LMFDB - Embedded newform 432.2.s.e.143.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [432,2,Mod(143,432)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(432, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 0, 1]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("432.143");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 432 = 2^{4} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	432.s (of order $6$, degree $2$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$3.44953736732$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 3^{2} $
Twist minimal:	no (minimal twist has level 144)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			143.1
Root			$-0.866025 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	432.143
Dual form			432.2.s.e.287.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.50000 + 0.866025i) q^{5} +(-2.59808 - 1.50000i) q^{7} +O(q^{10})$	$q+(-1.50000 + 0.866025i) q^{5} +(-2.59808 - 1.50000i) q^{7} +(2.59808 - 4.50000i) q^{11} +(-0.500000 - 0.866025i) q^{13} -3.46410i q^{17} -6.00000i q^{19} +(-2.59808 - 4.50000i) q^{23} +(-1.00000 + 1.73205i) q^{25} +(7.50000 + 4.33013i) q^{29} +(-2.59808 + 1.50000i) q^{31} +5.19615 q^{35} -4.00000 q^{37} +(-4.50000 + 2.59808i) q^{41} +(-2.59808 - 1.50000i) q^{43} +(-2.59808 + 4.50000i) q^{47} +(1.00000 + 1.73205i) q^{49} -10.3923i q^{53} +9.00000i q^{55} +(2.59808 + 4.50000i) q^{59} +(3.50000 - 6.06218i) q^{61} +(1.50000 + 0.866025i) q^{65} +(7.79423 - 4.50000i) q^{67} -10.3923 q^{71} +4.00000 q^{73} +(-13.5000 + 7.79423i) q^{77} +(12.9904 + 7.50000i) q^{79} +(-2.59808 + 4.50000i) q^{83} +(3.00000 + 5.19615i) q^{85} +3.46410i q^{89} +3.00000i q^{91} +(5.19615 + 9.00000i) q^{95} +(-0.500000 + 0.866025i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 6 q^{5}+O(q^{10}) $ 4 * q - 6 * q^5	$ 4 q - 6 q^{5} - 2 q^{13} - 4 q^{25} + 30 q^{29} - 16 q^{37} - 18 q^{41} + 4 q^{49} + 14 q^{61} + 6 q^{65} + 16 q^{73} - 54 q^{77} + 12 q^{85} - 2 q^{97}+O(q^{100}) $ 4 * q - 6 * q^5 - 2 * q^13 - 4 * q^25 + 30 * q^29 - 16 * q^37 - 18 * q^41 + 4 * q^49 + 14 * q^61 + 6 * q^65 + 16 * q^73 - 54 * q^77 + 12 * q^85 - 2 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/432\mathbb{Z}\right)^\times$.

$n$	$271$	$325$	$353$
$\chi(n)$	$-1$	$1$	$e\left(\frac{1}{6}\right)$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−1.50000	+	0.866025i	−0.670820	+	0.387298i	−0.796387	−	0.604787i	$-0.793258\pi$
$5$	−1.50000	+	0.866025i	−0.670820	+	0.387298i	0.125567	+	0.992085i	$0.459925\pi$
$6$		0			0
$7$	−2.59808	−	1.50000i	−0.981981	−	0.566947i	−0.0791130	−	0.996866i	$-0.525209\pi$
$7$	−2.59808	−	1.50000i	−0.981981	−	0.566947i	−0.902867	+	0.429919i	$0.858542\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	2.59808	−	4.50000i	0.783349	−	1.35680i	−0.146631	−	0.989191i	$-0.546843\pi$
$11$	2.59808	−	4.50000i	0.783349	−	1.35680i	0.929980	−	0.367610i	$-0.119824\pi$
$12$		0			0
$13$	−0.500000	−	0.866025i	−0.138675	−	0.240192i	0.788320	−	0.615265i	$-0.210951\pi$
$13$	−0.500000	−	0.866025i	−0.138675	−	0.240192i	−0.926995	+	0.375073i	$0.877618\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	3.46410i		−	0.840168i	−0.907485	−	0.420084i	$-0.862001\pi$
$17$		−	3.46410i		−	0.840168i	0.907485	−	0.420084i	$-0.137999\pi$
$18$		0			0
$19$		−	6.00000i		−	1.37649i	−0.725476	−	0.688247i	$-0.758380\pi$
$19$		−	6.00000i		−	1.37649i	0.725476	−	0.688247i	$-0.241620\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−2.59808	−	4.50000i	−0.541736	−	0.938315i	−0.998805	−	0.0488832i	$-0.984434\pi$
$23$	−2.59808	−	4.50000i	−0.541736	−	0.938315i	0.457068	−	0.889432i	$-0.348900\pi$
$24$		0			0
$25$	−1.00000	+	1.73205i	−0.200000	+	0.346410i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	7.50000	+	4.33013i	1.39272	+	0.804084i	0.993615	−	0.112823i	$-0.0359893\pi$
$29$	7.50000	+	4.33013i	1.39272	+	0.804084i	0.399100	+	0.916907i	$0.369323\pi$
$30$		0			0
$31$	−2.59808	+	1.50000i	−0.466628	+	0.269408i	−0.714827	−	0.699301i	$-0.753495\pi$
$31$	−2.59808	+	1.50000i	−0.466628	+	0.269408i	0.248199	+	0.968709i	$0.420161\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	5.19615			0.878310
$36$		0			0
$37$	−4.00000			−0.657596			−0.328798	−	0.944400i	$-0.606644\pi$
$37$	−4.00000			−0.657596			−0.328798	+	0.944400i	$0.606644\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−4.50000	+	2.59808i	−0.702782	+	0.405751i	−0.808383	−	0.588657i	$-0.799657\pi$
$41$	−4.50000	+	2.59808i	−0.702782	+	0.405751i	0.105601	+	0.994409i	$0.466323\pi$
$42$		0			0
$43$	−2.59808	−	1.50000i	−0.396203	−	0.228748i	0.288641	−	0.957437i	$-0.406796\pi$
$43$	−2.59808	−	1.50000i	−0.396203	−	0.228748i	−0.684844	+	0.728689i	$0.740130\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−2.59808	+	4.50000i	−0.378968	+	0.656392i	−0.990912	−	0.134509i	$-0.957054\pi$
$47$	−2.59808	+	4.50000i	−0.378968	+	0.656392i	0.611944	+	0.790901i	$0.290388\pi$
$48$		0			0
$49$	1.00000	+	1.73205i	0.142857	+	0.247436i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	10.3923i		−	1.42749i	−0.700404	−	0.713746i	$-0.746997\pi$
$53$		−	10.3923i		−	1.42749i	0.700404	−	0.713746i	$-0.253003\pi$
$54$		0			0
$55$			9.00000i			1.21356i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	2.59808	+	4.50000i	0.338241	+	0.585850i	0.984102	−	0.177605i	$-0.0568349\pi$
$59$	2.59808	+	4.50000i	0.338241	+	0.585850i	−0.645861	+	0.763455i	$0.723502\pi$
$60$		0			0
$61$	3.50000	−	6.06218i	0.448129	−	0.776182i	−0.550135	−	0.835076i	$-0.685424\pi$
$61$	3.50000	−	6.06218i	0.448129	−	0.776182i	0.998264	+	0.0588933i	$0.0187572\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	1.50000	+	0.866025i	0.186052	+	0.107417i
$66$		0			0
$67$	7.79423	−	4.50000i	0.952217	−	0.549762i	0.0584478	−	0.998290i	$-0.481385\pi$
$67$	7.79423	−	4.50000i	0.952217	−	0.549762i	0.893769	+	0.448528i	$0.148052\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−10.3923			−1.23334			−0.616670	−	0.787222i	$-0.711519\pi$
