.445855
$460$	0	0
$461$	27.0223	1.25855	0.629276	−	0.777182i	$-0.283351\pi$
$461$	27.0223	1.25855	0.629276	+	0.777182i	$0.283351\pi$
$462$	0	0
$463$	−33.4334	−1.55378	−0.776890	−	0.629636i	$-0.783204\pi$
$463$	−33.4334	−1.55378	−0.776890	+	0.629636i	$0.783204\pi$
$464$	0	0
$465$	4.26515	0.197791
$466$	0	0
$467$	16.1273	0.746282	0.373141	−	0.927775i	$-0.378281\pi$
$467$	16.1273	0.746282	0.373141	+	0.927775i	$0.378281\pi$
$468$	0	0
$469$	−6.60663	−0.305066
$470$	0	0
$471$	9.33244	0.430016
$472$	0	0
$473$	−19.0323	−0.875105
$474$	0	0
$475$	4.91130	0.225346
$476$	0	0
$477$	−0.0535202	−0.00245052
$478$	0	0
$479$	−40.7057	−1.85989	−0.929945	−	0.367697i	$-0.880146\pi$
$479$	−40.7057	−1.85989	−0.929945	+	0.367697i	$0.880146\pi$
$480$	0	0
$481$	58.1929	2.65337
$482$	0	0
$483$	−12.3451	−0.561720
$484$	0	0
$485$	3.81838	0.173384
$486$	0	0
$487$	13.1373	0.595310	0.297655	−	0.954674i	$-0.403796\pi$
$487$	13.1373	0.595310	0.297655	+	0.954674i	$0.403796\pi$
$488$	0	0
$489$	12.6734	0.573109
$490$	0	0
$491$	−7.43687	−0.335621	−0.167811	−	0.985819i	$-0.553670\pi$
$491$	−7.43687	−0.335621	−0.167811	+	0.985819i	$0.553670\pi$
$492$	0	0
$493$	1.39491	0.0628237
$494$	0	0
$495$	0.237768	0.0106869
$496$	0	0
$497$	5.08528	0.228106
$498$	0	0
$499$	25.3655	1.13551	0.567757	−	0.823196i	$-0.307811\pi$
$499$	25.3655	1.13551	0.567757	+	0.823196i	$0.307811\pi$
$500$	0	0
$501$	38.7013	1.72905
$502$	0	0
$503$	12.7232	0.567301	0.283651	−	0.958928i	$-0.408454\pi$
$503$	12.7232	0.567301	0.283651	+	0.958928i	$0.408454\pi$
$504$	0	0
$505$	1.44621	0.0643554
$506$	0	0
$507$	−50.0266	−2.22176
$508$	0	0
$509$	8.58516	0.380530	0.190265	−	0.981733i	$-0.439065\pi$
$509$	8.58516	0.380530	0.190265	+	0.981733i	$0.439065\pi$
$510$	0	0
$511$	−21.8150	−0.965040
$512$	0	0
$513$	−5.32346	−0.235037
$514$	0	0
$515$	0.652727	0.0287626
$516$	0	0
$517$	−0.0279049	−0.00122726
$518$	0	0
$519$	−23.5670	−1.03448
$520$	0	0
$521$	−16.9490	−0.742550	−0.371275	−	0.928523i	$-0.621079\pi$
$521$	−16.9490	−0.742550	−0.371275	+	0.928523i	$0.621079\pi$
$522$	0	0
$523$	−16.8689	−0.737625	−0.368812	−	0.929504i	$-0.620235\pi$
$523$	−16.8689	−0.737625	−0.368812	+	0.929504i	$0.620235\pi$
$524$	0	0
$525$	−12.4435	−0.543079
$526$	0	0
$527$	−15.2414	−0.663926
$528$	0	0
$529$	0.740732	0.0322057
$530$	0	0
$531$	1.91979	0.0833120
$532$	0	0
$533$	25.8180	1.11830
$534$	0	0
$535$	−0.640488	−0.0276907
$536$	0	0
$537$	14.2220	0.613725
$538$	0	0
$539$	24.0332	1.03519
$540$	0	0
