LMFDB - Embedded newform 531.4.a.c.1.3

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:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 41x^{5} - 7x^{4} + 484x^{3} + 63x^{2} - 1736x - 44 $ x^7 - 41x^5 - 7x^4 + 484x^3 + 63x^2 - 1736*x - 44
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 177)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-2.74916$ 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)

$f(q)$	$=$	$q-2.74916 q^{2} -0.442134 q^{4} -13.7745 q^{5} -35.6462 q^{7} +23.2088 q^{8} +O(q^{10})$	\(q-2.74916 q^{2} -0.442134 q^{4} -13.7745 q^{5} -35.6462 q^{7} +23.2088 q^{8} +37.8683 q^{10} +30.8537 q^{11} +14.6435 q^{13} +97.9970 q^{14} -60.2674 q^{16} +77.8827 q^{17} +98.6377 q^{19} +6.09019 q^{20} -84.8216 q^{22} +117.065 q^{23} +64.7375 q^{25} -40.2572 q^{26} +15.7604 q^{28} +54.2015 q^{29} -231.433 q^{31} -19.9854 q^{32} -214.112 q^{34} +491.009 q^{35} -258.311 q^{37} -271.171 q^{38} -319.690 q^{40} -326.921 q^{41} +250.920 q^{43} -13.6415 q^{44} -321.830 q^{46} -509.390 q^{47} +927.650 q^{49} -177.974 q^{50} -6.47437 q^{52} +313.730 q^{53} -424.994 q^{55} -827.304 q^{56} -149.008 q^{58} -59.0000 q^{59} +528.854 q^{61} +636.245 q^{62} +537.082 q^{64} -201.707 q^{65} -373.284 q^{67} -34.4346 q^{68} -1349.86 q^{70} +521.969 q^{71} +809.404 q^{73} +710.137 q^{74} -43.6111 q^{76} -1099.82 q^{77} -926.088 q^{79} +830.155 q^{80} +898.756 q^{82} +1311.77 q^{83} -1072.80 q^{85} -689.818 q^{86} +716.075 q^{88} -510.201 q^{89} -521.983 q^{91} -51.7585 q^{92} +1400.39 q^{94} -1358.69 q^{95} -856.449 q^{97} -2550.26 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + 26 q^{4} + 2 q^{5} - 59 q^{7} + 21 q^{8}+O(q^{10}) $ 7 * q + 26 * q^4 + 2 * q^5 - 59 * q^7 + 21 * q^8	$ 7 q + 26 q^{4} + 2 q^{5} - 59 q^{7} + 21 q^{8} - 71 q^{10} + 5 q^{11} - 67 q^{13} + 65 q^{14} - 94 q^{16} + 23 q^{17} - 176 q^{19} + 207 q^{20} - 704 q^{22} + 218 q^{23} - 183 q^{25} - 58 q^{26} - 938 q^{28} - 168 q^{29} - 604 q^{31} + 448 q^{32} - 610 q^{34} + 336 q^{35} - 505 q^{37} + 453 q^{38} - 1080 q^{40} + 265 q^{41} - 493 q^{43} - 504 q^{44} + 381 q^{46} + 244 q^{47} + 770 q^{49} - 1639 q^{50} + 160 q^{52} - 686 q^{53} - 116 q^{55} - 2190 q^{56} + 1584 q^{58} - 413 q^{59} - 838 q^{61} - 286 q^{62} + 205 q^{64} - 490 q^{65} - 1504 q^{67} - 3047 q^{68} + 1530 q^{70} + 1267 q^{71} - 666 q^{73} - 528 q^{74} - 64 q^{76} - 1109 q^{77} - 2741 q^{79} - 1213 q^{80} + 953 q^{82} + 2025 q^{83} - 1274 q^{85} - 4394 q^{86} - 1639 q^{88} - 616 q^{89} - 2415 q^{91} - 218 q^{92} + 900 q^{94} - 2554 q^{95} - 1298 q^{97} + 172 q^{98}+O(q^{100}) $ 7 * q + 26 * q^4 + 2 * q^5 - 59 * q^7 + 21 * q^8 - 71 * q^10 + 5 * q^11 - 67 * q^13 + 65 * q^14 - 94 * q^16 + 23 * q^17 - 176 * q^19 + 207 * q^20 - 704 * q^22 + 218 * q^23 - 183 * q^25 - 58 * q^26 - 938 * q^28 - 168 * q^29 - 604 * q^31 + 448 * q^32 - 610 * q^34 + 336 * q^35 - 505 * q^37 + 453 * q^38 - 1080 * q^40 + 265 * q^41 - 493 * q^43 - 504 * q^44 + 381 * q^46 + 244 * q^47 + 770 * q^49 - 1639 * q^50 + 160 * q^52 - 686 * q^53 - 116 * q^55 - 2190 * q^56 + 1584 * q^58 - 413 * q^59 - 838 * q^61 - 286 * q^62 + 205 * q^64 - 490 * q^65 - 1504 * q^67 - 3047 * q^68 + 1530 * q^70 + 1267 * q^71 - 666 * q^73 - 528 * q^74 - 64 * q^76 - 1109 * q^77 - 2741 * q^79 - 1213 * q^80 + 953 * q^82 + 2025 * q^83 - 1274 * q^85 - 4394 * q^86 - 1639 * q^88 - 616 * q^89 - 2415 * q^91 - 218 * q^92 + 900 * q^94 - 2554 * q^95 - 1298 * q^97 + 172 * 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 $

$2$	−2.74916	−0.971974	−0.485987	−	0.873966i	$-0.661540\pi$
$2$	−2.74916	−0.971974	−0.485987	+	0.873966i	$0.661540\pi$
$3$	0	0
$3$	0	0
$4$	−0.442134	−0.0552668
$5$	−13.7745	−1.23203	−0.616015	−	0.787734i	$-0.711254\pi$
$5$	−13.7745	−1.23203	−0.616015	+	0.787734i	$0.711254\pi$
$6$	0	0
$7$	−35.6462	−1.92471	−0.962357	−	0.271790i	$-0.912384\pi$
$7$	−35.6462	−1.92471	−0.962357	+	0.271790i	$0.912384\pi$
$8$	23.2088	1.02569
$9$	0	0
$10$	37.8683	1.19750
$11$	30.8537	0.845703	0.422851	−	0.906199i	$-0.361029\pi$
$11$	30.8537	0.845703	0.422851	+	0.906199i	$0.361029\pi$
$12$	0	0
$13$	14.6435	0.312413	0.156206	−	0.987724i	$-0.450074\pi$
$13$	14.6435	0.312413	0.156206	+	0.987724i	$0.450074\pi$
$14$	97.9970	1.87077
$15$	0	0
$16$	−60.2674	−0.941679
$17$	77.8827	1.11114	0.555568	−	0.831471i	$-0.312501\pi$
$17$	77.8827	1.11114	0.555568	+	0.831471i	$0.312501\pi$
$18$	0	0
$19$	98.6377	1.19100	0.595501	−	0.803355i	$-0.296954\pi$
$19$	98.6377	1.19100	0.595501	+	0.803355i	$0.296954\pi$
$20$	6.09019	0.0680904
$21$	0	0
$22$	−84.8216	−0.822001
$23$	117.065	1.06129	0.530647	−	0.847593i	$-0.321949\pi$
$23$	117.065	1.06129	0.530647	+	0.847593i	$0.321949\pi$
$24$	0	0
$25$	64.7375	0.517900
$26$	−40.2572	−0.303657
$27$	0	0
$28$	15.7604	0.106373
$29$	54.2015	0.347068	0.173534	−	0.984828i	$-0.444481\pi$
$29$	54.2015	0.347068	0.173534	+	0.984828i	$0.444481\pi$
$30$	0	0
$31$	−231.433	−1.34086	−0.670428	−	0.741975i	$-0.733890\pi$
$31$	−231.433	−1.34086	−0.670428	+	0.741975i	$0.733890\pi$
$32$	−19.9854	−0.110405
$33$	0	0
$34$	−214.112	−1.08000
$35$	491.009	2.37131
$36$	0	0
$37$	−258.311	−1.14773	−0.573866	−	0.818950i	$-0.694557\pi$
$37$	−258.311	−1.14773	−0.573866	+	0.818950i	$0.694557\pi$
$38$	−271.171	−1.15762
$39$	0	0
$40$	−319.690	−1.26368
