LMFDB - Embedded newform 441.2.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [441,2,Mod(1,441)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(441, 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("441.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 441 = 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	441.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:	$3.52140272914$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 21)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	441.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-2.00000 q^{2} +2.00000 q^{4} -2.00000 q^{5} +4.00000 q^{10} +2.00000 q^{11} -1.00000 q^{13} -4.00000 q^{16} -1.00000 q^{19} -4.00000 q^{20} -4.00000 q^{22} -1.00000 q^{25} +2.00000 q^{26} -4.00000 q^{29} -9.00000 q^{31} +8.00000 q^{32} +3.00000 q^{37} +2.00000 q^{38} -10.0000 q^{41} +5.00000 q^{43} +4.00000 q^{44} -6.00000 q^{47} +2.00000 q^{50} -2.00000 q^{52} -12.0000 q^{53} -4.00000 q^{55} +8.00000 q^{58} -12.0000 q^{59} -10.0000 q^{61} +18.0000 q^{62} -8.00000 q^{64} +2.00000 q^{65} -5.00000 q^{67} +6.00000 q^{71} +3.00000 q^{73} -6.00000 q^{74} -2.00000 q^{76} -1.00000 q^{79} +8.00000 q^{80} +20.0000 q^{82} +6.00000 q^{83} -10.0000 q^{86} +16.0000 q^{89} +12.0000 q^{94} +2.00000 q^{95} +6.00000 q^{97} +O(q^{100})$

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	$-0.750000\pi$
$2$	−2.00000	−1.41421	−0.707107	+	0.707107i	$0.750000\pi$
$3$	0	0
$3$	0	0
$4$	2.00000	1.00000
$5$	−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
$7$	0	0
$7$	0	0
$8$	0	0
$9$	0	0
$10$	4.00000	1.26491
$11$	2.00000	0.603023	0.301511	−	0.953463i	$-0.402509\pi$
$11$	2.00000	0.603023	0.301511	+	0.953463i	$0.402509\pi$
$12$	0	0
$13$	−1.00000	−0.277350	−0.138675	−	0.990338i	$-0.544284\pi$
$13$	−1.00000	−0.277350	−0.138675	+	0.990338i	$0.544284\pi$
$14$	0	0
$15$	0	0
$16$	−4.00000	−1.00000
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	0	0
$19$	−1.00000	−0.229416	−0.114708	−	0.993399i	$-0.536593\pi$
$19$	−1.00000	−0.229416	−0.114708	+	0.993399i	$0.536593\pi$
$20$	−4.00000	−0.894427
$21$	0	0
$22$	−4.00000	−0.852803
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	2.00000	0.392232
$27$	0	0
$28$	0	0
$29$	−4.00000	−0.742781	−0.371391	−	0.928477i	$-0.621119\pi$
$29$	−4.00000	−0.742781	−0.371391	+	0.928477i	$0.621119\pi$
$30$	0	0
$31$	−9.00000	−1.61645	−0.808224	−	0.588875i	$-0.799571\pi$
$31$	−9.00000	−1.61645	−0.808224	+	0.588875i	$0.799571\pi$
$32$	8.00000	1.41421
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	3.00000	0.493197	0.246598	−	0.969118i	$-0.420687\pi$
$37$	3.00000	0.493197	0.246598	+	0.969118i	$0.420687\pi$
$38$	2.00000	0.324443
$39$	0	0
$40$	0	0
$41$	−10.0000	−1.56174	−0.780869	−	0.624695i	$-0.785223\pi$
$41$	−10.0000	−1.56174	−0.780869	+	0.624695i	$0.785223\pi$
$42$	0	0
$43$	5.00000	0.762493	0.381246	−	0.924473i	$-0.375495\pi$
$43$	5.00000	0.762493	0.381246	+	0.924473i	$0.375495\pi$
$44$	4.00000	0.603023
$45$	0	0
$46$	0	0
$47$	−6.00000	−0.875190	−0.437595	−	0.899172i	$-0.644170\pi$
$47$	−6.00000	−0.875190	−0.437595	+	0.899172i	$0.644170\pi$
$48$	0	0
$49$	0	0
$50$	2.00000	0.282843
$51$	0	0
$52$	−2.00000	−0.277350
$53$	−12.0000	−1.64833	−0.824163	−	0.566352i	$-0.808354\pi$
$53$	−12.0000	−1.64833	−0.824163	+	0.566352i	$0.808354\pi$
$54$	0	0
$55$	−4.00000	−0.539360
$56$	0	0
$57$	0	0
$58$	8.00000	1.05045
$59$	−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
$61$	−10.0000	−1.28037	−0.640184	−	0.768221i	$-0.721142\pi$
$61$	−10.0000	−1.28037	−0.640184	+	0.768221i	$0.721142\pi$
$62$	18.0000	2.28600
$63$	0	0
$64$	−8.00000	−1.00000
$65$	2.00000	0.248069
$66$	0	0
$67$	−5.00000	−0.610847	−0.305424	−	0.952217i	$-0.598798\pi$
$67$	−5.00000	−0.610847	−0.305424	+	0.952217i	$0.598798\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	6.00000	0.712069	0.356034	−	0.934473i	$-0.384129\pi$
$71$	6.00000	0.712069	0.356034	+	0.934473i	$0.384129\pi$
$72$	0	0
$73$	3.00000	0.351123	0.175562	−	0.984468i	$-0.443826\pi$
$73$	3.00000	0.351123	0.175562	+	0.984468i	$0.443826\pi$
$74$	−6.00000	−0.697486
$75$	0	0
$76$	−2.00000	−0.229416
$77$	0	0
$78$	0	0
$79$	−1.00000	−0.112509	−0.0562544	−	0.998416i	$-0.517916\pi$
$79$	−1.00000	−0.112509	−0.0562544	+	0.998416i	$0.517916\pi$
$80$	8.00000	0.894427
$81$	0	0
$82$	20.0000	2.20863
$83$	6.00000	0.658586	0.329293	−	0.944228i	$-0.393190\pi$
$83$	6.00000	0.658586	0.329293	+	0.944228i	$0.393190\pi$
$84$	0	0
$85$	0	0
$86$	−10.0000	−1.07833
$87$	0	0
$88$	0	0
$89$	16.0000	1.69600	0.847998	−	0.529999i	$-0.177808\pi$
