LMFDB - Embedded newform 1350.3.g.j.1243.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1350,3,Mod(757,1350)] 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("1350.757"); S:= CuspForms(chi, 3); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1350, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 3, names="a")

Level:	$ N $	$=$	$ 1350 = 2 \cdot 3^{3} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1350.g (of order $4$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [4,4,0,0,0,0,0,-8,0,0,0,0,0,0,0,-16,24] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

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

Self dual:	no
Analytic conductor:	$36.7848356886$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(i)$
Coefficient field:	$\Q(i, \sqrt{6})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 9 $ x^4 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			1243.1
Root			$-1.22474 + 1.22474i$ of defining polynomial
Character	$\chi$	$=$	1350.1243
Dual form			1350.3.g.j.757.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(1.00000 - 1.00000i) q^{2} -2.00000i q^{4} +(-3.67423 + 3.67423i) q^{7} +(-2.00000 - 2.00000i) q^{8} +(3.67423 + 3.67423i) q^{13} +7.34847i q^{14} -4.00000 q^{16} +(6.00000 - 6.00000i) q^{17} +7.00000i q^{19} +(-12.0000 - 12.0000i) q^{23} +7.34847 q^{26} +(7.34847 + 7.34847i) q^{28} +44.0908i q^{29} +10.0000 q^{31} +(-4.00000 + 4.00000i) q^{32} -12.0000i q^{34} +(-25.7196 + 25.7196i) q^{37} +(7.00000 + 7.00000i) q^{38} +44.0908 q^{41} +(29.3939 + 29.3939i) q^{43} -24.0000 q^{46} +(-18.0000 + 18.0000i) q^{47} +22.0000i q^{49} +(7.34847 - 7.34847i) q^{52} +(30.0000 + 30.0000i) q^{53} +14.6969 q^{56} +(44.0908 + 44.0908i) q^{58} +44.0908i q^{59} +23.0000 q^{61} +(10.0000 - 10.0000i) q^{62} +8.00000i q^{64} +(-33.0681 + 33.0681i) q^{67} +(-12.0000 - 12.0000i) q^{68} +132.272 q^{71} +(33.0681 + 33.0681i) q^{73} +51.4393i q^{74} +14.0000 q^{76} -103.000i q^{79} +(44.0908 - 44.0908i) q^{82} +(42.0000 + 42.0000i) q^{83} +58.7878 q^{86} +44.0908i q^{89} -27.0000 q^{91} +(-24.0000 + 24.0000i) q^{92} +36.0000i q^{94} +(-91.8559 + 91.8559i) q^{97} +(22.0000 + 22.0000i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{2} - 8 q^{8} - 16 q^{16} + 24 q^{17} - 48 q^{23} + 40 q^{31} - 16 q^{32} + 28 q^{38} - 96 q^{46} - 72 q^{47} + 120 q^{53} + 92 q^{61} + 40 q^{62} - 48 q^{68} + 56 q^{76} + 168 q^{83} - 108 q^{91}+ \cdots + 88 q^{98}+O(q^{100}) $ 4 * q + 4 * q^2 - 8 * q^8 - 16 * q^16 + 24 * q^17 - 48 * q^23 + 40 * q^31 - 16 * q^32 + 28 * q^38 - 96 * q^46 - 72 * q^47 + 120 * q^53 + 92 * q^61 + 40 * q^62 - 48 * q^68 + 56 * q^76 + 168 * q^83 - 108 * q^91 - 96 * q^92 + 88 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1350\mathbb{Z}\right)^\times$.

$n$	$1001$	$1027$
$\chi(n)$	$1$	$e\left(\frac{3}{4}\right)$

Coefficient data

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

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

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

$2$	1.00000	−	1.00000i	0.500000	−	0.500000i
$2$	1.00000	−	1.00000i	0.500000	−	0.500000i
$3$		0			0
$3$		0			0
$4$		−	2.00000i		−	0.500000i
$5$		0			0
$5$		0			0
$6$		0			0
$7$	−3.67423	+	3.67423i	−0.524891	+	0.524891i	−0.919044	−	0.394154i	$-0.871038\pi$
$7$	−3.67423	+	3.67423i	−0.524891	+	0.524891i	0.394154	+	0.919044i	$0.371038\pi$
$8$	−2.00000	−	2.00000i	−0.250000	−	0.250000i
$9$		0			0
$10$		0			0
$11$		0			0			−	1.00000i	$-0.5\pi$
$11$		0			0				1.00000i	$0.5\pi$
$12$		0			0
$13$	3.67423	+	3.67423i	0.282633	+	0.282633i	0.834158	−	0.551525i	$-0.185954\pi$
$13$	3.67423	+	3.67423i	0.282633	+	0.282633i	−0.551525	+	0.834158i	$0.685954\pi$
$14$			7.34847i			0.524891i
$15$		0			0
$16$	−4.00000			−0.250000
$17$	6.00000	−	6.00000i	0.352941	−	0.352941i	−0.508262	−	0.861203i	$-0.669712\pi$
$17$	6.00000	−	6.00000i	0.352941	−	0.352941i	0.861203	+	0.508262i	$0.169712\pi$
$18$		0			0
$19$			7.00000i			0.368421i	0.982887	+	0.184211i	$0.0589728\pi$
$19$			7.00000i			0.368421i	−0.982887	+	0.184211i	$0.941027\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−12.0000	−	12.0000i	−0.521739	−	0.521739i	0.396357	−	0.918096i	$-0.370274\pi$
$23$	−12.0000	−	12.0000i	−0.521739	−	0.521739i	−0.918096	+	0.396357i	$0.870274\pi$
$24$		0			0
$25$		0			0
$26$	7.34847			0.282633
$27$		0			0
$28$	7.34847	+	7.34847i	0.262445	+	0.262445i
$29$			44.0908i			1.52037i	0.649705	+	0.760186i	$0.274892\pi$
$29$			44.0908i			1.52037i	−0.649705	+	0.760186i	$0.725108\pi$
$30$		0			0
$31$	10.0000			0.322581			0.161290	−	0.986907i	$-0.448434\pi$
$31$	10.0000			0.322581			0.161290	+	0.986907i	$0.448434\pi$
$32$	−4.00000	+	4.00000i	−0.125000	+	0.125000i
$33$		0			0
$34$		−	12.0000i		−	0.352941i
$35$		0			0
$36$		0			0
$37$	−25.7196	+	25.7196i	−0.695125	+	0.695125i	−0.963355	−	0.268230i	$-0.913561\pi$
$37$	−25.7196	+	25.7196i	−0.695125	+	0.695125i	0.268230	+	0.963355i	$0.413561\pi$
$38$	7.00000	+	7.00000i	0.184211	+	0.184211i
$39$		0			0
$40$		0			0
$41$	44.0908			1.07539			0.537693	−	0.843141i	$-0.319296\pi$
$41$	44.0908			1.07539			0.537693	+	0.843141i	$0.319296\pi$
$42$		0			0
$43$	29.3939	+	29.3939i	0.683579	+	0.683579i	0.960805	−	0.277226i	$-0.0894151\pi$
$43$	29.3939	+	29.3939i	0.683579	+	0.683579i	−0.277226	+	0.960805i	$0.589415\pi$
$44$		0			0
$45$		0			0
$46$	−24.0000			−0.521739
$47$	−18.0000	+	18.0000i	−0.382979	+	0.382979i	−0.872174	−	0.489195i	$-0.837290\pi$
$47$	−18.0000	+	18.0000i	−0.382979	+	0.382979i	0.489195	+	0.872174i	$0.337290\pi$
$48$		0			0
$49$			22.0000i			0.448980i
$50$		0			0
$51$		0			0
$52$	7.34847	−	7.34847i	0.141317	−	0.141317i
$53$	30.0000	+	30.0000i	0.566038	+	0.566038i	0.931016	−	0.364978i	$-0.118924\pi$
$53$	30.0000	+	30.0000i	0.566038	+	0.566038i	−0.364978	+	0.931016i	$0.618924\pi$
$54$		0			0
$55$		0			0
$56$	14.6969			0.262445
$57$		0			0
$58$	44.0908	+	44.0908i	0.760186	+	0.760186i
$59$			44.0908i			0.747302i	0.927569	+	0.373651i	$0.121894\pi$
$59$			44.0908i			0.747302i	−0.927569	+	0.373651i	$0.878106\pi$
$60$		0			0
$61$	23.0000			0.377049			0.188525	−	0.982068i	$-0.439629\pi$
$61$	23.0000			0.377049			0.188525	+	0.982068i	$0.439629\pi$
