LMFDB - Embedded newform 5625.2.a.t.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5625,2,Mod(1,5625)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5625, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("5625.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5625 = 3^{2} \cdot 5^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5625.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:	$44.9158511370$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	8.8.5444000000.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} - 2x^{6} + 20x^{5} - 4x^{4} - 30x^{3} + 7x^{2} + 12x + 1 $ x^8 - 4x^7 - 2x^6 + 20x^5 - 4x^4 - 30x^3 + 7x^2 + 12*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 75)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-0.0898194$ of defining polynomial
Character	$\chi$	$=$	5625.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.08982 q^{2} -0.812294 q^{4} -3.08724 q^{7} +3.06489 q^{8} +O(q^{10})$	$q-1.08982 q^{2} -0.812294 q^{4} -3.08724 q^{7} +3.06489 q^{8} +1.14831 q^{11} +4.07954 q^{13} +3.36454 q^{14} -1.71559 q^{16} -4.62758 q^{17} -5.96899 q^{19} -1.25145 q^{22} +2.32568 q^{23} -4.44596 q^{26} +2.50775 q^{28} +5.28417 q^{29} -0.589279 q^{31} -4.26010 q^{32} +5.04322 q^{34} +11.3997 q^{37} +6.50512 q^{38} -9.49200 q^{41} +2.42954 q^{43} -0.932764 q^{44} -2.53458 q^{46} -6.04998 q^{47} +2.53108 q^{49} -3.31379 q^{52} +3.24380 q^{53} -9.46207 q^{56} -5.75879 q^{58} -3.18640 q^{59} +13.7452 q^{61} +0.642208 q^{62} +8.07392 q^{64} +3.15873 q^{67} +3.75895 q^{68} -6.46551 q^{71} +7.20998 q^{73} -12.4236 q^{74} +4.84857 q^{76} -3.54511 q^{77} -12.3374 q^{79} +10.3446 q^{82} -12.3941 q^{83} -2.64776 q^{86} +3.51944 q^{88} -1.08404 q^{89} -12.5945 q^{91} -1.88914 q^{92} +6.59339 q^{94} +4.52132 q^{97} -2.75842 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 4 q^{2} + 4 q^{4} + 8 q^{7} - 12 q^{8}+O(q^{10}) $ 8 * q - 4 * q^2 + 4 * q^4 + 8 * q^7 - 12 * q^8	$ 8 q - 4 q^{2} + 4 q^{4} + 8 q^{7} - 12 q^{8} - 2 q^{11} + 16 q^{13} - 6 q^{14} - 16 q^{17} - 14 q^{19} + 12 q^{22} - 14 q^{23} - 6 q^{26} + 16 q^{28} - 2 q^{29} - 22 q^{31} + 2 q^{32} - 12 q^{34} + 28 q^{37} + 16 q^{38} - 8 q^{41} + 20 q^{43} - 22 q^{44} - 2 q^{46} - 10 q^{47} + 16 q^{52} - 44 q^{53} - 30 q^{56} + 8 q^{58} - 14 q^{59} - 20 q^{61} - 16 q^{62} + 6 q^{64} + 16 q^{67} + 2 q^{68} - 16 q^{71} + 24 q^{73} - 26 q^{74} - 16 q^{76} - 46 q^{77} - 30 q^{79} + 16 q^{82} - 12 q^{83} - 32 q^{86} + 32 q^{88} - 16 q^{89} - 12 q^{91} + 2 q^{92} + 14 q^{94} + 16 q^{97} - 4 q^{98}+O(q^{100}) $ 8 * q - 4 * q^2 + 4 * q^4 + 8 * q^7 - 12 * q^8 - 2 * q^11 + 16 * q^13 - 6 * q^14 - 16 * q^17 - 14 * q^19 + 12 * q^22 - 14 * q^23 - 6 * q^26 + 16 * q^28 - 2 * q^29 - 22 * q^31 + 2 * q^32 - 12 * q^34 + 28 * q^37 + 16 * q^38 - 8 * q^41 + 20 * q^43 - 22 * q^44 - 2 * q^46 - 10 * q^47 + 16 * q^52 - 44 * q^53 - 30 * q^56 + 8 * q^58 - 14 * q^59 - 20 * q^61 - 16 * q^62 + 6 * q^64 + 16 * q^67 + 2 * q^68 - 16 * q^71 + 24 * q^73 - 26 * q^74 - 16 * q^76 - 46 * q^77 - 30 * q^79 + 16 * q^82 - 12 * q^83 - 32 * q^86 + 32 * q^88 - 16 * q^89 - 12 * q^91 + 2 * q^92 + 14 * q^94 + 16 * q^97 - 4 * q^98

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 $

$2$	−1.08982	−0.770619	−0.385309	−	0.922787i	$-0.625905\pi$
$2$	−1.08982	−0.770619	−0.385309	+	0.922787i	$0.625905\pi$
$3$	0	0
$3$	0	0
$4$	−0.812294	−0.406147
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−3.08724	−1.16687	−0.583434	−	0.812160i	$-0.698291\pi$
$7$	−3.08724	−1.16687	−0.583434	+	0.812160i	$0.698291\pi$
$8$	3.06489	1.08360
$9$	0	0
$10$	0	0
$11$	1.14831	0.346228	0.173114	−	0.984902i	$-0.444617\pi$
$11$	1.14831	0.346228	0.173114	+	0.984902i	$0.444617\pi$
$12$	0	0
$13$	4.07954	1.13146	0.565731	−	0.824590i	$-0.308594\pi$
$13$	4.07954	1.13146	0.565731	+	0.824590i	$0.308594\pi$
$14$	3.36454	0.899211
$15$	0	0
$16$	−1.71559	−0.428898
$17$	−4.62758	−1.12235	−0.561176	−	0.827696i	$-0.689651\pi$
$17$	−4.62758	−1.12235	−0.561176	+	0.827696i	$0.689651\pi$
$18$	0	0
$19$	−5.96899	−1.36938	−0.684690	−	0.728834i	$-0.740062\pi$
$19$	−5.96899	−1.36938	−0.684690	+	0.728834i	$0.740062\pi$
$20$	0	0
$21$	0	0
$22$	−1.25145	−0.266810
$23$	2.32568	0.484939	0.242469	−	0.970159i	$-0.422043\pi$
$23$	2.32568	0.484939	0.242469	+	0.970159i	$0.422043\pi$
$24$	0	0
$25$	0	0
$26$	−4.44596	−0.871925
$27$	0	0
$28$	2.50775	0.473920
$29$	5.28417	0.981245	0.490623	−	0.871372i	$-0.336769\pi$
$29$	5.28417	0.981245	0.490623	+	0.871372i	$0.336769\pi$
$30$	0	0
$31$	−0.589279	−0.105838	−0.0529188	−	0.998599i	$-0.516852\pi$
$31$	−0.589279	−0.105838	−0.0529188	+	0.998599i	$0.516852\pi$
$32$	−4.26010	−0.753086
$33$	0	0
$34$	5.04322	0.864906
$35$	0	0
$36$	0	0
$37$	11.3997	1.87409	0.937047	−	0.349204i	$-0.113548\pi$
$37$	11.3997	1.87409	0.937047	+	0.349204i	$0.113548\pi$
$38$	6.50512	1.05527
$39$	0	0
$40$	0	0
$41$	−9.49200	−1.48240	−0.741201	−	0.671283i	$-0.765743\pi$
$41$	−9.49200	−1.48240	−0.741201	+	0.671283i	$0.765743\pi$
$42$	0	0
$43$	2.42954	0.370501	0.185250	−	0.982691i	$-0.440690\pi$
$43$	2.42954	0.370501	0.185250	+	0.982691i	$0.440690\pi$