$541$	0.340282	0.0146299	0.00731493	−	0.999973i	$-0.497672\pi$
$541$	0.340282	0.0146299	0.00731493	+	0.999973i	$0.497672\pi$
$542$	0	0
$543$	−41.9659	−1.80093
$544$	0	0
$545$	5.28997	0.226597
$546$	0	0
$547$	22.1007	0.944959	0.472479	−	0.881342i	$-0.343359\pi$
$547$	22.1007	0.944959	0.472479	+	0.881342i	$0.343359\pi$
$548$	0	0
$549$	0.227367	0.00970377
$550$	0	0
$551$	−0.777394	−0.0331181
$552$	0	0
$553$	1.50278	0.0639049
$554$	0	0
$555$	4.47313	0.189874
$556$	0	0
$557$	−7.74883	−0.328329	−0.164164	−	0.986433i	$-0.552493\pi$
$557$	−7.74883	−0.328329	−0.164164	+	0.986433i	$0.552493\pi$
$558$	0	0
$559$	−24.5290	−1.03747
$560$	0	0
$561$	15.3334	0.647378
$562$	0	0
$563$	−20.3011	−0.855588	−0.427794	−	0.903876i	$-0.640709\pi$
$563$	−20.3011	−0.855588	−0.427794	+	0.903876i	$0.640709\pi$
$564$	0	0
$565$	−1.67957	−0.0706600
$566$	0	0
$567$	12.7777	0.536612
$568$	0	0
$569$	−4.46628	−0.187236	−0.0936182	−	0.995608i	$-0.529843\pi$
$569$	−4.46628	−0.187236	−0.0936182	+	0.995608i	$0.529843\pi$
$570$	0	0
$571$	10.7130	0.448326	0.224163	−	0.974552i	$-0.428035\pi$
$571$	10.7130	0.448326	0.224163	+	0.974552i	$0.428035\pi$
$572$	0	0
$573$	−12.8215	−0.535625
$574$	0	0
$575$	23.9300	0.997952
$576$	0	0
$577$	15.9770	0.665130	0.332565	−	0.943080i	$-0.392086\pi$
$577$	15.9770	0.665130	0.332565	+	0.943080i	$0.392086\pi$
$578$	0	0
$579$	1.16349	0.0483529
$580$	0	0
$581$	19.1206	0.793257
$582$	0	0
$583$	−1.72225	−0.0713284
$584$	0	0
$585$	0.306439	0.0126697
$586$	0	0
$587$	30.2371	1.24802	0.624008	−	0.781418i	$-0.285503\pi$
$587$	30.2371	1.24802	0.624008	+	0.781418i	$0.285503\pi$
$588$	0	0
$589$	8.49414	0.349995
$590$	0	0
$591$	9.76048	0.401492
$592$	0	0
$593$	−16.9751	−0.697084	−0.348542	−	0.937293i	$-0.613323\pi$
$593$	−16.9751	−0.697084	−0.348542	+	0.937293i	$0.613323\pi$
$594$	0	0
$595$	−0.803096	−0.0329237
$596$	0	0
$597$	30.2743	1.23904
$598$	0	0
$599$	38.8215	1.58620	0.793101	−	0.609089i	$-0.208465\pi$
$599$	38.8215	1.58620	0.793101	+	0.609089i	$0.208465\pi$
$600$	0	0
$601$	−34.6862	−1.41488	−0.707439	−	0.706775i	$-0.750150\pi$
$601$	−34.6862	−1.41488	−0.707439	+	0.706775i	$0.750150\pi$
$602$	0	0
$603$	−0.692450	−0.0281988
$604$	0	0
$605$	4.37515	0.177875
$606$	0	0
$607$	13.9862	0.567682	0.283841	−	0.958871i	$-0.408391\pi$
$607$	13.9862	0.567682	0.283841	+	0.958871i	$0.408391\pi$
$608$	0	0
$609$	1.96964	0.0798139
$610$	0	0
$611$	−0.0359642	−0.00145495
$612$	0	0
$613$	−28.1666	−1.13764	−0.568820	−	0.822462i	$-0.692600\pi$
