LMFDB - Embedded newform 1800.2.d.g.1549.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1800,2,Mod(1549,1800)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1800.1549"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1800, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1800 = 2^{3} \cdot 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1800.d (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,2,0,0,0,0,0,-4,0,0,0,0,-12,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(14)] == traces)

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

Self dual:	no
Analytic conductor:	$14.3730723638$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 360)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1549.1
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	1800.1549
Dual form			1800.2.d.g.1549.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(1.00000 - 1.00000i) q^{2} -2.00000i q^{4} +4.00000i q^{7} +(-2.00000 - 2.00000i) q^{8} +2.00000i q^{11} -6.00000 q^{13} +(4.00000 + 4.00000i) q^{14} -4.00000 q^{16} -6.00000i q^{17} +4.00000i q^{19} +(2.00000 + 2.00000i) q^{22} +8.00000i q^{23} +(-6.00000 + 6.00000i) q^{26} +8.00000 q^{28} +6.00000i q^{29} -2.00000 q^{31} +(-4.00000 + 4.00000i) q^{32} +(-6.00000 - 6.00000i) q^{34} +2.00000 q^{37} +(4.00000 + 4.00000i) q^{38} +4.00000 q^{41} -4.00000 q^{43} +4.00000 q^{44} +(8.00000 + 8.00000i) q^{46} +4.00000i q^{47} -9.00000 q^{49} +12.0000i q^{52} -10.0000 q^{53} +(8.00000 - 8.00000i) q^{56} +(6.00000 + 6.00000i) q^{58} +2.00000i q^{59} -4.00000i q^{61} +(-2.00000 + 2.00000i) q^{62} +8.00000i q^{64} -12.0000 q^{67} -12.0000 q^{68} +8.00000 q^{71} +10.0000i q^{73} +(2.00000 - 2.00000i) q^{74} +8.00000 q^{76} -8.00000 q^{77} +2.00000 q^{79} +(4.00000 - 4.00000i) q^{82} +(-4.00000 + 4.00000i) q^{86} +(4.00000 - 4.00000i) q^{88} +4.00000 q^{89} -24.0000i q^{91} +16.0000 q^{92} +(4.00000 + 4.00000i) q^{94} -2.00000i q^{97} +(-9.00000 + 9.00000i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} - 4 q^{8} - 12 q^{13} + 8 q^{14} - 8 q^{16} + 4 q^{22} - 12 q^{26} + 16 q^{28} - 4 q^{31} - 8 q^{32} - 12 q^{34} + 4 q^{37} + 8 q^{38} + 8 q^{41} - 8 q^{43} + 8 q^{44} + 16 q^{46} - 18 q^{49}+ \cdots - 18 q^{98}+O(q^{100}) $ 2 * q + 2 * q^2 - 4 * q^8 - 12 * q^13 + 8 * q^14 - 8 * q^16 + 4 * q^22 - 12 * q^26 + 16 * q^28 - 4 * q^31 - 8 * q^32 - 12 * q^34 + 4 * q^37 + 8 * q^38 + 8 * q^41 - 8 * q^43 + 8 * q^44 + 16 * q^46 - 18 * q^49 - 20 * q^53 + 16 * q^56 + 12 * q^58 - 4 * q^62 - 24 * q^67 - 24 * q^68 + 16 * q^71 + 4 * q^74 + 16 * q^76 - 16 * q^77 + 4 * q^79 + 8 * q^82 - 8 * q^86 + 8 * q^88 + 8 * q^89 + 32 * q^92 + 8 * q^94 - 18 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1800\mathbb{Z}\right)^\times$.

$n$	$577$	$901$	$1001$	$1351$
$\chi(n)$	$-1$	$-1$	$1$	$1$

Coefficient data

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

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

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

$2$	1.00000	−	1.00000i	0.707107	−	0.707107i
$2$	1.00000	−	1.00000i	0.707107	−	0.707107i
$3$		0			0
$3$		0			0
$4$		−	2.00000i		−	1.00000i
$5$		0			0
$5$		0			0
$6$		0			0
$7$			4.00000i			1.51186i	0.654654	+	0.755929i	$0.272814\pi$
$7$			4.00000i			1.51186i	−0.654654	+	0.755929i	$0.727186\pi$
$8$	−2.00000	−	2.00000i	−0.707107	−	0.707107i
$9$		0			0
$10$		0			0
$11$			2.00000i			0.603023i	0.953463	+	0.301511i	$0.0974911\pi$
$11$			2.00000i			0.603023i	−0.953463	+	0.301511i	$0.902509\pi$
$12$		0			0
$13$	−6.00000			−1.66410			−0.832050	−	0.554700i	$-0.812833\pi$
$13$	−6.00000			−1.66410			−0.832050	+	0.554700i	$0.812833\pi$
$14$	4.00000	+	4.00000i	1.06904	+	1.06904i
$15$		0			0
$16$	−4.00000			−1.00000
$17$		−	6.00000i		−	1.45521i	−0.685994	−	0.727607i	$-0.740633\pi$
$17$		−	6.00000i		−	1.45521i	0.685994	−	0.727607i	$-0.259367\pi$
$18$		0			0
$19$			4.00000i			0.917663i	0.888523	+	0.458831i	$0.151732\pi$
$19$			4.00000i			0.917663i	−0.888523	+	0.458831i	$0.848268\pi$
$20$		0			0
$21$		0			0
$22$	2.00000	+	2.00000i	0.426401	+	0.426401i
$23$			8.00000i			1.66812i	0.551677	+	0.834058i	$0.313988\pi$
$23$			8.00000i			1.66812i	−0.551677	+	0.834058i	$0.686012\pi$
$24$		0			0
$25$		0			0
$26$	−6.00000	+	6.00000i	−1.17670	+	1.17670i
$27$		0			0
$28$	8.00000			1.51186
$29$			6.00000i			1.11417i	0.830455	+	0.557086i	$0.188081\pi$
$29$			6.00000i			1.11417i	−0.830455	+	0.557086i	$0.811919\pi$
$30$		0			0
$31$	−2.00000			−0.359211			−0.179605	−	0.983739i	$-0.557482\pi$
$31$	−2.00000			−0.359211			−0.179605	+	0.983739i	$0.557482\pi$
$32$	−4.00000	+	4.00000i	−0.707107	+	0.707107i
$33$		0			0
$34$	−6.00000	−	6.00000i	−1.02899	−	1.02899i
$35$		0			0
$36$		0			0
$37$	2.00000			0.328798			0.164399	−	0.986394i	$-0.447432\pi$
$37$	2.00000			0.328798			0.164399	+	0.986394i	$0.447432\pi$
$38$	4.00000	+	4.00000i	0.648886	+	0.648886i
$39$		0			0
$40$		0			0
$41$	4.00000			0.624695			0.312348	−	0.949968i	$-0.398885\pi$
$41$	4.00000			0.624695			0.312348	+	0.949968i	$0.398885\pi$
$42$		0			0
$43$	−4.00000			−0.609994			−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000			−0.609994			−0.304997	+	0.952353i	$0.598656\pi$
$44$	4.00000			0.603023
$45$		0			0
$46$	8.00000	+	8.00000i	1.17954	+	1.17954i
$47$			4.00000i			0.583460i	0.956501	+	0.291730i	$0.0942309\pi$
$47$			4.00000i			0.583460i	−0.956501	+	0.291730i	$0.905769\pi$
$48$		0			0
$49$	−9.00000			−1.28571
$50$		0			0
$51$		0			0
$52$			12.0000i			1.66410i
$53$	−10.0000			−1.37361			−0.686803	−	0.726844i	$-0.740986\pi$
$53$	−10.0000			−1.37361			−0.686803	+	0.726844i	$0.740986\pi$
$54$		0			0
$55$		0			0
$56$	8.00000	−	8.00000i	1.06904	−	1.06904i
$57$		0			0
$58$	6.00000	+	6.00000i	0.787839	+	0.787839i
$59$			2.00000i			0.260378i	0.991489	+	0.130189i	$0.0415584\pi$
$59$			2.00000i			0.260378i	−0.991489	+	0.130189i	$0.958442\pi$
$60$		0			0
$61$		−	4.00000i		−	0.512148i	−0.966657	−	0.256074i	$-0.917571\pi$
$61$		−	4.00000i		−	0.512148i	0.966657	−	0.256074i	$-0.0824290\pi$