$62$	10.0000	−	10.0000i	0.161290	−	0.161290i
$63$		0			0
$64$			8.00000i			0.125000i
$65$		0			0
$66$		0			0
$67$	−33.0681	+	33.0681i	−0.493554	+	0.493554i	−0.909424	−	0.415870i	$-0.863477\pi$
$67$	−33.0681	+	33.0681i	−0.493554	+	0.493554i	0.415870	+	0.909424i	$0.363477\pi$
$68$	−12.0000	−	12.0000i	−0.176471	−	0.176471i
$69$		0			0
$70$		0			0
$71$	132.272			1.86299			0.931496	−	0.363751i	$-0.118504\pi$
$71$	132.272			1.86299			0.931496	+	0.363751i	$0.118504\pi$
$72$		0			0
$73$	33.0681	+	33.0681i	0.452988	+	0.452988i	0.896345	−	0.443357i	$-0.146213\pi$
$73$	33.0681	+	33.0681i	0.452988	+	0.452988i	−0.443357	+	0.896345i	$0.646213\pi$
$74$			51.4393i			0.695125i
$75$		0			0
$76$	14.0000			0.184211
$77$		0			0
$78$		0			0
$79$		−	103.000i		−	1.30380i	−0.758306	−	0.651899i	$-0.773973\pi$
$79$		−	103.000i		−	1.30380i	0.758306	−	0.651899i	$-0.226027\pi$
$80$		0			0
$81$		0			0
$82$	44.0908	−	44.0908i	0.537693	−	0.537693i
$83$	42.0000	+	42.0000i	0.506024	+	0.506024i	0.913304	−	0.407279i	$-0.133523\pi$
$83$	42.0000	+	42.0000i	0.506024	+	0.506024i	−0.407279	+	0.913304i	$0.633523\pi$
$84$		0			0
$85$		0			0
$86$	58.7878			0.683579
$87$		0			0
$88$		0			0
$89$			44.0908i			0.495402i	0.968836	+	0.247701i	$0.0796751\pi$
$89$			44.0908i			0.495402i	−0.968836	+	0.247701i	$0.920325\pi$
$90$		0			0
$91$	−27.0000			−0.296703
$92$	−24.0000	+	24.0000i	−0.260870	+	0.260870i
$93$		0			0
$94$			36.0000i			0.382979i
$95$		0			0
$96$		0			0
$97$	−91.8559	+	91.8559i	−0.946968	+	0.946968i	−0.998663	−	0.0516952i	$-0.983538\pi$
$97$	−91.8559	+	91.8559i	−0.946968	+	0.946968i	0.0516952	+	0.998663i	$0.483538\pi$
$98$	22.0000	+	22.0000i	0.224490	+	0.224490i
$99$		0			0
$100$		0			0
$101$	176.363			1.74617			0.873085	−	0.487567i	$-0.162116\pi$
$101$	176.363			1.74617			0.873085	+	0.487567i	$0.162116\pi$
$102$		0			0
$103$	−55.1135	−	55.1135i	−0.535083	−	0.535083i	0.386998	−	0.922081i	$-0.373512\pi$
$103$	−55.1135	−	55.1135i	−0.535083	−	0.535083i	−0.922081	+	0.386998i	$0.873512\pi$
$104$		−	14.6969i		−	0.141317i
$105$		0			0
$106$	60.0000			0.566038
$107$	36.0000	−	36.0000i	0.336449	−	0.336449i	−0.518580	−	0.855029i	$-0.673539\pi$
$107$	36.0000	−	36.0000i	0.336449	−	0.336449i	0.855029	+	0.518580i	$0.173539\pi$
$108$		0			0
$109$			146.000i			1.33945i	0.742609	+	0.669725i	$0.233588\pi$
$109$			146.000i			1.33945i	−0.742609	+	0.669725i	$0.766412\pi$
$110$		0			0
$111$		0			0
$112$	14.6969	−	14.6969i	0.131223	−	0.131223i
$113$	−144.000	−	144.000i	−1.27434	−	1.27434i	−0.943790	−	0.330547i	$-0.892767\pi$
$113$	−144.000	−	144.000i	−1.27434	−	1.27434i	−0.330547	−	0.943790i	$-0.607233\pi$
$114$		0			0
$115$		0			0
$116$	88.1816			0.760186
$117$		0			0
$118$	44.0908	+	44.0908i	0.373651	+	0.373651i
$119$			44.0908i			0.370511i
$120$		0			0
$121$	−121.000			−1.00000
$122$	23.0000	−	23.0000i	0.188525	−	0.188525i
$123$		0			0
$124$		−	20.0000i		−	0.161290i
$125$		0			0
$126$		0			0
$127$	−58.7878	+	58.7878i	−0.462896	+	0.462896i	−0.899603	−	0.436708i	$-0.856144\pi$
$127$	−58.7878	+	58.7878i	−0.462896	+	0.462896i	0.436708	+	0.899603i	$0.356144\pi$
$128$	8.00000	+	8.00000i	0.0625000	+	0.0625000i
$129$		0			0
$130$		0			0
$131$	176.363			1.34628			0.673142	−	0.739513i	$-0.264944\pi$
$131$	176.363			1.34628			0.673142	+	0.739513i	$0.264944\pi$
$132$		0			0
$133$	−25.7196	−	25.7196i	−0.193381	−	0.193381i
$134$			66.1362i			0.493554i
$135$		0			0
$136$	−24.0000			−0.176471
$137$	−48.0000	+	48.0000i	−0.350365	+	0.350365i	−0.860245	−	0.509880i	$-0.829690\pi$
$137$	−48.0000	+	48.0000i	−0.350365	+	0.350365i	0.509880	+	0.860245i	$0.329690\pi$
$138$		0			0
$139$		−	89.0000i		−	0.640288i	−0.947369	−	0.320144i	$-0.896269\pi$
$139$		−	89.0000i		−	0.640288i	0.947369	−	0.320144i	$-0.103731\pi$
$140$		0			0
$141$		0			0
$142$	132.272	−	132.272i	0.931496	−	0.931496i
$143$		0			0
$144$		0			0
$145$		0			0
$146$	66.1362			0.452988
$147$		0			0
$148$	51.4393	+	51.4393i	0.347563	+	0.347563i
$149$			88.1816i			0.591823i	0.955215	+	0.295912i	$0.0956234\pi$
$149$			88.1816i			0.591823i	−0.955215	+	0.295912i	$0.904377\pi$
$150$		0			0
$151$	103.000			0.682119			0.341060	−	0.940042i	$-0.389214\pi$
$151$	103.000			0.682119			0.341060	+	0.940042i	$0.389214\pi$
$152$	14.0000	−	14.0000i	0.0921053	−	0.0921053i
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−29.3939	+	29.3939i	−0.187222	+	0.187222i	−0.794494	−	0.607272i	$-0.792264\pi$
$157$	−29.3939	+	29.3939i	−0.187222	+	0.187222i	0.607272	+	0.794494i	$0.292264\pi$
$158$	−103.000	−	103.000i	−0.651899	−	0.651899i
$159$		0			0
$160$		0			0
$161$	88.1816			0.547712
$162$		0			0
$163$	−172.689	−	172.689i	−1.05944	−	1.05944i	−0.998118	−	0.0613240i	$-0.980468\pi$
$163$	−172.689	−	172.689i	−1.05944	−	1.05944i	−0.0613240	−	0.998118i	$-0.519532\pi$
$164$		−	88.1816i		−	0.537693i
$165$		0			0
$166$	84.0000			0.506024
$167$	102.000	−	102.000i	0.610778	−	0.610778i	−0.332370	−	0.943149i	$-0.607848\pi$
$167$	102.000	−	102.000i	0.610778	−	0.610778i	0.943149	+	0.332370i	$0.107848\pi$
$168$		0			0
$169$		−	142.000i		−	0.840237i
$170$		0			0
$171$		0			0
$172$	58.7878	−	58.7878i	0.341789	−	0.341789i
$173$	192.000	+	192.000i	1.10983	+	1.10983i	0.993173	+	0.116654i	$0.0372168\pi$
$173$	192.000	+	192.000i	1.10983	+	1.10983i	0.116654	+	0.993173i	$0.462783\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$	44.0908	+	44.0908i	0.247701	+	0.247701i
$179$			220.454i			1.23159i	0.787908	+	0.615794i	$0.211164\pi$
$179$			220.454i			1.23159i	−0.787908	+	0.615794i	$0.788836\pi$
$180$		0			0
$181$	−47.0000			−0.259669			−0.129834	−	0.991536i	$-0.541445\pi$
$181$	−47.0000			−0.259669			−0.129834	+	0.991536i	$0.541445\pi$
$182$	−27.0000	+	27.0000i	−0.148352	+	0.148352i
$183$		0			0
$184$			48.0000i			0.260870i
$185$		0			0
$186$		0			0
$187$		0			0
$188$	36.0000	+	36.0000i	0.191489	+	0.191489i
$189$		0			0
$190$		0			0
$191$	132.272			0.692526			0.346263	−	0.938138i	$-0.387451\pi$
$191$	132.272			0.692526			0.346263	+	0.938138i	$0.387451\pi$
$192$		0			0
$193$	−121.250	−	121.250i	−0.628237	−	0.628237i	0.319387	−	0.947624i	$-0.396523\pi$
$193$	−121.250	−	121.250i	−0.628237	−	0.628237i	−0.947624	+	0.319387i	$0.896523\pi$
$194$			183.712i			0.946968i
