LMFDB - Embedded newform 441.4.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [441,4,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, 4, names="a")

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

chi := DirichletCharacter("441.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 441 = 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
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:	$26.0198423125$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 147)
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)

$f(q)$

$=$

$q-4.00000 q^{2} +8.00000 q^{4} -18.0000 q^{5} +O(q^{10})$

$q-4.00000 q^{2} +8.00000 q^{4} -18.0000 q^{5} +72.0000 q^{10} +50.0000 q^{11} -36.0000 q^{13} -64.0000 q^{16} -126.000 q^{17} -72.0000 q^{19} -144.000 q^{20} -200.000 q^{22} -14.0000 q^{23} +199.000 q^{25} +144.000 q^{26} -158.000 q^{29} -36.0000 q^{31} +256.000 q^{32} +504.000 q^{34} -162.000 q^{37} +288.000 q^{38} +270.000 q^{41} -324.000 q^{43} +400.000 q^{44} +56.0000 q^{46} +72.0000 q^{47} -796.000 q^{50} -288.000 q^{52} +22.0000 q^{53} -900.000 q^{55} +632.000 q^{58} -468.000 q^{59} +792.000 q^{61} +144.000 q^{62} -512.000 q^{64} +648.000 q^{65} +232.000 q^{67} -1008.00 q^{68} +734.000 q^{71} +180.000 q^{73} +648.000 q^{74} -576.000 q^{76} +236.000 q^{79} +1152.00 q^{80} -1080.00 q^{82} -36.0000 q^{83} +2268.00 q^{85} +1296.00 q^{86} -234.000 q^{89} -112.000 q^{92} -288.000 q^{94} +1296.00 q^{95} +468.000 q^{97} +O(q^{100})$

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 $

$2$	−4.00000	−1.41421	−0.707107	−	0.707107i	$-0.750000\pi$
$2$	−4.00000	−1.41421	−0.707107	+	0.707107i	$0.750000\pi$
$3$	0	0
$3$	0	0
$4$	8.00000	1.00000
$5$	−18.0000	−1.60997	−0.804984	−	0.593296i	$-0.797826\pi$
$5$	−18.0000	−1.60997	−0.804984	+	0.593296i	$0.797826\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	0	0
$10$	72.0000	2.27684
$11$	50.0000	1.37051	0.685253	−	0.728305i	$-0.259692\pi$
$11$	50.0000	1.37051	0.685253	+	0.728305i	$0.259692\pi$
$12$	0	0
$13$	−36.0000	−0.768046	−0.384023	−	0.923323i	$-0.625462\pi$
$13$	−36.0000	−0.768046	−0.384023	+	0.923323i	$0.625462\pi$
$14$	0	0
$15$	0	0
$16$	−64.0000	−1.00000
$17$	−126.000	−1.79762	−0.898808	−	0.438342i	$-0.855566\pi$
$17$	−126.000	−1.79762	−0.898808	+	0.438342i	$0.855566\pi$
$18$	0	0
$19$	−72.0000	−0.869365	−0.434682	−	0.900584i	$-0.643139\pi$
$19$	−72.0000	−0.869365	−0.434682	+	0.900584i	$0.643139\pi$
$20$	−144.000	−1.60997
$21$	0	0
$22$	−200.000	−1.93819
$23$	−14.0000	−0.126922	−0.0634609	−	0.997984i	$-0.520214\pi$
$23$	−14.0000	−0.126922	−0.0634609	+	0.997984i	$0.520214\pi$
$24$	0	0
$25$	199.000	1.59200
$26$	144.000	1.08618
$27$	0	0
$28$	0	0
$29$	−158.000	−1.01172	−0.505860	−	0.862616i	$-0.668825\pi$
$29$	−158.000	−1.01172	−0.505860	+	0.862616i	$0.668825\pi$
$30$	0	0
$31$	−36.0000	−0.208574	−0.104287	−	0.994547i	$-0.533256\pi$
$31$	−36.0000	−0.208574	−0.104287	+	0.994547i	$0.533256\pi$
$32$	256.000	1.41421
$33$	0	0
$34$	504.000	2.54221
$35$	0	0
$36$	0	0
$37$	−162.000	−0.719801	−0.359900	−	0.932991i	$-0.617189\pi$
$37$	−162.000	−0.719801	−0.359900	+	0.932991i	$0.617189\pi$
$38$	288.000	1.22947
$39$	0	0
$40$	0	0
$41$	270.000	1.02846	0.514231	−	0.857652i	$-0.328078\pi$
$41$	270.000	1.02846	0.514231	+	0.857652i	$0.328078\pi$
$42$	0	0
$43$	−324.000	−1.14906	−0.574529	−	0.818484i	$-0.694815\pi$
$43$	−324.000	−1.14906	−0.574529	+	0.818484i	$0.694815\pi$
$44$	400.000	1.37051
$45$	0	0
$46$	56.0000	0.179495
$47$	72.0000	0.223453	0.111726	−	0.993739i	$-0.464362\pi$
$47$	72.0000	0.223453	0.111726	+	0.993739i	$0.464362\pi$
$48$	0	0
$49$	0	0
$50$	−796.000	−2.25143
$51$	0	0
$52$	−288.000	−0.768046
$53$	22.0000	0.0570176	0.0285088	−	0.999594i	$-0.490924\pi$
$53$	22.0000	0.0570176	0.0285088	+	0.999594i	$0.490924\pi$
$54$	0	0
$55$	−900.000	−2.20647
$56$	0	0
$57$	0	0
$58$	632.000	1.43079
$59$	−468.000	−1.03268	−0.516342	−	0.856382i	$-0.672707\pi$
$59$	−468.000	−1.03268	−0.516342	+	0.856382i	$0.672707\pi$
$60$	0	0
$61$	792.000	1.66238	0.831190	−	0.555988i	$-0.187660\pi$
$61$	792.000	1.66238	0.831190	+	0.555988i	$0.187660\pi$
$62$	144.000	0.294968
$63$	0	0
$64$	−512.000	−1.00000
$65$	648.000	1.23653
$66$	0	0
$67$	232.000	0.423034	0.211517	−	0.977374i	$-0.432160\pi$
$67$	232.000	0.423034	0.211517	+	0.977374i	$0.432160\pi$
$68$	−1008.00	−1.79762
$69$	0	0
$70$	0	0
$71$	734.000	1.22690	0.613449	−	0.789734i	$-0.289782\pi$
$71$	734.000	1.22690	0.613449	+	0.789734i	$0.289782\pi$
$72$	0	0
$73$	180.000	0.288595	0.144297	−	0.989534i	$-0.453908\pi$
