LMFDB - Embedded newform 531.4.a.e.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [531,4,Mod(1,531)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(531, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("531.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 531 = 3^{2} \cdot 59 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	531.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:	$31.3300142130$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} - 45x^{6} + 47x^{5} + 654x^{4} - 157x^{3} - 2898x^{2} + 96x + 2432 $ x^8 - 2x^7 - 45x^6 + 47x^5 + 654x^4 - 157x^3 - 2898x^2 + 96*x + 2432
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 177)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$4.77847$ of defining polynomial
Character	$\chi$	$=$	531.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.77847 q^{2} +6.27681 q^{4} +5.74283 q^{5} -24.4226 q^{7} -6.51103 q^{8} +O(q^{10})$	\(q+3.77847 q^{2} +6.27681 q^{4} +5.74283 q^{5} -24.4226 q^{7} -6.51103 q^{8} +21.6991 q^{10} -9.31943 q^{11} +17.0721 q^{13} -92.2799 q^{14} -74.8162 q^{16} -19.6787 q^{17} -15.1270 q^{19} +36.0466 q^{20} -35.2132 q^{22} -108.469 q^{23} -92.0199 q^{25} +64.5065 q^{26} -153.296 q^{28} -237.137 q^{29} +184.628 q^{31} -230.602 q^{32} -74.3553 q^{34} -140.255 q^{35} +155.472 q^{37} -57.1569 q^{38} -37.3917 q^{40} -261.745 q^{41} +127.647 q^{43} -58.4963 q^{44} -409.848 q^{46} -555.816 q^{47} +253.463 q^{49} -347.694 q^{50} +107.158 q^{52} +220.668 q^{53} -53.5199 q^{55} +159.016 q^{56} -896.014 q^{58} -59.0000 q^{59} -436.694 q^{61} +697.610 q^{62} -272.793 q^{64} +98.0424 q^{65} +924.006 q^{67} -123.519 q^{68} -529.948 q^{70} +937.019 q^{71} +13.0348 q^{73} +587.446 q^{74} -94.9493 q^{76} +227.605 q^{77} +140.714 q^{79} -429.657 q^{80} -988.996 q^{82} +26.4265 q^{83} -113.011 q^{85} +482.310 q^{86} +60.6791 q^{88} +136.748 q^{89} -416.946 q^{91} -680.842 q^{92} -2100.13 q^{94} -86.8719 q^{95} -360.785 q^{97} +957.701 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 6 q^{2} + 34 q^{4} - 42 q^{5} + 53 q^{7} - 51 q^{8}+O(q^{10}) $ 8 * q - 6 * q^2 + 34 * q^4 - 42 * q^5 + 53 * q^7 - 51 * q^8	$ 8 q - 6 q^{2} + 34 q^{4} - 42 q^{5} + 53 q^{7} - 51 q^{8} + 21 q^{10} - 67 q^{11} + 33 q^{13} - 79 q^{14} - 30 q^{16} - 139 q^{17} + 64 q^{19} - 117 q^{20} - 84 q^{22} - 226 q^{23} + 96 q^{25} - 24 q^{26} + 34 q^{28} - 456 q^{29} + 124 q^{31} - 174 q^{32} - 114 q^{34} - 556 q^{35} + 127 q^{37} - 237 q^{38} - 188 q^{40} - 425 q^{41} - 115 q^{43} - 510 q^{44} - 711 q^{46} - 420 q^{47} + 171 q^{49} + 137 q^{50} - 922 q^{52} - 98 q^{53} - 616 q^{55} + 412 q^{56} - 1548 q^{58} - 472 q^{59} - 1254 q^{61} + 766 q^{62} - 2019 q^{64} + 734 q^{65} - 1010 q^{67} + 503 q^{68} - 2956 q^{70} + 17 q^{71} - 1180 q^{73} + 1228 q^{74} - 2008 q^{76} - 441 q^{77} - 873 q^{79} + 865 q^{80} - 3645 q^{82} - 759 q^{83} - 850 q^{85} + 1226 q^{86} - 3047 q^{88} - 988 q^{89} - 2111 q^{91} + 1062 q^{92} - 2240 q^{94} - 1822 q^{95} - 668 q^{97} + 1368 q^{98}+O(q^{100}) $ 8 * q - 6 * q^2 + 34 * q^4 - 42 * q^5 + 53 * q^7 - 51 * q^8 + 21 * q^10 - 67 * q^11 + 33 * q^13 - 79 * q^14 - 30 * q^16 - 139 * q^17 + 64 * q^19 - 117 * q^20 - 84 * q^22 - 226 * q^23 + 96 * q^25 - 24 * q^26 + 34 * q^28 - 456 * q^29 + 124 * q^31 - 174 * q^32 - 114 * q^34 - 556 * q^35 + 127 * q^37 - 237 * q^38 - 188 * q^40 - 425 * q^41 - 115 * q^43 - 510 * q^44 - 711 * q^46 - 420 * q^47 + 171 * q^49 + 137 * q^50 - 922 * q^52 - 98 * q^53 - 616 * q^55 + 412 * q^56 - 1548 * q^58 - 472 * q^59 - 1254 * q^61 + 766 * q^62 - 2019 * q^64 + 734 * q^65 - 1010 * q^67 + 503 * q^68 - 2956 * q^70 + 17 * q^71 - 1180 * q^73 + 1228 * q^74 - 2008 * q^76 - 441 * q^77 - 873 * q^79 + 865 * q^80 - 3645 * q^82 - 759 * q^83 - 850 * q^85 + 1226 * q^86 - 3047 * q^88 - 988 * q^89 - 2111 * q^91 + 1062 * q^92 - 2240 * q^94 - 1822 * q^95 - 668 * q^97 + 1368 * 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 $

$2$	3.77847	1.33589	0.667945	−	0.744211i	$-0.267174\pi$
$2$	3.77847	1.33589	0.667945	+	0.744211i	$0.267174\pi$
$3$	0	0
$3$	0	0
$4$	6.27681	0.784601
$5$	5.74283	0.513655	0.256827	−	0.966457i	$-0.417323\pi$
$5$	5.74283	0.513655	0.256827	+	0.966457i	$0.417323\pi$
$6$	0	0
$7$	−24.4226	−1.31870	−0.659348	−	0.751838i	$-0.729168\pi$
$7$	−24.4226	−1.31870	−0.659348	+	0.751838i	$0.729168\pi$
$8$	−6.51103	−0.287749
$9$	0	0
$10$	21.6991	0.686186
$11$	−9.31943	−0.255447	−0.127723	−	0.991810i	$-0.540767\pi$
$11$	−9.31943	−0.255447	−0.127723	+	0.991810i	$0.540767\pi$
$12$	0	0
$13$	17.0721	0.364227	0.182114	−	0.983277i	$-0.441706\pi$
$13$	17.0721	0.364227	0.182114	+	0.983277i	$0.441706\pi$
$14$	−92.2799	−1.76163
$15$	0	0
$16$	−74.8162	−1.16900
$17$	−19.6787	−0.280752	−0.140376	−	0.990098i	$-0.544831\pi$
$17$	−19.6787	−0.280752	−0.140376	+	0.990098i	$0.544831\pi$
$18$	0	0
$19$	−15.1270	−0.182651	−0.0913256	−	0.995821i	$-0.529110\pi$
$19$	−15.1270	−0.182651	−0.0913256	+	0.995821i	$0.529110\pi$
$20$	36.0466	0.403014
$21$	0	0
$22$	−35.2132	−0.341249
$23$	−108.469	−0.983367	−0.491683	−	0.870774i	$-0.663618\pi$
$23$	−108.469	−0.983367	−0.491683	+	0.870774i	$0.663618\pi$
$24$	0	0
$25$	−92.0199	−0.736159
$26$	64.5065	0.486568
$27$	0	0
$28$	−153.296	−1.03465
$29$	−237.137	−1.51846	−0.759228	−	0.650825i	$-0.774423\pi$
$29$	−237.137	−1.51846	−0.759228	+	0.650825i	$0.774423\pi$
$30$	0	0
