LMFDB - Embedded newform 3000.2.a.j.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3000,2,Mod(1,3000)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3000, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("3000.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3000 = 2^{3} \cdot 3 \cdot 5^{3} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3000.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:	$23.9551206064$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.47025.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 22x^{2} + 13x + 109 $ x^4 - x^3 - 22x^2 + 13x + 109
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-2.35902$ of defining polynomial
Character	$\chi$	$=$	3000.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +2.97706 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +2.97706 q^{7} +1.00000 q^{9} +5.59509 q^{11} +6.43501 q^{13} -1.61803 q^{17} -3.07599 q^{19} -2.97706 q^{21} +3.59509 q^{23} -1.00000 q^{27} +7.19894 q^{29} +5.69402 q^{31} -5.59509 q^{33} -9.27493 q^{37} -6.43501 q^{39} +11.4492 q^{41} -0.145898 q^{43} -12.9071 q^{47} +1.86287 q^{49} +1.61803 q^{51} +0.419089 q^{53} +3.07599 q^{57} +5.54379 q^{59} -11.8471 q^{61} +2.97706 q^{63} -6.91525 q^{67} -3.59509 q^{69} -6.81698 q^{71} -3.33500 q^{73} +16.6569 q^{77} -6.30329 q^{79} +1.00000 q^{81} -8.45903 q^{83} -7.19894 q^{87} -2.31206 q^{89} +19.1574 q^{91} -5.69402 q^{93} +9.44919 q^{97} +5.59509 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{3} - 3 q^{7} + 4 q^{9}+O(q^{10}) $ 4 * q - 4 * q^3 - 3 * q^7 + 4 * q^9	$ 4 q - 4 q^{3} - 3 q^{7} + 4 q^{9} + 3 q^{11} + 3 q^{13} - 2 q^{17} + 3 q^{21} - 5 q^{23} - 4 q^{27} + 15 q^{29} + 6 q^{31} - 3 q^{33} - 11 q^{37} - 3 q^{39} + 13 q^{41} - 14 q^{43} - 11 q^{47} + 19 q^{49} + 2 q^{51} + 11 q^{53} + 8 q^{59} + 13 q^{61} - 3 q^{63} - 15 q^{67} + 5 q^{69} - 9 q^{71} - 7 q^{73} + 45 q^{77} + 21 q^{79} + 4 q^{81} - 7 q^{83} - 15 q^{87} + 12 q^{89} + 17 q^{91} - 6 q^{93} + 5 q^{97} + 3 q^{99}+O(q^{100}) $ 4 * q - 4 * q^3 - 3 * q^7 + 4 * q^9 + 3 * q^11 + 3 * q^13 - 2 * q^17 + 3 * q^21 - 5 * q^23 - 4 * q^27 + 15 * q^29 + 6 * q^31 - 3 * q^33 - 11 * q^37 - 3 * q^39 + 13 * q^41 - 14 * q^43 - 11 * q^47 + 19 * q^49 + 2 * q^51 + 11 * q^53 + 8 * q^59 + 13 * q^61 - 3 * q^63 - 15 * q^67 + 5 * q^69 - 9 * q^71 - 7 * q^73 + 45 * q^77 + 21 * q^79 + 4 * q^81 - 7 * q^83 - 15 * q^87 + 12 * q^89 + 17 * q^91 - 6 * q^93 + 5 * q^97 + 3 * q^99

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 $

$2$	0	0
$2$	0	0
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	2.97706	1.12522	0.562611	−	0.826722i	$-0.309797\pi$
$7$	2.97706	1.12522	0.562611	+	0.826722i	$0.309797\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	5.59509	1.68698	0.843492	−	0.537142i	$-0.180496\pi$
$11$	5.59509	1.68698	0.843492	+	0.537142i	$0.180496\pi$
$12$	0	0
$13$	6.43501	1.78475	0.892376	−	0.451293i	$-0.149037\pi$
$13$	6.43501	1.78475	0.892376	+	0.451293i	$0.149037\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−1.61803	−0.392431	−0.196215	−	0.980561i	$-0.562865\pi$
$17$	−1.61803	−0.392431	−0.196215	+	0.980561i	$0.562865\pi$
$18$	0	0
$19$	−3.07599	−0.705681	−0.352840	−	0.935684i	$-0.614784\pi$
$19$	−3.07599	−0.705681	−0.352840	+	0.935684i	$0.614784\pi$
$20$	0	0
$21$	−2.97706	−0.649647
$22$	0	0
$23$	3.59509	0.749628	0.374814	−	0.927100i	$-0.377707\pi$
$23$	3.59509	0.749628	0.374814	+	0.927100i	$0.377707\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	7.19894	1.33681	0.668405	−	0.743797i	$-0.266977\pi$
$29$	7.19894	1.33681	0.668405	+	0.743797i	$0.266977\pi$
$30$	0	0
$31$	5.69402	1.02268	0.511338	−	0.859379i	$-0.329150\pi$
$31$	5.69402	1.02268	0.511338	+	0.859379i	$0.329150\pi$
$32$	0	0
$33$	−5.59509	−0.973980
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−9.27493	−1.52479	−0.762395	−	0.647112i	$-0.775977\pi$
$37$	−9.27493	−1.52479	−0.762395	+	0.647112i	$0.775977\pi$
$38$	0	0
$39$	−6.43501	−1.03043
$40$	0	0
$41$	11.4492	1.78806	0.894032	−	0.448004i	$-0.147865\pi$
$41$	11.4492	1.78806	0.894032	+	0.448004i	$0.147865\pi$
$42$	0	0
$43$	−0.145898	−0.0222492	−0.0111246	−	0.999938i	$-0.503541\pi$
$43$	−0.145898	−0.0222492	−0.0111246	+	0.999938i	$0.503541\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−12.9071	−1.88270	−0.941351	−	0.337430i	$-0.890442\pi$
$47$	−12.9071	−1.88270	−0.941351	+	0.337430i	$0.890442\pi$
$48$	0	0
$49$	1.86287	0.266124
$50$	0	0
$51$	1.61803	0.226570
$52$	0	0
$53$	0.419089	0.0575663	0.0287832	−	0.999586i	$-0.490837\pi$
$53$	0.419089	0.0575663	0.0287832	+	0.999586i	$0.490837\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	3.07599	0.407425
$58$	0	0
$59$	5.54379	0.721740	0.360870	−	0.932616i	$-0.382480\pi$
