LMFDB - Embedded newform 810.3.j.a.269.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [810,3,Mod(269,810)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(810, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([1, 3]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("810.269");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 810 = 2 \cdot 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	810.j (of order $6$, degree $2$, not minimal)

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:	no
Analytic conductor:	$22.0709014132$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{6})$
Coefficient field:	8.0.81622204416.3
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 14x^{6} + 165x^{4} - 434x^{2} + 961 $ x^8 - 14x^6 + 165x^4 - 434*x^2 + 961
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 270)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			269.1
Root			$1.43806 - 0.830265i$ of defining polynomial
Character	$\chi$	$=$	810.269
Dual form			810.3.j.a.539.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.707107 + 1.22474i) q^{2} +(-1.00000 - 1.73205i) q^{4} +(-4.24083 - 2.64865i) q^{5} +(-11.8534 - 6.84358i) q^{7} +2.82843 q^{8} +O(q^{10})$	\(q+(-0.707107 + 1.22474i) q^{2} +(-1.00000 - 1.73205i) q^{4} +(-4.24083 - 2.64865i) q^{5} +(-11.8534 - 6.84358i) q^{7} +2.82843 q^{8} +(6.24264 - 3.32106i) q^{10} +(-10.6621 - 6.15576i) q^{11} +(-14.7295 + 8.50411i) q^{13} +(16.7633 - 9.67828i) q^{14} +(-2.00000 + 3.46410i) q^{16} +6.89949 q^{17} -7.24264 q^{19} +(-0.346763 + 9.99399i) q^{20} +(15.0785 - 8.70556i) q^{22} +(-17.3995 - 30.1368i) q^{23} +(10.9693 + 22.4650i) q^{25} -24.0532i q^{26} +27.3743i q^{28} +(18.3035 + 10.5675i) q^{29} +(19.1066 + 33.0936i) q^{31} +(-2.82843 - 4.89898i) q^{32} +(-4.87868 + 8.45012i) q^{34} +(32.1421 + 60.4180i) q^{35} -21.5380i q^{37} +(5.12132 - 8.87039i) q^{38} +(-11.9949 - 7.49151i) q^{40} +(31.4928 - 18.1824i) q^{41} +(-5.40331 - 3.11960i) q^{43} +24.6230i q^{44} +49.2132 q^{46} +(20.1005 - 34.8151i) q^{47} +(69.1690 + 119.804i) q^{49} +(-35.2703 - 2.45051i) q^{50} +(29.4591 + 17.0082i) q^{52} +38.2721 q^{53} +(28.9117 + 54.3457i) q^{55} +(-33.5265 - 19.3566i) q^{56} +(-25.8851 + 14.9448i) q^{58} +(-36.0537 + 20.8156i) q^{59} +(7.52944 - 13.0414i) q^{61} -54.0416 q^{62} +8.00000 q^{64} +(84.9899 + 2.94891i) q^{65} +(-111.386 + 64.3089i) q^{67} +(-6.89949 - 11.9503i) q^{68} +(-96.7246 - 3.35607i) q^{70} -104.967i q^{71} +2.11232i q^{73} +(26.3786 + 15.2297i) q^{74} +(7.24264 + 12.5446i) q^{76} +(84.2548 + 145.934i) q^{77} +(22.0477 - 38.1878i) q^{79} +(17.6569 - 9.39338i) q^{80} +51.4275i q^{82} +(27.5122 - 47.6525i) q^{83} +(-29.2596 - 18.2743i) q^{85} +(7.64144 - 4.41179i) q^{86} +(-30.1569 - 17.4111i) q^{88} +68.1020i q^{89} +232.794 q^{91} +(-34.7990 + 60.2736i) q^{92} +(28.4264 + 49.2360i) q^{94} +(30.7148 + 19.1832i) q^{95} +(-87.6794 - 50.6217i) q^{97} -195.640 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{4} - 12 q^{5}+O(q^{10}) $ 8 * q - 8 * q^4 - 12 * q^5	$ 8 q - 8 q^{4} - 12 q^{5} + 16 q^{10} - 16 q^{16} - 24 q^{17} - 24 q^{19} - 24 q^{20} - 60 q^{23} + 12 q^{25} + 68 q^{31} - 56 q^{34} + 144 q^{35} + 24 q^{38} - 16 q^{40} + 224 q^{46} + 240 q^{47} + 180 q^{49} - 96 q^{50} + 408 q^{53} - 176 q^{55} + 196 q^{61} - 240 q^{62} + 64 q^{64} + 24 q^{65} + 24 q^{68} - 80 q^{70} + 24 q^{76} + 312 q^{77} - 180 q^{79} + 96 q^{80} - 108 q^{83} - 20 q^{85} + 912 q^{91} - 120 q^{92} - 112 q^{94} + 60 q^{95} - 1056 q^{98}+O(q^{100}) $ 8 * q - 8 * q^4 - 12 * q^5 + 16 * q^10 - 16 * q^16 - 24 * q^17 - 24 * q^19 - 24 * q^20 - 60 * q^23 + 12 * q^25 + 68 * q^31 - 56 * q^34 + 144 * q^35 + 24 * q^38 - 16 * q^40 + 224 * q^46 + 240 * q^47 + 180 * q^49 - 96 * q^50 + 408 * q^53 - 176 * q^55 + 196 * q^61 - 240 * q^62 + 64 * q^64 + 24 * q^65 + 24 * q^68 - 80 * q^70 + 24 * q^76 + 312 * q^77 - 180 * q^79 + 96 * q^80 - 108 * q^83 - 20 * q^85 + 912 * q^91 - 120 * q^92 - 112 * q^94 + 60 * q^95 - 1056 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/810\mathbb{Z}\right)^\times$.

$n$	$487$	$731$
$\chi(n)$	$-1$	$e\left(\frac{1}{6}\right)$

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 $

$2$	−0.707107	+	1.22474i	−0.353553	+	0.612372i
$2$	−0.707107	+	1.22474i	−0.353553	+	0.612372i
$3$		0			0
$3$		0			0
$4$	−1.00000	−	1.73205i	−0.250000	−	0.433013i
$5$	−4.24083	−	2.64865i	−0.848166	−	0.529730i
$5$	−4.24083	−	2.64865i	−0.848166	−	0.529730i
$6$		0			0
$7$	−11.8534	−	6.84358i	−1.69335	−	0.977654i	−0.951784	−	0.306769i	$-0.900752\pi$
$7$	−11.8534	−	6.84358i	−1.69335	−	0.977654i	−0.741562	−	0.670885i	$-0.765915\pi$
$8$	2.82843			0.353553
$9$		0			0
$10$	6.24264	−	3.32106i	0.624264	−	0.332106i
$11$	−10.6621	−	6.15576i	−0.969281	−	0.559615i	−0.0702641	−	0.997528i	$-0.522384\pi$
$11$	−10.6621	−	6.15576i	−0.969281	−	0.559615i	−0.899017	+	0.437914i	$0.855718\pi$
$12$		0			0
$13$	−14.7295	+	8.50411i	−1.13304	+	0.654162i	−0.944698	−	0.327941i	$-0.893645\pi$
$13$	−14.7295	+	8.50411i	−1.13304	+	0.654162i	−0.188344	+	0.982103i	$0.560312\pi$
$14$	16.7633	−	9.67828i	1.19738	−	0.691306i
$15$		0			0
$16$	−2.00000	+	3.46410i	−0.125000	+	0.216506i
$17$	6.89949			0.405853			0.202926	−	0.979194i	$-0.434955\pi$
$17$	6.89949			0.405853			0.202926	+	0.979194i	$0.434955\pi$
$18$		0			0
$19$	−7.24264			−0.381192			−0.190596	−	0.981669i	$-0.561042\pi$
$19$	−7.24264			−0.381192			−0.190596	+	0.981669i	$0.561042\pi$
$20$	−0.346763	+	9.99399i	−0.0173381	+	0.499699i
$21$		0			0
$22$	15.0785	−	8.70556i	0.685385	−	0.395707i
$23$	−17.3995	−	30.1368i	−0.756500	−	1.31030i	−0.944625	−	0.328151i	$-0.893574\pi$
$23$	−17.3995	−	30.1368i	−0.756500	−	1.31030i	0.188125	−	0.982145i	$-0.439759\pi$
$24$		0			0
$25$	10.9693	+	22.4650i	0.438773	+	0.898598i
$26$		−	24.0532i		−	0.925125i
$27$		0			0
$28$			27.3743i			0.977654i
$29$	18.3035	+	10.5675i	0.631156	+	0.364398i	0.781200	−	0.624281i	$-0.214608\pi$
$29$	18.3035	+	10.5675i	0.631156	+	0.364398i	−0.150044	+	0.988679i	$0.547941\pi$
$30$		0			0
