LMFDB - Embedded newform 1920.2.m.t.959.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1920,2,Mod(959,1920)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1920, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 1, 1])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1920.959"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 1920 = 2^{7} \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1920.m (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,2,0,4,0,-4,0,0,0,0,0,8,0,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(15)] == traces)

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

Self dual:	no
Analytic conductor:	$15.3312771881$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{5})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 3x^{2} + 1 $ x^4 + 3*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			959.4
Root			$1.61803i$ of defining polynomial
Character	$\chi$	$=$	1920.959
Dual form			1920.2.m.t.959.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(1.61803 + 0.618034i) q^{3} +(1.00000 + 2.00000i) q^{5} -3.23607 q^{7} +(2.23607 + 2.00000i) q^{9} -4.00000i q^{11} +6.47214 q^{13} +(0.381966 + 3.85410i) q^{15} +6.47214 q^{17} +2.47214 q^{19} +(-5.23607 - 2.00000i) q^{21} -1.23607i q^{23} +(-3.00000 + 4.00000i) q^{25} +(2.38197 + 4.61803i) q^{27} -4.47214 q^{29} +2.47214i q^{31} +(2.47214 - 6.47214i) q^{33} +(-3.23607 - 6.47214i) q^{35} -1.52786 q^{37} +(10.4721 + 4.00000i) q^{39} -4.00000i q^{41} -9.23607i q^{43} +(-1.76393 + 6.47214i) q^{45} +9.23607i q^{47} +3.47214 q^{49} +(10.4721 + 4.00000i) q^{51} +8.94427i q^{53} +(8.00000 - 4.00000i) q^{55} +(4.00000 + 1.52786i) q^{57} +8.94427i q^{59} +4.94427i q^{61} +(-7.23607 - 6.47214i) q^{63} +(6.47214 + 12.9443i) q^{65} +11.7082i q^{67} +(0.763932 - 2.00000i) q^{69} -4.94427i q^{73} +(-7.32624 + 4.61803i) q^{75} +12.9443i q^{77} -10.4721i q^{79} +(1.00000 + 8.94427i) q^{81} +8.18034 q^{83} +(6.47214 + 12.9443i) q^{85} +(-7.23607 - 2.76393i) q^{87} +8.00000i q^{89} -20.9443 q^{91} +(-1.52786 + 4.00000i) q^{93} +(2.47214 + 4.94427i) q^{95} -12.9443i q^{97} +(8.00000 - 8.94427i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{3} + 4 q^{5} - 4 q^{7} + 8 q^{13} + 6 q^{15} + 8 q^{17} - 8 q^{19} - 12 q^{21} - 12 q^{25} + 14 q^{27} - 8 q^{33} - 4 q^{35} - 24 q^{37} + 24 q^{39} - 16 q^{45} - 4 q^{49} + 24 q^{51} + 32 q^{55}+ \cdots + 32 q^{99}+O(q^{100}) $ 4 * q + 2 * q^3 + 4 * q^5 - 4 * q^7 + 8 * q^13 + 6 * q^15 + 8 * q^17 - 8 * q^19 - 12 * q^21 - 12 * q^25 + 14 * q^27 - 8 * q^33 - 4 * q^35 - 24 * q^37 + 24 * q^39 - 16 * q^45 - 4 * q^49 + 24 * q^51 + 32 * q^55 + 16 * q^57 - 20 * q^63 + 8 * q^65 + 12 * q^69 + 2 * q^75 + 4 * q^81 - 12 * q^83 + 8 * q^85 - 20 * q^87 - 48 * q^91 - 24 * q^93 - 8 * q^95 + 32 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1920\mathbb{Z}\right)^\times$.

$n$	$511$	$641$	$901$	$1537$
$\chi(n)$	$-1$	$-1$	$-1$	$-1$

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 $

$2$		0			0
$2$		0			0
$3$	1.61803	+	0.618034i	0.934172	+	0.356822i
$3$	1.61803	+	0.618034i	0.934172	+	0.356822i
$4$		0			0
$5$	1.00000	+	2.00000i	0.447214	+	0.894427i
$5$	1.00000	+	2.00000i	0.447214	+	0.894427i
$6$		0			0
$7$	−3.23607			−1.22312			−0.611559	−	0.791199i	$-0.709457\pi$
$7$	−3.23607			−1.22312			−0.611559	+	0.791199i	$0.709457\pi$
$8$		0			0
$9$	2.23607	+	2.00000i	0.745356	+	0.666667i
$10$		0			0
$11$		−	4.00000i		−	1.20605i	−0.797724	−	0.603023i	$-0.793963\pi$
$11$		−	4.00000i		−	1.20605i	0.797724	−	0.603023i	$-0.206037\pi$
$12$		0			0
$13$	6.47214			1.79505			0.897524	−	0.440966i	$-0.145364\pi$
$13$	6.47214			1.79505			0.897524	+	0.440966i	$0.145364\pi$
$14$		0			0
$15$	0.381966	+	3.85410i	0.0986232	+	0.995125i
$16$		0			0
$17$	6.47214			1.56972			0.784862	−	0.619671i	$-0.212734\pi$
$17$	6.47214			1.56972			0.784862	+	0.619671i	$0.212734\pi$
$18$		0			0
$19$	2.47214			0.567147			0.283573	−	0.958951i	$-0.408480\pi$
$19$	2.47214			0.567147			0.283573	+	0.958951i	$0.408480\pi$
$20$		0			0
$21$	−5.23607	−	2.00000i	−1.14260	−	0.436436i
$22$		0			0
$23$		−	1.23607i		−	0.257738i	−0.991662	−	0.128869i	$-0.958865\pi$
$23$		−	1.23607i		−	0.257738i	0.991662	−	0.128869i	$-0.0411347\pi$
$24$		0			0
$25$	−3.00000	+	4.00000i	−0.600000	+	0.800000i
$26$		0			0
$27$	2.38197	+	4.61803i	0.458410	+	0.888741i
$28$		0			0
$29$	−4.47214			−0.830455			−0.415227	−	0.909718i	$-0.636298\pi$
$29$	−4.47214			−0.830455			−0.415227	+	0.909718i	$0.636298\pi$
$30$		0			0
$31$			2.47214i			0.444009i	0.975046	+	0.222004i	$0.0712599\pi$
$31$			2.47214i			0.444009i	−0.975046	+	0.222004i	$0.928740\pi$
$32$		0			0
$33$	2.47214	−	6.47214i	0.430344	−	1.12665i
$34$		0			0
$35$	−3.23607	−	6.47214i	−0.546995	−	1.09399i
$36$		0			0
$37$	−1.52786			−0.251179			−0.125590	−	0.992082i	$-0.540082\pi$
$37$	−1.52786			−0.251179			−0.125590	+	0.992082i	$0.540082\pi$
$38$		0			0
$39$	10.4721	+	4.00000i	1.67688	+	0.640513i
$40$		0			0
$41$		−	4.00000i		−	0.624695i	−0.949968	−	0.312348i	$-0.898885\pi$
$41$		−	4.00000i		−	0.624695i	0.949968	−	0.312348i	$-0.101115\pi$
$42$		0			0
$43$		−	9.23607i		−	1.40849i	−0.709958	−	0.704244i	$-0.751286\pi$
$43$		−	9.23607i		−	1.40849i	0.709958	−	0.704244i	$-0.248714\pi$
$44$		0			0
$45$	−1.76393	+	6.47214i	−0.262951	+	0.964809i
$46$		0			0
$47$			9.23607i			1.34722i	0.739087	+	0.673609i	$0.235257\pi$
$47$			9.23607i			1.34722i	−0.739087	+	0.673609i	$0.764743\pi$
$48$		0			0
$49$	3.47214			0.496019
$50$		0			0
$51$	10.4721	+	4.00000i	1.46639	+	0.560112i
$52$		0			0
$53$			8.94427i			1.22859i	0.789076	+	0.614295i	$0.210560\pi$
$53$			8.94427i			1.22859i	−0.789076	+	0.614295i	$0.789440\pi$
$54$		0			0
$55$	8.00000	−	4.00000i	1.07872	−	0.539360i
$56$		0			0
$57$	4.00000	+	1.52786i	0.529813	+	0.202371i
$58$		0			0
$59$			8.94427i			1.16445i	0.813029	+	0.582223i	$0.197817\pi$
$59$			8.94427i			1.16445i	−0.813029	+	0.582223i	$0.802183\pi$
$60$		0			0
$61$			4.94427i			0.633049i	0.948584	+	0.316525i	$0.102516\pi$
$61$			4.94427i			0.633049i	−0.948584	+	0.316525i	$0.897484\pi$
