LMFDB - Embedded newform 6027.2.a.bm.1.11

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			1.11
Root			$-0.819942$ 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)

$f(q)$	$=$	$q+0.819942 q^{2} +1.00000 q^{3} -1.32769 q^{4} +0.332064 q^{5} +0.819942 q^{6} -2.72852 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+0.819942 q^{2} +1.00000 q^{3} -1.32769 q^{4} +0.332064 q^{5} +0.819942 q^{6} -2.72852 q^{8} +1.00000 q^{9} +0.272273 q^{10} +1.66451 q^{11} -1.32769 q^{12} +0.0164518 q^{13} +0.332064 q^{15} +0.418163 q^{16} +1.78787 q^{17} +0.819942 q^{18} -2.53514 q^{19} -0.440879 q^{20} +1.36480 q^{22} -2.87658 q^{23} -2.72852 q^{24} -4.88973 q^{25} +0.0134895 q^{26} +1.00000 q^{27} +0.605814 q^{29} +0.272273 q^{30} -7.60543 q^{31} +5.79990 q^{32} +1.66451 q^{33} +1.46595 q^{34} -1.32769 q^{36} +1.91702 q^{37} -2.07867 q^{38} +0.0164518 q^{39} -0.906041 q^{40} +1.00000 q^{41} -12.8625 q^{43} -2.20996 q^{44} +0.332064 q^{45} -2.35863 q^{46} -6.16676 q^{47} +0.418163 q^{48} -4.00930 q^{50} +1.78787 q^{51} -0.0218429 q^{52} +3.84096 q^{53} +0.819942 q^{54} +0.552722 q^{55} -2.53514 q^{57} +0.496733 q^{58} +1.41418 q^{59} -0.440879 q^{60} -4.42492 q^{61} -6.23601 q^{62} +3.91926 q^{64} +0.00546303 q^{65} +1.36480 q^{66} +4.72149 q^{67} -2.37374 q^{68} -2.87658 q^{69} +5.80768 q^{71} -2.72852 q^{72} +0.748580 q^{73} +1.57184 q^{74} -4.88973 q^{75} +3.36589 q^{76} +0.0134895 q^{78} +4.49461 q^{79} +0.138857 q^{80} +1.00000 q^{81} +0.819942 q^{82} +11.4541 q^{83} +0.593686 q^{85} -10.5465 q^{86} +0.605814 q^{87} -4.54164 q^{88} -5.00144 q^{89} +0.272273 q^{90} +3.81922 q^{92} -7.60543 q^{93} -5.05638 q^{94} -0.841828 q^{95} +5.79990 q^{96} +6.40794 q^{97} +1.66451 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	0.819942	0.579787	0.289893	−	0.957059i	$-0.406380\pi$
$2$	0.819942	0.579787	0.289893	+	0.957059i	$0.406380\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	−1.32769	−0.663847
$5$	0.332064	0.148503	0.0742517	−	0.997240i	$-0.476343\pi$
$5$	0.332064	0.148503	0.0742517	+	0.997240i	$0.476343\pi$
$6$	0.819942	0.334740
$7$	0	0
$7$	0	0
$8$	−2.72852	−0.964677
$9$	1.00000	0.333333
$10$	0.272273	0.0861003
$11$	1.66451	0.501868	0.250934	−	0.968004i	$-0.419262\pi$
$11$	1.66451	0.501868	0.250934	+	0.968004i	$0.419262\pi$
$12$	−1.32769	−0.383272
$13$	0.0164518	0.00456290	0.00228145	−	0.999997i	$-0.499274\pi$
$13$	0.0164518	0.00456290	0.00228145	+	0.999997i	$0.499274\pi$
$14$	0	0
$15$	0.332064	0.0857385
$16$	0.418163	0.104541
$17$	1.78787	0.433622	0.216811	−	0.976214i	$-0.430435\pi$
$17$	1.78787	0.433622	0.216811	+	0.976214i	$0.430435\pi$
$18$	0.819942	0.193262
$19$	−2.53514	−0.581601	−0.290801	−	0.956784i	$-0.593922\pi$
$19$	−2.53514	−0.581601	−0.290801	+	0.956784i	$0.593922\pi$
$20$	−0.440879	−0.0985836
$21$	0	0
$22$	1.36480	0.290976
$23$	−2.87658	−0.599809	−0.299904	−	0.953969i	$-0.596955\pi$
$23$	−2.87658	−0.599809	−0.299904	+	0.953969i	$0.596955\pi$
$24$	−2.72852	−0.556956
$25$	−4.88973	−0.977947
$26$	0.0134895	0.00264551
$27$	1.00000	0.192450
$28$	0	0
$29$	0.605814	0.112497	0.0562484	−	0.998417i	$-0.482086\pi$
$29$	0.605814	0.112497	0.0562484	+	0.998417i	$0.482086\pi$
$30$	0.272273	0.0497100
$31$	−7.60543	−1.36597	−0.682987	−	0.730430i	$-0.739320\pi$
$31$	−7.60543	−1.36597	−0.682987	+	0.730430i	$0.739320\pi$
$32$	5.79990	1.02529
$33$	1.66451	0.289754
$34$	1.46595	0.251408
$35$	0	0
$36$	−1.32769	−0.221282
$37$	1.91702	0.315155	0.157578	−	0.987507i	$-0.449632\pi$
$37$	1.91702	0.315155	0.157578	+	0.987507i	$0.449632\pi$
$38$	−2.07867	−0.337205
$39$	0.0164518	0.00263439
$40$	−0.906041	−0.143258
$41$	1.00000	0.156174
$41$	1.00000	0.156174
$42$	0	0
$43$	−12.8625	−1.96151	−0.980753	−	0.195251i	$-0.937448\pi$
$43$	−12.8625	−1.96151	−0.980753	+	0.195251i	$0.937448\pi$
$44$	−2.20996	−0.333164
$45$	0.332064	0.0495011
$46$	−2.35863	−0.347761
$47$	−6.16676	−0.899514	−0.449757	−	0.893151i	$-0.648489\pi$
$47$	−6.16676	−0.899514	−0.449757	+	0.893151i	$0.648489\pi$
$48$	0.418163	0.0603566
$49$	0	0
$50$	−4.00930	−0.567001
$51$	1.78787	0.250351
$52$	−0.0218429	−0.00302907
$53$	3.84096	0.527596	0.263798	−	0.964578i	$-0.415025\pi$
$53$	3.84096	0.527596	0.263798	+	0.964578i	$0.415025\pi$
$54$	0.819942	0.111580
$55$	0.552722	0.0745291
$56$	0	0
$57$	−2.53514	−0.335788
$58$	0.496733	0.0652242
$59$	1.41418	0.184110	0.0920551	−	0.995754i	$-0.470656\pi$
$59$	1.41418	0.184110	0.0920551	+	0.995754i	$0.470656\pi$
$60$	−0.440879	−0.0569172
$61$	−4.42492	−0.566552	−0.283276	−	0.959038i	$-0.591421\pi$
$61$	−4.42492	−0.566552	−0.283276	+	0.959038i	$0.591421\pi$
$62$	−6.23601	−0.791974
$63$	0	0
$64$	3.91926	0.489908
$65$	0.00546303	0.000677605 0
$66$	1.36480	0.167995
$67$	4.72149	0.576822	0.288411	−	0.957507i	$-0.406873\pi$
$67$	4.72149	0.576822	0.288411	+	0.957507i	$0.406873\pi$
$68$	−2.37374	−0.287858
$69$	−2.87658	−0.346300
$70$	0	0
$71$	5.80768	0.689244	0.344622	−	0.938741i	$-0.388007\pi$
$71$	5.80768	0.689244	0.344622	+	0.938741i	$0.388007\pi$
$72$	−2.72852	−0.321559
$73$	0.748580	0.0876147	0.0438073	−	0.999040i	$-0.486051\pi$
$73$	0.748580	0.0876147	0.0438073	+	0.999040i	$0.486051\pi$
$74$	1.57184	0.182723
$75$	−4.88973	−0.564618
$76$	3.36589	0.386094
$77$	0	0
$78$	0.0134895	0.00152738
$79$	4.49461	0.505683	0.252842	−	0.967508i	$-0.418635\pi$
$79$	4.49461	0.505683	0.252842	+	0.967508i	$0.418635\pi$
