LMFDB - Embedded newform 7569.2.a.bl.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7569,2,Mod(1,7569)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7569, 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("7569.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7569 = 3^{2} \cdot 29^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7569.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:	$60.4387692899$
Analytic rank:	$0$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - x^{8} - 14x^{7} + 9x^{6} + 70x^{5} - 23x^{4} - 141x^{3} + 14x^{2} + 84x - 7 $ x^9 - x^8 - 14x^7 + 9x^6 + 70x^5 - 23x^4 - 141x^3 + 14x^2 + 84*x - 7
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 87)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-1.15439$ of defining polynomial
Character	$\chi$	$=$	7569.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.15439 q^{2} -0.667390 q^{4} +3.44957 q^{5} +4.51754 q^{7} +3.07920 q^{8} +O(q^{10})$	\(q-1.15439 q^{2} -0.667390 q^{4} +3.44957 q^{5} +4.51754 q^{7} +3.07920 q^{8} -3.98215 q^{10} +0.554210 q^{11} +2.75998 q^{13} -5.21499 q^{14} -2.21981 q^{16} +2.99688 q^{17} -0.278196 q^{19} -2.30221 q^{20} -0.639773 q^{22} -0.927891 q^{23} +6.89956 q^{25} -3.18609 q^{26} -3.01496 q^{28} +9.12071 q^{31} -3.59588 q^{32} -3.45957 q^{34} +15.5836 q^{35} -6.87516 q^{37} +0.321146 q^{38} +10.6219 q^{40} +2.85317 q^{41} -11.2189 q^{43} -0.369874 q^{44} +1.07115 q^{46} -4.19552 q^{47} +13.4081 q^{49} -7.96477 q^{50} -1.84198 q^{52} +2.07886 q^{53} +1.91179 q^{55} +13.9104 q^{56} +14.2670 q^{59} -7.23144 q^{61} -10.5288 q^{62} +8.59066 q^{64} +9.52076 q^{65} -12.2857 q^{67} -2.00009 q^{68} -17.9895 q^{70} +2.38207 q^{71} -0.913120 q^{73} +7.93660 q^{74} +0.185665 q^{76} +2.50366 q^{77} +2.06809 q^{79} -7.65740 q^{80} -3.29367 q^{82} +10.9704 q^{83} +10.3380 q^{85} +12.9510 q^{86} +1.70652 q^{88} +6.80692 q^{89} +12.4683 q^{91} +0.619265 q^{92} +4.84325 q^{94} -0.959659 q^{95} -3.51641 q^{97} -15.4782 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q + q^{2} + 11 q^{4} + 5 q^{7} + 12 q^{8}+O(q^{10}) $ 9 * q + q^2 + 11 * q^4 + 5 * q^7 + 12 * q^8	$ 9 q + q^{2} + 11 q^{4} + 5 q^{7} + 12 q^{8} - 4 q^{10} + 3 q^{11} + 5 q^{13} + 15 q^{14} - 5 q^{16} + 16 q^{17} + q^{19} - 8 q^{20} + 24 q^{22} + 10 q^{23} + 9 q^{25} + 12 q^{26} - 24 q^{28} + 4 q^{31} + 25 q^{32} + 24 q^{34} + 44 q^{35} - 25 q^{37} + 10 q^{38} - 5 q^{40} + 34 q^{41} - 12 q^{43} - 23 q^{44} - 6 q^{46} + 8 q^{47} + 26 q^{49} + 27 q^{50} - 23 q^{52} + 32 q^{53} + 5 q^{55} - 14 q^{56} - 10 q^{59} - 51 q^{61} - 8 q^{62} - 8 q^{64} + 11 q^{65} + 7 q^{67} + 11 q^{68} + 14 q^{70} - 7 q^{71} - 17 q^{73} + 62 q^{74} + 6 q^{76} + 64 q^{77} - 13 q^{79} - 54 q^{80} + 37 q^{82} + 31 q^{83} - 42 q^{85} + 70 q^{86} - 29 q^{88} - 32 q^{89} + 45 q^{91} - 9 q^{92} + 38 q^{94} - 20 q^{95} - 16 q^{97} - 12 q^{98}+O(q^{100}) $ 9 * q + q^2 + 11 * q^4 + 5 * q^7 + 12 * q^8 - 4 * q^10 + 3 * q^11 + 5 * q^13 + 15 * q^14 - 5 * q^16 + 16 * q^17 + q^19 - 8 * q^20 + 24 * q^22 + 10 * q^23 + 9 * q^25 + 12 * q^26 - 24 * q^28 + 4 * q^31 + 25 * q^32 + 24 * q^34 + 44 * q^35 - 25 * q^37 + 10 * q^38 - 5 * q^40 + 34 * q^41 - 12 * q^43 - 23 * q^44 - 6 * q^46 + 8 * q^47 + 26 * q^49 + 27 * q^50 - 23 * q^52 + 32 * q^53 + 5 * q^55 - 14 * q^56 - 10 * q^59 - 51 * q^61 - 8 * q^62 - 8 * q^64 + 11 * q^65 + 7 * q^67 + 11 * q^68 + 14 * q^70 - 7 * q^71 - 17 * q^73 + 62 * q^74 + 6 * q^76 + 64 * q^77 - 13 * q^79 - 54 * q^80 + 37 * q^82 + 31 * q^83 - 42 * q^85 + 70 * q^86 - 29 * q^88 - 32 * q^89 + 45 * q^91 - 9 * q^92 + 38 * q^94 - 20 * q^95 - 16 * q^97 - 12 * 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 $

$2$	−1.15439	−0.816275	−0.408138	−	0.912920i	$-0.633822\pi$
$2$	−1.15439	−0.816275	−0.408138	+	0.912920i	$0.633822\pi$
$3$	0	0
$3$	0	0
$4$	−0.667390	−0.333695
$5$	3.44957	1.54270	0.771348	−	0.636413i	$-0.219583\pi$
$5$	3.44957	1.54270	0.771348	+	0.636413i	$0.219583\pi$
$6$	0	0
$7$	4.51754	1.70747	0.853734	−	0.520710i	$-0.174333\pi$
$7$	4.51754	1.70747	0.853734	+	0.520710i	$0.174333\pi$
$8$	3.07920	1.08866
$9$	0	0
$10$	−3.98215	−1.25926
$11$	0.554210	0.167101	0.0835503	−	0.996504i	$-0.473374\pi$
$11$	0.554210	0.167101	0.0835503	+	0.996504i	$0.473374\pi$
$12$	0	0
$13$	2.75998	0.765481	0.382741	−	0.923856i	$-0.374980\pi$
$13$	2.75998	0.765481	0.382741	+	0.923856i	$0.374980\pi$
$14$	−5.21499	−1.39376
$15$	0	0
$16$	−2.21981	−0.554953
$17$	2.99688	0.726851	0.363426	−	0.931623i	$-0.381607\pi$
$17$	2.99688	0.726851	0.363426	+	0.931623i	$0.381607\pi$
$18$	0	0
$19$	−0.278196	−0.0638226	−0.0319113	−	0.999491i	$-0.510159\pi$
$19$	−0.278196	−0.0638226	−0.0319113	+	0.999491i	$0.510159\pi$
$20$	−2.30221	−0.514790
$21$	0	0
$22$	−0.639773	−0.136400
$23$	−0.927891	−0.193479	−0.0967393	−	0.995310i	$-0.530841\pi$
$23$	−0.927891	−0.193479	−0.0967393	+	0.995310i	$0.530841\pi$
$24$	0	0
$25$	6.89956	1.37991
$26$	−3.18609	−0.624843
$27$	0	0
$28$	−3.01496	−0.569773
$29$	0	0
$29$	0	0
$30$	0	0
$31$	9.12071	1.63813	0.819064	−	0.573702i	$-0.194493\pi$
$31$	9.12071	1.63813	0.819064	+	0.573702i	$0.194493\pi$
$32$	−3.59588	−0.635668
$33$	0	0
$34$	−3.45957	−0.593311
$35$	15.5836	2.63410
$36$	0	0
$37$	−6.87516	−1.13027	−0.565135	−	0.824999i	$-0.691176\pi$
$37$	−6.87516	−1.13027	−0.565135	+	0.824999i	$0.691176\pi$
$38$	0.321146	0.0520968
