LMFDB - Embedded newform 336.2.k.c.209.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [336,2,Mod(209,336)] 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("336.209"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(336, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 336 = 2^{4} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	336.k (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

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

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

Self dual:	no
Analytic conductor:	$2.68297350792$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.342102016.5
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + x^{6} + 4x^{4} + 4x^{2} + 16 $ x^8 + x^6 + 4x^4 + 4x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	no (minimal twist has level 168)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			209.4
Root			$-1.17915 - 0.780776i$ of defining polynomial
Character	$\chi$	$=$	336.209
Dual form			336.2.k.c.209.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-0.848071 + 1.51022i) q^{3} +3.02045 q^{5} +(2.56155 + 0.662153i) q^{7} +(-1.56155 - 2.56155i) q^{9} -3.12311i q^{11} +1.69614i q^{13} +(-2.56155 + 4.56155i) q^{15} +1.32431 q^{17} +6.41273i q^{19} +(-3.17238 + 3.30697i) q^{21} -2.00000i q^{23} +4.12311 q^{25} +(5.19283 - 0.185917i) q^{27} +9.12311i q^{29} -7.36520i q^{31} +(4.71659 + 2.64861i) q^{33} +(7.73704 + 2.00000i) q^{35} -2.00000 q^{37} +(-2.56155 - 1.43845i) q^{39} -10.7575 q^{41} +4.00000 q^{43} +(-4.71659 - 7.73704i) q^{45} -8.68951 q^{47} +(6.12311 + 3.39228i) q^{49} +(-1.12311 + 2.00000i) q^{51} -9.12311i q^{53} -9.43318i q^{55} +(-9.68466 - 5.43845i) q^{57} -5.08842 q^{59} -5.08842i q^{61} +(-2.30386 - 7.59554i) q^{63} +5.12311i q^{65} +6.24621 q^{67} +(3.02045 + 1.69614i) q^{69} -4.87689i q^{71} -12.0818i q^{73} +(-3.49668 + 6.22681i) q^{75} +(2.06798 - 8.00000i) q^{77} -2.87689 q^{79} +(-4.12311 + 8.00000i) q^{81} -1.69614 q^{83} +4.00000 q^{85} +(-13.7779 - 7.73704i) q^{87} +11.5012 q^{89} +(-1.12311 + 4.34475i) q^{91} +(11.1231 + 6.24621i) q^{93} +19.3693i q^{95} -8.68951i q^{97} +(-8.00000 + 4.87689i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 4 q^{7} + 4 q^{9} - 4 q^{15} - 8 q^{21} - 16 q^{37} - 4 q^{39} + 32 q^{43} + 16 q^{49} + 24 q^{51} - 28 q^{57} - 32 q^{63} - 16 q^{67} - 56 q^{79} + 32 q^{85} + 24 q^{91} + 56 q^{93} - 64 q^{99}+O(q^{100}) $ 8 * q + 4 * q^7 + 4 * q^9 - 4 * q^15 - 8 * q^21 - 16 * q^37 - 4 * q^39 + 32 * q^43 + 16 * q^49 + 24 * q^51 - 28 * q^57 - 32 * q^63 - 16 * q^67 - 56 * q^79 + 32 * q^85 + 24 * q^91 + 56 * q^93 - 64 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/336\mathbb{Z}\right)^\times$.

$n$	$85$	$113$	$127$	$241$
$\chi(n)$	$1$	$-1$	$1$	$-1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	−0.848071	+	1.51022i	−0.489634	+	0.871928i
$3$	−0.848071	+	1.51022i	−0.489634	+	0.871928i
$4$		0			0
$5$	3.02045			1.35079			0.675393	−	0.737458i	$-0.263974\pi$
$5$	3.02045			1.35079			0.675393	+	0.737458i	$0.263974\pi$
$6$		0			0
$7$	2.56155	+	0.662153i	0.968176	+	0.250270i
$7$	2.56155	+	0.662153i	0.968176	+	0.250270i
$8$		0			0
$9$	−1.56155	−	2.56155i	−0.520518	−	0.853851i
$10$		0			0
$11$		−	3.12311i		−	0.941652i	−0.882226	−	0.470826i	$-0.843956\pi$
$11$		−	3.12311i		−	0.941652i	0.882226	−	0.470826i	$-0.156044\pi$
$12$		0			0
$13$			1.69614i			0.470425i	0.971944	+	0.235212i	$0.0755786\pi$
$13$			1.69614i			0.470425i	−0.971944	+	0.235212i	$0.924421\pi$
$14$		0			0
$15$	−2.56155	+	4.56155i	−0.661390	+	1.17779i
$16$		0			0
$17$	1.32431			0.321192			0.160596	−	0.987020i	$-0.448658\pi$
$17$	1.32431			0.321192			0.160596	+	0.987020i	$0.448658\pi$
$18$		0			0
$19$			6.41273i			1.47118i	0.677426	+	0.735591i	$0.263095\pi$
$19$			6.41273i			1.47118i	−0.677426	+	0.735591i	$0.736905\pi$
$20$		0			0
$21$	−3.17238	+	3.30697i	−0.692270	+	0.721639i
$22$		0			0
$23$		−	2.00000i		−	0.417029i	−0.978019	−	0.208514i	$-0.933137\pi$
$23$		−	2.00000i		−	0.417029i	0.978019	−	0.208514i	$-0.0668628\pi$
$24$		0			0
$25$	4.12311			0.824621
$26$		0			0
$27$	5.19283	−	0.185917i	0.999360	−	0.0357798i
$28$		0			0
$29$			9.12311i			1.69412i	0.531499	+	0.847059i	$0.321629\pi$
$29$			9.12311i			1.69412i	−0.531499	+	0.847059i	$0.678371\pi$
$30$		0			0
$31$		−	7.36520i		−	1.32283i	−0.750020	−	0.661415i	$-0.769956\pi$
$31$		−	7.36520i		−	1.32283i	0.750020	−	0.661415i	$-0.230044\pi$
$32$		0			0
$33$	4.71659	+	2.64861i	0.821053	+	0.461064i
$34$		0			0
$35$	7.73704	+	2.00000i	1.30780	+	0.338062i
$36$		0			0
$37$	−2.00000			−0.328798			−0.164399	−	0.986394i	$-0.552568\pi$
$37$	−2.00000			−0.328798			−0.164399	+	0.986394i	$0.552568\pi$
$38$		0			0
$39$	−2.56155	−	1.43845i	−0.410177	−	0.230336i
$40$		0			0
$41$	−10.7575			−1.68004			−0.840018	−	0.542558i	$-0.817456\pi$
$41$	−10.7575			−1.68004			−0.840018	+	0.542558i	$0.817456\pi$
$42$		0			0
$43$	4.00000			0.609994			0.304997	−	0.952353i	$-0.401344\pi$
$43$	4.00000			0.609994			0.304997	+	0.952353i	$0.401344\pi$
$44$		0			0
$45$	−4.71659	−	7.73704i	−0.703108	−	1.15337i
$46$		0			0
$47$	−8.68951			−1.26750			−0.633748	−	0.773540i	$-0.718484\pi$
$47$	−8.68951			−1.26750			−0.633748	+	0.773540i	$0.718484\pi$
$48$		0			0
$49$	6.12311	+	3.39228i	0.874729	+	0.484612i
$50$		0			0
$51$	−1.12311	+	2.00000i	−0.157266	+	0.280056i
$52$		0			0
$53$		−	9.12311i		−	1.25315i	−0.779359	−	0.626577i	$-0.784455\pi$
$53$		−	9.12311i		−	1.25315i	0.779359	−	0.626577i	$-0.215545\pi$
$54$		0			0
$55$		−	9.43318i		−	1.27197i
$56$		0			0
$57$	−9.68466	−	5.43845i	−1.28276	−	0.720340i
$58$		0			0
$59$	−5.08842			−0.662456			−0.331228	−	0.943551i	$-0.607463\pi$
$59$	−5.08842			−0.662456			−0.331228	+	0.943551i	$0.607463\pi$
$60$		0			0
$61$		−	5.08842i		−	0.651506i	−0.945455	−	0.325753i	$-0.894382\pi$
$61$		−	5.08842i		−	0.651506i	0.945455	−	0.325753i	$-0.105618\pi$
$62$		0			0
$63$	−2.30386	−	7.59554i	−0.290259	−	0.956948i
$64$		0			0
$65$			5.12311i			0.635443i
$66$		0			0
