LMFDB - Embedded newform 4851.2.a.bl.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4851,2,Mod(1,4851)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4851, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("4851.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4851 = 3^{2} \cdot 7^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4851.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$38.7354300205$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.229.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 4x - 1 $ x^3 - 4*x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1617)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.11491$ of defining polynomial
Character	$\chi$	$=$	4851.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.11491 q^{2} +2.47283 q^{4} -1.64207 q^{5} -1.00000 q^{8} +O(q^{10})$	$q-2.11491 q^{2} +2.47283 q^{4} -1.64207 q^{5} -1.00000 q^{8} +3.47283 q^{10} -1.00000 q^{11} +4.58774 q^{13} -2.83076 q^{16} -0.715853 q^{17} +1.64207 q^{19} -4.06058 q^{20} +2.11491 q^{22} -2.30359 q^{25} -9.70265 q^{26} +5.53341 q^{29} -5.17548 q^{31} +7.98680 q^{32} +1.51396 q^{34} -6.58774 q^{37} -3.47283 q^{38} +1.64207 q^{40} -5.17548 q^{41} +5.17548 q^{43} -2.47283 q^{44} -7.87189 q^{47} +4.87189 q^{50} +11.3447 q^{52} -2.71585 q^{53} +1.64207 q^{55} -11.7026 q^{58} +3.15604 q^{59} +13.1755 q^{61} +10.9457 q^{62} -11.2298 q^{64} -7.53341 q^{65} +4.21037 q^{67} -1.77018 q^{68} +1.85244 q^{71} -10.4791 q^{73} +13.9325 q^{74} +4.06058 q^{76} -12.4596 q^{79} +4.64832 q^{80} +10.9457 q^{82} +9.17548 q^{83} +1.17548 q^{85} -10.9457 q^{86} +1.00000 q^{88} -4.71585 q^{89} +16.6483 q^{94} -2.69641 q^{95} -13.6351 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 2 q^{4} - 4 q^{5} - 3 q^{8}+O(q^{10}) $ 3 * q + 2 * q^4 - 4 * q^5 - 3 * q^8	$ 3 q + 2 q^{4} - 4 q^{5} - 3 q^{8} + 5 q^{10} - 3 q^{11} + 2 q^{13} - 4 q^{16} - 4 q^{17} + 4 q^{19} + 5 q^{20} + 3 q^{25} - 11 q^{26} - 6 q^{29} + 8 q^{31} + 4 q^{32} - 10 q^{34} - 8 q^{37} - 5 q^{38} + 4 q^{40} + 8 q^{41} - 8 q^{43} - 2 q^{44} - 10 q^{47} + q^{50} + 15 q^{52} - 10 q^{53} + 4 q^{55} - 17 q^{58} - 6 q^{59} + 16 q^{61} + 22 q^{62} - 21 q^{64} + 8 q^{67} - 18 q^{68} + 2 q^{73} + 11 q^{74} - 5 q^{76} - 12 q^{79} - 15 q^{80} + 22 q^{82} + 4 q^{83} - 20 q^{85} - 22 q^{86} + 3 q^{88} - 16 q^{89} + 21 q^{94} - 18 q^{95} + 8 q^{97}+O(q^{100}) $ 3 * q + 2 * q^4 - 4 * q^5 - 3 * q^8 + 5 * q^10 - 3 * q^11 + 2 * q^13 - 4 * q^16 - 4 * q^17 + 4 * q^19 + 5 * q^20 + 3 * q^25 - 11 * q^26 - 6 * q^29 + 8 * q^31 + 4 * q^32 - 10 * q^34 - 8 * q^37 - 5 * q^38 + 4 * q^40 + 8 * q^41 - 8 * q^43 - 2 * q^44 - 10 * q^47 + q^50 + 15 * q^52 - 10 * q^53 + 4 * q^55 - 17 * q^58 - 6 * q^59 + 16 * q^61 + 22 * q^62 - 21 * q^64 + 8 * q^67 - 18 * q^68 + 2 * q^73 + 11 * q^74 - 5 * q^76 - 12 * q^79 - 15 * q^80 + 22 * q^82 + 4 * q^83 - 20 * q^85 - 22 * q^86 + 3 * q^88 - 16 * q^89 + 21 * q^94 - 18 * q^95 + 8 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−2.11491	−1.49547	−0.747733	−	0.664000i	$-0.768858\pi$
$2$	−2.11491	−1.49547	−0.747733	+	0.664000i	$0.768858\pi$
$3$	0	0
$3$	0	0
$4$	2.47283	1.23642
$5$	−1.64207	−0.734358	−0.367179	−	0.930150i	$-0.619676\pi$
$5$	−1.64207	−0.734358	−0.367179	+	0.930150i	$0.619676\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−1.00000	−0.353553
$9$	0	0
$10$	3.47283	1.09821
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	4.58774	1.27241	0.636205	−	0.771520i	$-0.280503\pi$
$13$	4.58774	1.27241	0.636205	+	0.771520i	$0.280503\pi$
$14$	0	0
$15$	0	0
$16$	−2.83076	−0.707690
$17$	−0.715853	−0.173620	−0.0868099	−	0.996225i	$-0.527667\pi$
$17$	−0.715853	−0.173620	−0.0868099	+	0.996225i	$0.527667\pi$
$18$	0	0
$19$	1.64207	0.376718	0.188359	−	0.982100i	$-0.439683\pi$
$19$	1.64207	0.376718	0.188359	+	0.982100i	$0.439683\pi$
$20$	−4.06058	−0.907972
$21$	0	0
$22$	2.11491	0.450900
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	0	0
$25$	−2.30359	−0.460719
$26$	−9.70265	−1.90285
$27$	0	0
$28$	0	0
$29$	5.53341	1.02753	0.513764	−	0.857931i	$-0.328251\pi$
$29$	5.53341	1.02753	0.513764	+	0.857931i	$0.328251\pi$
$30$	0	0
$31$	−5.17548	−0.929544	−0.464772	−	0.885430i	$-0.653864\pi$
$31$	−5.17548	−0.929544	−0.464772	+	0.885430i	$0.653864\pi$
$32$	7.98680	1.41188
$33$	0	0
$34$	1.51396	0.259642
$35$	0	0
$36$	0	0
$37$	−6.58774	−1.08302	−0.541509	−	0.840695i	$-0.682147\pi$
$37$	−6.58774	−1.08302	−0.541509	+	0.840695i	$0.682147\pi$
$38$	−3.47283	−0.563368
$39$	0	0
$40$	1.64207	0.259635
$41$	−5.17548	−0.808275	−0.404137	−	0.914698i	$-0.632428\pi$
$41$	−5.17548	−0.808275	−0.404137	+	0.914698i	$0.632428\pi$
$42$	0	0
$43$	5.17548	0.789254	0.394627	−	0.918841i	$-0.370874\pi$
$43$	5.17548	0.789254	0.394627	+	0.918841i	$0.370874\pi$
$44$	−2.47283	−0.372794
$45$	0	0
$46$	0	0
$47$	−7.87189	−1.14823	−0.574116	−	0.818774i	$-0.694654\pi$
$47$	−7.87189	−1.14823	−0.574116	+	0.818774i	$0.694654\pi$
$48$	0	0
$49$	0	0
