LMFDB - Embedded newform 8910.2.a.bs.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [8910,2,Mod(1,8910)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8910, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("8910.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 8910 = 2 \cdot 3^{4} \cdot 5 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8910.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,-4,0,4,4,0,2,-4,0,-4,-4,0,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$71.1467082010$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.4752.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 3x^{2} + 4x + 1 $ x^4 - 2x^3 - 3x^2 + 4*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-0.219687$ of defining polynomial
Character	$\chi$	$=$	8910.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +1.60020 q^{7} -1.00000 q^{8} -1.00000 q^{10} -1.00000 q^{11} +4.77162 q^{13} -1.60020 q^{14} +1.00000 q^{16} -3.93244 q^{17} -1.29268 q^{19} +1.00000 q^{20} +1.00000 q^{22} -4.33225 q^{23} +1.00000 q^{25} -4.77162 q^{26} +1.60020 q^{28} -6.23996 q^{29} +3.33225 q^{31} -1.00000 q^{32} +3.93244 q^{34} +1.60020 q^{35} -2.43937 q^{37} +1.29268 q^{38} -1.00000 q^{40} +1.69346 q^{41} -1.34709 q^{43} -1.00000 q^{44} +4.33225 q^{46} +3.35697 q^{47} -4.43937 q^{49} -1.00000 q^{50} +4.77162 q^{52} -1.19941 q^{53} -1.00000 q^{55} -1.60020 q^{56} +6.23996 q^{58} +1.26371 q^{59} -0.878747 q^{61} -3.33225 q^{62} +1.00000 q^{64} +4.77162 q^{65} -3.77162 q^{67} -3.93244 q^{68} -1.60020 q^{70} -7.15658 q^{71} -10.8755 q^{73} +2.43937 q^{74} -1.29268 q^{76} -1.60020 q^{77} -7.41139 q^{79} +1.00000 q^{80} -1.69346 q^{82} -1.73629 q^{83} -3.93244 q^{85} +1.34709 q^{86} +1.00000 q^{88} -5.78575 q^{89} +7.63553 q^{91} -4.33225 q^{92} -3.35697 q^{94} -1.29268 q^{95} +0.425248 q^{97} +4.43937 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{2} + 4 q^{4} + 4 q^{5} + 2 q^{7} - 4 q^{8} - 4 q^{10} - 4 q^{11} + 2 q^{13} - 2 q^{14} + 4 q^{16} - 4 q^{19} + 4 q^{20} + 4 q^{22} - 6 q^{23} + 4 q^{25} - 2 q^{26} + 2 q^{28} - 6 q^{29} + 2 q^{31}+ \cdots + 12 q^{98}+O(q^{100}) $ 4 * q - 4 * q^2 + 4 * q^4 + 4 * q^5 + 2 * q^7 - 4 * q^8 - 4 * q^10 - 4 * q^11 + 2 * q^13 - 2 * q^14 + 4 * q^16 - 4 * q^19 + 4 * q^20 + 4 * q^22 - 6 * q^23 + 4 * q^25 - 2 * q^26 + 2 * q^28 - 6 * q^29 + 2 * q^31 - 4 * q^32 + 2 * q^35 - 4 * q^37 + 4 * q^38 - 4 * q^40 - 12 * q^41 + 8 * q^43 - 4 * q^44 + 6 * q^46 - 6 * q^47 - 12 * q^49 - 4 * q^50 + 2 * q^52 - 18 * q^53 - 4 * q^55 - 2 * q^56 + 6 * q^58 + 8 * q^61 - 2 * q^62 + 4 * q^64 + 2 * q^65 + 2 * q^67 - 2 * q^70 - 18 * q^71 + 2 * q^73 + 4 * q^74 - 4 * q^76 - 2 * q^77 + 2 * q^79 + 4 * q^80 + 12 * q^82 - 12 * q^83 - 8 * q^86 + 4 * q^88 - 12 * q^89 + 4 * q^91 - 6 * q^92 + 6 * q^94 - 4 * q^95 - 10 * q^97 + 12 * q^98

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
$2$	−1.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	1.60020	0.604817	0.302409	−	0.953178i	$-0.402209\pi$
$7$	1.60020	0.604817	0.302409	+	0.953178i	$0.402209\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	−1.00000	−0.316228
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	4.77162	1.32341	0.661705	−	0.749765i	$-0.269833\pi$
$13$	4.77162	1.32341	0.661705	+	0.749765i	$0.269833\pi$
$14$	−1.60020	−0.427670
$15$	0	0
$16$	1.00000	0.250000
$17$	−3.93244	−0.953757	−0.476879	−	0.878969i	$-0.658232\pi$
$17$	−3.93244	−0.953757	−0.476879	+	0.878969i	$0.658232\pi$
$18$	0	0
$19$	−1.29268	−0.296560	−0.148280	−	0.988945i	$-0.547374\pi$
$19$	−1.29268	−0.296560	−0.148280	+	0.988945i	$0.547374\pi$
$20$	1.00000	0.223607
$21$	0	0
$22$	1.00000	0.213201
$23$	−4.33225	−0.903336	−0.451668	−	0.892186i	$-0.649171\pi$
$23$	−4.33225	−0.903336	−0.451668	+	0.892186i	$0.649171\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−4.77162	−0.935792
$27$	0	0
$28$	1.60020	0.302409
$29$	−6.23996	−1.15873	−0.579366	−	0.815068i	$-0.696700\pi$
$29$	−6.23996	−1.15873	−0.579366	+	0.815068i	$0.696700\pi$
$30$	0	0
$31$	3.33225	0.598489	0.299245	−	0.954176i	$-0.403265\pi$
$31$	3.33225	0.598489	0.299245	+	0.954176i	$0.403265\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	3.93244	0.674408
$35$	1.60020	0.270482
$36$	0	0
$37$	−2.43937	−0.401031	−0.200515	−	0.979691i	$-0.564262\pi$
$37$	−2.43937	−0.401031	−0.200515	+	0.979691i	$0.564262\pi$
$38$	1.29268	0.209700
$39$	0	0
$40$	−1.00000	−0.158114
$41$	1.69346	0.264474	0.132237	−	0.991218i	$-0.457784\pi$
$41$	1.69346	0.264474	0.132237	+	0.991218i	$0.457784\pi$
$42$	0	0
$43$	−1.34709	−0.205429	−0.102715	−	0.994711i	$-0.532753\pi$
$43$	−1.34709	−0.205429	−0.102715	+	0.994711i	$0.532753\pi$
$44$	−1.00000	−0.150756
$45$	0	0
$46$	4.33225	0.638755
$47$	3.35697	0.489665	0.244833	−	0.969565i	$-0.421267\pi$
$47$	3.35697	0.489665	0.244833	+	0.969565i	$0.421267\pi$
$48$	0	0
$49$	−4.43937	−0.634196
$50$	−1.00000	−0.141421
$51$	0	0
$52$	4.77162	0.661705
$53$	−1.19941	−0.164752	−0.0823760	−	0.996601i	$-0.526251\pi$
$53$	−1.19941	−0.164752	−0.0823760	+	0.996601i	$0.526251\pi$
$54$	0	0
$55$	−1.00000	−0.134840
$56$	−1.60020	−0.213835
$57$	0	0
$58$	6.23996	0.819347