$31$	19.1066	+	33.0936i	0.616342	+	1.06754i	0.990147	+	0.140029i	$0.0447194\pi$
$31$	19.1066	+	33.0936i	0.616342	+	1.06754i	−0.373805	+	0.927507i	$0.621947\pi$
$32$	−2.82843	−	4.89898i	−0.0883883	−	0.153093i
$33$		0			0
$34$	−4.87868	+	8.45012i	−0.143491	+	0.248533i
$35$	32.1421	+	60.4180i	0.918347	+	1.72623i
$36$		0			0
$37$		−	21.5380i		−	0.582108i	−0.956707	−	0.291054i	$-0.905994\pi$
$37$		−	21.5380i		−	0.582108i	0.956707	−	0.291054i	$-0.0940060\pi$
$38$	5.12132	−	8.87039i	0.134772	−	0.233431i
$39$		0			0
$40$	−11.9949	−	7.49151i	−0.299872	−	0.187288i
$41$	31.4928	−	18.1824i	0.768117	−	0.443473i	−0.0640853	−	0.997944i	$-0.520413\pi$
$41$	31.4928	−	18.1824i	0.768117	−	0.443473i	0.832203	+	0.554472i	$0.187080\pi$
$42$		0			0
$43$	−5.40331	−	3.11960i	−0.125658	−	0.0725489i	0.435853	−	0.900018i	$-0.356447\pi$
$43$	−5.40331	−	3.11960i	−0.125658	−	0.0725489i	−0.561512	+	0.827469i	$0.689780\pi$
$44$			24.6230i			0.559615i
$45$		0			0
$46$	49.2132			1.06985
$47$	20.1005	−	34.8151i	0.427670	−	0.740747i	−0.568995	−	0.822341i	$-0.692668\pi$
$47$	20.1005	−	34.8151i	0.427670	−	0.740747i	0.996666	+	0.0815940i	$0.0260011\pi$
$48$		0			0
$49$	69.1690	+	119.804i	1.41161	+	2.44499i
$50$	−35.2703	−	2.45051i	−0.705406	−	0.0490102i
$51$		0			0
$52$	29.4591	+	17.0082i	0.566521	+	0.327081i
$53$	38.2721			0.722115			0.361057	−	0.932544i	$-0.382416\pi$
$53$	38.2721			0.722115			0.361057	+	0.932544i	$0.382416\pi$
$54$		0			0
$55$	28.9117	+	54.3457i	0.525667	+	0.988103i
$56$	−33.5265	−	19.3566i	−0.598688	−	0.345653i
$57$		0			0
$58$	−25.8851	+	14.9448i	−0.446295	+	0.257668i
$59$	−36.0537	+	20.8156i	−0.611080	+	0.352807i	−0.773388	−	0.633933i	$-0.781440\pi$
$59$	−36.0537	+	20.8156i	−0.611080	+	0.352807i	0.162308	+	0.986740i	$0.448106\pi$
$60$		0			0
$61$	7.52944	−	13.0414i	0.123433	−	0.213793i	−0.797686	−	0.603073i	$-0.793943\pi$
$61$	7.52944	−	13.0414i	0.123433	−	0.213793i	0.921119	+	0.389280i	$0.127276\pi$
$62$	−54.0416			−0.871639
$63$		0			0
$64$	8.00000			0.125000
$65$	84.9899	+	2.94891i	1.30754	+	0.0453678i
$66$		0			0
$67$	−111.386	+	64.3089i	−1.66248	+	0.959834i	−0.690956	+	0.722897i	$0.742810\pi$
$67$	−111.386	+	64.3089i	−1.66248	+	0.959834i	−0.971525	+	0.236937i	$0.923856\pi$
$68$	−6.89949	−	11.9503i	−0.101463	−	0.175739i
$69$		0			0
$70$	−96.7246	−	3.35607i	−1.38178	−	0.0479438i
$71$		−	104.967i		−	1.47841i	−0.673478	−	0.739207i	$-0.735200\pi$
$71$		−	104.967i		−	1.47841i	0.673478	−	0.739207i	$-0.264800\pi$
$72$		0			0
$73$			2.11232i			0.0289359i	0.999895	+	0.0144680i	$0.00460546\pi$
$73$			2.11232i			0.0289359i	−0.999895	+	0.0144680i	$0.995395\pi$
$74$	26.3786	+	15.2297i	0.356467	+	0.205806i
$75$		0			0
$76$	7.24264	+	12.5446i	0.0952979	+	0.165061i
$77$	84.2548	+	145.934i	1.09422	+	1.89524i
$78$		0			0
$79$	22.0477	−	38.1878i	0.279085	−	0.483390i	−0.692072	−	0.721828i	$-0.743302\pi$
$79$	22.0477	−	38.1878i	0.279085	−	0.483390i	0.971158	+	0.238438i	$0.0766355\pi$
$80$	17.6569	−	9.39338i	0.220711	−	0.117417i
$81$		0			0
$82$			51.4275i			0.627165i
$83$	27.5122	−	47.6525i	0.331472	−	0.574127i	−0.651329	−	0.758796i	$-0.725788\pi$
$83$	27.5122	−	47.6525i	0.331472	−	0.574127i	0.982801	+	0.184669i	$0.0591214\pi$
$84$		0			0
$85$	−29.2596	−	18.2743i	−0.344231	−	0.214992i
$86$	7.64144	−	4.41179i	0.0888539	−	0.0512998i
$87$		0			0
$88$	−30.1569	−	17.4111i	−0.342693	−	0.197854i
$89$			68.1020i			0.765191i	0.923916	+	0.382595i	$0.124970\pi$
$89$			68.1020i			0.765191i	−0.923916	+	0.382595i	$0.875030\pi$
$90$		0			0
$91$	232.794			2.55818
$92$	−34.7990	+	60.2736i	−0.378250	+	0.655148i
$93$		0			0
$94$	28.4264	+	49.2360i	0.302409	+	0.523787i
$95$	30.7148	+	19.1832i	0.323314	+	0.201929i
$96$		0			0
$97$	−87.6794	−	50.6217i	−0.903911	−	0.521873i	−0.0254441	−	0.999676i	$-0.508100\pi$
$97$	−87.6794	−	50.6217i	−0.903911	−	0.521873i	−0.878467	+	0.477803i	$0.841433\pi$
$98$	−195.640			−1.99632
$99$		0			0
$100$	27.9411	−	41.4644i	0.279411	−	0.414644i
$101$	13.1893	+	7.61484i	0.130587	+	0.0753944i	0.563870	−	0.825863i	$-0.309312\pi$
$101$	13.1893	+	7.61484i	0.130587	+	0.0753944i	−0.433283	+	0.901258i	$0.642645\pi$
$102$		0			0
$103$	66.7640	−	38.5462i	0.648194	−	0.374235i	−0.139570	−	0.990212i	$-0.544572\pi$
$103$	66.7640	−	38.5462i	0.648194	−	0.374235i	0.787764	+	0.615977i	$0.211239\pi$
$104$	−41.6614	+	24.0532i	−0.400591	+	0.231281i
$105$		0			0
$106$	−27.0624	+	46.8735i	−0.255306	+	0.442203i
$107$	−78.4264			−0.732957			−0.366479	−	0.930427i	$-0.619437\pi$
$107$	−78.4264			−0.732957			−0.366479	+	0.930427i	$0.619437\pi$
$108$		0			0
$109$	−146.279			−1.34201			−0.671006	−	0.741452i	$-0.734137\pi$
$109$	−146.279			−1.34201			−0.671006	+	0.741452i	$0.734137\pi$
$110$	−87.0033	−	3.01876i	−0.790939	−	0.0274433i
$111$		0			0
$112$	47.4137	−	27.3743i	0.423336	−	0.244413i
$113$	−27.2132	−	47.1347i	−0.240825	−	0.417121i	0.720125	−	0.693845i	$-0.244085\pi$
$113$	−27.2132	−	47.1347i	−0.240825	−	0.417121i	−0.960949	+	0.276724i	$0.910751\pi$
$114$		0			0
$115$	−6.03350	+	173.890i	−0.0524652	+	1.51209i
$116$		−	42.2702i		−	0.364398i
$117$		0			0
$118$		−	58.8755i		−	0.498945i
$119$	−81.7826	−	47.2172i	−0.687249	−	0.396783i
$120$		0			0
$121$	15.2868	+	26.4775i	0.126337	+	0.218822i
$122$	10.6482	+	18.4433i	0.0872806	+	0.151174i
$123$		0			0
$124$	38.2132	−	66.1872i	0.308171	−	0.533768i
$125$	12.9828	−	124.324i	0.103862	−	0.994592i
$126$		0			0
$127$			159.215i			1.25366i	0.779154	+	0.626832i	$0.215649\pi$
$127$			159.215i			1.25366i	−0.779154	+	0.626832i	$0.784351\pi$
$128$	−5.65685	+	9.79796i	−0.0441942	+	0.0765466i
$129$		0			0
$130$	−63.7086	+	102.006i	−0.490066	+	0.784660i
$131$	−95.5252	+	55.1515i	−0.729200	+	0.421004i	−0.818130	−	0.575034i	$-0.804989\pi$
$131$	−95.5252	+	55.1515i	−0.729200	+	0.421004i	0.0889293	+	0.996038i	$0.471655\pi$
$132$		0			0
$133$	85.8501	+	49.5656i	0.645489	+	0.372673i
$134$		−	181.893i		−	1.35741i
$135$		0			0
$136$	19.5147			0.143491
$137$	−108.959	+	188.723i	−0.795324	+	1.37754i	0.127309	+	0.991863i	$0.459366\pi$
$137$	−108.959	+	188.723i	−0.795324	+	1.37754i	−0.922633	+	0.385679i	$0.873967\pi$
