LMFDB - Embedded newform 531.2.a.g.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [531,2,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, 2, names="a")

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

chi := DirichletCharacter("531.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 531 = 3^{2} \cdot 59 $
Weight:	$ k $	$=$	$ 2 $
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:	$4.24005634733$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.246832.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 5x^{3} + 6x^{2} + 7x - 2 $ x^5 - 2x^4 - 5x^3 + 6x^2 + 7x - 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-1.15351$ 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)

$f(q)$	$=$	$q+2.15351 q^{2} +2.63760 q^{4} +2.11101 q^{5} -1.09624 q^{7} +1.37308 q^{8} +O(q^{10})$	\(q+2.15351 q^{2} +2.63760 q^{4} +2.11101 q^{5} -1.09624 q^{7} +1.37308 q^{8} +4.54608 q^{10} -1.39257 q^{11} +6.23742 q^{13} -2.36076 q^{14} -2.31826 q^{16} +6.48881 q^{17} -2.66942 q^{19} +5.56801 q^{20} -2.99892 q^{22} -5.07756 q^{23} -0.543632 q^{25} +13.4323 q^{26} -2.89144 q^{28} -1.56801 q^{29} -7.25220 q^{31} -7.73856 q^{32} +13.9737 q^{34} -2.31417 q^{35} -2.28998 q^{37} -5.74861 q^{38} +2.89859 q^{40} +3.03897 q^{41} -5.01313 q^{43} -3.67305 q^{44} -10.9346 q^{46} +2.93882 q^{47} -5.79826 q^{49} -1.17072 q^{50} +16.4518 q^{52} +8.52551 q^{53} -2.93974 q^{55} -1.50522 q^{56} -3.37672 q^{58} +1.00000 q^{59} -6.92270 q^{61} -15.6177 q^{62} -12.0285 q^{64} +13.1673 q^{65} +1.04722 q^{67} +17.1149 q^{68} -4.98359 q^{70} +15.1100 q^{71} -0.393655 q^{73} -4.93149 q^{74} -7.04086 q^{76} +1.52659 q^{77} -6.27333 q^{79} -4.89387 q^{80} +6.54444 q^{82} -5.63407 q^{83} +13.6980 q^{85} -10.7958 q^{86} -1.91212 q^{88} -3.22447 q^{89} -6.83770 q^{91} -13.3926 q^{92} +6.32878 q^{94} -5.63517 q^{95} +2.04005 q^{97} -12.4866 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 3 q^{2} + 5 q^{4} + 8 q^{5} + 2 q^{7} + 9 q^{8}+O(q^{10}) $ 5 * q + 3 * q^2 + 5 * q^4 + 8 * q^5 + 2 * q^7 + 9 * q^8	$ 5 q + 3 q^{2} + 5 q^{4} + 8 q^{5} + 2 q^{7} + 9 q^{8} - 2 q^{10} + 10 q^{11} - 4 q^{13} + 6 q^{14} + q^{16} + 8 q^{17} - 6 q^{19} + 2 q^{22} + 4 q^{23} + 7 q^{25} + 2 q^{26} - 2 q^{28} + 20 q^{29} - 6 q^{31} + q^{32} + 4 q^{34} + 12 q^{35} - 18 q^{38} - 6 q^{40} + 8 q^{41} - 4 q^{43} + 30 q^{44} - 2 q^{46} - 2 q^{47} - 5 q^{49} - 19 q^{50} - 2 q^{52} + 20 q^{53} + 10 q^{55} - 14 q^{56} + 22 q^{58} + 5 q^{59} - 14 q^{61} - 6 q^{62} - 9 q^{64} + 2 q^{65} - 24 q^{67} - 8 q^{68} + 6 q^{70} + 12 q^{71} - 2 q^{73} - 22 q^{74} - 4 q^{76} + 2 q^{77} - 8 q^{79} - 28 q^{80} - 18 q^{82} - 18 q^{83} + 10 q^{85} - 14 q^{86} + 48 q^{88} + 4 q^{89} - 24 q^{91} - 50 q^{92} - 12 q^{94} - 4 q^{95} + 20 q^{97} - 3 q^{98}+O(q^{100}) $ 5 * q + 3 * q^2 + 5 * q^4 + 8 * q^5 + 2 * q^7 + 9 * q^8 - 2 * q^10 + 10 * q^11 - 4 * q^13 + 6 * q^14 + q^16 + 8 * q^17 - 6 * q^19 + 2 * q^22 + 4 * q^23 + 7 * q^25 + 2 * q^26 - 2 * q^28 + 20 * q^29 - 6 * q^31 + q^32 + 4 * q^34 + 12 * q^35 - 18 * q^38 - 6 * q^40 + 8 * q^41 - 4 * q^43 + 30 * q^44 - 2 * q^46 - 2 * q^47 - 5 * q^49 - 19 * q^50 - 2 * q^52 + 20 * q^53 + 10 * q^55 - 14 * q^56 + 22 * q^58 + 5 * q^59 - 14 * q^61 - 6 * q^62 - 9 * q^64 + 2 * q^65 - 24 * q^67 - 8 * q^68 + 6 * q^70 + 12 * q^71 - 2 * q^73 - 22 * q^74 - 4 * q^76 + 2 * q^77 - 8 * q^79 - 28 * q^80 - 18 * q^82 - 18 * q^83 + 10 * q^85 - 14 * q^86 + 48 * q^88 + 4 * q^89 - 24 * q^91 - 50 * q^92 - 12 * q^94 - 4 * q^95 + 20 * q^97 - 3 * 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 $

$2$	2.15351	1.52276	0.761380	−	0.648305i	$-0.224522\pi$
$2$	2.15351	1.52276	0.761380	+	0.648305i	$0.224522\pi$
$3$	0	0
$3$	0	0
$4$	2.63760	1.31880
$5$	2.11101	0.944073	0.472036	−	0.881579i	$-0.343519\pi$
$5$	2.11101	0.944073	0.472036	+	0.881579i	$0.343519\pi$
$6$	0	0
$7$	−1.09624	−0.414339	−0.207169	−	0.978305i	$-0.566425\pi$
$7$	−1.09624	−0.414339	−0.207169	+	0.978305i	$0.566425\pi$
$8$	1.37308	0.485458
$9$	0	0
$10$	4.54608	1.43760
$11$	−1.39257	−0.419876	−0.209938	−	0.977715i	$-0.567326\pi$
$11$	−1.39257	−0.419876	−0.209938	+	0.977715i	$0.567326\pi$
$12$	0	0
$13$	6.23742	1.72995	0.864975	−	0.501815i	$-0.167334\pi$
$13$	6.23742	1.72995	0.864975	+	0.501815i	$0.167334\pi$
$14$	−2.36076	−0.630939
$15$	0	0
$16$	−2.31826	−0.579565
$17$	6.48881	1.57377	0.786884	−	0.617101i	$-0.211693\pi$
$17$	6.48881	1.57377	0.786884	+	0.617101i	$0.211693\pi$
$18$	0	0
$19$	−2.66942	−0.612406	−0.306203	−	0.951966i	$-0.599059\pi$
$19$	−2.66942	−0.612406	−0.306203	+	0.951966i	$0.599059\pi$
$20$	5.56801	1.24504
$21$	0	0
$22$	−2.99892	−0.639371
$23$	−5.07756	−1.05874	−0.529372	−	0.848390i	$-0.677572\pi$
$23$	−5.07756	−1.05874	−0.529372	+	0.848390i	$0.677572\pi$
$24$	0	0
$25$	−0.543632	−0.108726
$26$	13.4323	2.63430
$27$	0	0
$28$	−2.89144	−0.546431
$29$	−1.56801	−0.291172	−0.145586	−	0.989346i	$-0.546507\pi$
$29$	−1.56801	−0.291172	−0.145586	+	0.989346i	$0.546507\pi$
$30$	0	0
$31$	−7.25220	−1.30253	−0.651267	−	0.758849i	$-0.725762\pi$
$31$	−7.25220	−1.30253	−0.651267	+	0.758849i	$0.725762\pi$
$32$	−7.73856	−1.36800
$33$	0	0
$34$	13.9737	2.39647
$35$	−2.31417	−0.391166
$36$	0	0
$37$	−2.28998	−0.376470	−0.188235	−	0.982124i	$-0.560277\pi$
$37$	−2.28998	−0.376470	−0.188235	+	0.982124i	$0.560277\pi$