$613$	−28.1666	−1.13764	−0.568820	+	0.822462i	$0.692600\pi$
$614$	0	0
$615$	1.98456	0.0800251
$616$	0	0
$617$	−47.8089	−1.92471	−0.962357	−	0.271789i	$-0.912385\pi$
$617$	−47.8089	−1.92471	−0.962357	+	0.271789i	$0.912385\pi$
$618$	0	0
$619$	8.54622	0.343502	0.171751	−	0.985140i	$-0.445058\pi$
$619$	8.54622	0.343502	0.171751	+	0.985140i	$0.445058\pi$
$620$	0	0
$621$	−25.9383	−1.04087
$622$	0	0
$623$	2.61850	0.104908
$624$	0	0
$625$	23.6773	0.947094
$626$	0	0
$627$	−8.54542	−0.341271
$628$	0	0
$629$	−15.9846	−0.637349
$630$	0	0
$631$	17.5956	0.700469	0.350235	−	0.936662i	$-0.386102\pi$
$631$	17.5956	0.700469	0.350235	+	0.936662i	$0.386102\pi$
$632$	0	0
$633$	23.2201	0.922915
$634$	0	0
$635$	−3.80253	−0.150899
$636$	0	0
$637$	30.9743	1.22725
$638$	0	0
$639$	0.532995	0.0210850
$640$	0	0
$641$	39.7978	1.57192	0.785960	−	0.618278i	$-0.212169\pi$
$641$	39.7978	1.57192	0.785960	+	0.618278i	$0.212169\pi$
$642$	0	0
$643$	−48.7078	−1.92085	−0.960424	−	0.278541i	$-0.910149\pi$
$643$	−48.7078	−1.92085	−0.960424	+	0.278541i	$0.910149\pi$
$644$	0	0
$645$	−1.88548	−0.0742406
$646$	0	0
$647$	−37.5355	−1.47567	−0.737837	−	0.674979i	$-0.764153\pi$
$647$	−37.5355	−1.47567	−0.737837	+	0.674979i	$0.764153\pi$
$648$	0	0
$649$	61.7779	2.42499
$650$	0	0
$651$	−21.5211	−0.843480
$652$	0	0
$653$	44.8155	1.75377	0.876883	−	0.480703i	$-0.159619\pi$
$653$	44.8155	1.75377	0.876883	+	0.480703i	$0.159619\pi$
$654$	0	0
$655$	−2.54441	−0.0994183
$656$	0	0
$657$	−2.28646	−0.0892035
$658$	0	0
$659$	−29.6377	−1.15452	−0.577260	−	0.816561i	$-0.695878\pi$
$659$	−29.6377	−1.15452	−0.577260	+	0.816561i	$0.695878\pi$
$660$	0	0
$661$	−16.5736	−0.644640	−0.322320	−	0.946631i	$-0.604463\pi$
$661$	−16.5736	−0.644640	−0.322320	+	0.946631i	$0.604463\pi$
$662$	0	0
$663$	19.7619	0.767488
$664$	0	0
$665$	0.447571	0.0173561
$666$	0	0
$667$	−3.78781	−0.146665
$668$	0	0
$669$	−0.375852	−0.0145313
$670$	0	0
$671$	7.31653	0.282451
$672$	0	0
$673$	−28.2342	−1.08835	−0.544174	−	0.838973i	$-0.683157\pi$
$673$	−28.2342	−1.08835	−0.544174	+	0.838973i	$0.683157\pi$
$674$	0	0
$675$	−26.1451	−1.00633
$676$	0	0
$677$	−28.2766	−1.08676	−0.543378	−	0.839488i	$-0.682855\pi$
$677$	−28.2766	−1.08676	−0.543378	+	0.839488i	$0.682855\pi$
$678$	0	0
$679$	−19.2668	−0.739394
$680$	0	0
$681$	−14.2641	−0.546603
$682$	0	0
$683$	−48.6007	−1.85965	−0.929827	−	0.367998i	$-0.880043\pi$
$683$	−48.6007	−1.85965	−0.929827	+	0.367998i	$0.880043\pi$
$684$	0	0
$685$	−3.97217	−0.151769
$686$	0	0
$687$	−8.47597	−0.323378
