LMFDB - Embedded newform 5292.2.i.h.1549.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5292,2,Mod(1549,5292)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5292, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 2, 2]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("5292.1549");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5292 = 2^{2} \cdot 3^{3} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5292.i (of order $3$, degree $2$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$42.2568327497$
Analytic rank:	$0$
Dimension:	$12$
Relative dimension:	$6$ over $\Q(\zeta_{3})$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 2x^{10} + 16x^{8} - 39x^{6} + 144x^{4} - 162x^{2} + 729 $ x^12 - 2x^10 + 16x^8 - 39x^6 + 144x^4 - 162*x^2 + 729
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 3^{2} $
Twist minimal:	no (minimal twist has level 1764)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			1549.1
Root			$0.965975 - 1.43767i$ of defining polynomial
Character	$\chi$	$=$	5292.1549
Dual form			5292.2.i.h.2125.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.95014 - 3.37774i) q^{5} +O(q^{10})$	$q+(-1.95014 - 3.37774i) q^{5} +(-2.13378 + 3.69582i) q^{11} +(-2.71221 + 4.69768i) q^{13} +(-2.49012 - 4.31301i) q^{17} +(-0.222091 + 0.384673i) q^{19} +(4.10607 + 7.11191i) q^{23} +(-5.10607 + 8.84396i) q^{25} +(-1.33850 - 2.31835i) q^{29} -0.851993 q^{31} +(4.90135 - 8.48939i) q^{37} +(2.69402 - 4.66618i) q^{41} +(4.31078 + 7.46649i) q^{43} +3.49246 q^{47} +(-2.83850 - 4.91642i) q^{53} +16.6447 q^{55} -1.89558 q^{59} +11.2566 q^{61} +21.1567 q^{65} +2.59057 q^{67} -0.141862 q^{71} +(-5.16595 - 8.94769i) q^{73} -0.409429 q^{79} +(2.08231 + 3.60666i) q^{83} +(-9.71213 + 16.8219i) q^{85} +(-1.20625 + 2.08929i) q^{89} +1.73243 q^{95} +(-3.80448 - 6.58955i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q+O(q^{10}) $ 12 * q	$ 12 q - 8 q^{11} - 12 q^{25} - 2 q^{29} + 6 q^{37} + 6 q^{43} - 20 q^{53} + 92 q^{65} + 24 q^{67} - 44 q^{71} - 12 q^{79} - 18 q^{85} + 56 q^{95}+O(q^{100}) $ 12 * q - 8 * q^11 - 12 * q^25 - 2 * q^29 + 6 * q^37 + 6 * q^43 - 20 * q^53 + 92 * q^65 + 24 * q^67 - 44 * q^71 - 12 * q^79 - 18 * q^85 + 56 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/5292\mathbb{Z}\right)^\times$.

$n$	$785$	$1081$	$2647$
$\chi(n)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{1}{3}\right)$	$1$

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
$5$	−1.95014	−	3.37774i	−0.872128	−	1.51057i	−0.859791	−	0.510645i	$-0.829407\pi$
$5$	−1.95014	−	3.37774i	−0.872128	−	1.51057i	−0.0123362	−	0.999924i	$-0.503927\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−2.13378	+	3.69582i	−0.643360	+	1.11433i	0.341318	+	0.939948i	$0.389127\pi$
$11$	−2.13378	+	3.69582i	−0.643360	+	1.11433i	−0.984678	+	0.174384i	$0.944207\pi$
$12$		0			0
$13$	−2.71221	+	4.69768i	−0.752231	+	1.30290i	0.194508	+	0.980901i	$0.437689\pi$
$13$	−2.71221	+	4.69768i	−0.752231	+	1.30290i	−0.946739	+	0.322001i	$0.895645\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−2.49012	−	4.31301i	−0.603942	−	1.04606i	−0.992218	−	0.124515i	$-0.960263\pi$
$17$	−2.49012	−	4.31301i	−0.603942	−	1.04606i	0.388276	−	0.921543i	$-0.373071\pi$
$18$		0			0
$19$	−0.222091	+	0.384673i	−0.0509512	+	0.0882501i	−0.890376	−	0.455226i	$-0.849559\pi$
$19$	−0.222091	+	0.384673i	−0.0509512	+	0.0882501i	0.839425	+	0.543476i	$0.182892\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	4.10607	+	7.11191i	0.856174	+	1.48294i	0.875552	+	0.483124i	$0.160498\pi$
$23$	4.10607	+	7.11191i	0.856174	+	1.48294i	−0.0193779	+	0.999812i	$0.506169\pi$
$24$		0			0
$25$	−5.10607	+	8.84396i	−1.02121	+	1.76879i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−1.33850	−	2.31835i	−0.248553	−	0.430506i	0.714572	−	0.699562i	$-0.246622\pi$
$29$	−1.33850	−	2.31835i	−0.248553	−	0.430506i	−0.963125	+	0.269056i	$0.913288\pi$
$30$		0			0
$31$	−0.851993			−0.153022			−0.0765112	−	0.997069i	$-0.524378\pi$
$31$	−0.851993			−0.153022			−0.0765112	+	0.997069i	$0.524378\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	4.90135	−	8.48939i	0.805777	−	1.39565i	−0.109988	−	0.993933i	$-0.535081\pi$
$37$	4.90135	−	8.48939i	0.805777	−	1.39565i	0.915765	−	0.401714i	$-0.131585\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	2.69402	−	4.66618i	0.420735	−	0.728735i	−0.575276	−	0.817959i	$-0.695105\pi$
$41$	2.69402	−	4.66618i	0.420735	−	0.728735i	0.996012	+	0.0892242i	$0.0284388\pi$
$42$		0			0
$43$	4.31078	+	7.46649i	0.657388	+	1.13863i	0.981289	+	0.192538i	$0.0616720\pi$
$43$	4.31078	+	7.46649i	0.657388	+	1.13863i	−0.323902	+	0.946091i	$0.604995\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	3.49246			0.509428			0.254714	−	0.967016i	$-0.418019\pi$
$47$	3.49246			0.509428			0.254714	+	0.967016i	$0.418019\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−2.83850	−	4.91642i	−0.389898	−	0.675323i	0.602538	−	0.798090i	$-0.294156\pi$
$53$	−2.83850	−	4.91642i	−0.389898	−	0.675323i	−0.992435	+	0.122768i	$0.960823\pi$
$54$		0			0
$55$	16.6447			2.24437
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−1.89558			−0.246783			−0.123392	−	0.992358i	$-0.539377\pi$
$59$	−1.89558			−0.246783			−0.123392	+	0.992358i	$0.539377\pi$
$60$		0			0
$61$	11.2566			1.44126			0.720632	−	0.693317i	$-0.243852\pi$
$61$	11.2566			1.44126			0.720632	+	0.693317i	$0.243852\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	21.1567			2.62416