$59$	5.54379	0.721740	0.360870	+	0.932616i	$0.382480\pi$
$60$	0	0
$61$	−11.8471	−1.51686	−0.758432	−	0.651753i	$-0.774034\pi$
$61$	−11.8471	−1.51686	−0.758432	+	0.651753i	$0.774034\pi$
$62$	0	0
$63$	2.97706	0.375074
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−6.91525	−0.844832	−0.422416	−	0.906402i	$-0.638818\pi$
$67$	−6.91525	−0.844832	−0.422416	+	0.906402i	$0.638818\pi$
$68$	0	0
$69$	−3.59509	−0.432798
$70$	0	0
$71$	−6.81698	−0.809027	−0.404513	−	0.914532i	$-0.632559\pi$
$71$	−6.81698	−0.809027	−0.404513	+	0.914532i	$0.632559\pi$
$72$	0	0
$73$	−3.33500	−0.390332	−0.195166	−	0.980770i	$-0.562525\pi$
$73$	−3.33500	−0.390332	−0.195166	+	0.980770i	$0.562525\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	16.6569	1.89823
$78$	0	0
$79$	−6.30329	−0.709176	−0.354588	−	0.935023i	$-0.615379\pi$
$79$	−6.30329	−0.709176	−0.354588	+	0.935023i	$0.615379\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−8.45903	−0.928500	−0.464250	−	0.885704i	$-0.653676\pi$
$83$	−8.45903	−0.928500	−0.464250	+	0.885704i	$0.653676\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−7.19894	−0.771808
$88$	0	0
$89$	−2.31206	−0.245078	−0.122539	−	0.992464i	$-0.539104\pi$
$89$	−2.31206	−0.245078	−0.122539	+	0.992464i	$0.539104\pi$
$90$	0	0
$91$	19.1574	2.00824
$92$	0	0
$93$	−5.69402	−0.590443
$94$	0	0
$95$	0	0
$96$	0	0
$97$	9.44919	0.959420	0.479710	−	0.877427i	$-0.340742\pi$
$97$	9.44919	0.959420	0.479710	+	0.877427i	$0.340742\pi$
$98$	0	0
$99$	5.59509	0.562328
$100$	0	0
$101$	16.2273	1.61468	0.807339	−	0.590089i	$-0.200907\pi$
$101$	16.2273	1.61468	0.807339	+	0.590089i	$0.200907\pi$
$102$	0	0
$103$	−10.2990	−1.01479	−0.507393	−	0.861715i	$-0.669391\pi$
$103$	−10.2990	−1.01479	−0.507393	+	0.861715i	$0.669391\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1.10877	0.107189	0.0535946	−	0.998563i	$-0.482932\pi$
$107$	1.10877	0.107189	0.0535946	+	0.998563i	$0.482932\pi$
$108$	0	0
$109$	−15.1821	−1.45418	−0.727090	−	0.686542i	$-0.759128\pi$
$109$	−15.1821	−1.45418	−0.727090	+	0.686542i	$0.759128\pi$
$110$	0	0
$111$	9.27493	0.880338
$112$	0	0
$113$	5.48090	0.515600	0.257800	−	0.966198i	$-0.417002\pi$
$113$	5.48090	0.515600	0.257800	+	0.966198i	$0.417002\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	6.43501	0.594917
$118$	0	0
$119$	−4.81698	−0.441572
$120$	0	0
$121$	20.3050	1.84591
$122$	0	0
$123$	−11.4492	−1.03234
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−3.39615	−0.301359	−0.150680	−	0.988583i	$-0.548146\pi$
$127$	−3.39615	−0.301359	−0.150680	+	0.988583i	$0.548146\pi$
$128$	0	0
$129$	0.145898	0.0128456
$130$	0	0
$131$	1.05130	0.0918528	0.0459264	−	0.998945i	$-0.485376\pi$
$131$	1.05130	0.0918528	0.0459264	+	0.998945i	$0.485376\pi$
$132$	0	0
$133$	−9.15740	−0.794047
$134$	0	0
$135$	0	0
$136$	0	0
$137$	4.11311	0.351407	0.175703	−	0.984443i	$-0.443780\pi$
$137$	4.11311	0.351407	0.175703	+	0.984443i	$0.443780\pi$
$138$	0	0
$139$	20.9559	1.77745	0.888726	−	0.458438i	$-0.151591\pi$
$139$	20.9559	1.77745	0.888726	+	0.458438i	$0.151591\pi$
$140$	0	0
$141$	12.9071	1.08698
$142$	0	0
$143$	36.0045	3.01085
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−1.86287	−0.153647
$148$	0	0
$149$	10.8781	0.891171	0.445585	−	0.895239i	$-0.352996\pi$
$149$	10.8781	0.891171	0.445585	+	0.895239i	$0.352996\pi$
$150$	0	0
$151$	15.9530	1.29824	0.649120	−	0.760686i	$-0.275137\pi$
$151$	15.9530	1.29824	0.649120	+	0.760686i	$0.275137\pi$
$152$	0	0
$153$	−1.61803	−0.130810
$154$	0	0
$155$	0	0
$156$	0	0
$157$	21.8744	1.74576	0.872882	−	0.487931i	$-0.162248\pi$
$157$	21.8744	1.74576	0.872882	+	0.487931i	$0.162248\pi$
$158$	0	0
$159$	−0.419089	−0.0332359
$160$	0	0
$161$	10.7028	0.843498
$162$	0	0
$163$	−18.1432	−1.42109	−0.710543	−	0.703654i	$-0.751551\pi$
$163$	−18.1432	−1.42109	−0.710543	+	0.703654i	$0.751551\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−5.18910	−0.401545	−0.200772	−	0.979638i	$-0.564345\pi$
$167$	−5.18910	−0.401545	−0.200772	+	0.979638i	$0.564345\pi$
$168$	0	0
$169$	28.4094	2.18534
$170$	0	0
$171$	−3.07599	−0.235227
$172$	0	0
$173$	18.9771	1.44280	0.721399	−	0.692519i	$-0.243499\pi$
$173$	18.9771	1.44280	0.721399	+	0.692519i	$0.243499\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−5.54379	−0.416697
$178$	0	0
$179$	−11.9744	−0.895007	−0.447503	−	0.894282i	$-0.647687\pi$
$179$	−11.9744	−0.895007	−0.447503	+	0.894282i	$0.647687\pi$
$180$	0	0
$181$	11.5881	0.861334	0.430667	−	0.902511i	$-0.358278\pi$
$181$	11.5881	0.861334	0.430667	+	0.902511i	$0.358278\pi$
$182$	0	0
$183$	11.8471	0.875761
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−9.05305	−0.662024
$188$	0	0
$189$	−2.97706	−0.216549
$190$	0	0
$191$	−12.8322	−0.928508	−0.464254	−	0.885702i	$-0.653678\pi$
