LMFDB - Embedded newform 2208.2.n.b.367.13

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2208,2,Mod(367,2208)] 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("2208.367"); 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, 0, 1])) N = Newforms(chi, 2, names="a")

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

Newform invariants

comment:select newform

sage:traces = [24,0,24] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	no
Analytic conductor:	$17.6309687663$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	no (minimal twist has level 552)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			367.13
Character	$\chi$	$=$	2208.367
Dual form			2208.2.n.b.367.14

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +0.901487 q^{5} -1.56211 q^{7} +1.00000 q^{9} -3.69478i q^{11} +2.36562i q^{13} +0.901487 q^{15} +0.473035i q^{17} -7.66256i q^{19} -1.56211 q^{21} +(4.73018 - 0.790796i) q^{23} -4.18732 q^{25} +1.00000 q^{27} -4.29403i q^{29} +1.83523i q^{31} -3.69478i q^{33} -1.40822 q^{35} +5.81865 q^{37} +2.36562i q^{39} +2.62473 q^{41} -7.06267i q^{43} +0.901487 q^{45} -4.27804i q^{47} -4.55981 q^{49} +0.473035i q^{51} -4.64065 q^{53} -3.33080i q^{55} -7.66256i q^{57} +7.81205 q^{59} +11.9981 q^{61} -1.56211 q^{63} +2.13258i q^{65} -5.35433i q^{67} +(4.73018 - 0.790796i) q^{69} +6.95616i q^{71} +11.2808 q^{73} -4.18732 q^{75} +5.77167i q^{77} -15.8447 q^{79} +1.00000 q^{81} -7.33508i q^{83} +0.426435i q^{85} -4.29403i q^{87} -4.91384i q^{89} -3.69537i q^{91} +1.83523i q^{93} -6.90769i q^{95} -15.0308i q^{97} -3.69478i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 24 q^{3} + 24 q^{9} + 24 q^{25} + 24 q^{27} + 56 q^{49} + 32 q^{73} + 24 q^{75} + 24 q^{81}+O(q^{100}) $ 24 * q + 24 * q^3 + 24 * q^9 + 24 * q^25 + 24 * q^27 + 56 * q^49 + 32 * q^73 + 24 * q^75 + 24 * q^81

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
$3$	1.00000			0.577350
$3$	1.00000			0.577350
$4$		0			0
$5$	0.901487			0.403157			0.201579	−	0.979472i	$-0.435393\pi$
$5$	0.901487			0.403157			0.201579	+	0.979472i	$0.435393\pi$
$6$		0			0
$7$	−1.56211			−0.590423			−0.295211	−	0.955432i	$-0.595390\pi$
$7$	−1.56211			−0.590423			−0.295211	+	0.955432i	$0.595390\pi$
$8$		0			0
$9$	1.00000			0.333333
$10$		0			0
$11$		−	3.69478i		−	1.11402i	−0.830506	−	0.557010i	$-0.811949\pi$
$11$		−	3.69478i		−	1.11402i	0.830506	−	0.557010i	$-0.188051\pi$
$12$		0			0
$13$			2.36562i			0.656106i	0.944659	+	0.328053i	$0.106392\pi$
$13$			2.36562i			0.656106i	−0.944659	+	0.328053i	$0.893608\pi$
$14$		0			0
$15$	0.901487			0.232763
$16$		0			0
$17$			0.473035i			0.114728i	0.998353	+	0.0573639i	$0.0182695\pi$
$17$			0.473035i			0.114728i	−0.998353	+	0.0573639i	$0.981730\pi$
$18$		0			0
$19$		−	7.66256i		−	1.75791i	−0.476904	−	0.878955i	$-0.658241\pi$
$19$		−	7.66256i		−	1.75791i	0.476904	−	0.878955i	$-0.341759\pi$
$20$		0			0
$21$	−1.56211			−0.340881
$22$		0			0
$23$	4.73018	−	0.790796i	0.986312	−	0.164892i
$23$	4.73018	−	0.790796i	0.986312	−	0.164892i
$24$		0			0
$25$	−4.18732			−0.837464
$26$		0			0
$27$	1.00000			0.192450
$28$		0			0
$29$		−	4.29403i		−	0.797382i	−0.917085	−	0.398691i	$-0.869465\pi$
$29$		−	4.29403i		−	0.797382i	0.917085	−	0.398691i	$-0.130535\pi$
$30$		0			0
$31$			1.83523i			0.329617i	0.986326	+	0.164809i	$0.0527006\pi$
$31$			1.83523i			0.329617i	−0.986326	+	0.164809i	$0.947299\pi$
$32$		0			0
$33$		−	3.69478i		−	0.643179i
$34$		0			0
$35$	−1.40822			−0.238033
$36$		0			0
$37$	5.81865			0.956580			0.478290	−	0.878202i	$-0.341257\pi$
$37$	5.81865			0.956580			0.478290	+	0.878202i	$0.341257\pi$
$38$		0			0
$39$			2.36562i			0.378803i
$40$		0			0
$41$	2.62473			0.409913			0.204957	−	0.978771i	$-0.434295\pi$
$41$	2.62473			0.409913			0.204957	+	0.978771i	$0.434295\pi$
$42$		0			0
$43$		−	7.06267i		−	1.07705i	−0.842610	−	0.538524i	$-0.818982\pi$
$43$		−	7.06267i		−	1.07705i	0.842610	−	0.538524i	$-0.181018\pi$
$44$		0			0
$45$	0.901487			0.134386
$46$		0			0
$47$		−	4.27804i		−	0.624016i	−0.950079	−	0.312008i	$-0.898998\pi$
$47$		−	4.27804i		−	0.624016i	0.950079	−	0.312008i	$-0.101002\pi$
$48$		0			0
$49$	−4.55981			−0.651401
$50$		0			0
$51$			0.473035i			0.0662381i
$52$		0			0
$53$	−4.64065			−0.637442			−0.318721	−	0.947849i	$-0.603253\pi$
$53$	−4.64065			−0.637442			−0.318721	+	0.947849i	$0.603253\pi$
$54$		0			0
$55$		−	3.33080i		−	0.449125i
$56$		0			0
$57$		−	7.66256i		−	1.01493i
$58$		0			0
$59$	7.81205			1.01704			0.508521	−	0.861050i	$-0.330192\pi$
$59$	7.81205			1.01704			0.508521	+	0.861050i	$0.330192\pi$
$60$		0			0
$61$	11.9981			1.53620			0.768098	−	0.640332i	$-0.221203\pi$
$61$	11.9981			1.53620			0.768098	+	0.640332i	$0.221203\pi$
$62$		0			0
$63$	−1.56211			−0.196808
$64$		0			0
$65$			2.13258i			0.264514i
$66$		0			0
$67$		−	5.35433i		−	0.654135i	−0.945001	−	0.327068i	$-0.893939\pi$
$67$		−	5.35433i		−	0.654135i	0.945001	−	0.327068i	$-0.106061\pi$
$68$		0			0
$69$	4.73018	−	0.790796i	0.569447	−	0.0952006i
$70$		0			0
$71$			6.95616i			0.825544i	0.910834	+	0.412772i	$0.135439\pi$
$71$			6.95616i			0.825544i	−0.910834	+	0.412772i	$0.864561\pi$
$72$		0			0
$73$	11.2808			1.32032			0.660158	−	0.751127i	$-0.270489\pi$
$73$	11.2808			1.32032			0.660158	+	0.751127i	$0.270489\pi$