$38$	−5.74861	−0.932548
$39$	0	0
$40$	2.89859	0.458307
$41$	3.03897	0.474607	0.237303	−	0.971436i	$-0.423736\pi$
$41$	3.03897	0.474607	0.237303	+	0.971436i	$0.423736\pi$
$42$	0	0
$43$	−5.01313	−0.764496	−0.382248	−	0.924060i	$-0.624850\pi$
$43$	−5.01313	−0.764496	−0.382248	+	0.924060i	$0.624850\pi$
$44$	−3.67305	−0.553733
$45$	0	0
$46$	−10.9346	−1.61221
$47$	2.93882	0.428671	0.214336	−	0.976760i	$-0.431241\pi$
$47$	2.93882	0.428671	0.214336	+	0.976760i	$0.431241\pi$
$48$	0	0
$49$	−5.79826	−0.828323
$50$	−1.17072	−0.165564
$51$	0	0
$52$	16.4518	2.28146
$53$	8.52551	1.17107	0.585534	−	0.810648i	$-0.300885\pi$
$53$	8.52551	1.17107	0.585534	+	0.810648i	$0.300885\pi$
$54$	0	0
$55$	−2.93974	−0.396394
$56$	−1.50522	−0.201144
$57$	0	0
$58$	−3.37672	−0.443385
$59$	1.00000	0.130189
$59$	1.00000	0.130189
$60$	0	0
$61$	−6.92270	−0.886360	−0.443180	−	0.896433i	$-0.646150\pi$
$61$	−6.92270	−0.886360	−0.443180	+	0.896433i	$0.646150\pi$
$62$	−15.6177	−1.98345
$63$	0	0
$64$	−12.0285	−1.50357
$65$	13.1673	1.63320
$66$	0	0
$67$	1.04722	0.127938	0.0639689	−	0.997952i	$-0.479624\pi$
$67$	1.04722	0.127938	0.0639689	+	0.997952i	$0.479624\pi$
$68$	17.1149	2.07549
$69$	0	0
$70$	−4.98359	−0.595653
$71$	15.1100	1.79323	0.896613	−	0.442814i	$-0.146020\pi$
$71$	15.1100	1.79323	0.896613	+	0.442814i	$0.146020\pi$
$72$	0	0
$73$	−0.393655	−0.0460738	−0.0230369	−	0.999735i	$-0.507334\pi$
$73$	−0.393655	−0.0460738	−0.0230369	+	0.999735i	$0.507334\pi$
$74$	−4.93149	−0.573274
$75$	0	0
$76$	−7.04086	−0.807642
$77$	1.52659	0.173971
$78$	0	0
$79$	−6.27333	−0.705805	−0.352902	−	0.935660i	$-0.614805\pi$
$79$	−6.27333	−0.705805	−0.352902	+	0.935660i	$0.614805\pi$
$80$	−4.89387	−0.547152
$81$	0	0
$82$	6.54444	0.722713
$83$	−5.63407	−0.618420	−0.309210	−	0.950994i	$-0.600065\pi$
$83$	−5.63407	−0.618420	−0.309210	+	0.950994i	$0.600065\pi$
$84$	0	0
$85$	13.6980	1.48575
$86$	−10.7958	−1.16414
$87$	0	0
$88$	−1.91212	−0.203832
$89$	−3.22447	−0.341793	−0.170897	−	0.985289i	$-0.554666\pi$
$89$	−3.22447	−0.341793	−0.170897	+	0.985289i	$0.554666\pi$
$90$	0	0
$91$	−6.83770	−0.716786
$92$	−13.3926	−1.39627
$93$	0	0
$94$	6.32878	0.652764
$95$	−5.63517	−0.578156
$96$	0	0
$97$	2.04005	0.207136	0.103568	−	0.994622i	$-0.466974\pi$
$97$	2.04005	0.207136	0.103568	+	0.994622i	$0.466974\pi$
$98$	−12.4866	−1.26134
$99$	0	0
$100$	−1.43388	−0.143388
$101$	−8.16783	−0.812729	−0.406365	−	0.913711i	$-0.633204\pi$
$101$	−8.16783	−0.812729	−0.406365	+	0.913711i	$0.633204\pi$
$102$	0	0
$103$	18.8669	1.85901	0.929504	−	0.368813i	$-0.120236\pi$
$103$	18.8669	1.85901	0.929504	+	0.368813i	$0.120236\pi$
$104$	8.56449	0.839817
$105$	0	0
$106$	18.3598	1.78326
$107$	9.35442	0.904326	0.452163	−	0.891935i	$-0.350653\pi$
$107$	9.35442	0.904326	0.452163	+	0.891935i	$0.350653\pi$
$108$	0	0
$109$	−19.6185	−1.87911	−0.939554	−	0.342400i	$-0.888760\pi$
$109$	−19.6185	−1.87911	−0.939554	+	0.342400i	$0.888760\pi$
$110$	−6.33075	−0.603613
$111$	0	0
$112$	2.54136	0.240136
$113$	−8.53075	−0.802505	−0.401253	−	0.915967i	$-0.631425\pi$
$113$	−8.53075	−0.802505	−0.401253	+	0.915967i	$0.631425\pi$
$114$	0	0
$115$	−10.7188	−0.999531
$116$	−4.13578	−0.383997
$117$	0	0
$118$	2.15351	0.198247
$119$	−7.11328	−0.652073
$120$	0	0
$121$	−9.06074	−0.823704
$122$	−14.9081	−1.34971
$123$	0	0
$124$	−19.1284	−1.71778
$125$	−11.7027	−1.04672
$126$	0	0
$127$	−1.71474	−0.152159	−0.0760793	−	0.997102i	$-0.524240\pi$
$127$	−1.71474	−0.152159	−0.0760793	+	0.997102i	$0.524240\pi$
$128$	−10.4264	−0.921576
$129$	0	0
$130$	28.3558	2.48697
$131$	−3.47586	−0.303687	−0.151844	−	0.988405i	$-0.548521\pi$
$131$	−3.47586	−0.303687	−0.151844	+	0.988405i	$0.548521\pi$
$132$	0	0
$133$	2.92632	0.253744
$134$	2.25519	0.194819
$135$	0	0
$136$	8.90967	0.763998
$137$	16.7854	1.43407	0.717037	−	0.697035i	$-0.245498\pi$
$137$	16.7854	1.43407	0.717037	+	0.697035i	$0.245498\pi$
$138$	0	0
$139$	19.8309	1.68204	0.841018	−	0.541008i	$-0.181957\pi$
$139$	19.8309	1.68204	0.841018	+	0.541008i	$0.181957\pi$
$140$	−6.10386	−0.515870
$141$	0	0
$142$	32.5395	2.73066
$143$	−8.68607	−0.726365
$144$	0	0
$145$	−3.31008	−0.274887
$146$	−0.847739	−0.0701594
$147$	0	0
$148$	−6.04005	−0.496489
$149$	18.8342	1.54296	0.771478	−	0.636257i	$-0.219518\pi$
$149$	18.8342	1.54296	0.771478	+	0.636257i	$0.219518\pi$
$150$	0	0
$151$	4.13992	0.336902	0.168451	−	0.985710i	$-0.446123\pi$
$151$	4.13992	0.336902	0.168451	+	0.985710i	$0.446123\pi$
$152$	−3.66533	−0.297297
$153$	0	0
$154$	3.28753	0.264916
$155$	−15.3095	−1.22969
$156$	0	0
$157$	20.2708	1.61778	0.808891	−	0.587958i	$-0.200068\pi$
$157$	20.2708	1.61778	0.808891	+	0.587958i	$0.200068\pi$
$158$	−13.5097	−1.07477
$159$	0	0
$160$	−16.3362	−1.29149
$161$	5.56621	0.438679
$162$	0	0
$163$	4.16213	0.326004	0.163002	−	0.986626i	$-0.447882\pi$
$163$	4.16213	0.326004	0.163002	+	0.986626i	$0.447882\pi$
$164$	8.01558	0.625912
$165$	0	0
$166$	−12.1330	−0.941705
$167$	−0.544372	−0.0421248	−0.0210624	−	0.999778i	$-0.506705\pi$
$167$	−0.544372	−0.0421248	−0.0210624	+	0.999778i	$0.506705\pi$
$168$	0	0
$169$	25.9055	1.99273
$170$	29.4987	2.26244
$171$	0	0
$172$	−13.2227	−1.00822
$173$	−2.14881	−0.163371	−0.0816854	−	0.996658i	$-0.526030\pi$
$173$	−2.14881	−0.163371	−0.0816854	+	0.996658i	$0.526030\pi$
$174$	0	0
$175$	0.595950	0.0450496
