LMFDB - Embedded newform 6027.2.a.bm.1.16

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6027,2,Mod(1,6027)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6027, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("6027.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6027 = 3 \cdot 7^{2} \cdot 41 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6027.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:	$48.1258372982$
Analytic rank:	$1$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 4 x^{15} - 14 x^{14} + 68 x^{13} + 64 x^{12} - 456 x^{11} - 54 x^{10} + 1532 x^{9} - 400 x^{8} + \cdots - 49 $ x^16 - 4x^15 - 14x^14 + 68x^13 + 64x^12 - 456x^11 - 54x^10 + 1532x^9 - 400x^8 - 2708x^7 + 1218x^6 + 2424x^5 - 1276x^4 - 960x^3 + 500x^2 + 112*x - 49
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.16
Root			$-2.14627$ of defining polynomial
Character	$\chi$	$=$	6027.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.14627 q^{2} +1.00000 q^{3} +2.60647 q^{4} -2.36369 q^{5} +2.14627 q^{6} +1.30166 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+2.14627 q^{2} +1.00000 q^{3} +2.60647 q^{4} -2.36369 q^{5} +2.14627 q^{6} +1.30166 q^{8} +1.00000 q^{9} -5.07312 q^{10} -2.70007 q^{11} +2.60647 q^{12} -3.97235 q^{13} -2.36369 q^{15} -2.41924 q^{16} +6.48870 q^{17} +2.14627 q^{18} +7.54442 q^{19} -6.16090 q^{20} -5.79507 q^{22} -1.03945 q^{23} +1.30166 q^{24} +0.587030 q^{25} -8.52574 q^{26} +1.00000 q^{27} -8.83199 q^{29} -5.07312 q^{30} -2.51706 q^{31} -7.79566 q^{32} -2.70007 q^{33} +13.9265 q^{34} +2.60647 q^{36} -6.41148 q^{37} +16.1924 q^{38} -3.97235 q^{39} -3.07671 q^{40} +1.00000 q^{41} -0.547170 q^{43} -7.03765 q^{44} -2.36369 q^{45} -2.23094 q^{46} -3.62288 q^{47} -2.41924 q^{48} +1.25992 q^{50} +6.48870 q^{51} -10.3538 q^{52} -8.41749 q^{53} +2.14627 q^{54} +6.38212 q^{55} +7.54442 q^{57} -18.9558 q^{58} -12.6629 q^{59} -6.16090 q^{60} -4.26490 q^{61} -5.40229 q^{62} -11.8931 q^{64} +9.38941 q^{65} -5.79507 q^{66} -9.56262 q^{67} +16.9126 q^{68} -1.03945 q^{69} -6.72417 q^{71} +1.30166 q^{72} +7.72007 q^{73} -13.7608 q^{74} +0.587030 q^{75} +19.6643 q^{76} -8.52574 q^{78} +14.0058 q^{79} +5.71834 q^{80} +1.00000 q^{81} +2.14627 q^{82} +8.73964 q^{83} -15.3373 q^{85} -1.17437 q^{86} -8.83199 q^{87} -3.51456 q^{88} -16.8526 q^{89} -5.07312 q^{90} -2.70930 q^{92} -2.51706 q^{93} -7.77568 q^{94} -17.8327 q^{95} -7.79566 q^{96} +7.45775 q^{97} -2.70007 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 4 q^{2} + 16 q^{3} + 12 q^{4} - 12 q^{5} - 4 q^{6} - 12 q^{8} + 16 q^{9}+O(q^{10}) $ 16 * q - 4 * q^2 + 16 * q^3 + 12 * q^4 - 12 * q^5 - 4 * q^6 - 12 * q^8 + 16 * q^9	$ 16 q - 4 q^{2} + 16 q^{3} + 12 q^{4} - 12 q^{5} - 4 q^{6} - 12 q^{8} + 16 q^{9} - 4 q^{10} - 4 q^{11} + 12 q^{12} - 12 q^{15} - 8 q^{17} - 4 q^{18} + 4 q^{19} - 20 q^{20} - 16 q^{22} - 12 q^{23} - 12 q^{24} - 8 q^{25} - 8 q^{26} + 16 q^{27} - 16 q^{29} - 4 q^{30} - 4 q^{31} - 48 q^{32} - 4 q^{33} + 16 q^{34} + 12 q^{36} - 48 q^{37} - 4 q^{38} + 56 q^{40} + 16 q^{41} - 16 q^{43} - 12 q^{45} - 4 q^{46} - 36 q^{47} - 8 q^{50} - 8 q^{51} - 60 q^{53} - 4 q^{54} + 8 q^{55} + 4 q^{57} - 36 q^{58} - 36 q^{59} - 20 q^{60} - 4 q^{61} - 12 q^{62} + 52 q^{64} - 36 q^{65} - 16 q^{66} - 52 q^{67} - 8 q^{68} - 12 q^{69} - 12 q^{71} - 12 q^{72} - 16 q^{73} + 4 q^{74} - 8 q^{75} + 16 q^{76} - 8 q^{78} - 36 q^{79} - 68 q^{80} + 16 q^{81} - 4 q^{82} - 32 q^{83} - 28 q^{85} - 8 q^{86} - 16 q^{87} - 36 q^{88} - 12 q^{89} - 4 q^{90} - 36 q^{92} - 4 q^{93} + 24 q^{94} - 20 q^{95} - 48 q^{96} + 48 q^{97} - 4 q^{99}+O(q^{100}) $ 16 * q - 4 * q^2 + 16 * q^3 + 12 * q^4 - 12 * q^5 - 4 * q^6 - 12 * q^8 + 16 * q^9 - 4 * q^10 - 4 * q^11 + 12 * q^12 - 12 * q^15 - 8 * q^17 - 4 * q^18 + 4 * q^19 - 20 * q^20 - 16 * q^22 - 12 * q^23 - 12 * q^24 - 8 * q^25 - 8 * q^26 + 16 * q^27 - 16 * q^29 - 4 * q^30 - 4 * q^31 - 48 * q^32 - 4 * q^33 + 16 * q^34 + 12 * q^36 - 48 * q^37 - 4 * q^38 + 56 * q^40 + 16 * q^41 - 16 * q^43 - 12 * q^45 - 4 * q^46 - 36 * q^47 - 8 * q^50 - 8 * q^51 - 60 * q^53 - 4 * q^54 + 8 * q^55 + 4 * q^57 - 36 * q^58 - 36 * q^59 - 20 * q^60 - 4 * q^61 - 12 * q^62 + 52 * q^64 - 36 * q^65 - 16 * q^66 - 52 * q^67 - 8 * q^68 - 12 * q^69 - 12 * q^71 - 12 * q^72 - 16 * q^73 + 4 * q^74 - 8 * q^75 + 16 * q^76 - 8 * q^78 - 36 * q^79 - 68 * q^80 + 16 * q^81 - 4 * q^82 - 32 * q^83 - 28 * q^85 - 8 * q^86 - 16 * q^87 - 36 * q^88 - 12 * q^89 - 4 * q^90 - 36 * q^92 - 4 * q^93 + 24 * q^94 - 20 * q^95 - 48 * q^96 + 48 * q^97 - 4 * q^99

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 $

$2$	2.14627	1.51764	0.758821	−	0.651299i	$-0.225776\pi$
$2$	2.14627	1.51764	0.758821	+	0.651299i	$0.225776\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	2.60647	1.30324
$5$	−2.36369	−1.05707	−0.528537	−	0.848910i	$-0.677259\pi$
$5$	−2.36369	−1.05707	−0.528537	+	0.848910i	$0.677259\pi$
$6$	2.14627	0.876211
$7$	0	0
$7$	0	0
$8$	1.30166	0.460205
$9$	1.00000	0.333333
$10$	−5.07312	−1.60426
$11$	−2.70007	−0.814101	−0.407050	−	0.913406i	$-0.633443\pi$
$11$	−2.70007	−0.814101	−0.407050	+	0.913406i	$0.633443\pi$
$12$	2.60647	0.752424
$13$	−3.97235	−1.10173	−0.550866	−	0.834593i	$-0.685703\pi$
$13$	−3.97235	−1.10173	−0.550866	+	0.834593i	$0.685703\pi$
$14$	0	0
$15$	−2.36369	−0.610302
$16$	−2.41924	−0.604810
$17$	6.48870	1.57374	0.786871	−	0.617118i	$-0.211700\pi$
$17$	6.48870	1.57374	0.786871	+	0.617118i	$0.211700\pi$
$18$	2.14627	0.505881
$19$	7.54442	1.73081	0.865405	−	0.501073i	$-0.167061\pi$
$19$	7.54442	1.73081	0.865405	+	0.501073i	$0.167061\pi$
$20$	−6.16090	−1.37762
$21$	0	0
$22$	−5.79507	−1.23551
$23$	−1.03945	−0.216740	−0.108370	−	0.994111i	$-0.534563\pi$
$23$	−1.03945	−0.216740	−0.108370	+	0.994111i	$0.534563\pi$
$24$	1.30166	0.265699
$25$	0.587030	0.117406
$26$	−8.52574	−1.67204
$27$	1.00000	0.192450
$28$	0	0
