LMFDB - Embedded newform 1728.2.i.b.577.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1728,2,Mod(577,1728)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1728, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 2]))

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

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

chi := DirichletCharacter("1728.577");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1728 = 2^{6} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1728.i (of order $3$, 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:	$13.7981494693$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 288)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			577.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	1728.577
Dual form			1728.2.i.b.1153.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.00000 + 3.46410i) q^{5} +(1.00000 + 1.73205i) q^{7} +O(q^{10})$	$q+(-2.00000 + 3.46410i) q^{5} +(1.00000 + 1.73205i) q^{7} +(2.50000 + 4.33013i) q^{11} +(-1.00000 + 1.73205i) q^{13} +3.00000 q^{17} -1.00000 q^{19} +(-3.00000 + 5.19615i) q^{23} +(-5.50000 - 9.52628i) q^{25} +(1.00000 + 1.73205i) q^{29} +(2.00000 - 3.46410i) q^{31} -8.00000 q^{35} +8.00000 q^{37} +(0.500000 - 0.866025i) q^{41} +(-3.50000 - 6.06218i) q^{43} +(1.00000 + 1.73205i) q^{47} +(1.50000 - 2.59808i) q^{49} -4.00000 q^{53} -20.0000 q^{55} +(-2.50000 + 4.33013i) q^{59} +(-4.00000 - 6.92820i) q^{65} +(-6.50000 + 11.2583i) q^{67} -8.00000 q^{71} +3.00000 q^{73} +(-5.00000 + 8.66025i) q^{77} +(-4.00000 - 6.92820i) q^{79} +(-6.00000 - 10.3923i) q^{83} +(-6.00000 + 10.3923i) q^{85} +10.0000 q^{89} -4.00000 q^{91} +(2.00000 - 3.46410i) q^{95} +(5.50000 + 9.52628i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{5} + 2 q^{7}+O(q^{10}) $ 2 * q - 4 * q^5 + 2 * q^7	$ 2 q - 4 q^{5} + 2 q^{7} + 5 q^{11} - 2 q^{13} + 6 q^{17} - 2 q^{19} - 6 q^{23} - 11 q^{25} + 2 q^{29} + 4 q^{31} - 16 q^{35} + 16 q^{37} + q^{41} - 7 q^{43} + 2 q^{47} + 3 q^{49} - 8 q^{53} - 40 q^{55} - 5 q^{59} - 8 q^{65} - 13 q^{67} - 16 q^{71} + 6 q^{73} - 10 q^{77} - 8 q^{79} - 12 q^{83} - 12 q^{85} + 20 q^{89} - 8 q^{91} + 4 q^{95} + 11 q^{97}+O(q^{100}) $ 2 * q - 4 * q^5 + 2 * q^7 + 5 * q^11 - 2 * q^13 + 6 * q^17 - 2 * q^19 - 6 * q^23 - 11 * q^25 + 2 * q^29 + 4 * q^31 - 16 * q^35 + 16 * q^37 + q^41 - 7 * q^43 + 2 * q^47 + 3 * q^49 - 8 * q^53 - 40 * q^55 - 5 * q^59 - 8 * q^65 - 13 * q^67 - 16 * q^71 + 6 * q^73 - 10 * q^77 - 8 * q^79 - 12 * q^83 - 12 * q^85 + 20 * q^89 - 8 * q^91 + 4 * q^95 + 11 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1728\mathbb{Z}\right)^\times$.

$n$	$325$	$703$	$1217$
$\chi(n)$	$1$	$1$	$e\left(\frac{1}{3}\right)$

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
$5$	−2.00000	+	3.46410i	−0.894427	+	1.54919i	−0.0599153	+	0.998203i	$0.519083\pi$
$5$	−2.00000	+	3.46410i	−0.894427	+	1.54919i	−0.834512	+	0.550990i	$0.814250\pi$
$6$		0			0
$7$	1.00000	+	1.73205i	0.377964	+	0.654654i	0.990766	−	0.135583i	$-0.0432908\pi$
$7$	1.00000	+	1.73205i	0.377964	+	0.654654i	−0.612801	+	0.790237i	$0.709957\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	2.50000	+	4.33013i	0.753778	+	1.30558i	0.945979	+	0.324227i	$0.105104\pi$
$11$	2.50000	+	4.33013i	0.753778	+	1.30558i	−0.192201	+	0.981356i	$0.561563\pi$
$12$		0			0
$13$	−1.00000	+	1.73205i	−0.277350	+	0.480384i	−0.970725	−	0.240192i	$-0.922790\pi$
$13$	−1.00000	+	1.73205i	−0.277350	+	0.480384i	0.693375	+	0.720577i	$0.256123\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	3.00000			0.727607			0.363803	−	0.931476i	$-0.381478\pi$
$17$	3.00000			0.727607			0.363803	+	0.931476i	$0.381478\pi$
$18$		0			0
$19$	−1.00000			−0.229416			−0.114708	−	0.993399i	$-0.536593\pi$
$19$	−1.00000			−0.229416			−0.114708	+	0.993399i	$0.536593\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−3.00000	+	5.19615i	−0.625543	+	1.08347i	0.362892	+	0.931831i	$0.381789\pi$
$23$	−3.00000	+	5.19615i	−0.625543	+	1.08347i	−0.988436	+	0.151642i	$0.951544\pi$
$24$		0			0
$25$	−5.50000	−	9.52628i	−1.10000	−	1.90526i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	1.00000	+	1.73205i	0.185695	+	0.321634i	0.943811	−	0.330487i	$-0.107213\pi$
$29$	1.00000	+	1.73205i	0.185695	+	0.321634i	−0.758115	+	0.652121i	$0.773880\pi$
$30$		0			0
$31$	2.00000	−	3.46410i	0.359211	−	0.622171i	−0.628619	−	0.777714i	$-0.716379\pi$
$31$	2.00000	−	3.46410i	0.359211	−	0.622171i	0.987829	+	0.155543i	$0.0497126\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−8.00000			−1.35225
$36$		0			0
$37$	8.00000			1.31519			0.657596	−	0.753371i	$-0.271573\pi$
$37$	8.00000			1.31519			0.657596	+	0.753371i	$0.271573\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	0.500000	−	0.866025i	0.0780869	−	0.135250i	−0.824338	−	0.566099i	$-0.808452\pi$
$41$	0.500000	−	0.866025i	0.0780869	−	0.135250i	0.902424	+	0.430848i	$0.141786\pi$
$42$		0			0
$43$	−3.50000	−	6.06218i	−0.533745	−	0.924473i	−0.999223	−	0.0394140i	$-0.987451\pi$
$43$	−3.50000	−	6.06218i	−0.533745	−	0.924473i	0.465478	−	0.885059i	$-0.345882\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	1.00000	+	1.73205i	0.145865	+	0.252646i	0.929695	−	0.368329i	$-0.120070\pi$
$47$	1.00000	+	1.73205i	0.145865	+	0.252646i	−0.783830	+	0.620975i	$0.786737\pi$
$48$		0			0
$49$	1.50000	−	2.59808i	0.214286	−	0.371154i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−4.00000			−0.549442			−0.274721	−	0.961524i	$-0.588586\pi$
$53$	−4.00000			−0.549442			−0.274721	+	0.961524i	$0.588586\pi$
$54$		0			0
$55$	−20.0000			−2.69680
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−2.50000	+	4.33013i	−0.325472	+	0.563735i	−0.981608	−	0.190909i	$-0.938857\pi$
$59$	−2.50000	+	4.33013i	−0.325472	+	0.563735i	0.656136	+	0.754643i	$0.272190\pi$
$60$		0			0
$61$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$61$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−4.00000	−	6.92820i	−0.496139	−	0.859338i
$66$		0			0
