LMFDB - Embedded newform 531.4.a.f.1.1

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			1.1
Root			$5.26363$ 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)

$f(q)$	$=$	$q-5.26363 q^{2} +19.7058 q^{4} -11.8799 q^{5} -3.36662 q^{7} -61.6149 q^{8} +O(q^{10})$	\(q-5.26363 q^{2} +19.7058 q^{4} -11.8799 q^{5} -3.36662 q^{7} -61.6149 q^{8} +62.5311 q^{10} +3.40311 q^{11} -54.8553 q^{13} +17.7206 q^{14} +166.672 q^{16} -68.7547 q^{17} +8.88097 q^{19} -234.102 q^{20} -17.9127 q^{22} -80.8007 q^{23} +16.1310 q^{25} +288.738 q^{26} -66.3419 q^{28} +235.255 q^{29} +145.469 q^{31} -384.378 q^{32} +361.899 q^{34} +39.9950 q^{35} -309.554 q^{37} -46.7461 q^{38} +731.976 q^{40} -37.6647 q^{41} -465.320 q^{43} +67.0609 q^{44} +425.305 q^{46} -271.410 q^{47} -331.666 q^{49} -84.9076 q^{50} -1080.97 q^{52} +82.0797 q^{53} -40.4284 q^{55} +207.434 q^{56} -1238.29 q^{58} +59.0000 q^{59} -736.743 q^{61} -765.695 q^{62} +689.850 q^{64} +651.673 q^{65} +768.441 q^{67} -1354.86 q^{68} -210.519 q^{70} +164.728 q^{71} +445.018 q^{73} +1629.38 q^{74} +175.006 q^{76} -11.4570 q^{77} +602.572 q^{79} -1980.03 q^{80} +198.253 q^{82} +779.626 q^{83} +816.795 q^{85} +2449.27 q^{86} -209.682 q^{88} -1393.87 q^{89} +184.677 q^{91} -1592.24 q^{92} +1428.60 q^{94} -105.505 q^{95} -269.591 q^{97} +1745.77 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−5.26363	−1.86097	−0.930487	−	0.366325i	$-0.880616\pi$
$2$	−5.26363	−1.86097	−0.930487	+	0.366325i	$0.880616\pi$
$3$	0	0
$3$	0	0
$4$	19.7058	2.46322
$5$	−11.8799	−1.06257	−0.531283	−	0.847194i	$-0.678290\pi$
$5$	−11.8799	−1.06257	−0.531283	+	0.847194i	$0.678290\pi$
$6$	0	0
$7$	−3.36662	−0.181781	−0.0908903	−	0.995861i	$-0.528971\pi$
$7$	−3.36662	−0.181781	−0.0908903	+	0.995861i	$0.528971\pi$
$8$	−61.6149	−2.72302
$9$	0	0
$10$	62.5311	1.97741
$11$	3.40311	0.0932796	0.0466398	−	0.998912i	$-0.485149\pi$
$11$	3.40311	0.0932796	0.0466398	+	0.998912i	$0.485149\pi$
$12$	0	0
$13$	−54.8553	−1.17032	−0.585158	−	0.810919i	$-0.698968\pi$
$13$	−54.8553	−1.17032	−0.585158	+	0.810919i	$0.698968\pi$
$14$	17.7206	0.338289
$15$	0	0
$16$	166.672	2.60424
$17$	−68.7547	−0.980909	−0.490455	−	0.871467i	$-0.663169\pi$
$17$	−68.7547	−0.980909	−0.490455	+	0.871467i	$0.663169\pi$
$18$	0	0
$19$	8.88097	0.107233	0.0536167	−	0.998562i	$-0.482925\pi$
$19$	8.88097	0.107233	0.0536167	+	0.998562i	$0.482925\pi$
$20$	−234.102	−2.61734
$21$	0	0
$22$	−17.9127	−0.173591
$23$	−80.8007	−0.732526	−0.366263	−	0.930511i	$-0.619363\pi$
$23$	−80.8007	−0.732526	−0.366263	+	0.930511i	$0.619363\pi$
$24$	0	0
$25$	16.1310	0.129048
$26$	288.738	2.17793
$27$	0	0
$28$	−66.3419	−0.447766
$29$	235.255	1.50640	0.753202	−	0.657789i	$-0.228508\pi$
$29$	235.255	1.50640	0.753202	+	0.657789i	$0.228508\pi$
$30$	0	0
$31$	145.469	0.842807	0.421404	−	0.906873i	$-0.361538\pi$
$31$	145.469	0.842807	0.421404	+	0.906873i	$0.361538\pi$
$32$	−384.378	−2.12341
$33$	0	0
$34$	361.899	1.82545
$35$	39.9950	0.193154
$36$	0	0
$37$	−309.554	−1.37542	−0.687709	−	0.725987i	$-0.741383\pi$
$37$	−309.554	−1.37542	−0.687709	+	0.725987i	$0.741383\pi$
$38$	−46.7461	−0.199558
$39$	0	0
$40$	731.976	2.89339
$41$	−37.6647	−0.143469	−0.0717346	−	0.997424i	$-0.522853\pi$
$41$	−37.6647	−0.143469	−0.0717346	+	0.997424i	$0.522853\pi$
$42$	0	0
$43$	−465.320	−1.65025	−0.825124	−	0.564952i	$-0.808895\pi$
$43$	−465.320	−1.65025	−0.825124	+	0.564952i	$0.808895\pi$
$44$	67.0609	0.229768
$45$	0	0
$46$	425.305	1.36321
$47$	−271.410	−0.842324	−0.421162	−	0.906986i	$-0.638378\pi$
$47$	−271.410	−0.842324	−0.421162	+	0.906986i	$0.638378\pi$
$48$	0	0
$49$	−331.666	−0.966956
$50$	−84.9076	−0.240155
$51$	0	0
$52$	−1080.97	−2.88275
$53$	82.0797	0.212727	0.106363	−	0.994327i	$-0.466079\pi$
$53$	82.0797	0.212727	0.106363	+	0.994327i	$0.466079\pi$
$54$	0	0
$55$	−40.4284	−0.0991158
$56$	207.434	0.494992
$57$	0	0
$58$	−1238.29	−2.80338
$59$	59.0000	0.130189
$59$	59.0000	0.130189
$60$	0	0
$61$	−736.743	−1.54640	−0.773199	−	0.634164i	$-0.781345\pi$
$61$	−736.743	−1.54640	−0.773199	+	0.634164i	$0.781345\pi$
$62$	−765.695	−1.56844
$63$	0	0
$64$	689.850	1.34736
$65$	651.673	1.24354
$66$	0	0
$67$	768.441	1.40119	0.700597	−	0.713557i	$-0.252917\pi$
$67$	768.441	1.40119	0.700597	+	0.713557i	$0.252917\pi$
$68$	−1354.86	−2.41620
$69$	0	0
$70$	−210.519	−0.359454
$71$	164.728	0.275347	0.137674	−	0.990478i	$-0.456037\pi$
$71$	164.728	0.275347	0.137674	+	0.990478i	$0.456037\pi$
$72$	0	0
$73$	445.018	0.713499	0.356749	−	0.934200i	$-0.383885\pi$
$73$	445.018	0.713499	0.356749	+	0.934200i	$0.383885\pi$
$74$	1629.38	2.55961
$75$	0	0
$76$	175.006	0.264140
$77$	−11.4570	−0.0169564
$78$	0	0
$79$	602.572	0.858160	0.429080	−	0.903266i	$-0.358838\pi$
$79$	602.572	0.858160	0.429080	+	0.903266i	$0.358838\pi$
$80$	−1980.03	−2.76718
$81$	0	0
$82$	198.253	0.266992