$195$		0			0
$196$	44.0000			0.224490
$197$	−42.0000	+	42.0000i	−0.213198	+	0.213198i	−0.805624	−	0.592427i	$-0.798170\pi$
$197$	−42.0000	+	42.0000i	−0.213198	+	0.213198i	0.592427	+	0.805624i	$0.298170\pi$
$198$		0			0
$199$			209.000i			1.05025i	0.851025	+	0.525126i	$0.175982\pi$
$199$			209.000i			1.05025i	−0.851025	+	0.525126i	$0.824018\pi$
$200$		0			0
$201$		0			0
$202$	176.363	−	176.363i	0.873085	−	0.873085i
$203$	−162.000	−	162.000i	−0.798030	−	0.798030i
$204$		0			0
$205$		0			0
$206$	−110.227			−0.535083
$207$		0			0
$208$	−14.6969	−	14.6969i	−0.0706584	−	0.0706584i
$209$		0			0
$210$		0			0
$211$	−55.0000			−0.260664			−0.130332	−	0.991470i	$-0.541604\pi$
$211$	−55.0000			−0.260664			−0.130332	+	0.991470i	$0.541604\pi$
$212$	60.0000	−	60.0000i	0.283019	−	0.283019i
$213$		0			0
$214$		−	72.0000i		−	0.336449i
$215$		0			0
$216$		0			0
$217$	−36.7423	+	36.7423i	−0.169320	+	0.169320i
$218$	146.000	+	146.000i	0.669725	+	0.669725i
$219$		0			0
$220$		0			0
$221$	44.0908			0.199506
$222$		0			0
$223$	−235.151	−	235.151i	−1.05449	−	1.05449i	−0.998427	−	0.0560615i	$-0.982146\pi$
$223$	−235.151	−	235.151i	−1.05449	−	1.05449i	−0.0560615	−	0.998427i	$-0.517854\pi$
$224$		−	29.3939i		−	0.131223i
$225$		0			0
$226$	−288.000			−1.27434
$227$	−222.000	+	222.000i	−0.977974	+	0.977974i	−0.999763	−	0.0217890i	$-0.993064\pi$
$227$	−222.000	+	222.000i	−0.977974	+	0.977974i	0.0217890	+	0.999763i	$0.493064\pi$
$228$		0			0
$229$			46.0000i			0.200873i	0.994943	+	0.100437i	$0.0320240\pi$
$229$			46.0000i			0.200873i	−0.994943	+	0.100437i	$0.967976\pi$
$230$		0			0
$231$		0			0
$232$	88.1816	−	88.1816i	0.380093	−	0.380093i
$233$	72.0000	+	72.0000i	0.309013	+	0.309013i	0.844527	−	0.535514i	$-0.179882\pi$
$233$	72.0000	+	72.0000i	0.309013	+	0.309013i	−0.535514	+	0.844527i	$0.679882\pi$
$234$		0			0
$235$		0			0
$236$	88.1816			0.373651
$237$		0			0
$238$	44.0908	+	44.0908i	0.185256	+	0.185256i
$239$			44.0908i			0.184480i	0.995737	+	0.0922402i	$0.0294028\pi$
$239$			44.0908i			0.184480i	−0.995737	+	0.0922402i	$0.970597\pi$
$240$		0			0
$241$	167.000			0.692946			0.346473	−	0.938060i	$-0.387379\pi$
$241$	167.000			0.692946			0.346473	+	0.938060i	$0.387379\pi$
$242$	−121.000	+	121.000i	−0.500000	+	0.500000i
$243$		0			0
$244$		−	46.0000i		−	0.188525i
$245$		0			0
$246$		0			0
$247$	−25.7196	+	25.7196i	−0.104128	+	0.104128i
$248$	−20.0000	−	20.0000i	−0.0806452	−	0.0806452i
$249$		0			0
$250$		0			0
$251$	−220.454			−0.878303			−0.439152	−	0.898413i	$-0.644721\pi$
$251$	−220.454			−0.878303			−0.439152	+	0.898413i	$0.644721\pi$
$252$		0			0
$253$		0			0
$254$			117.576i			0.462896i
$255$		0			0
$256$	16.0000			0.0625000
$257$	276.000	−	276.000i	1.07393	−	1.07393i	0.0768904	−	0.997040i	$-0.475501\pi$
$257$	276.000	−	276.000i	1.07393	−	1.07393i	0.997040	−	0.0768904i	$-0.0244992\pi$
$258$		0			0
$259$		−	189.000i		−	0.729730i
$260$		0			0
$261$		0			0
$262$	176.363	−	176.363i	0.673142	−	0.673142i
$263$	216.000	+	216.000i	0.821293	+	0.821293i	0.986293	−	0.165001i	$-0.0527626\pi$
$263$	216.000	+	216.000i	0.821293	+	0.821293i	−0.165001	+	0.986293i	$0.552763\pi$
$264$		0			0
$265$		0			0
$266$	−51.4393			−0.193381
$267$		0			0
$268$	66.1362	+	66.1362i	0.246777	+	0.246777i
$269$		−	44.0908i		−	0.163906i	−0.996636	−	0.0819532i	$-0.973884\pi$
$269$		−	44.0908i		−	0.163906i	0.996636	−	0.0819532i	$-0.0261158\pi$
$270$		0			0
$271$	281.000			1.03690			0.518450	−	0.855108i	$-0.326509\pi$
$271$	281.000			1.03690			0.518450	+	0.855108i	$0.326509\pi$
$272$	−24.0000	+	24.0000i	−0.0882353	+	0.0882353i
$273$		0			0
$274$			96.0000i			0.350365i
$275$		0			0
$276$		0			0
$277$	−205.757	+	205.757i	−0.742806	+	0.742806i	−0.973117	−	0.230311i	$-0.926026\pi$
$277$	−205.757	+	205.757i	−0.742806	+	0.742806i	0.230311	+	0.973117i	$0.426026\pi$
$278$	−89.0000	−	89.0000i	−0.320144	−	0.320144i
$279$		0			0
$280$		0			0
$281$	−264.545			−0.941441			−0.470720	−	0.882282i	$-0.656006\pi$
$281$	−264.545			−0.941441			−0.470720	+	0.882282i	$0.656006\pi$
$282$		0			0
$283$	−146.969	−	146.969i	−0.519326	−	0.519326i	0.398041	−	0.917368i	$-0.369690\pi$
$283$	−146.969	−	146.969i	−0.519326	−	0.519326i	−0.917368	+	0.398041i	$0.869690\pi$
$284$		−	264.545i		−	0.931496i
$285$		0			0
$286$		0			0
$287$	−162.000	+	162.000i	−0.564460	+	0.564460i
$288$		0			0
$289$			217.000i			0.750865i
$290$		0			0
$291$		0			0
$292$	66.1362	−	66.1362i	0.226494	−	0.226494i
$293$	−348.000	−	348.000i	−1.18771	−	1.18771i	−0.977698	−	0.210015i	$-0.932649\pi$
$293$	−348.000	−	348.000i	−1.18771	−	1.18771i	−0.210015	−	0.977698i	$-0.567351\pi$
$294$		0			0
$295$		0			0
$296$	102.879			0.347563
$297$		0			0
$298$	88.1816	+	88.1816i	0.295912	+	0.295912i
$299$		−	88.1816i		−	0.294922i
$300$		0			0
$301$	−216.000			−0.717608
$302$	103.000	−	103.000i	0.341060	−	0.341060i
$303$		0			0
$304$		−	28.0000i		−	0.0921053i
$305$		0			0
$306$		0			0
$307$	−88.1816	+	88.1816i	−0.287237	+	0.287237i	−0.835987	−	0.548750i	$-0.815104\pi$
$307$	−88.1816	+	88.1816i	−0.287237	+	0.287237i	0.548750	+	0.835987i	$0.315104\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−396.817			−1.27594			−0.637970	−	0.770061i	$-0.720226\pi$
$311$	−396.817			−1.27594			−0.637970	+	0.770061i	$0.720226\pi$
$312$		0			0
$313$	−62.4620	−	62.4620i	−0.199559	−	0.199559i	0.600252	−	0.799811i	$-0.295067\pi$
$313$	−62.4620	−	62.4620i	−0.199559	−	0.199559i	−0.799811	+	0.600252i	$0.795067\pi$
$314$			58.7878i			0.187222i
$315$		0			0
$316$	−206.000			−0.651899
$317$	−96.0000	+	96.0000i	−0.302839	+	0.302839i	−0.842124	−	0.539285i	$-0.818695\pi$
$317$	−96.0000	+	96.0000i	−0.302839	+	0.302839i	0.539285	+	0.842124i	$0.318695\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$	88.1816	−	88.1816i	0.273856	−	0.273856i
$323$	42.0000	+	42.0000i	0.130031	+	0.130031i
$324$		0			0
$325$		0			0
$326$	−345.378			−1.05944
$327$		0			0
$328$	−88.1816	−	88.1816i	−0.268846	−	0.268846i
$329$		−	132.272i		−	0.402044i
$330$		0			0
$331$	−175.000			−0.528701			−0.264350	−	0.964427i	$-0.585158\pi$
$331$	−175.000			−0.528701			−0.264350	+	0.964427i	$0.585158\pi$
$332$	84.0000	−	84.0000i	0.253012	−	0.253012i
$333$		0			0