$73$	180.000	0.288595	0.144297	+	0.989534i	$0.453908\pi$
$74$	648.000	1.01795
$75$	0	0
$76$	−576.000	−0.869365
$77$	0	0
$78$	0	0
$79$	236.000	0.336102	0.168051	−	0.985778i	$-0.446253\pi$
$79$	236.000	0.336102	0.168051	+	0.985778i	$0.446253\pi$
$80$	1152.00	1.60997
$81$	0	0
$82$	−1080.00	−1.45446
$83$	−36.0000	−0.0476086	−0.0238043	−	0.999717i	$-0.507578\pi$
$83$	−36.0000	−0.0476086	−0.0238043	+	0.999717i	$0.507578\pi$
$84$	0	0
$85$	2268.00	2.89411
$86$	1296.00	1.62501
$87$	0	0
$88$	0	0
$89$	−234.000	−0.278696	−0.139348	−	0.990243i	$-0.544501\pi$
$89$	−234.000	−0.278696	−0.139348	+	0.990243i	$0.544501\pi$
$90$	0	0
$91$	0	0
$92$	−112.000	−0.126922
$93$	0	0
$94$	−288.000	−0.316010
$95$	1296.00	1.39965
$96$	0	0
$97$	468.000	0.489878	0.244939	−	0.969538i	$-0.421232\pi$
$97$	468.000	0.489878	0.244939	+	0.969538i	$0.421232\pi$
$98$	0	0
$99$	0	0
$100$	1592.00	1.59200
$101$	666.000	0.656133	0.328067	−	0.944655i	$-0.393603\pi$
$101$	666.000	0.656133	0.328067	+	0.944655i	$0.393603\pi$
$102$	0	0
$103$	−252.000	−0.241071	−0.120535	−	0.992709i	$-0.538461\pi$
$103$	−252.000	−0.241071	−0.120535	+	0.992709i	$0.538461\pi$
$104$	0	0
$105$	0	0
$106$	−88.0000	−0.0806351
$107$	−670.000	−0.605340	−0.302670	−	0.953095i	$-0.597878\pi$
$107$	−670.000	−0.605340	−0.302670	+	0.953095i	$0.597878\pi$
$108$	0	0
$109$	162.000	0.142356	0.0711779	−	0.997464i	$-0.477324\pi$
$109$	162.000	0.142356	0.0711779	+	0.997464i	$0.477324\pi$
$110$	3600.00	3.12042
$111$	0	0
$112$	0	0
$113$	1390.00	1.15717	0.578585	−	0.815622i	$-0.303605\pi$
$113$	1390.00	1.15717	0.578585	+	0.815622i	$0.303605\pi$
$114$	0	0
$115$	252.000	0.204340
$116$	−1264.00	−1.01172
$117$	0	0
$118$	1872.00	1.46044
$119$	0	0
$120$	0	0
$121$	1169.00	0.878287
$122$	−3168.00	−2.35096
$123$	0	0
$124$	−288.000	−0.208574
$125$	−1332.00	−0.953102
$126$	0	0
$127$	916.000	0.640015	0.320007	−	0.947415i	$-0.396315\pi$
$127$	916.000	0.640015	0.320007	+	0.947415i	$0.396315\pi$
$128$	0	0
$129$	0	0
$130$	−2592.00	−1.74872
$131$	−2268.00	−1.51264	−0.756321	−	0.654201i	$-0.773005\pi$
$131$	−2268.00	−1.51264	−0.756321	+	0.654201i	$0.773005\pi$
$132$	0	0
$133$	0	0
$134$	−928.000	−0.598261
$135$	0	0
$136$	0	0
$137$	−806.000	−0.502637	−0.251318	−	0.967904i	$-0.580864\pi$
$137$	−806.000	−0.502637	−0.251318	+	0.967904i	$0.580864\pi$
$138$	0	0
$139$	2628.00	1.60363	0.801813	−	0.597575i	$-0.203869\pi$
$139$	2628.00	1.60363	0.801813	+	0.597575i	$0.203869\pi$
$140$	0	0
$141$	0	0
$142$	−2936.00	−1.73510
$143$	−1800.00	−1.05261
$144$	0	0
$145$	2844.00	1.62884
$146$	−720.000	−0.408134
$147$	0	0
$148$	−1296.00	−0.719801
$149$	2390.00	1.31407	0.657035	−	0.753860i	$-0.271810\pi$
$149$	2390.00	1.31407	0.657035	+	0.753860i	$0.271810\pi$
$150$	0	0
$151$	3240.00	1.74614	0.873071	−	0.487593i	$-0.162125\pi$
$151$	3240.00	1.74614	0.873071	+	0.487593i	$0.162125\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	648.000	0.335798
$156$	0	0
$157$	3024.00	1.53721	0.768603	−	0.639726i	$-0.220952\pi$
$157$	3024.00	1.53721	0.768603	+	0.639726i	$0.220952\pi$
$158$	−944.000	−0.475320
$159$	0	0
$160$	−4608.00	−2.27684
$161$	0	0
$162$	0	0
$163$	−1784.00	−0.857262	−0.428631	−	0.903480i	$-0.641004\pi$
$163$	−1784.00	−0.857262	−0.428631	+	0.903480i	$0.641004\pi$
$164$	2160.00	1.02846
$165$	0	0
$166$	144.000	0.0673287
$167$	3024.00	1.40122	0.700611	−	0.713543i	$-0.252911\pi$
$167$	3024.00	1.40122	0.700611	+	0.713543i	$0.252911\pi$
$168$	0	0
$169$	−901.000	−0.410105
$170$	−9072.00	−4.09289
$171$	0	0
$172$	−2592.00	−1.14906
$173$	−1566.00	−0.688213	−0.344106	−	0.938931i	$-0.611818\pi$
$173$	−1566.00	−0.688213	−0.344106	+	0.938931i	$0.611818\pi$
$174$	0	0
$175$	0	0
$176$	−3200.00	−1.37051
$177$	0	0
$178$	936.000	0.394136
$179$	−3802.00	−1.58757	−0.793784	−	0.608199i	$-0.791892\pi$
$179$	−3802.00	−1.58757	−0.793784	+	0.608199i	$0.791892\pi$
$180$	0	0
$181$	−468.000	−0.192189	−0.0960944	−	0.995372i	$-0.530635\pi$
$181$	−468.000	−0.192189	−0.0960944	+	0.995372i	$0.530635\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	2916.00	1.15886
$186$	0	0
$187$	−6300.00	−2.46365
$188$	576.000	0.223453
$189$	0	0
$190$	−5184.00	−1.97940
$191$	−482.000	−0.182598	−0.0912992	−	0.995824i	$-0.529102\pi$
$191$	−482.000	−0.182598	−0.0912992	+	0.995824i	$0.529102\pi$
$192$	0	0
$193$	−810.000	−0.302099	−0.151049	−	0.988526i	$-0.548265\pi$
$193$	−810.000	−0.302099	−0.151049	+	0.988526i	$0.548265\pi$
$194$	−1872.00	−0.692793
$195$	0	0
$196$	0	0
$197$	2462.00	0.890407	0.445204	−	0.895429i	$-0.353131\pi$
$197$	2462.00	0.890407	0.445204	+	0.895429i	$0.353131\pi$
