LMFDB - Embedded newform 927.2.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [927,2,Mod(1,927)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(927, 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("927.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 927 = 3^{2} \cdot 103 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	927.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:	$7.40213226737$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 309)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	927.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+1.00000 q^{2} -1.00000 q^{4} +1.00000 q^{5} -2.00000 q^{7} -3.00000 q^{8} +O(q^{10})$

$q+1.00000 q^{2} -1.00000 q^{4} +1.00000 q^{5} -2.00000 q^{7} -3.00000 q^{8} +1.00000 q^{10} +2.00000 q^{11} -5.00000 q^{13} -2.00000 q^{14} -1.00000 q^{16} -8.00000 q^{19} -1.00000 q^{20} +2.00000 q^{22} -1.00000 q^{23} -4.00000 q^{25} -5.00000 q^{26} +2.00000 q^{28} +2.00000 q^{29} +5.00000 q^{31} +5.00000 q^{32} -2.00000 q^{35} +2.00000 q^{37} -8.00000 q^{38} -3.00000 q^{40} -8.00000 q^{41} -11.0000 q^{43} -2.00000 q^{44} -1.00000 q^{46} +2.00000 q^{47} -3.00000 q^{49} -4.00000 q^{50} +5.00000 q^{52} -10.0000 q^{53} +2.00000 q^{55} +6.00000 q^{56} +2.00000 q^{58} +11.0000 q^{59} -5.00000 q^{61} +5.00000 q^{62} +7.00000 q^{64} -5.00000 q^{65} +11.0000 q^{67} -2.00000 q^{70} -16.0000 q^{71} +12.0000 q^{73} +2.00000 q^{74} +8.00000 q^{76} -4.00000 q^{77} +6.00000 q^{79} -1.00000 q^{80} -8.00000 q^{82} -1.00000 q^{83} -11.0000 q^{86} -6.00000 q^{88} +6.00000 q^{89} +10.0000 q^{91} +1.00000 q^{92} +2.00000 q^{94} -8.00000 q^{95} -7.00000 q^{97} -3.00000 q^{98} +O(q^{100})$

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	1.00000	0.707107	0.353553	−	0.935414i	$-0.384973\pi$
$2$	1.00000	0.707107	0.353553	+	0.935414i	$0.384973\pi$
$3$	0	0
$3$	0	0
$4$	−1.00000	−0.500000
$5$	1.00000	0.447214	0.223607	−	0.974679i	$-0.428217\pi$
$5$	1.00000	0.447214	0.223607	+	0.974679i	$0.428217\pi$
$6$	0	0
$7$	−2.00000	−0.755929	−0.377964	−	0.925820i	$-0.623376\pi$
$7$	−2.00000	−0.755929	−0.377964	+	0.925820i	$0.623376\pi$
$8$	−3.00000	−1.06066
$9$	0	0
$10$	1.00000	0.316228
$11$	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
$13$	−5.00000	−1.38675	−0.693375	−	0.720577i	$-0.743877\pi$
$13$	−5.00000	−1.38675	−0.693375	+	0.720577i	$0.743877\pi$
$14$	−2.00000	−0.534522
$15$	0	0
$16$	−1.00000	−0.250000
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	0	0
$19$	−8.00000	−1.83533	−0.917663	−	0.397360i	$-0.869927\pi$
$19$	−8.00000	−1.83533	−0.917663	+	0.397360i	$0.869927\pi$
$20$	−1.00000	−0.223607
$21$	0	0
$22$	2.00000	0.426401
$23$	−1.00000	−0.208514	−0.104257	−	0.994550i	$-0.533247\pi$
$23$	−1.00000	−0.208514	−0.104257	+	0.994550i	$0.533247\pi$
$24$	0	0
$25$	−4.00000	−0.800000
$26$	−5.00000	−0.980581
$27$	0	0
$28$	2.00000	0.377964
$29$	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
$31$	5.00000	0.898027	0.449013	−	0.893525i	$-0.351776\pi$
$31$	5.00000	0.898027	0.449013	+	0.893525i	$0.351776\pi$
$32$	5.00000	0.883883
$33$	0	0
$34$	0	0
$35$	−2.00000	−0.338062
$36$	0	0
$37$	2.00000	0.328798	0.164399	−	0.986394i	$-0.447432\pi$
$37$	2.00000	0.328798	0.164399	+	0.986394i	$0.447432\pi$
$38$	−8.00000	−1.29777
$39$	0	0
$40$	−3.00000	−0.474342
$41$	−8.00000	−1.24939	−0.624695	−	0.780869i	$-0.714777\pi$
$41$	−8.00000	−1.24939	−0.624695	+	0.780869i	$0.714777\pi$
$42$	0	0
$43$	−11.0000	−1.67748	−0.838742	−	0.544529i	$-0.816708\pi$
$43$	−11.0000	−1.67748	−0.838742	+	0.544529i	$0.816708\pi$
$44$	−2.00000	−0.301511
$45$	0	0
$46$	−1.00000	−0.147442
$47$	2.00000	0.291730	0.145865	−	0.989305i	$-0.453403\pi$
$47$	2.00000	0.291730	0.145865	+	0.989305i	$0.453403\pi$
$48$	0	0
$49$	−3.00000	−0.428571
$50$	−4.00000	−0.565685
$51$	0	0
$52$	5.00000	0.693375
$53$	−10.0000	−1.37361	−0.686803	−	0.726844i	$-0.740986\pi$
$53$	−10.0000	−1.37361	−0.686803	+	0.726844i	$0.740986\pi$
$54$	0	0
$55$	2.00000	0.269680
$56$	6.00000	0.801784
$57$	0	0
$58$	2.00000	0.262613
$59$	11.0000	1.43208	0.716039	−	0.698060i	$-0.245953\pi$
$59$	11.0000	1.43208	0.716039	+	0.698060i	$0.245953\pi$
$60$	0	0
$61$	−5.00000	−0.640184	−0.320092	−	0.947386i	$-0.603714\pi$
$61$	−5.00000	−0.640184	−0.320092	+	0.947386i	$0.603714\pi$
$62$	5.00000	0.635001
$63$	0	0
$64$	7.00000	0.875000
$65$	−5.00000	−0.620174
$66$	0	0
$67$	11.0000	1.34386	0.671932	−	0.740613i	$-0.265465\pi$
$67$	11.0000	1.34386	0.671932	+	0.740613i	$0.265465\pi$
$68$	0	0
$69$	0	0
$70$	−2.00000	−0.239046
$71$	−16.0000	−1.89885	−0.949425	−	0.313993i	$-0.898333\pi$
$71$	−16.0000	−1.89885	−0.949425	+	0.313993i	$0.898333\pi$