$334$		−	204.000i		−	0.610778i
$335$		0			0
$336$		0			0
$337$	−327.007	+	327.007i	−0.970347	+	0.970347i	−0.999573	−	0.0292260i	$-0.990696\pi$
$337$	−327.007	+	327.007i	−0.970347	+	0.970347i	0.0292260	+	0.999573i	$0.490696\pi$
$338$	−142.000	−	142.000i	−0.420118	−	0.420118i
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−260.871	−	260.871i	−0.760556	−	0.760556i
$344$		−	117.576i		−	0.341789i
$345$		0			0
$346$	384.000			1.10983
$347$	414.000	−	414.000i	1.19308	−	1.19308i	0.216887	−	0.976197i	$-0.430410\pi$
$347$	414.000	−	414.000i	1.19308	−	1.19308i	0.976197	−	0.216887i	$-0.0695903\pi$
$348$		0			0
$349$		−	167.000i		−	0.478510i	−0.970957	−	0.239255i	$-0.923097\pi$
$349$		−	167.000i		−	0.478510i	0.970957	−	0.239255i	$-0.0769032\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	288.000	+	288.000i	0.815864	+	0.815864i	0.985506	−	0.169642i	$-0.0542611\pi$
$353$	288.000	+	288.000i	0.815864	+	0.815864i	−0.169642	+	0.985506i	$0.554261\pi$
$354$		0			0
$355$		0			0
$356$	88.1816			0.247701
$357$		0			0
$358$	220.454	+	220.454i	0.615794	+	0.615794i
$359$		−	617.271i		−	1.71942i	−0.510783	−	0.859709i	$-0.670645\pi$
$359$		−	617.271i		−	1.71942i	0.510783	−	0.859709i	$-0.329355\pi$
$360$		0			0
$361$	312.000			0.864266
$362$	−47.0000	+	47.0000i	−0.129834	+	0.129834i
$363$		0			0
$364$			54.0000i			0.148352i
$365$		0			0
$366$		0			0
$367$	209.431	−	209.431i	0.570658	−	0.570658i	−0.361654	−	0.932312i	$-0.617788\pi$
$367$	209.431	−	209.431i	0.570658	−	0.570658i	0.932312	+	0.361654i	$0.117788\pi$
$368$	48.0000	+	48.0000i	0.130435	+	0.130435i
$369$		0			0
$370$		0			0
$371$	−220.454			−0.594216
$372$		0			0
$373$	356.401	+	356.401i	0.955498	+	0.955498i	0.999051	−	0.0435531i	$-0.0138678\pi$
$373$	356.401	+	356.401i	0.955498	+	0.955498i	−0.0435531	+	0.999051i	$0.513868\pi$
$374$		0			0
$375$		0			0
$376$	72.0000			0.191489
$377$	−162.000	+	162.000i	−0.429708	+	0.429708i
$378$		0			0
$379$			151.000i			0.398417i	0.979957	+	0.199208i	$0.0638371\pi$
$379$			151.000i			0.398417i	−0.979957	+	0.199208i	$0.936163\pi$
$380$		0			0
$381$		0			0
$382$	132.272	−	132.272i	0.346263	−	0.346263i
$383$	−492.000	−	492.000i	−1.28460	−	1.28460i	−0.938023	−	0.346572i	$-0.887346\pi$
$383$	−492.000	−	492.000i	−1.28460	−	1.28460i	−0.346572	−	0.938023i	$-0.612654\pi$
$384$		0			0
$385$		0			0
$386$	−242.499			−0.628237
$387$		0			0
$388$	183.712	+	183.712i	0.473484	+	0.473484i
$389$		−	220.454i		−	0.566720i	−0.959014	−	0.283360i	$-0.908551\pi$
$389$		−	220.454i		−	0.566720i	0.959014	−	0.283360i	$-0.0914491\pi$
$390$		0			0
$391$	−144.000			−0.368286
$392$	44.0000	−	44.0000i	0.112245	−	0.112245i
$393$		0			0
$394$			84.0000i			0.213198i
$395$		0			0
$396$		0			0
$397$	−117.576	+	117.576i	−0.296160	+	0.296160i	−0.839508	−	0.543348i	$-0.817157\pi$
$397$	−117.576	+	117.576i	−0.296160	+	0.296160i	0.543348	+	0.839508i	$0.317157\pi$
$398$	209.000	+	209.000i	0.525126	+	0.525126i
$399$		0			0
$400$		0			0
$401$	−308.636			−0.769665			−0.384833	−	0.922986i	$-0.625741\pi$
$401$	−308.636			−0.769665			−0.384833	+	0.922986i	$0.625741\pi$
$402$		0			0
$403$	36.7423	+	36.7423i	0.0911721	+	0.0911721i
$404$		−	352.727i		−	0.873085i
$405$		0			0
$406$	−324.000			−0.798030
$407$		0			0
$408$		0			0
$409$		−	287.000i		−	0.701711i	−0.936430	−	0.350856i	$-0.885891\pi$
$409$		−	287.000i		−	0.701711i	0.936430	−	0.350856i	$-0.114109\pi$
$410$		0			0
$411$		0			0
$412$	−110.227	+	110.227i	−0.267541	+	0.267541i
$413$	−162.000	−	162.000i	−0.392252	−	0.392252i
$414$		0			0
$415$		0			0
$416$	−29.3939			−0.0706584
$417$		0			0
$418$		0			0
$419$		−	573.181i		−	1.36797i	−0.729495	−	0.683986i	$-0.760245\pi$
$419$		−	573.181i		−	1.36797i	0.729495	−	0.683986i	$-0.239755\pi$
$420$		0			0
$421$	−311.000			−0.738717			−0.369359	−	0.929287i	$-0.620423\pi$
$421$	−311.000			−0.738717			−0.369359	+	0.929287i	$0.620423\pi$
$422$	−55.0000	+	55.0000i	−0.130332	+	0.130332i
$423$		0			0
$424$		−	120.000i		−	0.283019i
$425$		0			0
$426$		0			0
$427$	−84.5074	+	84.5074i	−0.197910	+	0.197910i
$428$	−72.0000	−	72.0000i	−0.168224	−	0.168224i
$429$		0			0
$430$		0			0
$431$	−264.545			−0.613793			−0.306897	−	0.951743i	$-0.599291\pi$
$431$	−264.545			−0.613793			−0.306897	+	0.951743i	$0.599291\pi$
$432$		0			0
$433$	499.696	+	499.696i	1.15403	+	1.15403i	0.985737	+	0.168296i	$0.0538263\pi$
$433$	499.696	+	499.696i	1.15403	+	1.15403i	0.168296	+	0.985737i	$0.446174\pi$
$434$			73.4847i			0.169320i
$435$		0			0
$436$	292.000			0.669725
$437$	84.0000	−	84.0000i	0.192220	−	0.192220i
$438$		0			0
$439$			590.000i			1.34396i	0.740568	+	0.671982i	$0.234557\pi$
$439$			590.000i			1.34396i	−0.740568	+	0.671982i	$0.765443\pi$
$440$		0			0
$441$		0			0
$442$	44.0908	−	44.0908i	0.0997530	−	0.0997530i
$443$	480.000	+	480.000i	1.08352	+	1.08352i	0.996178	+	0.0873432i	$0.0278377\pi$
$443$	480.000	+	480.000i	1.08352	+	1.08352i	0.0873432	+	0.996178i	$0.472162\pi$
$444$		0			0
$445$		0			0
$446$	−470.302			−1.05449
$447$		0			0
$448$	−29.3939	−	29.3939i	−0.0656113	−	0.0656113i
$449$		−	220.454i		−	0.490989i	−0.969398	−	0.245495i	$-0.921050\pi$
$449$		−	220.454i		−	0.490989i	0.969398	−	0.245495i	$-0.0789503\pi$
$450$		0			0
$451$		0			0
$452$	−288.000	+	288.000i	−0.637168	+	0.637168i
$453$		0			0
$454$			444.000i			0.977974i
$455$		0			0
$456$		0			0
$457$	−382.120	+	382.120i	−0.836150	+	0.836150i	−0.988350	−	0.152200i	$-0.951364\pi$
$457$	−382.120	+	382.120i	−0.836150	+	0.836150i	0.152200	+	0.988350i	$0.451364\pi$
$458$	46.0000	+	46.0000i	0.100437	+	0.100437i
$459$		0			0
$460$		0			0
$461$	−749.544			−1.62591			−0.812954	−	0.582327i	$-0.802142\pi$
$461$	−749.544			−1.62591			−0.812954	+	0.582327i	$0.802142\pi$
$462$		0			0
$463$	25.7196	+	25.7196i	0.0555500	+	0.0555500i	0.734336	−	0.678786i	$-0.237494\pi$
$463$	25.7196	+	25.7196i	0.0555500	+	0.0555500i	−0.678786	+	0.734336i	$0.737494\pi$
$464$		−	176.363i		−	0.380093i
$465$		0			0
$466$	144.000			0.309013
$467$	−114.000	+	114.000i	−0.244111	+	0.244111i	−0.818549	−	0.574437i	$-0.805221\pi$
$467$	−114.000	+	114.000i	−0.244111	+	0.244111i	0.574437	+	0.818549i	$0.305221\pi$
$468$		0			0
$469$		−	243.000i		−	0.518124i
$470$		0			0
$471$		0			0
