LMFDB - Embedded newform 189.2.a.d.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [189,2,Mod(1,189)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(189, 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("189.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 189 = 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	189.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:	$1.50917259820$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	189.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

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

$q+2.00000 q^{2} +2.00000 q^{4} +1.00000 q^{5} -1.00000 q^{7} +2.00000 q^{10} +4.00000 q^{11} -2.00000 q^{13} -2.00000 q^{14} -4.00000 q^{16} -3.00000 q^{17} -8.00000 q^{19} +2.00000 q^{20} +8.00000 q^{22} +6.00000 q^{23} -4.00000 q^{25} -4.00000 q^{26} -2.00000 q^{28} +4.00000 q^{29} +6.00000 q^{31} -8.00000 q^{32} -6.00000 q^{34} -1.00000 q^{35} -3.00000 q^{37} -16.0000 q^{38} -1.00000 q^{41} +11.0000 q^{43} +8.00000 q^{44} +12.0000 q^{46} -9.00000 q^{47} +1.00000 q^{49} -8.00000 q^{50} -4.00000 q^{52} -6.00000 q^{53} +4.00000 q^{55} +8.00000 q^{58} +15.0000 q^{59} +4.00000 q^{61} +12.0000 q^{62} -8.00000 q^{64} -2.00000 q^{65} -8.00000 q^{67} -6.00000 q^{68} -2.00000 q^{70} +12.0000 q^{71} +6.00000 q^{73} -6.00000 q^{74} -16.0000 q^{76} -4.00000 q^{77} -1.00000 q^{79} -4.00000 q^{80} -2.00000 q^{82} +9.00000 q^{83} -3.00000 q^{85} +22.0000 q^{86} -2.00000 q^{89} +2.00000 q^{91} +12.0000 q^{92} -18.0000 q^{94} -8.00000 q^{95} +12.0000 q^{97} +2.00000 q^{98} +O(q^{100})$

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 $

$2$	2.00000	1.41421	0.707107	−	0.707107i	$-0.250000\pi$
$2$	2.00000	1.41421	0.707107	+	0.707107i	$0.250000\pi$
$3$	0	0
$3$	0	0
$4$	2.00000	1.00000
$5$	1.00000	0.447214	0.223607	−	0.974679i	$-0.428217\pi$
$5$	1.00000	0.447214	0.223607	+	0.974679i	$0.428217\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	0	0
$10$	2.00000	0.632456
$11$	4.00000	1.20605	0.603023	−	0.797724i	$-0.293963\pi$
$11$	4.00000	1.20605	0.603023	+	0.797724i	$0.293963\pi$
$12$	0	0
$13$	−2.00000	−0.554700	−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−2.00000	−0.554700	−0.277350	+	0.960769i	$0.589456\pi$
$14$	−2.00000	−0.534522
$15$	0	0
$16$	−4.00000	−1.00000
$17$	−3.00000	−0.727607	−0.363803	−	0.931476i	$-0.618522\pi$
$17$	−3.00000	−0.727607	−0.363803	+	0.931476i	$0.618522\pi$
$18$	0	0
$19$	−8.00000	−1.83533	−0.917663	−	0.397360i	$-0.869927\pi$
$19$	−8.00000	−1.83533	−0.917663	+	0.397360i	$0.869927\pi$
$20$	2.00000	0.447214
$21$	0	0
$22$	8.00000	1.70561
$23$	6.00000	1.25109	0.625543	−	0.780189i	$-0.284877\pi$
$23$	6.00000	1.25109	0.625543	+	0.780189i	$0.284877\pi$
$24$	0	0
$25$	−4.00000	−0.800000
$26$	−4.00000	−0.784465
$27$	0	0
$28$	−2.00000	−0.377964
$29$	4.00000	0.742781	0.371391	−	0.928477i	$-0.378881\pi$
$29$	4.00000	0.742781	0.371391	+	0.928477i	$0.378881\pi$
$30$	0	0
$31$	6.00000	1.07763	0.538816	−	0.842424i	$-0.318872\pi$
$31$	6.00000	1.07763	0.538816	+	0.842424i	$0.318872\pi$
$32$	−8.00000	−1.41421
$33$	0	0
$34$	−6.00000	−1.02899
$35$	−1.00000	−0.169031
$36$	0	0
$37$	−3.00000	−0.493197	−0.246598	−	0.969118i	$-0.579313\pi$
$37$	−3.00000	−0.493197	−0.246598	+	0.969118i	$0.579313\pi$
$38$	−16.0000	−2.59554
$39$	0	0
$40$	0	0
$41$	−1.00000	−0.156174	−0.0780869	−	0.996947i	$-0.524881\pi$
$41$	−1.00000	−0.156174	−0.0780869	+	0.996947i	$0.524881\pi$
$42$	0	0
$43$	11.0000	1.67748	0.838742	−	0.544529i	$-0.183292\pi$
$43$	11.0000	1.67748	0.838742	+	0.544529i	$0.183292\pi$
$44$	8.00000	1.20605
$45$	0	0
$46$	12.0000	1.76930
$47$	−9.00000	−1.31278	−0.656392	−	0.754420i	$-0.727918\pi$
$47$	−9.00000	−1.31278	−0.656392	+	0.754420i	$0.727918\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−8.00000	−1.13137
$51$	0	0
$52$	−4.00000	−0.554700
$53$	−6.00000	−0.824163	−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000	−0.824163	−0.412082	+	0.911147i	$0.635198\pi$
$54$	0	0
$55$	4.00000	0.539360
$56$	0	0
$57$	0	0
$58$	8.00000	1.05045
$59$	15.0000	1.95283	0.976417	−	0.215894i	$-0.0692665\pi$
$59$	15.0000	1.95283	0.976417	+	0.215894i	$0.0692665\pi$
$60$	0	0
$61$	4.00000	0.512148	0.256074	−	0.966657i	$-0.417571\pi$
$61$	4.00000	0.512148	0.256074	+	0.966657i	$0.417571\pi$
$62$	12.0000	1.52400
$63$	0	0
$64$	−8.00000	−1.00000
$65$	−2.00000	−0.248069
$66$	0	0
$67$	−8.00000	−0.977356	−0.488678	−	0.872464i	$-0.662521\pi$
$67$	−8.00000	−0.977356	−0.488678	+	0.872464i	$0.662521\pi$
$68$	−6.00000	−0.727607
$69$	0	0
$70$	−2.00000	−0.239046
$71$	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	$-0.385800\pi$