$688$	0	0
$689$	−2.21966	−0.0845622
$690$	0	0
$691$	41.0963	1.56338	0.781689	−	0.623668i	$-0.214358\pi$
$691$	41.0963	1.56338	0.781689	+	0.623668i	$0.214358\pi$
$692$	0	0
$693$	−1.19974	−0.0455742
$694$	0	0
$695$	−5.20476	−0.197428
$696$	0	0
$697$	−7.09177	−0.268620
$698$	0	0
$699$	13.8562	0.524090
$700$	0	0
$701$	−22.2050	−0.838671	−0.419335	−	0.907831i	$-0.637737\pi$
$701$	−22.2050	−0.838671	−0.419335	+	0.907831i	$0.637737\pi$
$702$	0	0
$703$	8.90835	0.335985
$704$	0	0
$705$	−0.00276447	−0.000104116 0
$706$	0	0
$707$	−7.29730	−0.274443
$708$	0	0
$709$	−16.2568	−0.610537	−0.305269	−	0.952266i	$-0.598746\pi$
$709$	−16.2568	−0.610537	−0.305269	+	0.952266i	$0.598746\pi$
$710$	0	0
$711$	0.157509	0.00590705
$712$	0	0
$713$	41.3872	1.54996
$714$	0	0
$715$	9.86103	0.368782
$716$	0	0
$717$	−45.0593	−1.68277
$718$	0	0
$719$	−22.3862	−0.834863	−0.417431	−	0.908708i	$-0.637070\pi$
$719$	−22.3862	−0.834863	−0.417431	+	0.908708i	$0.637070\pi$
$720$	0	0
$721$	−3.29354	−0.122658
$722$	0	0
$723$	9.56602	0.355764
$724$	0	0
$725$	−3.81801	−0.141797
$726$	0	0
$727$	−32.5261	−1.20633	−0.603163	−	0.797618i	$-0.706093\pi$
$727$	−32.5261	−1.20633	−0.603163	+	0.797618i	$0.706093\pi$
$728$	0	0
$729$	28.2648	1.04685
$730$	0	0
$731$	6.73771	0.249203
$732$	0	0
$733$	2.18897	0.0808516	0.0404258	−	0.999183i	$-0.487129\pi$
$733$	2.18897	0.0808516	0.0404258	+	0.999183i	$0.487129\pi$
$734$	0	0
$735$	2.38091	0.0878213
$736$	0	0
$737$	−22.2827	−0.820792
$738$	0	0
$739$	−22.2262	−0.817603	−0.408801	−	0.912623i	$-0.634053\pi$
$739$	−22.2262	−0.817603	−0.408801	+	0.912623i	$0.634053\pi$
$740$	0	0
$741$	−11.0134	−0.404589
$742$	0	0
$743$	−43.9638	−1.61288	−0.806439	−	0.591318i	$-0.798608\pi$
$743$	−43.9638	−1.61288	−0.806439	+	0.591318i	$0.798608\pi$
$744$	0	0
$745$	−0.562671	−0.0206147
$746$	0	0
$747$	2.00406	0.0733248
$748$	0	0
$749$	3.23179	0.118087
$750$	0	0
$751$	−23.8736	−0.871158	−0.435579	−	0.900150i	$-0.643456\pi$
$751$	−23.8736	−0.871158	−0.435579	+	0.900150i	$0.643456\pi$
$752$	0	0
$753$	−10.4032	−0.379113
$754$	0	0
$755$	−0.721511	−0.0262585
$756$	0	0
$757$	−24.7452	−0.899381	−0.449690	−	0.893184i	$-0.648466\pi$
$757$	−24.7452	−0.899381	−0.449690	+	0.893184i	$0.648466\pi$
$758$	0	0
$759$	−41.6371	−1.51133
$760$	0	0
$761$	−8.00314	−0.290114	−0.145057	−	0.989423i	$-0.546336\pi$
$761$	−8.00314	−0.290114	−0.145057	+	0.989423i	$0.546336\pi$
$762$	0	0
$763$	−26.6922	−0.966322
$764$	0	0
$765$	−0.0841737	−0.00304331
$766$	0	0
