LMFDB - Embedded newform 3168.2.h.d.2287.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3168,2,Mod(2287,3168)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3168, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 0, 1])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3168.2287"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 3168 = 2^{5} \cdot 3^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3168.h (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,0,0,0,0,0,0,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	no
Analytic conductor:	$25.2966073603$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{2}, \sqrt{-7})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 6x^{2} + 16 $ x^4 + 6*x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 264)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2287.4
Root			$0.707107 + 1.87083i$ of defining polynomial
Character	$\chi$	$=$	3168.2287
Dual form			3168.2.h.d.2287.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+3.74166i q^{5} +1.41421 q^{7} +(-2.00000 - 2.64575i) q^{11} +4.24264 q^{13} -5.29150i q^{17} -5.29150i q^{19} -3.74166i q^{23} -9.00000 q^{25} -2.82843 q^{29} -7.48331i q^{31} +5.29150i q^{35} -7.48331i q^{37} -5.29150i q^{43} +3.74166i q^{47} -5.00000 q^{49} -3.74166i q^{53} +(9.89949 - 7.48331i) q^{55} +4.00000 q^{59} -12.7279 q^{61} +15.8745i q^{65} -2.00000 q^{67} -3.74166i q^{71} +10.5830i q^{73} +(-2.82843 - 3.74166i) q^{77} +1.41421 q^{79} -10.5830i q^{83} +19.7990 q^{85} -14.0000 q^{89} +6.00000 q^{91} +19.7990 q^{95} +16.0000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 8 q^{11} - 36 q^{25} - 20 q^{49} + 16 q^{59} - 8 q^{67} - 56 q^{89} + 24 q^{91} + 64 q^{97}+O(q^{100}) $ 4 * q - 8 * q^11 - 36 * q^25 - 20 * q^49 + 16 * q^59 - 8 * q^67 - 56 * q^89 + 24 * q^91 + 64 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3168\mathbb{Z}\right)^\times$.

$n$	$353$	$991$	$1189$	$1729$
$\chi(n)$	$1$	$-1$	$-1$	$-1$

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			0
$3$		0			0
$4$		0			0
$5$			3.74166i			1.67332i	0.547723	+	0.836660i	$0.315495\pi$
$5$			3.74166i			1.67332i	−0.547723	+	0.836660i	$0.684505\pi$
$6$		0			0
$7$	1.41421			0.534522			0.267261	−	0.963624i	$-0.413881\pi$
$7$	1.41421			0.534522			0.267261	+	0.963624i	$0.413881\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−2.00000	−	2.64575i	−0.603023	−	0.797724i
$11$	−2.00000	−	2.64575i	−0.603023	−	0.797724i
$12$		0			0
$13$	4.24264			1.17670			0.588348	−	0.808608i	$-0.299778\pi$
$13$	4.24264			1.17670			0.588348	+	0.808608i	$0.299778\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	5.29150i		−	1.28338i	−0.766965	−	0.641689i	$-0.778234\pi$
$17$		−	5.29150i		−	1.28338i	0.766965	−	0.641689i	$-0.221766\pi$
$18$		0			0
$19$		−	5.29150i		−	1.21395i	−0.794719	−	0.606977i	$-0.792382\pi$
$19$		−	5.29150i		−	1.21395i	0.794719	−	0.606977i	$-0.207618\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	3.74166i		−	0.780189i	−0.920775	−	0.390095i	$-0.872442\pi$
$23$		−	3.74166i		−	0.780189i	0.920775	−	0.390095i	$-0.127558\pi$
$24$		0			0
$25$	−9.00000			−1.80000
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−2.82843			−0.525226			−0.262613	−	0.964901i	$-0.584584\pi$
$29$	−2.82843			−0.525226			−0.262613	+	0.964901i	$0.584584\pi$
$30$		0			0
$31$		−	7.48331i		−	1.34404i	−0.740532	−	0.672022i	$-0.765426\pi$
$31$		−	7.48331i		−	1.34404i	0.740532	−	0.672022i	$-0.234574\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$			5.29150i			0.894427i
$36$		0			0
$37$		−	7.48331i		−	1.23025i	−0.788430	−	0.615125i	$-0.789106\pi$
$37$		−	7.48331i		−	1.23025i	0.788430	−	0.615125i	$-0.210894\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		1.00000			$0$
$41$		0			0		−1.00000			$\pi$
$42$		0			0
$43$		−	5.29150i		−	0.806947i	−0.914991	−	0.403473i	$-0.867803\pi$
$43$		−	5.29150i		−	0.806947i	0.914991	−	0.403473i	$-0.132197\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			3.74166i			0.545777i	0.962046	+	0.272888i	$0.0879790\pi$
$47$			3.74166i			0.545777i	−0.962046	+	0.272888i	$0.912021\pi$
$48$		0			0
$49$	−5.00000			−0.714286
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	3.74166i		−	0.513956i	−0.966417	−	0.256978i	$-0.917273\pi$
$53$		−	3.74166i		−	0.513956i	0.966417	−	0.256978i	$-0.0827268\pi$
$54$		0			0
$55$	9.89949	−	7.48331i	1.33485	−	1.00905i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	4.00000			0.520756			0.260378	−	0.965507i	$-0.416153\pi$
$59$	4.00000			0.520756			0.260378	+	0.965507i	$0.416153\pi$
$60$		0			0
$61$	−12.7279			−1.62964			−0.814822	−	0.579712i	$-0.803165\pi$
$61$	−12.7279			−1.62964			−0.814822	+	0.579712i	$0.803165\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$			15.8745i			1.96899i
$66$		0			0
$67$	−2.00000			−0.244339			−0.122169	−	0.992509i	$-0.538985\pi$
$67$	−2.00000			−0.244339			−0.122169	+	0.992509i	$0.538985\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	3.74166i		−	0.444053i	−0.975041	−	0.222027i	$-0.928733\pi$
