LMFDB - Embedded newform 2057.4.a.v.1.10

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2057,4,Mod(1,2057)] 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("2057.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2057, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 2057 = 11^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2057.a (trivial)

Newform invariants

comment:select newform

sage:traces = [52,1,20,235,40] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	yes
Analytic conductor:	$121.366928882$
Analytic rank:	$0$
Dimension:	$52$
Twist minimal:	no (minimal twist has level 187)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
Character	$\chi$	$=$	2057.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-3.88649 q^{2} +0.0923746 q^{3} +7.10482 q^{4} -3.54222 q^{5} -0.359013 q^{6} +27.1367 q^{7} +3.47912 q^{8} -26.9915 q^{9} +13.7668 q^{10} +0.656305 q^{12} +60.7507 q^{13} -105.466 q^{14} -0.327211 q^{15} -70.3601 q^{16} -17.0000 q^{17} +104.902 q^{18} -135.682 q^{19} -25.1668 q^{20} +2.50674 q^{21} -25.8969 q^{23} +0.321383 q^{24} -112.453 q^{25} -236.107 q^{26} -4.98744 q^{27} +192.801 q^{28} +16.7371 q^{29} +1.27170 q^{30} +64.6559 q^{31} +245.621 q^{32} +66.0704 q^{34} -96.1240 q^{35} -191.769 q^{36} +237.140 q^{37} +527.328 q^{38} +5.61183 q^{39} -12.3238 q^{40} -197.539 q^{41} -9.74242 q^{42} -54.5472 q^{43} +95.6096 q^{45} +100.648 q^{46} +303.980 q^{47} -6.49949 q^{48} +393.399 q^{49} +437.046 q^{50} -1.57037 q^{51} +431.623 q^{52} +65.9571 q^{53} +19.3837 q^{54} +94.4118 q^{56} -12.5336 q^{57} -65.0485 q^{58} +524.268 q^{59} -2.32477 q^{60} +33.7414 q^{61} -251.285 q^{62} -732.459 q^{63} -391.723 q^{64} -215.192 q^{65} -612.294 q^{67} -120.782 q^{68} -2.39222 q^{69} +373.585 q^{70} -833.552 q^{71} -93.9066 q^{72} -129.139 q^{73} -921.641 q^{74} -10.3878 q^{75} -963.997 q^{76} -21.8103 q^{78} +1037.89 q^{79} +249.231 q^{80} +728.309 q^{81} +767.732 q^{82} +764.711 q^{83} +17.8099 q^{84} +60.2177 q^{85} +211.997 q^{86} +1.54608 q^{87} +513.861 q^{89} -371.586 q^{90} +1648.57 q^{91} -183.993 q^{92} +5.97257 q^{93} -1181.42 q^{94} +480.616 q^{95} +22.6892 q^{96} -302.815 q^{97} -1528.94 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 52 q + q^{2} + 20 q^{3} + 235 q^{4} + 40 q^{5} - 24 q^{6} - 42 q^{7} + 45 q^{8} + 572 q^{9} + 33 q^{10} + 233 q^{12} + 12 q^{13} + 73 q^{14} + 400 q^{15} + 1223 q^{16} - 884 q^{17} + 201 q^{18} - 44 q^{19}+ \cdots - 4610 q^{98}+O(q^{100}) $ 52 * q + q^2 + 20 * q^3 + 235 * q^4 + 40 * q^5 - 24 * q^6 - 42 * q^7 + 45 * q^8 + 572 * q^9 + 33 * q^10 + 233 * q^12 + 12 * q^13 + 73 * q^14 + 400 * q^15 + 1223 * q^16 - 884 * q^17 + 201 * q^18 - 44 * q^19 + 655 * q^20 + 260 * q^21 + 572 * q^23 - 104 * q^24 + 1858 * q^25 + 465 * q^26 + 1070 * q^27 - 577 * q^28 + 322 * q^29 - 320 * q^30 + 1110 * q^31 + 481 * q^32 - 17 * q^34 + 102 * q^35 + 2507 * q^36 + 1678 * q^37 - 360 * q^38 - 1282 * q^39 + 1791 * q^40 - 826 * q^41 + 2133 * q^42 + 270 * q^43 + 710 * q^45 + 2158 * q^46 + 2464 * q^47 + 2201 * q^48 + 3224 * q^49 + 2379 * q^50 - 340 * q^51 - 3664 * q^52 + 992 * q^53 + 1202 * q^54 + 1731 * q^56 + 1016 * q^57 + 1358 * q^58 + 1442 * q^59 + 1444 * q^60 - 140 * q^61 + 464 * q^62 - 766 * q^63 + 8427 * q^64 - 1268 * q^65 + 5766 * q^67 - 3995 * q^68 + 2460 * q^69 + 2422 * q^70 + 2704 * q^71 + 5455 * q^72 + 4 * q^73 - 4008 * q^74 + 5204 * q^75 - 1935 * q^76 + 4092 * q^78 - 2180 * q^79 + 5040 * q^80 + 7192 * q^81 + 3197 * q^82 + 4200 * q^83 + 7951 * q^84 - 680 * q^85 + 3091 * q^86 - 752 * q^87 - 240 * q^89 - 4495 * q^90 + 5494 * q^91 + 6902 * q^92 + 6266 * q^93 + 5990 * q^94 + 3168 * q^95 - 9467 * q^96 + 5322 * q^97 - 4610 * q^98

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 $

$2$	−3.88649	−1.37408	−0.687041	−	0.726619i	$-0.741091\pi$
$2$	−3.88649	−1.37408	−0.687041	+	0.726619i	$0.741091\pi$
$3$	0.0923746	0.0177775	0.00888875	−	0.999960i	$-0.497171\pi$
$3$	0.0923746	0.0177775	0.00888875	+	0.999960i	$0.497171\pi$
$4$	7.10482	0.888102
$5$	−3.54222	−0.316826	−0.158413	−	0.987373i	$-0.550638\pi$
$5$	−3.54222	−0.316826	−0.158413	+	0.987373i	$0.550638\pi$
$6$	−0.359013	−0.0244278
$7$	27.1367	1.46524	0.732621	−	0.680637i	$-0.238297\pi$
$7$	27.1367	1.46524	0.732621	+	0.680637i	$0.238297\pi$
$8$	3.47912	0.153757
$9$	−26.9915	−0.999684
$10$	13.7668	0.435344
$11$	0	0
$11$	0	0
$12$	0.656305	0.0157882
$13$	60.7507	1.29609	0.648047	−	0.761600i	$-0.275586\pi$
$13$	60.7507	1.29609	0.648047	+	0.761600i	$0.275586\pi$
$14$	−105.466	−2.01336
$15$	−0.327211	−0.00563237
$16$	−70.3601	−1.09938
$17$	−17.0000	−0.242536
$17$	−17.0000	−0.242536
$18$	104.902	1.37365
$19$	−135.682	−1.63830	−0.819148	−	0.573582i	$-0.805553\pi$
$19$	−135.682	−1.63830	−0.819148	+	0.573582i	$0.805553\pi$
$20$	−25.1668	−0.281373
$21$	2.50674	0.0260484
$22$	0	0
$23$	−25.8969	−0.234777	−0.117389	−	0.993086i	$-0.537452\pi$
$23$	−25.8969	−0.234777	−0.117389	+	0.993086i	$0.537452\pi$
$24$	0.321383	0.00273341
$25$	−112.453	−0.899622
$26$	−236.107	−1.78094
$27$	−4.98744	−0.0355494
$28$	192.801	1.30128
$29$	16.7371	0.107172	0.0535862	−	0.998563i	$-0.482935\pi$
$29$	16.7371	0.107172	0.0535862	+	0.998563i	$0.482935\pi$
$30$	1.27170	0.00773934
$31$	64.6559	0.374598	0.187299	−	0.982303i	$-0.440027\pi$
$31$	64.6559	0.374598	0.187299	+	0.982303i	$0.440027\pi$
$32$	245.621	1.35688
$33$	0	0
$34$	66.0704	0.333264
$35$	−96.1240	−0.464226
$36$	−191.769	−0.887821
$37$	237.140	1.05366	0.526831	−	0.849970i	$-0.323380\pi$
$37$	237.140	1.05366	0.526831	+	0.849970i	$0.323380\pi$
$38$	527.328	2.25115
$39$	5.61183	0.0230413
$40$	−12.3238	−0.0487141
$41$	−197.539	−0.752447	−0.376224	−	0.926529i	$-0.622778\pi$
$41$	−197.539	−0.752447	−0.376224	+	0.926529i	$0.622778\pi$
$42$	−9.74242	−0.0357926
$43$	−54.5472	−0.193450	−0.0967252	−	0.995311i	$-0.530837\pi$
