LMFDB - Embedded newform 531.4.a.f.1.5

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:	$0$
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} - 49x^{6} + 89x^{5} + 648x^{4} - 1023x^{3} - 1476x^{2} + 1940x - 384 $ x^8 - 2x^7 - 49x^6 + 89x^5 + 648x^4 - 1023x^3 - 1476x^2 + 1940*x - 384
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			1.5
Root			$0.254436$ 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)

$f(q)$	$=$	$q-0.254436 q^{2} -7.93526 q^{4} +10.8225 q^{5} -23.2950 q^{7} +4.05451 q^{8} +O(q^{10})$	\(q-0.254436 q^{2} -7.93526 q^{4} +10.8225 q^{5} -23.2950 q^{7} +4.05451 q^{8} -2.75362 q^{10} -51.0608 q^{11} +51.8661 q^{13} +5.92710 q^{14} +62.4505 q^{16} +0.0421754 q^{17} -85.2082 q^{19} -85.8790 q^{20} +12.9917 q^{22} +8.50543 q^{23} -7.87442 q^{25} -13.1966 q^{26} +184.852 q^{28} +101.142 q^{29} +271.338 q^{31} -48.3257 q^{32} -0.0107310 q^{34} -252.110 q^{35} -9.54238 q^{37} +21.6800 q^{38} +43.8797 q^{40} +185.363 q^{41} +277.532 q^{43} +405.181 q^{44} -2.16409 q^{46} -309.877 q^{47} +199.659 q^{49} +2.00354 q^{50} -411.571 q^{52} -273.504 q^{53} -552.603 q^{55} -94.4499 q^{56} -25.7341 q^{58} +59.0000 q^{59} +752.893 q^{61} -69.0383 q^{62} -487.308 q^{64} +561.319 q^{65} +751.435 q^{67} -0.334673 q^{68} +64.1458 q^{70} +651.607 q^{71} +842.428 q^{73} +2.42793 q^{74} +676.149 q^{76} +1189.46 q^{77} -368.828 q^{79} +675.868 q^{80} -47.1631 q^{82} +545.888 q^{83} +0.456442 q^{85} -70.6143 q^{86} -207.026 q^{88} -634.328 q^{89} -1208.22 q^{91} -67.4928 q^{92} +78.8440 q^{94} -922.162 q^{95} +316.368 q^{97} -50.8004 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 2 q^{2} + 38 q^{4} + 12 q^{5} + 53 q^{7} - 3 q^{8}+O(q^{10}) $ 8 * q - 2 * q^2 + 38 * q^4 + 12 * q^5 + 53 * q^7 - 3 * q^8	$ 8 q - 2 q^{2} + 38 q^{4} + 12 q^{5} + 53 q^{7} - 3 q^{8} + 29 q^{10} + 27 q^{11} + 89 q^{13} + 37 q^{14} + 362 q^{16} - 79 q^{17} + 288 q^{19} - 457 q^{20} + 596 q^{22} - 202 q^{23} + 264 q^{25} - 270 q^{26} + 702 q^{28} + 114 q^{29} + 538 q^{31} - 316 q^{32} + 498 q^{34} + 196 q^{35} + 395 q^{37} - 397 q^{38} + 918 q^{40} + 39 q^{41} + 527 q^{43} - 64 q^{44} - 539 q^{46} - 860 q^{47} + 347 q^{49} + 591 q^{50} - 644 q^{52} + 812 q^{53} + 536 q^{55} + 2218 q^{56} - 1154 q^{58} + 472 q^{59} - 460 q^{61} + 2014 q^{62} - 451 q^{64} + 986 q^{65} + 1934 q^{67} + 69 q^{68} - 1028 q^{70} + 1687 q^{71} + 1980 q^{73} + 2400 q^{74} - 940 q^{76} + 821 q^{77} + 3319 q^{79} + 2119 q^{80} + 429 q^{82} - 2057 q^{83} + 566 q^{85} + 6690 q^{86} + 1189 q^{88} - 1668 q^{89} + 2427 q^{91} + 980 q^{92} + 332 q^{94} - 2146 q^{95} + 1956 q^{97} + 2026 q^{98}+O(q^{100}) $ 8 * q - 2 * q^2 + 38 * q^4 + 12 * q^5 + 53 * q^7 - 3 * q^8 + 29 * q^10 + 27 * q^11 + 89 * q^13 + 37 * q^14 + 362 * q^16 - 79 * q^17 + 288 * q^19 - 457 * q^20 + 596 * q^22 - 202 * q^23 + 264 * q^25 - 270 * q^26 + 702 * q^28 + 114 * q^29 + 538 * q^31 - 316 * q^32 + 498 * q^34 + 196 * q^35 + 395 * q^37 - 397 * q^38 + 918 * q^40 + 39 * q^41 + 527 * q^43 - 64 * q^44 - 539 * q^46 - 860 * q^47 + 347 * q^49 + 591 * q^50 - 644 * q^52 + 812 * q^53 + 536 * q^55 + 2218 * q^56 - 1154 * q^58 + 472 * q^59 - 460 * q^61 + 2014 * q^62 - 451 * q^64 + 986 * q^65 + 1934 * q^67 + 69 * q^68 - 1028 * q^70 + 1687 * q^71 + 1980 * q^73 + 2400 * q^74 - 940 * q^76 + 821 * q^77 + 3319 * q^79 + 2119 * q^80 + 429 * q^82 - 2057 * q^83 + 566 * q^85 + 6690 * q^86 + 1189 * q^88 - 1668 * q^89 + 2427 * q^91 + 980 * q^92 + 332 * q^94 - 2146 * q^95 + 1956 * q^97 + 2026 * 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 $

$2$	−0.254436	−0.0899568	−0.0449784	−	0.998988i	$-0.514322\pi$
$2$	−0.254436	−0.0899568	−0.0449784	+	0.998988i	$0.514322\pi$
$3$	0	0
$3$	0	0
$4$	−7.93526	−0.991908
$5$	10.8225	0.967990	0.483995	−	0.875071i	$-0.339185\pi$
$5$	10.8225	0.967990	0.483995	+	0.875071i	$0.339185\pi$
$6$	0	0
$7$	−23.2950	−1.25781	−0.628907	−	0.777481i	$-0.716497\pi$
$7$	−23.2950	−1.25781	−0.628907	+	0.777481i	$0.716497\pi$
$8$	4.05451	0.179186
$9$	0	0
$10$	−2.75362	−0.0870773
$11$	−51.0608	−1.39958	−0.699791	−	0.714347i	$-0.746724\pi$
$11$	−51.0608	−1.39958	−0.699791	+	0.714347i	$0.746724\pi$
$12$	0	0
$13$	51.8661	1.10654	0.553272	−	0.833001i	$-0.313379\pi$
$13$	51.8661	1.10654	0.553272	+	0.833001i	$0.313379\pi$
$14$	5.92710	0.113149
$15$	0	0
$16$	62.4505	0.975789
$17$	0.0421754	0.000601709 0	0.000300854	−	1.00000i	$-0.499904\pi$
$17$	0.0421754	0.000601709 0	0.000300854		1.00000i	$0.499904\pi$
$18$	0	0
$19$	−85.2082	−1.02885	−0.514424	−	0.857536i	$-0.671994\pi$
$19$	−85.2082	−1.02885	−0.514424	+	0.857536i	$0.671994\pi$
$20$	−85.8790	−0.960157
$21$	0	0
$22$	12.9917	0.125902
$23$	8.50543	0.0771089	0.0385544	−	0.999257i	$-0.487725\pi$
$23$	8.50543	0.0771089	0.0385544	+	0.999257i	$0.487725\pi$
$24$	0	0
$25$	−7.87442	−0.0629954
$26$	−13.1966	−0.0995411
$27$	0	0
$28$	184.852	1.24764
$29$	101.142	0.647639	0.323819	−	0.946119i	$-0.395033\pi$
$29$	101.142	0.647639	0.323819	+	0.946119i	$0.395033\pi$
$30$	0	0
$31$	271.338	1.57206	0.786029	−	0.618189i	$-0.212133\pi$
$31$	271.338	1.57206	0.786029	+	0.618189i	$0.212133\pi$
$32$	−48.3257	−0.266964
$33$	0	0
$34$	−0.0107310	−5.41278e−5 0
$35$	−252.110	−1.21755
$36$	0	0
$37$	−9.54238	−0.0423988	−0.0211994	−	0.999775i	$-0.506748\pi$
$37$	−9.54238	−0.0423988	−0.0211994	+	0.999775i	$0.506748\pi$
$38$	21.6800	0.0925518
$39$	0	0
$40$	43.8797	0.173450