$89$	16.0000	1.69600	0.847998	+	0.529999i	$0.177808\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	12.0000	1.23771
$95$	2.00000	0.205196
$96$	0	0
$97$	6.00000	0.609208	0.304604	−	0.952479i	$-0.401476\pi$
$97$	6.00000	0.609208	0.304604	+	0.952479i	$0.401476\pi$
$98$	0	0
$99$	0	0
$100$	−2.00000	−0.200000
$101$	2.00000	0.199007	0.0995037	−	0.995037i	$-0.468274\pi$
$101$	2.00000	0.199007	0.0995037	+	0.995037i	$0.468274\pi$
$102$	0	0
$103$	7.00000	0.689730	0.344865	−	0.938652i	$-0.387925\pi$
$103$	7.00000	0.689730	0.344865	+	0.938652i	$0.387925\pi$
$104$	0	0
$105$	0	0
$106$	24.0000	2.33109
$107$	8.00000	0.773389	0.386695	−	0.922208i	$-0.373617\pi$
$107$	8.00000	0.773389	0.386695	+	0.922208i	$0.373617\pi$
$108$	0	0
$109$	9.00000	0.862044	0.431022	−	0.902342i	$-0.358153\pi$
$109$	9.00000	0.862044	0.431022	+	0.902342i	$0.358153\pi$
$110$	8.00000	0.762770
$111$	0	0
$112$	0	0
$113$	−10.0000	−0.940721	−0.470360	−	0.882474i	$-0.655876\pi$
$113$	−10.0000	−0.940721	−0.470360	+	0.882474i	$0.655876\pi$
$114$	0	0
$115$	0	0
$116$	−8.00000	−0.742781
$117$	0	0
$118$	24.0000	2.20938
$119$	0	0
$120$	0	0
$121$	−7.00000	−0.636364
$122$	20.0000	1.81071
$123$	0	0
$124$	−18.0000	−1.61645
$125$	12.0000	1.07331
$126$	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
$131$	−14.0000	−1.22319	−0.611593	−	0.791173i	$-0.709471\pi$
$131$	−14.0000	−1.22319	−0.611593	+	0.791173i	$0.709471\pi$
$132$	0	0
$133$	0	0
$134$	10.0000	0.863868
$135$	0	0
$136$	0	0
$137$	12.0000	1.02523	0.512615	−	0.858619i	$-0.328677\pi$
$137$	12.0000	1.02523	0.512615	+	0.858619i	$0.328677\pi$
$138$	0	0
$139$	3.00000	0.254457	0.127228	−	0.991873i	$-0.459392\pi$
$139$	3.00000	0.254457	0.127228	+	0.991873i	$0.459392\pi$
$140$	0	0
$141$	0	0
$142$	−12.0000	−1.00702
$143$	−2.00000	−0.167248
$144$	0	0
$145$	8.00000	0.664364
$146$	−6.00000	−0.496564
$147$	0	0
$148$	6.00000	0.493197
$149$	12.0000	0.983078	0.491539	−	0.870855i	$-0.336434\pi$
$149$	12.0000	0.983078	0.491539	+	0.870855i	$0.336434\pi$
$150$	0	0
$151$	−16.0000	−1.30206	−0.651031	−	0.759051i	$-0.725663\pi$
$151$	−16.0000	−1.30206	−0.651031	+	0.759051i	$0.725663\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	18.0000	1.44579
$156$	0	0
$157$	14.0000	1.11732	0.558661	−	0.829396i	$-0.311315\pi$
$157$	14.0000	1.11732	0.558661	+	0.829396i	$0.311315\pi$
$158$	2.00000	0.159111
$159$	0	0
$160$	−16.0000	−1.26491
$161$	0	0
$162$	0	0
$163$	4.00000	0.313304	0.156652	−	0.987654i	$-0.449930\pi$
$163$	4.00000	0.313304	0.156652	+	0.987654i	$0.449930\pi$
$164$	−20.0000	−1.56174
$165$	0	0
$166$	−12.0000	−0.931381
$167$	−14.0000	−1.08335	−0.541676	−	0.840587i	$-0.682210\pi$
$167$	−14.0000	−1.08335	−0.541676	+	0.840587i	$0.682210\pi$
$168$	0	0
$169$	−12.0000	−0.923077
$170$	0	0
$171$	0	0
$172$	10.0000	0.762493
$173$	8.00000	0.608229	0.304114	−	0.952636i	$-0.401639\pi$
$173$	8.00000	0.608229	0.304114	+	0.952636i	$0.401639\pi$
$174$	0	0
$175$	0	0
$176$	−8.00000	−0.603023
$177$	0	0
$178$	−32.0000	−2.39850
$179$	−2.00000	−0.149487	−0.0747435	−	0.997203i	$-0.523814\pi$
$179$	−2.00000	−0.149487	−0.0747435	+	0.997203i	$0.523814\pi$
$180$	0	0
$181$	−13.0000	−0.966282	−0.483141	−	0.875542i	$-0.660504\pi$
$181$	−13.0000	−0.966282	−0.483141	+	0.875542i	$0.660504\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−6.00000	−0.441129
$186$	0	0
$187$	0	0
$188$	−12.0000	−0.875190
$189$	0	0
$190$	−4.00000	−0.290191
$191$	−10.0000	−0.723575	−0.361787	−	0.932261i	$-0.617833\pi$
$191$	−10.0000	−0.723575	−0.361787	+	0.932261i	$0.617833\pi$
$192$	0	0
$193$	11.0000	0.791797	0.395899	−	0.918294i	$-0.370433\pi$
$193$	11.0000	0.791797	0.395899	+	0.918294i	$0.370433\pi$
$194$	−12.0000	−0.861550
$195$	0	0
$196$	0	0
$197$	−16.0000	−1.13995	−0.569976	−	0.821661i	$-0.693048\pi$
$197$	−16.0000	−1.13995	−0.569976	+	0.821661i	$0.693048\pi$
$198$	0	0
$199$	0	0		−	1.00000i	$-0.5\pi$
$199$	0	0			1.00000i	$0.5\pi$
$200$	0	0
$201$	0	0
$202$	−4.00000	−0.281439
$203$	0	0
$204$	0	0
$205$	20.0000	1.39686
$206$	−14.0000	−0.975426
$207$	0	0
$208$	4.00000	0.277350
$209$	−2.00000	−0.138343
$210$	0	0
$211$	4.00000	0.275371	0.137686	−	0.990476i	$-0.456034\pi$
$211$	4.00000	0.275371	0.137686	+	0.990476i	$0.456034\pi$
$212$	−24.0000	−1.64833
$213$	0	0
$214$	−16.0000	−1.09374
$215$	−10.0000	−0.681994
$216$	0	0
$217$	0	0
$218$	−18.0000	−1.21911
$219$	0	0
$220$	−8.00000	−0.539360
$221$	0	0
$222$	0	0
$223$	−16.0000	−1.07144	−0.535720	−	0.844396i	$-0.679960\pi$
$223$	−16.0000	−1.07144	−0.535720	+	0.844396i	$0.679960\pi$
$224$	0	0
$225$	0	0
$226$	20.0000	1.33038
$227$	18.0000	1.19470	0.597351	−	0.801980i	$-0.296220\pi$
