LMFDB - Embedded newform 4900.2.a.bg.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4900,2,Mod(1,4900)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("4900.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4900, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 4900 = 2^{2} \cdot 5^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4900.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,0,0,0,0,0,0,-4,0,0,0,0,0,0,0,0,0,0,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(23)] == traces)

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

Self dual:	yes
Analytic conductor:	$39.1266969904$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{2}, \sqrt{5})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 6x^{2} + 4 $ x^4 - 6*x^2 + 4
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			1.4
Root			$2.28825$ of defining polynomial
Character	$\chi$	$=$	4900.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.28825 q^{3} +2.23607 q^{9} -5.47214 q^{11} -0.874032 q^{13} +4.57649 q^{17} -5.99070 q^{19} +3.47214 q^{23} -1.74806 q^{27} +0.236068 q^{29} -8.27895 q^{31} -12.5216 q^{33} -4.23607 q^{37} -2.00000 q^{39} -5.11667 q^{41} -3.76393 q^{43} +4.91034 q^{47} +10.4721 q^{51} -11.7082 q^{53} -13.7082 q^{57} +1.95440 q^{59} -7.53244 q^{61} +13.9443 q^{67} +7.94510 q^{69} +16.7082 q^{71} -7.53244 q^{73} -11.4721 q^{79} -10.7082 q^{81} +12.1877 q^{83} +0.540182 q^{87} -5.86319 q^{89} -18.9443 q^{93} +16.3516 q^{97} -12.2361 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{11} - 4 q^{23} - 8 q^{29} - 8 q^{37} - 8 q^{39} - 24 q^{43} + 24 q^{51} - 20 q^{53} - 28 q^{57} + 20 q^{67} + 40 q^{71} - 28 q^{79} - 16 q^{81} - 40 q^{93} - 40 q^{99}+O(q^{100}) $ 4 * q - 4 * q^11 - 4 * q^23 - 8 * q^29 - 8 * q^37 - 8 * q^39 - 24 * q^43 + 24 * q^51 - 20 * q^53 - 28 * q^57 + 20 * q^67 + 40 * q^71 - 28 * q^79 - 16 * q^81 - 40 * q^93 - 40 * 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
$3$	2.28825	1.32112	0.660560	−	0.750774i	$-0.270319\pi$
$3$	2.28825	1.32112	0.660560	+	0.750774i	$0.270319\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	2.23607	0.745356
$10$	0	0
$11$	−5.47214	−1.64991	−0.824956	−	0.565198i	$-0.808800\pi$
$11$	−5.47214	−1.64991	−0.824956	+	0.565198i	$0.808800\pi$
$12$	0	0
$13$	−0.874032	−0.242413	−0.121206	−	0.992627i	$-0.538676\pi$
$13$	−0.874032	−0.242413	−0.121206	+	0.992627i	$0.538676\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	4.57649	1.10996	0.554981	−	0.831863i	$-0.312725\pi$
$17$	4.57649	1.10996	0.554981	+	0.831863i	$0.312725\pi$
$18$	0	0
$19$	−5.99070	−1.37436	−0.687181	−	0.726486i	$-0.741152\pi$
$19$	−5.99070	−1.37436	−0.687181	+	0.726486i	$0.741152\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	3.47214	0.723990	0.361995	−	0.932180i	$-0.382096\pi$
$23$	3.47214	0.723990	0.361995	+	0.932180i	$0.382096\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−1.74806	−0.336415
$28$	0	0
$29$	0.236068	0.0438367	0.0219184	−	0.999760i	$-0.493023\pi$
$29$	0.236068	0.0438367	0.0219184	+	0.999760i	$0.493023\pi$
$30$	0	0
$31$	−8.27895	−1.48694	−0.743472	−	0.668767i	$-0.766822\pi$
$31$	−8.27895	−1.48694	−0.743472	+	0.668767i	$0.766822\pi$
$32$	0	0
$33$	−12.5216	−2.17973
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−4.23607	−0.696405	−0.348203	−	0.937419i	$-0.613208\pi$
$37$	−4.23607	−0.696405	−0.348203	+	0.937419i	$0.613208\pi$
$38$	0	0
$39$	−2.00000	−0.320256
$40$	0	0
$41$	−5.11667	−0.799090	−0.399545	−	0.916714i	$-0.630832\pi$
$41$	−5.11667	−0.799090	−0.399545	+	0.916714i	$0.630832\pi$
$42$	0	0
$43$	−3.76393	−0.573994	−0.286997	−	0.957931i	$-0.592657\pi$
$43$	−3.76393	−0.573994	−0.286997	+	0.957931i	$0.592657\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	4.91034	0.716247	0.358123	−	0.933674i	$-0.383417\pi$
$47$	4.91034	0.716247	0.358123	+	0.933674i	$0.383417\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	10.4721	1.46639
$52$	0	0
$53$	−11.7082	−1.60825	−0.804123	−	0.594463i	$-0.797365\pi$
$53$	−11.7082	−1.60825	−0.804123	+	0.594463i	$0.797365\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−13.7082	−1.81570
$58$	0	0
$59$	1.95440	0.254441	0.127220	−	0.991874i	$-0.459394\pi$
$59$	1.95440	0.254441	0.127220	+	0.991874i	$0.459394\pi$
$60$	0	0
$61$	−7.53244	−0.964430	−0.482215	−	0.876053i	$-0.660168\pi$
$61$	−7.53244	−0.964430	−0.482215	+	0.876053i	$0.660168\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	13.9443	1.70356	0.851782	−	0.523896i	$-0.175522\pi$
