LMFDB - Embedded newform 6027.2.a.bl.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	$-0.523657\pi$
$5$	−0.332064	−0.148503	−0.0742517	+	0.997240i	$0.523657\pi$
$6$	−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	$-0.500726\pi$
$13$	−0.0164518	−0.00456290	−0.00228145	+	0.999997i	$0.500726\pi$
$14$	0	0
$15$	0.332064	0.0857385
$16$	0.418163	0.104541
$17$	−1.78787	−0.433622	−0.216811	−	0.976214i	$-0.569565\pi$
$17$	−1.78787	−0.433622	−0.216811	+	0.976214i	$0.569565\pi$
$18$	0.819942	0.193262
$19$	2.53514	0.581601	0.290801	−	0.956784i	$-0.406078\pi$
$19$	2.53514	0.581601	0.290801	+	0.956784i	$0.406078\pi$
$20$	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	$-0.260680\pi$
$31$	7.60543	1.36597	0.682987	+	0.730430i	$0.260680\pi$
$32$	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	$-0.351511\pi$
$47$	6.16676	0.899514	0.449757	+	0.893151i	$0.351511\pi$
$48$	−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	$-0.529344\pi$
$59$	−1.41418	−0.184110	−0.0920551	+	0.995754i	$0.529344\pi$
$60$	−0.440879	−0.0569172
$61$	4.42492	0.566552	0.283276	−	0.959038i	$-0.408579\pi$
$61$	4.42492	0.566552	0.283276	+	0.959038i	$0.408579\pi$
$62$	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	$-0.513949\pi$
$73$	−0.748580	−0.0876147	−0.0438073	+	0.999040i	$0.513949\pi$
$74$	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	$-0.716383\pi$
$83$	−11.4541	−1.25725	−0.628626	+	0.777707i	$0.716383\pi$
$84$	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	$-0.414603\pi$
$89$	5.00144	0.530152	0.265076	+	0.964228i	$0.414603\pi$
$90$	−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	$-0.605470\pi$
$97$	−6.40794	−0.650628	−0.325314	+	0.945606i	$0.605470\pi$
$98$	0	0
$99$	1.66451	0.167289
$100$	6.49207	0.649207
$101$	17.5942	1.75069	0.875344	−	0.483501i	$-0.160635\pi$
$101$	17.5942	1.75069	0.875344	+	0.483501i	$0.160635\pi$
$102$	1.46595	0.145150
$103$	4.00896	0.395015	0.197507	−	0.980301i	$-0.436715\pi$
$103$	4.00896	0.395015	0.197507	+	0.980301i	$0.436715\pi$
$104$	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	$-0.637904\pi$
$131$	−9.60996	−0.839626	−0.419813	+	0.907611i	$0.637904\pi$
$132$	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	$-0.831945\pi$
$139$	−20.3689	−1.72767	−0.863836	+	0.503773i	$0.831945\pi$
$140$	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	$-0.301184\pi$
$157$	14.6543	1.16954	0.584771	+	0.811198i	$0.301184\pi$
$158$	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	$-0.619167\pi$
$167$	−9.45153	−0.731381	−0.365691	+	0.930736i	$0.619167\pi$
$168$	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	$-0.382100\pi$
$173$	9.52226	0.723964	0.361982	+	0.932185i	$0.382100\pi$
$174$	−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	$-0.462516\pi$
$181$	3.16124	0.234973	0.117487	+	0.993074i	$0.462516\pi$
$182$	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	$-0.463179\pi$
$199$	3.25632	0.230835	0.115417	+	0.993317i	$0.463179\pi$
$200$	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	$-0.577040\pi$
$223$	−7.15813	−0.479344	−0.239672	+	0.970854i	$0.577040\pi$
$224$	0	0
$225$	−4.88973	−0.325982
$226$	2.93816	0.195444
$227$	7.34618	0.487583	0.243791	−	0.969828i	$-0.421609\pi$
$227$	7.34618	0.487583	0.243791	+	0.969828i	$0.421609\pi$
$228$	3.36589	0.222912
$229$	9.96063	0.658217	0.329109	−	0.944292i	$-0.393252\pi$
$229$	9.96063	0.658217	0.329109	+	0.944292i	$0.393252\pi$
$230$	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	$-0.334946\pi$
$241$	15.3877	0.991212	0.495606	+	0.868547i	$0.334946\pi$
$242$	−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	$-0.461356\pi$
$251$	3.83734	0.242211	0.121105	+	0.992640i	$0.461356\pi$
$252$	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	$-0.875632\pi$
$257$	−29.6461	−1.84927	−0.924637	+	0.380850i	$0.875632\pi$
$258$	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	$-0.443805\pi$
$269$	5.76098	0.351253	0.175627	+	0.984457i	$0.443805\pi$
$270$	0.272273	0.0165700
$271$	−5.41966	−0.329221	−0.164611	−	0.986359i	$-0.552637\pi$
$271$	−5.41966	−0.329221	−0.164611	+	0.986359i	$0.552637\pi$