$41$	−326.921	−1.24528	−0.622639	−	0.782509i	$-0.713940\pi$
$41$	−326.921	−1.24528	−0.622639	+	0.782509i	$0.713940\pi$
$42$	0	0
$43$	250.920	0.889882	0.444941	−	0.895560i	$-0.353225\pi$
$43$	250.920	0.889882	0.444941	+	0.895560i	$0.353225\pi$
$44$	−13.6415	−0.0467393
$45$	0	0
$46$	−321.830	−1.03155
$47$	−509.390	−1.58090	−0.790449	−	0.612528i	$-0.790153\pi$
$47$	−509.390	−1.58090	−0.790449	+	0.612528i	$0.790153\pi$
$48$	0	0
$49$	927.650	2.70452
$50$	−177.974	−0.503385
$51$	0	0
$52$	−6.47437	−0.0172660
$53$	313.730	0.813097	0.406549	−	0.913629i	$-0.366732\pi$
$53$	313.730	0.813097	0.406549	+	0.913629i	$0.366732\pi$
$54$	0	0
$55$	−424.994	−1.04193
$56$	−827.304	−1.97416
$57$	0	0
$58$	−149.008	−0.337341
$59$	−59.0000	−0.130189
$59$	−59.0000	−0.130189
$60$	0	0
$61$	528.854	1.11005	0.555023	−	0.831835i	$-0.312709\pi$
$61$	528.854	1.11005	0.555023	+	0.831835i	$0.312709\pi$
$62$	636.245	1.30328
$63$	0	0
$64$	537.082	1.04899
$65$	−201.707	−0.384902
$66$	0	0
$67$	−373.284	−0.680656	−0.340328	−	0.940307i	$-0.610538\pi$
$67$	−373.284	−0.680656	−0.340328	+	0.940307i	$0.610538\pi$
$68$	−34.4346	−0.0614090
$69$	0	0
$70$	−1349.86	−2.30485
$71$	521.969	0.872483	0.436241	−	0.899830i	$-0.356309\pi$
$71$	521.969	0.872483	0.436241	+	0.899830i	$0.356309\pi$
$72$	0	0
$73$	809.404	1.29772	0.648860	−	0.760908i	$-0.275246\pi$
$73$	809.404	1.29772	0.648860	+	0.760908i	$0.275246\pi$
$74$	710.137	1.11556
$75$	0	0
$76$	−43.6111	−0.0658229
$77$	−1099.82	−1.62773
$78$	0	0
$79$	−926.088	−1.31890	−0.659449	−	0.751749i	$-0.729211\pi$
$79$	−926.088	−1.31890	−0.659449	+	0.751749i	$0.729211\pi$
$80$	830.155	1.16018
$81$	0	0
$82$	898.756	1.21038
$83$	1311.77	1.73476	0.867380	−	0.497646i	$-0.165802\pi$
$83$	1311.77	1.73476	0.867380	+	0.497646i	$0.165802\pi$
$84$	0	0
$85$	−1072.80	−1.36895
$86$	−689.818	−0.864942
$87$	0	0
$88$	716.075	0.867430
$89$	−510.201	−0.607654	−0.303827	−	0.952727i	$-0.598265\pi$
$89$	−510.201	−0.607654	−0.303827	+	0.952727i	$0.598265\pi$
$90$	0	0
$91$	−521.983	−0.601305
$92$	−51.7585	−0.0586543
$93$	0	0
$94$	1400.39	1.53659
$95$	−1358.69	−1.46735
$96$	0	0
$97$	−856.449	−0.896486	−0.448243	−	0.893912i	$-0.647950\pi$
$97$	−856.449	−0.896486	−0.448243	+	0.893912i	$0.647950\pi$
$98$	−2550.26	−2.62872
$99$	0	0
$100$	−28.6227	−0.0286227
$101$	−811.456	−0.799435	−0.399717	−	0.916638i	$-0.630892\pi$
$101$	−811.456	−0.799435	−0.399717	+	0.916638i	$0.630892\pi$
$102$	0	0
$103$	−419.340	−0.401153	−0.200577	−	0.979678i	$-0.564282\pi$
$103$	−419.340	−0.401153	−0.200577	+	0.979678i	$0.564282\pi$
$104$	339.856	0.320439
$105$	0	0
$106$	−862.494	−0.790309
$107$	1165.05	1.05261	0.526305	−	0.850296i	$-0.323577\pi$
$107$	1165.05	1.05261	0.526305	+	0.850296i	$0.323577\pi$
$108$	0	0
$109$	−2016.17	−1.77169	−0.885844	−	0.463984i	$-0.846419\pi$
$109$	−2016.17	−1.77169	−0.885844	+	0.463984i	$0.846419\pi$
$110$	1168.38	1.01273
$111$	0	0
$112$	2148.30	1.81246
$113$	731.760	0.609188	0.304594	−	0.952482i	$-0.401479\pi$
$113$	731.760	0.609188	0.304594	+	0.952482i	$0.401479\pi$
$114$	0	0
$115$	−1612.51	−1.30755
$116$	−23.9643	−0.0191813
$117$	0	0
$118$	162.200	0.126540
$119$	−2776.22	−2.13862
$120$	0	0
$121$	−379.051	−0.284787
$122$	−1453.90	−1.07894
$123$	0	0
$124$	102.324	0.0741048
$125$	830.087	0.593962
$126$	0	0
$127$	−1187.43	−0.829666	−0.414833	−	0.909898i	$-0.636160\pi$
$127$	−1187.43	−0.829666	−0.414833	+	0.909898i	$0.636160\pi$
$128$	−1316.64	−0.909185
$129$	0	0
$130$	554.523	0.374115
$131$	250.996	0.167402	0.0837008	−	0.996491i	$-0.473326\pi$
$131$	250.996	0.167402	0.0837008	+	0.996491i	$0.473326\pi$
$132$	0	0
$133$	−3516.06	−2.29234
$134$	1026.22	0.661580
$135$	0	0
$136$	1807.56	1.13968
$137$	−2474.51	−1.54315	−0.771576	−	0.636137i	$-0.780531\pi$
$137$	−2474.51	−1.54315	−0.771576	+	0.636137i	$0.780531\pi$
$138$	0	0
$139$	2067.65	1.26169	0.630847	−	0.775907i	$-0.282708\pi$
$139$	2067.65	1.26169	0.630847	+	0.775907i	$0.282708\pi$
$140$	−217.092	−0.131054
$141$	0	0
$142$	−1434.97	−0.848031
$143$	451.804	0.264208
$144$	0	0
$145$	−746.600	−0.427598
$146$	−2225.18	−1.26135
$147$	0	0
$148$	114.208	0.0634314
$149$	−70.2921	−0.0386480	−0.0193240	−	0.999813i	$-0.506151\pi$
$149$	−70.2921	−0.0386480	−0.0193240	+	0.999813i	$0.506151\pi$
$150$	0	0
$151$	2156.31	1.16211	0.581054	−	0.813865i	$-0.302641\pi$
$151$	2156.31	1.16211	0.581054	+	0.813865i	$0.302641\pi$
$152$	2289.26	1.22160
$153$	0	0
$154$	3023.57	1.58212
$155$	3187.88	1.65198
$156$	0	0
$157$	105.729	0.0537459	0.0268729	−	0.999639i	$-0.491445\pi$
$157$	105.729	0.0537459	0.0268729	+	0.999639i	$0.491445\pi$
$158$	2545.96	1.28194
$159$	0	0
$160$	275.289	0.136022
$161$	−4172.92	−2.04268
$162$	0	0
$163$	−1006.11	−0.483463	−0.241731	−	0.970343i	$-0.577715\pi$
$163$	−1006.11	−0.483463	−0.241731	+	0.970343i	$0.577715\pi$
$164$	144.543	0.0688226
$165$	0	0
$166$	−3606.25	−1.68614
$167$	668.739	0.309872	0.154936	−	0.987925i	$-0.450483\pi$
$167$	668.739	0.309872	0.154936	+	0.987925i	$0.450483\pi$
$168$	0	0
$169$	−1982.57	−0.902398
$170$	2949.29	1.33059
$171$	0	0
$172$	−110.940	−0.0491809
$173$	2879.96	1.26566	0.632831	−	0.774290i	$-0.281893\pi$
$173$	2879.96	1.26566	0.632831	+	0.774290i	$0.281893\pi$
$174$	0	0
$175$	−2307.64	−0.996809
$176$	−1859.47	−0.796380
$177$	0	0
$178$	1402.62	0.590624
$179$	−1700.00	−0.709856	−0.354928	−	0.934894i	$-0.615495\pi$
