LMFDB - Embedded newform 6762.2.a.bs.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6762,2,Mod(1,6762)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6762, 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("6762.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6762 = 2 \cdot 3 \cdot 7^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6762.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:	$53.9948418468$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ of defining polynomial
Character	$\chi$	$=$	6762.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -0.414214 q^{5} +1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -0.414214 q^{5} +1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +0.414214 q^{10} -0.585786 q^{11} -1.00000 q^{12} +4.41421 q^{13} +0.414214 q^{15} +1.00000 q^{16} -1.17157 q^{17} -1.00000 q^{18} -6.24264 q^{19} -0.414214 q^{20} +0.585786 q^{22} +1.00000 q^{23} +1.00000 q^{24} -4.82843 q^{25} -4.41421 q^{26} -1.00000 q^{27} -4.07107 q^{29} -0.414214 q^{30} -0.585786 q^{31} -1.00000 q^{32} +0.585786 q^{33} +1.17157 q^{34} +1.00000 q^{36} -7.58579 q^{37} +6.24264 q^{38} -4.41421 q^{39} +0.414214 q^{40} +3.82843 q^{41} +2.65685 q^{43} -0.585786 q^{44} -0.414214 q^{45} -1.00000 q^{46} +7.58579 q^{47} -1.00000 q^{48} +4.82843 q^{50} +1.17157 q^{51} +4.41421 q^{52} +9.41421 q^{53} +1.00000 q^{54} +0.242641 q^{55} +6.24264 q^{57} +4.07107 q^{58} +8.24264 q^{59} +0.414214 q^{60} +11.8995 q^{61} +0.585786 q^{62} +1.00000 q^{64} -1.82843 q^{65} -0.585786 q^{66} -9.31371 q^{67} -1.17157 q^{68} -1.00000 q^{69} +15.8995 q^{71} -1.00000 q^{72} -6.24264 q^{73} +7.58579 q^{74} +4.82843 q^{75} -6.24264 q^{76} +4.41421 q^{78} -5.89949 q^{79} -0.414214 q^{80} +1.00000 q^{81} -3.82843 q^{82} -2.48528 q^{83} +0.485281 q^{85} -2.65685 q^{86} +4.07107 q^{87} +0.585786 q^{88} -3.75736 q^{89} +0.414214 q^{90} +1.00000 q^{92} +0.585786 q^{93} -7.58579 q^{94} +2.58579 q^{95} +1.00000 q^{96} -4.65685 q^{97} -0.585786 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{5} + 2 q^{6} - 2 q^{8} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 + 2 * q^5 + 2 * q^6 - 2 * q^8 + 2 * q^9	$ 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{5} + 2 q^{6} - 2 q^{8} + 2 q^{9} - 2 q^{10} - 4 q^{11} - 2 q^{12} + 6 q^{13} - 2 q^{15} + 2 q^{16} - 8 q^{17} - 2 q^{18} - 4 q^{19} + 2 q^{20} + 4 q^{22} + 2 q^{23} + 2 q^{24} - 4 q^{25} - 6 q^{26} - 2 q^{27} + 6 q^{29} + 2 q^{30} - 4 q^{31} - 2 q^{32} + 4 q^{33} + 8 q^{34} + 2 q^{36} - 18 q^{37} + 4 q^{38} - 6 q^{39} - 2 q^{40} + 2 q^{41} - 6 q^{43} - 4 q^{44} + 2 q^{45} - 2 q^{46} + 18 q^{47} - 2 q^{48} + 4 q^{50} + 8 q^{51} + 6 q^{52} + 16 q^{53} + 2 q^{54} - 8 q^{55} + 4 q^{57} - 6 q^{58} + 8 q^{59} - 2 q^{60} + 4 q^{61} + 4 q^{62} + 2 q^{64} + 2 q^{65} - 4 q^{66} + 4 q^{67} - 8 q^{68} - 2 q^{69} + 12 q^{71} - 2 q^{72} - 4 q^{73} + 18 q^{74} + 4 q^{75} - 4 q^{76} + 6 q^{78} + 8 q^{79} + 2 q^{80} + 2 q^{81} - 2 q^{82} + 12 q^{83} - 16 q^{85} + 6 q^{86} - 6 q^{87} + 4 q^{88} - 16 q^{89} - 2 q^{90} + 2 q^{92} + 4 q^{93} - 18 q^{94} + 8 q^{95} + 2 q^{96} + 2 q^{97} - 4 q^{99}+O(q^{100}) $ 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 + 2 * q^5 + 2 * q^6 - 2 * q^8 + 2 * q^9 - 2 * q^10 - 4 * q^11 - 2 * q^12 + 6 * q^13 - 2 * q^15 + 2 * q^16 - 8 * q^17 - 2 * q^18 - 4 * q^19 + 2 * q^20 + 4 * q^22 + 2 * q^23 + 2 * q^24 - 4 * q^25 - 6 * q^26 - 2 * q^27 + 6 * q^29 + 2 * q^30 - 4 * q^31 - 2 * q^32 + 4 * q^33 + 8 * q^34 + 2 * q^36 - 18 * q^37 + 4 * q^38 - 6 * q^39 - 2 * q^40 + 2 * q^41 - 6 * q^43 - 4 * q^44 + 2 * q^45 - 2 * q^46 + 18 * q^47 - 2 * q^48 + 4 * q^50 + 8 * q^51 + 6 * q^52 + 16 * q^53 + 2 * q^54 - 8 * q^55 + 4 * q^57 - 6 * q^58 + 8 * q^59 - 2 * q^60 + 4 * q^61 + 4 * q^62 + 2 * q^64 + 2 * q^65 - 4 * q^66 + 4 * q^67 - 8 * q^68 - 2 * q^69 + 12 * q^71 - 2 * q^72 - 4 * q^73 + 18 * q^74 + 4 * q^75 - 4 * q^76 + 6 * q^78 + 8 * q^79 + 2 * q^80 + 2 * q^81 - 2 * q^82 + 12 * q^83 - 16 * q^85 + 6 * q^86 - 6 * q^87 + 4 * q^88 - 16 * q^89 - 2 * q^90 + 2 * q^92 + 4 * q^93 - 18 * q^94 + 8 * q^95 + 2 * q^96 + 2 * 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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	1.00000	0.500000
$5$	−0.414214	−0.185242	−0.0926210	−	0.995701i	$-0.529524\pi$
$5$	−0.414214	−0.185242	−0.0926210	+	0.995701i	$0.529524\pi$
$6$	1.00000	0.408248
$7$	0	0
$7$	0	0
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	0.414214	0.130986
$11$	−0.585786	−0.176621	−0.0883106	−	0.996093i	$-0.528147\pi$
$11$	−0.585786	−0.176621	−0.0883106	+	0.996093i	$0.528147\pi$
$12$	−1.00000	−0.288675
$13$	4.41421	1.22428	0.612141	−	0.790748i	$-0.290308\pi$
$13$	4.41421	1.22428	0.612141	+	0.790748i	$0.290308\pi$
$14$	0	0
$15$	0.414214	0.106949
$16$	1.00000	0.250000
$17$	−1.17157	−0.284148	−0.142074	−	0.989856i	$-0.545377\pi$
$17$	−1.17157	−0.284148	−0.142074	+	0.989856i	$0.545377\pi$
$18$	−1.00000	−0.235702
$19$	−6.24264	−1.43216	−0.716080	−	0.698018i	$-0.754065\pi$
$19$	−6.24264	−1.43216	−0.716080	+	0.698018i	$0.754065\pi$
$20$	−0.414214	−0.0926210
$21$	0	0
$22$	0.585786	0.124890
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	1.00000	0.204124
$25$	−4.82843	−0.965685
$26$	−4.41421	−0.865699
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−4.07107	−0.755978	−0.377989	−	0.925810i	$-0.623384\pi$
$29$	−4.07107	−0.755978	−0.377989	+	0.925810i	$0.623384\pi$
$30$	−0.414214	−0.0756247
$31$	−0.585786	−0.105210	−0.0526052	−	0.998615i	$-0.516752\pi$
$31$	−0.585786	−0.105210	−0.0526052	+	0.998615i	$0.516752\pi$
$32$	−1.00000	−0.176777
$33$	0.585786	0.101972
$34$	1.17157	0.200923
$35$	0	0
$36$	1.00000	0.166667
$37$	−7.58579	−1.24710	−0.623548	−	0.781785i	$-0.714309\pi$
$37$	−7.58579	−1.24710	−0.623548	+	0.781785i	$0.714309\pi$
$38$	6.24264	1.01269
$39$	−4.41421	−0.706840
$40$	0.414214	0.0654929