$138$		0			0
$139$	53.1249	+	92.0150i	0.382193	+	0.661979i	0.991376	−	0.131052i	$-0.0418355\pi$
$139$	53.1249	+	92.0150i	0.382193	+	0.661979i	−0.609182	+	0.793030i	$0.708502\pi$
$140$	72.5049	−	116.090i	0.517892	−	0.829213i
$141$		0			0
$142$	128.558	+	74.2232i	0.905340	+	0.522698i
$143$	209.397			1.46431
$144$		0			0
$145$	−49.6325	−	93.2948i	−0.342293	−	0.643413i
$146$	−2.58706	−	1.49364i	−0.0177196	−	0.0102304i
$147$		0			0
$148$	−37.3049	+	21.5380i	−0.252060	+	0.145527i
$149$	−12.2622	+	7.07960i	−0.0822968	+	0.0475141i	−0.540584	−	0.841290i	$-0.681797\pi$
$149$	−12.2622	+	7.07960i	−0.0822968	+	0.0475141i	0.458287	+	0.888804i	$0.348463\pi$
$150$		0			0
$151$	−36.6690	+	63.5127i	−0.242841	+	0.420614i	−0.961522	−	0.274726i	$-0.911413\pi$
$151$	−36.6690	+	63.5127i	−0.242841	+	0.420614i	0.718681	+	0.695340i	$0.244746\pi$
$152$	−20.4853			−0.134772
$153$		0			0
$154$	−238.309			−1.54746
$155$	6.62546	−	190.951i	0.0427449	−	1.23194i
$156$		0			0
$157$	60.7475	−	35.0726i	0.386927	−	0.223392i	−0.293901	−	0.955836i	$-0.594954\pi$
$157$	60.7475	−	35.0726i	0.386927	−	0.223392i	0.680828	+	0.732444i	$0.261620\pi$
$158$	31.1802	+	54.0057i	0.197343	+	0.341808i
$159$		0			0
$160$	−0.980793	+	28.2673i	−0.00612996	+	0.176670i
$161$			476.299i			2.95838i
$162$		0			0
$163$			9.05959i			0.0555803i	0.999614	+	0.0277902i	$0.00884702\pi$
$163$			9.05959i			0.0555803i	−0.999614	+	0.0277902i	$0.991153\pi$
$164$	−62.9856	−	36.3648i	−0.384059	−	0.221736i
$165$		0			0
$166$	38.9081	+	67.3908i	0.234386	+	0.405969i
$167$	31.9264	+	55.2982i	0.191176	+	0.331127i	0.945640	−	0.325215i	$-0.105437\pi$
$167$	31.9264	+	55.2982i	0.191176	+	0.331127i	−0.754464	+	0.656341i	$0.772103\pi$
$168$		0			0
$169$	60.1396	−	104.165i	0.355856	−	0.616360i
$170$	43.0711	−	22.9136i	0.253359	−	0.134786i
$171$		0			0
$172$			12.4784i			0.0725489i
$173$	−13.9939	+	24.2382i	−0.0808896	+	0.140105i	−0.903632	−	0.428309i	$-0.859109\pi$
$173$	−13.9939	+	24.2382i	−0.0808896	+	0.140105i	0.822743	+	0.568414i	$0.192443\pi$
$174$		0			0
$175$	23.7167	−	341.356i	0.135524	−	1.95060i
$176$	42.6484	−	24.6230i	0.242320	−	0.139904i
$177$		0			0
$178$	−83.4075	−	48.1554i	−0.468582	−	0.270536i
$179$			21.0660i			0.117687i	0.998267	+	0.0588435i	$0.0187413\pi$
$179$			21.0660i			0.117687i	−0.998267	+	0.0588435i	$0.981259\pi$
$180$		0			0
$181$	53.6030			0.296149			0.148075	−	0.988976i	$-0.452692\pi$
$181$	53.6030			0.296149			0.148075	+	0.988976i	$0.452692\pi$
$182$	−164.610	+	285.113i	−0.904452	+	1.56656i
$183$		0			0
$184$	−49.2132	−	85.2398i	−0.267463	−	0.463260i
$185$	−57.0466	+	91.3391i	−0.308360	+	0.493725i
$186$		0			0
$187$	−73.5630	−	42.4716i	−0.393385	−	0.227121i
$188$	−80.4020			−0.427670
$189$		0			0
$190$	−45.2132	+	24.0532i	−0.237964	+	0.126596i
$191$	193.144	+	111.512i	1.01123	+	0.583831i	0.911550	−	0.411189i	$-0.134886\pi$
$191$	193.144	+	111.512i	1.01123	+	0.583831i	0.0996752	+	0.995020i	$0.468220\pi$
$192$		0			0
$193$	−152.614	+	88.1118i	−0.790746	+	0.456538i	−0.840225	−	0.542237i	$-0.817577\pi$
$193$	−152.614	+	88.1118i	−0.790746	+	0.456538i	0.0494788	+	0.998775i	$0.484244\pi$
$194$	123.997	−	71.5899i	0.639162	−	0.369020i
$195$		0			0
$196$	138.338	−	239.609i	0.705807	−	1.22249i
$197$	277.586			1.40906			0.704532	−	0.709672i	$-0.251157\pi$
$197$	277.586			1.40906			0.704532	+	0.709672i	$0.251157\pi$
$198$		0			0
$199$	71.5736			0.359666			0.179833	−	0.983697i	$-0.442444\pi$
$199$	71.5736			0.359666			0.179833	+	0.983697i	$0.442444\pi$
$200$	31.0259	+	63.5405i	0.155130	+	0.317702i
$201$		0			0
$202$	−18.6525	+	10.7690i	−0.0923389	+	0.0533119i
$203$	−144.640	−	250.523i	−0.712510	−	1.23410i
$204$		0			0
$205$	−181.714	−	6.30497i	−0.886412	−	0.0307560i
$206$			109.025i			0.529248i
$207$		0			0
$208$		−	68.0328i		−	0.327081i
$209$	77.2217	+	44.5840i	0.369482	+	0.213320i
$210$		0			0
$211$	−159.746	−	276.689i	−0.757091	−	1.31132i	−0.944328	−	0.329005i	$-0.893287\pi$
$211$	−159.746	−	276.689i	−0.757091	−	1.31132i	0.187237	−	0.982315i	$-0.440047\pi$
$212$	−38.2721	−	66.2892i	−0.180529	−	0.312685i
$213$		0			0
$214$	55.4558	−	96.0523i	0.259139	−	0.448843i
$215$	14.6518	+	27.5412i	0.0681479	+	0.128099i
$216$		0			0
$217$		−	523.030i		−	2.41028i
$218$	103.435	−	179.155i	0.474473	−	0.821811i
$219$		0			0
$220$	65.2178	−	104.422i	0.296445	−	0.474646i
$221$	−101.626	+	58.6740i	−0.459848	+	0.265493i
$222$		0			0
$223$	−140.676	−	81.2193i	−0.630834	−	0.364212i	0.150241	−	0.988649i	$-0.451995\pi$
$223$	−140.676	−	81.2193i	−0.630834	−	0.364212i	−0.781075	+	0.624437i	$0.785328\pi$
$224$			77.4262i			0.345653i
$225$		0			0
$226$	76.9706			0.340578
$227$	−120.549	+	208.797i	−0.531052	+	0.919809i	0.468291	+	0.883574i	$0.344870\pi$
$227$	−120.549	+	208.797i	−0.531052	+	0.919809i	−0.999343	+	0.0362347i	$0.988464\pi$
$228$		0			0
$229$	−51.1102	−	88.5254i	−0.223189	−	0.386574i	0.732586	−	0.680675i	$-0.238313\pi$
$229$	−51.1102	−	88.5254i	−0.223189	−	0.386574i	−0.955774	+	0.294101i	$0.904980\pi$
$230$	−208.705	−	130.349i	−0.907413	−	0.566733i
$231$		0			0
$232$	51.7702	+	29.8895i	0.223147	+	0.128834i
$233$	241.966			1.03848			0.519239	−	0.854629i	$-0.326215\pi$
$233$	241.966			1.03848			0.519239	+	0.854629i	$0.326215\pi$
$234$		0			0
$235$	−177.456	+	94.4058i	−0.755131	+	0.401727i
$236$	72.1075	+	41.6313i	0.305540	+	0.176404i
$237$		0			0
$238$	115.658	−	66.7752i	0.485958	−	0.280568i
$239$	325.157	−	187.729i	1.36049	−	0.785478i	0.370799	−	0.928713i	$-0.379084\pi$
$239$	325.157	−	187.729i	1.36049	−	0.785478i	0.989689	+	0.143235i	$0.0457505\pi$
$240$		0			0
$241$	159.140	−	275.638i	0.660330	−	1.14373i	−0.320198	−	0.947350i	$-0.603750\pi$
$241$	159.140	−	275.638i	0.660330	−	1.14373i	0.980529	−	0.196375i	$-0.0629171\pi$
$242$	−43.2376			−0.178668
$243$		0			0
$244$	−30.1177			−0.123433
$245$	23.9853	−	691.274i	0.0978990	−	2.82153i
$246$		0			0
$247$	106.681	−	61.5922i	0.431906	−	0.249361i
$248$	54.0416	+	93.6028i	0.217910	+	0.377431i
$249$		0			0
$250$	143.085	+	103.811i	0.572340	+	0.415244i
$251$		−	85.5417i		−	0.340804i	−0.985375	−	0.170402i	$-0.945493\pi$
