LMFDB - Embedded newform 405.3.d.a.404.17

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			404.17
Root			$-0.346576 - 1.69702i$ of defining polynomial
Character	$\chi$	$=$	405.404
Dual form			405.3.d.a.404.18

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.84162 q^{2} +4.07481 q^{4} +(-4.81147 - 1.36005i) q^{5} -9.72928i q^{7} +0.212574 q^{8} +(-13.6724 - 3.86473i) q^{10} +0.428355i q^{11} -18.7519i q^{13} -27.6469i q^{14} -15.6952 q^{16} -2.85718 q^{17} +0.530568 q^{19} +(-19.6058 - 5.54192i) q^{20} +1.21722i q^{22} +21.7727 q^{23} +(21.3006 + 13.0876i) q^{25} -53.2858i q^{26} -39.6450i q^{28} -24.3218i q^{29} -12.6633 q^{31} -45.4500 q^{32} -8.11902 q^{34} +(-13.2323 + 46.8122i) q^{35} +14.5875i q^{37} +1.50767 q^{38} +(-1.02279 - 0.289110i) q^{40} +38.2436i q^{41} -57.7661i q^{43} +1.74546i q^{44} +61.8697 q^{46} +49.5700 q^{47} -45.6590 q^{49} +(60.5281 + 37.1901i) q^{50} -76.4104i q^{52} +44.5876 q^{53} +(0.582582 - 2.06102i) q^{55} -2.06819i q^{56} -69.1132i q^{58} +63.0846i q^{59} -22.1646 q^{61} -35.9842 q^{62} -66.3710 q^{64} +(-25.5035 + 90.2243i) q^{65} +32.5440i q^{67} -11.6425 q^{68} +(-37.6011 + 133.022i) q^{70} +89.8896i q^{71} -144.328i q^{73} +41.4522i q^{74} +2.16196 q^{76} +4.16759 q^{77} +50.2480 q^{79} +(75.5169 + 21.3462i) q^{80} +108.674i q^{82} -76.5112 q^{83} +(13.7472 + 3.88589i) q^{85} -164.149i q^{86} +0.0910570i q^{88} -28.9588i q^{89} -182.443 q^{91} +88.7195 q^{92} +140.859 q^{94} +(-2.55282 - 0.721597i) q^{95} +22.9900i q^{97} -129.746 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	2.84162			1.42081			0.710405	−	0.703793i	$-0.248512\pi$
$2$	2.84162			1.42081			0.710405	+	0.703793i	$0.248512\pi$
$3$		0			0
$3$		0			0
$4$	4.07481			1.01870
$5$	−4.81147	−	1.36005i	−0.962295	−	0.272009i
$5$	−4.81147	−	1.36005i	−0.962295	−	0.272009i
$6$		0			0
$7$		−	9.72928i		−	1.38990i	−0.719059	−	0.694949i	$-0.755427\pi$
$7$		−	9.72928i		−	1.38990i	0.719059	−	0.694949i	$-0.244573\pi$
$8$	0.212574			0.0265717
$9$		0			0
$10$	−13.6724	−	3.86473i	−1.36724	−	0.386473i
$11$			0.428355i			0.0389413i	0.999810	+	0.0194707i	$0.00619810\pi$
$11$			0.428355i			0.0389413i	−0.999810	+	0.0194707i	$0.993802\pi$
$12$		0			0
$13$		−	18.7519i		−	1.44245i	−0.692699	−	0.721227i	$-0.743578\pi$
$13$		−	18.7519i		−	1.44245i	0.692699	−	0.721227i	$-0.256422\pi$
$14$		−	27.6469i		−	1.97478i
$15$		0			0
$16$	−15.6952			−0.980948
$17$	−2.85718			−0.168069			−0.0840347	−	0.996463i	$-0.526781\pi$
$17$	−2.85718			−0.168069			−0.0840347	+	0.996463i	$0.526781\pi$
$18$		0			0
$19$	0.530568			0.0279246			0.0139623	−	0.999903i	$-0.495556\pi$
$19$	0.530568			0.0279246			0.0139623	+	0.999903i	$0.495556\pi$
$20$	−19.6058	−	5.54192i	−0.980291	−	0.277096i
$21$		0			0
$22$			1.21722i			0.0553283i
$23$	21.7727			0.946639			0.473319	−	0.880891i	$-0.343056\pi$
$23$	21.7727			0.946639			0.473319	+	0.880891i	$0.343056\pi$
$24$		0			0
$25$	21.3006	+	13.0876i	0.852022	+	0.523506i
$26$		−	53.2858i		−	2.04945i
$27$		0			0
$28$		−	39.6450i		−	1.41589i
$29$		−	24.3218i		−	0.838682i	−0.907829	−	0.419341i	$-0.862261\pi$
$29$		−	24.3218i		−	0.838682i	0.907829	−	0.419341i	$-0.137739\pi$
$30$		0			0
$31$	−12.6633			−0.408492			−0.204246	−	0.978920i	$-0.565474\pi$
$31$	−12.6633			−0.408492			−0.204246	+	0.978920i	$0.565474\pi$
$32$	−45.4500			−1.42031
$33$		0			0
$34$	−8.11902			−0.238795
$35$	−13.2323	+	46.8122i	−0.378065	+	1.33749i
$36$		0			0
$37$			14.5875i			0.394257i	0.980378	+	0.197129i	$0.0631617\pi$
$37$			14.5875i			0.394257i	−0.980378	+	0.197129i	$0.936838\pi$
$38$	1.50767			0.0396756
$39$		0			0
$40$	−1.02279	−	0.289110i	−0.0255698	−	0.00722776i
$41$			38.2436i			0.932771i	0.884581	+	0.466386i	$0.154444\pi$
$41$			38.2436i			0.932771i	−0.884581	+	0.466386i	$0.845556\pi$
$42$		0			0
$43$		−	57.7661i		−	1.34340i	−0.740824	−	0.671699i	$-0.765565\pi$
$43$		−	57.7661i		−	1.34340i	0.740824	−	0.671699i	$-0.234435\pi$
$44$			1.74546i			0.0396696i
$45$		0			0
$46$	61.8697			1.34499
$47$	49.5700			1.05468			0.527341	−	0.849654i	$-0.323189\pi$
$47$	49.5700			1.05468			0.527341	+	0.849654i	$0.323189\pi$
$48$		0			0
$49$	−45.6590			−0.931816
$50$	60.5281	+	37.1901i	1.21056	+	0.743803i
$51$		0			0
$52$		−	76.4104i		−	1.46943i
$53$	44.5876			0.841275			0.420637	−	0.907229i	$-0.361807\pi$
$53$	44.5876			0.841275			0.420637	+	0.907229i	$0.361807\pi$
$54$		0			0
$55$	0.582582	−	2.06102i	0.0105924	−	0.0374730i
$56$		−	2.06819i		−	0.0369320i
$57$		0			0
$58$		−	69.1132i		−	1.19161i
$59$			63.0846i			1.06923i	0.845096	+	0.534615i	$0.179543\pi$
$59$			63.0846i			1.06923i	−0.845096	+	0.534615i	$0.820457\pi$
$60$		0			0
$61$	−22.1646			−0.363353			−0.181677	−	0.983358i	$-0.558152\pi$
$61$	−22.1646			−0.363353			−0.181677	+	0.983358i	$0.558152\pi$
$62$	−35.9842			−0.580390
$63$		0			0
$64$	−66.3710			−1.03705
$65$	−25.5035	+	90.2243i	−0.392361	+	1.38807i
$66$		0			0
$67$			32.5440i			0.485731i	0.970060	+	0.242866i	$0.0780874\pi$
$67$			32.5440i			0.485731i	−0.970060	+	0.242866i	$0.921913\pi$
$68$	−11.6425			−0.171213
$69$		0			0
$70$	−37.6011	+	133.022i	−0.537159	+	1.90032i
$71$			89.8896i			1.26605i	0.774131	+	0.633025i	$0.218187\pi$
$71$			89.8896i			1.26605i	−0.774131	+	0.633025i	$0.781813\pi$
$72$		0			0
$73$		−	144.328i		−	1.97709i	−0.150926	−	0.988545i	$-0.548225\pi$
$73$		−	144.328i		−	1.97709i	0.150926	−	0.988545i	$-0.451775\pi$
$74$			41.4522i			0.560165i
$75$		0			0
$76$	2.16196			0.0284469
$77$	4.16759			0.0541245
$78$		0			0
$79$	50.2480			0.636051			0.318025	−	0.948082i	$-0.396980\pi$
$79$	50.2480			0.636051			0.318025	+	0.948082i	$0.396980\pi$
$80$	75.5169	+	21.3462i	0.943961	+	0.266827i
$81$		0			0
$82$			108.674i			1.32529i
$83$	−76.5112			−0.921822			−0.460911	−	0.887446i	$-0.652477\pi$
$83$	−76.5112			−0.921822			−0.460911	+	0.887446i	$0.652477\pi$
$84$		0			0
$85$	13.7472	+	3.88589i	0.161732	+	0.0457164i
$86$		−	164.149i		−	1.90871i
$87$		0			0
$88$			0.0910570i			0.00103474i