$43$	−54.5472	−0.193450	−0.0967252	+	0.995311i	$0.530837\pi$
$44$	0	0
$45$	95.6096	0.316725
$46$	100.648	0.322603
$47$	303.980	0.943406	0.471703	−	0.881758i	$-0.343640\pi$
$47$	303.980	0.943406	0.471703	+	0.881758i	$0.343640\pi$
$48$	−6.49949	−0.0195442
$49$	393.399	1.14694
$50$	437.046	1.23615
$51$	−1.57037	−0.00431168
$52$	431.623	1.15106
$53$	65.9571	0.170942	0.0854708	−	0.996341i	$-0.472761\pi$
$53$	65.9571	0.170942	0.0854708	+	0.996341i	$0.472761\pi$
$54$	19.3837	0.0488478
$55$	0	0
$56$	94.4118	0.225291
$57$	−12.5336	−0.0291248
$58$	−65.0485	−0.147264
$59$	524.268	1.15685	0.578423	−	0.815737i	$-0.303668\pi$
$59$	524.268	1.15685	0.578423	+	0.815737i	$0.303668\pi$
$60$	−2.32477	−0.00500212
$61$	33.7414	0.0708221	0.0354111	−	0.999373i	$-0.488726\pi$
$61$	33.7414	0.0708221	0.0354111	+	0.999373i	$0.488726\pi$
$62$	−251.285	−0.514729
$63$	−732.459	−1.46478
$64$	−391.723	−0.765084
$65$	−215.192	−0.410636
$66$	0	0
$67$	−612.294	−1.11647	−0.558236	−	0.829682i	$-0.688522\pi$
$67$	−612.294	−1.11647	−0.558236	+	0.829682i	$0.688522\pi$
$68$	−120.782	−0.215396
$69$	−2.39222	−0.00417375
$70$	373.585	0.637885
$71$	−833.552	−1.39330	−0.696651	−	0.717410i	$-0.745327\pi$
$71$	−833.552	−1.39330	−0.696651	+	0.717410i	$0.745327\pi$
$72$	−93.9066	−0.153708
$73$	−129.139	−0.207049	−0.103524	−	0.994627i	$-0.533012\pi$
$73$	−129.139	−0.207049	−0.103524	+	0.994627i	$0.533012\pi$
$74$	−921.641	−1.44782
$75$	−10.3878	−0.0159930
$76$	−963.997	−1.45497
$77$	0	0
$78$	−21.8103	−0.0316607
$79$	1037.89	1.47813	0.739063	−	0.673636i	$-0.235269\pi$
$79$	1037.89	1.47813	0.739063	+	0.673636i	$0.235269\pi$
$80$	249.231	0.348311
$81$	728.309	0.999052
$82$	767.732	1.03392
$83$	764.711	1.01130	0.505650	−	0.862738i	$-0.331253\pi$
$83$	764.711	1.01130	0.505650	+	0.862738i	$0.331253\pi$
$84$	17.8099	0.0231336
$85$	60.2177	0.0768415
$86$	211.997	0.265817
$87$	1.54608	0.00190526
$88$	0	0
$89$	513.861	0.612013	0.306007	−	0.952029i	$-0.401007\pi$
$89$	513.861	0.612013	0.306007	+	0.952029i	$0.401007\pi$
$90$	−371.586	−0.435207
$91$	1648.57	1.89909
$92$	−183.993	−0.208506
$93$	5.97257	0.00665942
$94$	−1181.42	−1.29632
$95$	480.616	0.519054
$96$	22.6892	0.0241219
$97$	−302.815	−0.316972	−0.158486	−	0.987361i	$-0.550661\pi$
$97$	−302.815	−0.316972	−0.158486	+	0.987361i	$0.550661\pi$
$98$	−1528.94	−1.57598
$99$	0	0
$100$	−798.956	−0.798956
$101$	−948.841	−0.934784	−0.467392	−	0.884050i	$-0.654806\pi$
$101$	−948.841	−0.934784	−0.467392	+	0.884050i	$0.654806\pi$
$102$	6.10323	0.00592460
$103$	2064.62	1.97508	0.987541	−	0.157360i	$-0.0502983\pi$
$103$	2064.62	1.97508	0.987541	+	0.157360i	$0.0502983\pi$
$104$	211.359	0.199283
$105$	−8.87942	−0.00825279
$106$	−256.342	−0.234888
$107$	−1896.97	−1.71390	−0.856951	−	0.515398i	$-0.827644\pi$
$107$	−1896.97	−1.71390	−0.856951	+	0.515398i	$0.827644\pi$
$108$	−35.4349	−0.0315715
$109$	842.737	0.740547	0.370273	−	0.928923i	$-0.379264\pi$
$109$	842.737	0.740547	0.370273	+	0.928923i	$0.379264\pi$
$110$	0	0
$111$	21.9057	0.0187315
$112$	−1909.34	−1.61085
$113$	−1568.35	−1.30564	−0.652821	−	0.757512i	$-0.726415\pi$
$113$	−1568.35	−1.30564	−0.652821	+	0.757512i	$0.726415\pi$
$114$	48.7117	0.0400199
$115$	91.7324	0.0743834
$116$	118.914	0.0951800
$117$	−1639.75	−1.29568
$118$	−2037.56	−1.58960
$119$	−461.323	−0.355373
$120$	−1.13841	−0.000866016 0
$121$	0	0
$122$	−131.136	−0.0973154
$123$	−18.2476	−0.0133766
$124$	459.368	0.332681
$125$	841.109	0.601849
$126$	2846.69	2.01273
$127$	−281.186	−0.196467	−0.0982333	−	0.995163i	$-0.531319\pi$
$127$	−281.186	−0.196467	−0.0982333	+	0.995163i	$0.531319\pi$
$128$	−442.540	−0.305589
$129$	−5.03878	−0.00343907
$130$	836.343	0.564247
$131$	−57.4316	−0.0383040	−0.0191520	−	0.999817i	$-0.506097\pi$
$131$	−57.4316	−0.0383040	−0.0191520	+	0.999817i	$0.506097\pi$
$132$	0	0
$133$	−3681.96	−2.40050
$134$	2379.68	1.53412
$135$	17.6666	0.0112630
$136$	−59.1451	−0.0372915
$137$	−462.328	−0.288316	−0.144158	−	0.989555i	$-0.546047\pi$
$137$	−462.328	−0.288316	−0.144158	+	0.989555i	$0.546047\pi$
$138$	9.29733	0.00573508
$139$	−1375.37	−0.839262	−0.419631	−	0.907695i	$-0.637840\pi$
$139$	−1375.37	−0.839262	−0.419631	+	0.907695i	$0.637840\pi$
$140$	−682.943	−0.412280
$141$	28.0801	0.0167714
$142$	3239.59	1.91451
$143$	0	0
$144$	1899.12	1.09903
$145$	−59.2864	−0.0339549
$146$	501.897	0.284502
$147$	36.3401	0.0203897
$148$	1684.83	0.935760
$149$	571.352	0.314141	0.157071	−	0.987587i	$-0.449795\pi$
$149$	571.352	0.314141	0.157071	+	0.987587i	$0.449795\pi$
$150$	40.3720	0.0219757
$151$	2399.07	1.29294	0.646468	−	0.762941i	$-0.276245\pi$
$151$	2399.07	1.29294	0.646468	+	0.762941i	$0.276245\pi$
$152$	−472.055	−0.251899
$153$	458.855	0.242459
$154$	0	0
$155$	−229.025	−0.118682
$156$	39.8710	0.0204631
$157$	3894.35	1.97964	0.989818	−	0.142335i	$-0.0454612\pi$
$157$	3894.35	1.97964	0.989818	+	0.142335i	$0.0454612\pi$
$158$	−4033.76	−2.03107
$159$	6.09276	0.00303892
$160$	−870.043	−0.429893
$161$	−702.755	−0.344006
$162$	−2830.57	−1.37278
$163$	−941.872	−0.452596	−0.226298	−	0.974058i	$-0.572662\pi$
$163$	−941.872	−0.452596	−0.226298	+	0.974058i	$0.572662\pi$
$164$	−1403.48	−0.668250
$165$	0	0
$166$	−2972.04	−1.38961
$167$	2407.24	1.11544	0.557718	−	0.830031i	$-0.311677\pi$
$167$	2407.24	1.11544	0.557718	+	0.830031i	$0.311677\pi$
$168$	8.72125	0.00400512
$169$	1493.65	0.679860
$170$	−234.036	−0.105587
$171$	3662.26	1.63778
$172$	−387.548	−0.171804
$173$	−780.435	−0.342979	−0.171490	−	0.985186i	$-0.554858\pi$
$173$	−780.435	−0.342979	−0.171490	+	0.985186i	$0.554858\pi$
$174$	−6.00883	−0.00261798
$175$	−3051.59	−1.31816
$176$	0	0
$177$	48.4291	0.0205658
$178$	−1997.12	−0.840957
$179$	−1562.43	−0.652410	−0.326205	−	0.945299i	$-0.605770\pi$
$179$	−1562.43	−0.652410	−0.326205	+	0.945299i	$0.605770\pi$
$180$	679.289	0.281284
