LMFDB - Embedded newform 5625.2.a.bd.1.5

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			1.5
Root			$1.08982$ 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	$-0.374095\pi$
$2$	1.08982	0.770619	0.385309	+	0.922787i	$0.374095\pi$
$3$	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	$-0.301709\pi$
$7$	3.08724	1.16687	0.583434	+	0.812160i	$0.301709\pi$
$8$	−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	$-0.691406\pi$
$13$	−4.07954	−1.13146	−0.565731	+	0.824590i	$0.691406\pi$
$14$	3.36454	0.899211
$15$	0	0
$16$	−1.71559	−0.428898
$17$	4.62758	1.12235	0.561176	−	0.827696i	$-0.310349\pi$
$17$	4.62758	1.12235	0.561176	+	0.827696i	$0.310349\pi$
$18$	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	$-0.577957\pi$
$23$	−2.32568	−0.484939	−0.242469	+	0.970159i	$0.577957\pi$
$24$	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	$-0.886452\pi$
$37$	−11.3997	−1.87409	−0.937047	+	0.349204i	$0.886452\pi$
$38$	−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	$-0.559310\pi$
$43$	−2.42954	−0.370501	−0.185250	+	0.982691i	$0.559310\pi$
$44$	−0.932764	−0.140619
$45$	0	0
$46$	−2.53458	−0.373703
$47$	6.04998	0.882480	0.441240	−	0.897389i	$-0.354539\pi$
$47$	6.04998	0.882480	0.441240	+	0.897389i	$0.354539\pi$
$48$	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	$-0.571515\pi$
$53$	−3.24380	−0.445570	−0.222785	+	0.974868i	$0.571515\pi$
$54$	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	$-0.561806\pi$
$67$	−3.15873	−0.385901	−0.192950	+	0.981209i	$0.561806\pi$
$68$	−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	$-0.638648\pi$
$73$	−7.20998	−0.843864	−0.421932	+	0.906627i	$0.638648\pi$
$74$	−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	$-0.261887\pi$
$83$	12.3941	1.36043	0.680214	+	0.733013i	$0.261887\pi$
$84$	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	$-0.573721\pi$
$97$	−4.52132	−0.459071	−0.229535	+	0.973300i	$0.573721\pi$
$98$	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	$-0.566445\pi$
$103$	−4.20634	−0.414463	−0.207231	+	0.978292i	$0.566445\pi$
$104$	12.5034	1.22606
$105$	0	0
$106$	−3.53515	−0.343364
$107$	−4.01195	−0.387849	−0.193925	−	0.981016i	$-0.562122\pi$
$107$	−4.01195	−0.387849	−0.193925	+	0.981016i	$0.562122\pi$
$108$	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	$-0.443489\pi$
$113$	3.75465	0.353208	0.176604	+	0.984282i	$0.443489\pi$
$114$	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	$-0.668117\pi$
$127$	−11.3583	−1.00788	−0.503942	+	0.863738i	$0.668117\pi$
$128$	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	$-0.811012\pi$
$137$	−19.4032	−1.65773	−0.828863	+	0.559452i	$0.811012\pi$
$138$	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	$-0.505593\pi$
$157$	−0.440336	−0.0351426	−0.0175713	+	0.999846i	$0.505593\pi$
$158$	−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	$-0.540513\pi$
$163$	−3.24109	−0.253862	−0.126931	+	0.991912i	$0.540513\pi$
$164$	7.71029	0.602073
$165$	0	0
$166$	13.5073	1.04837
$167$	−15.9605	−1.23506	−0.617530	−	0.786547i	$-0.711867\pi$
$167$	−15.9605	−1.23506	−0.617530	+	0.786547i	$0.711867\pi$
$168$	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	$-0.493202\pi$
$173$	0.561789	0.0427121	0.0213560	+	0.999772i	$0.493202\pi$
$174$	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	$-0.476778\pi$
$193$	2.02523	0.145780	0.0728898	+	0.997340i	$0.476778\pi$
$194$	−4.92742	−0.353768
$195$	0	0
$196$	−2.05598	−0.146856
$197$	11.5454	0.822578	0.411289	−	0.911505i	$-0.365079\pi$