$767$	79.6201	2.87491
$768$	0	0
$769$	−4.72828	−0.170506	−0.0852531	−	0.996359i	$-0.527170\pi$
$769$	−4.72828	−0.170506	−0.0852531	+	0.996359i	$0.527170\pi$
$770$	0	0
$771$	11.2883	0.406537
$772$	0	0
$773$	49.0439	1.76399	0.881994	−	0.471261i	$-0.156201\pi$
$773$	49.0439	1.76399	0.881994	+	0.471261i	$0.156201\pi$
$774$	0	0
$775$	41.7173	1.49853
$776$	0	0
$777$	−22.5706	−0.809716
$778$	0	0
$779$	3.95229	0.141606
$780$	0	0
$781$	17.1515	0.613728
$782$	0	0
$783$	4.13843	0.147895
$784$	0	0
$785$	−1.64858	−0.0588404
$786$	0	0
$787$	−38.5666	−1.37475	−0.687375	−	0.726302i	$-0.741237\pi$
$787$	−38.5666	−1.37475	−0.687375	+	0.726302i	$0.741237\pi$
$788$	0	0
$789$	−48.8746	−1.73998
$790$	0	0
$791$	8.47480	0.301329
$792$	0	0
$793$	9.42963	0.334856
$794$	0	0
$795$	−0.170619	−0.00605123
$796$	0	0
$797$	−46.9589	−1.66337	−0.831685	−	0.555248i	$-0.812623\pi$
$797$	−46.9589	−1.66337	−0.831685	+	0.555248i	$0.812623\pi$
$798$	0	0
$799$	0.00987876	0.000349485 0
$800$	0	0
$801$	0.274449	0.00969717
$802$	0	0
$803$	−73.5772	−2.59648
$804$	0	0
$805$	2.18076	0.0768618
$806$	0	0
$807$	30.5332	1.07482
$808$	0	0
$809$	6.62375	0.232879	0.116439	−	0.993198i	$-0.462852\pi$
$809$	6.62375	0.232879	0.116439	+	0.993198i	$0.462852\pi$
$810$	0	0
$811$	16.2920	0.572090	0.286045	−	0.958216i	$-0.407659\pi$
$811$	16.2920	0.572090	0.286045	+	0.958216i	$0.407659\pi$
$812$	0	0
$813$	−33.5220	−1.17567
$814$	0	0
$815$	−2.23876	−0.0784204
$816$	0	0
$817$	−3.75497	−0.131370
$818$	0	0
$819$	−1.54623	−0.0540298
$820$	0	0
$821$	12.9720	0.452725	0.226362	−	0.974043i	$-0.427317\pi$
$821$	12.9720	0.452725	0.226362	+	0.974043i	$0.427317\pi$
$822$	0	0
$823$	−41.1024	−1.43274	−0.716369	−	0.697722i	$-0.754197\pi$
$823$	−41.1024	−1.43274	−0.716369	+	0.697722i	$0.754197\pi$
$824$	0	0
$825$	−41.9691	−1.46118
$826$	0	0
$827$	8.90876	0.309788	0.154894	−	0.987931i	$-0.450496\pi$
$827$	8.90876	0.309788	0.154894	+	0.987931i	$0.450496\pi$
$828$	0	0
$829$	−6.61588	−0.229779	−0.114890	−	0.993378i	$-0.536651\pi$
$829$	−6.61588	−0.229779	−0.114890	+	0.993378i	$0.536651\pi$
$830$	0	0
$831$	41.4909	1.43930
$832$	0	0
$833$	−8.50813	−0.294789
$834$	0	0
$835$	−6.83661	−0.236591
$836$	0	0
$837$	−45.2182	−1.56297
$838$	0	0
$839$	−29.3161	−1.01211	−0.506053	−	0.862502i	$-0.668896\pi$
$839$	−29.3161	−1.01211	−0.506053	+	0.862502i	$0.668896\pi$
$840$	0	0
$841$	−28.3957	−0.979161
$842$	0	0
$843$	−5.76750	−0.198643
$844$	0	0
$845$	8.83724	0.304010
$846$	0	0
$847$	−22.0762	−0.758547
$848$	0	0
