LMFDB - Embedded newform 405.3.d.a.404.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [405,3,Mod(404,405)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(405, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1])) N = Newforms(chi, 3, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("405.404"); S:= CuspForms(chi, 3); N := Newforms(S);

Level:	$ N $	$=$	$ 405 = 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	405.d (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [20] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$11.0354507066$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 3 x^{18} - 19 x^{16} + 66 x^{14} + 109 x^{12} - 813 x^{10} + 981 x^{8} + 5346 x^{6} + \cdots + 59049 $ x^20 - 3x^18 - 19x^16 + 66x^14 + 109x^12 - 813x^10 + 981x^8 + 5346x^6 - 13851x^4 - 19683*x^2 + 59049
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{10}\cdot 3^{20} $
Twist minimal:	no (minimal twist has level 45)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			404.1
Root			$-1.44078 + 0.961330i$ of defining polynomial
Character	$\chi$	$=$	405.404
Dual form			405.3.d.a.404.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-3.68728 q^{2} +9.59601 q^{4} +(-1.59117 - 4.74006i) q^{5} -2.56739i q^{7} -20.6340 q^{8} +(5.86709 + 17.4779i) q^{10} +9.83010i q^{11} +12.0633i q^{13} +9.46669i q^{14} +37.6993 q^{16} +4.28451 q^{17} +7.16698 q^{19} +(-15.2689 - 45.4857i) q^{20} -36.2463i q^{22} +0.510539 q^{23} +(-19.9364 + 15.0845i) q^{25} -44.4808i q^{26} -24.6367i q^{28} +30.5521i q^{29} -19.2272 q^{31} -56.4718 q^{32} -15.7982 q^{34} +(-12.1696 + 4.08516i) q^{35} +1.31851i q^{37} -26.4267 q^{38} +(32.8323 + 97.8066i) q^{40} -34.6104i q^{41} -51.9467i q^{43} +94.3297i q^{44} -1.88250 q^{46} +50.9313 q^{47} +42.4085 q^{49} +(73.5108 - 55.6207i) q^{50} +115.760i q^{52} +86.6349 q^{53} +(46.5953 - 15.6414i) q^{55} +52.9757i q^{56} -112.654i q^{58} +105.962i q^{59} +31.3200 q^{61} +70.8961 q^{62} +57.4297 q^{64} +(57.1809 - 19.1948i) q^{65} +78.0541i q^{67} +41.1142 q^{68} +(44.8727 - 15.0631i) q^{70} +72.6762i q^{71} -30.3097i q^{73} -4.86172i q^{74} +68.7744 q^{76} +25.2377 q^{77} +115.280 q^{79} +(-59.9861 - 178.697i) q^{80} +127.618i q^{82} +60.0138 q^{83} +(-6.81739 - 20.3089i) q^{85} +191.542i q^{86} -202.835i q^{88} -71.2992i q^{89} +30.9713 q^{91} +4.89914 q^{92} -187.798 q^{94} +(-11.4039 - 33.9719i) q^{95} +127.964i q^{97} -156.372 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 36 q^{4} + 4 q^{10} + 52 q^{16} - 8 q^{19} - 4 q^{25} - 56 q^{31} + 8 q^{34} + 68 q^{40} + 116 q^{46} + 80 q^{49} + 36 q^{55} + 100 q^{61} + 140 q^{64} + 108 q^{70} + 192 q^{76} + 256 q^{79} + 148 q^{85}+ \cdots - 436 q^{94}+O(q^{100}) $ 20 * q + 36 * q^4 + 4 * q^10 + 52 * q^16 - 8 * q^19 - 4 * q^25 - 56 * q^31 + 8 * q^34 + 68 * q^40 + 116 * q^46 + 80 * q^49 + 36 * q^55 + 100 * q^61 + 140 * q^64 + 108 * q^70 + 192 * q^76 + 256 * q^79 + 148 * q^85 - 288 * q^91 - 436 * q^94

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/405\mathbb{Z}\right)^\times$.

$n$	$82$	$326$
$\chi(n)$	$-1$	$-1$

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 $

$2$	−3.68728			−1.84364			−0.921819	−	0.387620i	$-0.873297\pi$
$2$	−3.68728			−1.84364			−0.921819	+	0.387620i	$0.873297\pi$
$3$		0			0
$3$		0			0
$4$	9.59601			2.39900
$5$	−1.59117	−	4.74006i	−0.318234	−	0.948012i
$5$	−1.59117	−	4.74006i	−0.318234	−	0.948012i
$6$		0			0
$7$		−	2.56739i		−	0.366771i	−0.983041	−	0.183385i	$-0.941294\pi$
$7$		−	2.56739i		−	0.366771i	0.983041	−	0.183385i	$-0.0587056\pi$
$8$	−20.6340			−2.57925
$9$		0			0
$10$	5.86709	+	17.4779i	0.586709	+	1.74779i
$11$			9.83010i			0.893645i	0.894623	+	0.446823i	$0.147444\pi$
$11$			9.83010i			0.893645i	−0.894623	+	0.446823i	$0.852556\pi$
$12$		0			0
$13$			12.0633i			0.927948i	0.885849	+	0.463974i	$0.153577\pi$
$13$			12.0633i			0.927948i	−0.885849	+	0.463974i	$0.846423\pi$
$14$			9.46669i			0.676192i
$15$		0			0
$16$	37.6993			2.35621
$17$	4.28451			0.252030			0.126015	−	0.992028i	$-0.459781\pi$
$17$	4.28451			0.252030			0.126015	+	0.992028i	$0.459781\pi$
$18$		0			0
$19$	7.16698			0.377210			0.188605	−	0.982053i	$-0.439603\pi$
$19$	7.16698			0.377210			0.188605	+	0.982053i	$0.439603\pi$
$20$	−15.2689	−	45.4857i	−0.763444	−	2.27428i
$21$		0			0
$22$		−	36.2463i		−	1.64756i
$23$	0.510539			0.0221973			0.0110987	−	0.999938i	$-0.496467\pi$
$23$	0.510539			0.0221973			0.0110987	+	0.999938i	$0.496467\pi$
$24$		0			0
$25$	−19.9364	+	15.0845i	−0.797454	+	0.603380i
$26$		−	44.4808i		−	1.71080i
$27$		0			0
$28$		−	24.6367i		−	0.879884i
$29$			30.5521i			1.05352i	0.850014	+	0.526760i	$0.176594\pi$
$29$			30.5521i			1.05352i	−0.850014	+	0.526760i	$0.823406\pi$
$30$		0			0
$31$	−19.2272			−0.620233			−0.310116	−	0.950699i	$-0.600368\pi$
$31$	−19.2272			−0.620233			−0.310116	+	0.950699i	$0.600368\pi$
$32$	−56.4718			−1.76474
$33$		0			0
$34$	−15.7982			−0.464653
$35$	−12.1696	+	4.08516i	−0.347703	+	0.116719i
$36$		0			0
$37$			1.31851i			0.0356355i	0.999841	+	0.0178177i	$0.00567186\pi$
$37$			1.31851i			0.0356355i	−0.999841	+	0.0178177i	$0.994328\pi$
$38$	−26.4267			−0.695438
$39$		0			0
$40$	32.8323	+	97.8066i	0.820807	+	2.44516i
$41$		−	34.6104i		−	0.844156i	−0.906559	−	0.422078i	$-0.861301\pi$
$41$		−	34.6104i		−	0.844156i	0.906559	−	0.422078i	$-0.138699\pi$
$42$		0			0
$43$		−	51.9467i		−	1.20806i	−0.796960	−	0.604032i	$-0.793560\pi$
$43$		−	51.9467i		−	1.20806i	0.796960	−	0.604032i	$-0.206440\pi$
$44$			94.3297i			2.14386i
$45$		0			0
$46$	−1.88250			−0.0409239
$47$	50.9313			1.08364			0.541822	−	0.840493i	$-0.317735\pi$
$47$	50.9313			1.08364			0.541822	+	0.840493i	$0.317735\pi$
$48$		0			0
$49$	42.4085			0.865479
$50$	73.5108	−	55.6207i	1.47022	−	1.11241i
$51$		0			0
$52$			115.760i			2.22615i
$53$	86.6349			1.63462			0.817311	−	0.576197i	$-0.195464\pi$
$53$	86.6349			1.63462			0.817311	+	0.576197i	$0.195464\pi$
$54$		0			0
$55$	46.5953	−	15.6414i	0.847186	−	0.284388i
$56$			52.9757i			0.945995i
$57$		0			0
$58$		−	112.654i		−	1.94231i
$59$			105.962i			1.79596i	0.440033	+	0.897982i	$0.354967\pi$
$59$			105.962i			1.79596i	−0.440033	+	0.897982i	$0.645033\pi$
$60$		0			0
