LMFDB - Embedded newform 507.4.a.p.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [507,4,Mod(1,507)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(507, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("507.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 507 = 3 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	507.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:	$29.9139683729$
Analytic rank:	$0$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - 56x^{7} - 27x^{6} + 945x^{5} + 763x^{4} - 4139x^{3} - 2478x^{2} + 63x + 27 $ x^9 - 56x^7 - 27x^6 + 945x^5 + 763x^4 - 4139x^3 - 2478x^2 + 63*x + 27
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 13^{2} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$-4.14324$ of defining polynomial
Character	$\chi$	$=$	507.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.69820 q^{2} +3.00000 q^{3} +14.0731 q^{4} +4.47249 q^{5} +14.0946 q^{6} +27.2096 q^{7} +28.5326 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q+4.69820 q^{2} +3.00000 q^{3} +14.0731 q^{4} +4.47249 q^{5} +14.0946 q^{6} +27.2096 q^{7} +28.5326 q^{8} +9.00000 q^{9} +21.0127 q^{10} -5.99207 q^{11} +42.2193 q^{12} +127.836 q^{14} +13.4175 q^{15} +21.4672 q^{16} +105.037 q^{17} +42.2838 q^{18} -156.462 q^{19} +62.9418 q^{20} +81.6287 q^{21} -28.1519 q^{22} -175.423 q^{23} +85.5978 q^{24} -104.997 q^{25} +27.0000 q^{27} +382.923 q^{28} +204.886 q^{29} +63.0380 q^{30} +31.9570 q^{31} -127.404 q^{32} -17.9762 q^{33} +493.483 q^{34} +121.695 q^{35} +126.658 q^{36} +344.140 q^{37} -735.088 q^{38} +127.612 q^{40} +46.5921 q^{41} +383.508 q^{42} -173.286 q^{43} -84.3269 q^{44} +40.2524 q^{45} -824.175 q^{46} -265.613 q^{47} +64.4016 q^{48} +397.361 q^{49} -493.296 q^{50} +315.110 q^{51} +172.912 q^{53} +126.851 q^{54} -26.7995 q^{55} +776.360 q^{56} -469.385 q^{57} +962.596 q^{58} -137.566 q^{59} +188.825 q^{60} -58.9384 q^{61} +150.140 q^{62} +244.886 q^{63} -770.306 q^{64} -84.4558 q^{66} +211.668 q^{67} +1478.19 q^{68} -526.270 q^{69} +571.746 q^{70} -436.317 q^{71} +256.793 q^{72} +1159.11 q^{73} +1616.84 q^{74} -314.990 q^{75} -2201.90 q^{76} -163.042 q^{77} -1017.51 q^{79} +96.0118 q^{80} +81.0000 q^{81} +218.899 q^{82} -150.251 q^{83} +1148.77 q^{84} +469.775 q^{85} -814.131 q^{86} +614.659 q^{87} -170.969 q^{88} +565.984 q^{89} +189.114 q^{90} -2468.75 q^{92} +95.8709 q^{93} -1247.90 q^{94} -699.773 q^{95} -382.211 q^{96} -286.741 q^{97} +1866.88 q^{98} -53.9286 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q + 6 q^{2} + 27 q^{3} + 44 q^{4} + 33 q^{5} + 18 q^{6} + 83 q^{7} + 87 q^{8} + 81 q^{9}+O(q^{10}) $ 9 * q + 6 * q^2 + 27 * q^3 + 44 * q^4 + 33 * q^5 + 18 * q^6 + 83 * q^7 + 87 * q^8 + 81 * q^9	\( 9 q + 6 q^{2} + 27 q^{3} + 44 q^{4} + 33 q^{5} + 18 q^{6} + 83 q^{7} + 87 q^{8} + 81 q^{9} - 54 q^{10} + 85 q^{11} + 132 q^{12} + 158 q^{14} + 99 q^{15} + 216 q^{16} + 178 q^{17} + 54 q^{18} + 352 q^{19} + 402 q^{20} + 249 q^{21} - 630 q^{22} + 150 q^{23} + 261 q^{24} - 20 q^{25} + 243 q^{27} + 940 q^{28} - 97 q^{29} - 162 q^{30} + 717 q^{31} + 707 q^{32} + 255 q^{33} + 632 q^{34} - 418 q^{35} + 396 q^{36} + 1108 q^{37} - 660 q^{38} - 1506 q^{40} + 334 q^{41} + 474 q^{42} + 242 q^{43} - 307 q^{44} + 297 q^{45} + 979 q^{46} - 184 q^{47} + 648 q^{48} - 38 q^{49} - 2031 q^{50} + 534 q^{51} - 151 q^{53} + 162 q^{54} + 2064 q^{55} + 2276 q^{56} + 1056 q^{57} + 1161 q^{58} + 537 q^{59} + 1206 q^{60} - 1340 q^{61} + 347 q^{62} + 747 q^{63} + 893 q^{64} - 1890 q^{66} + 2308 q^{67} + 2785 q^{68} + 450 q^{69} - 1420 q^{70} + 96 q^{71} + 783 q^{72} + 2505 q^{73} - 1191 q^{74} - 60 q^{75} + 2409 q^{76} - 2142 q^{77} - 1591 q^{79} - 2671 q^{80} + 729 q^{81} + 1517 q^{82} + 1539 q^{83} + 2820 q^{84} + 4296 q^{85} - 3763 q^{86} - 291 q^{87} - 3716 q^{88} - 592 q^{89} - 486 q^{90} + 515 q^{92} + 2151 q^{93} - 692 q^{94} + 4158 q^{95} + 2121 q^{96} + 1445 q^{97} + 1457 q^{98} + 765 q^{99}+O(q^{100}) \) 9 * q + 6 * q^2 + 27 * q^3 + 44 * q^4 + 33 * q^5 + 18 * q^6 + 83 * q^7 + 87 * q^8 + 81 * q^9 - 54 * q^10 + 85 * q^11 + 132 * q^12 + 158 * q^14 + 99 * q^15 + 216 * q^16 + 178 * q^17 + 54 * q^18 + 352 * q^19 + 402 * q^20 + 249 * q^21 - 630 * q^22 + 150 * q^23 + 261 * q^24 - 20 * q^25 + 243 * q^27 + 940 * q^28 - 97 * q^29 - 162 * q^30 + 717 * q^31 + 707 * q^32 + 255 * q^33 + 632 * q^34 - 418 * q^35 + 396 * q^36 + 1108 * q^37 - 660 * q^38 - 1506 * q^40 + 334 * q^41 + 474 * q^42 + 242 * q^43 - 307 * q^44 + 297 * q^45 + 979 * q^46 - 184 * q^47 + 648 * q^48 - 38 * q^49 - 2031 * q^50 + 534 * q^51 - 151 * q^53 + 162 * q^54 + 2064 * q^55 + 2276 * q^56 + 1056 * q^57 + 1161 * q^58 + 537 * q^59 + 1206 * q^60 - 1340 * q^61 + 347 * q^62 + 747 * q^63 + 893 * q^64 - 1890 * q^66 + 2308 * q^67 + 2785 * q^68 + 450 * q^69 - 1420 * q^70 + 96 * q^71 + 783 * q^72 + 2505 * q^73 - 1191 * q^74 - 60 * q^75 + 2409 * q^76 - 2142 * q^77 - 1591 * q^79 - 2671 * q^80 + 729 * q^81 + 1517 * q^82 + 1539 * q^83 + 2820 * q^84 + 4296 * q^85 - 3763 * q^86 - 291 * q^87 - 3716 * q^88 - 592 * q^89 - 486 * q^90 + 515 * q^92 + 2151 * q^93 - 692 * q^94 + 4158 * q^95 + 2121 * q^96 + 1445 * q^97 + 1457 * q^98 + 765 * 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 $

$2$	4.69820	1.66106	0.830532	−	0.556970i	$-0.188036\pi$
$2$	4.69820	1.66106	0.830532	+	0.556970i	$0.188036\pi$
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	14.0731	1.75914
$5$	4.47249	0.400032	0.200016	−	0.979793i	$-0.435901\pi$
$5$	4.47249	0.400032	0.200016	+	0.979793i	$0.435901\pi$
$6$	14.0946	0.959016
$7$	27.2096	1.46918	0.734589	−	0.678512i	$-0.237375\pi$
$7$	27.2096	1.46918	0.734589	+	0.678512i	$0.237375\pi$
$8$	28.5326	1.26098
$9$	9.00000	0.333333
$10$	21.0127	0.664479
$11$	−5.99207	−0.164243	−0.0821216	−	0.996622i	$-0.526170\pi$
$11$	−5.99207	−0.164243	−0.0821216	+	0.996622i	$0.526170\pi$
$12$	42.2193	1.01564
$13$	0	0
$13$	0	0
$14$	127.836	2.44040
$15$	13.4175	0.230958
$16$	21.4672	0.335425
$17$	105.037	1.49854	0.749268	−	0.662267i	$-0.230406\pi$
$17$	105.037	1.49854	0.749268	+	0.662267i	$0.230406\pi$
$18$	42.2838	0.553688
$19$	−156.462	−1.88920	−0.944599	−	0.328227i	$-0.893549\pi$