$31$	184.628	1.06968	0.534841	−	0.844953i	$-0.320372\pi$
$31$	184.628	1.06968	0.534841	+	0.844953i	$0.320372\pi$
$32$	−230.602	−1.27391
$33$	0	0
$34$	−74.3553	−0.375054
$35$	−140.255	−0.677354
$36$	0	0
$37$	155.472	0.690795	0.345398	−	0.938456i	$-0.387744\pi$
$37$	155.472	0.690795	0.345398	+	0.938456i	$0.387744\pi$
$38$	−57.1569	−0.244002
$39$	0	0
$40$	−37.3917	−0.147804
$41$	−261.745	−0.997019	−0.498509	−	0.866884i	$-0.666119\pi$
$41$	−261.745	−0.997019	−0.498509	+	0.866884i	$0.666119\pi$
$42$	0	0
$43$	127.647	0.452697	0.226349	−	0.974046i	$-0.427321\pi$
$43$	127.647	0.452697	0.226349	+	0.974046i	$0.427321\pi$
$44$	−58.4963	−0.200424
$45$	0	0
$46$	−409.848	−1.31367
$47$	−555.816	−1.72498	−0.862490	−	0.506075i	$-0.831096\pi$
$47$	−555.816	−1.72498	−0.862490	+	0.506075i	$0.831096\pi$
$48$	0	0
$49$	253.463	0.738959
$50$	−347.694	−0.983427
$51$	0	0
$52$	107.158	0.285773
$53$	220.668	0.571908	0.285954	−	0.958243i	$-0.407689\pi$
$53$	220.668	0.571908	0.285954	+	0.958243i	$0.407689\pi$
$54$	0	0
$55$	−53.5199	−0.131211
$56$	159.016	0.379454
$57$	0	0
$58$	−896.014	−2.02849
$59$	−59.0000	−0.130189
$59$	−59.0000	−0.130189
$60$	0	0
$61$	−436.694	−0.916606	−0.458303	−	0.888796i	$-0.651543\pi$
$61$	−436.694	−0.916606	−0.458303	+	0.888796i	$0.651543\pi$
$62$	697.610	1.42898
$63$	0	0
$64$	−272.793	−0.532799
$65$	98.0424	0.187087
$66$	0	0
$67$	924.006	1.68486	0.842428	−	0.538809i	$-0.181126\pi$
$67$	924.006	1.68486	0.842428	+	0.538809i	$0.181126\pi$
$68$	−123.519	−0.220278
$69$	0	0
$70$	−529.948	−0.904870
$71$	937.019	1.56625	0.783125	−	0.621865i	$-0.213625\pi$
$71$	937.019	1.56625	0.783125	+	0.621865i	$0.213625\pi$
$72$	0	0
$73$	13.0348	0.0208988	0.0104494	−	0.999945i	$-0.496674\pi$
$73$	13.0348	0.0208988	0.0104494	+	0.999945i	$0.496674\pi$
$74$	587.446	0.922826
$75$	0	0
$76$	−94.9493	−0.143308
$77$	227.605	0.336857
$78$	0	0
$79$	140.714	0.200400	0.100200	−	0.994967i	$-0.468052\pi$
$79$	140.714	0.200400	0.100200	+	0.994967i	$0.468052\pi$
$80$	−429.657	−0.600463
$81$	0	0
$82$	−988.996	−1.33191
$83$	26.4265	0.0349480	0.0174740	−	0.999847i	$-0.494438\pi$
$83$	26.4265	0.0349480	0.0174740	+	0.999847i	$0.494438\pi$
$84$	0	0
$85$	−113.011	−0.144210
$86$	482.310	0.604753
$87$	0	0
$88$	60.6791	0.0735047
$89$	136.748	0.162869	0.0814343	−	0.996679i	$-0.474050\pi$
$89$	136.748	0.162869	0.0814343	+	0.996679i	$0.474050\pi$
$90$	0	0
$91$	−416.946	−0.480305
$92$	−680.842	−0.771551
$93$	0	0
$94$	−2100.13	−2.30438
$95$	−86.8719	−0.0938196
$96$	0	0
$97$	−360.785	−0.377651	−0.188826	−	0.982011i	$-0.560468\pi$
$97$	−360.785	−0.377651	−0.188826	+	0.982011i	$0.560468\pi$
$98$	957.701	0.987168
$99$	0	0
$100$	−577.591	−0.577591
$101$	−421.340	−0.415098	−0.207549	−	0.978225i	$-0.566549\pi$
$101$	−421.340	−0.415098	−0.207549	+	0.978225i	$0.566549\pi$
$102$	0	0
$103$	−1389.28	−1.32903	−0.664515	−	0.747275i	$-0.731362\pi$
$103$	−1389.28	−1.32903	−0.664515	+	0.747275i	$0.731362\pi$
$104$	−111.157	−0.104806
$105$	0	0
$106$	833.788	0.764006
$107$	339.530	0.306763	0.153382	−	0.988167i	$-0.450984\pi$
$107$	339.530	0.306763	0.153382	+	0.988167i	$0.450984\pi$
$108$	0	0
$109$	297.022	0.261005	0.130503	−	0.991448i	$-0.458341\pi$
$109$	297.022	0.261005	0.130503	+	0.991448i	$0.458341\pi$
$110$	−202.223	−0.175284
$111$	0	0
$112$	1827.20	1.54156
$113$	−1553.81	−1.29354	−0.646769	−	0.762686i	$-0.723880\pi$
$113$	−1553.81	−1.29354	−0.646769	+	0.762686i	$0.723880\pi$
$114$	0	0
$115$	−622.922	−0.505111
$116$	−1488.46	−1.19138
$117$	0	0
$118$	−222.930	−0.173918
$119$	480.605	0.370227
$120$	0	0
$121$	−1244.15	−0.934747
$122$	−1650.03	−1.22448
$123$	0	0
$124$	1158.87	0.839273
$125$	−1246.31	−0.891786
$126$	0	0
$127$	1320.50	0.922642	0.461321	−	0.887233i	$-0.347376\pi$
$127$	1320.50	0.922642	0.461321	+	0.887233i	$0.347376\pi$
$128$	814.078	0.562148
$129$	0	0
$130$	370.450	0.249928
$131$	1738.41	1.15943	0.579716	−	0.814819i	$-0.303164\pi$
$131$	1738.41	1.15943	0.579716	+	0.814819i	$0.303164\pi$
$132$	0	0
$133$	369.441	0.240861
$134$	3491.33	2.25078
$135$	0	0
$136$	128.129	0.0807863
$137$	776.721	0.484378	0.242189	−	0.970229i	$-0.422135\pi$
$137$	776.721	0.484378	0.242189	+	0.970229i	$0.422135\pi$
$138$	0	0
$139$	855.546	0.522061	0.261030	−	0.965331i	$-0.415938\pi$
$139$	855.546	0.522061	0.261030	+	0.965331i	$0.415938\pi$
$140$	−880.353	−0.531453
$141$	0	0
$142$	3540.49	2.09234
$143$	−159.103	−0.0930407
$144$	0	0
$145$	−1361.84	−0.779962
$146$	49.2517	0.0279185
$147$	0	0
$148$	975.867	0.541999
$149$	−2001.36	−1.10039	−0.550194	−	0.835037i	$-0.685446\pi$
$149$	−2001.36	−1.10039	−0.550194	+	0.835037i	$0.685446\pi$
$150$	0	0
$151$	1656.52	0.892750	0.446375	−	0.894846i	$-0.352715\pi$
$151$	1656.52	0.892750	0.446375	+	0.894846i	$0.352715\pi$
$152$	98.4924	0.0525578
$153$	0	0
$154$	859.996	0.450003
$155$	1060.29	0.549447
$156$	0	0
$157$	649.443	0.330135	0.165068	−	0.986282i	$-0.447216\pi$
$157$	649.443	0.330135	0.165068	+	0.986282i	$0.447216\pi$
$158$	531.685	0.267713
$159$	0	0
$160$	−1324.31	−0.654349
$161$	2649.10	1.29676
$162$	0	0
$163$	508.548	0.244372	0.122186	−	0.992507i	$-0.461010\pi$
$163$	508.548	0.244372	0.122186	+	0.992507i	$0.461010\pi$
$164$	−1642.93	−0.782262
$165$	0	0
$166$	99.8517	0.0466867
$167$	−1419.46	−0.657733	−0.328867	−	0.944376i	$-0.606667\pi$
$167$	−1419.46	−0.657733	−0.328867	+	0.944376i	$0.606667\pi$
$168$	0	0
$169$	−1905.54	−0.867338
$170$	−427.010	−0.192648
$171$	0	0