$472$	88.1816	−	88.1816i	0.186825	−	0.186825i
$473$		0			0
$474$		0			0
$475$		0			0
$476$	88.1816			0.185256
$477$		0			0
$478$	44.0908	+	44.0908i	0.0922402	+	0.0922402i
$479$			132.272i			0.276143i	0.990422	+	0.138071i	$0.0440904\pi$
$479$			132.272i			0.276143i	−0.990422	+	0.138071i	$0.955910\pi$
$480$		0			0
$481$	−189.000			−0.392931
$482$	167.000	−	167.000i	0.346473	−	0.346473i
$483$		0			0
$484$			242.000i			0.500000i
$485$		0			0
$486$		0			0
$487$	−62.4620	+	62.4620i	−0.128259	+	0.128259i	−0.768322	−	0.640063i	$-0.778908\pi$
$487$	−62.4620	+	62.4620i	−0.128259	+	0.128259i	0.640063	+	0.768322i	$0.278908\pi$
$488$	−46.0000	−	46.0000i	−0.0942623	−	0.0942623i
$489$		0			0
$490$		0			0
$491$	−88.1816			−0.179596			−0.0897980	−	0.995960i	$-0.528622\pi$
$491$	−88.1816			−0.179596			−0.0897980	+	0.995960i	$0.528622\pi$
$492$		0			0
$493$	264.545	+	264.545i	0.536602	+	0.536602i
$494$			51.4393i			0.104128i
$495$		0			0
$496$	−40.0000			−0.0806452
$497$	−486.000	+	486.000i	−0.977867	+	0.977867i
$498$		0			0
$499$		−	586.000i		−	1.17435i	−0.809460	−	0.587174i	$-0.800240\pi$
$499$		−	586.000i		−	1.17435i	0.809460	−	0.587174i	$-0.199760\pi$
$500$		0			0
$501$		0			0
$502$	−220.454	+	220.454i	−0.439152	+	0.439152i
$503$	312.000	+	312.000i	0.620278	+	0.620278i	0.945603	−	0.325324i	$-0.105473\pi$
$503$	312.000	+	312.000i	0.620278	+	0.620278i	−0.325324	+	0.945603i	$0.605473\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$	117.576	+	117.576i	0.231448	+	0.231448i
$509$		−	264.545i		−	0.519735i	−0.965644	−	0.259867i	$-0.916321\pi$
$509$		−	264.545i		−	0.519735i	0.965644	−	0.259867i	$-0.0836788\pi$
$510$		0			0
$511$	−243.000			−0.475538
$512$	16.0000	−	16.0000i	0.0312500	−	0.0312500i
$513$		0			0
$514$		−	552.000i		−	1.07393i
$515$		0			0
$516$		0			0
$517$		0			0
$518$	−189.000	−	189.000i	−0.364865	−	0.364865i
$519$		0			0
$520$		0			0
$521$	−617.271			−1.18478			−0.592391	−	0.805651i	$-0.701816\pi$
$521$	−617.271			−1.18478			−0.592391	+	0.805651i	$0.701816\pi$
$522$		0			0
$523$	−297.613	−	297.613i	−0.569050	−	0.569050i	0.362812	−	0.931862i	$-0.381817\pi$
$523$	−297.613	−	297.613i	−0.569050	−	0.569050i	−0.931862	+	0.362812i	$0.881817\pi$
$524$		−	352.727i		−	0.673142i
$525$		0			0
$526$	432.000			0.821293
$527$	60.0000	−	60.0000i	0.113852	−	0.113852i
$528$		0			0
$529$		−	241.000i		−	0.455577i
$530$		0			0
$531$		0			0
$532$	−51.4393	+	51.4393i	−0.0966904	+	0.0966904i
$533$	162.000	+	162.000i	0.303940	+	0.303940i
$534$		0			0
$535$		0			0
$536$	132.272			0.246777
$537$		0			0
$538$	−44.0908	−	44.0908i	−0.0819532	−	0.0819532i
$539$		0			0
$540$		0			0
$541$	1055.00			1.95009			0.975046	−	0.222002i	$-0.0712592\pi$
$541$	1055.00			1.95009			0.975046	+	0.222002i	$0.0712592\pi$
$542$	281.000	−	281.000i	0.518450	−	0.518450i
$543$		0			0
$544$			48.0000i			0.0882353i
$545$		0			0
$546$		0			0
$547$	385.795	−	385.795i	0.705292	−	0.705292i	−0.260250	−	0.965541i	$-0.583805\pi$
$547$	385.795	−	385.795i	0.705292	−	0.705292i	0.965541	+	0.260250i	$0.0838048\pi$
$548$	96.0000	+	96.0000i	0.175182	+	0.175182i
$549$		0			0
$550$		0			0
$551$	−308.636			−0.560137
$552$		0			0
$553$	378.446	+	378.446i	0.684351	+	0.684351i
$554$			411.514i			0.742806i
$555$		0			0
$556$	−178.000			−0.320144
$557$	342.000	−	342.000i	0.614004	−	0.614004i	−0.329983	−	0.943987i	$-0.607043\pi$
$557$	342.000	−	342.000i	0.614004	−	0.614004i	0.943987	+	0.329983i	$0.107043\pi$
$558$		0			0
$559$			216.000i			0.386404i
$560$		0			0
$561$		0			0
$562$	−264.545	+	264.545i	−0.470720	+	0.470720i
$563$	−432.000	−	432.000i	−0.767318	−	0.767318i	0.210316	−	0.977634i	$-0.432551\pi$
$563$	−432.000	−	432.000i	−0.767318	−	0.767318i	−0.977634	+	0.210316i	$0.932551\pi$
$564$		0			0
$565$		0			0
$566$	−293.939			−0.519326
$567$		0			0
$568$	−264.545	−	264.545i	−0.465748	−	0.465748i
$569$		−	220.454i		−	0.387441i	−0.981057	−	0.193721i	$-0.937944\pi$
$569$		−	220.454i		−	0.387441i	0.981057	−	0.193721i	$-0.0620555\pi$
$570$		0			0
$571$	−847.000			−1.48336			−0.741681	−	0.670752i	$-0.765971\pi$
$571$	−847.000			−1.48336			−0.741681	+	0.670752i	$0.765971\pi$
$572$		0			0
$573$		0			0
$574$			324.000i			0.564460i
$575$		0			0
$576$		0			0
$577$	−209.431	+	209.431i	−0.362966	+	0.362966i	−0.864904	−	0.501938i	$-0.832621\pi$
$577$	−209.431	+	209.431i	−0.362966	+	0.362966i	0.501938	+	0.864904i	$0.332621\pi$
$578$	217.000	+	217.000i	0.375433	+	0.375433i
$579$		0			0
$580$		0			0
$581$	−308.636			−0.531215
$582$		0			0
$583$		0			0
$584$		−	132.272i		−	0.226494i
$585$		0			0
$586$	−696.000			−1.18771
$587$	−84.0000	+	84.0000i	−0.143101	+	0.143101i	−0.775028	−	0.631927i	$-0.782264\pi$
$587$	−84.0000	+	84.0000i	−0.143101	+	0.143101i	0.631927	+	0.775028i	$0.282264\pi$
$588$		0			0
$589$			70.0000i			0.118846i
$590$		0			0
$591$		0			0
$592$	102.879	−	102.879i	0.173781	−	0.173781i
$593$	78.0000	+	78.0000i	0.131535	+	0.131535i	0.769809	−	0.638274i	$-0.220351\pi$
$593$	78.0000	+	78.0000i	0.131535	+	0.131535i	−0.638274	+	0.769809i	$0.720351\pi$
$594$		0			0
$595$		0			0
$596$	176.363			0.295912
$597$		0			0
$598$	−88.1816	−	88.1816i	−0.147461	−	0.147461i
$599$			573.181i			0.956896i	0.878116	+	0.478448i	$0.158800\pi$
$599$			573.181i			0.956896i	−0.878116	+	0.478448i	$0.841200\pi$
$600$		0			0
$601$	166.000			0.276206			0.138103	−	0.990418i	$-0.455899\pi$
$601$	166.000			0.276206			0.138103	+	0.990418i	$0.455899\pi$
$602$	−216.000	+	216.000i	−0.358804	+	0.358804i
$603$		0			0
$604$		−	206.000i		−	0.341060i
$605$		0			0
$606$		0			0
$607$	349.052	−	349.052i	0.575045	−	0.575045i	−0.358489	−	0.933534i	$-0.616708\pi$
$607$	349.052	−	349.052i	0.575045	−	0.575045i	0.933534	+	0.358489i	$0.116708\pi$
$608$	−28.0000	−	28.0000i	−0.0460526	−	0.0460526i
$609$		0			0
$610$		0			0
$611$	−132.272			−0.216485
$612$		0			0
$613$	407.840	+	407.840i	0.665318	+	0.665318i	0.956629	−	0.291310i	$-0.0940912\pi$
$613$	407.840	+	407.840i	0.665318	+	0.665318i	−0.291310	+	0.956629i	$0.594091\pi$
$614$			176.363i			0.287237i
$615$		0			0
$616$		0			0
$617$	276.000	−	276.000i	0.447326	−	0.447326i	−0.447139	−	0.894465i	$-0.647557\pi$
$617$	276.000	−	276.000i	0.447326	−	0.447326i	0.894465	+	0.447139i	$0.147557\pi$