$251$		−	85.5417i		−	0.340804i	0.985375	−	0.170402i	$-0.0545066\pi$
$252$		0			0
$253$			428.429i			1.69339i
$254$	−194.998	−	112.582i	−0.767709	−	0.443237i
$255$		0			0
$256$	−8.00000	−	13.8564i	−0.0312500	−	0.0541266i
$257$	107.114	+	185.526i	0.416785	+	0.721893i	0.995614	−	0.0935562i	$-0.0298235\pi$
$257$	107.114	+	185.526i	0.416785	+	0.721893i	−0.578829	+	0.815449i	$0.696490\pi$
$258$		0			0
$259$	−147.397	+	255.299i	−0.569100	+	0.985711i
$260$	−79.8823	−	150.156i	−0.307239	−	0.577522i
$261$		0			0
$262$		−	155.992i		−	0.595389i
$263$	53.4386	−	92.5584i	0.203189	−	0.351933i	−0.746365	−	0.665536i	$-0.768203\pi$
$263$	53.4386	−	92.5584i	0.203189	−	0.351933i	0.949554	+	0.313603i	$0.101536\pi$
$264$		0			0
$265$	−162.305	−	101.369i	−0.612473	−	0.382526i
$266$	−121.410	+	70.0963i	−0.456430	+	0.263520i
$267$		0			0
$268$	222.772	+	128.618i	0.831241	+	0.479917i
$269$		−	365.927i		−	1.36032i	−0.733062	−	0.680161i	$-0.761910\pi$
$269$		−	365.927i		−	1.36032i	0.733062	−	0.680161i	$-0.238090\pi$
$270$		0			0
$271$	−92.8894			−0.342765			−0.171383	−	0.985205i	$-0.554823\pi$
$271$	−92.8894			−0.342765			−0.171383	+	0.985205i	$0.554823\pi$
$272$	−13.7990	+	23.9006i	−0.0507316	+	0.0878697i
$273$		0			0
$274$	−154.092	−	266.895i	−0.562379	−	0.974069i
$275$	21.3331	−	307.048i	0.0775747	−	1.11654i
$276$		0			0
$277$	195.841	+	113.069i	0.707006	+	0.408190i	0.809951	−	0.586497i	$-0.199493\pi$
$277$	195.841	+	113.069i	0.707006	+	0.408190i	−0.102946	+	0.994687i	$0.532827\pi$
$278$	−150.260			−0.540503
$279$		0			0
$280$	90.9117	+	170.888i	0.324685	+	0.610314i
$281$	−54.9106	−	31.7026i	−0.195411	−	0.112821i	0.399102	−	0.916907i	$-0.369322\pi$
$281$	−54.9106	−	31.7026i	−0.195411	−	0.112821i	−0.594513	+	0.804086i	$0.702655\pi$
$282$		0			0
$283$	314.459	−	181.553i	1.11116	−	0.641531i	0.172033	−	0.985091i	$-0.444966\pi$
$283$	314.459	−	181.553i	1.11116	−	0.641531i	0.939131	+	0.343560i	$0.111633\pi$
$284$	−181.809	+	104.967i	−0.640172	+	0.369604i
$285$		0			0
$286$	−148.066	+	256.458i	−0.517713	+	0.896706i
$287$	−497.730			−1.73425
$288$		0			0
$289$	−241.397			−0.835284
$290$	149.358	+	5.18229i	0.515027	+	0.0178700i
$291$		0			0
$292$	3.65865	−	2.11232i	0.0125296	−	0.00723398i
$293$	113.717	+	196.963i	0.388112	+	0.672229i	0.992196	−	0.124691i	$-0.0397941\pi$
$293$	113.717	+	196.963i	0.388112	+	0.672229i	−0.604084	+	0.796921i	$0.706461\pi$
$294$		0			0
$295$	208.031	+	7.21808i	0.705190	+	0.0244681i
$296$		−	60.9187i		−	0.205806i
$297$		0			0
$298$		−	20.0241i		−	0.0671950i
$299$	512.573	+	295.934i	1.71429	+	0.989747i
$300$		0			0
$301$	42.6985	+	73.9559i	0.141855	+	0.245701i
$302$	−51.8579	−	89.8205i	−0.171715	−	0.297419i
$303$		0			0
$304$	14.4853	−	25.0892i	0.0476490	−	0.0825304i
$305$	−66.4731	+	35.3634i	−0.217945	+	0.115946i
$306$		0			0
$307$			340.164i			1.10803i	0.832508	+	0.554013i	$0.186904\pi$
$307$			340.164i			1.10803i	−0.832508	+	0.554013i	$0.813096\pi$
$308$	168.510	−	291.867i	0.547109	−	0.947621i
$309$		0			0
$310$	229.181	+	143.137i	0.739295	+	0.461733i
$311$	334.951	−	193.384i	1.07701	−	0.621815i	0.146925	−	0.989148i	$-0.453062\pi$
$311$	334.951	−	193.384i	1.07701	−	0.621815i	0.930089	+	0.367333i	$0.119729\pi$
$312$		0			0
$313$	325.974	+	188.201i	1.04145	+	0.601282i	0.920244	−	0.391345i	$-0.127990\pi$
$313$	325.974	+	188.201i	1.04145	+	0.601282i	0.121207	+	0.992627i	$0.461323\pi$
$314$			99.2002i			0.315924i
$315$		0			0
$316$	−88.1909			−0.279085
$317$	−177.700	+	307.785i	−0.560566	+	0.970929i	0.436881	+	0.899519i	$0.356083\pi$
$317$	−177.700	+	307.785i	−0.560566	+	0.970929i	−0.997447	+	0.0714099i	$0.977250\pi$
$318$		0			0
$319$	−130.103	−	225.344i	−0.407845	−	0.706408i
$320$	−33.9267	−	21.1892i	−0.106021	−	0.0662162i
$321$		0			0
$322$	−583.345	−	336.794i	−1.81163	−	1.04594i
$323$	−49.9706			−0.154708
$324$		0			0
$325$	−352.617	−	237.614i	−1.08498	−	0.731121i
$326$	−11.0957	−	6.40610i	−0.0340359	−	0.0196506i
$327$		0			0
$328$	89.0751	−	51.4275i	0.271570	−	0.156791i
$329$	−476.519	+	275.119i	−1.44839	+	0.836227i
$330$		0			0
$331$	222.095	−	384.681i	0.670983	−	1.16218i	−0.306642	−	0.951825i	$-0.599206\pi$
$331$	222.095	−	384.681i	0.670983	−	1.16218i	0.977626	−	0.210352i	$-0.0674611\pi$
$332$	−110.049			−0.331472
$333$		0			0
$334$	−90.3015			−0.270364
$335$	642.702	+	22.2999i	1.91851	+	0.0665669i
$336$		0			0
$337$	352.547	−	203.543i	1.04613	−	0.603985i	0.124569	−	0.992211i	$-0.460245\pi$
$337$	352.547	−	203.543i	1.04613	−	0.603985i	0.921564	+	0.388226i	$0.126912\pi$
$338$	85.0503	+	147.311i	0.251628	+	0.435832i
$339$		0			0
$340$	−2.39249	+	68.9535i	−0.00703673	+	0.202804i
$341$		−	470.463i		−	1.37966i
$342$		0			0
$343$		−	1222.78i		−	3.56497i
$344$	−15.2829	−	8.82357i	−0.0444270	−	0.0256499i
$345$		0			0
$346$	−19.7904	−	34.2779i	−0.0571976	−	0.0990691i
$347$	244.706	+	423.843i	0.705204	+	1.22145i	0.966618	+	0.256222i	$0.0824777\pi$
$347$	244.706	+	423.843i	0.705204	+	1.22145i	−0.261414	+	0.965227i	$0.584189\pi$
$348$		0			0
$349$	−40.8970	+	70.8356i	−0.117183	+	0.202967i	−0.918650	−	0.395071i	$-0.870720\pi$
$349$	−40.8970	+	70.8356i	−0.117183	+	0.202967i	0.801467	+	0.598039i	$0.204053\pi$
$350$	401.304	+	270.422i	1.14658	+	0.772634i
$351$		0			0
$352$			69.6445i			0.197854i
$353$	125.693	−	217.707i	0.356072	−	0.616735i	−0.631229	−	0.775597i	$-0.717449\pi$
$353$	125.693	−	217.707i	0.356072	−	0.616735i	0.987301	+	0.158862i	$0.0507824\pi$
$354$		0			0
$355$	−278.022	+	445.149i	−0.783160	+	1.25394i
$356$	117.956	−	68.1020i	0.331337	−	0.191298i
$357$		0			0
$358$	−25.8004	−	14.8959i	−0.0720682	−	0.0416086i
$359$			518.500i			1.44429i	0.691742	+	0.722145i	$0.256844\pi$
$359$			518.500i			1.44429i	−0.691742	+	0.722145i	$0.743156\pi$
$360$		0			0
$361$	−308.544			−0.854693
$362$	−37.9031	+	65.6500i	−0.104705	+	0.181354i
$363$		0			0
$364$	−232.794	−	403.211i	−0.639544	−	1.10772i
$365$	5.59480	−	8.95801i	0.0153282	−	0.0245425i
$366$		0			0
$367$	296.166	+	170.992i	0.806992	+	0.465917i	0.845910	−	0.533325i	$-0.179058\pi$