$61$	31.3200			0.513443			0.256721	−	0.966485i	$-0.417358\pi$
$61$	31.3200			0.513443			0.256721	+	0.966485i	$0.417358\pi$
$62$	70.8961			1.14348
$63$		0			0
$64$	57.4297			0.897339
$65$	57.1809	−	19.1948i	0.879706	−	0.295305i
$66$		0			0
$67$			78.0541i			1.16499i	0.812835	+	0.582493i	$0.197923\pi$
$67$			78.0541i			1.16499i	−0.812835	+	0.582493i	$0.802077\pi$
$68$	41.1142			0.604621
$69$		0			0
$70$	44.8727	−	15.0631i	0.641039	−	0.215187i
$71$			72.6762i			1.02361i	0.859102	+	0.511804i	$0.171023\pi$
$71$			72.6762i			1.02361i	−0.859102	+	0.511804i	$0.828977\pi$
$72$		0			0
$73$		−	30.3097i		−	0.415201i	−0.978214	−	0.207601i	$-0.933435\pi$
$73$		−	30.3097i		−	0.415201i	0.978214	−	0.207601i	$-0.0665654\pi$
$74$		−	4.86172i		−	0.0656989i
$75$		0			0
$76$	68.7744			0.904927
$77$	25.2377			0.327763
$78$		0			0
$79$	115.280			1.45923			0.729617	−	0.683856i	$-0.239698\pi$
$79$	115.280			1.45923			0.729617	+	0.683856i	$0.239698\pi$
$80$	−59.9861	−	178.697i	−0.749826	−	2.23371i
$81$		0			0
$82$			127.618i			1.55632i
$83$	60.0138			0.723057			0.361529	−	0.932361i	$-0.382255\pi$
$83$	60.0138			0.723057			0.361529	+	0.932361i	$0.382255\pi$
$84$		0			0
$85$	−6.81739	−	20.3089i	−0.0802046	−	0.238928i
$86$			191.542i			2.22723i
$87$		0			0
$88$		−	202.835i		−	2.30494i
$89$		−	71.2992i		−	0.801115i	−0.916272	−	0.400558i	$-0.868816\pi$
$89$		−	71.2992i		−	0.801115i	0.916272	−	0.400558i	$-0.131184\pi$
$90$		0			0
$91$	30.9713			0.340344
$92$	4.89914			0.0532515
$93$		0			0
$94$	−187.798			−1.99785
$95$	−11.4039	−	33.9719i	−0.120041	−	0.357599i
$96$		0			0
$97$			127.964i			1.31921i	0.751611	+	0.659607i	$0.229277\pi$
$97$			127.964i			1.31921i	−0.751611	+	0.659607i	$0.770723\pi$
$98$	−156.372			−1.59563
$99$		0			0
$100$	−191.309	+	144.751i	−1.91309	+	1.44751i
$101$		−	86.0766i		−	0.852244i	−0.904666	−	0.426122i	$-0.859879\pi$
$101$		−	86.0766i		−	0.852244i	0.904666	−	0.426122i	$-0.140121\pi$
$102$		0			0
$103$		−	40.8833i		−	0.396925i	−0.980108	−	0.198463i	$-0.936405\pi$
$103$		−	40.8833i		−	0.396925i	0.980108	−	0.198463i	$-0.0635948\pi$
$104$		−	248.915i		−	2.39341i
$105$		0			0
$106$	−319.447			−3.01365
$107$	1.66026			0.0155165			0.00775823	−	0.999970i	$-0.497530\pi$
$107$	1.66026			0.0155165			0.00775823	+	0.999970i	$0.497530\pi$
$108$		0			0
$109$	−148.641			−1.36368			−0.681839	−	0.731502i	$-0.738820\pi$
$109$	−148.641			−1.36368			−0.681839	+	0.731502i	$0.738820\pi$
$110$	−171.810	+	57.6740i	−1.56191	+	0.524309i
$111$		0			0
$112$		−	96.7891i		−	0.864188i
$113$	131.457			1.16333			0.581667	−	0.813427i	$-0.302401\pi$
$113$	131.457			1.16333			0.581667	+	0.813427i	$0.302401\pi$
$114$		0			0
$115$	−0.812355	−	2.41999i	−0.00706395	−	0.0210434i
$116$			293.178i			2.52740i
$117$		0			0
$118$		−	390.711i		−	3.31111i
$119$		−	11.0000i		−	0.0924373i
$120$		0			0
$121$	24.3692			0.201398
$122$	−115.486			−0.946603
$123$		0			0
$124$	−184.505			−1.48794
$125$	103.224	+	70.4975i	0.825788	+	0.563980i
$126$		0			0
$127$			101.150i			0.796460i	0.917286	+	0.398230i	$0.130375\pi$
$127$			101.150i			0.796460i	−0.917286	+	0.398230i	$0.869625\pi$
$128$	14.1279			0.110374
$129$		0			0
$130$	−210.842	+	70.7765i	−1.62186	+	0.544435i
$131$		−	85.7969i		−	0.654938i	−0.944862	−	0.327469i	$-0.893804\pi$
$131$		−	85.7969i		−	0.654938i	0.944862	−	0.327469i	$-0.106196\pi$
$132$		0			0
$133$		−	18.4005i		−	0.138349i
$134$		−	287.807i		−	2.14781i
$135$		0			0
$136$	−88.4068			−0.650050
$137$	−69.7262			−0.508951			−0.254475	−	0.967079i	$-0.581903\pi$
$137$	−69.7262			−0.508951			−0.254475	+	0.967079i	$0.581903\pi$
$138$		0			0
$139$	138.917			0.999406			0.499703	−	0.866197i	$-0.333442\pi$
$139$	138.917			0.999406			0.499703	+	0.866197i	$0.333442\pi$
$140$	−116.780	+	39.2013i	−0.834140	+	0.280009i
$141$		0			0
$142$		−	267.977i		−	1.88716i
$143$	−118.584			−0.829256
$144$		0			0
$145$	144.819	−	48.6136i	0.998751	−	0.335266i
$146$			111.760i			0.765481i
$147$		0			0
$148$			12.6524i			0.0854895i
$149$			47.7568i			0.320515i	0.987075	+	0.160258i	$0.0512325\pi$
$149$			47.7568i			0.320515i	−0.987075	+	0.160258i	$0.948767\pi$
$150$		0			0
$151$	55.4914			0.367493			0.183746	−	0.982974i	$-0.441178\pi$
$151$	55.4914			0.367493			0.183746	+	0.982974i	$0.441178\pi$
$152$	−147.884			−0.972920
$153$		0			0
$154$	−93.0585			−0.604276
$155$	30.5938	+	91.1382i	0.197379	+	0.587988i
$156$		0			0
$157$			11.0721i			0.0705232i	0.999378	+	0.0352616i	$0.0112264\pi$
$157$			11.0721i			0.0705232i	−0.999378	+	0.0352616i	$0.988774\pi$
$158$	−425.068			−2.69030
$159$		0			0
$160$	89.8562	+	267.680i	0.561602	+	1.67300i
$161$		−	1.31076i		−	0.00814134i
$162$		0			0
$163$			216.230i			1.32656i	0.748370	+	0.663282i	$0.230837\pi$
$163$			216.230i			1.32656i	−0.748370	+	0.663282i	$0.769163\pi$
$164$		−	332.122i		−	2.02513i
$165$		0			0
$166$	−221.287			−1.33306
$167$	−61.9468			−0.370939			−0.185469	−	0.982650i	$-0.559381\pi$
$167$	−61.9468			−0.370939			−0.185469	+	0.982650i	$0.559381\pi$
$168$		0			0
$169$	23.4763			0.138913
$170$	25.1376	+	74.8844i	0.147868	+	0.440496i
$171$		0			0
$172$		−	498.481i		−	2.89815i
$173$	−333.471			−1.92758			−0.963788	−	0.266670i	$-0.914077\pi$
$173$	−333.471			−1.92758			−0.963788	+	0.266670i	$0.914077\pi$
$174$		0			0
$175$	38.7278	+	51.1845i	0.221302	+	0.292483i
$176$			370.588i			2.10561i
$177$		0			0
$178$			262.900i			1.47697i
$179$		−	193.521i		−	1.08112i	−0.841304	−	0.540562i	$-0.818212\pi$
$179$		−	193.521i		−	1.08112i	0.841304	−	0.540562i	$-0.181788\pi$
$180$		0			0
$181$	243.865			1.34732			0.673661	−	0.739041i	$-0.264721\pi$
$181$	243.865			1.34732			0.673661	+	0.739041i	$0.264721\pi$
$182$	−114.200			−0.627471
$183$		0			0
$184$	−10.5345			−0.0572526
$185$	6.24983	−	2.09798i	0.0337828	−	0.0113404i
$186$		0			0
$187$			42.1172i			0.225226i
$188$	488.737			2.59966
$189$		0			0
$190$	42.0493	+	125.264i	0.221312	+	0.659284i
$191$			141.565i			0.741178i	0.928797	+	0.370589i	$0.120844\pi$
$191$			141.565i			0.741178i	−0.928797	+	0.370589i	$0.879156\pi$
$192$		0			0
$193$		−	221.422i		−	1.14727i	−0.819112	−	0.573633i	$-0.805534\pi$
$193$		−	221.422i		−	1.14727i	0.819112	−	0.573633i	$-0.194466\pi$