$198$	0	0
$199$	−4536.00	−1.61582	−0.807911	−	0.589305i	$-0.799402\pi$
$199$	−4536.00	−1.61582	−0.807911	+	0.589305i	$0.799402\pi$
$200$	0	0
$201$	0	0
$202$	−2664.00	−0.927913
$203$	0	0
$204$	0	0
$205$	−4860.00	−1.65579
$206$	1008.00	0.340926
$207$	0	0
$208$	2304.00	0.768046
$209$	−3600.00	−1.19147
$210$	0	0
$211$	2916.00	0.951402	0.475701	−	0.879607i	$-0.342195\pi$
$211$	2916.00	0.951402	0.475701	+	0.879607i	$0.342195\pi$
$212$	176.000	0.0570176
$213$	0	0
$214$	2680.00	0.856080
$215$	5832.00	1.84995
$216$	0	0
$217$	0	0
$218$	−648.000	−0.201322
$219$	0	0
$220$	−7200.00	−2.20647
$221$	4536.00	1.38065
$222$	0	0
$223$	−1080.00	−0.324315	−0.162157	−	0.986765i	$-0.551845\pi$
$223$	−1080.00	−0.324315	−0.162157	+	0.986765i	$0.551845\pi$
$224$	0	0
$225$	0	0
$226$	−5560.00	−1.63649
$227$	1332.00	0.389462	0.194731	−	0.980857i	$-0.437617\pi$
$227$	1332.00	0.389462	0.194731	+	0.980857i	$0.437617\pi$
$228$	0	0
$229$	1620.00	0.467479	0.233739	−	0.972299i	$-0.424904\pi$
$229$	1620.00	0.467479	0.233739	+	0.972299i	$0.424904\pi$
$230$	−1008.00	−0.288981
$231$	0	0
$232$	0	0
$233$	−6718.00	−1.88889	−0.944444	−	0.328673i	$-0.893399\pi$
$233$	−6718.00	−1.88889	−0.944444	+	0.328673i	$0.893399\pi$
$234$	0	0
$235$	−1296.00	−0.359752
$236$	−3744.00	−1.03268
$237$	0	0
$238$	0	0
$239$	3578.00	0.968375	0.484187	−	0.874964i	$-0.339115\pi$
$239$	3578.00	0.968375	0.484187	+	0.874964i	$0.339115\pi$
$240$	0	0
$241$	−756.000	−0.202067	−0.101034	−	0.994883i	$-0.532215\pi$
$241$	−756.000	−0.202067	−0.101034	+	0.994883i	$0.532215\pi$
$242$	−4676.00	−1.24209
$243$	0	0
$244$	6336.00	1.66238
$245$	0	0
$246$	0	0
$247$	2592.00	0.667713
$248$	0	0
$249$	0	0
$250$	5328.00	1.34789
$251$	6516.00	1.63859	0.819295	−	0.573372i	$-0.194365\pi$
$251$	6516.00	1.63859	0.819295	+	0.573372i	$0.194365\pi$
$252$	0	0
$253$	−700.000	−0.173947
$254$	−3664.00	−0.905117
$255$	0	0
$256$	4096.00	1.00000
$257$	6030.00	1.46358	0.731792	−	0.681528i	$-0.238684\pi$
$257$	6030.00	1.46358	0.731792	+	0.681528i	$0.238684\pi$
$258$	0	0
$259$	0	0
$260$	5184.00	1.23653
$261$	0	0
$262$	9072.00	2.13920
$263$	−590.000	−0.138331	−0.0691653	−	0.997605i	$-0.522034\pi$
$263$	−590.000	−0.138331	−0.0691653	+	0.997605i	$0.522034\pi$
$264$	0	0
$265$	−396.000	−0.0917966
$266$	0	0
$267$	0	0
$268$	1856.00	0.423034
$269$	990.000	0.224392	0.112196	−	0.993686i	$-0.464212\pi$
$269$	990.000	0.224392	0.112196	+	0.993686i	$0.464212\pi$
$270$	0	0
$271$	−3420.00	−0.766606	−0.383303	−	0.923623i	$-0.625213\pi$
$271$	−3420.00	−0.766606	−0.383303	+	0.923623i	$0.625213\pi$
$272$	8064.00	1.79762
$273$	0	0
$274$	3224.00	0.710836
$275$	9950.00	2.18185
$276$	0	0
$277$	−2734.00	−0.593033	−0.296516	−	0.955028i	$-0.595825\pi$
$277$	−2734.00	−0.593033	−0.296516	+	0.955028i	$0.595825\pi$
$278$	−10512.0	−2.26787
$279$	0	0
$280$	0	0
$281$	−598.000	−0.126953	−0.0634763	−	0.997983i	$-0.520219\pi$
$281$	−598.000	−0.126953	−0.0634763	+	0.997983i	$0.520219\pi$
$282$	0	0
$283$	3600.00	0.756176	0.378088	−	0.925770i	$-0.376582\pi$
$283$	3600.00	0.756176	0.378088	+	0.925770i	$0.376582\pi$
$284$	5872.00	1.22690
$285$	0	0
$286$	7200.00	1.48862
$287$	0	0
$288$	0	0
$289$	10963.0	2.23143
$290$	−11376.0	−2.30352
$291$	0	0
$292$	1440.00	0.288595
$293$	−7902.00	−1.57556	−0.787781	−	0.615955i	$-0.788770\pi$
$293$	−7902.00	−1.57556	−0.787781	+	0.615955i	$0.788770\pi$
$294$	0	0
$295$	8424.00	1.66259
$296$	0	0
$297$	0	0
$298$	−9560.00	−1.85838
$299$	504.000	0.0974818
$300$	0	0
$301$	0	0
$302$	−12960.0	−2.46942
$303$	0	0
$304$	4608.00	0.869365
$305$	−14256.0	−2.67638
$306$	0	0
$307$	−10224.0	−1.90070	−0.950349	−	0.311185i	$-0.899274\pi$
$307$	−10224.0	−1.90070	−0.950349	+	0.311185i	$0.899274\pi$
$308$	0	0
$309$	0	0
$310$	−2592.00	−0.474889
$311$	−3888.00	−0.708901	−0.354451	−	0.935075i	$-0.615332\pi$
$311$	−3888.00	−0.708901	−0.354451	+	0.935075i	$0.615332\pi$
$312$	0	0
$313$	5112.00	0.923154	0.461577	−	0.887100i	$-0.347284\pi$
$313$	5112.00	0.923154	0.461577	+	0.887100i	$0.347284\pi$
$314$	−12096.0	−2.17394
$315$	0	0
$316$	1888.00	0.336102
$317$	10102.0	1.78986	0.894929	−	0.446209i	$-0.147226\pi$
$317$	10102.0	1.78986	0.894929	+	0.446209i	$0.147226\pi$
$318$	0	0
$319$	−7900.00	−1.38657
$320$	9216.00	1.60997
$321$	0	0
$322$	0	0
$323$	9072.00	1.56279
$324$	0	0
$325$	−7164.00	−1.22273
$326$	7136.00	1.21235
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	5508.00	0.914644	0.457322	−	0.889301i	$-0.348809\pi$
$331$	5508.00	0.914644	0.457322	+	0.889301i	$0.348809\pi$
$332$	−288.000	−0.0476086
$333$	0	0
$334$	−12096.0	−1.98163