$74$		0			0
$75$	−4.18732			−0.483510
$76$		0			0
$77$			5.77167i			0.657742i
$78$		0			0
$79$	−15.8447			−1.78267			−0.891335	−	0.453345i	$-0.850230\pi$
$79$	−15.8447			−1.78267			−0.891335	+	0.453345i	$0.850230\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$		−	7.33508i		−	0.805129i	−0.915392	−	0.402565i	$-0.868119\pi$
$83$		−	7.33508i		−	0.805129i	0.915392	−	0.402565i	$-0.131881\pi$
$84$		0			0
$85$			0.426435i			0.0462533i
$86$		0			0
$87$		−	4.29403i		−	0.460369i
$88$		0			0
$89$		−	4.91384i		−	0.520866i	−0.965492	−	0.260433i	$-0.916135\pi$
$89$		−	4.91384i		−	0.520866i	0.965492	−	0.260433i	$-0.0838653\pi$
$90$		0			0
$91$		−	3.69537i		−	0.387380i
$92$		0			0
$93$			1.83523i			0.190305i
$94$		0			0
$95$		−	6.90769i		−	0.708714i
$96$		0			0
$97$		−	15.0308i		−	1.52614i	−0.646314	−	0.763071i	$-0.723691\pi$
$97$		−	15.0308i		−	1.52614i	0.646314	−	0.763071i	$-0.276309\pi$
$98$		0			0
$99$		−	3.69478i		−	0.371340i
$100$		0			0
$101$		−	1.47763i		−	0.147030i	−0.997294	−	0.0735149i	$-0.976578\pi$
$101$		−	1.47763i		−	0.147030i	0.997294	−	0.0735149i	$-0.0234217\pi$
$102$		0			0
$103$	−3.82331			−0.376722			−0.188361	−	0.982100i	$-0.560317\pi$
$103$	−3.82331			−0.376722			−0.188361	+	0.982100i	$0.560317\pi$
$104$		0			0
$105$	−1.40822			−0.137429
$106$		0			0
$107$			8.57550i			0.829024i	0.910044	+	0.414512i	$0.136048\pi$
$107$			8.57550i			0.829024i	−0.910044	+	0.414512i	$0.863952\pi$
$108$		0			0
$109$	15.2790			1.46346			0.731732	−	0.681592i	$-0.238712\pi$
$109$	15.2790			1.46346			0.731732	+	0.681592i	$0.238712\pi$
$110$		0			0
$111$	5.81865			0.552282
$112$		0			0
$113$			5.89000i			0.554085i	0.960858	+	0.277042i	$0.0893542\pi$
$113$			5.89000i			0.554085i	−0.960858	+	0.277042i	$0.910646\pi$
$114$		0			0
$115$	4.26420	−	0.712892i	0.397639	−	0.0664775i
$116$		0			0
$117$			2.36562i			0.218702i
$118$		0			0
$119$		−	0.738933i		−	0.0677379i
$120$		0			0
$121$	−2.65143			−0.241039
$122$		0			0
$123$	2.62473			0.236664
$124$		0			0
$125$	−8.28225			−0.740787
$126$		0			0
$127$			18.7735i			1.66588i	0.553364	+	0.832939i	$0.313344\pi$
$127$			18.7735i			1.66588i	−0.553364	+	0.832939i	$0.686656\pi$
$128$		0			0
$129$		−	7.06267i		−	0.621834i
$130$		0			0
$131$	−6.40382			−0.559505			−0.279752	−	0.960072i	$-0.590252\pi$
$131$	−6.40382			−0.559505			−0.279752	+	0.960072i	$0.590252\pi$
$132$		0			0
$133$			11.9698i			1.03791i
$134$		0			0
$135$	0.901487			0.0775876
$136$		0			0
$137$			3.16726i			0.270597i	0.990805	+	0.135299i	$0.0431994\pi$
$137$			3.16726i			0.270597i	−0.990805	+	0.135299i	$0.956801\pi$
$138$		0			0
$139$	−4.84346			−0.410817			−0.205408	−	0.978676i	$-0.565852\pi$
$139$	−4.84346			−0.410817			−0.205408	+	0.978676i	$0.565852\pi$
$140$		0			0
$141$		−	4.27804i		−	0.360276i
$142$		0			0
$143$	8.74047			0.730915
$144$		0			0
$145$		−	3.87102i		−	0.321470i
$146$		0			0
$147$	−4.55981			−0.376087
$148$		0			0
$149$	−3.34176			−0.273767			−0.136884	−	0.990587i	$-0.543709\pi$
$149$	−3.34176			−0.273767			−0.136884	+	0.990587i	$0.543709\pi$
$150$		0			0
$151$		−	17.3731i		−	1.41380i	−0.707314	−	0.706900i	$-0.750093\pi$
$151$		−	17.3731i		−	1.41380i	0.707314	−	0.706900i	$-0.249907\pi$
$152$		0			0
$153$			0.473035i			0.0382426i
$154$		0			0
$155$			1.65444i			0.132888i
$156$		0			0
$157$	4.88151			0.389587			0.194793	−	0.980844i	$-0.437596\pi$
$157$	4.88151			0.389587			0.194793	+	0.980844i	$0.437596\pi$
$158$		0			0
$159$	−4.64065			−0.368027
$160$		0			0
$161$	−7.38908	+	1.23531i	−0.582341	+	0.0973561i
$162$		0			0
$163$	22.1556			1.73536			0.867680	−	0.497124i	$-0.165611\pi$
$163$	22.1556			1.73536			0.867680	+	0.497124i	$0.165611\pi$
$164$		0			0
$165$		−	3.33080i		−	0.259302i
$166$		0			0
$167$			21.3490i			1.65203i	0.563645	+	0.826017i	$0.309399\pi$
$167$			21.3490i			1.65203i	−0.563645	+	0.826017i	$0.690601\pi$
$168$		0			0
$169$	7.40382			0.569525
$170$		0			0
$171$		−	7.66256i		−	0.585970i
$172$		0			0
$173$		−	16.6902i		−	1.26893i	−0.772950	−	0.634467i	$-0.781219\pi$
$173$		−	16.6902i		−	1.26893i	0.772950	−	0.634467i	$-0.218781\pi$
$174$		0			0
$175$	6.54106			0.494458
$176$		0			0
$177$	7.81205			0.587189
$178$		0			0
$179$	−13.5240			−1.01083			−0.505416	−	0.862876i	$-0.668661\pi$
$179$	−13.5240			−1.01083			−0.505416	+	0.862876i	$0.668661\pi$
$180$		0			0
$181$	−15.6631			−1.16423			−0.582115	−	0.813106i	$-0.697775\pi$
$181$	−15.6631			−1.16423			−0.582115	+	0.813106i	$0.697775\pi$
$182$		0			0
$183$	11.9981			0.886924
$184$		0			0
$185$	5.24544			0.385652
$186$		0			0
$187$	1.74776			0.127809
$188$		0			0
$189$	−1.56211			−0.113627
$190$		0			0
$191$	−10.5542			−0.763677			−0.381839	−	0.924229i	$-0.624709\pi$
$191$	−10.5542			−0.763677			−0.381839	+	0.924229i	$0.624709\pi$
$192$		0			0
$193$	0.464113			0.0334076			0.0167038	−	0.999860i	$-0.494683\pi$
$193$	0.464113			0.0334076			0.0167038	+	0.999860i	$0.494683\pi$
$194$		0			0
$195$			2.13258i			0.152717i
$196$		0			0
$197$			10.0657i			0.717151i	0.933501	+	0.358576i	$0.116738\pi$
$197$			10.0657i			0.717151i	−0.933501	+	0.358576i	$0.883262\pi$
$198$		0			0
$199$	8.22327			0.582932			0.291466	−	0.956581i	$-0.405857\pi$
$199$	8.22327			0.582932			0.291466	+	0.956581i	$0.405857\pi$
$200$		0			0
$201$		−	5.35433i		−	0.377665i