$194$		−	471.838i		−	2.43215i
$195$		0			0
$196$	406.952			2.07629
$197$	−28.4424			−0.144378			−0.0721889	−	0.997391i	$-0.522998\pi$
$197$	−28.4424			−0.144378			−0.0721889	+	0.997391i	$0.522998\pi$
$198$		0			0
$199$	153.875			0.773244			0.386622	−	0.922238i	$-0.373642\pi$
$199$	153.875			0.773244			0.386622	+	0.922238i	$0.373642\pi$
$200$	411.367	−	311.254i	2.05684	−	1.55627i
$201$		0			0
$202$			317.388i			1.57123i
$203$	78.4393			0.386401
$204$		0			0
$205$	−164.055	+	55.0711i	−0.800271	+	0.268639i
$206$			150.748i			0.731786i
$207$		0			0
$208$			454.779i			2.18644i
$209$			70.4521i			0.337092i
$210$		0			0
$211$	180.178			0.853925			0.426962	−	0.904269i	$-0.359584\pi$
$211$	180.178			0.853925			0.426962	+	0.904269i	$0.359584\pi$
$212$	831.350			3.92146
$213$		0			0
$214$	−6.12184			−0.0286067
$215$	−246.231	+	82.6561i	−1.14526	+	0.384447i
$216$		0			0
$217$			49.3638i			0.227483i
$218$	548.080			2.51413
$219$		0			0
$220$	447.128	−	150.095i	2.03240	−	0.682248i
$221$			51.6855i			0.233871i
$222$		0			0
$223$		−	215.403i		−	0.965931i	−0.875639	−	0.482966i	$-0.839560\pi$
$223$		−	215.403i		−	0.965931i	0.875639	−	0.482966i	$-0.160440\pi$
$224$			144.985i			0.647256i
$225$		0			0
$226$	−484.717			−2.14477
$227$	50.4159			0.222097			0.111048	−	0.993815i	$-0.464579\pi$
$227$	50.4159			0.222097			0.111048	+	0.993815i	$0.464579\pi$
$228$		0			0
$229$	−6.16704			−0.0269303			−0.0134652	−	0.999909i	$-0.504286\pi$
$229$	−6.16704			−0.0269303			−0.0134652	+	0.999909i	$0.504286\pi$
$230$	2.99538	+	8.92316i	0.0130234	+	0.0387963i
$231$		0			0
$232$		−	630.413i		−	2.71730i
$233$	52.5336			0.225466			0.112733	−	0.993625i	$-0.464040\pi$
$233$	52.5336			0.225466			0.112733	+	0.993625i	$0.464040\pi$
$234$		0			0
$235$	−81.0403	−	241.417i	−0.344853	−	1.02731i
$236$			1016.81i			4.30852i
$237$		0			0
$238$			40.5602i			0.170421i
$239$			97.5840i			0.408301i	0.978939	+	0.204151i	$0.0654432\pi$
$239$			97.5840i			0.408301i	−0.978939	+	0.204151i	$0.934557\pi$
$240$		0			0
$241$	−142.729			−0.592238			−0.296119	−	0.955151i	$-0.595693\pi$
$241$	−142.729			−0.592238			−0.296119	+	0.955151i	$0.595693\pi$
$242$	−89.8560			−0.371306
$243$		0			0
$244$	300.547			1.23175
$245$	−67.4791	−	201.019i	−0.275425	−	0.820485i
$246$		0			0
$247$			86.4576i			0.350031i
$248$	396.735			1.59974
$249$		0			0
$250$	−380.614	−	259.944i	−1.52245	−	1.03978i
$251$		−	254.631i		−	1.01447i	−0.861809	−	0.507233i	$-0.830668\pi$
$251$		−	254.631i		−	1.01447i	0.861809	−	0.507233i	$-0.169332\pi$
$252$		0			0
$253$			5.01865i			0.0198366i
$254$		−	372.969i		−	1.46838i
$255$		0			0
$256$	−281.812			−1.10083
$257$	−142.736			−0.555394			−0.277697	−	0.960669i	$-0.589571\pi$
$257$	−142.736			−0.555394			−0.277697	+	0.960669i	$0.589571\pi$
$258$		0			0
$259$	3.38514			0.0130700
$260$	548.708	−	184.193i	2.11042	−	0.708436i
$261$		0			0
$262$			316.357i			1.20747i
$263$	−252.469			−0.959957			−0.479979	−	0.877280i	$-0.659356\pi$
$263$	−252.469			−0.959957			−0.479979	+	0.877280i	$0.659356\pi$
$264$		0			0
$265$	−137.851	−	410.655i	−0.520192	−	1.54964i
$266$			67.8477i			0.255066i
$267$		0			0
$268$			749.008i			2.79481i
$269$			388.672i			1.44488i	0.691435	+	0.722439i	$0.256979\pi$
$269$			388.672i			1.44488i	−0.691435	+	0.722439i	$0.743021\pi$
$270$		0			0
$271$	−163.253			−0.602410			−0.301205	−	0.953559i	$-0.597389\pi$
$271$	−163.253			−0.602410			−0.301205	+	0.953559i	$0.597389\pi$
$272$	161.523			0.593836
$273$		0			0
$274$	257.100			0.938321
$275$	−148.282	−	195.976i	−0.539207	−	0.712641i
$276$		0			0
$277$			483.823i			1.74666i	0.487133	+	0.873328i	$0.338043\pi$
$277$			483.823i			1.74666i	−0.487133	+	0.873328i	$0.661957\pi$
$278$	−512.227			−1.84254
$279$		0			0
$280$	251.108	−	84.2934i	0.896814	−	0.301048i
$281$		−	267.658i		−	0.952519i	−0.879305	−	0.476260i	$-0.841992\pi$
$281$		−	267.658i		−	0.952519i	0.879305	−	0.476260i	$-0.158008\pi$
$282$		0			0
$283$			491.796i			1.73780i	0.494991	+	0.868898i	$0.335172\pi$
$283$			491.796i			1.73780i	−0.494991	+	0.868898i	$0.664828\pi$
$284$			697.401i			2.45564i
$285$		0			0
$286$	437.250			1.52885
$287$	−88.8586			−0.309612
$288$		0			0
$289$	−270.643			−0.936481
$290$	−533.987	+	179.252i	−1.84133	+	0.618110i
$291$		0			0
$292$		−	290.852i		−	0.996069i
$293$	−322.146			−1.09947			−0.549737	−	0.835338i	$-0.685272\pi$
$293$	−322.146			−1.09947			−0.549737	+	0.835338i	$0.685272\pi$
$294$		0			0
$295$	502.266	−	168.603i	1.70260	−	0.571537i
$296$		−	27.2062i		−	0.0919129i
$297$		0			0
$298$		−	176.092i		−	0.590914i
$299$			6.15879i			0.0205980i
$300$		0			0
$301$	−133.368			−0.443082
$302$	−204.612			−0.677523
$303$		0			0
$304$	270.191			0.888785
$305$	−49.8355	−	148.459i	−0.163395	−	0.486750i
$306$		0			0
$307$			426.031i			1.38772i	0.720109	+	0.693861i	$0.244092\pi$
$307$			426.031i			1.38772i	−0.720109	+	0.693861i	$0.755908\pi$
$308$	242.182			0.786304
$309$		0			0
$310$	−112.808	−	336.052i	−0.363896	−	1.08404i
$311$		−	1.16797i		−	0.00375554i	−0.999998	−	0.00187777i	$-0.999402\pi$
$311$		−	1.16797i		−	0.00375554i	0.999998	−	0.00187777i	$-0.000597714\pi$
$312$		0			0
$313$		−	260.950i		−	0.833705i	−0.908974	−	0.416852i	$-0.863133\pi$
$313$		−	260.950i		−	0.833705i	0.908974	−	0.416852i	$-0.136867\pi$
$314$		−	40.8260i		−	0.130019i
$315$		0			0
$316$	1106.22			3.50071
$317$	131.036			0.413361			0.206681	−	0.978408i	$-0.433734\pi$
$317$	131.036			0.413361			0.206681	+	0.978408i	$0.433734\pi$
$318$		0			0
$319$	−300.330			−0.941474
$320$	−91.3805	−	272.220i	−0.285564	−	0.850689i
$321$		0			0
$322$			4.83312i			0.0150097i
$323$	30.7071			0.0950683
$324$		0			0
$325$	−181.969	−	240.499i	−0.559905	−	0.739996i
$326$		−	797.300i		−	2.44570i
$327$		0			0
$328$			714.152i			2.17729i
$329$		−	130.761i		−	0.397449i
$330$		0			0
$331$	193.484			0.584543			0.292272	−	0.956335i	$-0.405589\pi$
$331$	193.484			0.584543			0.292272	+	0.956335i	$0.405589\pi$
$332$	575.893			1.73462
$333$		0			0
$334$	228.415			0.683877
$335$	369.981	−	124.197i	1.10442	−	0.370738i
$336$		0			0
$337$			463.595i			1.37565i	0.725875	+	0.687827i	$0.241435\pi$
$337$			463.595i			1.37565i	−0.725875	+	0.687827i	$0.758565\pi$
$338$	−86.5638			−0.256106