$83$	779.626	1.03102	0.515512	−	0.856882i	$-0.327602\pi$
$83$	779.626	1.03102	0.515512	+	0.856882i	$0.327602\pi$
$84$	0	0
$85$	816.795	1.04228
$86$	2449.27	3.07107
$87$	0	0
$88$	−209.682	−0.254002
$89$	−1393.87	−1.66011	−0.830056	−	0.557681i	$-0.811691\pi$
$89$	−1393.87	−1.66011	−0.830056	+	0.557681i	$0.811691\pi$
$90$	0	0
$91$	184.677	0.212741
$92$	−1592.24	−1.80437
$93$	0	0
$94$	1428.60	1.56754
$95$	−105.505	−0.113943
$96$	0	0
$97$	−269.591	−0.282194	−0.141097	−	0.989996i	$-0.545063\pi$
$97$	−269.591	−0.282194	−0.141097	+	0.989996i	$0.545063\pi$
$98$	1745.77	1.79948
$99$	0	0
$100$	317.874	0.317874
$101$	−936.949	−0.923069	−0.461534	−	0.887122i	$-0.652701\pi$
$101$	−936.949	−0.923069	−0.461534	+	0.887122i	$0.652701\pi$
$102$	0	0
$103$	1461.76	1.39836	0.699182	−	0.714943i	$-0.253547\pi$
$103$	1461.76	1.39836	0.699182	+	0.714943i	$0.253547\pi$
$104$	3379.90	3.18679
$105$	0	0
$106$	−432.037	−0.395879
$107$	−830.314	−0.750182	−0.375091	−	0.926988i	$-0.622389\pi$
$107$	−830.314	−0.750182	−0.375091	+	0.926988i	$0.622389\pi$
$108$	0	0
$109$	1476.02	1.29703	0.648517	−	0.761200i	$-0.275390\pi$
$109$	1476.02	1.29703	0.648517	+	0.761200i	$0.275390\pi$
$110$	212.800	0.184452
$111$	0	0
$112$	−561.120	−0.473401
$113$	2355.58	1.96101	0.980507	−	0.196482i	$-0.0629518\pi$
$113$	2355.58	1.96101	0.980507	+	0.196482i	$0.0629518\pi$
$114$	0	0
$115$	959.900	0.778358
$116$	4635.88	3.71061
$117$	0	0
$118$	−310.554	−0.242278
$119$	231.471	0.178310
$120$	0	0
$121$	−1319.42	−0.991299
$122$	3877.94	2.87781
$123$	0	0
$124$	2866.58	2.07602
$125$	1293.35	0.925445
$126$	0	0
$127$	2512.25	1.75533	0.877663	−	0.479279i	$-0.159102\pi$
$127$	2512.25	1.75533	0.877663	+	0.479279i	$0.159102\pi$
$128$	−556.092	−0.384001
$129$	0	0
$130$	−3430.16	−2.31419
$131$	907.845	0.605487	0.302744	−	0.953072i	$-0.402097\pi$
$131$	907.845	0.605487	0.302744	+	0.953072i	$0.402097\pi$
$132$	0	0
$133$	−29.8989	−0.0194929
$134$	−4044.79	−2.60758
$135$	0	0
$136$	4236.31	2.67103
$137$	−326.883	−0.203850	−0.101925	−	0.994792i	$-0.532500\pi$
$137$	−326.883	−0.203850	−0.101925	+	0.994792i	$0.532500\pi$
$138$	0	0
$139$	3008.75	1.83597	0.917983	−	0.396620i	$-0.129817\pi$
$139$	3008.75	1.83597	0.917983	+	0.396620i	$0.129817\pi$
$140$	788.132	0.475781
$141$	0	0
$142$	−867.069	−0.512414
$143$	−186.678	−0.109167
$144$	0	0
$145$	−2794.79	−1.60065
$146$	−2342.41	−1.32780
$147$	0	0
$148$	−6100.01	−3.38796
$149$	144.752	0.0795873	0.0397937	−	0.999208i	$-0.487330\pi$
$149$	144.752	0.0795873	0.0397937	+	0.999208i	$0.487330\pi$
$150$	0	0
$151$	−767.404	−0.413579	−0.206789	−	0.978385i	$-0.566302\pi$
$151$	−767.404	−0.413579	−0.206789	+	0.978385i	$0.566302\pi$
$152$	−547.200	−0.291998
$153$	0	0
$154$	60.3052	0.0315554
$155$	−1728.15	−0.895539
$156$	0	0
$157$	3523.01	1.79087	0.895436	−	0.445190i	$-0.146864\pi$
$157$	3523.01	1.79087	0.895436	+	0.445190i	$0.146864\pi$
$158$	−3171.71	−1.59701
$159$	0	0
$160$	4566.36	2.25626
$161$	272.025	0.133159
$162$	0	0
$163$	−1448.84	−0.696209	−0.348104	−	0.937456i	$-0.613175\pi$
$163$	−1448.84	−0.696209	−0.348104	+	0.937456i	$0.613175\pi$
$164$	−742.212	−0.353397
$165$	0	0
$166$	−4103.66	−1.91871
$167$	584.113	0.270659	0.135329	−	0.990801i	$-0.456791\pi$
$167$	584.113	0.270659	0.135329	+	0.990801i	$0.456791\pi$
$168$	0	0
$169$	812.102	0.369641
$170$	−4299.31	−1.93966
$171$	0	0
$172$	−9169.49	−4.06493
$173$	−2756.29	−1.21131	−0.605656	−	0.795727i	$-0.707089\pi$
$173$	−2756.29	−1.21131	−0.605656	+	0.795727i	$0.707089\pi$
$174$	0	0
$175$	−54.3070	−0.0234584
$176$	567.201	0.242923
$177$	0	0
$178$	7336.81	3.08942
$179$	3014.07	1.25856	0.629280	−	0.777179i	$-0.283350\pi$
$179$	3014.07	1.25856	0.629280	+	0.777179i	$0.283350\pi$
$180$	0	0
$181$	−1361.03	−0.558921	−0.279461	−	0.960157i	$-0.590156\pi$
$181$	−1361.03	−0.558921	−0.279461	+	0.960157i	$0.590156\pi$
$182$	−972.071	−0.395905
$183$	0	0
$184$	4978.52	1.99468
$185$	3677.46	1.46147
$186$	0	0
$187$	−233.979	−0.0914988
$188$	−5348.34	−2.07483
$189$	0	0
$190$	555.337	0.212044
$191$	−182.879	−0.0692810	−0.0346405	−	0.999400i	$-0.511029\pi$
$191$	−182.879	−0.0692810	−0.0346405	+	0.999400i	$0.511029\pi$
$192$	0	0
$193$	4052.56	1.51145	0.755725	−	0.654889i	$-0.227285\pi$
$193$	4052.56	1.51145	0.755725	+	0.654889i	$0.227285\pi$
$194$	1419.03	0.525155
$195$	0	0
$196$	−6535.73	−2.38183
$197$	−1243.86	−0.449856	−0.224928	−	0.974375i	$-0.572215\pi$
$197$	−1243.86	−0.449856	−0.224928	+	0.974375i	$0.572215\pi$
$198$	0	0
$199$	−191.567	−0.0682404	−0.0341202	−	0.999418i	$-0.510863\pi$
$199$	−191.567	−0.0682404	−0.0341202	+	0.999418i	$0.510863\pi$
$200$	−993.910	−0.351400
$201$	0	0
$202$	4931.75	1.71781
$203$	−792.014	−0.273835
$204$	0	0
$205$	447.451	0.152446
$206$	−7694.17	−2.60232
$207$	0	0
$208$	−9142.81	−3.04779
$209$	30.2229	0.0100027