$62$	−2.00000	+	2.00000i	−0.254000	+	0.254000i
$63$		0			0
$64$			8.00000i			1.00000i
$65$		0			0
$66$		0			0
$67$	−12.0000			−1.46603			−0.733017	−	0.680211i	$-0.761888\pi$
$67$	−12.0000			−1.46603			−0.733017	+	0.680211i	$0.761888\pi$
$68$	−12.0000			−1.45521
$69$		0			0
$70$		0			0
$71$	8.00000			0.949425			0.474713	−	0.880141i	$-0.342552\pi$
$71$	8.00000			0.949425			0.474713	+	0.880141i	$0.342552\pi$
$72$		0			0
$73$			10.0000i			1.17041i	0.810885	+	0.585206i	$0.198986\pi$
$73$			10.0000i			1.17041i	−0.810885	+	0.585206i	$0.801014\pi$
$74$	2.00000	−	2.00000i	0.232495	−	0.232495i
$75$		0			0
$76$	8.00000			0.917663
$77$	−8.00000			−0.911685
$78$		0			0
$79$	2.00000			0.225018			0.112509	−	0.993651i	$-0.464111\pi$
$79$	2.00000			0.225018			0.112509	+	0.993651i	$0.464111\pi$
$80$		0			0
$81$		0			0
$82$	4.00000	−	4.00000i	0.441726	−	0.441726i
$83$		0			0			−	1.00000i	$-0.5\pi$
$83$		0			0				1.00000i	$0.5\pi$
$84$		0			0
$85$		0			0
$86$	−4.00000	+	4.00000i	−0.431331	+	0.431331i
$87$		0			0
$88$	4.00000	−	4.00000i	0.426401	−	0.426401i
$89$	4.00000			0.423999			0.212000	−	0.977270i	$-0.432002\pi$
$89$	4.00000			0.423999			0.212000	+	0.977270i	$0.432002\pi$
$90$		0			0
$91$		−	24.0000i		−	2.51588i
$92$	16.0000			1.66812
$93$		0			0
$94$	4.00000	+	4.00000i	0.412568	+	0.412568i
$95$		0			0
$96$		0			0
$97$		−	2.00000i		−	0.203069i	−0.994832	−	0.101535i	$-0.967625\pi$
$97$		−	2.00000i		−	0.203069i	0.994832	−	0.101535i	$-0.0323753\pi$
$98$	−9.00000	+	9.00000i	−0.909137	+	0.909137i
$99$		0			0
$100$		0			0
$101$			10.0000i			0.995037i	0.867453	+	0.497519i	$0.165755\pi$
$101$			10.0000i			0.995037i	−0.867453	+	0.497519i	$0.834245\pi$
$102$		0			0
$103$		−	8.00000i		−	0.788263i	−0.919054	−	0.394132i	$-0.871045\pi$
$103$		−	8.00000i		−	0.788263i	0.919054	−	0.394132i	$-0.128955\pi$
$104$	12.0000	+	12.0000i	1.17670	+	1.17670i
$105$		0			0
$106$	−10.0000	+	10.0000i	−0.971286	+	0.971286i
$107$	20.0000			1.93347			0.966736	−	0.255774i	$-0.0823304\pi$
$107$	20.0000			1.93347			0.966736	+	0.255774i	$0.0823304\pi$
$108$		0			0
$109$		−	8.00000i		−	0.766261i	−0.923694	−	0.383131i	$-0.874846\pi$
$109$		−	8.00000i		−	0.766261i	0.923694	−	0.383131i	$-0.125154\pi$
$110$		0			0
$111$		0			0
$112$		−	16.0000i		−	1.51186i
$113$			6.00000i			0.564433i	0.959351	+	0.282216i	$0.0910696\pi$
$113$			6.00000i			0.564433i	−0.959351	+	0.282216i	$0.908930\pi$
$114$		0			0
$115$		0			0
$116$	12.0000			1.11417
$117$		0			0
$118$	2.00000	+	2.00000i	0.184115	+	0.184115i
$119$	24.0000			2.20008
$120$		0			0
$121$	7.00000			0.636364
$122$	−4.00000	−	4.00000i	−0.362143	−	0.362143i
$123$		0			0
$124$			4.00000i			0.359211i
$125$		0			0
$126$		0			0
$127$		0			0		1.00000			$0$
$127$		0			0		−1.00000			$\pi$
$128$	8.00000	+	8.00000i	0.707107	+	0.707107i
$129$		0			0
$130$		0			0
$131$		−	18.0000i		−	1.57267i	−0.617802	−	0.786334i	$-0.711977\pi$
$131$		−	18.0000i		−	1.57267i	0.617802	−	0.786334i	$-0.288023\pi$
$132$		0			0
$133$	−16.0000			−1.38738
$134$	−12.0000	+	12.0000i	−1.03664	+	1.03664i
$135$		0			0
$136$	−12.0000	+	12.0000i	−1.02899	+	1.02899i
$137$			2.00000i			0.170872i	0.996344	+	0.0854358i	$0.0272282\pi$
$137$			2.00000i			0.170872i	−0.996344	+	0.0854358i	$0.972772\pi$
$138$		0			0
$139$		−	4.00000i		−	0.339276i	−0.985506	−	0.169638i	$-0.945740\pi$
$139$		−	4.00000i		−	0.339276i	0.985506	−	0.169638i	$-0.0542598\pi$
$140$		0			0
$141$		0			0
$142$	8.00000	−	8.00000i	0.671345	−	0.671345i
$143$		−	12.0000i		−	1.00349i
$144$		0			0
$145$		0			0
$146$	10.0000	+	10.0000i	0.827606	+	0.827606i
$147$		0			0
$148$		−	4.00000i		−	0.328798i
$149$			6.00000i			0.491539i	0.969328	+	0.245770i	$0.0790407\pi$
$149$			6.00000i			0.491539i	−0.969328	+	0.245770i	$0.920959\pi$
$150$		0			0
$151$	−2.00000			−0.162758			−0.0813788	−	0.996683i	$-0.525932\pi$
$151$	−2.00000			−0.162758			−0.0813788	+	0.996683i	$0.525932\pi$
$152$	8.00000	−	8.00000i	0.648886	−	0.648886i
$153$		0			0
$154$	−8.00000	+	8.00000i	−0.644658	+	0.644658i
$155$		0			0
$156$		0			0
$157$	−14.0000			−1.11732			−0.558661	−	0.829396i	$-0.688685\pi$
$157$	−14.0000			−1.11732			−0.558661	+	0.829396i	$0.688685\pi$
$158$	2.00000	−	2.00000i	0.159111	−	0.159111i
$159$		0			0
$160$		0			0
$161$	−32.0000			−2.52195
$162$		0			0
$163$	−16.0000			−1.25322			−0.626608	−	0.779334i	$-0.715557\pi$
$163$	−16.0000			−1.25322			−0.626608	+	0.779334i	$0.715557\pi$
$164$		−	8.00000i		−	0.624695i
$165$		0			0
$166$		0			0
$167$			12.0000i			0.928588i	0.885681	+	0.464294i	$0.153692\pi$
$167$			12.0000i			0.928588i	−0.885681	+	0.464294i	$0.846308\pi$
$168$		0			0
$169$	23.0000			1.76923
$170$		0			0
$171$		0			0
$172$			8.00000i			0.609994i
$173$	14.0000			1.06440			0.532200	−	0.846619i	$-0.321365\pi$
$173$	14.0000			1.06440			0.532200	+	0.846619i	$0.321365\pi$
$174$		0			0
$175$		0			0
$176$		−	8.00000i		−	0.603023i
$177$		0			0
$178$	4.00000	−	4.00000i	0.299813	−	0.299813i
$179$			6.00000i			0.448461i	0.974536	+	0.224231i	$0.0719869\pi$
$179$			6.00000i			0.448461i	−0.974536	+	0.224231i	$0.928013\pi$
$180$		0			0
$181$		−	16.0000i		−	1.18927i	−0.803996	−	0.594635i	$-0.797296\pi$
$181$		−	16.0000i		−	1.18927i	0.803996	−	0.594635i	$-0.202704\pi$
$182$	−24.0000	−	24.0000i	−1.77900	−	1.77900i
$183$		0			0
$184$	16.0000	−	16.0000i	1.17954	−	1.17954i
$185$		0			0
$186$		0			0
$187$	12.0000			0.877527
$188$	8.00000			0.583460
$189$		0			0
$190$		0			0
$191$	16.0000			1.15772			0.578860	−	0.815427i	$-0.303498\pi$
$191$	16.0000			1.15772			0.578860	+	0.815427i	$0.303498\pi$
$192$		0			0
$193$		−	26.0000i		−	1.87152i	−0.352636	−	0.935760i	$-0.614715\pi$
$193$		−	26.0000i		−	1.87152i	0.352636	−	0.935760i	$-0.385285\pi$
$194$	−2.00000	−	2.00000i	−0.143592	−	0.143592i
$195$		0			0
$196$			18.0000i			1.28571i
$197$	−14.0000			−0.997459			−0.498729	−	0.866758i	$-0.666200\pi$
$197$	−14.0000			−0.997459			−0.498729	+	0.866758i	$0.666200\pi$
$198$		0			0
$199$	6.00000			0.425329			0.212664	−	0.977125i	$-0.431786\pi$
$199$	6.00000			0.425329			0.212664	+	0.977125i	$0.431786\pi$