$67$	−6.50000	+	11.2583i	−0.794101	+	1.37542i	0.129307	+	0.991605i	$0.458725\pi$
$67$	−6.50000	+	11.2583i	−0.794101	+	1.37542i	−0.923408	+	0.383819i	$0.874609\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−8.00000			−0.949425			−0.474713	−	0.880141i	$-0.657448\pi$
$71$	−8.00000			−0.949425			−0.474713	+	0.880141i	$0.657448\pi$
$72$		0			0
$73$	3.00000			0.351123			0.175562	−	0.984468i	$-0.443826\pi$
$73$	3.00000			0.351123			0.175562	+	0.984468i	$0.443826\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−5.00000	+	8.66025i	−0.569803	+	0.986928i
$78$		0			0
$79$	−4.00000	−	6.92820i	−0.450035	−	0.779484i	0.548352	−	0.836247i	$-0.315255\pi$
$79$	−4.00000	−	6.92820i	−0.450035	−	0.779484i	−0.998388	+	0.0567635i	$0.981922\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−6.00000	−	10.3923i	−0.658586	−	1.14070i	−0.980982	−	0.194099i	$-0.937822\pi$
$83$	−6.00000	−	10.3923i	−0.658586	−	1.14070i	0.322396	−	0.946605i	$-0.395512\pi$
$84$		0			0
$85$	−6.00000	+	10.3923i	−0.650791	+	1.12720i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	10.0000			1.06000			0.529999	−	0.847998i	$-0.322192\pi$
$89$	10.0000			1.06000			0.529999	+	0.847998i	$0.322192\pi$
$90$		0			0
$91$	−4.00000			−0.419314
$92$		0			0
$93$		0			0
$94$		0			0
$95$	2.00000	−	3.46410i	0.205196	−	0.355409i
$96$		0			0
$97$	5.50000	+	9.52628i	0.558440	+	0.967247i	0.997627	+	0.0688512i	$0.0219334\pi$
$97$	5.50000	+	9.52628i	0.558440	+	0.967247i	−0.439187	+	0.898396i	$0.644733\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−6.00000	−	10.3923i	−0.597022	−	1.03407i	−0.993258	−	0.115924i	$-0.963017\pi$
$101$	−6.00000	−	10.3923i	−0.597022	−	1.03407i	0.396236	−	0.918149i	$-0.370316\pi$
$102$		0			0
$103$	−3.00000	+	5.19615i	−0.295599	+	0.511992i	−0.975124	−	0.221660i	$-0.928852\pi$
$103$	−3.00000	+	5.19615i	−0.295599	+	0.511992i	0.679525	+	0.733652i	$0.262186\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	5.00000			0.483368			0.241684	−	0.970355i	$-0.422300\pi$
$107$	5.00000			0.483368			0.241684	+	0.970355i	$0.422300\pi$
$108$		0			0
$109$	−8.00000			−0.766261			−0.383131	−	0.923694i	$-0.625154\pi$
$109$	−8.00000			−0.766261			−0.383131	+	0.923694i	$0.625154\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−1.00000	+	1.73205i	−0.0940721	+	0.162938i	−0.909221	−	0.416314i	$-0.863322\pi$
$113$	−1.00000	+	1.73205i	−0.0940721	+	0.162938i	0.815149	+	0.579252i	$0.196655\pi$
$114$		0			0
$115$	−12.0000	−	20.7846i	−1.11901	−	1.93817i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	3.00000	+	5.19615i	0.275010	+	0.476331i
$120$		0			0
$121$	−7.00000	+	12.1244i	−0.636364	+	1.10221i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	24.0000			2.14663
$126$		0			0
$127$	2.00000			0.177471			0.0887357	−	0.996055i	$-0.471717\pi$
$127$	2.00000			0.177471			0.0887357	+	0.996055i	$0.471717\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−8.00000	+	13.8564i	−0.698963	+	1.21064i	0.269863	+	0.962899i	$0.413022\pi$
$131$	−8.00000	+	13.8564i	−0.698963	+	1.21064i	−0.968826	+	0.247741i	$0.920312\pi$
$132$		0			0
$133$	−1.00000	−	1.73205i	−0.0867110	−	0.150188i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−1.50000	−	2.59808i	−0.128154	−	0.221969i	0.794808	−	0.606861i	$-0.207572\pi$
$137$	−1.50000	−	2.59808i	−0.128154	−	0.221969i	−0.922961	+	0.384893i	$0.874238\pi$
$138$		0			0
$139$	−6.50000	+	11.2583i	−0.551323	+	0.954919i	0.446857	+	0.894606i	$0.352543\pi$
$139$	−6.50000	+	11.2583i	−0.551323	+	0.954919i	−0.998179	+	0.0603135i	$0.980790\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−10.0000			−0.836242
$144$		0			0
$145$	−8.00000			−0.664364
$146$		0			0
$147$		0			0
$148$		0			0
$149$	9.00000	−	15.5885i	0.737309	−	1.27706i	−0.216394	−	0.976306i	$-0.569430\pi$
$149$	9.00000	−	15.5885i	0.737309	−	1.27706i	0.953703	−	0.300750i	$-0.0972370\pi$
$150$		0			0
$151$	9.00000	+	15.5885i	0.732410	+	1.26857i	0.955851	+	0.293853i	$0.0949377\pi$
$151$	9.00000	+	15.5885i	0.732410	+	1.26857i	−0.223441	+	0.974717i	$0.571729\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	8.00000	+	13.8564i	0.642575	+	1.11297i
$156$		0			0
$157$	10.0000	−	17.3205i	0.798087	−	1.38233i	−0.122774	−	0.992435i	$-0.539179\pi$
$157$	10.0000	−	17.3205i	0.798087	−	1.38233i	0.920860	−	0.389892i	$-0.127488\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−12.0000			−0.945732
$162$		0			0
$163$	20.0000			1.56652			0.783260	−	0.621694i	$-0.213555\pi$
$163$	20.0000			1.56652			0.783260	+	0.621694i	$0.213555\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−8.00000	+	13.8564i	−0.619059	+	1.07224i	0.370599	+	0.928793i	$0.379152\pi$
$167$	−8.00000	+	13.8564i	−0.619059	+	1.07224i	−0.989658	+	0.143448i	$0.954181\pi$
$168$		0			0
$169$	4.50000	+	7.79423i	0.346154	+	0.599556i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−9.00000	−	15.5885i	−0.684257	−	1.18517i	−0.973670	−	0.227964i	$-0.926793\pi$
$173$	−9.00000	−	15.5885i	−0.684257	−	1.18517i	0.289412	−	0.957205i	$-0.406540\pi$
$174$		0			0
$175$	11.0000	−	19.0526i	0.831522	−	1.44024i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	12.0000			0.896922			0.448461	−	0.893802i	$-0.351972\pi$
$179$	12.0000			0.896922			0.448461	+	0.893802i	$0.351972\pi$
$180$		0			0
$181$	−2.00000			−0.148659			−0.0743294	−	0.997234i	$-0.523682\pi$
$181$	−2.00000			−0.148659			−0.0743294	+	0.997234i	$0.523682\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−16.0000	+	27.7128i	−1.17634	+	2.03749i
$186$		0			0
$187$	7.50000	+	12.9904i	0.548454	+	0.949951i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−3.00000	−	5.19615i	−0.217072	−	0.375980i	0.736839	−	0.676068i	$-0.236317\pi$
$191$	−3.00000	−	5.19615i	−0.217072	−	0.375980i	−0.953912	+	0.300088i	$0.902984\pi$
$192$		0			0
$193$	1.50000	−	2.59808i	0.107972	−	0.187014i	−0.806976	−	0.590584i	$-0.798898\pi$
$193$	1.50000	−	2.59808i	0.107972	−	0.187014i	0.914949	+	0.403570i	$0.132231\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	4.00000			0.284988			0.142494	−	0.989796i	$-0.454488\pi$