$227$	18.0000	1.19470	0.597351	+	0.801980i	$0.296220\pi$
$228$	0	0
$229$	19.0000	1.25556	0.627778	−	0.778393i	$-0.283965\pi$
$229$	19.0000	1.25556	0.627778	+	0.778393i	$0.283965\pi$
$230$	0	0
$231$	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
$235$	12.0000	0.782794
$236$	−24.0000	−1.56227
$237$	0	0
$238$	0	0
$239$	−6.00000	−0.388108	−0.194054	−	0.980991i	$-0.562164\pi$
$239$	−6.00000	−0.388108	−0.194054	+	0.980991i	$0.562164\pi$
$240$	0	0
$241$	−14.0000	−0.901819	−0.450910	−	0.892570i	$-0.648900\pi$
$241$	−14.0000	−0.901819	−0.450910	+	0.892570i	$0.648900\pi$
$242$	14.0000	0.899954
$243$	0	0
$244$	−20.0000	−1.28037
$245$	0	0
$246$	0	0
$247$	1.00000	0.0636285
$248$	0	0
$249$	0	0
$250$	−24.0000	−1.51789
$251$	−8.00000	−0.504956	−0.252478	−	0.967603i	$-0.581245\pi$
$251$	−8.00000	−0.504956	−0.252478	+	0.967603i	$0.581245\pi$
$252$	0	0
$253$	0	0
$254$	30.0000	1.88237
$255$	0	0
$256$	16.0000	1.00000
$257$	26.0000	1.62184	0.810918	−	0.585160i	$-0.198968\pi$
$257$	26.0000	1.62184	0.810918	+	0.585160i	$0.198968\pi$
$258$	0	0
$259$	0	0
$260$	4.00000	0.248069
$261$	0	0
$262$	28.0000	1.72985
$263$	−4.00000	−0.246651	−0.123325	−	0.992366i	$-0.539356\pi$
$263$	−4.00000	−0.246651	−0.123325	+	0.992366i	$0.539356\pi$
$264$	0	0
$265$	24.0000	1.47431
$266$	0	0
$267$	0	0
$268$	−10.0000	−0.610847
$269$	6.00000	0.365826	0.182913	−	0.983129i	$-0.441447\pi$
$269$	6.00000	0.365826	0.182913	+	0.983129i	$0.441447\pi$
$270$	0	0
$271$	−16.0000	−0.971931	−0.485965	−	0.873978i	$-0.661532\pi$
$271$	−16.0000	−0.971931	−0.485965	+	0.873978i	$0.661532\pi$
$272$	0	0
$273$	0	0
$274$	−24.0000	−1.44989
$275$	−2.00000	−0.120605
$276$	0	0
$277$	13.0000	0.781094	0.390547	−	0.920583i	$-0.372286\pi$
$277$	13.0000	0.781094	0.390547	+	0.920583i	$0.372286\pi$
$278$	−6.00000	−0.359856
$279$	0	0
$280$	0	0
$281$	4.00000	0.238620	0.119310	−	0.992857i	$-0.461932\pi$
$281$	4.00000	0.238620	0.119310	+	0.992857i	$0.461932\pi$
$282$	0	0
$283$	11.0000	0.653882	0.326941	−	0.945045i	$-0.393982\pi$
$283$	11.0000	0.653882	0.326941	+	0.945045i	$0.393982\pi$
$284$	12.0000	0.712069
$285$	0	0
$286$	4.00000	0.236525
$287$	0	0
$288$	0	0
$289$	−17.0000	−1.00000
$290$	−16.0000	−0.939552
$291$	0	0
$292$	6.00000	0.351123
$293$	8.00000	0.467365	0.233682	−	0.972313i	$-0.424922\pi$
$293$	8.00000	0.467365	0.233682	+	0.972313i	$0.424922\pi$
$294$	0	0
$295$	24.0000	1.39733
$296$	0	0
$297$	0	0
$298$	−24.0000	−1.39028
$299$	0	0
$300$	0	0
$301$	0	0
$302$	32.0000	1.84139
$303$	0	0
$304$	4.00000	0.229416
$305$	20.0000	1.14520
$306$	0	0
$307$	17.0000	0.970241	0.485121	−	0.874447i	$-0.338776\pi$
$307$	17.0000	0.970241	0.485121	+	0.874447i	$0.338776\pi$
$308$	0	0
$309$	0	0
$310$	−36.0000	−2.04466
$311$	−6.00000	−0.340229	−0.170114	−	0.985424i	$-0.554414\pi$
$311$	−6.00000	−0.340229	−0.170114	+	0.985424i	$0.554414\pi$
$312$	0	0
$313$	1.00000	0.0565233	0.0282617	−	0.999601i	$-0.491003\pi$
$313$	1.00000	0.0565233	0.0282617	+	0.999601i	$0.491003\pi$
$314$	−28.0000	−1.58013
$315$	0	0
$316$	−2.00000	−0.112509
$317$	−24.0000	−1.34797	−0.673987	−	0.738743i	$-0.735420\pi$
$317$	−24.0000	−1.34797	−0.673987	+	0.738743i	$0.735420\pi$
$318$	0	0
$319$	−8.00000	−0.447914
$320$	16.0000	0.894427
$321$	0	0
$322$	0	0
$323$	0	0
$324$	0	0
$325$	1.00000	0.0554700
$326$	−8.00000	−0.443079
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−25.0000	−1.37412	−0.687062	−	0.726599i	$-0.741100\pi$
$331$	−25.0000	−1.37412	−0.687062	+	0.726599i	$0.741100\pi$
$332$	12.0000	0.658586
$333$	0	0
$334$	28.0000	1.53209
$335$	10.0000	0.546358
$336$	0	0
$337$	13.0000	0.708155	0.354078	−	0.935216i	$-0.384795\pi$
$337$	13.0000	0.708155	0.354078	+	0.935216i	$0.384795\pi$
$338$	24.0000	1.30543
$339$	0	0
$340$	0	0
$341$	−18.0000	−0.974755
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	−16.0000	−0.860165
$347$	−32.0000	−1.71785	−0.858925	−	0.512101i	$-0.828867\pi$
$347$	−32.0000	−1.71785	−0.858925	+	0.512101i	$0.828867\pi$
$348$	0	0
$349$	14.0000	0.749403	0.374701	−	0.927146i	$-0.377745\pi$
$349$	14.0000	0.749403	0.374701	+	0.927146i	$0.377745\pi$
$350$	0	0
$351$	0	0
$352$	16.0000	0.852803
$353$	34.0000	1.80964	0.904819	−	0.425797i	$-0.140006\pi$
$353$	34.0000	1.80964	0.904819	+	0.425797i	$0.140006\pi$
$354$	0	0
$355$	−12.0000	−0.636894
$356$	32.0000	1.69600
$357$	0	0
$358$	4.00000	0.211407
$359$	−20.0000	−1.05556	−0.527780	−	0.849381i	$-0.676975\pi$
$359$	−20.0000	−1.05556	−0.527780	+	0.849381i	$0.676975\pi$
$360$	0	0
$361$	−18.0000	−0.947368
$362$	26.0000	1.36653
$363$	0	0
$364$	0	0
$365$	−6.00000	−0.314054
$366$	0	0
