LMFDB - Embedded newform 3380.2.a.n.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3380,2,Mod(1,3380)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3380, 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("3380.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 3380 = 2^{2} \cdot 5 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3380.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,0,0,0,3,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$26.9894358832$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.756.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 6x - 2 $ x^3 - 6*x - 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 260)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.26180$ of defining polynomial
Character	$\chi$	$=$	3380.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-2.26180 q^{3} +1.00000 q^{5} +1.11575 q^{7} +2.11575 q^{9} +5.37755 q^{11} -2.26180 q^{15} -7.90116 q^{19} -2.52360 q^{21} +6.49330 q^{23} +1.00000 q^{25} +2.00000 q^{27} +3.63935 q^{29} +3.14605 q^{31} -12.1630 q^{33} +1.11575 q^{35} +7.40786 q^{37} +4.75510 q^{41} +4.78541 q^{43} +2.11575 q^{45} -6.16296 q^{47} -5.75510 q^{49} +0.292106 q^{53} +5.37755 q^{55} +17.8709 q^{57} -11.3776 q^{59} -5.34725 q^{61} +2.36065 q^{63} -11.8709 q^{67} -14.6866 q^{69} -3.43816 q^{71} +4.59214 q^{73} -2.26180 q^{75} +6.00000 q^{77} -10.8157 q^{79} -10.8709 q^{81} +7.40786 q^{83} -8.23150 q^{87} +14.8157 q^{89} -7.11575 q^{93} -7.90116 q^{95} +3.76850 q^{97} +11.3776 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{5} + 3 q^{9} + 6 q^{11} + 6 q^{21} + 6 q^{23} + 3 q^{25} + 6 q^{27} - 6 q^{29} + 6 q^{31} - 6 q^{33} + 12 q^{37} - 6 q^{41} - 6 q^{43} + 3 q^{45} + 12 q^{47} + 3 q^{49} - 6 q^{53} + 6 q^{55}+ \cdots + 24 q^{99}+O(q^{100}) $ 3 * q + 3 * q^5 + 3 * q^9 + 6 * q^11 + 6 * q^21 + 6 * q^23 + 3 * q^25 + 6 * q^27 - 6 * q^29 + 6 * q^31 - 6 * q^33 + 12 * q^37 - 6 * q^41 - 6 * q^43 + 3 * q^45 + 12 * q^47 + 3 * q^49 - 6 * q^53 + 6 * q^55 + 30 * q^57 - 24 * q^59 - 6 * q^61 + 24 * q^63 - 12 * q^67 + 24 * q^73 + 18 * q^77 - 12 * q^79 - 9 * q^81 + 12 * q^83 - 18 * q^87 + 24 * q^89 - 18 * q^93 + 18 * q^97 + 24 * q^99

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 $

$2$	0	0
$2$	0	0
$3$	−2.26180	−1.30585	−0.652926	−	0.757422i	$-0.726459\pi$
$3$	−2.26180	−1.30585	−0.652926	+	0.757422i	$0.726459\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	1.11575	0.421714	0.210857	−	0.977517i	$-0.432375\pi$
$7$	1.11575	0.421714	0.210857	+	0.977517i	$0.432375\pi$
$8$	0	0
$9$	2.11575	0.705250
$10$	0	0
$11$	5.37755	1.62139	0.810696	−	0.585467i	$-0.199089\pi$
$11$	5.37755	1.62139	0.810696	+	0.585467i	$0.199089\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	−2.26180	−0.583995
$16$	0	0
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	0	0
$19$	−7.90116	−1.81265	−0.906325	−	0.422582i	$-0.861124\pi$
$19$	−7.90116	−1.81265	−0.906325	+	0.422582i	$0.861124\pi$
$20$	0	0
$21$	−2.52360	−0.550696
$22$	0	0
$23$	6.49330	1.35395	0.676973	−	0.736007i	$-0.263291\pi$
$23$	6.49330	1.35395	0.676973	+	0.736007i	$0.263291\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	2.00000	0.384900
$28$	0	0
$29$	3.63935	0.675811	0.337906	−	0.941180i	$-0.390282\pi$
$29$	3.63935	0.675811	0.337906	+	0.941180i	$0.390282\pi$
$30$	0	0
$31$	3.14605	0.565048	0.282524	−	0.959260i	$-0.408828\pi$
$31$	3.14605	0.565048	0.282524	+	0.959260i	$0.408828\pi$
$32$	0	0
$33$	−12.1630	−2.11730
$34$	0	0
$35$	1.11575	0.188596
$36$	0	0
$37$	7.40786	1.21784	0.608922	−	0.793230i	$-0.291602\pi$
$37$	7.40786	1.21784	0.608922	+	0.793230i	$0.291602\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	4.75510	0.742622	0.371311	−	0.928508i	$-0.378908\pi$
$41$	4.75510	0.742622	0.371311	+	0.928508i	$0.378908\pi$
$42$	0	0
$43$	4.78541	0.729768	0.364884	−	0.931053i	$-0.381109\pi$
$43$	4.78541	0.729768	0.364884	+	0.931053i	$0.381109\pi$
$44$	0	0
$45$	2.11575	0.315397
$46$	0	0
$47$	−6.16296	−0.898960	−0.449480	−	0.893290i	$-0.648391\pi$
$47$	−6.16296	−0.898960	−0.449480	+	0.893290i	$0.648391\pi$
$48$	0	0
$49$	−5.75510	−0.822158
$50$	0	0
$51$	0	0
$52$	0	0
$53$	0.292106	0.0401238	0.0200619	−	0.999799i	$-0.493614\pi$
$53$	0.292106	0.0401238	0.0200619	+	0.999799i	$0.493614\pi$
$54$	0	0
$55$	5.37755	0.725109
$56$	0	0
$57$	17.8709	2.36705
$58$	0	0
$59$	−11.3776	−1.48123	−0.740616	−	0.671929i	$-0.765466\pi$
$59$	−11.3776	−1.48123	−0.740616	+	0.671929i	$0.765466\pi$
$60$	0	0
$61$	−5.34725	−0.684645	−0.342322	−	0.939583i	$-0.611214\pi$
$61$	−5.34725	−0.684645	−0.342322	+	0.939583i	$0.611214\pi$
$62$	0	0
$63$	2.36065	0.297413
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−11.8709	−1.45026	−0.725128	−	0.688614i	$-0.758219\pi$
