LMFDB - Embedded newform 2592.2.i.bg.865.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2592,2,Mod(865,2592)] 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("2592.865"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2592, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 4])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2592 = 2^{5} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2592.i (of order $3$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,0,0,0,0,0,0,0,-4,0,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	no
Analytic conductor:	$20.6972242039$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{3})$
Coefficient field:	8.0.49787136.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 3x^{6} + 5x^{4} + 12x^{2} + 16 $ x^8 + 3x^6 + 5x^4 + 12*x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4}\cdot 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			865.4
Root			$1.09445 + 0.895644i$ of defining polynomial
Character	$\chi$	$=$	2592.865
Dual form			2592.2.i.bg.1729.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.32288 + 2.29129i) q^{5} +(2.18890 - 3.79129i) q^{7} +(1.79129 - 3.10260i) q^{11} +(3.29129 + 5.70068i) q^{13} -1.73205 q^{17} +2.55040 q^{19} +(-3.79129 - 6.56670i) q^{23} +(-1.00000 + 1.73205i) q^{25} +(3.05493 - 5.29129i) q^{29} +(4.37780 + 7.58258i) q^{31} +11.5826 q^{35} +2.58258 q^{37} +(-0.913701 - 1.58258i) q^{41} +(1.27520 - 2.20871i) q^{43} +(-4.00000 + 6.92820i) q^{47} +(-6.08258 - 10.5353i) q^{49} -1.82740 q^{53} +9.47860 q^{55} +(-4.00000 - 6.92820i) q^{59} +(0.708712 - 1.22753i) q^{61} +(-8.70793 + 15.0826i) q^{65} +(1.27520 + 2.20871i) q^{67} +0.417424 q^{71} -6.16515 q^{73} +(-7.84190 - 13.5826i) q^{77} +(4.73930 - 8.20871i) q^{79} +(-7.58258 + 13.1334i) q^{83} +(-2.29129 - 3.96863i) q^{85} +12.3151 q^{89} +28.8172 q^{91} +(3.37386 + 5.84370i) q^{95} +(2.58258 - 4.47315i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 4 q^{11} + 8 q^{13} - 12 q^{23} - 8 q^{25} + 56 q^{35} - 16 q^{37} - 32 q^{47} - 12 q^{49} - 32 q^{59} + 24 q^{61} + 40 q^{71} + 24 q^{73} - 24 q^{83} - 28 q^{95} - 16 q^{97}+O(q^{100}) $ 8 * q - 4 * q^11 + 8 * q^13 - 12 * q^23 - 8 * q^25 + 56 * q^35 - 16 * q^37 - 32 * q^47 - 12 * q^49 - 32 * q^59 + 24 * q^61 + 40 * q^71 + 24 * q^73 - 24 * q^83 - 28 * q^95 - 16 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2592\mathbb{Z}\right)^\times$.

$n$	$325$	$1217$	$2431$
$\chi(n)$	$1$	$e\left(\frac{2}{3}\right)$	$1$

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
$5$	1.32288	+	2.29129i	0.591608	+	1.02470i	0.994016	+	0.109235i	$0.0348400\pi$
$5$	1.32288	+	2.29129i	0.591608	+	1.02470i	−0.402408	+	0.915460i	$0.631827\pi$
$6$		0			0
$7$	2.18890	−	3.79129i	0.827327	−	1.43297i	−0.0728011	−	0.997346i	$-0.523194\pi$
$7$	2.18890	−	3.79129i	0.827327	−	1.43297i	0.900128	−	0.435626i	$-0.143473\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	1.79129	−	3.10260i	0.540094	−	0.935470i	−0.458804	−	0.888537i	$-0.651722\pi$
$11$	1.79129	−	3.10260i	0.540094	−	0.935470i	0.998898	−	0.0469323i	$-0.0149445\pi$
$12$		0			0
$13$	3.29129	+	5.70068i	0.912839	+	1.58108i	0.810035	+	0.586382i	$0.199448\pi$
$13$	3.29129	+	5.70068i	0.912839	+	1.58108i	0.102804	+	0.994702i	$0.467218\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−1.73205			−0.420084			−0.210042	−	0.977692i	$-0.567360\pi$
$17$	−1.73205			−0.420084			−0.210042	+	0.977692i	$0.567360\pi$
$18$		0			0
$19$	2.55040			0.585102			0.292551	−	0.956250i	$-0.405496\pi$
$19$	2.55040			0.585102			0.292551	+	0.956250i	$0.405496\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−3.79129	−	6.56670i	−0.790538	−	1.36925i	−0.925634	−	0.378420i	$-0.876468\pi$
$23$	−3.79129	−	6.56670i	−0.790538	−	1.36925i	0.135096	−	0.990833i	$-0.456866\pi$
$24$		0			0
$25$	−1.00000	+	1.73205i	−0.200000	+	0.346410i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	3.05493	−	5.29129i	0.567286	−	0.982567i	−0.429547	−	0.903044i	$-0.641327\pi$
$29$	3.05493	−	5.29129i	0.567286	−	0.982567i	0.996833	−	0.0795232i	$-0.0253398\pi$
$30$		0			0
$31$	4.37780	+	7.58258i	0.786276	+	1.36187i	0.928234	+	0.371998i	$0.121327\pi$
$31$	4.37780	+	7.58258i	0.786276	+	1.36187i	−0.141957	+	0.989873i	$0.545339\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	11.5826			1.95781
$36$		0			0
$37$	2.58258			0.424573			0.212286	−	0.977207i	$-0.431909\pi$
$37$	2.58258			0.424573			0.212286	+	0.977207i	$0.431909\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−0.913701	−	1.58258i	−0.142696	−	0.247157i	0.785815	−	0.618462i	$-0.212244\pi$
$41$	−0.913701	−	1.58258i	−0.142696	−	0.247157i	−0.928511	+	0.371305i	$0.878910\pi$
$42$		0			0
$43$	1.27520	−	2.20871i	0.194466	−	0.336825i	−0.752259	−	0.658867i	$-0.771036\pi$
$43$	1.27520	−	2.20871i	0.194466	−	0.336825i	0.946725	+	0.322042i	$0.104369\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−4.00000	+	6.92820i	−0.583460	+	1.01058i	0.411606	+	0.911362i	$0.364968\pi$
$47$	−4.00000	+	6.92820i	−0.583460	+	1.01058i	−0.995066	+	0.0992202i	$0.968365\pi$
$48$		0			0
$49$	−6.08258	−	10.5353i	−0.868939	−	1.50505i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−1.82740			−0.251013			−0.125506	−	0.992093i	$-0.540056\pi$
$53$	−1.82740			−0.251013			−0.125506	+	0.992093i	$0.540056\pi$
$54$		0			0
$55$	9.47860			1.27809
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−4.00000	−	6.92820i	−0.520756	−	0.901975i	−0.999709	−	0.0241347i	$-0.992317\pi$
$59$	−4.00000	−	6.92820i	−0.520756	−	0.901975i	0.478953	−	0.877841i	$-0.341016\pi$
$60$		0			0
$61$	0.708712	−	1.22753i	0.0907413	−	0.157169i	−0.817082	−	0.576522i	$-0.804410\pi$
$61$	0.708712	−	1.22753i	0.0907413	−	0.157169i	0.907823	+	0.419353i	$0.137743\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−8.70793	+	15.0826i	−1.08009	+	1.87076i
$66$		0			0
$67$	1.27520	+	2.20871i	0.155791	+	0.269837i	0.933347	−	0.358976i	$-0.116874\pi$
