LMFDB - Embedded newform 192.6.a.n.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [192,6,Mod(1,192)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(192, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("192.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 192 = 2^{6} \cdot 3 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	192.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:	$30.7936934041$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 24)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	192.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+9.00000 q^{3} +34.0000 q^{5} -240.000 q^{7} +81.0000 q^{9} +O(q^{10})$

$q+9.00000 q^{3} +34.0000 q^{5} -240.000 q^{7} +81.0000 q^{9} +124.000 q^{11} -46.0000 q^{13} +306.000 q^{15} +1954.00 q^{17} +1924.00 q^{19} -2160.00 q^{21} +2840.00 q^{23} -1969.00 q^{25} +729.000 q^{27} +8922.00 q^{29} -4648.00 q^{31} +1116.00 q^{33} -8160.00 q^{35} +4362.00 q^{37} -414.000 q^{39} -2886.00 q^{41} -11332.0 q^{43} +2754.00 q^{45} +7008.00 q^{47} +40793.0 q^{49} +17586.0 q^{51} +22594.0 q^{53} +4216.00 q^{55} +17316.0 q^{57} +28.0000 q^{59} +6386.00 q^{61} -19440.0 q^{63} -1564.00 q^{65} +39076.0 q^{67} +25560.0 q^{69} -54872.0 q^{71} +21034.0 q^{73} -17721.0 q^{75} -29760.0 q^{77} +26632.0 q^{79} +6561.00 q^{81} -56188.0 q^{83} +66436.0 q^{85} +80298.0 q^{87} +64410.0 q^{89} +11040.0 q^{91} -41832.0 q^{93} +65416.0 q^{95} -116158. q^{97} +10044.0 q^{99} +O(q^{100})$

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
$3$	9.00000	0.577350
$3$	9.00000	0.577350
$4$	0	0
$5$	34.0000	0.608210	0.304105	−	0.952638i	$-0.401643\pi$
$5$	34.0000	0.608210	0.304105	+	0.952638i	$0.401643\pi$
$6$	0	0
$7$	−240.000	−1.85125	−0.925627	−	0.378436i	$-0.876462\pi$
$7$	−240.000	−1.85125	−0.925627	+	0.378436i	$0.876462\pi$
$8$	0	0
$9$	81.0000	0.333333
$10$	0	0
$11$	124.000	0.308987	0.154493	−	0.987994i	$-0.450625\pi$
$11$	124.000	0.308987	0.154493	+	0.987994i	$0.450625\pi$
$12$	0	0
$13$	−46.0000	−0.0754917	−0.0377459	−	0.999287i	$-0.512018\pi$
$13$	−46.0000	−0.0754917	−0.0377459	+	0.999287i	$0.512018\pi$
$14$	0	0
$15$	306.000	0.351150
$16$	0	0
$17$	1954.00	1.63984	0.819921	−	0.572476i	$-0.194017\pi$
$17$	1954.00	1.63984	0.819921	+	0.572476i	$0.194017\pi$
$18$	0	0
$19$	1924.00	1.22270	0.611352	−	0.791359i	$-0.290626\pi$
$19$	1924.00	1.22270	0.611352	+	0.791359i	$0.290626\pi$
$20$	0	0
$21$	−2160.00	−1.06882
$22$	0	0
$23$	2840.00	1.11943	0.559717	−	0.828684i	$-0.310910\pi$
$23$	2840.00	1.11943	0.559717	+	0.828684i	$0.310910\pi$
$24$	0	0
$25$	−1969.00	−0.630080
$26$	0	0
$27$	729.000	0.192450
$28$	0	0
$29$	8922.00	1.97000	0.985002	−	0.172541i	$-0.0551979\pi$
$29$	8922.00	1.97000	0.985002	+	0.172541i	$0.0551979\pi$
$30$	0	0
$31$	−4648.00	−0.868684	−0.434342	−	0.900748i	$-0.643019\pi$
$31$	−4648.00	−0.868684	−0.434342	+	0.900748i	$0.643019\pi$
$32$	0	0
$33$	1116.00	0.178394
$34$	0	0
$35$	−8160.00	−1.12595
$36$	0	0
$37$	4362.00	0.523819	0.261910	−	0.965092i	$-0.415648\pi$
$37$	4362.00	0.523819	0.261910	+	0.965092i	$0.415648\pi$
$38$	0	0
$39$	−414.000	−0.0435852
$40$	0	0
$41$	−2886.00	−0.268125	−0.134062	−	0.990973i	$-0.542802\pi$
$41$	−2886.00	−0.268125	−0.134062	+	0.990973i	$0.542802\pi$
$42$	0	0
$43$	−11332.0	−0.934621	−0.467310	−	0.884093i	$-0.654777\pi$
$43$	−11332.0	−0.934621	−0.467310	+	0.884093i	$0.654777\pi$
$44$	0	0
$45$	2754.00	0.202737
$46$	0	0
$47$	7008.00	0.462753	0.231377	−	0.972864i	$-0.425677\pi$
$47$	7008.00	0.462753	0.231377	+	0.972864i	$0.425677\pi$
$48$	0	0
$49$	40793.0	2.42714
$50$	0	0
$51$	17586.0	0.946764
$52$	0	0
$53$	22594.0	1.10485	0.552425	−	0.833562i	$-0.313703\pi$
$53$	22594.0	1.10485	0.552425	+	0.833562i	$0.313703\pi$
$54$	0	0
$55$	4216.00	0.187929
$56$	0	0
$57$	17316.0	0.705928
$58$	0	0
$59$	28.0000	0.00104720	0.000523598	−	1.00000i	$-0.499833\pi$
$59$	28.0000	0.00104720	0.000523598		1.00000i	$0.499833\pi$
$60$	0	0
$61$	6386.00	0.219738	0.109869	−	0.993946i	$-0.464957\pi$
$61$	6386.00	0.219738	0.109869	+	0.993946i	$0.464957\pi$
$62$	0	0
$63$	−19440.0	−0.617085
$64$	0	0
$65$	−1564.00	−0.0459149
$66$	0	0
$67$	39076.0	1.06346	0.531732	−	0.846912i	$-0.321541\pi$
$67$	39076.0	1.06346	0.531732	+	0.846912i	$0.321541\pi$
$68$	0	0
$69$	25560.0	0.646306
$70$	0	0
$71$	−54872.0	−1.29183	−0.645914	−	0.763410i	$-0.723524\pi$
$71$	−54872.0	−1.29183	−0.645914	+	0.763410i	$0.723524\pi$
$72$	0	0
$73$	21034.0	0.461971	0.230986	−	0.972957i	$-0.425805\pi$
$73$	21034.0	0.461971	0.230986	+	0.972957i	$0.425805\pi$
$74$	0	0
$75$	−17721.0	−0.363777
$76$	0	0
$77$	−29760.0	−0.572013
$78$	0	0