$191$	−12.8322	−0.928508	−0.464254	+	0.885702i	$0.653678\pi$
$192$	0	0
$193$	−9.15024	−0.658648	−0.329324	−	0.944217i	$-0.606821\pi$
$193$	−9.15024	−0.658648	−0.329324	+	0.944217i	$0.606821\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	8.45187	0.602171	0.301086	−	0.953597i	$-0.402651\pi$
$197$	8.45187	0.602171	0.301086	+	0.953597i	$0.402651\pi$
$198$	0	0
$199$	−18.0231	−1.27762	−0.638811	−	0.769364i	$-0.720574\pi$
$199$	−18.0231	−1.27762	−0.638811	+	0.769364i	$0.720574\pi$
$200$	0	0
$201$	6.91525	0.487764
$202$	0	0
$203$	21.4317	1.50421
$204$	0	0
$205$	0	0
$206$	0	0
$207$	3.59509	0.249876
$208$	0	0
$209$	−17.2104	−1.19047
$210$	0	0
$211$	7.11245	0.489641	0.244821	−	0.969568i	$-0.421271\pi$
$211$	7.11245	0.489641	0.244821	+	0.969568i	$0.421271\pi$
$212$	0	0
$213$	6.81698	0.467092
$214$	0	0
$215$	0	0
$216$	0	0
$217$	16.9514	1.15074
$218$	0	0
$219$	3.33500	0.225359
$220$	0	0
$221$	−10.4121	−0.700392
$222$	0	0
$223$	−24.7383	−1.65660	−0.828300	−	0.560285i	$-0.810692\pi$
$223$	−24.7383	−1.65660	−0.828300	+	0.560285i	$0.810692\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−20.2581	−1.34458	−0.672288	−	0.740290i	$-0.734688\pi$
$227$	−20.2581	−1.34458	−0.672288	+	0.740290i	$0.734688\pi$
$228$	0	0
$229$	2.20002	0.145382	0.0726908	−	0.997355i	$-0.476841\pi$
$229$	2.20002	0.145382	0.0726908	+	0.997355i	$0.476841\pi$
$230$	0	0
$231$	−16.6569	−1.09594
$232$	0	0
$233$	1.55189	0.101667	0.0508337	−	0.998707i	$-0.483812\pi$
$233$	1.55189	0.101667	0.0508337	+	0.998707i	$0.483812\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	6.30329	0.409443
$238$	0	0
$239$	−5.98690	−0.387260	−0.193630	−	0.981075i	$-0.562026\pi$
$239$	−5.98690	−0.387260	−0.193630	+	0.981075i	$0.562026\pi$
$240$	0	0
$241$	−17.4368	−1.12320	−0.561600	−	0.827409i	$-0.689814\pi$
$241$	−17.4368	−1.12320	−0.561600	+	0.827409i	$0.689814\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−19.7940	−1.25946
$248$	0	0
$249$	8.45903	0.536069
$250$	0	0
$251$	15.4580	0.975698	0.487849	−	0.872928i	$-0.337782\pi$
$251$	15.4580	0.975698	0.487849	+	0.872928i	$0.337782\pi$
$252$	0	0
$253$	20.1149	1.26461
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−9.74881	−0.608114	−0.304057	−	0.952654i	$-0.598341\pi$
$257$	−9.74881	−0.608114	−0.304057	+	0.952654i	$0.598341\pi$
$258$	0	0
$259$	−27.6120	−1.71573
$260$	0	0
$261$	7.19894	0.445603
$262$	0	0
$263$	7.81590	0.481949	0.240975	−	0.970531i	$-0.422533\pi$
$263$	7.81590	0.481949	0.240975	+	0.970531i	$0.422533\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	2.31206	0.141496
$268$	0	0
$269$	14.3640	0.875790	0.437895	−	0.899026i	$-0.355724\pi$
$269$	14.3640	0.875790	0.437895	+	0.899026i	$0.355724\pi$
$270$	0	0
$271$	−7.68418	−0.466781	−0.233390	−	0.972383i	$-0.574982\pi$
$271$	−7.68418	−0.466781	−0.233390	+	0.972383i	$0.574982\pi$
$272$	0	0
$273$	−19.1574	−1.15946
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−4.92307	−0.295799	−0.147899	−	0.989002i	$-0.547251\pi$
$277$	−4.92307	−0.295799	−0.147899	+	0.989002i	$0.547251\pi$
$278$	0	0
$279$	5.69402	0.340892
$280$	0	0
$281$	−15.1371	−0.903006	−0.451503	−	0.892270i	$-0.649112\pi$
$281$	−15.1371	−0.903006	−0.451503	+	0.892270i	$0.649112\pi$
$282$	0	0
$283$	20.8914	1.24186	0.620931	−	0.783865i	$-0.286755\pi$
$283$	20.8914	1.24186	0.620931	+	0.783865i	$0.286755\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	34.0849	2.01197
$288$	0	0
$289$	−14.3820	−0.845998
$290$	0	0
$291$	−9.44919	−0.553921
$292$	0	0
$293$	13.2820	0.775940	0.387970	−	0.921672i	$-0.373176\pi$
$293$	13.2820	0.775940	0.387970	+	0.921672i	$0.373176\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−5.59509	−0.324660
$298$	0	0
$299$	23.1345	1.33790
$300$	0	0
$301$	−0.434347	−0.0250353
$302$	0	0
$303$	−16.2273	−0.932234
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−0.650819	−0.0371442	−0.0185721	−	0.999828i	$-0.505912\pi$
$307$	−0.650819	−0.0371442	−0.0185721	+	0.999828i	$0.505912\pi$
$308$	0	0
$309$	10.2990	0.585887
$310$	0	0
$311$	25.6128	1.45237	0.726183	−	0.687501i	$-0.241292\pi$
$311$	25.6128	1.45237	0.726183	+	0.687501i	$0.241292\pi$
$312$	0	0
$313$	−18.8372	−1.06474	−0.532372	−	0.846511i	$-0.678699\pi$
$313$	−18.8372	−1.06474	−0.532372	+	0.846511i	$0.678699\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−5.57649	−0.313207	−0.156603	−	0.987662i	$-0.550054\pi$
$317$	−5.57649	−0.313207	−0.156603	+	0.987662i	$0.550054\pi$
$318$	0	0
$319$	40.2787	2.25518
$320$	0	0
$321$	−1.10877	−0.0618858
$322$	0	0
$323$	4.97706	0.276931
$324$	0	0
$325$	0	0
$326$	0	0
$327$	15.1821	0.839571
$328$	0	0
$329$	−38.4253	−2.11846
$330$	0	0
$331$	−13.3088	−0.731518	−0.365759	−	0.930710i	$-0.619190\pi$
$331$	−13.3088	−0.731518	−0.365759	+	0.930710i	$0.619190\pi$
$332$	0	0
$333$	−9.27493	−0.508263