$67$	13.9443	1.70356	0.851782	+	0.523896i	$0.175522\pi$
$68$	0	0
$69$	7.94510	0.956478
$70$	0	0
$71$	16.7082	1.98290	0.991449	−	0.130491i	$-0.0416554\pi$
$71$	16.7082	1.98290	0.991449	+	0.130491i	$0.0416554\pi$
$72$	0	0
$73$	−7.53244	−0.881605	−0.440803	−	0.897604i	$-0.645306\pi$
$73$	−7.53244	−0.881605	−0.440803	+	0.897604i	$0.645306\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−11.4721	−1.29072	−0.645358	−	0.763880i	$-0.723292\pi$
$79$	−11.4721	−1.29072	−0.645358	+	0.763880i	$0.723292\pi$
$80$	0	0
$81$	−10.7082	−1.18980
$82$	0	0
$83$	12.1877	1.33778	0.668889	−	0.743362i	$-0.266770\pi$
$83$	12.1877	1.33778	0.668889	+	0.743362i	$0.266770\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0.540182	0.0579135
$88$	0	0
$89$	−5.86319	−0.621496	−0.310748	−	0.950492i	$-0.600580\pi$
$89$	−5.86319	−0.621496	−0.310748	+	0.950492i	$0.600580\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−18.9443	−1.96443
$94$	0	0
$95$	0	0
$96$	0	0
$97$	16.3516	1.66025	0.830125	−	0.557577i	$-0.188269\pi$
$97$	16.3516	1.66025	0.830125	+	0.557577i	$0.188269\pi$
$98$	0	0
$99$	−12.2361	−1.22977
$100$	0	0
$101$	−3.36861	−0.335189	−0.167595	−	0.985856i	$-0.553600\pi$
$101$	−3.36861	−0.335189	−0.167595	+	0.985856i	$0.553600\pi$
$102$	0	0
$103$	−18.3848	−1.81151	−0.905753	−	0.423806i	$-0.860694\pi$
$103$	−18.3848	−1.81151	−0.905753	+	0.423806i	$0.860694\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−12.4721	−1.20573	−0.602863	−	0.797844i	$-0.705974\pi$
$107$	−12.4721	−1.20573	−0.602863	+	0.797844i	$0.705974\pi$
$108$	0	0
$109$	−16.4164	−1.57241	−0.786203	−	0.617968i	$-0.787956\pi$
$109$	−16.4164	−1.57241	−0.786203	+	0.617968i	$0.787956\pi$
$110$	0	0
$111$	−9.69316	−0.920034
$112$	0	0
$113$	−13.7639	−1.29480	−0.647401	−	0.762150i	$-0.724144\pi$
$113$	−13.7639	−1.29480	−0.647401	+	0.762150i	$0.724144\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−1.95440	−0.180684
$118$	0	0
$119$	0	0
$120$	0	0
$121$	18.9443	1.72221
$122$	0	0
$123$	−11.7082	−1.05569
$124$	0	0
$125$	0	0
$126$	0	0
$127$	11.1803	0.992095	0.496047	−	0.868295i	$-0.334784\pi$
$127$	11.1803	0.992095	0.496047	+	0.868295i	$0.334784\pi$
$128$	0	0
$129$	−8.61280	−0.758315
$130$	0	0
$131$	2.16073	0.188784	0.0943918	−	0.995535i	$-0.469909\pi$
$131$	2.16073	0.188784	0.0943918	+	0.995535i	$0.469909\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−10.0000	−0.854358	−0.427179	−	0.904167i	$-0.640493\pi$
$137$	−10.0000	−0.854358	−0.427179	+	0.904167i	$0.640493\pi$
$138$	0	0
$139$	−10.3609	−0.878797	−0.439399	−	0.898292i	$-0.644808\pi$
$139$	−10.3609	−0.878797	−0.439399	+	0.898292i	$0.644808\pi$
$140$	0	0
$141$	11.2361	0.946248
$142$	0	0
$143$	4.78282	0.399960
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	6.70820	0.549557	0.274779	−	0.961507i	$-0.411395\pi$
$149$	6.70820	0.549557	0.274779	+	0.961507i	$0.411395\pi$
$150$	0	0
$151$	8.23607	0.670242	0.335121	−	0.942175i	$-0.391223\pi$
$151$	8.23607	0.670242	0.335121	+	0.942175i	$0.391223\pi$
$152$	0	0
$153$	10.2333	0.827317
$154$	0	0
$155$	0	0
$156$	0	0
$157$	9.02546	0.720310	0.360155	−	0.932892i	$-0.382724\pi$
$157$	9.02546	0.720310	0.360155	+	0.932892i	$0.382724\pi$
$158$	0	0
$159$	−26.7912	−2.12468
$160$	0	0
$161$	0	0
$162$	0	0
$163$	2.18034	0.170777	0.0853887	−	0.996348i	$-0.472787\pi$
$163$	2.18034	0.170777	0.0853887	+	0.996348i	$0.472787\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	4.11512	0.318438	0.159219	−	0.987243i	$-0.449102\pi$
$167$	4.11512	0.318438	0.159219	+	0.987243i	$0.449102\pi$
$168$	0	0
$169$	−12.2361	−0.941236
$170$	0	0
$171$	−13.3956	−1.02439
$172$	0	0
$173$	−18.3848	−1.39777	−0.698884	−	0.715235i	$-0.746320\pi$
$173$	−18.3848	−1.39777	−0.698884	+	0.715235i	$0.746320\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	4.47214	0.336146
$178$	0	0
$179$	1.70820	0.127677	0.0638386	−	0.997960i	$-0.479666\pi$
$179$	1.70820	0.127677	0.0638386	+	0.997960i	$0.479666\pi$
$180$	0	0
$181$	3.62365	0.269344	0.134672	−	0.990890i	$-0.457002\pi$
$181$	3.62365	0.269344	0.134672	+	0.990890i	$0.457002\pi$
$182$	0	0
$183$	−17.2361	−1.27413
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−25.0432	−1.83134
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−12.1803	−0.881338	−0.440669	−	0.897670i	$-0.645259\pi$
$191$	−12.1803	−0.881338	−0.440669	+	0.897670i	$0.645259\pi$
$192$	0	0
$193$	15.6525	1.12669	0.563345	−	0.826222i	$-0.309514\pi$