$79$	26632.0	0.480105	0.240052	−	0.970760i	$-0.422835\pi$
$79$	26632.0	0.480105	0.240052	+	0.970760i	$0.422835\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	0	0
$83$	−56188.0	−0.895258	−0.447629	−	0.894219i	$-0.647732\pi$
$83$	−56188.0	−0.895258	−0.447629	+	0.894219i	$0.647732\pi$
$84$	0	0
$85$	66436.0	0.997370
$86$	0	0
$87$	80298.0	1.13738
$88$	0	0
$89$	64410.0	0.861942	0.430971	−	0.902366i	$-0.358171\pi$
$89$	64410.0	0.861942	0.430971	+	0.902366i	$0.358171\pi$
$90$	0	0
$91$	11040.0	0.139754
$92$	0	0
$93$	−41832.0	−0.501535
$94$	0	0
$95$	65416.0	0.743661
$96$	0	0
$97$	−116158.	−1.25349	−0.626743	−	0.779226i	$-0.715613\pi$
$97$	−116158.	−1.25349	−0.626743	+	0.779226i	$0.715613\pi$
$98$	0	0
$99$	10044.0	0.102996
$100$	0	0
$101$	66834.0	0.651920	0.325960	−	0.945384i	$-0.394313\pi$
$101$	66834.0	0.651920	0.325960	+	0.945384i	$0.394313\pi$
$102$	0	0
$103$	64000.0	0.594411	0.297206	−	0.954814i	$-0.403945\pi$
$103$	64000.0	0.594411	0.297206	+	0.954814i	$0.403945\pi$
$104$	0	0
$105$	−73440.0	−0.650069
$106$	0	0
$107$	15084.0	0.127367	0.0636835	−	0.997970i	$-0.479715\pi$
$107$	15084.0	0.127367	0.0636835	+	0.997970i	$0.479715\pi$
$108$	0	0
$109$	39698.0	0.320039	0.160019	−	0.987114i	$-0.448844\pi$
$109$	39698.0	0.320039	0.160019	+	0.987114i	$0.448844\pi$
$110$	0	0
$111$	39258.0	0.302427
$112$	0	0
$113$	155154.	1.14305	0.571527	−	0.820583i	$-0.306351\pi$
$113$	155154.	1.14305	0.571527	+	0.820583i	$0.306351\pi$
$114$	0	0
$115$	96560.0	0.680852
$116$	0	0
$117$	−3726.00	−0.0251639
$118$	0	0
$119$	−468960.	−3.03577
$120$	0	0
$121$	−145675.	−0.904527
$122$	0	0
$123$	−25974.0	−0.154802
$124$	0	0
$125$	−173196.	−0.991432
$126$	0	0
$127$	52072.0	0.286480	0.143240	−	0.989688i	$-0.454248\pi$
$127$	52072.0	0.286480	0.143240	+	0.989688i	$0.454248\pi$
$128$	0	0
$129$	−101988.	−0.539604
$130$	0	0
$131$	−159964.	−0.814412	−0.407206	−	0.913336i	$-0.633497\pi$
$131$	−159964.	−0.814412	−0.407206	+	0.913336i	$0.633497\pi$
$132$	0	0
$133$	−461760.	−2.26353
$134$	0	0
$135$	24786.0	0.117050
$136$	0	0
$137$	−262278.	−1.19388	−0.596940	−	0.802286i	$-0.703617\pi$
$137$	−262278.	−1.19388	−0.596940	+	0.802286i	$0.703617\pi$
$138$	0	0
$139$	−253524.	−1.11297	−0.556483	−	0.830859i	$-0.687850\pi$
$139$	−253524.	−1.11297	−0.556483	+	0.830859i	$0.687850\pi$
$140$	0	0
$141$	63072.0	0.267171
$142$	0	0
$143$	−5704.00	−0.0233260
$144$	0	0
$145$	303348.	1.19818
$146$	0	0
$147$	367137.	1.40131
$148$	0	0
$149$	−355630.	−1.31230	−0.656149	−	0.754631i	$-0.727816\pi$
$149$	−355630.	−1.31230	−0.656149	+	0.754631i	$0.727816\pi$
$150$	0	0
$151$	−1024.00	−0.00365475	−0.00182737	−	0.999998i	$-0.500582\pi$
$151$	−1024.00	−0.00365475	−0.00182737	+	0.999998i	$0.500582\pi$
$152$	0	0
$153$	158274.	0.546614
$154$	0	0
$155$	−158032.	−0.528343
$156$	0	0
$157$	59954.0	0.194119	0.0970597	−	0.995279i	$-0.469056\pi$
$157$	59954.0	0.194119	0.0970597	+	0.995279i	$0.469056\pi$
$158$	0	0
$159$	203346.	0.637886
$160$	0	0
$161$	−681600.	−2.07236
$162$	0	0
$163$	341556.	1.00692	0.503458	−	0.864020i	$-0.332061\pi$
$163$	341556.	1.00692	0.503458	+	0.864020i	$0.332061\pi$
$164$	0	0
$165$	37944.0	0.108501
$166$	0	0
$167$	5016.00	0.0139177	0.00695883	−	0.999976i	$-0.497785\pi$
$167$	5016.00	0.0139177	0.00695883	+	0.999976i	$0.497785\pi$
$168$	0	0
$169$	−369177.	−0.994301
$170$	0	0
$171$	155844.	0.407568
$172$	0	0
$173$	228666.	0.580880	0.290440	−	0.956893i	$-0.406198\pi$
$173$	228666.	0.580880	0.290440	+	0.956893i	$0.406198\pi$
$174$	0	0
$175$	472560.	1.16644
$176$	0	0
$177$	252.000	0.000604599 0
$178$	0	0
$179$	−161388.	−0.376477	−0.188239	−	0.982123i	$-0.560278\pi$
$179$	−161388.	−0.376477	−0.188239	+	0.982123i	$0.560278\pi$
$180$	0	0
$181$	291690.	0.661797	0.330899	−	0.943666i	$-0.392648\pi$
$181$	291690.	0.661797	0.330899	+	0.943666i	$0.392648\pi$
$182$	0	0
$183$	57474.0	0.126866
$184$	0	0
$185$	148308.	0.318592
$186$	0	0
$187$	242296.	0.506690
$188$	0	0
$189$	−174960.	−0.356274
$190$	0	0
$191$	−55680.0	−0.110437	−0.0552187	−	0.998474i	$-0.517586\pi$
$191$	−55680.0	−0.110437	−0.0552187	+	0.998474i	$0.517586\pi$
$192$	0	0
$193$	−176254.	−0.340601	−0.170300	−	0.985392i	$-0.554474\pi$
$193$	−176254.	−0.340601	−0.170300	+	0.985392i	$0.554474\pi$
$194$	0	0
$195$	−14076.0	−0.0265090
$196$	0	0
$197$	374610.	0.687723	0.343862	−	0.939020i	$-0.388265\pi$
$197$	374610.	0.687723	0.343862	+	0.939020i	$0.388265\pi$
$198$	0	0
$199$	−637760.	−1.14163	−0.570814	−	0.821079i	$-0.693372\pi$
$199$	−637760.	−1.14163	−0.570814	+	0.821079i	$0.693372\pi$
$200$	0	0
$201$	351684.	0.613992
$202$	0	0
$203$	−2.14128e6	−3.64698
$204$	0	0