$176$	3.22835	0.243346
$177$	0	0
$178$	−6.94393	−0.520470
$179$	15.7304	1.17574	0.587872	−	0.808954i	$-0.299966\pi$
$179$	15.7304	1.17574	0.587872	+	0.808954i	$0.299966\pi$
$180$	0	0
$181$	−3.18613	−0.236823	−0.118412	−	0.992965i	$-0.537780\pi$
$181$	−3.18613	−0.236823	−0.118412	+	0.992965i	$0.537780\pi$
$182$	−14.7250	−1.09149
$183$	0	0
$184$	−6.97190	−0.513975
$185$	−4.83417	−0.355415
$186$	0	0
$187$	−9.03614	−0.660788
$188$	7.75144	0.565332
$189$	0	0
$190$	−12.1354	−0.880393
$191$	6.71163	0.485636	0.242818	−	0.970072i	$-0.421928\pi$
$191$	6.71163	0.485636	0.242818	+	0.970072i	$0.421928\pi$
$192$	0	0
$193$	13.4512	0.968239	0.484120	−	0.875002i	$-0.339140\pi$
$193$	13.4512	0.968239	0.484120	+	0.875002i	$0.339140\pi$
$194$	4.39326	0.315418
$195$	0	0
$196$	−15.2935	−1.09239
$197$	−14.5352	−1.03559	−0.517795	−	0.855505i	$-0.673247\pi$
$197$	−14.5352	−1.03559	−0.517795	+	0.855505i	$0.673247\pi$
$198$	0	0
$199$	21.8949	1.55209	0.776045	−	0.630677i	$-0.217223\pi$
$199$	21.8949	1.55209	0.776045	+	0.630677i	$0.217223\pi$
$200$	−0.746451	−0.0527821
$201$	0	0
$202$	−17.5895	−1.23759
$203$	1.71891	0.120644
$204$	0	0
$205$	6.41529	0.448063
$206$	40.6300	2.83082
$207$	0	0
$208$	−14.4600	−1.00262
$209$	3.71736	0.257135
$210$	0	0
$211$	18.8449	1.29734	0.648669	−	0.761071i	$-0.275326\pi$
$211$	18.8449	1.29734	0.648669	+	0.761071i	$0.275326\pi$
$212$	22.4869	1.54441
$213$	0	0
$214$	20.1448	1.37707
$215$	−10.5828	−0.721740
$216$	0	0
$217$	7.95013	0.539690
$218$	−42.2485	−2.86143
$219$	0	0
$220$	−7.75385	−0.522765
$221$	40.4735	2.72254
$222$	0	0
$223$	−21.6651	−1.45080	−0.725400	−	0.688328i	$-0.758345\pi$
$223$	−21.6651	−1.45080	−0.725400	+	0.688328i	$0.758345\pi$
$224$	8.48330	0.566814
$225$	0	0
$226$	−18.3710	−1.22202
$227$	17.4203	1.15622	0.578112	−	0.815957i	$-0.303790\pi$
$227$	17.4203	1.15622	0.578112	+	0.815957i	$0.303790\pi$
$228$	0	0
$229$	−23.9150	−1.58035	−0.790173	−	0.612883i	$-0.790009\pi$
$229$	−23.9150	−1.58035	−0.790173	+	0.612883i	$0.790009\pi$
$230$	−23.0830	−1.52205
$231$	0	0
$232$	−2.15300	−0.141351
$233$	1.42711	0.0934930	0.0467465	−	0.998907i	$-0.485115\pi$
$233$	1.42711	0.0934930	0.0467465	+	0.998907i	$0.485115\pi$
$234$	0	0
$235$	6.20388	0.404697
$236$	2.63760	0.171693
$237$	0	0
$238$	−15.3185	−0.992952
$239$	27.4489	1.77552	0.887762	−	0.460303i	$-0.152259\pi$
$239$	27.4489	1.77552	0.887762	+	0.460303i	$0.152259\pi$
$240$	0	0
$241$	−19.3569	−1.24689	−0.623444	−	0.781868i	$-0.714267\pi$
$241$	−19.3569	−1.24689	−0.623444	+	0.781868i	$0.714267\pi$
$242$	−19.5124	−1.25430
$243$	0	0
$244$	−18.2593	−1.16893
$245$	−12.2402	−0.781998
$246$	0	0
$247$	−16.6503	−1.05943
$248$	−9.95786	−0.632325
$249$	0	0
$250$	−25.2018	−1.59390
$251$	−27.8353	−1.75695	−0.878474	−	0.477790i	$-0.841438\pi$
$251$	−27.8353	−1.75695	−0.878474	+	0.477790i	$0.841438\pi$
$252$	0	0
$253$	7.07087	0.444542
$254$	−3.69271	−0.231701
$255$	0	0
$256$	1.60362	0.100226
$257$	7.88924	0.492117	0.246059	−	0.969255i	$-0.420864\pi$
$257$	7.88924	0.492117	0.246059	+	0.969255i	$0.420864\pi$
$258$	0	0
$259$	2.51036	0.155986
$260$	34.7300	2.15386
$261$	0	0
$262$	−7.48529	−0.462443
$263$	−17.0382	−1.05062	−0.525311	−	0.850911i	$-0.676051\pi$
$263$	−17.0382	−1.05062	−0.525311	+	0.850911i	$0.676051\pi$
$264$	0	0
$265$	17.9974	1.10557
$266$	6.30185	0.386391
$267$	0	0
$268$	2.76214	0.168724
$269$	13.1291	0.800494	0.400247	−	0.916407i	$-0.368924\pi$
$269$	13.1291	0.800494	0.400247	+	0.916407i	$0.368924\pi$
$270$	0	0
$271$	−13.7399	−0.834639	−0.417319	−	0.908760i	$-0.637030\pi$
$271$	−13.7399	−0.834639	−0.417319	+	0.908760i	$0.637030\pi$
$272$	−15.0427	−0.912101
$273$	0	0
$274$	36.1475	2.18375
$275$	0.757047	0.0456517
$276$	0	0
$277$	−22.1961	−1.33363	−0.666816	−	0.745222i	$-0.732344\pi$
$277$	−22.1961	−1.33363	−0.666816	+	0.745222i	$0.732344\pi$
$278$	42.7060	2.56134
$279$	0	0
$280$	−3.17754	−0.189895
$281$	0.776826	0.0463416	0.0231708	−	0.999732i	$-0.492624\pi$
$281$	0.776826	0.0463416	0.0231708	+	0.999732i	$0.492624\pi$
$282$	0	0
$283$	−16.4114	−0.975558	−0.487779	−	0.872967i	$-0.662193\pi$
$283$	−16.4114	−0.975558	−0.487779	+	0.872967i	$0.662193\pi$
$284$	39.8542	2.36491
$285$	0	0
$286$	−18.7055	−1.10608
$287$	−3.33143	−0.196648
$288$	0	0
$289$	25.1047	1.47674
$290$	−7.12829	−0.418587
$291$	0	0
$292$	−1.03830	−0.0607621
$293$	21.0273	1.22843	0.614214	−	0.789140i	$-0.289473\pi$
$293$	21.0273	1.22843	0.614214	+	0.789140i	$0.289473\pi$
$294$	0	0
$295$	2.11101	0.122908
$296$	−3.14433	−0.182760
$297$	0	0
$298$	40.5596	2.34955
$299$	−31.6709	−1.83157
$300$	0	0
$301$	5.49559	0.316760
$302$	8.91536	0.513021
$303$	0	0
$304$	6.18840	0.354929
$305$	−14.6139	−0.836789
$306$	0	0
$307$	19.6118	1.11930	0.559651	−	0.828728i	$-0.310935\pi$
$307$	19.6118	1.11930	0.559651	+	0.828728i	$0.310935\pi$
$308$	4.02654	0.229433
$309$	0	0
$310$	−32.9691	−1.87252
$311$	−34.4576	−1.95391	−0.976956	−	0.213442i	$-0.931533\pi$
$311$	−34.4576	−1.95391	−0.976956	+	0.213442i	$0.931533\pi$
$312$	0	0
$313$	−8.20876	−0.463987	−0.231993	−	0.972717i	$-0.574525\pi$
$313$	−8.20876	−0.463987	−0.231993	+	0.972717i	$0.574525\pi$
$314$	43.6533	2.46350
$315$	0	0
$316$	−16.5465	−0.930816
$317$	−11.4125	−0.640989	−0.320494	−	0.947250i	$-0.603849\pi$
$317$	−11.4125	−0.640989	−0.320494	+	0.947250i	$0.603849\pi$
$318$	0	0
