LMFDB - Embedded newform 3168.2.b.f.2177.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3168,2,Mod(2177,3168)] 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("3168.2177"); S:= CuspForms(chi, 2); N := Newforms(S);

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

Level:	$ N $	$=$	$ 3168 = 2^{5} \cdot 3^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3168.b (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,0,0,0,0,0,0,6,0,0,0,0,0,-4,0,0,0,0,0,0,0,10] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(25)] == traces)

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

Self dual:	no
Analytic conductor:	$25.2966073603$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-2}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 2 $ x^2 + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2177.1
Root			$-1.41421i$ of defining polynomial
Character	$\chi$	$=$	3168.2177
Dual form			3168.2.b.f.2177.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-2.82843i q^{7} +(3.00000 + 1.41421i) q^{11} +4.24264i q^{13} -2.00000 q^{17} +1.41421i q^{19} +1.41421i q^{23} +5.00000 q^{25} -4.00000 q^{29} +6.00000 q^{31} -6.00000 q^{37} +2.00000 q^{41} +1.41421i q^{43} +4.24264i q^{47} -1.00000 q^{49} +8.48528i q^{53} +11.3137i q^{59} -4.24264i q^{61} +12.0000 q^{67} +12.7279i q^{71} -8.48528i q^{73} +(4.00000 - 8.48528i) q^{77} -11.3137i q^{79} +6.00000 q^{83} -4.24264i q^{89} +12.0000 q^{91} -12.0000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 6 q^{11} - 4 q^{17} + 10 q^{25} - 8 q^{29} + 12 q^{31} - 12 q^{37} + 4 q^{41} - 2 q^{49} + 24 q^{67} + 8 q^{77} + 12 q^{83} + 24 q^{91} - 24 q^{97}+O(q^{100}) $ 2 * q + 6 * q^11 - 4 * q^17 + 10 * q^25 - 8 * q^29 + 12 * q^31 - 12 * q^37 + 4 * q^41 - 2 * q^49 + 24 * q^67 + 8 * q^77 + 12 * q^83 + 24 * q^91 - 24 * 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
$5$		0			0		1.00000			$0$
$5$		0			0		−1.00000			$\pi$
$6$		0			0
$7$		−	2.82843i		−	1.06904i	−0.845154	−	0.534522i	$-0.820491\pi$
$7$		−	2.82843i		−	1.06904i	0.845154	−	0.534522i	$-0.179509\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	3.00000	+	1.41421i	0.904534	+	0.426401i
$11$	3.00000	+	1.41421i	0.904534	+	0.426401i
$12$		0			0
$13$			4.24264i			1.17670i	0.808608	+	0.588348i	$0.200222\pi$
$13$			4.24264i			1.17670i	−0.808608	+	0.588348i	$0.799778\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−2.00000			−0.485071			−0.242536	−	0.970143i	$-0.577979\pi$
$17$	−2.00000			−0.485071			−0.242536	+	0.970143i	$0.577979\pi$
$18$		0			0
$19$			1.41421i			0.324443i	0.986754	+	0.162221i	$0.0518659\pi$
$19$			1.41421i			0.324443i	−0.986754	+	0.162221i	$0.948134\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			1.41421i			0.294884i	0.989071	+	0.147442i	$0.0471040\pi$
$23$			1.41421i			0.294884i	−0.989071	+	0.147442i	$0.952896\pi$
$24$		0			0
$25$	5.00000			1.00000
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−4.00000			−0.742781			−0.371391	−	0.928477i	$-0.621119\pi$
$29$	−4.00000			−0.742781			−0.371391	+	0.928477i	$0.621119\pi$
$30$		0			0
$31$	6.00000			1.07763			0.538816	−	0.842424i	$-0.318872\pi$
$31$	6.00000			1.07763			0.538816	+	0.842424i	$0.318872\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−6.00000			−0.986394			−0.493197	−	0.869918i	$-0.664172\pi$
$37$	−6.00000			−0.986394			−0.493197	+	0.869918i	$0.664172\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	2.00000			0.312348			0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000			0.312348			0.156174	+	0.987730i	$0.450084\pi$
$42$		0			0
$43$			1.41421i			0.215666i	0.994169	+	0.107833i	$0.0343911\pi$
$43$			1.41421i			0.215666i	−0.994169	+	0.107833i	$0.965609\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			4.24264i			0.618853i	0.950923	+	0.309426i	$0.100137\pi$
$47$			4.24264i			0.618853i	−0.950923	+	0.309426i	$0.899863\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$		0			0
$51$		0			0
$52$		0			0
$53$			8.48528i			1.16554i	0.812636	+	0.582772i	$0.198032\pi$
$53$			8.48528i			1.16554i	−0.812636	+	0.582772i	$0.801968\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$			11.3137i			1.47292i	0.676481	+	0.736460i	$0.263504\pi$
$59$			11.3137i			1.47292i	−0.676481	+	0.736460i	$0.736496\pi$
$60$		0			0
$61$		−	4.24264i		−	0.543214i	−0.962408	−	0.271607i	$-0.912445\pi$
$61$		−	4.24264i		−	0.543214i	0.962408	−	0.271607i	$-0.0875552\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$	12.0000			1.46603			0.733017	−	0.680211i	$-0.238112\pi$
$67$	12.0000			1.46603			0.733017	+	0.680211i	$0.238112\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			12.7279i			1.51053i	0.655422	+	0.755263i	$0.272491\pi$
$71$			12.7279i			1.51053i	−0.655422	+	0.755263i	$0.727509\pi$