$59$	1.26371	0.164521	0.0822605	−	0.996611i	$-0.473786\pi$
$59$	1.26371	0.164521	0.0822605	+	0.996611i	$0.473786\pi$
$60$	0	0
$61$	−0.878747	−0.112512	−0.0562560	−	0.998416i	$-0.517916\pi$
$61$	−0.878747	−0.112512	−0.0562560	+	0.998416i	$0.517916\pi$
$62$	−3.33225	−0.423196
$63$	0	0
$64$	1.00000	0.125000
$65$	4.77162	0.591847
$66$	0	0
$67$	−3.77162	−0.460777	−0.230388	−	0.973099i	$-0.574000\pi$
$67$	−3.77162	−0.460777	−0.230388	+	0.973099i	$0.574000\pi$
$68$	−3.93244	−0.476879
$69$	0	0
$70$	−1.60020	−0.191260
$71$	−7.15658	−0.849330	−0.424665	−	0.905351i	$-0.639608\pi$
$71$	−7.15658	−0.849330	−0.424665	+	0.905351i	$0.639608\pi$
$72$	0	0
$73$	−10.8755	−1.27288	−0.636440	−	0.771326i	$-0.719594\pi$
$73$	−10.8755	−1.27288	−0.636440	+	0.771326i	$0.719594\pi$
$74$	2.43937	0.283571
$75$	0	0
$76$	−1.29268	−0.148280
$77$	−1.60020	−0.182359
$78$	0	0
$79$	−7.41139	−0.833846	−0.416923	−	0.908942i	$-0.636892\pi$
$79$	−7.41139	−0.833846	−0.416923	+	0.908942i	$0.636892\pi$
$80$	1.00000	0.111803
$81$	0	0
$82$	−1.69346	−0.187011
$83$	−1.73629	−0.190583	−0.0952913	−	0.995449i	$-0.530378\pi$
$83$	−1.73629	−0.190583	−0.0952913	+	0.995449i	$0.530378\pi$
$84$	0	0
$85$	−3.93244	−0.426533
$86$	1.34709	0.145260
$87$	0	0
$88$	1.00000	0.106600
$89$	−5.78575	−0.613288	−0.306644	−	0.951824i	$-0.599206\pi$
$89$	−5.78575	−0.613288	−0.306644	+	0.951824i	$0.599206\pi$
$90$	0	0
$91$	7.63553	0.800421
$92$	−4.33225	−0.451668
$93$	0	0
$94$	−3.35697	−0.346245
$95$	−1.29268	−0.132626
$96$	0	0
$97$	0.425248	0.0431774	0.0215887	−	0.999767i	$-0.493128\pi$
$97$	0.425248	0.0431774	0.0215887	+	0.999767i	$0.493128\pi$
$98$	4.43937	0.448444
$99$	0	0
$100$	1.00000	0.100000
$101$	9.00734	0.896264	0.448132	−	0.893967i	$-0.352089\pi$
$101$	9.00734	0.896264	0.448132	+	0.893967i	$0.352089\pi$
$102$	0	0
$103$	−16.8327	−1.65857	−0.829285	−	0.558825i	$-0.811252\pi$
$103$	−16.8327	−1.65857	−0.829285	+	0.558825i	$0.811252\pi$
$104$	−4.77162	−0.467896
$105$	0	0
$106$	1.19941	0.116497
$107$	−8.47470	−0.819281	−0.409640	−	0.912247i	$-0.634346\pi$
$107$	−8.47470	−0.819281	−0.409640	+	0.912247i	$0.634346\pi$
$108$	0	0
$109$	5.17142	0.495333	0.247666	−	0.968845i	$-0.420336\pi$
$109$	5.17142	0.495333	0.247666	+	0.968845i	$0.420336\pi$
$110$	1.00000	0.0953463
$111$	0	0
$112$	1.60020	0.151204
$113$	−15.4757	−1.45583	−0.727915	−	0.685668i	$-0.759510\pi$
$113$	−15.4757	−1.45583	−0.727915	+	0.685668i	$0.759510\pi$
$114$	0	0
$115$	−4.33225	−0.403984
$116$	−6.23996	−0.579366
$117$	0	0
$118$	−1.26371	−0.116334
$119$	−6.29268	−0.576849
$120$	0	0
$121$	1.00000	0.0909091
$122$	0.878747	0.0795581
$123$	0	0
$124$	3.33225	0.299245
$125$	1.00000	0.0894427
$126$	0	0
$127$	18.5110	1.64259	0.821293	−	0.570506i	$-0.193253\pi$
$127$	18.5110	1.64259	0.821293	+	0.570506i	$0.193253\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	−4.77162	−0.418499
$131$	−12.0073	−1.04909	−0.524543	−	0.851384i	$-0.675764\pi$
$131$	−12.0073	−1.04909	−0.524543	+	0.851384i	$0.675764\pi$
$132$	0	0
$133$	−2.06854	−0.179365
$134$	3.77162	0.325818
$135$	0	0
$136$	3.93244	0.337204
$137$	10.4039	0.888864	0.444432	−	0.895813i	$-0.353406\pi$
$137$	10.4039	0.888864	0.444432	+	0.895813i	$0.353406\pi$
$138$	0	0
$139$	3.10811	0.263626	0.131813	−	0.991275i	$-0.457920\pi$
$139$	3.10811	0.263626	0.131813	+	0.991275i	$0.457920\pi$
$140$	1.60020	0.135241
$141$	0	0
$142$	7.15658	0.600567
$143$	−4.77162	−0.399023
$144$	0	0
$145$	−6.23996	−0.518201
$146$	10.8755	0.900062
$147$	0	0
$148$	−2.43937	−0.200515
$149$	−6.05272	−0.495858	−0.247929	−	0.968778i	$-0.579750\pi$
$149$	−6.05272	−0.495858	−0.247929	+	0.968778i	$0.579750\pi$
$150$	0	0
$151$	−0.0289668	−0.00235729	−0.00117864	−	0.999999i	$-0.500375\pi$
$151$	−0.0289668	−0.00235729	−0.00117864	+	0.999999i	$0.500375\pi$
$152$	1.29268	0.104850
$153$	0	0
$154$	1.60020	0.128947
$155$	3.33225	0.267652
$156$	0	0
$157$	−15.0462	−1.20082	−0.600409	−	0.799693i	$-0.704995\pi$
$157$	−15.0462	−1.20082	−0.600409	+	0.799693i	$0.704995\pi$
$158$	7.41139	0.589618
$159$	0	0
$160$	−1.00000	−0.0790569
$161$	−6.93244	−0.546353
$162$	0	0
$163$	12.5037	0.979363	0.489682	−	0.871901i	$-0.337113\pi$
$163$	12.5037	0.979363	0.489682	+	0.871901i	$0.337113\pi$
$164$	1.69346	0.132237
$165$	0	0
$166$	1.73629	0.134762
$167$	5.02473	0.388825	0.194413	−	0.980920i	$-0.437720\pi$
$167$	5.02473	0.388825	0.194413	+	0.980920i	$0.437720\pi$
$168$	0	0
$169$	9.76836	0.751412
$170$	3.93244	0.301605
$171$	0	0
$172$	−1.34709	−0.102715
$173$	4.96777	0.377693	0.188846	−	0.982007i	$-0.439525\pi$
$173$	4.96777	0.377693	0.188846	+	0.982007i	$0.439525\pi$
$174$	0	0
$175$	1.60020	0.120963
$176$	−1.00000	−0.0753778
$177$	0	0
$178$	5.78575	0.433660
$179$	−11.1460	−0.833090	−0.416545	−	0.909115i	$-0.636759\pi$
$179$	−11.1460	−0.833090	−0.416545	+	0.909115i	$0.636759\pi$
$180$	0	0
$181$	−4.50693	−0.334998	−0.167499	−	0.985872i	$-0.553569\pi$
$181$	−4.50693	−0.334998	−0.167499	+	0.985872i	$0.553569\pi$
$182$	−7.63553	−0.565983
$183$	0	0
$184$	4.33225	0.319377
$185$	−2.43937	−0.179346
$186$	0	0
$187$	3.93244	0.287569
$188$	3.35697	0.244833
$189$	0	0
$190$	1.29268	0.0937807