$335$	−4176.00	−0.681072
$336$	0	0
$337$	−9234.00	−1.49261	−0.746303	−	0.665607i	$-0.768173\pi$
$337$	−9234.00	−1.49261	−0.746303	+	0.665607i	$0.768173\pi$
$338$	3604.00	0.579976
$339$	0	0
$340$	18144.0	2.89411
$341$	−1800.00	−0.285852
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	6264.00	0.973280
$347$	6494.00	1.00466	0.502329	−	0.864677i	$-0.332477\pi$
$347$	6494.00	1.00466	0.502329	+	0.864677i	$0.332477\pi$
$348$	0	0
$349$	10080.0	1.54605	0.773023	−	0.634378i	$-0.218744\pi$
$349$	10080.0	1.54605	0.773023	+	0.634378i	$0.218744\pi$
$350$	0	0
$351$	0	0
$352$	12800.0	1.93819
$353$	738.000	0.111274	0.0556371	−	0.998451i	$-0.482281\pi$
$353$	738.000	0.111274	0.0556371	+	0.998451i	$0.482281\pi$
$354$	0	0
$355$	−13212.0	−1.97527
$356$	−1872.00	−0.278696
$357$	0	0
$358$	15208.0	2.24516
$359$	−194.000	−0.0285207	−0.0142603	−	0.999898i	$-0.504539\pi$
$359$	−194.000	−0.0285207	−0.0142603	+	0.999898i	$0.504539\pi$
$360$	0	0
$361$	−1675.00	−0.244205
$362$	1872.00	0.271796
$363$	0	0
$364$	0	0
$365$	−3240.00	−0.464628
$366$	0	0
$367$	−4752.00	−0.675892	−0.337946	−	0.941165i	$-0.609732\pi$
$367$	−4752.00	−0.675892	−0.337946	+	0.941165i	$0.609732\pi$
$368$	896.000	0.126922
$369$	0	0
$370$	−11664.0	−1.63887
$371$	0	0
$372$	0	0
$373$	−2306.00	−0.320108	−0.160054	−	0.987108i	$-0.551167\pi$
$373$	−2306.00	−0.320108	−0.160054	+	0.987108i	$0.551167\pi$
$374$	25200.0	3.48412
$375$	0	0
$376$	0	0
$377$	5688.00	0.777047
$378$	0	0
$379$	−7452.00	−1.00998	−0.504991	−	0.863124i	$-0.668504\pi$
$379$	−7452.00	−1.00998	−0.504991	+	0.863124i	$0.668504\pi$
$380$	10368.0	1.39965
$381$	0	0
$382$	1928.00	0.258233
$383$	1152.00	0.153693	0.0768465	−	0.997043i	$-0.475515\pi$
$383$	1152.00	0.153693	0.0768465	+	0.997043i	$0.475515\pi$
$384$	0	0
$385$	0	0
$386$	3240.00	0.427232
$387$	0	0
$388$	3744.00	0.489878
$389$	−1894.00	−0.246863	−0.123431	−	0.992353i	$-0.539390\pi$
$389$	−1894.00	−0.246863	−0.123431	+	0.992353i	$0.539390\pi$
$390$	0	0
$391$	1764.00	0.228157
$392$	0	0
$393$	0	0
$394$	−9848.00	−1.25923
$395$	−4248.00	−0.541114
$396$	0	0
$397$	9216.00	1.16508	0.582541	−	0.812801i	$-0.302058\pi$
$397$	9216.00	1.16508	0.582541	+	0.812801i	$0.302058\pi$
$398$	18144.0	2.28512
$399$	0	0
$400$	−12736.0	−1.59200
$401$	11650.0	1.45081	0.725403	−	0.688324i	$-0.241653\pi$
$401$	11650.0	1.45081	0.725403	+	0.688324i	$0.241653\pi$
$402$	0	0
$403$	1296.00	0.160194
$404$	5328.00	0.656133
$405$	0	0
$406$	0	0
$407$	−8100.00	−0.986492
$408$	0	0
$409$	−7524.00	−0.909628	−0.454814	−	0.890586i	$-0.650294\pi$
$409$	−7524.00	−0.909628	−0.454814	+	0.890586i	$0.650294\pi$
$410$	19440.0	2.34164
$411$	0	0
$412$	−2016.00	−0.241071
$413$	0	0
$414$	0	0
$415$	648.000	0.0766484
$416$	−9216.00	−1.08618
$417$	0	0
$418$	14400.0	1.68499
$419$	3852.00	0.449123	0.224561	−	0.974460i	$-0.427905\pi$
$419$	3852.00	0.449123	0.224561	+	0.974460i	$0.427905\pi$
$420$	0	0
$421$	10402.0	1.20419	0.602093	−	0.798426i	$-0.294334\pi$
$421$	10402.0	1.20419	0.602093	+	0.798426i	$0.294334\pi$
$422$	−11664.0	−1.34548
$423$	0	0
$424$	0	0
$425$	−25074.0	−2.86181
$426$	0	0
$427$	0	0
$428$	−5360.00	−0.605340
$429$	0	0
$430$	−23328.0	−2.61622
$431$	10390.0	1.16118	0.580590	−	0.814196i	$-0.302822\pi$
$431$	10390.0	1.16118	0.580590	+	0.814196i	$0.302822\pi$
$432$	0	0
$433$	11232.0	1.24659	0.623297	−	0.781985i	$-0.285793\pi$
$433$	11232.0	1.24659	0.623297	+	0.781985i	$0.285793\pi$
$434$	0	0
$435$	0	0
$436$	1296.00	0.142356
$437$	1008.00	0.110341
$438$	0	0
$439$	14616.0	1.58903	0.794514	−	0.607245i	$-0.207725\pi$
$439$	14616.0	1.58903	0.794514	+	0.607245i	$0.207725\pi$
$440$	0	0
$441$	0	0
$442$	−18144.0	−1.95254
$443$	−11938.0	−1.28034	−0.640171	−	0.768232i	$-0.721137\pi$
$443$	−11938.0	−1.28034	−0.640171	+	0.768232i	$0.721137\pi$
$444$	0	0
$445$	4212.00	0.448692
$446$	4320.00	0.458650
$447$	0	0
$448$	0	0
$449$	−8186.00	−0.860404	−0.430202	−	0.902733i	$-0.641558\pi$
$449$	−8186.00	−0.860404	−0.430202	+	0.902733i	$0.641558\pi$
$450$	0	0
$451$	13500.0	1.40951
$452$	11120.0	1.15717
$453$	0	0
$454$	−5328.00	−0.550783
$455$	0	0
$456$	0	0
$457$	2106.00	0.215568	0.107784	−	0.994174i	$-0.465625\pi$
$457$	2106.00	0.215568	0.107784	+	0.994174i	$0.465625\pi$
$458$	−6480.00	−0.661115
$459$	0	0
$460$	2016.00	0.204340
$461$	−9486.00	−0.958367	−0.479183	−	0.877715i	$-0.659067\pi$
$461$	−9486.00	−0.958367	−0.479183	+	0.877715i	$0.659067\pi$
$462$	0	0
$463$	−12652.0	−1.26995	−0.634977	−	0.772531i	$-0.718990\pi$
$463$	−12652.0	−1.26995	−0.634977	+	0.772531i	$0.718990\pi$
$464$	10112.0	1.01172