$319$	2.18356	0.122256
$320$	−25.3924	−1.41948
$321$	0	0
$322$	11.9869	0.668003
$323$	−17.3213	−0.963785
$324$	0	0
$325$	−3.39086	−0.188091
$326$	8.96320	0.496425
$327$	0	0
$328$	4.17275	0.230401
$329$	−3.22165	−0.177615
$330$	0	0
$331$	5.61289	0.308512	0.154256	−	0.988031i	$-0.450702\pi$
$331$	5.61289	0.308512	0.154256	+	0.988031i	$0.450702\pi$
$332$	−14.8604	−0.815572
$333$	0	0
$334$	−1.17231	−0.0641460
$335$	2.21068	0.120783
$336$	0	0
$337$	5.94854	0.324038	0.162019	−	0.986788i	$-0.448199\pi$
$337$	5.94854	0.324038	0.162019	+	0.986788i	$0.448199\pi$
$338$	55.7876	3.03445
$339$	0	0
$340$	36.1297	1.95941
$341$	10.0992	0.546903
$342$	0	0
$343$	14.0299	0.757545
$344$	−6.88344	−0.371130
$345$	0	0
$346$	−4.62748	−0.248775
$347$	−12.6802	−0.680708	−0.340354	−	0.940297i	$-0.610547\pi$
$347$	−12.6802	−0.680708	−0.340354	+	0.940297i	$0.610547\pi$
$348$	0	0
$349$	−0.624399	−0.0334233	−0.0167117	−	0.999860i	$-0.505320\pi$
$349$	−0.624399	−0.0334233	−0.0167117	+	0.999860i	$0.505320\pi$
$350$	1.28338	0.0685997
$351$	0	0
$352$	10.7765	0.574390
$353$	13.4958	0.718311	0.359155	−	0.933278i	$-0.383065\pi$
$353$	13.4958	0.718311	0.359155	+	0.933278i	$0.383065\pi$
$354$	0	0
$355$	31.8974	1.69294
$356$	−8.50487	−0.450757
$357$	0	0
$358$	33.8755	1.79038
$359$	−9.47711	−0.500183	−0.250091	−	0.968222i	$-0.580461\pi$
$359$	−9.47711	−0.500183	−0.250091	+	0.968222i	$0.580461\pi$
$360$	0	0
$361$	−11.8742	−0.624959
$362$	−6.86137	−0.360626
$363$	0	0
$364$	−18.0351	−0.945298
$365$	−0.831009	−0.0434970
$366$	0	0
$367$	31.9206	1.66624	0.833120	−	0.553092i	$-0.186552\pi$
$367$	31.9206	1.66624	0.833120	+	0.553092i	$0.186552\pi$
$368$	11.7711	0.613611
$369$	0	0
$370$	−10.4104	−0.541212
$371$	−9.34598	−0.485219
$372$	0	0
$373$	−34.6247	−1.79280	−0.896398	−	0.443249i	$-0.853826\pi$
$373$	−34.6247	−1.79280	−0.896398	+	0.443249i	$0.853826\pi$
$374$	−19.4594	−1.00622
$375$	0	0
$376$	4.03524	0.208102
$377$	−9.78032	−0.503712
$378$	0	0
$379$	19.1592	0.984141	0.492071	−	0.870555i	$-0.336240\pi$
$379$	19.1592	0.984141	0.492071	+	0.870555i	$0.336240\pi$
$380$	−14.8633	−0.762473
$381$	0	0
$382$	14.4535	0.739508
$383$	12.0369	0.615055	0.307527	−	0.951539i	$-0.400498\pi$
$383$	12.0369	0.615055	0.307527	+	0.951539i	$0.400498\pi$
$384$	0	0
$385$	3.22265	0.164241
$386$	28.9673	1.47440
$387$	0	0
$388$	5.38083	0.273171
$389$	35.7718	1.81370	0.906851	−	0.421452i	$-0.138479\pi$
$389$	35.7718	1.81370	0.906851	+	0.421452i	$0.138479\pi$
$390$	0	0
$391$	−32.9473	−1.66622
$392$	−7.96149	−0.402116
$393$	0	0
$394$	−31.3017	−1.57696
$395$	−13.2431	−0.666331
$396$	0	0
$397$	−6.07608	−0.304950	−0.152475	−	0.988307i	$-0.548724\pi$
$397$	−6.07608	−0.304950	−0.152475	+	0.988307i	$0.548724\pi$
$398$	47.1509	2.36346
$399$	0	0
$400$	1.26028	0.0630140
$401$	−12.6527	−0.631846	−0.315923	−	0.948785i	$-0.602314\pi$
$401$	−12.6527	−0.631846	−0.315923	+	0.948785i	$0.602314\pi$
$402$	0	0
$403$	−45.2350	−2.25332
$404$	−21.5435	−1.07183
$405$	0	0
$406$	3.70169	0.183712
$407$	3.18896	0.158071
$408$	0	0
$409$	27.6114	1.36530	0.682649	−	0.730747i	$-0.260828\pi$
$409$	27.6114	1.36530	0.682649	+	0.730747i	$0.260828\pi$
$410$	13.8154	0.682293
$411$	0	0
$412$	49.7633	2.45166
$413$	−1.09624	−0.0539423
$414$	0	0
$415$	−11.8936	−0.583833
$416$	−48.2687	−2.36657
$417$	0	0
$418$	8.00536	0.391555
$419$	37.3028	1.82236	0.911179	−	0.412010i	$-0.135173\pi$
$419$	37.3028	1.82236	0.911179	+	0.412010i	$0.135173\pi$
$420$	0	0
$421$	−17.1206	−0.834408	−0.417204	−	0.908813i	$-0.636990\pi$
$421$	−17.1206	−0.834408	−0.417204	+	0.908813i	$0.636990\pi$
$422$	40.5827	1.97554
$423$	0	0
$424$	11.7062	0.568504
$425$	−3.52752	−0.171110
$426$	0	0
$427$	7.58892	0.367254
$428$	24.6732	1.19263
$429$	0	0
$430$	−22.7901	−1.09904
$431$	6.95204	0.334868	0.167434	−	0.985883i	$-0.446452\pi$
$431$	6.95204	0.334868	0.167434	+	0.985883i	$0.446452\pi$
$432$	0	0
$433$	0.780679	0.0375171	0.0187585	−	0.999824i	$-0.494029\pi$
$433$	0.780679	0.0375171	0.0187585	+	0.999824i	$0.494029\pi$
$434$	17.1207	0.821819
$435$	0	0
$436$	−51.7457	−2.47817
$437$	13.5541	0.648381
$438$	0	0
$439$	29.3855	1.40249	0.701246	−	0.712919i	$-0.252627\pi$
$439$	29.3855	1.40249	0.701246	+	0.712919i	$0.252627\pi$
$440$	−4.03650	−0.192432
$441$	0	0
$442$	87.1600	4.14578
$443$	−25.0656	−1.19090	−0.595452	−	0.803391i	$-0.703027\pi$
$443$	−25.0656	−1.19090	−0.595452	+	0.803391i	$0.703027\pi$
$444$	0	0
$445$	−6.80690	−0.322678
$446$	−46.6559	−2.20922
$447$	0	0
$448$	13.1861	0.622986
$449$	−34.3303	−1.62014	−0.810072	−	0.586330i	$-0.800572\pi$
$449$	−34.3303	−1.62014	−0.810072	+	0.586330i	$0.800572\pi$
$450$	0	0
$451$	−4.23198	−0.199276
$452$	−22.5007	−1.05835
$453$	0	0
$454$	37.5147	1.76065
$455$	−14.4345	−0.676698
$456$	0	0
$457$	−2.23763	−0.104672	−0.0523360	−	0.998630i	$-0.516667\pi$
$457$	−2.23763	−0.104672	−0.0523360	+	0.998630i	$0.516667\pi$
$458$	−51.5011	−2.40649
$459$	0	0
$460$	−28.2719	−1.31818
$461$	35.3414	1.64601	0.823006	−	0.568033i	$-0.192295\pi$
$461$	35.3414	1.64601	0.823006	+	0.568033i	$0.192295\pi$
$462$	0	0
$463$	−5.69731	−0.264776	−0.132388	−	0.991198i	$-0.542265\pi$
$463$	−5.69731	−0.264776	−0.132388	+	0.991198i	$0.542265\pi$
$464$	3.63505	0.168753
$465$	0	0
$466$	3.07329	0.142368
$467$	−28.5160	−1.31956	−0.659781	−	0.751458i	$-0.729351\pi$