$193$	15.6525	1.12669	0.563345	+	0.826222i	$0.309514\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−19.4721	−1.38733	−0.693666	−	0.720297i	$-0.744006\pi$
$197$	−19.4721	−1.38733	−0.693666	+	0.720297i	$0.744006\pi$
$198$	0	0
$199$	21.6746	1.53647	0.768235	−	0.640168i	$-0.221135\pi$
$199$	21.6746	1.53647	0.768235	+	0.640168i	$0.221135\pi$
$200$	0	0
$201$	31.9079	2.25061
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	7.76393	0.539631
$208$	0	0
$209$	32.7820	2.26757
$210$	0	0
$211$	9.52786	0.655925	0.327963	−	0.944691i	$-0.393638\pi$
$211$	9.52786	0.655925	0.327963	+	0.944691i	$0.393638\pi$
$212$	0	0
$213$	38.2325	2.61965
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−17.2361	−1.16471
$220$	0	0
$221$	−4.00000	−0.269069
$222$	0	0
$223$	11.8539	0.793795	0.396898	−	0.917863i	$-0.370087\pi$
$223$	11.8539	0.793795	0.396898	+	0.917863i	$0.370087\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−6.40337	−0.425006	−0.212503	−	0.977160i	$-0.568162\pi$
$227$	−6.40337	−0.425006	−0.212503	+	0.977160i	$0.568162\pi$
$228$	0	0
$229$	5.57804	0.368607	0.184304	−	0.982869i	$-0.440997\pi$
$229$	5.57804	0.368607	0.184304	+	0.982869i	$0.440997\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−10.5279	−0.689703	−0.344852	−	0.938657i	$-0.612071\pi$
$233$	−10.5279	−0.689703	−0.344852	+	0.938657i	$0.612071\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−26.2511	−1.70519
$238$	0	0
$239$	8.65248	0.559682	0.279841	−	0.960046i	$-0.409718\pi$
$239$	8.65248	0.559682	0.279841	+	0.960046i	$0.409718\pi$
$240$	0	0
$241$	1.41421	0.0910975	0.0455488	−	0.998962i	$-0.485496\pi$
$241$	1.41421	0.0910975	0.0455488	+	0.998962i	$0.485496\pi$
$242$	0	0
$243$	−19.2588	−1.23545
$244$	0	0
$245$	0	0
$246$	0	0
$247$	5.23607	0.333163
$248$	0	0
$249$	27.8885	1.76736
$250$	0	0
$251$	−1.74806	−0.110337	−0.0551684	−	0.998477i	$-0.517570\pi$
$251$	−1.74806	−0.110337	−0.0551684	+	0.998477i	$0.517570\pi$
$252$	0	0
$253$	−19.0000	−1.19452
$254$	0	0
$255$	0	0
$256$	0	0
$257$	8.27895	0.516427	0.258213	−	0.966088i	$-0.416866\pi$
$257$	8.27895	0.516427	0.258213	+	0.966088i	$0.416866\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0.527864	0.0326740
$262$	0	0
$263$	10.7082	0.660296	0.330148	−	0.943929i	$-0.392901\pi$
$263$	10.7082	0.660296	0.330148	+	0.943929i	$0.392901\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−13.4164	−0.821071
$268$	0	0
$269$	32.2418	1.96582	0.982908	−	0.184099i	$-0.0589367\pi$
$269$	32.2418	1.96582	0.982908	+	0.184099i	$0.0589367\pi$
$270$	0	0
$271$	24.8369	1.50873	0.754366	−	0.656454i	$-0.227945\pi$
$271$	24.8369	1.50873	0.754366	+	0.656454i	$0.227945\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−20.9443	−1.25842	−0.629210	−	0.777236i	$-0.716621\pi$
$277$	−20.9443	−1.25842	−0.629210	+	0.777236i	$0.716621\pi$
$278$	0	0
$279$	−18.5123	−1.10830
$280$	0	0
$281$	−24.1246	−1.43915	−0.719577	−	0.694413i	$-0.755664\pi$
$281$	−24.1246	−1.43915	−0.719577	+	0.694413i	$0.755664\pi$
$282$	0	0
$283$	10.5672	0.628155	0.314077	−	0.949397i	$-0.398305\pi$
$283$	10.5672	0.628155	0.314077	+	0.949397i	$0.398305\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	3.94427	0.232016
$290$	0	0
$291$	37.4164	2.19339
$292$	0	0
$293$	−19.8477	−1.15951	−0.579757	−	0.814789i	$-0.696853\pi$
$293$	−19.8477	−1.15951	−0.579757	+	0.814789i	$0.696853\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	9.56564	0.555055
$298$	0	0
$299$	−3.03476	−0.175505
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−7.70820	−0.442825
$304$	0	0
$305$	0	0
$306$	0	0
$307$	18.7186	1.06833	0.534164	−	0.845381i	$-0.320626\pi$
$307$	18.7186	1.06833	0.534164	+	0.845381i	$0.320626\pi$
$308$	0	0
$309$	−42.0689	−2.39322
$310$	0	0
$311$	9.40802	0.533480	0.266740	−	0.963769i	$-0.414054\pi$
$311$	9.40802	0.533480	0.266740	+	0.963769i	$0.414054\pi$
$312$	0	0
$313$	−3.57494	−0.202068	−0.101034	−	0.994883i	$-0.532215\pi$
$313$	−3.57494	−0.202068	−0.101034	+	0.994883i	$0.532215\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	0.416408	0.0233878	0.0116939	−	0.999932i	$-0.496278\pi$
$317$	0.416408	0.0233878	0.0116939	+	0.999932i	$0.496278\pi$
$318$	0	0
$319$	−1.29180	−0.0723267
$320$	0	0
$321$	−28.5393	−1.59291
$322$	0	0
$323$	−27.4164	−1.52549
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−37.5648	−2.07734
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−17.3607	−0.954229	−0.477115	−	0.878841i	$-0.658317\pi$
$331$	−17.3607	−0.954229	−0.477115	+	0.878841i	$0.658317\pi$