$67$	1.27520	+	2.20871i	0.155791	+	0.269837i	−0.777556	+	0.628814i	$0.783541\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	0.417424			0.0495392			0.0247696	−	0.999693i	$-0.492115\pi$
$71$	0.417424			0.0495392			0.0247696	+	0.999693i	$0.492115\pi$
$72$		0			0
$73$	−6.16515			−0.721576			−0.360788	−	0.932648i	$-0.617492\pi$
$73$	−6.16515			−0.721576			−0.360788	+	0.932648i	$0.617492\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−7.84190	−	13.5826i	−0.893668	−	1.54788i
$78$		0			0
$79$	4.73930	−	8.20871i	0.533213	−	0.923552i	−0.466034	−	0.884767i	$-0.654318\pi$
$79$	4.73930	−	8.20871i	0.533213	−	0.923552i	0.999248	−	0.0387857i	$-0.0123490\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−7.58258	+	13.1334i	−0.832296	+	1.44158i	0.0639175	+	0.997955i	$0.479641\pi$
$83$	−7.58258	+	13.1334i	−0.832296	+	1.44158i	−0.896213	+	0.443623i	$0.853693\pi$
$84$		0			0
$85$	−2.29129	−	3.96863i	−0.248525	−	0.430458i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	12.3151			1.30539			0.652697	−	0.757619i	$-0.273638\pi$
$89$	12.3151			1.30539			0.652697	+	0.757619i	$0.273638\pi$
$90$		0			0
$91$	28.8172			3.02086
$92$		0			0
$93$		0			0
$94$		0			0
$95$	3.37386	+	5.84370i	0.346151	+	0.599551i
$96$		0			0
$97$	2.58258	−	4.47315i	0.262221	−	0.454180i	−0.704611	−	0.709594i	$-0.748878\pi$
$97$	2.58258	−	4.47315i	0.262221	−	0.454180i	0.966832	+	0.255414i	$0.0822118\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	6.92820	−	12.0000i	0.689382	−	1.19404i	−0.282656	−	0.959221i	$-0.591216\pi$
$101$	6.92820	−	12.0000i	0.689382	−	1.19404i	0.972038	−	0.234823i	$-0.0754512\pi$
$102$		0			0
$103$	−6.20520	−	10.7477i	−0.611417	−	1.05901i	−0.991002	−	0.133848i	$-0.957267\pi$
$103$	−6.20520	−	10.7477i	−0.611417	−	1.05901i	0.379585	−	0.925157i	$-0.376067\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−7.16515			−0.692681			−0.346341	−	0.938109i	$-0.612576\pi$
$107$	−7.16515			−0.692681			−0.346341	+	0.938109i	$0.612576\pi$
$108$		0			0
$109$	−14.5826			−1.39676			−0.698379	−	0.715728i	$-0.746095\pi$
$109$	−14.5826			−1.39676			−0.698379	+	0.715728i	$0.746095\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	9.62163	+	16.6652i	0.905127	+	1.56773i	0.820747	+	0.571292i	$0.193558\pi$
$113$	9.62163	+	16.6652i	0.905127	+	1.56773i	0.0843798	+	0.996434i	$0.473109\pi$
$114$		0			0
$115$	10.0308	−	17.3739i	0.935377	−	1.62012i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−3.79129	+	6.56670i	−0.347547	+	0.601969i
$120$		0			0
$121$	−0.917424	−	1.58903i	−0.0834022	−	0.144457i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	7.93725			0.709930
$126$		0			0
$127$	18.2342			1.61802			0.809012	−	0.587792i	$-0.200003\pi$
$127$	18.2342			1.61802			0.809012	+	0.587792i	$0.200003\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	1.79129	+	3.10260i	0.156506	+	0.271076i	0.933606	−	0.358301i	$-0.116644\pi$
$131$	1.79129	+	3.10260i	0.156506	+	0.271076i	−0.777101	+	0.629376i	$0.783310\pi$
$132$		0			0
$133$	5.58258	−	9.66930i	0.484071	−	0.838435i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−4.33013	+	7.50000i	−0.369948	+	0.640768i	−0.989557	−	0.144142i	$-0.953958\pi$
$137$	−4.33013	+	7.50000i	−0.369948	+	0.640768i	0.619609	+	0.784910i	$0.287291\pi$
$138$		0			0
$139$	4.37780	+	7.58258i	0.371320	+	0.643146i	0.989769	−	0.142680i	$-0.0455718\pi$
$139$	4.37780	+	7.58258i	0.371320	+	0.643146i	−0.618449	+	0.785825i	$0.712239\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	23.5826			1.97207
$144$		0			0
$145$	16.1652			1.34244
$146$		0			0
$147$		0			0
$148$		0			0
$149$	0.504525	+	0.873864i	0.0413323	+	0.0715897i	0.885952	−	0.463778i	$-0.153506\pi$
$149$	0.504525	+	0.873864i	0.0413323	+	0.0715897i	−0.844619	+	0.535368i	$0.820173\pi$
$150$		0			0
$151$	−6.20520	+	10.7477i	−0.504972	+	0.874638i	0.495011	+	0.868887i	$0.335164\pi$
$151$	−6.20520	+	10.7477i	−0.504972	+	0.874638i	−0.999983	+	0.00575103i	$0.998169\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−11.5826	+	20.0616i	−0.930335	+	1.61139i
$156$		0			0
$157$	10.2913	+	17.8250i	0.821334	+	1.42259i	0.904689	+	0.426073i	$0.140103\pi$
$157$	10.2913	+	17.8250i	0.821334	+	1.42259i	−0.0833548	+	0.996520i	$0.526563\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−33.1950			−2.61613
$162$		0			0
$163$	−3.65480			−0.286266			−0.143133	−	0.989703i	$-0.545718\pi$
$163$	−3.65480			−0.286266			−0.143133	+	0.989703i	$0.545718\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−7.79129	−	13.4949i	−0.602908	−	1.04427i	−0.992378	−	0.123228i	$-0.960675\pi$
$167$	−7.79129	−	13.4949i	−0.602908	−	1.04427i	0.389470	−	0.921039i	$-0.372658\pi$
$168$		0			0
$169$	−15.1652	+	26.2668i	−1.16655	+	2.02052i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	1.22753	−	2.12614i	0.0933270	−	0.161647i	−0.815582	−	0.578641i	$-0.803583\pi$
$173$	1.22753	−	2.12614i	0.0933270	−	0.161647i	0.908909	+	0.416994i	$0.136916\pi$
$174$		0			0
$175$	4.37780	+	7.58258i	0.330931	+	0.573189i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0			−	1.00000i	$-0.5\pi$
$179$		0			0				1.00000i	$0.5\pi$
$180$		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
$185$	3.41643	+	5.91742i	0.251181	+	0.435058i
$186$		0			0
$187$	−3.10260	+	5.37386i	−0.226885	+	0.392976i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	3.79129	−	6.56670i	0.274328	−	0.475150i	−0.695637	−	0.718393i	$-0.744878\pi$
$191$	3.79129	−	6.56670i	0.274328	−	0.475150i	0.969965	+	0.243243i	$0.0782113\pi$
$192$		0			0
$193$	6.08258	+	10.5353i	0.437833	+	0.758350i	0.997522	−	0.0703532i	$-0.0224126\pi$
$193$	6.08258	+	10.5353i	0.437833	+	0.758350i	−0.559689	+	0.828703i	$0.689079\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	14.6748			1.04553			0.522767	−	0.852476i	$-0.324900\pi$
$197$	14.6748			1.04553			0.522767	+	0.852476i	$0.324900\pi$
$198$		0			0
$199$	17.5112			1.24134			0.620668	−	0.784073i	$-0.286861\pi$
