LMFDB - Embedded newform 198.2.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [198,2,Mod(1,198)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(198, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("198.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 198 = 2 \cdot 3^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	198.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:	$1.58103796002$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 66)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	198.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{2} +1.00000 q^{4} -2.00000 q^{5} -4.00000 q^{7} -1.00000 q^{8} +O(q^{10})$

$q-1.00000 q^{2} +1.00000 q^{4} -2.00000 q^{5} -4.00000 q^{7} -1.00000 q^{8} +2.00000 q^{10} +1.00000 q^{11} -6.00000 q^{13} +4.00000 q^{14} +1.00000 q^{16} -2.00000 q^{17} +4.00000 q^{19} -2.00000 q^{20} -1.00000 q^{22} -4.00000 q^{23} -1.00000 q^{25} +6.00000 q^{26} -4.00000 q^{28} -6.00000 q^{29} -1.00000 q^{32} +2.00000 q^{34} +8.00000 q^{35} +6.00000 q^{37} -4.00000 q^{38} +2.00000 q^{40} +6.00000 q^{41} +4.00000 q^{43} +1.00000 q^{44} +4.00000 q^{46} +12.0000 q^{47} +9.00000 q^{49} +1.00000 q^{50} -6.00000 q^{52} -2.00000 q^{53} -2.00000 q^{55} +4.00000 q^{56} +6.00000 q^{58} -12.0000 q^{59} -14.0000 q^{61} +1.00000 q^{64} +12.0000 q^{65} +4.00000 q^{67} -2.00000 q^{68} -8.00000 q^{70} +12.0000 q^{71} -6.00000 q^{73} -6.00000 q^{74} +4.00000 q^{76} -4.00000 q^{77} -4.00000 q^{79} -2.00000 q^{80} -6.00000 q^{82} -4.00000 q^{83} +4.00000 q^{85} -4.00000 q^{86} -1.00000 q^{88} -10.0000 q^{89} +24.0000 q^{91} -4.00000 q^{92} -12.0000 q^{94} -8.00000 q^{95} -14.0000 q^{97} -9.00000 q^{98} +O(q^{100})$

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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	−2.00000	−0.894427	−0.447214	−	0.894427i	$-0.647584\pi$
$5$	−2.00000	−0.894427	−0.447214	+	0.894427i	$0.647584\pi$
$6$	0	0
$7$	−4.00000	−1.51186	−0.755929	−	0.654654i	$-0.772814\pi$
$7$	−4.00000	−1.51186	−0.755929	+	0.654654i	$0.772814\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	2.00000	0.632456
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−6.00000	−1.66410	−0.832050	−	0.554700i	$-0.812833\pi$
$13$	−6.00000	−1.66410	−0.832050	+	0.554700i	$0.812833\pi$
$14$	4.00000	1.06904
$15$	0	0
$16$	1.00000	0.250000
$17$	−2.00000	−0.485071	−0.242536	−	0.970143i	$-0.577979\pi$
$17$	−2.00000	−0.485071	−0.242536	+	0.970143i	$0.577979\pi$
$18$	0	0
$19$	4.00000	0.917663	0.458831	−	0.888523i	$-0.348268\pi$
$19$	4.00000	0.917663	0.458831	+	0.888523i	$0.348268\pi$
$20$	−2.00000	−0.447214
$21$	0	0
$22$	−1.00000	−0.213201
$23$	−4.00000	−0.834058	−0.417029	−	0.908893i	$-0.636929\pi$
$23$	−4.00000	−0.834058	−0.417029	+	0.908893i	$0.636929\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	6.00000	1.17670
$27$	0	0
$28$	−4.00000	−0.755929
$29$	−6.00000	−1.11417	−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000	−1.11417	−0.557086	+	0.830455i	$0.688081\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	2.00000	0.342997
$35$	8.00000	1.35225
$36$	0	0
$37$	6.00000	0.986394	0.493197	−	0.869918i	$-0.335828\pi$
$37$	6.00000	0.986394	0.493197	+	0.869918i	$0.335828\pi$
$38$	−4.00000	−0.648886
$39$	0	0
$40$	2.00000	0.316228
$41$	6.00000	0.937043	0.468521	−	0.883452i	$-0.344787\pi$
$41$	6.00000	0.937043	0.468521	+	0.883452i	$0.344787\pi$
$42$	0	0
$43$	4.00000	0.609994	0.304997	−	0.952353i	$-0.401344\pi$
$43$	4.00000	0.609994	0.304997	+	0.952353i	$0.401344\pi$
$44$	1.00000	0.150756
$45$	0	0
$46$	4.00000	0.589768
$47$	12.0000	1.75038	0.875190	−	0.483779i	$-0.160736\pi$
$47$	12.0000	1.75038	0.875190	+	0.483779i	$0.160736\pi$
$48$	0	0
$49$	9.00000	1.28571
$50$	1.00000	0.141421
$51$	0	0
$52$	−6.00000	−0.832050
$53$	−2.00000	−0.274721	−0.137361	−	0.990521i	$-0.543862\pi$
$53$	−2.00000	−0.274721	−0.137361	+	0.990521i	$0.543862\pi$
$54$	0	0
$55$	−2.00000	−0.269680
$56$	4.00000	0.534522
$57$	0	0
$58$	6.00000	0.787839
$59$	−12.0000	−1.56227	−0.781133	−	0.624364i	$-0.785358\pi$
$59$	−12.0000	−1.56227	−0.781133	+	0.624364i	$0.785358\pi$
$60$	0	0
$61$	−14.0000	−1.79252	−0.896258	−	0.443533i	$-0.853725\pi$
$61$	−14.0000	−1.79252	−0.896258	+	0.443533i	$0.853725\pi$
$62$	0	0
$63$	0	0
$64$	1.00000	0.125000
$65$	12.0000	1.48842
$66$	0	0
$67$	4.00000	0.488678	0.244339	−	0.969690i	$-0.421429\pi$
$67$	4.00000	0.488678	0.244339	+	0.969690i	$0.421429\pi$
$68$	−2.00000	−0.242536
$69$	0	0
$70$	−8.00000	−0.956183
$71$	12.0000	1.42414	0.712069	−	0.702109i	$-0.247758\pi$
$71$	12.0000	1.42414	0.712069	+	0.702109i	$0.247758\pi$
$72$	0	0
$73$	−6.00000	−0.702247	−0.351123	−	0.936329i	$-0.614200\pi$
$73$	−6.00000	−0.702247	−0.351123	+	0.936329i	$0.614200\pi$
$74$	−6.00000	−0.697486
$75$	0	0
$76$	4.00000	0.458831
$77$	−4.00000	−0.455842
$78$	0	0