$80$	0.138857	0.0155246
$81$	1.00000	0.111111
$82$	0.819942	0.0905475
$83$	11.4541	1.25725	0.628626	−	0.777707i	$-0.283617\pi$
$83$	11.4541	1.25725	0.628626	+	0.777707i	$0.283617\pi$
$84$	0	0
$85$	0.593686	0.0643943
$86$	−10.5465	−1.13726
$87$	0.605814	0.0649501
$88$	−4.54164	−0.484140
$89$	−5.00144	−0.530152	−0.265076	−	0.964228i	$-0.585397\pi$
$89$	−5.00144	−0.530152	−0.265076	+	0.964228i	$0.585397\pi$
$90$	0.272273	0.0287001
$91$	0	0
$92$	3.81922	0.398181
$93$	−7.60543	−0.788646
$94$	−5.05638	−0.521526
$95$	−0.841828	−0.0863697
$96$	5.79990	0.591950
$97$	6.40794	0.650628	0.325314	−	0.945606i	$-0.394530\pi$
$97$	6.40794	0.650628	0.325314	+	0.945606i	$0.394530\pi$
$98$	0	0
$99$	1.66451	0.167289
$100$	6.49207	0.649207
$101$	−17.5942	−1.75069	−0.875344	−	0.483501i	$-0.839365\pi$
$101$	−17.5942	−1.75069	−0.875344	+	0.483501i	$0.839365\pi$
$102$	1.46595	0.145150
$103$	−4.00896	−0.395015	−0.197507	−	0.980301i	$-0.563285\pi$
$103$	−4.00896	−0.395015	−0.197507	+	0.980301i	$0.563285\pi$
$104$	−0.0448889	−0.00440172
$105$	0	0
$106$	3.14937	0.305893
$107$	−11.8977	−1.15019	−0.575097	−	0.818085i	$-0.695036\pi$
$107$	−11.8977	−1.15019	−0.575097	+	0.818085i	$0.695036\pi$
$108$	−1.32769	−0.127757
$109$	−14.1105	−1.35154	−0.675772	−	0.737110i	$-0.736190\pi$
$109$	−14.1105	−1.35154	−0.675772	+	0.737110i	$0.736190\pi$
$110$	0.453200	0.0432110
$111$	1.91702	0.181955
$112$	0	0
$113$	3.58338	0.337096	0.168548	−	0.985693i	$-0.446092\pi$
$113$	3.58338	0.337096	0.168548	+	0.985693i	$0.446092\pi$
$114$	−2.07867	−0.194685
$115$	−0.955208	−0.0890736
$116$	−0.804336	−0.0746807
$117$	0.0164518	0.00152097
$118$	1.15954	0.106745
$119$	0	0
$120$	−0.906041	−0.0827099
$121$	−8.22941	−0.748129
$122$	−3.62818	−0.328480
$123$	1.00000	0.0901670
$124$	10.0977	0.906799
$125$	−3.28402	−0.293732
$126$	0	0
$127$	−16.6099	−1.47389	−0.736944	−	0.675954i	$-0.763732\pi$
$127$	−16.6099	−1.47389	−0.736944	+	0.675954i	$0.763732\pi$
$128$	−8.38624	−0.741246
$129$	−12.8625	−1.13248
$130$	0.00447937	0.000392867 0
$131$	9.60996	0.839626	0.419813	−	0.907611i	$-0.362096\pi$
$131$	9.60996	0.839626	0.419813	+	0.907611i	$0.362096\pi$
$132$	−2.20996	−0.192352
$133$	0	0
$134$	3.87135	0.334434
$135$	0.332064	0.0285795
$136$	−4.87823	−0.418305
$137$	−5.93129	−0.506744	−0.253372	−	0.967369i	$-0.581540\pi$
$137$	−5.93129	−0.506744	−0.253372	+	0.967369i	$0.581540\pi$
$138$	−2.35863	−0.200780
$139$	20.3689	1.72767	0.863836	−	0.503773i	$-0.168055\pi$
$139$	20.3689	1.72767	0.863836	+	0.503773i	$0.168055\pi$
$140$	0	0
$141$	−6.16676	−0.519335
$142$	4.76196	0.399615
$143$	0.0273841	0.00228997
$144$	0.418163	0.0348469
$145$	0.201169	0.0167062
$146$	0.613793	0.0507978
$147$	0	0
$148$	−2.54521	−0.209215
$149$	−15.7679	−1.29176	−0.645878	−	0.763440i	$-0.723509\pi$
$149$	−15.7679	−1.29176	−0.645878	+	0.763440i	$0.723509\pi$
$150$	−4.00930	−0.327358
$151$	12.4627	1.01420	0.507102	−	0.861886i	$-0.330717\pi$
$151$	12.4627	1.01420	0.507102	+	0.861886i	$0.330717\pi$
$152$	6.91718	0.561057
$153$	1.78787	0.144541
$154$	0	0
$155$	−2.52549	−0.202852
$156$	−0.0218429	−0.00174883
$157$	−14.6543	−1.16954	−0.584771	−	0.811198i	$-0.698816\pi$
$157$	−14.6543	−1.16954	−0.584771	+	0.811198i	$0.698816\pi$
$158$	3.68532	0.293188
$159$	3.84096	0.304608
$160$	1.92594	0.152259
$161$	0	0
$162$	0.819942	0.0644207
$163$	−10.4822	−0.821030	−0.410515	−	0.911854i	$-0.634651\pi$
$163$	−10.4822	−0.821030	−0.410515	+	0.911854i	$0.634651\pi$
$164$	−1.32769	−0.103676
$165$	0.552722	0.0430294
$166$	9.39172	0.728939
$167$	9.45153	0.731381	0.365691	−	0.930736i	$-0.380833\pi$
$167$	9.45153	0.731381	0.365691	+	0.930736i	$0.380833\pi$
$168$	0	0
$169$	−12.9997	−0.999979
$170$	0.486788	0.0373349
$171$	−2.53514	−0.193867
$172$	17.0774	1.30214
$173$	−9.52226	−0.723964	−0.361982	−	0.932185i	$-0.617900\pi$
$173$	−9.52226	−0.723964	−0.361982	+	0.932185i	$0.617900\pi$
$174$	0.496733	0.0376572
$175$	0	0
$176$	0.696035	0.0524656
$177$	1.41418	0.106296
$178$	−4.10089	−0.307375
$179$	−22.8105	−1.70494	−0.852469	−	0.522778i	$-0.824896\pi$
$179$	−22.8105	−1.70494	−0.852469	+	0.522778i	$0.824896\pi$
$180$	−0.440879	−0.0328612
$181$	−3.16124	−0.234973	−0.117487	−	0.993074i	$-0.537484\pi$
$181$	−3.16124	−0.234973	−0.117487	+	0.993074i	$0.537484\pi$
$182$	0	0
$183$	−4.42492	−0.327099
$184$	7.84880	0.578621
$185$	0.636571	0.0468016
$186$	−6.23601	−0.457246
$187$	2.97592	0.217621
$188$	8.18757	0.597140
$189$	0	0
$190$	−0.690250	−0.0500760
$191$	5.66282	0.409747	0.204874	−	0.978788i	$-0.434322\pi$
$191$	5.66282	0.409747	0.204874	+	0.978788i	$0.434322\pi$
$192$	3.91926	0.282848
$193$	−20.1887	−1.45321	−0.726606	−	0.687055i	$-0.758903\pi$
$193$	−20.1887	−1.45321	−0.726606	+	0.687055i	$0.758903\pi$
$194$	5.25414	0.377225
$195$	0.00546303	0.000391216 0
$196$	0	0
$197$	0.477679	0.0340332	0.0170166	−	0.999855i	$-0.494583\pi$
$197$	0.477679	0.0340332	0.0170166	+	0.999855i	$0.494583\pi$
$198$	1.36480	0.0969921
$199$	−3.25632	−0.230835	−0.115417	−	0.993317i	$-0.536821\pi$
$199$	−3.25632	−0.230835	−0.115417	+	0.993317i	$0.536821\pi$
$200$	13.3417	0.943402
$201$	4.72149	0.333029
$202$	−14.4262	−1.01503
$203$	0	0
$204$	−2.37374	−0.166195
$205$	0.332064	0.0231923
$206$	−3.28712	−0.229024
$207$	−2.87658	−0.199936
$208$	0.00687951	0.000477008 0
$209$	−4.21976	−0.291887
$210$	0	0