$73$	6.00000	0.702247	0.351123	+	0.936329i	$0.385800\pi$
$74$	−6.00000	−0.697486
$75$	0	0
$76$	−16.0000	−1.83533
$77$	−4.00000	−0.455842
$78$	0	0
$79$	−1.00000	−0.112509	−0.0562544	−	0.998416i	$-0.517916\pi$
$79$	−1.00000	−0.112509	−0.0562544	+	0.998416i	$0.517916\pi$
$80$	−4.00000	−0.447214
$81$	0	0
$82$	−2.00000	−0.220863
$83$	9.00000	0.987878	0.493939	−	0.869496i	$-0.335557\pi$
$83$	9.00000	0.987878	0.493939	+	0.869496i	$0.335557\pi$
$84$	0	0
$85$	−3.00000	−0.325396
$86$	22.0000	2.37232
$87$	0	0
$88$	0	0
$89$	−2.00000	−0.212000	−0.106000	−	0.994366i	$-0.533804\pi$
$89$	−2.00000	−0.212000	−0.106000	+	0.994366i	$0.533804\pi$
$90$	0	0
$91$	2.00000	0.209657
$92$	12.0000	1.25109
$93$	0	0
$94$	−18.0000	−1.85656
$95$	−8.00000	−0.820783
$96$	0	0
$97$	12.0000	1.21842	0.609208	−	0.793011i	$-0.291488\pi$
$97$	12.0000	1.21842	0.609208	+	0.793011i	$0.291488\pi$
$98$	2.00000	0.202031
$99$	0	0
$100$	−8.00000	−0.800000
$101$	−10.0000	−0.995037	−0.497519	−	0.867453i	$-0.665755\pi$
$101$	−10.0000	−0.995037	−0.497519	+	0.867453i	$0.665755\pi$
$102$	0	0
$103$	2.00000	0.197066	0.0985329	−	0.995134i	$-0.468585\pi$
$103$	2.00000	0.197066	0.0985329	+	0.995134i	$0.468585\pi$
$104$	0	0
$105$	0	0
$106$	−12.0000	−1.16554
$107$	−2.00000	−0.193347	−0.0966736	−	0.995316i	$-0.530820\pi$
$107$	−2.00000	−0.193347	−0.0966736	+	0.995316i	$0.530820\pi$
$108$	0	0
$109$	−9.00000	−0.862044	−0.431022	−	0.902342i	$-0.641847\pi$
$109$	−9.00000	−0.862044	−0.431022	+	0.902342i	$0.641847\pi$
$110$	8.00000	0.762770
$111$	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
$115$	6.00000	0.559503
$116$	8.00000	0.742781
$117$	0	0
$118$	30.0000	2.76172
$119$	3.00000	0.275010
$120$	0	0
$121$	5.00000	0.454545
$122$	8.00000	0.724286
$123$	0	0
$124$	12.0000	1.07763
$125$	−9.00000	−0.804984
$126$	0	0
$127$	−15.0000	−1.33103	−0.665517	−	0.746382i	$-0.731789\pi$
$127$	−15.0000	−1.33103	−0.665517	+	0.746382i	$0.731789\pi$
$128$	0	0
$129$	0	0
$130$	−4.00000	−0.350823
$131$	4.00000	0.349482	0.174741	−	0.984614i	$-0.444091\pi$
$131$	4.00000	0.349482	0.174741	+	0.984614i	$0.444091\pi$
$132$	0	0
$133$	8.00000	0.693688
$134$	−16.0000	−1.38219
$135$	0	0
$136$	0	0
$137$	18.0000	1.53784	0.768922	−	0.639343i	$-0.220793\pi$
$137$	18.0000	1.53784	0.768922	+	0.639343i	$0.220793\pi$
$138$	0	0
$139$	0	0		−	1.00000i	$-0.5\pi$
$139$	0	0			1.00000i	$0.5\pi$
$140$	−2.00000	−0.169031
$141$	0	0
$142$	24.0000	2.01404
$143$	−8.00000	−0.668994
$144$	0	0
$145$	4.00000	0.332182
$146$	12.0000	0.993127
$147$	0	0
$148$	−6.00000	−0.493197
$149$	−12.0000	−0.983078	−0.491539	−	0.870855i	$-0.663566\pi$
$149$	−12.0000	−0.983078	−0.491539	+	0.870855i	$0.663566\pi$
$150$	0	0
$151$	5.00000	0.406894	0.203447	−	0.979086i	$-0.434786\pi$
$151$	5.00000	0.406894	0.203447	+	0.979086i	$0.434786\pi$
$152$	0	0
$153$	0	0
$154$	−8.00000	−0.644658
$155$	6.00000	0.481932
$156$	0	0
$157$	−8.00000	−0.638470	−0.319235	−	0.947676i	$-0.603426\pi$
$157$	−8.00000	−0.638470	−0.319235	+	0.947676i	$0.603426\pi$
$158$	−2.00000	−0.159111
$159$	0	0
$160$	−8.00000	−0.632456
$161$	−6.00000	−0.472866
$162$	0	0
$163$	−11.0000	−0.861586	−0.430793	−	0.902451i	$-0.641766\pi$
$163$	−11.0000	−0.861586	−0.430793	+	0.902451i	$0.641766\pi$
$164$	−2.00000	−0.156174
$165$	0	0
$166$	18.0000	1.39707
$167$	−17.0000	−1.31550	−0.657750	−	0.753237i	$-0.728492\pi$
$167$	−17.0000	−1.31550	−0.657750	+	0.753237i	$0.728492\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	−6.00000	−0.460179
$171$	0	0
$172$	22.0000	1.67748
$173$	−22.0000	−1.67263	−0.836315	−	0.548250i	$-0.815294\pi$
$173$	−22.0000	−1.67263	−0.836315	+	0.548250i	$0.815294\pi$
$174$	0	0
$175$	4.00000	0.302372
$176$	−16.0000	−1.20605
$177$	0	0
$178$	−4.00000	−0.299813
$179$	2.00000	0.149487	0.0747435	−	0.997203i	$-0.476186\pi$
$179$	2.00000	0.149487	0.0747435	+	0.997203i	$0.476186\pi$
$180$	0	0
$181$	16.0000	1.18927	0.594635	−	0.803996i	$-0.297296\pi$
$181$	16.0000	1.18927	0.594635	+	0.803996i	$0.297296\pi$
$182$	4.00000	0.296500
$183$	0	0
$184$	0	0
$185$	−3.00000	−0.220564
$186$	0	0
$187$	−12.0000	−0.877527
$188$	−18.0000	−1.31278
$189$	0	0
$190$	−16.0000	−1.16076
$191$	−2.00000	−0.144715	−0.0723575	−	0.997379i	$-0.523052\pi$
$191$	−2.00000	−0.144715	−0.0723575	+	0.997379i	$0.523052\pi$
$192$	0	0
$193$	−25.0000	−1.79954	−0.899770	−	0.436365i	$-0.856266\pi$
$193$	−25.0000	−1.79954	−0.899770	+	0.436365i	$0.856266\pi$
$194$	24.0000	1.72310
$195$	0	0
$196$	2.00000	0.142857
$197$	4.00000	0.284988	0.142494	−	0.989796i	$-0.454488\pi$
$197$	4.00000	0.284988	0.142494	+	0.989796i	$0.454488\pi$
$198$	0	0