$41$	3.82843	0.597900	0.298950	−	0.954269i	$-0.403364\pi$
$41$	3.82843	0.597900	0.298950	+	0.954269i	$0.403364\pi$
$42$	0	0
$43$	2.65685	0.405166	0.202583	−	0.979265i	$-0.435066\pi$
$43$	2.65685	0.405166	0.202583	+	0.979265i	$0.435066\pi$
$44$	−0.585786	−0.0883106
$45$	−0.414214	−0.0617473
$46$	−1.00000	−0.147442
$47$	7.58579	1.10650	0.553250	−	0.833015i	$-0.313387\pi$
$47$	7.58579	1.10650	0.553250	+	0.833015i	$0.313387\pi$
$48$	−1.00000	−0.144338
$49$	0	0
$50$	4.82843	0.682843
$51$	1.17157	0.164053
$52$	4.41421	0.612141
$53$	9.41421	1.29314	0.646571	−	0.762854i	$-0.276202\pi$
$53$	9.41421	1.29314	0.646571	+	0.762854i	$0.276202\pi$
$54$	1.00000	0.136083
$55$	0.242641	0.0327177
$56$	0	0
$57$	6.24264	0.826858
$58$	4.07107	0.534557
$59$	8.24264	1.07310	0.536550	−	0.843868i	$-0.319727\pi$
$59$	8.24264	1.07310	0.536550	+	0.843868i	$0.319727\pi$
$60$	0.414214	0.0534747
$61$	11.8995	1.52357	0.761787	−	0.647827i	$-0.224322\pi$
$61$	11.8995	1.52357	0.761787	+	0.647827i	$0.224322\pi$
$62$	0.585786	0.0743950
$63$	0	0
$64$	1.00000	0.125000
$65$	−1.82843	−0.226788
$66$	−0.585786	−0.0721053
$67$	−9.31371	−1.13785	−0.568925	−	0.822389i	$-0.692641\pi$
$67$	−9.31371	−1.13785	−0.568925	+	0.822389i	$0.692641\pi$
$68$	−1.17157	−0.142074
$69$	−1.00000	−0.120386
$70$	0	0
$71$	15.8995	1.88692	0.943461	−	0.331482i	$-0.107549\pi$
$71$	15.8995	1.88692	0.943461	+	0.331482i	$0.107549\pi$
$72$	−1.00000	−0.117851
$73$	−6.24264	−0.730646	−0.365323	−	0.930881i	$-0.619041\pi$
$73$	−6.24264	−0.730646	−0.365323	+	0.930881i	$0.619041\pi$
$74$	7.58579	0.881830
$75$	4.82843	0.557539
$76$	−6.24264	−0.716080
$77$	0	0
$78$	4.41421	0.499811
$79$	−5.89949	−0.663745	−0.331873	−	0.943324i	$-0.607680\pi$
$79$	−5.89949	−0.663745	−0.331873	+	0.943324i	$0.607680\pi$
$80$	−0.414214	−0.0463105
$81$	1.00000	0.111111
$82$	−3.82843	−0.422779
$83$	−2.48528	−0.272795	−0.136398	−	0.990654i	$-0.543552\pi$
$83$	−2.48528	−0.272795	−0.136398	+	0.990654i	$0.543552\pi$
$84$	0	0
$85$	0.485281	0.0526362
$86$	−2.65685	−0.286496
$87$	4.07107	0.436464
$88$	0.585786	0.0624450
$89$	−3.75736	−0.398279	−0.199140	−	0.979971i	$-0.563815\pi$
$89$	−3.75736	−0.398279	−0.199140	+	0.979971i	$0.563815\pi$
$90$	0.414214	0.0436619
$91$	0	0
$92$	1.00000	0.104257
$93$	0.585786	0.0607432
$94$	−7.58579	−0.782414
$95$	2.58579	0.265296
$96$	1.00000	0.102062
$97$	−4.65685	−0.472832	−0.236416	−	0.971652i	$-0.575973\pi$
$97$	−4.65685	−0.472832	−0.236416	+	0.971652i	$0.575973\pi$
$98$	0	0
$99$	−0.585786	−0.0588738
$100$	−4.82843	−0.482843
$101$	−0.585786	−0.0582879	−0.0291440	−	0.999575i	$-0.509278\pi$
$101$	−0.585786	−0.0582879	−0.0291440	+	0.999575i	$0.509278\pi$
$102$	−1.17157	−0.116003
$103$	−5.72792	−0.564389	−0.282194	−	0.959357i	$-0.591062\pi$
$103$	−5.72792	−0.564389	−0.282194	+	0.959357i	$0.591062\pi$
$104$	−4.41421	−0.432849
$105$	0	0
$106$	−9.41421	−0.914389
$107$	−1.65685	−0.160174	−0.0800871	−	0.996788i	$-0.525520\pi$
$107$	−1.65685	−0.160174	−0.0800871	+	0.996788i	$0.525520\pi$
$108$	−1.00000	−0.0962250
$109$	−17.5858	−1.68441	−0.842206	−	0.539155i	$-0.818744\pi$
$109$	−17.5858	−1.68441	−0.842206	+	0.539155i	$0.818744\pi$
$110$	−0.242641	−0.0231349
$111$	7.58579	0.720011
$112$	0	0
$113$	−7.48528	−0.704156	−0.352078	−	0.935971i	$-0.614525\pi$
$113$	−7.48528	−0.704156	−0.352078	+	0.935971i	$0.614525\pi$
$114$	−6.24264	−0.584677
$115$	−0.414214	−0.0386256
$116$	−4.07107	−0.377989
$117$	4.41421	0.408094
$118$	−8.24264	−0.758797
$119$	0	0
$120$	−0.414214	−0.0378124
$121$	−10.6569	−0.968805
$122$	−11.8995	−1.07733
$123$	−3.82843	−0.345198
$124$	−0.585786	−0.0526052
$125$	4.07107	0.364127
$126$	0	0
$127$	21.7279	1.92804	0.964021	−	0.265827i	$-0.0856451\pi$
$127$	21.7279	1.92804	0.964021	+	0.265827i	$0.0856451\pi$
$128$	−1.00000	−0.0883883
$129$	−2.65685	−0.233923
$130$	1.82843	0.160364
$131$	11.7574	1.02725	0.513623	−	0.858016i	$-0.328303\pi$
$131$	11.7574	1.02725	0.513623	+	0.858016i	$0.328303\pi$
$132$	0.585786	0.0509862
$133$	0	0
$134$	9.31371	0.804582
$135$	0.414214	0.0356498
$136$	1.17157	0.100462
$137$	−15.1421	−1.29368	−0.646840	−	0.762626i	$-0.723910\pi$
$137$	−15.1421	−1.29368	−0.646840	+	0.762626i	$0.723910\pi$
$138$	1.00000	0.0851257
$139$	9.34315	0.792475	0.396238	−	0.918148i	$-0.370316\pi$
$139$	9.34315	0.792475	0.396238	+	0.918148i	$0.370316\pi$
$140$	0	0
$141$	−7.58579	−0.638838
$142$	−15.8995	−1.33426
$143$	−2.58579	−0.216234
$144$	1.00000	0.0833333
$145$	1.68629	0.140039
$146$	6.24264	0.516645
$147$	0	0
$148$	−7.58579	−0.623548
$149$	−4.72792	−0.387326	−0.193663	−	0.981068i	$-0.562037\pi$
$149$	−4.72792	−0.387326	−0.193663	+	0.981068i	$0.562037\pi$
$150$	−4.82843	−0.394239
$151$	20.0711	1.63336	0.816680	−	0.577091i	$-0.195812\pi$
$151$	20.0711	1.63336	0.816680	+	0.577091i	$0.195812\pi$
$152$	6.24264	0.506345
$153$	−1.17157	−0.0947161
$154$	0	0
$155$	0.242641	0.0194894
$156$	−4.41421	−0.353420
$157$	23.0711	1.84127	0.920636	−	0.390422i	$-0.127671\pi$
$157$	23.0711	1.84127	0.920636	+	0.390422i	$0.127671\pi$
$158$	5.89949	0.469339
$159$	−9.41421	−0.746596
$160$	0.414214	0.0327465
$161$	0	0
$162$	−1.00000	−0.0785674
$163$	−14.8284	−1.16145	−0.580726	−	0.814099i	$-0.697231\pi$
$163$	−14.8284	−1.16145	−0.580726	+	0.814099i	$0.697231\pi$
$164$	3.82843	0.298950
$165$	−0.242641	−0.0188896
$166$	2.48528	0.192895
$167$	3.51472	0.271977	0.135989	−	0.990710i	$-0.456579\pi$
$167$	3.51472	0.271977	0.135989	+	0.990710i	$0.456579\pi$
$168$	0	0
$169$	6.48528	0.498868
$170$	−0.485281	−0.0372194
$171$	−6.24264	−0.477387
$172$	2.65685	0.202583
$173$	0.828427	0.0629841	0.0314921	−	0.999504i	$-0.489974\pi$
$173$	0.828427	0.0629841	0.0314921	+	0.999504i	$0.489974\pi$
$174$	−4.07107	−0.308627
$175$	0	0
$176$	−0.585786	−0.0441553
$177$	−8.24264	−0.619555
$178$	3.75736	0.281626
$179$	4.65685	0.348070	0.174035	−	0.984740i	$-0.444319\pi$
