LMFDB - Embedded newform 4925.2.a.j.1.9

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4925,2,Mod(1,4925)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("4925.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4925, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 4925 = 5^{2} \cdot 197 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4925.a (trivial)

Newform invariants

comment:select newform

sage:traces = [10,2,3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	yes
Analytic conductor:	$39.3263229955$
Analytic rank:	$1$
Dimension:	$10$
Coefficient field:	10.10.21886214112361.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 2x^{9} - 9x^{8} + 16x^{7} + 26x^{6} - 38x^{5} - 27x^{4} + 32x^{3} + 6x^{2} - 7x + 1 $ x^10 - 2x^9 - 9x^8 + 16x^7 + 26x^6 - 38x^5 - 27x^4 + 32x^3 + 6x^2 - 7*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 985)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Root			$2.24278$ of defining polynomial
Character	$\chi$	$=$	4925.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.24278 q^{2} -1.69515 q^{3} +3.03007 q^{4} -3.80186 q^{6} +3.86416 q^{7} +2.31023 q^{8} -0.126460 q^{9} -2.20820 q^{11} -5.13643 q^{12} -1.54718 q^{13} +8.66648 q^{14} -0.878807 q^{16} -5.49863 q^{17} -0.283622 q^{18} +0.753438 q^{19} -6.55035 q^{21} -4.95250 q^{22} +5.79633 q^{23} -3.91619 q^{24} -3.46999 q^{26} +5.29982 q^{27} +11.7087 q^{28} -8.47619 q^{29} -8.33139 q^{31} -6.59143 q^{32} +3.74323 q^{33} -12.3322 q^{34} -0.383183 q^{36} -10.8352 q^{37} +1.68980 q^{38} +2.62271 q^{39} -1.03866 q^{41} -14.6910 q^{42} +4.39227 q^{43} -6.69099 q^{44} +12.9999 q^{46} +6.20938 q^{47} +1.48971 q^{48} +7.93177 q^{49} +9.32101 q^{51} -4.68808 q^{52} +5.32664 q^{53} +11.8864 q^{54} +8.92710 q^{56} -1.27719 q^{57} -19.0102 q^{58} +0.409456 q^{59} -10.7844 q^{61} -18.6855 q^{62} -0.488662 q^{63} -13.0255 q^{64} +8.39524 q^{66} -2.86371 q^{67} -16.6612 q^{68} -9.82567 q^{69} -14.9588 q^{71} -0.292151 q^{72} -4.86429 q^{73} -24.3010 q^{74} +2.28297 q^{76} -8.53283 q^{77} +5.88217 q^{78} +7.99199 q^{79} -8.60463 q^{81} -2.32949 q^{82} +0.733605 q^{83} -19.8480 q^{84} +9.85091 q^{86} +14.3684 q^{87} -5.10143 q^{88} +1.13084 q^{89} -5.97857 q^{91} +17.5633 q^{92} +14.1230 q^{93} +13.9263 q^{94} +11.1735 q^{96} -17.3030 q^{97} +17.7892 q^{98} +0.279248 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 2 q^{2} + 3 q^{3} + 2 q^{4} - 9 q^{6} + 6 q^{7} + 6 q^{8} - 5 q^{9} - 11 q^{11} + 5 q^{13} - 9 q^{14} - 2 q^{16} + 4 q^{17} - 15 q^{18} - 28 q^{19} - 7 q^{21} + 12 q^{22} + 24 q^{23} - 3 q^{24} + 7 q^{26}+ \cdots - 16 q^{99}+O(q^{100}) $ 10 * q + 2 * q^2 + 3 * q^3 + 2 * q^4 - 9 * q^6 + 6 * q^7 + 6 * q^8 - 5 * q^9 - 11 * q^11 + 5 * q^13 - 9 * q^14 - 2 * q^16 + 4 * q^17 - 15 * q^18 - 28 * q^19 - 7 * q^21 + 12 * q^22 + 24 * q^23 - 3 * q^24 + 7 * q^26 + 9 * q^27 - 3 * q^28 - 16 * q^29 - 34 * q^31 + 4 * q^32 + 10 * q^33 + q^34 - q^36 - 5 * q^37 - 13 * q^38 - 15 * q^39 - 15 * q^41 - 7 * q^42 + 5 * q^43 - 11 * q^44 + 6 * q^46 + 14 * q^47 - 12 * q^48 + 4 * q^49 - q^51 + 7 * q^52 + 15 * q^53 - 4 * q^54 - 5 * q^56 - 9 * q^57 - 28 * q^58 - 49 * q^59 - 16 * q^61 - 20 * q^62 - 8 * q^63 - 16 * q^64 + 40 * q^66 + 2 * q^67 - 19 * q^68 - 16 * q^69 - 64 * q^71 - q^72 - 18 * q^73 - 37 * q^74 - 19 * q^76 - 8 * q^77 - 4 * q^78 - 11 * q^79 - 14 * q^81 - 10 * q^82 - 4 * q^83 - 11 * q^84 + 5 * q^86 + q^87 - 19 * q^88 - 5 * q^89 - 30 * q^91 + 32 * q^92 - 41 * q^93 - 37 * q^94 + 26 * q^96 - 20 * q^97 + 37 * q^98 - 16 * q^99

Coefficient data

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

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

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

$2$	2.24278	1.58589	0.792943	−	0.609295i	$-0.208548\pi$
$2$	2.24278	1.58589	0.792943	+	0.609295i	$0.208548\pi$
$3$	−1.69515	−0.978696	−0.489348	−	0.872088i	$-0.662765\pi$
$3$	−1.69515	−0.978696	−0.489348	+	0.872088i	$0.662765\pi$
$4$	3.03007	1.51504
$5$	0	0
$5$	0	0
$6$	−3.80186	−1.55210
$7$	3.86416	1.46052	0.730258	−	0.683171i	$-0.239400\pi$
$7$	3.86416	1.46052	0.730258	+	0.683171i	$0.239400\pi$
$8$	2.31023	0.816789
$9$	−0.126460	−0.0421533
$10$	0	0
$11$	−2.20820	−0.665796	−0.332898	−	0.942963i	$-0.608026\pi$
$11$	−2.20820	−0.665796	−0.332898	+	0.942963i	$0.608026\pi$
$12$	−5.13643	−1.48276
$13$	−1.54718	−0.429111	−0.214556	−	0.976712i	$-0.568830\pi$
$13$	−1.54718	−0.429111	−0.214556	+	0.976712i	$0.568830\pi$
$14$	8.66648	2.31621
$15$	0	0
$16$	−0.878807	−0.219702
$17$	−5.49863	−1.33361	−0.666807	−	0.745230i	$-0.732339\pi$
$17$	−5.49863	−1.33361	−0.666807	+	0.745230i	$0.732339\pi$
$18$	−0.283622	−0.0668504
$19$	0.753438	0.172851	0.0864253	−	0.996258i	$-0.472456\pi$
$19$	0.753438	0.172851	0.0864253	+	0.996258i	$0.472456\pi$
$20$	0	0
$21$	−6.55035	−1.42940
$22$	−4.95250	−1.05588
$23$	5.79633	1.20862	0.604310	−	0.796750i	$-0.293449\pi$
$23$	5.79633	1.20862	0.604310	+	0.796750i	$0.293449\pi$
$24$	−3.91619	−0.799388
$25$	0	0
$26$	−3.46999	−0.680522
$27$	5.29982	1.01995
$28$	11.7087	2.21274
$29$	−8.47619	−1.57399	−0.786994	−	0.616961i	$-0.788364\pi$
$29$	−8.47619	−1.57399	−0.786994	+	0.616961i	$0.788364\pi$
$30$	0	0
$31$	−8.33139	−1.49636	−0.748181	−	0.663495i	$-0.769072\pi$
$31$	−8.33139	−1.49636	−0.748181	+	0.663495i	$0.769072\pi$
$32$	−6.59143	−1.16521
$33$	3.74323	0.651612
$34$	−12.3322	−2.11496
$35$	0	0
$36$	−0.383183	−0.0638638
$37$	−10.8352	−1.78129	−0.890646	−	0.454696i	$-0.849748\pi$
$37$	−10.8352	−1.78129	−0.890646	+	0.454696i	$0.849748\pi$