$205$	−98124.0	−0.163076
$206$	0	0
$207$	230040.	0.373145
$208$	0	0
$209$	238576.	0.377799
$210$	0	0
$211$	904628.	1.39883	0.699413	−	0.714717i	$-0.253445\pi$
$211$	904628.	1.39883	0.699413	+	0.714717i	$0.253445\pi$
$212$	0	0
$213$	−493848.	−0.745838
$214$	0	0
$215$	−385288.	−0.568446
$216$	0	0
$217$	1.11552e6	1.60816
$218$	0	0
$219$	189306.	0.266719
$220$	0	0
$221$	−89884.0	−0.123795
$222$	0	0
$223$	619048.	0.833609	0.416804	−	0.908996i	$-0.363150\pi$
$223$	619048.	0.833609	0.416804	+	0.908996i	$0.363150\pi$
$224$	0	0
$225$	−159489.	−0.210027
$226$	0	0
$227$	1.46975e6	1.89312	0.946560	−	0.322527i	$-0.104532\pi$
$227$	1.46975e6	1.89312	0.946560	+	0.322527i	$0.104532\pi$
$228$	0	0
$229$	3290.00	0.00414579	0.00207289	−	0.999998i	$-0.499340\pi$
$229$	3290.00	0.00414579	0.00207289	+	0.999998i	$0.499340\pi$
$230$	0	0
$231$	−267840.	−0.330252
$232$	0	0
$233$	935402.	1.12878	0.564389	−	0.825509i	$-0.309112\pi$
$233$	935402.	1.12878	0.564389	+	0.825509i	$0.309112\pi$
$234$	0	0
$235$	238272.	0.281451
$236$	0	0
$237$	239688.	0.277189
$238$	0	0
$239$	−875600.	−0.991542	−0.495771	−	0.868453i	$-0.665114\pi$
$239$	−875600.	−0.991542	−0.495771	+	0.868453i	$0.665114\pi$
$240$	0	0
$241$	−959214.	−1.06383	−0.531916	−	0.846797i	$-0.678528\pi$
$241$	−959214.	−1.06383	−0.531916	+	0.846797i	$0.678528\pi$
$242$	0	0
$243$	59049.0	0.0641500
$244$	0	0
$245$	1.38696e6	1.47621
$246$	0	0
$247$	−88504.0	−0.0923040
$248$	0	0
$249$	−505692.	−0.516878
$250$	0	0
$251$	−318868.	−0.319467	−0.159734	−	0.987160i	$-0.551064\pi$
$251$	−318868.	−0.319467	−0.159734	+	0.987160i	$0.551064\pi$
$252$	0	0
$253$	352160.	0.345891
$254$	0	0
$255$	597924.	0.575832
$256$	0	0
$257$	1.71469e6	1.61940	0.809698	−	0.586847i	$-0.199631\pi$
$257$	1.71469e6	1.61940	0.809698	+	0.586847i	$0.199631\pi$
$258$	0	0
$259$	−1.04688e6	−0.969723
$260$	0	0
$261$	722682.	0.656668
$262$	0	0
$263$	−1.11028e6	−0.989790	−0.494895	−	0.868953i	$-0.664794\pi$
$263$	−1.11028e6	−0.989790	−0.494895	+	0.868953i	$0.664794\pi$
$264$	0	0
$265$	768196.	0.671982
$266$	0	0
$267$	579690.	0.497643
$268$	0	0
$269$	398378.	0.335672	0.167836	−	0.985815i	$-0.446322\pi$
$269$	398378.	0.335672	0.167836	+	0.985815i	$0.446322\pi$
$270$	0	0
$271$	1.44198e6	1.19271	0.596355	−	0.802721i	$-0.296615\pi$
$271$	1.44198e6	1.19271	0.596355	+	0.802721i	$0.296615\pi$
$272$	0	0
$273$	99360.0	0.0806873
$274$	0	0
$275$	−244156.	−0.194686
$276$	0	0
$277$	−117238.	−0.0918056	−0.0459028	−	0.998946i	$-0.514616\pi$
$277$	−117238.	−0.0918056	−0.0459028	+	0.998946i	$0.514616\pi$
$278$	0	0
$279$	−376488.	−0.289561
$280$	0	0
$281$	−1.67514e6	−1.26557	−0.632784	−	0.774328i	$-0.718088\pi$
$281$	−1.67514e6	−1.26557	−0.632784	+	0.774328i	$0.718088\pi$
$282$	0	0
$283$	−1.92468e6	−1.42854	−0.714269	−	0.699872i	$-0.753240\pi$
$283$	−1.92468e6	−1.42854	−0.714269	+	0.699872i	$0.753240\pi$
$284$	0	0
$285$	588744.	0.429353
$286$	0	0
$287$	692640.	0.496367
$288$	0	0
$289$	2.39826e6	1.68908
$290$	0	0
$291$	−1.04542e6	−0.723701
$292$	0	0
$293$	−1.28062e6	−0.871469	−0.435734	−	0.900075i	$-0.643511\pi$
$293$	−1.28062e6	−0.871469	−0.435734	+	0.900075i	$0.643511\pi$
$294$	0	0
$295$	952.000	0.000636916 0
$296$	0	0
$297$	90396.0	0.0594645
$298$	0	0
$299$	−130640.	−0.0845081
$300$	0	0
$301$	2.71968e6	1.73022
$302$	0	0
$303$	601506.	0.376386
$304$	0	0
$305$	217124.	0.133647
$306$	0	0
$307$	2.26319e6	1.37049	0.685243	−	0.728314i	$-0.259696\pi$
$307$	2.26319e6	1.37049	0.685243	+	0.728314i	$0.259696\pi$
$308$	0	0
$309$	576000.	0.343183
$310$	0	0
$311$	247848.	0.145306	0.0726532	−	0.997357i	$-0.476853\pi$
$311$	247848.	0.145306	0.0726532	+	0.997357i	$0.476853\pi$
$312$	0	0
$313$	−1.82391e6	−1.05231	−0.526154	−	0.850390i	$-0.676366\pi$
$313$	−1.82391e6	−1.05231	−0.526154	+	0.850390i	$0.676366\pi$
$314$	0	0
$315$	−660960.	−0.375317
$316$	0	0
$317$	−2.85629e6	−1.59645	−0.798224	−	0.602361i	$-0.794227\pi$
$317$	−2.85629e6	−1.59645	−0.798224	+	0.602361i	$0.794227\pi$
$318$	0	0
$319$	1.10633e6	0.608705
$320$	0	0
$321$	135756.	0.0735354
$322$	0	0
$323$	3.75950e6	2.00504
$324$	0	0
$325$	90574.0	0.0475658
$326$	0	0
$327$	357282.	0.184774
$328$	0	0
$329$	−1.68192e6	−0.856674
$330$	0	0
$331$	147148.	0.0738218	0.0369109	−	0.999319i	$-0.488248\pi$
$331$	147148.	0.0738218	0.0369109	+	0.999319i	$0.488248\pi$
$332$	0	0
$333$	353322.	0.174606
$334$	0	0
$335$	1.32858e6	0.646810
$336$	0	0
$337$	−3.24728e6	−1.55756	−0.778780	−	0.627297i	$-0.784161\pi$
$337$	−3.24728e6	−1.55756	−0.778780	+	0.627297i	$0.784161\pi$
$338$	0	0
$339$	1.39639e6	0.659943
$340$	0	0
$341$	−576352.	−0.268412
$342$	0	0
$343$	−5.75664e6	−2.64201