$72$	0	0
$73$	12.0000	1.40449	0.702247	−	0.711934i	$-0.252180\pi$
$73$	12.0000	1.40449	0.702247	+	0.711934i	$0.252180\pi$
$74$	2.00000	0.232495
$75$	0	0
$76$	8.00000	0.917663
$77$	−4.00000	−0.455842
$78$	0	0
$79$	6.00000	0.675053	0.337526	−	0.941316i	$-0.390410\pi$
$79$	6.00000	0.675053	0.337526	+	0.941316i	$0.390410\pi$
$80$	−1.00000	−0.111803
$81$	0	0
$82$	−8.00000	−0.883452
$83$	−1.00000	−0.109764	−0.0548821	−	0.998493i	$-0.517478\pi$
$83$	−1.00000	−0.109764	−0.0548821	+	0.998493i	$0.517478\pi$
$84$	0	0
$85$	0	0
$86$	−11.0000	−1.18616
$87$	0	0
$88$	−6.00000	−0.639602
$89$	6.00000	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	0	0
$91$	10.0000	1.04828
$92$	1.00000	0.104257
$93$	0	0
$94$	2.00000	0.206284
$95$	−8.00000	−0.820783
$96$	0	0
$97$	−7.00000	−0.710742	−0.355371	−	0.934725i	$-0.615646\pi$
$97$	−7.00000	−0.710742	−0.355371	+	0.934725i	$0.615646\pi$
$98$	−3.00000	−0.303046
$99$	0	0
$100$	4.00000	0.400000
$101$	9.00000	0.895533	0.447767	−	0.894150i	$-0.352219\pi$
$101$	9.00000	0.895533	0.447767	+	0.894150i	$0.352219\pi$
$102$	0	0
$103$	1.00000	0.0985329
$103$	1.00000	0.0985329
$104$	15.0000	1.47087
$105$	0	0
$106$	−10.0000	−0.971286
$107$	11.0000	1.06341	0.531705	−	0.846930i	$-0.321551\pi$
$107$	11.0000	1.06341	0.531705	+	0.846930i	$0.321551\pi$
$108$	0	0
$109$	6.00000	0.574696	0.287348	−	0.957826i	$-0.407226\pi$
$109$	6.00000	0.574696	0.287348	+	0.957826i	$0.407226\pi$
$110$	2.00000	0.190693
$111$	0	0
$112$	2.00000	0.188982
$113$	1.00000	0.0940721	0.0470360	−	0.998893i	$-0.485022\pi$
$113$	1.00000	0.0940721	0.0470360	+	0.998893i	$0.485022\pi$
$114$	0	0
$115$	−1.00000	−0.0932505
$116$	−2.00000	−0.185695
$117$	0	0
$118$	11.0000	1.01263
$119$	0	0
$120$	0	0
$121$	−7.00000	−0.636364
$122$	−5.00000	−0.452679
$123$	0	0
$124$	−5.00000	−0.449013
$125$	−9.00000	−0.804984
$126$	0	0
$127$	−7.00000	−0.621150	−0.310575	−	0.950549i	$-0.600522\pi$
$127$	−7.00000	−0.621150	−0.310575	+	0.950549i	$0.600522\pi$
$128$	−3.00000	−0.265165
$129$	0	0
$130$	−5.00000	−0.438529
$131$	17.0000	1.48530	0.742648	−	0.669681i	$-0.233569\pi$
$131$	17.0000	1.48530	0.742648	+	0.669681i	$0.233569\pi$
$132$	0	0
$133$	16.0000	1.38738
$134$	11.0000	0.950255
$135$	0	0
$136$	0	0
$137$	−6.00000	−0.512615	−0.256307	−	0.966595i	$-0.582506\pi$
$137$	−6.00000	−0.512615	−0.256307	+	0.966595i	$0.582506\pi$
$138$	0	0
$139$	−8.00000	−0.678551	−0.339276	−	0.940687i	$-0.610182\pi$
$139$	−8.00000	−0.678551	−0.339276	+	0.940687i	$0.610182\pi$
$140$	2.00000	0.169031
$141$	0	0
$142$	−16.0000	−1.34269
$143$	−10.0000	−0.836242
$144$	0	0
$145$	2.00000	0.166091
$146$	12.0000	0.993127
$147$	0	0
$148$	−2.00000	−0.164399
$149$	−2.00000	−0.163846	−0.0819232	−	0.996639i	$-0.526106\pi$
$149$	−2.00000	−0.163846	−0.0819232	+	0.996639i	$0.526106\pi$
$150$	0	0
$151$	−8.00000	−0.651031	−0.325515	−	0.945537i	$-0.605538\pi$
$151$	−8.00000	−0.651031	−0.325515	+	0.945537i	$0.605538\pi$
$152$	24.0000	1.94666
$153$	0	0
$154$	−4.00000	−0.322329
$155$	5.00000	0.401610
$156$	0	0
$157$	−8.00000	−0.638470	−0.319235	−	0.947676i	$-0.603426\pi$
$157$	−8.00000	−0.638470	−0.319235	+	0.947676i	$0.603426\pi$
$158$	6.00000	0.477334
$159$	0	0
$160$	5.00000	0.395285
$161$	2.00000	0.157622
$162$	0	0
$163$	−10.0000	−0.783260	−0.391630	−	0.920123i	$-0.628089\pi$
$163$	−10.0000	−0.783260	−0.391630	+	0.920123i	$0.628089\pi$
$164$	8.00000	0.624695
$165$	0	0
$166$	−1.00000	−0.0776151
$167$	12.0000	0.928588	0.464294	−	0.885681i	$-0.346308\pi$
$167$	12.0000	0.928588	0.464294	+	0.885681i	$0.346308\pi$
$168$	0	0
$169$	12.0000	0.923077
$170$	0	0
$171$	0	0
$172$	11.0000	0.838742
$173$	−6.00000	−0.456172	−0.228086	−	0.973641i	$-0.573247\pi$
$173$	−6.00000	−0.456172	−0.228086	+	0.973641i	$0.573247\pi$
$174$	0	0
$175$	8.00000	0.604743
$176$	−2.00000	−0.150756
$177$	0	0
$178$	6.00000	0.449719
$179$	20.0000	1.49487	0.747435	−	0.664335i	$-0.231285\pi$
$179$	20.0000	1.49487	0.747435	+	0.664335i	$0.231285\pi$
$180$	0	0
$181$	12.0000	0.891953	0.445976	−	0.895045i	$-0.352856\pi$
$181$	12.0000	0.891953	0.445976	+	0.895045i	$0.352856\pi$
$182$	10.0000	0.741249
$183$	0	0
$184$	3.00000	0.221163
$185$	2.00000	0.147043
$186$	0	0
$187$	0	0
$188$	−2.00000	−0.145865
$189$	0	0
$190$	−8.00000	−0.580381
$191$	20.0000	1.44715	0.723575	−	0.690246i	$-0.242498\pi$
$191$	20.0000	1.44715	0.723575	+	0.690246i	$0.242498\pi$
$192$	0	0
$193$	−8.00000	−0.575853	−0.287926	−	0.957653i	$-0.592966\pi$
$193$	−8.00000	−0.575853	−0.287926	+	0.957653i	$0.592966\pi$
$194$	−7.00000	−0.502571
$195$	0	0
$196$	3.00000	0.214286
$197$	−11.0000	−0.783718	−0.391859	−	0.920025i	$-0.628168\pi$