$67$	6.24621			0.763096			0.381548	−	0.924349i	$-0.375391\pi$
$67$	6.24621			0.763096			0.381548	+	0.924349i	$0.375391\pi$
$68$		0			0
$69$	3.02045	+	1.69614i	0.363619	+	0.204191i
$70$		0			0
$71$		−	4.87689i		−	0.578781i	−0.957211	−	0.289390i	$-0.906547\pi$
$71$		−	4.87689i		−	0.578781i	0.957211	−	0.289390i	$-0.0934526\pi$
$72$		0			0
$73$		−	12.0818i		−	1.41407i	−0.707180	−	0.707033i	$-0.750033\pi$
$73$		−	12.0818i		−	1.41407i	0.707180	−	0.707033i	$-0.249967\pi$
$74$		0			0
$75$	−3.49668	+	6.22681i	−0.403762	+	0.719010i
$76$		0			0
$77$	2.06798	−	8.00000i	0.235668	−	0.911685i
$78$		0			0
$79$	−2.87689			−0.323676			−0.161838	−	0.986817i	$-0.551742\pi$
$79$	−2.87689			−0.323676			−0.161838	+	0.986817i	$0.551742\pi$
$80$		0			0
$81$	−4.12311	+	8.00000i	−0.458123	+	0.888889i
$82$		0			0
$83$	−1.69614			−0.186176			−0.0930878	−	0.995658i	$-0.529674\pi$
$83$	−1.69614			−0.186176			−0.0930878	+	0.995658i	$0.529674\pi$
$84$		0			0
$85$	4.00000			0.433861
$86$		0			0
$87$	−13.7779	−	7.73704i	−1.47715	−	0.829497i
$88$		0			0
$89$	11.5012			1.21912			0.609560	−	0.792740i	$-0.291346\pi$
$89$	11.5012			1.21912			0.609560	+	0.792740i	$0.291346\pi$
$90$		0			0
$91$	−1.12311	+	4.34475i	−0.117733	+	0.455454i
$92$		0			0
$93$	11.1231	+	6.24621i	1.15341	+	0.647702i
$94$		0			0
$95$			19.3693i			1.98725i
$96$		0			0
$97$		−	8.68951i		−	0.882286i	−0.897437	−	0.441143i	$-0.854573\pi$
$97$		−	8.68951i		−	0.882286i	0.897437	−	0.441143i	$-0.145427\pi$
$98$		0			0
$99$	−8.00000	+	4.87689i	−0.804030	+	0.490146i
$100$		0			0
$101$	−3.76412			−0.374544			−0.187272	−	0.982308i	$-0.559965\pi$
$101$	−3.76412			−0.374544			−0.187272	+	0.982308i	$0.559965\pi$
$102$		0			0
$103$			4.71659i			0.464739i	0.972628	+	0.232370i	$0.0746479\pi$
$103$			4.71659i			0.464739i	−0.972628	+	0.232370i	$0.925352\pi$
$104$		0			0
$105$	−9.58200	+	9.98852i	−0.935108	+	0.974779i
$106$		0			0
$107$			7.12311i			0.688617i	0.938857	+	0.344308i	$0.111887\pi$
$107$			7.12311i			0.688617i	−0.938857	+	0.344308i	$0.888113\pi$
$108$		0			0
$109$	−12.2462			−1.17297			−0.586487	−	0.809959i	$-0.699490\pi$
$109$	−12.2462			−1.17297			−0.586487	+	0.809959i	$0.699490\pi$
$110$		0			0
$111$	1.69614	−	3.02045i	0.160991	−	0.286688i
$112$		0			0
$113$		−	7.36932i		−	0.693247i	−0.938004	−	0.346624i	$-0.887328\pi$
$113$		−	7.36932i		−	0.693247i	0.938004	−	0.346624i	$-0.112672\pi$
$114$		0			0
$115$		−	6.04090i		−	0.563316i
$116$		0			0
$117$	4.34475	−	2.64861i	0.401673	−	0.244864i
$118$		0			0
$119$	3.39228	+	0.876894i	0.310970	+	0.0803848i
$120$		0			0
$121$	1.24621			0.113292
$122$		0			0
$123$	9.12311	−	16.2462i	0.822603	−	1.46487i
$124$		0			0
$125$	−2.64861			−0.236899
$126$		0			0
$127$		0			0			−	1.00000i	$-0.5\pi$
$127$		0			0				1.00000i	$0.5\pi$
$128$		0			0
$129$	−3.39228	+	6.04090i	−0.298674	+	0.531871i
$130$		0			0
$131$	13.7779			1.20378			0.601892	−	0.798578i	$-0.294414\pi$
$131$	13.7779			1.20378			0.601892	+	0.798578i	$0.294414\pi$
$132$		0			0
$133$	−4.24621	+	16.4265i	−0.368193	+	1.42436i
$134$		0			0
$135$	15.6847	−	0.561553i	1.34992	−	0.0483308i
$136$		0			0
$137$			2.24621i			0.191907i	0.995386	+	0.0959534i	$0.0305900\pi$
$137$			2.24621i			0.191907i	−0.995386	+	0.0959534i	$0.969410\pi$
$138$		0			0
$139$			6.41273i			0.543921i	0.962308	+	0.271960i	$0.0876720\pi$
$139$			6.41273i			0.543921i	−0.962308	+	0.271960i	$0.912328\pi$
$140$		0			0
$141$	7.36932	−	13.1231i	0.620608	−	1.10516i
$142$		0			0
$143$	5.29723			0.442976
$144$		0			0
$145$			27.5559i			2.28839i
$146$		0			0
$147$	−10.3159	+	6.37037i	−0.850844	+	0.525419i
$148$		0			0
$149$		−	3.36932i		−	0.276025i	−0.990430	−	0.138013i	$-0.955929\pi$
$149$		−	3.36932i		−	0.276025i	0.990430	−	0.138013i	$-0.0440714\pi$
$150$		0			0
$151$	−12.4924			−1.01662			−0.508309	−	0.861174i	$-0.669729\pi$
$151$	−12.4924			−1.01662			−0.508309	+	0.861174i	$0.669729\pi$
$152$		0			0
$153$	−2.06798	−	3.39228i	−0.167186	−	0.274250i
$154$		0			0
$155$		−	22.2462i		−	1.78686i
$156$		0			0
$157$		−	13.7779i		−	1.09960i	−0.835297	−	0.549799i	$-0.814704\pi$
$157$		−	13.7779i		−	1.09960i	0.835297	−	0.549799i	$-0.185296\pi$
$158$		0			0
$159$	13.7779	+	7.73704i	1.09266	+	0.613587i
$160$		0			0
$161$	1.32431	−	5.12311i	0.104370	−	0.403757i
$162$		0			0
$163$	−14.2462			−1.11585			−0.557925	−	0.829892i	$-0.688402\pi$
$163$	−14.2462			−1.11585			−0.557925	+	0.829892i	$0.688402\pi$
$164$		0			0
$165$	14.2462	+	8.00000i	1.10907	+	0.622799i
$166$		0			0
$167$	−3.39228			−0.262503			−0.131251	−	0.991349i	$-0.541899\pi$
$167$	−3.39228			−0.262503			−0.131251	+	0.991349i	$0.541899\pi$
$168$		0			0
$169$	10.1231			0.778700
$170$		0			0
$171$	16.4265	−	10.0138i	1.25617	−	0.765776i
$172$		0			0
$173$	−15.8459			−1.20474			−0.602371	−	0.798216i	$-0.705777\pi$
$173$	−15.8459			−1.20474			−0.602371	+	0.798216i	$0.705777\pi$
$174$		0			0
$175$	10.5616	+	2.73013i	0.798378	+	0.206378i
$176$		0			0
$177$	4.31534	−	7.68466i	0.324361	−	0.577614i
$178$		0			0
$179$			4.87689i			0.364516i	0.983251	+	0.182258i	$0.0583406\pi$
$179$			4.87689i			0.364516i	−0.983251	+	0.182258i	$0.941659\pi$
$180$		0			0
$181$			15.6829i			1.16570i	0.812580	+	0.582850i	$0.198062\pi$
$181$			15.6829i			1.16570i	−0.812580	+	0.582850i	$0.801938\pi$
$182$		0			0
$183$	7.68466	+	4.31534i	0.568066	+	0.318999i
$184$		0			0
$185$	−6.04090			−0.444135
$186$		0			0
$187$		−	4.13595i		−	0.302451i
$188$		0			0
$189$	13.4248	+	2.96221i	0.976511	+	0.215469i
$190$		0			0
$191$		−	17.3693i		−	1.25680i	−0.777891	−	0.628400i	$-0.783710\pi$
$191$		−	17.3693i		−	1.25680i	0.777891	−	0.628400i	$-0.216290\pi$
$192$		0			0
$193$	7.12311			0.512732			0.256366	−	0.966580i	$-0.417475\pi$
$193$	7.12311			0.512732			0.256366	+	0.966580i	$0.417475\pi$
$194$		0			0
$195$	−7.73704	−	4.34475i	−0.554061	−	0.311134i
$196$		0			0
$197$			17.1231i			1.21997i	0.792413	+	0.609985i	$0.208825\pi$
$197$			17.1231i			1.21997i	−0.792413	+	0.609985i	$0.791175\pi$
$198$		0			0
$199$			10.0138i			0.709861i	0.934893	+	0.354930i	$0.115495\pi$