$79$	−4.00000	−0.450035	−0.225018	−	0.974355i	$-0.572244\pi$
$79$	−4.00000	−0.450035	−0.225018	+	0.974355i	$0.572244\pi$
$80$	−2.00000	−0.223607
$81$	0	0
$82$	−6.00000	−0.662589
$83$	−4.00000	−0.439057	−0.219529	−	0.975606i	$-0.570452\pi$
$83$	−4.00000	−0.439057	−0.219529	+	0.975606i	$0.570452\pi$
$84$	0	0
$85$	4.00000	0.433861
$86$	−4.00000	−0.431331
$87$	0	0
$88$	−1.00000	−0.106600
$89$	−10.0000	−1.06000	−0.529999	−	0.847998i	$-0.677808\pi$
$89$	−10.0000	−1.06000	−0.529999	+	0.847998i	$0.677808\pi$
$90$	0	0
$91$	24.0000	2.51588
$92$	−4.00000	−0.417029
$93$	0	0
$94$	−12.0000	−1.23771
$95$	−8.00000	−0.820783
$96$	0	0
$97$	−14.0000	−1.42148	−0.710742	−	0.703452i	$-0.751641\pi$
$97$	−14.0000	−1.42148	−0.710742	+	0.703452i	$0.751641\pi$
$98$	−9.00000	−0.909137
$99$	0	0
$100$	−1.00000	−0.100000
$101$	−14.0000	−1.39305	−0.696526	−	0.717532i	$-0.745272\pi$
$101$	−14.0000	−1.39305	−0.696526	+	0.717532i	$0.745272\pi$
$102$	0	0
$103$	0	0		−	1.00000i	$-0.5\pi$
$103$	0	0			1.00000i	$0.5\pi$
$104$	6.00000	0.588348
$105$	0	0
$106$	2.00000	0.194257
$107$	−4.00000	−0.386695	−0.193347	−	0.981130i	$-0.561934\pi$
$107$	−4.00000	−0.386695	−0.193347	+	0.981130i	$0.561934\pi$
$108$	0	0
$109$	−6.00000	−0.574696	−0.287348	−	0.957826i	$-0.592774\pi$
$109$	−6.00000	−0.574696	−0.287348	+	0.957826i	$0.592774\pi$
$110$	2.00000	0.190693
$111$	0	0
$112$	−4.00000	−0.377964
$113$	−2.00000	−0.188144	−0.0940721	−	0.995565i	$-0.529988\pi$
$113$	−2.00000	−0.188144	−0.0940721	+	0.995565i	$0.529988\pi$
$114$	0	0
$115$	8.00000	0.746004
$116$	−6.00000	−0.557086
$117$	0	0
$118$	12.0000	1.10469
$119$	8.00000	0.733359
$120$	0	0
$121$	1.00000	0.0909091
$122$	14.0000	1.26750
$123$	0	0
$124$	0	0
$125$	12.0000	1.07331
$126$	0	0
$127$	12.0000	1.06483	0.532414	−	0.846484i	$-0.321285\pi$
$127$	12.0000	1.06483	0.532414	+	0.846484i	$0.321285\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	−12.0000	−1.05247
$131$	−4.00000	−0.349482	−0.174741	−	0.984614i	$-0.555909\pi$
$131$	−4.00000	−0.349482	−0.174741	+	0.984614i	$0.555909\pi$
$132$	0	0
$133$	−16.0000	−1.38738
$134$	−4.00000	−0.345547
$135$	0	0
$136$	2.00000	0.171499
$137$	−2.00000	−0.170872	−0.0854358	−	0.996344i	$-0.527228\pi$
$137$	−2.00000	−0.170872	−0.0854358	+	0.996344i	$0.527228\pi$
$138$	0	0
$139$	−4.00000	−0.339276	−0.169638	−	0.985506i	$-0.554260\pi$
$139$	−4.00000	−0.339276	−0.169638	+	0.985506i	$0.554260\pi$
$140$	8.00000	0.676123
$141$	0	0
$142$	−12.0000	−1.00702
$143$	−6.00000	−0.501745
$144$	0	0
$145$	12.0000	0.996546
$146$	6.00000	0.496564
$147$	0	0
$148$	6.00000	0.493197
$149$	10.0000	0.819232	0.409616	−	0.912258i	$-0.365663\pi$
$149$	10.0000	0.819232	0.409616	+	0.912258i	$0.365663\pi$
$150$	0	0
$151$	4.00000	0.325515	0.162758	−	0.986666i	$-0.447961\pi$
$151$	4.00000	0.325515	0.162758	+	0.986666i	$0.447961\pi$
$152$	−4.00000	−0.324443
$153$	0	0
$154$	4.00000	0.322329
$155$	0	0
$156$	0	0
$157$	−10.0000	−0.798087	−0.399043	−	0.916932i	$-0.630658\pi$
$157$	−10.0000	−0.798087	−0.399043	+	0.916932i	$0.630658\pi$
$158$	4.00000	0.318223
$159$	0	0
$160$	2.00000	0.158114
$161$	16.0000	1.26098
$162$	0	0
$163$	−20.0000	−1.56652	−0.783260	−	0.621694i	$-0.786445\pi$
$163$	−20.0000	−1.56652	−0.783260	+	0.621694i	$0.786445\pi$
$164$	6.00000	0.468521
$165$	0	0
$166$	4.00000	0.310460
$167$	0	0		−	1.00000i	$-0.5\pi$
$167$	0	0			1.00000i	$0.5\pi$
$168$	0	0
$169$	23.0000	1.76923
$170$	−4.00000	−0.306786
$171$	0	0
$172$	4.00000	0.304997
$173$	10.0000	0.760286	0.380143	−	0.924928i	$-0.375875\pi$
$173$	10.0000	0.760286	0.380143	+	0.924928i	$0.375875\pi$
$174$	0	0
$175$	4.00000	0.302372
$176$	1.00000	0.0753778
$177$	0	0
$178$	10.0000	0.749532
$179$	−20.0000	−1.49487	−0.747435	−	0.664335i	$-0.768715\pi$
$179$	−20.0000	−1.49487	−0.747435	+	0.664335i	$0.768715\pi$
$180$	0	0
$181$	−2.00000	−0.148659	−0.0743294	−	0.997234i	$-0.523682\pi$
$181$	−2.00000	−0.148659	−0.0743294	+	0.997234i	$0.523682\pi$
$182$	−24.0000	−1.77900
$183$	0	0
$184$	4.00000	0.294884
$185$	−12.0000	−0.882258
$186$	0	0
$187$	−2.00000	−0.146254
$188$	12.0000	0.875190
$189$	0	0
$190$	8.00000	0.580381
$191$	12.0000	0.868290	0.434145	−	0.900843i	$-0.357051\pi$
$191$	12.0000	0.868290	0.434145	+	0.900843i	$0.357051\pi$
$192$	0	0
$193$	10.0000	0.719816	0.359908	−	0.932988i	$-0.382808\pi$
$193$	10.0000	0.719816	0.359908	+	0.932988i	$0.382808\pi$
$194$	14.0000	1.00514
$195$	0	0
$196$	9.00000	0.642857
$197$	2.00000	0.142494	0.0712470	−	0.997459i	$-0.477302\pi$
$197$	2.00000	0.142494	0.0712470	+	0.997459i	$0.477302\pi$
$198$	0	0
$199$	−16.0000	−1.13421	−0.567105	−	0.823646i	$-0.691937\pi$
$199$	−16.0000	−1.13421	−0.567105	+	0.823646i	$0.691937\pi$
$200$	1.00000	0.0707107
$201$	0	0
$202$	14.0000	0.985037
$203$	24.0000	1.68447
$204$	0	0
$205$	−12.0000	−0.838116
$206$	0	0
$207$	0	0
$208$	−6.00000	−0.416025