$211$	0.492222	0.0338860	0.0169430	−	0.999856i	$-0.494607\pi$
$211$	0.492222	0.0338860	0.0169430	+	0.999856i	$0.494607\pi$
$212$	−5.09962	−0.350244
$213$	5.80768	0.397935
$214$	−9.75543	−0.666868
$215$	−4.27115	−0.291290
$216$	−2.72852	−0.185652
$217$	0	0
$218$	−11.5698	−0.783608
$219$	0.748580	0.0505844
$220$	−0.733847	−0.0494759
$221$	0.0294136	0.00197857
$222$	1.57184	0.105495
$223$	7.15813	0.479344	0.239672	−	0.970854i	$-0.422960\pi$
$223$	7.15813	0.479344	0.239672	+	0.970854i	$0.422960\pi$
$224$	0	0
$225$	−4.88973	−0.325982
$226$	2.93816	0.195444
$227$	−7.34618	−0.487583	−0.243791	−	0.969828i	$-0.578391\pi$
$227$	−7.34618	−0.487583	−0.243791	+	0.969828i	$0.578391\pi$
$228$	3.36589	0.222912
$229$	−9.96063	−0.658217	−0.329109	−	0.944292i	$-0.606748\pi$
$229$	−9.96063	−0.658217	−0.329109	+	0.944292i	$0.606748\pi$
$230$	−0.783215	−0.0516437
$231$	0	0
$232$	−1.65297	−0.108523
$233$	−21.3065	−1.39583	−0.697917	−	0.716179i	$-0.745889\pi$
$233$	−21.3065	−1.39583	−0.697917	+	0.716179i	$0.745889\pi$
$234$	0.0134895	0.000881835 0
$235$	−2.04776	−0.133581
$236$	−1.87760	−0.122221
$237$	4.49461	0.291956
$238$	0	0
$239$	−10.7115	−0.692868	−0.346434	−	0.938074i	$-0.612608\pi$
$239$	−10.7115	−0.692868	−0.346434	+	0.938074i	$0.612608\pi$
$240$	0.138857	0.00896315
$241$	−15.3877	−0.991212	−0.495606	−	0.868547i	$-0.665054\pi$
$241$	−15.3877	−0.991212	−0.495606	+	0.868547i	$0.665054\pi$
$242$	−6.74764	−0.433755
$243$	1.00000	0.0641500
$244$	5.87494	0.376104
$245$	0	0
$246$	0.819942	0.0522776
$247$	−0.0417075	−0.00265379
$248$	20.7515	1.31772
$249$	11.4541	0.725875
$250$	−2.69271	−0.170302
$251$	−3.83734	−0.242211	−0.121105	−	0.992640i	$-0.538644\pi$
$251$	−3.83734	−0.242211	−0.121105	+	0.992640i	$0.538644\pi$
$252$	0	0
$253$	−4.78809	−0.301025
$254$	−13.6191	−0.854541
$255$	0.593686	0.0371780
$256$	−14.7148	−0.919672
$257$	29.6461	1.84927	0.924637	−	0.380850i	$-0.124368\pi$
$257$	29.6461	1.84927	0.924637	+	0.380850i	$0.124368\pi$
$258$	−10.5465	−0.656595
$259$	0	0
$260$	−0.00725323	−0.000449827 0
$261$	0.605814	0.0374990
$262$	7.87961	0.486804
$263$	−15.0006	−0.924976	−0.462488	−	0.886625i	$-0.653043\pi$
$263$	−15.0006	−0.924976	−0.462488	+	0.886625i	$0.653043\pi$
$264$	−4.54164	−0.279518
$265$	1.27544	0.0783498
$266$	0	0
$267$	−5.00144	−0.306083
$268$	−6.26870	−0.382922
$269$	−5.76098	−0.351253	−0.175627	−	0.984457i	$-0.556195\pi$
$269$	−5.76098	−0.351253	−0.175627	+	0.984457i	$0.556195\pi$
$270$	0.272273	0.0165700
$271$	5.41966	0.329221	0.164611	−	0.986359i	$-0.447363\pi$
$271$	5.41966	0.329221	0.164611	+	0.986359i	$0.447363\pi$
$272$	0.747619	0.0453311
$273$	0	0
$274$	−4.86331	−0.293803
$275$	−8.13900	−0.490800
$276$	3.81922	0.229890
$277$	−20.4003	−1.22573	−0.612867	−	0.790186i	$-0.709984\pi$
$277$	−20.4003	−1.22573	−0.612867	+	0.790186i	$0.709984\pi$
$278$	16.7014	1.00168
$279$	−7.60543	−0.455325
$280$	0	0
$281$	5.03670	0.300464	0.150232	−	0.988651i	$-0.451998\pi$
$281$	5.03670	0.300464	0.150232	+	0.988651i	$0.451998\pi$
$282$	−5.05638	−0.301103
$283$	0.733053	0.0435755	0.0217877	−	0.999763i	$-0.493064\pi$
$283$	0.733053	0.0435755	0.0217877	+	0.999763i	$0.493064\pi$
$284$	−7.71082	−0.457553
$285$	−0.841828	−0.0498656
$286$	0.0224534	0.00132769
$287$	0	0
$288$	5.79990	0.341763
$289$	−13.8035	−0.811972
$290$	0.164947	0.00968601
$291$	6.40794	0.375640
$292$	−0.993886	−0.0581628
$293$	9.92734	0.579961	0.289981	−	0.957033i	$-0.406351\pi$
$293$	9.92734	0.579961	0.289981	+	0.957033i	$0.406351\pi$
$294$	0	0
$295$	0.469597	0.0273410
$296$	−5.23061	−0.304023
$297$	1.66451	0.0965845
$298$	−12.9288	−0.748943
$299$	−0.0473248	−0.00273686
$300$	6.49207	0.374820
$301$	0	0
$302$	10.2187	0.588022
$303$	−17.5942	−1.01076
$304$	−1.06010	−0.0608010
$305$	−1.46935	−0.0841349
$306$	1.46595	0.0838027
$307$	18.0797	1.03186	0.515930	−	0.856631i	$-0.327446\pi$
$307$	18.0797	1.03186	0.515930	+	0.856631i	$0.327446\pi$
$308$	0	0
$309$	−4.00896	−0.228062
$310$	−2.07075	−0.117611
$311$	6.53524	0.370579	0.185290	−	0.982684i	$-0.440678\pi$
$311$	6.53524	0.370579	0.185290	+	0.982684i	$0.440678\pi$
$312$	−0.0448889	−0.00254133
$313$	6.62531	0.374485	0.187242	−	0.982314i	$-0.440045\pi$
$313$	6.62531	0.374485	0.187242	+	0.982314i	$0.440045\pi$
$314$	−12.0157	−0.678085
$315$	0	0
$316$	−5.96747	−0.335696
$317$	5.00888	0.281327	0.140663	−	0.990057i	$-0.455077\pi$
$317$	5.00888	0.281327	0.140663	+	0.990057i	$0.455077\pi$
$318$	3.14937	0.176608
$319$	1.00838	0.0564586
$320$	1.30144	0.0727529
$321$	−11.8977	−0.664065
$322$	0	0
$323$	−4.53250	−0.252195
$324$	−1.32769	−0.0737608
$325$	−0.0804447	−0.00446227
$326$	−8.59480	−0.476022
$327$	−14.1105	−0.780315
$328$	−2.72852	−0.150657
$329$	0	0
$330$	0.453200	0.0249479
$331$	−2.06115	−0.113291	−0.0566456	−	0.998394i	$-0.518040\pi$
$331$	−2.06115	−0.113291	−0.0566456	+	0.998394i	$0.518040\pi$
$332$	−15.2076	−0.834624
$333$	1.91702	0.105052
$334$	7.74971	0.424045
$335$	1.56784	0.0856600
$336$	0	0
$337$	12.5592	0.684141	0.342070	−	0.939674i	$-0.388872\pi$
$337$	12.5592	0.684141	0.342070	+	0.939674i	$0.388872\pi$
$338$	−10.6590	−0.579775
$339$	3.58338	0.194622
$340$	−0.788233	−0.0427480
$341$	−12.6593	−0.685539
$342$	−2.07867	−0.112402
$343$	0	0
$344$	35.0954	1.89222
$345$	−0.955208	−0.0514267
$346$	−7.80770	−0.419745
$347$	11.4809	0.616325	0.308163	−	0.951334i	$-0.400286\pi$