$67$	−11.8709	−1.45026	−0.725128	+	0.688614i	$0.758219\pi$
$68$	0	0
$69$	−14.6866	−1.76805
$70$	0	0
$71$	−3.43816	−0.408034	−0.204017	−	0.978967i	$-0.565400\pi$
$71$	−3.43816	−0.408034	−0.204017	+	0.978967i	$0.565400\pi$
$72$	0	0
$73$	4.59214	0.537470	0.268735	−	0.963214i	$-0.413394\pi$
$73$	4.59214	0.537470	0.268735	+	0.963214i	$0.413394\pi$
$74$	0	0
$75$	−2.26180	−0.261170
$76$	0	0
$77$	6.00000	0.683763
$78$	0	0
$79$	−10.8157	−1.21686	−0.608431	−	0.793607i	$-0.708201\pi$
$79$	−10.8157	−1.21686	−0.608431	+	0.793607i	$0.708201\pi$
$80$	0	0
$81$	−10.8709	−1.20787
$82$	0	0
$83$	7.40786	0.813118	0.406559	−	0.913625i	$-0.366729\pi$
$83$	7.40786	0.813118	0.406559	+	0.913625i	$0.366729\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−8.23150	−0.882509
$88$	0	0
$89$	14.8157	1.57046	0.785231	−	0.619203i	$-0.212544\pi$
$89$	14.8157	1.57046	0.785231	+	0.619203i	$0.212544\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−7.11575	−0.737869
$94$	0	0
$95$	−7.90116	−0.810642
$96$	0	0
$97$	3.76850	0.382633	0.191317	−	0.981528i	$-0.438724\pi$
$97$	3.76850	0.382633	0.191317	+	0.981528i	$0.438724\pi$
$98$	0	0
$99$	11.3776	1.14349
$100$	0	0
$101$	0.292106	0.0290656	0.0145328	−	0.999894i	$-0.495374\pi$
$101$	0.292106	0.0290656	0.0145328	+	0.999894i	$0.495374\pi$
$102$	0	0
$103$	−4.03030	−0.397118	−0.198559	−	0.980089i	$-0.563626\pi$
$103$	−4.03030	−0.397118	−0.198559	+	0.980089i	$0.563626\pi$
$104$	0	0
$105$	−2.52360	−0.246279
$106$	0	0
$107$	5.50670	0.532353	0.266176	−	0.963924i	$-0.414240\pi$
$107$	5.50670	0.532353	0.266176	+	0.963924i	$0.414240\pi$
$108$	0	0
$109$	−1.27871	−0.122478	−0.0612390	−	0.998123i	$-0.519505\pi$
$109$	−1.27871	−0.122478	−0.0612390	+	0.998123i	$0.519505\pi$
$110$	0	0
$111$	−16.7551	−1.59032
$112$	0	0
$113$	19.2787	1.81359	0.906794	−	0.421574i	$-0.138522\pi$
$113$	19.2787	1.81359	0.906794	+	0.421574i	$0.138522\pi$
$114$	0	0
$115$	6.49330	0.605503
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	17.9181	1.62891
$122$	0	0
$123$	−10.7551	−0.969755
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−7.96970	−0.707196	−0.353598	−	0.935397i	$-0.615042\pi$
$127$	−7.96970	−0.707196	−0.353598	+	0.935397i	$0.615042\pi$
$128$	0	0
$129$	−10.8236	−0.952969
$130$	0	0
$131$	7.27871	0.635944	0.317972	−	0.948100i	$-0.396998\pi$
$131$	7.27871	0.635944	0.317972	+	0.948100i	$0.396998\pi$
$132$	0	0
$133$	−8.81571	−0.764419
$134$	0	0
$135$	2.00000	0.172133
$136$	0	0
$137$	4.75510	0.406256	0.203128	−	0.979152i	$-0.434889\pi$
$137$	4.75510	0.406256	0.203128	+	0.979152i	$0.434889\pi$
$138$	0	0
$139$	−16.3259	−1.38475	−0.692373	−	0.721540i	$-0.743435\pi$
$139$	−16.3259	−1.38475	−0.692373	+	0.721540i	$0.743435\pi$
$140$	0	0
$141$	13.9394	1.17391
$142$	0	0
$143$	0	0
$144$	0	0
$145$	3.63935	0.302232
$146$	0	0
$147$	13.0169	1.07362
$148$	0	0
$149$	8.81571	0.722211	0.361106	−	0.932525i	$-0.382399\pi$
$149$	8.81571	0.722211	0.361106	+	0.932525i	$0.382399\pi$
$150$	0	0
$151$	19.9012	1.61953	0.809767	−	0.586752i	$-0.199594\pi$
$151$	19.9012	1.61953	0.809767	+	0.586752i	$0.199594\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	3.14605	0.252697
$156$	0	0
$157$	18.7551	1.49682	0.748410	−	0.663236i	$-0.230818\pi$
$157$	18.7551	1.49682	0.748410	+	0.663236i	$0.230818\pi$
$158$	0	0
$159$	−0.660685	−0.0523958
$160$	0	0
$161$	7.24490	0.570978
$162$	0	0
$163$	13.4417	1.05283	0.526416	−	0.850227i	$-0.323535\pi$
$163$	13.4417	1.05283	0.526416	+	0.850227i	$0.323535\pi$
$164$	0	0
$165$	−12.1630	−0.946885
$166$	0	0
$167$	−4.91806	−0.380571	−0.190286	−	0.981729i	$-0.560941\pi$
$167$	−4.91806	−0.380571	−0.190286	+	0.981729i	$0.560941\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	−16.7169	−1.27837
$172$	0	0
$173$	19.5708	1.48794	0.743971	−	0.668212i	$-0.232940\pi$
$173$	19.5708	1.48794	0.743971	+	0.668212i	$0.232940\pi$
$174$	0	0
$175$	1.11575	0.0843427
$176$	0	0
$177$	25.7338	1.93427
$178$	0	0
$179$	12.9866	0.970664	0.485332	−	0.874330i	$-0.338699\pi$
$179$	12.9866	0.970664	0.485332	+	0.874330i	$0.338699\pi$
$180$	0	0
$181$	−9.40786	−0.699280	−0.349640	−	0.936884i	$-0.613696\pi$
$181$	−9.40786	−0.699280	−0.349640	+	0.936884i	$0.613696\pi$
$182$	0	0
$183$	12.0944	0.894045
$184$	0	0
$185$	7.40786	0.544636
$186$	0	0
$187$	0	0
$188$	0	0
$189$	2.23150	0.162318
$190$	0	0
$191$	26.5574	1.92163	0.960814	−	0.277195i	$-0.0894049\pi$
$191$	26.5574	1.92163	0.960814	+	0.277195i	$0.0894049\pi$
$192$	0	0
$193$	21.8023	1.56936	0.784682	−	0.619898i	$-0.212826\pi$
$193$	21.8023	1.56936	0.784682	+	0.619898i	$0.212826\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	7.57081	0.539398	0.269699	−	0.962945i	$-0.413076\pi$