$210$	0	0
$211$	3016.73	0.984268	0.492134	−	0.870520i	$-0.336217\pi$
$211$	3016.73	0.984268	0.492134	+	0.870520i	$0.336217\pi$
$212$	1617.44	0.523993
$213$	0	0
$214$	4370.46	1.39607
$215$	5527.94	1.75350
$216$	0	0
$217$	−489.740	−0.153206
$218$	−7769.20	−2.41375
$219$	0	0
$220$	−796.674	−0.244144
$221$	3771.56	1.14797
$222$	0	0
$223$	−3416.65	−1.02599	−0.512995	−	0.858392i	$-0.671464\pi$
$223$	−3416.65	−1.02599	−0.512995	+	0.858392i	$0.671464\pi$
$224$	1294.06	0.385994
$225$	0	0
$226$	−12398.9	−3.64940
$227$	−442.483	−0.129377	−0.0646886	−	0.997906i	$-0.520605\pi$
$227$	−442.483	−0.129377	−0.0646886	+	0.997906i	$0.520605\pi$
$228$	0	0
$229$	−5469.58	−1.57834	−0.789170	−	0.614175i	$-0.789489\pi$
$229$	−5469.58	−1.57834	−0.789170	+	0.614175i	$0.789489\pi$
$230$	−5052.56	−1.44850
$231$	0	0
$232$	−14495.2	−4.10197
$233$	−3053.40	−0.858519	−0.429259	−	0.903181i	$-0.641225\pi$
$233$	−3053.40	−0.858519	−0.429259	+	0.903181i	$0.641225\pi$
$234$	0	0
$235$	3224.31	0.895025
$236$	1162.64	0.320684
$237$	0	0
$238$	−1218.38	−0.331830
$239$	4582.19	1.24015	0.620077	−	0.784541i	$-0.287101\pi$
$239$	4582.19	1.24015	0.620077	+	0.784541i	$0.287101\pi$
$240$	0	0
$241$	6410.62	1.71346	0.856731	−	0.515764i	$-0.172492\pi$
$241$	6410.62	1.71346	0.856731	+	0.515764i	$0.172492\pi$
$242$	6944.93	1.84478
$243$	0	0
$244$	−14518.1	−3.80912
$245$	3940.14	1.02746
$246$	0	0
$247$	−487.168	−0.125497
$248$	−8963.06	−2.29498
$249$	0	0
$250$	−6807.70	−1.72223
$251$	−1936.29	−0.486922	−0.243461	−	0.969911i	$-0.578283\pi$
$251$	−1936.29	−0.486922	−0.243461	+	0.969911i	$0.578283\pi$
$252$	0	0
$253$	−274.973	−0.0683297
$254$	−13223.6	−3.26661
$255$	0	0
$256$	−2591.74	−0.632749
$257$	341.077	0.0827852	0.0413926	−	0.999143i	$-0.486821\pi$
$257$	341.077	0.0827852	0.0413926	+	0.999143i	$0.486821\pi$
$258$	0	0
$259$	1042.15	0.250024
$260$	12841.7	3.06312
$261$	0	0
$262$	−4778.56	−1.12680
$263$	−5497.72	−1.28899	−0.644494	−	0.764609i	$-0.722932\pi$
$263$	−5497.72	−1.28899	−0.644494	+	0.764609i	$0.722932\pi$
$264$	0	0
$265$	−975.095	−0.226036
$266$	157.377	0.0362758
$267$	0	0
$268$	15142.7	3.45145
$269$	−104.441	−0.0236725	−0.0118363	−	0.999930i	$-0.503768\pi$
$269$	−104.441	−0.0236725	−0.0118363	+	0.999930i	$0.503768\pi$
$270$	0	0
$271$	4199.52	0.941339	0.470669	−	0.882310i	$-0.344012\pi$
$271$	4199.52	0.941339	0.470669	+	0.882310i	$0.344012\pi$
$272$	−11459.4	−2.55453
$273$	0	0
$274$	1720.59	0.379360
$275$	54.8955	0.0120375
$276$	0	0
$277$	1330.67	0.288636	0.144318	−	0.989531i	$-0.453901\pi$
$277$	1330.67	0.288636	0.144318	+	0.989531i	$0.453901\pi$
$278$	−15837.0	−3.41668
$279$	0	0
$280$	−2464.29	−0.525962
$281$	4164.03	0.884004	0.442002	−	0.897014i	$-0.354268\pi$
$281$	4164.03	0.884004	0.442002	+	0.897014i	$0.354268\pi$
$282$	0	0
$283$	−985.954	−0.207099	−0.103549	−	0.994624i	$-0.533020\pi$
$283$	−985.954	−0.207099	−0.103549	+	0.994624i	$0.533020\pi$
$284$	3246.10	0.678242
$285$	0	0
$286$	982.605	0.203156
$287$	126.803	0.0260799
$288$	0	0
$289$	−185.798	−0.0378175
$290$	14710.8	2.97878
$291$	0	0
$292$	8769.43	1.75751
$293$	1679.74	0.334920	0.167460	−	0.985879i	$-0.446443\pi$
$293$	1679.74	0.334920	0.167460	+	0.985879i	$0.446443\pi$
$294$	0	0
$295$	−700.912	−0.138334
$296$	19073.2	3.74529
$297$	0	0
$298$	−761.918	−0.148110
$299$	4432.34	0.857288
$300$	0	0
$301$	1566.56	0.299983
$302$	4039.33	0.769660
$303$	0	0
$304$	1480.20	0.279262
$305$	8752.40	1.64315
$306$	0	0
$307$	−2907.22	−0.540469	−0.270234	−	0.962795i	$-0.587101\pi$
$307$	−2907.22	−0.540469	−0.270234	+	0.962795i	$0.587101\pi$
$308$	−225.769	−0.0417674
$309$	0	0
$310$	9096.35	1.66657
$311$	1187.35	0.216490	0.108245	−	0.994124i	$-0.465477\pi$
$311$	1187.35	0.216490	0.108245	+	0.994124i	$0.465477\pi$
$312$	0	0
$313$	3620.47	0.653806	0.326903	−	0.945058i	$-0.393995\pi$
$313$	3620.47	0.653806	0.326903	+	0.945058i	$0.393995\pi$
$314$	−18543.8	−3.33277
$315$	0	0
$316$	11874.1	2.11384
$317$	3700.69	0.655683	0.327842	−	0.944733i	$-0.393679\pi$
$317$	3700.69	0.655683	0.327842	+	0.944733i	$0.393679\pi$
$318$	0	0
$319$	800.597	0.140517
$320$	−8195.32	−1.43166
$321$	0	0
$322$	−1431.84	−0.247805
$323$	−610.608	−0.105186
$324$	0	0
$325$	−884.871	−0.151027
$326$	7626.16	1.29563
$327$	0	0
$328$	2320.71	0.390669
$329$	913.735	0.153118
$330$	0	0
$331$	−1595.64	−0.264967	−0.132484	−	0.991185i	$-0.542295\pi$
$331$	−1595.64	−0.264967	−0.132484	+	0.991185i	$0.542295\pi$
$332$	15363.1	2.53964
$333$	0	0
$334$	−3074.55	−0.503689
$335$	−9128.97	−1.48886
$336$	0	0
$337$	−5677.39	−0.917707	−0.458853	−	0.888512i	$-0.651740\pi$
$337$	−5677.39	−0.917707	−0.458853	+	0.888512i	$0.651740\pi$
$338$	−4274.60	−0.687893
$339$	0	0
$340$	16095.6	2.56737
$341$	495.047	0.0786167
$342$	0	0
$343$	2271.34	0.357554
$344$	28670.6	4.49365
$345$	0	0
$346$	14508.1	2.25422