$334$	0	0
$335$	0	0
$336$	0	0
$337$	16.2122	0.883134	0.441567	−	0.897228i	$-0.354423\pi$
$337$	16.2122	0.883134	0.441567	+	0.897228i	$0.354423\pi$
$338$	0	0
$339$	−5.48090	−0.297682
$340$	0	0
$341$	31.8586	1.72524
$342$	0	0
$343$	−15.2935	−0.825774
$344$	0	0
$345$	0	0
$346$	0	0
$347$	7.18651	0.385792	0.192896	−	0.981219i	$-0.438212\pi$
$347$	7.18651	0.385792	0.192896	+	0.981219i	$0.438212\pi$
$348$	0	0
$349$	19.6390	1.05125	0.525625	−	0.850717i	$-0.323832\pi$
$349$	19.6390	1.05125	0.525625	+	0.850717i	$0.323832\pi$
$350$	0	0
$351$	−6.43501	−0.343476
$352$	0	0
$353$	10.6099	0.564710	0.282355	−	0.959310i	$-0.408884\pi$
$353$	10.6099	0.564710	0.282355	+	0.959310i	$0.408884\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	4.81698	0.254942
$358$	0	0
$359$	−30.3040	−1.59938	−0.799691	−	0.600412i	$-0.795003\pi$
$359$	−30.3040	−1.59938	−0.799691	+	0.600412i	$0.795003\pi$
$360$	0	0
$361$	−9.53828	−0.502015
$362$	0	0
$363$	−20.3050	−1.06574
$364$	0	0
$365$	0	0
$366$	0	0
$367$	1.16549	0.0608383	0.0304191	−	0.999537i	$-0.490316\pi$
$367$	1.16549	0.0608383	0.0304191	+	0.999537i	$0.490316\pi$
$368$	0	0
$369$	11.4492	0.596021
$370$	0	0
$371$	1.24765	0.0647749
$372$	0	0
$373$	2.49400	0.129135	0.0645673	−	0.997913i	$-0.479433\pi$
$373$	2.49400	0.129135	0.0645673	+	0.997913i	$0.479433\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	46.3253	2.38587
$378$	0	0
$379$	1.83492	0.0942534	0.0471267	−	0.998889i	$-0.484994\pi$
$379$	1.83492	0.0942534	0.0471267	+	0.998889i	$0.484994\pi$
$380$	0	0
$381$	3.39615	0.173990
$382$	0	0
$383$	−16.0646	−0.820864	−0.410432	−	0.911891i	$-0.634622\pi$
$383$	−16.0646	−0.820864	−0.410432	+	0.911891i	$0.634622\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−0.145898	−0.00741641
$388$	0	0
$389$	−25.1203	−1.27365	−0.636824	−	0.771009i	$-0.719752\pi$
$389$	−25.1203	−1.27365	−0.636824	+	0.771009i	$0.719752\pi$
$390$	0	0
$391$	−5.81698	−0.294177
$392$	0	0
$393$	−1.05130	−0.0530312
$394$	0	0
$395$	0	0
$396$	0	0
$397$	17.6798	0.887326	0.443663	−	0.896194i	$-0.353679\pi$
$397$	17.6798	0.887326	0.443663	+	0.896194i	$0.353679\pi$
$398$	0	0
$399$	9.15740	0.458443
$400$	0	0
$401$	2.43219	0.121458	0.0607289	−	0.998154i	$-0.480657\pi$
$401$	2.43219	0.121458	0.0607289	+	0.998154i	$0.480657\pi$
$402$	0	0
$403$	36.6411	1.82522
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−51.8941	−2.57230
$408$	0	0
$409$	−20.3509	−1.00629	−0.503144	−	0.864202i	$-0.667824\pi$
$409$	−20.3509	−1.00629	−0.503144	+	0.864202i	$0.667824\pi$
$410$	0	0
$411$	−4.11311	−0.202885
$412$	0	0
$413$	16.5042	0.812117
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−20.9559	−1.02621
$418$	0	0
$419$	24.0158	1.17325	0.586625	−	0.809859i	$-0.300456\pi$
$419$	24.0158	1.17325	0.586625	+	0.809859i	$0.300456\pi$
$420$	0	0
$421$	−32.6241	−1.59000	−0.795001	−	0.606608i	$-0.792530\pi$
$421$	−32.6241	−1.59000	−0.795001	+	0.606608i	$0.792530\pi$
$422$	0	0
$423$	−12.9071	−0.627567
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−35.2694	−1.70681
$428$	0	0
$429$	−36.0045	−1.73831
$430$	0	0
$431$	−8.72399	−0.420220	−0.210110	−	0.977678i	$-0.567382\pi$
$431$	−8.72399	−0.420220	−0.210110	+	0.977678i	$0.567382\pi$
$432$	0	0
$433$	−33.2307	−1.59697	−0.798483	−	0.602018i	$-0.794364\pi$
$433$	−33.2307	−1.59697	−0.798483	+	0.602018i	$0.794364\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−11.0585	−0.528998
$438$	0	0
$439$	22.1581	1.05755	0.528773	−	0.848763i	$-0.322652\pi$
$439$	22.1581	1.05755	0.528773	+	0.848763i	$0.322652\pi$
$440$	0	0
$441$	1.86287	0.0887079
$442$	0	0
$443$	−2.95085	−0.140199	−0.0700996	−	0.997540i	$-0.522332\pi$
$443$	−2.95085	−0.140199	−0.0700996	+	0.997540i	$0.522332\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−10.8781	−0.514518
$448$	0	0
$449$	7.48698	0.353333	0.176666	−	0.984271i	$-0.443469\pi$
$449$	7.48698	0.353333	0.176666	+	0.984271i	$0.443469\pi$
$450$	0	0
$451$	64.0593	3.01643
$452$	0	0
$453$	−15.9530	−0.749539
$454$	0	0
$455$	0	0
$456$	0	0
$457$	28.7258	1.34374	0.671868	−	0.740671i	$-0.265492\pi$
$457$	28.7258	1.34374	0.671868	+	0.740671i	$0.265492\pi$
$458$	0	0
$459$	1.61803	0.0755234
$460$	0	0
$461$	−16.0034	−0.745353	−0.372676	−	0.927961i	$-0.621560\pi$
$461$	−16.0034	−0.745353	−0.372676	+	0.927961i	$0.621560\pi$
$462$	0	0
$463$	−16.4852	−0.766134	−0.383067	−	0.923721i	$-0.625132\pi$
$463$	−16.4852	−0.766134	−0.383067	+	0.923721i	$0.625132\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−7.68700	−0.355712	−0.177856	−	0.984057i	$-0.556916\pi$
$467$	−7.68700	−0.355712	−0.177856	+	0.984057i	$0.556916\pi$
$468$	0	0
$469$	−20.5871	−0.950623
$470$	0	0
$471$	−21.8744	−1.00792
$472$	0	0
$473$	−0.816313	−0.0375341
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0.419089	0.0191888