$197$	7.57081	0.539398	0.269699	+	0.962945i	$0.413076\pi$
$198$	0	0
$199$	−17.5708	−1.24556	−0.622781	−	0.782396i	$-0.713997\pi$
$199$	−17.5708	−1.24556	−0.622781	+	0.782396i	$0.713997\pi$
$200$	0	0
$201$	26.8495	1.89382
$202$	0	0
$203$	4.06061	0.284999
$204$	0	0
$205$	4.75510	0.332111
$206$	0	0
$207$	13.7382	0.954871
$208$	0	0
$209$	−42.4889	−2.93902
$210$	0	0
$211$	−0.0606069	−0.00417235	−0.00208618	−	0.999998i	$-0.500664\pi$
$211$	−0.0606069	−0.00417235	−0.00208618	+	0.999998i	$0.500664\pi$
$212$	0	0
$213$	7.77643	0.532833
$214$	0	0
$215$	4.78541	0.326362
$216$	0	0
$217$	3.51021	0.238288
$218$	0	0
$219$	−10.3865	−0.701856
$220$	0	0
$221$	0	0
$222$	0	0
$223$	22.6260	1.51515	0.757573	−	0.652750i	$-0.226385\pi$
$223$	22.6260	1.51515	0.757573	+	0.652750i	$0.226385\pi$
$224$	0	0
$225$	2.11575	0.141050
$226$	0	0
$227$	−4.59214	−0.304791	−0.152396	−	0.988320i	$-0.548699\pi$
$227$	−4.59214	−0.304791	−0.152396	+	0.988320i	$0.548699\pi$
$228$	0	0
$229$	−0.986602	−0.0651965	−0.0325983	−	0.999469i	$-0.510378\pi$
$229$	−0.986602	−0.0651965	−0.0325983	+	0.999469i	$0.510378\pi$
$230$	0	0
$231$	−13.5708	−0.892894
$232$	0	0
$233$	−13.2787	−0.869917	−0.434959	−	0.900450i	$-0.643237\pi$
$233$	−13.2787	−0.869917	−0.434959	+	0.900450i	$0.643237\pi$
$234$	0	0
$235$	−6.16296	−0.402027
$236$	0	0
$237$	24.4630	1.58904
$238$	0	0
$239$	25.6429	1.65870	0.829349	−	0.558730i	$-0.188711\pi$
$239$	25.6429	1.65870	0.829349	+	0.558730i	$0.188711\pi$
$240$	0	0
$241$	14.5236	0.935548	0.467774	−	0.883848i	$-0.345056\pi$
$241$	14.5236	0.935548	0.467774	+	0.883848i	$0.345056\pi$
$242$	0	0
$243$	18.5877	1.19240
$244$	0	0
$245$	−5.75510	−0.367680
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−16.7551	−1.06181
$250$	0	0
$251$	−20.2653	−1.27914	−0.639568	−	0.768735i	$-0.720887\pi$
$251$	−20.2653	−1.27914	−0.639568	+	0.768735i	$0.720887\pi$
$252$	0	0
$253$	34.9181	2.19528
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−6.58421	−0.410712	−0.205356	−	0.978687i	$-0.565835\pi$
$257$	−6.58421	−0.410712	−0.205356	+	0.978687i	$0.565835\pi$
$258$	0	0
$259$	8.26531	0.513581
$260$	0	0
$261$	7.69996	0.476616
$262$	0	0
$263$	−18.7854	−1.15836	−0.579179	−	0.815200i	$-0.696627\pi$
$263$	−18.7854	−1.15836	−0.579179	+	0.815200i	$0.696627\pi$
$264$	0	0
$265$	0.292106	0.0179439
$266$	0	0
$267$	−33.5102	−2.05079
$268$	0	0
$269$	−14.8495	−0.905391	−0.452696	−	0.891665i	$-0.649538\pi$
$269$	−14.8495	−0.905391	−0.452696	+	0.891665i	$0.649538\pi$
$270$	0	0
$271$	−20.8878	−1.26884	−0.634420	−	0.772988i	$-0.718761\pi$
$271$	−20.8878	−1.26884	−0.634420	+	0.772988i	$0.718761\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	5.37755	0.324279
$276$	0	0
$277$	−22.2653	−1.33779	−0.668896	−	0.743356i	$-0.733233\pi$
$277$	−22.2653	−1.33779	−0.668896	+	0.743356i	$0.733233\pi$
$278$	0	0
$279$	6.65626	0.398500
$280$	0	0
$281$	7.57081	0.451637	0.225818	−	0.974169i	$-0.427494\pi$
$281$	7.57081	0.451637	0.225818	+	0.974169i	$0.427494\pi$
$282$	0	0
$283$	−0.724800	−0.0430849	−0.0215424	−	0.999768i	$-0.506858\pi$
$283$	−0.724800	−0.0430849	−0.0215424	+	0.999768i	$0.506858\pi$
$284$	0	0
$285$	17.8709	1.05858
$286$	0	0
$287$	5.30550	0.313174
$288$	0	0
$289$	−17.0000	−1.00000
$290$	0	0
$291$	−8.52360	−0.499663
$292$	0	0
$293$	10.2236	0.597267	0.298634	−	0.954368i	$-0.403469\pi$
$293$	10.2236	0.597267	0.298634	+	0.954368i	$0.403469\pi$
$294$	0	0
$295$	−11.3776	−0.662427
$296$	0	0
$297$	10.7551	0.624074
$298$	0	0
$299$	0	0
$300$	0	0
$301$	5.33931	0.307753
$302$	0	0
$303$	−0.660685	−0.0379554
$304$	0	0
$305$	−5.34725	−0.306183
$306$	0	0
$307$	21.6394	1.23502	0.617512	−	0.786562i	$-0.288141\pi$
$307$	21.6394	1.23502	0.617512	+	0.786562i	$0.288141\pi$
$308$	0	0
$309$	9.11575	0.518577
$310$	0	0
$311$	−0.986602	−0.0559451	−0.0279725	−	0.999609i	$-0.508905\pi$
$311$	−0.986602	−0.0559451	−0.0279725	+	0.999609i	$0.508905\pi$
$312$	0	0
$313$	25.8361	1.46034	0.730172	−	0.683263i	$-0.239440\pi$
$313$	25.8361	1.46034	0.730172	+	0.683263i	$0.239440\pi$
$314$	0	0
$315$	2.36065	0.133007
$316$	0	0
$317$	−26.4283	−1.48436	−0.742180	−	0.670201i	$-0.766208\pi$
$317$	−26.4283	−1.48436	−0.742180	+	0.670201i	$0.766208\pi$
$318$	0	0
$319$	19.5708	1.09576
$320$	0	0
$321$	−12.4551	−0.695174
$322$	0	0
$323$	0	0
$324$	0	0
$325$	0	0
$326$	0	0
$327$	2.89218	0.159938
$328$	0	0
$329$	−6.87632	−0.379104
$330$	0	0
$331$	−14.8878	−0.818305	−0.409153	−	0.912466i	$-0.634176\pi$
$331$	−14.8878	−0.818305	−0.409153	+	0.912466i	$0.634176\pi$
$332$	0	0
$333$	15.6732	0.858884
$334$	0	0
$335$	−11.8709	−0.648574
$336$	0	0
$337$	20.0000	1.08947	0.544735	−	0.838608i	$-0.316630\pi$
$337$	20.0000	1.08947	0.544735	+	0.838608i	$0.316630\pi$