$191$	−14.2805	−1.03330	−0.516651	−	0.856196i	$-0.672821\pi$
$191$	−14.2805	−1.03330	−0.516651	+	0.856196i	$0.672821\pi$
$192$	0	0
$193$	8.18979	0.589514	0.294757	−	0.955572i	$-0.404761\pi$
$193$	8.18979	0.589514	0.294757	+	0.955572i	$0.404761\pi$
$194$	−0.425248	−0.0305311
$195$	0	0
$196$	−4.43937	−0.317098
$197$	−8.08903	−0.576319	−0.288160	−	0.957582i	$-0.593043\pi$
$197$	−8.08903	−0.576319	−0.288160	+	0.957582i	$0.593043\pi$
$198$	0	0
$199$	4.45024	0.315469	0.157735	−	0.987482i	$-0.449581\pi$
$199$	4.45024	0.315469	0.157735	+	0.987482i	$0.449581\pi$
$200$	−1.00000	−0.0707107
$201$	0	0
$202$	−9.00734	−0.633754
$203$	−9.98516	−0.700821
$204$	0	0
$205$	1.69346	0.118276
$206$	16.8327	1.17279
$207$	0	0
$208$	4.77162	0.330852
$209$	1.29268	0.0894163
$210$	0	0
$211$	−6.49307	−0.447001	−0.223501	−	0.974704i	$-0.571748\pi$
$211$	−6.49307	−0.447001	−0.223501	+	0.974704i	$0.571748\pi$
$212$	−1.19941	−0.0823760
$213$	0	0
$214$	8.47470	0.579319
$215$	−1.34709	−0.0918706
$216$	0	0
$217$	5.33225	0.361976
$218$	−5.17142	−0.350253
$219$	0	0
$220$	−1.00000	−0.0674200
$221$	−18.7641	−1.26221
$222$	0	0
$223$	19.1251	1.28071	0.640355	−	0.768079i	$-0.278787\pi$
$223$	19.1251	1.28071	0.640355	+	0.768079i	$0.278787\pi$
$224$	−1.60020	−0.106918
$225$	0	0
$226$	15.4757	1.02943
$227$	1.07816	0.0715600	0.0357800	−	0.999360i	$-0.488608\pi$
$227$	1.07816	0.0715600	0.0357800	+	0.999360i	$0.488608\pi$
$228$	0	0
$229$	−6.03461	−0.398779	−0.199389	−	0.979920i	$-0.563896\pi$
$229$	−6.03461	−0.398779	−0.199389	+	0.979920i	$0.563896\pi$
$230$	4.33225	0.285660
$231$	0	0
$232$	6.23996	0.409673
$233$	−5.54578	−0.363316	−0.181658	−	0.983362i	$-0.558146\pi$
$233$	−5.54578	−0.363316	−0.181658	+	0.983362i	$0.558146\pi$
$234$	0	0
$235$	3.35697	0.218985
$236$	1.26371	0.0822605
$237$	0	0
$238$	6.29268	0.407894
$239$	8.00636	0.517889	0.258944	−	0.965892i	$-0.416625\pi$
$239$	8.00636	0.517889	0.258944	+	0.965892i	$0.416625\pi$
$240$	0	0
$241$	18.0726	1.16416	0.582080	−	0.813132i	$-0.302239\pi$
$241$	18.0726	1.16416	0.582080	+	0.813132i	$0.302239\pi$
$242$	−1.00000	−0.0642824
$243$	0	0
$244$	−0.878747	−0.0562560
$245$	−4.43937	−0.283621
$246$	0	0
$247$	−6.16816	−0.392471
$248$	−3.33225	−0.211598
$249$	0	0
$250$	−1.00000	−0.0632456
$251$	−0.0312489	−0.00197241	−0.000986205	−	1.00000i	$-0.500314\pi$
$251$	−0.0312489	−0.00197241	−0.000986205		1.00000i	$0.500314\pi$
$252$	0	0
$253$	4.33225	0.272366
$254$	−18.5110	−1.16148
$255$	0	0
$256$	1.00000	0.0625000
$257$	11.6860	0.728950	0.364475	−	0.931213i	$-0.381248\pi$
$257$	11.6860	0.728950	0.364475	+	0.931213i	$0.381248\pi$
$258$	0	0
$259$	−3.90348	−0.242550
$260$	4.77162	0.295923
$261$	0	0
$262$	12.0073	0.741816
$263$	20.4717	1.26234	0.631170	−	0.775645i	$-0.282575\pi$
$263$	20.4717	1.26234	0.631170	+	0.775645i	$0.282575\pi$
$264$	0	0
$265$	−1.19941	−0.0736793
$266$	2.06854	0.126830
$267$	0	0
$268$	−3.77162	−0.230388
$269$	−20.2322	−1.23358	−0.616789	−	0.787128i	$-0.711567\pi$
$269$	−20.2322	−1.23358	−0.616789	+	0.787128i	$0.711567\pi$
$270$	0	0
$271$	−10.6085	−0.644421	−0.322211	−	0.946668i	$-0.604426\pi$
$271$	−10.6085	−0.644421	−0.322211	+	0.946668i	$0.604426\pi$
$272$	−3.93244	−0.238439
$273$	0	0
$274$	−10.4039	−0.628522
$275$	−1.00000	−0.0603023
$276$	0	0
$277$	−28.0446	−1.68504	−0.842519	−	0.538666i	$-0.818928\pi$
$277$	−28.0446	−1.68504	−0.842519	+	0.538666i	$0.818928\pi$
$278$	−3.10811	−0.186412
$279$	0	0
$280$	−1.60020	−0.0956300
$281$	−10.8251	−0.645769	−0.322884	−	0.946438i	$-0.604652\pi$
$281$	−10.8251	−0.645769	−0.322884	+	0.946438i	$0.604652\pi$
$282$	0	0
$283$	−24.9076	−1.48060	−0.740300	−	0.672276i	$-0.765317\pi$
$283$	−24.9076	−1.48060	−0.740300	+	0.672276i	$0.765317\pi$
$284$	−7.15658	−0.424665
$285$	0	0
$286$	4.77162	0.282152
$287$	2.70987	0.159958
$288$	0	0
$289$	−1.53590	−0.0903470
$290$	6.23996	0.366423
$291$	0	0
$292$	−10.8755	−0.636440
$293$	−21.1844	−1.23761	−0.618803	−	0.785546i	$-0.712382\pi$
$293$	−21.1844	−1.23761	−0.618803	+	0.785546i	$0.712382\pi$
$294$	0	0
$295$	1.26371	0.0735761
$296$	2.43937	0.141786
$297$	0	0
$298$	6.05272	0.350624
$299$	−20.6718	−1.19548
$300$	0	0
$301$	−2.15560	−0.124247
$302$	0.0289668	0.00166685
$303$	0	0
$304$	−1.29268	−0.0741401
$305$	−0.878747	−0.0503169
$306$	0	0
$307$	23.5842	1.34602	0.673011	−	0.739632i	$-0.265001\pi$
$307$	23.5842	1.34602	0.673011	+	0.739632i	$0.265001\pi$
$308$	−1.60020	−0.0911796
$309$	0	0
$310$	−3.33225	−0.189259
$311$	−4.50622	−0.255524	−0.127762	−	0.991805i	$-0.540779\pi$
$311$	−4.50622	−0.255524	−0.127762	+	0.991805i	$0.540779\pi$
$312$	0	0
$313$	13.0620	0.738309	0.369154	−	0.929368i	$-0.379647\pi$
$313$	13.0620	0.738309	0.369154	+	0.929368i	$0.379647\pi$
$314$	15.0462	0.849106
$315$	0	0
$316$	−7.41139	−0.416923
$317$	28.1913	1.58338	0.791691	−	0.610921i	$-0.209201\pi$
$317$	28.1913	1.58338	0.791691	+	0.610921i	$0.209201\pi$
$318$	0	0
$319$	6.23996	0.349371
$320$	1.00000	0.0559017
$321$	0	0
$322$	6.93244	0.386330
$323$	5.08338	0.282847
$324$	0	0
$325$	4.77162	0.264682
$326$	−12.5037	−0.692514
$327$	0	0
$328$	−1.69346	−0.0935057
$329$	5.37182	0.296158
$330$	0	0