$332$	0	0
$333$	−9.47214	−0.519070
$334$	0	0
$335$	0	0
$336$	0	0
$337$	17.1246	0.932837	0.466419	−	0.884564i	$-0.345544\pi$
$337$	17.1246	0.932837	0.466419	+	0.884564i	$0.345544\pi$
$338$	0	0
$339$	−31.4953	−1.71059
$340$	0	0
$341$	45.3035	2.45332
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−25.3607	−1.36143	−0.680716	−	0.732547i	$-0.738331\pi$
$347$	−25.3607	−1.36143	−0.680716	+	0.732547i	$0.738331\pi$
$348$	0	0
$349$	34.9427	1.87044	0.935219	−	0.354069i	$-0.115202\pi$
$349$	34.9427	1.87044	0.935219	+	0.354069i	$0.115202\pi$
$350$	0	0
$351$	1.52786	0.0815513
$352$	0	0
$353$	−11.7751	−0.626724	−0.313362	−	0.949634i	$-0.601455\pi$
$353$	−11.7751	−0.626724	−0.313362	+	0.949634i	$0.601455\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−29.0689	−1.53420	−0.767099	−	0.641529i	$-0.778300\pi$
$359$	−29.0689	−1.53420	−0.767099	+	0.641529i	$0.778300\pi$
$360$	0	0
$361$	16.8885	0.888871
$362$	0	0
$363$	43.3491	2.27524
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−5.78437	−0.301942	−0.150971	−	0.988538i	$-0.548240\pi$
$367$	−5.78437	−0.301942	−0.150971	+	0.988538i	$0.548240\pi$
$368$	0	0
$369$	−11.4412	−0.595607
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−30.1246	−1.55979	−0.779897	−	0.625908i	$-0.784728\pi$
$373$	−30.1246	−1.55979	−0.779897	+	0.625908i	$0.784728\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−0.206331	−0.0106266
$378$	0	0
$379$	10.1246	0.520066	0.260033	−	0.965600i	$-0.416266\pi$
$379$	10.1246	0.520066	0.260033	+	0.965600i	$0.416266\pi$
$380$	0	0
$381$	25.5834	1.31068
$382$	0	0
$383$	21.5958	1.10349	0.551746	−	0.834012i	$-0.313962\pi$
$383$	21.5958	1.10349	0.551746	+	0.834012i	$0.313962\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−8.41641	−0.427830
$388$	0	0
$389$	−3.18034	−0.161250	−0.0806248	−	0.996745i	$-0.525692\pi$
$389$	−3.18034	−0.161250	−0.0806248	+	0.996745i	$0.525692\pi$
$390$	0	0
$391$	15.8902	0.803602
$392$	0	0
$393$	4.94427	0.249406
$394$	0	0
$395$	0	0
$396$	0	0
$397$	7.73877	0.388398	0.194199	−	0.980962i	$-0.437789\pi$
$397$	7.73877	0.388398	0.194199	+	0.980962i	$0.437789\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	0.708204	0.0353660	0.0176830	−	0.999844i	$-0.494371\pi$
$401$	0.708204	0.0353660	0.0176830	+	0.999844i	$0.494371\pi$
$402$	0	0
$403$	7.23607	0.360454
$404$	0	0
$405$	0	0
$406$	0	0
$407$	23.1803	1.14901
$408$	0	0
$409$	−9.89949	−0.489499	−0.244749	−	0.969586i	$-0.578706\pi$
$409$	−9.89949	−0.489499	−0.244749	+	0.969586i	$0.578706\pi$
$410$	0	0
$411$	−22.8825	−1.12871
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−23.7082	−1.16100
$418$	0	0
$419$	−22.4211	−1.09534	−0.547671	−	0.836694i	$-0.684485\pi$
$419$	−22.4211	−1.09534	−0.547671	+	0.836694i	$0.684485\pi$
$420$	0	0
$421$	30.5967	1.49119	0.745597	−	0.666397i	$-0.232164\pi$
$421$	30.5967	1.49119	0.745597	+	0.666397i	$0.232164\pi$
$422$	0	0
$423$	10.9799	0.533859
$424$	0	0
$425$	0	0
$426$	0	0
$427$	0	0
$428$	0	0
$429$	10.9443	0.528394
$430$	0	0
$431$	14.6525	0.705785	0.352892	−	0.935664i	$-0.385198\pi$
$431$	14.6525	0.705785	0.352892	+	0.935664i	$0.385198\pi$
$432$	0	0
$433$	2.41577	0.116094	0.0580471	−	0.998314i	$-0.481513\pi$
$433$	2.41577	0.116094	0.0580471	+	0.998314i	$0.481513\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−20.8005	−0.995025
$438$	0	0
$439$	23.7565	1.13384	0.566918	−	0.823774i	$-0.308136\pi$
$439$	23.7565	1.13384	0.566918	+	0.823774i	$0.308136\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−39.1246	−1.85887	−0.929433	−	0.368990i	$-0.879704\pi$
$443$	−39.1246	−1.85887	−0.929433	+	0.368990i	$0.879704\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	15.3500	0.726031
$448$	0	0
$449$	9.76393	0.460788	0.230394	−	0.973097i	$-0.425998\pi$
$449$	9.76393	0.460788	0.230394	+	0.973097i	$0.425998\pi$
$450$	0	0
$451$	27.9991	1.31843
$452$	0	0
$453$	18.8461	0.885469
$454$	0	0
$455$	0	0
$456$	0	0
$457$	25.6525	1.19997	0.599986	−	0.800010i	$-0.295173\pi$
$457$	25.6525	1.19997	0.599986	+	0.800010i	$0.295173\pi$
$458$	0	0
$459$	−8.00000	−0.373408
$460$	0	0
$461$	17.0193	0.792666	0.396333	−	0.918107i	$-0.370282\pi$
$461$	17.0193	0.792666	0.396333	+	0.918107i	$0.370282\pi$
$462$	0	0
$463$	−13.4164	−0.623513	−0.311757	−	0.950162i	$-0.600917\pi$
$463$	−13.4164	−0.623513	−0.311757	+	0.950162i	$0.600917\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−17.5595	−0.812555	−0.406277	−	0.913750i	$-0.633173\pi$