$467$	−28.5160	−1.31956	−0.659781	+	0.751458i	$0.729351\pi$
$468$	0	0
$469$	−1.14800	−0.0530096
$470$	13.3601	0.616256
$471$	0	0
$472$	1.37308	0.0632012
$473$	6.98115	0.320994
$474$	0	0
$475$	1.45118	0.0665847
$476$	−18.7620	−0.859955
$477$	0	0
$478$	59.1115	2.70370
$479$	−8.67563	−0.396400	−0.198200	−	0.980162i	$-0.563510\pi$
$479$	−8.67563	−0.396400	−0.198200	+	0.980162i	$0.563510\pi$
$480$	0	0
$481$	−14.2836	−0.651274
$482$	−41.6853	−1.89871
$483$	0	0
$484$	−23.8986	−1.08630
$485$	4.30656	0.195551
$486$	0	0
$487$	21.5198	0.975156	0.487578	−	0.873080i	$-0.337881\pi$
$487$	21.5198	0.975156	0.487578	+	0.873080i	$0.337881\pi$
$488$	−9.50543	−0.430290
$489$	0	0
$490$	−26.3594	−1.19080
$491$	−13.1169	−0.591957	−0.295979	−	0.955195i	$-0.595646\pi$
$491$	−13.1169	−0.591957	−0.295979	+	0.955195i	$0.595646\pi$
$492$	0	0
$493$	−10.1745	−0.458236
$494$	−35.8565	−1.61326
$495$	0	0
$496$	16.8125	0.754903
$497$	−16.5642	−0.743004
$498$	0	0
$499$	−27.5880	−1.23501	−0.617505	−	0.786567i	$-0.711856\pi$
$499$	−27.5880	−1.23501	−0.617505	+	0.786567i	$0.711856\pi$
$500$	−30.8670	−1.38041
$501$	0	0
$502$	−59.9436	−2.67541
$503$	−28.2351	−1.25894	−0.629470	−	0.777025i	$-0.716728\pi$
$503$	−28.2351	−1.25894	−0.629470	+	0.777025i	$0.716728\pi$
$504$	0	0
$505$	−17.2424	−0.767276
$506$	15.2272	0.676931
$507$	0	0
$508$	−4.52280	−0.200667
$509$	−21.1253	−0.936363	−0.468181	−	0.883632i	$-0.655091\pi$
$509$	−21.1253	−0.936363	−0.468181	+	0.883632i	$0.655091\pi$
$510$	0	0
$511$	0.431539	0.0190902
$512$	24.3063	1.07420
$513$	0	0
$514$	16.9896	0.749377
$515$	39.8282	1.75504
$516$	0	0
$517$	−4.09252	−0.179989
$518$	5.40608	0.237530
$519$	0	0
$520$	18.0797	0.792849
$521$	8.65236	0.379067	0.189533	−	0.981874i	$-0.439303\pi$
$521$	8.65236	0.379067	0.189533	+	0.981874i	$0.439303\pi$
$522$	0	0
$523$	−21.6819	−0.948085	−0.474043	−	0.880502i	$-0.657206\pi$
$523$	−21.6819	−0.948085	−0.474043	+	0.880502i	$0.657206\pi$
$524$	−9.16793	−0.400503
$525$	0	0
$526$	−36.6920	−1.59985
$527$	−47.0581	−2.04988
$528$	0	0
$529$	2.78158	0.120938
$530$	38.7577	1.68353
$531$	0	0
$532$	7.71845	0.334637
$533$	18.9553	0.821046
$534$	0	0
$535$	19.7473	0.853749
$536$	1.43791	0.0621084
$537$	0	0
$538$	28.2736	1.21896
$539$	8.07450	0.347793
$540$	0	0
$541$	−33.6531	−1.44686	−0.723430	−	0.690398i	$-0.757436\pi$
$541$	−33.6531	−1.44686	−0.723430	+	0.690398i	$0.757436\pi$
$542$	−29.5890	−1.27096
$543$	0	0
$544$	−50.2140	−2.15291
$545$	−41.4148	−1.77402
$546$	0	0
$547$	−0.162107	−0.00693119	−0.00346559	−	0.999994i	$-0.501103\pi$
$547$	−0.162107	−0.00693119	−0.00346559	+	0.999994i	$0.501103\pi$
$548$	44.2732	1.89126
$549$	0	0
$550$	1.63031	0.0695166
$551$	4.18566	0.178315
$552$	0	0
$553$	6.87706	0.292442
$554$	−47.7995	−2.03080
$555$	0	0
$556$	52.3060	2.21827
$557$	30.2877	1.28333	0.641666	−	0.766984i	$-0.278243\pi$
$557$	30.2877	1.28333	0.641666	+	0.766984i	$0.278243\pi$
$558$	0	0
$559$	−31.2690	−1.32254
$560$	5.36485	0.226706
$561$	0	0
$562$	1.67290	0.0705671
$563$	−18.7555	−0.790450	−0.395225	−	0.918584i	$-0.629333\pi$
$563$	−18.7555	−0.790450	−0.395225	+	0.918584i	$0.629333\pi$
$564$	0	0
$565$	−18.0085	−0.757624
$566$	−35.3422	−1.48554
$567$	0	0
$568$	20.7473	0.870536
$569$	−4.23290	−0.177452	−0.0887261	−	0.996056i	$-0.528280\pi$
$569$	−4.23290	−0.177452	−0.0887261	+	0.996056i	$0.528280\pi$
$570$	0	0
$571$	13.7100	0.573746	0.286873	−	0.957969i	$-0.407384\pi$
$571$	13.7100	0.573746	0.286873	+	0.957969i	$0.407384\pi$
$572$	−22.9104	−0.957931
$573$	0	0
$574$	−7.17426	−0.299448
$575$	2.76032	0.115113
$576$	0	0
$577$	20.1181	0.837529	0.418765	−	0.908095i	$-0.362463\pi$
$577$	20.1181	0.837529	0.418765	+	0.908095i	$0.362463\pi$
$578$	54.0631	2.24873
$579$	0	0
$580$	−8.73067	−0.362521
$581$	6.17628	0.256235
$582$	0	0
$583$	−11.8724	−0.491704
$584$	−0.540520	−0.0223669
$585$	0	0
$586$	45.2825	1.87060
$587$	−12.9025	−0.532544	−0.266272	−	0.963898i	$-0.585792\pi$
$587$	−12.9025	−0.532544	−0.266272	+	0.963898i	$0.585792\pi$
$588$	0	0
$589$	19.3591	0.797679
$590$	4.54608	0.187159
$591$	0	0
$592$	5.30876	0.218189
$593$	−22.1016	−0.907603	−0.453802	−	0.891103i	$-0.649933\pi$
$593$	−22.1016	−0.907603	−0.453802	+	0.891103i	$0.649933\pi$
$594$	0	0
$595$	−15.0162	−0.615605
$596$	49.6770	2.03485
$597$	0	0
$598$	−68.2035	−2.78905
$599$	44.4096	1.81453	0.907263	−	0.420564i	$-0.138168\pi$
$599$	44.4096	1.81453	0.907263	+	0.420564i	$0.138168\pi$
$600$	0	0
$601$	−19.5324	−0.796741	−0.398371	−	0.917225i	$-0.630424\pi$
$601$	−19.5324	−0.796741	−0.398371	+	0.917225i	$0.630424\pi$
$602$	11.8348	0.482350
$603$	0	0
$604$	10.9195	0.444307
$605$	−19.1273	−0.777636
$606$	0	0
$607$	−24.0502	−0.976167	−0.488084	−	0.872797i	$-0.662304\pi$
$607$	−24.0502	−0.976167	−0.488084	+	0.872797i	$0.662304\pi$
$608$	20.6574	0.837770
$609$	0	0
$610$	−31.4711	−1.27423
$611$	18.3307	0.741579
$612$	0	0
$613$	−3.46718	−0.140038	−0.0700190	−	0.997546i	$-0.522306\pi$
$613$	−3.46718	−0.140038	−0.0700190	+	0.997546i	$0.522306\pi$
$614$	42.2341	1.70443
$615$	0	0
$616$	2.09613	0.0844556
$617$	−31.8929	−1.28396	−0.641980	−	0.766722i	$-0.721887\pi$
$617$	−31.8929	−1.28396	−0.641980	+	0.766722i	$0.721887\pi$
$618$	0	0
$619$	8.81235	0.354198	0.177099	−	0.984193i	$-0.443329\pi$
$619$	8.81235	0.354198	0.177099	+	0.984193i	$0.443329\pi$