$41$	185.363	0.706071	0.353035	−	0.935610i	$-0.385150\pi$
$41$	185.363	0.706071	0.353035	+	0.935610i	$0.385150\pi$
$42$	0	0
$43$	277.532	0.984263	0.492131	−	0.870521i	$-0.336218\pi$
$43$	277.532	0.984263	0.492131	+	0.870521i	$0.336218\pi$
$44$	405.181	1.38826
$45$	0	0
$46$	−2.16409	−0.00693647
$47$	−309.877	−0.961708	−0.480854	−	0.876801i	$-0.659673\pi$
$47$	−309.877	−0.961708	−0.480854	+	0.876801i	$0.659673\pi$
$48$	0	0
$49$	199.659	0.582096
$50$	2.00354	0.00566686
$51$	0	0
$52$	−411.571	−1.09759
$53$	−273.504	−0.708842	−0.354421	−	0.935086i	$-0.615322\pi$
$53$	−273.504	−0.708842	−0.354421	+	0.935086i	$0.615322\pi$
$54$	0	0
$55$	−552.603	−1.35478
$56$	−94.4499	−0.225382
$57$	0	0
$58$	−25.7341	−0.0582595
$59$	59.0000	0.130189
$59$	59.0000	0.130189
$60$	0	0
$61$	752.893	1.58030	0.790148	−	0.612916i	$-0.210004\pi$
$61$	752.893	1.58030	0.790148	+	0.612916i	$0.210004\pi$
$62$	−69.0383	−0.141417
$63$	0	0
$64$	−487.308	−0.951774
$65$	561.319	1.07112
$66$	0	0
$67$	751.435	1.37018	0.685092	−	0.728456i	$-0.259762\pi$
$67$	751.435	1.37018	0.685092	+	0.728456i	$0.259762\pi$
$68$	−0.334673	−0.000596840 0
$69$	0	0
$70$	64.1458	0.109527
$71$	651.607	1.08918	0.544588	−	0.838704i	$-0.316686\pi$
$71$	651.607	1.08918	0.544588	+	0.838704i	$0.316686\pi$
$72$	0	0
$73$	842.428	1.35067	0.675334	−	0.737512i	$-0.263999\pi$
$73$	842.428	1.35067	0.675334	+	0.737512i	$0.263999\pi$
$74$	2.42793	0.00381406
$75$	0	0
$76$	676.149	1.02052
$77$	1189.46	1.76041
$78$	0	0
$79$	−368.828	−0.525271	−0.262635	−	0.964895i	$-0.584592\pi$
$79$	−368.828	−0.525271	−0.262635	+	0.964895i	$0.584592\pi$
$80$	675.868	0.944554
$81$	0	0
$82$	−47.1631	−0.0635158
$83$	545.888	0.721916	0.360958	−	0.932582i	$-0.382450\pi$
$83$	545.888	0.721916	0.360958	+	0.932582i	$0.382450\pi$
$84$	0	0
$85$	0.456442	0.000582448 0
$86$	−70.6143	−0.0885411
$87$	0	0
$88$	−207.026	−0.250785
$89$	−634.328	−0.755490	−0.377745	−	0.925910i	$-0.623300\pi$
$89$	−634.328	−0.755490	−0.377745	+	0.925910i	$0.623300\pi$
$90$	0	0
$91$	−1208.22	−1.39183
$92$	−67.4928	−0.0764849
$93$	0	0
$94$	78.8440	0.0865121
$95$	−922.162	−0.995914
$96$	0	0
$97$	316.368	0.331158	0.165579	−	0.986197i	$-0.447051\pi$
$97$	316.368	0.331158	0.165579	+	0.986197i	$0.447051\pi$
$98$	−50.8004	−0.0523635
$99$	0	0
$100$	62.4856	0.0624856
$101$	1615.25	1.59132	0.795662	−	0.605740i	$-0.207123\pi$
$101$	1615.25	1.59132	0.795662	+	0.605740i	$0.207123\pi$
$102$	0	0
$103$	1261.07	1.20638	0.603190	−	0.797598i	$-0.293896\pi$
$103$	1261.07	1.20638	0.603190	+	0.797598i	$0.293896\pi$
$104$	210.291	0.198277
$105$	0	0
$106$	69.5892	0.0637651
$107$	−733.783	−0.662967	−0.331483	−	0.943461i	$-0.607549\pi$
$107$	−733.783	−0.662967	−0.331483	+	0.943461i	$0.607549\pi$
$108$	0	0
$109$	582.848	0.512171	0.256086	−	0.966654i	$-0.417567\pi$
$109$	582.848	0.512171	0.256086	+	0.966654i	$0.417567\pi$
$110$	140.602	0.121872
$111$	0	0
$112$	−1454.79	−1.22736
$113$	1661.78	1.38342	0.691712	−	0.722174i	$-0.256857\pi$
$113$	1661.78	1.38342	0.691712	+	0.722174i	$0.256857\pi$
$114$	0	0
$115$	92.0496	0.0746406
$116$	−802.585	−0.642398
$117$	0	0
$118$	−15.0117	−0.0117114
$119$	−0.982479	−0.000756838 0
$120$	0	0
$121$	1276.20	0.958831
$122$	−191.563	−0.142158
$123$	0	0
$124$	−2153.14	−1.55934
$125$	−1438.03	−1.02897
$126$	0	0
$127$	1560.58	1.09039	0.545193	−	0.838310i	$-0.316456\pi$
$127$	1560.58	1.09039	0.545193	+	0.838310i	$0.316456\pi$
$128$	510.595	0.352583
$129$	0	0
$130$	−142.820	−0.0963548
$131$	−2608.59	−1.73980	−0.869898	−	0.493232i	$-0.835815\pi$
$131$	−2608.59	−1.73980	−0.869898	+	0.493232i	$0.835815\pi$
$132$	0	0
$133$	1984.93	1.29410
$134$	−191.192	−0.123257
$135$	0	0
$136$	0.171001	0.000107818 0
$137$	2284.40	1.42459	0.712296	−	0.701879i	$-0.247655\pi$
$137$	2284.40	1.42459	0.712296	+	0.701879i	$0.247655\pi$
$138$	0	0
$139$	−288.324	−0.175937	−0.0879687	−	0.996123i	$-0.528038\pi$
$139$	−288.324	−0.175937	−0.0879687	+	0.996123i	$0.528038\pi$
$140$	2000.56	1.20770
$141$	0	0
$142$	−165.792	−0.0979787
$143$	−2648.32	−1.54870
$144$	0	0
$145$	1094.60	0.626908
$146$	−214.344	−0.121502
$147$	0	0
$148$	75.7212	0.0420557
$149$	−591.503	−0.325220	−0.162610	−	0.986690i	$-0.551991\pi$
$149$	−591.503	−0.325220	−0.162610	+	0.986690i	$0.551991\pi$
$150$	0	0
$151$	−2398.78	−1.29278	−0.646391	−	0.763007i	$-0.723722\pi$
$151$	−2398.78	−1.29278	−0.646391	+	0.763007i	$0.723722\pi$
$152$	−345.477	−0.184355
$153$	0	0
$154$	−302.642	−0.158361
$155$	2936.55	1.52174
$156$	0	0
$157$	135.478	0.0688684	0.0344342	−	0.999407i	$-0.489037\pi$
$157$	135.478	0.0688684	0.0344342	+	0.999407i	$0.489037\pi$
$158$	93.8432	0.0472517
$159$	0	0
$160$	−523.003	−0.258419
$161$	−198.134	−0.0969886
$162$	0	0
$163$	1649.77	0.792763	0.396381	−	0.918086i	$-0.370266\pi$
$163$	1649.77	0.792763	0.396381	+	0.918086i	$0.370266\pi$
$164$	−1470.91	−0.700357
$165$	0	0
$166$	−138.894	−0.0649412
$167$	−3500.76	−1.62214	−0.811070	−	0.584949i	$-0.801114\pi$
$167$	−3500.76	−1.62214	−0.811070	+	0.584949i	$0.801114\pi$
$168$	0	0
$169$	493.092	0.224439
$170$	−0.116135	−5.23951e−5 0
$171$	0	0
$172$	−2202.29	−0.976298
$173$	−2524.80	−1.10958	−0.554790	−	0.831991i	$-0.687201\pi$
$173$	−2524.80	−1.10958	−0.554790	+	0.831991i	$0.687201\pi$
$174$	0	0
$175$	183.435	0.0792364
$176$	−3188.77	−1.36570
$177$	0	0
$178$	161.396	0.0679614
$179$	−2717.45	−1.13470	−0.567351	−	0.823476i	$-0.692032\pi$