$200$		0			0
$201$		0			0
$202$	10.0000	+	10.0000i	0.703598	+	0.703598i
$203$	−24.0000			−1.68447
$204$		0			0
$205$		0			0
$206$	−8.00000	−	8.00000i	−0.557386	−	0.557386i
$207$		0			0
$208$	24.0000			1.66410
$209$	−8.00000			−0.553372
$210$		0			0
$211$			12.0000i			0.826114i	0.910705	+	0.413057i	$0.135539\pi$
$211$			12.0000i			0.826114i	−0.910705	+	0.413057i	$0.864461\pi$
$212$			20.0000i			1.37361i
$213$		0			0
$214$	20.0000	−	20.0000i	1.36717	−	1.36717i
$215$		0			0
$216$		0			0
$217$		−	8.00000i		−	0.543075i
$218$	−8.00000	−	8.00000i	−0.541828	−	0.541828i
$219$		0			0
$220$		0			0
$221$			36.0000i			2.42162i
$222$		0			0
$223$			16.0000i			1.07144i	0.844396	+	0.535720i	$0.179960\pi$
$223$			16.0000i			1.07144i	−0.844396	+	0.535720i	$0.820040\pi$
$224$	−16.0000	−	16.0000i	−1.06904	−	1.06904i
$225$		0			0
$226$	6.00000	+	6.00000i	0.399114	+	0.399114i
$227$	−12.0000			−0.796468			−0.398234	−	0.917284i	$-0.630377\pi$
$227$	−12.0000			−0.796468			−0.398234	+	0.917284i	$0.630377\pi$
$228$		0			0
$229$			20.0000i			1.32164i	0.750546	+	0.660819i	$0.229791\pi$
$229$			20.0000i			1.32164i	−0.750546	+	0.660819i	$0.770209\pi$
$230$		0			0
$231$		0			0
$232$	12.0000	−	12.0000i	0.787839	−	0.787839i
$233$			6.00000i			0.393073i	0.980497	+	0.196537i	$0.0629694\pi$
$233$			6.00000i			0.393073i	−0.980497	+	0.196537i	$0.937031\pi$
$234$		0			0
$235$		0			0
$236$	4.00000			0.260378
$237$		0			0
$238$	24.0000	−	24.0000i	1.55569	−	1.55569i
$239$	−16.0000			−1.03495			−0.517477	−	0.855697i	$-0.673129\pi$
$239$	−16.0000			−1.03495			−0.517477	+	0.855697i	$0.673129\pi$
$240$		0			0
$241$	−18.0000			−1.15948			−0.579741	−	0.814801i	$-0.696846\pi$
$241$	−18.0000			−1.15948			−0.579741	+	0.814801i	$0.696846\pi$
$242$	7.00000	−	7.00000i	0.449977	−	0.449977i
$243$		0			0
$244$	−8.00000			−0.512148
$245$		0			0
$246$		0			0
$247$		−	24.0000i		−	1.52708i
$248$	4.00000	+	4.00000i	0.254000	+	0.254000i
$249$		0			0
$250$		0			0
$251$		−	18.0000i		−	1.13615i	−0.822977	−	0.568075i	$-0.807688\pi$
$251$		−	18.0000i		−	1.13615i	0.822977	−	0.568075i	$-0.192312\pi$
$252$		0			0
$253$	−16.0000			−1.00591
$254$		0			0
$255$		0			0
$256$	16.0000			1.00000
$257$		−	22.0000i		−	1.37232i	−0.727450	−	0.686161i	$-0.759294\pi$
$257$		−	22.0000i		−	1.37232i	0.727450	−	0.686161i	$-0.240706\pi$
$258$		0			0
$259$			8.00000i			0.497096i
$260$		0			0
$261$		0			0
$262$	−18.0000	−	18.0000i	−1.11204	−	1.11204i
$263$		−	12.0000i		−	0.739952i	−0.929041	−	0.369976i	$-0.879366\pi$
$263$		−	12.0000i		−	0.739952i	0.929041	−	0.369976i	$-0.120634\pi$
$264$		0			0
$265$		0			0
$266$	−16.0000	+	16.0000i	−0.981023	+	0.981023i
$267$		0			0
$268$			24.0000i			1.46603i
$269$		−	2.00000i		−	0.121942i	−0.998140	−	0.0609711i	$-0.980580\pi$
$269$		−	2.00000i		−	0.121942i	0.998140	−	0.0609711i	$-0.0194197\pi$
$270$		0			0
$271$	−2.00000			−0.121491			−0.0607457	−	0.998153i	$-0.519348\pi$
$271$	−2.00000			−0.121491			−0.0607457	+	0.998153i	$0.519348\pi$
$272$			24.0000i			1.45521i
$273$		0			0
$274$	2.00000	+	2.00000i	0.120824	+	0.120824i
$275$		0			0
$276$		0			0
$277$	−2.00000			−0.120168			−0.0600842	−	0.998193i	$-0.519137\pi$
$277$	−2.00000			−0.120168			−0.0600842	+	0.998193i	$0.519137\pi$
$278$	−4.00000	−	4.00000i	−0.239904	−	0.239904i
$279$		0			0
$280$		0			0
$281$	−28.0000			−1.67034			−0.835170	−	0.549992i	$-0.814631\pi$
$281$	−28.0000			−1.67034			−0.835170	+	0.549992i	$0.814631\pi$
$282$		0			0
$283$	−28.0000			−1.66443			−0.832214	−	0.554455i	$-0.812927\pi$
$283$	−28.0000			−1.66443			−0.832214	+	0.554455i	$0.812927\pi$
$284$		−	16.0000i		−	0.949425i
$285$		0			0
$286$	−12.0000	−	12.0000i	−0.709575	−	0.709575i
$287$			16.0000i			0.944450i
$288$		0			0
$289$	−19.0000			−1.11765
$290$		0			0
$291$		0			0
$292$	20.0000			1.17041
$293$	22.0000			1.28525			0.642627	−	0.766179i	$-0.277845\pi$
$293$	22.0000			1.28525			0.642627	+	0.766179i	$0.277845\pi$
$294$		0			0
$295$		0			0
$296$	−4.00000	−	4.00000i	−0.232495	−	0.232495i
$297$		0			0
$298$	6.00000	+	6.00000i	0.347571	+	0.347571i
$299$		−	48.0000i		−	2.77591i
$300$		0			0
$301$		−	16.0000i		−	0.922225i
$302$	−2.00000	+	2.00000i	−0.115087	+	0.115087i
$303$		0			0
$304$		−	16.0000i		−	0.917663i
$305$		0			0
$306$		0			0
$307$	−16.0000			−0.913168			−0.456584	−	0.889680i	$-0.650927\pi$
$307$	−16.0000			−0.913168			−0.456584	+	0.889680i	$0.650927\pi$
$308$			16.0000i			0.911685i
$309$		0			0
$310$		0			0
$311$	32.0000			1.81455			0.907277	−	0.420534i	$-0.138157\pi$
$311$	32.0000			1.81455			0.907277	+	0.420534i	$0.138157\pi$
$312$		0			0
$313$			18.0000i			1.01742i	0.860938	+	0.508710i	$0.169877\pi$
$313$			18.0000i			1.01742i	−0.860938	+	0.508710i	$0.830123\pi$
$314$	−14.0000	+	14.0000i	−0.790066	+	0.790066i
$315$		0			0
$316$		−	4.00000i		−	0.225018i
$317$	6.00000			0.336994			0.168497	−	0.985702i	$-0.446109\pi$
$317$	6.00000			0.336994			0.168497	+	0.985702i	$0.446109\pi$
$318$		0			0
$319$	−12.0000			−0.671871
$320$		0			0
$321$		0			0
$322$	−32.0000	+	32.0000i	−1.78329	+	1.78329i
$323$	24.0000			1.33540
$324$		0			0
$325$		0			0
$326$	−16.0000	+	16.0000i	−0.886158	+	0.886158i
$327$		0			0
$328$	−8.00000	−	8.00000i	−0.441726	−	0.441726i
$329$	−16.0000			−0.882109
$330$		0			0
$331$		−	4.00000i		−	0.219860i	−0.993939	−	0.109930i	$-0.964937\pi$
$331$		−	4.00000i		−	0.219860i	0.993939	−	0.109930i	$-0.0350627\pi$
$332$		0			0
$333$		0			0
$334$	12.0000	+	12.0000i	0.656611	+	0.656611i
$335$		0			0
$336$		0			0
$337$			18.0000i			0.980522i	0.871576	+	0.490261i	$0.163099\pi$
$337$			18.0000i			0.980522i	−0.871576	+	0.490261i	$0.836901\pi$
$338$	23.0000	−	23.0000i	1.25104	−	1.25104i
$339$		0			0
$340$		0			0
$341$		−	4.00000i		−	0.216612i
$342$		0			0
$343$		−	8.00000i		−	0.431959i
$344$	8.00000	+	8.00000i	0.431331	+	0.431331i
$345$		0			0
$346$	14.0000	−	14.0000i	0.752645	−	0.752645i
$347$	20.0000			1.07366			0.536828	−	0.843692i	$-0.319622\pi$
$347$	20.0000			1.07366			0.536828	+	0.843692i	$0.319622\pi$