$465$	0	0
$466$	26872.0	2.67129
$467$	3708.00	0.367421	0.183711	−	0.982980i	$-0.441189\pi$
$467$	3708.00	0.367421	0.183711	+	0.982980i	$0.441189\pi$
$468$	0	0
$469$	0	0
$470$	5184.00	0.508766
$471$	0	0
$472$	0	0
$473$	−16200.0	−1.57479
$474$	0	0
$475$	−14328.0	−1.38403
$476$	0	0
$477$	0	0
$478$	−14312.0	−1.36949
$479$	8064.00	0.769214	0.384607	−	0.923080i	$-0.374337\pi$
$479$	8064.00	0.769214	0.384607	+	0.923080i	$0.374337\pi$
$480$	0	0
$481$	5832.00	0.552841
$482$	3024.00	0.285766
$483$	0	0
$484$	9352.00	0.878287
$485$	−8424.00	−0.788689
$486$	0	0
$487$	−11664.0	−1.08531	−0.542655	−	0.839955i	$-0.682581\pi$
$487$	−11664.0	−1.08531	−0.542655	+	0.839955i	$0.682581\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	9814.00	0.902036	0.451018	−	0.892515i	$-0.351061\pi$
$491$	9814.00	0.902036	0.451018	+	0.892515i	$0.351061\pi$
$492$	0	0
$493$	19908.0	1.81868
$494$	−10368.0	−0.944288
$495$	0	0
$496$	2304.00	0.208574
$497$	0	0
$498$	0	0
$499$	−15228.0	−1.36613	−0.683065	−	0.730358i	$-0.739353\pi$
$499$	−15228.0	−1.36613	−0.683065	+	0.730358i	$0.739353\pi$
$500$	−10656.0	−0.953102
$501$	0	0
$502$	−26064.0	−2.31732
$503$	11088.0	0.982882	0.491441	−	0.870911i	$-0.336470\pi$
$503$	11088.0	0.982882	0.491441	+	0.870911i	$0.336470\pi$
$504$	0	0
$505$	−11988.0	−1.05635
$506$	2800.00	0.245998
$507$	0	0
$508$	7328.00	0.640015
$509$	5814.00	0.506289	0.253144	−	0.967429i	$-0.418535\pi$
$509$	5814.00	0.506289	0.253144	+	0.967429i	$0.418535\pi$
$510$	0	0
$511$	0	0
$512$	−16384.0	−1.41421
$513$	0	0
$514$	−24120.0	−2.06982
$515$	4536.00	0.388117
$516$	0	0
$517$	3600.00	0.306243
$518$	0	0
$519$	0	0
$520$	0	0
$521$	11682.0	0.982337	0.491169	−	0.871065i	$-0.336570\pi$
$521$	11682.0	0.982337	0.491169	+	0.871065i	$0.336570\pi$
$522$	0	0
$523$	−2988.00	−0.249820	−0.124910	−	0.992168i	$-0.539864\pi$
$523$	−2988.00	−0.249820	−0.124910	+	0.992168i	$0.539864\pi$
$524$	−18144.0	−1.51264
$525$	0	0
$526$	2360.00	0.195629
$527$	4536.00	0.374936
$528$	0	0
$529$	−11971.0	−0.983891
$530$	1584.00	0.129820
$531$	0	0
$532$	0	0
$533$	−9720.00	−0.789906
$534$	0	0
$535$	12060.0	0.974578
$536$	0	0
$537$	0	0
$538$	−3960.00	−0.317338
$539$	0	0
$540$	0	0
$541$	7130.00	0.566622	0.283311	−	0.959028i	$-0.408567\pi$
$541$	7130.00	0.566622	0.283311	+	0.959028i	$0.408567\pi$
$542$	13680.0	1.08414
$543$	0	0
$544$	−32256.0	−2.54221
$545$	−2916.00	−0.229188
$546$	0	0
$547$	−5488.00	−0.428976	−0.214488	−	0.976727i	$-0.568808\pi$
$547$	−5488.00	−0.428976	−0.214488	+	0.976727i	$0.568808\pi$
$548$	−6448.00	−0.502637
$549$	0	0
$550$	−39800.0	−3.08560
$551$	11376.0	0.879553
$552$	0	0
$553$	0	0
$554$	10936.0	0.838675
$555$	0	0
$556$	21024.0	1.60363
$557$	−5746.00	−0.437102	−0.218551	−	0.975826i	$-0.570133\pi$
$557$	−5746.00	−0.437102	−0.218551	+	0.975826i	$0.570133\pi$
$558$	0	0
$559$	11664.0	0.882531
$560$	0	0
$561$	0	0
$562$	2392.00	0.179538
$563$	−13068.0	−0.978243	−0.489121	−	0.872216i	$-0.662682\pi$
$563$	−13068.0	−0.978243	−0.489121	+	0.872216i	$0.662682\pi$
$564$	0	0
$565$	−25020.0	−1.86301
$566$	−14400.0	−1.06939
$567$	0	0
$568$	0	0
$569$	1130.00	0.0832549	0.0416275	−	0.999133i	$-0.486746\pi$
$569$	1130.00	0.0832549	0.0416275	+	0.999133i	$0.486746\pi$
$570$	0	0
$571$	16864.0	1.23597	0.617983	−	0.786192i	$-0.287950\pi$
$571$	16864.0	1.23597	0.617983	+	0.786192i	$0.287950\pi$
$572$	−14400.0	−1.05261
$573$	0	0
$574$	0	0
$575$	−2786.00	−0.202060
$576$	0	0
$577$	−2088.00	−0.150649	−0.0753246	−	0.997159i	$-0.523999\pi$
$577$	−2088.00	−0.150649	−0.0753246	+	0.997159i	$0.523999\pi$
$578$	−43852.0	−3.15571
$579$	0	0
$580$	22752.0	1.62884
$581$	0	0
$582$	0	0
$583$	1100.00	0.0781430
$584$	0	0
$585$	0	0
$586$	31608.0	2.22818
$587$	−10260.0	−0.721423	−0.360712	−	0.932677i	$-0.617466\pi$
$587$	−10260.0	−0.721423	−0.360712	+	0.932677i	$0.617466\pi$
$588$	0	0
$589$	2592.00	0.181327
$590$	−33696.0	−2.35126
$591$	0	0
$592$	10368.0	0.719801
$593$	−3582.00	−0.248052	−0.124026	−	0.992279i	$-0.539581\pi$
$593$	−3582.00	−0.248052	−0.124026	+	0.992279i	$0.539581\pi$
$594$	0	0
$595$	0	0
$596$	19120.0	1.31407
$597$	0	0
$598$	−2016.00	−0.137860
$599$	−7034.00	−0.479802	−0.239901	−	0.970797i	$-0.577115\pi$
$599$	−7034.00	−0.479802	−0.239901	+	0.970797i	$0.577115\pi$
$600$	0	0
$601$	−18072.0	−1.22658	−0.613288	−	0.789859i	$-0.710154\pi$
$601$	−18072.0	−1.22658	−0.613288	+	0.789859i	$0.710154\pi$
$602$	0	0
$603$	0	0
$604$	25920.0	1.74614
$605$	−21042.0	−1.41401
$606$	0	0
$607$	28584.0	1.91135	0.955674	−	0.294425i	$-0.0951283\pi$
$607$	28584.0	1.91135	0.955674	+	0.294425i	$0.0951283\pi$