$29$	−8.83199	−1.64006	−0.820030	−	0.572321i	$-0.806043\pi$
$29$	−8.83199	−1.64006	−0.820030	+	0.572321i	$0.806043\pi$
$30$	−5.07312	−0.926220
$31$	−2.51706	−0.452077	−0.226038	−	0.974118i	$-0.572578\pi$
$31$	−2.51706	−0.452077	−0.226038	+	0.974118i	$0.572578\pi$
$32$	−7.79566	−1.37809
$33$	−2.70007	−0.470021
$34$	13.9265	2.38838
$35$	0	0
$36$	2.60647	0.434412
$37$	−6.41148	−1.05404	−0.527020	−	0.849853i	$-0.676691\pi$
$37$	−6.41148	−1.05404	−0.527020	+	0.849853i	$0.676691\pi$
$38$	16.1924	2.62675
$39$	−3.97235	−0.636086
$40$	−3.07671	−0.486471
$41$	1.00000	0.156174
$41$	1.00000	0.156174
$42$	0	0
$43$	−0.547170	−0.0834426	−0.0417213	−	0.999129i	$-0.513284\pi$
$43$	−0.547170	−0.0834426	−0.0417213	+	0.999129i	$0.513284\pi$
$44$	−7.03765	−1.06097
$45$	−2.36369	−0.352358
$46$	−2.23094	−0.328934
$47$	−3.62288	−0.528451	−0.264226	−	0.964461i	$-0.585116\pi$
$47$	−3.62288	−0.528451	−0.264226	+	0.964461i	$0.585116\pi$
$48$	−2.41924	−0.349188
$49$	0	0
$50$	1.25992	0.178180
$51$	6.48870	0.908600
$52$	−10.3538	−1.43582
$53$	−8.41749	−1.15623	−0.578116	−	0.815955i	$-0.696212\pi$
$53$	−8.41749	−1.15623	−0.578116	+	0.815955i	$0.696212\pi$
$54$	2.14627	0.292070
$55$	6.38212	0.860565
$56$	0	0
$57$	7.54442	0.999283
$58$	−18.9558	−2.48902
$59$	−12.6629	−1.64857	−0.824283	−	0.566178i	$-0.808421\pi$
$59$	−12.6629	−1.64857	−0.824283	+	0.566178i	$0.808421\pi$
$60$	−6.16090	−0.795368
$61$	−4.26490	−0.546065	−0.273032	−	0.962005i	$-0.588027\pi$
$61$	−4.26490	−0.546065	−0.273032	+	0.962005i	$0.588027\pi$
$62$	−5.40229	−0.686091
$63$	0	0
$64$	−11.8931	−1.48664
$65$	9.38941	1.16461
$66$	−5.79507	−0.713324
$67$	−9.56262	−1.16826	−0.584130	−	0.811660i	$-0.698564\pi$
$67$	−9.56262	−1.16826	−0.584130	+	0.811660i	$0.698564\pi$
$68$	16.9126	2.05096
$69$	−1.03945	−0.125135
$70$	0	0
$71$	−6.72417	−0.798012	−0.399006	−	0.916948i	$-0.630645\pi$
$71$	−6.72417	−0.798012	−0.399006	+	0.916948i	$0.630645\pi$
$72$	1.30166	0.153402
$73$	7.72007	0.903566	0.451783	−	0.892128i	$-0.350788\pi$
$73$	7.72007	0.903566	0.451783	+	0.892128i	$0.350788\pi$
$74$	−13.7608	−1.59966
$75$	0.587030	0.0677844
$76$	19.6643	2.25566
$77$	0	0
$78$	−8.52574	−0.965350
$79$	14.0058	1.57577	0.787887	−	0.615820i	$-0.211175\pi$
$79$	14.0058	1.57577	0.787887	+	0.615820i	$0.211175\pi$
$80$	5.71834	0.639330
$81$	1.00000	0.111111
$82$	2.14627	0.237016
$83$	8.73964	0.959301	0.479650	−	0.877460i	$-0.340764\pi$
$83$	8.73964	0.959301	0.479650	+	0.877460i	$0.340764\pi$
$84$	0	0
$85$	−15.3373	−1.66356
$86$	−1.17437	−0.126636
$87$	−8.83199	−0.946889
$88$	−3.51456	−0.374653
$89$	−16.8526	−1.78637	−0.893185	−	0.449688i	$-0.851535\pi$
$89$	−16.8526	−1.78637	−0.893185	+	0.449688i	$0.851535\pi$
$90$	−5.07312	−0.534753
$91$	0	0
$92$	−2.70930	−0.282464
$93$	−2.51706	−0.261007
$94$	−7.77568	−0.802000
$95$	−17.8327	−1.82959
$96$	−7.79566	−0.795641
$97$	7.45775	0.757220	0.378610	−	0.925556i	$-0.376402\pi$
$97$	7.45775	0.757220	0.378610	+	0.925556i	$0.376402\pi$
$98$	0	0
$99$	−2.70007	−0.271367
$100$	1.53008	0.153008
$101$	−5.04947	−0.502441	−0.251221	−	0.967930i	$-0.580832\pi$
$101$	−5.04947	−0.502441	−0.251221	+	0.967930i	$0.580832\pi$
$102$	13.9265	1.37893
$103$	−10.1965	−1.00469	−0.502347	−	0.864666i	$-0.667530\pi$
$103$	−10.1965	−1.00469	−0.502347	+	0.864666i	$0.667530\pi$
$104$	−5.17064	−0.507023
$105$	0	0
$106$	−18.0662	−1.75475
$107$	17.2088	1.66364	0.831818	−	0.555049i	$-0.187300\pi$
$107$	17.2088	1.66364	0.831818	+	0.555049i	$0.187300\pi$
$108$	2.60647	0.250808
$109$	−1.53956	−0.147463	−0.0737316	−	0.997278i	$-0.523491\pi$
$109$	−1.53956	−0.147463	−0.0737316	+	0.997278i	$0.523491\pi$
$110$	13.6978	1.30603
$111$	−6.41148	−0.608551
$112$	0	0
$113$	8.66798	0.815415	0.407708	−	0.913113i	$-0.366328\pi$
$113$	8.66798	0.815415	0.407708	+	0.913113i	$0.366328\pi$
$114$	16.1924	1.51655
$115$	2.45694	0.229111
$116$	−23.0203	−2.13739
$117$	−3.97235	−0.367244
$118$	−27.1779	−2.50193
$119$	0	0
$120$	−3.07671	−0.280864
$121$	−3.70964	−0.337240
$122$	−9.15363	−0.828731
$123$	1.00000	0.0901670
$124$	−6.56065	−0.589163
$125$	10.4309	0.932967
$126$	0	0
$127$	−0.809245	−0.0718088	−0.0359044	−	0.999355i	$-0.511431\pi$
$127$	−0.809245	−0.0718088	−0.0359044	+	0.999355i	$0.511431\pi$
$128$	−9.93449	−0.878093
$129$	−0.547170	−0.0481756
$130$	20.1522	1.76747
$131$	−22.1985	−1.93949	−0.969746	−	0.244118i	$-0.921502\pi$
$131$	−22.1985	−1.93949	−0.969746	+	0.244118i	$0.921502\pi$
$132$	−7.03765	−0.612549
$133$	0	0
$134$	−20.5240	−1.77300
$135$	−2.36369	−0.203434
$136$	8.44606	0.724244
$137$	14.3569	1.22660	0.613298	−	0.789851i	$-0.289842\pi$
$137$	14.3569	1.22660	0.613298	+	0.789851i	$0.289842\pi$
$138$	−2.23094	−0.189910
$139$	−14.6198	−1.24004	−0.620018	−	0.784587i	$-0.712875\pi$
$139$	−14.6198	−1.24004	−0.620018	+	0.784587i	$0.712875\pi$
$140$	0	0
$141$	−3.62288	−0.305101
$142$	−14.4319	−1.21110
$143$	10.7256	0.896921
$144$	−2.41924	−0.201603
$145$	20.8761	1.73366
$146$	16.5694	1.37129
$147$	0	0
$148$	−16.7114	−1.37366
$149$	−20.7890	−1.70310	−0.851551	−	0.524272i	$-0.824337\pi$
$149$	−20.7890	−1.70310	−0.851551	+	0.524272i	$0.824337\pi$
$150$	1.25992	0.102872
$151$	−9.41227	−0.765960	−0.382980	−	0.923757i	$-0.625102\pi$
$151$	−9.41227	−0.765960	−0.382980	+	0.923757i	$0.625102\pi$
$152$	9.82025	0.796527
$153$	6.48870	0.524581
$154$	0	0
$155$	5.94954	0.477879
$156$	−10.3538	−0.828970
$157$	16.2722	1.29866	0.649331	−	0.760506i	$-0.275049\pi$
$157$	16.2722	1.29866	0.649331	+	0.760506i	$0.275049\pi$
$158$	30.0602	2.39146
$159$	−8.41749	−0.667551
$160$	18.4265	1.45674
$161$	0	0
$162$	2.14627	0.168627
$163$	−4.56932	−0.357897	−0.178948	−	0.983858i	$-0.557269\pi$
$163$	−4.56932	−0.357897	−0.178948	+	0.983858i	$0.557269\pi$
$164$	2.60647	0.203531
$165$	6.38212	0.496847
$166$	18.7576	1.45587
$167$	−2.84726	−0.220327	−0.110164	−	0.993913i	$-0.535137\pi$
$167$	−2.84726	−0.220327	−0.110164	+	0.993913i	$0.535137\pi$