$179$	4.65685	0.348070	0.174035	+	0.984740i	$0.444319\pi$
$180$	−0.414214	−0.0308737
$181$	6.48528	0.482047	0.241024	−	0.970519i	$-0.422517\pi$
$181$	6.48528	0.482047	0.241024	+	0.970519i	$0.422517\pi$
$182$	0	0
$183$	−11.8995	−0.879636
$184$	−1.00000	−0.0737210
$185$	3.14214	0.231014
$186$	−0.585786	−0.0429519
$187$	0.686292	0.0501866
$188$	7.58579	0.553250
$189$	0	0
$190$	−2.58579	−0.187593
$191$	−2.00000	−0.144715	−0.0723575	−	0.997379i	$-0.523052\pi$
$191$	−2.00000	−0.144715	−0.0723575	+	0.997379i	$0.523052\pi$
$192$	−1.00000	−0.0721688
$193$	−22.3137	−1.60618	−0.803088	−	0.595861i	$-0.796811\pi$
$193$	−22.3137	−1.60618	−0.803088	+	0.595861i	$0.796811\pi$
$194$	4.65685	0.334343
$195$	1.82843	0.130936
$196$	0	0
$197$	23.7279	1.69054	0.845272	−	0.534336i	$-0.179438\pi$
$197$	23.7279	1.69054	0.845272	+	0.534336i	$0.179438\pi$
$198$	0.585786	0.0416300
$199$	−19.7279	−1.39848	−0.699238	−	0.714889i	$-0.746477\pi$
$199$	−19.7279	−1.39848	−0.699238	+	0.714889i	$0.746477\pi$
$200$	4.82843	0.341421
$201$	9.31371	0.656938
$202$	0.585786	0.0412158
$203$	0	0
$204$	1.17157	0.0820265
$205$	−1.58579	−0.110756
$206$	5.72792	0.399083
$207$	1.00000	0.0695048
$208$	4.41421	0.306071
$209$	3.65685	0.252950
$210$	0	0
$211$	−6.92893	−0.477007	−0.238504	−	0.971142i	$-0.576657\pi$
$211$	−6.92893	−0.477007	−0.238504	+	0.971142i	$0.576657\pi$
$212$	9.41421	0.646571
$213$	−15.8995	−1.08942
$214$	1.65685	0.113260
$215$	−1.10051	−0.0750538
$216$	1.00000	0.0680414
$217$	0	0
$218$	17.5858	1.19106
$219$	6.24264	0.421839
$220$	0.242641	0.0163588
$221$	−5.17157	−0.347878
$222$	−7.58579	−0.509125
$223$	11.6569	0.780601	0.390300	−	0.920688i	$-0.372371\pi$
$223$	11.6569	0.780601	0.390300	+	0.920688i	$0.372371\pi$
$224$	0	0
$225$	−4.82843	−0.321895
$226$	7.48528	0.497914
$227$	21.4853	1.42603	0.713014	−	0.701150i	$-0.247330\pi$
$227$	21.4853	1.42603	0.713014	+	0.701150i	$0.247330\pi$
$228$	6.24264	0.413429
$229$	6.34315	0.419167	0.209583	−	0.977791i	$-0.432789\pi$
$229$	6.34315	0.419167	0.209583	+	0.977791i	$0.432789\pi$
$230$	0.414214	0.0273124
$231$	0	0
$232$	4.07107	0.267279
$233$	−15.7574	−1.03230	−0.516149	−	0.856499i	$-0.672635\pi$
$233$	−15.7574	−1.03230	−0.516149	+	0.856499i	$0.672635\pi$
$234$	−4.41421	−0.288566
$235$	−3.14214	−0.204970
$236$	8.24264	0.536550
$237$	5.89949	0.383213
$238$	0	0
$239$	21.6569	1.40087	0.700433	−	0.713718i	$-0.252990\pi$
$239$	21.6569	1.40087	0.700433	+	0.713718i	$0.252990\pi$
$240$	0.414214	0.0267374
$241$	23.6274	1.52198	0.760988	−	0.648766i	$-0.224715\pi$
$241$	23.6274	1.52198	0.760988	+	0.648766i	$0.224715\pi$
$242$	10.6569	0.685049
$243$	−1.00000	−0.0641500
$244$	11.8995	0.761787
$245$	0	0
$246$	3.82843	0.244092
$247$	−27.5563	−1.75337
$248$	0.585786	0.0371975
$249$	2.48528	0.157498
$250$	−4.07107	−0.257477
$251$	−9.00000	−0.568075	−0.284037	−	0.958813i	$-0.591674\pi$
$251$	−9.00000	−0.568075	−0.284037	+	0.958813i	$0.591674\pi$
$252$	0	0
$253$	−0.585786	−0.0368281
$254$	−21.7279	−1.36333
$255$	−0.485281	−0.0303895
$256$	1.00000	0.0625000
$257$	−3.31371	−0.206703	−0.103352	−	0.994645i	$-0.532957\pi$
$257$	−3.31371	−0.206703	−0.103352	+	0.994645i	$0.532957\pi$
$258$	2.65685	0.165409
$259$	0	0
$260$	−1.82843	−0.113394
$261$	−4.07107	−0.251993
$262$	−11.7574	−0.726372
$263$	1.72792	0.106548	0.0532741	−	0.998580i	$-0.483034\pi$
$263$	1.72792	0.106548	0.0532741	+	0.998580i	$0.483034\pi$
$264$	−0.585786	−0.0360527
$265$	−3.89949	−0.239544
$266$	0	0
$267$	3.75736	0.229947
$268$	−9.31371	−0.568925
$269$	−19.2132	−1.17145	−0.585725	−	0.810510i	$-0.699190\pi$
$269$	−19.2132	−1.17145	−0.585725	+	0.810510i	$0.699190\pi$
$270$	−0.414214	−0.0252082
$271$	−0.242641	−0.0147394	−0.00736969	−	0.999973i	$-0.502346\pi$
$271$	−0.242641	−0.0147394	−0.00736969	+	0.999973i	$0.502346\pi$
$272$	−1.17157	−0.0710370
$273$	0	0
$274$	15.1421	0.914770
$275$	2.82843	0.170561
$276$	−1.00000	−0.0601929
$277$	10.5858	0.636038	0.318019	−	0.948084i	$-0.396982\pi$
$277$	10.5858	0.636038	0.318019	+	0.948084i	$0.396982\pi$
$278$	−9.34315	−0.560365
$279$	−0.585786	−0.0350701
$280$	0	0
$281$	27.2843	1.62764	0.813822	−	0.581115i	$-0.197383\pi$
$281$	27.2843	1.62764	0.813822	+	0.581115i	$0.197383\pi$
$282$	7.58579	0.451727
$283$	5.41421	0.321842	0.160921	−	0.986967i	$-0.448554\pi$
$283$	5.41421	0.321842	0.160921	+	0.986967i	$0.448554\pi$
$284$	15.8995	0.943461
$285$	−2.58579	−0.153169
$286$	2.58579	0.152901
$287$	0	0
$288$	−1.00000	−0.0589256
$289$	−15.6274	−0.919260
$290$	−1.68629	−0.0990224
$291$	4.65685	0.272990
$292$	−6.24264	−0.365323
$293$	−10.4853	−0.612557	−0.306278	−	0.951942i	$-0.599084\pi$
$293$	−10.4853	−0.612557	−0.306278	+	0.951942i	$0.599084\pi$
$294$	0	0
$295$	−3.41421	−0.198783
$296$	7.58579	0.440915
$297$	0.585786	0.0339908
$298$	4.72792	0.273881
$299$	4.41421	0.255281
$300$	4.82843	0.278769
$301$	0	0
$302$	−20.0711	−1.15496
$303$	0.585786	0.0336526
$304$	−6.24264	−0.358040
$305$	−4.92893	−0.282230
$306$	1.17157	0.0669744
$307$	19.9706	1.13978	0.569890	−	0.821721i	$-0.306986\pi$
$307$	19.9706	1.13978	0.569890	+	0.821721i	$0.306986\pi$
$308$	0	0
$309$	5.72792	0.325850
$310$	−0.242641	−0.0137811
$311$	−6.00000	−0.340229	−0.170114	−	0.985424i	$-0.554414\pi$
$311$	−6.00000	−0.340229	−0.170114	+	0.985424i	$0.554414\pi$
$312$	4.41421	0.249906
$313$	0.343146	0.0193957	0.00969787	−	0.999953i	$-0.496913\pi$
$313$	0.343146	0.0193957	0.00969787	+	0.999953i	$0.496913\pi$
$314$	−23.0711	−1.30198
$315$	0	0
$316$	−5.89949	−0.331873
$317$	23.7279	1.33269	0.666346	−	0.745642i	$-0.267857\pi$
$317$	23.7279	1.33269	0.666346	+	0.745642i	$0.267857\pi$
$318$	9.41421	0.527923
$319$	2.38478	0.133522
$320$	−0.414214	−0.0231552
$321$	1.65685	0.0924766
$322$	0	0
$323$	7.31371	0.406946
$324$	1.00000	0.0555556
$325$	−21.3137	−1.18227
$326$	14.8284	0.821271
$327$	17.5858	0.972496
$328$	−3.82843	−0.211390