$199$	17.5112			1.24134			0.620668	+	0.784073i	$0.286861\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−13.3739	−	23.1642i	−0.938661	−	1.62581i
$204$		0			0
$205$	2.41742	−	4.18710i	0.168840	−	0.292440i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	4.56850	−	7.91288i	0.316010	−	0.547345i
$210$		0			0
$211$	5.65300	+	9.79129i	0.389169	+	0.674060i	0.992338	−	0.123553i	$-0.0394290\pi$
$211$	5.65300	+	9.79129i	0.389169	+	0.674060i	−0.603169	+	0.797613i	$0.706096\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	6.74773			0.460191
$216$		0			0
$217$	38.3303			2.60203
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−5.70068	−	9.87386i	−0.383469	−	0.664188i
$222$		0			0
$223$	8.39410	−	14.5390i	0.562111	−	0.973604i	−0.435201	−	0.900333i	$-0.643323\pi$
$223$	8.39410	−	14.5390i	0.562111	−	0.973604i	0.997312	−	0.0732713i	$-0.0233439\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	5.37386	−	9.30780i	0.356676	−	0.617781i	−0.630727	−	0.776004i	$-0.717243\pi$
$227$	5.37386	−	9.30780i	0.356676	−	0.617781i	0.987403	+	0.158224i	$0.0505767\pi$
$228$		0			0
$229$	−1.87386	−	3.24563i	−0.123828	−	0.214477i	0.797446	−	0.603390i	$-0.206184\pi$
$229$	−1.87386	−	3.24563i	−0.123828	−	0.214477i	−0.921274	+	0.388913i	$0.872851\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−10.6784			−0.699562			−0.349781	−	0.936831i	$-0.613744\pi$
$233$	−10.6784			−0.699562			−0.349781	+	0.936831i	$0.613744\pi$
$234$		0			0
$235$	−21.1660			−1.38072
$236$		0			0
$237$		0			0
$238$		0			0
$239$	4.41742	+	7.65120i	0.285739	+	0.494915i	0.972788	−	0.231696i	$-0.0744275\pi$
$239$	4.41742	+	7.65120i	0.285739	+	0.494915i	−0.687049	+	0.726611i	$0.741094\pi$
$240$		0			0
$241$	−0.500000	+	0.866025i	−0.0322078	+	0.0557856i	−0.881680	−	0.471848i	$-0.843587\pi$
$241$	−0.500000	+	0.866025i	−0.0322078	+	0.0557856i	0.849472	+	0.527633i	$0.176921\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	16.0930	−	27.8739i	1.02814	−	1.78080i
$246$		0			0
$247$	8.39410	+	14.5390i	0.534104	+	0.925095i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	4.41742			0.278825			0.139413	−	0.990234i	$-0.455479\pi$
$251$	4.41742			0.278825			0.139413	+	0.990234i	$0.455479\pi$
$252$		0			0
$253$	−27.1652			−1.70786
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−1.77973	−	3.08258i	−0.111016	−	0.192286i	0.805164	−	0.593052i	$-0.202077\pi$
$257$	−1.77973	−	3.08258i	−0.111016	−	0.192286i	−0.916180	+	0.400766i	$0.868744\pi$
$258$		0			0
$259$	5.65300	−	9.79129i	0.351260	−	0.608401i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	3.16515	−	5.48220i	0.195172	−	0.338047i	−0.751785	−	0.659408i	$-0.770807\pi$
$263$	3.16515	−	5.48220i	0.195172	−	0.338047i	0.946957	+	0.321361i	$0.104140\pi$
$264$		0			0
$265$	−2.41742	−	4.18710i	−0.148501	−	0.257212i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−5.91915			−0.360897			−0.180449	−	0.983584i	$-0.557755\pi$
$269$	−5.91915			−0.360897			−0.180449	+	0.983584i	$0.557755\pi$
$270$		0			0
$271$	−13.1334			−0.797798			−0.398899	−	0.916995i	$-0.630608\pi$
$271$	−13.1334			−0.797798			−0.398899	+	0.916995i	$0.630608\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	3.58258	+	6.20520i	0.216037	+	0.374188i
$276$		0			0
$277$	10.1652	−	17.6066i	0.610765	−	1.05788i	−0.380347	−	0.924844i	$-0.624195\pi$
$277$	10.1652	−	17.6066i	0.610765	−	1.05788i	0.991112	−	0.133032i	$-0.0424712\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−12.2674	+	21.2477i	−0.731811	+	1.26753i	0.224298	+	0.974521i	$0.427991\pi$
$281$	−12.2674	+	21.2477i	−0.731811	+	1.26753i	−0.956109	+	0.293013i	$0.905342\pi$
$282$		0			0
$283$	−8.75560	−	15.1652i	−0.520467	−	0.901475i	−0.999717	−	0.0237963i	$-0.992425\pi$
$283$	−8.75560	−	15.1652i	−0.520467	−	0.901475i	0.479250	−	0.877678i	$-0.340909\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−8.00000			−0.472225
$288$		0			0
$289$	−14.0000			−0.823529
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−7.43273	−	12.8739i	−0.434225	−	0.752099i	0.563007	−	0.826452i	$-0.309644\pi$
$293$	−7.43273	−	12.8739i	−0.434225	−	0.752099i	−0.997232	+	0.0743527i	$0.976311\pi$
$294$		0			0
$295$	10.5830	−	18.3303i	0.616166	−	1.06723i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	24.9564	−	43.2258i	1.44327	−	2.49981i
$300$		0			0
$301$	−5.58258	−	9.66930i	−0.321774	−	0.557329i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	3.75015			0.214733
$306$		0			0
$307$	12.4104			0.708299			0.354150	−	0.935189i	$-0.384770\pi$
$307$	12.4104			0.708299			0.354150	+	0.935189i	$0.384770\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−7.58258	−	13.1334i	−0.429968	−	0.744727i	0.566902	−	0.823786i	$-0.308142\pi$
$311$	−7.58258	−	13.1334i	−0.429968	−	0.744727i	−0.996870	+	0.0790584i	$0.974809\pi$
$312$		0			0
$313$	−9.50000	+	16.4545i	−0.536972	+	0.930062i	0.462093	+	0.886831i	$0.347098\pi$
$313$	−9.50000	+	16.4545i	−0.536972	+	0.930062i	−0.999065	+	0.0432311i	$0.986235\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−13.6379	+	23.6216i	−0.765983	+	1.32672i	0.173743	+	0.984791i	$0.444414\pi$
$317$	−13.6379	+	23.6216i	−0.765983	+	1.32672i	−0.939726	+	0.341930i	$0.888920\pi$
$318$		0			0
$319$	−10.9445	−	18.9564i	−0.612775	−	1.06136i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−4.41742			−0.245792
$324$		0			0
$325$	−13.1652			−0.730271
$326$		0			0
$327$		0			0
$328$		0			0
$329$	17.5112	+	30.3303i	0.965424	+	1.67216i
$330$		0			0
$331$	0.552200	−	0.956439i	0.0303517	−	0.0525707i	−0.850451	−	0.526055i	$-0.823671\pi$
$331$	0.552200	−	0.956439i	0.0303517	−	0.0525707i	0.880802	+	0.473484i	$0.157004\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−3.37386	+	5.84370i	−0.184334	+	0.319276i
$336$		0			0
$337$	−8.58258	−	14.8655i	−0.467523	−	0.809773i	0.531789	−	0.846877i	$-0.321520\pi$
$337$	−8.58258	−	14.8655i	−0.467523	−	0.809773i	−0.999311	+	0.0371040i	$0.988187\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	31.3676			1.69865