$338$	0	0
$339$	−43.6046	−2.36828
$340$	0	0
$341$	16.9181	0.916164
$342$	0	0
$343$	−14.2315	−0.768429
$344$	0	0
$345$	−14.6866	−0.790698
$346$	0	0
$347$	−26.0641	−1.39919	−0.699597	−	0.714537i	$-0.746637\pi$
$347$	−26.0641	−1.39919	−0.699597	+	0.714537i	$0.746637\pi$
$348$	0	0
$349$	−10.7551	−0.575707	−0.287854	−	0.957674i	$-0.592942\pi$
$349$	−10.7551	−0.575707	−0.287854	+	0.957674i	$0.592942\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−35.6126	−1.89547	−0.947733	−	0.319066i	$-0.896631\pi$
$353$	−35.6126	−1.89547	−0.947733	+	0.319066i	$0.896631\pi$
$354$	0	0
$355$	−3.43816	−0.182479
$356$	0	0
$357$	0	0
$358$	0	0
$359$	6.07205	0.320470	0.160235	−	0.987079i	$-0.448775\pi$
$359$	6.07205	0.320470	0.160235	+	0.987079i	$0.448775\pi$
$360$	0	0
$361$	43.4283	2.28570
$362$	0	0
$363$	−40.5271	−2.12712
$364$	0	0
$365$	4.59214	0.240364
$366$	0	0
$367$	26.7854	1.39819	0.699093	−	0.715030i	$-0.253587\pi$
$367$	26.7854	1.39819	0.699093	+	0.715030i	$0.253587\pi$
$368$	0	0
$369$	10.0606	0.523734
$370$	0	0
$371$	0.325917	0.0169208
$372$	0	0
$373$	10.8763	0.563154	0.281577	−	0.959539i	$-0.409142\pi$
$373$	10.8763	0.563154	0.281577	+	0.959539i	$0.409142\pi$
$374$	0	0
$375$	−2.26180	−0.116799
$376$	0	0
$377$	0	0
$378$	0	0
$379$	5.96176	0.306235	0.153118	−	0.988208i	$-0.451069\pi$
$379$	5.96176	0.306235	0.153118	+	0.988208i	$0.451069\pi$
$380$	0	0
$381$	18.0259	0.923494
$382$	0	0
$383$	−19.4079	−0.991695	−0.495848	−	0.868410i	$-0.665143\pi$
$383$	−19.4079	−0.991695	−0.495848	+	0.868410i	$0.665143\pi$
$384$	0	0
$385$	6.00000	0.305788
$386$	0	0
$387$	10.1247	0.514669
$388$	0	0
$389$	8.55742	0.433878	0.216939	−	0.976185i	$-0.430393\pi$
$389$	8.55742	0.433878	0.216939	+	0.976185i	$0.430393\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	−16.4630	−0.830448
$394$	0	0
$395$	−10.8157	−0.544197
$396$	0	0
$397$	14.9449	0.750061	0.375030	−	0.927012i	$-0.377632\pi$
$397$	14.9449	0.750061	0.375030	+	0.927012i	$0.377632\pi$
$398$	0	0
$399$	19.9394	0.998218
$400$	0	0
$401$	27.1416	1.35539	0.677694	−	0.735344i	$-0.262979\pi$
$401$	27.1416	1.35539	0.677694	+	0.735344i	$0.262979\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	−10.8709	−0.540177
$406$	0	0
$407$	39.8361	1.97460
$408$	0	0
$409$	33.2181	1.64253	0.821265	−	0.570547i	$-0.193269\pi$
$409$	33.2181	1.64253	0.821265	+	0.570547i	$0.193269\pi$
$410$	0	0
$411$	−10.7551	−0.530510
$412$	0	0
$413$	−12.6945	−0.624655
$414$	0	0
$415$	7.40786	0.363637
$416$	0	0
$417$	36.9260	1.80827
$418$	0	0
$419$	−17.7079	−0.865087	−0.432544	−	0.901613i	$-0.642384\pi$
$419$	−17.7079	−0.865087	−0.432544	+	0.901613i	$0.642384\pi$
$420$	0	0
$421$	22.2047	1.08219	0.541096	−	0.840961i	$-0.318010\pi$
$421$	22.2047	1.08219	0.541096	+	0.840961i	$0.318010\pi$
$422$	0	0
$423$	−13.0393	−0.633991
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−5.96619	−0.288724
$428$	0	0
$429$	0	0
$430$	0	0
$431$	10.6831	0.514585	0.257292	−	0.966334i	$-0.417170\pi$
$431$	10.6831	0.514585	0.257292	+	0.966334i	$0.417170\pi$
$432$	0	0
$433$	2.00000	0.0961139	0.0480569	−	0.998845i	$-0.484697\pi$
$433$	2.00000	0.0961139	0.0480569	+	0.998845i	$0.484697\pi$
$434$	0	0
$435$	−8.23150	−0.394670
$436$	0	0
$437$	−51.3046	−2.45423
$438$	0	0
$439$	5.24490	0.250325	0.125163	−	0.992136i	$-0.460055\pi$
$439$	5.24490	0.250325	0.125163	+	0.992136i	$0.460055\pi$
$440$	0	0
$441$	−12.1764	−0.579826
$442$	0	0
$443$	25.4799	1.21059	0.605293	−	0.796002i	$-0.293056\pi$
$443$	25.4799	1.21059	0.605293	+	0.796002i	$0.293056\pi$
$444$	0	0
$445$	14.8157	0.702332
$446$	0	0
$447$	−19.9394	−0.943101
$448$	0	0
$449$	13.9394	0.657841	0.328920	−	0.944358i	$-0.393315\pi$
$449$	13.9394	0.657841	0.328920	+	0.944358i	$0.393315\pi$
$450$	0	0
$451$	25.5708	1.20408
$452$	0	0
$453$	−45.0125	−2.11487
$454$	0	0
$455$	0	0
$456$	0	0
$457$	12.6945	0.593823	0.296912	−	0.954905i	$-0.404043\pi$
$457$	12.6945	0.593823	0.296912	+	0.954905i	$0.404043\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−23.0810	−1.07499	−0.537495	−	0.843267i	$-0.680629\pi$
$461$	−23.0810	−1.07499	−0.537495	+	0.843267i	$0.680629\pi$
$462$	0	0
$463$	−6.16296	−0.286417	−0.143208	−	0.989693i	$-0.545742\pi$
$463$	−6.16296	−0.286417	−0.143208	+	0.989693i	$0.545742\pi$
$464$	0	0
$465$	−7.11575	−0.329985
$466$	0	0
$467$	23.5067	1.08776	0.543880	−	0.839163i	$-0.316955\pi$
$467$	23.5067	1.08776	0.543880	+	0.839163i	$0.316955\pi$
$468$	0	0
$469$	−13.2449	−0.611593
$470$	0	0
$471$	−42.4203	−1.95463
$472$	0	0
$473$	25.7338	1.18324
$474$	0	0
$475$	−7.90116	−0.362530
$476$	0	0
$477$	0.618022	0.0282973
$478$	0	0
$479$	33.2137	1.51757	0.758786	−	0.651340i	$-0.225793\pi$