$172$	801.215	0.355187
$173$	929.169	0.408343	0.204172	−	0.978935i	$-0.434550\pi$
$173$	929.169	0.408343	0.204172	+	0.978935i	$0.434550\pi$
$174$	0	0
$175$	2247.36	0.970770
$176$	697.244	0.298618
$177$	0	0
$178$	516.699	0.217574
$179$	2212.72	0.923948	0.461974	−	0.886893i	$-0.347141\pi$
$179$	2212.72	0.923948	0.461974	+	0.886893i	$0.347141\pi$
$180$	0	0
$181$	−46.5937	−0.0191342	−0.00956709	−	0.999954i	$-0.503045\pi$
$181$	−46.5937	−0.0191342	−0.00956709	+	0.999954i	$0.503045\pi$
$182$	−1575.41	−0.641635
$183$	0	0
$184$	706.247	0.282963
$185$	892.849	0.354830
$186$	0	0
$187$	183.394	0.0717172
$188$	−3488.75	−1.35342
$189$	0	0
$190$	−328.242	−0.125333
$191$	2035.98	0.771302	0.385651	−	0.922645i	$-0.373977\pi$
$191$	2035.98	0.771302	0.385651	+	0.922645i	$0.373977\pi$
$192$	0	0
$193$	−1530.18	−0.570697	−0.285349	−	0.958424i	$-0.592109\pi$
$193$	−1530.18	−0.570697	−0.285349	+	0.958424i	$0.592109\pi$
$194$	−1363.21	−0.504500
$195$	0	0
$196$	1590.94	0.579788
$197$	−2461.81	−0.890337	−0.445169	−	0.895447i	$-0.646856\pi$
$197$	−2461.81	−0.890337	−0.445169	+	0.895447i	$0.646856\pi$
$198$	0	0
$199$	1442.27	0.513768	0.256884	−	0.966442i	$-0.417304\pi$
$199$	1442.27	0.513768	0.256884	+	0.966442i	$0.417304\pi$
$200$	599.144	0.211829
$201$	0	0
$202$	−1592.02	−0.554525
$203$	5791.50	2.00238
$204$	0	0
$205$	−1503.16	−0.512123
$206$	−5249.36	−1.77544
$207$	0	0
$208$	−1277.27	−0.425783
$209$	140.975	0.0466577
$210$	0	0
$211$	−732.346	−0.238942	−0.119471	−	0.992838i	$-0.538120\pi$
$211$	−732.346	−0.238942	−0.119471	+	0.992838i	$0.538120\pi$
$212$	1385.09	0.448720
$213$	0	0
$214$	1282.90	0.409802
$215$	733.055	0.232530
$216$	0	0
$217$	−4509.09	−1.41058
$218$	1122.29	0.348674
$219$	0	0
$220$	−335.934	−0.102949
$221$	−335.957	−0.102258
$222$	0	0
$223$	2928.26	0.879332	0.439666	−	0.898161i	$-0.355097\pi$
$223$	2928.26	0.879332	0.439666	+	0.898161i	$0.355097\pi$
$224$	5631.90	1.67990
$225$	0	0
$226$	−5871.00	−1.72802
$227$	−3909.72	−1.14316	−0.571580	−	0.820546i	$-0.693669\pi$
$227$	−3909.72	−1.14316	−0.571580	+	0.820546i	$0.693669\pi$
$228$	0	0
$229$	5580.68	1.61040	0.805200	−	0.593003i	$-0.202058\pi$
$229$	5580.68	1.61040	0.805200	+	0.593003i	$0.202058\pi$
$230$	−2353.69	−0.674772
$231$	0	0
$232$	1544.01	0.436935
$233$	3793.50	1.06661	0.533306	−	0.845923i	$-0.320950\pi$
$233$	3793.50	1.06661	0.533306	+	0.845923i	$0.320950\pi$
$234$	0	0
$235$	−3191.96	−0.886043
$236$	−370.332	−0.102146
$237$	0	0
$238$	1815.95	0.494582
$239$	−4995.54	−1.35203	−0.676013	−	0.736889i	$-0.736294\pi$
$239$	−4995.54	−1.35203	−0.676013	+	0.736889i	$0.736294\pi$
$240$	0	0
$241$	1674.51	0.447572	0.223786	−	0.974638i	$-0.428158\pi$
$241$	1674.51	0.447572	0.223786	+	0.974638i	$0.428158\pi$
$242$	−4700.97	−1.24872
$243$	0	0
$244$	−2741.05	−0.719170
$245$	1455.60	0.379570
$246$	0	0
$247$	−258.250	−0.0665266
$248$	−1202.12	−0.307800
$249$	0	0
$250$	−4709.14	−1.19133
$251$	−4489.35	−1.12895	−0.564473	−	0.825452i	$-0.690920\pi$
$251$	−4489.35	−1.12895	−0.564473	+	0.825452i	$0.690920\pi$
$252$	0	0
$253$	1010.87	0.251198
$254$	4989.47	1.23255
$255$	0	0
$256$	5258.31	1.28377
$257$	−5099.27	−1.23768	−0.618840	−	0.785517i	$-0.712397\pi$
$257$	−5099.27	−1.23768	−0.618840	+	0.785517i	$0.712397\pi$
$258$	0	0
$259$	−3797.03	−0.910949
$260$	615.393	0.146789
$261$	0	0
$262$	6568.52	1.54887
$263$	−7949.30	−1.86378	−0.931892	−	0.362737i	$-0.881842\pi$
$263$	−7949.30	−1.86378	−0.931892	+	0.362737i	$0.881842\pi$
$264$	0	0
$265$	1267.26	0.293763
$266$	1395.92	0.321764
$267$	0	0
$268$	5799.81	1.32194
$269$	3831.43	0.868426	0.434213	−	0.900810i	$-0.357027\pi$
$269$	3831.43	0.868426	0.434213	+	0.900810i	$0.357027\pi$
$270$	0	0
$271$	−6824.86	−1.52982	−0.764909	−	0.644139i	$-0.777216\pi$
$271$	−6824.86	−1.52982	−0.764909	+	0.644139i	$0.777216\pi$
$272$	1472.28	0.328200
$273$	0	0
$274$	2934.82	0.647075
$275$	857.573	0.188049
$276$	0	0
$277$	−40.1056	−0.00869931	−0.00434966	−	0.999991i	$-0.501385\pi$
$277$	−40.1056	−0.00869931	−0.00434966	+	0.999991i	$0.501385\pi$
$278$	3232.65	0.697416
$279$	0	0
$280$	913.203	0.194908
$281$	2405.83	0.510746	0.255373	−	0.966843i	$-0.417802\pi$
$281$	2405.83	0.510746	0.255373	+	0.966843i	$0.417802\pi$
$282$	0	0
$283$	8232.15	1.72915	0.864577	−	0.502501i	$-0.167587\pi$
$283$	8232.15	1.72915	0.864577	+	0.502501i	$0.167587\pi$
$284$	5881.49	1.22888
$285$	0	0
$286$	−601.164	−0.124292
$287$	6392.50	1.31476
$288$	0	0
$289$	−4525.75	−0.921178
$290$	−5145.66	−1.04194
$291$	0	0
$292$	81.8171	0.0163972
$293$	−1902.22	−0.379280	−0.189640	−	0.981854i	$-0.560732\pi$
$293$	−1902.22	−0.379280	−0.189640	+	0.981854i	$0.560732\pi$
$294$	0	0
$295$	−338.827	−0.0668721
$296$	−1012.28	−0.198776
$297$	0	0
$298$	−7562.07	−1.47000
$299$	−1851.80	−0.358169
$300$	0	0
$301$	−3117.47	−0.596970
$302$	6259.09	1.19262
$303$	0	0
$304$	1131.74	0.213520
$305$	−2507.86	−0.470819
$306$	0	0
$307$	9226.20	1.71520	0.857601	−	0.514315i	$-0.171954\pi$
$307$	9226.20	1.71520	0.857601	+	0.514315i	$0.171954\pi$
$308$	1428.63	0.264298
$309$	0	0
$310$	4006.26	0.734000
$311$	2395.89	0.436844	0.218422	−	0.975854i	$-0.429909\pi$
$311$	2395.89	0.436844	0.218422	+	0.975854i	$0.429909\pi$
$312$	0	0
$313$	−10071.9	−1.81884	−0.909421	−	0.415878i	$-0.863474\pi$
$313$	−10071.9	−1.81884	−0.909421	+	0.415878i	$0.863474\pi$
$314$	2453.90	0.441024
$315$	0	0
$316$	883.238	0.157234