$181$	297.891	0.122332	0.0611658	−	0.998128i	$-0.480518\pi$
$181$	297.891	0.122332	0.0611658	+	0.998128i	$0.480518\pi$
$182$	−6407.16	−2.60951
$183$	3.11685	0.00125904
$184$	−90.0984	−0.0360986
$185$	−840.000	−0.333827
$186$	−23.2123	−0.00915060
$187$	0	0
$188$	2159.72	0.837840
$189$	−135.343	−0.0520885
$190$	−1867.91	−0.713223
$191$	−2183.13	−0.827045	−0.413522	−	0.910494i	$-0.635702\pi$
$191$	−2183.13	−0.827045	−0.413522	+	0.910494i	$0.635702\pi$
$192$	−36.1853	−0.0136013
$193$	1275.35	0.475657	0.237828	−	0.971307i	$-0.423564\pi$
$193$	1275.35	0.475657	0.237828	+	0.971307i	$0.423564\pi$
$194$	1176.89	0.435545
$195$	−19.8783	−0.00730008
$196$	2795.03	1.01860
$197$	−5271.02	−1.90632	−0.953160	−	0.302467i	$-0.902190\pi$
$197$	−5271.02	−1.90632	−0.953160	+	0.302467i	$0.902190\pi$
$198$	0	0
$199$	3500.22	1.24685	0.623427	−	0.781881i	$-0.285740\pi$
$199$	3500.22	1.24685	0.623427	+	0.781881i	$0.285740\pi$
$200$	−391.237	−0.138323
$201$	−56.5604	−0.0198481
$202$	3687.66	1.28447
$203$	454.189	0.157033
$204$	−11.1572	−0.00382921
$205$	699.725	0.238395
$206$	−8024.15	−2.71393
$207$	698.995	0.234703
$208$	−4274.43	−1.42490
$209$	0	0
$210$	34.5098	0.0113400
$211$	2247.97	0.733444	0.366722	−	0.930331i	$-0.380480\pi$
$211$	2247.97	0.733444	0.366722	+	0.930331i	$0.380480\pi$
$212$	468.613	0.151814
$213$	−76.9991	−0.0247694
$214$	7372.58	2.35504
$215$	193.218	0.0612900
$216$	−17.3519	−0.00546597
$217$	1754.55	0.548877
$218$	−3275.29	−1.01757
$219$	−11.9291	−0.00368081
$220$	0	0
$221$	−1032.76	−0.314349
$222$	−85.1363	−0.0257386
$223$	−3860.13	−1.15916	−0.579581	−	0.814914i	$-0.696784\pi$
$223$	−3860.13	−1.15916	−0.579581	+	0.814914i	$0.696784\pi$
$224$	6665.34	1.98815
$225$	3035.26	0.899337
$226$	6095.37	1.79406
$227$	4753.15	1.38977	0.694885	−	0.719121i	$-0.255455\pi$
$227$	4753.15	1.38977	0.694885	+	0.719121i	$0.255455\pi$
$228$	−89.0489	−0.0258658
$229$	299.717	0.0864886	0.0432443	−	0.999065i	$-0.486231\pi$
$229$	299.717	0.0864886	0.0432443	+	0.999065i	$0.486231\pi$
$230$	−356.517	−0.102209
$231$	0	0
$232$	58.2303	0.0164785
$233$	1630.62	0.458477	0.229239	−	0.973370i	$-0.426376\pi$
$233$	1630.62	0.458477	0.229239	+	0.973370i	$0.426376\pi$
$234$	6372.88	1.78038
$235$	−1076.76	−0.298895
$236$	3724.83	1.02740
$237$	95.8749	0.0262774
$238$	1792.93	0.488312
$239$	−1871.45	−0.506504	−0.253252	−	0.967400i	$-0.581500\pi$
$239$	−1871.45	−0.506504	−0.253252	+	0.967400i	$0.581500\pi$
$240$	23.0226	0.00619210
$241$	−624.356	−0.166881	−0.0834405	−	0.996513i	$-0.526591\pi$
$241$	−624.356	−0.166881	−0.0834405	+	0.996513i	$0.526591\pi$
$242$	0	0
$243$	201.938	0.0533101
$244$	239.727	0.0628973
$245$	−1393.50	−0.363378
$246$	70.9190	0.0183806
$247$	−8242.79	−2.12339
$248$	224.946	0.0575971
$249$	70.6399	0.0179784
$250$	−3268.96	−0.826990
$251$	−5375.52	−1.35179	−0.675895	−	0.736998i	$-0.736243\pi$
$251$	−5375.52	−1.35179	−0.675895	+	0.736998i	$0.736243\pi$
$252$	−5203.98	−1.30087
$253$	0	0
$254$	1092.83	0.269961
$255$	5.56259	0.00136605
$256$	4853.71	1.18499
$257$	−4064.72	−0.986578	−0.493289	−	0.869866i	$-0.664205\pi$
$257$	−4064.72	−0.986578	−0.493289	+	0.869866i	$0.664205\pi$
$258$	19.5832	0.00472556
$259$	6435.18	1.54387
$260$	−1528.90	−0.364686
$261$	−451.758	−0.107138
$262$	223.207	0.0526328
$263$	−6009.25	−1.40892	−0.704461	−	0.709743i	$-0.748811\pi$
$263$	−6009.25	−1.40892	−0.704461	+	0.709743i	$0.748811\pi$
$264$	0	0
$265$	−233.634	−0.0541586
$266$	14309.9	3.29849
$267$	47.4677	0.0108801
$268$	−4350.24	−0.991541
$269$	7996.94	1.81257	0.906286	−	0.422664i	$-0.138905\pi$
$269$	7996.94	1.81257	0.906286	+	0.422664i	$0.138905\pi$
$270$	−68.6611	−0.0154762
$271$	5583.01	1.25145	0.625726	−	0.780043i	$-0.284803\pi$
$271$	5583.01	1.25145	0.625726	+	0.780043i	$0.284803\pi$
$272$	1196.12	0.266638
$273$	152.286	0.0337611
$274$	1796.83	0.396171
$275$	0	0
$276$	−16.9963	−0.00370672
$277$	259.590	0.0563078	0.0281539	−	0.999604i	$-0.491037\pi$
$277$	259.590	0.0563078	0.0281539	+	0.999604i	$0.491037\pi$
$278$	5345.36	1.15321
$279$	−1745.16	−0.374480
$280$	−334.427	−0.0713780
$281$	1528.98	0.324595	0.162298	−	0.986742i	$-0.448110\pi$
$281$	1528.98	0.324595	0.162298	+	0.986742i	$0.448110\pi$
$282$	−109.133	−0.0230453
$283$	−3261.08	−0.684986	−0.342493	−	0.939520i	$-0.611271\pi$
$283$	−3261.08	−0.684986	−0.342493	+	0.939520i	$0.611271\pi$
$284$	−5922.23	−1.23739
$285$	44.3967	0.00922749
$286$	0	0
$287$	−5360.54	−1.10252
$288$	−6629.67	−1.35645
$289$	289.000	0.0588235
$290$	230.416	0.0466569
$291$	−27.9725	−0.00563497
$292$	−917.507	−0.183880
$293$	−1082.89	−0.215915	−0.107957	−	0.994156i	$-0.534431\pi$
$293$	−1082.89	−0.215915	−0.107957	+	0.994156i	$0.534431\pi$
$294$	−141.235	−0.0280171
$295$	−1857.07	−0.366518
$296$	825.037	0.162008
$297$	0	0
$298$	−2220.56	−0.431656
$299$	−1573.26	−0.304293
$300$	−73.8033	−0.0142034
$301$	−1480.23	−0.283452
$302$	−9323.96	−1.77660
$303$	−87.6488	−0.0166181
$304$	9546.61	1.80110
$305$	−119.520	−0.0224383
$306$	−1783.34	−0.333159
$307$	7355.28	1.36739	0.683694	−	0.729769i	$-0.260372\pi$
$307$	7355.28	1.36739	0.683694	+	0.729769i	$0.260372\pi$
$308$	0	0
$309$	190.719	0.0351120
$310$	890.105	0.163079
$311$	9763.57	1.78020	0.890098	−	0.455768i	$-0.150635\pi$
$311$	9763.57	1.78020	0.890098	+	0.455768i	$0.150635\pi$
$312$	19.5242	0.00354276
$313$	6636.33	1.19843	0.599213	−	0.800590i	$-0.295480\pi$
$313$	6636.33	1.19843	0.599213	+	0.800590i	$0.295480\pi$
$314$	−15135.4	−2.72018
$315$	2594.53	0.464079
$316$	7374.03	1.31273
$317$	7537.27	1.33544	0.667721	−	0.744412i	$-0.267270\pi$
$317$	7537.27	1.33544	0.667721	+	0.744412i	$0.267270\pi$
$318$	−23.6795	−0.00417572
$319$	0	0
$320$	1387.57	0.242398
$321$	−175.232	−0.0304689
$322$	2731.25	0.472692
$323$	2306.60	0.397345
$324$	5174.50	0.887260
$325$	−6831.59	−1.16599
$326$	3660.58	0.621904
$327$	77.8475	0.0131651