$197$	4.00000			0.284988			0.142494	+	0.989796i	$0.454488\pi$
$198$		0			0
$199$	2.00000			0.141776			0.0708881	−	0.997484i	$-0.477417\pi$
$199$	2.00000			0.141776			0.0708881	+	0.997484i	$0.477417\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−2.00000	+	3.46410i	−0.140372	+	0.243132i
$204$		0			0
$205$	2.00000	+	3.46410i	0.139686	+	0.241943i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−2.50000	−	4.33013i	−0.172929	−	0.299521i
$210$		0			0
$211$	14.0000	−	24.2487i	0.963800	−	1.66935i	0.250994	−	0.967989i	$-0.419243\pi$
$211$	14.0000	−	24.2487i	0.963800	−	1.66935i	0.712806	−	0.701361i	$-0.247424\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	28.0000			1.90958
$216$		0			0
$217$	8.00000			0.543075
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−3.00000	+	5.19615i	−0.201802	+	0.349531i
$222$		0			0
$223$	7.00000	+	12.1244i	0.468755	+	0.811907i	0.999362	−	0.0357107i	$-0.0113695\pi$
$223$	7.00000	+	12.1244i	0.468755	+	0.811907i	−0.530607	+	0.847618i	$0.678036\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−1.50000	−	2.59808i	−0.0995585	−	0.172440i	0.811943	−	0.583736i	$-0.198410\pi$
$227$	−1.50000	−	2.59808i	−0.0995585	−	0.172440i	−0.911502	+	0.411296i	$0.865076\pi$
$228$		0			0
$229$	7.00000	−	12.1244i	0.462573	−	0.801200i	−0.536515	−	0.843891i	$-0.680260\pi$
$229$	7.00000	−	12.1244i	0.462573	−	0.801200i	0.999088	+	0.0426906i	$0.0135930\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	21.0000			1.37576			0.687878	−	0.725826i	$-0.258542\pi$
$233$	21.0000			1.37576			0.687878	+	0.725826i	$0.258542\pi$
$234$		0			0
$235$	−8.00000			−0.521862
$236$		0			0
$237$		0			0
$238$		0			0
$239$	3.00000	−	5.19615i	0.194054	−	0.336111i	−0.752536	−	0.658551i	$-0.771170\pi$
$239$	3.00000	−	5.19615i	0.194054	−	0.336111i	0.946590	+	0.322440i	$0.104503\pi$
$240$		0			0
$241$	11.5000	+	19.9186i	0.740780	+	1.28307i	0.952141	+	0.305661i	$0.0988773\pi$
$241$	11.5000	+	19.9186i	0.740780	+	1.28307i	−0.211360	+	0.977408i	$0.567789\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	6.00000	+	10.3923i	0.383326	+	0.663940i
$246$		0			0
$247$	1.00000	−	1.73205i	0.0636285	−	0.110208i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−21.0000			−1.32551			−0.662754	−	0.748837i	$-0.730613\pi$
$251$	−21.0000			−1.32551			−0.662754	+	0.748837i	$0.730613\pi$
$252$		0			0
$253$	−30.0000			−1.88608
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−2.50000	+	4.33013i	−0.155946	+	0.270106i	−0.933403	−	0.358830i	$-0.883176\pi$
$257$	−2.50000	+	4.33013i	−0.155946	+	0.270106i	0.777457	+	0.628936i	$0.216509\pi$
$258$		0			0
$259$	8.00000	+	13.8564i	0.497096	+	0.860995i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	3.00000	+	5.19615i	0.184988	+	0.320408i	0.943572	−	0.331166i	$-0.107442\pi$
$263$	3.00000	+	5.19615i	0.184988	+	0.320408i	−0.758585	+	0.651575i	$0.774109\pi$
$264$		0			0
$265$	8.00000	−	13.8564i	0.491436	−	0.851192i
$266$		0			0
$267$		0			0
$268$		0			0
$269$		0			0			−	1.00000i	$-0.5\pi$
$269$		0			0				1.00000i	$0.5\pi$
$270$		0			0
$271$	−12.0000			−0.728948			−0.364474	−	0.931214i	$-0.618751\pi$
$271$	−12.0000			−0.728948			−0.364474	+	0.931214i	$0.618751\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	27.5000	−	47.6314i	1.65831	−	2.87228i
$276$		0			0
$277$	−7.00000	−	12.1244i	−0.420589	−	0.728482i	0.575408	−	0.817867i	$-0.304843\pi$
$277$	−7.00000	−	12.1244i	−0.420589	−	0.728482i	−0.995997	+	0.0893846i	$0.971510\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−9.00000	−	15.5885i	−0.536895	−	0.929929i	−0.999069	−	0.0431402i	$-0.986264\pi$
$281$	−9.00000	−	15.5885i	−0.536895	−	0.929929i	0.462174	−	0.886789i	$-0.347070\pi$
$282$		0			0
$283$	−14.0000	+	24.2487i	−0.832214	+	1.44144i	0.0640654	+	0.997946i	$0.479593\pi$
$283$	−14.0000	+	24.2487i	−0.832214	+	1.44144i	−0.896279	+	0.443491i	$0.853740\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	2.00000			0.118056
$288$		0			0
$289$	−8.00000			−0.470588
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−15.0000	+	25.9808i	−0.876309	+	1.51781i	−0.0209480	+	0.999781i	$0.506668\pi$
$293$	−15.0000	+	25.9808i	−0.876309	+	1.51781i	−0.855361	+	0.518032i	$0.826665\pi$
$294$		0			0
$295$	−10.0000	−	17.3205i	−0.582223	−	1.00844i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−6.00000	−	10.3923i	−0.346989	−	0.601003i
$300$		0			0
$301$	7.00000	−	12.1244i	0.403473	−	0.698836i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$	9.00000			0.513657			0.256829	−	0.966457i	$-0.417322\pi$
$307$	9.00000			0.513657			0.256829	+	0.966457i	$0.417322\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	9.00000	−	15.5885i	0.510343	−	0.883940i	−0.489585	−	0.871956i	$-0.662852\pi$
$311$	9.00000	−	15.5885i	0.510343	−	0.883940i	0.999928	−	0.0119847i	$-0.00381495\pi$
$312$		0			0
$313$	−6.50000	−	11.2583i	−0.367402	−	0.636358i	0.621757	−	0.783210i	$-0.286419\pi$
$313$	−6.50000	−	11.2583i	−0.367402	−	0.636358i	−0.989158	+	0.146852i	$0.953086\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−9.00000	−	15.5885i	−0.505490	−	0.875535i	−0.999980	−	0.00635137i	$-0.997978\pi$
$317$	−9.00000	−	15.5885i	−0.505490	−	0.875535i	0.494489	−	0.869184i	$-0.335355\pi$
$318$		0			0
$319$	−5.00000	+	8.66025i	−0.279946	+	0.484881i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−3.00000			−0.166924
$324$		0			0
$325$	22.0000			1.22034
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−2.00000	+	3.46410i	−0.110264	+	0.190982i
$330$		0			0
$331$	−2.00000	−	3.46410i	−0.109930	−	0.190404i	0.805812	−	0.592172i	$-0.201729\pi$
$331$	−2.00000	−	3.46410i	−0.109930	−	0.190404i	−0.915742	+	0.401768i	$0.868396\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−26.0000	−	45.0333i	−1.42053	−	2.46043i
$336$		0			0
$337$	12.5000	−	21.6506i	0.680918	−	1.17939i	−0.293783	−	0.955872i	$-0.594914\pi$
$337$	12.5000	−	21.6506i	0.680918	−	1.17939i	0.974701	−	0.223513i	$-0.0717525\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	20.0000			1.08306