$344$	0	0
$345$	869040.	0.393090
$346$	0	0
$347$	1.55675e6	0.694056	0.347028	−	0.937855i	$-0.387191\pi$
$347$	1.55675e6	0.694056	0.347028	+	0.937855i	$0.387191\pi$
$348$	0	0
$349$	−4.03217e6	−1.77205	−0.886024	−	0.463639i	$-0.846544\pi$
$349$	−4.03217e6	−1.77205	−0.886024	+	0.463639i	$0.846544\pi$
$350$	0	0
$351$	−33534.0	−0.0145284
$352$	0	0
$353$	1.79399e6	0.766271	0.383135	−	0.923692i	$-0.374844\pi$
$353$	1.79399e6	0.766271	0.383135	+	0.923692i	$0.374844\pi$
$354$	0	0
$355$	−1.86565e6	−0.785704
$356$	0	0
$357$	−4.22064e6	−1.75270
$358$	0	0
$359$	1.55278e6	0.635876	0.317938	−	0.948111i	$-0.397010\pi$
$359$	1.55278e6	0.635876	0.317938	+	0.948111i	$0.397010\pi$
$360$	0	0
$361$	1.22568e6	0.495003
$362$	0	0
$363$	−1.31108e6	−0.522229
$364$	0	0
$365$	715156.	0.280976
$366$	0	0
$367$	−3.11545e6	−1.20741	−0.603706	−	0.797207i	$-0.706310\pi$
$367$	−3.11545e6	−1.20741	−0.603706	+	0.797207i	$0.706310\pi$
$368$	0	0
$369$	−233766.	−0.0893749
$370$	0	0
$371$	−5.42256e6	−2.04536
$372$	0	0
$373$	630682.	0.234714	0.117357	−	0.993090i	$-0.462558\pi$
$373$	630682.	0.234714	0.117357	+	0.993090i	$0.462558\pi$
$374$	0	0
$375$	−1.55876e6	−0.572403
$376$	0	0
$377$	−410412.	−0.148719
$378$	0	0
$379$	−48404.0	−0.0173094	−0.00865472	−	0.999963i	$-0.502755\pi$
$379$	−48404.0	−0.0173094	−0.00865472	+	0.999963i	$0.502755\pi$
$380$	0	0
$381$	468648.	0.165400
$382$	0	0
$383$	1.74182e6	0.606747	0.303373	−	0.952872i	$-0.401887\pi$
$383$	1.74182e6	0.606747	0.303373	+	0.952872i	$0.401887\pi$
$384$	0	0
$385$	−1.01184e6	−0.347904
$386$	0	0
$387$	−917892.	−0.311540
$388$	0	0
$389$	3.06819e6	1.02804	0.514019	−	0.857779i	$-0.328156\pi$
$389$	3.06819e6	1.02804	0.514019	+	0.857779i	$0.328156\pi$
$390$	0	0
$391$	5.54936e6	1.83570
$392$	0	0
$393$	−1.43968e6	−0.470201
$394$	0	0
$395$	905488.	0.292005
$396$	0	0
$397$	−5.35984e6	−1.70677	−0.853386	−	0.521280i	$-0.825455\pi$
$397$	−5.35984e6	−1.70677	−0.853386	+	0.521280i	$0.825455\pi$
$398$	0	0
$399$	−4.15584e6	−1.30685
$400$	0	0
$401$	−2.76473e6	−0.858603	−0.429302	−	0.903161i	$-0.641240\pi$
$401$	−2.76473e6	−0.858603	−0.429302	+	0.903161i	$0.641240\pi$
$402$	0	0
$403$	213808.	0.0655785
$404$	0	0
$405$	223074.	0.0675789
$406$	0	0
$407$	540888.	0.161853
$408$	0	0
$409$	−1.20893e6	−0.357350	−0.178675	−	0.983908i	$-0.557181\pi$
$409$	−1.20893e6	−0.357350	−0.178675	+	0.983908i	$0.557181\pi$
$410$	0	0
$411$	−2.36050e6	−0.689287
$412$	0	0
$413$	−6720.00	−0.00193863
$414$	0	0
$415$	−1.91039e6	−0.544505
$416$	0	0
$417$	−2.28172e6	−0.642571
$418$	0	0
$419$	4.38008e6	1.21884	0.609421	−	0.792847i	$-0.291402\pi$
$419$	4.38008e6	1.21884	0.609421	+	0.792847i	$0.291402\pi$
$420$	0	0
$421$	922810.	0.253751	0.126875	−	0.991919i	$-0.459505\pi$
$421$	922810.	0.253751	0.126875	+	0.991919i	$0.459505\pi$
$422$	0	0
$423$	567648.	0.154251
$424$	0	0
$425$	−3.84743e6	−1.03323
$426$	0	0
$427$	−1.53264e6	−0.406790
$428$	0	0
$429$	−51336.0	−0.0134672
$430$	0	0
$431$	6.12678e6	1.58869	0.794345	−	0.607466i	$-0.207814\pi$
$431$	6.12678e6	1.58869	0.794345	+	0.607466i	$0.207814\pi$
$432$	0	0
$433$	−1.76315e6	−0.451928	−0.225964	−	0.974136i	$-0.572553\pi$
$433$	−1.76315e6	−0.451928	−0.225964	+	0.974136i	$0.572553\pi$
$434$	0	0
$435$	2.73013e6	0.691768
$436$	0	0
$437$	5.46416e6	1.36874
$438$	0	0
$439$	3.85906e6	0.955696	0.477848	−	0.878443i	$-0.341417\pi$
$439$	3.85906e6	0.955696	0.477848	+	0.878443i	$0.341417\pi$
$440$	0	0
$441$	3.30423e6	0.809048
$442$	0	0
$443$	4.39396e6	1.06377	0.531884	−	0.846817i	$-0.321484\pi$
$443$	4.39396e6	1.06377	0.531884	+	0.846817i	$0.321484\pi$
$444$	0	0
$445$	2.18994e6	0.524242
$446$	0	0
$447$	−3.20067e6	−0.757656
$448$	0	0
$449$	−793390.	−0.185725	−0.0928626	−	0.995679i	$-0.529602\pi$
$449$	−793390.	−0.185725	−0.0928626	+	0.995679i	$0.529602\pi$
$450$	0	0
$451$	−357864.	−0.0828470
$452$	0	0
$453$	−9216.00	−0.00211007
$454$	0	0
$455$	375360.	0.0850001
$456$	0	0
$457$	7.04302e6	1.57750	0.788748	−	0.614717i	$-0.210730\pi$
$457$	7.04302e6	1.57750	0.788748	+	0.614717i	$0.210730\pi$
$458$	0	0
$459$	1.42447e6	0.315588
$460$	0	0
$461$	−7.43005e6	−1.62832	−0.814160	−	0.580641i	$-0.802802\pi$
$461$	−7.43005e6	−1.62832	−0.814160	+	0.580641i	$0.802802\pi$
$462$	0	0
$463$	−4.10567e6	−0.890086	−0.445043	−	0.895509i	$-0.646812\pi$
$463$	−4.10567e6	−0.890086	−0.445043	+	0.895509i	$0.646812\pi$
$464$	0	0
$465$	−1.42229e6	−0.305039
$466$	0	0
$467$	−3.39817e6	−0.721030	−0.360515	−	0.932753i	$-0.617399\pi$
$467$	−3.39817e6	−0.721030	−0.360515	+	0.932753i	$0.617399\pi$
$468$	0	0
$469$	−9.37824e6	−1.96874
$470$	0	0
$471$	539586.	0.112075
$472$	0	0
$473$	−1.40517e6	−0.288786