$39$	0	0
$40$	10.6219	1.67948
$41$	2.85317	0.445591	0.222795	−	0.974865i	$-0.428482\pi$
$41$	2.85317	0.445591	0.222795	+	0.974865i	$0.428482\pi$
$42$	0	0
$43$	−11.2189	−1.71087	−0.855433	−	0.517914i	$-0.826709\pi$
$43$	−11.2189	−1.71087	−0.855433	+	0.517914i	$0.826709\pi$
$44$	−0.369874	−0.0557606
$45$	0	0
$46$	1.07115	0.157932
$47$	−4.19552	−0.611979	−0.305989	−	0.952035i	$-0.598987\pi$
$47$	−4.19552	−0.611979	−0.305989	+	0.952035i	$0.598987\pi$
$48$	0	0
$49$	13.4081	1.91545
$50$	−7.96477	−1.12639
$51$	0	0
$52$	−1.84198	−0.255437
$53$	2.07886	0.285554	0.142777	−	0.989755i	$-0.454397\pi$
$53$	2.07886	0.285554	0.142777	+	0.989755i	$0.454397\pi$
$54$	0	0
$55$	1.91179	0.257785
$56$	13.9104	1.85886
$57$	0	0
$58$	0	0
$59$	14.2670	1.85741	0.928703	−	0.370824i	$-0.120925\pi$
$59$	14.2670	1.85741	0.928703	+	0.370824i	$0.120925\pi$
$60$	0	0
$61$	−7.23144	−0.925892	−0.462946	−	0.886387i	$-0.653208\pi$
$61$	−7.23144	−0.925892	−0.462946	+	0.886387i	$0.653208\pi$
$62$	−10.5288	−1.33716
$63$	0	0
$64$	8.59066	1.07383
$65$	9.52076	1.18091
$66$	0	0
$67$	−12.2857	−1.50094	−0.750469	−	0.660905i	$-0.770172\pi$
$67$	−12.2857	−1.50094	−0.750469	+	0.660905i	$0.770172\pi$
$68$	−2.00009	−0.242547
$69$	0	0
$70$	−17.9895	−2.15015
$71$	2.38207	0.282699	0.141350	−	0.989960i	$-0.454856\pi$
$71$	2.38207	0.282699	0.141350	+	0.989960i	$0.454856\pi$
$72$	0	0
$73$	−0.913120	−0.106873	−0.0534363	−	0.998571i	$-0.517017\pi$
$73$	−0.913120	−0.106873	−0.0534363	+	0.998571i	$0.517017\pi$
$74$	7.93660	0.922611
$75$	0	0
$76$	0.185665	0.0212973
$77$	2.50366	0.285319
$78$	0	0
$79$	2.06809	0.232678	0.116339	−	0.993210i	$-0.462884\pi$
$79$	2.06809	0.232678	0.116339	+	0.993210i	$0.462884\pi$
$80$	−7.65740	−0.856124
$81$	0	0
$82$	−3.29367	−0.363725
$83$	10.9704	1.20416	0.602080	−	0.798436i	$-0.294339\pi$
$83$	10.9704	1.20416	0.602080	+	0.798436i	$0.294339\pi$
$84$	0	0
$85$	10.3380	1.12131
$86$	12.9510	1.39654
$87$	0	0
$88$	1.70652	0.181916
$89$	6.80692	0.721532	0.360766	−	0.932656i	$-0.382515\pi$
$89$	6.80692	0.721532	0.360766	+	0.932656i	$0.382515\pi$
$90$	0	0
$91$	12.4683	1.30703
$92$	0.619265	0.0645628
$93$	0	0
$94$	4.84325	0.499543
$95$	−0.959659	−0.0984590
$96$	0	0
$97$	−3.51641	−0.357037	−0.178518	−	0.983937i	$-0.557130\pi$
$97$	−3.51641	−0.357037	−0.178518	+	0.983937i	$0.557130\pi$
$98$	−15.4782	−1.56353
$99$	0	0
$100$	−4.60470	−0.460470
$101$	12.6707	1.26078	0.630390	−	0.776279i	$-0.282895\pi$
$101$	12.6707	1.26078	0.630390	+	0.776279i	$0.282895\pi$
$102$	0	0
$103$	2.01754	0.198794	0.0993972	−	0.995048i	$-0.468309\pi$
$103$	2.01754	0.198794	0.0993972	+	0.995048i	$0.468309\pi$
$104$	8.49854	0.833350
$105$	0	0
$106$	−2.39981	−0.233091
$107$	0.572045	0.0553017	0.0276508	−	0.999618i	$-0.491197\pi$
$107$	0.572045	0.0553017	0.0276508	+	0.999618i	$0.491197\pi$
$108$	0	0
$109$	19.4684	1.86474	0.932368	−	0.361510i	$-0.117739\pi$
$109$	19.4684	1.86474	0.932368	+	0.361510i	$0.117739\pi$
$110$	−2.20694	−0.210424
$111$	0	0
$112$	−10.0281	−0.947564
$113$	−2.43686	−0.229241	−0.114620	−	0.993409i	$-0.536565\pi$
$113$	−2.43686	−0.229241	−0.114620	+	0.993409i	$0.536565\pi$
$114$	0	0
$115$	−3.20083	−0.298479
$116$	0	0
$117$	0	0
$118$	−16.4697	−1.51615
$119$	13.5385	1.24108
$120$	0	0
$121$	−10.6929	−0.972077
$122$	8.34789	0.755782
$123$	0	0
$124$	−6.08707	−0.546635
$125$	6.55269	0.586090
$126$	0	0
$127$	−9.18908	−0.815399	−0.407699	−	0.913116i	$-0.633669\pi$
$127$	−9.18908	−0.815399	−0.407699	+	0.913116i	$0.633669\pi$
$128$	−2.72519	−0.240875
$129$	0	0
$130$	−10.9906	−0.963944
$131$	13.0939	1.14402	0.572010	−	0.820247i	$-0.306164\pi$
$131$	13.0939	1.14402	0.572010	+	0.820247i	$0.306164\pi$
$132$	0	0
$133$	−1.25676	−0.108975
$134$	14.1825	1.22518
$135$	0	0
$136$	9.22801	0.791295
$137$	7.60048	0.649353	0.324676	−	0.945825i	$-0.394745\pi$
$137$	7.60048	0.649353	0.324676	+	0.945825i	$0.394745\pi$
$138$	0	0
$139$	8.85308	0.750908	0.375454	−	0.926841i	$-0.377487\pi$
$139$	8.85308	0.750908	0.375454	+	0.926841i	$0.377487\pi$
$140$	−10.4003	−0.878987
$141$	0	0
$142$	−2.74983	−0.230760
$143$	1.52961	0.127912
$144$	0	0
$145$	0	0
$146$	1.05409	0.0872375
$147$	0	0
$148$	4.58841	0.377165
$149$	−7.97006	−0.652932	−0.326466	−	0.945209i	$-0.605858\pi$
$149$	−7.97006	−0.652932	−0.326466	+	0.945209i	$0.605858\pi$
$150$	0	0
$151$	−0.105522	−0.00858723	−0.00429361	−	0.999991i	$-0.501367\pi$
$151$	−0.105522	−0.00858723	−0.00429361	+	0.999991i	$0.501367\pi$
$152$	−0.856623	−0.0694813
$153$	0	0
$154$	−2.89020	−0.232899
$155$	31.4626	2.52714
$156$	0	0
$157$	−13.5037	−1.07771	−0.538857	−	0.842397i	$-0.681144\pi$
$157$	−13.5037	−1.07771	−0.538857	+	0.842397i	$0.681144\pi$
$158$	−2.38738	−0.189930
$159$	0	0
$160$	−12.4043	−0.980643
$161$	−4.19178	−0.330358
$162$	0	0
$163$	−2.61379	−0.204728	−0.102364	−	0.994747i	$-0.532641\pi$
$163$	−2.61379	−0.204728	−0.102364	+	0.994747i	$0.532641\pi$
$164$	−1.90418	−0.148691
$165$	0	0
$166$	−12.6641	−0.982926
$167$	10.2188	0.790752	0.395376	−	0.918519i	$-0.370614\pi$
$167$	10.2188	0.790752	0.395376	+	0.918519i	$0.370614\pi$
$168$	0	0
$169$	−5.38250	−0.414038
$170$	−11.9340	−0.915298
$171$	0	0
$172$	7.48738	0.570907
$173$	−23.0919	−1.75564	−0.877822	−	0.478987i	$-0.841004\pi$
$173$	−23.0919	−1.75564	−0.877822	+	0.478987i	$0.841004\pi$
$174$	0	0
$175$	31.1690	2.35616
$176$	−1.23024	−0.0927329