$209$	4.00000	0.276686
$210$	0	0
$211$	−4.00000	−0.275371	−0.137686	−	0.990476i	$-0.543966\pi$
$211$	−4.00000	−0.275371	−0.137686	+	0.990476i	$0.543966\pi$
$212$	−2.00000	−0.137361
$213$	0	0
$214$	4.00000	0.273434
$215$	−8.00000	−0.545595
$216$	0	0
$217$	0	0
$218$	6.00000	0.406371
$219$	0	0
$220$	−2.00000	−0.134840
$221$	12.0000	0.807207
$222$	0	0
$223$	0	0		−	1.00000i	$-0.5\pi$
$223$	0	0			1.00000i	$0.5\pi$
$224$	4.00000	0.267261
$225$	0	0
$226$	2.00000	0.133038
$227$	−12.0000	−0.796468	−0.398234	−	0.917284i	$-0.630377\pi$
$227$	−12.0000	−0.796468	−0.398234	+	0.917284i	$0.630377\pi$
$228$	0	0
$229$	14.0000	0.925146	0.462573	−	0.886581i	$-0.346926\pi$
$229$	14.0000	0.925146	0.462573	+	0.886581i	$0.346926\pi$
$230$	−8.00000	−0.527504
$231$	0	0
$232$	6.00000	0.393919
$233$	−10.0000	−0.655122	−0.327561	−	0.944830i	$-0.606227\pi$
$233$	−10.0000	−0.655122	−0.327561	+	0.944830i	$0.606227\pi$
$234$	0	0
$235$	−24.0000	−1.56559
$236$	−12.0000	−0.781133
$237$	0	0
$238$	−8.00000	−0.518563
$239$	−8.00000	−0.517477	−0.258738	−	0.965947i	$-0.583307\pi$
$239$	−8.00000	−0.517477	−0.258738	+	0.965947i	$0.583307\pi$
$240$	0	0
$241$	10.0000	0.644157	0.322078	−	0.946713i	$-0.395619\pi$
$241$	10.0000	0.644157	0.322078	+	0.946713i	$0.395619\pi$
$242$	−1.00000	−0.0642824
$243$	0	0
$244$	−14.0000	−0.896258
$245$	−18.0000	−1.14998
$246$	0	0
$247$	−24.0000	−1.52708
$248$	0	0
$249$	0	0
$250$	−12.0000	−0.758947
$251$	4.00000	0.252478	0.126239	−	0.992000i	$-0.459709\pi$
$251$	4.00000	0.252478	0.126239	+	0.992000i	$0.459709\pi$
$252$	0	0
$253$	−4.00000	−0.251478
$254$	−12.0000	−0.752947
$255$	0	0
$256$	1.00000	0.0625000
$257$	−2.00000	−0.124757	−0.0623783	−	0.998053i	$-0.519869\pi$
$257$	−2.00000	−0.124757	−0.0623783	+	0.998053i	$0.519869\pi$
$258$	0	0
$259$	−24.0000	−1.49129
$260$	12.0000	0.744208
$261$	0	0
$262$	4.00000	0.247121
$263$	24.0000	1.47990	0.739952	−	0.672660i	$-0.234848\pi$
$263$	24.0000	1.47990	0.739952	+	0.672660i	$0.234848\pi$
$264$	0	0
$265$	4.00000	0.245718
$266$	16.0000	0.981023
$267$	0	0
$268$	4.00000	0.244339
$269$	−26.0000	−1.58525	−0.792624	−	0.609711i	$-0.791286\pi$
$269$	−26.0000	−1.58525	−0.792624	+	0.609711i	$0.791286\pi$
$270$	0	0
$271$	20.0000	1.21491	0.607457	−	0.794353i	$-0.292190\pi$
$271$	20.0000	1.21491	0.607457	+	0.794353i	$0.292190\pi$
$272$	−2.00000	−0.121268
$273$	0	0
$274$	2.00000	0.120824
$275$	−1.00000	−0.0603023
$276$	0	0
$277$	26.0000	1.56219	0.781094	−	0.624413i	$-0.214662\pi$
$277$	26.0000	1.56219	0.781094	+	0.624413i	$0.214662\pi$
$278$	4.00000	0.239904
$279$	0	0
$280$	−8.00000	−0.478091
$281$	22.0000	1.31241	0.656205	−	0.754583i	$-0.272161\pi$
$281$	22.0000	1.31241	0.656205	+	0.754583i	$0.272161\pi$
$282$	0	0
$283$	4.00000	0.237775	0.118888	−	0.992908i	$-0.462067\pi$
$283$	4.00000	0.237775	0.118888	+	0.992908i	$0.462067\pi$
$284$	12.0000	0.712069
$285$	0	0
$286$	6.00000	0.354787
$287$	−24.0000	−1.41668
$288$	0	0
$289$	−13.0000	−0.764706
$290$	−12.0000	−0.704664
$291$	0	0
$292$	−6.00000	−0.351123
$293$	−22.0000	−1.28525	−0.642627	−	0.766179i	$-0.722155\pi$
$293$	−22.0000	−1.28525	−0.642627	+	0.766179i	$0.722155\pi$
$294$	0	0
$295$	24.0000	1.39733
$296$	−6.00000	−0.348743
$297$	0	0
$298$	−10.0000	−0.579284
$299$	24.0000	1.38796
$300$	0	0
$301$	−16.0000	−0.922225
$302$	−4.00000	−0.230174
$303$	0	0
$304$	4.00000	0.229416
$305$	28.0000	1.60328
$306$	0	0
$307$	4.00000	0.228292	0.114146	−	0.993464i	$-0.463587\pi$
$307$	4.00000	0.228292	0.114146	+	0.993464i	$0.463587\pi$
$308$	−4.00000	−0.227921
$309$	0	0
$310$	0	0
$311$	−4.00000	−0.226819	−0.113410	−	0.993548i	$-0.536177\pi$
$311$	−4.00000	−0.226819	−0.113410	+	0.993548i	$0.536177\pi$
$312$	0	0
$313$	26.0000	1.46961	0.734803	−	0.678280i	$-0.237274\pi$
$313$	26.0000	1.46961	0.734803	+	0.678280i	$0.237274\pi$
$314$	10.0000	0.564333
$315$	0	0
$316$	−4.00000	−0.225018
$317$	−18.0000	−1.01098	−0.505490	−	0.862832i	$-0.668688\pi$
$317$	−18.0000	−1.01098	−0.505490	+	0.862832i	$0.668688\pi$
$318$	0	0
$319$	−6.00000	−0.335936
$320$	−2.00000	−0.111803
$321$	0	0
$322$	−16.0000	−0.891645
$323$	−8.00000	−0.445132
$324$	0	0
$325$	6.00000	0.332820
$326$	20.0000	1.10770
$327$	0	0
$328$	−6.00000	−0.331295
$329$	−48.0000	−2.64633
$330$	0	0
$331$	20.0000	1.09930	0.549650	−	0.835395i	$-0.314761\pi$
$331$	20.0000	1.09930	0.549650	+	0.835395i	$0.314761\pi$
$332$	−4.00000	−0.219529
$333$	0	0
$334$	0	0
$335$	−8.00000	−0.437087
$336$	0	0
$337$	18.0000	0.980522	0.490261	−	0.871576i	$-0.336901\pi$
$337$	18.0000	0.980522	0.490261	+	0.871576i	$0.336901\pi$
$338$	−23.0000	−1.25104
$339$	0	0
$340$	4.00000	0.216930
$341$	0	0
$342$	0	0
$343$	−8.00000	−0.431959
$344$	−4.00000	−0.215666
$345$	0	0
$346$	−10.0000	−0.537603
$347$	4.00000	0.214731	0.107366	−	0.994220i	$-0.465758\pi$
$347$	4.00000	0.214731	0.107366	+	0.994220i	$0.465758\pi$