$38$	1.68980	0.274121
$39$	2.62271	0.419970
$40$	0	0
$41$	−1.03866	−0.162212	−0.0811059	−	0.996705i	$-0.525845\pi$
$41$	−1.03866	−0.162212	−0.0811059	+	0.996705i	$0.525845\pi$
$42$	−14.6910	−2.26687
$43$	4.39227	0.669815	0.334908	−	0.942251i	$-0.391295\pi$
$43$	4.39227	0.669815	0.334908	+	0.942251i	$0.391295\pi$
$44$	−6.69099	−1.00870
$45$	0	0
$46$	12.9999	1.91673
$47$	6.20938	0.905731	0.452865	−	0.891579i	$-0.350402\pi$
$47$	6.20938	0.905731	0.452865	+	0.891579i	$0.350402\pi$
$48$	1.48971	0.215021
$49$	7.93177	1.13311
$50$	0	0
$51$	9.32101	1.30520
$52$	−4.68808	−0.650119
$53$	5.32664	0.731670	0.365835	−	0.930680i	$-0.380783\pi$
$53$	5.32664	0.731670	0.365835	+	0.930680i	$0.380783\pi$
$54$	11.8864	1.61753
$55$	0	0
$56$	8.92710	1.19293
$57$	−1.27719	−0.169168
$58$	−19.0102	−2.49617
$59$	0.409456	0.0533066	0.0266533	−	0.999645i	$-0.491515\pi$
$59$	0.409456	0.0533066	0.0266533	+	0.999645i	$0.491515\pi$
$60$	0	0
$61$	−10.7844	−1.38081	−0.690403	−	0.723425i	$-0.742567\pi$
$61$	−10.7844	−1.38081	−0.690403	+	0.723425i	$0.742567\pi$
$62$	−18.6855	−2.37306
$63$	−0.488662	−0.0615657
$64$	−13.0255	−1.62819
$65$	0	0
$66$	8.39524	1.03338
$67$	−2.86371	−0.349857	−0.174929	−	0.984581i	$-0.555969\pi$
$67$	−2.86371	−0.349857	−0.174929	+	0.984581i	$0.555969\pi$
$68$	−16.6612	−2.02047
$69$	−9.82567	−1.18287
$70$	0	0
$71$	−14.9588	−1.77529	−0.887644	−	0.460531i	$-0.847659\pi$
$71$	−14.9588	−1.77529	−0.887644	+	0.460531i	$0.847659\pi$
$72$	−0.292151	−0.0344304
$73$	−4.86429	−0.569322	−0.284661	−	0.958628i	$-0.591881\pi$
$73$	−4.86429	−0.569322	−0.284661	+	0.958628i	$0.591881\pi$
$74$	−24.3010	−2.82493
$75$	0	0
$76$	2.28297	0.261875
$77$	−8.53283	−0.972406
$78$	5.88217	0.666024
$79$	7.99199	0.899169	0.449585	−	0.893238i	$-0.351572\pi$
$79$	7.99199	0.899169	0.449585	+	0.893238i	$0.351572\pi$
$80$	0	0
$81$	−8.60463	−0.956070
$82$	−2.32949	−0.257250
$83$	0.733605	0.0805236	0.0402618	−	0.999189i	$-0.487181\pi$
$83$	0.733605	0.0805236	0.0402618	+	0.999189i	$0.487181\pi$
$84$	−19.8480	−2.16560
$85$	0	0
$86$	9.85091	1.06225
$87$	14.3684	1.54046
$88$	−5.10143	−0.543815
$89$	1.13084	0.119869	0.0599346	−	0.998202i	$-0.480911\pi$
$89$	1.13084	0.119869	0.0599346	+	0.998202i	$0.480911\pi$
$90$	0	0
$91$	−5.97857	−0.626724
$92$	17.5633	1.83110
$93$	14.1230	1.46448
$94$	13.9263	1.43639
$95$	0	0
$96$	11.1735	1.14039
$97$	−17.3030	−1.75686	−0.878428	−	0.477876i	$-0.841407\pi$
$97$	−17.3030	−1.75686	−0.878428	+	0.477876i	$0.841407\pi$
$98$	17.7892	1.79698
$99$	0.279248	0.0280655
$100$	0	0
$101$	7.88818	0.784903	0.392452	−	0.919773i	$-0.371627\pi$
$101$	7.88818	0.784903	0.392452	+	0.919773i	$0.371627\pi$
$102$	20.9050	2.06990
$103$	12.0867	1.19094	0.595471	−	0.803377i	$-0.296965\pi$
$103$	12.0867	1.19094	0.595471	+	0.803377i	$0.296965\pi$
$104$	−3.57434	−0.350493
$105$	0	0
$106$	11.9465	1.16035
$107$	6.21333	0.600665	0.300332	−	0.953835i	$-0.402902\pi$
$107$	6.21333	0.600665	0.300332	+	0.953835i	$0.402902\pi$
$108$	16.0589	1.54526
$109$	1.21143	0.116034	0.0580169	−	0.998316i	$-0.481522\pi$
$109$	1.21143	0.116034	0.0580169	+	0.998316i	$0.481522\pi$
$110$	0	0
$111$	18.3673	1.74335
$112$	−3.39585	−0.320878
$113$	14.0267	1.31952	0.659760	−	0.751477i	$-0.270658\pi$
$113$	14.0267	1.31952	0.659760	+	0.751477i	$0.270658\pi$
$114$	−2.86446	−0.268282
$115$	0	0
$116$	−25.6835	−2.38465
$117$	0.195657	0.0180885
$118$	0.918320	0.0845382
$119$	−21.2476	−1.94777
$120$	0	0
$121$	−6.12387	−0.556716
$122$	−24.1872	−2.18980
$123$	1.76069	0.158756
$124$	−25.2447	−2.26704
$125$	0	0
$126$	−1.09596	−0.0976362
$127$	6.14524	0.545302	0.272651	−	0.962113i	$-0.412100\pi$
$127$	6.14524	0.545302	0.272651	+	0.962113i	$0.412100\pi$
$128$	−16.0306	−1.41691
$129$	−7.44557	−0.655546
$130$	0	0
$131$	−17.3305	−1.51418	−0.757088	−	0.653313i	$-0.773379\pi$
$131$	−17.3305	−1.51418	−0.757088	+	0.653313i	$0.773379\pi$
$132$	11.3422	0.987216
$133$	2.91141	0.252451
$134$	−6.42267	−0.554834
$135$	0	0
$136$	−12.7031	−1.08928
$137$	17.6950	1.51179	0.755895	−	0.654693i	$-0.227202\pi$
$137$	17.6950	1.51179	0.755895	+	0.654693i	$0.227202\pi$
$138$	−22.0368	−1.87590
$139$	19.1818	1.62698	0.813490	−	0.581579i	$-0.197565\pi$
$139$	19.1818	1.62698	0.813490	+	0.581579i	$0.197565\pi$
$140$	0	0
$141$	−10.5258	−0.886435
$142$	−33.5494	−2.81540
$143$	3.41648	0.285701
$144$	0.111134	0.00926116
$145$	0	0
$146$	−10.9095	−0.902880
$147$	−13.4455	−1.10897
$148$	−32.8314	−2.69872
$149$	−12.4189	−1.01740	−0.508698	−	0.860945i	$-0.669873\pi$
$149$	−12.4189	−1.01740	−0.508698	+	0.860945i	$0.669873\pi$
$150$	0	0
$151$	11.6634	0.949155	0.474578	−	0.880214i	$-0.342601\pi$
$151$	11.6634	0.949155	0.474578	+	0.880214i	$0.342601\pi$
$152$	1.74061	0.141182
$153$	0.695357	0.0562163
$154$	−19.1373	−1.54213
$155$	0	0
$156$	7.94700	0.636269
$157$	−6.54955	−0.522711	−0.261355	−	0.965243i	$-0.584169\pi$
$157$	−6.54955	−0.522711	−0.261355	+	0.965243i	$0.584169\pi$
$158$	17.9243	1.42598
$159$	−9.02946	−0.716083
$160$	0	0
$161$	22.3980	1.76521
$162$	−19.2983	−1.51622
$163$	−12.9926	−1.01766	−0.508831	−	0.860867i	$-0.669922\pi$
$163$	−12.9926	−1.01766	−0.508831	+	0.860867i	$0.669922\pi$
$164$	−3.14722	−0.245757
$165$	0	0
$166$	1.64532	0.127701
$167$	−7.57091	−0.585855	−0.292927	−	0.956135i	$-0.594629\pi$
$167$	−7.57091	−0.585855	−0.292927	+	0.956135i	$0.594629\pi$
$168$	−15.1328	−1.16752
$169$	−10.6062	−0.815863
$170$	0	0
$171$	−0.0952798	−0.00728623
$172$	13.3089	1.01479
$173$	−22.3315	−1.69783	−0.848917	−	0.528526i	$-0.822745\pi$