$179$	−2717.45	−1.13470	−0.567351	+	0.823476i	$0.692032\pi$
$180$	0	0
$181$	2116.30	0.869077	0.434539	−	0.900653i	$-0.356911\pi$
$181$	2116.30	0.869077	0.434539	+	0.900653i	$0.356911\pi$
$182$	307.416	0.125204
$183$	0	0
$184$	34.4853	0.0138168
$185$	−103.272	−0.0410416
$186$	0	0
$187$	−2.15351	−0.000842141 0
$188$	2458.96	0.953925
$189$	0	0
$190$	234.631	0.0895892
$191$	162.049	0.0613899	0.0306950	−	0.999529i	$-0.490228\pi$
$191$	162.049	0.0613899	0.0306950	+	0.999529i	$0.490228\pi$
$192$	0	0
$193$	−1737.71	−0.648099	−0.324050	−	0.946040i	$-0.605044\pi$
$193$	−1737.71	−0.648099	−0.324050	+	0.946040i	$0.605044\pi$
$194$	−80.4956	−0.0297899
$195$	0	0
$196$	−1584.35	−0.577385
$197$	1898.32	0.686548	0.343274	−	0.939235i	$-0.388464\pi$
$197$	1898.32	0.686548	0.343274	+	0.939235i	$0.388464\pi$
$198$	0	0
$199$	−925.010	−0.329509	−0.164754	−	0.986335i	$-0.552683\pi$
$199$	−925.010	−0.329509	−0.164754	+	0.986335i	$0.552683\pi$
$200$	−31.9269	−0.0112879
$201$	0	0
$202$	−410.979	−0.143150
$203$	−2356.10	−0.814609
$204$	0	0
$205$	2006.09	0.683469
$206$	−320.862	−0.108522
$207$	0	0
$208$	3239.06	1.07975
$209$	4350.80	1.43996
$210$	0	0
$211$	2553.68	0.833186	0.416593	−	0.909093i	$-0.363224\pi$
$211$	2553.68	0.833186	0.416593	+	0.909093i	$0.363224\pi$
$212$	2170.32	0.703106
$213$	0	0
$214$	186.701	0.0596383
$215$	3003.58	0.952756
$216$	0	0
$217$	−6320.84	−1.97736
$218$	−148.298	−0.0460733
$219$	0	0
$220$	4385.05	1.34382
$221$	2.18748	0.000665817 0
$222$	0	0
$223$	−2867.05	−0.860949	−0.430475	−	0.902603i	$-0.641654\pi$
$223$	−2867.05	−0.860949	−0.430475	+	0.902603i	$0.641654\pi$
$224$	1125.75	0.335792
$225$	0	0
$226$	−422.816	−0.124448
$227$	5544.65	1.62120	0.810598	−	0.585603i	$-0.199142\pi$
$227$	5544.65	1.62120	0.810598	+	0.585603i	$0.199142\pi$
$228$	0	0
$229$	−3010.94	−0.868857	−0.434429	−	0.900706i	$-0.643050\pi$
$229$	−3010.94	−0.868857	−0.434429	+	0.900706i	$0.643050\pi$
$230$	−23.4208	−0.00671443
$231$	0	0
$232$	410.079	0.116048
$233$	−132.563	−0.0372724	−0.0186362	−	0.999826i	$-0.505932\pi$
$233$	−132.563	−0.0372724	−0.0186362	+	0.999826i	$0.505932\pi$
$234$	0	0
$235$	−3353.63	−0.930923
$236$	−468.180	−0.129135
$237$	0	0
$238$	0.249978	6.80827e−5 0
$239$	−7104.31	−1.92276	−0.961380	−	0.275224i	$-0.911248\pi$
$239$	−7104.31	−1.92276	−0.961380	+	0.275224i	$0.911248\pi$
$240$	0	0
$241$	124.665	0.0333211	0.0166605	−	0.999861i	$-0.494697\pi$
$241$	124.665	0.0333211	0.0166605	+	0.999861i	$0.494697\pi$
$242$	−324.713	−0.0862534
$243$	0	0
$244$	−5974.40	−1.56751
$245$	2160.80	0.563463
$246$	0	0
$247$	−4419.42	−1.13846
$248$	1100.14	0.281690
$249$	0	0
$250$	365.886	0.0925627
$251$	6840.81	1.72027	0.860135	−	0.510066i	$-0.170379\pi$
$251$	6840.81	1.72027	0.860135	+	0.510066i	$0.170379\pi$
$252$	0	0
$253$	−434.294	−0.107920
$254$	−397.068	−0.0980877
$255$	0	0
$256$	3768.55	0.920056
$257$	374.942	0.0910049	0.0455024	−	0.998964i	$-0.485511\pi$
$257$	374.942	0.0910049	0.0455024	+	0.998964i	$0.485511\pi$
$258$	0	0
$259$	222.290	0.0533298
$260$	−4454.21	−1.06246
$261$	0	0
$262$	663.719	0.156506
$263$	−7174.93	−1.68222	−0.841112	−	0.540861i	$-0.818099\pi$
$263$	−7174.93	−1.68222	−0.841112	+	0.540861i	$0.818099\pi$
$264$	0	0
$265$	−2959.98	−0.686152
$266$	−505.038	−0.116413
$267$	0	0
$268$	−5962.83	−1.35910
$269$	2997.57	0.679425	0.339713	−	0.940529i	$-0.389670\pi$
$269$	2997.57	0.679425	0.339713	+	0.940529i	$0.389670\pi$
$270$	0	0
$271$	5993.77	1.34353	0.671763	−	0.740766i	$-0.265537\pi$
$271$	5993.77	1.34353	0.671763	+	0.740766i	$0.265537\pi$
$272$	2.63388	0.000587141 0
$273$	0	0
$274$	−581.233	−0.128152
$275$	402.074	0.0881672
$276$	0	0
$277$	−8759.33	−1.89999	−0.949994	−	0.312267i	$-0.898912\pi$
$277$	−8759.33	−1.89999	−0.949994	+	0.312267i	$0.898912\pi$
$278$	73.3600	0.0158268
$279$	0	0
$280$	−1022.18	−0.218168
$281$	4991.90	1.05976	0.529878	−	0.848074i	$-0.322238\pi$
$281$	4991.90	1.05976	0.529878	+	0.848074i	$0.322238\pi$
$282$	0	0
$283$	−470.303	−0.0987866	−0.0493933	−	0.998779i	$-0.515729\pi$
$283$	−470.303	−0.0987866	−0.0493933	+	0.998779i	$0.515729\pi$
$284$	−5170.67	−1.08036
$285$	0	0
$286$	673.829	0.139316
$287$	−4318.05	−0.888105
$288$	0	0
$289$	−4913.00	−1.00000
$290$	−278.506	−0.0563946
$291$	0	0
$292$	−6684.89	−1.33974
$293$	−1777.56	−0.354425	−0.177212	−	0.984173i	$-0.556708\pi$
$293$	−1777.56	−0.354425	−0.177212	+	0.984173i	$0.556708\pi$
$294$	0	0
$295$	638.525	0.126022
$296$	−38.6896	−0.00759726
$297$	0	0
$298$	150.500	0.0292558
$299$	441.143	0.0853243
$300$	0	0
$301$	−6465.13	−1.23802
$302$	610.336	0.116294
$303$	0	0
$304$	−5321.29	−1.00394
$305$	8148.15	1.52971
$306$	0	0
$307$	4138.06	0.769289	0.384645	−	0.923065i	$-0.374324\pi$
$307$	4138.06	0.769289	0.384645	+	0.923065i	$0.374324\pi$
$308$	−9438.70	−1.74617
$309$	0	0
$310$	−747.164	−0.136891
$311$	−7356.20	−1.34126	−0.670630	−	0.741792i	$-0.733976\pi$
$311$	−7356.20	−1.34126	−0.670630	+	0.741792i	$0.733976\pi$
$312$	0	0
$313$	6696.92	1.20937	0.604685	−	0.796465i	$-0.293299\pi$
$313$	6696.92	1.20937	0.604685	+	0.796465i	$0.293299\pi$
$314$	−34.4706	−0.00619518
$315$	0	0
$316$	2926.75	0.521020
$317$	−7828.34	−1.38701	−0.693507	−	0.720450i	$-0.743935\pi$
$317$	−7828.34	−1.38701	−0.693507	+	0.720450i	$0.743935\pi$
$318$	0	0
$319$	−5164.37	−0.906424
$320$	−5273.87	−0.921307
$321$	0	0
$322$	50.4125	0.00872478
$323$	−3.59369	−0.000619066 0
$324$	0	0