$44$	−0.932764	−0.140619
$45$	0	0
$46$	−2.53458	−0.373703
$47$	−6.04998	−0.882480	−0.441240	−	0.897389i	$-0.645461\pi$
$47$	−6.04998	−0.882480	−0.441240	+	0.897389i	$0.645461\pi$
$48$	0	0
$49$	2.53108	0.361583
$50$	0	0
$51$	0	0
$52$	−3.31379	−0.459539
$53$	3.24380	0.445570	0.222785	−	0.974868i	$-0.428485\pi$
$53$	3.24380	0.445570	0.222785	+	0.974868i	$0.428485\pi$
$54$	0	0
$55$	0	0
$56$	−9.46207	−1.26442
$57$	0	0
$58$	−5.75879	−0.756166
$59$	−3.18640	−0.414833	−0.207417	−	0.978253i	$-0.566506\pi$
$59$	−3.18640	−0.414833	−0.207417	+	0.978253i	$0.566506\pi$
$60$	0	0
$61$	13.7452	1.75989	0.879946	−	0.475074i	$-0.157579\pi$
$61$	13.7452	1.75989	0.879946	+	0.475074i	$0.157579\pi$
$62$	0.642208	0.0815605
$63$	0	0
$64$	8.07392	1.00924
$65$	0	0
$66$	0	0
$67$	3.15873	0.385901	0.192950	−	0.981209i	$-0.438194\pi$
$67$	3.15873	0.385901	0.192950	+	0.981209i	$0.438194\pi$
$68$	3.75895	0.455840
$69$	0	0
$70$	0	0
$71$	−6.46551	−0.767315	−0.383657	−	0.923475i	$-0.625336\pi$
$71$	−6.46551	−0.767315	−0.383657	+	0.923475i	$0.625336\pi$
$72$	0	0
$73$	7.20998	0.843864	0.421932	−	0.906627i	$-0.361352\pi$
$73$	7.20998	0.843864	0.421932	+	0.906627i	$0.361352\pi$
$74$	−12.4236	−1.44421
$75$	0	0
$76$	4.84857	0.556169
$77$	−3.54511	−0.404003
$78$	0	0
$79$	−12.3374	−1.38807	−0.694033	−	0.719944i	$-0.744168\pi$
$79$	−12.3374	−1.38807	−0.694033	+	0.719944i	$0.744168\pi$
$80$	0	0
$81$	0	0
$82$	10.3446	1.14237
$83$	−12.3941	−1.36043	−0.680214	−	0.733013i	$-0.738113\pi$
$83$	−12.3941	−1.36043	−0.680214	+	0.733013i	$0.738113\pi$
$84$	0	0
$85$	0	0
$86$	−2.64776	−0.285515
$87$	0	0
$88$	3.51944	0.375174
$89$	−1.08404	−0.114907	−0.0574537	−	0.998348i	$-0.518298\pi$
$89$	−1.08404	−0.114907	−0.0574537	+	0.998348i	$0.518298\pi$
$90$	0	0
$91$	−12.5945	−1.32027
$92$	−1.88914	−0.196956
$93$	0	0
$94$	6.59339	0.680056
$95$	0	0
$96$	0	0
$97$	4.52132	0.459071	0.229535	−	0.973300i	$-0.426279\pi$
$97$	4.52132	0.459071	0.229535	+	0.973300i	$0.426279\pi$
$98$	−2.75842	−0.278642
$99$	0	0
$100$	0	0
$101$	6.61332	0.658050	0.329025	−	0.944321i	$-0.393280\pi$
$101$	6.61332	0.658050	0.329025	+	0.944321i	$0.393280\pi$
$102$	0	0
$103$	4.20634	0.414463	0.207231	−	0.978292i	$-0.433555\pi$
$103$	4.20634	0.414463	0.207231	+	0.978292i	$0.433555\pi$
$104$	12.5034	1.22606
$105$	0	0
$106$	−3.53515	−0.343364
$107$	4.01195	0.387849	0.193925	−	0.981016i	$-0.437878\pi$
$107$	4.01195	0.387849	0.193925	+	0.981016i	$0.437878\pi$
$108$	0	0
$109$	9.09364	0.871013	0.435506	−	0.900186i	$-0.356569\pi$
$109$	9.09364	0.871013	0.435506	+	0.900186i	$0.356569\pi$
$110$	0	0
$111$	0	0
$112$	5.29645	0.500468
$113$	−3.75465	−0.353208	−0.176604	−	0.984282i	$-0.556511\pi$
$113$	−3.75465	−0.353208	−0.176604	+	0.984282i	$0.556511\pi$
$114$	0	0
$115$	0	0
$116$	−4.29230	−0.398530
$117$	0	0
$118$	3.47260	0.319678
$119$	14.2865	1.30964
$120$	0	0
$121$	−9.68139	−0.880126
$122$	−14.9798	−1.35621
$123$	0	0
$124$	0.478668	0.0429856
$125$	0	0
$126$	0	0
$127$	11.3583	1.00788	0.503942	−	0.863738i	$-0.331883\pi$
$127$	11.3583	1.00788	0.503942	+	0.863738i	$0.331883\pi$
$128$	−0.278920	−0.0246532
$129$	0	0
$130$	0	0
$131$	2.07849	0.181599	0.0907993	−	0.995869i	$-0.471058\pi$
$131$	2.07849	0.181599	0.0907993	+	0.995869i	$0.471058\pi$
$132$	0	0
$133$	18.4277	1.59789
$134$	−3.44245	−0.297382
$135$	0	0
$136$	−14.1830	−1.21618
$137$	19.4032	1.65773	0.828863	−	0.559452i	$-0.188988\pi$
$137$	19.4032	1.65773	0.828863	+	0.559452i	$0.188988\pi$
$138$	0	0
$139$	−17.1603	−1.45552	−0.727758	−	0.685834i	$-0.759438\pi$
$139$	−17.1603	−1.45552	−0.727758	+	0.685834i	$0.759438\pi$
$140$	0	0
$141$	0	0
$142$	7.04624	0.591307
$143$	4.68458	0.391744
$144$	0	0
$145$	0	0
$146$	−7.85758	−0.650298
$147$	0	0
$148$	−9.25988	−0.761157
$149$	−0.210127	−0.0172143	−0.00860714	−	0.999963i	$-0.502740\pi$
$149$	−0.210127	−0.0172143	−0.00860714	+	0.999963i	$0.502740\pi$
$150$	0	0
$151$	−4.05924	−0.330336	−0.165168	−	0.986265i	$-0.552817\pi$
$151$	−4.05924	−0.330336	−0.165168	+	0.986265i	$0.552817\pi$
$152$	−18.2943	−1.48386
$153$	0	0
$154$	3.86353	0.311332
$155$	0	0
$156$	0	0
$157$	0.440336	0.0351426	0.0175713	−	0.999846i	$-0.494407\pi$
$157$	0.440336	0.0351426	0.0175713	+	0.999846i	$0.494407\pi$
$158$	13.4455	1.06967
$159$	0	0
$160$	0	0
$161$	−7.17996	−0.565860
$162$	0	0
$163$	3.24109	0.253862	0.126931	−	0.991912i	$-0.459487\pi$
$163$	3.24109	0.253862	0.126931	+	0.991912i	$0.459487\pi$
$164$	7.71029	0.602073
$165$	0	0
$166$	13.5073	1.04837
$167$	15.9605	1.23506	0.617530	−	0.786547i	$-0.288133\pi$
$167$	15.9605	1.23506	0.617530	+	0.786547i	$0.288133\pi$
$168$	0	0
$169$	3.64267	0.280205
$170$	0	0
$171$	0	0
$172$	−1.97350	−0.150478
$173$	−0.561789	−0.0427121	−0.0213560	−	0.999772i	$-0.506798\pi$
$173$	−0.561789	−0.0427121	−0.0213560	+	0.999772i	$0.506798\pi$
$174$	0	0
$175$	0	0
$176$	−1.97003	−0.148497
$177$	0	0
$178$	1.18140	0.0885499
$179$	−15.9387	−1.19131	−0.595656	−	0.803239i	$-0.703108\pi$
$179$	−15.9387	−1.19131	−0.595656	+	0.803239i	$0.703108\pi$
$180$	0	0
$181$	−21.4875	−1.59715	−0.798577	−	0.601892i	$-0.794414\pi$