$179$	−1700.00	−0.709856	−0.354928	+	0.934894i	$0.615495\pi$
$180$	0	0
$181$	−1750.62	−0.718910	−0.359455	−	0.933162i	$-0.617037\pi$
$181$	−1750.62	−0.718910	−0.359455	+	0.933162i	$0.617037\pi$
$182$	1435.01	0.584452
$183$	0	0
$184$	2716.93	1.08856
$185$	3558.11	1.41404
$186$	0	0
$187$	2402.97	0.939691
$188$	225.219	0.0873711
$189$	0	0
$190$	3735.24	1.42623
$191$	−1504.78	−0.570062	−0.285031	−	0.958518i	$-0.592004\pi$
$191$	−1504.78	−0.570062	−0.285031	+	0.958518i	$0.592004\pi$
$192$	0	0
$193$	−803.437	−0.299651	−0.149826	−	0.988712i	$-0.547871\pi$
$193$	−803.437	−0.299651	−0.149826	+	0.988712i	$0.547871\pi$
$194$	2354.51	0.871361
$195$	0	0
$196$	−410.146	−0.149470
$197$	1329.45	0.480808	0.240404	−	0.970673i	$-0.422720\pi$
$197$	1329.45	0.480808	0.240404	+	0.970673i	$0.422720\pi$
$198$	0	0
$199$	−2857.91	−1.01805	−0.509025	−	0.860752i	$-0.669994\pi$
$199$	−2857.91	−1.01805	−0.509025	+	0.860752i	$0.669994\pi$
$200$	1502.48	0.531206
$201$	0	0
$202$	2230.82	0.777030
$203$	−1932.08	−0.668006
$204$	0	0
$205$	4503.18	1.53422
$206$	1152.83	0.389911
$207$	0	0
$208$	−882.524	−0.294192
$209$	3043.33	1.00723
$210$	0	0
$211$	−3121.15	−1.01834	−0.509168	−	0.860667i	$-0.670047\pi$
$211$	−3121.15	−1.01834	−0.509168	+	0.860667i	$0.670047\pi$
$212$	−138.711	−0.0449373
$213$	0	0
$214$	−3202.89	−1.02311
$215$	−3456.30	−1.09636
$216$	0	0
$217$	8249.69	2.58076
$218$	5542.76	1.72203
$219$	0	0
$220$	187.905	0.0575842
$221$	1140.47	0.347133
$222$	0	0
$223$	−2100.41	−0.630734	−0.315367	−	0.948970i	$-0.602128\pi$
$223$	−2100.41	−0.630734	−0.315367	+	0.948970i	$0.602128\pi$
$224$	712.402	0.212497
$225$	0	0
$226$	−2011.72	−0.592114
$227$	−2601.37	−0.760613	−0.380306	−	0.924861i	$-0.624181\pi$
$227$	−2601.37	−0.760613	−0.380306	+	0.924861i	$0.624181\pi$
$228$	0	0
$229$	−1502.73	−0.433637	−0.216819	−	0.976212i	$-0.569568\pi$
$229$	−1502.73	−0.433637	−0.216819	+	0.976212i	$0.569568\pi$
$230$	4433.06	1.27090
$231$	0	0
$232$	1257.95	0.355984
$233$	1672.77	0.470329	0.235164	−	0.971956i	$-0.424437\pi$
$233$	1672.77	0.470329	0.235164	+	0.971956i	$0.424437\pi$
$234$	0	0
$235$	7016.60	1.94771
$236$	26.0859	0.00719512
$237$	0	0
$238$	7632.27	2.07868
$239$	−3489.49	−0.944420	−0.472210	−	0.881486i	$-0.656544\pi$
$239$	−3489.49	−0.944420	−0.472210	+	0.881486i	$0.656544\pi$
$240$	0	0
$241$	−872.061	−0.233089	−0.116544	−	0.993185i	$-0.537182\pi$
$241$	−872.061	−0.233089	−0.116544	+	0.993185i	$0.537182\pi$
$242$	1042.07	0.276806
$243$	0	0
$244$	−233.825	−0.0613487
$245$	−12777.9	−3.33205
$246$	0	0
$247$	1444.40	0.372084
$248$	−5371.26	−1.37531
$249$	0	0
$250$	−2282.04	−0.577316
$251$	463.531	0.116565	0.0582825	−	0.998300i	$-0.481438\pi$
$251$	463.531	0.116565	0.0582825	+	0.998300i	$0.481438\pi$
$252$	0	0
$253$	3611.88	0.897538
$254$	3264.44	0.806413
$255$	0	0
$256$	−677.006	−0.165285
$257$	−2168.52	−0.526338	−0.263169	−	0.964750i	$-0.584768\pi$
$257$	−2168.52	−0.526338	−0.263169	+	0.964750i	$0.584768\pi$
$258$	0	0
$259$	9207.80	2.20905
$260$	89.1814	0.0212723
$261$	0	0
$262$	−690.027	−0.162710
$263$	−567.749	−0.133114	−0.0665569	−	0.997783i	$-0.521201\pi$
$263$	−567.749	−0.133114	−0.0665569	+	0.997783i	$0.521201\pi$
$264$	0	0
$265$	−4321.48	−1.00176
$266$	9666.20	2.22809
$267$	0	0
$268$	165.042	0.0376177
$269$	326.225	0.0739415	0.0369708	−	0.999316i	$-0.488229\pi$
$269$	326.225	0.0739415	0.0369708	+	0.999316i	$0.488229\pi$
$270$	0	0
$271$	1478.00	0.331300	0.165650	−	0.986185i	$-0.447028\pi$
$271$	1478.00	0.331300	0.165650	+	0.986185i	$0.447028\pi$
$272$	−4693.79	−1.04633
$273$	0	0
$274$	6802.82	1.49990
$275$	1997.39	0.437989
$276$	0	0
$277$	8647.79	1.87580	0.937898	−	0.346911i	$-0.112769\pi$
$277$	8647.79	1.87580	0.937898	+	0.346911i	$0.112769\pi$
$278$	−5684.29	−1.22633
$279$	0	0
$280$	11395.7	2.43223
$281$	4653.34	0.987882	0.493941	−	0.869495i	$-0.335556\pi$
$281$	4653.34	0.987882	0.493941	+	0.869495i	$0.335556\pi$
$282$	0	0
$283$	−2594.93	−0.545062	−0.272531	−	0.962147i	$-0.587861\pi$
$283$	−2594.93	−0.545062	−0.272531	+	0.962147i	$0.587861\pi$
$284$	−230.780	−0.0482193
$285$	0	0
$286$	−1242.08	−0.256803
$287$	11653.5	2.39680
$288$	0	0
$289$	1152.71	0.234625
$290$	2052.52	0.415614
$291$	0	0
$292$	−357.865	−0.0717208
$293$	7299.73	1.45548	0.727739	−	0.685854i	$-0.240571\pi$
$293$	7299.73	1.45548	0.727739	+	0.685854i	$0.240571\pi$
$294$	0	0
$295$	812.697	0.160397
$296$	−5995.07	−1.17722
$297$	0	0
$298$	193.244	0.0375648
$299$	1714.24	0.331561
$300$	0	0
$301$	−8944.34	−1.71277
$302$	−5928.04	−1.12954
$303$	0	0
$304$	−5944.64	−1.12154
$305$	−7284.72	−1.36761
$306$	0	0
$307$	3677.41	0.683651	0.341826	−	0.939763i	$-0.388955\pi$
$307$	3677.41	0.683651	0.341826	+	0.939763i	$0.388955\pi$
$308$	486.266	0.0899597
$309$	0	0
$310$	−8763.97	−1.60568
$311$	−3389.46	−0.618002	−0.309001	−	0.951062i	$-0.599995\pi$
$311$	−3389.46	−0.618002	−0.309001	+	0.951062i	$0.599995\pi$
$312$	0	0
$313$	−4847.47	−0.875384	−0.437692	−	0.899125i	$-0.644204\pi$
$313$	−4847.47	−0.875384	−0.437692	+	0.899125i	$0.644204\pi$
$314$	−290.666	−0.0522396
$315$	0	0
$316$	409.455	0.0728913
$317$	−10460.9	−1.85344	−0.926721	−	0.375750i	$-0.877385\pi$
$317$	−10460.9	−1.85344	−0.926721	+	0.375750i	$0.877385\pi$
$318$	0	0
$319$	1672.31	0.293516
$320$	−7398.05	−1.29239
$321$	0	0