$618$		0			0
$619$			833.000i			1.34572i	0.739770	+	0.672859i	$0.234934\pi$
$619$			833.000i			1.34572i	−0.739770	+	0.672859i	$0.765066\pi$
$620$		0			0
$621$		0			0
$622$	−396.817	+	396.817i	−0.637970	+	0.637970i
$623$	−162.000	−	162.000i	−0.260032	−	0.260032i
$624$		0			0
$625$		0			0
$626$	−124.924			−0.199559
$627$		0			0
$628$	58.7878	+	58.7878i	0.0936111	+	0.0936111i
$629$			308.636i			0.490677i
$630$		0			0
$631$	1001.00			1.58637			0.793185	−	0.608980i	$-0.208421\pi$
$631$	1001.00			1.58637			0.793185	+	0.608980i	$0.208421\pi$
$632$	−206.000	+	206.000i	−0.325949	+	0.325949i
$633$		0			0
$634$			192.000i			0.302839i
$635$		0			0
$636$		0			0
$637$	−80.8332	+	80.8332i	−0.126897	+	0.126897i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	881.816			1.37569			0.687844	−	0.725858i	$-0.258557\pi$
$641$	881.816			1.37569			0.687844	+	0.725858i	$0.258557\pi$
$642$		0			0
$643$	−705.453	−	705.453i	−1.09713	−	1.09713i	−0.994745	−	0.102382i	$-0.967353\pi$
$643$	−705.453	−	705.453i	−1.09713	−	1.09713i	−0.102382	−	0.994745i	$-0.532647\pi$
$644$		−	176.363i		−	0.273856i
$645$		0			0
$646$	84.0000			0.130031
$647$	210.000	−	210.000i	0.324575	−	0.324575i	−0.525944	−	0.850519i	$-0.676288\pi$
$647$	210.000	−	210.000i	0.324575	−	0.324575i	0.850519	+	0.525944i	$0.176288\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$	−345.378	+	345.378i	−0.529721	+	0.529721i
$653$	−348.000	−	348.000i	−0.532925	−	0.532925i	0.388517	−	0.921442i	$-0.372988\pi$
$653$	−348.000	−	348.000i	−0.532925	−	0.532925i	−0.921442	+	0.388517i	$0.872988\pi$
$654$		0			0
$655$		0			0
$656$	−176.363			−0.268846
$657$		0			0
$658$	−132.272	−	132.272i	−0.201022	−	0.201022i
$659$		−	440.908i		−	0.669056i	−0.942386	−	0.334528i	$-0.891423\pi$
$659$		−	440.908i		−	0.669056i	0.942386	−	0.334528i	$-0.108577\pi$
$660$		0			0
$661$	−431.000			−0.652042			−0.326021	−	0.945362i	$-0.605708\pi$
$661$	−431.000			−0.652042			−0.326021	+	0.945362i	$0.605708\pi$
$662$	−175.000	+	175.000i	−0.264350	+	0.264350i
$663$		0			0
$664$		−	168.000i		−	0.253012i
$665$		0			0
$666$		0			0
$667$	529.090	−	529.090i	0.793238	−	0.793238i
$668$	−204.000	−	204.000i	−0.305389	−	0.305389i
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	180.037	+	180.037i	0.267515	+	0.267515i	0.828098	−	0.560583i	$-0.189423\pi$
$673$	180.037	+	180.037i	0.267515	+	0.267515i	−0.560583	+	0.828098i	$0.689423\pi$
$674$			654.014i			0.970347i
$675$		0			0
$676$	−284.000			−0.420118
$677$	234.000	−	234.000i	0.345643	−	0.345643i	−0.512841	−	0.858484i	$-0.671407\pi$
$677$	234.000	−	234.000i	0.345643	−	0.345643i	0.858484	+	0.512841i	$0.171407\pi$
$678$		0			0
$679$		−	675.000i		−	0.994109i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−222.000	−	222.000i	−0.325037	−	0.325037i	0.525659	−	0.850695i	$-0.323819\pi$
$683$	−222.000	−	222.000i	−0.325037	−	0.325037i	−0.850695	+	0.525659i	$0.823819\pi$
$684$		0			0
$685$		0			0
$686$	−521.741			−0.760556
$687$		0			0
$688$	−117.576	−	117.576i	−0.170895	−	0.170895i
$689$			220.454i			0.319962i
$690$		0			0
$691$	−878.000			−1.27062			−0.635311	−	0.772256i	$-0.719128\pi$
$691$	−878.000			−1.27062			−0.635311	+	0.772256i	$0.719128\pi$
$692$	384.000	−	384.000i	0.554913	−	0.554913i
$693$		0			0
$694$		−	828.000i		−	1.19308i
$695$		0			0
$696$		0			0
$697$	264.545	−	264.545i	0.379548	−	0.379548i
$698$	−167.000	−	167.000i	−0.239255	−	0.239255i
$699$		0			0
$700$		0			0
$701$	88.1816			0.125794			0.0628970	−	0.998020i	$-0.479966\pi$
$701$	88.1816			0.125794			0.0628970	+	0.998020i	$0.479966\pi$
$702$		0			0
$703$	−180.037	−	180.037i	−0.256099	−	0.256099i
$704$		0			0
$705$		0			0
$706$	576.000			0.815864
$707$	−648.000	+	648.000i	−0.916549	+	0.916549i
$708$		0			0
$709$			553.000i			0.779972i	0.920821	+	0.389986i	$0.127520\pi$
$709$			553.000i			0.779972i	−0.920821	+	0.389986i	$0.872480\pi$
$710$		0			0
$711$		0			0
$712$	88.1816	−	88.1816i	0.123851	−	0.123851i
$713$	−120.000	−	120.000i	−0.168303	−	0.168303i
$714$		0			0
$715$		0			0
$716$	440.908			0.615794
$717$		0			0
$718$	−617.271	−	617.271i	−0.859709	−	0.859709i
$719$			661.362i			0.919836i	0.887961	+	0.459918i	$0.152121\pi$
$719$			661.362i			0.919836i	−0.887961	+	0.459918i	$0.847879\pi$
$720$		0			0
$721$	405.000			0.561720
$722$	312.000	−	312.000i	0.432133	−	0.432133i
$723$		0			0
$724$			94.0000i			0.129834i
$725$		0			0
$726$		0			0
$727$	440.908	−	440.908i	0.606476	−	0.606476i	−0.335547	−	0.942023i	$-0.608921\pi$
$727$	440.908	−	440.908i	0.606476	−	0.606476i	0.942023	+	0.335547i	$0.108921\pi$
$728$	54.0000	+	54.0000i	0.0741758	+	0.0741758i
$729$		0			0
$730$		0			0
$731$	352.727			0.482526
$732$		0			0
$733$	558.484	+	558.484i	0.761915	+	0.761915i	0.976668	−	0.214753i	$-0.0688948\pi$
$733$	558.484	+	558.484i	0.761915	+	0.761915i	−0.214753	+	0.976668i	$0.568895\pi$
$734$		−	418.863i		−	0.570658i
$735$		0			0
$736$	96.0000			0.130435
$737$		0			0
$738$		0			0
$739$		−	110.000i		−	0.148850i	−0.997227	−	0.0744249i	$-0.976288\pi$
$739$		−	110.000i		−	0.148850i	0.997227	−	0.0744249i	$-0.0237121\pi$
$740$		0			0
$741$		0			0
$742$	−220.454	+	220.454i	−0.297108	+	0.297108i
$743$	378.000	+	378.000i	0.508748	+	0.508748i	0.914142	−	0.405394i	$-0.132866\pi$
$743$	378.000	+	378.000i	0.508748	+	0.508748i	−0.405394	+	0.914142i	$0.632866\pi$
$744$		0			0
$745$		0			0
$746$	712.802			0.955498
$747$		0			0
$748$		0			0
$749$			264.545i			0.353197i
$750$		0			0
$751$	703.000			0.936085			0.468043	−	0.883706i	$-0.344959\pi$
$751$	703.000			0.936085			0.468043	+	0.883706i	$0.344959\pi$
$752$	72.0000	−	72.0000i	0.0957447	−	0.0957447i
$753$		0			0
$754$			324.000i			0.429708i
$755$		0			0
$756$		0			0
$757$	1003.07	−	1003.07i	1.32505	−	1.32505i	0.415428	−	0.909626i	$-0.363632\pi$
$757$	1003.07	−	1003.07i	1.32505	−	1.32505i	0.909626	−	0.415428i	$-0.136368\pi$
$758$	151.000	+	151.000i	0.199208	+	0.199208i
$759$		0			0
$760$		0			0
$761$	617.271			0.811132			0.405566	−	0.914066i	$-0.367074\pi$
$761$	617.271			0.811132			0.405566	+	0.914066i	$0.367074\pi$
$762$		0			0
$763$	−536.438	−	536.438i	−0.703065	−	0.703065i
$764$		−	264.545i		−	0.346263i
$765$		0			0