$348$		0			0
$349$			20.0000i			1.07058i	0.844670	+	0.535288i	$0.179797\pi$
$349$			20.0000i			1.07058i	−0.844670	+	0.535288i	$0.820203\pi$
$350$		0			0
$351$		0			0
$352$	−8.00000	−	8.00000i	−0.426401	−	0.426401i
$353$		−	6.00000i		−	0.319348i	−0.987170	−	0.159674i	$-0.948956\pi$
$353$		−	6.00000i		−	0.319348i	0.987170	−	0.159674i	$-0.0510443\pi$
$354$		0			0
$355$		0			0
$356$		−	8.00000i		−	0.423999i
$357$		0			0
$358$	6.00000	+	6.00000i	0.317110	+	0.317110i
$359$	−24.0000			−1.26667			−0.633336	−	0.773877i	$-0.718315\pi$
$359$	−24.0000			−1.26667			−0.633336	+	0.773877i	$0.718315\pi$
$360$		0			0
$361$	3.00000			0.157895
$362$	−16.0000	−	16.0000i	−0.840941	−	0.840941i
$363$		0			0
$364$	−48.0000			−2.51588
$365$		0			0
$366$		0			0
$367$			12.0000i			0.626395i	0.949688	+	0.313197i	$0.101400\pi$
$367$			12.0000i			0.626395i	−0.949688	+	0.313197i	$0.898600\pi$
$368$		−	32.0000i		−	1.66812i
$369$		0			0
$370$		0			0
$371$		−	40.0000i		−	2.07670i
$372$		0			0
$373$	26.0000			1.34623			0.673114	−	0.739538i	$-0.264956\pi$
$373$	26.0000			1.34623			0.673114	+	0.739538i	$0.264956\pi$
$374$	12.0000	−	12.0000i	0.620505	−	0.620505i
$375$		0			0
$376$	8.00000	−	8.00000i	0.412568	−	0.412568i
$377$		−	36.0000i		−	1.85409i
$378$		0			0
$379$		−	12.0000i		−	0.616399i	−0.951322	−	0.308199i	$-0.900274\pi$
$379$		−	12.0000i		−	0.616399i	0.951322	−	0.308199i	$-0.0997264\pi$
$380$		0			0
$381$		0			0
$382$	16.0000	−	16.0000i	0.818631	−	0.818631i
$383$		0			0		1.00000			$0$
$383$		0			0		−1.00000			$\pi$
$384$		0			0
$385$		0			0
$386$	−26.0000	−	26.0000i	−1.32337	−	1.32337i
$387$		0			0
$388$	−4.00000			−0.203069
$389$			10.0000i			0.507020i	0.967333	+	0.253510i	$0.0815851\pi$
$389$			10.0000i			0.507020i	−0.967333	+	0.253510i	$0.918415\pi$
$390$		0			0
$391$	48.0000			2.42746
$392$	18.0000	+	18.0000i	0.909137	+	0.909137i
$393$		0			0
$394$	−14.0000	+	14.0000i	−0.705310	+	0.705310i
$395$		0			0
$396$		0			0
$397$	−2.00000			−0.100377			−0.0501886	−	0.998740i	$-0.515982\pi$
$397$	−2.00000			−0.100377			−0.0501886	+	0.998740i	$0.515982\pi$
$398$	6.00000	−	6.00000i	0.300753	−	0.300753i
$399$		0			0
$400$		0			0
$401$	24.0000			1.19850			0.599251	−	0.800561i	$-0.295465\pi$
$401$	24.0000			1.19850			0.599251	+	0.800561i	$0.295465\pi$
$402$		0			0
$403$	12.0000			0.597763
$404$	20.0000			0.995037
$405$		0			0
$406$	−24.0000	+	24.0000i	−1.19110	+	1.19110i
$407$			4.00000i			0.198273i
$408$		0			0
$409$	30.0000			1.48340			0.741702	−	0.670729i	$-0.234019\pi$
$409$	30.0000			1.48340			0.741702	+	0.670729i	$0.234019\pi$
$410$		0			0
$411$		0			0
$412$	−16.0000			−0.788263
$413$	−8.00000			−0.393654
$414$		0			0
$415$		0			0
$416$	24.0000	−	24.0000i	1.17670	−	1.17670i
$417$		0			0
$418$	−8.00000	+	8.00000i	−0.391293	+	0.391293i
$419$			14.0000i			0.683945i	0.939710	+	0.341972i	$0.111095\pi$
$419$			14.0000i			0.683945i	−0.939710	+	0.341972i	$0.888905\pi$
$420$		0			0
$421$			4.00000i			0.194948i	0.995238	+	0.0974740i	$0.0310763\pi$
$421$			4.00000i			0.194948i	−0.995238	+	0.0974740i	$0.968924\pi$
$422$	12.0000	+	12.0000i	0.584151	+	0.584151i
$423$		0			0
$424$	20.0000	+	20.0000i	0.971286	+	0.971286i
$425$		0			0
$426$		0			0
$427$	16.0000			0.774294
$428$		−	40.0000i		−	1.93347i
$429$		0			0
$430$		0			0
$431$		0			0			−	1.00000i	$-0.5\pi$
$431$		0			0				1.00000i	$0.5\pi$
$432$		0			0
$433$		−	10.0000i		−	0.480569i	−0.970702	−	0.240285i	$-0.922759\pi$
$433$		−	10.0000i		−	0.480569i	0.970702	−	0.240285i	$-0.0772408\pi$
$434$	−8.00000	−	8.00000i	−0.384012	−	0.384012i
$435$		0			0
$436$	−16.0000			−0.766261
$437$	−32.0000			−1.53077
$438$		0			0
$439$	−26.0000			−1.24091			−0.620456	−	0.784241i	$-0.713053\pi$
$439$	−26.0000			−1.24091			−0.620456	+	0.784241i	$0.713053\pi$
$440$		0			0
$441$		0			0
$442$	36.0000	+	36.0000i	1.71235	+	1.71235i
$443$	4.00000			0.190046			0.0950229	−	0.995475i	$-0.469708\pi$
$443$	4.00000			0.190046			0.0950229	+	0.995475i	$0.469708\pi$
$444$		0			0
$445$		0			0
$446$	16.0000	+	16.0000i	0.757622	+	0.757622i
$447$		0			0
$448$	−32.0000			−1.51186
$449$	−24.0000			−1.13263			−0.566315	−	0.824189i	$-0.691631\pi$
$449$	−24.0000			−1.13263			−0.566315	+	0.824189i	$0.691631\pi$
$450$		0			0
$451$			8.00000i			0.376705i
$452$	12.0000			0.564433
$453$		0			0
$454$	−12.0000	+	12.0000i	−0.563188	+	0.563188i
$455$		0			0
$456$		0			0
$457$			18.0000i			0.842004i	0.907060	+	0.421002i	$0.138322\pi$
$457$			18.0000i			0.842004i	−0.907060	+	0.421002i	$0.861678\pi$
$458$	20.0000	+	20.0000i	0.934539	+	0.934539i
$459$		0			0
$460$		0			0
$461$		−	2.00000i		−	0.0931493i	−0.998915	−	0.0465746i	$-0.985169\pi$
$461$		−	2.00000i		−	0.0931493i	0.998915	−	0.0465746i	$-0.0148305\pi$
$462$		0			0
$463$			20.0000i			0.929479i	0.885448	+	0.464739i	$0.153852\pi$
$463$			20.0000i			0.929479i	−0.885448	+	0.464739i	$0.846148\pi$
$464$		−	24.0000i		−	1.11417i
$465$		0			0
$466$	6.00000	+	6.00000i	0.277945	+	0.277945i
$467$	32.0000			1.48078			0.740392	−	0.672176i	$-0.234640\pi$
$467$	32.0000			1.48078			0.740392	+	0.672176i	$0.234640\pi$
$468$		0			0
$469$		−	48.0000i		−	2.21643i
$470$		0			0
$471$		0			0
$472$	4.00000	−	4.00000i	0.184115	−	0.184115i
$473$		−	8.00000i		−	0.367840i
$474$		0			0
$475$		0			0
$476$		−	48.0000i		−	2.20008i
$477$		0			0
$478$	−16.0000	+	16.0000i	−0.731823	+	0.731823i
$479$	32.0000			1.46212			0.731059	−	0.682315i	$-0.239027\pi$
$479$	32.0000			1.46212			0.731059	+	0.682315i	$0.239027\pi$
$480$		0			0
$481$	−12.0000			−0.547153
$482$	−18.0000	+	18.0000i	−0.819878	+	0.819878i
$483$		0			0
$484$		−	14.0000i		−	0.636364i
$485$		0			0
$486$		0			0
$487$			24.0000i			1.08754i	0.839233	+	0.543772i	$0.183004\pi$
$487$			24.0000i			1.08754i	−0.839233	+	0.543772i	$0.816996\pi$
$488$	−8.00000	+	8.00000i	−0.362143	+	0.362143i
$489$		0			0
$490$		0			0
$491$			34.0000i			1.53440i	0.641409	+	0.767199i	$0.278350\pi$
$491$			34.0000i			1.53440i	−0.641409	+	0.767199i	$0.721650\pi$
$492$		0			0
$493$	36.0000			1.62136
$494$	−24.0000	−	24.0000i	−1.07981	−	1.07981i