$71$	−10.3923			−1.23334			−0.616670	+	0.787222i	$0.711519\pi$
$72$		0			0
$73$	4.00000			0.468165			0.234082	−	0.972217i	$-0.424791\pi$
$73$	4.00000			0.468165			0.234082	+	0.972217i	$0.424791\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−13.5000	+	7.79423i	−1.53847	+	0.888235i
$78$		0			0
$79$	12.9904	+	7.50000i	1.46153	+	0.843816i	0.999082	−	0.0428296i	$-0.0136373\pi$
$79$	12.9904	+	7.50000i	1.46153	+	0.843816i	0.462450	+	0.886646i	$0.346971\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−2.59808	+	4.50000i	−0.285176	+	0.493939i	−0.972652	−	0.232268i	$-0.925385\pi$
$83$	−2.59808	+	4.50000i	−0.285176	+	0.493939i	0.687476	+	0.726207i	$0.258719\pi$
$84$		0			0
$85$	3.00000	+	5.19615i	0.325396	+	0.563602i
$86$		0			0
$87$		0			0
$88$		0			0
$89$			3.46410i			0.367194i	0.983002	+	0.183597i	$0.0587741\pi$
$89$			3.46410i			0.367194i	−0.983002	+	0.183597i	$0.941226\pi$
$90$		0			0
$91$			3.00000i			0.314485i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	5.19615	+	9.00000i	0.533114	+	0.923381i
$96$		0			0
$97$	−0.500000	+	0.866025i	−0.0507673	+	0.0879316i	−0.890292	−	0.455389i	$-0.849500\pi$
$97$	−0.500000	+	0.866025i	−0.0507673	+	0.0879316i	0.839525	+	0.543321i	$0.182833\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	4.50000	+	2.59808i	0.447767	+	0.258518i	0.706887	−	0.707327i	$-0.250099\pi$
$101$	4.50000	+	2.59808i	0.447767	+	0.258518i	−0.259120	+	0.965845i	$0.583432\pi$
$102$		0			0
$103$	−7.79423	+	4.50000i	−0.767988	+	0.443398i	−0.832156	−	0.554541i	$-0.812894\pi$
$103$	−7.79423	+	4.50000i	−0.767988	+	0.443398i	0.0641683	+	0.997939i	$0.479561\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0			−	1.00000i	$-0.5\pi$
$107$		0			0				1.00000i	$0.5\pi$
$108$		0			0
$109$	4.00000			0.383131			0.191565	−	0.981480i	$-0.438644\pi$
$109$	4.00000			0.383131			0.191565	+	0.981480i	$0.438644\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	10.5000	−	6.06218i	0.987757	−	0.570282i	0.0831539	−	0.996537i	$-0.473501\pi$
$113$	10.5000	−	6.06218i	0.987757	−	0.570282i	0.904603	+	0.426255i	$0.140167\pi$
$114$		0			0
$115$	7.79423	+	4.50000i	0.726816	+	0.419627i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−5.19615	+	9.00000i	−0.476331	+	0.825029i
$120$		0			0
$121$	−8.00000	−	13.8564i	−0.727273	−	1.25967i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		−	12.1244i		−	1.08444i
$126$		0			0
$127$			6.00000i			0.532414i	0.963916	+	0.266207i	$0.0857705\pi$
$127$			6.00000i			0.532414i	−0.963916	+	0.266207i	$0.914230\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	2.59808	+	4.50000i	0.226995	+	0.393167i	0.956916	−	0.290365i	$-0.0937766\pi$
$131$	2.59808	+	4.50000i	0.226995	+	0.393167i	−0.729921	+	0.683531i	$0.760443\pi$
$132$		0			0
$133$	−9.00000	+	15.5885i	−0.780399	+	1.35169i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	1.50000	+	0.866025i	0.128154	+	0.0739895i	0.562706	−	0.826657i	$-0.309760\pi$
$137$	1.50000	+	0.866025i	0.128154	+	0.0739895i	−0.434553	+	0.900646i	$0.643094\pi$
$138$		0			0
$139$	2.59808	−	1.50000i	0.220366	−	0.127228i	−0.385754	−	0.922602i	$-0.626059\pi$
$139$	2.59808	−	1.50000i	0.220366	−	0.127228i	0.606120	+	0.795373i	$0.292725\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−5.19615			−0.434524
$144$		0			0
$145$	−15.0000			−1.24568
$146$		0			0
$147$		0			0
$148$		0			0
$149$	1.50000	−	0.866025i	0.122885	−	0.0709476i	−0.437298	−	0.899317i	$-0.644064\pi$
$149$	1.50000	−	0.866025i	0.122885	−	0.0709476i	0.560182	+	0.828369i	$0.310731\pi$
$150$		0			0
$151$	−7.79423	−	4.50000i	−0.634285	−	0.366205i	0.148124	−	0.988969i	$-0.452676\pi$
$151$	−7.79423	−	4.50000i	−0.634285	−	0.366205i	−0.782410	+	0.622764i	$0.786010\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	2.59808	−	4.50000i	0.208683	−	0.361449i
$156$		0			0
$157$	0.500000	+	0.866025i	0.0399043	+	0.0691164i	0.885288	−	0.465044i	$-0.153961\pi$
$157$	0.500000	+	0.866025i	0.0399043	+	0.0691164i	−0.845383	+	0.534160i	$0.820628\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			15.5885i			1.22854i
$162$		0			0
$163$		−	6.00000i		−	0.469956i	−0.972001	−	0.234978i	$-0.924498\pi$
$163$		−	6.00000i		−	0.469956i	0.972001	−	0.234978i	$-0.0755019\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−7.79423	−	13.5000i	−0.603136	−	1.04466i	−0.992343	−	0.123511i	$-0.960584\pi$
$167$	−7.79423	−	13.5000i	−0.603136	−	1.04466i	0.389208	−	0.921150i	$-0.372749\pi$
$168$		0			0
$169$	6.00000	−	10.3923i	0.461538	−	0.799408i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−1.50000	−	0.866025i	−0.114043	−	0.0658427i	0.441894	−	0.897067i	$-0.354307\pi$
$173$	−1.50000	−	0.866025i	−0.114043	−	0.0658427i	−0.555936	+	0.831225i	$0.687640\pi$
$174$		0			0
$175$	5.19615	−	3.00000i	0.392792	−	0.226779i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	20.7846			1.55351			0.776757	−	0.629800i	$-0.216863\pi$
$179$	20.7846			1.55351			0.776757	+	0.629800i	$0.216863\pi$
$180$		0			0
$181$	16.0000			1.18927			0.594635	−	0.803996i	$-0.297296\pi$
$181$	16.0000			1.18927			0.594635	+	0.803996i	$0.297296\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	6.00000	−	3.46410i	0.441129	−	0.254686i
$186$		0			0
$187$	−15.5885	−	9.00000i	−1.13994	−	0.658145i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	7.79423	−	13.5000i	0.563971	−	0.976826i	−0.433174	−	0.901310i	$-0.642606\pi$
$191$	7.79423	−	13.5000i	0.563971	−	0.976826i	0.997145	−	0.0755154i	$-0.0240602\pi$
$192$		0			0
$193$	9.50000	+	16.4545i	0.683825	+	1.18442i	0.973805	+	0.227387i	$0.0730182\pi$
$193$	9.50000	+	16.4545i	0.683825	+	1.18442i	−0.289980	+	0.957033i	$0.593649\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			6.92820i			0.493614i	0.969065	+	0.246807i	$0.0793814\pi$
$197$			6.92820i			0.493614i	−0.969065	+	0.246807i	$0.920619\pi$
$198$		0			0
$199$		−	12.0000i		−	0.850657i	−0.905039	−	0.425329i	$-0.860158\pi$