$849$	13.9546	0.478920
$850$	0	0
$851$	43.4054	1.48792
$852$	0	0
$853$	−21.5016	−0.736202	−0.368101	−	0.929786i	$-0.619992\pi$
$853$	−21.5016	−0.736202	−0.368101	+	0.929786i	$0.619992\pi$
$854$	0	0
$855$	0.0469105	0.00160431
$856$	0	0
$857$	−23.7326	−0.810689	−0.405344	−	0.914164i	$-0.632848\pi$
$857$	−23.7326	−0.810689	−0.405344	+	0.914164i	$0.632848\pi$
$858$	0	0
$859$	4.19942	0.143282	0.0716412	−	0.997430i	$-0.477176\pi$
$859$	4.19942	0.143282	0.0716412	+	0.997430i	$0.477176\pi$
$860$	0	0
$861$	−10.0137	−0.341267
$862$	0	0
$863$	37.9750	1.29268	0.646342	−	0.763048i	$-0.276298\pi$
$863$	37.9750	1.29268	0.646342	+	0.763048i	$0.276298\pi$
$864$	0	0
$865$	4.16312	0.141550
$866$	0	0
$867$	23.2332	0.789041
$868$	0	0
$869$	5.06855	0.171939
$870$	0	0
$871$	−28.7182	−0.973077
$872$	0	0
$873$	−2.01939	−0.0683459
$874$	0	0
$875$	4.43601	0.149964
$876$	0	0
$877$	−6.33380	−0.213877	−0.106939	−	0.994266i	$-0.534105\pi$
$877$	−6.33380	−0.213877	−0.106939	+	0.994266i	$0.534105\pi$
$878$	0	0
$879$	25.5824	0.862873
$880$	0	0
$881$	24.5356	0.826627	0.413313	−	0.910589i	$-0.364371\pi$
$881$	24.5356	0.826627	0.413313	+	0.910589i	$0.364371\pi$
$882$	0	0
$883$	22.9651	0.772837	0.386418	−	0.922324i	$-0.373712\pi$
$883$	22.9651	0.772837	0.386418	+	0.922324i	$0.373712\pi$
$884$	0	0
$885$	6.12018	0.205728
$886$	0	0
$887$	22.5521	0.757224	0.378612	−	0.925555i	$-0.376401\pi$
$887$	22.5521	0.757224	0.378612	+	0.925555i	$0.376401\pi$
$888$	0	0
$889$	19.1868	0.643506
$890$	0	0
$891$	43.0962	1.44378
$892$	0	0
$893$	−0.00550550	−0.000184235 0
$894$	0	0
$895$	−2.51233	−0.0839779
$896$	0	0
$897$	−53.6624	−1.79173
$898$	0	0
$899$	−6.60329	−0.220232
$900$	0	0
$901$	0.609703	0.0203122
$902$	0	0
$903$	9.51377	0.316599
$904$	0	0
$905$	7.41330	0.246427
$906$	0	0
$907$	−2.85034	−0.0946438	−0.0473219	−	0.998880i	$-0.515069\pi$
$907$	−2.85034	−0.0946438	−0.0473219	+	0.998880i	$0.515069\pi$
$908$	0	0
$909$	−0.764841	−0.0253682
$910$	0	0
$911$	15.8014	0.523525	0.261763	−	0.965132i	$-0.415696\pi$
$911$	15.8014	0.523525	0.261763	+	0.965132i	$0.415696\pi$
$912$	0	0
$913$	64.4896	2.13429
$914$	0	0
$915$	0.724830	0.0239621
$916$	0	0
$917$	12.8386	0.423969
$918$	0	0
$919$	−25.6359	−0.845651	−0.422825	−	0.906211i	$-0.638962\pi$
$919$	−25.6359	−0.845651	−0.422825	+	0.906211i	$0.638962\pi$
$920$	0	0
$921$	46.2627	1.52441
$922$	0	0
$923$	22.1050	0.727596
$924$	0	0
$925$	43.7516	1.43854
$926$	0	0
$927$	−0.345201	−0.0113379
$928$	0	0
$929$	−1.87077	−0.0613778	−0.0306889	−	0.999529i	$-0.509770\pi$
