LMFDB - Embedded newform 2208.2.j.d.47.20

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2208,2,Mod(47,2208)] 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("2208.47"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2208, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 1, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2208 = 2^{5} \cdot 3 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2208.j (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [42,0,-2,0,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	no
Analytic conductor:	$17.6309687663$
Analytic rank:	$0$
Dimension:	$42$
Twist minimal:	no (minimal twist has level 552)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			47.20
Character	$\chi$	$=$	2208.47
Dual form			2208.2.j.d.47.19

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-0.262136 + 1.71210i) q^{3} -0.130526 q^{5} -3.77863i q^{7} +(-2.86257 - 0.897607i) q^{9} -4.36656i q^{11} +5.50350i q^{13} +(0.0342155 - 0.223473i) q^{15} -0.402065i q^{17} +2.08168 q^{19} +(6.46939 + 0.990517i) q^{21} -1.00000 q^{23} -4.98296 q^{25} +(2.28718 - 4.66571i) q^{27} -8.77100 q^{29} +7.75722i q^{31} +(7.47599 + 1.14463i) q^{33} +0.493208i q^{35} +10.1015i q^{37} +(-9.42255 - 1.44267i) q^{39} -2.80694i q^{41} -3.07905 q^{43} +(0.373638 + 0.117161i) q^{45} -3.41142 q^{47} -7.27804 q^{49} +(0.688374 + 0.105396i) q^{51} -0.336698 q^{53} +0.569948i q^{55} +(-0.545685 + 3.56405i) q^{57} +9.68031i q^{59} +2.78101i q^{61} +(-3.39173 + 10.8166i) q^{63} -0.718348i q^{65} -12.8230 q^{67} +(0.262136 - 1.71210i) q^{69} -8.19053 q^{71} +0.417051 q^{73} +(1.30622 - 8.53133i) q^{75} -16.4996 q^{77} -3.09607i q^{79} +(7.38860 + 5.13893i) q^{81} +3.59045i q^{83} +0.0524797i q^{85} +(2.29920 - 15.0168i) q^{87} +11.1036i q^{89} +20.7957 q^{91} +(-13.2811 - 2.03345i) q^{93} -0.271713 q^{95} +4.53325 q^{97} +(-3.91946 + 12.4996i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 42 q - 2 q^{3} + 8 q^{5} + 2 q^{9} + 8 q^{15} - 4 q^{19} + 8 q^{21} - 42 q^{23} + 22 q^{25} + 16 q^{27} + 12 q^{33} - 8 q^{39} - 28 q^{43} - 8 q^{45} - 50 q^{49} - 28 q^{51} + 24 q^{53} - 8 q^{57} + 16 q^{63}+ \cdots + 20 q^{97}+O(q^{100}) $ 42 * q - 2 * q^3 + 8 * q^5 + 2 * q^9 + 8 * q^15 - 4 * q^19 + 8 * q^21 - 42 * q^23 + 22 * q^25 + 16 * q^27 + 12 * q^33 - 8 * q^39 - 28 * q^43 - 8 * q^45 - 50 * q^49 - 28 * q^51 + 24 * q^53 - 8 * q^57 + 16 * q^63 + 4 * q^67 + 2 * q^69 + 4 * q^73 + 6 * q^75 + 32 * q^77 + 18 * q^81 + 48 * q^87 + 8 * q^91 - 22 * q^93 + 16 * q^95 + 20 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2208\mathbb{Z}\right)^\times$.

$n$	$97$	$415$	$737$	$1381$
$\chi(n)$	$1$	$-1$	$-1$	$-1$

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
$3$	−0.262136	+	1.71210i	−0.151345	+	0.988481i
$3$	−0.262136	+	1.71210i	−0.151345	+	0.988481i
$4$		0			0
$5$	−0.130526			−0.0583728			−0.0291864	−	0.999574i	$-0.509292\pi$
$5$	−0.130526			−0.0583728			−0.0291864	+	0.999574i	$0.509292\pi$
$6$		0			0
$7$		−	3.77863i		−	1.42819i	−0.700050	−	0.714094i	$-0.746839\pi$
$7$		−	3.77863i		−	1.42819i	0.700050	−	0.714094i	$-0.253161\pi$
$8$		0			0
$9$	−2.86257	−	0.897607i	−0.954190	−	0.299202i
$10$		0			0
$11$		−	4.36656i		−	1.31657i	−0.752770	−	0.658284i	$-0.771283\pi$
$11$		−	4.36656i		−	1.31657i	0.752770	−	0.658284i	$-0.228717\pi$
$12$		0			0
$13$			5.50350i			1.52640i	0.646164	+	0.763199i	$0.276372\pi$
$13$			5.50350i			1.52640i	−0.646164	+	0.763199i	$0.723628\pi$
$14$		0			0
$15$	0.0342155	−	0.223473i	0.00883441	−	0.0577004i
$16$		0			0
$17$		−	0.402065i		−	0.0975150i	−0.998811	−	0.0487575i	$-0.984474\pi$
$17$		−	0.402065i		−	0.0975150i	0.998811	−	0.0487575i	$-0.0155261\pi$
$18$		0			0
$19$	2.08168			0.477571			0.238785	−	0.971072i	$-0.423251\pi$
$19$	2.08168			0.477571			0.238785	+	0.971072i	$0.423251\pi$
$20$		0			0
$21$	6.46939	+	0.990517i	1.41174	+	0.216148i
$22$		0			0
$23$	−1.00000			−0.208514
$23$	−1.00000			−0.208514
$24$		0			0
$25$	−4.98296			−0.996593
$26$		0			0
$27$	2.28718	−	4.66571i	0.440167	−	0.897916i
$28$		0			0
$29$	−8.77100			−1.62873			−0.814367	−	0.580350i	$-0.802916\pi$
$29$	−8.77100			−1.62873			−0.814367	+	0.580350i	$0.802916\pi$
$30$		0			0
$31$			7.75722i			1.39324i	0.717442	+	0.696619i	$0.245313\pi$
$31$			7.75722i			1.39324i	−0.717442	+	0.696619i	$0.754687\pi$
$32$		0			0
$33$	7.47599	+	1.14463i	1.30140	+	0.199255i
$34$		0			0
$35$			0.493208i			0.0833673i
$36$		0			0
$37$			10.1015i			1.66068i	0.557254	+	0.830342i	$0.311855\pi$
$37$			10.1015i			1.66068i	−0.557254	+	0.830342i	$0.688145\pi$
$38$		0			0
$39$	−9.42255	−	1.44267i	−1.50882	−	0.231012i
$40$		0			0
$41$		−	2.80694i		−	0.438371i	−0.975683	−	0.219185i	$-0.929660\pi$
$41$		−	2.80694i		−	0.438371i	0.975683	−	0.219185i	$-0.0703399\pi$
$42$		0			0
$43$	−3.07905			−0.469550			−0.234775	−	0.972050i	$-0.575435\pi$
$43$	−3.07905			−0.469550			−0.234775	+	0.972050i	$0.575435\pi$
$44$		0			0
$45$	0.373638	+	0.117161i	0.0556987	+	0.0174653i
$46$		0			0
$47$	−3.41142			−0.497607			−0.248803	−	0.968554i	$-0.580037\pi$
$47$	−3.41142			−0.497607			−0.248803	+	0.968554i	$0.580037\pi$
$48$		0			0
$49$	−7.27804			−1.03972
$50$		0			0
$51$	0.688374	+	0.105396i	0.0963917	+	0.0147584i
$52$		0			0
$53$	−0.336698			−0.0462490			−0.0231245	−	0.999733i	$-0.507361\pi$
$53$	−0.336698			−0.0462490			−0.0231245	+	0.999733i	$0.507361\pi$
$54$		0			0
$55$			0.569948i			0.0768518i
$56$		0			0
$57$	−0.545685	+	3.56405i	−0.0722777	+	0.472070i
$58$		0			0
$59$			9.68031i			1.26027i	0.776486	+	0.630134i	$0.217000\pi$
$59$			9.68031i			1.26027i	−0.776486	+	0.630134i	$0.783000\pi$
$60$		0			0
$61$			2.78101i			0.356072i	0.984024	+	0.178036i	$0.0569744\pi$
$61$			2.78101i			0.356072i	−0.984024	+	0.178036i	$0.943026\pi$
$62$		0			0
$63$	−3.39173	+	10.8166i	−0.427317	+	1.36276i
$64$		0			0