$177$	0	0
$178$	−7.85783	−0.588969
$179$	−14.7887	−1.10536	−0.552678	−	0.833395i	$-0.686394\pi$
$179$	−14.7887	−1.10536	−0.552678	+	0.833395i	$0.686394\pi$
$180$	0	0
$181$	2.88931	0.214761	0.107380	−	0.994218i	$-0.465754\pi$
$181$	2.88931	0.214761	0.107380	+	0.994218i	$0.465754\pi$
$182$	−14.3933	−1.06690
$183$	0	0
$184$	−2.85716	−0.210633
$185$	−23.7164	−1.74366
$186$	0	0
$187$	1.66090	0.121457
$188$	2.80004	0.204214
$189$	0	0
$190$	1.10782	0.0803696
$191$	4.66376	0.337458	0.168729	−	0.985663i	$-0.446034\pi$
$191$	4.66376	0.337458	0.168729	+	0.985663i	$0.446034\pi$
$192$	0	0
$193$	−14.5839	−1.04977	−0.524887	−	0.851172i	$-0.675892\pi$
$193$	−14.5839	−1.04977	−0.524887	+	0.851172i	$0.675892\pi$
$194$	4.05929	0.291440
$195$	0	0
$196$	−8.94844	−0.639175
$197$	−2.89152	−0.206012	−0.103006	−	0.994681i	$-0.532846\pi$
$197$	−2.89152	−0.206012	−0.103006	+	0.994681i	$0.532846\pi$
$198$	0	0
$199$	4.05076	0.287151	0.143575	−	0.989639i	$-0.454140\pi$
$199$	4.05076	0.287151	0.143575	+	0.989639i	$0.454140\pi$
$200$	21.2451	1.50226
$201$	0	0
$202$	−14.6269	−1.02914
$203$	0	0
$204$	0	0
$205$	9.84224	0.687412
$206$	−2.32903	−0.162271
$207$	0	0
$208$	−6.12664	−0.424806
$209$	−0.154179	−0.0106648
$210$	0	0
$211$	−13.1686	−0.906565	−0.453283	−	0.891367i	$-0.649747\pi$
$211$	−13.1686	−0.906565	−0.453283	+	0.891367i	$0.649747\pi$
$212$	−1.38741	−0.0952879
$213$	0	0
$214$	−0.660361	−0.0451414
$215$	−38.7004	−2.63935
$216$	0	0
$217$	41.2031	2.79705
$218$	−22.4741	−1.52214
$219$	0	0
$220$	−1.27591	−0.0860217
$221$	8.27135	0.556391
$222$	0	0
$223$	−15.2764	−1.02299	−0.511493	−	0.859287i	$-0.670907\pi$
$223$	−15.2764	−1.02299	−0.511493	+	0.859287i	$0.670907\pi$
$224$	−16.2445	−1.08538
$225$	0	0
$226$	2.81309	0.187124
$227$	14.7503	0.979014	0.489507	−	0.871999i	$-0.337177\pi$
$227$	14.7503	0.979014	0.489507	+	0.871999i	$0.337177\pi$
$228$	0	0
$229$	10.3737	0.685515	0.342758	−	0.939424i	$-0.388639\pi$
$229$	10.3737	0.685515	0.342758	+	0.939424i	$0.388639\pi$
$230$	3.69500	0.243641
$231$	0	0
$232$	0	0
$233$	−0.280226	−0.0183582	−0.00917911	−	0.999958i	$-0.502922\pi$
$233$	−0.280226	−0.0183582	−0.00917911	+	0.999958i	$0.502922\pi$
$234$	0	0
$235$	−14.4727	−0.944098
$236$	−9.52166	−0.619807
$237$	0	0
$238$	−15.6287	−1.01306
$239$	−6.43890	−0.416498	−0.208249	−	0.978076i	$-0.566776\pi$
$239$	−6.43890	−0.416498	−0.208249	+	0.978076i	$0.566776\pi$
$240$	0	0
$241$	−15.4251	−0.993617	−0.496809	−	0.867860i	$-0.665495\pi$
$241$	−15.4251	−0.993617	−0.496809	+	0.867860i	$0.665495\pi$
$242$	12.3437	0.793483
$243$	0	0
$244$	4.82619	0.308965
$245$	46.2523	2.95495
$246$	0	0
$247$	−0.767817	−0.0488550
$248$	28.0845	1.78337
$249$	0	0
$250$	−7.56434	−0.478411
$251$	3.29145	0.207754	0.103877	−	0.994590i	$-0.466875\pi$
$251$	3.29145	0.207754	0.103877	+	0.994590i	$0.466875\pi$
$252$	0	0
$253$	−0.514246	−0.0323304
$254$	10.6078	0.665590
$255$	0	0
$256$	−14.0354	−0.877212
$257$	0.395643	0.0246796	0.0123398	−	0.999924i	$-0.496072\pi$
$257$	0.395643	0.0246796	0.0123398	+	0.999924i	$0.496072\pi$
$258$	0	0
$259$	−31.0588	−1.92990
$260$	−6.35406	−0.394062
$261$	0	0
$262$	−15.1154	−0.933835
$263$	8.14356	0.502153	0.251077	−	0.967967i	$-0.419215\pi$
$263$	8.14356	0.502153	0.251077	+	0.967967i	$0.419215\pi$
$264$	0	0
$265$	7.17120	0.440523
$266$	1.45079	0.0889537
$267$	0	0
$268$	8.19936	0.500856
$269$	−8.97754	−0.547370	−0.273685	−	0.961819i	$-0.588243\pi$
$269$	−8.97754	−0.547370	−0.273685	+	0.961819i	$0.588243\pi$
$270$	0	0
$271$	5.03959	0.306133	0.153066	−	0.988216i	$-0.451085\pi$
$271$	5.03959	0.306133	0.153066	+	0.988216i	$0.451085\pi$
$272$	−6.65252	−0.403368
$273$	0	0
$274$	−8.77389	−0.530050
$275$	3.82381	0.230584
$276$	0	0
$277$	2.03513	0.122279	0.0611395	−	0.998129i	$-0.480527\pi$
$277$	2.03513	0.122279	0.0611395	+	0.998129i	$0.480527\pi$
$278$	−10.2199	−0.612948
$279$	0	0
$280$	47.9850	2.86765
$281$	−23.8243	−1.42124	−0.710618	−	0.703578i	$-0.751584\pi$
$281$	−23.8243	−1.42124	−0.710618	+	0.703578i	$0.751584\pi$
$282$	0	0
$283$	−14.9858	−0.890816	−0.445408	−	0.895328i	$-0.646941\pi$
$283$	−14.9858	−0.890816	−0.445408	+	0.895328i	$0.646941\pi$
$284$	−1.58977	−0.0943353
$285$	0	0
$286$	−1.76576	−0.104412
$287$	12.8893	0.760832
$288$	0	0
$289$	−8.01868	−0.471687
$290$	0	0
$291$	0	0
$292$	0.609407	0.0356629
$293$	−30.3700	−1.77424	−0.887118	−	0.461542i	$-0.847296\pi$
$293$	−30.3700	−1.77424	−0.887118	+	0.461542i	$0.847296\pi$
$294$	0	0
$295$	49.2151	2.86541
$296$	−21.1700	−1.23048
$297$	0	0
$298$	9.20053	0.532972
$299$	−2.56096	−0.148104
$300$	0	0
$301$	−50.6818	−2.92125
$302$	0.121813	0.00700954
$303$	0	0
$304$	0.617544	0.0354185
$305$	−24.9454	−1.42837
$306$	0	0
$307$	2.10479	0.120127	0.0600635	−	0.998195i	$-0.480870\pi$
$307$	2.10479	0.120127	0.0600635	+	0.998195i	$0.480870\pi$
$308$	−1.67092	−0.0952094
$309$	0	0
$310$	−36.3200	−2.06284
$311$	−16.8761	−0.956958	−0.478479	−	0.878099i	$-0.658812\pi$
$311$	−16.8761	−0.956958	−0.478479	+	0.878099i	$0.658812\pi$
$312$	0	0
$313$	14.8295	0.838213	0.419106	−	0.907937i	$-0.362343\pi$
$313$	14.8295	0.838213	0.419106	+	0.907937i	$0.362343\pi$
$314$	15.5885	0.879712
$315$	0	0
$316$	−1.38022	−0.0776436
$317$	5.50623	0.309260	0.154630	−	0.987972i	$-0.450581\pi$
$317$	5.50623	0.309260	0.154630	+	0.987972i	$0.450581\pi$
$318$	0	0
$319$	0	0
$320$	29.6341	1.65660
$321$	0	0