$173$	−22.3315	−1.69783	−0.848917	+	0.528526i	$0.822745\pi$
$174$	32.2252	2.44299
$175$	0	0
$176$	1.94058	0.146276
$177$	−0.694090	−0.0521710
$178$	2.53624	0.190099
$179$	−1.85486	−0.138639	−0.0693194	−	0.997595i	$-0.522083\pi$
$179$	−1.85486	−0.138639	−0.0693194	+	0.997595i	$0.522083\pi$
$180$	0	0
$181$	−23.3521	−1.73575	−0.867874	−	0.496785i	$-0.834514\pi$
$181$	−23.3521	−1.73575	−0.867874	+	0.496785i	$0.834514\pi$
$182$	−13.4086	−0.993914
$183$	18.2813	1.35139
$184$	13.3909	0.987187
$185$	0	0
$186$	31.6747	2.32250
$187$	12.1420	0.887915
$188$	18.8149	1.37221
$189$	20.4794	1.48966
$190$	0	0
$191$	−2.83410	−0.205068	−0.102534	−	0.994729i	$-0.532695\pi$
$191$	−2.83410	−0.205068	−0.102534	+	0.994729i	$0.532695\pi$
$192$	22.0802	1.59350
$193$	−15.9231	−1.14617	−0.573084	−	0.819497i	$-0.694253\pi$
$193$	−15.9231	−1.14617	−0.573084	+	0.819497i	$0.694253\pi$
$194$	−38.8069	−2.78617
$195$	0	0
$196$	24.0338	1.71670
$197$	−1.00000	−0.0712470
$197$	−1.00000	−0.0712470
$198$	0.626293	0.0445087
$199$	13.1038	0.928900	0.464450	−	0.885599i	$-0.346252\pi$
$199$	13.1038	0.928900	0.464450	+	0.885599i	$0.346252\pi$
$200$	0	0
$201$	4.85442	0.342404
$202$	17.6915	1.24477
$203$	−32.7534	−2.29884
$204$	28.2433	1.97743
$205$	0	0
$206$	27.1079	1.88870
$207$	−0.733005	−0.0509473
$208$	1.35967	0.0942765
$209$	−1.66374	−0.115083
$210$	0	0
$211$	2.18803	0.150630	0.0753152	−	0.997160i	$-0.476004\pi$
$211$	2.18803	0.150630	0.0753152	+	0.997160i	$0.476004\pi$
$212$	16.1401	1.10851
$213$	25.3575	1.73747
$214$	13.9351	0.952586
$215$	0	0
$216$	12.2438	0.833085
$217$	−32.1939	−2.18546
$218$	2.71697	0.184016
$219$	8.24571	0.557193
$220$	0	0
$221$	8.50739	0.572269
$222$	41.1938	2.76475
$223$	−8.88901	−0.595252	−0.297626	−	0.954682i	$-0.596195\pi$
$223$	−8.88901	−0.595252	−0.297626	+	0.954682i	$0.596195\pi$
$224$	−25.4704	−1.70181
$225$	0	0
$226$	31.4588	2.09261
$227$	14.2949	0.948786	0.474393	−	0.880313i	$-0.342668\pi$
$227$	14.2949	0.948786	0.474393	+	0.880313i	$0.342668\pi$
$228$	−3.86998	−0.256296
$229$	−2.10860	−0.139340	−0.0696701	−	0.997570i	$-0.522195\pi$
$229$	−2.10860	−0.139340	−0.0696701	+	0.997570i	$0.522195\pi$
$230$	0	0
$231$	14.4644	0.951690
$232$	−19.5819	−1.28562
$233$	−25.6904	−1.68303	−0.841516	−	0.540232i	$-0.818337\pi$
$233$	−25.6904	−1.68303	−0.841516	+	0.540232i	$0.818337\pi$
$234$	0.438816	0.0286863
$235$	0	0
$236$	1.24068	0.0807614
$237$	−13.5476	−0.880014
$238$	−47.6538	−3.08894
$239$	14.8887	0.963070	0.481535	−	0.876427i	$-0.340079\pi$
$239$	14.8887	0.963070	0.481535	+	0.876427i	$0.340079\pi$
$240$	0	0
$241$	−16.6389	−1.07181	−0.535903	−	0.844279i	$-0.680029\pi$
$241$	−16.6389	−1.07181	−0.535903	+	0.844279i	$0.680029\pi$
$242$	−13.7345	−0.882888
$243$	−1.31332	−0.0842497
$244$	−32.6776	−2.09197
$245$	0	0
$246$	3.94885	0.251769
$247$	−1.16571	−0.0741721
$248$	−19.2474	−1.22221
$249$	−1.24357	−0.0788081
$250$	0	0
$251$	−15.7523	−0.994277	−0.497139	−	0.867671i	$-0.665616\pi$
$251$	−15.7523	−0.994277	−0.497139	+	0.867671i	$0.665616\pi$
$252$	−1.48068	−0.0932742
$253$	−12.7994	−0.804694
$254$	13.7824	0.864786
$255$	0	0
$256$	−9.90200	−0.618875
$257$	−11.0447	−0.688951	−0.344475	−	0.938795i	$-0.611943\pi$
$257$	−11.0447	−0.688951	−0.344475	+	0.938795i	$0.611943\pi$
$258$	−16.6988	−1.03962
$259$	−41.8689	−2.60161
$260$	0	0
$261$	1.07190	0.0663489
$262$	−38.8686	−2.40131
$263$	9.41071	0.580289	0.290144	−	0.956983i	$-0.406297\pi$
$263$	9.41071	0.580289	0.290144	+	0.956983i	$0.406297\pi$
$264$	8.64770	0.532229
$265$	0	0
$266$	6.52966	0.400359
$267$	−1.91695	−0.117316
$268$	−8.67724	−0.530047
$269$	15.1379	0.922973	0.461487	−	0.887147i	$-0.347316\pi$
$269$	15.1379	0.922973	0.461487	+	0.887147i	$0.347316\pi$
$270$	0	0
$271$	15.1694	0.921477	0.460739	−	0.887536i	$-0.347585\pi$
$271$	15.1694	0.921477	0.460739	+	0.887536i	$0.347585\pi$
$272$	4.83223	0.292997
$273$	10.1346	0.613373
$274$	39.6861	2.39753
$275$	0	0
$276$	−29.7725	−1.79209
$277$	8.93015	0.536561	0.268280	−	0.963341i	$-0.413545\pi$
$277$	8.93015	0.536561	0.268280	+	0.963341i	$0.413545\pi$
$278$	43.0206	2.58021
$279$	1.05359	0.0630766
$280$	0	0
$281$	13.9052	0.829512	0.414756	−	0.909933i	$-0.363867\pi$
$281$	13.9052	0.829512	0.414756	+	0.909933i	$0.363867\pi$
$282$	−23.6072	−1.40579
$283$	−8.97483	−0.533498	−0.266749	−	0.963766i	$-0.585950\pi$
$283$	−8.97483	−0.533498	−0.266749	+	0.963766i	$0.585950\pi$
$284$	−45.3264	−2.68962
$285$	0	0
$286$	7.66242	0.453089
$287$	−4.01356	−0.236913
$288$	0.833552	0.0491175
$289$	13.2349	0.778526
$290$	0	0
$291$	29.3312	1.71943
$292$	−14.7391	−0.862543
$293$	2.76823	0.161721	0.0808607	−	0.996725i	$-0.474233\pi$
$293$	2.76823	0.161721	0.0808607	+	0.996725i	$0.474233\pi$
$294$	−30.1554	−1.75870
$295$	0	0
$296$	−25.0317	−1.45494
$297$	−11.7030	−0.679080
$298$	−27.8529	−1.61348
$299$	−8.96799	−0.518632
$300$	0	0
$301$	16.9725	0.978276
$302$	26.1585	1.50525
$303$	−13.3717	−0.768182
$304$	−0.662127	−0.0379756
$305$	0	0
$306$	1.55953	0.0891527
$307$	4.53630	0.258900	0.129450	−	0.991586i	$-0.458679\pi$
$307$	4.53630	0.258900	0.129450	+	0.991586i	$0.458679\pi$
$308$	−25.8551	−1.47323
$309$	−20.4889	−1.16557
$310$	0	0
$311$	−22.6660	−1.28527	−0.642637	−	0.766171i	$-0.722159\pi$
$311$	−22.6660	−1.28527	−0.642637	+	0.766171i	$0.722159\pi$
$312$	6.05906	0.343027
$313$	−1.97222	−0.111476	−0.0557382	−	0.998445i	$-0.517751\pi$
$313$	−1.97222	−0.111476	−0.0557382	+	0.998445i	$0.517751\pi$
$314$	−14.6892	−0.828960
$315$	0	0
$316$	24.2163	1.36227
$317$	30.0838	1.68968	0.844838	−	0.535023i	$-0.179697\pi$
