LMFDB - Embedded newform 1323.2.a.c.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1323,2,Mod(1,1323)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1323, 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("1323.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1323 = 3^{3} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1323.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:	$10.5642081874$
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	$\chi$	$=$	1323.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

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

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
$5$	3.00000	1.34164	0.670820	−	0.741620i	$-0.265942\pi$
$5$	3.00000	1.34164	0.670820	+	0.741620i	$0.265942\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	0	0
$10$	−6.00000	−1.89737
$11$	−2.00000	−0.603023	−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−2.00000	−0.603023	−0.301511	+	0.953463i	$0.597491\pi$
$12$	0	0
$13$	−6.00000	−1.66410	−0.832050	−	0.554700i	$-0.812833\pi$
$13$	−6.00000	−1.66410	−0.832050	+	0.554700i	$0.812833\pi$
$14$	0	0
$15$	0	0
$16$	−4.00000	−1.00000
$17$	3.00000	0.727607	0.363803	−	0.931476i	$-0.381478\pi$
$17$	3.00000	0.727607	0.363803	+	0.931476i	$0.381478\pi$
$18$	0	0
$19$	6.00000	1.37649	0.688247	−	0.725476i	$-0.258380\pi$
$19$	6.00000	1.37649	0.688247	+	0.725476i	$0.258380\pi$
$20$	6.00000	1.34164
$21$	0	0
$22$	4.00000	0.852803
$23$	8.00000	1.66812	0.834058	−	0.551677i	$-0.186012\pi$
$23$	8.00000	1.66812	0.834058	+	0.551677i	$0.186012\pi$
$24$	0	0
$25$	4.00000	0.800000
$26$	12.0000	2.35339
$27$	0	0
$28$	0	0
$29$	2.00000	0.371391	0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000	0.371391	0.185695	+	0.982607i	$0.440546\pi$
$30$	0	0
$31$	−6.00000	−1.07763	−0.538816	−	0.842424i	$-0.681128\pi$
$31$	−6.00000	−1.07763	−0.538816	+	0.842424i	$0.681128\pi$
$32$	8.00000	1.41421
$33$	0	0
$34$	−6.00000	−1.02899
$35$	0	0
$36$	0	0
$37$	9.00000	1.47959	0.739795	−	0.672832i	$-0.234922\pi$
$37$	9.00000	1.47959	0.739795	+	0.672832i	$0.234922\pi$
$38$	−12.0000	−1.94666
$39$	0	0
$40$	0	0
$41$	9.00000	1.40556	0.702782	−	0.711405i	$-0.251941\pi$
$41$	9.00000	1.40556	0.702782	+	0.711405i	$0.251941\pi$
$42$	0	0
$43$	−9.00000	−1.37249	−0.686244	−	0.727372i	$-0.740742\pi$
$43$	−9.00000	−1.37249	−0.686244	+	0.727372i	$0.740742\pi$
$44$	−4.00000	−0.603023
$45$	0	0
$46$	−16.0000	−2.35907
$47$	−3.00000	−0.437595	−0.218797	−	0.975770i	$-0.570213\pi$
$47$	−3.00000	−0.437595	−0.218797	+	0.975770i	$0.570213\pi$
$48$	0	0
$49$	0	0
$50$	−8.00000	−1.13137
$51$	0	0
$52$	−12.0000	−1.66410
$53$	−4.00000	−0.549442	−0.274721	−	0.961524i	$-0.588586\pi$
$53$	−4.00000	−0.549442	−0.274721	+	0.961524i	$0.588586\pi$
$54$	0	0
$55$	−6.00000	−0.809040
$56$	0	0
$57$	0	0
$58$	−4.00000	−0.525226
$59$	−3.00000	−0.390567	−0.195283	−	0.980747i	$-0.562563\pi$
$59$	−3.00000	−0.390567	−0.195283	+	0.980747i	$0.562563\pi$
$60$	0	0
$61$	6.00000	0.768221	0.384111	−	0.923287i	$-0.374508\pi$
$61$	6.00000	0.768221	0.384111	+	0.923287i	$0.374508\pi$
$62$	12.0000	1.52400
$63$	0	0
$64$	−8.00000	−1.00000
$65$	−18.0000	−2.23263
$66$	0	0
$67$	4.00000	0.488678	0.244339	−	0.969690i	$-0.421429\pi$
$67$	4.00000	0.488678	0.244339	+	0.969690i	$0.421429\pi$
$68$	6.00000	0.727607
$69$	0	0
$70$	0	0
$71$	4.00000	0.474713	0.237356	−	0.971423i	$-0.423719\pi$
$71$	4.00000	0.474713	0.237356	+	0.971423i	$0.423719\pi$
$72$	0	0
$73$	12.0000	1.40449	0.702247	−	0.711934i	$-0.252180\pi$
$73$	12.0000	1.40449	0.702247	+	0.711934i	$0.252180\pi$
$74$	−18.0000	−2.09246
$75$	0	0
$76$	12.0000	1.37649
$77$	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	$0.517916\pi$
$80$	−12.0000	−1.34164
$81$	0	0
$82$	−18.0000	−1.98777
$83$	15.0000	1.64646	0.823232	−	0.567705i	$-0.192169\pi$
$83$	15.0000	1.64646	0.823232	+	0.567705i	$0.192169\pi$
$84$	0	0
$85$	9.00000	0.976187
$86$	18.0000	1.94099
$87$	0	0
$88$	0	0
$89$	−6.00000	−0.635999	−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000	−0.635999	−0.317999	+	0.948091i	$0.603011\pi$
$90$	0	0
$91$	0	0
$92$	16.0000	1.66812
$93$	0	0
$94$	6.00000	0.618853
$95$	18.0000	1.84676
$96$	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
$100$	8.00000	0.800000