$62$		0			0
$63$	−7.23607	−	6.47214i	−0.911659	−	0.815412i
$64$		0			0
$65$	6.47214	+	12.9443i	0.802770	+	1.60554i
$66$		0			0
$67$			11.7082i			1.43038i	0.698928	+	0.715192i	$0.253661\pi$
$67$			11.7082i			1.43038i	−0.698928	+	0.715192i	$0.746339\pi$
$68$		0			0
$69$	0.763932	−	2.00000i	0.0919666	−	0.240772i
$70$		0			0
$71$		0			0			−	1.00000i	$-0.5\pi$
$71$		0			0				1.00000i	$0.5\pi$
$72$		0			0
$73$		−	4.94427i		−	0.578683i	−0.957226	−	0.289342i	$-0.906564\pi$
$73$		−	4.94427i		−	0.578683i	0.957226	−	0.289342i	$-0.0934364\pi$
$74$		0			0
$75$	−7.32624	+	4.61803i	−0.845961	+	0.533245i
$76$		0			0
$77$			12.9443i			1.47514i
$78$		0			0
$79$		−	10.4721i		−	1.17821i	−0.808057	−	0.589104i	$-0.799481\pi$
$79$		−	10.4721i		−	1.17821i	0.808057	−	0.589104i	$-0.200519\pi$
$80$		0			0
$81$	1.00000	+	8.94427i	0.111111	+	0.993808i
$82$		0			0
$83$	8.18034			0.897909			0.448954	−	0.893555i	$-0.351797\pi$
$83$	8.18034			0.897909			0.448954	+	0.893555i	$0.351797\pi$
$84$		0			0
$85$	6.47214	+	12.9443i	0.702002	+	1.40400i
$86$		0			0
$87$	−7.23607	−	2.76393i	−0.775788	−	0.296325i
$88$		0			0
$89$			8.00000i			0.847998i	0.905663	+	0.423999i	$0.139374\pi$
$89$			8.00000i			0.847998i	−0.905663	+	0.423999i	$0.860626\pi$
$90$		0			0
$91$	−20.9443			−2.19556
$92$		0			0
$93$	−1.52786	+	4.00000i	−0.158432	+	0.414781i
$94$		0			0
$95$	2.47214	+	4.94427i	0.253636	+	0.507272i
$96$		0			0
$97$		−	12.9443i		−	1.31429i	−0.753763	−	0.657146i	$-0.771764\pi$
$97$		−	12.9443i		−	1.31429i	0.753763	−	0.657146i	$-0.228236\pi$
$98$		0			0
$99$	8.00000	−	8.94427i	0.804030	−	0.898933i
$100$		0			0
$101$	−3.52786			−0.351036			−0.175518	−	0.984476i	$-0.556160\pi$
$101$	−3.52786			−0.351036			−0.175518	+	0.984476i	$0.556160\pi$
$102$		0			0
$103$	1.70820			0.168314			0.0841572	−	0.996452i	$-0.473180\pi$
$103$	1.70820			0.168314			0.0841572	+	0.996452i	$0.473180\pi$
$104$		0			0
$105$	−1.23607	−	12.4721i	−0.120628	−	1.21716i
$106$		0			0
$107$	−9.70820			−0.938527			−0.469264	−	0.883058i	$-0.655481\pi$
$107$	−9.70820			−0.938527			−0.469264	+	0.883058i	$0.655481\pi$
$108$		0			0
$109$		−	12.9443i		−	1.23984i	−0.784666	−	0.619918i	$-0.787166\pi$
$109$		−	12.9443i		−	1.23984i	0.784666	−	0.619918i	$-0.212834\pi$
$110$		0			0
$111$	−2.47214	−	0.944272i	−0.234645	−	0.0896263i
$112$		0			0
$113$	14.4721			1.36142			0.680712	−	0.732551i	$-0.261671\pi$
$113$	14.4721			1.36142			0.680712	+	0.732551i	$0.261671\pi$
$114$		0			0
$115$	2.47214	−	1.23607i	0.230528	−	0.115264i
$116$		0			0
$117$	14.4721	+	12.9443i	1.33795	+	1.19670i
$118$		0			0
$119$	−20.9443			−1.91996
$120$		0			0
$121$	−5.00000			−0.454545
$122$		0			0
$123$	2.47214	−	6.47214i	0.222905	−	0.583573i
$124$		0			0
$125$	−11.0000	−	2.00000i	−0.983870	−	0.178885i
$126$		0			0
$127$	11.2361			0.997040			0.498520	−	0.866878i	$-0.333877\pi$
$127$	11.2361			0.997040			0.498520	+	0.866878i	$0.333877\pi$
$128$		0			0
$129$	5.70820	−	14.9443i	0.502579	−	1.31577i
$130$		0			0
$131$		−	0.944272i		−	0.0825014i	−0.999149	−	0.0412507i	$-0.986866\pi$
$131$		−	0.944272i		−	0.0825014i	0.999149	−	0.0412507i	$-0.0131342\pi$
$132$		0			0
$133$	−8.00000			−0.693688
$134$		0			0
$135$	−6.85410	+	9.38197i	−0.589907	+	0.807471i
$136$		0			0
$137$	−1.52786			−0.130534			−0.0652671	−	0.997868i	$-0.520790\pi$
$137$	−1.52786			−0.130534			−0.0652671	+	0.997868i	$0.520790\pi$
$138$		0			0
$139$	−23.4164			−1.98615			−0.993077	−	0.117466i	$-0.962523\pi$
$139$	−23.4164			−1.98615			−0.993077	+	0.117466i	$0.962523\pi$
$140$		0			0
$141$	−5.70820	+	14.9443i	−0.480717	+	1.25853i
$142$		0			0
$143$		−	25.8885i		−	2.16491i
$144$		0			0
$145$	−4.47214	−	8.94427i	−0.371391	−	0.742781i
$146$		0			0
$147$	5.61803	+	2.14590i	0.463368	+	0.176991i
$148$		0			0
$149$	−11.8885			−0.973947			−0.486974	−	0.873417i	$-0.661899\pi$
$149$	−11.8885			−0.973947			−0.486974	+	0.873417i	$0.661899\pi$
$150$		0			0
$151$		−	10.4721i		−	0.852210i	−0.904674	−	0.426105i	$-0.859885\pi$
$151$		−	10.4721i		−	0.852210i	0.904674	−	0.426105i	$-0.140115\pi$
$152$		0			0
$153$	14.4721	+	12.9443i	1.17000	+	1.04648i
$154$		0			0
$155$	−4.94427	+	2.47214i	−0.397133	+	0.198567i
$156$		0			0
$157$	−1.52786			−0.121937			−0.0609684	−	0.998140i	$-0.519419\pi$
$157$	−1.52786			−0.121937			−0.0609684	+	0.998140i	$0.519419\pi$
$158$		0			0
$159$	−5.52786	+	14.4721i	−0.438388	+	1.14772i
$160$		0			0
$161$			4.00000i			0.315244i
$162$		0			0
$163$		−	14.7639i		−	1.15640i	−0.815895	−	0.578200i	$-0.803755\pi$
$163$		−	14.7639i		−	1.15640i	0.815895	−	0.578200i	$-0.196245\pi$
$164$		0			0
$165$	15.4164	−	1.52786i	1.20017	−	0.118944i
$166$		0			0
$167$			1.23607i			0.0956498i	0.998856	+	0.0478249i	$0.0152289\pi$
$167$			1.23607i			0.0956498i	−0.998856	+	0.0478249i	$0.984771\pi$
$168$		0			0
$169$	28.8885			2.22220
$170$		0			0
$171$	5.52786	+	4.94427i	0.422726	+	0.378098i
$172$		0			0
$173$			4.00000i			0.304114i	0.988372	+	0.152057i	$0.0485898\pi$
$173$			4.00000i			0.304114i	−0.988372	+	0.152057i	$0.951410\pi$
$174$		0			0
$175$	9.70820	−	12.9443i	0.733871	−	0.978495i
$176$		0			0
$177$	−5.52786	+	14.4721i	−0.415500	+	1.08779i
$178$		0			0
$179$		−	0.944272i		−	0.0705782i	−0.999377	−	0.0352891i	$-0.988765\pi$
$179$		−	0.944272i		−	0.0705782i	0.999377	−	0.0352891i	$-0.0112352\pi$
$180$		0			0
$181$		−	0.944272i		−	0.0701872i	−0.999384	−	0.0350936i	$-0.988827\pi$
$181$		−	0.944272i		−	0.0701872i	0.999384	−	0.0350936i	$-0.0111729\pi$
$182$		0			0
$183$	−3.05573	+	8.00000i	−0.225886	+	0.591377i
$184$		0			0
$185$	−1.52786	−	3.05573i	−0.112331	−	0.224662i
$186$		0			0
$187$		−	25.8885i		−	1.89316i
$188$		0			0
$189$	−7.70820	−	14.9443i	−0.560689	−	1.08704i
$190$		0			0
$191$	4.94427			0.357755			0.178877	−	0.983871i	$-0.442753\pi$
$191$	4.94427			0.357755			0.178877	+	0.983871i	$0.442753\pi$
$192$		0			0
$193$			20.9443i			1.50760i	0.657103	+	0.753801i	$0.271782\pi$
$193$			20.9443i			1.50760i	−0.657103	+	0.753801i	$0.728218\pi$
$194$		0			0
$195$	2.47214	+	24.9443i	0.177033	+	1.78630i