$342$		0			0
$343$	20.0000			1.07990
$344$		0			0
$345$		0			0
$346$		0			0
$347$	0.500000	−	0.866025i	0.0268414	−	0.0464907i	−0.852293	−	0.523065i	$-0.824788\pi$
$347$	0.500000	−	0.866025i	0.0268414	−	0.0464907i	0.879134	+	0.476575i	$0.158122\pi$
$348$		0			0
$349$	8.00000	+	13.8564i	0.428230	+	0.741716i	0.996716	−	0.0809766i	$-0.0258039\pi$
$349$	8.00000	+	13.8564i	0.428230	+	0.741716i	−0.568486	+	0.822693i	$0.692471\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	17.5000	+	30.3109i	0.931431	+	1.61329i	0.780878	+	0.624684i	$0.214772\pi$
$353$	17.5000	+	30.3109i	0.931431	+	1.61329i	0.150553	+	0.988602i	$0.451894\pi$
$354$		0			0
$355$	16.0000	−	27.7128i	0.849192	−	1.47084i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	14.0000			0.738892			0.369446	−	0.929252i	$-0.379548\pi$
$359$	14.0000			0.738892			0.369446	+	0.929252i	$0.379548\pi$
$360$		0			0
$361$	−18.0000			−0.947368
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−6.00000	+	10.3923i	−0.314054	+	0.543958i
$366$		0			0
$367$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$367$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−4.00000	−	6.92820i	−0.207670	−	0.359694i
$372$		0			0
$373$	−11.0000	+	19.0526i	−0.569558	+	0.986504i	0.427051	+	0.904227i	$0.359552\pi$
$373$	−11.0000	+	19.0526i	−0.569558	+	0.986504i	−0.996610	+	0.0822766i	$0.973781\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−4.00000			−0.206010
$378$		0			0
$379$	−1.00000			−0.0513665			−0.0256833	−	0.999670i	$-0.508176\pi$
$379$	−1.00000			−0.0513665			−0.0256833	+	0.999670i	$0.508176\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	10.0000	−	17.3205i	0.510976	−	0.885037i	−0.488943	−	0.872316i	$-0.662617\pi$
$383$	10.0000	−	17.3205i	0.510976	−	0.885037i	0.999919	−	0.0127209i	$-0.00404928\pi$
$384$		0			0
$385$	−20.0000	−	34.6410i	−1.01929	−	1.76547i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	3.00000	+	5.19615i	0.152106	+	0.263455i	0.932002	−	0.362454i	$-0.118061\pi$
$389$	3.00000	+	5.19615i	0.152106	+	0.263455i	−0.779895	+	0.625910i	$0.784728\pi$
$390$		0			0
$391$	−9.00000	+	15.5885i	−0.455150	+	0.788342i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	32.0000			1.61009
$396$		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
$401$	2.50000	−	4.33013i	0.124844	−	0.216236i	−0.796828	−	0.604206i	$-0.793490\pi$
$401$	2.50000	−	4.33013i	0.124844	−	0.216236i	0.921672	+	0.387970i	$0.126824\pi$
$402$		0			0
$403$	4.00000	+	6.92820i	0.199254	+	0.345118i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	20.0000	+	34.6410i	0.991363	+	1.71709i
$408$		0			0
$409$	−12.5000	+	21.6506i	−0.618085	+	1.07056i	0.371750	+	0.928333i	$0.378758\pi$
$409$	−12.5000	+	21.6506i	−0.618085	+	1.07056i	−0.989835	+	0.142222i	$0.954575\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−10.0000			−0.492068
$414$		0			0
$415$	48.0000			2.35623
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−2.00000	+	3.46410i	−0.0977064	+	0.169232i	−0.910735	−	0.412991i	$-0.864484\pi$
$419$	−2.00000	+	3.46410i	−0.0977064	+	0.169232i	0.813029	+	0.582224i	$0.197817\pi$
$420$		0			0
$421$	4.00000	+	6.92820i	0.194948	+	0.337660i	0.946883	−	0.321577i	$-0.104213\pi$
$421$	4.00000	+	6.92820i	0.194948	+	0.337660i	−0.751935	+	0.659237i	$0.770879\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−16.5000	−	28.5788i	−0.800368	−	1.38628i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−30.0000			−1.44505			−0.722525	−	0.691345i	$-0.757018\pi$
$431$	−30.0000			−1.44505			−0.722525	+	0.691345i	$0.757018\pi$
$432$		0			0
$433$	1.00000			0.0480569			0.0240285	−	0.999711i	$-0.492351\pi$
$433$	1.00000			0.0480569			0.0240285	+	0.999711i	$0.492351\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	3.00000	−	5.19615i	0.143509	−	0.248566i
$438$		0			0
$439$	4.00000	+	6.92820i	0.190910	+	0.330665i	0.945552	−	0.325471i	$-0.105523\pi$
$439$	4.00000	+	6.92820i	0.190910	+	0.330665i	−0.754642	+	0.656136i	$0.772190\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	13.5000	+	23.3827i	0.641404	+	1.11094i	0.985119	+	0.171871i	$0.0549812\pi$
$443$	13.5000	+	23.3827i	0.641404	+	1.11094i	−0.343715	+	0.939074i	$0.611685\pi$
$444$		0			0
$445$	−20.0000	+	34.6410i	−0.948091	+	1.64214i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	31.0000			1.46298			0.731490	−	0.681852i	$-0.238825\pi$
$449$	31.0000			1.46298			0.731490	+	0.681852i	$0.238825\pi$
$450$		0			0
$451$	5.00000			0.235441
$452$		0			0
$453$		0			0
$454$		0			0
$455$	8.00000	−	13.8564i	0.375046	−	0.649598i
$456$		0			0
$457$	3.50000	+	6.06218i	0.163723	+	0.283577i	0.936201	−	0.351465i	$-0.114316\pi$
$457$	3.50000	+	6.06218i	0.163723	+	0.283577i	−0.772478	+	0.635042i	$0.780983\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	9.00000	+	15.5885i	0.419172	+	0.726027i	0.995856	−	0.0909401i	$-0.0289872\pi$
$461$	9.00000	+	15.5885i	0.419172	+	0.726027i	−0.576685	+	0.816967i	$0.695654\pi$
$462$		0			0
$463$	8.00000	−	13.8564i	0.371792	−	0.643962i	−0.618050	−	0.786139i	$-0.712077\pi$
$463$	8.00000	−	13.8564i	0.371792	−	0.643962i	0.989841	+	0.142177i	$0.0454103\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	7.00000			0.323921			0.161961	−	0.986797i	$-0.448218\pi$
$467$	7.00000			0.323921			0.161961	+	0.986797i	$0.448218\pi$
$468$		0			0
$469$	−26.0000			−1.20057
$470$		0			0
$471$		0			0
$472$		0			0
$473$	17.5000	−	30.3109i	0.804651	−	1.39370i
$474$		0			0
$475$	5.50000	+	9.52628i	0.252357	+	0.437096i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	9.00000	+	15.5885i	0.411220	+	0.712255i	0.995023	−	0.0996406i	$-0.0317693\pi$
$479$	9.00000	+	15.5885i	0.411220	+	0.712255i	−0.583803	+	0.811895i	$0.698436\pi$
$480$		0			0
$481$	−8.00000	+	13.8564i	−0.364769	+	0.631798i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−44.0000			−1.99794
$486$		0			0
$487$	22.0000			0.996915			0.498458	−	0.866914i	$-0.333900\pi$