$101$	6.00000	0.597022	0.298511	−	0.954406i	$-0.403510\pi$
$101$	6.00000	0.597022	0.298511	+	0.954406i	$0.403510\pi$
$102$	0	0
$103$	12.0000	1.18240	0.591198	−	0.806527i	$-0.298655\pi$
$103$	12.0000	1.18240	0.591198	+	0.806527i	$0.298655\pi$
$104$	0	0
$105$	0	0
$106$	8.00000	0.777029
$107$	4.00000	0.386695	0.193347	−	0.981130i	$-0.438066\pi$
$107$	4.00000	0.386695	0.193347	+	0.981130i	$0.438066\pi$
$108$	0	0
$109$	−9.00000	−0.862044	−0.431022	−	0.902342i	$-0.641847\pi$
$109$	−9.00000	−0.862044	−0.431022	+	0.902342i	$0.641847\pi$
$110$	12.0000	1.14416
$111$	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
$115$	24.0000	2.23801
$116$	4.00000	0.371391
$117$	0	0
$118$	6.00000	0.552345
$119$	0	0
$120$	0	0
$121$	−7.00000	−0.636364
$122$	−12.0000	−1.08643
$123$	0	0
$124$	−12.0000	−1.07763
$125$	−3.00000	−0.268328
$126$	0	0
$127$	−11.0000	−0.976092	−0.488046	−	0.872818i	$-0.662290\pi$
$127$	−11.0000	−0.976092	−0.488046	+	0.872818i	$0.662290\pi$
$128$	0	0
$129$	0	0
$130$	36.0000	3.15741
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	0	0
$133$	0	0
$134$	−8.00000	−0.691095
$135$	0	0
$136$	0	0
$137$	2.00000	0.170872	0.0854358	−	0.996344i	$-0.472772\pi$
$137$	2.00000	0.170872	0.0854358	+	0.996344i	$0.472772\pi$
$138$	0	0
$139$	6.00000	0.508913	0.254457	−	0.967084i	$-0.418103\pi$
$139$	6.00000	0.508913	0.254457	+	0.967084i	$0.418103\pi$
$140$	0	0
$141$	0	0
$142$	−8.00000	−0.671345
$143$	12.0000	1.00349
$144$	0	0
$145$	6.00000	0.498273
$146$	−24.0000	−1.98625
$147$	0	0
$148$	18.0000	1.47959
$149$	10.0000	0.819232	0.409616	−	0.912258i	$-0.365663\pi$
$149$	10.0000	0.819232	0.409616	+	0.912258i	$0.365663\pi$
$150$	0	0
$151$	9.00000	0.732410	0.366205	−	0.930534i	$-0.380657\pi$
$151$	9.00000	0.732410	0.366205	+	0.930534i	$0.380657\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−18.0000	−1.44579
$156$	0	0
$157$	18.0000	1.43656	0.718278	−	0.695756i	$-0.244931\pi$
$157$	18.0000	1.43656	0.718278	+	0.695756i	$0.244931\pi$
$158$	2.00000	0.159111
$159$	0	0
$160$	24.0000	1.89737
$161$	0	0
$162$	0	0
$163$	−11.0000	−0.861586	−0.430793	−	0.902451i	$-0.641766\pi$
$163$	−11.0000	−0.861586	−0.430793	+	0.902451i	$0.641766\pi$
$164$	18.0000	1.40556
$165$	0	0
$166$	−30.0000	−2.32845
$167$	9.00000	0.696441	0.348220	−	0.937413i	$-0.386786\pi$
$167$	9.00000	0.696441	0.348220	+	0.937413i	$0.386786\pi$
$168$	0	0
$169$	23.0000	1.76923
$170$	−18.0000	−1.38054
$171$	0	0
$172$	−18.0000	−1.37249
$173$	18.0000	1.36851	0.684257	−	0.729241i	$-0.260127\pi$
$173$	18.0000	1.36851	0.684257	+	0.729241i	$0.260127\pi$
$174$	0	0
$175$	0	0
$176$	8.00000	0.603023
$177$	0	0
$178$	12.0000	0.899438
$179$	−14.0000	−1.04641	−0.523205	−	0.852207i	$-0.675264\pi$
$179$	−14.0000	−1.04641	−0.523205	+	0.852207i	$0.675264\pi$
$180$	0	0
$181$	12.0000	0.891953	0.445976	−	0.895045i	$-0.352856\pi$
$181$	12.0000	0.891953	0.445976	+	0.895045i	$0.352856\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	27.0000	1.98508
$186$	0	0
$187$	−6.00000	−0.438763
$188$	−6.00000	−0.437595
$189$	0	0
$190$	−36.0000	−2.61171
$191$	−16.0000	−1.15772	−0.578860	−	0.815427i	$-0.696502\pi$
$191$	−16.0000	−1.15772	−0.578860	+	0.815427i	$0.696502\pi$
$192$	0	0
$193$	−9.00000	−0.647834	−0.323917	−	0.946085i	$-0.605000\pi$
$193$	−9.00000	−0.647834	−0.323917	+	0.946085i	$0.605000\pi$
$194$	−12.0000	−0.861550
$195$	0	0
$196$	0	0
$197$	22.0000	1.56744	0.783718	−	0.621117i	$-0.213321\pi$
$197$	22.0000	1.56744	0.783718	+	0.621117i	$0.213321\pi$
$198$	0	0
$199$	0	0		−	1.00000i	$-0.5\pi$
$199$	0	0			1.00000i	$0.5\pi$
$200$	0	0
$201$	0	0
$202$	−12.0000	−0.844317
$203$	0	0
$204$	0	0
$205$	27.0000	1.88576
$206$	−24.0000	−1.67216
$207$	0	0
$208$	24.0000	1.66410
$209$	−12.0000	−0.830057
$210$	0	0
$211$	0	0		−	1.00000i	$-0.5\pi$
$211$	0	0			1.00000i	$0.5\pi$
$212$	−8.00000	−0.549442
$213$	0	0
$214$	−8.00000	−0.546869
$215$	−27.0000	−1.84138
$216$	0	0
$217$	0	0
$218$	18.0000	1.21911
$219$	0	0
$220$	−12.0000	−0.809040
$221$	−18.0000	−1.21081
$222$	0	0
$223$	−18.0000	−1.20537	−0.602685	−	0.797980i	$-0.705902\pi$
$223$	−18.0000	−1.20537	−0.602685	+	0.797980i	$0.705902\pi$
$224$	0	0
$225$	0	0
$226$	20.0000	1.33038
$227$	12.0000	0.796468	0.398234	−	0.917284i	$-0.369623\pi$
$227$	12.0000	0.796468	0.398234	+	0.917284i	$0.369623\pi$