$347$	11.4809	0.616325	0.308163	+	0.951334i	$0.400286\pi$
$348$	−0.804336	−0.0431169
$349$	15.2242	0.814931	0.407466	−	0.913221i	$-0.366413\pi$
$349$	15.2242	0.814931	0.407466	+	0.913221i	$0.366413\pi$
$350$	0	0
$351$	0.0164518	0.000878130 0
$352$	9.65398	0.514559
$353$	20.4119	1.08642	0.543209	−	0.839598i	$-0.317209\pi$
$353$	20.4119	1.08642	0.543209	+	0.839598i	$0.317209\pi$
$354$	1.15954	0.0616291
$355$	1.92852	0.102355
$356$	6.64039	0.351940
$357$	0	0
$358$	−18.7033	−0.988501
$359$	30.1992	1.59385	0.796926	−	0.604077i	$-0.206458\pi$
$359$	30.1992	1.59385	0.796926	+	0.604077i	$0.206458\pi$
$360$	−0.906041	−0.0477526
$361$	−12.5731	−0.661740
$362$	−2.59203	−0.136234
$363$	−8.22941	−0.431932
$364$	0	0
$365$	0.248576	0.0130111
$366$	−3.62818	−0.189648
$367$	5.89363	0.307645	0.153823	−	0.988098i	$-0.450842\pi$
$367$	5.89363	0.307645	0.153823	+	0.988098i	$0.450842\pi$
$368$	−1.20288	−0.0627044
$369$	1.00000	0.0520579
$370$	0.521951	0.0271350
$371$	0	0
$372$	10.0977	0.523541
$373$	23.0679	1.19441	0.597206	−	0.802088i	$-0.296278\pi$
$373$	23.0679	1.19441	0.597206	+	0.802088i	$0.296278\pi$
$374$	2.44008	0.126174
$375$	−3.28402	−0.169586
$376$	16.8261	0.867740
$377$	0.00996671	0.000513311 0
$378$	0	0
$379$	15.9801	0.820843	0.410421	−	0.911896i	$-0.365382\pi$
$379$	15.9801	0.820843	0.410421	+	0.911896i	$0.365382\pi$
$380$	1.11769	0.0573363
$381$	−16.6099	−0.850950
$382$	4.64319	0.237566
$383$	−28.4963	−1.45609	−0.728046	−	0.685529i	$-0.759571\pi$
$383$	−28.4963	−1.45609	−0.728046	+	0.685529i	$0.759571\pi$
$384$	−8.38624	−0.427959
$385$	0	0
$386$	−16.5535	−0.842553
$387$	−12.8625	−0.653836
$388$	−8.50779	−0.431918
$389$	4.42493	0.224353	0.112177	−	0.993688i	$-0.464218\pi$
$389$	4.42493	0.224353	0.112177	+	0.993688i	$0.464218\pi$
$390$	0.00447937	0.000226822 0
$391$	−5.14295	−0.260090
$392$	0	0
$393$	9.60996	0.484758
$394$	0.391669	0.0197320
$395$	1.49250	0.0750956
$396$	−2.20996	−0.111055
$397$	31.0599	1.55885	0.779426	−	0.626494i	$-0.215511\pi$
$397$	31.0599	1.55885	0.779426	+	0.626494i	$0.215511\pi$
$398$	−2.67000	−0.133835
$399$	0	0
$400$	−2.04470	−0.102235
$401$	−2.27093	−0.113405	−0.0567025	−	0.998391i	$-0.518059\pi$
$401$	−2.27093	−0.113405	−0.0567025	+	0.998391i	$0.518059\pi$
$402$	3.87135	0.193086
$403$	−0.125123	−0.00623280
$404$	23.3597	1.16219
$405$	0.332064	0.0165004
$406$	0	0
$407$	3.19089	0.158166
$408$	−4.87823	−0.241508
$409$	5.31380	0.262751	0.131375	−	0.991333i	$-0.458061\pi$
$409$	5.31380	0.262751	0.131375	+	0.991333i	$0.458061\pi$
$410$	0.272273	0.0134466
$411$	−5.93129	−0.292569
$412$	5.32268	0.262229
$413$	0	0
$414$	−2.35863	−0.115920
$415$	3.80350	0.186706
$416$	0.0954186	0.00467828
$417$	20.3689	0.997472
$418$	−3.45996	−0.169232
$419$	16.3430	0.798410	0.399205	−	0.916862i	$-0.369286\pi$
$419$	16.3430	0.798410	0.399205	+	0.916862i	$0.369286\pi$
$420$	0	0
$421$	−7.54033	−0.367493	−0.183747	−	0.982974i	$-0.558823\pi$
$421$	−7.54033	−0.367493	−0.183747	+	0.982974i	$0.558823\pi$
$422$	0.403594	0.0196466
$423$	−6.16676	−0.299838
$424$	−10.4801	−0.508960
$425$	−8.74219	−0.424059
$426$	4.76196	0.230718
$427$	0	0
$428$	15.7965	0.763554
$429$	0.0273841	0.00132212
$430$	−3.50210	−0.168886
$431$	8.22298	0.396087	0.198044	−	0.980193i	$-0.436541\pi$
$431$	8.22298	0.396087	0.198044	+	0.980193i	$0.436541\pi$
$432$	0.418163	0.0201189
$433$	6.10083	0.293187	0.146594	−	0.989197i	$-0.453169\pi$
$433$	6.10083	0.293187	0.146594	+	0.989197i	$0.453169\pi$
$434$	0	0
$435$	0.201169	0.00964531
$436$	18.7345	0.897220
$437$	7.29254	0.348850
$438$	0.613793	0.0293281
$439$	−16.8750	−0.805398	−0.402699	−	0.915333i	$-0.631928\pi$
$439$	−16.8750	−0.805398	−0.402699	+	0.915333i	$0.631928\pi$
$440$	−1.50811	−0.0718965
$441$	0	0
$442$	0.0241174	0.00114715
$443$	−24.6875	−1.17294	−0.586468	−	0.809972i	$-0.699482\pi$
$443$	−24.6875	−1.17294	−0.586468	+	0.809972i	$0.699482\pi$
$444$	−2.54521	−0.120790
$445$	−1.66080	−0.0787293
$446$	5.86925	0.277917
$447$	−15.7679	−0.745796
$448$	0	0
$449$	1.81494	0.0856523	0.0428262	−	0.999083i	$-0.486364\pi$
$449$	1.81494	0.0856523	0.0428262	+	0.999083i	$0.486364\pi$
$450$	−4.00930	−0.189000
$451$	1.66451	0.0783786
$452$	−4.75763	−0.223780
$453$	12.4627	0.585551
$454$	−6.02344	−0.282694
$455$	0	0
$456$	6.91718	0.323927
$457$	28.1049	1.31469	0.657345	−	0.753589i	$-0.271679\pi$
$457$	28.1049	1.31469	0.657345	+	0.753589i	$0.271679\pi$
$458$	−8.16714	−0.381625
$459$	1.78787	0.0834505
$460$	1.26822	0.0591313
$461$	27.5428	1.28280	0.641399	−	0.767207i	$-0.278354\pi$
$461$	27.5428	1.28280	0.641399	+	0.767207i	$0.278354\pi$
$462$	0	0
$463$	−23.8028	−1.10621	−0.553104	−	0.833112i	$-0.686557\pi$
$463$	−23.8028	−1.10621	−0.553104	+	0.833112i	$0.686557\pi$
$464$	0.253329	0.0117605
$465$	−2.52549	−0.117117
$466$	−17.4701	−0.809286
$467$	−4.10944	−0.190162	−0.0950811	−	0.995470i	$-0.530311\pi$
$467$	−4.10944	−0.190162	−0.0950811	+	0.995470i	$0.530311\pi$
$468$	−0.0218429	−0.00100969
$469$	0	0
$470$	−1.67904	−0.0774484
$471$	−14.6543	−0.675235
$472$	−3.85861	−0.177607
$473$	−21.4097	−0.984417
$474$	3.68532	0.169272
$475$	12.3962	0.568775
$476$	0	0
$477$	3.84096	0.175865
$478$	−8.78279	−0.401716
$479$	11.3384	0.518063	0.259031	−	0.965869i	$-0.416597\pi$
$479$	11.3384	0.518063	0.259031	+	0.965869i	$0.416597\pi$
$480$	1.92594	0.0879066
$481$	0.0315383	0.00143802
$482$	−12.6171	−0.574692
$483$	0	0