$50$	4.87189	0.688989
$51$	0	0
$52$	11.3447	1.57323
$53$	−2.71585	−0.373051	−0.186526	−	0.982450i	$-0.559723\pi$
$53$	−2.71585	−0.373051	−0.186526	+	0.982450i	$0.559723\pi$
$54$	0	0
$55$	1.64207	0.221417
$56$	0	0
$57$	0	0
$58$	−11.7026	−1.53663
$59$	3.15604	0.410881	0.205440	−	0.978670i	$-0.434137\pi$
$59$	3.15604	0.410881	0.205440	+	0.978670i	$0.434137\pi$
$60$	0	0
$61$	13.1755	1.68695	0.843474	−	0.537170i	$-0.180507\pi$
$61$	13.1755	1.68695	0.843474	+	0.537170i	$0.180507\pi$
$62$	10.9457	1.39010
$63$	0	0
$64$	−11.2298	−1.40373
$65$	−7.53341	−0.934404
$66$	0	0
$67$	4.21037	0.514378	0.257189	−	0.966361i	$-0.417204\pi$
$67$	4.21037	0.514378	0.257189	+	0.966361i	$0.417204\pi$
$68$	−1.77018	−0.214666
$69$	0	0
$70$	0	0
$71$	1.85244	0.219844	0.109922	−	0.993940i	$-0.464940\pi$
$71$	1.85244	0.219844	0.109922	+	0.993940i	$0.464940\pi$
$72$	0	0
$73$	−10.4791	−1.22648	−0.613242	−	0.789895i	$-0.710135\pi$
$73$	−10.4791	−1.22648	−0.613242	+	0.789895i	$0.710135\pi$
$74$	13.9325	1.61962
$75$	0	0
$76$	4.06058	0.465780
$77$	0	0
$78$	0	0
$79$	−12.4596	−1.40182	−0.700909	−	0.713251i	$-0.747222\pi$
$79$	−12.4596	−1.40182	−0.700909	+	0.713251i	$0.747222\pi$
$80$	4.64832	0.519698
$81$	0	0
$82$	10.9457	1.20875
$83$	9.17548	1.00714	0.503570	−	0.863954i	$-0.332020\pi$
$83$	9.17548	1.00714	0.503570	+	0.863954i	$0.332020\pi$
$84$	0	0
$85$	1.17548	0.127499
$86$	−10.9457	−1.18030
$87$	0	0
$88$	1.00000	0.106600
$89$	−4.71585	−0.499879	−0.249940	−	0.968261i	$-0.580411\pi$
$89$	−4.71585	−0.499879	−0.249940	+	0.968261i	$0.580411\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	16.6483	1.71714
$95$	−2.69641	−0.276645
$96$	0	0
$97$	−13.6351	−1.38444	−0.692218	−	0.721688i	$-0.743366\pi$
$97$	−13.6351	−1.38444	−0.692218	+	0.721688i	$0.743366\pi$
$98$	0	0
$99$	0	0
$100$	−5.69641	−0.569641
$101$	14.6072	1.45347	0.726735	−	0.686918i	$-0.241037\pi$
$101$	14.6072	1.45347	0.726735	+	0.686918i	$0.241037\pi$
$102$	0	0
$103$	17.8913	1.76289	0.881443	−	0.472291i	$-0.156573\pi$
$103$	17.8913	1.76289	0.881443	+	0.472291i	$0.156573\pi$
$104$	−4.58774	−0.449865
$105$	0	0
$106$	5.74378	0.557885
$107$	−11.1560	−1.07849	−0.539247	−	0.842147i	$-0.681291\pi$
$107$	−11.1560	−1.07849	−0.539247	+	0.842147i	$0.681291\pi$
$108$	0	0
$109$	−0.108664	−0.0104082	−0.00520408	−	0.999986i	$-0.501657\pi$
$109$	−0.108664	−0.0104082	−0.00520408	+	0.999986i	$0.501657\pi$
$110$	−3.47283	−0.331122
$111$	0	0
$112$	0	0
$113$	13.0668	1.22922	0.614611	−	0.788830i	$-0.289313\pi$
$113$	13.0668	1.22922	0.614611	+	0.788830i	$0.289313\pi$
$114$	0	0
$115$	0	0
$116$	13.6832	1.27045
$117$	0	0
$118$	−6.67472	−0.614458
$119$	0	0
$120$	0	0
$121$	1.00000	0.0909091
$122$	−27.8649	−2.52277
$123$	0	0
$124$	−12.7981	−1.14930
$125$	11.9930	1.07269
$126$	0	0
$127$	−20.4596	−1.81550	−0.907749	−	0.419513i	$-0.862201\pi$
$127$	−20.4596	−1.81550	−0.907749	+	0.419513i	$0.862201\pi$
$128$	7.77643	0.687346
$129$	0	0
$130$	15.9325	1.39737
$131$	15.0668	1.31639	0.658197	−	0.752846i	$-0.271319\pi$
$131$	15.0668	1.31639	0.658197	+	0.752846i	$0.271319\pi$
$132$	0	0
$133$	0	0
$134$	−8.90454	−0.769235
$135$	0	0
$136$	0.715853	0.0613839
$137$	14.4596	1.23537	0.617685	−	0.786426i	$-0.288071\pi$
$137$	14.4596	1.23537	0.617685	+	0.786426i	$0.288071\pi$
$138$	0	0
$139$	−5.13659	−0.435680	−0.217840	−	0.975985i	$-0.569901\pi$
$139$	−5.13659	−0.435680	−0.217840	+	0.975985i	$0.569901\pi$
$140$	0	0
$141$	0	0
$142$	−3.91774	−0.328770
$143$	−4.58774	−0.383646
$144$	0	0
$145$	−9.08627	−0.754573
$146$	22.1623	1.83416
$147$	0	0
$148$	−16.2904	−1.33906
$149$	−16.8176	−1.37775	−0.688874	−	0.724881i	$-0.741895\pi$
$149$	−16.8176	−1.37775	−0.688874	+	0.724881i	$0.741895\pi$
$150$	0	0
$151$	4.71585	0.383771	0.191885	−	0.981417i	$-0.438540\pi$
$151$	4.71585	0.383771	0.191885	+	0.981417i	$0.438540\pi$
$152$	−1.64207	−0.133190
$153$	0	0
$154$	0	0
$155$	8.49852	0.682618
$156$	0	0
$157$	−11.0279	−0.880124	−0.440062	−	0.897967i	$-0.645044\pi$
$157$	−11.0279	−0.880124	−0.440062	+	0.897967i	$0.645044\pi$
$158$	26.3510	2.09637
$159$	0	0
$160$	−13.1149	−1.03682
$161$	0	0
$162$	0	0
$163$	5.64207	0.441921	0.220961	−	0.975283i	$-0.429081\pi$
$163$	5.64207	0.441921	0.220961	+	0.975283i	$0.429081\pi$
$164$	−12.7981	−0.999364
$165$	0	0
$166$	−19.4053	−1.50614
$167$	−16.0000	−1.23812	−0.619059	−	0.785345i	$-0.712486\pi$
$167$	−16.0000	−1.23812	−0.619059	+	0.785345i	$0.712486\pi$
$168$	0	0
$169$	8.04737	0.619029
$170$	−2.48604	−0.190670
$171$	0	0
$172$	12.7981	0.975847
$173$	15.0279	1.14255	0.571276	−	0.820758i	$-0.306449\pi$
$173$	15.0279	1.14255	0.571276	+	0.820758i	$0.306449\pi$
$174$	0	0
$175$	0	0
$176$	2.83076	0.213377
$177$	0	0
$178$	9.97359	0.747552
$179$	−9.89134	−0.739313	−0.369657	−	0.929168i	$-0.620525\pi$
$179$	−9.89134	−0.739313	−0.369657	+	0.929168i	$0.620525\pi$
$180$	0	0
$181$	6.82452	0.507262	0.253631	−	0.967301i	$-0.418375\pi$
$181$	6.82452	0.507262	0.253631	+	0.967301i	$0.418375\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	10.8176	0.795323