$348$	0	0
$349$	−6.00000	−0.321173	−0.160586	−	0.987022i	$-0.551338\pi$
$349$	−6.00000	−0.321173	−0.160586	+	0.987022i	$0.551338\pi$
$350$	−4.00000	−0.213809
$351$	0	0
$352$	−1.00000	−0.0533002
$353$	−18.0000	−0.958043	−0.479022	−	0.877803i	$-0.659008\pi$
$353$	−18.0000	−0.958043	−0.479022	+	0.877803i	$0.659008\pi$
$354$	0	0
$355$	−24.0000	−1.27379
$356$	−10.0000	−0.529999
$357$	0	0
$358$	20.0000	1.05703
$359$	0	0		−	1.00000i	$-0.5\pi$
$359$	0	0			1.00000i	$0.5\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	2.00000	0.105118
$363$	0	0
$364$	24.0000	1.25794
$365$	12.0000	0.628109
$366$	0	0
$367$	−16.0000	−0.835193	−0.417597	−	0.908633i	$-0.637127\pi$
$367$	−16.0000	−0.835193	−0.417597	+	0.908633i	$0.637127\pi$
$368$	−4.00000	−0.208514
$369$	0	0
$370$	12.0000	0.623850
$371$	8.00000	0.415339
$372$	0	0
$373$	−14.0000	−0.724893	−0.362446	−	0.932005i	$-0.618058\pi$
$373$	−14.0000	−0.724893	−0.362446	+	0.932005i	$0.618058\pi$
$374$	2.00000	0.103418
$375$	0	0
$376$	−12.0000	−0.618853
$377$	36.0000	1.85409
$378$	0	0
$379$	−28.0000	−1.43826	−0.719132	−	0.694874i	$-0.755460\pi$
$379$	−28.0000	−1.43826	−0.719132	+	0.694874i	$0.755460\pi$
$380$	−8.00000	−0.410391
$381$	0	0
$382$	−12.0000	−0.613973
$383$	4.00000	0.204390	0.102195	−	0.994764i	$-0.467413\pi$
$383$	4.00000	0.204390	0.102195	+	0.994764i	$0.467413\pi$
$384$	0	0
$385$	8.00000	0.407718
$386$	−10.0000	−0.508987
$387$	0	0
$388$	−14.0000	−0.710742
$389$	30.0000	1.52106	0.760530	−	0.649303i	$-0.224939\pi$
$389$	30.0000	1.52106	0.760530	+	0.649303i	$0.224939\pi$
$390$	0	0
$391$	8.00000	0.404577
$392$	−9.00000	−0.454569
$393$	0	0
$394$	−2.00000	−0.100759
$395$	8.00000	0.402524
$396$	0	0
$397$	22.0000	1.10415	0.552074	−	0.833795i	$-0.313837\pi$
$397$	22.0000	1.10415	0.552074	+	0.833795i	$0.313837\pi$
$398$	16.0000	0.802008
$399$	0	0
$400$	−1.00000	−0.0500000
$401$	38.0000	1.89763	0.948815	−	0.315833i	$-0.102284\pi$
$401$	38.0000	1.89763	0.948815	+	0.315833i	$0.102284\pi$
$402$	0	0
$403$	0	0
$404$	−14.0000	−0.696526
$405$	0	0
$406$	−24.0000	−1.19110
$407$	6.00000	0.297409
$408$	0	0
$409$	−14.0000	−0.692255	−0.346128	−	0.938187i	$-0.612504\pi$
$409$	−14.0000	−0.692255	−0.346128	+	0.938187i	$0.612504\pi$
$410$	12.0000	0.592638
$411$	0	0
$412$	0	0
$413$	48.0000	2.36193
$414$	0	0
$415$	8.00000	0.392705
$416$	6.00000	0.294174
$417$	0	0
$418$	−4.00000	−0.195646
$419$	12.0000	0.586238	0.293119	−	0.956076i	$-0.405307\pi$
$419$	12.0000	0.586238	0.293119	+	0.956076i	$0.405307\pi$
$420$	0	0
$421$	−10.0000	−0.487370	−0.243685	−	0.969854i	$-0.578356\pi$
$421$	−10.0000	−0.487370	−0.243685	+	0.969854i	$0.578356\pi$
$422$	4.00000	0.194717
$423$	0	0
$424$	2.00000	0.0971286
$425$	2.00000	0.0970143
$426$	0	0
$427$	56.0000	2.71003
$428$	−4.00000	−0.193347
$429$	0	0
$430$	8.00000	0.385794
$431$	0	0		−	1.00000i	$-0.5\pi$
$431$	0	0			1.00000i	$0.5\pi$
$432$	0	0
$433$	2.00000	0.0961139	0.0480569	−	0.998845i	$-0.484697\pi$
$433$	2.00000	0.0961139	0.0480569	+	0.998845i	$0.484697\pi$
$434$	0	0
$435$	0	0
$436$	−6.00000	−0.287348
$437$	−16.0000	−0.765384
$438$	0	0
$439$	4.00000	0.190910	0.0954548	−	0.995434i	$-0.469569\pi$
$439$	4.00000	0.190910	0.0954548	+	0.995434i	$0.469569\pi$
$440$	2.00000	0.0953463
$441$	0	0
$442$	−12.0000	−0.570782
$443$	−28.0000	−1.33032	−0.665160	−	0.746701i	$-0.731637\pi$
$443$	−28.0000	−1.33032	−0.665160	+	0.746701i	$0.731637\pi$
$444$	0	0
$445$	20.0000	0.948091
$446$	0	0
$447$	0	0
$448$	−4.00000	−0.188982
$449$	22.0000	1.03824	0.519122	−	0.854700i	$-0.326259\pi$
$449$	22.0000	1.03824	0.519122	+	0.854700i	$0.326259\pi$
$450$	0	0
$451$	6.00000	0.282529
$452$	−2.00000	−0.0940721
$453$	0	0
$454$	12.0000	0.563188
$455$	−48.0000	−2.25027
$456$	0	0
$457$	34.0000	1.59045	0.795226	−	0.606313i	$-0.207352\pi$
$457$	34.0000	1.59045	0.795226	+	0.606313i	$0.207352\pi$
$458$	−14.0000	−0.654177
$459$	0	0
$460$	8.00000	0.373002
$461$	−14.0000	−0.652045	−0.326023	−	0.945362i	$-0.605709\pi$
$461$	−14.0000	−0.652045	−0.326023	+	0.945362i	$0.605709\pi$
$462$	0	0
$463$	−24.0000	−1.11537	−0.557687	−	0.830051i	$-0.688311\pi$
$463$	−24.0000	−1.11537	−0.557687	+	0.830051i	$0.688311\pi$
$464$	−6.00000	−0.278543
$465$	0	0
$466$	10.0000	0.463241
$467$	−12.0000	−0.555294	−0.277647	−	0.960683i	$-0.589555\pi$
$467$	−12.0000	−0.555294	−0.277647	+	0.960683i	$0.589555\pi$
$468$	0	0
$469$	−16.0000	−0.738811
$470$	24.0000	1.10704
$471$	0	0
$472$	12.0000	0.552345
$473$	4.00000	0.183920
$474$	0	0
$475$	−4.00000	−0.183533
$476$	8.00000	0.366679
$477$	0	0
$478$	8.00000	0.365911
$479$	0	0		−	1.00000i	$-0.5\pi$
$479$	0	0			1.00000i	$0.5\pi$
$480$	0	0
$481$	−36.0000	−1.64146
$482$	−10.0000	−0.455488
$483$	0	0
$484$	1.00000	0.0454545
$485$	28.0000	1.27141
$486$	0	0
$487$	−16.0000	−0.725029	−0.362515	−	0.931978i	$-0.618082\pi$