$199$			10.0138i			0.709861i	−0.934893	+	0.354930i	$0.884505\pi$
$200$		0			0
$201$	−5.29723	+	9.43318i	−0.373638	+	0.665365i
$202$		0			0
$203$	−6.04090	+	23.3693i	−0.423988	+	1.64020i
$204$		0			0
$205$	−32.4924			−2.26937
$206$		0			0
$207$	−5.12311	+	3.12311i	−0.356080	+	0.217071i
$208$		0			0
$209$	20.0276			1.38534
$210$		0			0
$211$	22.2462			1.53149			0.765746	−	0.643143i	$-0.222370\pi$
$211$	22.2462			1.53149			0.765746	+	0.643143i	$0.222370\pi$
$212$		0			0
$213$	7.36520	+	4.13595i	0.504655	+	0.283391i
$214$		0			0
$215$	12.0818			0.823971
$216$		0			0
$217$	4.87689	−	18.8664i	0.331065	−	1.28073i
$218$		0			0
$219$	18.2462	+	10.2462i	1.23296	+	0.692375i
$220$		0			0
$221$			2.24621i			0.151097i
$222$		0			0
$223$			20.1907i			1.35207i	0.736871	+	0.676033i	$0.236303\pi$
$223$			20.1907i			1.35207i	−0.736871	+	0.676033i	$0.763697\pi$
$224$		0			0
$225$	−6.43845	−	10.5616i	−0.429230	−	0.704104i
$226$		0			0
$227$	−19.0752			−1.26606			−0.633031	−	0.774126i	$-0.718190\pi$
$227$	−19.0752			−1.26606			−0.633031	+	0.774126i	$0.718190\pi$
$228$		0			0
$229$		−	3.60109i		−	0.237966i	−0.992896	−	0.118983i	$-0.962037\pi$
$229$		−	3.60109i		−	0.237966i	0.992896	−	0.118983i	$-0.0379635\pi$
$230$		0			0
$231$	10.3280	+	9.90767i	0.679533	+	0.651877i
$232$		0			0
$233$		−	8.00000i		−	0.524097i	−0.965055	−	0.262049i	$-0.915602\pi$
$233$		−	8.00000i		−	0.524097i	0.965055	−	0.262049i	$-0.0843981\pi$
$234$		0			0
$235$	−26.2462			−1.71211
$236$		0			0
$237$	2.43981	−	4.34475i	0.158483	−	0.282222i
$238$		0			0
$239$		−	4.24621i		−	0.274665i	−0.990525	−	0.137332i	$-0.956147\pi$
$239$		−	4.24621i		−	0.274665i	0.990525	−	0.137332i	$-0.0438528\pi$
$240$		0			0
$241$			15.4741i			0.996773i	0.866955	+	0.498386i	$0.166074\pi$
$241$			15.4741i			0.996773i	−0.866955	+	0.498386i	$0.833926\pi$
$242$		0			0
$243$	−8.58511	−	13.0114i	−0.550735	−	0.834680i
$244$		0			0
$245$	18.4945	+	10.2462i	1.18157	+	0.654606i
$246$		0			0
$247$	−10.8769			−0.692080
$248$		0			0
$249$	1.43845	−	2.56155i	0.0911579	−	0.162332i
$250$		0			0
$251$	−6.99337			−0.441418			−0.220709	−	0.975340i	$-0.570837\pi$
$251$	−6.99337			−0.441418			−0.220709	+	0.975340i	$0.570837\pi$
$252$		0			0
$253$	−6.24621			−0.392696
$254$		0			0
$255$	−3.39228	+	6.04090i	−0.212433	+	0.378296i
$256$		0			0
$257$	−0.580639			−0.0362192			−0.0181096	−	0.999836i	$-0.505765\pi$
$257$	−0.580639			−0.0362192			−0.0181096	+	0.999836i	$0.505765\pi$
$258$		0			0
$259$	−5.12311	−	1.32431i	−0.318334	−	0.0822884i
$260$		0			0
$261$	23.3693	−	14.2462i	1.44652	−	0.881818i
$262$		0			0
$263$		−	9.36932i		−	0.577737i	−0.957369	−	0.288868i	$-0.906721\pi$
$263$		−	9.36932i		−	0.577737i	0.957369	−	0.288868i	$-0.0932790\pi$
$264$		0			0
$265$		−	27.5559i		−	1.69274i
$266$		0			0
$267$	−9.75379	+	17.3693i	−0.596922	+	1.06298i
$268$		0			0
$269$	−9.06134			−0.552480			−0.276240	−	0.961089i	$-0.589088\pi$
$269$	−9.06134			−0.552480			−0.276240	+	0.961089i	$0.589088\pi$
$270$		0			0
$271$		−	5.46026i		−	0.331687i	−0.986152	−	0.165844i	$-0.946965\pi$
$271$		−	5.46026i		−	0.331687i	0.986152	−	0.165844i	$-0.0530347\pi$
$272$		0			0
$273$	−5.60908	−	5.38080i	−0.339477	−	0.325661i
$274$		0			0
$275$		−	12.8769i		−	0.776506i
$276$		0			0
$277$	−12.2462			−0.735804			−0.367902	−	0.929865i	$-0.619924\pi$
$277$	−12.2462			−0.735804			−0.367902	+	0.929865i	$0.619924\pi$
$278$		0			0
$279$	−18.8664	+	11.5012i	−1.12950	+	0.688556i
$280$		0			0
$281$			28.4924i			1.69972i	0.527012	+	0.849858i	$0.323312\pi$
$281$			28.4924i			1.69972i	−0.527012	+	0.849858i	$0.676688\pi$
$282$		0			0
$283$		−	12.4536i		−	0.740291i	−0.928974	−	0.370146i	$-0.879308\pi$
$283$		−	12.4536i		−	0.740291i	0.928974	−	0.370146i	$-0.120692\pi$
$284$		0			0
$285$	−29.2520	−	16.4265i	−1.73274	−	0.973025i
$286$		0			0
$287$	−27.5559	−	7.12311i	−1.62657	−	0.420464i
$288$		0			0
$289$	−15.2462			−0.896836
$290$		0			0
$291$	13.1231	+	7.36932i	0.769290	+	0.431997i
$292$		0			0
$293$	25.2791			1.47682			0.738410	−	0.674352i	$-0.235577\pi$
$293$	25.2791			1.47682			0.738410	+	0.674352i	$0.235577\pi$
$294$		0			0
$295$	−15.3693			−0.894836
$296$		0			0
$297$	−0.580639	−	16.2177i	−0.0336921	−	0.941049i
$298$		0			0
$299$	3.39228			0.196181
$300$		0			0
$301$	10.2462	+	2.64861i	0.590582	+	0.152664i
$302$		0			0
$303$	3.19224	−	5.68466i	0.183389	−	0.326575i
$304$		0			0
$305$		−	15.3693i		−	0.880045i
$306$		0			0
$307$		−	24.5354i		−	1.40031i	−0.713991	−	0.700155i	$-0.753114\pi$
$307$		−	24.5354i		−	1.40031i	0.713991	−	0.700155i	$-0.246886\pi$
$308$		0			0
$309$	−7.12311	−	4.00000i	−0.405219	−	0.227552i
$310$		0			0
$311$	18.8664			1.06981			0.534906	−	0.844911i	$-0.320347\pi$
$311$	18.8664			1.06981			0.534906	+	0.844911i	$0.320347\pi$
$312$		0			0
$313$			3.39228i			0.191743i	0.995394	+	0.0958716i	$0.0305638\pi$
$313$			3.39228i			0.191743i	−0.995394	+	0.0958716i	$0.969436\pi$
$314$		0			0
$315$	−6.95869	−	22.9419i	−0.392078	−	1.29263i
$316$		0			0
$317$			13.6155i			0.764724i	0.924013	+	0.382362i	$0.124889\pi$
$317$			13.6155i			0.764724i	−0.924013	+	0.382362i	$0.875111\pi$
$318$		0			0
$319$	28.4924			1.59527
$320$		0			0
$321$	−10.7575	−	6.04090i	−0.600424	−	0.337170i
$322$		0			0
$323$			8.49242i			0.472531i
$324$		0			0
$325$			6.99337i			0.387922i
$326$		0			0
$327$	10.3857	−	18.4945i	0.574328	−	1.02275i
$328$		0			0
$329$	−22.2586	−	5.75379i	−1.22716	−	0.317217i
$330$		0			0
$331$	−22.2462			−1.22276			−0.611381	−	0.791336i	$-0.709386\pi$
$331$	−22.2462			−1.22276			−0.611381	+	0.791336i	$0.709386\pi$
$332$		0			0
$333$	3.12311	+	5.12311i	0.171145	+	0.280744i
$334$		0			0
$335$	18.8664			1.03078
$336$		0			0
$337$	25.3693			1.38195			0.690977	−	0.722876i	$-0.257180\pi$
$337$	25.3693			1.38195			0.690977	+	0.722876i	$0.257180\pi$
$338$		0			0
$339$	11.1293	+	6.24970i	0.604462	+	0.339437i
$340$		0			0
$341$	−23.0023			−1.24564
$342$		0			0
$343$	13.4384	+	12.7439i	0.725608	+	0.688108i
$344$		0			0
$345$	9.12311	+	5.12311i	0.491171	+	0.275819i
$346$		0			0