$322$	11472.0	1.98544
$323$	7682.17	1.32337
$324$	0	0
$325$	947.980	0.161798
$326$	2765.95	0.469913
$327$	0	0
$328$	−7587.42	−1.27727
$329$	18157.8	3.04277
$330$	0	0
$331$	−2859.84	−0.474898	−0.237449	−	0.971400i	$-0.576311\pi$
$331$	−2859.84	−0.474898	−0.237449	+	0.971400i	$0.576311\pi$
$332$	−579.977	−0.0958747
$333$	0	0
$334$	−1838.47	−0.301187
$335$	5141.82	0.838589
$336$	0	0
$337$	4699.94	0.759710	0.379855	−	0.925046i	$-0.375974\pi$
$337$	4699.94	0.759710	0.379855	+	0.925046i	$0.375974\pi$
$338$	5450.39	0.877108
$339$	0	0
$340$	474.320	0.0756577
$341$	−7140.55	−1.13397
$342$	0	0
$343$	−20840.6	−3.28071
$344$	5823.54	0.912745
$345$	0	0
$346$	−7917.47	−1.23019
$347$	−951.938	−0.147270	−0.0736350	−	0.997285i	$-0.523460\pi$
$347$	−951.938	−0.147270	−0.0736350	+	0.997285i	$0.523460\pi$
$348$	0	0
$349$	−10647.9	−1.63315	−0.816575	−	0.577239i	$-0.804130\pi$
$349$	−10647.9	−1.63315	−0.816575	+	0.577239i	$0.804130\pi$
$350$	6344.08	0.968872
$351$	0	0
$352$	−616.622	−0.0933695
$353$	−11818.1	−1.78191	−0.890957	−	0.454088i	$-0.849965\pi$
$353$	−11818.1	−1.78191	−0.890957	+	0.454088i	$0.849965\pi$
$354$	0	0
$355$	−7189.87	−1.07493
$356$	225.578	0.0335831
$357$	0	0
$358$	4673.58	0.689962
$359$	−8794.89	−1.29297	−0.646485	−	0.762926i	$-0.723762\pi$
$359$	−8794.89	−1.29297	−0.646485	+	0.762926i	$0.723762\pi$
$360$	0	0
$361$	2870.40	0.418486
$362$	4812.74	0.698762
$363$	0	0
$364$	230.787	0.0332322
$365$	−11149.2	−1.59883
$366$	0	0
$367$	13284.1	1.88945	0.944723	−	0.327869i	$-0.106331\pi$
$367$	13284.1	1.88945	0.944723	+	0.327869i	$0.106331\pi$
$368$	−7055.21	−0.999397
$369$	0	0
$370$	−9781.80	−1.37441
$371$	−11183.3	−1.56498
$372$	0	0
$373$	−11897.3	−1.65153	−0.825763	−	0.564018i	$-0.809255\pi$
$373$	−11897.3	−1.65153	−0.825763	+	0.564018i	$0.809255\pi$
$374$	−6606.13	−0.913355
$375$	0	0
$376$	−11822.3	−1.62151
$377$	793.697	0.108428
$378$	0	0
$379$	1743.93	0.236357	0.118179	−	0.992992i	$-0.462294\pi$
$379$	1743.93	0.236357	0.118179	+	0.992992i	$0.462294\pi$
$380$	600.722	0.0810958
$381$	0	0
$382$	4136.87	0.554085
$383$	−1572.59	−0.209806	−0.104903	−	0.994482i	$-0.533453\pi$
$383$	−1572.59	−0.209806	−0.104903	+	0.994482i	$0.533453\pi$
$384$	0	0
$385$	15149.4	2.00542
$386$	2208.77	0.291253
$387$	0	0
$388$	378.665	0.0495459
$389$	3545.77	0.462153	0.231077	−	0.972936i	$-0.425775\pi$
$389$	3545.77	0.462153	0.231077	+	0.972936i	$0.425775\pi$
$390$	0	0
$391$	9117.34	1.17924
$392$	21529.6	2.77400
$393$	0	0
$394$	−3654.86	−0.467333
$395$	12756.4	1.62492
$396$	0	0
$397$	7176.14	0.907205	0.453602	−	0.891204i	$-0.350139\pi$
$397$	7176.14	0.907205	0.453602	+	0.891204i	$0.350139\pi$
$398$	7856.85	0.989518
$399$	0	0
$400$	−3901.56	−0.487695
$401$	−7431.68	−0.925487	−0.462744	−	0.886492i	$-0.653135\pi$
$401$	−7431.68	−0.925487	−0.462744	+	0.886492i	$0.653135\pi$
$402$	0	0
$403$	−3388.97	−0.418900
$404$	358.773	0.0441822
$405$	0	0
$406$	5311.58	0.649284
$407$	−7969.84	−0.970639
$408$	0	0
$409$	−5038.83	−0.609179	−0.304589	−	0.952484i	$-0.598519\pi$
$409$	−5038.83	−0.609179	−0.304589	+	0.952484i	$0.598519\pi$
$410$	−12379.9	−1.49122
$411$	0	0
$412$	185.405	0.0221705
$413$	2103.12	0.250576
$414$	0	0
$415$	−18069.0	−2.13728
$416$	−292.655	−0.0344918
$417$	0	0
$418$	−8366.60	−0.979005
$419$	3543.16	0.413114	0.206557	−	0.978435i	$-0.433774\pi$
$419$	3543.16	0.413114	0.206557	+	0.978435i	$0.433774\pi$
$420$	0	0
$421$	13131.9	1.52022	0.760109	−	0.649796i	$-0.225145\pi$
$421$	13131.9	1.52022	0.760109	+	0.649796i	$0.225145\pi$
$422$	8580.54	0.989797
$423$	0	0
$424$	7281.29	0.833987
$425$	5041.93	0.575458
$426$	0	0
$427$	−18851.6	−2.13652
$428$	−515.107	−0.0581744
$429$	0	0
$430$	9501.92	1.06564
$431$	9117.45	1.01896	0.509480	−	0.860482i	$-0.329838\pi$
$431$	9117.45	1.01896	0.509480	+	0.860482i	$0.329838\pi$
$432$	0	0
$433$	−2554.40	−0.283502	−0.141751	−	0.989902i	$-0.545273\pi$
$433$	−2554.40	−0.283502	−0.141751	+	0.989902i	$0.545273\pi$
$434$	−22679.7	−2.50843
$435$	0	0
$436$	891.417	0.0979155
$437$	11547.0	1.26400
$438$	0	0
$439$	1523.94	0.165681	0.0828404	−	0.996563i	$-0.473601\pi$
$439$	1523.94	0.165681	0.0828404	+	0.996563i	$0.473601\pi$
$440$	−9863.59	−1.06870
$441$	0	0
$442$	−3135.34	−0.337404
$443$	−10110.8	−1.08437	−0.542186	−	0.840258i	$-0.682403\pi$
$443$	−10110.8	−1.08437	−0.542186	+	0.840258i	$0.682403\pi$
$444$	0	0
$445$	7027.78	0.748649
$446$	5774.35	0.613057
$447$	0	0
$448$	−19144.9	−2.01900
$449$	−17927.7	−1.88432	−0.942162	−	0.335158i	$-0.891210\pi$
$449$	−17927.7	−1.88432	−0.942162	+	0.335158i	$0.891210\pi$
$450$	0	0
$451$	−10086.7	−1.05314
$452$	−323.536	−0.0336678
$453$	0	0
$454$	7151.58	0.739296
$455$	7190.07	0.740826
$456$	0	0
$457$	−8911.04	−0.912125	−0.456062	−	0.889948i	$-0.650741\pi$
$457$	−8911.04	−0.912125	−0.456062	+	0.889948i	$0.650741\pi$
$458$	4131.23	0.421484
$459$	0	0
$460$	712.948	0.0722639
$461$	−16724.3	−1.68965	−0.844825	−	0.535042i	$-0.820296\pi$
$461$	−16724.3	−1.68965	−0.844825	+	0.535042i	$0.820296\pi$
$462$	0	0
$463$	−14050.4	−1.41031	−0.705157	−	0.709051i	$-0.749123\pi$
$463$	−14050.4	−1.41031	−0.705157	+	0.709051i	$0.749123\pi$
$464$	−3266.58	−0.326826
$465$	0	0
$466$	−4598.70	−0.457147
$467$	−10526.3	−1.04304	−0.521520	−	0.853239i	$-0.674635\pi$
$467$	−10526.3	−1.04304	−0.521520	+	0.853239i	$0.674635\pi$
$468$	0	0