$367$	296.166	+	170.992i	0.806992	+	0.465917i	−0.0389180	+	0.999242i	$0.512391\pi$
$368$	139.196			0.378250
$369$		0			0
$370$	−71.5290	−	134.454i	−0.193322	−	0.363389i
$371$	−453.655	−	261.918i	−1.22279	−	0.705978i
$372$		0			0
$373$	−95.5252	+	55.1515i	−0.256100	+	0.147859i	−0.622554	−	0.782577i	$-0.713905\pi$
$373$	−95.5252	+	55.1515i	−0.256100	+	0.147859i	0.366454	+	0.930436i	$0.380572\pi$
$374$	104.034	−	60.0640i	0.278165	−	0.160599i
$375$		0			0
$376$	56.8528	−	98.4720i	0.151204	−	0.261894i
$377$	−359.470			−0.953502
$378$		0			0
$379$	324.345			0.855792			0.427896	−	0.903828i	$-0.359255\pi$
$379$	324.345			0.855792			0.427896	+	0.903828i	$0.359255\pi$
$380$	2.51148	−	72.3828i	0.00660915	−	0.190481i
$381$		0			0
$382$	−273.147	+	157.701i	−0.715044	+	0.412831i
$383$	−286.240	−	495.782i	−0.747363	−	1.29447i	−0.949082	−	0.315028i	$-0.897986\pi$
$383$	−286.240	−	495.782i	−0.747363	−	1.29447i	0.201719	−	0.979443i	$-0.435347\pi$
$384$		0			0
$385$	29.2164	−	842.042i	0.0758869	−	2.18712i
$386$		−	249.218i		−	0.645642i
$387$		0			0
$388$			202.487i			0.521873i
$389$	−624.981	−	360.833i	−1.60664	−	0.927592i	−0.990116	−	0.140253i	$-0.955208\pi$
$389$	−624.981	−	360.833i	−1.60664	−	0.927592i	−0.616520	−	0.787339i	$-0.711458\pi$
$390$		0			0
$391$	−120.048	−	207.929i	−0.307027	−	0.531787i
$392$	195.640	+	338.858i	0.499081	+	0.864433i
$393$		0			0
$394$	−196.283	+	339.972i	−0.498180	+	0.862873i
$395$	−194.647	+	103.551i	−0.492777	+	0.262155i
$396$		0			0
$397$			379.321i			0.955468i	0.878505	+	0.477734i	$0.158542\pi$
$397$			379.321i			0.955468i	−0.878505	+	0.477734i	$0.841458\pi$
$398$	−50.6102	+	87.6594i	−0.127161	+	0.220250i
$399$		0			0
$400$	−99.7595	−	6.93108i	−0.249399	−	0.0173277i
$401$	−303.952	+	175.487i	−0.757985	+	0.437623i	−0.828572	−	0.559883i	$-0.810846\pi$
$401$	−303.952	+	175.487i	−0.757985	+	0.437623i	0.0705866	+	0.997506i	$0.477513\pi$
$402$		0			0
$403$	−562.863	−	324.969i	−1.39668	−	0.806375i
$404$		−	30.4593i		−	0.0753944i
$405$		0			0
$406$	409.103			1.00764
$407$	−132.583	+	229.640i	−0.325756	+	0.564227i
$408$		0			0
$409$	266.713	+	461.961i	0.652111	+	1.12949i	0.982610	+	0.185682i	$0.0594495\pi$
$409$	266.713	+	461.961i	0.652111	+	1.12949i	−0.330499	+	0.943806i	$0.607217\pi$
$410$	136.214	−	218.096i	0.332228	−	0.531940i
$411$		0			0
$412$	−133.528	−	77.0924i	−0.324097	−	0.187118i
$413$	569.813			1.37969
$414$		0			0
$415$	−242.889	+	129.216i	−0.585276	+	0.311364i
$416$	83.3229	+	48.1065i	0.200295	+	0.115641i
$417$		0			0
$418$	−109.208	+	63.0513i	−0.261263	+	0.150840i
$419$	−207.634	+	119.878i	−0.495547	+	0.286104i	−0.726873	−	0.686772i	$-0.759027\pi$
$419$	−207.634	+	119.878i	−0.495547	+	0.286104i	0.231326	+	0.972876i	$0.425694\pi$
$420$		0			0
$421$	−212.471	+	368.010i	−0.504681	+	0.874133i	0.495305	+	0.868719i	$0.335056\pi$
$421$	−212.471	+	368.010i	−0.504681	+	0.874133i	−0.999985	+	0.00541323i	$0.998277\pi$
$422$	451.831			1.07069
$423$		0			0
$424$	108.250			0.255306
$425$	75.6827	+	154.997i	0.178077	+	0.364698i
$426$		0			0
$427$	−178.499	+	103.057i	−0.418031	+	0.241350i
$428$	78.4264	+	135.839i	0.183239	+	0.317380i
$429$		0			0
$430$	−44.0913	−	1.52984i	−0.102538	−	0.00355777i
$431$		−	544.039i		−	1.26227i	−0.775673	−	0.631135i	$-0.782589\pi$
$431$		−	544.039i		−	1.26227i	0.775673	−	0.631135i	$-0.217411\pi$
$432$		0			0
$433$			602.735i			1.39200i	0.718043	+	0.695999i	$0.245038\pi$
$433$			602.735i			1.39200i	−0.718043	+	0.695999i	$0.754962\pi$
$434$	640.578	+	369.838i	1.47599	+	0.852161i
$435$		0			0
$436$	146.279	+	253.363i	0.335503	+	0.581108i
$437$	126.018	+	218.270i	0.288371	+	0.499474i
$438$		0			0
$439$	−201.143	+	348.390i	−0.458185	+	0.793600i	−0.998865	−	0.0476287i	$-0.984834\pi$
$439$	−201.143	+	348.390i	−0.458185	+	0.793600i	0.540680	+	0.841228i	$0.318167\pi$
$440$	81.7746	+	153.713i	0.185851	+	0.349347i
$441$		0			0
$442$		−	165.955i		−	0.375464i
$443$	−219.257	+	379.765i	−0.494938	+	0.857257i	−0.999983	−	0.00583572i	$-0.998142\pi$
$443$	−219.257	+	379.765i	−0.494938	+	0.857257i	0.505045	+	0.863093i	$0.331476\pi$
$444$		0			0
$445$	180.378	−	288.809i	0.405344	−	0.649009i
$446$	198.946	−	114.861i	0.446067	−	0.257537i
$447$		0			0
$448$	−94.8274	−	54.7486i	−0.211668	−	0.122207i
$449$			670.866i			1.49413i	0.664749	+	0.747067i	$0.268538\pi$
$449$			670.866i			1.49413i	−0.664749	+	0.747067i	$0.731462\pi$
$450$		0			0
$451$	−447.706			−0.992695
$452$	−54.4264	+	94.2693i	−0.120412	+	0.208560i
$453$		0			0
$454$	−170.482	−	295.283i	−0.375510	−	0.650403i
$455$	−987.240	−	616.589i	−2.16976	−	1.35514i
$456$		0			0
$457$	601.913	+	347.514i	1.31710	+	0.760425i	0.983260	−	0.182206i	$-0.0583238\pi$
$457$	601.913	+	347.514i	1.31710	+	0.760425i	0.333835	+	0.942632i	$0.391657\pi$
$458$	144.561			0.315636
$459$		0			0
$460$	307.220	−	163.440i	0.667870	−	0.355304i
$461$	−11.3352	−	6.54436i	−0.0245882	−	0.0141960i	0.487656	−	0.873036i	$-0.337852\pi$
$461$	−11.3352	−	6.54436i	−0.0245882	−	0.0141960i	−0.512244	+	0.858840i	$0.671186\pi$
$462$		0			0
$463$	−202.894	+	117.141i	−0.438215	+	0.253004i	−0.702840	−	0.711348i	$-0.748085\pi$
$463$	−202.894	+	117.141i	−0.438215	+	0.253004i	0.264625	+	0.964351i	$0.414752\pi$
$464$	−73.2141	+	42.2702i	−0.157789	+	0.0910995i
$465$		0			0
$466$	−171.095	+	296.346i	−0.367158	+	0.635936i
$467$	−742.882			−1.59075			−0.795377	−	0.606115i	$-0.792727\pi$
$467$	−742.882			−1.59075			−0.795377	+	0.606115i	$0.792727\pi$
$468$		0			0
$469$	1760.41			3.75354
$470$	9.85722	−	284.093i	0.0209728	−	0.604453i
$471$		0			0
$472$	−101.975	+	58.8755i	−0.216049	+	0.124736i
$473$	38.4071	+	66.5230i	0.0811989	+	0.140641i
$474$		0			0
$475$	−79.4468	−	162.706i	−0.167256	−	0.342538i
$476$			188.869i			0.396783i
$477$		0			0
$478$			530.978i			1.11083i
$479$	−387.469	−	223.705i	−0.808913	−	0.467026i	0.0376655	−	0.999290i	$-0.488008\pi$
$479$	−387.469	−	223.705i	−0.808913	−	0.467026i	−0.846578	+	0.532264i	$0.821341\pi$
$480$		0			0
$481$	183.161	+	317.245i	0.380793	+	0.659553i
$482$	225.057	+	389.811i	0.466924	+	0.808736i