$317$	−1068.39	−0.189296	−0.0946478	−	0.995511i	$-0.530172\pi$
$317$	−1068.39	−0.189296	−0.0946478	+	0.995511i	$0.530172\pi$
$318$	0	0
$319$	2209.98	0.387885
$320$	−1566.60	−0.273674
$321$	0	0
$322$	10009.6	1.73233
$323$	297.680	0.0512797
$324$	0	0
$325$	−1570.98	−0.268129
$326$	1921.53	0.326454
$327$	0	0
$328$	1704.23	0.286892
$329$	13574.5	2.27472
$330$	0	0
$331$	−9649.80	−1.60242	−0.801210	−	0.598384i	$-0.795810\pi$
$331$	−9649.80	−1.60242	−0.801210	+	0.598384i	$0.795810\pi$
$332$	165.874	0.0274203
$333$	0	0
$334$	−5363.40	−0.878659
$335$	5306.41	0.865434
$336$	0	0
$337$	−6617.95	−1.06974	−0.534871	−	0.844934i	$-0.679640\pi$
$337$	−6617.95	−1.06974	−0.534871	+	0.844934i	$0.679640\pi$
$338$	−7200.03	−1.15867
$339$	0	0
$340$	−709.351	−0.113147
$341$	−1720.63	−0.273247
$342$	0	0
$343$	2186.73	0.344234
$344$	−831.113	−0.130263
$345$	0	0
$346$	3510.83	0.545502
$347$	−3888.20	−0.601525	−0.300763	−	0.953699i	$-0.597241\pi$
$347$	−3888.20	−0.601525	−0.300763	+	0.953699i	$0.597241\pi$
$348$	0	0
$349$	−8651.21	−1.32690	−0.663451	−	0.748220i	$-0.730909\pi$
$349$	−8651.21	−1.32690	−0.663451	+	0.748220i	$0.730909\pi$
$350$	8491.59	1.29684
$351$	0	0
$352$	2149.08	0.325416
$353$	−1996.55	−0.301036	−0.150518	−	0.988607i	$-0.548094\pi$
$353$	−1996.55	−0.301036	−0.150518	+	0.988607i	$0.548094\pi$
$354$	0	0
$355$	5381.14	0.804511
$356$	858.343	0.127787
$357$	0	0
$358$	8360.70	1.23429
$359$	−11674.1	−1.71626	−0.858128	−	0.513436i	$-0.828373\pi$
$359$	−11674.1	−1.71626	−0.858128	+	0.513436i	$0.828373\pi$
$360$	0	0
$361$	−6630.17	−0.966639
$362$	−176.053	−0.0255611
$363$	0	0
$364$	−2617.09	−0.376848
$365$	74.8569	0.0107348
$366$	0	0
$367$	11122.7	1.58201	0.791005	−	0.611810i	$-0.209558\pi$
$367$	11122.7	1.58201	0.791005	+	0.611810i	$0.209558\pi$
$368$	8115.27	1.14956
$369$	0	0
$370$	3373.60	0.474014
$371$	−5389.30	−0.754173
$372$	0	0
$373$	7447.39	1.03381	0.516905	−	0.856043i	$-0.327084\pi$
$373$	7447.39	1.03381	0.516905	+	0.856043i	$0.327084\pi$
$374$	692.949	0.0958063
$375$	0	0
$376$	3618.93	0.496362
$377$	−4048.43	−0.553063
$378$	0	0
$379$	−2736.87	−0.370933	−0.185467	−	0.982651i	$-0.559380\pi$
$379$	−2736.87	−0.370933	−0.185467	+	0.982651i	$0.559380\pi$
$380$	−545.278	−0.0736110
$381$	0	0
$382$	7692.90	1.03037
$383$	3711.27	0.495136	0.247568	−	0.968871i	$-0.420369\pi$
$383$	3711.27	0.495136	0.247568	+	0.968871i	$0.420369\pi$
$384$	0	0
$385$	1307.10	0.173028
$386$	−5781.72	−0.762388
$387$	0	0
$388$	−2264.58	−0.296306
$389$	−68.7597	−0.00896209	−0.00448105	−	0.999990i	$-0.501426\pi$
$389$	−68.7597	−0.00896209	−0.00448105	+	0.999990i	$0.501426\pi$
$390$	0	0
$391$	2134.54	0.276082
$392$	−1650.30	−0.212635
$393$	0	0
$394$	−9301.85	−1.18939
$395$	808.100	0.102936
$396$	0	0
$397$	−9361.13	−1.18343	−0.591715	−	0.806147i	$-0.701549\pi$
$397$	−9361.13	−1.18343	−0.591715	+	0.806147i	$0.701549\pi$
$398$	5449.57	0.686337
$399$	0	0
$400$	6884.57	0.860572
$401$	95.5960	0.0119048	0.00595241	−	0.999982i	$-0.498105\pi$
$401$	95.5960	0.0119048	0.00595241	+	0.999982i	$0.498105\pi$
$402$	0	0
$403$	3151.99	0.389607
$404$	−2644.67	−0.325686
$405$	0	0
$406$	21883.0	2.67496
$407$	−1448.91	−0.176461
$408$	0	0
$409$	−11054.7	−1.33648	−0.668240	−	0.743946i	$-0.732952\pi$
$409$	−11054.7	−1.33648	−0.668240	+	0.743946i	$0.732952\pi$
$410$	−5679.64	−0.684140
$411$	0	0
$412$	−8720.26	−1.04276
$413$	1440.93	0.171680
$414$	0	0
$415$	151.763	0.0179512
$416$	−3936.87	−0.463992
$417$	0	0
$418$	532.670	0.0623295
$419$	4854.35	0.565992	0.282996	−	0.959121i	$-0.408672\pi$
$419$	4854.35	0.565992	0.282996	+	0.959121i	$0.408672\pi$
$420$	0	0
$421$	−5480.81	−0.634485	−0.317242	−	0.948344i	$-0.602757\pi$
$421$	−5480.81	−0.634485	−0.317242	+	0.948344i	$0.602757\pi$
$422$	−2767.14	−0.319200
$423$	0	0
$424$	−1436.78	−0.164566
$425$	1810.83	0.206678
$426$	0	0
$427$	10665.2	1.20872
$428$	2131.17	0.240687
$429$	0	0
$430$	2769.82	0.310634
$431$	−7366.56	−0.823282	−0.411641	−	0.911346i	$-0.635044\pi$
$431$	−7366.56	−0.823282	−0.411641	+	0.911346i	$0.635044\pi$
$432$	0	0
$433$	228.362	0.0253450	0.0126725	−	0.999920i	$-0.495966\pi$
$433$	228.362	0.0253450	0.0126725	+	0.999920i	$0.495966\pi$
$434$	−17037.4	−1.88439
$435$	0	0
$436$	1864.35	0.204785
$437$	1640.82	0.179613
$438$	0	0
$439$	−16897.8	−1.83710	−0.918550	−	0.395305i	$-0.870639\pi$
$439$	−16897.8	−1.83710	−0.918550	+	0.395305i	$0.870639\pi$
$440$	348.470	0.0377560
$441$	0	0
$442$	−1269.40	−0.136605
$443$	−8663.70	−0.929176	−0.464588	−	0.885527i	$-0.653798\pi$
$443$	−8663.70	−0.929176	−0.464588	+	0.885527i	$0.653798\pi$
$444$	0	0
$445$	785.323	0.0836582
$446$	11064.3	1.17469
$447$	0	0
$448$	6662.31	0.702599
$449$	12014.8	1.26284	0.631418	−	0.775442i	$-0.282473\pi$
$449$	12014.8	1.26284	0.631418	+	0.775442i	$0.282473\pi$
$450$	0	0
$451$	2439.32	0.254685
$452$	−9752.94	−1.01491
$453$	0	0
$454$	−14772.8	−1.52714
$455$	−2394.45	−0.246711
$456$	0	0
$457$	15590.7	1.59585	0.797925	−	0.602756i	$-0.205931\pi$
$457$	15590.7	1.59585	0.797925	+	0.602756i	$0.205931\pi$
$458$	21086.4	2.15132
$459$	0	0
$460$	−3909.96	−0.396310
$461$	−1475.12	−0.149031	−0.0745155	−	0.997220i	$-0.523741\pi$
$461$	−1475.12	−0.149031	−0.0745155	+	0.997220i	$0.523741\pi$
$462$	0	0
$463$	1335.14	0.134015	0.0670077	−	0.997752i	$-0.478655\pi$
$463$	1335.14	0.134015	0.0670077	+	0.997752i	$0.478655\pi$
$464$	17741.7	1.77508
$465$	0	0
$466$	14333.6	1.42488