$228$	0	0
$229$	0	0		−	1.00000i	$-0.5\pi$
$229$	0	0			1.00000i	$0.5\pi$
$230$	−48.0000	−3.16503
$231$	0	0
$232$	0	0
$233$	22.0000	1.44127	0.720634	−	0.693316i	$-0.243851\pi$
$233$	22.0000	1.44127	0.720634	+	0.693316i	$0.243851\pi$
$234$	0	0
$235$	−9.00000	−0.587095
$236$	−6.00000	−0.390567
$237$	0	0
$238$	0	0
$239$	10.0000	0.646846	0.323423	−	0.946254i	$-0.395166\pi$
$239$	10.0000	0.646846	0.323423	+	0.946254i	$0.395166\pi$
$240$	0	0
$241$	−18.0000	−1.15948	−0.579741	−	0.814801i	$-0.696846\pi$
$241$	−18.0000	−1.15948	−0.579741	+	0.814801i	$0.696846\pi$
$242$	14.0000	0.899954
$243$	0	0
$244$	12.0000	0.768221
$245$	0	0
$246$	0	0
$247$	−36.0000	−2.29063
$248$	0	0
$249$	0	0
$250$	6.00000	0.379473
$251$	3.00000	0.189358	0.0946792	−	0.995508i	$-0.469817\pi$
$251$	3.00000	0.189358	0.0946792	+	0.995508i	$0.469817\pi$
$252$	0	0
$253$	−16.0000	−1.00591
$254$	22.0000	1.38040
$255$	0	0
$256$	16.0000	1.00000
$257$	−6.00000	−0.374270	−0.187135	−	0.982334i	$-0.559920\pi$
$257$	−6.00000	−0.374270	−0.187135	+	0.982334i	$0.559920\pi$
$258$	0	0
$259$	0	0
$260$	−36.0000	−2.23263
$261$	0	0
$262$	0	0
$263$	2.00000	0.123325	0.0616626	−	0.998097i	$-0.480360\pi$
$263$	2.00000	0.123325	0.0616626	+	0.998097i	$0.480360\pi$
$264$	0	0
$265$	−12.0000	−0.737154
$266$	0	0
$267$	0	0
$268$	8.00000	0.488678
$269$	15.0000	0.914566	0.457283	−	0.889321i	$-0.348823\pi$
$269$	15.0000	0.914566	0.457283	+	0.889321i	$0.348823\pi$
$270$	0	0
$271$	−12.0000	−0.728948	−0.364474	−	0.931214i	$-0.618751\pi$
$271$	−12.0000	−0.728948	−0.364474	+	0.931214i	$0.618751\pi$
$272$	−12.0000	−0.727607
$273$	0	0
$274$	−4.00000	−0.241649
$275$	−8.00000	−0.482418
$276$	0	0
$277$	−19.0000	−1.14160	−0.570800	−	0.821089i	$-0.693367\pi$
$277$	−19.0000	−1.14160	−0.570800	+	0.821089i	$0.693367\pi$
$278$	−12.0000	−0.719712
$279$	0	0
$280$	0	0
$281$	−20.0000	−1.19310	−0.596550	−	0.802576i	$-0.703462\pi$
$281$	−20.0000	−1.19310	−0.596550	+	0.802576i	$0.703462\pi$
$282$	0	0
$283$	6.00000	0.356663	0.178331	−	0.983970i	$-0.442930\pi$
$283$	6.00000	0.356663	0.178331	+	0.983970i	$0.442930\pi$
$284$	8.00000	0.474713
$285$	0	0
$286$	−24.0000	−1.41915
$287$	0	0
$288$	0	0
$289$	−8.00000	−0.470588
$290$	−12.0000	−0.704664
$291$	0	0
$292$	24.0000	1.40449
$293$	21.0000	1.22683	0.613417	−	0.789760i	$-0.289795\pi$
$293$	21.0000	1.22683	0.613417	+	0.789760i	$0.289795\pi$
$294$	0	0
$295$	−9.00000	−0.524000
$296$	0	0
$297$	0	0
$298$	−20.0000	−1.15857
$299$	−48.0000	−2.77591
$300$	0	0
$301$	0	0
$302$	−18.0000	−1.03578
$303$	0	0
$304$	−24.0000	−1.37649
$305$	18.0000	1.03068
$306$	0	0
$307$	−12.0000	−0.684876	−0.342438	−	0.939540i	$-0.611253\pi$
$307$	−12.0000	−0.684876	−0.342438	+	0.939540i	$0.611253\pi$
$308$	0	0
$309$	0	0
$310$	36.0000	2.04466
$311$	−27.0000	−1.53103	−0.765515	−	0.643418i	$-0.777516\pi$
$311$	−27.0000	−1.53103	−0.765515	+	0.643418i	$0.777516\pi$
$312$	0	0
$313$	24.0000	1.35656	0.678280	−	0.734803i	$-0.262726\pi$
$313$	24.0000	1.35656	0.678280	+	0.734803i	$0.262726\pi$
$314$	−36.0000	−2.03160
$315$	0	0
$316$	−2.00000	−0.112509
$317$	20.0000	1.12331	0.561656	−	0.827371i	$-0.310164\pi$
$317$	20.0000	1.12331	0.561656	+	0.827371i	$0.310164\pi$
$318$	0	0
$319$	−4.00000	−0.223957
$320$	−24.0000	−1.34164
$321$	0	0
$322$	0	0
$323$	18.0000	1.00155
$324$	0	0
$325$	−24.0000	−1.33128
$326$	22.0000	1.21847
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−9.00000	−0.494685	−0.247342	−	0.968928i	$-0.579557\pi$
$331$	−9.00000	−0.494685	−0.247342	+	0.968928i	$0.579557\pi$
$332$	30.0000	1.64646
$333$	0	0
$334$	−18.0000	−0.984916
$335$	12.0000	0.655630
$336$	0	0
$337$	27.0000	1.47078	0.735392	−	0.677642i	$-0.236998\pi$
$337$	27.0000	1.47078	0.735392	+	0.677642i	$0.236998\pi$
$338$	−46.0000	−2.50207
$339$	0	0
$340$	18.0000	0.976187
$341$	12.0000	0.649836
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	−36.0000	−1.93537
$347$	−26.0000	−1.39575	−0.697877	−	0.716218i	$-0.745872\pi$
$347$	−26.0000	−1.39575	−0.697877	+	0.716218i	$0.745872\pi$
$348$	0	0
$349$	−12.0000	−0.642345	−0.321173	−	0.947021i	$-0.604077\pi$
$349$	−12.0000	−0.642345	−0.321173	+	0.947021i	$0.604077\pi$
$350$	0	0
$351$	0	0
$352$	−16.0000	−0.852803
$353$	3.00000	0.159674	0.0798369	−	0.996808i	$-0.474560\pi$