$181$	−21.4875	−1.59715	−0.798577	+	0.601892i	$0.794414\pi$
$182$	13.7258	1.01742
$183$	0	0
$184$	7.12797	0.525481
$185$	0	0
$186$	0	0
$187$	−5.31389	−0.388590
$188$	4.91436	0.358417
$189$	0	0
$190$	0	0
$191$	18.3939	1.33093	0.665467	−	0.746427i	$-0.268232\pi$
$191$	18.3939	1.33093	0.665467	+	0.746427i	$0.268232\pi$
$192$	0	0
$193$	−2.02523	−0.145780	−0.0728898	−	0.997340i	$-0.523222\pi$
$193$	−2.02523	−0.145780	−0.0728898	+	0.997340i	$0.523222\pi$
$194$	−4.92742	−0.353768
$195$	0	0
$196$	−2.05598	−0.146856
$197$	−11.5454	−0.822578	−0.411289	−	0.911505i	$-0.634921\pi$
$197$	−11.5454	−0.822578	−0.411289	+	0.911505i	$0.634921\pi$
$198$	0	0
$199$	22.9779	1.62886	0.814431	−	0.580260i	$-0.197049\pi$
$199$	22.9779	1.62886	0.814431	+	0.580260i	$0.197049\pi$
$200$	0	0
$201$	0	0
$202$	−7.20733	−0.507106
$203$	−16.3135	−1.14498
$204$	0	0
$205$	0	0
$206$	−4.58415	−0.319393
$207$	0	0
$208$	−6.99883	−0.485282
$209$	−6.85425	−0.474118
$210$	0	0
$211$	−6.88140	−0.473735	−0.236868	−	0.971542i	$-0.576121\pi$
$211$	−6.88140	−0.473735	−0.236868	+	0.971542i	$0.576121\pi$
$212$	−2.63492	−0.180967
$213$	0	0
$214$	−4.37230	−0.298884
$215$	0	0
$216$	0	0
$217$	1.81925	0.123499
$218$	−9.91043	−0.671219
$219$	0	0
$220$	0	0
$221$	−18.8784	−1.26990
$222$	0	0
$223$	8.12631	0.544178	0.272089	−	0.962272i	$-0.412286\pi$
$223$	8.12631	0.544178	0.272089	+	0.962272i	$0.412286\pi$
$224$	13.1520	0.878753
$225$	0	0
$226$	4.09189	0.272188
$227$	−25.6300	−1.70112	−0.850561	−	0.525876i	$-0.823738\pi$
$227$	−25.6300	−1.70112	−0.850561	+	0.525876i	$0.823738\pi$
$228$	0	0
$229$	1.23314	0.0814884	0.0407442	−	0.999170i	$-0.487027\pi$
$229$	1.23314	0.0814884	0.0407442	+	0.999170i	$0.487027\pi$
$230$	0	0
$231$	0	0
$232$	16.1954	1.06328
$233$	2.99987	0.196528	0.0982641	−	0.995160i	$-0.468671\pi$
$233$	2.99987	0.196528	0.0982641	+	0.995160i	$0.468671\pi$
$234$	0	0
$235$	0	0
$236$	2.58829	0.168483
$237$	0	0
$238$	−15.5697	−1.00923
$239$	−22.1341	−1.43174	−0.715869	−	0.698235i	$-0.753969\pi$
$239$	−22.1341	−1.43174	−0.715869	+	0.698235i	$0.753969\pi$
$240$	0	0
$241$	−13.4403	−0.865764	−0.432882	−	0.901451i	$-0.642503\pi$
$241$	−13.4403	−0.865764	−0.432882	+	0.901451i	$0.642503\pi$
$242$	10.5510	0.678242
$243$	0	0
$244$	−11.1651	−0.714774
$245$	0	0
$246$	0	0
$247$	−24.3507	−1.54940
$248$	−1.80608	−0.114686
$249$	0	0
$250$	0	0
$251$	−18.8799	−1.19169	−0.595843	−	0.803101i	$-0.703182\pi$
$251$	−18.8799	−1.19169	−0.595843	+	0.803101i	$0.703182\pi$
$252$	0	0
$253$	2.67060	0.167899
$254$	−12.3785	−0.776694
$255$	0	0
$256$	−15.8439	−0.990242
$257$	−7.06320	−0.440590	−0.220295	−	0.975433i	$-0.570702\pi$
$257$	−7.06320	−0.440590	−0.220295	+	0.975433i	$0.570702\pi$
$258$	0	0
$259$	−35.1936	−2.18682
$260$	0	0
$261$	0	0
$262$	−2.26518	−0.139943
$263$	−20.4356	−1.26011	−0.630057	−	0.776549i	$-0.716969\pi$
$263$	−20.4356	−1.26011	−0.630057	+	0.776549i	$0.716969\pi$
$264$	0	0
$265$	0	0
$266$	−20.0829	−1.23136
$267$	0	0
$268$	−2.56582	−0.156732
$269$	−21.1923	−1.29212	−0.646060	−	0.763287i	$-0.723584\pi$
$269$	−21.1923	−1.29212	−0.646060	+	0.763287i	$0.723584\pi$
$270$	0	0
$271$	9.95317	0.604612	0.302306	−	0.953211i	$-0.402244\pi$
$271$	9.95317	0.604612	0.302306	+	0.953211i	$0.402244\pi$
$272$	7.93903	0.481375
$273$	0	0
$274$	−21.1460	−1.27747
$275$	0	0
$276$	0	0
$277$	−18.7349	−1.12567	−0.562837	−	0.826568i	$-0.690290\pi$
$277$	−18.7349	−1.12567	−0.562837	+	0.826568i	$0.690290\pi$
$278$	18.7016	1.12165
$279$	0	0
$280$	0	0
$281$	−29.0916	−1.73546	−0.867730	−	0.497035i	$-0.834422\pi$
$281$	−29.0916	−1.73546	−0.867730	+	0.497035i	$0.834422\pi$
$282$	0	0
$283$	7.18007	0.426811	0.213405	−	0.976964i	$-0.431545\pi$
$283$	7.18007	0.426811	0.213405	+	0.976964i	$0.431545\pi$
$284$	5.25189	0.311643
$285$	0	0
$286$	−5.10534	−0.301885
$287$	29.3041	1.72977
$288$	0	0
$289$	4.41447	0.259675
$290$	0	0
$291$	0	0
$292$	−5.85662	−0.342733
$293$	1.79825	0.105055	0.0525276	−	0.998619i	$-0.483272\pi$
$293$	1.79825	0.105055	0.0525276	+	0.998619i	$0.483272\pi$
$294$	0	0
$295$	0	0
$296$	34.9388	2.03077
$297$	0	0
$298$	0.229001	0.0132656
$299$	9.48773	0.548689
$300$	0	0
$301$	−7.50057	−0.432326
$302$	4.42384	0.254563
$303$	0	0
$304$	10.2404	0.587324
$305$	0	0
$306$	0	0
$307$	−5.98864	−0.341790	−0.170895	−	0.985289i	$-0.554666\pi$
$307$	−5.98864	−0.341790	−0.170895	+	0.985289i	$0.554666\pi$
$308$	2.87967	0.164085
$309$	0	0
$310$	0	0
$311$	−23.8684	−1.35345	−0.676726	−	0.736235i	$-0.736602\pi$
$311$	−23.8684	−1.35345	−0.676726	+	0.736235i	$0.736602\pi$
$312$	0	0
$313$	−5.75913	−0.325525	−0.162763	−	0.986665i	$-0.552041\pi$
$313$	−5.75913	−0.325525	−0.162763	+	0.986665i	$0.552041\pi$
$314$	−0.479887	−0.0270816
$315$	0	0
$316$	10.0216	0.563758
$317$	−12.2481	−0.687922	−0.343961	−	0.938984i	$-0.611769\pi$
$317$	−12.2481	−0.687922	−0.343961	+	0.938984i	$0.611769\pi$
$318$	0	0
$319$	6.06786	0.339735
$320$	0	0
$321$	0	0
$322$	7.82486	0.436062
$323$	27.6220	1.53693
$324$	0	0
$325$	0	0
$326$	−3.53220	−0.195631
$327$	0	0
$328$	−29.0920	−1.60633
$329$	18.6778	1.02974
$330$	0	0
$331$	−6.02899	−0.331383	−0.165692	−	0.986178i	$-0.552986\pi$