$483$		0			0
$484$	30.5736	−	52.9550i	0.0631686	−	0.109411i
$485$	237.754	+	446.910i	0.490215	+	0.921464i
$486$		0			0
$487$			50.3166i			0.103319i	0.998665	+	0.0516597i	$0.0164511\pi$
$487$			50.3166i			0.103319i	−0.998665	+	0.0516597i	$0.983549\pi$
$488$	21.2965	−	36.8866i	0.0436403	−	0.0755872i
$489$		0			0
$490$	829.675	+	518.181i	1.69321	+	1.05751i
$491$	684.707	−	395.316i	1.39452	−	0.805124i	0.400704	−	0.916208i	$-0.368766\pi$
$491$	684.707	−	395.316i	1.39452	−	0.805124i	0.993811	+	0.111084i	$0.0354322\pi$
$492$		0			0
$493$	126.285	+	72.9107i	0.256156	+	0.147892i
$494$			174.209i			0.352650i
$495$		0			0
$496$	−152.853			−0.308171
$497$	−718.352	+	1244.22i	−1.44538	+	2.50347i
$498$		0			0
$499$	300.195	+	519.953i	0.601593	+	1.04199i	0.992580	+	0.121593i	$0.0388004\pi$
$499$	300.195	+	519.953i	0.601593	+	1.04199i	−0.390987	+	0.920396i	$0.627866\pi$
$500$	−228.318	+	101.837i	−0.456636	+	0.203674i
$501$		0			0
$502$	104.767	+	60.4871i	0.208699	+	0.120492i
$503$	−433.368			−0.861566			−0.430783	−	0.902456i	$-0.641763\pi$
$503$	−433.368			−0.861566			−0.430783	+	0.902456i	$0.641763\pi$
$504$		0			0
$505$	−35.7645	−	67.2270i	−0.0708208	−	0.133123i
$506$	−524.716	−	302.945i	−1.03699	−	0.598705i
$507$		0			0
$508$	275.769	−	159.215i	0.542852	−	0.313416i
$509$	832.870	−	480.857i	1.63629	−	0.944710i	0.654190	−	0.756331i	$-0.273010\pi$
$509$	832.870	−	480.857i	1.63629	−	0.944710i	0.982096	−	0.188380i	$-0.0603235\pi$
$510$		0			0
$511$	14.4558	−	25.0383i	0.0282893	−	0.0489985i
$512$	22.6274			0.0441942
$513$		0			0
$514$	−302.963			−0.589423
$515$	−385.230	−	13.3664i	−0.748020	−	0.0259542i
$516$		0			0
$517$	−428.627	+	247.468i	−0.829065	+	0.478661i
$518$	−208.451	−	361.047i	−0.402415	−	0.697003i
$519$		0			0
$520$	240.388	+	8.34077i	0.462284	+	0.0160399i
$521$			5.90542i			0.0113348i	0.999984	+	0.00566739i	$0.00180400\pi$
$521$			5.90542i			0.0113348i	−0.999984	+	0.00566739i	$0.998196\pi$
$522$		0			0
$523$			165.846i			0.317104i	0.987351	+	0.158552i	$0.0506826\pi$
$523$			165.846i			0.317104i	−0.987351	+	0.158552i	$0.949317\pi$
$524$	191.050	+	110.303i	0.364600	+	0.210502i
$525$		0			0
$526$	75.5736	+	130.897i	0.143676	+	0.248854i
$527$	131.826	+	228.329i	0.250144	+	0.433262i
$528$		0			0
$529$	−340.985	+	590.603i	−0.644584	+	1.11645i
$530$	238.919	−	127.104i	0.450790	−	0.239819i
$531$		0			0
$532$		−	198.262i		−	0.372673i
$533$	−309.250	+	535.636i	−0.580206	+	1.00495i
$534$		0			0
$535$	332.593	+	207.724i	0.621670	+	0.388269i
$536$	−315.048	+	181.893i	−0.587776	+	0.339353i
$537$		0			0
$538$	448.167	+	258.749i	0.833024	+	0.480947i
$539$		−	1703.15i		−	3.15984i
$540$		0			0
$541$	216.985			0.401081			0.200541	−	0.979685i	$-0.435730\pi$
$541$	216.985			0.401081			0.200541	+	0.979685i	$0.435730\pi$
$542$	65.6827	−	113.766i	0.121186	−	0.209900i
$543$		0			0
$544$	−19.5147	−	33.8005i	−0.0358726	−	0.0621332i
$545$	620.346	+	387.442i	1.13825	+	0.710903i
$546$		0			0
$547$	−87.1611	−	50.3225i	−0.159344	−	0.0919973i	0.418208	−	0.908351i	$-0.362658\pi$
$547$	−87.1611	−	50.3225i	−0.159344	−	0.0919973i	−0.577552	+	0.816354i	$0.695992\pi$
$548$	435.838			0.795324
$549$		0			0
$550$	360.971	+	243.243i	0.656310	+	0.442260i
$551$	−132.566	−	76.5369i	−0.240591	−	0.138906i
$552$		0			0
$553$	−522.682	+	301.771i	−0.945175	+	0.545697i
$554$	−276.960	+	159.903i	−0.499928	+	0.288634i
$555$		0			0
$556$	106.250	−	184.030i	0.191097	−	0.330989i
$557$	116.662			0.209447			0.104723	−	0.994501i	$-0.466604\pi$
$557$	116.662			0.209447			0.104723	+	0.994501i	$0.466604\pi$
$558$		0			0
$559$	106.118			0.189835
$560$	−273.578	−	9.49239i	−0.488533	−	0.0169507i
$561$		0			0
$562$	77.6553	−	44.8343i	0.138177	−	0.0797763i
$563$	−266.953	−	462.377i	−0.474162	−	0.821273i	0.525400	−	0.850855i	$-0.323916\pi$
$563$	−266.953	−	462.377i	−0.474162	−	0.821273i	−0.999562	+	0.0295822i	$0.990582\pi$
$564$		0			0
$565$	−9.43653	+	271.968i	−0.0167018	+	0.481360i
$566$			513.510i			0.907262i
$567$		0			0
$568$		−	296.893i		−	0.522698i
$569$	97.5590	+	56.3257i	0.171457	+	0.0989907i	0.583273	−	0.812276i	$-0.301772\pi$
$569$	97.5590	+	56.3257i	0.171457	+	0.0989907i	−0.411816	+	0.911267i	$0.635105\pi$
$570$		0			0
$571$	430.768	+	746.111i	0.754409	+	1.30668i	0.945668	+	0.325135i	$0.105410\pi$
$571$	430.768	+	746.111i	0.754409	+	1.30668i	−0.191258	+	0.981540i	$0.561257\pi$
$572$	−209.397	−	362.686i	−0.366079	−	0.634067i
$573$		0			0
$574$	351.948	−	609.592i	0.613150	−	1.06201i
$575$	486.161	−	721.459i	0.845498	−	1.25471i
$576$		0			0
$577$			37.3256i			0.0646891i	0.999477	+	0.0323446i	$0.0102974\pi$
$577$			37.3256i			0.0646891i	−0.999477	+	0.0323446i	$0.989703\pi$
$578$	170.693	−	295.650i	0.295317	−	0.511505i
$579$		0			0
$580$	−111.959	+	179.261i	−0.193033	+	0.309070i
$581$	−652.227	+	376.564i	−1.12259	+	0.648130i
$582$		0			0
$583$	−408.060	−	235.594i	−0.699932	−	0.404106i
$584$			5.97455i			0.0102304i
$585$		0			0
$586$	−321.640			−0.548873
$587$	269.051	−	466.011i	0.458350	−	0.793885i	−0.540524	−	0.841328i	$-0.681774\pi$
$587$	269.051	−	466.011i	0.458350	−	0.793885i	0.998874	+	0.0474434i	$0.0151074\pi$
$588$		0			0
$589$	−138.382	−	239.685i	−0.234944	−	0.406936i
$590$	−155.941	+	249.681i	−0.264306	+	0.423188i
$591$		0			0
$592$	74.6098	+	43.0760i	0.126030	+	0.0727635i
$593$	−884.169			−1.49101			−0.745505	−	0.666500i	$-0.767792\pi$
$593$	−884.169			−1.49101			−0.745505	+	0.666500i	$0.767792\pi$
$594$		0			0
$595$	221.765	+	416.854i	0.372713	+	0.700595i
$596$	24.5244	+	14.1592i	0.0411484	+	0.0237570i
$597$		0			0
$598$	−724.888	+	418.514i	−1.21219	+	0.699857i
$599$	−38.7750	+	22.3868i	−0.0647330	+	0.0373736i	−0.532017	−	0.846734i	$-0.678566\pi$
$599$	−38.7750	+	22.3868i	−0.0647330	+	0.0373736i	0.467284	+	0.884107i	$0.345232\pi$
$600$		0			0
$601$	−197.287	+	341.711i	−0.328264	+	0.568570i	−0.982168	−	0.188008i	$-0.939797\pi$
$601$	−197.287	+	341.711i	−0.328264	+	0.568570i	0.653903	+	0.756578i	$0.273130\pi$
$602$	−120.770			−0.200614
$603$		0			0
$604$	146.676			0.242841
$605$	5.30089	−	152.776i	0.00876181	−	0.252522i