$199$	−12.0000	−0.850657	−0.425329	−	0.905039i	$-0.639842\pi$
$199$	−12.0000	−0.850657	−0.425329	+	0.905039i	$0.639842\pi$
$200$	0	0
$201$	0	0
$202$	−20.0000	−1.40720
$203$	−4.00000	−0.280745
$204$	0	0
$205$	−1.00000	−0.0698430
$206$	4.00000	0.278693
$207$	0	0
$208$	8.00000	0.554700
$209$	−32.0000	−2.21349
$210$	0	0
$211$	4.00000	0.275371	0.137686	−	0.990476i	$-0.456034\pi$
$211$	4.00000	0.275371	0.137686	+	0.990476i	$0.456034\pi$
$212$	−12.0000	−0.824163
$213$	0	0
$214$	−4.00000	−0.273434
$215$	11.0000	0.750194
$216$	0	0
$217$	−6.00000	−0.407307
$218$	−18.0000	−1.21911
$219$	0	0
$220$	8.00000	0.539360
$221$	6.00000	0.403604
$222$	0	0
$223$	−2.00000	−0.133930	−0.0669650	−	0.997755i	$-0.521332\pi$
$223$	−2.00000	−0.133930	−0.0669650	+	0.997755i	$0.521332\pi$
$224$	8.00000	0.534522
$225$	0	0
$226$	−4.00000	−0.266076
$227$	24.0000	1.59294	0.796468	−	0.604681i	$-0.206699\pi$
$227$	24.0000	1.59294	0.796468	+	0.604681i	$0.206699\pi$
$228$	0	0
$229$	−4.00000	−0.264327	−0.132164	−	0.991228i	$-0.542192\pi$
$229$	−4.00000	−0.264327	−0.132164	+	0.991228i	$0.542192\pi$
$230$	12.0000	0.791257
$231$	0	0
$232$	0	0
$233$	−6.00000	−0.393073	−0.196537	−	0.980497i	$-0.562969\pi$
$233$	−6.00000	−0.393073	−0.196537	+	0.980497i	$0.562969\pi$
$234$	0	0
$235$	−9.00000	−0.587095
$236$	30.0000	1.95283
$237$	0	0
$238$	6.00000	0.388922
$239$	18.0000	1.16432	0.582162	−	0.813073i	$-0.302207\pi$
$239$	18.0000	1.16432	0.582162	+	0.813073i	$0.302207\pi$
$240$	0	0
$241$	−10.0000	−0.644157	−0.322078	−	0.946713i	$-0.604381\pi$
$241$	−10.0000	−0.644157	−0.322078	+	0.946713i	$0.604381\pi$
$242$	10.0000	0.642824
$243$	0	0
$244$	8.00000	0.512148
$245$	1.00000	0.0638877
$246$	0	0
$247$	16.0000	1.01806
$248$	0	0
$249$	0	0
$250$	−18.0000	−1.13842
$251$	25.0000	1.57799	0.788993	−	0.614402i	$-0.210603\pi$
$251$	25.0000	1.57799	0.788993	+	0.614402i	$0.210603\pi$
$252$	0	0
$253$	24.0000	1.50887
$254$	−30.0000	−1.88237
$255$	0	0
$256$	16.0000	1.00000
$257$	−10.0000	−0.623783	−0.311891	−	0.950118i	$-0.600963\pi$
$257$	−10.0000	−0.623783	−0.311891	+	0.950118i	$0.600963\pi$
$258$	0	0
$259$	3.00000	0.186411
$260$	−4.00000	−0.248069
$261$	0	0
$262$	8.00000	0.494242
$263$	−14.0000	−0.863277	−0.431638	−	0.902047i	$-0.642064\pi$
$263$	−14.0000	−0.863277	−0.431638	+	0.902047i	$0.642064\pi$
$264$	0	0
$265$	−6.00000	−0.368577
$266$	16.0000	0.981023
$267$	0	0
$268$	−16.0000	−0.977356
$269$	21.0000	1.28039	0.640196	−	0.768211i	$-0.278853\pi$
$269$	21.0000	1.28039	0.640196	+	0.768211i	$0.278853\pi$
$270$	0	0
$271$	−2.00000	−0.121491	−0.0607457	−	0.998153i	$-0.519348\pi$
$271$	−2.00000	−0.121491	−0.0607457	+	0.998153i	$0.519348\pi$
$272$	12.0000	0.727607
$273$	0	0
$274$	36.0000	2.17484
$275$	−16.0000	−0.964836
$276$	0	0
$277$	13.0000	0.781094	0.390547	−	0.920583i	$-0.372286\pi$
$277$	13.0000	0.781094	0.390547	+	0.920583i	$0.372286\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−10.0000	−0.596550	−0.298275	−	0.954480i	$-0.596411\pi$
$281$	−10.0000	−0.596550	−0.298275	+	0.954480i	$0.596411\pi$
$282$	0	0
$283$	28.0000	1.66443	0.832214	−	0.554455i	$-0.187073\pi$
$283$	28.0000	1.66443	0.832214	+	0.554455i	$0.187073\pi$
$284$	24.0000	1.42414
$285$	0	0
$286$	−16.0000	−0.946100
$287$	1.00000	0.0590281
$288$	0	0
$289$	−8.00000	−0.470588
$290$	8.00000	0.469776
$291$	0	0
$292$	12.0000	0.702247
$293$	−1.00000	−0.0584206	−0.0292103	−	0.999573i	$-0.509299\pi$
$293$	−1.00000	−0.0584206	−0.0292103	+	0.999573i	$0.509299\pi$
$294$	0	0
$295$	15.0000	0.873334
$296$	0	0
$297$	0	0
$298$	−24.0000	−1.39028
$299$	−12.0000	−0.693978
$300$	0	0
$301$	−11.0000	−0.634029
$302$	10.0000	0.575435
$303$	0	0
$304$	32.0000	1.83533
$305$	4.00000	0.229039
$306$	0	0
$307$	−2.00000	−0.114146	−0.0570730	−	0.998370i	$-0.518177\pi$
$307$	−2.00000	−0.114146	−0.0570730	+	0.998370i	$0.518177\pi$
$308$	−8.00000	−0.455842
$309$	0	0
$310$	12.0000	0.681554
$311$	3.00000	0.170114	0.0850572	−	0.996376i	$-0.472893\pi$
$311$	3.00000	0.170114	0.0850572	+	0.996376i	$0.472893\pi$
$312$	0	0
$313$	−16.0000	−0.904373	−0.452187	−	0.891923i	$-0.649356\pi$
$313$	−16.0000	−0.904373	−0.452187	+	0.891923i	$0.649356\pi$
$314$	−16.0000	−0.902932
$315$	0	0
$316$	−2.00000	−0.112509
$317$	18.0000	1.01098	0.505490	−	0.862832i	$-0.331312\pi$
$317$	18.0000	1.01098	0.505490	+	0.862832i	$0.331312\pi$
$318$	0	0
$319$	16.0000	0.895828
$320$	−8.00000	−0.447214
$321$	0	0
$322$	−12.0000	−0.668734
$323$	24.0000	1.33540
$324$	0	0
$325$	8.00000	0.443760
$326$	−22.0000	−1.21847
$327$	0	0
$328$	0	0
$329$	9.00000	0.496186
$330$	0	0
$331$	−1.00000	−0.0549650	−0.0274825	−	0.999622i	$-0.508749\pi$
$331$	−1.00000	−0.0549650	−0.0274825	+	0.999622i	$0.508749\pi$