$331$	22.9019	1.25880	0.629401	−	0.777080i	$-0.283300\pi$
$331$	22.9019	1.25880	0.629401	+	0.777080i	$0.283300\pi$
$332$	−1.73629	−0.0952913
$333$	0	0
$334$	−5.02473	−0.274941
$335$	−3.77162	−0.206066
$336$	0	0
$337$	0.934724	0.0509177	0.0254588	−	0.999676i	$-0.491895\pi$
$337$	0.934724	0.0509177	0.0254588	+	0.999676i	$0.491895\pi$
$338$	−9.76836	−0.531329
$339$	0	0
$340$	−3.93244	−0.213267
$341$	−3.33225	−0.180451
$342$	0	0
$343$	−18.3052	−0.988390
$344$	1.34709	0.0726301
$345$	0	0
$346$	−4.96777	−0.267069
$347$	13.9009	0.746241	0.373121	−	0.927783i	$-0.378288\pi$
$347$	13.9009	0.746241	0.373121	+	0.927783i	$0.378288\pi$
$348$	0	0
$349$	−5.55808	−0.297517	−0.148759	−	0.988874i	$-0.547528\pi$
$349$	−5.55808	−0.297517	−0.148759	+	0.988874i	$0.547528\pi$
$350$	−1.60020	−0.0855341
$351$	0	0
$352$	1.00000	0.0533002
$353$	16.3898	0.872339	0.436169	−	0.899865i	$-0.356335\pi$
$353$	16.3898	0.872339	0.436169	+	0.899865i	$0.356335\pi$
$354$	0	0
$355$	−7.15658	−0.379832
$356$	−5.78575	−0.306644
$357$	0	0
$358$	11.1460	0.589083
$359$	15.1946	0.801940	0.400970	−	0.916091i	$-0.368673\pi$
$359$	15.1946	0.801940	0.400970	+	0.916091i	$0.368673\pi$
$360$	0	0
$361$	−17.3290	−0.912052
$362$	4.50693	0.236879
$363$	0	0
$364$	7.63553	0.400210
$365$	−10.8755	−0.569249
$366$	0	0
$367$	11.7684	0.614303	0.307152	−	0.951661i	$-0.400624\pi$
$367$	11.7684	0.614303	0.307152	+	0.951661i	$0.400624\pi$
$368$	−4.33225	−0.225834
$369$	0	0
$370$	2.43937	0.126817
$371$	−1.91930	−0.0996449
$372$	0	0
$373$	−10.1587	−0.525998	−0.262999	−	0.964796i	$-0.584712\pi$
$373$	−10.1587	−0.525998	−0.262999	+	0.964796i	$0.584712\pi$
$374$	−3.93244	−0.203342
$375$	0	0
$376$	−3.35697	−0.173123
$377$	−29.7747	−1.53348
$378$	0	0
$379$	21.5795	1.10847	0.554233	−	0.832361i	$-0.313011\pi$
$379$	21.5795	1.10847	0.554233	+	0.832361i	$0.313011\pi$
$380$	−1.29268	−0.0663129
$381$	0	0
$382$	14.2805	0.730654
$383$	−21.7701	−1.11240	−0.556199	−	0.831049i	$-0.687741\pi$
$383$	−21.7701	−1.11240	−0.556199	+	0.831049i	$0.687741\pi$
$384$	0	0
$385$	−1.60020	−0.0815535
$386$	−8.18979	−0.416849
$387$	0	0
$388$	0.425248	0.0215887
$389$	−13.0897	−0.663676	−0.331838	−	0.943336i	$-0.607669\pi$
$389$	−13.0897	−0.663676	−0.331838	+	0.943336i	$0.607669\pi$
$390$	0	0
$391$	17.0363	0.861563
$392$	4.43937	0.224222
$393$	0	0
$394$	8.08903	0.407519
$395$	−7.41139	−0.372907
$396$	0	0
$397$	−1.86250	−0.0934761	−0.0467380	−	0.998907i	$-0.514883\pi$
$397$	−1.86250	−0.0934761	−0.0467380	+	0.998907i	$0.514883\pi$
$398$	−4.45024	−0.223070
$399$	0	0
$400$	1.00000	0.0500000
$401$	−8.87451	−0.443172	−0.221586	−	0.975141i	$-0.571123\pi$
$401$	−8.87451	−0.443172	−0.221586	+	0.975141i	$0.571123\pi$
$402$	0	0
$403$	15.9002	0.792046
$404$	9.00734	0.448132
$405$	0	0
$406$	9.98516	0.495555
$407$	2.43937	0.120915
$408$	0	0
$409$	−21.6362	−1.06984	−0.534922	−	0.844902i	$-0.679659\pi$
$409$	−21.6362	−1.06984	−0.534922	+	0.844902i	$0.679659\pi$
$410$	−1.69346	−0.0836340
$411$	0	0
$412$	−16.8327	−0.829285
$413$	2.02218	0.0995052
$414$	0	0
$415$	−1.73629	−0.0852311
$416$	−4.77162	−0.233948
$417$	0	0
$418$	−1.29268	−0.0632269
$419$	10.1395	0.495345	0.247673	−	0.968844i	$-0.420334\pi$
$419$	10.1395	0.495345	0.247673	+	0.968844i	$0.420334\pi$
$420$	0	0
$421$	−12.0791	−0.588701	−0.294351	−	0.955698i	$-0.595103\pi$
$421$	−12.0791	−0.588701	−0.294351	+	0.955698i	$0.595103\pi$
$422$	6.49307	0.316078
$423$	0	0
$424$	1.19941	0.0582486
$425$	−3.93244	−0.190751
$426$	0	0
$427$	−1.40617	−0.0680492
$428$	−8.47470	−0.409640
$429$	0	0
$430$	1.34709	0.0649624
$431$	10.7706	0.518804	0.259402	−	0.965769i	$-0.416475\pi$
$431$	10.7706	0.518804	0.259402	+	0.965769i	$0.416475\pi$
$432$	0	0
$433$	25.7446	1.23721	0.618604	−	0.785703i	$-0.287699\pi$
$433$	25.7446	1.23721	0.618604	+	0.785703i	$0.287699\pi$
$434$	−5.33225	−0.255956
$435$	0	0
$436$	5.17142	0.247666
$437$	5.60020	0.267894
$438$	0	0
$439$	13.6963	0.653689	0.326844	−	0.945078i	$-0.394015\pi$
$439$	13.6963	0.653689	0.326844	+	0.945078i	$0.394015\pi$
$440$	1.00000	0.0476731
$441$	0	0
$442$	18.7641	0.892518
$443$	−4.43373	−0.210653	−0.105326	−	0.994438i	$-0.533589\pi$
$443$	−4.43373	−0.210653	−0.105326	+	0.994438i	$0.533589\pi$
$444$	0	0
$445$	−5.78575	−0.274271
$446$	−19.1251	−0.905598
$447$	0	0
$448$	1.60020	0.0756021
$449$	−10.5807	−0.499333	−0.249667	−	0.968332i	$-0.580321\pi$
$449$	−10.5807	−0.499333	−0.249667	+	0.968332i	$0.580321\pi$
$450$	0	0
$451$	−1.69346	−0.0797419
$452$	−15.4757	−0.727915
$453$	0	0
$454$	−1.07816	−0.0506006
$455$	7.63553	0.357959
$456$	0	0
$457$	−11.1211	−0.520223	−0.260111	−	0.965579i	$-0.583759\pi$
$457$	−11.1211	−0.520223	−0.260111	+	0.965579i	$0.583759\pi$
$458$	6.03461	0.281979
$459$	0	0
$460$	−4.33225	−0.201992
$461$	39.0734	1.81983	0.909916	−	0.414793i	$-0.136146\pi$
$461$	39.0734	1.81983	0.909916	+	0.414793i	$0.136146\pi$
$462$	0	0
$463$	−18.3462	−0.852621	−0.426310	−	0.904577i	$-0.640187\pi$
$463$	−18.3462	−0.852621	−0.426310	+	0.904577i	$0.640187\pi$
$464$	−6.23996	−0.289683
$465$	0	0
$466$	5.54578	0.256904
$467$	−14.0377	−0.649588	−0.324794	−	0.945785i	$-0.605295\pi$