$606$		0			0
$607$	186.345	−	107.586i	0.306993	−	0.177243i	−0.338587	−	0.940935i	$-0.609949\pi$
$607$	186.345	−	107.586i	0.306993	−	0.177243i	0.645580	+	0.763692i	$0.276616\pi$
$608$	20.4853	+	35.4815i	0.0336929	+	0.0583578i
$609$		0			0
$610$	3.69241	−	106.418i	0.00605313	−	0.174456i
$611$			683.747i			1.11906i
$612$		0			0
$613$			333.937i			0.544758i	0.962190	+	0.272379i	$0.0878105\pi$
$613$			333.937i			0.544758i	−0.962190	+	0.272379i	$0.912189\pi$
$614$	−416.614	−	240.532i	−0.678525	−	0.391747i
$615$		0			0
$616$	238.309	+	412.763i	0.386865	+	0.670069i
$617$	32.7117	+	56.6584i	0.0530174	+	0.0918288i	0.891316	−	0.453382i	$-0.149783\pi$
$617$	32.7117	+	56.6584i	0.0530174	+	0.0918288i	−0.838299	+	0.545211i	$0.816449\pi$
$618$		0			0
$619$	9.96342	−	17.2571i	0.0160960	−	0.0278791i	−0.857865	−	0.513875i	$-0.828210\pi$
$619$	9.96342	−	17.2571i	0.0160960	−	0.0278791i	0.873961	+	0.485996i	$0.161543\pi$
$620$	−337.362	+	179.475i	−0.544133	+	0.289477i
$621$		0			0
$622$			546.973i			0.879379i
$623$	466.061	−	807.241i	0.748091	−	1.29573i
$624$		0			0
$625$	−384.348	+	492.850i	−0.614957	+	0.788560i
$626$	−460.997	+	266.157i	−0.736417	+	0.425171i
$627$		0			0
$628$	−121.495	−	70.1452i	−0.193463	−	0.111696i
$629$		−	148.601i		−	0.236250i
$630$		0			0
$631$	−210.625			−0.333796			−0.166898	−	0.985974i	$-0.553375\pi$
$631$	−210.625			−0.333796			−0.166898	+	0.985974i	$0.553375\pi$
$632$	62.3604	−	108.011i	0.0986715	−	0.170904i
$633$		0			0
$634$	−251.305	−	435.273i	−0.396380	−	0.686551i
$635$	421.706	−	675.206i	0.664103	−	1.06332i
$636$		0			0
$637$	−2037.66	−	1176.44i	−3.19883	−	1.84685i
$638$	367.986			0.576780
$639$		0			0
$640$	49.9411	−	26.5685i	0.0780330	−	0.0415132i
$641$	1019.97	+	588.881i	1.59122	+	0.918692i	0.993098	+	0.117287i	$0.0374196\pi$
$641$	1019.97	+	588.881i	1.59122	+	0.918692i	0.598122	+	0.801405i	$0.295914\pi$
$642$		0			0
$643$	−32.5995	+	18.8213i	−0.0506990	+	0.0292711i	−0.525135	−	0.851019i	$-0.675985\pi$
$643$	−32.5995	+	18.8213i	−0.0506990	+	0.0292711i	0.474436	+	0.880290i	$0.342652\pi$
$644$	824.974	−	476.299i	1.28102	−	0.739595i
$645$		0			0
$646$	35.3345	−	61.2012i	0.0546974	−	0.0947387i
$647$	−695.382			−1.07478			−0.537389	−	0.843334i	$-0.680589\pi$
$647$	−695.382			−1.07478			−0.537389	+	0.843334i	$0.680589\pi$
$648$		0			0
$649$	512.544			0.789744
$650$	540.355	−	263.848i	0.831315	−	0.405919i
$651$		0			0
$652$	15.6917	−	9.05959i	0.0240670	−	0.0138951i
$653$	462.148	+	800.464i	0.707731	+	1.22583i	0.965697	+	0.259672i	$0.0836144\pi$
$653$	462.148	+	800.464i	0.707731	+	1.22583i	−0.257966	+	0.966154i	$0.583052\pi$
$654$		0			0
$655$	551.183	+	19.1245i	0.841501	+	0.0291977i
$656$			145.459i			0.221736i
$657$		0			0
$658$		−	778.153i		−	1.18260i
$659$	−58.6043	−	33.8352i	−0.0889291	−	0.0513433i	0.454876	−	0.890555i	$-0.349684\pi$
$659$	−58.6043	−	33.8352i	−0.0889291	−	0.0513433i	−0.543805	+	0.839212i	$0.683017\pi$
$660$		0			0
$661$	−516.434	−	894.489i	−0.781291	−	1.35324i	−0.931190	−	0.364535i	$-0.881228\pi$
$661$	−516.434	−	894.489i	−0.781291	−	1.35324i	0.149898	−	0.988701i	$-0.452105\pi$
$662$	314.090	+	544.021i	0.474457	+	0.821783i
$663$		0			0
$664$	77.8162	−	134.782i	0.117193	−	0.202984i
$665$	−232.794	−	437.586i	−0.350066	−	0.658024i
$666$		0			0
$667$		−	735.480i		−	1.10267i
$668$	63.8528	−	110.596i	0.0955880	−	0.165563i
$669$		0			0
$670$	−481.771	+	771.377i	−0.719061	+	1.15131i
$671$	−160.559	+	92.6988i	−0.239283	+	0.138150i
$672$		0			0
$673$	−8.80797	−	5.08528i	−0.0130876	−	0.00755614i	0.493442	−	0.869779i	$-0.335739\pi$
$673$	−8.80797	−	5.08528i	−0.0130876	−	0.00755614i	−0.506530	+	0.862223i	$0.669072\pi$
$674$			575.707i			0.854164i
$675$		0			0
$676$	−240.558			−0.355856
$677$	−288.468	+	499.641i	−0.426098	+	0.738023i	−0.996522	−	0.0833273i	$-0.973445\pi$
$677$	−288.468	+	499.641i	−0.426098	+	0.738023i	0.570425	+	0.821350i	$0.306779\pi$
$678$		0			0
$679$	692.867	+	1200.08i	1.02042	+	1.76742i
$680$	−82.7586	−	51.6876i	−0.121704	−	0.0760112i
$681$		0			0
$682$	576.197	+	332.667i	0.844863	+	0.487782i
$683$	−526.009			−0.770145			−0.385073	−	0.922886i	$-0.625824\pi$
$683$	−526.009			−0.770145			−0.385073	+	0.922886i	$0.625824\pi$
$684$		0			0
$685$	961.940	−	511.748i	1.40429	−	0.747078i
$686$	1497.60	+	864.639i	2.18309	+	1.26041i
$687$		0			0
$688$	21.6132	−	12.4784i	0.0314146	−	0.0181372i
$689$	−563.730	+	325.470i	−0.818186	+	0.472380i
$690$		0			0
$691$	−211.452	+	366.245i	−0.306008	+	0.530022i	−0.977485	−	0.211003i	$-0.932327\pi$
$691$	−211.452	+	366.245i	−0.306008	+	0.530022i	0.671477	+	0.741025i	$0.265660\pi$
$692$	55.9756			0.0808896
$693$		0			0
$694$	−692.132			−0.997308
$695$	18.4217	−	530.929i	0.0265061	−	0.763927i
$696$		0			0
$697$	217.284	−	125.449i	0.311742	−	0.179985i
$698$	−57.8370	−	100.177i	−0.0828611	−	0.143520i
$699$		0			0
$700$	−614.962	+	300.277i	−0.878518	+	0.428968i
$701$			881.146i			1.25698i	0.777816	+	0.628492i	$0.216328\pi$
$701$			881.146i			1.25698i	−0.777816	+	0.628492i	$0.783672\pi$
$702$		0			0
$703$			155.992i			0.221895i
$704$	−85.2967	−	49.2461i	−0.121160	−	0.0699518i
$705$		0			0
$706$	177.757	+	307.885i	0.251781	+	0.436097i
$707$	−104.225	−	180.524i	−0.147419	−	0.255338i
$708$		0			0
$709$	−50.7136	+	87.8386i	−0.0715284	+	0.123891i	−0.899571	−	0.436774i	$-0.856121\pi$
$709$	−50.7136	+	87.8386i	−0.0715284	+	0.123891i	0.828043	+	0.560665i	$0.189454\pi$
$710$	−348.603	−	655.274i	−0.490990	−	0.922921i
$711$		0			0
$712$			192.621i			0.270536i
$713$	664.890	−	1151.62i	0.932525	−	1.61518i
$714$		0			0
$715$	−888.017	−	554.619i	−1.24198	−	0.775691i
$716$	36.4873	−	21.0660i	0.0509599	−	0.0294217i
$717$		0			0
$718$	−635.030	−	366.635i	−0.884443	−	0.510634i
$719$			36.2956i			0.0504807i	0.999681	+	0.0252404i	$0.00803511\pi$
$719$			36.2956i			0.0504807i	−0.999681	+	0.0252404i	$0.991965\pi$
$720$		0			0
$721$	−1055.18			−1.46349
$722$	218.174	−	377.888i	0.302180	−	0.523390i
$723$		0			0
$724$	−53.6030	−	92.8432i	−0.0740373	−	0.128236i
$725$	−36.6223	+	527.107i	−0.0505135	+	0.727044i