$479$	33.2137	1.51757	0.758786	+	0.651340i	$0.225793\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	−16.3865	−0.745613
$484$	0	0
$485$	3.76850	0.171119
$486$	0	0
$487$	−18.7472	−0.849515	−0.424758	−	0.905307i	$-0.639641\pi$
$487$	−18.7472	−0.849515	−0.424758	+	0.905307i	$0.639641\pi$
$488$	0	0
$489$	−30.4024	−1.37484
$490$	0	0
$491$	−26.5574	−1.19852	−0.599260	−	0.800555i	$-0.704538\pi$
$491$	−26.5574	−1.19852	−0.599260	+	0.800555i	$0.704538\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	11.3776	0.511383
$496$	0	0
$497$	−3.83612	−0.172074
$498$	0	0
$499$	−26.8539	−1.20215	−0.601074	−	0.799193i	$-0.705260\pi$
$499$	−26.8539	−1.20215	−0.601074	+	0.799193i	$0.705260\pi$
$500$	0	0
$501$	11.1237	0.496970
$502$	0	0
$503$	−20.4665	−0.912556	−0.456278	−	0.889837i	$-0.650818\pi$
$503$	−20.4665	−0.912556	−0.456278	+	0.889837i	$0.650818\pi$
$504$	0	0
$505$	0.292106	0.0129985
$506$	0	0
$507$	0	0
$508$	0	0
$509$	26.9598	1.19497	0.597486	−	0.801879i	$-0.296166\pi$
$509$	26.9598	1.19497	0.597486	+	0.801879i	$0.296166\pi$
$510$	0	0
$511$	5.12368	0.226658
$512$	0	0
$513$	−15.8023	−0.697689
$514$	0	0
$515$	−4.03030	−0.177596
$516$	0	0
$517$	−33.1416	−1.45757
$518$	0	0
$519$	−44.2653	−1.94303
$520$	0	0
$521$	−8.36065	−0.366287	−0.183143	−	0.983086i	$-0.558627\pi$
$521$	−8.36065	−0.366287	−0.183143	+	0.983086i	$0.558627\pi$
$522$	0	0
$523$	−18.7248	−0.818778	−0.409389	−	0.912360i	$-0.634258\pi$
$523$	−18.7248	−0.818778	−0.409389	+	0.912360i	$0.634258\pi$
$524$	0	0
$525$	−2.52360	−0.110139
$526$	0	0
$527$	0	0
$528$	0	0
$529$	19.1630	0.833172
$530$	0	0
$531$	−24.0720	−1.04464
$532$	0	0
$533$	0	0
$534$	0	0
$535$	5.50670	0.238075
$536$	0	0
$537$	−29.3731	−1.26754
$538$	0	0
$539$	−30.9484	−1.33304
$540$	0	0
$541$	−42.9439	−1.84630	−0.923152	−	0.384435i	$-0.874396\pi$
$541$	−42.9439	−1.84630	−0.923152	+	0.384435i	$0.874396\pi$
$542$	0	0
$543$	21.2787	0.913157
$544$	0	0
$545$	−1.27871	−0.0547738
$546$	0	0
$547$	14.0909	0.602484	0.301242	−	0.953548i	$-0.402599\pi$
$547$	14.0909	0.602484	0.301242	+	0.953548i	$0.402599\pi$
$548$	0	0
$549$	−11.3134	−0.482846
$550$	0	0
$551$	−28.7551	−1.22501
$552$	0	0
$553$	−12.0676	−0.513167
$554$	0	0
$555$	−16.7551	−0.711215
$556$	0	0
$557$	−0.713359	−0.0302260	−0.0151130	−	0.999886i	$-0.504811\pi$
$557$	−0.713359	−0.0302260	−0.0151130	+	0.999886i	$0.504811\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	12.2012	0.514219	0.257110	−	0.966382i	$-0.417230\pi$
$563$	12.2012	0.514219	0.257110	+	0.966382i	$0.417230\pi$
$564$	0	0
$565$	19.2787	0.811061
$566$	0	0
$567$	−12.1291	−0.509376
$568$	0	0
$569$	−16.6260	−0.696996	−0.348498	−	0.937309i	$-0.613308\pi$
$569$	−16.6260	−0.696996	−0.348498	+	0.937309i	$0.613308\pi$
$570$	0	0
$571$	20.3259	0.850613	0.425307	−	0.905049i	$-0.360166\pi$
$571$	20.3259	0.850613	0.425307	+	0.905049i	$0.360166\pi$
$572$	0	0
$573$	−60.0676	−2.50936
$574$	0	0
$575$	6.49330	0.270789
$576$	0	0
$577$	19.1496	0.797207	0.398603	−	0.917123i	$-0.369495\pi$
$577$	19.1496	0.797207	0.398603	+	0.917123i	$0.369495\pi$
$578$	0	0
$579$	−49.3125	−2.04936
$580$	0	0
$581$	8.26531	0.342903
$582$	0	0
$583$	1.57081	0.0650564
$584$	0	0
$585$	0	0
$586$	0	0
$587$	30.4889	1.25841	0.629205	−	0.777239i	$-0.283380\pi$
$587$	30.4889	1.25841	0.629205	+	0.777239i	$0.283380\pi$
$588$	0	0
$589$	−24.8575	−1.02423
$590$	0	0
$591$	−17.1237	−0.704374
$592$	0	0
$593$	−42.3259	−1.73812	−0.869059	−	0.494709i	$-0.835275\pi$
$593$	−42.3259	−1.73812	−0.869059	+	0.494709i	$0.835275\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	39.7417	1.62652
$598$	0	0
$599$	15.5440	0.635111	0.317556	−	0.948240i	$-0.397138\pi$
$599$	15.5440	0.635111	0.317556	+	0.948240i	$0.397138\pi$
$600$	0	0
$601$	25.1416	1.02555	0.512774	−	0.858524i	$-0.328618\pi$
$601$	25.1416	1.02555	0.512774	+	0.858524i	$0.328618\pi$
$602$	0	0
$603$	−25.1157	−1.02279
$604$	0	0
$605$	17.9181	0.728473
$606$	0	0
$607$	−10.0909	−0.409577	−0.204789	−	0.978806i	$-0.565651\pi$
$607$	−10.0909	−0.409577	−0.204789	+	0.978806i	$0.565651\pi$
$608$	0	0
$609$	−9.18429	−0.372166
$610$	0	0
$611$	0	0
$612$	0	0
$613$	37.0204	1.49524	0.747620	−	0.664127i	$-0.231196\pi$
$613$	37.0204	1.49524	0.747620	+	0.664127i	$0.231196\pi$
$614$	0	0
$615$	−10.7551	−0.433688
$616$	0	0
$617$	8.63389	0.347587	0.173794	−	0.984782i	$-0.444397\pi$
$617$	8.63389	0.347587	0.173794	+	0.984782i	$0.444397\pi$
$618$	0	0
$619$	−9.47197	−0.380711	−0.190355	−	0.981715i	$-0.560964\pi$
$619$	−9.47197	−0.380711	−0.190355	+	0.981715i	$0.560964\pi$
$620$	0	0
$621$	12.9866	0.521134
$622$	0	0
$623$	16.5306	0.662285
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	96.1014	3.83792
$628$	0	0
$629$	0	0
$630$	0	0
$631$	18.0382	0.718091	0.359045	−	0.933320i	$-0.383102\pi$