$467$	−14.0377	−0.649588	−0.324794	+	0.945785i	$0.605295\pi$
$468$	0	0
$469$	−6.03533	−0.278686
$470$	−3.35697	−0.154846
$471$	0	0
$472$	−1.26371	−0.0581670
$473$	1.34709	0.0619392
$474$	0	0
$475$	−1.29268	−0.0593121
$476$	−6.29268	−0.288424
$477$	0	0
$478$	−8.00636	−0.366203
$479$	6.98121	0.318979	0.159490	−	0.987200i	$-0.449015\pi$
$479$	6.98121	0.318979	0.159490	+	0.987200i	$0.449015\pi$
$480$	0	0
$481$	−11.6398	−0.530728
$482$	−18.0726	−0.823185
$483$	0	0
$484$	1.00000	0.0454545
$485$	0.425248	0.0193095
$486$	0	0
$487$	12.5779	0.569957	0.284978	−	0.958534i	$-0.408014\pi$
$487$	12.5779	0.569957	0.284978	+	0.958534i	$0.408014\pi$
$488$	0.878747	0.0397790
$489$	0	0
$490$	4.43937	0.200550
$491$	−0.944023	−0.0426032	−0.0213016	−	0.999773i	$-0.506781\pi$
$491$	−0.944023	−0.0426032	−0.0213016	+	0.999773i	$0.506781\pi$
$492$	0	0
$493$	24.5383	1.10515
$494$	6.16816	0.277519
$495$	0	0
$496$	3.33225	0.149622
$497$	−11.4519	−0.513689
$498$	0	0
$499$	0.990376	0.0443353	0.0221677	−	0.999754i	$-0.492943\pi$
$499$	0.990376	0.0443353	0.0221677	+	0.999754i	$0.492943\pi$
$500$	1.00000	0.0447214
$501$	0	0
$502$	0.0312489	0.00139471
$503$	−27.3700	−1.22037	−0.610183	−	0.792260i	$-0.708904\pi$
$503$	−27.3700	−1.22037	−0.610183	+	0.792260i	$0.708904\pi$
$504$	0	0
$505$	9.00734	0.400821
$506$	−4.33225	−0.192592
$507$	0	0
$508$	18.5110	0.821293
$509$	−39.3317	−1.74335	−0.871673	−	0.490089i	$-0.836964\pi$
$509$	−39.3317	−1.74335	−0.871673	+	0.490089i	$0.836964\pi$
$510$	0	0
$511$	−17.4029	−0.769859
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	−11.6860	−0.515446
$515$	−16.8327	−0.741735
$516$	0	0
$517$	−3.35697	−0.147640
$518$	3.90348	0.171509
$519$	0	0
$520$	−4.77162	−0.209249
$521$	−31.7699	−1.39186	−0.695932	−	0.718107i	$-0.745009\pi$
$521$	−31.7699	−1.39186	−0.695932	+	0.718107i	$0.745009\pi$
$522$	0	0
$523$	39.4279	1.72406	0.862031	−	0.506856i	$-0.169192\pi$
$523$	39.4279	1.72406	0.862031	+	0.506856i	$0.169192\pi$
$524$	−12.0073	−0.524543
$525$	0	0
$526$	−20.4717	−0.892609
$527$	−13.1039	−0.570813
$528$	0	0
$529$	−4.23164	−0.183984
$530$	1.19941	0.0520992
$531$	0	0
$532$	−2.06854	−0.0896824
$533$	8.08055	0.350007
$534$	0	0
$535$	−8.47470	−0.366393
$536$	3.77162	0.162909
$537$	0	0
$538$	20.2322	0.872272
$539$	4.43937	0.191217
$540$	0	0
$541$	34.0082	1.46213	0.731063	−	0.682310i	$-0.239025\pi$
$541$	34.0082	1.46213	0.731063	+	0.682310i	$0.239025\pi$
$542$	10.6085	0.455675
$543$	0	0
$544$	3.93244	0.168602
$545$	5.17142	0.221519
$546$	0	0
$547$	−1.90419	−0.0814174	−0.0407087	−	0.999171i	$-0.512962\pi$
$547$	−1.90419	−0.0814174	−0.0407087	+	0.999171i	$0.512962\pi$
$548$	10.4039	0.444432
$549$	0	0
$550$	1.00000	0.0426401
$551$	8.06625	0.343634
$552$	0	0
$553$	−11.8597	−0.504324
$554$	28.0446	1.19150
$555$	0	0
$556$	3.10811	0.131813
$557$	20.0890	0.851199	0.425600	−	0.904912i	$-0.360063\pi$
$557$	20.0890	0.851199	0.425600	+	0.904912i	$0.360063\pi$
$558$	0	0
$559$	−6.42779	−0.271867
$560$	1.60020	0.0676206
$561$	0	0
$562$	10.8251	0.456627
$563$	26.7802	1.12865	0.564325	−	0.825552i	$-0.309136\pi$
$563$	26.7802	1.12865	0.564325	+	0.825552i	$0.309136\pi$
$564$	0	0
$565$	−15.4757	−0.651067
$566$	24.9076	1.04694
$567$	0	0
$568$	7.15658	0.300284
$569$	−26.4460	−1.10867	−0.554337	−	0.832292i	$-0.687028\pi$
$569$	−26.4460	−1.10867	−0.554337	+	0.832292i	$0.687028\pi$
$570$	0	0
$571$	−24.4263	−1.02221	−0.511105	−	0.859518i	$-0.670764\pi$
$571$	−24.4263	−1.02221	−0.511105	+	0.859518i	$0.670764\pi$
$572$	−4.77162	−0.199511
$573$	0	0
$574$	−2.70987	−0.113108
$575$	−4.33225	−0.180667
$576$	0	0
$577$	5.33861	0.222249	0.111125	−	0.993806i	$-0.464555\pi$
$577$	5.33861	0.222249	0.111125	+	0.993806i	$0.464555\pi$
$578$	1.53590	0.0638850
$579$	0	0
$580$	−6.23996	−0.259100
$581$	−2.77840	−0.115268
$582$	0	0
$583$	1.19941	0.0496746
$584$	10.8755	0.450031
$585$	0	0
$586$	21.1844	0.875120
$587$	19.2037	0.792620	0.396310	−	0.918117i	$-0.370291\pi$
$587$	19.2037	0.792620	0.396310	+	0.918117i	$0.370291\pi$
$588$	0	0
$589$	−4.30752	−0.177488
$590$	−1.26371	−0.0520261
$591$	0	0
$592$	−2.43937	−0.100258
$593$	−7.48785	−0.307489	−0.153745	−	0.988111i	$-0.549133\pi$
$593$	−7.48785	−0.307489	−0.153745	+	0.988111i	$0.549133\pi$
$594$	0	0
$595$	−6.29268	−0.257975
$596$	−6.05272	−0.247929
$597$	0	0
$598$	20.6718	0.845334
$599$	46.2159	1.88833	0.944165	−	0.329472i	$-0.106871\pi$
$599$	46.2159	1.88833	0.944165	+	0.329472i	$0.106871\pi$
$600$	0	0
$601$	−14.3309	−0.584571	−0.292286	−	0.956331i	$-0.594416\pi$
$601$	−14.3309	−0.584571	−0.292286	+	0.956331i	$0.594416\pi$
$602$	2.15560	0.0878559
$603$	0	0
$604$	−0.0289668	−0.00117864
$605$	1.00000	0.0406558
$606$	0	0
$607$	−29.7406	−1.20714	−0.603568	−	0.797312i	$-0.706255\pi$
$607$	−29.7406	−1.20714	−0.603568	+	0.797312i	$0.706255\pi$
$608$	1.29268	0.0524250
$609$	0	0
$610$	0.878747	0.0355794
$611$	16.0182	0.648027
$612$	0	0
$613$	13.3693	0.539980	0.269990	−	0.962863i	$-0.412980\pi$
$613$	13.3693	0.539980	0.269990	+	0.962863i	$0.412980\pi$
$614$	−23.5842	−0.951782
$615$	0	0
$616$	1.60020	0.0644737
$617$	31.1713	1.25491	0.627454	−	0.778654i	$-0.284097\pi$