$89$		−	28.9588i		−	0.325379i	−0.986677	−	0.162690i	$-0.947983\pi$
$89$		−	28.9588i		−	0.325379i	0.986677	−	0.162690i	$-0.0520169\pi$
$90$		0			0
$91$	−182.443			−2.00486
$92$	88.7195			0.964343
$93$		0			0
$94$	140.859			1.49850
$95$	−2.55282	−	0.721597i	−0.0268717	−	0.00759576i
$96$		0			0
$97$			22.9900i			0.237010i	0.992953	+	0.118505i	$0.0378102\pi$
$97$			22.9900i			0.237010i	−0.992953	+	0.118505i	$0.962190\pi$
$98$	−129.746			−1.32393
$99$		0			0
$100$	86.7956	+	53.3296i	0.867956	+	0.533296i
$101$		−	57.4276i		−	0.568590i	−0.958737	−	0.284295i	$-0.908241\pi$
$101$		−	57.4276i		−	0.568590i	0.958737	−	0.284295i	$-0.0917595\pi$
$102$		0			0
$103$		−	86.1903i		−	0.836799i	−0.908263	−	0.418400i	$-0.862591\pi$
$103$		−	86.1903i		−	0.836799i	0.908263	−	0.418400i	$-0.137409\pi$
$104$		−	3.98617i		−	0.0383285i
$105$		0			0
$106$	126.701			1.19529
$107$	97.0919			0.907401			0.453701	−	0.891154i	$-0.350104\pi$
$107$	97.0919			0.907401			0.453701	+	0.891154i	$0.350104\pi$
$108$		0			0
$109$	164.495			1.50913			0.754565	−	0.656225i	$-0.227848\pi$
$109$	164.495			1.50913			0.754565	+	0.656225i	$0.227848\pi$
$110$	1.65548	−	5.85663i	0.0150498	−	0.0532421i
$111$		0			0
$112$			152.703i			1.36342i
$113$	−42.8047			−0.378803			−0.189401	−	0.981900i	$-0.560655\pi$
$113$	−42.8047			−0.378803			−0.189401	+	0.981900i	$0.560655\pi$
$114$		0			0
$115$	−104.759	−	29.6119i	−0.910946	−	0.257494i
$116$		−	99.1065i		−	0.854366i
$117$		0			0
$118$			179.262i			1.51917i
$119$			27.7983i			0.233599i
$120$		0			0
$121$	120.817			0.998484
$122$	−62.9832			−0.516256
$123$		0			0
$124$	−51.6003			−0.416132
$125$	−84.6872	−	91.9406i	−0.677498	−	0.735525i
$126$		0			0
$127$			78.7223i			0.619861i	0.950759	+	0.309931i	$0.100306\pi$
$127$			78.7223i			0.619861i	−0.950759	+	0.309931i	$0.899694\pi$
$128$	−6.80117			−0.0531342
$129$		0			0
$130$	−72.4711	+	256.383i	−0.557470	+	1.97218i
$131$		−	182.783i		−	1.39529i	−0.716444	−	0.697644i	$-0.754232\pi$
$131$		−	182.783i		−	1.39529i	0.716444	−	0.697644i	$-0.245768\pi$
$132$		0			0
$133$		−	5.16205i		−	0.0388124i
$134$			92.4776i			0.690132i
$135$		0			0
$136$	−0.607362			−0.00446589
$137$	−5.56392			−0.0406125			−0.0203063	−	0.999794i	$-0.506464\pi$
$137$	−5.56392			−0.0406125			−0.0203063	+	0.999794i	$0.506464\pi$
$138$		0			0
$139$	−169.920			−1.22245			−0.611224	−	0.791458i	$-0.709323\pi$
$139$	−169.920			−1.22245			−0.611224	+	0.791458i	$0.709323\pi$
$140$	−53.9190	+	190.751i	−0.385135	+	1.36250i
$141$		0			0
$142$			255.432i			1.79882i
$143$	8.03247			0.0561711
$144$		0			0
$145$	−33.0787	+	117.024i	−0.228129	+	0.807059i
$146$		−	410.124i		−	2.80907i
$147$		0			0
$148$			59.4413i			0.401631i
$149$			80.7344i			0.541842i	0.962602	+	0.270921i	$0.0873282\pi$
$149$			80.7344i			0.541842i	−0.962602	+	0.270921i	$0.912672\pi$
$150$		0			0
$151$	88.2744			0.584598			0.292299	−	0.956327i	$-0.405580\pi$
$151$	88.2744			0.584598			0.292299	+	0.956327i	$0.405580\pi$
$152$	0.112785			0.000742006
$153$		0			0
$154$	11.8427			0.0769006
$155$	60.9289	+	17.2226i	0.393090	+	0.111114i
$156$		0			0
$157$		−	62.7465i		−	0.399659i	−0.979831	−	0.199830i	$-0.935961\pi$
$157$		−	62.7465i		−	0.399659i	0.979831	−	0.199830i	$-0.0640389\pi$
$158$	142.786			0.903707
$159$		0			0
$160$	218.682	+	61.8141i	1.36676	+	0.386338i
$161$		−	211.833i		−	1.31573i
$162$		0			0
$163$		−	241.437i		−	1.48121i	−0.671941	−	0.740605i	$-0.734539\pi$
$163$		−	241.437i		−	1.48121i	0.671941	−	0.740605i	$-0.265461\pi$
$164$			155.835i			0.950216i
$165$		0			0
$166$	−217.416			−1.30973
$167$	133.174			0.797451			0.398726	−	0.917070i	$-0.369453\pi$
$167$	133.174			0.797451			0.398726	+	0.917070i	$0.369453\pi$
$168$		0			0
$169$	−182.634			−1.08068
$170$	39.0644	+	11.0422i	0.229791	+	0.0649543i
$171$		0			0
$172$		−	235.386i		−	1.36852i
$173$	67.0423			0.387527			0.193764	−	0.981048i	$-0.437930\pi$
$173$	67.0423			0.387527			0.193764	+	0.981048i	$0.437930\pi$
$174$		0			0
$175$	127.333	−	207.239i	0.727620	−	1.18422i
$176$		−	6.72310i		−	0.0381994i
$177$		0			0
$178$		−	82.2898i		−	0.462302i
$179$			37.9613i			0.212074i	0.994362	+	0.106037i	$0.0338162\pi$
$179$			37.9613i			0.212074i	−0.994362	+	0.106037i	$0.966184\pi$
$180$		0			0
$181$	278.012			1.53598			0.767990	−	0.640462i	$-0.221257\pi$
$181$	278.012			1.53598			0.767990	+	0.640462i	$0.221257\pi$
$182$	−518.433			−2.84853
$183$		0			0
$184$	4.62831			0.0251538
$185$	19.8397	−	70.1875i	0.107242	−	0.379392i
$186$		0			0
$187$		−	1.22389i		−	0.00654485i
$188$	201.988			1.07441
$189$		0			0
$190$	−7.25413	−	2.05051i	−0.0381796	−	0.0107921i
$191$			58.3058i			0.305266i	0.988283	+	0.152633i	$0.0487752\pi$
$191$			58.3058i			0.305266i	−0.988283	+	0.152633i	$0.951225\pi$
$192$		0			0
$193$			126.567i			0.655789i	0.944714	+	0.327895i	$0.106339\pi$
$193$			126.567i			0.655789i	−0.944714	+	0.327895i	$0.893661\pi$
$194$			65.3287i			0.336746i
$195$		0			0
$196$	−186.052			−0.949243
$197$	−335.557			−1.70333			−0.851667	−	0.524083i	$-0.824408\pi$
$197$	−335.557			−1.70333			−0.851667	+	0.524083i	$0.824408\pi$
$198$		0			0
$199$	−127.161			−0.638998			−0.319499	−	0.947587i	$-0.603515\pi$
$199$	−127.161			−0.638998			−0.319499	+	0.947587i	$0.603515\pi$
$200$	4.52794	+	2.78209i	0.0226397	+	0.0139105i
$201$		0			0
$202$		−	163.187i		−	0.807858i
$203$	−236.633			−1.16568
$204$		0			0
$205$	52.0131	−	184.008i	0.253722	−	0.897601i
$206$		−	244.920i		−	1.18893i
$207$		0			0
$208$			294.314i			1.41497i
$209$			0.227271i			0.00108742i
$210$		0			0
$211$	97.7160			0.463109			0.231555	−	0.972822i	$-0.425619\pi$
$211$	97.7160			0.463109			0.231555	+	0.972822i	$0.425619\pi$
$212$	181.686			0.857008
$213$		0			0
$214$	275.898			1.28925
$215$	−78.5646	+	277.940i	−0.365417	+	1.29275i
$216$		0			0
$217$			123.204i			0.567762i
$218$	467.433			2.14419
$219$		0			0
$220$	2.37391	−	8.39825i	0.0107905	−	0.0381739i
$221$			53.5776i			0.242432i
$222$		0			0