$317$	30.0838	1.68968	0.844838	+	0.535023i	$0.179697\pi$
$318$	−20.2511	−1.13563
$319$	18.7171	1.04795
$320$	0	0
$321$	−10.5325	−0.587869
$322$	50.2338	2.79942
$323$	−4.14288	−0.230516
$324$	−26.0726	−1.44848
$325$	0	0
$326$	−29.1396	−1.61390
$327$	−2.05355	−0.113562
$328$	−2.39955	−0.132493
$329$	23.9941	1.32284
$330$	0	0
$331$	23.2300	1.27684	0.638418	−	0.769690i	$-0.279589\pi$
$331$	23.2300	1.27684	0.638418	+	0.769690i	$0.279589\pi$
$332$	2.22288	0.121996
$333$	1.37022	0.0750875
$334$	−16.9799	−0.929099
$335$	0	0
$336$	5.75649	0.314042
$337$	24.7804	1.34988	0.674938	−	0.737875i	$-0.264170\pi$
$337$	24.7804	1.34988	0.674938	+	0.737875i	$0.264170\pi$
$338$	−23.7875	−1.29387
$339$	−23.7774	−1.29141
$340$	0	0
$341$	18.3973	0.996271
$342$	−0.213692	−0.0115551
$343$	3.60050	0.194409
$344$	10.1471	0.547098
$345$	0	0
$346$	−50.0848	−2.69257
$347$	−32.2025	−1.72872	−0.864360	−	0.502873i	$-0.832276\pi$
$347$	−32.2025	−1.72872	−0.864360	+	0.502873i	$0.832276\pi$
$348$	43.5374	2.33385
$349$	−9.48794	−0.507878	−0.253939	−	0.967220i	$-0.581726\pi$
$349$	−9.48794	−0.507878	−0.253939	+	0.967220i	$0.581726\pi$
$350$	0	0
$351$	−8.19980	−0.437673
$352$	14.5552	0.775793
$353$	33.6633	1.79172	0.895858	−	0.444340i	$-0.146562\pi$
$353$	33.6633	1.79172	0.895858	+	0.444340i	$0.146562\pi$
$354$	−1.55669	−0.0827372
$355$	0	0
$356$	3.42654	0.181606
$357$	36.0179	1.90627
$358$	−4.16005	−0.219865
$359$	18.4847	0.975585	0.487793	−	0.872960i	$-0.337802\pi$
$359$	18.4847	0.975585	0.487793	+	0.872960i	$0.337802\pi$
$360$	0	0
$361$	−18.4323	−0.970123
$362$	−52.3736	−2.75270
$363$	10.3809	0.544856
$364$	−18.1155	−0.949510
$365$	0	0
$366$	41.0009	2.14315
$367$	23.8288	1.24385	0.621926	−	0.783076i	$-0.286350\pi$
$367$	23.8288	1.24385	0.621926	+	0.783076i	$0.286350\pi$
$368$	−5.09386	−0.265536
$369$	0.131349	0.00683777
$370$	0	0
$371$	20.5830	1.06862
$372$	42.7936	2.21875
$373$	−4.38690	−0.227145	−0.113572	−	0.993530i	$-0.536229\pi$
$373$	−4.38690	−0.227145	−0.113572	+	0.993530i	$0.536229\pi$
$374$	27.2320	1.40813
$375$	0	0
$376$	14.3451	0.739791
$377$	13.1142	0.675416
$378$	45.9308	2.36243
$379$	−2.49316	−0.128065	−0.0640324	−	0.997948i	$-0.520396\pi$
$379$	−2.49316	−0.128065	−0.0640324	+	0.997948i	$0.520396\pi$
$380$	0	0
$381$	−10.4171	−0.533685
$382$	−6.35627	−0.325215
$383$	−30.5816	−1.56265	−0.781324	−	0.624125i	$-0.785455\pi$
$383$	−30.5816	−1.56265	−0.781324	+	0.624125i	$0.785455\pi$
$384$	27.1742	1.38673
$385$	0	0
$386$	−35.7120	−1.81769
$387$	−0.555447	−0.0282349
$388$	−52.4294	−2.66170
$389$	−12.2757	−0.622404	−0.311202	−	0.950344i	$-0.600732\pi$
$389$	−12.2757	−0.622404	−0.311202	+	0.950344i	$0.600732\pi$
$390$	0	0
$391$	−31.8719	−1.61183
$392$	18.3242	0.925511
$393$	29.3779	1.48192
$394$	−2.24278	−0.112990
$395$	0	0
$396$	0.846143	0.0425203
$397$	−13.4565	−0.675364	−0.337682	−	0.941260i	$-0.609643\pi$
$397$	−13.4565	−0.675364	−0.337682	+	0.941260i	$0.609643\pi$
$398$	29.3889	1.47313
$399$	−4.93528	−0.247073
$400$	0	0
$401$	17.0003	0.848955	0.424478	−	0.905438i	$-0.360458\pi$
$401$	17.0003	0.848955	0.424478	+	0.905438i	$0.360458\pi$
$402$	10.8874	0.543014
$403$	12.8902	0.642106
$404$	23.9018	1.18916
$405$	0	0
$406$	−73.4587	−3.64569
$407$	23.9262	1.18598
$408$	21.5337	1.06608
$409$	−1.19427	−0.0590527	−0.0295263	−	0.999564i	$-0.509400\pi$
$409$	−1.19427	−0.0590527	−0.0295263	+	0.999564i	$0.509400\pi$
$410$	0	0
$411$	−29.9958	−1.47958
$412$	36.6237	1.80432
$413$	1.58220	0.0778552
$414$	−1.64397	−0.0807967
$415$	0	0
$416$	10.1981	0.500005
$417$	−32.5161	−1.59232
$418$	−3.73140	−0.182509
$419$	32.3524	1.58052	0.790259	−	0.612772i	$-0.209946\pi$
$419$	32.3524	1.58052	0.790259	+	0.612772i	$0.209946\pi$
$420$	0	0
$421$	23.3730	1.13913	0.569564	−	0.821947i	$-0.307112\pi$
$421$	23.3730	1.13913	0.569564	+	0.821947i	$0.307112\pi$
$422$	4.90728	0.238883
$423$	−0.785238	−0.0381796
$424$	12.3057	0.597620
$425$	0	0
$426$	56.8713	2.75543
$427$	−41.6729	−2.01669
$428$	18.8268	0.910029
$429$	−5.79146	−0.279614
$430$	0	0
$431$	32.2361	1.55276	0.776380	−	0.630265i	$-0.217054\pi$
$431$	32.2361	1.55276	0.776380	+	0.630265i	$0.217054\pi$
$432$	−4.65752	−0.224085
$433$	13.1015	0.629616	0.314808	−	0.949155i	$-0.398060\pi$
$433$	13.1015	0.629616	0.314808	+	0.949155i	$0.398060\pi$
$434$	−72.2038	−3.46589
$435$	0	0
$436$	3.67071	0.175795
$437$	4.36718	0.208911
$438$	18.4933	0.883646
$439$	−12.5909	−0.600933	−0.300466	−	0.953792i	$-0.597142\pi$
$439$	−12.5909	−0.600933	−0.300466	+	0.953792i	$0.597142\pi$
$440$	0	0
$441$	−1.00305	−0.0477644
$442$	19.0802	0.907553
$443$	−8.21481	−0.390297	−0.195149	−	0.980774i	$-0.562519\pi$
$443$	−8.21481	−0.390297	−0.195149	+	0.980774i	$0.562519\pi$
$444$	55.6542	2.64123
$445$	0	0
$446$	−19.9361	−0.944003
$447$	21.0519	0.995723
$448$	−50.3328	−2.37800
$449$	20.5177	0.968292	0.484146	−	0.874987i	$-0.339130\pi$
$449$	20.5177	0.968292	0.484146	+	0.874987i	$0.339130\pi$
$450$	0	0
$451$	2.29357	0.108000
$452$	42.5019	1.99912
$453$	−19.7713	−0.928935
$454$	32.0604	1.50467
$455$	0	0
$456$	−2.95060	−0.138175
$457$	−27.9176	−1.30593	−0.652965	−	0.757388i	$-0.726475\pi$
$457$	−27.9176	−1.30593	−0.652965	+	0.757388i	$0.726475\pi$
$458$	−4.72913	−0.220978
$459$	−29.1418	−1.36022
$460$	0	0
$461$	−16.1654	−0.752899	−0.376450	−	0.926437i	$-0.622855\pi$
$461$	−16.1654	−0.752899	−0.376450	+	0.926437i	$0.622855\pi$
$462$	32.4406	1.50927
$463$	1.24537	0.0578773	0.0289386	−	0.999581i	$-0.490787\pi$
$463$	1.24537	0.0578773	0.0289386	+	0.999581i	$0.490787\pi$