$367$	9.00000	0.469796	0.234898	−	0.972020i	$-0.424524\pi$
$367$	9.00000	0.469796	0.234898	+	0.972020i	$0.424524\pi$
$368$	0	0
$369$	0	0
$370$	12.0000	0.623850
$371$	0	0
$372$	0	0
$373$	23.0000	1.19089	0.595447	−	0.803394i	$-0.296975\pi$
$373$	23.0000	1.19089	0.595447	+	0.803394i	$0.296975\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	4.00000	0.206010
$378$	0	0
$379$	3.00000	0.154100	0.0770498	−	0.997027i	$-0.475450\pi$
$379$	3.00000	0.154100	0.0770498	+	0.997027i	$0.475450\pi$
$380$	4.00000	0.205196
$381$	0	0
$382$	20.0000	1.02329
$383$	−12.0000	−0.613171	−0.306586	−	0.951843i	$-0.599187\pi$
$383$	−12.0000	−0.613171	−0.306586	+	0.951843i	$0.599187\pi$
$384$	0	0
$385$	0	0
$386$	−22.0000	−1.11977
$387$	0	0
$388$	12.0000	0.609208
$389$	6.00000	0.304212	0.152106	−	0.988364i	$-0.451394\pi$
$389$	6.00000	0.304212	0.152106	+	0.988364i	$0.451394\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	0	0
$394$	32.0000	1.61214
$395$	2.00000	0.100631
$396$	0	0
$397$	9.00000	0.451697	0.225849	−	0.974162i	$-0.427485\pi$
$397$	9.00000	0.451697	0.225849	+	0.974162i	$0.427485\pi$
$398$	0	0
$399$	0	0
$400$	4.00000	0.200000
$401$	36.0000	1.79775	0.898877	−	0.438201i	$-0.144384\pi$
$401$	36.0000	1.79775	0.898877	+	0.438201i	$0.144384\pi$
$402$	0	0
$403$	9.00000	0.448322
$404$	4.00000	0.199007
$405$	0	0
$406$	0	0
$407$	6.00000	0.297409
$408$	0	0
$409$	−5.00000	−0.247234	−0.123617	−	0.992330i	$-0.539449\pi$
$409$	−5.00000	−0.247234	−0.123617	+	0.992330i	$0.539449\pi$
$410$	−40.0000	−1.97546
$411$	0	0
$412$	14.0000	0.689730
$413$	0	0
$414$	0	0
$415$	−12.0000	−0.589057
$416$	−8.00000	−0.392232
$417$	0	0
$418$	4.00000	0.195646
$419$	30.0000	1.46560	0.732798	−	0.680446i	$-0.238214\pi$
$419$	30.0000	1.46560	0.732798	+	0.680446i	$0.238214\pi$
$420$	0	0
$421$	−7.00000	−0.341159	−0.170580	−	0.985344i	$-0.554564\pi$
$421$	−7.00000	−0.341159	−0.170580	+	0.985344i	$0.554564\pi$
$422$	−8.00000	−0.389434
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	0	0
$428$	16.0000	0.773389
$429$	0	0
$430$	20.0000	0.964486
$431$	18.0000	0.867029	0.433515	−	0.901146i	$-0.357273\pi$
$431$	18.0000	0.867029	0.433515	+	0.901146i	$0.357273\pi$
$432$	0	0
$433$	−31.0000	−1.48976	−0.744882	−	0.667196i	$-0.767494\pi$
$433$	−31.0000	−1.48976	−0.744882	+	0.667196i	$0.767494\pi$
$434$	0	0
$435$	0	0
$436$	18.0000	0.862044
$437$	0	0
$438$	0	0
$439$	0	0		−	1.00000i	$-0.5\pi$
$439$	0	0			1.00000i	$0.5\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−12.0000	−0.570137	−0.285069	−	0.958507i	$-0.592016\pi$
$443$	−12.0000	−0.570137	−0.285069	+	0.958507i	$0.592016\pi$
$444$	0	0
$445$	−32.0000	−1.51695
$446$	32.0000	1.51524
$447$	0	0
$448$	0	0
$449$	18.0000	0.849473	0.424736	−	0.905317i	$-0.360367\pi$
$449$	18.0000	0.849473	0.424736	+	0.905317i	$0.360367\pi$
$450$	0	0
$451$	−20.0000	−0.941763
$452$	−20.0000	−0.940721
$453$	0	0
$454$	−36.0000	−1.68956
$455$	0	0
$456$	0	0
$457$	−11.0000	−0.514558	−0.257279	−	0.966337i	$-0.582826\pi$
$457$	−11.0000	−0.514558	−0.257279	+	0.966337i	$0.582826\pi$
$458$	−38.0000	−1.77562
$459$	0	0
$460$	0	0
$461$	20.0000	0.931493	0.465746	−	0.884918i	$-0.345786\pi$
$461$	20.0000	0.931493	0.465746	+	0.884918i	$0.345786\pi$
$462$	0	0
$463$	−17.0000	−0.790057	−0.395029	−	0.918669i	$-0.629265\pi$
$463$	−17.0000	−0.790057	−0.395029	+	0.918669i	$0.629265\pi$
$464$	16.0000	0.742781
$465$	0	0
$466$	12.0000	0.555889
$467$	6.00000	0.277647	0.138823	−	0.990317i	$-0.455668\pi$
$467$	6.00000	0.277647	0.138823	+	0.990317i	$0.455668\pi$
$468$	0	0
$469$	0	0
$470$	−24.0000	−1.10704
$471$	0	0
$472$	0	0
$473$	10.0000	0.459800
$474$	0	0
$475$	1.00000	0.0458831
$476$	0	0
$477$	0	0
$478$	12.0000	0.548867
$479$	−28.0000	−1.27935	−0.639676	−	0.768644i	$-0.720932\pi$
$479$	−28.0000	−1.27935	−0.639676	+	0.768644i	$0.720932\pi$
$480$	0	0
$481$	−3.00000	−0.136788
$482$	28.0000	1.27537
$483$	0	0
$484$	−14.0000	−0.636364
$485$	−12.0000	−0.544892
$486$	0	0
$487$	31.0000	1.40474	0.702372	−	0.711810i	$-0.252124\pi$
$487$	31.0000	1.40474	0.702372	+	0.711810i	$0.252124\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	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
$493$	0	0
$494$	−2.00000	−0.0899843
$495$	0	0
$496$	36.0000	1.61645
$497$	0	0
$498$	0	0
$499$	37.0000	1.65635	0.828174	−	0.560471i	$-0.189380\pi$
$499$	37.0000	1.65635	0.828174	+	0.560471i	$0.189380\pi$
$500$	24.0000	1.07331
$501$	0	0
$502$	16.0000	0.714115
$503$	−42.0000	−1.87269	−0.936344	−	0.351085i	$-0.885813\pi$
$503$	−42.0000	−1.87269	−0.936344	+	0.351085i	$0.885813\pi$
$504$	0	0
$505$	−4.00000	−0.177998
$506$	0	0
$507$	0	0