$484$	10.9262	0.496643
$485$	2.12784	0.0966204
$486$	0.819942	0.0371933
$487$	9.89485	0.448378	0.224189	−	0.974546i	$-0.428027\pi$
$487$	9.89485	0.448378	0.224189	+	0.974546i	$0.428027\pi$
$488$	12.0735	0.546540
$489$	−10.4822	−0.474022
$490$	0	0
$491$	25.2342	1.13880	0.569402	−	0.822059i	$-0.307175\pi$
$491$	25.2342	1.13880	0.569402	+	0.822059i	$0.307175\pi$
$492$	−1.32769	−0.0598571
$493$	1.08312	0.0487811
$494$	−0.0341978	−0.00153863
$495$	0.552722	0.0248430
$496$	−3.18030	−0.142800
$497$	0	0
$498$	9.39172	0.420853
$499$	−1.10126	−0.0492993	−0.0246497	−	0.999696i	$-0.507847\pi$
$499$	−1.10126	−0.0492993	−0.0246497	+	0.999696i	$0.507847\pi$
$500$	4.36018	0.194993
$501$	9.45153	0.422263
$502$	−3.14640	−0.140431
$503$	−25.1272	−1.12036	−0.560182	−	0.828369i	$-0.689269\pi$
$503$	−25.1272	−1.12036	−0.560182	+	0.828369i	$0.689269\pi$
$504$	0	0
$505$	−5.84239	−0.259983
$506$	−3.92596	−0.174530
$507$	−12.9997	−0.577338
$508$	22.0528	0.978437
$509$	−7.92927	−0.351459	−0.175729	−	0.984439i	$-0.556228\pi$
$509$	−7.92927	−0.351459	−0.175729	+	0.984439i	$0.556228\pi$
$510$	0.486788	0.0215553
$511$	0	0
$512$	4.70723	0.208032
$513$	−2.53514	−0.111929
$514$	24.3081	1.07218
$515$	−1.33123	−0.0586610
$516$	17.0774	0.751791
$517$	−10.2646	−0.451437
$518$	0	0
$519$	−9.52226	−0.417981
$520$	−0.0149060	−0.000653670 0
$521$	−14.0312	−0.614716	−0.307358	−	0.951594i	$-0.599445\pi$
$521$	−14.0312	−0.614716	−0.307358	+	0.951594i	$0.599445\pi$
$522$	0.496733	0.0217414
$523$	19.4953	0.852469	0.426234	−	0.904613i	$-0.359840\pi$
$523$	19.4953	0.852469	0.426234	+	0.904613i	$0.359840\pi$
$524$	−12.7591	−0.557384
$525$	0	0
$526$	−12.2996	−0.536289
$527$	−13.5975	−0.592316
$528$	0.696035	0.0302910
$529$	−14.7253	−0.640230
$530$	1.04579	0.0454262
$531$	1.41418	0.0613701
$532$	0	0
$533$	0.0164518	0.000712605 0
$534$	−4.10089	−0.177463
$535$	−3.95080	−0.170808
$536$	−12.8827	−0.556447
$537$	−22.8105	−0.984347
$538$	−4.72367	−0.203652
$539$	0	0
$540$	−0.440879	−0.0189724
$541$	−13.5263	−0.581541	−0.290771	−	0.956793i	$-0.593912\pi$
$541$	−13.5263	−0.581541	−0.290771	+	0.956793i	$0.593912\pi$
$542$	4.44381	0.190878
$543$	−3.16124	−0.135662
$544$	10.3695	0.444587
$545$	−4.68560	−0.200709
$546$	0	0
$547$	21.4272	0.916162	0.458081	−	0.888911i	$-0.348537\pi$
$547$	21.4272	0.916162	0.458081	+	0.888911i	$0.348537\pi$
$548$	7.87494	0.336401
$549$	−4.42492	−0.188851
$550$	−6.67351	−0.284559
$551$	−1.53582	−0.0654283
$552$	7.84880	0.334067
$553$	0	0
$554$	−16.7270	−0.710664
$555$	0.636571	0.0270209
$556$	−27.0437	−1.14691
$557$	−10.4798	−0.444045	−0.222022	−	0.975042i	$-0.571266\pi$
$557$	−10.4798	−0.444045	−0.222022	+	0.975042i	$0.571266\pi$
$558$	−6.23601	−0.263991
$559$	−0.211610	−0.00895015
$560$	0	0
$561$	2.97592	0.125643
$562$	4.12980	0.174205
$563$	−31.3895	−1.32291	−0.661454	−	0.749986i	$-0.730060\pi$
$563$	−31.3895	−1.32291	−0.661454	+	0.749986i	$0.730060\pi$
$564$	8.18757	0.344759
$565$	1.18991	0.0500598
$566$	0.601061	0.0252645
$567$	0	0
$568$	−15.8463	−0.664898
$569$	6.23334	0.261315	0.130658	−	0.991428i	$-0.458291\pi$
$569$	6.23334	0.261315	0.130658	+	0.991428i	$0.458291\pi$
$570$	−0.690250	−0.0289114
$571$	−23.9382	−1.00178	−0.500890	−	0.865511i	$-0.666994\pi$
$571$	−23.9382	−1.00178	−0.500890	+	0.865511i	$0.666994\pi$
$572$	−0.0363577	−0.00152019
$573$	5.66282	0.236568
$574$	0	0
$575$	14.0657	0.586581
$576$	3.91926	0.163303
$577$	16.8790	0.702683	0.351341	−	0.936247i	$-0.385726\pi$
$577$	16.8790	0.702683	0.351341	+	0.936247i	$0.385726\pi$
$578$	−11.3181	−0.470771
$579$	−20.1887	−0.839012
$580$	−0.267091	−0.0110903
$581$	0	0
$582$	5.25414	0.217791
$583$	6.39331	0.264784
$584$	−2.04251	−0.0845198
$585$	0.00546303	0.000225868 0
$586$	8.13984	0.336254
$587$	14.5227	0.599416	0.299708	−	0.954031i	$-0.403111\pi$
$587$	14.5227	0.599416	0.299708	+	0.954031i	$0.403111\pi$
$588$	0	0
$589$	19.2808	0.794453
$590$	0.385042	0.0158519
$591$	0.477679	0.0196491
$592$	0.801624	0.0329465
$593$	−7.13395	−0.292956	−0.146478	−	0.989214i	$-0.546794\pi$
$593$	−7.13395	−0.292956	−0.146478	+	0.989214i	$0.546794\pi$
$594$	1.36480	0.0559984
$595$	0	0
$596$	20.9350	0.857529
$597$	−3.25632	−0.133272
$598$	−0.0388036	−0.00158680
$599$	5.13045	0.209624	0.104812	−	0.994492i	$-0.466576\pi$
$599$	5.13045	0.209624	0.104812	+	0.994492i	$0.466576\pi$
$600$	13.3417	0.544674
$601$	37.6827	1.53711	0.768554	−	0.639785i	$-0.220977\pi$
$601$	37.6827	1.53711	0.768554	+	0.639785i	$0.220977\pi$
$602$	0	0
$603$	4.72149	0.192274
$604$	−16.5467	−0.673276
$605$	−2.73269	−0.111100
$606$	−14.4262	−0.586025
$607$	−15.7766	−0.640354	−0.320177	−	0.947358i	$-0.603742\pi$
$607$	−15.7766	−0.640354	−0.320177	+	0.947358i	$0.603742\pi$
$608$	−14.7036	−0.596309
$609$	0	0
$610$	−1.20478	−0.0487803
$611$	−0.101454	−0.00410439
$612$	−2.37374	−0.0959528
$613$	28.8922	1.16694	0.583472	−	0.812133i	$-0.301694\pi$
$613$	28.8922	1.16694	0.583472	+	0.812133i	$0.301694\pi$
$614$	14.8243	0.598259
$615$	0.332064	0.0133901
$616$	0	0
$617$	−23.0012	−0.925993	−0.462996	−	0.886360i	$-0.653226\pi$
$617$	−23.0012	−0.925993	−0.462996	+	0.886360i	$0.653226\pi$
$618$	−3.28712	−0.132227
$619$	13.3883	0.538121	0.269060	−	0.963123i	$-0.413287\pi$
$619$	13.3883	0.538121	0.269060	+	0.963123i	$0.413287\pi$
$620$	3.35307	0.134663
$621$	−2.87658	−0.115433
$622$	5.35852	0.214857
$623$	0	0