$71$		−	3.74166i		−	0.444053i	0.975041	−	0.222027i	$-0.0712672\pi$
$72$		0			0
$73$			10.5830i			1.23865i	0.785136	+	0.619324i	$0.212593\pi$
$73$			10.5830i			1.23865i	−0.785136	+	0.619324i	$0.787407\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−2.82843	−	3.74166i	−0.322329	−	0.426401i
$78$		0			0
$79$	1.41421			0.159111			0.0795557	−	0.996830i	$-0.474650\pi$
$79$	1.41421			0.159111			0.0795557	+	0.996830i	$0.474650\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		−	10.5830i		−	1.16164i	−0.814034	−	0.580818i	$-0.802733\pi$
$83$		−	10.5830i		−	1.16164i	0.814034	−	0.580818i	$-0.197267\pi$
$84$		0			0
$85$	19.7990			2.14750
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−14.0000			−1.48400			−0.741999	−	0.670402i	$-0.766122\pi$
$89$	−14.0000			−1.48400			−0.741999	+	0.670402i	$0.766122\pi$
$90$		0			0
$91$	6.00000			0.628971
$92$		0			0
$93$		0			0
$94$		0			0
$95$	19.7990			2.03133
$96$		0			0
$97$	16.0000			1.62455			0.812277	−	0.583272i	$-0.198228\pi$
$97$	16.0000			1.62455			0.812277	+	0.583272i	$0.198228\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	11.3137			1.12576			0.562878	−	0.826540i	$-0.309694\pi$
$101$	11.3137			1.12576			0.562878	+	0.826540i	$0.309694\pi$
$102$		0			0
$103$			7.48331i			0.737353i	0.929558	+	0.368676i	$0.120189\pi$
$103$			7.48331i			0.737353i	−0.929558	+	0.368676i	$0.879811\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	15.8745i		−	1.53465i	−0.641260	−	0.767323i	$-0.721588\pi$
$107$		−	15.8745i		−	1.53465i	0.641260	−	0.767323i	$-0.278412\pi$
$108$		0			0
$109$	1.41421			0.135457			0.0677285	−	0.997704i	$-0.478425\pi$
$109$	1.41421			0.135457			0.0677285	+	0.997704i	$0.478425\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	6.00000			0.564433			0.282216	−	0.959351i	$-0.408930\pi$
$113$	6.00000			0.564433			0.282216	+	0.959351i	$0.408930\pi$
$114$		0			0
$115$	14.0000			1.30551
$116$		0			0
$117$		0			0
$118$		0			0
$119$		−	7.48331i		−	0.685994i
$120$		0			0
$121$	−3.00000	+	10.5830i	−0.272727	+	0.962091i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		−	14.9666i		−	1.33866i
$126$		0			0
$127$	−7.07107			−0.627456			−0.313728	−	0.949513i	$-0.601578\pi$
$127$	−7.07107			−0.627456			−0.313728	+	0.949513i	$0.601578\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$			21.1660i			1.84928i	0.380839	+	0.924641i	$0.375635\pi$
$131$			21.1660i			1.84928i	−0.380839	+	0.924641i	$0.624365\pi$
$132$		0			0
$133$		−	7.48331i		−	0.648886i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−2.00000			−0.170872			−0.0854358	−	0.996344i	$-0.527228\pi$
$137$	−2.00000			−0.170872			−0.0854358	+	0.996344i	$0.527228\pi$
$138$		0			0
$139$			15.8745i			1.34646i	0.739434	+	0.673229i	$0.235093\pi$
$139$			15.8745i			1.34646i	−0.739434	+	0.673229i	$0.764907\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−8.48528	−	11.2250i	−0.709575	−	0.938679i
$144$		0			0
$145$		−	10.5830i		−	0.878871i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	16.9706			1.39028			0.695141	−	0.718873i	$-0.255342\pi$
$149$	16.9706			1.39028			0.695141	+	0.718873i	$0.255342\pi$
$150$		0			0
$151$	4.24264			0.345261			0.172631	−	0.984987i	$-0.444773\pi$
$151$	4.24264			0.345261			0.172631	+	0.984987i	$0.444773\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	28.0000			2.24901
$156$		0			0
$157$		−	14.9666i		−	1.19447i	−0.802067	−	0.597234i	$-0.796267\pi$
$157$		−	14.9666i		−	1.19447i	0.802067	−	0.597234i	$-0.203733\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		−	5.29150i		−	0.417029i
$162$		0			0
$163$	4.00000			0.313304			0.156652	−	0.987654i	$-0.449930\pi$
$163$	4.00000			0.313304			0.156652	+	0.987654i	$0.449930\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−14.1421			−1.09435			−0.547176	−	0.837018i	$-0.684297\pi$
$167$	−14.1421			−1.09435			−0.547176	+	0.837018i	$0.684297\pi$
$168$		0			0
$169$	5.00000			0.384615
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−2.82843			−0.215041			−0.107521	−	0.994203i	$-0.534291\pi$
$173$	−2.82843			−0.215041			−0.107521	+	0.994203i	$0.534291\pi$
$174$		0			0
$175$	−12.7279			−0.962140
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−16.0000			−1.19590			−0.597948	−	0.801535i	$-0.704017\pi$
$179$	−16.0000			−1.19590			−0.597948	+	0.801535i	$0.704017\pi$
$180$		0			0
$181$			22.4499i			1.66869i	0.551242	+	0.834346i	$0.314154\pi$
$181$			22.4499i			1.66869i	−0.551242	+	0.834346i	$0.685846\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	28.0000			2.05860
$186$		0			0
$187$	−14.0000	+	10.5830i	−1.02378	+	0.773906i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	3.74166i		−	0.270737i	−0.990795	−	0.135368i	$-0.956778\pi$
$191$		−	3.74166i		−	0.270737i	0.990795	−	0.135368i	$-0.0432218\pi$
$192$		0			0
$193$		0			0		1.00000			$0$
$193$		0			0		−1.00000			$\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	16.9706			1.20910			0.604551	−	0.796566i	$-0.293352\pi$
