LMFDB - Embedded newform 531.4.a.e.1.2

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			1.2
Root			$-3.08481$ 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)

$f(q)$	$=$	$q-4.08481 q^{2} +8.68564 q^{4} +7.45529 q^{5} +34.0237 q^{7} -2.80073 q^{8} +O(q^{10})$	\(q-4.08481 q^{2} +8.68564 q^{4} +7.45529 q^{5} +34.0237 q^{7} -2.80073 q^{8} -30.4534 q^{10} -30.1014 q^{11} -5.30717 q^{13} -138.980 q^{14} -58.0447 q^{16} -63.2506 q^{17} -86.0442 q^{19} +64.7540 q^{20} +122.958 q^{22} -83.0398 q^{23} -69.4186 q^{25} +21.6788 q^{26} +295.518 q^{28} +27.8076 q^{29} -48.3051 q^{31} +259.507 q^{32} +258.366 q^{34} +253.657 q^{35} -358.130 q^{37} +351.474 q^{38} -20.8802 q^{40} +139.953 q^{41} -366.669 q^{43} -261.450 q^{44} +339.202 q^{46} +130.162 q^{47} +814.613 q^{49} +283.562 q^{50} -46.0962 q^{52} -608.850 q^{53} -224.415 q^{55} -95.2911 q^{56} -113.588 q^{58} -59.0000 q^{59} -361.223 q^{61} +197.317 q^{62} -595.679 q^{64} -39.5665 q^{65} -519.397 q^{67} -549.372 q^{68} -1036.14 q^{70} +733.270 q^{71} +1132.20 q^{73} +1462.89 q^{74} -747.349 q^{76} -1024.16 q^{77} +280.538 q^{79} -432.740 q^{80} -571.680 q^{82} -239.288 q^{83} -471.552 q^{85} +1497.77 q^{86} +84.3058 q^{88} -1214.72 q^{89} -180.570 q^{91} -721.255 q^{92} -531.686 q^{94} -641.485 q^{95} +149.047 q^{97} -3327.54 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−4.08481	−1.44420	−0.722099	−	0.691790i	$-0.756822\pi$
$2$	−4.08481	−1.44420	−0.722099	+	0.691790i	$0.756822\pi$
$3$	0	0
$3$	0	0
$4$	8.68564	1.08571
$5$	7.45529	0.666822	0.333411	−	0.942782i	$-0.391800\pi$
$5$	7.45529	0.666822	0.333411	+	0.942782i	$0.391800\pi$
$6$	0	0
$7$	34.0237	1.83711	0.918554	−	0.395296i	$-0.129358\pi$
$7$	34.0237	1.83711	0.918554	+	0.395296i	$0.129358\pi$
$8$	−2.80073	−0.123776
$9$	0	0
$10$	−30.4534	−0.963022
$11$	−30.1014	−0.825083	−0.412542	−	0.910939i	$-0.635359\pi$
$11$	−30.1014	−0.825083	−0.412542	+	0.910939i	$0.635359\pi$
$12$	0	0
$13$	−5.30717	−0.113226	−0.0566132	−	0.998396i	$-0.518030\pi$
$13$	−5.30717	−0.113226	−0.0566132	+	0.998396i	$0.518030\pi$
$14$	−138.980	−2.65315
$15$	0	0
$16$	−58.0447	−0.906949
$17$	−63.2506	−0.902384	−0.451192	−	0.892427i	$-0.649001\pi$
$17$	−63.2506	−0.902384	−0.451192	+	0.892427i	$0.649001\pi$
$18$	0	0
$19$	−86.0442	−1.03894	−0.519471	−	0.854488i	$-0.673871\pi$
$19$	−86.0442	−1.03894	−0.519471	+	0.854488i	$0.673871\pi$
$20$	64.7540	0.723972
$21$	0	0
$22$	122.958	1.19158
$23$	−83.0398	−0.752826	−0.376413	−	0.926452i	$-0.622843\pi$
$23$	−83.0398	−0.752826	−0.376413	+	0.926452i	$0.622843\pi$
$24$	0	0
$25$	−69.4186	−0.555349
$26$	21.6788	0.163521
$27$	0	0
$28$	295.518	1.99456
$29$	27.8076	0.178060	0.0890299	−	0.996029i	$-0.471623\pi$
$29$	27.8076	0.178060	0.0890299	+	0.996029i	$0.471623\pi$
$30$	0	0
$31$	−48.3051	−0.279866	−0.139933	−	0.990161i	$-0.544689\pi$
$31$	−48.3051	−0.279866	−0.139933	+	0.990161i	$0.544689\pi$
$32$	259.507	1.43359
$33$	0	0
$34$	258.366	1.30322
$35$	253.657	1.22502
$36$	0	0
$37$	−358.130	−1.59125	−0.795625	−	0.605789i	$-0.792857\pi$
$37$	−358.130	−1.59125	−0.795625	+	0.605789i	$0.792857\pi$
$38$	351.474	1.50044
$39$	0	0
$40$	−20.8802	−0.0825363
$41$	139.953	0.533096	0.266548	−	0.963822i	$-0.414117\pi$
$41$	139.953	0.533096	0.266548	+	0.963822i	$0.414117\pi$
$42$	0	0
$43$	−366.669	−1.30038	−0.650192	−	0.759770i	$-0.725312\pi$
$43$	−366.669	−1.30038	−0.650192	+	0.759770i	$0.725312\pi$
$44$	−261.450	−0.895797
$45$	0	0
$46$	339.202	1.08723
$47$	130.162	0.403959	0.201979	−	0.979390i	$-0.435263\pi$
$47$	130.162	0.403959	0.201979	+	0.979390i	$0.435263\pi$
$48$	0	0
$49$	814.613	2.37497
$50$	283.562	0.802034
$51$	0	0
$52$	−46.0962	−0.122931
$53$	−608.850	−1.57796	−0.788982	−	0.614417i	$-0.789391\pi$
$53$	−608.850	−1.57796	−0.788982	+	0.614417i	$0.789391\pi$
$54$	0	0
$55$	−224.415	−0.550183
$56$	−95.2911	−0.227389
$57$	0	0
$58$	−113.588	−0.257153
$59$	−59.0000	−0.130189
$59$	−59.0000	−0.130189
$60$	0	0
$61$	−361.223	−0.758194	−0.379097	−	0.925357i	$-0.623765\pi$
$61$	−361.223	−0.758194	−0.379097	+	0.925357i	$0.623765\pi$
$62$	197.317	0.404182
$63$	0	0
$64$	−595.679	−1.16344
$65$	−39.5665	−0.0755018
$66$	0	0
$67$	−519.397	−0.947082	−0.473541	−	0.880772i	$-0.657024\pi$
$67$	−519.397	−0.947082	−0.473541	+	0.880772i	$0.657024\pi$
$68$	−549.372	−0.979723
$69$	0	0
$70$	−1036.14	−1.76917
$71$	733.270	1.22568	0.612839	−	0.790208i	$-0.290027\pi$
$71$	733.270	1.22568	0.612839	+	0.790208i	$0.290027\pi$
$72$	0	0
$73$	1132.20	1.81525	0.907626	−	0.419779i	$-0.137893\pi$
$73$	1132.20	1.81525	0.907626	+	0.419779i	$0.137893\pi$
$74$	1462.89	2.29808
$75$	0	0
$76$	−747.349	−1.12798
$77$	−1024.16	−1.51577
$78$	0	0
$79$	280.538	0.399532	0.199766	−	0.979844i	$-0.435982\pi$
$79$	280.538	0.399532	0.199766	+	0.979844i	$0.435982\pi$
$80$	−432.740	−0.604773