$328$	−687.261	−0.115694
$329$	8249.01	1.38232
$330$	0	0
$331$	4014.50	0.666637	0.333319	−	0.942814i	$-0.391832\pi$
$331$	4014.50	0.666637	0.333319	+	0.942814i	$0.391832\pi$
$332$	5433.13	0.898138
$333$	−6400.75	−1.05333
$334$	−9355.71	−1.53270
$335$	2168.88	0.353727
$336$	−176.375	−0.0286370
$337$	−2971.40	−0.480303	−0.240152	−	0.970735i	$-0.577197\pi$
$337$	−2971.40	−0.480303	−0.240152	+	0.970735i	$0.577197\pi$
$338$	−5805.07	−0.934184
$339$	−144.875	−0.0232111
$340$	427.836	0.0682431
$341$	0	0
$342$	−14233.3	−2.25044
$343$	1367.66	0.215296
$344$	−189.776	−0.0297443
$345$	8.47375	0.00132235
$346$	3033.15	0.471281
$347$	−9948.22	−1.53905	−0.769523	−	0.638620i	$-0.779506\pi$
$347$	−9948.22	−1.53905	−0.769523	+	0.638620i	$0.779506\pi$
$348$	10.9846	0.00169206
$349$	−3102.38	−0.475835	−0.237918	−	0.971285i	$-0.576465\pi$
$349$	−3102.38	−0.475835	−0.237918	+	0.971285i	$0.576465\pi$
$350$	11860.0	1.81127
$351$	−302.991	−0.0460754
$352$	0	0
$353$	−824.026	−0.124245	−0.0621225	−	0.998069i	$-0.519787\pi$
$353$	−824.026	−0.124245	−0.0621225	+	0.998069i	$0.519787\pi$
$354$	−188.219	−0.0282591
$355$	2952.62	0.441434
$356$	3650.89	0.543530
$357$	−42.6146	−0.00631766
$358$	6072.36	0.896465
$359$	−607.542	−0.0893172	−0.0446586	−	0.999002i	$-0.514220\pi$
$359$	−607.542	−0.0893172	−0.0446586	+	0.999002i	$0.514220\pi$
$360$	332.638	0.0486987
$361$	11550.6	1.68401
$362$	−1157.75	−0.168094
$363$	0	0
$364$	11712.8	1.68659
$365$	457.438	0.0655983
$366$	−12.1136	−0.00173003
$367$	7719.03	1.09790	0.548951	−	0.835855i	$-0.315027\pi$
$367$	7719.03	1.09790	0.548951	+	0.835855i	$0.315027\pi$
$368$	1822.11	0.258109
$369$	5331.86	0.752210
$370$	3264.65	0.458706
$371$	1789.86	0.250471
$372$	42.4340	0.00591425
$373$	8024.53	1.11393	0.556963	−	0.830537i	$-0.311967\pi$
$373$	8024.53	1.11393	0.556963	+	0.830537i	$0.311967\pi$
$374$	0	0
$375$	77.6971	0.0106994
$376$	1057.58	0.145055
$377$	1016.79	0.138905
$378$	526.008	0.0715739
$379$	−14358.9	−1.94608	−0.973040	−	0.230635i	$-0.925920\pi$
$379$	−14358.9	−1.94608	−0.973040	+	0.230635i	$0.925920\pi$
$380$	3414.69	0.460973
$381$	−25.9745	−0.00349269
$382$	8484.71	1.13643
$383$	7822.17	1.04359	0.521794	−	0.853072i	$-0.325263\pi$
$383$	7822.17	1.04359	0.521794	+	0.853072i	$0.325263\pi$
$384$	−40.8794	−0.00543260
$385$	0	0
$386$	−4956.64	−0.653591
$387$	1472.31	0.193389
$388$	−2151.45	−0.281503
$389$	−9317.39	−1.21442	−0.607212	−	0.794540i	$-0.707712\pi$
$389$	−9317.39	−1.21442	−0.607212	+	0.794540i	$0.707712\pi$
$390$	77.2569	0.0100309
$391$	440.247	0.0569418
$392$	1368.68	0.176349
$393$	−5.30522	−0.000680949 0
$394$	20485.8	2.61944
$395$	−3676.44	−0.468308
$396$	0	0
$397$	11913.9	1.50616	0.753078	−	0.657931i	$-0.228568\pi$
$397$	11913.9	1.50616	0.753078	+	0.657931i	$0.228568\pi$
$398$	−13603.6	−1.71328
$399$	−340.120	−0.0426749
$400$	7912.18	0.989023
$401$	6795.80	0.846299	0.423149	−	0.906060i	$-0.360925\pi$
$401$	6795.80	0.846299	0.423149	+	0.906060i	$0.360925\pi$
$402$	219.822	0.0272729
$403$	3927.90	0.485515
$404$	−6741.34	−0.830184
$405$	−2579.83	−0.316525
$406$	−1765.20	−0.215777
$407$	0	0
$408$	−5.46350	−0.000662950 0
$409$	−7470.56	−0.903167	−0.451584	−	0.892229i	$-0.649141\pi$
$409$	−7470.56	−0.903167	−0.451584	+	0.892229i	$0.649141\pi$
$410$	−2719.47	−0.327574
$411$	−42.7074	−0.00512555
$412$	14668.8	1.75408
$413$	14226.9	1.69506
$414$	−2716.64	−0.322501
$415$	−2708.77	−0.320406
$416$	14921.7	1.75864
$417$	−127.049	−0.0149200
$418$	0	0
$419$	13399.6	1.56232	0.781160	−	0.624331i	$-0.214628\pi$
$419$	13399.6	1.56232	0.781160	+	0.624331i	$0.214628\pi$
$420$	−63.0866	−0.00732932
$421$	15500.2	1.79438	0.897189	−	0.441646i	$-0.145605\pi$
$421$	15500.2	1.79438	0.897189	+	0.441646i	$0.145605\pi$
$422$	−8736.72	−1.00781
$423$	−8204.87	−0.943107
$424$	229.473	0.0262834
$425$	1911.70	0.218190
$426$	299.256	0.0340352
$427$	915.630	0.103772
$428$	−13477.7	−1.52212
$429$	0	0
$430$	−750.940	−0.0842175
$431$	−10265.3	−1.14724	−0.573620	−	0.819121i	$-0.694461\pi$
$431$	−10265.3	−1.14724	−0.573620	+	0.819121i	$0.694461\pi$
$432$	350.917	0.0390822
$433$	−7146.13	−0.793120	−0.396560	−	0.918009i	$-0.629796\pi$
$433$	−7146.13	−0.793120	−0.396560	+	0.918009i	$0.629796\pi$
$434$	−6819.03	−0.754202
$435$	−5.47656	−0.000603634 0
$436$	5987.49	0.657681
$437$	3513.75	0.384634
$438$	46.3625	0.00505773
$439$	10796.9	1.17382	0.586911	−	0.809652i	$-0.300344\pi$
$439$	10796.9	1.17382	0.586911	+	0.809652i	$0.300344\pi$
$440$	0	0
$441$	−10618.4	−1.14657
$442$	4013.82	0.431941
$443$	12646.3	1.35630	0.678152	−	0.734922i	$-0.262781\pi$
$443$	12646.3	1.35630	0.678152	+	0.734922i	$0.262781\pi$
$444$	155.636	0.0166355
$445$	−1820.21	−0.193901
$446$	15002.4	1.59278
$447$	52.7785	0.00558465
$448$	−10630.1	−1.12103
$449$	−4626.37	−0.486263	−0.243132	−	0.969993i	$-0.578175\pi$
$449$	−4626.37	−0.486263	−0.243132	+	0.969993i	$0.578175\pi$
$450$	−11796.5	−1.23576
$451$	0	0
$452$	−11142.8	−1.15954
$453$	221.613	0.0229852
$454$	−18473.1	−1.90966
$455$	−5839.60	−0.601681
$456$	−43.6059	−0.00447814
$457$	5160.37	0.528210	0.264105	−	0.964494i	$-0.414923\pi$
$457$	5160.37	0.528210	0.264105	+	0.964494i	$0.414923\pi$
$458$	−1164.85	−0.118842
$459$	84.7865	0.00862200
$460$	651.742	0.0660601
$461$	8319.71	0.840537	0.420269	−	0.907400i	$-0.361936\pi$
$461$	8319.71	0.840537	0.420269	+	0.907400i	$0.361936\pi$
$462$	0	0
$463$	15571.7	1.56302	0.781510	−	0.623892i	$-0.214450\pi$
$463$	15571.7	1.56302	0.781510	+	0.623892i	$0.214450\pi$
$464$	−1177.62	−0.117823
$465$	−21.1561	−0.00210988
$466$	−6337.37	−0.629985
$467$	10198.0	1.01050	0.505252	−	0.862972i	$-0.331400\pi$
$467$	10198.0	1.01050	0.505252	+	0.862972i	$0.331400\pi$
$468$	−11650.1	−1.15070
$469$	−16615.6	−1.63590
$470$	4184.83	0.410706
$471$	359.739	0.0351930
$472$	1823.99	0.177873
$473$	0	0
$474$	−372.617	−0.0361073