$620$	−40.3803	−1.62171
$621$	0	0
$622$	−74.2048	−2.97534
$623$	3.53479	0.141618
$624$	0	0
$625$	−21.9863	−0.879452
$626$	−17.6777	−0.706541
$627$	0	0
$628$	53.4662	2.13353
$629$	−14.8592	−0.592476
$630$	0	0
$631$	4.11493	0.163813	0.0819065	−	0.996640i	$-0.473899\pi$
$631$	4.11493	0.163813	0.0819065	+	0.996640i	$0.473899\pi$
$632$	−8.61379	−0.342638
$633$	0	0
$634$	−24.5769	−0.976073
$635$	−3.61984	−0.143649
$636$	0	0
$637$	−36.1662	−1.43296
$638$	4.70232	0.186167
$639$	0	0
$640$	−22.0103	−0.870035
$641$	−43.6648	−1.72465	−0.862327	−	0.506351i	$-0.830994\pi$
$641$	−43.6648	−1.72465	−0.862327	+	0.506351i	$0.830994\pi$
$642$	0	0
$643$	−25.0167	−0.986561	−0.493281	−	0.869870i	$-0.664202\pi$
$643$	−25.0167	−0.986561	−0.493281	+	0.869870i	$0.664202\pi$
$644$	14.6814	0.578530
$645$	0	0
$646$	−37.3017	−1.46761
$647$	0.907084	0.0356612	0.0178306	−	0.999841i	$-0.494324\pi$
$647$	0.907084	0.0356612	0.0178306	+	0.999841i	$0.494324\pi$
$648$	0	0
$649$	−1.39257	−0.0546633
$650$	−7.30225	−0.286418
$651$	0	0
$652$	10.9781	0.429934
$653$	14.5632	0.569902	0.284951	−	0.958542i	$-0.408023\pi$
$653$	14.5632	0.569902	0.284951	+	0.958542i	$0.408023\pi$
$654$	0	0
$655$	−7.33758	−0.286703
$656$	−7.04511	−0.275065
$657$	0	0
$658$	−6.93784	−0.270465
$659$	−29.6707	−1.15581	−0.577904	−	0.816105i	$-0.696129\pi$
$659$	−29.6707	−1.15581	−0.577904	+	0.816105i	$0.696129\pi$
$660$	0	0
$661$	34.3832	1.33735	0.668677	−	0.743553i	$-0.266861\pi$
$661$	34.3832	1.33735	0.668677	+	0.743553i	$0.266861\pi$
$662$	12.0874	0.469790
$663$	0	0
$664$	−7.73604	−0.300216
$665$	6.17748	0.239553
$666$	0	0
$667$	7.96164	0.308276
$668$	−1.43584	−0.0555542
$669$	0	0
$670$	4.76073	0.183923
$671$	9.64036	0.372162
$672$	0	0
$673$	16.3217	0.629155	0.314577	−	0.949232i	$-0.398137\pi$
$673$	16.3217	0.629155	0.314577	+	0.949232i	$0.398137\pi$
$674$	12.8102	0.493432
$675$	0	0
$676$	68.3283	2.62801
$677$	30.0619	1.15537	0.577687	−	0.816258i	$-0.303955\pi$
$677$	30.0619	1.15537	0.577687	+	0.816258i	$0.303955\pi$
$678$	0	0
$679$	−2.23638	−0.0858243
$680$	18.8084	0.721269
$681$	0	0
$682$	21.7487	0.832802
$683$	−24.5873	−0.940808	−0.470404	−	0.882451i	$-0.655892\pi$
$683$	−24.5873	−0.940808	−0.470404	+	0.882451i	$0.655892\pi$
$684$	0	0
$685$	35.4342	1.35387
$686$	30.2136	1.15356
$687$	0	0
$688$	11.6217	0.443075
$689$	53.1772	2.02589
$690$	0	0
$691$	10.7921	0.410549	0.205275	−	0.978704i	$-0.434191\pi$
$691$	10.7921	0.410549	0.205275	+	0.978704i	$0.434191\pi$
$692$	−5.66770	−0.215454
$693$	0	0
$694$	−27.3069	−1.03656
$695$	41.8633	1.58796
$696$	0	0
$697$	19.7193	0.746921
$698$	−1.34465	−0.0508957
$699$	0	0
$700$	1.57188	0.0594114
$701$	26.8572	1.01438	0.507192	−	0.861833i	$-0.330683\pi$
$701$	26.8572	1.01438	0.507192	+	0.861833i	$0.330683\pi$
$702$	0	0
$703$	6.11290	0.230552
$704$	16.7506	0.631312
$705$	0	0
$706$	29.0634	1.09382
$707$	8.95388	0.336745
$708$	0	0
$709$	39.0954	1.46826	0.734129	−	0.679010i	$-0.237591\pi$
$709$	39.0954	1.46826	0.734129	+	0.679010i	$0.237591\pi$
$710$	68.6913	2.57794
$711$	0	0
$712$	−4.42746	−0.165926
$713$	36.8234	1.37905
$714$	0	0
$715$	−18.3364	−0.685742
$716$	41.4905	1.55057
$717$	0	0
$718$	−20.4091	−0.761659
$719$	−41.0530	−1.53102	−0.765510	−	0.643424i	$-0.777513\pi$
$719$	−41.0530	−1.53102	−0.765510	+	0.643424i	$0.777513\pi$
$720$	0	0
$721$	−20.6826	−0.770259
$722$	−25.5712	−0.951663
$723$	0	0
$724$	−8.40375	−0.312323
$725$	0.852419	0.0316580
$726$	0	0
$727$	−28.2244	−1.04678	−0.523392	−	0.852092i	$-0.675334\pi$
$727$	−28.2244	−1.04678	−0.523392	+	0.852092i	$0.675334\pi$
$728$	−9.38872	−0.347969
$729$	0	0
$730$	−1.78959	−0.0662355
$731$	−32.5293	−1.20314
$732$	0	0
$733$	18.0881	0.668099	0.334049	−	0.942556i	$-0.391585\pi$
$733$	18.0881	0.668099	0.334049	+	0.942556i	$0.391585\pi$
$734$	68.7412	2.53729
$735$	0	0
$736$	39.2930	1.44836
$737$	−1.45832	−0.0537181
$738$	0	0
$739$	−1.41118	−0.0519113	−0.0259556	−	0.999663i	$-0.508263\pi$
$739$	−1.41118	−0.0519113	−0.0259556	+	0.999663i	$0.508263\pi$
$740$	−12.7506	−0.468722
$741$	0	0
$742$	−20.1267	−0.738873
$743$	26.5689	0.974719	0.487359	−	0.873202i	$-0.337960\pi$
$743$	26.5689	0.974719	0.487359	+	0.873202i	$0.337960\pi$
$744$	0	0
$745$	39.7591	1.45666
$746$	−74.5645	−2.73000
$747$	0	0
$748$	−23.8337	−0.871448
$749$	−10.2547	−0.374697
$750$	0	0
$751$	19.6508	0.717066	0.358533	−	0.933517i	$-0.383277\pi$
$751$	19.6508	0.717066	0.358533	+	0.933517i	$0.383277\pi$
$752$	−6.81295	−0.248443
$753$	0	0
$754$	−21.0620	−0.767033
$755$	8.73942	0.318060
$756$	0	0
$757$	23.8108	0.865418	0.432709	−	0.901534i	$-0.357558\pi$
$757$	23.8108	0.865418	0.432709	+	0.901534i	$0.357558\pi$
$758$	41.2595	1.49861
$759$	0	0
$760$	−7.73755	−0.280670
$761$	−32.7124	−1.18582	−0.592911	−	0.805268i	$-0.702021\pi$
$761$	−32.7124	−1.18582	−0.592911	+	0.805268i	$0.702021\pi$
$762$	0	0
$763$	21.5065	0.778588
$764$	17.7026	0.640458
$765$	0	0
$766$	25.9215	0.936582
$767$	6.23742	0.225220
$768$	0	0
$769$	14.2572	0.514127	0.257064	−	0.966394i	$-0.417245\pi$
$769$	14.2572	0.514127	0.257064	+	0.966394i	$0.417245\pi$
$770$	6.94001	0.250100
$771$	0	0
$772$	35.4789	1.27692
$773$	13.7948	0.496166	0.248083	−	0.968739i	$-0.420199\pi$
$773$	13.7948	0.496166	0.248083	+	0.968739i	$0.420199\pi$
$774$	0	0
$775$	3.94253	0.141620
$776$	2.80115	0.100556
$777$	0	0