$467$	−17.5595	−0.812555	−0.406277	+	0.913750i	$0.633173\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	20.6525	0.951616
$472$	0	0
$473$	20.5967	0.947039
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−26.1803	−1.19872
$478$	0	0
$479$	27.4102	1.25241	0.626203	−	0.779660i	$-0.284608\pi$
$479$	27.4102	1.25241	0.626203	+	0.779660i	$0.284608\pi$
$480$	0	0
$481$	3.70246	0.168818
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	31.4721	1.42614	0.713069	−	0.701094i	$-0.247304\pi$
$487$	31.4721	1.42614	0.713069	+	0.701094i	$0.247304\pi$
$488$	0	0
$489$	4.98915	0.225617
$490$	0	0
$491$	1.76393	0.0796051	0.0398026	−	0.999208i	$-0.487327\pi$
$491$	1.76393	0.0796051	0.0398026	+	0.999208i	$0.487327\pi$
$492$	0	0
$493$	1.08036	0.0486571
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−26.9443	−1.20619	−0.603096	−	0.797669i	$-0.706066\pi$
$499$	−26.9443	−1.20619	−0.603096	+	0.797669i	$0.706066\pi$
$500$	0	0
$501$	9.41641	0.420694
$502$	0	0
$503$	12.3153	0.549110	0.274555	−	0.961571i	$-0.411469\pi$
$503$	12.3153	0.549110	0.274555	+	0.961571i	$0.411469\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−27.9991	−1.24348
$508$	0	0
$509$	−15.0162	−0.665580	−0.332790	−	0.943001i	$-0.607990\pi$
$509$	−15.0162	−0.665580	−0.332790	+	0.943001i	$0.607990\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	10.4721	0.462356
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−26.8701	−1.18174
$518$	0	0
$519$	−42.0689	−1.84662
$520$	0	0
$521$	−34.5300	−1.51279	−0.756394	−	0.654117i	$-0.773041\pi$
$521$	−34.5300	−1.51279	−0.756394	+	0.654117i	$0.773041\pi$
$522$	0	0
$523$	3.90879	0.170919	0.0854597	−	0.996342i	$-0.472764\pi$
$523$	3.90879	0.170919	0.0854597	+	0.996342i	$0.472764\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−37.8885	−1.65045
$528$	0	0
$529$	−10.9443	−0.475838
$530$	0	0
$531$	4.37016	0.189649
$532$	0	0
$533$	4.47214	0.193710
$534$	0	0
$535$	0	0
$536$	0	0
$537$	3.90879	0.168677
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−10.7082	−0.460382	−0.230191	−	0.973146i	$-0.573935\pi$
$541$	−10.7082	−0.460382	−0.230191	+	0.973146i	$0.573935\pi$
$542$	0	0
$543$	8.29180	0.355835
$544$	0	0
$545$	0	0
$546$	0	0
$547$	3.29180	0.140747	0.0703735	−	0.997521i	$-0.477581\pi$
$547$	3.29180	0.140747	0.0703735	+	0.997521i	$0.477581\pi$
$548$	0	0
$549$	−16.8430	−0.718844
$550$	0	0
$551$	−1.41421	−0.0602475
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−14.3475	−0.607924	−0.303962	−	0.952684i	$-0.598310\pi$
$557$	−14.3475	−0.607924	−0.303962	+	0.952684i	$0.598310\pi$
$558$	0	0
$559$	3.28980	0.139144
$560$	0	0
$561$	−57.3050	−2.41942
$562$	0	0
$563$	−41.6799	−1.75660	−0.878299	−	0.478112i	$-0.841321\pi$
$563$	−41.6799	−1.75660	−0.878299	+	0.478112i	$0.841321\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	23.9443	1.00380	0.501898	−	0.864927i	$-0.332635\pi$
$569$	23.9443	1.00380	0.501898	+	0.864927i	$0.332635\pi$
$570$	0	0
$571$	4.34752	0.181938	0.0909691	−	0.995854i	$-0.471004\pi$
$571$	4.34752	0.181938	0.0909691	+	0.995854i	$0.471004\pi$
$572$	0	0
$573$	−27.8716	−1.16435
$574$	0	0
$575$	0	0
$576$	0	0
$577$	39.5192	1.64520	0.822602	−	0.568617i	$-0.192521\pi$
$577$	39.5192	1.64520	0.822602	+	0.568617i	$0.192521\pi$
$578$	0	0
$579$	35.8167	1.48849
$580$	0	0
$581$	0	0
$582$	0	0
$583$	64.0689	2.65346
$584$	0	0
$585$	0	0
$586$	0	0
$587$	36.8971	1.52291	0.761453	−	0.648221i	$-0.224487\pi$
$587$	36.8971	1.52291	0.761453	+	0.648221i	$0.224487\pi$
$588$	0	0
$589$	49.5967	2.04360
$590$	0	0
$591$	−44.5570	−1.83283
$592$	0	0
$593$	−30.5424	−1.25423	−0.627113	−	0.778928i	$-0.715764\pi$
$593$	−30.5424	−1.25423	−0.627113	+	0.778928i	$0.715764\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	49.5967	2.02986
$598$	0	0
$599$	8.52786	0.348439	0.174220	−	0.984707i	$-0.444260\pi$
$599$	8.52786	0.348439	0.174220	+	0.984707i	$0.444260\pi$
$600$	0	0
$601$	−32.4481	−1.32359	−0.661793	−	0.749687i	$-0.730204\pi$
$601$	−32.4481	−1.32359	−0.661793	+	0.749687i	$0.730204\pi$
$602$	0	0
$603$	31.1803	1.26976
$604$	0	0
$605$	0	0
$606$	0	0
$607$	5.03786	0.204480	0.102240	−	0.994760i	$-0.467399\pi$
$607$	5.03786	0.204480	0.102240	+	0.994760i	$0.467399\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−4.29180	−0.173627
$612$	0	0
$613$	−26.8885	−1.08602	−0.543009	−	0.839727i	$-0.682715\pi$
$613$	−26.8885	−1.08602	−0.543009	+	0.839727i	$0.682715\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	15.3607	0.618398	0.309199	−	0.950997i	$-0.399939\pi$