$168$	0	0
$169$	2.77959	0.213815
$170$	−32.9179	−2.52469
$171$	7.54442	0.576937
$172$	−1.42618	−0.108746
$173$	14.2178	1.08096	0.540479	−	0.841357i	$-0.318243\pi$
$173$	14.2178	1.08096	0.540479	+	0.841357i	$0.318243\pi$
$174$	−18.9558	−1.43704
$175$	0	0
$176$	6.53211	0.492377
$177$	−12.6629	−0.951800
$178$	−36.1702	−2.71107
$179$	−0.557078	−0.0416379	−0.0208190	−	0.999783i	$-0.506627\pi$
$179$	−0.557078	−0.0416379	−0.0208190	+	0.999783i	$0.506627\pi$
$180$	−6.16090	−0.459206
$181$	4.64100	0.344963	0.172481	−	0.985013i	$-0.444822\pi$
$181$	4.64100	0.344963	0.172481	+	0.985013i	$0.444822\pi$
$182$	0	0
$183$	−4.26490	−0.315271
$184$	−1.35301	−0.0997450
$185$	15.1547	1.11420
$186$	−5.40229	−0.396115
$187$	−17.5199	−1.28118
$188$	−9.44294	−0.688697
$189$	0	0
$190$	−38.2737	−2.77667
$191$	13.5352	0.979371	0.489686	−	0.871899i	$-0.337112\pi$
$191$	13.5352	0.979371	0.489686	+	0.871899i	$0.337112\pi$
$192$	−11.8931	−0.858311
$193$	−1.23342	−0.0887838	−0.0443919	−	0.999014i	$-0.514135\pi$
$193$	−1.23342	−0.0887838	−0.0443919	+	0.999014i	$0.514135\pi$
$194$	16.0063	1.14919
$195$	9.38941	0.672390
$196$	0	0
$197$	8.98219	0.639955	0.319977	−	0.947425i	$-0.396325\pi$
$197$	8.98219	0.639955	0.319977	+	0.947425i	$0.396325\pi$
$198$	−5.79507	−0.411838
$199$	−8.55964	−0.606777	−0.303389	−	0.952867i	$-0.598118\pi$
$199$	−8.55964	−0.606777	−0.303389	+	0.952867i	$0.598118\pi$
$200$	0.764111	0.0540308
$201$	−9.56262	−0.674495
$202$	−10.8375	−0.762526
$203$	0	0
$204$	16.9126	1.18412
$205$	−2.36369	−0.165087
$206$	−21.8845	−1.52477
$207$	−1.03945	−0.0722468
$208$	9.61008	0.666339
$209$	−20.3705	−1.40905
$210$	0	0
$211$	−14.7292	−1.01400	−0.507000	−	0.861946i	$-0.669245\pi$
$211$	−14.7292	−1.01400	−0.507000	+	0.861946i	$0.669245\pi$
$212$	−21.9400	−1.50684
$213$	−6.72417	−0.460732
$214$	36.9347	2.52480
$215$	1.29334	0.0882051
$216$	1.30166	0.0885665
$217$	0	0
$218$	−3.30431	−0.223796
$219$	7.72007	0.521674
$220$	16.6348	1.12152
$221$	−25.7754	−1.73384
$222$	−13.7608	−0.923562
$223$	13.2746	0.888934	0.444467	−	0.895795i	$-0.353393\pi$
$223$	13.2746	0.888934	0.444467	+	0.895795i	$0.353393\pi$
$224$	0	0
$225$	0.587030	0.0391353
$226$	18.6038	1.23751
$227$	−17.4048	−1.15520	−0.577600	−	0.816320i	$-0.696011\pi$
$227$	−17.4048	−1.15520	−0.577600	+	0.816320i	$0.696011\pi$
$228$	19.6643	1.30230
$229$	−3.08034	−0.203555	−0.101777	−	0.994807i	$-0.532453\pi$
$229$	−3.08034	−0.203555	−0.101777	+	0.994807i	$0.532453\pi$
$230$	5.27325	0.347708
$231$	0	0
$232$	−11.4962	−0.754763
$233$	−18.9481	−1.24133	−0.620665	−	0.784076i	$-0.713137\pi$
$233$	−18.9481	−1.24133	−0.620665	+	0.784076i	$0.713137\pi$
$234$	−8.52574	−0.557345
$235$	8.56336	0.558612
$236$	−33.0054	−2.14847
$237$	14.0058	0.909773
$238$	0	0
$239$	9.97167	0.645014	0.322507	−	0.946567i	$-0.395474\pi$
$239$	9.97167	0.645014	0.322507	+	0.946567i	$0.395474\pi$
$240$	5.71834	0.369117
$241$	23.0166	1.48263	0.741314	−	0.671158i	$-0.234203\pi$
$241$	23.0166	1.48263	0.741314	+	0.671158i	$0.234203\pi$
$242$	−7.96189	−0.511809
$243$	1.00000	0.0641500
$244$	−11.1164	−0.711652
$245$	0	0
$246$	2.14627	0.136841
$247$	−29.9691	−1.90689
$248$	−3.27634	−0.208048
$249$	8.73964	0.553852
$250$	22.3875	1.41591
$251$	30.1597	1.90366	0.951832	−	0.306619i	$-0.0991977\pi$
$251$	30.1597	1.90366	0.951832	+	0.306619i	$0.0991977\pi$
$252$	0	0
$253$	2.80659	0.176449
$254$	−1.73686	−0.108980
$255$	−15.3373	−0.960458
$256$	2.46411	0.154007
$257$	−6.59194	−0.411194	−0.205597	−	0.978637i	$-0.565914\pi$
$257$	−6.59194	−0.411194	−0.205597	+	0.978637i	$0.565914\pi$
$258$	−1.17437	−0.0731134
$259$	0	0
$260$	24.4733	1.51777
$261$	−8.83199	−0.546686
$262$	−47.6440	−2.94345
$263$	16.3547	1.00847	0.504237	−	0.863565i	$-0.331774\pi$
$263$	16.3547	1.00847	0.504237	+	0.863565i	$0.331774\pi$
$264$	−3.51456	−0.216306
$265$	19.8963	1.22222
$266$	0	0
$267$	−16.8526	−1.03136
$268$	−24.9247	−1.52252
$269$	−21.2223	−1.29395	−0.646973	−	0.762513i	$-0.723965\pi$
$269$	−21.2223	−1.29395	−0.646973	+	0.762513i	$0.723965\pi$
$270$	−5.07312	−0.308740
$271$	−1.96725	−0.119502	−0.0597509	−	0.998213i	$-0.519031\pi$
$271$	−1.96725	−0.119502	−0.0597509	+	0.998213i	$0.519031\pi$
$272$	−15.6977	−0.951815
$273$	0	0
$274$	30.8139	1.86153
$275$	−1.58502	−0.0955803
$276$	−2.70930	−0.163081
$277$	10.1496	0.609827	0.304914	−	0.952380i	$-0.401372\pi$
$277$	10.1496	0.609827	0.304914	+	0.952380i	$0.401372\pi$
$278$	−31.3781	−1.88193
$279$	−2.51706	−0.150692
$280$	0	0
$281$	26.3025	1.56907	0.784537	−	0.620082i	$-0.212901\pi$
$281$	26.3025	1.56907	0.784537	+	0.620082i	$0.212901\pi$
$282$	−7.77568	−0.463035
$283$	23.4341	1.39301	0.696506	−	0.717551i	$-0.254737\pi$
$283$	23.4341	1.39301	0.696506	+	0.717551i	$0.254737\pi$
$284$	−17.5264	−1.04000
$285$	−17.8327	−1.05632
$286$	23.0201	1.36121
$287$	0	0
$288$	−7.79566	−0.459364
$289$	25.1033	1.47666
$290$	44.8057	2.63108
$291$	7.45775	0.437181
$292$	20.1222	1.17756
$293$	11.0295	0.644352	0.322176	−	0.946680i	$-0.395586\pi$
$293$	11.0295	0.644352	0.322176	+	0.946680i	$0.395586\pi$
$294$	0	0
$295$	29.9311	1.74266
$296$	−8.34554	−0.485075
$297$	−2.70007	−0.156674
$298$	−44.6188	−2.58470
$299$	4.12907	0.238790
$300$	1.53008	0.0883391
$301$	0	0
$302$	−20.2013	−1.16245
$303$	−5.04947	−0.290085
$304$	−18.2518	−1.04681
$305$	10.0809	0.577231
$306$	13.9265	0.796125
$307$	19.8917	1.13528	0.567639	−	0.823278i	$-0.307857\pi$
$307$	19.8917	1.13528	0.567639	+	0.823278i	$0.307857\pi$
$308$	0	0
$309$	−10.1965	−0.580060
$310$	12.7693	0.725249
$311$	16.8715	0.956696	0.478348	−	0.878170i	$-0.341236\pi$
$311$	16.8715	0.956696	0.478348	+	0.878170i	$0.341236\pi$
$312$	−5.17064	−0.292730
$313$	20.8946	1.18104	0.590518	−	0.807025i	$-0.298924\pi$
$313$	20.8946	1.18104	0.590518	+	0.807025i	$0.298924\pi$
$314$	34.9245	1.97090
$315$	0	0
$316$	36.5057	2.05361
$317$	−32.5637	−1.82896	−0.914479	−	0.404634i	$-0.867399\pi$