$508$	−30.0000	−1.33103
$509$	2.00000	0.0886484	0.0443242	−	0.999017i	$-0.485887\pi$
$509$	2.00000	0.0886484	0.0443242	+	0.999017i	$0.485887\pi$
$510$	0	0
$511$	0	0
$512$	−32.0000	−1.41421
$513$	0	0
$514$	−52.0000	−2.29362
$515$	−14.0000	−0.616914
$516$	0	0
$517$	−12.0000	−0.527759
$518$	0	0
$519$	0	0
$520$	0	0
$521$	12.0000	0.525730	0.262865	−	0.964833i	$-0.415333\pi$
$521$	12.0000	0.525730	0.262865	+	0.964833i	$0.415333\pi$
$522$	0	0
$523$	−31.0000	−1.35554	−0.677768	−	0.735276i	$-0.737052\pi$
$523$	−31.0000	−1.35554	−0.677768	+	0.735276i	$0.737052\pi$
$524$	−28.0000	−1.22319
$525$	0	0
$526$	8.00000	0.348817
$527$	0	0
$528$	0	0
$529$	−23.0000	−1.00000
$530$	−48.0000	−2.08499
$531$	0	0
$532$	0	0
$533$	10.0000	0.433148
$534$	0	0
$535$	−16.0000	−0.691740
$536$	0	0
$537$	0	0
$538$	−12.0000	−0.517357
$539$	0	0
$540$	0	0
$541$	−19.0000	−0.816874	−0.408437	−	0.912787i	$-0.633926\pi$
$541$	−19.0000	−0.816874	−0.408437	+	0.912787i	$0.633926\pi$
$542$	32.0000	1.37452
$543$	0	0
$544$	0	0
$545$	−18.0000	−0.771035
$546$	0	0
$547$	28.0000	1.19719	0.598597	−	0.801050i	$-0.295725\pi$
$547$	28.0000	1.19719	0.598597	+	0.801050i	$0.295725\pi$
$548$	24.0000	1.02523
$549$	0	0
$550$	4.00000	0.170561
$551$	4.00000	0.170406
$552$	0	0
$553$	0	0
$554$	−26.0000	−1.10463
$555$	0	0
$556$	6.00000	0.254457
$557$	2.00000	0.0847427	0.0423714	−	0.999102i	$-0.486509\pi$
$557$	2.00000	0.0847427	0.0423714	+	0.999102i	$0.486509\pi$
$558$	0	0
$559$	−5.00000	−0.211477
$560$	0	0
$561$	0	0
$562$	−8.00000	−0.337460
$563$	−26.0000	−1.09577	−0.547885	−	0.836554i	$-0.684567\pi$
$563$	−26.0000	−1.09577	−0.547885	+	0.836554i	$0.684567\pi$
$564$	0	0
$565$	20.0000	0.841406
$566$	−22.0000	−0.924729
$567$	0	0
$568$	0	0
$569$	26.0000	1.08998	0.544988	−	0.838444i	$-0.316534\pi$
$569$	26.0000	1.08998	0.544988	+	0.838444i	$0.316534\pi$
$570$	0	0
$571$	−19.0000	−0.795125	−0.397563	−	0.917575i	$-0.630144\pi$
$571$	−19.0000	−0.795125	−0.397563	+	0.917575i	$0.630144\pi$
$572$	−4.00000	−0.167248
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	17.0000	0.707719	0.353860	−	0.935299i	$-0.384869\pi$
$577$	17.0000	0.707719	0.353860	+	0.935299i	$0.384869\pi$
$578$	34.0000	1.41421
$579$	0	0
$580$	16.0000	0.664364
$581$	0	0
$582$	0	0
$583$	−24.0000	−0.993978
$584$	0	0
$585$	0	0
$586$	−16.0000	−0.660954
$587$	16.0000	0.660391	0.330195	−	0.943913i	$-0.392885\pi$
$587$	16.0000	0.660391	0.330195	+	0.943913i	$0.392885\pi$
$588$	0	0
$589$	9.00000	0.370839
$590$	−48.0000	−1.97613
$591$	0	0
$592$	−12.0000	−0.493197
$593$	−6.00000	−0.246390	−0.123195	−	0.992382i	$-0.539314\pi$
$593$	−6.00000	−0.246390	−0.123195	+	0.992382i	$0.539314\pi$
$594$	0	0
$595$	0	0
$596$	24.0000	0.983078
$597$	0	0
$598$	0	0
$599$	−12.0000	−0.490307	−0.245153	−	0.969484i	$-0.578838\pi$
$599$	−12.0000	−0.490307	−0.245153	+	0.969484i	$0.578838\pi$
$600$	0	0
$601$	9.00000	0.367118	0.183559	−	0.983009i	$-0.441238\pi$
$601$	9.00000	0.367118	0.183559	+	0.983009i	$0.441238\pi$
$602$	0	0
$603$	0	0
$604$	−32.0000	−1.30206
$605$	14.0000	0.569181
$606$	0	0
$607$	−23.0000	−0.933541	−0.466771	−	0.884378i	$-0.654583\pi$
$607$	−23.0000	−0.933541	−0.466771	+	0.884378i	$0.654583\pi$
$608$	−8.00000	−0.324443
$609$	0	0
$610$	−40.0000	−1.61955
$611$	6.00000	0.242734
$612$	0	0
$613$	34.0000	1.37325	0.686624	−	0.727013i	$-0.259092\pi$
$613$	34.0000	1.37325	0.686624	+	0.727013i	$0.259092\pi$
$614$	−34.0000	−1.37213
$615$	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
$619$	29.0000	1.16561	0.582804	−	0.812613i	$-0.301955\pi$
$619$	29.0000	1.16561	0.582804	+	0.812613i	$0.301955\pi$
$620$	36.0000	1.44579
$621$	0	0
$622$	12.0000	0.481156
$623$	0	0
$624$	0	0
$625$	−19.0000	−0.760000
$626$	−2.00000	−0.0799361
$627$	0	0
$628$	28.0000	1.11732
$629$	0	0
$630$	0	0
$631$	8.00000	0.318475	0.159237	−	0.987240i	$-0.449096\pi$
$631$	8.00000	0.318475	0.159237	+	0.987240i	$0.449096\pi$
$632$	0	0
$633$	0	0
$634$	48.0000	1.90632
$635$	30.0000	1.19051
$636$	0	0
$637$	0	0
$638$	16.0000	0.633446
$639$	0	0
$640$	0	0
$641$	0	0		−	1.00000i	$-0.5\pi$
$641$	0	0			1.00000i	$0.5\pi$
$642$	0	0
$643$	19.0000	0.749287	0.374643	−	0.927169i	$-0.377765\pi$
$643$	19.0000	0.749287	0.374643	+	0.927169i	$0.377765\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	2.00000	0.0786281	0.0393141	−	0.999227i	$-0.487483\pi$
$647$	2.00000	0.0786281	0.0393141	+	0.999227i	$0.487483\pi$
$648$	0	0
$649$	−24.0000	−0.942082
$650$	−2.00000	−0.0784465
$651$	0	0
$652$	8.00000	0.313304
$653$	−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
$655$	28.0000	1.09405
$656$	40.0000	1.56174