$624$	0.00687951	0.000275401 0
$625$	23.3582	0.934327
$626$	5.43237	0.217121
$627$	−4.21976	−0.168521
$628$	19.4565	0.776397
$629$	3.42737	0.136658
$630$	0	0
$631$	−0.0230585	−0.000917944 0	−0.000458972	−	1.00000i	$-0.500146\pi$
$631$	−0.0230585	−0.000917944 0	−0.000458972		1.00000i	$0.500146\pi$
$632$	−12.2636	−0.487821
$633$	0.492222	0.0195641
$634$	4.10699	0.163109
$635$	−5.51554	−0.218877
$636$	−5.09962	−0.202213
$637$	0	0
$638$	0.826815	0.0327339
$639$	5.80768	0.229748
$640$	−2.78477	−0.110078
$641$	−1.14808	−0.0453465	−0.0226732	−	0.999743i	$-0.507218\pi$
$641$	−1.14808	−0.0453465	−0.0226732	+	0.999743i	$0.507218\pi$
$642$	−9.75543	−0.385016
$643$	6.32451	0.249414	0.124707	−	0.992194i	$-0.460201\pi$
$643$	6.32451	0.249414	0.124707	+	0.992194i	$0.460201\pi$
$644$	0	0
$645$	−4.27115	−0.168177
$646$	−3.71639	−0.146219
$647$	18.0069	0.707925	0.353963	−	0.935260i	$-0.384834\pi$
$647$	18.0069	0.707925	0.353963	+	0.935260i	$0.384834\pi$
$648$	−2.72852	−0.107186
$649$	2.35391	0.0923990
$650$	−0.0659600	−0.00258716
$651$	0	0
$652$	13.9172	0.545038
$653$	−1.68150	−0.0658021	−0.0329010	−	0.999459i	$-0.510475\pi$
$653$	−1.68150	−0.0658021	−0.0329010	+	0.999459i	$0.510475\pi$
$654$	−11.5698	−0.452416
$655$	3.19112	0.124687
$656$	0.418163	0.0163265
$657$	0.748580	0.0292049
$658$	0	0
$659$	38.1251	1.48514	0.742571	−	0.669767i	$-0.233606\pi$
$659$	38.1251	1.48514	0.742571	+	0.669767i	$0.233606\pi$
$660$	−0.733847	−0.0285649
$661$	−7.33130	−0.285154	−0.142577	−	0.989784i	$-0.545539\pi$
$661$	−7.33130	−0.285154	−0.142577	+	0.989784i	$0.545539\pi$
$662$	−1.69002	−0.0656847
$663$	0.0294136	0.00114233
$664$	−31.2528	−1.21284
$665$	0	0
$666$	1.57184	0.0609076
$667$	−1.74267	−0.0674766
$668$	−12.5487	−0.485526
$669$	7.15813	0.276749
$670$	1.28554	0.0496646
$671$	−7.36530	−0.284334
$672$	0	0
$673$	11.6171	0.447806	0.223903	−	0.974611i	$-0.428120\pi$
$673$	11.6171	0.447806	0.223903	+	0.974611i	$0.428120\pi$
$674$	10.2978	0.396656
$675$	−4.88973	−0.188206
$676$	17.2597	0.663834
$677$	33.6999	1.29519	0.647597	−	0.761983i	$-0.275774\pi$
$677$	33.6999	1.29519	0.647597	+	0.761983i	$0.275774\pi$
$678$	2.93816	0.112839
$679$	0	0
$680$	−1.61988	−0.0621196
$681$	−7.34618	−0.281506
$682$	−10.3799	−0.397466
$683$	17.5803	0.672693	0.336346	−	0.941738i	$-0.390809\pi$
$683$	17.5803	0.672693	0.336346	+	0.941738i	$0.390809\pi$
$684$	3.36589	0.128698
$685$	−1.96956	−0.0752532
$686$	0	0
$687$	−9.96063	−0.380022
$688$	−5.37860	−0.205057
$689$	0.0631905	0.00240737
$690$	−0.783215	−0.0298165
$691$	8.94418	0.340252	0.170126	−	0.985422i	$-0.445582\pi$
$691$	8.94418	0.340252	0.170126	+	0.985422i	$0.445582\pi$
$692$	12.6427	0.480602
$693$	0	0
$694$	9.41365	0.357337
$695$	6.76379	0.256565
$696$	−1.65297	−0.0626558
$697$	1.78787	0.0677203
$698$	12.4829	0.472486
$699$	−21.3065	−0.805885
$700$	0	0
$701$	39.6138	1.49619	0.748097	−	0.663590i	$-0.230968\pi$
$701$	39.6138	1.49619	0.748097	+	0.663590i	$0.230968\pi$
$702$	0.0134895	0.000509128 0
$703$	−4.85990	−0.183295
$704$	6.52364	0.245869
$705$	−2.04776	−0.0771229
$706$	16.7366	0.629890
$707$	0	0
$708$	−1.87760	−0.0705644
$709$	17.1546	0.644256	0.322128	−	0.946696i	$-0.395602\pi$
$709$	17.1546	0.644256	0.322128	+	0.946696i	$0.395602\pi$
$710$	1.58127	0.0593441
$711$	4.49461	0.168561
$712$	13.6465	0.511425
$713$	21.8776	0.819324
$714$	0	0
$715$	0.00909325	0.000340068 0
$716$	30.2854	1.13182
$717$	−10.7115	−0.400028
$718$	24.7616	0.924094
$719$	11.0220	0.411053	0.205526	−	0.978652i	$-0.434109\pi$
$719$	11.0220	0.411053	0.205526	+	0.978652i	$0.434109\pi$
$720$	0.138857	0.00517488
$721$	0	0
$722$	−10.3092	−0.383668
$723$	−15.3877	−0.572276
$724$	4.19716	0.155986
$725$	−2.96227	−0.110016
$726$	−6.74764	−0.250429
$727$	25.4401	0.943520	0.471760	−	0.881727i	$-0.343619\pi$
$727$	25.4401	0.943520	0.471760	+	0.881727i	$0.343619\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0.203818	0.00754365
$731$	−22.9964	−0.850552
$732$	5.87494	0.217144
$733$	−32.5187	−1.20110	−0.600552	−	0.799586i	$-0.705053\pi$
$733$	−32.5187	−1.20110	−0.600552	+	0.799586i	$0.705053\pi$
$734$	4.83244	0.178369
$735$	0	0
$736$	−16.6839	−0.614977
$737$	7.85896	0.289489
$738$	0.819942	0.0301825
$739$	19.0598	0.701126	0.350563	−	0.936539i	$-0.385990\pi$
$739$	19.0598	0.701126	0.350563	+	0.936539i	$0.385990\pi$
$740$	−0.845172	−0.0310691
$741$	−0.0417075	−0.00153216
$742$	0	0
$743$	15.1611	0.556207	0.278104	−	0.960551i	$-0.410294\pi$
$743$	15.1611	0.556207	0.278104	+	0.960551i	$0.410294\pi$
$744$	20.7515	0.760788
$745$	−5.23594	−0.191830
$746$	18.9144	0.692504
$747$	11.4541	0.419084
$748$	−3.95111	−0.144467
$749$	0	0
$750$	−2.69271	−0.0983238
$751$	−1.92281	−0.0701645	−0.0350822	−	0.999384i	$-0.511169\pi$
$751$	−1.92281	−0.0701645	−0.0350822	+	0.999384i	$0.511169\pi$
$752$	−2.57871	−0.0940358
$753$	−3.83734	−0.139840
$754$	0.00817212	0.000297611 0
$755$	4.13842	0.150613
$756$	0	0
$757$	−15.5738	−0.566041	−0.283021	−	0.959114i	$-0.591336\pi$
$757$	−15.5738	−0.566041	−0.283021	+	0.959114i	$0.591336\pi$
$758$	13.1028	0.475914
$759$	−4.78809	−0.173797
$760$	2.29694	0.0833189
$761$	−52.7733	−1.91303	−0.956515	−	0.291683i	$-0.905785\pi$
$761$	−52.7733	−1.91303	−0.956515	+	0.291683i	$0.905785\pi$
$762$	−13.6191	−0.493369
$763$	0	0
$764$	−7.51850	−0.272010
$765$	0.593686	0.0214648
$766$	−23.3653	−0.844222