$197$	16.9706			1.20910			0.604551	+	0.796566i	$0.293352\pi$
$198$		0			0
$199$		0			0		1.00000			$0$
$199$		0			0		−1.00000			$\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−4.00000			−0.280745
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−14.0000	+	10.5830i	−0.968400	+	0.732042i
$210$		0			0
$211$		−	15.8745i		−	1.09285i	−0.837509	−	0.546423i	$-0.815989\pi$
$211$		−	15.8745i		−	1.09285i	0.837509	−	0.546423i	$-0.184011\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	19.7990			1.35028
$216$		0			0
$217$		−	10.5830i		−	0.718421i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		−	22.4499i		−	1.51015i
$222$		0			0
$223$			14.9666i			1.00224i	0.865378	+	0.501120i	$0.167078\pi$
$223$			14.9666i			1.00224i	−0.865378	+	0.501120i	$0.832922\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			5.29150i			0.351209i	0.984461	+	0.175605i	$0.0561880\pi$
$227$			5.29150i			0.351209i	−0.984461	+	0.175605i	$0.943812\pi$
$228$		0			0
$229$		0			0		1.00000			$0$
$229$		0			0		−1.00000			$\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		1.00000			$0$
$233$		0			0		−1.00000			$\pi$
$234$		0			0
$235$	−14.0000			−0.913259
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−2.82843			−0.182956			−0.0914779	−	0.995807i	$-0.529159\pi$
$239$	−2.82843			−0.182956			−0.0914779	+	0.995807i	$0.529159\pi$
$240$		0			0
$241$		0			0		1.00000			$0$
$241$		0			0		−1.00000			$\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		−	18.7083i		−	1.19523i
$246$		0			0
$247$		−	22.4499i		−	1.42846i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−28.0000			−1.76734			−0.883672	−	0.468106i	$-0.844936\pi$
$251$	−28.0000			−1.76734			−0.883672	+	0.468106i	$0.844936\pi$
$252$		0			0
$253$	−9.89949	+	7.48331i	−0.622376	+	0.470472i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	6.00000			0.374270			0.187135	−	0.982334i	$-0.440080\pi$
$257$	6.00000			0.374270			0.187135	+	0.982334i	$0.440080\pi$
$258$		0			0
$259$		−	10.5830i		−	0.657596i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	25.4558			1.56967			0.784837	−	0.619702i	$-0.212746\pi$
$263$	25.4558			1.56967			0.784837	+	0.619702i	$0.212746\pi$
$264$		0			0
$265$	14.0000			0.860013
$266$		0			0
$267$		0			0
$268$		0			0
$269$		−	11.2250i		−	0.684399i	−0.939627	−	0.342199i	$-0.888828\pi$
$269$		−	11.2250i		−	0.684399i	0.939627	−	0.342199i	$-0.111172\pi$
$270$		0			0
$271$	12.7279			0.773166			0.386583	−	0.922255i	$-0.373655\pi$
$271$	12.7279			0.773166			0.386583	+	0.922255i	$0.373655\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	18.0000	+	23.8118i	1.08544	+	1.43590i
$276$		0			0
$277$	4.24264			0.254916			0.127458	−	0.991844i	$-0.459318\pi$
$277$	4.24264			0.254916			0.127458	+	0.991844i	$0.459318\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	15.8745i		−	0.946994i	−0.880795	−	0.473497i	$-0.842992\pi$
$281$		−	15.8745i		−	0.946994i	0.880795	−	0.473497i	$-0.157008\pi$
$282$		0			0
$283$		−	5.29150i		−	0.314547i	−0.987555	−	0.157274i	$-0.949730\pi$
$283$		−	5.29150i		−	0.314547i	0.987555	−	0.157274i	$-0.0502704\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−11.0000			−0.647059
$290$		0			0
$291$		0			0
$292$		0			0
$293$	19.7990			1.15667			0.578335	−	0.815800i	$-0.303703\pi$
$293$	19.7990			1.15667			0.578335	+	0.815800i	$0.303703\pi$
$294$		0			0
$295$			14.9666i			0.871391i
$296$		0			0
$297$		0			0
$298$		0			0
$299$		−	15.8745i		−	0.918046i
$300$		0			0
$301$		−	7.48331i		−	0.431331i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		−	47.6235i		−	2.72692i
$306$		0			0
$307$			26.4575i			1.51001i	0.655719	+	0.755005i	$0.272366\pi$
$307$			26.4575i			1.51001i	−0.655719	+	0.755005i	$0.727634\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	18.7083i		−	1.06085i	−0.847732	−	0.530425i	$-0.822032\pi$
$311$		−	18.7083i		−	1.06085i	0.847732	−	0.530425i	$-0.177968\pi$
$312$		0			0
$313$	−22.0000			−1.24351			−0.621757	−	0.783210i	$-0.713581\pi$
$313$	−22.0000			−1.24351			−0.621757	+	0.783210i	$0.713581\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			11.2250i			0.630457i	0.949016	+	0.315229i	$0.102081\pi$
$317$			11.2250i			0.630457i	−0.949016	+	0.315229i	$0.897919\pi$
$318$		0			0
$319$	5.65685	+	7.48331i	0.316723	+	0.418985i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−28.0000			−1.55796
$324$		0			0
$325$	−38.1838			−2.11805
$326$		0			0
$327$		0			0
$328$		0			0
$329$			5.29150i			0.291730i
$330$		0			0
$331$	−4.00000			−0.219860			−0.109930	−	0.993939i	$-0.535063\pi$
$331$	−4.00000			−0.219860			−0.109930	+	0.993939i	$0.535063\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		−	7.48331i		−	0.408857i
$336$		0			0
$337$		−	21.1660i		−	1.15299i	−0.817102	−	0.576493i	$-0.804421\pi$
$337$		−	21.1660i		−	1.15299i	0.817102	−	0.576493i	$-0.195579\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−19.7990	+	14.9666i	−1.07218	+	0.810488i
