LMFDB - Embedded newform 136.2.s.a.75.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [136,2,Mod(3,136)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(136, base_ring=CyclotomicField(16)) chi = DirichletCharacter(H, H._module([8, 8, 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("136.3"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 136 = 2^{3} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	136.s (of order $16$, degree $8$, minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	no
Analytic conductor:	$1.08596546749$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\Q(\zeta_{16})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 1 $ x^8 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{16}]$

Embedding invariants

Embedding label			75.1
Root			$-0.382683 - 0.923880i$ of defining polynomial
Character	$\chi$	$=$	136.75
Dual form			136.2.s.a.107.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-0.541196 + 1.30656i) q^{2} +(1.39635 - 2.08979i) q^{3} +(-1.41421 - 1.41421i) q^{4} +(1.97474 + 2.95541i) q^{6} +(2.61313 - 1.08239i) q^{8} +(-1.26937 - 3.06453i) q^{9} +(4.72705 - 3.15851i) q^{11} +(-4.93015 + 0.980668i) q^{12} +4.00000i q^{16} +(-3.76118 + 1.68925i) q^{17} +4.69098 q^{18} +(-2.83730 + 6.84984i) q^{19} +(1.56854 + 7.88556i) q^{22} +(1.38687 - 6.97229i) q^{24} +(-4.61940 + 1.91342i) q^{25} +(-0.781491 - 0.155448i) q^{27} +(-5.22625 - 2.16478i) q^{32} -14.2889i q^{33} +(-0.171573 - 5.82843i) q^{34} +(-2.53874 + 6.12906i) q^{36} +(-7.41421 - 7.41421i) q^{38} +(0.252982 - 1.27183i) q^{41} +(4.46447 + 10.7782i) q^{43} +(-11.1519 - 2.21824i) q^{44} +(8.35916 + 5.58541i) q^{48} +(-6.46716 - 2.67878i) q^{49} -7.07107i q^{50} +(-1.72176 + 10.2189i) q^{51} +(0.626043 - 0.936940i) q^{54} +(10.3529 + 15.4942i) q^{57} +(9.94975 - 4.12132i) q^{59} +(5.65685 - 5.65685i) q^{64} +(18.6694 + 7.73312i) q^{66} -16.1815i q^{67} +(7.70806 + 2.93015i) q^{68} +(-6.63405 - 6.63405i) q^{72} +(2.96960 + 14.9292i) q^{73} +(-2.45167 + 12.3254i) q^{75} +(13.6997 - 5.67459i) q^{76} +(5.62038 - 5.62038i) q^{81} +(1.52481 + 1.01884i) q^{82} +(-14.3640 - 5.94975i) q^{83} -16.4985 q^{86} +(8.93362 - 13.3701i) q^{88} +(8.56628 + 8.56628i) q^{89} +(-11.8216 + 7.89897i) q^{96} +(-14.4112 + 2.86657i) q^{97} +(7.00000 - 7.00000i) q^{98} +(-15.6797 - 10.4769i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{3} + 8 q^{6} + 16 q^{9} - 16 q^{12} - 24 q^{22} + 32 q^{24} - 32 q^{27} - 24 q^{34} + 32 q^{36} - 48 q^{38} - 32 q^{41} + 64 q^{43} - 16 q^{44} + 32 q^{48} - 40 q^{51} + 40 q^{54} - 40 q^{57}+ \cdots - 48 q^{99}+O(q^{100}) $ 8 * q - 8 * q^3 + 8 * q^6 + 16 * q^9 - 16 * q^12 - 24 * q^22 + 32 * q^24 - 32 * q^27 - 24 * q^34 + 32 * q^36 - 48 * q^38 - 32 * q^41 + 64 * q^43 - 16 * q^44 + 32 * q^48 - 40 * q^51 + 40 * q^54 - 40 * q^57 + 40 * q^59 + 72 * q^66 - 48 * q^72 + 48 * q^73 + 56 * q^81 - 64 * q^83 - 32 * q^96 + 56 * q^98 - 48 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/136\mathbb{Z}\right)^\times$.

$n$	$69$	$103$	$105$
$\chi(n)$	$-1$	$-1$	$e\left(\frac{11}{16}\right)$

Coefficient data

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

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

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

$2$	−0.541196	+	1.30656i	−0.382683	+	0.923880i
$2$	−0.541196	+	1.30656i	−0.382683	+	0.923880i
$3$	1.39635	−	2.08979i	0.806185	−	1.20654i	−0.169102	−	0.985599i	$-0.554087\pi$
$3$	1.39635	−	2.08979i	0.806185	−	1.20654i	0.975287	−	0.220942i	$-0.0709133\pi$
$4$	−1.41421	−	1.41421i	−0.707107	−	0.707107i
$5$		0			0		−0.195090	−	0.980785i	$-0.562500\pi$
$5$		0			0		0.195090	+	0.980785i	$0.437500\pi$
$6$	1.97474	+	2.95541i	0.806185	+	1.20654i
$7$		0			0		0.195090	−	0.980785i	$-0.437500\pi$
$7$		0			0		−0.195090	+	0.980785i	$0.562500\pi$
$8$	2.61313	−	1.08239i	0.923880	−	0.382683i
$9$	−1.26937	−	3.06453i	−0.423124	−	1.02151i
$10$		0			0
$11$	4.72705	−	3.15851i	1.42526	−	0.952327i	0.426401	−	0.904534i	$-0.359781\pi$
$11$	4.72705	−	3.15851i	1.42526	−	0.952327i	0.998857	−	0.0477934i	$-0.0152189\pi$
$12$	−4.93015	+	0.980668i	−1.42321	+	0.283094i
$13$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$13$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$14$		0			0
$15$		0			0
$16$			4.00000i			1.00000i
$17$	−3.76118	+	1.68925i	−0.912219	+	0.409702i
$17$	−3.76118	+	1.68925i	−0.912219	+	0.409702i
$18$	4.69098			1.10568
$19$	−2.83730	+	6.84984i	−0.650921	+	1.57146i	0.160524	+	0.987032i	$0.448682\pi$
$19$	−2.83730	+	6.84984i	−0.650921	+	1.57146i	−0.811445	+	0.584429i	$0.801318\pi$
$20$		0			0
$21$		0			0
$22$	1.56854	+	7.88556i	0.334413	+	1.68121i
$23$		0			0		−0.555570	−	0.831470i	$-0.687500\pi$
$23$		0			0		0.555570	+	0.831470i	$0.312500\pi$
$24$	1.38687	−	6.97229i	0.283094	−	1.42321i
$25$	−4.61940	+	1.91342i	−0.923880	+	0.382683i
$26$		0			0
$27$	−0.781491	−	0.155448i	−0.150398	−	0.0299160i
$28$		0			0
$29$		0			0		0.980785	−	0.195090i	$-0.0625000\pi$
$29$		0			0		−0.980785	+	0.195090i	$0.937500\pi$
$30$		0			0
$31$		0			0		−0.831470	−	0.555570i	$-0.812500\pi$
$31$		0			0		0.831470	+	0.555570i	$0.187500\pi$
$32$	−5.22625	−	2.16478i	−0.923880	−	0.382683i
$33$		−	14.2889i		−	2.48738i
$34$	−0.171573	−	5.82843i	−0.0294245	−	0.999567i
$35$		0			0
$36$	−2.53874	+	6.12906i	−0.423124	+	1.02151i
$37$		0			0		0.555570	−	0.831470i	$-0.312500\pi$
$37$		0			0		−0.555570	+	0.831470i	$0.687500\pi$
$38$	−7.41421	−	7.41421i	−1.20274	−	1.20274i
$39$		0			0
$40$		0			0
$41$	0.252982	−	1.27183i	0.0395091	−	0.198626i	−0.955990	−	0.293400i	$-0.905213\pi$
$41$	0.252982	−	1.27183i	0.0395091	−	0.198626i	0.995499	+	0.0947747i	$0.0302131\pi$
$42$		0			0
$43$	4.46447	+	10.7782i	0.680825	+	1.64366i	0.762493	+	0.646997i	$0.223975\pi$
$43$	4.46447	+	10.7782i	0.680825	+	1.64366i	−0.0816682	+	0.996660i	$0.526025\pi$
$44$	−11.1519	−	2.21824i	−1.68121	−	0.334413i
$45$		0			0
$46$		0			0