$317$	−32.5637	−1.82896	−0.914479	+	0.404634i	$0.867399\pi$
$318$	−18.0662	−1.01310
$319$	23.8470	1.33517
$320$	28.1116	1.57149
$321$	17.2088	0.960501
$322$	0	0
$323$	48.9535	2.72385
$324$	2.60647	0.144804
$325$	−2.33189	−0.129350
$326$	−9.80699	−0.543159
$327$	−1.53956	−0.0851379
$328$	1.30166	0.0718719
$329$	0	0
$330$	13.6978	0.754036
$331$	13.7235	0.754314	0.377157	−	0.926149i	$-0.376902\pi$
$331$	13.7235	0.754314	0.377157	+	0.926149i	$0.376902\pi$
$332$	22.7797	1.25020
$333$	−6.41148	−0.351347
$334$	−6.11098	−0.334378
$335$	22.6031	1.23494
$336$	0	0
$337$	16.8326	0.916928	0.458464	−	0.888713i	$-0.348400\pi$
$337$	16.8326	0.916928	0.458464	+	0.888713i	$0.348400\pi$
$338$	5.96576	0.324495
$339$	8.66798	0.470780
$340$	−39.9762	−2.16802
$341$	6.79622	0.368036
$342$	16.1924	0.875583
$343$	0	0
$344$	−0.712227	−0.0384007
$345$	2.45694	0.132277
$346$	30.5152	1.64051
$347$	−30.4692	−1.63567	−0.817837	−	0.575450i	$-0.804827\pi$
$347$	−30.4692	−1.63567	−0.817837	+	0.575450i	$0.804827\pi$
$348$	−23.0203	−1.23402
$349$	28.7979	1.54151	0.770757	−	0.637129i	$-0.219878\pi$
$349$	28.7979	1.54151	0.770757	+	0.637129i	$0.219878\pi$
$350$	0	0
$351$	−3.97235	−0.212029
$352$	21.0488	1.12190
$353$	15.7831	0.840049	0.420025	−	0.907513i	$-0.362021\pi$
$353$	15.7831	0.840049	0.420025	+	0.907513i	$0.362021\pi$
$354$	−27.1779	−1.44449
$355$	15.8938	0.843558
$356$	−43.9258	−2.32806
$357$	0	0
$358$	−1.19564	−0.0631915
$359$	−22.2166	−1.17255	−0.586273	−	0.810114i	$-0.699405\pi$
$359$	−22.2166	−1.17255	−0.586273	+	0.810114i	$0.699405\pi$
$360$	−3.07671	−0.162157
$361$	37.9183	1.99570
$362$	9.96084	0.523530
$363$	−3.70964	−0.194706
$364$	0	0
$365$	−18.2478	−0.955136
$366$	−9.15363	−0.478468
$367$	−5.65678	−0.295282	−0.147641	−	0.989041i	$-0.547168\pi$
$367$	−5.65678	−0.295282	−0.147641	+	0.989041i	$0.547168\pi$
$368$	2.51468	0.131087
$369$	1.00000	0.0520579
$370$	32.5262	1.69096
$371$	0	0
$372$	−6.56065	−0.340154
$373$	16.1337	0.835370	0.417685	−	0.908592i	$-0.362842\pi$
$373$	16.1337	0.835370	0.417685	+	0.908592i	$0.362842\pi$
$374$	−37.6025	−1.94438
$375$	10.4309	0.538649
$376$	−4.71574	−0.243196
$377$	35.0838	1.80691
$378$	0	0
$379$	−3.92347	−0.201535	−0.100768	−	0.994910i	$-0.532130\pi$
$379$	−3.92347	−0.201535	−0.100768	+	0.994910i	$0.532130\pi$
$380$	−46.4804	−2.38440
$381$	−0.809245	−0.0414589
$382$	29.0501	1.48633
$383$	−4.82783	−0.246691	−0.123345	−	0.992364i	$-0.539362\pi$
$383$	−4.82783	−0.246691	−0.123345	+	0.992364i	$0.539362\pi$
$384$	−9.93449	−0.506967
$385$	0	0
$386$	−2.64726	−0.134742
$387$	−0.547170	−0.0278142
$388$	19.4384	0.986836
$389$	−18.7165	−0.948966	−0.474483	−	0.880265i	$-0.657365\pi$
$389$	−18.7165	−0.948966	−0.474483	+	0.880265i	$0.657365\pi$
$390$	20.1522	1.02045
$391$	−6.74469	−0.341093
$392$	0	0
$393$	−22.1985	−1.11977
$394$	19.2782	0.971222
$395$	−33.1053	−1.66571
$396$	−7.03765	−0.353655
$397$	−21.8834	−1.09830	−0.549149	−	0.835725i	$-0.685048\pi$
$397$	−21.8834	−1.09830	−0.549149	+	0.835725i	$0.685048\pi$
$398$	−18.3713	−0.920870
$399$	0	0
$400$	−1.42017	−0.0710084
$401$	−31.3215	−1.56412	−0.782060	−	0.623203i	$-0.785831\pi$
$401$	−31.3215	−1.56412	−0.782060	+	0.623203i	$0.785831\pi$
$402$	−20.5240	−1.02364
$403$	9.99864	0.498068
$404$	−13.1613	−0.654800
$405$	−2.36369	−0.117453
$406$	0	0
$407$	17.3114	0.858095
$408$	8.44606	0.418142
$409$	−17.6392	−0.872200	−0.436100	−	0.899898i	$-0.643641\pi$
$409$	−17.6392	−0.872200	−0.436100	+	0.899898i	$0.643641\pi$
$410$	−5.07312	−0.250543
$411$	14.3569	0.708176
$412$	−26.5770	−1.30935
$413$	0	0
$414$	−2.23094	−0.109645
$415$	−20.6578	−1.01405
$416$	30.9671	1.51829
$417$	−14.6198	−0.715935
$418$	−43.7205	−2.13844
$419$	7.89174	0.385537	0.192768	−	0.981244i	$-0.438253\pi$
$419$	7.89174	0.385537	0.192768	+	0.981244i	$0.438253\pi$
$420$	0	0
$421$	−13.7342	−0.669364	−0.334682	−	0.942331i	$-0.608629\pi$
$421$	−13.7342	−0.669364	−0.334682	+	0.942331i	$0.608629\pi$
$422$	−31.6128	−1.53889
$423$	−3.62288	−0.176150
$424$	−10.9567	−0.532104
$425$	3.80906	0.184767
$426$	−14.4319	−0.699227
$427$	0	0
$428$	44.8542	2.16811
$429$	10.7256	0.517838
$430$	2.77586	0.133864
$431$	−3.47384	−0.167329	−0.0836646	−	0.996494i	$-0.526662\pi$
$431$	−3.47384	−0.167329	−0.0836646	+	0.996494i	$0.526662\pi$
$432$	−2.41924	−0.116396
$433$	−13.4315	−0.645476	−0.322738	−	0.946488i	$-0.604603\pi$
$433$	−13.4315	−0.645476	−0.322738	+	0.946488i	$0.604603\pi$
$434$	0	0
$435$	20.8761	1.00093
$436$	−4.01282	−0.192179
$437$	−7.84206	−0.375136
$438$	16.5694	0.791714
$439$	16.1538	0.770980	0.385490	−	0.922712i	$-0.374032\pi$
$439$	16.1538	0.770980	0.385490	+	0.922712i	$0.374032\pi$
$440$	8.30733	0.396036
$441$	0	0
$442$	−55.3210	−2.63135
$443$	30.6303	1.45529	0.727645	−	0.685954i	$-0.240615\pi$
$443$	30.6303	1.45529	0.727645	+	0.685954i	$0.240615\pi$
$444$	−16.7114	−0.793086
$445$	39.8343	1.88833
$446$	28.4909	1.34908
$447$	−20.7890	−0.983286
$448$	0	0
$449$	−19.0932	−0.901063	−0.450531	−	0.892761i	$-0.648765\pi$
$449$	−19.0932	−0.901063	−0.450531	+	0.892761i	$0.648765\pi$
$450$	1.25992	0.0593934
$451$	−2.70007	−0.127141
$452$	22.5929	1.06268
$453$	−9.41227	−0.442227
$454$	−37.3555	−1.75318
$455$	0	0
$456$	9.82025	0.459875
$457$	23.5772	1.10290	0.551448	−	0.834209i	$-0.314076\pi$
$457$	23.5772	1.10290	0.551448	+	0.834209i	$0.314076\pi$
$458$	−6.61125	−0.308923
$459$	6.48870	0.302867
$460$	6.40395	0.298586
$461$	−9.36375	−0.436113	−0.218057	−	0.975936i	$-0.569972\pi$
$461$	−9.36375	−0.436113	−0.218057	+	0.975936i	$0.569972\pi$
$462$	0	0
$463$	−7.27610	−0.338149	−0.169074	−	0.985603i	$-0.554078\pi$
$463$	−7.27610	−0.338149	−0.169074	+	0.985603i	$0.554078\pi$
$464$	21.3667	0.991925
$465$	5.94954	0.275904
$466$	−40.6677	−1.88389
$467$	−10.0800	−0.466445	−0.233222	−	0.972423i	$-0.574927\pi$
$467$	−10.0800	−0.466445	−0.233222	+	0.972423i	$0.574927\pi$
$468$	−10.3538	−0.478606