$657$	0	0
$658$	0	0
$659$	−36.0000	−1.40236	−0.701180	−	0.712984i	$-0.747343\pi$
$659$	−36.0000	−1.40236	−0.701180	+	0.712984i	$0.747343\pi$
$660$	0	0
$661$	41.0000	1.59472	0.797358	−	0.603507i	$-0.206231\pi$
$661$	41.0000	1.59472	0.797358	+	0.603507i	$0.206231\pi$
$662$	50.0000	1.94331
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0	0
$668$	−28.0000	−1.08335
$669$	0	0
$670$	−20.0000	−0.772667
$671$	−20.0000	−0.772091
$672$	0	0
$673$	−41.0000	−1.58043	−0.790217	−	0.612827i	$-0.790032\pi$
$673$	−41.0000	−1.58043	−0.790217	+	0.612827i	$0.790032\pi$
$674$	−26.0000	−1.00148
$675$	0	0
$676$	−24.0000	−0.923077
$677$	12.0000	0.461197	0.230599	−	0.973049i	$-0.425932\pi$
$677$	12.0000	0.461197	0.230599	+	0.973049i	$0.425932\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	36.0000	1.37851
$683$	12.0000	0.459167	0.229584	−	0.973289i	$-0.426264\pi$
$683$	12.0000	0.459167	0.229584	+	0.973289i	$0.426264\pi$
$684$	0	0
$685$	−24.0000	−0.916993
$686$	0	0
$687$	0	0
$688$	−20.0000	−0.762493
$689$	12.0000	0.457164
$690$	0	0
$691$	37.0000	1.40755	0.703773	−	0.710425i	$-0.251497\pi$
$691$	37.0000	1.40755	0.703773	+	0.710425i	$0.251497\pi$
$692$	16.0000	0.608229
$693$	0	0
$694$	64.0000	2.42941
$695$	−6.00000	−0.227593
$696$	0	0
$697$	0	0
$698$	−28.0000	−1.05982
$699$	0	0
$700$	0	0
$701$	0	0		−	1.00000i	$-0.5\pi$
$701$	0	0			1.00000i	$0.5\pi$
$702$	0	0
$703$	−3.00000	−0.113147
$704$	−16.0000	−0.603023
$705$	0	0
$706$	−68.0000	−2.55921
$707$	0	0
$708$	0	0
$709$	30.0000	1.12667	0.563337	−	0.826227i	$-0.309517\pi$
$709$	30.0000	1.12667	0.563337	+	0.826227i	$0.309517\pi$
$710$	24.0000	0.900704
$711$	0	0
$712$	0	0
$713$	0	0
$714$	0	0
$715$	4.00000	0.149592
$716$	−4.00000	−0.149487
$717$	0	0
$718$	40.0000	1.49279
$719$	−18.0000	−0.671287	−0.335643	−	0.941989i	$-0.608954\pi$
$719$	−18.0000	−0.671287	−0.335643	+	0.941989i	$0.608954\pi$
$720$	0	0
$721$	0	0
$722$	36.0000	1.33978
$723$	0	0
$724$	−26.0000	−0.966282
$725$	4.00000	0.148556
$726$	0	0
$727$	13.0000	0.482143	0.241072	−	0.970507i	$-0.422501\pi$
$727$	13.0000	0.482143	0.241072	+	0.970507i	$0.422501\pi$
$728$	0	0
$729$	0	0
$730$	12.0000	0.444140
$731$	0	0
$732$	0	0
$733$	15.0000	0.554038	0.277019	−	0.960864i	$-0.410654\pi$
$733$	15.0000	0.554038	0.277019	+	0.960864i	$0.410654\pi$
$734$	−18.0000	−0.664392
$735$	0	0
$736$	0	0
$737$	−10.0000	−0.368355
$738$	0	0
$739$	−15.0000	−0.551784	−0.275892	−	0.961189i	$-0.588973\pi$
$739$	−15.0000	−0.551784	−0.275892	+	0.961189i	$0.588973\pi$
$740$	−12.0000	−0.441129
$741$	0	0
$742$	0	0
$743$	−42.0000	−1.54083	−0.770415	−	0.637542i	$-0.779951\pi$
$743$	−42.0000	−1.54083	−0.770415	+	0.637542i	$0.779951\pi$
$744$	0	0
$745$	−24.0000	−0.879292
$746$	−46.0000	−1.68418
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	13.0000	0.474377	0.237188	−	0.971464i	$-0.423774\pi$
$751$	13.0000	0.474377	0.237188	+	0.971464i	$0.423774\pi$
$752$	24.0000	0.875190
$753$	0	0
$754$	−8.00000	−0.291343
$755$	32.0000	1.16460
$756$	0	0
$757$	−22.0000	−0.799604	−0.399802	−	0.916602i	$-0.630921\pi$
$757$	−22.0000	−0.799604	−0.399802	+	0.916602i	$0.630921\pi$
$758$	−6.00000	−0.217930
$759$	0	0
$760$	0	0
$761$	−48.0000	−1.74000	−0.869999	−	0.493053i	$-0.835881\pi$
$761$	−48.0000	−1.74000	−0.869999	+	0.493053i	$0.835881\pi$
$762$	0	0
$763$	0	0
$764$	−20.0000	−0.723575
$765$	0	0
$766$	24.0000	0.867155
$767$	12.0000	0.433295
$768$	0	0
$769$	49.0000	1.76699	0.883493	−	0.468445i	$-0.155186\pi$
$769$	49.0000	1.76699	0.883493	+	0.468445i	$0.155186\pi$
$770$	0	0
$771$	0	0
$772$	22.0000	0.791797
$773$	−34.0000	−1.22290	−0.611448	−	0.791285i	$-0.709412\pi$
$773$	−34.0000	−1.22290	−0.611448	+	0.791285i	$0.709412\pi$
$774$	0	0
$775$	9.00000	0.323290
$776$	0	0
$777$	0	0
$778$	−12.0000	−0.430221
$779$	10.0000	0.358287
$780$	0	0
$781$	12.0000	0.429394
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−28.0000	−0.999363
$786$	0	0
$787$	−40.0000	−1.42585	−0.712923	−	0.701242i	$-0.752629\pi$
$787$	−40.0000	−1.42585	−0.712923	+	0.701242i	$0.752629\pi$
$788$	−32.0000	−1.13995
$789$	0	0
$790$	−4.00000	−0.142314
$791$	0	0
$792$	0	0
$793$	10.0000	0.355110
$794$	−18.0000	−0.638796
$795$	0	0
$796$	0	0
$797$	−8.00000	−0.283375	−0.141687	−	0.989911i	$-0.545253\pi$
$797$	−8.00000	−0.283375	−0.141687	+	0.989911i	$0.545253\pi$
$798$	0	0
$799$	0	0
$800$	−8.00000	−0.282843
$801$	0	0
$802$	−72.0000	−2.54241
$803$	6.00000	0.211735
$804$	0	0
$805$	0	0
$806$	−18.0000	−0.634023
$807$	0	0
$808$	0	0
$809$	−30.0000	−1.05474	−0.527372	−	0.849635i	$-0.676823\pi$
$809$	−30.0000	−1.05474	−0.527372	+	0.849635i	$0.676823\pi$