$342$		0			0
$343$	−16.9706			−0.916324
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	10.5830i		−	0.568125i	−0.958806	−	0.284063i	$-0.908318\pi$
$347$		−	10.5830i		−	0.568125i	0.958806	−	0.284063i	$-0.0916824\pi$
$348$		0			0
$349$	9.89949			0.529908			0.264954	−	0.964261i	$-0.414643\pi$
$349$	9.89949			0.529908			0.264954	+	0.964261i	$0.414643\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−14.0000			−0.745145			−0.372572	−	0.928003i	$-0.621524\pi$
$353$	−14.0000			−0.745145			−0.372572	+	0.928003i	$0.621524\pi$
$354$		0			0
$355$	14.0000			0.743043
$356$		0			0
$357$		0			0
$358$		0			0
$359$	11.3137			0.597115			0.298557	−	0.954392i	$-0.403495\pi$
$359$	11.3137			0.597115			0.298557	+	0.954392i	$0.403495\pi$
$360$		0			0
$361$	−9.00000			−0.473684
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−39.5980			−2.07265
$366$		0			0
$367$		0			0		1.00000			$0$
$367$		0			0		−1.00000			$\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		−	5.29150i		−	0.274721i
$372$		0			0
$373$	21.2132			1.09838			0.549189	−	0.835698i	$-0.314937\pi$
$373$	21.2132			1.09838			0.549189	+	0.835698i	$0.314937\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−12.0000			−0.618031
$378$		0			0
$379$	−20.0000			−1.02733			−0.513665	−	0.857991i	$-0.671713\pi$
$379$	−20.0000			−1.02733			−0.513665	+	0.857991i	$0.671713\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		−	3.74166i		−	0.191190i	−0.995420	−	0.0955949i	$-0.969525\pi$
$383$		−	3.74166i		−	0.191190i	0.995420	−	0.0955949i	$-0.0304753\pi$
$384$		0			0
$385$	14.0000	−	10.5830i	0.713506	−	0.539360i
$386$		0			0
$387$		0			0
$388$		0			0
$389$			18.7083i			0.948548i	0.880377	+	0.474274i	$0.157289\pi$
$389$			18.7083i			0.948548i	−0.880377	+	0.474274i	$0.842711\pi$
$390$		0			0
$391$	−19.7990			−1.00128
$392$		0			0
$393$		0			0
$394$		0			0
$395$			5.29150i			0.266244i
$396$		0			0
$397$			22.4499i			1.12673i	0.826208	+	0.563365i	$0.190494\pi$
$397$			22.4499i			1.12673i	−0.826208	+	0.563365i	$0.809506\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	10.0000			0.499376			0.249688	−	0.968326i	$-0.419672\pi$
$401$	10.0000			0.499376			0.249688	+	0.968326i	$0.419672\pi$
$402$		0			0
$403$		−	31.7490i		−	1.58153i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−19.7990	+	14.9666i	−0.981399	+	0.741868i
$408$		0			0
$409$			21.1660i			1.04659i	0.852151	+	0.523296i	$0.175298\pi$
$409$			21.1660i			1.04659i	−0.852151	+	0.523296i	$0.824702\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	5.65685			0.278356
$414$		0			0
$415$	39.5980			1.94379
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0			−	1.00000i	$-0.5\pi$
$419$		0			0				1.00000i	$0.5\pi$
$420$		0			0
$421$			22.4499i			1.09414i	0.837086	+	0.547072i	$0.184257\pi$
$421$			22.4499i			1.09414i	−0.837086	+	0.547072i	$0.815743\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$			47.6235i			2.31008i
$426$		0			0
$427$	−18.0000			−0.871081
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−14.1421			−0.681203			−0.340601	−	0.940208i	$-0.610631\pi$
$431$	−14.1421			−0.681203			−0.340601	+	0.940208i	$0.610631\pi$
$432$		0			0
$433$	14.0000			0.672797			0.336399	−	0.941720i	$-0.390791\pi$
$433$	14.0000			0.672797			0.336399	+	0.941720i	$0.390791\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−19.7990			−0.947114
$438$		0			0
$439$	−32.5269			−1.55242			−0.776212	−	0.630471i	$-0.782862\pi$
$439$	−32.5269			−1.55242			−0.776212	+	0.630471i	$0.782862\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	36.0000			1.71041			0.855206	−	0.518289i	$-0.173431\pi$
$443$	36.0000			1.71041			0.855206	+	0.518289i	$0.173431\pi$
$444$		0			0
$445$		−	52.3832i		−	2.48320i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	30.0000			1.41579			0.707894	−	0.706319i	$-0.249646\pi$
$449$	30.0000			1.41579			0.707894	+	0.706319i	$0.249646\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$			22.4499i			1.05247i
$456$		0			0
$457$		−	31.7490i		−	1.48516i	−0.669759	−	0.742578i	$-0.733603\pi$
$457$		−	31.7490i		−	1.48516i	0.669759	−	0.742578i	$-0.266397\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	25.4558			1.18560			0.592798	−	0.805351i	$-0.298023\pi$
$461$	25.4558			1.18560			0.592798	+	0.805351i	$0.298023\pi$
$462$		0			0
$463$		−	7.48331i		−	0.347779i	−0.984765	−	0.173890i	$-0.944366\pi$
$463$		−	7.48331i		−	0.347779i	0.984765	−	0.173890i	$-0.0556336\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0			−	1.00000i	$-0.5\pi$
$467$		0			0				1.00000i	$0.5\pi$
$468$		0			0
$469$	−2.82843			−0.130605
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−14.0000	+	10.5830i	−0.643721	+	0.486607i
$474$		0			0
$475$			47.6235i			2.18512i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	8.48528			0.387702			0.193851	−	0.981031i	$-0.437902\pi$
$479$	8.48528			0.387702			0.193851	+	0.981031i	$0.437902\pi$
$480$		0			0
$481$		−	31.7490i		−	1.44763i