$322$	4.83894	0.269663
$323$	−0.833723	−0.0463896
$324$	0	0
$325$	19.0427	1.05630
$326$	3.01733	0.167114
$327$	0	0
$328$	8.78550	0.485098
$329$	−18.9534	−1.04493
$330$	0	0
$331$	−5.88833	−0.323652	−0.161826	−	0.986819i	$-0.551738\pi$
$331$	−5.88833	−0.323652	−0.161826	+	0.986819i	$0.551738\pi$
$332$	−7.32154	−0.401822
$333$	0	0
$334$	−11.7964	−0.645471
$335$	−42.3805	−2.31549
$336$	0	0
$337$	17.4297	0.949457	0.474729	−	0.880132i	$-0.342546\pi$
$337$	17.4297	0.949457	0.474729	+	0.880132i	$0.342546\pi$
$338$	6.21349	0.337969
$339$	0	0
$340$	−6.89946	−0.374176
$341$	5.05479	0.273732
$342$	0	0
$343$	28.9489	1.56309
$344$	−34.5452	−1.86255
$345$	0	0
$346$	26.6570	1.43309
$347$	2.81562	0.151150	0.0755751	−	0.997140i	$-0.475921\pi$
$347$	2.81562	0.151150	0.0755751	+	0.997140i	$0.475921\pi$
$348$	0	0
$349$	23.4949	1.25765	0.628826	−	0.777546i	$-0.283536\pi$
$349$	23.4949	1.25765	0.628826	+	0.777546i	$0.283536\pi$
$350$	−35.9811	−1.92327
$351$	0	0
$352$	−1.99287	−0.106220
$353$	−16.1562	−0.859910	−0.429955	−	0.902850i	$-0.641471\pi$
$353$	−16.1562	−0.859910	−0.429955	+	0.902850i	$0.641471\pi$
$354$	0	0
$355$	8.21712	0.436119
$356$	−4.54287	−0.240772
$357$	0	0
$358$	17.0718	0.902274
$359$	8.64024	0.456014	0.228007	−	0.973659i	$-0.426779\pi$
$359$	8.64024	0.456014	0.228007	+	0.973659i	$0.426779\pi$
$360$	0	0
$361$	−18.9226	−0.995927
$362$	−3.33538	−0.175304
$363$	0	0
$364$	−8.32123	−0.436151
$365$	−3.14988	−0.164872
$366$	0	0
$367$	13.4463	0.701892	0.350946	−	0.936396i	$-0.385860\pi$
$367$	13.4463	0.701892	0.350946	+	0.936396i	$0.385860\pi$
$368$	2.05974	0.107371
$369$	0	0
$370$	27.3779	1.42331
$371$	9.39134	0.487574
$372$	0	0
$373$	−3.19324	−0.165340	−0.0826698	−	0.996577i	$-0.526345\pi$
$373$	−3.19324	−0.165340	−0.0826698	+	0.996577i	$0.526345\pi$
$374$	−1.91733	−0.0991425
$375$	0	0
$376$	−12.9188	−0.666238
$377$	0	0
$378$	0	0
$379$	−16.5019	−0.847646	−0.423823	−	0.905745i	$-0.639312\pi$
$379$	−16.5019	−0.847646	−0.423823	+	0.905745i	$0.639312\pi$
$380$	0.640467	0.0328553
$381$	0	0
$382$	−5.38378	−0.275458
$383$	−1.57818	−0.0806413	−0.0403206	−	0.999187i	$-0.512838\pi$
$383$	−1.57818	−0.0806413	−0.0403206	+	0.999187i	$0.512838\pi$
$384$	0	0
$385$	8.63657	0.440160
$386$	16.8355	0.856904
$387$	0	0
$388$	2.34681	0.119141
$389$	14.0108	0.710375	0.355188	−	0.934795i	$-0.384417\pi$
$389$	14.0108	0.710375	0.355188	+	0.934795i	$0.384417\pi$
$390$	0	0
$391$	−2.78078	−0.140630
$392$	41.2863	2.08527
$393$	0	0
$394$	3.33794	0.168163
$395$	7.13403	0.358952
$396$	0	0
$397$	23.9158	1.20030	0.600150	−	0.799887i	$-0.295107\pi$
$397$	23.9158	1.20030	0.600150	+	0.799887i	$0.295107\pi$
$398$	−4.67615	−0.234394
$399$	0	0
$400$	−15.3157	−0.765786
$401$	35.8795	1.79174	0.895869	−	0.444318i	$-0.146554\pi$
$401$	35.8795	1.79174	0.895869	+	0.444318i	$0.146554\pi$
$402$	0	0
$403$	25.1730	1.25396
$404$	−8.45629	−0.420716
$405$	0	0
$406$	0	0
$407$	−3.81028	−0.188869
$408$	0	0
$409$	15.0182	0.742600	0.371300	−	0.928513i	$-0.378912\pi$
$409$	15.0182	0.742600	0.371300	+	0.928513i	$0.378912\pi$
$410$	−11.3618	−0.561117
$411$	0	0
$412$	−1.34649	−0.0663367
$413$	64.4517	3.17146
$414$	0	0
$415$	37.8433	1.85765
$416$	−9.92456	−0.486592
$417$	0	0
$418$	0.177983	0.00870541
$419$	−36.9542	−1.80533	−0.902666	−	0.430343i	$-0.858393\pi$
$419$	−36.9542	−1.80533	−0.902666	+	0.430343i	$0.858393\pi$
$420$	0	0
$421$	1.28657	0.0627036	0.0313518	−	0.999508i	$-0.490019\pi$
$421$	1.28657	0.0627036	0.0313518	+	0.999508i	$0.490019\pi$
$422$	15.2017	0.740007
$423$	0	0
$424$	6.40124	0.310872
$425$	20.6772	1.00299
$426$	0	0
$427$	−32.6683	−1.58093
$428$	−0.381777	−0.0184539
$429$	0	0
$430$	44.6753	2.15443
$431$	25.2826	1.21782	0.608910	−	0.793239i	$-0.291607\pi$
$431$	25.2826	1.21782	0.608910	+	0.793239i	$0.291607\pi$
$432$	0	0
$433$	−10.3588	−0.497812	−0.248906	−	0.968528i	$-0.580071\pi$
$433$	−10.3588	−0.497812	−0.248906	+	0.968528i	$0.580071\pi$
$434$	−47.5644	−2.28316
$435$	0	0
$436$	−12.9930	−0.622253
$437$	0.258136	0.0123483
$438$	0	0
$439$	−13.3949	−0.639306	−0.319653	−	0.947535i	$-0.603566\pi$
$439$	−13.3949	−0.639306	−0.319653	+	0.947535i	$0.603566\pi$
$440$	5.88678	0.280641
$441$	0	0
$442$	−9.54834	−0.454168
$443$	−30.7933	−1.46303	−0.731517	−	0.681823i	$-0.761187\pi$
$443$	−30.7933	−1.46303	−0.731517	+	0.681823i	$0.761187\pi$
$444$	0	0
$445$	23.4810	1.11311
$446$	17.6349	0.835038
$447$	0	0
$448$	38.8086	1.83353
$449$	−6.14934	−0.290205	−0.145103	−	0.989417i	$-0.546351\pi$
$449$	−6.14934	−0.290205	−0.145103	+	0.989417i	$0.546351\pi$
$450$	0	0
$451$	1.58126	0.0744585
$452$	1.62634	0.0764965
$453$	0	0
$454$	−17.0276	−0.799145
$455$	43.0104	2.01636
$456$	0	0
$457$	−17.0452	−0.797341	−0.398670	−	0.917094i	$-0.630528\pi$
$457$	−17.0452	−0.797341	−0.398670	+	0.917094i	$0.630528\pi$
$458$	−11.9753	−0.559569
$459$	0	0
$460$	2.13620	0.0996008
$461$	16.5645	0.771484	0.385742	−	0.922607i	$-0.373946\pi$
$461$	16.5645	0.771484	0.385742	+	0.922607i	$0.373946\pi$
$462$	0	0
$463$	28.9072	1.34343	0.671715	−	0.740809i	$-0.265558\pi$
$463$	28.9072	1.34343	0.671715	+	0.740809i	$0.265558\pi$
$464$	0	0
$465$	0	0
$466$	0.323489	0.0149854
$467$	−22.6915	−1.05004	−0.525019	−	0.851090i	$-0.675942\pi$
$467$	−22.6915	−1.05004	−0.525019	+	0.851090i	$0.675942\pi$
$468$	0	0
$469$	−55.5011	−2.56280
$470$	16.7072	0.770644
$471$	0	0