$467$	13910.9	1.37841	0.689205	−	0.724566i	$-0.257960\pi$
$467$	13910.9	1.37841	0.689205	+	0.724566i	$0.257960\pi$
$468$	0	0
$469$	−22566.6	−2.22181
$470$	−12060.7	−1.18366
$471$	0	0
$472$	384.151	0.0374618
$473$	−1189.60	−0.115640
$474$	0	0
$475$	1391.99	0.134460
$476$	3016.66	0.290480
$477$	0	0
$478$	−18875.5	−1.80616
$479$	−15483.9	−1.47699	−0.738495	−	0.674259i	$-0.764463\pi$
$479$	−15483.9	−1.47699	−0.738495	+	0.674259i	$0.764463\pi$
$480$	0	0
$481$	2654.24	0.251607
$482$	6327.09	0.597906
$483$	0	0
$484$	−7809.28	−0.733403
$485$	−2071.93	−0.193982
$486$	0	0
$487$	−12901.8	−1.20049	−0.600243	−	0.799817i	$-0.704930\pi$
$487$	−12901.8	−1.20049	−0.600243	+	0.799817i	$0.704930\pi$
$488$	2843.33	0.263753
$489$	0	0
$490$	5499.92	0.507063
$491$	4348.11	0.399649	0.199824	−	0.979832i	$-0.435963\pi$
$491$	4348.11	0.399649	0.199824	+	0.979832i	$0.435963\pi$
$492$	0	0
$493$	4666.55	0.426310
$494$	−975.790	−0.0888722
$495$	0	0
$496$	−13813.1	−1.25046
$497$	−22884.4	−2.06541
$498$	0	0
$499$	−12051.7	−1.08118	−0.540589	−	0.841287i	$-0.681798\pi$
$499$	−12051.7	−1.08118	−0.540589	+	0.841287i	$0.681798\pi$
$500$	−7822.84	−0.699696
$501$	0	0
$502$	−16962.9	−1.50815
$503$	−16522.2	−1.46459	−0.732293	−	0.680990i	$-0.761550\pi$
$503$	−16522.2	−1.46459	−0.732293	+	0.680990i	$0.761550\pi$
$504$	0	0
$505$	−2419.69	−0.213217
$506$	3819.55	0.335573
$507$	0	0
$508$	8288.53	0.723906
$509$	15147.5	1.31906	0.659530	−	0.751678i	$-0.270755\pi$
$509$	15147.5	1.31906	0.659530	+	0.751678i	$0.270755\pi$
$510$	0	0
$511$	−318.344	−0.0275592
$512$	13355.7	1.15282
$513$	0	0
$514$	−19267.4	−1.65340
$515$	−7978.42	−0.682662
$516$	0	0
$517$	5179.88	0.440640
$518$	−14346.9	−1.21693
$519$	0	0
$520$	−638.357	−0.0538342
$521$	−14442.1	−1.21443	−0.607215	−	0.794538i	$-0.707713\pi$
$521$	−14442.1	−1.21443	−0.607215	+	0.794538i	$0.707713\pi$
$522$	0	0
$523$	−4796.97	−0.401065	−0.200532	−	0.979687i	$-0.564267\pi$
$523$	−4796.97	−0.401065	−0.200532	+	0.979687i	$0.564267\pi$
$524$	10911.7	0.909691
$525$	0	0
$526$	−30036.2	−2.48981
$527$	−3633.24	−0.300315
$528$	0	0
$529$	−401.383	−0.0329895
$530$	4788.31	0.392435
$531$	0	0
$532$	2318.91	0.188980
$533$	−4468.55	−0.363142
$534$	0	0
$535$	1949.87	0.157570
$536$	−6016.23	−0.484816
$537$	0	0
$538$	14476.9	1.16012
$539$	−2362.13	−0.188765
$540$	0	0
$541$	−833.411	−0.0662313	−0.0331157	−	0.999452i	$-0.510543\pi$
$541$	−833.411	−0.0662313	−0.0331157	+	0.999452i	$0.510543\pi$
$542$	−25787.5	−2.04367
$543$	0	0
$544$	4537.95	0.357653
$545$	1705.75	0.134066
$546$	0	0
$547$	−19682.1	−1.53848	−0.769239	−	0.638961i	$-0.779365\pi$
$547$	−19682.1	−1.53848	−0.769239	+	0.638961i	$0.779365\pi$
$548$	4875.33	0.380043
$549$	0	0
$550$	3240.31	0.251213
$551$	3587.17	0.277348
$552$	0	0
$553$	−3436.61	−0.264267
$554$	−151.538	−0.0116213
$555$	0	0
$556$	5370.10	0.409609
$557$	10287.7	0.782594	0.391297	−	0.920264i	$-0.372026\pi$
$557$	10287.7	0.782594	0.391297	+	0.920264i	$0.372026\pi$
$558$	0	0
$559$	2179.21	0.164885
$560$	10493.3	0.791829
$561$	0	0
$562$	9090.34	0.682301
$563$	8172.20	0.611754	0.305877	−	0.952071i	$-0.401050\pi$
$563$	8172.20	0.611754	0.305877	+	0.952071i	$0.401050\pi$
$564$	0	0
$565$	−8923.25	−0.664431
$566$	31104.9	2.30996
$567$	0	0
$568$	−6100.96	−0.450687
$569$	−9993.92	−0.736322	−0.368161	−	0.929762i	$-0.620012\pi$
$569$	−9993.92	−0.736322	−0.368161	+	0.929762i	$0.620012\pi$
$570$	0	0
$571$	−8399.41	−0.615594	−0.307797	−	0.951452i	$-0.599592\pi$
$571$	−8399.41	−0.615594	−0.307797	+	0.951452i	$0.599592\pi$
$572$	−998.656	−0.0729998
$573$	0	0
$574$	24153.9	1.75638
$575$	9981.34	0.723914
$576$	0	0
$577$	−9807.97	−0.707645	−0.353822	−	0.935313i	$-0.615118\pi$
$577$	−9807.97	−0.707645	−0.353822	+	0.935313i	$0.615118\pi$
$578$	−17100.4	−1.23059
$579$	0	0
$580$	−8547.99	−0.611959
$581$	−645.404	−0.0460858
$582$	0	0
$583$	−2056.50	−0.146092
$584$	−84.8702	−0.00601362
$585$	0	0
$586$	−7187.48	−0.506676
$587$	6955.54	0.489073	0.244537	−	0.969640i	$-0.421364\pi$
$587$	6955.54	0.489073	0.244537	+	0.969640i	$0.421364\pi$
$588$	0	0
$589$	−2792.87	−0.195379
$590$	−1280.25	−0.0893338
$591$	0	0
$592$	−11631.8	−0.807542
$593$	13988.6	0.968710	0.484355	−	0.874872i	$-0.339054\pi$
$593$	13988.6	0.968710	0.484355	+	0.874872i	$0.339054\pi$
$594$	0	0
$595$	2760.03	0.190169
$596$	−12562.1	−0.863365
$597$	0	0
$598$	−6996.98	−0.478475
$599$	−26538.4	−1.81023	−0.905117	−	0.425163i	$-0.860217\pi$
$599$	−26538.4	−1.81023	−0.905117	+	0.425163i	$0.860217\pi$
$600$	0	0
$601$	−16540.3	−1.12261	−0.561307	−	0.827608i	$-0.689701\pi$
$601$	−16540.3	−1.12261	−0.561307	+	0.827608i	$0.689701\pi$
$602$	−11779.3	−0.797486
$603$	0	0
$604$	10397.6	0.700453
$605$	−7144.93	−0.480137
$606$	0	0
$607$	2733.68	0.182795	0.0913974	−	0.995814i	$-0.470867\pi$
$607$	2733.68	0.182795	0.0913974	+	0.995814i	$0.470867\pi$
$608$	3488.32	0.232681
$609$	0	0
$610$	−9475.87	−0.628962
$611$	−9488.95	−0.628285
$612$	0	0
$613$	9935.81	0.654655	0.327328	−	0.944911i	$-0.393852\pi$
$613$	9935.81	0.654655	0.327328	+	0.944911i	$0.393852\pi$
$614$	34860.9	2.29132
$615$	0	0
$616$	−1481.94	−0.0969303
$617$	−4126.63	−0.269257	−0.134629	−	0.990896i	$-0.542984\pi$
$617$	−4126.63	−0.269257	−0.134629	+	0.990896i	$0.542984\pi$
$618$	0	0
$619$	13867.2	0.900436	0.450218	−	0.892919i	$-0.351346\pi$
$619$	13867.2	0.900436	0.450218	+	0.892919i	$0.351346\pi$
$620$	6655.21	0.431096
$621$	0	0