$202$		0			0
$203$			6.70776i			0.470792i
$204$		0			0
$205$	2.36616			0.165259
$206$		0			0
$207$	4.73018	−	0.790796i	0.328771	−	0.0549641i
$208$		0			0
$209$	−28.3115			−1.95835
$210$		0			0
$211$	−16.1286			−1.11034			−0.555168	−	0.831738i	$-0.687346\pi$
$211$	−16.1286			−1.11034			−0.555168	+	0.831738i	$0.687346\pi$
$212$		0			0
$213$			6.95616i			0.476628i
$214$		0			0
$215$		−	6.36691i		−	0.434220i
$216$		0			0
$217$		−	2.86683i		−	0.194613i
$218$		0			0
$219$	11.2808			0.762285
$220$		0			0
$221$	−1.11902			−0.0752736
$222$		0			0
$223$		−	6.77440i		−	0.453647i	−0.973936	−	0.226824i	$-0.927166\pi$
$223$		−	6.77440i		−	0.453647i	0.973936	−	0.226824i	$-0.0728341\pi$
$224$		0			0
$225$	−4.18732			−0.279155
$226$		0			0
$227$			23.8069i			1.58012i	0.613030	+	0.790059i	$0.289950\pi$
$227$			23.8069i			1.58012i	−0.613030	+	0.790059i	$0.710050\pi$
$228$		0			0
$229$	−10.4208			−0.688628			−0.344314	−	0.938854i	$-0.611889\pi$
$229$	−10.4208			−0.688628			−0.344314	+	0.938854i	$0.611889\pi$
$230$		0			0
$231$			5.77167i			0.379748i
$232$		0			0
$233$	2.52840			0.165641			0.0828205	−	0.996564i	$-0.473607\pi$
$233$	2.52840			0.165641			0.0828205	+	0.996564i	$0.473607\pi$
$234$		0			0
$235$		−	3.85660i		−	0.251577i
$236$		0			0
$237$	−15.8447			−1.02922
$238$		0			0
$239$		−	12.3970i		−	0.801895i	−0.916101	−	0.400948i	$-0.868681\pi$
$239$		−	12.3970i		−	0.801895i	0.916101	−	0.400948i	$-0.131319\pi$
$240$		0			0
$241$		−	6.64297i		−	0.427911i	−0.976843	−	0.213956i	$-0.931365\pi$
$241$		−	6.64297i		−	0.427911i	0.976843	−	0.213956i	$-0.0686348\pi$
$242$		0			0
$243$	1.00000			0.0641500
$244$		0			0
$245$	−4.11061			−0.262617
$246$		0			0
$247$	18.1267			1.15338
$248$		0			0
$249$		−	7.33508i		−	0.464842i
$250$		0			0
$251$			14.0093i			0.884259i	0.896951	+	0.442130i	$0.145777\pi$
$251$			14.0093i			0.884259i	−0.896951	+	0.442130i	$0.854223\pi$
$252$		0			0
$253$	−2.92182	−	17.4770i	−0.183693	−	1.09877i
$254$		0			0
$255$			0.426435i			0.0267044i
$256$		0			0
$257$	17.0620			1.06430			0.532150	−	0.846650i	$-0.321384\pi$
$257$	17.0620			1.06430			0.532150	+	0.846650i	$0.321384\pi$
$258$		0			0
$259$	−9.08938			−0.564787
$260$		0			0
$261$		−	4.29403i		−	0.265794i
$262$		0			0
$263$	12.5846			0.775999			0.388000	−	0.921660i	$-0.373166\pi$
$263$	12.5846			0.775999			0.388000	+	0.921660i	$0.373166\pi$
$264$		0			0
$265$	−4.18348			−0.256989
$266$		0			0
$267$		−	4.91384i		−	0.300722i
$268$		0			0
$269$			5.33445i			0.325247i	0.986688	+	0.162624i	$0.0519956\pi$
$269$			5.33445i			0.325247i	−0.986688	+	0.162624i	$0.948004\pi$
$270$		0			0
$271$		−	5.16603i		−	0.313814i	−0.987613	−	0.156907i	$-0.949848\pi$
$271$		−	5.16603i		−	0.313814i	0.987613	−	0.156907i	$-0.0501523\pi$
$272$		0			0
$273$		−	3.69537i		−	0.223654i
$274$		0			0
$275$			15.4713i			0.932952i
$276$		0			0
$277$		−	26.4195i		−	1.58739i	−0.608313	−	0.793697i	$-0.708154\pi$
$277$		−	26.4195i		−	1.58739i	0.608313	−	0.793697i	$-0.291846\pi$
$278$		0			0
$279$			1.83523i			0.109872i
$280$		0			0
$281$			20.8795i			1.24557i	0.782395	+	0.622783i	$0.213998\pi$
$281$			20.8795i			1.24557i	−0.782395	+	0.622783i	$0.786002\pi$
$282$		0			0
$283$		−	11.8930i		−	0.706966i	−0.935441	−	0.353483i	$-0.884997\pi$
$283$		−	11.8930i		−	0.706966i	0.935441	−	0.353483i	$-0.115003\pi$
$284$		0			0
$285$		−	6.90769i		−	0.409176i
$286$		0			0
$287$	−4.10011			−0.242022
$288$		0			0
$289$	16.7762			0.986838
$290$		0			0
$291$		−	15.0308i		−	0.881119i
$292$		0			0
$293$	−29.0826			−1.69902			−0.849512	−	0.527570i	$-0.823103\pi$
$293$	−29.0826			−1.69902			−0.849512	+	0.527570i	$0.823103\pi$
$294$		0			0
$295$	7.04246			0.410028
$296$		0			0
$297$		−	3.69478i		−	0.214393i
$298$		0			0
$299$	1.87072	+	11.1898i	0.108187	+	0.647125i
$300$		0			0
$301$			11.0327i			0.635913i
$302$		0			0
$303$		−	1.47763i		−	0.0848877i
$304$		0			0
$305$	10.8161			0.619329
$306$		0			0
$307$	−0.564207			−0.0322010			−0.0161005	−	0.999870i	$-0.505125\pi$
$307$	−0.564207			−0.0322010			−0.0161005	+	0.999870i	$0.505125\pi$
$308$		0			0
$309$	−3.82331			−0.217500
$310$		0			0
$311$			22.4646i			1.27385i	0.770924	+	0.636927i	$0.219795\pi$
$311$			22.4646i			1.27385i	−0.770924	+	0.636927i	$0.780205\pi$
$312$		0			0
$313$		−	5.00245i		−	0.282755i	−0.989956	−	0.141378i	$-0.954847\pi$
$313$		−	5.00245i		−	0.282755i	0.989956	−	0.141378i	$-0.0451532\pi$
$314$		0			0
$315$	−1.40822			−0.0793444
$316$		0			0
$317$			19.1167i			1.07370i	0.843678	+	0.536849i	$0.180386\pi$
$317$			19.1167i			1.07370i	−0.843678	+	0.536849i	$0.819614\pi$
$318$		0			0
$319$	−15.8655			−0.888299
$320$		0			0
$321$			8.57550i			0.478638i
$322$		0			0
$323$	3.62465			0.201681
$324$		0			0
$325$		−	9.90563i		−	0.549465i
$326$		0			0
$327$	15.2790			0.844932
$328$		0			0
$329$			6.68277i			0.368433i
$330$		0			0
$331$	9.65359			0.530609			0.265305	−	0.964165i	$-0.414527\pi$
$331$	9.65359			0.530609			0.265305	+	0.964165i	$0.414527\pi$
$332$		0			0
$333$	5.81865			0.318860
$334$		0			0
$335$		−	4.82686i		−	0.263719i
$336$		0			0
$337$			15.5246i			0.845678i	0.906205	+	0.422839i	$0.138966\pi$
$337$			15.5246i			0.845678i	−0.906205	+	0.422839i	$0.861034\pi$
$338$		0			0
$339$			5.89000i			0.319901i
$340$		0			0