$186$	0	0
$187$	0.715853	0.0523483
$188$	−19.4659	−1.41969
$189$	0	0
$190$	5.70265	0.413714
$191$	−11.0279	−0.797953	−0.398976	−	0.916961i	$-0.630634\pi$
$191$	−11.0279	−0.797953	−0.398976	+	0.916961i	$0.630634\pi$
$192$	0	0
$193$	−7.85244	−0.565231	−0.282616	−	0.959233i	$-0.591202\pi$
$193$	−7.85244	−0.565231	−0.282616	+	0.959233i	$0.591202\pi$
$194$	28.8370	2.07038
$195$	0	0
$196$	0	0
$197$	−19.8913	−1.41720	−0.708599	−	0.705611i	$-0.750673\pi$
$197$	−19.8913	−1.41720	−0.708599	+	0.705611i	$0.750673\pi$
$198$	0	0
$199$	23.5264	1.66775	0.833873	−	0.551957i	$-0.186119\pi$
$199$	23.5264	1.66775	0.833873	+	0.551957i	$0.186119\pi$
$200$	2.30359	0.162889
$201$	0	0
$202$	−30.8929	−2.17361
$203$	0	0
$204$	0	0
$205$	8.49852	0.593563
$206$	−37.8385	−2.63633
$207$	0	0
$208$	−12.9868	−0.900472
$209$	−1.64207	−0.113585
$210$	0	0
$211$	−4.25622	−0.293010	−0.146505	−	0.989210i	$-0.546803\pi$
$211$	−4.25622	−0.293010	−0.146505	+	0.989210i	$0.546803\pi$
$212$	−6.71585	−0.461247
$213$	0	0
$214$	23.5940	1.61285
$215$	−8.49852	−0.579595
$216$	0	0
$217$	0	0
$218$	0.229815	0.0155650
$219$	0	0
$220$	4.06058	0.273764
$221$	−3.28415	−0.220916
$222$	0	0
$223$	−3.74378	−0.250702	−0.125351	−	0.992112i	$-0.540006\pi$
$223$	−3.74378	−0.250702	−0.125351	+	0.992112i	$0.540006\pi$
$224$	0	0
$225$	0	0
$226$	−27.6351	−1.83826
$227$	15.4876	1.02795	0.513973	−	0.857807i	$-0.328173\pi$
$227$	15.4876	1.02795	0.513973	+	0.857807i	$0.328173\pi$
$228$	0	0
$229$	−8.00000	−0.528655	−0.264327	−	0.964433i	$-0.585150\pi$
$229$	−8.00000	−0.528655	−0.264327	+	0.964433i	$0.585150\pi$
$230$	0	0
$231$	0	0
$232$	−5.53341	−0.363286
$233$	5.54037	0.362962	0.181481	−	0.983394i	$-0.441911\pi$
$233$	5.54037	0.362962	0.181481	+	0.983394i	$0.441911\pi$
$234$	0	0
$235$	12.9262	0.843214
$236$	7.80435	0.508020
$237$	0	0
$238$	0	0
$239$	1.26470	0.0818067	0.0409033	−	0.999163i	$-0.486976\pi$
$239$	1.26470	0.0818067	0.0409033	+	0.999163i	$0.486976\pi$
$240$	0	0
$241$	−23.8719	−1.53772	−0.768862	−	0.639415i	$-0.779177\pi$
$241$	−23.8719	−1.53772	−0.768862	+	0.639415i	$0.779177\pi$
$242$	−2.11491	−0.135951
$243$	0	0
$244$	32.5808	2.08577
$245$	0	0
$246$	0	0
$247$	7.53341	0.479339
$248$	5.17548	0.328643
$249$	0	0
$250$	−25.3642	−1.60417
$251$	−22.0194	−1.38986	−0.694928	−	0.719080i	$-0.744564\pi$
$251$	−22.0194	−1.38986	−0.694928	+	0.719080i	$0.744564\pi$
$252$	0	0
$253$	0	0
$254$	43.2702	2.71502
$255$	0	0
$256$	6.01320	0.375825
$257$	−27.3161	−1.70393	−0.851965	−	0.523599i	$-0.824589\pi$
$257$	−27.3161	−1.70393	−0.851965	+	0.523599i	$0.824589\pi$
$258$	0	0
$259$	0	0
$260$	−18.6289	−1.15531
$261$	0	0
$262$	−31.8649	−1.96862
$263$	11.8719	0.732052	0.366026	−	0.930605i	$-0.380718\pi$
$263$	11.8719	0.732052	0.366026	+	0.930605i	$0.380718\pi$
$264$	0	0
$265$	4.45963	0.273953
$266$	0	0
$267$	0	0
$268$	10.4115	0.635986
$269$	−5.63511	−0.343579	−0.171789	−	0.985134i	$-0.554955\pi$
$269$	−5.63511	−0.343579	−0.171789	+	0.985134i	$0.554955\pi$
$270$	0	0
$271$	−5.68097	−0.345094	−0.172547	−	0.985001i	$-0.555200\pi$
$271$	−5.68097	−0.345094	−0.172547	+	0.985001i	$0.555200\pi$
$272$	2.02641	0.122869
$273$	0	0
$274$	−30.5808	−1.84745
$275$	2.30359	0.138912
$276$	0	0
$277$	−25.4876	−1.53140	−0.765699	−	0.643199i	$-0.777607\pi$
$277$	−25.4876	−1.53140	−0.765699	+	0.643199i	$0.777607\pi$
$278$	10.8634	0.651544
$279$	0	0
$280$	0	0
$281$	−17.5723	−1.04828	−0.524138	−	0.851633i	$-0.675612\pi$
$281$	−17.5723	−1.04828	−0.524138	+	0.851633i	$0.675612\pi$
$282$	0	0
$283$	25.1296	1.49380	0.746901	−	0.664936i	$-0.231541\pi$
$283$	25.1296	1.49380	0.746901	+	0.664936i	$0.231541\pi$
$284$	4.58078	0.271819
$285$	0	0
$286$	9.70265	0.573730
$287$	0	0
$288$	0	0
$289$	−16.4876	−0.969856
$290$	19.2166	1.12844
$291$	0	0
$292$	−25.9130	−1.51644
$293$	14.3510	0.838392	0.419196	−	0.907896i	$-0.362312\pi$
$293$	14.3510	0.838392	0.419196	+	0.907896i	$0.362312\pi$
$294$	0	0
$295$	−5.18244	−0.301734
$296$	6.58774	0.382905
$297$	0	0
$298$	35.5676	2.06037
$299$	0	0
$300$	0	0
$301$	0	0
$302$	−9.97359	−0.573916
$303$	0	0
$304$	−4.64832	−0.266599
$305$	−21.6351	−1.23882
$306$	0	0
$307$	13.1366	0.749745	0.374872	−	0.927076i	$-0.377687\pi$
$307$	13.1366	0.749745	0.374872	+	0.927076i	$0.377687\pi$
$308$	0	0
$309$	0	0
$310$	−17.9736	−1.02083
$311$	14.3510	0.813769	0.406884	−	0.913480i	$-0.366615\pi$
$311$	14.3510	0.813769	0.406884	+	0.913480i	$0.366615\pi$
$312$	0	0
$313$	23.7438	1.34208	0.671039	−	0.741422i	$-0.265848\pi$
$313$	23.7438	1.34208	0.671039	+	0.741422i	$0.265848\pi$
$314$	23.3230	1.31620
$315$	0	0
$316$	−30.8106	−1.73323
$317$	6.45963	0.362809	0.181404	−	0.983409i	$-0.441936\pi$
$317$	6.45963	0.362809	0.181404	+	0.983409i	$0.441936\pi$
$318$	0	0
$319$	−5.53341	−0.309811
$320$	18.4402	1.03084
$321$	0	0
$322$	0	0
$323$	−1.17548	−0.0654056
$324$	0	0
$325$	−10.5683	−0.586224
$326$	−11.9325	−0.660878
$327$	0	0
$328$	5.17548	0.285768