$329$	0	0
$330$	0.242641	0.0133569
$331$	14.1421	0.777322	0.388661	−	0.921381i	$-0.372938\pi$
$331$	14.1421	0.777322	0.388661	+	0.921381i	$0.372938\pi$
$332$	−2.48528	−0.136398
$333$	−7.58579	−0.415699
$334$	−3.51472	−0.192317
$335$	3.85786	0.210778
$336$	0	0
$337$	−6.24264	−0.340058	−0.170029	−	0.985439i	$-0.554386\pi$
$337$	−6.24264	−0.340058	−0.170029	+	0.985439i	$0.554386\pi$
$338$	−6.48528	−0.352753
$339$	7.48528	0.406545
$340$	0.485281	0.0263181
$341$	0.343146	0.0185824
$342$	6.24264	0.337563
$343$	0	0
$344$	−2.65685	−0.143248
$345$	0.414214	0.0223005
$346$	−0.828427	−0.0445365
$347$	−5.82843	−0.312886	−0.156443	−	0.987687i	$-0.550003\pi$
$347$	−5.82843	−0.312886	−0.156443	+	0.987687i	$0.550003\pi$
$348$	4.07107	0.218232
$349$	14.8284	0.793748	0.396874	−	0.917873i	$-0.370095\pi$
$349$	14.8284	0.793748	0.396874	+	0.917873i	$0.370095\pi$
$350$	0	0
$351$	−4.41421	−0.235613
$352$	0.585786	0.0312225
$353$	18.5147	0.985439	0.492720	−	0.870188i	$-0.336003\pi$
$353$	18.5147	0.985439	0.492720	+	0.870188i	$0.336003\pi$
$354$	8.24264	0.438091
$355$	−6.58579	−0.349537
$356$	−3.75736	−0.199140
$357$	0	0
$358$	−4.65685	−0.246122
$359$	−15.0416	−0.793867	−0.396933	−	0.917847i	$-0.629926\pi$
$359$	−15.0416	−0.793867	−0.396933	+	0.917847i	$0.629926\pi$
$360$	0.414214	0.0218310
$361$	19.9706	1.05108
$362$	−6.48528	−0.340859
$363$	10.6569	0.559340
$364$	0	0
$365$	2.58579	0.135346
$366$	11.8995	0.621997
$367$	21.7279	1.13419	0.567094	−	0.823653i	$-0.308068\pi$
$367$	21.7279	1.13419	0.567094	+	0.823653i	$0.308068\pi$
$368$	1.00000	0.0521286
$369$	3.82843	0.199300
$370$	−3.14214	−0.163352
$371$	0	0
$372$	0.585786	0.0303716
$373$	6.00000	0.310668	0.155334	−	0.987862i	$-0.450355\pi$
$373$	6.00000	0.310668	0.155334	+	0.987862i	$0.450355\pi$
$374$	−0.686292	−0.0354873
$375$	−4.07107	−0.210229
$376$	−7.58579	−0.391207
$377$	−17.9706	−0.925531
$378$	0	0
$379$	−9.00000	−0.462299	−0.231149	−	0.972918i	$-0.574249\pi$
$379$	−9.00000	−0.462299	−0.231149	+	0.972918i	$0.574249\pi$
$380$	2.58579	0.132648
$381$	−21.7279	−1.11316
$382$	2.00000	0.102329
$383$	31.5563	1.61245	0.806227	−	0.591606i	$-0.201506\pi$
$383$	31.5563	1.61245	0.806227	+	0.591606i	$0.201506\pi$
$384$	1.00000	0.0510310
$385$	0	0
$386$	22.3137	1.13574
$387$	2.65685	0.135055
$388$	−4.65685	−0.236416
$389$	18.0000	0.912636	0.456318	−	0.889817i	$-0.349168\pi$
$389$	18.0000	0.912636	0.456318	+	0.889817i	$0.349168\pi$
$390$	−1.82843	−0.0925860
$391$	−1.17157	−0.0592490
$392$	0	0
$393$	−11.7574	−0.593080
$394$	−23.7279	−1.19540
$395$	2.44365	0.122953
$396$	−0.585786	−0.0294369
$397$	20.4853	1.02813	0.514063	−	0.857752i	$-0.328140\pi$
$397$	20.4853	1.02813	0.514063	+	0.857752i	$0.328140\pi$
$398$	19.7279	0.988871
$399$	0	0
$400$	−4.82843	−0.241421
$401$	−1.17157	−0.0585056	−0.0292528	−	0.999572i	$-0.509313\pi$
$401$	−1.17157	−0.0585056	−0.0292528	+	0.999572i	$0.509313\pi$
$402$	−9.31371	−0.464526
$403$	−2.58579	−0.128807
$404$	−0.585786	−0.0291440
$405$	−0.414214	−0.0205824
$406$	0	0
$407$	4.44365	0.220264
$408$	−1.17157	−0.0580015
$409$	4.10051	0.202757	0.101378	−	0.994848i	$-0.467675\pi$
$409$	4.10051	0.202757	0.101378	+	0.994848i	$0.467675\pi$
$410$	1.58579	0.0783164
$411$	15.1421	0.746906
$412$	−5.72792	−0.282194
$413$	0	0
$414$	−1.00000	−0.0491473
$415$	1.02944	0.0505331
$416$	−4.41421	−0.216425
$417$	−9.34315	−0.457536
$418$	−3.65685	−0.178863
$419$	32.1421	1.57025	0.785123	−	0.619340i	$-0.212600\pi$
$419$	32.1421	1.57025	0.785123	+	0.619340i	$0.212600\pi$
$420$	0	0
$421$	−19.2426	−0.937829	−0.468914	−	0.883244i	$-0.655355\pi$
$421$	−19.2426	−0.937829	−0.468914	+	0.883244i	$0.655355\pi$
$422$	6.92893	0.337295
$423$	7.58579	0.368834
$424$	−9.41421	−0.457195
$425$	5.65685	0.274398
$426$	15.8995	0.770333
$427$	0	0
$428$	−1.65685	−0.0800871
$429$	2.58579	0.124843
$430$	1.10051	0.0530711
$431$	27.7279	1.33561	0.667804	−	0.744338i	$-0.267235\pi$
$431$	27.7279	1.33561	0.667804	+	0.744338i	$0.267235\pi$
$432$	−1.00000	−0.0481125
$433$	9.14214	0.439343	0.219672	−	0.975574i	$-0.429501\pi$
$433$	9.14214	0.439343	0.219672	+	0.975574i	$0.429501\pi$
$434$	0	0
$435$	−1.68629	−0.0808515
$436$	−17.5858	−0.842206
$437$	−6.24264	−0.298626
$438$	−6.24264	−0.298285
$439$	8.58579	0.409777	0.204889	−	0.978785i	$-0.434317\pi$
$439$	8.58579	0.409777	0.204889	+	0.978785i	$0.434317\pi$
$440$	−0.242641	−0.0115674
$441$	0	0
$442$	5.17157	0.245987
$443$	4.17157	0.198197	0.0990987	−	0.995078i	$-0.468404\pi$
$443$	4.17157	0.198197	0.0990987	+	0.995078i	$0.468404\pi$
$444$	7.58579	0.360005
$445$	1.55635	0.0737780
$446$	−11.6569	−0.551968
$447$	4.72792	0.223623
$448$	0	0
$449$	−23.4558	−1.10695	−0.553475	−	0.832866i	$-0.686699\pi$
$449$	−23.4558	−1.10695	−0.553475	+	0.832866i	$0.686699\pi$
$450$	4.82843	0.227614
$451$	−2.24264	−0.105602
$452$	−7.48528	−0.352078
$453$	−20.0711	−0.943021
$454$	−21.4853	−1.00835
$455$	0	0
$456$	−6.24264	−0.292338
$457$	−7.31371	−0.342121	−0.171060	−	0.985261i	$-0.554719\pi$
$457$	−7.31371	−0.342121	−0.171060	+	0.985261i	$0.554719\pi$
$458$	−6.34315	−0.296396
$459$	1.17157	0.0546843
$460$	−0.414214	−0.0193128
$461$	−38.5269	−1.79438	−0.897189	−	0.441648i	$-0.854394\pi$
$461$	−38.5269	−1.79438	−0.897189	+	0.441648i	$0.854394\pi$
$462$	0	0
$463$	12.2721	0.570332	0.285166	−	0.958478i	$-0.407951\pi$
$463$	12.2721	0.570332	0.285166	+	0.958478i	$0.407951\pi$
$464$	−4.07107	−0.188995
$465$	−0.242641	−0.0112522
$466$	15.7574	0.729946
$467$	37.7696	1.74777	0.873883	−	0.486136i	$-0.161594\pi$
$467$	37.7696	1.74777	0.873883	+	0.486136i	$0.161594\pi$
$468$	4.41421	0.204047
$469$	0	0
$470$	3.14214	0.144936
$471$	−23.0711	−1.06306
$472$	−8.24264	−0.379398
$473$	−1.55635	−0.0715610
$474$	−5.89949	−0.270973
$475$	30.1421	1.38302
$476$	0	0
$477$	9.41421	0.431047
$478$	−21.6569	−0.990561
$479$	−14.9706	−0.684022	−0.342011	−	0.939696i	$-0.611108\pi$