$617$	31.1713	1.25491	0.627454	+	0.778654i	$0.284097\pi$
$618$	0	0
$619$	−6.12197	−0.246063	−0.123031	−	0.992403i	$-0.539262\pi$
$619$	−6.12197	−0.246063	−0.123031	+	0.992403i	$0.539262\pi$
$620$	3.33225	0.133826
$621$	0	0
$622$	4.50622	0.180683
$623$	−9.25833	−0.370927
$624$	0	0
$625$	1.00000	0.0400000
$626$	−13.0620	−0.522063
$627$	0	0
$628$	−15.0462	−0.600409
$629$	9.59270	0.382486
$630$	0	0
$631$	−34.4972	−1.37331	−0.686655	−	0.726983i	$-0.740922\pi$
$631$	−34.4972	−1.37331	−0.686655	+	0.726983i	$0.740922\pi$
$632$	7.41139	0.294809
$633$	0	0
$634$	−28.1913	−1.11962
$635$	18.5110	0.734587
$636$	0	0
$637$	−21.1830	−0.839301
$638$	−6.23996	−0.247042
$639$	0	0
$640$	−1.00000	−0.0395285
$641$	2.74167	0.108290	0.0541448	−	0.998533i	$-0.482757\pi$
$641$	2.74167	0.108290	0.0541448	+	0.998533i	$0.482757\pi$
$642$	0	0
$643$	6.96947	0.274849	0.137424	−	0.990512i	$-0.456118\pi$
$643$	6.96947	0.274849	0.137424	+	0.990512i	$0.456118\pi$
$644$	−6.93244	−0.273176
$645$	0	0
$646$	−5.08338	−0.200003
$647$	14.2934	0.561931	0.280966	−	0.959718i	$-0.409345\pi$
$647$	14.2934	0.561931	0.280966	+	0.959718i	$0.409345\pi$
$648$	0	0
$649$	−1.26371	−0.0496050
$650$	−4.77162	−0.187158
$651$	0	0
$652$	12.5037	0.489682
$653$	31.6841	1.23990	0.619948	−	0.784643i	$-0.287154\pi$
$653$	31.6841	1.23990	0.619948	+	0.784643i	$0.287154\pi$
$654$	0	0
$655$	−12.0073	−0.469166
$656$	1.69346	0.0661185
$657$	0	0
$658$	−5.37182	−0.209415
$659$	−39.1715	−1.52591	−0.762953	−	0.646454i	$-0.776251\pi$
$659$	−39.1715	−1.52591	−0.762953	+	0.646454i	$0.776251\pi$
$660$	0	0
$661$	−16.7629	−0.652000	−0.326000	−	0.945370i	$-0.605701\pi$
$661$	−16.7629	−0.652000	−0.326000	+	0.945370i	$0.605701\pi$
$662$	−22.9019	−0.890108
$663$	0	0
$664$	1.73629	0.0673811
$665$	−2.06854	−0.0802144
$666$	0	0
$667$	27.0330	1.04672
$668$	5.02473	0.194413
$669$	0	0
$670$	3.77162	0.145710
$671$	0.878747	0.0339237
$672$	0	0
$673$	7.55482	0.291217	0.145608	−	0.989342i	$-0.453486\pi$
$673$	7.55482	0.291217	0.145608	+	0.989342i	$0.453486\pi$
$674$	−0.934724	−0.0360042
$675$	0	0
$676$	9.76836	0.375706
$677$	−34.0157	−1.30733	−0.653664	−	0.756785i	$-0.726769\pi$
$677$	−34.0157	−1.30733	−0.653664	+	0.756785i	$0.726769\pi$
$678$	0	0
$679$	0.680481	0.0261145
$680$	3.93244	0.150802
$681$	0	0
$682$	3.33225	0.127598
$683$	−29.1225	−1.11434	−0.557171	−	0.830398i	$-0.688113\pi$
$683$	−29.1225	−1.11434	−0.557171	+	0.830398i	$0.688113\pi$
$684$	0	0
$685$	10.4039	0.397512
$686$	18.3052	0.698897
$687$	0	0
$688$	−1.34709	−0.0513573
$689$	−5.72314	−0.218034
$690$	0	0
$691$	−21.7231	−0.826387	−0.413194	−	0.910643i	$-0.635587\pi$
$691$	−21.7231	−0.826387	−0.413194	+	0.910643i	$0.635587\pi$
$692$	4.96777	0.188846
$693$	0	0
$694$	−13.9009	−0.527672
$695$	3.10811	0.117897
$696$	0	0
$697$	−6.65943	−0.252244
$698$	5.55808	0.210377
$699$	0	0
$700$	1.60020	0.0604817
$701$	11.2004	0.423033	0.211516	−	0.977374i	$-0.432160\pi$
$701$	11.2004	0.423033	0.211516	+	0.977374i	$0.432160\pi$
$702$	0	0
$703$	3.15332	0.118930
$704$	−1.00000	−0.0376889
$705$	0	0
$706$	−16.3898	−0.616837
$707$	14.4135	0.542076
$708$	0	0
$709$	−16.3287	−0.613238	−0.306619	−	0.951832i	$-0.599198\pi$
$709$	−16.3287	−0.613238	−0.306619	+	0.951832i	$0.599198\pi$
$710$	7.15658	0.268582
$711$	0	0
$712$	5.78575	0.216830
$713$	−14.4361	−0.540637
$714$	0	0
$715$	−4.77162	−0.178448
$716$	−11.1460	−0.416545
$717$	0	0
$718$	−15.1946	−0.567057
$719$	−28.8637	−1.07644	−0.538218	−	0.842806i	$-0.680902\pi$
$719$	−28.8637	−1.07644	−0.538218	+	0.842806i	$0.680902\pi$
$720$	0	0
$721$	−26.9355	−1.00313
$722$	17.3290	0.644918
$723$	0	0
$724$	−4.50693	−0.167499
$725$	−6.23996	−0.231746
$726$	0	0
$727$	−38.9156	−1.44330	−0.721650	−	0.692258i	$-0.756616\pi$
$727$	−38.9156	−1.44330	−0.721650	+	0.692258i	$0.756616\pi$
$728$	−7.63553	−0.282991
$729$	0	0
$730$	10.8755	0.402520
$731$	5.29735	0.195929
$732$	0	0
$733$	16.5660	0.611880	0.305940	−	0.952051i	$-0.401029\pi$
$733$	16.5660	0.611880	0.305940	+	0.952051i	$0.401029\pi$
$734$	−11.7684	−0.434378
$735$	0	0
$736$	4.33225	0.159689
$737$	3.77162	0.138929
$738$	0	0
$739$	−46.0815	−1.69514	−0.847568	−	0.530687i	$-0.821934\pi$
$739$	−46.0815	−1.69514	−0.847568	+	0.530687i	$0.821934\pi$
$740$	−2.43937	−0.0896732
$741$	0	0
$742$	1.91930	0.0704596
$743$	12.6985	0.465864	0.232932	−	0.972493i	$-0.425168\pi$
$743$	12.6985	0.465864	0.232932	+	0.972493i	$0.425168\pi$
$744$	0	0
$745$	−6.05272	−0.221754
$746$	10.1587	0.371937
$747$	0	0
$748$	3.93244	0.143784
$749$	−13.5612	−0.495515
$750$	0	0
$751$	−28.0957	−1.02522	−0.512612	−	0.858620i	$-0.671322\pi$
$751$	−28.0957	−1.02522	−0.512612	+	0.858620i	$0.671322\pi$
$752$	3.35697	0.122416
$753$	0	0
$754$	29.7747	1.08433
$755$	−0.0289668	−0.00105421
$756$	0	0
$757$	10.8771	0.395333	0.197667	−	0.980269i	$-0.436664\pi$
$757$	10.8771	0.395333	0.197667	+	0.980269i	$0.436664\pi$
$758$	−21.5795	−0.783805
$759$	0	0
$760$	1.29268	0.0468903
$761$	−7.36529	−0.266992	−0.133496	−	0.991049i	$-0.542620\pi$
$761$	−7.36529	−0.266992	−0.133496	+	0.991049i	$0.542620\pi$
$762$	0	0
$763$	8.27529	0.299586
$764$	−14.2805	−0.516651
$765$	0	0
$766$	21.7701	0.786584