$474$	0	0
$475$	−3.78836e6	−0.770401
$476$	0	0
$477$	1.83011e6	0.368283
$478$	0	0
$479$	2.78133e6	0.553877	0.276939	−	0.960888i	$-0.410680\pi$
$479$	2.78133e6	0.553877	0.276939	+	0.960888i	$0.410680\pi$
$480$	0	0
$481$	−200652.	−0.0395440
$482$	0	0
$483$	−6.13440e6	−1.19648
$484$	0	0
$485$	−3.94937e6	−0.762384
$486$	0	0
$487$	−2.06734e6	−0.394994	−0.197497	−	0.980304i	$-0.563281\pi$
$487$	−2.06734e6	−0.394994	−0.197497	+	0.980304i	$0.563281\pi$
$488$	0	0
$489$	3.07400e6	0.581343
$490$	0	0
$491$	7.65976e6	1.43387	0.716937	−	0.697138i	$-0.245543\pi$
$491$	7.65976e6	1.43387	0.716937	+	0.697138i	$0.245543\pi$
$492$	0	0
$493$	1.74336e7	3.23050
$494$	0	0
$495$	341496.	0.0626430
$496$	0	0
$497$	1.31693e7	2.39150
$498$	0	0
$499$	386580.	0.0695005	0.0347503	−	0.999396i	$-0.488936\pi$
$499$	386580.	0.0695005	0.0347503	+	0.999396i	$0.488936\pi$
$500$	0	0
$501$	45144.0	0.00803537
$502$	0	0
$503$	−2.57326e6	−0.453485	−0.226743	−	0.973955i	$-0.572808\pi$
$503$	−2.57326e6	−0.453485	−0.226743	+	0.973955i	$0.572808\pi$
$504$	0	0
$505$	2.27236e6	0.396504
$506$	0	0
$507$	−3.32259e6	−0.574060
$508$	0	0
$509$	−360678.	−0.0617057	−0.0308528	−	0.999524i	$-0.509822\pi$
$509$	−360678.	−0.0617057	−0.0308528	+	0.999524i	$0.509822\pi$
$510$	0	0
$511$	−5.04816e6	−0.855226
$512$	0	0
$513$	1.40260e6	0.235309
$514$	0	0
$515$	2.17600e6	0.361527
$516$	0	0
$517$	868992.	0.142985
$518$	0	0
$519$	2.05799e6	0.335371
$520$	0	0
$521$	−1.55908e6	−0.251636	−0.125818	−	0.992053i	$-0.540156\pi$
$521$	−1.55908e6	−0.251636	−0.125818	+	0.992053i	$0.540156\pi$
$522$	0	0
$523$	9.18220e6	1.46789	0.733944	−	0.679210i	$-0.237678\pi$
$523$	9.18220e6	1.46789	0.733944	+	0.679210i	$0.237678\pi$
$524$	0	0
$525$	4.25304e6	0.673444
$526$	0	0
$527$	−9.08219e6	−1.42451
$528$	0	0
$529$	1.62926e6	0.253134
$530$	0	0
$531$	2268.00	0.000349065 0
$532$	0	0
$533$	132756.	0.0202412
$534$	0	0
$535$	512856.	0.0774660
$536$	0	0
$537$	−1.45249e6	−0.217359
$538$	0	0
$539$	5.05833e6	0.749955
$540$	0	0
$541$	6.67773e6	0.980925	0.490462	−	0.871462i	$-0.336828\pi$
$541$	6.67773e6	0.980925	0.490462	+	0.871462i	$0.336828\pi$
$542$	0	0
$543$	2.62521e6	0.382089
$544$	0	0
$545$	1.34973e6	0.194651
$546$	0	0
$547$	−8.89656e6	−1.27132	−0.635658	−	0.771971i	$-0.719271\pi$
$547$	−8.89656e6	−1.27132	−0.635658	+	0.771971i	$0.719271\pi$
$548$	0	0
$549$	517266.	0.0732459
$550$	0	0
$551$	1.71659e7	2.40873
$552$	0	0
$553$	−6.39168e6	−0.888796
$554$	0	0
$555$	1.33477e6	0.183939
$556$	0	0
$557$	4.46070e6	0.609207	0.304603	−	0.952479i	$-0.401476\pi$
$557$	4.46070e6	0.609207	0.304603	+	0.952479i	$0.401476\pi$
$558$	0	0
$559$	521272.	0.0705562
$560$	0	0
$561$	2.18066e6	0.292538
$562$	0	0
$563$	−6.37660e6	−0.847849	−0.423924	−	0.905698i	$-0.639348\pi$
$563$	−6.37660e6	−0.847849	−0.423924	+	0.905698i	$0.639348\pi$
$564$	0	0
$565$	5.27524e6	0.695218
$566$	0	0
$567$	−1.57464e6	−0.205695
$568$	0	0
$569$	5.51143e6	0.713648	0.356824	−	0.934172i	$-0.383860\pi$
$569$	5.51143e6	0.713648	0.356824	+	0.934172i	$0.383860\pi$
$570$	0	0
$571$	−1.35431e6	−0.173831	−0.0869155	−	0.996216i	$-0.527701\pi$
$571$	−1.35431e6	−0.173831	−0.0869155	+	0.996216i	$0.527701\pi$
$572$	0	0
$573$	−501120.	−0.0637610
$574$	0	0
$575$	−5.59196e6	−0.705333
$576$	0	0
$577$	−5.00736e6	−0.626137	−0.313068	−	0.949731i	$-0.601357\pi$
$577$	−5.00736e6	−0.626137	−0.313068	+	0.949731i	$0.601357\pi$
$578$	0	0
$579$	−1.58629e6	−0.196646
$580$	0	0
$581$	1.34851e7	1.65735
$582$	0	0
$583$	2.80166e6	0.341384
$584$	0	0
$585$	−126684.	−0.0153050
$586$	0	0
$587$	−2.69964e6	−0.323378	−0.161689	−	0.986842i	$-0.551694\pi$
$587$	−2.69964e6	−0.323378	−0.161689	+	0.986842i	$0.551694\pi$
$588$	0	0
$589$	−8.94275e6	−1.06214
$590$	0	0
$591$	3.37149e6	0.397057
$592$	0	0
$593$	1.31035e7	1.53021	0.765103	−	0.643908i	$-0.222688\pi$
$593$	1.31035e7	1.53021	0.765103	+	0.643908i	$0.222688\pi$
$594$	0	0
$595$	−1.59446e7	−1.84639
$596$	0	0
$597$	−5.73984e6	−0.659119
$598$	0	0
$599$	−5.22804e6	−0.595349	−0.297675	−	0.954667i	$-0.596211\pi$
$599$	−5.22804e6	−0.595349	−0.297675	+	0.954667i	$0.596211\pi$
$600$	0	0
$601$	1.02248e7	1.15470	0.577351	−	0.816496i	$-0.304087\pi$
$601$	1.02248e7	1.15470	0.577351	+	0.816496i	$0.304087\pi$
$602$	0	0
$603$	3.16516e6	0.354488
$604$	0	0
$605$	−4.95295e6	−0.550143
$606$	0	0
$607$	8.81684e6	0.971273	0.485636	−	0.874161i	$-0.338588\pi$
$607$	8.81684e6	0.971273	0.485636	+	0.874161i	$0.338588\pi$
$608$	0	0
$609$	−1.92715e7	−2.10558
$610$	0	0
$611$	−322368.	−0.0349340
$612$	0	0
$613$	−1.13600e7	−1.22103	−0.610514	−	0.792006i	$-0.709037\pi$
$613$	−1.13600e7	−1.22103	−0.610514	+	0.792006i	$0.709037\pi$