$347$		−	8.87689i		−	0.476537i	−0.971199	−	0.238268i	$-0.923420\pi$
$347$		−	8.87689i		−	0.476537i	0.971199	−	0.238268i	$-0.0765798\pi$
$348$		0			0
$349$			3.60109i			0.192762i	0.995345	+	0.0963809i	$0.0307267\pi$
$349$			3.60109i			0.192762i	−0.995345	+	0.0963809i	$0.969273\pi$
$350$		0			0
$351$	0.315342	+	8.80776i	0.0168317	+	0.470124i
$352$		0			0
$353$	−0.580639			−0.0309043			−0.0154521	−	0.999881i	$-0.504919\pi$
$353$	−0.580639			−0.0309043			−0.0154521	+	0.999881i	$0.504919\pi$
$354$		0			0
$355$		−	14.7304i		−	0.781809i
$356$		0			0
$357$	−4.20120	+	4.37944i	−0.222351	+	0.231784i
$358$		0			0
$359$			24.2462i			1.27967i	0.768514	+	0.639833i	$0.220997\pi$
$359$			24.2462i			1.27967i	−0.768514	+	0.639833i	$0.779003\pi$
$360$		0			0
$361$	−22.1231			−1.16437
$362$		0			0
$363$	−1.05688	+	1.88206i	−0.0554716	+	0.0987824i
$364$		0			0
$365$		−	36.4924i		−	1.91010i
$366$		0			0
$367$		−	3.97292i		−	0.207385i	−0.994609	−	0.103692i	$-0.966934\pi$
$367$		−	3.97292i		−	0.207385i	0.994609	−	0.103692i	$-0.0330657\pi$
$368$		0			0
$369$	16.7984	+	27.5559i	0.874489	+	1.43450i
$370$		0			0
$371$	6.04090	−	23.3693i	0.313628	−	1.21327i
$372$		0			0
$373$	10.4924			0.543277			0.271639	−	0.962399i	$-0.412434\pi$
$373$	10.4924			0.543277			0.271639	+	0.962399i	$0.412434\pi$
$374$		0			0
$375$	2.24621	−	4.00000i	0.115994	−	0.206559i
$376$		0			0
$377$	−15.4741			−0.796955
$378$		0			0
$379$	−0.492423			−0.0252940			−0.0126470	−	0.999920i	$-0.504026\pi$
$379$	−0.492423			−0.0252940			−0.0126470	+	0.999920i	$0.504026\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	1.90495			0.0973382			0.0486691	−	0.998815i	$-0.484502\pi$
$383$	1.90495			0.0973382			0.0486691	+	0.998815i	$0.484502\pi$
$384$		0			0
$385$	6.24621	−	24.1636i	0.318336	−	1.23149i
$386$		0			0
$387$	−6.24621	−	10.2462i	−0.317513	−	0.520844i
$388$		0			0
$389$		−	3.36932i		−	0.170831i	−0.996345	−	0.0854156i	$-0.972778\pi$
$389$		−	3.36932i		−	0.170831i	0.996345	−	0.0854156i	$-0.0272218\pi$
$390$		0			0
$391$		−	2.64861i		−	0.133946i
$392$		0			0
$393$	−11.6847	+	20.8078i	−0.589413	+	1.04961i
$394$		0			0
$395$	−8.68951			−0.437217
$396$		0			0
$397$		−	37.9415i		−	1.90423i	−0.305740	−	0.952115i	$-0.598904\pi$
$397$		−	37.9415i		−	1.90423i	0.305740	−	0.952115i	$-0.401096\pi$
$398$		0			0
$399$	−21.2067	−	20.3436i	−1.06166	−	1.01845i
$400$		0			0
$401$			29.1231i			1.45434i	0.686458	+	0.727169i	$0.259164\pi$
$401$			29.1231i			1.45434i	−0.686458	+	0.727169i	$0.740836\pi$
$402$		0			0
$403$	12.4924			0.622292
$404$		0			0
$405$	−12.4536	+	24.1636i	−0.618826	+	1.20070i
$406$		0			0
$407$			6.24621i			0.309613i
$408$		0			0
$409$			34.3404i			1.69802i	0.528373	+	0.849012i	$0.322802\pi$
$409$			34.3404i			1.69802i	−0.528373	+	0.849012i	$0.677198\pi$
$410$		0			0
$411$	−3.39228	−	1.90495i	−0.167329	−	0.0939640i
$412$		0			0
$413$	−13.0343	−	3.36932i	−0.641374	−	0.165793i
$414$		0			0
$415$	−5.12311			−0.251483
$416$		0			0
$417$	−9.68466	−	5.43845i	−0.474260	−	0.266322i
$418$		0			0
$419$	−8.48071			−0.414310			−0.207155	−	0.978308i	$-0.566420\pi$
$419$	−8.48071			−0.414310			−0.207155	+	0.978308i	$0.566420\pi$
$420$		0			0
$421$	−16.7386			−0.815791			−0.407896	−	0.913029i	$-0.633737\pi$
$421$	−16.7386			−0.815791			−0.407896	+	0.913029i	$0.633737\pi$
$422$		0			0
$423$	13.5691	+	22.2586i	0.659754	+	1.08225i
$424$		0			0
$425$	5.46026			0.264861
$426$		0			0
$427$	3.36932	−	13.0343i	0.163053	−	0.630772i
$428$		0			0
$429$	−4.49242	+	8.00000i	−0.216896	+	0.386244i
$430$		0			0
$431$			6.00000i			0.289010i	0.989504	+	0.144505i	$0.0461589\pi$
$431$			6.00000i			0.289010i	−0.989504	+	0.144505i	$0.953841\pi$
$432$		0			0
$433$			15.4741i			0.743637i	0.928306	+	0.371818i	$0.121266\pi$
$433$			15.4741i			0.743637i	−0.928306	+	0.371818i	$0.878734\pi$
$434$		0			0
$435$	−41.6155	−	23.3693i	−1.99531	−	1.12047i
$436$		0			0
$437$	12.8255			0.613525
$438$		0			0
$439$			8.10887i			0.387015i	0.981099	+	0.193508i	$0.0619864\pi$
$439$			8.10887i			0.387015i	−0.981099	+	0.193508i	$0.938014\pi$
$440$		0			0
$441$	−0.872043	−	20.9819i	−0.0415259	−	0.999137i
$442$		0			0
$443$			23.1231i			1.09861i	0.835621	+	0.549306i	$0.185108\pi$
$443$			23.1231i			1.09861i	−0.835621	+	0.549306i	$0.814892\pi$
$444$		0			0
$445$	34.7386			1.64677
$446$		0			0
$447$	5.08842	+	2.85742i	0.240674	+	0.135151i
$448$		0			0
$449$			10.2462i			0.483549i	0.970333	+	0.241774i	$0.0777294\pi$
$449$			10.2462i			0.483549i	−0.970333	+	0.241774i	$0.922271\pi$
$450$		0			0
$451$			33.5968i			1.58201i
$452$		0			0
$453$	10.5945	−	18.8664i	0.497771	−	0.886419i
$454$		0			0
$455$	−3.39228	+	13.1231i	−0.159033	+	0.615221i
$456$		0			0
$457$	11.6155			0.543351			0.271676	−	0.962389i	$-0.412422\pi$
$457$	11.6155			0.543351			0.271676	+	0.962389i	$0.412422\pi$
$458$		0			0
$459$	6.87689	−	0.246211i	0.320986	−	0.0114922i
$460$		0			0
$461$	13.1973			0.614659			0.307330	−	0.951603i	$-0.400565\pi$
$461$	13.1973			0.614659			0.307330	+	0.951603i	$0.400565\pi$
$462$		0			0
$463$	17.6155			0.818663			0.409332	−	0.912386i	$-0.365762\pi$
$463$	17.6155			0.818663			0.409332	+	0.912386i	$0.365762\pi$
$464$		0			0
$465$	33.5968	+	18.8664i	1.55801	+	0.874906i
$466$		0			0
$467$	−8.48071			−0.392440			−0.196220	−	0.980560i	$-0.562867\pi$
$467$	−8.48071			−0.392440			−0.196220	+	0.980560i	$0.562867\pi$
$468$		0			0
$469$	16.0000	+	4.13595i	0.738811	+	0.190980i
$470$		0			0
$471$	20.8078	+	11.6847i	0.958771	+	0.538401i
$472$		0			0
$473$		−	12.4924i		−	0.574402i
$474$		0			0
$475$			26.4404i			1.21317i
$476$		0			0
$477$	−23.3693	+	14.2462i	−1.07001	+	0.652289i
$478$		0			0
$479$	15.4741			0.707028			0.353514	−	0.935429i	$-0.384987\pi$
$479$	15.4741			0.707028			0.353514	+	0.935429i	$0.384987\pi$
$480$		0			0
$481$		−	3.39228i		−	0.154675i
$482$		0			0
$483$	6.61393	+	6.34475i	0.300944	+	0.288696i
$484$		0			0
$485$		−	26.2462i		−	1.19178i
$486$		0			0
$487$	−8.00000			−0.362515			−0.181257	−	0.983436i	$-0.558017\pi$