$197$	−11.0000	−0.783718	−0.391859	+	0.920025i	$0.628168\pi$
$198$	0	0
$199$	−1.00000	−0.0708881	−0.0354441	−	0.999372i	$-0.511285\pi$
$199$	−1.00000	−0.0708881	−0.0354441	+	0.999372i	$0.511285\pi$
$200$	12.0000	0.848528
$201$	0	0
$202$	9.00000	0.633238
$203$	−4.00000	−0.280745
$204$	0	0
$205$	−8.00000	−0.558744
$206$	1.00000	0.0696733
$207$	0	0
$208$	5.00000	0.346688
$209$	−16.0000	−1.10674
$210$	0	0
$211$	4.00000	0.275371	0.137686	−	0.990476i	$-0.456034\pi$
$211$	4.00000	0.275371	0.137686	+	0.990476i	$0.456034\pi$
$212$	10.0000	0.686803
$213$	0	0
$214$	11.0000	0.751945
$215$	−11.0000	−0.750194
$216$	0	0
$217$	−10.0000	−0.678844
$218$	6.00000	0.406371
$219$	0	0
$220$	−2.00000	−0.134840
$221$	0	0
$222$	0	0
$223$	−4.00000	−0.267860	−0.133930	−	0.990991i	$-0.542760\pi$
$223$	−4.00000	−0.267860	−0.133930	+	0.990991i	$0.542760\pi$
$224$	−10.0000	−0.668153
$225$	0	0
$226$	1.00000	0.0665190
$227$	−24.0000	−1.59294	−0.796468	−	0.604681i	$-0.793301\pi$
$227$	−24.0000	−1.59294	−0.796468	+	0.604681i	$0.793301\pi$
$228$	0	0
$229$	3.00000	0.198246	0.0991228	−	0.995075i	$-0.468396\pi$
$229$	3.00000	0.198246	0.0991228	+	0.995075i	$0.468396\pi$
$230$	−1.00000	−0.0659380
$231$	0	0
$232$	−6.00000	−0.393919
$233$	2.00000	0.131024	0.0655122	−	0.997852i	$-0.479132\pi$
$233$	2.00000	0.131024	0.0655122	+	0.997852i	$0.479132\pi$
$234$	0	0
$235$	2.00000	0.130466
$236$	−11.0000	−0.716039
$237$	0	0
$238$	0	0
$239$	20.0000	1.29369	0.646846	−	0.762620i	$-0.276088\pi$
$239$	20.0000	1.29369	0.646846	+	0.762620i	$0.276088\pi$
$240$	0	0
$241$	2.00000	0.128831	0.0644157	−	0.997923i	$-0.479482\pi$
$241$	2.00000	0.128831	0.0644157	+	0.997923i	$0.479482\pi$
$242$	−7.00000	−0.449977
$243$	0	0
$244$	5.00000	0.320092
$245$	−3.00000	−0.191663
$246$	0	0
$247$	40.0000	2.54514
$248$	−15.0000	−0.952501
$249$	0	0
$250$	−9.00000	−0.569210
$251$	18.0000	1.13615	0.568075	−	0.822977i	$-0.307688\pi$
$251$	18.0000	1.13615	0.568075	+	0.822977i	$0.307688\pi$
$252$	0	0
$253$	−2.00000	−0.125739
$254$	−7.00000	−0.439219
$255$	0	0
$256$	−17.0000	−1.06250
$257$	7.00000	0.436648	0.218324	−	0.975876i	$-0.429941\pi$
$257$	7.00000	0.436648	0.218324	+	0.975876i	$0.429941\pi$
$258$	0	0
$259$	−4.00000	−0.248548
$260$	5.00000	0.310087
$261$	0	0
$262$	17.0000	1.05026
$263$	−30.0000	−1.84988	−0.924940	−	0.380114i	$-0.875885\pi$
$263$	−30.0000	−1.84988	−0.924940	+	0.380114i	$0.875885\pi$
$264$	0	0
$265$	−10.0000	−0.614295
$266$	16.0000	0.981023
$267$	0	0
$268$	−11.0000	−0.671932
$269$	−24.0000	−1.46331	−0.731653	−	0.681677i	$-0.761251\pi$
$269$	−24.0000	−1.46331	−0.731653	+	0.681677i	$0.761251\pi$
$270$	0	0
$271$	−25.0000	−1.51864	−0.759321	−	0.650716i	$-0.774469\pi$
$271$	−25.0000	−1.51864	−0.759321	+	0.650716i	$0.774469\pi$
$272$	0	0
$273$	0	0
$274$	−6.00000	−0.362473
$275$	−8.00000	−0.482418
$276$	0	0
$277$	−8.00000	−0.480673	−0.240337	−	0.970690i	$-0.577258\pi$
$277$	−8.00000	−0.480673	−0.240337	+	0.970690i	$0.577258\pi$
$278$	−8.00000	−0.479808
$279$	0	0
$280$	6.00000	0.358569
$281$	−21.0000	−1.25275	−0.626377	−	0.779520i	$-0.715463\pi$
$281$	−21.0000	−1.25275	−0.626377	+	0.779520i	$0.715463\pi$
$282$	0	0
$283$	13.0000	0.772770	0.386385	−	0.922338i	$-0.373724\pi$
$283$	13.0000	0.772770	0.386385	+	0.922338i	$0.373724\pi$
$284$	16.0000	0.949425
$285$	0	0
$286$	−10.0000	−0.591312
$287$	16.0000	0.944450
$288$	0	0
$289$	−17.0000	−1.00000
$290$	2.00000	0.117444
$291$	0	0
$292$	−12.0000	−0.702247
$293$	−9.00000	−0.525786	−0.262893	−	0.964825i	$-0.584677\pi$
$293$	−9.00000	−0.525786	−0.262893	+	0.964825i	$0.584677\pi$
$294$	0	0
$295$	11.0000	0.640445
$296$	−6.00000	−0.348743
$297$	0	0
$298$	−2.00000	−0.115857
$299$	5.00000	0.289157
$300$	0	0
$301$	22.0000	1.26806
$302$	−8.00000	−0.460348
$303$	0	0
$304$	8.00000	0.458831
$305$	−5.00000	−0.286299
$306$	0	0
$307$	−7.00000	−0.399511	−0.199756	−	0.979846i	$-0.564015\pi$
$307$	−7.00000	−0.399511	−0.199756	+	0.979846i	$0.564015\pi$
$308$	4.00000	0.227921
$309$	0	0
$310$	5.00000	0.283981
$311$	33.0000	1.87126	0.935629	−	0.352985i	$-0.114833\pi$
$311$	33.0000	1.87126	0.935629	+	0.352985i	$0.114833\pi$
$312$	0	0
$313$	−22.0000	−1.24351	−0.621757	−	0.783210i	$-0.713581\pi$
$313$	−22.0000	−1.24351	−0.621757	+	0.783210i	$0.713581\pi$
$314$	−8.00000	−0.451466
$315$	0	0
$316$	−6.00000	−0.337526
$317$	−26.0000	−1.46031	−0.730153	−	0.683284i	$-0.760551\pi$
$317$	−26.0000	−1.46031	−0.730153	+	0.683284i	$0.760551\pi$
$318$	0	0
$319$	4.00000	0.223957
$320$	7.00000	0.391312
$321$	0	0
$322$	2.00000	0.111456
$323$	0	0
$324$	0	0
$325$	20.0000	1.10940
$326$	−10.0000	−0.553849
$327$	0	0
$328$	24.0000	1.32518
$329$	−4.00000	−0.220527