$767$	0.0232657	0.000840076 0
$768$	−14.7148	−0.530973
$769$	23.4101	0.844189	0.422095	−	0.906552i	$-0.361295\pi$
$769$	23.4101	0.844189	0.422095	+	0.906552i	$0.361295\pi$
$770$	0	0
$771$	29.6461	1.06768
$772$	26.8044	0.964710
$773$	−5.06047	−0.182012	−0.0910062	−	0.995850i	$-0.529008\pi$
$773$	−5.06047	−0.182012	−0.0910062	+	0.995850i	$0.529008\pi$
$774$	−10.5465	−0.379085
$775$	37.1885	1.33585
$776$	−17.4842	−0.627646
$777$	0	0
$778$	3.62819	0.130077
$779$	−2.53514	−0.0908309
$780$	−0.00725323	−0.000259707 0
$781$	9.66692	0.345910
$782$	−4.21692	−0.150797
$783$	0.605814	0.0216500
$784$	0	0
$785$	−4.86617	−0.173681
$786$	7.87961	0.281056
$787$	−36.6471	−1.30633	−0.653164	−	0.757216i	$-0.726559\pi$
$787$	−36.6471	−1.30633	−0.653164	+	0.757216i	$0.726559\pi$
$788$	−0.634211	−0.0225928
$789$	−15.0006	−0.534035
$790$	1.22376	0.0435395
$791$	0	0
$792$	−4.54164	−0.161380
$793$	−0.0727976	−0.00258512
$794$	25.4673	0.903802
$795$	1.27544	0.0452353
$796$	4.32340	0.153239
$797$	−2.67790	−0.0948562	−0.0474281	−	0.998875i	$-0.515103\pi$
$797$	−2.67790	−0.0948562	−0.0474281	+	0.998875i	$0.515103\pi$
$798$	0	0
$799$	−11.0253	−0.390049
$800$	−28.3600	−1.00268
$801$	−5.00144	−0.176717
$802$	−1.86203	−0.0657507
$803$	1.24602	0.0439710
$804$	−6.26870	−0.221080
$805$	0	0
$806$	−0.102593	−0.00361370
$807$	−5.76098	−0.202796
$808$	48.0061	1.68885
$809$	−26.0424	−0.915603	−0.457801	−	0.889055i	$-0.651363\pi$
$809$	−26.0424	−0.915603	−0.457801	+	0.889055i	$0.651363\pi$
$810$	0.272273	0.00956670
$811$	−0.407542	−0.0143108	−0.00715538	−	0.999974i	$-0.502278\pi$
$811$	−0.407542	−0.0143108	−0.00715538	+	0.999974i	$0.502278\pi$
$812$	0	0
$813$	5.41966	0.190076
$814$	2.61634	0.0917027
$815$	−3.48076	−0.121926
$816$	0.747619	0.0261719
$817$	32.6082	1.14081
$818$	4.35701	0.152339
$819$	0	0
$820$	−0.440879	−0.0153962
$821$	−38.1013	−1.32974	−0.664871	−	0.746958i	$-0.731514\pi$
$821$	−38.1013	−1.32974	−0.664871	+	0.746958i	$0.731514\pi$
$822$	−4.86331	−0.169628
$823$	−3.55328	−0.123860	−0.0619298	−	0.998081i	$-0.519725\pi$
$823$	−3.55328	−0.123860	−0.0619298	+	0.998081i	$0.519725\pi$
$824$	10.9385	0.381062
$825$	−8.13900	−0.283364
$826$	0	0
$827$	12.3979	0.431118	0.215559	−	0.976491i	$-0.430843\pi$
$827$	12.3979	0.431118	0.215559	+	0.976491i	$0.430843\pi$
$828$	3.81922	0.132727
$829$	25.6106	0.889492	0.444746	−	0.895657i	$-0.353294\pi$
$829$	25.6106	0.889492	0.444746	+	0.895657i	$0.353294\pi$
$830$	3.11865	0.108250
$831$	−20.4003	−0.707678
$832$	0.0644787	0.00223540
$833$	0	0
$834$	16.7014	0.578321
$835$	3.13851	0.108613
$836$	5.60256	0.193768
$837$	−7.60543	−0.262882
$838$	13.4004	0.462908
$839$	−20.2672	−0.699703	−0.349851	−	0.936805i	$-0.613768\pi$
$839$	−20.2672	−0.699703	−0.349851	+	0.936805i	$0.613768\pi$
$840$	0	0
$841$	−28.6330	−0.987344
$842$	−6.18264	−0.213068
$843$	5.03670	0.173473
$844$	−0.653521	−0.0224951
$845$	−4.31674	−0.148500
$846$	−5.05638	−0.173842
$847$	0	0
$848$	1.60615	0.0551553
$849$	0.733053	0.0251583
$850$	−7.16810	−0.245864
$851$	−5.51445	−0.189033
$852$	−7.71082	−0.264168
$853$	17.4438	0.597264	0.298632	−	0.954368i	$-0.403470\pi$
$853$	17.4438	0.597264	0.298632	+	0.954368i	$0.403470\pi$
$854$	0	0
$855$	−0.841828	−0.0287899
$856$	32.4631	1.10957
$857$	38.4532	1.31354	0.656769	−	0.754092i	$-0.271923\pi$
$857$	38.4532	1.31354	0.656769	+	0.754092i	$0.271923\pi$
$858$	0.0224534	0.000766545 0
$859$	−1.73122	−0.0590686	−0.0295343	−	0.999564i	$-0.509402\pi$
$859$	−1.73122	−0.0590686	−0.0295343	+	0.999564i	$0.509402\pi$
$860$	5.67079	0.193372
$861$	0	0
$862$	6.74237	0.229646
$863$	−22.5174	−0.766502	−0.383251	−	0.923644i	$-0.625196\pi$
$863$	−22.5174	−0.766502	−0.383251	+	0.923644i	$0.625196\pi$
$864$	5.79990	0.197317
$865$	−3.16200	−0.107511
$866$	5.00233	0.169986
$867$	−13.8035	−0.468792
$868$	0	0
$869$	7.48131	0.253786
$870$	0.164947	0.00559222
$871$	0.0776769	0.00263198
$872$	38.5009	1.30380
$873$	6.40794	0.216876
$874$	5.97946	0.202258
$875$	0	0
$876$	−0.993886	−0.0335803
$877$	−4.48494	−0.151446	−0.0757228	−	0.997129i	$-0.524126\pi$
$877$	−4.48494	−0.151446	−0.0757228	+	0.997129i	$0.524126\pi$
$878$	−13.8365	−0.466959
$879$	9.92734	0.334841
$880$	0.231128	0.00779132
$881$	−40.1306	−1.35203	−0.676016	−	0.736887i	$-0.736295\pi$
$881$	−40.1306	−1.35203	−0.676016	+	0.736887i	$0.736295\pi$
$882$	0	0
$883$	−15.5050	−0.521784	−0.260892	−	0.965368i	$-0.584017\pi$
$883$	−15.5050	−0.521784	−0.260892	+	0.965368i	$0.584017\pi$
$884$	−0.0390522	−0.00131347
$885$	0.469597	0.0157853
$886$	−20.2423	−0.680053
$887$	−38.3024	−1.28607	−0.643034	−	0.765838i	$-0.722325\pi$
$887$	−38.3024	−1.28607	−0.643034	+	0.765838i	$0.722325\pi$
$888$	−5.23061	−0.175528
$889$	0	0
$890$	−1.36176	−0.0456462
$891$	1.66451	0.0557631
$892$	−9.50381	−0.318211
$893$	15.6336	0.523158
$894$	−12.9288	−0.432403
$895$	−7.57454	−0.253189
$896$	0	0
$897$	−0.0473248	−0.00158013
$898$	1.48815	0.0496601
$899$	−4.60747	−0.153668
$900$	6.49207	0.216402
$901$	6.86713	0.228777
$902$	1.36480	0.0454429
$903$	0	0
$904$	−9.77730	−0.325188
$905$	−1.04973	−0.0348943
$906$	10.2187	0.339494
$907$	−47.6553	−1.58237	−0.791184	−	0.611578i	$-0.790535\pi$
$907$	−47.6553	−1.58237	−0.791184	+	0.611578i	$0.790535\pi$
$908$	9.75348	0.323681
$909$	−17.5942	−0.583562
$910$	0	0
$911$	−49.7822	−1.64936	−0.824679	−	0.565601i	$-0.808644\pi$