$199$		−	12.0000i		−	0.850657i	0.905039	−	0.425329i	$-0.139842\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−12.9904	−	22.5000i	−0.911746	−	1.57919i
$204$		0			0
$205$	4.50000	−	7.79423i	0.314294	−	0.544373i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−27.0000	−	15.5885i	−1.86763	−	1.07828i
$210$		0			0
$211$	−2.59808	+	1.50000i	−0.178859	+	0.103264i	−0.586756	−	0.809763i	$-0.699595\pi$
$211$	−2.59808	+	1.50000i	−0.178859	+	0.103264i	0.407898	+	0.913028i	$0.366262\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	5.19615			0.354375
$216$		0			0
$217$	9.00000			0.610960
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−3.00000	+	1.73205i	−0.201802	+	0.116510i
$222$		0			0
$223$	−12.9904	−	7.50000i	−0.869900	−	0.502237i	−0.00258516	−	0.999997i	$-0.500823\pi$
$223$	−12.9904	−	7.50000i	−0.869900	−	0.502237i	−0.867315	+	0.497760i	$0.834156\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	2.59808	−	4.50000i	0.172440	−	0.298675i	−0.766832	−	0.641848i	$-0.778168\pi$
$227$	2.59808	−	4.50000i	0.172440	−	0.298675i	0.939272	+	0.343172i	$0.111501\pi$
$228$		0			0
$229$	−9.50000	−	16.4545i	−0.627778	−	1.08734i	−0.987997	−	0.154475i	$-0.950631\pi$
$229$	−9.50000	−	16.4545i	−0.627778	−	1.08734i	0.360219	−	0.932868i	$-0.382702\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			13.8564i			0.907763i	0.891062	+	0.453882i	$0.149961\pi$
$233$			13.8564i			0.907763i	−0.891062	+	0.453882i	$0.850039\pi$
$234$		0			0
$235$		−	9.00000i		−	0.587095i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	7.79423	+	13.5000i	0.504167	+	0.873242i	0.999988	+	0.00481804i	$0.00153363\pi$
$239$	7.79423	+	13.5000i	0.504167	+	0.873242i	−0.495822	+	0.868424i	$0.665133\pi$
$240$		0			0
$241$	−3.50000	+	6.06218i	−0.225455	+	0.390499i	−0.956456	−	0.291877i	$-0.905720\pi$
$241$	−3.50000	+	6.06218i	−0.225455	+	0.390499i	0.731001	+	0.682376i	$0.239053\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−3.00000	−	1.73205i	−0.191663	−	0.110657i
$246$		0			0
$247$	−5.19615	+	3.00000i	−0.330623	+	0.190885i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−20.7846			−1.31191			−0.655956	−	0.754799i	$-0.727735\pi$
$251$	−20.7846			−1.31191			−0.655956	+	0.754799i	$0.727735\pi$
$252$		0			0
$253$	−27.0000			−1.69748
$254$		0			0
$255$		0			0
$256$		0			0
$257$	16.5000	−	9.52628i	1.02924	−	0.594233i	0.112474	−	0.993655i	$-0.464122\pi$
$257$	16.5000	−	9.52628i	1.02924	−	0.594233i	0.916767	+	0.399422i	$0.130789\pi$
$258$		0			0
$259$	10.3923	+	6.00000i	0.645746	+	0.372822i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	12.9904	−	22.5000i	0.801021	−	1.38741i	−0.117923	−	0.993023i	$-0.537624\pi$
$263$	12.9904	−	22.5000i	0.801021	−	1.38741i	0.918945	−	0.394387i	$-0.129043\pi$
$264$		0			0
$265$	9.00000	+	15.5885i	0.552866	+	0.957591i
$266$		0			0
$267$		0			0
$268$		0			0
$269$			3.46410i			0.211210i	0.994408	+	0.105605i	$0.0336779\pi$
$269$			3.46410i			0.211210i	−0.994408	+	0.105605i	$0.966322\pi$
$270$		0			0
$271$		−	18.0000i		−	1.09342i	−0.837321	−	0.546711i	$-0.815880\pi$
$271$		−	18.0000i		−	1.09342i	0.837321	−	0.546711i	$-0.184120\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	5.19615	+	9.00000i	0.313340	+	0.542720i
$276$		0			0
$277$	8.50000	−	14.7224i	0.510716	−	0.884585i	−0.489207	−	0.872167i	$-0.662714\pi$
$277$	8.50000	−	14.7224i	0.510716	−	0.884585i	0.999923	−	0.0124177i	$-0.00395278\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−10.5000	−	6.06218i	−0.626377	−	0.361639i	0.152970	−	0.988231i	$-0.451116\pi$
$281$	−10.5000	−	6.06218i	−0.626377	−	0.361639i	−0.779348	+	0.626592i	$0.784449\pi$
$282$		0			0
$283$	−28.5788	+	16.5000i	−1.69884	+	0.980823i	−0.751968	+	0.659200i	$0.770895\pi$
$283$	−28.5788	+	16.5000i	−1.69884	+	0.980823i	−0.946868	+	0.321624i	$0.895771\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	15.5885			0.920158
$288$		0			0
$289$	5.00000			0.294118
$290$		0			0
$291$		0			0
$292$		0			0
$293$	16.5000	−	9.52628i	0.963940	−	0.556531i	0.0665568	−	0.997783i	$-0.478799\pi$
$293$	16.5000	−	9.52628i	0.963940	−	0.556531i	0.897384	+	0.441251i	$0.145465\pi$
$294$		0			0
$295$	−7.79423	−	4.50000i	−0.453798	−	0.262000i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−2.59808	+	4.50000i	−0.150251	+	0.260242i
$300$		0			0
$301$	4.50000	+	7.79423i	0.259376	+	0.449252i
$302$		0			0
$303$		0			0
$304$		0			0
$305$			12.1244i			0.694239i
$306$		0			0
$307$			18.0000i			1.02731i	0.857996	+	0.513657i	$0.171710\pi$
$307$			18.0000i			1.02731i	−0.857996	+	0.513657i	$0.828290\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−2.59808	−	4.50000i	−0.147323	−	0.255172i	0.782914	−	0.622130i	$-0.213732\pi$
$311$	−2.59808	−	4.50000i	−0.147323	−	0.255172i	−0.930237	+	0.366958i	$0.880399\pi$
$312$		0			0
$313$	−5.50000	+	9.52628i	−0.310878	+	0.538457i	−0.978553	−	0.205996i	$-0.933957\pi$
$313$	−5.50000	+	9.52628i	−0.310878	+	0.538457i	0.667674	+	0.744453i	$0.267290\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	19.5000	+	11.2583i	1.09523	+	0.632331i	0.934964	−	0.354743i	$-0.115432\pi$
$317$	19.5000	+	11.2583i	1.09523	+	0.632331i	0.160265	+	0.987074i	$0.448765\pi$
$318$		0			0
$319$	38.9711	−	22.5000i	2.18197	−	1.25976i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−20.7846			−1.15649
$324$		0			0
$325$	2.00000			0.110940
$326$		0			0
$327$		0			0
$328$		0			0
$329$	13.5000	−	7.79423i	0.744279	−	0.429710i
$330$		0			0
$331$	2.59808	+	1.50000i	0.142803	+	0.0824475i	0.569699	−	0.821853i	$-0.307060\pi$
$331$	2.59808	+	1.50000i	0.142803	+	0.0824475i	−0.426896	+	0.904301i	$0.640393\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−7.79423	+	13.5000i	−0.425844	+	0.737584i
$336$		0			0
$337$	−15.5000	−	26.8468i	−0.844339	−	1.46244i	−0.886194	−	0.463314i	$-0.846660\pi$
$337$	−15.5000	−	26.8468i	−0.844339	−	1.46244i	0.0418554	−	0.999124i	$-0.486673\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			15.5885i			0.844162i
$342$		0			0