$631$	18.0382	0.718091	0.359045	+	0.933320i	$0.383102\pi$
$632$	0	0
$633$	0.137081	0.00544847
$634$	0	0
$635$	−7.96970	−0.316268
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−7.27428	−0.287766
$640$	0	0
$641$	−17.4158	−0.687882	−0.343941	−	0.938991i	$-0.611762\pi$
$641$	−17.4158	−0.687882	−0.343941	+	0.938991i	$0.611762\pi$
$642$	0	0
$643$	15.3472	0.605236	0.302618	−	0.953112i	$-0.402139\pi$
$643$	15.3472	0.605236	0.302618	+	0.953112i	$0.402139\pi$
$644$	0	0
$645$	−10.8236	−0.426181
$646$	0	0
$647$	3.05072	0.119936	0.0599680	−	0.998200i	$-0.480900\pi$
$647$	3.05072	0.119936	0.0599680	+	0.998200i	$0.480900\pi$
$648$	0	0
$649$	−61.1834	−2.40166
$650$	0	0
$651$	−7.93939	−0.311169
$652$	0	0
$653$	−25.5708	−1.00066	−0.500332	−	0.865834i	$-0.666789\pi$
$653$	−25.5708	−1.00066	−0.500332	+	0.865834i	$0.666789\pi$
$654$	0	0
$655$	7.27871	0.284403
$656$	0	0
$657$	9.71583	0.379051
$658$	0	0
$659$	6.29211	0.245106	0.122553	−	0.992462i	$-0.460892\pi$
$659$	6.29211	0.245106	0.122553	+	0.992462i	$0.460892\pi$
$660$	0	0
$661$	−39.5102	−1.53677	−0.768384	−	0.639989i	$-0.778939\pi$
$661$	−39.5102	−1.53677	−0.768384	+	0.639989i	$0.778939\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−8.81571	−0.341859
$666$	0	0
$667$	23.6314	0.915012
$668$	0	0
$669$	−51.1754	−1.97856
$670$	0	0
$671$	−28.7551	−1.11008
$672$	0	0
$673$	−1.30550	−0.0503235	−0.0251617	−	0.999683i	$-0.508010\pi$
$673$	−1.30550	−0.0503235	−0.0251617	+	0.999683i	$0.508010\pi$
$674$	0	0
$675$	2.00000	0.0769800
$676$	0	0
$677$	5.01340	0.192681	0.0963403	−	0.995348i	$-0.469286\pi$
$677$	5.01340	0.192681	0.0963403	+	0.995348i	$0.469286\pi$
$678$	0	0
$679$	4.20470	0.161362
$680$	0	0
$681$	10.3865	0.398012
$682$	0	0
$683$	−19.4079	−0.742621	−0.371310	−	0.928509i	$-0.621091\pi$
$683$	−19.4079	−0.742621	−0.371310	+	0.928509i	$0.621091\pi$
$684$	0	0
$685$	4.75510	0.181683
$686$	0	0
$687$	2.23150	0.0851370
$688$	0	0
$689$	0	0
$690$	0	0
$691$	−2.56184	−0.0974570	−0.0487285	−	0.998812i	$-0.515517\pi$
$691$	−2.56184	−0.0974570	−0.0487285	+	0.998812i	$0.515517\pi$
$692$	0	0
$693$	12.6945	0.482224
$694$	0	0
$695$	−16.3259	−0.619277
$696$	0	0
$697$	0	0
$698$	0	0
$699$	30.0338	1.13598
$700$	0	0
$701$	−38.8495	−1.46733	−0.733663	−	0.679513i	$-0.762191\pi$
$701$	−38.8495	−1.46733	−0.733663	+	0.679513i	$0.762191\pi$
$702$	0	0
$703$	−58.5306	−2.20752
$704$	0	0
$705$	13.9394	0.524988
$706$	0	0
$707$	0.325917	0.0122574
$708$	0	0
$709$	35.3393	1.32720	0.663598	−	0.748089i	$-0.269029\pi$
$709$	35.3393	1.32720	0.663598	+	0.748089i	$0.269029\pi$
$710$	0	0
$711$	−22.8833	−0.858192
$712$	0	0
$713$	20.4283	0.765045
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−57.9991	−2.16602
$718$	0	0
$719$	−6.11028	−0.227875	−0.113938	−	0.993488i	$-0.536346\pi$
$719$	−6.11028	−0.227875	−0.113938	+	0.993488i	$0.536346\pi$
$720$	0	0
$721$	−4.49681	−0.167470
$722$	0	0
$723$	−32.8495	−1.22169
$724$	0	0
$725$	3.63935	0.135162
$726$	0	0
$727$	−1.05072	−0.0389689	−0.0194845	−	0.999810i	$-0.506202\pi$
$727$	−1.05072	−0.0389689	−0.0194845	+	0.999810i	$0.506202\pi$
$728$	0	0
$729$	−9.42919	−0.349229
$730$	0	0
$731$	0	0
$732$	0	0
$733$	−31.5708	−1.16609	−0.583047	−	0.812438i	$-0.698140\pi$
$733$	−31.5708	−1.16609	−0.583047	+	0.812438i	$0.698140\pi$
$734$	0	0
$735$	13.0169	0.480136
$736$	0	0
$737$	−63.8361	−2.35143
$738$	0	0
$739$	−31.8673	−1.17226	−0.586130	−	0.810217i	$-0.699349\pi$
$739$	−31.8673	−1.17226	−0.586130	+	0.810217i	$0.699349\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	39.8550	1.46214	0.731069	−	0.682304i	$-0.239022\pi$
$743$	39.8550	1.46214	0.731069	+	0.682304i	$0.239022\pi$
$744$	0	0
$745$	8.81571	0.322983
$746$	0	0
$747$	15.6732	0.573451
$748$	0	0
$749$	6.14410	0.224500
$750$	0	0
$751$	−9.24490	−0.337351	−0.168676	−	0.985672i	$-0.553949\pi$
$751$	−9.24490	−0.337351	−0.168676	+	0.985672i	$0.553949\pi$
$752$	0	0
$753$	45.8361	1.67036
$754$	0	0
$755$	19.9012	0.724277
$756$	0	0
$757$	−22.5912	−0.821092	−0.410546	−	0.911840i	$-0.634662\pi$
$757$	−22.5912	−0.821092	−0.410546	+	0.911840i	$0.634662\pi$
$758$	0	0
$759$	−78.9778	−2.86671
$760$	0	0
$761$	−16.2047	−0.587420	−0.293710	−	0.955895i	$-0.594890\pi$
$761$	−16.2047	−0.587420	−0.293710	+	0.955895i	$0.594890\pi$
$762$	0	0
$763$	−1.42672	−0.0516506
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−16.7889	−0.605424	−0.302712	−	0.953082i	$-0.597892\pi$
$769$	−16.7889	−0.605424	−0.302712	+	0.953082i	$0.597892\pi$
$770$	0	0
$771$	14.8922	0.536329
$772$	0	0
$773$	−16.5921	−0.596778	−0.298389	−	0.954444i	$-0.596449\pi$
$773$	−16.5921	−0.596778	−0.298389	+	0.954444i	$0.596449\pi$
$774$	0	0
$775$	3.14605	0.113010