$72$		0			0
$73$		−	8.48528i		−	0.993127i	−0.868000	−	0.496564i	$-0.834595\pi$
$73$		−	8.48528i		−	0.993127i	0.868000	−	0.496564i	$-0.165405\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	4.00000	−	8.48528i	0.455842	−	0.966988i
$78$		0			0
$79$		−	11.3137i		−	1.27289i	−0.771321	−	0.636446i	$-0.780404\pi$
$79$		−	11.3137i		−	1.27289i	0.771321	−	0.636446i	$-0.219596\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	6.00000			0.658586			0.329293	−	0.944228i	$-0.393190\pi$
$83$	6.00000			0.658586			0.329293	+	0.944228i	$0.393190\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	4.24264i		−	0.449719i	−0.974391	−	0.224860i	$-0.927808\pi$
$89$		−	4.24264i		−	0.449719i	0.974391	−	0.224860i	$-0.0721923\pi$
$90$		0			0
$91$	12.0000			1.25794
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−12.0000			−1.21842			−0.609208	−	0.793011i	$-0.708512\pi$
$97$	−12.0000			−1.21842			−0.609208	+	0.793011i	$0.708512\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	10.0000			0.995037			0.497519	−	0.867453i	$-0.334245\pi$
$101$	10.0000			0.995037			0.497519	+	0.867453i	$0.334245\pi$
$102$		0			0
$103$	−6.00000			−0.591198			−0.295599	−	0.955312i	$-0.595519\pi$
$103$	−6.00000			−0.591198			−0.295599	+	0.955312i	$0.595519\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	12.0000			1.16008			0.580042	−	0.814587i	$-0.303036\pi$
$107$	12.0000			1.16008			0.580042	+	0.814587i	$0.303036\pi$
$108$		0			0
$109$		−	4.24264i		−	0.406371i	−0.979140	−	0.203186i	$-0.934871\pi$
$109$		−	4.24264i		−	0.406371i	0.979140	−	0.203186i	$-0.0651295\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	4.24264i		−	0.399114i	−0.979886	−	0.199557i	$-0.936050\pi$
$113$		−	4.24264i		−	0.399114i	0.979886	−	0.199557i	$-0.0639503\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$			5.65685i			0.518563i
$120$		0			0
$121$	7.00000	+	8.48528i	0.636364	+	0.771389i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$		−	11.3137i		−	1.00393i	−0.864888	−	0.501965i	$-0.832611\pi$
$127$		−	11.3137i		−	1.00393i	0.864888	−	0.501965i	$-0.167389\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	12.0000			1.04844			0.524222	−	0.851581i	$-0.324356\pi$
$131$	12.0000			1.04844			0.524222	+	0.851581i	$0.324356\pi$
$132$		0			0
$133$	4.00000			0.346844
$134$		0			0
$135$		0			0
$136$		0			0
$137$			4.24264i			0.362473i	0.983440	+	0.181237i	$0.0580100\pi$
$137$			4.24264i			0.362473i	−0.983440	+	0.181237i	$0.941990\pi$
$138$		0			0
$139$			7.07107i			0.599760i	0.953977	+	0.299880i	$0.0969467\pi$
$139$			7.07107i			0.599760i	−0.953977	+	0.299880i	$0.903053\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−6.00000	+	12.7279i	−0.501745	+	1.06436i
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$	10.0000			0.819232			0.409616	−	0.912258i	$-0.365663\pi$
$149$	10.0000			0.819232			0.409616	+	0.912258i	$0.365663\pi$
$150$		0			0
$151$		−	5.65685i		−	0.460348i	−0.973149	−	0.230174i	$-0.926070\pi$
$151$		−	5.65685i		−	0.460348i	0.973149	−	0.230174i	$-0.0739296\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	14.0000			1.11732			0.558661	−	0.829396i	$-0.311315\pi$
$157$	14.0000			1.11732			0.558661	+	0.829396i	$0.311315\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	4.00000			0.315244
$162$		0			0
$163$	24.0000			1.87983			0.939913	−	0.341415i	$-0.110906\pi$
$163$	24.0000			1.87983			0.939913	+	0.341415i	$0.110906\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	12.0000			0.928588			0.464294	−	0.885681i	$-0.346308\pi$
$167$	12.0000			0.928588			0.464294	+	0.885681i	$0.346308\pi$
$168$		0			0
$169$	−5.00000			−0.384615
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−16.0000			−1.21646			−0.608229	−	0.793762i	$-0.708120\pi$
$173$	−16.0000			−1.21646			−0.608229	+	0.793762i	$0.708120\pi$
$174$		0			0
$175$		−	14.1421i		−	1.06904i
$176$		0			0
$177$		0			0
$178$		0			0
$179$			8.48528i			0.634220i	0.948389	+	0.317110i	$0.102712\pi$
$179$			8.48528i			0.634220i	−0.948389	+	0.317110i	$0.897288\pi$
$180$		0			0
$181$	2.00000			0.148659			0.0743294	−	0.997234i	$-0.476318\pi$
$181$	2.00000			0.148659			0.0743294	+	0.997234i	$0.476318\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−6.00000	−	2.82843i	−0.438763	−	0.206835i
$188$		0			0
$189$		0			0
$190$		0			0
$191$			21.2132i			1.53493i	0.641089	+	0.767467i	$0.278483\pi$
$191$			21.2132i			1.53493i	−0.641089	+	0.767467i	$0.721517\pi$
$192$		0			0
$193$			25.4558i			1.83235i	0.400776	+	0.916176i	$0.368740\pi$
$193$			25.4558i			1.83235i	−0.400776	+	0.916176i	$0.631260\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−10.0000			−0.712470			−0.356235	−	0.934396i	$-0.615940\pi$
$197$	−10.0000			−0.712470			−0.356235	+	0.934396i	$0.615940\pi$
$198$		0			0