$223$		−	93.4261i		−	0.418951i	−0.977814	−	0.209476i	$-0.932824\pi$
$223$		−	93.4261i		−	0.418951i	0.977814	−	0.209476i	$-0.0671757\pi$
$224$			442.196i			1.97409i
$225$		0			0
$226$	−121.635			−0.538207
$227$	197.990			0.872202			0.436101	−	0.899898i	$-0.356359\pi$
$227$	197.990			0.872202			0.436101	+	0.899898i	$0.356359\pi$
$228$		0			0
$229$	181.239			0.791439			0.395719	−	0.918372i	$-0.370495\pi$
$229$	181.239			0.791439			0.395719	+	0.918372i	$0.370495\pi$
$230$	−297.685	−	84.1457i	−1.29428	−	0.365851i
$231$		0			0
$232$		−	5.17017i		−	0.0222852i
$233$	178.525			0.766202			0.383101	−	0.923706i	$-0.374856\pi$
$233$	178.525			0.766202			0.383101	+	0.923706i	$0.374856\pi$
$234$		0			0
$235$	−238.505	−	67.4175i	−1.01491	−	0.286883i
$236$			257.058i			1.08923i
$237$		0			0
$238$			78.9923i			0.331900i
$239$		−	15.2601i		−	0.0638499i	−0.999490	−	0.0319250i	$-0.989836\pi$
$239$		−	15.2601i		−	0.0638499i	0.999490	−	0.0319250i	$-0.0101638\pi$
$240$		0			0
$241$	−319.166			−1.32434			−0.662170	−	0.749354i	$-0.730364\pi$
$241$	−319.166			−1.32434			−0.662170	+	0.749354i	$0.730364\pi$
$242$	343.315			1.41866
$243$		0			0
$244$	−90.3163			−0.370149
$245$	219.687	+	62.0983i	0.896682	+	0.253462i
$246$		0			0
$247$		−	9.94917i		−	0.0402800i
$248$	−2.69188			−0.0108543
$249$		0			0
$250$	−240.649	−	261.260i	−0.962596	−	1.04504i
$251$		−	95.3404i		−	0.379842i	−0.981799	−	0.189921i	$-0.939177\pi$
$251$		−	95.3404i		−	0.379842i	0.981799	−	0.189921i	$-0.0608232\pi$
$252$		0			0
$253$			9.32644i			0.0368634i
$254$			223.699i			0.880705i
$255$		0			0
$256$	246.158			0.961554
$257$	7.93274			0.0308667			0.0154334	−	0.999881i	$-0.495087\pi$
$257$	7.93274			0.0308667			0.0154334	+	0.999881i	$0.495087\pi$
$258$		0			0
$259$	141.926			0.547977
$260$	−103.922	+	367.647i	−0.399699	+	1.41403i
$261$		0			0
$262$		−	519.399i		−	1.98244i
$263$	−410.379			−1.56038			−0.780188	−	0.625545i	$-0.784877\pi$
$263$	−410.379			−1.56038			−0.780188	+	0.625545i	$0.784877\pi$
$264$		0			0
$265$	−214.532	−	60.6411i	−0.809554	−	0.228834i
$266$		−	14.6686i		−	0.0551451i
$267$		0			0
$268$			132.610i			0.494815i
$269$		−	201.081i		−	0.747513i	−0.927527	−	0.373757i	$-0.878070\pi$
$269$		−	201.081i		−	0.747513i	0.927527	−	0.373757i	$-0.121930\pi$
$270$		0			0
$271$	121.070			0.446751			0.223376	−	0.974732i	$-0.428292\pi$
$271$	121.070			0.446751			0.223376	+	0.974732i	$0.428292\pi$
$272$	44.8439			0.164867
$273$		0			0
$274$	−15.8105			−0.0577027
$275$	−5.60616	+	9.12419i	−0.0203860	+	0.0331789i
$276$		0			0
$277$			335.008i			1.20941i	0.796448	+	0.604707i	$0.206710\pi$
$277$			335.008i			1.20941i	−0.796448	+	0.604707i	$0.793290\pi$
$278$	−482.849			−1.73687
$279$		0			0
$280$	−2.81284	+	9.95105i	−0.0100458	+	0.0355395i
$281$			138.870i			0.494199i	0.968990	+	0.247099i	$0.0794774\pi$
$281$			138.870i			0.494199i	−0.968990	+	0.247099i	$0.920523\pi$
$282$		0			0
$283$		−	96.8045i		−	0.342066i	−0.985265	−	0.171033i	$-0.945290\pi$
$283$		−	96.8045i		−	0.342066i	0.985265	−	0.171033i	$-0.0547104\pi$
$284$			366.283i			1.28973i
$285$		0			0
$286$	22.8252			0.0798085
$287$	372.083			1.29646
$288$		0			0
$289$	−280.837			−0.971753
$290$	−93.9972	+	332.536i	−0.324128	+	1.14668i
$291$		0			0
$292$		−	588.107i		−	2.01407i
$293$	385.079			1.31426			0.657131	−	0.753776i	$-0.271770\pi$
$293$	385.079			1.31426			0.657131	+	0.753776i	$0.271770\pi$
$294$		0			0
$295$	85.7979	−	303.530i	0.290840	−	1.02891i
$296$			3.10093i			0.0104761i
$297$		0			0
$298$			229.417i			0.769854i
$299$		−	408.280i		−	1.36548i
$300$		0			0
$301$	−562.023			−1.86719
$302$	250.842			0.830603
$303$		0			0
$304$	−8.32736			−0.0273926
$305$	106.644	+	30.1448i	0.349653	+	0.0988354i
$306$		0			0
$307$		−	207.311i		−	0.675281i	−0.941275	−	0.337641i	$-0.890371\pi$
$307$		−	207.311i		−	0.675281i	0.941275	−	0.337641i	$-0.109629\pi$
$308$	16.9821			0.0551367
$309$		0			0
$310$	173.137	+	48.9401i	0.558506	+	0.157871i
$311$		−	162.163i		−	0.521425i	−0.965417	−	0.260712i	$-0.916043\pi$
$311$		−	162.163i		−	0.521425i	0.965417	−	0.260712i	$-0.0839574\pi$
$312$		0			0
$313$			226.814i			0.724645i	0.932053	+	0.362322i	$0.118016\pi$
$313$			226.814i			0.724645i	−0.932053	+	0.362322i	$0.881984\pi$
$314$		−	178.302i		−	0.567840i
$315$		0			0
$316$	204.751			0.647946
$317$	−488.776			−1.54188			−0.770941	−	0.636907i	$-0.780213\pi$
$317$	−488.776			−1.54188			−0.770941	+	0.636907i	$0.780213\pi$
$318$		0			0
$319$	10.4183			0.0326594
$320$	319.342	+	90.2676i	0.997945	+	0.282086i
$321$		0			0
$322$		−	601.948i		−	1.86940i
$323$	−1.51593			−0.00469328
$324$		0			0
$325$	245.418	−	399.426i	0.755134	−	1.22900i
$326$		−	686.073i		−	2.10452i
$327$		0			0
$328$			8.12960i			0.0247854i
$329$		−	482.281i		−	1.46590i
$330$		0			0
$331$	237.646			0.717964			0.358982	−	0.933345i	$-0.383124\pi$
$331$	237.646			0.717964			0.358982	+	0.933345i	$0.383124\pi$
$332$	−311.768			−0.939061
$333$		0			0
$334$	378.431			1.13303
$335$	44.2613	−	156.584i	0.132123	−	0.467416i
$336$		0			0
$337$			282.035i			0.836900i	0.908240	+	0.418450i	$0.137427\pi$
$337$			282.035i			0.836900i	−0.908240	+	0.418450i	$0.862573\pi$
$338$	−518.977			−1.53543
$339$		0			0
$340$	56.0174	+	15.8343i	0.164757	+	0.0465714i
$341$		−	5.42437i		−	0.0159072i
$342$		0			0
$343$		−	32.5057i		−	0.0947688i
$344$		−	12.2796i		−	0.0356964i
$345$		0			0
$346$	190.509			0.550603
$347$	−439.971			−1.26793			−0.633963	−	0.773363i	$-0.718573\pi$
$347$	−439.971			−1.26793			−0.633963	+	0.773363i	$0.718573\pi$
$348$		0			0
$349$	−97.2581			−0.278677			−0.139338	−	0.990245i	$-0.544498\pi$
$349$	−97.2581			−0.278677			−0.139338	+	0.990245i	$0.544498\pi$
$350$	361.833	−	588.895i	1.03381	−	1.68256i
$351$		0			0
$352$		−	19.4687i		−	0.0553089i
$353$	−98.1956			−0.278174			−0.139087	−	0.990280i	$-0.544417\pi$
$353$	−98.1956			−0.278174			−0.139087	+	0.990280i	$0.544417\pi$
$354$		0			0
$355$	122.254	−	432.501i	0.344377	−	1.21831i
$356$		−	118.001i		−	0.331465i