$19$	−156.462	−1.88920	−0.944599	+	0.328227i	$0.893549\pi$
$20$	62.9418	0.703711
$21$	81.6287	0.848231
$22$	−28.1519	−0.272819
$23$	−175.423	−1.59036	−0.795181	−	0.606372i	$-0.792624\pi$
$23$	−175.423	−1.59036	−0.795181	+	0.606372i	$0.792624\pi$
$24$	85.5978	0.728024
$25$	−104.997	−0.839975
$26$	0	0
$27$	27.0000	0.192450
$28$	382.923	2.58449
$29$	204.886	1.31195	0.655973	−	0.754785i	$-0.272259\pi$
$29$	204.886	1.31195	0.655973	+	0.754785i	$0.272259\pi$
$30$	63.0380	0.383637
$31$	31.9570	0.185150	0.0925748	−	0.995706i	$-0.470490\pi$
$31$	31.9570	0.185150	0.0925748	+	0.995706i	$0.470490\pi$
$32$	−127.404	−0.703813
$33$	−17.9762	−0.0948259
$34$	493.483	2.48916
$35$	121.695	0.587718
$36$	126.658	0.586379
$37$	344.140	1.52909	0.764545	−	0.644571i	$-0.222964\pi$
$37$	344.140	1.52909	0.764545	+	0.644571i	$0.222964\pi$
$38$	−735.088	−3.13808
$39$	0	0
$40$	127.612	0.504430
$41$	46.5921	0.177475	0.0887373	−	0.996055i	$-0.471717\pi$
$41$	46.5921	0.177475	0.0887373	+	0.996055i	$0.471717\pi$
$42$	383.508	1.40897
$43$	−173.286	−0.614554	−0.307277	−	0.951620i	$-0.599418\pi$
$43$	−173.286	−0.614554	−0.307277	+	0.951620i	$0.599418\pi$
$44$	−84.3269	−0.288926
$45$	40.2524	0.133344
$46$	−824.175	−2.64169
$47$	−265.613	−0.824334	−0.412167	−	0.911108i	$-0.635228\pi$
$47$	−265.613	−0.824334	−0.412167	+	0.911108i	$0.635228\pi$
$48$	64.4016	0.193658
$49$	397.361	1.15849
$50$	−493.296	−1.39525
$51$	315.110	0.865180
$52$	0	0
$53$	172.912	0.448137	0.224068	−	0.974573i	$-0.428066\pi$
$53$	172.912	0.448137	0.224068	+	0.974573i	$0.428066\pi$
$54$	126.851	0.319672
$55$	−26.7995	−0.0657025
$56$	776.360	1.85260
$57$	−469.385	−1.09073
$58$	962.596	2.17923
$59$	−137.566	−0.303551	−0.151775	−	0.988415i	$-0.548499\pi$
$59$	−137.566	−0.303551	−0.151775	+	0.988415i	$0.548499\pi$
$60$	188.825	0.406288
$61$	−58.9384	−0.123710	−0.0618548	−	0.998085i	$-0.519702\pi$
$61$	−58.9384	−0.123710	−0.0618548	+	0.998085i	$0.519702\pi$
$62$	150.140	0.307546
$63$	244.886	0.489726
$64$	−770.306	−1.50450
$65$	0	0
$66$	−84.4558	−0.157512
$67$	211.668	0.385961	0.192980	−	0.981203i	$-0.438185\pi$
$67$	211.668	0.385961	0.192980	+	0.981203i	$0.438185\pi$
$68$	1478.19	2.63613
$69$	−526.270	−0.918196
$70$	571.746	0.976238
$71$	−436.317	−0.729314	−0.364657	−	0.931142i	$-0.618814\pi$
$71$	−436.317	−0.729314	−0.364657	+	0.931142i	$0.618814\pi$
$72$	256.793	0.420325
$73$	1159.11	1.85840	0.929199	−	0.369579i	$-0.120498\pi$
$73$	1159.11	1.85840	0.929199	+	0.369579i	$0.120498\pi$
$74$	1616.84	2.53992
$75$	−314.990	−0.484960
$76$	−2201.90	−3.32336
$77$	−163.042	−0.241303
$78$	0	0
$79$	−1017.51	−1.44910	−0.724548	−	0.689224i	$-0.757952\pi$
$79$	−1017.51	−1.44910	−0.724548	+	0.689224i	$0.757952\pi$
$80$	96.0118	0.134181
$81$	81.0000	0.111111
$82$	218.899	0.294797
$83$	−150.251	−0.198701	−0.0993504	−	0.995053i	$-0.531676\pi$
$83$	−150.251	−0.198701	−0.0993504	+	0.995053i	$0.531676\pi$
$84$	1148.77	1.49215
$85$	469.775	0.599462
$86$	−814.131	−1.02081
$87$	614.659	0.757452
$88$	−170.969	−0.207107
$89$	565.984	0.674092	0.337046	−	0.941488i	$-0.390572\pi$
$89$	565.984	0.674092	0.337046	+	0.941488i	$0.390572\pi$
$90$	189.114	0.221493
$91$	0	0
$92$	−2468.75	−2.79766
$93$	95.8709	0.106896
$94$	−1247.90	−1.36927
$95$	−699.773	−0.755739
$96$	−382.211	−0.406346
$97$	−286.741	−0.300145	−0.150073	−	0.988675i	$-0.547951\pi$
$97$	−286.741	−0.300145	−0.150073	+	0.988675i	$0.547951\pi$
$98$	1866.88	1.92432
$99$	−53.9286	−0.0547478
$100$	−1477.63	−1.47763
$101$	−1218.57	−1.20052	−0.600258	−	0.799807i	$-0.704935\pi$
$101$	−1218.57	−1.20052	−0.600258	+	0.799807i	$0.704935\pi$
$102$	1480.45	1.43712
$103$	74.2485	0.0710283	0.0355142	−	0.999369i	$-0.488693\pi$
$103$	74.2485	0.0710283	0.0355142	+	0.999369i	$0.488693\pi$
$104$	0	0
$105$	365.084	0.339319
$106$	812.374	0.744384
$107$	−1253.46	−1.13249	−0.566246	−	0.824237i	$-0.691605\pi$
$107$	−1253.46	−1.13249	−0.566246	+	0.824237i	$0.691605\pi$
$108$	379.973	0.338546
$109$	722.643	0.635015	0.317507	−	0.948256i	$-0.397154\pi$
$109$	722.643	0.635015	0.317507	+	0.948256i	$0.397154\pi$
$110$	−125.909	−0.109136
$111$	1032.42	0.882820
$112$	584.113	0.492799
$113$	−855.913	−0.712544	−0.356272	−	0.934382i	$-0.615952\pi$
$113$	−855.913	−0.712544	−0.356272	+	0.934382i	$0.615952\pi$
$114$	−2205.26	−1.81177
$115$	−784.580	−0.636195
$116$	2883.38	2.30789
$117$	0	0
$118$	−646.310	−0.504218
$119$	2858.00	2.20162
$120$	382.836	0.291233
$121$	−1295.10	−0.973024
$122$	−276.905	−0.205490
$123$	139.776	0.102465
$124$	449.733	0.325703
$125$	−1028.66	−0.736048
$126$	1150.52	0.813467
$127$	726.104	0.507333	0.253667	−	0.967292i	$-0.418363\pi$
$127$	726.104	0.507333	0.253667	+	0.967292i	$0.418363\pi$
$128$	−2599.82	−1.79526
$129$	−519.857	−0.354813
$130$	0	0
$131$	−1456.91	−0.971685	−0.485843	−	0.874046i	$-0.661487\pi$
$131$	−1456.91	−0.971685	−0.485843	+	0.874046i	$0.661487\pi$
$132$	−252.981	−0.166812
$133$	−4257.25	−2.77557
$134$	994.460	0.641106
$135$	120.757	0.0769862
$136$	2996.97	1.88962
$137$	1806.80	1.12675	0.563377	−	0.826200i	$-0.309502\pi$
$137$	1806.80	1.12675	0.563377	+	0.826200i	$0.309502\pi$
$138$	−2472.52	−1.52518
$139$	1229.28	0.750117	0.375059	−	0.927001i	$-0.377623\pi$
$139$	1229.28	0.750117	0.375059	+	0.927001i	$0.377623\pi$
$140$	1712.62	1.03388
$141$	−796.840	−0.475929
$142$	−2049.90	−1.21144
$143$	0	0
$144$	193.205	0.111808
$145$	916.352	0.524820
$146$	5445.71	3.08692
$147$	1192.08	0.668853
$148$	4843.12	2.68988
$149$	446.127	0.245290	0.122645	−	0.992451i	$-0.460862\pi$
$149$	446.127	0.245290	0.122645	+	0.992451i	$0.460862\pi$
$150$	−1479.89	−0.805549
$151$	207.208	0.111671	0.0558355	−	0.998440i	$-0.482218\pi$
$151$	207.208	0.111671	0.0558355	+	0.998440i	$0.482218\pi$
$152$	−4464.26	−2.38223
$153$	945.329	0.499512
$154$	−766.002	−0.400819
$155$	142.927	0.0740658
$156$	0	0
$157$	1096.60	0.557442	0.278721	−	0.960372i	$-0.410089\pi$
$157$	1096.60	0.557442	0.278721	+	0.960372i	$0.410089\pi$
$158$	−4780.46	−2.40704
$159$	518.735	0.258732
$160$	−569.812	−0.281547
$161$	−4773.20	−2.33653
$162$	380.554	0.184563
$163$	−3154.25	−1.51571	−0.757853	−	0.652425i	$-0.773752\pi$
$163$	−3154.25	−1.51571	−0.757853	+	0.652425i	$0.773752\pi$