$347$	−4193.13	−0.648700	−0.324350	−	0.945937i	$-0.605146\pi$
$347$	−4193.13	−0.648700	−0.324350	+	0.945937i	$0.605146\pi$
$348$	0	0
$349$	3131.79	0.480346	0.240173	−	0.970730i	$-0.422796\pi$
$349$	3131.79	0.480346	0.240173	+	0.970730i	$0.422796\pi$
$350$	285.852	0.0436555
$351$	0	0
$352$	−1308.08	−0.198071
$353$	−3984.82	−0.600823	−0.300412	−	0.953810i	$-0.597124\pi$
$353$	−3984.82	−0.600823	−0.300412	+	0.953810i	$0.597124\pi$
$354$	0	0
$355$	−1956.95	−0.292575
$356$	−27467.3	−4.08922
$357$	0	0
$358$	−15864.9	−2.34215
$359$	4378.11	0.643643	0.321822	−	0.946800i	$-0.395705\pi$
$359$	4378.11	0.643643	0.321822	+	0.946800i	$0.395705\pi$
$360$	0	0
$361$	−6780.13	−0.988501
$362$	7163.97	1.04014
$363$	0	0
$364$	3639.20	0.524028
$365$	−5286.75	−0.758140
$366$	0	0
$367$	−8615.60	−1.22542	−0.612712	−	0.790306i	$-0.709922\pi$
$367$	−8615.60	−1.22542	−0.612712	+	0.790306i	$0.709922\pi$
$368$	−13467.2	−1.90768
$369$	0	0
$370$	−19356.8	−2.71976
$371$	−276.331	−0.0386695
$372$	0	0
$373$	−8463.20	−1.17482	−0.587410	−	0.809290i	$-0.699852\pi$
$373$	−8463.20	−1.17482	−0.587410	+	0.809290i	$0.699852\pi$
$374$	1231.58	0.170277
$375$	0	0
$376$	16722.9	2.29366
$377$	−12905.0	−1.76297
$378$	0	0
$379$	5731.49	0.776800	0.388400	−	0.921491i	$-0.373028\pi$
$379$	5731.49	0.776800	0.388400	+	0.921491i	$0.373028\pi$
$380$	−2079.05	−0.280666
$381$	0	0
$382$	962.608	0.128930
$383$	12082.4	1.61197	0.805983	−	0.591939i	$-0.201637\pi$
$383$	12082.4	1.61197	0.805983	+	0.591939i	$0.201637\pi$
$384$	0	0
$385$	136.107	0.0180173
$386$	−21331.2	−2.81277
$387$	0	0
$388$	−5312.50	−0.695106
$389$	14859.0	1.93671	0.968357	−	0.249571i	$-0.0802895\pi$
$389$	14859.0	1.93671	0.968357	+	0.249571i	$0.0802895\pi$
$390$	0	0
$391$	5555.42	0.718541
$392$	20435.5	2.63304
$393$	0	0
$394$	6547.24	0.837171
$395$	−7158.47	−0.911852
$396$	0	0
$397$	11650.0	1.47279	0.736396	−	0.676551i	$-0.236526\pi$
$397$	11650.0	1.47279	0.736396	+	0.676551i	$0.236526\pi$
$398$	1008.34	0.126994
$399$	0	0
$400$	2688.58	0.336072
$401$	5665.33	0.705518	0.352759	−	0.935714i	$-0.385243\pi$
$401$	5665.33	0.705518	0.352759	+	0.935714i	$0.385243\pi$
$402$	0	0
$403$	−7979.75	−0.986352
$404$	−18463.3	−2.27372
$405$	0	0
$406$	4168.87	0.509599
$407$	−1053.45	−0.128298
$408$	0	0
$409$	−7654.36	−0.925388	−0.462694	−	0.886518i	$-0.653117\pi$
$409$	−7654.36	−0.925388	−0.462694	+	0.886518i	$0.653117\pi$
$410$	−2355.22	−0.283697
$411$	0	0
$412$	28805.1	3.44448
$413$	−198.631	−0.0236658
$414$	0	0
$415$	−9261.84	−1.09553
$416$	21085.2	2.48506
$417$	0	0
$418$	−159.082	−0.0186147
$419$	−8887.25	−1.03621	−0.518103	−	0.855318i	$-0.673362\pi$
$419$	−8887.25	−1.03621	−0.518103	+	0.855318i	$0.673362\pi$
$420$	0	0
$421$	−8283.80	−0.958974	−0.479487	−	0.877549i	$-0.659177\pi$
$421$	−8283.80	−0.958974	−0.479487	+	0.877549i	$0.659177\pi$
$422$	−15879.0	−1.83170
$423$	0	0
$424$	−5057.33	−0.579258
$425$	−1109.08	−0.126584
$426$	0	0
$427$	2480.33	0.281105
$428$	−16362.0	−1.84787
$429$	0	0
$430$	−29097.0	−3.26321
$431$	5882.75	0.657453	0.328726	−	0.944425i	$-0.393381\pi$
$431$	5882.75	0.657453	0.328726	+	0.944425i	$0.393381\pi$
$432$	0	0
$433$	−14513.8	−1.61083	−0.805415	−	0.592711i	$-0.798058\pi$
$433$	−14513.8	−1.61083	−0.805415	+	0.592711i	$0.798058\pi$
$434$	2577.81	0.285112
$435$	0	0
$436$	29086.1	3.19488
$437$	−717.588	−0.0785512
$438$	0	0
$439$	3287.31	0.357392	0.178696	−	0.983904i	$-0.442812\pi$
$439$	3287.31	0.357392	0.178696	+	0.983904i	$0.442812\pi$
$440$	2490.99	0.269894
$441$	0	0
$442$	−19852.1	−2.13635
$443$	−12974.8	−1.39153	−0.695767	−	0.718267i	$-0.744936\pi$
$443$	−12974.8	−1.39153	−0.695767	+	0.718267i	$0.744936\pi$
$444$	0	0
$445$	16559.0	1.76398
$446$	17984.0	1.90934
$447$	0	0
$448$	−2322.47	−0.244925
$449$	−15485.8	−1.62766	−0.813831	−	0.581101i	$-0.802622\pi$
$449$	−15485.8	−1.62766	−0.813831	+	0.581101i	$0.802622\pi$
$450$	0	0
$451$	−128.177	−0.0133827
$452$	46418.6	4.83042
$453$	0	0
$454$	2329.06	0.240767
$455$	−2193.94	−0.226051
$456$	0	0
$457$	4088.09	0.418452	0.209226	−	0.977867i	$-0.432906\pi$
$457$	4088.09	0.418452	0.209226	+	0.977867i	$0.432906\pi$
$458$	28789.8	2.93725
$459$	0	0
$460$	18915.6	1.91727
$461$	7983.57	0.806577	0.403289	−	0.915073i	$-0.367867\pi$
$461$	7983.57	0.806577	0.403289	+	0.915073i	$0.367867\pi$
$462$	0	0
$463$	9829.43	0.986635	0.493318	−	0.869849i	$-0.335784\pi$
$463$	9829.43	0.986635	0.493318	+	0.869849i	$0.335784\pi$
$464$	39210.3	3.92304
$465$	0	0
$466$	16072.0	1.59768
$467$	7096.19	0.703153	0.351577	−	0.936159i	$-0.385646\pi$
$467$	7096.19	0.703153	0.351577	+	0.936159i	$0.385646\pi$
$468$	0	0
$469$	−2587.05	−0.254710
$470$	−16971.6	−1.66562
$471$	0	0
$472$	−3635.28	−0.354507
$473$	−1583.53	−0.153934
$474$	0	0
$475$	143.259	0.0138383
$476$	4561.32	0.439218
$477$	0	0
$478$	−24118.9	−2.30790