$487$	−16.0000	−0.725029	−0.362515	+	0.931978i	$0.618082\pi$
$488$	14.0000	0.633750
$489$	0	0
$490$	18.0000	0.813157
$491$	28.0000	1.26362	0.631811	−	0.775122i	$-0.282312\pi$
$491$	28.0000	1.26362	0.631811	+	0.775122i	$0.282312\pi$
$492$	0	0
$493$	12.0000	0.540453
$494$	24.0000	1.07981
$495$	0	0
$496$	0	0
$497$	−48.0000	−2.15309
$498$	0	0
$499$	−4.00000	−0.179065	−0.0895323	−	0.995984i	$-0.528537\pi$
$499$	−4.00000	−0.179065	−0.0895323	+	0.995984i	$0.528537\pi$
$500$	12.0000	0.536656
$501$	0	0
$502$	−4.00000	−0.178529
$503$	−32.0000	−1.42681	−0.713405	−	0.700752i	$-0.752848\pi$
$503$	−32.0000	−1.42681	−0.713405	+	0.700752i	$0.752848\pi$
$504$	0	0
$505$	28.0000	1.24598
$506$	4.00000	0.177822
$507$	0	0
$508$	12.0000	0.532414
$509$	22.0000	0.975133	0.487566	−	0.873086i	$-0.337885\pi$
$509$	22.0000	0.975133	0.487566	+	0.873086i	$0.337885\pi$
$510$	0	0
$511$	24.0000	1.06170
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	2.00000	0.0882162
$515$	0	0
$516$	0	0
$517$	12.0000	0.527759
$518$	24.0000	1.05450
$519$	0	0
$520$	−12.0000	−0.526235
$521$	−18.0000	−0.788594	−0.394297	−	0.918983i	$-0.629012\pi$
$521$	−18.0000	−0.788594	−0.394297	+	0.918983i	$0.629012\pi$
$522$	0	0
$523$	20.0000	0.874539	0.437269	−	0.899331i	$-0.355946\pi$
$523$	20.0000	0.874539	0.437269	+	0.899331i	$0.355946\pi$
$524$	−4.00000	−0.174741
$525$	0	0
$526$	−24.0000	−1.04645
$527$	0	0
$528$	0	0
$529$	−7.00000	−0.304348
$530$	−4.00000	−0.173749
$531$	0	0
$532$	−16.0000	−0.693688
$533$	−36.0000	−1.55933
$534$	0	0
$535$	8.00000	0.345870
$536$	−4.00000	−0.172774
$537$	0	0
$538$	26.0000	1.12094
$539$	9.00000	0.387657
$540$	0	0
$541$	−38.0000	−1.63375	−0.816874	−	0.576816i	$-0.804295\pi$
$541$	−38.0000	−1.63375	−0.816874	+	0.576816i	$0.804295\pi$
$542$	−20.0000	−0.859074
$543$	0	0
$544$	2.00000	0.0857493
$545$	12.0000	0.514024
$546$	0	0
$547$	−28.0000	−1.19719	−0.598597	−	0.801050i	$-0.704275\pi$
$547$	−28.0000	−1.19719	−0.598597	+	0.801050i	$0.704275\pi$
$548$	−2.00000	−0.0854358
$549$	0	0
$550$	1.00000	0.0426401
$551$	−24.0000	−1.02243
$552$	0	0
$553$	16.0000	0.680389
$554$	−26.0000	−1.10463
$555$	0	0
$556$	−4.00000	−0.169638
$557$	−30.0000	−1.27114	−0.635570	−	0.772043i	$-0.719235\pi$
$557$	−30.0000	−1.27114	−0.635570	+	0.772043i	$0.719235\pi$
$558$	0	0
$559$	−24.0000	−1.01509
$560$	8.00000	0.338062
$561$	0	0
$562$	−22.0000	−0.928014
$563$	20.0000	0.842900	0.421450	−	0.906852i	$-0.361521\pi$
$563$	20.0000	0.842900	0.421450	+	0.906852i	$0.361521\pi$
$564$	0	0
$565$	4.00000	0.168281
$566$	−4.00000	−0.168133
$567$	0	0
$568$	−12.0000	−0.503509
$569$	−10.0000	−0.419222	−0.209611	−	0.977785i	$-0.567220\pi$
$569$	−10.0000	−0.419222	−0.209611	+	0.977785i	$0.567220\pi$
$570$	0	0
$571$	−20.0000	−0.836974	−0.418487	−	0.908223i	$-0.637439\pi$
$571$	−20.0000	−0.836974	−0.418487	+	0.908223i	$0.637439\pi$
$572$	−6.00000	−0.250873
$573$	0	0
$574$	24.0000	1.00174
$575$	4.00000	0.166812
$576$	0	0
$577$	−46.0000	−1.91501	−0.957503	−	0.288425i	$-0.906868\pi$
$577$	−46.0000	−1.91501	−0.957503	+	0.288425i	$0.906868\pi$
$578$	13.0000	0.540729
$579$	0	0
$580$	12.0000	0.498273
$581$	16.0000	0.663792
$582$	0	0
$583$	−2.00000	−0.0828315
$584$	6.00000	0.248282
$585$	0	0
$586$	22.0000	0.908812
$587$	36.0000	1.48588	0.742940	−	0.669359i	$-0.233431\pi$
$587$	36.0000	1.48588	0.742940	+	0.669359i	$0.233431\pi$
$588$	0	0
$589$	0	0
$590$	−24.0000	−0.988064
$591$	0	0
$592$	6.00000	0.246598
$593$	30.0000	1.23195	0.615976	−	0.787765i	$-0.288762\pi$
$593$	30.0000	1.23195	0.615976	+	0.787765i	$0.288762\pi$
$594$	0	0
$595$	−16.0000	−0.655936
$596$	10.0000	0.409616
$597$	0	0
$598$	−24.0000	−0.981433
$599$	−36.0000	−1.47092	−0.735460	−	0.677568i	$-0.763034\pi$
$599$	−36.0000	−1.47092	−0.735460	+	0.677568i	$0.763034\pi$
$600$	0	0
$601$	10.0000	0.407909	0.203954	−	0.978980i	$-0.434621\pi$
$601$	10.0000	0.407909	0.203954	+	0.978980i	$0.434621\pi$
$602$	16.0000	0.652111
$603$	0	0
$604$	4.00000	0.162758
$605$	−2.00000	−0.0813116
$606$	0	0
$607$	28.0000	1.13648	0.568242	−	0.822861i	$-0.307624\pi$
$607$	28.0000	1.13648	0.568242	+	0.822861i	$0.307624\pi$
$608$	−4.00000	−0.162221
$609$	0	0
$610$	−28.0000	−1.13369
$611$	−72.0000	−2.91281
$612$	0	0
$613$	−6.00000	−0.242338	−0.121169	−	0.992632i	$-0.538664\pi$
$613$	−6.00000	−0.242338	−0.121169	+	0.992632i	$0.538664\pi$
$614$	−4.00000	−0.161427
$615$	0	0
$616$	4.00000	0.161165
$617$	22.0000	0.885687	0.442843	−	0.896599i	$-0.353970\pi$
$617$	22.0000	0.885687	0.442843	+	0.896599i	$0.353970\pi$
$618$	0	0
$619$	−4.00000	−0.160774	−0.0803868	−	0.996764i	$-0.525616\pi$
$619$	−4.00000	−0.160774	−0.0803868	+	0.996764i	$0.525616\pi$
$620$	0	0
$621$	0	0
$622$	4.00000	0.160385
$623$	40.0000	1.60257
$624$	0	0
$625$	−19.0000	−0.760000
$626$	−26.0000	−1.03917
$627$	0	0
$628$	−10.0000	−0.399043
$629$	−12.0000	−0.478471
$630$	0	0