$47$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$47$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$48$	8.35916	+	5.58541i	1.20654	+	0.806185i
$49$	−6.46716	−	2.67878i	−0.923880	−	0.382683i
$50$		−	7.07107i		−	1.00000i
$51$	−1.72176	+	10.2189i	−0.241095	+	1.43093i
$52$		0			0
$53$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$53$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$54$	0.626043	−	0.936940i	0.0851937	−	0.127501i
$55$		0			0
$56$		0			0
$57$	10.3529	+	15.4942i	1.37127	+	2.05225i
$58$		0			0
$59$	9.94975	−	4.12132i	1.29535	−	0.536550i	0.374772	−	0.927117i	$-0.377721\pi$
$59$	9.94975	−	4.12132i	1.29535	−	0.536550i	0.920575	+	0.390567i	$0.127721\pi$
$60$		0			0
$61$		0			0		−0.980785	−	0.195090i	$-0.937500\pi$
$61$		0			0		0.980785	+	0.195090i	$0.0625000\pi$
$62$		0			0
$63$		0			0
$64$	5.65685	−	5.65685i	0.707107	−	0.707107i
$65$		0			0
$66$	18.6694	+	7.73312i	2.29804	+	0.951881i
$67$		−	16.1815i		−	1.97688i	−0.151601	−	0.988442i	$-0.548443\pi$
$67$		−	16.1815i		−	1.97688i	0.151601	−	0.988442i	$-0.451557\pi$
$68$	7.70806	+	2.93015i	0.934740	+	0.355333i
$69$		0			0
$70$		0			0
$71$		0			0		0.555570	−	0.831470i	$-0.312500\pi$
$71$		0			0		−0.555570	+	0.831470i	$0.687500\pi$
$72$	−6.63405	−	6.63405i	−0.781830	−	0.781830i
$73$	2.96960	+	14.9292i	0.347565	+	1.74733i	0.619486	+	0.785007i	$0.287341\pi$
$73$	2.96960	+	14.9292i	0.347565	+	1.74733i	−0.271921	+	0.962319i	$0.587659\pi$
$74$		0			0
$75$	−2.45167	+	12.3254i	−0.283094	+	1.42321i
$76$	13.6997	−	5.67459i	1.57146	−	0.650921i
$77$		0			0
$78$		0			0
$79$		0			0		0.831470	−	0.555570i	$-0.187500\pi$
$79$		0			0		−0.831470	+	0.555570i	$0.812500\pi$
$80$		0			0
$81$	5.62038	−	5.62038i	0.624487	−	0.624487i
$82$	1.52481	+	1.01884i	0.168387	+	0.112512i
$83$	−14.3640	−	5.94975i	−1.57665	−	0.653070i	−0.588771	−	0.808300i	$-0.700388\pi$
$83$	−14.3640	−	5.94975i	−1.57665	−	0.653070i	−0.987878	+	0.155230i	$0.950388\pi$
$84$		0			0
$85$		0			0
$86$	−16.4985			−1.77908
$87$		0			0
$88$	8.93362	−	13.3701i	0.952327	−	1.42526i
$89$	8.56628	+	8.56628i	0.908024	+	0.908024i	0.996113	−	0.0880885i	$-0.0280758\pi$
$89$	8.56628	+	8.56628i	0.908024	+	0.908024i	−0.0880885	+	0.996113i	$0.528076\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$	−11.8216	+	7.89897i	−1.20654	+	0.806185i
$97$	−14.4112	+	2.86657i	−1.46324	+	0.291057i	−0.861550	−	0.507673i	$-0.830506\pi$
$97$	−14.4112	+	2.86657i	−1.46324	+	0.291057i	−0.601690	+	0.798730i	$0.705506\pi$
$98$	7.00000	−	7.00000i	0.707107	−	0.707107i
$99$	−15.6797	−	10.4769i	−1.57587	−	1.05296i
$100$	9.23880	+	3.82683i	0.923880	+	0.382683i
$101$		0			0			−	1.00000i	$-0.5\pi$
$101$		0			0				1.00000i	$0.5\pi$
$102$	−12.4198	−	7.77999i	−1.22974	−	0.770334i
$103$		0			0		1.00000			$0$
$103$		0			0		−1.00000			$\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−3.56001	−	17.8974i	−0.344159	−	1.73021i	−0.634180	−	0.773185i	$-0.718662\pi$
$107$	−3.56001	−	17.8974i	−0.344159	−	1.73021i	0.290021	−	0.957020i	$-0.406338\pi$
$108$	0.885359	+	1.32503i	0.0851937	+	0.127501i
$109$		0			0		0.195090	−	0.980785i	$-0.437500\pi$
$109$		0			0		−0.195090	+	0.980785i	$0.562500\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−6.30644	+	4.21383i	−0.593260	+	0.396404i	−0.815643	−	0.578556i	$-0.803617\pi$
$113$	−6.30644	+	4.21383i	−0.593260	+	0.396404i	0.222383	+	0.974959i	$0.428617\pi$
$114$	−25.8470	+	5.14129i	−2.42079	+	0.481526i
$115$		0			0
$116$		0			0
$117$		0			0
$118$			15.2304i			1.40207i
$119$		0			0
$120$		0			0
$121$	8.15927	−	19.6982i	0.741751	−	1.79075i
$122$		0			0
$123$	−2.30460	−	2.30460i	−0.207798	−	0.207798i
$124$		0			0
$125$		0			0
$126$		0			0
$127$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$127$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$128$	4.32957	+	10.4525i	0.382683	+	0.923880i
$129$	28.7581	+	5.72034i	2.53201	+	0.503648i
$130$		0			0
$131$	22.3849	−	4.45262i	1.95577	−	0.389028i	0.962900	−	0.269858i	$-0.0869767\pi$
$131$	22.3849	−	4.45262i	1.95577	−	0.389028i	0.992874	−	0.119170i	$-0.0380233\pi$
$132$	−20.2076	+	20.2076i	−1.75885	+	1.75885i
$133$		0			0
$134$	21.1421	+	8.75736i	1.82640	+	0.756521i
$135$		0			0
$136$	−8.00000	+	8.48528i	−0.685994	+	0.727607i
$137$	−11.7574			−1.00450			−0.502249	−	0.864723i	$-0.667494\pi$
$137$	−11.7574			−1.00450			−0.502249	+	0.864723i	$0.667494\pi$
$138$		0			0
$139$	2.23909	−	3.35103i	0.189917	−	0.284231i	−0.724276	−	0.689510i	$-0.757826\pi$
$139$	2.23909	−	3.35103i	0.189917	−	0.284231i	0.914193	+	0.405279i	$0.132826\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	12.2581	−	5.07748i	1.02151	−	0.423124i
$145$		0			0
$146$	−21.1130	−	4.19964i	−1.74733	−	0.347565i
$147$	−14.6285	+	9.77447i	−1.20654	+	0.806185i
$148$		0			0
$149$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$149$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$150$	−14.7770	−	9.87371i	−1.20654	−	0.806185i
$151$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$151$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$152$			20.9706i			1.70094i
$153$	9.95108	+	9.38196i	0.804497	+	0.758487i
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$157$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$	4.30165	+	10.3851i	0.337970	+	0.815931i
$163$	−24.8697	−	4.94689i	−1.94794	−	0.387470i	−0.996928	−	0.0783260i	$-0.975042\pi$
$163$	−24.8697	−	4.94689i	−1.94794	−	0.387470i	−0.951015	−	0.309144i	$-0.899958\pi$
$164$	−2.15640	+	1.44086i	−0.168387	+	0.112512i
$165$		0			0
$166$	15.5474	−	15.5474i	1.20672	−	1.20672i
$167$		0			0		−0.831470	−	0.555570i	$-0.812500\pi$
$167$		0			0		0.831470	+	0.555570i	$0.187500\pi$
$168$		0			0
$169$		−	13.0000i		−	1.00000i
$170$		0			0
$171$	24.5931			1.88068
$172$	8.92893	−	21.5563i	0.680825	−	1.64366i
$173$		0			0		0.555570	−	0.831470i	$-0.312500\pi$
$173$		0			0		−0.555570	+	0.831470i	$0.687500\pi$
$174$		0			0
$175$		0			0
$176$	12.6341	+	18.9082i	0.952327	+	1.42526i