$339$		0			0
$340$	−65.4198	−	194.884i	−0.192411	−	0.573188i
$341$		−	189.005i		−	0.554268i
$342$		0			0
$343$		−	234.682i		−	0.684203i
$344$			1071.87i			3.11590i
$345$		0			0
$346$	1229.60			3.55375
$347$	−9.71868			−0.0280077			−0.0140039	−	0.999902i	$-0.504458\pi$
$347$	−9.71868			−0.0280077			−0.0140039	+	0.999902i	$0.504458\pi$
$348$		0			0
$349$	−49.3358			−0.141363			−0.0706816	−	0.997499i	$-0.522517\pi$
$349$	−49.3358			−0.141363			−0.0706816	+	0.997499i	$0.522517\pi$
$350$	−142.800	−	188.731i	−0.408001	−	0.539232i
$351$		0			0
$352$		−	555.123i		−	1.57705i
$353$	−95.3620			−0.270147			−0.135074	−	0.990836i	$-0.543127\pi$
$353$	−95.3620			−0.270147			−0.135074	+	0.990836i	$0.543127\pi$
$354$		0			0
$355$	344.489	−	115.640i	0.970393	−	0.325747i
$356$		−	684.188i		−	1.92188i
$357$		0			0
$358$			713.566i			1.99320i
$359$			539.284i			1.50219i	0.660197	+	0.751093i	$0.270473\pi$
$359$			539.284i			1.50219i	−0.660197	+	0.751093i	$0.729527\pi$
$360$		0			0
$361$	−309.634			−0.857713
$362$	−899.199			−2.48397
$363$		0			0
$364$	297.201			0.816486
$365$	−143.670	+	48.2279i	−0.393616	+	0.132131i
$366$		0			0
$367$			303.089i			0.825854i	0.910764	+	0.412927i	$0.135494\pi$
$367$			303.089i			0.825854i	−0.910764	+	0.412927i	$0.864506\pi$
$368$	19.2470			0.0523016
$369$		0			0
$370$	−23.0448	+	7.73582i	−0.0622833	+	0.0209076i
$371$		−	222.426i		−	0.599531i
$372$		0			0
$373$		−	130.665i		−	0.350308i	−0.984541	−	0.175154i	$-0.943958\pi$
$373$		−	130.665i		−	0.350308i	0.984541	−	0.175154i	$-0.0560424\pi$
$374$		−	155.298i		−	0.415235i
$375$		0			0
$376$	−1050.92			−2.79499
$377$	−368.560			−0.977612
$378$		0			0
$379$	545.141			1.43837			0.719183	−	0.694821i	$-0.244516\pi$
$379$	545.141			1.43837			0.719183	+	0.694821i	$0.244516\pi$
$380$	−109.432	−	325.995i	−0.287979	−	0.857882i
$381$		0			0
$382$		−	521.989i		−	1.36646i
$383$	1.24650			0.00325456			0.00162728	−	0.999999i	$-0.499482\pi$
$383$	1.24650			0.00325456			0.00162728	+	0.999999i	$0.499482\pi$
$384$		0			0
$385$	−40.1575	−	119.628i	−0.104305	−	0.310723i
$386$			816.445i			2.11514i
$387$		0			0
$388$			1227.94i			3.16480i
$389$			398.373i			1.02410i	0.858957	+	0.512048i	$0.171113\pi$
$389$			398.373i			1.02410i	−0.858957	+	0.512048i	$0.828887\pi$
$390$		0			0
$391$	2.18741			0.00559440
$392$	−875.058			−2.23229
$393$		0			0
$394$	104.875			0.266180
$395$	−183.429	−	546.432i	−0.464378	−	1.38337i
$396$		0			0
$397$		−	685.998i		−	1.72795i	−0.503531	−	0.863977i	$-0.667966\pi$
$397$		−	685.998i		−	1.72795i	0.503531	−	0.863977i	$-0.332034\pi$
$398$	−567.381			−1.42558
$399$		0			0
$400$	−751.587	+	568.675i	−1.87897	+	1.42169i
$401$			51.8705i			0.129353i	0.997906	+	0.0646764i	$0.0206015\pi$
$401$			51.8705i			0.129353i	−0.997906	+	0.0646764i	$0.979398\pi$
$402$		0			0
$403$		−	231.944i		−	0.575543i
$404$		−	825.992i		−	2.04453i
$405$		0			0
$406$	−289.227			−0.712383
$407$	−12.9611			−0.0318454
$408$		0			0
$409$	−271.296			−0.663317			−0.331658	−	0.943400i	$-0.607608\pi$
$409$	−271.296			−0.663317			−0.331658	+	0.943400i	$0.607608\pi$
$410$	604.918	−	203.062i	1.47541	−	0.495274i
$411$		0			0
$412$		−	392.316i		−	0.952224i
$413$	272.046			0.658707
$414$		0			0
$415$	−95.4921	−	284.469i	−0.230101	−	0.685467i
$416$		−	681.237i		−	1.63759i
$417$		0			0
$418$		−	259.777i		−	0.621475i
$419$			27.6513i			0.0659935i	0.999455	+	0.0329967i	$0.0105051\pi$
$419$			27.6513i			0.0659935i	−0.999455	+	0.0329967i	$0.989495\pi$
$420$		0			0
$421$	−437.226			−1.03854			−0.519271	−	0.854610i	$-0.673796\pi$
$421$	−437.226			−1.03854			−0.519271	+	0.854610i	$0.673796\pi$
$422$	−664.367			−1.57433
$423$		0			0
$424$	−1787.63			−4.21610
$425$	−85.4176	+	64.6297i	−0.200983	+	0.152070i
$426$		0			0
$427$		−	80.4108i		−	0.188316i
$428$	15.9319			0.0372240
$429$		0			0
$430$	907.920	−	304.776i	2.11144	−	0.708781i
$431$		−	54.3602i		−	0.126126i	−0.998010	−	0.0630629i	$-0.979913\pi$
$431$		−	54.3602i		−	0.126126i	0.998010	−	0.0630629i	$-0.0200869\pi$
$432$		0			0
$433$			526.426i			1.21576i	0.794028	+	0.607882i	$0.207981\pi$
$433$			526.426i			1.21576i	−0.794028	+	0.607882i	$0.792019\pi$
$434$		−	182.018i		−	0.419397i
$435$		0			0
$436$	−1426.36			−3.27147
$437$	3.65903			0.00837306
$438$		0			0
$439$	−416.248			−0.948173			−0.474087	−	0.880478i	$-0.657222\pi$
$439$	−416.248			−0.948173			−0.474087	+	0.880478i	$0.657222\pi$
$440$	−961.448	+	322.744i	−2.18511	+	0.733510i
$441$		0			0
$442$		−	190.579i		−	0.431173i
$443$	427.165			0.964254			0.482127	−	0.876101i	$-0.339864\pi$
$443$	427.165			0.964254			0.482127	+	0.876101i	$0.339864\pi$
$444$		0			0
$445$	−337.963	+	113.449i	−0.759467	+	0.254942i
$446$			794.249i			1.78083i
$447$		0			0
$448$		−	147.445i		−	0.329118i
$449$			236.730i			0.527239i	0.964627	+	0.263620i	$0.0849164\pi$
$449$			236.730i			0.527239i	−0.964627	+	0.263620i	$0.915084\pi$
$450$		0			0
$451$	340.224			0.754376
$452$	1261.46			2.79084
$453$		0			0
$454$	−185.898			−0.409466
$455$	−49.2806	−	146.806i	−0.108309	−	0.322650i
$456$		0			0
$457$		−	697.781i		−	1.52687i	−0.645883	−	0.763437i	$-0.723511\pi$
$457$		−	697.781i		−	1.52687i	0.645883	−	0.763437i	$-0.276489\pi$
$458$	22.7396			0.0496497
$459$		0			0
$460$	−7.79536	−	23.2222i	−0.0169464	−	0.0504831i
$461$			383.752i			0.832435i	0.909265	+	0.416217i	$0.136644\pi$
$461$			383.752i			0.832435i	−0.909265	+	0.416217i	$0.863356\pi$
$462$		0			0
$463$		−	270.112i		−	0.583395i	−0.956511	−	0.291698i	$-0.905780\pi$
$463$		−	270.112i		−	0.583395i	0.956511	−	0.291698i	$-0.0942201\pi$
$464$			1151.79i			2.48232i
$465$		0			0
$466$	−193.706			−0.415678
$467$	777.952			1.66585			0.832926	−	0.553385i	$-0.186664\pi$
$467$	777.952			1.66585			0.832926	+	0.553385i	$0.186664\pi$
$468$		0			0
$469$	200.396			0.427283
$470$	298.818	+	890.173i	0.635783	+	1.89398i
$471$		0			0
$472$		−	2186.42i		−	4.63225i
$473$	510.641			1.07958
$474$		0			0
$475$	−142.884	+	108.110i	−0.300807	+	0.227601i
$476$		−	105.556i		−	0.221757i
$477$		0			0
$478$		−	359.819i		−	0.752760i
$479$		−	72.5647i		−	0.151492i	−0.997127	−	0.0757460i	$-0.975866\pi$
$479$		−	72.5647i		−	0.151492i	0.997127	−	0.0757460i	$-0.0241338\pi$