$482$		0			0
$483$		0			0
$484$		0			0
$485$			59.8665i			2.71840i
$486$		0			0
$487$		−	29.9333i		−	1.35641i	−0.734875	−	0.678203i	$-0.762759\pi$
$487$		−	29.9333i		−	1.35641i	0.734875	−	0.678203i	$-0.237241\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			15.8745i			0.716407i	0.933644	+	0.358203i	$0.116611\pi$
$491$			15.8745i			0.716407i	−0.933644	+	0.358203i	$0.883389\pi$
$492$		0			0
$493$			14.9666i			0.674063i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	5.29150i		−	0.237356i
$498$		0			0
$499$	−20.0000			−0.895323			−0.447661	−	0.894203i	$-0.647743\pi$
$499$	−20.0000			−0.895323			−0.447661	+	0.894203i	$0.647743\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−14.1421			−0.630567			−0.315283	−	0.948998i	$-0.602100\pi$
$503$	−14.1421			−0.630567			−0.315283	+	0.948998i	$0.602100\pi$
$504$		0			0
$505$			42.3320i			1.88375i
$506$		0			0
$507$		0			0
$508$		0			0
$509$			41.1582i			1.82431i	0.409849	+	0.912153i	$0.365581\pi$
$509$			41.1582i			1.82431i	−0.409849	+	0.912153i	$0.634419\pi$
$510$		0			0
$511$			14.9666i			0.662085i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−28.0000			−1.23383
$516$		0			0
$517$	9.89949	−	7.48331i	0.435379	−	0.329116i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−14.0000			−0.613351			−0.306676	−	0.951814i	$-0.599217\pi$
$521$	−14.0000			−0.613351			−0.306676	+	0.951814i	$0.599217\pi$
$522$		0			0
$523$			5.29150i			0.231381i	0.993285	+	0.115691i	$0.0369081\pi$
$523$			5.29150i			0.231381i	−0.993285	+	0.115691i	$0.963092\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−39.5980			−1.72492
$528$		0			0
$529$	9.00000			0.391304
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$	59.3970			2.56795
$536$		0			0
$537$		0			0
$538$		0			0
$539$	10.0000	+	13.2288i	0.430730	+	0.569803i
$540$		0			0
$541$	41.0122			1.76325			0.881626	−	0.471949i	$-0.156449\pi$
$541$	41.0122			1.76325			0.881626	+	0.471949i	$0.156449\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$			5.29150i			0.226663i
$546$		0			0
$547$			5.29150i			0.226248i	0.993581	+	0.113124i	$0.0360858\pi$
$547$			5.29150i			0.226248i	−0.993581	+	0.113124i	$0.963914\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$			14.9666i			0.637600i
$552$		0			0
$553$	2.00000			0.0850487
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−8.48528			−0.359533			−0.179766	−	0.983709i	$-0.557534\pi$
$557$	−8.48528			−0.359533			−0.179766	+	0.983709i	$0.557534\pi$
$558$		0			0
$559$		−	22.4499i		−	0.949531i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		−	15.8745i		−	0.669031i	−0.942390	−	0.334515i	$-0.891427\pi$
$563$		−	15.8745i		−	0.669031i	0.942390	−	0.334515i	$-0.108573\pi$
$564$		0			0
$565$			22.4499i			0.944476i
$566$		0			0
$567$		0			0
$568$		0			0
$569$			5.29150i			0.221831i	0.993830	+	0.110916i	$0.0353783\pi$
$569$			5.29150i			0.221831i	−0.993830	+	0.110916i	$0.964622\pi$
$570$		0			0
$571$			26.4575i			1.10721i	0.832779	+	0.553606i	$0.186749\pi$
$571$			26.4575i			1.10721i	−0.832779	+	0.553606i	$0.813251\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			33.6749i			1.40434i
$576$		0			0
$577$	32.0000			1.33218			0.666089	−	0.745873i	$-0.267967\pi$
$577$	32.0000			1.33218			0.666089	+	0.745873i	$0.267967\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		−	14.9666i		−	0.620920i
$582$		0			0
$583$	−9.89949	+	7.48331i	−0.409995	+	0.309927i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	16.0000			0.660391			0.330195	−	0.943913i	$-0.392885\pi$
$587$	16.0000			0.660391			0.330195	+	0.943913i	$0.392885\pi$
$588$		0			0
$589$	−39.5980			−1.63161
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	15.8745i		−	0.651888i	−0.945389	−	0.325944i	$-0.894318\pi$
$593$		−	15.8745i		−	0.651888i	0.945389	−	0.325944i	$-0.105682\pi$
$594$		0			0
$595$	28.0000			1.14789
$596$		0			0
$597$		0			0
$598$		0			0
$599$			3.74166i			0.152880i	0.997074	+	0.0764400i	$0.0243554\pi$
$599$			3.74166i			0.152880i	−0.997074	+	0.0764400i	$0.975645\pi$
$600$		0			0
$601$			21.1660i			0.863380i	0.902022	+	0.431690i	$0.142082\pi$
$601$			21.1660i			0.863380i	−0.902022	+	0.431690i	$0.857918\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−39.5980	−	11.2250i	−1.60989	−	0.456360i
$606$		0			0
$607$	29.6985			1.20542			0.602712	−	0.797959i	$-0.294087\pi$
$607$	29.6985			1.20542			0.602712	+	0.797959i	$0.294087\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			15.8745i			0.642214i
$612$		0			0
$613$	−1.41421			−0.0571195			−0.0285598	−	0.999592i	$-0.509092\pi$
$613$	−1.41421			−0.0571195			−0.0285598	+	0.999592i	$0.509092\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−6.00000			−0.241551			−0.120775	−	0.992680i	$-0.538538\pi$
$617$	−6.00000			−0.241551			−0.120775	+	0.992680i	$0.538538\pi$
$618$		0			0
$619$	42.0000			1.68812			0.844061	−	0.536247i	$-0.180158\pi$
$619$	42.0000			1.68812			0.844061	+	0.536247i	$0.180158\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−19.7990			−0.793230