$778$	77.0349	2.76183
$779$	−8.11227	−0.290652
$780$	0	0
$781$	−21.0418	−0.752934
$782$	−70.9523	−2.53725
$783$	0	0
$784$	13.4419	0.480067
$785$	42.7918	1.52730
$786$	0	0
$787$	24.1930	0.862388	0.431194	−	0.902259i	$-0.358093\pi$
$787$	24.1930	0.862388	0.431194	+	0.902259i	$0.358093\pi$
$788$	−38.3381	−1.36574
$789$	0	0
$790$	−28.5191	−1.01466
$791$	9.35173	0.332509
$792$	0	0
$793$	−43.1798	−1.53336
$794$	−13.0849	−0.464366
$795$	0	0
$796$	57.7501	2.04690
$797$	19.4755	0.689859	0.344929	−	0.938629i	$-0.387903\pi$
$797$	19.4755	0.689859	0.344929	+	0.938629i	$0.387903\pi$
$798$	0	0
$799$	19.0695	0.674629
$800$	4.20693	0.148737
$801$	0	0
$802$	−27.2477	−0.962151
$803$	0.548193	0.0193453
$804$	0	0
$805$	11.7503	0.414145
$806$	−97.4140	−3.43126
$807$	0	0
$808$	−11.2151	−0.394546
$809$	−19.6358	−0.690357	−0.345179	−	0.938537i	$-0.612182\pi$
$809$	−19.6358	−0.690357	−0.345179	+	0.938537i	$0.612182\pi$
$810$	0	0
$811$	−42.1136	−1.47881	−0.739405	−	0.673261i	$-0.764893\pi$
$811$	−42.1136	−1.47881	−0.739405	+	0.673261i	$0.764893\pi$
$812$	4.53380	0.159105
$813$	0	0
$814$	6.86745	0.240704
$815$	8.78631	0.307771
$816$	0	0
$817$	13.3821	0.468182
$818$	59.4615	2.07902
$819$	0	0
$820$	16.9210	0.590906
$821$	20.0009	0.698037	0.349019	−	0.937116i	$-0.386515\pi$
$821$	20.0009	0.698037	0.349019	+	0.937116i	$0.386515\pi$
$822$	0	0
$823$	16.9437	0.590620	0.295310	−	0.955401i	$-0.404577\pi$
$823$	16.9437	0.590620	0.295310	+	0.955401i	$0.404577\pi$
$824$	25.9057	0.902469
$825$	0	0
$826$	−2.36076	−0.0821413
$827$	33.5188	1.16556	0.582782	−	0.812628i	$-0.301964\pi$
$827$	33.5188	1.16556	0.582782	+	0.812628i	$0.301964\pi$
$828$	0	0
$829$	35.0043	1.21575	0.607876	−	0.794032i	$-0.292022\pi$
$829$	35.0043	1.21575	0.607876	+	0.794032i	$0.292022\pi$
$830$	−25.6129	−0.889038
$831$	0	0
$832$	−75.0271	−2.60110
$833$	−37.6238	−1.30359
$834$	0	0
$835$	−1.14918	−0.0397689
$836$	9.80491	0.339110
$837$	0	0
$838$	80.3318	2.77502
$839$	−24.4827	−0.845237	−0.422619	−	0.906308i	$-0.638889\pi$
$839$	−24.4827	−0.845237	−0.422619	+	0.906308i	$0.638889\pi$
$840$	0	0
$841$	−26.5414	−0.915219
$842$	−36.8694	−1.27060
$843$	0	0
$844$	49.7054	1.71093
$845$	54.6867	1.88128
$846$	0	0
$847$	9.93273	0.341293
$848$	−19.7643	−0.678710
$849$	0	0
$850$	−7.59656	−0.260560
$851$	11.6275	0.398585
$852$	0	0
$853$	−56.8488	−1.94647	−0.973233	−	0.229821i	$-0.926186\pi$
$853$	−56.8488	−1.94647	−0.973233	+	0.229821i	$0.926186\pi$
$854$	16.3428	0.559239
$855$	0	0
$856$	12.8444	0.439012
$857$	−43.6475	−1.49097	−0.745485	−	0.666523i	$-0.767782\pi$
$857$	−43.6475	−1.49097	−0.745485	+	0.666523i	$0.767782\pi$
$858$	0	0
$859$	−35.0392	−1.19552	−0.597761	−	0.801674i	$-0.703943\pi$
$859$	−35.0392	−1.19552	−0.597761	+	0.801674i	$0.703943\pi$
$860$	−27.9132	−0.951831
$861$	0	0
$862$	14.9713	0.509924
$863$	−30.4689	−1.03717	−0.518587	−	0.855025i	$-0.673542\pi$
$863$	−30.4689	−1.03717	−0.518587	+	0.855025i	$0.673542\pi$
$864$	0	0
$865$	−4.53616	−0.154234
$866$	1.68120	0.0571295
$867$	0	0
$868$	20.9693	0.711744
$869$	8.73607	0.296351
$870$	0	0
$871$	6.53193	0.221326
$872$	−26.9378	−0.912227
$873$	0	0
$874$	29.1889	0.987330
$875$	12.8289	0.433696
$876$	0	0
$877$	17.9852	0.607315	0.303658	−	0.952781i	$-0.401792\pi$
$877$	17.9852	0.607315	0.303658	+	0.952781i	$0.401792\pi$
$878$	63.2819	2.13566
$879$	0	0
$880$	6.81507	0.229736
$881$	−27.8993	−0.939950	−0.469975	−	0.882680i	$-0.655737\pi$
$881$	−27.8993	−0.939950	−0.469975	+	0.882680i	$0.655737\pi$
$882$	0	0
$883$	19.7435	0.664423	0.332211	−	0.943205i	$-0.392205\pi$
$883$	19.7435	0.664423	0.332211	+	0.943205i	$0.392205\pi$
$884$	106.753	3.59049
$885$	0	0
$886$	−53.9791	−1.81346
$887$	−34.0507	−1.14331	−0.571655	−	0.820494i	$-0.693698\pi$
$887$	−34.0507	−1.14331	−0.571655	+	0.820494i	$0.693698\pi$
$888$	0	0
$889$	1.87976	0.0630452
$890$	−14.6587	−0.491361
$891$	0	0
$892$	−57.1438	−1.91332
$893$	−7.84494	−0.262521
$894$	0	0
$895$	33.2070	1.10999
$896$	11.4299	0.381845
$897$	0	0
$898$	−73.9305	−2.46709
$899$	11.3715	0.379261
$900$	0	0
$901$	55.3204	1.84299
$902$	−9.11361	−0.303450
$903$	0	0
$904$	−11.7134	−0.389582
$905$	−6.72596	−0.223579
$906$	0	0
$907$	−53.9117	−1.79011	−0.895054	−	0.445957i	$-0.852864\pi$
$907$	−53.9117	−1.79011	−0.895054	+	0.445957i	$0.852864\pi$
$908$	45.9478	1.52483
$909$	0	0
$910$	−31.0847	−1.03045
$911$	−1.45383	−0.0481675	−0.0240837	−	0.999710i	$-0.507667\pi$
$911$	−1.45383	−0.0481675	−0.0240837	+	0.999710i	$0.507667\pi$
$912$	0	0
$913$	7.84585	0.259660
$914$	−4.81876	−0.159391
$915$	0	0
$916$	−63.0782	−2.08416
$917$	3.81037	0.125829
$918$	0	0
$919$	51.8918	1.71175	0.855877	−	0.517180i	$-0.173018\pi$
$919$	51.8918	1.71175	0.855877	+	0.517180i	$0.173018\pi$
$920$	−14.7178	−0.485230
$921$	0	0
$922$	76.1080	2.50648
$923$	94.2475	3.10219
$924$	0	0
$925$	1.24490	0.0409322
$926$	−12.2692	−0.403191
$927$	0	0
$928$	12.1341	0.398322
$929$	13.8429	0.454172	0.227086	−	0.973875i	$-0.427080\pi$
$929$	13.8429	0.454172	0.227086	+	0.973875i	$0.427080\pi$
$930$	0	0
$931$	15.4780	0.507270
$932$	3.76415	0.123299
$933$	0	0
$934$	−61.4095	−2.00938
$935$	−19.0754	−0.623832
$936$	0	0
$937$	−21.1929	−0.692343	−0.346171	−	0.938171i	$-0.612518\pi$
$937$	−21.1929	−0.692343	−0.346171	+	0.938171i	$0.612518\pi$
$938$	−2.47222	−0.0807210
$939$	0	0