$66$		0			0
$67$	2.59057			0.316489			0.158244	−	0.987400i	$-0.449417\pi$
$67$	2.59057			0.316489			0.158244	+	0.987400i	$0.449417\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−0.141862			−0.0168359			−0.00841794	−	0.999965i	$-0.502680\pi$
$71$	−0.141862			−0.0168359			−0.00841794	+	0.999965i	$0.502680\pi$
$72$		0			0
$73$	−5.16595	−	8.94769i	−0.604629	−	1.04725i	−0.992110	−	0.125370i	$-0.959988\pi$
$73$	−5.16595	−	8.94769i	−0.604629	−	1.04725i	0.387481	−	0.921878i	$-0.373345\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−0.409429			−0.0460644			−0.0230322	−	0.999735i	$-0.507332\pi$
$79$	−0.409429			−0.0460644			−0.0230322	+	0.999735i	$0.507332\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	2.08231	+	3.60666i	0.228563	+	0.395882i	0.957382	−	0.288823i	$-0.0932641\pi$
$83$	2.08231	+	3.60666i	0.228563	+	0.395882i	−0.728820	+	0.684706i	$0.759931\pi$
$84$		0			0
$85$	−9.71213	+	16.8219i	−1.05343	+	1.82459i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−1.20625	+	2.08929i	−0.127863	+	0.221464i	−0.922848	−	0.385164i	$-0.874145\pi$
$89$	−1.20625	+	2.08929i	−0.127863	+	0.221464i	0.794986	+	0.606628i	$0.207478\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$	1.73243			0.177744
$96$		0			0
$97$	−3.80448	−	6.58955i	−0.386286	−	0.669067i	0.605661	−	0.795723i	$-0.292909\pi$
$97$	−3.80448	−	6.58955i	−0.386286	−	0.669067i	−0.991947	+	0.126656i	$0.959576\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−2.56185	+	4.43726i	−0.254914	+	0.441524i	−0.964872	−	0.262720i	$-0.915380\pi$
$101$	−2.56185	+	4.43726i	−0.254914	+	0.441524i	0.709958	+	0.704244i	$0.248714\pi$
$102$		0			0
$103$	−6.48031	−	11.2242i	−0.638524	−	1.10596i	−0.985757	−	0.168177i	$-0.946212\pi$
$103$	−6.48031	−	11.2242i	−0.638524	−	1.10596i	0.347233	−	0.937779i	$-0.387121\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−5.76757	+	9.98972i	−0.557572	+	0.965743i	0.440127	+	0.897936i	$0.354934\pi$
$107$	−5.76757	+	9.98972i	−0.557572	+	0.965743i	−0.997698	+	0.0678070i	$0.978400\pi$
$108$		0			0
$109$	0.204714	+	0.354576i	0.0196081	+	0.0339622i	0.875663	−	0.482923i	$-0.160425\pi$
$109$	0.204714	+	0.354576i	0.0196081	+	0.0339622i	−0.856055	+	0.516885i	$0.827091\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	6.94456	−	12.0283i	0.653290	−	1.13153i	−0.329030	−	0.944319i	$-0.606722\pi$
$113$	6.94456	−	12.0283i	0.653290	−	1.13153i	0.982320	−	0.187211i	$-0.0599449\pi$
$114$		0			0
$115$	16.0148	−	27.7384i	1.49339	−	2.58662i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−3.60607	−	6.24589i	−0.327824	−	0.567808i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	20.3287			1.81826
$126$		0			0
$127$	11.6216			1.03125			0.515623	−	0.856815i	$-0.327560\pi$
$127$	11.6216			1.03125			0.515623	+	0.856815i	$0.327560\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−2.02775	−	3.51216i	−0.177165	−	0.306859i	0.763743	−	0.645520i	$-0.223359\pi$
$131$	−2.02775	−	3.51216i	−0.177165	−	0.306859i	−0.940908	+	0.338661i	$0.890026\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	4.57835	−	7.92993i	0.391155	−	0.677500i	−0.601448	−	0.798912i	$-0.705409\pi$
$137$	4.57835	−	7.92993i	0.391155	−	0.677500i	0.992602	+	0.121413i	$0.0387424\pi$
$138$		0			0
$139$	−5.23869	+	9.07369i	−0.444340	+	0.769620i	−0.998006	−	0.0631191i	$-0.979895\pi$
$139$	−5.23869	+	9.07369i	−0.444340	+	0.769620i	0.553666	+	0.832739i	$0.313229\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−11.5745	−	20.0477i	−0.967910	−	1.67647i
$144$		0			0
$145$	−5.22051	+	9.04219i	−0.433540	+	0.750913i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−8.13378	−	14.0881i	−0.666346	−	1.15414i	−0.978919	−	0.204251i	$-0.934524\pi$
$149$	−8.13378	−	14.0881i	−0.666346	−	1.15414i	0.312573	−	0.949894i	$-0.398809\pi$
$150$		0			0
$151$	3.60607	−	6.24589i	0.293457	−	0.508283i	−0.681167	−	0.732128i	$-0.738527\pi$
$151$	3.60607	−	6.24589i	0.293457	−	0.508283i	0.974625	+	0.223844i	$0.0718608\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	1.66150	+	2.87781i	0.133455	+	0.231151i
$156$		0			0
$157$	−1.18706			−0.0947376			−0.0473688	−	0.998877i	$-0.515084\pi$
$157$	−1.18706			−0.0947376			−0.0473688	+	0.998877i	$0.515084\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−10.5074	+	18.1994i	−0.823004	+	1.42549i	0.0804306	+	0.996760i	$0.474370\pi$
$163$	−10.5074	+	18.1994i	−0.823004	+	1.42549i	−0.903435	+	0.428725i	$0.858963\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	10.0322	−	17.3763i	0.776315	−	1.34462i	−0.157738	−	0.987481i	$-0.550420\pi$
$167$	10.0322	−	17.3763i	0.776315	−	1.34462i	0.934053	−	0.357136i	$-0.116247\pi$
$168$		0			0
$169$	−8.21213	−	14.2238i	−0.631702	−	1.09414i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	6.06019			0.460748			0.230374	−	0.973102i	$-0.426005\pi$
$173$	6.06019			0.460748			0.230374	+	0.973102i	$0.426005\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	4.41685	+	7.65020i	0.330131	+	0.571803i	0.982537	−	0.186066i	$-0.0595739\pi$
$179$	4.41685	+	7.65020i	0.330131	+	0.571803i	−0.652407	+	0.757869i	$0.726241\pi$
$180$		0			0
$181$	24.1809			1.79735			0.898676	−	0.438614i	$-0.144530\pi$
$181$	24.1809			1.79735			0.898676	+	0.438614i	$0.144530\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−38.2332			−2.81096
$186$		0			0
$187$	21.2535			1.55421
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−20.2270			−1.46357			−0.731786	−	0.681535i	$-0.761313\pi$