$631$	−8.00000	−0.318475	−0.159237	−	0.987240i	$-0.550904\pi$
$631$	−8.00000	−0.318475	−0.159237	+	0.987240i	$0.550904\pi$
$632$	4.00000	0.159111
$633$	0	0
$634$	18.0000	0.714871
$635$	−24.0000	−0.952411
$636$	0	0
$637$	−54.0000	−2.13956
$638$	6.00000	0.237542
$639$	0	0
$640$	2.00000	0.0790569
$641$	−42.0000	−1.65890	−0.829450	−	0.558581i	$-0.811346\pi$
$641$	−42.0000	−1.65890	−0.829450	+	0.558581i	$0.811346\pi$
$642$	0	0
$643$	−28.0000	−1.10421	−0.552106	−	0.833774i	$-0.686176\pi$
$643$	−28.0000	−1.10421	−0.552106	+	0.833774i	$0.686176\pi$
$644$	16.0000	0.630488
$645$	0	0
$646$	8.00000	0.314756
$647$	28.0000	1.10079	0.550397	−	0.834903i	$-0.314476\pi$
$647$	28.0000	1.10079	0.550397	+	0.834903i	$0.314476\pi$
$648$	0	0
$649$	−12.0000	−0.471041
$650$	−6.00000	−0.235339
$651$	0	0
$652$	−20.0000	−0.783260
$653$	−18.0000	−0.704394	−0.352197	−	0.935926i	$-0.614565\pi$
$653$	−18.0000	−0.704394	−0.352197	+	0.935926i	$0.614565\pi$
$654$	0	0
$655$	8.00000	0.312586
$656$	6.00000	0.234261
$657$	0	0
$658$	48.0000	1.87123
$659$	36.0000	1.40236	0.701180	−	0.712984i	$-0.252657\pi$
$659$	36.0000	1.40236	0.701180	+	0.712984i	$0.252657\pi$
$660$	0	0
$661$	−18.0000	−0.700119	−0.350059	−	0.936727i	$-0.613839\pi$
$661$	−18.0000	−0.700119	−0.350059	+	0.936727i	$0.613839\pi$
$662$	−20.0000	−0.777322
$663$	0	0
$664$	4.00000	0.155230
$665$	32.0000	1.24091
$666$	0	0
$667$	24.0000	0.929284
$668$	0	0
$669$	0	0
$670$	8.00000	0.309067
$671$	−14.0000	−0.540464
$672$	0	0
$673$	26.0000	1.00223	0.501113	−	0.865382i	$-0.332924\pi$
$673$	26.0000	1.00223	0.501113	+	0.865382i	$0.332924\pi$
$674$	−18.0000	−0.693334
$675$	0	0
$676$	23.0000	0.884615
$677$	−46.0000	−1.76792	−0.883962	−	0.467559i	$-0.845134\pi$
$677$	−46.0000	−1.76792	−0.883962	+	0.467559i	$0.845134\pi$
$678$	0	0
$679$	56.0000	2.14908
$680$	−4.00000	−0.153393
$681$	0	0
$682$	0	0
$683$	−12.0000	−0.459167	−0.229584	−	0.973289i	$-0.573736\pi$
$683$	−12.0000	−0.459167	−0.229584	+	0.973289i	$0.573736\pi$
$684$	0	0
$685$	4.00000	0.152832
$686$	8.00000	0.305441
$687$	0	0
$688$	4.00000	0.152499
$689$	12.0000	0.457164
$690$	0	0
$691$	−12.0000	−0.456502	−0.228251	−	0.973602i	$-0.573301\pi$
$691$	−12.0000	−0.456502	−0.228251	+	0.973602i	$0.573301\pi$
$692$	10.0000	0.380143
$693$	0	0
$694$	−4.00000	−0.151838
$695$	8.00000	0.303457
$696$	0	0
$697$	−12.0000	−0.454532
$698$	6.00000	0.227103
$699$	0	0
$700$	4.00000	0.151186
$701$	−30.0000	−1.13308	−0.566542	−	0.824033i	$-0.691719\pi$
$701$	−30.0000	−1.13308	−0.566542	+	0.824033i	$0.691719\pi$
$702$	0	0
$703$	24.0000	0.905177
$704$	1.00000	0.0376889
$705$	0	0
$706$	18.0000	0.677439
$707$	56.0000	2.10610
$708$	0	0
$709$	−18.0000	−0.676004	−0.338002	−	0.941145i	$-0.609751\pi$
$709$	−18.0000	−0.676004	−0.338002	+	0.941145i	$0.609751\pi$
$710$	24.0000	0.900704
$711$	0	0
$712$	10.0000	0.374766
$713$	0	0
$714$	0	0
$715$	12.0000	0.448775
$716$	−20.0000	−0.747435
$717$	0	0
$718$	0	0
$719$	−12.0000	−0.447524	−0.223762	−	0.974644i	$-0.571834\pi$
$719$	−12.0000	−0.447524	−0.223762	+	0.974644i	$0.571834\pi$
$720$	0	0
$721$	0	0
$722$	3.00000	0.111648
$723$	0	0
$724$	−2.00000	−0.0743294
$725$	6.00000	0.222834
$726$	0	0
$727$	0	0		−	1.00000i	$-0.5\pi$
$727$	0	0			1.00000i	$0.5\pi$
$728$	−24.0000	−0.889499
$729$	0	0
$730$	−12.0000	−0.444140
$731$	−8.00000	−0.295891
$732$	0	0
$733$	−22.0000	−0.812589	−0.406294	−	0.913742i	$-0.633179\pi$
$733$	−22.0000	−0.812589	−0.406294	+	0.913742i	$0.633179\pi$
$734$	16.0000	0.590571
$735$	0	0
$736$	4.00000	0.147442
$737$	4.00000	0.147342
$738$	0	0
$739$	44.0000	1.61857	0.809283	−	0.587419i	$-0.199856\pi$
$739$	44.0000	1.61857	0.809283	+	0.587419i	$0.199856\pi$
$740$	−12.0000	−0.441129
$741$	0	0
$742$	−8.00000	−0.293689
$743$	32.0000	1.17397	0.586983	−	0.809599i	$-0.300316\pi$
$743$	32.0000	1.17397	0.586983	+	0.809599i	$0.300316\pi$
$744$	0	0
$745$	−20.0000	−0.732743
$746$	14.0000	0.512576
$747$	0	0
$748$	−2.00000	−0.0731272
$749$	16.0000	0.584627
$750$	0	0
$751$	32.0000	1.16770	0.583848	−	0.811863i	$-0.301546\pi$
$751$	32.0000	1.16770	0.583848	+	0.811863i	$0.301546\pi$
$752$	12.0000	0.437595
$753$	0	0
$754$	−36.0000	−1.31104
$755$	−8.00000	−0.291150
$756$	0	0
$757$	38.0000	1.38113	0.690567	−	0.723269i	$-0.257361\pi$
$757$	38.0000	1.38113	0.690567	+	0.723269i	$0.257361\pi$
$758$	28.0000	1.01701
$759$	0	0
$760$	8.00000	0.290191
$761$	−42.0000	−1.52250	−0.761249	−	0.648459i	$-0.775414\pi$
$761$	−42.0000	−1.52250	−0.761249	+	0.648459i	$0.775414\pi$
$762$	0	0
$763$	24.0000	0.868858
$764$	12.0000	0.434145
$765$	0	0
$766$	−4.00000	−0.144526
$767$	72.0000	2.59977
$768$	0	0
$769$	10.0000	0.360609	0.180305	−	0.983611i	$-0.442292\pi$
$769$	10.0000	0.360609	0.180305	+	0.983611i	$0.442292\pi$
$770$	−8.00000	−0.288300
$771$	0	0
$772$	10.0000	0.359908
$773$	−18.0000	−0.647415	−0.323708	−	0.946157i	$-0.604929\pi$