$164$	655.694	0.312202
$165$	−80.3984	−0.0379334
$166$	−705.908	−0.330055
$167$	3679.65	1.70503	0.852514	−	0.522705i	$-0.175077\pi$
$167$	3679.65	1.70503	0.852514	+	0.522705i	$0.175077\pi$
$168$	2329.08	1.06960
$169$	0	0
$170$	2207.10	0.995745
$171$	−1408.15	−0.629733
$172$	−2438.66	−1.08108
$173$	−3666.99	−1.61154	−0.805768	−	0.592231i	$-0.798247\pi$
$173$	−3666.99	−1.61154	−0.805768	+	0.592231i	$0.798247\pi$
$174$	2887.79	1.25818
$175$	−2856.92	−1.23407
$176$	−128.633	−0.0550913
$177$	−412.697	−0.175255
$178$	2659.11	1.11971
$179$	4173.50	1.74269	0.871347	−	0.490666i	$-0.163247\pi$
$179$	4173.50	1.74269	0.871347	+	0.490666i	$0.163247\pi$
$180$	566.476	0.234570
$181$	−499.197	−0.205000	−0.102500	−	0.994733i	$-0.532684\pi$
$181$	−499.197	−0.205000	−0.102500	+	0.994733i	$0.532684\pi$
$182$	0	0
$183$	−176.815	−0.0714238
$184$	−5005.29	−2.00541
$185$	1539.16	0.611685
$186$	450.421	0.177562
$187$	−629.386	−0.246124
$188$	−3738.00	−1.45012
$189$	734.659	0.282744
$190$	−3287.68	−1.25533
$191$	−3086.68	−1.16934	−0.584671	−	0.811271i	$-0.698776\pi$
$191$	−3086.68	−1.16934	−0.584671	+	0.811271i	$0.698776\pi$
$192$	−2310.92	−0.868625
$193$	1644.14	0.613201	0.306601	−	0.951838i	$-0.400808\pi$
$193$	1644.14	0.613201	0.306601	+	0.951838i	$0.400808\pi$
$194$	−1347.17	−0.498561
$195$	0	0
$196$	5592.10	2.03794
$197$	−1371.21	−0.495911	−0.247956	−	0.968771i	$-0.579759\pi$
$197$	−1371.21	−0.495911	−0.247956	+	0.968771i	$0.579759\pi$
$198$	−253.367	−0.0909396
$199$	−4627.33	−1.64836	−0.824178	−	0.566332i	$-0.808362\pi$
$199$	−4627.33	−1.64836	−0.824178	+	0.566332i	$0.808362\pi$
$200$	−2995.83	−1.05919
$201$	635.004	0.222835
$202$	−5725.08	−1.99413
$203$	5574.87	1.92748
$204$	4434.57	1.52197
$205$	208.383	0.0709955
$206$	348.834	0.117983
$207$	−1578.81	−0.530121
$208$	0	0
$209$	937.528	0.310288
$210$	1715.24	0.563631
$211$	5088.31	1.66016	0.830080	−	0.557644i	$-0.188294\pi$
$211$	5088.31	1.66016	0.830080	+	0.557644i	$0.188294\pi$
$212$	2433.40	0.788334
$213$	−1308.95	−0.421069
$214$	−5889.01	−1.88114
$215$	−775.019	−0.245841
$216$	770.380	0.242675
$217$	869.535	0.272018
$218$	3395.12	1.05480
$219$	3477.32	1.07295
$220$	−377.151	−0.115580
$221$	0	0
$222$	4850.52	1.46642
$223$	4744.82	1.42483	0.712415	−	0.701759i	$-0.247602\pi$
$223$	4744.82	1.42483	0.712415	+	0.701759i	$0.247602\pi$
$224$	−3466.60	−1.03403
$225$	−944.971	−0.279992
$226$	−4021.25	−1.18358
$227$	1145.52	0.334937	0.167469	−	0.985877i	$-0.446441\pi$
$227$	1145.52	0.334937	0.167469	+	0.985877i	$0.446441\pi$
$228$	−6605.70	−1.91874
$229$	1348.47	0.389123	0.194561	−	0.980890i	$-0.437672\pi$
$229$	1348.47	0.389123	0.194561	+	0.980890i	$0.437672\pi$
$230$	−3686.12	−1.05676
$231$	−489.125	−0.139316
$232$	5845.94	1.65433
$233$	−952.002	−0.267673	−0.133836	−	0.991003i	$-0.542730\pi$
$233$	−952.002	−0.267673	−0.133836	+	0.991003i	$0.542730\pi$
$234$	0	0
$235$	−1187.95	−0.329760
$236$	−1935.97	−0.533987
$237$	−3052.52	−0.836636
$238$	13427.5	3.65703
$239$	−3069.12	−0.830647	−0.415323	−	0.909674i	$-0.636332\pi$
$239$	−3069.12	−0.830647	−0.415323	+	0.909674i	$0.636332\pi$
$240$	288.036	0.0774692
$241$	2508.17	0.670396	0.335198	−	0.942148i	$-0.391197\pi$
$241$	2508.17	0.670396	0.335198	+	0.942148i	$0.391197\pi$
$242$	−6084.62	−1.61626
$243$	243.000	0.0641500
$244$	−829.446	−0.217622
$245$	1777.19	0.463432
$246$	656.697	0.170201
$247$	0	0
$248$	911.815	0.233469
$249$	−450.752	−0.114720
$250$	−4832.85	−1.22262
$251$	3405.91	0.856490	0.428245	−	0.903663i	$-0.359132\pi$
$251$	3405.91	0.856490	0.428245	+	0.903663i	$0.359132\pi$
$252$	3446.31	0.861495
$253$	1051.15	0.261206
$254$	3411.38	0.842714
$255$	1409.33	0.346100
$256$	−6052.04	−1.47755
$257$	−4733.95	−1.14901	−0.574506	−	0.818501i	$-0.694806\pi$
$257$	−4733.95	−1.14901	−0.574506	+	0.818501i	$0.694806\pi$
$258$	−2442.39	−0.589367
$259$	9363.91	2.24651
$260$	0	0
$261$	1843.98	0.437315
$262$	−6844.85	−1.61403
$263$	−1866.11	−0.437526	−0.218763	−	0.975778i	$-0.570202\pi$
$263$	−1866.11	−0.437526	−0.218763	+	0.975778i	$0.570202\pi$
$264$	−512.908	−0.119573
$265$	773.346	0.179269
$266$	−20001.4	−4.61040
$267$	1697.95	0.389187
$268$	2978.83	0.678958
$269$	1580.19	0.358164	0.179082	−	0.983834i	$-0.442687\pi$
$269$	1580.19	0.358164	0.179082	+	0.983834i	$0.442687\pi$
$270$	567.342	0.127879
$271$	4922.09	1.10330	0.551652	−	0.834074i	$-0.313998\pi$
$271$	4922.09	1.10330	0.551652	+	0.834074i	$0.313998\pi$
$272$	2254.84	0.502646
$273$	0	0
$274$	8488.71	1.87161
$275$	629.148	0.137960
$276$	−7406.25	−1.61523
$277$	2687.08	0.582856	0.291428	−	0.956593i	$-0.405870\pi$
$277$	2687.08	0.582856	0.291428	+	0.956593i	$0.405870\pi$
$278$	5775.41	1.24599
$279$	287.613	0.0617165
$280$	3472.26	0.741098
$281$	883.753	0.187617	0.0938083	−	0.995590i	$-0.470096\pi$
$281$	883.753	0.187617	0.0938083	+	0.995590i	$0.470096\pi$
$282$	−3743.71	−0.790550
$283$	−469.776	−0.0986759	−0.0493379	−	0.998782i	$-0.515711\pi$
$283$	−469.776	−0.0986759	−0.0493379	+	0.998782i	$0.515711\pi$
$284$	−6140.33	−1.28296
$285$	−2099.32	−0.436326
$286$	0	0
$287$	1267.75	0.260742
$288$	−1146.63	−0.234604
$289$	6119.68	1.24561
$290$	4305.20	0.871760
$291$	−860.222	−0.173289
$292$	16312.2	3.26918
$293$	−3403.76	−0.678668	−0.339334	−	0.940666i	$-0.610202\pi$
$293$	−3403.76	−0.678668	−0.339334	+	0.940666i	$0.610202\pi$
$294$	5600.65	1.11101
$295$	−615.261	−0.121430
$296$	9819.22	1.92814
$297$	−161.786	−0.0316086
$298$	2095.99	0.407442
$299$	0	0
$300$	−4432.89	−0.853110
$301$	−4715.03	−0.902889
$302$	973.504	0.185493
$303$	−3655.70	−0.693118
$304$	−3358.79	−0.633684
$305$	−263.602	−0.0494878
$306$	4441.35	0.829722
$307$	−888.862	−0.165244	−0.0826222	−	0.996581i	$-0.526329\pi$
$307$	−888.862	−0.165244	−0.0826222	+	0.996581i	$0.526329\pi$
$308$	−2294.50	−0.424484
$309$	222.745	0.0410082
$310$	671.501	0.123028
$311$	−1218.36	−0.222145	−0.111072	−	0.993812i	$-0.535429\pi$
$311$	−1218.36	−0.222145	−0.111072	+	0.993812i	$0.535429\pi$
$312$	0	0
$313$	−4870.41	−0.879526	−0.439763	−	0.898114i	$-0.644938\pi$
$313$	−4870.41	−0.879526	−0.439763	+	0.898114i	$0.644938\pi$
$314$	5152.06	0.925948
$315$	1095.25	0.195906
$316$	−14319.5	−2.54916