$353$	3.00000	0.159674	0.0798369	+	0.996808i	$0.474560\pi$
$354$	0	0
$355$	12.0000	0.636894
$356$	−12.0000	−0.635999
$357$	0	0
$358$	28.0000	1.47985
$359$	−4.00000	−0.211112	−0.105556	−	0.994413i	$-0.533662\pi$
$359$	−4.00000	−0.211112	−0.105556	+	0.994413i	$0.533662\pi$
$360$	0	0
$361$	17.0000	0.894737
$362$	−24.0000	−1.26141
$363$	0	0
$364$	0	0
$365$	36.0000	1.88433
$366$	0	0
$367$	−36.0000	−1.87918	−0.939592	−	0.342296i	$-0.888796\pi$
$367$	−36.0000	−1.87918	−0.939592	+	0.342296i	$0.888796\pi$
$368$	−32.0000	−1.66812
$369$	0	0
$370$	−54.0000	−2.80733
$371$	0	0
$372$	0	0
$373$	−5.00000	−0.258890	−0.129445	−	0.991587i	$-0.541320\pi$
$373$	−5.00000	−0.258890	−0.129445	+	0.991587i	$0.541320\pi$
$374$	12.0000	0.620505
$375$	0	0
$376$	0	0
$377$	−12.0000	−0.618031
$378$	0	0
$379$	9.00000	0.462299	0.231149	−	0.972918i	$-0.425751\pi$
$379$	9.00000	0.462299	0.231149	+	0.972918i	$0.425751\pi$
$380$	36.0000	1.84676
$381$	0	0
$382$	32.0000	1.63726
$383$	−3.00000	−0.153293	−0.0766464	−	0.997058i	$-0.524421\pi$
$383$	−3.00000	−0.153293	−0.0766464	+	0.997058i	$0.524421\pi$
$384$	0	0
$385$	0	0
$386$	18.0000	0.916176
$387$	0	0
$388$	12.0000	0.609208
$389$	34.0000	1.72387	0.861934	−	0.507020i	$-0.169253\pi$
$389$	34.0000	1.72387	0.861934	+	0.507020i	$0.169253\pi$
$390$	0	0
$391$	24.0000	1.21373
$392$	0	0
$393$	0	0
$394$	−44.0000	−2.21669
$395$	−3.00000	−0.150946
$396$	0	0
$397$	−12.0000	−0.602263	−0.301131	−	0.953583i	$-0.597364\pi$
$397$	−12.0000	−0.602263	−0.301131	+	0.953583i	$0.597364\pi$
$398$	0	0
$399$	0	0
$400$	−16.0000	−0.800000
$401$	−28.0000	−1.39825	−0.699127	−	0.714998i	$-0.746428\pi$
$401$	−28.0000	−1.39825	−0.699127	+	0.714998i	$0.746428\pi$
$402$	0	0
$403$	36.0000	1.79329
$404$	12.0000	0.597022
$405$	0	0
$406$	0	0
$407$	−18.0000	−0.892227
$408$	0	0
$409$	−12.0000	−0.593362	−0.296681	−	0.954977i	$-0.595880\pi$
$409$	−12.0000	−0.593362	−0.296681	+	0.954977i	$0.595880\pi$
$410$	−54.0000	−2.66687
$411$	0	0
$412$	24.0000	1.18240
$413$	0	0
$414$	0	0
$415$	45.0000	2.20896
$416$	−48.0000	−2.35339
$417$	0	0
$418$	24.0000	1.17388
$419$	−21.0000	−1.02592	−0.512959	−	0.858413i	$-0.671451\pi$
$419$	−21.0000	−1.02592	−0.512959	+	0.858413i	$0.671451\pi$
$420$	0	0
$421$	−26.0000	−1.26716	−0.633581	−	0.773676i	$-0.718416\pi$
$421$	−26.0000	−1.26716	−0.633581	+	0.773676i	$0.718416\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	12.0000	0.582086
$426$	0	0
$427$	0	0
$428$	8.00000	0.386695
$429$	0	0
$430$	54.0000	2.60411
$431$	−4.00000	−0.192673	−0.0963366	−	0.995349i	$-0.530713\pi$
$431$	−4.00000	−0.192673	−0.0963366	+	0.995349i	$0.530713\pi$
$432$	0	0
$433$	−18.0000	−0.865025	−0.432512	−	0.901628i	$-0.642373\pi$
$433$	−18.0000	−0.865025	−0.432512	+	0.901628i	$0.642373\pi$
$434$	0	0
$435$	0	0
$436$	−18.0000	−0.862044
$437$	48.0000	2.29615
$438$	0	0
$439$	−12.0000	−0.572729	−0.286364	−	0.958121i	$-0.592447\pi$
$439$	−12.0000	−0.572729	−0.286364	+	0.958121i	$0.592447\pi$
$440$	0	0
$441$	0	0
$442$	36.0000	1.71235
$443$	−2.00000	−0.0950229	−0.0475114	−	0.998871i	$-0.515129\pi$
$443$	−2.00000	−0.0950229	−0.0475114	+	0.998871i	$0.515129\pi$
$444$	0	0
$445$	−18.0000	−0.853282
$446$	36.0000	1.70465
$447$	0	0
$448$	0	0
$449$	−22.0000	−1.03824	−0.519122	−	0.854700i	$-0.673741\pi$
$449$	−22.0000	−1.03824	−0.519122	+	0.854700i	$0.673741\pi$
$450$	0	0
$451$	−18.0000	−0.847587
$452$	−20.0000	−0.940721
$453$	0	0
$454$	−24.0000	−1.12638
$455$	0	0
$456$	0	0
$457$	18.0000	0.842004	0.421002	−	0.907060i	$-0.361678\pi$
$457$	18.0000	0.842004	0.421002	+	0.907060i	$0.361678\pi$
$458$	0	0
$459$	0	0
$460$	48.0000	2.23801
$461$	−3.00000	−0.139724	−0.0698620	−	0.997557i	$-0.522256\pi$
$461$	−3.00000	−0.139724	−0.0698620	+	0.997557i	$0.522256\pi$
$462$	0	0
$463$	5.00000	0.232370	0.116185	−	0.993228i	$-0.462933\pi$
$463$	5.00000	0.232370	0.116185	+	0.993228i	$0.462933\pi$
$464$	−8.00000	−0.371391
$465$	0	0
$466$	−44.0000	−2.03826
$467$	12.0000	0.555294	0.277647	−	0.960683i	$-0.410445\pi$
$467$	12.0000	0.555294	0.277647	+	0.960683i	$0.410445\pi$
$468$	0	0
$469$	0	0
$470$	18.0000	0.830278
$471$	0	0
$472$	0	0
$473$	18.0000	0.827641
$474$	0	0
$475$	24.0000	1.10120
$476$	0	0
$477$	0	0
$478$	−20.0000	−0.914779
$479$	15.0000	0.685367	0.342684	−	0.939451i	$-0.388664\pi$