$475$	15257.8	1.47385
$476$	−3277.62	−0.315608
$477$	−1780.28	−0.170888
$478$	7273.39	0.695978
$479$	−4312.62	−0.411375	−0.205687	−	0.978618i	$-0.565943\pi$
$479$	−4312.62	−0.411375	−0.205687	+	0.978618i	$0.565943\pi$
$480$	−80.3699	−0.00764243
$481$	14406.4	1.36565
$482$	2426.55	0.229308
$483$	−64.9168	−0.00611556
$484$	0	0
$485$	1072.64	0.100425
$486$	−784.831	−0.0732524
$487$	1945.35	0.181011	0.0905053	−	0.995896i	$-0.471152\pi$
$487$	1945.35	0.181011	0.0905053	+	0.995896i	$0.471152\pi$
$488$	117.391	0.0108894
$489$	−87.0051	−0.00804602
$490$	5415.84	0.499312
$491$	20406.8	1.87565	0.937827	−	0.347102i	$-0.112834\pi$
$491$	20406.8	1.87565	0.937827	+	0.347102i	$0.112834\pi$
$492$	−129.646	−0.0118798
$493$	−284.530	−0.0259931
$494$	32035.5	2.91771
$495$	0	0
$496$	−4549.20	−0.411825
$497$	−22619.8	−2.04152
$498$	−274.541	−0.0247038
$499$	11310.4	1.01467	0.507336	−	0.861748i	$-0.330630\pi$
$499$	11310.4	1.01467	0.507336	+	0.861748i	$0.330630\pi$
$500$	5975.93	0.534503
$501$	222.368	0.0198297
$502$	20891.9	1.85747
$503$	−9801.79	−0.868867	−0.434434	−	0.900704i	$-0.643051\pi$
$503$	−9801.79	−0.868867	−0.434434	+	0.900704i	$0.643051\pi$
$504$	−2548.31	−0.225220
$505$	3361.00	0.296163
$506$	0	0
$507$	137.976	0.0120862
$508$	−1997.78	−0.174482
$509$	16863.2	1.46847	0.734233	−	0.678898i	$-0.237542\pi$
$509$	16863.2	1.46847	0.734233	+	0.678898i	$0.237542\pi$
$510$	−21.6190	−0.00187707
$511$	−3504.40	−0.303376
$512$	−15323.6	−1.32268
$513$	676.707	0.0582404
$514$	15797.5	1.35564
$515$	−7313.35	−0.625757
$516$	−35.7996	−0.00305424
$517$	0	0
$518$	−25010.3	−2.12141
$519$	−72.0924	−0.00609731
$520$	−748.680	−0.0631381
$521$	6029.40	0.507011	0.253506	−	0.967334i	$-0.418416\pi$
$521$	6029.40	0.507011	0.253506	+	0.967334i	$0.418416\pi$
$522$	1755.75	0.147217
$523$	−13254.9	−1.10821	−0.554106	−	0.832446i	$-0.686940\pi$
$523$	−13254.9	−1.10821	−0.554106	+	0.832446i	$0.686940\pi$
$524$	−408.041	−0.0340178
$525$	−281.890	−0.0234337
$526$	23354.9	1.93597
$527$	−1099.15	−0.0908534
$528$	0	0
$529$	−11496.4	−0.944880
$530$	908.018	0.0744184
$531$	−14150.8	−1.15648
$532$	−26159.7	−2.13189
$533$	−12000.6	−0.975243
$534$	−184.483	−0.0149501
$535$	6719.50	0.543008
$536$	−2130.25	−0.171665
$537$	−144.329	−0.0115982
$538$	−31080.0	−2.49062
$539$	0	0
$540$	125.518	0.0100027
$541$	13839.5	1.09982	0.549912	−	0.835222i	$-0.314661\pi$
$541$	13839.5	1.09982	0.549912	+	0.835222i	$0.314661\pi$
$542$	−21698.3	−1.71960
$543$	27.5175	0.00217475
$544$	−4175.56	−0.329091
$545$	−2985.16	−0.234624
$546$	−591.860	−0.0463906
$547$	16896.3	1.32072	0.660360	−	0.750949i	$-0.270404\pi$
$547$	16896.3	1.32072	0.660360	+	0.750949i	$0.270404\pi$
$548$	−3284.76	−0.256054
$549$	−910.731	−0.0707997
$550$	0	0
$551$	−2270.92	−0.175580
$552$	−8.32281	−0.000641743 0
$553$	28164.9	2.16581
$554$	−1008.89	−0.0773715
$555$	−77.5947	−0.00593462
$556$	−9771.75	−0.745350
$557$	6741.52	0.512832	0.256416	−	0.966567i	$-0.417458\pi$
$557$	6741.52	0.512832	0.256416	+	0.966567i	$0.417458\pi$
$558$	6782.54	0.514566
$559$	−3313.78	−0.250730
$560$	6763.29	0.510359
$561$	0	0
$562$	−5942.36	−0.446020
$563$	−15869.8	−1.18798	−0.593989	−	0.804473i	$-0.702448\pi$
$563$	−15869.8	−1.18798	−0.593989	+	0.804473i	$0.702448\pi$
$564$	199.504	0.0148947
$565$	5555.42	0.413661
$566$	12674.2	0.941227
$567$	19763.9	1.46385
$568$	−2900.03	−0.214230
$569$	−10677.0	−0.786651	−0.393325	−	0.919399i	$-0.628675\pi$
$569$	−10677.0	−0.786651	−0.393325	+	0.919399i	$0.628675\pi$
$570$	−172.547	−0.0126793
$571$	5406.95	0.396277	0.198138	−	0.980174i	$-0.436510\pi$
$571$	5406.95	0.396277	0.198138	+	0.980174i	$0.436510\pi$
$572$	0	0
$573$	−201.666	−0.0147028
$574$	20833.7	1.51495
$575$	2912.18	0.211211
$576$	10573.2	0.764842
$577$	1779.45	0.128387	0.0641935	−	0.997937i	$-0.479553\pi$
$577$	1779.45	0.128387	0.0641935	+	0.997937i	$0.479553\pi$
$578$	−1123.20	−0.0808284
$579$	117.810	0.00845599
$580$	−421.219	−0.0301554
$581$	20751.7	1.48180
$582$	108.715	0.00774291
$583$	0	0
$584$	−449.289	−0.0318352
$585$	5808.36	0.410506
$586$	4208.64	0.296685
$587$	14326.9	1.00739	0.503693	−	0.863883i	$-0.331974\pi$
$587$	14326.9	1.00739	0.503693	+	0.863883i	$0.331974\pi$
$588$	258.190	0.0181081
$589$	−8772.65	−0.613703
$590$	7217.49	0.503626
$591$	−486.909	−0.0338896
$592$	−16685.2	−1.15837
$593$	17239.6	1.19384	0.596920	−	0.802301i	$-0.296391\pi$
$593$	17239.6	1.19384	0.596920	+	0.802301i	$0.296391\pi$
$594$	0	0
$595$	1634.11	0.112591
$596$	4059.35	0.278989
$597$	323.332	0.0221660
$598$	6114.45	0.418124
$599$	−27264.5	−1.85976	−0.929882	−	0.367858i	$-0.880091\pi$
$599$	−27264.5	−1.85976	−0.929882	+	0.367858i	$0.880091\pi$
$600$	−36.1403	−0.00245904
$601$	23932.1	1.62431	0.812155	−	0.583441i	$-0.198294\pi$
$601$	23932.1	1.62431	0.812155	+	0.583441i	$0.198294\pi$
$602$	5752.90	0.389486
$603$	16526.7	1.11612
$604$	17044.9	1.14826
$605$	0	0
$606$	340.646	0.0228347
$607$	−8230.35	−0.550346	−0.275173	−	0.961395i	$-0.588735\pi$
$607$	−8230.35	−0.550346	−0.275173	+	0.961395i	$0.588735\pi$
$608$	−33326.4	−2.22297
$609$	41.9555	0.00279166
$610$	464.512	0.0308320
$611$	18467.0	1.22274
$612$	3260.08	0.215328
$613$	29025.8	1.91246	0.956232	−	0.292608i	$-0.0945231\pi$
$613$	29025.8	1.91246	0.956232	+	0.292608i	$0.0945231\pi$
$614$	−28586.2	−1.87890
$615$	64.6368	0.00423806
$616$	0	0
$617$	20432.5	1.33320	0.666598	−	0.745418i	$-0.267750\pi$
$617$	20432.5	1.33320	0.666598	+	0.745418i	$0.267750\pi$
$618$	−741.228	−0.0482468
$619$	16427.5	1.06668	0.533341	−	0.845900i	$-0.320936\pi$
$619$	16427.5	1.06668	0.533341	+	0.845900i	$0.320936\pi$
$620$	−1627.18	−0.105402
$621$	129.159	0.00834619
$622$	−37946.0	−2.44614
$623$	13944.5	0.896748
$624$	−394.849	−0.0253311
$625$	11077.2	0.708941
$626$	−25792.0	−1.64674
$627$	0	0
$628$	27668.6	1.75812
$629$	−4031.37	−0.255551