$331$	−6.02899	−0.331383	−0.165692	+	0.986178i	$0.552986\pi$
$332$	10.0676	0.552534
$333$	0	0
$334$	−17.3941	−0.951761
$335$	0	0
$336$	0	0
$337$	−5.53083	−0.301284	−0.150642	−	0.988588i	$-0.548134\pi$
$337$	−5.53083	−0.301284	−0.150642	+	0.988588i	$0.548134\pi$
$338$	−3.96985	−0.215931
$339$	0	0
$340$	0	0
$341$	−0.676675	−0.0366440
$342$	0	0
$343$	13.7967	0.744949
$344$	7.44626	0.401476
$345$	0	0
$346$	0.612249	0.0329147
$347$	−25.9445	−1.39277	−0.696387	−	0.717666i	$-0.745210\pi$
$347$	−25.9445	−1.39277	−0.696387	+	0.717666i	$0.745210\pi$
$348$	0	0
$349$	−19.0025	−1.01718	−0.508591	−	0.861008i	$-0.669833\pi$
$349$	−19.0025	−1.01718	−0.508591	+	0.861008i	$0.669833\pi$
$350$	0	0
$351$	0	0
$352$	−4.89191	−0.260740
$353$	8.19134	0.435981	0.217990	−	0.975951i	$-0.430050\pi$
$353$	8.19134	0.435981	0.217990	+	0.975951i	$0.430050\pi$
$354$	0	0
$355$	0	0
$356$	0.880555	0.0466693
$357$	0	0
$358$	17.3703	0.918048
$359$	26.7857	1.41369	0.706847	−	0.707367i	$-0.250117\pi$
$359$	26.7857	1.41369	0.706847	+	0.707367i	$0.250117\pi$
$360$	0	0
$361$	16.6288	0.875202
$362$	23.4175	1.23080
$363$	0	0
$364$	10.2305	0.536222
$365$	0	0
$366$	0	0
$367$	−18.8533	−0.984135	−0.492068	−	0.870557i	$-0.663759\pi$
$367$	−18.8533	−0.984135	−0.492068	+	0.870557i	$0.663759\pi$
$368$	−3.98992	−0.207989
$369$	0	0
$370$	0	0
$371$	−10.0144	−0.519921
$372$	0	0
$373$	17.8477	0.924118	0.462059	−	0.886849i	$-0.347111\pi$
$373$	17.8477	0.924118	0.462059	+	0.886849i	$0.347111\pi$
$374$	5.79118	0.299455
$375$	0	0
$376$	−18.5425	−0.956259
$377$	21.5570	1.11024
$378$	0	0
$379$	12.5653	0.645438	0.322719	−	0.946495i	$-0.395403\pi$
$379$	12.5653	0.645438	0.322719	+	0.946495i	$0.395403\pi$
$380$	0	0
$381$	0	0
$382$	−20.0460	−1.02564
$383$	7.53739	0.385143	0.192571	−	0.981283i	$-0.438317\pi$
$383$	7.53739	0.385143	0.192571	+	0.981283i	$0.438317\pi$
$384$	0	0
$385$	0	0
$386$	2.20714	0.112340
$387$	0	0
$388$	−3.67264	−0.186450
$389$	13.7331	0.696296	0.348148	−	0.937440i	$-0.386811\pi$
$389$	13.7331	0.696296	0.348148	+	0.937440i	$0.386811\pi$
$390$	0	0
$391$	−10.7623	−0.544272
$392$	7.75749	0.391812
$393$	0	0
$394$	12.5824	0.633894
$395$	0	0
$396$	0	0
$397$	−13.0022	−0.652562	−0.326281	−	0.945273i	$-0.605796\pi$
$397$	−13.0022	−0.652562	−0.326281	+	0.945273i	$0.605796\pi$
$398$	−25.0418	−1.25523
$399$	0	0
$400$	0	0
$401$	6.47047	0.323120	0.161560	−	0.986863i	$-0.448347\pi$
$401$	6.47047	0.323120	0.161560	+	0.986863i	$0.448347\pi$
$402$	0	0
$403$	−2.40399	−0.119751
$404$	−5.37196	−0.267265
$405$	0	0
$406$	17.7788	0.882346
$407$	13.0903	0.648864
$408$	0	0
$409$	−1.11200	−0.0549850	−0.0274925	−	0.999622i	$-0.508752\pi$
$409$	−1.11200	−0.0549850	−0.0274925	+	0.999622i	$0.508752\pi$
$410$	0	0
$411$	0	0
$412$	−3.41678	−0.168333
$413$	9.83718	0.484056
$414$	0	0
$415$	0	0
$416$	−17.3793	−0.852088
$417$	0	0
$418$	7.46989	0.365364
$419$	−11.7932	−0.576136	−0.288068	−	0.957610i	$-0.593013\pi$
$419$	−11.7932	−0.576136	−0.288068	+	0.957610i	$0.593013\pi$
$420$	0	0
$421$	−10.1257	−0.493494	−0.246747	−	0.969080i	$-0.579362\pi$
$421$	−10.1257	−0.493494	−0.246747	+	0.969080i	$0.579362\pi$
$422$	7.49949	0.365069
$423$	0	0
$424$	9.94189	0.482821
$425$	0	0
$426$	0	0
$427$	−42.4348	−2.05356
$428$	−3.25888	−0.157524
$429$	0	0
$430$	0	0
$431$	5.69259	0.274202	0.137101	−	0.990557i	$-0.456221\pi$
$431$	5.69259	0.274202	0.137101	+	0.990557i	$0.456221\pi$
$432$	0	0
$433$	−33.0488	−1.58822	−0.794112	−	0.607771i	$-0.792064\pi$
$433$	−33.0488	−1.58822	−0.794112	+	0.607771i	$0.792064\pi$
$434$	−1.98265	−0.0951704
$435$	0	0
$436$	−7.38671	−0.353759
$437$	−13.8820	−0.664065
$438$	0	0
$439$	11.4637	0.547131	0.273566	−	0.961853i	$-0.411797\pi$
$439$	11.4637	0.547131	0.273566	+	0.961853i	$0.411797\pi$
$440$	0	0
$441$	0	0
$442$	20.5740	0.978607
$443$	−17.8993	−0.850422	−0.425211	−	0.905094i	$-0.639800\pi$
$443$	−17.8993	−0.850422	−0.425211	+	0.905094i	$0.639800\pi$
$444$	0	0
$445$	0	0
$446$	−8.85621	−0.419353
$447$	0	0
$448$	−24.9262	−1.17765
$449$	6.82040	0.321874	0.160937	−	0.986965i	$-0.448548\pi$
$449$	6.82040	0.321874	0.160937	+	0.986965i	$0.448548\pi$
$450$	0	0
$451$	−10.8998	−0.513249
$452$	3.04988	0.143454
$453$	0	0
$454$	27.9321	1.31092
$455$	0	0
$456$	0	0
$457$	2.76381	0.129286	0.0646429	−	0.997908i	$-0.479409\pi$
$457$	2.76381	0.129286	0.0646429	+	0.997908i	$0.479409\pi$
$458$	−1.34390	−0.0627965
$459$	0	0
$460$	0	0
$461$	8.96463	0.417525	0.208762	−	0.977966i	$-0.433057\pi$
$461$	8.96463	0.417525	0.208762	+	0.977966i	$0.433057\pi$
$462$	0	0
$463$	13.1617	0.611675	0.305837	−	0.952084i	$-0.401064\pi$
$463$	13.1617	0.611675	0.305837	+	0.952084i	$0.401064\pi$
$464$	−9.06547	−0.420854
$465$	0	0
$466$	−3.26932	−0.151448
$467$	21.8005	1.00880	0.504402	−	0.863469i	$-0.331713\pi$
$467$	21.8005	1.00880	0.504402	+	0.863469i	$0.331713\pi$
$468$	0	0
$469$	−9.75179	−0.450296
$470$	0	0
$471$	0	0
$472$	−9.76596	−0.449515
$473$	2.78986	0.128278
$474$	0	0
$475$	0	0
$476$	−11.6048	−0.531905
$477$	0	0
$478$	24.1222	1.10332
$479$	−42.7490	−1.95325	−0.976626	−	0.214947i	$-0.931042\pi$