$330$	0	0
$331$	−31.0000	−1.70391	−0.851957	−	0.523612i	$-0.824584\pi$
$331$	−31.0000	−1.70391	−0.851957	+	0.523612i	$0.824584\pi$
$332$	1.00000	0.0548821
$333$	0	0
$334$	12.0000	0.656611
$335$	11.0000	0.600994
$336$	0	0
$337$	17.0000	0.926049	0.463025	−	0.886345i	$-0.346764\pi$
$337$	17.0000	0.926049	0.463025	+	0.886345i	$0.346764\pi$
$338$	12.0000	0.652714
$339$	0	0
$340$	0	0
$341$	10.0000	0.541530
$342$	0	0
$343$	20.0000	1.07990
$344$	33.0000	1.77924
$345$	0	0
$346$	−6.00000	−0.322562
$347$	−27.0000	−1.44944	−0.724718	−	0.689046i	$-0.758030\pi$
$347$	−27.0000	−1.44944	−0.724718	+	0.689046i	$0.758030\pi$
$348$	0	0
$349$	−4.00000	−0.214115	−0.107058	−	0.994253i	$-0.534143\pi$
$349$	−4.00000	−0.214115	−0.107058	+	0.994253i	$0.534143\pi$
$350$	8.00000	0.427618
$351$	0	0
$352$	10.0000	0.533002
$353$	−27.0000	−1.43706	−0.718532	−	0.695493i	$-0.755186\pi$
$353$	−27.0000	−1.43706	−0.718532	+	0.695493i	$0.755186\pi$
$354$	0	0
$355$	−16.0000	−0.849192
$356$	−6.00000	−0.317999
$357$	0	0
$358$	20.0000	1.05703
$359$	27.0000	1.42501	0.712503	−	0.701669i	$-0.247562\pi$
$359$	27.0000	1.42501	0.712503	+	0.701669i	$0.247562\pi$
$360$	0	0
$361$	45.0000	2.36842
$362$	12.0000	0.630706
$363$	0	0
$364$	−10.0000	−0.524142
$365$	12.0000	0.628109
$366$	0	0
$367$	−4.00000	−0.208798	−0.104399	−	0.994535i	$-0.533292\pi$
$367$	−4.00000	−0.208798	−0.104399	+	0.994535i	$0.533292\pi$
$368$	1.00000	0.0521286
$369$	0	0
$370$	2.00000	0.103975
$371$	20.0000	1.03835
$372$	0	0
$373$	10.0000	0.517780	0.258890	−	0.965907i	$-0.416643\pi$
$373$	10.0000	0.517780	0.258890	+	0.965907i	$0.416643\pi$
$374$	0	0
$375$	0	0
$376$	−6.00000	−0.309426
$377$	−10.0000	−0.515026
$378$	0	0
$379$	−23.0000	−1.18143	−0.590715	−	0.806880i	$-0.701154\pi$
$379$	−23.0000	−1.18143	−0.590715	+	0.806880i	$0.701154\pi$
$380$	8.00000	0.410391
$381$	0	0
$382$	20.0000	1.02329
$383$	6.00000	0.306586	0.153293	−	0.988181i	$-0.451012\pi$
$383$	6.00000	0.306586	0.153293	+	0.988181i	$0.451012\pi$
$384$	0	0
$385$	−4.00000	−0.203859
$386$	−8.00000	−0.407189
$387$	0	0
$388$	7.00000	0.355371
$389$	30.0000	1.52106	0.760530	−	0.649303i	$-0.224939\pi$
$389$	30.0000	1.52106	0.760530	+	0.649303i	$0.224939\pi$
$390$	0	0
$391$	0	0
$392$	9.00000	0.454569
$393$	0	0
$394$	−11.0000	−0.554172
$395$	6.00000	0.301893
$396$	0	0
$397$	8.00000	0.401508	0.200754	−	0.979642i	$-0.435661\pi$
$397$	8.00000	0.401508	0.200754	+	0.979642i	$0.435661\pi$
$398$	−1.00000	−0.0501255
$399$	0	0
$400$	4.00000	0.200000
$401$	−16.0000	−0.799002	−0.399501	−	0.916733i	$-0.630817\pi$
$401$	−16.0000	−0.799002	−0.399501	+	0.916733i	$0.630817\pi$
$402$	0	0
$403$	−25.0000	−1.24534
$404$	−9.00000	−0.447767
$405$	0	0
$406$	−4.00000	−0.198517
$407$	4.00000	0.198273
$408$	0	0
$409$	−34.0000	−1.68119	−0.840596	−	0.541663i	$-0.817795\pi$
$409$	−34.0000	−1.68119	−0.840596	+	0.541663i	$0.817795\pi$
$410$	−8.00000	−0.395092
$411$	0	0
$412$	−1.00000	−0.0492665
$413$	−22.0000	−1.08255
$414$	0	0
$415$	−1.00000	−0.0490881
$416$	−25.0000	−1.22573
$417$	0	0
$418$	−16.0000	−0.782586
$419$	−12.0000	−0.586238	−0.293119	−	0.956076i	$-0.594693\pi$
$419$	−12.0000	−0.586238	−0.293119	+	0.956076i	$0.594693\pi$
$420$	0	0
$421$	−3.00000	−0.146211	−0.0731055	−	0.997324i	$-0.523291\pi$
$421$	−3.00000	−0.146211	−0.0731055	+	0.997324i	$0.523291\pi$
$422$	4.00000	0.194717
$423$	0	0
$424$	30.0000	1.45693
$425$	0	0
$426$	0	0
$427$	10.0000	0.483934
$428$	−11.0000	−0.531705
$429$	0	0
$430$	−11.0000	−0.530467
$431$	−27.0000	−1.30054	−0.650272	−	0.759701i	$-0.725345\pi$
$431$	−27.0000	−1.30054	−0.650272	+	0.759701i	$0.725345\pi$
$432$	0	0
$433$	16.0000	0.768911	0.384455	−	0.923144i	$-0.374389\pi$
$433$	16.0000	0.768911	0.384455	+	0.923144i	$0.374389\pi$
$434$	−10.0000	−0.480015
$435$	0	0
$436$	−6.00000	−0.287348
$437$	8.00000	0.382692
$438$	0	0
$439$	−24.0000	−1.14546	−0.572729	−	0.819745i	$-0.694115\pi$
$439$	−24.0000	−1.14546	−0.572729	+	0.819745i	$0.694115\pi$
$440$	−6.00000	−0.286039
$441$	0	0
$442$	0	0
$443$	4.00000	0.190046	0.0950229	−	0.995475i	$-0.469708\pi$
$443$	4.00000	0.190046	0.0950229	+	0.995475i	$0.469708\pi$
$444$	0	0
$445$	6.00000	0.284427
$446$	−4.00000	−0.189405
$447$	0	0
$448$	−14.0000	−0.661438
$449$	−27.0000	−1.27421	−0.637104	−	0.770778i	$-0.719868\pi$
$449$	−27.0000	−1.27421	−0.637104	+	0.770778i	$0.719868\pi$
$450$	0	0
$451$	−16.0000	−0.753411
$452$	−1.00000	−0.0470360
$453$	0	0
$454$	−24.0000	−1.12638
$455$	10.0000	0.468807
$456$	0	0
$457$	−22.0000	−1.02912	−0.514558	−	0.857455i	$-0.672044\pi$
$457$	−22.0000	−1.02912	−0.514558	+	0.857455i	$0.672044\pi$
$458$	3.00000	0.140181
$459$	0	0