$479$	8405.79	0.801817	0.400909	−	0.916118i	$-0.368695\pi$
$479$	8405.79	0.801817	0.400909	+	0.916118i	$0.368695\pi$
$480$	0	0
$481$	16980.7	1.60967
$482$	−33743.1	−3.18871
$483$	0	0
$484$	−26000.2	−2.44179
$485$	3202.70	0.299850
$486$	0	0
$487$	−33.3955	−0.00310738	−0.00155369	−	0.999999i	$-0.500495\pi$
$487$	−33.3955	−0.00310738	−0.00155369	+	0.999999i	$0.500495\pi$
$488$	45394.3	4.21087
$489$	0	0
$490$	−20739.4	−1.91207
$491$	3539.38	0.325316	0.162658	−	0.986683i	$-0.447993\pi$
$491$	3539.38	0.325316	0.162658	+	0.986683i	$0.447993\pi$
$492$	0	0
$493$	−16174.9	−1.47765
$494$	2564.27	0.233547
$495$	0	0
$496$	24245.6	2.19487
$497$	−554.578	−0.0500528
$498$	0	0
$499$	3739.04	0.335435	0.167718	−	0.985835i	$-0.446360\pi$
$499$	3739.04	0.335435	0.167718	+	0.985835i	$0.446360\pi$
$500$	25486.4	2.27958
$501$	0	0
$502$	10191.9	0.906148
$503$	−18545.4	−1.64393	−0.821965	−	0.569539i	$-0.807122\pi$
$503$	−18545.4	−1.64393	−0.821965	+	0.569539i	$0.807122\pi$
$504$	0	0
$505$	11130.8	0.980822
$506$	1447.36	0.127160
$507$	0	0
$508$	49505.9	4.32376
$509$	−18221.3	−1.58673	−0.793363	−	0.608749i	$-0.791672\pi$
$509$	−18221.3	−1.58673	−0.793363	+	0.608749i	$0.791672\pi$
$510$	0	0
$511$	−1498.21	−0.129700
$512$	18090.7	1.56153
$513$	0	0
$514$	−1795.30	−0.154061
$515$	−17365.5	−1.48586
$516$	0	0
$517$	−923.637	−0.0785716
$518$	−5485.50	−0.465288
$519$	0	0
$520$	−40152.7	−3.38618
$521$	−15891.9	−1.33634	−0.668172	−	0.744007i	$-0.732923\pi$
$521$	−15891.9	−1.33634	−0.668172	+	0.744007i	$0.732923\pi$
$522$	0	0
$523$	15241.9	1.27434	0.637170	−	0.770723i	$-0.280105\pi$
$523$	15241.9	1.27434	0.637170	+	0.770723i	$0.280105\pi$
$524$	17889.8	1.49145
$525$	0	0
$526$	28938.0	2.39877
$527$	−10001.7	−0.826717
$528$	0	0
$529$	−5638.25	−0.463405
$530$	5132.54	0.420647
$531$	0	0
$532$	−589.180	−0.0480154
$533$	2066.11	0.167904
$534$	0	0
$535$	9864.01	0.797118
$536$	−47347.4	−3.81548
$537$	0	0
$538$	549.741	0.0440539
$539$	−1128.69	−0.0901972
$540$	0	0
$541$	14342.6	1.13981	0.569904	−	0.821711i	$-0.306980\pi$
$541$	14342.6	1.13981	0.569904	+	0.821711i	$0.306980\pi$
$542$	−22104.7	−1.75181
$543$	0	0
$544$	26427.8	2.08287
$545$	−17534.9	−1.37819
$546$	0	0
$547$	4343.76	0.339536	0.169768	−	0.985484i	$-0.445698\pi$
$547$	4343.76	0.339536	0.169768	+	0.985484i	$0.445698\pi$
$548$	−6441.48	−0.502129
$549$	0	0
$550$	−288.950	−0.0224016
$551$	2089.29	0.161537
$552$	0	0
$553$	−2028.63	−0.155997
$554$	−7004.15	−0.537144
$555$	0	0
$556$	59289.9	4.52239
$557$	7199.47	0.547669	0.273834	−	0.961777i	$-0.411708\pi$
$557$	7199.47	0.547669	0.273834	+	0.961777i	$0.411708\pi$
$558$	0	0
$559$	25525.3	1.93131
$560$	6666.03	0.503020
$561$	0	0
$562$	−21917.9	−1.64511
$563$	−9629.26	−0.720826	−0.360413	−	0.932793i	$-0.617364\pi$
$563$	−9629.26	−0.720826	−0.360413	+	0.932793i	$0.617364\pi$
$564$	0	0
$565$	−27984.0	−2.08371
$566$	5189.69	0.385405
$567$	0	0
$568$	−10149.7	−0.749776
$569$	−17896.5	−1.31856	−0.659279	−	0.751899i	$-0.729138\pi$
$569$	−17896.5	−1.31856	−0.659279	+	0.751899i	$0.729138\pi$
$570$	0	0
$571$	−14247.6	−1.04421	−0.522105	−	0.852881i	$-0.674853\pi$
$571$	−14247.6	−1.04421	−0.522105	+	0.852881i	$0.674853\pi$
$572$	−3678.64	−0.268902
$573$	0	0
$574$	−667.443	−0.0485340
$575$	−1303.40	−0.0945311
$576$	0	0
$577$	−14178.5	−1.02298	−0.511490	−	0.859289i	$-0.670906\pi$
$577$	−14178.5	−1.02298	−0.511490	+	0.859289i	$0.670906\pi$
$578$	977.969	0.0703774
$579$	0	0
$580$	−55073.6	−3.94277
$581$	−2624.71	−0.187420
$582$	0	0
$583$	279.326	0.0198430
$584$	−27419.7	−1.94287
$585$	0	0
$586$	−8841.54	−0.623278
$587$	17769.4	1.24944	0.624720	−	0.780848i	$-0.285213\pi$
$587$	17769.4	1.24944	0.624720	+	0.780848i	$0.285213\pi$
$588$	0	0
$589$	1291.91	0.0903771
$590$	3689.34	0.257437
$591$	0	0
$592$	−51593.9	−3.58192
$593$	12282.0	0.850527	0.425263	−	0.905070i	$-0.360181\pi$
$593$	12282.0	0.850527	0.425263	+	0.905070i	$0.360181\pi$
$594$	0	0
$595$	−2749.84	−0.189466
$596$	2852.44	0.196041
$597$	0	0
$598$	−23330.2	−1.59539
$599$	−22571.6	−1.53965	−0.769827	−	0.638253i	$-0.779657\pi$
$599$	−22571.6	−1.53965	−0.769827	+	0.638253i	$0.779657\pi$
$600$	0	0
$601$	−5500.85	−0.373352	−0.186676	−	0.982422i	$-0.559771\pi$
$601$	−5500.85	−0.373352	−0.186676	+	0.982422i	$0.559771\pi$
$602$	−8245.77	−0.558260
$603$	0	0
$604$	−15122.3	−1.01874
$605$	15674.5	1.05332
$606$	0	0
$607$	377.433	0.0252381	0.0126190	−	0.999920i	$-0.495983\pi$
$607$	377.433	0.0252381	0.0126190	+	0.999920i	$0.495983\pi$
$608$	−3413.65	−0.227700
$609$	0	0
$610$	−46069.4	−3.05786
$611$	14888.3	0.985786
$612$	0	0
$613$	12016.8	0.791766	0.395883	−	0.918301i	$-0.370439\pi$
$613$	12016.8	0.791766	0.395883	+	0.918301i	$0.370439\pi$
$614$	15302.5	1.00580
$615$	0	0
$616$	705.920	0.0461726
$617$	15545.9	1.01435	0.507176	−	0.861843i	$-0.330689\pi$