$196$		0			0
$197$		−	8.94427i		−	0.637253i	−0.947880	−	0.318626i	$-0.896778\pi$
$197$		−	8.94427i		−	0.637253i	0.947880	−	0.318626i	$-0.103222\pi$
$198$		0			0
$199$		−	13.5279i		−	0.958965i	−0.877551	−	0.479482i	$-0.840824\pi$
$199$		−	13.5279i		−	0.958965i	0.877551	−	0.479482i	$-0.159176\pi$
$200$		0			0
$201$	−7.23607	+	18.9443i	−0.510393	+	1.33623i
$202$		0			0
$203$	14.4721			1.01574
$204$		0			0
$205$	8.00000	−	4.00000i	0.558744	−	0.279372i
$206$		0			0
$207$	2.47214	−	2.76393i	0.171825	−	0.192107i
$208$		0			0
$209$		−	9.88854i		−	0.684005i
$210$		0			0
$211$	7.41641			0.510567			0.255283	−	0.966866i	$-0.417831\pi$
$211$	7.41641			0.510567			0.255283	+	0.966866i	$0.417831\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	18.4721	−	9.23607i	1.25979	−	0.629895i
$216$		0			0
$217$		−	8.00000i		−	0.543075i
$218$		0			0
$219$	3.05573	−	8.00000i	0.206487	−	0.540590i
$220$		0			0
$221$	41.8885			2.81773
$222$		0			0
$223$	−21.1246			−1.41461			−0.707304	−	0.706909i	$-0.750089\pi$
$223$	−21.1246			−1.41461			−0.707304	+	0.706909i	$0.750089\pi$
$224$		0			0
$225$	−14.7082	+	2.94427i	−0.980547	+	0.196285i
$226$		0			0
$227$	−12.7639			−0.847172			−0.423586	−	0.905856i	$-0.639229\pi$
$227$	−12.7639			−0.847172			−0.423586	+	0.905856i	$0.639229\pi$
$228$		0			0
$229$		−	24.9443i		−	1.64836i	−0.566325	−	0.824182i	$-0.691635\pi$
$229$		−	24.9443i		−	1.64836i	0.566325	−	0.824182i	$-0.308365\pi$
$230$		0			0
$231$	−8.00000	+	20.9443i	−0.526361	+	1.37803i
$232$		0			0
$233$	−9.52786			−0.624191			−0.312096	−	0.950051i	$-0.601031\pi$
$233$	−9.52786			−0.624191			−0.312096	+	0.950051i	$0.601031\pi$
$234$		0			0
$235$	−18.4721	+	9.23607i	−1.20499	+	0.602495i
$236$		0			0
$237$	6.47214	−	16.9443i	0.420410	−	1.10065i
$238$		0			0
$239$	11.0557			0.715136			0.357568	−	0.933887i	$-0.383606\pi$
$239$	11.0557			0.715136			0.357568	+	0.933887i	$0.383606\pi$
$240$		0			0
$241$	−12.4721			−0.803401			−0.401700	−	0.915771i	$-0.631581\pi$
$241$	−12.4721			−0.803401			−0.401700	+	0.915771i	$0.631581\pi$
$242$		0			0
$243$	−3.90983	+	15.0902i	−0.250816	+	0.968035i
$244$		0			0
$245$	3.47214	+	6.94427i	0.221827	+	0.443653i
$246$		0			0
$247$	16.0000			1.01806
$248$		0			0
$249$	13.2361	+	5.05573i	0.838802	+	0.320394i
$250$		0			0
$251$			21.8885i			1.38159i	0.723049	+	0.690796i	$0.242740\pi$
$251$			21.8885i			1.38159i	−0.723049	+	0.690796i	$0.757260\pi$
$252$		0			0
$253$	−4.94427			−0.310844
$254$		0			0
$255$	2.47214	+	24.9443i	0.154811	+	1.56207i
$256$		0			0
$257$	4.58359			0.285917			0.142958	−	0.989729i	$-0.454339\pi$
$257$	4.58359			0.285917			0.142958	+	0.989729i	$0.454339\pi$
$258$		0			0
$259$	4.94427			0.307222
$260$		0			0
$261$	−10.0000	−	8.94427i	−0.618984	−	0.553637i
$262$		0			0
$263$			14.1803i			0.874397i	0.899365	+	0.437199i	$0.144029\pi$
$263$			14.1803i			0.874397i	−0.899365	+	0.437199i	$0.855971\pi$
$264$		0			0
$265$	−17.8885	+	8.94427i	−1.09888	+	0.549442i
$266$		0			0
$267$	−4.94427	+	12.9443i	−0.302585	+	0.792177i
$268$		0			0
$269$	−19.8885			−1.21263			−0.606313	−	0.795226i	$-0.707352\pi$
$269$	−19.8885			−1.21263			−0.606313	+	0.795226i	$0.707352\pi$
$270$		0			0
$271$		−	26.4721i		−	1.60807i	−0.594583	−	0.804034i	$-0.702683\pi$
$271$		−	26.4721i		−	1.60807i	0.594583	−	0.804034i	$-0.297317\pi$
$272$		0			0
$273$	−33.8885	−	12.9443i	−2.05103	−	0.783423i
$274$		0			0
$275$	16.0000	+	12.0000i	0.964836	+	0.723627i
$276$		0			0
$277$	−11.4164			−0.685945			−0.342973	−	0.939345i	$-0.611434\pi$
$277$	−11.4164			−0.685945			−0.342973	+	0.939345i	$0.611434\pi$
$278$		0			0
$279$	−4.94427	+	5.52786i	−0.296006	+	0.330945i
$280$		0			0
$281$		−	12.0000i		−	0.715860i	−0.933748	−	0.357930i	$-0.883483\pi$
$281$		−	12.0000i		−	0.715860i	0.933748	−	0.357930i	$-0.116517\pi$
$282$		0			0
$283$			3.70820i			0.220430i	0.993908	+	0.110215i	$0.0351539\pi$
$283$			3.70820i			0.220430i	−0.993908	+	0.110215i	$0.964846\pi$
$284$		0			0
$285$	0.944272	+	9.52786i	0.0559338	+	0.564382i
$286$		0			0
$287$			12.9443i			0.764076i
$288$		0			0
$289$	24.8885			1.46403
$290$		0			0
$291$	8.00000	−	20.9443i	0.468968	−	1.22777i
$292$		0			0
$293$		−	13.8885i		−	0.811377i	−0.914011	−	0.405689i	$-0.867032\pi$
$293$		−	13.8885i		−	0.811377i	0.914011	−	0.405689i	$-0.132968\pi$
$294$		0			0
$295$	−17.8885	+	8.94427i	−1.04151	+	0.520756i
$296$		0			0
$297$	18.4721	−	9.52786i	1.07186	−	0.552863i
$298$		0			0
$299$		−	8.00000i		−	0.462652i
$300$		0			0
$301$			29.8885i			1.72275i
$302$		0			0
$303$	−5.70820	−	2.18034i	−0.327928	−	0.125257i
$304$		0			0
$305$	−9.88854	+	4.94427i	−0.566216	+	0.283108i
$306$		0			0
$307$		−	1.81966i		−	0.103853i	−0.998651	−	0.0519267i	$-0.983464\pi$
$307$		−	1.81966i		−	0.103853i	0.998651	−	0.0519267i	$-0.0165362\pi$
$308$		0			0
$309$	2.76393	+	1.05573i	0.157235	+	0.0600583i
$310$		0			0
$311$	16.0000			0.907277			0.453638	−	0.891186i	$-0.350126\pi$
$311$	16.0000			0.907277			0.453638	+	0.891186i	$0.350126\pi$
$312$		0			0
$313$		−	24.0000i		−	1.35656i	−0.734803	−	0.678280i	$-0.762726\pi$
$313$		−	24.0000i		−	1.35656i	0.734803	−	0.678280i	$-0.237274\pi$
$314$		0			0
$315$	5.70820	−	20.9443i	0.321621	−	1.18008i
$316$		0			0
$317$		−	16.9443i		−	0.951685i	−0.879530	−	0.475843i	$-0.842143\pi$
$317$		−	16.9443i		−	0.951685i	0.879530	−	0.475843i	$-0.157857\pi$
$318$		0			0
$319$			17.8885i			1.00157i
$320$		0			0
$321$	−15.7082	−	6.00000i	−0.876746	−	0.334887i
$322$		0			0
$323$	16.0000			0.890264
$324$		0			0
$325$	−19.4164	+	25.8885i	−1.07703	+	1.43604i
$326$		0			0
$327$	8.00000	−	20.9443i	0.442401	−	1.15822i
$328$		0			0
$329$		−	29.8885i		−	1.64781i
$330$		0			0
$331$	2.47214			0.135881			0.0679404	−	0.997689i	$-0.478357\pi$
$331$	2.47214			0.135881			0.0679404	+	0.997689i	$0.478357\pi$
$332$		0			0
$333$	−3.41641	−	3.05573i	−0.187218	−	0.167453i
$334$		0			0
$335$	−23.4164	+	11.7082i	−1.27938	+	0.639688i
$336$		0			0
$337$			17.8885i			0.974451i	0.873276	+	0.487226i	$0.161991\pi$
$337$			17.8885i			0.974451i	−0.873276	+	0.487226i	$0.838009\pi$