$487$	−8.00000			−0.362515			−0.181257	+	0.983436i	$0.558017\pi$
$488$		0			0
$489$	12.0818	−	21.5150i	0.546358	−	0.972941i
$490$		0			0
$491$			28.8769i			1.30320i	0.758564	+	0.651598i	$0.225901\pi$
$491$			28.8769i			1.30320i	−0.758564	+	0.651598i	$0.774099\pi$
$492$		0			0
$493$			12.0818i			0.544137i
$494$		0			0
$495$	−24.1636	+	14.7304i	−1.08607	+	0.662082i
$496$		0			0
$497$	3.22925	−	12.4924i	0.144852	−	0.560362i
$498$		0			0
$499$	36.9848			1.65567			0.827835	−	0.560972i	$-0.189573\pi$
$499$	36.9848			1.65567			0.827835	+	0.560972i	$0.189573\pi$
$500$		0			0
$501$	2.87689	−	5.12311i	0.128530	−	0.228883i
$502$		0			0
$503$	5.29723			0.236192			0.118096	−	0.993002i	$-0.462321\pi$
$503$	5.29723			0.236192			0.118096	+	0.993002i	$0.462321\pi$
$504$		0			0
$505$	−11.3693			−0.505928
$506$		0			0
$507$	−8.58511	+	15.2882i	−0.381278	+	0.678971i
$508$		0			0
$509$	25.6967			1.13899			0.569493	−	0.821996i	$-0.307140\pi$
$509$	25.6967			1.13899			0.569493	+	0.821996i	$0.307140\pi$
$510$		0			0
$511$	8.00000	−	30.9481i	0.353899	−	1.36907i
$512$		0			0
$513$	1.19224	+	33.3002i	0.0526385	+	1.47024i
$514$		0			0
$515$			14.2462i			0.627763i
$516$		0			0
$517$			27.1383i			1.19354i
$518$		0			0
$519$	13.4384	−	23.9309i	0.589882	−	1.05045i
$520$		0			0
$521$	26.9752			1.18181			0.590903	−	0.806743i	$-0.298772\pi$
$521$	26.9752			1.18181			0.590903	+	0.806743i	$0.298772\pi$
$522$		0			0
$523$		−	2.27678i		−	0.0995566i	−0.998760	−	0.0497783i	$-0.984149\pi$
$523$		−	2.27678i		−	0.0995566i	0.998760	−	0.0497783i	$-0.0158515\pi$
$524$		0			0
$525$	−13.0800	+	13.6350i	−0.570860	+	0.595079i
$526$		0			0
$527$		−	9.75379i		−	0.424882i
$528$		0			0
$529$	19.0000			0.826087
$530$		0			0
$531$	7.94584	+	13.0343i	0.344820	+	0.565639i
$532$		0			0
$533$		−	18.2462i		−	0.790331i
$534$		0			0
$535$			21.5150i			0.930173i
$536$		0			0
$537$	−7.36520	−	4.13595i	−0.317832	−	0.178479i
$538$		0			0
$539$	10.5945	−	19.1231i	0.456335	−	0.823690i
$540$		0			0
$541$	−4.24621			−0.182559			−0.0912794	−	0.995825i	$-0.529096\pi$
$541$	−4.24621			−0.182559			−0.0912794	+	0.995825i	$0.529096\pi$
$542$		0			0
$543$	−23.6847	−	13.3002i	−1.01641	−	0.570766i
$544$		0			0
$545$	−36.9890			−1.58444
$546$		0			0
$547$	38.2462			1.63529			0.817645	−	0.575723i	$-0.195279\pi$
$547$	38.2462			1.63529			0.817645	+	0.575723i	$0.195279\pi$
$548$		0			0
$549$	−13.0343	+	7.94584i	−0.556289	+	0.339120i
$550$		0			0
$551$	−58.5040			−2.49235
$552$		0			0
$553$	−7.36932	−	1.90495i	−0.313375	−	0.0810065i
$554$		0			0
$555$	5.12311	−	9.12311i	0.217464	−	0.387254i
$556$		0			0
$557$			29.6155i			1.25485i	0.778677	+	0.627425i	$0.215891\pi$
$557$			29.6155i			1.25485i	−0.778677	+	0.627425i	$0.784109\pi$
$558$		0			0
$559$			6.78456i			0.286956i
$560$		0			0
$561$	6.24621	+	3.50758i	0.263715	+	0.148090i
$562$		0			0
$563$	20.5625			0.866606			0.433303	−	0.901248i	$-0.357348\pi$
$563$	20.5625			0.866606			0.433303	+	0.901248i	$0.357348\pi$
$564$		0			0
$565$		−	22.2586i		−	0.936428i
$566$		0			0
$567$	−15.8588	+	17.7623i	−0.666006	+	0.745946i
$568$		0			0
$569$		−	46.1080i		−	1.93295i	−0.256769	−	0.966473i	$-0.582658\pi$
$569$		−	46.1080i		−	1.93295i	0.256769	−	0.966473i	$-0.417342\pi$
$570$		0			0
$571$	−26.7386			−1.11898			−0.559488	−	0.828838i	$-0.689002\pi$
$571$	−26.7386			−1.11898			−0.559488	+	0.828838i	$0.689002\pi$
$572$		0			0
$573$	26.2316	+	14.7304i	1.09584	+	0.615372i
$574$		0			0
$575$		−	8.24621i		−	0.343891i
$576$		0			0
$577$		−	13.5691i		−	0.564890i	−0.959283	−	0.282445i	$-0.908854\pi$
$577$		−	13.5691i		−	0.564890i	0.959283	−	0.282445i	$-0.0911455\pi$
$578$		0			0
$579$	−6.04090	+	10.7575i	−0.251051	+	0.447066i
$580$		0			0
$581$	−4.34475	−	1.12311i	−0.180251	−	0.0465943i
$582$		0			0
$583$	−28.4924			−1.18004
$584$		0			0
$585$	13.1231	−	8.00000i	0.542574	−	0.330759i
$586$		0			0
$587$	48.1184			1.98606			0.993029	−	0.117873i	$-0.0376076\pi$
$587$	48.1184			1.98606			0.993029	+	0.117873i	$0.0376076\pi$
$588$		0			0
$589$	47.2311			1.94612
$590$		0			0
$591$	−25.8597	−	14.5216i	−1.06373	−	0.597339i
$592$		0			0
$593$	16.7984			0.689827			0.344913	−	0.938635i	$-0.387908\pi$
$593$	16.7984			0.689827			0.344913	+	0.938635i	$0.387908\pi$
$594$		0			0
$595$	10.2462	+	2.64861i	0.420054	+	0.108583i
$596$		0			0
$597$	−15.1231	−	8.49242i	−0.618948	−	0.347572i
$598$		0			0
$599$		−	41.3693i		−	1.69030i	−0.534526	−	0.845152i	$-0.679510\pi$
$599$		−	41.3693i		−	1.69030i	0.534526	−	0.845152i	$-0.320490\pi$
$600$		0			0
$601$		−	5.29723i		−	0.216078i	−0.994147	−	0.108039i	$-0.965543\pi$
$601$		−	5.29723i		−	0.216078i	0.994147	−	0.108039i	$-0.0344572\pi$
$602$		0			0
$603$	−9.75379	−	16.0000i	−0.397205	−	0.651570i
$604$		0			0
$605$	3.76412			0.153033
$606$		0			0
$607$		−	14.5674i		−	0.591272i	−0.955301	−	0.295636i	$-0.904469\pi$
$607$		−	14.5674i		−	0.591272i	0.955301	−	0.295636i	$-0.0955315\pi$
$608$		0			0
$609$	−30.1698	−	28.9419i	−1.22254	−	1.17279i
$610$		0			0
$611$		−	14.7386i		−	0.596261i
$612$		0			0
$613$	−5.50758			−0.222449			−0.111224	−	0.993795i	$-0.535477\pi$
$613$	−5.50758			−0.222449			−0.111224	+	0.993795i	$0.535477\pi$
$614$		0			0
$615$	27.5559	−	49.0708i	1.11116	−	1.97873i
$616$		0			0
$617$		−	16.6307i		−	0.669526i	−0.942302	−	0.334763i	$-0.891344\pi$
$617$		−	16.6307i		−	0.669526i	0.942302	−	0.334763i	$-0.108656\pi$
$618$		0			0
$619$			15.1022i			0.607010i	0.952830	+	0.303505i	$0.0981570\pi$
$619$			15.1022i			0.607010i	−0.952830	+	0.303505i	$0.901843\pi$
$620$		0			0
$621$	−0.371834	−	10.3857i	−0.0149212	−	0.416762i
$622$		0			0
$623$	29.4608	+	7.61553i	1.18032	+	0.305110i
$624$		0			0
$625$	−28.6155			−1.14462
$626$		0			0
$627$	−16.9848	+	30.2462i	−0.678309	+	1.20792i
$628$		0			0
$629$	−2.64861			−0.105607
$630$		0			0
$631$	−37.1231			−1.47785			−0.738924	−	0.673789i	$-0.764666\pi$
$631$	−37.1231			−1.47785			−0.738924	+	0.673789i	$0.764666\pi$
$632$		0			0
$633$	−18.8664	+	33.5968i	−0.749870	+	1.33535i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−5.75379	+	10.3857i	−0.227973	+	0.411494i