$199$	24.0000			1.70131			0.850657	−	0.525720i	$-0.176204\pi$
$199$	24.0000			1.70131			0.850657	+	0.525720i	$0.176204\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$			11.3137i			0.794067i
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−2.00000	+	4.24264i	−0.138343	+	0.293470i
$210$		0			0
$211$			24.0416i			1.65509i	0.561396	+	0.827547i	$0.310264\pi$
$211$			24.0416i			1.65509i	−0.561396	+	0.827547i	$0.689736\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		−	16.9706i		−	1.15204i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		−	8.48528i		−	0.570782i
$222$		0			0
$223$		0			0			−	1.00000i	$-0.5\pi$
$223$		0			0				1.00000i	$0.5\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	12.0000			0.796468			0.398234	−	0.917284i	$-0.369623\pi$
$227$	12.0000			0.796468			0.398234	+	0.917284i	$0.369623\pi$
$228$		0			0
$229$	−6.00000			−0.396491			−0.198246	−	0.980152i	$-0.563524\pi$
$229$	−6.00000			−0.396491			−0.198246	+	0.980152i	$0.563524\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−14.0000			−0.917170			−0.458585	−	0.888650i	$-0.651644\pi$
$233$	−14.0000			−0.917170			−0.458585	+	0.888650i	$0.651644\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−24.0000			−1.55243			−0.776215	−	0.630468i	$-0.782863\pi$
$239$	−24.0000			−1.55243			−0.776215	+	0.630468i	$0.782863\pi$
$240$		0			0
$241$		0			0		1.00000			$0$
$241$		0			0		−1.00000			$\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−6.00000			−0.381771
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	8.48528i		−	0.535586i	−0.963476	−	0.267793i	$-0.913706\pi$
$251$		−	8.48528i		−	0.535586i	0.963476	−	0.267793i	$-0.0862944\pi$
$252$		0			0
$253$	−2.00000	+	4.24264i	−0.125739	+	0.266733i
$254$		0			0
$255$		0			0
$256$		0			0
$257$			21.2132i			1.32324i	0.749838	+	0.661622i	$0.230131\pi$
$257$			21.2132i			1.32324i	−0.749838	+	0.661622i	$0.769869\pi$
$258$		0			0
$259$			16.9706i			1.05450i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−12.0000			−0.739952			−0.369976	−	0.929041i	$-0.620634\pi$
$263$	−12.0000			−0.739952			−0.369976	+	0.929041i	$0.620634\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$			8.48528i			0.517357i	0.965964	+	0.258678i	$0.0832870\pi$
$269$			8.48528i			0.517357i	−0.965964	+	0.258678i	$0.916713\pi$
$270$		0			0
$271$		−	22.6274i		−	1.37452i	−0.726413	−	0.687259i	$-0.758814\pi$
$271$		−	22.6274i		−	1.37452i	0.726413	−	0.687259i	$-0.241186\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	15.0000	+	7.07107i	0.904534	+	0.426401i
$276$		0			0
$277$		−	4.24264i		−	0.254916i	−0.991844	−	0.127458i	$-0.959318\pi$
$277$		−	4.24264i		−	0.254916i	0.991844	−	0.127458i	$-0.0406817\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	10.0000			0.596550			0.298275	−	0.954480i	$-0.403589\pi$
$281$	10.0000			0.596550			0.298275	+	0.954480i	$0.403589\pi$
$282$		0			0
$283$		−	9.89949i		−	0.588464i	−0.955734	−	0.294232i	$-0.904936\pi$
$283$		−	9.89949i		−	0.588464i	0.955734	−	0.294232i	$-0.0950638\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	5.65685i		−	0.333914i
$288$		0			0
$289$	−13.0000			−0.764706
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−28.0000			−1.63578			−0.817889	−	0.575376i	$-0.804856\pi$
$293$	−28.0000			−1.63578			−0.817889	+	0.575376i	$0.804856\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−6.00000			−0.346989
$300$		0			0
$301$	4.00000			0.230556
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$			7.07107i			0.403567i	0.979430	+	0.201784i	$0.0646738\pi$
$307$			7.07107i			0.403567i	−0.979430	+	0.201784i	$0.935326\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			1.41421i			0.0801927i	0.999196	+	0.0400963i	$0.0127665\pi$
$311$			1.41421i			0.0801927i	−0.999196	+	0.0400963i	$0.987234\pi$
$312$		0			0
$313$		0			0			−	1.00000i	$-0.5\pi$
$313$		0			0				1.00000i	$0.5\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	8.48528i		−	0.476581i	−0.971194	−	0.238290i	$-0.923413\pi$
$317$		−	8.48528i		−	0.476581i	0.971194	−	0.238290i	$-0.0765870\pi$
$318$		0			0
$319$	−12.0000	−	5.65685i	−0.671871	−	0.316723i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	2.82843i		−	0.157378i
$324$		0			0
$325$			21.2132i			1.17670i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	12.0000			0.661581
$330$		0			0
$331$	12.0000			0.659580			0.329790	−	0.944054i	$-0.393022\pi$
$331$	12.0000			0.659580			0.329790	+	0.944054i	$0.393022\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$		−	25.4558i		−	1.38667i	−0.720616	−	0.693334i	$-0.756141\pi$
$337$		−	25.4558i		−	1.38667i	0.720616	−	0.693334i	$-0.243859\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	18.0000	+	8.48528i	0.974755	+	0.459504i
$342$		0			0