$479$	−42.7490	−1.95325	−0.976626	+	0.214947i	$0.931042\pi$
$480$	0	0
$481$	46.5054	2.12046
$482$	14.6475	0.667174
$483$	0	0
$484$	7.86413	0.357460
$485$	0	0
$486$	0	0
$487$	11.3497	0.514305	0.257153	−	0.966371i	$-0.417216\pi$
$487$	11.3497	0.514305	0.257153	+	0.966371i	$0.417216\pi$
$488$	42.1275	1.90702
$489$	0	0
$490$	0	0
$491$	−16.6546	−0.751614	−0.375807	−	0.926698i	$-0.622634\pi$
$491$	−16.6546	−0.751614	−0.375807	+	0.926698i	$0.622634\pi$
$492$	0	0
$493$	−24.4529	−1.10130
$494$	26.5379	1.19400
$495$	0	0
$496$	1.01096	0.0453936
$497$	19.9606	0.895356
$498$	0	0
$499$	−12.2321	−0.547584	−0.273792	−	0.961789i	$-0.588278\pi$
$499$	−12.2321	−0.547584	−0.273792	+	0.961789i	$0.588278\pi$
$500$	0	0
$501$	0	0
$502$	20.5756	0.918336
$503$	−1.30967	−0.0583951	−0.0291976	−	0.999574i	$-0.509295\pi$
$503$	−1.30967	−0.0583951	−0.0291976	+	0.999574i	$0.509295\pi$
$504$	0	0
$505$	0	0
$506$	−2.91048	−0.129386
$507$	0	0
$508$	−9.22625	−0.409349
$509$	−17.7318	−0.785946	−0.392973	−	0.919550i	$-0.628553\pi$
$509$	−17.7318	−0.785946	−0.392973	+	0.919550i	$0.628553\pi$
$510$	0	0
$511$	−22.2590	−0.984679
$512$	17.8248	0.787752
$513$	0	0
$514$	7.69761	0.339527
$515$	0	0
$516$	0	0
$517$	−6.94725	−0.305540
$518$	38.3546	1.68521
$519$	0	0
$520$	0	0
$521$	−26.2616	−1.15054	−0.575270	−	0.817964i	$-0.695103\pi$
$521$	−26.2616	−1.15054	−0.575270	+	0.817964i	$0.695103\pi$
$522$	0	0
$523$	2.55674	0.111798	0.0558991	−	0.998436i	$-0.482197\pi$
$523$	2.55674	0.111798	0.0558991	+	0.998436i	$0.482197\pi$
$524$	−1.68834	−0.0737557
$525$	0	0
$526$	22.2711	0.971068
$527$	2.72693	0.118787
$528$	0	0
$529$	−17.5912	−0.764835
$530$	0	0
$531$	0	0
$532$	−14.9687	−0.648977
$533$	−38.7230	−1.67728
$534$	0	0
$535$	0	0
$536$	9.68118	0.418163
$537$	0	0
$538$	23.0958	0.995732
$539$	2.90646	0.125190
$540$	0	0
$541$	9.38916	0.403672	0.201836	−	0.979419i	$-0.435309\pi$
$541$	9.38916	0.403672	0.201836	+	0.979419i	$0.435309\pi$
$542$	−10.8472	−0.465925
$543$	0	0
$544$	19.7139	0.845228
$545$	0	0
$546$	0	0
$547$	−34.8549	−1.49029	−0.745144	−	0.666903i	$-0.767619\pi$
$547$	−34.8549	−1.49029	−0.745144	+	0.666903i	$0.767619\pi$
$548$	−15.7611	−0.673280
$549$	0	0
$550$	0	0
$551$	−31.5411	−1.34370
$552$	0	0
$553$	38.0886	1.61969
$554$	20.4177	0.867465
$555$	0	0
$556$	13.9392	0.591153
$557$	14.1466	0.599411	0.299705	−	0.954032i	$-0.403112\pi$
$557$	14.1466	0.599411	0.299705	+	0.954032i	$0.403112\pi$
$558$	0	0
$559$	9.91139	0.419207
$560$	0	0
$561$	0	0
$562$	31.7046	1.33738
$563$	−38.1854	−1.60932	−0.804661	−	0.593734i	$-0.797653\pi$
$563$	−38.1854	−1.60932	−0.804661	+	0.593734i	$0.797653\pi$
$564$	0	0
$565$	0	0
$566$	−7.82498	−0.328908
$567$	0	0
$568$	−19.8161	−0.831465
$569$	−29.6301	−1.24216	−0.621080	−	0.783747i	$-0.713306\pi$
$569$	−29.6301	−1.24216	−0.621080	+	0.783747i	$0.713306\pi$
$570$	0	0
$571$	32.6504	1.36638	0.683188	−	0.730242i	$-0.260593\pi$
$571$	32.6504	1.36638	0.683188	+	0.730242i	$0.260593\pi$
$572$	−3.80525	−0.159106
$573$	0	0
$574$	−31.9362	−1.33299
$575$	0	0
$576$	0	0
$577$	22.6433	0.942653	0.471326	−	0.881959i	$-0.343775\pi$
$577$	22.6433	0.942653	0.471326	+	0.881959i	$0.343775\pi$
$578$	−4.81097	−0.200110
$579$	0	0
$580$	0	0
$581$	38.2636	1.58744
$582$	0	0
$583$	3.72488	0.154269
$584$	22.0978	0.914414
$585$	0	0
$586$	−1.95977	−0.0809575
$587$	−41.5667	−1.71564	−0.857820	−	0.513950i	$-0.828182\pi$
$587$	−41.5667	−1.71564	−0.857820	+	0.513950i	$0.828182\pi$
$588$	0	0
$589$	3.51740	0.144932
$590$	0	0
$591$	0	0
$592$	−19.5572	−0.803795
$593$	−2.09050	−0.0858465	−0.0429233	−	0.999078i	$-0.513667\pi$
$593$	−2.09050	−0.0858465	−0.0429233	+	0.999078i	$0.513667\pi$
$594$	0	0
$595$	0	0
$596$	0.170685	0.00699152
$597$	0	0
$598$	−10.3399	−0.422830
$599$	−17.9768	−0.734511	−0.367255	−	0.930120i	$-0.619702\pi$
$599$	−17.9768	−0.734511	−0.367255	+	0.930120i	$0.619702\pi$
$600$	0	0
$601$	−1.11000	−0.0452778	−0.0226389	−	0.999744i	$-0.507207\pi$
$601$	−1.11000	−0.0452778	−0.0226389	+	0.999744i	$0.507207\pi$
$602$	8.17427	0.333158
$603$	0	0
$604$	3.29730	0.134165
$605$	0	0
$606$	0	0
$607$	−12.2310	−0.496441	−0.248220	−	0.968704i	$-0.579846\pi$
$607$	−12.2310	−0.496441	−0.248220	+	0.968704i	$0.579846\pi$
$608$	25.4285	1.03126
$609$	0	0
$610$	0	0
$611$	−24.6812	−0.998493
$612$	0	0
$613$	−12.9776	−0.524160	−0.262080	−	0.965046i	$-0.584409\pi$
$613$	−12.9776	−0.524160	−0.262080	+	0.965046i	$0.584409\pi$
$614$	6.52654	0.263390
$615$	0	0
$616$	−10.8654	−0.437779
$617$	−20.4921	−0.824981	−0.412490	−	0.910962i	$-0.635341\pi$
$617$	−20.4921	−0.824981	−0.412490	+	0.910962i	$0.635341\pi$
$618$	0	0
$619$	−41.0456	−1.64976	−0.824881	−	0.565307i	$-0.808758\pi$
$619$	−41.0456	−1.64976	−0.824881	+	0.565307i	$0.808758\pi$
$620$	0	0
$621$	0	0
$622$	26.0122	1.04300
$623$	3.34668	0.134082
$624$	0	0
$625$	0	0
$626$	6.27641	0.250856
$627$	0	0
$628$	−0.357682	−0.0142731
$629$	−52.7528	−2.10339
$630$	0	0
$631$	14.0872	0.560804	0.280402	−	0.959883i	$-0.409532\pi$
$631$	14.0872	0.560804	0.280402	+	0.959883i	$0.409532\pi$
$632$	−37.8128	−1.50411
$633$	0	0
$634$	13.3482	0.530126
$635$	0	0
$636$	0	0