$357$		0			0
$358$			107.872i			0.301317i
$359$		−	500.068i		−	1.39295i	−0.717582	−	0.696474i	$-0.754751\pi$
$359$		−	500.068i		−	1.39295i	0.717582	−	0.696474i	$-0.245249\pi$
$360$		0			0
$361$	−360.718			−0.999220
$362$	790.005			2.18234
$363$		0			0
$364$	−743.419			−2.04236
$365$	−196.292	+	694.428i	−0.537787	+	1.90254i
$366$		0			0
$367$		−	341.818i		−	0.931385i	−0.884947	−	0.465692i	$-0.845805\pi$
$367$		−	341.818i		−	0.931385i	0.884947	−	0.465692i	$-0.154195\pi$
$368$	−341.726			−0.928604
$369$		0			0
$370$	56.3769	−	199.446i	0.152370	−	0.539044i
$371$		−	433.805i		−	1.16929i
$372$		0			0
$373$			611.043i			1.63819i	0.573661	+	0.819093i	$0.305523\pi$
$373$			611.043i			1.63819i	−0.573661	+	0.819093i	$0.694477\pi$
$374$		−	3.47782i		−	0.00929899i
$375$		0			0
$376$	10.5373			0.0280247
$377$	−456.080			−1.20976
$378$		0			0
$379$	187.436			0.494555			0.247277	−	0.968945i	$-0.420464\pi$
$379$	187.436			0.494555			0.247277	+	0.968945i	$0.420464\pi$
$380$	−10.4022	−	2.94037i	−0.0273743	−	0.00773782i
$381$		0			0
$382$			165.683i			0.433725i
$383$	−147.818			−0.385947			−0.192973	−	0.981204i	$-0.561813\pi$
$383$	−147.818			−0.385947			−0.192973	+	0.981204i	$0.561813\pi$
$384$		0			0
$385$	−20.0522	−	5.66811i	−0.0520837	−	0.0147224i
$386$			359.656i			0.931752i
$387$		0			0
$388$			93.6797i			0.241442i
$389$		−	459.444i		−	1.18109i	−0.807005	−	0.590545i	$-0.798913\pi$
$389$		−	459.444i		−	1.18109i	0.807005	−	0.590545i	$-0.201087\pi$
$390$		0			0
$391$	−62.2085			−0.159101
$392$	−9.70591			−0.0247600
$393$		0			0
$394$	−953.525			−2.42011
$395$	−241.767	−	68.3396i	−0.612068	−	0.173012i
$396$		0			0
$397$			718.905i			1.81084i	0.424512	+	0.905422i	$0.360446\pi$
$397$			718.905i			1.81084i	−0.424512	+	0.905422i	$0.639554\pi$
$398$	−361.342			−0.907895
$399$		0			0
$400$	−334.316	−	205.413i	−0.835790	−	0.513532i
$401$			75.7434i			0.188886i	0.995530	+	0.0944431i	$0.0301070\pi$
$401$			75.7434i			0.188886i	−0.995530	+	0.0944431i	$0.969893\pi$
$402$		0			0
$403$			237.460i			0.589231i
$404$		−	234.006i		−	0.579223i
$405$		0			0
$406$	−672.422			−1.65621
$407$	−6.24863			−0.0153529
$408$		0			0
$409$	510.069			1.24711			0.623556	−	0.781778i	$-0.285687\pi$
$409$	510.069			1.24711			0.623556	+	0.781778i	$0.285687\pi$
$410$	147.801	−	522.881i	0.360491	−	1.27532i
$411$		0			0
$412$		−	351.209i		−	0.852449i
$413$	613.768			1.48612
$414$		0			0
$415$	368.132	+	104.059i	0.887064	+	0.250744i
$416$			852.275i			2.04874i
$417$		0			0
$418$			0.645819i			0.00154502i
$419$		−	697.172i		−	1.66389i	−0.554855	−	0.831947i	$-0.687226\pi$
$419$		−	697.172i		−	1.66389i	0.554855	−	0.831947i	$-0.312774\pi$
$420$		0			0
$421$	756.820			1.79767			0.898836	−	0.438285i	$-0.144414\pi$
$421$	756.820			1.79767			0.898836	+	0.438285i	$0.144414\pi$
$422$	277.672			0.657990
$423$		0			0
$424$	9.47815			0.0223541
$425$	−60.8595	−	37.3938i	−0.143199	−	0.0879853i
$426$		0			0
$427$			215.645i			0.505024i
$428$	395.631			0.924371
$429$		0			0
$430$	−223.251	+	789.801i	−0.519188	+	1.83675i
$431$			497.875i			1.15516i	0.816333	+	0.577581i	$0.196003\pi$
$431$			497.875i			1.15516i	−0.816333	+	0.577581i	$0.803997\pi$
$432$		0			0
$433$			421.645i			0.973775i	0.873465	+	0.486888i	$0.161868\pi$
$433$			421.645i			0.973775i	−0.873465	+	0.486888i	$0.838132\pi$
$434$			350.100i			0.806683i
$435$		0			0
$436$	670.286			1.53735
$437$	11.5519			0.0264346
$438$		0			0
$439$	457.487			1.04211			0.521055	−	0.853523i	$-0.325539\pi$
$439$	457.487			1.04211			0.521055	+	0.853523i	$0.325539\pi$
$440$	0.123842	−	0.438118i	0.000281458	−	0.000995724i
$441$		0			0
$442$			152.247i			0.344450i
$443$	232.271			0.524314			0.262157	−	0.965025i	$-0.415566\pi$
$443$	232.271			0.524314			0.262157	+	0.965025i	$0.415566\pi$
$444$		0			0
$445$	−39.3852	+	139.334i	−0.0885062	+	0.313111i
$446$		−	265.481i		−	0.595250i
$447$		0			0
$448$			645.743i			1.44139i
$449$			678.345i			1.51079i	0.655269	+	0.755395i	$0.272555\pi$
$449$			678.345i			1.51079i	−0.655269	+	0.755395i	$0.727445\pi$
$450$		0			0
$451$	−16.3818			−0.0363234
$452$	−174.421			−0.385887
$453$		0			0
$454$	562.612			1.23923
$455$	877.818	+	248.130i	1.92927	+	0.545342i
$456$		0			0
$457$			735.553i			1.60953i	0.593596	+	0.804763i	$0.297708\pi$
$457$			735.553i			1.60953i	−0.593596	+	0.804763i	$0.702292\pi$
$458$	515.014			1.12448
$459$		0			0
$460$	−426.872	−	120.663i	−0.927982	−	0.262310i
$461$			398.085i			0.863525i	0.901987	+	0.431763i	$0.142108\pi$
$461$			398.085i			0.863525i	−0.901987	+	0.431763i	$0.857892\pi$
$462$		0			0
$463$			160.591i			0.346849i	0.984847	+	0.173424i	$0.0554832\pi$
$463$			160.591i			0.346849i	−0.984847	+	0.173424i	$0.944517\pi$
$464$			381.734i			0.822703i
$465$		0			0
$466$	507.301			1.08863
$467$	−406.660			−0.870792			−0.435396	−	0.900239i	$-0.643392\pi$
$467$	−406.660			−0.870792			−0.435396	+	0.900239i	$0.643392\pi$
$468$		0			0
$469$	316.630			0.675117
$470$	−677.740	−	191.575i	−1.44200	−	0.407606i
$471$		0			0
$472$			13.4101i			0.0284113i
$473$	24.7444			0.0523137
$474$		0			0
$475$	11.3014	+	6.94389i	0.0237924	+	0.0146187i
$476$			113.273i			0.237968i
$477$		0			0
$478$		−	43.3635i		−	0.0907186i
$479$			844.606i			1.76327i	0.471932	+	0.881635i	$0.343557\pi$
$479$			844.606i			1.76327i	−0.471932	+	0.881635i	$0.656443\pi$
$480$		0			0
$481$	273.544			0.568698
$482$	−906.948			−1.88164
$483$		0			0
$484$	492.304			1.01716
$485$	31.2674	−	110.616i	0.0644689	−	0.228073i
$486$		0			0
$487$			225.518i			0.463075i	0.972826	+	0.231538i	$0.0743757\pi$
$487$			225.518i			0.463075i	−0.972826	+	0.231538i	$0.925624\pi$
$488$	−4.71160			−0.00965493
$489$		0			0
$490$	624.267	+	176.460i	1.27401	+	0.360122i
$491$			765.150i			1.55835i	0.626807	+	0.779175i	$0.284362\pi$
$491$			765.150i			1.55835i	−0.626807	+	0.779175i	$0.715638\pi$
$492$		0			0
$493$			69.4916i			0.140957i
$494$		−	28.2718i		−	0.0572303i
$495$		0			0
$496$	198.752			0.400710