$325$	−408.415	−0.0697071
$326$	−419.762	−0.0713144
$327$	0	0
$328$	751.557	0.126518
$329$	7218.60	1.20965
$330$	0	0
$331$	−791.925	−0.131505	−0.0657525	−	0.997836i	$-0.520945\pi$
$331$	−791.925	−0.131505	−0.0657525	+	0.997836i	$0.520945\pi$
$332$	−4331.77	−0.716074
$333$	0	0
$334$	890.721	0.145922
$335$	8132.37	1.32633
$336$	0	0
$337$	3609.06	0.583378	0.291689	−	0.956513i	$-0.405783\pi$
$337$	3609.06	0.583378	0.291689	+	0.956513i	$0.405783\pi$
$338$	−125.460	−0.0201898
$339$	0	0
$340$	−3.62199	−0.000577735 0
$341$	−13854.8	−2.20023
$342$	0	0
$343$	3339.14	0.525646
$344$	1125.26	0.176366
$345$	0	0
$346$	642.401	0.0998142
$347$	4501.38	0.696389	0.348194	−	0.937422i	$-0.386795\pi$
$347$	4501.38	0.696389	0.348194	+	0.937422i	$0.386795\pi$
$348$	0	0
$349$	4505.21	0.690998	0.345499	−	0.938419i	$-0.387710\pi$
$349$	4505.21	0.690998	0.345499	+	0.938419i	$0.387710\pi$
$350$	−46.6725	−0.00712785
$351$	0	0
$352$	2467.55	0.373639
$353$	6156.62	0.928282	0.464141	−	0.885761i	$-0.346363\pi$
$353$	6156.62	0.928282	0.464141	+	0.885761i	$0.346363\pi$
$354$	0	0
$355$	7051.99	1.05431
$356$	5033.56	0.749376
$357$	0	0
$358$	691.417	0.102074
$359$	−8811.87	−1.29547	−0.647734	−	0.761867i	$-0.724283\pi$
$359$	−8811.87	−1.29547	−0.647734	+	0.761867i	$0.724283\pi$
$360$	0	0
$361$	401.436	0.0585269
$362$	−538.462	−0.0781794
$363$	0	0
$364$	9587.56	1.38056
$365$	9117.14	1.30743
$366$	0	0
$367$	13290.6	1.89036	0.945181	−	0.326546i	$-0.105885\pi$
$367$	13290.6	1.89036	0.945181	+	0.326546i	$0.105885\pi$
$368$	531.168	0.0752420
$369$	0	0
$370$	26.2761	0.00369197
$371$	6371.28	0.891591
$372$	0	0
$373$	10006.7	1.38908	0.694541	−	0.719453i	$-0.255608\pi$
$373$	10006.7	1.38908	0.694541	+	0.719453i	$0.255608\pi$
$374$	0.547931	7.57563e−5 0
$375$	0	0
$376$	−1256.40	−0.172324
$377$	5245.82	0.716641
$378$	0	0
$379$	−3895.86	−0.528013	−0.264006	−	0.964521i	$-0.585044\pi$
$379$	−3895.86	−0.528013	−0.264006	+	0.964521i	$0.585044\pi$
$380$	7317.60	0.987855
$381$	0	0
$382$	−41.2312	−0.00552244
$383$	3038.64	0.405398	0.202699	−	0.979241i	$-0.435029\pi$
$383$	3038.64	0.405398	0.202699	+	0.979241i	$0.435029\pi$
$384$	0	0
$385$	12872.9	1.70406
$386$	442.136	0.0583009
$387$	0	0
$388$	−2510.47	−0.328478
$389$	13537.3	1.76444	0.882222	−	0.470834i	$-0.156047\pi$
$389$	13537.3	1.76444	0.882222	+	0.470834i	$0.156047\pi$
$390$	0	0
$391$	0.358720	4.63971e−5 0
$392$	809.518	0.104303
$393$	0	0
$394$	−483.002	−0.0617597
$395$	−3991.63	−0.508457
$396$	0	0
$397$	−3491.32	−0.441371	−0.220686	−	0.975345i	$-0.570829\pi$
$397$	−3491.32	−0.441371	−0.220686	+	0.975345i	$0.570829\pi$
$398$	235.356	0.0296415
$399$	0	0
$400$	−491.761	−0.0614702
$401$	−12667.7	−1.57754	−0.788769	−	0.614689i	$-0.789281\pi$
$401$	−12667.7	−1.57754	−0.788769	+	0.614689i	$0.789281\pi$
$402$	0	0
$403$	14073.3	1.73955
$404$	−12817.5	−1.57845
$405$	0	0
$406$	599.477	0.0732796
$407$	487.241	0.0593407
$408$	0	0
$409$	10431.3	1.26111	0.630557	−	0.776143i	$-0.282826\pi$
$409$	10431.3	1.26111	0.630557	+	0.776143i	$0.282826\pi$
$410$	−510.421	−0.0614827
$411$	0	0
$412$	−10006.9	−1.19662
$413$	−1374.41	−0.163753
$414$	0	0
$415$	5907.85	0.698808
$416$	−2506.47	−0.295408
$417$	0	0
$418$	−1107.00	−0.129534
$419$	6955.06	0.810923	0.405462	−	0.914112i	$-0.367111\pi$
$419$	6955.06	0.810923	0.405462	+	0.914112i	$0.367111\pi$
$420$	0	0
$421$	−2977.09	−0.344643	−0.172321	−	0.985041i	$-0.555127\pi$
$421$	−2977.09	−0.344643	−0.172321	+	0.985041i	$0.555127\pi$
$422$	−649.748	−0.0749507
$423$	0	0
$424$	−1108.92	−0.127014
$425$	−0.332107	−3.79049e−5 0
$426$	0	0
$427$	−17538.7	−1.98772
$428$	5822.76	0.657602
$429$	0	0
$430$	−764.220	−0.0857069
$431$	9721.43	1.08646	0.543230	−	0.839584i	$-0.317201\pi$
$431$	9721.43	1.08646	0.543230	+	0.839584i	$0.317201\pi$
$432$	0	0
$433$	814.588	0.0904079	0.0452040	−	0.998978i	$-0.485606\pi$
$433$	814.588	0.0904079	0.0452040	+	0.998978i	$0.485606\pi$
$434$	1608.25	0.177877
$435$	0	0
$436$	−4625.05	−0.508027
$437$	−724.732	−0.0793333
$438$	0	0
$439$	15373.9	1.67143	0.835716	−	0.549162i	$-0.185053\pi$
$439$	15373.9	1.67143	0.835716	+	0.549162i	$0.185053\pi$
$440$	−2240.53	−0.242757
$441$	0	0
$442$	−0.556573	−5.98947e−5 0
$443$	−5462.95	−0.585897	−0.292949	−	0.956128i	$-0.594636\pi$
$443$	−5462.95	−0.585897	−0.292949	+	0.956128i	$0.594636\pi$
$444$	0	0
$445$	−6864.99	−0.731307
$446$	729.480	0.0774482
$447$	0	0
$448$	11351.9	1.19715
$449$	9408.54	0.988901	0.494451	−	0.869206i	$-0.335369\pi$
$449$	9408.54	0.988901	0.494451	+	0.869206i	$0.335369\pi$
$450$	0	0
$451$	−9464.80	−0.988204
$452$	−13186.6	−1.37223
$453$	0	0
$454$	−1410.76	−0.145838
$455$	−13075.9	−1.34727
$456$	0	0
$457$	1254.04	0.128362	0.0641809	−	0.997938i	$-0.479557\pi$
$457$	1254.04	0.128362	0.0641809	+	0.997938i	$0.479557\pi$
$458$	766.091	0.0781596
$459$	0	0
$460$	−730.438	−0.0740366
$461$	15647.0	1.58081	0.790407	−	0.612582i	$-0.209869\pi$
$461$	15647.0	1.58081	0.790407	+	0.612582i	$0.209869\pi$
$462$	0	0
$463$	18387.3	1.84564	0.922821	−	0.385228i	$-0.125877\pi$
$463$	18387.3	1.84564	0.922821	+	0.385228i	$0.125877\pi$
$464$	6316.34	0.631959
$465$	0	0
$466$	33.7287	0.00335290
$467$	−13135.8	−1.30161	−0.650805	−	0.759245i	$-0.725569\pi$
$467$	−13135.8	−1.30161	−0.650805	+	0.759245i	$0.725569\pi$
$468$	0	0
$469$	−17504.7	−1.72344
$470$	853.286	0.0837429
$471$	0	0
$472$	239.216	0.0233280
$473$	−14171.0	−1.37756
$474$	0	0