$767$	6.02995	0.217729
$768$	0	0
$769$	−42.9885	−1.55021	−0.775103	−	0.631835i	$-0.782302\pi$
$769$	−42.9885	−1.55021	−0.775103	+	0.631835i	$0.782302\pi$
$770$	1.60020	0.0576670
$771$	0	0
$772$	8.18979	0.294757
$773$	−46.7197	−1.68039	−0.840196	−	0.542283i	$-0.817560\pi$
$773$	−46.7197	−1.68039	−0.840196	+	0.542283i	$0.817560\pi$
$774$	0	0
$775$	3.33225	0.119698
$776$	−0.425248	−0.0152655
$777$	0	0
$778$	13.0897	0.469290
$779$	−2.18910	−0.0784325
$780$	0	0
$781$	7.15658	0.256083
$782$	−17.0363	−0.609217
$783$	0	0
$784$	−4.43937	−0.158549
$785$	−15.0462	−0.537022
$786$	0	0
$787$	29.8643	1.06455	0.532274	−	0.846572i	$-0.321338\pi$
$787$	29.8643	1.06455	0.532274	+	0.846572i	$0.321338\pi$
$788$	−8.08903	−0.288160
$789$	0	0
$790$	7.41139	0.263685
$791$	−24.7641	−0.880511
$792$	0	0
$793$	−4.19305	−0.148900
$794$	1.86250	0.0660976
$795$	0	0
$796$	4.45024	0.157735
$797$	−44.4966	−1.57615	−0.788075	−	0.615580i	$-0.788922\pi$
$797$	−44.4966	−1.57615	−0.788075	+	0.615580i	$0.788922\pi$
$798$	0	0
$799$	−13.2011	−0.467022
$800$	−1.00000	−0.0353553
$801$	0	0
$802$	8.87451	0.313370
$803$	10.8755	0.383788
$804$	0	0
$805$	−6.93244	−0.244336
$806$	−15.9002	−0.560061
$807$	0	0
$808$	−9.00734	−0.316877
$809$	22.5167	0.791644	0.395822	−	0.918327i	$-0.370460\pi$
$809$	22.5167	0.791644	0.395822	+	0.918327i	$0.370460\pi$
$810$	0	0
$811$	−13.8926	−0.487836	−0.243918	−	0.969796i	$-0.578433\pi$
$811$	−13.8926	−0.487836	−0.243918	+	0.969796i	$0.578433\pi$
$812$	−9.98516	−0.350410
$813$	0	0
$814$	−2.43937	−0.0855000
$815$	12.5037	0.437984
$816$	0	0
$817$	1.74135	0.0609221
$818$	21.6362	0.756493
$819$	0	0
$820$	1.69346	0.0591382
$821$	53.4183	1.86431	0.932156	−	0.362057i	$-0.117925\pi$
$821$	53.4183	1.86431	0.932156	+	0.362057i	$0.117925\pi$
$822$	0	0
$823$	20.0514	0.698948	0.349474	−	0.936946i	$-0.386360\pi$
$823$	20.0514	0.698948	0.349474	+	0.936946i	$0.386360\pi$
$824$	16.8327	0.586393
$825$	0	0
$826$	−2.02218	−0.0703608
$827$	17.9704	0.624894	0.312447	−	0.949935i	$-0.398851\pi$
$827$	17.9704	0.624894	0.312447	+	0.949935i	$0.398851\pi$
$828$	0	0
$829$	−56.4430	−1.96035	−0.980174	−	0.198139i	$-0.936510\pi$
$829$	−56.4430	−1.96035	−0.980174	+	0.198139i	$0.936510\pi$
$830$	1.73629	0.0602675
$831$	0	0
$832$	4.77162	0.165426
$833$	17.4576	0.604869
$834$	0	0
$835$	5.02473	0.173888
$836$	1.29268	0.0447082
$837$	0	0
$838$	−10.1395	−0.350262
$839$	−20.1061	−0.694141	−0.347071	−	0.937839i	$-0.612824\pi$
$839$	−20.1061	−0.694141	−0.347071	+	0.937839i	$0.612824\pi$
$840$	0	0
$841$	9.93711	0.342659
$842$	12.0791	0.416275
$843$	0	0
$844$	−6.49307	−0.223501
$845$	9.76836	0.336042
$846$	0	0
$847$	1.60020	0.0549834
$848$	−1.19941	−0.0411880
$849$	0	0
$850$	3.93244	0.134882
$851$	10.5680	0.362265
$852$	0	0
$853$	2.32308	0.0795406	0.0397703	−	0.999209i	$-0.487337\pi$
$853$	2.32308	0.0795406	0.0397703	+	0.999209i	$0.487337\pi$
$854$	1.40617	0.0481181
$855$	0	0
$856$	8.47470	0.289659
$857$	−0.973969	−0.0332701	−0.0166351	−	0.999862i	$-0.505295\pi$
$857$	−0.973969	−0.0332701	−0.0166351	+	0.999862i	$0.505295\pi$
$858$	0	0
$859$	−2.89613	−0.0988148	−0.0494074	−	0.998779i	$-0.515733\pi$
$859$	−2.89613	−0.0988148	−0.0494074	+	0.998779i	$0.515733\pi$
$860$	−1.34709	−0.0459353
$861$	0	0
$862$	−10.7706	−0.366850
$863$	−47.2949	−1.60994	−0.804969	−	0.593317i	$-0.797818\pi$
$863$	−47.2949	−1.60994	−0.804969	+	0.593317i	$0.797818\pi$
$864$	0	0
$865$	4.96777	0.168909
$866$	−25.7446	−0.874837
$867$	0	0
$868$	5.33225	0.180988
$869$	7.41139	0.251414
$870$	0	0
$871$	−17.9967	−0.609796
$872$	−5.17142	−0.175127
$873$	0	0
$874$	−5.60020	−0.189429
$875$	1.60020	0.0540965
$876$	0	0
$877$	18.9304	0.639235	0.319617	−	0.947547i	$-0.396446\pi$
$877$	18.9304	0.639235	0.319617	+	0.947547i	$0.396446\pi$
$878$	−13.6963	−0.462228
$879$	0	0
$880$	−1.00000	−0.0337100
$881$	21.6526	0.729494	0.364747	−	0.931107i	$-0.381155\pi$
$881$	21.6526	0.729494	0.364747	+	0.931107i	$0.381155\pi$
$882$	0	0
$883$	9.98190	0.335918	0.167959	−	0.985794i	$-0.446282\pi$
$883$	9.98190	0.335918	0.167959	+	0.985794i	$0.446282\pi$
$884$	−18.7641	−0.631106
$885$	0	0
$886$	4.43373	0.148954
$887$	37.5435	1.26059	0.630294	−	0.776357i	$-0.282934\pi$
$887$	37.5435	1.26059	0.630294	+	0.776357i	$0.282934\pi$
$888$	0	0
$889$	29.6212	0.993465
$890$	5.78575	0.193939
$891$	0	0
$892$	19.1251	0.640355
$893$	−4.33948	−0.145215
$894$	0	0
$895$	−11.1460	−0.372569
$896$	−1.60020	−0.0534588
$897$	0	0
$898$	10.5807	0.353082
$899$	−20.7931	−0.693488
$900$	0	0
$901$	4.71662	0.157133
$902$	1.69346	0.0563860
$903$	0	0
$904$	15.4757	0.514714
$905$	−4.50693	−0.149815
$906$	0	0
$907$	−4.17097	−0.138495	−0.0692474	−	0.997600i	$-0.522060\pi$
$907$	−4.17097	−0.138495	−0.0692474	+	0.997600i	$0.522060\pi$
$908$	1.07816	0.0357800
$909$	0	0
$910$	−7.63553	−0.253115
$911$	−47.6865	−1.57992	−0.789962	−	0.613155i	$-0.789900\pi$
$911$	−47.6865	−1.57992	−0.789962	+	0.613155i	$0.789900\pi$
$912$	0	0
$913$	1.73629	0.0574628
$914$	11.1211	0.367853
$915$	0	0
$916$	−6.03461	−0.199389
$917$	−19.2141	−0.634505
$918$	0	0
$919$	−29.3482	−0.968107	−0.484053	−	0.875038i	$-0.660836\pi$