$495$		0			0
$496$	8.00000			0.359211
$497$			32.0000i			1.43540i
$498$		0			0
$499$		−	20.0000i		−	0.895323i	−0.894203	−	0.447661i	$-0.852257\pi$
$499$		−	20.0000i		−	0.895323i	0.894203	−	0.447661i	$-0.147743\pi$
$500$		0			0
$501$		0			0
$502$	−18.0000	−	18.0000i	−0.803379	−	0.803379i
$503$			12.0000i			0.535054i	0.963550	+	0.267527i	$0.0862064\pi$
$503$			12.0000i			0.535054i	−0.963550	+	0.267527i	$0.913794\pi$
$504$		0			0
$505$		0			0
$506$	−16.0000	+	16.0000i	−0.711287	+	0.711287i
$507$		0			0
$508$		0			0
$509$		−	30.0000i		−	1.32973i	−0.746965	−	0.664863i	$-0.768490\pi$
$509$		−	30.0000i		−	1.32973i	0.746965	−	0.664863i	$-0.231510\pi$
$510$		0			0
$511$	−40.0000			−1.76950
$512$	16.0000	−	16.0000i	0.707107	−	0.707107i
$513$		0			0
$514$	−22.0000	−	22.0000i	−0.970378	−	0.970378i
$515$		0			0
$516$		0			0
$517$	−8.00000			−0.351840
$518$	8.00000	+	8.00000i	0.351500	+	0.351500i
$519$		0			0
$520$		0			0
$521$	−24.0000			−1.05146			−0.525730	−	0.850652i	$-0.676208\pi$
$521$	−24.0000			−1.05146			−0.525730	+	0.850652i	$0.676208\pi$
$522$		0			0
$523$	40.0000			1.74908			0.874539	−	0.484955i	$-0.161164\pi$
$523$	40.0000			1.74908			0.874539	+	0.484955i	$0.161164\pi$
$524$	−36.0000			−1.57267
$525$		0			0
$526$	−12.0000	−	12.0000i	−0.523225	−	0.523225i
$527$			12.0000i			0.522728i
$528$		0			0
$529$	−41.0000			−1.78261
$530$		0			0
$531$		0			0
$532$			32.0000i			1.38738i
$533$	−24.0000			−1.03956
$534$		0			0
$535$		0			0
$536$	24.0000	+	24.0000i	1.03664	+	1.03664i
$537$		0			0
$538$	−2.00000	−	2.00000i	−0.0862261	−	0.0862261i
$539$		−	18.0000i		−	0.775315i
$540$		0			0
$541$			32.0000i			1.37579i	0.725811	+	0.687894i	$0.241464\pi$
$541$			32.0000i			1.37579i	−0.725811	+	0.687894i	$0.758536\pi$
$542$	−2.00000	+	2.00000i	−0.0859074	+	0.0859074i
$543$		0			0
$544$	24.0000	+	24.0000i	1.02899	+	1.02899i
$545$		0			0
$546$		0			0
$547$	28.0000			1.19719			0.598597	−	0.801050i	$-0.295725\pi$
$547$	28.0000			1.19719			0.598597	+	0.801050i	$0.295725\pi$
$548$	4.00000			0.170872
$549$		0			0
$550$		0			0
$551$	−24.0000			−1.02243
$552$		0			0
$553$			8.00000i			0.340195i
$554$	−2.00000	+	2.00000i	−0.0849719	+	0.0849719i
$555$		0			0
$556$	−8.00000			−0.339276
$557$	−30.0000			−1.27114			−0.635570	−	0.772043i	$-0.719235\pi$
$557$	−30.0000			−1.27114			−0.635570	+	0.772043i	$0.719235\pi$
$558$		0			0
$559$	24.0000			1.01509
$560$		0			0
$561$		0			0
$562$	−28.0000	+	28.0000i	−1.18111	+	1.18111i
$563$	36.0000			1.51722			0.758610	−	0.651546i	$-0.225879\pi$
$563$	36.0000			1.51722			0.758610	+	0.651546i	$0.225879\pi$
$564$		0			0
$565$		0			0
$566$	−28.0000	+	28.0000i	−1.17693	+	1.17693i
$567$		0			0
$568$	−16.0000	−	16.0000i	−0.671345	−	0.671345i
$569$	−32.0000			−1.34151			−0.670755	−	0.741679i	$-0.734030\pi$
$569$	−32.0000			−1.34151			−0.670755	+	0.741679i	$0.734030\pi$
$570$		0			0
$571$			20.0000i			0.836974i	0.908223	+	0.418487i	$0.137439\pi$
$571$			20.0000i			0.836974i	−0.908223	+	0.418487i	$0.862561\pi$
$572$	−24.0000			−1.00349
$573$		0			0
$574$	16.0000	+	16.0000i	0.667827	+	0.667827i
$575$		0			0
$576$		0			0
$577$		−	10.0000i		−	0.416305i	−0.978096	−	0.208153i	$-0.933255\pi$
$577$		−	10.0000i		−	0.416305i	0.978096	−	0.208153i	$-0.0667451\pi$
$578$	−19.0000	+	19.0000i	−0.790296	+	0.790296i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		−	20.0000i		−	0.828315i
$584$	20.0000	−	20.0000i	0.827606	−	0.827606i
$585$		0			0
$586$	22.0000	−	22.0000i	0.908812	−	0.908812i
$587$	12.0000			0.495293			0.247647	−	0.968850i	$-0.420343\pi$
$587$	12.0000			0.495293			0.247647	+	0.968850i	$0.420343\pi$
$588$		0			0
$589$		−	8.00000i		−	0.329634i
$590$		0			0
$591$		0			0
$592$	−8.00000			−0.328798
$593$		−	26.0000i		−	1.06769i	−0.845582	−	0.533846i	$-0.820746\pi$
$593$		−	26.0000i		−	1.06769i	0.845582	−	0.533846i	$-0.179254\pi$
$594$		0			0
$595$		0			0
$596$	12.0000			0.491539
$597$		0			0
$598$	−48.0000	−	48.0000i	−1.96287	−	1.96287i
$599$		0			0			−	1.00000i	$-0.5\pi$
$599$		0			0				1.00000i	$0.5\pi$
$600$		0			0
$601$	22.0000			0.897399			0.448699	−	0.893683i	$-0.351887\pi$
$601$	22.0000			0.897399			0.448699	+	0.893683i	$0.351887\pi$
$602$	−16.0000	−	16.0000i	−0.652111	−	0.652111i
$603$		0			0
$604$			4.00000i			0.162758i
$605$		0			0
$606$		0			0
$607$			4.00000i			0.162355i	0.996700	+	0.0811775i	$0.0258681\pi$
$607$			4.00000i			0.162355i	−0.996700	+	0.0811775i	$0.974132\pi$
$608$	−16.0000	−	16.0000i	−0.648886	−	0.648886i
$609$		0			0
$610$		0			0
$611$		−	24.0000i		−	0.970936i
$612$		0			0
$613$	−26.0000			−1.05013			−0.525065	−	0.851062i	$-0.675959\pi$
$613$	−26.0000			−1.05013			−0.525065	+	0.851062i	$0.675959\pi$
$614$	−16.0000	+	16.0000i	−0.645707	+	0.645707i
$615$		0			0
$616$	16.0000	+	16.0000i	0.644658	+	0.644658i
$617$		−	6.00000i		−	0.241551i	−0.992680	−	0.120775i	$-0.961462\pi$
$617$		−	6.00000i		−	0.241551i	0.992680	−	0.120775i	$-0.0385381\pi$
$618$		0			0
$619$			20.0000i			0.803868i	0.915669	+	0.401934i	$0.131662\pi$
$619$			20.0000i			0.803868i	−0.915669	+	0.401934i	$0.868338\pi$
$620$		0			0
$621$		0			0
$622$	32.0000	−	32.0000i	1.28308	−	1.28308i
$623$			16.0000i			0.641026i
$624$		0			0
$625$		0			0
$626$	18.0000	+	18.0000i	0.719425	+	0.719425i
$627$		0			0
$628$			28.0000i			1.11732i
$629$		−	12.0000i		−	0.478471i
$630$		0			0
$631$	−30.0000			−1.19428			−0.597141	−	0.802137i	$-0.703697\pi$
$631$	−30.0000			−1.19428			−0.597141	+	0.802137i	$0.703697\pi$
$632$	−4.00000	−	4.00000i	−0.159111	−	0.159111i
$633$		0			0
$634$	6.00000	−	6.00000i	0.238290	−	0.238290i
$635$		0			0
$636$		0			0
$637$	54.0000			2.13956
$638$	−12.0000	+	12.0000i	−0.475085	+	0.475085i
$639$		0			0
$640$		0			0
$641$	−4.00000			−0.157991			−0.0789953	−	0.996875i	$-0.525171\pi$
$641$	−4.00000			−0.157991			−0.0789953	+	0.996875i	$0.525171\pi$
$642$		0			0
$643$	24.0000			0.946468			0.473234	−	0.880937i	$-0.343087\pi$
$643$	24.0000			0.946468			0.473234	+	0.880937i	$0.343087\pi$
$644$			64.0000i			2.52195i
$645$		0			0