$617$	15.3607	0.618398	0.309199	+	0.950997i	$0.399939\pi$
$618$	0	0
$619$	26.1723	1.05195	0.525976	−	0.850500i	$-0.323700\pi$
$619$	26.1723	1.05195	0.525976	+	0.850500i	$0.323700\pi$
$620$	0	0
$621$	−6.06952	−0.243561
$622$	0	0
$623$	0	0
$624$	0	0
$625$	0	0
$626$	0	0
$627$	75.0132	2.99574
$628$	0	0
$629$	−19.3863	−0.772984
$630$	0	0
$631$	−1.00000	−0.0398094	−0.0199047	−	0.999802i	$-0.506336\pi$
$631$	−1.00000	−0.0398094	−0.0199047	+	0.999802i	$0.506336\pi$
$632$	0	0
$633$	21.8021	0.866555
$634$	0	0
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0	0
$639$	37.3607	1.47797
$640$	0	0
$641$	−39.3607	−1.55465	−0.777327	−	0.629097i	$-0.783425\pi$
$641$	−39.3607	−1.55465	−0.777327	+	0.629097i	$0.783425\pi$
$642$	0	0
$643$	−26.3786	−1.04027	−0.520135	−	0.854084i	$-0.674118\pi$
$643$	−26.3786	−1.04027	−0.520135	+	0.854084i	$0.674118\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−12.4729	−0.490360	−0.245180	−	0.969478i	$-0.578847\pi$
$647$	−12.4729	−0.490360	−0.245180	+	0.969478i	$0.578847\pi$
$648$	0	0
$649$	−10.6947	−0.419804
$650$	0	0
$651$	0	0
$652$	0	0
$653$	35.3050	1.38159	0.690795	−	0.723051i	$-0.257261\pi$
$653$	35.3050	1.38159	0.690795	+	0.723051i	$0.257261\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−16.8430	−0.657110
$658$	0	0
$659$	16.4721	0.641663	0.320832	−	0.947136i	$-0.396038\pi$
$659$	16.4721	0.641663	0.320832	+	0.947136i	$0.396038\pi$
$660$	0	0
$661$	44.0957	1.71512	0.857561	−	0.514382i	$-0.171979\pi$
$661$	44.0957	1.71512	0.857561	+	0.514382i	$0.171979\pi$
$662$	0	0
$663$	−9.15298	−0.355472
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0.819660	0.0317374
$668$	0	0
$669$	27.1246	1.04870
$670$	0	0
$671$	41.2185	1.59122
$672$	0	0
$673$	8.29180	0.319625	0.159813	−	0.987147i	$-0.448911\pi$
$673$	8.29180	0.319625	0.159813	+	0.987147i	$0.448911\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	30.7788	1.18293	0.591464	−	0.806332i	$-0.298550\pi$
$677$	30.7788	1.18293	0.591464	+	0.806332i	$0.298550\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−14.6525	−0.561484
$682$	0	0
$683$	−9.47214	−0.362441	−0.181221	−	0.983442i	$-0.558005\pi$
$683$	−9.47214	−0.362441	−0.181221	+	0.983442i	$0.558005\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	12.7639	0.486974
$688$	0	0
$689$	10.2333	0.389859
$690$	0	0
$691$	−27.0764	−1.03003	−0.515017	−	0.857180i	$-0.672214\pi$
$691$	−27.0764	−1.03003	−0.515017	+	0.857180i	$0.672214\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−23.4164	−0.886960
$698$	0	0
$699$	−24.0903	−0.911180
$700$	0	0
$701$	34.2492	1.29358	0.646788	−	0.762670i	$-0.276112\pi$
$701$	34.2492	1.29358	0.646788	+	0.762670i	$0.276112\pi$
$702$	0	0
$703$	25.3770	0.957113
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−15.8885	−0.596707	−0.298353	−	0.954455i	$-0.596437\pi$
$709$	−15.8885	−0.596707	−0.298353	+	0.954455i	$0.596437\pi$
$710$	0	0
$711$	−25.6525	−0.962043
$712$	0	0
$713$	−28.7456	−1.07653
$714$	0	0
$715$	0	0
$716$	0	0
$717$	19.7990	0.739407
$718$	0	0
$719$	−23.6777	−0.883028	−0.441514	−	0.897254i	$-0.645559\pi$
$719$	−23.6777	−0.883028	−0.441514	+	0.897254i	$0.645559\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	3.23607	0.120351
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−37.9473	−1.40739	−0.703694	−	0.710503i	$-0.748468\pi$
$727$	−37.9473	−1.40739	−0.703694	+	0.710503i	$0.748468\pi$
$728$	0	0
$729$	−11.9443	−0.442380
$730$	0	0
$731$	−17.2256	−0.637112
$732$	0	0
$733$	19.1313	0.706630	0.353315	−	0.935504i	$-0.385054\pi$
$733$	19.1313	0.706630	0.353315	+	0.935504i	$0.385054\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−76.3050	−2.81073
$738$	0	0
$739$	−18.0557	−0.664191	−0.332095	−	0.943246i	$-0.607756\pi$
$739$	−18.0557	−0.664191	−0.332095	+	0.943246i	$0.607756\pi$
$740$	0	0
$741$	11.9814	0.440148
$742$	0	0
$743$	46.1803	1.69419	0.847096	−	0.531440i	$-0.178349\pi$
$743$	46.1803	1.69419	0.847096	+	0.531440i	$0.178349\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	27.2526	0.997121
$748$	0	0
$749$	0	0
$750$	0	0
$751$	4.94427	0.180419	0.0902095	−	0.995923i	$-0.471246\pi$
$751$	4.94427	0.180419	0.0902095	+	0.995923i	$0.471246\pi$
$752$	0	0
$753$	−4.00000	−0.145768
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−49.7214	−1.80715	−0.903577	−	0.428426i	$-0.859068\pi$
$757$	−49.7214	−1.80715	−0.903577	+	0.428426i	$0.859068\pi$
$758$	0	0
$759$	−43.4767	−1.57810
$760$	0	0