$497$	874.562			1.75968
$498$		0			0
$499$	−204.049			−0.408916			−0.204458	−	0.978875i	$-0.565543\pi$
$499$	−204.049			−0.408916			−0.204458	+	0.978875i	$0.565543\pi$
$500$	−345.084	−	374.640i	−0.690168	−	0.749280i
$501$		0			0
$502$		−	270.921i		−	0.539684i
$503$	−585.545			−1.16411			−0.582053	−	0.813151i	$-0.697750\pi$
$503$	−585.545			−1.16411			−0.582053	+	0.813151i	$0.697750\pi$
$504$		0			0
$505$	−78.1041	+	276.311i	−0.154662	+	0.547151i
$506$			26.5022i			0.0523759i
$507$		0			0
$508$			320.778i			0.631454i
$509$		−	11.9351i		−	0.0234482i	−0.999931	−	0.0117241i	$-0.996268\pi$
$509$		−	11.9351i		−	0.0234482i	0.999931	−	0.0117241i	$-0.00373198\pi$
$510$		0			0
$511$	−1404.20			−2.74795
$512$	726.692			1.41932
$513$		0			0
$514$	22.5418			0.0438557
$515$	−117.223	+	414.702i	−0.227617	+	0.805247i
$516$		0			0
$517$			21.2336i			0.0410707i
$518$	403.300			0.778572
$519$		0			0
$520$	−5.42137	+	19.1793i	−0.0104257	+	0.0368833i
$521$			541.869i			1.04006i	0.854149	+	0.520028i	$0.174078\pi$
$521$			541.869i			1.04006i	−0.854149	+	0.520028i	$0.825922\pi$
$522$		0			0
$523$		−	356.100i		−	0.680880i	−0.940266	−	0.340440i	$-0.889424\pi$
$523$		−	356.100i		−	0.680880i	0.940266	−	0.340440i	$-0.110576\pi$
$524$		−	744.805i		−	1.42138i
$525$		0			0
$526$	−1166.14			−2.21700
$527$	36.1812			0.0686550
$528$		0			0
$529$	−54.9498			−0.103875
$530$	−609.618	−	172.319i	−1.15022	−	0.325130i
$531$		0			0
$532$		−	21.0344i		−	0.0395383i
$533$	717.141			1.34548
$534$		0			0
$535$	−467.155	−	132.049i	−0.873187	−	0.246821i
$536$			6.91800i			0.0129067i
$537$		0			0
$538$		−	571.396i		−	1.06207i
$539$		−	19.5582i		−	0.0362862i
$540$		0			0
$541$	378.892			0.700355			0.350178	−	0.936683i	$-0.386121\pi$
$541$	378.892			0.700355			0.350178	+	0.936683i	$0.386121\pi$
$542$	344.034			0.634749
$543$		0			0
$544$	129.859			0.238711
$545$	−791.464	−	223.721i	−1.45223	−	0.410497i
$546$		0			0
$547$		−	559.346i		−	1.02257i	−0.859411	−	0.511285i	$-0.829170\pi$
$547$		−	559.346i		−	1.02257i	0.859411	−	0.511285i	$-0.170830\pi$
$548$	−22.6719			−0.0413721
$549$		0			0
$550$	−15.9306	+	25.9275i	−0.0289647	+	0.0471409i
$551$		−	12.9044i		−	0.0234199i
$552$		0			0
$553$		−	488.877i		−	0.884045i
$554$			951.965i			1.71835i
$555$		0			0
$556$	−692.392			−1.24531
$557$	471.971			0.847345			0.423673	−	0.905815i	$-0.360741\pi$
$557$	471.971			0.847345			0.423673	+	0.905815i	$0.360741\pi$
$558$		0			0
$559$	−1083.23			−1.93779
$560$	207.683	−	734.726i	0.370862	−	1.31201i
$561$		0			0
$562$			394.615i			0.702162i
$563$	494.577			0.878467			0.439234	−	0.898373i	$-0.355250\pi$
$563$	494.577			0.878467			0.439234	+	0.898373i	$0.355250\pi$
$564$		0			0
$565$	205.954	+	58.2164i	0.364520	+	0.103038i
$566$		−	275.082i		−	0.486010i
$567$		0			0
$568$			19.1082i			0.0336412i
$569$			165.474i			0.290815i	0.989372	+	0.145408i	$0.0464494\pi$
$569$			165.474i			0.290815i	−0.989372	+	0.145408i	$0.953551\pi$
$570$		0			0
$571$	51.0603			0.0894226			0.0447113	−	0.999000i	$-0.485763\pi$
$571$	51.0603			0.0894226			0.0447113	+	0.999000i	$0.485763\pi$
$572$	32.7308			0.0572216
$573$		0			0
$574$	1057.32			1.84202
$575$	463.770	+	284.953i	0.806557	+	0.495571i
$576$		0			0
$577$			707.833i			1.22675i	0.789793	+	0.613373i	$0.210188\pi$
$577$			707.833i			1.22675i	−0.789793	+	0.613373i	$0.789812\pi$
$578$	−798.031			−1.38068
$579$		0			0
$580$	−134.789	+	476.848i	−0.232396	+	0.822152i
$581$			744.399i			1.28124i
$582$		0			0
$583$			19.0993i			0.0327604i
$584$		−	30.6803i		−	0.0525347i
$585$		0			0
$586$	1094.25			1.86732
$587$	−452.501			−0.770871			−0.385436	−	0.922735i	$-0.625949\pi$
$587$	−452.501			−0.770871			−0.385436	+	0.922735i	$0.625949\pi$
$588$		0			0
$589$	−6.71872			−0.0114070
$590$	243.805	−	862.517i	0.413229	−	1.46189i
$591$		0			0
$592$		−	228.954i		−	0.386746i
$593$	227.811			0.384168			0.192084	−	0.981379i	$-0.438475\pi$
$593$	227.811			0.384168			0.192084	+	0.981379i	$0.438475\pi$
$594$		0			0
$595$	37.8070	−	133.751i	0.0635411	−	0.224791i
$596$			328.977i			0.551975i
$597$		0			0
$598$		−	1160.18i		−	1.94009i
$599$			559.580i			0.934191i	0.884207	+	0.467096i	$0.154700\pi$
$599$			559.580i			0.934191i	−0.884207	+	0.467096i	$0.845300\pi$
$600$		0			0
$601$	670.543			1.11571			0.557856	−	0.829938i	$-0.311624\pi$
$601$	670.543			1.11571			0.557856	+	0.829938i	$0.311624\pi$
$602$	−1597.06			−2.65292
$603$		0			0
$604$	359.701			0.595532
$605$	−581.305	−	164.316i	−0.960835	−	0.271597i
$606$		0			0
$607$			998.017i			1.64418i	0.569357	+	0.822090i	$0.307192\pi$
$607$			998.017i			1.64418i	−0.569357	+	0.822090i	$0.692808\pi$
$608$	−24.1143			−0.0396618
$609$		0			0
$610$	303.042	+	85.6601i	0.496791	+	0.140426i
$611$		−	929.533i		−	1.52133i
$612$		0			0
$613$		−	229.314i		−	0.374086i	−0.982352	−	0.187043i	$-0.940110\pi$
$613$		−	229.314i		−	0.374086i	0.982352	−	0.187043i	$-0.0598903\pi$
$614$		−	589.100i		−	0.959447i
$615$		0			0
$616$	0.885920			0.00143818
$617$	497.435			0.806215			0.403108	−	0.915153i	$-0.367930\pi$
$617$	497.435			0.806215			0.403108	+	0.915153i	$0.367930\pi$
$618$		0			0
$619$	−1057.73			−1.70878			−0.854390	−	0.519633i	$-0.826069\pi$
$619$	−1057.73			−1.70878			−0.854390	+	0.519633i	$0.826069\pi$
$620$	248.274	+	70.1788i	0.400441	+	0.113192i
$621$		0			0
$622$		−	460.806i		−	0.740846i
$623$	−281.748			−0.452244
$624$		0			0
$625$	282.427	+	557.548i	0.451883	+	0.892077i
$626$			644.519i			1.02958i
$627$		0			0
$628$		−	255.680i		−	0.407134i
$629$		−	41.6792i		−	0.0662626i
$630$		0			0
$631$	638.591			1.01203			0.506015	−	0.862524i	$-0.331118\pi$
$631$	638.591			1.01203			0.506015	+	0.862524i	$0.331118\pi$
$632$	10.6814			0.0169010
$633$		0			0
$634$	−1388.92			−2.19072
$635$	107.066	−	378.770i	0.168608	−	0.596489i
$636$		0			0
$637$			856.193i			1.34410i
$638$	29.6050			0.0464028
$639$		0			0
$640$	32.7237	+	9.24991i	0.0511307	+	0.0144530i
$641$			884.263i			1.37951i	0.724045	+	0.689753i	$0.242281\pi$