$630$	−10083.6	−0.637683
$631$	−26174.8	−1.65135	−0.825674	−	0.564147i	$-0.809205\pi$
$631$	−26174.8	−1.65135	−0.825674	+	0.564147i	$0.809205\pi$
$632$	3610.95	0.227272
$633$	207.655	0.0130388
$634$	−29293.5	−1.83501
$635$	996.023	0.0622456
$636$	43.2880	0.00269887
$637$	23899.3	1.48654
$638$	0	0
$639$	22498.8	1.39286
$640$	1567.57	0.0968183
$641$	16197.4	0.998062	0.499031	−	0.866584i	$-0.333689\pi$
$641$	16197.4	0.998062	0.499031	+	0.866584i	$0.333689\pi$
$642$	681.039	0.0418668
$643$	−17872.6	−1.09616	−0.548078	−	0.836427i	$-0.684640\pi$
$643$	−17872.6	−1.09616	−0.548078	+	0.836427i	$0.684640\pi$
$644$	−4992.95	−0.305512
$645$	17.8484	0.00108958
$646$	−8964.57	−0.545985
$647$	−22749.9	−1.38237	−0.691184	−	0.722679i	$-0.742910\pi$
$647$	−22749.9	−1.38237	−0.691184	+	0.722679i	$0.742910\pi$
$648$	2533.88	0.153611
$649$	0	0
$650$	26550.9	1.60217
$651$	162.076	0.00975767
$652$	−6691.83	−0.401951
$653$	−654.228	−0.0392066	−0.0196033	−	0.999808i	$-0.506240\pi$
$653$	−654.228	−0.0392066	−0.0196033	+	0.999808i	$0.506240\pi$
$654$	−302.554	−0.0180899
$655$	203.435	0.0121357
$656$	13898.8	0.827223
$657$	3485.64	0.206983
$658$	−32059.7	−1.89942
$659$	−4629.40	−0.273650	−0.136825	−	0.990595i	$-0.543690\pi$
$659$	−4629.40	−0.273650	−0.136825	+	0.990595i	$0.543690\pi$
$660$	0	0
$661$	8457.38	0.497661	0.248830	−	0.968547i	$-0.419954\pi$
$661$	8457.38	0.497661	0.248830	+	0.968547i	$0.419954\pi$
$662$	−15602.3	−0.916014
$663$	−95.4011	−0.00558834
$664$	2660.52	0.155494
$665$	13042.3	0.760540
$666$	24876.4	1.44736
$667$	−433.438	−0.0251616
$668$	17103.0	0.990620
$669$	−356.578	−0.0206070
$670$	−8429.33	−0.486050
$671$	0	0
$672$	615.708	0.0353444
$673$	−3677.31	−0.210624	−0.105312	−	0.994439i	$-0.533584\pi$
$673$	−3677.31	−0.210624	−0.105312	+	0.994439i	$0.533584\pi$
$674$	11548.3	0.659977
$675$	560.851	0.0319810
$676$	10612.1	0.603785
$677$	2759.09	0.156633	0.0783163	−	0.996929i	$-0.475046\pi$
$677$	2759.09	0.156633	0.0783163	+	0.996929i	$0.475046\pi$
$678$	563.057	0.0318939
$679$	−8217.40	−0.464440
$680$	209.505	0.0118149
$681$	439.071	0.0247066
$682$	0	0
$683$	6291.53	0.352472	0.176236	−	0.984348i	$-0.443608\pi$
$683$	6291.53	0.352472	0.176236	+	0.984348i	$0.443608\pi$
$684$	26019.7	1.45451
$685$	1637.67	0.0913460
$686$	−5315.38	−0.295834
$687$	27.6863	0.00153755
$688$	3837.95	0.212675
$689$	4006.94	0.221556
$690$	−32.9332	−0.00181702
$691$	−1086.66	−0.0598243	−0.0299121	−	0.999553i	$-0.509523\pi$
$691$	−1086.66	−0.0598243	−0.0299121	+	0.999553i	$0.509523\pi$
$692$	−5544.85	−0.304600
$693$	0	0
$694$	38663.7	2.11477
$695$	4871.86	0.265900
$696$	5.37901	0.000292946 0
$697$	3358.16	0.182495
$698$	12057.4	0.653837
$699$	150.628	0.00815058
$700$	−21681.0	−1.17066
$701$	−29306.8	−1.57904	−0.789518	−	0.613728i	$-0.789669\pi$
$701$	−29306.8	−1.57904	−0.789518	+	0.613728i	$0.789669\pi$
$702$	1177.57	0.0633114
$703$	−32175.6	−1.72621
$704$	0	0
$705$	−99.4657	−0.00531361
$706$	3202.57	0.170723
$707$	−25748.4	−1.36969
$708$	344.080	0.0182646
$709$	−24677.6	−1.30717	−0.653587	−	0.756851i	$-0.726737\pi$
$709$	−24677.6	−1.30717	−0.653587	+	0.756851i	$0.726737\pi$
$710$	−11475.3	−0.606566
$711$	−28014.2	−1.47766
$712$	1787.79	0.0941013
$713$	−1674.39	−0.0879471
$714$	165.621	0.00868098
$715$	0	0
$716$	−11100.8	−0.579407
$717$	−172.875	−0.00900437
$718$	2361.21	0.122729
$719$	−11580.6	−0.600673	−0.300337	−	0.953833i	$-0.597099\pi$
$719$	−11580.6	−0.600673	−0.300337	+	0.953833i	$0.597099\pi$
$720$	−6727.10	−0.348201
$721$	56027.0	2.89397
$722$	−44891.5	−2.31397
$723$	−57.6747	−0.00296673
$724$	2116.46	0.108643
$725$	−1882.13	−0.0964145
$726$	0	0
$727$	−5861.34	−0.299016	−0.149508	−	0.988760i	$-0.547769\pi$
$727$	−5861.34	−0.299016	−0.149508	+	0.988760i	$0.547769\pi$
$728$	5735.59	0.291999
$729$	−19645.7	−0.998104
$730$	−1777.83	−0.0901375
$731$	927.302	0.0469186
$732$	22.1447	0.00111816
$733$	−7296.78	−0.367685	−0.183842	−	0.982956i	$-0.558854\pi$
$733$	−7296.78	−0.367685	−0.183842	+	0.982956i	$0.558854\pi$
$734$	−29999.9	−1.50861
$735$	−128.724	−0.00645996
$736$	−6360.82	−0.318564
$737$	0	0
$738$	−20722.2	−1.03360
$739$	27276.0	1.35773	0.678867	−	0.734262i	$-0.262471\pi$
$739$	27276.0	1.35773	0.678867	+	0.734262i	$0.262471\pi$
$740$	−5968.05	−0.296473
$741$	−761.425	−0.0377485
$742$	−6956.26	−0.344168
$743$	−11833.4	−0.584288	−0.292144	−	0.956374i	$-0.594369\pi$
$743$	−11833.4	−0.584288	−0.292144	+	0.956374i	$0.594369\pi$
$744$	20.7793	0.00102393
$745$	−2023.85	−0.0995279
$746$	−31187.3	−1.53063
$747$	−20640.7	−1.01098
$748$	0	0
$749$	−51477.6	−2.51128
$750$	−301.969	−0.0147018
$751$	16146.2	0.784532	0.392266	−	0.919852i	$-0.371691\pi$
$751$	16146.2	0.784532	0.392266	+	0.919852i	$0.371691\pi$
$752$	−21388.1	−1.03716
$753$	−496.561	−0.0240315
$754$	−3951.75	−0.190867
$755$	−8498.02	−0.409635
$756$	−961.584	−0.0462599
$757$	−34539.2	−1.65832	−0.829160	−	0.559012i	$-0.811181\pi$
$757$	−34539.2	−1.65832	−0.829160	+	0.559012i	$0.811181\pi$
$758$	55805.6	2.67408
$759$	0	0
$760$	1672.12	0.0798081
$761$	35841.1	1.70728	0.853639	−	0.520866i	$-0.174391\pi$
$761$	35841.1	1.70728	0.853639	+	0.520866i	$0.174391\pi$
$762$	100.950	0.00479924
$763$	22869.1	1.08508
$764$	−15510.7	−0.734500
$765$	−1625.36	−0.0768172
$766$	−30400.8	−1.43398
$767$	31849.7	1.49938
$768$	448.360	0.0210661
$769$	−11467.7	−0.537758	−0.268879	−	0.963174i	$-0.586653\pi$
$769$	−11467.7	−0.537758	−0.268879	+	0.963174i	$0.586653\pi$
$770$	0	0
$771$	−375.477	−0.0175389
$772$	9061.13	0.422432
$773$	−4411.61	−0.205271	−0.102636	−	0.994719i	$-0.532728\pi$
$773$	−4411.61	−0.205271	−0.102636	+	0.994719i	$0.532728\pi$
$774$	−5722.12	−0.265733
$775$	−7270.73	−0.336997
$776$	−1053.53	−0.0487366
$777$	594.447	0.0274462
$778$	36212.0	1.66872
$779$	26802.5	1.23273
$780$	−141.232	−0.00648322
$781$	0	0
$782$	−1711.02	−0.0782428