$338$		0			0
$339$	23.4164	+	8.94427i	1.27180	+	0.485786i
$340$		0			0
$341$	9.88854			0.535495
$342$		0			0
$343$	11.4164			0.616428
$344$		0			0
$345$	4.76393	−	0.472136i	0.256481	−	0.0254189i
$346$		0			0
$347$	−4.76393			−0.255741			−0.127871	−	0.991791i	$-0.540814\pi$
$347$	−4.76393			−0.255741			−0.127871	+	0.991791i	$0.540814\pi$
$348$		0			0
$349$		−	18.8328i		−	1.00810i	−0.863675	−	0.504049i	$-0.831843\pi$
$349$		−	18.8328i		−	1.00810i	0.863675	−	0.504049i	$-0.168157\pi$
$350$		0			0
$351$	15.4164	+	29.8885i	0.822867	+	1.59533i
$352$		0			0
$353$	−11.4164			−0.607634			−0.303817	−	0.952730i	$-0.598261\pi$
$353$	−11.4164			−0.607634			−0.303817	+	0.952730i	$0.598261\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	−33.8885	−	12.9443i	−1.79357	−	0.685084i
$358$		0			0
$359$	30.8328			1.62729			0.813647	−	0.581359i	$-0.197479\pi$
$359$	30.8328			1.62729			0.813647	+	0.581359i	$0.197479\pi$
$360$		0			0
$361$	−12.8885			−0.678344
$362$		0			0
$363$	−8.09017	−	3.09017i	−0.424624	−	0.162192i
$364$		0			0
$365$	9.88854	−	4.94427i	0.517590	−	0.258795i
$366$		0			0
$367$	14.6525			0.764853			0.382427	−	0.923986i	$-0.375089\pi$
$367$	14.6525			0.764853			0.382427	+	0.923986i	$0.375089\pi$
$368$		0			0
$369$	8.00000	−	8.94427i	0.416463	−	0.465620i
$370$		0			0
$371$		−	28.9443i		−	1.50271i
$372$		0			0
$373$	30.4721			1.57779			0.788894	−	0.614530i	$-0.210654\pi$
$373$	30.4721			1.57779			0.788894	+	0.614530i	$0.210654\pi$
$374$		0			0
$375$	−16.5623	−	10.0344i	−0.855274	−	0.518176i
$376$		0			0
$377$	−28.9443			−1.49071
$378$		0			0
$379$	−8.58359			−0.440910			−0.220455	−	0.975397i	$-0.570754\pi$
$379$	−8.58359			−0.440910			−0.220455	+	0.975397i	$0.570754\pi$
$380$		0			0
$381$	18.1803	+	6.94427i	0.931407	+	0.355766i
$382$		0			0
$383$		−	22.1803i		−	1.13336i	−0.823937	−	0.566681i	$-0.808227\pi$
$383$		−	22.1803i		−	1.13336i	0.823937	−	0.566681i	$-0.191773\pi$
$384$		0			0
$385$	−25.8885	+	12.9443i	−1.31940	+	0.659701i
$386$		0			0
$387$	18.4721	−	20.6525i	0.938991	−	1.04982i
$388$		0			0
$389$	18.0000			0.912636			0.456318	−	0.889817i	$-0.349168\pi$
$389$	18.0000			0.912636			0.456318	+	0.889817i	$0.349168\pi$
$390$		0			0
$391$		−	8.00000i		−	0.404577i
$392$		0			0
$393$	0.583592	−	1.52786i	0.0294383	−	0.0770705i
$394$		0			0
$395$	20.9443	−	10.4721i	1.05382	−	0.526910i
$396$		0			0
$397$	−3.41641			−0.171465			−0.0857323	−	0.996318i	$-0.527323\pi$
$397$	−3.41641			−0.171465			−0.0857323	+	0.996318i	$0.527323\pi$
$398$		0			0
$399$	−12.9443	−	4.94427i	−0.648024	−	0.247523i
$400$		0			0
$401$		0			0		1.00000			$0$
$401$		0			0		−1.00000			$\pi$
$402$		0			0
$403$			16.0000i			0.797017i
$404$		0			0
$405$	−16.8885	+	10.9443i	−0.839198	+	0.543825i
$406$		0			0
$407$			6.11146i			0.302934i
$408$		0			0
$409$	−4.47214			−0.221133			−0.110566	−	0.993869i	$-0.535266\pi$
$409$	−4.47214			−0.221133			−0.110566	+	0.993869i	$0.535266\pi$
$410$		0			0
$411$	−2.47214	−	0.944272i	−0.121941	−	0.0465775i
$412$		0			0
$413$		−	28.9443i		−	1.42425i
$414$		0			0
$415$	8.18034	+	16.3607i	0.401557	+	0.803114i
$416$		0			0
$417$	−37.8885	−	14.4721i	−1.85541	−	0.708704i
$418$		0			0
$419$		−	4.00000i		−	0.195413i	−0.995215	−	0.0977064i	$-0.968849\pi$
$419$		−	4.00000i		−	0.195413i	0.995215	−	0.0977064i	$-0.0311506\pi$
$420$		0			0
$421$		−	4.94427i		−	0.240969i	−0.992715	−	0.120485i	$-0.961555\pi$
$421$		−	4.94427i		−	0.240969i	0.992715	−	0.120485i	$-0.0384448\pi$
$422$		0			0
$423$	−18.4721	+	20.6525i	−0.898146	+	1.00416i
$424$		0			0
$425$	−19.4164	+	25.8885i	−0.941834	+	1.25578i
$426$		0			0
$427$		−	16.0000i		−	0.774294i
$428$		0			0
$429$	16.0000	−	41.8885i	0.772487	−	2.02240i
$430$		0			0
$431$	−35.7771			−1.72332			−0.861661	−	0.507485i	$-0.830575\pi$
$431$	−35.7771			−1.72332			−0.861661	+	0.507485i	$0.830575\pi$
$432$		0			0
$433$		−	3.05573i		−	0.146849i	−0.997301	−	0.0734245i	$-0.976607\pi$
$433$		−	3.05573i		−	0.146849i	0.997301	−	0.0734245i	$-0.0233928\pi$
$434$		0			0
$435$	−1.70820	−	17.2361i	−0.0819021	−	0.826406i
$436$		0			0
$437$		−	3.05573i		−	0.146175i
$438$		0			0
$439$		−	7.41641i		−	0.353966i	−0.984214	−	0.176983i	$-0.943366\pi$
$439$		−	7.41641i		−	0.353966i	0.984214	−	0.176983i	$-0.0566337\pi$
$440$		0			0
$441$	7.76393	+	6.94427i	0.369711	+	0.330680i
$442$		0			0
$443$	−16.1803			−0.768751			−0.384376	−	0.923177i	$-0.625583\pi$
$443$	−16.1803			−0.768751			−0.384376	+	0.923177i	$0.625583\pi$
$444$		0			0
$445$	−16.0000	+	8.00000i	−0.758473	+	0.379236i
$446$		0			0
$447$	−19.2361	−	7.34752i	−0.909835	−	0.347526i
$448$		0			0
$449$			2.11146i			0.0996458i	0.998758	+	0.0498229i	$0.0158657\pi$
$449$			2.11146i			0.0996458i	−0.998758	+	0.0498229i	$0.984134\pi$
$450$		0			0
$451$	−16.0000			−0.753411
$452$		0			0
$453$	6.47214	−	16.9443i	0.304087	−	0.796111i
$454$		0			0
$455$	−20.9443	−	41.8885i	−0.981883	−	1.96377i
$456$		0			0
$457$			41.8885i			1.95946i	0.200314	+	0.979732i	$0.435804\pi$
$457$			41.8885i			1.95946i	−0.200314	+	0.979732i	$0.564196\pi$
$458$		0			0
$459$	15.4164	+	29.8885i	0.719576	+	1.39508i
$460$		0			0
$461$	−20.4721			−0.953482			−0.476741	−	0.879044i	$-0.658182\pi$
$461$	−20.4721			−0.953482			−0.476741	+	0.879044i	$0.658182\pi$
$462$		0			0
$463$	22.2918			1.03599			0.517994	−	0.855384i	$-0.326679\pi$
$463$	22.2918			1.03599			0.517994	+	0.855384i	$0.326679\pi$
$464$		0			0
$465$	−9.52786	+	0.944272i	−0.441844	+	0.0437896i
$466$		0			0
$467$	−19.2361			−0.890139			−0.445070	−	0.895496i	$-0.646821\pi$
$467$	−19.2361			−0.890139			−0.445070	+	0.895496i	$0.646821\pi$
$468$		0			0
$469$		−	37.8885i		−	1.74953i
$470$		0			0
$471$	−2.47214	−	0.944272i	−0.113910	−	0.0435098i
$472$		0			0
$473$	−36.9443			−1.69870
$474$		0			0
$475$	−7.41641	+	9.88854i	−0.340288	+	0.453718i
$476$		0			0
$477$	−17.8885	+	20.0000i	−0.819060	+	0.915737i
$478$		0			0
$479$	14.8328			0.677729			0.338864	−	0.940835i	$-0.389957\pi$
$479$	14.8328			0.677729			0.338864	+	0.940835i	$0.389957\pi$
$480$		0			0