$469$	0	0
$470$	18.3793	0.847773
$471$	16.2722	0.749783
$472$	−16.4827	−0.758678
$473$	1.47740	0.0679307
$474$	30.0602	1.38071
$475$	4.42880	0.203207
$476$	0	0
$477$	−8.41749	−0.385411
$478$	21.4019	0.978900
$479$	−34.2087	−1.56304	−0.781518	−	0.623882i	$-0.785554\pi$
$479$	−34.2087	−1.56304	−0.781518	+	0.623882i	$0.785554\pi$
$480$	18.4265	0.841052
$481$	25.4687	1.16127
$482$	49.3998	2.25010
$483$	0	0
$484$	−9.66908	−0.439504
$485$	−17.6278	−0.800437
$486$	2.14627	0.0973568
$487$	−7.80193	−0.353539	−0.176770	−	0.984252i	$-0.556565\pi$
$487$	−7.80193	−0.353539	−0.176770	+	0.984252i	$0.556565\pi$
$488$	−5.55144	−0.251302
$489$	−4.56932	−0.206632
$490$	0	0
$491$	−0.678699	−0.0306292	−0.0153146	−	0.999883i	$-0.504875\pi$
$491$	−0.678699	−0.0306292	−0.0153146	+	0.999883i	$0.504875\pi$
$492$	2.60647	0.117509
$493$	−57.3082	−2.58103
$494$	−64.3218	−2.89398
$495$	6.38212	0.286855
$496$	6.08937	0.273421
$497$	0	0
$498$	18.7576	0.840550
$499$	−1.92512	−0.0861802	−0.0430901	−	0.999071i	$-0.513720\pi$
$499$	−1.92512	−0.0861802	−0.0430901	+	0.999071i	$0.513720\pi$
$500$	27.1878	1.21588
$501$	−2.84726	−0.127206
$502$	64.7309	2.88908
$503$	13.4830	0.601179	0.300590	−	0.953754i	$-0.402817\pi$
$503$	13.4830	0.601179	0.300590	+	0.953754i	$0.402817\pi$
$504$	0	0
$505$	11.9354	0.531118
$506$	6.02369	0.267786
$507$	2.77959	0.123446
$508$	−2.10927	−0.0935839
$509$	42.4828	1.88302	0.941510	−	0.336986i	$-0.109408\pi$
$509$	42.4828	1.88302	0.941510	+	0.336986i	$0.109408\pi$
$510$	−32.9179	−1.45763
$511$	0	0
$512$	25.1576	1.11182
$513$	7.54442	0.333094
$514$	−14.1481	−0.624045
$515$	24.1014	1.06204
$516$	−1.42618	−0.0627843
$517$	9.78202	0.430213
$518$	0	0
$519$	14.2178	0.624092
$520$	12.2218	0.535961
$521$	5.58156	0.244533	0.122266	−	0.992497i	$-0.460984\pi$
$521$	5.58156	0.244533	0.122266	+	0.992497i	$0.460984\pi$
$522$	−18.9558	−0.829674
$523$	−33.2749	−1.45501	−0.727505	−	0.686103i	$-0.759320\pi$
$523$	−33.2749	−1.45501	−0.727505	+	0.686103i	$0.759320\pi$
$524$	−57.8598	−2.52762
$525$	0	0
$526$	35.1016	1.53050
$527$	−16.3324	−0.711452
$528$	6.53211	0.284274
$529$	−21.9195	−0.953024
$530$	42.7029	1.85490
$531$	−12.6629	−0.549522
$532$	0	0
$533$	−3.97235	−0.172062
$534$	−36.1702	−1.56524
$535$	−40.6762	−1.75859
$536$	−12.4472	−0.537639
$537$	−0.557078	−0.0240397
$538$	−45.5488	−1.96375
$539$	0	0
$540$	−6.16090	−0.265123
$541$	−9.69717	−0.416914	−0.208457	−	0.978032i	$-0.566844\pi$
$541$	−9.69717	−0.416914	−0.208457	+	0.978032i	$0.566844\pi$
$542$	−4.22225	−0.181361
$543$	4.64100	0.199164
$544$	−50.5837	−2.16876
$545$	3.63904	0.155879
$546$	0	0
$547$	−18.9270	−0.809258	−0.404629	−	0.914481i	$-0.632599\pi$
$547$	−18.9270	−0.809258	−0.404629	+	0.914481i	$0.632599\pi$
$548$	37.4210	1.59855
$549$	−4.26490	−0.182022
$550$	−3.40188	−0.145057
$551$	−66.6323	−2.83863
$552$	−1.35301	−0.0575878
$553$	0	0
$554$	21.7837	0.925500
$555$	15.1547	0.643283
$556$	−38.1062	−1.61606
$557$	−1.78182	−0.0754983	−0.0377491	−	0.999287i	$-0.512019\pi$
$557$	−1.78182	−0.0754983	−0.0377491	+	0.999287i	$0.512019\pi$
$558$	−5.40229	−0.228697
$559$	2.17355	0.0919315
$560$	0	0
$561$	−17.5199	−0.739692
$562$	56.4522	2.38129
$563$	−17.8823	−0.753650	−0.376825	−	0.926284i	$-0.622984\pi$
$563$	−17.8823	−0.753650	−0.376825	+	0.926284i	$0.622984\pi$
$564$	−9.44294	−0.397620
$565$	−20.4884	−0.861955
$566$	50.2959	2.11409
$567$	0	0
$568$	−8.75256	−0.367249
$569$	0.523555	0.0219486	0.0109743	−	0.999940i	$-0.496507\pi$
$569$	0.523555	0.0219486	0.0109743	+	0.999940i	$0.496507\pi$
$570$	−38.2737	−1.60311
$571$	−0.00376852	−0.000157708 0	−7.88539e−5	−	1.00000i	$-0.500025\pi$
$571$	−0.00376852	−0.000157708 0	−7.88539e−5		1.00000i	$0.500025\pi$
$572$	27.9560	1.16890
$573$	13.5352	0.565440
$574$	0	0
$575$	−0.610189	−0.0254466
$576$	−11.8931	−0.495546
$577$	43.4350	1.80822	0.904111	−	0.427297i	$-0.140534\pi$
$577$	43.4350	1.80822	0.904111	+	0.427297i	$0.140534\pi$
$578$	53.8784	2.24105
$579$	−1.23342	−0.0512593
$580$	54.4130	2.25938
$581$	0	0
$582$	16.0063	0.663484
$583$	22.7278	0.941289
$584$	10.0489	0.415825
$585$	9.38941	0.388204
$586$	23.6724	0.977896
$587$	−44.5640	−1.83935	−0.919677	−	0.392675i	$-0.871550\pi$
$587$	−44.5640	−1.83935	−0.919677	+	0.392675i	$0.871550\pi$
$588$	0	0
$589$	−18.9898	−0.782459
$590$	64.2402	2.64473
$591$	8.98219	0.369478
$592$	15.5109	0.637495
$593$	−15.3565	−0.630617	−0.315308	−	0.948989i	$-0.602108\pi$
$593$	−15.3565	−0.630617	−0.315308	+	0.948989i	$0.602108\pi$
$594$	−5.79507	−0.237775
$595$	0	0
$596$	−54.1860	−2.21954
$597$	−8.55964	−0.350323
$598$	8.86209	0.362398
$599$	−29.7817	−1.21685	−0.608423	−	0.793613i	$-0.708198\pi$
$599$	−29.7817	−1.21685	−0.608423	+	0.793613i	$0.708198\pi$
$600$	0.764111	0.0311947
$601$	−25.9030	−1.05660	−0.528302	−	0.849057i	$-0.677171\pi$
$601$	−25.9030	−1.05660	−0.528302	+	0.849057i	$0.677171\pi$
$602$	0	0
$603$	−9.56262	−0.389420
$604$	−24.5328	−0.998227
$605$	8.76844	0.356488
$606$	−10.8375	−0.440245
$607$	−23.4600	−0.952211	−0.476105	−	0.879388i	$-0.657952\pi$
$607$	−23.4600	−0.952211	−0.476105	+	0.879388i	$0.657952\pi$
$608$	−58.8138	−2.38521
$609$	0	0
$610$	21.6363	0.876030
$611$	14.3914	0.582212
$612$	16.9126	0.683653
$613$	−48.3231	−1.95175	−0.975875	−	0.218330i	$-0.929939\pi$
$613$	−48.3231	−1.95175	−0.975875	+	0.218330i	$0.929939\pi$
$614$	42.6929	1.72294
$615$	−2.36369	−0.0953132
$616$	0	0
$617$	16.9455	0.682201	0.341100	−	0.940027i	$-0.389200\pi$
$617$	16.9455	0.682201	0.341100	+	0.940027i	$0.389200\pi$
$618$	−21.8845	−0.880324
$619$	9.43946	0.379404	0.189702	−	0.981842i	$-0.439248\pi$
$619$	9.43946	0.379404	0.189702	+	0.981842i	$0.439248\pi$
$620$	15.5073	0.622789
$621$	−1.03945	−0.0417117
$622$	36.2108	1.45192
$623$	0	0
$624$	9.61008	0.384711
$625$	−27.5905	−1.10362
$626$	44.8455	1.79239
$627$	−20.3705	−0.813517
$628$	42.4130	1.69246
$629$	−41.6022	−1.65879
$630$	0	0