$646$	24.0000	−	24.0000i	0.944267	−	0.944267i
$647$			12.0000i			0.471769i	0.971781	+	0.235884i	$0.0757987\pi$
$647$			12.0000i			0.471769i	−0.971781	+	0.235884i	$0.924201\pi$
$648$		0			0
$649$	−4.00000			−0.157014
$650$		0			0
$651$		0			0
$652$			32.0000i			1.25322i
$653$	6.00000			0.234798			0.117399	−	0.993085i	$-0.462544\pi$
$653$	6.00000			0.234798			0.117399	+	0.993085i	$0.462544\pi$
$654$		0			0
$655$		0			0
$656$	−16.0000			−0.624695
$657$		0			0
$658$	−16.0000	+	16.0000i	−0.623745	+	0.623745i
$659$			18.0000i			0.701180i	0.936529	+	0.350590i	$0.114019\pi$
$659$			18.0000i			0.701180i	−0.936529	+	0.350590i	$0.885981\pi$
$660$		0			0
$661$			28.0000i			1.08907i	0.838737	+	0.544537i	$0.183295\pi$
$661$			28.0000i			1.08907i	−0.838737	+	0.544537i	$0.816705\pi$
$662$	−4.00000	−	4.00000i	−0.155464	−	0.155464i
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−48.0000			−1.85857
$668$	24.0000			0.928588
$669$		0			0
$670$		0			0
$671$	8.00000			0.308837
$672$		0			0
$673$		−	2.00000i		−	0.0770943i	−0.999257	−	0.0385472i	$-0.987727\pi$
$673$		−	2.00000i		−	0.0770943i	0.999257	−	0.0385472i	$-0.0122730\pi$
$674$	18.0000	+	18.0000i	0.693334	+	0.693334i
$675$		0			0
$676$		−	46.0000i		−	1.76923i
$677$	14.0000			0.538064			0.269032	−	0.963131i	$-0.413296\pi$
$677$	14.0000			0.538064			0.269032	+	0.963131i	$0.413296\pi$
$678$		0			0
$679$	8.00000			0.307012
$680$		0			0
$681$		0			0
$682$	−4.00000	−	4.00000i	−0.153168	−	0.153168i
$683$	−24.0000			−0.918334			−0.459167	−	0.888350i	$-0.651852\pi$
$683$	−24.0000			−0.918334			−0.459167	+	0.888350i	$0.651852\pi$
$684$		0			0
$685$		0			0
$686$	−8.00000	−	8.00000i	−0.305441	−	0.305441i
$687$		0			0
$688$	16.0000			0.609994
$689$	60.0000			2.28582
$690$		0			0
$691$		−	12.0000i		−	0.456502i	−0.973602	−	0.228251i	$-0.926699\pi$
$691$		−	12.0000i		−	0.456502i	0.973602	−	0.228251i	$-0.0733006\pi$
$692$		−	28.0000i		−	1.06440i
$693$		0			0
$694$	20.0000	−	20.0000i	0.759190	−	0.759190i
$695$		0			0
$696$		0			0
$697$		−	24.0000i		−	0.909065i
$698$	20.0000	+	20.0000i	0.757011	+	0.757011i
$699$		0			0
$700$		0			0
$701$		−	2.00000i		−	0.0755390i	−0.999286	−	0.0377695i	$-0.987975\pi$
$701$		−	2.00000i		−	0.0755390i	0.999286	−	0.0377695i	$-0.0120253\pi$
$702$		0			0
$703$			8.00000i			0.301726i
$704$	−16.0000			−0.603023
$705$		0			0
$706$	−6.00000	−	6.00000i	−0.225813	−	0.225813i
$707$	−40.0000			−1.50435
$708$		0			0
$709$			44.0000i			1.65245i	0.563337	+	0.826227i	$0.309517\pi$
$709$			44.0000i			1.65245i	−0.563337	+	0.826227i	$0.690483\pi$
$710$		0			0
$711$		0			0
$712$	−8.00000	−	8.00000i	−0.299813	−	0.299813i
$713$		−	16.0000i		−	0.599205i
$714$		0			0
$715$		0			0
$716$	12.0000			0.448461
$717$		0			0
$718$	−24.0000	+	24.0000i	−0.895672	+	0.895672i
$719$	24.0000			0.895049			0.447524	−	0.894272i	$-0.352306\pi$
$719$	24.0000			0.895049			0.447524	+	0.894272i	$0.352306\pi$
$720$		0			0
$721$	32.0000			1.19174
$722$	3.00000	−	3.00000i	0.111648	−	0.111648i
$723$		0			0
$724$	−32.0000			−1.18927
$725$		0			0
$726$		0			0
$727$		−	40.0000i		−	1.48352i	−0.670667	−	0.741759i	$-0.733992\pi$
$727$		−	40.0000i		−	1.48352i	0.670667	−	0.741759i	$-0.266008\pi$
$728$	−48.0000	+	48.0000i	−1.77900	+	1.77900i
$729$		0			0
$730$		0			0
$731$			24.0000i			0.887672i
$732$		0			0
$733$	6.00000			0.221615			0.110808	−	0.993842i	$-0.464656\pi$
$733$	6.00000			0.221615			0.110808	+	0.993842i	$0.464656\pi$
$734$	12.0000	+	12.0000i	0.442928	+	0.442928i
$735$		0			0
$736$	−32.0000	−	32.0000i	−1.17954	−	1.17954i
$737$		−	24.0000i		−	0.884051i
$738$		0			0
$739$			12.0000i			0.441427i	0.975339	+	0.220714i	$0.0708386\pi$
$739$			12.0000i			0.441427i	−0.975339	+	0.220714i	$0.929161\pi$
$740$		0			0
$741$		0			0
$742$	−40.0000	−	40.0000i	−1.46845	−	1.46845i
$743$		−	48.0000i		−	1.76095i	−0.474093	−	0.880475i	$-0.657224\pi$
$743$		−	48.0000i		−	1.76095i	0.474093	−	0.880475i	$-0.342776\pi$
$744$		0			0
$745$		0			0
$746$	26.0000	−	26.0000i	0.951928	−	0.951928i
$747$		0			0
$748$		−	24.0000i		−	0.877527i
$749$			80.0000i			2.92314i
$750$		0			0
$751$	−10.0000			−0.364905			−0.182453	−	0.983215i	$-0.558404\pi$
$751$	−10.0000			−0.364905			−0.182453	+	0.983215i	$0.558404\pi$
$752$		−	16.0000i		−	0.583460i
$753$		0			0
$754$	−36.0000	−	36.0000i	−1.31104	−	1.31104i
$755$		0			0
$756$		0			0
$757$	−10.0000			−0.363456			−0.181728	−	0.983349i	$-0.558169\pi$
$757$	−10.0000			−0.363456			−0.181728	+	0.983349i	$0.558169\pi$
$758$	−12.0000	−	12.0000i	−0.435860	−	0.435860i
$759$		0			0
$760$		0			0
$761$	−8.00000			−0.290000			−0.145000	−	0.989432i	$-0.546318\pi$
$761$	−8.00000			−0.290000			−0.145000	+	0.989432i	$0.546318\pi$
$762$		0			0
$763$	32.0000			1.15848
$764$		−	32.0000i		−	1.15772i
$765$		0			0
$766$		0			0
$767$		−	12.0000i		−	0.433295i
$768$		0			0
$769$	18.0000			0.649097			0.324548	−	0.945869i	$-0.394788\pi$
$769$	18.0000			0.649097			0.324548	+	0.945869i	$0.394788\pi$
$770$		0			0
$771$		0			0
$772$	−52.0000			−1.87152
$773$	6.00000			0.215805			0.107903	−	0.994161i	$-0.465587\pi$
$773$	6.00000			0.215805			0.107903	+	0.994161i	$0.465587\pi$
$774$		0			0
$775$		0			0
$776$	−4.00000	+	4.00000i	−0.143592	+	0.143592i
$777$		0			0
$778$	10.0000	+	10.0000i	0.358517	+	0.358517i
$779$			16.0000i			0.573259i
$780$		0			0
$781$			16.0000i			0.572525i
$782$	48.0000	−	48.0000i	1.71648	−	1.71648i
$783$		0			0
$784$	36.0000			1.28571
$785$		0			0
$786$		0			0
$787$	32.0000			1.14068			0.570338	−	0.821410i	$-0.306812\pi$
$787$	32.0000			1.14068			0.570338	+	0.821410i	$0.306812\pi$
$788$			28.0000i			0.997459i
$789$		0			0
$790$		0			0
$791$	−24.0000			−0.853342
$792$		0			0
$793$			24.0000i			0.852265i
$794$	−2.00000	+	2.00000i	−0.0709773	+	0.0709773i
$795$		0			0
$796$		−	12.0000i		−	0.425329i
$797$	18.0000			0.637593			0.318796	−	0.947823i	$-0.396721\pi$
$797$	18.0000			0.637593			0.318796	+	0.947823i	$0.396721\pi$
$798$		0			0
$799$	24.0000			0.849059
$800$		0			0
$801$		0			0
$802$	24.0000	−	24.0000i	0.847469	−	0.847469i