$919$	−29.3482	−0.968107	−0.484053	+	0.875038i	$0.660836\pi$
$920$	4.33225	0.142830
$921$	0	0
$922$	−39.0734	−1.28682
$923$	−34.1485	−1.12401
$924$	0	0
$925$	−2.43937	−0.0802061
$926$	18.3462	0.602894
$927$	0	0
$928$	6.23996	0.204837
$929$	−5.13864	−0.168593	−0.0842966	−	0.996441i	$-0.526864\pi$
$929$	−5.13864	−0.168593	−0.0842966	+	0.996441i	$0.526864\pi$
$930$	0	0
$931$	5.73868	0.188078
$932$	−5.54578	−0.181658
$933$	0	0
$934$	14.0377	0.459328
$935$	3.93244	0.128605
$936$	0	0
$937$	−20.7436	−0.677665	−0.338832	−	0.940847i	$-0.610032\pi$
$937$	−20.7436	−0.677665	−0.338832	+	0.940847i	$0.610032\pi$
$938$	6.03533	0.197061
$939$	0	0
$940$	3.35697	0.109492
$941$	−33.1262	−1.07988	−0.539941	−	0.841703i	$-0.681553\pi$
$941$	−33.1262	−1.07988	−0.539941	+	0.841703i	$0.681553\pi$
$942$	0	0
$943$	−7.33649	−0.238909
$944$	1.26371	0.0411303
$945$	0	0
$946$	−1.34709	−0.0437976
$947$	−21.2645	−0.691004	−0.345502	−	0.938418i	$-0.612291\pi$
$947$	−21.2645	−0.691004	−0.345502	+	0.938418i	$0.612291\pi$
$948$	0	0
$949$	−51.8937	−1.68454
$950$	1.29268	0.0419400
$951$	0	0
$952$	6.29268	0.203947
$953$	13.8508	0.448670	0.224335	−	0.974512i	$-0.427979\pi$
$953$	13.8508	0.448670	0.224335	+	0.974512i	$0.427979\pi$
$954$	0	0
$955$	−14.2805	−0.462106
$956$	8.00636	0.258944
$957$	0	0
$958$	−6.98121	−0.225553
$959$	16.6483	0.537600
$960$	0	0
$961$	−19.8961	−0.641811
$962$	11.6398	0.375281
$963$	0	0
$964$	18.0726	0.582080
$965$	8.18979	0.263639
$966$	0	0
$967$	−32.3170	−1.03924	−0.519622	−	0.854396i	$-0.673927\pi$
$967$	−32.3170	−1.03924	−0.519622	+	0.854396i	$0.673927\pi$
$968$	−1.00000	−0.0321412
$969$	0	0
$970$	−0.425248	−0.0136539
$971$	−36.1086	−1.15878	−0.579391	−	0.815050i	$-0.696709\pi$
$971$	−36.1086	−1.15878	−0.579391	+	0.815050i	$0.696709\pi$
$972$	0	0
$973$	4.97358	0.159446
$974$	−12.5779	−0.403020
$975$	0	0
$976$	−0.878747	−0.0281280
$977$	29.8174	0.953943	0.476971	−	0.878919i	$-0.341735\pi$
$977$	29.8174	0.953943	0.476971	+	0.878919i	$0.341735\pi$
$978$	0	0
$979$	5.78575	0.184913
$980$	−4.43937	−0.141811
$981$	0	0
$982$	0.944023	0.0301250
$983$	36.3472	1.15930	0.579648	−	0.814867i	$-0.303190\pi$
$983$	36.3472	1.15930	0.579648	+	0.814867i	$0.303190\pi$
$984$	0	0
$985$	−8.08903	−0.257738
$986$	−24.5383	−0.781458
$987$	0	0
$988$	−6.16816	−0.196235
$989$	5.83592	0.185571
$990$	0	0
$991$	26.2117	0.832642	0.416321	−	0.909218i	$-0.363319\pi$
$991$	26.2117	0.832642	0.416321	+	0.909218i	$0.363319\pi$
$992$	−3.33225	−0.105799
$993$	0	0
$994$	11.4519	0.363233
$995$	4.45024	0.141082
$996$	0	0
$997$	10.7048	0.339024	0.169512	−	0.985528i	$-0.445781\pi$
$997$	10.7048	0.339024	0.169512	+	0.985528i	$0.445781\pi$
$998$	−0.990376	−0.0313498
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8910.2.a.bs.1.3	✓	4
3.2	odd	2		8910.2.a.bu.1.3	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8910.2.a.bs.1.3	✓	4	1.1	even	1	trivial
8910.2.a.bu.1.3	yes	4	3.2	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 8910 = 2 \cdot 3^{4} \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8910.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +1.60020 q^{7} -1.00000 q^{8} -1.00000 q^{10} -1.00000 q^{11} +4.77162 q^{13} -1.60020 q^{14} +1.00000 q^{16} -3.93244 q^{17} -1.29268 q^{19} +1.00000 q^{20} +1.00000 q^{22} -4.33225 q^{23} +1.00000 q^{25} -4.77162 q^{26} +1.60020 q^{28} -6.23996 q^{29} +3.33225 q^{31} -1.00000 q^{32} +3.93244 q^{34} +1.60020 q^{35} -2.43937 q^{37} +1.29268 q^{38} -1.00000 q^{40} +1.69346 q^{41} -1.34709 q^{43} -1.00000 q^{44} +4.33225 q^{46} +3.35697 q^{47} -4.43937 q^{49} -1.00000 q^{50} +4.77162 q^{52} -1.19941 q^{53} -1.00000 q^{55} -1.60020 q^{56} +6.23996 q^{58} +1.26371 q^{59} -0.878747 q^{61} -3.33225 q^{62} +1.00000 q^{64} +4.77162 q^{65} -3.77162 q^{67} -3.93244 q^{68} -1.60020 q^{70} -7.15658 q^{71} -10.8755 q^{73} +2.43937 q^{74} -1.29268 q^{76} -1.60020 q^{77} -7.41139 q^{79} +1.00000 q^{80} -1.69346 q^{82} -1.73629 q^{83} -3.93244 q^{85} +1.34709 q^{86} +1.00000 q^{88} -5.78575 q^{89} +7.63553 q^{91} -4.33225 q^{92} -3.35697 q^{94} -1.29268 q^{95} +0.425248 q^{97} +4.43937 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{2} + 4 q^{4} + 4 q^{5} + 2 q^{7} - 4 q^{8} - 4 q^{10} - 4 q^{11} + 2 q^{13} - 2 q^{14} + 4 q^{16} - 4 q^{19} + 4 q^{20} + 4 q^{22} - 6 q^{23} + 4 q^{25} - 2 q^{26} + 2 q^{28} - 6 q^{29} + 2 q^{31}+ \cdots + 12 q^{98}+O(q^{100}) \) 4 * q - 4 * q^2 + 4 * q^4 + 4 * q^5 + 2 * q^7 - 4 * q^8 - 4 * q^10 - 4 * q^11 + 2 * q^13 - 2 * q^14 + 4 * q^16 - 4 * q^19 + 4 * q^20 + 4 * q^22 - 6 * q^23 + 4 * q^25 - 2 * q^26 + 2 * q^28 - 6 * q^29 + 2 * q^31 - 4 * q^32 + 2 * q^35 - 4 * q^37 + 4 * q^38 - 4 * q^40 - 12 * q^41 + 8 * q^43 - 4 * q^44 + 6 * q^46 - 6 * q^47 - 12 * q^49 - 4 * q^50 + 2 * q^52 - 18 * q^53 - 4 * q^55 - 2 * q^56 + 6 * q^58 + 8 * q^61 - 2 * q^62 + 4 * q^64 + 2 * q^65 + 2 * q^67 - 2 * q^70 - 18 * q^71 + 2 * q^73 + 4 * q^74 - 4 * q^76 - 2 * q^77 + 2 * q^79 + 4 * q^80 + 12 * q^82 - 12 * q^83 - 8 * q^86 + 4 * q^88 - 12 * q^89 + 4 * q^91 - 6 * q^92 + 6 * q^94 - 4 * q^95 - 10 * q^97 + 12 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more