$479$	15.0000	0.685367	0.342684	+	0.939451i	$0.388664\pi$
$480$	0	0
$481$	−54.0000	−2.46219
$482$	36.0000	1.63976
$483$	0	0
$484$	−14.0000	−0.636364
$485$	18.0000	0.817338
$486$	0	0
$487$	0	0		−	1.00000i	$-0.5\pi$
$487$	0	0			1.00000i	$0.5\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−10.0000	−0.451294	−0.225647	−	0.974209i	$-0.572450\pi$
$491$	−10.0000	−0.451294	−0.225647	+	0.974209i	$0.572450\pi$
$492$	0	0
$493$	6.00000	0.270226
$494$	72.0000	3.23943
$495$	0	0
$496$	24.0000	1.07763
$497$	0	0
$498$	0	0
$499$	9.00000	0.402895	0.201448	−	0.979499i	$-0.435435\pi$
$499$	9.00000	0.402895	0.201448	+	0.979499i	$0.435435\pi$
$500$	−6.00000	−0.268328
$501$	0	0
$502$	−6.00000	−0.267793
$503$	33.0000	1.47140	0.735699	−	0.677309i	$-0.236854\pi$
$503$	33.0000	1.47140	0.735699	+	0.677309i	$0.236854\pi$
$504$	0	0
$505$	18.0000	0.800989
$506$	32.0000	1.42257
$507$	0	0
$508$	−22.0000	−0.976092
$509$	3.00000	0.132973	0.0664863	−	0.997787i	$-0.478821\pi$
$509$	3.00000	0.132973	0.0664863	+	0.997787i	$0.478821\pi$
$510$	0	0
$511$	0	0
$512$	−32.0000	−1.41421
$513$	0	0
$514$	12.0000	0.529297
$515$	36.0000	1.58635
$516$	0	0
$517$	6.00000	0.263880
$518$	0	0
$519$	0	0
$520$	0	0
$521$	15.0000	0.657162	0.328581	−	0.944476i	$-0.393430\pi$
$521$	15.0000	0.657162	0.328581	+	0.944476i	$0.393430\pi$
$522$	0	0
$523$	6.00000	0.262362	0.131181	−	0.991358i	$-0.458123\pi$
$523$	6.00000	0.262362	0.131181	+	0.991358i	$0.458123\pi$
$524$	0	0
$525$	0	0
$526$	−4.00000	−0.174408
$527$	−18.0000	−0.784092
$528$	0	0
$529$	41.0000	1.78261
$530$	24.0000	1.04249
$531$	0	0
$532$	0	0
$533$	−54.0000	−2.33900
$534$	0	0
$535$	12.0000	0.518805
$536$	0	0
$537$	0	0
$538$	−30.0000	−1.29339
$539$	0	0
$540$	0	0
$541$	−7.00000	−0.300954	−0.150477	−	0.988614i	$-0.548081\pi$
$541$	−7.00000	−0.300954	−0.150477	+	0.988614i	$0.548081\pi$
$542$	24.0000	1.03089
$543$	0	0
$544$	24.0000	1.02899
$545$	−27.0000	−1.15655
$546$	0	0
$547$	−37.0000	−1.58201	−0.791003	−	0.611812i	$-0.790441\pi$
$547$	−37.0000	−1.58201	−0.791003	+	0.611812i	$0.790441\pi$
$548$	4.00000	0.170872
$549$	0	0
$550$	16.0000	0.682242
$551$	12.0000	0.511217
$552$	0	0
$553$	0	0
$554$	38.0000	1.61447
$555$	0	0
$556$	12.0000	0.508913
$557$	−32.0000	−1.35588	−0.677942	−	0.735116i	$-0.737128\pi$
$557$	−32.0000	−1.35588	−0.677942	+	0.735116i	$0.737128\pi$
$558$	0	0
$559$	54.0000	2.28396
$560$	0	0
$561$	0	0
$562$	40.0000	1.68730
$563$	−36.0000	−1.51722	−0.758610	−	0.651546i	$-0.774121\pi$
$563$	−36.0000	−1.51722	−0.758610	+	0.651546i	$0.774121\pi$
$564$	0	0
$565$	−30.0000	−1.26211
$566$	−12.0000	−0.504398
$567$	0	0
$568$	0	0
$569$	−20.0000	−0.838444	−0.419222	−	0.907884i	$-0.637697\pi$
$569$	−20.0000	−0.838444	−0.419222	+	0.907884i	$0.637697\pi$
$570$	0	0
$571$	−41.0000	−1.71580	−0.857898	−	0.513820i	$-0.828230\pi$
$571$	−41.0000	−1.71580	−0.857898	+	0.513820i	$0.828230\pi$
$572$	24.0000	1.00349
$573$	0	0
$574$	0	0
$575$	32.0000	1.33449
$576$	0	0
$577$	12.0000	0.499567	0.249783	−	0.968302i	$-0.419641\pi$
$577$	12.0000	0.499567	0.249783	+	0.968302i	$0.419641\pi$
$578$	16.0000	0.665512
$579$	0	0
$580$	12.0000	0.498273
$581$	0	0
$582$	0	0
$583$	8.00000	0.331326
$584$	0	0
$585$	0	0
$586$	−42.0000	−1.73500
$587$	36.0000	1.48588	0.742940	−	0.669359i	$-0.233431\pi$
$587$	36.0000	1.48588	0.742940	+	0.669359i	$0.233431\pi$
$588$	0	0
$589$	−36.0000	−1.48335
$590$	18.0000	0.741048
$591$	0	0
$592$	−36.0000	−1.47959
$593$	39.0000	1.60154	0.800769	−	0.598973i	$-0.204424\pi$
$593$	39.0000	1.60154	0.800769	+	0.598973i	$0.204424\pi$
$594$	0	0
$595$	0	0
$596$	20.0000	0.819232
$597$	0	0
$598$	96.0000	3.92573
$599$	−46.0000	−1.87951	−0.939755	−	0.341850i	$-0.888947\pi$
$599$	−46.0000	−1.87951	−0.939755	+	0.341850i	$0.888947\pi$
$600$	0	0
$601$	30.0000	1.22373	0.611863	−	0.790964i	$-0.290420\pi$
$601$	30.0000	1.22373	0.611863	+	0.790964i	$0.290420\pi$
$602$	0	0
$603$	0	0
$604$	18.0000	0.732410
$605$	−21.0000	−0.853771
$606$	0	0
$607$	−42.0000	−1.70473	−0.852364	−	0.522949i	$-0.824832\pi$
$607$	−42.0000	−1.70473	−0.852364	+	0.522949i	$0.824832\pi$
$608$	48.0000	1.94666
$609$	0	0
$610$	−36.0000	−1.45760
$611$	18.0000	0.728202
$612$	0	0
$613$	−2.00000	−0.0807792	−0.0403896	−	0.999184i	$-0.512860\pi$