$177$	5.28067	−	26.5477i	0.396919	−	1.99545i
$178$	−15.8284	+	6.55635i	−1.18639	+	0.491419i
$179$	0.486803	+	1.17525i	0.0363854	+	0.0878421i	0.941028	−	0.338330i	$-0.109862\pi$
$179$	0.486803	+	1.17525i	0.0363854	+	0.0878421i	−0.904642	+	0.426172i	$0.859862\pi$
$180$		0			0
$181$		0			0		0.831470	−	0.555570i	$-0.187500\pi$
$181$		0			0		−0.831470	+	0.555570i	$0.812500\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−12.4438	+	19.8649i	−0.909978	+	1.45266i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$191$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$192$	−3.92267	−	19.7206i	−0.283094	−	1.42321i
$193$	−10.1483	−	15.1880i	−0.730492	−	1.09326i	−0.991775	−	0.127996i	$-0.959146\pi$
$193$	−10.1483	−	15.1880i	−0.730492	−	1.09326i	0.261283	−	0.965262i	$-0.415854\pi$
$194$	4.05395	−	20.3806i	0.291057	−	1.46324i
$195$		0			0
$196$	5.35757	+	12.9343i	0.382683	+	0.923880i
$197$		0			0		−0.980785	−	0.195090i	$-0.937500\pi$
$197$		0			0		0.980785	+	0.195090i	$0.0625000\pi$
$198$	22.1745	−	14.8165i	1.57587	−	1.05296i
$199$		0			0		0.980785	−	0.195090i	$-0.0625000\pi$
$199$		0			0		−0.980785	+	0.195090i	$0.937500\pi$
$200$	−10.0000	+	10.0000i	−0.707107	+	0.707107i
$201$	−33.8159	−	22.5951i	−2.38519	−	1.59373i
$202$		0			0
$203$		0			0
$204$	16.8866	−	12.0167i	1.18230	−	0.841338i
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	8.22327	+	41.3412i	0.568815	+	2.85963i
$210$		0			0
$211$	−2.61183	+	13.1305i	−0.179806	+	0.903943i	0.780535	+	0.625112i	$0.214947\pi$
$211$	−2.61183	+	13.1305i	−0.179806	+	0.903943i	−0.960340	+	0.278831i	$0.910053\pi$
$212$		0			0
$213$		0			0
$214$	25.3107	+	5.03462i	1.73021	+	0.344159i
$215$		0			0
$216$	−2.21039	+	0.439674i	−0.150398	+	0.0299160i
$217$		0			0
$218$		0			0
$219$	35.3454	+	14.6406i	2.38842	+	0.989317i
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$223$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$224$		0			0
$225$	11.7275	+	11.7275i	0.781830	+	0.781830i
$226$	−2.09261	−	10.5203i	−0.139198	−	0.699798i
$227$	−1.71040	−	2.55979i	−0.113523	−	0.169899i	0.770357	−	0.637613i	$-0.220078\pi$
$227$	−1.71040	−	2.55979i	−0.113523	−	0.169899i	−0.883880	+	0.467714i	$0.845078\pi$
$228$	7.27088	−	36.5532i	0.481526	−	2.42079i
$229$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$229$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	27.7758	−	5.52495i	1.81965	−	0.361952i	0.836971	−	0.547248i	$-0.184325\pi$
$233$	27.7758	−	5.52495i	1.81965	−	0.361952i	0.982683	+	0.185296i	$0.0593245\pi$
$234$		0			0
$235$		0			0
$236$	−19.8995	−	8.24264i	−1.29535	−	0.536550i
$237$		0			0
$238$		0			0
$239$		0			0		1.00000			$0$
$239$		0			0		−1.00000			$\pi$
$240$		0			0
$241$	−6.77233	+	10.1355i	−0.436244	+	0.652885i	−0.982829	−	0.184518i	$-0.940928\pi$
$241$	−6.77233	+	10.1355i	−0.436244	+	0.652885i	0.546585	+	0.837404i	$0.315928\pi$
$242$	21.3212	+	21.3212i	1.37058	+	1.37058i
$243$	−4.36373	−	21.9379i	−0.279933	−	1.40732i
$244$		0			0
$245$		0			0
$246$	4.25834	−	1.76386i	0.271502	−	0.112460i
$247$		0			0
$248$		0			0
$249$	−32.4909	+	21.7097i	−2.05903	+	1.37580i
$250$		0			0
$251$	12.3387	−	12.3387i	0.778813	−	0.778813i	−0.200816	−	0.979629i	$-0.564359\pi$
$251$	12.3387	−	12.3387i	0.778813	−	0.778813i	0.979629	+	0.200816i	$0.0643592\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−16.0000			−1.00000
$257$	−12.2635	+	29.6066i	−0.764973	+	1.84681i	−0.352865	+	0.935674i	$0.614792\pi$
$257$	−12.2635	+	29.6066i	−0.764973	+	1.84681i	−0.412108	+	0.911135i	$0.635208\pi$
$258$	−23.0378	+	34.4784i	−1.43427	+	2.14653i
$259$		0			0
$260$		0			0
$261$		0			0
$262$	−6.29696	+	31.6570i	−0.389028	+	1.95577i
$263$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$263$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$264$	−15.4662	−	37.3388i	−0.951881	−	2.29804i
$265$		0			0
$266$		0			0
$267$	29.8633	−	5.94018i	1.82760	−	0.363533i
$268$	−22.8841	+	22.8841i	−1.39787	+	1.39787i
$269$		0			0		−0.831470	−	0.555570i	$-0.812500\pi$
$269$		0			0		0.831470	+	0.555570i	$0.187500\pi$
$270$		0			0
$271$		0			0			−	1.00000i	$-0.5\pi$
$271$		0			0				1.00000i	$0.5\pi$
$272$	−6.75699	−	15.0447i	−0.409702	−	0.912219i
$273$		0			0
$274$	6.36304	−	15.3617i	0.384405	−	0.928036i
$275$	−15.7926	+	23.6352i	−0.952327	+	1.42526i
$276$		0			0
$277$		0			0		−0.195090	−	0.980785i	$-0.562500\pi$
$277$		0			0		0.195090	+	0.980785i	$0.437500\pi$
$278$	3.16655	+	4.73907i	0.189917	+	0.284231i
$279$		0			0
$280$		0			0
$281$	−12.5244	−	30.2367i	−0.747145	−	1.80377i	−0.573969	−	0.818877i	$-0.694597\pi$
$281$	−12.5244	−	30.2367i	−0.747145	−	1.80377i	−0.173176	−	0.984891i	$-0.555403\pi$
$282$		0			0
$283$	−3.45045	+	2.30552i	−0.205108	+	0.137049i	−0.653882	−	0.756596i	$-0.726861\pi$
$283$	−3.45045	+	2.30552i	−0.205108	+	0.137049i	0.448774	+	0.893645i	$0.351861\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$			18.7639i			1.10568i
$289$	11.2929	−	12.7071i	0.664288	−	0.747477i
$290$		0			0
$291$	−14.1326	+	34.1192i	−0.828470	+	2.00010i
$292$	16.9134	−	25.3127i	0.989781	−	1.48131i
$293$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$293$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$294$	−4.85406	−	24.4030i	−0.283094	−	1.42321i
$295$		0			0
$296$		0			0
$297$	−4.18513	+	1.73354i	−0.242846	+	0.100590i
$298$		0			0
$299$		0			0
$300$	20.8979	−	13.9635i	1.20654	−	0.806185i
$301$		0			0
$302$		0			0
$303$		0			0
$304$	−27.3994	−	11.3492i	−1.57146	−	0.650921i
$305$		0			0
$306$	−17.6436	+	7.92422i	−1.00862	+	0.452998i
$307$	8.48528			0.484281			0.242140	−	0.970241i	$-0.422151\pi$
$307$	8.48528			0.484281			0.242140	+	0.970241i	$0.422151\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		−0.195090	−	0.980785i	$-0.562500\pi$
$311$		0			0		0.195090	+	0.980785i	$0.437500\pi$
$312$		0			0
$313$	−0.621610	+	3.12504i	−0.0351355	+	0.176638i	−0.994368	−	0.105979i	$-0.966202\pi$