$329$	0	0
$330$	0	0
$331$	31.2702	1.71877	0.859384	−	0.511332i	$-0.170848\pi$
$331$	31.2702	1.71877	0.859384	+	0.511332i	$0.170848\pi$
$332$	22.6894	1.24525
$333$	0	0
$334$	33.8385	1.85156
$335$	−6.91373	−0.377738
$336$	0	0
$337$	−16.3121	−0.888575	−0.444288	−	0.895884i	$-0.646543\pi$
$337$	−16.3121	−0.888575	−0.444288	+	0.895884i	$0.646543\pi$
$338$	−17.0194	−0.925736
$339$	0	0
$340$	2.90677	0.157642
$341$	5.17548	0.280268
$342$	0	0
$343$	0	0
$344$	−5.17548	−0.279043
$345$	0	0
$346$	−31.7827	−1.70865
$347$	6.86341	0.368447	0.184224	−	0.982884i	$-0.441023\pi$
$347$	6.86341	0.368447	0.184224	+	0.982884i	$0.441023\pi$
$348$	0	0
$349$	23.8580	1.27709	0.638544	−	0.769585i	$-0.279537\pi$
$349$	23.8580	1.27709	0.638544	+	0.769585i	$0.279537\pi$
$350$	0	0
$351$	0	0
$352$	−7.98680	−0.425698
$353$	−11.7368	−0.624688	−0.312344	−	0.949969i	$-0.601114\pi$
$353$	−11.7368	−0.624688	−0.312344	+	0.949969i	$0.601114\pi$
$354$	0	0
$355$	−3.04185	−0.161444
$356$	−11.6615	−0.618059
$357$	0	0
$358$	20.9193	1.10562
$359$	27.7827	1.46631	0.733157	−	0.680060i	$-0.238046\pi$
$359$	27.7827	1.46631	0.733157	+	0.680060i	$0.238046\pi$
$360$	0	0
$361$	−16.3036	−0.858084
$362$	−14.4332	−0.758593
$363$	0	0
$364$	0	0
$365$	17.2074	0.900677
$366$	0	0
$367$	−26.5544	−1.38613	−0.693064	−	0.720877i	$-0.743739\pi$
$367$	−26.5544	−1.38613	−0.693064	+	0.720877i	$0.743739\pi$
$368$	0	0
$369$	0	0
$370$	−22.8781	−1.18938
$371$	0	0
$372$	0	0
$373$	5.06682	0.262350	0.131175	−	0.991359i	$-0.458125\pi$
$373$	5.06682	0.262350	0.131175	+	0.991359i	$0.458125\pi$
$374$	−1.51396	−0.0782851
$375$	0	0
$376$	7.87189	0.405962
$377$	25.3859	1.30744
$378$	0	0
$379$	−30.5613	−1.56983	−0.784915	−	0.619603i	$-0.787294\pi$
$379$	−30.5613	−1.56983	−0.784915	+	0.619603i	$0.787294\pi$
$380$	−6.66776	−0.342049
$381$	0	0
$382$	23.3230	1.19331
$383$	−9.13659	−0.466858	−0.233429	−	0.972374i	$-0.574995\pi$
$383$	−9.13659	−0.466858	−0.233429	+	0.972374i	$0.574995\pi$
$384$	0	0
$385$	0	0
$386$	16.6072	0.845284
$387$	0	0
$388$	−33.7174	−1.71174
$389$	−21.2702	−1.07844	−0.539222	−	0.842164i	$-0.681281\pi$
$389$	−21.2702	−1.07844	−0.539222	+	0.842164i	$0.681281\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	0	0
$394$	42.0683	2.11937
$395$	20.4596	1.02944
$396$	0	0
$397$	28.9582	1.45337	0.726684	−	0.686972i	$-0.241060\pi$
$397$	28.9582	1.45337	0.726684	+	0.686972i	$0.241060\pi$
$398$	−49.7563	−2.49406
$399$	0	0
$400$	6.52092	0.326046
$401$	−32.3510	−1.61553	−0.807765	−	0.589505i	$-0.799323\pi$
$401$	−32.3510	−1.61553	−0.807765	+	0.589505i	$0.799323\pi$
$402$	0	0
$403$	−23.7438	−1.18276
$404$	36.1212	1.79709
$405$	0	0
$406$	0	0
$407$	6.58774	0.326542
$408$	0	0
$409$	2.31207	0.114325	0.0571623	−	0.998365i	$-0.481795\pi$
$409$	2.31207	0.114325	0.0571623	+	0.998365i	$0.481795\pi$
$410$	−17.9736	−0.887652
$411$	0	0
$412$	44.2423	2.17966
$413$	0	0
$414$	0	0
$415$	−15.0668	−0.739601
$416$	36.6414	1.79649
$417$	0	0
$418$	3.47283	0.169862
$419$	−38.4263	−1.87725	−0.938623	−	0.344945i	$-0.887898\pi$
$419$	−38.4263	−1.87725	−0.938623	+	0.344945i	$0.887898\pi$
$420$	0	0
$421$	15.5070	0.755765	0.377883	−	0.925854i	$-0.376652\pi$
$421$	15.5070	0.755765	0.377883	+	0.925854i	$0.376652\pi$
$422$	9.00152	0.438187
$423$	0	0
$424$	2.71585	0.131893
$425$	1.64903	0.0799899
$426$	0	0
$427$	0	0
$428$	−27.5870	−1.33347
$429$	0	0
$430$	17.9736	0.866764
$431$	−37.9666	−1.82879	−0.914394	−	0.404825i	$-0.867332\pi$
$431$	−37.9666	−1.82879	−0.914394	+	0.404825i	$0.867332\pi$
$432$	0	0
$433$	10.1087	0.485791	0.242896	−	0.970052i	$-0.421903\pi$
$433$	10.1087	0.485791	0.242896	+	0.970052i	$0.421903\pi$
$434$	0	0
$435$	0	0
$436$	−0.268709	−0.0128688
$437$	0	0
$438$	0	0
$439$	−22.5224	−1.07494	−0.537469	−	0.843284i	$-0.680619\pi$
$439$	−22.5224	−1.07494	−0.537469	+	0.843284i	$0.680619\pi$
$440$	−1.64207	−0.0782828
$441$	0	0
$442$	6.94567	0.330372
$443$	−20.6630	−0.981731	−0.490865	−	0.871235i	$-0.663319\pi$
$443$	−20.6630	−0.981731	−0.490865	+	0.871235i	$0.663319\pi$
$444$	0	0
$445$	7.74378	0.367090
$446$	7.91774	0.374916
$447$	0	0
$448$	0	0
$449$	−40.8106	−1.92597	−0.962986	−	0.269553i	$-0.913124\pi$
$449$	−40.8106	−1.92597	−0.962986	+	0.269553i	$0.913124\pi$
$450$	0	0
$451$	5.17548	0.243704
$452$	32.3121	1.51983
$453$	0	0
$454$	−32.7547	−1.53726
$455$	0	0
$456$	0	0
$457$	−28.3899	−1.32802	−0.664011	−	0.747723i	$-0.731147\pi$
$457$	−28.3899	−1.32802	−0.664011	+	0.747723i	$0.731147\pi$
$458$	16.9193	0.790585
$459$	0	0
$460$	0	0
$461$	−21.4178	−0.997526	−0.498763	−	0.866739i	$-0.666212\pi$
$461$	−21.4178	−0.997526	−0.498763	+	0.866739i	$0.666212\pi$
$462$	0	0
$463$	20.7478	0.964231	0.482116	−	0.876108i	$-0.339868\pi$
$463$	20.7478	0.964231	0.482116	+	0.876108i	$0.339868\pi$
$464$	−15.6638	−0.727172
$465$	0	0
$466$	−11.7174	−0.542797
$467$	−26.8998	−1.24477	−0.622387	−	0.782709i	$-0.713837\pi$
$467$	−26.8998	−1.24477	−0.622387	+	0.782709i	$0.713837\pi$
$468$	0	0
$469$	0	0