$773$	−18.0000	−0.647415	−0.323708	+	0.946157i	$0.604929\pi$
$774$	0	0
$775$	0	0
$776$	14.0000	0.502571
$777$	0	0
$778$	−30.0000	−1.07555
$779$	24.0000	0.859889
$780$	0	0
$781$	12.0000	0.429394
$782$	−8.00000	−0.286079
$783$	0	0
$784$	9.00000	0.321429
$785$	20.0000	0.713831
$786$	0	0
$787$	−20.0000	−0.712923	−0.356462	−	0.934310i	$-0.616017\pi$
$787$	−20.0000	−0.712923	−0.356462	+	0.934310i	$0.616017\pi$
$788$	2.00000	0.0712470
$789$	0	0
$790$	−8.00000	−0.284627
$791$	8.00000	0.284447
$792$	0	0
$793$	84.0000	2.98293
$794$	−22.0000	−0.780751
$795$	0	0
$796$	−16.0000	−0.567105
$797$	54.0000	1.91278	0.956389	−	0.292096i	$-0.0943526\pi$
$797$	54.0000	1.91278	0.956389	+	0.292096i	$0.0943526\pi$
$798$	0	0
$799$	−24.0000	−0.849059
$800$	1.00000	0.0353553
$801$	0	0
$802$	−38.0000	−1.34183
$803$	−6.00000	−0.211735
$804$	0	0
$805$	−32.0000	−1.12785
$806$	0	0
$807$	0	0
$808$	14.0000	0.492518
$809$	−10.0000	−0.351581	−0.175791	−	0.984428i	$-0.556248\pi$
$809$	−10.0000	−0.351581	−0.175791	+	0.984428i	$0.556248\pi$
$810$	0	0
$811$	−28.0000	−0.983213	−0.491606	−	0.870817i	$-0.663590\pi$
$811$	−28.0000	−0.983213	−0.491606	+	0.870817i	$0.663590\pi$
$812$	24.0000	0.842235
$813$	0	0
$814$	−6.00000	−0.210300
$815$	40.0000	1.40114
$816$	0	0
$817$	16.0000	0.559769
$818$	14.0000	0.489499
$819$	0	0
$820$	−12.0000	−0.419058
$821$	2.00000	0.0698005	0.0349002	−	0.999391i	$-0.488889\pi$
$821$	2.00000	0.0698005	0.0349002	+	0.999391i	$0.488889\pi$
$822$	0	0
$823$	−40.0000	−1.39431	−0.697156	−	0.716919i	$-0.745552\pi$
$823$	−40.0000	−1.39431	−0.697156	+	0.716919i	$0.745552\pi$
$824$	0	0
$825$	0	0
$826$	−48.0000	−1.67013
$827$	52.0000	1.80822	0.904109	−	0.427303i	$-0.140536\pi$
$827$	52.0000	1.80822	0.904109	+	0.427303i	$0.140536\pi$
$828$	0	0
$829$	46.0000	1.59765	0.798823	−	0.601566i	$-0.205456\pi$
$829$	46.0000	1.59765	0.798823	+	0.601566i	$0.205456\pi$
$830$	−8.00000	−0.277684
$831$	0	0
$832$	−6.00000	−0.208013
$833$	−18.0000	−0.623663
$834$	0	0
$835$	0	0
$836$	4.00000	0.138343
$837$	0	0
$838$	−12.0000	−0.414533
$839$	−12.0000	−0.414286	−0.207143	−	0.978311i	$-0.566417\pi$
$839$	−12.0000	−0.414286	−0.207143	+	0.978311i	$0.566417\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	10.0000	0.344623
$843$	0	0
$844$	−4.00000	−0.137686
$845$	−46.0000	−1.58245
$846$	0	0
$847$	−4.00000	−0.137442
$848$	−2.00000	−0.0686803
$849$	0	0
$850$	−2.00000	−0.0685994
$851$	−24.0000	−0.822709
$852$	0	0
$853$	−6.00000	−0.205436	−0.102718	−	0.994711i	$-0.532754\pi$
$853$	−6.00000	−0.205436	−0.102718	+	0.994711i	$0.532754\pi$
$854$	−56.0000	−1.91628
$855$	0	0
$856$	4.00000	0.136717
$857$	−26.0000	−0.888143	−0.444072	−	0.895991i	$-0.646466\pi$
$857$	−26.0000	−0.888143	−0.444072	+	0.895991i	$0.646466\pi$
$858$	0	0
$859$	4.00000	0.136478	0.0682391	−	0.997669i	$-0.478262\pi$
$859$	4.00000	0.136478	0.0682391	+	0.997669i	$0.478262\pi$
$860$	−8.00000	−0.272798
$861$	0	0
$862$	0	0
$863$	−20.0000	−0.680808	−0.340404	−	0.940279i	$-0.610564\pi$
$863$	−20.0000	−0.680808	−0.340404	+	0.940279i	$0.610564\pi$
$864$	0	0
$865$	−20.0000	−0.680020
$866$	−2.00000	−0.0679628
$867$	0	0
$868$	0	0
$869$	−4.00000	−0.135691
$870$	0	0
$871$	−24.0000	−0.813209
$872$	6.00000	0.203186
$873$	0	0
$874$	16.0000	0.541208
$875$	−48.0000	−1.62270
$876$	0	0
$877$	42.0000	1.41824	0.709120	−	0.705088i	$-0.249093\pi$
$877$	42.0000	1.41824	0.709120	+	0.705088i	$0.249093\pi$
$878$	−4.00000	−0.134993
$879$	0	0
$880$	−2.00000	−0.0674200
$881$	14.0000	0.471672	0.235836	−	0.971793i	$-0.424217\pi$
$881$	14.0000	0.471672	0.235836	+	0.971793i	$0.424217\pi$
$882$	0	0
$883$	4.00000	0.134611	0.0673054	−	0.997732i	$-0.478560\pi$
$883$	4.00000	0.134611	0.0673054	+	0.997732i	$0.478560\pi$
$884$	12.0000	0.403604
$885$	0	0
$886$	28.0000	0.940678
$887$	0	0		−	1.00000i	$-0.5\pi$
$887$	0	0			1.00000i	$0.5\pi$
$888$	0	0
$889$	−48.0000	−1.60987
$890$	−20.0000	−0.670402
$891$	0	0
$892$	0	0
$893$	48.0000	1.60626
$894$	0	0
$895$	40.0000	1.33705
$896$	4.00000	0.133631
$897$	0	0
$898$	−22.0000	−0.734150
$899$	0	0
$900$	0	0
$901$	4.00000	0.133259
$902$	−6.00000	−0.199778
$903$	0	0
$904$	2.00000	0.0665190
$905$	4.00000	0.132964
$906$	0	0
$907$	28.0000	0.929725	0.464862	−	0.885383i	$-0.346104\pi$
$907$	28.0000	0.929725	0.464862	+	0.885383i	$0.346104\pi$
$908$	−12.0000	−0.398234
$909$	0	0
$910$	48.0000	1.59118
$911$	−12.0000	−0.397578	−0.198789	−	0.980042i	$-0.563701\pi$
$911$	−12.0000	−0.397578	−0.198789	+	0.980042i	$0.563701\pi$
$912$	0	0
$913$	−4.00000	−0.132381
$914$	−34.0000	−1.12462
$915$	0	0
$916$	14.0000	0.462573
$917$	16.0000	0.528367
$918$	0	0
$919$	−4.00000	−0.131948	−0.0659739	−	0.997821i	$-0.521015\pi$
$919$	−4.00000	−0.131948	−0.0659739	+	0.997821i	$0.521015\pi$
$920$	−8.00000	−0.263752
$921$	0	0
$922$	14.0000	0.461065
$923$	−72.0000	−2.36991