$197$	11.5454	0.822578	0.411289	+	0.911505i	$0.365079\pi$
$198$	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	$-0.587714\pi$
$223$	−8.12631	−0.544178	−0.272089	+	0.962272i	$0.587714\pi$
$224$	13.1520	0.878753
$225$	0	0
$226$	4.09189	0.272188
$227$	25.6300	1.70112	0.850561	−	0.525876i	$-0.176262\pi$
$227$	25.6300	1.70112	0.850561	+	0.525876i	$0.176262\pi$
$228$	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	$-0.531329\pi$
$233$	−2.99987	−0.196528	−0.0982641	+	0.995160i	$0.531329\pi$
$234$	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	$-0.429298\pi$
$257$	7.06320	0.440590	0.220295	+	0.975433i	$0.429298\pi$
$258$	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	$-0.283031\pi$
$263$	20.4356	1.26011	0.630057	+	0.776549i	$0.283031\pi$
$264$	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	$-0.309710\pi$
$277$	18.7349	1.12567	0.562837	+	0.826568i	$0.309710\pi$
$278$	−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	$-0.568455\pi$
$283$	−7.18007	−0.426811	−0.213405	+	0.976964i	$0.568455\pi$
$284$	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	$-0.516728\pi$
$293$	−1.79825	−0.105055	−0.0525276	+	0.998619i	$0.516728\pi$
$294$	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	$-0.445334\pi$
$307$	5.98864	0.341790	0.170895	+	0.985289i	$0.445334\pi$
$308$	−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	$-0.447959\pi$
$313$	5.75913	0.325525	0.162763	+	0.986665i	$0.447959\pi$
$314$	−0.479887	−0.0270816
$315$	0	0
$316$	10.0216	0.563758
$317$	12.2481	0.687922	0.343961	−	0.938984i	$-0.388231\pi$
$317$	12.2481	0.687922	0.343961	+	0.938984i	$0.388231\pi$
$318$	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	$-0.451866\pi$
$337$	5.53083	0.301284	0.150642	+	0.988588i	$0.451866\pi$
$338$	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	$-0.254790\pi$
$347$	25.9445	1.39277	0.696387	+	0.717666i	$0.254790\pi$
$348$	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	$-0.569950\pi$
$353$	−8.19134	−0.435981	−0.217990	+	0.975951i	$0.569950\pi$
$354$	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	$-0.336241\pi$
$367$	18.8533	0.984135	0.492068	+	0.870557i	$0.336241\pi$
$368$	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	$-0.652889\pi$
$373$	−17.8477	−0.924118	−0.462059	+	0.886849i	$0.652889\pi$
$374$	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	$-0.561683\pi$
$383$	−7.53739	−0.385143	−0.192571	+	0.981283i	$0.561683\pi$
$384$	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	$-0.394204\pi$
$397$	13.0022	0.652562	0.326281	+	0.945273i	$0.394204\pi$
$398$	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	$-0.207936\pi$
$433$	33.0488	1.58822	0.794112	+	0.607771i	$0.207936\pi$
$434$	−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	$-0.360200\pi$
$443$	17.8993	0.850422	0.425211	+	0.905094i	$0.360200\pi$
$444$	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	$-0.520591\pi$
$457$	−2.76381	−0.129286	−0.0646429	+	0.997908i	$0.520591\pi$
$458$	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	$-0.598936\pi$
$463$	−13.1617	−0.611675	−0.305837	+	0.952084i	$0.598936\pi$
$464$	−9.06547	−0.420854
$465$	0	0
$466$	−3.26932	−0.151448
$467$	−21.8005	−1.00880	−0.504402	−	0.863469i	$-0.668287\pi$
$467$	−21.8005	−1.00880	−0.504402	+	0.863469i	$0.668287\pi$
$468$	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	$-0.582784\pi$
$487$	−11.3497	−0.514305	−0.257153	+	0.966371i	$0.582784\pi$