$332$	18.0000	0.987878
$333$	0	0
$334$	−34.0000	−1.86040
$335$	−8.00000	−0.437087
$336$	0	0
$337$	7.00000	0.381314	0.190657	−	0.981657i	$-0.438938\pi$
$337$	7.00000	0.381314	0.190657	+	0.981657i	$0.438938\pi$
$338$	−18.0000	−0.979071
$339$	0	0
$340$	−6.00000	−0.325396
$341$	24.0000	1.29967
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	0	0
$346$	−44.0000	−2.36545
$347$	2.00000	0.107366	0.0536828	−	0.998558i	$-0.482904\pi$
$347$	2.00000	0.107366	0.0536828	+	0.998558i	$0.482904\pi$
$348$	0	0
$349$	−20.0000	−1.07058	−0.535288	−	0.844670i	$-0.679797\pi$
$349$	−20.0000	−1.07058	−0.535288	+	0.844670i	$0.679797\pi$
$350$	8.00000	0.427618
$351$	0	0
$352$	−32.0000	−1.70561
$353$	25.0000	1.33062	0.665308	−	0.746569i	$-0.268300\pi$
$353$	25.0000	1.33062	0.665308	+	0.746569i	$0.268300\pi$
$354$	0	0
$355$	12.0000	0.636894
$356$	−4.00000	−0.212000
$357$	0	0
$358$	4.00000	0.211407
$359$	−34.0000	−1.79445	−0.897226	−	0.441572i	$-0.854421\pi$
$359$	−34.0000	−1.79445	−0.897226	+	0.441572i	$0.854421\pi$
$360$	0	0
$361$	45.0000	2.36842
$362$	32.0000	1.68188
$363$	0	0
$364$	4.00000	0.209657
$365$	6.00000	0.314054
$366$	0	0
$367$	−30.0000	−1.56599	−0.782994	−	0.622030i	$-0.786308\pi$
$367$	−30.0000	−1.56599	−0.782994	+	0.622030i	$0.786308\pi$
$368$	−24.0000	−1.25109
$369$	0	0
$370$	−6.00000	−0.311925
$371$	6.00000	0.311504
$372$	0	0
$373$	−13.0000	−0.673114	−0.336557	−	0.941663i	$-0.609263\pi$
$373$	−13.0000	−0.673114	−0.336557	+	0.941663i	$0.609263\pi$
$374$	−24.0000	−1.24101
$375$	0	0
$376$	0	0
$377$	−8.00000	−0.412021
$378$	0	0
$379$	−3.00000	−0.154100	−0.0770498	−	0.997027i	$-0.524550\pi$
$379$	−3.00000	−0.154100	−0.0770498	+	0.997027i	$0.524550\pi$
$380$	−16.0000	−0.820783
$381$	0	0
$382$	−4.00000	−0.204658
$383$	−21.0000	−1.07305	−0.536525	−	0.843884i	$-0.680263\pi$
$383$	−21.0000	−1.07305	−0.536525	+	0.843884i	$0.680263\pi$
$384$	0	0
$385$	−4.00000	−0.203859
$386$	−50.0000	−2.54493
$387$	0	0
$388$	24.0000	1.21842
$389$	24.0000	1.21685	0.608424	−	0.793612i	$-0.291802\pi$
$389$	24.0000	1.21685	0.608424	+	0.793612i	$0.291802\pi$
$390$	0	0
$391$	−18.0000	−0.910299
$392$	0	0
$393$	0	0
$394$	8.00000	0.403034
$395$	−1.00000	−0.0503155
$396$	0	0
$397$	6.00000	0.301131	0.150566	−	0.988600i	$-0.451890\pi$
$397$	6.00000	0.301131	0.150566	+	0.988600i	$0.451890\pi$
$398$	−24.0000	−1.20301
$399$	0	0
$400$	16.0000	0.800000
$401$	0	0		−	1.00000i	$-0.5\pi$
$401$	0	0			1.00000i	$0.5\pi$
$402$	0	0
$403$	−12.0000	−0.597763
$404$	−20.0000	−0.995037
$405$	0	0
$406$	−8.00000	−0.397033
$407$	−12.0000	−0.594818
$408$	0	0
$409$	32.0000	1.58230	0.791149	−	0.611623i	$-0.209483\pi$
$409$	32.0000	1.58230	0.791149	+	0.611623i	$0.209483\pi$
$410$	−2.00000	−0.0987730
$411$	0	0
$412$	4.00000	0.197066
$413$	−15.0000	−0.738102
$414$	0	0
$415$	9.00000	0.441793
$416$	16.0000	0.784465
$417$	0	0
$418$	−64.0000	−3.13034
$419$	33.0000	1.61216	0.806078	−	0.591810i	$-0.201586\pi$
$419$	33.0000	1.61216	0.806078	+	0.591810i	$0.201586\pi$
$420$	0	0
$421$	14.0000	0.682318	0.341159	−	0.940006i	$-0.389181\pi$
$421$	14.0000	0.682318	0.341159	+	0.940006i	$0.389181\pi$
$422$	8.00000	0.389434
$423$	0	0
$424$	0	0
$425$	12.0000	0.582086
$426$	0	0
$427$	−4.00000	−0.193574
$428$	−4.00000	−0.193347
$429$	0	0
$430$	22.0000	1.06093
$431$	−24.0000	−1.15604	−0.578020	−	0.816023i	$-0.696174\pi$
$431$	−24.0000	−1.15604	−0.578020	+	0.816023i	$0.696174\pi$
$432$	0	0
$433$	−2.00000	−0.0961139	−0.0480569	−	0.998845i	$-0.515303\pi$
$433$	−2.00000	−0.0961139	−0.0480569	+	0.998845i	$0.515303\pi$
$434$	−12.0000	−0.576018
$435$	0	0
$436$	−18.0000	−0.862044
$437$	−48.0000	−2.29615
$438$	0	0
$439$	24.0000	1.14546	0.572729	−	0.819745i	$-0.305885\pi$
$439$	24.0000	1.14546	0.572729	+	0.819745i	$0.305885\pi$
$440$	0	0
$441$	0	0
$442$	12.0000	0.570782
$443$	24.0000	1.14027	0.570137	−	0.821549i	$-0.306890\pi$
$443$	24.0000	1.14027	0.570137	+	0.821549i	$0.306890\pi$
$444$	0	0
$445$	−2.00000	−0.0948091
$446$	−4.00000	−0.189405
$447$	0	0
$448$	8.00000	0.377964
$449$	6.00000	0.283158	0.141579	−	0.989927i	$-0.454782\pi$
$449$	6.00000	0.283158	0.141579	+	0.989927i	$0.454782\pi$
$450$	0	0
$451$	−4.00000	−0.188353
$452$	−4.00000	−0.188144
$453$	0	0
$454$	48.0000	2.25275
$455$	2.00000	0.0937614
$456$	0	0
$457$	−14.0000	−0.654892	−0.327446	−	0.944870i	$-0.606188\pi$
$457$	−14.0000	−0.654892	−0.327446	+	0.944870i	$0.606188\pi$
$458$	−8.00000	−0.373815
$459$	0	0
$460$	12.0000	0.559503
$461$	−1.00000	−0.0465746	−0.0232873	−	0.999729i	$-0.507413\pi$
$461$	−1.00000	−0.0465746	−0.0232873	+	0.999729i	$0.507413\pi$
$462$	0	0