$622$	9052.79	0.583576
$623$	−3339.75	−0.214774
$624$	0	0
$625$	4345.14	0.278089
$626$	−38056.3	−2.42977
$627$	0	0
$628$	4076.43	0.259024
$629$	−3059.49	−0.193942
$630$	0	0
$631$	2950.74	0.186160	0.0930801	−	0.995659i	$-0.470329\pi$
$631$	2950.74	0.186160	0.0930801	+	0.995659i	$0.470329\pi$
$632$	−916.196	−0.0576651
$633$	0	0
$634$	−4036.87	−0.252878
$635$	7583.42	0.473919
$636$	0	0
$637$	4327.15	0.269149
$638$	8350.34	0.518171
$639$	0	0
$640$	4675.11	0.288750
$641$	11795.4	0.726820	0.363410	−	0.931629i	$-0.381612\pi$
$641$	11795.4	0.726820	0.363410	+	0.931629i	$0.381612\pi$
$642$	0	0
$643$	19304.7	1.18398	0.591992	−	0.805944i	$-0.298342\pi$
$643$	19304.7	1.18398	0.591992	+	0.805944i	$0.298342\pi$
$644$	16627.9	1.01744
$645$	0	0
$646$	1124.77	0.0685041
$647$	−21361.3	−1.29799	−0.648995	−	0.760793i	$-0.724810\pi$
$647$	−21361.3	−1.29799	−0.648995	+	0.760793i	$0.724810\pi$
$648$	0	0
$649$	549.846	0.0332563
$650$	−5935.88	−0.358191
$651$	0	0
$652$	3192.06	0.191734
$653$	−5795.08	−0.347288	−0.173644	−	0.984809i	$-0.555554\pi$
$653$	−5795.08	−0.347288	−0.173644	+	0.984809i	$0.555554\pi$
$654$	0	0
$655$	9983.39	0.595547
$656$	19582.8	1.16552
$657$	0	0
$658$	51290.6	3.03878
$659$	−11950.9	−0.706434	−0.353217	−	0.935542i	$-0.614912\pi$
$659$	−11950.9	−0.706434	−0.353217	+	0.935542i	$0.614912\pi$
$660$	0	0
$661$	17906.5	1.05368	0.526838	−	0.849965i	$-0.323377\pi$
$661$	17906.5	1.05368	0.526838	+	0.849965i	$0.323377\pi$
$662$	−36461.4	−2.14065
$663$	0	0
$664$	−172.064	−0.0100563
$665$	2121.64	0.123720
$666$	0	0
$667$	25722.1	1.49320
$668$	−8909.70	−0.516058
$669$	0	0
$670$	20050.1	1.15612
$671$	4069.74	0.234144
$672$	0	0
$673$	12825.6	0.734608	0.367304	−	0.930101i	$-0.380281\pi$
$673$	12825.6	0.734608	0.367304	+	0.930101i	$0.380281\pi$
$674$	−25005.7	−1.42906
$675$	0	0
$676$	−11960.7	−0.680514
$677$	−11752.0	−0.667156	−0.333578	−	0.942722i	$-0.608256\pi$
$677$	−11752.0	−0.667156	−0.333578	+	0.942722i	$0.608256\pi$
$678$	0	0
$679$	8811.31	0.498007
$680$	735.821	0.0414962
$681$	0	0
$682$	−6501.33	−0.365027
$683$	10731.7	0.601223	0.300612	−	0.953747i	$-0.402809\pi$
$683$	10731.7	0.601223	0.300612	+	0.953747i	$0.402809\pi$
$684$	0	0
$685$	4460.58	0.248803
$686$	8262.47	0.459858
$687$	0	0
$688$	−9550.05	−0.529204
$689$	3767.28	0.208305
$690$	0	0
$691$	28804.7	1.58579	0.792895	−	0.609358i	$-0.208573\pi$
$691$	28804.7	1.58579	0.792895	+	0.609358i	$0.208573\pi$
$692$	5832.21	0.320387
$693$	0	0
$694$	−14691.4	−0.803572
$695$	4913.26	0.268159
$696$	0	0
$697$	5150.81	0.279915
$698$	−32688.3	−1.77259
$699$	0	0
$700$	14106.3	0.761667
$701$	1663.10	0.0896070	0.0448035	−	0.998996i	$-0.485734\pi$
$701$	1663.10	0.0896070	0.0448035	+	0.998996i	$0.485734\pi$
$702$	0	0
$703$	−2351.83	−0.126175
$704$	2542.28	0.136102
$705$	0	0
$706$	−7543.90	−0.402151
$707$	10290.2	0.547388
$708$	0	0
$709$	32357.6	1.71399	0.856993	−	0.515328i	$-0.172330\pi$
$709$	32357.6	1.71399	0.856993	+	0.515328i	$0.172330\pi$
$710$	20332.5	1.07474
$711$	0	0
$712$	−890.372	−0.0468653
$713$	−20026.5	−1.05189
$714$	0	0
$715$	−913.699	−0.0477908
$716$	13888.8	0.724931
$717$	0	0
$718$	−44110.2	−2.29273
$719$	9360.97	0.485543	0.242771	−	0.970084i	$-0.421943\pi$
$719$	9360.97	0.485543	0.242771	+	0.970084i	$0.421943\pi$
$720$	0	0
$721$	33929.9	1.75259
$722$	−25051.9	−1.29132
$723$	0	0
$724$	−292.460	−0.0150127
$725$	21821.3	1.11783
$726$	0	0
$727$	−18574.0	−0.947556	−0.473778	−	0.880644i	$-0.657110\pi$
$727$	−18574.0	−0.947556	−0.473778	+	0.880644i	$0.657110\pi$
$728$	2714.74	0.138208
$729$	0	0
$730$	282.844	0.0143405
$731$	−2511.93	−0.127096
$732$	0	0
$733$	26673.5	1.34408	0.672038	−	0.740516i	$-0.265419\pi$
$733$	26673.5	1.34408	0.672038	+	0.740516i	$0.265419\pi$
$734$	42026.6	2.11339
$735$	0	0
$736$	25013.3	1.25272
$737$	−8611.21	−0.430391
$738$	0	0
$739$	25484.9	1.26857	0.634287	−	0.773098i	$-0.281294\pi$
$739$	25484.9	1.26857	0.634287	+	0.773098i	$0.281294\pi$
$740$	5604.24	0.278400
$741$	0	0
$742$	−20363.3	−1.00749
$743$	14039.9	0.693237	0.346619	−	0.938006i	$-0.387330\pi$
$743$	14039.9	0.693237	0.346619	+	0.938006i	$0.387330\pi$
$744$	0	0
$745$	−11493.5	−0.565219
$746$	28139.7	1.38106
$747$	0	0
$748$	1151.13	0.0562694
$749$	−8292.21	−0.404527
$750$	0	0
$751$	34000.2	1.65205	0.826023	−	0.563637i	$-0.190598\pi$
$751$	34000.2	1.65205	0.826023	+	0.563637i	$0.190598\pi$
$752$	41584.0	2.01650
$753$	0	0
$754$	−15296.9	−0.738832
$755$	9513.09	0.458565
$756$	0	0
$757$	−37552.9	−1.80302	−0.901508	−	0.432763i	$-0.857539\pi$
$757$	−37552.9	−1.80302	−0.901508	+	0.432763i	$0.857539\pi$
$758$	−10341.2	−0.495526
$759$	0	0
$760$	565.625	0.0269966
$761$	18976.0	0.903917	0.451959	−	0.892039i	$-0.350725\pi$
$761$	18976.0	0.903917	0.451959	+	0.892039i	$0.350725\pi$
$762$	0	0
$763$	−7254.05	−0.344186
$764$	12779.5	0.605164
$765$	0	0
$766$	14022.9	0.661447
$767$	−1007.26	−0.0474184
$768$	0	0
$769$	−14782.6	−0.693206	−0.346603	−	0.938012i	$-0.612665\pi$
$769$	−14782.6	−0.693206	−0.346603	+	0.938012i	$0.612665\pi$
$770$	4938.82	0.231146
$771$	0	0
$772$	−9604.62	−0.447769
$773$	32518.7	1.51309	0.756543	−	0.653944i	$-0.226887\pi$
$773$	32518.7	1.51309	0.756543	+	0.653944i	$0.226887\pi$
$774$	0	0
$775$	−16989.4	−0.787456
$776$	2349.08	0.108669
$777$	0	0
$778$	−259.806	−0.0119724
$779$	3959.43	0.182107
$780$	0	0
$781$	−8732.48	−0.400093
$782$	8065.28	0.368816
$783$	0	0
$784$	−18963.1	−0.863845