$475$	670.965	0.0648126
$476$	7.79623	0.000750713 0
$477$	0	0
$478$	1807.59	0.172965
$479$	4918.54	0.469173	0.234587	−	0.972095i	$-0.424626\pi$
$479$	4918.54	0.469173	0.234587	+	0.972095i	$0.424626\pi$
$480$	0	0
$481$	−494.926	−0.0469162
$482$	−31.7193	−0.00299745
$483$	0	0
$484$	−10127.0	−0.951072
$485$	3423.88	0.320558
$486$	0	0
$487$	12272.0	1.14189	0.570944	−	0.820989i	$-0.306577\pi$
$487$	12272.0	1.14189	0.570944	+	0.820989i	$0.306577\pi$
$488$	3052.61	0.283166
$489$	0	0
$490$	−549.786	−0.0506873
$491$	9165.57	0.842437	0.421218	−	0.906959i	$-0.361603\pi$
$491$	9165.57	0.842437	0.421218	+	0.906959i	$0.361603\pi$
$492$	0	0
$493$	4.26569	0.000389690 0
$494$	1124.46	0.102413
$495$	0	0
$496$	16945.2	1.53400
$497$	−15179.2	−1.36998
$498$	0	0
$499$	−4308.93	−0.386561	−0.193281	−	0.981144i	$-0.561913\pi$
$499$	−4308.93	−0.386561	−0.193281	+	0.981144i	$0.561913\pi$
$500$	11411.1	1.02064
$501$	0	0
$502$	−1740.55	−0.154750
$503$	10004.0	0.886795	0.443397	−	0.896325i	$-0.353773\pi$
$503$	10004.0	0.886795	0.443397	+	0.896325i	$0.353773\pi$
$504$	0	0
$505$	17481.0	1.54039
$506$	110.500	0.00970816
$507$	0	0
$508$	−12383.6	−1.08156
$509$	11329.0	0.986538	0.493269	−	0.869877i	$-0.335802\pi$
$509$	11329.0	0.986538	0.493269	+	0.869877i	$0.335802\pi$
$510$	0	0
$511$	−19624.4	−1.69889
$512$	−5043.61	−0.435348
$513$	0	0
$514$	−95.3989	−0.00818650
$515$	13647.9	1.16776
$516$	0	0
$517$	15822.6	1.34599
$518$	−56.5586	−0.00479738
$519$	0	0
$520$	2275.87	0.191930
$521$	578.251	0.0486251	0.0243125	−	0.999704i	$-0.492260\pi$
$521$	578.251	0.0486251	0.0243125	+	0.999704i	$0.492260\pi$
$522$	0	0
$523$	−22559.8	−1.88618	−0.943088	−	0.332543i	$-0.892093\pi$
$523$	−22559.8	−1.88618	−0.943088	+	0.332543i	$0.892093\pi$
$524$	20699.8	1.72572
$525$	0	0
$526$	1825.56	0.151327
$527$	11.4438	0.000945922 0
$528$	0	0
$529$	−12094.7	−0.994054
$530$	753.126	0.0617240
$531$	0	0
$532$	−15750.9	−1.28363
$533$	9614.07	0.781298
$534$	0	0
$535$	−7941.33	−0.641745
$536$	3046.70	0.245517
$537$	0	0
$538$	−762.691	−0.0611189
$539$	−10194.7	−0.814691
$540$	0	0
$541$	−8649.61	−0.687386	−0.343693	−	0.939082i	$-0.611678\pi$
$541$	−8649.61	−0.687386	−0.343693	+	0.939082i	$0.611678\pi$
$542$	−1525.03	−0.120859
$543$	0	0
$544$	−2.03816	−0.000160635 0
$545$	6307.84	0.495777
$546$	0	0
$547$	−9720.33	−0.759801	−0.379901	−	0.925027i	$-0.624042\pi$
$547$	−9720.33	−0.759801	−0.379901	+	0.925027i	$0.624042\pi$
$548$	−18127.3	−1.41306
$549$	0	0
$550$	−102.302	−0.00793124
$551$	−8618.09	−0.666322
$552$	0	0
$553$	8591.87	0.660693
$554$	2228.69	0.170917
$555$	0	0
$556$	2287.92	0.174514
$557$	−14288.6	−1.08694	−0.543472	−	0.839428i	$-0.682890\pi$
$557$	−14288.6	−1.08694	−0.543472	+	0.839428i	$0.682890\pi$
$558$	0	0
$559$	14394.5	1.08913
$560$	−15744.4	−1.18807
$561$	0	0
$562$	−1270.12	−0.0953323
$563$	2067.53	0.154771	0.0773856	−	0.997001i	$-0.475343\pi$
$563$	2067.53	0.154771	0.0773856	+	0.997001i	$0.475343\pi$
$564$	0	0
$565$	17984.5	1.33914
$566$	119.662	0.00888652
$567$	0	0
$568$	2641.94	0.195165
$569$	3196.60	0.235516	0.117758	−	0.993042i	$-0.462429\pi$
$569$	3196.60	0.235516	0.117758	+	0.993042i	$0.462429\pi$
$570$	0	0
$571$	−15856.8	−1.16215	−0.581073	−	0.813852i	$-0.697367\pi$
$571$	−15856.8	−1.16215	−0.581073	+	0.813852i	$0.697367\pi$
$572$	21015.1	1.53617
$573$	0	0
$574$	1098.67	0.0798911
$575$	−66.9753	−0.00485750
$576$	0	0
$577$	18096.9	1.30569	0.652846	−	0.757491i	$-0.273575\pi$
$577$	18096.9	1.30569	0.652846	+	0.757491i	$0.273575\pi$
$578$	1250.04	0.0899567
$579$	0	0
$580$	−8685.94	−0.621835
$581$	−12716.5	−0.908036
$582$	0	0
$583$	13965.3	0.992082
$584$	3415.63	0.242020
$585$	0	0
$586$	452.277	0.0318829
$587$	−1903.98	−0.133877	−0.0669383	−	0.997757i	$-0.521323\pi$
$587$	−1903.98	−0.133877	−0.0669383	+	0.997757i	$0.521323\pi$
$588$	0	0
$589$	−23120.3	−1.61741
$590$	−162.464	−0.0113365
$591$	0	0
$592$	−595.926	−0.0413723
$593$	−16287.4	−1.12790	−0.563948	−	0.825810i	$-0.690718\pi$
$593$	−16287.4	−1.12790	−0.563948	+	0.825810i	$0.690718\pi$
$594$	0	0
$595$	−10.6328	−0.000732611 0
$596$	4693.73	0.322588
$597$	0	0
$598$	−112.243	−0.00767550
$599$	13497.9	0.920715	0.460357	−	0.887734i	$-0.347721\pi$
$599$	13497.9	0.920715	0.460357	+	0.887734i	$0.347721\pi$
$600$	0	0
$601$	13860.8	0.940755	0.470378	−	0.882465i	$-0.344118\pi$
$601$	13860.8	0.940755	0.470378	+	0.882465i	$0.344118\pi$
$602$	1644.96	0.111368
$603$	0	0
$604$	19034.9	1.28232
$605$	13811.7	0.928139
$606$	0	0
$607$	13140.0	0.878641	0.439321	−	0.898330i	$-0.355219\pi$
$607$	13140.0	0.878641	0.439321	+	0.898330i	$0.355219\pi$
$608$	4117.75	0.274666
$609$	0	0
$610$	−2073.18	−0.137608
$611$	−16072.1	−1.06417
$612$	0	0
$613$	−10495.5	−0.691535	−0.345768	−	0.938320i	$-0.612381\pi$
$613$	−10495.5	−0.691535	−0.345768	+	0.938320i	$0.612381\pi$
$614$	−1052.87	−0.0692028
$615$	0	0
$616$	4822.69	0.315441
$617$	−9157.76	−0.597533	−0.298766	−	0.954326i	$-0.596575\pi$
$617$	−9157.76	−0.597533	−0.298766	+	0.954326i	$0.596575\pi$
$618$	0	0
$619$	15266.3	0.991286	0.495643	−	0.868526i	$-0.334932\pi$
$619$	15266.3	0.991286	0.495643	+	0.868526i	$0.334932\pi$
$620$	−23302.3	−1.50942
$621$	0	0
$622$	1871.68	0.120655
$623$	14776.7	0.950266
$624$	0	0
$625$	−14578.7	−0.933036
$626$	−1703.94	−0.108791
$627$	0	0
$628$	−1075.06	−0.0683111
$629$	−0.402454	−2.55117e−5 0
$630$	0	0
$631$	6869.98	0.433422	0.216711	−	0.976236i	$-0.430467\pi$