$631$	11.9744	0.476694	0.238347	−	0.971180i	$-0.423394\pi$
$631$	11.9744	0.476694	0.238347	+	0.971180i	$0.423394\pi$
$632$	18.2307	0.725179
$633$	−14.7292	−0.585433
$634$	−69.8904	−2.77570
$635$	1.91280	0.0759073
$636$	−21.9400	−0.869977
$637$	0	0
$638$	51.1820	2.02632
$639$	−6.72417	−0.266004
$640$	23.4820	0.928209
$641$	16.4679	0.650444	0.325222	−	0.945638i	$-0.394561\pi$
$641$	16.4679	0.650444	0.325222	+	0.945638i	$0.394561\pi$
$642$	36.9347	1.45770
$643$	26.1074	1.02957	0.514787	−	0.857318i	$-0.327871\pi$
$643$	26.1074	1.02957	0.514787	+	0.857318i	$0.327871\pi$
$644$	0	0
$645$	1.29334	0.0509252
$646$	105.067	4.13382
$647$	−6.19131	−0.243405	−0.121703	−	0.992567i	$-0.538835\pi$
$647$	−6.19131	−0.243405	−0.121703	+	0.992567i	$0.538835\pi$
$648$	1.30166	0.0511339
$649$	34.1906	1.34210
$650$	−5.00487	−0.196307
$651$	0	0
$652$	−11.9098	−0.466424
$653$	−34.9689	−1.36844	−0.684219	−	0.729276i	$-0.739857\pi$
$653$	−34.9689	−1.36844	−0.684219	+	0.729276i	$0.739857\pi$
$654$	−3.30431	−0.129209
$655$	52.4704	2.05019
$656$	−2.41924	−0.0944555
$657$	7.72007	0.301189
$658$	0	0
$659$	−9.85808	−0.384016	−0.192008	−	0.981393i	$-0.561500\pi$
$659$	−9.85808	−0.384016	−0.192008	+	0.981393i	$0.561500\pi$
$660$	16.6348	0.647510
$661$	20.0383	0.779401	0.389700	−	0.920942i	$-0.372579\pi$
$661$	20.0383	0.779401	0.389700	+	0.920942i	$0.372579\pi$
$662$	29.4544	1.14478
$663$	−25.7754	−1.00103
$664$	11.3760	0.441475
$665$	0	0
$666$	−13.7608	−0.533219
$667$	9.18042	0.355467
$668$	−7.42130	−0.287139
$669$	13.2746	0.513226
$670$	48.5123	1.87419
$671$	11.5155	0.444552
$672$	0	0
$673$	23.8533	0.919478	0.459739	−	0.888054i	$-0.347943\pi$
$673$	23.8533	0.919478	0.459739	+	0.888054i	$0.347943\pi$
$674$	36.1272	1.39157
$675$	0.587030	0.0225948
$676$	7.24494	0.278652
$677$	−21.0909	−0.810588	−0.405294	−	0.914186i	$-0.632831\pi$
$677$	−21.0909	−0.810588	−0.405294	+	0.914186i	$0.632831\pi$
$678$	18.6038	0.714476
$679$	0	0
$680$	−19.9639	−0.765579
$681$	−17.4048	−0.666955
$682$	14.5865	0.558547
$683$	−15.0526	−0.575970	−0.287985	−	0.957635i	$-0.592985\pi$
$683$	−15.0526	−0.575970	−0.287985	+	0.957635i	$0.592985\pi$
$684$	19.6643	0.751885
$685$	−33.9354	−1.29660
$686$	0	0
$687$	−3.08034	−0.117522
$688$	1.32374	0.0504670
$689$	33.4373	1.27386
$690$	5.27325	0.200749
$691$	28.1067	1.06923	0.534615	−	0.845096i	$-0.320457\pi$
$691$	28.1067	1.06923	0.534615	+	0.845096i	$0.320457\pi$
$692$	37.0583	1.40874
$693$	0	0
$694$	−65.3952	−2.48237
$695$	34.5567	1.31081
$696$	−11.4962	−0.435763
$697$	6.48870	0.245777
$698$	61.8080	2.33947
$699$	−18.9481	−0.716682
$700$	0	0
$701$	19.3952	0.732545	0.366272	−	0.930508i	$-0.380634\pi$
$701$	19.3952	0.732545	0.366272	+	0.930508i	$0.380634\pi$
$702$	−8.52574	−0.321783
$703$	−48.3709	−1.82434
$704$	32.1122	1.21027
$705$	8.56336	0.322515
$706$	33.8748	1.27489
$707$	0	0
$708$	−33.0054	−1.24042
$709$	16.5021	0.619749	0.309874	−	0.950777i	$-0.399713\pi$
$709$	16.5021	0.619749	0.309874	+	0.950777i	$0.399713\pi$
$710$	34.1125	1.28022
$711$	14.0058	0.525258
$712$	−21.9363	−0.822097
$713$	2.61636	0.0979834
$714$	0	0
$715$	−25.3520	−0.948113
$716$	−1.45201	−0.0542641
$717$	9.97167	0.372399
$718$	−47.6827	−1.77950
$719$	−44.6880	−1.66658	−0.833291	−	0.552834i	$-0.813546\pi$
$719$	−44.6880	−1.66658	−0.833291	+	0.552834i	$0.813546\pi$
$720$	5.71834	0.213110
$721$	0	0
$722$	81.3830	3.02876
$723$	23.0166	0.855996
$724$	12.0966	0.449568
$725$	−5.18464	−0.192553
$726$	−7.96189	−0.295493
$727$	3.76452	0.139618	0.0698092	−	0.997560i	$-0.477761\pi$
$727$	3.76452	0.139618	0.0698092	+	0.997560i	$0.477761\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	−39.1648	−1.44955
$731$	−3.55042	−0.131317
$732$	−11.1164	−0.410872
$733$	2.11688	0.0781888	0.0390944	−	0.999236i	$-0.487553\pi$
$733$	2.11688	0.0781888	0.0390944	+	0.999236i	$0.487553\pi$
$734$	−12.1410	−0.448132
$735$	0	0
$736$	8.10320	0.298688
$737$	25.8197	0.951081
$738$	2.14627	0.0790053
$739$	−23.9700	−0.881750	−0.440875	−	0.897568i	$-0.645332\pi$
$739$	−23.9700	−0.881750	−0.440875	+	0.897568i	$0.645332\pi$
$740$	39.5005	1.45207
$741$	−29.9691	−1.10094
$742$	0	0
$743$	33.2447	1.21963	0.609814	−	0.792544i	$-0.291244\pi$
$743$	33.2447	1.21963	0.609814	+	0.792544i	$0.291244\pi$
$744$	−3.27634	−0.120117
$745$	49.1388	1.80030
$746$	34.6272	1.26779
$747$	8.73964	0.319767
$748$	−45.6652	−1.66969
$749$	0	0
$750$	22.3875	0.817476
$751$	20.3773	0.743579	0.371790	−	0.928317i	$-0.378744\pi$
$751$	20.3773	0.743579	0.371790	+	0.928317i	$0.378744\pi$
$752$	8.76462	0.319613
$753$	30.1597	1.09908
$754$	75.2993	2.74224
$755$	22.2477	0.809676
$756$	0	0
$757$	−13.4178	−0.487679	−0.243839	−	0.969816i	$-0.578407\pi$
$757$	−13.4178	−0.487679	−0.243839	+	0.969816i	$0.578407\pi$
$758$	−8.42083	−0.305858
$759$	2.80659	0.101873
$760$	−23.2120	−0.841989
$761$	−8.17818	−0.296459	−0.148229	−	0.988953i	$-0.547357\pi$
$761$	−8.17818	−0.296459	−0.148229	+	0.988953i	$0.547357\pi$
$762$	−1.73686	−0.0629197
$763$	0	0
$764$	35.2791	1.27635
$765$	−15.3373	−0.554521
$766$	−10.3618	−0.374388
$767$	50.3014	1.81628
$768$	2.46411	0.0889160
$769$	2.57529	0.0928673	0.0464336	−	0.998921i	$-0.485214\pi$
$769$	2.57529	0.0928673	0.0464336	+	0.998921i	$0.485214\pi$
$770$	0	0
$771$	−6.59194	−0.237403
$772$	−3.21489	−0.115706
$773$	2.59611	0.0933756	0.0466878	−	0.998910i	$-0.485133\pi$
$773$	2.59611	0.0933756	0.0466878	+	0.998910i	$0.485133\pi$
$774$	−1.17437	−0.0422120
$775$	−1.47759	−0.0530765
$776$	9.70742	0.348476
$777$	0	0
$778$	−40.1707	−1.44019
$779$	7.54442	0.270307
$780$	24.4733	0.876283
$781$	18.1557	0.649662
$782$	−14.4759	−0.517658
$783$	−8.83199	−0.315630
$784$	0	0
$785$	−38.4624	−1.37278
$786$	−47.6440	−1.69940
$787$	−1.96310	−0.0699769	−0.0349884	−	0.999388i	$-0.511139\pi$
$787$	−1.96310	−0.0699769	−0.0349884	+	0.999388i	$0.511139\pi$
$788$	23.4118	0.834013
$789$	16.3547	0.582243