$463$	−23.0000	−1.06890	−0.534450	−	0.845200i	$-0.679481\pi$
$463$	−23.0000	−1.06890	−0.534450	+	0.845200i	$0.679481\pi$
$464$	−16.0000	−0.742781
$465$	0	0
$466$	−12.0000	−0.555889
$467$	−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
$469$	8.00000	0.369406
$470$	−18.0000	−0.830278
$471$	0	0
$472$	0	0
$473$	44.0000	2.02312
$474$	0	0
$475$	32.0000	1.46826
$476$	6.00000	0.275010
$477$	0	0
$478$	36.0000	1.64660
$479$	17.0000	0.776750	0.388375	−	0.921501i	$-0.373037\pi$
$479$	17.0000	0.776750	0.388375	+	0.921501i	$0.373037\pi$
$480$	0	0
$481$	6.00000	0.273576
$482$	−20.0000	−0.910975
$483$	0	0
$484$	10.0000	0.454545
$485$	12.0000	0.544892
$486$	0	0
$487$	16.0000	0.725029	0.362515	−	0.931978i	$-0.381918\pi$
$487$	16.0000	0.725029	0.362515	+	0.931978i	$0.381918\pi$
$488$	0	0
$489$	0	0
$490$	2.00000	0.0903508
$491$	26.0000	1.17336	0.586682	−	0.809818i	$-0.300434\pi$
$491$	26.0000	1.17336	0.586682	+	0.809818i	$0.300434\pi$
$492$	0	0
$493$	−12.0000	−0.540453
$494$	32.0000	1.43975
$495$	0	0
$496$	−24.0000	−1.07763
$497$	−12.0000	−0.538274
$498$	0	0
$499$	13.0000	0.581960	0.290980	−	0.956729i	$-0.406019\pi$
$499$	13.0000	0.581960	0.290980	+	0.956729i	$0.406019\pi$
$500$	−18.0000	−0.804984
$501$	0	0
$502$	50.0000	2.23161
$503$	−21.0000	−0.936344	−0.468172	−	0.883637i	$-0.655087\pi$
$503$	−21.0000	−0.936344	−0.468172	+	0.883637i	$0.655087\pi$
$504$	0	0
$505$	−10.0000	−0.444994
$506$	48.0000	2.13386
$507$	0	0
$508$	−30.0000	−1.33103
$509$	41.0000	1.81729	0.908647	−	0.417566i	$-0.137117\pi$
$509$	41.0000	1.81729	0.908647	+	0.417566i	$0.137117\pi$
$510$	0	0
$511$	−6.00000	−0.265424
$512$	32.0000	1.41421
$513$	0	0
$514$	−20.0000	−0.882162
$515$	2.00000	0.0881305
$516$	0	0
$517$	−36.0000	−1.58328
$518$	6.00000	0.263625
$519$	0	0
$520$	0	0
$521$	−15.0000	−0.657162	−0.328581	−	0.944476i	$-0.606570\pi$
$521$	−15.0000	−0.657162	−0.328581	+	0.944476i	$0.606570\pi$
$522$	0	0
$523$	−26.0000	−1.13690	−0.568450	−	0.822718i	$-0.692457\pi$
$523$	−26.0000	−1.13690	−0.568450	+	0.822718i	$0.692457\pi$
$524$	8.00000	0.349482
$525$	0	0
$526$	−28.0000	−1.22086
$527$	−18.0000	−0.784092
$528$	0	0
$529$	13.0000	0.565217
$530$	−12.0000	−0.521247
$531$	0	0
$532$	16.0000	0.693688
$533$	2.00000	0.0866296
$534$	0	0
$535$	−2.00000	−0.0864675
$536$	0	0
$537$	0	0
$538$	42.0000	1.81075
$539$	4.00000	0.172292
$540$	0	0
$541$	29.0000	1.24681	0.623404	−	0.781900i	$-0.285749\pi$
$541$	29.0000	1.24681	0.623404	+	0.781900i	$0.285749\pi$
$542$	−4.00000	−0.171815
$543$	0	0
$544$	24.0000	1.02899
$545$	−9.00000	−0.385518
$546$	0	0
$547$	−29.0000	−1.23995	−0.619975	−	0.784621i	$-0.712857\pi$
$547$	−29.0000	−1.23995	−0.619975	+	0.784621i	$0.712857\pi$
$548$	36.0000	1.53784
$549$	0	0
$550$	−32.0000	−1.36448
$551$	−32.0000	−1.36325
$552$	0	0
$553$	1.00000	0.0425243
$554$	26.0000	1.10463
$555$	0	0
$556$	0	0
$557$	4.00000	0.169485	0.0847427	−	0.996403i	$-0.472993\pi$
$557$	4.00000	0.169485	0.0847427	+	0.996403i	$0.472993\pi$
$558$	0	0
$559$	−22.0000	−0.930501
$560$	4.00000	0.169031
$561$	0	0
$562$	−20.0000	−0.843649
$563$	4.00000	0.168580	0.0842900	−	0.996441i	$-0.473138\pi$
$563$	4.00000	0.168580	0.0842900	+	0.996441i	$0.473138\pi$
$564$	0	0
$565$	−2.00000	−0.0841406
$566$	56.0000	2.35386
$567$	0	0
$568$	0	0
$569$	−2.00000	−0.0838444	−0.0419222	−	0.999121i	$-0.513348\pi$
$569$	−2.00000	−0.0838444	−0.0419222	+	0.999121i	$0.513348\pi$
$570$	0	0
$571$	11.0000	0.460336	0.230168	−	0.973151i	$-0.426072\pi$
$571$	11.0000	0.460336	0.230168	+	0.973151i	$0.426072\pi$
$572$	−16.0000	−0.668994
$573$	0	0
$574$	2.00000	0.0834784
$575$	−24.0000	−1.00087
$576$	0	0
$577$	−20.0000	−0.832611	−0.416305	−	0.909225i	$-0.636675\pi$
$577$	−20.0000	−0.832611	−0.416305	+	0.909225i	$0.636675\pi$
$578$	−16.0000	−0.665512
$579$	0	0
$580$	8.00000	0.332182
$581$	−9.00000	−0.373383
$582$	0	0
$583$	−24.0000	−0.993978
$584$	0	0
$585$	0	0
$586$	−2.00000	−0.0826192
$587$	−20.0000	−0.825488	−0.412744	−	0.910847i	$-0.635430\pi$
$587$	−20.0000	−0.825488	−0.412744	+	0.910847i	$0.635430\pi$
$588$	0	0
$589$	−48.0000	−1.97781
$590$	30.0000	1.23508
$591$	0	0
$592$	12.0000	0.493197
$593$	33.0000	1.35515	0.677574	−	0.735455i	$-0.263031\pi$
$593$	33.0000	1.35515	0.677574	+	0.735455i	$0.263031\pi$
$594$	0	0
$595$	3.00000	0.122988
$596$	−24.0000	−0.983078
$597$	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
$601$	−18.0000	−0.734235	−0.367118	−	0.930175i	$-0.619655\pi$
$601$	−18.0000	−0.734235	−0.367118	+	0.930175i	$0.619655\pi$
$602$	−22.0000	−0.896653
$603$	0	0
$604$	10.0000	0.406894
$605$	5.00000	0.203279
$606$	0	0
$607$	14.0000	0.568242	0.284121	−	0.958788i	$-0.408298\pi$