$65$		−	0.718348i		−	0.0891001i
$66$		0			0
$67$	−12.8230			−1.56658			−0.783292	−	0.621654i	$-0.786461\pi$
$67$	−12.8230			−1.56658			−0.783292	+	0.621654i	$0.786461\pi$
$68$		0			0
$69$	0.262136	−	1.71210i	0.0315575	−	0.206113i
$70$		0			0
$71$	−8.19053			−0.972037			−0.486018	−	0.873949i	$-0.661551\pi$
$71$	−8.19053			−0.972037			−0.486018	+	0.873949i	$0.661551\pi$
$72$		0			0
$73$	0.417051			0.0488121			0.0244061	−	0.999702i	$-0.492231\pi$
$73$	0.417051			0.0488121			0.0244061	+	0.999702i	$0.492231\pi$
$74$		0			0
$75$	1.30622	−	8.53133i	0.150829	−	0.985113i
$76$		0			0
$77$	−16.4996			−1.88031
$78$		0			0
$79$		−	3.09607i		−	0.348335i	−0.984716	−	0.174168i	$-0.944277\pi$
$79$		−	3.09607i		−	0.348335i	0.984716	−	0.174168i	$-0.0557235\pi$
$80$		0			0
$81$	7.38860	+	5.13893i	0.820956	+	0.570992i
$82$		0			0
$83$			3.59045i			0.394103i	0.980393	+	0.197052i	$0.0631367\pi$
$83$			3.59045i			0.394103i	−0.980393	+	0.197052i	$0.936863\pi$
$84$		0			0
$85$			0.0524797i			0.00569222i
$86$		0			0
$87$	2.29920	−	15.0168i	0.246500	−	1.60997i
$88$		0			0
$89$			11.1036i			1.17698i	0.808506	+	0.588488i	$0.200276\pi$
$89$			11.1036i			1.17698i	−0.808506	+	0.588488i	$0.799724\pi$
$90$		0			0
$91$	20.7957			2.17998
$92$		0			0
$93$	−13.2811	−	2.03345i	−1.37719	−	0.210859i
$94$		0			0
$95$	−0.271713			−0.0278772
$96$		0			0
$97$	4.53325			0.460281			0.230141	−	0.973157i	$-0.426081\pi$
$97$	4.53325			0.460281			0.230141	+	0.973157i	$0.426081\pi$
$98$		0			0
$99$	−3.91946	+	12.4996i	−0.393920	+	1.25626i
$100$		0			0
$101$	3.79039			0.377158			0.188579	−	0.982058i	$-0.439612\pi$
$101$	3.79039			0.377158			0.188579	+	0.982058i	$0.439612\pi$
$102$		0			0
$103$		−	6.81074i		−	0.671082i	−0.942026	−	0.335541i	$-0.891081\pi$
$103$		−	6.81074i		−	0.671082i	0.942026	−	0.335541i	$-0.108919\pi$
$104$		0			0
$105$	−0.844421	−	0.129288i	−0.0824070	−	0.0126172i
$106$		0			0
$107$			6.74888i			0.652439i	0.945294	+	0.326219i	$0.105775\pi$
$107$			6.74888i			0.652439i	−0.945294	+	0.326219i	$0.894225\pi$
$108$		0			0
$109$			18.3211i			1.75485i	0.479716	+	0.877424i	$0.340740\pi$
$109$			18.3211i			1.75485i	−0.479716	+	0.877424i	$0.659260\pi$
$110$		0			0
$111$	−17.2948	−	2.64798i	−1.64155	−	0.251335i
$112$		0			0
$113$		−	14.3459i		−	1.34955i	−0.738023	−	0.674775i	$-0.764241\pi$
$113$		−	14.3459i		−	1.34955i	0.738023	−	0.674775i	$-0.235759\pi$
$114$		0			0
$115$	0.130526			0.0121716
$116$		0			0
$117$	4.93999	−	15.7542i	0.456702	−	1.45647i
$118$		0			0
$119$	−1.51925			−0.139270
$120$		0			0
$121$	−8.06685			−0.733350
$122$		0			0
$123$	4.80576	+	0.735802i	0.433321	+	0.0663450i
$124$		0			0
$125$	1.30303			0.116547
$126$		0			0
$127$		−	13.5190i		−	1.19962i	−0.800143	−	0.599810i	$-0.795243\pi$
$127$		−	13.5190i		−	1.19962i	0.800143	−	0.599810i	$-0.204757\pi$
$128$		0			0
$129$	0.807131	−	5.27164i	0.0710639	−	0.464142i
$130$		0			0
$131$		−	21.6396i		−	1.89066i	−0.326114	−	0.945331i	$-0.605739\pi$
$131$		−	21.6396i		−	1.89066i	0.326114	−	0.945331i	$-0.394261\pi$
$132$		0			0
$133$		−	7.86591i		−	0.682061i
$134$		0			0
$135$	−0.298535	+	0.608994i	−0.0256938	+	0.0524139i
$136$		0			0
$137$		−	14.9588i		−	1.27801i	−0.769201	−	0.639007i	$-0.779345\pi$
$137$		−	14.9588i		−	1.27801i	0.769201	−	0.639007i	$-0.220655\pi$
$138$		0			0
$139$	−11.5627			−0.980736			−0.490368	−	0.871515i	$-0.663138\pi$
$139$	−11.5627			−0.980736			−0.490368	+	0.871515i	$0.663138\pi$
$140$		0			0
$141$	0.894258	−	5.84069i	0.0753101	−	0.491875i
$142$		0			0
$143$	24.0314			2.00961
$144$		0			0
$145$	1.14484			0.0950738
$146$		0			0
$147$	1.90784	−	12.4607i	0.157356	−	1.02774i
$148$		0			0
$149$	−13.1430			−1.07671			−0.538356	−	0.842717i	$-0.680955\pi$
$149$	−13.1430			−1.07671			−0.538356	+	0.842717i	$0.680955\pi$
$150$		0			0
$151$			1.10155i			0.0896425i	0.998995	+	0.0448213i	$0.0142718\pi$
$151$			1.10155i			0.0896425i	−0.998995	+	0.0448213i	$0.985728\pi$
$152$		0			0
$153$	−0.360896	+	1.15094i	−0.0291767	+	0.0930478i
$154$		0			0
$155$		−	1.01252i		−	0.0813272i
$156$		0			0
$157$		−	17.3372i		−	1.38366i	−0.722062	−	0.691828i	$-0.756805\pi$
$157$		−	17.3372i		−	1.38366i	0.722062	−	0.691828i	$-0.243195\pi$
$158$		0			0
$159$	0.0882607	−	0.576460i	0.00699953	−	0.0457162i
$160$		0			0
$161$			3.77863i			0.297798i
$162$		0			0
$163$	10.4421			0.817889			0.408944	−	0.912559i	$-0.365897\pi$
$163$	10.4421			0.817889			0.408944	+	0.912559i	$0.365897\pi$
$164$		0			0
$165$	−0.975807	−	0.149404i	−0.0759665	−	0.0116311i
$166$		0			0
$167$	1.67012			0.129238			0.0646188	−	0.997910i	$-0.479417\pi$
$167$	1.67012			0.129238			0.0646188	+	0.997910i	$0.479417\pi$
$168$		0			0
$169$	−17.2886			−1.32989
$170$		0			0
$171$	−5.95896	−	1.86853i	−0.455693	−	0.142890i
$172$		0			0
$173$	−8.55215			−0.650208			−0.325104	−	0.945678i	$-0.605399\pi$
$173$	−8.55215			−0.650208			−0.325104	+	0.945678i	$0.605399\pi$
$174$		0			0
$175$			18.8288i			1.42332i
$176$		0			0
$177$	−16.5736	−	2.53756i	−1.24575	−	0.190735i
$178$		0			0
$179$		−	14.7284i		−	1.10085i	−0.834885	−	0.550425i	$-0.814466\pi$
$179$		−	14.7284i		−	1.10085i	0.834885	−	0.550425i	$-0.185534\pi$
$180$		0			0
$181$			21.7258i			1.61487i	0.589959	+	0.807433i	$0.299144\pi$
$181$			21.7258i			1.61487i	−0.589959	+	0.807433i	$0.700856\pi$
$182$		0			0
$183$	−4.76137	−	0.729005i	−0.351971	−	0.0538896i
$184$		0			0
$185$		−	1.31851i		−	0.0969388i
$186$		0			0
$187$	−1.75564			−0.128385
$188$		0			0
$189$	−17.6300	−	8.64240i	−1.28239	−	0.628642i
$190$		0			0
$191$	−10.7127			−0.775145			−0.387572	−	0.921839i	$-0.626686\pi$
$191$	−10.7127			−0.775145			−0.387572	+	0.921839i	$0.626686\pi$
$192$		0			0
$193$	14.8172			1.06657			0.533283	−	0.845937i	$-0.320958\pi$
$193$	14.8172			1.06657			0.533283	+	0.845937i	$0.320958\pi$
$194$		0			0
$195$	1.22988	+	0.188305i	0.0880738	+	0.0134848i