$624$		0			0
$625$	11.0000			0.440000
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−39.5980			−1.57887
$630$		0			0
$631$		−	7.48331i		−	0.297906i	−0.988844	−	0.148953i	$-0.952410\pi$
$631$		−	7.48331i		−	0.297906i	0.988844	−	0.148953i	$-0.0475903\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		−	26.4575i		−	1.04993i
$636$		0			0
$637$	−21.2132			−0.840498
$638$		0			0
$639$		0			0
$640$		0			0
$641$	2.00000			0.0789953			0.0394976	−	0.999220i	$-0.487424\pi$
$641$	2.00000			0.0789953			0.0394976	+	0.999220i	$0.487424\pi$
$642$		0			0
$643$	26.0000			1.02534			0.512670	−	0.858586i	$-0.328656\pi$
$643$	26.0000			1.02534			0.512670	+	0.858586i	$0.328656\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			48.6415i			1.91230i	0.292883	+	0.956148i	$0.405385\pi$
$647$			48.6415i			1.91230i	−0.292883	+	0.956148i	$0.594615\pi$
$648$		0			0
$649$	−8.00000	−	10.5830i	−0.314027	−	0.415419i
$650$		0			0
$651$		0			0
$652$		0			0
$653$			11.2250i			0.439267i	0.975582	+	0.219634i	$0.0704862\pi$
$653$			11.2250i			0.439267i	−0.975582	+	0.219634i	$0.929514\pi$
$654$		0			0
$655$	−79.1960			−3.09444
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	15.8745i		−	0.618383i	−0.951000	−	0.309192i	$-0.899942\pi$
$659$		−	15.8745i		−	0.618383i	0.951000	−	0.309192i	$-0.100058\pi$
$660$		0			0
$661$		−	44.8999i		−	1.74640i	−0.487359	−	0.873202i	$-0.662040\pi$
$661$		−	44.8999i		−	1.74640i	0.487359	−	0.873202i	$-0.337960\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	28.0000			1.08579
$666$		0			0
$667$			10.5830i			0.409776i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	25.4558	+	33.6749i	0.982712	+	1.30001i
$672$		0			0
$673$			10.5830i			0.407945i	0.978977	+	0.203972i	$0.0653853\pi$
$673$			10.5830i			0.407945i	−0.978977	+	0.203972i	$0.934615\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−39.5980			−1.52187			−0.760937	−	0.648826i	$-0.775260\pi$
$677$	−39.5980			−1.52187			−0.760937	+	0.648826i	$0.775260\pi$
$678$		0			0
$679$	22.6274			0.868361
$680$		0			0
$681$		0			0
$682$		0			0
$683$	12.0000			0.459167			0.229584	−	0.973289i	$-0.426264\pi$
$683$	12.0000			0.459167			0.229584	+	0.973289i	$0.426264\pi$
$684$		0			0
$685$		−	7.48331i		−	0.285923i
$686$		0			0
$687$		0			0
$688$		0			0
$689$		−	15.8745i		−	0.604771i
$690$		0			0
$691$	−34.0000			−1.29342			−0.646710	−	0.762736i	$-0.723856\pi$
$691$	−34.0000			−1.29342			−0.646710	+	0.762736i	$0.723856\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−59.3970			−2.25306
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$	42.4264			1.60242			0.801212	−	0.598381i	$-0.204189\pi$
$701$	42.4264			1.60242			0.801212	+	0.598381i	$0.204189\pi$
$702$		0			0
$703$	−39.5980			−1.49347
$704$		0			0
$705$		0			0
$706$		0			0
$707$	16.0000			0.601742
$708$		0			0
$709$			14.9666i			0.562084i	0.959696	+	0.281042i	$0.0906800\pi$
$709$			14.9666i			0.562084i	−0.959696	+	0.281042i	$0.909320\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−28.0000			−1.04861
$714$		0			0
$715$	42.0000	−	31.7490i	1.57071	−	1.18735i
$716$		0			0
$717$		0			0
$718$		0			0
$719$		−	33.6749i		−	1.25586i	−0.778269	−	0.627931i	$-0.783902\pi$
$719$		−	33.6749i		−	1.25586i	0.778269	−	0.627931i	$-0.216098\pi$
$720$		0			0
$721$			10.5830i			0.394132i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	25.4558			0.945406
$726$		0			0
$727$			14.9666i			0.555082i	0.960714	+	0.277541i	$0.0895194\pi$
$727$			14.9666i			0.555082i	−0.960714	+	0.277541i	$0.910481\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−28.0000			−1.03562
$732$		0			0
$733$	12.7279			0.470117			0.235058	−	0.971981i	$-0.424472\pi$
$733$	12.7279			0.470117			0.235058	+	0.971981i	$0.424472\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	4.00000	+	5.29150i	0.147342	+	0.194915i
$738$		0			0
$739$		−	37.0405i		−	1.36256i	−0.732024	−	0.681279i	$-0.761424\pi$
$739$		−	37.0405i		−	1.36256i	0.732024	−	0.681279i	$-0.238576\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	5.65685			0.207530			0.103765	−	0.994602i	$-0.466911\pi$
$743$	5.65685			0.207530			0.103765	+	0.994602i	$0.466911\pi$
$744$		0			0
$745$			63.4980i			2.32639i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		−	22.4499i		−	0.820303i
$750$		0			0
$751$			7.48331i			0.273070i	0.990635	+	0.136535i	$0.0435966\pi$
$751$			7.48331i			0.273070i	−0.990635	+	0.136535i	$0.956403\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$			15.8745i			0.577732i
$756$		0			0
$757$		−	22.4499i		−	0.815957i	−0.912992	−	0.407979i	$-0.866234\pi$
$757$		−	22.4499i		−	0.815957i	0.912992	−	0.407979i	$-0.133766\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	15.8745i		−	0.575450i	−0.957713	−	0.287725i	$-0.907101\pi$
$761$		−	15.8745i		−	0.575450i	0.957713	−	0.287725i	$-0.0928990\pi$
$762$		0			0
$763$	2.00000			0.0724049
$764$		0			0
$765$		0			0
$766$		0			0
$767$	16.9706			0.612772