$481$	−9.88854			−0.450879
$482$		0			0
$483$	−2.47214	+	6.47214i	−0.112486	+	0.294492i
$484$		0			0
$485$	25.8885	−	12.9443i	1.17554	−	0.587769i
$486$		0			0
$487$	−6.65248			−0.301452			−0.150726	−	0.988576i	$-0.548161\pi$
$487$	−6.65248			−0.301452			−0.150726	+	0.988576i	$0.548161\pi$
$488$		0			0
$489$	9.12461	−	23.8885i	0.412629	−	1.08028i
$490$		0			0
$491$		−	0.944272i		−	0.0426144i	−0.999773	−	0.0213072i	$-0.993217\pi$
$491$		−	0.944272i		−	0.0426144i	0.999773	−	0.0213072i	$-0.00678280\pi$
$492$		0			0
$493$	−28.9443			−1.30358
$494$		0			0
$495$	25.8885	+	7.05573i	1.16360	+	0.317131i
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−28.3607			−1.26960			−0.634799	−	0.772677i	$-0.718917\pi$
$499$	−28.3607			−1.26960			−0.634799	+	0.772677i	$0.718917\pi$
$500$		0			0
$501$	−0.763932	+	2.00000i	−0.0341300	+	0.0893534i
$502$		0			0
$503$			37.0132i			1.65034i	0.564888	+	0.825168i	$0.308919\pi$
$503$			37.0132i			1.65034i	−0.564888	+	0.825168i	$0.691081\pi$
$504$		0			0
$505$	−3.52786	−	7.05573i	−0.156988	−	0.313976i
$506$		0			0
$507$	46.7426	+	17.8541i	2.07591	+	0.792929i
$508$		0			0
$509$	−5.41641			−0.240078			−0.120039	−	0.992769i	$-0.538302\pi$
$509$	−5.41641			−0.240078			−0.120039	+	0.992769i	$0.538302\pi$
$510$		0			0
$511$			16.0000i			0.707798i
$512$		0			0
$513$	5.88854	+	11.4164i	0.259986	+	0.504047i
$514$		0			0
$515$	1.70820	+	3.41641i	0.0752725	+	0.150545i
$516$		0			0
$517$	36.9443			1.62481
$518$		0			0
$519$	−2.47214	+	6.47214i	−0.108515	+	0.284095i
$520$		0			0
$521$		−	8.00000i		−	0.350486i	−0.984525	−	0.175243i	$-0.943929\pi$
$521$		−	8.00000i		−	0.350486i	0.984525	−	0.175243i	$-0.0560712\pi$
$522$		0			0
$523$		−	3.70820i		−	0.162148i	−0.996708	−	0.0810742i	$-0.974165\pi$
$523$		−	3.70820i		−	0.162148i	0.996708	−	0.0810742i	$-0.0258351\pi$
$524$		0			0
$525$	23.7082	−	14.9443i	1.03471	−	0.652221i
$526$		0			0
$527$			16.0000i			0.696971i
$528$		0			0
$529$	21.4721			0.933571
$530$		0			0
$531$	−17.8885	+	20.0000i	−0.776297	+	0.867926i
$532$		0			0
$533$		−	25.8885i		−	1.12136i
$534$		0			0
$535$	−9.70820	−	19.4164i	−0.419722	−	0.839445i
$536$		0			0
$537$	0.583592	−	1.52786i	0.0251839	−	0.0659322i
$538$		0			0
$539$		−	13.8885i		−	0.598222i
$540$		0			0
$541$		−	7.05573i		−	0.303349i	−0.988430	−	0.151675i	$-0.951533\pi$
$541$		−	7.05573i		−	0.303349i	0.988430	−	0.151675i	$-0.0484666\pi$
$542$		0			0
$543$	0.583592	−	1.52786i	0.0250443	−	0.0655669i
$544$		0			0
$545$	25.8885	−	12.9443i	1.10894	−	0.554472i
$546$		0			0
$547$			14.7639i			0.631260i	0.948882	+	0.315630i	$0.102216\pi$
$547$			14.7639i			0.631260i	−0.948882	+	0.315630i	$0.897784\pi$
$548$		0			0
$549$	−9.88854	+	11.0557i	−0.422033	+	0.471847i
$550$		0			0
$551$	−11.0557			−0.470990
$552$		0			0
$553$			33.8885i			1.44109i
$554$		0			0
$555$	−0.583592	−	5.88854i	−0.0247721	−	0.249955i
$556$		0			0
$557$		−	5.88854i		−	0.249506i	−0.992188	−	0.124753i	$-0.960186\pi$
$557$		−	5.88854i		−	0.249506i	0.992188	−	0.124753i	$-0.0398138\pi$
$558$		0			0
$559$		−	59.7771i		−	2.52830i
$560$		0			0
$561$	16.0000	−	41.8885i	0.675521	−	1.76854i
$562$		0			0
$563$	−29.1246			−1.22746			−0.613728	−	0.789518i	$-0.710331\pi$
$563$	−29.1246			−1.22746			−0.613728	+	0.789518i	$0.710331\pi$
$564$		0			0
$565$	14.4721	+	28.9443i	0.608847	+	1.21769i
$566$		0			0
$567$	−3.23607	−	28.9443i	−0.135902	−	1.21555i
$568$		0			0
$569$		−	31.7771i		−	1.33216i	−0.745878	−	0.666082i	$-0.767970\pi$
$569$		−	31.7771i		−	1.33216i	0.745878	−	0.666082i	$-0.232030\pi$
$570$		0			0
$571$	18.4721			0.773035			0.386517	−	0.922282i	$-0.373678\pi$
$571$	18.4721			0.773035			0.386517	+	0.922282i	$0.373678\pi$
$572$		0			0
$573$	8.00000	+	3.05573i	0.334205	+	0.127655i
$574$		0			0
$575$	4.94427	+	3.70820i	0.206190	+	0.154643i
$576$		0			0
$577$			8.00000i			0.333044i	0.986038	+	0.166522i	$0.0532537\pi$
$577$			8.00000i			0.333044i	−0.986038	+	0.166522i	$0.946746\pi$
$578$		0			0
$579$	−12.9443	+	33.8885i	−0.537946	+	1.40836i
$580$		0			0
$581$	−26.4721			−1.09825
$582$		0			0
$583$	35.7771			1.48174
$584$		0			0
$585$	−11.4164	+	41.8885i	−0.472010	+	1.73188i
$586$		0			0
$587$	−36.7639			−1.51741			−0.758705	−	0.651434i	$-0.774168\pi$
$587$	−36.7639			−1.51741			−0.758705	+	0.651434i	$0.774168\pi$
$588$		0			0
$589$			6.11146i			0.251818i
$590$		0			0
$591$	5.52786	−	14.4721i	0.227386	−	0.595304i
$592$		0			0
$593$	30.4721			1.25134			0.625670	−	0.780088i	$-0.284826\pi$
$593$	30.4721			1.25134			0.625670	+	0.780088i	$0.284826\pi$
$594$		0			0
$595$	−20.9443	−	41.8885i	−0.858631	−	1.71726i
$596$		0			0
$597$	8.36068	−	21.8885i	0.342180	−	0.895838i
$598$		0			0
$599$	−4.94427			−0.202017			−0.101009	−	0.994886i	$-0.532207\pi$
$599$	−4.94427			−0.202017			−0.101009	+	0.994886i	$0.532207\pi$
$600$		0			0
$601$	4.47214			0.182422			0.0912111	−	0.995832i	$-0.470926\pi$
$601$	4.47214			0.182422			0.0912111	+	0.995832i	$0.470926\pi$
$602$		0			0
$603$	−23.4164	+	26.1803i	−0.953590	+	1.06615i
$604$		0			0
$605$	−5.00000	−	10.0000i	−0.203279	−	0.406558i
$606$		0			0
$607$	−48.1803			−1.95558			−0.977790	−	0.209588i	$-0.932788\pi$
$607$	−48.1803			−1.95558			−0.977790	+	0.209588i	$0.932788\pi$
$608$		0			0
$609$	23.4164	+	8.94427i	0.948881	+	0.362440i
$610$		0			0
$611$			59.7771i			2.41832i
$612$		0			0
$613$	−19.4164			−0.784221			−0.392111	−	0.919918i	$-0.628255\pi$
$613$	−19.4164			−0.784221			−0.392111	+	0.919918i	$0.628255\pi$
$614$		0			0
$615$	15.4164	−	1.52786i	0.621650	−	0.0616094i
$616$		0			0
$617$	−35.4164			−1.42581			−0.712905	−	0.701260i	$-0.752621\pi$
$617$	−35.4164			−1.42581			−0.712905	+	0.701260i	$0.752621\pi$
$618$		0			0
$619$	34.4721			1.38555			0.692776	−	0.721153i	$-0.256387\pi$
$619$	34.4721			1.38555			0.692776	+	0.721153i	$0.256387\pi$
$620$		0			0
$621$	5.70820	−	2.94427i	0.229062	−	0.118150i
$622$		0			0
$623$		−	25.8885i		−	1.03720i
$624$		0			0
$625$	−7.00000	−	24.0000i	−0.280000	−	0.960000i
$626$		0			0
$627$	6.11146	−	16.0000i	0.244068	−	0.638978i
$628$		0			0
$629$	−9.88854			−0.394282