$766$	−984.000			−1.28460
$767$	−162.000	+	162.000i	−0.211213	+	0.211213i
$768$		0			0
$769$			359.000i			0.466840i	0.972376	+	0.233420i	$0.0749917\pi$
$769$			359.000i			0.466840i	−0.972376	+	0.233420i	$0.925008\pi$
$770$		0			0
$771$		0			0
$772$	−242.499	+	242.499i	−0.314119	+	0.314119i
$773$	546.000	+	546.000i	0.706339	+	0.706339i	0.965763	−	0.259424i	$-0.0835329\pi$
$773$	546.000	+	546.000i	0.706339	+	0.706339i	−0.259424	+	0.965763i	$0.583533\pi$
$774$		0			0
$775$		0			0
$776$	367.423			0.473484
$777$		0			0
$778$	−220.454	−	220.454i	−0.283360	−	0.283360i
$779$			308.636i			0.396195i
$780$		0			0
$781$		0			0
$782$	−144.000	+	144.000i	−0.184143	+	0.184143i
$783$		0			0
$784$		−	88.0000i		−	0.112245i
$785$		0			0
$786$		0			0
$787$	672.385	−	672.385i	0.854365	−	0.854365i	−0.136303	−	0.990667i	$-0.543522\pi$
$787$	672.385	−	672.385i	0.854365	−	0.854365i	0.990667	+	0.136303i	$0.0435220\pi$
$788$	84.0000	+	84.0000i	0.106599	+	0.106599i
$789$		0			0
$790$		0			0
$791$	1058.18			1.33777
$792$		0			0
$793$	84.5074	+	84.5074i	0.106567	+	0.106567i
$794$			235.151i			0.296160i
$795$		0			0
$796$	418.000			0.525126
$797$	−204.000	+	204.000i	−0.255960	+	0.255960i	−0.823409	−	0.567449i	$-0.807930\pi$
$797$	−204.000	+	204.000i	−0.255960	+	0.255960i	0.567449	+	0.823409i	$0.307930\pi$
$798$		0			0
$799$			216.000i			0.270338i
$800$		0			0
$801$		0			0
$802$	−308.636	+	308.636i	−0.384833	+	0.384833i
$803$		0			0
$804$		0			0
$805$		0			0
$806$	73.4847			0.0911721
$807$		0			0
$808$	−352.727	−	352.727i	−0.436543	−	0.436543i
$809$			88.1816i			0.109001i	0.998514	+	0.0545004i	$0.0173566\pi$
$809$			88.1816i			0.109001i	−0.998514	+	0.0545004i	$0.982643\pi$
$810$		0			0
$811$	754.000			0.929716			0.464858	−	0.885385i	$-0.346105\pi$
$811$	754.000			0.929716			0.464858	+	0.885385i	$0.346105\pi$
$812$	−324.000	+	324.000i	−0.399015	+	0.399015i
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−205.757	+	205.757i	−0.251845	+	0.251845i
$818$	−287.000	−	287.000i	−0.350856	−	0.350856i
$819$		0			0
$820$		0			0
$821$	1014.09			1.23519			0.617594	−	0.786497i	$-0.288108\pi$
$821$	1014.09			1.23519			0.617594	+	0.786497i	$0.288108\pi$
$822$		0			0
$823$	−819.354	−	819.354i	−0.995570	−	0.995570i	0.00441998	−	0.999990i	$-0.498593\pi$
$823$	−819.354	−	819.354i	−0.995570	−	0.995570i	−0.999990	+	0.00441998i	$0.998593\pi$
$824$			220.454i			0.267541i
$825$		0			0
$826$	−324.000			−0.392252
$827$	402.000	−	402.000i	0.486094	−	0.486094i	−0.420977	−	0.907071i	$-0.638313\pi$
$827$	402.000	−	402.000i	0.486094	−	0.486094i	0.907071	+	0.420977i	$0.138313\pi$
$828$		0			0
$829$		−	335.000i		−	0.404101i	−0.979375	−	0.202051i	$-0.935239\pi$
$829$		−	335.000i		−	0.404101i	0.979375	−	0.202051i	$-0.0647606\pi$
$830$		0			0
$831$		0			0
$832$	−29.3939	+	29.3939i	−0.0353292	+	0.0353292i
$833$	132.000	+	132.000i	0.158463	+	0.158463i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$	−573.181	−	573.181i	−0.683986	−	0.683986i
$839$		−	176.363i		−	0.210207i	−0.994461	−	0.105103i	$-0.966483\pi$
$839$		−	176.363i		−	0.210207i	0.994461	−	0.105103i	$-0.0335173\pi$
$840$		0			0
$841$	−1103.00			−1.31153
$842$	−311.000	+	311.000i	−0.369359	+	0.369359i
$843$		0			0
$844$			110.000i			0.130332i
$845$		0			0
$846$		0			0
$847$	444.582	−	444.582i	0.524891	−	0.524891i
$848$	−120.000	−	120.000i	−0.141509	−	0.141509i
$849$		0			0
$850$		0			0
$851$	617.271			0.725348
$852$		0			0
$853$	1003.07	+	1003.07i	1.17593	+	1.17593i	0.980772	+	0.195155i	$0.0625210\pi$
$853$	1003.07	+	1003.07i	1.17593	+	1.17593i	0.195155	+	0.980772i	$0.437479\pi$
$854$			169.015i			0.197910i
$855$		0			0
$856$	−144.000			−0.168224
$857$	114.000	−	114.000i	0.133022	−	0.133022i	−0.637461	−	0.770483i	$-0.720015\pi$
$857$	114.000	−	114.000i	0.133022	−	0.133022i	0.770483	+	0.637461i	$0.220015\pi$
$858$		0			0
$859$		−	919.000i		−	1.06985i	−0.844900	−	0.534924i	$-0.820340\pi$
$859$		−	919.000i		−	1.06985i	0.844900	−	0.534924i	$-0.179660\pi$
$860$		0			0
$861$		0			0
$862$	−264.545	+	264.545i	−0.306897	+	0.306897i
$863$	102.000	+	102.000i	0.118192	+	0.118192i	0.763729	−	0.645537i	$-0.223366\pi$
$863$	102.000	+	102.000i	0.118192	+	0.118192i	−0.645537	+	0.763729i	$0.723366\pi$
$864$		0			0
$865$		0			0
$866$	999.392			1.15403
$867$		0			0
$868$	73.4847	+	73.4847i	0.0846598	+	0.0846598i
$869$		0			0
$870$		0			0
$871$	−243.000			−0.278990
$872$	292.000	−	292.000i	0.334862	−	0.334862i
$873$		0			0
$874$		−	168.000i		−	0.192220i
$875$		0			0
$876$		0			0
$877$	−113.901	+	113.901i	−0.129876	+	0.129876i	−0.769057	−	0.639181i	$-0.779274\pi$
$877$	−113.901	+	113.901i	−0.129876	+	0.129876i	0.639181	+	0.769057i	$0.279274\pi$
$878$	590.000	+	590.000i	0.671982	+	0.671982i
$879$		0			0
$880$		0			0
$881$	1058.18			1.20111			0.600556	−	0.799583i	$-0.294946\pi$
$881$	1058.18			1.20111			0.600556	+	0.799583i	$0.294946\pi$
$882$		0			0
$883$	−62.4620	−	62.4620i	−0.0707384	−	0.0707384i	0.670852	−	0.741591i	$-0.265928\pi$
$883$	−62.4620	−	62.4620i	−0.0707384	−	0.0707384i	−0.741591	+	0.670852i	$0.765928\pi$
$884$		−	88.1816i		−	0.0997530i
$885$		0			0
$886$	960.000			1.08352
$887$	96.0000	−	96.0000i	0.108230	−	0.108230i	−0.650918	−	0.759148i	$-0.725616\pi$
$887$	96.0000	−	96.0000i	0.108230	−	0.108230i	0.759148	+	0.650918i	$0.225616\pi$
$888$		0			0
$889$		−	432.000i		−	0.485939i
$890$		0			0
$891$		0			0
$892$	−470.302	+	470.302i	−0.527244	+	0.527244i
$893$	−126.000	−	126.000i	−0.141097	−	0.141097i
$894$		0			0
$895$		0			0
$896$	−58.7878			−0.0656113
$897$		0			0
$898$	−220.454	−	220.454i	−0.245495	−	0.245495i
$899$			440.908i			0.490443i
$900$		0			0
$901$	360.000			0.399556
$902$		0			0
$903$		0			0
$904$			576.000i			0.637168i
$905$		0			0
$906$		0			0
$907$	907.536	−	907.536i	1.00059	−	1.00059i	0.000591078	−	1.00000i	$-0.499812\pi$
$907$	907.536	−	907.536i	1.00059	−	1.00059i	1.00000		0.000591078i	$-0.000188146\pi$
$908$	444.000	+	444.000i	0.488987	+	0.488987i
$909$		0			0
$910$		0			0
$911$	−88.1816			−0.0967965			−0.0483983	−	0.998828i	$-0.515412\pi$
$911$	−88.1816			−0.0967965			−0.0483983	+	0.998828i	$0.515412\pi$
$912$		0			0
$913$		0			0
$914$			764.241i			0.836150i
$915$		0			0