$343$		−	16.9706i		−	0.916324i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	6.00000			0.322097			0.161048	−	0.986947i	$-0.448512\pi$
$347$	6.00000			0.322097			0.161048	+	0.986947i	$0.448512\pi$
$348$		0			0
$349$		−	12.7279i		−	0.681310i	−0.940188	−	0.340655i	$-0.889351\pi$
$349$		−	12.7279i		−	0.681310i	0.940188	−	0.340655i	$-0.110649\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			12.7279i			0.677439i	0.940887	+	0.338719i	$0.109994\pi$
$353$			12.7279i			0.677439i	−0.940887	+	0.338719i	$0.890006\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$	12.0000			0.633336			0.316668	−	0.948536i	$-0.397436\pi$
$359$	12.0000			0.633336			0.316668	+	0.948536i	$0.397436\pi$
$360$		0			0
$361$	17.0000			0.894737
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−24.0000			−1.25279			−0.626395	−	0.779506i	$-0.715470\pi$
$367$	−24.0000			−1.25279			−0.626395	+	0.779506i	$0.715470\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	24.0000			1.24602
$372$		0			0
$373$			4.24264i			0.219676i	0.993950	+	0.109838i	$0.0350331\pi$
$373$			4.24264i			0.219676i	−0.993950	+	0.109838i	$0.964967\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	16.9706i		−	0.874028i
$378$		0			0
$379$	12.0000			0.616399			0.308199	−	0.951322i	$-0.400274\pi$
$379$	12.0000			0.616399			0.308199	+	0.951322i	$0.400274\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		−	1.41421i		−	0.0722629i	−0.999347	−	0.0361315i	$-0.988496\pi$
$383$		−	1.41421i		−	0.0722629i	0.999347	−	0.0361315i	$-0.0115035\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$		−	16.9706i		−	0.860442i	−0.902724	−	0.430221i	$-0.858436\pi$
$389$		−	16.9706i		−	0.860442i	0.902724	−	0.430221i	$-0.141564\pi$
$390$		0			0
$391$		−	2.82843i		−	0.143040i
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−18.0000			−0.903394			−0.451697	−	0.892171i	$-0.649181\pi$
$397$	−18.0000			−0.903394			−0.451697	+	0.892171i	$0.649181\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	29.6985i		−	1.48307i	−0.670913	−	0.741536i	$-0.734098\pi$
$401$		−	29.6985i		−	1.48307i	0.670913	−	0.741536i	$-0.265902\pi$
$402$		0			0
$403$			25.4558i			1.26805i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−18.0000	−	8.48528i	−0.892227	−	0.420600i
$408$		0			0
$409$			16.9706i			0.839140i	0.907723	+	0.419570i	$0.137819\pi$
$409$			16.9706i			0.839140i	−0.907723	+	0.419570i	$0.862181\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	32.0000			1.57462
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$		−	25.4558i		−	1.24360i	−0.783176	−	0.621800i	$-0.786402\pi$
$419$		−	25.4558i		−	1.24360i	0.783176	−	0.621800i	$-0.213598\pi$
$420$		0			0
$421$	−10.0000			−0.487370			−0.243685	−	0.969854i	$-0.578356\pi$
$421$	−10.0000			−0.487370			−0.243685	+	0.969854i	$0.578356\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−10.0000			−0.485071
$426$		0			0
$427$	−12.0000			−0.580721
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0			−	1.00000i	$-0.5\pi$
$431$		0			0				1.00000i	$0.5\pi$
$432$		0			0
$433$	24.0000			1.15337			0.576683	−	0.816968i	$-0.304347\pi$
$433$	24.0000			1.15337			0.576683	+	0.816968i	$0.304347\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−2.00000			−0.0956730
$438$		0			0
$439$			14.1421i			0.674967i	0.941331	+	0.337484i	$0.109576\pi$
$439$			14.1421i			0.674967i	−0.941331	+	0.337484i	$0.890424\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		−	22.6274i		−	1.07506i	−0.843244	−	0.537531i	$-0.819357\pi$
$443$		−	22.6274i		−	1.07506i	0.843244	−	0.537531i	$-0.180643\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	4.24264i		−	0.200223i	−0.994976	−	0.100111i	$-0.968080\pi$
$449$		−	4.24264i		−	0.200223i	0.994976	−	0.100111i	$-0.0319199\pi$
$450$		0			0
$451$	6.00000	+	2.82843i	0.282529	+	0.133185i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$			16.9706i			0.793849i	0.917851	+	0.396925i	$0.129923\pi$
$457$			16.9706i			0.793849i	−0.917851	+	0.396925i	$0.870077\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	16.0000			0.745194			0.372597	−	0.927993i	$-0.378467\pi$
$461$	16.0000			0.745194			0.372597	+	0.927993i	$0.378467\pi$
$462$		0			0
$463$	−18.0000			−0.836531			−0.418265	−	0.908325i	$-0.637362\pi$
$463$	−18.0000			−0.836531			−0.418265	+	0.908325i	$0.637362\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	19.7990i		−	0.916188i	−0.888904	−	0.458094i	$-0.848532\pi$
$467$		−	19.7990i		−	0.916188i	0.888904	−	0.458094i	$-0.151468\pi$
$468$		0			0
$469$		−	33.9411i		−	1.56726i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−2.00000	+	4.24264i	−0.0919601	+	0.195077i
$474$		0			0
$475$			7.07107i			0.324443i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−24.0000			−1.09659			−0.548294	−	0.836286i	$-0.684723\pi$