$342$		0			0
$343$	−22.6120			−1.22093
$344$		0			0
$345$		0			0
$346$		0			0
$347$	17.7913	+	30.8154i	0.955086	+	1.65426i	0.734170	+	0.678965i	$0.237571\pi$
$347$	17.7913	+	30.8154i	0.955086	+	1.65426i	0.220916	+	0.975293i	$0.429095\pi$
$348$		0			0
$349$	−0.582576	+	1.00905i	−0.0311846	+	0.0540132i	−0.881196	−	0.472750i	$-0.843261\pi$
$349$	−0.582576	+	1.00905i	−0.0311846	+	0.0540132i	0.850012	+	0.526763i	$0.176595\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−15.6838	+	27.1652i	−0.834765	+	1.44586i	0.0594569	+	0.998231i	$0.481063\pi$
$353$	−15.6838	+	27.1652i	−0.834765	+	1.44586i	−0.894222	+	0.447624i	$0.852270\pi$
$354$		0			0
$355$	0.552200	+	0.956439i	0.0293078	+	0.0507625i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−21.9129			−1.15652			−0.578259	−	0.815853i	$-0.696268\pi$
$359$	−21.9129			−1.15652			−0.578259	+	0.815853i	$0.696268\pi$
$360$		0			0
$361$	−12.4955			−0.657655
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−8.15573	−	14.1261i	−0.426890	−	0.739396i
$366$		0			0
$367$	−13.1334	+	22.7477i	−0.685558	+	1.18742i	0.287703	+	0.957720i	$0.407108\pi$
$367$	−13.1334	+	22.7477i	−0.685558	+	1.18742i	−0.973261	+	0.229702i	$0.926225\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−4.00000	+	6.92820i	−0.207670	+	0.359694i
$372$		0			0
$373$	−7.00000	−	12.1244i	−0.362446	−	0.627775i	0.625917	−	0.779890i	$-0.284725\pi$
$373$	−7.00000	−	12.1244i	−0.362446	−	0.627775i	−0.988363	+	0.152115i	$0.951392\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	40.2186			2.07136
$378$		0			0
$379$	−5.10080			−0.262011			−0.131005	−	0.991382i	$-0.541820\pi$
$379$	−5.10080			−0.262011			−0.131005	+	0.991382i	$0.541820\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−7.79129	−	13.4949i	−0.398116	−	0.689558i	0.595377	−	0.803446i	$-0.297003\pi$
$383$	−7.79129	−	13.4949i	−0.398116	−	0.689558i	−0.993494	+	0.113889i	$0.963669\pi$
$384$		0			0
$385$	20.7477	−	35.9361i	1.05740	−	1.83147i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	16.5975	−	28.7477i	0.841527	−	1.45757i	−0.0470764	−	0.998891i	$-0.514990\pi$
$389$	16.5975	−	28.7477i	0.841527	−	1.45757i	0.888603	−	0.458676i	$-0.151676\pi$
$390$		0			0
$391$	6.56670	+	11.3739i	0.332092	+	0.575201i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	25.0780			1.26181
$396$		0			0
$397$	17.7477			0.890733			0.445366	−	0.895348i	$-0.353073\pi$
$397$	17.7477			0.890733			0.445366	+	0.895348i	$0.353073\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	0.0476751	+	0.0825757i	0.00238078	+	0.00412363i	0.867213	−	0.497937i	$-0.165909\pi$
$401$	0.0476751	+	0.0825757i	0.00238078	+	0.00412363i	−0.864833	+	0.502060i	$0.832576\pi$
$402$		0			0
$403$	−28.8172	+	49.9129i	−1.43549	+	2.48634i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	4.62614	−	8.01270i	0.229309	−	0.397175i
$408$		0			0
$409$	−8.08258	−	13.9994i	−0.399658	−	0.692227i	0.594026	−	0.804446i	$-0.297538\pi$
$409$	−8.08258	−	13.9994i	−0.399658	−	0.692227i	−0.993684	+	0.112219i	$0.964204\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−35.0224			−1.72334
$414$		0			0
$415$	−40.1232			−1.96957
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−3.58258	−	6.20520i	−0.175020	−	0.303144i	0.765148	−	0.643854i	$-0.222666\pi$
$419$	−3.58258	−	6.20520i	−0.175020	−	0.303144i	−0.940168	+	0.340710i	$0.889332\pi$
$420$		0			0
$421$	−0.291288	+	0.504525i	−0.0141965	+	0.0245891i	−0.873036	−	0.487655i	$-0.837852\pi$
$421$	−0.291288	+	0.504525i	−0.0141965	+	0.0245891i	0.858840	+	0.512244i	$0.171186\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	1.73205	−	3.00000i	0.0840168	−	0.145521i
$426$		0			0
$427$	−3.10260	−	5.37386i	−0.150145	−	0.260059i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−39.1652			−1.88652			−0.943259	−	0.332057i	$-0.892258\pi$
$431$	−39.1652			−1.88652			−0.943259	+	0.332057i	$0.892258\pi$
$432$		0			0
$433$	−23.3303			−1.12118			−0.560591	−	0.828093i	$-0.689426\pi$
$433$	−23.3303			−1.12118			−0.560591	+	0.828093i	$0.689426\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−9.66930	−	16.7477i	−0.462546	−	0.801152i
$438$		0			0
$439$	−10.5830	+	18.3303i	−0.505099	+	0.874858i	0.494883	+	0.868959i	$0.335211\pi$
$439$	−10.5830	+	18.3303i	−0.505099	+	0.874858i	−0.999983	+	0.00589819i	$0.998123\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−18.3303	+	31.7490i	−0.870899	+	1.50844i	−0.00983083	+	0.999952i	$0.503129\pi$
$443$	−18.3303	+	31.7490i	−0.870899	+	1.50844i	−0.861068	+	0.508490i	$0.830204\pi$
$444$		0			0
$445$	16.2913	+	28.2173i	0.772281	+	1.33763i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−29.5402			−1.39409			−0.697044	−	0.717028i	$-0.745502\pi$
$449$	−29.5402			−1.39409			−0.697044	+	0.717028i	$0.745502\pi$
$450$		0			0
$451$	−6.54680			−0.308277
$452$		0			0
$453$		0			0
$454$		0			0
$455$	38.1216	+	66.0285i	1.78717	+	3.09547i
$456$		0			0
$457$	−4.08258	+	7.07123i	−0.190975	+	0.330778i	−0.945574	−	0.325408i	$-0.894498\pi$
$457$	−4.08258	+	7.07123i	−0.190975	+	0.330778i	0.754599	+	0.656187i	$0.227832\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	2.74110	−	4.74773i	0.127666	−	0.221124i	−0.795106	−	0.606470i	$-0.792585\pi$
$461$	2.74110	−	4.74773i	0.127666	−	0.221124i	0.922772	+	0.385347i	$0.125918\pi$
$462$		0			0
$463$	−1.82740	−	3.16515i	−0.0849265	−	0.147097i	0.820433	−	0.571742i	$-0.193732\pi$
$463$	−1.82740	−	3.16515i	−0.0849265	−	0.147097i	−0.905360	+	0.424645i	$0.860399\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	0.834849			0.0386322			0.0193161	−	0.999813i	$-0.493851\pi$
$467$	0.834849			0.0386322			0.0193161	+	0.999813i	$0.493851\pi$
$468$		0			0
$469$	11.1652			0.515559
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−4.56850	−	7.91288i	−0.210060	−	0.363835i
$474$		0			0