$768$		0			0
$769$		−	10.5830i		−	0.381633i	−0.981626	−	0.190816i	$-0.938886\pi$
$769$		−	10.5830i		−	0.381633i	0.981626	−	0.190816i	$-0.0611135\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	41.1582i		−	1.48036i	−0.672410	−	0.740179i	$-0.734741\pi$
$773$		−	41.1582i		−	1.48036i	0.672410	−	0.740179i	$-0.265259\pi$
$774$		0			0
$775$			67.3498i			2.41928i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$	−9.89949	+	7.48331i	−0.354232	+	0.267774i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	56.0000			1.99873
$786$		0			0
$787$		−	26.4575i		−	0.943108i	−0.881837	−	0.471554i	$-0.843693\pi$
$787$		−	26.4575i		−	0.943108i	0.881837	−	0.471554i	$-0.156307\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	8.48528			0.301702
$792$		0			0
$793$	−54.0000			−1.91760
$794$		0			0
$795$		0			0
$796$		0			0
$797$			41.1582i			1.45790i	0.684567	+	0.728950i	$0.259991\pi$
$797$			41.1582i			1.45790i	−0.684567	+	0.728950i	$0.740009\pi$
$798$		0			0
$799$	19.7990			0.700438
$800$		0			0
$801$		0			0
$802$		0			0
$803$	28.0000	−	21.1660i	0.988099	−	0.746932i
$804$		0			0
$805$	19.7990			0.697823
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0		1.00000			$0$
$809$		0			0		−1.00000			$\pi$
$810$		0			0
$811$		−	37.0405i		−	1.30067i	−0.759648	−	0.650334i	$-0.774629\pi$
$811$		−	37.0405i		−	1.30067i	0.759648	−	0.650334i	$-0.225371\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$			14.9666i			0.524258i
$816$		0			0
$817$	−28.0000			−0.979596
$818$		0			0
$819$		0			0
$820$		0			0
$821$	11.3137			0.394851			0.197426	−	0.980318i	$-0.436742\pi$
$821$	11.3137			0.394851			0.197426	+	0.980318i	$0.436742\pi$
$822$		0			0
$823$			44.8999i			1.56511i	0.622580	+	0.782556i	$0.286084\pi$
$823$			44.8999i			1.56511i	−0.622580	+	0.782556i	$0.713916\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			31.7490i			1.10402i	0.833837	+	0.552011i	$0.186139\pi$
$827$			31.7490i			1.10402i	−0.833837	+	0.552011i	$0.813861\pi$
$828$		0			0
$829$			44.8999i			1.55944i	0.626130	+	0.779719i	$0.284638\pi$
$829$			44.8999i			1.55944i	−0.626130	+	0.779719i	$0.715362\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$			26.4575i			0.916698i
$834$		0			0
$835$		−	52.9150i		−	1.83120i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	18.7083i		−	0.645882i	−0.946419	−	0.322941i	$-0.895328\pi$
$839$		−	18.7083i		−	0.645882i	0.946419	−	0.322941i	$-0.104672\pi$
$840$		0			0
$841$	−21.0000			−0.724138
$842$		0			0
$843$		0			0
$844$		0			0
$845$			18.7083i			0.643585i
$846$		0			0
$847$	−4.24264	+	14.9666i	−0.145779	+	0.514259i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−28.0000			−0.959828
$852$		0			0
$853$	−49.4975			−1.69476			−0.847381	−	0.530986i	$-0.821822\pi$
$853$	−49.4975			−1.69476			−0.847381	+	0.530986i	$0.821822\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			21.1660i			0.723017i	0.932369	+	0.361509i	$0.117738\pi$
$857$			21.1660i			0.723017i	−0.932369	+	0.361509i	$0.882262\pi$
$858$		0			0
$859$	−20.0000			−0.682391			−0.341196	−	0.939992i	$-0.610832\pi$
$859$	−20.0000			−0.682391			−0.341196	+	0.939992i	$0.610832\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			48.6415i			1.65578i	0.560892	+	0.827889i	$0.310458\pi$
$863$			48.6415i			1.65578i	−0.560892	+	0.827889i	$0.689542\pi$
$864$		0			0
$865$		−	10.5830i		−	0.359833i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−2.82843	−	3.74166i	−0.0959478	−	0.126927i
$870$		0			0
$871$	−8.48528			−0.287513
$872$		0			0
$873$		0			0
$874$		0			0
$875$		−	21.1660i		−	0.715542i
$876$		0			0
$877$	38.1838			1.28937			0.644687	−	0.764447i	$-0.276988\pi$
$877$	38.1838			1.28937			0.644687	+	0.764447i	$0.276988\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	30.0000			1.01073			0.505363	−	0.862907i	$-0.331359\pi$
$881$	30.0000			1.01073			0.505363	+	0.862907i	$0.331359\pi$
$882$		0			0
$883$	−6.00000			−0.201916			−0.100958	−	0.994891i	$-0.532191\pi$
$883$	−6.00000			−0.201916			−0.100958	+	0.994891i	$0.532191\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	16.9706			0.569816			0.284908	−	0.958555i	$-0.408037\pi$
$887$	16.9706			0.569816			0.284908	+	0.958555i	$0.408037\pi$
$888$		0			0
$889$	−10.0000			−0.335389
$890$		0			0
$891$		0			0
$892$		0			0
$893$	19.7990			0.662548
$894$		0			0
$895$		−	59.8665i		−	2.00112i
$896$		0			0
$897$		0			0
$898$		0			0
$899$			21.1660i			0.705926i
$900$		0			0
$901$	−19.7990			−0.659600
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−84.0000			−2.79225
$906$		0			0
$907$	−38.0000			−1.26177			−0.630885	−	0.775877i	$-0.717308\pi$
$907$	−38.0000			−1.26177			−0.630885	+	0.775877i	$0.717308\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	18.7083i		−	0.619833i	−0.950764	−	0.309917i	$-0.899699\pi$
$911$		−	18.7083i		−	0.619833i	0.950764	−	0.309917i	$-0.100301\pi$
$912$		0			0
$913$	−28.0000	+	21.1660i	−0.926665	+	0.700493i
$914$		0			0
$915$		0			0
$916$		0			0