$470$	−27.3378	−1.26100
$471$	0	0
$472$	−3.15604	−0.145268
$473$	−5.17548	−0.237969
$474$	0	0
$475$	−3.78267	−0.173561
$476$	0	0
$477$	0	0
$478$	−2.67472	−0.122339
$479$	−30.8106	−1.40777	−0.703886	−	0.710313i	$-0.748553\pi$
$479$	−30.8106	−1.40777	−0.703886	+	0.710313i	$0.748553\pi$
$480$	0	0
$481$	−30.2229	−1.37804
$482$	50.4868	2.29961
$483$	0	0
$484$	2.47283	0.112402
$485$	22.3899	1.01667
$486$	0	0
$487$	−23.4876	−1.06432	−0.532161	−	0.846643i	$-0.678620\pi$
$487$	−23.4876	−1.06432	−0.532161	+	0.846643i	$0.678620\pi$
$488$	−13.1755	−0.596426
$489$	0	0
$490$	0	0
$491$	16.6266	0.750350	0.375175	−	0.926954i	$-0.377583\pi$
$491$	16.6266	0.750350	0.375175	+	0.926954i	$0.377583\pi$
$492$	0	0
$493$	−3.96111	−0.178399
$494$	−15.9325	−0.716835
$495$	0	0
$496$	14.6506	0.657829
$497$	0	0
$498$	0	0
$499$	−7.99304	−0.357818	−0.178909	−	0.983866i	$-0.557257\pi$
$499$	−7.99304	−0.357818	−0.178909	+	0.983866i	$0.557257\pi$
$500$	29.6568	1.32629
$501$	0	0
$502$	46.5691	2.07848
$503$	−24.6630	−1.09967	−0.549835	−	0.835273i	$-0.685310\pi$
$503$	−24.6630	−1.09967	−0.549835	+	0.835273i	$0.685310\pi$
$504$	0	0
$505$	−23.9861	−1.06737
$506$	0	0
$507$	0	0
$508$	−50.5933	−2.24471
$509$	−15.0668	−0.667825	−0.333912	−	0.942604i	$-0.608369\pi$
$509$	−15.0668	−0.667825	−0.333912	+	0.942604i	$0.608369\pi$
$510$	0	0
$511$	0	0
$512$	−28.2702	−1.24938
$513$	0	0
$514$	57.7710	2.54817
$515$	−29.3789	−1.29459
$516$	0	0
$517$	7.87189	0.346205
$518$	0	0
$519$	0	0
$520$	7.53341	0.330362
$521$	42.3051	1.85342	0.926710	−	0.375777i	$-0.122624\pi$
$521$	42.3051	1.85342	0.926710	+	0.375777i	$0.122624\pi$
$522$	0	0
$523$	5.84548	0.255605	0.127803	−	0.991800i	$-0.459208\pi$
$523$	5.84548	0.255605	0.127803	+	0.991800i	$0.459208\pi$
$524$	37.2577	1.62761
$525$	0	0
$526$	−25.1079	−1.09476
$527$	3.70488	0.161387
$528$	0	0
$529$	−23.0000	−1.00000
$530$	−9.43171	−0.409687
$531$	0	0
$532$	0	0
$533$	−23.7438	−1.02846
$534$	0	0
$535$	18.3190	0.792001
$536$	−4.21037	−0.181860
$537$	0	0
$538$	11.9177	0.513810
$539$	0	0
$540$	0	0
$541$	−40.2982	−1.73255	−0.866276	−	0.499565i	$-0.833493\pi$
$541$	−40.2982	−1.73255	−0.866276	+	0.499565i	$0.833493\pi$
$542$	12.0147	0.516076
$543$	0	0
$544$	−5.71737	−0.245130
$545$	0.178435	0.00764331
$546$	0	0
$547$	2.98903	0.127802	0.0639009	−	0.997956i	$-0.479646\pi$
$547$	2.98903	0.127802	0.0639009	+	0.997956i	$0.479646\pi$
$548$	35.7563	1.52743
$549$	0	0
$550$	−4.87189	−0.207738
$551$	9.08627	0.387088
$552$	0	0
$553$	0	0
$554$	53.9038	2.29015
$555$	0	0
$556$	−12.7019	−0.538682
$557$	−21.1127	−0.894573	−0.447286	−	0.894391i	$-0.647610\pi$
$557$	−21.1127	−0.894573	−0.447286	+	0.894391i	$0.647610\pi$
$558$	0	0
$559$	23.7438	1.00425
$560$	0	0
$561$	0	0
$562$	37.1638	1.56766
$563$	−29.8774	−1.25918	−0.629591	−	0.776926i	$-0.716778\pi$
$563$	−29.8774	−1.25918	−0.629591	+	0.776926i	$0.716778\pi$
$564$	0	0
$565$	−21.4567	−0.902689
$566$	−53.1468	−2.23393
$567$	0	0
$568$	−1.85244	−0.0777267
$569$	−0.186452	−0.00781648	−0.00390824	−	0.999992i	$-0.501244\pi$
$569$	−0.186452	−0.00781648	−0.00390824	+	0.999992i	$0.501244\pi$
$570$	0	0
$571$	3.06682	0.128342	0.0641712	−	0.997939i	$-0.479560\pi$
$571$	3.06682	0.128342	0.0641712	+	0.997939i	$0.479560\pi$
$572$	−11.3447	−0.474347
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	18.8495	0.784715	0.392357	−	0.919813i	$-0.371660\pi$
$577$	18.8495	0.784715	0.392357	+	0.919813i	$0.371660\pi$
$578$	34.8697	1.45039
$579$	0	0
$580$	−22.4688	−0.932967
$581$	0	0
$582$	0	0
$583$	2.71585	0.112479
$584$	10.4791	0.433627
$585$	0	0
$586$	−30.3510	−1.25379
$587$	8.11419	0.334908	0.167454	−	0.985880i	$-0.446445\pi$
$587$	8.11419	0.334908	0.167454	+	0.985880i	$0.446445\pi$
$588$	0	0
$589$	−8.49852	−0.350176
$590$	10.9604	0.451232
$591$	0	0
$592$	18.6483	0.766441
$593$	−6.36489	−0.261375	−0.130687	−	0.991424i	$-0.541718\pi$
$593$	−6.36489	−0.261375	−0.130687	+	0.991424i	$0.541718\pi$
$594$	0	0
$595$	0	0
$596$	−41.5870	−1.70347
$597$	0	0
$598$	0	0
$599$	−0.676959	−0.0276598	−0.0138299	−	0.999904i	$-0.504402\pi$
$599$	−0.676959	−0.0276598	−0.0138299	+	0.999904i	$0.504402\pi$
$600$	0	0
$601$	−22.6964	−0.925806	−0.462903	−	0.886409i	$-0.653192\pi$
$601$	−22.6964	−0.925806	−0.462903	+	0.886409i	$0.653192\pi$
$602$	0	0
$603$	0	0
$604$	11.6615	0.474501
$605$	−1.64207	−0.0667598
$606$	0	0
$607$	14.6141	0.593170	0.296585	−	0.955006i	$-0.404152\pi$
$607$	14.6141	0.593170	0.296585	+	0.955006i	$0.404152\pi$
$608$	13.1149	0.531880
$609$	0	0
$610$	45.7563	1.85262
$611$	−36.1142	−1.46102
$612$	0	0
$613$	−28.3899	−1.14666	−0.573328	−	0.819326i	$-0.694348\pi$
$613$	−28.3899	−1.14666	−0.573328	+	0.819326i	$0.694348\pi$
$614$	−27.7827	−1.12122
$615$	0	0
$616$	0	0
$617$	21.5404	0.867183	0.433591	−	0.901110i	$-0.357246\pi$
$617$	21.5404	0.867183	0.433591	+	0.901110i	$0.357246\pi$
$618$	0	0
$619$	23.7827	0.955906	0.477953	−	0.878385i	$-0.341379\pi$
$619$	23.7827	0.955906	0.477953	+	0.878385i	$0.341379\pi$