$343$			15.0000i			0.809924i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−2.59808	−	4.50000i	−0.139472	−	0.241573i	0.787825	−	0.615899i	$-0.211207\pi$
$347$	−2.59808	−	4.50000i	−0.139472	−	0.241573i	−0.927297	+	0.374327i	$0.877874\pi$
$348$		0			0
$349$	−17.5000	+	30.3109i	−0.936754	+	1.62250i	−0.165277	+	0.986247i	$0.552852\pi$
$349$	−17.5000	+	30.3109i	−0.936754	+	1.62250i	−0.771477	+	0.636257i	$0.780482\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−22.5000	−	12.9904i	−1.19755	−	0.691408i	−0.237545	−	0.971377i	$-0.576343\pi$
$353$	−22.5000	−	12.9904i	−1.19755	−	0.691408i	−0.960009	+	0.279968i	$0.909676\pi$
$354$		0			0
$355$	15.5885	−	9.00000i	0.827349	−	0.477670i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	20.7846			1.09697			0.548485	−	0.836160i	$-0.315205\pi$
$359$	20.7846			1.09697			0.548485	+	0.836160i	$0.315205\pi$
$360$		0			0
$361$	−17.0000			−0.894737
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−6.00000	+	3.46410i	−0.314054	+	0.181319i
$366$		0			0
$367$	23.3827	+	13.5000i	1.22057	+	0.704694i	0.965039	−	0.262108i	$-0.0844175\pi$
$367$	23.3827	+	13.5000i	1.22057	+	0.704694i	0.255528	+	0.966802i	$0.417751\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−15.5885	+	27.0000i	−0.809312	+	1.40177i
$372$		0			0
$373$	−2.50000	−	4.33013i	−0.129445	−	0.224205i	0.794017	−	0.607896i	$-0.207986\pi$
$373$	−2.50000	−	4.33013i	−0.129445	−	0.224205i	−0.923462	+	0.383691i	$0.874653\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	8.66025i		−	0.446026i
$378$		0			0
$379$			24.0000i			1.23280i	0.787434	+	0.616399i	$0.211409\pi$
$379$			24.0000i			1.23280i	−0.787434	+	0.616399i	$0.788591\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	2.59808	+	4.50000i	0.132755	+	0.229939i	0.924738	−	0.380605i	$-0.124284\pi$
$383$	2.59808	+	4.50000i	0.132755	+	0.229939i	−0.791982	+	0.610544i	$0.790951\pi$
$384$		0			0
$385$	13.5000	−	23.3827i	0.688024	−	1.19169i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−13.5000	−	7.79423i	−0.684477	−	0.395183i	0.117063	−	0.993125i	$-0.462652\pi$
$389$	−13.5000	−	7.79423i	−0.684477	−	0.395183i	−0.801540	+	0.597941i	$0.795986\pi$
$390$		0			0
$391$	−15.5885	+	9.00000i	−0.788342	+	0.455150i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−25.9808			−1.30723
$396$		0			0
$397$	32.0000			1.60603			0.803017	−	0.595956i	$-0.203227\pi$
$397$	32.0000			1.60603			0.803017	+	0.595956i	$0.203227\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	7.50000	−	4.33013i	0.374532	−	0.216236i	−0.300904	−	0.953654i	$-0.597289\pi$
$401$	7.50000	−	4.33013i	0.374532	−	0.216236i	0.675437	+	0.737418i	$0.263955\pi$
$402$		0			0
$403$	2.59808	+	1.50000i	0.129419	+	0.0747203i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−10.3923	+	18.0000i	−0.515127	+	0.892227i
$408$		0			0
$409$	8.50000	+	14.7224i	0.420298	+	0.727977i	0.995968	−	0.0897044i	$-0.0285922\pi$
$409$	8.50000	+	14.7224i	0.420298	+	0.727977i	−0.575670	+	0.817682i	$0.695259\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		−	15.5885i		−	0.767058i
$414$		0			0
$415$		−	9.00000i		−	0.441793i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	18.1865	+	31.5000i	0.888470	+	1.53888i	0.841684	+	0.539971i	$0.181565\pi$
$419$	18.1865	+	31.5000i	0.888470	+	1.53888i	0.0467865	+	0.998905i	$0.485102\pi$
$420$		0			0
$421$	5.50000	−	9.52628i	0.268054	−	0.464282i	−0.700306	−	0.713843i	$-0.746953\pi$
$421$	5.50000	−	9.52628i	0.268054	−	0.464282i	0.968359	+	0.249561i	$0.0802862\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	6.00000	+	3.46410i	0.291043	+	0.168034i
$426$		0			0
$427$	−18.1865	+	10.5000i	−0.880108	+	0.508131i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	20.7846			1.00116			0.500580	−	0.865690i	$-0.333120\pi$
$431$	20.7846			1.00116			0.500580	+	0.865690i	$0.333120\pi$
$432$		0			0
$433$	32.0000			1.53782			0.768911	−	0.639356i	$-0.220799\pi$
$433$	32.0000			1.53782			0.768911	+	0.639356i	$0.220799\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−27.0000	+	15.5885i	−1.29159	+	0.745697i
$438$		0			0
$439$	23.3827	+	13.5000i	1.11599	+	0.644320i	0.940375	−	0.340138i	$-0.110474\pi$
$439$	23.3827	+	13.5000i	1.11599	+	0.644320i	0.175619	+	0.984458i	$0.443807\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−2.59808	+	4.50000i	−0.123438	+	0.213801i	−0.921121	−	0.389275i	$-0.872725\pi$
$443$	−2.59808	+	4.50000i	−0.123438	+	0.213801i	0.797683	+	0.603077i	$0.206059\pi$
$444$		0			0
$445$	−3.00000	−	5.19615i	−0.142214	−	0.246321i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			31.1769i			1.47133i	0.677346	+	0.735665i	$0.263130\pi$
$449$			31.1769i			1.47133i	−0.677346	+	0.735665i	$0.736870\pi$
$450$		0			0
$451$			27.0000i			1.27138i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−2.59808	−	4.50000i	−0.121800	−	0.210963i
$456$		0			0
$457$	−12.5000	+	21.6506i	−0.584725	+	1.01277i	0.410184	+	0.912003i	$0.365464\pi$
$457$	−12.5000	+	21.6506i	−0.584725	+	1.01277i	−0.994910	+	0.100771i	$0.967869\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−4.50000	−	2.59808i	−0.209586	−	0.121004i	0.391533	−	0.920164i	$-0.371945\pi$
$461$	−4.50000	−	2.59808i	−0.209586	−	0.121004i	−0.601119	+	0.799160i	$0.705278\pi$
$462$		0			0
$463$	−2.59808	+	1.50000i	−0.120743	+	0.0697109i	−0.559155	−	0.829063i	$-0.688874\pi$
$463$	−2.59808	+	1.50000i	−0.120743	+	0.0697109i	0.438412	+	0.898774i	$0.355541\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0			−	1.00000i	$-0.5\pi$
$467$		0			0				1.00000i	$0.5\pi$
$468$		0			0
$469$	−27.0000			−1.24674
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−13.5000	+	7.79423i	−0.620731	+	0.358379i
$474$		0			0
$475$	10.3923	+	6.00000i	0.476832	+	0.275299i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−7.79423	+	13.5000i	−0.356127	+	0.616831i	−0.987310	−	0.158803i	$-0.949236\pi$
$479$	−7.79423	+	13.5000i	−0.356127	+	0.616831i	0.631183	+	0.775634i	$0.282570\pi$
$480$		0			0