$191$	−20.2270			−1.46357			−0.731786	+	0.681535i	$0.761313\pi$
$192$		0			0
$193$	−12.3933			−0.892087			−0.446044	−	0.895011i	$-0.647167\pi$
$193$	−12.3933			−0.892087			−0.446044	+	0.895011i	$0.647167\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−14.3933			−1.02548			−0.512739	−	0.858544i	$-0.671369\pi$
$197$	−14.3933			−1.02548			−0.512739	+	0.858544i	$0.671369\pi$
$198$		0			0
$199$	5.25688	+	9.10518i	0.372650	+	0.645449i	0.989972	−	0.141261i	$-0.0451157\pi$
$199$	5.25688	+	9.10518i	0.372650	+	0.645449i	−0.617322	+	0.786711i	$0.711782\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$	−21.0148			−1.46774
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−0.947789	−	1.64162i	−0.0655599	−	0.113553i
$210$		0			0
$211$	11.7121	−	20.2860i	0.806296	−	1.39655i	−0.109116	−	0.994029i	$-0.534802\pi$
$211$	11.7121	−	20.2860i	0.806296	−	1.39655i	0.915412	−	0.402517i	$-0.131865\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	16.8132	−	29.1214i	1.14665	−	1.98606i
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$	27.0148			1.81721
$222$		0			0
$223$	12.6727	+	21.9497i	0.848625	+	1.46986i	0.882436	+	0.470433i	$0.155902\pi$
$223$	12.6727	+	21.9497i	0.848625	+	1.46986i	−0.0338111	+	0.999428i	$0.510764\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	9.01267	−	15.6104i	0.598192	−	1.03610i	−0.394896	−	0.918726i	$-0.629219\pi$
$227$	9.01267	−	15.6104i	0.598192	−	1.03610i	0.993088	−	0.117373i	$-0.0374474\pi$
$228$		0			0
$229$	−2.71221	−	4.69768i	−0.179228	−	0.310431i	0.762389	−	0.647120i	$-0.224027\pi$
$229$	−2.71221	−	4.69768i	−0.179228	−	0.310431i	−0.941616	+	0.336688i	$0.890693\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−1.42165	+	2.46237i	−0.0931356	+	0.161316i	−0.908829	−	0.417169i	$-0.863022\pi$
$233$	−1.42165	+	2.46237i	−0.0931356	+	0.161316i	0.815693	+	0.578484i	$0.196356\pi$
$234$		0			0
$235$	−6.81078	−	11.7966i	−0.444286	−	0.769526i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	6.09057	−	10.5492i	0.393966	−	0.682370i	−0.599002	−	0.800747i	$-0.704436\pi$
$239$	6.09057	−	10.5492i	0.393966	−	0.682370i	0.992969	+	0.118378i	$0.0377693\pi$
$240$		0			0
$241$	9.31237	−	16.1295i	0.599863	−	1.03899i	−0.392978	−	0.919548i	$-0.628555\pi$
$241$	9.31237	−	16.1295i	0.599863	−	1.03899i	0.992841	−	0.119445i	$-0.0381115\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−1.20471	−	2.08663i	−0.0766541	−	0.132769i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	7.14015			0.450682			0.225341	−	0.974280i	$-0.427650\pi$
$251$	7.14015			0.450682			0.225341	+	0.974280i	$0.427650\pi$
$252$		0			0
$253$	−35.0458			−2.20331
$254$		0			0
$255$		0			0
$256$		0			0
$257$	0.672148	+	1.16419i	0.0419274	+	0.0726204i	0.886228	−	0.463250i	$-0.153317\pi$
$257$	0.672148	+	1.16419i	0.0419274	+	0.0726204i	−0.844300	+	0.535871i	$0.819984\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−12.1493	+	21.0432i	−0.749157	+	1.29758i	0.199071	+	0.979985i	$0.436208\pi$
$263$	−12.1493	+	21.0432i	−0.749157	+	1.29758i	−0.948227	+	0.317592i	$0.897126\pi$
$264$		0			0
$265$	−11.0709	+	19.1754i	−0.680081	+	1.17794i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−2.69989	−	4.67636i	−0.164615	−	0.285122i	0.771903	−	0.635740i	$-0.219305\pi$
$269$	−2.69989	−	4.67636i	−0.164615	−	0.285122i	−0.936519	+	0.350618i	$0.885972\pi$
$270$		0			0
$271$	6.78589	−	11.7535i	0.412213	−	0.713974i	−0.582918	−	0.812531i	$-0.698089\pi$
$271$	6.78589	−	11.7535i	0.412213	−	0.713974i	0.995131	+	0.0985565i	$0.0314225\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−21.7905	−	37.7422i	−1.31402	−	2.27594i
$276$		0			0
$277$	14.5074	−	25.1276i	0.871666	−	1.50977i	0.0113940	−	0.999935i	$-0.496373\pi$
$277$	14.5074	−	25.1276i	0.871666	−	1.50977i	0.860272	−	0.509835i	$-0.170294\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−0.885857	−	1.53435i	−0.0528458	−	0.0915316i	0.838392	−	0.545067i	$-0.183496\pi$
$281$	−0.885857	−	1.53435i	−0.0528458	−	0.0915316i	−0.891238	+	0.453536i	$0.850163\pi$
$282$		0			0
$283$	−24.2536			−1.44173			−0.720864	−	0.693076i	$-0.756255\pi$
$283$	−24.2536			−1.44173			−0.720864	+	0.693076i	$0.756255\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−3.90135	+	6.75734i	−0.229491	+	0.397490i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−7.52491	+	13.0335i	−0.439610	+	0.761426i	−0.997659	−	0.0683813i	$-0.978217\pi$
$293$	−7.52491	+	13.0335i	−0.439610	+	0.761426i	0.558050	+	0.829808i	$0.311550\pi$
$294$		0			0
$295$	3.69664	+	6.40276i	0.215226	+	0.372783i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−44.5460			−2.57616
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−21.9520	−	38.0219i	−1.25697	−	2.17713i
$306$		0			0
$307$	4.01912			0.229383			0.114692	−	0.993401i	$-0.463412\pi$
$307$	4.01912			0.229383			0.114692	+	0.993401i	$0.463412\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	1.59688			0.0905510			0.0452755	−	0.998975i	$-0.485583\pi$
$311$	1.59688			0.0905510			0.0452755	+	0.998975i	$0.485583\pi$
$312$		0			0
$313$	4.87111			0.275332			0.137666	−	0.990479i	$-0.456040\pi$
$313$	4.87111			0.275332			0.137666	+	0.990479i	$0.456040\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	8.37844			0.470580			0.235290	−	0.971925i	$-0.424396\pi$
$317$	8.37844			0.470580			0.235290	+	0.971925i	$0.424396\pi$
$318$		0			0
$319$	11.4243			0.639636
$320$		0			0
$321$		0			0
$322$		0			0