$460$	1.00000	0.0466252
$461$	−4.00000	−0.186299	−0.0931493	−	0.995652i	$-0.529693\pi$
$461$	−4.00000	−0.186299	−0.0931493	+	0.995652i	$0.529693\pi$
$462$	0	0
$463$	39.0000	1.81248	0.906242	−	0.422760i	$-0.138939\pi$
$463$	39.0000	1.81248	0.906242	+	0.422760i	$0.138939\pi$
$464$	−2.00000	−0.0928477
$465$	0	0
$466$	2.00000	0.0926482
$467$	−36.0000	−1.66588	−0.832941	−	0.553362i	$-0.813345\pi$
$467$	−36.0000	−1.66588	−0.832941	+	0.553362i	$0.813345\pi$
$468$	0	0
$469$	−22.0000	−1.01587
$470$	2.00000	0.0922531
$471$	0	0
$472$	−33.0000	−1.51895
$473$	−22.0000	−1.01156
$474$	0	0
$475$	32.0000	1.46826
$476$	0	0
$477$	0	0
$478$	20.0000	0.914779
$479$	−6.00000	−0.274147	−0.137073	−	0.990561i	$-0.543770\pi$
$479$	−6.00000	−0.274147	−0.137073	+	0.990561i	$0.543770\pi$
$480$	0	0
$481$	−10.0000	−0.455961
$482$	2.00000	0.0910975
$483$	0	0
$484$	7.00000	0.318182
$485$	−7.00000	−0.317854
$486$	0	0
$487$	−8.00000	−0.362515	−0.181257	−	0.983436i	$-0.558017\pi$
$487$	−8.00000	−0.362515	−0.181257	+	0.983436i	$0.558017\pi$
$488$	15.0000	0.679018
$489$	0	0
$490$	−3.00000	−0.135526
$491$	19.0000	0.857458	0.428729	−	0.903433i	$-0.358962\pi$
$491$	19.0000	0.857458	0.428729	+	0.903433i	$0.358962\pi$
$492$	0	0
$493$	0	0
$494$	40.0000	1.79969
$495$	0	0
$496$	−5.00000	−0.224507
$497$	32.0000	1.43540
$498$	0	0
$499$	13.0000	0.581960	0.290980	−	0.956729i	$-0.406019\pi$
$499$	13.0000	0.581960	0.290980	+	0.956729i	$0.406019\pi$
$500$	9.00000	0.402492
$501$	0	0
$502$	18.0000	0.803379
$503$	27.0000	1.20387	0.601935	−	0.798545i	$-0.294397\pi$
$503$	27.0000	1.20387	0.601935	+	0.798545i	$0.294397\pi$
$504$	0	0
$505$	9.00000	0.400495
$506$	−2.00000	−0.0889108
$507$	0	0
$508$	7.00000	0.310575
$509$	30.0000	1.32973	0.664863	−	0.746965i	$-0.268490\pi$
$509$	30.0000	1.32973	0.664863	+	0.746965i	$0.268490\pi$
$510$	0	0
$511$	−24.0000	−1.06170
$512$	−11.0000	−0.486136
$513$	0	0
$514$	7.00000	0.308757
$515$	1.00000	0.0440653
$516$	0	0
$517$	4.00000	0.175920
$518$	−4.00000	−0.175750
$519$	0	0
$520$	15.0000	0.657794
$521$	−39.0000	−1.70862	−0.854311	−	0.519763i	$-0.826020\pi$
$521$	−39.0000	−1.70862	−0.854311	+	0.519763i	$0.826020\pi$
$522$	0	0
$523$	−8.00000	−0.349816	−0.174908	−	0.984585i	$-0.555963\pi$
$523$	−8.00000	−0.349816	−0.174908	+	0.984585i	$0.555963\pi$
$524$	−17.0000	−0.742648
$525$	0	0
$526$	−30.0000	−1.30806
$527$	0	0
$528$	0	0
$529$	−22.0000	−0.956522
$530$	−10.0000	−0.434372
$531$	0	0
$532$	−16.0000	−0.693688
$533$	40.0000	1.73259
$534$	0	0
$535$	11.0000	0.475571
$536$	−33.0000	−1.42538
$537$	0	0
$538$	−24.0000	−1.03471
$539$	−6.00000	−0.258438
$540$	0	0
$541$	−19.0000	−0.816874	−0.408437	−	0.912787i	$-0.633926\pi$
$541$	−19.0000	−0.816874	−0.408437	+	0.912787i	$0.633926\pi$
$542$	−25.0000	−1.07384
$543$	0	0
$544$	0	0
$545$	6.00000	0.257012
$546$	0	0
$547$	8.00000	0.342055	0.171028	−	0.985266i	$-0.445291\pi$
$547$	8.00000	0.342055	0.171028	+	0.985266i	$0.445291\pi$
$548$	6.00000	0.256307
$549$	0	0
$550$	−8.00000	−0.341121
$551$	−16.0000	−0.681623
$552$	0	0
$553$	−12.0000	−0.510292
$554$	−8.00000	−0.339887
$555$	0	0
$556$	8.00000	0.339276
$557$	9.00000	0.381342	0.190671	−	0.981654i	$-0.438934\pi$
$557$	9.00000	0.381342	0.190671	+	0.981654i	$0.438934\pi$
$558$	0	0
$559$	55.0000	2.32625
$560$	2.00000	0.0845154
$561$	0	0
$562$	−21.0000	−0.885832
$563$	−12.0000	−0.505740	−0.252870	−	0.967500i	$-0.581374\pi$
$563$	−12.0000	−0.505740	−0.252870	+	0.967500i	$0.581374\pi$
$564$	0	0
$565$	1.00000	0.0420703
$566$	13.0000	0.546431
$567$	0	0
$568$	48.0000	2.01404
$569$	−9.00000	−0.377300	−0.188650	−	0.982044i	$-0.560411\pi$
$569$	−9.00000	−0.377300	−0.188650	+	0.982044i	$0.560411\pi$
$570$	0	0
$571$	−32.0000	−1.33916	−0.669579	−	0.742741i	$-0.733526\pi$
$571$	−32.0000	−1.33916	−0.669579	+	0.742741i	$0.733526\pi$
$572$	10.0000	0.418121
$573$	0	0
$574$	16.0000	0.667827
$575$	4.00000	0.166812
$576$	0	0
$577$	−8.00000	−0.333044	−0.166522	−	0.986038i	$-0.553254\pi$
$577$	−8.00000	−0.333044	−0.166522	+	0.986038i	$0.553254\pi$
$578$	−17.0000	−0.707107
$579$	0	0
$580$	−2.00000	−0.0830455
$581$	2.00000	0.0829740
$582$	0	0
$583$	−20.0000	−0.828315
$584$	−36.0000	−1.48969
$585$	0	0
$586$	−9.00000	−0.371787
$587$	33.0000	1.36206	0.681028	−	0.732257i	$-0.261533\pi$
$587$	33.0000	1.36206	0.681028	+	0.732257i	$0.261533\pi$
$588$	0	0
$589$	−40.0000	−1.64817
$590$	11.0000	0.452863
$591$	0	0
$592$	−2.00000	−0.0821995
$593$	6.00000	0.246390	0.123195	−	0.992382i	$-0.460686\pi$
$593$	6.00000	0.246390	0.123195	+	0.992382i	$0.460686\pi$
$594$	0	0
$595$	0	0
$596$	2.00000	0.0819232
$597$	0	0
$598$	5.00000	0.204465
$599$	−26.0000	−1.06233	−0.531166	−	0.847268i	$-0.678246\pi$