$478$	0	0
$479$	16.0318	0.732514	0.366257	−	0.930514i	$-0.380639\pi$
$479$	16.0318	0.732514	0.366257	+	0.930514i	$0.380639\pi$
$480$	0	0
$481$	−59.6843	−2.72137
$482$	0	0
$483$	−10.7028	−0.486994
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−18.6066	−0.843145	−0.421573	−	0.906795i	$-0.638522\pi$
$487$	−18.6066	−0.843145	−0.421573	+	0.906795i	$0.638522\pi$
$488$	0	0
$489$	18.1432	0.820465
$490$	0	0
$491$	3.62788	0.163724	0.0818619	−	0.996644i	$-0.473913\pi$
$491$	3.62788	0.163724	0.0818619	+	0.996644i	$0.473913\pi$
$492$	0	0
$493$	−11.6481	−0.524606
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−20.2945	−0.910334
$498$	0	0
$499$	9.10435	0.407567	0.203783	−	0.979016i	$-0.434676\pi$
$499$	9.10435	0.407567	0.203783	+	0.979016i	$0.434676\pi$
$500$	0	0
$501$	5.18910	0.231832
$502$	0	0
$503$	1.80714	0.0805763	0.0402881	−	0.999188i	$-0.487172\pi$
$503$	1.80714	0.0805763	0.0402881	+	0.999188i	$0.487172\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−28.4094	−1.26171
$508$	0	0
$509$	25.9576	1.15055	0.575275	−	0.817960i	$-0.304895\pi$
$509$	25.9576	1.15055	0.575275	+	0.817960i	$0.304895\pi$
$510$	0	0
$511$	−9.92849	−0.439210
$512$	0	0
$513$	3.07599	0.135808
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−72.2167	−3.17609
$518$	0	0
$519$	−18.9771	−0.833000
$520$	0	0
$521$	−35.4236	−1.55193	−0.775967	−	0.630773i	$-0.782738\pi$
$521$	−35.4236	−1.55193	−0.775967	+	0.630773i	$0.782738\pi$
$522$	0	0
$523$	23.7691	1.03935	0.519675	−	0.854364i	$-0.326053\pi$
$523$	23.7691	1.03935	0.519675	+	0.854364i	$0.326053\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−9.21312	−0.401330
$528$	0	0
$529$	−10.0753	−0.438058
$530$	0	0
$531$	5.54379	0.240580
$532$	0	0
$533$	73.6757	3.19125
$534$	0	0
$535$	0	0
$536$	0	0
$537$	11.9744	0.516732
$538$	0	0
$539$	10.4229	0.448946
$540$	0	0
$541$	0.931170	0.0400341	0.0200171	−	0.999800i	$-0.493628\pi$
$541$	0.931170	0.0400341	0.0200171	+	0.999800i	$0.493628\pi$
$542$	0	0
$543$	−11.5881	−0.497292
$544$	0	0
$545$	0	0
$546$	0	0
$547$	28.6968	1.22698	0.613492	−	0.789701i	$-0.289764\pi$
$547$	28.6968	1.22698	0.613492	+	0.789701i	$0.289764\pi$
$548$	0	0
$549$	−11.8471	−0.505621
$550$	0	0
$551$	−22.1439	−0.943361
$552$	0	0
$553$	−18.7653	−0.797980
$554$	0	0
$555$	0	0
$556$	0	0
$557$	28.5273	1.20874	0.604371	−	0.796703i	$-0.293424\pi$
$557$	28.5273	1.20874	0.604371	+	0.796703i	$0.293424\pi$
$558$	0	0
$559$	−0.938856	−0.0397094
$560$	0	0
$561$	9.05305	0.382220
$562$	0	0
$563$	35.6411	1.50209	0.751047	−	0.660249i	$-0.229549\pi$
$563$	35.6411	1.50209	0.751047	+	0.660249i	$0.229549\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	2.97706	0.125025
$568$	0	0
$569$	28.8728	1.21041	0.605205	−	0.796070i	$-0.293091\pi$
$569$	28.8728	1.21041	0.605205	+	0.796070i	$0.293091\pi$
$570$	0	0
$571$	−16.7217	−0.699782	−0.349891	−	0.936790i	$-0.613781\pi$
$571$	−16.7217	−0.699782	−0.349891	+	0.936790i	$0.613781\pi$
$572$	0	0
$573$	12.8322	0.536074
$574$	0	0
$575$	0	0
$576$	0	0
$577$	11.5022	0.478844	0.239422	−	0.970916i	$-0.423042\pi$
$577$	11.5022	0.478844	0.239422	+	0.970916i	$0.423042\pi$
$578$	0	0
$579$	9.15024	0.380271
$580$	0	0
$581$	−25.1830	−1.04477
$582$	0	0
$583$	2.34484	0.0971135
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−22.5701	−0.931569	−0.465785	−	0.884898i	$-0.654228\pi$
$587$	−22.5701	−0.931569	−0.465785	+	0.884898i	$0.654228\pi$
$588$	0	0
$589$	−17.5148	−0.721683
$590$	0	0
$591$	−8.45187	−0.347664
$592$	0	0
$593$	−47.2237	−1.93924	−0.969622	−	0.244608i	$-0.921341\pi$
$593$	−47.2237	−1.93924	−0.969622	+	0.244608i	$0.921341\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	18.0231	0.737636
$598$	0	0
$599$	−2.79672	−0.114271	−0.0571354	−	0.998366i	$-0.518197\pi$
$599$	−2.79672	−0.114271	−0.0571354	+	0.998366i	$0.518197\pi$
$600$	0	0
$601$	−14.2935	−0.583042	−0.291521	−	0.956564i	$-0.594161\pi$
$601$	−14.2935	−0.583042	−0.291521	+	0.956564i	$0.594161\pi$
$602$	0	0
$603$	−6.91525	−0.281611
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−12.8876	−0.523090	−0.261545	−	0.965191i	$-0.584232\pi$
$607$	−12.8876	−0.523090	−0.261545	+	0.965191i	$0.584232\pi$
$608$	0	0
$609$	−21.4317	−0.868455
$610$	0	0
$611$	−83.0577	−3.36015
$612$	0	0
$613$	14.1722	0.572411	0.286206	−	0.958168i	$-0.407606\pi$
$613$	14.1722	0.572411	0.286206	+	0.958168i	$0.407606\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−10.7558	−0.433014	−0.216507	−	0.976281i	$-0.569466\pi$
$617$	−10.7558	−0.433014	−0.216507	+	0.976281i	$0.569466\pi$
$618$	0	0
$619$	−40.2143	−1.61635	−0.808175	−	0.588942i	$-0.799545\pi$
$619$	−40.2143	−1.61635	−0.808175	+	0.588942i	$0.799545\pi$
$620$	0	0
$621$	−3.59509	−0.144266
$622$	0	0
$623$	−6.88313	−0.275767