$479$	−14.9706	−0.684022	−0.342011	+	0.939696i	$0.611108\pi$
$480$	−0.414214	−0.0189062
$481$	−33.4853	−1.52680
$482$	−23.6274	−1.07620
$483$	0	0
$484$	−10.6569	−0.484402
$485$	1.92893	0.0875883
$486$	1.00000	0.0453609
$487$	−13.7279	−0.622072	−0.311036	−	0.950398i	$-0.600676\pi$
$487$	−13.7279	−0.622072	−0.311036	+	0.950398i	$0.600676\pi$
$488$	−11.8995	−0.538665
$489$	14.8284	0.670565
$490$	0	0
$491$	−9.51472	−0.429393	−0.214697	−	0.976681i	$-0.568876\pi$
$491$	−9.51472	−0.429393	−0.214697	+	0.976681i	$0.568876\pi$
$492$	−3.82843	−0.172599
$493$	4.76955	0.214810
$494$	27.5563	1.23982
$495$	0.242641	0.0109059
$496$	−0.585786	−0.0263026
$497$	0	0
$498$	−2.48528	−0.111368
$499$	31.9411	1.42988	0.714941	−	0.699185i	$-0.246454\pi$
$499$	31.9411	1.42988	0.714941	+	0.699185i	$0.246454\pi$
$500$	4.07107	0.182064
$501$	−3.51472	−0.157026
$502$	9.00000	0.401690
$503$	20.1005	0.896237	0.448119	−	0.893974i	$-0.352094\pi$
$503$	20.1005	0.896237	0.448119	+	0.893974i	$0.352094\pi$
$504$	0	0
$505$	0.242641	0.0107974
$506$	0.585786	0.0260414
$507$	−6.48528	−0.288021
$508$	21.7279	0.964021
$509$	16.2426	0.719942	0.359971	−	0.932963i	$-0.382787\pi$
$509$	16.2426	0.719942	0.359971	+	0.932963i	$0.382787\pi$
$510$	0.485281	0.0214886
$511$	0	0
$512$	−1.00000	−0.0441942
$513$	6.24264	0.275619
$514$	3.31371	0.146161
$515$	2.37258	0.104548
$516$	−2.65685	−0.116961
$517$	−4.44365	−0.195432
$518$	0	0
$519$	−0.828427	−0.0363639
$520$	1.82843	0.0801818
$521$	−24.8284	−1.08775	−0.543877	−	0.839165i	$-0.683044\pi$
$521$	−24.8284	−1.08775	−0.543877	+	0.839165i	$0.683044\pi$
$522$	4.07107	0.178186
$523$	−18.9706	−0.829525	−0.414762	−	0.909930i	$-0.636135\pi$
$523$	−18.9706	−0.829525	−0.414762	+	0.909930i	$0.636135\pi$
$524$	11.7574	0.513623
$525$	0	0
$526$	−1.72792	−0.0753410
$527$	0.686292	0.0298953
$528$	0.585786	0.0254931
$529$	1.00000	0.0434783
$530$	3.89949	0.169383
$531$	8.24264	0.357700
$532$	0	0
$533$	16.8995	0.731998
$534$	−3.75736	−0.162597
$535$	0.686292	0.0296710
$536$	9.31371	0.402291
$537$	−4.65685	−0.200958
$538$	19.2132	0.828340
$539$	0	0
$540$	0.414214	0.0178249
$541$	22.6274	0.972829	0.486414	−	0.873728i	$-0.338305\pi$
$541$	22.6274	0.972829	0.486414	+	0.873728i	$0.338305\pi$
$542$	0.242641	0.0104223
$543$	−6.48528	−0.278310
$544$	1.17157	0.0502308
$545$	7.28427	0.312024
$546$	0	0
$547$	22.9706	0.982150	0.491075	−	0.871117i	$-0.336604\pi$
$547$	22.9706	0.982150	0.491075	+	0.871117i	$0.336604\pi$
$548$	−15.1421	−0.646840
$549$	11.8995	0.507858
$550$	−2.82843	−0.120605
$551$	25.4142	1.08268
$552$	1.00000	0.0425628
$553$	0	0
$554$	−10.5858	−0.449747
$555$	−3.14214	−0.133376
$556$	9.34315	0.396238
$557$	24.0000	1.01691	0.508456	−	0.861088i	$-0.330216\pi$
$557$	24.0000	1.01691	0.508456	+	0.861088i	$0.330216\pi$
$558$	0.585786	0.0247983
$559$	11.7279	0.496038
$560$	0	0
$561$	−0.686292	−0.0289752
$562$	−27.2843	−1.15092
$563$	−7.34315	−0.309477	−0.154738	−	0.987955i	$-0.549453\pi$
$563$	−7.34315	−0.309477	−0.154738	+	0.987955i	$0.549453\pi$
$564$	−7.58579	−0.319419
$565$	3.10051	0.130439
$566$	−5.41421	−0.227576
$567$	0	0
$568$	−15.8995	−0.667128
$569$	1.14214	0.0478808	0.0239404	−	0.999713i	$-0.492379\pi$
$569$	1.14214	0.0478808	0.0239404	+	0.999713i	$0.492379\pi$
$570$	2.58579	0.108307
$571$	−39.3137	−1.64523	−0.822614	−	0.568601i	$-0.807485\pi$
$571$	−39.3137	−1.64523	−0.822614	+	0.568601i	$0.807485\pi$
$572$	−2.58579	−0.108117
$573$	2.00000	0.0835512
$574$	0	0
$575$	−4.82843	−0.201359
$576$	1.00000	0.0416667
$577$	40.1421	1.67114	0.835569	−	0.549385i	$-0.185138\pi$
$577$	40.1421	1.67114	0.835569	+	0.549385i	$0.185138\pi$
$578$	15.6274	0.650015
$579$	22.3137	0.927326
$580$	1.68629	0.0700194
$581$	0	0
$582$	−4.65685	−0.193033
$583$	−5.51472	−0.228396
$584$	6.24264	0.258322
$585$	−1.82843	−0.0755962
$586$	10.4853	0.433143
$587$	−5.55635	−0.229335	−0.114668	−	0.993404i	$-0.536580\pi$
$587$	−5.55635	−0.229335	−0.114668	+	0.993404i	$0.536580\pi$
$588$	0	0
$589$	3.65685	0.150678
$590$	3.41421	0.140561
$591$	−23.7279	−0.976036
$592$	−7.58579	−0.311774
$593$	27.6274	1.13452	0.567261	−	0.823538i	$-0.308003\pi$
$593$	27.6274	1.13452	0.567261	+	0.823538i	$0.308003\pi$
$594$	−0.585786	−0.0240351
$595$	0	0
$596$	−4.72792	−0.193663
$597$	19.7279	0.807410
$598$	−4.41421	−0.180511
$599$	−2.38478	−0.0974393	−0.0487197	−	0.998812i	$-0.515514\pi$
$599$	−2.38478	−0.0974393	−0.0487197	+	0.998812i	$0.515514\pi$
$600$	−4.82843	−0.197120
$601$	−11.3137	−0.461496	−0.230748	−	0.973014i	$-0.574117\pi$
$601$	−11.3137	−0.461496	−0.230748	+	0.973014i	$0.574117\pi$
$602$	0	0
$603$	−9.31371	−0.379284
$604$	20.0711	0.816680
$605$	4.41421	0.179463
$606$	−0.585786	−0.0237959
$607$	−18.4853	−0.750294	−0.375147	−	0.926965i	$-0.622408\pi$
$607$	−18.4853	−0.750294	−0.375147	+	0.926965i	$0.622408\pi$
$608$	6.24264	0.253173
$609$	0	0
$610$	4.92893	0.199567
$611$	33.4853	1.35467
$612$	−1.17157	−0.0473580
$613$	−25.2426	−1.01954	−0.509770	−	0.860311i	$-0.670270\pi$
$613$	−25.2426	−1.01954	−0.509770	+	0.860311i	$0.670270\pi$
$614$	−19.9706	−0.805946
$615$	1.58579	0.0639451
$616$	0	0
$617$	28.6274	1.15250	0.576248	−	0.817275i	$-0.304516\pi$
$617$	28.6274	1.15250	0.576248	+	0.817275i	$0.304516\pi$
$618$	−5.72792	−0.230411
$619$	−28.7279	−1.15467	−0.577336	−	0.816506i	$-0.695908\pi$
$619$	−28.7279	−1.15467	−0.577336	+	0.816506i	$0.695908\pi$
$620$	0.242641	0.00974468
$621$	−1.00000	−0.0401286
$622$	6.00000	0.240578
$623$	0	0
$624$	−4.41421	−0.176710
$625$	22.4558	0.898234
$626$	−0.343146	−0.0137149
$627$	−3.65685	−0.146041
$628$	23.0711	0.920636
$629$	8.88730	0.354360
$630$	0	0
$631$	28.9289	1.15164	0.575821	−	0.817576i	$-0.304682\pi$
$631$	28.9289	1.15164	0.575821	+	0.817576i	$0.304682\pi$
$632$	5.89949	0.234669
$633$	6.92893	0.275400
$634$	−23.7279	−0.942356
$635$	−9.00000	−0.357154