$613$	−2.00000	−0.0807792	−0.0403896	+	0.999184i	$0.512860\pi$
$614$	24.0000	0.968561
$615$	0	0
$616$	0	0
$617$	−8.00000	−0.322068	−0.161034	−	0.986949i	$-0.551483\pi$
$617$	−8.00000	−0.322068	−0.161034	+	0.986949i	$0.551483\pi$
$618$	0	0
$619$	6.00000	0.241160	0.120580	−	0.992704i	$-0.461525\pi$
$619$	6.00000	0.241160	0.120580	+	0.992704i	$0.461525\pi$
$620$	−36.0000	−1.44579
$621$	0	0
$622$	54.0000	2.16520
$623$	0	0
$624$	0	0
$625$	−29.0000	−1.16000
$626$	−48.0000	−1.91847
$627$	0	0
$628$	36.0000	1.43656
$629$	27.0000	1.07656
$630$	0	0
$631$	25.0000	0.995234	0.497617	−	0.867397i	$-0.334208\pi$
$631$	25.0000	0.995234	0.497617	+	0.867397i	$0.334208\pi$
$632$	0	0
$633$	0	0
$634$	−40.0000	−1.58860
$635$	−33.0000	−1.30957
$636$	0	0
$637$	0	0
$638$	8.00000	0.316723
$639$	0	0
$640$	0	0
$641$	−38.0000	−1.50091	−0.750455	−	0.660922i	$-0.770166\pi$
$641$	−38.0000	−1.50091	−0.750455	+	0.660922i	$0.770166\pi$
$642$	0	0
$643$	−12.0000	−0.473234	−0.236617	−	0.971603i	$-0.576039\pi$
$643$	−12.0000	−0.473234	−0.236617	+	0.971603i	$0.576039\pi$
$644$	0	0
$645$	0	0
$646$	−36.0000	−1.41640
$647$	−48.0000	−1.88707	−0.943537	−	0.331266i	$-0.892524\pi$
$647$	−48.0000	−1.88707	−0.943537	+	0.331266i	$0.892524\pi$
$648$	0	0
$649$	6.00000	0.235521
$650$	48.0000	1.88271
$651$	0	0
$652$	−22.0000	−0.861586
$653$	−10.0000	−0.391330	−0.195665	−	0.980671i	$-0.562687\pi$
$653$	−10.0000	−0.391330	−0.195665	+	0.980671i	$0.562687\pi$
$654$	0	0
$655$	0	0
$656$	−36.0000	−1.40556
$657$	0	0
$658$	0	0
$659$	20.0000	0.779089	0.389545	−	0.921008i	$-0.372632\pi$
$659$	20.0000	0.779089	0.389545	+	0.921008i	$0.372632\pi$
$660$	0	0
$661$	−30.0000	−1.16686	−0.583432	−	0.812162i	$-0.698291\pi$
$661$	−30.0000	−1.16686	−0.583432	+	0.812162i	$0.698291\pi$
$662$	18.0000	0.699590
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	16.0000	0.619522
$668$	18.0000	0.696441
$669$	0	0
$670$	−24.0000	−0.927201
$671$	−12.0000	−0.463255
$672$	0	0
$673$	18.0000	0.693849	0.346925	−	0.937893i	$-0.387226\pi$
$673$	18.0000	0.693849	0.346925	+	0.937893i	$0.387226\pi$
$674$	−54.0000	−2.08000
$675$	0	0
$676$	46.0000	1.76923
$677$	30.0000	1.15299	0.576497	−	0.817099i	$-0.304419\pi$
$677$	30.0000	1.15299	0.576497	+	0.817099i	$0.304419\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	−24.0000	−0.919007
$683$	14.0000	0.535695	0.267848	−	0.963461i	$-0.413688\pi$
$683$	14.0000	0.535695	0.267848	+	0.963461i	$0.413688\pi$
$684$	0	0
$685$	6.00000	0.229248
$686$	0	0
$687$	0	0
$688$	36.0000	1.37249
$689$	24.0000	0.914327
$690$	0	0
$691$	−18.0000	−0.684752	−0.342376	−	0.939563i	$-0.611232\pi$
$691$	−18.0000	−0.684752	−0.342376	+	0.939563i	$0.611232\pi$
$692$	36.0000	1.36851
$693$	0	0
$694$	52.0000	1.97389
$695$	18.0000	0.682779
$696$	0	0
$697$	27.0000	1.02270
$698$	24.0000	0.908413
$699$	0	0
$700$	0	0
$701$	14.0000	0.528773	0.264386	−	0.964417i	$-0.414831\pi$
$701$	14.0000	0.528773	0.264386	+	0.964417i	$0.414831\pi$
$702$	0	0
$703$	54.0000	2.03665
$704$	16.0000	0.603023
$705$	0	0
$706$	−6.00000	−0.225813
$707$	0	0
$708$	0	0
$709$	27.0000	1.01401	0.507003	−	0.861944i	$-0.330753\pi$
$709$	27.0000	1.01401	0.507003	+	0.861944i	$0.330753\pi$
$710$	−24.0000	−0.900704
$711$	0	0
$712$	0	0
$713$	−48.0000	−1.79761
$714$	0	0
$715$	36.0000	1.34632
$716$	−28.0000	−1.04641
$717$	0	0
$718$	8.00000	0.298557
$719$	−51.0000	−1.90198	−0.950990	−	0.309223i	$-0.899931\pi$
$719$	−51.0000	−1.90198	−0.950990	+	0.309223i	$0.899931\pi$
$720$	0	0
$721$	0	0
$722$	−34.0000	−1.26535
$723$	0	0
$724$	24.0000	0.891953
$725$	8.00000	0.297113
$726$	0	0
$727$	−18.0000	−0.667583	−0.333792	−	0.942647i	$-0.608328\pi$
$727$	−18.0000	−0.667583	−0.333792	+	0.942647i	$0.608328\pi$
$728$	0	0
$729$	0	0
$730$	−72.0000	−2.66484
$731$	−27.0000	−0.998631
$732$	0	0
$733$	48.0000	1.77292	0.886460	−	0.462805i	$-0.153157\pi$
$733$	48.0000	1.77292	0.886460	+	0.462805i	$0.153157\pi$
$734$	72.0000	2.65757
$735$	0	0
$736$	64.0000	2.35907
$737$	−8.00000	−0.294684
$738$	0	0
$739$	20.0000	0.735712	0.367856	−	0.929883i	$-0.380092\pi$
$739$	20.0000	0.735712	0.367856	+	0.929883i	$0.380092\pi$
$740$	54.0000	1.98508
$741$	0	0
$742$	0	0
$743$	−10.0000	−0.366864	−0.183432	−	0.983032i	$-0.558721\pi$
$743$	−10.0000	−0.366864	−0.183432	+	0.983032i	$0.558721\pi$