$313$	−0.621610	+	3.12504i	−0.0351355	+	0.176638i	0.959233	+	0.282617i	$0.0912024\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0		0.831470	−	0.555570i	$-0.187500\pi$
$317$		0			0		−0.831470	+	0.555570i	$0.812500\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$	−42.3728	−	17.5514i	−2.36502	−	0.979623i
$322$		0			0
$323$	−0.899495	−	30.5563i	−0.0500492	−	1.70020i
$324$	−15.8968			−0.883158
$325$		0			0
$326$	19.9228	−	29.8166i	1.10342	−	1.65139i
$327$		0			0
$328$	−0.715541	−	3.59727i	−0.0395091	−	0.198626i
$329$		0			0
$330$		0			0
$331$	13.1924	−	5.46447i	0.725119	−	0.300354i	0.0105746	−	0.999944i	$-0.496634\pi$
$331$	13.1924	−	5.46447i	0.725119	−	0.300354i	0.714545	+	0.699590i	$0.246634\pi$
$332$	11.8995	+	28.7279i	0.653070	+	1.57665i
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	29.9497	+	20.0118i	1.63147	+	1.09011i	0.923312	+	0.384052i	$0.125472\pi$
$337$	29.9497	+	20.0118i	1.63147	+	1.09011i	0.708154	+	0.706058i	$0.249528\pi$
$338$	16.9853	+	7.03555i	0.923880	+	0.382683i
$339$			19.0631i			1.03537i
$340$		0			0
$341$		0			0
$342$	−13.3097	+	32.1325i	−0.719707	+	1.73753i
$343$		0			0
$344$	23.3324	+	23.3324i	1.25800	+	1.25800i
$345$		0			0
$346$		0			0
$347$	−4.95859	+	24.9285i	−0.266191	+	1.33823i	0.583998	+	0.811755i	$0.301488\pi$
$347$	−4.95859	+	24.9285i	−0.266191	+	1.33823i	−0.850189	+	0.526477i	$0.823512\pi$
$348$		0			0
$349$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$349$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$350$		0			0
$351$		0			0
$352$	−31.5422	+	6.27414i	−1.68121	+	0.334413i
$353$	−6.66413	+	6.66413i	−0.354696	+	0.354696i	−0.861853	−	0.507157i	$-0.830696\pi$
$353$	−6.66413	+	6.66413i	−0.354696	+	0.354696i	0.507157	+	0.861853i	$0.330696\pi$
$354$	31.8284	+	21.2670i	1.69166	+	1.13033i
$355$		0			0
$356$		−	24.2291i		−	1.28414i
$357$		0			0
$358$	−1.79899			−0.0950796
$359$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$359$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$360$		0			0
$361$	−25.4350	−	25.4350i	−1.33869	−	1.33869i
$362$		0			0
$363$	−29.7719	−	44.5568i	−1.56262	−	2.33863i
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		−0.980785	−	0.195090i	$-0.937500\pi$
$367$		0			0		0.980785	+	0.195090i	$0.0625000\pi$
$368$		0			0
$369$	−4.21868	+	0.839147i	−0.219616	+	0.0436843i
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0			−	1.00000i	$-0.5\pi$
$373$		0			0				1.00000i	$0.5\pi$
$374$	−19.2202	−	27.0093i	−0.993853	−	1.39662i
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	7.08373	+	35.6123i	0.363867	+	1.82928i	0.536011	+	0.844211i	$0.319930\pi$
$379$	7.08373	+	35.6123i	0.363867	+	1.82928i	−0.172145	+	0.985072i	$0.555070\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$383$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$384$	27.8891	+	5.54750i	1.42321	+	0.283094i
$385$		0			0
$386$	25.3363	−	5.03971i	1.28959	−	0.256515i
$387$	27.3630	−	27.3630i	1.39094	−	1.39094i
$388$	24.4345	+	16.3266i	1.24047	+	0.828859i
$389$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$389$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$390$		0			0
$391$		0			0
$392$	−19.7990			−1.00000
$393$	21.9521	−	52.9971i	1.10734	−	2.67335i
$394$		0			0
$395$		0			0
$396$	7.35797	+	36.9910i	0.369752	+	1.85887i
$397$		0			0		−0.555570	−	0.831470i	$-0.687500\pi$
$397$		0			0		0.555570	+	0.831470i	$0.312500\pi$
$398$		0			0
$399$		0			0
$400$	−7.65367	−	18.4776i	−0.382683	−	0.923880i
$401$	1.80511	+	0.359058i	0.0901427	+	0.0179305i	0.239956	−	0.970784i	$-0.422867\pi$
$401$	1.80511	+	0.359058i	0.0901427	+	0.0179305i	−0.149813	+	0.988714i	$0.547867\pi$
$402$	47.8229	−	31.9543i	2.38519	−	1.59373i
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$	6.56163	+	28.5668i	0.324849	+	1.41427i
$409$	33.3141			1.64727			0.823637	−	0.567117i	$-0.191941\pi$
$409$	33.3141			1.64727			0.823637	+	0.567117i	$0.191941\pi$
$410$		0			0
$411$	−16.4174	+	24.5704i	−0.809812	+	1.21197i
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−3.87640	−	9.35845i	−0.189828	−	0.458285i
$418$	−58.4652	−	11.6295i	−2.85963	−	0.568815i
$419$	32.9051	−	21.9865i	1.60752	−	1.07411i	0.661529	−	0.749919i	$-0.269908\pi$
$419$	32.9051	−	21.9865i	1.60752	−	1.07411i	0.945991	−	0.324192i	$-0.105092\pi$
$420$		0			0
$421$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$421$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$422$	−15.7424	−	10.5187i	−0.766326	−	0.512043i
$423$		0			0
$424$		0			0
$425$	14.1421	−	15.0000i	0.685994	−	0.727607i
$426$		0			0
$427$		0			0
$428$	−20.2761	+	30.3453i	−0.980083	+	1.46680i
$429$		0			0
$430$		0			0
$431$		0			0		−0.555570	−	0.831470i	$-0.687500\pi$
$431$		0			0		0.555570	+	0.831470i	$0.312500\pi$
$432$	0.621793	−	3.12597i	0.0299160	−	0.150398i
$433$	35.9113	−	14.8749i	1.72578	−	0.714843i	0.726158	−	0.687528i	$-0.241304\pi$
$433$	35.9113	−	14.8749i	1.72578	−	0.714843i	0.999627	−	0.0273152i	$-0.00869578\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$	−38.2576	+	38.2576i	−1.82802	+	1.82802i
$439$		0			0		−0.831470	−	0.555570i	$-0.812500\pi$
$439$		0			0		0.831470	+	0.555570i	$0.187500\pi$
$440$		0			0
$441$			23.2192i			1.10568i
$442$		0			0
$443$	13.4596			0.639484			0.319742	−	0.947505i	$-0.396404\pi$
$443$	13.4596			0.639484			0.319742	+	0.947505i	$0.396404\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$	2.68092	−	13.4779i	0.126521	−	0.636062i	−0.864531	−	0.502580i	$-0.832384\pi$
$449$	2.68092	−	13.4779i	0.126521	−	0.636062i	0.991051	−	0.133482i	$-0.0426157\pi$
$450$	−21.6695	+	8.97581i	−1.02151	+	0.423124i
$451$	−2.82122	−	6.81103i	−0.132846	−	0.320719i
$452$	14.8779	+	2.95940i	0.699798	+	0.139198i
$453$		0			0
$454$	4.27018	−	0.849392i	0.200410	−	0.0398639i
$455$		0			0
$456$	43.8241	+	29.2823i	2.05225	+	1.37127i
$457$	−5.18779	−	2.14885i	−0.242675	−	0.100519i	0.258031	−	0.966137i	$-0.416926\pi$
$457$	−5.18779	−	2.14885i	−0.242675	−	0.100519i	−0.500706	+	0.865617i	$0.666926\pi$
$458$		0			0