$638$		0			0
$639$	−12.4924	+	7.61553i	−0.494193	+	0.301266i
$640$		0			0
$641$			27.8617i			1.10047i	0.835009	+	0.550236i	$0.185462\pi$
$641$			27.8617i			1.10047i	−0.835009	+	0.550236i	$0.814538\pi$
$642$		0			0
$643$		−	17.7509i		−	0.700025i	−0.936745	−	0.350013i	$-0.886177\pi$
$643$		−	17.7509i		−	0.700025i	0.936745	−	0.350013i	$-0.113823\pi$
$644$		0			0
$645$	−10.2462	+	18.2462i	−0.403444	+	0.718444i
$646$		0			0
$647$	−7.20217			−0.283147			−0.141573	−	0.989928i	$-0.545216\pi$
$647$	−7.20217			−0.283147			−0.141573	+	0.989928i	$0.545216\pi$
$648$		0			0
$649$			15.8917i			0.623803i
$650$		0			0
$651$	24.3565	+	23.3652i	0.954605	+	0.915754i
$652$		0			0
$653$			35.3693i			1.38411i	0.721846	+	0.692054i	$0.243294\pi$
$653$			35.3693i			1.38411i	−0.721846	+	0.692054i	$0.756706\pi$
$654$		0			0
$655$	41.6155			1.62605
$656$		0			0
$657$	−30.9481	+	18.8664i	−1.20740	+	0.736047i
$658$		0			0
$659$			15.1231i			0.589113i	0.955634	+	0.294556i	$0.0951718\pi$
$659$			15.1231i			0.589113i	−0.955634	+	0.294556i	$0.904828\pi$
$660$		0			0
$661$		−	19.0752i		−	0.741938i	−0.928645	−	0.370969i	$-0.879026\pi$
$661$		−	19.0752i		−	0.741938i	0.928645	−	0.370969i	$-0.120974\pi$
$662$		0			0
$663$	−3.39228	−	1.90495i	−0.131745	−	0.0739820i
$664$		0			0
$665$	−12.8255	+	49.6155i	−0.497350	+	1.92401i
$666$		0			0
$667$	18.2462			0.706496
$668$		0			0
$669$	−30.4924	−	17.1231i	−1.17891	−	0.662018i
$670$		0			0
$671$	−15.8917			−0.613492
$672$		0			0
$673$	14.4924			0.558642			0.279321	−	0.960198i	$-0.409891\pi$
$673$	14.4924			0.558642			0.279321	+	0.960198i	$0.409891\pi$
$674$		0			0
$675$	21.4106	−	0.766556i	0.824093	−	0.0295047i
$676$		0			0
$677$	−43.4018			−1.66807			−0.834033	−	0.551715i	$-0.813974\pi$
$677$	−43.4018			−1.66807			−0.834033	+	0.551715i	$0.813974\pi$
$678$		0			0
$679$	5.75379	−	22.2586i	0.220810	−	0.854208i
$680$		0			0
$681$	16.1771	−	28.8078i	0.619907	−	1.10392i
$682$		0			0
$683$			1.36932i			0.0523955i	0.999657	+	0.0261977i	$0.00833995\pi$
$683$			1.36932i			0.0523955i	−0.999657	+	0.0261977i	$0.991660\pi$
$684$		0			0
$685$			6.78456i			0.259225i
$686$		0			0
$687$	5.43845	+	3.05398i	0.207490	+	0.116516i
$688$		0			0
$689$	15.4741			0.589515
$690$		0			0
$691$			1.11550i			0.0424357i	0.999775	+	0.0212179i	$0.00675436\pi$
$691$			1.11550i			0.0424357i	−0.999775	+	0.0212179i	$0.993246\pi$
$692$		0			0
$693$	−23.7217	+	7.19519i	−0.901112	+	0.273323i
$694$		0			0
$695$			19.3693i			0.734720i
$696$		0			0
$697$	−14.2462			−0.539614
$698$		0			0
$699$	12.0818	+	6.78456i	0.456975	+	0.256616i
$700$		0			0
$701$		−	1.12311i		−	0.0424191i	−0.999775	−	0.0212096i	$-0.993248\pi$
$701$		−	1.12311i		−	0.0424191i	0.999775	−	0.0212096i	$-0.00675172\pi$
$702$		0			0
$703$		−	12.8255i		−	0.483721i
$704$		0			0
$705$	22.2586	−	39.6377i	0.838309	−	1.49284i
$706$		0			0
$707$	−9.64198	−	2.49242i	−0.362624	−	0.0937372i
$708$		0			0
$709$	0.246211			0.00924666			0.00462333	−	0.999989i	$-0.498528\pi$
$709$	0.246211			0.00924666			0.00462333	+	0.999989i	$0.498528\pi$
$710$		0			0
$711$	4.49242	+	7.36932i	0.168479	+	0.276371i
$712$		0			0
$713$	−14.7304			−0.551658
$714$		0			0
$715$	16.0000			0.598366
$716$		0			0
$717$	6.41273	+	3.60109i	0.239488	+	0.134485i
$718$		0			0
$719$	4.87962			0.181979			0.0909895	−	0.995852i	$-0.470997\pi$
$719$	4.87962			0.181979			0.0909895	+	0.995852i	$0.470997\pi$
$720$		0			0
$721$	−3.12311	+	12.0818i	−0.116311	+	0.449949i
$722$		0			0
$723$	−23.3693	−	13.1231i	−0.869114	−	0.488054i
$724$		0			0
$725$			37.6155i			1.39701i
$726$		0			0
$727$		−	46.5853i		−	1.72775i	−0.503705	−	0.863876i	$-0.668030\pi$
$727$		−	46.5853i		−	1.72775i	0.503705	−	0.863876i	$-0.331970\pi$
$728$		0			0
$729$	26.9309	−	1.93087i	0.997440	−	0.0715137i
$730$		0			0
$731$	5.29723			0.195925
$732$		0			0
$733$		−	31.1570i		−	1.15081i	−0.817869	−	0.575404i	$-0.804845\pi$
$733$		−	31.1570i		−	1.15081i	0.817869	−	0.575404i	$-0.195155\pi$
$734$		0			0
$735$	−31.1587	+	19.2414i	−1.14931	+	0.709728i
$736$		0			0
$737$		−	19.5076i		−	0.718571i
$738$		0			0
$739$	−40.4924			−1.48954			−0.744769	−	0.667322i	$-0.767440\pi$
$739$	−40.4924			−1.48954			−0.744769	+	0.667322i	$0.767440\pi$
$740$		0			0
$741$	9.22437	−	16.4265i	0.338866	−	0.603444i
$742$		0			0
$743$			30.0000i			1.10059i	0.834969	+	0.550297i	$0.185485\pi$
$743$			30.0000i			1.10059i	−0.834969	+	0.550297i	$0.814515\pi$
$744$		0			0
$745$		−	10.1768i		−	0.372851i
$746$		0			0
$747$	2.64861	+	4.34475i	0.0969077	+	0.158966i
$748$		0			0
$749$	−4.71659	+	18.2462i	−0.172340	+	0.666702i
$750$		0			0
$751$	−16.0000			−0.583848			−0.291924	−	0.956441i	$-0.594295\pi$
$751$	−16.0000			−0.583848			−0.291924	+	0.956441i	$0.594295\pi$
$752$		0			0
$753$	5.93087	−	10.5616i	0.216133	−	0.384884i
$754$		0			0
$755$	−37.7327			−1.37323
$756$		0			0
$757$	10.4924			0.381354			0.190677	−	0.981653i	$-0.438932\pi$
$757$	10.4924			0.381354			0.190677	+	0.981653i	$0.438932\pi$
$758$		0			0
$759$	5.29723	−	9.43318i	0.192277	−	0.342403i
$760$		0			0
$761$	37.5697			1.36190			0.680950	−	0.732330i	$-0.261567\pi$
$761$	37.5697			1.36190			0.680950	+	0.732330i	$0.261567\pi$
$762$		0			0
$763$	−31.3693	−	8.10887i	−1.13565	−	0.293561i
$764$		0			0
$765$	−6.24621	−	10.2462i	−0.225832	−	0.370453i
$766$		0			0
$767$		−	8.63068i		−	0.311636i
$768$		0			0
$769$		−	35.8278i		−	1.29198i	−0.763345	−	0.645991i	$-0.776444\pi$
$769$		−	35.8278i		−	1.29198i	0.763345	−	0.645991i	$-0.223556\pi$
$770$		0			0
$771$	0.492423	−	0.876894i	0.0177342	−	0.0315806i
$772$		0			0
$773$	−3.76412			−0.135386			−0.0676929	−	0.997706i	$-0.521564\pi$
$773$	−3.76412			−0.135386			−0.0676929	+	0.997706i	$0.521564\pi$
$774$		0			0
$775$		−	30.3675i		−	1.09083i
$776$		0			0
$777$	6.34475	−	6.61393i	0.227617	−	0.237273i
$778$		0			0
$779$		−	68.9848i		−	2.47164i
$780$		0			0
$781$	−15.2311			−0.545010
$782$		0			0
$783$	1.69614	+	47.3747i	0.0606151	+	1.69303i
$784$		0			0
$785$		−	41.6155i		−	1.48532i