$624$	0	0
$625$	0	0
$626$	0	0
$627$	17.2104	0.687319
$628$	0	0
$629$	15.0072	0.598375
$630$	0	0
$631$	−3.16710	−0.126080	−0.0630401	−	0.998011i	$-0.520080\pi$
$631$	−3.16710	−0.126080	−0.0630401	+	0.998011i	$0.520080\pi$
$632$	0	0
$633$	−7.11245	−0.282694
$634$	0	0
$635$	0	0
$636$	0	0
$637$	11.9876	0.474965
$638$	0	0
$639$	−6.81698	−0.269676
$640$	0	0
$641$	−8.93559	−0.352935	−0.176467	−	0.984306i	$-0.556467\pi$
$641$	−8.93559	−0.352935	−0.176467	+	0.984306i	$0.556467\pi$
$642$	0	0
$643$	−17.5289	−0.691274	−0.345637	−	0.938368i	$-0.612337\pi$
$643$	−17.5289	−0.691274	−0.345637	+	0.938368i	$0.612337\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−0.0600668	−0.00236147	−0.00118073	−	0.999999i	$-0.500376\pi$
$647$	−0.0600668	−0.00236147	−0.00118073	+	0.999999i	$0.500376\pi$
$648$	0	0
$649$	31.0180	1.21756
$650$	0	0
$651$	−16.9514	−0.664379
$652$	0	0
$653$	5.63396	0.220474	0.110237	−	0.993905i	$-0.464839\pi$
$653$	5.63396	0.220474	0.110237	+	0.993905i	$0.464839\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−3.33500	−0.130111
$658$	0	0
$659$	−39.1355	−1.52450	−0.762252	−	0.647280i	$-0.775906\pi$
$659$	−39.1355	−1.52450	−0.762252	+	0.647280i	$0.775906\pi$
$660$	0	0
$661$	−13.8720	−0.539556	−0.269778	−	0.962922i	$-0.586950\pi$
$661$	−13.8720	−0.539556	−0.269778	+	0.962922i	$0.586950\pi$
$662$	0	0
$663$	10.4121	0.404371
$664$	0	0
$665$	0	0
$666$	0	0
$667$	25.8809	1.00211
$668$	0	0
$669$	24.7383	0.956438
$670$	0	0
$671$	−66.2855	−2.55892
$672$	0	0
$673$	18.5323	0.714369	0.357185	−	0.934034i	$-0.383737\pi$
$673$	18.5323	0.714369	0.357185	+	0.934034i	$0.383737\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	35.6361	1.36961	0.684804	−	0.728728i	$-0.259888\pi$
$677$	35.6361	1.36961	0.684804	+	0.728728i	$0.259888\pi$
$678$	0	0
$679$	28.1308	1.07956
$680$	0	0
$681$	20.2581	0.776291
$682$	0	0
$683$	30.4749	1.16609	0.583044	−	0.812440i	$-0.301861\pi$
$683$	30.4749	1.16609	0.583044	+	0.812440i	$0.301861\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−2.20002	−0.0839361
$688$	0	0
$689$	2.69685	0.102742
$690$	0	0
$691$	16.5208	0.628479	0.314240	−	0.949344i	$-0.398250\pi$
$691$	16.5208	0.628479	0.314240	+	0.949344i	$0.398250\pi$
$692$	0	0
$693$	16.6569	0.632743
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−18.5252	−0.701691
$698$	0	0
$699$	−1.55189	−0.0586977
$700$	0	0
$701$	6.68728	0.252575	0.126287	−	0.991994i	$-0.459694\pi$
$701$	6.68728	0.252575	0.126287	+	0.991994i	$0.459694\pi$
$702$	0	0
$703$	28.5296	1.07601
$704$	0	0
$705$	0	0
$706$	0	0
$707$	48.3096	1.81687
$708$	0	0
$709$	−4.84628	−0.182006	−0.0910029	−	0.995851i	$-0.529007\pi$
$709$	−4.84628	−0.182006	−0.0910029	+	0.995851i	$0.529007\pi$
$710$	0	0
$711$	−6.30329	−0.236392
$712$	0	0
$713$	20.4705	0.766627
$714$	0	0
$715$	0	0
$716$	0	0
$717$	5.98690	0.223585
$718$	0	0
$719$	36.6303	1.36608	0.683041	−	0.730380i	$-0.260657\pi$
$719$	36.6303	1.36608	0.683041	+	0.730380i	$0.260657\pi$
$720$	0	0
$721$	−30.6606	−1.14186
$722$	0	0
$723$	17.4368	0.648480
$724$	0	0
$725$	0	0
$726$	0	0
$727$	2.36143	0.0875807	0.0437903	−	0.999041i	$-0.486057\pi$
$727$	2.36143	0.0875807	0.0437903	+	0.999041i	$0.486057\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	0.236068	0.00873129
$732$	0	0
$733$	−3.19126	−0.117872	−0.0589359	−	0.998262i	$-0.518771\pi$
$733$	−3.19126	−0.117872	−0.0589359	+	0.998262i	$0.518771\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−38.6914	−1.42522
$738$	0	0
$739$	−53.3412	−1.96219	−0.981094	−	0.193530i	$-0.938006\pi$
$739$	−53.3412	−1.96219	−0.981094	+	0.193530i	$0.938006\pi$
$740$	0	0
$741$	19.7940	0.727152
$742$	0	0
$743$	37.2497	1.36656	0.683280	−	0.730157i	$-0.260553\pi$
$743$	37.2497	1.36656	0.683280	+	0.730157i	$0.260553\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−8.45903	−0.309500
$748$	0	0
$749$	3.30089	0.120612
$750$	0	0
$751$	39.5540	1.44334	0.721672	−	0.692235i	$-0.243374\pi$
$751$	39.5540	1.44334	0.721672	+	0.692235i	$0.243374\pi$
$752$	0	0
$753$	−15.4580	−0.563319
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−17.0039	−0.618017	−0.309009	−	0.951059i	$-0.599997\pi$
$757$	−17.0039	−0.618017	−0.309009	+	0.951059i	$0.599997\pi$
$758$	0	0
$759$	−20.1149	−0.730123
$760$	0	0
$761$	40.6662	1.47415	0.737075	−	0.675811i	$-0.236207\pi$
$761$	40.6662	1.47415	0.737075	+	0.675811i	$0.236207\pi$
$762$	0	0
$763$	−45.1979	−1.63627
$764$	0	0
$765$	0	0
$766$	0	0
$767$	35.6743	1.28813
$768$	0	0
$769$	4.31197	0.155494	0.0777468	−	0.996973i	$-0.475227\pi$
$769$	4.31197	0.155494	0.0777468	+	0.996973i	$0.475227\pi$
$770$	0	0
$771$	9.74881	0.351095
$772$	0	0
$773$	21.1022	0.758993	0.379497	−	0.925193i	$-0.376097\pi$
$773$	21.1022	0.758993	0.379497	+	0.925193i	$0.376097\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	27.6120	0.990575