$472$	43.9310	2.02209
$473$	−6.21762	−0.285887
$474$	0	0
$475$	−1.91943	−0.0880697
$476$	−9.03548	−0.414140
$477$	0	0
$478$	7.43299	0.339977
$479$	−6.52355	−0.298068	−0.149034	−	0.988832i	$-0.547616\pi$
$479$	−6.52355	−0.298068	−0.149034	+	0.988832i	$0.547616\pi$
$480$	0	0
$481$	−18.9753	−0.865200
$482$	17.8065	0.811065
$483$	0	0
$484$	7.13630	0.324377
$485$	−12.1301	−0.550800
$486$	0	0
$487$	−23.4649	−1.06329	−0.531647	−	0.846966i	$-0.678427\pi$
$487$	−23.4649	−1.06329	−0.531647	+	0.846966i	$0.678427\pi$
$488$	−22.2671	−1.00798
$489$	0	0
$490$	−53.3931	−2.41205
$491$	29.9858	1.35324	0.676619	−	0.736333i	$-0.263444\pi$
$491$	29.9858	1.35324	0.676619	+	0.736333i	$0.263444\pi$
$492$	0	0
$493$	0	0
$494$	0.886358	0.0398791
$495$	0	0
$496$	−20.2463	−0.909084
$497$	10.7611	0.482700
$498$	0	0
$499$	15.5126	0.694441	0.347220	−	0.937784i	$-0.387126\pi$
$499$	15.5126	0.694441	0.347220	+	0.937784i	$0.387126\pi$
$500$	−4.37320	−0.195575
$501$	0	0
$502$	−3.79960	−0.169585
$503$	14.6872	0.654871	0.327435	−	0.944874i	$-0.393816\pi$
$503$	14.6872	0.654871	0.327435	+	0.944874i	$0.393816\pi$
$504$	0	0
$505$	43.7085	1.94500
$506$	0.593639	0.0263905
$507$	0	0
$508$	6.13270	0.272094
$509$	−32.6885	−1.44889	−0.724445	−	0.689332i	$-0.757904\pi$
$509$	−32.6885	−1.44889	−0.724445	+	0.689332i	$0.757904\pi$
$510$	0	0
$511$	−4.12505	−0.182482
$512$	21.6527	0.956922
$513$	0	0
$514$	−0.456726	−0.0201453
$515$	6.95967	0.306679
$516$	0	0
$517$	−2.32520	−0.102262
$518$	35.8538	1.57533
$519$	0	0
$520$	29.3163	1.28561
$521$	33.5048	1.46787	0.733937	−	0.679218i	$-0.237681\pi$
$521$	33.5048	1.46787	0.733937	+	0.679218i	$0.237681\pi$
$522$	0	0
$523$	−23.3879	−1.02268	−0.511341	−	0.859378i	$-0.670851\pi$
$523$	−23.3879	−1.02268	−0.511341	+	0.859378i	$0.670851\pi$
$524$	−8.73874	−0.381754
$525$	0	0
$526$	−9.40082	−0.409895
$527$	27.3337	1.19068
$528$	0	0
$529$	−22.1390	−0.962566
$530$	−8.27834	−0.359588
$531$	0	0
$532$	0.838750	0.0363644
$533$	7.87471	0.341092
$534$	0	0
$535$	1.97331	0.0853137
$536$	−37.8302	−1.63401
$537$	0	0
$538$	10.3636	0.446805
$539$	7.43091	0.320072
$540$	0	0
$541$	−14.6065	−0.627981	−0.313990	−	0.949426i	$-0.601666\pi$
$541$	−14.6065	−0.627981	−0.313990	+	0.949426i	$0.601666\pi$
$542$	−5.81763	−0.249889
$543$	0	0
$544$	−10.7764	−0.462036
$545$	67.1578	2.87672
$546$	0	0
$547$	12.3265	0.527044	0.263522	−	0.964653i	$-0.415116\pi$
$547$	12.3265	0.527044	0.263522	+	0.964653i	$0.415116\pi$
$548$	−5.07248	−0.216686
$549$	0	0
$550$	−4.41415	−0.188220
$551$	0	0
$552$	0	0
$553$	9.34267	0.397291
$554$	−2.34933	−0.0998134
$555$	0	0
$556$	−5.90845	−0.250574
$557$	24.8902	1.05463	0.527316	−	0.849669i	$-0.323199\pi$
$557$	24.8902	1.05463	0.527316	+	0.849669i	$0.323199\pi$
$558$	0	0
$559$	−30.9640	−1.30964
$560$	−34.5926	−1.46180
$561$	0	0
$562$	27.5024	1.16012
$563$	−16.2217	−0.683662	−0.341831	−	0.939762i	$-0.611047\pi$
$563$	−16.2217	−0.683662	−0.341831	+	0.939762i	$0.611047\pi$
$564$	0	0
$565$	−8.40615	−0.353649
$566$	17.2995	0.727151
$567$	0	0
$568$	7.33486	0.307764
$569$	44.8298	1.87936	0.939681	−	0.342052i	$-0.111122\pi$
$569$	44.8298	1.87936	0.939681	+	0.342052i	$0.111122\pi$
$570$	0	0
$571$	−16.9942	−0.711183	−0.355592	−	0.934641i	$-0.615721\pi$
$571$	−16.9942	−0.711183	−0.355592	+	0.934641i	$0.615721\pi$
$572$	−1.02085	−0.0426837
$573$	0	0
$574$	−14.8793	−0.621048
$575$	−6.40204	−0.266984
$576$	0	0
$577$	−5.54668	−0.230911	−0.115456	−	0.993313i	$-0.536833\pi$
$577$	−5.54668	−0.230911	−0.115456	+	0.993313i	$0.536833\pi$
$578$	9.25667	0.385027
$579$	0	0
$580$	0	0
$581$	49.5592	2.05606
$582$	0	0
$583$	1.15213	0.0477162
$584$	−2.81168	−0.116348
$585$	0	0
$586$	35.0588	1.44827
$587$	19.1436	0.790140	0.395070	−	0.918651i	$-0.370720\pi$
$587$	19.1436	0.790140	0.395070	+	0.918651i	$0.370720\pi$
$588$	0	0
$589$	−2.53735	−0.104550
$590$	−56.8133	−2.33897
$591$	0	0
$592$	15.2616	0.627246
$593$	−31.7165	−1.30244	−0.651220	−	0.758889i	$-0.725742\pi$
$593$	−31.7165	−1.30244	−0.651220	+	0.758889i	$0.725742\pi$
$594$	0	0
$595$	46.7022	1.91460
$596$	5.31913	0.217880
$597$	0	0
$598$	2.95634	0.120894
$599$	47.1580	1.92683	0.963413	−	0.268023i	$-0.0863703\pi$
$599$	47.1580	1.92683	0.963413	+	0.268023i	$0.0863703\pi$
$600$	0	0
$601$	14.8050	0.603907	0.301954	−	0.953323i	$-0.402361\pi$
$601$	14.8050	0.603907	0.301954	+	0.953323i	$0.402361\pi$
$602$	58.5064	2.38454
$603$	0	0
$604$	0.0704241	0.00286551
$605$	−36.8858	−1.49962
$606$	0	0
$607$	11.7032	0.475017	0.237508	−	0.971385i	$-0.423669\pi$
$607$	11.7032	0.475017	0.237508	+	0.971385i	$0.423669\pi$
$608$	1.00036	0.0405700
$609$	0	0
$610$	28.7967	1.16594
$611$	−11.5795	−0.468458
$612$	0	0
$613$	−45.9483	−1.85583	−0.927917	−	0.372787i	$-0.878402\pi$
$613$	−45.9483	−1.85583	−0.927917	+	0.372787i	$0.878402\pi$
$614$	−2.42975	−0.0980566
$615$	0	0
$616$	7.70928	0.310616
$617$	41.1355	1.65605	0.828026	−	0.560689i	$-0.189464\pi$
$617$	41.1355	1.65605	0.828026	+	0.560689i	$0.189464\pi$
$618$	0	0
$619$	−2.97032	−0.119387	−0.0596936	−	0.998217i	$-0.519012\pi$
$619$	−2.97032	−0.119387	−0.0596936	+	0.998217i	$0.519012\pi$
$620$	−20.9978	−0.843292
$621$	0	0
$622$	19.4816	0.781141
$623$	30.7505	1.23199
$624$	0	0
$625$	−11.8938	−0.475753
$626$	−17.1190	−0.684212
$627$	0	0
$628$	9.01225	0.359628
$629$	−20.6041	−0.821538
$630$	0	0
$631$	21.5126	0.856403	0.428202	−	0.903683i	$-0.359147\pi$