$488$	−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	$-0.490705\pi$
$503$	1.30967	0.0583951	0.0291976	+	0.999574i	$0.490705\pi$
$504$	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	$-0.517803\pi$
$523$	−2.55674	−0.111798	−0.0558991	+	0.998436i	$0.517803\pi$
$524$	−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	$-0.232381\pi$
$547$	34.8549	1.49029	0.745144	+	0.666903i	$0.232381\pi$
$548$	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	$-0.596888\pi$
$557$	−14.1466	−0.599411	−0.299705	+	0.954032i	$0.596888\pi$
$558$	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	$-0.202347\pi$
$563$	38.1854	1.60932	0.804661	+	0.593734i	$0.202347\pi$
$564$	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	$-0.656225\pi$
$577$	−22.6433	−0.942653	−0.471326	+	0.881959i	$0.656225\pi$
$578$	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	$-0.171818\pi$
$587$	41.5667	1.71564	0.857820	+	0.513950i	$0.171818\pi$
$588$	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	$-0.486333\pi$
$593$	2.09050	0.0858465	0.0429233	+	0.999078i	$0.486333\pi$
$594$	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	$-0.420154\pi$
$607$	12.2310	0.496441	0.248220	+	0.968704i	$0.420154\pi$
$608$	−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	$-0.415591\pi$
$613$	12.9776	0.524160	0.262080	+	0.965046i	$0.415591\pi$
$614$	6.52654	0.263390
$615$	0	0
$616$	−10.8654	−0.437779
$617$	20.4921	0.824981	0.412490	−	0.910962i	$-0.364659\pi$
$617$	20.4921	0.824981	0.412490	+	0.910962i	$0.364659\pi$
$618$	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	$-0.431429\pi$
$643$	10.8408	0.427521	0.213761	+	0.976886i	$0.431429\pi$
$644$	5.83223	0.229822
$645$	0	0
$646$	−30.1029	−1.18438
$647$	−34.6388	−1.36179	−0.680896	−	0.732380i	$-0.738410\pi$
$647$	−34.6388	−1.36179	−0.680896	+	0.732380i	$0.738410\pi$
$648$	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	$-0.525907\pi$
$653$	−4.15506	−0.162600	−0.0813000	+	0.996690i	$0.525907\pi$
$654$	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	$-0.205458\pi$
$673$	41.4463	1.59764	0.798819	+	0.601571i	$0.205458\pi$
$674$	6.02761	0.232175
$675$	0	0
$676$	−2.95892	−0.113804
$677$	18.1837	0.698856	0.349428	−	0.936963i	$-0.386376\pi$
$677$	18.1837	0.698856	0.349428	+	0.936963i	$0.386376\pi$
$678$	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	$-0.316777\pi$
$683$	28.4523	1.08870	0.544348	+	0.838859i	$0.316777\pi$
$684$	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	$-0.579385\pi$
$727$	−13.3099	−0.493638	−0.246819	+	0.969062i	$0.579385\pi$
$728$	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	$-0.321630\pi$
$733$	28.7794	1.06299	0.531496	+	0.847061i	$0.321630\pi$
$734$	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	$-0.329589\pi$
$743$	27.8114	1.02030	0.510151	+	0.860085i	$0.329589\pi$
$744$	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	$-0.580472\pi$
$757$	−13.7637	−0.500251	−0.250126	+	0.968213i	$0.580472\pi$
$758$	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	$-0.349424\pi$
$773$	25.3341	0.911204	0.455602	+	0.890184i	$0.349424\pi$
$774$	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	$-0.258127\pi$
$787$	38.6480	1.37765	0.688826	+	0.724927i	$0.258127\pi$
$788$	−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	$-0.125917\pi$
$797$	52.1020	1.84555	0.922774	+	0.385342i	$0.125917\pi$
$798$	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	$-0.664458\pi$
$823$	−28.3425	−0.987959	−0.493979	+	0.869474i	$0.664458\pi$