$272$	−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	$-0.506936\pi$
$283$	−0.733053	−0.0435755	−0.0217877	+	0.999763i	$0.506936\pi$
$284$	−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	$-0.593649\pi$
$293$	−9.92734	−0.579961	−0.289981	+	0.957033i	$0.593649\pi$
$294$	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	$-0.672554\pi$
$307$	−18.0797	−1.03186	−0.515930	+	0.856631i	$0.672554\pi$
$308$	0	0
$309$	−4.00896	−0.228062
$310$	−2.07075	−0.117611
$311$	−6.53524	−0.370579	−0.185290	−	0.982684i	$-0.559322\pi$
$311$	−6.53524	−0.370579	−0.185290	+	0.982684i	$0.559322\pi$
$312$	−0.0448889	−0.00254133
$313$	−6.62531	−0.374485	−0.187242	−	0.982314i	$-0.559955\pi$
$313$	−6.62531	−0.374485	−0.187242	+	0.982314i	$0.559955\pi$
$314$	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	$-0.633587\pi$
$349$	−15.2242	−0.814931	−0.407466	+	0.913221i	$0.633587\pi$
$350$	0	0
$351$	0.0164518	0.000878130 0
$352$	9.65398	0.514559
$353$	−20.4119	−1.08642	−0.543209	−	0.839598i	$-0.682791\pi$
$353$	−20.4119	−1.08642	−0.543209	+	0.839598i	$0.682791\pi$
$354$	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	$-0.549158\pi$
$367$	−5.89363	−0.307645	−0.153823	+	0.988098i	$0.549158\pi$
$368$	−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	$-0.240429\pi$
$383$	28.4963	1.45609	0.728046	+	0.685529i	$0.240429\pi$
$384$	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	$-0.784489\pi$
$397$	−31.0599	−1.55885	−0.779426	+	0.626494i	$0.784489\pi$
$398$	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	$-0.541939\pi$
$409$	−5.31380	−0.262751	−0.131375	+	0.991333i	$0.541939\pi$
$410$	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	$-0.630714\pi$
$419$	−16.3430	−0.798410	−0.399205	+	0.916862i	$0.630714\pi$
$420$	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	$-0.546831\pi$
$433$	−6.10083	−0.293187	−0.146594	+	0.989197i	$0.546831\pi$
$434$	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	$-0.368072\pi$
$439$	16.8750	0.805398	0.402699	+	0.915333i	$0.368072\pi$
$440$	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	$-0.721646\pi$
$461$	−27.5428	−1.28280	−0.641399	+	0.767207i	$0.721646\pi$
$462$	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	$-0.469689\pi$
$467$	4.10944	0.190162	0.0950811	+	0.995470i	$0.469689\pi$
$468$	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	$-0.583403\pi$
$479$	−11.3384	−0.518063	−0.259031	+	0.965869i	$0.583403\pi$
$480$	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	$-0.310731\pi$
$503$	25.1272	1.12036	0.560182	+	0.828369i	$0.310731\pi$
$504$	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	$-0.443772\pi$
$509$	7.92927	0.351459	0.175729	+	0.984439i	$0.443772\pi$
$510$	−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	$-0.400555\pi$
$521$	14.0312	0.614716	0.307358	+	0.951594i	$0.400555\pi$
$522$	0.496733	0.0217414
$523$	−19.4953	−0.852469	−0.426234	−	0.904613i	$-0.640160\pi$
$523$	−19.4953	−0.852469	−0.426234	+	0.904613i	$0.640160\pi$
$524$	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	$-0.269940\pi$
$563$	31.3895	1.32291	0.661454	+	0.749986i	$0.269940\pi$
$564$	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	$-0.614274\pi$
$577$	−16.8790	−0.702683	−0.351341	+	0.936247i	$0.614274\pi$
$578$	−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	$-0.596889\pi$
$587$	−14.5227	−0.599416	−0.299708	+	0.954031i	$0.596889\pi$
$588$	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	$-0.453206\pi$
$593$	7.13395	0.292956	0.146478	+	0.989214i	$0.453206\pi$
$594$	−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	$-0.779023\pi$
$601$	−37.6827	−1.53711	−0.768554	+	0.639785i	$0.779023\pi$
$602$	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	$-0.396258\pi$
$607$	15.7766	0.640354	0.320177	+	0.947358i	$0.396258\pi$
$608$	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	$-0.586713\pi$
$619$	−13.3883	−0.538121	−0.269060	+	0.963123i	$0.586713\pi$
$620$	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	$-0.539799\pi$
$643$	−6.32451	−0.249414	−0.124707	+	0.992194i	$0.539799\pi$
$644$	0	0
$645$	−4.27115	−0.168177