$924$	0	0
$925$	−6.00000	−0.197279
$926$	24.0000	0.788689
$927$	0	0
$928$	6.00000	0.196960
$929$	−42.0000	−1.37798	−0.688988	−	0.724773i	$-0.741945\pi$
$929$	−42.0000	−1.37798	−0.688988	+	0.724773i	$0.741945\pi$
$930$	0	0
$931$	36.0000	1.17985
$932$	−10.0000	−0.327561
$933$	0	0
$934$	12.0000	0.392652
$935$	4.00000	0.130814
$936$	0	0
$937$	−38.0000	−1.24141	−0.620703	−	0.784046i	$-0.713153\pi$
$937$	−38.0000	−1.24141	−0.620703	+	0.784046i	$0.713153\pi$
$938$	16.0000	0.522419
$939$	0	0
$940$	−24.0000	−0.782794
$941$	42.0000	1.36916	0.684580	−	0.728937i	$-0.259985\pi$
$941$	42.0000	1.36916	0.684580	+	0.728937i	$0.259985\pi$
$942$	0	0
$943$	−24.0000	−0.781548
$944$	−12.0000	−0.390567
$945$	0	0
$946$	−4.00000	−0.130051
$947$	−12.0000	−0.389948	−0.194974	−	0.980808i	$-0.562462\pi$
$947$	−12.0000	−0.389948	−0.194974	+	0.980808i	$0.562462\pi$
$948$	0	0
$949$	36.0000	1.16861
$950$	4.00000	0.129777
$951$	0	0
$952$	−8.00000	−0.259281
$953$	6.00000	0.194359	0.0971795	−	0.995267i	$-0.469018\pi$
$953$	6.00000	0.194359	0.0971795	+	0.995267i	$0.469018\pi$
$954$	0	0
$955$	−24.0000	−0.776622
$956$	−8.00000	−0.258738
$957$	0	0
$958$	0	0
$959$	8.00000	0.258333
$960$	0	0
$961$	−31.0000	−1.00000
$962$	36.0000	1.16069
$963$	0	0
$964$	10.0000	0.322078
$965$	−20.0000	−0.643823
$966$	0	0
$967$	−44.0000	−1.41494	−0.707472	−	0.706741i	$-0.750165\pi$
$967$	−44.0000	−1.41494	−0.707472	+	0.706741i	$0.750165\pi$
$968$	−1.00000	−0.0321412
$969$	0	0
$970$	−28.0000	−0.899026
$971$	−12.0000	−0.385098	−0.192549	−	0.981287i	$-0.561675\pi$
$971$	−12.0000	−0.385098	−0.192549	+	0.981287i	$0.561675\pi$
$972$	0	0
$973$	16.0000	0.512936
$974$	16.0000	0.512673
$975$	0	0
$976$	−14.0000	−0.448129
$977$	−26.0000	−0.831814	−0.415907	−	0.909407i	$-0.636536\pi$
$977$	−26.0000	−0.831814	−0.415907	+	0.909407i	$0.636536\pi$
$978$	0	0
$979$	−10.0000	−0.319601
$980$	−18.0000	−0.574989
$981$	0	0
$982$	−28.0000	−0.893516
$983$	−36.0000	−1.14822	−0.574111	−	0.818778i	$-0.694652\pi$
$983$	−36.0000	−1.14822	−0.574111	+	0.818778i	$0.694652\pi$
$984$	0	0
$985$	−4.00000	−0.127451
$986$	−12.0000	−0.382158
$987$	0	0
$988$	−24.0000	−0.763542
$989$	−16.0000	−0.508770
$990$	0	0
$991$	32.0000	1.01651	0.508257	−	0.861206i	$-0.330290\pi$
$991$	32.0000	1.01651	0.508257	+	0.861206i	$0.330290\pi$
$992$	0	0
$993$	0	0
$994$	48.0000	1.52247
$995$	32.0000	1.01447
$996$	0	0
$997$	−14.0000	−0.443384	−0.221692	−	0.975117i	$-0.571158\pi$
$997$	−14.0000	−0.443384	−0.221692	+	0.975117i	$0.571158\pi$
$998$	4.00000	0.126618
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	198.2.a.a.1.1		1
3.2	odd	2		66.2.a.b.1.1	✓	1
4.3	odd	2		1584.2.a.f.1.1		1
5.2	odd	4		4950.2.c.p.199.1		2
5.3	odd	4		4950.2.c.p.199.2		2
5.4	even	2		4950.2.a.bu.1.1		1
7.6	odd	2		9702.2.a.x.1.1		1
8.3	odd	2		6336.2.a.cj.1.1		1
8.5	even	2		6336.2.a.bw.1.1		1
9.2	odd	6		1782.2.e.e.595.1		2
9.4	even	3		1782.2.e.v.1189.1		2
9.5	odd	6		1782.2.e.e.1189.1		2
9.7	even	3		1782.2.e.v.595.1		2
11.10	odd	2		2178.2.a.g.1.1		1
12.11	even	2		528.2.a.j.1.1		1
15.2	even	4		1650.2.c.e.199.2		2
15.8	even	4		1650.2.c.e.199.1		2
15.14	odd	2		1650.2.a.k.1.1		1
21.20	even	2		3234.2.a.t.1.1		1
24.5	odd	2		2112.2.a.r.1.1		1
24.11	even	2		2112.2.a.e.1.1		1
33.2	even	10		726.2.e.o.565.1		4
33.5	odd	10		726.2.e.g.487.1		4
33.8	even	10		726.2.e.o.493.1		4
33.14	odd	10		726.2.e.g.493.1		4
33.17	even	10		726.2.e.o.487.1		4
33.20	odd	10		726.2.e.g.565.1		4
33.26	odd	10		726.2.e.g.511.1		4
33.29	even	10		726.2.e.o.511.1		4
33.32	even	2		726.2.a.c.1.1		1
132.131	odd	2		5808.2.a.bc.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
66.2.a.b.1.1	✓	1	3.2	odd	2
198.2.a.a.1.1		1	1.1	even	1	trivial
528.2.a.j.1.1		1	12.11	even	2
726.2.a.c.1.1		1	33.32	even	2
726.2.e.g.487.1		4	33.5	odd	10
726.2.e.g.493.1		4	33.14	odd	10
726.2.e.g.511.1		4	33.26	odd	10
726.2.e.g.565.1		4	33.20	odd	10
726.2.e.o.487.1		4	33.17	even	10
726.2.e.o.493.1		4	33.8	even	10
726.2.e.o.511.1		4	33.29	even	10
726.2.e.o.565.1		4	33.2	even	10
1584.2.a.f.1.1		1	4.3	odd	2
1650.2.a.k.1.1		1	15.14	odd	2
1650.2.c.e.199.1		2	15.8	even	4
1650.2.c.e.199.2		2	15.2	even	4
1782.2.e.e.595.1		2	9.2	odd	6
1782.2.e.e.1189.1		2	9.5	odd	6
1782.2.e.v.595.1		2	9.7	even	3
1782.2.e.v.1189.1		2	9.4	even	3
2112.2.a.e.1.1		1	24.11	even	2
2112.2.a.r.1.1		1	24.5	odd	2
2178.2.a.g.1.1		1	11.10	odd	2
3234.2.a.t.1.1		1	21.20	even	2
4950.2.a.bu.1.1		1	5.4	even	2
4950.2.c.p.199.1		2	5.2	odd	4
4950.2.c.p.199.2		2	5.3	odd	4
5808.2.a.bc.1.1		1	132.131	odd	2
6336.2.a.bw.1.1		1	8.5	even	2
6336.2.a.cj.1.1		1	8.3	odd	2
9702.2.a.x.1.1		1	7.6	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 198 = 2 \cdot 3^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	198.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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