$599$	−26.0000	−1.06233	−0.531166	+	0.847268i	$0.678246\pi$
$600$	0	0
$601$	30.0000	1.22373	0.611863	−	0.790964i	$-0.290420\pi$
$601$	30.0000	1.22373	0.611863	+	0.790964i	$0.290420\pi$
$602$	22.0000	0.896653
$603$	0	0
$604$	8.00000	0.325515
$605$	−7.00000	−0.284590
$606$	0	0
$607$	−16.0000	−0.649420	−0.324710	−	0.945814i	$-0.605267\pi$
$607$	−16.0000	−0.649420	−0.324710	+	0.945814i	$0.605267\pi$
$608$	−40.0000	−1.62221
$609$	0	0
$610$	−5.00000	−0.202444
$611$	−10.0000	−0.404557
$612$	0	0
$613$	−17.0000	−0.686624	−0.343312	−	0.939222i	$-0.611549\pi$
$613$	−17.0000	−0.686624	−0.343312	+	0.939222i	$0.611549\pi$
$614$	−7.00000	−0.282497
$615$	0	0
$616$	12.0000	0.483494
$617$	25.0000	1.00646	0.503231	−	0.864152i	$-0.332144\pi$
$617$	25.0000	1.00646	0.503231	+	0.864152i	$0.332144\pi$
$618$	0	0
$619$	2.00000	0.0803868	0.0401934	−	0.999192i	$-0.487203\pi$
$619$	2.00000	0.0803868	0.0401934	+	0.999192i	$0.487203\pi$
$620$	−5.00000	−0.200805
$621$	0	0
$622$	33.0000	1.32318
$623$	−12.0000	−0.480770
$624$	0	0
$625$	11.0000	0.440000
$626$	−22.0000	−0.879297
$627$	0	0
$628$	8.00000	0.319235
$629$	0	0
$630$	0	0
$631$	14.0000	0.557331	0.278666	−	0.960388i	$-0.410108\pi$
$631$	14.0000	0.557331	0.278666	+	0.960388i	$0.410108\pi$
$632$	−18.0000	−0.716002
$633$	0	0
$634$	−26.0000	−1.03259
$635$	−7.00000	−0.277787
$636$	0	0
$637$	15.0000	0.594322
$638$	4.00000	0.158362
$639$	0	0
$640$	−3.00000	−0.118585
$641$	−24.0000	−0.947943	−0.473972	−	0.880540i	$-0.657180\pi$
$641$	−24.0000	−0.947943	−0.473972	+	0.880540i	$0.657180\pi$
$642$	0	0
$643$	−12.0000	−0.473234	−0.236617	−	0.971603i	$-0.576039\pi$
$643$	−12.0000	−0.473234	−0.236617	+	0.971603i	$0.576039\pi$
$644$	−2.00000	−0.0788110
$645$	0	0
$646$	0	0
$647$	41.0000	1.61188	0.805938	−	0.592000i	$-0.201661\pi$
$647$	41.0000	1.61188	0.805938	+	0.592000i	$0.201661\pi$
$648$	0	0
$649$	22.0000	0.863576
$650$	20.0000	0.784465
$651$	0	0
$652$	10.0000	0.391630
$653$	−35.0000	−1.36966	−0.684828	−	0.728705i	$-0.740123\pi$
$653$	−35.0000	−1.36966	−0.684828	+	0.728705i	$0.740123\pi$
$654$	0	0
$655$	17.0000	0.664245
$656$	8.00000	0.312348
$657$	0	0
$658$	−4.00000	−0.155936
$659$	−27.0000	−1.05177	−0.525885	−	0.850555i	$-0.676266\pi$
$659$	−27.0000	−1.05177	−0.525885	+	0.850555i	$0.676266\pi$
$660$	0	0
$661$	26.0000	1.01128	0.505641	−	0.862744i	$-0.331256\pi$
$661$	26.0000	1.01128	0.505641	+	0.862744i	$0.331256\pi$
$662$	−31.0000	−1.20485
$663$	0	0
$664$	3.00000	0.116423
$665$	16.0000	0.620453
$666$	0	0
$667$	−2.00000	−0.0774403
$668$	−12.0000	−0.464294
$669$	0	0
$670$	11.0000	0.424967
$671$	−10.0000	−0.386046
$672$	0	0
$673$	−35.0000	−1.34915	−0.674575	−	0.738206i	$-0.735673\pi$
$673$	−35.0000	−1.34915	−0.674575	+	0.738206i	$0.735673\pi$
$674$	17.0000	0.654816
$675$	0	0
$676$	−12.0000	−0.461538
$677$	34.0000	1.30673	0.653363	−	0.757045i	$-0.273358\pi$
$677$	34.0000	1.30673	0.653363	+	0.757045i	$0.273358\pi$
$678$	0	0
$679$	14.0000	0.537271
$680$	0	0
$681$	0	0
$682$	10.0000	0.382920
$683$	36.0000	1.37750	0.688751	−	0.724998i	$-0.258159\pi$
$683$	36.0000	1.37750	0.688751	+	0.724998i	$0.258159\pi$
$684$	0	0
$685$	−6.00000	−0.229248
$686$	20.0000	0.763604
$687$	0	0
$688$	11.0000	0.419371
$689$	50.0000	1.90485
$690$	0	0
$691$	−8.00000	−0.304334	−0.152167	−	0.988355i	$-0.548625\pi$
$691$	−8.00000	−0.304334	−0.152167	+	0.988355i	$0.548625\pi$
$692$	6.00000	0.228086
$693$	0	0
$694$	−27.0000	−1.02491
$695$	−8.00000	−0.303457
$696$	0	0
$697$	0	0
$698$	−4.00000	−0.151402
$699$	0	0
$700$	−8.00000	−0.302372
$701$	42.0000	1.58632	0.793159	−	0.609015i	$-0.208435\pi$
$701$	42.0000	1.58632	0.793159	+	0.609015i	$0.208435\pi$
$702$	0	0
$703$	−16.0000	−0.603451
$704$	14.0000	0.527645
$705$	0	0
$706$	−27.0000	−1.01616
$707$	−18.0000	−0.676960
$708$	0	0
$709$	34.0000	1.27690	0.638448	−	0.769665i	$-0.279577\pi$
$709$	34.0000	1.27690	0.638448	+	0.769665i	$0.279577\pi$
$710$	−16.0000	−0.600469
$711$	0	0
$712$	−18.0000	−0.674579
$713$	−5.00000	−0.187251
$714$	0	0
$715$	−10.0000	−0.373979
$716$	−20.0000	−0.747435
$717$	0	0
$718$	27.0000	1.00763
$719$	−2.00000	−0.0745874	−0.0372937	−	0.999304i	$-0.511874\pi$
$719$	−2.00000	−0.0745874	−0.0372937	+	0.999304i	$0.511874\pi$
$720$	0	0
$721$	−2.00000	−0.0744839
$722$	45.0000	1.67473
$723$	0	0
$724$	−12.0000	−0.445976
$725$	−8.00000	−0.297113
$726$	0	0
$727$	37.0000	1.37225	0.686127	−	0.727482i	$-0.259309\pi$
$727$	37.0000	1.37225	0.686127	+	0.727482i	$0.259309\pi$
$728$	−30.0000	−1.11187
$729$	0	0
$730$	12.0000	0.444140
$731$	0	0
$732$	0	0
$733$	−26.0000	−0.960332	−0.480166	−	0.877178i	$-0.659424\pi$