$783$	−83.4752	−0.00380991
$784$	−27679.6	−1.26091
$785$	−13794.6	−0.627200
$786$	20.6187	0.000935680 0
$787$	−23844.7	−1.08001	−0.540007	−	0.841660i	$-0.681578\pi$
$787$	−23844.7	−1.08001	−0.540007	+	0.841660i	$0.681578\pi$
$788$	−37449.7	−1.69301
$789$	−555.103	−0.0250471
$790$	14288.4	0.643494
$791$	−42559.7	−1.91308
$792$	0	0
$793$	2049.82	0.0917921
$794$	−46303.5	−2.06958
$795$	−21.5819	−0.000962806 0
$796$	24868.4	1.10733
$797$	19939.2	0.886178	0.443089	−	0.896478i	$-0.353883\pi$
$797$	19939.2	0.886178	0.443089	+	0.896478i	$0.353883\pi$
$798$	1321.87	0.0586388
$799$	−5167.66	−0.228809
$800$	−27620.7	−1.22068
$801$	−13869.9	−0.611820
$802$	−26411.8	−1.16288
$803$	0	0
$804$	−401.852	−0.0176271
$805$	2489.31	0.108990
$806$	−15265.7	−0.667137
$807$	738.714	0.0322230
$808$	−3301.13	−0.143729
$809$	37266.3	1.61955	0.809773	−	0.586743i	$-0.199590\pi$
$809$	37266.3	1.61955	0.809773	+	0.586743i	$0.199590\pi$
$810$	10026.5	0.434932
$811$	−27286.7	−1.18146	−0.590731	−	0.806869i	$-0.701161\pi$
$811$	−27286.7	−1.18146	−0.590731	+	0.806869i	$0.701161\pi$
$812$	3226.93	0.139462
$813$	515.729	0.0222477
$814$	0	0
$815$	3336.31	0.143394
$816$	110.491	0.00474016
$817$	7401.08	0.316929
$818$	29034.3	1.24103
$819$	−44497.4	−1.89849
$820$	4971.42	0.211719
$821$	−16982.5	−0.721918	−0.360959	−	0.932582i	$-0.617551\pi$
$821$	−16982.5	−0.721918	−0.360959	+	0.932582i	$0.617551\pi$
$822$	165.982	0.00704293
$823$	20829.2	0.882214	0.441107	−	0.897455i	$-0.354586\pi$
$823$	20829.2	0.882214	0.441107	+	0.897455i	$0.354586\pi$
$824$	7183.08	0.303683
$825$	0	0
$826$	−55292.7	−2.32915
$827$	1906.45	0.0801616	0.0400808	−	0.999196i	$-0.487238\pi$
$827$	1906.45	0.0801616	0.0400808	+	0.999196i	$0.487238\pi$
$828$	4966.23	0.208440
$829$	22740.5	0.952725	0.476362	−	0.879249i	$-0.341955\pi$
$829$	22740.5	0.952725	0.476362	+	0.879249i	$0.341955\pi$
$830$	10527.6	0.440264
$831$	23.9795	0.00100101
$832$	−23797.5	−0.991621
$833$	−6687.78	−0.278173
$834$	493.776	0.0205013
$835$	−8526.96	−0.353398
$836$	0	0
$837$	−322.468	−0.0133167
$838$	−52077.4	−2.14676
$839$	2199.29	0.0904982	0.0452491	−	0.998976i	$-0.485592\pi$
$839$	2199.29	0.0904982	0.0452491	+	0.998976i	$0.485592\pi$
$840$	−30.8926	−0.00126892
$841$	−24108.9	−0.988514
$842$	−60241.4	−2.46562
$843$	141.239	0.00577049
$844$	15971.4	0.651373
$845$	−5290.84	−0.215397
$846$	31888.2	1.29591
$847$	0	0
$848$	−4640.75	−0.187929
$849$	−301.241	−0.0121773
$850$	−7429.79	−0.299811
$851$	−6141.18	−0.247376
$852$	−547.064	−0.0219978
$853$	−12977.4	−0.520914	−0.260457	−	0.965486i	$-0.583873\pi$
$853$	−12977.4	−0.520914	−0.260457	+	0.965486i	$0.583873\pi$
$854$	−3558.59	−0.142591
$855$	−12972.5	−0.518890
$856$	−6599.81	−0.263524
$857$	47924.2	1.91022	0.955111	−	0.296247i	$-0.0957352\pi$
$857$	47924.2	1.91022	0.955111	+	0.296247i	$0.0957352\pi$
$858$	0	0
$859$	−27458.0	−1.09063	−0.545316	−	0.838230i	$-0.683590\pi$
$859$	−27458.0	−1.09063	−0.545316	+	0.838230i	$0.683590\pi$
$860$	1372.78	0.0544318
$861$	−495.178	−0.0196000
$862$	39895.9	1.57640
$863$	−11600.5	−0.457574	−0.228787	−	0.973477i	$-0.573476\pi$
$863$	−11600.5	−0.457574	−0.228787	+	0.973477i	$0.573476\pi$
$864$	−1225.02	−0.0482362
$865$	2764.47	0.108665
$866$	27773.4	1.08981
$867$	26.6963	0.00104574
$868$	12465.7	0.487459
$869$	0	0
$870$	21.2846	0.000829443 0
$871$	−37197.3	−1.44705
$872$	2931.98	0.113864
$873$	8173.43	0.316871
$874$	−13656.1	−0.528519
$875$	22824.9	0.881854
$876$	−84.7544	−0.00326893
$877$	−30437.2	−1.17194	−0.585970	−	0.810332i	$-0.699287\pi$
$877$	−30437.2	−1.17194	−0.585970	+	0.810332i	$0.699287\pi$
$878$	−41962.0	−1.61293
$879$	−100.032	−0.00383843
$880$	0	0
$881$	6135.42	0.234628	0.117314	−	0.993095i	$-0.462572\pi$
$881$	6135.42	0.234628	0.117314	+	0.993095i	$0.462572\pi$
$882$	41268.4	1.57549
$883$	−29999.6	−1.14334	−0.571669	−	0.820484i	$-0.693704\pi$
$883$	−29999.6	−1.14334	−0.571669	+	0.820484i	$0.693704\pi$
$884$	−7337.59	−0.279174
$885$	−171.546	−0.00651578
$886$	−49149.6	−1.86367
$887$	−21669.8	−0.820295	−0.410148	−	0.912019i	$-0.634523\pi$
$887$	−21669.8	−0.820295	−0.410148	+	0.912019i	$0.634523\pi$
$888$	76.2125	0.00288010
$889$	−7630.46	−0.287871
$890$	7074.22	0.266437
$891$	0	0
$892$	−27425.5	−1.02945
$893$	−41244.7	−1.54558
$894$	−205.123	−0.00767376
$895$	5534.46	0.206700
$896$	−12009.0	−0.447761
$897$	−145.329	−0.00540958
$898$	17980.4	0.668165
$899$	1082.15	0.0401466
$900$	21565.0	0.798703
$901$	−1121.27	−0.0414594
$902$	0	0
$903$	−136.736	−0.00503907
$904$	−5456.47	−0.200752
$905$	−1055.19	−0.0387578
$906$	−861.298	−0.0315836
$907$	50578.2	1.85162	0.925811	−	0.377986i	$-0.123383\pi$
$907$	50578.2	1.85162	0.925811	+	0.377986i	$0.123383\pi$
$908$	33770.3	1.23426
$909$	25610.6	0.934489
$910$	22695.6	0.826759
$911$	−47399.5	−1.72384	−0.861918	−	0.507047i	$-0.830737\pi$
$911$	−47399.5	−1.72384	−0.861918	+	0.507047i	$0.830737\pi$
$912$	881.865	0.0320191
$913$	0	0
$914$	−20055.7	−0.725804
$915$	−11.0406	−0.000398896 0
$916$	2129.44	0.0768107
$917$	−1558.50	−0.0561246
$918$	−329.522	−0.0118473
$919$	−13702.7	−0.491852	−0.245926	−	0.969289i	$-0.579092\pi$
$919$	−13702.7	−0.491852	−0.245926	+	0.969289i	$0.579092\pi$
$920$	319.148	0.0114370
$921$	679.442	0.0243088
$922$	−32334.5	−1.15497
$923$	−50638.9	−1.80585
$924$	0	0
$925$	−26667.0	−0.947897
$926$	−60519.3	−2.14772
$927$	−55727.2	−1.97446
$928$	4110.98	0.145420
$929$	13908.1	0.491184	0.245592	−	0.969373i	$-0.421018\pi$
$929$	13908.1	0.491184	0.245592	+	0.969373i	$0.421018\pi$
$930$	82.2231	0.00289914
$931$	−53377.2	−1.87902
$932$	11585.2	0.407174
$933$	901.906	0.0316475
$934$	−39634.3	−1.38851
$935$	0	0
$936$	−5704.90	−0.199220
$937$	22326.6	0.778420	0.389210	−	0.921149i	$-0.372748\pi$
$937$	22326.6	0.778420	0.389210	+	0.921149i	$0.372748\pi$
$938$	64576.5	2.24786
$939$	613.028	0.0213050