$607$	14.0000	0.568242	0.284121	+	0.958788i	$0.408298\pi$
$608$	64.0000	2.59554
$609$	0	0
$610$	8.00000	0.323911
$611$	18.0000	0.728202
$612$	0	0
$613$	−26.0000	−1.05013	−0.525065	−	0.851062i	$-0.675959\pi$
$613$	−26.0000	−1.05013	−0.525065	+	0.851062i	$0.675959\pi$
$614$	−4.00000	−0.161427
$615$	0	0
$616$	0	0
$617$	6.00000	0.241551	0.120775	−	0.992680i	$-0.461462\pi$
$617$	6.00000	0.241551	0.120775	+	0.992680i	$0.461462\pi$
$618$	0	0
$619$	−44.0000	−1.76851	−0.884255	−	0.467005i	$-0.845333\pi$
$619$	−44.0000	−1.76851	−0.884255	+	0.467005i	$0.845333\pi$
$620$	12.0000	0.481932
$621$	0	0
$622$	6.00000	0.240578
$623$	2.00000	0.0801283
$624$	0	0
$625$	11.0000	0.440000
$626$	−32.0000	−1.27898
$627$	0	0
$628$	−16.0000	−0.638470
$629$	9.00000	0.358854
$630$	0	0
$631$	−19.0000	−0.756378	−0.378189	−	0.925728i	$-0.623453\pi$
$631$	−19.0000	−0.756378	−0.378189	+	0.925728i	$0.623453\pi$
$632$	0	0
$633$	0	0
$634$	36.0000	1.42974
$635$	−15.0000	−0.595257
$636$	0	0
$637$	−2.00000	−0.0792429
$638$	32.0000	1.26689
$639$	0	0
$640$	0	0
$641$	−12.0000	−0.473972	−0.236986	−	0.971513i	$-0.576159\pi$
$641$	−12.0000	−0.473972	−0.236986	+	0.971513i	$0.576159\pi$
$642$	0	0
$643$	26.0000	1.02534	0.512670	−	0.858586i	$-0.328656\pi$
$643$	26.0000	1.02534	0.512670	+	0.858586i	$0.328656\pi$
$644$	−12.0000	−0.472866
$645$	0	0
$646$	48.0000	1.88853
$647$	8.00000	0.314512	0.157256	−	0.987558i	$-0.449735\pi$
$647$	8.00000	0.314512	0.157256	+	0.987558i	$0.449735\pi$
$648$	0	0
$649$	60.0000	2.35521
$650$	16.0000	0.627572
$651$	0	0
$652$	−22.0000	−0.861586
$653$	42.0000	1.64359	0.821794	−	0.569785i	$-0.192974\pi$
$653$	42.0000	1.64359	0.821794	+	0.569785i	$0.192974\pi$
$654$	0	0
$655$	4.00000	0.156293
$656$	4.00000	0.156174
$657$	0	0
$658$	18.0000	0.701713
$659$	−12.0000	−0.467454	−0.233727	−	0.972302i	$-0.575092\pi$
$659$	−12.0000	−0.467454	−0.233727	+	0.972302i	$0.575092\pi$
$660$	0	0
$661$	−14.0000	−0.544537	−0.272268	−	0.962221i	$-0.587774\pi$
$661$	−14.0000	−0.544537	−0.272268	+	0.962221i	$0.587774\pi$
$662$	−2.00000	−0.0777322
$663$	0	0
$664$	0	0
$665$	8.00000	0.310227
$666$	0	0
$667$	24.0000	0.929284
$668$	−34.0000	−1.31550
$669$	0	0
$670$	−16.0000	−0.618134
$671$	16.0000	0.617673
$672$	0	0
$673$	−2.00000	−0.0770943	−0.0385472	−	0.999257i	$-0.512273\pi$
$673$	−2.00000	−0.0770943	−0.0385472	+	0.999257i	$0.512273\pi$
$674$	14.0000	0.539260
$675$	0	0
$676$	−18.0000	−0.692308
$677$	−30.0000	−1.15299	−0.576497	−	0.817099i	$-0.695581\pi$
$677$	−30.0000	−1.15299	−0.576497	+	0.817099i	$0.695581\pi$
$678$	0	0
$679$	−12.0000	−0.460518
$680$	0	0
$681$	0	0
$682$	48.0000	1.83801
$683$	−42.0000	−1.60709	−0.803543	−	0.595247i	$-0.797054\pi$
$683$	−42.0000	−1.60709	−0.803543	+	0.595247i	$0.797054\pi$
$684$	0	0
$685$	18.0000	0.687745
$686$	−2.00000	−0.0763604
$687$	0	0
$688$	−44.0000	−1.67748
$689$	12.0000	0.457164
$690$	0	0
$691$	50.0000	1.90209	0.951045	−	0.309053i	$-0.100012\pi$
$691$	50.0000	1.90209	0.951045	+	0.309053i	$0.100012\pi$
$692$	−44.0000	−1.67263
$693$	0	0
$694$	4.00000	0.151838
$695$	0	0
$696$	0	0
$697$	3.00000	0.113633
$698$	−40.0000	−1.51402
$699$	0	0
$700$	8.00000	0.302372
$701$	−18.0000	−0.679851	−0.339925	−	0.940452i	$-0.610402\pi$
$701$	−18.0000	−0.679851	−0.339925	+	0.940452i	$0.610402\pi$
$702$	0	0
$703$	24.0000	0.905177
$704$	−32.0000	−1.20605
$705$	0	0
$706$	50.0000	1.88177
$707$	10.0000	0.376089
$708$	0	0
$709$	15.0000	0.563337	0.281668	−	0.959512i	$-0.409112\pi$
$709$	15.0000	0.563337	0.281668	+	0.959512i	$0.409112\pi$
$710$	24.0000	0.900704
$711$	0	0
$712$	0	0
$713$	36.0000	1.34821
$714$	0	0
$715$	−8.00000	−0.299183
$716$	4.00000	0.149487
$717$	0	0
$718$	−68.0000	−2.53774
$719$	−33.0000	−1.23069	−0.615346	−	0.788257i	$-0.710984\pi$
$719$	−33.0000	−1.23069	−0.615346	+	0.788257i	$0.710984\pi$
$720$	0	0
$721$	−2.00000	−0.0744839
$722$	90.0000	3.34945
$723$	0	0
$724$	32.0000	1.18927
$725$	−16.0000	−0.594225
$726$	0	0
$727$	−4.00000	−0.148352	−0.0741759	−	0.997245i	$-0.523633\pi$
$727$	−4.00000	−0.148352	−0.0741759	+	0.997245i	$0.523633\pi$
$728$	0	0
$729$	0	0
$730$	12.0000	0.444140
$731$	−33.0000	−1.22055
$732$	0	0
$733$	0	0		−	1.00000i	$-0.5\pi$
$733$	0	0			1.00000i	$0.5\pi$
$734$	−60.0000	−2.21464
$735$	0	0
$736$	−48.0000	−1.76930
$737$	−32.0000	−1.17874
$738$	0	0
$739$	12.0000	0.441427	0.220714	−	0.975339i	$-0.429161\pi$
$739$	12.0000	0.441427	0.220714	+	0.975339i	$0.429161\pi$
$740$	−6.00000	−0.220564
$741$	0	0
$742$	12.0000	0.440534
$743$	−36.0000	−1.32071	−0.660356	−	0.750953i	$-0.729595\pi$
$743$	−36.0000	−1.32071	−0.660356	+	0.750953i	$0.729595\pi$
$744$	0	0