$940$	16.3634	0.533714
$941$	56.0903	1.82849	0.914246	−	0.405160i	$-0.132784\pi$
$941$	56.0903	1.82849	0.914246	+	0.405160i	$0.132784\pi$
$942$	0	0
$943$	−15.4305	−0.502487
$944$	−2.31826	−0.0754529
$945$	0	0
$946$	15.0340	0.488797
$947$	−4.13650	−0.134418	−0.0672091	−	0.997739i	$-0.521409\pi$
$947$	−4.13650	−0.134418	−0.0672091	+	0.997739i	$0.521409\pi$
$948$	0	0
$949$	−2.45539	−0.0797053
$950$	3.12513	0.101393
$951$	0	0
$952$	−9.76711	−0.316554
$953$	47.9570	1.55348	0.776740	−	0.629822i	$-0.216872\pi$
$953$	47.9570	1.55348	0.776740	+	0.629822i	$0.216872\pi$
$954$	0	0
$955$	14.1683	0.458476
$956$	72.3994	2.34156
$957$	0	0
$958$	−18.6831	−0.603622
$959$	−18.4008	−0.594192
$960$	0	0
$961$	21.5944	0.696592
$962$	−30.7598	−0.991735
$963$	0	0
$964$	−51.0558	−1.64440
$965$	28.3957	0.914089
$966$	0	0
$967$	−21.7913	−0.700760	−0.350380	−	0.936608i	$-0.613948\pi$
$967$	−21.7913	−0.700760	−0.350380	+	0.936608i	$0.613948\pi$
$968$	−12.4411	−0.399873
$969$	0	0
$970$	9.27423	0.297777
$971$	−35.8613	−1.15084	−0.575422	−	0.817857i	$-0.695162\pi$
$971$	−35.8613	−1.15084	−0.575422	+	0.817857i	$0.695162\pi$
$972$	0	0
$973$	−21.7394	−0.696933
$974$	46.3431	1.48493
$975$	0	0
$976$	16.0486	0.513703
$977$	−31.7008	−1.01420	−0.507100	−	0.861887i	$-0.669283\pi$
$977$	−31.7008	−1.01420	−0.507100	+	0.861887i	$0.669283\pi$
$978$	0	0
$979$	4.49031	0.143511
$980$	−32.2848	−1.03130
$981$	0	0
$982$	−28.2474	−0.901410
$983$	−25.0289	−0.798297	−0.399148	−	0.916886i	$-0.630694\pi$
$983$	−25.0289	−0.798297	−0.399148	+	0.916886i	$0.630694\pi$
$984$	0	0
$985$	−30.6840	−0.977672
$986$	−21.9109	−0.697785
$987$	0	0
$988$	−43.9168	−1.39718
$989$	25.4545	0.809405
$990$	0	0
$991$	−16.4805	−0.523520	−0.261760	−	0.965133i	$-0.584303\pi$
$991$	−16.4805	−0.523520	−0.261760	+	0.965133i	$0.584303\pi$
$992$	56.1215	1.78186
$993$	0	0
$994$	−35.6711	−1.13142
$995$	46.2205	1.46529
$996$	0	0
$997$	20.6217	0.653097	0.326549	−	0.945180i	$-0.394114\pi$
$997$	20.6217	0.653097	0.326549	+	0.945180i	$0.394114\pi$
$998$	−59.4110	−1.88062
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	531.2.a.g.1.4	yes	5
3.2	odd	2		531.2.a.e.1.2	✓	5
4.3	odd	2		8496.2.a.bz.1.3		5
12.11	even	2		8496.2.a.bu.1.3		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
531.2.a.e.1.2	✓	5	3.2	odd	2
531.2.a.g.1.4	yes	5	1.1	even	1	trivial
8496.2.a.bu.1.3		5	12.11	even	2
8496.2.a.bz.1.3		5	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+2.15351 q^{2} +2.63760 q^{4} +2.11101 q^{5} -1.09624 q^{7} +1.37308 q^{8} +O(q^{10})\)	\(q+2.15351 q^{2} +2.63760 q^{4} +2.11101 q^{5} -1.09624 q^{7} +1.37308 q^{8} +4.54608 q^{10} -1.39257 q^{11} +6.23742 q^{13} -2.36076 q^{14} -2.31826 q^{16} +6.48881 q^{17} -2.66942 q^{19} +5.56801 q^{20} -2.99892 q^{22} -5.07756 q^{23} -0.543632 q^{25} +13.4323 q^{26} -2.89144 q^{28} -1.56801 q^{29} -7.25220 q^{31} -7.73856 q^{32} +13.9737 q^{34} -2.31417 q^{35} -2.28998 q^{37} -5.74861 q^{38} +2.89859 q^{40} +3.03897 q^{41} -5.01313 q^{43} -3.67305 q^{44} -10.9346 q^{46} +2.93882 q^{47} -5.79826 q^{49} -1.17072 q^{50} +16.4518 q^{52} +8.52551 q^{53} -2.93974 q^{55} -1.50522 q^{56} -3.37672 q^{58} +1.00000 q^{59} -6.92270 q^{61} -15.6177 q^{62} -12.0285 q^{64} +13.1673 q^{65} +1.04722 q^{67} +17.1149 q^{68} -4.98359 q^{70} +15.1100 q^{71} -0.393655 q^{73} -4.93149 q^{74} -7.04086 q^{76} +1.52659 q^{77} -6.27333 q^{79} -4.89387 q^{80} +6.54444 q^{82} -5.63407 q^{83} +13.6980 q^{85} -10.7958 q^{86} -1.91212 q^{88} -3.22447 q^{89} -6.83770 q^{91} -13.3926 q^{92} +6.32878 q^{94} -5.63517 q^{95} +2.04005 q^{97} -12.4866 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 3 q^{2} + 5 q^{4} + 8 q^{5} + 2 q^{7} + 9 q^{8}+O(q^{10}) \) 5 * q + 3 * q^2 + 5 * q^4 + 8 * q^5 + 2 * q^7 + 9 * q^8	\( 5 q + 3 q^{2} + 5 q^{4} + 8 q^{5} + 2 q^{7} + 9 q^{8} - 2 q^{10} + 10 q^{11} - 4 q^{13} + 6 q^{14} + q^{16} + 8 q^{17} - 6 q^{19} + 2 q^{22} + 4 q^{23} + 7 q^{25} + 2 q^{26} - 2 q^{28} + 20 q^{29} - 6 q^{31} + q^{32} + 4 q^{34} + 12 q^{35} - 18 q^{38} - 6 q^{40} + 8 q^{41} - 4 q^{43} + 30 q^{44} - 2 q^{46} - 2 q^{47} - 5 q^{49} - 19 q^{50} - 2 q^{52} + 20 q^{53} + 10 q^{55} - 14 q^{56} + 22 q^{58} + 5 q^{59} - 14 q^{61} - 6 q^{62} - 9 q^{64} + 2 q^{65} - 24 q^{67} - 8 q^{68} + 6 q^{70} + 12 q^{71} - 2 q^{73} - 22 q^{74} - 4 q^{76} + 2 q^{77} - 8 q^{79} - 28 q^{80} - 18 q^{82} - 18 q^{83} + 10 q^{85} - 14 q^{86} + 48 q^{88} + 4 q^{89} - 24 q^{91} - 50 q^{92} - 12 q^{94} - 4 q^{95} + 20 q^{97} - 3 q^{98}+O(q^{100}) \) 5 * q + 3 * q^2 + 5 * q^4 + 8 * q^5 + 2 * q^7 + 9 * q^8 - 2 * q^10 + 10 * q^11 - 4 * q^13 + 6 * q^14 + q^16 + 8 * q^17 - 6 * q^19 + 2 * q^22 + 4 * q^23 + 7 * q^25 + 2 * q^26 - 2 * q^28 + 20 * q^29 - 6 * q^31 + q^32 + 4 * q^34 + 12 * q^35 - 18 * q^38 - 6 * q^40 + 8 * q^41 - 4 * q^43 + 30 * q^44 - 2 * q^46 - 2 * q^47 - 5 * q^49 - 19 * q^50 - 2 * q^52 + 20 * q^53 + 10 * q^55 - 14 * q^56 + 22 * q^58 + 5 * q^59 - 14 * q^61 - 6 * q^62 - 9 * q^64 + 2 * q^65 - 24 * q^67 - 8 * q^68 + 6 * q^70 + 12 * q^71 - 2 * q^73 - 22 * q^74 - 4 * q^76 + 2 * q^77 - 8 * q^79 - 28 * q^80 - 18 * q^82 - 18 * q^83 + 10 * q^85 - 14 * q^86 + 48 * q^88 + 4 * q^89 - 24 * q^91 - 50 * q^92 - 12 * q^94 - 4 * q^95 + 20 * q^97 - 3 * q^98

Introduction

L-functions

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