$196$		0			0
$197$	−17.5258			−1.24866			−0.624331	−	0.781160i	$-0.714628\pi$
$197$	−17.5258			−1.24866			−0.624331	+	0.781160i	$0.714628\pi$
$198$		0			0
$199$			23.3044i			1.65201i	0.563666	+	0.826003i	$0.309391\pi$
$199$			23.3044i			1.65201i	−0.563666	+	0.826003i	$0.690609\pi$
$200$		0			0
$201$	3.36139	−	21.9543i	0.237094	−	1.54854i
$202$		0			0
$203$			33.1424i			2.32614i
$204$		0			0
$205$			0.366378i			0.0255889i
$206$		0			0
$207$	2.86257	+	0.897607i	0.198962	+	0.0623880i
$208$		0			0
$209$		−	9.08979i		−	0.628754i
$210$		0			0
$211$	−16.0748			−1.10664			−0.553319	−	0.832970i	$-0.686639\pi$
$211$	−16.0748			−1.10664			−0.553319	+	0.832970i	$0.686639\pi$
$212$		0			0
$213$	2.14704	−	14.0230i	0.147112	−	0.960840i
$214$		0			0
$215$	0.401894			0.0274090
$216$		0			0
$217$	29.3117			1.98980
$218$		0			0
$219$	−0.109324	+	0.714033i	−0.00738745	+	0.0482499i
$220$		0			0
$221$	2.21276			0.148847
$222$		0			0
$223$			16.7322i			1.12047i	0.828333	+	0.560237i	$0.189290\pi$
$223$			16.7322i			1.12047i	−0.828333	+	0.560237i	$0.810710\pi$
$224$		0			0
$225$	14.2641	+	4.47274i	0.950938	+	0.298183i
$226$		0			0
$227$		−	5.60178i		−	0.371803i	−0.982568	−	0.185902i	$-0.940479\pi$
$227$		−	5.60178i		−	0.371803i	0.982568	−	0.185902i	$-0.0595206\pi$
$228$		0			0
$229$		−	17.9554i		−	1.18653i	−0.805008	−	0.593264i	$-0.797839\pi$
$229$		−	17.9554i		−	1.18653i	0.805008	−	0.593264i	$-0.202161\pi$
$230$		0			0
$231$	4.32515	−	28.2490i	0.284574	−	1.85865i
$232$		0			0
$233$			24.5310i			1.60708i	0.595252	+	0.803539i	$0.297052\pi$
$233$			24.5310i			1.60708i	−0.595252	+	0.803539i	$0.702948\pi$
$234$		0			0
$235$	0.445278			0.0290467
$236$		0			0
$237$	5.30078	+	0.811594i	0.344323	+	0.0527187i
$238$		0			0
$239$	3.73158			0.241376			0.120688	−	0.992691i	$-0.461490\pi$
$239$	3.73158			0.241376			0.120688	+	0.992691i	$0.461490\pi$
$240$		0			0
$241$	8.92696			0.575036			0.287518	−	0.957775i	$-0.407170\pi$
$241$	8.92696			0.575036			0.287518	+	0.957775i	$0.407170\pi$
$242$		0			0
$243$	−10.7352	+	11.3029i	−0.688662	+	0.725083i
$244$		0			0
$245$	0.949971			0.0606914
$246$		0			0
$247$			11.4566i			0.728963i
$248$		0			0
$249$	−6.14721	−	0.941188i	−0.389564	−	0.0596454i
$250$		0			0
$251$		−	15.0476i		−	0.949798i	−0.880040	−	0.474899i	$-0.842485\pi$
$251$		−	15.0476i		−	0.949798i	0.880040	−	0.474899i	$-0.157515\pi$
$252$		0			0
$253$			4.36656i			0.274523i
$254$		0			0
$255$	−0.0898505	−	0.0137568i	−0.00562666	−	0.000861487i
$256$		0			0
$257$		−	7.84239i		−	0.489195i	−0.969625	−	0.244597i	$-0.921344\pi$
$257$		−	7.84239i		−	0.489195i	0.969625	−	0.244597i	$-0.0786558\pi$
$258$		0			0
$259$	38.1700			2.37177
$260$		0			0
$261$	25.1076	+	7.87292i	1.55412	+	0.487321i
$262$		0			0
$263$	−18.6358			−1.14913			−0.574566	−	0.818458i	$-0.694829\pi$
$263$	−18.6358			−1.14913			−0.574566	+	0.818458i	$0.694829\pi$
$264$		0			0
$265$	0.0439477			0.00269968
$266$		0			0
$267$	−19.0104	−	2.91065i	−1.16342	−	0.178129i
$268$		0			0
$269$	5.72570			0.349102			0.174551	−	0.984648i	$-0.444153\pi$
$269$	5.72570			0.349102			0.174551	+	0.984648i	$0.444153\pi$
$270$		0			0
$271$			10.0447i			0.610173i	0.952325	+	0.305086i	$0.0986853\pi$
$271$			10.0447i			0.610173i	−0.952325	+	0.305086i	$0.901315\pi$
$272$		0			0
$273$	−5.45131	+	35.6043i	−0.329928	+	2.15487i
$274$		0			0
$275$			21.7584i			1.31208i
$276$		0			0
$277$		−	12.6894i		−	0.762431i	−0.924486	−	0.381215i	$-0.875506\pi$
$277$		−	12.6894i		−	0.762431i	0.924486	−	0.381215i	$-0.124494\pi$
$278$		0			0
$279$	6.96294	−	22.2056i	0.416860	−	1.32941i
$280$		0			0
$281$			25.8591i			1.54262i	0.636458	+	0.771311i	$0.280399\pi$
$281$			25.8591i			1.54262i	−0.636458	+	0.771311i	$0.719601\pi$
$282$		0			0
$283$	−21.0898			−1.25366			−0.626828	−	0.779158i	$-0.715647\pi$
$283$	−21.0898			−1.25366			−0.626828	+	0.779158i	$0.715647\pi$
$284$		0			0
$285$	0.0712259	−	0.465199i	0.00421906	−	0.0275560i
$286$		0			0
$287$	−10.6064			−0.626076
$288$		0			0
$289$	16.8383			0.990491
$290$		0			0
$291$	−1.18833	+	7.76137i	−0.0696611	+	0.454979i
$292$		0			0
$293$	17.5822			1.02717			0.513583	−	0.858040i	$-0.328318\pi$
$293$	17.5822			1.02717			0.513583	+	0.858040i	$0.328318\pi$
$294$		0			0
$295$		−	1.26353i		−	0.0735654i
$296$		0			0
$297$	−20.3731	−	9.98710i	−1.18217	−	0.579510i
$298$		0			0
$299$		−	5.50350i		−	0.318276i
$300$		0			0
$301$			11.6346i			0.670606i
$302$		0			0
$303$	−0.993599	+	6.48952i	−0.0570808	+	0.372813i
$304$		0			0
$305$		−	0.362993i		−	0.0207849i
$306$		0			0
$307$	12.0490			0.687674			0.343837	−	0.939029i	$-0.388273\pi$
$307$	12.0490			0.687674			0.343837	+	0.939029i	$0.388273\pi$
$308$		0			0
$309$	11.6607	+	1.78534i	0.663352	+	0.101565i
$310$		0			0
$311$	8.92121			0.505876			0.252938	−	0.967483i	$-0.418603\pi$
$311$	8.92121			0.505876			0.252938	+	0.967483i	$0.418603\pi$
$312$		0			0
$313$	0.481564			0.0272196			0.0136098	−	0.999907i	$-0.495668\pi$
$313$	0.481564			0.0272196			0.0136098	+	0.999907i	$0.495668\pi$
$314$		0			0
$315$	0.442707	−	1.41184i	0.0249437	−	0.0795483i
$316$		0			0
$317$	−6.46721			−0.363235			−0.181617	−	0.983369i	$-0.558133\pi$
$317$	−6.46721			−0.363235			−0.181617	+	0.983369i	$0.558133\pi$
$318$		0			0
$319$			38.2991i			2.14434i
$320$		0			0
$321$	−11.5548	−	1.76913i	−0.644923	−	0.0987431i
$322$		0			0
$323$		−	0.836971i		−	0.0465703i
$324$		0			0
$325$		−	27.4238i		−	1.52120i
$326$		0			0
$327$	−31.3676	−	4.80264i	−1.73463	−	0.265587i
$328$		0			0
$329$			12.8905i			0.710676i
$330$		0			0
$331$	−25.1117			−1.38026			−0.690132	−	0.723683i	$-0.742448\pi$
$331$	−25.1117			−1.38026			−0.690132	+	0.723683i	$0.742448\pi$
$332$		0			0
$333$	9.06722	−	28.9164i	0.496881	−	1.58461i
$334$		0			0
$335$	1.67374			0.0914459
$336$		0			0
$337$	−10.0318			−0.546465			−0.273232	−	0.961948i	$-0.588093\pi$