$475$	−2.55040	+	4.41742i	−0.117020	+	0.202685i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−11.3739	+	19.7001i	−0.519685	+	0.900121i	0.480053	+	0.877239i	$0.340617\pi$
$479$	−11.3739	+	19.7001i	−0.519685	+	0.900121i	−0.999738	+	0.0228815i	$0.992716\pi$
$480$		0			0
$481$	8.50000	+	14.7224i	0.387567	+	0.671285i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	13.6657			0.620528
$486$		0			0
$487$	−18.9572			−0.859033			−0.429517	−	0.903059i	$-0.641316\pi$
$487$	−18.9572			−0.859033			−0.429517	+	0.903059i	$0.641316\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	1.79129	+	3.10260i	0.0808397	+	0.140018i	0.903611	−	0.428354i	$-0.140906\pi$
$491$	1.79129	+	3.10260i	0.0808397	+	0.140018i	−0.822771	+	0.568373i	$0.807573\pi$
$492$		0			0
$493$	−5.29129	+	9.16478i	−0.238308	+	0.412761i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	0.913701	−	1.58258i	0.0409851	−	0.0709882i
$498$		0			0
$499$	−16.9590	−	29.3739i	−0.759189	−	1.31495i	−0.943264	−	0.332043i	$-0.892262\pi$
$499$	−16.9590	−	29.3739i	−0.759189	−	1.31495i	0.184075	−	0.982912i	$-0.441071\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−15.1652			−0.676181			−0.338090	−	0.941114i	$-0.609781\pi$
$503$	−15.1652			−0.676181			−0.338090	+	0.941114i	$0.609781\pi$
$504$		0			0
$505$	36.6606			1.63138
$506$		0			0
$507$		0			0
$508$		0			0
$509$	16.5975	+	28.7477i	0.735672	+	1.27422i	0.954428	+	0.298441i	$0.0964666\pi$
$509$	16.5975	+	28.7477i	0.735672	+	1.27422i	−0.218756	+	0.975779i	$0.570200\pi$
$510$		0			0
$511$	−13.4949	+	23.3739i	−0.596980	+	1.03400i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	16.4174	−	28.4358i	0.723438	−	1.25303i
$516$		0			0
$517$	14.3303	+	24.8208i	0.630246	+	1.09162i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−5.48220			−0.240180			−0.120090	−	0.992763i	$-0.538318\pi$
$521$	−5.48220			−0.240180			−0.120090	+	0.992763i	$0.538318\pi$
$522$		0			0
$523$	−21.5076			−0.940462			−0.470231	−	0.882543i	$-0.655829\pi$
$523$	−21.5076			−0.940462			−0.470231	+	0.882543i	$0.655829\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−7.58258	−	13.1334i	−0.330302	−	0.572100i
$528$		0			0
$529$	−17.2477	+	29.8739i	−0.749901	+	1.29887i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	6.01450	−	10.4174i	0.260517	−	0.451229i
$534$		0			0
$535$	−9.47860	−	16.4174i	−0.409796	−	0.709787i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−43.5826			−1.87723
$540$		0			0
$541$	2.91288			0.125234			0.0626172	−	0.998038i	$-0.480055\pi$
$541$	2.91288			0.125234			0.0626172	+	0.998038i	$0.480055\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−19.2909	−	33.4129i	−0.826333	−	1.43125i
$546$		0			0
$547$	−6.92820	+	12.0000i	−0.296229	+	0.513083i	−0.975270	−	0.221017i	$-0.929062\pi$
$547$	−6.92820	+	12.0000i	−0.296229	+	0.513083i	0.679041	+	0.734100i	$0.262396\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	7.79129	−	13.4949i	0.331920	−	0.574902i
$552$		0			0
$553$	−20.7477	−	35.9361i	−0.882283	−	1.52816i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−33.8227			−1.43311			−0.716556	−	0.697529i	$-0.754283\pi$
$557$	−33.8227			−1.43311			−0.716556	+	0.697529i	$0.754283\pi$
$558$		0			0
$559$	16.7882			0.710066
$560$		0			0
$561$		0			0
$562$		0			0
$563$	12.0000	+	20.7846i	0.505740	+	0.875967i	0.999978	+	0.00664037i	$0.00211371\pi$
$563$	12.0000	+	20.7846i	0.505740	+	0.875967i	−0.494238	+	0.869326i	$0.664553\pi$
$564$		0			0
$565$	−25.4564	+	44.0918i	−1.07096	+	1.85496i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	10.3446	−	17.9174i	0.433669	−	0.751138i	−0.563517	−	0.826105i	$-0.690552\pi$
$569$	10.3446	−	17.9174i	0.433669	−	0.751138i	0.997186	+	0.0749673i	$0.0238852\pi$
$570$		0			0
$571$	−10.5830	−	18.3303i	−0.442885	−	0.767099i	0.555017	−	0.831839i	$-0.312712\pi$
$571$	−10.5830	−	18.3303i	−0.442885	−	0.767099i	−0.997902	+	0.0647396i	$0.979378\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	15.1652			0.632431
$576$		0			0
$577$	18.1652			0.756225			0.378113	−	0.925760i	$-0.376573\pi$
$577$	18.1652			0.756225			0.378113	+	0.925760i	$0.376573\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	33.1950	+	57.4955i	1.37716	+	2.38531i
$582$		0			0
$583$	−3.27340	+	5.66970i	−0.135570	+	0.234815i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−2.20871	+	3.82560i	−0.0911633	+	0.157899i	−0.908001	−	0.418968i	$-0.862392\pi$
$587$	−2.20871	+	3.82560i	−0.0911633	+	0.157899i	0.816838	+	0.576868i	$0.195725\pi$
$588$		0			0
$589$	11.1652	+	19.3386i	0.460052	+	0.796834i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−12.1244			−0.497888			−0.248944	−	0.968518i	$-0.580083\pi$
$593$	−12.1244			−0.497888			−0.248944	+	0.968518i	$0.580083\pi$
$594$		0			0
$595$	−20.0616			−0.822446
$596$		0			0
$597$		0			0
$598$		0			0
$599$	15.5826	+	26.9898i	0.636687	+	1.10277i	0.986155	+	0.165826i	$0.0530289\pi$
$599$	15.5826	+	26.9898i	0.636687	+	1.10277i	−0.349468	+	0.936948i	$0.613638\pi$
$600$		0			0
$601$	7.24773	−	12.5534i	0.295641	−	0.512065i	−0.679493	−	0.733682i	$-0.737800\pi$
$601$	7.24773	−	12.5534i	0.295641	−	0.512065i	0.975134	+	0.221617i	$0.0711335\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	2.42728	−	4.20417i	0.0986828	−	0.170924i
$606$		0			0
$607$	−15.3223	−	26.5390i	−0.621913	−	1.07719i	−0.989129	−	0.147050i	$-0.953022\pi$
$607$	−15.3223	−	26.5390i	−0.621913	−	1.07719i	0.367216	−	0.930136i	$-0.380311\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−52.6606			−2.13042
$612$		0			0
$613$	30.0000			1.21169			0.605844	−	0.795583i	$-0.292835\pi$
$613$	30.0000			1.21169			0.605844	+	0.795583i	$0.292835\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	5.33918	+	9.24773i	0.214947	+	0.372299i	0.953256	−	0.302163i	$-0.0977087\pi$
$617$	5.33918	+	9.24773i	0.214947	+	0.372299i	−0.738309	+	0.674463i	$0.764375\pi$
$618$		0			0
$619$	20.0616	−	34.7477i	0.806344	−	1.39663i	−0.109036	−	0.994038i	$-0.534776\pi$