$479$	−24.0000			−1.09659			−0.548294	+	0.836286i	$0.684723\pi$
$480$		0			0
$481$		−	25.4558i		−	1.16069i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$		0			0			−	1.00000i	$-0.5\pi$
$487$		0			0				1.00000i	$0.5\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−36.0000			−1.62466			−0.812329	−	0.583200i	$-0.801800\pi$
$491$	−36.0000			−1.62466			−0.812329	+	0.583200i	$0.801800\pi$
$492$		0			0
$493$	8.00000			0.360302
$494$		0			0
$495$		0			0
$496$		0			0
$497$	36.0000			1.61482
$498$		0			0
$499$	−12.0000			−0.537194			−0.268597	−	0.963253i	$-0.586560\pi$
$499$	−12.0000			−0.537194			−0.268597	+	0.963253i	$0.586560\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0			−	1.00000i	$-0.5\pi$
$503$		0			0				1.00000i	$0.5\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$		0			0
$509$			42.4264i			1.88052i	0.340461	+	0.940259i	$0.389417\pi$
$509$			42.4264i			1.88052i	−0.340461	+	0.940259i	$0.610583\pi$
$510$		0			0
$511$	−24.0000			−1.06170
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$	−6.00000	+	12.7279i	−0.263880	+	0.559773i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	29.6985i		−	1.30111i	−0.759457	−	0.650557i	$-0.774535\pi$
$521$		−	29.6985i		−	1.30111i	0.759457	−	0.650557i	$-0.225465\pi$
$522$		0			0
$523$		−	9.89949i		−	0.432875i	−0.976297	−	0.216437i	$-0.930556\pi$
$523$		−	9.89949i		−	0.432875i	0.976297	−	0.216437i	$-0.0694437\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−12.0000			−0.522728
$528$		0			0
$529$	21.0000			0.913043
$530$		0			0
$531$		0			0
$532$		0			0
$533$			8.48528i			0.367538i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−3.00000	−	1.41421i	−0.129219	−	0.0609145i
$540$		0			0
$541$			21.2132i			0.912027i	0.889973	+	0.456013i	$0.150723\pi$
$541$			21.2132i			0.912027i	−0.889973	+	0.456013i	$0.849277\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$			26.8701i			1.14888i	0.818546	+	0.574440i	$0.194780\pi$
$547$			26.8701i			1.14888i	−0.818546	+	0.574440i	$0.805220\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	5.65685i		−	0.240990i
$552$		0			0
$553$	−32.0000			−1.36078
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−16.0000			−0.677942			−0.338971	−	0.940797i	$-0.610079\pi$
$557$	−16.0000			−0.677942			−0.338971	+	0.940797i	$0.610079\pi$
$558$		0			0
$559$	−6.00000			−0.253773
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−18.0000			−0.758610			−0.379305	−	0.925272i	$-0.623837\pi$
$563$	−18.0000			−0.758610			−0.379305	+	0.925272i	$0.623837\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−34.0000			−1.42535			−0.712677	−	0.701492i	$-0.752517\pi$
$569$	−34.0000			−1.42535			−0.712677	+	0.701492i	$0.752517\pi$
$570$		0			0
$571$		−	1.41421i		−	0.0591830i	−0.999562	−	0.0295915i	$-0.990579\pi$
$571$		−	1.41421i		−	0.0591830i	0.999562	−	0.0295915i	$-0.00942064\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			7.07107i			0.294884i
$576$		0			0
$577$	−20.0000			−0.832611			−0.416305	−	0.909225i	$-0.636675\pi$
$577$	−20.0000			−0.832611			−0.416305	+	0.909225i	$0.636675\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		−	16.9706i		−	0.704058i
$582$		0			0
$583$	−12.0000	+	25.4558i	−0.496989	+	1.05427i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	28.2843i		−	1.16742i	−0.811963	−	0.583708i	$-0.801601\pi$
$587$		−	28.2843i		−	1.16742i	0.811963	−	0.583708i	$-0.198399\pi$
$588$		0			0
$589$			8.48528i			0.349630i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−34.0000			−1.39621			−0.698106	−	0.715994i	$-0.745974\pi$
$593$	−34.0000			−1.39621			−0.698106	+	0.715994i	$0.745974\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	24.0416i		−	0.982314i	−0.871071	−	0.491157i	$-0.836574\pi$
$599$		−	24.0416i		−	0.982314i	0.871071	−	0.491157i	$-0.163426\pi$
$600$		0			0
$601$		−	33.9411i		−	1.38449i	−0.721664	−	0.692244i	$-0.756622\pi$
$601$		−	33.9411i		−	1.38449i	0.721664	−	0.692244i	$-0.243378\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		0			0
$606$		0			0
$607$		−	22.6274i		−	0.918419i	−0.888328	−	0.459209i	$-0.848133\pi$
$607$		−	22.6274i		−	0.918419i	0.888328	−	0.459209i	$-0.151867\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−18.0000			−0.728202
$612$		0			0
$613$		−	4.24264i		−	0.171359i	−0.996323	−	0.0856793i	$-0.972694\pi$
$613$		−	4.24264i		−	0.171359i	0.996323	−	0.0856793i	$-0.0273061\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			46.6690i			1.87883i	0.342789	+	0.939413i	$0.388629\pi$
$617$			46.6690i			1.87883i	−0.342789	+	0.939413i	$0.611371\pi$
$618$		0			0
$619$	−36.0000			−1.44696			−0.723481	−	0.690344i	$-0.757459\pi$
$619$	−36.0000			−1.44696			−0.723481	+	0.690344i	$0.757459\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−12.0000			−0.480770