$617$	15545.9	1.01435	0.507176	+	0.861843i	$0.330689\pi$
$618$	0	0
$619$	2904.00	0.188565	0.0942825	−	0.995545i	$-0.469944\pi$
$619$	2904.00	0.188565	0.0942825	+	0.995545i	$0.469944\pi$
$620$	−34054.6	−2.20591
$621$	0	0
$622$	−6249.76	−0.402882
$623$	4692.63	0.301776
$624$	0	0
$625$	−17381.2	−1.11239
$626$	−19056.8	−1.21672
$627$	0	0
$628$	69423.7	4.41132
$629$	21283.3	1.34916
$630$	0	0
$631$	−16236.4	−1.02434	−0.512172	−	0.858883i	$-0.671159\pi$
$631$	−16236.4	−1.02434	−0.512172	+	0.858883i	$0.671159\pi$
$632$	−37127.4	−2.33679
$633$	0	0
$634$	−19479.1	−1.22021
$635$	−29845.2	−1.86515
$636$	0	0
$637$	18193.6	1.13164
$638$	−4214.05	−0.261498
$639$	0	0
$640$	6606.30	0.408026
$641$	28390.2	1.74937	0.874684	−	0.484694i	$-0.161069\pi$
$641$	28390.2	1.74937	0.874684	+	0.484694i	$0.161069\pi$
$642$	0	0
$643$	23941.9	1.46839	0.734196	−	0.678937i	$-0.237559\pi$
$643$	23941.9	1.46839	0.734196	+	0.678937i	$0.237559\pi$
$644$	5360.47	0.328000
$645$	0	0
$646$	3214.01	0.195749
$647$	−3874.64	−0.235437	−0.117718	−	0.993047i	$-0.537558\pi$
$647$	−3874.64	−0.235437	−0.117718	+	0.993047i	$0.537558\pi$
$648$	0	0
$649$	200.783	0.0121440
$650$	4657.63	0.281057
$651$	0	0
$652$	−28550.6	−1.71492
$653$	23410.1	1.40292	0.701460	−	0.712709i	$-0.252532\pi$
$653$	23410.1	1.40292	0.701460	+	0.712709i	$0.252532\pi$
$654$	0	0
$655$	−10785.1	−0.643371
$656$	−6277.63	−0.373629
$657$	0	0
$658$	−4809.56	−0.284949
$659$	−22240.7	−1.31468	−0.657342	−	0.753593i	$-0.728319\pi$
$659$	−22240.7	−1.31468	−0.657342	+	0.753593i	$0.728319\pi$
$660$	0	0
$661$	30096.9	1.77100	0.885502	−	0.464635i	$-0.153815\pi$
$661$	30096.9	1.77100	0.885502	+	0.464635i	$0.153815\pi$
$662$	8398.84	0.493097
$663$	0	0
$664$	−48036.5	−2.80750
$665$	355.194	0.0207125
$666$	0	0
$667$	−19008.7	−1.10348
$668$	11510.4	0.666693
$669$	0	0
$670$	48051.5	2.77073
$671$	−2507.21	−0.144247
$672$	0	0
$673$	17650.2	1.01094	0.505471	−	0.862844i	$-0.331319\pi$
$673$	17650.2	1.01094	0.505471	+	0.862844i	$0.331319\pi$
$674$	29883.7	1.70783
$675$	0	0
$676$	16003.1	0.910509
$677$	8151.07	0.462734	0.231367	−	0.972866i	$-0.425680\pi$
$677$	8151.07	0.462734	0.231367	+	0.972866i	$0.425680\pi$
$678$	0	0
$679$	907.611	0.0512974
$680$	−50326.7	−2.83815
$681$	0	0
$682$	−2605.74	−0.146304
$683$	23902.8	1.33912	0.669558	−	0.742760i	$-0.266484\pi$
$683$	23902.8	1.33912	0.669558	+	0.742760i	$0.266484\pi$
$684$	0	0
$685$	3883.32	0.216605
$686$	−11955.5	−0.665399
$687$	0	0
$688$	−77555.6	−4.29764
$689$	−4502.50	−0.248958
$690$	0	0
$691$	−25203.7	−1.38754	−0.693772	−	0.720194i	$-0.744053\pi$
$691$	−25203.7	−1.38754	−0.693772	+	0.720194i	$0.744053\pi$
$692$	−54314.9	−2.98373
$693$	0	0
$694$	22071.1	1.20721
$695$	−35743.6	−1.95084
$696$	0	0
$697$	2589.62	0.140730
$698$	−16484.6	−0.893912
$699$	0	0
$700$	−1070.16	−0.0577833
$701$	6102.41	0.328794	0.164397	−	0.986394i	$-0.447432\pi$
$701$	6102.41	0.328794	0.164397	+	0.986394i	$0.447432\pi$
$702$	0	0
$703$	−2749.14	−0.147491
$704$	2347.63	0.125682
$705$	0	0
$706$	20974.6	1.11812
$707$	3154.35	0.167796
$708$	0	0
$709$	−483.971	−0.0256360	−0.0128180	−	0.999918i	$-0.504080\pi$
$709$	−483.971	−0.0256360	−0.0128180	+	0.999918i	$0.504080\pi$
$710$	10300.7	0.544474
$711$	0	0
$712$	85883.1	4.52051
$713$	−11754.0	−0.617378
$714$	0	0
$715$	2217.71	0.115997
$716$	59394.6	3.10011
$717$	0	0
$718$	−23044.8	−1.19780
$719$	−21253.9	−1.10242	−0.551208	−	0.834368i	$-0.685833\pi$
$719$	−21253.9	−1.10242	−0.551208	+	0.834368i	$0.685833\pi$
$720$	0	0
$721$	−4921.20	−0.254195
$722$	35688.1	1.83957
$723$	0	0
$724$	−26820.2	−1.37675
$725$	3794.90	0.194399
$726$	0	0
$727$	−26369.1	−1.34522	−0.672610	−	0.739997i	$-0.734827\pi$
$727$	−26369.1	−1.34522	−0.672610	+	0.739997i	$0.734827\pi$
$728$	−11378.9	−0.579297
$729$	0	0
$730$	27827.5	1.41088
$731$	31992.9	1.61874
$732$	0	0
$733$	−6909.57	−0.348173	−0.174086	−	0.984730i	$-0.555697\pi$
$733$	−6909.57	−0.348173	−0.174086	+	0.984730i	$0.555697\pi$
$734$	45349.3	2.28048
$735$	0	0
$736$	31058.0	1.55545
$737$	2615.09	0.130703
$738$	0	0
$739$	−34884.6	−1.73647	−0.868235	−	0.496153i	$-0.834746\pi$
$739$	−34884.6	−1.73647	−0.868235	+	0.496153i	$0.834746\pi$
$740$	72467.3	3.59993
$741$	0	0
$742$	1454.50	0.0719630
$743$	−34839.2	−1.72022	−0.860112	−	0.510106i	$-0.829606\pi$
$743$	−34839.2	−1.72022	−0.860112	+	0.510106i	$0.829606\pi$
$744$	0	0
$745$	−1719.63	−0.0845668
$746$	44547.1	2.18631
$747$	0	0
$748$	−4610.75	−0.225382
$749$	2795.35	0.136368
$750$	0	0
$751$	32677.5	1.58777	0.793887	−	0.608065i	$-0.208054\pi$
$751$	32677.5	1.58777	0.793887	+	0.608065i	$0.208054\pi$
$752$	−45236.3	−2.19362
$753$	0	0
$754$	67927.0	3.28084
$755$	9116.65	0.439455
$756$	0	0
$757$	−263.405	−0.0126468	−0.00632339	−	0.999980i	$-0.502013\pi$
$757$	−263.405	−0.0126468	−0.00632339	+	0.999980i	$0.502013\pi$
$758$	−30168.5	−1.44560