$630$		0			0
$631$		−	13.5279i		−	0.538536i	−0.963065	−	0.269268i	$-0.913218\pi$
$631$		−	13.5279i		−	0.538536i	0.963065	−	0.269268i	$-0.0867817\pi$
$632$		0			0
$633$	12.0000	+	4.58359i	0.476957	+	0.182181i
$634$		0			0
$635$	11.2361	+	22.4721i	0.445890	+	0.891780i
$636$		0			0
$637$	22.4721			0.890378
$638$		0			0
$639$		0			0
$640$		0			0
$641$			13.8885i			0.548564i	0.961649	+	0.274282i	$0.0884403\pi$
$641$			13.8885i			0.548564i	−0.961649	+	0.274282i	$0.911560\pi$
$642$		0			0
$643$		−	21.5967i		−	0.851693i	−0.904796	−	0.425846i	$-0.859976\pi$
$643$		−	21.5967i		−	0.851693i	0.904796	−	0.425846i	$-0.140024\pi$
$644$		0			0
$645$	35.5967	−	3.52786i	1.40162	−	0.138910i
$646$		0			0
$647$			20.2918i			0.797753i	0.917005	+	0.398876i	$0.130600\pi$
$647$			20.2918i			0.797753i	−0.917005	+	0.398876i	$0.869400\pi$
$648$		0			0
$649$	35.7771			1.40437
$650$		0			0
$651$	4.94427	−	12.9443i	0.193781	−	0.507326i
$652$		0			0
$653$			32.9443i			1.28921i	0.764516	+	0.644604i	$0.222978\pi$
$653$			32.9443i			1.28921i	−0.764516	+	0.644604i	$0.777022\pi$
$654$		0			0
$655$	1.88854	−	0.944272i	0.0737915	−	0.0368958i
$656$		0			0
$657$	9.88854	−	11.0557i	0.385789	−	0.431325i
$658$		0			0
$659$		−	7.05573i		−	0.274852i	−0.990512	−	0.137426i	$-0.956117\pi$
$659$		−	7.05573i		−	0.274852i	0.990512	−	0.137426i	$-0.0438830\pi$
$660$		0			0
$661$		−	3.05573i		−	0.118854i	−0.998233	−	0.0594270i	$-0.981073\pi$
$661$		−	3.05573i		−	0.118854i	0.998233	−	0.0594270i	$-0.0189274\pi$
$662$		0			0
$663$	67.7771	+	25.8885i	2.63224	+	1.00543i
$664$		0			0
$665$	−8.00000	−	16.0000i	−0.310227	−	0.620453i
$666$		0			0
$667$			5.52786i			0.214040i
$668$		0			0
$669$	−34.1803	−	13.0557i	−1.32149	−	0.504764i
$670$		0			0
$671$	19.7771			0.763486
$672$		0			0
$673$		−	28.9443i		−	1.11572i	−0.829935	−	0.557860i	$-0.811623\pi$
$673$		−	28.9443i		−	1.11572i	0.829935	−	0.557860i	$-0.188377\pi$
$674$		0			0
$675$	−25.6180	−	4.32624i	−0.986039	−	0.166517i
$676$		0			0
$677$			24.9443i			0.958686i	0.877628	+	0.479343i	$0.159125\pi$
$677$			24.9443i			0.958686i	−0.877628	+	0.479343i	$0.840875\pi$
$678$		0			0
$679$			41.8885i			1.60753i
$680$		0			0
$681$	−20.6525	−	7.88854i	−0.791405	−	0.302290i
$682$		0			0
$683$	11.2361			0.429936			0.214968	−	0.976621i	$-0.431035\pi$
$683$	11.2361			0.429936			0.214968	+	0.976621i	$0.431035\pi$
$684$		0			0
$685$	−1.52786	−	3.05573i	−0.0583767	−	0.116753i
$686$		0			0
$687$	15.4164	−	40.3607i	0.588173	−	1.53986i
$688$		0			0
$689$			57.8885i			2.20538i
$690$		0			0
$691$	−17.3050			−0.658311			−0.329156	−	0.944276i	$-0.606764\pi$
$691$	−17.3050			−0.658311			−0.329156	+	0.944276i	$0.606764\pi$
$692$		0			0
$693$	−25.8885	+	28.9443i	−0.983424	+	1.09950i
$694$		0			0
$695$	−23.4164	−	46.8328i	−0.888235	−	1.77647i
$696$		0			0
$697$		−	25.8885i		−	0.980599i
$698$		0			0
$699$	−15.4164	−	5.88854i	−0.583102	−	0.222725i
$700$		0			0
$701$	10.0000			0.377695			0.188847	−	0.982006i	$-0.439525\pi$
$701$	10.0000			0.377695			0.188847	+	0.982006i	$0.439525\pi$
$702$		0			0
$703$	−3.77709			−0.142456
$704$		0			0
$705$	−35.5967	+	3.52786i	−1.34065	+	0.132867i
$706$		0			0
$707$	11.4164			0.429358
$708$		0			0
$709$		−	32.9443i		−	1.23725i	−0.785687	−	0.618624i	$-0.787690\pi$
$709$		−	32.9443i		−	1.23725i	0.785687	−	0.618624i	$-0.212310\pi$
$710$		0			0
$711$	20.9443	−	23.4164i	0.785472	−	0.878184i
$712$		0			0
$713$	3.05573			0.114438
$714$		0			0
$715$	51.7771	−	25.8885i	1.93635	−	0.968177i
$716$		0			0
$717$	17.8885	+	6.83282i	0.668060	+	0.255176i
$718$		0			0
$719$	−25.8885			−0.965480			−0.482740	−	0.875764i	$-0.660358\pi$
$719$	−25.8885			−0.965480			−0.482740	+	0.875764i	$0.660358\pi$
$720$		0			0
$721$	−5.52786			−0.205868
$722$		0			0
$723$	−20.1803	−	7.70820i	−0.750515	−	0.286671i
$724$		0			0
$725$	13.4164	−	17.8885i	0.498273	−	0.664364i
$726$		0			0
$727$	−12.7639			−0.473388			−0.236694	−	0.971584i	$-0.576064\pi$
$727$	−12.7639			−0.473388			−0.236694	+	0.971584i	$0.576064\pi$
$728$		0			0
$729$	−15.6525	+	22.0000i	−0.579721	+	0.814815i
$730$		0			0
$731$		−	59.7771i		−	2.21094i
$732$		0			0
$733$	−1.52786			−0.0564329			−0.0282165	−	0.999602i	$-0.508983\pi$
$733$	−1.52786			−0.0564329			−0.0282165	+	0.999602i	$0.508983\pi$
$734$		0			0
$735$	1.32624	+	13.3820i	0.0489190	+	0.493601i
$736$		0			0
$737$	46.8328			1.72511
$738$		0			0
$739$	−18.4721			−0.679509			−0.339754	−	0.940514i	$-0.610344\pi$
$739$	−18.4721			−0.679509			−0.339754	+	0.940514i	$0.610344\pi$
$740$		0			0
$741$	25.8885	+	9.88854i	0.951039	+	0.363265i
$742$		0			0
$743$			47.4853i			1.74207i	0.491224	+	0.871033i	$0.336550\pi$
$743$			47.4853i			1.74207i	−0.491224	+	0.871033i	$0.663450\pi$
$744$		0			0
$745$	−11.8885	−	23.7771i	−0.435563	−	0.871125i
$746$		0			0
$747$	18.2918	+	16.3607i	0.669262	+	0.598606i
$748$		0			0
$749$	31.4164			1.14793
$750$		0			0
$751$		−	17.3050i		−	0.631467i	−0.948848	−	0.315733i	$-0.897750\pi$
$751$		−	17.3050i		−	0.631467i	0.948848	−	0.315733i	$-0.102250\pi$
$752$		0			0
$753$	−13.5279	+	35.4164i	−0.492983	+	1.29065i
$754$		0			0
$755$	20.9443	−	10.4721i	0.762240	−	0.381120i
$756$		0			0
$757$	−27.4164			−0.996466			−0.498233	−	0.867043i	$-0.666018\pi$
$757$	−27.4164			−0.996466			−0.498233	+	0.867043i	$0.666018\pi$
$758$		0			0
$759$	−8.00000	−	3.05573i	−0.290382	−	0.110916i
$760$		0			0
$761$			9.88854i			0.358459i	0.983807	+	0.179230i	$0.0573605\pi$
$761$			9.88854i			0.358459i	−0.983807	+	0.179230i	$0.942639\pi$
$762$		0			0
$763$			41.8885i			1.51647i
$764$		0			0
$765$	−11.4164	+	41.8885i	−0.412761	+	1.51448i
$766$		0			0
$767$			57.8885i			2.09023i
$768$		0			0
$769$	27.8885			1.00569			0.502843	−	0.864378i	$-0.332287\pi$
$769$	27.8885			1.00569			0.502843	+	0.864378i	$0.332287\pi$
$770$		0			0
$771$	7.41641	+	2.83282i	0.267095	+	0.102021i
$772$		0			0
$773$			52.7214i			1.89626i	0.317890	+	0.948128i	$0.397026\pi$
$773$			52.7214i			1.89626i	−0.317890	+	0.948128i	$0.602974\pi$
$774$		0			0
$775$	−9.88854	−	7.41641i	−0.355207	−	0.266405i