$614$	0	0
$615$	−883116.	−0.0941521
$616$	0	0
$617$	−4.77356e6	−0.504812	−0.252406	−	0.967621i	$-0.581222\pi$
$617$	−4.77356e6	−0.504812	−0.252406	+	0.967621i	$0.581222\pi$
$618$	0	0
$619$	2.55931e6	0.268470	0.134235	−	0.990950i	$-0.457142\pi$
$619$	2.55931e6	0.268470	0.134235	+	0.990950i	$0.457142\pi$
$620$	0	0
$621$	2.07036e6	0.215435
$622$	0	0
$623$	−1.54584e7	−1.59567
$624$	0	0
$625$	264461.	0.0270808
$626$	0	0
$627$	2.14718e6	0.218122
$628$	0	0
$629$	8.52335e6	0.858981
$630$	0	0
$631$	−8.41981e6	−0.841839	−0.420919	−	0.907098i	$-0.638292\pi$
$631$	−8.41981e6	−0.841839	−0.420919	+	0.907098i	$0.638292\pi$
$632$	0	0
$633$	8.14165e6	0.807613
$634$	0	0
$635$	1.77045e6	0.174240
$636$	0	0
$637$	−1.87648e6	−0.183229
$638$	0	0
$639$	−4.44463e6	−0.430610
$640$	0	0
$641$	−1.21494e7	−1.16791	−0.583957	−	0.811785i	$-0.698496\pi$
$641$	−1.21494e7	−1.16791	−0.583957	+	0.811785i	$0.698496\pi$
$642$	0	0
$643$	1.08968e7	1.03937	0.519685	−	0.854358i	$-0.326049\pi$
$643$	1.08968e7	1.03937	0.519685	+	0.854358i	$0.326049\pi$
$644$	0	0
$645$	−3.46759e6	−0.328193
$646$	0	0
$647$	1.32166e7	1.24124	0.620622	−	0.784110i	$-0.286880\pi$
$647$	1.32166e7	1.24124	0.620622	+	0.784110i	$0.286880\pi$
$648$	0	0
$649$	3472.00	0.000323570 0
$650$	0	0
$651$	1.00397e7	0.928469
$652$	0	0
$653$	−1.65915e7	−1.52266	−0.761329	−	0.648365i	$-0.775453\pi$
$653$	−1.65915e7	−1.52266	−0.761329	+	0.648365i	$0.775453\pi$
$654$	0	0
$655$	−5.43878e6	−0.495334
$656$	0	0
$657$	1.70375e6	0.153990
$658$	0	0
$659$	2.29372e6	0.205743	0.102872	−	0.994695i	$-0.467197\pi$
$659$	2.29372e6	0.205743	0.102872	+	0.994695i	$0.467197\pi$
$660$	0	0
$661$	719194.	0.0640239	0.0320120	−	0.999487i	$-0.489809\pi$
$661$	719194.	0.0640239	0.0320120	+	0.999487i	$0.489809\pi$
$662$	0	0
$663$	−808956.	−0.0714728
$664$	0	0
$665$	−1.56998e7	−1.37671
$666$	0	0
$667$	2.53385e7	2.20529
$668$	0	0
$669$	5.57143e6	0.481284
$670$	0	0
$671$	791864.	0.0678960
$672$	0	0
$673$	8.64695e6	0.735911	0.367955	−	0.929843i	$-0.380058\pi$
$673$	8.64695e6	0.735911	0.367955	+	0.929843i	$0.380058\pi$
$674$	0	0
$675$	−1.43540e6	−0.121259
$676$	0	0
$677$	1.69592e7	1.42211	0.711056	−	0.703135i	$-0.248217\pi$
$677$	1.69592e7	1.42211	0.711056	+	0.703135i	$0.248217\pi$
$678$	0	0
$679$	2.78779e7	2.32052
$680$	0	0
$681$	1.32277e7	1.09299
$682$	0	0
$683$	−1.87105e7	−1.53473	−0.767367	−	0.641209i	$-0.778433\pi$
$683$	−1.87105e7	−1.53473	−0.767367	+	0.641209i	$0.778433\pi$
$684$	0	0
$685$	−8.91745e6	−0.726130
$686$	0	0
$687$	29610.0	0.00239357
$688$	0	0
$689$	−1.03932e6	−0.0834071
$690$	0	0
$691$	1.16204e7	0.925820	0.462910	−	0.886405i	$-0.346805\pi$
$691$	1.16204e7	0.925820	0.462910	+	0.886405i	$0.346805\pi$
$692$	0	0
$693$	−2.41056e6	−0.190671
$694$	0	0
$695$	−8.61982e6	−0.676918
$696$	0	0
$697$	−5.63924e6	−0.439682
$698$	0	0
$699$	8.41862e6	0.651700
$700$	0	0
$701$	−2.23497e7	−1.71781	−0.858907	−	0.512132i	$-0.828856\pi$
$701$	−2.23497e7	−1.71781	−0.858907	+	0.512132i	$0.828856\pi$
$702$	0	0
$703$	8.39249e6	0.640475
$704$	0	0
$705$	2.14445e6	0.162496
$706$	0	0
$707$	−1.60402e7	−1.20687
$708$	0	0
$709$	−1.02353e7	−0.764687	−0.382344	−	0.924020i	$-0.624883\pi$
$709$	−1.02353e7	−0.764687	−0.382344	+	0.924020i	$0.624883\pi$
$710$	0	0
$711$	2.15719e6	0.160035
$712$	0	0
$713$	−1.32003e7	−0.972435
$714$	0	0
$715$	−193936.	−0.0141871
$716$	0	0
$717$	−7.88040e6	−0.572467
$718$	0	0
$719$	−1.70339e7	−1.22883	−0.614416	−	0.788982i	$-0.710608\pi$
$719$	−1.70339e7	−1.22883	−0.614416	+	0.788982i	$0.710608\pi$
$720$	0	0
$721$	−1.53600e7	−1.10041
$722$	0	0
$723$	−8.63293e6	−0.614203
$724$	0	0
$725$	−1.75674e7	−1.24126
$726$	0	0
$727$	−1.62280e7	−1.13875	−0.569377	−	0.822077i	$-0.692815\pi$
$727$	−1.62280e7	−1.13875	−0.569377	+	0.822077i	$0.692815\pi$
$728$	0	0
$729$	531441.	0.0370370
$730$	0	0
$731$	−2.21427e7	−1.53263
$732$	0	0
$733$	2.17495e7	1.49517	0.747583	−	0.664168i	$-0.231214\pi$
$733$	2.17495e7	1.49517	0.747583	+	0.664168i	$0.231214\pi$
$734$	0	0
$735$	1.24827e7	0.852293
$736$	0	0
$737$	4.84542e6	0.328597
$738$	0	0
$739$	−1.96200e7	−1.32156	−0.660781	−	0.750578i	$-0.729775\pi$
$739$	−1.96200e7	−1.32156	−0.660781	+	0.750578i	$0.729775\pi$
$740$	0	0
$741$	−796536.	−0.0532917
$742$	0	0
$743$	1.74018e7	1.15644	0.578218	−	0.815882i	$-0.303748\pi$
$743$	1.74018e7	1.15644	0.578218	+	0.815882i	$0.303748\pi$
$744$	0	0
$745$	−1.20914e7	−0.798154
$746$	0	0
$747$	−4.55123e6	−0.298419
$748$	0	0
$749$	−3.62016e6	−0.235789
$750$	0	0
$751$	−2.62693e7	−1.69961	−0.849803	−	0.527101i	$-0.823279\pi$
$751$	−2.62693e7	−1.69961	−0.849803	+	0.527101i	$0.823279\pi$