$459$	3.20192	−	0.735463i	0.149453	−	0.0343285i
$460$		0			0
$461$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$461$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$462$		0			0
$463$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$463$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$464$		0			0
$465$		0			0
$466$	−7.81346	+	39.2809i	−0.361952	+	1.81965i
$467$	−39.9238	+	16.5370i	−1.84745	+	0.765240i	−0.916760	+	0.399439i	$0.869205\pi$
$467$	−39.9238	+	16.5370i	−1.84745	+	0.765240i	−0.930693	+	0.365801i	$0.880795\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$	21.5391	−	21.5391i	0.991416	−	0.991416i
$473$	55.1467	+	36.8479i	2.53565	+	1.69427i
$474$		0			0
$475$		−	37.0711i		−	1.70094i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		0.555570	−	0.831470i	$-0.312500\pi$
$479$		0			0		−0.555570	+	0.831470i	$0.687500\pi$
$480$		0			0
$481$		0			0
$482$	−9.57752	−	14.3338i	−0.436244	−	0.652885i
$483$		0			0
$484$	−39.3964	+	16.3185i	−1.79075	+	0.741751i
$485$		0			0
$486$	31.0249	+	6.17124i	1.40732	+	0.279933i
$487$		0			0		0.831470	−	0.555570i	$-0.187500\pi$
$487$		0			0		−0.831470	+	0.555570i	$0.812500\pi$
$488$		0			0
$489$	−45.0648	+	45.0648i	−2.03790	+	2.03790i
$490$		0			0
$491$	−18.1990	−	7.53828i	−0.821311	−	0.340198i	−0.0678537	−	0.997695i	$-0.521615\pi$
$491$	−18.1990	−	7.53828i	−0.821311	−	0.340198i	−0.753457	+	0.657497i	$0.771615\pi$
$492$			6.51838i			0.293871i
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$	−10.7812	−	54.2006i	−0.483116	−	2.42879i
$499$	−6.62811	−	9.91966i	−0.296715	−	0.444065i	0.652919	−	0.757428i	$-0.273544\pi$
$499$	−6.62811	−	9.91966i	−0.296715	−	0.444065i	−0.949633	+	0.313363i	$0.898544\pi$
$500$		0			0
$501$		0			0
$502$	9.44365	+	22.7990i	0.421491	+	1.01757i
$503$		0			0		−0.980785	−	0.195090i	$-0.937500\pi$
$503$		0			0		0.980785	+	0.195090i	$0.0625000\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−27.1673	−	18.1526i	−1.20654	−	0.806185i
$508$		0			0
$509$		0			0			−	1.00000i	$-0.5\pi$
$509$		0			0				1.00000i	$0.5\pi$
$510$		0			0
$511$		0			0
$512$	8.65914	−	20.9050i	0.382683	−	0.923880i
$513$	3.28212	−	4.91204i	0.144909	−	0.216872i
$514$	−32.0460	−	32.0460i	−1.41349	−	1.41349i
$515$		0			0
$516$	−32.5803	−	48.7599i	−1.43427	−	2.14653i
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−28.5149	+	19.0531i	−1.24926	+	0.834730i	−0.991325	−	0.131432i	$-0.958042\pi$
$521$	−28.5149	+	19.0531i	−1.24926	+	0.834730i	−0.257936	+	0.966162i	$0.583042\pi$
$522$		0			0
$523$	6.34711	−	6.34711i	0.277540	−	0.277540i	−0.554587	−	0.832126i	$-0.687124\pi$
$523$	6.34711	−	6.34711i	0.277540	−	0.277540i	0.832126	+	0.554587i	$0.187124\pi$
$524$	−37.9539	−	25.3600i	−1.65803	−	1.10786i
$525$		0			0
$526$		0			0
$527$		0			0
$528$	57.1558			2.48738
$529$	−8.80172	+	21.2492i	−0.382683	+	0.923880i
$530$		0			0
$531$	−25.2598	−	25.2598i	−1.09618	−	1.09618i
$532$		0			0
$533$		0			0
$534$	−8.40068	+	42.2331i	−0.363533	+	1.82760i
$535$		0			0
$536$	−17.5147	−	42.2843i	−0.756521	−	1.82640i
$537$	3.13577	+	0.623743i	0.135318	+	0.0269165i
$538$		0			0
$539$	−39.0315	+	7.76386i	−1.68121	+	0.334413i
$540$		0			0
$541$		0			0		−0.831470	−	0.555570i	$-0.812500\pi$
$541$		0			0		0.831470	+	0.555570i	$0.187500\pi$
$542$		0			0
$543$		0			0
$544$	23.3137	−	0.686292i	0.999567	−	0.0294245i
$545$		0			0
$546$		0			0
$547$	9.60936	−	14.3814i	0.410867	−	0.614905i	−0.567104	−	0.823646i	$-0.691936\pi$
$547$	9.60936	−	14.3814i	0.410867	−	0.614905i	0.977971	+	0.208741i	$0.0669364\pi$
$548$	16.6274	+	16.6274i	0.710288	+	0.710288i
$549$		0			0
$550$	−22.3341	−	33.4253i	−0.952327	−	1.42526i
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$	−7.90562	+	1.57253i	−0.335273	+	0.0666900i
$557$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$557$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$	24.1375	+	53.7432i	1.01909	+	2.26904i
$562$	46.2843			1.95238
$563$	−5.22183	+	12.6066i	−0.220074	+	0.531305i	−0.994900	−	0.100870i	$-0.967837\pi$
$563$	−5.22183	+	12.6066i	−0.220074	+	0.531305i	0.774826	+	0.632175i	$0.217837\pi$
$564$		0			0
$565$		0			0
$566$	−1.14493	−	5.75597i	−0.0481251	−	0.241941i
$567$		0			0
$568$		0			0
$569$	−27.8492	+	11.5355i	−1.16750	+	0.483595i	−0.880366	−	0.474295i	$-0.842703\pi$
$569$	−27.8492	+	11.5355i	−1.16750	+	0.483595i	−0.287135	+	0.957890i	$0.592703\pi$
$570$		0			0
$571$	46.5861	+	9.26656i	1.94957	+	0.387793i	0.996332	+	0.0855694i	$0.0272709\pi$
$571$	46.5861	+	9.26656i	1.94957	+	0.387793i	0.953237	+	0.302224i	$0.0977291\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	−24.5163	−	10.1550i	−1.02151	−	0.423124i
$577$			33.9411i			1.41299i	0.707719	+	0.706494i	$0.249724\pi$
$577$			33.9411i			1.41299i	−0.707719	+	0.706494i	$0.750276\pi$
$578$	10.4910	+	21.6319i	0.436367	+	0.899769i
$579$	−45.9104			−1.90797
$580$		0			0
$581$		0			0
$582$	−36.9304	−	36.9304i	−1.53081	−	1.53081i
$583$		0			0
$584$	23.9191	+	35.7975i	0.989781	+	1.48131i
$585$		0			0
$586$		0			0
$587$	−2.29610	−	5.54328i	−0.0947702	−	0.228796i	0.869385	−	0.494136i	$-0.164516\pi$
$587$	−2.29610	−	5.54328i	−0.0947702	−	0.228796i	−0.964155	+	0.265341i	$0.914516\pi$
$588$	34.5111	+	6.86468i	1.42321	+	0.283094i
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−31.3640	−	12.9914i	−1.28796	−	0.533492i	−0.369586	−	0.929197i	$-0.620500\pi$
$593$	−31.3640	−	12.9914i	−1.28796	−	0.533492i	−0.918378	+	0.395705i	$0.870500\pi$
$594$		−	6.40632i		−	0.262855i
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$599$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$600$	6.93437	+	34.8614i	0.283094	+	1.42321i
$601$	26.9046	+	40.2655i	1.09746	+	1.64246i	0.678804	+	0.734319i	$0.262498\pi$
$601$	26.9046	+	40.2655i	1.09746	+	1.64246i	0.418655	+	0.908145i	$0.362502\pi$
$602$		0			0
$603$	−49.5887	+	20.5403i	−2.01941	+	0.836466i
$604$		0			0
$605$		0			0
$606$		0			0
$607$		0			0		0.980785	−	0.195090i	$-0.0625000\pi$