$636$	−9.41421	−0.373298
$637$	0	0
$638$	−2.38478	−0.0944142
$639$	15.8995	0.628974
$640$	0.414214	0.0163732
$641$	40.6569	1.60585	0.802925	−	0.596081i	$-0.203276\pi$
$641$	40.6569	1.60585	0.802925	+	0.596081i	$0.203276\pi$
$642$	−1.65685	−0.0653908
$643$	−5.79899	−0.228690	−0.114345	−	0.993441i	$-0.536477\pi$
$643$	−5.79899	−0.228690	−0.114345	+	0.993441i	$0.536477\pi$
$644$	0	0
$645$	1.10051	0.0433323
$646$	−7.31371	−0.287754
$647$	26.8284	1.05473	0.527367	−	0.849638i	$-0.323179\pi$
$647$	26.8284	1.05473	0.527367	+	0.849638i	$0.323179\pi$
$648$	−1.00000	−0.0392837
$649$	−4.82843	−0.189532
$650$	21.3137	0.835992
$651$	0	0
$652$	−14.8284	−0.580726
$653$	−14.4142	−0.564072	−0.282036	−	0.959404i	$-0.591010\pi$
$653$	−14.4142	−0.564072	−0.282036	+	0.959404i	$0.591010\pi$
$654$	−17.5858	−0.687659
$655$	−4.87006	−0.190289
$656$	3.82843	0.149475
$657$	−6.24264	−0.243549
$658$	0	0
$659$	−12.5858	−0.490273	−0.245136	−	0.969489i	$-0.578833\pi$
$659$	−12.5858	−0.490273	−0.245136	+	0.969489i	$0.578833\pi$
$660$	−0.242641	−0.00944478
$661$	36.1421	1.40577	0.702883	−	0.711305i	$-0.251896\pi$
$661$	36.1421	1.40577	0.702883	+	0.711305i	$0.251896\pi$
$662$	−14.1421	−0.549650
$663$	5.17157	0.200847
$664$	2.48528	0.0964476
$665$	0	0
$666$	7.58579	0.293943
$667$	−4.07107	−0.157632
$668$	3.51472	0.135989
$669$	−11.6569	−0.450680
$670$	−3.85786	−0.149042
$671$	−6.97056	−0.269096
$672$	0	0
$673$	−28.1127	−1.08366	−0.541832	−	0.840487i	$-0.682269\pi$
$673$	−28.1127	−1.08366	−0.541832	+	0.840487i	$0.682269\pi$
$674$	6.24264	0.240458
$675$	4.82843	0.185846
$676$	6.48528	0.249434
$677$	−14.2843	−0.548989	−0.274495	−	0.961589i	$-0.588511\pi$
$677$	−14.2843	−0.548989	−0.274495	+	0.961589i	$0.588511\pi$
$678$	−7.48528	−0.287470
$679$	0	0
$680$	−0.485281	−0.0186097
$681$	−21.4853	−0.823318
$682$	−0.343146	−0.0131397
$683$	11.1716	0.427468	0.213734	−	0.976892i	$-0.431437\pi$
$683$	11.1716	0.427468	0.213734	+	0.976892i	$0.431437\pi$
$684$	−6.24264	−0.238693
$685$	6.27208	0.239644
$686$	0	0
$687$	−6.34315	−0.242006
$688$	2.65685	0.101292
$689$	41.5563	1.58317
$690$	−0.414214	−0.0157688
$691$	39.2843	1.49444	0.747222	−	0.664574i	$-0.231387\pi$
$691$	39.2843	1.49444	0.747222	+	0.664574i	$0.231387\pi$
$692$	0.828427	0.0314921
$693$	0	0
$694$	5.82843	0.221244
$695$	−3.87006	−0.146800
$696$	−4.07107	−0.154313
$697$	−4.48528	−0.169892
$698$	−14.8284	−0.561264
$699$	15.7574	0.595998
$700$	0	0
$701$	−3.79899	−0.143486	−0.0717429	−	0.997423i	$-0.522856\pi$
$701$	−3.79899	−0.143486	−0.0717429	+	0.997423i	$0.522856\pi$
$702$	4.41421	0.166604
$703$	47.3553	1.78604
$704$	−0.585786	−0.0220777
$705$	3.14214	0.118340
$706$	−18.5147	−0.696811
$707$	0	0
$708$	−8.24264	−0.309777
$709$	−2.20101	−0.0826607	−0.0413303	−	0.999146i	$-0.513160\pi$
$709$	−2.20101	−0.0826607	−0.0413303	+	0.999146i	$0.513160\pi$
$710$	6.58579	0.247160
$711$	−5.89949	−0.221248
$712$	3.75736	0.140813
$713$	−0.585786	−0.0219379
$714$	0	0
$715$	1.07107	0.0400557
$716$	4.65685	0.174035
$717$	−21.6569	−0.808790
$718$	15.0416	0.561349
$719$	−30.8995	−1.15236	−0.576178	−	0.817324i	$-0.695457\pi$
$719$	−30.8995	−1.15236	−0.576178	+	0.817324i	$0.695457\pi$
$720$	−0.414214	−0.0154368
$721$	0	0
$722$	−19.9706	−0.743227
$723$	−23.6274	−0.878713
$724$	6.48528	0.241024
$725$	19.6569	0.730037
$726$	−10.6569	−0.395513
$727$	28.6274	1.06173	0.530866	−	0.847456i	$-0.321867\pi$
$727$	28.6274	1.06173	0.530866	+	0.847456i	$0.321867\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	−2.58579	−0.0957042
$731$	−3.11270	−0.115127
$732$	−11.8995	−0.439818
$733$	−0.727922	−0.0268864	−0.0134432	−	0.999910i	$-0.504279\pi$
$733$	−0.727922	−0.0268864	−0.0134432	+	0.999910i	$0.504279\pi$
$734$	−21.7279	−0.801992
$735$	0	0
$736$	−1.00000	−0.0368605
$737$	5.45584	0.200969
$738$	−3.82843	−0.140926
$739$	−7.27208	−0.267508	−0.133754	−	0.991015i	$-0.542703\pi$
$739$	−7.27208	−0.267508	−0.133754	+	0.991015i	$0.542703\pi$
$740$	3.14214	0.115507
$741$	27.5563	1.01231
$742$	0	0
$743$	37.9411	1.39192	0.695962	−	0.718078i	$-0.254978\pi$
$743$	37.9411	1.39192	0.695962	+	0.718078i	$0.254978\pi$
$744$	−0.585786	−0.0214760
$745$	1.95837	0.0717491
$746$	−6.00000	−0.219676
$747$	−2.48528	−0.0909317
$748$	0.686292	0.0250933
$749$	0	0
$750$	4.07107	0.148654
$751$	−3.07107	−0.112065	−0.0560324	−	0.998429i	$-0.517845\pi$
$751$	−3.07107	−0.112065	−0.0560324	+	0.998429i	$0.517845\pi$
$752$	7.58579	0.276625
$753$	9.00000	0.327978
$754$	17.9706	0.654449
$755$	−8.31371	−0.302567
$756$	0	0
$757$	29.4558	1.07059	0.535295	−	0.844665i	$-0.320200\pi$
$757$	29.4558	1.07059	0.535295	+	0.844665i	$0.320200\pi$
$758$	9.00000	0.326895
$759$	0.585786	0.0212627
$760$	−2.58579	−0.0937963
$761$	−7.31371	−0.265122	−0.132561	−	0.991175i	$-0.542320\pi$
$761$	−7.31371	−0.265122	−0.132561	+	0.991175i	$0.542320\pi$
$762$	21.7279	0.787120
$763$	0	0
$764$	−2.00000	−0.0723575
$765$	0.485281	0.0175454
$766$	−31.5563	−1.14018
$767$	36.3848	1.31378
$768$	−1.00000	−0.0360844
$769$	−31.6863	−1.14264	−0.571318	−	0.820728i	$-0.693568\pi$
$769$	−31.6863	−1.14264	−0.571318	+	0.820728i	$0.693568\pi$
$770$	0	0
$771$	3.31371	0.119340
$772$	−22.3137	−0.803088
$773$	49.7279	1.78859	0.894295	−	0.447479i	$-0.147678\pi$
$773$	49.7279	1.78859	0.894295	+	0.447479i	$0.147678\pi$
$774$	−2.65685	−0.0954987
$775$	2.82843	0.101600
$776$	4.65685	0.167171
$777$	0	0
$778$	−18.0000	−0.645331
$779$	−23.8995	−0.856288
$780$	1.82843	0.0654682
$781$	−9.31371	−0.333271
$782$	1.17157	0.0418954
$783$	4.07107	0.145488
$784$	0	0
$785$	−9.55635	−0.341081
$786$	11.7574	0.419371
$787$	−13.7574	−0.490397	−0.245198	−	0.969473i	$-0.578853\pi$
$787$	−13.7574	−0.490397	−0.245198	+	0.969473i	$0.578853\pi$
$788$	23.7279	0.845272
$789$	−1.72792	−0.0615157
$790$	−2.44365	−0.0869412
$791$	0	0
$792$	0.585786	0.0208150
$793$	52.5269	1.86529