$469$	13306.2	1.31007
$470$	−19289.7	−1.89313
$471$	0	0
$472$	−1369.32	−0.133534
$473$	7741.80	0.752576
$474$	0	0
$475$	6385.56	0.616820
$476$	1227.46	0.118195
$477$	0	0
$478$	9593.16	0.917952
$479$	6393.78	0.609894	0.304947	−	0.952369i	$-0.401361\pi$
$479$	6393.78	0.609894	0.304947	+	0.952369i	$0.401361\pi$
$480$	0	0
$481$	−3782.56	−0.358566
$482$	2397.43	0.226556
$483$	0	0
$484$	167.592	0.0157393
$485$	11797.2	1.10450
$486$	0	0
$487$	14196.4	1.32095	0.660473	−	0.750850i	$-0.270356\pi$
$487$	14196.4	1.32095	0.660473	+	0.750850i	$0.270356\pi$
$488$	12274.1	1.13857
$489$	0	0
$490$	35128.6	3.23867
$491$	5867.49	0.539299	0.269650	−	0.962958i	$-0.413092\pi$
$491$	5867.49	0.539299	0.269650	+	0.962958i	$0.413092\pi$
$492$	0	0
$493$	4221.36	0.385640
$494$	−3970.87	−0.361656
$495$	0	0
$496$	13947.9	1.26266
$497$	−18606.2	−1.67928
$498$	0	0
$499$	−8345.43	−0.748683	−0.374341	−	0.927291i	$-0.622131\pi$
$499$	−8345.43	−0.748683	−0.374341	+	0.927291i	$0.622131\pi$
$500$	−367.010	−0.0328264
$501$	0	0
$502$	−1274.32	−0.113298
$503$	1607.53	0.142498	0.0712488	−	0.997459i	$-0.477302\pi$
$503$	1607.53	0.142498	0.0712488	+	0.997459i	$0.477302\pi$
$504$	0	0
$505$	11177.4	0.984929
$506$	−9929.64	−0.872384
$507$	0	0
$508$	525.004	0.0458530
$509$	9106.68	0.793018	0.396509	−	0.918031i	$-0.370221\pi$
$509$	9106.68	0.793018	0.396509	+	0.918031i	$0.370221\pi$
$510$	0	0
$511$	−28852.2	−2.49774
$512$	12394.3	1.06984
$513$	0	0
$514$	5961.61	0.511586
$515$	5776.21	0.494233
$516$	0	0
$517$	−15716.5	−1.33697
$518$	−25313.7	−2.14714
$519$	0	0
$520$	−4681.36	−0.394791
$521$	7301.03	0.613942	0.306971	−	0.951719i	$-0.400684\pi$
$521$	7301.03	0.613942	0.306971	+	0.951719i	$0.400684\pi$
$522$	0	0
$523$	4874.89	0.407579	0.203790	−	0.979015i	$-0.434674\pi$
$523$	4874.89	0.407579	0.203790	+	0.979015i	$0.434674\pi$
$524$	−110.974	−0.00925175
$525$	0	0
$526$	1560.83	0.129383
$527$	−18024.6	−1.48987
$528$	0	0
$529$	1537.22	0.126343
$530$	11880.4	0.973685
$531$	0	0
$532$	1554.57	0.126690
$533$	−4787.25	−0.389041
$534$	0	0
$535$	−16047.9	−1.29685
$536$	−8663.47	−0.698143
$537$	0	0
$538$	−896.843	−0.0718692
$539$	28621.4	2.28722
$540$	0	0
$541$	−1382.28	−0.109850	−0.0549249	−	0.998490i	$-0.517492\pi$
$541$	−1382.28	−0.109850	−0.0549249	+	0.998490i	$0.517492\pi$
$542$	−4063.26	−0.322015
$543$	0	0
$544$	−1556.51	−0.122675
$545$	27771.8	2.18277
$546$	0	0
$547$	−20117.5	−1.57251	−0.786255	−	0.617902i	$-0.787983\pi$
$547$	−20117.5	−1.57251	−0.786255	+	0.617902i	$0.787983\pi$
$548$	1094.07	0.0852851
$549$	0	0
$550$	−5491.14	−0.425714
$551$	5346.31	0.413358
$552$	0	0
$553$	33011.5	2.53850
$554$	−23774.1	−1.82322
$555$	0	0
$556$	−914.178	−0.0697298
$557$	12926.3	0.983311	0.491655	−	0.870790i	$-0.336392\pi$
$557$	12926.3	0.983311	0.491655	+	0.870790i	$0.336392\pi$
$558$	0	0
$559$	3674.33	0.278010
$560$	−29591.9	−2.23301
$561$	0	0
$562$	−12792.8	−0.960195
$563$	−1483.56	−0.111056	−0.0555279	−	0.998457i	$-0.517684\pi$
$563$	−1483.56	−0.111056	−0.0555279	+	0.998457i	$0.517684\pi$
$564$	0	0
$565$	−10079.6	−0.750538
$566$	7133.87	0.529786
$567$	0	0
$568$	12114.2	0.894899
$569$	−535.139	−0.0394274	−0.0197137	−	0.999806i	$-0.506275\pi$
$569$	−535.139	−0.0394274	−0.0197137	+	0.999806i	$0.506275\pi$
$570$	0	0
$571$	11303.3	0.828420	0.414210	−	0.910181i	$-0.364058\pi$
$571$	11303.3	0.828420	0.414210	+	0.910181i	$0.364058\pi$
$572$	−199.758	−0.0146019
$573$	0	0
$574$	−32037.2	−2.32963
$575$	7578.50	0.549644
$576$	0	0
$577$	−22867.6	−1.64990	−0.824950	−	0.565206i	$-0.808797\pi$
$577$	−22867.6	−1.64990	−0.824950	+	0.565206i	$0.808797\pi$
$578$	−3168.99	−0.228049
$579$	0	0
$580$	330.097	0.0236320
$581$	−46759.5	−3.33892
$582$	0	0
$583$	9679.73	0.687639
$584$	18785.3	1.33106
$585$	0	0
$586$	−20068.1	−1.41469
$587$	17589.4	1.23679	0.618393	−	0.785869i	$-0.287784\pi$
$587$	17589.4	1.23679	0.618393	+	0.785869i	$0.287784\pi$
$588$	0	0
$589$	−22828.0	−1.59696
$590$	−2234.23	−0.155901
$591$	0	0
$592$	15567.7	1.08079
$593$	16110.5	1.11565	0.557825	−	0.829959i	$-0.311636\pi$
$593$	16110.5	1.11565	0.557825	+	0.829959i	$0.311636\pi$
$594$	0	0
$595$	38241.1	2.63485
$596$	31.0785	0.00213595
$597$	0	0
$598$	−4712.70	−0.322269
$599$	−22231.7	−1.51646	−0.758231	−	0.651986i	$-0.773936\pi$
$599$	−22231.7	−1.51646	−0.758231	+	0.651986i	$0.773936\pi$
$600$	0	0
$601$	−16668.9	−1.13135	−0.565673	−	0.824629i	$-0.691384\pi$
$601$	−16668.9	−1.13135	−0.565673	+	0.824629i	$0.691384\pi$
$602$	24589.4	1.66477
$603$	0	0
$604$	−953.380	−0.0642259
$605$	5221.25	0.350866
$606$	0	0
$607$	11293.3	0.755155	0.377577	−	0.925978i	$-0.376757\pi$
$607$	11293.3	0.755155	0.377577	+	0.925978i	$0.376757\pi$
$608$	−1971.31	−0.131492
$609$	0	0
$610$	20026.8	1.32928
$611$	−7459.23	−0.493892
$612$	0	0
$613$	−9187.06	−0.605321	−0.302661	−	0.953098i	$-0.597875\pi$
$613$	−9187.06	−0.605321	−0.302661	+	0.953098i	$0.597875\pi$
$614$	−10109.8	−0.664491
$615$	0	0
$616$	−25525.3	−1.66955
$617$	5405.24	0.352685	0.176342	−	0.984329i	$-0.443573\pi$
$617$	5405.24	0.352685	0.176342	+	0.984329i	$0.443573\pi$
$618$	0	0
$619$	26890.8	1.74610	0.873049	−	0.487633i	$-0.162140\pi$
$619$	26890.8	1.74610	0.873049	+	0.487633i	$0.162140\pi$
$620$	−1409.47	−0.0912994
$621$	0	0
$622$	9318.16	0.600682
$623$	18186.7	1.16956
$624$	0	0
$625$	−19526.2	−1.24968