$341$	6.78078			0.367200
$342$		0			0
$343$	18.0577			0.975025
$344$		0			0
$345$	4.26420	−	0.712892i	0.229577	−	0.0383808i
$346$		0			0
$347$	24.2199			1.30019			0.650095	−	0.759853i	$-0.274729\pi$
$347$	24.2199			1.30019			0.650095	+	0.759853i	$0.274729\pi$
$348$		0			0
$349$			2.99062i			0.160084i	0.996791	+	0.0800421i	$0.0255055\pi$
$349$			2.99062i			0.160084i	−0.996791	+	0.0800421i	$0.974495\pi$
$350$		0			0
$351$			2.36562i			0.126268i
$352$		0			0
$353$	4.59122			0.244366			0.122183	−	0.992508i	$-0.461011\pi$
$353$	4.59122			0.244366			0.122183	+	0.992508i	$0.461011\pi$
$354$		0			0
$355$			6.27089i			0.332824i
$356$		0			0
$357$		−	0.738933i		−	0.0391085i
$358$		0			0
$359$	6.08015			0.320898			0.160449	−	0.987044i	$-0.448706\pi$
$359$	6.08015			0.320898			0.160449	+	0.987044i	$0.448706\pi$
$360$		0			0
$361$	−39.7148			−2.09025
$362$		0			0
$363$	−2.65143			−0.139164
$364$		0			0
$365$	10.1695			0.532295
$366$		0			0
$367$	−21.8559			−1.14087			−0.570433	−	0.821344i	$-0.693225\pi$
$367$	−21.8559			−1.14087			−0.570433	+	0.821344i	$0.693225\pi$
$368$		0			0
$369$	2.62473			0.136638
$370$		0			0
$371$	7.24921			0.376360
$372$		0			0
$373$	−6.69692			−0.346754			−0.173377	−	0.984856i	$-0.555468\pi$
$373$	−6.69692			−0.346754			−0.173377	+	0.984856i	$0.555468\pi$
$374$		0			0
$375$	−8.28225			−0.427694
$376$		0			0
$377$	10.1581			0.523167
$378$		0			0
$379$		−	26.1881i		−	1.34519i	−0.740011	−	0.672595i	$-0.765180\pi$
$379$		−	26.1881i		−	1.34519i	0.740011	−	0.672595i	$-0.234820\pi$
$380$		0			0
$381$			18.7735i			0.961796i
$382$		0			0
$383$	−8.67994			−0.443524			−0.221762	−	0.975101i	$-0.571181\pi$
$383$	−8.67994			−0.443524			−0.221762	+	0.975101i	$0.571181\pi$
$384$		0			0
$385$			5.20308i			0.265174i
$386$		0			0
$387$		−	7.06267i		−	0.359016i
$388$		0			0
$389$	27.5178			1.39521			0.697604	−	0.716484i	$-0.254250\pi$
$389$	27.5178			1.39521			0.697604	+	0.716484i	$0.254250\pi$
$390$		0			0
$391$	0.374074	+	2.23754i	0.0189177	+	0.113157i
$392$		0			0
$393$	−6.40382			−0.323030
$394$		0			0
$395$	−14.2838			−0.718696
$396$		0			0
$397$			33.4878i			1.68070i	0.542042	+	0.840351i	$0.317651\pi$
$397$			33.4878i			1.68070i	−0.542042	+	0.840351i	$0.682349\pi$
$398$		0			0
$399$			11.9698i			0.599238i
$400$		0			0
$401$			27.6454i			1.38054i	0.723550	+	0.690272i	$0.242509\pi$
$401$			27.6454i			1.38054i	−0.723550	+	0.690272i	$0.757491\pi$
$402$		0			0
$403$	−4.34147			−0.216264
$404$		0			0
$405$	0.901487			0.0447952
$406$		0			0
$407$		−	21.4987i		−	1.06565i
$408$		0			0
$409$	−3.78504			−0.187158			−0.0935790	−	0.995612i	$-0.529831\pi$
$409$	−3.78504			−0.187158			−0.0935790	+	0.995612i	$0.529831\pi$
$410$		0			0
$411$			3.16726i			0.156229i
$412$		0			0
$413$	−12.2033			−0.600485
$414$		0			0
$415$		−	6.61248i		−	0.324594i
$416$		0			0
$417$	−4.84346			−0.237185
$418$		0			0
$419$			5.09760i			0.249034i	0.992217	+	0.124517i	$0.0397381\pi$
$419$			5.09760i			0.249034i	−0.992217	+	0.124517i	$0.960262\pi$
$420$		0			0
$421$	19.3072			0.940977			0.470489	−	0.882406i	$-0.344078\pi$
$421$	19.3072			0.940977			0.470489	+	0.882406i	$0.344078\pi$
$422$		0			0
$423$		−	4.27804i		−	0.208005i
$424$		0			0
$425$		−	1.98075i		−	0.0960804i
$426$		0			0
$427$	−18.7423			−0.907005
$428$		0			0
$429$	8.74047			0.421994
$430$		0			0
$431$	−38.3713			−1.84828			−0.924140	−	0.382054i	$-0.875217\pi$
$431$	−38.3713			−1.84828			−0.924140	+	0.382054i	$0.875217\pi$
$432$		0			0
$433$			4.61645i			0.221853i	0.993829	+	0.110926i	$0.0353817\pi$
$433$			4.61645i			0.221853i	−0.993829	+	0.110926i	$0.964618\pi$
$434$		0			0
$435$		−	3.87102i		−	0.185601i
$436$		0			0
$437$	−6.05951	−	36.2453i	−0.289866	−	1.73385i
$438$		0			0
$439$			5.78060i			0.275893i	0.990440	+	0.137946i	$0.0440502\pi$
$439$			5.78060i			0.275893i	−0.990440	+	0.137946i	$0.955950\pi$
$440$		0			0
$441$	−4.55981			−0.217134
$442$		0			0
$443$	−37.7822			−1.79508			−0.897542	−	0.440928i	$-0.854649\pi$
$443$	−37.7822			−1.79508			−0.897542	+	0.440928i	$0.854649\pi$
$444$		0			0
$445$		−	4.42976i		−	0.209991i
$446$		0			0
$447$	−3.34176			−0.158060
$448$		0			0
$449$	−18.7113			−0.883042			−0.441521	−	0.897251i	$-0.645561\pi$
$449$	−18.7113			−0.883042			−0.441521	+	0.897251i	$0.645561\pi$
$450$		0			0
$451$		−	9.69780i		−	0.456651i
$452$		0			0
$453$		−	17.3731i		−	0.816257i
$454$		0			0
$455$		−	3.33133i		−	0.156175i
$456$		0			0
$457$			17.6970i			0.827829i	0.910316	+	0.413915i	$0.135839\pi$
$457$			17.6970i			0.827829i	−0.910316	+	0.413915i	$0.864161\pi$
$458$		0			0
$459$			0.473035i			0.0220794i
$460$		0			0
$461$		−	4.75304i		−	0.221371i	−0.993855	−	0.110686i	$-0.964695\pi$
$461$		−	4.75304i		−	0.221371i	0.993855	−	0.110686i	$-0.0353047\pi$
$462$		0			0
$463$			10.7668i			0.500378i	0.968197	+	0.250189i	$0.0804927\pi$
$463$			10.7668i			0.500378i	−0.968197	+	0.250189i	$0.919507\pi$
$464$		0			0
$465$			1.65444i			0.0767227i
$466$		0			0
$467$			38.6391i			1.78800i	0.448063	+	0.894002i	$0.352114\pi$
$467$			38.6391i			1.78800i	−0.448063	+	0.894002i	$0.647886\pi$
$468$		0			0
$469$			8.36406i			0.386216i
$470$		0			0
$471$	4.88151			0.224928
$472$		0			0
$473$	−26.0951			−1.19985
$474$		0			0
$475$			32.0856i			1.47219i
$476$		0			0
$477$	−4.64065			−0.212481