$917$			29.9333i			0.988483i
$918$		0			0
$919$	−46.6690			−1.53947			−0.769735	−	0.638364i	$-0.779612\pi$
$919$	−46.6690			−1.53947			−0.769735	+	0.638364i	$0.779612\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		−	15.8745i		−	0.522516i
$924$		0			0
$925$			67.3498i			2.21445i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−50.0000			−1.64045			−0.820223	−	0.572043i	$-0.806151\pi$
$929$	−50.0000			−1.64045			−0.820223	+	0.572043i	$0.806151\pi$
$930$		0			0
$931$			26.4575i			0.867110i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−39.5980	−	52.3832i	−1.29499	−	1.71311i
$936$		0			0
$937$		−	31.7490i		−	1.03720i	−0.855018	−	0.518598i	$-0.826454\pi$
$937$		−	31.7490i		−	1.03720i	0.855018	−	0.518598i	$-0.173546\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	11.3137			0.368816			0.184408	−	0.982850i	$-0.440963\pi$
$941$	11.3137			0.368816			0.184408	+	0.982850i	$0.440963\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−36.0000			−1.16984			−0.584921	−	0.811090i	$-0.698875\pi$
$947$	−36.0000			−1.16984			−0.584921	+	0.811090i	$0.698875\pi$
$948$		0			0
$949$			44.8999i			1.45751i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	21.1660i		−	0.685634i	−0.939402	−	0.342817i	$-0.888619\pi$
$953$		−	21.1660i		−	0.685634i	0.939402	−	0.342817i	$-0.111381\pi$
$954$		0			0
$955$	14.0000			0.453029
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−2.82843			−0.0913347
$960$		0			0
$961$	−25.0000			−0.806452
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	15.5563			0.500258			0.250129	−	0.968212i	$-0.419527\pi$
$967$	15.5563			0.500258			0.250129	+	0.968212i	$0.419527\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	36.0000			1.15529			0.577647	−	0.816286i	$-0.303971\pi$
$971$	36.0000			1.15529			0.577647	+	0.816286i	$0.303971\pi$
$972$		0			0
$973$			22.4499i			0.719712i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−18.0000			−0.575871			−0.287936	−	0.957650i	$-0.592969\pi$
$977$	−18.0000			−0.575871			−0.287936	+	0.957650i	$0.592969\pi$
$978$		0			0
$979$	28.0000	+	37.0405i	0.894884	+	1.18382i
$980$		0			0
$981$		0			0
$982$		0			0
$983$			11.2250i			0.358021i	0.983847	+	0.179011i	$0.0572896\pi$
$983$			11.2250i			0.358021i	−0.983847	+	0.179011i	$0.942710\pi$
$984$		0			0
$985$			63.4980i			2.02322i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−19.7990			−0.629571
$990$		0			0
$991$			37.4166i			1.18858i	0.804252	+	0.594288i	$0.202566\pi$
$991$			37.4166i			1.18858i	−0.804252	+	0.594288i	$0.797434\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	29.6985			0.940560			0.470280	−	0.882517i	$-0.344153\pi$
$997$	29.6985			0.940560			0.470280	+	0.882517i	$0.344153\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3168.2.h.d.2287.4		4
3.2	odd	2		1056.2.h.d.175.2		4
4.3	odd	2		792.2.h.e.307.2		4
8.3	odd	2	inner	3168.2.h.d.2287.1		4
8.5	even	2		792.2.h.e.307.3		4
11.10	odd	2	inner	3168.2.h.d.2287.3		4
12.11	even	2		264.2.h.d.43.3	yes	4
24.5	odd	2		264.2.h.d.43.2	yes	4
24.11	even	2		1056.2.h.d.175.3		4
33.32	even	2		1056.2.h.d.175.1		4
44.43	even	2		792.2.h.e.307.4		4
88.21	odd	2		792.2.h.e.307.1		4
88.43	even	2	inner	3168.2.h.d.2287.2		4
132.131	odd	2		264.2.h.d.43.1	✓	4
264.131	odd	2		1056.2.h.d.175.4		4
264.197	even	2		264.2.h.d.43.4	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
264.2.h.d.43.1	✓	4	132.131	odd	2
264.2.h.d.43.2	yes	4	24.5	odd	2
264.2.h.d.43.3	yes	4	12.11	even	2
264.2.h.d.43.4	yes	4	264.197	even	2
792.2.h.e.307.1		4	88.21	odd	2
792.2.h.e.307.2		4	4.3	odd	2
792.2.h.e.307.3		4	8.5	even	2
792.2.h.e.307.4		4	44.43	even	2
1056.2.h.d.175.1		4	33.32	even	2
1056.2.h.d.175.2		4	3.2	odd	2
1056.2.h.d.175.3		4	24.11	even	2
1056.2.h.d.175.4		4	264.131	odd	2
3168.2.h.d.2287.1		4	8.3	odd	2	inner
3168.2.h.d.2287.2		4	88.43	even	2	inner
3168.2.h.d.2287.3		4	11.10	odd	2	inner
3168.2.h.d.2287.4		4	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:	\( N \)	\(=\)	\( 3168 = 2^{5} \cdot 3^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3168.h (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+3.74166i q^{5} +1.41421 q^{7} +(-2.00000 - 2.64575i) q^{11} +4.24264 q^{13} -5.29150i q^{17} -5.29150i q^{19} -3.74166i q^{23} -9.00000 q^{25} -2.82843 q^{29} -7.48331i q^{31} +5.29150i q^{35} -7.48331i q^{37} -5.29150i q^{43} +3.74166i q^{47} -5.00000 q^{49} -3.74166i q^{53} +(9.89949 - 7.48331i) q^{55} +4.00000 q^{59} -12.7279 q^{61} +15.8745i q^{65} -2.00000 q^{67} -3.74166i q^{71} +10.5830i q^{73} +(-2.82843 - 3.74166i) q^{77} +1.41421 q^{79} -10.5830i q^{83} +19.7990 q^{85} -14.0000 q^{89} +6.00000 q^{91} +19.7990 q^{95} +16.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 8 q^{11} - 36 q^{25} - 20 q^{49} + 16 q^{59} - 8 q^{67} - 56 q^{89} + 24 q^{91} + 64 q^{97}+O(q^{100}) \) 4 * q - 8 * q^11 - 36 * q^25 - 20 * q^49 + 16 * q^59 - 8 * q^67 - 56 * q^89 + 24 * q^91 + 64 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more