$794$	−20.4853	−0.726995
$795$	3.89949	0.138301
$796$	−19.7279	−0.699238
$797$	−20.5563	−0.728143	−0.364072	−	0.931371i	$-0.618614\pi$
$797$	−20.5563	−0.728143	−0.364072	+	0.931371i	$0.618614\pi$
$798$	0	0
$799$	−8.88730	−0.314410
$800$	4.82843	0.170711
$801$	−3.75736	−0.132760
$802$	1.17157	0.0413697
$803$	3.65685	0.129048
$804$	9.31371	0.328469
$805$	0	0
$806$	2.58579	0.0910804
$807$	19.2132	0.676337
$808$	0.585786	0.0206079
$809$	−7.89949	−0.277731	−0.138866	−	0.990311i	$-0.544346\pi$
$809$	−7.89949	−0.277731	−0.138866	+	0.990311i	$0.544346\pi$
$810$	0.414214	0.0145540
$811$	26.1716	0.919008	0.459504	−	0.888176i	$-0.348027\pi$
$811$	26.1716	0.919008	0.459504	+	0.888176i	$0.348027\pi$
$812$	0	0
$813$	0.242641	0.00850978
$814$	−4.44365	−0.155750
$815$	6.14214	0.215150
$816$	1.17157	0.0410133
$817$	−16.5858	−0.580263
$818$	−4.10051	−0.143371
$819$	0	0
$820$	−1.58579	−0.0553781
$821$	45.4558	1.58642	0.793210	−	0.608948i	$-0.208408\pi$
$821$	45.4558	1.58642	0.793210	+	0.608948i	$0.208408\pi$
$822$	−15.1421	−0.528143
$823$	10.4142	0.363017	0.181508	−	0.983389i	$-0.441902\pi$
$823$	10.4142	0.363017	0.181508	+	0.983389i	$0.441902\pi$
$824$	5.72792	0.199542
$825$	−2.82843	−0.0984732
$826$	0	0
$827$	−19.9411	−0.693421	−0.346710	−	0.937972i	$-0.612701\pi$
$827$	−19.9411	−0.693421	−0.346710	+	0.937972i	$0.612701\pi$
$828$	1.00000	0.0347524
$829$	31.4558	1.09251	0.546253	−	0.837620i	$-0.316054\pi$
$829$	31.4558	1.09251	0.546253	+	0.837620i	$0.316054\pi$
$830$	−1.02944	−0.0357323
$831$	−10.5858	−0.367217
$832$	4.41421	0.153035
$833$	0	0
$834$	9.34315	0.323527
$835$	−1.45584	−0.0503816
$836$	3.65685	0.126475
$837$	0.585786	0.0202477
$838$	−32.1421	−1.11033
$839$	−24.3848	−0.841856	−0.420928	−	0.907094i	$-0.638295\pi$
$839$	−24.3848	−0.841856	−0.420928	+	0.907094i	$0.638295\pi$
$840$	0	0
$841$	−12.4264	−0.428497
$842$	19.2426	0.663145
$843$	−27.2843	−0.939720
$844$	−6.92893	−0.238504
$845$	−2.68629	−0.0924112
$846$	−7.58579	−0.260805
$847$	0	0
$848$	9.41421	0.323285
$849$	−5.41421	−0.185815
$850$	−5.65685	−0.194029
$851$	−7.58579	−0.260037
$852$	−15.8995	−0.544708
$853$	20.0711	0.687220	0.343610	−	0.939112i	$-0.388350\pi$
$853$	20.0711	0.687220	0.343610	+	0.939112i	$0.388350\pi$
$854$	0	0
$855$	2.58579	0.0884320
$856$	1.65685	0.0566301
$857$	−9.00000	−0.307434	−0.153717	−	0.988115i	$-0.549124\pi$
$857$	−9.00000	−0.307434	−0.153717	+	0.988115i	$0.549124\pi$
$858$	−2.58579	−0.0882773
$859$	−15.6274	−0.533201	−0.266600	−	0.963807i	$-0.585900\pi$
$859$	−15.6274	−0.533201	−0.266600	+	0.963807i	$0.585900\pi$
$860$	−1.10051	−0.0375269
$861$	0	0
$862$	−27.7279	−0.944417
$863$	10.3431	0.352085	0.176042	−	0.984383i	$-0.443670\pi$
$863$	10.3431	0.352085	0.176042	+	0.984383i	$0.443670\pi$
$864$	1.00000	0.0340207
$865$	−0.343146	−0.0116673
$866$	−9.14214	−0.310662
$867$	15.6274	0.530735
$868$	0	0
$869$	3.45584	0.117231
$870$	1.68629	0.0571706
$871$	−41.1127	−1.39305
$872$	17.5858	0.595530
$873$	−4.65685	−0.157611
$874$	6.24264	0.211160
$875$	0	0
$876$	6.24264	0.210919
$877$	13.8579	0.467947	0.233973	−	0.972243i	$-0.424827\pi$
$877$	13.8579	0.467947	0.233973	+	0.972243i	$0.424827\pi$
$878$	−8.58579	−0.289756
$879$	10.4853	0.353660
$880$	0.242641	0.00817942
$881$	−49.1127	−1.65465	−0.827324	−	0.561724i	$-0.810138\pi$
$881$	−49.1127	−1.65465	−0.827324	+	0.561724i	$0.810138\pi$
$882$	0	0
$883$	10.5858	0.356240	0.178120	−	0.984009i	$-0.442998\pi$
$883$	10.5858	0.356240	0.178120	+	0.984009i	$0.442998\pi$
$884$	−5.17157	−0.173939
$885$	3.41421	0.114768
$886$	−4.17157	−0.140147
$887$	48.0833	1.61448	0.807239	−	0.590225i	$-0.200961\pi$
$887$	48.0833	1.61448	0.807239	+	0.590225i	$0.200961\pi$
$888$	−7.58579	−0.254562
$889$	0	0
$890$	−1.55635	−0.0521689
$891$	−0.585786	−0.0196246
$892$	11.6569	0.390300
$893$	−47.3553	−1.58469
$894$	−4.72792	−0.158125
$895$	−1.92893	−0.0644771
$896$	0	0
$897$	−4.41421	−0.147386
$898$	23.4558	0.782732
$899$	2.38478	0.0795367
$900$	−4.82843	−0.160948
$901$	−11.0294	−0.367444
$902$	2.24264	0.0746718
$903$	0	0
$904$	7.48528	0.248957
$905$	−2.68629	−0.0892954
$906$	20.0711	0.666817
$907$	13.8284	0.459165	0.229583	−	0.973289i	$-0.426264\pi$
$907$	13.8284	0.459165	0.229583	+	0.973289i	$0.426264\pi$
$908$	21.4853	0.713014
$909$	−0.585786	−0.0194293
$910$	0	0
$911$	7.78680	0.257988	0.128994	−	0.991645i	$-0.458825\pi$
$911$	7.78680	0.257988	0.128994	+	0.991645i	$0.458825\pi$
$912$	6.24264	0.206714
$913$	1.45584	0.0481814
$914$	7.31371	0.241916
$915$	4.92893	0.162945
$916$	6.34315	0.209583
$917$	0	0
$918$	−1.17157	−0.0386677
$919$	−58.4264	−1.92731	−0.963655	−	0.267151i	$-0.913918\pi$
$919$	−58.4264	−1.92731	−0.963655	+	0.267151i	$0.913918\pi$
$920$	0.414214	0.0136562
$921$	−19.9706	−0.658052
$922$	38.5269	1.26882
$923$	70.1838	2.31013
$924$	0	0
$925$	36.6274	1.20430
$926$	−12.2721	−0.403286
$927$	−5.72792	−0.188130
$928$	4.07107	0.133639
$929$	55.4853	1.82041	0.910207	−	0.414155i	$-0.135923\pi$
$929$	55.4853	1.82041	0.910207	+	0.414155i	$0.135923\pi$
$930$	0.242641	0.00795650
$931$	0	0
$932$	−15.7574	−0.516149
$933$	6.00000	0.196431
$934$	−37.7696	−1.23586
$935$	−0.284271	−0.00929666
$936$	−4.41421	−0.144283
$937$	10.6569	0.348144	0.174072	−	0.984733i	$-0.444307\pi$
$937$	10.6569	0.348144	0.174072	+	0.984733i	$0.444307\pi$
$938$	0	0
$939$	−0.343146	−0.0111981
$940$	−3.14214	−0.102485
$941$	50.0122	1.63035	0.815175	−	0.579214i	$-0.196640\pi$
$941$	50.0122	1.63035	0.815175	+	0.579214i	$0.196640\pi$
$942$	23.0711	0.751696
$943$	3.82843	0.124671
$944$	8.24264	0.268275
$945$	0	0
$946$	1.55635	0.0506013
$947$	−2.02944	−0.0659478	−0.0329739	−	0.999456i	$-0.510498\pi$
$947$	−2.02944	−0.0659478	−0.0329739	+	0.999456i	$0.510498\pi$
$948$	5.89949	0.191607
$949$	−27.5563	−0.894517
$950$	−30.1421	−0.977940
$951$	−23.7279	−0.769431
$952$	0	0
$953$	−15.3137	−0.496060	−0.248030	−	0.968752i	$-0.579783\pi$