$619$	20.0616	−	34.7477i	0.806344	−	1.39663i	0.915380	−	0.402591i	$-0.131890\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	26.9564	−	46.6899i	1.07999	−	1.87059i
$624$		0			0
$625$	15.5000	+	26.8468i	0.620000	+	1.07387i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−4.47315			−0.178356
$630$		0			0
$631$	−22.6120			−0.900170			−0.450085	−	0.892986i	$-0.648606\pi$
$631$	−22.6120			−0.900170			−0.450085	+	0.892986i	$0.648606\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	24.1216	+	41.7798i	0.957236	+	1.65798i
$636$		0			0
$637$	40.0390	−	69.3496i	1.58640	−	2.74773i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−9.62163	+	16.6652i	−0.380032	+	0.658234i	−0.991066	−	0.133370i	$-0.957420\pi$
$641$	−9.62163	+	16.6652i	−0.380032	+	0.658234i	0.611035	+	0.791604i	$0.290754\pi$
$642$		0			0
$643$	13.1334	+	22.7477i	0.517931	+	0.897083i	0.999783	+	0.0208302i	$0.00663093\pi$
$643$	13.1334	+	22.7477i	0.517931	+	0.897083i	−0.481852	+	0.876253i	$0.660036\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	13.9129			0.546972			0.273486	−	0.961876i	$-0.411823\pi$
$647$	13.9129			0.546972			0.273486	+	0.961876i	$0.411823\pi$
$648$		0			0
$649$	−28.6606			−1.12503
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−12.9427	−	22.4174i	−0.506487	−	0.877262i	−0.999972	−	0.00750707i	$-0.997610\pi$
$653$	−12.9427	−	22.4174i	−0.506487	−	0.877262i	0.493485	−	0.869755i	$-0.335723\pi$
$654$		0			0
$655$	−4.73930	+	8.20871i	−0.185180	+	0.320741i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	17.3739	−	30.0924i	0.676790	−	1.17223i	−0.299153	−	0.954205i	$-0.596704\pi$
$659$	17.3739	−	30.0924i	0.676790	−	1.17223i	0.975942	−	0.218029i	$-0.0699627\pi$
$660$		0			0
$661$	5.87386	+	10.1738i	0.228467	+	0.395716i	0.957354	−	0.288918i	$-0.0932954\pi$
$661$	5.87386	+	10.1738i	0.228467	+	0.395716i	−0.728887	+	0.684634i	$0.759962\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	29.5402			1.14552
$666$		0			0
$667$	−46.3284			−1.79384
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−2.53901	−	4.39770i	−0.0980176	−	0.169771i
$672$		0			0
$673$	10.9174	−	18.9095i	0.420836	−	0.728909i	−0.575186	−	0.818023i	$-0.695070\pi$
$673$	10.9174	−	18.9095i	0.420836	−	0.728909i	0.996021	+	0.0891139i	$0.0284035\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−18.4249	+	31.9129i	−0.708127	+	1.22651i	0.257424	+	0.966298i	$0.417126\pi$
$677$	−18.4249	+	31.9129i	−0.708127	+	1.22651i	−0.965551	+	0.260213i	$0.916207\pi$
$678$		0			0
$679$	−11.3060	−	19.5826i	−0.433885	−	0.751510i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	24.0000			0.918334			0.459167	−	0.888350i	$-0.348148\pi$
$683$	24.0000			0.918334			0.459167	+	0.888350i	$0.348148\pi$
$684$		0			0
$685$	−22.9129			−0.875456
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−6.01450	−	10.4174i	−0.229134	−	0.396872i
$690$		0			0
$691$	−8.20340	+	14.2087i	−0.312072	+	0.540525i	−0.978811	−	0.204767i	$-0.934356\pi$
$691$	−8.20340	+	14.2087i	−0.312072	+	0.540525i	0.666739	+	0.745292i	$0.267690\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−11.5826	+	20.0616i	−0.439352	+	0.760980i
$696$		0			0
$697$	1.58258	+	2.74110i	0.0599443	+	0.103827i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	27.2759			1.03020			0.515098	−	0.857132i	$-0.327756\pi$
$701$	27.2759			1.03020			0.515098	+	0.857132i	$0.327756\pi$
$702$		0			0
$703$	6.58660			0.248418
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−30.3303	−	52.5336i	−1.14069	−	1.97573i
$708$		0			0
$709$	−17.8739	+	30.9584i	−0.671267	+	1.16267i	0.306278	+	0.951942i	$0.400916\pi$
$709$	−17.8739	+	30.9584i	−0.671267	+	1.16267i	−0.977545	+	0.210727i	$0.932417\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	33.1950	−	57.4955i	1.24316	−	2.15322i
$714$		0			0
$715$	31.1968	+	54.0345i	1.16669	+	2.02077i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−2.08712			−0.0778365			−0.0389182	−	0.999242i	$-0.512391\pi$
$719$	−2.08712			−0.0778365			−0.0389182	+	0.999242i	$0.512391\pi$
$720$		0			0
$721$	−54.3303			−2.02337
$722$		0			0
$723$		0			0
$724$		0			0
$725$	6.10985	+	10.5826i	0.226914	+	0.393027i
$726$		0			0
$727$	−0.361500	+	0.626136i	−0.0134073	+	0.0232221i	−0.872651	−	0.488344i	$-0.837601\pi$
$727$	−0.361500	+	0.626136i	−0.0134073	+	0.0232221i	0.859244	+	0.511566i	$0.170934\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−2.20871	+	3.82560i	−0.0816922	+	0.141495i
$732$		0			0
$733$	−16.1652	−	27.9989i	−0.597073	−	1.03416i	−0.993251	−	0.115988i	$-0.962997\pi$
$733$	−16.1652	−	27.9989i	−0.597073	−	1.03416i	0.396177	−	0.918174i	$-0.370337\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	9.13701			0.336566
$738$		0			0
$739$	−42.3320			−1.55721			−0.778604	−	0.627515i	$-0.784072\pi$
$739$	−42.3320			−1.55721			−0.778604	+	0.627515i	$0.784072\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−15.1652	−	26.2668i	−0.556355	−	0.963636i	−0.997797	−	0.0663457i	$-0.978866\pi$
$743$	−15.1652	−	26.2668i	−0.556355	−	0.963636i	0.441441	−	0.897290i	$-0.354467\pi$
$744$		0			0
$745$	−1.33485	+	2.31203i	−0.0489051	+	0.0847061i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−15.6838	+	27.1652i	−0.573074	+	0.992593i
$750$		0			0
$751$	10.2215	+	17.7042i	0.372988	+	0.646034i	0.990024	−	0.140901i	$-0.0449999\pi$
$751$	10.2215	+	17.7042i	0.372988	+	0.646034i	−0.617036	+	0.786935i	$0.711667\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−32.8348			−1.19498
$756$		0			0
$757$	18.0000			0.654221			0.327111	−	0.944986i	$-0.393925\pi$
$757$	18.0000			0.654221			0.327111	+	0.944986i	$0.393925\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−11.2583	−	19.5000i	−0.408114	−	0.706874i	0.586564	−	0.809903i	$-0.300480\pi$
$761$	−11.2583	−	19.5000i	−0.408114	−	0.706874i	−0.994678	+	0.103028i	$0.967147\pi$
$762$		0			0
$763$	−31.9198	+	55.2867i	−1.15557	+	2.00151i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	26.3303	−	45.6054i	0.950732	−	1.64672i