$641$			884.263i			1.37951i	−0.724045	+	0.689753i	$0.757719\pi$
$642$		0			0
$643$		−	1021.23i		−	1.58822i	−0.607773	−	0.794111i	$-0.707937\pi$
$643$		−	1021.23i		−	1.58822i	0.607773	−	0.794111i	$-0.292063\pi$
$644$		−	863.178i		−	1.34034i
$645$		0			0
$646$	−4.30769			−0.00666826
$647$	488.043			0.754317			0.377159	−	0.926149i	$-0.376901\pi$
$647$	488.043			0.754317			0.377159	+	0.926149i	$0.376901\pi$
$648$		0			0
$649$	−27.0226			−0.0416373
$650$	697.386	−	1135.02i	1.07290	−	1.74618i
$651$		0			0
$652$		−	983.810i		−	1.50891i
$653$	−611.295			−0.936134			−0.468067	−	0.883693i	$-0.655049\pi$
$653$	−611.295			−0.936134			−0.468067	+	0.883693i	$0.655049\pi$
$654$		0			0
$655$	−248.593	+	879.455i	−0.379531	+	1.34268i
$656$		−	600.240i		−	0.915001i
$657$		0			0
$658$		−	1370.46i		−	2.08276i
$659$			145.131i			0.220230i	0.993919	+	0.110115i	$0.0351219\pi$
$659$			145.131i			0.220230i	−0.993919	+	0.110115i	$0.964878\pi$
$660$		0			0
$661$	830.256			1.25606			0.628030	−	0.778189i	$-0.283861\pi$
$661$	830.256			1.25606			0.628030	+	0.778189i	$0.283861\pi$
$662$	675.300			1.02009
$663$		0			0
$664$	−16.2643			−0.0244944
$665$	−7.02063	+	24.8371i	−0.0105573	+	0.0373490i
$666$		0			0
$667$		−	529.550i		−	0.793929i
$668$	542.660			0.812365
$669$		0			0
$670$	125.774	−	444.954i	0.187722	−	0.664110i
$671$		−	9.49429i		−	0.0141495i
$672$		0			0
$673$		−	271.416i		−	0.403293i	−0.979458	−	0.201647i	$-0.935371\pi$
$673$		−	271.416i		−	0.403293i	0.979458	−	0.201647i	$-0.0646293\pi$
$674$			801.438i			1.18908i
$675$		0			0
$676$	−744.199			−1.10089
$677$	−895.939			−1.32340			−0.661698	−	0.749770i	$-0.730164\pi$
$677$	−895.939			−1.32340			−0.661698	+	0.749770i	$0.730164\pi$
$678$		0			0
$679$	223.676			0.329420
$680$	2.92230	+	0.826040i	0.00429751	+	0.00121476i
$681$		0			0
$682$		−	15.4140i		−	0.0226012i
$683$	−255.548			−0.374155			−0.187078	−	0.982345i	$-0.559902\pi$
$683$	−255.548			−0.374155			−0.187078	+	0.982345i	$0.559902\pi$
$684$		0			0
$685$	26.7706	+	7.56719i	0.0390812	+	0.0110470i
$686$		−	92.3689i		−	0.134648i
$687$		0			0
$688$			906.650i			1.31780i
$689$		−	836.102i		−	1.21350i
$690$		0			0
$691$	122.595			0.177416			0.0887081	−	0.996058i	$-0.471726\pi$
$691$	122.595			0.177416			0.0887081	+	0.996058i	$0.471726\pi$
$692$	273.184			0.394775
$693$		0			0
$694$	−1250.23			−1.80148
$695$	817.567	+	231.099i	1.17636	+	0.332517i
$696$		0			0
$697$		−	109.269i		−	0.156770i
$698$	−276.371			−0.395947
$699$		0			0
$700$	518.859	−	844.459i	0.741228	−	1.20637i
$701$			595.027i			0.848826i	0.905469	+	0.424413i	$0.139520\pi$
$701$			595.027i			0.848826i	−0.905469	+	0.424413i	$0.860480\pi$
$702$		0			0
$703$			7.73968i			0.0110095i
$704$		−	28.4303i		−	0.0403840i
$705$		0			0
$706$	−279.035			−0.395233
$707$	−558.729			−0.790282
$708$		0			0
$709$	560.984			0.791232			0.395616	−	0.918416i	$-0.370531\pi$
$709$	560.984			0.791232			0.395616	+	0.918416i	$0.370531\pi$
$710$	347.399	−	1229.00i	0.489295	−	1.73099i
$711$		0			0
$712$		−	6.15588i		−	0.00864589i
$713$	−275.713			−0.386695
$714$		0			0
$715$	−38.6480	−	10.9245i	−0.0540532	−	0.0152791i
$716$			154.685i			0.216040i
$717$		0			0
$718$		−	1421.00i		−	1.97911i
$719$		−	1218.37i		−	1.69453i	−0.531168	−	0.847266i	$-0.678247\pi$
$719$		−	1218.37i		−	1.69453i	0.531168	−	0.847266i	$-0.321753\pi$
$720$		0			0
$721$	−838.570			−1.16307
$722$	−1025.03			−1.41970
$723$		0			0
$724$	1132.85			1.56471
$725$	318.315	−	518.067i	0.439055	−	0.714575i
$726$		0			0
$727$			794.343i			1.09263i	0.837579	+	0.546316i	$0.183970\pi$
$727$			794.343i			1.09263i	−0.837579	+	0.546316i	$0.816030\pi$
$728$	−38.7825			−0.0532727
$729$		0			0
$730$	−557.788	+	1973.30i	−0.764093	+	2.70315i
$731$			165.048i			0.225784i
$732$		0			0
$733$			786.822i			1.07343i	0.843764	+	0.536714i	$0.180334\pi$
$733$			786.822i			1.07343i	−0.843764	+	0.536714i	$0.819666\pi$
$734$		−	971.318i		−	1.32332i
$735$		0			0
$736$	−989.570			−1.34452
$737$	−13.9404			−0.0189150
$738$		0			0
$739$	1398.66			1.89264			0.946320	−	0.323231i	$-0.104769\pi$
$739$	1398.66			1.89264			0.946320	+	0.323231i	$0.104769\pi$
$740$	80.8430	−	286.000i	0.109247	−	0.386487i
$741$		0			0
$742$		−	1232.71i		−	1.66133i
$743$	−548.629			−0.738398			−0.369199	−	0.929350i	$-0.620368\pi$
$743$	−548.629			−0.738398			−0.369199	+	0.929350i	$0.620368\pi$
$744$		0			0
$745$	109.802	−	388.451i	0.147386	−	0.521411i
$746$			1736.35i			2.32755i
$747$		0			0
$748$		−	4.98710i		−	0.00666725i
$749$		−	944.635i		−	1.26120i
$750$		0			0
$751$	−267.361			−0.356007			−0.178003	−	0.984030i	$-0.556964\pi$
$751$	−267.361			−0.356007			−0.178003	+	0.984030i	$0.556964\pi$
$752$	−778.010			−1.03459
$753$		0			0
$754$	−1296.01			−1.71884
$755$	−424.730	−	120.057i	−0.562556	−	0.159016i
$756$		0			0
$757$		−	471.386i		−	0.622703i	−0.950295	−	0.311351i	$-0.899218\pi$
$757$		−	471.386i		−	0.622703i	0.950295	−	0.311351i	$-0.100782\pi$
$758$	532.623			0.702669
$759$		0			0
$760$	−0.542662	−	0.153393i	−0.000714029	−	0.000201833i
$761$		−	1060.01i		−	1.39292i	−0.717595	−	0.696461i	$-0.754757\pi$
$761$		−	1060.01i		−	1.39292i	0.717595	−	0.696461i	$-0.245243\pi$
$762$		0			0
$763$		−	1600.42i		−	2.09754i
$764$			237.585i			0.310975i
$765$		0			0
$766$	−420.042			−0.548357
$767$	1182.96			1.54232
$768$		0			0
$769$	433.026			0.563103			0.281551	−	0.959546i	$-0.409151\pi$
$769$	433.026			0.563103			0.281551	+	0.959546i	$0.409151\pi$
$770$	−56.9808	−	16.1066i	−0.0740011	−	0.0209177i
$771$		0			0
$772$			515.737i			0.668054i
$773$	−1088.28			−1.40787			−0.703936	−	0.710264i	$-0.748576\pi$
$773$	−1088.28			−1.40787			−0.703936	+	0.710264i	$0.748576\pi$
$774$		0			0
$775$	−269.734	−	165.732i	−0.348044	−	0.213848i
$776$			4.88707i			0.00629776i
$777$		0			0
$778$		−	1305.56i		−	1.67810i
$779$			20.2909i			0.0260473i
$780$		0			0
$781$	−38.5046			−0.0493017
$782$	−176.773			−0.226052
$783$		0			0
$784$	716.626			0.914063