$744$	0	0
$745$	30.0000	1.09911
$746$	10.0000	0.366126
$747$	0	0
$748$	−12.0000	−0.438763
$749$	0	0
$750$	0	0
$751$	0	0		−	1.00000i	$-0.5\pi$
$751$	0	0			1.00000i	$0.5\pi$
$752$	12.0000	0.437595
$753$	0	0
$754$	24.0000	0.874028
$755$	27.0000	0.982631
$756$	0	0
$757$	−45.0000	−1.63555	−0.817776	−	0.575536i	$-0.804793\pi$
$757$	−45.0000	−1.63555	−0.817776	+	0.575536i	$0.804793\pi$
$758$	−18.0000	−0.653789
$759$	0	0
$760$	0	0
$761$	21.0000	0.761249	0.380625	−	0.924730i	$-0.375709\pi$
$761$	21.0000	0.761249	0.380625	+	0.924730i	$0.375709\pi$
$762$	0	0
$763$	0	0
$764$	−32.0000	−1.15772
$765$	0	0
$766$	6.00000	0.216789
$767$	18.0000	0.649942
$768$	0	0
$769$	36.0000	1.29819	0.649097	−	0.760706i	$-0.275147\pi$
$769$	36.0000	1.29819	0.649097	+	0.760706i	$0.275147\pi$
$770$	0	0
$771$	0	0
$772$	−18.0000	−0.647834
$773$	−27.0000	−0.971123	−0.485561	−	0.874203i	$-0.661385\pi$
$773$	−27.0000	−0.971123	−0.485561	+	0.874203i	$0.661385\pi$
$774$	0	0
$775$	−24.0000	−0.862105
$776$	0	0
$777$	0	0
$778$	−68.0000	−2.43792
$779$	54.0000	1.93475
$780$	0	0
$781$	−8.00000	−0.286263
$782$	−48.0000	−1.71648
$783$	0	0
$784$	0	0
$785$	54.0000	1.92734
$786$	0	0
$787$	−48.0000	−1.71102	−0.855508	−	0.517790i	$-0.826755\pi$
$787$	−48.0000	−1.71102	−0.855508	+	0.517790i	$0.826755\pi$
$788$	44.0000	1.56744
$789$	0	0
$790$	6.00000	0.213470
$791$	0	0
$792$	0	0
$793$	−36.0000	−1.27840
$794$	24.0000	0.851728
$795$	0	0
$796$	0	0
$797$	30.0000	1.06265	0.531327	−	0.847167i	$-0.321693\pi$
$797$	30.0000	1.06265	0.531327	+	0.847167i	$0.321693\pi$
$798$	0	0
$799$	−9.00000	−0.318397
$800$	32.0000	1.13137
$801$	0	0
$802$	56.0000	1.97743
$803$	−24.0000	−0.846942
$804$	0	0
$805$	0	0
$806$	−72.0000	−2.53609
$807$	0	0
$808$	0	0
$809$	4.00000	0.140633	0.0703163	−	0.997525i	$-0.477599\pi$
$809$	4.00000	0.140633	0.0703163	+	0.997525i	$0.477599\pi$
$810$	0	0
$811$	−42.0000	−1.47482	−0.737410	−	0.675446i	$-0.763951\pi$
$811$	−42.0000	−1.47482	−0.737410	+	0.675446i	$0.763951\pi$
$812$	0	0
$813$	0	0
$814$	36.0000	1.26180
$815$	−33.0000	−1.15594
$816$	0	0
$817$	−54.0000	−1.88922
$818$	24.0000	0.839140
$819$	0	0
$820$	54.0000	1.88576
$821$	−20.0000	−0.698005	−0.349002	−	0.937122i	$-0.613479\pi$
$821$	−20.0000	−0.698005	−0.349002	+	0.937122i	$0.613479\pi$
$822$	0	0
$823$	−31.0000	−1.08059	−0.540296	−	0.841475i	$-0.681688\pi$
$823$	−31.0000	−1.08059	−0.540296	+	0.841475i	$0.681688\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−22.0000	−0.765015	−0.382507	−	0.923952i	$-0.624939\pi$
$827$	−22.0000	−0.765015	−0.382507	+	0.923952i	$0.624939\pi$
$828$	0	0
$829$	−18.0000	−0.625166	−0.312583	−	0.949890i	$-0.601194\pi$
$829$	−18.0000	−0.625166	−0.312583	+	0.949890i	$0.601194\pi$
$830$	−90.0000	−3.12395
$831$	0	0
$832$	48.0000	1.66410
$833$	0	0
$834$	0	0
$835$	27.0000	0.934374
$836$	−24.0000	−0.830057
$837$	0	0
$838$	42.0000	1.45087
$839$	−3.00000	−0.103572	−0.0517858	−	0.998658i	$-0.516491\pi$
$839$	−3.00000	−0.103572	−0.0517858	+	0.998658i	$0.516491\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	52.0000	1.79204
$843$	0	0
$844$	0	0
$845$	69.0000	2.37367
$846$	0	0
$847$	0	0
$848$	16.0000	0.549442
$849$	0	0
$850$	−24.0000	−0.823193
$851$	72.0000	2.46813
$852$	0	0
$853$	12.0000	0.410872	0.205436	−	0.978671i	$-0.434139\pi$
$853$	12.0000	0.410872	0.205436	+	0.978671i	$0.434139\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	9.00000	0.307434	0.153717	−	0.988115i	$-0.450876\pi$
$857$	9.00000	0.307434	0.153717	+	0.988115i	$0.450876\pi$
$858$	0	0
$859$	0	0		−	1.00000i	$-0.5\pi$
$859$	0	0			1.00000i	$0.5\pi$
$860$	−54.0000	−1.84138
$861$	0	0
$862$	8.00000	0.272481
$863$	32.0000	1.08929	0.544646	−	0.838666i	$-0.316664\pi$
$863$	32.0000	1.08929	0.544646	+	0.838666i	$0.316664\pi$
$864$	0	0
$865$	54.0000	1.83606
$866$	36.0000	1.22333
$867$	0	0
$868$	0	0
$869$	2.00000	0.0678454
$870$	0	0
$871$	−24.0000	−0.813209
$872$	0	0
$873$	0	0
$874$	−96.0000	−3.24725
$875$	0	0
$876$	0	0
$877$	−31.0000	−1.04680	−0.523398	−	0.852088i	$-0.675336\pi$
$877$	−31.0000	−1.04680	−0.523398	+	0.852088i	$0.675336\pi$
$878$	24.0000	0.809961
$879$	0	0
$880$	24.0000	0.809040
$881$	−42.0000	−1.41502	−0.707508	−	0.706705i	$-0.750181\pi$
$881$	−42.0000	−1.41502	−0.707508	+	0.706705i	$0.750181\pi$