$337$	−10.0318			−0.546465			−0.273232	+	0.961948i	$0.588093\pi$
$338$		0			0
$339$	24.5616	+	3.76059i	1.33401	+	0.204247i
$340$		0			0
$341$	33.8724			1.83429
$342$		0			0
$343$			1.05062i			0.0567283i
$344$		0			0
$345$	−0.0342155	+	0.223473i	−0.00184210	+	0.0120314i
$346$		0			0
$347$		−	12.4062i		−	0.665998i	−0.942927	−	0.332999i	$-0.891939\pi$
$347$		−	12.4062i		−	0.665998i	0.942927	−	0.332999i	$-0.108061\pi$
$348$		0			0
$349$			17.9711i			0.961969i	0.876729	+	0.480985i	$0.159721\pi$
$349$			17.9711i			0.961969i	−0.876729	+	0.480985i	$0.840279\pi$
$350$		0			0
$351$	25.6777	+	12.5875i	1.37058	+	0.671870i
$352$		0			0
$353$			25.5729i			1.36111i	0.732698	+	0.680554i	$0.238261\pi$
$353$			25.5729i			1.36111i	−0.732698	+	0.680554i	$0.761739\pi$
$354$		0			0
$355$	1.06907			0.0567405
$356$		0			0
$357$	0.398252	−	2.60111i	0.0210777	−	0.137665i
$358$		0			0
$359$	20.5959			1.08701			0.543506	−	0.839405i	$-0.317096\pi$
$359$	20.5959			1.08701			0.543506	+	0.839405i	$0.317096\pi$
$360$		0			0
$361$	−14.6666			−0.771926
$362$		0			0
$363$	2.11462	−	13.8113i	0.110989	−	0.724903i
$364$		0			0
$365$	−0.0544358			−0.00284930
$366$		0			0
$367$			0.995463i			0.0519628i	0.999662	+	0.0259814i	$0.00827106\pi$
$367$			0.995463i			0.0519628i	−0.999662	+	0.0259814i	$0.991729\pi$
$368$		0			0
$369$	−2.51953	+	8.03507i	−0.131162	+	0.418289i
$370$		0			0
$371$			1.27226i			0.0660522i
$372$		0			0
$373$			19.8991i			1.03034i	0.857089	+	0.515168i	$0.172271\pi$
$373$			19.8991i			1.03034i	−0.857089	+	0.515168i	$0.827729\pi$
$374$		0			0
$375$	−0.341572	+	2.23092i	−0.0176387	+	0.115204i
$376$		0			0
$377$		−	48.2713i		−	2.48610i
$378$		0			0
$379$	13.7477			0.706172			0.353086	−	0.935591i	$-0.385132\pi$
$379$	13.7477			0.706172			0.353086	+	0.935591i	$0.385132\pi$
$380$		0			0
$381$	23.1459	+	3.54383i	1.18580	+	0.181556i
$382$		0			0
$383$	−5.57295			−0.284764			−0.142382	−	0.989812i	$-0.545476\pi$
$383$	−5.57295			−0.284764			−0.142382	+	0.989812i	$0.545476\pi$
$384$		0			0
$385$	2.15362			0.109759
$386$		0			0
$387$	8.81399	+	2.76378i	0.448040	+	0.140491i
$388$		0			0
$389$	−14.5658			−0.738516			−0.369258	−	0.929327i	$-0.620388\pi$
$389$	−14.5658			−0.738516			−0.369258	+	0.929327i	$0.620388\pi$
$390$		0			0
$391$			0.402065i			0.0203333i
$392$		0			0
$393$	37.0492	+	5.67253i	1.86888	+	0.286141i
$394$		0			0
$395$			0.404117i			0.0203333i
$396$		0			0
$397$		−	1.50196i		−	0.0753812i	−0.999289	−	0.0376906i	$-0.988000\pi$
$397$		−	1.50196i		−	0.0753812i	0.999289	−	0.0376906i	$-0.0120001\pi$
$398$		0			0
$399$	13.4672	+	2.06194i	0.674204	+	0.103226i
$400$		0			0
$401$		−	21.6674i		−	1.08202i	−0.841016	−	0.541010i	$-0.818042\pi$
$401$		−	21.6674i		−	1.08202i	0.841016	−	0.541010i	$-0.181958\pi$
$402$		0			0
$403$	−42.6919			−2.12663
$404$		0			0
$405$	−0.964402	−	0.670761i	−0.0479215	−	0.0333304i
$406$		0			0
$407$	44.1090			2.18640
$408$		0			0
$409$	7.53712			0.372687			0.186343	−	0.982485i	$-0.440336\pi$
$409$	7.53712			0.372687			0.186343	+	0.982485i	$0.440336\pi$
$410$		0			0
$411$	25.6109	+	3.92124i	1.26329	+	0.193421i
$412$		0			0
$413$	36.5783			1.79990
$414$		0			0
$415$		−	0.468646i		−	0.0230049i
$416$		0			0
$417$	3.03101	−	19.7965i	0.148429	−	0.969439i
$418$		0			0
$419$		−	4.47808i		−	0.218768i	−0.994000	−	0.109384i	$-0.965112\pi$
$419$		−	4.47808i		−	0.218768i	0.994000	−	0.109384i	$-0.0348879\pi$
$420$		0			0
$421$			8.65241i			0.421693i	0.977519	+	0.210846i	$0.0676220\pi$
$421$			8.65241i			0.421693i	−0.977519	+	0.210846i	$0.932378\pi$
$422$		0			0
$423$	9.76543	+	3.06212i	0.474811	+	0.148885i
$424$		0			0
$425$			2.00347i			0.0971827i
$426$		0			0
$427$	10.5084			0.508538
$428$		0			0
$429$	−6.29950	+	41.1441i	−0.304143	+	1.98646i
$430$		0			0
$431$	12.6268			0.608214			0.304107	−	0.952638i	$-0.401642\pi$
$431$	12.6268			0.608214			0.304107	+	0.952638i	$0.401642\pi$
$432$		0			0
$433$	−30.6380			−1.47237			−0.736185	−	0.676781i	$-0.763375\pi$
$433$	−30.6380			−1.47237			−0.736185	+	0.676781i	$0.763375\pi$
$434$		0			0
$435$	−0.300104	+	1.96008i	−0.0143889	+	0.0939787i
$436$		0			0
$437$	−2.08168			−0.0995804
$438$		0			0
$439$		−	30.6982i		−	1.46514i	−0.680689	−	0.732572i	$-0.738319\pi$
$439$		−	30.6982i		−	1.46514i	0.680689	−	0.732572i	$-0.261681\pi$
$440$		0			0
$441$	20.8339	+	6.53283i	0.992091	+	0.311087i
$442$		0			0
$443$		−	1.20435i		−	0.0572206i	−0.999591	−	0.0286103i	$-0.990892\pi$
$443$		−	1.20435i		−	0.0572206i	0.999591	−	0.0286103i	$-0.00910818\pi$
$444$		0			0
$445$		−	1.44930i		−	0.0687034i
$446$		0			0
$447$	3.44525	−	22.5020i	0.162955	−	1.06431i
$448$		0			0
$449$		−	30.8882i		−	1.45771i	−0.684670	−	0.728853i	$-0.740054\pi$
$449$		−	30.8882i		−	1.45771i	0.684670	−	0.728853i	$-0.259946\pi$
$450$		0			0
$451$	−12.2567			−0.577145
$452$		0			0
$453$	−1.88596	−	0.288755i	−0.0886099	−	0.0135669i
$454$		0			0
$455$	−2.71437			−0.127252
$456$		0			0
$457$	−2.41262			−0.112858			−0.0564289	−	0.998407i	$-0.517971\pi$
$457$	−2.41262			−0.112858			−0.0564289	+	0.998407i	$0.517971\pi$
$458$		0			0
$459$	−1.87592	−	0.919593i	−0.0875602	−	0.0429229i
$460$		0			0
$461$	26.7626			1.24646			0.623230	−	0.782039i	$-0.285820\pi$
$461$	26.7626			1.24646			0.623230	+	0.782039i	$0.285820\pi$
$462$		0			0
$463$			4.03126i			0.187349i	0.995603	+	0.0936743i	$0.0298613\pi$
$463$			4.03126i			0.187349i	−0.995603	+	0.0936743i	$0.970139\pi$
$464$		0			0
$465$	1.73353	+	0.265417i	0.0803904	+	0.0123084i
$466$		0			0
$467$			19.4906i			0.901918i	0.892545	+	0.450959i	$0.148918\pi$
$467$			19.4906i			0.901918i	−0.892545	+	0.450959i	$0.851082\pi$
$468$		0			0
$469$			48.4535i			2.23738i
$470$		0			0
$471$	29.6830	+	4.54470i	1.36772	+	0.209409i
$472$		0			0
$473$			13.4448i			0.618195i
$474$		0			0
$475$	−10.3729			−0.475944