$317$	4340.93	0.769120	0.384560	−	0.923100i	$-0.374353\pi$
$317$	4340.93	0.769120	0.384560	+	0.923100i	$0.374353\pi$
$318$	2437.12	0.429770
$319$	−1227.69	−0.215478
$320$	−3445.19	−0.601849
$321$	−3760.38	−0.653844
$322$	−22425.4	−3.88112
$323$	−16434.2	−2.83103
$324$	1139.92	0.195460
$325$	0	0
$326$	−14819.3	−2.51769
$327$	2167.93	0.366626
$328$	1329.39	0.223791
$329$	−7227.23	−1.21109
$330$	−377.728	−0.0630098
$331$	907.289	0.150662	0.0753310	−	0.997159i	$-0.475999\pi$
$331$	907.289	0.150662	0.0753310	+	0.997159i	$0.475999\pi$
$332$	−2114.49	−0.349542
$333$	3097.26	0.509697
$334$	17287.7	2.83216
$335$	946.684	0.154397
$336$	1752.34	0.284518
$337$	8660.88	1.39997	0.699983	−	0.714160i	$-0.253191\pi$
$337$	8660.88	1.39997	0.699983	+	0.714160i	$0.253191\pi$
$338$	0	0
$339$	−2567.74	−0.411388
$340$	6611.19	1.05454
$341$	−191.488	−0.0304096
$342$	−6615.79	−1.04603
$343$	1479.14	0.232846
$344$	−4944.29	−0.774937
$345$	−2353.74	−0.367308
$346$	−17228.2	−2.67687
$347$	−347.605	−0.0537763	−0.0268882	−	0.999638i	$-0.508560\pi$
$347$	−347.605	−0.0537763	−0.0268882	+	0.999638i	$0.508560\pi$
$348$	8650.15	1.33246
$349$	10970.2	1.68258	0.841292	−	0.540581i	$-0.181795\pi$
$349$	10970.2	1.68258	0.841292	+	0.540581i	$0.181795\pi$
$350$	−13422.4	−2.04988
$351$	0	0
$352$	763.411	0.115596
$353$	−10384.8	−1.56580	−0.782901	−	0.622146i	$-0.786261\pi$
$353$	−10384.8	−1.56580	−0.782901	+	0.622146i	$0.786261\pi$
$354$	−1938.93	−0.291110
$355$	−1951.42	−0.291749
$356$	7965.15	1.18582
$357$	8574.00	1.27110
$358$	19608.0	2.89473
$359$	−8665.80	−1.27399	−0.636996	−	0.770867i	$-0.719823\pi$
$359$	−8665.80	−1.27399	−0.636996	+	0.770867i	$0.719823\pi$
$360$	1148.51	0.168143
$361$	17621.2	2.56907
$362$	−2345.33	−0.340518
$363$	−3885.29	−0.561776
$364$	0	0
$365$	5184.09	0.743419
$366$	−830.714	−0.118640
$367$	1234.22	0.175547	0.0877734	−	0.996140i	$-0.472025\pi$
$367$	1234.22	0.175547	0.0877734	+	0.996140i	$0.472025\pi$
$368$	−3765.85	−0.533447
$369$	419.329	0.0591582
$370$	7231.31	1.01605
$371$	4704.85	0.658393
$372$	1349.20	0.188045
$373$	−427.483	−0.0593410	−0.0296705	−	0.999560i	$-0.509446\pi$
$373$	−427.483	−0.0593410	−0.0296705	+	0.999560i	$0.509446\pi$
$374$	−2956.98	−0.408829
$375$	−3085.98	−0.424958
$376$	−7578.64	−1.03946
$377$	0	0
$378$	3451.57	0.469656
$379$	124.241	0.0168386	0.00841929	−	0.999965i	$-0.497320\pi$
$379$	124.241	0.0168386	0.00841929	+	0.999965i	$0.497320\pi$
$380$	−9847.98	−1.32945
$381$	2178.31	0.292909
$382$	−14501.8	−1.94235
$383$	9341.21	1.24625	0.623125	−	0.782122i	$-0.285863\pi$
$383$	9341.21	1.24625	0.623125	+	0.782122i	$0.285863\pi$
$384$	−7799.46	−1.03650
$385$	−729.202	−0.0965288
$386$	7724.51	1.01857
$387$	−1559.57	−0.204851
$388$	−4035.33	−0.527997
$389$	11368.9	1.48182	0.740908	−	0.671607i	$-0.234396\pi$
$389$	11368.9	1.48182	0.740908	+	0.671607i	$0.234396\pi$
$390$	0	0
$391$	−18425.9	−2.38321
$392$	11337.7	1.46082
$393$	−4370.73	−0.561003
$394$	−6442.21	−0.823741
$395$	−4550.80	−0.579685
$396$	−758.942	−0.0963088
$397$	−12077.8	−1.52687	−0.763436	−	0.645883i	$-0.776489\pi$
$397$	−12077.8	−1.52687	−0.763436	+	0.645883i	$0.776489\pi$
$398$	−21740.1	−2.73802
$399$	−12771.8	−1.60248
$400$	−2253.99	−0.281748
$401$	−4856.74	−0.604823	−0.302411	−	0.953178i	$-0.597792\pi$
$401$	−4856.74	−0.604823	−0.302411	+	0.953178i	$0.597792\pi$
$402$	2983.38	0.370143
$403$	0	0
$404$	−17149.0	−2.11187
$405$	362.272	0.0444480
$406$	26191.8	3.20167
$407$	−2062.11	−0.251143
$408$	8990.90	1.09097
$409$	2981.80	0.360490	0.180245	−	0.983622i	$-0.442311\pi$
$409$	2981.80	0.360490	0.180245	+	0.983622i	$0.442311\pi$
$410$	979.023	0.117928
$411$	5420.40	0.650532
$412$	1044.91	0.124949
$413$	−3743.10	−0.445971
$414$	−7417.57	−0.880565
$415$	−671.995	−0.0794866
$416$	0	0
$417$	3687.84	0.433080
$418$	4404.70	0.515409
$419$	7774.01	0.906408	0.453204	−	0.891407i	$-0.350281\pi$
$419$	7774.01	0.906408	0.453204	+	0.891407i	$0.350281\pi$
$420$	5137.86	0.596909
$421$	3959.71	0.458396	0.229198	−	0.973380i	$-0.426390\pi$
$421$	3959.71	0.458396	0.229198	+	0.973380i	$0.426390\pi$
$422$	23905.9	2.75763
$423$	−2390.52	−0.274778
$424$	4933.62	0.565089
$425$	−11028.5	−1.25873
$426$	−6149.71	−0.699424
$427$	−1603.69	−0.181752
$428$	−17640.1	−1.99221
$429$	0	0
$430$	−3641.19	−0.408358
$431$	4375.12	0.488961	0.244480	−	0.969654i	$-0.421383\pi$
$431$	4375.12	0.488961	0.244480	+	0.969654i	$0.421383\pi$
$432$	579.614	0.0645525
$433$	8992.74	0.998068	0.499034	−	0.866582i	$-0.333688\pi$
$433$	8992.74	0.998068	0.499034	+	0.866582i	$0.333688\pi$
$434$	4085.25	0.451839
$435$	2749.06	0.303005
$436$	10169.8	1.11708
$437$	27447.0	3.00451
$438$	16337.1	1.78223
$439$	−195.465	−0.0212506	−0.0106253	−	0.999944i	$-0.503382\pi$
$439$	−195.465	−0.0212506	−0.0106253	+	0.999944i	$0.503382\pi$
$440$	−764.659	−0.0828493
$441$	3576.25	0.386162
$442$	0	0
$443$	7369.97	0.790424	0.395212	−	0.918590i	$-0.370671\pi$
$443$	7369.97	0.790424	0.395212	+	0.918590i	$0.370671\pi$
$444$	14529.4	1.55300
$445$	2531.36	0.269658
$446$	22292.1	2.36673
$447$	1338.38	0.141618
$448$	−20959.7	−2.21038
$449$	−164.281	−0.0172670	−0.00863351	−	0.999963i	$-0.502748\pi$
$449$	−164.281	−0.0172670	−0.00863351	+	0.999963i	$0.502748\pi$
$450$	−4439.67	−0.465084
$451$	−279.183	−0.0291490
$452$	−12045.3	−1.25346
$453$	621.623	0.0644733
$454$	5381.88	0.556352
$455$	0	0
$456$	−13392.8	−1.37538
$457$	11687.0	1.19627	0.598133	−	0.801397i	$-0.295909\pi$
$457$	11687.0	1.19627	0.598133	+	0.801397i	$0.295909\pi$
$458$	6335.36	0.646358
$459$	2835.99	0.288393
$460$	−11041.5	−1.11915
$461$	1057.53	0.106842	0.0534211	−	0.998572i	$-0.482987\pi$
$461$	1057.53	0.106842	0.0534211	+	0.998572i	$0.482987\pi$
$462$	−2298.01	−0.231413
$463$	8554.74	0.858688	0.429344	−	0.903141i	$-0.358745\pi$
$463$	8554.74	0.858688	0.429344	+	0.903141i	$0.358745\pi$
$464$	4398.33	0.440059
$465$	428.782	0.0427619
$466$	−4472.70	−0.444622
$467$	−7705.26	−0.763506	−0.381753	−	0.924264i	$-0.624679\pi$
$467$	−7705.26	−0.763506	−0.381753	+	0.924264i	$0.624679\pi$
$468$	0	0
$469$	5759.40	0.567046
$470$	−5581.24	−0.547752
$471$	3289.81	0.321839
$472$	−3925.10	−0.382770
$473$	1038.34	0.100936
$474$	−14341.4	−1.38971