$608$	−18432.0	−1.22947
$609$	0	0
$610$	57024.0	3.78497
$611$	−2592.00	−0.171622
$612$	0	0
$613$	−10910.0	−0.718843	−0.359421	−	0.933175i	$-0.617026\pi$
$613$	−10910.0	−0.718843	−0.359421	+	0.933175i	$0.617026\pi$
$614$	40896.0	2.68799
$615$	0	0
$616$	0	0
$617$	5522.00	0.360304	0.180152	−	0.983639i	$-0.442341\pi$
$617$	5522.00	0.360304	0.180152	+	0.983639i	$0.442341\pi$
$618$	0	0
$619$	2412.00	0.156618	0.0783089	−	0.996929i	$-0.475048\pi$
$619$	2412.00	0.156618	0.0783089	+	0.996929i	$0.475048\pi$
$620$	5184.00	0.335798
$621$	0	0
$622$	15552.0	1.00254
$623$	0	0
$624$	0	0
$625$	−899.000	−0.0575360
$626$	−20448.0	−1.30554
$627$	0	0
$628$	24192.0	1.53721
$629$	20412.0	1.29393
$630$	0	0
$631$	24676.0	1.55679	0.778396	−	0.627773i	$-0.216034\pi$
$631$	24676.0	1.55679	0.778396	+	0.627773i	$0.216034\pi$
$632$	0	0
$633$	0	0
$634$	−40408.0	−2.53124
$635$	−16488.0	−1.03040
$636$	0	0
$637$	0	0
$638$	31600.0	1.96090
$639$	0	0
$640$	0	0
$641$	27482.0	1.69341	0.846703	−	0.532065i	$-0.178584\pi$
$641$	27482.0	1.69341	0.846703	+	0.532065i	$0.178584\pi$
$642$	0	0
$643$	−22752.0	−1.39541	−0.697707	−	0.716383i	$-0.745796\pi$
$643$	−22752.0	−1.39541	−0.697707	+	0.716383i	$0.745796\pi$
$644$	0	0
$645$	0	0
$646$	−36288.0	−2.21011
$647$	−14832.0	−0.901246	−0.450623	−	0.892714i	$-0.648798\pi$
$647$	−14832.0	−0.901246	−0.450623	+	0.892714i	$0.648798\pi$
$648$	0	0
$649$	−23400.0	−1.41530
$650$	28656.0	1.72920
$651$	0	0
$652$	−14272.0	−0.857262
$653$	−2822.00	−0.169117	−0.0845585	−	0.996419i	$-0.526948\pi$
$653$	−2822.00	−0.169117	−0.0845585	+	0.996419i	$0.526948\pi$
$654$	0	0
$655$	40824.0	2.43531
$656$	−17280.0	−1.02846
$657$	0	0
$658$	0	0
$659$	15826.0	0.935498	0.467749	−	0.883861i	$-0.345065\pi$
$659$	15826.0	0.935498	0.467749	+	0.883861i	$0.345065\pi$
$660$	0	0
$661$	−23832.0	−1.40236	−0.701178	−	0.712986i	$-0.747342\pi$
$661$	−23832.0	−1.40236	−0.701178	+	0.712986i	$0.747342\pi$
$662$	−22032.0	−1.29350
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	2212.00	0.128409
$668$	24192.0	1.40122
$669$	0	0
$670$	16704.0	0.963182
$671$	39600.0	2.27830
$672$	0	0
$673$	13770.0	0.788699	0.394350	−	0.918961i	$-0.370970\pi$
$673$	13770.0	0.788699	0.394350	+	0.918961i	$0.370970\pi$
$674$	36936.0	2.11086
$675$	0	0
$676$	−7208.00	−0.410105
$677$	−8334.00	−0.473119	−0.236560	−	0.971617i	$-0.576020\pi$
$677$	−8334.00	−0.473119	−0.236560	+	0.971617i	$0.576020\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	7200.00	0.404255
$683$	18598.0	1.04192	0.520961	−	0.853580i	$-0.325574\pi$
$683$	18598.0	1.04192	0.520961	+	0.853580i	$0.325574\pi$
$684$	0	0
$685$	14508.0	0.809229
$686$	0	0
$687$	0	0
$688$	20736.0	1.14906
$689$	−792.000	−0.0437922
$690$	0	0
$691$	8964.00	0.493497	0.246749	−	0.969080i	$-0.420638\pi$
$691$	8964.00	0.493497	0.246749	+	0.969080i	$0.420638\pi$
$692$	−12528.0	−0.688213
$693$	0	0
$694$	−25976.0	−1.42080
$695$	−47304.0	−2.58179
$696$	0	0
$697$	−34020.0	−1.84878
$698$	−40320.0	−2.18644
$699$	0	0
$700$	0	0
$701$	−3542.00	−0.190841	−0.0954205	−	0.995437i	$-0.530420\pi$
$701$	−3542.00	−0.190841	−0.0954205	+	0.995437i	$0.530420\pi$
$702$	0	0
$703$	11664.0	0.625770
$704$	−25600.0	−1.37051
$705$	0	0
$706$	−2952.00	−0.157365
$707$	0	0
$708$	0	0
$709$	−486.000	−0.0257435	−0.0128717	−	0.999917i	$-0.504097\pi$
$709$	−486.000	−0.0257435	−0.0128717	+	0.999917i	$0.504097\pi$
$710$	52848.0	2.79345
$711$	0	0
$712$	0	0
$713$	504.000	0.0264726
$714$	0	0
$715$	32400.0	1.69467
$716$	−30416.0	−1.58757
$717$	0	0
$718$	776.000	0.0403343
$719$	26928.0	1.39672	0.698362	−	0.715744i	$-0.253912\pi$
$719$	26928.0	1.39672	0.698362	+	0.715744i	$0.253912\pi$
$720$	0	0
$721$	0	0
$722$	6700.00	0.345358
$723$	0	0
$724$	−3744.00	−0.192189
$725$	−31442.0	−1.61066
$726$	0	0
$727$	20628.0	1.05234	0.526169	−	0.850380i	$-0.323628\pi$
$727$	20628.0	1.05234	0.526169	+	0.850380i	$0.323628\pi$
$728$	0	0
$729$	0	0
$730$	12960.0	0.657084
$731$	40824.0	2.06557
$732$	0	0
$733$	−9756.00	−0.491604	−0.245802	−	0.969320i	$-0.579051\pi$
$733$	−9756.00	−0.491604	−0.245802	+	0.969320i	$0.579051\pi$
$734$	19008.0	0.955856
$735$	0	0
$736$	−3584.00	−0.179495
$737$	11600.0	0.579771
$738$	0	0
$739$	19064.0	0.948959	0.474479	−	0.880267i	$-0.342636\pi$
$739$	19064.0	0.948959	0.474479	+	0.880267i	$0.342636\pi$
$740$	23328.0	1.15886
$741$	0	0
$742$	0	0
$743$	3766.00	0.185950	0.0929752	−	0.995668i	$-0.470362\pi$
$743$	3766.00	0.185950	0.0929752	+	0.995668i	$0.470362\pi$
$744$	0	0
$745$	−43020.0	−2.11561
$746$	9224.00	0.452701
$747$	0	0
$748$	−50400.0	−2.46365