$916$	92.0000			0.100437
$917$	−648.000	+	648.000i	−0.706652	+	0.706652i
$918$		0			0
$919$			754.000i			0.820457i	0.911983	+	0.410229i	$0.134551\pi$
$919$			754.000i			0.820457i	−0.911983	+	0.410229i	$0.865449\pi$
$920$		0			0
$921$		0			0
$922$	−749.544	+	749.544i	−0.812954	+	0.812954i
$923$	486.000	+	486.000i	0.526544	+	0.526544i
$924$		0			0
$925$		0			0
$926$	51.4393			0.0555500
$927$		0			0
$928$	−176.363	−	176.363i	−0.190047	−	0.190047i
$929$			1278.63i			1.37635i	0.725542	+	0.688177i	$0.241589\pi$
$929$			1278.63i			1.37635i	−0.725542	+	0.688177i	$0.758411\pi$
$930$		0			0
$931$	−154.000			−0.165414
$932$	144.000	−	144.000i	0.154506	−	0.154506i
$933$		0			0
$934$			228.000i			0.244111i
$935$		0			0
$936$		0			0
$937$	−1083.90	+	1083.90i	−1.15678	+	1.15678i	−0.171611	+	0.985165i	$0.554897\pi$
$937$	−1083.90	+	1083.90i	−1.15678	+	1.15678i	−0.985165	+	0.171611i	$0.945103\pi$
$938$	−243.000	−	243.000i	−0.259062	−	0.259062i
$939$		0			0
$940$		0			0
$941$	969.998			1.03082			0.515408	−	0.856945i	$-0.327640\pi$
$941$	969.998			1.03082			0.515408	+	0.856945i	$0.327640\pi$
$942$		0			0
$943$	−529.090	−	529.090i	−0.561071	−	0.561071i
$944$		−	176.363i		−	0.186825i
$945$		0			0
$946$		0			0
$947$	798.000	−	798.000i	0.842661	−	0.842661i	−0.146543	−	0.989204i	$-0.546815\pi$
$947$	798.000	−	798.000i	0.842661	−	0.842661i	0.989204	+	0.146543i	$0.0468147\pi$
$948$		0			0
$949$			243.000i			0.256059i
$950$		0			0
$951$		0			0
$952$	88.1816	−	88.1816i	0.0926278	−	0.0926278i
$953$	−738.000	−	738.000i	−0.774397	−	0.774397i	0.204475	−	0.978872i	$-0.434451\pi$
$953$	−738.000	−	738.000i	−0.774397	−	0.774397i	−0.978872	+	0.204475i	$0.934451\pi$
$954$		0			0
$955$		0			0
$956$	88.1816			0.0922402
$957$		0			0
$958$	132.272	+	132.272i	0.138071	+	0.138071i
$959$		−	352.727i		−	0.367807i
$960$		0			0
$961$	−861.000			−0.895942
$962$	−189.000	+	189.000i	−0.196466	+	0.196466i
$963$		0			0
$964$		−	334.000i		−	0.346473i
$965$		0			0
$966$		0			0
$967$	1201.47	−	1201.47i	1.24248	−	1.24248i	0.283506	−	0.958970i	$-0.408502\pi$
$967$	1201.47	−	1201.47i	1.24248	−	1.24248i	0.958970	−	0.283506i	$-0.0914976\pi$
$968$	242.000	+	242.000i	0.250000	+	0.250000i
$969$		0			0
$970$		0			0
$971$	749.544			0.771930			0.385965	−	0.922513i	$-0.373869\pi$
$971$	749.544			0.771930			0.385965	+	0.922513i	$0.373869\pi$
$972$		0			0
$973$	327.007	+	327.007i	0.336081	+	0.336081i
$974$			124.924i			0.128259i
$975$		0			0
$976$	−92.0000			−0.0942623
$977$	−594.000	+	594.000i	−0.607984	+	0.607984i	−0.942419	−	0.334435i	$-0.891454\pi$
$977$	−594.000	+	594.000i	−0.607984	+	0.607984i	0.334435	+	0.942419i	$0.391454\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		0			0
$982$	−88.1816	+	88.1816i	−0.0897980	+	0.0897980i
$983$	−768.000	−	768.000i	−0.781282	−	0.781282i	0.198765	−	0.980047i	$-0.436307\pi$
$983$	−768.000	−	768.000i	−0.781282	−	0.781282i	−0.980047	+	0.198765i	$0.936307\pi$
$984$		0			0
$985$		0			0
$986$	529.090			0.536602
$987$		0			0
$988$	51.4393	+	51.4393i	0.0520641	+	0.0520641i
$989$		−	705.453i		−	0.713299i
$990$		0			0
$991$	−7.00000			−0.00706357			−0.00353179	−	0.999994i	$-0.501124\pi$
$991$	−7.00000			−0.00706357			−0.00353179	+	0.999994i	$0.501124\pi$
$992$	−40.0000	+	40.0000i	−0.0403226	+	0.0403226i
$993$		0			0
$994$			972.000i			0.977867i
$995$		0			0
$996$		0			0
$997$	440.908	−	440.908i	0.442235	−	0.442235i	−0.450528	−	0.892762i	$-0.648764\pi$
$997$	440.908	−	440.908i	0.442235	−	0.442235i	0.892762	+	0.450528i	$0.148764\pi$
$998$	−586.000	−	586.000i	−0.587174	−	0.587174i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1350.3.g.j.1243.1	yes	4
3.2	odd	2		1350.3.g.c.1243.1	yes	4
5.2	odd	4	inner	1350.3.g.j.757.1	yes	4
5.3	odd	4		1350.3.g.c.757.2	yes	4
5.4	even	2		1350.3.g.c.1243.2	yes	4
15.2	even	4		1350.3.g.c.757.1	✓	4
15.8	even	4	inner	1350.3.g.j.757.2	yes	4
15.14	odd	2	inner	1350.3.g.j.1243.2	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1350.3.g.c.757.1	✓	4	15.2	even	4
1350.3.g.c.757.2	yes	4	5.3	odd	4
1350.3.g.c.1243.1	yes	4	3.2	odd	2
1350.3.g.c.1243.2	yes	4	5.4	even	2
1350.3.g.j.757.1	yes	4	5.2	odd	4	inner
1350.3.g.j.757.2	yes	4	15.8	even	4	inner
1350.3.g.j.1243.1	yes	4	1.1	even	1	trivial
1350.3.g.j.1243.2	yes	4	15.14	odd	2	inner

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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:	\( N \)	\(=\)	\( 1350 = 2 \cdot 3^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1350.g (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(1.00000 - 1.00000i) q^{2} -2.00000i q^{4} +(-3.67423 + 3.67423i) q^{7} +(-2.00000 - 2.00000i) q^{8} +(3.67423 + 3.67423i) q^{13} +7.34847i q^{14} -4.00000 q^{16} +(6.00000 - 6.00000i) q^{17} +7.00000i q^{19} +(-12.0000 - 12.0000i) q^{23} +7.34847 q^{26} +(7.34847 + 7.34847i) q^{28} +44.0908i q^{29} +10.0000 q^{31} +(-4.00000 + 4.00000i) q^{32} -12.0000i q^{34} +(-25.7196 + 25.7196i) q^{37} +(7.00000 + 7.00000i) q^{38} +44.0908 q^{41} +(29.3939 + 29.3939i) q^{43} -24.0000 q^{46} +(-18.0000 + 18.0000i) q^{47} +22.0000i q^{49} +(7.34847 - 7.34847i) q^{52} +(30.0000 + 30.0000i) q^{53} +14.6969 q^{56} +(44.0908 + 44.0908i) q^{58} +44.0908i q^{59} +23.0000 q^{61} +(10.0000 - 10.0000i) q^{62} +8.00000i q^{64} +(-33.0681 + 33.0681i) q^{67} +(-12.0000 - 12.0000i) q^{68} +132.272 q^{71} +(33.0681 + 33.0681i) q^{73} +51.4393i q^{74} +14.0000 q^{76} -103.000i q^{79} +(44.0908 - 44.0908i) q^{82} +(42.0000 + 42.0000i) q^{83} +58.7878 q^{86} +44.0908i q^{89} -27.0000 q^{91} +(-24.0000 + 24.0000i) q^{92} +36.0000i q^{94} +(-91.8559 + 91.8559i) q^{97} +(22.0000 + 22.0000i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{2} - 8 q^{8} - 16 q^{16} + 24 q^{17} - 48 q^{23} + 40 q^{31} - 16 q^{32} + 28 q^{38} - 96 q^{46} - 72 q^{47} + 120 q^{53} + 92 q^{61} + 40 q^{62} - 48 q^{68} + 56 q^{76} + 168 q^{83} - 108 q^{91}+ \cdots + 88 q^{98}+O(q^{100}) \) 4 * q + 4 * q^2 - 8 * q^8 - 16 * q^16 + 24 * q^17 - 48 * q^23 + 40 * q^31 - 16 * q^32 + 28 * q^38 - 96 * q^46 - 72 * q^47 + 120 * q^53 + 92 * q^61 + 40 * q^62 - 48 * q^68 + 56 * q^76 + 168 * q^83 - 108 * q^91 - 96 * q^92 + 88 * q^98

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