$323$	2.21213			0.123086
$324$		0			0
$325$	−27.6974	−	47.9733i	−1.53638	−	2.66108i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	25.8337			1.41995			0.709974	−	0.704228i	$-0.248707\pi$
$331$	25.8337			1.41995			0.709974	+	0.704228i	$0.248707\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−5.05197	−	8.75026i	−0.276018	−	0.478078i
$336$		0			0
$337$	9.81820	−	17.0056i	0.534831	−	0.926355i	−0.464340	−	0.885657i	$-0.653709\pi$
$337$	9.81820	−	17.0056i	0.534831	−	0.926355i	0.999171	−	0.0406980i	$-0.0129582\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	1.81797	−	3.14881i	0.0984485	−	0.170518i
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	22.4094			1.20300			0.601501	−	0.798872i	$-0.294570\pi$
$347$	22.4094			1.20300			0.601501	+	0.798872i	$0.294570\pi$
$348$		0			0
$349$	5.38804	+	9.33236i	0.288415	+	0.499550i	0.973432	−	0.228978i	$-0.0735383\pi$
$349$	5.38804	+	9.33236i	0.288415	+	0.499550i	−0.685016	+	0.728528i	$0.740205\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	5.70106	−	9.87453i	0.303437	−	0.525568i	−0.673475	−	0.739210i	$-0.735199\pi$
$353$	5.70106	−	9.87453i	0.303437	−	0.525568i	0.976912	+	0.213642i	$0.0685325\pi$
$354$		0			0
$355$	0.276650	+	0.479171i	0.0146830	+	0.0254318i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−9.38171	+	16.2496i	−0.495148	+	0.857621i	−0.999984	−	0.00559386i	$-0.998219\pi$
$359$	−9.38171	+	16.2496i	−0.495148	+	0.857621i	0.504837	+	0.863215i	$0.331553\pi$
$360$		0			0
$361$	9.40135	+	16.2836i	0.494808	+	0.857033i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−20.1486	+	34.8984i	−1.05463	+	1.82667i
$366$		0			0
$367$	0.833806	−	1.44420i	0.0435243	−	0.0753864i	−0.843443	−	0.537219i	$-0.819475\pi$
$367$	0.833806	−	1.44420i	0.0435243	−	0.0753864i	0.886967	+	0.461833i	$0.152808\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	12.5229	+	21.6903i	0.648412	+	1.12308i	0.983502	+	0.180896i	$0.0578998\pi$
$373$	12.5229	+	21.6903i	0.648412	+	1.12308i	−0.335090	+	0.942186i	$0.608767\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	14.5211			0.747876
$378$		0			0
$379$	5.21213			0.267729			0.133865	−	0.991000i	$-0.457261\pi$
$379$	5.21213			0.267729			0.133865	+	0.991000i	$0.457261\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−11.2385	−	19.4656i	−0.574258	−	0.994644i	−0.996122	−	0.0879849i	$-0.971957\pi$
$383$	−11.2385	−	19.4656i	−0.574258	−	0.994644i	0.421864	−	0.906659i	$-0.361376\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−5.20471	+	9.01483i	−0.263889	+	0.457070i	−0.967272	−	0.253741i	$-0.918339\pi$
$389$	−5.20471	+	9.01483i	−0.263889	+	0.457070i	0.703383	+	0.710811i	$0.251672\pi$
$390$		0			0
$391$	20.4492	−	35.4190i	1.03416	−	1.79121i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	0.798442	+	1.38294i	0.0401740	+	0.0695834i
$396$		0			0
$397$	1.31436	−	2.27654i	0.0659659	−	0.114256i	−0.831156	−	0.556039i	$-0.812320\pi$
$397$	1.31436	−	2.27654i	0.0659659	−	0.114256i	0.897122	+	0.441783i	$0.145654\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−5.59865	−	9.69714i	−0.279583	−	0.484252i	0.691698	−	0.722187i	$-0.256863\pi$
$401$	−5.59865	−	9.69714i	−0.279583	−	0.484252i	−0.971281	+	0.237935i	$0.923530\pi$
$402$		0			0
$403$	2.31078	−	4.00239i	0.115108	−	0.199373i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	20.9168	+	36.2290i	1.03681	+	1.79581i
$408$		0			0
$409$	31.7309			1.56899			0.784495	−	0.620135i	$-0.212922\pi$
$409$	31.7309			1.56899			0.784495	+	0.620135i	$0.212922\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$	8.12156	−	14.0670i	0.398672	−	0.690520i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	19.0449	−	32.9867i	0.930403	−	1.61151i	0.147770	−	0.989022i	$-0.452791\pi$
$419$	19.0449	−	32.9867i	0.930403	−	1.61151i	0.782633	−	0.622483i	$-0.213876\pi$
$420$		0			0
$421$	−0.614143	−	1.06373i	−0.0299315	−	0.0518429i	0.850672	−	0.525697i	$-0.176196\pi$
$421$	−0.614143	−	1.06373i	−0.0299315	−	0.0518429i	−0.880603	+	0.473855i	$0.842862\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	50.8588			2.46701
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−4.66150	−	8.07396i	−0.224537	−	0.388909i	0.731644	−	0.681687i	$-0.238754\pi$
$431$	−4.66150	−	8.07396i	−0.224537	−	0.388909i	−0.956180	+	0.292778i	$0.905420\pi$
$432$		0			0
$433$	15.3849			0.739350			0.369675	−	0.929161i	$-0.379469\pi$
$433$	15.3849			0.739350			0.369675	+	0.929161i	$0.379469\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−3.64768			−0.174492
$438$		0			0
$439$	5.08935			0.242901			0.121451	−	0.992597i	$-0.461245\pi$
$439$	5.08935			0.242901			0.121451	+	0.992597i	$0.461245\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	34.7865			1.65276			0.826379	−	0.563114i	$-0.190397\pi$
$443$	34.7865			1.65276			0.826379	+	0.563114i	$0.190397\pi$
$444$		0			0
$445$	9.40943			0.446050
$446$		0			0
$447$		0			0
$448$		0			0
$449$	23.8337			1.12478			0.562391	−	0.826872i	$-0.309882\pi$
$449$	23.8337			1.12478			0.562391	+	0.826872i	$0.309882\pi$
$450$		0			0
$451$	11.4969	+	19.9132i	0.541369	+	0.937678i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	16.8337			0.787447			0.393723	−	0.919229i	$-0.371187\pi$
$457$	16.8337			0.787447			0.393723	+	0.919229i	$0.371187\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	6.13780	+	10.6310i	0.285866	+	0.495134i	0.972819	−	0.231568i	$-0.0743855\pi$
$461$	6.13780	+	10.6310i	0.285866	+	0.495134i	−0.686953	+	0.726702i	$0.741052\pi$