$776$		0			0
$777$	8.00000	+	3.05573i	0.286998	+	0.109624i
$778$		0			0
$779$		−	9.88854i		−	0.354294i
$780$		0			0
$781$		0			0
$782$		0			0
$783$	−10.6525	−	20.6525i	−0.380688	−	0.738059i
$784$		0			0
$785$	−1.52786	−	3.05573i	−0.0545318	−	0.109064i
$786$		0			0
$787$			14.1803i			0.505475i	0.967535	+	0.252737i	$0.0813308\pi$
$787$			14.1803i			0.505475i	−0.967535	+	0.252737i	$0.918669\pi$
$788$		0			0
$789$	−8.76393	+	22.9443i	−0.312004	+	0.816838i
$790$		0			0
$791$	−46.8328			−1.66518
$792$		0			0
$793$			32.0000i			1.13635i
$794$		0			0
$795$	−34.4721	+	3.41641i	−1.22260	+	0.121168i
$796$		0			0
$797$		−	23.0557i		−	0.816676i	−0.912831	−	0.408338i	$-0.866109\pi$
$797$		−	23.0557i		−	0.816676i	0.912831	−	0.408338i	$-0.133891\pi$
$798$		0			0
$799$			59.7771i			2.11476i
$800$		0			0
$801$	−16.0000	+	17.8885i	−0.565332	+	0.632061i
$802$		0			0
$803$	−19.7771			−0.697918
$804$		0			0
$805$	−8.00000	+	4.00000i	−0.281963	+	0.140981i
$806$		0			0
$807$	−32.1803	−	12.2918i	−1.13280	−	0.432692i
$808$		0			0
$809$		−	40.0000i		−	1.40633i	−0.711029	−	0.703163i	$-0.751771\pi$
$809$		−	40.0000i		−	1.40633i	0.711029	−	0.703163i	$-0.248229\pi$
$810$		0			0
$811$	−8.58359			−0.301411			−0.150705	−	0.988579i	$-0.548154\pi$
$811$	−8.58359			−0.301411			−0.150705	+	0.988579i	$0.548154\pi$
$812$		0			0
$813$	16.3607	−	42.8328i	0.573794	−	1.50221i
$814$		0			0
$815$	29.5279	−	14.7639i	1.03432	−	0.517158i
$816$		0			0
$817$		−	22.8328i		−	0.798819i
$818$		0			0
$819$	−46.8328	−	41.8885i	−1.63647	−	1.46370i
$820$		0			0
$821$	4.11146			0.143491			0.0717454	−	0.997423i	$-0.477143\pi$
$821$	4.11146			0.143491			0.0717454	+	0.997423i	$0.477143\pi$
$822$		0			0
$823$	−14.2918			−0.498181			−0.249090	−	0.968480i	$-0.580132\pi$
$823$	−14.2918			−0.498181			−0.249090	+	0.968480i	$0.580132\pi$
$824$		0			0
$825$	18.4721	+	29.3050i	0.643117	+	1.02027i
$826$		0			0
$827$	24.5410			0.853375			0.426687	−	0.904399i	$-0.359680\pi$
$827$	24.5410			0.853375			0.426687	+	0.904399i	$0.359680\pi$
$828$		0			0
$829$			20.9443i			0.727425i	0.931511	+	0.363712i	$0.118491\pi$
$829$			20.9443i			0.727425i	−0.931511	+	0.363712i	$0.881509\pi$
$830$		0			0
$831$	−18.4721	−	7.05573i	−0.640791	−	0.244760i
$832$		0			0
$833$	22.4721			0.778613
$834$		0			0
$835$	−2.47214	+	1.23607i	−0.0855518	+	0.0427759i
$836$		0			0
$837$	−11.4164	+	5.88854i	−0.394609	+	0.203538i
$838$		0			0
$839$	−30.8328			−1.06447			−0.532234	−	0.846598i	$-0.678647\pi$
$839$	−30.8328			−1.06447			−0.532234	+	0.846598i	$0.678647\pi$
$840$		0			0
$841$	−9.00000			−0.310345
$842$		0			0
$843$	7.41641	−	19.4164i	0.255435	−	0.668737i
$844$		0			0
$845$	28.8885	+	57.7771i	0.993796	+	1.98759i
$846$		0			0
$847$	16.1803			0.555963
$848$		0			0
$849$	−2.29180	+	6.00000i	−0.0786542	+	0.205919i
$850$		0			0
$851$			1.88854i			0.0647384i
$852$		0			0
$853$	−41.5279			−1.42189			−0.710943	−	0.703249i	$-0.751732\pi$
$853$	−41.5279			−1.42189			−0.710943	+	0.703249i	$0.751732\pi$
$854$		0			0
$855$	−4.36068	+	16.0000i	−0.149132	+	0.547188i
$856$		0			0
$857$	40.3607			1.37869			0.689347	−	0.724431i	$-0.257897\pi$
$857$	40.3607			1.37869			0.689347	+	0.724431i	$0.257897\pi$
$858$		0			0
$859$	39.4164			1.34487			0.672435	−	0.740156i	$-0.265248\pi$
$859$	39.4164			1.34487			0.672435	+	0.740156i	$0.265248\pi$
$860$		0			0
$861$	−8.00000	+	20.9443i	−0.272639	+	0.713779i
$862$		0			0
$863$		−	38.7639i		−	1.31954i	−0.751468	−	0.659770i	$-0.770654\pi$
$863$		−	38.7639i		−	1.31954i	0.751468	−	0.659770i	$-0.229346\pi$
$864$		0			0
$865$	−8.00000	+	4.00000i	−0.272008	+	0.136004i
$866$		0			0
$867$	40.2705	+	15.3820i	1.36766	+	0.522399i
$868$		0			0
$869$	−41.8885			−1.42097
$870$		0			0
$871$			75.7771i			2.56761i
$872$		0			0
$873$	25.8885	−	28.9443i	0.876194	−	0.979615i
$874$		0			0
$875$	35.5967	+	6.47214i	1.20339	+	0.218798i
$876$		0			0
$877$	−33.5279			−1.13216			−0.566078	−	0.824352i	$-0.691540\pi$
$877$	−33.5279			−1.13216			−0.566078	+	0.824352i	$0.691540\pi$
$878$		0			0
$879$	8.58359	−	22.4721i	0.289517	−	0.757966i
$880$		0			0
$881$			29.8885i			1.00697i	0.864004	+	0.503485i	$0.167949\pi$
$881$			29.8885i			1.00697i	−0.864004	+	0.503485i	$0.832051\pi$
$882$		0			0
$883$			1.23607i			0.0415970i	0.999784	+	0.0207985i	$0.00662085\pi$
$883$			1.23607i			0.0415970i	−0.999784	+	0.0207985i	$0.993379\pi$
$884$		0			0
$885$	−34.4721	+	3.41641i	−1.15877	+	0.114841i
$886$		0			0
$887$		−	21.5967i		−	0.725148i	−0.931955	−	0.362574i	$-0.881898\pi$
$887$		−	21.5967i		−	0.725148i	0.931955	−	0.362574i	$-0.118102\pi$
$888$		0			0
$889$	−36.3607			−1.21950
$890$		0			0
$891$	35.7771	−	4.00000i	1.19858	−	0.134005i
$892$		0			0
$893$			22.8328i			0.764071i
$894$		0			0
$895$	1.88854	−	0.944272i	0.0631271	−	0.0315635i
$896$		0			0
$897$	4.94427	−	12.9443i	0.165084	−	0.432197i
$898$		0			0
$899$		−	11.0557i		−	0.368729i
$900$		0			0
$901$			57.8885i			1.92855i
$902$		0			0
$903$	−18.4721	+	48.3607i	−0.614714	+	1.60934i
$904$		0			0
$905$	1.88854	−	0.944272i	0.0627773	−	0.0313887i
$906$		0			0
$907$			55.4853i			1.84236i	0.389138	+	0.921179i	$0.372773\pi$
$907$			55.4853i			1.84236i	−0.389138	+	0.921179i	$0.627227\pi$
$908$		0			0
$909$	−7.88854	−	7.05573i	−0.261646	−	0.234024i
$910$		0			0
$911$	57.8885			1.91793			0.958967	−	0.283519i	$-0.0915020\pi$
$911$	57.8885			1.91793			0.958967	+	0.283519i	$0.0915020\pi$
$912$		0			0
$913$		−	32.7214i		−	1.08292i
$914$		0			0
$915$	−19.0557	+	1.88854i	−0.629963	+	0.0624333i
$916$		0			0
$917$			3.05573i			0.100909i
$918$		0			0
$919$		−	42.4721i		−	1.40103i	−0.713639	−	0.700513i	$-0.752954\pi$
$919$		−	42.4721i		−	1.40103i	0.713639	−	0.700513i	$-0.247046\pi$
$920$		0			0
$921$	1.12461	−	2.94427i	0.0370572	−	0.0970171i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	4.58359	−	6.11146i	0.150708	−	0.200943i
$926$		0			0
$927$	3.81966	+	3.41641i	0.125454	+	0.112210i
$928$		0			0
$929$		−	53.8885i		−	1.76803i	−0.467463	−	0.884013i	$-0.654832\pi$
$929$		−	53.8885i		−	1.76803i	0.467463	−	0.884013i	$-0.345168\pi$