$624$		0			0
$625$	25.0000			1.00000
$626$		0			0
$627$		0			0
$628$		0			0
$629$	12.0000			0.478471
$630$		0			0
$631$	30.0000			1.19428			0.597141	−	0.802137i	$-0.296303\pi$
$631$	30.0000			1.19428			0.597141	+	0.802137i	$0.296303\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$		−	4.24264i		−	0.168100i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	12.7279i		−	0.502723i	−0.967893	−	0.251361i	$-0.919122\pi$
$641$		−	12.7279i		−	0.502723i	0.967893	−	0.251361i	$-0.0808782\pi$
$642$		0			0
$643$	−12.0000			−0.473234			−0.236617	−	0.971603i	$-0.576039\pi$
$643$	−12.0000			−0.473234			−0.236617	+	0.971603i	$0.576039\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	43.8406i		−	1.72355i	−0.507288	−	0.861776i	$-0.669352\pi$
$647$		−	43.8406i		−	1.72355i	0.507288	−	0.861776i	$-0.330648\pi$
$648$		0			0
$649$	−16.0000	+	33.9411i	−0.628055	+	1.33231i
$650$		0			0
$651$		0			0
$652$		0			0
$653$			16.9706i			0.664109i	0.943260	+	0.332055i	$0.107742\pi$
$653$			16.9706i			0.664109i	−0.943260	+	0.332055i	$0.892258\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$	30.0000			1.16863			0.584317	−	0.811525i	$-0.301362\pi$
$659$	30.0000			1.16863			0.584317	+	0.811525i	$0.301362\pi$
$660$		0			0
$661$	−14.0000			−0.544537			−0.272268	−	0.962221i	$-0.587774\pi$
$661$	−14.0000			−0.544537			−0.272268	+	0.962221i	$0.587774\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		−	5.65685i		−	0.219034i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	6.00000	−	12.7279i	0.231627	−	0.491356i
$672$		0			0
$673$			25.4558i			0.981251i	0.871371	+	0.490625i	$0.163232\pi$
$673$			25.4558i			0.981251i	−0.871371	+	0.490625i	$0.836768\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−14.0000			−0.538064			−0.269032	−	0.963131i	$-0.586704\pi$
$677$	−14.0000			−0.538064			−0.269032	+	0.963131i	$0.586704\pi$
$678$		0			0
$679$			33.9411i			1.30254i
$680$		0			0
$681$		0			0
$682$		0			0
$683$			8.48528i			0.324680i	0.986735	+	0.162340i	$0.0519042\pi$
$683$			8.48528i			0.324680i	−0.986735	+	0.162340i	$0.948096\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−36.0000			−1.37149
$690$		0			0
$691$	−24.0000			−0.913003			−0.456502	−	0.889723i	$-0.650898\pi$
$691$	−24.0000			−0.913003			−0.456502	+	0.889723i	$0.650898\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−4.00000			−0.151511
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−20.0000			−0.755390			−0.377695	−	0.925930i	$-0.623283\pi$
$701$	−20.0000			−0.755390			−0.377695	+	0.925930i	$0.623283\pi$
$702$		0			0
$703$		−	8.48528i		−	0.320028i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		−	28.2843i		−	1.06374i
$708$		0			0
$709$	30.0000			1.12667			0.563337	−	0.826227i	$-0.309517\pi$
$709$	30.0000			1.12667			0.563337	+	0.826227i	$0.309517\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			8.48528i			0.317776i
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		−	38.1838i		−	1.42401i	−0.702172	−	0.712007i	$-0.747786\pi$
$719$		−	38.1838i		−	1.42401i	0.702172	−	0.712007i	$-0.252214\pi$
$720$		0			0
$721$			16.9706i			0.632017i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−20.0000			−0.742781
$726$		0			0
$727$	−48.0000			−1.78022			−0.890111	−	0.455744i	$-0.849373\pi$
$727$	−48.0000			−1.78022			−0.890111	+	0.455744i	$0.849373\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		−	2.82843i		−	0.104613i
$732$		0			0
$733$		−	38.1838i		−	1.41035i	−0.709034	−	0.705175i	$-0.750869\pi$
$733$		−	38.1838i		−	1.41035i	0.709034	−	0.705175i	$-0.249131\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	36.0000	+	16.9706i	1.32608	+	0.625119i
$738$		0			0
$739$			9.89949i			0.364159i	0.983284	+	0.182079i	$0.0582828\pi$
$739$			9.89949i			0.364159i	−0.983284	+	0.182079i	$0.941717\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	36.0000			1.32071			0.660356	−	0.750953i	$-0.270405\pi$
$743$	36.0000			1.32071			0.660356	+	0.750953i	$0.270405\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		−	33.9411i		−	1.24018i
$750$		0			0
$751$	−18.0000			−0.656829			−0.328415	−	0.944534i	$-0.606514\pi$
$751$	−18.0000			−0.656829			−0.328415	+	0.944534i	$0.606514\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	34.0000			1.23575			0.617876	−	0.786276i	$-0.287994\pi$
$757$	34.0000			1.23575			0.617876	+	0.786276i	$0.287994\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	10.0000			0.362500			0.181250	−	0.983437i	$-0.441986\pi$
$761$	10.0000			0.362500			0.181250	+	0.983437i	$0.441986\pi$
$762$		0			0
$763$	−12.0000			−0.434429
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−48.0000			−1.73318
$768$		0			0
$769$		−	25.4558i		−	0.917961i	−0.888446	−	0.458981i	$-0.848215\pi$