$761$	−49.9287	−1.80992	−0.904958	−	0.425501i	$-0.860098\pi$
$761$	−49.9287	−1.80992	−0.904958	+	0.425501i	$0.860098\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−1.70820	−0.0616797
$768$	0	0
$769$	16.6854	0.601692	0.300846	−	0.953673i	$-0.402731\pi$
$769$	16.6854	0.601692	0.300846	+	0.953673i	$0.402731\pi$
$770$	0	0
$771$	18.9443	0.682261
$772$	0	0
$773$	−6.94355	−0.249742	−0.124871	−	0.992173i	$-0.539852\pi$
$773$	−6.94355	−0.249742	−0.124871	+	0.992173i	$0.539852\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	30.6525	1.09824
$780$	0	0
$781$	−91.4296	−3.27161
$782$	0	0
$783$	−0.412662	−0.0147473
$784$	0	0
$785$	0	0
$786$	0	0
$787$	38.4875	1.37193	0.685966	−	0.727634i	$-0.259380\pi$
$787$	38.4875	1.37193	0.685966	+	0.727634i	$0.259380\pi$
$788$	0	0
$789$	24.5030	0.872330
$790$	0	0
$791$	0	0
$792$	0	0
$793$	6.58359	0.233790
$794$	0	0
$795$	0	0
$796$	0	0
$797$	14.3972	0.509974	0.254987	−	0.966944i	$-0.417929\pi$
$797$	14.3972	0.509974	0.254987	+	0.966944i	$0.417929\pi$
$798$	0	0
$799$	22.4721	0.795007
$800$	0	0
$801$	−13.1105	−0.463236
$802$	0	0
$803$	41.2185	1.45457
$804$	0	0
$805$	0	0
$806$	0	0
$807$	73.7771	2.59708
$808$	0	0
$809$	11.8754	0.417516	0.208758	−	0.977967i	$-0.433058\pi$
$809$	11.8754	0.417516	0.208758	+	0.977967i	$0.433058\pi$
$810$	0	0
$811$	−10.1545	−0.356574	−0.178287	−	0.983979i	$-0.557056\pi$
$811$	−10.1545	−0.356574	−0.178287	+	0.983979i	$0.557056\pi$
$812$	0	0
$813$	56.8328	1.99321
$814$	0	0
$815$	0	0
$816$	0	0
$817$	22.5486	0.788876
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−19.5279	−0.681527	−0.340764	−	0.940149i	$-0.610686\pi$
$821$	−19.5279	−0.681527	−0.340764	+	0.940149i	$0.610686\pi$
$822$	0	0
$823$	16.0557	0.559667	0.279834	−	0.960048i	$-0.409721\pi$
$823$	16.0557	0.559667	0.279834	+	0.960048i	$0.409721\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	47.6525	1.65704	0.828519	−	0.559960i	$-0.189184\pi$
$827$	47.6525	1.65704	0.828519	+	0.559960i	$0.189184\pi$
$828$	0	0
$829$	−43.3491	−1.50558	−0.752789	−	0.658262i	$-0.771292\pi$
$829$	−43.3491	−1.50558	−0.752789	+	0.658262i	$0.771292\pi$
$830$	0	0
$831$	−47.9256	−1.66252
$832$	0	0
$833$	0	0
$834$	0	0
$835$	0	0
$836$	0	0
$837$	14.4721	0.500230
$838$	0	0
$839$	−23.5803	−0.814081	−0.407040	−	0.913410i	$-0.633439\pi$
$839$	−23.5803	−0.814081	−0.407040	+	0.913410i	$0.633439\pi$
$840$	0	0
$841$	−28.9443	−0.998078
$842$	0	0
$843$	−55.2030	−1.90129
$844$	0	0
$845$	0	0
$846$	0	0
$847$	0	0
$848$	0	0
$849$	24.1803	0.829867
$850$	0	0
$851$	−14.7082	−0.504191
$852$	0	0
$853$	0.255039	0.00873237	0.00436619	−	0.999990i	$-0.498610\pi$
$853$	0.255039	0.00873237	0.00436619	+	0.999990i	$0.498610\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	6.03941	0.206302	0.103151	−	0.994666i	$-0.467107\pi$
$857$	6.03941	0.206302	0.103151	+	0.994666i	$0.467107\pi$
$858$	0	0
$859$	−10.4096	−0.355170	−0.177585	−	0.984105i	$-0.556828\pi$
$859$	−10.4096	−0.355170	−0.177585	+	0.984105i	$0.556828\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	9.87539	0.336162	0.168081	−	0.985773i	$-0.446243\pi$
$863$	9.87539	0.336162	0.168081	+	0.985773i	$0.446243\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	9.02546	0.306521
$868$	0	0
$869$	62.7771	2.12957
$870$	0	0
$871$	−12.1877	−0.412966
$872$	0	0
$873$	36.5632	1.23748
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−44.4296	−1.50028	−0.750140	−	0.661279i	$-0.770014\pi$
$877$	−44.4296	−1.50028	−0.750140	+	0.661279i	$0.770014\pi$
$878$	0	0
$879$	−45.4164	−1.53186
$880$	0	0
$881$	−28.7456	−0.968465	−0.484233	−	0.874939i	$-0.660901\pi$
$881$	−28.7456	−0.968465	−0.484233	+	0.874939i	$0.660901\pi$
$882$	0	0
$883$	6.05573	0.203791	0.101896	−	0.994795i	$-0.467509\pi$
$883$	6.05573	0.203791	0.101896	+	0.994795i	$0.467509\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	48.4658	1.62732	0.813661	−	0.581339i	$-0.197471\pi$
$887$	48.4658	1.62732	0.813661	+	0.581339i	$0.197471\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	58.5967	1.96306
$892$	0	0
$893$	−29.4164	−0.984383
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−6.94427	−0.231862
$898$	0	0
$899$	−1.95440	−0.0651827
$900$	0	0
$901$	−53.5825	−1.78509
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−27.7771	−0.922323	−0.461162	−	0.887316i	$-0.652567\pi$
$907$	−27.7771	−0.922323	−0.461162	+	0.887316i	$0.652567\pi$
$908$	0	0
$909$	−7.53244	−0.249835
$910$	0	0
$911$	−7.11146	−0.235613	−0.117807	−	0.993037i	$-0.537586\pi$