$626$	13326.5	0.850850
$627$	0	0
$628$	−46.7465	−0.00297036
$629$	−20117.9	−1.27529
$630$	0	0
$631$	13088.1	0.825718	0.412859	−	0.910795i	$-0.364530\pi$
$631$	13088.1	0.825718	0.412859	+	0.910795i	$0.364530\pi$
$632$	−21493.3	−1.35278
$633$	0	0
$634$	28758.6	1.80150
$635$	16356.3	1.02217
$636$	0	0
$637$	13584.0	0.844926
$638$	−4597.46	−0.285290
$639$	0	0
$640$	18136.1	1.12014
$641$	−11372.4	−0.700754	−0.350377	−	0.936609i	$-0.613947\pi$
$641$	−11372.4	−0.700754	−0.350377	+	0.936609i	$0.613947\pi$
$642$	0	0
$643$	28882.8	1.77142	0.885712	−	0.464236i	$-0.153671\pi$
$643$	28882.8	1.77142	0.885712	+	0.464236i	$0.153671\pi$
$644$	1844.99	0.112893
$645$	0	0
$646$	−21119.5	−1.28628
$647$	−3509.03	−0.213222	−0.106611	−	0.994301i	$-0.534000\pi$
$647$	−3509.03	−0.213222	−0.106611	+	0.994301i	$0.534000\pi$
$648$	0	0
$649$	−1820.37	−0.110101
$650$	−2606.15	−0.157264
$651$	0	0
$652$	444.835	0.0267194
$653$	12551.5	0.752185	0.376092	−	0.926582i	$-0.377268\pi$
$653$	12551.5	0.752185	0.376092	+	0.926582i	$0.377268\pi$
$654$	0	0
$655$	−3457.35	−0.206244
$656$	19702.7	1.17265
$657$	0	0
$658$	−49918.7	−2.95750
$659$	26042.1	1.53938	0.769692	−	0.638415i	$-0.220410\pi$
$659$	26042.1	1.53938	0.769692	+	0.638415i	$0.220410\pi$
$660$	0	0
$661$	8841.43	0.520260	0.260130	−	0.965574i	$-0.416235\pi$
$661$	8841.43	0.520260	0.260130	+	0.965574i	$0.416235\pi$
$662$	7862.16	0.461588
$663$	0	0
$664$	30444.5	1.77933
$665$	48432.0	2.82423
$666$	0	0
$667$	6345.10	0.368341
$668$	−295.673	−0.0171256
$669$	0	0
$670$	−14135.7	−0.815087
$671$	16317.1	0.938770
$672$	0	0
$673$	5539.99	0.317312	0.158656	−	0.987334i	$-0.449284\pi$
$673$	5539.99	0.317312	0.158656	+	0.987334i	$0.449284\pi$
$674$	−12920.9	−0.738418
$675$	0	0
$676$	876.562	0.0498727
$677$	−6105.80	−0.346625	−0.173312	−	0.984867i	$-0.555447\pi$
$677$	−6105.80	−0.346625	−0.173312	+	0.984867i	$0.555447\pi$
$678$	0	0
$679$	30529.1	1.72548
$680$	−24898.3	−1.40413
$681$	0	0
$682$	19630.5	1.10218
$683$	−6169.93	−0.345660	−0.172830	−	0.984952i	$-0.555291\pi$
$683$	−6169.93	−0.345660	−0.172830	+	0.984952i	$0.555291\pi$
$684$	0	0
$685$	34085.2	1.90121
$686$	57294.0	3.18877
$687$	0	0
$688$	−15122.3	−0.837983
$689$	4594.09	0.254022
$690$	0	0
$691$	−24137.5	−1.32885	−0.664423	−	0.747357i	$-0.731323\pi$
$691$	−24137.5	−1.32885	−0.664423	+	0.747357i	$0.731323\pi$
$692$	−1273.33	−0.0699491
$693$	0	0
$694$	2617.03	0.143143
$695$	−28480.8	−1.55445
$696$	0	0
$697$	−25461.5	−1.38368
$698$	29272.8	1.58738
$699$	0	0
$700$	1020.29	0.0550904
$701$	−1381.52	−0.0744354	−0.0372177	−	0.999307i	$-0.511849\pi$
$701$	−1381.52	−0.0744354	−0.0372177	+	0.999307i	$0.511849\pi$
$702$	0	0
$703$	−25479.2	−1.36695
$704$	16571.0	0.887133
$705$	0	0
$706$	32489.9	1.73197
$707$	28925.3	1.53868
$708$	0	0
$709$	11440.3	0.605995	0.302998	−	0.952991i	$-0.402013\pi$
$709$	11440.3	0.605995	0.302998	+	0.952991i	$0.402013\pi$
$710$	19766.1	1.04480
$711$	0	0
$712$	−11841.1	−0.623266
$713$	−27092.7	−1.42304
$714$	0	0
$715$	−6223.39	−0.325513
$716$	751.630	0.0392315
$717$	0	0
$718$	24178.5	1.25673
$719$	−23773.0	−1.23308	−0.616539	−	0.787324i	$-0.711466\pi$
$719$	−23773.0	−1.23308	−0.616539	+	0.787324i	$0.711466\pi$
$720$	0	0
$721$	14947.9	0.772105
$722$	−7891.17	−0.406757
$723$	0	0
$724$	774.010	0.0397319
$725$	3508.87	0.179746
$726$	0	0
$727$	−38328.7	−1.95534	−0.977670	−	0.210146i	$-0.932606\pi$
$727$	−38328.7	−1.95534	−0.977670	+	0.210146i	$0.932606\pi$
$728$	−12114.6	−0.616753
$729$	0	0
$730$	30650.8	1.55402
$731$	19542.3	0.988781
$732$	0	0
$733$	−23440.6	−1.18117	−0.590586	−	0.806975i	$-0.701103\pi$
$733$	−23440.6	−1.18117	−0.590586	+	0.806975i	$0.701103\pi$
$734$	−36520.2	−1.83649
$735$	0	0
$736$	−2339.59	−0.117172
$737$	−11517.2	−0.575633
$738$	0	0
$739$	−20289.0	−1.00994	−0.504968	−	0.863138i	$-0.668496\pi$
$739$	−20289.0	−1.00994	−0.504968	+	0.863138i	$0.668496\pi$
$740$	−1573.16	−0.0781495
$741$	0	0
$742$	30744.6	1.52112
$743$	21978.5	1.08521	0.542607	−	0.839987i	$-0.317437\pi$
$743$	21978.5	1.08521	0.542607	+	0.839987i	$0.317437\pi$
$744$	0	0
$745$	968.240	0.0476155
$746$	32707.6	1.60524
$747$	0	0
$748$	−1062.43	−0.0519337
$749$	−41529.4	−2.02597
$750$	0	0
$751$	−11321.9	−0.550122	−0.275061	−	0.961427i	$-0.588698\pi$
$751$	−11321.9	−0.550122	−0.275061	+	0.961427i	$0.588698\pi$
$752$	30699.6	1.48870
$753$	0	0
$754$	−2182.00	−0.105389
$755$	−29702.2	−1.43175
$756$	0	0
$757$	−29868.4	−1.43406	−0.717031	−	0.697042i	$-0.754499\pi$
$757$	−29868.4	−1.43406	−0.717031	+	0.697042i	$0.754499\pi$
$758$	−4794.33	−0.229733
$759$	0	0
$760$	−31533.4	−1.50505
$761$	−25711.9	−1.22478	−0.612388	−	0.790557i	$-0.709791\pi$
$761$	−25711.9	−1.22478	−0.612388	+	0.790557i	$0.709791\pi$
$762$	0	0
$763$	71868.7	3.40999
$764$	665.313	0.0315055
$765$	0	0
$766$	4323.31	0.203926
$767$	−863.964	−0.0406727
$768$	0	0
$769$	−16360.5	−0.767199	−0.383600	−	0.923500i	$-0.625316\pi$
$769$	−16360.5	−0.767199	−0.383600	+	0.923500i	$0.625316\pi$
$770$	−41648.2	−1.94922
$771$	0	0
$772$	355.227	0.0165608
$773$	8952.62	0.416563	0.208282	−	0.978069i	$-0.433213\pi$
$773$	8952.62	0.416563	0.208282	+	0.978069i	$0.433213\pi$
$774$	0	0
$775$	−14982.4	−0.694429
$776$	−19877.1	−0.919519
$777$	0	0
$778$	−9747.88	−0.449201
$779$	−32246.7	−1.48313
$780$	0	0
$781$	16104.6	0.737861
$782$	−25065.0	−1.14619