$478$		0			0
$479$	−37.7337			−1.72410			−0.862049	−	0.506826i	$-0.830819\pi$
$479$	−37.7337			−1.72410			−0.862049	+	0.506826i	$0.830819\pi$
$480$		0			0
$481$			13.7647i			0.627618i
$482$		0			0
$483$	−7.38908	+	1.23531i	−0.336215	+	0.0562086i
$484$		0			0
$485$		−	13.5500i		−	0.615275i
$486$		0			0
$487$			12.5644i			0.569347i	0.958625	+	0.284673i	$0.0918851\pi$
$487$			12.5644i			0.569347i	−0.958625	+	0.284673i	$0.908115\pi$
$488$		0			0
$489$	22.1556			1.00191
$490$		0			0
$491$	−24.1152			−1.08830			−0.544152	−	0.838987i	$-0.683149\pi$
$491$	−24.1152			−1.08830			−0.544152	+	0.838987i	$0.683149\pi$
$492$		0			0
$493$	2.03123			0.0914818
$494$		0			0
$495$		−	3.33080i		−	0.149708i
$496$		0			0
$497$		−	10.8663i		−	0.487420i
$498$		0			0
$499$	38.6256			1.72912			0.864560	−	0.502530i	$-0.167597\pi$
$499$	38.6256			1.72912			0.864560	+	0.502530i	$0.167597\pi$
$500$		0			0
$501$			21.3490i			0.953802i
$502$		0			0
$503$	28.0677			1.25148			0.625738	−	0.780033i	$-0.284798\pi$
$503$	28.0677			1.25148			0.625738	+	0.780033i	$0.284798\pi$
$504$		0			0
$505$		−	1.33207i		−	0.0592762i
$506$		0			0
$507$	7.40382			0.328815
$508$		0			0
$509$			27.0543i			1.19916i	0.800315	+	0.599580i	$0.204666\pi$
$509$			27.0543i			1.19916i	−0.800315	+	0.599580i	$0.795334\pi$
$510$		0			0
$511$	−17.6218			−0.779545
$512$		0			0
$513$		−	7.66256i		−	0.338310i
$514$		0			0
$515$	−3.44666			−0.151878
$516$		0			0
$517$	−15.8064			−0.695166
$518$		0			0
$519$		−	16.6902i		−	0.732620i
$520$		0			0
$521$			13.8241i			0.605644i	0.953047	+	0.302822i	$0.0979287\pi$
$521$			13.8241i			0.605644i	−0.953047	+	0.302822i	$0.902071\pi$
$522$		0			0
$523$		−	8.04855i		−	0.351939i	−0.984396	−	0.175969i	$-0.943694\pi$
$523$		−	8.04855i		−	0.351939i	0.984396	−	0.175969i	$-0.0563060\pi$
$524$		0			0
$525$	6.54106			0.285475
$526$		0			0
$527$	−0.868128			−0.0378162
$528$		0			0
$529$	21.7493	−	7.48122i	0.945621	−	0.325270i
$530$		0			0
$531$	7.81205			0.339014
$532$		0			0
$533$			6.20911i			0.268947i
$534$		0			0
$535$			7.73070i			0.334227i
$536$		0			0
$537$	−13.5240			−0.583604
$538$		0			0
$539$			16.8475i			0.725674i
$540$		0			0
$541$		−	4.11616i		−	0.176968i	−0.996078	−	0.0884838i	$-0.971798\pi$
$541$		−	4.11616i		−	0.176968i	0.996078	−	0.0884838i	$-0.0282022\pi$
$542$		0			0
$543$	−15.6631			−0.672169
$544$		0			0
$545$	13.7738			0.590006
$546$		0			0
$547$	6.49890			0.277873			0.138936	−	0.990301i	$-0.455632\pi$
$547$	6.49890			0.277873			0.138936	+	0.990301i	$0.455632\pi$
$548$		0			0
$549$	11.9981			0.512066
$550$		0			0
$551$	−32.9033			−1.40173
$552$		0			0
$553$	24.7512			1.05253
$554$		0			0
$555$	5.24544			0.222656
$556$		0			0
$557$	18.6206			0.788980			0.394490	−	0.918900i	$-0.370921\pi$
$557$	18.6206			0.788980			0.394490	+	0.918900i	$0.370921\pi$
$558$		0			0
$559$	16.7076			0.706658
$560$		0			0
$561$	1.74776			0.0737905
$562$		0			0
$563$			3.49532i			0.147310i	0.997284	+	0.0736551i	$0.0234664\pi$
$563$			3.49532i			0.147310i	−0.997284	+	0.0736551i	$0.976534\pi$
$564$		0			0
$565$			5.30976i			0.223383i
$566$		0			0
$567$	−1.56211			−0.0656025
$568$		0			0
$569$			21.8934i			0.917819i	0.888483	+	0.458909i	$0.151760\pi$
$569$			21.8934i			0.917819i	−0.888483	+	0.458909i	$0.848240\pi$
$570$		0			0
$571$			28.9265i			1.21053i	0.796022	+	0.605267i	$0.206934\pi$
$571$			28.9265i			1.21053i	−0.796022	+	0.605267i	$0.793066\pi$
$572$		0			0
$573$	−10.5542			−0.440909
$574$		0			0
$575$	−19.8068	+	3.31131i	−0.826001	+	0.138091i
$576$		0			0
$577$	−23.6246			−0.983507			−0.491753	−	0.870735i	$-0.663644\pi$
$577$	−23.6246			−0.983507			−0.491753	+	0.870735i	$0.663644\pi$
$578$		0			0
$579$	0.464113			0.0192879
$580$		0			0
$581$			11.4582i			0.475367i
$582$		0			0
$583$			17.1462i			0.710123i
$584$		0			0
$585$			2.13258i			0.0881713i
$586$		0			0
$587$	29.1236			1.20206			0.601030	−	0.799226i	$-0.294757\pi$
$587$	29.1236			1.20206			0.601030	+	0.799226i	$0.294757\pi$
$588$		0			0
$589$	14.0626			0.579438
$590$		0			0
$591$			10.0657i			0.414048i
$592$		0			0
$593$	−10.2451			−0.420717			−0.210359	−	0.977624i	$-0.567463\pi$
$593$	−10.2451			−0.420717			−0.210359	+	0.977624i	$0.567463\pi$
$594$		0			0
$595$		−	0.666138i		−	0.0273090i
$596$		0			0
$597$	8.22327			0.336556
$598$		0			0
$599$			1.56887i			0.0641022i	0.999486	+	0.0320511i	$0.0102039\pi$
$599$			1.56887i			0.0641022i	−0.999486	+	0.0320511i	$0.989796\pi$
$600$		0			0
$601$	7.22050			0.294530			0.147265	−	0.989097i	$-0.452953\pi$
$601$	7.22050			0.294530			0.147265	+	0.989097i	$0.452953\pi$
$602$		0			0
$603$		−	5.35433i		−	0.218045i
$604$		0			0
$605$	−2.39023			−0.0971768
$606$		0			0
$607$		−	32.2388i		−	1.30853i	−0.756264	−	0.654266i	$-0.772978\pi$
$607$		−	32.2388i		−	1.30853i	0.756264	−	0.654266i	$-0.227022\pi$
$608$		0			0
$609$			6.70776i			0.271812i
$610$		0			0
$611$	10.1202			0.409421
$612$		0			0
$613$	−20.2345			−0.817262			−0.408631	−	0.912700i	$-0.633994\pi$
$613$	−20.2345			−0.817262			−0.408631	+	0.912700i	$0.633994\pi$
$614$		0			0
$615$	2.36616			0.0954126
$616$		0			0
$617$			7.12572i			0.286871i	0.989660	+	0.143435i	$0.0458149\pi$
$617$			7.12572i			0.286871i	−0.989660	+	0.143435i	$0.954185\pi$
$618$		0			0