$786$		0			0
$787$			23.3741i			0.833198i	0.909090	+	0.416599i	$0.136778\pi$
$787$			23.3741i			0.833198i	−0.909090	+	0.416599i	$0.863222\pi$
$788$		0			0
$789$	14.1498	+	7.94584i	0.503745	+	0.282879i
$790$		0			0
$791$	4.87962	−	18.8769i	0.173499	−	0.671185i
$792$		0			0
$793$	8.63068			0.306485
$794$		0			0
$795$	41.6155	+	23.3693i	1.47595	+	0.828824i
$796$		0			0
$797$	−48.6990			−1.72501			−0.862504	−	0.506051i	$-0.831105\pi$
$797$	−48.6990			−1.72501			−0.862504	+	0.506051i	$0.831105\pi$
$798$		0			0
$799$	−11.5076			−0.407109
$800$		0			0
$801$	−17.9597	−	29.4608i	−0.634573	−	1.04095i
$802$		0			0
$803$	−37.7327			−1.33156
$804$		0			0
$805$	4.00000	−	15.4741i	0.140981	−	0.545389i
$806$		0			0
$807$	7.68466	−	13.6847i	0.270513	−	0.481723i
$808$		0			0
$809$			10.8769i			0.382411i	0.981550	+	0.191206i	$0.0612397\pi$
$809$			10.8769i			0.382411i	−0.981550	+	0.191206i	$0.938760\pi$
$810$		0			0
$811$			41.1708i			1.44570i	0.691004	+	0.722851i	$0.257169\pi$
$811$			41.1708i			1.44570i	−0.691004	+	0.722851i	$0.742831\pi$
$812$		0			0
$813$	8.24621	+	4.63068i	0.289207	+	0.162405i
$814$		0			0
$815$	−43.0299			−1.50727
$816$		0			0
$817$			25.6509i			0.897412i
$818$		0			0
$819$	12.8831	−	3.90767i	0.450172	−	0.136545i
$820$		0			0
$821$		−	14.8769i		−	0.519207i	−0.965715	−	0.259604i	$-0.916408\pi$
$821$		−	14.8769i		−	0.519207i	0.965715	−	0.259604i	$-0.0835919\pi$
$822$		0			0
$823$	21.1231			0.736305			0.368153	−	0.929765i	$-0.379990\pi$
$823$	21.1231			0.736305			0.368153	+	0.929765i	$0.379990\pi$
$824$		0			0
$825$	19.4470	+	10.9205i	0.677057	+	0.380204i
$826$		0			0
$827$		−	33.8617i		−	1.17749i	−0.808320	−	0.588744i	$-0.799623\pi$
$827$		−	33.8617i		−	1.17749i	0.808320	−	0.588744i	$-0.200377\pi$
$828$		0			0
$829$			22.0498i			0.765822i	0.923785	+	0.382911i	$0.125078\pi$
$829$			22.0498i			0.765822i	−0.923785	+	0.382911i	$0.874922\pi$
$830$		0			0
$831$	10.3857	−	18.4945i	0.360274	−	0.641568i
$832$		0			0
$833$	8.10887	+	4.49242i	0.280956	+	0.155653i
$834$		0			0
$835$	−10.2462			−0.354585
$836$		0			0
$837$	−1.36932	−	38.2462i	−0.0473305	−	1.32198i
$838$		0			0
$839$	−47.9096			−1.65402			−0.827011	−	0.562186i	$-0.809960\pi$
$839$	−47.9096			−1.65402			−0.827011	+	0.562186i	$0.809960\pi$
$840$		0			0
$841$	−54.2311			−1.87004
$842$		0			0
$843$	−43.0299	−	24.1636i	−1.48203	−	0.832238i
$844$		0			0
$845$	30.5763			1.05186
$846$		0			0
$847$	3.19224	+	0.825183i	0.109687	+	0.0283536i
$848$		0			0
$849$	18.8078	+	10.5616i	0.645481	+	0.362472i
$850$		0			0
$851$			4.00000i			0.137118i
$852$		0			0
$853$		−	6.57576i		−	0.225150i	−0.993643	−	0.112575i	$-0.964090\pi$
$853$		−	6.57576i		−	0.225150i	0.993643	−	0.112575i	$-0.0359098\pi$
$854$		0			0
$855$	49.6155	−	30.2462i	1.69682	−	1.03440i
$856$		0			0
$857$	6.62153			0.226187			0.113094	−	0.993584i	$-0.463924\pi$
$857$	6.62153			0.226187			0.113094	+	0.993584i	$0.463924\pi$
$858$		0			0
$859$			44.5631i			1.52047i	0.649647	+	0.760236i	$0.274917\pi$
$859$			44.5631i			1.52047i	−0.649647	+	0.760236i	$0.725083\pi$
$860$		0			0
$861$	34.1268	−	35.5746i	1.16304	−	1.21238i
$862$		0			0
$863$			35.1231i			1.19560i	0.801644	+	0.597802i	$0.203959\pi$
$863$			35.1231i			1.19560i	−0.801644	+	0.597802i	$0.796041\pi$
$864$		0			0
$865$	−47.8617			−1.62735
$866$		0			0
$867$	12.9299	−	23.0252i	0.439121	−	0.781977i
$868$		0			0
$869$			8.98485i			0.304790i
$870$		0			0
$871$			10.5945i			0.358979i
$872$		0			0
$873$	−22.2586	+	13.5691i	−0.753341	+	0.459245i
$874$		0			0
$875$	−6.78456	−	1.75379i	−0.229360	−	0.0592889i
$876$		0			0
$877$	28.7386			0.970435			0.485217	−	0.874394i	$-0.338740\pi$
$877$	28.7386			0.970435			0.485217	+	0.874394i	$0.338740\pi$
$878$		0			0
$879$	−21.4384	+	38.1771i	−0.723101	+	1.28768i
$880$		0			0
$881$	34.1774			1.15147			0.575733	−	0.817638i	$-0.304717\pi$
$881$	34.1774			1.15147			0.575733	+	0.817638i	$0.304717\pi$
$882$		0			0
$883$	22.2462			0.748645			0.374322	−	0.927299i	$-0.377875\pi$
$883$	22.2462			0.748645			0.374322	+	0.927299i	$0.377875\pi$
$884$		0			0
$885$	13.0343	−	23.2111i	0.438142	−	0.780233i
$886$		0			0
$887$	34.3404			1.15304			0.576519	−	0.817083i	$-0.304411\pi$
$887$	34.3404			1.15304			0.576519	+	0.817083i	$0.304411\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	24.9848	+	12.8769i	0.837024	+	0.431392i
$892$		0			0
$893$		−	55.7235i		−	1.86472i
$894$		0			0
$895$			14.7304i			0.492383i
$896$		0			0
$897$	−2.87689	+	5.12311i	−0.0960567	+	0.171056i
$898$		0			0
$899$	67.1935			2.24103
$900$		0			0
$901$		−	12.0818i		−	0.402503i
$902$		0			0
$903$	−12.6895	+	13.2279i	−0.422280	+	0.440196i
$904$		0			0
$905$			47.3693i			1.57461i
$906$		0			0
$907$	−16.4924			−0.547622			−0.273811	−	0.961784i	$-0.588284\pi$
$907$	−16.4924			−0.547622			−0.273811	+	0.961784i	$0.588284\pi$
$908$		0			0
$909$	5.87787	+	9.64198i	0.194957	+	0.319804i
$910$		0			0
$911$			6.00000i			0.198789i	0.995048	+	0.0993944i	$0.0316906\pi$
$911$			6.00000i			0.198789i	−0.995048	+	0.0993944i	$0.968309\pi$
$912$		0			0
$913$			5.29723i			0.175313i
$914$		0			0
$915$	23.2111	+	13.0343i	0.767336	+	0.430899i
$916$		0			0
$917$	35.2929	+	9.12311i	1.16547	+	0.301271i
$918$		0			0
$919$	6.38447			0.210604			0.105302	−	0.994440i	$-0.466419\pi$
$919$	6.38447			0.210604			0.105302	+	0.994440i	$0.466419\pi$
$920$		0			0
$921$	37.0540	+	20.8078i	1.22097	+	0.685639i
$922$		0			0
$923$	8.27190			0.272273
$924$		0			0
$925$	−8.24621			−0.271134
$926$		0			0
$927$	12.0818	−	7.36520i	0.396818	−	0.241905i
$928$		0			0
$929$	−50.8128			−1.66711			−0.833556	−	0.552435i	$-0.813699\pi$
$929$	−50.8128			−1.66711			−0.833556	+	0.552435i	$0.813699\pi$
$930$		0			0
$931$	−21.7538	+	39.2658i	−0.712952	+	1.28689i
$932$		0			0
$933$	−16.0000	+	28.4924i	−0.523816	+	0.932800i
$934$		0			0
$935$		−	12.4924i		−	0.408546i
$936$		0			0
$937$			18.8664i			0.616337i	0.951332	+	0.308168i	$0.0997161\pi$
$937$			18.8664i			0.616337i	−0.951332	+	0.308168i	$0.900284\pi$
$938$		0			0