$487$	22.0000			0.996915			0.498458	+	0.866914i	$0.333900\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−3.50000	+	6.06218i	−0.157953	+	0.273582i	−0.934130	−	0.356932i	$-0.883823\pi$
$491$	−3.50000	+	6.06218i	−0.157953	+	0.273582i	0.776178	+	0.630514i	$0.217156\pi$
$492$		0			0
$493$	3.00000	+	5.19615i	0.135113	+	0.234023i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−8.00000	−	13.8564i	−0.358849	−	0.621545i
$498$		0			0
$499$	10.5000	−	18.1865i	0.470045	−	0.814141i	−0.529369	−	0.848392i	$-0.677571\pi$
$499$	10.5000	−	18.1865i	0.470045	−	0.814141i	0.999413	+	0.0342508i	$0.0109045\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−24.0000			−1.07011			−0.535054	−	0.844818i	$-0.679709\pi$
$503$	−24.0000			−1.07011			−0.535054	+	0.844818i	$0.679709\pi$
$504$		0			0
$505$	48.0000			2.13597
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−12.0000	+	20.7846i	−0.531891	+	0.921262i	0.467416	+	0.884037i	$0.345185\pi$
$509$	−12.0000	+	20.7846i	−0.531891	+	0.921262i	−0.999307	+	0.0372243i	$0.988148\pi$
$510$		0			0
$511$	3.00000	+	5.19615i	0.132712	+	0.229864i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−12.0000	−	20.7846i	−0.528783	−	0.915879i
$516$		0			0
$517$	−5.00000	+	8.66025i	−0.219900	+	0.380878i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	3.00000			0.131432			0.0657162	−	0.997838i	$-0.479067\pi$
$521$	3.00000			0.131432			0.0657162	+	0.997838i	$0.479067\pi$
$522$		0			0
$523$	12.0000			0.524723			0.262362	−	0.964970i	$-0.415499\pi$
$523$	12.0000			0.524723			0.262362	+	0.964970i	$0.415499\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	6.00000	−	10.3923i	0.261364	−	0.452696i
$528$		0			0
$529$	−6.50000	−	11.2583i	−0.282609	−	0.489493i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	1.00000	+	1.73205i	0.0433148	+	0.0750234i
$534$		0			0
$535$	−10.0000	+	17.3205i	−0.432338	+	0.748831i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	15.0000			0.646096
$540$		0			0
$541$	32.0000			1.37579			0.687894	−	0.725811i	$-0.258536\pi$
$541$	32.0000			1.37579			0.687894	+	0.725811i	$0.258536\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	16.0000	−	27.7128i	0.685365	−	1.18709i
$546$		0			0
$547$	16.5000	+	28.5788i	0.705489	+	1.22194i	0.966515	+	0.256611i	$0.0826059\pi$
$547$	16.5000	+	28.5788i	0.705489	+	1.22194i	−0.261026	+	0.965332i	$0.584061\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−1.00000	−	1.73205i	−0.0426014	−	0.0737878i
$552$		0			0
$553$	8.00000	−	13.8564i	0.340195	−	0.589234i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	10.0000			0.423714			0.211857	−	0.977301i	$-0.432049\pi$
$557$	10.0000			0.423714			0.211857	+	0.977301i	$0.432049\pi$
$558$		0			0
$559$	14.0000			0.592137
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−7.50000	+	12.9904i	−0.316087	+	0.547479i	−0.979668	−	0.200625i	$-0.935703\pi$
$563$	−7.50000	+	12.9904i	−0.316087	+	0.547479i	0.663581	+	0.748105i	$0.269036\pi$
$564$		0			0
$565$	−4.00000	−	6.92820i	−0.168281	−	0.291472i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−5.50000	−	9.52628i	−0.230572	−	0.399362i	0.727405	−	0.686209i	$-0.240726\pi$
$569$	−5.50000	−	9.52628i	−0.230572	−	0.399362i	−0.957977	+	0.286846i	$0.907393\pi$
$570$		0			0
$571$	6.50000	−	11.2583i	0.272017	−	0.471146i	−0.697362	−	0.716720i	$-0.745643\pi$
$571$	6.50000	−	11.2583i	0.272017	−	0.471146i	0.969378	+	0.245573i	$0.0789761\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	66.0000			2.75239
$576$		0			0
$577$	3.00000			0.124892			0.0624458	−	0.998048i	$-0.480110\pi$
$577$	3.00000			0.124892			0.0624458	+	0.998048i	$0.480110\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	12.0000	−	20.7846i	0.497844	−	0.862291i
$582$		0			0
$583$	−10.0000	−	17.3205i	−0.414158	−	0.717342i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−4.50000	−	7.79423i	−0.185735	−	0.321702i	0.758089	−	0.652151i	$-0.226133\pi$
$587$	−4.50000	−	7.79423i	−0.185735	−	0.321702i	−0.943824	+	0.330449i	$0.892800\pi$
$588$		0			0
$589$	−2.00000	+	3.46410i	−0.0824086	+	0.142736i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−6.00000			−0.246390			−0.123195	−	0.992382i	$-0.539314\pi$
$593$	−6.00000			−0.246390			−0.123195	+	0.992382i	$0.539314\pi$
$594$		0			0
$595$	−24.0000			−0.983904
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−20.0000	+	34.6410i	−0.817178	+	1.41539i	0.0905757	+	0.995890i	$0.471129\pi$
$599$	−20.0000	+	34.6410i	−0.817178	+	1.41539i	−0.907754	+	0.419504i	$0.862204\pi$
$600$		0			0
$601$	−9.50000	−	16.4545i	−0.387513	−	0.671192i	0.604601	−	0.796528i	$-0.293332\pi$
$601$	−9.50000	−	16.4545i	−0.387513	−	0.671192i	−0.992114	+	0.125336i	$0.959999\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−28.0000	−	48.4974i	−1.13836	−	1.97170i
$606$		0			0
$607$	−10.0000	+	17.3205i	−0.405887	+	0.703018i	−0.994424	−	0.105453i	$-0.966371\pi$
$607$	−10.0000	+	17.3205i	−0.405887	+	0.703018i	0.588537	+	0.808470i	$0.299704\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−4.00000			−0.161823
$612$		0			0
$613$	−12.0000			−0.484675			−0.242338	−	0.970192i	$-0.577914\pi$
$613$	−12.0000			−0.484675			−0.242338	+	0.970192i	$0.577914\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−22.5000	+	38.9711i	−0.905816	+	1.56892i	−0.0859976	+	0.996295i	$0.527408\pi$
$617$	−22.5000	+	38.9711i	−0.905816	+	1.56892i	−0.819818	+	0.572624i	$0.805926\pi$
$618$		0			0
$619$	10.5000	+	18.1865i	0.422031	+	0.730978i	0.996138	−	0.0878015i	$-0.0279841\pi$
$619$	10.5000	+	18.1865i	0.422031	+	0.730978i	−0.574107	+	0.818780i	$0.694651\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	10.0000	+	17.3205i	0.400642	+	0.693932i
$624$		0			0
$625$	−20.5000	+	35.5070i	−0.820000	+	1.42028i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	24.0000			0.956943
$630$		0			0
$631$	−28.0000			−1.11466			−0.557331	−	0.830290i	$-0.688175\pi$
$631$	−28.0000			−1.11466			−0.557331	+	0.830290i	$0.688175\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−4.00000	+	6.92820i	−0.158735	+	0.274937i
$636$		0			0