$733$	−26.0000	−0.960332	−0.480166	+	0.877178i	$0.659424\pi$
$734$	−4.00000	−0.147643
$735$	0	0
$736$	−5.00000	−0.184302
$737$	22.0000	0.810380
$738$	0	0
$739$	24.0000	0.882854	0.441427	−	0.897297i	$-0.354472\pi$
$739$	24.0000	0.882854	0.441427	+	0.897297i	$0.354472\pi$
$740$	−2.00000	−0.0735215
$741$	0	0
$742$	20.0000	0.734223
$743$	−36.0000	−1.32071	−0.660356	−	0.750953i	$-0.729595\pi$
$743$	−36.0000	−1.32071	−0.660356	+	0.750953i	$0.729595\pi$
$744$	0	0
$745$	−2.00000	−0.0732743
$746$	10.0000	0.366126
$747$	0	0
$748$	0	0
$749$	−22.0000	−0.803863
$750$	0	0
$751$	30.0000	1.09472	0.547358	−	0.836899i	$-0.315634\pi$
$751$	30.0000	1.09472	0.547358	+	0.836899i	$0.315634\pi$
$752$	−2.00000	−0.0729325
$753$	0	0
$754$	−10.0000	−0.364179
$755$	−8.00000	−0.291150
$756$	0	0
$757$	−31.0000	−1.12671	−0.563357	−	0.826214i	$-0.690490\pi$
$757$	−31.0000	−1.12671	−0.563357	+	0.826214i	$0.690490\pi$
$758$	−23.0000	−0.835398
$759$	0	0
$760$	24.0000	0.870572
$761$	27.0000	0.978749	0.489375	−	0.872074i	$-0.337225\pi$
$761$	27.0000	0.978749	0.489375	+	0.872074i	$0.337225\pi$
$762$	0	0
$763$	−12.0000	−0.434429
$764$	−20.0000	−0.723575
$765$	0	0
$766$	6.00000	0.216789
$767$	−55.0000	−1.98593
$768$	0	0
$769$	26.0000	0.937584	0.468792	−	0.883309i	$-0.344689\pi$
$769$	26.0000	0.937584	0.468792	+	0.883309i	$0.344689\pi$
$770$	−4.00000	−0.144150
$771$	0	0
$772$	8.00000	0.287926
$773$	18.0000	0.647415	0.323708	−	0.946157i	$-0.395071\pi$
$773$	18.0000	0.647415	0.323708	+	0.946157i	$0.395071\pi$
$774$	0	0
$775$	−20.0000	−0.718421
$776$	21.0000	0.753856
$777$	0	0
$778$	30.0000	1.07555
$779$	64.0000	2.29304
$780$	0	0
$781$	−32.0000	−1.14505
$782$	0	0
$783$	0	0
$784$	3.00000	0.107143
$785$	−8.00000	−0.285532
$786$	0	0
$787$	−2.00000	−0.0712923	−0.0356462	−	0.999364i	$-0.511349\pi$
$787$	−2.00000	−0.0712923	−0.0356462	+	0.999364i	$0.511349\pi$
$788$	11.0000	0.391859
$789$	0	0
$790$	6.00000	0.213470
$791$	−2.00000	−0.0711118
$792$	0	0
$793$	25.0000	0.887776
$794$	8.00000	0.283909
$795$	0	0
$796$	1.00000	0.0354441
$797$	−18.0000	−0.637593	−0.318796	−	0.947823i	$-0.603279\pi$
$797$	−18.0000	−0.637593	−0.318796	+	0.947823i	$0.603279\pi$
$798$	0	0
$799$	0	0
$800$	−20.0000	−0.707107
$801$	0	0
$802$	−16.0000	−0.564980
$803$	24.0000	0.846942
$804$	0	0
$805$	2.00000	0.0704907
$806$	−25.0000	−0.880587
$807$	0	0
$808$	−27.0000	−0.949857
$809$	3.00000	0.105474	0.0527372	−	0.998608i	$-0.483205\pi$
$809$	3.00000	0.105474	0.0527372	+	0.998608i	$0.483205\pi$
$810$	0	0
$811$	−21.0000	−0.737410	−0.368705	−	0.929547i	$-0.620199\pi$
$811$	−21.0000	−0.737410	−0.368705	+	0.929547i	$0.620199\pi$
$812$	4.00000	0.140372
$813$	0	0
$814$	4.00000	0.140200
$815$	−10.0000	−0.350285
$816$	0	0
$817$	88.0000	3.07873
$818$	−34.0000	−1.18878
$819$	0	0
$820$	8.00000	0.279372
$821$	24.0000	0.837606	0.418803	−	0.908077i	$-0.362450\pi$
$821$	24.0000	0.837606	0.418803	+	0.908077i	$0.362450\pi$
$822$	0	0
$823$	−13.0000	−0.453152	−0.226576	−	0.973994i	$-0.572753\pi$
$823$	−13.0000	−0.453152	−0.226576	+	0.973994i	$0.572753\pi$
$824$	−3.00000	−0.104510
$825$	0	0
$826$	−22.0000	−0.765478
$827$	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
$829$	50.0000	1.73657	0.868286	−	0.496064i	$-0.165222\pi$
$829$	50.0000	1.73657	0.868286	+	0.496064i	$0.165222\pi$
$830$	−1.00000	−0.0347105
$831$	0	0
$832$	−35.0000	−1.21341
$833$	0	0
$834$	0	0
$835$	12.0000	0.415277
$836$	16.0000	0.553372
$837$	0	0
$838$	−12.0000	−0.414533
$839$	48.0000	1.65714	0.828572	−	0.559883i	$-0.189154\pi$
$839$	48.0000	1.65714	0.828572	+	0.559883i	$0.189154\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	−3.00000	−0.103387
$843$	0	0
$844$	−4.00000	−0.137686
$845$	12.0000	0.412813
$846$	0	0
$847$	14.0000	0.481046
$848$	10.0000	0.343401
$849$	0	0
$850$	0	0
$851$	−2.00000	−0.0685591
$852$	0	0
$853$	−35.0000	−1.19838	−0.599189	−	0.800608i	$-0.704510\pi$
$853$	−35.0000	−1.19838	−0.599189	+	0.800608i	$0.704510\pi$
$854$	10.0000	0.342193
$855$	0	0
$856$	−33.0000	−1.12792
$857$	14.0000	0.478231	0.239115	−	0.970991i	$-0.423143\pi$
$857$	14.0000	0.478231	0.239115	+	0.970991i	$0.423143\pi$
$858$	0	0
$859$	29.0000	0.989467	0.494734	−	0.869045i	$-0.335266\pi$
$859$	29.0000	0.989467	0.494734	+	0.869045i	$0.335266\pi$
$860$	11.0000	0.375097
$861$	0	0
$862$	−27.0000	−0.919624
$863$	0	0		−	1.00000i	$-0.5\pi$
$863$	0	0			1.00000i	$0.5\pi$
$864$	0	0
$865$	−6.00000	−0.204006
$866$	16.0000	0.543702
$867$	0	0
$868$	10.0000	0.339422
$869$	12.0000	0.407072
$870$	0	0
$871$	−55.0000	−1.86360
$872$	−18.0000	−0.609557
$873$	0	0
$874$	8.00000	0.270604
$875$	18.0000	0.608511
$876$	0	0
$877$	42.0000	1.41824	0.709120	−	0.705088i	$-0.249093\pi$