$620$	21.0154	0.844000
$621$	0	0
$622$	−30.3510	−1.21696
$623$	0	0
$624$	0	0
$625$	−8.17548	−0.327019
$626$	−50.2159	−2.00703
$627$	0	0
$628$	−27.2702	−1.08820
$629$	4.71585	0.188033
$630$	0	0
$631$	−40.4068	−1.60857	−0.804285	−	0.594244i	$-0.797451\pi$
$631$	−40.4068	−1.60857	−0.804285	+	0.594244i	$0.797451\pi$
$632$	12.4596	0.495617
$633$	0	0
$634$	−13.6615	−0.542568
$635$	33.5962	1.33323
$636$	0	0
$637$	0	0
$638$	11.7026	0.463312
$639$	0	0
$640$	−12.7695	−0.504758
$641$	−13.4876	−0.532726	−0.266363	−	0.963873i	$-0.585822\pi$
$641$	−13.4876	−0.532726	−0.266363	+	0.963873i	$0.585822\pi$
$642$	0	0
$643$	26.8245	1.05786	0.528928	−	0.848667i	$-0.322594\pi$
$643$	26.8245	1.05786	0.528928	+	0.848667i	$0.322594\pi$
$644$	0	0
$645$	0	0
$646$	2.48604	0.0978118
$647$	−42.4791	−1.67002	−0.835012	−	0.550231i	$-0.814540\pi$
$647$	−42.4791	−1.67002	−0.835012	+	0.550231i	$0.814540\pi$
$648$	0	0
$649$	−3.15604	−0.123885
$650$	22.3510	0.876677
$651$	0	0
$652$	13.9519	0.546399
$653$	−7.39281	−0.289303	−0.144652	−	0.989483i	$-0.546206\pi$
$653$	−7.39281	−0.289303	−0.144652	+	0.989483i	$0.546206\pi$
$654$	0	0
$655$	−24.7408	−0.966704
$656$	14.6506	0.572008
$657$	0	0
$658$	0	0
$659$	−18.2368	−0.710404	−0.355202	−	0.934790i	$-0.615588\pi$
$659$	−18.2368	−0.710404	−0.355202	+	0.934790i	$0.615588\pi$
$660$	0	0
$661$	8.16451	0.317563	0.158781	−	0.987314i	$-0.449243\pi$
$661$	8.16451	0.317563	0.158781	+	0.987314i	$0.449243\pi$
$662$	−66.1336	−2.57036
$663$	0	0
$664$	−9.17548	−0.356078
$665$	0	0
$666$	0	0
$667$	0	0
$668$	−39.5653	−1.53083
$669$	0	0
$670$	14.6219	0.564894
$671$	−13.1755	−0.508634
$672$	0	0
$673$	43.6212	1.68147	0.840737	−	0.541444i	$-0.182122\pi$
$673$	43.6212	1.68147	0.840737	+	0.541444i	$0.182122\pi$
$674$	34.4985	1.32883
$675$	0	0
$676$	19.8998	0.765377
$677$	23.1057	0.888025	0.444012	−	0.896021i	$-0.353555\pi$
$677$	23.1057	0.888025	0.444012	+	0.896021i	$0.353555\pi$
$678$	0	0
$679$	0	0
$680$	−1.17548	−0.0450777
$681$	0	0
$682$	−10.9457	−0.419131
$683$	−27.5653	−1.05476	−0.527379	−	0.849630i	$-0.676825\pi$
$683$	−27.5653	−1.05476	−0.527379	+	0.849630i	$0.676825\pi$
$684$	0	0
$685$	−23.7438	−0.907203
$686$	0	0
$687$	0	0
$688$	−14.6506	−0.558547
$689$	−12.4596	−0.474674
$690$	0	0
$691$	−5.85244	−0.222637	−0.111319	−	0.993785i	$-0.535507\pi$
$691$	−5.85244	−0.222637	−0.111319	+	0.993785i	$0.535507\pi$
$692$	37.1616	1.41267
$693$	0	0
$694$	−14.5155	−0.551000
$695$	8.43466	0.319945
$696$	0	0
$697$	3.70488	0.140332
$698$	−50.4574	−1.90984
$699$	0	0
$700$	0	0
$701$	−9.32304	−0.352126	−0.176063	−	0.984379i	$-0.556336\pi$
$701$	−9.32304	−0.352126	−0.176063	+	0.984379i	$0.556336\pi$
$702$	0	0
$703$	−10.8176	−0.407992
$704$	11.2298	0.423240
$705$	0	0
$706$	24.8223	0.934199
$707$	0	0
$708$	0	0
$709$	41.5629	1.56093	0.780463	−	0.625202i	$-0.214983\pi$
$709$	41.5629	1.56093	0.780463	+	0.625202i	$0.214983\pi$
$710$	6.43322	0.241435
$711$	0	0
$712$	4.71585	0.176734
$713$	0	0
$714$	0	0
$715$	7.53341	0.281734
$716$	−24.4596	−0.914099
$717$	0	0
$718$	−58.7578	−2.19282
$719$	31.4372	1.17241	0.586205	−	0.810162i	$-0.300621\pi$
$719$	31.4372	1.17241	0.586205	+	0.810162i	$0.300621\pi$
$720$	0	0
$721$	0	0
$722$	34.4806	1.28323
$723$	0	0
$724$	16.8759	0.627188
$725$	−12.7467	−0.473402
$726$	0	0
$727$	−14.2732	−0.529363	−0.264681	−	0.964336i	$-0.585267\pi$
$727$	−14.2732	−0.529363	−0.264681	+	0.964336i	$0.585267\pi$
$728$	0	0
$729$	0	0
$730$	−36.3921	−1.34693
$731$	−3.70488	−0.137030
$732$	0	0
$733$	−2.31207	−0.0853983	−0.0426992	−	0.999088i	$-0.513596\pi$
$733$	−2.31207	−0.0853983	−0.0426992	+	0.999088i	$0.513596\pi$
$734$	56.1600	2.07291
$735$	0	0
$736$	0	0
$737$	−4.21037	−0.155091
$738$	0	0
$739$	−26.0558	−0.958480	−0.479240	−	0.877684i	$-0.659088\pi$
$739$	−26.0558	−0.958480	−0.479240	+	0.877684i	$0.659088\pi$
$740$	26.7500	0.983350
$741$	0	0
$742$	0	0
$743$	−5.04737	−0.185170	−0.0925851	−	0.995705i	$-0.529513\pi$
$743$	−5.04737	−0.185170	−0.0925851	+	0.995705i	$0.529513\pi$
$744$	0	0
$745$	27.6157	1.01176
$746$	−10.7159	−0.392335
$747$	0	0
$748$	1.77018	0.0647244
$749$	0	0
$750$	0	0
$751$	−40.4387	−1.47563	−0.737815	−	0.675002i	$-0.764143\pi$
$751$	−40.4387	−1.47563	−0.737815	+	0.675002i	$0.764143\pi$
$752$	22.2834	0.812593
$753$	0	0
$754$	−53.6887	−1.95523
$755$	−7.74378	−0.281825
$756$	0	0
$757$	−24.4263	−0.887788	−0.443894	−	0.896079i	$-0.646403\pi$
$757$	−24.4263	−0.887788	−0.443894	+	0.896079i	$0.646403\pi$
$758$	64.6344	2.34763
$759$	0	0
$760$	2.69641	0.0978089
$761$	−5.18940	−0.188116	−0.0940579	−	0.995567i	$-0.529984\pi$
$761$	−5.18940	−0.188116	−0.0940579	+	0.995567i	$0.529984\pi$
$762$	0	0
$763$	0	0
$764$	−27.2702	−0.986602
$765$	0	0
$766$	19.3230	0.698170
$767$	14.4791	0.522809
$768$	0	0
$769$	−28.1670	−1.01573	−0.507864	−	0.861437i	$-0.669565\pi$
$769$	−28.1670	−1.01573	−0.507864	+	0.861437i	$0.669565\pi$
$770$	0	0
$771$	0	0
$772$	−19.4178	−0.698861