$752$	0	0
$753$	−2.86981e6	−0.184445
$754$	0	0
$755$	−34816.0	−0.00222286
$756$	0	0
$757$	5.70356e6	0.361748	0.180874	−	0.983506i	$-0.442107\pi$
$757$	5.70356e6	0.361748	0.180874	+	0.983506i	$0.442107\pi$
$758$	0	0
$759$	3.16944e6	0.199700
$760$	0	0
$761$	−2.13762e7	−1.33804	−0.669020	−	0.743244i	$-0.733286\pi$
$761$	−2.13762e7	−1.33804	−0.669020	+	0.743244i	$0.733286\pi$
$762$	0	0
$763$	−9.52752e6	−0.592473
$764$	0	0
$765$	5.38132e6	0.332457
$766$	0	0
$767$	−1288.00	−7.90547e−5 0
$768$	0	0
$769$	−2.01523e6	−0.122888	−0.0614439	−	0.998111i	$-0.519571\pi$
$769$	−2.01523e6	−0.122888	−0.0614439	+	0.998111i	$0.519571\pi$
$770$	0	0
$771$	1.54322e7	0.934958
$772$	0	0
$773$	1.27674e7	0.768520	0.384260	−	0.923225i	$-0.374457\pi$
$773$	1.27674e7	0.768520	0.384260	+	0.923225i	$0.374457\pi$
$774$	0	0
$775$	9.15191e6	0.547340
$776$	0	0
$777$	−9.42192e6	−0.559870
$778$	0	0
$779$	−5.55266e6	−0.327837
$780$	0	0
$781$	−6.80413e6	−0.399158
$782$	0	0
$783$	6.50414e6	0.379128
$784$	0	0
$785$	2.03844e6	0.118065
$786$	0	0
$787$	2.72384e7	1.56764	0.783818	−	0.620990i	$-0.213269\pi$
$787$	2.72384e7	1.56764	0.783818	+	0.620990i	$0.213269\pi$
$788$	0	0
$789$	−9.99252e6	−0.571456
$790$	0	0
$791$	−3.72370e7	−2.11608
$792$	0	0
$793$	−293756.	−0.0165884
$794$	0	0
$795$	6.91376e6	0.387969
$796$	0	0
$797$	7.66724e6	0.427556	0.213778	−	0.976882i	$-0.431423\pi$
$797$	7.66724e6	0.427556	0.213778	+	0.976882i	$0.431423\pi$
$798$	0	0
$799$	1.36936e7	0.758843
$800$	0	0
$801$	5.21721e6	0.287314
$802$	0	0
$803$	2.60822e6	0.142743
$804$	0	0
$805$	−2.31744e7	−1.26043
$806$	0	0
$807$	3.58540e6	0.193800
$808$	0	0
$809$	−1.05541e7	−0.566956	−0.283478	−	0.958979i	$-0.591488\pi$
$809$	−1.05541e7	−0.566956	−0.283478	+	0.958979i	$0.591488\pi$
$810$	0	0
$811$	1.32883e6	0.0709442	0.0354721	−	0.999371i	$-0.488707\pi$
$811$	1.32883e6	0.0709442	0.0354721	+	0.999371i	$0.488707\pi$
$812$	0	0
$813$	1.29778e7	0.688611
$814$	0	0
$815$	1.16129e7	0.612416
$816$	0	0
$817$	−2.18028e7	−1.14276
$818$	0	0
$819$	894240.	0.0465848
$820$	0	0
$821$	6.15933e6	0.318915	0.159458	−	0.987205i	$-0.449025\pi$
$821$	6.15933e6	0.318915	0.159458	+	0.987205i	$0.449025\pi$
$822$	0	0
$823$	1.00734e7	0.518414	0.259207	−	0.965822i	$-0.416539\pi$
$823$	1.00734e7	0.518414	0.259207	+	0.965822i	$0.416539\pi$
$824$	0	0
$825$	−2.19740e6	−0.112402
$826$	0	0
$827$	6.49152e6	0.330052	0.165026	−	0.986289i	$-0.447229\pi$
$827$	6.49152e6	0.330052	0.165026	+	0.986289i	$0.447229\pi$
$828$	0	0
$829$	1.93536e7	0.978082	0.489041	−	0.872261i	$-0.337347\pi$
$829$	1.93536e7	0.978082	0.489041	+	0.872261i	$0.337347\pi$
$830$	0	0
$831$	−1.05514e6	−0.0530040
$832$	0	0
$833$	7.97095e7	3.98013
$834$	0	0
$835$	170544.	0.00846487
$836$	0	0
$837$	−3.38839e6	−0.167178
$838$	0	0
$839$	−2.78622e7	−1.36650	−0.683251	−	0.730183i	$-0.739435\pi$
$839$	−2.78622e7	−1.36650	−0.683251	+	0.730183i	$0.739435\pi$
$840$	0	0
$841$	5.90909e7	2.88092
$842$	0	0
$843$	−1.50763e7	−0.730677
$844$	0	0
$845$	−1.25520e7	−0.604744
$846$	0	0
$847$	3.49620e7	1.67451
$848$	0	0
$849$	−1.73221e7	−0.824766
$850$	0	0
$851$	1.23881e7	0.586381
$852$	0	0
$853$	−1.07651e7	−0.506577	−0.253288	−	0.967391i	$-0.581512\pi$
$853$	−1.07651e7	−0.506577	−0.253288	+	0.967391i	$0.581512\pi$
$854$	0	0
$855$	5.29870e6	0.247887
$856$	0	0
$857$	1.22439e7	0.569465	0.284733	−	0.958607i	$-0.408095\pi$
$857$	1.22439e7	0.569465	0.284733	+	0.958607i	$0.408095\pi$
$858$	0	0
$859$	1.38664e6	0.0641179	0.0320590	−	0.999486i	$-0.489794\pi$
$859$	1.38664e6	0.0641179	0.0320590	+	0.999486i	$0.489794\pi$
$860$	0	0
$861$	6.23376e6	0.286578
$862$	0	0
$863$	−1.09856e7	−0.502109	−0.251055	−	0.967973i	$-0.580777\pi$
$863$	−1.09856e7	−0.502109	−0.251055	+	0.967973i	$0.580777\pi$
$864$	0	0
$865$	7.77464e6	0.353297
$866$	0	0
$867$	2.15843e7	0.975194
$868$	0	0
$869$	3.30237e6	0.148346
$870$	0	0
$871$	−1.79750e6	−0.0802828
$872$	0	0
$873$	−9.40880e6	−0.417829
$874$	0	0
$875$	4.15670e7	1.83539
$876$	0	0
$877$	−8.17798e6	−0.359044	−0.179522	−	0.983754i	$-0.557455\pi$
$877$	−8.17798e6	−0.359044	−0.179522	+	0.983754i	$0.557455\pi$
$878$	0	0
$879$	−1.15256e7	−0.503143
$880$	0	0
$881$	4.66520e6	0.202503	0.101251	−	0.994861i	$-0.467715\pi$
$881$	4.66520e6	0.202503	0.101251	+	0.994861i	$0.467715\pi$
$882$	0	0
$883$	−3.82201e7	−1.64964	−0.824822	−	0.565393i	$-0.808724\pi$
$883$	−3.82201e7	−1.64964	−0.824822	+	0.565393i	$0.808724\pi$
$884$	0	0
$885$	8568.00	0.000367723 0
$886$	0	0
$887$	−7.72172e6	−0.329538	−0.164769	−	0.986332i	$-0.552688\pi$
$887$	−7.72172e6	−0.329538	−0.164769	+	0.986332i	$0.552688\pi$
$888$	0	0
$889$	−1.24973e7	−0.530348