$953$	−15.3137	−0.496060	−0.248030	+	0.968752i	$0.579783\pi$
$954$	−9.41421	−0.304796
$955$	0.828427	0.0268073
$956$	21.6569	0.700433
$957$	−2.38478	−0.0770889
$958$	14.9706	0.483677
$959$	0	0
$960$	0.414214	0.0133687
$961$	−30.6569	−0.988931
$962$	33.4853	1.07961
$963$	−1.65685	−0.0533914
$964$	23.6274	0.760988
$965$	9.24264	0.297531
$966$	0	0
$967$	−22.8284	−0.734113	−0.367056	−	0.930199i	$-0.619634\pi$
$967$	−22.8284	−0.734113	−0.367056	+	0.930199i	$0.619634\pi$
$968$	10.6569	0.342524
$969$	−7.31371	−0.234950
$970$	−1.92893	−0.0619343
$971$	61.5980	1.97677	0.988387	−	0.151960i	$-0.0485585\pi$
$971$	61.5980	1.97677	0.988387	+	0.151960i	$0.0485585\pi$
$972$	−1.00000	−0.0320750
$973$	0	0
$974$	13.7279	0.439871
$975$	21.3137	0.682585
$976$	11.8995	0.380894
$977$	−53.4853	−1.71115	−0.855573	−	0.517682i	$-0.826795\pi$
$977$	−53.4853	−1.71115	−0.855573	+	0.517682i	$0.826795\pi$
$978$	−14.8284	−0.474161
$979$	2.20101	0.0703446
$980$	0	0
$981$	−17.5858	−0.561471
$982$	9.51472	0.303627
$983$	16.5858	0.529004	0.264502	−	0.964385i	$-0.414792\pi$
$983$	16.5858	0.529004	0.264502	+	0.964385i	$0.414792\pi$
$984$	3.82843	0.122046
$985$	−9.82843	−0.313160
$986$	−4.76955	−0.151893
$987$	0	0
$988$	−27.5563	−0.876684
$989$	2.65685	0.0844831
$990$	−0.242641	−0.00771163
$991$	−39.1127	−1.24246	−0.621228	−	0.783630i	$-0.713366\pi$
$991$	−39.1127	−1.24246	−0.621228	+	0.783630i	$0.713366\pi$
$992$	0.585786	0.0185987
$993$	−14.1421	−0.448787
$994$	0	0
$995$	8.17157	0.259056
$996$	2.48528	0.0787492
$997$	−0.828427	−0.0262366	−0.0131183	−	0.999914i	$-0.504176\pi$
$997$	−0.828427	−0.0262366	−0.0131183	+	0.999914i	$0.504176\pi$
$998$	−31.9411	−1.01108
$999$	7.58579	0.240004

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6762.2.a.bs.1.1	✓	2
7.6	odd	2		6762.2.a.bu.1.2	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6762.2.a.bs.1.1	✓	2	1.1	even	1	trivial
6762.2.a.bu.1.2	yes	2	7.6	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 6762 = 2 \cdot 3 \cdot 7^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6762.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -0.414214 q^{5} +1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -0.414214 q^{5} +1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +0.414214 q^{10} -0.585786 q^{11} -1.00000 q^{12} +4.41421 q^{13} +0.414214 q^{15} +1.00000 q^{16} -1.17157 q^{17} -1.00000 q^{18} -6.24264 q^{19} -0.414214 q^{20} +0.585786 q^{22} +1.00000 q^{23} +1.00000 q^{24} -4.82843 q^{25} -4.41421 q^{26} -1.00000 q^{27} -4.07107 q^{29} -0.414214 q^{30} -0.585786 q^{31} -1.00000 q^{32} +0.585786 q^{33} +1.17157 q^{34} +1.00000 q^{36} -7.58579 q^{37} +6.24264 q^{38} -4.41421 q^{39} +0.414214 q^{40} +3.82843 q^{41} +2.65685 q^{43} -0.585786 q^{44} -0.414214 q^{45} -1.00000 q^{46} +7.58579 q^{47} -1.00000 q^{48} +4.82843 q^{50} +1.17157 q^{51} +4.41421 q^{52} +9.41421 q^{53} +1.00000 q^{54} +0.242641 q^{55} +6.24264 q^{57} +4.07107 q^{58} +8.24264 q^{59} +0.414214 q^{60} +11.8995 q^{61} +0.585786 q^{62} +1.00000 q^{64} -1.82843 q^{65} -0.585786 q^{66} -9.31371 q^{67} -1.17157 q^{68} -1.00000 q^{69} +15.8995 q^{71} -1.00000 q^{72} -6.24264 q^{73} +7.58579 q^{74} +4.82843 q^{75} -6.24264 q^{76} +4.41421 q^{78} -5.89949 q^{79} -0.414214 q^{80} +1.00000 q^{81} -3.82843 q^{82} -2.48528 q^{83} +0.485281 q^{85} -2.65685 q^{86} +4.07107 q^{87} +0.585786 q^{88} -3.75736 q^{89} +0.414214 q^{90} +1.00000 q^{92} +0.585786 q^{93} -7.58579 q^{94} +2.58579 q^{95} +1.00000 q^{96} -4.65685 q^{97} -0.585786 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{5} + 2 q^{6} - 2 q^{8} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 + 2 * q^5 + 2 * q^6 - 2 * q^8 + 2 * q^9	\( 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{5} + 2 q^{6} - 2 q^{8} + 2 q^{9} - 2 q^{10} - 4 q^{11} - 2 q^{12} + 6 q^{13} - 2 q^{15} + 2 q^{16} - 8 q^{17} - 2 q^{18} - 4 q^{19} + 2 q^{20} + 4 q^{22} + 2 q^{23} + 2 q^{24} - 4 q^{25} - 6 q^{26} - 2 q^{27} + 6 q^{29} + 2 q^{30} - 4 q^{31} - 2 q^{32} + 4 q^{33} + 8 q^{34} + 2 q^{36} - 18 q^{37} + 4 q^{38} - 6 q^{39} - 2 q^{40} + 2 q^{41} - 6 q^{43} - 4 q^{44} + 2 q^{45} - 2 q^{46} + 18 q^{47} - 2 q^{48} + 4 q^{50} + 8 q^{51} + 6 q^{52} + 16 q^{53} + 2 q^{54} - 8 q^{55} + 4 q^{57} - 6 q^{58} + 8 q^{59} - 2 q^{60} + 4 q^{61} + 4 q^{62} + 2 q^{64} + 2 q^{65} - 4 q^{66} + 4 q^{67} - 8 q^{68} - 2 q^{69} + 12 q^{71} - 2 q^{72} - 4 q^{73} + 18 q^{74} + 4 q^{75} - 4 q^{76} + 6 q^{78} + 8 q^{79} + 2 q^{80} + 2 q^{81} - 2 q^{82} + 12 q^{83} - 16 q^{85} + 6 q^{86} - 6 q^{87} + 4 q^{88} - 16 q^{89} - 2 q^{90} + 2 q^{92} + 4 q^{93} - 18 q^{94} + 8 q^{95} + 2 q^{96} + 2 q^{97} - 4 q^{99}+O(q^{100}) \) 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 + 2 * q^5 + 2 * q^6 - 2 * q^8 + 2 * q^9 - 2 * q^10 - 4 * q^11 - 2 * q^12 + 6 * q^13 - 2 * q^15 + 2 * q^16 - 8 * q^17 - 2 * q^18 - 4 * q^19 + 2 * q^20 + 4 * q^22 + 2 * q^23 + 2 * q^24 - 4 * q^25 - 6 * q^26 - 2 * q^27 + 6 * q^29 + 2 * q^30 - 4 * q^31 - 2 * q^32 + 4 * q^33 + 8 * q^34 + 2 * q^36 - 18 * q^37 + 4 * q^38 - 6 * q^39 - 2 * q^40 + 2 * q^41 - 6 * q^43 - 4 * q^44 + 2 * q^45 - 2 * q^46 + 18 * q^47 - 2 * q^48 + 4 * q^50 + 8 * q^51 + 6 * q^52 + 16 * q^53 + 2 * q^54 - 8 * q^55 + 4 * q^57 - 6 * q^58 + 8 * q^59 - 2 * q^60 + 4 * q^61 + 4 * q^62 + 2 * q^64 + 2 * q^65 - 4 * q^66 + 4 * q^67 - 8 * q^68 - 2 * q^69 + 12 * q^71 - 2 * q^72 - 4 * q^73 + 18 * q^74 + 4 * q^75 - 4 * q^76 + 6 * q^78 + 8 * q^79 + 2 * q^80 + 2 * q^81 - 2 * q^82 + 12 * q^83 - 16 * q^85 + 6 * q^86 - 6 * q^87 + 4 * q^88 - 16 * q^89 - 2 * q^90 + 2 * q^92 + 4 * q^93 - 18 * q^94 + 8 * q^95 + 2 * q^96 + 2 * q^97 - 4 * q^99

Introduction

L-functions

Modular forms

Varieties

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Groups

Database

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