$776$	0	0
$777$	−18.6945	−0.670661
$778$	0	0
$779$	−37.5708	−1.34611
$780$	0	0
$781$	−18.4889	−0.661584
$782$	0	0
$783$	7.27871	0.260120
$784$	0	0
$785$	18.7551	0.669398
$786$	0	0
$787$	31.1496	1.11036	0.555181	−	0.831730i	$-0.312649\pi$
$787$	31.1496	1.11036	0.555181	+	0.831730i	$0.312649\pi$
$788$	0	0
$789$	42.4889	1.51264
$790$	0	0
$791$	21.5102	0.764815
$792$	0	0
$793$	0	0
$794$	0	0
$795$	−0.660685	−0.0234321
$796$	0	0
$797$	0.402391	0.0142534	0.00712670	−	0.999975i	$-0.497731\pi$
$797$	0.402391	0.0142534	0.00712670	+	0.999975i	$0.497731\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	31.3463	1.10757
$802$	0	0
$803$	24.6945	0.871450
$804$	0	0
$805$	7.24490	0.255349
$806$	0	0
$807$	33.5867	1.18231
$808$	0	0
$809$	−55.3652	−1.94654	−0.973268	−	0.229671i	$-0.926235\pi$
$809$	−55.3652	−1.94654	−0.973268	+	0.229671i	$0.926235\pi$
$810$	0	0
$811$	−27.1461	−0.953227	−0.476613	−	0.879113i	$-0.658136\pi$
$811$	−27.1461	−0.953227	−0.476613	+	0.879113i	$0.658136\pi$
$812$	0	0
$813$	47.2440	1.65692
$814$	0	0
$815$	13.4417	0.470841
$816$	0	0
$817$	−37.8102	−1.32281
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−27.8361	−0.971487	−0.485744	−	0.874101i	$-0.661451\pi$
$821$	−27.8361	−0.971487	−0.485744	+	0.874101i	$0.661451\pi$
$822$	0	0
$823$	−25.2146	−0.878925	−0.439463	−	0.898261i	$-0.644831\pi$
$823$	−25.2146	−0.878925	−0.439463	+	0.898261i	$0.644831\pi$
$824$	0	0
$825$	−12.1630	−0.423460
$826$	0	0
$827$	1.03928	0.0361391	0.0180696	−	0.999837i	$-0.494248\pi$
$827$	1.03928	0.0361391	0.0180696	+	0.999837i	$0.494248\pi$
$828$	0	0
$829$	−6.97867	−0.242379	−0.121190	−	0.992629i	$-0.538671\pi$
$829$	−6.97867	−0.242379	−0.121190	+	0.992629i	$0.538671\pi$
$830$	0	0
$831$	50.3597	1.74696
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−4.91806	−0.170197
$836$	0	0
$837$	6.29211	0.217487
$838$	0	0
$839$	−13.3169	−0.459752	−0.229876	−	0.973220i	$-0.573832\pi$
$839$	−13.3169	−0.459752	−0.229876	+	0.973220i	$0.573832\pi$
$840$	0	0
$841$	−15.7551	−0.543279
$842$	0	0
$843$	−17.1237	−0.589771
$844$	0	0
$845$	0	0
$846$	0	0
$847$	19.9921	0.686936
$848$	0	0
$849$	1.63935	0.0562625
$850$	0	0
$851$	48.1014	1.64890
$852$	0	0
$853$	30.5653	1.04654	0.523269	−	0.852168i	$-0.324712\pi$
$853$	30.5653	1.04654	0.523269	+	0.852168i	$0.324712\pi$
$854$	0	0
$855$	−16.7169	−0.571705
$856$	0	0
$857$	52.4203	1.79064	0.895322	−	0.445419i	$-0.146945\pi$
$857$	52.4203	1.79064	0.895322	+	0.445419i	$0.146945\pi$
$858$	0	0
$859$	19.1416	0.653104	0.326552	−	0.945179i	$-0.394113\pi$
$859$	19.1416	0.653104	0.326552	+	0.945179i	$0.394113\pi$
$860$	0	0
$861$	−12.0000	−0.408959
$862$	0	0
$863$	−36.8575	−1.25464	−0.627321	−	0.778761i	$-0.715849\pi$
$863$	−36.8575	−1.25464	−0.627321	+	0.778761i	$0.715849\pi$
$864$	0	0
$865$	19.5708	0.665428
$866$	0	0
$867$	38.4506	1.30585
$868$	0	0
$869$	−58.1620	−1.97301
$870$	0	0
$871$	0	0
$872$	0	0
$873$	7.97320	0.269852
$874$	0	0
$875$	1.11575	0.0377192
$876$	0	0
$877$	37.5370	1.26753	0.633767	−	0.773524i	$-0.281508\pi$
$877$	37.5370	1.26753	0.633767	+	0.773524i	$0.281508\pi$
$878$	0	0
$879$	−23.1237	−0.779942
$880$	0	0
$881$	15.8212	0.533029	0.266514	−	0.963831i	$-0.414128\pi$
$881$	15.8212	0.533029	0.266514	+	0.963831i	$0.414128\pi$
$882$	0	0
$883$	−49.0507	−1.65069	−0.825344	−	0.564630i	$-0.809019\pi$
$883$	−49.0507	−1.65069	−0.825344	+	0.564630i	$0.809019\pi$
$884$	0	0
$885$	25.7338	0.865031
$886$	0	0
$887$	12.0909	0.405973	0.202987	−	0.979182i	$-0.434935\pi$
$887$	12.0909	0.405973	0.202987	+	0.979182i	$0.434935\pi$
$888$	0	0
$889$	−8.89218	−0.298234
$890$	0	0
$891$	−58.4586	−1.95844
$892$	0	0
$893$	48.6945	1.62950
$894$	0	0
$895$	12.9866	0.434094
$896$	0	0
$897$	0	0
$898$	0	0
$899$	11.4496	0.381866
$900$	0	0
$901$	0	0
$902$	0	0
$903$	−12.0765	−0.401880
$904$	0	0
$905$	−9.40786	−0.312728
$906$	0	0
$907$	−9.54051	−0.316787	−0.158394	−	0.987376i	$-0.550632\pi$
$907$	−9.54051	−0.316787	−0.158394	+	0.987376i	$0.550632\pi$
$908$	0	0
$909$	0.618022	0.0204985
$910$	0	0
$911$	−26.7392	−0.885910	−0.442955	−	0.896544i	$-0.646070\pi$
$911$	−26.7392	−0.885910	−0.442955	+	0.896544i	$0.646070\pi$
$912$	0	0
$913$	39.8361	1.31838
$914$	0	0
$915$	12.0944	0.399829
$916$	0	0
$917$	8.12121	0.268186
$918$	0	0
$919$	51.0810	1.68501	0.842504	−	0.538691i	$-0.181081\pi$
$919$	51.0810	1.68501	0.842504	+	0.538691i	$0.181081\pi$
$920$	0	0
$921$	−48.9439	−1.61276
$922$	0	0
$923$	0	0
$924$	0	0
$925$	7.40786	0.243569
$926$	0	0
$927$	−8.52711	−0.280067
$928$	0	0
$929$	−11.1237	−0.364956	−0.182478	−	0.983210i	$-0.558412\pi$
$929$	−11.1237	−0.364956	−0.182478	+	0.983210i	$0.558412\pi$
$930$	0	0
$931$	45.4720	1.49028
$932$	0	0
$933$	2.23150	0.0730560
$934$	0	0