$877$	42.0000	1.41824	0.709120	+	0.705088i	$0.249093\pi$
$878$	−24.0000	−0.809961
$879$	0	0
$880$	−2.00000	−0.0674200
$881$	53.0000	1.78562	0.892808	−	0.450438i	$-0.148732\pi$
$881$	53.0000	1.78562	0.892808	+	0.450438i	$0.148732\pi$
$882$	0	0
$883$	−14.0000	−0.471138	−0.235569	−	0.971858i	$-0.575695\pi$
$883$	−14.0000	−0.471138	−0.235569	+	0.971858i	$0.575695\pi$
$884$	0	0
$885$	0	0
$886$	4.00000	0.134383
$887$	1.00000	0.0335767	0.0167884	−	0.999859i	$-0.494656\pi$
$887$	1.00000	0.0335767	0.0167884	+	0.999859i	$0.494656\pi$
$888$	0	0
$889$	14.0000	0.469545
$890$	6.00000	0.201120
$891$	0	0
$892$	4.00000	0.133930
$893$	−16.0000	−0.535420
$894$	0	0
$895$	20.0000	0.668526
$896$	6.00000	0.200446
$897$	0	0
$898$	−27.0000	−0.901002
$899$	10.0000	0.333519
$900$	0	0
$901$	0	0
$902$	−16.0000	−0.532742
$903$	0	0
$904$	−3.00000	−0.0997785
$905$	12.0000	0.398893
$906$	0	0
$907$	28.0000	0.929725	0.464862	−	0.885383i	$-0.346104\pi$
$907$	28.0000	0.929725	0.464862	+	0.885383i	$0.346104\pi$
$908$	24.0000	0.796468
$909$	0	0
$910$	10.0000	0.331497
$911$	−12.0000	−0.397578	−0.198789	−	0.980042i	$-0.563701\pi$
$911$	−12.0000	−0.397578	−0.198789	+	0.980042i	$0.563701\pi$
$912$	0	0
$913$	−2.00000	−0.0661903
$914$	−22.0000	−0.727695
$915$	0	0
$916$	−3.00000	−0.0991228
$917$	−34.0000	−1.12278
$918$	0	0
$919$	20.0000	0.659739	0.329870	−	0.944027i	$-0.392995\pi$
$919$	20.0000	0.659739	0.329870	+	0.944027i	$0.392995\pi$
$920$	3.00000	0.0989071
$921$	0	0
$922$	−4.00000	−0.131733
$923$	80.0000	2.63323
$924$	0	0
$925$	−8.00000	−0.263038
$926$	39.0000	1.28162
$927$	0	0
$928$	10.0000	0.328266
$929$	−10.0000	−0.328089	−0.164045	−	0.986453i	$-0.552454\pi$
$929$	−10.0000	−0.328089	−0.164045	+	0.986453i	$0.552454\pi$
$930$	0	0
$931$	24.0000	0.786568
$932$	−2.00000	−0.0655122
$933$	0	0
$934$	−36.0000	−1.17796
$935$	0	0
$936$	0	0
$937$	−10.0000	−0.326686	−0.163343	−	0.986569i	$-0.552228\pi$
$937$	−10.0000	−0.326686	−0.163343	+	0.986569i	$0.552228\pi$
$938$	−22.0000	−0.718325
$939$	0	0
$940$	−2.00000	−0.0652328
$941$	−12.0000	−0.391189	−0.195594	−	0.980685i	$-0.562664\pi$
$941$	−12.0000	−0.391189	−0.195594	+	0.980685i	$0.562664\pi$
$942$	0	0
$943$	8.00000	0.260516
$944$	−11.0000	−0.358020
$945$	0	0
$946$	−22.0000	−0.715282
$947$	6.00000	0.194974	0.0974869	−	0.995237i	$-0.468920\pi$
$947$	6.00000	0.194974	0.0974869	+	0.995237i	$0.468920\pi$
$948$	0	0
$949$	−60.0000	−1.94768
$950$	32.0000	1.03822
$951$	0	0
$952$	0	0
$953$	−34.0000	−1.10137	−0.550684	−	0.834714i	$-0.685633\pi$
$953$	−34.0000	−1.10137	−0.550684	+	0.834714i	$0.685633\pi$
$954$	0	0
$955$	20.0000	0.647185
$956$	−20.0000	−0.646846
$957$	0	0
$958$	−6.00000	−0.193851
$959$	12.0000	0.387500
$960$	0	0
$961$	−6.00000	−0.193548
$962$	−10.0000	−0.322413
$963$	0	0
$964$	−2.00000	−0.0644157
$965$	−8.00000	−0.257529
$966$	0	0
$967$	11.0000	0.353736	0.176868	−	0.984235i	$-0.443403\pi$
$967$	11.0000	0.353736	0.176868	+	0.984235i	$0.443403\pi$
$968$	21.0000	0.674966
$969$	0	0
$970$	−7.00000	−0.224756
$971$	34.0000	1.09111	0.545556	−	0.838074i	$-0.316319\pi$
$971$	34.0000	1.09111	0.545556	+	0.838074i	$0.316319\pi$
$972$	0	0
$973$	16.0000	0.512936
$974$	−8.00000	−0.256337
$975$	0	0
$976$	5.00000	0.160046
$977$	4.00000	0.127971	0.0639857	−	0.997951i	$-0.479619\pi$
$977$	4.00000	0.127971	0.0639857	+	0.997951i	$0.479619\pi$
$978$	0	0
$979$	12.0000	0.383522
$980$	3.00000	0.0958315
$981$	0	0
$982$	19.0000	0.606314
$983$	−23.0000	−0.733586	−0.366793	−	0.930303i	$-0.619544\pi$
$983$	−23.0000	−0.733586	−0.366793	+	0.930303i	$0.619544\pi$
$984$	0	0
$985$	−11.0000	−0.350489
$986$	0	0
$987$	0	0
$988$	−40.0000	−1.27257
$989$	11.0000	0.349780
$990$	0	0
$991$	54.0000	1.71537	0.857683	−	0.514178i	$-0.171903\pi$
$991$	54.0000	1.71537	0.857683	+	0.514178i	$0.171903\pi$
$992$	25.0000	0.793751
$993$	0	0
$994$	32.0000	1.01498
$995$	−1.00000	−0.0317021
$996$	0	0
$997$	−32.0000	−1.01345	−0.506725	−	0.862108i	$-0.669144\pi$
$997$	−32.0000	−1.01345	−0.506725	+	0.862108i	$0.669144\pi$
$998$	13.0000	0.411508
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	927.2.a.a.1.1		1
3.2	odd	2		309.2.a.a.1.1	✓	1
12.11	even	2		4944.2.a.c.1.1		1
15.14	odd	2		7725.2.a.k.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
309.2.a.a.1.1	✓	1	3.2	odd	2
927.2.a.a.1.1		1	1.1	even	1	trivial
4944.2.a.c.1.1		1	12.11	even	2
7725.2.a.k.1.1		1	15.14	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 927 = 3^{2} \cdot 103 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	927.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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