$768$		0			0
$769$	−1.91742	−	3.32108i	−0.0691441	−	0.119761i	0.829381	−	0.558684i	$-0.188693\pi$
$769$	−1.91742	−	3.32108i	−0.0691441	−	0.119761i	−0.898525	+	0.438923i	$0.855360\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	37.6682			1.35483			0.677415	−	0.735601i	$-0.263100\pi$
$773$	37.6682			1.35483			0.677415	+	0.735601i	$0.263100\pi$
$774$		0			0
$775$	−17.5112			−0.629021
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−2.33030	−	4.03620i	−0.0834918	−	0.144612i
$780$		0			0
$781$	0.747727	−	1.29510i	0.0267558	−	0.0463424i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−27.2282	+	47.1606i	−0.971816	+	1.68323i
$786$		0			0
$787$	−16.2360	−	28.1216i	−0.578751	−	1.00243i	−0.995623	−	0.0934611i	$-0.970207\pi$
$787$	−16.2360	−	28.1216i	−0.578751	−	1.00243i	0.416872	−	0.908965i	$-0.363126\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	84.2432			2.99534
$792$		0			0
$793$	9.33030			0.331329
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−1.51358	−	2.62159i	−0.0536136	−	0.0928615i	0.837973	−	0.545712i	$-0.183741\pi$
$797$	−1.51358	−	2.62159i	−0.0536136	−	0.0928615i	−0.891587	+	0.452850i	$0.850407\pi$
$798$		0			0
$799$	6.92820	−	12.0000i	0.245102	−	0.424529i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−11.0436	+	19.1280i	−0.389719	+	0.675013i
$804$		0			0
$805$	−43.9129	−	76.0593i	−1.54773	−	2.68074i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	34.5457			1.21456			0.607280	−	0.794488i	$-0.292260\pi$
$809$	34.5457			1.21456			0.607280	+	0.794488i	$0.292260\pi$
$810$		0			0
$811$	−17.5112			−0.614902			−0.307451	−	0.951564i	$-0.599476\pi$
$811$	−17.5112			−0.614902			−0.307451	+	0.951564i	$0.599476\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−4.83485	−	8.37420i	−0.169357	−	0.293336i
$816$		0			0
$817$	3.25227	−	5.63310i	0.113783	−	0.197077i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−8.34643	+	14.4564i	−0.291292	+	0.504533i	−0.974116	−	0.226050i	$-0.927419\pi$
$821$	−8.34643	+	14.4564i	−0.291292	+	0.504533i	0.682823	+	0.730584i	$0.260752\pi$
$822$		0			0
$823$	22.6120	+	39.1652i	0.788205	+	1.36521i	0.927066	+	0.374899i	$0.122323\pi$
$823$	22.6120	+	39.1652i	0.788205	+	1.36521i	−0.138860	+	0.990312i	$0.544344\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−18.7477			−0.651922			−0.325961	−	0.945383i	$-0.605688\pi$
$827$	−18.7477			−0.651922			−0.325961	+	0.945383i	$0.605688\pi$
$828$		0			0
$829$	−28.3303			−0.983952			−0.491976	−	0.870609i	$-0.663725\pi$
$829$	−28.3303			−0.983952			−0.491976	+	0.870609i	$0.663725\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	10.5353	+	18.2477i	0.365028	+	0.632246i
$834$		0			0
$835$	20.6138	−	35.7042i	0.713370	−	1.23559i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	8.62614	−	14.9409i	0.297807	−	0.515817i	−0.677827	−	0.735222i	$-0.737078\pi$
$839$	8.62614	−	14.9409i	0.297807	−	0.515817i	0.975634	+	0.219404i	$0.0704114\pi$
$840$		0			0
$841$	−4.16515	−	7.21425i	−0.143626	−	0.248767i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−80.2464			−2.76056
$846$		0			0
$847$	−8.03260			−0.276004
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−9.79129	−	16.9590i	−0.335641	−	0.581347i
$852$		0			0
$853$	−18.1652	+	31.4630i	−0.621963	+	1.07727i	0.367157	+	0.930159i	$0.380331\pi$
$853$	−18.1652	+	31.4630i	−0.621963	+	1.07727i	−0.989120	+	0.147112i	$0.953002\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	17.3682	−	30.0826i	0.593286	−	1.02760i	−0.400500	−	0.916297i	$-0.631164\pi$
$857$	17.3682	−	30.0826i	0.593286	−	1.02760i	0.993786	−	0.111305i	$-0.0355029\pi$
$858$		0			0
$859$	−3.27340	−	5.66970i	−0.111687	−	0.193448i	0.804764	−	0.593596i	$-0.202292\pi$
$859$	−3.27340	−	5.66970i	−0.111687	−	0.193448i	−0.916451	+	0.400148i	$0.868959\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−51.4083			−1.74996			−0.874980	−	0.484159i	$-0.839126\pi$
$863$	−51.4083			−1.74996			−0.874980	+	0.484159i	$0.839126\pi$
$864$		0			0
$865$	6.49545			0.220852
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−16.9789	−	29.4083i	−0.575970	−	0.997609i
$870$		0			0
$871$	−8.39410	+	14.5390i	−0.284423	+	0.492636i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	17.3739	−	30.0924i	0.587344	−	1.01731i
$876$		0			0
$877$	−12.0390	−	20.8522i	−0.406529	−	0.704128i	0.587969	−	0.808883i	$-0.299928\pi$
$877$	−12.0390	−	20.8522i	−0.406529	−	0.704128i	−0.994498	+	0.104755i	$0.966594\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−33.1950			−1.11837			−0.559184	−	0.829043i	$-0.688886\pi$
$881$	−33.1950			−1.11837			−0.559184	+	0.829043i	$0.688886\pi$
$882$		0			0
$883$	43.7780			1.47325			0.736624	−	0.676303i	$-0.236419\pi$
$883$	43.7780			1.47325			0.736624	+	0.676303i	$0.236419\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	8.20871	+	14.2179i	0.275622	+	0.477391i	0.970292	−	0.241938i	$-0.0777830\pi$
$887$	8.20871	+	14.2179i	0.275622	+	0.477391i	−0.694670	+	0.719328i	$0.744450\pi$
$888$		0			0
$889$	39.9129	−	69.1311i	1.33863	−	2.31858i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−10.2016	+	17.6697i	−0.341384	+	0.591294i
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$	53.4955			1.78417
$900$		0			0
$901$	3.16515			0.105446
$902$		0			0
$903$		0			0
$904$		0			0
$905$	2.64575	+	4.58258i	0.0879477	+	0.152330i
$906$		0			0
$907$	14.9608	−	25.9129i	0.496765	−	0.860423i	−0.503228	−	0.864154i	$-0.667854\pi$
$907$	14.9608	−	25.9129i	0.496765	−	0.860423i	0.999993	+	0.00373091i	$0.00118759\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−15.7913	+	27.3513i	−0.523189	+	0.906189i	0.476447	+	0.879203i	$0.341924\pi$
$911$	−15.7913	+	27.3513i	−0.523189	+	0.906189i	−0.999636	+	0.0269863i	$0.991409\pi$
$912$		0			0
$913$	27.1652	+	47.0514i	0.899035	+	1.55717i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	15.6838			0.517925
$918$		0			0
$919$	14.5794			0.480930			0.240465	−	0.970658i	$-0.422700\pi$