$637$	3.00000	+	5.19615i	0.118864	+	0.205879i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	2.50000	+	4.33013i	0.0987441	+	0.171030i	0.911165	−	0.412042i	$-0.135184\pi$
$641$	2.50000	+	4.33013i	0.0987441	+	0.171030i	−0.812421	+	0.583071i	$0.801851\pi$
$642$		0			0
$643$	−7.50000	+	12.9904i	−0.295771	+	0.512291i	−0.975164	−	0.221484i	$-0.928910\pi$
$643$	−7.50000	+	12.9904i	−0.295771	+	0.512291i	0.679393	+	0.733775i	$0.262243\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−6.00000			−0.235884			−0.117942	−	0.993020i	$-0.537630\pi$
$647$	−6.00000			−0.235884			−0.117942	+	0.993020i	$0.537630\pi$
$648$		0			0
$649$	−25.0000			−0.981336
$650$		0			0
$651$		0			0
$652$		0			0
$653$	11.0000	−	19.0526i	0.430463	−	0.745584i	−0.566450	−	0.824096i	$-0.691684\pi$
$653$	11.0000	−	19.0526i	0.430463	−	0.745584i	0.996913	+	0.0785119i	$0.0250169\pi$
$654$		0			0
$655$	−32.0000	−	55.4256i	−1.25034	−	2.16566i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	6.00000	+	10.3923i	0.233727	+	0.404827i	0.958902	−	0.283738i	$-0.0915745\pi$
$659$	6.00000	+	10.3923i	0.233727	+	0.404827i	−0.725175	+	0.688565i	$0.758241\pi$
$660$		0			0
$661$	16.0000	−	27.7128i	0.622328	−	1.07790i	−0.366723	−	0.930330i	$-0.619520\pi$
$661$	16.0000	−	27.7128i	0.622328	−	1.07790i	0.989051	−	0.147573i	$-0.0471463\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	8.00000			0.310227
$666$		0			0
$667$	−12.0000			−0.464642
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−1.00000	−	1.73205i	−0.0385472	−	0.0667657i	0.846108	−	0.533011i	$-0.178940\pi$
$673$	−1.00000	−	1.73205i	−0.0385472	−	0.0667657i	−0.884655	+	0.466246i	$0.845606\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	14.0000	+	24.2487i	0.538064	+	0.931954i	0.999008	+	0.0445248i	$0.0141774\pi$
$677$	14.0000	+	24.2487i	0.538064	+	0.931954i	−0.460945	+	0.887429i	$0.652489\pi$
$678$		0			0
$679$	−11.0000	+	19.0526i	−0.422141	+	0.731170i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−15.0000			−0.573959			−0.286980	−	0.957937i	$-0.592651\pi$
$683$	−15.0000			−0.573959			−0.286980	+	0.957937i	$0.592651\pi$
$684$		0			0
$685$	12.0000			0.458496
$686$		0			0
$687$		0			0
$688$		0			0
$689$	4.00000	−	6.92820i	0.152388	−	0.263944i
$690$		0			0
$691$	−4.00000	−	6.92820i	−0.152167	−	0.263561i	0.779857	−	0.625958i	$-0.215292\pi$
$691$	−4.00000	−	6.92820i	−0.152167	−	0.263561i	−0.932024	+	0.362397i	$0.881959\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−26.0000	−	45.0333i	−0.986236	−	1.70821i
$696$		0			0
$697$	1.50000	−	2.59808i	0.0568166	−	0.0984092i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	42.0000			1.58632			0.793159	−	0.609015i	$-0.208435\pi$
$701$	42.0000			1.58632			0.793159	+	0.609015i	$0.208435\pi$
$702$		0			0
$703$	−8.00000			−0.301726
$704$		0			0
$705$		0			0
$706$		0			0
$707$	12.0000	−	20.7846i	0.451306	−	0.781686i
$708$		0			0
$709$	−10.0000	−	17.3205i	−0.375558	−	0.650485i	0.614852	−	0.788642i	$-0.289216\pi$
$709$	−10.0000	−	17.3205i	−0.375558	−	0.650485i	−0.990410	+	0.138157i	$0.955882\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	12.0000	+	20.7846i	0.449404	+	0.778390i
$714$		0			0
$715$	20.0000	−	34.6410i	0.747958	−	1.29550i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−32.0000			−1.19340			−0.596699	−	0.802465i	$-0.703521\pi$
$719$	−32.0000			−1.19340			−0.596699	+	0.802465i	$0.703521\pi$
$720$		0			0
$721$	−12.0000			−0.446903
$722$		0			0
$723$		0			0
$724$		0			0
$725$	11.0000	−	19.0526i	0.408530	−	0.707594i
$726$		0			0
$727$	15.0000	+	25.9808i	0.556319	+	0.963573i	0.997800	+	0.0663022i	$0.0211201\pi$
$727$	15.0000	+	25.9808i	0.556319	+	0.963573i	−0.441480	+	0.897271i	$0.645547\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−10.5000	−	18.1865i	−0.388357	−	0.672653i
$732$		0			0
$733$	−17.0000	+	29.4449i	−0.627909	+	1.08757i	0.360061	+	0.932929i	$0.382756\pi$
$733$	−17.0000	+	29.4449i	−0.627909	+	1.08757i	−0.987971	+	0.154642i	$0.950578\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−65.0000			−2.39431
$738$		0			0
$739$	−49.0000			−1.80249			−0.901247	−	0.433306i	$-0.857347\pi$
$739$	−49.0000			−1.80249			−0.901247	+	0.433306i	$0.857347\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−3.00000	+	5.19615i	−0.110059	+	0.190628i	−0.915794	−	0.401648i	$-0.868437\pi$
$743$	−3.00000	+	5.19615i	−0.110059	+	0.190628i	0.805735	+	0.592277i	$0.201771\pi$
$744$		0			0
$745$	36.0000	+	62.3538i	1.31894	+	2.28447i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	5.00000	+	8.66025i	0.182696	+	0.316439i
$750$		0			0
$751$	26.0000	−	45.0333i	0.948753	−	1.64329i	0.200698	−	0.979653i	$-0.435679\pi$
$751$	26.0000	−	45.0333i	0.948753	−	1.64329i	0.748056	−	0.663636i	$-0.230988\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−72.0000			−2.62035
$756$		0			0
$757$	−18.0000			−0.654221			−0.327111	−	0.944986i	$-0.606075\pi$
$757$	−18.0000			−0.654221			−0.327111	+	0.944986i	$0.606075\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−15.0000	+	25.9808i	−0.543750	+	0.941802i	0.454935	+	0.890525i	$0.349663\pi$
$761$	−15.0000	+	25.9808i	−0.543750	+	0.941802i	−0.998684	+	0.0512772i	$0.983671\pi$
$762$		0			0
$763$	−8.00000	−	13.8564i	−0.289619	−	0.501636i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−5.00000	−	8.66025i	−0.180540	−	0.312704i
$768$		0			0
$769$	−5.00000	+	8.66025i	−0.180305	+	0.312297i	−0.941984	−	0.335657i	$-0.891042\pi$
$769$	−5.00000	+	8.66025i	−0.180305	+	0.312297i	0.761680	+	0.647954i	$0.224375\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−42.0000			−1.51064			−0.755318	−	0.655359i	$-0.772517\pi$
$773$	−42.0000			−1.51064			−0.755318	+	0.655359i	$0.772517\pi$
$774$		0			0
$775$	−44.0000			−1.58053
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−0.500000	+	0.866025i	−0.0179144	+	0.0310286i
$780$		0			0
$781$	−20.0000	−	34.6410i	−0.715656	−	1.23955i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	40.0000	+	69.2820i	1.42766	+	2.47278i
$786$		0			0