$607$		0			0		−0.980785	+	0.195090i	$0.937500\pi$
$608$	29.6569	−	29.6569i	1.20274	−	1.20274i
$609$		0			0
$610$		0			0
$611$		0			0
$612$	−0.804845	−	27.3410i	−0.0325339	−	1.10520i
$613$		0			0		1.00000			$0$
$613$		0			0		−1.00000			$\pi$
$614$	−4.59220	+	11.0866i	−0.185326	+	0.447417i
$615$		0			0
$616$		0			0
$617$	−9.15823	−	46.0415i	−0.368697	−	1.85356i	−0.505310	−	0.862938i	$-0.668622\pi$
$617$	−9.15823	−	46.0415i	−0.368697	−	1.85356i	0.136613	−	0.990624i	$-0.456378\pi$
$618$		0			0
$619$	−9.10751	+	45.7865i	−0.366062	+	1.84032i	0.156452	+	0.987685i	$0.449994\pi$
$619$	−9.10751	+	45.7865i	−0.366062	+	1.84032i	−0.522514	+	0.852631i	$0.675006\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	17.6777	−	17.6777i	0.707107	−	0.707107i
$626$	−3.74665	−	2.50343i	−0.149746	−	0.100057i
$627$	97.8769	+	40.5420i	3.90883	+	1.61909i
$628$		0			0
$629$		0			0
$630$		0			0
$631$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$631$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$632$		0			0
$633$	23.7930	+	23.7930i	0.945688	+	0.945688i
$634$		0			0
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	0.180035	−	0.0358112i	0.00711096	−	0.00141446i	−0.191534	−	0.981486i	$-0.561346\pi$
$641$	0.180035	−	0.0358112i	0.00711096	−	0.00141446i	0.198645	+	0.980072i	$0.436346\pi$
$642$	45.8640	−	45.8640i	1.81011	−	1.81011i
$643$	17.2308	+	11.5132i	0.679515	+	0.454037i	0.846828	−	0.531866i	$-0.178509\pi$
$643$	17.2308	+	11.5132i	0.679515	+	0.454037i	−0.167313	+	0.985904i	$0.553509\pi$
$644$		0			0
$645$		0			0
$646$	40.4106	+	15.3617i	1.58993	+	0.604399i
$647$		0			0		1.00000			$0$
$647$		0			0		−1.00000			$\pi$
$648$	8.60331	−	20.7702i	0.337970	−	0.815931i
$649$	34.0157	−	50.9081i	1.33523	−	1.99832i
$650$		0			0
$651$		0			0
$652$	28.1751	+	42.1670i	1.10342	+	1.65139i
$653$		0			0		0.195090	−	0.980785i	$-0.437500\pi$
$653$		0			0		−0.195090	+	0.980785i	$0.562500\pi$
$654$		0			0
$655$		0			0
$656$	5.08730	+	1.01193i	0.198626	+	0.0395091i
$657$	41.9814	−	28.0511i	1.63785	−	1.09438i
$658$		0			0
$659$	−12.7279	+	12.7279i	−0.495809	+	0.495809i	−0.910131	−	0.414321i	$-0.864019\pi$
$659$	−12.7279	+	12.7279i	−0.495809	+	0.495809i	0.414321	+	0.910131i	$0.364019\pi$
$660$		0			0
$661$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$661$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$662$			20.1940i			0.784863i
$663$		0			0
$664$	−43.9748			−1.70655
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−19.5866	−	3.89602i	−0.755008	−	0.150180i	−0.197444	−	0.980314i	$-0.563264\pi$
$673$	−19.5866	−	3.89602i	−0.755008	−	0.150180i	−0.557564	+	0.830134i	$0.688264\pi$
$674$	−42.3553	+	28.3009i	−1.63147	+	1.09011i
$675$	3.90746	−	0.777242i	0.150398	−	0.0299160i
$676$	−18.3848	+	18.3848i	−0.707107	+	0.707107i
$677$		0			0		−0.831470	−	0.555570i	$-0.812500\pi$
$677$		0			0		0.831470	+	0.555570i	$0.187500\pi$
$678$	−24.9072	−	10.3169i	−0.956555	−	0.396218i
$679$		0			0
$680$		0			0
$681$	−7.73773			−0.296511
$682$		0			0
$683$	9.79828	−	14.6642i	0.374921	−	0.561108i	−0.595247	−	0.803543i	$-0.702946\pi$
$683$	9.79828	−	14.6642i	0.374921	−	0.561108i	0.970168	+	0.242434i	$0.0779459\pi$
$684$	−34.7799	−	34.7799i	−1.32984	−	1.32984i
$685$		0			0
$686$		0			0
$687$		0			0
$688$	−43.1127	+	17.8579i	−1.64366	+	0.680825i
$689$		0			0
$690$		0			0
$691$	43.5608	−	29.1064i	1.65713	−	1.10726i	0.782169	−	0.623066i	$-0.214113\pi$
$691$	43.5608	−	29.1064i	1.65713	−	1.10726i	0.874961	−	0.484193i	$-0.160887\pi$
$692$		0			0
$693$		0			0
$694$	−29.8871	−	19.9699i	−1.13450	−	0.758048i
$695$		0			0
$696$		0			0
$697$	1.19692	+	5.21091i	0.0453365	+	0.197377i
$698$		0			0
$699$	27.2388	−	65.7604i	1.03027	−	2.48729i
$700$		0			0
$701$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$701$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$702$		0			0
$703$		0			0
$704$	8.87298	−	44.6075i	0.334413	−	1.68121i
$705$		0			0
$706$	−5.10051	−	12.3137i	−0.191960	−	0.463433i
$707$		0			0
$708$	−45.0121	+	30.0761i	−1.69166	+	1.13033i
$709$		0			0		0.980785	−	0.195090i	$-0.0625000\pi$
$709$		0			0		−0.980785	+	0.195090i	$0.937500\pi$
$710$		0			0
$711$		0			0
$712$	31.6569	+	13.1127i	1.18639	+	0.491419i
$713$		0			0
$714$		0			0
$715$		0			0
$716$	0.973606	−	2.35049i	0.0363854	−	0.0878421i
$717$		0			0
$718$		0			0
$719$		0			0		−0.195090	−	0.980785i	$-0.562500\pi$
$719$		0			0		0.195090	+	0.980785i	$0.437500\pi$
$720$		0			0
$721$		0			0
$722$	46.9978	−	19.4671i	1.74908	−	0.724491i
$723$	11.7245	+	28.3055i	0.436040	+	1.05269i
$724$		0			0
$725$		0			0
$726$	74.3287	−	14.7849i	2.75860	−	0.548719i
$727$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$727$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$728$		0			0
$729$	−29.9088	−	12.3886i	−1.10773	−	0.458839i
$730$		0			0
$731$	−34.9986	−	32.9970i	−1.29447	−	1.22044i
$732$		0			0
$733$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$733$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−51.1094	−	76.4907i	−1.88264	−	2.81757i
$738$	1.18673	−	5.96611i	0.0436843	−	0.219616i
$739$	−39.1969	+	16.2359i	−1.44188	+	0.597247i	−0.960253	−	0.279129i	$-0.909954\pi$
$739$	−39.1969	+	16.2359i	−1.44188	+	0.597247i	−0.481627	+	0.876376i	$0.659954\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		0.980785	−	0.195090i	$-0.0625000\pi$
$743$		0			0		−0.980785	+	0.195090i	$0.937500\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$			51.5713i			1.88689i
$748$	45.6913	−	10.4950i	1.67064	−	0.383737i
$749$		0			0
$750$		0			0
$751$		0			0		0.555570	−	0.831470i	$-0.312500\pi$
$751$		0			0		−0.555570	+	0.831470i	$0.687500\pi$
$752$		0			0
$753$	−8.55613	−	43.0146i	−0.311803	−	1.56754i
$754$		0			0
$755$		0			0
$756$		0			0
$757$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$757$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$758$	−50.3634	−	10.0179i	−1.82928	−	0.363867i
$759$		0			0
$760$		0			0