$475$	16428.0	1.58688
$476$	40220.9	3.87295
$477$	1556.21	0.149379
$478$	−14419.3	−1.37976
$479$	−4508.59	−0.430069	−0.215034	−	0.976606i	$-0.568986\pi$
$479$	−4508.59	−0.430069	−0.215034	+	0.976606i	$0.568986\pi$
$480$	−1709.44	−0.162551
$481$	0	0
$482$	11783.9	1.11357
$483$	−14319.6	−1.34899
$484$	−18226.0	−1.71168
$485$	−1282.45	−0.120068
$486$	1141.66	0.106557
$487$	6725.73	0.625815	0.312908	−	0.949784i	$-0.398697\pi$
$487$	6725.73	0.625815	0.312908	+	0.949784i	$0.398697\pi$
$488$	−1681.67	−0.155995
$489$	−9462.76	−0.875094
$490$	8349.61	0.769790
$491$	11517.5	1.05861	0.529303	−	0.848433i	$-0.322454\pi$
$491$	11517.5	1.05861	0.529303	+	0.848433i	$0.322454\pi$
$492$	1967.08	0.180250
$493$	21520.5	1.96600
$494$	0	0
$495$	−241.195	−0.0219008
$496$	686.026	0.0621038
$497$	−11872.0	−1.07149
$498$	−2117.72	−0.190557
$499$	19907.9	1.78598	0.892988	−	0.450080i	$-0.148605\pi$
$499$	19907.9	1.78598	0.892988	+	0.450080i	$0.148605\pi$
$500$	−14476.4	−1.29481
$501$	11038.9	0.984398
$502$	16001.6	1.42269
$503$	−5735.48	−0.508415	−0.254207	−	0.967150i	$-0.581815\pi$
$503$	−5735.48	−0.508415	−0.254207	+	0.967150i	$0.581815\pi$
$504$	6987.24	0.617533
$505$	−5450.04	−0.480244
$506$	4938.51	0.433881
$507$	0	0
$508$	10218.5	0.892469
$509$	9253.84	0.805834	0.402917	−	0.915237i	$-0.367996\pi$
$509$	9253.84	0.805834	0.402917	+	0.915237i	$0.367996\pi$
$510$	6621.29	0.574894
$511$	31538.8	2.73032
$512$	−7635.12	−0.659039
$513$	−4224.46	−0.363576
$514$	−22241.1	−1.90858
$515$	332.076	0.0284136
$516$	−7315.99	−0.624164
$517$	1591.57	0.135391
$518$	43993.5	3.73159
$519$	−11001.0	−0.930421
$520$	0	0
$521$	3887.42	0.326892	0.163446	−	0.986552i	$-0.447739\pi$
$521$	3887.42	0.326892	0.163446	+	0.986552i	$0.447739\pi$
$522$	8663.37	0.726409
$523$	4782.27	0.399836	0.199918	−	0.979813i	$-0.435932\pi$
$523$	4782.27	0.399836	0.199918	+	0.979813i	$0.435932\pi$
$524$	−20503.2	−1.70933
$525$	−8570.76	−0.712492
$526$	−8767.37	−0.726759
$527$	3356.65	0.277453
$528$	−385.898	−0.0318070
$529$	18606.4	1.52925
$530$	3633.34	0.297777
$531$	−1238.09	−0.101184
$532$	−59912.7	−4.88261
$533$	0	0
$534$	7977.32	0.646465
$535$	−5606.09	−0.453033
$536$	6039.44	0.486687
$537$	12520.5	1.00615
$538$	7424.06	0.594933
$539$	−2381.01	−0.190274
$540$	1699.43	0.135429
$541$	−14872.6	−1.18192	−0.590962	−	0.806699i	$-0.701252\pi$
$541$	−14872.6	−1.18192	−0.590962	+	0.806699i	$0.701252\pi$
$542$	23125.0	1.83266
$543$	−1497.59	−0.118357
$544$	−13382.0	−1.05469
$545$	3232.01	0.254026
$546$	0	0
$547$	16965.3	1.32611	0.663055	−	0.748570i	$-0.269259\pi$
$547$	16965.3	1.32611	0.663055	+	0.748570i	$0.269259\pi$
$548$	25427.3	1.98212
$549$	−530.446	−0.0412366
$550$	2955.86	0.229161
$551$	−32056.8	−2.47852
$552$	−15015.9	−1.15782
$553$	−27686.0	−2.12898
$554$	12624.5	0.968162
$555$	4617.49	0.353156
$556$	17299.8	1.31956
$557$	1934.05	0.147125	0.0735623	−	0.997291i	$-0.476563\pi$
$557$	1934.05	0.147125	0.0735623	+	0.997291i	$0.476563\pi$
$558$	1351.26	0.102515
$559$	0	0
$560$	2612.44	0.197135
$561$	−1888.16	−0.142100
$562$	4152.05	0.311643
$563$	6592.13	0.493473	0.246736	−	0.969083i	$-0.420642\pi$
$563$	6592.13	0.493473	0.246736	+	0.969083i	$0.420642\pi$
$564$	−11214.0	−0.837225
$565$	−3828.06	−0.285040
$566$	−2207.10	−0.163907
$567$	2203.98	0.163242
$568$	−12449.3	−0.919646
$569$	−14477.7	−1.06667	−0.533336	−	0.845903i	$-0.679062\pi$
$569$	−14477.7	−1.06667	−0.533336	+	0.845903i	$0.679062\pi$
$570$	−9863.03	−0.724766
$571$	−4998.21	−0.366320	−0.183160	−	0.983083i	$-0.558633\pi$
$571$	−4998.21	−0.366320	−0.183160	+	0.983083i	$0.558633\pi$
$572$	0	0
$573$	−9260.04	−0.675120
$574$	5956.15	0.433109
$575$	18418.9	1.33586
$576$	−6932.75	−0.501501
$577$	9291.73	0.670398	0.335199	−	0.942147i	$-0.391196\pi$
$577$	9291.73	0.670398	0.335199	+	0.942147i	$0.391196\pi$
$578$	28751.5	2.06904
$579$	4932.42	0.354032
$580$	12895.9	0.923230
$581$	−4088.26	−0.291927
$582$	−4041.50	−0.287844
$583$	−1036.10	−0.0736034
$584$	33072.3	2.34339
$585$	0	0
$586$	−15991.5	−1.12731
$587$	−5602.64	−0.393945	−0.196972	−	0.980409i	$-0.563111\pi$
$587$	−5602.64	−0.393945	−0.196972	+	0.980409i	$0.563111\pi$
$588$	16776.3	1.17660
$589$	−5000.04	−0.349784
$590$	−2890.62	−0.201703
$591$	−4113.62	−0.286314
$592$	7387.73	0.512895
$593$	−10885.8	−0.753839	−0.376919	−	0.926246i	$-0.623017\pi$
$593$	−10885.8	−0.753839	−0.376919	+	0.926246i	$0.623017\pi$
$594$	−760.102	−0.0525040
$595$	12782.4	0.880717
$596$	6278.38	0.431498
$597$	−13882.0	−0.951678
$598$	0	0
$599$	20403.2	1.39174	0.695872	−	0.718166i	$-0.255018\pi$
$599$	20403.2	1.39174	0.695872	+	0.718166i	$0.255018\pi$
$600$	−8987.50	−0.611522
$601$	−6312.19	−0.428419	−0.214209	−	0.976788i	$-0.568717\pi$
$601$	−6312.19	−0.428419	−0.214209	+	0.976788i	$0.568717\pi$
$602$	−22152.2	−1.49976
$603$	1905.01	0.128654
$604$	2916.05	0.196445
$605$	−5792.30	−0.389241
$606$	−17175.2	−1.15131
$607$	−21848.6	−1.46097	−0.730484	−	0.682930i	$-0.760705\pi$
$607$	−21848.6	−1.46097	−0.730484	+	0.682930i	$0.760705\pi$
$608$	19933.8	1.32964
$609$	16724.6	1.11283
$610$	−1238.45	−0.0822025
$611$	0	0
$612$	13303.7	0.878710
$613$	−1335.14	−0.0879704	−0.0439852	−	0.999032i	$-0.514005\pi$
$613$	−1335.14	−0.0879704	−0.0439852	+	0.999032i	$0.514005\pi$
$614$	−4176.05	−0.274482
$615$	625.148	0.0409893
$616$	−4652.00	−0.304277
$617$	18908.3	1.23374	0.616871	−	0.787064i	$-0.288400\pi$
$617$	18908.3	1.23374	0.616871	+	0.787064i	$0.288400\pi$
$618$	1046.50	0.0681173
$619$	6722.49	0.436510	0.218255	−	0.975892i	$-0.429964\pi$
$619$	6722.49	0.436510	0.218255	+	0.975892i	$0.429964\pi$
$620$	2011.43	0.130292
$621$	−4736.43	−0.306065
$622$	−5724.11	−0.368997
$623$	15400.2	0.990362
$624$	0	0
$625$	8523.93	0.545532
$626$	−22882.2	−1.46095
$627$	2812.59	0.179145
$628$	15432.6	0.980617
$629$	36147.3	2.29140
$630$	5145.71	0.325413
$631$	−8098.44	−0.510925	−0.255463	−	0.966819i	$-0.582228\pi$
$631$	−8098.44	−0.510925	−0.255463	+	0.966819i	$0.582228\pi$
$632$	−29032.2	−1.82727
$633$	15264.9	0.958494
$634$	20394.6	1.27756
$635$	3247.50	0.202950
$636$	7300.21	0.455145
$637$	0	0
$638$	−5767.94	−0.357923
$639$	−3926.85	−0.243105
$640$	−11627.7	−0.718163