$476$		0			0
$477$	0.963820	+	0.302222i	0.0441303	+	0.0138378i
$478$		0			0
$479$	11.7760			0.538061			0.269030	−	0.963132i	$-0.413297\pi$
$479$	11.7760			0.538061			0.269030	+	0.963132i	$0.413297\pi$
$480$		0			0
$481$	−55.5939			−2.53486
$482$		0			0
$483$	−6.46939	−	0.990517i	−0.294367	−	0.0450701i
$484$		0			0
$485$	−0.591705			−0.0268679
$486$		0			0
$487$			7.24525i			0.328314i	0.986434	+	0.164157i	$0.0524903\pi$
$487$			7.24525i			0.328314i	−0.986434	+	0.164157i	$0.947510\pi$
$488$		0			0
$489$	−2.73726	+	17.8779i	−0.123783	+	0.808468i
$490$		0			0
$491$			16.2051i			0.731328i	0.930747	+	0.365664i	$0.119158\pi$
$491$			16.2051i			0.731328i	−0.930747	+	0.365664i	$0.880842\pi$
$492$		0			0
$493$			3.52651i			0.158826i
$494$		0			0
$495$	0.511589	−	1.63152i	0.0229942	−	0.0733312i
$496$		0			0
$497$			30.9490i			1.38825i
$498$		0			0
$499$	−5.30706			−0.237576			−0.118788	−	0.992920i	$-0.537901\pi$
$499$	−5.30706			−0.237576			−0.118788	+	0.992920i	$0.537901\pi$
$500$		0			0
$501$	−0.437799	+	2.85941i	−0.0195594	+	0.127749i
$502$		0			0
$503$	−0.131455			−0.00586131			−0.00293065	−	0.999996i	$-0.500933\pi$
$503$	−0.131455			−0.00586131			−0.00293065	+	0.999996i	$0.500933\pi$
$504$		0			0
$505$	−0.494742			−0.0220157
$506$		0			0
$507$	4.53196	−	29.5997i	0.201272	−	1.31457i
$508$		0			0
$509$	−35.3633			−1.56745			−0.783725	−	0.621108i	$-0.786683\pi$
$509$	−35.3633			−1.56745			−0.783725	+	0.621108i	$0.786683\pi$
$510$		0			0
$511$		−	1.57588i		−	0.0697129i
$512$		0			0
$513$	4.76118	−	9.71252i	0.210211	−	0.428818i
$514$		0			0
$515$			0.888976i			0.0391730i
$516$		0			0
$517$			14.8962i			0.655133i
$518$		0			0
$519$	2.24183	−	14.6421i	0.0984055	−	0.642719i
$520$		0			0
$521$			0.362476i			0.0158804i	0.999968	+	0.00794019i	$0.00252747\pi$
$521$			0.362476i			0.0158804i	−0.999968	+	0.00794019i	$0.997473\pi$
$522$		0			0
$523$	0.701562			0.0306772			0.0153386	−	0.999882i	$-0.495117\pi$
$523$	0.701562			0.0306772			0.0153386	+	0.999882i	$0.495117\pi$
$524$		0			0
$525$	−32.2367	−	4.93571i	−1.40693	−	0.215412i
$526$		0			0
$527$	3.11890			0.135862
$528$		0			0
$529$	1.00000			0.0434783
$530$		0			0
$531$	8.68912	−	27.7105i	0.377076	−	1.20254i
$532$		0			0
$533$	15.4480			0.669128
$534$		0			0
$535$		−	0.880901i		−	0.0380847i
$536$		0			0
$537$	25.2164	+	3.86084i	1.08817	+	0.166608i
$538$		0			0
$539$			31.7800i			1.36886i
$540$		0			0
$541$		−	8.62101i		−	0.370646i	−0.982678	−	0.185323i	$-0.940667\pi$
$541$		−	8.62101i		−	0.370646i	0.982678	−	0.185323i	$-0.0593332\pi$
$542$		0			0
$543$	−37.1967	−	5.69513i	−1.59627	−	0.244401i
$544$		0			0
$545$		−	2.39138i		−	0.102435i
$546$		0			0
$547$	7.45918			0.318931			0.159466	−	0.987203i	$-0.449023\pi$
$547$	7.45918			0.318931			0.159466	+	0.987203i	$0.449023\pi$
$548$		0			0
$549$	2.49626	−	7.96084i	0.106538	−	0.339760i
$550$		0			0
$551$	−18.2584			−0.777836
$552$		0			0
$553$	−11.6989			−0.497488
$554$		0			0
$555$	2.25742	+	0.345630i	0.0958222	+	0.0146712i
$556$		0			0
$557$	24.0703			1.01989			0.509945	−	0.860207i	$-0.329666\pi$
$557$	24.0703			1.01989			0.509945	+	0.860207i	$0.329666\pi$
$558$		0			0
$559$		−	16.9456i		−	0.716720i
$560$		0			0
$561$	0.460217	−	3.00583i	0.0194304	−	0.126906i
$562$		0			0
$563$		−	3.51983i		−	0.148343i	−0.997246	−	0.0741715i	$-0.976369\pi$
$563$		−	3.51983i		−	0.148343i	0.997246	−	0.0741715i	$-0.0236312\pi$
$564$		0			0
$565$			1.87251i			0.0787771i
$566$		0			0
$567$	19.4181	−	27.9188i	0.815484	−	1.17248i
$568$		0			0
$569$		−	34.0590i		−	1.42783i	−0.700233	−	0.713914i	$-0.746921\pi$
$569$		−	34.0590i		−	1.42783i	0.700233	−	0.713914i	$-0.253079\pi$
$570$		0			0
$571$	9.16056			0.383357			0.191679	−	0.981458i	$-0.438607\pi$
$571$	9.16056			0.383357			0.191679	+	0.981458i	$0.438607\pi$
$572$		0			0
$573$	2.80819	−	18.3412i	0.117314	−	0.766216i
$574$		0			0
$575$	4.98296			0.207804
$576$		0			0
$577$	14.7661			0.614720			0.307360	−	0.951593i	$-0.400554\pi$
$577$	14.7661			0.614720			0.307360	+	0.951593i	$0.400554\pi$
$578$		0			0
$579$	−3.88413	+	25.3686i	−0.161419	+	1.05428i
$580$		0			0
$581$	13.5670			0.562853
$582$		0			0
$583$			1.47021i			0.0608899i
$584$		0			0
$585$	−0.644795	+	2.05632i	−0.0266590	+	0.0850184i
$586$		0			0
$587$		−	0.974726i		−	0.0402313i	−0.999798	−	0.0201156i	$-0.993597\pi$
$587$		−	0.974726i		−	0.0402313i	0.999798	−	0.0201156i	$-0.00640344\pi$
$588$		0			0
$589$			16.1481i			0.665370i
$590$		0			0
$591$	4.59415	−	30.0059i	0.188978	−	1.23428i
$592$		0			0
$593$			20.2807i			0.832828i	0.909175	+	0.416414i	$0.136713\pi$
$593$			20.2807i			0.832828i	−0.909175	+	0.416414i	$0.863287\pi$
$594$		0			0
$595$	0.198301			0.00812956
$596$		0			0
$597$	−39.8995	−	6.10894i	−1.63298	−	0.250022i
$598$		0			0
$599$	31.5805			1.29034			0.645172	−	0.764037i	$-0.276786\pi$
$599$	31.5805			1.29034			0.645172	+	0.764037i	$0.276786\pi$
$600$		0			0
$601$	7.95503			0.324492			0.162246	−	0.986750i	$-0.448126\pi$
$601$	7.95503			0.324492			0.162246	+	0.986750i	$0.448126\pi$
$602$		0			0
$603$	36.7069	+	11.5101i	1.49482	+	0.468726i
$604$		0			0
$605$	1.05293			0.0428077
$606$		0			0
$607$			5.15628i			0.209287i	0.994510	+	0.104643i	$0.0333701\pi$
$607$			5.15628i			0.209287i	−0.994510	+	0.104643i	$0.966630\pi$
$608$		0			0
$609$	−56.7430	−	8.68782i	−2.29934	−	0.352048i
$610$		0			0
$611$		−	18.7748i		−	0.759546i
$612$		0			0
$613$		−	34.3625i		−	1.38789i	−0.720029	−	0.693944i	$-0.755871\pi$
$613$		−	34.3625i		−	1.38789i	0.720029	−	0.693944i	$-0.244129\pi$
$614$		0			0
$615$	−0.627275	−	0.0960410i	−0.0252942	−	0.00387275i
$616$		0			0
$617$			0.319112i			0.0128470i	0.999979	+	0.00642349i	$0.00204467\pi$
$617$			0.319112i			0.0128470i	−0.999979	+	0.00642349i	$0.997955\pi$
$618$		0			0
$619$	−1.90822			−0.0766978			−0.0383489	−	0.999264i	$-0.512210\pi$