$790$	−71.0530	−2.52795
$791$	0	0
$792$	−3.51456	−0.124884
$793$	16.9417	0.601617
$794$	−46.9677	−1.66682
$795$	19.8963	0.705651
$796$	−22.3105	−0.790774
$797$	17.6653	0.625735	0.312868	−	0.949797i	$-0.398710\pi$
$797$	17.6653	0.625735	0.312868	+	0.949797i	$0.398710\pi$
$798$	0	0
$799$	−23.5078	−0.831646
$800$	−4.57629	−0.161796
$801$	−16.8526	−0.595457
$802$	−67.2244	−2.37377
$803$	−20.8447	−0.735594
$804$	−24.9247	−0.879027
$805$	0	0
$806$	21.4598	0.755889
$807$	−21.2223	−0.747060
$808$	−6.57268	−0.231226
$809$	−27.2660	−0.958620	−0.479310	−	0.877646i	$-0.659113\pi$
$809$	−27.2660	−0.958620	−0.479310	+	0.877646i	$0.659113\pi$
$810$	−5.07312	−0.178251
$811$	−27.2425	−0.956613	−0.478306	−	0.878193i	$-0.658749\pi$
$811$	−27.2425	−0.956613	−0.478306	+	0.878193i	$0.658749\pi$
$812$	0	0
$813$	−1.96725	−0.0689944
$814$	37.1550	1.30228
$815$	10.8005	0.378323
$816$	−15.6977	−0.549531
$817$	−4.12808	−0.144423
$818$	−37.8584	−1.32369
$819$	0	0
$820$	−6.16090	−0.215148
$821$	3.83844	0.133963	0.0669813	−	0.997754i	$-0.478663\pi$
$821$	3.83844	0.133963	0.0669813	+	0.997754i	$0.478663\pi$
$822$	30.8139	1.07476
$823$	3.75854	0.131015	0.0655073	−	0.997852i	$-0.479133\pi$
$823$	3.75854	0.131015	0.0655073	+	0.997852i	$0.479133\pi$
$824$	−13.2724	−0.462365
$825$	−1.58502	−0.0551833
$826$	0	0
$827$	22.8573	0.794826	0.397413	−	0.917640i	$-0.369908\pi$
$827$	22.8573	0.794826	0.397413	+	0.917640i	$0.369908\pi$
$828$	−2.70930	−0.0941547
$829$	−39.5690	−1.37429	−0.687144	−	0.726521i	$-0.741136\pi$
$829$	−39.5690	−1.37429	−0.687144	+	0.726521i	$0.741136\pi$
$830$	−44.3372	−1.53897
$831$	10.1496	0.352084
$832$	47.2436	1.63788
$833$	0	0
$834$	−31.3781	−1.08653
$835$	6.73003	0.232902
$836$	−53.0950	−1.83633
$837$	−2.51706	−0.0870022
$838$	16.9378	0.585106
$839$	19.9208	0.687743	0.343871	−	0.939017i	$-0.388262\pi$
$839$	19.9208	0.687743	0.343871	+	0.939017i	$0.388262\pi$
$840$	0	0
$841$	49.0040	1.68979
$842$	−29.4773	−1.01585
$843$	26.3025	0.905906
$844$	−38.3913	−1.32148
$845$	−6.57010	−0.226018
$846$	−7.77568	−0.267333
$847$	0	0
$848$	20.3640	0.699301
$849$	23.4341	0.804256
$850$	8.17528	0.280410
$851$	6.66442	0.228453
$852$	−17.5264	−0.600443
$853$	−7.90604	−0.270698	−0.135349	−	0.990798i	$-0.543216\pi$
$853$	−7.90604	−0.270698	−0.135349	+	0.990798i	$0.543216\pi$
$854$	0	0
$855$	−17.8327	−0.609865
$856$	22.3999	0.765614
$857$	−35.6747	−1.21863	−0.609313	−	0.792930i	$-0.708555\pi$
$857$	−35.6747	−1.21863	−0.609313	+	0.792930i	$0.708555\pi$
$858$	23.0201	0.785892
$859$	34.1715	1.16592	0.582958	−	0.812503i	$-0.301895\pi$
$859$	34.1715	1.16592	0.582958	+	0.812503i	$0.301895\pi$
$860$	3.37106	0.114952
$861$	0	0
$862$	−7.45581	−0.253946
$863$	−51.5126	−1.75351	−0.876755	−	0.480937i	$-0.840297\pi$
$863$	−51.5126	−1.75351	−0.876755	+	0.480937i	$0.840297\pi$
$864$	−7.79566	−0.265214
$865$	−33.6064	−1.14265
$866$	−28.8276	−0.979602
$867$	25.1033	0.852552
$868$	0	0
$869$	−37.8166	−1.28284
$870$	44.8057	1.51906
$871$	37.9861	1.28711
$872$	−2.00398	−0.0678633
$873$	7.45775	0.252407
$874$	−16.8312	−0.569323
$875$	0	0
$876$	20.1222	0.679865
$877$	22.3831	0.755822	0.377911	−	0.925842i	$-0.376642\pi$
$877$	22.3831	0.755822	0.377911	+	0.925842i	$0.376642\pi$
$878$	34.6705	1.17007
$879$	11.0295	0.372017
$880$	−15.4399	−0.520479
$881$	22.3157	0.751835	0.375918	−	0.926653i	$-0.377328\pi$
$881$	22.3157	0.751835	0.375918	+	0.926653i	$0.377328\pi$
$882$	0	0
$883$	27.6469	0.930392	0.465196	−	0.885208i	$-0.345984\pi$
$883$	27.6469	0.930392	0.465196	+	0.885208i	$0.345984\pi$
$884$	−67.1830	−2.25961
$885$	29.9311	1.00612
$886$	65.7409	2.20861
$887$	9.84010	0.330398	0.165199	−	0.986260i	$-0.447173\pi$
$887$	9.84010	0.330398	0.165199	+	0.986260i	$0.447173\pi$
$888$	−8.34554	−0.280058
$889$	0	0
$890$	85.4951	2.86580
$891$	−2.70007	−0.0904556
$892$	34.5999	1.15849
$893$	−27.3325	−0.914649
$894$	−44.6188	−1.49228
$895$	1.31676	0.0440144
$896$	0	0
$897$	4.12907	0.137865
$898$	−40.9791	−1.36749
$899$	22.2306	0.741433
$900$	1.53008	0.0510026
$901$	−54.6186	−1.81961
$902$	−5.79507	−0.192955
$903$	0	0
$904$	11.2827	0.375258
$905$	−10.9699	−0.364651
$906$	−20.2013	−0.671142
$907$	10.7705	0.357627	0.178814	−	0.983883i	$-0.442774\pi$
$907$	10.7705	0.357627	0.178814	+	0.983883i	$0.442774\pi$
$908$	−45.3653	−1.50550
$909$	−5.04947	−0.167480
$910$	0	0
$911$	42.6611	1.41343	0.706713	−	0.707500i	$-0.250177\pi$
$911$	42.6611	1.41343	0.706713	+	0.707500i	$0.250177\pi$
$912$	−18.2518	−0.604377
$913$	−23.5976	−0.780967
$914$	50.6031	1.67380
$915$	10.0809	0.333264
$916$	−8.02884	−0.265280
$917$	0	0
$918$	13.9265	0.459643
$919$	−33.1114	−1.09224	−0.546122	−	0.837706i	$-0.683896\pi$
$919$	−33.1114	−1.09224	−0.546122	+	0.837706i	$0.683896\pi$
$920$	3.19809	0.105438
$921$	19.8917	0.655453
$922$	−20.0971	−0.661864
$923$	26.7108	0.879196
$924$	0	0
$925$	−3.76373	−0.123751
$926$	−15.6165	−0.513189
$927$	−10.1965	−0.334898
$928$	68.8512	2.26015
$929$	36.9948	1.21376	0.606881	−	0.794793i	$-0.292421\pi$
$929$	36.9948	1.21376	0.606881	+	0.794793i	$0.292421\pi$
$930$	12.7693	0.418723
$931$	0	0
$932$	−49.3877	−1.61775
$933$	16.8715	0.552348
$934$	−21.6343	−0.707896
$935$	41.4117	1.35431
$936$	−5.17064	−0.169008
$937$	10.1650	0.332076	0.166038	−	0.986119i	$-0.446903\pi$
$937$	10.1650	0.332076	0.166038	+	0.986119i	$0.446903\pi$
$938$	0	0
$939$	20.8946	0.681871
$940$	22.3202	0.728004
$941$	21.1350	0.688982	0.344491	−	0.938790i	$-0.388051\pi$
$941$	21.1350	0.688982	0.344491	+	0.938790i	$0.388051\pi$
$942$	34.9245	1.13790
$943$	−1.03945	−0.0338492
$944$	30.6346	0.997070
$945$	0	0
$946$	3.17089	0.103095
$947$	−33.0070	−1.07258	−0.536292	−	0.844032i	$-0.680176\pi$
$947$	−33.0070	−1.07258	−0.536292	+	0.844032i	$0.680176\pi$
$948$	36.5057	1.18565
$949$	−30.6668	−0.995488
$950$	9.50541	0.308396
$951$	−32.5637	−1.05595
$952$	0	0
$953$	10.9557	0.354889	0.177445	−	0.984131i	$-0.443217\pi$