$480$		0			0
$481$	−15.9056			−0.0330678
$482$	526.283			1.09187
$483$		0			0
$484$	233.847			0.483155
$485$	606.556	−	203.612i	1.25063	−	0.419819i
$486$		0			0
$487$			729.487i			1.49792i	0.662616	+	0.748959i	$0.269446\pi$
$487$			729.487i			1.49792i	−0.662616	+	0.748959i	$0.730554\pi$
$488$	−646.258			−1.32430
$489$		0			0
$490$	248.814	+	741.212i	0.507784	+	1.51268i
$491$			3.81570i			0.00777127i	0.999992	+	0.00388564i	$0.00123684\pi$
$491$			3.81570i			0.00777127i	−0.999992	+	0.00388564i	$0.998763\pi$
$492$		0			0
$493$			130.901i			0.265519i
$494$		−	318.793i		−	0.645330i
$495$		0			0
$496$	−724.853			−1.46140
$497$	186.588			0.375429
$498$		0			0
$499$	205.303			0.411428			0.205714	−	0.978612i	$-0.434048\pi$
$499$	205.303			0.411428			0.205714	+	0.978612i	$0.434048\pi$
$500$	990.534	+	676.495i	1.98107	+	1.35299i
$501$		0			0
$502$			938.895i			1.87031i
$503$	224.016			0.445360			0.222680	−	0.974892i	$-0.428519\pi$
$503$	224.016			0.445360			0.222680	+	0.974892i	$0.428519\pi$
$504$		0			0
$505$	−408.008	+	136.963i	−0.807937	+	0.271213i
$506$		−	18.5051i		−	0.0365714i
$507$		0			0
$508$			970.640i			1.91071i
$509$		−	460.810i		−	0.905324i	−0.891682	−	0.452662i	$-0.850474\pi$
$509$		−	460.810i		−	0.905324i	0.891682	−	0.452662i	$-0.149526\pi$
$510$		0			0
$511$	−77.8169			−0.152284
$512$	982.608			1.91916
$513$		0			0
$514$	526.309			1.02395
$515$	−193.789	+	65.0523i	−0.376290	+	0.126315i
$516$		0			0
$517$			500.659i			0.968393i
$518$	−12.4819			−0.0240964
$519$		0			0
$520$	−1179.87	+	396.066i	−2.26898	+	0.761665i
$521$		−	182.438i		−	0.350168i	−0.984554	−	0.175084i	$-0.943980\pi$
$521$		−	182.438i		−	0.350168i	0.984554	−	0.175084i	$-0.0560197\pi$
$522$		0			0
$523$		−	431.339i		−	0.824741i	−0.911016	−	0.412370i	$-0.864701\pi$
$523$		−	431.339i		−	0.824741i	0.911016	−	0.412370i	$-0.135299\pi$
$524$		−	823.308i		−	1.57120i
$525$		0			0
$526$	930.922			1.76981
$527$	−82.3793			−0.156317
$528$		0			0
$529$	−528.739			−0.999507
$530$	508.295	+	1514.20i	0.959046	+	2.85698i
$531$		0			0
$532$		−	176.571i		−	0.331901i
$533$	417.516			0.783333
$534$		0			0
$535$	−2.64176	−	7.86974i	−0.00493786	−	0.0147098i
$536$		−	1610.57i		−	3.00480i
$537$		0			0
$538$		−	1433.14i		−	2.66383i
$539$			416.879i			0.773431i
$540$		0			0
$541$	−649.924			−1.20134			−0.600669	−	0.799498i	$-0.705099\pi$
$541$	−649.924			−1.20134			−0.600669	+	0.799498i	$0.705099\pi$
$542$	601.959			1.11063
$543$		0			0
$544$	−241.954			−0.444769
$545$	236.513	+	704.567i	0.433969	+	1.29278i
$546$		0			0
$547$			425.316i			0.777542i	0.921334	+	0.388771i	$0.127100\pi$
$547$			425.316i			0.777542i	−0.921334	+	0.388771i	$0.872900\pi$
$548$	−669.094			−1.22097
$549$		0			0
$550$	546.757	+	722.619i	0.994103	+	1.31385i
$551$			218.966i			0.397398i
$552$		0			0
$553$		−	295.968i		−	0.535205i
$554$		−	1783.99i		−	3.22020i
$555$		0			0
$556$	1333.05			2.39758
$557$	−325.885			−0.585073			−0.292536	−	0.956254i	$-0.594499\pi$
$557$	−325.885			−0.585073			−0.292536	+	0.956254i	$0.594499\pi$
$558$		0			0
$559$	626.650			1.12102
$560$	−458.786	+	154.008i	−0.819261	+	0.275014i
$561$		0			0
$562$			986.929i			1.75610i
$563$	706.763			1.25535			0.627676	−	0.778475i	$-0.284006\pi$
$563$	706.763			1.25535			0.627676	+	0.778475i	$0.284006\pi$
$564$		0			0
$565$	−209.170	−	623.113i	−0.370213	−	1.10285i
$566$		−	1813.39i		−	3.20387i
$567$		0			0
$568$		−	1499.60i		−	2.64015i
$569$			560.236i			0.984597i	0.870426	+	0.492299i	$0.163843\pi$
$569$			560.236i			0.984597i	−0.870426	+	0.492299i	$0.836157\pi$
$570$		0			0
$571$	−210.692			−0.368988			−0.184494	−	0.982834i	$-0.559065\pi$
$571$	−210.692			−0.368988			−0.184494	+	0.982834i	$0.559065\pi$
$572$	−1137.93			−1.98939
$573$		0			0
$574$	327.646			0.570812
$575$	−10.1783	+	7.70122i	−0.0177014	+	0.0133934i
$576$		0			0
$577$			502.258i			0.870464i	0.900318	+	0.435232i	$0.143334\pi$
$577$			502.258i			0.870464i	−0.900318	+	0.435232i	$0.856666\pi$
$578$	997.935			1.72653
$579$		0			0
$580$	1389.68	−	466.497i	2.39600	−	0.804304i
$581$		−	154.079i		−	0.265196i
$582$		0			0
$583$			851.630i			1.46077i
$584$			625.411i			1.07091i
$585$		0			0
$586$	1187.84			2.02703
$587$	−409.772			−0.698079			−0.349040	−	0.937108i	$-0.613492\pi$
$587$	−409.772			−0.698079			−0.349040	+	0.937108i	$0.613492\pi$
$588$		0			0
$589$	−137.801			−0.233958
$590$	−1851.99	+	621.687i	−3.13897	+	1.05371i
$591$		0			0
$592$			49.7070i			0.0839646i
$593$	−944.139			−1.59214			−0.796070	−	0.605204i	$-0.793092\pi$
$593$	−944.139			−1.59214			−0.796070	+	0.605204i	$0.793092\pi$
$594$		0			0
$595$	−52.1409	+	17.5029i	−0.0876317	+	0.0294167i
$596$			458.275i			0.768917i
$597$		0			0
$598$		−	22.7092i		−	0.0379752i
$599$			532.342i			0.888717i	0.895849	+	0.444359i	$0.146568\pi$
$599$			532.342i			0.888717i	−0.895849	+	0.444359i	$0.853432\pi$
$600$		0			0
$601$	515.566			0.857846			0.428923	−	0.903341i	$-0.358893\pi$
$601$	515.566			0.857846			0.428923	+	0.903341i	$0.358893\pi$
$602$	491.764			0.816883
$603$		0			0
$604$	532.496			0.881615
$605$	−38.7756	−	115.512i	−0.0640918	−	0.190928i
$606$		0			0
$607$		−	171.153i		−	0.281965i	−0.990012	−	0.140983i	$-0.954974\pi$
$607$		−	171.153i		−	0.281965i	0.990012	−	0.140983i	$-0.0450262\pi$
$608$	−404.732			−0.665678
$609$		0			0
$610$	183.757	+	547.409i	0.301241	+	0.897391i
$611$			614.400i			1.00557i
$612$		0			0
$613$		−	875.826i		−	1.42875i	−0.699761	−	0.714377i	$-0.746710\pi$
$613$		−	875.826i		−	1.42875i	0.699761	−	0.714377i	$-0.253290\pi$
$614$		−	1570.89i		−	2.55846i
$615$		0			0
$616$	−520.756			−0.845384
$617$	1131.75			1.83427			0.917137	−	0.398571i	$-0.130494\pi$
$617$	1131.75			1.83427			0.917137	+	0.398571i	$0.130494\pi$
$618$		0			0
$619$	520.375			0.840670			0.420335	−	0.907369i	$-0.361913\pi$
$619$	520.375			0.840670			0.420335	+	0.907369i	$0.361913\pi$
$620$	293.578	+	874.563i	0.473513	+	1.41058i
$621$		0			0
$622$			4.30664i			0.00692386i
$623$	−183.053			−0.293826
$624$		0			0
$625$	169.916	−	601.459i	0.271866	−	0.962335i
$626$			962.193i			1.53705i
$627$		0			0
$628$			106.248i			0.169185i
$629$			5.64918i			0.00898121i
$630$		0			0
$631$	−607.475			−0.962718			−0.481359	−	0.876523i	$-0.659857\pi$