$641$	−10955.9	−0.675090	−0.337545	−	0.941309i	$-0.609597\pi$
$641$	−10955.9	−0.675090	−0.337545	+	0.941309i	$0.609597\pi$
$642$	−17667.0	−1.08608
$643$	−28125.0	−1.72495	−0.862473	−	0.506104i	$-0.831085\pi$
$643$	−28125.0	−1.72495	−0.862473	+	0.506104i	$0.831085\pi$
$644$	−67173.7	−4.11027
$645$	−2325.06	−0.141936
$646$	−77211.1	−4.70253
$647$	−29001.4	−1.76223	−0.881115	−	0.472901i	$-0.843207\pi$
$647$	−29001.4	−1.76223	−0.881115	+	0.472901i	$0.843207\pi$
$648$	2311.14	0.140108
$649$	824.302	0.0498562
$650$	0	0
$651$	2608.61	0.157050
$652$	−44390.1	−2.66634
$653$	19506.3	1.16898	0.584488	−	0.811402i	$-0.301295\pi$
$653$	19506.3	1.16898	0.584488	+	0.811402i	$0.301295\pi$
$654$	10185.4	0.608989
$655$	−6516.01	−0.388705
$656$	1000.20	0.0595294
$657$	10432.0	0.619466
$658$	−33955.0	−2.01171
$659$	5985.86	0.353833	0.176917	−	0.984226i	$-0.443388\pi$
$659$	5985.86	0.353833	0.176917	+	0.984226i	$0.443388\pi$
$660$	−1131.45	−0.0667300
$661$	−280.836	−0.0165254	−0.00826268	−	0.999966i	$-0.502630\pi$
$661$	−280.836	−0.0165254	−0.00826268	+	0.999966i	$0.502630\pi$
$662$	4262.63	0.250259
$663$	0	0
$664$	−4287.04	−0.250557
$665$	−19040.5	−1.11032
$666$	14551.6	0.846639
$667$	−35941.8	−2.08647
$668$	51784.0	2.99938
$669$	14234.5	0.822626
$670$	4447.71	0.256463
$671$	353.163	0.0203185
$672$	−10399.8	−0.596996
$673$	−14868.9	−0.851640	−0.425820	−	0.904808i	$-0.640014\pi$
$673$	−14868.9	−0.851640	−0.425820	+	0.904808i	$0.640014\pi$
$674$	40690.6	2.32543
$675$	−2834.91	−0.161653
$676$	0	0
$677$	−20769.5	−1.17908	−0.589540	−	0.807739i	$-0.700691\pi$
$677$	−20769.5	−1.17908	−0.589540	+	0.807739i	$0.700691\pi$
$678$	−12063.7	−0.683341
$679$	−7802.09	−0.440967
$680$	13403.9	0.755907
$681$	3436.56	0.193376
$682$	−899.650	−0.0505123
$683$	14980.8	0.839275	0.419638	−	0.907692i	$-0.362157\pi$
$683$	14980.8	0.839275	0.419638	+	0.907692i	$0.362157\pi$
$684$	−19817.1	−1.10779
$685$	8080.90	0.450738
$686$	6949.30	0.386772
$687$	4045.40	0.224660
$688$	−3719.96	−0.206137
$689$	0	0
$690$	−11058.3	−0.610122
$691$	9472.45	0.521489	0.260745	−	0.965408i	$-0.416032\pi$
$691$	9472.45	0.521489	0.260745	+	0.965408i	$0.416032\pi$
$692$	−51605.8	−2.83491
$693$	−1467.37	−0.0804342
$694$	−1633.12	−0.0893260
$695$	5497.95	0.300071
$696$	17537.8	0.955128
$697$	4893.87	0.265952
$698$	51540.3	2.79488
$699$	−2856.01	−0.154541
$700$	−40205.7	−2.17090
$701$	1035.34	0.0557834	0.0278917	−	0.999611i	$-0.491121\pi$
$701$	1035.34	0.0557834	0.0278917	+	0.999611i	$0.491121\pi$
$702$	0	0
$703$	−53844.8	−2.88875
$704$	4615.72	0.247105
$705$	−3563.86	−0.190387
$706$	−48790.0	−2.60090
$707$	−33156.7	−1.76377
$708$	−5807.92	−0.308298
$709$	16800.4	0.889917	0.444958	−	0.895551i	$-0.353219\pi$
$709$	16800.4	0.889917	0.444958	+	0.895551i	$0.353219\pi$
$710$	−9168.18	−0.484614
$711$	−9157.57	−0.483032
$712$	16149.0	0.850013
$713$	−5606.00	−0.294455
$714$	40282.4	2.11139
$715$	0	0
$716$	58734.1	3.06564
$717$	−9207.35	−0.479574
$718$	−40713.7	−2.11618
$719$	14873.1	0.771449	0.385724	−	0.922614i	$-0.373952\pi$
$719$	14873.1	0.771449	0.385724	+	0.922614i	$0.373952\pi$
$720$	864.107	0.0447269
$721$	2020.27	0.104353
$722$	82788.1	4.26739
$723$	7524.51	0.387053
$724$	−7025.24	−0.360623
$725$	−21512.4	−1.10200
$726$	−18253.9	−0.933146
$727$	−2318.92	−0.118300	−0.0591500	−	0.998249i	$-0.518839\pi$
$727$	−2318.92	−0.118300	−0.0591500	+	0.998249i	$0.518839\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	24355.9	1.23487
$731$	−18201.3	−0.920931
$732$	−2488.34	−0.125644
$733$	−3350.95	−0.168854	−0.0844270	−	0.996430i	$-0.526906\pi$
$733$	−3350.95	−0.168854	−0.0844270	+	0.996430i	$0.526906\pi$
$734$	5798.61	0.291595
$735$	5331.58	0.267562
$736$	22349.6	1.11932
$737$	−1268.33	−0.0633915
$738$	1970.09	0.0982656
$739$	−29348.1	−1.46088	−0.730438	−	0.682979i	$-0.760684\pi$
$739$	−29348.1	−1.46088	−0.730438	+	0.682979i	$0.760684\pi$
$740$	21660.8	1.07604
$741$	0	0
$742$	22104.3	1.09363
$743$	23622.2	1.16637	0.583187	−	0.812338i	$-0.301805\pi$
$743$	23622.2	1.16637	0.583187	+	0.812338i	$0.301805\pi$
$744$	2735.45	0.134793
$745$	1995.30	0.0981236
$746$	−2008.40	−0.0985693
$747$	−1352.26	−0.0662336
$748$	−8857.41	−0.432966
$749$	−34106.1	−1.66383
$750$	−14498.5	−0.705882
$751$	29751.4	1.44560	0.722798	−	0.691059i	$-0.242856\pi$
$751$	29751.4	1.44560	0.722798	+	0.691059i	$0.242856\pi$
$752$	−5701.97	−0.276502
$753$	10217.7	0.494495
$754$	0	0
$755$	926.735	0.0446720
$756$	10338.9	0.497385
$757$	−36907.2	−1.77201	−0.886006	−	0.463673i	$-0.846531\pi$
$757$	−36907.2	−1.77201	−0.886006	+	0.463673i	$0.846531\pi$
$758$	583.709	0.0279700
$759$	3153.45	0.150807
$760$	−19966.4	−0.952968
$761$	−20208.2	−0.962613	−0.481306	−	0.876552i	$-0.659838\pi$
$761$	−20208.2	−0.962613	−0.481306	+	0.876552i	$0.659838\pi$
$762$	10234.2	0.486541
$763$	19662.8	0.932950
$764$	−43439.1	−2.05703
$765$	4227.98	0.199821
$766$	43886.9	2.07010
$767$	0	0
$768$	−18156.1	−0.853063
$769$	24751.0	1.16066	0.580329	−	0.814382i	$-0.302924\pi$
$769$	24751.0	1.16066	0.580329	+	0.814382i	$0.302924\pi$
$770$	−3425.94	−0.160341
$771$	−14201.9	−0.663382
$772$	23138.2	1.07871
$773$	−30673.4	−1.42722	−0.713612	−	0.700541i	$-0.752942\pi$
$773$	−30673.4	−1.42722	−0.713612	+	0.700541i	$0.752942\pi$
$774$	−7327.18	−0.340271
$775$	−3355.38	−0.155521
$776$	−8181.46	−0.378476
$777$	28091.7	1.29702
$778$	53413.4	2.46139
$779$	−7289.87	−0.335285
$780$	0	0
$781$	2614.44	0.119785
$782$	−86568.5	−3.95867
$783$	5531.93	0.252484
$784$	8530.23	0.388585
$785$	4904.55	0.222995
$786$	−20534.6	−0.931862
$787$	29009.9	1.31397	0.656983	−	0.753905i	$-0.271832\pi$
$787$	29009.9	1.31397	0.656983	+	0.753905i	$0.271832\pi$
$788$	−19297.1	−0.872375
$789$	−5598.33	−0.252606
$790$	−21380.6	−0.962894
$791$	−23289.0	−1.04685
$792$	−1538.72	−0.0690355
$793$	0	0
$794$	−56744.0	−2.53623
$795$	2320.04	0.103501
$796$	−65120.8	−2.89968
$797$	6778.24	0.301252	0.150626	−	0.988591i	$-0.451871\pi$
$797$	6778.24	0.301252	0.150626	+	0.988591i	$0.451871\pi$
$798$	−60004.3	−2.66182
$799$	−27899.1	−1.23529
$800$	13377.0	0.591185
$801$	5093.86	0.224697
$802$	−22817.9	−1.00465
$803$	−6945.44	−0.305229
$804$	8936.48	0.391997
$805$	−21348.1	−0.934685