$953$	10.9557	0.354889	0.177445	+	0.984131i	$0.443217\pi$
$954$	−18.0662	−0.584915
$955$	−31.9930	−1.03527
$956$	25.9909	0.840606
$957$	23.8470	0.770863
$958$	−73.4212	−2.37213
$959$	0	0
$960$	28.1116	0.907298
$961$	−24.6644	−0.795626
$962$	54.6626	1.76239
$963$	17.2088	0.554545
$964$	59.9921	1.93222
$965$	2.91543	0.0938510
$966$	0	0
$967$	−10.4174	−0.335002	−0.167501	−	0.985872i	$-0.553570\pi$
$967$	−10.4174	−0.335002	−0.167501	+	0.985872i	$0.553570\pi$
$968$	−4.82868	−0.155200
$969$	48.9535	1.57261
$970$	−37.8340	−1.21478
$971$	−12.6885	−0.407192	−0.203596	−	0.979055i	$-0.565263\pi$
$971$	−12.6885	−0.407192	−0.203596	+	0.979055i	$0.565263\pi$
$972$	2.60647	0.0836027
$973$	0	0
$974$	−16.7450	−0.536546
$975$	−2.33189	−0.0746803
$976$	10.3178	0.330266
$977$	−44.9608	−1.43842	−0.719212	−	0.694791i	$-0.755497\pi$
$977$	−44.9608	−1.43842	−0.719212	+	0.694791i	$0.755497\pi$
$978$	−9.80699	−0.313593
$979$	45.5031	1.45429
$980$	0	0
$981$	−1.53956	−0.0491544
$982$	−1.45667	−0.0464842
$983$	−28.4540	−0.907543	−0.453771	−	0.891118i	$-0.649922\pi$
$983$	−28.4540	−0.907543	−0.453771	+	0.891118i	$0.649922\pi$
$984$	1.30166	0.0414953
$985$	−21.2311	−0.676480
$986$	−122.999	−3.91708
$987$	0	0
$988$	−78.1137	−2.48513
$989$	0.568756	0.0180854
$990$	13.6978	0.435343
$991$	18.2267	0.578990	0.289495	−	0.957180i	$-0.406513\pi$
$991$	18.2267	0.578990	0.289495	+	0.957180i	$0.406513\pi$
$992$	19.6221	0.623003
$993$	13.7235	0.435504
$994$	0	0
$995$	20.2323	0.641408
$996$	22.7797	0.721801
$997$	−50.7037	−1.60580	−0.802900	−	0.596113i	$-0.796711\pi$
$997$	−50.7037	−1.60580	−0.802900	+	0.596113i	$0.796711\pi$
$998$	−4.13183	−0.130791
$999$	−6.41148	−0.202850

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6027.2.a.bm.1.16	yes	16
7.6	odd	2		6027.2.a.bl.1.16	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6027.2.a.bl.1.16	✓	16	7.6	odd	2
6027.2.a.bm.1.16	yes	16	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:	\( N \)	\(=\)	\( 6027 = 3 \cdot 7^{2} \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6027.a (trivial)

\(f(q)\)	\(=\)	\(q+2.14627 q^{2} +1.00000 q^{3} +2.60647 q^{4} -2.36369 q^{5} +2.14627 q^{6} +1.30166 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+2.14627 q^{2} +1.00000 q^{3} +2.60647 q^{4} -2.36369 q^{5} +2.14627 q^{6} +1.30166 q^{8} +1.00000 q^{9} -5.07312 q^{10} -2.70007 q^{11} +2.60647 q^{12} -3.97235 q^{13} -2.36369 q^{15} -2.41924 q^{16} +6.48870 q^{17} +2.14627 q^{18} +7.54442 q^{19} -6.16090 q^{20} -5.79507 q^{22} -1.03945 q^{23} +1.30166 q^{24} +0.587030 q^{25} -8.52574 q^{26} +1.00000 q^{27} -8.83199 q^{29} -5.07312 q^{30} -2.51706 q^{31} -7.79566 q^{32} -2.70007 q^{33} +13.9265 q^{34} +2.60647 q^{36} -6.41148 q^{37} +16.1924 q^{38} -3.97235 q^{39} -3.07671 q^{40} +1.00000 q^{41} -0.547170 q^{43} -7.03765 q^{44} -2.36369 q^{45} -2.23094 q^{46} -3.62288 q^{47} -2.41924 q^{48} +1.25992 q^{50} +6.48870 q^{51} -10.3538 q^{52} -8.41749 q^{53} +2.14627 q^{54} +6.38212 q^{55} +7.54442 q^{57} -18.9558 q^{58} -12.6629 q^{59} -6.16090 q^{60} -4.26490 q^{61} -5.40229 q^{62} -11.8931 q^{64} +9.38941 q^{65} -5.79507 q^{66} -9.56262 q^{67} +16.9126 q^{68} -1.03945 q^{69} -6.72417 q^{71} +1.30166 q^{72} +7.72007 q^{73} -13.7608 q^{74} +0.587030 q^{75} +19.6643 q^{76} -8.52574 q^{78} +14.0058 q^{79} +5.71834 q^{80} +1.00000 q^{81} +2.14627 q^{82} +8.73964 q^{83} -15.3373 q^{85} -1.17437 q^{86} -8.83199 q^{87} -3.51456 q^{88} -16.8526 q^{89} -5.07312 q^{90} -2.70930 q^{92} -2.51706 q^{93} -7.77568 q^{94} -17.8327 q^{95} -7.79566 q^{96} +7.45775 q^{97} -2.70007 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 4 q^{2} + 16 q^{3} + 12 q^{4} - 12 q^{5} - 4 q^{6} - 12 q^{8} + 16 q^{9}+O(q^{10}) \) 16 * q - 4 * q^2 + 16 * q^3 + 12 * q^4 - 12 * q^5 - 4 * q^6 - 12 * q^8 + 16 * q^9	\( 16 q - 4 q^{2} + 16 q^{3} + 12 q^{4} - 12 q^{5} - 4 q^{6} - 12 q^{8} + 16 q^{9} - 4 q^{10} - 4 q^{11} + 12 q^{12} - 12 q^{15} - 8 q^{17} - 4 q^{18} + 4 q^{19} - 20 q^{20} - 16 q^{22} - 12 q^{23} - 12 q^{24} - 8 q^{25} - 8 q^{26} + 16 q^{27} - 16 q^{29} - 4 q^{30} - 4 q^{31} - 48 q^{32} - 4 q^{33} + 16 q^{34} + 12 q^{36} - 48 q^{37} - 4 q^{38} + 56 q^{40} + 16 q^{41} - 16 q^{43} - 12 q^{45} - 4 q^{46} - 36 q^{47} - 8 q^{50} - 8 q^{51} - 60 q^{53} - 4 q^{54} + 8 q^{55} + 4 q^{57} - 36 q^{58} - 36 q^{59} - 20 q^{60} - 4 q^{61} - 12 q^{62} + 52 q^{64} - 36 q^{65} - 16 q^{66} - 52 q^{67} - 8 q^{68} - 12 q^{69} - 12 q^{71} - 12 q^{72} - 16 q^{73} + 4 q^{74} - 8 q^{75} + 16 q^{76} - 8 q^{78} - 36 q^{79} - 68 q^{80} + 16 q^{81} - 4 q^{82} - 32 q^{83} - 28 q^{85} - 8 q^{86} - 16 q^{87} - 36 q^{88} - 12 q^{89} - 4 q^{90} - 36 q^{92} - 4 q^{93} + 24 q^{94} - 20 q^{95} - 48 q^{96} + 48 q^{97} - 4 q^{99}+O(q^{100}) \) 16 * q - 4 * q^2 + 16 * q^3 + 12 * q^4 - 12 * q^5 - 4 * q^6 - 12 * q^8 + 16 * q^9 - 4 * q^10 - 4 * q^11 + 12 * q^12 - 12 * q^15 - 8 * q^17 - 4 * q^18 + 4 * q^19 - 20 * q^20 - 16 * q^22 - 12 * q^23 - 12 * q^24 - 8 * q^25 - 8 * q^26 + 16 * q^27 - 16 * q^29 - 4 * q^30 - 4 * q^31 - 48 * q^32 - 4 * q^33 + 16 * q^34 + 12 * q^36 - 48 * q^37 - 4 * q^38 + 56 * q^40 + 16 * q^41 - 16 * q^43 - 12 * q^45 - 4 * q^46 - 36 * q^47 - 8 * q^50 - 8 * q^51 - 60 * q^53 - 4 * q^54 + 8 * q^55 + 4 * q^57 - 36 * q^58 - 36 * q^59 - 20 * q^60 - 4 * q^61 - 12 * q^62 + 52 * q^64 - 36 * q^65 - 16 * q^66 - 52 * q^67 - 8 * q^68 - 12 * q^69 - 12 * q^71 - 12 * q^72 - 16 * q^73 + 4 * q^74 - 8 * q^75 + 16 * q^76 - 8 * q^78 - 36 * q^79 - 68 * q^80 + 16 * q^81 - 4 * q^82 - 32 * q^83 - 28 * q^85 - 8 * q^86 - 16 * q^87 - 36 * q^88 - 12 * q^89 - 4 * q^90 - 36 * q^92 - 4 * q^93 + 24 * q^94 - 20 * q^95 - 48 * q^96 + 48 * q^97 - 4 * q^99

Introduction

L-functions

Modular forms

Varieties

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Groups

Database

Properties

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