$619$	−1.90822			−0.0766978			−0.0383489	+	0.999264i	$0.512210\pi$
$620$		0			0
$621$	−2.28718	+	4.66571i	−0.0917812	+	0.187228i
$622$		0			0
$623$	41.9563			1.68094
$624$		0			0
$625$	24.7447			0.989789
$626$		0			0
$627$	15.5626	+	2.38277i	0.621512	+	0.0951585i
$628$		0			0
$629$	4.06147			0.161942
$630$		0			0
$631$		−	37.3783i		−	1.48801i	−0.668175	−	0.744004i	$-0.732924\pi$
$631$		−	37.3783i		−	1.48801i	0.668175	−	0.744004i	$-0.267076\pi$
$632$		0			0
$633$	4.21380	−	27.5217i	0.167484	−	1.09389i
$634$		0			0
$635$			1.76458i			0.0700252i
$636$		0			0
$637$		−	40.0547i		−	1.58703i
$638$		0			0
$639$	23.4459	+	7.35188i	0.927507	+	0.290836i
$640$		0			0
$641$			20.2501i			0.799831i	0.916552	+	0.399915i	$0.130961\pi$
$641$			20.2501i			0.799831i	−0.916552	+	0.399915i	$0.869039\pi$
$642$		0			0
$643$	28.2916			1.11571			0.557856	−	0.829938i	$-0.311624\pi$
$643$	28.2916			1.11571			0.557856	+	0.829938i	$0.311624\pi$
$644$		0			0
$645$	−0.105351	+	0.688083i	−0.00414820	+	0.0270933i
$646$		0			0
$647$	−47.8762			−1.88221			−0.941104	−	0.338118i	$-0.890210\pi$
$647$	−47.8762			−1.88221			−0.941104	+	0.338118i	$0.890210\pi$
$648$		0			0
$649$	42.2696			1.65923
$650$		0			0
$651$	−7.68365	+	50.1845i	−0.301146	+	1.96688i
$652$		0			0
$653$	−9.67576			−0.378642			−0.189321	−	0.981915i	$-0.560629\pi$
$653$	−9.67576			−0.378642			−0.189321	+	0.981915i	$0.560629\pi$
$654$		0			0
$655$			2.82452i			0.110363i
$656$		0			0
$657$	−1.19384	−	0.374348i	−0.0465760	−	0.0146047i
$658$		0			0
$659$		−	18.2791i		−	0.712052i	−0.934476	−	0.356026i	$-0.884131\pi$
$659$		−	18.2791i		−	0.712052i	0.934476	−	0.356026i	$-0.115869\pi$
$660$		0			0
$661$		−	1.38156i		−	0.0537364i	−0.999639	−	0.0268682i	$-0.991447\pi$
$661$		−	1.38156i		−	0.0537364i	0.999639	−	0.0268682i	$-0.00855344\pi$
$662$		0			0
$663$	−0.580046	+	3.78847i	−0.0225271	+	0.147132i
$664$		0			0
$665$			1.02670i			0.0398138i
$666$		0			0
$667$	8.77100			0.339615
$668$		0			0
$669$	−28.6472	−	4.38613i	−1.10757	−	0.169578i
$670$		0			0
$671$	12.1435			0.468793
$672$		0			0
$673$	−21.5643			−0.831243			−0.415621	−	0.909538i	$-0.636436\pi$
$673$	−21.5643			−0.831243			−0.415621	+	0.909538i	$0.636436\pi$
$674$		0			0
$675$	−11.3969	+	23.2490i	−0.438668	+	0.894856i
$676$		0			0
$677$	−24.7287			−0.950401			−0.475201	−	0.879878i	$-0.657624\pi$
$677$	−24.7287			−0.950401			−0.475201	+	0.879878i	$0.657624\pi$
$678$		0			0
$679$		−	17.1295i		−	0.657368i
$680$		0			0
$681$	9.59081	+	1.46843i	0.367520	+	0.0562704i
$682$		0			0
$683$		−	35.4360i		−	1.35592i	−0.735098	−	0.677961i	$-0.762864\pi$
$683$		−	35.4360i		−	1.35592i	0.735098	−	0.677961i	$-0.237136\pi$
$684$		0			0
$685$			1.95250i			0.0746013i
$686$		0			0
$687$	30.7415	+	4.70677i	1.17286	+	0.179575i
$688$		0			0
$689$		−	1.85302i		−	0.0705943i
$690$		0			0
$691$	−47.3180			−1.80006			−0.900032	−	0.435824i	$-0.856457\pi$
$691$	−47.3180			−1.80006			−0.900032	+	0.435824i	$0.856457\pi$
$692$		0			0
$693$	47.2313	+	14.8102i	1.79417	+	0.562592i
$694$		0			0
$695$	1.50923			0.0572483
$696$		0			0
$697$	−1.12857			−0.0427477
$698$		0			0
$699$	−41.9995	−	6.43046i	−1.58857	−	0.243222i
$700$		0			0
$701$	30.6619			1.15808			0.579042	−	0.815298i	$-0.303427\pi$
$701$	30.6619			1.15808			0.579042	+	0.815298i	$0.303427\pi$
$702$		0			0
$703$			21.0282i			0.793094i
$704$		0			0
$705$	−0.116724	+	0.762360i	−0.00439606	+	0.0287121i
$706$		0			0
$707$		−	14.3225i		−	0.538652i
$708$		0			0
$709$			19.6131i			0.736586i	0.929710	+	0.368293i	$0.120058\pi$
$709$			19.6131i			0.736586i	−0.929710	+	0.368293i	$0.879942\pi$
$710$		0			0
$711$	−2.77906	+	8.86272i	−0.104223	+	0.332378i
$712$		0			0
$713$		−	7.75722i		−	0.290510i
$714$		0			0
$715$	−3.13671			−0.117306
$716$		0			0
$717$	−0.978182	+	6.38883i	−0.0365309	+	0.238595i
$718$		0			0
$719$	−42.9444			−1.60156			−0.800778	−	0.598961i	$-0.795580\pi$
$719$	−42.9444			−1.60156			−0.800778	+	0.598961i	$0.795580\pi$
$720$		0			0
$721$	−25.7353			−0.958432
$722$		0			0
$723$	−2.34008	+	15.2838i	−0.0870285	+	0.568412i
$724$		0			0
$725$	43.7056			1.62318
$726$		0			0
$727$		−	51.8477i		−	1.92292i	−0.274937	−	0.961462i	$-0.588657\pi$
$727$		−	51.8477i		−	1.92292i	0.274937	−	0.961462i	$-0.411343\pi$
$728$		0			0
$729$	−16.5376	−	21.3426i	−0.612505	−	0.790466i
$730$		0			0
$731$			1.23798i			0.0457882i
$732$		0			0
$733$		−	25.2940i		−	0.934254i	−0.884190	−	0.467127i	$-0.845289\pi$
$733$		−	25.2940i		−	0.934254i	0.884190	−	0.467127i	$-0.154711\pi$
$734$		0			0
$735$	−0.249022	+	1.62644i	−0.00918532	+	0.0599923i
$736$		0			0
$737$			55.9926i			2.06251i
$738$		0			0
$739$	−2.11141			−0.0776696			−0.0388348	−	0.999246i	$-0.512365\pi$
$739$	−2.11141			−0.0776696			−0.0388348	+	0.999246i	$0.512365\pi$
$740$		0			0
$741$	−19.6148	−	3.00318i	−0.720566	−	0.110325i
$742$		0			0
$743$	6.40799			0.235087			0.117543	−	0.993068i	$-0.462498\pi$
$743$	6.40799			0.235087			0.117543	+	0.993068i	$0.462498\pi$
$744$		0			0
$745$	1.71549			0.0628508
$746$		0			0
$747$	3.22282	−	10.2779i	0.117917	−	0.376049i
$748$		0			0
$749$	25.5015			0.931805
$750$		0			0
$751$			24.3046i			0.886888i	0.896302	+	0.443444i	$0.146244\pi$
$751$			24.3046i			0.886888i	−0.896302	+	0.443444i	$0.853756\pi$
$752$		0			0
$753$	25.7630	+	3.94453i	0.938857	+	0.143747i
$754$		0			0
$755$		−	0.143780i		−	0.00523269i
$756$		0			0
$757$		−	18.6833i		−	0.679054i	−0.940596	−	0.339527i	$-0.889733\pi$
$757$		−	18.6833i		−	0.679054i	0.940596	−	0.339527i	$-0.110267\pi$
$758$		0			0
$759$	−7.47599	−	1.14463i	−0.271361	−	0.0415476i
$760$		0			0
$761$			30.2497i			1.09655i	0.836297	+	0.548276i	$0.184716\pi$
$761$			30.2497i			1.09655i	−0.836297	+	0.548276i	$0.815284\pi$
$762$		0			0
$763$	69.2288			2.50625
$764$		0			0
$765$	0.0471062	−	0.150227i	0.00170313	−	0.00543146i
$766$		0			0