$930$		0			0
$931$	8.58359			0.281316
$932$		0			0
$933$	25.8885	+	9.88854i	0.847553	+	0.323736i
$934$		0			0
$935$	51.7771	−	25.8885i	1.69329	−	0.846646i
$936$		0			0
$937$		−	38.8328i		−	1.26861i	−0.773082	−	0.634306i	$-0.781286\pi$
$937$		−	38.8328i		−	1.26861i	0.773082	−	0.634306i	$-0.218714\pi$
$938$		0			0
$939$	14.8328	−	38.8328i	0.484051	−	1.26726i
$940$		0			0
$941$	−53.4164			−1.74133			−0.870663	−	0.491881i	$-0.836310\pi$
$941$	−53.4164			−1.74133			−0.870663	+	0.491881i	$0.836310\pi$
$942$		0			0
$943$	−4.94427			−0.161008
$944$		0			0
$945$	22.1803	−	30.3607i	0.721527	−	0.987633i
$946$		0			0
$947$	−50.0689			−1.62702			−0.813510	−	0.581551i	$-0.802446\pi$
$947$	−50.0689			−1.62702			−0.813510	+	0.581551i	$0.802446\pi$
$948$		0			0
$949$		−	32.0000i		−	1.03876i
$950$		0			0
$951$	10.4721	−	27.4164i	0.339582	−	0.889038i
$952$		0			0
$953$	−1.52786			−0.0494924			−0.0247462	−	0.999694i	$-0.507878\pi$
$953$	−1.52786			−0.0494924			−0.0247462	+	0.999694i	$0.507878\pi$
$954$		0			0
$955$	4.94427	+	9.88854i	0.159993	+	0.319986i
$956$		0			0
$957$	−11.0557	+	28.9443i	−0.357381	+	0.935635i
$958$		0			0
$959$	4.94427			0.159659
$960$		0			0
$961$	24.8885			0.802856
$962$		0			0
$963$	−21.7082	−	19.4164i	−0.699537	−	0.625685i
$964$		0			0
$965$	−41.8885	+	20.9443i	−1.34844	+	0.674220i
$966$		0			0
$967$	−43.5967			−1.40198			−0.700988	−	0.713173i	$-0.747258\pi$
$967$	−43.5967			−1.40198			−0.700988	+	0.713173i	$0.747258\pi$
$968$		0			0
$969$	25.8885	+	9.88854i	0.831660	+	0.317666i
$970$		0			0
$971$			28.0000i			0.898563i	0.893390	+	0.449281i	$0.148320\pi$
$971$			28.0000i			0.898563i	−0.893390	+	0.449281i	$0.851680\pi$
$972$		0			0
$973$	75.7771			2.42930
$974$		0			0
$975$	−47.4164	+	29.8885i	−1.51854	+	0.957199i
$976$		0			0
$977$	12.5836			0.402585			0.201292	−	0.979531i	$-0.435486\pi$
$977$	12.5836			0.402585			0.201292	+	0.979531i	$0.435486\pi$
$978$		0			0
$979$	32.0000			1.02272
$980$		0			0
$981$	25.8885	−	28.9443i	0.826558	−	0.924119i
$982$		0			0
$983$			11.7082i			0.373434i	0.982414	+	0.186717i	$0.0597847\pi$
$983$			11.7082i			0.373434i	−0.982414	+	0.186717i	$0.940215\pi$
$984$		0			0
$985$	17.8885	−	8.94427i	0.569976	−	0.284988i
$986$		0			0
$987$	18.4721	−	48.3607i	0.587975	−	1.53934i
$988$		0			0
$989$	−11.4164			−0.363021
$990$		0			0
$991$			51.1935i			1.62621i	0.582114	+	0.813107i	$0.302226\pi$
$991$			51.1935i			1.62621i	−0.582114	+	0.813107i	$0.697774\pi$
$992$		0			0
$993$	4.00000	+	1.52786i	0.126936	+	0.0484853i
$994$		0			0
$995$	27.0557	−	13.5279i	0.857724	−	0.428862i
$996$		0			0
$997$	42.2492			1.33805			0.669023	−	0.743242i	$-0.266713\pi$
$997$	42.2492			1.33805			0.669023	+	0.743242i	$0.266713\pi$
$998$		0			0
$999$	−3.63932	−	7.05573i	−0.115143	−	0.223233i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1920.2.m.t.959.4	yes	4
3.2	odd	2		1920.2.m.d.959.2	yes	4
4.3	odd	2		1920.2.m.h.959.1	yes	4
5.4	even	2		1920.2.m.g.959.1	yes	4
8.3	odd	2		1920.2.m.o.959.4	yes	4
8.5	even	2		1920.2.m.c.959.1	✓	4
12.11	even	2		1920.2.m.p.959.3	yes	4
15.14	odd	2		1920.2.m.o.959.3	yes	4
20.19	odd	2		1920.2.m.s.959.4	yes	4
24.5	odd	2		1920.2.m.s.959.3	yes	4
24.11	even	2		1920.2.m.g.959.2	yes	4
40.19	odd	2		1920.2.m.d.959.1	yes	4
40.29	even	2		1920.2.m.p.959.4	yes	4
60.59	even	2		1920.2.m.c.959.2	yes	4
120.29	odd	2		1920.2.m.h.959.2	yes	4
120.59	even	2	inner	1920.2.m.t.959.3	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1920.2.m.c.959.1	✓	4	8.5	even	2
1920.2.m.c.959.2	yes	4	60.59	even	2
1920.2.m.d.959.1	yes	4	40.19	odd	2
1920.2.m.d.959.2	yes	4	3.2	odd	2
1920.2.m.g.959.1	yes	4	5.4	even	2
1920.2.m.g.959.2	yes	4	24.11	even	2
1920.2.m.h.959.1	yes	4	4.3	odd	2
1920.2.m.h.959.2	yes	4	120.29	odd	2
1920.2.m.o.959.3	yes	4	15.14	odd	2
1920.2.m.o.959.4	yes	4	8.3	odd	2
1920.2.m.p.959.3	yes	4	12.11	even	2
1920.2.m.p.959.4	yes	4	40.29	even	2
1920.2.m.s.959.3	yes	4	24.5	odd	2
1920.2.m.s.959.4	yes	4	20.19	odd	2
1920.2.m.t.959.3	yes	4	120.59	even	2	inner
1920.2.m.t.959.4	yes	4	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 1920 = 2^{7} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1920.m (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(1.61803 + 0.618034i) q^{3} +(1.00000 + 2.00000i) q^{5} -3.23607 q^{7} +(2.23607 + 2.00000i) q^{9} -4.00000i q^{11} +6.47214 q^{13} +(0.381966 + 3.85410i) q^{15} +6.47214 q^{17} +2.47214 q^{19} +(-5.23607 - 2.00000i) q^{21} -1.23607i q^{23} +(-3.00000 + 4.00000i) q^{25} +(2.38197 + 4.61803i) q^{27} -4.47214 q^{29} +2.47214i q^{31} +(2.47214 - 6.47214i) q^{33} +(-3.23607 - 6.47214i) q^{35} -1.52786 q^{37} +(10.4721 + 4.00000i) q^{39} -4.00000i q^{41} -9.23607i q^{43} +(-1.76393 + 6.47214i) q^{45} +9.23607i q^{47} +3.47214 q^{49} +(10.4721 + 4.00000i) q^{51} +8.94427i q^{53} +(8.00000 - 4.00000i) q^{55} +(4.00000 + 1.52786i) q^{57} +8.94427i q^{59} +4.94427i q^{61} +(-7.23607 - 6.47214i) q^{63} +(6.47214 + 12.9443i) q^{65} +11.7082i q^{67} +(0.763932 - 2.00000i) q^{69} -4.94427i q^{73} +(-7.32624 + 4.61803i) q^{75} +12.9443i q^{77} -10.4721i q^{79} +(1.00000 + 8.94427i) q^{81} +8.18034 q^{83} +(6.47214 + 12.9443i) q^{85} +(-7.23607 - 2.76393i) q^{87} +8.00000i q^{89} -20.9443 q^{91} +(-1.52786 + 4.00000i) q^{93} +(2.47214 + 4.94427i) q^{95} -12.9443i q^{97} +(8.00000 - 8.94427i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{3} + 4 q^{5} - 4 q^{7} + 8 q^{13} + 6 q^{15} + 8 q^{17} - 8 q^{19} - 12 q^{21} - 12 q^{25} + 14 q^{27} - 8 q^{33} - 4 q^{35} - 24 q^{37} + 24 q^{39} - 16 q^{45} - 4 q^{49} + 24 q^{51} + 32 q^{55}+ \cdots + 32 q^{99}+O(q^{100}) \) 4 * q + 2 * q^3 + 4 * q^5 - 4 * q^7 + 8 * q^13 + 6 * q^15 + 8 * q^17 - 8 * q^19 - 12 * q^21 - 12 * q^25 + 14 * q^27 - 8 * q^33 - 4 * q^35 - 24 * q^37 + 24 * q^39 - 16 * q^45 - 4 * q^49 + 24 * q^51 + 32 * q^55 + 16 * q^57 - 20 * q^63 + 8 * q^65 + 12 * q^69 + 2 * q^75 + 4 * q^81 - 12 * q^83 + 8 * q^85 - 20 * q^87 - 48 * q^91 - 24 * q^93 - 8 * q^95 + 32 * q^99

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