$773$	−38.2662	−1.37634	−0.688170	−	0.725549i	$-0.741586\pi$
$773$	−38.2662	−1.37634	−0.688170	+	0.725549i	$0.741586\pi$
$774$	0	0
$775$	11.9222	0.428259
$776$	13.6351	0.489472
$777$	0	0
$778$	44.9846	1.61277
$779$	−8.49852	−0.304491
$780$	0	0
$781$	−1.85244	−0.0662856
$782$	0	0
$783$	0	0
$784$	0	0
$785$	18.1087	0.646326
$786$	0	0
$787$	−13.4247	−0.478540	−0.239270	−	0.970953i	$-0.576908\pi$
$787$	−13.4247	−0.478540	−0.239270	+	0.970953i	$0.576908\pi$
$788$	−49.1880	−1.75225
$789$	0	0
$790$	−43.2702	−1.53949
$791$	0	0
$792$	0	0
$793$	60.4457	2.14649
$794$	−61.2438	−2.17346
$795$	0	0
$796$	58.1770	2.06203
$797$	22.3440	0.791465	0.395733	−	0.918366i	$-0.370491\pi$
$797$	22.3440	0.791465	0.395733	+	0.918366i	$0.370491\pi$
$798$	0	0
$799$	5.63511	0.199356
$800$	−18.3983	−0.650479
$801$	0	0
$802$	68.4193	2.41597
$803$	10.4791	0.369799
$804$	0	0
$805$	0	0
$806$	50.2159	1.76878
$807$	0	0
$808$	−14.6072	−0.513879
$809$	−31.8455	−1.11963	−0.559814	−	0.828618i	$-0.689127\pi$
$809$	−31.8455	−1.11963	−0.559814	+	0.828618i	$0.689127\pi$
$810$	0	0
$811$	8.70889	0.305811	0.152905	−	0.988241i	$-0.451137\pi$
$811$	8.70889	0.305811	0.152905	+	0.988241i	$0.451137\pi$
$812$	0	0
$813$	0	0
$814$	−13.9325	−0.488333
$815$	−9.26470	−0.324528
$816$	0	0
$817$	8.49852	0.297326
$818$	−4.88982	−0.170968
$819$	0	0
$820$	21.0154	0.733891
$821$	−6.45267	−0.225200	−0.112600	−	0.993640i	$-0.535918\pi$
$821$	−6.45267	−0.225200	−0.112600	+	0.993640i	$0.535918\pi$
$822$	0	0
$823$	25.3719	0.884410	0.442205	−	0.896914i	$-0.354196\pi$
$823$	25.3719	0.884410	0.442205	+	0.896914i	$0.354196\pi$
$824$	−17.8913	−0.623274
$825$	0	0
$826$	0	0
$827$	−14.0194	−0.487504	−0.243752	−	0.969838i	$-0.578378\pi$
$827$	−14.0194	−0.487504	−0.243752	+	0.969838i	$0.578378\pi$
$828$	0	0
$829$	−48.9970	−1.70174	−0.850869	−	0.525378i	$-0.823924\pi$
$829$	−48.9970	−1.70174	−0.850869	+	0.525378i	$0.823924\pi$
$830$	31.8649	1.10605
$831$	0	0
$832$	−51.5195	−1.78612
$833$	0	0
$834$	0	0
$835$	26.2732	0.909221
$836$	−4.06058	−0.140438
$837$	0	0
$838$	81.2680	2.80736
$839$	13.2647	0.457948	0.228974	−	0.973433i	$-0.426463\pi$
$839$	13.2647	0.457948	0.228974	+	0.973433i	$0.426463\pi$
$840$	0	0
$841$	1.61862	0.0558144
$842$	−32.7959	−1.13022
$843$	0	0
$844$	−10.5249	−0.362283
$845$	−13.2144	−0.454588
$846$	0	0
$847$	0	0
$848$	7.68793	0.264005
$849$	0	0
$850$	−3.48755	−0.119622
$851$	0	0
$852$	0	0
$853$	−4.25622	−0.145730	−0.0728651	−	0.997342i	$-0.523214\pi$
$853$	−4.25622	−0.145730	−0.0728651	+	0.997342i	$0.523214\pi$
$854$	0	0
$855$	0	0
$856$	11.1560	0.381305
$857$	23.5264	0.803648	0.401824	−	0.915717i	$-0.368376\pi$
$857$	23.5264	0.803648	0.401824	+	0.915717i	$0.368376\pi$
$858$	0	0
$859$	−31.0279	−1.05866	−0.529330	−	0.848416i	$-0.677556\pi$
$859$	−31.0279	−1.05866	−0.529330	+	0.848416i	$0.677556\pi$
$860$	−21.0154	−0.716620
$861$	0	0
$862$	80.2959	2.73489
$863$	46.3121	1.57648	0.788241	−	0.615367i	$-0.210992\pi$
$863$	46.3121	1.57648	0.788241	+	0.615367i	$0.210992\pi$
$864$	0	0
$865$	−24.6770	−0.839042
$866$	−21.3789	−0.726484
$867$	0	0
$868$	0	0
$869$	12.4596	0.422664
$870$	0	0
$871$	19.3161	0.654500
$872$	0.108664	0.00367984
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−12.8245	−0.433053	−0.216527	−	0.976277i	$-0.569473\pi$
$877$	−12.8245	−0.433053	−0.216527	+	0.976277i	$0.569473\pi$
$878$	47.6329	1.60753
$879$	0	0
$880$	−4.64832	−0.156695
$881$	56.9512	1.91873	0.959367	−	0.282160i	$-0.0910510\pi$
$881$	56.9512	1.91873	0.959367	+	0.282160i	$0.0910510\pi$
$882$	0	0
$883$	8.00696	0.269456	0.134728	−	0.990883i	$-0.456984\pi$
$883$	8.00696	0.269456	0.134728	+	0.990883i	$0.456984\pi$
$884$	−8.12115	−0.273144
$885$	0	0
$886$	43.7004	1.46814
$887$	−35.7827	−1.20146	−0.600732	−	0.799450i	$-0.705124\pi$
$887$	−35.7827	−1.20146	−0.600732	+	0.799450i	$0.705124\pi$
$888$	0	0
$889$	0	0
$890$	−16.3774	−0.548971
$891$	0	0
$892$	−9.25774	−0.309972
$893$	−12.9262	−0.432559
$894$	0	0
$895$	16.2423	0.542920
$896$	0	0
$897$	0	0
$898$	86.3106	2.88022
$899$	−28.6381	−0.955133
$900$	0	0
$901$	1.94415	0.0647690
$902$	−10.9457	−0.364451
$903$	0	0
$904$	−13.0668	−0.434596
$905$	−11.2064	−0.372512
$906$	0	0
$907$	5.43171	0.180357	0.0901784	−	0.995926i	$-0.471256\pi$
$907$	5.43171	0.180357	0.0901784	+	0.995926i	$0.471256\pi$
$908$	38.2982	1.27097
$909$	0	0
$910$	0	0
$911$	−16.6770	−0.552532	−0.276266	−	0.961081i	$-0.589097\pi$
$911$	−16.6770	−0.552532	−0.276266	+	0.961081i	$0.589097\pi$
$912$	0	0
$913$	−9.17548	−0.303664
$914$	60.0419	1.98601
$915$	0	0
$916$	−19.7827	−0.653638
$917$	0	0
$918$	0	0
$919$	14.4038	0.475137	0.237568	−	0.971371i	$-0.423650\pi$
$919$	14.4038	0.475137	0.237568	+	0.971371i	$0.423650\pi$
$920$	0	0
$921$	0	0
$922$	45.2966	1.49177
$923$	8.49852	0.279732
$924$	0	0
$925$	15.1755	0.498967
$926$	−43.8796	−1.44197
$927$	0	0
$928$	44.1942	1.45075
$929$	−1.89830	−0.0622811	−0.0311405	−	0.999515i	$-0.509914\pi$
$929$	−1.89830	−0.0622811	−0.0311405	+	0.999515i	$0.509914\pi$