$749$	0	0
$750$	0	0
$751$	−11664.0	−0.566745	−0.283372	−	0.959010i	$-0.591453\pi$
$751$	−11664.0	−0.566745	−0.283372	+	0.959010i	$0.591453\pi$
$752$	−4608.00	−0.223453
$753$	0	0
$754$	−22752.0	−1.09891
$755$	−58320.0	−2.81123
$756$	0	0
$757$	−34182.0	−1.64117	−0.820585	−	0.571524i	$-0.806352\pi$
$757$	−34182.0	−1.64117	−0.820585	+	0.571524i	$0.806352\pi$
$758$	29808.0	1.42833
$759$	0	0
$760$	0	0
$761$	4734.00	0.225502	0.112751	−	0.993623i	$-0.464034\pi$
$761$	4734.00	0.225502	0.112751	+	0.993623i	$0.464034\pi$
$762$	0	0
$763$	0	0
$764$	−3856.00	−0.182598
$765$	0	0
$766$	−4608.00	−0.217355
$767$	16848.0	0.793150
$768$	0	0
$769$	−30240.0	−1.41805	−0.709026	−	0.705182i	$-0.750865\pi$
$769$	−30240.0	−1.41805	−0.709026	+	0.705182i	$0.750865\pi$
$770$	0	0
$771$	0	0
$772$	−6480.00	−0.302099
$773$	−27702.0	−1.28897	−0.644484	−	0.764618i	$-0.722928\pi$
$773$	−27702.0	−1.28897	−0.644484	+	0.764618i	$0.722928\pi$
$774$	0	0
$775$	−7164.00	−0.332050
$776$	0	0
$777$	0	0
$778$	7576.00	0.349117
$779$	−19440.0	−0.894108
$780$	0	0
$781$	36700.0	1.68147
$782$	−7056.00	−0.322662
$783$	0	0
$784$	0	0
$785$	−54432.0	−2.47486
$786$	0	0
$787$	−22644.0	−1.02563	−0.512815	−	0.858499i	$-0.671397\pi$
$787$	−22644.0	−1.02563	−0.512815	+	0.858499i	$0.671397\pi$
$788$	19696.0	0.890407
$789$	0	0
$790$	16992.0	0.765251
$791$	0	0
$792$	0	0
$793$	−28512.0	−1.27679
$794$	−36864.0	−1.64768
$795$	0	0
$796$	−36288.0	−1.61582
$797$	−30150.0	−1.33998	−0.669992	−	0.742368i	$-0.733703\pi$
$797$	−30150.0	−1.33998	−0.669992	+	0.742368i	$0.733703\pi$
$798$	0	0
$799$	−9072.00	−0.401682
$800$	50944.0	2.25143
$801$	0	0
$802$	−46600.0	−2.05175
$803$	9000.00	0.395521
$804$	0	0
$805$	0	0
$806$	−5184.00	−0.226549
$807$	0	0
$808$	0	0
$809$	11318.0	0.491866	0.245933	−	0.969287i	$-0.420906\pi$
$809$	11318.0	0.491866	0.245933	+	0.969287i	$0.420906\pi$
$810$	0	0
$811$	−29628.0	−1.28284	−0.641418	−	0.767192i	$-0.721653\pi$
$811$	−29628.0	−1.28284	−0.641418	+	0.767192i	$0.721653\pi$
$812$	0	0
$813$	0	0
$814$	32400.0	1.39511
$815$	32112.0	1.38016
$816$	0	0
$817$	23328.0	0.998952
$818$	30096.0	1.28641
$819$	0	0
$820$	−38880.0	−1.65579
$821$	−17770.0	−0.755393	−0.377696	−	0.925930i	$-0.623284\pi$
$821$	−17770.0	−0.755393	−0.377696	+	0.925930i	$0.623284\pi$
$822$	0	0
$823$	7868.00	0.333246	0.166623	−	0.986021i	$-0.446714\pi$
$823$	7868.00	0.333246	0.166623	+	0.986021i	$0.446714\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−35726.0	−1.50219	−0.751097	−	0.660192i	$-0.770475\pi$
$827$	−35726.0	−1.50219	−0.751097	+	0.660192i	$0.770475\pi$
$828$	0	0
$829$	27108.0	1.13571	0.567853	−	0.823130i	$-0.307774\pi$
$829$	27108.0	1.13571	0.567853	+	0.823130i	$0.307774\pi$
$830$	−2592.00	−0.108397
$831$	0	0
$832$	18432.0	0.768046
$833$	0	0
$834$	0	0
$835$	−54432.0	−2.25592
$836$	−28800.0	−1.19147
$837$	0	0
$838$	−15408.0	−0.635156
$839$	−23256.0	−0.956956	−0.478478	−	0.878099i	$-0.658811\pi$
$839$	−23256.0	−0.956956	−0.478478	+	0.878099i	$0.658811\pi$
$840$	0	0
$841$	575.000	0.0235762
$842$	−41608.0	−1.70298
$843$	0	0
$844$	23328.0	0.951402
$845$	16218.0	0.660256
$846$	0	0
$847$	0	0
$848$	−1408.00	−0.0570176
$849$	0	0
$850$	100296.	4.04721
$851$	2268.00	0.0913584
$852$	0	0
$853$	35280.0	1.41614	0.708068	−	0.706144i	$-0.249567\pi$
$853$	35280.0	1.41614	0.708068	+	0.706144i	$0.249567\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	19710.0	0.785625	0.392813	−	0.919619i	$-0.371502\pi$
$857$	19710.0	0.785625	0.392813	+	0.919619i	$0.371502\pi$
$858$	0	0
$859$	3888.00	0.154432	0.0772159	−	0.997014i	$-0.475397\pi$
$859$	3888.00	0.154432	0.0772159	+	0.997014i	$0.475397\pi$
$860$	46656.0	1.84995
$861$	0	0
$862$	−41560.0	−1.64216
$863$	36634.0	1.44500	0.722500	−	0.691370i	$-0.242993\pi$
$863$	36634.0	1.44500	0.722500	+	0.691370i	$0.242993\pi$
$864$	0	0
$865$	28188.0	1.10800
$866$	−44928.0	−1.76295
$867$	0	0
$868$	0	0
$869$	11800.0	0.460630
$870$	0	0
$871$	−8352.00	−0.324910
$872$	0	0
$873$	0	0
$874$	−4032.00	−0.156046
$875$	0	0
$876$	0	0
$877$	1226.00	0.0472053	0.0236027	−	0.999721i	$-0.492486\pi$
$877$	1226.00	0.0472053	0.0236027	+	0.999721i	$0.492486\pi$
$878$	−58464.0	−2.24723
$879$	0	0
$880$	57600.0	2.20647
$881$	38538.0	1.47376	0.736878	−	0.676026i	$-0.236299\pi$
$881$	38538.0	1.47376	0.736878	+	0.676026i	$0.236299\pi$
$882$	0	0
$883$	−37260.0	−1.42004	−0.710022	−	0.704180i	$-0.751315\pi$
$883$	−37260.0	−1.42004	−0.710022	+	0.704180i	$0.751315\pi$
$884$	36288.0	1.38065
$885$	0	0
$886$	47752.0	1.81068
$887$	−26640.0	−1.00844	−0.504219	−	0.863576i	$-0.668219\pi$