$464$	7.44893	0.345808
$465$	0	0
$466$	−57.6179	−2.66910
$467$	1.38654	0.0641613	0.0320807	−	0.999485i	$-0.489787\pi$
$467$	1.38654	0.0641613	0.0320807	+	0.999485i	$0.489787\pi$
$468$	0.592854	0.0274047
$469$	−11.0658	−0.510973
$470$	0	0
$471$	11.1025	0.511575
$472$	0.945936	0.0435402
$473$	−9.69899	−0.445960
$474$	−30.3844	−1.39560
$475$	0	0
$476$	−64.3818	−2.95094
$477$	−0.673607	−0.0308423
$478$	33.3921	1.52732
$479$	25.0648	1.14524	0.572621	−	0.819820i	$-0.305927\pi$
$479$	25.0648	1.14524	0.572621	+	0.819820i	$0.305927\pi$
$480$	0	0
$481$	16.7640	0.764373
$482$	−37.3175	−1.69976
$483$	−37.9680	−1.72760
$484$	−18.5558	−0.843445
$485$	0	0
$486$	−2.94550	−0.133610
$487$	−29.6378	−1.34302	−0.671508	−	0.740998i	$-0.734353\pi$
$487$	−29.6378	−1.34302	−0.671508	+	0.740998i	$0.734353\pi$
$488$	−24.9145	−1.12783
$489$	22.0245	0.995981
$490$	0	0
$491$	−10.2768	−0.463785	−0.231893	−	0.972741i	$-0.574492\pi$
$491$	−10.2768	−0.463785	−0.231893	+	0.972741i	$0.574492\pi$
$492$	5.33502	0.240521
$493$	46.6074	2.09909
$494$	−2.61443	−0.117629
$495$	0	0
$496$	7.32168	0.328753
$497$	−57.8034	−2.59284
$498$	−2.78906	−0.124981
$499$	42.0398	1.88196	0.940979	−	0.338464i	$-0.109907\pi$
$499$	42.0398	1.88196	0.940979	+	0.338464i	$0.109907\pi$
$500$	0	0
$501$	12.8339	0.573374
$502$	−35.3290	−1.57681
$503$	25.7320	1.14734	0.573668	−	0.819088i	$-0.305520\pi$
$503$	25.7320	1.14734	0.573668	+	0.819088i	$0.305520\pi$
$504$	−1.12892	−0.0502862
$505$	0	0
$506$	−28.7064	−1.27615
$507$	17.9792	0.798483
$508$	18.6205	0.826152
$509$	−29.0968	−1.28969	−0.644846	−	0.764312i	$-0.723079\pi$
$509$	−29.0968	−1.28969	−0.644846	+	0.764312i	$0.723079\pi$
$510$	0	0
$511$	−18.7964	−0.831504
$512$	9.85308	0.435449
$513$	3.99309	0.176299
$514$	−24.7709	−1.09260
$515$	0	0
$516$	−22.5606	−0.993175
$517$	−13.7115	−0.603032
$518$	−93.9029	−4.12586
$519$	37.8553	1.66166
$520$	0	0
$521$	−25.7092	−1.12634	−0.563171	−	0.826341i	$-0.690419\pi$
$521$	−25.7092	−1.12634	−0.563171	+	0.826341i	$0.690419\pi$
$522$	2.40404	0.105222
$523$	−20.6858	−0.904529	−0.452264	−	0.891884i	$-0.649384\pi$
$523$	−20.6858	−0.904529	−0.452264	+	0.891884i	$0.649384\pi$
$524$	−52.5128	−2.29403
$525$	0	0
$526$	21.1062	0.920272
$527$	45.8112	1.99557
$528$	−3.28957	−0.143160
$529$	10.5975	0.460761
$530$	0	0
$531$	−0.0517798	−0.00224705
$532$	8.82178	0.382473
$533$	1.60700	0.0696069
$534$	−4.29931	−0.186049
$535$	0	0
$536$	−6.61581	−0.285760
$537$	3.14427	0.135685
$538$	33.9510	1.46373
$539$	−17.5149	−0.754420
$540$	0	0
$541$	−9.09467	−0.391010	−0.195505	−	0.980703i	$-0.562635\pi$
$541$	−9.09467	−0.391010	−0.195505	+	0.980703i	$0.562635\pi$
$542$	34.0217	1.46136
$543$	39.5853	1.69877
$544$	36.2438	1.55394
$545$	0	0
$546$	22.7297	0.972740
$547$	32.8100	1.40286	0.701428	−	0.712741i	$-0.252546\pi$
$547$	32.8100	1.40286	0.701428	+	0.712741i	$0.252546\pi$
$548$	53.6172	2.29042
$549$	1.36380	0.0582056
$550$	0	0
$551$	−6.38628	−0.272065
$552$	−22.6995	−0.966156
$553$	30.8824	1.31325
$554$	20.0284	0.850924
$555$	0	0
$556$	58.1223	2.46493
$557$	41.0031	1.73736	0.868679	−	0.495375i	$-0.164969\pi$
$557$	41.0031	1.73736	0.868679	+	0.495375i	$0.164969\pi$
$558$	2.36297	0.100032
$559$	−6.79565	−0.287425
$560$	0	0
$561$	−20.5826	−0.868999
$562$	31.1862	1.31551
$563$	7.96647	0.335747	0.167873	−	0.985809i	$-0.446310\pi$
$563$	7.96647	0.335747	0.167873	+	0.985809i	$0.446310\pi$
$564$	−31.8940	−1.34298
$565$	0	0
$566$	−20.1286	−0.846068
$567$	−33.2497	−1.39636
$568$	−34.5583	−1.45003
$569$	39.1633	1.64181	0.820905	−	0.571065i	$-0.193470\pi$
$569$	39.1633	1.64181	0.820905	+	0.571065i	$0.193470\pi$
$570$	0	0
$571$	−17.5561	−0.734700	−0.367350	−	0.930083i	$-0.619735\pi$
$571$	−17.5561	−0.734700	−0.367350	+	0.930083i	$0.619735\pi$
$572$	10.3522	0.432847
$573$	4.80423	0.200700
$574$	−9.00155	−0.375717
$575$	0	0
$576$	1.64721	0.0686337
$577$	−16.3195	−0.679391	−0.339695	−	0.940536i	$-0.610324\pi$
$577$	−16.3195	−0.679391	−0.339695	+	0.940536i	$0.610324\pi$
$578$	29.6831	1.23465
$579$	26.9920	1.12175
$580$	0	0
$581$	2.83477	0.117606
$582$	65.7836	2.72682
$583$	−11.7623	−0.487143
$584$	−11.2376	−0.465016
$585$	0	0
$586$	6.20853	0.256472
$587$	−8.69995	−0.359085	−0.179543	−	0.983750i	$-0.557462\pi$
$587$	−8.69995	−0.359085	−0.179543	+	0.983750i	$0.557462\pi$
$588$	−40.7410	−1.68013
$589$	−6.27719	−0.258647
$590$	0	0
$591$	1.69515	0.0697292
$592$	9.52203	0.391353
$593$	3.41542	0.140255	0.0701273	−	0.997538i	$-0.477659\pi$
$593$	3.41542	0.140255	0.0701273	+	0.997538i	$0.477659\pi$
$594$	−26.2474	−1.07694
$595$	0	0
$596$	−37.6302	−1.54139
$597$	−22.2129	−0.909112
$598$	−20.1132	−0.822492
$599$	26.6727	1.08982	0.544909	−	0.838495i	$-0.316564\pi$
$599$	26.6727	1.08982	0.544909	+	0.838495i	$0.316564\pi$
$600$	0	0
$601$	29.3801	1.19844	0.599220	−	0.800584i	$-0.295477\pi$
$601$	29.3801	1.19844	0.599220	+	0.800584i	$0.295477\pi$
$602$	38.0655	1.55144
$603$	0.362144	0.0147477
$604$	35.3410	1.43800
$605$	0	0
$606$	−29.9897	−1.21825
$607$	−23.9432	−0.971823	−0.485912	−	0.874008i	$-0.661512\pi$
$607$	−23.9432	−0.971823	−0.485912	+	0.874008i	$0.661512\pi$
$608$	−4.96623	−0.201407
$609$	55.5219	2.24986
$610$	0	0
$611$	−9.60704	−0.388659
$612$	2.10698	0.0851697
$613$	28.2699	1.14181	0.570906	−	0.821015i	$-0.306592\pi$
$613$	28.2699	1.14181	0.570906	+	0.821015i	$0.306592\pi$
$614$	10.1739	0.410586
$615$	0	0
$616$	−19.7128	−0.794250
$617$	9.44553	0.380263	0.190131	−	0.981759i	$-0.439109\pi$
$617$	9.44553	0.380263	0.190131	+	0.981759i	$0.439109\pi$