$631$	6869.98	0.433422	0.216711	+	0.976236i	$0.430467\pi$
$632$	−1495.42	−0.0941210
$633$	0	0
$634$	1991.81	0.124771
$635$	16889.3	1.05548
$636$	0	0
$637$	10355.5	0.644114
$638$	1314.00	0.0815390
$639$	0	0
$640$	5525.89	0.341297
$641$	−14205.2	−0.875306	−0.437653	−	0.899144i	$-0.644190\pi$
$641$	−14205.2	−0.875306	−0.437653	+	0.899144i	$0.644190\pi$
$642$	0	0
$643$	32050.2	1.96569	0.982843	−	0.184442i	$-0.0590479\pi$
$643$	32050.2	1.96569	0.982843	+	0.184442i	$0.0590479\pi$
$644$	1572.25	0.0962038
$645$	0	0
$646$	0.914366	5.56892e−5 0
$647$	−27275.5	−1.65736	−0.828679	−	0.559725i	$-0.810907\pi$
$647$	−27275.5	−1.65736	−0.828679	+	0.559725i	$0.810907\pi$
$648$	0	0
$649$	−3012.59	−0.182210
$650$	103.916	0.00627063
$651$	0	0
$652$	−13091.4	−0.786347
$653$	6694.40	0.401182	0.200591	−	0.979675i	$-0.435714\pi$
$653$	6694.40	0.401182	0.200591	+	0.979675i	$0.435714\pi$
$654$	0	0
$655$	−28231.3	−1.68411
$656$	11576.0	0.688976
$657$	0	0
$658$	−1836.67	−0.108816
$659$	−1828.61	−0.108092	−0.0540460	−	0.998538i	$-0.517212\pi$
$659$	−1828.61	−0.108092	−0.0540460	+	0.998538i	$0.517212\pi$
$660$	0	0
$661$	3994.07	0.235025	0.117512	−	0.993071i	$-0.462508\pi$
$661$	3994.07	0.235025	0.117512	+	0.993071i	$0.462508\pi$
$662$	201.494	0.0118298
$663$	0	0
$664$	2213.31	0.129357
$665$	21481.8	1.25267
$666$	0	0
$667$	860.253	0.0499387
$668$	27779.5	1.60901
$669$	0	0
$670$	−2069.17	−0.119312
$671$	−38443.3	−2.21175
$672$	0	0
$673$	1511.93	0.0865980	0.0432990	−	0.999062i	$-0.486213\pi$
$673$	1511.93	0.0865980	0.0432990	+	0.999062i	$0.486213\pi$
$674$	−918.276	−0.0524788
$675$	0	0
$676$	−3912.81	−0.222623
$677$	−105.944	−0.00601442	−0.00300721	−	0.999995i	$-0.500957\pi$
$677$	−105.944	−0.00601442	−0.00300721	+	0.999995i	$0.500957\pi$
$678$	0	0
$679$	−7369.81	−0.416535
$680$	1.85065	0.000104366 0
$681$	0	0
$682$	3525.15	0.197925
$683$	5370.44	0.300870	0.150435	−	0.988620i	$-0.451933\pi$
$683$	5370.44	0.300870	0.150435	+	0.988620i	$0.451933\pi$
$684$	0	0
$685$	24722.8	1.37899
$686$	−849.597	−0.0472854
$687$	0	0
$688$	17332.0	0.960433
$689$	−14185.6	−0.784364
$690$	0	0
$691$	−11938.4	−0.657246	−0.328623	−	0.944461i	$-0.606584\pi$
$691$	−11938.4	−0.657246	−0.328623	+	0.944461i	$0.606584\pi$
$692$	20035.0	1.10060
$693$	0	0
$694$	−1145.31	−0.0626449
$695$	−3120.37	−0.170306
$696$	0	0
$697$	7.81778	0.000424849 0
$698$	−1146.29	−0.0621599
$699$	0	0
$700$	−1455.60	−0.0785952
$701$	−20482.4	−1.10358	−0.551791	−	0.833982i	$-0.686055\pi$
$701$	−20482.4	−1.10358	−0.551791	+	0.833982i	$0.686055\pi$
$702$	0	0
$703$	813.089	0.0436219
$704$	24882.3	1.33209
$705$	0	0
$706$	−1566.47	−0.0835053
$707$	−37627.4	−2.00159
$708$	0	0
$709$	−9508.25	−0.503653	−0.251826	−	0.967772i	$-0.581031\pi$
$709$	−9508.25	−0.503653	−0.251826	+	0.967772i	$0.581031\pi$
$710$	−1794.28	−0.0948424
$711$	0	0
$712$	−2571.89	−0.135373
$713$	2307.85	0.121220
$714$	0	0
$715$	−28661.4	−1.49913
$716$	21563.7	1.12552
$717$	0	0
$718$	2242.06	0.116536
$719$	−7683.75	−0.398547	−0.199274	−	0.979944i	$-0.563858\pi$
$719$	−7683.75	−0.398547	−0.199274	+	0.979944i	$0.563858\pi$
$720$	0	0
$721$	−29376.7	−1.51740
$722$	−102.140	−0.00526489
$723$	0	0
$724$	−16793.4	−0.862044
$725$	−796.432	−0.0407982
$726$	0	0
$727$	−28207.9	−1.43903	−0.719515	−	0.694477i	$-0.755636\pi$
$727$	−28207.9	−1.43903	−0.719515	+	0.694477i	$0.755636\pi$
$728$	−4898.75	−0.249395
$729$	0	0
$730$	−2319.73	−0.117612
$731$	11.7051	0.000592239 0
$732$	0	0
$733$	−21546.7	−1.08574	−0.542870	−	0.839817i	$-0.682662\pi$
$733$	−21546.7	−1.08574	−0.542870	+	0.839817i	$0.682662\pi$
$734$	−3381.61	−0.170051
$735$	0	0
$736$	−411.031	−0.0205853
$737$	−38368.9	−1.91769
$738$	0	0
$739$	13254.6	0.659780	0.329890	−	0.944019i	$-0.392988\pi$
$739$	13254.6	0.659780	0.329890	+	0.944019i	$0.392988\pi$
$740$	819.490	0.0407095
$741$	0	0
$742$	−1621.08	−0.0802046
$743$	13311.7	0.657280	0.328640	−	0.944455i	$-0.393410\pi$
$743$	13311.7	0.657280	0.328640	+	0.944455i	$0.393410\pi$
$744$	0	0
$745$	−6401.51	−0.314810
$746$	−2546.07	−0.124957
$747$	0	0
$748$	17.0887	0.000835326 0
$749$	17093.5	0.833889
$750$	0	0
$751$	30067.0	1.46093	0.730467	−	0.682948i	$-0.239302\pi$
$751$	30067.0	1.46093	0.730467	+	0.682948i	$0.239302\pi$
$752$	−19352.0	−0.938423
$753$	0	0
$754$	−1334.73	−0.0644667
$755$	−25960.7	−1.25140
$756$	0	0
$757$	−40542.2	−1.94654	−0.973269	−	0.229666i	$-0.926236\pi$
$757$	−40542.2	−1.94654	−0.973269	+	0.229666i	$0.926236\pi$
$758$	991.247	0.0474983
$759$	0	0
$760$	−3738.91	−0.178453
$761$	35970.2	1.71343	0.856713	−	0.515793i	$-0.172502\pi$
$761$	35970.2	1.71343	0.856713	+	0.515793i	$0.172502\pi$
$762$	0	0
$763$	−13577.5	−0.644216
$764$	−1285.90	−0.0608931
$765$	0	0
$766$	−773.140	−0.0364683
$767$	3060.10	0.144060
$768$	0	0
$769$	29540.6	1.38525	0.692627	−	0.721296i	$-0.256453\pi$
$769$	29540.6	1.38525	0.692627	+	0.721296i	$0.256453\pi$
$770$	−3275.34	−0.153292
$771$	0	0
$772$	13789.2	0.642854
$773$	−31936.1	−1.48598	−0.742989	−	0.669304i	$-0.766592\pi$
$773$	−31936.1	−1.48598	−0.742989	+	0.669304i	$0.766592\pi$
$774$	0	0
$775$	−2136.63	−0.0990324
$776$	1282.72	0.0593388
$777$	0	0
$778$	−3444.38	−0.158724
$779$	−15794.5	−0.726439
$780$	0	0
$781$	−33271.6	−1.52439
$782$	−0.0912714	−4.17373e−6 0
$783$	0	0
$784$	12468.8	0.568003
$785$	1466.21	0.0666640
$786$	0	0
$787$	19554.7	0.885705	0.442853	−	0.896594i	$-0.353966\pi$