$803$	−20.0000			−0.705785
$804$		0			0
$805$		0			0
$806$	12.0000	−	12.0000i	0.422682	−	0.422682i
$807$		0			0
$808$	20.0000	−	20.0000i	0.703598	−	0.703598i
$809$	44.0000			1.54696			0.773479	−	0.633822i	$-0.218515\pi$
$809$	44.0000			1.54696			0.773479	+	0.633822i	$0.218515\pi$
$810$		0			0
$811$			52.0000i			1.82597i	0.407997	+	0.912983i	$0.366228\pi$
$811$			52.0000i			1.82597i	−0.407997	+	0.912983i	$0.633772\pi$
$812$			48.0000i			1.68447i
$813$		0			0
$814$	4.00000	+	4.00000i	0.140200	+	0.140200i
$815$		0			0
$816$		0			0
$817$		−	16.0000i		−	0.559769i
$818$	30.0000	−	30.0000i	1.04893	−	1.04893i
$819$		0			0
$820$		0			0
$821$			46.0000i			1.60541i	0.596376	+	0.802706i	$0.296607\pi$
$821$			46.0000i			1.60541i	−0.596376	+	0.802706i	$0.703393\pi$
$822$		0			0
$823$		−	48.0000i		−	1.67317i	−0.547833	−	0.836587i	$-0.684547\pi$
$823$		−	48.0000i		−	1.67317i	0.547833	−	0.836587i	$-0.315453\pi$
$824$	−16.0000	+	16.0000i	−0.557386	+	0.557386i
$825$		0			0
$826$	−8.00000	+	8.00000i	−0.278356	+	0.278356i
$827$	16.0000			0.556375			0.278187	−	0.960527i	$-0.410266\pi$
$827$	16.0000			0.556375			0.278187	+	0.960527i	$0.410266\pi$
$828$		0			0
$829$		−	8.00000i		−	0.277851i	−0.990303	−	0.138926i	$-0.955635\pi$
$829$		−	8.00000i		−	0.277851i	0.990303	−	0.138926i	$-0.0443649\pi$
$830$		0			0
$831$		0			0
$832$		−	48.0000i		−	1.66410i
$833$			54.0000i			1.87099i
$834$		0			0
$835$		0			0
$836$			16.0000i			0.553372i
$837$		0			0
$838$	14.0000	+	14.0000i	0.483622	+	0.483622i
$839$	−16.0000			−0.552381			−0.276191	−	0.961103i	$-0.589072\pi$
$839$	−16.0000			−0.552381			−0.276191	+	0.961103i	$0.589072\pi$
$840$		0			0
$841$	−7.00000			−0.241379
$842$	4.00000	+	4.00000i	0.137849	+	0.137849i
$843$		0			0
$844$	24.0000			0.826114
$845$		0			0
$846$		0			0
$847$			28.0000i			0.962091i
$848$	40.0000			1.37361
$849$		0			0
$850$		0			0
$851$			16.0000i			0.548473i
$852$		0			0
$853$	46.0000			1.57501			0.787505	−	0.616308i	$-0.211372\pi$
$853$	46.0000			1.57501			0.787505	+	0.616308i	$0.211372\pi$
$854$	16.0000	−	16.0000i	0.547509	−	0.547509i
$855$		0			0
$856$	−40.0000	−	40.0000i	−1.36717	−	1.36717i
$857$			6.00000i			0.204956i	0.994735	+	0.102478i	$0.0326771\pi$
$857$			6.00000i			0.204956i	−0.994735	+	0.102478i	$0.967323\pi$
$858$		0			0
$859$			28.0000i			0.955348i	0.878537	+	0.477674i	$0.158520\pi$
$859$			28.0000i			0.955348i	−0.878537	+	0.477674i	$0.841480\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			36.0000i			1.22545i	0.790295	+	0.612727i	$0.209928\pi$
$863$			36.0000i			1.22545i	−0.790295	+	0.612727i	$0.790072\pi$
$864$		0			0
$865$		0			0
$866$	−10.0000	−	10.0000i	−0.339814	−	0.339814i
$867$		0			0
$868$	−16.0000			−0.543075
$869$			4.00000i			0.135691i
$870$		0			0
$871$	72.0000			2.43963
$872$	−16.0000	+	16.0000i	−0.541828	+	0.541828i
$873$		0			0
$874$	−32.0000	+	32.0000i	−1.08242	+	1.08242i
$875$		0			0
$876$		0			0
$877$	10.0000			0.337676			0.168838	−	0.985644i	$-0.445999\pi$
$877$	10.0000			0.337676			0.168838	+	0.985644i	$0.445999\pi$
$878$	−26.0000	+	26.0000i	−0.877457	+	0.877457i
$879$		0			0
$880$		0			0
$881$	12.0000			0.404290			0.202145	−	0.979356i	$-0.435209\pi$
$881$	12.0000			0.404290			0.202145	+	0.979356i	$0.435209\pi$
$882$		0			0
$883$	8.00000			0.269221			0.134611	−	0.990899i	$-0.457022\pi$
$883$	8.00000			0.269221			0.134611	+	0.990899i	$0.457022\pi$
$884$	72.0000			2.42162
$885$		0			0
$886$	4.00000	−	4.00000i	0.134383	−	0.134383i
$887$			20.0000i			0.671534i	0.941945	+	0.335767i	$0.108996\pi$
$887$			20.0000i			0.671534i	−0.941945	+	0.335767i	$0.891004\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$	32.0000			1.07144
$893$	−16.0000			−0.535420
$894$		0			0
$895$		0			0
$896$	−32.0000	+	32.0000i	−1.06904	+	1.06904i
$897$		0			0
$898$	−24.0000	+	24.0000i	−0.800890	+	0.800890i
$899$		−	12.0000i		−	0.400222i
$900$		0			0
$901$			60.0000i			1.99889i
$902$	8.00000	+	8.00000i	0.266371	+	0.266371i
$903$		0			0
$904$	12.0000	−	12.0000i	0.399114	−	0.399114i
$905$		0			0
$906$		0			0
$907$	4.00000			0.132818			0.0664089	−	0.997792i	$-0.478846\pi$
$907$	4.00000			0.132818			0.0664089	+	0.997792i	$0.478846\pi$
$908$			24.0000i			0.796468i
$909$		0			0
$910$		0			0
$911$	48.0000			1.59031			0.795155	−	0.606406i	$-0.207389\pi$
$911$	48.0000			1.59031			0.795155	+	0.606406i	$0.207389\pi$
$912$		0			0
$913$		0			0
$914$	18.0000	+	18.0000i	0.595387	+	0.595387i
$915$		0			0
$916$	40.0000			1.32164
$917$	72.0000			2.37765
$918$		0			0
$919$	14.0000			0.461817			0.230909	−	0.972975i	$-0.425830\pi$
$919$	14.0000			0.461817			0.230909	+	0.972975i	$0.425830\pi$
$920$		0			0
$921$		0			0
$922$	−2.00000	−	2.00000i	−0.0658665	−	0.0658665i
$923$	−48.0000			−1.57994
$924$		0			0
$925$		0			0
$926$	20.0000	+	20.0000i	0.657241	+	0.657241i
$927$		0			0
$928$	−24.0000	−	24.0000i	−0.787839	−	0.787839i
$929$	−16.0000			−0.524943			−0.262471	−	0.964940i	$-0.584538\pi$
$929$	−16.0000			−0.524943			−0.262471	+	0.964940i	$0.584538\pi$
$930$		0			0
$931$		−	36.0000i		−	1.17985i
$932$	12.0000			0.393073
$933$		0			0
$934$	32.0000	−	32.0000i	1.04707	−	1.04707i
$935$		0			0
$936$		0			0
$937$		−	22.0000i		−	0.718709i	−0.933201	−	0.359354i	$-0.882997\pi$
$937$		−	22.0000i		−	0.718709i	0.933201	−	0.359354i	$-0.117003\pi$
$938$	−48.0000	−	48.0000i	−1.56726	−	1.56726i
$939$		0			0
$940$		0			0
$941$		−	46.0000i		−	1.49956i	−0.661689	−	0.749779i	$-0.730160\pi$
$941$		−	46.0000i		−	1.49956i	0.661689	−	0.749779i	$-0.269840\pi$
$942$		0			0
$943$			32.0000i			1.04206i
$944$		−	8.00000i		−	0.260378i
$945$		0			0
$946$	−8.00000	−	8.00000i	−0.260102	−	0.260102i
$947$	−48.0000			−1.55979			−0.779895	−	0.625910i	$-0.784728\pi$
$947$	−48.0000			−1.55979			−0.779895	+	0.625910i	$0.784728\pi$
$948$		0			0
$949$		−	60.0000i		−	1.94768i
$950$		0			0
$951$		0			0
$952$	−48.0000	−	48.0000i	−1.55569	−	1.55569i
$953$			46.0000i			1.49009i	0.667016	+	0.745043i	$0.267571\pi$
$953$			46.0000i			1.49009i	−0.667016	+	0.745043i	$0.732429\pi$