$810$	0	0
$811$	−32.0000	−1.12367	−0.561836	−	0.827249i	$-0.689905\pi$
$811$	−32.0000	−1.12367	−0.561836	+	0.827249i	$0.689905\pi$
$812$	0	0
$813$	0	0
$814$	−12.0000	−0.420600
$815$	−8.00000	−0.280228
$816$	0	0
$817$	−5.00000	−0.174928
$818$	10.0000	0.349642
$819$	0	0
$820$	40.0000	1.39686
$821$	−2.00000	−0.0698005	−0.0349002	−	0.999391i	$-0.511111\pi$
$821$	−2.00000	−0.0698005	−0.0349002	+	0.999391i	$0.511111\pi$
$822$	0	0
$823$	0	0		−	1.00000i	$-0.5\pi$
$823$	0	0			1.00000i	$0.5\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	30.0000	1.04320	0.521601	−	0.853189i	$-0.325335\pi$
$827$	30.0000	1.04320	0.521601	+	0.853189i	$0.325335\pi$
$828$	0	0
$829$	−41.0000	−1.42399	−0.711994	−	0.702185i	$-0.752208\pi$
$829$	−41.0000	−1.42399	−0.711994	+	0.702185i	$0.752208\pi$
$830$	24.0000	0.833052
$831$	0	0
$832$	8.00000	0.277350
$833$	0	0
$834$	0	0
$835$	28.0000	0.968980
$836$	−4.00000	−0.138343
$837$	0	0
$838$	−60.0000	−2.07267
$839$	−44.0000	−1.51905	−0.759524	−	0.650479i	$-0.774568\pi$
$839$	−44.0000	−1.51905	−0.759524	+	0.650479i	$0.774568\pi$
$840$	0	0
$841$	−13.0000	−0.448276
$842$	14.0000	0.482472
$843$	0	0
$844$	8.00000	0.275371
$845$	24.0000	0.825625
$846$	0	0
$847$	0	0
$848$	48.0000	1.64833
$849$	0	0
$850$	0	0
$851$	0	0
$852$	0	0
$853$	−35.0000	−1.19838	−0.599189	−	0.800608i	$-0.704510\pi$
$853$	−35.0000	−1.19838	−0.599189	+	0.800608i	$0.704510\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−32.0000	−1.09310	−0.546550	−	0.837427i	$-0.684059\pi$
$857$	−32.0000	−1.09310	−0.546550	+	0.837427i	$0.684059\pi$
$858$	0	0
$859$	40.0000	1.36478	0.682391	−	0.730987i	$-0.260940\pi$
$859$	40.0000	1.36478	0.682391	+	0.730987i	$0.260940\pi$
$860$	−20.0000	−0.681994
$861$	0	0
$862$	−36.0000	−1.22616
$863$	54.0000	1.83818	0.919091	−	0.394046i	$-0.128925\pi$
$863$	54.0000	1.83818	0.919091	+	0.394046i	$0.128925\pi$
$864$	0	0
$865$	−16.0000	−0.544016
$866$	62.0000	2.10685
$867$	0	0
$868$	0	0
$869$	−2.00000	−0.0678454
$870$	0	0
$871$	5.00000	0.169419
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−38.0000	−1.28317	−0.641584	−	0.767052i	$-0.721723\pi$
$877$	−38.0000	−1.28317	−0.641584	+	0.767052i	$0.721723\pi$
$878$	0	0
$879$	0	0
$880$	16.0000	0.539360
$881$	24.0000	0.808581	0.404290	−	0.914631i	$-0.367519\pi$
$881$	24.0000	0.808581	0.404290	+	0.914631i	$0.367519\pi$
$882$	0	0
$883$	−13.0000	−0.437485	−0.218742	−	0.975783i	$-0.570195\pi$
$883$	−13.0000	−0.437485	−0.218742	+	0.975783i	$0.570195\pi$
$884$	0	0
$885$	0	0
$886$	24.0000	0.806296
$887$	−34.0000	−1.14161	−0.570804	−	0.821086i	$-0.693368\pi$
$887$	−34.0000	−1.14161	−0.570804	+	0.821086i	$0.693368\pi$
$888$	0	0
$889$	0	0
$890$	64.0000	2.14528
$891$	0	0
$892$	−32.0000	−1.07144
$893$	6.00000	0.200782
$894$	0	0
$895$	4.00000	0.133705
$896$	0	0
$897$	0	0
$898$	−36.0000	−1.20134
$899$	36.0000	1.20067
$900$	0	0
$901$	0	0
$902$	40.0000	1.33185
$903$	0	0
$904$	0	0
$905$	26.0000	0.864269
$906$	0	0
$907$	−37.0000	−1.22856	−0.614282	−	0.789086i	$-0.710554\pi$
$907$	−37.0000	−1.22856	−0.614282	+	0.789086i	$0.710554\pi$
$908$	36.0000	1.19470
$909$	0	0
$910$	0	0
$911$	24.0000	0.795155	0.397578	−	0.917568i	$-0.369851\pi$
$911$	24.0000	0.795155	0.397578	+	0.917568i	$0.369851\pi$
$912$	0	0
$913$	12.0000	0.397142
$914$	22.0000	0.727695
$915$	0	0
$916$	38.0000	1.25556
$917$	0	0
$918$	0	0
$919$	23.0000	0.758700	0.379350	−	0.925253i	$-0.376148\pi$
$919$	23.0000	0.758700	0.379350	+	0.925253i	$0.376148\pi$
$920$	0	0
$921$	0	0
$922$	−40.0000	−1.31733
$923$	−6.00000	−0.197492
$924$	0	0
$925$	−3.00000	−0.0986394
$926$	34.0000	1.11731
$927$	0	0
$928$	−32.0000	−1.05045
$929$	14.0000	0.459325	0.229663	−	0.973270i	$-0.426238\pi$
$929$	14.0000	0.459325	0.229663	+	0.973270i	$0.426238\pi$
$930$	0	0
$931$	0	0
$932$	−12.0000	−0.393073
$933$	0	0
$934$	−12.0000	−0.392652
$935$	0	0
$936$	0	0
$937$	−15.0000	−0.490029	−0.245014	−	0.969519i	$-0.578793\pi$
$937$	−15.0000	−0.490029	−0.245014	+	0.969519i	$0.578793\pi$
$938$	0	0
$939$	0	0
$940$	24.0000	0.782794
$941$	−4.00000	−0.130396	−0.0651981	−	0.997872i	$-0.520768\pi$
$941$	−4.00000	−0.130396	−0.0651981	+	0.997872i	$0.520768\pi$
$942$	0	0
$943$	0	0
$944$	48.0000	1.56227
$945$	0	0
$946$	−20.0000	−0.650256
$947$	10.0000	0.324956	0.162478	−	0.986712i	$-0.448051\pi$
$947$	10.0000	0.324956	0.162478	+	0.986712i	$0.448051\pi$
$948$	0	0
$949$	−3.00000	−0.0973841
$950$	−2.00000	−0.0648886
$951$	0	0
$952$	0	0
$953$	−44.0000	−1.42530	−0.712650	−	0.701520i	$-0.752505\pi$
$953$	−44.0000	−1.42530	−0.712650	+	0.701520i	$0.752505\pi$
$954$	0	0
$955$	20.0000	0.647185