$81$	0	0
$82$	−571.680	−0.769896
$83$	−239.288	−0.316450	−0.158225	−	0.987403i	$-0.550577\pi$
$83$	−239.288	−0.316450	−0.158225	+	0.987403i	$0.550577\pi$
$84$	0	0
$85$	−471.552	−0.601729
$86$	1497.77	1.87801
$87$	0	0
$88$	84.3058	0.102125
$89$	−1214.72	−1.44674	−0.723372	−	0.690459i	$-0.757409\pi$
$89$	−1214.72	−1.44674	−0.723372	+	0.690459i	$0.757409\pi$
$90$	0	0
$91$	−180.570	−0.208009
$92$	−721.255	−0.817348
$93$	0	0
$94$	−531.686	−0.583396
$95$	−641.485	−0.692789
$96$	0	0
$97$	149.047	0.156015	0.0780074	−	0.996953i	$-0.475144\pi$
$97$	149.047	0.156015	0.0780074	+	0.996953i	$0.475144\pi$
$98$	−3327.54	−3.42992
$99$	0	0
$100$	−602.946	−0.602946
$101$	491.208	0.483931	0.241965	−	0.970285i	$-0.422208\pi$
$101$	491.208	0.483931	0.241965	+	0.970285i	$0.422208\pi$
$102$	0	0
$103$	−1710.99	−1.63679	−0.818393	−	0.574658i	$-0.805135\pi$
$103$	−1710.99	−1.63679	−0.818393	+	0.574658i	$0.805135\pi$
$104$	14.8639	0.0140147
$105$	0	0
$106$	2487.04	2.27889
$107$	−19.6195	−0.0177261	−0.00886304	−	0.999961i	$-0.502821\pi$
$107$	−19.6195	−0.0177261	−0.00886304	+	0.999961i	$0.502821\pi$
$108$	0	0
$109$	227.070	0.199535	0.0997677	−	0.995011i	$-0.468190\pi$
$109$	227.070	0.199535	0.0997677	+	0.995011i	$0.468190\pi$
$110$	916.691	0.794573
$111$	0	0
$112$	−1974.90	−1.66616
$113$	406.168	0.338134	0.169067	−	0.985605i	$-0.445925\pi$
$113$	406.168	0.338134	0.169067	+	0.985605i	$0.445925\pi$
$114$	0	0
$115$	−619.086	−0.502001
$116$	241.527	0.193320
$117$	0	0
$118$	241.004	0.188018
$119$	−2152.02	−1.65778
$120$	0	0
$121$	−424.905	−0.319238
$122$	1475.52	1.09498
$123$	0	0
$124$	−419.561	−0.303852
$125$	−1449.45	−1.03714
$126$	0	0
$127$	1305.00	0.911810	0.455905	−	0.890028i	$-0.349316\pi$
$127$	1305.00	0.911810	0.455905	+	0.890028i	$0.349316\pi$
$128$	357.176	0.246642
$129$	0	0
$130$	161.621	0.109040
$131$	−1256.06	−0.837727	−0.418863	−	0.908049i	$-0.637571\pi$
$131$	−1256.06	−0.837727	−0.418863	+	0.908049i	$0.637571\pi$
$132$	0	0
$133$	−2927.54	−1.90865
$134$	2121.64	1.36777
$135$	0	0
$136$	177.148	0.111693
$137$	1061.86	0.662198	0.331099	−	0.943596i	$-0.392581\pi$
$137$	1061.86	0.662198	0.331099	+	0.943596i	$0.392581\pi$
$138$	0	0
$139$	−1006.82	−0.614368	−0.307184	−	0.951650i	$-0.599387\pi$
$139$	−1006.82	−0.614368	−0.307184	+	0.951650i	$0.599387\pi$
$140$	2203.17	1.33001
$141$	0	0
$142$	−2995.27	−1.77012
$143$	159.753	0.0934212
$144$	0	0
$145$	207.313	0.118734
$146$	−4624.80	−2.62158
$147$	0	0
$148$	−3110.59	−1.72763
$149$	−971.154	−0.533960	−0.266980	−	0.963702i	$-0.586026\pi$
$149$	−971.154	−0.533960	−0.266980	+	0.963702i	$0.586026\pi$
$150$	0	0
$151$	3599.13	1.93969	0.969843	−	0.243729i	$-0.0783708\pi$
$151$	3599.13	1.93969	0.969843	+	0.243729i	$0.0783708\pi$
$152$	240.986	0.128596
$153$	0	0
$154$	4183.50	2.18907
$155$	−360.129	−0.186621
$156$	0	0
$157$	1208.64	0.614393	0.307197	−	0.951646i	$-0.400609\pi$
$157$	1208.64	0.614393	0.307197	+	0.951646i	$0.400609\pi$
$158$	−1145.94	−0.577003
$159$	0	0
$160$	1934.70	0.955948
$161$	−2825.32	−1.38302
$162$	0	0
$163$	686.989	0.330118	0.165059	−	0.986284i	$-0.447219\pi$
$163$	686.989	0.330118	0.165059	+	0.986284i	$0.447219\pi$
$164$	1215.58	0.578785
$165$	0	0
$166$	977.447	0.457016
$167$	−3111.46	−1.44175	−0.720874	−	0.693066i	$-0.756260\pi$
$167$	−3111.46	−1.44175	−0.720874	+	0.693066i	$0.756260\pi$
$168$	0	0
$169$	−2168.83	−0.987180
$170$	1926.20	0.869015
$171$	0	0
$172$	−3184.76	−1.41183
$173$	−574.070	−0.252288	−0.126144	−	0.992012i	$-0.540260\pi$
$173$	−574.070	−0.252288	−0.126144	+	0.992012i	$0.540260\pi$
$174$	0	0
$175$	−2361.88	−1.02024
$176$	1747.23	0.748308
$177$	0	0
$178$	4961.90	2.08938
$179$	−3296.00	−1.37628	−0.688142	−	0.725576i	$-0.741574\pi$
$179$	−3296.00	−1.37628	−0.688142	+	0.725576i	$0.741574\pi$
$180$	0	0
$181$	2022.64	0.830616	0.415308	−	0.909681i	$-0.363674\pi$
$181$	2022.64	0.830616	0.415308	+	0.909681i	$0.363674\pi$
$182$	737.592	0.300406
$183$	0	0
$184$	232.572	0.0931816
$185$	−2669.97	−1.06108
$186$	0	0
$187$	1903.93	0.744542
$188$	1130.54	0.438580
$189$	0	0
$190$	2620.34	1.00052
$191$	3437.29	1.30216	0.651082	−	0.759007i	$-0.274315\pi$
$191$	3437.29	1.30216	0.651082	+	0.759007i	$0.274315\pi$
$192$	0	0
$193$	4382.77	1.63460	0.817301	−	0.576210i	$-0.195469\pi$
$193$	4382.77	1.63460	0.817301	+	0.576210i	$0.195469\pi$
$194$	−608.828	−0.225316
$195$	0	0
$196$	7075.44	2.57851
$197$	5031.35	1.81964	0.909819	−	0.415004i	$-0.136220\pi$
$197$	5031.35	1.81964	0.909819	+	0.415004i	$0.136220\pi$
$198$	0	0
$199$	4357.76	1.55233	0.776164	−	0.630532i	$-0.217163\pi$
$199$	4357.76	1.55233	0.776164	+	0.630532i	$0.217163\pi$
$200$	194.423	0.0687387
$201$	0	0
$202$	−2006.49	−0.698891
$203$	946.116	0.327115
$204$	0	0
$205$	1043.39	0.355480
$206$	6989.07	2.36384
$207$	0	0
$208$	308.053	0.102691