$929$	−1.87077	−0.0613778	−0.0306889	+	0.999529i	$0.509770\pi$
$930$	0	0
$931$	4.74164	0.155401
$932$	0	0
$933$	−41.3367	−1.35330
$934$	0	0
$935$	−2.70866	−0.0885827
$936$	0	0
$937$	−43.2420	−1.41265	−0.706327	−	0.707886i	$-0.749649\pi$
$937$	−43.2420	−1.41265	−0.706327	+	0.707886i	$0.749649\pi$
$938$	0	0
$939$	−31.2195	−1.01881
$940$	0	0
$941$	−22.4139	−0.730671	−0.365335	−	0.930876i	$-0.619046\pi$
$941$	−22.4139	−0.730671	−0.365335	+	0.930876i	$0.619046\pi$
$942$	0	0
$943$	19.2573	0.627105
$944$	0	0
$945$	−2.38263	−0.0775068
$946$	0	0
$947$	−7.53734	−0.244931	−0.122465	−	0.992473i	$-0.539080\pi$
$947$	−7.53734	−0.244931	−0.122465	+	0.992473i	$0.539080\pi$
$948$	0	0
$949$	−94.8271	−3.07822
$950$	0	0
$951$	7.72378	0.250461
$952$	0	0
$953$	−34.5755	−1.12001	−0.560005	−	0.828489i	$-0.689201\pi$
$953$	−34.5755	−1.12001	−0.560005	+	0.828489i	$0.689201\pi$
$954$	0	0
$955$	2.26493	0.0732913
$956$	0	0
$957$	6.64316	0.214743
$958$	0	0
$959$	20.0428	0.647217
$960$	0	0
$961$	41.1504	1.32743
$962$	0	0
$963$	0.338728	0.0109154
$964$	0	0
$965$	−0.205531	−0.00661627
$966$	0	0
$967$	36.8336	1.18449	0.592244	−	0.805759i	$-0.298242\pi$
$967$	36.8336	1.18449	0.592244	+	0.805759i	$0.298242\pi$
$968$	0	0
$969$	3.02521	0.0971837
$970$	0	0
$971$	−25.7864	−0.827526	−0.413763	−	0.910385i	$-0.635786\pi$
$971$	−25.7864	−0.827526	−0.413763	+	0.910385i	$0.635786\pi$
$972$	0	0
$973$	26.2622	0.841929
$974$	0	0
$975$	−54.0903	−1.73227
$976$	0	0
$977$	25.5548	0.817569	0.408785	−	0.912631i	$-0.365953\pi$
$977$	25.5548	0.817569	0.408785	+	0.912631i	$0.365953\pi$
$978$	0	0
$979$	8.83160	0.282259
$980$	0	0
$981$	−2.79765	−0.0893220
$982$	0	0
$983$	−35.7367	−1.13982	−0.569912	−	0.821705i	$-0.693023\pi$
$983$	−35.7367	−1.13982	−0.569912	+	0.821705i	$0.693023\pi$
$984$	0	0
$985$	−1.72420	−0.0549375
$986$	0	0
$987$	0.0139490	0.000444001 0
$988$	0	0
$989$	−18.2959	−0.581776
$990$	0	0
$991$	4.80564	0.152656	0.0763280	−	0.997083i	$-0.475680\pi$
$991$	4.80564	0.152656	0.0763280	+	0.997083i	$0.475680\pi$
$992$	0	0
$993$	45.6394	1.44832
$994$	0	0
$995$	−5.34798	−0.169542
$996$	0	0
$997$	19.7339	0.624980	0.312490	−	0.949921i	$-0.398837\pi$
$997$	19.7339	0.624980	0.312490	+	0.949921i	$0.398837\pi$
$998$	0	0
$999$	−47.4232	−1.50041

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6004.2.a.h.1.11	✓	31