$481$	2.00000	+	3.46410i	0.0911922	+	0.157949i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		−	1.73205i		−	0.0786484i
$486$		0			0
$487$		−	30.0000i		−	1.35943i	−0.733476	−	0.679715i	$-0.762104\pi$
$487$		−	30.0000i		−	1.35943i	0.733476	−	0.679715i	$-0.237896\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−7.79423	−	13.5000i	−0.351749	−	0.609246i	0.634807	−	0.772670i	$-0.281079\pi$
$491$	−7.79423	−	13.5000i	−0.351749	−	0.609246i	−0.986556	+	0.163424i	$0.947746\pi$
$492$		0			0
$493$	15.0000	−	25.9808i	0.675566	−	1.17011i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	27.0000	+	15.5885i	1.21112	+	0.699238i
$498$		0			0
$499$	23.3827	−	13.5000i	1.04675	−	0.604343i	0.125014	−	0.992155i	$-0.460102\pi$
$499$	23.3827	−	13.5000i	1.04675	−	0.604343i	0.921739	+	0.387812i	$0.126769\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−31.1769			−1.39011			−0.695055	−	0.718957i	$-0.744620\pi$
$503$	−31.1769			−1.39011			−0.695055	+	0.718957i	$0.744620\pi$
$504$		0			0
$505$	−9.00000			−0.400495
$506$		0			0
$507$		0			0
$508$		0			0
$509$	13.5000	−	7.79423i	0.598377	−	0.345473i	−0.170026	−	0.985440i	$-0.554385\pi$
$509$	13.5000	−	7.79423i	0.598377	−	0.345473i	0.768403	+	0.639966i	$0.221052\pi$
$510$		0			0
$511$	−10.3923	−	6.00000i	−0.459728	−	0.265424i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	7.79423	−	13.5000i	0.343455	−	0.594881i
$516$		0			0
$517$	13.5000	+	23.3827i	0.593729	+	1.02837i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	41.5692i		−	1.82118i	−0.413310	−	0.910590i	$-0.635627\pi$
$521$		−	41.5692i		−	1.82118i	0.413310	−	0.910590i	$-0.364373\pi$
$522$		0			0
$523$		0			0		1.00000			$0$
$523$		0			0		−1.00000			$\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	5.19615	+	9.00000i	0.226348	+	0.392046i
$528$		0			0
$529$	−2.00000	+	3.46410i	−0.0869565	+	0.150613i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	4.50000	+	2.59808i	0.194917	+	0.112535i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$	10.3923			0.447628
$540$		0			0
$541$	−14.0000			−0.601907			−0.300954	−	0.953639i	$-0.597305\pi$
$541$	−14.0000			−0.601907			−0.300954	+	0.953639i	$0.597305\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−6.00000	+	3.46410i	−0.257012	+	0.148386i
$546$		0			0
$547$	−7.79423	−	4.50000i	−0.333257	−	0.192406i	0.324029	−	0.946047i	$-0.394962\pi$
$547$	−7.79423	−	4.50000i	−0.333257	−	0.192406i	−0.657286	+	0.753641i	$0.728296\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	25.9808	−	45.0000i	1.10682	−	1.91706i
$552$		0			0
$553$	−22.5000	−	38.9711i	−0.956797	−	1.65722i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	45.0333i		−	1.90812i	−0.299611	−	0.954062i	$-0.596857\pi$
$557$		−	45.0333i		−	1.90812i	0.299611	−	0.954062i	$-0.403143\pi$
$558$		0			0
$559$			3.00000i			0.126886i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−12.9904	−	22.5000i	−0.547479	−	0.948262i	−0.998446	−	0.0557214i	$-0.982254\pi$
$563$	−12.9904	−	22.5000i	−0.547479	−	0.948262i	0.450967	−	0.892541i	$-0.351079\pi$
$564$		0			0
$565$	−10.5000	+	18.1865i	−0.441738	+	0.765113i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	16.5000	+	9.52628i	0.691716	+	0.399362i	0.804255	−	0.594285i	$-0.202565\pi$
$569$	16.5000	+	9.52628i	0.691716	+	0.399362i	−0.112539	+	0.993647i	$0.535898\pi$
$570$		0			0
$571$	−2.59808	+	1.50000i	−0.108726	+	0.0627730i	−0.553377	−	0.832931i	$-0.686661\pi$
$571$	−2.59808	+	1.50000i	−0.108726	+	0.0627730i	0.444651	+	0.895704i	$0.353328\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	10.3923			0.433389
$576$		0			0
$577$	−40.0000			−1.66522			−0.832611	−	0.553858i	$-0.813155\pi$
$577$	−40.0000			−1.66522			−0.832611	+	0.553858i	$0.813155\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	13.5000	−	7.79423i	0.560074	−	0.323359i
$582$		0			0
$583$	−46.7654	−	27.0000i	−1.93682	−	1.11823i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	7.79423	−	13.5000i	0.321702	−	0.557205i	−0.659137	−	0.752023i	$-0.729078\pi$
$587$	7.79423	−	13.5000i	0.321702	−	0.557205i	0.980839	+	0.194818i	$0.0624117\pi$
$588$		0			0
$589$	9.00000	+	15.5885i	0.370839	+	0.642311i
$590$		0			0
$591$		0			0
$592$		0			0
$593$			24.2487i			0.995775i	0.867242	+	0.497888i	$0.165891\pi$
$593$			24.2487i			0.995775i	−0.867242	+	0.497888i	$0.834109\pi$
$594$		0			0
$595$		−	18.0000i		−	0.737928i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	12.9904	+	22.5000i	0.530773	+	0.919325i	0.999355	+	0.0359054i	$0.0114315\pi$
$599$	12.9904	+	22.5000i	0.530773	+	0.919325i	−0.468583	+	0.883420i	$0.655235\pi$
$600$		0			0
$601$	3.50000	−	6.06218i	0.142768	−	0.247281i	−0.785770	−	0.618519i	$-0.787733\pi$
$601$	3.50000	−	6.06218i	0.142768	−	0.247281i	0.928538	+	0.371237i	$0.121066\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	24.0000	+	13.8564i	0.975739	+	0.563343i
$606$		0			0
$607$	2.59808	−	1.50000i	0.105453	−	0.0608831i	−0.446346	−	0.894860i	$-0.647275\pi$
$607$	2.59808	−	1.50000i	0.105453	−	0.0608831i	0.551799	+	0.833977i	$0.313942\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	5.19615			0.210214
$612$		0			0
$613$	20.0000			0.807792			0.403896	−	0.914805i	$-0.367656\pi$
$613$	20.0000			0.807792			0.403896	+	0.914805i	$0.367656\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−28.5000	+	16.4545i	−1.14737	+	0.662433i	−0.948244	−	0.317542i	$-0.897143\pi$
$617$	−28.5000	+	16.4545i	−1.14737	+	0.662433i	−0.199123	+	0.979975i	$0.563809\pi$
$618$		0			0
$619$	−7.79423	−	4.50000i	−0.313276	−	0.180870i	0.335115	−	0.942177i	$-0.391225\pi$
$619$	−7.79423	−	4.50000i	−0.313276	−	0.180870i	−0.648392	+	0.761307i	$0.724558\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	5.19615	−	9.00000i	0.208179	−	0.360577i
$624$		0			0
$625$	5.50000	+	9.52628i	0.220000	+	0.381051i
$626$		0			0
$627$		0			0
$628$		0			0
$629$			13.8564i			0.552491i
$630$		0			0
$631$			6.00000i			0.238856i	0.992843	+	0.119428i	$0.0381061\pi$