$209$	2590.05	0.857213
$210$	0	0
$211$	−4436.30	−1.44743	−0.723715	−	0.690099i	$-0.757567\pi$
$211$	−4436.30	−1.44743	−0.723715	+	0.690099i	$0.757567\pi$
$212$	−5288.26	−1.71320
$213$	0	0
$214$	80.1420	0.0256000
$215$	−2733.63	−0.867124
$216$	0	0
$217$	−1643.52	−0.514144
$218$	−927.536	−0.288168
$219$	0	0
$220$	−1949.19	−0.597337
$221$	335.682	0.102174
$222$	0	0
$223$	−518.147	−0.155595	−0.0777975	−	0.996969i	$-0.524789\pi$
$223$	−518.147	−0.155595	−0.0777975	+	0.996969i	$0.524789\pi$
$224$	8829.40	2.63366
$225$	0	0
$226$	−1659.12	−0.488332
$227$	5358.31	1.56671	0.783355	−	0.621574i	$-0.213507\pi$
$227$	5358.31	1.56671	0.783355	+	0.621574i	$0.213507\pi$
$228$	0	0
$229$	2805.75	0.809647	0.404823	−	0.914395i	$-0.367333\pi$
$229$	2805.75	0.809647	0.404823	+	0.914395i	$0.367333\pi$
$230$	2528.85	0.724988
$231$	0	0
$232$	−77.8813	−0.0220395
$233$	4221.06	1.18683	0.593414	−	0.804897i	$-0.297780\pi$
$233$	4221.06	1.18683	0.593414	+	0.804897i	$0.297780\pi$
$234$	0	0
$235$	970.394	0.269368
$236$	−512.453	−0.141347
$237$	0	0
$238$	8790.59	2.39416
$239$	−5412.00	−1.46474	−0.732371	−	0.680905i	$-0.761586\pi$
$239$	−5412.00	−1.46474	−0.732371	+	0.680905i	$0.761586\pi$
$240$	0	0
$241$	−3003.56	−0.802806	−0.401403	−	0.915902i	$-0.631477\pi$
$241$	−3003.56	−0.802806	−0.401403	+	0.915902i	$0.631477\pi$
$242$	1735.66	0.461042
$243$	0	0
$244$	−3137.45	−0.823175
$245$	6073.18	1.58368
$246$	0	0
$247$	456.651	0.117636
$248$	135.289	0.0346407
$249$	0	0
$250$	5920.71	1.49784
$251$	900.091	0.226348	0.113174	−	0.993575i	$-0.463898\pi$
$251$	900.091	0.226348	0.113174	+	0.993575i	$0.463898\pi$
$252$	0	0
$253$	2499.62	0.621144
$254$	−5330.67	−1.31683
$255$	0	0
$256$	3306.44	0.807236
$257$	−3419.24	−0.829907	−0.414953	−	0.909843i	$-0.636202\pi$
$257$	−3419.24	−0.829907	−0.414953	+	0.909843i	$0.636202\pi$
$258$	0	0
$259$	−12184.9	−2.92330
$260$	−343.660	−0.0819727
$261$	0	0
$262$	5130.75	1.20984
$263$	−1273.15	−0.298501	−0.149250	−	0.988799i	$-0.547686\pi$
$263$	−1273.15	−0.298501	−0.149250	+	0.988799i	$0.547686\pi$
$264$	0	0
$265$	−4539.16	−1.05222
$266$	11958.4	2.75646
$267$	0	0
$268$	−4511.30	−1.02825
$269$	−3437.51	−0.779141	−0.389570	−	0.920997i	$-0.627376\pi$
$269$	−3437.51	−0.779141	−0.389570	+	0.920997i	$0.627376\pi$
$270$	0	0
$271$	−2409.95	−0.540199	−0.270100	−	0.962832i	$-0.587057\pi$
$271$	−2409.95	−0.540199	−0.270100	+	0.962832i	$0.587057\pi$
$272$	3671.36	0.818416
$273$	0	0
$274$	−4337.51	−0.956344
$275$	2089.60	0.458209
$276$	0	0
$277$	−7279.05	−1.57890	−0.789450	−	0.613814i	$-0.789634\pi$
$277$	−7279.05	−1.57890	−0.789450	+	0.613814i	$0.789634\pi$
$278$	4112.66	0.887269
$279$	0	0
$280$	−710.423	−0.151628
$281$	−6298.57	−1.33716	−0.668579	−	0.743641i	$-0.733097\pi$
$281$	−6298.57	−1.33716	−0.668579	+	0.743641i	$0.733097\pi$
$282$	0	0
$283$	−3561.67	−0.748124	−0.374062	−	0.927404i	$-0.622035\pi$
$283$	−3561.67	−0.748124	−0.374062	+	0.927404i	$0.622035\pi$
$284$	6368.92	1.33072
$285$	0	0
$286$	−652.561	−0.134919
$287$	4761.71	0.979355
$288$	0	0
$289$	−912.361	−0.185703
$290$	−846.835	−0.171475
$291$	0	0
$292$	9833.84	1.97083
$293$	4374.10	0.872142	0.436071	−	0.899912i	$-0.356370\pi$
$293$	4374.10	0.872142	0.436071	+	0.899912i	$0.356370\pi$
$294$	0	0
$295$	−439.862	−0.0868128
$296$	1003.02	0.196958
$297$	0	0
$298$	3966.98	0.771144
$299$	440.706	0.0852398
$300$	0	0
$301$	−12475.4	−2.38895
$302$	−14701.7	−2.80129
$303$	0	0
$304$	4994.41	0.942267
$305$	−2693.02	−0.505580
$306$	0	0
$307$	5264.64	0.978726	0.489363	−	0.872080i	$-0.337229\pi$
$307$	5264.64	0.978726	0.489363	+	0.872080i	$0.337229\pi$
$308$	−8895.50	−1.64568
$309$	0	0
$310$	1471.06	0.269517
$311$	−3889.00	−0.709083	−0.354542	−	0.935040i	$-0.615363\pi$
$311$	−3889.00	−0.709083	−0.354542	+	0.935040i	$0.615363\pi$
$312$	0	0
$313$	−8815.40	−1.59194	−0.795968	−	0.605339i	$-0.793038\pi$
$313$	−8815.40	−1.59194	−0.795968	+	0.605339i	$0.793038\pi$
$314$	−4937.05	−0.887305
$315$	0	0
$316$	2436.65	0.433774
$317$	10172.7	1.80239	0.901196	−	0.433412i	$-0.142691\pi$
$317$	10172.7	1.80239	0.901196	+	0.433412i	$0.142691\pi$
$318$	0	0
$319$	−837.047	−0.146914
$320$	−4440.96	−0.775804
$321$	0	0
$322$	11540.9	1.99736
$323$	5442.35	0.937524
$324$	0	0
$325$	368.416	0.0628802
$326$	−2806.22	−0.476755
$327$	0	0
$328$	−391.969	−0.0659844
$329$	4428.59	0.742115
$330$	0	0
$331$	−2899.09	−0.481415	−0.240707	−	0.970598i	$-0.577379\pi$
$331$	−2899.09	−0.481415	−0.240707	+	0.970598i	$0.577379\pi$
$332$	−2078.37	−0.343571
$333$	0	0
$334$	12709.7	2.08217
$335$	−3872.26	−0.631534
$336$	0	0
$337$	−11201.0	−1.81055	−0.905277	−	0.424821i	$-0.860337\pi$
$337$	−11201.0	−1.81055	−0.905277	+	0.424821i	$0.860337\pi$
$338$	8859.27	1.42568
$339$	0	0
$340$	−4095.73	−0.653300
$341$	1454.05	0.230913
$342$	0	0
$343$	16046.0	2.52596