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6004.2.a.h.1.11	✓	31	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:	\( N \)	\(=\)	\( 6004 = 2^{2} \cdot 19 \cdot 79 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6004.a (trivial)

\(f(q)\)	\(=\)	\(q-1.68597 q^{3} +0.297828 q^{5} -1.50278 q^{7} -0.157509 q^{9} +O(q^{10})\)	\(q-1.68597 q^{3} +0.297828 q^{5} -1.50278 q^{7} -0.157509 q^{9} -5.06855 q^{11} -6.53241 q^{13} -0.502128 q^{15} +1.79434 q^{17} -1.00000 q^{19} +2.53365 q^{21} -4.87245 q^{23} -4.91130 q^{25} +5.32346 q^{27} +0.777394 q^{29} -8.49414 q^{31} +8.54542 q^{33} -0.447571 q^{35} -8.90835 q^{37} +11.0134 q^{39} -3.95229 q^{41} +3.75497 q^{43} -0.0469105 q^{45} +0.00550550 q^{47} -4.74164 q^{49} -3.02521 q^{51} +0.339792 q^{53} -1.50956 q^{55} +1.68597 q^{57} -12.1885 q^{59} -1.44351 q^{61} +0.236702 q^{63} -1.94553 q^{65} +4.39626 q^{67} +8.21479 q^{69} -3.38390 q^{71} +14.5164 q^{73} +8.28030 q^{75} +7.61694 q^{77} -1.00000 q^{79} -8.50266 q^{81} -12.7235 q^{83} +0.534406 q^{85} -1.31066 q^{87} -1.74243 q^{89} +9.81680 q^{91} +14.3209 q^{93} -0.297828 q^{95} +12.8208 q^{97} +0.798342 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 31 q - 4 q^{3} + 10 q^{5} + 11 q^{7} + 47 q^{9}+O(q^{10}) \) 31 * q - 4 * q^3 + 10 * q^5 + 11 * q^7 + 47 * q^9	\( 31 q - 4 q^{3} + 10 q^{5} + 11 q^{7} + 47 q^{9} - 4 q^{11} + 11 q^{13} + 5 q^{15} + 14 q^{17} - 31 q^{19} + 22 q^{21} + 15 q^{23} + 59 q^{25} + 5 q^{27} + 34 q^{29} - 12 q^{31} + 10 q^{33} + 8 q^{35} + 16 q^{37} + 18 q^{39} + 27 q^{41} + 2 q^{43} + 22 q^{45} + 30 q^{47} + 62 q^{49} - 14 q^{51} + 35 q^{53} + 8 q^{55} + 4 q^{57} - 16 q^{59} + 37 q^{61} + 31 q^{63} + 80 q^{65} + 16 q^{67} + q^{69} + 19 q^{71} + 38 q^{73} + 21 q^{75} + 44 q^{77} - 31 q^{79} + 55 q^{81} - 12 q^{83} + 66 q^{85} + 58 q^{87} + 16 q^{89} - 42 q^{91} + 10 q^{93} - 10 q^{95} - 12 q^{97} + 12 q^{99}+O(q^{100}) \) 31 * q - 4 * q^3 + 10 * q^5 + 11 * q^7 + 47 * q^9 - 4 * q^11 + 11 * q^13 + 5 * q^15 + 14 * q^17 - 31 * q^19 + 22 * q^21 + 15 * q^23 + 59 * q^25 + 5 * q^27 + 34 * q^29 - 12 * q^31 + 10 * q^33 + 8 * q^35 + 16 * q^37 + 18 * q^39 + 27 * q^41 + 2 * q^43 + 22 * q^45 + 30 * q^47 + 62 * q^49 - 14 * q^51 + 35 * q^53 + 8 * q^55 + 4 * q^57 - 16 * q^59 + 37 * q^61 + 31 * q^63 + 80 * q^65 + 16 * q^67 + q^69 + 19 * q^71 + 38 * q^73 + 21 * q^75 + 44 * q^77 - 31 * q^79 + 55 * q^81 - 12 * q^83 + 66 * q^85 + 58 * q^87 + 16 * q^89 - 42 * q^91 + 10 * q^93 - 10 * q^95 - 12 * q^97 + 12 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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