$882$	0	0
$883$	45.0000	1.51437	0.757185	−	0.653200i	$-0.226574\pi$
$883$	45.0000	1.51437	0.757185	+	0.653200i	$0.226574\pi$
$884$	−36.0000	−1.21081
$885$	0	0
$886$	4.00000	0.134383
$887$	−15.0000	−0.503651	−0.251825	−	0.967773i	$-0.581031\pi$
$887$	−15.0000	−0.503651	−0.251825	+	0.967773i	$0.581031\pi$
$888$	0	0
$889$	0	0
$890$	36.0000	1.20672
$891$	0	0
$892$	−36.0000	−1.20537
$893$	−18.0000	−0.602347
$894$	0	0
$895$	−42.0000	−1.40391
$896$	0	0
$897$	0	0
$898$	44.0000	1.46830
$899$	−12.0000	−0.400222
$900$	0	0
$901$	−12.0000	−0.399778
$902$	36.0000	1.19867
$903$	0	0
$904$	0	0
$905$	36.0000	1.19668
$906$	0	0
$907$	−27.0000	−0.896520	−0.448260	−	0.893903i	$-0.647956\pi$
$907$	−27.0000	−0.896520	−0.448260	+	0.893903i	$0.647956\pi$
$908$	24.0000	0.796468
$909$	0	0
$910$	0	0
$911$	−2.00000	−0.0662630	−0.0331315	−	0.999451i	$-0.510548\pi$
$911$	−2.00000	−0.0662630	−0.0331315	+	0.999451i	$0.510548\pi$
$912$	0	0
$913$	−30.0000	−0.992855
$914$	−36.0000	−1.19077
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−27.0000	−0.890648	−0.445324	−	0.895370i	$-0.646911\pi$
$919$	−27.0000	−0.890648	−0.445324	+	0.895370i	$0.646911\pi$
$920$	0	0
$921$	0	0
$922$	6.00000	0.197599
$923$	−24.0000	−0.789970
$924$	0	0
$925$	36.0000	1.18367
$926$	−10.0000	−0.328620
$927$	0	0
$928$	16.0000	0.525226
$929$	9.00000	0.295280	0.147640	−	0.989041i	$-0.452832\pi$
$929$	9.00000	0.295280	0.147640	+	0.989041i	$0.452832\pi$
$930$	0	0
$931$	0	0
$932$	44.0000	1.44127
$933$	0	0
$934$	−24.0000	−0.785304
$935$	−18.0000	−0.588663
$936$	0	0
$937$	−18.0000	−0.588034	−0.294017	−	0.955800i	$-0.594992\pi$
$937$	−18.0000	−0.588034	−0.294017	+	0.955800i	$0.594992\pi$
$938$	0	0
$939$	0	0
$940$	−18.0000	−0.587095
$941$	−9.00000	−0.293392	−0.146696	−	0.989182i	$-0.546864\pi$
$941$	−9.00000	−0.293392	−0.146696	+	0.989182i	$0.546864\pi$
$942$	0	0
$943$	72.0000	2.34464
$944$	12.0000	0.390567
$945$	0	0
$946$	−36.0000	−1.17046
$947$	−56.0000	−1.81976	−0.909878	−	0.414876i	$-0.863825\pi$
$947$	−56.0000	−1.81976	−0.909878	+	0.414876i	$0.863825\pi$
$948$	0	0
$949$	−72.0000	−2.33722
$950$	−48.0000	−1.55733
$951$	0	0
$952$	0	0
$953$	40.0000	1.29573	0.647864	−	0.761756i	$-0.275663\pi$
$953$	40.0000	1.29573	0.647864	+	0.761756i	$0.275663\pi$
$954$	0	0
$955$	−48.0000	−1.55324
$956$	20.0000	0.646846
$957$	0	0
$958$	−30.0000	−0.969256
$959$	0	0
$960$	0	0
$961$	5.00000	0.161290
$962$	108.000	3.48206
$963$	0	0
$964$	−36.0000	−1.15948
$965$	−27.0000	−0.869161
$966$	0	0
$967$	32.0000	1.02905	0.514525	−	0.857475i	$-0.327968\pi$
$967$	32.0000	1.02905	0.514525	+	0.857475i	$0.327968\pi$
$968$	0	0
$969$	0	0
$970$	−36.0000	−1.15589
$971$	27.0000	0.866471	0.433236	−	0.901281i	$-0.357372\pi$
$971$	27.0000	0.866471	0.433236	+	0.901281i	$0.357372\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	−24.0000	−0.768221
$977$	−8.00000	−0.255943	−0.127971	−	0.991778i	$-0.540847\pi$
$977$	−8.00000	−0.255943	−0.127971	+	0.991778i	$0.540847\pi$
$978$	0	0
$979$	12.0000	0.383522
$980$	0	0
$981$	0	0
$982$	20.0000	0.638226
$983$	33.0000	1.05254	0.526268	−	0.850319i	$-0.323591\pi$
$983$	33.0000	1.05254	0.526268	+	0.850319i	$0.323591\pi$
$984$	0	0
$985$	66.0000	2.10293
$986$	−12.0000	−0.382158
$987$	0	0
$988$	−72.0000	−2.29063
$989$	−72.0000	−2.28947
$990$	0	0
$991$	−9.00000	−0.285894	−0.142947	−	0.989730i	$-0.545658\pi$
$991$	−9.00000	−0.285894	−0.142947	+	0.989730i	$0.545658\pi$
$992$	−48.0000	−1.52400
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	18.0000	0.570066	0.285033	−	0.958518i	$-0.407995\pi$
$997$	18.0000	0.570066	0.285033	+	0.958518i	$0.407995\pi$
$998$	−18.0000	−0.569780
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1323.2.a.c.1.1	yes	1
3.2	odd	2		1323.2.a.q.1.1	yes	1
7.6	odd	2		1323.2.a.a.1.1	✓	1
21.20	even	2		1323.2.a.s.1.1	yes	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1323.2.a.a.1.1	✓	1	7.6	odd	2
1323.2.a.c.1.1	yes	1	1.1	even	1	trivial
1323.2.a.q.1.1	yes	1	3.2	odd	2
1323.2.a.s.1.1	yes	1	21.20	even	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:	\( N \)	\(=\)	\( 1323 = 3^{3} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1323.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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