$344$	1026.94	0.160956
$345$	0	0
$346$	2344.97	0.364353
$347$	−9034.82	−1.39774	−0.698868	−	0.715251i	$-0.746313\pi$
$347$	−9034.82	−1.39774	−0.698868	+	0.715251i	$0.746313\pi$
$348$	0	0
$349$	2985.35	0.457886	0.228943	−	0.973440i	$-0.426473\pi$
$349$	2985.35	0.457886	0.228943	+	0.973440i	$0.426473\pi$
$350$	9647.82	1.47342
$351$	0	0
$352$	−7811.54	−1.18283
$353$	−10272.8	−1.54892	−0.774458	−	0.632626i	$-0.781977\pi$
$353$	−10272.8	−1.54892	−0.774458	+	0.632626i	$0.781977\pi$
$354$	0	0
$355$	5466.74	0.817308
$356$	−10550.6	−1.57074
$357$	0	0
$358$	13463.5	1.98763
$359$	4648.82	0.683442	0.341721	−	0.939802i	$-0.388990\pi$
$359$	4648.82	0.683442	0.341721	+	0.939802i	$0.388990\pi$
$360$	0	0
$361$	544.604	0.0793999
$362$	−8262.08	−1.19957
$363$	0	0
$364$	−1568.36	−0.225837
$365$	8440.84	1.21045
$366$	0	0
$367$	−1646.71	−0.234216	−0.117108	−	0.993119i	$-0.537362\pi$
$367$	−1646.71	−0.234216	−0.117108	+	0.993119i	$0.537362\pi$
$368$	4820.03	0.682775
$369$	0	0
$370$	10906.3	1.53241
$371$	−20715.4	−2.89889
$372$	0	0
$373$	6528.88	0.906307	0.453154	−	0.891432i	$-0.350299\pi$
$373$	6528.88	0.906307	0.453154	+	0.891432i	$0.350299\pi$
$374$	−7777.19	−1.07527
$375$	0	0
$376$	−364.548	−0.0500003
$377$	−147.579	−0.0201611
$378$	0	0
$379$	6421.70	0.870344	0.435172	−	0.900347i	$-0.356688\pi$
$379$	6421.70	0.870344	0.435172	+	0.900347i	$0.356688\pi$
$380$	−5571.71	−0.752165
$381$	0	0
$382$	−14040.7	−1.88058
$383$	3048.13	0.406663	0.203332	−	0.979110i	$-0.434823\pi$
$383$	3048.13	0.406663	0.203332	+	0.979110i	$0.434823\pi$
$384$	0	0
$385$	−7635.42	−1.01075
$386$	−17902.7	−2.36069
$387$	0	0
$388$	1294.57	0.169386
$389$	−8129.80	−1.05963	−0.529816	−	0.848112i	$-0.677739\pi$
$389$	−8129.80	−1.05963	−0.529816	+	0.848112i	$0.677739\pi$
$390$	0	0
$391$	5252.32	0.679338
$392$	−2281.51	−0.293963
$393$	0	0
$394$	−20552.1	−2.62792
$395$	2091.49	0.266416
$396$	0	0
$397$	−5072.35	−0.641244	−0.320622	−	0.947207i	$-0.603892\pi$
$397$	−5072.35	−0.641244	−0.320622	+	0.947207i	$0.603892\pi$
$398$	−17800.6	−2.24187
$399$	0	0
$400$	4029.39	0.503673
$401$	−9539.59	−1.18799	−0.593996	−	0.804468i	$-0.702450\pi$
$401$	−9539.59	−1.18799	−0.593996	+	0.804468i	$0.702450\pi$
$402$	0	0
$403$	256.363	0.0316883
$404$	4266.46	0.525406
$405$	0	0
$406$	−3864.70	−0.472419
$407$	10780.2	1.31291
$408$	0	0
$409$	10696.8	1.29321	0.646605	−	0.762825i	$-0.276188\pi$
$409$	10696.8	1.29321	0.646605	+	0.762825i	$0.276188\pi$
$410$	−4262.04	−0.513383
$411$	0	0
$412$	−14861.1	−1.77707
$413$	−2007.40	−0.239171
$414$	0	0
$415$	−1783.96	−0.211015
$416$	−1377.25	−0.162320
$417$	0	0
$418$	−10579.9	−1.23799
$419$	2134.24	0.248841	0.124421	−	0.992230i	$-0.460293\pi$
$419$	2134.24	0.248841	0.124421	+	0.992230i	$0.460293\pi$
$420$	0	0
$421$	6780.72	0.784970	0.392485	−	0.919758i	$-0.371616\pi$
$421$	6780.72	0.784970	0.392485	+	0.919758i	$0.371616\pi$
$422$	18121.4	2.09037
$423$	0	0
$424$	1705.22	0.195314
$425$	4390.77	0.501138
$426$	0	0
$427$	−12290.1	−1.39288
$428$	−170.408	−0.0192453
$429$	0	0
$430$	11166.3	1.25230
$431$	16691.4	1.86542	0.932708	−	0.360633i	$-0.117439\pi$
$431$	16691.4	1.86542	0.932708	+	0.360633i	$0.117439\pi$
$432$	0	0
$433$	−2902.98	−0.322190	−0.161095	−	0.986939i	$-0.551503\pi$
$433$	−2902.98	−0.322190	−0.161095	+	0.986939i	$0.551503\pi$
$434$	6713.46	0.742526
$435$	0	0
$436$	1972.25	0.216637
$437$	7145.10	0.782143
$438$	0	0
$439$	8526.93	0.927034	0.463517	−	0.886088i	$-0.346587\pi$
$439$	8526.93	0.927034	0.463517	+	0.886088i	$0.346587\pi$
$440$	628.524	0.0680993
$441$	0	0
$442$	−1371.19	−0.147559
$443$	3174.04	0.340413	0.170207	−	0.985408i	$-0.445556\pi$
$443$	3174.04	0.340413	0.170207	+	0.985408i	$0.445556\pi$
$444$	0	0
$445$	−9056.10	−0.964720
$446$	2116.53	0.224710
$447$	0	0
$448$	−20267.2	−2.13736
$449$	2111.09	0.221889	0.110945	−	0.993827i	$-0.464612\pi$
$449$	2111.09	0.221889	0.110945	+	0.993827i	$0.464612\pi$
$450$	0	0
$451$	−4212.77	−0.439849
$452$	3527.83	0.367114
$453$	0	0
$454$	−21887.6	−2.26264
$455$	−1346.20	−0.138705
$456$	0	0
$457$	−6995.71	−0.716074	−0.358037	−	0.933707i	$-0.616554\pi$
$457$	−6995.71	−0.716074	−0.358037	+	0.933707i	$0.616554\pi$
$458$	−11460.9	−1.16929
$459$	0	0
$460$	−5377.16	−0.545025
$461$	−426.893	−0.0431288	−0.0215644	−	0.999767i	$-0.506865\pi$
$461$	−426.893	−0.0431288	−0.0215644	+	0.999767i	$0.506865\pi$
$462$	0	0
$463$	2901.91	0.291281	0.145641	−	0.989338i	$-0.453476\pi$
$463$	2901.91	0.291281	0.145641	+	0.989338i	$0.453476\pi$
$464$	−1614.08	−0.161491
$465$	0	0
$466$	−17242.2	−1.71401
$467$	−8424.34	−0.834758	−0.417379	−	0.908733i	$-0.637051\pi$
$467$	−8424.34	−0.834758	−0.417379	+	0.908733i	$0.637051\pi$
$468$	0	0
$469$	−17671.8	−1.73989
$470$	−3963.87	−0.389021
$471$	0	0
$472$	165.243	0.0161142
$473$	11037.3	1.07293
$474$	0	0