$759$	0	0
$760$	6500.66	0.310268
$761$	23504.7	1.11964	0.559819	−	0.828615i	$-0.310870\pi$
$761$	23504.7	1.11964	0.559819	+	0.828615i	$0.310870\pi$
$762$	0	0
$763$	−4969.19	−0.235776
$764$	−3603.78	−0.170655
$765$	0	0
$766$	−63597.4	−2.99983
$767$	−3236.46	−0.152362
$768$	0	0
$769$	−14106.7	−0.661508	−0.330754	−	0.943717i	$-0.607303\pi$
$769$	−14106.7	−0.661508	−0.330754	+	0.943717i	$0.607303\pi$
$770$	−716.418	−0.0335297
$771$	0	0
$772$	79858.9	3.72304
$773$	8023.79	0.373345	0.186672	−	0.982422i	$-0.440230\pi$
$773$	8023.79	0.373345	0.186672	+	0.982422i	$0.440230\pi$
$774$	0	0
$775$	2346.56	0.108763
$776$	16610.8	0.768419
$777$	0	0
$778$	−78212.3	−3.60417
$779$	−334.499	−0.0153847
$780$	0	0
$781$	560.588	0.0256843
$782$	−29241.7	−1.33719
$783$	0	0
$784$	−55279.3	−2.51819
$785$	−41852.9	−1.90292
$786$	0	0
$787$	22881.4	1.03638	0.518192	−	0.855264i	$-0.326605\pi$
$787$	22881.4	1.03638	0.518192	+	0.855264i	$0.326605\pi$
$788$	−24511.3	−1.10810
$789$	0	0
$790$	37679.5	1.69693
$791$	−7930.36	−0.356474
$792$	0	0
$793$	40414.2	1.80978
$794$	−61321.4	−2.74083
$795$	0	0
$796$	−3774.98	−0.168091
$797$	3384.94	0.150440	0.0752200	−	0.997167i	$-0.476034\pi$
$797$	3384.94	0.150440	0.0752200	+	0.997167i	$0.476034\pi$
$798$	0	0
$799$	18660.7	0.826243
$800$	−6200.40	−0.274022
$801$	0	0
$802$	−29820.2	−1.31295
$803$	1514.44	0.0665549
$804$	0	0
$805$	−3231.62	−0.141490
$806$	42002.4	1.83557
$807$	0	0
$808$	57730.0	2.51353
$809$	19080.0	0.829193	0.414597	−	0.910005i	$-0.363923\pi$
$809$	19080.0	0.829193	0.414597	+	0.910005i	$0.363923\pi$
$810$	0	0
$811$	−38157.7	−1.65216	−0.826078	−	0.563556i	$-0.809433\pi$
$811$	−38157.7	−1.65216	−0.826078	+	0.563556i	$0.809433\pi$
$812$	−15607.3	−0.674516
$813$	0	0
$814$	5544.95	0.238760
$815$	17212.0	0.739768
$816$	0	0
$817$	−4132.49	−0.176962
$818$	40289.7	1.72212
$819$	0	0
$820$	8817.38	0.375507
$821$	24776.7	1.05324	0.526621	−	0.850100i	$-0.323459\pi$
$821$	24776.7	1.05324	0.526621	+	0.850100i	$0.323459\pi$
$822$	0	0
$823$	−29084.0	−1.23184	−0.615921	−	0.787808i	$-0.711216\pi$
$823$	−29084.0	−1.23184	−0.615921	+	0.787808i	$0.711216\pi$
$824$	−90066.2	−3.80777
$825$	0	0
$826$	1045.52	0.0440414
$827$	7505.76	0.315600	0.157800	−	0.987471i	$-0.449560\pi$
$827$	7505.76	0.315600	0.157800	+	0.987471i	$0.449560\pi$
$828$	0	0
$829$	−4101.70	−0.171843	−0.0859216	−	0.996302i	$-0.527383\pi$
$829$	−4101.70	−0.171843	−0.0859216	+	0.996302i	$0.527383\pi$
$830$	48750.9	2.03876
$831$	0	0
$832$	−37841.9	−1.57684
$833$	22803.6	0.948496
$834$	0	0
$835$	−6939.18	−0.287593
$836$	595.565	0.0246388
$837$	0	0
$838$	46779.2	1.92835
$839$	33368.3	1.37307	0.686533	−	0.727099i	$-0.259132\pi$
$839$	33368.3	1.37307	0.686533	+	0.727099i	$0.259132\pi$
$840$	0	0
$841$	30955.8	1.26925
$842$	43602.9	1.78462
$843$	0	0
$844$	59447.1	2.42447
$845$	−9647.66	−0.392769
$846$	0	0
$847$	4441.98	0.180199
$848$	13680.3	0.553992
$849$	0	0
$850$	5837.79	0.235570
$851$	25012.2	1.00753
$852$	0	0
$853$	9085.16	0.364678	0.182339	−	0.983236i	$-0.441633\pi$
$853$	9085.16	0.364678	0.182339	+	0.983236i	$0.441633\pi$
$854$	−13055.6	−0.523129
$855$	0	0
$856$	51159.7	2.04276
$857$	−40753.2	−1.62439	−0.812196	−	0.583385i	$-0.801728\pi$
$857$	−40753.2	−1.62439	−0.812196	+	0.583385i	$0.801728\pi$
$858$	0	0
$859$	−20281.5	−0.805585	−0.402792	−	0.915291i	$-0.631960\pi$
$859$	−20281.5	−0.805585	−0.402792	+	0.915291i	$0.631960\pi$
$860$	108932.	4.31926
$861$	0	0
$862$	−30964.6	−1.22350
$863$	−20235.1	−0.798157	−0.399079	−	0.916917i	$-0.630670\pi$
$863$	−20235.1	−0.798157	−0.399079	+	0.916917i	$0.630670\pi$
$864$	0	0
$865$	32744.4	1.28710
$866$	76395.3	2.99771
$867$	0	0
$868$	−9650.70	−0.377380
$869$	2050.62	0.0800488
$870$	0	0
$871$	−42153.0	−1.63984
$872$	−90944.6	−3.53185
$873$	0	0
$874$	3777.12	0.146182
$875$	−4354.21	−0.168228
$876$	0	0
$877$	32647.0	1.25703	0.628513	−	0.777799i	$-0.283664\pi$
$877$	32647.0	1.25703	0.628513	+	0.777799i	$0.283664\pi$
$878$	−17303.2	−0.665096
$879$	0	0
$880$	−6738.27	−0.258121
$881$	30042.8	1.14888	0.574442	−	0.818545i	$-0.305219\pi$
$881$	30042.8	1.14888	0.574442	+	0.818545i	$0.305219\pi$
$882$	0	0
$883$	−21269.9	−0.810634	−0.405317	−	0.914176i	$-0.632839\pi$
$883$	−21269.9	−0.810634	−0.405317	+	0.914176i	$0.632839\pi$
$884$	74321.5	2.82772
$885$	0	0
$886$	68294.4	2.58961
$887$	1012.07	0.0383112	0.0191556	−	0.999817i	$-0.493902\pi$
$887$	1012.07	0.0383112	0.0191556	+	0.999817i	$0.493902\pi$
$888$	0	0
$889$	−8457.80	−0.319084
$890$	−87160.3	−3.28272
$891$	0	0
$892$	−67327.7	−2.52724
$893$	−2410.38	−0.0903252
$894$	0	0
$895$	−35806.7	−1.33730
$896$	1872.15	0.0698038
$897$	0	0
$898$	81511.5	3.02904
$899$	34222.3	1.26961
$900$	0	0
$901$	−5643.36	−0.208665
$902$	674.676	0.0249049
$903$	0	0
$904$	−145139.	−5.33988
$905$	16168.9	0.593891
$906$	0	0