$631$	−607.475			−0.962718			−0.481359	+	0.876523i	$0.659857\pi$
$632$	−2378.68			−3.76374
$633$		0			0
$634$	−483.164			−0.762089
$635$	479.459	−	160.947i	0.755053	−	0.253461i
$636$		0			0
$637$			511.587i			0.803119i
$638$	1107.40			1.73574
$639$		0			0
$640$	−22.4799	−	66.9670i	−0.0351248	−	0.104636i
$641$		−	186.198i		−	0.290481i	−0.989396	−	0.145240i	$-0.953604\pi$
$641$		−	186.198i		−	0.290481i	0.989396	−	0.145240i	$-0.0463955\pi$
$642$		0			0
$643$		−	797.522i		−	1.24031i	−0.784478	−	0.620157i	$-0.787069\pi$
$643$		−	797.522i		−	1.24031i	0.784478	−	0.620157i	$-0.212931\pi$
$644$		−	12.5780i		−	0.0195311i
$645$		0			0
$646$	−113.225			−0.175271
$647$	849.489			1.31297			0.656483	−	0.754341i	$-0.272043\pi$
$647$	849.489			1.31297			0.656483	+	0.754341i	$0.272043\pi$
$648$		0			0
$649$	−1041.61			−1.60495
$650$	670.970	+	886.785i	1.03226	+	1.36428i
$651$		0			0
$652$			2074.94i			3.18243i
$653$	−1041.19			−1.59447			−0.797235	−	0.603669i	$-0.793705\pi$
$653$	−1041.19			−1.59447			−0.797235	+	0.603669i	$0.793705\pi$
$654$		0			0
$655$	−406.683	+	136.518i	−0.620889	+	0.208424i
$656$		−	1304.79i		−	1.98901i
$657$		0			0
$658$			482.151i			0.732752i
$659$		−	642.380i		−	0.974781i	−0.873184	−	0.487390i	$-0.837949\pi$
$659$		−	642.380i		−	0.974781i	0.873184	−	0.487390i	$-0.162051\pi$
$660$		0			0
$661$	−789.517			−1.19443			−0.597214	−	0.802082i	$-0.703726\pi$
$661$	−789.517			−1.19443			−0.597214	+	0.802082i	$0.703726\pi$
$662$	−713.429			−1.07769
$663$		0			0
$664$	−1238.33			−1.86495
$665$	−87.2194	+	29.2783i	−0.131157	+	0.0440275i
$666$		0			0
$667$			15.5980i			0.0233854i
$668$	−594.442			−0.889883
$669$		0			0
$670$	−1364.22	+	457.950i	−2.03615	+	0.683508i
$671$			307.879i			0.458836i
$672$		0			0
$673$		−	1015.93i		−	1.50956i	−0.655978	−	0.754780i	$-0.727744\pi$
$673$		−	1015.93i		−	1.50956i	0.655978	−	0.754780i	$-0.272256\pi$
$674$		−	1709.40i		−	2.53621i
$675$		0			0
$676$	225.279			0.333253
$677$	743.622			1.09841			0.549204	−	0.835688i	$-0.314931\pi$
$677$	743.622			1.09841			0.549204	+	0.835688i	$0.314931\pi$
$678$		0			0
$679$	328.533			0.483849
$680$	140.670	+	419.054i	0.206868	+	0.616255i
$681$		0			0
$682$			696.915i			1.02187i
$683$	152.482			0.223254			0.111627	−	0.993750i	$-0.464394\pi$
$683$	152.482			0.223254			0.111627	+	0.993750i	$0.464394\pi$
$684$		0			0
$685$	110.946	+	330.507i	0.161965	+	0.482491i
$686$			865.336i			1.26142i
$687$		0			0
$688$		−	1958.36i		−	2.84645i
$689$			1045.10i			1.51684i
$690$		0			0
$691$	777.171			1.12471			0.562353	−	0.826898i	$-0.309896\pi$
$691$	777.171			1.12471			0.562353	+	0.826898i	$0.309896\pi$
$692$	−3199.99			−4.62426
$693$		0			0
$694$	35.8355			0.0516361
$695$	−221.041	−	658.477i	−0.318045	−	0.947449i
$696$		0			0
$697$		−	148.289i		−	0.212753i
$698$	181.915			0.260623
$699$		0			0
$700$	371.633	+	491.167i	0.530904	+	0.701667i
$701$			461.657i			0.658569i	0.944231	+	0.329284i	$0.106808\pi$
$701$			461.657i			0.658569i	−0.944231	+	0.329284i	$0.893192\pi$
$702$		0			0
$703$			9.44975i			0.0134420i
$704$			564.540i			0.801903i
$705$		0			0
$706$	351.626			0.498054
$707$	−220.993			−0.312578
$708$		0			0
$709$	−358.455			−0.505578			−0.252789	−	0.967521i	$-0.581348\pi$
$709$	−358.455			−0.505578			−0.252789	+	0.967521i	$0.581348\pi$
$710$	−1270.23	+	426.397i	−1.78905	+	0.600560i
$711$		0			0
$712$			1471.19i			2.06628i
$713$	−9.81624			−0.0137675
$714$		0			0
$715$	188.687	+	562.093i	0.263897	+	0.786145i
$716$		−	1857.03i		−	2.59362i
$717$		0			0
$718$		−	1988.49i		−	2.76949i
$719$		−	283.414i		−	0.394178i	−0.980386	−	0.197089i	$-0.936851\pi$
$719$		−	283.414i		−	0.394178i	0.980386	−	0.197089i	$-0.0631488\pi$
$720$		0			0
$721$	−104.964			−0.145580
$722$	1141.71			1.58131
$723$		0			0
$724$	2340.13			3.23223
$725$	−460.863	−	609.098i	−0.635673	−	0.840135i
$726$		0			0
$727$		−	155.359i		−	0.213699i	−0.994275	−	0.106850i	$-0.965924\pi$
$727$		−	155.359i		−	0.213699i	0.994275	−	0.106850i	$-0.0340763\pi$
$728$	−639.063			−0.877833
$729$		0			0
$730$	529.750	−	177.830i	0.725685	−	0.243602i
$731$		−	222.566i		−	0.304468i
$732$		0			0
$733$			104.932i			0.143154i	0.997435	+	0.0715768i	$0.0228031\pi$
$733$			104.932i			0.143154i	−0.997435	+	0.0715768i	$0.977197\pi$
$734$		−	1117.57i		−	1.52258i
$735$		0			0
$736$	−28.8311			−0.0391726
$737$	−767.279			−1.04108
$738$		0			0
$739$	−627.375			−0.848951			−0.424476	−	0.905439i	$-0.639542\pi$
$739$	−627.375			−0.848951			−0.424476	+	0.905439i	$0.639542\pi$
$740$	59.9734	−	20.1322i	0.0810451	−	0.0272057i
$741$		0			0
$742$			820.146i			1.10532i
$743$	−386.590			−0.520310			−0.260155	−	0.965567i	$-0.583774\pi$
$743$	−386.590			−0.520310			−0.260155	+	0.965567i	$0.583774\pi$
$744$		0			0
$745$	226.370	−	75.9892i	0.303852	−	0.101999i
$746$			481.798i			0.645842i
$747$		0			0
$748$			404.157i			0.540317i
$749$		−	4.26254i		−	0.00569098i
$750$		0			0
$751$	905.203			1.20533			0.602665	−	0.797994i	$-0.294106\pi$
$751$	905.203			1.20533			0.602665	+	0.797994i	$0.294106\pi$
$752$	1920.08			2.55329
$753$		0			0
$754$	1358.98			1.80236
$755$	−88.2962	−	263.032i	−0.116949	−	0.348387i
$756$		0			0
$757$		−	332.222i		−	0.438867i	−0.975627	−	0.219433i	$-0.929579\pi$
$757$		−	332.222i		−	0.438867i	0.975627	−	0.219433i	$-0.0704209\pi$
$758$	−2010.08			−2.65183
$759$		0			0
$760$	235.308	+	700.978i	0.309616	+	0.922340i
$761$		−	458.100i		−	0.601971i	−0.953629	−	0.300986i	$-0.902684\pi$
$761$		−	458.100i		−	0.601971i	0.953629	−	0.300986i	$-0.0973156\pi$
$762$		0			0
$763$			381.620i			0.500157i
$764$			1358.46i			1.77809i
$765$		0			0
$766$	−4.59618			−0.00600023
$767$	−1278.25			−1.66656
$768$		0			0
$769$	−916.392			−1.19167			−0.595833	−	0.803108i	$-0.703178\pi$
$769$	−916.392			−1.19167			−0.595833	+	0.803108i	$0.703178\pi$
$770$	148.072	+	441.103i	0.192301	+	0.572861i
$771$		0			0
$772$		−	2124.77i		−	2.75229i
$773$	1186.06			1.53436			0.767178	−	0.641434i	$-0.221660\pi$
$773$	1186.06			1.53436			0.767178	+	0.641434i	$0.221660\pi$
$774$		0			0
$775$	383.321	−	290.033i	0.494607	−	0.374236i
$776$		−	2640.41i		−	3.40259i
$777$		0			0
$778$		−	1468.91i		−	1.88806i