$767$	−53.2756			−1.92367
$768$		0			0
$769$	47.4784			1.71211			0.856057	−	0.516882i	$-0.172907\pi$
$769$	47.4784			1.71211			0.856057	+	0.516882i	$0.172907\pi$
$770$		0			0
$771$	13.4270	+	2.05578i	0.483560	+	0.0740370i
$772$		0			0
$773$	30.2729			1.08884			0.544421	−	0.838812i	$-0.316750\pi$
$773$	30.2729			1.08884			0.544421	+	0.838812i	$0.316750\pi$
$774$		0			0
$775$		−	38.6539i		−	1.38849i
$776$		0			0
$777$	−10.0057	+	65.3508i	−0.358954	+	2.34445i
$778$		0			0
$779$		−	5.84316i		−	0.209353i
$780$		0			0
$781$			35.7644i			1.27975i
$782$		0			0
$783$	−20.0608	+	40.9229i	−0.716916	+	1.46247i
$784$		0			0
$785$			2.26294i			0.0807679i
$786$		0			0
$787$	11.7940			0.420411			0.210205	−	0.977657i	$-0.432587\pi$
$787$	11.7940			0.420411			0.210205	+	0.977657i	$0.432587\pi$
$788$		0			0
$789$	4.88512	−	31.9064i	0.173915	−	1.13590i
$790$		0			0
$791$	−54.2079			−1.92741
$792$		0			0
$793$	−15.3053			−0.543508
$794$		0			0
$795$	−0.0115203	+	0.0752428i	−0.000408582	+	0.00266859i
$796$		0			0
$797$	13.6252			0.482629			0.241314	−	0.970447i	$-0.422421\pi$
$797$	13.6252			0.482629			0.241314	+	0.970447i	$0.422421\pi$
$798$		0			0
$799$			1.37161i			0.0485241i
$800$		0			0
$801$	9.96664	−	31.7847i	0.352154	−	1.12306i
$802$		0			0
$803$		−	1.82108i		−	0.0642645i
$804$		0			0
$805$		−	0.493208i		−	0.0173833i
$806$		0			0
$807$	−1.50091	+	9.80297i	−0.0528347	+	0.345081i
$808$		0			0
$809$			5.37668i			0.189034i	0.995523	+	0.0945170i	$0.0301307\pi$
$809$			5.37668i			0.189034i	−0.995523	+	0.0945170i	$0.969869\pi$
$810$		0			0
$811$	20.2058			0.709521			0.354761	−	0.934957i	$-0.384562\pi$
$811$	20.2058			0.709521			0.354761	+	0.934957i	$0.384562\pi$
$812$		0			0
$813$	−17.1975	−	2.63308i	−0.603144	−	0.0923463i
$814$		0			0
$815$	−1.36296			−0.0477425
$816$		0			0
$817$	−6.40960			−0.224244
$818$		0			0
$819$	−59.5291	−	18.6664i	−2.08012	−	0.652256i
$820$		0			0
$821$	−26.7962			−0.935193			−0.467596	−	0.883942i	$-0.654880\pi$
$821$	−26.7962			−0.935193			−0.467596	+	0.883942i	$0.654880\pi$
$822$		0			0
$823$			1.87513i			0.0653629i	0.999466	+	0.0326815i	$0.0104047\pi$
$823$			1.87513i			0.0653629i	−0.999466	+	0.0326815i	$0.989595\pi$
$824$		0			0
$825$	−37.2526	−	5.70367i	−1.29697	−	0.198576i
$826$		0			0
$827$			53.8718i			1.87331i	0.350260	+	0.936653i	$0.386093\pi$
$827$			53.8718i			1.87331i	−0.350260	+	0.936653i	$0.613907\pi$
$828$		0			0
$829$			36.9767i			1.28425i	0.766599	+	0.642127i	$0.221948\pi$
$829$			36.9767i			1.28425i	−0.766599	+	0.642127i	$0.778052\pi$
$830$		0			0
$831$	21.7255	+	3.32635i	0.753648	+	0.115390i
$832$		0			0
$833$			2.92624i			0.101388i
$834$		0			0
$835$	−0.217993			−0.00754396
$836$		0			0
$837$	36.1929	+	17.7421i	1.25101	+	0.613258i
$838$		0			0
$839$	−12.1566			−0.419693			−0.209847	−	0.977734i	$-0.567296\pi$
$839$	−12.1566			−0.419693			−0.209847	+	0.977734i	$0.567296\pi$
$840$		0			0
$841$	47.9305			1.65278
$842$		0			0
$843$	−44.2733	−	6.77860i	−1.52485	−	0.233468i
$844$		0			0
$845$	2.25660			0.0776294
$846$		0			0
$847$			30.4817i			1.04736i
$848$		0			0
$849$	5.52840	−	36.1078i	0.189734	−	1.23922i
$850$		0			0
$851$		−	10.1015i		−	0.346277i
$852$		0			0
$853$		−	19.4838i		−	0.667114i	−0.942730	−	0.333557i	$-0.891751\pi$
$853$		−	19.4838i		−	0.667114i	0.942730	−	0.333557i	$-0.108249\pi$
$854$		0			0
$855$	0.777797	+	0.243891i	0.0266001	+	0.00834091i
$856$		0			0
$857$		−	23.9656i		−	0.818649i	−0.912389	−	0.409324i	$-0.865764\pi$
$857$		−	23.9656i		−	0.818649i	0.912389	−	0.409324i	$-0.134236\pi$
$858$		0			0
$859$	3.22143			0.109914			0.0549568	−	0.998489i	$-0.482498\pi$
$859$	3.22143			0.109914			0.0549568	+	0.998489i	$0.482498\pi$
$860$		0			0
$861$	2.78032	−	18.1592i	0.0947532	−	0.618864i
$862$		0			0
$863$	−28.5105			−0.970509			−0.485255	−	0.874373i	$-0.661273\pi$
$863$	−28.5105			−0.970509			−0.485255	+	0.874373i	$0.661273\pi$
$864$		0			0
$865$	1.11627			0.0379545
$866$		0			0
$867$	−4.41394	+	28.8289i	−0.149905	+	0.979081i
$868$		0			0
$869$	−13.5192			−0.458607
$870$		0			0
$871$		−	70.5717i		−	2.39123i
$872$		0			0
$873$	−12.9767	−	4.06908i	−0.439196	−	0.137717i
$874$		0			0
$875$		−	4.92368i		−	0.166451i
$876$		0			0
$877$			26.6860i			0.901123i	0.892746	+	0.450561i	$0.148776\pi$
$877$			26.6860i			0.901123i	−0.892746	+	0.450561i	$0.851224\pi$
$878$		0			0
$879$	−4.60894	+	30.1025i	−0.155456	+	1.01533i
$880$		0			0
$881$			7.40195i			0.249378i	0.992196	+	0.124689i	$0.0397933\pi$
$881$			7.40195i			0.249378i	−0.992196	+	0.124689i	$0.960207\pi$
$882$		0			0
$883$	7.09453			0.238750			0.119375	−	0.992849i	$-0.461911\pi$
$883$	7.09453			0.238750			0.119375	+	0.992849i	$0.461911\pi$
$884$		0			0
$885$	2.16329	+	0.331217i	0.0727180	+	0.0111337i
$886$		0			0
$887$	−27.4157			−0.920529			−0.460264	−	0.887782i	$-0.652245\pi$
$887$	−27.4157			−0.920529			−0.460264	+	0.887782i	$0.652245\pi$
$888$		0			0
$889$	−51.0834			−1.71328
$890$		0			0
$891$	22.4394	−	32.2628i	0.751749	−	1.08084i
$892$		0			0
$893$	−7.10150			−0.237643
$894$		0			0
$895$			1.92243i			0.0642597i
$896$		0			0
$897$	9.42255	+	1.44267i	0.314610	+	0.0481693i
$898$		0			0
$899$		−	68.0386i		−	2.26921i
$900$		0			0
$901$			0.135374i			0.00450997i
$902$		0			0
$903$	−19.9196	−	3.04985i	−0.662881	−	0.101493i
$904$		0			0
$905$		−	2.83577i		−	0.0942643i
$906$		0			0
$907$	−17.7447			−0.589203			−0.294601	−	0.955620i	$-0.595187\pi$
$907$	−17.7447			−0.589203			−0.294601	+	0.955620i	$0.595187\pi$
$908$		0			0
$909$	−10.8502	−	3.40228i	−0.359880	−	0.112846i
$910$		0			0
$911$	40.5517			1.34354			0.671769	−	0.740761i	$-0.265535\pi$
$911$	40.5517			1.34354			0.671769	+	0.740761i	$0.265535\pi$
$912$		0			0
$913$	15.6779			0.518864
$914$		0			0
$915$	0.621481	+	0.0951538i	0.0205455	+	0.00314569i
$916$		0			0
$917$	−81.7681			−2.70022