$637$	10.3256	0.409117
$638$	−6.61287	−0.261806
$639$	0	0
$640$	0	0
$641$	28.4358	1.12315	0.561573	−	0.827427i	$-0.310197\pi$
$641$	28.4358	1.12315	0.561573	+	0.827427i	$0.310197\pi$
$642$	0	0
$643$	−10.8408	−0.427521	−0.213761	−	0.976886i	$-0.568571\pi$
$643$	−10.8408	−0.427521	−0.213761	+	0.976886i	$0.568571\pi$
$644$	5.83223	0.229822
$645$	0	0
$646$	−30.1029	−1.18438
$647$	34.6388	1.36179	0.680896	−	0.732380i	$-0.261590\pi$
$647$	34.6388	1.36179	0.680896	+	0.732380i	$0.261590\pi$
$648$	0	0
$649$	−3.65897	−0.143627
$650$	0	0
$651$	0	0
$652$	−2.63272	−0.103105
$653$	4.15506	0.162600	0.0813000	−	0.996690i	$-0.474093\pi$
$653$	4.15506	0.162600	0.0813000	+	0.996690i	$0.474093\pi$
$654$	0	0
$655$	0	0
$656$	16.2844	0.635799
$657$	0	0
$658$	−20.3554	−0.793536
$659$	4.01424	0.156373	0.0781863	−	0.996939i	$-0.475087\pi$
$659$	4.01424	0.156373	0.0781863	+	0.996939i	$0.475087\pi$
$660$	0	0
$661$	19.6667	0.764944	0.382472	−	0.923967i	$-0.375073\pi$
$661$	19.6667	0.764944	0.382472	+	0.923967i	$0.375073\pi$
$662$	6.57051	0.255370
$663$	0	0
$664$	−37.9866	−1.47416
$665$	0	0
$666$	0	0
$667$	12.2893	0.475844
$668$	−12.9646	−0.501616
$669$	0	0
$670$	0	0
$671$	15.7837	0.609324
$672$	0	0
$673$	−41.4463	−1.59764	−0.798819	−	0.601571i	$-0.794542\pi$
$673$	−41.4463	−1.59764	−0.798819	+	0.601571i	$0.794542\pi$
$674$	6.02761	0.232175
$675$	0	0
$676$	−2.95892	−0.113804
$677$	−18.1837	−0.698856	−0.349428	−	0.936963i	$-0.613624\pi$
$677$	−18.1837	−0.698856	−0.349428	+	0.936963i	$0.613624\pi$
$678$	0	0
$679$	−13.9584	−0.535675
$680$	0	0
$681$	0	0
$682$	0.737453	0.0282385
$683$	−28.4523	−1.08870	−0.544348	−	0.838859i	$-0.683223\pi$
$683$	−28.4523	−1.08870	−0.544348	+	0.838859i	$0.683223\pi$
$684$	0	0
$685$	0	0
$686$	−15.0359	−0.574072
$687$	0	0
$688$	−4.16809	−0.158907
$689$	13.2332	0.504145
$690$	0	0
$691$	24.4175	0.928884	0.464442	−	0.885603i	$-0.346255\pi$
$691$	24.4175	0.928884	0.464442	+	0.885603i	$0.346255\pi$
$692$	0.456338	0.0173474
$693$	0	0
$694$	28.2748	1.07330
$695$	0	0
$696$	0	0
$697$	43.9250	1.66378
$698$	20.7093	0.783859
$699$	0	0
$700$	0	0
$701$	25.0371	0.945638	0.472819	−	0.881159i	$-0.343236\pi$
$701$	25.0371	0.945638	0.472819	+	0.881159i	$0.343236\pi$
$702$	0	0
$703$	−68.0445	−2.56635
$704$	9.27136	0.349428
$705$	0	0
$706$	−8.92708	−0.335975
$707$	−20.4170	−0.767858
$708$	0	0
$709$	36.5633	1.37316	0.686582	−	0.727052i	$-0.259110\pi$
$709$	36.5633	1.37316	0.686582	+	0.727052i	$0.259110\pi$
$710$	0	0
$711$	0	0
$712$	−3.32245	−0.124514
$713$	−1.37048	−0.0513248
$714$	0	0
$715$	0	0
$716$	12.9469	0.483848
$717$	0	0
$718$	−29.1915	−1.08942
$719$	7.23571	0.269847	0.134923	−	0.990856i	$-0.456921\pi$
$719$	7.23571	0.269847	0.134923	+	0.990856i	$0.456921\pi$
$720$	0	0
$721$	−12.9860	−0.483624
$722$	−18.1224	−0.674447
$723$	0	0
$724$	17.4542	0.648679
$725$	0	0
$726$	0	0
$727$	13.3099	0.493638	0.246819	−	0.969062i	$-0.420615\pi$
$727$	13.3099	0.493638	0.246819	+	0.969062i	$0.420615\pi$
$728$	−38.6009	−1.43065
$729$	0	0
$730$	0	0
$731$	−11.2429	−0.415832
$732$	0	0
$733$	−28.7794	−1.06299	−0.531496	−	0.847061i	$-0.678370\pi$
$733$	−28.7794	−1.06299	−0.531496	+	0.847061i	$0.678370\pi$
$734$	20.5467	0.758393
$735$	0	0
$736$	−9.90764	−0.365201
$737$	3.62720	0.133610
$738$	0	0
$739$	40.3208	1.48322	0.741612	−	0.670829i	$-0.234061\pi$
$739$	40.3208	1.48322	0.741612	+	0.670829i	$0.234061\pi$
$740$	0	0
$741$	0	0
$742$	10.9139	0.400661
$743$	−27.8114	−1.02030	−0.510151	−	0.860085i	$-0.670411\pi$
$743$	−27.8114	−1.02030	−0.510151	+	0.860085i	$0.670411\pi$
$744$	0	0
$745$	0	0
$746$	−19.4507	−0.712142
$747$	0	0
$748$	4.31644	0.157825
$749$	−12.3859	−0.452569
$750$	0	0
$751$	−11.4124	−0.416443	−0.208221	−	0.978082i	$-0.566767\pi$
$751$	−11.4124	−0.416443	−0.208221	+	0.978082i	$0.566767\pi$
$752$	10.3793	0.378494
$753$	0	0
$754$	−23.4932	−0.855573
$755$	0	0
$756$	0	0
$757$	13.7637	0.500251	0.250126	−	0.968213i	$-0.419528\pi$
$757$	13.7637	0.500251	0.250126	+	0.968213i	$0.419528\pi$
$758$	−13.6939	−0.497386
$759$	0	0
$760$	0	0
$761$	14.7409	0.534359	0.267179	−	0.963647i	$-0.413908\pi$
$761$	14.7409	0.534359	0.267179	+	0.963647i	$0.413908\pi$
$762$	0	0
$763$	−28.0743	−1.01636
$764$	−14.9412	−0.540555
$765$	0	0
$766$	−8.21440	−0.296798
$767$	−12.9990	−0.469368
$768$	0	0
$769$	−2.43753	−0.0878997	−0.0439498	−	0.999034i	$-0.513994\pi$
$769$	−2.43753	−0.0878997	−0.0439498	+	0.999034i	$0.513994\pi$
$770$	0	0
$771$	0	0
$772$	1.64509	0.0592079
$773$	−25.3341	−0.911204	−0.455602	−	0.890184i	$-0.650576\pi$
$773$	−25.3341	−0.911204	−0.455602	+	0.890184i	$0.650576\pi$
$774$	0	0
$775$	0	0
$776$	13.8574	0.497450
$777$	0	0
$778$	−14.9666	−0.536579
$779$	56.6577	2.02997
$780$	0	0
$781$	−7.42441	−0.265666
$782$	11.7289	0.419426
$783$	0	0
$784$	−4.34230	−0.155082
$785$	0	0
$786$	0	0
$787$	−38.6480	−1.37765	−0.688826	−	0.724927i	$-0.741873\pi$
$787$	−38.6480	−1.37765	−0.688826	+	0.724927i	$0.741873\pi$
$788$	9.37829	0.334088
$789$	0	0
$790$	0	0
$791$	11.5915	0.412147
$792$	0	0
$793$	56.0741	1.99125
$794$	14.1700	0.502876
$795$	0	0
$796$	−18.6648	−0.661557