$769$		−	25.4558i		−	0.917961i	0.888446	−	0.458981i	$-0.151785\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			8.48528i			0.305194i	0.988288	+	0.152597i	$0.0487637\pi$
$773$			8.48528i			0.305194i	−0.988288	+	0.152597i	$0.951236\pi$
$774$		0			0
$775$	30.0000			1.07763
$776$		0			0
$777$		0			0
$778$		0			0
$779$			2.82843i			0.101339i
$780$		0			0
$781$	−18.0000	+	38.1838i	−0.644091	+	1.36632i
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$			35.3553i			1.26028i	0.776481	+	0.630141i	$0.217003\pi$
$787$			35.3553i			1.26028i	−0.776481	+	0.630141i	$0.782997\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−12.0000			−0.426671
$792$		0			0
$793$	18.0000			0.639199
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		1.00000			$0$
$797$		0			0		−1.00000			$\pi$
$798$		0			0
$799$		−	8.48528i		−	0.300188i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	12.0000	−	25.4558i	0.423471	−	0.898317i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	46.0000			1.61727			0.808637	−	0.588308i	$-0.200206\pi$
$809$	46.0000			1.61727			0.808637	+	0.588308i	$0.200206\pi$
$810$		0			0
$811$		−	43.8406i		−	1.53945i	−0.638374	−	0.769726i	$-0.720393\pi$
$811$		−	43.8406i		−	1.53945i	0.638374	−	0.769726i	$-0.279607\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−2.00000			−0.0699711
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−10.0000			−0.349002			−0.174501	−	0.984657i	$-0.555831\pi$
$821$	−10.0000			−0.349002			−0.174501	+	0.984657i	$0.555831\pi$
$822$		0			0
$823$	−24.0000			−0.836587			−0.418294	−	0.908312i	$-0.637372\pi$
$823$	−24.0000			−0.836587			−0.418294	+	0.908312i	$0.637372\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−30.0000			−1.04320			−0.521601	−	0.853189i	$-0.674665\pi$
$827$	−30.0000			−1.04320			−0.521601	+	0.853189i	$0.674665\pi$
$828$		0			0
$829$	6.00000			0.208389			0.104194	−	0.994557i	$-0.466774\pi$
$829$	6.00000			0.208389			0.104194	+	0.994557i	$0.466774\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	2.00000			0.0692959
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	35.3553i		−	1.22060i	−0.792170	−	0.610301i	$-0.791049\pi$
$839$		−	35.3553i		−	1.22060i	0.792170	−	0.610301i	$-0.208951\pi$
$840$		0			0
$841$	−13.0000			−0.448276
$842$		0			0
$843$		0			0
$844$		0			0
$845$		0			0
$846$		0			0
$847$	24.0000	−	19.7990i	0.824650	−	0.680301i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		−	8.48528i		−	0.290872i
$852$		0			0
$853$			55.1543i			1.88845i	0.329304	+	0.944224i	$0.393186\pi$
$853$			55.1543i			1.88845i	−0.329304	+	0.944224i	$0.606814\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	14.0000			0.478231			0.239115	−	0.970991i	$-0.423143\pi$
$857$	14.0000			0.478231			0.239115	+	0.970991i	$0.423143\pi$
$858$		0			0
$859$	−48.0000			−1.63774			−0.818869	−	0.573980i	$-0.805399\pi$
$859$	−48.0000			−1.63774			−0.818869	+	0.573980i	$0.805399\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	18.3848i		−	0.625825i	−0.949782	−	0.312913i	$-0.898695\pi$
$863$		−	18.3848i		−	0.625825i	0.949782	−	0.312913i	$-0.101305\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$	16.0000	−	33.9411i	0.542763	−	1.15137i
$870$		0			0
$871$			50.9117i			1.72508i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$		−	46.6690i		−	1.57590i	−0.615738	−	0.787951i	$-0.711142\pi$
$877$		−	46.6690i		−	1.57590i	0.615738	−	0.787951i	$-0.288858\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			12.7279i			0.428815i	0.976744	+	0.214407i	$0.0687820\pi$
$881$			12.7279i			0.428815i	−0.976744	+	0.214407i	$0.931218\pi$
$882$		0			0
$883$	36.0000			1.21150			0.605748	−	0.795656i	$-0.292874\pi$
$883$	36.0000			1.21150			0.605748	+	0.795656i	$0.292874\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	12.0000			0.402921			0.201460	−	0.979497i	$-0.435431\pi$
$887$	12.0000			0.402921			0.201460	+	0.979497i	$0.435431\pi$
$888$		0			0
$889$	−32.0000			−1.07325
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−6.00000			−0.200782
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−24.0000			−0.800445
$900$		0			0
$901$		−	16.9706i		−	0.565371i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−36.0000			−1.19536			−0.597680	−	0.801735i	$-0.703911\pi$
$907$	−36.0000			−1.19536			−0.597680	+	0.801735i	$0.703911\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			46.6690i			1.54621i	0.634275	+	0.773107i	$0.281299\pi$
$911$			46.6690i			1.54621i	−0.634275	+	0.773107i	$0.718701\pi$
$912$		0			0
$913$	18.0000	+	8.48528i	0.595713	+	0.280822i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		−	33.9411i		−	1.12083i
$918$		0			0
$919$			5.65685i			0.186602i	0.995638	+	0.0933012i	$0.0297420\pi$
$919$			5.65685i			0.186602i	−0.995638	+	0.0933012i	$0.970258\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−54.0000			−1.77743