$785$	−85.3381	+	301.903i	−0.108711	+	0.384590i
$786$		0			0
$787$		−	322.934i		−	0.410336i	−0.978727	−	0.205168i	$-0.934226\pi$
$787$		−	322.934i		−	0.410336i	0.978727	−	0.205168i	$-0.0657740\pi$
$788$	−1367.33			−1.73519
$789$		0			0
$790$	−687.010	−	194.195i	−0.869633	−	0.245817i
$791$			416.459i			0.526497i
$792$		0			0
$793$			415.628i			0.524121i
$794$			2042.86i			2.57287i
$795$		0			0
$796$	−518.155			−0.650949
$797$	1445.43			1.81358			0.906792	−	0.421578i	$-0.138524\pi$
$797$	1445.43			1.81358			0.906792	+	0.421578i	$0.138524\pi$
$798$		0			0
$799$	−141.630			−0.177260
$800$	−968.111	−	594.834i	−1.21014	−	0.743542i
$801$		0			0
$802$			215.234i			0.268372i
$803$	61.8234			0.0769905
$804$		0			0
$805$	−288.102	+	1019.23i	−0.357891	+	1.26612i
$806$			674.772i			0.837186i
$807$		0			0
$808$		−	12.2076i		−	0.0151084i
$809$		−	981.839i		−	1.21365i	−0.794837	−	0.606823i	$-0.792444\pi$
$809$		−	981.839i		−	1.21365i	0.794837	−	0.606823i	$-0.207556\pi$
$810$		0			0
$811$	−600.049			−0.739888			−0.369944	−	0.929054i	$-0.620623\pi$
$811$	−600.049			−0.739888			−0.369944	+	0.929054i	$0.620623\pi$
$812$	−964.235			−1.18748
$813$		0			0
$814$	−17.7562			−0.0218136
$815$	−328.366	+	1161.67i	−0.402903	+	1.42536i
$816$		0			0
$817$		−	30.6489i		−	0.0375139i
$818$	1449.42			1.77191
$819$		0			0
$820$	211.943	−	749.798i	0.258467	−	0.914388i
$821$		−	1035.02i		−	1.26068i	−0.776318	−	0.630342i	$-0.782915\pi$
$821$		−	1035.02i		−	1.26068i	0.776318	−	0.630342i	$-0.217085\pi$
$822$		0			0
$823$		−	1408.02i		−	1.71084i	−0.517933	−	0.855421i	$-0.673298\pi$
$823$		−	1408.02i		−	1.71084i	0.517933	−	0.855421i	$-0.326702\pi$
$824$		−	18.3218i		−	0.0222352i
$825$		0			0
$826$	1744.10			2.11150
$827$	1535.05			1.85617			0.928086	−	0.372367i	$-0.121454\pi$
$827$	1535.05			1.85617			0.928086	+	0.372367i	$0.121454\pi$
$828$		0			0
$829$	−185.325			−0.223552			−0.111776	−	0.993733i	$-0.535654\pi$
$829$	−185.325			−0.223552			−0.111776	+	0.993733i	$0.535654\pi$
$830$	1046.09	+	295.695i	1.26035	+	0.356260i
$831$		0			0
$832$			1244.58i			1.49589i
$833$	130.456			0.156610
$834$		0			0
$835$	−640.765	−	181.123i	−0.767383	−	0.216914i
$836$			0.926087i			0.00110776i
$837$		0			0
$838$		−	1981.10i		−	2.36408i
$839$		−	1293.87i		−	1.54216i	−0.636737	−	0.771081i	$-0.719716\pi$
$839$		−	1293.87i		−	1.54216i	0.636737	−	0.771081i	$-0.280284\pi$
$840$		0			0
$841$	249.452			0.296613
$842$	2150.59			2.55415
$843$		0			0
$844$	398.174			0.471770
$845$	878.739	+	248.391i	1.03993	+	0.293954i
$846$		0			0
$847$		−	1175.46i		−	1.38779i
$848$	−699.809			−0.825247
$849$		0			0
$850$	−172.940	−	106.259i	−0.203458	−	0.125010i
$851$			317.610i			0.373219i
$852$		0			0
$853$		−	387.663i		−	0.454471i	−0.973840	−	0.227235i	$-0.927031\pi$
$853$		−	387.663i		−	0.454471i	0.973840	−	0.227235i	$-0.0729686\pi$
$854$			612.782i			0.717543i
$855$		0			0
$856$	20.6392			0.0241112
$857$	79.1392			0.0923444			0.0461722	−	0.998933i	$-0.485298\pi$
$857$	79.1392			0.0923444			0.0461722	+	0.998933i	$0.485298\pi$
$858$		0			0
$859$	−659.058			−0.767239			−0.383620	−	0.923491i	$-0.625323\pi$
$859$	−659.058			−0.767239			−0.383620	+	0.923491i	$0.625323\pi$
$860$	−320.136	+	1132.55i	−0.372251	+	1.31692i
$861$		0			0
$862$			1414.77i			1.64127i
$863$	−1070.61			−1.24057			−0.620286	−	0.784375i	$-0.712984\pi$
$863$	−1070.61			−1.24057			−0.620286	+	0.784375i	$0.712984\pi$
$864$		0			0
$865$	−322.572	−	91.1805i	−0.372916	−	0.105411i
$866$			1198.15i			1.38355i
$867$		0			0
$868$			502.034i			0.578381i
$869$			21.5240i			0.0247687i
$870$		0			0
$871$	610.262			0.700645
$872$	34.9674			0.0401002
$873$		0			0
$874$	32.8261			0.0375585
$875$	−894.516	+	823.946i	−1.02230	+	0.941653i
$876$		0			0
$877$		−	1072.49i		−	1.22290i	−0.791281	−	0.611452i	$-0.790586\pi$
$877$		−	1072.49i		−	1.22290i	0.791281	−	0.611452i	$-0.209414\pi$
$878$	1300.00			1.48064
$879$		0			0
$880$	−9.14373	+	32.3480i	−0.0103906	+	0.0367591i
$881$			1433.62i			1.62726i	0.581381	+	0.813632i	$0.302513\pi$
$881$			1433.62i			1.62726i	−0.581381	+	0.813632i	$0.697487\pi$
$882$		0			0
$883$			731.816i			0.828784i	0.910099	+	0.414392i	$0.136006\pi$
$883$			731.816i			0.828784i	−0.910099	+	0.414392i	$0.863994\pi$
$884$			218.318i			0.246966i
$885$		0			0
$886$	660.026			0.744951
$887$	−1284.96			−1.44866			−0.724331	−	0.689452i	$-0.757851\pi$
$887$	−1284.96			−1.44866			−0.724331	+	0.689452i	$0.757851\pi$
$888$		0			0
$889$	765.912			0.861543
$890$	−111.918	+	395.935i	−0.125750	+	0.444871i
$891$		0			0
$892$		−	380.693i		−	0.426786i
$893$	26.3003			0.0294516
$894$		0			0
$895$	51.6291	−	182.650i	0.0576861	−	0.204078i
$896$			66.1706i			0.0738511i
$897$		0			0
$898$			1927.60i			2.14655i
$899$			307.993i			0.342595i
$900$		0			0
$901$	−127.395			−0.141392
$902$	−46.5510			−0.0516086
$903$		0			0
$904$	−9.09916			−0.0100654
$905$	−1337.65	−	378.109i	−1.47806	−	0.417801i
$906$		0			0
$907$		−	1012.85i		−	1.11671i	−0.829603	−	0.558353i	$-0.811433\pi$
$907$		−	1012.85i		−	1.11671i	0.829603	−	0.558353i	$-0.188567\pi$
$908$	806.770			0.888514
$909$		0			0
$910$	2494.43	+	705.092i	2.74113	+	0.774827i
$911$			636.509i			0.698693i	0.936994	+	0.349347i	$0.113596\pi$
$911$			636.509i			0.698693i	−0.936994	+	0.349347i	$0.886404\pi$
$912$		0			0
$913$		−	32.7739i		−	0.0358970i
$914$			2090.16i			2.28683i
$915$		0			0
$916$	738.516			0.806240
$917$	−1778.35			−1.93931
$918$		0			0
$919$	1142.09			1.24275			0.621377	−	0.783511i	$-0.286573\pi$
$919$	1142.09			1.24275			0.621377	+	0.783511i	$0.286573\pi$
$920$	−22.2690	−	6.29471i	−0.0242054	−	0.00684207i
$921$		0			0
$922$			1131.21i			1.22691i
$923$	1685.60			1.82622
$924$		0			0
$925$	−190.916	+	310.722i	−0.206396	+	0.335916i
$926$			456.338i			0.492806i
$927$		0			0
$928$			1105.42i			1.19119i
$929$		−	786.538i		−	0.846651i	−0.905978	−	0.423325i	$-0.860863\pi$
$929$		−	786.538i		−	0.846651i	0.905978	−	0.423325i	$-0.139137\pi$
$930$		0			0
$931$	−24.2252			−0.0260206