$806$	0	0
$807$	4740.58	0.206786
$808$	−34768.9	−1.51382
$809$	−34862.9	−1.51510	−0.757549	−	0.652778i	$-0.773603\pi$
$809$	−34862.9	−1.51510	−0.757549	+	0.652778i	$0.773603\pi$
$810$	1702.03	0.0738310
$811$	22665.4	0.981370	0.490685	−	0.871337i	$-0.336747\pi$
$811$	22665.4	0.981370	0.490685	+	0.871337i	$0.336747\pi$
$812$	78455.6	3.39070
$813$	14766.3	0.636993
$814$	−9688.21	−0.417164
$815$	−14107.4	−0.606331
$816$	6764.52	0.290203
$817$	27112.6	1.16101
$818$	14009.1	0.598797
$819$	0	0
$820$	2932.59	0.124891
$821$	20748.7	0.882016	0.441008	−	0.897503i	$-0.354621\pi$
$821$	20748.7	0.882016	0.441008	+	0.897503i	$0.354621\pi$
$822$	25466.1	1.08058
$823$	6141.41	0.260117	0.130058	−	0.991506i	$-0.458484\pi$
$823$	6141.41	0.260117	0.130058	+	0.991506i	$0.458484\pi$
$824$	2118.50	0.0895650
$825$	1887.44	0.0796513
$826$	−17585.8	−0.740786
$827$	−28383.5	−1.19346	−0.596730	−	0.802442i	$-0.703533\pi$
$827$	−28383.5	−1.19346	−0.596730	+	0.802442i	$0.703533\pi$
$828$	−22218.8	−0.932555
$829$	−908.734	−0.0380720	−0.0190360	−	0.999819i	$-0.506060\pi$
$829$	−908.734	−0.0380720	−0.0190360	+	0.999819i	$0.506060\pi$
$830$	−3157.17	−0.132032
$831$	8061.25	0.336512
$832$	0	0
$833$	41737.4	1.73603
$834$	17326.2	0.719375
$835$	16457.2	0.682065
$836$	13193.9	0.545839
$837$	862.838	0.0356321
$838$	36523.8	1.50560
$839$	−27820.3	−1.14477	−0.572385	−	0.819985i	$-0.693982\pi$
$839$	−27820.3	−1.14477	−0.572385	+	0.819985i	$0.693982\pi$
$840$	10416.8	0.427873
$841$	17589.3	0.721200
$842$	18603.5	0.761425
$843$	2651.26	0.108321
$844$	71608.3	2.92045
$845$	0	0
$846$	−11231.1	−0.456424
$847$	−35239.0	−1.42955
$848$	3711.93	0.150316
$849$	−1409.33	−0.0569706
$850$	−51814.1	−2.09084
$851$	−60370.3	−2.43181
$852$	−18421.0	−0.740719
$853$	5802.11	0.232896	0.116448	−	0.993197i	$-0.462849\pi$
$853$	5802.11	0.232896	0.116448	+	0.993197i	$0.462849\pi$
$854$	−7534.45	−0.301901
$855$	−6297.96	−0.251913
$856$	−35764.5	−1.42804
$857$	−43311.1	−1.72635	−0.863173	−	0.504909i	$-0.831526\pi$
$857$	−43311.1	−1.72635	−0.863173	+	0.504909i	$0.831526\pi$
$858$	0	0
$859$	−16698.2	−0.663254	−0.331627	−	0.943411i	$-0.607598\pi$
$859$	−16698.2	−0.663254	−0.331627	+	0.943411i	$0.607598\pi$
$860$	−10906.9	−0.432468
$861$	3803.25	0.150539
$862$	20555.2	0.812196
$863$	19429.2	0.766369	0.383185	−	0.923672i	$-0.374827\pi$
$863$	19429.2	0.766369	0.383185	+	0.923672i	$0.374827\pi$
$864$	−3439.90	−0.135449
$865$	−16400.6	−0.644666
$866$	42249.7	1.65786
$867$	18359.0	0.719153
$868$	12237.0	0.478517
$869$	6096.98	0.238004
$870$	12915.6	0.503311
$871$	0	0
$872$	20618.9	0.800738
$873$	−2580.67	−0.100048
$874$	128952.	4.99068
$875$	−27989.4	−1.08139
$876$	48936.6	1.88746
$877$	11619.7	0.447400	0.223700	−	0.974658i	$-0.428186\pi$
$877$	11619.7	0.447400	0.223700	+	0.974658i	$0.428186\pi$
$878$	−918.332	−0.0352986
$879$	−10211.3	−0.391829
$880$	−575.309	−0.0220383
$881$	−51102.0	−1.95422	−0.977112	−	0.212726i	$-0.931766\pi$
$881$	−51102.0	−1.95422	−0.977112	+	0.212726i	$0.931766\pi$
$882$	16801.9	0.641441
$883$	−37838.5	−1.44209	−0.721046	−	0.692888i	$-0.756338\pi$
$883$	−37838.5	−1.44209	−0.721046	+	0.692888i	$0.756338\pi$
$884$	0	0
$885$	−1845.78	−0.0701077
$886$	34625.6	1.31295
$887$	−9626.92	−0.364420	−0.182210	−	0.983260i	$-0.558325\pi$
$887$	−9626.92	−0.364420	−0.182210	+	0.983260i	$0.558325\pi$
$888$	29457.7	1.11321
$889$	19757.0	0.745364
$890$	11892.8	0.447920
$891$	−485.357	−0.0182493
$892$	66774.3	2.50647
$893$	41558.3	1.55733
$894$	6287.98	0.235237
$895$	18666.0	0.697133
$896$	−70740.0	−2.63757
$897$	0	0
$898$	−771.825	−0.0286816
$899$	6547.54	0.242906
$900$	−13298.7	−0.492543
$901$	18162.0	0.671549
$902$	−1311.66	−0.0484184
$903$	−14145.1	−0.521283
$904$	−24421.4	−0.898500
$905$	−2232.65	−0.0820065
$906$	2920.51	0.107094
$907$	17066.0	0.624772	0.312386	−	0.949955i	$-0.398872\pi$
$907$	17066.0	0.624772	0.312386	+	0.949955i	$0.398872\pi$
$908$	16121.0	0.589200
$909$	−10967.1	−0.400172
$910$	0	0
$911$	−37423.9	−1.36104	−0.680521	−	0.732729i	$-0.738246\pi$
$911$	−37423.9	−1.36104	−0.680521	+	0.732729i	$0.738246\pi$
$912$	−10076.4	−0.365858
$913$	900.312	0.0326353
$914$	54907.7	1.98708
$915$	−790.805	−0.0285718
$916$	18977.1	0.684520
$917$	−39641.9	−1.42758
$918$	13324.0	0.479040
$919$	−2783.88	−0.0999256	−0.0499628	−	0.998751i	$-0.515910\pi$
$919$	−2783.88	−0.0999256	−0.0499628	+	0.998751i	$0.515910\pi$
$920$	−22386.1	−0.802227
$921$	−2666.59	−0.0954039
$922$	4968.50	0.177472
$923$	0	0
$924$	−6883.50	−0.245076
$925$	−36133.6	−1.28440
$926$	40191.9	1.42634
$927$	668.236	0.0236761
$928$	−26103.3	−0.923363
$929$	26585.2	0.938894	0.469447	−	0.882961i	$-0.344453\pi$
$929$	26585.2	0.938894	0.469447	+	0.882961i	$0.344453\pi$
$930$	2014.50	0.0710303
$931$	−62171.8	−2.18861
$932$	−13397.6	−0.470873
$933$	−3655.09	−0.128255
$934$	−36200.9	−1.26823
$935$	−2814.92	−0.0984576
$936$	0	0
$937$	34474.0	1.20194	0.600970	−	0.799272i	$-0.294781\pi$
$937$	34474.0	1.20194	0.600970	+	0.799272i	$0.294781\pi$
$938$	27058.8	0.941900
$939$	−14611.2	−0.507795
$940$	−16718.2	−0.580092
$941$	−41994.3	−1.45481	−0.727404	−	0.686209i	$-0.759274\pi$
$941$	−41994.3	−1.45481	−0.727404	+	0.686209i	$0.759274\pi$
$942$	15456.2	0.534596
$943$	−8173.34	−0.282249
$944$	−2953.15	−0.101819
$945$	3285.75	0.113106
$946$	4878.32	0.167662
$947$	−49352.0	−1.69348	−0.846739	−	0.532008i	$-0.821438\pi$
$947$	−49352.0	−1.69348	−0.846739	+	0.532008i	$0.821438\pi$
$948$	−42958.5	−1.47176
$949$	0	0
$950$	77181.9	2.63591
$951$	13022.8	0.444052
$952$	81546.2	2.77618
$953$	−51144.5	−1.73844	−0.869220	−	0.494425i	$-0.835379\pi$
$953$	−51144.5	−1.73844	−0.869220	+	0.494425i	$0.835379\pi$
$954$	7311.36	0.248128
$955$	−13805.2	−0.467774
$956$	−43192.0	−1.46122
$957$	−3683.07	−0.124406
$958$	−21182.3	−0.714372
$959$	49162.3	1.65540
$960$	−10335.6	−0.347478
$961$	−28769.8	−0.965720
$962$	0	0
$963$	−11281.1	−0.377497
$964$	35297.7	1.17932
$965$	7353.41	0.245300
$966$	−67276.3	−2.24077
$967$	−24895.1	−0.827892	−0.413946	−	0.910301i	$-0.635850\pi$
$967$	−24895.1	−0.827892	−0.413946	+	0.910301i	$0.635850\pi$
$968$	−36952.4	−1.22696
$969$	−49302.6	−1.63450
$970$	−6025.19	−0.199440