$631$	21.5126	0.856403	0.428202	+	0.903683i	$0.359147\pi$
$632$	6.36807	0.253308
$633$	0	0
$634$	−6.35632	−0.252442
$635$	−31.6984	−1.25791
$636$	0	0
$637$	37.0062	1.46624
$638$	0	0
$639$	0	0
$640$	−9.40075	−0.371597
$641$	32.2082	1.27215	0.636075	−	0.771628i	$-0.280557\pi$
$641$	32.2082	1.27215	0.636075	+	0.771628i	$0.280557\pi$
$642$	0	0
$643$	39.0467	1.53985	0.769926	−	0.638133i	$-0.220293\pi$
$643$	39.0467	1.53985	0.769926	+	0.638133i	$0.220293\pi$
$644$	2.79755	0.110239
$645$	0	0
$646$	0.962439	0.0378666
$647$	4.12296	0.162090	0.0810452	−	0.996710i	$-0.474174\pi$
$647$	4.12296	0.162090	0.0810452	+	0.996710i	$0.474174\pi$
$648$	0	0
$649$	7.90692	0.310374
$650$	−21.9826	−0.862229
$651$	0	0
$652$	1.74442	0.0683167
$653$	15.2804	0.597966	0.298983	−	0.954258i	$-0.403353\pi$
$653$	15.2804	0.597966	0.298983	+	0.954258i	$0.403353\pi$
$654$	0	0
$655$	45.1684	1.76488
$656$	−6.33351	−0.247282
$657$	0	0
$658$	21.8796	0.852954
$659$	34.5146	1.34450	0.672249	−	0.740326i	$-0.265329\pi$
$659$	34.5146	1.34450	0.672249	+	0.740326i	$0.265329\pi$
$660$	0	0
$661$	11.9611	0.465233	0.232616	−	0.972569i	$-0.425271\pi$
$661$	11.9611	0.465233	0.232616	+	0.972569i	$0.425271\pi$
$662$	6.79741	0.264189
$663$	0	0
$664$	33.7801	1.31092
$665$	−4.33529	−0.168116
$666$	0	0
$667$	0	0
$668$	−6.81990	−0.263870
$669$	0	0
$670$	48.9235	1.89008
$671$	−4.00774	−0.154717
$672$	0	0
$673$	−17.6881	−0.681825	−0.340912	−	0.940095i	$-0.610736\pi$
$673$	−17.6881	−0.681825	−0.340912	+	0.940095i	$0.610736\pi$
$674$	−20.1206	−0.775018
$675$	0	0
$676$	3.59223	0.138163
$677$	−29.7747	−1.14433	−0.572166	−	0.820138i	$-0.693897\pi$
$677$	−29.7747	−1.14433	−0.572166	+	0.820138i	$0.693897\pi$
$678$	0	0
$679$	−15.8855	−0.609629
$680$	31.8327	1.22073
$681$	0	0
$682$	−5.83518	−0.223441
$683$	16.9703	0.649349	0.324675	−	0.945826i	$-0.394745\pi$
$683$	16.9703	0.649349	0.324675	+	0.945826i	$0.394745\pi$
$684$	0	0
$685$	26.2184	1.00175
$686$	−33.4183	−1.27592
$687$	0	0
$688$	24.9038	0.949450
$689$	5.73763	0.218586
$690$	0	0
$691$	−24.6905	−0.939271	−0.469636	−	0.882860i	$-0.655615\pi$
$691$	−24.6905	−0.939271	−0.469636	+	0.882860i	$0.655615\pi$
$692$	15.4113	0.585849
$693$	0	0
$694$	−3.25031	−0.123380
$695$	30.5394	1.15842
$696$	0	0
$697$	8.55063	0.323878
$698$	−27.1222	−1.02659
$699$	0	0
$700$	−20.8019	−0.786238
$701$	−25.2940	−0.955340	−0.477670	−	0.878539i	$-0.658519\pi$
$701$	−25.2940	−0.955340	−0.477670	+	0.878539i	$0.658519\pi$
$702$	0	0
$703$	1.91264	0.0721367
$704$	4.76103	0.179438
$705$	0	0
$706$	18.6506	0.701923
$707$	57.2403	2.15274
$708$	0	0
$709$	1.99466	0.0749111	0.0374555	−	0.999298i	$-0.488075\pi$
$709$	1.99466	0.0749111	0.0374555	+	0.999298i	$0.488075\pi$
$710$	−9.48573	−0.355993
$711$	0	0
$712$	20.9599	0.785505
$713$	−8.46302	−0.316943
$714$	0	0
$715$	5.27650	0.197330
$716$	9.86979	0.368852
$717$	0	0
$718$	−9.97418	−0.372233
$719$	−6.32608	−0.235923	−0.117961	−	0.993018i	$-0.537636\pi$
$719$	−6.32608	−0.235923	−0.117961	+	0.993018i	$0.537636\pi$
$720$	0	0
$721$	9.11432	0.339435
$722$	21.8440	0.812950
$723$	0	0
$724$	−1.92830	−0.0716646
$725$	0	0
$726$	0	0
$727$	18.1899	0.674626	0.337313	−	0.941393i	$-0.390482\pi$
$727$	18.1899	0.674626	0.337313	+	0.941393i	$0.390482\pi$
$728$	38.3925	1.42292
$729$	0	0
$730$	3.63618	0.134581
$731$	−33.6217	−1.24354
$732$	0	0
$733$	−36.2229	−1.33792	−0.668962	−	0.743296i	$-0.733261\pi$
$733$	−36.2229	−1.33792	−0.668962	+	0.743296i	$0.733261\pi$
$734$	−15.5223	−0.572937
$735$	0	0
$736$	3.33658	0.122988
$737$	−6.80886	−0.250808
$738$	0	0
$739$	−4.16599	−0.153248	−0.0766242	−	0.997060i	$-0.524414\pi$
$739$	−4.16599	−0.153248	−0.0766242	+	0.997060i	$0.524414\pi$
$740$	15.8281	0.581851
$741$	0	0
$742$	−10.8412	−0.397995
$743$	−24.5985	−0.902431	−0.451216	−	0.892415i	$-0.649009\pi$
$743$	−24.5985	−0.902431	−0.451216	+	0.892415i	$0.649009\pi$
$744$	0	0
$745$	−27.4933	−1.00728
$746$	3.68623	0.134963
$747$	0	0
$748$	−1.10847	−0.0405297
$749$	2.58423	0.0944258
$750$	0	0
$751$	26.1808	0.955350	0.477675	−	0.878537i	$-0.341480\pi$
$751$	26.1808	0.955350	0.477675	+	0.878537i	$0.341480\pi$
$752$	9.31325	0.339619
$753$	0	0
$754$	0	0
$755$	−0.364005	−0.0132475
$756$	0	0
$757$	−52.2657	−1.89963	−0.949815	−	0.312814i	$-0.898729\pi$
$757$	−52.2657	−1.89963	−0.949815	+	0.312814i	$0.898729\pi$
$758$	19.0496	0.691912
$759$	0	0
$760$	−2.95498	−0.107189
$761$	−4.90272	−0.177724	−0.0888618	−	0.996044i	$-0.528323\pi$
$761$	−4.90272	−0.177724	−0.0888618	+	0.996044i	$0.528323\pi$
$762$	0	0
$763$	87.9493	3.18398
$764$	−3.11254	−0.112608
$765$	0	0
$766$	1.82183	0.0658255
$767$	39.3767	1.42181
$768$	0	0
$769$	29.0468	1.04746	0.523728	−	0.851886i	$-0.324541\pi$
$769$	29.0468	1.04746	0.523728	+	0.851886i	$0.324541\pi$
$770$	−9.96995	−0.359292
$771$	0	0
$772$	9.73316	0.350304
$773$	−53.5664	−1.92665	−0.963325	−	0.268337i	$-0.913526\pi$
$773$	−53.5664	−1.92665	−0.963325	+	0.268337i	$0.913526\pi$
$774$	0	0
$775$	62.9289	2.26047
$776$	−10.8277	−0.388692
$777$	0	0
$778$	−16.1739	−0.579862
$779$	−0.793743	−0.0284388
$780$	0	0
$781$	1.32016	0.0472392
$782$	3.21010	0.114793
$783$	0	0
$784$	−29.7635	−1.06298
$785$	−46.5821	−1.66259
$786$	0	0
$787$	−42.8418	−1.52715	−0.763573	−	0.645722i	$-0.776557\pi$
$787$	−42.8418	−1.52715	−0.763573	+	0.645722i	$0.776557\pi$
$788$	1.92977	0.0687453