$631$			6.00000i			0.238856i	−0.992843	+	0.119428i	$0.961894\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−5.19615	−	9.00000i	−0.206203	−	0.357154i
$636$		0			0
$637$	1.00000	−	1.73205i	0.0396214	−	0.0686264i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	10.5000	+	6.06218i	0.414725	+	0.239442i	0.692818	−	0.721113i	$-0.256369\pi$
$641$	10.5000	+	6.06218i	0.414725	+	0.239442i	−0.278093	+	0.960554i	$0.589702\pi$
$642$		0			0
$643$	−23.3827	+	13.5000i	−0.922123	+	0.532388i	−0.884312	−	0.466897i	$-0.845372\pi$
$643$	−23.3827	+	13.5000i	−0.922123	+	0.532388i	−0.0378113	+	0.999285i	$0.512039\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−20.7846			−0.817127			−0.408564	−	0.912730i	$-0.633970\pi$
$647$	−20.7846			−0.817127			−0.408564	+	0.912730i	$0.633970\pi$
$648$		0			0
$649$	27.0000			1.05984
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−31.5000	+	18.1865i	−1.23269	+	0.711694i	−0.967590	−	0.252527i	$-0.918738\pi$
$653$	−31.5000	+	18.1865i	−1.23269	+	0.711694i	−0.265100	+	0.964221i	$0.585405\pi$
$654$		0			0
$655$	−7.79423	−	4.50000i	−0.304546	−	0.175830i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	2.59808	−	4.50000i	0.101207	−	0.175295i	−0.810975	−	0.585080i	$-0.801063\pi$
$659$	2.59808	−	4.50000i	0.101207	−	0.175295i	0.912182	+	0.409785i	$0.134396\pi$
$660$		0			0
$661$	0.500000	+	0.866025i	0.0194477	+	0.0336845i	0.875585	−	0.483063i	$-0.160476\pi$
$661$	0.500000	+	0.866025i	0.0194477	+	0.0336845i	−0.856138	+	0.516748i	$0.827143\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		−	31.1769i		−	1.20899i
$666$		0			0
$667$		−	45.0000i		−	1.74241i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−18.1865	−	31.5000i	−0.702083	−	1.21604i
$672$		0			0
$673$	3.50000	−	6.06218i	0.134915	−	0.233680i	−0.790650	−	0.612268i	$-0.790257\pi$
$673$	3.50000	−	6.06218i	0.134915	−	0.233680i	0.925565	+	0.378589i	$0.123591\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−13.5000	−	7.79423i	−0.518847	−	0.299557i	0.217616	−	0.976035i	$-0.430172\pi$
$677$	−13.5000	−	7.79423i	−0.518847	−	0.299557i	−0.736463	+	0.676478i	$0.763505\pi$
$678$		0			0
$679$	2.59808	−	1.50000i	0.0997050	−	0.0575647i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	41.5692			1.59060			0.795301	−	0.606215i	$-0.207313\pi$
$683$	41.5692			1.59060			0.795301	+	0.606215i	$0.207313\pi$
$684$		0			0
$685$	−3.00000			−0.114624
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−9.00000	+	5.19615i	−0.342873	+	0.197958i
$690$		0			0
$691$	−12.9904	−	7.50000i	−0.494177	−	0.285313i	0.232128	−	0.972685i	$-0.425431\pi$
$691$	−12.9904	−	7.50000i	−0.494177	−	0.285313i	−0.726306	+	0.687372i	$0.758764\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−2.59808	+	4.50000i	−0.0985506	+	0.170695i
$696$		0			0
$697$	9.00000	+	15.5885i	0.340899	+	0.590455i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	17.3205i		−	0.654187i	−0.944992	−	0.327093i	$-0.893931\pi$
$701$		−	17.3205i		−	0.654187i	0.944992	−	0.327093i	$-0.106069\pi$
$702$		0			0
$703$			24.0000i			0.905177i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−7.79423	−	13.5000i	−0.293132	−	0.507720i
$708$		0			0
$709$	14.5000	−	25.1147i	0.544559	−	0.943204i	−0.454076	−	0.890963i	$-0.650030\pi$
$709$	14.5000	−	25.1147i	0.544559	−	0.943204i	0.998635	−	0.0522406i	$-0.0166363\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	13.5000	+	7.79423i	0.505579	+	0.291896i
$714$		0			0
$715$	7.79423	−	4.50000i	0.291488	−	0.168290i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	20.7846			0.775135			0.387568	−	0.921841i	$-0.373315\pi$
$719$	20.7846			0.775135			0.387568	+	0.921841i	$0.373315\pi$
$720$		0			0
$721$	27.0000			1.00553
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−15.0000	+	8.66025i	−0.557086	+	0.321634i
$726$		0			0
$727$	23.3827	+	13.5000i	0.867216	+	0.500687i	0.866422	−	0.499312i	$-0.166414\pi$
$727$	23.3827	+	13.5000i	0.867216	+	0.500687i	0.000793791		1.00000i	$0.499747\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−5.19615	+	9.00000i	−0.192187	+	0.332877i
$732$		0			0
$733$	−6.50000	−	11.2583i	−0.240083	−	0.415836i	0.720655	−	0.693294i	$-0.243841\pi$
$733$	−6.50000	−	11.2583i	−0.240083	−	0.415836i	−0.960738	+	0.277458i	$0.910508\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	46.7654i		−	1.72262i
$738$		0			0
$739$		−	48.0000i		−	1.76571i	−0.469647	−	0.882854i	$-0.655619\pi$
$739$		−	48.0000i		−	1.76571i	0.469647	−	0.882854i	$-0.344381\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−18.1865	−	31.5000i	−0.667199	−	1.15562i	−0.978684	−	0.205372i	$-0.934160\pi$
$743$	−18.1865	−	31.5000i	−0.667199	−	1.15562i	0.311485	−	0.950251i	$-0.399174\pi$
$744$		0			0
$745$	−1.50000	+	2.59808i	−0.0549557	+	0.0951861i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−18.1865	+	10.5000i	−0.663636	+	0.383150i	−0.793661	−	0.608360i	$-0.791828\pi$
$751$	−18.1865	+	10.5000i	−0.663636	+	0.383150i	0.130025	+	0.991511i	$0.458494\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	15.5885			0.567322
$756$		0			0
$757$	−2.00000			−0.0726912			−0.0363456	−	0.999339i	$-0.511572\pi$
$757$	−2.00000			−0.0726912			−0.0363456	+	0.999339i	$0.511572\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−10.5000	+	6.06218i	−0.380625	+	0.219754i	−0.678090	−	0.734979i	$-0.737192\pi$
$761$	−10.5000	+	6.06218i	−0.380625	+	0.219754i	0.297465	+	0.954733i	$0.403859\pi$
$762$		0			0
$763$	−10.3923	−	6.00000i	−0.376227	−	0.217215i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	2.59808	−	4.50000i	0.0938111	−	0.162486i
$768$		0			0
$769$	−17.5000	−	30.3109i	−0.631066	−	1.09304i	−0.987334	−	0.158655i	$-0.949284\pi$
$769$	−17.5000	−	30.3109i	−0.631066	−	1.09304i	0.356268	−	0.934384i	$-0.384049\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			38.1051i			1.37055i	0.728286	+	0.685273i	$0.240317\pi$
$773$			38.1051i			1.37055i	−0.728286	+	0.685273i	$0.759683\pi$
$774$		0			0
$775$		−	6.00000i		−	0.215526i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	15.5885	+	27.0000i	0.558514	+	0.967375i