$787$	−18.0000	+	31.1769i	−0.641631	+	1.11134i	0.343438	+	0.939175i	$0.388408\pi$
$787$	−18.0000	+	31.1769i	−0.641631	+	1.11134i	−0.985069	+	0.172162i	$0.944925\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−4.00000			−0.142224
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−12.0000	+	20.7846i	−0.425062	+	0.736229i	−0.996426	−	0.0844678i	$-0.973081\pi$
$797$	−12.0000	+	20.7846i	−0.425062	+	0.736229i	0.571364	+	0.820696i	$0.306414\pi$
$798$		0			0
$799$	3.00000	+	5.19615i	0.106132	+	0.183827i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	7.50000	+	12.9904i	0.264669	+	0.458421i
$804$		0			0
$805$	24.0000	−	41.5692i	0.845889	−	1.46512i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−17.0000			−0.597688			−0.298844	−	0.954302i	$-0.596601\pi$
$809$	−17.0000			−0.597688			−0.298844	+	0.954302i	$0.596601\pi$
$810$		0			0
$811$	−55.0000			−1.93131			−0.965656	−	0.259825i	$-0.916335\pi$
$811$	−55.0000			−1.93131			−0.965656	+	0.259825i	$0.916335\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−40.0000	+	69.2820i	−1.40114	+	2.42684i
$816$		0			0
$817$	3.50000	+	6.06218i	0.122449	+	0.212089i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	25.0000	+	43.3013i	0.872506	+	1.51122i	0.859396	+	0.511311i	$0.170840\pi$
$821$	25.0000	+	43.3013i	0.872506	+	1.51122i	0.0131101	+	0.999914i	$0.495827\pi$
$822$		0			0
$823$	−14.0000	+	24.2487i	−0.488009	+	0.845257i	−0.999905	−	0.0137907i	$-0.995610\pi$
$823$	−14.0000	+	24.2487i	−0.488009	+	0.845257i	0.511896	+	0.859048i	$0.328943\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	28.0000			0.973655			0.486828	−	0.873498i	$-0.338154\pi$
$827$	28.0000			0.973655			0.486828	+	0.873498i	$0.338154\pi$
$828$		0			0
$829$	−14.0000			−0.486240			−0.243120	−	0.969996i	$-0.578171\pi$
$829$	−14.0000			−0.486240			−0.243120	+	0.969996i	$0.578171\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	4.50000	−	7.79423i	0.155916	−	0.270054i
$834$		0			0
$835$	−32.0000	−	55.4256i	−1.10741	−	1.91808i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−2.00000	−	3.46410i	−0.0690477	−	0.119594i	0.829435	−	0.558604i	$-0.188663\pi$
$839$	−2.00000	−	3.46410i	−0.0690477	−	0.119594i	−0.898482	+	0.439010i	$0.855329\pi$
$840$		0			0
$841$	12.5000	−	21.6506i	0.431034	−	0.746574i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−36.0000			−1.23844
$846$		0			0
$847$	−28.0000			−0.962091
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−24.0000	+	41.5692i	−0.822709	+	1.42497i
$852$		0			0
$853$	15.0000	+	25.9808i	0.513590	+	0.889564i	0.999876	+	0.0157644i	$0.00501816\pi$
$853$	15.0000	+	25.9808i	0.513590	+	0.889564i	−0.486286	+	0.873800i	$0.661649\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−5.00000	−	8.66025i	−0.170797	−	0.295829i	0.767902	−	0.640567i	$-0.221301\pi$
$857$	−5.00000	−	8.66025i	−0.170797	−	0.295829i	−0.938699	+	0.344739i	$0.887967\pi$
$858$		0			0
$859$	2.50000	−	4.33013i	0.0852989	−	0.147742i	−0.820220	−	0.572049i	$-0.806149\pi$
$859$	2.50000	−	4.33013i	0.0852989	−	0.147742i	0.905519	+	0.424307i	$0.139482\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	28.0000			0.953131			0.476566	−	0.879139i	$-0.341881\pi$
$863$	28.0000			0.953131			0.476566	+	0.879139i	$0.341881\pi$
$864$		0			0
$865$	72.0000			2.44807
$866$		0			0
$867$		0			0
$868$		0			0
$869$	20.0000	−	34.6410i	0.678454	−	1.17512i
$870$		0			0
$871$	−13.0000	−	22.5167i	−0.440488	−	0.762948i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	24.0000	+	41.5692i	0.811348	+	1.40530i
$876$		0			0
$877$	20.0000	−	34.6410i	0.675352	−	1.16974i	−0.301014	−	0.953620i	$-0.597325\pi$
$877$	20.0000	−	34.6410i	0.675352	−	1.16974i	0.976366	−	0.216124i	$-0.0693416\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−30.0000			−1.01073			−0.505363	−	0.862907i	$-0.668641\pi$
$881$	−30.0000			−1.01073			−0.505363	+	0.862907i	$0.668641\pi$
$882$		0			0
$883$	13.0000			0.437485			0.218742	−	0.975783i	$-0.429805\pi$
$883$	13.0000			0.437485			0.218742	+	0.975783i	$0.429805\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	18.0000	−	31.1769i	0.604381	−	1.04682i	−0.387768	−	0.921757i	$-0.626754\pi$
$887$	18.0000	−	31.1769i	0.604381	−	1.04682i	0.992149	−	0.125061i	$-0.0399128\pi$
$888$		0			0
$889$	2.00000	+	3.46410i	0.0670778	+	0.116182i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−1.00000	−	1.73205i	−0.0334637	−	0.0579609i
$894$		0			0
$895$	−24.0000	+	41.5692i	−0.802232	+	1.38951i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	8.00000			0.266815
$900$		0			0
$901$	−12.0000			−0.399778
$902$		0			0
$903$		0			0
$904$		0			0
$905$	4.00000	−	6.92820i	0.132964	−	0.230301i
$906$		0			0
$907$	3.50000	+	6.06218i	0.116216	+	0.201291i	0.918265	−	0.395966i	$-0.129590\pi$
$907$	3.50000	+	6.06218i	0.116216	+	0.201291i	−0.802049	+	0.597258i	$0.796257\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−12.0000	−	20.7846i	−0.397578	−	0.688625i	0.595849	−	0.803097i	$-0.296816\pi$
$911$	−12.0000	−	20.7846i	−0.397578	−	0.688625i	−0.993426	+	0.114472i	$0.963482\pi$
$912$		0			0
$913$	30.0000	−	51.9615i	0.992855	−	1.71968i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−32.0000			−1.05673
$918$		0			0
$919$	54.0000			1.78130			0.890648	−	0.454694i	$-0.150251\pi$
$919$	54.0000			1.78130			0.890648	+	0.454694i	$0.150251\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	8.00000	−	13.8564i	0.263323	−	0.456089i
$924$		0			0
$925$	−44.0000	−	76.2102i	−1.44671	−	2.50578i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−3.00000	−	5.19615i	−0.0984268	−	0.170480i	0.812607	−	0.582812i	$-0.198048\pi$
$929$	−3.00000	−	5.19615i	−0.0984268	−	0.170480i	−0.911034	+	0.412332i	$0.864714\pi$
$930$		0			0
$931$	−1.50000	+	2.59808i	−0.0491605	+	0.0851485i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−60.0000			−1.96221
$936$		0			0
$937$	−58.0000			−1.89478			−0.947389	−	0.320085i	$-0.896288\pi$
$937$	−58.0000			−1.89478			−0.947389	+	0.320085i	$0.896288\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	14.0000	−	24.2487i	0.456387	−	0.790485i	−0.542380	−	0.840133i	$-0.682477\pi$