$778$	0	0
$779$	−35.2176	−1.26180
$780$	0	0
$781$	−38.1416	−1.36481
$782$	0	0
$783$	−7.19894	−0.257269
$784$	0	0
$785$	0	0
$786$	0	0
$787$	35.6296	1.27006	0.635029	−	0.772488i	$-0.280988\pi$
$787$	35.6296	1.27006	0.635029	+	0.772488i	$0.280988\pi$
$788$	0	0
$789$	−7.81590	−0.278253
$790$	0	0
$791$	16.3169	0.580164
$792$	0	0
$793$	−76.2361	−2.70722
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−35.1353	−1.24455	−0.622277	−	0.782797i	$-0.713792\pi$
$797$	−35.1353	−1.24455	−0.622277	+	0.782797i	$0.713792\pi$
$798$	0	0
$799$	20.8842	0.738830
$800$	0	0
$801$	−2.31206	−0.0816926
$802$	0	0
$803$	−18.6596	−0.658484
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−14.3640	−0.505638
$808$	0	0
$809$	−0.0908359	−0.00319362	−0.00159681	−	0.999999i	$-0.500508\pi$
$809$	−0.0908359	−0.00319362	−0.00159681	+	0.999999i	$0.500508\pi$
$810$	0	0
$811$	−3.09125	−0.108548	−0.0542742	−	0.998526i	$-0.517285\pi$
$811$	−3.09125	−0.108548	−0.0542742	+	0.998526i	$0.517285\pi$
$812$	0	0
$813$	7.68418	0.269496
$814$	0	0
$815$	0	0
$816$	0	0
$817$	0.448781	0.0157009
$818$	0	0
$819$	19.1574	0.669414
$820$	0	0
$821$	26.8171	0.935924	0.467962	−	0.883749i	$-0.344988\pi$
$821$	26.8171	0.935924	0.467962	+	0.883749i	$0.344988\pi$
$822$	0	0
$823$	−8.66133	−0.301915	−0.150957	−	0.988540i	$-0.548236\pi$
$823$	−8.66133	−0.301915	−0.150957	+	0.988540i	$0.548236\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−54.9171	−1.90966	−0.954828	−	0.297160i	$-0.903961\pi$
$827$	−54.9171	−1.90966	−0.954828	+	0.297160i	$0.903961\pi$
$828$	0	0
$829$	27.2963	0.948039	0.474019	−	0.880514i	$-0.342803\pi$
$829$	27.2963	0.948039	0.474019	+	0.880514i	$0.342803\pi$
$830$	0	0
$831$	4.92307	0.170779
$832$	0	0
$833$	−3.01418	−0.104435
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−5.69402	−0.196814
$838$	0	0
$839$	26.1547	0.902961	0.451481	−	0.892281i	$-0.350896\pi$
$839$	26.1547	0.902961	0.451481	+	0.892281i	$0.350896\pi$
$840$	0	0
$841$	22.8248	0.787062
$842$	0	0
$843$	15.1371	0.521351
$844$	0	0
$845$	0	0
$846$	0	0
$847$	60.4492	2.07706
$848$	0	0
$849$	−20.8914	−0.716990
$850$	0	0
$851$	−33.3442	−1.14303
$852$	0	0
$853$	−15.7614	−0.539660	−0.269830	−	0.962908i	$-0.586967\pi$
$853$	−15.7614	−0.539660	−0.269830	+	0.962908i	$0.586967\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−8.76868	−0.299532	−0.149766	−	0.988721i	$-0.547852\pi$
$857$	−8.76868	−0.299532	−0.149766	+	0.988721i	$0.547852\pi$
$858$	0	0
$859$	−39.6170	−1.35171	−0.675857	−	0.737032i	$-0.736227\pi$
$859$	−39.6170	−1.35171	−0.675857	+	0.737032i	$0.736227\pi$
$860$	0	0
$861$	−34.0849	−1.16161
$862$	0	0
$863$	−15.7727	−0.536911	−0.268455	−	0.963292i	$-0.586513\pi$
$863$	−15.7727	−0.536911	−0.268455	+	0.963292i	$0.586513\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	14.3820	0.488437
$868$	0	0
$869$	−35.2675	−1.19637
$870$	0	0
$871$	−44.4997	−1.50781
$872$	0	0
$873$	9.44919	0.319807
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−52.2422	−1.76409	−0.882047	−	0.471162i	$-0.843835\pi$
$877$	−52.2422	−1.76409	−0.882047	+	0.471162i	$0.843835\pi$
$878$	0	0
$879$	−13.2820	−0.447989
$880$	0	0
$881$	13.3902	0.451127	0.225564	−	0.974228i	$-0.427578\pi$
$881$	13.3902	0.451127	0.225564	+	0.974228i	$0.427578\pi$
$882$	0	0
$883$	−57.4075	−1.93192	−0.965958	−	0.258698i	$-0.916707\pi$
$883$	−57.4075	−1.93192	−0.965958	+	0.258698i	$0.916707\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	36.0967	1.21201	0.606004	−	0.795462i	$-0.292772\pi$
$887$	36.0967	1.21201	0.606004	+	0.795462i	$0.292772\pi$
$888$	0	0
$889$	−10.1105	−0.339096
$890$	0	0
$891$	5.59509	0.187443
$892$	0	0
$893$	39.7023	1.32859
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−23.1345	−0.772437
$898$	0	0
$899$	40.9910	1.36713
$900$	0	0
$901$	−0.678101	−0.0225908
$902$	0	0
$903$	0.434347	0.0144542
$904$	0	0
$905$	0	0
$906$	0	0
$907$	7.07317	0.234861	0.117430	−	0.993081i	$-0.462534\pi$
$907$	7.07317	0.234861	0.117430	+	0.993081i	$0.462534\pi$
$908$	0	0
$909$	16.2273	0.538226
$910$	0	0
$911$	−15.6951	−0.520002	−0.260001	−	0.965608i	$-0.583723\pi$
$911$	−15.6951	−0.520002	−0.260001	+	0.965608i	$0.583723\pi$
$912$	0	0
$913$	−47.3291	−1.56636
$914$	0	0
$915$	0	0
$916$	0	0
$917$	3.12979	0.103355
$918$	0	0
$919$	20.4509	0.674614	0.337307	−	0.941395i	$-0.390484\pi$
$919$	20.4509	0.674614	0.337307	+	0.941395i	$0.390484\pi$
$920$	0	0
$921$	0.650819	0.0214452
$922$	0	0
$923$	−43.8673	−1.44391
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−10.2990	−0.338262
$928$	0	0
$929$	20.3367	0.667227	0.333613	−	0.942710i	$-0.391732\pi$
$929$	20.3367	0.667227	0.333613	+	0.942710i	$0.391732\pi$
$930$	0	0
$931$	−5.73016	−0.187798
$932$	0	0