$745$	−12.0000	−0.439646
$746$	−26.0000	−0.951928
$747$	0	0
$748$	−24.0000	−0.877527
$749$	2.00000	0.0730784
$750$	0	0
$751$	16.0000	0.583848	0.291924	−	0.956441i	$-0.405705\pi$
$751$	16.0000	0.583848	0.291924	+	0.956441i	$0.405705\pi$
$752$	36.0000	1.31278
$753$	0	0
$754$	−16.0000	−0.582686
$755$	5.00000	0.181969
$756$	0	0
$757$	23.0000	0.835949	0.417975	−	0.908459i	$-0.362740\pi$
$757$	23.0000	0.835949	0.417975	+	0.908459i	$0.362740\pi$
$758$	−6.00000	−0.217930
$759$	0	0
$760$	0	0
$761$	3.00000	0.108750	0.0543750	−	0.998521i	$-0.482683\pi$
$761$	3.00000	0.108750	0.0543750	+	0.998521i	$0.482683\pi$
$762$	0	0
$763$	9.00000	0.325822
$764$	−4.00000	−0.144715
$765$	0	0
$766$	−42.0000	−1.51752
$767$	−30.0000	−1.08324
$768$	0	0
$769$	44.0000	1.58668	0.793340	−	0.608778i	$-0.208340\pi$
$769$	44.0000	1.58668	0.793340	+	0.608778i	$0.208340\pi$
$770$	−8.00000	−0.288300
$771$	0	0
$772$	−50.0000	−1.79954
$773$	−49.0000	−1.76241	−0.881204	−	0.472737i	$-0.843266\pi$
$773$	−49.0000	−1.76241	−0.881204	+	0.472737i	$0.843266\pi$
$774$	0	0
$775$	−24.0000	−0.862105
$776$	0	0
$777$	0	0
$778$	48.0000	1.72088
$779$	8.00000	0.286630
$780$	0	0
$781$	48.0000	1.71758
$782$	−36.0000	−1.28736
$783$	0	0
$784$	−4.00000	−0.142857
$785$	−8.00000	−0.285532
$786$	0	0
$787$	−2.00000	−0.0712923	−0.0356462	−	0.999364i	$-0.511349\pi$
$787$	−2.00000	−0.0712923	−0.0356462	+	0.999364i	$0.511349\pi$
$788$	8.00000	0.284988
$789$	0	0
$790$	−2.00000	−0.0711568
$791$	2.00000	0.0711118
$792$	0	0
$793$	−8.00000	−0.284088
$794$	12.0000	0.425864
$795$	0	0
$796$	−24.0000	−0.850657
$797$	−2.00000	−0.0708436	−0.0354218	−	0.999372i	$-0.511277\pi$
$797$	−2.00000	−0.0708436	−0.0354218	+	0.999372i	$0.511277\pi$
$798$	0	0
$799$	27.0000	0.955191
$800$	32.0000	1.13137
$801$	0	0
$802$	0	0
$803$	24.0000	0.846942
$804$	0	0
$805$	−6.00000	−0.211472
$806$	−24.0000	−0.845364
$807$	0	0
$808$	0	0
$809$	42.0000	1.47664	0.738321	−	0.674450i	$-0.235619\pi$
$809$	42.0000	1.47664	0.738321	+	0.674450i	$0.235619\pi$
$810$	0	0
$811$	−46.0000	−1.61528	−0.807639	−	0.589677i	$-0.799255\pi$
$811$	−46.0000	−1.61528	−0.807639	+	0.589677i	$0.799255\pi$
$812$	−8.00000	−0.280745
$813$	0	0
$814$	−24.0000	−0.841200
$815$	−11.0000	−0.385313
$816$	0	0
$817$	−88.0000	−3.07873
$818$	64.0000	2.23771
$819$	0	0
$820$	−2.00000	−0.0698430
$821$	8.00000	0.279202	0.139601	−	0.990208i	$-0.455418\pi$
$821$	8.00000	0.279202	0.139601	+	0.990208i	$0.455418\pi$
$822$	0	0
$823$	−3.00000	−0.104573	−0.0522867	−	0.998632i	$-0.516651\pi$
$823$	−3.00000	−0.104573	−0.0522867	+	0.998632i	$0.516651\pi$
$824$	0	0
$825$	0	0
$826$	−30.0000	−1.04383
$827$	18.0000	0.625921	0.312961	−	0.949766i	$-0.398679\pi$
$827$	18.0000	0.625921	0.312961	+	0.949766i	$0.398679\pi$
$828$	0	0
$829$	−16.0000	−0.555703	−0.277851	−	0.960624i	$-0.589622\pi$
$829$	−16.0000	−0.555703	−0.277851	+	0.960624i	$0.589622\pi$
$830$	18.0000	0.624789
$831$	0	0
$832$	16.0000	0.554700
$833$	−3.00000	−0.103944
$834$	0	0
$835$	−17.0000	−0.588309
$836$	−64.0000	−2.21349
$837$	0	0
$838$	66.0000	2.27993
$839$	55.0000	1.89881	0.949405	−	0.314053i	$-0.101687\pi$
$839$	55.0000	1.89881	0.949405	+	0.314053i	$0.101687\pi$
$840$	0	0
$841$	−13.0000	−0.448276
$842$	28.0000	0.964944
$843$	0	0
$844$	8.00000	0.275371
$845$	−9.00000	−0.309609
$846$	0	0
$847$	−5.00000	−0.171802
$848$	24.0000	0.824163
$849$	0	0
$850$	24.0000	0.823193
$851$	−18.0000	−0.617032
$852$	0	0
$853$	8.00000	0.273915	0.136957	−	0.990577i	$-0.456268\pi$
$853$	8.00000	0.273915	0.136957	+	0.990577i	$0.456268\pi$
$854$	−8.00000	−0.273754
$855$	0	0
$856$	0	0
$857$	−17.0000	−0.580709	−0.290354	−	0.956919i	$-0.593773\pi$
$857$	−17.0000	−0.580709	−0.290354	+	0.956919i	$0.593773\pi$
$858$	0	0
$859$	2.00000	0.0682391	0.0341196	−	0.999418i	$-0.489137\pi$
$859$	2.00000	0.0682391	0.0341196	+	0.999418i	$0.489137\pi$
$860$	22.0000	0.750194
$861$	0	0
$862$	−48.0000	−1.63489
$863$	−6.00000	−0.204242	−0.102121	−	0.994772i	$-0.532563\pi$
$863$	−6.00000	−0.204242	−0.102121	+	0.994772i	$0.532563\pi$
$864$	0	0
$865$	−22.0000	−0.748022
$866$	−4.00000	−0.135926
$867$	0	0
$868$	−12.0000	−0.407307
$869$	−4.00000	−0.135691
$870$	0	0
$871$	16.0000	0.542139
$872$	0	0
$873$	0	0
$874$	−96.0000	−3.24725
$875$	9.00000	0.304256
$876$	0	0
$877$	37.0000	1.24940	0.624701	−	0.780864i	$-0.285221\pi$
$877$	37.0000	1.24940	0.624701	+	0.780864i	$0.285221\pi$
$878$	48.0000	1.61992
$879$	0	0
$880$	−16.0000	−0.539360
$881$	−30.0000	−1.01073	−0.505363	−	0.862907i	$-0.668641\pi$
$881$	−30.0000	−1.01073	−0.505363	+	0.862907i	$0.668641\pi$
$882$	0	0
$883$	41.0000	1.37976	0.689880	−	0.723924i	$-0.257663\pi$