$911$	−49.7822	−1.64936	−0.824679	+	0.565601i	$0.808644\pi$
$912$	−1.06010	−0.0351035
$913$	19.0655	0.630975
$914$	23.0444	0.762240
$915$	−1.46935	−0.0485753
$916$	13.2247	0.436956
$917$	0	0
$918$	1.46595	0.0483835
$919$	58.0170	1.91381	0.956903	−	0.290409i	$-0.0937914\pi$
$919$	58.0170	1.91381	0.956903	+	0.290409i	$0.0937914\pi$
$920$	2.60630	0.0859272
$921$	18.0797	0.595745
$922$	22.5835	0.743749
$923$	0.0955465	0.00314495
$924$	0	0
$925$	−9.37369	−0.308205
$926$	−19.5169	−0.641365
$927$	−4.00896	−0.131672
$928$	3.51366	0.115342
$929$	57.4678	1.88546	0.942728	−	0.333561i	$-0.108250\pi$
$929$	57.4678	1.88546	0.942728	+	0.333561i	$0.108250\pi$
$930$	−2.07075	−0.0679026
$931$	0	0
$932$	28.2885	0.926620
$933$	6.53524	0.213954
$934$	−3.36950	−0.110254
$935$	0.988194	0.0323174
$936$	−0.0448889	−0.00146724
$937$	−3.15232	−0.102982	−0.0514909	−	0.998673i	$-0.516397\pi$
$937$	−3.15232	−0.102982	−0.0514909	+	0.998673i	$0.516397\pi$
$938$	0	0
$939$	6.62531	0.216209
$940$	2.71879	0.0886773
$941$	−6.38432	−0.208123	−0.104062	−	0.994571i	$-0.533184\pi$
$941$	−6.38432	−0.208123	−0.104062	+	0.994571i	$0.533184\pi$
$942$	−12.0157	−0.391493
$943$	−2.87658	−0.0936744
$944$	0.591356	0.0192470
$945$	0	0
$946$	−17.5547	−0.570752
$947$	4.84026	0.157287	0.0786436	−	0.996903i	$-0.474941\pi$
$947$	4.84026	0.157287	0.0786436	+	0.996903i	$0.474941\pi$
$948$	−5.96747	−0.193814
$949$	0.0123155	0.000399777 0
$950$	10.1641	0.329768
$951$	5.00888	0.162424
$952$	0	0
$953$	−47.9739	−1.55403	−0.777014	−	0.629484i	$-0.783266\pi$
$953$	−47.9739	−1.55403	−0.777014	+	0.629484i	$0.783266\pi$
$954$	3.14937	0.101964
$955$	1.88042	0.0608488
$956$	14.2216	0.459959
$957$	1.00838	0.0325964
$958$	9.29680	0.300366
$959$	0	0
$960$	1.30144	0.0420039
$961$	26.8425	0.865887
$962$	0.0258596	0.000833745 0
$963$	−11.8977	−0.383398
$964$	20.4302	0.658013
$965$	−6.70392	−0.215807
$966$	0	0
$967$	5.25867	0.169107	0.0845537	−	0.996419i	$-0.473054\pi$
$967$	5.25867	0.169107	0.0845537	+	0.996419i	$0.473054\pi$
$968$	22.4541	0.721702
$969$	−4.53250	−0.145605
$970$	1.74471	0.0560193
$971$	−56.4143	−1.81042	−0.905210	−	0.424964i	$-0.860287\pi$
$971$	−56.4143	−1.81042	−0.905210	+	0.424964i	$0.860287\pi$
$972$	−1.32769	−0.0425858
$973$	0	0
$974$	8.11320	0.259964
$975$	−0.0804447	−0.00257629
$976$	−1.85033	−0.0592277
$977$	6.39439	0.204575	0.102287	−	0.994755i	$-0.467384\pi$
$977$	6.39439	0.204575	0.102287	+	0.994755i	$0.467384\pi$
$978$	−8.59480	−0.274832
$979$	−8.32494	−0.266066
$980$	0	0
$981$	−14.1105	−0.450515
$982$	20.6906	0.660264
$983$	25.9542	0.827809	0.413905	−	0.910320i	$-0.364165\pi$
$983$	25.9542	0.827809	0.413905	+	0.910320i	$0.364165\pi$
$984$	−2.72852	−0.0869820
$985$	0.158620	0.00505404
$986$	0.888092	0.0282826
$987$	0	0
$988$	0.0553749	0.00176171
$989$	36.9999	1.17653
$990$	0.453200	0.0144037
$991$	48.0216	1.52546	0.762728	−	0.646719i	$-0.223859\pi$
$991$	48.0216	1.52546	0.762728	+	0.646719i	$0.223859\pi$
$992$	−44.1107	−1.40052
$993$	−2.06115	−0.0654086
$994$	0	0
$995$	−1.08131	−0.0342797
$996$	−15.2076	−0.481870
$997$	48.7312	1.54333	0.771667	−	0.636027i	$-0.219423\pi$
$997$	48.7312	1.54333	0.771667	+	0.636027i	$0.219423\pi$
$998$	−0.902973	−0.0285831
$999$	1.91702	0.0606517

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6027.2.a.bm.1.11	yes	16
7.6	odd	2		6027.2.a.bl.1.11	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6027.2.a.bl.1.11	✓	16	7.6	odd	2
6027.2.a.bm.1.11	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)

\(f(q)\)	\(=\)	\(q+0.819942 q^{2} +1.00000 q^{3} -1.32769 q^{4} +0.332064 q^{5} +0.819942 q^{6} -2.72852 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+0.819942 q^{2} +1.00000 q^{3} -1.32769 q^{4} +0.332064 q^{5} +0.819942 q^{6} -2.72852 q^{8} +1.00000 q^{9} +0.272273 q^{10} +1.66451 q^{11} -1.32769 q^{12} +0.0164518 q^{13} +0.332064 q^{15} +0.418163 q^{16} +1.78787 q^{17} +0.819942 q^{18} -2.53514 q^{19} -0.440879 q^{20} +1.36480 q^{22} -2.87658 q^{23} -2.72852 q^{24} -4.88973 q^{25} +0.0134895 q^{26} +1.00000 q^{27} +0.605814 q^{29} +0.272273 q^{30} -7.60543 q^{31} +5.79990 q^{32} +1.66451 q^{33} +1.46595 q^{34} -1.32769 q^{36} +1.91702 q^{37} -2.07867 q^{38} +0.0164518 q^{39} -0.906041 q^{40} +1.00000 q^{41} -12.8625 q^{43} -2.20996 q^{44} +0.332064 q^{45} -2.35863 q^{46} -6.16676 q^{47} +0.418163 q^{48} -4.00930 q^{50} +1.78787 q^{51} -0.0218429 q^{52} +3.84096 q^{53} +0.819942 q^{54} +0.552722 q^{55} -2.53514 q^{57} +0.496733 q^{58} +1.41418 q^{59} -0.440879 q^{60} -4.42492 q^{61} -6.23601 q^{62} +3.91926 q^{64} +0.00546303 q^{65} +1.36480 q^{66} +4.72149 q^{67} -2.37374 q^{68} -2.87658 q^{69} +5.80768 q^{71} -2.72852 q^{72} +0.748580 q^{73} +1.57184 q^{74} -4.88973 q^{75} +3.36589 q^{76} +0.0134895 q^{78} +4.49461 q^{79} +0.138857 q^{80} +1.00000 q^{81} +0.819942 q^{82} +11.4541 q^{83} +0.593686 q^{85} -10.5465 q^{86} +0.605814 q^{87} -4.54164 q^{88} -5.00144 q^{89} +0.272273 q^{90} +3.81922 q^{92} -7.60543 q^{93} -5.05638 q^{94} -0.841828 q^{95} +5.79990 q^{96} +6.40794 q^{97} +1.66451 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 4 q^{2} + 16 q^{3} + 12 q^{4} - 12 q^{5} - 4 q^{6} - 12 q^{8} + 16 q^{9}+O(q^{10}) \) 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

Fields

Representations

Groups

Database

Properties

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