$907$	−9343.76	−0.342067	−0.171033	−	0.985265i	$-0.554711\pi$
$907$	−9343.76	−0.342067	−0.171033	+	0.985265i	$0.554711\pi$
$908$	−8719.47	−0.318685
$909$	0	0
$910$	11548.1	0.420675
$911$	27773.6	1.01008	0.505039	−	0.863096i	$-0.331478\pi$
$911$	27773.6	1.01008	0.505039	+	0.863096i	$0.331478\pi$
$912$	0	0
$913$	2653.15	0.0961735
$914$	−21518.2	−0.778729
$915$	0	0
$916$	−107782.	−3.88780
$917$	−3056.37	−0.110066
$918$	0	0
$919$	16876.6	0.605776	0.302888	−	0.953026i	$-0.402049\pi$
$919$	16876.6	0.605776	0.302888	+	0.953026i	$0.402049\pi$
$920$	−59144.1	−2.11948
$921$	0	0
$922$	−42022.6	−1.50102
$923$	−9036.22	−0.322244
$924$	0	0
$925$	−4993.42	−0.177495
$926$	−51738.4	−1.83610
$927$	0	0
$928$	−90426.8	−3.19871
$929$	−10461.8	−0.369474	−0.184737	−	0.982788i	$-0.559143\pi$
$929$	−10461.8	−0.369474	−0.184737	+	0.982788i	$0.559143\pi$
$930$	0	0
$931$	−2945.51	−0.103690
$932$	−60169.6	−2.11472
$933$	0	0
$934$	−37351.7	−1.30855
$935$	2779.64	0.0972235
$936$	0	0
$937$	−13284.7	−0.463173	−0.231586	−	0.972814i	$-0.574392\pi$
$937$	−13284.7	−0.463173	−0.231586	+	0.972814i	$0.574392\pi$
$938$	13617.3	0.474008
$939$	0	0
$940$	63537.6	2.20465
$941$	−2290.28	−0.0793422	−0.0396711	−	0.999213i	$-0.512631\pi$
$941$	−2290.28	−0.0793422	−0.0396711	+	0.999213i	$0.512631\pi$
$942$	0	0
$943$	3043.33	0.105095
$944$	9833.62	0.339044
$945$	0	0
$946$	8335.13	0.286468
$947$	−18255.7	−0.626431	−0.313216	−	0.949682i	$-0.601406\pi$
$947$	−18255.7	−0.626431	−0.313216	+	0.949682i	$0.601406\pi$
$948$	0	0
$949$	−24411.6	−0.835020
$950$	−754.062	−0.0257526
$951$	0	0
$952$	−14262.1	−0.485542
$953$	7649.51	0.260013	0.130006	−	0.991513i	$-0.458500\pi$
$953$	7649.51	0.260013	0.130006	+	0.991513i	$0.458500\pi$
$954$	0	0
$955$	2172.58	0.0736157
$956$	90295.6	3.05478
$957$	0	0
$958$	−44245.0	−1.49216
$959$	1100.49	0.0370560
$960$	0	0
$961$	−8629.73	−0.289676
$962$	−89380.1	−2.99556
$963$	0	0
$964$	126326.	4.22064
$965$	−48143.9	−1.60602
$966$	0	0
$967$	22498.5	0.748194	0.374097	−	0.927390i	$-0.377953\pi$
$967$	22498.5	0.748194	0.374097	+	0.927390i	$0.377953\pi$
$968$	81295.8	2.69933
$969$	0	0
$970$	−16857.8	−0.558013
$971$	−16368.8	−0.540988	−0.270494	−	0.962722i	$-0.587187\pi$
$971$	−16368.8	−0.540988	−0.270494	+	0.962722i	$0.587187\pi$
$972$	0	0
$973$	−10129.3	−0.333743
$974$	175.782	0.00578276
$975$	0	0
$976$	−122794.	−4.02719
$977$	−45793.1	−1.49954	−0.749770	−	0.661699i	$-0.769836\pi$
$977$	−45793.1	−1.49954	−0.749770	+	0.661699i	$0.769836\pi$
$978$	0	0
$979$	−4743.49	−0.154854
$980$	77643.6	2.53085
$981$	0	0
$982$	−18630.0	−0.605404
$983$	−29019.5	−0.941587	−0.470793	−	0.882244i	$-0.656032\pi$
$983$	−29019.5	−0.941587	−0.470793	+	0.882244i	$0.656032\pi$
$984$	0	0
$985$	14776.9	0.478002
$986$	85138.5	2.74986
$987$	0	0
$988$	−9600.03	−0.309127
$989$	37598.2	1.20885
$990$	0	0
$991$	24513.2	0.785760	0.392880	−	0.919590i	$-0.371479\pi$
$991$	24513.2	0.785760	0.392880	+	0.919590i	$0.371479\pi$
$992$	−55915.1	−1.78962
$993$	0	0
$994$	2919.09	0.0931469
$995$	2275.79	0.0725100
$996$	0	0
$997$	29685.7	0.942983	0.471491	−	0.881871i	$-0.343716\pi$
$997$	29685.7	0.942983	0.471491	+	0.881871i	$0.343716\pi$
$998$	−19680.9	−0.624236
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	531.4.a.f.1.1		8
3.2	odd	2		177.4.a.c.1.8	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.4.a.c.1.8	✓	8	3.2	odd	2
531.4.a.f.1.1		8	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-5.26363 q^{2} +19.7058 q^{4} -11.8799 q^{5} -3.36662 q^{7} -61.6149 q^{8} +O(q^{10})\)	\(q-5.26363 q^{2} +19.7058 q^{4} -11.8799 q^{5} -3.36662 q^{7} -61.6149 q^{8} +62.5311 q^{10} +3.40311 q^{11} -54.8553 q^{13} +17.7206 q^{14} +166.672 q^{16} -68.7547 q^{17} +8.88097 q^{19} -234.102 q^{20} -17.9127 q^{22} -80.8007 q^{23} +16.1310 q^{25} +288.738 q^{26} -66.3419 q^{28} +235.255 q^{29} +145.469 q^{31} -384.378 q^{32} +361.899 q^{34} +39.9950 q^{35} -309.554 q^{37} -46.7461 q^{38} +731.976 q^{40} -37.6647 q^{41} -465.320 q^{43} +67.0609 q^{44} +425.305 q^{46} -271.410 q^{47} -331.666 q^{49} -84.9076 q^{50} -1080.97 q^{52} +82.0797 q^{53} -40.4284 q^{55} +207.434 q^{56} -1238.29 q^{58} +59.0000 q^{59} -736.743 q^{61} -765.695 q^{62} +689.850 q^{64} +651.673 q^{65} +768.441 q^{67} -1354.86 q^{68} -210.519 q^{70} +164.728 q^{71} +445.018 q^{73} +1629.38 q^{74} +175.006 q^{76} -11.4570 q^{77} +602.572 q^{79} -1980.03 q^{80} +198.253 q^{82} +779.626 q^{83} +816.795 q^{85} +2449.27 q^{86} -209.682 q^{88} -1393.87 q^{89} +184.677 q^{91} -1592.24 q^{92} +1428.60 q^{94} -105.505 q^{95} -269.591 q^{97} +1745.77 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 2 q^{2} + 38 q^{4} + 12 q^{5} + 53 q^{7} - 3 q^{8}+O(q^{10}) \) 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

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more