$779$		−	248.052i		−	0.318424i
$780$		0			0
$781$	−714.414			−0.914742
$782$	−8.06559			−0.0103141
$783$		0			0
$784$	1598.77			2.03925
$785$	52.4826	−	17.6177i	0.0668568	−	0.0224429i
$786$		0			0
$787$		−	882.479i		−	1.12132i	−0.828046	−	0.560660i	$-0.810547\pi$
$787$		−	882.479i		−	1.12132i	0.828046	−	0.560660i	$-0.189453\pi$
$788$	−272.934			−0.346363
$789$		0			0
$790$	676.355	+	2014.85i	0.856146	+	2.55044i
$791$		−	337.501i		−	0.426677i
$792$		0			0
$793$			377.823i			0.476448i
$794$			2529.46i			3.18572i
$795$		0			0
$796$	1476.59			1.85501
$797$	−143.803			−0.180431			−0.0902153	−	0.995922i	$-0.528756\pi$
$797$	−143.803			−0.180431			−0.0902153	+	0.995922i	$0.528756\pi$
$798$		0			0
$799$	218.216			0.273111
$800$	1125.84	−	851.848i	1.40730	−	1.06481i
$801$		0			0
$802$		−	191.261i		−	0.238480i
$803$	297.947			0.371043
$804$		0			0
$805$	−6.21306	+	2.08563i	−0.00771809	+	0.00259085i
$806$			855.242i			1.06109i
$807$		0			0
$808$			1776.11i			2.19815i
$809$		−	52.3078i		−	0.0646574i	−0.999477	−	0.0323287i	$-0.989708\pi$
$809$		−	52.3078i		−	0.0646574i	0.999477	−	0.0323287i	$-0.0102923\pi$
$810$		0			0
$811$	1518.65			1.87256			0.936281	−	0.351253i	$-0.114244\pi$
$811$	1518.65			1.87256			0.936281	+	0.351253i	$0.114244\pi$
$812$	752.704			0.926976
$813$		0			0
$814$	47.7911			0.0587115
$815$	1024.94	−	344.059i	1.25760	−	0.422158i
$816$		0			0
$817$		−	372.301i		−	0.455693i
$818$	1000.35			1.22292
$819$		0			0
$820$	−1574.28	+	528.462i	−1.91985	+	0.644466i
$821$		−	1097.03i		−	1.33622i	−0.744064	−	0.668108i	$-0.767104\pi$
$821$		−	1097.03i		−	1.33622i	0.744064	−	0.668108i	$-0.232896\pi$
$822$		0			0
$823$			957.100i			1.16294i	0.813568	+	0.581470i	$0.197522\pi$
$823$			957.100i			1.16294i	−0.813568	+	0.581470i	$0.802478\pi$
$824$			843.587i			1.02377i
$825$		0			0
$826$	−1003.11			−1.21442
$827$	−171.626			−0.207529			−0.103764	−	0.994602i	$-0.533089\pi$
$827$	−171.626			−0.207529			−0.103764	+	0.994602i	$0.533089\pi$
$828$		0			0
$829$	203.896			0.245954			0.122977	−	0.992410i	$-0.460756\pi$
$829$	203.896			0.245954			0.122977	+	0.992410i	$0.460756\pi$
$830$	352.106	+	1048.92i	0.424224	+	1.26375i
$831$		0			0
$832$			692.793i			0.832684i
$833$	181.700			0.218127
$834$		0			0
$835$	98.5679	+	293.631i	0.118045	+	0.351654i
$836$			676.059i			0.808683i
$837$		0			0
$838$		−	101.958i		−	0.121668i
$839$			1556.65i			1.85536i	0.373377	+	0.927680i	$0.378200\pi$
$839$			1556.65i			1.85536i	−0.373377	+	0.927680i	$0.621800\pi$
$840$		0			0
$841$	−92.4311			−0.109906
$842$	1612.17			1.91469
$843$		0			0
$844$	1728.99			2.04857
$845$	−37.3549	−	111.279i	−0.0442069	−	0.131692i
$846$		0			0
$847$		−	62.5654i		−	0.0738670i
$848$	3266.08			3.85151
$849$		0			0
$850$	314.958	−	238.308i	0.370539	−	0.280362i
$851$			0.673152i			0.000791013i
$852$		0			0
$853$			563.929i			0.661112i	0.943786	+	0.330556i	$0.107236\pi$
$853$			563.929i			0.661112i	−0.943786	+	0.330556i	$0.892764\pi$
$854$			296.497i			0.347186i
$855$		0			0
$856$	−34.2579			−0.0400209
$857$	953.906			1.11308			0.556538	−	0.830822i	$-0.312129\pi$
$857$	953.906			1.11308			0.556538	+	0.830822i	$0.312129\pi$
$858$		0			0
$859$	845.456			0.984233			0.492116	−	0.870529i	$-0.336223\pi$
$859$	845.456			0.984233			0.492116	+	0.870529i	$0.336223\pi$
$860$	−2362.83	+	793.168i	−2.74748	+	0.922289i
$861$		0			0
$862$			200.441i			0.232530i
$863$	−1109.42			−1.28554			−0.642768	−	0.766061i	$-0.722214\pi$
$863$	−1109.42			−1.28554			−0.642768	+	0.766061i	$0.722214\pi$
$864$		0			0
$865$	530.609	+	1580.67i	0.613420	+	1.82737i
$866$		−	1941.08i		−	2.24143i
$867$		0			0
$868$			473.696i			0.545733i
$869$			1133.21i			1.30404i
$870$		0			0
$871$	−941.592			−1.08105
$872$	3067.06			3.51727
$873$		0			0
$874$	−13.4918			−0.0154369
$875$	180.995	−	265.016i	0.206851	−	0.302875i
$876$		0			0
$877$			1014.66i			1.15697i	0.815693	+	0.578485i	$0.196356\pi$
$877$			1014.66i			1.15697i	−0.815693	+	0.578485i	$0.803644\pi$
$878$	1534.82			1.74809
$879$		0			0
$880$	1756.61	−	589.669i	1.99615	−	0.670078i
$881$		−	1226.86i		−	1.39257i	−0.717764	−	0.696287i	$-0.754834\pi$
$881$		−	1226.86i		−	1.39257i	0.717764	−	0.696287i	$-0.245166\pi$
$882$		0			0
$883$		−	659.407i		−	0.746780i	−0.927674	−	0.373390i	$-0.878195\pi$
$883$		−	659.407i		−	0.746780i	0.927674	−	0.373390i	$-0.121805\pi$
$884$			495.974i			0.561057i
$885$		0			0
$886$	−1575.07			−1.77774
$887$	−771.967			−0.870313			−0.435156	−	0.900355i	$-0.643307\pi$
$887$	−771.967			−0.870313			−0.435156	+	0.900355i	$0.643307\pi$
$888$		0			0
$889$	259.693			0.292118
$890$	1246.16	−	418.319i	1.40018	−	0.470021i
$891$		0			0
$892$		−	2067.01i		−	2.31727i
$893$	365.024			0.408761
$894$		0			0
$895$	−917.302	+	307.925i	−1.02492	+	0.344050i
$896$		−	36.2718i		−	0.0404820i
$897$		0			0
$898$		−	872.891i		−	0.972038i
$899$		−	587.432i		−	0.653428i
$900$		0			0
$901$	371.189			0.411974
$902$	−1254.50			−1.39080
$903$		0			0
$904$	−2712.48			−3.00053
$905$	−388.031	−	1155.94i	−0.428764	−	1.27728i
$906$		0			0
$907$		−	121.471i		−	0.133926i	−0.997755	−	0.0669629i	$-0.978669\pi$
$907$		−	121.471i		−	0.133926i	0.997755	−	0.0669629i	$-0.0213309\pi$
$908$	483.792			0.532810
$909$		0			0
$910$	181.711	+	541.314i	0.199683	+	0.594850i
$911$			1155.23i			1.26809i	0.773295	+	0.634046i	$0.218607\pi$
$911$			1155.23i			1.26809i	−0.773295	+	0.634046i	$0.781393\pi$
$912$		0			0
$913$			589.941i			0.646157i
$914$			2572.91i			2.81500i
$915$		0			0
$916$	−59.1790			−0.0646059
$917$	−220.275			−0.240212
$918$		0			0
$919$	−994.576			−1.08224			−0.541119	−	0.840946i	$-0.681999\pi$
$919$	−994.576			−1.08224			−0.541119	+	0.840946i	$0.681999\pi$
$920$	16.7621	+	49.9341i	0.0182197	+	0.0542762i
$921$		0			0
$922$		−	1415.00i		−	1.53471i
$923$	−876.716			−0.949855
$924$		0			0
$925$	−19.8891	−	26.2863i	−0.0215017	−	0.0284176i
$926$			995.977i			1.07557i
$927$		0			0
$928$		−	1725.33i		−	1.85919i
$929$		−	161.931i		−	0.174307i	−0.996195	−	0.0871534i	$-0.972223\pi$
$929$		−	161.931i		−	0.174307i	0.996195	−	0.0871534i	$-0.0277770\pi$
$930$		0			0
$931$	303.941			0.326467
$932$	504.113			0.540893
$933$		0			0
$934$	−2868.53			−3.07123