$940$	−7650.21	−0.265449
$941$	−9071.95	−0.314280	−0.157140	−	0.987576i	$-0.550227\pi$
$941$	−9071.95	−0.314280	−0.157140	+	0.987576i	$0.550227\pi$
$942$	−1398.12	−0.0483581
$943$	5115.64	0.176658
$944$	−36887.6	−1.27181
$945$	479.413	0.0165030
$946$	0	0
$947$	36042.0	1.23675	0.618377	−	0.785882i	$-0.287790\pi$
$947$	36042.0	1.23675	0.618377	+	0.785882i	$0.287790\pi$
$948$	681.174	0.0233370
$949$	−7845.28	−0.268355
$950$	−59299.4	−2.02519
$951$	696.252	0.0237408
$952$	−1605.00	−0.0546411
$953$	14367.5	0.488361	0.244180	−	0.969730i	$-0.421481\pi$
$953$	14367.5	0.488361	0.244180	+	0.969730i	$0.421481\pi$
$954$	6919.04	0.234814
$955$	7733.11	0.262029
$956$	−13296.3	−0.449827
$957$	0	0
$958$	16760.9	0.565263
$959$	−12546.0	−0.422454
$960$	128.176	0.00430924
$961$	−25610.6	−0.859676
$962$	−55990.4	−1.87651
$963$	51202.1	1.71336
$964$	−4435.94	−0.148207
$965$	−4517.57	−0.150700
$966$	252.299	0.00840328
$967$	28043.1	0.932579	0.466290	−	0.884632i	$-0.345590\pi$
$967$	28043.1	0.932579	0.466290	+	0.884632i	$0.345590\pi$
$968$	0	0
$969$	213.071	0.00706381
$970$	−4168.80	−0.137992
$971$	51541.9	1.70346	0.851729	−	0.523982i	$-0.175554\pi$
$971$	51541.9	1.70346	0.851729	+	0.523982i	$0.175554\pi$
$972$	1434.73	0.0473448
$973$	−37323.0	−1.22972
$974$	−7560.58	−0.248723
$975$	−631.065	−0.0207285
$976$	−2374.05	−0.0778602
$977$	−11802.0	−0.386469	−0.193235	−	0.981153i	$-0.561898\pi$
$977$	−11802.0	−0.386469	−0.193235	+	0.981153i	$0.561898\pi$
$978$	338.144	0.0110559
$979$	0	0
$980$	−9900.59	−0.322717
$981$	−22746.7	−0.740313
$982$	−79310.9	−2.57730
$983$	−14482.2	−0.469899	−0.234949	−	0.972008i	$-0.575492\pi$
$983$	−14482.2	−0.469899	−0.234949	+	0.972008i	$0.575492\pi$
$984$	−63.4855	−0.00205675
$985$	18671.1	0.603971
$986$	1105.82	0.0357167
$987$	761.999	0.0245742
$988$	−58563.5	−1.88578
$989$	1412.60	0.0454177
$990$	0	0
$991$	6229.88	0.199696	0.0998479	−	0.995003i	$-0.468164\pi$
$991$	6229.88	0.199696	0.0998479	+	0.995003i	$0.468164\pi$
$992$	15880.9	0.508284
$993$	370.838	0.0118511
$994$	87911.8	2.80522
$995$	−12398.5	−0.395035
$996$	501.884	0.0159667
$997$	−55005.4	−1.74728	−0.873639	−	0.486574i	$-0.838246\pi$
$997$	−55005.4	−1.74728	−0.873639	+	0.486574i	$0.838246\pi$
$998$	−43957.6	−1.39424
$999$	−1182.72	−0.0374571

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2057.4.a.v.1.10		52
11.3	even	5		187.4.g.b.86.6	✓	104
11.4	even	5		187.4.g.b.137.6	yes	104
11.10	odd	2		2057.4.a.u.1.43		52

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
187.4.g.b.86.6	✓	104	11.3	even	5
187.4.g.b.137.6	yes	104	11.4	even	5
2057.4.a.u.1.43		52	11.10	odd	2
2057.4.a.v.1.10		52	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2057 = 11^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2057.a (trivial)

\(f(q)\)	\(=\)	\(q-3.88649 q^{2} +0.0923746 q^{3} +7.10482 q^{4} -3.54222 q^{5} -0.359013 q^{6} +27.1367 q^{7} +3.47912 q^{8} -26.9915 q^{9} +13.7668 q^{10} +0.656305 q^{12} +60.7507 q^{13} -105.466 q^{14} -0.327211 q^{15} -70.3601 q^{16} -17.0000 q^{17} +104.902 q^{18} -135.682 q^{19} -25.1668 q^{20} +2.50674 q^{21} -25.8969 q^{23} +0.321383 q^{24} -112.453 q^{25} -236.107 q^{26} -4.98744 q^{27} +192.801 q^{28} +16.7371 q^{29} +1.27170 q^{30} +64.6559 q^{31} +245.621 q^{32} +66.0704 q^{34} -96.1240 q^{35} -191.769 q^{36} +237.140 q^{37} +527.328 q^{38} +5.61183 q^{39} -12.3238 q^{40} -197.539 q^{41} -9.74242 q^{42} -54.5472 q^{43} +95.6096 q^{45} +100.648 q^{46} +303.980 q^{47} -6.49949 q^{48} +393.399 q^{49} +437.046 q^{50} -1.57037 q^{51} +431.623 q^{52} +65.9571 q^{53} +19.3837 q^{54} +94.4118 q^{56} -12.5336 q^{57} -65.0485 q^{58} +524.268 q^{59} -2.32477 q^{60} +33.7414 q^{61} -251.285 q^{62} -732.459 q^{63} -391.723 q^{64} -215.192 q^{65} -612.294 q^{67} -120.782 q^{68} -2.39222 q^{69} +373.585 q^{70} -833.552 q^{71} -93.9066 q^{72} -129.139 q^{73} -921.641 q^{74} -10.3878 q^{75} -963.997 q^{76} -21.8103 q^{78} +1037.89 q^{79} +249.231 q^{80} +728.309 q^{81} +767.732 q^{82} +764.711 q^{83} +17.8099 q^{84} +60.2177 q^{85} +211.997 q^{86} +1.54608 q^{87} +513.861 q^{89} -371.586 q^{90} +1648.57 q^{91} -183.993 q^{92} +5.97257 q^{93} -1181.42 q^{94} +480.616 q^{95} +22.6892 q^{96} -302.815 q^{97} -1528.94 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 52 q + q^{2} + 20 q^{3} + 235 q^{4} + 40 q^{5} - 24 q^{6} - 42 q^{7} + 45 q^{8} + 572 q^{9} + 33 q^{10} + 233 q^{12} + 12 q^{13} + 73 q^{14} + 400 q^{15} + 1223 q^{16} - 884 q^{17} + 201 q^{18} - 44 q^{19}+ \cdots - 4610 q^{98}+O(q^{100}) \) 52 * q + q^2 + 20 * q^3 + 235 * q^4 + 40 * q^5 - 24 * q^6 - 42 * q^7 + 45 * q^8 + 572 * q^9 + 33 * q^10 + 233 * q^12 + 12 * q^13 + 73 * q^14 + 400 * q^15 + 1223 * q^16 - 884 * q^17 + 201 * q^18 - 44 * q^19 + 655 * q^20 + 260 * q^21 + 572 * q^23 - 104 * q^24 + 1858 * q^25 + 465 * q^26 + 1070 * q^27 - 577 * q^28 + 322 * q^29 - 320 * q^30 + 1110 * q^31 + 481 * q^32 - 17 * q^34 + 102 * q^35 + 2507 * q^36 + 1678 * q^37 - 360 * q^38 - 1282 * q^39 + 1791 * q^40 - 826 * q^41 + 2133 * q^42 + 270 * q^43 + 710 * q^45 + 2158 * q^46 + 2464 * q^47 + 2201 * q^48 + 3224 * q^49 + 2379 * q^50 - 340 * q^51 - 3664 * q^52 + 992 * q^53 + 1202 * q^54 + 1731 * q^56 + 1016 * q^57 + 1358 * q^58 + 1442 * q^59 + 1444 * q^60 - 140 * q^61 + 464 * q^62 - 766 * q^63 + 8427 * q^64 - 1268 * q^65 + 5766 * q^67 - 3995 * q^68 + 2460 * q^69 + 2422 * q^70 + 2704 * q^71 + 5455 * q^72 + 4 * q^73 - 4008 * q^74 + 5204 * q^75 - 1935 * q^76 + 4092 * q^78 - 2180 * q^79 + 5040 * q^80 + 7192 * q^81 + 3197 * q^82 + 4200 * q^83 + 7951 * q^84 - 680 * q^85 + 3091 * q^86 - 752 * q^87 - 240 * q^89 - 4495 * q^90 + 5494 * q^91 + 6902 * q^92 + 6266 * q^93 + 5990 * q^94 + 3168 * q^95 - 9467 * q^96 + 5322 * q^97 - 4610 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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