$618$	−45.9521	−1.84846
$619$	−15.4898	−0.622587	−0.311293	−	0.950314i	$-0.600762\pi$
$619$	−15.4898	−0.622587	−0.311293	+	0.950314i	$0.600762\pi$
$620$	0	0
$621$	30.7196	1.23273
$622$	−50.8350	−2.03830
$623$	4.36977	0.175071
$624$	−2.30486	−0.0922681
$625$	0	0
$626$	−4.42325	−0.176789
$627$	2.82029	0.112632
$628$	−19.8456	−0.791926
$629$	59.5787	2.37556
$630$	0	0
$631$	−21.7509	−0.865888	−0.432944	−	0.901421i	$-0.642525\pi$
$631$	−21.7509	−0.865888	−0.432944	+	0.901421i	$0.642525\pi$
$632$	18.4633	0.734432
$633$	−3.70905	−0.147421
$634$	67.4714	2.67963
$635$	0	0
$636$	−27.3599	−1.08489
$637$	−12.2719	−0.486230
$638$	41.9783	1.66194
$639$	1.89169	0.0748343
$640$	0	0
$641$	24.1709	0.954691	0.477346	−	0.878716i	$-0.341599\pi$
$641$	24.1709	0.954691	0.477346	+	0.878716i	$0.341599\pi$
$642$	−23.6222	−0.932293
$643$	32.9439	1.29918	0.649590	−	0.760285i	$-0.274941\pi$
$643$	32.9439	1.29918	0.649590	+	0.760285i	$0.274941\pi$
$644$	67.8675	2.67436
$645$	0	0
$646$	−9.29157	−0.365572
$647$	15.2866	0.600978	0.300489	−	0.953785i	$-0.402850\pi$
$647$	15.2866	0.600978	0.300489	+	0.953785i	$0.402850\pi$
$648$	−19.8786	−0.780907
$649$	−0.904158	−0.0354913
$650$	0	0
$651$	54.5735	2.13890
$652$	−39.3686	−1.54179
$653$	4.44815	0.174069	0.0870347	−	0.996205i	$-0.472261\pi$
$653$	4.44815	0.174069	0.0870347	+	0.996205i	$0.472261\pi$
$654$	−4.60567	−0.180096
$655$	0	0
$656$	0.912784	0.0356382
$657$	0.615138	0.0239988
$658$	53.8134	2.09787
$659$	−31.6128	−1.23146	−0.615730	−	0.787958i	$-0.711139\pi$
$659$	−31.6128	−1.23146	−0.615730	+	0.787958i	$0.711139\pi$
$660$	0	0
$661$	17.6169	0.685218	0.342609	−	0.939478i	$-0.388689\pi$
$661$	17.6169	0.685218	0.342609	+	0.939478i	$0.388689\pi$
$662$	52.0998	2.02492
$663$	−14.4213	−0.560077
$664$	1.69479	0.0657708
$665$	0	0
$666$	3.07310	0.119080
$667$	−49.1308	−1.90235
$668$	−22.9404	−0.887591
$669$	15.0682	0.582571
$670$	0	0
$671$	23.8142	0.919335
$672$	43.1761	1.66556
$673$	−16.4231	−0.633064	−0.316532	−	0.948582i	$-0.602518\pi$
$673$	−16.4231	−0.633064	−0.316532	+	0.948582i	$0.602518\pi$
$674$	55.5771	2.14075
$675$	0	0
$676$	−32.1376	−1.23606
$677$	24.4906	0.941250	0.470625	−	0.882333i	$-0.344028\pi$
$677$	24.4906	0.941250	0.470625	+	0.882333i	$0.344028\pi$
$678$	−53.3274	−2.04803
$679$	−66.8617	−2.56592
$680$	0	0
$681$	−24.2320	−0.928574
$682$	41.2612	1.57997
$683$	21.2479	0.813029	0.406515	−	0.913644i	$-0.366744\pi$
$683$	21.2479	0.813029	0.406515	+	0.913644i	$0.366744\pi$
$684$	−0.288705	−0.0110389
$685$	0	0
$686$	8.07514	0.308310
$687$	3.57440	0.136372
$688$	−3.85996	−0.147160
$689$	−8.24128	−0.313968
$690$	0	0
$691$	−35.8032	−1.36202	−0.681008	−	0.732276i	$-0.738458\pi$
$691$	−35.8032	−1.36202	−0.681008	+	0.732276i	$0.738458\pi$
$692$	−67.6662	−2.57228
$693$	1.07906	0.0409902
$694$	−72.2232	−2.74156
$695$	0	0
$696$	33.1943	1.25823
$697$	5.71122	0.216328
$698$	−21.2794	−0.805436
$699$	43.5491	1.64718
$700$	0	0
$701$	10.6147	0.400913	0.200456	−	0.979703i	$-0.435758\pi$
$701$	10.6147	0.400913	0.200456	+	0.979703i	$0.435758\pi$
$702$	−18.3904	−0.694099
$703$	−8.16364	−0.307897
$704$	28.7629	1.08404
$705$	0	0
$706$	75.4994	2.84146
$707$	30.4812	1.14636
$708$	−2.10314	−0.0790409
$709$	−26.7085	−1.00306	−0.501529	−	0.865141i	$-0.667229\pi$
$709$	−26.7085	−1.00306	−0.501529	+	0.865141i	$0.667229\pi$
$710$	0	0
$711$	−1.01067	−0.0379030
$712$	2.61251	0.0979078
$713$	−48.2915	−1.80853
$714$	80.7804	3.02313
$715$	0	0
$716$	−5.62036	−0.210043
$717$	−25.2386	−0.942554
$718$	41.4572	1.54717
$719$	−6.73003	−0.250988	−0.125494	−	0.992094i	$-0.540052\pi$
$719$	−6.73003	−0.250988	−0.125494	+	0.992094i	$0.540052\pi$
$720$	0	0
$721$	46.7052	1.73939
$722$	−41.3397	−1.53850
$723$	28.2055	1.04897
$724$	−70.7585	−2.62972
$725$	0	0
$726$	23.2821	0.864079
$727$	20.4337	0.757845	0.378923	−	0.925428i	$-0.376295\pi$
$727$	20.4337	0.757845	0.378923	+	0.925428i	$0.376295\pi$
$728$	−13.8119	−0.511901
$729$	28.0402	1.03852
$730$	0	0
$731$	−24.1515	−0.893275
$732$	55.3936	2.04741
$733$	−13.1771	−0.486709	−0.243354	−	0.969937i	$-0.578248\pi$
$733$	−13.1771	−0.486709	−0.243354	+	0.969937i	$0.578248\pi$
$734$	53.4428	1.97261
$735$	0	0
$736$	−38.2061	−1.40830
$737$	6.32362	0.232934
$738$	0.294588	0.0108439
$739$	−21.4638	−0.789558	−0.394779	−	0.918776i	$-0.629179\pi$
$739$	−21.4638	−0.789558	−0.394779	+	0.918776i	$0.629179\pi$
$740$	0	0
$741$	1.97605	0.0725920
$742$	46.1632	1.69470
$743$	−40.8748	−1.49955	−0.749775	−	0.661693i	$-0.769838\pi$
$743$	−40.8748	−1.49955	−0.749775	+	0.661693i	$0.769838\pi$
$744$	32.6273	1.19617
$745$	0	0
$746$	−9.83886	−0.360226
$747$	−0.0927717	−0.00339434
$748$	36.7913	1.34522
$749$	24.0093	0.877281
$750$	0	0
$751$	23.3466	0.851929	0.425965	−	0.904740i	$-0.359935\pi$
$751$	23.3466	0.851929	0.425965	+	0.904740i	$0.359935\pi$
$752$	−5.45684	−0.198991
$753$	26.7026	0.973096
$754$	29.4123	1.07113
$755$	0	0
$756$	62.0540	2.25688
$757$	17.9899	0.653854	0.326927	−	0.945050i	$-0.393987\pi$
$757$	17.9899	0.653854	0.326927	+	0.945050i	$0.393987\pi$
$758$	−5.59161	−0.203096
$759$	21.6970	0.787551
$760$	0	0
$761$	−16.2485	−0.589009	−0.294505	−	0.955650i	$-0.595155\pi$
$761$	−16.2485	−0.589009	−0.294505	+	0.955650i	$0.595155\pi$
$762$	−23.3633	−0.846363
$763$	4.68116	0.169469
$764$	−8.58753	−0.310686
$765$	0	0
$766$	−68.5880	−2.47818
$767$	−0.633503	−0.0228745
$768$	16.7854	0.605691
$769$	−23.6922	−0.854362	−0.427181	−	0.904166i	$-0.640493\pi$
$769$	−23.6922	−0.854362	−0.427181	+	0.904166i	$0.640493\pi$
$770$	0	0
$771$	18.7225	0.674274