$462$		0			0
$463$	6.10607	−	10.5760i	0.283773	−	0.491509i	−0.688538	−	0.725200i	$-0.741747\pi$
$463$	6.10607	−	10.5760i	0.283773	−	0.491509i	0.972311	+	0.233691i	$0.0750805\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−16.8791	+	29.2354i	−0.781071	+	1.35285i	0.150248	+	0.988648i	$0.451993\pi$
$467$	−16.8791	+	29.2354i	−0.781071	+	1.35285i	−0.931318	+	0.364206i	$0.881340\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−36.7931			−1.69175
$474$		0			0
$475$	−2.26802	−	3.92833i	−0.104064	−	0.180244i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−11.9593	+	20.7141i	−0.546434	+	0.946451i	0.452082	+	0.891977i	$0.350682\pi$
$479$	−11.9593	+	20.7141i	−0.546434	+	0.946451i	−0.998515	+	0.0544741i	$0.982652\pi$
$480$		0			0
$481$	26.5870	+	46.0500i	1.21226	+	2.09970i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−14.8385	+	25.7010i	−0.673781	+	1.16702i
$486$		0			0
$487$	5.19664	+	9.00084i	0.235482	+	0.407867i	0.959413	−	0.282006i	$-0.0909998\pi$
$487$	5.19664	+	9.00084i	0.235482	+	0.407867i	−0.723931	+	0.689873i	$0.757666\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	2.68922	−	4.65787i	0.121363	−	0.210207i	−0.798943	−	0.601407i	$-0.794607\pi$
$491$	2.68922	−	4.65787i	0.121363	−	0.210207i	0.920305	+	0.391201i	$0.127940\pi$
$492$		0			0
$493$	−6.66603	+	11.5459i	−0.300223	+	0.520001i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	9.62156	+	16.6650i	0.430720	+	0.746029i	0.996935	−	0.0782282i	$-0.0249263\pi$
$499$	9.62156	+	16.6650i	0.430720	+	0.746029i	−0.566215	+	0.824257i	$0.691593\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	22.3334			0.995798			0.497899	−	0.867235i	$-0.334105\pi$
$503$	22.3334			0.995798			0.497899	+	0.867235i	$0.334105\pi$
$504$		0			0
$505$	19.9838			0.889269
$506$		0			0
$507$		0			0
$508$		0			0
$509$	20.1131	+	34.8369i	0.891497	+	1.54412i	0.838081	+	0.545545i	$0.183677\pi$
$509$	20.1131	+	34.8369i	0.891497	+	1.54412i	0.0534152	+	0.998572i	$0.482989\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−25.2750	+	43.7776i	−1.11375	+	1.92907i
$516$		0			0
$517$	−7.45216	+	12.9075i	−0.327746	+	0.567672i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−18.3497	−	31.7826i	−0.803914	−	1.39242i	−0.917022	−	0.398837i	$-0.869414\pi$
$521$	−18.3497	−	31.7826i	−0.803914	−	1.39242i	0.113108	−	0.993583i	$-0.463919\pi$
$522$		0			0
$523$	14.1727	−	24.5479i	0.619731	−	1.07341i	−0.369804	−	0.929110i	$-0.620575\pi$
$523$	14.1727	−	24.5479i	0.619731	−	1.07341i	0.989535	−	0.144296i	$-0.0460916\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	2.12156	+	3.67465i	0.0924166	+	0.160070i
$528$		0			0
$529$	−22.2195	+	38.4854i	−0.966067	+	1.67328i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	14.6135	+	25.3113i	0.632980	+	1.09635i
$534$		0			0
$535$	44.9902			1.94509
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	12.3027	−	21.3089i	0.528934	−	0.916141i	−0.470496	−	0.882402i	$-0.655925\pi$
$541$	12.3027	−	21.3089i	0.528934	−	0.916141i	0.999431	−	0.0337394i	$-0.0107416\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	0.798442	−	1.38294i	0.0342015	−	0.0592388i
$546$		0			0
$547$	−0.106065	−	0.183711i	−0.00453503	−	0.00785491i	0.863749	−	0.503922i	$-0.168110\pi$
$547$	−0.106065	−	0.183711i	−0.00453503	−	0.00785491i	−0.868284	+	0.496067i	$0.834777\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	1.18907			0.0506563
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	9.40877	+	16.2965i	0.398662	+	0.690503i	0.993561	−	0.113297i	$-0.0361412\pi$
$557$	9.40877	+	16.2965i	0.398662	+	0.690503i	−0.594899	+	0.803801i	$0.702808\pi$
$558$		0			0
$559$	−46.7669			−1.97803
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−12.8781			−0.542749			−0.271375	−	0.962474i	$-0.587478\pi$
$563$	−12.8781			−0.542749			−0.271375	+	0.962474i	$0.587478\pi$
$564$		0			0
$565$	−54.1714			−2.27901
$566$		0			0
$567$		0			0
$568$		0			0
$569$	27.0000			1.13190			0.565949	−	0.824440i	$-0.308510\pi$
$569$	27.0000			1.13190			0.565949	+	0.824440i	$0.308510\pi$
$570$		0			0
$571$	−10.8498			−0.454052			−0.227026	−	0.973889i	$-0.572900\pi$
$571$	−10.8498			−0.454052			−0.227026	+	0.973889i	$0.572900\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−83.8634			−3.49734
$576$		0			0
$577$	−10.6085	−	18.3745i	−0.441640	−	0.764942i	0.556172	−	0.831067i	$-0.312270\pi$
$577$	−10.6085	−	18.3745i	−0.441640	−	0.764942i	−0.997811	+	0.0661250i	$0.978936\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	24.2270			1.00338
$584$		0			0
$585$		0			0
$586$		0			0
$587$	7.92684	+	13.7297i	0.327176	+	0.566685i	0.981950	−	0.189139i	$-0.0605698\pi$
$587$	7.92684	+	13.7297i	0.327176	+	0.566685i	−0.654775	+	0.755824i	$0.727236\pi$
$588$		0			0
$589$	0.189220	−	0.327739i	0.00779668	−	0.0135042i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	12.2580	−	21.2314i	0.503375	−	0.871871i	−0.496618	−	0.867969i	$-0.665425\pi$
$593$	12.2580	−	21.2314i	0.503375	−	0.871871i	0.999992	−	0.00390123i	$-0.00124180\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−4.42426			−0.180770			−0.0903852	−	0.995907i	$-0.528810\pi$
$599$	−4.42426			−0.180770			−0.0903852	+	0.995907i	$0.528810\pi$
$600$		0			0
$601$	4.47075	+	7.74357i	0.182366	+	0.315867i	0.942686	−	0.333682i	$-0.108291\pi$
$601$	4.47075	+	7.74357i	0.182366	+	0.315867i	−0.760320	+	0.649549i	$0.774958\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−14.0646	+	24.3607i	−0.571809	+	0.990402i