$783$	0	0
$784$	−55907.1	−2.54679
$785$	−1456.37	−0.0662166
$786$	0	0
$787$	−1427.86	−0.0646731	−0.0323365	−	0.999477i	$-0.510295\pi$
$787$	−1427.86	−0.0646731	−0.0323365	+	0.999477i	$0.510295\pi$
$788$	−587.794	−0.0265727
$789$	0	0
$790$	−35069.4	−1.57938
$791$	−26084.5	−1.17251
$792$	0	0
$793$	7744.25	0.346793
$794$	−19728.3	−0.881779
$795$	0	0
$796$	1263.58	0.0562644
$797$	−33475.6	−1.48779	−0.743894	−	0.668297i	$-0.767023\pi$
$797$	−33475.6	−1.48779	−0.743894	+	0.668297i	$0.767023\pi$
$798$	0	0
$799$	−39672.7	−1.75659
$800$	−1293.80	−0.0571785
$801$	0	0
$802$	20430.9	0.899549
$803$	24973.1	1.09749
$804$	0	0
$805$	57480.0	2.51665
$806$	9316.82	0.407160
$807$	0	0
$808$	−18832.9	−0.819974
$809$	−18354.9	−0.797679	−0.398839	−	0.917021i	$-0.630587\pi$
$809$	−18354.9	−0.797679	−0.398839	+	0.917021i	$0.630587\pi$
$810$	0	0
$811$	18590.6	0.804937	0.402469	−	0.915434i	$-0.368152\pi$
$811$	18590.6	0.804937	0.402469	+	0.915434i	$0.368152\pi$
$812$	854.237	0.0369185
$813$	0	0
$814$	21910.3	0.943436
$815$	13858.7	0.595641
$816$	0	0
$817$	24750.2	1.05985
$818$	13852.5	0.592106
$819$	0	0
$820$	−1991.01	−0.0847915
$821$	−11083.4	−0.471151	−0.235575	−	0.971856i	$-0.575697\pi$
$821$	−11083.4	−0.471151	−0.235575	+	0.971856i	$0.575697\pi$
$822$	0	0
$823$	−11533.7	−0.488506	−0.244253	−	0.969712i	$-0.578543\pi$
$823$	−11533.7	−0.488506	−0.244253	+	0.969712i	$0.578543\pi$
$824$	−9732.36	−0.411460
$825$	0	0
$826$	−5781.82	−0.243554
$827$	25859.7	1.08734	0.543670	−	0.839299i	$-0.317034\pi$
$827$	25859.7	1.08734	0.543670	+	0.839299i	$0.317034\pi$
$828$	0	0
$829$	−20734.3	−0.868674	−0.434337	−	0.900750i	$-0.643017\pi$
$829$	−20734.3	−0.868674	−0.434337	+	0.900750i	$0.643017\pi$
$830$	49674.4	2.07738
$831$	0	0
$832$	7864.74	0.327717
$833$	72247.9	3.00509
$834$	0	0
$835$	−9211.56	−0.381772
$836$	−1345.56	−0.0556666
$837$	0	0
$838$	−9740.71	−0.401536
$839$	−8282.44	−0.340812	−0.170406	−	0.985374i	$-0.554508\pi$
$839$	−8282.44	−0.340812	−0.170406	+	0.985374i	$0.554508\pi$
$840$	0	0
$841$	−21451.2	−0.879544
$842$	−36101.8	−1.47761
$843$	0	0
$844$	1379.97	0.0562802
$845$	27308.9	1.11178
$846$	0	0
$847$	13511.7	0.548133
$848$	−18907.7	−0.765676
$849$	0	0
$850$	−13861.1	−0.559330
$851$	−30239.2	−1.21808
$852$	0	0
$853$	27976.9	1.12299	0.561495	−	0.827480i	$-0.310226\pi$
$853$	27976.9	1.12299	0.561495	+	0.827480i	$0.310226\pi$
$854$	51826.1	2.07664
$855$	0	0
$856$	27039.3	1.07965
$857$	30334.1	1.20910	0.604548	−	0.796569i	$-0.293354\pi$
$857$	30334.1	1.20910	0.604548	+	0.796569i	$0.293354\pi$
$858$	0	0
$859$	35079.4	1.39336	0.696679	−	0.717383i	$-0.254660\pi$
$859$	35079.4	1.39336	0.696679	+	0.717383i	$0.254660\pi$
$860$	1528.15	0.0605924
$861$	0	0
$862$	−25065.3	−0.990403
$863$	8904.72	0.351240	0.175620	−	0.984458i	$-0.443807\pi$
$863$	8904.72	0.351240	0.175620	+	0.984458i	$0.443807\pi$
$864$	0	0
$865$	−39670.1	−1.55934
$866$	7022.44	0.275557
$867$	0	0
$868$	−3647.47	−0.142631
$869$	−28573.2	−1.11540
$870$	0	0
$871$	−5466.17	−0.212645
$872$	−46792.8	−1.81720
$873$	0	0
$874$	−31744.6	−1.22858
$875$	−29589.4	−1.14321
$876$	0	0
$877$	5487.15	0.211275	0.105637	−	0.994405i	$-0.466312\pi$
$877$	5487.15	0.211275	0.105637	+	0.994405i	$0.466312\pi$
$878$	−4189.56	−0.161037
$879$	0	0
$880$	25613.3	0.981165
$881$	12650.9	0.483792	0.241896	−	0.970302i	$-0.422231\pi$
$881$	12650.9	0.483792	0.241896	+	0.970302i	$0.422231\pi$
$882$	0	0
$883$	31973.4	1.21856	0.609282	−	0.792954i	$-0.291458\pi$
$883$	31973.4	1.21856	0.609282	+	0.792954i	$0.291458\pi$
$884$	−504.242	−0.0191849
$885$	0	0
$886$	27796.1	1.05398
$887$	48252.3	1.82655	0.913277	−	0.407338i	$-0.133543\pi$
$887$	48252.3	1.82655	0.913277	+	0.407338i	$0.133543\pi$
$888$	0	0
$889$	42327.4	1.59687
$890$	−19320.5	−0.727667
$891$	0	0
$892$	928.662	0.0348586
$893$	−50245.1	−1.88285
$894$	0	0
$895$	23416.8	0.874565
$896$	46933.2	1.74992
$897$	0	0
$898$	49286.1	1.83151
$899$	−12544.0	−0.465368
$900$	0	0
$901$	24434.1	0.903462
$902$	27729.9	1.02362
$903$	0	0
$904$	16983.2	0.624839
$905$	24114.0	0.885719
$906$	0	0
$907$	33861.7	1.23964	0.619822	−	0.784742i	$-0.287205\pi$
$907$	33861.7	1.23964	0.619822	+	0.784742i	$0.287205\pi$
$908$	1150.16	0.0420366
$909$	0	0
$910$	−19766.6	−0.720063
$911$	−45845.4	−1.66732	−0.833659	−	0.552280i	$-0.813758\pi$
$911$	−45845.4	−1.66732	−0.833659	+	0.552280i	$0.813758\pi$
$912$	0	0
$913$	40472.8	1.46709
$914$	24497.9	0.886561
$915$	0	0
$916$	664.407	0.0239657
$917$	−8947.05	−0.322200
$918$	0	0
$919$	−14126.6	−0.507066	−0.253533	−	0.967327i	$-0.581593\pi$
$919$	−14126.6	−0.507066	−0.253533	+	0.967327i	$0.581593\pi$
$920$	−37424.5	−1.34114
$921$	0	0
$922$	45977.8	1.64230
$923$	7643.42	0.272575
$924$	0	0
$925$	−16722.4	−0.594410
$926$	38626.6	1.37079
$927$	0	0
$928$	−1083.24	−0.0383179
$929$	−27790.5	−0.981461	−0.490730	−	0.871311i	$-0.663270\pi$
$929$	−27790.5	−0.981461	−0.490730	+	0.871311i	$0.663270\pi$
$930$	0	0
$931$	91501.3	3.22109
$932$	−739.588	−0.0259936
$933$	0	0
$934$	28938.5	1.01381
$935$	−33099.7	−1.15773
$936$	0	0
$937$	−6472.88	−0.225677	−0.112839	−	0.993613i	$-0.535994\pi$
$937$	−6472.88	−0.225677	−0.112839	+	0.993613i	$0.535994\pi$
$938$	−36580.7	−1.27335
$939$	0	0
$940$	−3102.28	−0.107644
$941$	12312.7	0.426550	0.213275	−	0.976992i	$-0.431587\pi$