$475$	5973.07	0.576975
$476$	−18691.7	−1.79986
$477$	0	0
$478$	22107.0	2.11538
$479$	10436.0	0.995476	0.497738	−	0.867327i	$-0.334164\pi$
$479$	10436.0	0.995476	0.497738	+	0.867327i	$0.334164\pi$
$480$	0	0
$481$	1900.66	0.180172
$482$	12269.0	1.15941
$483$	0	0
$484$	−3690.58	−0.346598
$485$	1111.19	0.104034
$486$	0	0
$487$	6183.28	0.575341	0.287671	−	0.957729i	$-0.407119\pi$
$487$	6183.28	0.575341	0.287671	+	0.957729i	$0.407119\pi$
$488$	1011.69	0.0938460
$489$	0	0
$490$	−24807.8	−2.28714
$491$	4039.61	0.371293	0.185647	−	0.982617i	$-0.440562\pi$
$491$	4039.61	0.371293	0.185647	+	0.982617i	$0.440562\pi$
$492$	0	0
$493$	−1758.84	−0.160678
$494$	−1865.33	−0.169889
$495$	0	0
$496$	2803.86	0.253824
$497$	24948.6	2.25170
$498$	0	0
$499$	−17134.6	−1.53718	−0.768589	−	0.639743i	$-0.779041\pi$
$499$	−17134.6	−1.53718	−0.768589	+	0.639743i	$0.779041\pi$
$500$	−12589.4	−1.12603
$501$	0	0
$502$	−3676.70	−0.326890
$503$	−19616.1	−1.73884	−0.869422	−	0.494069i	$-0.835509\pi$
$503$	−19616.1	−1.73884	−0.869422	+	0.494069i	$0.835509\pi$
$504$	0	0
$505$	3662.10	0.322695
$506$	−10210.4	−0.897055
$507$	0	0
$508$	11334.8	0.989957
$509$	−15281.4	−1.33072	−0.665358	−	0.746524i	$-0.731721\pi$
$509$	−15281.4	−1.33072	−0.665358	+	0.746524i	$0.731721\pi$
$510$	0	0
$511$	38521.5	3.33481
$512$	−16363.6	−1.41245
$513$	0	0
$514$	13966.9	1.19855
$515$	−12755.9	−1.09144
$516$	0	0
$517$	−3918.05	−0.333299
$518$	49773.1	4.22182
$519$	0	0
$520$	110.815	0.00934530
$521$	−20973.2	−1.76363	−0.881816	−	0.471593i	$-0.843679\pi$
$521$	−20973.2	−1.76363	−0.881816	+	0.471593i	$0.843679\pi$
$522$	0	0
$523$	7587.00	0.634334	0.317167	−	0.948370i	$-0.397268\pi$
$523$	7587.00	0.634334	0.317167	+	0.948370i	$0.397268\pi$
$524$	−10909.7	−0.909525
$525$	0	0
$526$	5200.57	0.431094
$527$	3055.33	0.252547
$528$	0	0
$529$	−5271.39	−0.433253
$530$	18541.6	1.51961
$531$	0	0
$532$	−25427.6	−2.07223
$533$	−742.753	−0.0603606
$534$	0	0
$535$	−146.269	−0.0118201
$536$	1454.69	0.117226
$537$	0	0
$538$	14041.6	1.12523
$539$	−24521.0	−1.95954
$540$	0	0
$541$	1078.04	0.0856721	0.0428360	−	0.999082i	$-0.486361\pi$
$541$	1078.04	0.0856721	0.0428360	+	0.999082i	$0.486361\pi$
$542$	9844.18	0.780154
$543$	0	0
$544$	−16414.0	−1.29365
$545$	1692.87	0.133054
$546$	0	0
$547$	−318.365	−0.0248854	−0.0124427	−	0.999923i	$-0.503961\pi$
$547$	−318.365	−0.0248854	−0.0124427	+	0.999923i	$0.503961\pi$
$548$	9222.97	0.718952
$549$	0	0
$550$	−8535.61	−0.661744
$551$	−2392.68	−0.184994
$552$	0	0
$553$	9544.95	0.733983
$554$	29733.5	2.28024
$555$	0	0
$556$	−8744.87	−0.667023
$557$	7357.32	0.559676	0.279838	−	0.960047i	$-0.409719\pi$
$557$	7357.32	0.559676	0.279838	+	0.960047i	$0.409719\pi$
$558$	0	0
$559$	1945.98	0.147238
$560$	−14723.4	−1.11103
$561$	0	0
$562$	25728.5	1.93112
$563$	10907.4	0.816505	0.408253	−	0.912869i	$-0.366138\pi$
$563$	10907.4	0.816505	0.408253	+	0.912869i	$0.366138\pi$
$564$	0	0
$565$	3028.10	0.225475
$566$	14548.7	1.08044
$567$	0	0
$568$	−2053.69	−0.151709
$569$	8712.06	0.641878	0.320939	−	0.947100i	$-0.396002\pi$
$569$	8712.06	0.641878	0.320939	+	0.947100i	$0.396002\pi$
$570$	0	0
$571$	120.132	0.00880446	0.00440223	−	0.999990i	$-0.498599\pi$
$571$	120.132	0.00880446	0.00440223	+	0.999990i	$0.498599\pi$
$572$	1387.56	0.101428
$573$	0	0
$574$	−19450.7	−1.41438
$575$	5764.51	0.418081
$576$	0	0
$577$	2169.26	0.156512	0.0782560	−	0.996933i	$-0.475065\pi$
$577$	2169.26	0.156512	0.0782560	+	0.996933i	$0.475065\pi$
$578$	3726.82	0.268192
$579$	0	0
$580$	1800.65	0.128910
$581$	−8141.48	−0.581352
$582$	0	0
$583$	18327.3	1.30195
$584$	−3170.97	−0.224684
$585$	0	0
$586$	−17867.3	−1.25954
$587$	1300.42	0.0914378	0.0457189	−	0.998954i	$-0.485442\pi$
$587$	1300.42	0.0914378	0.0457189	+	0.998954i	$0.485442\pi$
$588$	0	0
$589$	4156.37	0.290765
$590$	1796.75	0.125375
$591$	0	0
$592$	20787.6	1.44318
$593$	13449.8	0.931395	0.465698	−	0.884944i	$-0.345803\pi$
$593$	13449.8	0.931395	0.465698	+	0.884944i	$0.345803\pi$
$594$	0	0
$595$	−16043.9	−1.10544
$596$	−8435.10	−0.579724
$597$	0	0
$598$	−1800.20	−0.123103
$599$	11679.1	0.796652	0.398326	−	0.917244i	$-0.369591\pi$
$599$	11679.1	0.796652	0.398326	+	0.917244i	$0.369591\pi$
$600$	0	0
$601$	−7992.33	−0.542452	−0.271226	−	0.962516i	$-0.587429\pi$
$601$	−7992.33	−0.542452	−0.271226	+	0.962516i	$0.587429\pi$
$602$	50959.8	3.45011
$603$	0	0
$604$	31260.7	2.10593
$605$	−3167.79	−0.212875
$606$	0	0
$607$	9263.57	0.619434	0.309717	−	0.950829i	$-0.399766\pi$
$607$	9263.57	0.619434	0.309717	+	0.950829i	$0.399766\pi$
$608$	−22329.1	−1.48942
$609$	0	0
$610$	11000.5	0.730157
$611$	−690.791	−0.0457388
$612$	0	0
$613$	−11447.6	−0.754267	−0.377134	−	0.926159i	$-0.623090\pi$
$613$	−11447.6	−0.754267	−0.377134	+	0.926159i	$0.623090\pi$
$614$	−21505.0	−1.41347
$615$	0	0
$616$	2868.40	0.187615