$606$		0			0
$607$	−18.9127	−	32.7578i	−0.767643	−	1.32960i	−0.938838	−	0.344359i	$-0.888096\pi$
$607$	−18.9127	−	32.7578i	−0.767643	−	1.32960i	0.171195	−	0.985237i	$-0.445237\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−9.47228	+	16.4065i	−0.383208	+	0.663735i
$612$		0			0
$613$	9.33369	+	16.1664i	0.376984	+	0.652956i	0.990622	−	0.136632i	$-0.0436278\pi$
$613$	9.33369	+	16.1664i	0.376984	+	0.652956i	−0.613638	+	0.789588i	$0.710294\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	14.9365	−	25.8708i	0.601320	−	1.04152i	−0.391301	−	0.920263i	$-0.627975\pi$
$617$	14.9365	−	25.8708i	0.601320	−	1.04152i	0.992621	−	0.121255i	$-0.0386917\pi$
$618$		0			0
$619$	1.90789	−	3.30456i	0.0766846	−	0.132822i	−0.825133	−	0.564938i	$-0.808900\pi$
$619$	1.90789	−	3.30456i	0.0766846	−	0.132822i	0.901818	+	0.432117i	$0.142233\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−14.1135	−	24.4453i	−0.564539	−	0.977811i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−48.8197			−1.94657
$630$		0			0
$631$	20.0458			0.798012			0.399006	−	0.916948i	$-0.369355\pi$
$631$	20.0458			0.798012			0.399006	+	0.916948i	$0.369355\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−22.6636	−	39.2546i	−0.899379	−	1.55777i
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−4.75207	+	8.23083i	−0.187696	+	0.325098i	−0.944482	−	0.328564i	$-0.893435\pi$
$641$	−4.75207	+	8.23083i	−0.187696	+	0.325098i	0.756786	+	0.653663i	$0.226769\pi$
$642$		0			0
$643$	15.0611	−	26.0866i	0.593952	−	1.02876i	−0.399741	−	0.916628i	$-0.630900\pi$
$643$	15.0611	−	26.0866i	0.593952	−	1.02876i	0.993694	−	0.112128i	$-0.0357666\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	10.6267	+	18.4060i	0.417780	+	0.723616i	0.995716	−	0.0924659i	$-0.0294749\pi$
$647$	10.6267	+	18.4060i	0.417780	+	0.723616i	−0.577936	+	0.816082i	$0.696142\pi$
$648$		0			0
$649$	4.04475	−	7.00572i	0.158770	−	0.274999i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−0.225016	−	0.389739i	−0.00880556	−	0.0152517i	0.861589	−	0.507606i	$-0.169470\pi$
$653$	−0.225016	−	0.389739i	−0.00880556	−	0.0152517i	−0.870395	+	0.492355i	$0.836136\pi$
$654$		0			0
$655$	−7.90877	+	13.6984i	−0.309021	+	0.535240i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−8.68441	−	15.0418i	−0.338297	−	0.585947i	0.645816	−	0.763493i	$-0.276517\pi$
$659$	−8.68441	−	15.0418i	−0.338297	−	0.585947i	−0.984112	+	0.177546i	$0.943184\pi$
$660$		0			0
$661$	−34.0096			−1.32282			−0.661411	−	0.750024i	$-0.730042\pi$
$661$	−34.0096			−1.32282			−0.661411	+	0.750024i	$0.730042\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	10.9919	−	19.0386i	0.425609	−	0.737176i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−24.0192	+	41.6025i	−0.927252	+	1.60605i
$672$		0			0
$673$	−10.5825	−	18.3294i	−0.407925	−	0.706547i	0.586732	−	0.809781i	$-0.300414\pi$
$673$	−10.5825	−	18.3294i	−0.407925	−	0.706547i	−0.994657	+	0.103234i	$0.967081\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−17.7149			−0.680839			−0.340420	−	0.940274i	$-0.610569\pi$
$677$	−17.7149			−0.680839			−0.340420	+	0.940274i	$0.610569\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−16.2676	−	28.1763i	−0.622461	−	1.07813i	−0.989026	−	0.147742i	$-0.952800\pi$
$683$	−16.2676	−	28.1763i	−0.622461	−	1.07813i	0.366565	−	0.930393i	$-0.380534\pi$
$684$		0			0
$685$	−35.7136			−1.36455
$686$		0			0
$687$		0			0
$688$		0			0
$689$	30.7944			1.17317
$690$		0			0
$691$	−31.3820			−1.19383			−0.596914	−	0.802305i	$-0.703607\pi$
$691$	−31.3820			−1.19383			−0.596914	+	0.802305i	$0.703607\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	40.8647			1.55009
$696$		0			0
$697$	−26.8337			−1.01640
$698$		0			0
$699$		0			0
$700$		0			0
$701$	11.7163			0.442518			0.221259	−	0.975215i	$-0.428983\pi$
$701$	11.7163			0.442518			0.221259	+	0.975215i	$0.428983\pi$
$702$		0			0
$703$	2.17709	+	3.77084i	0.0821106	+	0.142220i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	7.18114			0.269693			0.134847	−	0.990866i	$-0.456946\pi$
$709$	7.18114			0.269693			0.134847	+	0.990866i	$0.456946\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−3.49834	−	6.05930i	−0.131014	−	0.226922i
$714$		0			0
$715$	−45.1438	+	78.1914i	−1.68828	+	2.92419i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−7.82360	+	13.5509i	−0.291771	+	0.505362i	−0.974229	−	0.225563i	$-0.927578\pi$
$719$	−7.82360	+	13.5509i	−0.291771	+	0.505362i	0.682458	+	0.730925i	$0.260911\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$	27.3378			1.01530
$726$		0			0
$727$	22.5678	+	39.0886i	0.836995	+	1.44972i	0.892396	+	0.451253i	$0.149023\pi$
$727$	22.5678	+	39.0886i	0.836995	+	1.44972i	−0.0554015	+	0.998464i	$0.517644\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	21.4687	−	37.1848i	0.794048	−	1.37533i
$732$		0			0
$733$	20.6167	+	35.7092i	0.761495	+	1.31895i	0.942080	+	0.335388i	$0.108867\pi$
$733$	20.6167	+	35.7092i	0.761495	+	1.31895i	−0.180585	+	0.983559i	$0.557799\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−5.52772	+	9.57429i	−0.203616	+	0.352673i
$738$		0			0
$739$	−16.6209	−	28.7882i	−0.611410	−	1.05899i	−0.991003	−	0.133839i	$-0.957269\pi$
$739$	−16.6209	−	28.7882i	−0.611410	−	1.05899i	0.379593	−	0.925153i	$-0.376064\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	16.3263	−	28.2779i	0.598953	−	1.03742i	−0.394023	−	0.919101i	$-0.628917\pi$