$797$	−52.1020	−1.84555	−0.922774	−	0.385342i	$-0.874083\pi$
$797$	−52.1020	−1.84555	−0.922774	+	0.385342i	$0.874083\pi$
$798$	0	0
$799$	27.9968	0.990454
$800$	0	0
$801$	0	0
$802$	−7.05165	−0.249002
$803$	8.27929	0.292170
$804$	0	0
$805$	0	0
$806$	2.61991	0.0922825
$807$	0	0
$808$	20.2691	0.713065
$809$	−31.7938	−1.11781	−0.558905	−	0.829231i	$-0.688778\pi$
$809$	−31.7938	−1.11781	−0.558905	+	0.829231i	$0.688778\pi$
$810$	0	0
$811$	−31.1213	−1.09282	−0.546409	−	0.837518i	$-0.684006\pi$
$811$	−31.1213	−1.09282	−0.546409	+	0.837518i	$0.684006\pi$
$812$	13.2514	0.465032
$813$	0	0
$814$	−14.2661	−0.500027
$815$	0	0
$816$	0	0
$817$	−14.5019	−0.507356
$818$	1.21188	0.0423725
$819$	0	0
$820$	0	0
$821$	47.0263	1.64123	0.820615	−	0.571482i	$-0.193631\pi$
$821$	47.0263	1.64123	0.820615	+	0.571482i	$0.193631\pi$
$822$	0	0
$823$	28.3425	0.987959	0.493979	−	0.869474i	$-0.335542\pi$
$823$	28.3425	0.987959	0.493979	+	0.869474i	$0.335542\pi$
$824$	12.8920	0.449113
$825$	0	0
$826$	−10.7208	−0.373023
$827$	14.1680	0.492670	0.246335	−	0.969185i	$-0.420774\pi$
$827$	14.1680	0.492670	0.246335	+	0.969185i	$0.420774\pi$
$828$	0	0
$829$	1.51657	0.0526728	0.0263364	−	0.999653i	$-0.491616\pi$
$829$	1.51657	0.0526728	0.0263364	+	0.999653i	$0.491616\pi$
$830$	0	0
$831$	0	0
$832$	32.9379	1.14192
$833$	−11.7128	−0.405823
$834$	0	0
$835$	0	0
$836$	5.56766	0.192562
$837$	0	0
$838$	12.8525	0.443981
$839$	−10.4919	−0.362220	−0.181110	−	0.983463i	$-0.557969\pi$
$839$	−10.4919	−0.362220	−0.181110	+	0.983463i	$0.557969\pi$
$840$	0	0
$841$	−1.07758	−0.0371578
$842$	11.0351	0.380296
$843$	0	0
$844$	5.58972	0.192406
$845$	0	0
$846$	0	0
$847$	29.8888	1.02699
$848$	−5.56503	−0.191104
$849$	0	0
$850$	0	0
$851$	26.5120	0.908820
$852$	0	0
$853$	14.8571	0.508698	0.254349	−	0.967112i	$-0.418139\pi$
$853$	14.8571	0.508698	0.254349	+	0.967112i	$0.418139\pi$
$854$	46.2462	1.58251
$855$	0	0
$856$	12.2962	0.420275
$857$	−26.4088	−0.902109	−0.451054	−	0.892496i	$-0.648952\pi$
$857$	−26.4088	−0.902109	−0.451054	+	0.892496i	$0.648952\pi$
$858$	0	0
$859$	−9.13013	−0.311516	−0.155758	−	0.987795i	$-0.549782\pi$
$859$	−9.13013	−0.311516	−0.155758	+	0.987795i	$0.549782\pi$
$860$	0	0
$861$	0	0
$862$	−6.20389	−0.211305
$863$	12.4796	0.424810	0.212405	−	0.977182i	$-0.431870\pi$
$863$	12.4796	0.424810	0.212405	+	0.977182i	$0.431870\pi$
$864$	0	0
$865$	0	0
$866$	36.0172	1.22392
$867$	0	0
$868$	−1.47776	−0.0501586
$869$	−14.1671	−0.480587
$870$	0	0
$871$	12.8862	0.436632
$872$	27.8710	0.943832
$873$	0	0
$874$	15.1289	0.511741
$875$	0	0
$876$	0	0
$877$	40.8121	1.37813	0.689064	−	0.724701i	$-0.258022\pi$
$877$	40.8121	1.37813	0.689064	+	0.724701i	$0.258022\pi$
$878$	−12.4933	−0.421629
$879$	0	0
$880$	0	0
$881$	3.64199	0.122702	0.0613509	−	0.998116i	$-0.480459\pi$
$881$	3.64199	0.122702	0.0613509	+	0.998116i	$0.480459\pi$
$882$	0	0
$883$	0.332855	0.0112015	0.00560073	−	0.999984i	$-0.498217\pi$
$883$	0.332855	0.0112015	0.00560073	+	0.999984i	$0.498217\pi$
$884$	15.3348	0.515765
$885$	0	0
$886$	19.5070	0.655351
$887$	1.14274	0.0383694	0.0191847	−	0.999816i	$-0.493893\pi$
$887$	1.14274	0.0383694	0.0191847	+	0.999816i	$0.493893\pi$
$888$	0	0
$889$	−35.0658	−1.17607
$890$	0	0
$891$	0	0
$892$	−6.60095	−0.221016
$893$	36.1123	1.20845
$894$	0	0
$895$	0	0
$896$	0.861093	0.0287671
$897$	0	0
$898$	−7.43300	−0.248042
$899$	−3.11385	−0.103853
$900$	0	0
$901$	−15.0109	−0.500086
$902$	11.8788	0.395520
$903$	0	0
$904$	−11.5076	−0.382737
$905$	0	0
$906$	0	0
$907$	−18.4202	−0.611632	−0.305816	−	0.952091i	$-0.598929\pi$
$907$	−18.4202	−0.611632	−0.305816	+	0.952091i	$0.598929\pi$
$908$	20.8191	0.690906
$909$	0	0
$910$	0	0
$911$	−18.2506	−0.604670	−0.302335	−	0.953202i	$-0.597766\pi$
$911$	−18.2506	−0.604670	−0.302335	+	0.953202i	$0.597766\pi$
$912$	0	0
$913$	−14.2323	−0.471019
$914$	−3.01206	−0.0996300
$915$	0	0
$916$	−1.00167	−0.0330963
$917$	−6.41681	−0.211902
$918$	0	0
$919$	10.8382	0.357520	0.178760	−	0.983893i	$-0.442791\pi$
$919$	10.8382	0.357520	0.178760	+	0.983893i	$0.442791\pi$
$920$	0	0
$921$	0	0
$922$	−9.76983	−0.321752
$923$	−26.3763	−0.868187
$924$	0	0
$925$	0	0
$926$	−14.3438	−0.471368
$927$	0	0
$928$	−22.5111	−0.738962
$929$	52.3491	1.71752	0.858759	−	0.512379i	$-0.171236\pi$
$929$	52.3491	1.71752	0.858759	+	0.512379i	$0.171236\pi$
$930$	0	0
$931$	−15.1080	−0.495144
$932$	−2.43678	−0.0798193
$933$	0	0
$934$	−23.7586	−0.777404
$935$	0	0
$936$	0	0
$937$	−0.00131225	−4.28694e−5 0	−2.14347e−5	−	1.00000i	$-0.500007\pi$
$937$	−0.00131225	−4.28694e−5 0	−2.14347e−5		1.00000i	$0.500007\pi$
$938$	10.6277	0.347006
$939$	0	0
$940$	0	0
$941$	21.7222	0.708124	0.354062	−	0.935222i	$-0.384800\pi$
$941$	21.7222	0.708124	0.354062	+	0.935222i	$0.384800\pi$
$942$	0	0
$943$	−22.0754	−0.718874
$944$	5.46655	0.177921
$945$	0	0
$946$	−3.04044	−0.0988533
$947$	33.9266	1.10247	0.551234	−	0.834351i	$-0.314157\pi$
$947$	33.9266	1.10247	0.551234	+	0.834351i	$0.314157\pi$
$948$	0	0
$949$	29.4134	0.954800
$950$	0	0
$951$	0	0
$952$	43.7865	1.41913
$953$	8.30308	0.268963	0.134482	−	0.990916i	$-0.457063\pi$