$919$	14.5794			0.480930			0.240465	+	0.970658i	$0.422700\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	1.37386	+	2.37960i	0.0452213	+	0.0783255i
$924$		0			0
$925$	−2.58258	+	4.47315i	−0.0849146	+	0.147076i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	7.79423	−	13.5000i	0.255720	−	0.442921i	−0.709371	−	0.704836i	$-0.751021\pi$
$929$	7.79423	−	13.5000i	0.255720	−	0.442921i	0.965091	+	0.261915i	$0.0843540\pi$
$930$		0			0
$931$	−15.5130	−	26.8693i	−0.508418	−	0.880606i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−16.4174			−0.536907
$936$		0			0
$937$	−16.1652			−0.528092			−0.264046	−	0.964510i	$-0.585057\pi$
$937$	−16.1652			−0.528092			−0.264046	+	0.964510i	$0.585057\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−14.4563	−	25.0390i	−0.471261	−	0.816249i	0.528198	−	0.849121i	$-0.322868\pi$
$941$	−14.4563	−	25.0390i	−0.471261	−	0.816249i	−0.999460	+	0.0328725i	$0.989534\pi$
$942$		0			0
$943$	−6.92820	+	12.0000i	−0.225613	+	0.390774i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−8.53901	+	14.7900i	−0.277481	+	0.480611i	−0.970758	−	0.240060i	$-0.922833\pi$
$947$	−8.53901	+	14.7900i	−0.277481	+	0.480611i	0.693277	+	0.720671i	$0.256166\pi$
$948$		0			0
$949$	−20.2913	−	35.1455i	−0.658683	−	1.14087i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	17.0345			0.551800			0.275900	−	0.961186i	$-0.411024\pi$
$953$	17.0345			0.551800			0.275900	+	0.961186i	$0.411024\pi$
$954$		0			0
$955$	20.0616			0.649178
$956$		0			0
$957$		0			0
$958$		0			0
$959$	18.9564	+	32.8335i	0.612135	+	1.06025i
$960$		0			0
$961$	−22.8303	+	39.5432i	−0.736461	+	1.27559i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−16.0930	+	27.8739i	−0.518051	+	0.897291i
$966$		0			0
$967$	7.28970	+	12.6261i	0.234421	+	0.406029i	0.959104	−	0.283053i	$-0.0913472\pi$
$967$	7.28970	+	12.6261i	0.234421	+	0.406029i	−0.724683	+	0.689082i	$0.758014\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	26.7477			0.858375			0.429188	−	0.903215i	$-0.358800\pi$
$971$	26.7477			0.858375			0.429188	+	0.903215i	$0.358800\pi$
$972$		0			0
$973$	38.3303			1.22881
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−14.7701	−	25.5826i	−0.472538	−	0.818459i	0.526969	−	0.849885i	$-0.323329\pi$
$977$	−14.7701	−	25.5826i	−0.472538	−	0.818459i	−0.999506	+	0.0314257i	$0.989995\pi$
$978$		0			0
$979$	22.0598	−	38.2087i	0.705035	−	1.22116i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−1.25227	+	2.16900i	−0.0399413	+	0.0691804i	−0.885305	−	0.465011i	$-0.846050\pi$
$983$	−1.25227	+	2.16900i	−0.0399413	+	0.0691804i	0.845364	+	0.534191i	$0.179384\pi$
$984$		0			0
$985$	19.4129	+	33.6241i	0.618546	+	1.07135i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−19.3386			−0.614932
$990$		0			0
$991$	18.2342			0.579229			0.289614	−	0.957143i	$-0.406473\pi$
$991$	18.2342			0.579229			0.289614	+	0.957143i	$0.406473\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	23.1652	+	40.1232i	0.734385	+	1.27199i
$996$		0			0
$997$	−18.8739	+	32.6905i	−0.597741	+	1.03532i	0.395412	+	0.918504i	$0.370602\pi$
$997$	−18.8739	+	32.6905i	−0.597741	+	1.03532i	−0.993154	+	0.116815i	$0.962732\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2592.2.i.bg.865.4		8
3.2	odd	2		2592.2.i.bh.865.2		8
4.3	odd	2		2592.2.i.bh.865.3		8
9.2	odd	6		2592.2.a.v.1.3	yes	4
9.4	even	3	inner	2592.2.i.bg.1729.4		8
9.5	odd	6		2592.2.i.bh.1729.2		8
9.7	even	3		2592.2.a.w.1.1	yes	4
12.11	even	2	inner	2592.2.i.bg.865.1		8
36.7	odd	6		2592.2.a.v.1.2	✓	4
36.11	even	6		2592.2.a.w.1.4	yes	4
36.23	even	6	inner	2592.2.i.bg.1729.1		8
36.31	odd	6		2592.2.i.bh.1729.3		8
72.11	even	6		5184.2.a.cd.1.2		4
72.29	odd	6		5184.2.a.ce.1.1		4
72.43	odd	6		5184.2.a.ce.1.4		4
72.61	even	6		5184.2.a.cd.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2592.2.a.v.1.2	✓	4	36.7	odd	6
2592.2.a.v.1.3	yes	4	9.2	odd	6
2592.2.a.w.1.1	yes	4	9.7	even	3
2592.2.a.w.1.4	yes	4	36.11	even	6
2592.2.i.bg.865.1		8	12.11	even	2	inner
2592.2.i.bg.865.4		8	1.1	even	1	trivial
2592.2.i.bg.1729.1		8	36.23	even	6	inner
2592.2.i.bg.1729.4		8	9.4	even	3	inner
2592.2.i.bh.865.2		8	3.2	odd	2
2592.2.i.bh.865.3		8	4.3	odd	2
2592.2.i.bh.1729.2		8	9.5	odd	6
2592.2.i.bh.1729.3		8	36.31	odd	6
5184.2.a.cd.1.2		4	72.11	even	6
5184.2.a.cd.1.3		4	72.61	even	6
5184.2.a.ce.1.1		4	72.29	odd	6
5184.2.a.ce.1.4		4	72.43	odd	6

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:	\( N \)	\(=\)	\( 2592 = 2^{5} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2592.i (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(1.32288 + 2.29129i) q^{5} +(2.18890 - 3.79129i) q^{7} +(1.79129 - 3.10260i) q^{11} +(3.29129 + 5.70068i) q^{13} -1.73205 q^{17} +2.55040 q^{19} +(-3.79129 - 6.56670i) q^{23} +(-1.00000 + 1.73205i) q^{25} +(3.05493 - 5.29129i) q^{29} +(4.37780 + 7.58258i) q^{31} +11.5826 q^{35} +2.58258 q^{37} +(-0.913701 - 1.58258i) q^{41} +(1.27520 - 2.20871i) q^{43} +(-4.00000 + 6.92820i) q^{47} +(-6.08258 - 10.5353i) q^{49} -1.82740 q^{53} +9.47860 q^{55} +(-4.00000 - 6.92820i) q^{59} +(0.708712 - 1.22753i) q^{61} +(-8.70793 + 15.0826i) q^{65} +(1.27520 + 2.20871i) q^{67} +0.417424 q^{71} -6.16515 q^{73} +(-7.84190 - 13.5826i) q^{77} +(4.73930 - 8.20871i) q^{79} +(-7.58258 + 13.1334i) q^{83} +(-2.29129 - 3.96863i) q^{85} +12.3151 q^{89} +28.8172 q^{91} +(3.37386 + 5.84370i) q^{95} +(2.58258 - 4.47315i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 4 q^{11} + 8 q^{13} - 12 q^{23} - 8 q^{25} + 56 q^{35} - 16 q^{37} - 32 q^{47} - 12 q^{49} - 32 q^{59} + 24 q^{61} + 40 q^{71} + 24 q^{73} - 24 q^{83} - 28 q^{95} - 16 q^{97}+O(q^{100}) \) 8 * q - 4 * q^11 + 8 * q^13 - 12 * q^23 - 8 * q^25 + 56 * q^35 - 16 * q^37 - 32 * q^47 - 12 * q^49 - 32 * q^59 + 24 * q^61 + 40 * q^71 + 24 * q^73 - 24 * q^83 - 28 * q^95 - 16 * q^97

Introduction

L-functions

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