$941$	12312.7	0.426550	0.213275	+	0.976992i	$0.431587\pi$
$942$	0	0
$943$	−38271.0	−1.32161
$944$	3555.78	0.122596
$945$	0	0
$946$	−21283.4	−0.731484
$947$	−4368.65	−0.149907	−0.0749536	−	0.997187i	$-0.523881\pi$
$947$	−4368.65	−0.149907	−0.0749536	+	0.997187i	$0.523881\pi$
$948$	0	0
$949$	11852.5	0.405424
$950$	−17554.9	−0.599533
$951$	0	0
$952$	−64432.6	−2.19356
$953$	−2917.34	−0.0991627	−0.0495814	−	0.998770i	$-0.515789\pi$
$953$	−2917.34	−0.0991627	−0.0495814	+	0.998770i	$0.515789\pi$
$954$	0	0
$955$	20727.6	0.702333
$956$	1542.82	0.0521951
$957$	0	0
$958$	−17577.5	−0.592801
$959$	88206.9	2.97012
$960$	0	0
$961$	23770.1	0.797895
$962$	10398.9	0.348516
$963$	0	0
$964$	385.568	0.0128821
$965$	11067.0	0.369179
$966$	0	0
$967$	42945.5	1.42816	0.714082	−	0.700062i	$-0.246845\pi$
$967$	42945.5	1.42816	0.714082	+	0.700062i	$0.246845\pi$
$968$	−8797.31	−0.292104
$969$	0	0
$970$	−32432.3	−1.07354
$971$	−43742.9	−1.44570	−0.722851	−	0.691004i	$-0.757169\pi$
$971$	−43742.9	−1.44570	−0.722851	+	0.691004i	$0.757169\pi$
$972$	0	0
$973$	−73703.7	−2.42840
$974$	−39028.1	−1.28392
$975$	0	0
$976$	−31872.7	−1.04531
$977$	29917.5	0.979679	0.489839	−	0.871813i	$-0.337055\pi$
$977$	29917.5	0.979679	0.489839	+	0.871813i	$0.337055\pi$
$978$	0	0
$979$	−15741.6	−0.513895
$980$	5649.57	0.184152
$981$	0	0
$982$	−16130.6	−0.524185
$983$	48495.4	1.57351	0.786756	−	0.617264i	$-0.211759\pi$
$983$	48495.4	1.57351	0.786756	+	0.617264i	$0.211759\pi$
$984$	0	0
$985$	−18312.5	−0.592370
$986$	−11605.2	−0.374832
$987$	0	0
$988$	−638.617	−0.0205639
$989$	29373.9	0.944426
$990$	0	0
$991$	−14192.8	−0.454944	−0.227472	−	0.973785i	$-0.573046\pi$
$991$	−14192.8	−0.454944	−0.227472	+	0.973785i	$0.573046\pi$
$992$	4625.27	0.148037
$993$	0	0
$994$	51151.4	1.63222
$995$	39366.4	1.25427
$996$	0	0
$997$	−49366.8	−1.56817	−0.784083	−	0.620656i	$-0.786866\pi$
$997$	−49366.8	−1.56817	−0.784083	+	0.620656i	$0.786866\pi$
$998$	22942.9	0.727700
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	531.4.a.c.1.3		7
3.2	odd	2		177.4.a.b.1.5	✓	7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.4.a.b.1.5	✓	7	3.2	odd	2
531.4.a.c.1.3		7	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)

\(f(q)\)	\(=\)	\(q-2.74916 q^{2} -0.442134 q^{4} -13.7745 q^{5} -35.6462 q^{7} +23.2088 q^{8} +O(q^{10})\)	\(q-2.74916 q^{2} -0.442134 q^{4} -13.7745 q^{5} -35.6462 q^{7} +23.2088 q^{8} +37.8683 q^{10} +30.8537 q^{11} +14.6435 q^{13} +97.9970 q^{14} -60.2674 q^{16} +77.8827 q^{17} +98.6377 q^{19} +6.09019 q^{20} -84.8216 q^{22} +117.065 q^{23} +64.7375 q^{25} -40.2572 q^{26} +15.7604 q^{28} +54.2015 q^{29} -231.433 q^{31} -19.9854 q^{32} -214.112 q^{34} +491.009 q^{35} -258.311 q^{37} -271.171 q^{38} -319.690 q^{40} -326.921 q^{41} +250.920 q^{43} -13.6415 q^{44} -321.830 q^{46} -509.390 q^{47} +927.650 q^{49} -177.974 q^{50} -6.47437 q^{52} +313.730 q^{53} -424.994 q^{55} -827.304 q^{56} -149.008 q^{58} -59.0000 q^{59} +528.854 q^{61} +636.245 q^{62} +537.082 q^{64} -201.707 q^{65} -373.284 q^{67} -34.4346 q^{68} -1349.86 q^{70} +521.969 q^{71} +809.404 q^{73} +710.137 q^{74} -43.6111 q^{76} -1099.82 q^{77} -926.088 q^{79} +830.155 q^{80} +898.756 q^{82} +1311.77 q^{83} -1072.80 q^{85} -689.818 q^{86} +716.075 q^{88} -510.201 q^{89} -521.983 q^{91} -51.7585 q^{92} +1400.39 q^{94} -1358.69 q^{95} -856.449 q^{97} -2550.26 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 26 q^{4} + 2 q^{5} - 59 q^{7} + 21 q^{8}+O(q^{10}) \) 7 * q + 26 * q^4 + 2 * q^5 - 59 * q^7 + 21 * q^8	\( 7 q + 26 q^{4} + 2 q^{5} - 59 q^{7} + 21 q^{8} - 71 q^{10} + 5 q^{11} - 67 q^{13} + 65 q^{14} - 94 q^{16} + 23 q^{17} - 176 q^{19} + 207 q^{20} - 704 q^{22} + 218 q^{23} - 183 q^{25} - 58 q^{26} - 938 q^{28} - 168 q^{29} - 604 q^{31} + 448 q^{32} - 610 q^{34} + 336 q^{35} - 505 q^{37} + 453 q^{38} - 1080 q^{40} + 265 q^{41} - 493 q^{43} - 504 q^{44} + 381 q^{46} + 244 q^{47} + 770 q^{49} - 1639 q^{50} + 160 q^{52} - 686 q^{53} - 116 q^{55} - 2190 q^{56} + 1584 q^{58} - 413 q^{59} - 838 q^{61} - 286 q^{62} + 205 q^{64} - 490 q^{65} - 1504 q^{67} - 3047 q^{68} + 1530 q^{70} + 1267 q^{71} - 666 q^{73} - 528 q^{74} - 64 q^{76} - 1109 q^{77} - 2741 q^{79} - 1213 q^{80} + 953 q^{82} + 2025 q^{83} - 1274 q^{85} - 4394 q^{86} - 1639 q^{88} - 616 q^{89} - 2415 q^{91} - 218 q^{92} + 900 q^{94} - 2554 q^{95} - 1298 q^{97} + 172 q^{98}+O(q^{100}) \) 7 * q + 26 * q^4 + 2 * q^5 - 59 * q^7 + 21 * q^8 - 71 * q^10 + 5 * q^11 - 67 * q^13 + 65 * q^14 - 94 * q^16 + 23 * q^17 - 176 * q^19 + 207 * q^20 - 704 * q^22 + 218 * q^23 - 183 * q^25 - 58 * q^26 - 938 * q^28 - 168 * q^29 - 604 * q^31 + 448 * q^32 - 610 * q^34 + 336 * q^35 - 505 * q^37 + 453 * q^38 - 1080 * q^40 + 265 * q^41 - 493 * q^43 - 504 * q^44 + 381 * q^46 + 244 * q^47 + 770 * q^49 - 1639 * q^50 + 160 * q^52 - 686 * q^53 - 116 * q^55 - 2190 * q^56 + 1584 * q^58 - 413 * q^59 - 838 * q^61 - 286 * q^62 + 205 * q^64 - 490 * q^65 - 1504 * q^67 - 3047 * q^68 + 1530 * q^70 + 1267 * q^71 - 666 * q^73 - 528 * q^74 - 64 * q^76 - 1109 * q^77 - 2741 * q^79 - 1213 * q^80 + 953 * q^82 + 2025 * q^83 - 1274 * q^85 - 4394 * q^86 - 1639 * q^88 - 616 * q^89 - 2415 * q^91 - 218 * q^92 + 900 * q^94 - 2554 * q^95 - 1298 * q^97 + 172 * q^98

Introduction

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