$911$	−7.11146	−0.235613	−0.117807	+	0.993037i	$0.537586\pi$
$912$	0	0
$913$	−66.6930	−2.20722
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	18.7082	0.617127	0.308563	−	0.951204i	$-0.400152\pi$
$919$	18.7082	0.617127	0.308563	+	0.951204i	$0.400152\pi$
$920$	0	0
$921$	42.8328	1.41139
$922$	0	0
$923$	−14.6035	−0.480680
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−41.1096	−1.35022
$928$	0	0
$929$	27.9991	0.918622	0.459311	−	0.888276i	$-0.348096\pi$
$929$	27.9991	0.918622	0.459311	+	0.888276i	$0.348096\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	21.5279	0.704791
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−55.6157	−1.81689	−0.908443	−	0.418009i	$-0.862728\pi$
$937$	−55.6157	−1.81689	−0.908443	+	0.418009i	$0.862728\pi$
$938$	0	0
$939$	−8.18034	−0.266955
$940$	0	0
$941$	6.78593	0.221215	0.110607	−	0.993864i	$-0.464720\pi$
$941$	6.78593	0.221215	0.110607	+	0.993864i	$0.464720\pi$
$942$	0	0
$943$	−17.7658	−0.578534
$944$	0	0
$945$	0	0
$946$	0	0
$947$	15.7771	0.512686	0.256343	−	0.966586i	$-0.417482\pi$
$947$	15.7771	0.512686	0.256343	+	0.966586i	$0.417482\pi$
$948$	0	0
$949$	6.58359	0.213712
$950$	0	0
$951$	0.952843	0.0308981
$952$	0	0
$953$	−55.9017	−1.81083	−0.905417	−	0.424524i	$-0.860442\pi$
$953$	−55.9017	−1.81083	−0.905417	+	0.424524i	$0.860442\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−2.95595	−0.0955522
$958$	0	0
$959$	0	0
$960$	0	0
$961$	37.5410	1.21100
$962$	0	0
$963$	−27.8885	−0.898696
$964$	0	0
$965$	0	0
$966$	0	0
$967$	4.58359	0.147398	0.0736992	−	0.997281i	$-0.476520\pi$
$967$	4.58359	0.147398	0.0736992	+	0.997281i	$0.476520\pi$
$968$	0	0
$969$	−62.7355	−2.01535
$970$	0	0
$971$	−12.4729	−0.400274	−0.200137	−	0.979768i	$-0.564139\pi$
$971$	−12.4729	−0.400274	−0.200137	+	0.979768i	$0.564139\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	6.59675	0.211049	0.105524	−	0.994417i	$-0.466348\pi$
$977$	6.59675	0.211049	0.105524	+	0.994417i	$0.466348\pi$
$978$	0	0
$979$	32.0841	1.02541
$980$	0	0
$981$	−36.7082	−1.17200
$982$	0	0
$983$	−42.4751	−1.35475	−0.677373	−	0.735640i	$-0.736882\pi$
$983$	−42.4751	−1.35475	−0.677373	+	0.735640i	$0.736882\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−13.0689	−0.415566
$990$	0	0
$991$	4.34752	0.138104	0.0690518	−	0.997613i	$-0.478003\pi$
$991$	4.34752	0.138104	0.0690518	+	0.997613i	$0.478003\pi$
$992$	0	0
$993$	−39.7255	−1.26065
$994$	0	0
$995$	0	0
$996$	0	0
$997$	9.23179	0.292374	0.146187	−	0.989257i	$-0.453300\pi$
$997$	9.23179	0.292374	0.146187	+	0.989257i	$0.453300\pi$
$998$	0	0
$999$	7.40492	0.234281

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4900.2.a.bg.1.4	yes	4
5.2	odd	4		4900.2.e.u.2549.2		8
5.3	odd	4		4900.2.e.u.2549.8		8
5.4	even	2		4900.2.a.bi.1.1	yes	4
7.6	odd	2	inner	4900.2.a.bg.1.1	✓	4
35.13	even	4		4900.2.e.u.2549.1		8
35.27	even	4		4900.2.e.u.2549.7		8
35.34	odd	2		4900.2.a.bi.1.4	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4900.2.a.bg.1.1	✓	4	7.6	odd	2	inner
4900.2.a.bg.1.4	yes	4	1.1	even	1	trivial
4900.2.a.bi.1.1	yes	4	5.4	even	2
4900.2.a.bi.1.4	yes	4	35.34	odd	2
4900.2.e.u.2549.1		8	35.13	even	4
4900.2.e.u.2549.2		8	5.2	odd	4
4900.2.e.u.2549.7		8	35.27	even	4
4900.2.e.u.2549.8		8	5.3	odd	4

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:	\( N \)	\(=\)	\( 4900 = 2^{2} \cdot 5^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4900.a (trivial)

\(f(q)\)	\(=\)	\(q+2.28825 q^{3} +2.23607 q^{9} -5.47214 q^{11} -0.874032 q^{13} +4.57649 q^{17} -5.99070 q^{19} +3.47214 q^{23} -1.74806 q^{27} +0.236068 q^{29} -8.27895 q^{31} -12.5216 q^{33} -4.23607 q^{37} -2.00000 q^{39} -5.11667 q^{41} -3.76393 q^{43} +4.91034 q^{47} +10.4721 q^{51} -11.7082 q^{53} -13.7082 q^{57} +1.95440 q^{59} -7.53244 q^{61} +13.9443 q^{67} +7.94510 q^{69} +16.7082 q^{71} -7.53244 q^{73} -11.4721 q^{79} -10.7082 q^{81} +12.1877 q^{83} +0.540182 q^{87} -5.86319 q^{89} -18.9443 q^{93} +16.3516 q^{97} -12.2361 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{11} - 4 q^{23} - 8 q^{29} - 8 q^{37} - 8 q^{39} - 24 q^{43} + 24 q^{51} - 20 q^{53} - 28 q^{57} + 20 q^{67} + 40 q^{71} - 28 q^{79} - 16 q^{81} - 40 q^{93} - 40 q^{99}+O(q^{100}) \) 4 * q - 4 * q^11 - 4 * q^23 - 8 * q^29 - 8 * q^37 - 8 * q^39 - 24 * q^43 + 24 * q^51 - 20 * q^53 - 28 * q^57 + 20 * q^67 + 40 * q^71 - 28 * q^79 - 16 * q^81 - 40 * q^93 - 40 * q^99

Introduction

L-functions

Modular forms

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