$646$	−3.71639	−0.146219
$647$	−18.0069	−0.707925	−0.353963	−	0.935260i	$-0.615166\pi$
$647$	−18.0069	−0.707925	−0.353963	+	0.935260i	$0.615166\pi$
$648$	−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	$-0.454461\pi$
$661$	7.33130	0.285154	0.142577	+	0.989784i	$0.454461\pi$
$662$	−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	$-0.724226\pi$
$677$	−33.6999	−1.29519	−0.647597	+	0.761983i	$0.724226\pi$
$678$	−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	$-0.554418\pi$
$691$	−8.94418	−0.340252	−0.170126	+	0.985422i	$0.554418\pi$
$692$	−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	$-0.565891\pi$
$719$	−11.0220	−0.411053	−0.205526	+	0.978652i	$0.565891\pi$
$720$	−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	$-0.656381\pi$
$727$	−25.4401	−0.943520	−0.471760	+	0.881727i	$0.656381\pi$
$728$	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	$-0.294947\pi$
$733$	32.5187	1.20110	0.600552	+	0.799586i	$0.294947\pi$
$734$	−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	$-0.0942153\pi$
$761$	52.7733	1.91303	0.956515	+	0.291683i	$0.0942153\pi$
$762$	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	$-0.638705\pi$
$769$	−23.4101	−0.844189	−0.422095	+	0.906552i	$0.638705\pi$
$770$	0	0
$771$	29.6461	1.06768
$772$	26.8044	0.964710
$773$	5.06047	0.182012	0.0910062	−	0.995850i	$-0.470992\pi$
$773$	5.06047	0.182012	0.0910062	+	0.995850i	$0.470992\pi$
$774$	−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	$-0.273441\pi$
$787$	36.6471	1.30633	0.653164	+	0.757216i	$0.273441\pi$
$788$	−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	$-0.484897\pi$
$797$	2.67790	0.0948562	0.0474281	+	0.998875i	$0.484897\pi$
$798$	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	$-0.497722\pi$
$811$	0.407542	0.0143108	0.00715538	+	0.999974i	$0.497722\pi$
$812$	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	$-0.646706\pi$
$829$	−25.6106	−0.889492	−0.444746	+	0.895657i	$0.646706\pi$
$830$	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	$-0.386232\pi$
$839$	20.2672	0.699703	0.349851	+	0.936805i	$0.386232\pi$
$840$	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	$-0.596530\pi$
$853$	−17.4438	−0.597264	−0.298632	+	0.954368i	$0.596530\pi$
$854$	0	0
$855$	−0.841828	−0.0287899
$856$	32.4631	1.10957
$857$	−38.4532	−1.31354	−0.656769	−	0.754092i	$-0.728077\pi$
$857$	−38.4532	−1.31354	−0.656769	+	0.754092i	$0.728077\pi$
$858$	0.0224534	0.000766545 0
$859$	1.73122	0.0590686	0.0295343	−	0.999564i	$-0.490598\pi$
$859$	1.73122	0.0590686	0.0295343	+	0.999564i	$0.490598\pi$
$860$	−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	$-0.263705\pi$
$881$	40.1306	1.35203	0.676016	+	0.736887i	$0.263705\pi$
$882$	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	$-0.277675\pi$
$887$	38.3024	1.28607	0.643034	+	0.765838i	$0.277675\pi$
$888$	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	$-0.891750\pi$
$929$	−57.4678	−1.88546	−0.942728	+	0.333561i	$0.891750\pi$
$930$	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	$-0.483603\pi$
$937$	3.15232	0.102982	0.0514909	+	0.998673i	$0.483603\pi$
$938$	0	0
$939$	6.62531	0.216209
$940$	2.71879	0.0886773
$941$	6.38432	0.208123	0.104062	−	0.994571i	$-0.466816\pi$
$941$	6.38432	0.208123	0.104062	+	0.994571i	$0.466816\pi$
$942$	−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	$-0.139713\pi$
$971$	56.4143	1.81042	0.905210	+	0.424964i	$0.139713\pi$
$972$	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	$-0.635835\pi$
$983$	−25.9542	−0.827809	−0.413905	+	0.910320i	$0.635835\pi$
$984$	−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	$-0.780577\pi$
$997$	−48.7312	−1.54333	−0.771667	+	0.636027i	$0.780577\pi$
$998$	−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.bl.1.11	✓	16
7.6	odd	2		6027.2.a.bm.1.11	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6027.2.a.bl.1.11	✓	16	1.1	even	1	trivial
6027.2.a.bm.1.11	yes	16	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:	\( N \)	\(=\)	\( 6027 = 3 \cdot 7^{2} \cdot 41 \)
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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