$743$	16.3263	−	28.2779i	0.598953	−	1.03742i	0.992976	−	0.118316i	$-0.0377497\pi$
$744$		0			0
$745$	−31.7240	+	54.9475i	−1.16228	+	2.01312i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−20.0384	−	34.7075i	−0.731212	−	1.26650i	−0.956366	−	0.292173i	$-0.905622\pi$
$751$	−20.0384	−	34.7075i	−0.731212	−	1.26650i	0.225154	−	0.974323i	$-0.427712\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−28.1293			−1.02373
$756$		0			0
$757$	21.8337			0.793559			0.396779	−	0.917914i	$-0.370128\pi$
$757$	21.8337			0.793559			0.396779	+	0.917914i	$0.370128\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−4.56067	−	7.89932i	−0.165324	−	0.286350i	0.771446	−	0.636295i	$-0.219534\pi$
$761$	−4.56067	−	7.89932i	−0.165324	−	0.286350i	−0.936770	+	0.349945i	$0.886200\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$	5.14120	−	8.90482i	0.185638	−	0.321534i
$768$		0			0
$769$	14.3654	−	24.8815i	0.518028	−	0.897250i	−0.481753	−	0.876307i	$-0.660000\pi$
$769$	14.3654	−	24.8815i	0.518028	−	0.897250i	0.999781	−	0.0209433i	$-0.00666695\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	19.7283	+	34.1705i	0.709578	+	1.22903i	0.965014	+	0.262200i	$0.0844479\pi$
$773$	19.7283	+	34.1705i	0.709578	+	1.22903i	−0.255435	+	0.966826i	$0.582219\pi$
$774$		0			0
$775$	4.35033	−	7.53499i	0.156268	−	0.270665i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	1.19664	+	2.07264i	0.0428740	+	0.0742599i
$780$		0			0
$781$	0.302702	−	0.524295i	0.0108315	−	0.0187608i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	2.31493	+	4.00957i	0.0826232	+	0.143108i
$786$		0			0
$787$	−11.0307			−0.393202			−0.196601	−	0.980484i	$-0.562990\pi$
$787$	−11.0307			−0.393202			−0.196601	+	0.980484i	$0.562990\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−30.5303	+	52.8801i	−1.08416	+	1.87783i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−1.15757	+	2.00497i	−0.0410032	+	0.0710196i	−0.885799	−	0.464070i	$-0.846389\pi$
$797$	−1.15757	+	2.00497i	−0.0410032	+	0.0710196i	0.844796	+	0.535089i	$0.179722\pi$
$798$		0			0
$799$	−8.69664	−	15.0630i	−0.307665	−	0.532891i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	44.0921			1.55598
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−3.70471	−	6.41675i	−0.130251	−	0.225601i	0.793522	−	0.608541i	$-0.208245\pi$
$809$	−3.70471	−	6.41675i	−0.130251	−	0.225601i	−0.923773	+	0.382940i	$0.874912\pi$
$810$		0			0
$811$	−12.8879			−0.452555			−0.226277	−	0.974063i	$-0.572656\pi$
$811$	−12.8879			−0.452555			−0.226277	+	0.974063i	$0.572656\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	81.9636			2.87106
$816$		0			0
$817$	−3.82955			−0.133979
$818$		0			0
$819$		0			0
$820$		0			0
$821$	25.8743			0.903019			0.451510	−	0.892266i	$-0.350886\pi$
$821$	25.8743			0.903019			0.451510	+	0.892266i	$0.350886\pi$
$822$		0			0
$823$	44.0620			1.53590			0.767952	−	0.640507i	$-0.221276\pi$
$823$	44.0620			1.53590			0.767952	+	0.640507i	$0.221276\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	6.81753			0.237069			0.118534	−	0.992950i	$-0.462180\pi$
$827$	6.81753			0.237069			0.118534	+	0.992950i	$0.462180\pi$
$828$		0			0
$829$	4.27759	+	7.40900i	0.148567	+	0.257325i	0.930698	−	0.365789i	$-0.119201\pi$
$829$	4.27759	+	7.40900i	0.148567	+	0.257325i	−0.782131	+	0.623114i	$0.785867\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$	−78.2566			−2.70818
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−18.0971	−	31.3451i	−0.624781	−	1.08215i	−0.988583	−	0.150676i	$-0.951855\pi$
$839$	−18.0971	−	31.3451i	−0.624781	−	1.08215i	0.363803	−	0.931476i	$-0.381478\pi$
$840$		0			0
$841$	10.9168	−	18.9085i	0.376443	−	0.652018i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−32.0296	+	55.4768i	−1.10185	+	1.90846i
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$	80.5011			2.75954
$852$		0			0
$853$	−3.97889	−	6.89164i	−0.136235	−	0.235965i	0.789834	−	0.613321i	$-0.210167\pi$
$853$	−3.97889	−	6.89164i	−0.136235	−	0.235965i	−0.926068	+	0.377356i	$0.876833\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−22.7899	+	39.4733i	−0.778489	+	1.34838i	0.154324	+	0.988020i	$0.450680\pi$
$857$	−22.7899	+	39.4733i	−0.778489	+	1.34838i	−0.932813	+	0.360362i	$0.882653\pi$
$858$		0			0
$859$	−10.4518	−	18.1030i	−0.356609	−	0.617666i	0.630783	−	0.775960i	$-0.282734\pi$
$859$	−10.4518	−	18.1030i	−0.356609	−	0.617666i	−0.987392	+	0.158294i	$0.949401\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−3.88586	+	6.73050i	−0.132276	+	0.229109i	−0.924554	−	0.381052i	$-0.875562\pi$
$863$	−3.88586	+	6.73050i	−0.132276	+	0.229109i	0.792278	+	0.610161i	$0.208895\pi$
$864$		0			0
$865$	−11.8182	−	20.4697i	−0.401831	−	0.695991i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	0.873633	−	1.51318i	0.0296360	−	0.0513310i
$870$		0			0
$871$	−7.02616	+	12.1697i	−0.238072	+	0.412354i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	19.6886	+	34.1016i	0.664835	+	1.15153i	0.979330	+	0.202269i	$0.0648317\pi$
$877$	19.6886	+	34.1016i	0.664835	+	1.15153i	−0.314495	+	0.949259i	$0.601835\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−26.7967			−0.902805			−0.451402	−	0.892320i	$-0.649076\pi$
$881$	−26.7967			−0.902805			−0.451402	+	0.892320i	$0.649076\pi$
$882$		0			0
$883$	−43.0755			−1.44961			−0.724803	−	0.688956i	$-0.758069\pi$
$883$	−43.0755			−1.44961			−0.724803	+	0.688956i	$0.758069\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	18.6253	+	32.2600i	0.625377	+	1.08318i	0.988468	+	0.151431i	$0.0483881\pi$