$824$	12.8920	0.449113
$825$	0	0
$826$	−10.7208	−0.373023
$827$	−14.1680	−0.492670	−0.246335	−	0.969185i	$-0.579226\pi$
$827$	−14.1680	−0.492670	−0.246335	+	0.969185i	$0.579226\pi$
$828$	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	$-0.581861\pi$
$853$	−14.8571	−0.508698	−0.254349	+	0.967112i	$0.581861\pi$
$854$	46.2462	1.58251
$855$	0	0
$856$	12.2962	0.420275
$857$	26.4088	0.902109	0.451054	−	0.892496i	$-0.351048\pi$
$857$	26.4088	0.902109	0.451054	+	0.892496i	$0.351048\pi$
$858$	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	$-0.568130\pi$
$863$	−12.4796	−0.424810	−0.212405	+	0.977182i	$0.568130\pi$
$864$	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	$-0.741978\pi$
$877$	−40.8121	−1.37813	−0.689064	+	0.724701i	$0.741978\pi$
$878$	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	$-0.501783\pi$
$883$	−0.332855	−0.0112015	−0.00560073	+	0.999984i	$0.501783\pi$
$884$	15.3348	0.515765
$885$	0	0
$886$	19.5070	0.655351
$887$	−1.14274	−0.0383694	−0.0191847	−	0.999816i	$-0.506107\pi$
$887$	−1.14274	−0.0383694	−0.0191847	+	0.999816i	$0.506107\pi$
$888$	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	$-0.401071\pi$
$907$	18.4202	0.611632	0.305816	+	0.952091i	$0.401071\pi$
$908$	−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	$-0.499993\pi$
$937$	0.00131225	4.28694e−5 0	2.14347e−5		1.00000i	$0.499993\pi$
$938$	−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	$-0.685843\pi$
$947$	−33.9266	−1.10247	−0.551234	+	0.834351i	$0.685843\pi$
$948$	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	$-0.542937\pi$
$953$	−8.30308	−0.268963	−0.134482	+	0.990916i	$0.542937\pi$
$954$	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	$-0.647457\pi$
$967$	−27.7915	−0.893714	−0.446857	+	0.894605i	$0.647457\pi$
$968$	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	$-0.248646\pi$
$977$	44.3917	1.42022	0.710109	+	0.704092i	$0.248646\pi$
$978$	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	$-0.746048\pi$
$983$	−43.7857	−1.39655	−0.698274	+	0.715831i	$0.746048\pi$
$984$	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	$-0.599335\pi$
$997$	−19.3890	−0.614056	−0.307028	+	0.951700i	$0.599335\pi$
$998$	−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.bd.1.5		8
3.2	odd	2		1875.2.a.m.1.4		8
5.4	even	2		5625.2.a.t.1.4		8
15.2	even	4		1875.2.b.h.1249.6		16
15.8	even	4		1875.2.b.h.1249.11		16
15.14	odd	2		1875.2.a.p.1.5		8
25.8	odd	20		225.2.m.b.64.3		16
25.22	odd	20		225.2.m.b.109.3		16
75.8	even	20		75.2.i.a.64.2	yes	16
75.17	even	20		375.2.i.c.199.3		16
75.29	odd	10		375.2.g.d.76.2		16
75.44	odd	10		375.2.g.d.301.2		16
75.47	even	20		75.2.i.a.34.2	✓	16
75.53	even	20		375.2.i.c.49.3		16
75.56	odd	10		375.2.g.e.301.3		16
75.71	odd	10		375.2.g.e.76.3		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
75.2.i.a.34.2	✓	16	75.47	even	20
75.2.i.a.64.2	yes	16	75.8	even	20
225.2.m.b.64.3		16	25.8	odd	20
225.2.m.b.109.3		16	25.22	odd	20
375.2.g.d.76.2		16	75.29	odd	10
375.2.g.d.301.2		16	75.44	odd	10
375.2.g.e.76.3		16	75.71	odd	10
375.2.g.e.301.3		16	75.56	odd	10
375.2.i.c.49.3		16	75.53	even	20
375.2.i.c.199.3		16	75.17	even	20
1875.2.a.m.1.4		8	3.2	odd	2
1875.2.a.p.1.5		8	15.14	odd	2
1875.2.b.h.1249.6		16	15.2	even	4
1875.2.b.h.1249.11		16	15.8	even	4
5625.2.a.t.1.4		8	5.4	even	2
5625.2.a.bd.1.5		8	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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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