$787$	19554.7	0.885705	0.442853	+	0.896594i	$0.353966\pi$
$788$	−15063.7	−0.680993
$789$	0	0
$790$	1015.61	0.0457391
$791$	−38711.2	−1.74009
$792$	0	0
$793$	39049.6	1.74867
$794$	888.319	0.0397043
$795$	0	0
$796$	7340.19	0.326842
$797$	19917.3	0.885202	0.442601	−	0.896719i	$-0.354056\pi$
$797$	19917.3	0.885202	0.442601	+	0.896719i	$0.354056\pi$
$798$	0	0
$799$	−13.0692	−0.000578668 0
$800$	380.537	0.0168175
$801$	0	0
$802$	3223.11	0.141910
$803$	−43015.1	−1.89037
$804$	0	0
$805$	−2144.30	−0.0938840
$806$	−3580.75	−0.156484
$807$	0	0
$808$	6549.06	0.285142
$809$	23172.3	1.00704	0.503520	−	0.863984i	$-0.332038\pi$
$809$	23172.3	1.00704	0.503520	+	0.863984i	$0.332038\pi$
$810$	0	0
$811$	−9763.58	−0.422744	−0.211372	−	0.977406i	$-0.567793\pi$
$811$	−9763.58	−0.422744	−0.211372	+	0.977406i	$0.567793\pi$
$812$	18696.3	0.808017
$813$	0	0
$814$	−123.972	−0.00533809
$815$	17854.6	0.767386
$816$	0	0
$817$	−23648.0	−1.01266
$818$	−2654.11	−0.113446
$819$	0	0
$820$	−15918.8	−0.677939
$821$	−24006.4	−1.02050	−0.510249	−	0.860026i	$-0.670447\pi$
$821$	−24006.4	−1.02050	−0.510249	+	0.860026i	$0.670447\pi$
$822$	0	0
$823$	36909.5	1.56329	0.781643	−	0.623726i	$-0.214382\pi$
$823$	36909.5	1.56329	0.781643	+	0.623726i	$0.214382\pi$
$824$	5113.02	0.216166
$825$	0	0
$826$	349.699	0.0147307
$827$	17834.8	0.749912	0.374956	−	0.927043i	$-0.377658\pi$
$827$	17834.8	0.749912	0.374956	+	0.927043i	$0.377658\pi$
$828$	0	0
$829$	−7164.05	−0.300142	−0.150071	−	0.988675i	$-0.547950\pi$
$829$	−7164.05	−0.300142	−0.150071	+	0.988675i	$0.547950\pi$
$830$	−1503.17	−0.0628625
$831$	0	0
$832$	−25274.8	−1.05318
$833$	8.42070	0.000350252 0
$834$	0	0
$835$	−37886.9	−1.57021
$836$	−34524.7	−1.42830
$837$	0	0
$838$	−1769.62	−0.0729480
$839$	30792.5	1.26708	0.633538	−	0.773712i	$-0.281602\pi$
$839$	30792.5	1.26708	0.633538	+	0.773712i	$0.281602\pi$
$840$	0	0
$841$	−14159.4	−0.580564
$842$	757.480	0.0310030
$843$	0	0
$844$	−20264.1	−0.826444
$845$	5336.47	0.217254
$846$	0	0
$847$	−29729.2	−1.20603
$848$	−17080.4	−0.691680
$849$	0	0
$850$	0.0845001	3.40980e−6 0
$851$	−81.1620	−0.00326933
$852$	0	0
$853$	31914.3	1.28104	0.640519	−	0.767942i	$-0.278719\pi$
$853$	31914.3	1.28104	0.640519	+	0.767942i	$0.278719\pi$
$854$	4462.47	0.178809
$855$	0	0
$856$	−2975.13	−0.118794
$857$	48527.1	1.93425	0.967126	−	0.254296i	$-0.0818438\pi$
$857$	48527.1	1.93425	0.967126	+	0.254296i	$0.0818438\pi$
$858$	0	0
$859$	−5570.19	−0.221249	−0.110624	−	0.993862i	$-0.535285\pi$
$859$	−5570.19	−0.221249	−0.110624	+	0.993862i	$0.535285\pi$
$860$	−23834.2	−0.945047
$861$	0	0
$862$	−2473.48	−0.0977345
$863$	45442.4	1.79244	0.896221	−	0.443607i	$-0.146301\pi$
$863$	45442.4	1.79244	0.896221	+	0.443607i	$0.146301\pi$
$864$	0	0
$865$	−27324.6	−1.07406
$866$	−207.261	−0.00813280
$867$	0	0
$868$	50157.5	1.96136
$869$	18832.7	0.735160
$870$	0	0
$871$	38974.0	1.51617
$872$	2363.16	0.0917737
$873$	0	0
$874$	184.398	0.00713657
$875$	33498.9	1.29425
$876$	0	0
$877$	−23890.4	−0.919865	−0.459932	−	0.887954i	$-0.652126\pi$
$877$	−23890.4	−0.919865	−0.459932	+	0.887954i	$0.652126\pi$
$878$	−3911.69	−0.150357
$879$	0	0
$880$	−34510.3	−1.32198
$881$	9587.92	0.366657	0.183329	−	0.983052i	$-0.441313\pi$
$881$	9587.92	0.366657	0.183329	+	0.983052i	$0.441313\pi$
$882$	0	0
$883$	−50410.3	−1.92122	−0.960612	−	0.277893i	$-0.910364\pi$
$883$	−50410.3	−1.92122	−0.960612	+	0.277893i	$0.910364\pi$
$884$	−17.3582	−0.000660429 0
$885$	0	0
$886$	1389.97	0.0527054
$887$	−7413.01	−0.280614	−0.140307	−	0.990108i	$-0.544809\pi$
$887$	−7413.01	−0.280614	−0.140307	+	0.990108i	$0.544809\pi$
$888$	0	0
$889$	−36353.8	−1.37150
$890$	1746.70	0.0657860
$891$	0	0
$892$	22750.8	0.853982
$893$	26404.1	0.989450
$894$	0	0
$895$	−29409.5	−1.09838
$896$	−11894.3	−0.443484
$897$	0	0
$898$	−2393.87	−0.0889584
$899$	27443.6	1.01813
$900$	0	0
$901$	−11.5351	−0.000426516 0
$902$	2408.19	0.0888957
$903$	0	0
$904$	6737.69	0.247890
$905$	22903.5	0.841258
$906$	0	0
$907$	−13964.2	−0.511216	−0.255608	−	0.966781i	$-0.582276\pi$
$907$	−13964.2	−0.511216	−0.255608	+	0.966781i	$0.582276\pi$
$908$	−43998.3	−1.60808
$909$	0	0
$910$	3326.99	0.121196
$911$	−48214.6	−1.75348	−0.876740	−	0.480965i	$-0.840286\pi$
$911$	−48214.6	−1.75348	−0.876740	+	0.480965i	$0.840286\pi$
$912$	0	0
$913$	−27873.5	−1.01038
$914$	−319.072	−0.0115470
$915$	0	0
$916$	23892.6	0.861826
$917$	60767.1	2.18834
$918$	0	0
$919$	−41258.5	−1.48095	−0.740475	−	0.672084i	$-0.765399\pi$
$919$	−41258.5	−1.48095	−0.740475	+	0.672084i	$0.765399\pi$
$920$	373.216	0.0133745
$921$	0	0
$922$	−3981.17	−0.142205
$923$	33796.3	1.20522
$924$	0	0
$925$	75.1407	0.00267093
$926$	−4678.41	−0.166028
$927$	0	0
$928$	−4887.74	−0.172897
$929$	−2682.75	−0.0947451	−0.0473726	−	0.998877i	$-0.515085\pi$
$929$	−2682.75	−0.0947451	−0.0473726	+	0.998877i	$0.515085\pi$
$930$	0	0
$931$	−17012.6	−0.598888
$932$	1051.92	0.0369708
$933$	0	0
$934$	3342.22	0.117089
$935$	−23.3063	−0.000815184 0
$936$	0	0
$937$	12243.8	0.426883	0.213441	−	0.976956i	$-0.431533\pi$
$937$	12243.8	0.426883	0.213441	+	0.976956i	$0.431533\pi$
$938$	4453.83	0.155035
$939$	0	0
$940$	26612.0	0.923390
$941$	−29701.6	−1.02895	−0.514476	−	0.857505i	$-0.672014\pi$
$941$	−29701.6	−1.02895	−0.514476	+	0.857505i	$0.672014\pi$
$942$	0	0
$943$	1576.59	0.0544443
$944$	3684.58	0.127037
$945$	0	0
$946$	3605.62	0.123921