$785$	3729.64	0.169575
$786$	0	0
$787$	−7602.30	−0.344336	−0.172168	−	0.985068i	$-0.555077\pi$
$787$	−7602.30	−0.344336	−0.172168	+	0.985068i	$0.555077\pi$
$788$	−15452.3	−0.698559
$789$	0	0
$790$	3053.38	0.137512
$791$	37948.0	1.70578
$792$	0	0
$793$	−7455.30	−0.333853
$794$	−35370.7	−1.58093
$795$	0	0
$796$	9052.85	0.403103
$797$	7713.91	0.342836	0.171418	−	0.985198i	$-0.445165\pi$
$797$	7713.91	0.342836	0.171418	+	0.985198i	$0.445165\pi$
$798$	0	0
$799$	10937.7	0.484292
$800$	21220.0	0.937799
$801$	0	0
$802$	361.206	0.0159035
$803$	−121.477	−0.00533853
$804$	0	0
$805$	15213.4	0.666088
$806$	11909.7	0.520472
$807$	0	0
$808$	2743.36	0.119444
$809$	−25690.3	−1.11647	−0.558234	−	0.829683i	$-0.688521\pi$
$809$	−25690.3	−1.11647	−0.558234	+	0.829683i	$0.688521\pi$
$810$	0	0
$811$	28467.8	1.23260	0.616302	−	0.787510i	$-0.288630\pi$
$811$	28467.8	1.23260	0.616302	+	0.787510i	$0.288630\pi$
$812$	36352.1	1.57107
$813$	0	0
$814$	−5474.66	−0.235733
$815$	2920.51	0.125523
$816$	0	0
$817$	−1930.92	−0.0826857
$818$	−41769.8	−1.78539
$819$	0	0
$820$	−9435.05	−0.401812
$821$	38119.4	1.62043	0.810217	−	0.586130i	$-0.199349\pi$
$821$	38119.4	1.62043	0.810217	+	0.586130i	$0.199349\pi$
$822$	0	0
$823$	−26656.3	−1.12902	−0.564508	−	0.825427i	$-0.690934\pi$
$823$	−26656.3	−1.12902	−0.564508	+	0.825427i	$0.690934\pi$
$824$	9045.66	0.382428
$825$	0	0
$826$	5444.52	0.229345
$827$	19016.7	0.799606	0.399803	−	0.916601i	$-0.369078\pi$
$827$	19016.7	0.799606	0.399803	+	0.916601i	$0.369078\pi$
$828$	0	0
$829$	−36260.4	−1.51915	−0.759575	−	0.650420i	$-0.774593\pi$
$829$	−36260.4	−1.51915	−0.759575	+	0.650420i	$0.774593\pi$
$830$	573.432	0.0239808
$831$	0	0
$832$	−4657.16	−0.194060
$833$	−4987.82	−0.207464
$834$	0	0
$835$	−8151.75	−0.337848
$836$	884.873	0.0366076
$837$	0	0
$838$	18342.0	0.756102
$839$	−18034.3	−0.742088	−0.371044	−	0.928615i	$-0.621000\pi$
$839$	−18034.3	−0.742088	−0.371044	+	0.928615i	$0.621000\pi$
$840$	0	0
$841$	31845.0	1.30571
$842$	−20709.0	−0.847602
$843$	0	0
$844$	−4596.79	−0.187474
$845$	−10943.2	−0.445512
$846$	0	0
$847$	30385.3	1.23265
$848$	−16509.6	−0.668562
$849$	0	0
$850$	6842.17	0.276099
$851$	−16864.0	−0.679305
$852$	0	0
$853$	−17485.1	−0.701851	−0.350926	−	0.936403i	$-0.614133\pi$
$853$	−17485.1	−0.701851	−0.350926	+	0.936403i	$0.614133\pi$
$854$	40298.1	1.61472
$855$	0	0
$856$	−2210.69	−0.0882709
$857$	−3337.89	−0.133046	−0.0665228	−	0.997785i	$-0.521191\pi$
$857$	−3337.89	−0.133046	−0.0665228	+	0.997785i	$0.521191\pi$
$858$	0	0
$859$	−41566.7	−1.65103	−0.825517	−	0.564377i	$-0.809117\pi$
$859$	−41566.7	−1.65103	−0.825517	+	0.564377i	$0.809117\pi$
$860$	4601.24	0.182443
$861$	0	0
$862$	−27834.3	−1.09981
$863$	−21883.2	−0.863165	−0.431583	−	0.902073i	$-0.642045\pi$
$863$	−21883.2	−0.863165	−0.431583	+	0.902073i	$0.642045\pi$
$864$	0	0
$865$	5336.06	0.209747
$866$	862.859	0.0338581
$867$	0	0
$868$	−28302.7	−1.10675
$869$	−1311.38	−0.0511916
$870$	0	0
$871$	15774.8	0.613671
$872$	−1933.92	−0.0751041
$873$	0	0
$874$	6199.77	0.239943
$875$	30438.1	1.17599
$876$	0	0
$877$	−18985.8	−0.731019	−0.365509	−	0.930808i	$-0.619105\pi$
$877$	−18985.8	−0.731019	−0.365509	+	0.930808i	$0.619105\pi$
$878$	−63847.7	−2.45416
$879$	0	0
$880$	4004.16	0.153386
$881$	47938.8	1.83326	0.916628	−	0.399742i	$-0.130900\pi$
$881$	47938.8	1.83326	0.916628	+	0.399742i	$0.130900\pi$
$882$	0	0
$883$	45963.2	1.75174	0.875868	−	0.482550i	$-0.160290\pi$
$883$	45963.2	1.75174	0.875868	+	0.482550i	$0.160290\pi$
$884$	−2108.74	−0.0802314
$885$	0	0
$886$	−32735.5	−1.24128
$887$	25211.9	0.954378	0.477189	−	0.878801i	$-0.341656\pi$
$887$	25211.9	0.954378	0.477189	+	0.878801i	$0.341656\pi$
$888$	0	0
$889$	−32250.1	−1.21668
$890$	2967.32	0.111758
$891$	0	0
$892$	18380.2	0.689925
$893$	8407.83	0.315070
$894$	0	0
$895$	12707.3	0.474590
$896$	−19881.9	−0.741303
$897$	0	0
$898$	45397.5	1.68701
$899$	−43782.1	−1.62426
$900$	0	0
$901$	−4342.47	−0.160565
$902$	9216.88	0.340231
$903$	0	0
$904$	10116.9	0.372215
$905$	−267.580	−0.00982836
$906$	0	0
$907$	−9607.36	−0.351717	−0.175858	−	0.984415i	$-0.556270\pi$
$907$	−9607.36	−0.351717	−0.175858	+	0.984415i	$0.556270\pi$
$908$	−24540.6	−0.896924
$909$	0	0
$910$	−9047.34	−0.329579
$911$	44171.3	1.60643	0.803217	−	0.595686i	$-0.203120\pi$
$911$	44171.3	1.60643	0.803217	+	0.595686i	$0.203120\pi$
$912$	0	0
$913$	−246.280	−0.00892736
$914$	58909.1	2.13188
$915$	0	0
$916$	35028.8	1.26352
$917$	−42456.4	−1.52894
$918$	0	0
$919$	33636.3	1.20736	0.603678	−	0.797228i	$-0.293701\pi$
$919$	33636.3	1.20736	0.603678	+	0.797228i	$0.293701\pi$
$920$	4055.86	0.145345
$921$	0	0
$922$	−5573.70	−0.199089
$923$	15996.9	0.570471
$924$	0	0
$925$	−14306.5	−0.508535
$926$	5044.78	0.179030
$927$	0	0
$928$	54684.3	1.93437
$929$	−11890.9	−0.419945	−0.209972	−	0.977707i	$-0.567337\pi$
$929$	−11890.9	−0.419945	−0.209972	+	0.977707i	$0.567337\pi$
$930$	0	0
$931$	−3834.14	−0.134972
$932$	23811.1	0.836864
$933$	0	0
$934$	52561.7	1.84140
$935$	1053.20	0.0368379
$936$	0	0
$937$	−18281.5	−0.637385	−0.318692	−	0.947858i	$-0.603244\pi$
$937$	−18281.5	−0.637385	−0.318692	+	0.947858i	$0.603244\pi$
$938$	−85267.2	−2.96810
$939$	0	0
$940$	−20035.3	−0.695190
$941$	−40341.0	−1.39753	−0.698767	−	0.715349i	$-0.746268\pi$
$941$	−40341.0	−1.39753	−0.698767	+	0.715349i	$0.746268\pi$
$942$	0	0
$943$	28391.4	0.980435
$944$	4414.15	0.152191
$945$	0	0