$789$	0	0
$790$	−8.23544	−0.293004
$791$	−11.0086	−0.391422
$792$	0	0
$793$	−19.9587	−0.708753
$794$	−27.6081	−0.979775
$795$	0	0
$796$	−2.70344	−0.0958208
$797$	−24.6827	−0.874307	−0.437154	−	0.899387i	$-0.644013\pi$
$797$	−24.6827	−0.874307	−0.437154	+	0.899387i	$0.644013\pi$
$798$	0	0
$799$	−12.5735	−0.444818
$800$	−24.8100	−0.877166
$801$	0	0
$802$	−41.4189	−1.46255
$803$	−0.506060	−0.0178585
$804$	0	0
$805$	−14.4599	−0.509643
$806$	−29.0594	−1.02357
$807$	0	0
$808$	39.0156	1.37256
$809$	−30.2713	−1.06428	−0.532141	−	0.846656i	$-0.678612\pi$
$809$	−30.2713	−1.06428	−0.532141	+	0.846656i	$0.678612\pi$
$810$	0	0
$811$	33.6697	1.18230	0.591151	−	0.806561i	$-0.298674\pi$
$811$	33.6697	1.18230	0.591151	+	0.806561i	$0.298674\pi$
$812$	0	0
$813$	0	0
$814$	4.39854	0.154169
$815$	−9.01647	−0.315833
$816$	0	0
$817$	3.12106	0.109192
$818$	−17.3368	−0.606166
$819$	0	0
$820$	−6.56861	−0.229386
$821$	39.4748	1.37768	0.688840	−	0.724913i	$-0.258120\pi$
$821$	39.4748	1.37768	0.688840	+	0.724913i	$0.258120\pi$
$822$	0	0
$823$	30.7129	1.07058	0.535292	−	0.844667i	$-0.320202\pi$
$823$	30.7129	1.07058	0.535292	+	0.844667i	$0.320202\pi$
$824$	6.21242	0.216420
$825$	0	0
$826$	−74.4022	−2.58879
$827$	−50.6699	−1.76196	−0.880982	−	0.473150i	$-0.843117\pi$
$827$	−50.6699	−1.76196	−0.880982	+	0.473150i	$0.843117\pi$
$828$	0	0
$829$	17.7320	0.615858	0.307929	−	0.951409i	$-0.400364\pi$
$829$	17.7320	0.615858	0.307929	+	0.951409i	$0.400364\pi$
$830$	−43.6858	−1.51636
$831$	0	0
$832$	23.7101	0.821999
$833$	40.1826	1.39224
$834$	0	0
$835$	35.2504	1.21989
$836$	0.102898	0.00355879
$837$	0	0
$838$	42.6595	1.47365
$839$	−23.4750	−0.810447	−0.405223	−	0.914218i	$-0.632806\pi$
$839$	−23.4750	−0.810447	−0.405223	+	0.914218i	$0.632806\pi$
$840$	0	0
$841$	0	0
$842$	−1.48520	−0.0511834
$843$	0	0
$844$	8.78860	0.302516
$845$	−18.5673	−0.638736
$846$	0	0
$847$	−48.3053	−1.65979
$848$	−4.61469	−0.158469
$849$	0	0
$850$	−23.8695	−0.818717
$851$	6.37939	0.218683
$852$	0	0
$853$	−24.8185	−0.849771	−0.424885	−	0.905247i	$-0.639686\pi$
$853$	−24.8185	−0.849771	−0.424885	+	0.905247i	$0.639686\pi$
$854$	37.7119	1.29047
$855$	0	0
$856$	1.76144	0.0602048
$857$	42.3415	1.44636	0.723180	−	0.690660i	$-0.242680\pi$
$857$	42.3415	1.44636	0.723180	+	0.690660i	$0.242680\pi$
$858$	0	0
$859$	22.2450	0.758990	0.379495	−	0.925194i	$-0.376098\pi$
$859$	22.2450	0.758990	0.379495	+	0.925194i	$0.376098\pi$
$860$	25.8283	0.880737
$861$	0	0
$862$	−29.1859	−0.994076
$863$	−36.5854	−1.24538	−0.622691	−	0.782468i	$-0.713961\pi$
$863$	−36.5854	−1.24538	−0.622691	+	0.782468i	$0.713961\pi$
$864$	0	0
$865$	−79.6572	−2.70843
$866$	11.9581	0.406352
$867$	0	0
$868$	−27.4986	−0.933362
$869$	1.14616	0.0388807
$870$	0	0
$871$	−33.9083	−1.14894
$872$	59.9472	2.03007
$873$	0	0
$874$	−0.297989	−0.0100796
$875$	29.6020	1.00073
$876$	0	0
$877$	36.0234	1.21642	0.608211	−	0.793775i	$-0.291887\pi$
$877$	36.0234	1.21642	0.608211	+	0.793775i	$0.291887\pi$
$878$	15.4630	0.521849
$879$	0	0
$880$	−4.24381	−0.143059
$881$	20.8449	0.702281	0.351141	−	0.936323i	$-0.385794\pi$
$881$	20.8449	0.702281	0.351141	+	0.936323i	$0.385794\pi$
$882$	0	0
$883$	27.9120	0.939314	0.469657	−	0.882849i	$-0.344378\pi$
$883$	27.9120	0.939314	0.469657	+	0.882849i	$0.344378\pi$
$884$	−5.52021	−0.185665
$885$	0	0
$886$	35.5474	1.19424
$887$	54.6365	1.83451	0.917257	−	0.398296i	$-0.130398\pi$
$887$	54.6365	1.83451	0.917257	+	0.398296i	$0.130398\pi$
$888$	0	0
$889$	−41.5120	−1.39227
$890$	−27.1062	−0.908600
$891$	0	0
$892$	10.1953	0.341365
$893$	1.16718	0.0390581
$894$	0	0
$895$	−51.0146	−1.70523
$896$	−12.3111	−0.411286
$897$	0	0
$898$	7.09872	0.236887
$899$	0	0
$900$	0	0
$901$	6.23012	0.207555
$902$	−1.82538	−0.0607786
$903$	0	0
$904$	−7.50360	−0.249566
$905$	9.96689	0.331311
$906$	0	0
$907$	16.1102	0.534931	0.267465	−	0.963567i	$-0.413814\pi$
$907$	16.1102	0.534931	0.267465	+	0.963567i	$0.413814\pi$
$908$	−9.84422	−0.326692
$909$	0	0
$910$	−49.6506	−1.64590
$911$	−42.9222	−1.42207	−0.711037	−	0.703154i	$-0.751774\pi$
$911$	−42.9222	−1.42207	−0.711037	+	0.703154i	$0.751774\pi$
$912$	0	0
$913$	6.07991	0.201216
$914$	19.6768	0.650849
$915$	0	0
$916$	−6.92332	−0.228753
$917$	59.1522	1.95338
$918$	0	0
$919$	−19.3983	−0.639892	−0.319946	−	0.947436i	$-0.603665\pi$
$919$	−19.3983	−0.639892	−0.319946	+	0.947436i	$0.603665\pi$
$920$	−9.85599	−0.324942
$921$	0	0
$922$	−19.1218	−0.629743
$923$	6.57446	0.216401
$924$	0	0
$925$	−47.4356	−1.55967
$926$	−33.3701	−1.09661
$927$	0	0
$928$	0	0
$929$	−15.9251	−0.522484	−0.261242	−	0.965273i	$-0.584132\pi$
$929$	−15.9251	−0.522484	−0.261242	+	0.965273i	$0.584132\pi$
$930$	0	0
$931$	−3.73009	−0.122249
$932$	0.187020	0.00612604
$933$	0	0
$934$	26.1948	0.857120
$935$	5.72941	0.187372
$936$	0	0
$937$	−40.1366	−1.31121	−0.655603	−	0.755105i	$-0.727586\pi$
$937$	−40.1366	−1.31121	−0.655603	+	0.755105i	$0.727586\pi$
$938$	64.0698	2.09195
$939$	0	0
$940$	9.65896	0.315041
$941$	−2.46839	−0.0804673	−0.0402337	−	0.999190i	$-0.512810\pi$
$941$	−2.46839	−0.0804673	−0.0402337	+	0.999190i	$0.512810\pi$
$942$	0	0
$943$	−2.64743	−0.0862123
$944$	−31.6701	−1.03077
$945$	0	0
$946$	7.17755	0.233362
$947$	−26.3388	−0.855896	−0.427948	−	0.903803i	$-0.640763\pi$
$947$	−26.3388	−0.855896	−0.427948	+	0.903803i	$0.640763\pi$
$948$	0	0
$949$	−2.52020	−0.0818090