$780$		0			0
$781$	−27.0000	+	46.7654i	−0.966136	+	1.67340i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−1.50000	−	0.866025i	−0.0535373	−	0.0309098i
$786$		0			0
$787$	−7.79423	+	4.50000i	−0.277834	+	0.160408i	−0.632443	−	0.774607i	$-0.717948\pi$
$787$	−7.79423	+	4.50000i	−0.277834	+	0.160408i	0.354608	+	0.935015i	$0.384614\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−36.3731			−1.29328
$792$		0			0
$793$	−7.00000			−0.248577
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−16.5000	+	9.52628i	−0.584460	+	0.337438i	−0.762904	−	0.646512i	$-0.776227\pi$
$797$	−16.5000	+	9.52628i	−0.584460	+	0.337438i	0.178444	+	0.983950i	$0.442894\pi$
$798$		0			0
$799$	15.5885	+	9.00000i	0.551480	+	0.318397i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	10.3923	−	18.0000i	0.366736	−	0.635206i
$804$		0			0
$805$	−13.5000	−	23.3827i	−0.475812	−	0.824131i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	17.3205i		−	0.608957i	−0.952519	−	0.304478i	$-0.901518\pi$
$809$		−	17.3205i		−	0.608957i	0.952519	−	0.304478i	$-0.0984821\pi$
$810$		0			0
$811$			30.0000i			1.05344i	0.850038	+	0.526721i	$0.176579\pi$
$811$			30.0000i			1.05344i	−0.850038	+	0.526721i	$0.823421\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	5.19615	+	9.00000i	0.182013	+	0.315256i
$816$		0			0
$817$	−9.00000	+	15.5885i	−0.314870	+	0.545371i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	1.50000	+	0.866025i	0.0523504	+	0.0302245i	0.525947	−	0.850518i	$-0.323711\pi$
$821$	1.50000	+	0.866025i	0.0523504	+	0.0302245i	−0.473596	+	0.880742i	$0.657044\pi$
$822$		0			0
$823$	18.1865	−	10.5000i	0.633943	−	0.366007i	−0.148335	−	0.988937i	$-0.547391\pi$
$823$	18.1865	−	10.5000i	0.633943	−	0.366007i	0.782277	+	0.622930i	$0.214058\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0			−	1.00000i	$-0.5\pi$
$827$		0			0				1.00000i	$0.5\pi$
$828$		0			0
$829$	16.0000			0.555703			0.277851	−	0.960624i	$-0.410378\pi$
$829$	16.0000			0.555703			0.277851	+	0.960624i	$0.410378\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	6.00000	−	3.46410i	0.207888	−	0.120024i
$834$		0			0
$835$	23.3827	+	13.5000i	0.809191	+	0.467187i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−12.9904	+	22.5000i	−0.448478	+	0.776786i	−0.998287	−	0.0585039i	$-0.981367\pi$
$839$	−12.9904	+	22.5000i	−0.448478	+	0.776786i	0.549809	+	0.835290i	$0.314700\pi$
$840$		0			0
$841$	23.0000	+	39.8372i	0.793103	+	1.37370i
$842$		0			0
$843$		0			0
$844$		0			0
$845$			20.7846i			0.715012i
$846$		0			0
$847$			48.0000i			1.64930i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	10.3923	+	18.0000i	0.356244	+	0.617032i
$852$		0			0
$853$	9.50000	−	16.4545i	0.325274	−	0.563391i	−0.656294	−	0.754505i	$-0.727877\pi$
$853$	9.50000	−	16.4545i	0.325274	−	0.563391i	0.981568	+	0.191115i	$0.0612102\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	37.5000	+	21.6506i	1.28098	+	0.739572i	0.977027	−	0.213117i	$-0.0683615\pi$
$857$	37.5000	+	21.6506i	1.28098	+	0.739572i	0.303949	+	0.952688i	$0.401695\pi$
$858$		0			0
$859$	49.3634	−	28.5000i	1.68426	−	0.972407i	0.725485	−	0.688238i	$-0.241615\pi$
$859$	49.3634	−	28.5000i	1.68426	−	0.972407i	0.958774	−	0.284170i	$-0.0917179\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−41.5692			−1.41503			−0.707516	−	0.706697i	$-0.750184\pi$
$863$	−41.5692			−1.41503			−0.707516	+	0.706697i	$0.750184\pi$
$864$		0			0
$865$	3.00000			0.102003
$866$		0			0
$867$		0			0
$868$		0			0
$869$	67.5000	−	38.9711i	2.28978	−	1.32201i
$870$		0			0
$871$	−7.79423	−	4.50000i	−0.264097	−	0.152477i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−18.1865	+	31.5000i	−0.614817	+	1.06489i
$876$		0			0
$877$	−8.50000	−	14.7224i	−0.287025	−	0.497141i	0.686074	−	0.727532i	$-0.259333\pi$
$877$	−8.50000	−	14.7224i	−0.287025	−	0.497141i	−0.973098	+	0.230391i	$0.925999\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	10.3923i		−	0.350126i	−0.984557	−	0.175063i	$-0.943987\pi$
$881$		−	10.3923i		−	0.350126i	0.984557	−	0.175063i	$-0.0560129\pi$
$882$		0			0
$883$			6.00000i			0.201916i	0.994891	+	0.100958i	$0.0321908\pi$
$883$			6.00000i			0.201916i	−0.994891	+	0.100958i	$0.967809\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	28.5788	+	49.5000i	0.959583	+	1.66205i	0.723512	+	0.690312i	$0.242527\pi$
$887$	28.5788	+	49.5000i	0.959583	+	1.66205i	0.236071	+	0.971736i	$0.424140\pi$
$888$		0			0
$889$	9.00000	−	15.5885i	0.301850	−	0.522820i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	27.0000	+	15.5885i	0.903521	+	0.521648i
$894$		0			0
$895$	−31.1769	+	18.0000i	−1.04213	+	0.601674i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−25.9808			−0.866507
$900$		0			0
$901$	−36.0000			−1.19933
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−24.0000	+	13.8564i	−0.797787	+	0.460603i
$906$		0			0
$907$	−2.59808	−	1.50000i	−0.0862677	−	0.0498067i	0.456246	−	0.889854i	$-0.349194\pi$
$907$	−2.59808	−	1.50000i	−0.0862677	−	0.0498067i	−0.542513	+	0.840047i	$0.682527\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−18.1865	+	31.5000i	−0.602547	+	1.04364i	0.389887	+	0.920863i	$0.372514\pi$
$911$	−18.1865	+	31.5000i	−0.602547	+	1.04364i	−0.992434	+	0.122779i	$0.960819\pi$
$912$		0			0
$913$	13.5000	+	23.3827i	0.446785	+	0.773854i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		−	15.5885i		−	0.514776i
$918$		0			0
$919$			54.0000i			1.78130i	0.454694	+	0.890648i	$0.349749\pi$
$919$			54.0000i			1.78130i	−0.454694	+	0.890648i	$0.650251\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	5.19615	+	9.00000i	0.171033	+	0.296239i
$924$		0			0
$925$	4.00000	−	6.92820i	0.131519	−	0.227798i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−46.5000	−	26.8468i	−1.52562	−	0.880815i	−0.999538	−	0.0303776i	$-0.990329\pi$
$929$	−46.5000	−	26.8468i	−1.52562	−	0.880815i	−0.526077	−	0.850437i	$-0.676338\pi$
$930$		0			0