$933$	−25.6128	−0.838524
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−60.3183	−1.97051	−0.985256	−	0.171086i	$-0.945273\pi$
$937$	−60.3183	−1.97051	−0.985256	+	0.171086i	$0.945273\pi$
$938$	0	0
$939$	18.8372	0.614730
$940$	0	0
$941$	−6.18625	−0.201666	−0.100833	−	0.994903i	$-0.532151\pi$
$941$	−6.18625	−0.201666	−0.100833	+	0.994903i	$0.532151\pi$
$942$	0	0
$943$	41.1609	1.34038
$944$	0	0
$945$	0	0
$946$	0	0
$947$	32.3809	1.05224	0.526120	−	0.850411i	$-0.323646\pi$
$947$	32.3809	1.05224	0.526120	+	0.850411i	$0.323646\pi$
$948$	0	0
$949$	−21.4608	−0.696646
$950$	0	0
$951$	5.57649	0.180830
$952$	0	0
$953$	−38.2037	−1.23754	−0.618770	−	0.785573i	$-0.712369\pi$
$953$	−38.2037	−1.23754	−0.618770	+	0.785573i	$0.712369\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−40.2787	−1.30203
$958$	0	0
$959$	12.2450	0.395411
$960$	0	0
$961$	1.42191	0.0458681
$962$	0	0
$963$	1.10877	0.0357298
$964$	0	0
$965$	0	0
$966$	0	0
$967$	31.7778	1.02191	0.510953	−	0.859609i	$-0.329293\pi$
$967$	31.7778	1.02191	0.510953	+	0.859609i	$0.329293\pi$
$968$	0	0
$969$	−4.97706	−0.159886
$970$	0	0
$971$	0.870691	0.0279418	0.0139709	−	0.999902i	$-0.495553\pi$
$971$	0.870691	0.0279418	0.0139709	+	0.999902i	$0.495553\pi$
$972$	0	0
$973$	62.3868	2.00003
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−14.3197	−0.458128	−0.229064	−	0.973411i	$-0.573567\pi$
$977$	−14.3197	−0.458128	−0.229064	+	0.973411i	$0.573567\pi$
$978$	0	0
$979$	−12.9362	−0.413442
$980$	0	0
$981$	−15.1821	−0.484727
$982$	0	0
$983$	12.4302	0.396461	0.198231	−	0.980155i	$-0.436481\pi$
$983$	12.4302	0.396461	0.198231	+	0.980155i	$0.436481\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	38.4253	1.22309
$988$	0	0
$989$	−0.524517	−0.0166787
$990$	0	0
$991$	27.1963	0.863918	0.431959	−	0.901893i	$-0.357823\pi$
$991$	27.1963	0.863918	0.431959	+	0.901893i	$0.357823\pi$
$992$	0	0
$993$	13.3088	0.422342
$994$	0	0
$995$	0	0
$996$	0	0
$997$	47.1334	1.49273	0.746364	−	0.665537i	$-0.231798\pi$
$997$	47.1334	1.49273	0.746364	+	0.665537i	$0.231798\pi$
$998$	0	0
$999$	9.27493	0.293446

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3000.2.a.j.1.4	✓	4
3.2	odd	2		9000.2.a.t.1.4		4
4.3	odd	2		6000.2.a.bl.1.1		4
5.2	odd	4		3000.2.f.h.1249.8		8
5.3	odd	4		3000.2.f.h.1249.1		8
5.4	even	2		3000.2.a.o.1.1	yes	4
15.14	odd	2		9000.2.a.y.1.1		4
20.3	even	4		6000.2.f.p.1249.8		8
20.7	even	4		6000.2.f.p.1249.1		8
20.19	odd	2		6000.2.a.bc.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3000.2.a.j.1.4	✓	4	1.1	even	1	trivial
3000.2.a.o.1.1	yes	4	5.4	even	2
3000.2.f.h.1249.1		8	5.3	odd	4
3000.2.f.h.1249.8		8	5.2	odd	4
6000.2.a.bc.1.4		4	20.19	odd	2
6000.2.a.bl.1.1		4	4.3	odd	2
6000.2.f.p.1249.1		8	20.7	even	4
6000.2.f.p.1249.8		8	20.3	even	4
9000.2.a.t.1.4		4	3.2	odd	2
9000.2.a.y.1.1		4	15.14	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 3000 = 2^{3} \cdot 3 \cdot 5^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3000.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +2.97706 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +2.97706 q^{7} +1.00000 q^{9} +5.59509 q^{11} +6.43501 q^{13} -1.61803 q^{17} -3.07599 q^{19} -2.97706 q^{21} +3.59509 q^{23} -1.00000 q^{27} +7.19894 q^{29} +5.69402 q^{31} -5.59509 q^{33} -9.27493 q^{37} -6.43501 q^{39} +11.4492 q^{41} -0.145898 q^{43} -12.9071 q^{47} +1.86287 q^{49} +1.61803 q^{51} +0.419089 q^{53} +3.07599 q^{57} +5.54379 q^{59} -11.8471 q^{61} +2.97706 q^{63} -6.91525 q^{67} -3.59509 q^{69} -6.81698 q^{71} -3.33500 q^{73} +16.6569 q^{77} -6.30329 q^{79} +1.00000 q^{81} -8.45903 q^{83} -7.19894 q^{87} -2.31206 q^{89} +19.1574 q^{91} -5.69402 q^{93} +9.44919 q^{97} +5.59509 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{3} - 3 q^{7} + 4 q^{9}+O(q^{10}) \) 4 * q - 4 * q^3 - 3 * q^7 + 4 * q^9	\( 4 q - 4 q^{3} - 3 q^{7} + 4 q^{9} + 3 q^{11} + 3 q^{13} - 2 q^{17} + 3 q^{21} - 5 q^{23} - 4 q^{27} + 15 q^{29} + 6 q^{31} - 3 q^{33} - 11 q^{37} - 3 q^{39} + 13 q^{41} - 14 q^{43} - 11 q^{47} + 19 q^{49} + 2 q^{51} + 11 q^{53} + 8 q^{59} + 13 q^{61} - 3 q^{63} - 15 q^{67} + 5 q^{69} - 9 q^{71} - 7 q^{73} + 45 q^{77} + 21 q^{79} + 4 q^{81} - 7 q^{83} - 15 q^{87} + 12 q^{89} + 17 q^{91} - 6 q^{93} + 5 q^{97} + 3 q^{99}+O(q^{100}) \) 4 * q - 4 * q^3 - 3 * q^7 + 4 * q^9 + 3 * q^11 + 3 * q^13 - 2 * q^17 + 3 * q^21 - 5 * q^23 - 4 * q^27 + 15 * q^29 + 6 * q^31 - 3 * q^33 - 11 * q^37 - 3 * q^39 + 13 * q^41 - 14 * q^43 - 11 * q^47 + 19 * q^49 + 2 * q^51 + 11 * q^53 + 8 * q^59 + 13 * q^61 - 3 * q^63 - 15 * q^67 + 5 * q^69 - 9 * q^71 - 7 * q^73 + 45 * q^77 + 21 * q^79 + 4 * q^81 - 7 * q^83 - 15 * q^87 + 12 * q^89 + 17 * q^91 - 6 * q^93 + 5 * q^97 + 3 * q^99

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