$887$	−26640.0	−1.00844	−0.504219	+	0.863576i	$0.668219\pi$
$888$	0	0
$889$	0	0
$890$	−16848.0	−0.634546
$891$	0	0
$892$	−8640.00	−0.324315
$893$	−5184.00	−0.194262
$894$	0	0
$895$	68436.0	2.55594
$896$	0	0
$897$	0	0
$898$	32744.0	1.21679
$899$	5688.00	0.211018
$900$	0	0
$901$	−2772.00	−0.102496
$902$	−54000.0	−1.99335
$903$	0	0
$904$	0	0
$905$	8424.00	0.309418
$906$	0	0
$907$	−12636.0	−0.462593	−0.231296	−	0.972883i	$-0.574297\pi$
$907$	−12636.0	−0.462593	−0.231296	+	0.972883i	$0.574297\pi$
$908$	10656.0	0.389462
$909$	0	0
$910$	0	0
$911$	33638.0	1.22336	0.611678	−	0.791107i	$-0.290495\pi$
$911$	33638.0	1.22336	0.611678	+	0.791107i	$0.290495\pi$
$912$	0	0
$913$	−1800.00	−0.0652479
$914$	−8424.00	−0.304859
$915$	0	0
$916$	12960.0	0.467479
$917$	0	0
$918$	0	0
$919$	36936.0	1.32580	0.662898	−	0.748710i	$-0.269326\pi$
$919$	36936.0	1.32580	0.662898	+	0.748710i	$0.269326\pi$
$920$	0	0
$921$	0	0
$922$	37944.0	1.35534
$923$	−26424.0	−0.942315
$924$	0	0
$925$	−32238.0	−1.14592
$926$	50608.0	1.79598
$927$	0	0
$928$	−40448.0	−1.43079
$929$	22302.0	0.787626	0.393813	−	0.919191i	$-0.371156\pi$
$929$	22302.0	0.787626	0.393813	+	0.919191i	$0.371156\pi$
$930$	0	0
$931$	0	0
$932$	−53744.0	−1.88889
$933$	0	0
$934$	−14832.0	−0.519612
$935$	113400.	3.96639
$936$	0	0
$937$	13824.0	0.481975	0.240987	−	0.970528i	$-0.422529\pi$
$937$	13824.0	0.481975	0.240987	+	0.970528i	$0.422529\pi$
$938$	0	0
$939$	0	0
$940$	−10368.0	−0.359752
$941$	−13554.0	−0.469551	−0.234776	−	0.972050i	$-0.575436\pi$
$941$	−13554.0	−0.469551	−0.234776	+	0.972050i	$0.575436\pi$
$942$	0	0
$943$	−3780.00	−0.130534
$944$	29952.0	1.03268
$945$	0	0
$946$	64800.0	2.22709
$947$	−44878.0	−1.53996	−0.769978	−	0.638070i	$-0.779733\pi$
$947$	−44878.0	−1.53996	−0.769978	+	0.638070i	$0.779733\pi$
$948$	0	0
$949$	−6480.00	−0.221654
$950$	57312.0	1.95731
$951$	0	0
$952$	0	0
$953$	−38362.0	−1.30395	−0.651976	−	0.758239i	$-0.726060\pi$
$953$	−38362.0	−1.30395	−0.651976	+	0.758239i	$0.726060\pi$
$954$	0	0
$955$	8676.00	0.293978
$956$	28624.0	0.968375
$957$	0	0
$958$	−32256.0	−1.08783
$959$	0	0
$960$	0	0
$961$	−28495.0	−0.956497
$962$	−23328.0	−0.781835
$963$	0	0
$964$	−6048.00	−0.202067
$965$	14580.0	0.486370
$966$	0	0
$967$	26444.0	0.879402	0.439701	−	0.898144i	$-0.355084\pi$
$967$	26444.0	0.879402	0.439701	+	0.898144i	$0.355084\pi$
$968$	0	0
$969$	0	0
$970$	33696.0	1.11537
$971$	−17820.0	−0.588951	−0.294475	−	0.955659i	$-0.595145\pi$
$971$	−17820.0	−0.588951	−0.294475	+	0.955659i	$0.595145\pi$
$972$	0	0
$973$	0	0
$974$	46656.0	1.53486
$975$	0	0
$976$	−50688.0	−1.66238
$977$	−34438.0	−1.12771	−0.563853	−	0.825875i	$-0.690682\pi$
$977$	−34438.0	−1.12771	−0.563853	+	0.825875i	$0.690682\pi$
$978$	0	0
$979$	−11700.0	−0.381955
$980$	0	0
$981$	0	0
$982$	−39256.0	−1.27567
$983$	−26064.0	−0.845689	−0.422845	−	0.906202i	$-0.638968\pi$
$983$	−26064.0	−0.845689	−0.422845	+	0.906202i	$0.638968\pi$
$984$	0	0
$985$	−44316.0	−1.43353
$986$	−79632.0	−2.57201
$987$	0	0
$988$	20736.0	0.667713
$989$	4536.00	0.145841
$990$	0	0
$991$	33696.0	1.08011	0.540055	−	0.841630i	$-0.318403\pi$
$991$	33696.0	1.08011	0.540055	+	0.841630i	$0.318403\pi$
$992$	−9216.00	−0.294968
$993$	0	0
$994$	0	0
$995$	81648.0	2.60142
$996$	0	0
$997$	36072.0	1.14585	0.572925	−	0.819608i	$-0.305809\pi$
$997$	36072.0	1.14585	0.572925	+	0.819608i	$0.305809\pi$
$998$	60912.0	1.93200
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.4.a.a.1.1		1
3.2	odd	2		147.4.a.h.1.1	yes	1
7.2	even	3		441.4.e.o.361.1		2
7.3	odd	6		441.4.e.l.226.1		2
7.4	even	3		441.4.e.o.226.1		2
7.5	odd	6		441.4.e.l.361.1		2
7.6	odd	2		441.4.a.c.1.1		1
12.11	even	2		2352.4.a.s.1.1		1
21.2	odd	6		147.4.e.a.67.1		2
21.5	even	6		147.4.e.d.67.1		2
21.11	odd	6		147.4.e.a.79.1		2
21.17	even	6		147.4.e.d.79.1		2
21.20	even	2		147.4.a.f.1.1	✓	1
84.83	odd	2		2352.4.a.t.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
147.4.a.f.1.1	✓	1	21.20	even	2
147.4.a.h.1.1	yes	1	3.2	odd	2
147.4.e.a.67.1		2	21.2	odd	6
147.4.e.a.79.1		2	21.11	odd	6
147.4.e.d.67.1		2	21.5	even	6
147.4.e.d.79.1		2	21.17	even	6
441.4.a.a.1.1		1	1.1	even	1	trivial
441.4.a.c.1.1		1	7.6	odd	2
441.4.e.l.226.1		2	7.3	odd	6
441.4.e.l.361.1		2	7.5	odd	6
441.4.e.o.226.1		2	7.4	even	3
441.4.e.o.361.1		2	7.2	even	3
2352.4.a.s.1.1		1	12.11	even	2
2352.4.a.t.1.1		1	84.83	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:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	441.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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