$772$	−48.2481	−1.73649
$773$	50.3310	1.81028	0.905140	−	0.425114i	$-0.139766\pi$
$773$	50.3310	1.81028	0.905140	+	0.425114i	$0.139766\pi$
$774$	−1.24575	−0.0447774
$775$	0	0
$776$	−39.9739	−1.43498
$777$	70.9742	2.54618
$778$	−27.5318	−0.987062
$779$	−0.782568	−0.0280384
$780$	0	0
$781$	33.0320	1.18198
$782$	−71.4817	−2.55618
$783$	−44.9223	−1.60539
$784$	−6.97049	−0.248946
$785$	0	0
$786$	65.8882	2.35016
$787$	−24.2358	−0.863914	−0.431957	−	0.901894i	$-0.642177\pi$
$787$	−24.2358	−0.863914	−0.431957	+	0.901894i	$0.642177\pi$
$788$	−3.03007	−0.107942
$789$	−15.9526	−0.567927
$790$	0	0
$791$	54.2014	1.92718
$792$	0.645127	0.0229236
$793$	16.6855	0.592520
$794$	−30.1801	−1.07105
$795$	0	0
$796$	39.7053	1.40732
$797$	−7.38173	−0.261474	−0.130737	−	0.991417i	$-0.541734\pi$
$797$	−7.38173	−0.261474	−0.130737	+	0.991417i	$0.541734\pi$
$798$	−11.0688	−0.391830
$799$	−34.1431	−1.20790
$800$	0	0
$801$	−0.143007	−0.00505289
$802$	38.1280	1.34635
$803$	10.7413	0.379052
$804$	14.7092	0.518755
$805$	0	0
$806$	28.9099	1.01831
$807$	−25.6610	−0.903311
$808$	18.2235	0.641100
$809$	24.8099	0.872269	0.436135	−	0.899881i	$-0.356347\pi$
$809$	24.8099	0.872269	0.436135	+	0.899881i	$0.356347\pi$
$810$	0	0
$811$	−40.8030	−1.43279	−0.716394	−	0.697696i	$-0.754209\pi$
$811$	−40.8030	−1.43279	−0.716394	+	0.697696i	$0.754209\pi$
$812$	−99.2451	−3.48282
$813$	−25.7145	−0.901846
$814$	53.6612	1.88083
$815$	0	0
$816$	−8.19137	−0.286755
$817$	3.30930	0.115778
$818$	−2.67848	−0.0936509
$819$	0.756050	0.0264185
$820$	0	0
$821$	−7.12363	−0.248616	−0.124308	−	0.992244i	$-0.539671\pi$
$821$	−7.12363	−0.248616	−0.124308	+	0.992244i	$0.539671\pi$
$822$	−67.2740	−2.34645
$823$	30.5815	1.06600	0.533002	−	0.846114i	$-0.321064\pi$
$823$	30.5815	1.06600	0.533002	+	0.846114i	$0.321064\pi$
$824$	27.9231	0.972748
$825$	0	0
$826$	3.54854	0.123469
$827$	20.5024	0.712938	0.356469	−	0.934307i	$-0.383980\pi$
$827$	20.5024	0.712938	0.356469	+	0.934307i	$0.383980\pi$
$828$	−2.22106	−0.0771871
$829$	−3.71448	−0.129009	−0.0645046	−	0.997917i	$-0.520547\pi$
$829$	−3.71448	−0.129009	−0.0645046	+	0.997917i	$0.520547\pi$
$830$	0	0
$831$	−15.1380	−0.525130
$832$	20.1529	0.698675
$833$	−43.6139	−1.51113
$834$	−72.9265	−2.52524
$835$	0	0
$836$	−5.04125	−0.174355
$837$	−44.1549	−1.52622
$838$	72.5594	2.50652
$839$	−17.7716	−0.613545	−0.306772	−	0.951783i	$-0.599249\pi$
$839$	−17.7716	−0.613545	−0.306772	+	0.951783i	$0.599249\pi$
$840$	0	0
$841$	42.8457	1.47744
$842$	52.4204	1.80653
$843$	−23.5714	−0.811841
$844$	6.62990	0.228210
$845$	0	0
$846$	−1.76112	−0.0605485
$847$	−23.6637	−0.813093
$848$	−4.68109	−0.160749
$849$	15.2137	0.522133
$850$	0	0
$851$	−62.8043	−2.15291
$852$	76.8351	2.63233
$853$	8.21343	0.281223	0.140611	−	0.990065i	$-0.455093\pi$
$853$	8.21343	0.281223	0.140611	+	0.990065i	$0.455093\pi$
$854$	−93.4632	−3.19824
$855$	0	0
$856$	14.3542	0.490616
$857$	47.5813	1.62535	0.812674	−	0.582719i	$-0.198011\pi$
$857$	47.5813	1.62535	0.812674	+	0.582719i	$0.198011\pi$
$858$	−12.9890	−0.443436
$859$	−35.1151	−1.19811	−0.599056	−	0.800707i	$-0.704457\pi$
$859$	−35.1151	−1.19811	−0.599056	+	0.800707i	$0.704457\pi$
$860$	0	0
$861$	6.80360	0.231866
$862$	72.2986	2.46250
$863$	4.51395	0.153657	0.0768284	−	0.997044i	$-0.475521\pi$
$863$	4.51395	0.153657	0.0768284	+	0.997044i	$0.475521\pi$
$864$	−34.9334	−1.18846
$865$	0	0
$866$	29.3837	0.998499
$867$	−22.4352	−0.761941
$868$	−97.5497	−3.31105
$869$	−17.6479	−0.598663
$870$	0	0
$871$	4.43068	0.150128
$872$	2.79867	0.0947751
$873$	2.18814	0.0740573
$874$	9.79463	0.331308
$875$	0	0
$876$	24.9851	0.844168
$877$	−50.7760	−1.71458	−0.857292	−	0.514830i	$-0.827855\pi$
$877$	−50.7760	−1.71458	−0.857292	+	0.514830i	$0.827855\pi$
$878$	−28.2387	−0.953011
$879$	−4.69256	−0.158276
$880$	0	0
$881$	27.2985	0.919711	0.459855	−	0.887994i	$-0.347901\pi$
$881$	27.2985	0.919711	0.459855	+	0.887994i	$0.347901\pi$
$882$	−2.24963	−0.0757489
$883$	10.0819	0.339283	0.169641	−	0.985506i	$-0.445739\pi$
$883$	10.0819	0.339283	0.169641	+	0.985506i	$0.445739\pi$
$884$	25.7780	0.867008
$885$	0	0
$886$	−18.4240	−0.618967
$887$	49.1148	1.64912	0.824558	−	0.565778i	$-0.191424\pi$
$887$	49.1148	1.64912	0.824558	+	0.565778i	$0.191424\pi$
$888$	42.4326	1.42394
$889$	23.7462	0.796422
$890$	0	0
$891$	19.0007	0.636547
$892$	−26.9344	−0.901829
$893$	4.67838	0.156556
$894$	47.2149	1.57910
$895$	0	0
$896$	−61.9447	−2.06943
$897$	15.2021	0.507584
$898$	46.0168	1.53560
$899$	70.6184	2.35526
$900$	0	0
$901$	−29.2892	−0.975765
$902$	5.14398	0.171276
$903$	−28.7709	−0.957435
$904$	32.4048	1.07777
$905$	0	0
$906$	−44.3426	−1.47319
$907$	−38.0037	−1.26189	−0.630946	−	0.775827i	$-0.717333\pi$
$907$	−38.0037	−1.26189	−0.630946	+	0.775827i	$0.717333\pi$
$908$	43.3146	1.43745
$909$	−0.997540	−0.0330863
$910$	0	0
$911$	21.0475	0.697334	0.348667	−	0.937247i	$-0.386634\pi$
$911$	21.0475	0.697334	0.348667	+	0.937247i	$0.386634\pi$
$912$	1.12241	0.0371665
$913$	−1.61994	−0.0536123
$914$	−62.6131	−2.07106
$915$	0	0
$916$	−6.38921	−0.211105
$917$	−66.9681	−2.21148
$918$	−65.3587	−2.15716
$919$	−22.6286	−0.746448	−0.373224	−	0.927741i	$-0.621748\pi$
$919$	−22.6286	−0.746448	−0.373224	+	0.927741i	$0.621748\pi$
$920$	0	0
$921$	−7.68972	−0.253385
$922$	−36.2556	−1.19401
$923$	23.1441	0.761796
$924$	43.8283	1.44185
$925$	0	0
$926$	2.79309	0.0917868
$927$	−1.52849	−0.0502022
$928$	55.8702	1.83403
$929$	37.8492	1.24179	0.620896	−	0.783893i	$-0.286769\pi$
$929$	37.8492	1.24179	0.620896	+	0.783893i	$0.286769\pi$