$947$	−11974.6	−0.410899	−0.205450	−	0.978668i	$-0.565866\pi$
$947$	−11974.6	−0.410899	−0.205450	+	0.978668i	$0.565866\pi$
$948$	0	0
$949$	43693.5	1.49457
$950$	−170.718	−0.00583033
$951$	0	0
$952$	−3.98347	−0.000135614 0
$953$	−23437.4	−0.796655	−0.398327	−	0.917243i	$-0.630409\pi$
$953$	−23437.4	−0.796655	−0.398327	+	0.917243i	$0.630409\pi$
$954$	0	0
$955$	1753.77	0.0594248
$956$	56374.6	1.90720
$957$	0	0
$958$	−1251.46	−0.0422053
$959$	−53215.1	−1.79187
$960$	0	0
$961$	43833.6	1.47137
$962$	125.927	0.00422043
$963$	0	0
$964$	−989.249	−0.0330514
$965$	−18806.3	−0.627353
$966$	0	0
$967$	8006.86	0.266270	0.133135	−	0.991098i	$-0.457496\pi$
$967$	8006.86	0.266270	0.133135	+	0.991098i	$0.457496\pi$
$968$	5174.38	0.171809
$969$	0	0
$970$	−871.160	−0.0288363
$971$	13457.8	0.444780	0.222390	−	0.974958i	$-0.428614\pi$
$971$	13457.8	0.444780	0.222390	+	0.974958i	$0.428614\pi$
$972$	0	0
$973$	6716.51	0.221297
$974$	−3122.45	−0.102721
$975$	0	0
$976$	47018.5	1.54204
$977$	−25366.7	−0.830659	−0.415329	−	0.909671i	$-0.636334\pi$
$977$	−25366.7	−0.830659	−0.415329	+	0.909671i	$0.636334\pi$
$978$	0	0
$979$	32389.3	1.05737
$980$	−17146.5	−0.558903
$981$	0	0
$982$	−2332.05	−0.0757829
$983$	−5778.72	−0.187500	−0.0937500	−	0.995596i	$-0.529885\pi$
$983$	−5778.72	−0.187500	−0.0937500	+	0.995596i	$0.529885\pi$
$984$	0	0
$985$	20544.5	0.664572
$986$	−1.08535	−3.50552e−5 0
$987$	0	0
$988$	35069.2	1.12925
$989$	2360.53	0.0758954
$990$	0	0
$991$	24959.9	0.800079	0.400039	−	0.916498i	$-0.368996\pi$
$991$	24959.9	0.800079	0.400039	+	0.916498i	$0.368996\pi$
$992$	−13112.6	−0.419684
$993$	0	0
$994$	3862.14	0.123239
$995$	−10010.9	−0.318961
$996$	0	0
$997$	29815.3	0.947101	0.473550	−	0.880767i	$-0.342972\pi$
$997$	29815.3	0.947101	0.473550	+	0.880767i	$0.342972\pi$
$998$	1096.35	0.0347738
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	531.4.a.f.1.5		8
3.2	odd	2		177.4.a.c.1.4	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.4.a.c.1.4	✓	8	3.2	odd	2
531.4.a.f.1.5		8	1.1	even	1	trivial

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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)

\(f(q)\)	\(=\)	\(q-0.254436 q^{2} -7.93526 q^{4} +10.8225 q^{5} -23.2950 q^{7} +4.05451 q^{8} +O(q^{10})\)	\(q-0.254436 q^{2} -7.93526 q^{4} +10.8225 q^{5} -23.2950 q^{7} +4.05451 q^{8} -2.75362 q^{10} -51.0608 q^{11} +51.8661 q^{13} +5.92710 q^{14} +62.4505 q^{16} +0.0421754 q^{17} -85.2082 q^{19} -85.8790 q^{20} +12.9917 q^{22} +8.50543 q^{23} -7.87442 q^{25} -13.1966 q^{26} +184.852 q^{28} +101.142 q^{29} +271.338 q^{31} -48.3257 q^{32} -0.0107310 q^{34} -252.110 q^{35} -9.54238 q^{37} +21.6800 q^{38} +43.8797 q^{40} +185.363 q^{41} +277.532 q^{43} +405.181 q^{44} -2.16409 q^{46} -309.877 q^{47} +199.659 q^{49} +2.00354 q^{50} -411.571 q^{52} -273.504 q^{53} -552.603 q^{55} -94.4499 q^{56} -25.7341 q^{58} +59.0000 q^{59} +752.893 q^{61} -69.0383 q^{62} -487.308 q^{64} +561.319 q^{65} +751.435 q^{67} -0.334673 q^{68} +64.1458 q^{70} +651.607 q^{71} +842.428 q^{73} +2.42793 q^{74} +676.149 q^{76} +1189.46 q^{77} -368.828 q^{79} +675.868 q^{80} -47.1631 q^{82} +545.888 q^{83} +0.456442 q^{85} -70.6143 q^{86} -207.026 q^{88} -634.328 q^{89} -1208.22 q^{91} -67.4928 q^{92} +78.8440 q^{94} -922.162 q^{95} +316.368 q^{97} -50.8004 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 2 q^{2} + 38 q^{4} + 12 q^{5} + 53 q^{7} - 3 q^{8}+O(q^{10}) \) 8 * q - 2 * q^2 + 38 * q^4 + 12 * q^5 + 53 * q^7 - 3 * q^8	\( 8 q - 2 q^{2} + 38 q^{4} + 12 q^{5} + 53 q^{7} - 3 q^{8} + 29 q^{10} + 27 q^{11} + 89 q^{13} + 37 q^{14} + 362 q^{16} - 79 q^{17} + 288 q^{19} - 457 q^{20} + 596 q^{22} - 202 q^{23} + 264 q^{25} - 270 q^{26} + 702 q^{28} + 114 q^{29} + 538 q^{31} - 316 q^{32} + 498 q^{34} + 196 q^{35} + 395 q^{37} - 397 q^{38} + 918 q^{40} + 39 q^{41} + 527 q^{43} - 64 q^{44} - 539 q^{46} - 860 q^{47} + 347 q^{49} + 591 q^{50} - 644 q^{52} + 812 q^{53} + 536 q^{55} + 2218 q^{56} - 1154 q^{58} + 472 q^{59} - 460 q^{61} + 2014 q^{62} - 451 q^{64} + 986 q^{65} + 1934 q^{67} + 69 q^{68} - 1028 q^{70} + 1687 q^{71} + 1980 q^{73} + 2400 q^{74} - 940 q^{76} + 821 q^{77} + 3319 q^{79} + 2119 q^{80} + 429 q^{82} - 2057 q^{83} + 566 q^{85} + 6690 q^{86} + 1189 q^{88} - 1668 q^{89} + 2427 q^{91} + 980 q^{92} + 332 q^{94} - 2146 q^{95} + 1956 q^{97} + 2026 q^{98}+O(q^{100}) \) 8 * q - 2 * q^2 + 38 * q^4 + 12 * q^5 + 53 * q^7 - 3 * q^8 + 29 * q^10 + 27 * q^11 + 89 * q^13 + 37 * q^14 + 362 * q^16 - 79 * q^17 + 288 * q^19 - 457 * q^20 + 596 * q^22 - 202 * q^23 + 264 * q^25 - 270 * q^26 + 702 * q^28 + 114 * q^29 + 538 * q^31 - 316 * q^32 + 498 * q^34 + 196 * q^35 + 395 * q^37 - 397 * q^38 + 918 * q^40 + 39 * q^41 + 527 * q^43 - 64 * q^44 - 539 * q^46 - 860 * q^47 + 347 * q^49 + 591 * q^50 - 644 * q^52 + 812 * q^53 + 536 * q^55 + 2218 * q^56 - 1154 * q^58 + 472 * q^59 - 460 * q^61 + 2014 * q^62 - 451 * q^64 + 986 * q^65 + 1934 * q^67 + 69 * q^68 - 1028 * q^70 + 1687 * q^71 + 1980 * q^73 + 2400 * q^74 - 940 * q^76 + 821 * q^77 + 3319 * q^79 + 2119 * q^80 + 429 * q^82 - 2057 * q^83 + 566 * q^85 + 6690 * q^86 + 1189 * q^88 - 1668 * q^89 + 2427 * q^91 + 980 * q^92 + 332 * q^94 - 2146 * q^95 + 1956 * q^97 + 2026 * q^98

Introduction

L-functions

Modular forms

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Database

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