$946$	−4494.85	−0.154482
$947$	44404.6	1.52371	0.761856	−	0.647746i	$-0.224288\pi$
$947$	44404.6	1.52371	0.761856	+	0.647746i	$0.224288\pi$
$948$	0	0
$949$	222.532	0.00761191
$950$	5259.57	0.179624
$951$	0	0
$952$	−3129.23	−0.106533
$953$	−34851.1	−1.18461	−0.592307	−	0.805713i	$-0.701783\pi$
$953$	−34851.1	−1.18461	−0.592307	+	0.805713i	$0.701783\pi$
$954$	0	0
$955$	11692.3	0.396183
$956$	−31356.0	−1.06080
$957$	0	0
$958$	−58505.5	−1.97310
$959$	−18969.5	−0.638747
$960$	0	0
$961$	4296.41	0.144218
$962$	10028.9	0.336119
$963$	0	0
$964$	10510.6	0.351165
$965$	−8787.55	−0.293141
$966$	0	0
$967$	2971.34	0.0988128	0.0494064	−	0.998779i	$-0.484267\pi$
$967$	2971.34	0.0988128	0.0494064	+	0.998779i	$0.484267\pi$
$968$	8100.68	0.268973
$969$	0	0
$970$	−7828.71	−0.259139
$971$	−34741.0	−1.14819	−0.574094	−	0.818789i	$-0.694646\pi$
$971$	−34741.0	−1.14819	−0.574094	+	0.818789i	$0.694646\pi$
$972$	0	0
$973$	−20894.6	−0.688440
$974$	−48749.1	−1.60372
$975$	0	0
$976$	32671.8	1.07151
$977$	31701.1	1.03808	0.519042	−	0.854749i	$-0.326289\pi$
$977$	31701.1	1.03808	0.519042	+	0.854749i	$0.326289\pi$
$978$	0	0
$979$	−1274.42	−0.0416042
$980$	9136.49	0.297811
$981$	0	0
$982$	16429.2	0.533887
$983$	47493.5	1.54100	0.770502	−	0.637437i	$-0.220005\pi$
$983$	47493.5	1.54100	0.770502	+	0.637437i	$0.220005\pi$
$984$	0	0
$985$	−14137.7	−0.457326
$986$	17632.4	0.569503
$987$	0	0
$988$	−1620.99	−0.0521968
$989$	−13845.8	−0.445167
$990$	0	0
$991$	−48621.1	−1.55853	−0.779263	−	0.626697i	$-0.784406\pi$
$991$	−48621.1	−1.55853	−0.779263	+	0.626697i	$0.784406\pi$
$992$	−42575.5	−1.36268
$993$	0	0
$994$	−86468.0	−2.75915
$995$	8282.71	0.263899
$996$	0	0
$997$	37128.0	1.17939	0.589697	−	0.807625i	$-0.299247\pi$
$997$	37128.0	1.17939	0.589697	+	0.807625i	$0.299247\pi$
$998$	−45536.9	−1.44433
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	531.4.a.e.1.7		8
3.2	odd	2		177.4.a.d.1.2	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.4.a.d.1.2	✓	8	3.2	odd	2
531.4.a.e.1.7		8	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 531 = 3^{2} \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	531.a (trivial)

\(f(q)\)	\(=\)	\(q+3.77847 q^{2} +6.27681 q^{4} +5.74283 q^{5} -24.4226 q^{7} -6.51103 q^{8} +O(q^{10})\)	\(q+3.77847 q^{2} +6.27681 q^{4} +5.74283 q^{5} -24.4226 q^{7} -6.51103 q^{8} +21.6991 q^{10} -9.31943 q^{11} +17.0721 q^{13} -92.2799 q^{14} -74.8162 q^{16} -19.6787 q^{17} -15.1270 q^{19} +36.0466 q^{20} -35.2132 q^{22} -108.469 q^{23} -92.0199 q^{25} +64.5065 q^{26} -153.296 q^{28} -237.137 q^{29} +184.628 q^{31} -230.602 q^{32} -74.3553 q^{34} -140.255 q^{35} +155.472 q^{37} -57.1569 q^{38} -37.3917 q^{40} -261.745 q^{41} +127.647 q^{43} -58.4963 q^{44} -409.848 q^{46} -555.816 q^{47} +253.463 q^{49} -347.694 q^{50} +107.158 q^{52} +220.668 q^{53} -53.5199 q^{55} +159.016 q^{56} -896.014 q^{58} -59.0000 q^{59} -436.694 q^{61} +697.610 q^{62} -272.793 q^{64} +98.0424 q^{65} +924.006 q^{67} -123.519 q^{68} -529.948 q^{70} +937.019 q^{71} +13.0348 q^{73} +587.446 q^{74} -94.9493 q^{76} +227.605 q^{77} +140.714 q^{79} -429.657 q^{80} -988.996 q^{82} +26.4265 q^{83} -113.011 q^{85} +482.310 q^{86} +60.6791 q^{88} +136.748 q^{89} -416.946 q^{91} -680.842 q^{92} -2100.13 q^{94} -86.8719 q^{95} -360.785 q^{97} +957.701 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 6 q^{2} + 34 q^{4} - 42 q^{5} + 53 q^{7} - 51 q^{8}+O(q^{10}) \) 8 * q - 6 * q^2 + 34 * q^4 - 42 * q^5 + 53 * q^7 - 51 * q^8	\( 8 q - 6 q^{2} + 34 q^{4} - 42 q^{5} + 53 q^{7} - 51 q^{8} + 21 q^{10} - 67 q^{11} + 33 q^{13} - 79 q^{14} - 30 q^{16} - 139 q^{17} + 64 q^{19} - 117 q^{20} - 84 q^{22} - 226 q^{23} + 96 q^{25} - 24 q^{26} + 34 q^{28} - 456 q^{29} + 124 q^{31} - 174 q^{32} - 114 q^{34} - 556 q^{35} + 127 q^{37} - 237 q^{38} - 188 q^{40} - 425 q^{41} - 115 q^{43} - 510 q^{44} - 711 q^{46} - 420 q^{47} + 171 q^{49} + 137 q^{50} - 922 q^{52} - 98 q^{53} - 616 q^{55} + 412 q^{56} - 1548 q^{58} - 472 q^{59} - 1254 q^{61} + 766 q^{62} - 2019 q^{64} + 734 q^{65} - 1010 q^{67} + 503 q^{68} - 2956 q^{70} + 17 q^{71} - 1180 q^{73} + 1228 q^{74} - 2008 q^{76} - 441 q^{77} - 873 q^{79} + 865 q^{80} - 3645 q^{82} - 759 q^{83} - 850 q^{85} + 1226 q^{86} - 3047 q^{88} - 988 q^{89} - 2111 q^{91} + 1062 q^{92} - 2240 q^{94} - 1822 q^{95} - 668 q^{97} + 1368 q^{98}+O(q^{100}) \) 8 * q - 6 * q^2 + 34 * q^4 - 42 * q^5 + 53 * q^7 - 51 * q^8 + 21 * q^10 - 67 * q^11 + 33 * q^13 - 79 * q^14 - 30 * q^16 - 139 * q^17 + 64 * q^19 - 117 * q^20 - 84 * q^22 - 226 * q^23 + 96 * q^25 - 24 * q^26 + 34 * q^28 - 456 * q^29 + 124 * q^31 - 174 * q^32 - 114 * q^34 - 556 * q^35 + 127 * q^37 - 237 * q^38 - 188 * q^40 - 425 * q^41 - 115 * q^43 - 510 * q^44 - 711 * q^46 - 420 * q^47 + 171 * q^49 + 137 * q^50 - 922 * q^52 - 98 * q^53 - 616 * q^55 + 412 * q^56 - 1548 * q^58 - 472 * q^59 - 1254 * q^61 + 766 * q^62 - 2019 * q^64 + 734 * q^65 - 1010 * q^67 + 503 * q^68 - 2956 * q^70 + 17 * q^71 - 1180 * q^73 + 1228 * q^74 - 2008 * q^76 - 441 * q^77 - 873 * q^79 + 865 * q^80 - 3645 * q^82 - 759 * q^83 - 850 * q^85 + 1226 * q^86 - 3047 * q^88 - 988 * q^89 - 2111 * q^91 + 1062 * q^92 - 2240 * q^94 - 1822 * q^95 - 668 * q^97 + 1368 * q^98

Introduction

L-functions

Modular forms

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Database

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