$930$	0	0
$931$	5.97610	0.195859
$932$	−77.8437	−2.54986
$933$	38.4224	1.25789
$934$	3.10970	0.101753
$935$	0	0
$936$	0.452012	0.0147745
$937$	1.43591	0.0469093	0.0234546	−	0.999725i	$-0.492533\pi$
$937$	1.43591	0.0469093	0.0234546	+	0.999725i	$0.492533\pi$
$938$	−24.8182	−0.810345
$939$	3.34321	0.109101
$940$	0	0
$941$	11.6012	0.378187	0.189094	−	0.981959i	$-0.439445\pi$
$941$	11.6012	0.378187	0.189094	+	0.981959i	$0.439445\pi$
$942$	24.9004	0.811300
$943$	−6.02044	−0.196052
$944$	−0.359832	−0.0117115
$945$	0	0
$946$	−21.7527	−0.707242
$947$	−15.1666	−0.492848	−0.246424	−	0.969162i	$-0.579256\pi$
$947$	−15.1666	−0.492848	−0.246424	+	0.969162i	$0.579256\pi$
$948$	−41.0503	−1.33325
$949$	7.52595	0.244303
$950$	0	0
$951$	−50.9966	−1.65368
$952$	−49.0868	−1.59091
$953$	−37.6905	−1.22092	−0.610458	−	0.792049i	$-0.709015\pi$
$953$	−37.6905	−1.22092	−0.610458	+	0.792049i	$0.709015\pi$
$954$	−1.51075	−0.0489125
$955$	0	0
$956$	45.1139	1.45909
$957$	−31.7283	−1.02563
$958$	56.2150	1.81622
$959$	68.3765	2.20799
$960$	0	0
$961$	38.4120	1.23910
$962$	37.5980	1.21221
$963$	−0.785737	−0.0253200
$964$	−50.4171	−1.62383
$965$	0	0
$966$	−85.1539	−2.73978
$967$	28.4723	0.915608	0.457804	−	0.889053i	$-0.348636\pi$
$967$	28.4723	0.915608	0.457804	+	0.889053i	$0.348636\pi$
$968$	−14.1475	−0.454719
$969$	7.02281	0.225605
$970$	0	0
$971$	−50.5220	−1.62133	−0.810664	−	0.585512i	$-0.800893\pi$
$971$	−50.5220	−1.62133	−0.810664	+	0.585512i	$0.800893\pi$
$972$	−3.97946	−0.127641
$973$	74.1217	2.37623
$974$	−66.4710	−2.12987
$975$	0	0
$976$	9.47744	0.303366
$977$	−61.4170	−1.96490	−0.982452	−	0.186516i	$-0.940281\pi$
$977$	−61.4170	−1.96490	−0.982452	+	0.186516i	$0.940281\pi$
$978$	49.3961	1.57951
$979$	−2.49712	−0.0798084
$980$	0	0
$981$	−0.153197	−0.00489121
$982$	−23.0486	−0.735511
$983$	13.1906	0.420713	0.210357	−	0.977625i	$-0.432537\pi$
$983$	13.1906	0.420713	0.210357	+	0.977625i	$0.432537\pi$
$984$	4.06760	0.129670
$985$	0	0
$986$	104.530	3.32892
$987$	−40.6736	−1.29465
$988$	−3.53218	−0.112373
$989$	25.4591	0.809552
$990$	0	0
$991$	−5.65610	−0.179672	−0.0898359	−	0.995957i	$-0.528634\pi$
$991$	−5.65610	−0.179672	−0.0898359	+	0.995957i	$0.528634\pi$
$992$	54.9157	1.74358
$993$	−39.3783	−1.24963
$994$	−129.640	−4.11194
$995$	0	0
$996$	−3.76811	−0.119397
$997$	−36.3926	−1.15257	−0.576283	−	0.817250i	$-0.695497\pi$
$997$	−36.3926	−1.15257	−0.576283	+	0.817250i	$0.695497\pi$
$998$	94.2860	2.98457
$999$	−57.4246	−1.81683

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4925.2.a.j.1.9		10
5.4	even	2		985.2.a.e.1.2	✓	10
15.14	odd	2		8865.2.a.t.1.9		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
985.2.a.e.1.2	✓	10	5.4	even	2
4925.2.a.j.1.9		10	1.1	even	1	trivial
8865.2.a.t.1.9		10	15.14	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 4925 = 5^{2} \cdot 197 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4925.a (trivial)

\(f(q)\)	\(=\)	\(q+2.24278 q^{2} -1.69515 q^{3} +3.03007 q^{4} -3.80186 q^{6} +3.86416 q^{7} +2.31023 q^{8} -0.126460 q^{9} -2.20820 q^{11} -5.13643 q^{12} -1.54718 q^{13} +8.66648 q^{14} -0.878807 q^{16} -5.49863 q^{17} -0.283622 q^{18} +0.753438 q^{19} -6.55035 q^{21} -4.95250 q^{22} +5.79633 q^{23} -3.91619 q^{24} -3.46999 q^{26} +5.29982 q^{27} +11.7087 q^{28} -8.47619 q^{29} -8.33139 q^{31} -6.59143 q^{32} +3.74323 q^{33} -12.3322 q^{34} -0.383183 q^{36} -10.8352 q^{37} +1.68980 q^{38} +2.62271 q^{39} -1.03866 q^{41} -14.6910 q^{42} +4.39227 q^{43} -6.69099 q^{44} +12.9999 q^{46} +6.20938 q^{47} +1.48971 q^{48} +7.93177 q^{49} +9.32101 q^{51} -4.68808 q^{52} +5.32664 q^{53} +11.8864 q^{54} +8.92710 q^{56} -1.27719 q^{57} -19.0102 q^{58} +0.409456 q^{59} -10.7844 q^{61} -18.6855 q^{62} -0.488662 q^{63} -13.0255 q^{64} +8.39524 q^{66} -2.86371 q^{67} -16.6612 q^{68} -9.82567 q^{69} -14.9588 q^{71} -0.292151 q^{72} -4.86429 q^{73} -24.3010 q^{74} +2.28297 q^{76} -8.53283 q^{77} +5.88217 q^{78} +7.99199 q^{79} -8.60463 q^{81} -2.32949 q^{82} +0.733605 q^{83} -19.8480 q^{84} +9.85091 q^{86} +14.3684 q^{87} -5.10143 q^{88} +1.13084 q^{89} -5.97857 q^{91} +17.5633 q^{92} +14.1230 q^{93} +13.9263 q^{94} +11.1735 q^{96} -17.3030 q^{97} +17.7892 q^{98} +0.279248 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 2 q^{2} + 3 q^{3} + 2 q^{4} - 9 q^{6} + 6 q^{7} + 6 q^{8} - 5 q^{9} - 11 q^{11} + 5 q^{13} - 9 q^{14} - 2 q^{16} + 4 q^{17} - 15 q^{18} - 28 q^{19} - 7 q^{21} + 12 q^{22} + 24 q^{23} - 3 q^{24} + 7 q^{26}+ \cdots - 16 q^{99}+O(q^{100}) \) 10 * q + 2 * q^2 + 3 * q^3 + 2 * q^4 - 9 * q^6 + 6 * q^7 + 6 * q^8 - 5 * q^9 - 11 * q^11 + 5 * q^13 - 9 * q^14 - 2 * q^16 + 4 * q^17 - 15 * q^18 - 28 * q^19 - 7 * q^21 + 12 * q^22 + 24 * q^23 - 3 * q^24 + 7 * q^26 + 9 * q^27 - 3 * q^28 - 16 * q^29 - 34 * q^31 + 4 * q^32 + 10 * q^33 + q^34 - q^36 - 5 * q^37 - 13 * q^38 - 15 * q^39 - 15 * q^41 - 7 * q^42 + 5 * q^43 - 11 * q^44 + 6 * q^46 + 14 * q^47 - 12 * q^48 + 4 * q^49 - q^51 + 7 * q^52 + 15 * q^53 - 4 * q^54 - 5 * q^56 - 9 * q^57 - 28 * q^58 - 49 * q^59 - 16 * q^61 - 20 * q^62 - 8 * q^63 - 16 * q^64 + 40 * q^66 + 2 * q^67 - 19 * q^68 - 16 * q^69 - 64 * q^71 - q^72 - 18 * q^73 - 37 * q^74 - 19 * q^76 - 8 * q^77 - 4 * q^78 - 11 * q^79 - 14 * q^81 - 10 * q^82 - 4 * q^83 - 11 * q^84 + 5 * q^86 + q^87 - 19 * q^88 - 5 * q^89 - 30 * q^91 + 32 * q^92 - 41 * q^93 - 37 * q^94 + 26 * q^96 - 20 * q^97 + 37 * q^98 - 16 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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