$931$	10.3923	−	6.00000i	0.340594	−	0.196642i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	31.1769			1.01959
$936$		0			0
$937$	−40.0000			−1.30674			−0.653372	−	0.757037i	$-0.726646\pi$
$937$	−40.0000			−1.30674			−0.653372	+	0.757037i	$0.726646\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	10.5000	−	6.06218i	0.342290	−	0.197621i	−0.318994	−	0.947757i	$-0.603345\pi$
$941$	10.5000	−	6.06218i	0.342290	−	0.197621i	0.661284	+	0.750135i	$0.270012\pi$
$942$		0			0
$943$	23.3827	+	13.5000i	0.761445	+	0.439620i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	12.9904	−	22.5000i	0.422131	−	0.731152i	−0.574017	−	0.818843i	$-0.694616\pi$
$947$	12.9904	−	22.5000i	0.422131	−	0.731152i	0.996148	+	0.0876916i	$0.0279490\pi$
$948$		0			0
$949$	−2.00000	−	3.46410i	−0.0649227	−	0.112449i
$950$		0			0
$951$		0			0
$952$		0			0
$953$			24.2487i			0.785493i	0.919647	+	0.392746i	$0.128475\pi$
$953$			24.2487i			0.785493i	−0.919647	+	0.392746i	$0.871525\pi$
$954$		0			0
$955$			27.0000i			0.873699i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−2.59808	−	4.50000i	−0.0838963	−	0.145313i
$960$		0			0
$961$	−11.0000	+	19.0526i	−0.354839	+	0.614599i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−28.5000	−	16.4545i	−0.917447	−	0.529689i
$966$		0			0
$967$	−23.3827	+	13.5000i	−0.751936	+	0.434131i	−0.826393	−	0.563094i	$-0.809611\pi$
$967$	−23.3827	+	13.5000i	−0.751936	+	0.434131i	0.0744567	+	0.997224i	$0.476278\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	31.1769			1.00051			0.500257	−	0.865877i	$-0.333239\pi$
$971$	31.1769			1.00051			0.500257	+	0.865877i	$0.333239\pi$
$972$		0			0
$973$	−9.00000			−0.288527
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−13.5000	+	7.79423i	−0.431903	+	0.249359i	−0.700157	−	0.713989i	$-0.746887\pi$
$977$	−13.5000	+	7.79423i	−0.431903	+	0.249359i	0.268254	+	0.963348i	$0.413553\pi$
$978$		0			0
$979$	15.5885	+	9.00000i	0.498209	+	0.287641i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−28.5788	+	49.5000i	−0.911523	+	1.57880i	−0.0996103	+	0.995027i	$0.531760\pi$
$983$	−28.5788	+	49.5000i	−0.911523	+	1.57880i	−0.811913	+	0.583778i	$0.801574\pi$
$984$		0			0
$985$	−6.00000	−	10.3923i	−0.191176	−	0.331126i
$986$		0			0
$987$		0			0
$988$		0			0
$989$			15.5885i			0.495684i
$990$		0			0
$991$		−	54.0000i		−	1.71537i	−0.514178	−	0.857683i	$-0.671903\pi$
$991$		−	54.0000i		−	1.71537i	0.514178	−	0.857683i	$-0.328097\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	10.3923	+	18.0000i	0.329458	+	0.570638i
$996$		0			0
$997$	−5.50000	+	9.52628i	−0.174187	+	0.301700i	−0.939880	−	0.341506i	$-0.889063\pi$
$997$	−5.50000	+	9.52628i	−0.174187	+	0.301700i	0.765693	+	0.643206i	$0.222396\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	432.2.s.e.143.1		4
3.2	odd	2		144.2.s.e.47.2	yes	4
4.3	odd	2	inner	432.2.s.e.143.2		4
8.3	odd	2		1728.2.s.e.575.2		4
8.5	even	2		1728.2.s.e.575.1		4
9.2	odd	6		1296.2.c.f.1295.4		4
9.4	even	3		144.2.s.e.95.1	yes	4
9.5	odd	6	inner	432.2.s.e.287.2		4
9.7	even	3		1296.2.c.f.1295.2		4
12.11	even	2		144.2.s.e.47.1	✓	4
24.5	odd	2		576.2.s.e.191.1		4
24.11	even	2		576.2.s.e.191.2		4
36.7	odd	6		1296.2.c.f.1295.1		4
36.11	even	6		1296.2.c.f.1295.3		4
36.23	even	6	inner	432.2.s.e.287.1		4
36.31	odd	6		144.2.s.e.95.2	yes	4
72.5	odd	6		1728.2.s.e.1151.2		4
72.11	even	6		5184.2.c.f.5183.1		4
72.13	even	6		576.2.s.e.383.2		4
72.29	odd	6		5184.2.c.f.5183.2		4
72.43	odd	6		5184.2.c.f.5183.3		4
72.59	even	6		1728.2.s.e.1151.1		4
72.61	even	6		5184.2.c.f.5183.4		4
72.67	odd	6		576.2.s.e.383.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
144.2.s.e.47.1	✓	4	12.11	even	2
144.2.s.e.47.2	yes	4	3.2	odd	2
144.2.s.e.95.1	yes	4	9.4	even	3
144.2.s.e.95.2	yes	4	36.31	odd	6
432.2.s.e.143.1		4	1.1	even	1	trivial
432.2.s.e.143.2		4	4.3	odd	2	inner
432.2.s.e.287.1		4	36.23	even	6	inner
432.2.s.e.287.2		4	9.5	odd	6	inner
576.2.s.e.191.1		4	24.5	odd	2
576.2.s.e.191.2		4	24.11	even	2
576.2.s.e.383.1		4	72.67	odd	6
576.2.s.e.383.2		4	72.13	even	6
1296.2.c.f.1295.1		4	36.7	odd	6
1296.2.c.f.1295.2		4	9.7	even	3
1296.2.c.f.1295.3		4	36.11	even	6
1296.2.c.f.1295.4		4	9.2	odd	6
1728.2.s.e.575.1		4	8.5	even	2
1728.2.s.e.575.2		4	8.3	odd	2
1728.2.s.e.1151.1		4	72.59	even	6
1728.2.s.e.1151.2		4	72.5	odd	6
5184.2.c.f.5183.1		4	72.11	even	6
5184.2.c.f.5183.2		4	72.29	odd	6
5184.2.c.f.5183.3		4	72.43	odd	6
5184.2.c.f.5183.4		4	72.61	even	6

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 432 = 2^{4} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	432.s (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.50000 + 0.866025i) q^{5} +(-2.59808 - 1.50000i) q^{7} +O(q^{10})\)	\(q+(-1.50000 + 0.866025i) q^{5} +(-2.59808 - 1.50000i) q^{7} +(2.59808 - 4.50000i) q^{11} +(-0.500000 - 0.866025i) q^{13} -3.46410i q^{17} -6.00000i q^{19} +(-2.59808 - 4.50000i) q^{23} +(-1.00000 + 1.73205i) q^{25} +(7.50000 + 4.33013i) q^{29} +(-2.59808 + 1.50000i) q^{31} +5.19615 q^{35} -4.00000 q^{37} +(-4.50000 + 2.59808i) q^{41} +(-2.59808 - 1.50000i) q^{43} +(-2.59808 + 4.50000i) q^{47} +(1.00000 + 1.73205i) q^{49} -10.3923i q^{53} +9.00000i q^{55} +(2.59808 + 4.50000i) q^{59} +(3.50000 - 6.06218i) q^{61} +(1.50000 + 0.866025i) q^{65} +(7.79423 - 4.50000i) q^{67} -10.3923 q^{71} +4.00000 q^{73} +(-13.5000 + 7.79423i) q^{77} +(12.9904 + 7.50000i) q^{79} +(-2.59808 + 4.50000i) q^{83} +(3.00000 + 5.19615i) q^{85} +3.46410i q^{89} +3.00000i q^{91} +(5.19615 + 9.00000i) q^{95} +(-0.500000 + 0.866025i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 6 q^{5}+O(q^{10}) \) 4 * q - 6 * q^5	\( 4 q - 6 q^{5} - 2 q^{13} - 4 q^{25} + 30 q^{29} - 16 q^{37} - 18 q^{41} + 4 q^{49} + 14 q^{61} + 6 q^{65} + 16 q^{73} - 54 q^{77} + 12 q^{85} - 2 q^{97}+O(q^{100}) \) 4 * q - 6 * q^5 - 2 * q^13 - 4 * q^25 + 30 * q^29 - 16 * q^37 - 18 * q^41 + 4 * q^49 + 14 * q^61 + 6 * q^65 + 16 * q^73 - 54 * q^77 + 12 * q^85 - 2 * q^97

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