$953$	8.30308	0.268963	0.134482	+	0.990916i	$0.457063\pi$
$954$	0	0
$955$	0	0
$956$	17.9794	0.581496
$957$	0	0
$958$	46.5887	1.50521
$959$	−59.9024	−1.93435
$960$	0	0
$961$	−30.6528	−0.988798
$962$	−50.6825	−1.63407
$963$	0	0
$964$	10.9174	0.351627
$965$	0	0
$966$	0	0
$967$	27.7915	0.893714	0.446857	−	0.894605i	$-0.352543\pi$
$967$	27.7915	0.893714	0.446857	+	0.894605i	$0.352543\pi$
$968$	−29.6724	−0.953707
$969$	0	0
$970$	0	0
$971$	31.0461	0.996317	0.498158	−	0.867086i	$-0.334010\pi$
$971$	31.0461	0.996317	0.498158	+	0.867086i	$0.334010\pi$
$972$	0	0
$973$	52.9780	1.69840
$974$	−12.3692	−0.396333
$975$	0	0
$976$	−23.5811	−0.754814
$977$	−44.3917	−1.42022	−0.710109	−	0.704092i	$-0.751354\pi$
$977$	−44.3917	−1.42022	−0.710109	+	0.704092i	$0.751354\pi$
$978$	0	0
$979$	−1.24481	−0.0397842
$980$	0	0
$981$	0	0
$982$	18.1506	0.579208
$983$	43.7857	1.39655	0.698274	−	0.715831i	$-0.253952\pi$
$983$	43.7857	1.39655	0.698274	+	0.715831i	$0.253952\pi$
$984$	0	0
$985$	0	0
$986$	26.6492	0.848685
$987$	0	0
$988$	19.7800	0.629284
$989$	5.65033	0.179670
$990$	0	0
$991$	34.4351	1.09387	0.546933	−	0.837176i	$-0.315795\pi$
$991$	34.4351	1.09387	0.546933	+	0.837176i	$0.315795\pi$
$992$	2.51039	0.0797049
$993$	0	0
$994$	−21.7535	−0.689978
$995$	0	0
$996$	0	0
$997$	19.3890	0.614056	0.307028	−	0.951700i	$-0.400665\pi$
$997$	19.3890	0.614056	0.307028	+	0.951700i	$0.400665\pi$
$998$	13.3308	0.421979
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5625.2.a.t.1.4		8
3.2	odd	2		1875.2.a.p.1.5		8
5.4	even	2		5625.2.a.bd.1.5		8
15.2	even	4		1875.2.b.h.1249.11		16
15.8	even	4		1875.2.b.h.1249.6		16
15.14	odd	2		1875.2.a.m.1.4		8
25.3	odd	20		225.2.m.b.109.3		16
25.17	odd	20		225.2.m.b.64.3		16
75.8	even	20		375.2.i.c.199.3		16
75.17	even	20		75.2.i.a.64.2	yes	16
75.29	odd	10		375.2.g.e.76.3		16
75.44	odd	10		375.2.g.e.301.3		16
75.47	even	20		375.2.i.c.49.3		16
75.53	even	20		75.2.i.a.34.2	✓	16
75.56	odd	10		375.2.g.d.301.2		16
75.71	odd	10		375.2.g.d.76.2		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
75.2.i.a.34.2	✓	16	75.53	even	20
75.2.i.a.64.2	yes	16	75.17	even	20
225.2.m.b.64.3		16	25.17	odd	20
225.2.m.b.109.3		16	25.3	odd	20
375.2.g.d.76.2		16	75.71	odd	10
375.2.g.d.301.2		16	75.56	odd	10
375.2.g.e.76.3		16	75.29	odd	10
375.2.g.e.301.3		16	75.44	odd	10
375.2.i.c.49.3		16	75.47	even	20
375.2.i.c.199.3		16	75.8	even	20
1875.2.a.m.1.4		8	15.14	odd	2
1875.2.a.p.1.5		8	3.2	odd	2
1875.2.b.h.1249.6		16	15.8	even	4
1875.2.b.h.1249.11		16	15.2	even	4
5625.2.a.t.1.4		8	1.1	even	1	trivial
5625.2.a.bd.1.5		8	5.4	even	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:	\( N \)	\(=\)	\( 5625 = 3^{2} \cdot 5^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5625.a (trivial)

\(f(q)\)	\(=\)	\(q-1.08982 q^{2} -0.812294 q^{4} -3.08724 q^{7} +3.06489 q^{8} +O(q^{10})\)	\(q-1.08982 q^{2} -0.812294 q^{4} -3.08724 q^{7} +3.06489 q^{8} +1.14831 q^{11} +4.07954 q^{13} +3.36454 q^{14} -1.71559 q^{16} -4.62758 q^{17} -5.96899 q^{19} -1.25145 q^{22} +2.32568 q^{23} -4.44596 q^{26} +2.50775 q^{28} +5.28417 q^{29} -0.589279 q^{31} -4.26010 q^{32} +5.04322 q^{34} +11.3997 q^{37} +6.50512 q^{38} -9.49200 q^{41} +2.42954 q^{43} -0.932764 q^{44} -2.53458 q^{46} -6.04998 q^{47} +2.53108 q^{49} -3.31379 q^{52} +3.24380 q^{53} -9.46207 q^{56} -5.75879 q^{58} -3.18640 q^{59} +13.7452 q^{61} +0.642208 q^{62} +8.07392 q^{64} +3.15873 q^{67} +3.75895 q^{68} -6.46551 q^{71} +7.20998 q^{73} -12.4236 q^{74} +4.84857 q^{76} -3.54511 q^{77} -12.3374 q^{79} +10.3446 q^{82} -12.3941 q^{83} -2.64776 q^{86} +3.51944 q^{88} -1.08404 q^{89} -12.5945 q^{91} -1.88914 q^{92} +6.59339 q^{94} +4.52132 q^{97} -2.75842 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 4 q^{2} + 4 q^{4} + 8 q^{7} - 12 q^{8}+O(q^{10}) \) 8 * q - 4 * q^2 + 4 * q^4 + 8 * q^7 - 12 * q^8	\( 8 q - 4 q^{2} + 4 q^{4} + 8 q^{7} - 12 q^{8} - 2 q^{11} + 16 q^{13} - 6 q^{14} - 16 q^{17} - 14 q^{19} + 12 q^{22} - 14 q^{23} - 6 q^{26} + 16 q^{28} - 2 q^{29} - 22 q^{31} + 2 q^{32} - 12 q^{34} + 28 q^{37} + 16 q^{38} - 8 q^{41} + 20 q^{43} - 22 q^{44} - 2 q^{46} - 10 q^{47} + 16 q^{52} - 44 q^{53} - 30 q^{56} + 8 q^{58} - 14 q^{59} - 20 q^{61} - 16 q^{62} + 6 q^{64} + 16 q^{67} + 2 q^{68} - 16 q^{71} + 24 q^{73} - 26 q^{74} - 16 q^{76} - 46 q^{77} - 30 q^{79} + 16 q^{82} - 12 q^{83} - 32 q^{86} + 32 q^{88} - 16 q^{89} - 12 q^{91} + 2 q^{92} + 14 q^{94} + 16 q^{97} - 4 q^{98}+O(q^{100}) \) 8 * q - 4 * q^2 + 4 * q^4 + 8 * q^7 - 12 * q^8 - 2 * q^11 + 16 * q^13 - 6 * q^14 - 16 * q^17 - 14 * q^19 + 12 * q^22 - 14 * q^23 - 6 * q^26 + 16 * q^28 - 2 * q^29 - 22 * q^31 + 2 * q^32 - 12 * q^34 + 28 * q^37 + 16 * q^38 - 8 * q^41 + 20 * q^43 - 22 * q^44 - 2 * q^46 - 10 * q^47 + 16 * q^52 - 44 * q^53 - 30 * q^56 + 8 * q^58 - 14 * q^59 - 20 * q^61 - 16 * q^62 + 6 * q^64 + 16 * q^67 + 2 * q^68 - 16 * q^71 + 24 * q^73 - 26 * q^74 - 16 * q^76 - 46 * q^77 - 30 * q^79 + 16 * q^82 - 12 * q^83 - 32 * q^86 + 32 * q^88 - 16 * q^89 - 12 * q^91 + 2 * q^92 + 14 * q^94 + 16 * q^97 - 4 * q^98

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