$617$	27000.2	1.76173	0.880863	−	0.473371i	$-0.156963\pi$
$617$	27000.2	1.76173	0.880863	+	0.473371i	$0.156963\pi$
$618$	0	0
$619$	22497.9	1.46085	0.730425	−	0.682993i	$-0.239322\pi$
$619$	22497.9	1.46085	0.730425	+	0.682993i	$0.239322\pi$
$620$	−3127.95	−0.202615
$621$	0	0
$622$	15885.8	1.02406
$623$	−41329.3	−2.65782
$624$	0	0
$625$	−2128.72	−0.136238
$626$	36009.2	2.29907
$627$	0	0
$628$	10497.8	0.667050
$629$	22652.0	1.43592
$630$	0	0
$631$	20510.4	1.29399	0.646994	−	0.762495i	$-0.276026\pi$
$631$	20510.4	1.29399	0.646994	+	0.762495i	$0.276026\pi$
$632$	−785.710	−0.0494523
$633$	0	0
$634$	−41553.7	−2.60301
$635$	9729.14	0.608015
$636$	0	0
$637$	−4323.29	−0.268909
$638$	3419.17	0.212173
$639$	0	0
$640$	2662.85	0.164466
$641$	−9877.54	−0.608642	−0.304321	−	0.952570i	$-0.598430\pi$
$641$	−9877.54	−0.608642	−0.304321	+	0.952570i	$0.598430\pi$
$642$	0	0
$643$	9789.87	0.600427	0.300214	−	0.953872i	$-0.402942\pi$
$643$	9789.87	0.600427	0.300214	+	0.953872i	$0.402942\pi$
$644$	−24539.8	−1.50156
$645$	0	0
$646$	−22230.9	−1.35397
$647$	9704.28	0.589667	0.294834	−	0.955549i	$-0.404736\pi$
$647$	9704.28	0.589667	0.294834	+	0.955549i	$0.404736\pi$
$648$	0	0
$649$	1775.98	0.107417
$650$	−1504.91	−0.0908114
$651$	0	0
$652$	5966.95	0.358410
$653$	−19355.5	−1.15994	−0.579970	−	0.814638i	$-0.696936\pi$
$653$	−19355.5	−1.15994	−0.579970	+	0.814638i	$0.696936\pi$
$654$	0	0
$655$	−9364.27	−0.558614
$656$	−8123.52	−0.483491
$657$	0	0
$658$	−18089.9	−1.07176
$659$	9598.00	0.567352	0.283676	−	0.958920i	$-0.408446\pi$
$659$	9598.00	0.567352	0.283676	+	0.958920i	$0.408446\pi$
$660$	0	0
$661$	−30099.9	−1.77118	−0.885591	−	0.464465i	$-0.846247\pi$
$661$	−30099.9	−1.77118	−0.885591	+	0.464465i	$0.846247\pi$
$662$	11842.2	0.695258
$663$	0	0
$664$	670.181	0.0391688
$665$	−21825.7	−1.27273
$666$	0	0
$667$	−2309.14	−0.134048
$668$	−27025.0	−1.56531
$669$	0	0
$670$	15817.4	0.912060
$671$	10873.3	0.625573
$672$	0	0
$673$	4334.61	0.248272	0.124136	−	0.992265i	$-0.460384\pi$
$673$	4334.61	0.248272	0.124136	+	0.992265i	$0.460384\pi$
$674$	45753.9	2.61480
$675$	0	0
$676$	−18837.7	−1.07179
$677$	−26966.0	−1.53085	−0.765427	−	0.643523i	$-0.777472\pi$
$677$	−26966.0	−1.53085	−0.765427	+	0.643523i	$0.777472\pi$
$678$	0	0
$679$	5071.13	0.286616
$680$	1320.69	0.0744795
$681$	0	0
$682$	−5939.52	−0.333484
$683$	−4741.14	−0.265615	−0.132807	−	0.991142i	$-0.542399\pi$
$683$	−4741.14	−0.265615	−0.132807	+	0.991142i	$0.542399\pi$
$684$	0	0
$685$	7916.50	0.441568
$686$	−65544.9	−3.64798
$687$	0	0
$688$	21283.2	1.17938
$689$	3231.27	0.178667
$690$	0	0
$691$	7335.54	0.403845	0.201923	−	0.979401i	$-0.435281\pi$
$691$	7335.54	0.403845	0.201923	+	0.979401i	$0.435281\pi$
$692$	−4986.17	−0.273910
$693$	0	0
$694$	36905.5	2.01861
$695$	−7506.12	−0.409674
$696$	0	0
$697$	−8852.09	−0.481057
$698$	−12194.6	−0.661277
$699$	0	0
$700$	−20514.4	−1.10768
$701$	−8441.52	−0.454824	−0.227412	−	0.973799i	$-0.573026\pi$
$701$	−8441.52	−0.454824	−0.227412	+	0.973799i	$0.573026\pi$
$702$	0	0
$703$	30815.0	1.65322
$704$	17930.8	0.959932
$705$	0	0
$706$	41962.5	2.23694
$707$	16712.7	0.889033
$708$	0	0
$709$	9530.08	0.504809	0.252405	−	0.967622i	$-0.418779\pi$
$709$	9530.08	0.504809	0.252405	+	0.967622i	$0.418779\pi$
$710$	−22330.6	−1.18035
$711$	0	0
$712$	3402.10	0.179072
$713$	4011.25	0.210691
$714$	0	0
$715$	1191.01	0.0622953
$716$	−28627.9	−1.49424
$717$	0	0
$718$	−18989.5	−0.987024
$719$	8740.89	0.453380	0.226690	−	0.973967i	$-0.427210\pi$
$719$	8740.89	0.453380	0.226690	+	0.973967i	$0.427210\pi$
$720$	0	0
$721$	−58214.3	−3.00695
$722$	−2224.60	−0.114669
$723$	0	0
$724$	17567.9	0.901804
$725$	−1930.36	−0.0988853
$726$	0	0
$727$	19162.5	0.977575	0.488788	−	0.872403i	$-0.337439\pi$
$727$	19162.5	0.977575	0.488788	+	0.872403i	$0.337439\pi$
$728$	505.726	0.0257465
$729$	0	0
$730$	−34479.2	−1.74813
$731$	23192.0	1.17345
$732$	0	0
$733$	14807.2	0.746134	0.373067	−	0.927804i	$-0.378306\pi$
$733$	14807.2	0.746134	0.373067	+	0.927804i	$0.378306\pi$
$734$	6726.48	0.338255
$735$	0	0
$736$	−21549.4	−1.07924
$737$	15634.6	0.781421
$738$	0	0
$739$	−22838.8	−1.13686	−0.568430	−	0.822732i	$-0.692449\pi$
$739$	−22838.8	−1.13686	−0.568430	+	0.822732i	$0.692449\pi$
$740$	−23190.4	−1.15202
$741$	0	0
$742$	84618.2	4.18657
$743$	2638.81	0.130294	0.0651472	−	0.997876i	$-0.479248\pi$
$743$	2638.81	0.130294	0.0651472	+	0.997876i	$0.479248\pi$
$744$	0	0
$745$	−7240.24	−0.356056
$746$	−26669.2	−1.30889
$747$	0	0
$748$	16536.9	0.808353
$749$	−667.529	−0.0325647
$750$	0	0
$751$	−6880.34	−0.334311	−0.167155	−	0.985931i	$-0.553458\pi$
$751$	−6880.34	−0.334311	−0.167155	+	0.985931i	$0.553458\pi$
$752$	−7555.21	−0.366370
$753$	0	0
$754$	602.833	0.0291166
$755$	26832.5	1.29342
$756$	0	0
$757$	41277.3	1.98183	0.990916	−	0.134479i	$-0.0429359\pi$