$918$		0			0
$919$			42.2700i			1.39436i	0.716896	+	0.697180i	$0.245562\pi$
$919$			42.2700i			1.39436i	−0.716896	+	0.697180i	$0.754438\pi$
$920$		0			0
$921$	−3.15849	+	20.6291i	−0.104076	+	0.679753i
$922$		0			0
$923$		−	45.0766i		−	1.48371i
$924$		0			0
$925$		−	50.3356i		−	1.65503i
$926$		0			0
$927$	−6.11337	+	19.4962i	−0.200789	+	0.640340i
$928$		0			0
$929$			28.1066i			0.922147i	0.887362	+	0.461073i	$0.152536\pi$
$929$			28.1066i			0.922147i	−0.887362	+	0.461073i	$0.847464\pi$
$930$		0			0
$931$	−15.1506			−0.496540
$932$		0			0
$933$	−2.33858	+	15.2740i	−0.0765615	+	0.500049i
$934$		0			0
$935$	0.229156			0.00749420
$936$		0			0
$937$	19.6450			0.641773			0.320887	−	0.947118i	$-0.396019\pi$
$937$	19.6450			0.641773			0.320887	+	0.947118i	$0.396019\pi$
$938$		0			0
$939$	−0.126235	+	0.824485i	−0.00411954	+	0.0269061i
$940$		0			0
$941$	39.9096			1.30101			0.650507	−	0.759500i	$-0.274556\pi$
$941$	39.9096			1.30101			0.650507	+	0.759500i	$0.274556\pi$
$942$		0			0
$943$			2.80694i			0.0914066i
$944$		0			0
$945$	2.30116	+	1.12805i	0.0748568	+	0.0366956i
$946$		0			0
$947$			40.6728i			1.32169i	0.750523	+	0.660845i	$0.229802\pi$
$947$			40.6728i			1.32169i	−0.750523	+	0.660845i	$0.770198\pi$
$948$		0			0
$949$			2.29524i			0.0745067i
$950$		0			0
$951$	1.69529	−	11.0725i	0.0549736	−	0.359051i
$952$		0			0
$953$		−	10.4493i		−	0.338484i	−0.985574	−	0.169242i	$-0.945868\pi$
$953$		−	10.4493i		−	0.338484i	0.985574	−	0.169242i	$-0.0541320\pi$
$954$		0			0
$955$	1.39828			0.0452474
$956$		0			0
$957$	−65.5719	−	10.0396i	−2.11964	−	0.324534i
$958$		0			0
$959$	−56.5237			−1.82525
$960$		0			0
$961$	−29.1744			−0.941111
$962$		0			0
$963$	6.05784	−	19.3191i	0.195211	−	0.622550i
$964$		0			0
$965$	−1.93403			−0.0622585
$966$		0			0
$967$		−	10.6501i		−	0.342484i	−0.985229	−	0.171242i	$-0.945222\pi$
$967$		−	10.6501i		−	0.342484i	0.985229	−	0.171242i	$-0.0547779\pi$
$968$		0			0
$969$	1.43298	+	0.219401i	0.0460339	+	0.00704816i
$970$		0			0
$971$		−	27.4861i		−	0.882070i	−0.897490	−	0.441035i	$-0.854611\pi$
$971$		−	27.4861i		−	0.882070i	0.897490	−	0.441035i	$-0.145389\pi$
$972$		0			0
$973$			43.6912i			1.40068i
$974$		0			0
$975$	46.9522	+	7.18877i	1.50367	+	0.230225i
$976$		0			0
$977$		−	49.0762i		−	1.57009i	−0.619441	−	0.785043i	$-0.712641\pi$
$977$		−	49.0762i		−	1.57009i	0.619441	−	0.785043i	$-0.287359\pi$
$978$		0			0
$979$	48.4844			1.54957
$980$		0			0
$981$	16.4452	−	52.4455i	0.525055	−	1.67446i
$982$		0			0
$983$	−26.9525			−0.859651			−0.429825	−	0.902912i	$-0.641425\pi$
$983$	−26.9525			−0.859651			−0.429825	+	0.902912i	$0.641425\pi$
$984$		0			0
$985$	2.28756			0.0728879
$986$		0			0
$987$	−22.0698	−	3.37907i	−0.702490	−	0.107557i
$988$		0			0
$989$	3.07905			0.0979080
$990$		0			0
$991$			44.4537i			1.41212i	0.708153	+	0.706059i	$0.249529\pi$
$991$			44.4537i			1.41212i	−0.708153	+	0.706059i	$0.750471\pi$
$992$		0			0
$993$	6.58269	−	42.9937i	0.208895	−	1.36437i
$994$		0			0
$995$		−	3.04182i		−	0.0964323i
$996$		0			0
$997$		−	10.5458i		−	0.333989i	−0.985958	−	0.166994i	$-0.946594\pi$
$997$		−	10.5458i		−	0.333989i	0.985958	−	0.166994i	$-0.0534062\pi$
$998$		0			0
$999$	47.1308	+	23.1040i	1.49115	+	0.730979i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2208.2.j.d.47.20		42
3.2	odd	2		2208.2.j.c.47.19		42
4.3	odd	2		552.2.j.d.323.19	yes	42
8.3	odd	2		2208.2.j.c.47.20		42
8.5	even	2		552.2.j.c.323.23	✓	42
12.11	even	2		552.2.j.c.323.24	yes	42
24.5	odd	2		552.2.j.d.323.20	yes	42
24.11	even	2	inner	2208.2.j.d.47.19		42

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
552.2.j.c.323.23	✓	42	8.5	even	2
552.2.j.c.323.24	yes	42	12.11	even	2
552.2.j.d.323.19	yes	42	4.3	odd	2
552.2.j.d.323.20	yes	42	24.5	odd	2
2208.2.j.c.47.19		42	3.2	odd	2
2208.2.j.c.47.20		42	8.3	odd	2
2208.2.j.d.47.19		42	24.11	even	2	inner
2208.2.j.d.47.20		42	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2208 = 2^{5} \cdot 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2208.j (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.262136 + 1.71210i) q^{3} -0.130526 q^{5} -3.77863i q^{7} +(-2.86257 - 0.897607i) q^{9} -4.36656i q^{11} +5.50350i q^{13} +(0.0342155 - 0.223473i) q^{15} -0.402065i q^{17} +2.08168 q^{19} +(6.46939 + 0.990517i) q^{21} -1.00000 q^{23} -4.98296 q^{25} +(2.28718 - 4.66571i) q^{27} -8.77100 q^{29} +7.75722i q^{31} +(7.47599 + 1.14463i) q^{33} +0.493208i q^{35} +10.1015i q^{37} +(-9.42255 - 1.44267i) q^{39} -2.80694i q^{41} -3.07905 q^{43} +(0.373638 + 0.117161i) q^{45} -3.41142 q^{47} -7.27804 q^{49} +(0.688374 + 0.105396i) q^{51} -0.336698 q^{53} +0.569948i q^{55} +(-0.545685 + 3.56405i) q^{57} +9.68031i q^{59} +2.78101i q^{61} +(-3.39173 + 10.8166i) q^{63} -0.718348i q^{65} -12.8230 q^{67} +(0.262136 - 1.71210i) q^{69} -8.19053 q^{71} +0.417051 q^{73} +(1.30622 - 8.53133i) q^{75} -16.4996 q^{77} -3.09607i q^{79} +(7.38860 + 5.13893i) q^{81} +3.59045i q^{83} +0.0524797i q^{85} +(2.29920 - 15.0168i) q^{87} +11.1036i q^{89} +20.7957 q^{91} +(-13.2811 - 2.03345i) q^{93} -0.271713 q^{95} +4.53325 q^{97} +(-3.91946 + 12.4996i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 42 q - 2 q^{3} + 8 q^{5} + 2 q^{9} + 8 q^{15} - 4 q^{19} + 8 q^{21} - 42 q^{23} + 22 q^{25} + 16 q^{27} + 12 q^{33} - 8 q^{39} - 28 q^{43} - 8 q^{45} - 50 q^{49} - 28 q^{51} + 24 q^{53} - 8 q^{57} + 16 q^{63}+ \cdots + 20 q^{97}+O(q^{100}) \) 42 * q - 2 * q^3 + 8 * q^5 + 2 * q^9 + 8 * q^15 - 4 * q^19 + 8 * q^21 - 42 * q^23 + 22 * q^25 + 16 * q^27 + 12 * q^33 - 8 * q^39 - 28 * q^43 - 8 * q^45 - 50 * q^49 - 28 * q^51 + 24 * q^53 - 8 * q^57 + 16 * q^63 + 4 * q^67 + 2 * q^69 + 4 * q^73 + 6 * q^75 + 32 * q^77 + 18 * q^81 + 48 * q^87 + 8 * q^91 - 22 * q^93 + 16 * q^95 + 20 * q^97

Introduction

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