$883$	41.0000	1.37976	0.689880	+	0.723924i	$0.257663\pi$
$884$	12.0000	0.403604
$885$	0	0
$886$	48.0000	1.61259
$887$	−13.0000	−0.436497	−0.218249	−	0.975893i	$-0.570034\pi$
$887$	−13.0000	−0.436497	−0.218249	+	0.975893i	$0.570034\pi$
$888$	0	0
$889$	15.0000	0.503084
$890$	−4.00000	−0.134080
$891$	0	0
$892$	−4.00000	−0.133930
$893$	72.0000	2.40939
$894$	0	0
$895$	2.00000	0.0668526
$896$	0	0
$897$	0	0
$898$	12.0000	0.400445
$899$	24.0000	0.800445
$900$	0	0
$901$	18.0000	0.599667
$902$	−8.00000	−0.266371
$903$	0	0
$904$	0	0
$905$	16.0000	0.531858
$906$	0	0
$907$	17.0000	0.564476	0.282238	−	0.959344i	$-0.408923\pi$
$907$	17.0000	0.564476	0.282238	+	0.959344i	$0.408923\pi$
$908$	48.0000	1.59294
$909$	0	0
$910$	4.00000	0.132599
$911$	−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
$913$	36.0000	1.19143
$914$	−28.0000	−0.926158
$915$	0	0
$916$	−8.00000	−0.264327
$917$	−4.00000	−0.132092
$918$	0	0
$919$	−43.0000	−1.41844	−0.709220	−	0.704988i	$-0.750953\pi$
$919$	−43.0000	−1.41844	−0.709220	+	0.704988i	$0.750953\pi$
$920$	0	0
$921$	0	0
$922$	−2.00000	−0.0658665
$923$	−24.0000	−0.789970
$924$	0	0
$925$	12.0000	0.394558
$926$	−46.0000	−1.51165
$927$	0	0
$928$	−32.0000	−1.05045
$929$	35.0000	1.14831	0.574156	−	0.818746i	$-0.305330\pi$
$929$	35.0000	1.14831	0.574156	+	0.818746i	$0.305330\pi$
$930$	0	0
$931$	−8.00000	−0.262189
$932$	−12.0000	−0.393073
$933$	0	0
$934$	−24.0000	−0.785304
$935$	−12.0000	−0.392442
$936$	0	0
$937$	−12.0000	−0.392023	−0.196011	−	0.980602i	$-0.562799\pi$
$937$	−12.0000	−0.392023	−0.196011	+	0.980602i	$0.562799\pi$
$938$	16.0000	0.522419
$939$	0	0
$940$	−18.0000	−0.587095
$941$	17.0000	0.554184	0.277092	−	0.960843i	$-0.410629\pi$
$941$	17.0000	0.554184	0.277092	+	0.960843i	$0.410629\pi$
$942$	0	0
$943$	−6.00000	−0.195387
$944$	−60.0000	−1.95283
$945$	0	0
$946$	88.0000	2.86113
$947$	−28.0000	−0.909878	−0.454939	−	0.890523i	$-0.650339\pi$
$947$	−28.0000	−0.909878	−0.454939	+	0.890523i	$0.650339\pi$
$948$	0	0
$949$	−12.0000	−0.389536
$950$	64.0000	2.07643
$951$	0	0
$952$	0	0
$953$	2.00000	0.0647864	0.0323932	−	0.999475i	$-0.489687\pi$
$953$	2.00000	0.0647864	0.0323932	+	0.999475i	$0.489687\pi$
$954$	0	0
$955$	−2.00000	−0.0647185
$956$	36.0000	1.16432
$957$	0	0
$958$	34.0000	1.09849
$959$	−18.0000	−0.581250
$960$	0	0
$961$	5.00000	0.161290
$962$	12.0000	0.386896
$963$	0	0
$964$	−20.0000	−0.644157
$965$	−25.0000	−0.804778
$966$	0	0
$967$	40.0000	1.28631	0.643157	−	0.765735i	$-0.277624\pi$
$967$	40.0000	1.28631	0.643157	+	0.765735i	$0.277624\pi$
$968$	0	0
$969$	0	0
$970$	24.0000	0.770594
$971$	45.0000	1.44412	0.722059	−	0.691831i	$-0.243196\pi$
$971$	45.0000	1.44412	0.722059	+	0.691831i	$0.243196\pi$
$972$	0	0
$973$	0	0
$974$	32.0000	1.02535
$975$	0	0
$976$	−16.0000	−0.512148
$977$	12.0000	0.383914	0.191957	−	0.981403i	$-0.438517\pi$
$977$	12.0000	0.383914	0.191957	+	0.981403i	$0.438517\pi$
$978$	0	0
$979$	−8.00000	−0.255681
$980$	2.00000	0.0638877
$981$	0	0
$982$	52.0000	1.65939
$983$	−21.0000	−0.669796	−0.334898	−	0.942254i	$-0.608702\pi$
$983$	−21.0000	−0.669796	−0.334898	+	0.942254i	$0.608702\pi$
$984$	0	0
$985$	4.00000	0.127451
$986$	−24.0000	−0.764316
$987$	0	0
$988$	32.0000	1.01806
$989$	66.0000	2.09868
$990$	0	0
$991$	11.0000	0.349427	0.174713	−	0.984619i	$-0.444100\pi$
$991$	11.0000	0.349427	0.174713	+	0.984619i	$0.444100\pi$
$992$	−48.0000	−1.52400
$993$	0	0
$994$	−24.0000	−0.761234
$995$	−12.0000	−0.380426
$996$	0	0
$997$	40.0000	1.26681	0.633406	−	0.773819i	$-0.281656\pi$
$997$	40.0000	1.26681	0.633406	+	0.773819i	$0.281656\pi$
$998$	26.0000	0.823016
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	189.2.a.d.1.1	yes	1
3.2	odd	2		189.2.a.a.1.1	✓	1
4.3	odd	2		3024.2.a.u.1.1		1
5.4	even	2		4725.2.a.c.1.1		1
7.6	odd	2		1323.2.a.r.1.1		1
9.2	odd	6		567.2.f.h.190.1		2
9.4	even	3		567.2.f.a.379.1		2
9.5	odd	6		567.2.f.h.379.1		2
9.7	even	3		567.2.f.a.190.1		2
12.11	even	2		3024.2.a.l.1.1		1
15.14	odd	2		4725.2.a.s.1.1		1
21.20	even	2		1323.2.a.b.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
189.2.a.a.1.1	✓	1	3.2	odd	2
189.2.a.d.1.1	yes	1	1.1	even	1	trivial
567.2.f.a.190.1		2	9.7	even	3
567.2.f.a.379.1		2	9.4	even	3
567.2.f.h.190.1		2	9.2	odd	6
567.2.f.h.379.1		2	9.5	odd	6
1323.2.a.b.1.1		1	21.20	even	2
1323.2.a.r.1.1		1	7.6	odd	2
3024.2.a.l.1.1		1	12.11	even	2
3024.2.a.u.1.1		1	4.3	odd	2
4725.2.a.c.1.1		1	5.4	even	2
4725.2.a.s.1.1		1	15.14	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:	\( N \)	\(=\)	\( 189 = 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	189.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more