$971$	42942.9	1.41926	0.709630	−	0.704574i	$-0.248862\pi$
$971$	42942.9	1.41926	0.709630	+	0.704574i	$0.248862\pi$
$972$	3419.76	0.112849
$973$	33448.2	1.10206
$974$	31598.8	1.03952
$975$	0	0
$976$	−1265.24	−0.0414953
$977$	42555.7	1.39353	0.696764	−	0.717301i	$-0.254623\pi$
$977$	42555.7	1.39353	0.696764	+	0.717301i	$0.254623\pi$
$978$	−44458.0	−1.45359
$979$	−3391.41	−0.110715
$980$	25010.6	0.815240
$981$	6503.78	0.211672
$982$	54111.3	1.75841
$983$	−4345.38	−0.140993	−0.0704965	−	0.997512i	$-0.522458\pi$
$983$	−4345.38	−0.140993	−0.0704965	+	0.997512i	$0.522458\pi$
$984$	3988.18	0.129206
$985$	−6132.71	−0.198380
$986$	101108.	3.26565
$987$	−21681.7	−0.699225
$988$	0	0
$989$	30398.4	0.977363
$990$	−1133.18	−0.0363787
$991$	7934.89	0.254349	0.127175	−	0.991880i	$-0.459409\pi$
$991$	7934.89	0.254349	0.127175	+	0.991880i	$0.459409\pi$
$992$	−4071.43	−0.130311
$993$	2721.87	0.0869848
$994$	−55777.0	−1.77982
$995$	−20695.7	−0.659395
$996$	−6343.48	−0.201808
$997$	2725.62	0.0865810	0.0432905	−	0.999063i	$-0.486216\pi$
$997$	2725.62	0.0865810	0.0432905	+	0.999063i	$0.486216\pi$
$998$	93531.5	2.96662
$999$	9291.79	0.294273

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	507.4.a.p.1.8	yes	9
3.2	odd	2		1521.4.a.bf.1.2		9
13.5	odd	4		507.4.b.k.337.3		18
13.8	odd	4		507.4.b.k.337.16		18
13.12	even	2		507.4.a.o.1.2	✓	9
39.38	odd	2		1521.4.a.bi.1.8		9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
507.4.a.o.1.2	✓	9	13.12	even	2
507.4.a.p.1.8	yes	9	1.1	even	1	trivial
507.4.b.k.337.3		18	13.5	odd	4
507.4.b.k.337.16		18	13.8	odd	4
1521.4.a.bf.1.2		9	3.2	odd	2
1521.4.a.bi.1.8		9	39.38	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:	\( N \)	\(=\)	\( 507 = 3 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	507.a (trivial)

\(f(q)\)	\(=\)	\(q+4.69820 q^{2} +3.00000 q^{3} +14.0731 q^{4} +4.47249 q^{5} +14.0946 q^{6} +27.2096 q^{7} +28.5326 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q+4.69820 q^{2} +3.00000 q^{3} +14.0731 q^{4} +4.47249 q^{5} +14.0946 q^{6} +27.2096 q^{7} +28.5326 q^{8} +9.00000 q^{9} +21.0127 q^{10} -5.99207 q^{11} +42.2193 q^{12} +127.836 q^{14} +13.4175 q^{15} +21.4672 q^{16} +105.037 q^{17} +42.2838 q^{18} -156.462 q^{19} +62.9418 q^{20} +81.6287 q^{21} -28.1519 q^{22} -175.423 q^{23} +85.5978 q^{24} -104.997 q^{25} +27.0000 q^{27} +382.923 q^{28} +204.886 q^{29} +63.0380 q^{30} +31.9570 q^{31} -127.404 q^{32} -17.9762 q^{33} +493.483 q^{34} +121.695 q^{35} +126.658 q^{36} +344.140 q^{37} -735.088 q^{38} +127.612 q^{40} +46.5921 q^{41} +383.508 q^{42} -173.286 q^{43} -84.3269 q^{44} +40.2524 q^{45} -824.175 q^{46} -265.613 q^{47} +64.4016 q^{48} +397.361 q^{49} -493.296 q^{50} +315.110 q^{51} +172.912 q^{53} +126.851 q^{54} -26.7995 q^{55} +776.360 q^{56} -469.385 q^{57} +962.596 q^{58} -137.566 q^{59} +188.825 q^{60} -58.9384 q^{61} +150.140 q^{62} +244.886 q^{63} -770.306 q^{64} -84.4558 q^{66} +211.668 q^{67} +1478.19 q^{68} -526.270 q^{69} +571.746 q^{70} -436.317 q^{71} +256.793 q^{72} +1159.11 q^{73} +1616.84 q^{74} -314.990 q^{75} -2201.90 q^{76} -163.042 q^{77} -1017.51 q^{79} +96.0118 q^{80} +81.0000 q^{81} +218.899 q^{82} -150.251 q^{83} +1148.77 q^{84} +469.775 q^{85} -814.131 q^{86} +614.659 q^{87} -170.969 q^{88} +565.984 q^{89} +189.114 q^{90} -2468.75 q^{92} +95.8709 q^{93} -1247.90 q^{94} -699.773 q^{95} -382.211 q^{96} -286.741 q^{97} +1866.88 q^{98} -53.9286 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + 6 q^{2} + 27 q^{3} + 44 q^{4} + 33 q^{5} + 18 q^{6} + 83 q^{7} + 87 q^{8} + 81 q^{9}+O(q^{10}) \) 9 * q + 6 * q^2 + 27 * q^3 + 44 * q^4 + 33 * q^5 + 18 * q^6 + 83 * q^7 + 87 * q^8 + 81 * q^9	\( 9 q + 6 q^{2} + 27 q^{3} + 44 q^{4} + 33 q^{5} + 18 q^{6} + 83 q^{7} + 87 q^{8} + 81 q^{9} - 54 q^{10} + 85 q^{11} + 132 q^{12} + 158 q^{14} + 99 q^{15} + 216 q^{16} + 178 q^{17} + 54 q^{18} + 352 q^{19} + 402 q^{20} + 249 q^{21} - 630 q^{22} + 150 q^{23} + 261 q^{24} - 20 q^{25} + 243 q^{27} + 940 q^{28} - 97 q^{29} - 162 q^{30} + 717 q^{31} + 707 q^{32} + 255 q^{33} + 632 q^{34} - 418 q^{35} + 396 q^{36} + 1108 q^{37} - 660 q^{38} - 1506 q^{40} + 334 q^{41} + 474 q^{42} + 242 q^{43} - 307 q^{44} + 297 q^{45} + 979 q^{46} - 184 q^{47} + 648 q^{48} - 38 q^{49} - 2031 q^{50} + 534 q^{51} - 151 q^{53} + 162 q^{54} + 2064 q^{55} + 2276 q^{56} + 1056 q^{57} + 1161 q^{58} + 537 q^{59} + 1206 q^{60} - 1340 q^{61} + 347 q^{62} + 747 q^{63} + 893 q^{64} - 1890 q^{66} + 2308 q^{67} + 2785 q^{68} + 450 q^{69} - 1420 q^{70} + 96 q^{71} + 783 q^{72} + 2505 q^{73} - 1191 q^{74} - 60 q^{75} + 2409 q^{76} - 2142 q^{77} - 1591 q^{79} - 2671 q^{80} + 729 q^{81} + 1517 q^{82} + 1539 q^{83} + 2820 q^{84} + 4296 q^{85} - 3763 q^{86} - 291 q^{87} - 3716 q^{88} - 592 q^{89} - 486 q^{90} + 515 q^{92} + 2151 q^{93} - 692 q^{94} + 4158 q^{95} + 2121 q^{96} + 1445 q^{97} + 1457 q^{98} + 765 q^{99}+O(q^{100}) \) 9 * q + 6 * q^2 + 27 * q^3 + 44 * q^4 + 33 * q^5 + 18 * q^6 + 83 * q^7 + 87 * q^8 + 81 * q^9 - 54 * q^10 + 85 * q^11 + 132 * q^12 + 158 * q^14 + 99 * q^15 + 216 * q^16 + 178 * q^17 + 54 * q^18 + 352 * q^19 + 402 * q^20 + 249 * q^21 - 630 * q^22 + 150 * q^23 + 261 * q^24 - 20 * q^25 + 243 * q^27 + 940 * q^28 - 97 * q^29 - 162 * q^30 + 717 * q^31 + 707 * q^32 + 255 * q^33 + 632 * q^34 - 418 * q^35 + 396 * q^36 + 1108 * q^37 - 660 * q^38 - 1506 * q^40 + 334 * q^41 + 474 * q^42 + 242 * q^43 - 307 * q^44 + 297 * q^45 + 979 * q^46 - 184 * q^47 + 648 * q^48 - 38 * q^49 - 2031 * q^50 + 534 * q^51 - 151 * q^53 + 162 * q^54 + 2064 * q^55 + 2276 * q^56 + 1056 * q^57 + 1161 * q^58 + 537 * q^59 + 1206 * q^60 - 1340 * q^61 + 347 * q^62 + 747 * q^63 + 893 * q^64 - 1890 * q^66 + 2308 * q^67 + 2785 * q^68 + 450 * q^69 - 1420 * q^70 + 96 * q^71 + 783 * q^72 + 2505 * q^73 - 1191 * q^74 - 60 * q^75 + 2409 * q^76 - 2142 * q^77 - 1591 * q^79 - 2671 * q^80 + 729 * q^81 + 1517 * q^82 + 1539 * q^83 + 2820 * q^84 + 4296 * q^85 - 3763 * q^86 - 291 * q^87 - 3716 * q^88 - 592 * q^89 - 486 * q^90 + 515 * q^92 + 2151 * q^93 - 692 * q^94 + 4158 * q^95 + 2121 * q^96 + 1445 * q^97 + 1457 * q^98 + 765 * q^99

Introduction

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