$619$			41.0670i			1.65062i	0.564679	+	0.825311i	$0.309000\pi$
$619$			41.0670i			1.65062i	−0.564679	+	0.825311i	$0.691000\pi$
$620$		0			0
$621$	4.73018	−	0.790796i	0.189816	−	0.0317335i
$622$		0			0
$623$			7.67597i			0.307531i
$624$		0			0
$625$	13.4703			0.538811
$626$		0			0
$627$	−28.3115			−1.13065
$628$		0			0
$629$			2.75242i			0.109746i
$630$		0			0
$631$	11.4174			0.454518			0.227259	−	0.973834i	$-0.427024\pi$
$631$	11.4174			0.454518			0.227259	+	0.973834i	$0.427024\pi$
$632$		0			0
$633$	−16.1286			−0.641053
$634$		0			0
$635$			16.9241i			0.671611i
$636$		0			0
$637$		−	10.7868i		−	0.427388i
$638$		0			0
$639$			6.95616i			0.275181i
$640$		0			0
$641$			43.3405i			1.71185i	0.517103	+	0.855923i	$0.327010\pi$
$641$			43.3405i			1.71185i	−0.517103	+	0.855923i	$0.672990\pi$
$642$		0			0
$643$			26.6168i			1.04966i	0.851206	+	0.524832i	$0.175872\pi$
$643$			26.6168i			1.04966i	−0.851206	+	0.524832i	$0.824128\pi$
$644$		0			0
$645$		−	6.36691i		−	0.250697i
$646$		0			0
$647$		−	36.4289i		−	1.43217i	−0.698014	−	0.716084i	$-0.745933\pi$
$647$		−	36.4289i		−	1.43217i	0.698014	−	0.716084i	$-0.254067\pi$
$648$		0			0
$649$		−	28.8638i		−	1.13300i
$650$		0			0
$651$		−	2.86683i		−	0.112360i
$652$		0			0
$653$		−	39.6850i		−	1.55300i	−0.630120	−	0.776498i	$-0.716994\pi$
$653$		−	39.6850i		−	1.55300i	0.630120	−	0.776498i	$-0.283006\pi$
$654$		0			0
$655$	−5.77296			−0.225568
$656$		0			0
$657$	11.2808			0.440105
$658$		0			0
$659$		−	28.1729i		−	1.09746i	−0.836000	−	0.548730i	$-0.815112\pi$
$659$		−	28.1729i		−	1.09746i	0.836000	−	0.548730i	$-0.184888\pi$
$660$		0			0
$661$	−6.71763			−0.261285			−0.130643	−	0.991430i	$-0.541704\pi$
$661$	−6.71763			−0.261285			−0.130643	+	0.991430i	$0.541704\pi$
$662$		0			0
$663$	−1.11902			−0.0434592
$664$		0			0
$665$			10.7906i			0.418441i
$666$		0			0
$667$	−3.39570	−	20.3116i	−0.131482	−	0.786467i
$668$		0			0
$669$		−	6.77440i		−	0.261913i
$670$		0			0
$671$		−	44.3303i		−	1.71135i
$672$		0			0
$673$	6.37456			0.245721			0.122861	−	0.992424i	$-0.460793\pi$
$673$	6.37456			0.245721			0.122861	+	0.992424i	$0.460793\pi$
$674$		0			0
$675$	−4.18732			−0.161170
$676$		0			0
$677$	51.0634			1.96252			0.981262	−	0.192678i	$-0.0617172\pi$
$677$	51.0634			1.96252			0.981262	+	0.192678i	$0.0617172\pi$
$678$		0			0
$679$			23.4797i			0.901069i
$680$		0			0
$681$			23.8069i			0.912282i
$682$		0			0
$683$	13.7454			0.525955			0.262977	−	0.964802i	$-0.415296\pi$
$683$	13.7454			0.525955			0.262977	+	0.964802i	$0.415296\pi$
$684$		0			0
$685$			2.85524i			0.109093i
$686$		0			0
$687$	−10.4208			−0.397580
$688$		0			0
$689$		−	10.9780i		−	0.418229i
$690$		0			0
$691$	−20.1619			−0.766995			−0.383497	−	0.923542i	$-0.625280\pi$
$691$	−20.1619			−0.766995			−0.383497	+	0.923542i	$0.625280\pi$
$692$		0			0
$693$			5.77167i			0.219247i
$694$		0			0
$695$	−4.36631			−0.165624
$696$		0			0
$697$			1.24159i			0.0470284i
$698$		0			0
$699$	2.52840			0.0956328
$700$		0			0
$701$	5.51359			0.208246			0.104123	−	0.994564i	$-0.466797\pi$
$701$	5.51359			0.208246			0.104123	+	0.994564i	$0.466797\pi$
$702$		0			0
$703$		−	44.5857i		−	1.68158i
$704$		0			0
$705$		−	3.85660i		−	0.145248i
$706$		0			0
$707$			2.30823i			0.0868098i
$708$		0			0
$709$	−24.3221			−0.913437			−0.456719	−	0.889611i	$-0.650975\pi$
$709$	−24.3221			−0.913437			−0.456719	+	0.889611i	$0.650975\pi$
$710$		0			0
$711$	−15.8447			−0.594223
$712$		0			0
$713$	1.45129	+	8.68098i	0.0543513	+	0.325105i
$714$		0			0
$715$	7.87942			0.294674
$716$		0			0
$717$		−	12.3970i		−	0.462975i
$718$		0			0
$719$			48.2017i			1.79762i	0.438338	+	0.898810i	$0.355567\pi$
$719$			48.2017i			1.79762i	−0.438338	+	0.898810i	$0.644433\pi$
$720$		0			0
$721$	5.97243			0.222425
$722$		0			0
$723$		−	6.64297i		−	0.247055i
$724$		0			0
$725$			17.9805i			0.667779i
$726$		0			0
$727$	23.1995			0.860420			0.430210	−	0.902729i	$-0.358439\pi$
$727$	23.1995			0.860420			0.430210	+	0.902729i	$0.358439\pi$
$728$		0			0
$729$	1.00000			0.0370370
$730$		0			0
$731$	3.34089			0.123567
$732$		0			0
$733$	−0.566445			−0.0209221			−0.0104611	−	0.999945i	$-0.503330\pi$
$733$	−0.566445			−0.0209221			−0.0104611	+	0.999945i	$0.503330\pi$
$734$		0			0
$735$	−4.11061			−0.151622
$736$		0			0
$737$	−19.7831			−0.728720
$738$		0			0
$739$	−26.9657			−0.991950			−0.495975	−	0.868337i	$-0.665189\pi$
$739$	−26.9657			−0.991950			−0.495975	+	0.868337i	$0.665189\pi$
$740$		0			0
$741$	18.1267			0.665902
$742$		0			0
$743$	37.7515			1.38497			0.692485	−	0.721433i	$-0.256516\pi$
$743$	37.7515			1.38497			0.692485	+	0.721433i	$0.256516\pi$
$744$		0			0
$745$	−3.01255			−0.110371
$746$		0			0
$747$		−	7.33508i		−	0.268376i
$748$		0			0
$749$		−	13.3959i		−	0.489475i
$750$		0			0
$751$	−17.1412			−0.625490			−0.312745	−	0.949837i	$-0.601248\pi$
$751$	−17.1412			−0.625490			−0.312745	+	0.949837i	$0.601248\pi$
$752$		0			0
$753$			14.0093i			0.510527i
$754$		0			0
$755$		−	15.6616i		−	0.569983i
$756$		0			0
$757$	−32.8391			−1.19356			−0.596779	−	0.802406i	$-0.703553\pi$
$757$	−32.8391			−1.19356			−0.596779	+	0.802406i	$0.703553\pi$
$758$		0			0
$759$	−2.92182	−	17.4770i	−0.106055	−	0.634375i
$760$		0			0
$761$	−21.8078			−0.790532			−0.395266	−	0.918567i	$-0.629348\pi$