$939$	−5.12311	−	2.87689i	−0.167186	−	0.0938839i
$940$		0			0
$941$	10.2226			0.333248			0.166624	−	0.986021i	$-0.446713\pi$
$941$	10.2226			0.333248			0.166624	+	0.986021i	$0.446713\pi$
$942$		0			0
$943$			21.5150i			0.700624i
$944$		0			0
$945$	40.5489	+	8.94720i	1.31906	+	0.291053i
$946$		0			0
$947$			51.6155i			1.67728i	0.544687	+	0.838640i	$0.316649\pi$
$947$			51.6155i			1.67728i	−0.544687	+	0.838640i	$0.683351\pi$
$948$		0			0
$949$	20.4924			0.665212
$950$		0			0
$951$	−20.5625	−	11.5469i	−0.666785	−	0.374435i
$952$		0			0
$953$		−	22.7386i		−	0.736577i	−0.929712	−	0.368288i	$-0.879944\pi$
$953$		−	22.7386i		−	0.736577i	0.929712	−	0.368288i	$-0.120056\pi$
$954$		0			0
$955$		−	52.4631i		−	1.69767i
$956$		0			0
$957$	−24.1636	+	43.0299i	−0.781098	+	1.39096i
$958$		0			0
$959$	−1.48734	+	5.75379i	−0.0480286	+	0.185800i
$960$		0			0
$961$	−23.2462			−0.749878
$962$		0			0
$963$	18.2462	−	11.1231i	0.587976	−	0.358437i
$964$		0			0
$965$	21.5150			0.692591
$966$		0			0
$967$	−28.4924			−0.916255			−0.458127	−	0.888887i	$-0.651480\pi$
$967$	−28.4924			−0.916255			−0.458127	+	0.888887i	$0.651480\pi$
$968$		0			0
$969$	−12.8255	−	7.20217i	−0.412013	−	0.231367i
$970$		0			0
$971$	25.8597			0.829878			0.414939	−	0.909849i	$-0.363803\pi$
$971$	25.8597			0.829878			0.414939	+	0.909849i	$0.363803\pi$
$972$		0			0
$973$	−4.24621	+	16.4265i	−0.136127	+	0.526611i
$974$		0			0
$975$	−10.5616	−	5.93087i	−0.338240	−	0.189940i
$976$		0			0
$977$			37.7538i			1.20785i	0.797041	+	0.603925i	$0.206398\pi$
$977$			37.7538i			1.20785i	−0.797041	+	0.603925i	$0.793602\pi$
$978$		0			0
$979$		−	35.9193i		−	1.14799i
$980$		0			0
$981$	19.1231	+	31.3693i	0.610554	+	1.00155i
$982$		0			0
$983$	7.20217			0.229714			0.114857	−	0.993382i	$-0.463359\pi$
$983$	7.20217			0.229714			0.114857	+	0.993382i	$0.463359\pi$
$984$		0			0
$985$			51.7194i			1.64792i
$986$		0			0
$987$	27.5664	−	28.7359i	0.877448	−	0.914674i
$988$		0			0
$989$		−	8.00000i		−	0.254385i
$990$		0			0
$991$	−13.1231			−0.416869			−0.208435	−	0.978036i	$-0.566837\pi$
$991$	−13.1231			−0.416869			−0.208435	+	0.978036i	$0.566837\pi$
$992$		0			0
$993$	18.8664	−	33.5968i	0.598706	−	1.06616i
$994$		0			0
$995$			30.2462i			0.958869i
$996$		0			0
$997$		−	8.48071i		−	0.268587i	−0.990942	−	0.134293i	$-0.957124\pi$
$997$		−	8.48071i		−	0.268587i	0.990942	−	0.134293i	$-0.0428764\pi$
$998$		0			0
$999$	−10.3857	+	0.371834i	−0.328587	+	0.0117643i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	336.2.k.c.209.4		8
3.2	odd	2	inner	336.2.k.c.209.6		8
4.3	odd	2		168.2.k.a.41.5	yes	8
7.6	odd	2	inner	336.2.k.c.209.5		8
8.3	odd	2		1344.2.k.f.1217.4		8
8.5	even	2		1344.2.k.i.1217.5		8
12.11	even	2		168.2.k.a.41.3	✓	8
21.20	even	2	inner	336.2.k.c.209.3		8
24.5	odd	2		1344.2.k.i.1217.3		8
24.11	even	2		1344.2.k.f.1217.6		8
28.3	even	6		1176.2.u.a.1097.8		16
28.11	odd	6		1176.2.u.a.1097.1		16
28.19	even	6		1176.2.u.a.521.3		16
28.23	odd	6		1176.2.u.a.521.6		16
28.27	even	2		168.2.k.a.41.4	yes	8
56.13	odd	2		1344.2.k.i.1217.4		8
56.27	even	2		1344.2.k.f.1217.5		8
84.11	even	6		1176.2.u.a.1097.3		16
84.23	even	6		1176.2.u.a.521.8		16
84.47	odd	6		1176.2.u.a.521.1		16
84.59	odd	6		1176.2.u.a.1097.6		16
84.83	odd	2		168.2.k.a.41.6	yes	8
168.83	odd	2		1344.2.k.f.1217.3		8
168.125	even	2		1344.2.k.i.1217.6		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
168.2.k.a.41.3	✓	8	12.11	even	2
168.2.k.a.41.4	yes	8	28.27	even	2
168.2.k.a.41.5	yes	8	4.3	odd	2
168.2.k.a.41.6	yes	8	84.83	odd	2
336.2.k.c.209.3		8	21.20	even	2	inner
336.2.k.c.209.4		8	1.1	even	1	trivial
336.2.k.c.209.5		8	7.6	odd	2	inner
336.2.k.c.209.6		8	3.2	odd	2	inner
1176.2.u.a.521.1		16	84.47	odd	6
1176.2.u.a.521.3		16	28.19	even	6
1176.2.u.a.521.6		16	28.23	odd	6
1176.2.u.a.521.8		16	84.23	even	6
1176.2.u.a.1097.1		16	28.11	odd	6
1176.2.u.a.1097.3		16	84.11	even	6
1176.2.u.a.1097.6		16	84.59	odd	6
1176.2.u.a.1097.8		16	28.3	even	6
1344.2.k.f.1217.3		8	168.83	odd	2
1344.2.k.f.1217.4		8	8.3	odd	2
1344.2.k.f.1217.5		8	56.27	even	2
1344.2.k.f.1217.6		8	24.11	even	2
1344.2.k.i.1217.3		8	24.5	odd	2
1344.2.k.i.1217.4		8	56.13	odd	2
1344.2.k.i.1217.5		8	8.5	even	2
1344.2.k.i.1217.6		8	168.125	even	2

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:	\( N \)	\(=\)	\( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	336.k (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.848071 + 1.51022i) q^{3} +3.02045 q^{5} +(2.56155 + 0.662153i) q^{7} +(-1.56155 - 2.56155i) q^{9} -3.12311i q^{11} +1.69614i q^{13} +(-2.56155 + 4.56155i) q^{15} +1.32431 q^{17} +6.41273i q^{19} +(-3.17238 + 3.30697i) q^{21} -2.00000i q^{23} +4.12311 q^{25} +(5.19283 - 0.185917i) q^{27} +9.12311i q^{29} -7.36520i q^{31} +(4.71659 + 2.64861i) q^{33} +(7.73704 + 2.00000i) q^{35} -2.00000 q^{37} +(-2.56155 - 1.43845i) q^{39} -10.7575 q^{41} +4.00000 q^{43} +(-4.71659 - 7.73704i) q^{45} -8.68951 q^{47} +(6.12311 + 3.39228i) q^{49} +(-1.12311 + 2.00000i) q^{51} -9.12311i q^{53} -9.43318i q^{55} +(-9.68466 - 5.43845i) q^{57} -5.08842 q^{59} -5.08842i q^{61} +(-2.30386 - 7.59554i) q^{63} +5.12311i q^{65} +6.24621 q^{67} +(3.02045 + 1.69614i) q^{69} -4.87689i q^{71} -12.0818i q^{73} +(-3.49668 + 6.22681i) q^{75} +(2.06798 - 8.00000i) q^{77} -2.87689 q^{79} +(-4.12311 + 8.00000i) q^{81} -1.69614 q^{83} +4.00000 q^{85} +(-13.7779 - 7.73704i) q^{87} +11.5012 q^{89} +(-1.12311 + 4.34475i) q^{91} +(11.1231 + 6.24621i) q^{93} +19.3693i q^{95} -8.68951i q^{97} +(-8.00000 + 4.87689i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 4 q^{7} + 4 q^{9} - 4 q^{15} - 8 q^{21} - 16 q^{37} - 4 q^{39} + 32 q^{43} + 16 q^{49} + 24 q^{51} - 28 q^{57} - 32 q^{63} - 16 q^{67} - 56 q^{79} + 32 q^{85} + 24 q^{91} + 56 q^{93} - 64 q^{99}+O(q^{100}) \) 8 * q + 4 * q^7 + 4 * q^9 - 4 * q^15 - 8 * q^21 - 16 * q^37 - 4 * q^39 + 32 * q^43 + 16 * q^49 + 24 * q^51 - 28 * q^57 - 32 * q^63 - 16 * q^67 - 56 * q^79 + 32 * q^85 + 24 * q^91 + 56 * q^93 - 64 * q^99

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