$924$		0			0
$925$	−30.0000			−0.986394
$926$		0			0
$927$		0			0
$928$		0			0
$929$		−	4.24264i		−	0.139197i	−0.997575	−	0.0695983i	$-0.977828\pi$
$929$		−	4.24264i		−	0.139197i	0.997575	−	0.0695983i	$-0.0221717\pi$
$930$		0			0
$931$		−	1.41421i		−	0.0463490i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$			16.9706i			0.554404i	0.960812	+	0.277202i	$0.0894071\pi$
$937$			16.9706i			0.554404i	−0.960812	+	0.277202i	$0.910593\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	38.0000			1.23876			0.619382	−	0.785090i	$-0.287383\pi$
$941$	38.0000			1.23876			0.619382	+	0.785090i	$0.287383\pi$
$942$		0			0
$943$			2.82843i			0.0921063i
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		1.00000			$0$
$947$		0			0		−1.00000			$\pi$
$948$		0			0
$949$	36.0000			1.16861
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−34.0000			−1.10137			−0.550684	−	0.834714i	$-0.685633\pi$
$953$	−34.0000			−1.10137			−0.550684	+	0.834714i	$0.685633\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$	12.0000			0.387500
$960$		0			0
$961$	5.00000			0.161290
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$			56.5685i			1.81912i	0.415571	+	0.909561i	$0.363582\pi$
$967$			56.5685i			1.81912i	−0.415571	+	0.909561i	$0.636418\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			56.5685i			1.81537i	0.419651	+	0.907685i	$0.362152\pi$
$971$			56.5685i			1.81537i	−0.419651	+	0.907685i	$0.637848\pi$
$972$		0			0
$973$	20.0000			0.641171
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	38.1838i		−	1.22161i	−0.791782	−	0.610803i	$-0.790847\pi$
$977$		−	38.1838i		−	1.22161i	0.791782	−	0.610803i	$-0.209153\pi$
$978$		0			0
$979$	6.00000	−	12.7279i	0.191761	−	0.406786i
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	15.5563i		−	0.496170i	−0.968738	−	0.248085i	$-0.920199\pi$
$983$		−	15.5563i		−	0.496170i	0.968738	−	0.248085i	$-0.0798013\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−2.00000			−0.0635963
$990$		0			0
$991$	18.0000			0.571789			0.285894	−	0.958261i	$-0.407709\pi$
$991$	18.0000			0.571789			0.285894	+	0.958261i	$0.407709\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$		−	46.6690i		−	1.47802i	−0.673693	−	0.739012i	$-0.735293\pi$
$997$		−	46.6690i		−	1.47802i	0.673693	−	0.739012i	$-0.264707\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3168.2.b.f.2177.1	yes	2
3.2	odd	2		3168.2.b.c.2177.1	yes	2
4.3	odd	2		3168.2.b.b.2177.2	yes	2
8.3	odd	2		6336.2.b.l.2177.2		2
8.5	even	2		6336.2.b.d.2177.1		2
11.10	odd	2		3168.2.b.c.2177.2	yes	2
12.11	even	2		3168.2.b.g.2177.2	yes	2
24.5	odd	2		6336.2.b.m.2177.1		2
24.11	even	2		6336.2.b.e.2177.2		2
33.32	even	2	inner	3168.2.b.f.2177.2	yes	2
44.43	even	2		3168.2.b.g.2177.1	yes	2
88.21	odd	2		6336.2.b.m.2177.2		2
88.43	even	2		6336.2.b.e.2177.1		2
132.131	odd	2		3168.2.b.b.2177.1	✓	2
264.131	odd	2		6336.2.b.l.2177.1		2
264.197	even	2		6336.2.b.d.2177.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3168.2.b.b.2177.1	✓	2	132.131	odd	2
3168.2.b.b.2177.2	yes	2	4.3	odd	2
3168.2.b.c.2177.1	yes	2	3.2	odd	2
3168.2.b.c.2177.2	yes	2	11.10	odd	2
3168.2.b.f.2177.1	yes	2	1.1	even	1	trivial
3168.2.b.f.2177.2	yes	2	33.32	even	2	inner
3168.2.b.g.2177.1	yes	2	44.43	even	2
3168.2.b.g.2177.2	yes	2	12.11	even	2
6336.2.b.d.2177.1		2	8.5	even	2
6336.2.b.d.2177.2		2	264.197	even	2
6336.2.b.e.2177.1		2	88.43	even	2
6336.2.b.e.2177.2		2	24.11	even	2
6336.2.b.l.2177.1		2	264.131	odd	2
6336.2.b.l.2177.2		2	8.3	odd	2
6336.2.b.m.2177.1		2	24.5	odd	2
6336.2.b.m.2177.2		2	88.21	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 3168 = 2^{5} \cdot 3^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3168.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.82843i q^{7} +(3.00000 + 1.41421i) q^{11} +4.24264i q^{13} -2.00000 q^{17} +1.41421i q^{19} +1.41421i q^{23} +5.00000 q^{25} -4.00000 q^{29} +6.00000 q^{31} -6.00000 q^{37} +2.00000 q^{41} +1.41421i q^{43} +4.24264i q^{47} -1.00000 q^{49} +8.48528i q^{53} +11.3137i q^{59} -4.24264i q^{61} +12.0000 q^{67} +12.7279i q^{71} -8.48528i q^{73} +(4.00000 - 8.48528i) q^{77} -11.3137i q^{79} +6.00000 q^{83} -4.24264i q^{89} +12.0000 q^{91} -12.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 6 q^{11} - 4 q^{17} + 10 q^{25} - 8 q^{29} + 12 q^{31} - 12 q^{37} + 4 q^{41} - 2 q^{49} + 24 q^{67} + 8 q^{77} + 12 q^{83} + 24 q^{91} - 24 q^{97}+O(q^{100}) \) 2 * q + 6 * q^11 - 4 * q^17 + 10 * q^25 - 8 * q^29 + 12 * q^31 - 12 * q^37 + 4 * q^41 - 2 * q^49 + 24 * q^67 + 8 * q^77 + 12 * q^83 + 24 * q^91 - 24 * q^97

Introduction

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Database

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