$950$	2.21577	0.0718891
$951$	0	0
$952$	41.6879	1.35111
$953$	−47.1256	−1.52655	−0.763275	−	0.646074i	$-0.776410\pi$
$953$	−47.1256	−1.52655	−0.763275	+	0.646074i	$0.776410\pi$
$954$	0	0
$955$	16.0880	0.520595
$956$	4.29726	0.138983
$957$	0	0
$958$	7.53070	0.243306
$959$	34.3354	1.10875
$960$	0	0
$961$	52.1874	1.68346
$962$	21.9049	0.706241
$963$	0	0
$964$	10.2945	0.331565
$965$	−50.3083	−1.61948
$966$	0	0
$967$	−28.5319	−0.917524	−0.458762	−	0.888559i	$-0.651707\pi$
$967$	−28.5319	−0.917524	−0.458762	+	0.888559i	$0.651707\pi$
$968$	−32.9254	−1.05826
$969$	0	0
$970$	14.0028	0.449604
$971$	−0.490076	−0.0157273	−0.00786364	−	0.999969i	$-0.502503\pi$
$971$	−0.490076	−0.0157273	−0.00786364	+	0.999969i	$0.502503\pi$
$972$	0	0
$973$	39.9941	1.28215
$974$	27.0876	0.867941
$975$	0	0
$976$	16.0524	0.513826
$977$	18.1436	0.580465	0.290232	−	0.956956i	$-0.406267\pi$
$977$	18.1436	0.580465	0.290232	+	0.956956i	$0.406267\pi$
$978$	0	0
$979$	3.77246	0.120568
$980$	−30.8683	−0.986053
$981$	0	0
$982$	−34.6152	−1.10462
$983$	55.6548	1.77511	0.887556	−	0.460699i	$-0.152401\pi$
$983$	55.6548	1.77511	0.887556	+	0.460699i	$0.152401\pi$
$984$	0	0
$985$	−9.97452	−0.317815
$986$	0	0
$987$	0	0
$988$	0.512433	0.0163027
$989$	10.4099	0.331016
$990$	0	0
$991$	20.2428	0.643034	0.321517	−	0.946904i	$-0.395807\pi$
$991$	20.2428	0.643034	0.321517	+	0.946904i	$0.395807\pi$
$992$	−32.7970	−1.04131
$993$	0	0
$994$	−12.4224	−0.394016
$995$	13.9734	0.442987
$996$	0	0
$997$	29.2318	0.925780	0.462890	−	0.886416i	$-0.346812\pi$
$997$	29.2318	0.925780	0.462890	+	0.886416i	$0.346812\pi$
$998$	−17.9076	−0.566855
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7569.2.a.bl.1.4		9
3.2	odd	2		2523.2.a.p.1.6		9
29.7	even	7		261.2.k.b.136.2		18
29.25	even	7		261.2.k.b.190.2		18
29.28	even	2		7569.2.a.bk.1.6		9
87.65	odd	14		87.2.g.b.49.2	yes	18
87.83	odd	14		87.2.g.b.16.2	✓	18
87.86	odd	2		2523.2.a.q.1.4		9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
87.2.g.b.16.2	✓	18	87.83	odd	14
87.2.g.b.49.2	yes	18	87.65	odd	14
261.2.k.b.136.2		18	29.7	even	7
261.2.k.b.190.2		18	29.25	even	7
2523.2.a.p.1.6		9	3.2	odd	2
2523.2.a.q.1.4		9	87.86	odd	2
7569.2.a.bk.1.6		9	29.28	even	2
7569.2.a.bl.1.4		9	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:	\( N \)	\(=\)	\( 7569 = 3^{2} \cdot 29^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7569.a (trivial)

\(f(q)\)	\(=\)	\(q-1.15439 q^{2} -0.667390 q^{4} +3.44957 q^{5} +4.51754 q^{7} +3.07920 q^{8} +O(q^{10})\)	\(q-1.15439 q^{2} -0.667390 q^{4} +3.44957 q^{5} +4.51754 q^{7} +3.07920 q^{8} -3.98215 q^{10} +0.554210 q^{11} +2.75998 q^{13} -5.21499 q^{14} -2.21981 q^{16} +2.99688 q^{17} -0.278196 q^{19} -2.30221 q^{20} -0.639773 q^{22} -0.927891 q^{23} +6.89956 q^{25} -3.18609 q^{26} -3.01496 q^{28} +9.12071 q^{31} -3.59588 q^{32} -3.45957 q^{34} +15.5836 q^{35} -6.87516 q^{37} +0.321146 q^{38} +10.6219 q^{40} +2.85317 q^{41} -11.2189 q^{43} -0.369874 q^{44} +1.07115 q^{46} -4.19552 q^{47} +13.4081 q^{49} -7.96477 q^{50} -1.84198 q^{52} +2.07886 q^{53} +1.91179 q^{55} +13.9104 q^{56} +14.2670 q^{59} -7.23144 q^{61} -10.5288 q^{62} +8.59066 q^{64} +9.52076 q^{65} -12.2857 q^{67} -2.00009 q^{68} -17.9895 q^{70} +2.38207 q^{71} -0.913120 q^{73} +7.93660 q^{74} +0.185665 q^{76} +2.50366 q^{77} +2.06809 q^{79} -7.65740 q^{80} -3.29367 q^{82} +10.9704 q^{83} +10.3380 q^{85} +12.9510 q^{86} +1.70652 q^{88} +6.80692 q^{89} +12.4683 q^{91} +0.619265 q^{92} +4.84325 q^{94} -0.959659 q^{95} -3.51641 q^{97} -15.4782 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + q^{2} + 11 q^{4} + 5 q^{7} + 12 q^{8}+O(q^{10}) \) 9 * q + q^2 + 11 * q^4 + 5 * q^7 + 12 * q^8	\( 9 q + q^{2} + 11 q^{4} + 5 q^{7} + 12 q^{8} - 4 q^{10} + 3 q^{11} + 5 q^{13} + 15 q^{14} - 5 q^{16} + 16 q^{17} + q^{19} - 8 q^{20} + 24 q^{22} + 10 q^{23} + 9 q^{25} + 12 q^{26} - 24 q^{28} + 4 q^{31} + 25 q^{32} + 24 q^{34} + 44 q^{35} - 25 q^{37} + 10 q^{38} - 5 q^{40} + 34 q^{41} - 12 q^{43} - 23 q^{44} - 6 q^{46} + 8 q^{47} + 26 q^{49} + 27 q^{50} - 23 q^{52} + 32 q^{53} + 5 q^{55} - 14 q^{56} - 10 q^{59} - 51 q^{61} - 8 q^{62} - 8 q^{64} + 11 q^{65} + 7 q^{67} + 11 q^{68} + 14 q^{70} - 7 q^{71} - 17 q^{73} + 62 q^{74} + 6 q^{76} + 64 q^{77} - 13 q^{79} - 54 q^{80} + 37 q^{82} + 31 q^{83} - 42 q^{85} + 70 q^{86} - 29 q^{88} - 32 q^{89} + 45 q^{91} - 9 q^{92} + 38 q^{94} - 20 q^{95} - 16 q^{97} - 12 q^{98}+O(q^{100}) \) 9 * q + q^2 + 11 * q^4 + 5 * q^7 + 12 * q^8 - 4 * q^10 + 3 * q^11 + 5 * q^13 + 15 * q^14 - 5 * q^16 + 16 * q^17 + q^19 - 8 * q^20 + 24 * q^22 + 10 * q^23 + 9 * q^25 + 12 * q^26 - 24 * q^28 + 4 * q^31 + 25 * q^32 + 24 * q^34 + 44 * q^35 - 25 * q^37 + 10 * q^38 - 5 * q^40 + 34 * q^41 - 12 * q^43 - 23 * q^44 - 6 * q^46 + 8 * q^47 + 26 * q^49 + 27 * q^50 - 23 * q^52 + 32 * q^53 + 5 * q^55 - 14 * q^56 - 10 * q^59 - 51 * q^61 - 8 * q^62 - 8 * q^64 + 11 * q^65 + 7 * q^67 + 11 * q^68 + 14 * q^70 - 7 * q^71 - 17 * q^73 + 62 * q^74 + 6 * q^76 + 64 * q^77 - 13 * q^79 - 54 * q^80 + 37 * q^82 + 31 * q^83 - 42 * q^85 + 70 * q^86 - 29 * q^88 - 32 * q^89 + 45 * q^91 - 9 * q^92 + 38 * q^94 - 20 * q^95 - 16 * q^97 - 12 * q^98

Introduction

L-functions

Modular forms

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Database

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