$932$	727.456			0.780532
$933$		0			0
$934$	−1155.57			−1.23723
$935$	−1.66454	+	5.88870i	−0.00178026	+	0.00629807i
$936$		0			0
$937$			811.089i			0.865623i	0.901484	+	0.432812i	$0.142478\pi$
$937$			811.089i			0.865623i	−0.901484	+	0.432812i	$0.857522\pi$
$938$	899.741			0.959213
$939$		0			0
$940$	−971.861	−	274.713i	−1.03389	−	0.292248i
$941$			826.195i			0.877996i	0.898488	+	0.438998i	$0.144667\pi$
$941$			826.195i			0.877996i	−0.898488	+	0.438998i	$0.855333\pi$
$942$		0			0
$943$			832.667i			0.882998i
$944$		−	990.124i		−	1.04886i
$945$		0			0
$946$	70.3142			0.0743279
$947$	−917.450			−0.968796			−0.484398	−	0.874848i	$-0.660961\pi$
$947$	−917.450			−0.968796			−0.484398	+	0.874848i	$0.660961\pi$
$948$		0			0
$949$	−2706.42			−2.85186
$950$	32.1143	+	19.7319i	0.0338045	+	0.0207704i
$951$		0			0
$952$			5.90919i			0.00620714i
$953$	192.772			0.202279			0.101140	−	0.994872i	$-0.467751\pi$
$953$	192.772			0.202279			0.101140	+	0.994872i	$0.467751\pi$
$954$		0			0
$955$	79.2986	−	280.537i	0.0830352	−	0.293756i
$956$		−	62.1821i		−	0.0650440i
$957$		0			0
$958$			2400.05i			2.50527i
$959$			54.1330i			0.0564473i
$960$		0			0
$961$	−800.642			−0.833134
$962$	777.308			0.808012
$963$		0			0
$964$	−1300.54			−1.34911
$965$	172.137	−	608.975i	0.178381	−	0.631063i
$966$		0			0
$967$			857.697i			0.886967i	0.896283	+	0.443483i	$0.146257\pi$
$967$			857.697i			0.886967i	−0.896283	+	0.443483i	$0.853743\pi$
$968$	25.6824			0.0265314
$969$		0			0
$970$	88.8501	−	314.328i	0.0915980	−	0.324049i
$971$		−	546.278i		−	0.562594i	−0.959621	−	0.281297i	$-0.909235\pi$
$971$		−	546.278i		−	0.562594i	0.959621	−	0.281297i	$-0.0907645\pi$
$972$		0			0
$973$			1653.20i			1.69908i
$974$			640.836i			0.657942i
$975$		0			0
$976$	347.877			0.356431
$977$	795.327			0.814050			0.407025	−	0.913417i	$-0.366566\pi$
$977$	795.327			0.814050			0.407025	+	0.913417i	$0.366566\pi$
$978$		0			0
$979$	12.4046			0.0126707
$980$	895.182	+	253.039i	0.913451	+	0.258203i
$981$		0			0
$982$			2174.27i			2.21412i
$983$	1079.42			1.09809			0.549044	−	0.835794i	$-0.314992\pi$
$983$	1079.42			1.09809			0.549044	+	0.835794i	$0.314992\pi$
$984$		0			0
$985$	1614.52	+	456.373i	1.63911	+	0.463322i
$986$			197.469i			0.200273i
$987$		0			0
$988$		−	40.5409i		−	0.0410333i
$989$		−	1257.72i		−	1.27171i
$990$		0			0
$991$	168.017			0.169543			0.0847716	−	0.996400i	$-0.472984\pi$
$991$	168.017			0.169543			0.0847716	+	0.996400i	$0.472984\pi$
$992$	575.545			0.580187
$993$		0			0
$994$	2485.17			2.50017
$995$	611.830	+	172.944i	0.614905	+	0.173813i
$996$		0			0
$997$		−	874.234i		−	0.876864i	−0.898764	−	0.438432i	$-0.855534\pi$
$997$		−	874.234i		−	0.876864i	0.898764	−	0.438432i	$-0.144466\pi$
$998$	−579.829			−0.580991
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	405.3.d.a.404.17		20
3.2	odd	2	inner	405.3.d.a.404.4		20
5.4	even	2	inner	405.3.d.a.404.3		20
9.2	odd	6		135.3.h.a.44.9		20
9.4	even	3		135.3.h.a.89.2		20
9.5	odd	6		45.3.h.a.29.9	yes	20
9.7	even	3		45.3.h.a.14.2	✓	20
15.14	odd	2	inner	405.3.d.a.404.18		20
45.2	even	12		675.3.j.e.476.2		20
45.4	even	6		135.3.h.a.89.9		20
45.7	odd	12		225.3.j.e.176.9		20
45.13	odd	12		675.3.j.e.251.9		20
45.14	odd	6		45.3.h.a.29.2	yes	20
45.22	odd	12		675.3.j.e.251.2		20
45.23	even	12		225.3.j.e.101.2		20
45.29	odd	6		135.3.h.a.44.2		20
45.32	even	12		225.3.j.e.101.9		20
45.34	even	6		45.3.h.a.14.9	yes	20
45.38	even	12		675.3.j.e.476.9		20
45.43	odd	12		225.3.j.e.176.2		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
45.3.h.a.14.2	✓	20	9.7	even	3
45.3.h.a.14.9	yes	20	45.34	even	6
45.3.h.a.29.2	yes	20	45.14	odd	6
45.3.h.a.29.9	yes	20	9.5	odd	6
135.3.h.a.44.2		20	45.29	odd	6
135.3.h.a.44.9		20	9.2	odd	6
135.3.h.a.89.2		20	9.4	even	3
135.3.h.a.89.9		20	45.4	even	6
225.3.j.e.101.2		20	45.23	even	12
225.3.j.e.101.9		20	45.32	even	12
225.3.j.e.176.2		20	45.43	odd	12
225.3.j.e.176.9		20	45.7	odd	12
405.3.d.a.404.3		20	5.4	even	2	inner
405.3.d.a.404.4		20	3.2	odd	2	inner
405.3.d.a.404.17		20	1.1	even	1	trivial
405.3.d.a.404.18		20	15.14	odd	2	inner
675.3.j.e.251.2		20	45.22	odd	12
675.3.j.e.251.9		20	45.13	odd	12
675.3.j.e.476.2		20	45.2	even	12
675.3.j.e.476.9		20	45.38	even	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)

\(f(q)\)	\(=\)	\(q+2.84162 q^{2} +4.07481 q^{4} +(-4.81147 - 1.36005i) q^{5} -9.72928i q^{7} +0.212574 q^{8} +(-13.6724 - 3.86473i) q^{10} +0.428355i q^{11} -18.7519i q^{13} -27.6469i q^{14} -15.6952 q^{16} -2.85718 q^{17} +0.530568 q^{19} +(-19.6058 - 5.54192i) q^{20} +1.21722i q^{22} +21.7727 q^{23} +(21.3006 + 13.0876i) q^{25} -53.2858i q^{26} -39.6450i q^{28} -24.3218i q^{29} -12.6633 q^{31} -45.4500 q^{32} -8.11902 q^{34} +(-13.2323 + 46.8122i) q^{35} +14.5875i q^{37} +1.50767 q^{38} +(-1.02279 - 0.289110i) q^{40} +38.2436i q^{41} -57.7661i q^{43} +1.74546i q^{44} +61.8697 q^{46} +49.5700 q^{47} -45.6590 q^{49} +(60.5281 + 37.1901i) q^{50} -76.4104i q^{52} +44.5876 q^{53} +(0.582582 - 2.06102i) q^{55} -2.06819i q^{56} -69.1132i q^{58} +63.0846i q^{59} -22.1646 q^{61} -35.9842 q^{62} -66.3710 q^{64} +(-25.5035 + 90.2243i) q^{65} +32.5440i q^{67} -11.6425 q^{68} +(-37.6011 + 133.022i) q^{70} +89.8896i q^{71} -144.328i q^{73} +41.4522i q^{74} +2.16196 q^{76} +4.16759 q^{77} +50.2480 q^{79} +(75.5169 + 21.3462i) q^{80} +108.674i q^{82} -76.5112 q^{83} +(13.7472 + 3.88589i) q^{85} -164.149i q^{86} +0.0910570i q^{88} -28.9588i q^{89} -182.443 q^{91} +88.7195 q^{92} +140.859 q^{94} +(-2.55282 - 0.721597i) q^{95} +22.9900i q^{97} -129.746 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 36 q^{4} + 4 q^{10} + 52 q^{16} - 8 q^{19} - 4 q^{25} - 56 q^{31} + 8 q^{34} + 68 q^{40} + 116 q^{46} + 80 q^{49} + 36 q^{55} + 100 q^{61} + 140 q^{64} + 108 q^{70} + 192 q^{76} + 256 q^{79} + 148 q^{85}+ \cdots - 436 q^{94}+O(q^{100}) \) 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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