$935$	0	0
$936$	0	0
$937$	15.3055	0.500009	0.250005	−	0.968245i	$-0.419568\pi$
$937$	15.3055	0.500009	0.250005	+	0.968245i	$0.419568\pi$
$938$	0	0
$939$	−58.4362	−1.90699
$940$	0	0
$941$	−28.7124	−0.935999	−0.467999	−	0.883729i	$-0.655025\pi$
$941$	−28.7124	−0.935999	−0.467999	+	0.883729i	$0.655025\pi$
$942$	0	0
$943$	30.8763	1.00547
$944$	0	0
$945$	2.23150	0.0725907
$946$	0	0
$947$	−14.1024	−0.458265	−0.229132	−	0.973395i	$-0.573589\pi$
$947$	−14.1024	−0.458265	−0.229132	+	0.973395i	$0.573589\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	59.7755	1.93835
$952$	0	0
$953$	−15.8361	−0.512982	−0.256491	−	0.966547i	$-0.582566\pi$
$953$	−15.8361	−0.512982	−0.256491	+	0.966547i	$0.582566\pi$
$954$	0	0
$955$	26.5574	0.859378
$956$	0	0
$957$	−44.2653	−1.43089
$958$	0	0
$959$	5.30550	0.171324
$960$	0	0
$961$	−21.1024	−0.680721
$962$	0	0
$963$	11.6508	0.375442
$964$	0	0
$965$	21.8023	0.701841
$966$	0	0
$967$	−0.713359	−0.0229401	−0.0114700	−	0.999934i	$-0.503651\pi$
$967$	−0.713359	−0.0229401	−0.0114700	+	0.999934i	$0.503651\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−11.8182	−0.379263	−0.189632	−	0.981855i	$-0.560729\pi$
$971$	−11.8182	−0.379263	−0.189632	+	0.981855i	$0.560729\pi$
$972$	0	0
$973$	−18.2156	−0.583966
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−16.9181	−0.541257	−0.270628	−	0.962684i	$-0.587231\pi$
$977$	−16.9181	−0.541257	−0.270628	+	0.962684i	$0.587231\pi$
$978$	0	0
$979$	79.6722	2.54634
$980$	0	0
$981$	−2.70543	−0.0863776
$982$	0	0
$983$	−12.5315	−0.399694	−0.199847	−	0.979827i	$-0.564045\pi$
$983$	−12.5315	−0.399694	−0.199847	+	0.979827i	$0.564045\pi$
$984$	0	0
$985$	7.57081	0.241226
$986$	0	0
$987$	15.5529	0.495053
$988$	0	0
$989$	31.0731	0.988067
$990$	0	0
$991$	−49.6543	−1.57732	−0.788660	−	0.614829i	$-0.789225\pi$
$991$	−49.6543	−1.57732	−0.788660	+	0.614829i	$0.789225\pi$
$992$	0	0
$993$	33.6732	1.06859
$994$	0	0
$995$	−17.5708	−0.557032
$996$	0	0
$997$	−54.5912	−1.72892	−0.864461	−	0.502700i	$-0.832340\pi$
$997$	−54.5912	−1.72892	−0.864461	+	0.502700i	$0.832340\pi$
$998$	0	0
$999$	14.8157	0.468748

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3380.2.a.n.1.1		3
13.5	odd	4		260.2.f.a.181.1	✓	6
13.8	odd	4		260.2.f.a.181.2	yes	6
13.12	even	2		3380.2.a.m.1.1		3
39.5	even	4		2340.2.c.d.181.5		6
39.8	even	4		2340.2.c.d.181.2		6
52.31	even	4		1040.2.k.c.961.5		6
52.47	even	4		1040.2.k.c.961.6		6
65.8	even	4		1300.2.d.c.649.2		6
65.18	even	4		1300.2.d.d.649.2		6
65.34	odd	4		1300.2.f.e.701.6		6
65.44	odd	4		1300.2.f.e.701.5		6
65.47	even	4		1300.2.d.d.649.5		6
65.57	even	4		1300.2.d.c.649.5		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
260.2.f.a.181.1	✓	6	13.5	odd	4
260.2.f.a.181.2	yes	6	13.8	odd	4
1040.2.k.c.961.5		6	52.31	even	4
1040.2.k.c.961.6		6	52.47	even	4
1300.2.d.c.649.2		6	65.8	even	4
1300.2.d.c.649.5		6	65.57	even	4
1300.2.d.d.649.2		6	65.18	even	4
1300.2.d.d.649.5		6	65.47	even	4
1300.2.f.e.701.5		6	65.44	odd	4
1300.2.f.e.701.6		6	65.34	odd	4
2340.2.c.d.181.2		6	39.8	even	4
2340.2.c.d.181.5		6	39.5	even	4
3380.2.a.m.1.1		3	13.12	even	2
3380.2.a.n.1.1		3	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 3380 = 2^{2} \cdot 5 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3380.a (trivial)

\(f(q)\)	\(=\)	\(q-2.26180 q^{3} +1.00000 q^{5} +1.11575 q^{7} +2.11575 q^{9} +5.37755 q^{11} -2.26180 q^{15} -7.90116 q^{19} -2.52360 q^{21} +6.49330 q^{23} +1.00000 q^{25} +2.00000 q^{27} +3.63935 q^{29} +3.14605 q^{31} -12.1630 q^{33} +1.11575 q^{35} +7.40786 q^{37} +4.75510 q^{41} +4.78541 q^{43} +2.11575 q^{45} -6.16296 q^{47} -5.75510 q^{49} +0.292106 q^{53} +5.37755 q^{55} +17.8709 q^{57} -11.3776 q^{59} -5.34725 q^{61} +2.36065 q^{63} -11.8709 q^{67} -14.6866 q^{69} -3.43816 q^{71} +4.59214 q^{73} -2.26180 q^{75} +6.00000 q^{77} -10.8157 q^{79} -10.8709 q^{81} +7.40786 q^{83} -8.23150 q^{87} +14.8157 q^{89} -7.11575 q^{93} -7.90116 q^{95} +3.76850 q^{97} +11.3776 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{5} + 3 q^{9} + 6 q^{11} + 6 q^{21} + 6 q^{23} + 3 q^{25} + 6 q^{27} - 6 q^{29} + 6 q^{31} - 6 q^{33} + 12 q^{37} - 6 q^{41} - 6 q^{43} + 3 q^{45} + 12 q^{47} + 3 q^{49} - 6 q^{53} + 6 q^{55}+ \cdots + 24 q^{99}+O(q^{100}) \) 3 * q + 3 * q^5 + 3 * q^9 + 6 * q^11 + 6 * q^21 + 6 * q^23 + 3 * q^25 + 6 * q^27 - 6 * q^29 + 6 * q^31 - 6 * q^33 + 12 * q^37 - 6 * q^41 - 6 * q^43 + 3 * q^45 + 12 * q^47 + 3 * q^49 - 6 * q^53 + 6 * q^55 + 30 * q^57 - 24 * q^59 - 6 * q^61 + 24 * q^63 - 12 * q^67 + 24 * q^73 + 18 * q^77 - 12 * q^79 - 9 * q^81 + 12 * q^83 - 18 * q^87 + 24 * q^89 - 18 * q^93 + 18 * q^97 + 24 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more