$761$	−21.3431	+	21.3431i	−0.773688	+	0.773688i	−0.978749	−	0.205061i	$-0.934261\pi$
$761$	−21.3431	+	21.3431i	−0.773688	+	0.773688i	0.205061	+	0.978749i	$0.434261\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	−22.3417	+	33.4366i	−0.806185	+	1.20654i
$769$	−14.4558	−	14.4558i	−0.521291	−	0.521291i	0.396670	−	0.917961i	$-0.370166\pi$
$769$	−14.4558	−	14.4558i	−0.521291	−	0.521291i	−0.917961	+	0.396670i	$0.870166\pi$
$770$		0			0
$771$	44.7475	+	66.9693i	1.61154	+	2.41184i
$772$	−7.12723	+	35.8310i	−0.256515	+	1.28959i
$773$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$773$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$774$	20.9427	+	50.5602i	0.752771	+	1.81735i
$775$		0			0
$776$	−34.5556	+	23.0893i	−1.24047	+	0.828859i
$777$		0			0
$778$		0			0
$779$	7.99402	+	5.34143i	0.286415	+	0.191377i
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	10.7151	−	25.8686i	0.382683	−	0.923880i
$785$		0			0
$786$	57.3636	+	57.3636i	2.04609	+	2.04609i
$787$	−8.59823	−	43.2262i	−0.306494	−	1.54085i	−0.760195	−	0.649695i	$-0.774897\pi$
$787$	−8.59823	−	43.2262i	−0.306494	−	1.54085i	0.453701	−	0.891154i	$-0.350103\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$	−52.3132	−	10.4057i	−1.85887	−	0.369752i
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$797$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$798$		0			0
$799$		0			0
$800$	28.2843			1.00000
$801$	15.3779	−	37.1254i	0.543350	−	1.31176i
$802$	−1.44605	+	2.16416i	−0.0510618	+	0.0764193i
$803$	61.1914	+	61.1914i	2.15940	+	2.15940i
$804$	15.8687	+	79.7772i	0.559645	+	2.81352i
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−16.5956	−	3.30106i	−0.583469	−	0.116059i	−0.105474	−	0.994422i	$-0.533636\pi$
$809$	−16.5956	−	3.30106i	−0.583469	−	0.116059i	−0.477994	+	0.878363i	$0.658636\pi$
$810$		0			0
$811$	−28.4358	+	5.65623i	−0.998515	+	0.198617i	−0.667180	−	0.744896i	$-0.732499\pi$
$811$	−28.4358	+	5.65623i	−0.998515	+	0.198617i	−0.331335	+	0.943513i	$0.607499\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$	−40.8754	−	6.88704i	−1.43093	−	0.241095i
$817$	−86.4958			−3.02610
$818$	−18.0294	+	43.5269i	−0.630384	+	1.52188i
$819$		0			0
$820$		0			0
$821$		0			0		−0.195090	−	0.980785i	$-0.562500\pi$
$821$		0			0		0.195090	+	0.980785i	$0.437500\pi$
$822$	−23.2177	−	34.7478i	−0.809812	−	1.21197i
$823$		0			0		0.195090	−	0.980785i	$-0.437500\pi$
$823$		0			0		−0.195090	+	0.980785i	$0.562500\pi$
$824$		0			0
$825$	27.3407	+	66.0063i	0.951881	+	2.29804i
$826$		0			0
$827$	40.8250	−	27.2784i	1.41962	−	0.948563i	0.420476	−	0.907304i	$-0.361863\pi$
$827$	40.8250	−	27.2784i	1.41962	−	0.948563i	0.999148	−	0.0412591i	$-0.0131369\pi$
$828$		0			0
$829$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$829$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	28.8492	−	0.849242i	0.999567	−	0.0294245i
$834$	14.3253			0.496044
$835$		0			0
$836$	46.8358	−	70.0947i	1.61985	−	2.42428i
$837$		0			0
$838$	10.9186	+	54.8916i	0.377178	+	1.89620i
$839$		0			0		−0.555570	−	0.831470i	$-0.687500\pi$
$839$		0			0		0.555570	+	0.831470i	$0.312500\pi$
$840$		0			0
$841$	26.7925	−	11.0978i	0.923880	−	0.382683i
$842$		0			0
$843$	−80.6768	−	16.0476i	−2.77866	−	0.552709i
$844$	22.2631	−	14.8757i	0.766326	−	0.512043i
$845$		0			0
$846$		0			0
$847$		0			0
$848$		0			0
$849$			10.4300i			0.357958i
$850$	11.9448	+	26.5955i	0.409702	+	0.912219i
$851$		0			0
$852$		0			0
$853$		0			0		0.555570	−	0.831470i	$-0.312500\pi$
$853$		0			0		−0.555570	+	0.831470i	$0.687500\pi$
$854$		0			0
$855$		0			0
$856$	−28.6747	−	42.9148i	−0.980083	−	1.46680i
$857$	−2.18243	+	10.9718i	−0.0745505	+	0.374791i	−0.999992	−	0.00410087i	$-0.998695\pi$
$857$	−2.18243	+	10.9718i	−0.0745505	+	0.374791i	0.925441	+	0.378892i	$0.123695\pi$
$858$		0			0
$859$	−5.49390	−	13.2635i	−0.187450	−	0.452543i	0.802018	−	0.597300i	$-0.203760\pi$
$859$	−5.49390	−	13.2635i	−0.187450	−	0.452543i	−0.989467	+	0.144757i	$0.953760\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$863$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$864$	3.74776	+	2.50417i	0.127501	+	0.0851937i
$865$		0			0
$866$			54.9706i			1.86798i
$867$	−10.7863	−	41.3434i	−0.366323	−	1.40410i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	27.0779	+	40.5250i	0.916449	+	1.37156i
$874$		0			0
$875$		0			0
$876$	−29.2811	−	70.6909i	−0.989317	−	2.38842i
$877$		0			0		−0.980785	−	0.195090i	$-0.937500\pi$
$877$		0			0		0.980785	+	0.195090i	$0.0625000\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−47.4231	−	31.6871i	−1.59773	−	1.06757i	−0.952921	−	0.303218i	$-0.901939\pi$
$881$	−47.4231	−	31.6871i	−1.59773	−	1.06757i	−0.644804	−	0.764348i	$-0.723061\pi$
$882$	−30.3373	−	12.5661i	−1.02151	−	0.423124i
$883$		−	55.6410i		−	1.87247i	−0.351376	−	0.936235i	$-0.614286\pi$
$883$		−	55.6410i		−	1.87247i	0.351376	−	0.936235i	$-0.385714\pi$
$884$		0			0
$885$		0			0
$886$	−7.28427	+	17.5858i	−0.244720	+	0.590806i
$887$		0			0		0.555570	−	0.831470i	$-0.312500\pi$
$887$		0			0		−0.555570	+	0.831470i	$0.687500\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	8.81577	−	44.3199i	0.295339	−	1.48477i
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$	16.1588	+	10.7970i	0.539227	+	0.360300i
$899$		0			0
$900$		−	33.1703i		−	1.10568i
$901$		0			0
$902$	10.4259			0.347143
$903$		0			0
$904$	−11.9185	+	17.8373i	−0.396404	+	0.593260i
$905$		0			0
$906$		0			0
$907$	24.3512	+	36.4442i	0.808570	+	1.21011i	0.974592	+	0.223988i	$0.0719076\pi$
$907$	24.3512	+	36.4442i	0.808570	+	1.21011i	−0.166022	+	0.986122i	$0.553092\pi$
$908$	−1.20122	+	6.03895i	−0.0398639	+	0.200410i
$909$		0			0
$910$		0			0
$911$		0			0		−0.980785	−	0.195090i	$-0.937500\pi$
$911$		0			0		0.980785	+	0.195090i	$0.0625000\pi$
$912$	−61.9766	+	41.4114i	−2.05225	+	1.37127i
$913$	−86.6915	+	17.2440i	−2.86907	+	0.570693i
$914$	5.61522	−	5.61522i	0.185735	−	0.185735i
$915$		0			0
$916$		0			0
$917$		0			0