$956$	−12.0000	−0.388108
$957$	0	0
$958$	56.0000	1.80928
$959$	0	0
$960$	0	0
$961$	50.0000	1.61290
$962$	6.00000	0.193448
$963$	0	0
$964$	−28.0000	−0.901819
$965$	−22.0000	−0.708205
$966$	0	0
$967$	19.0000	0.610999	0.305499	−	0.952192i	$-0.401177\pi$
$967$	19.0000	0.610999	0.305499	+	0.952192i	$0.401177\pi$
$968$	0	0
$969$	0	0
$970$	24.0000	0.770594
$971$	36.0000	1.15529	0.577647	−	0.816286i	$-0.303971\pi$
$971$	36.0000	1.15529	0.577647	+	0.816286i	$0.303971\pi$
$972$	0	0
$973$	0	0
$974$	−62.0000	−1.98661
$975$	0	0
$976$	40.0000	1.28037
$977$	18.0000	0.575871	0.287936	−	0.957650i	$-0.407031\pi$
$977$	18.0000	0.575871	0.287936	+	0.957650i	$0.407031\pi$
$978$	0	0
$979$	32.0000	1.02272
$980$	0	0
$981$	0	0
$982$	−56.0000	−1.78703
$983$	36.0000	1.14822	0.574111	−	0.818778i	$-0.305348\pi$
$983$	36.0000	1.14822	0.574111	+	0.818778i	$0.305348\pi$
$984$	0	0
$985$	32.0000	1.01960
$986$	0	0
$987$	0	0
$988$	2.00000	0.0636285
$989$	0	0
$990$	0	0
$991$	17.0000	0.540023	0.270011	−	0.962857i	$-0.412973\pi$
$991$	17.0000	0.540023	0.270011	+	0.962857i	$0.412973\pi$
$992$	−72.0000	−2.28600
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−19.0000	−0.601736	−0.300868	−	0.953666i	$-0.597276\pi$
$997$	−19.0000	−0.601736	−0.300868	+	0.953666i	$0.597276\pi$
$998$	−74.0000	−2.34243
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.2.a.a.1.1		1
3.2	odd	2		147.2.a.b.1.1		1
4.3	odd	2		7056.2.a.m.1.1		1
7.2	even	3		441.2.e.e.361.1		2
7.3	odd	6		63.2.e.b.37.1		2
7.4	even	3		441.2.e.e.226.1		2
7.5	odd	6		63.2.e.b.46.1		2
7.6	odd	2		441.2.a.b.1.1		1
12.11	even	2		2352.2.a.w.1.1		1
15.14	odd	2		3675.2.a.c.1.1		1
21.2	odd	6		147.2.e.a.67.1		2
21.5	even	6		21.2.e.a.4.1	✓	2
21.11	odd	6		147.2.e.a.79.1		2
21.17	even	6		21.2.e.a.16.1	yes	2
21.20	even	2		147.2.a.c.1.1		1
24.5	odd	2		9408.2.a.bz.1.1		1
24.11	even	2		9408.2.a.k.1.1		1
28.3	even	6		1008.2.s.d.289.1		2
28.19	even	6		1008.2.s.d.865.1		2
28.27	even	2		7056.2.a.bp.1.1		1
63.5	even	6		567.2.h.f.298.1		2
63.31	odd	6		567.2.g.f.541.1		2
63.38	even	6		567.2.h.f.352.1		2
63.40	odd	6		567.2.h.a.298.1		2
63.47	even	6		567.2.g.a.109.1		2
63.52	odd	6		567.2.h.a.352.1		2
63.59	even	6		567.2.g.a.541.1		2
63.61	odd	6		567.2.g.f.109.1		2
84.11	even	6		2352.2.q.c.961.1		2
84.23	even	6		2352.2.q.c.1537.1		2
84.47	odd	6		336.2.q.f.193.1		2
84.59	odd	6		336.2.q.f.289.1		2
84.83	odd	2		2352.2.a.d.1.1		1
105.17	odd	12		525.2.r.e.499.1		4
105.38	odd	12		525.2.r.e.499.2		4
105.47	odd	12		525.2.r.e.424.2		4
105.59	even	6		525.2.i.e.226.1		2
105.68	odd	12		525.2.r.e.424.1		4
105.89	even	6		525.2.i.e.151.1		2
105.104	even	2		3675.2.a.a.1.1		1
168.5	even	6		1344.2.q.m.193.1		2
168.59	odd	6		1344.2.q.c.961.1		2
168.83	odd	2		9408.2.a.cv.1.1		1
168.101	even	6		1344.2.q.m.961.1		2
168.125	even	2		9408.2.a.bg.1.1		1
168.131	odd	6		1344.2.q.c.193.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
21.2.e.a.4.1	✓	2	21.5	even	6
21.2.e.a.16.1	yes	2	21.17	even	6
63.2.e.b.37.1		2	7.3	odd	6
63.2.e.b.46.1		2	7.5	odd	6
147.2.a.b.1.1		1	3.2	odd	2
147.2.a.c.1.1		1	21.20	even	2
147.2.e.a.67.1		2	21.2	odd	6
147.2.e.a.79.1		2	21.11	odd	6
336.2.q.f.193.1		2	84.47	odd	6
336.2.q.f.289.1		2	84.59	odd	6
441.2.a.a.1.1		1	1.1	even	1	trivial
441.2.a.b.1.1		1	7.6	odd	2
441.2.e.e.226.1		2	7.4	even	3
441.2.e.e.361.1		2	7.2	even	3
525.2.i.e.151.1		2	105.89	even	6
525.2.i.e.226.1		2	105.59	even	6
525.2.r.e.424.1		4	105.68	odd	12
525.2.r.e.424.2		4	105.47	odd	12
525.2.r.e.499.1		4	105.17	odd	12
525.2.r.e.499.2		4	105.38	odd	12
567.2.g.a.109.1		2	63.47	even	6
567.2.g.a.541.1		2	63.59	even	6
567.2.g.f.109.1		2	63.61	odd	6
567.2.g.f.541.1		2	63.31	odd	6
567.2.h.a.298.1		2	63.40	odd	6
567.2.h.a.352.1		2	63.52	odd	6
567.2.h.f.298.1		2	63.5	even	6
567.2.h.f.352.1		2	63.38	even	6
1008.2.s.d.289.1		2	28.3	even	6
1008.2.s.d.865.1		2	28.19	even	6
1344.2.q.c.193.1		2	168.131	odd	6
1344.2.q.c.961.1		2	168.59	odd	6
1344.2.q.m.193.1		2	168.5	even	6
1344.2.q.m.961.1		2	168.101	even	6
2352.2.a.d.1.1		1	84.83	odd	2
2352.2.a.w.1.1		1	12.11	even	2
2352.2.q.c.961.1		2	84.11	even	6
2352.2.q.c.1537.1		2	84.23	even	6
3675.2.a.a.1.1		1	105.104	even	2
3675.2.a.c.1.1		1	15.14	odd	2
7056.2.a.m.1.1		1	4.3	odd	2
7056.2.a.bp.1.1		1	28.27	even	2
9408.2.a.k.1.1		1	24.11	even	2
9408.2.a.bg.1.1		1	168.125	even	2
9408.2.a.bz.1.1		1	24.5	odd	2
9408.2.a.cv.1.1		1	168.83	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:	\( N \)	\(=\)	\( 441 = 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	441.a (trivial)

Introduction

L-functions

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