$954$		0			0
$955$		0			0
$956$			32.0000i			1.03495i
$957$		0			0
$958$	32.0000	−	32.0000i	1.03387	−	1.03387i
$959$	−8.00000			−0.258333
$960$		0			0
$961$	−27.0000			−0.870968
$962$	−12.0000	+	12.0000i	−0.386896	+	0.386896i
$963$		0			0
$964$			36.0000i			1.15948i
$965$		0			0
$966$		0			0
$967$			8.00000i			0.257263i	0.991692	+	0.128631i	$0.0410584\pi$
$967$			8.00000i			0.257263i	−0.991692	+	0.128631i	$0.958942\pi$
$968$	−14.0000	−	14.0000i	−0.449977	−	0.449977i
$969$		0			0
$970$		0			0
$971$		−	42.0000i		−	1.34784i	−0.738802	−	0.673922i	$-0.764608\pi$
$971$		−	42.0000i		−	1.34784i	0.738802	−	0.673922i	$-0.235392\pi$
$972$		0			0
$973$	16.0000			0.512936
$974$	24.0000	+	24.0000i	0.769010	+	0.769010i
$975$		0			0
$976$			16.0000i			0.512148i
$977$			34.0000i			1.08776i	0.839164	+	0.543878i	$0.183045\pi$
$977$			34.0000i			1.08776i	−0.839164	+	0.543878i	$0.816955\pi$
$978$		0			0
$979$			8.00000i			0.255681i
$980$		0			0
$981$		0			0
$982$	34.0000	+	34.0000i	1.08498	+	1.08498i
$983$			16.0000i			0.510321i	0.966899	+	0.255160i	$0.0821283\pi$
$983$			16.0000i			0.510321i	−0.966899	+	0.255160i	$0.917872\pi$
$984$		0			0
$985$		0			0
$986$	36.0000	−	36.0000i	1.14647	−	1.14647i
$987$		0			0
$988$	−48.0000			−1.52708
$989$		−	32.0000i		−	1.01754i
$990$		0			0
$991$	−46.0000			−1.46124			−0.730619	−	0.682785i	$-0.760768\pi$
$991$	−46.0000			−1.46124			−0.730619	+	0.682785i	$0.760768\pi$
$992$	8.00000	−	8.00000i	0.254000	−	0.254000i
$993$		0			0
$994$	32.0000	+	32.0000i	1.01498	+	1.01498i
$995$		0			0
$996$		0			0
$997$	46.0000			1.45683			0.728417	−	0.685134i	$-0.240256\pi$
$997$	46.0000			1.45683			0.728417	+	0.685134i	$0.240256\pi$
$998$	−20.0000	−	20.0000i	−0.633089	−	0.633089i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1800.2.d.g.1549.1		2
3.2	odd	2		1800.2.d.a.1549.2		2
4.3	odd	2		7200.2.d.b.2449.1		2
5.2	odd	4		360.2.k.c.181.2	yes	2
5.3	odd	4		1800.2.k.c.901.1		2
5.4	even	2		1800.2.d.e.1549.2		2
8.3	odd	2		7200.2.d.i.2449.1		2
8.5	even	2		1800.2.d.e.1549.1		2
12.11	even	2		7200.2.d.a.2449.1		2
15.2	even	4		360.2.k.a.181.1	✓	2
15.8	even	4		1800.2.k.i.901.2		2
15.14	odd	2		1800.2.d.j.1549.1		2
20.3	even	4		7200.2.k.a.3601.1		2
20.7	even	4		1440.2.k.c.721.2		2
20.19	odd	2		7200.2.d.i.2449.2		2
24.5	odd	2		1800.2.d.j.1549.2		2
24.11	even	2		7200.2.d.h.2449.1		2
40.3	even	4		7200.2.k.a.3601.2		2
40.13	odd	4		1800.2.k.c.901.2		2
40.19	odd	2		7200.2.d.b.2449.2		2
40.27	even	4		1440.2.k.c.721.1		2
40.29	even	2	inner	1800.2.d.g.1549.2		2
40.37	odd	4		360.2.k.c.181.1	yes	2
60.23	odd	4		7200.2.k.c.3601.2		2
60.47	odd	4		1440.2.k.b.721.1		2
60.59	even	2		7200.2.d.h.2449.2		2
120.29	odd	2		1800.2.d.a.1549.1		2
120.53	even	4		1800.2.k.i.901.1		2
120.59	even	2		7200.2.d.a.2449.2		2
120.77	even	4		360.2.k.a.181.2	yes	2
120.83	odd	4		7200.2.k.c.3601.1		2
120.107	odd	4		1440.2.k.b.721.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
360.2.k.a.181.1	✓	2	15.2	even	4
360.2.k.a.181.2	yes	2	120.77	even	4
360.2.k.c.181.1	yes	2	40.37	odd	4
360.2.k.c.181.2	yes	2	5.2	odd	4
1440.2.k.b.721.1		2	60.47	odd	4
1440.2.k.b.721.2		2	120.107	odd	4
1440.2.k.c.721.1		2	40.27	even	4
1440.2.k.c.721.2		2	20.7	even	4
1800.2.d.a.1549.1		2	120.29	odd	2
1800.2.d.a.1549.2		2	3.2	odd	2
1800.2.d.e.1549.1		2	8.5	even	2
1800.2.d.e.1549.2		2	5.4	even	2
1800.2.d.g.1549.1		2	1.1	even	1	trivial
1800.2.d.g.1549.2		2	40.29	even	2	inner
1800.2.d.j.1549.1		2	15.14	odd	2
1800.2.d.j.1549.2		2	24.5	odd	2
1800.2.k.c.901.1		2	5.3	odd	4
1800.2.k.c.901.2		2	40.13	odd	4
1800.2.k.i.901.1		2	120.53	even	4
1800.2.k.i.901.2		2	15.8	even	4
7200.2.d.a.2449.1		2	12.11	even	2
7200.2.d.a.2449.2		2	120.59	even	2
7200.2.d.b.2449.1		2	4.3	odd	2
7200.2.d.b.2449.2		2	40.19	odd	2
7200.2.d.h.2449.1		2	24.11	even	2
7200.2.d.h.2449.2		2	60.59	even	2
7200.2.d.i.2449.1		2	8.3	odd	2
7200.2.d.i.2449.2		2	20.19	odd	2
7200.2.k.a.3601.1		2	20.3	even	4
7200.2.k.a.3601.2		2	40.3	even	4
7200.2.k.c.3601.1		2	120.83	odd	4
7200.2.k.c.3601.2		2	60.23	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1800 = 2^{3} \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1800.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(1.00000 - 1.00000i) q^{2} -2.00000i q^{4} +4.00000i q^{7} +(-2.00000 - 2.00000i) q^{8} +2.00000i q^{11} -6.00000 q^{13} +(4.00000 + 4.00000i) q^{14} -4.00000 q^{16} -6.00000i q^{17} +4.00000i q^{19} +(2.00000 + 2.00000i) q^{22} +8.00000i q^{23} +(-6.00000 + 6.00000i) q^{26} +8.00000 q^{28} +6.00000i q^{29} -2.00000 q^{31} +(-4.00000 + 4.00000i) q^{32} +(-6.00000 - 6.00000i) q^{34} +2.00000 q^{37} +(4.00000 + 4.00000i) q^{38} +4.00000 q^{41} -4.00000 q^{43} +4.00000 q^{44} +(8.00000 + 8.00000i) q^{46} +4.00000i q^{47} -9.00000 q^{49} +12.0000i q^{52} -10.0000 q^{53} +(8.00000 - 8.00000i) q^{56} +(6.00000 + 6.00000i) q^{58} +2.00000i q^{59} -4.00000i q^{61} +(-2.00000 + 2.00000i) q^{62} +8.00000i q^{64} -12.0000 q^{67} -12.0000 q^{68} +8.00000 q^{71} +10.0000i q^{73} +(2.00000 - 2.00000i) q^{74} +8.00000 q^{76} -8.00000 q^{77} +2.00000 q^{79} +(4.00000 - 4.00000i) q^{82} +(-4.00000 + 4.00000i) q^{86} +(4.00000 - 4.00000i) q^{88} +4.00000 q^{89} -24.0000i q^{91} +16.0000 q^{92} +(4.00000 + 4.00000i) q^{94} -2.00000i q^{97} +(-9.00000 + 9.00000i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} - 4 q^{8} - 12 q^{13} + 8 q^{14} - 8 q^{16} + 4 q^{22} - 12 q^{26} + 16 q^{28} - 4 q^{31} - 8 q^{32} - 12 q^{34} + 4 q^{37} + 8 q^{38} + 8 q^{41} - 8 q^{43} + 8 q^{44} + 16 q^{46} - 18 q^{49}+ \cdots - 18 q^{98}+O(q^{100}) \) 2 * q + 2 * q^2 - 4 * q^8 - 12 * q^13 + 8 * q^14 - 8 * q^16 + 4 * q^22 - 12 * q^26 + 16 * q^28 - 4 * q^31 - 8 * q^32 - 12 * q^34 + 4 * q^37 + 8 * q^38 + 8 * q^41 - 8 * q^43 + 8 * q^44 + 16 * q^46 - 18 * q^49 - 20 * q^53 + 16 * q^56 + 12 * q^58 - 4 * q^62 - 24 * q^67 - 24 * q^68 + 16 * q^71 + 4 * q^74 + 16 * q^76 - 16 * q^77 + 4 * q^79 + 8 * q^82 - 8 * q^86 + 8 * q^88 + 8 * q^89 + 32 * q^92 + 8 * q^94 - 18 * q^98

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