$930$	0	0
$931$	0	0
$932$	13.7004	0.448772
$933$	0	0
$934$	56.8906	1.86152
$935$	−1.17548	−0.0384424
$936$	0	0
$937$	−13.1755	−0.430424	−0.215212	−	0.976567i	$-0.569044\pi$
$937$	−13.1755	−0.430424	−0.215212	+	0.976567i	$0.569044\pi$
$938$	0	0
$939$	0	0
$940$	31.9644	1.04256
$941$	−0.381842	−0.0124477	−0.00622385	−	0.999981i	$-0.501981\pi$
$941$	−0.381842	−0.0124477	−0.00622385	+	0.999981i	$0.501981\pi$
$942$	0	0
$943$	0	0
$944$	−8.93398	−0.290776
$945$	0	0
$946$	10.9457	0.355874
$947$	52.2284	1.69719	0.848597	−	0.529040i	$-0.177448\pi$
$947$	52.2284	1.69719	0.848597	+	0.529040i	$0.177448\pi$
$948$	0	0
$949$	−48.0753	−1.56059
$950$	8.00000	0.259554
$951$	0	0
$952$	0	0
$953$	45.7757	1.48282	0.741410	−	0.671052i	$-0.234157\pi$
$953$	45.7757	1.48282	0.741410	+	0.671052i	$0.234157\pi$
$954$	0	0
$955$	18.1087	0.585983
$956$	3.12739	0.101147
$957$	0	0
$958$	65.1616	2.10527
$959$	0	0
$960$	0	0
$961$	−4.21438	−0.135948
$962$	63.9185	2.06082
$963$	0	0
$964$	−59.0312	−1.90127
$965$	12.8943	0.415082
$966$	0	0
$967$	−15.3230	−0.492756	−0.246378	−	0.969174i	$-0.579240\pi$
$967$	−15.3230	−0.492756	−0.246378	+	0.969174i	$0.579240\pi$
$968$	−1.00000	−0.0321412
$969$	0	0
$970$	−47.3525	−1.52040
$971$	−12.0753	−0.387515	−0.193757	−	0.981049i	$-0.562067\pi$
$971$	−12.0753	−0.387515	−0.193757	+	0.981049i	$0.562067\pi$
$972$	0	0
$973$	0	0
$974$	49.6740	1.59166
$975$	0	0
$976$	−37.2966	−1.19384
$977$	39.3650	1.25940	0.629698	−	0.776840i	$-0.283178\pi$
$977$	39.3650	1.25940	0.629698	+	0.776840i	$0.283178\pi$
$978$	0	0
$979$	4.71585	0.150719
$980$	0	0
$981$	0	0
$982$	−35.1638	−1.12212
$983$	−10.5683	−0.337076	−0.168538	−	0.985695i	$-0.553905\pi$
$983$	−10.5683	−0.337076	−0.168538	+	0.985695i	$0.553905\pi$
$984$	0	0
$985$	32.6630	1.04073
$986$	8.37737	0.266790
$987$	0	0
$988$	18.6289	0.592663
$989$	0	0
$990$	0	0
$991$	42.8036	1.35970	0.679851	−	0.733350i	$-0.262044\pi$
$991$	42.8036	1.35970	0.679851	+	0.733350i	$0.262044\pi$
$992$	−41.3355	−1.31240
$993$	0	0
$994$	0	0
$995$	−38.6322	−1.22472
$996$	0	0
$997$	−10.3121	−0.326587	−0.163293	−	0.986578i	$-0.552212\pi$
$997$	−10.3121	−0.326587	−0.163293	+	0.986578i	$0.552212\pi$
$998$	16.9045	0.535104
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4851.2.a.bl.1.1		3
3.2	odd	2		1617.2.a.r.1.3	yes	3
7.6	odd	2		4851.2.a.bm.1.1		3
21.20	even	2		1617.2.a.q.1.3	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1617.2.a.q.1.3	✓	3	21.20	even	2
1617.2.a.r.1.3	yes	3	3.2	odd	2
4851.2.a.bl.1.1		3	1.1	even	1	trivial
4851.2.a.bm.1.1		3	7.6	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 4851 = 3^{2} \cdot 7^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4851.a (trivial)

\(f(q)\)	\(=\)	\(q-2.11491 q^{2} +2.47283 q^{4} -1.64207 q^{5} -1.00000 q^{8} +O(q^{10})\)	\(q-2.11491 q^{2} +2.47283 q^{4} -1.64207 q^{5} -1.00000 q^{8} +3.47283 q^{10} -1.00000 q^{11} +4.58774 q^{13} -2.83076 q^{16} -0.715853 q^{17} +1.64207 q^{19} -4.06058 q^{20} +2.11491 q^{22} -2.30359 q^{25} -9.70265 q^{26} +5.53341 q^{29} -5.17548 q^{31} +7.98680 q^{32} +1.51396 q^{34} -6.58774 q^{37} -3.47283 q^{38} +1.64207 q^{40} -5.17548 q^{41} +5.17548 q^{43} -2.47283 q^{44} -7.87189 q^{47} +4.87189 q^{50} +11.3447 q^{52} -2.71585 q^{53} +1.64207 q^{55} -11.7026 q^{58} +3.15604 q^{59} +13.1755 q^{61} +10.9457 q^{62} -11.2298 q^{64} -7.53341 q^{65} +4.21037 q^{67} -1.77018 q^{68} +1.85244 q^{71} -10.4791 q^{73} +13.9325 q^{74} +4.06058 q^{76} -12.4596 q^{79} +4.64832 q^{80} +10.9457 q^{82} +9.17548 q^{83} +1.17548 q^{85} -10.9457 q^{86} +1.00000 q^{88} -4.71585 q^{89} +16.6483 q^{94} -2.69641 q^{95} -13.6351 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 2 q^{4} - 4 q^{5} - 3 q^{8}+O(q^{10}) \) 3 * q + 2 * q^4 - 4 * q^5 - 3 * q^8	\( 3 q + 2 q^{4} - 4 q^{5} - 3 q^{8} + 5 q^{10} - 3 q^{11} + 2 q^{13} - 4 q^{16} - 4 q^{17} + 4 q^{19} + 5 q^{20} + 3 q^{25} - 11 q^{26} - 6 q^{29} + 8 q^{31} + 4 q^{32} - 10 q^{34} - 8 q^{37} - 5 q^{38} + 4 q^{40} + 8 q^{41} - 8 q^{43} - 2 q^{44} - 10 q^{47} + q^{50} + 15 q^{52} - 10 q^{53} + 4 q^{55} - 17 q^{58} - 6 q^{59} + 16 q^{61} + 22 q^{62} - 21 q^{64} + 8 q^{67} - 18 q^{68} + 2 q^{73} + 11 q^{74} - 5 q^{76} - 12 q^{79} - 15 q^{80} + 22 q^{82} + 4 q^{83} - 20 q^{85} - 22 q^{86} + 3 q^{88} - 16 q^{89} + 21 q^{94} - 18 q^{95} + 8 q^{97}+O(q^{100}) \) 3 * q + 2 * q^4 - 4 * q^5 - 3 * q^8 + 5 * q^10 - 3 * q^11 + 2 * q^13 - 4 * q^16 - 4 * q^17 + 4 * q^19 + 5 * q^20 + 3 * q^25 - 11 * q^26 - 6 * q^29 + 8 * q^31 + 4 * q^32 - 10 * q^34 - 8 * q^37 - 5 * q^38 + 4 * q^40 + 8 * q^41 - 8 * q^43 - 2 * q^44 - 10 * q^47 + q^50 + 15 * q^52 - 10 * q^53 + 4 * q^55 - 17 * q^58 - 6 * q^59 + 16 * q^61 + 22 * q^62 - 21 * q^64 + 8 * q^67 - 18 * q^68 + 2 * q^73 + 11 * q^74 - 5 * q^76 - 12 * q^79 - 15 * q^80 + 22 * q^82 + 4 * q^83 - 20 * q^85 - 22 * q^86 + 3 * q^88 - 16 * q^89 + 21 * q^94 - 18 * q^95 + 8 * q^97

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