$757$	41277.3	1.98183	0.990916	+	0.134479i	$0.0429359\pi$
$758$	−26231.4	−1.25695
$759$	0	0
$760$	1796.62	0.0857504
$761$	−16764.5	−0.798570	−0.399285	−	0.916827i	$-0.630742\pi$
$761$	−16764.5	−0.798570	−0.399285	+	0.916827i	$0.630742\pi$
$762$	0	0
$763$	7725.76	0.366568
$764$	29855.1	1.41377
$765$	0	0
$766$	−12451.0	−0.587302
$767$	313.123	0.0147408
$768$	0	0
$769$	−29667.0	−1.39118	−0.695591	−	0.718438i	$-0.744858\pi$
$769$	−29667.0	−1.39118	−0.695591	+	0.718438i	$0.744858\pi$
$770$	31189.2	1.45972
$771$	0	0
$772$	38067.1	1.77470
$773$	1557.94	0.0724904	0.0362452	−	0.999343i	$-0.488460\pi$
$773$	1557.94	0.0724904	0.0362452	+	0.999343i	$0.488460\pi$
$774$	0	0
$775$	3353.27	0.155423
$776$	−417.440	−0.0193108
$777$	0	0
$778$	33208.7	1.53032
$779$	−12042.1	−0.553856
$780$	0	0
$781$	−22072.5	−1.01129
$782$	−21454.7	−0.981098
$783$	0	0
$784$	−47284.0	−2.15397
$785$	9010.74	0.409691
$786$	0	0
$787$	8898.56	0.403049	0.201524	−	0.979483i	$-0.435410\pi$
$787$	8898.56	0.403049	0.201524	+	0.979483i	$0.435410\pi$
$788$	43700.5	1.97559
$789$	0	0
$790$	−8543.35	−0.384758
$791$	13819.4	0.621188
$792$	0	0
$793$	1917.07	0.0858476
$794$	20719.6	0.926083
$795$	0	0
$796$	37849.9	1.68537
$797$	−25724.7	−1.14331	−0.571654	−	0.820495i	$-0.693698\pi$
$797$	−25724.7	−1.14331	−0.571654	+	0.820495i	$0.693698\pi$
$798$	0	0
$799$	−8232.81	−0.364526
$800$	−18014.6	−0.796142
$801$	0	0
$802$	38967.4	1.71569
$803$	−34080.7	−1.49773
$804$	0	0
$805$	−21063.6	−0.922229
$806$	−1047.19	−0.0457641
$807$	0	0
$808$	−1375.74	−0.0598989
$809$	25187.9	1.09463	0.547317	−	0.836926i	$-0.315649\pi$
$809$	25187.9	1.09463	0.547317	+	0.836926i	$0.315649\pi$
$810$	0	0
$811$	−1555.98	−0.0673709	−0.0336854	−	0.999432i	$-0.510724\pi$
$811$	−1555.98	−0.0673709	−0.0336854	+	0.999432i	$0.510724\pi$
$812$	8217.63	0.355151
$813$	0	0
$814$	−44035.1	−1.89611
$815$	5121.71	0.220129
$816$	0	0
$817$	31549.8	1.35102
$818$	−43694.3	−1.86765
$819$	0	0
$820$	9062.50	0.385947
$821$	10402.0	0.442182	0.221091	−	0.975253i	$-0.429038\pi$
$821$	10402.0	0.442182	0.221091	+	0.975253i	$0.429038\pi$
$822$	0	0
$823$	−3211.80	−0.136035	−0.0680173	−	0.997684i	$-0.521667\pi$
$823$	−3211.80	−0.136035	−0.0680173	+	0.997684i	$0.521667\pi$
$824$	4792.02	0.202595
$825$	0	0
$826$	8199.84	0.345410
$827$	14355.4	0.603612	0.301806	−	0.953369i	$-0.402411\pi$
$827$	14355.4	0.603612	0.301806	+	0.953369i	$0.402411\pi$
$828$	0	0
$829$	40266.3	1.68698	0.843490	−	0.537145i	$-0.180497\pi$
$829$	40266.3	1.68698	0.843490	+	0.537145i	$0.180497\pi$
$830$	7287.15	0.304748
$831$	0	0
$832$	3161.37	0.131732
$833$	−51524.8	−2.14313
$834$	0	0
$835$	−23196.8	−0.961389
$836$	22496.3	0.930681
$837$	0	0
$838$	−8717.96	−0.359376
$839$	32142.4	1.32262	0.661310	−	0.750113i	$-0.270001\pi$
$839$	32142.4	1.32262	0.661310	+	0.750113i	$0.270001\pi$
$840$	0	0
$841$	−23615.7	−0.968295
$842$	−27697.9	−1.13365
$843$	0	0
$844$	−38532.2	−1.57148
$845$	−16169.3	−0.658273
$846$	0	0
$847$	−14456.9	−0.586474
$848$	35340.6	1.43113
$849$	0	0
$850$	−17935.4	−0.723742
$851$	29739.1	1.19793
$852$	0	0
$853$	3961.96	0.159033	0.0795164	−	0.996834i	$-0.474662\pi$
$853$	3961.96	0.159033	0.0795164	+	0.996834i	$0.474662\pi$
$854$	50202.8	2.01160
$855$	0	0
$856$	54.9489	0.00219406
$857$	38708.1	1.54287	0.771437	−	0.636306i	$-0.219538\pi$
$857$	38708.1	1.54287	0.771437	+	0.636306i	$0.219538\pi$
$858$	0	0
$859$	−6725.04	−0.267119	−0.133560	−	0.991041i	$-0.542641\pi$
$859$	−6725.04	−0.267119	−0.133560	+	0.991041i	$0.542641\pi$
$860$	−23743.3	−0.941442
$861$	0	0
$862$	−68180.9	−2.69403
$863$	−22856.4	−0.901554	−0.450777	−	0.892637i	$-0.648853\pi$
$863$	−22856.4	−0.901554	−0.450777	+	0.892637i	$0.648853\pi$
$864$	0	0
$865$	−4279.86	−0.168231
$866$	11858.1	0.465306
$867$	0	0
$868$	−14275.0	−0.558209
$869$	−8444.59	−0.329647
$870$	0	0
$871$	2756.53	0.107235
$872$	−635.960	−0.0246976
$873$	0	0
$874$	−29186.3	−1.12957
$875$	−49315.6	−1.90534
$876$	0	0
$877$	25523.5	0.982744	0.491372	−	0.870950i	$-0.336496\pi$
$877$	25523.5	0.982744	0.491372	+	0.870950i	$0.336496\pi$
$878$	−34830.8	−1.33882
$879$	0	0
$880$	13026.1	0.498988
$881$	−37925.5	−1.45033	−0.725166	−	0.688574i	$-0.758237\pi$
$881$	−37925.5	−1.45033	−0.725166	+	0.688574i	$0.758237\pi$
$882$	0	0
$883$	−44752.5	−1.70560	−0.852798	−	0.522241i	$-0.825096\pi$
$883$	−44752.5	−1.70560	−0.852798	+	0.522241i	$0.825096\pi$
$884$	2915.61	0.110931
$885$	0	0
$886$	−12965.3	−0.491624
$887$	−27284.8	−1.03285	−0.516423	−	0.856334i	$-0.672737\pi$
$887$	−27284.8	−1.03285	−0.516423	+	0.856334i	$0.672737\pi$
$888$	0	0
$889$	44400.9	1.67509
$890$	36992.4	1.39325
$891$	0	0
$892$	−4500.44	−0.168930
$893$	−11199.7	−0.419689
$894$	0	0
$895$	−24572.7	−0.917736
$896$	12152.5	0.453109
$897$	0	0
$898$	−8623.38	−0.320452