$761$	−21.8078			−0.790532			−0.395266	+	0.918567i	$0.629348\pi$
$762$		0			0
$763$	−23.8675			−0.864063
$764$		0			0
$765$			0.426435i			0.0154178i
$766$		0			0
$767$			18.4804i			0.667287i
$768$		0			0
$769$		−	5.34560i		−	0.192767i	−0.995344	−	0.0963837i	$-0.969272\pi$
$769$		−	5.34560i		−	0.192767i	0.995344	−	0.0963837i	$-0.0307276\pi$
$770$		0			0
$771$	17.0620			0.614474
$772$		0			0
$773$	−11.9945			−0.431413			−0.215707	−	0.976458i	$-0.569205\pi$
$773$	−11.9945			−0.431413			−0.215707	+	0.976458i	$0.569205\pi$
$774$		0			0
$775$		−	7.68470i		−	0.276043i
$776$		0			0
$777$	−9.08938			−0.326080
$778$		0			0
$779$		−	20.1121i		−	0.720591i
$780$		0			0
$781$	25.7015			0.919673
$782$		0			0
$783$		−	4.29403i		−	0.153456i
$784$		0			0
$785$	4.40062			0.157065
$786$		0			0
$787$		−	14.2110i		−	0.506566i	−0.967392	−	0.253283i	$-0.918490\pi$
$787$		−	14.2110i		−	0.506566i	0.967392	−	0.253283i	$-0.0815104\pi$
$788$		0			0
$789$	12.5846			0.448023
$790$		0			0
$791$		−	9.20084i		−	0.327144i
$792$		0			0
$793$			28.3829i			1.00791i
$794$		0			0
$795$	−4.18348			−0.148373
$796$		0			0
$797$	−51.6166			−1.82835			−0.914177	−	0.405315i	$-0.867162\pi$
$797$	−51.6166			−1.82835			−0.914177	+	0.405315i	$0.867162\pi$
$798$		0			0
$799$	2.02366			0.0715920
$800$		0			0
$801$		−	4.91384i		−	0.173622i
$802$		0			0
$803$		−	41.6801i		−	1.47086i
$804$		0			0
$805$	−6.66115	+	1.11362i	−0.234775	+	0.0392498i
$806$		0			0
$807$			5.33445i			0.187782i
$808$		0			0
$809$	29.5286			1.03817			0.519084	−	0.854723i	$-0.326273\pi$
$809$	29.5286			1.03817			0.519084	+	0.854723i	$0.326273\pi$
$810$		0			0
$811$	−28.3984			−0.997203			−0.498602	−	0.866831i	$-0.666153\pi$
$811$	−28.3984			−0.997203			−0.498602	+	0.866831i	$0.666153\pi$
$812$		0			0
$813$		−	5.16603i		−	0.181181i
$814$		0			0
$815$	19.9730			0.699623
$816$		0			0
$817$	−54.1181			−1.89335
$818$		0			0
$819$		−	3.69537i		−	0.129127i
$820$		0			0
$821$		−	2.10263i		−	0.0733822i	−0.999327	−	0.0366911i	$-0.988318\pi$
$821$		−	2.10263i		−	0.0733822i	0.999327	−	0.0366911i	$-0.0116818\pi$
$822$		0			0
$823$			46.6676i			1.62673i	0.581754	+	0.813365i	$0.302367\pi$
$823$			46.6676i			1.62673i	−0.581754	+	0.813365i	$0.697633\pi$
$824$		0			0
$825$			15.4713i			0.538640i
$826$		0			0
$827$		−	51.2016i		−	1.78046i	−0.455515	−	0.890228i	$-0.650545\pi$
$827$		−	51.2016i		−	1.78046i	0.455515	−	0.890228i	$-0.349455\pi$
$828$		0			0
$829$		−	47.1124i		−	1.63628i	−0.575017	−	0.818141i	$-0.695005\pi$
$829$		−	47.1124i		−	1.63628i	0.575017	−	0.818141i	$-0.304995\pi$
$830$		0			0
$831$		−	26.4195i		−	0.916483i
$832$		0			0
$833$		−	2.15695i		−	0.0747338i
$834$		0			0
$835$			19.2458i			0.666030i
$836$		0			0
$837$			1.83523i			0.0634348i
$838$		0			0
$839$	29.3728			1.01406			0.507031	−	0.861928i	$-0.330743\pi$
$839$	29.3728			1.01406			0.507031	+	0.861928i	$0.330743\pi$
$840$		0			0
$841$	10.5613			0.364182
$842$		0			0
$843$			20.8795i			0.719128i
$844$		0			0
$845$	6.67445			0.229608
$846$		0			0
$847$	4.14184			0.142315
$848$		0			0
$849$		−	11.8930i		−	0.408167i
$850$		0			0
$851$	27.5233	−	4.60136i	0.943486	−	0.157733i
$852$		0			0
$853$			11.6850i			0.400086i	0.979787	+	0.200043i	$0.0641082\pi$
$853$			11.6850i			0.400086i	−0.979787	+	0.200043i	$0.935892\pi$
$854$		0			0
$855$		−	6.90769i		−	0.236238i
$856$		0			0
$857$	30.6664			1.04754			0.523772	−	0.851859i	$-0.324525\pi$
$857$	30.6664			1.04754			0.523772	+	0.851859i	$0.324525\pi$
$858$		0			0
$859$	19.3065			0.658731			0.329365	−	0.944203i	$-0.393165\pi$
$859$	19.3065			0.658731			0.329365	+	0.944203i	$0.393165\pi$
$860$		0			0
$861$	−4.10011			−0.139732
$862$		0			0
$863$			22.0488i			0.750550i	0.926913	+	0.375275i	$0.122452\pi$
$863$			22.0488i			0.750550i	−0.926913	+	0.375275i	$0.877548\pi$
$864$		0			0
$865$		−	15.0460i		−	0.511580i
$866$		0			0
$867$	16.7762			0.569751
$868$		0			0
$869$			58.5428i			1.98593i
$870$		0			0
$871$	12.6663			0.429182
$872$		0			0
$873$		−	15.0308i		−	0.508714i
$874$		0			0
$875$	12.9378			0.437377
$876$		0			0
$877$			38.7191i			1.30745i	0.756732	+	0.653725i	$0.226795\pi$
$877$			38.7191i			1.30745i	−0.756732	+	0.653725i	$0.773205\pi$
$878$		0			0
$879$	−29.0826			−0.980931
$880$		0			0
$881$			34.6478i			1.16731i	0.812001	+	0.583656i	$0.198378\pi$
$881$			34.6478i			1.16731i	−0.812001	+	0.583656i	$0.801622\pi$
$882$		0			0
$883$	38.6865			1.30190			0.650952	−	0.759119i	$-0.274370\pi$
$883$	38.6865			1.30190			0.650952	+	0.759119i	$0.274370\pi$
$884$		0			0
$885$	7.04246			0.236730
$886$		0			0
$887$			16.9840i			0.570265i	0.958488	+	0.285133i	$0.0920377\pi$
$887$			16.9840i			0.570265i	−0.958488	+	0.285133i	$0.907962\pi$
$888$		0			0
$889$		−	29.3263i		−	0.983573i
$890$		0			0
$891$		−	3.69478i		−	0.123780i
$892$		0			0
$893$	−32.7807			−1.09696
$894$		0			0
$895$	−12.1917			−0.407524
$896$		0			0
$897$	1.87072	+	11.1898i	0.0624617	+	0.373618i
$898$		0			0
$899$	7.88054			0.262831
$900$		0			0
$901$		−	2.19519i		−	0.0731323i
$902$		0			0
$903$			11.0327i			0.367145i
$904$		0			0
$905$	−14.1201			−0.469368
$906$		0			0
$907$		−	7.88616i		−	0.261856i	−0.991392	−	0.130928i	$-0.958204\pi$
$907$		−	7.88616i		−	0.261856i	0.991392	−	0.130928i	$-0.0417956\pi$
$908$		0			0
$909$		−	1.47763i		−	0.0490100i