$918$	−0.771936	+	4.58154i	−0.0254777	+	0.151213i
$919$		0			0		1.00000			$0$
$919$		0			0		−1.00000			$\pi$
$920$		0			0
$921$	11.8484	−	17.7325i	0.390420	−	0.584304i
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$	48.6247	−	32.4900i	1.59532	−	1.06596i	0.640850	−	0.767666i	$-0.278582\pi$
$929$	48.6247	−	32.4900i	1.59532	−	1.06596i	0.954474	−	0.298295i	$-0.0964179\pi$
$930$		0			0
$931$	36.6985	−	36.6985i	1.20274	−	1.20274i
$932$	−47.0944	−	31.4675i	−1.54263	−	1.03075i
$933$		0			0
$934$		−	61.1127i		−	1.99967i
$935$		0			0
$936$		0			0
$937$	−19.4831	+	47.0363i	−0.636484	+	1.53661i	0.194849	+	0.980833i	$0.437578\pi$
$937$	−19.4831	+	47.0363i	−0.636484	+	1.53661i	−0.831333	+	0.555775i	$0.812422\pi$
$938$		0			0
$939$	5.66270	+	5.66270i	0.184795	+	0.184795i
$940$		0			0
$941$		0			0		−0.555570	−	0.831470i	$-0.687500\pi$
$941$		0			0		0.555570	+	0.831470i	$0.312500\pi$
$942$		0			0
$943$		0			0
$944$	16.4853	+	39.7990i	0.536550	+	1.29535i
$945$		0			0
$946$	−77.9893	+	52.1108i	−2.53565	+	1.69427i
$947$	18.5755	−	3.69489i	0.603622	−	0.120068i	0.116187	−	0.993227i	$-0.462933\pi$
$947$	18.5755	−	3.69489i	0.603622	−	0.120068i	0.487435	+	0.873160i	$0.337933\pi$
$948$		0			0
$949$		0			0
$950$	48.4357	+	20.0627i	1.57146	+	0.650921i
$951$		0			0
$952$		0			0
$953$	57.8827			1.87501			0.937503	−	0.347978i	$-0.113132\pi$
$953$	57.8827			1.87501			0.937503	+	0.347978i	$0.113132\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	11.8632	+	28.6403i	0.382683	+	0.923880i
$962$		0			0
$963$	−50.3281	+	33.6282i	−1.62180	+	1.08365i
$964$	23.9113	−	4.75625i	0.770131	−	0.153189i
$965$		0			0
$966$		0			0
$967$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$967$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$968$		−	60.3054i		−	1.93829i
$969$	−65.1124	−	40.7877i	−2.09171	−	1.31029i
$970$		0			0
$971$	−3.09188	+	7.46447i	−0.0992233	+	0.239546i	−0.965694	−	0.259681i	$-0.916383\pi$
$971$	−3.09188	+	7.46447i	−0.0992233	+	0.239546i	0.866471	+	0.499227i	$0.166383\pi$
$972$	−24.8537	+	37.1962i	−0.797182	+	1.19307i
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	50.9914	−	21.1213i	1.63136	−	0.675731i	0.635975	−	0.771709i	$-0.280598\pi$
$977$	50.9914	−	21.1213i	1.63136	−	0.675731i	0.995383	+	0.0959785i	$0.0305980\pi$
$978$	−34.4911	−	83.2689i	−1.10290	−	2.66265i
$979$	67.5499	+	13.4365i	2.15891	+	0.429433i
$980$		0			0
$981$		0			0
$982$	19.6985	−	19.6985i	0.628604	−	0.628604i
$983$		0			0		−0.831470	−	0.555570i	$-0.812500\pi$
$983$		0			0		0.831470	+	0.555570i	$0.187500\pi$
$984$	−8.51668	−	3.52772i	−0.271502	−	0.112460i
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0		−0.195090	−	0.980785i	$-0.562500\pi$
$991$		0			0		0.195090	+	0.980785i	$0.437500\pi$
$992$		0			0
$993$	7.00165	−	35.1996i	0.222190	−	1.11703i
$994$		0			0
$995$		0			0
$996$	76.6512	+	15.2469i	2.42879	+	0.483116i
$997$		0			0		0.831470	−	0.555570i	$-0.187500\pi$
$997$		0			0		−0.831470	+	0.555570i	$0.812500\pi$
$998$	16.5478	−	3.29156i	0.523810	−	0.104192i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	136.2.s.a.75.1	✓	8
4.3	odd	2		544.2.cc.b.143.1		8
8.3	odd	2	CM	136.2.s.a.75.1	✓	8
8.5	even	2		544.2.cc.b.143.1		8
17.5	odd	16	inner	136.2.s.a.107.1	yes	8
68.39	even	16		544.2.cc.b.175.1		8
136.5	odd	16		544.2.cc.b.175.1		8
136.107	even	16	inner	136.2.s.a.107.1	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
136.2.s.a.75.1	✓	8	1.1	even	1	trivial
136.2.s.a.75.1	✓	8	8.3	odd	2	CM
136.2.s.a.107.1	yes	8	17.5	odd	16	inner
136.2.s.a.107.1	yes	8	136.107	even	16	inner
544.2.cc.b.143.1		8	4.3	odd	2
544.2.cc.b.143.1		8	8.5	even	2
544.2.cc.b.175.1		8	68.39	even	16
544.2.cc.b.175.1		8	136.5	odd	16

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:	\( N \)	\(=\)	\( 136 = 2^{3} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	136.s (of order \(16\), degree \(8\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.541196 + 1.30656i) q^{2} +(1.39635 - 2.08979i) q^{3} +(-1.41421 - 1.41421i) q^{4} +(1.97474 + 2.95541i) q^{6} +(2.61313 - 1.08239i) q^{8} +(-1.26937 - 3.06453i) q^{9} +(4.72705 - 3.15851i) q^{11} +(-4.93015 + 0.980668i) q^{12} +4.00000i q^{16} +(-3.76118 + 1.68925i) q^{17} +4.69098 q^{18} +(-2.83730 + 6.84984i) q^{19} +(1.56854 + 7.88556i) q^{22} +(1.38687 - 6.97229i) q^{24} +(-4.61940 + 1.91342i) q^{25} +(-0.781491 - 0.155448i) q^{27} +(-5.22625 - 2.16478i) q^{32} -14.2889i q^{33} +(-0.171573 - 5.82843i) q^{34} +(-2.53874 + 6.12906i) q^{36} +(-7.41421 - 7.41421i) q^{38} +(0.252982 - 1.27183i) q^{41} +(4.46447 + 10.7782i) q^{43} +(-11.1519 - 2.21824i) q^{44} +(8.35916 + 5.58541i) q^{48} +(-6.46716 - 2.67878i) q^{49} -7.07107i q^{50} +(-1.72176 + 10.2189i) q^{51} +(0.626043 - 0.936940i) q^{54} +(10.3529 + 15.4942i) q^{57} +(9.94975 - 4.12132i) q^{59} +(5.65685 - 5.65685i) q^{64} +(18.6694 + 7.73312i) q^{66} -16.1815i q^{67} +(7.70806 + 2.93015i) q^{68} +(-6.63405 - 6.63405i) q^{72} +(2.96960 + 14.9292i) q^{73} +(-2.45167 + 12.3254i) q^{75} +(13.6997 - 5.67459i) q^{76} +(5.62038 - 5.62038i) q^{81} +(1.52481 + 1.01884i) q^{82} +(-14.3640 - 5.94975i) q^{83} -16.4985 q^{86} +(8.93362 - 13.3701i) q^{88} +(8.56628 + 8.56628i) q^{89} +(-11.8216 + 7.89897i) q^{96} +(-14.4112 + 2.86657i) q^{97} +(7.00000 - 7.00000i) q^{98} +(-15.6797 - 10.4769i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{3} + 8 q^{6} + 16 q^{9} - 16 q^{12} - 24 q^{22} + 32 q^{24} - 32 q^{27} - 24 q^{34} + 32 q^{36} - 48 q^{38} - 32 q^{41} + 64 q^{43} - 16 q^{44} + 32 q^{48} - 40 q^{51} + 40 q^{54} - 40 q^{57}+ \cdots - 48 q^{99}+O(q^{100}) \) 8 * q - 8 * q^3 + 8 * q^6 + 16 * q^9 - 16 * q^12 - 24 * q^22 + 32 * q^24 - 32 * q^27 - 24 * q^34 + 32 * q^36 - 48 * q^38 - 32 * q^41 + 64 * q^43 - 16 * q^44 + 32 * q^48 - 40 * q^51 + 40 * q^54 - 40 * q^57 + 40 * q^59 + 72 * q^66 - 48 * q^72 + 48 * q^73 + 56 * q^81 - 64 * q^83 - 32 * q^96 + 56 * q^98 - 48 * q^99

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