$941$	14.0000	−	24.2487i	0.456387	−	0.790485i	0.998767	+	0.0496480i	$0.0158099\pi$
$942$		0			0
$943$	3.00000	+	5.19615i	0.0976934	+	0.169210i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	13.5000	+	23.3827i	0.438691	+	0.759835i	0.997589	−	0.0694014i	$-0.0221089\pi$
$947$	13.5000	+	23.3827i	0.438691	+	0.759835i	−0.558898	+	0.829237i	$0.688776\pi$
$948$		0			0
$949$	−3.00000	+	5.19615i	−0.0973841	+	0.168674i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	39.0000			1.26333			0.631667	−	0.775240i	$-0.282371\pi$
$953$	39.0000			1.26333			0.631667	+	0.775240i	$0.282371\pi$
$954$		0			0
$955$	24.0000			0.776622
$956$		0			0
$957$		0			0
$958$		0			0
$959$	3.00000	−	5.19615i	0.0968751	−	0.167793i
$960$		0			0
$961$	7.50000	+	12.9904i	0.241935	+	0.419045i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	6.00000	+	10.3923i	0.193147	+	0.334540i
$966$		0			0
$967$	21.0000	−	36.3731i	0.675314	−	1.16968i	−0.301062	−	0.953604i	$-0.597341\pi$
$967$	21.0000	−	36.3731i	0.675314	−	1.16968i	0.976377	−	0.216075i	$-0.0693254\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−20.0000			−0.641831			−0.320915	−	0.947108i	$-0.603990\pi$
$971$	−20.0000			−0.641831			−0.320915	+	0.947108i	$0.603990\pi$
$972$		0			0
$973$	−26.0000			−0.833522
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−5.50000	+	9.52628i	−0.175961	+	0.304773i	−0.940493	−	0.339812i	$-0.889636\pi$
$977$	−5.50000	+	9.52628i	−0.175961	+	0.304773i	0.764533	+	0.644585i	$0.222970\pi$
$978$		0			0
$979$	25.0000	+	43.3013i	0.799003	+	1.38391i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	18.0000	+	31.1769i	0.574111	+	0.994389i	0.996138	+	0.0878058i	$0.0279855\pi$
$983$	18.0000	+	31.1769i	0.574111	+	0.994389i	−0.422027	+	0.906583i	$0.638681\pi$
$984$		0			0
$985$	−8.00000	+	13.8564i	−0.254901	+	0.441502i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	42.0000			1.33552
$990$		0			0
$991$	−16.0000			−0.508257			−0.254128	−	0.967170i	$-0.581789\pi$
$991$	−16.0000			−0.508257			−0.254128	+	0.967170i	$0.581789\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−4.00000	+	6.92820i	−0.126809	+	0.219639i
$996$		0			0
$997$	30.0000	+	51.9615i	0.950110	+	1.64564i	0.745182	+	0.666861i	$0.232362\pi$
$997$	30.0000	+	51.9615i	0.950110	+	1.64564i	0.204927	+	0.978777i	$0.434304\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1728.2.i.b.577.1		2
3.2	odd	2		576.2.i.b.193.1		2
4.3	odd	2		1728.2.i.a.577.1		2
8.3	odd	2		864.2.i.a.577.1		2
8.5	even	2		864.2.i.b.577.1		2
9.2	odd	6		576.2.i.b.385.1		2
9.4	even	3		5184.2.a.be.1.1		1
9.5	odd	6		5184.2.a.a.1.1		1
9.7	even	3	inner	1728.2.i.b.1153.1		2
12.11	even	2		576.2.i.h.193.1		2
24.5	odd	2		288.2.i.b.193.1	yes	2
24.11	even	2		288.2.i.a.193.1	yes	2
36.7	odd	6		1728.2.i.a.1153.1		2
36.11	even	6		576.2.i.h.385.1		2
36.23	even	6		5184.2.a.b.1.1		1
36.31	odd	6		5184.2.a.bf.1.1		1
72.5	odd	6		2592.2.a.g.1.1		1
72.11	even	6		288.2.i.a.97.1	✓	2
72.13	even	6		2592.2.a.a.1.1		1
72.29	odd	6		288.2.i.b.97.1	yes	2
72.43	odd	6		864.2.i.a.289.1		2
72.59	even	6		2592.2.a.h.1.1		1
72.61	even	6		864.2.i.b.289.1		2
72.67	odd	6		2592.2.a.b.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
288.2.i.a.97.1	✓	2	72.11	even	6
288.2.i.a.193.1	yes	2	24.11	even	2
288.2.i.b.97.1	yes	2	72.29	odd	6
288.2.i.b.193.1	yes	2	24.5	odd	2
576.2.i.b.193.1		2	3.2	odd	2
576.2.i.b.385.1		2	9.2	odd	6
576.2.i.h.193.1		2	12.11	even	2
576.2.i.h.385.1		2	36.11	even	6
864.2.i.a.289.1		2	72.43	odd	6
864.2.i.a.577.1		2	8.3	odd	2
864.2.i.b.289.1		2	72.61	even	6
864.2.i.b.577.1		2	8.5	even	2
1728.2.i.a.577.1		2	4.3	odd	2
1728.2.i.a.1153.1		2	36.7	odd	6
1728.2.i.b.577.1		2	1.1	even	1	trivial
1728.2.i.b.1153.1		2	9.7	even	3	inner
2592.2.a.a.1.1		1	72.13	even	6
2592.2.a.b.1.1		1	72.67	odd	6
2592.2.a.g.1.1		1	72.5	odd	6
2592.2.a.h.1.1		1	72.59	even	6
5184.2.a.a.1.1		1	9.5	odd	6
5184.2.a.b.1.1		1	36.23	even	6
5184.2.a.be.1.1		1	9.4	even	3
5184.2.a.bf.1.1		1	36.31	odd	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:	\( N \)	\(=\)	\( 1728 = 2^{6} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1728.i (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-2.00000 + 3.46410i) q^{5} +(1.00000 + 1.73205i) q^{7} +O(q^{10})\)	\(q+(-2.00000 + 3.46410i) q^{5} +(1.00000 + 1.73205i) q^{7} +(2.50000 + 4.33013i) q^{11} +(-1.00000 + 1.73205i) q^{13} +3.00000 q^{17} -1.00000 q^{19} +(-3.00000 + 5.19615i) q^{23} +(-5.50000 - 9.52628i) q^{25} +(1.00000 + 1.73205i) q^{29} +(2.00000 - 3.46410i) q^{31} -8.00000 q^{35} +8.00000 q^{37} +(0.500000 - 0.866025i) q^{41} +(-3.50000 - 6.06218i) q^{43} +(1.00000 + 1.73205i) q^{47} +(1.50000 - 2.59808i) q^{49} -4.00000 q^{53} -20.0000 q^{55} +(-2.50000 + 4.33013i) q^{59} +(-4.00000 - 6.92820i) q^{65} +(-6.50000 + 11.2583i) q^{67} -8.00000 q^{71} +3.00000 q^{73} +(-5.00000 + 8.66025i) q^{77} +(-4.00000 - 6.92820i) q^{79} +(-6.00000 - 10.3923i) q^{83} +(-6.00000 + 10.3923i) q^{85} +10.0000 q^{89} -4.00000 q^{91} +(2.00000 - 3.46410i) q^{95} +(5.50000 + 9.52628i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{5} + 2 q^{7}+O(q^{10}) \) 2 * q - 4 * q^5 + 2 * q^7	\( 2 q - 4 q^{5} + 2 q^{7} + 5 q^{11} - 2 q^{13} + 6 q^{17} - 2 q^{19} - 6 q^{23} - 11 q^{25} + 2 q^{29} + 4 q^{31} - 16 q^{35} + 16 q^{37} + q^{41} - 7 q^{43} + 2 q^{47} + 3 q^{49} - 8 q^{53} - 40 q^{55} - 5 q^{59} - 8 q^{65} - 13 q^{67} - 16 q^{71} + 6 q^{73} - 10 q^{77} - 8 q^{79} - 12 q^{83} - 12 q^{85} + 20 q^{89} - 8 q^{91} + 4 q^{95} + 11 q^{97}+O(q^{100}) \) 2 * q - 4 * q^5 + 2 * q^7 + 5 * q^11 - 2 * q^13 + 6 * q^17 - 2 * q^19 - 6 * q^23 - 11 * q^25 + 2 * q^29 + 4 * q^31 - 16 * q^35 + 16 * q^37 + q^41 - 7 * q^43 + 2 * q^47 + 3 * q^49 - 8 * q^53 - 40 * q^55 - 5 * q^59 - 8 * q^65 - 13 * q^67 - 16 * q^71 + 6 * q^73 - 10 * q^77 - 8 * q^79 - 12 * q^83 - 12 * q^85 + 20 * q^89 - 8 * q^91 + 4 * q^95 + 11 * q^97

Introduction

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