$890$	0	0
$891$	813564.	0.0343319
$892$	0	0
$893$	1.34834e7	0.565810
$894$	0	0
$895$	−5.48719e6	−0.228977
$896$	0	0
$897$	−1.17576e6	−0.0487908
$898$	0	0
$899$	−4.14695e7	−1.71131
$900$	0	0
$901$	4.41487e7	1.81178
$902$	0	0
$903$	2.44771e7	0.998944
$904$	0	0
$905$	9.91746e6	0.402512
$906$	0	0
$907$	−4.33137e7	−1.74826	−0.874131	−	0.485689i	$-0.838569\pi$
$907$	−4.33137e7	−1.74826	−0.874131	+	0.485689i	$0.838569\pi$
$908$	0	0
$909$	5.41355e6	0.217307
$910$	0	0
$911$	3.44456e6	0.137511	0.0687556	−	0.997634i	$-0.478097\pi$
$911$	3.44456e6	0.137511	0.0687556	+	0.997634i	$0.478097\pi$
$912$	0	0
$913$	−6.96731e6	−0.276623
$914$	0	0
$915$	1.95412e6	0.0771610
$916$	0	0
$917$	3.83914e7	1.50768
$918$	0	0
$919$	−4.37073e7	−1.70712	−0.853562	−	0.520991i	$-0.825563\pi$
$919$	−4.37073e7	−1.70712	−0.853562	+	0.520991i	$0.825563\pi$
$920$	0	0
$921$	2.03687e7	0.791251
$922$	0	0
$923$	2.52411e6	0.0975224
$924$	0	0
$925$	−8.58878e6	−0.330048
$926$	0	0
$927$	5.18400e6	0.198137
$928$	0	0
$929$	−4.13022e7	−1.57012	−0.785062	−	0.619418i	$-0.787369\pi$
$929$	−4.13022e7	−1.57012	−0.785062	+	0.619418i	$0.787369\pi$
$930$	0	0
$931$	7.84857e7	2.96768
$932$	0	0
$933$	2.23063e6	0.0838926
$934$	0	0
$935$	8.23806e6	0.308174
$936$	0	0
$937$	9.57460e6	0.356264	0.178132	−	0.984007i	$-0.442995\pi$
$937$	9.57460e6	0.356264	0.178132	+	0.984007i	$0.442995\pi$
$938$	0	0
$939$	−1.64152e7	−0.607550
$940$	0	0
$941$	−8.71623e6	−0.320889	−0.160444	−	0.987045i	$-0.551293\pi$
$941$	−8.71623e6	−0.320889	−0.160444	+	0.987045i	$0.551293\pi$
$942$	0	0
$943$	−8.19624e6	−0.300148
$944$	0	0
$945$	−5.94864e6	−0.216690
$946$	0	0
$947$	1.30605e7	0.473244	0.236622	−	0.971602i	$-0.423960\pi$
$947$	1.30605e7	0.473244	0.236622	+	0.971602i	$0.423960\pi$
$948$	0	0
$949$	−967564.	−0.0348750
$950$	0	0
$951$	−2.57066e7	−0.921710
$952$	0	0
$953$	1.13875e7	0.406158	0.203079	−	0.979162i	$-0.434905\pi$
$953$	1.13875e7	0.406158	0.203079	+	0.979162i	$0.434905\pi$
$954$	0	0
$955$	−1.89312e6	−0.0671691
$956$	0	0
$957$	9.95695e6	0.351436
$958$	0	0
$959$	6.29467e7	2.21017
$960$	0	0
$961$	−7.02525e6	−0.245388
$962$	0	0
$963$	1.22180e6	0.0424557
$964$	0	0
$965$	−5.99264e6	−0.207157
$966$	0	0
$967$	4.62711e7	1.59127	0.795634	−	0.605778i	$-0.207138\pi$
$967$	4.62711e7	1.59127	0.795634	+	0.605778i	$0.207138\pi$
$968$	0	0
$969$	3.38355e7	1.15761
$970$	0	0
$971$	−1.63206e7	−0.555506	−0.277753	−	0.960653i	$-0.589590\pi$
$971$	−1.63206e7	−0.555506	−0.277753	+	0.960653i	$0.589590\pi$
$972$	0	0
$973$	6.08458e7	2.06038
$974$	0	0
$975$	815166.	0.0274621
$976$	0	0
$977$	−1.95213e7	−0.654294	−0.327147	−	0.944973i	$-0.606087\pi$
$977$	−1.95213e7	−0.654294	−0.327147	+	0.944973i	$0.606087\pi$
$978$	0	0
$979$	7.98684e6	0.266329
$980$	0	0
$981$	3.21554e6	0.106680
$982$	0	0
$983$	−4.33962e7	−1.43241	−0.716207	−	0.697888i	$-0.754123\pi$
$983$	−4.33962e7	−1.43241	−0.716207	+	0.697888i	$0.754123\pi$
$984$	0	0
$985$	1.27367e7	0.418281
$986$	0	0
$987$	−1.51373e7	−0.494601
$988$	0	0
$989$	−3.21829e7	−1.04625
$990$	0	0
$991$	3.83518e7	1.24051	0.620257	−	0.784399i	$-0.287028\pi$
$991$	3.83518e7	1.24051	0.620257	+	0.784399i	$0.287028\pi$
$992$	0	0
$993$	1.32433e6	0.0426210
$994$	0	0
$995$	−2.16838e7	−0.694350
$996$	0	0
$997$	7.82206e6	0.249220	0.124610	−	0.992206i	$-0.460232\pi$
$997$	7.82206e6	0.249220	0.124610	+	0.992206i	$0.460232\pi$
$998$	0	0
$999$	3.17990e6	0.100809

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	192.6.a.n.1.1		1
3.2	odd	2		576.6.a.k.1.1		1
4.3	odd	2		192.6.a.f.1.1		1
8.3	odd	2		48.6.a.d.1.1		1
8.5	even	2		24.6.a.a.1.1	✓	1
12.11	even	2		576.6.a.l.1.1		1
16.3	odd	4		768.6.d.a.385.1		2
16.5	even	4		768.6.d.r.385.1		2
16.11	odd	4		768.6.d.a.385.2		2
16.13	even	4		768.6.d.r.385.2		2
24.5	odd	2		72.6.a.e.1.1		1
24.11	even	2		144.6.a.i.1.1		1
40.13	odd	4		600.6.f.f.49.1		2
40.29	even	2		600.6.a.i.1.1		1
40.37	odd	4		600.6.f.f.49.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
24.6.a.a.1.1	✓	1	8.5	even	2
48.6.a.d.1.1		1	8.3	odd	2
72.6.a.e.1.1		1	24.5	odd	2
144.6.a.i.1.1		1	24.11	even	2
192.6.a.f.1.1		1	4.3	odd	2
192.6.a.n.1.1		1	1.1	even	1	trivial
576.6.a.k.1.1		1	3.2	odd	2
576.6.a.l.1.1		1	12.11	even	2
600.6.a.i.1.1		1	40.29	even	2
600.6.f.f.49.1		2	40.13	odd	4
600.6.f.f.49.2		2	40.37	odd	4
768.6.d.a.385.1		2	16.3	odd	4
768.6.d.a.385.2		2	16.11	odd	4
768.6.d.r.385.1		2	16.5	even	4
768.6.d.r.385.2		2	16.13	even	4

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:	\( N \)	\(=\)	\( 192 = 2^{6} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	192.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more