$935$	199.638	−	67.0156i	0.213517	−	0.0716745i
$936$		0			0
$937$		−	660.489i		−	0.704898i	−0.935831	−	0.352449i	$-0.885349\pi$
$937$		−	660.489i		−	0.704898i	0.935831	−	0.352449i	$-0.114651\pi$
$938$	−738.914			−0.787755
$939$		0			0
$940$	−777.664	−	2316.64i	−0.827302	−	2.46451i
$941$		−	1591.25i		−	1.69102i	−0.533961	−	0.845509i	$-0.679297\pi$
$941$		−	1591.25i		−	1.69102i	0.533961	−	0.845509i	$-0.320703\pi$
$942$		0			0
$943$		−	17.6700i		−	0.0187380i
$944$			3994.69i			4.23167i
$945$		0			0
$946$	−1882.88			−1.99035
$947$	−1563.36			−1.65085			−0.825426	−	0.564511i	$-0.809065\pi$
$947$	−1563.36			−1.65085			−0.825426	+	0.564511i	$0.809065\pi$
$948$		0			0
$949$	365.635			0.385285
$950$	526.851	−	398.633i	0.554580	−	0.419613i
$951$		0			0
$952$			226.975i			0.238419i
$953$	−996.612			−1.04576			−0.522881	−	0.852405i	$-0.675143\pi$
$953$	−996.612			−1.04576			−0.522881	+	0.852405i	$0.675143\pi$
$954$		0			0
$955$	671.027	−	225.254i	0.702646	−	0.235868i
$956$			936.417i			0.979516i
$957$		0			0
$958$			267.566i			0.279297i
$959$			179.015i			0.186668i
$960$		0			0
$961$	−591.314			−0.615311
$962$	58.6484			0.0609651
$963$		0			0
$964$	−1369.63			−1.42078
$965$	−1049.56	+	352.321i	−1.08762	+	0.365099i
$966$		0			0
$967$		−	649.999i		−	0.672181i	−0.941830	−	0.336091i	$-0.890895\pi$
$967$		−	649.999i		−	0.672181i	0.941830	−	0.336091i	$-0.109105\pi$
$968$	−502.835			−0.519458
$969$		0			0
$970$	−2236.54	+	750.774i	−2.30571	+	0.773994i
$971$			1717.75i			1.76905i	0.466495	+	0.884524i	$0.345517\pi$
$971$			1717.75i			1.76905i	−0.466495	+	0.884524i	$0.654483\pi$
$972$		0			0
$973$		−	356.656i		−	0.366553i
$974$		−	2689.82i		−	2.76162i
$975$		0			0
$976$	1180.74			1.20978
$977$	−282.121			−0.288763			−0.144381	−	0.989522i	$-0.546119\pi$
$977$	−282.121			−0.288763			−0.144381	+	0.989522i	$0.546119\pi$
$978$		0			0
$979$	700.878			0.715913
$980$	−647.530	−	1928.98i	−0.660745	−	1.96834i
$981$		0			0
$982$		−	14.0695i		−	0.0143274i
$983$	384.535			0.391185			0.195592	−	0.980685i	$-0.437337\pi$
$983$	384.535			0.391185			0.195592	+	0.980685i	$0.437337\pi$
$984$		0			0
$985$	45.2568	+	134.819i	0.0459459	+	0.136872i
$986$		−	482.668i		−	0.489521i
$987$		0			0
$988$			829.648i			0.839725i
$989$		−	26.5208i		−	0.0268158i
$990$		0			0
$991$	399.250			0.402876			0.201438	−	0.979501i	$-0.435439\pi$
$991$	399.250			0.402876			0.201438	+	0.979501i	$0.435439\pi$
$992$	1085.80			1.09455
$993$		0			0
$994$	−688.003			−0.692156
$995$	−244.842	−	729.379i	−0.246072	−	0.733044i
$996$		0			0
$997$		−	1452.17i		−	1.45654i	−0.685288	−	0.728272i	$-0.740324\pi$
$997$		−	1452.17i		−	1.45654i	0.685288	−	0.728272i	$-0.259676\pi$
$998$	−757.008			−0.758525
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	405.3.d.a.404.1		20
3.2	odd	2	inner	405.3.d.a.404.20		20
5.4	even	2	inner	405.3.d.a.404.19		20
9.2	odd	6		45.3.h.a.14.1	✓	20
9.4	even	3		45.3.h.a.29.10	yes	20
9.5	odd	6		135.3.h.a.89.1		20
9.7	even	3		135.3.h.a.44.10		20
15.14	odd	2	inner	405.3.d.a.404.2		20
45.2	even	12		225.3.j.e.176.10		20
45.4	even	6		45.3.h.a.29.1	yes	20
45.7	odd	12		675.3.j.e.476.1		20
45.13	odd	12		225.3.j.e.101.1		20
45.14	odd	6		135.3.h.a.89.10		20
45.22	odd	12		225.3.j.e.101.10		20
45.23	even	12		675.3.j.e.251.10		20
45.29	odd	6		45.3.h.a.14.10	yes	20
45.32	even	12		675.3.j.e.251.1		20
45.34	even	6		135.3.h.a.44.1		20
45.38	even	12		225.3.j.e.176.1		20
45.43	odd	12		675.3.j.e.476.10		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
45.3.h.a.14.1	✓	20	9.2	odd	6
45.3.h.a.14.10	yes	20	45.29	odd	6
45.3.h.a.29.1	yes	20	45.4	even	6
45.3.h.a.29.10	yes	20	9.4	even	3
135.3.h.a.44.1		20	45.34	even	6
135.3.h.a.44.10		20	9.7	even	3
135.3.h.a.89.1		20	9.5	odd	6
135.3.h.a.89.10		20	45.14	odd	6
225.3.j.e.101.1		20	45.13	odd	12
225.3.j.e.101.10		20	45.22	odd	12
225.3.j.e.176.1		20	45.38	even	12
225.3.j.e.176.10		20	45.2	even	12
405.3.d.a.404.1		20	1.1	even	1	trivial
405.3.d.a.404.2		20	15.14	odd	2	inner
405.3.d.a.404.19		20	5.4	even	2	inner
405.3.d.a.404.20		20	3.2	odd	2	inner
675.3.j.e.251.1		20	45.32	even	12
675.3.j.e.251.10		20	45.23	even	12
675.3.j.e.476.1		20	45.7	odd	12
675.3.j.e.476.10		20	45.43	odd	12

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 405 = 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	405.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-3.68728 q^{2} +9.59601 q^{4} +(-1.59117 - 4.74006i) q^{5} -2.56739i q^{7} -20.6340 q^{8} +(5.86709 + 17.4779i) q^{10} +9.83010i q^{11} +12.0633i q^{13} +9.46669i q^{14} +37.6993 q^{16} +4.28451 q^{17} +7.16698 q^{19} +(-15.2689 - 45.4857i) q^{20} -36.2463i q^{22} +0.510539 q^{23} +(-19.9364 + 15.0845i) q^{25} -44.4808i q^{26} -24.6367i q^{28} +30.5521i q^{29} -19.2272 q^{31} -56.4718 q^{32} -15.7982 q^{34} +(-12.1696 + 4.08516i) q^{35} +1.31851i q^{37} -26.4267 q^{38} +(32.8323 + 97.8066i) q^{40} -34.6104i q^{41} -51.9467i q^{43} +94.3297i q^{44} -1.88250 q^{46} +50.9313 q^{47} +42.4085 q^{49} +(73.5108 - 55.6207i) q^{50} +115.760i q^{52} +86.6349 q^{53} +(46.5953 - 15.6414i) q^{55} +52.9757i q^{56} -112.654i q^{58} +105.962i q^{59} +31.3200 q^{61} +70.8961 q^{62} +57.4297 q^{64} +(57.1809 - 19.1948i) q^{65} +78.0541i q^{67} +41.1142 q^{68} +(44.8727 - 15.0631i) q^{70} +72.6762i q^{71} -30.3097i q^{73} -4.86172i q^{74} +68.7744 q^{76} +25.2377 q^{77} +115.280 q^{79} +(-59.9861 - 178.697i) q^{80} +127.618i q^{82} +60.0138 q^{83} +(-6.81739 - 20.3089i) q^{85} +191.542i q^{86} -202.835i q^{88} -71.2992i q^{89} +30.9713 q^{91} +4.89914 q^{92} -187.798 q^{94} +(-11.4039 - 33.9719i) q^{95} +127.964i q^{97} -156.372 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 36 q^{4} + 4 q^{10} + 52 q^{16} - 8 q^{19} - 4 q^{25} - 56 q^{31} + 8 q^{34} + 68 q^{40} + 116 q^{46} + 80 q^{49} + 36 q^{55} + 100 q^{61} + 140 q^{64} + 108 q^{70} + 192 q^{76} + 256 q^{79} + 148 q^{85}+ \cdots - 436 q^{94}+O(q^{100}) \) 20 * q + 36 * q^4 + 4 * q^10 + 52 * q^16 - 8 * q^19 - 4 * q^25 - 56 * q^31 + 8 * q^34 + 68 * q^40 + 116 * q^46 + 80 * q^49 + 36 * q^55 + 100 * q^61 + 140 * q^64 + 108 * q^70 + 192 * q^76 + 256 * q^79 + 148 * q^85 - 288 * q^91 - 436 * q^94

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