$726$		0			0
$727$	457.049	+	263.878i	0.628679	+	0.362968i	0.780240	−	0.625480i	$-0.215097\pi$
$727$	457.049	+	263.878i	0.628679	+	0.362968i	−0.151561	+	0.988448i	$0.548430\pi$
$728$	658.441			0.904452
$729$		0			0
$730$	7.01515	+	13.1865i	0.00960980	+	0.0180637i
$731$	−37.2801	−	21.5237i	−0.0509988	−	0.0294442i
$732$		0			0
$733$	1130.97	−	652.966i	1.54293	−	0.890813i	0.544282	−	0.838902i	$-0.316802\pi$
$733$	1130.97	−	652.966i	1.54293	−	0.890813i	0.998652	−	0.0519106i	$-0.0165311\pi$
$734$	−418.842	+	241.819i	−0.570630	+	0.329453i
$735$		0			0
$736$	−98.4264	+	170.480i	−0.133732	+	0.231630i
$737$	1583.48			2.14855
$738$		0			0
$739$	754.875			1.02148			0.510741	−	0.859735i	$-0.329371\pi$
$739$	754.875			1.02148			0.510741	+	0.859735i	$0.329371\pi$
$740$	215.251	+	7.46858i	0.290879	+	0.0100927i
$741$		0			0
$742$	641.565	−	370.408i	0.864643	−	0.499202i
$743$	−656.294	−	1136.73i	−0.883302	−	1.52992i	−0.847647	−	0.530560i	$-0.821982\pi$
$743$	−656.294	−	1136.73i	−0.883302	−	1.52992i	−0.0356548	−	0.999364i	$-0.511352\pi$
$744$		0			0
$745$	70.7534	+	2.45494i	0.0949710	+	0.00329522i
$746$		−	155.992i		−	0.209105i
$747$		0			0
$748$			169.887i			0.227121i
$749$	929.621	+	536.717i	1.24115	+	0.716578i
$750$		0			0
$751$	−616.216	−	1067.32i	−0.820528	−	1.42120i	−0.905290	−	0.424795i	$-0.860346\pi$
$751$	−616.216	−	1067.32i	−0.820528	−	1.42120i	0.0847620	−	0.996401i	$-0.472987\pi$
$752$	80.4020	+	139.260i	0.106918	+	0.185187i
$753$		0			0
$754$	254.184	−	440.259i	0.337114	−	0.583898i
$755$	323.730	−	172.223i	0.428781	−	0.228110i
$756$		0			0
$757$			401.164i			0.529939i	0.964257	+	0.264970i	$0.0853619\pi$
$757$			401.164i			0.529939i	−0.964257	+	0.264970i	$0.914638\pi$
$758$	−229.347	+	397.240i	−0.302568	+	0.524064i
$759$		0			0
$760$	86.8746	+	54.2583i	0.114309	+	0.0713925i
$761$	454.776	−	262.565i	0.597603	−	0.345027i	−0.170495	−	0.985359i	$-0.554537\pi$
$761$	454.776	−	262.565i	0.597603	−	0.345027i	0.768098	+	0.640332i	$0.221203\pi$
$762$		0			0
$763$	1733.91	+	1001.07i	2.27249	+	1.31202i
$764$		−	446.047i		−	0.583831i
$765$		0			0
$766$	809.609			1.05693
$767$	354.037	−	613.209i	0.461586	−	0.799491i
$768$		0			0
$769$	271.375	+	470.035i	0.352894	+	0.611229i	0.986755	−	0.162217i	$-0.0518645\pi$
$769$	271.375	+	470.035i	0.352894	+	0.611229i	−0.633862	+	0.773446i	$0.718531\pi$
$770$	1010.63	+	631.196i	1.31250	+	0.819735i
$771$		0			0
$772$	305.228	+	176.224i	0.395373	+	0.228269i
$773$	−522.640			−0.676119			−0.338059	−	0.941125i	$-0.609770\pi$
$773$	−522.640			−0.676119			−0.338059	+	0.941125i	$0.609770\pi$
$774$		0			0
$775$	−533.860	+	792.243i	−0.688852	+	1.02225i
$776$	−247.995	−	143.180i	−0.319581	−	0.184510i
$777$		0			0
$778$	883.857	−	510.295i	1.13606	−	0.655906i
$779$	−228.091	+	131.688i	−0.292800	+	0.169048i
$780$		0			0
$781$	−646.154	+	1119.17i	−0.827342	+	1.43300i

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:	\( N \)	\(=\)	\( 810 = 2 \cdot 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	810.j (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.707107 + 1.22474i) q^{2} +(-1.00000 - 1.73205i) q^{4} +(-4.24083 - 2.64865i) q^{5} +(-11.8534 - 6.84358i) q^{7} +2.82843 q^{8} +O(q^{10})\)	\(q+(-0.707107 + 1.22474i) q^{2} +(-1.00000 - 1.73205i) q^{4} +(-4.24083 - 2.64865i) q^{5} +(-11.8534 - 6.84358i) q^{7} +2.82843 q^{8} +(6.24264 - 3.32106i) q^{10} +(-10.6621 - 6.15576i) q^{11} +(-14.7295 + 8.50411i) q^{13} +(16.7633 - 9.67828i) q^{14} +(-2.00000 + 3.46410i) q^{16} +6.89949 q^{17} -7.24264 q^{19} +(-0.346763 + 9.99399i) q^{20} +(15.0785 - 8.70556i) q^{22} +(-17.3995 - 30.1368i) q^{23} +(10.9693 + 22.4650i) q^{25} -24.0532i q^{26} +27.3743i q^{28} +(18.3035 + 10.5675i) q^{29} +(19.1066 + 33.0936i) q^{31} +(-2.82843 - 4.89898i) q^{32} +(-4.87868 + 8.45012i) q^{34} +(32.1421 + 60.4180i) q^{35} -21.5380i q^{37} +(5.12132 - 8.87039i) q^{38} +(-11.9949 - 7.49151i) q^{40} +(31.4928 - 18.1824i) q^{41} +(-5.40331 - 3.11960i) q^{43} +24.6230i q^{44} +49.2132 q^{46} +(20.1005 - 34.8151i) q^{47} +(69.1690 + 119.804i) q^{49} +(-35.2703 - 2.45051i) q^{50} +(29.4591 + 17.0082i) q^{52} +38.2721 q^{53} +(28.9117 + 54.3457i) q^{55} +(-33.5265 - 19.3566i) q^{56} +(-25.8851 + 14.9448i) q^{58} +(-36.0537 + 20.8156i) q^{59} +(7.52944 - 13.0414i) q^{61} -54.0416 q^{62} +8.00000 q^{64} +(84.9899 + 2.94891i) q^{65} +(-111.386 + 64.3089i) q^{67} +(-6.89949 - 11.9503i) q^{68} +(-96.7246 - 3.35607i) q^{70} -104.967i q^{71} +2.11232i q^{73} +(26.3786 + 15.2297i) q^{74} +(7.24264 + 12.5446i) q^{76} +(84.2548 + 145.934i) q^{77} +(22.0477 - 38.1878i) q^{79} +(17.6569 - 9.39338i) q^{80} +51.4275i q^{82} +(27.5122 - 47.6525i) q^{83} +(-29.2596 - 18.2743i) q^{85} +(7.64144 - 4.41179i) q^{86} +(-30.1569 - 17.4111i) q^{88} +68.1020i q^{89} +232.794 q^{91} +(-34.7990 + 60.2736i) q^{92} +(28.4264 + 49.2360i) q^{94} +(30.7148 + 19.1832i) q^{95} +(-87.6794 - 50.6217i) q^{97} -195.640 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{4} - 12 q^{5}+O(q^{10}) \) 8 * q - 8 * q^4 - 12 * q^5	\( 8 q - 8 q^{4} - 12 q^{5} + 16 q^{10} - 16 q^{16} - 24 q^{17} - 24 q^{19} - 24 q^{20} - 60 q^{23} + 12 q^{25} + 68 q^{31} - 56 q^{34} + 144 q^{35} + 24 q^{38} - 16 q^{40} + 224 q^{46} + 240 q^{47} + 180 q^{49} - 96 q^{50} + 408 q^{53} - 176 q^{55} + 196 q^{61} - 240 q^{62} + 64 q^{64} + 24 q^{65} + 24 q^{68} - 80 q^{70} + 24 q^{76} + 312 q^{77} - 180 q^{79} + 96 q^{80} - 108 q^{83} - 20 q^{85} + 912 q^{91} - 120 q^{92} - 112 q^{94} + 60 q^{95} - 1056 q^{98}+O(q^{100}) \) 8 * q - 8 * q^4 - 12 * q^5 + 16 * q^10 - 16 * q^16 - 24 * q^17 - 24 * q^19 - 24 * q^20 - 60 * q^23 + 12 * q^25 + 68 * q^31 - 56 * q^34 + 144 * q^35 + 24 * q^38 - 16 * q^40 + 224 * q^46 + 240 * q^47 + 180 * q^49 - 96 * q^50 + 408 * q^53 - 176 * q^55 + 196 * q^61 - 240 * q^62 + 64 * q^64 + 24 * q^65 + 24 * q^68 - 80 * q^70 + 24 * q^76 + 312 * q^77 - 180 * q^79 + 96 * q^80 - 108 * q^83 - 20 * q^85 + 912 * q^91 - 120 * q^92 - 112 * q^94 + 60 * q^95 - 1056 * q^98

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