$910$		0			0
$911$	−23.1173			−0.765909			−0.382954	−	0.923767i	$-0.625093\pi$
$911$	−23.1173			−0.765909			−0.382954	+	0.923767i	$0.625093\pi$
$912$		0			0
$913$	−27.1015			−0.896930
$914$		0			0
$915$	10.8161			0.357570
$916$		0			0
$917$	10.0035			0.330344
$918$		0			0
$919$	40.9707			1.35150			0.675749	−	0.737132i	$-0.263820\pi$
$919$	40.9707			1.35150			0.675749	+	0.737132i	$0.263820\pi$
$920$		0			0
$921$	−0.564207			−0.0185913
$922$		0			0
$923$	−16.4557			−0.541645
$924$		0			0
$925$	−24.3646			−0.801102
$926$		0			0
$927$	−3.82331			−0.125574
$928$		0			0
$929$	32.6873			1.07243			0.536217	−	0.844080i	$-0.319853\pi$
$929$	32.6873			1.07243			0.536217	+	0.844080i	$0.319853\pi$
$930$		0			0
$931$			34.9398i			1.14511i
$932$		0			0
$933$			22.4646i			0.735459i
$934$		0			0
$935$	1.57558			0.0515271
$936$		0			0
$937$		−	34.2523i		−	1.11897i	−0.828839	−	0.559487i	$-0.810998\pi$
$937$		−	34.2523i		−	1.11897i	0.828839	−	0.559487i	$-0.189002\pi$
$938$		0			0
$939$		−	5.00245i		−	0.163249i
$940$		0			0
$941$	−1.43382			−0.0467413			−0.0233706	−	0.999727i	$-0.507440\pi$
$941$	−1.43382			−0.0467413			−0.0233706	+	0.999727i	$0.507440\pi$
$942$		0			0
$943$	12.4154	−	2.07562i	0.404302	−	0.0675915i
$944$		0			0
$945$	−1.40822			−0.0458095
$946$		0			0
$947$	−38.9980			−1.26727			−0.633633	−	0.773634i	$-0.718437\pi$
$947$	−38.9980			−1.26727			−0.633633	+	0.773634i	$0.718437\pi$
$948$		0			0
$949$			26.6861i			0.866268i
$950$		0			0
$951$			19.1167i			0.619900i
$952$		0			0
$953$		−	57.9473i		−	1.87710i	−0.345146	−	0.938549i	$-0.612171\pi$
$953$		−	57.9473i		−	1.87710i	0.345146	−	0.938549i	$-0.387829\pi$
$954$		0			0
$955$	−9.51450			−0.307882
$956$		0			0
$957$	−15.8655			−0.512860
$958$		0			0
$959$		−	4.94761i		−	0.159767i
$960$		0			0
$961$	27.6319			0.891353
$962$		0			0
$963$			8.57550i			0.276341i
$964$		0			0
$965$	0.418392			0.0134685
$966$		0			0
$967$		−	8.58256i		−	0.275997i	−0.990432	−	0.137998i	$-0.955933\pi$
$967$		−	8.58256i		−	0.275997i	0.990432	−	0.137998i	$-0.0440668\pi$
$968$		0			0
$969$	3.62465			0.116441
$970$		0			0
$971$		−	2.14726i		−	0.0689089i	−0.999406	−	0.0344545i	$-0.989031\pi$
$971$		−	2.14726i		−	0.0689089i	0.999406	−	0.0344545i	$-0.0109694\pi$
$972$		0			0
$973$	7.56602			0.242555
$974$		0			0
$975$		−	9.90563i		−	0.317234i
$976$		0			0
$977$		−	38.6295i		−	1.23587i	−0.786231	−	0.617933i	$-0.787970\pi$
$977$		−	38.6295i		−	1.23587i	0.786231	−	0.617933i	$-0.212030\pi$
$978$		0			0
$979$	−18.1556			−0.580255
$980$		0			0
$981$	15.2790			0.487822
$982$		0			0
$983$	25.5149			0.813799			0.406899	−	0.913473i	$-0.366610\pi$
$983$	25.5149			0.813799			0.406899	+	0.913473i	$0.366610\pi$
$984$		0			0
$985$			9.07410i			0.289125i
$986$		0			0
$987$			6.68277i			0.212715i
$988$		0			0
$989$	−5.58513	−	33.4078i	−0.177597	−	1.06230i
$990$		0			0
$991$			26.3998i			0.838616i	0.907844	+	0.419308i	$0.137727\pi$
$991$			26.3998i			0.838616i	−0.907844	+	0.419308i	$0.862273\pi$
$992$		0			0
$993$	9.65359			0.306347
$994$		0			0
$995$	7.41317			0.235013
$996$		0			0
$997$		−	32.7755i		−	1.03801i	−0.854771	−	0.519006i	$-0.826302\pi$
$997$		−	32.7755i		−	1.03801i	0.854771	−	0.519006i	$-0.173698\pi$
$998$		0			0
$999$	5.81865			0.184094

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2208.2.n.b.367.13		24
4.3	odd	2		552.2.n.b.91.16	yes	24
8.3	odd	2	inner	2208.2.n.b.367.11		24
8.5	even	2		552.2.n.b.91.13	✓	24
23.22	odd	2	inner	2208.2.n.b.367.12		24
92.91	even	2		552.2.n.b.91.15	yes	24
184.45	odd	2		552.2.n.b.91.14	yes	24
184.91	even	2	inner	2208.2.n.b.367.14		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
552.2.n.b.91.13	✓	24	8.5	even	2
552.2.n.b.91.14	yes	24	184.45	odd	2
552.2.n.b.91.15	yes	24	92.91	even	2
552.2.n.b.91.16	yes	24	4.3	odd	2
2208.2.n.b.367.11		24	8.3	odd	2	inner
2208.2.n.b.367.12		24	23.22	odd	2	inner
2208.2.n.b.367.13		24	1.1	even	1	trivial
2208.2.n.b.367.14		24	184.91	even	2	inner

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.n (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +0.901487 q^{5} -1.56211 q^{7} +1.00000 q^{9} -3.69478i q^{11} +2.36562i q^{13} +0.901487 q^{15} +0.473035i q^{17} -7.66256i q^{19} -1.56211 q^{21} +(4.73018 - 0.790796i) q^{23} -4.18732 q^{25} +1.00000 q^{27} -4.29403i q^{29} +1.83523i q^{31} -3.69478i q^{33} -1.40822 q^{35} +5.81865 q^{37} +2.36562i q^{39} +2.62473 q^{41} -7.06267i q^{43} +0.901487 q^{45} -4.27804i q^{47} -4.55981 q^{49} +0.473035i q^{51} -4.64065 q^{53} -3.33080i q^{55} -7.66256i q^{57} +7.81205 q^{59} +11.9981 q^{61} -1.56211 q^{63} +2.13258i q^{65} -5.35433i q^{67} +(4.73018 - 0.790796i) q^{69} +6.95616i q^{71} +11.2808 q^{73} -4.18732 q^{75} +5.77167i q^{77} -15.8447 q^{79} +1.00000 q^{81} -7.33508i q^{83} +0.426435i q^{85} -4.29403i q^{87} -4.91384i q^{89} -3.69537i q^{91} +1.83523i q^{93} -6.90769i q^{95} -15.0308i q^{97} -3.69478i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 24 q^{3} + 24 q^{9} + 24 q^{25} + 24 q^{27} + 56 q^{49} + 32 q^{73} + 24 q^{75} + 24 q^{81}+O(q^{100}) \) 24 * q + 24 * q^3 + 24 * q^9 + 24 * q^25 + 24 * q^27 + 56 * q^49 + 32 * q^73 + 24 * q^75 + 24 * q^81

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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