$899$	−1343.25	−0.0498329
$900$	0	0
$901$	38510.2	1.42393
$902$	17208.4	0.635228
$903$	0	0
$904$	−1137.57	−0.0418528
$905$	15079.4	0.553873
$906$	0	0
$907$	22037.7	0.806779	0.403390	−	0.915028i	$-0.367832\pi$
$907$	22037.7	0.806779	0.403390	+	0.915028i	$0.367832\pi$
$908$	46540.3	1.70099
$909$	0	0
$910$	5498.96	0.200317
$911$	−53768.1	−1.95545	−0.977726	−	0.209884i	$-0.932691\pi$
$911$	−53768.1	−1.95545	−0.977726	+	0.209884i	$0.932691\pi$
$912$	0	0
$913$	7202.92	0.261097
$914$	28576.1	1.03415
$915$	0	0
$916$	24369.7	0.879038
$917$	−42735.7	−1.53899
$918$	0	0
$919$	12821.6	0.460224	0.230112	−	0.973164i	$-0.426091\pi$
$919$	12821.6	0.460224	0.230112	+	0.973164i	$0.426091\pi$
$920$	1733.89	0.0621355
$921$	0	0
$922$	1743.77	0.0622865
$923$	−3891.59	−0.138779
$924$	0	0
$925$	24860.9	0.883699
$926$	−11853.7	−0.420667
$927$	0	0
$928$	7216.26	0.255265
$929$	20227.5	0.714361	0.357181	−	0.934035i	$-0.383738\pi$
$929$	20227.5	0.714361	0.357181	+	0.934035i	$0.383738\pi$
$930$	0	0
$931$	−70092.7	−2.46745
$932$	36662.7	1.28855
$933$	0	0
$934$	34411.8	1.20555
$935$	14194.4	0.496476
$936$	0	0
$937$	−43813.7	−1.52757	−0.763784	−	0.645471i	$-0.776661\pi$
$937$	−43813.7	−1.52757	−0.763784	+	0.645471i	$0.776661\pi$
$938$	72186.0	2.51275
$939$	0	0
$940$	8428.50	0.292455
$941$	−36622.2	−1.26870	−0.634351	−	0.773045i	$-0.718733\pi$
$941$	−36622.2	−1.26870	−0.634351	+	0.773045i	$0.718733\pi$
$942$	0	0
$943$	−11621.7	−0.401329
$944$	3424.64	0.118075
$945$	0	0
$946$	−45085.1	−1.54952
$947$	15181.8	0.520952	0.260476	−	0.965480i	$-0.416121\pi$
$947$	15181.8	0.520952	0.260476	+	0.965480i	$0.416121\pi$
$948$	0	0
$949$	−6008.75	−0.205535
$950$	−24398.8	−0.833266
$951$	0	0
$952$	6027.22	0.205193
$953$	17530.8	0.595885	0.297942	−	0.954584i	$-0.403700\pi$
$953$	17530.8	0.595885	0.297942	+	0.954584i	$0.403700\pi$
$954$	0	0
$955$	25626.0	0.868311
$956$	−47006.7	−1.59028
$957$	0	0
$958$	−42629.0	−1.43766
$959$	36128.5	1.21653
$960$	0	0
$961$	−27457.6	−0.921675
$962$	−7763.82	−0.260203
$963$	0	0
$964$	−26087.9	−0.871611
$965$	32674.8	1.08999
$966$	0	0
$967$	50857.1	1.69127	0.845633	−	0.533765i	$-0.179223\pi$
$967$	50857.1	1.69127	0.845633	+	0.533765i	$0.179223\pi$
$968$	1190.04	0.0395139
$969$	0	0
$970$	−4538.99	−0.150246
$971$	24239.2	0.801105	0.400553	−	0.916274i	$-0.368818\pi$
$971$	24239.2	0.801105	0.400553	+	0.916274i	$0.368818\pi$
$972$	0	0
$973$	−34255.7	−1.12866
$974$	−25257.5	−0.830906
$975$	0	0
$976$	20967.1	0.687643
$977$	52188.5	1.70896	0.854482	−	0.519481i	$-0.173875\pi$
$977$	52188.5	1.70896	0.854482	+	0.519481i	$0.173875\pi$
$978$	0	0
$979$	36564.8	1.19368
$980$	52749.5	1.71941
$981$	0	0
$982$	−16501.0	−0.536221
$983$	6206.63	0.201384	0.100692	−	0.994918i	$-0.467894\pi$
$983$	6206.63	0.201384	0.100692	+	0.994918i	$0.467894\pi$
$984$	0	0
$985$	37510.2	1.21337
$986$	7184.54	0.232051
$987$	0	0
$988$	3966.31	0.127718
$989$	30448.2	0.978964
$990$	0	0
$991$	11627.8	0.372724	0.186362	−	0.982481i	$-0.440330\pi$
$991$	11627.8	0.372724	0.186362	+	0.982481i	$0.440330\pi$
$992$	−12535.5	−0.401213
$993$	0	0
$994$	−101910.	−3.25190
$995$	32488.3	1.03513
$996$	0	0
$997$	27404.9	0.870533	0.435266	−	0.900302i	$-0.356654\pi$
$997$	27404.9	0.870533	0.435266	+	0.900302i	$0.356654\pi$
$998$	69991.7	2.21999
$999$	0	0

Twists

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

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

\(f(q)\)	\(=\)	\(q-4.08481 q^{2} +8.68564 q^{4} +7.45529 q^{5} +34.0237 q^{7} -2.80073 q^{8} +O(q^{10})\)	\(q-4.08481 q^{2} +8.68564 q^{4} +7.45529 q^{5} +34.0237 q^{7} -2.80073 q^{8} -30.4534 q^{10} -30.1014 q^{11} -5.30717 q^{13} -138.980 q^{14} -58.0447 q^{16} -63.2506 q^{17} -86.0442 q^{19} +64.7540 q^{20} +122.958 q^{22} -83.0398 q^{23} -69.4186 q^{25} +21.6788 q^{26} +295.518 q^{28} +27.8076 q^{29} -48.3051 q^{31} +259.507 q^{32} +258.366 q^{34} +253.657 q^{35} -358.130 q^{37} +351.474 q^{38} -20.8802 q^{40} +139.953 q^{41} -366.669 q^{43} -261.450 q^{44} +339.202 q^{46} +130.162 q^{47} +814.613 q^{49} +283.562 q^{50} -46.0962 q^{52} -608.850 q^{53} -224.415 q^{55} -95.2911 q^{56} -113.588 q^{58} -59.0000 q^{59} -361.223 q^{61} +197.317 q^{62} -595.679 q^{64} -39.5665 q^{65} -519.397 q^{67} -549.372 q^{68} -1036.14 q^{70} +733.270 q^{71} +1132.20 q^{73} +1462.89 q^{74} -747.349 q^{76} -1024.16 q^{77} +280.538 q^{79} -432.740 q^{80} -571.680 q^{82} -239.288 q^{83} -471.552 q^{85} +1497.77 q^{86} +84.3058 q^{88} -1214.72 q^{89} -180.570 q^{91} -721.255 q^{92} -531.686 q^{94} -641.485 q^{95} +149.047 q^{97} -3327.54 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 6 q^{2} + 34 q^{4} - 42 q^{5} + 53 q^{7} - 51 q^{8}+O(q^{10}) \) 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

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more