$887$	18.6253	+	32.2600i	0.625377	+	1.08318i	−0.363091	+	0.931754i	$0.618279\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−0.775645	+	1.34346i	−0.0259560	+	0.0449571i
$894$		0			0
$895$	17.2269	−	29.8379i	0.575832	−	0.997370i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	1.14039	+	1.97521i	0.0380342	+	0.0658771i
$900$		0			0
$901$	−14.1364	+	24.4849i	−0.470951	+	0.815711i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−47.1560	−	81.6766i	−1.56752	−	2.71502i
$906$		0			0
$907$	4.89327	−	8.47540i	0.162478	−	0.281421i	−0.773279	−	0.634067i	$-0.781385\pi$
$907$	4.89327	−	8.47540i	0.162478	−	0.281421i	0.935757	+	0.352646i	$0.114718\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−14.2979	−	24.7647i	−0.473710	−	0.820490i	0.525837	−	0.850586i	$-0.323752\pi$
$911$	−14.2979	−	24.7647i	−0.473710	−	0.820490i	−0.999547	+	0.0300951i	$0.990419\pi$
$912$		0			0
$913$	−17.7728			−0.588192
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	8.61348	−	14.9190i	0.284133	−	0.492132i	−0.688266	−	0.725459i	$-0.741628\pi$
$919$	8.61348	−	14.9190i	0.284133	−	0.492132i	0.972398	+	0.233326i	$0.0749611\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	0.384758	−	0.666420i	0.0126645	−	0.0219355i
$924$		0			0
$925$	50.0532	+	86.6948i	1.64574	+	2.85051i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−26.8597			−0.881240			−0.440620	−	0.897694i	$-0.645241\pi$
$929$	−26.8597			−0.881240			−0.440620	+	0.897694i	$0.645241\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−41.4472	−	71.7886i	−1.35547	−	2.34774i
$936$		0			0
$937$	−51.5653			−1.68456			−0.842282	−	0.539037i	$-0.818788\pi$
$937$	−51.5653			−1.68456			−0.842282	+	0.539037i	$0.818788\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−41.8004			−1.36266			−0.681328	−	0.731979i	$-0.738597\pi$
$941$	−41.8004			−1.36266			−0.681328	+	0.731979i	$0.738597\pi$
$942$		0			0
$943$	44.2473			1.44089
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−9.50415			−0.308843			−0.154422	−	0.988005i	$-0.549351\pi$
$947$	−9.50415			−0.308843			−0.154422	+	0.988005i	$0.549351\pi$
$948$		0			0
$949$	56.0445			1.81928
$950$		0			0
$951$		0			0
$952$		0			0
$953$	58.4539			1.89351			0.946754	−	0.321957i	$-0.104341\pi$
$953$	58.4539			1.89351			0.946754	+	0.321957i	$0.104341\pi$
$954$		0			0
$955$	39.4453	+	68.3213i	1.27642	+	2.21083i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−30.2741			−0.976584
$962$		0			0
$963$		0			0
$964$		0			0
$965$	24.1686	+	41.8612i	0.778014	+	1.34756i
$966$		0			0
$967$	−5.09799	+	8.82997i	−0.163940	+	0.283953i	−0.936278	−	0.351259i	$-0.885754\pi$
$967$	−5.09799	+	8.82997i	−0.163940	+	0.283953i	0.772338	+	0.635212i	$0.219087\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	19.4238	−	33.6429i	0.623338	−	1.07965i	−0.365522	−	0.930803i	$-0.619109\pi$
$971$	19.4238	−	33.6429i	0.623338	−	1.07965i	0.988860	−	0.148850i	$-0.0475572\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	42.1322			1.34793			0.673965	−	0.738763i	$-0.264590\pi$
$977$	42.1322			1.34793			0.673965	+	0.738763i	$0.264590\pi$
$978$		0			0
$979$	−5.14776	−	8.91619i	−0.164523	−	0.284963i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	17.0284	−	29.4941i	0.543123	−	0.940716i	−0.455600	−	0.890185i	$-0.650575\pi$
$983$	17.0284	−	29.4941i	0.543123	−	0.940716i	0.998722	−	0.0505312i	$-0.0160914\pi$
$984$		0			0
$985$	28.0688	+	48.6167i	0.894348	+	1.54906i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−35.4007	+	61.3158i	−1.12568	+	1.94973i
$990$		0			0
$991$	−20.5980	−	35.6768i	−0.654317	−	1.13331i	−0.982065	−	0.188544i	$-0.939623\pi$
$991$	−20.5980	−	35.6768i	−0.654317	−	1.13331i	0.327748	−	0.944765i	$-0.393710\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	20.5033	−	35.5127i	0.649997	−	1.12583i
$996$		0			0
$997$	−15.7199	+	27.2277i	−0.497856	+	0.862311i	−0.999997	−	0.00247444i	$-0.999212\pi$
$997$	−15.7199	+	27.2277i	−0.497856	+	0.862311i	0.502141	+	0.864786i	$0.332546\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5292.2.i.h.1549.1		12
3.2	odd	2		1764.2.i.h.373.5		12
7.2	even	3		5292.2.j.f.1765.1		12
7.3	odd	6		5292.2.l.h.361.1		12
7.4	even	3		5292.2.l.h.361.6		12
7.5	odd	6		5292.2.j.f.1765.6		12
7.6	odd	2	inner	5292.2.i.h.1549.6		12
9.2	odd	6		1764.2.l.h.961.4		12
9.7	even	3		5292.2.l.h.3313.6		12
21.2	odd	6		1764.2.j.f.589.1	✓	12
21.5	even	6		1764.2.j.f.589.6	yes	12
21.11	odd	6		1764.2.l.h.949.4		12
21.17	even	6		1764.2.l.h.949.3		12
21.20	even	2		1764.2.i.h.373.2		12
63.2	odd	6		1764.2.j.f.1177.1	yes	12
63.11	odd	6		1764.2.i.h.1537.5		12
63.16	even	3		5292.2.j.f.3529.1		12
63.20	even	6		1764.2.l.h.961.3		12
63.25	even	3	inner	5292.2.i.h.2125.1		12
63.34	odd	6		5292.2.l.h.3313.1		12
63.38	even	6		1764.2.i.h.1537.2		12
63.47	even	6		1764.2.j.f.1177.6	yes	12
63.52	odd	6	inner	5292.2.i.h.2125.6		12
63.61	odd	6		5292.2.j.f.3529.6		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1764.2.i.h.373.2		12	21.20	even	2
1764.2.i.h.373.5		12	3.2	odd	2
1764.2.i.h.1537.2		12	63.38	even	6
1764.2.i.h.1537.5		12	63.11	odd	6
1764.2.j.f.589.1	✓	12	21.2	odd	6
1764.2.j.f.589.6	yes	12	21.5	even	6
1764.2.j.f.1177.1	yes	12	63.2	odd	6
1764.2.j.f.1177.6	yes	12	63.47	even	6
1764.2.l.h.949.3		12	21.17	even	6
1764.2.l.h.949.4		12	21.11	odd	6
1764.2.l.h.961.3		12	63.20	even	6
1764.2.l.h.961.4		12	9.2	odd	6
5292.2.i.h.1549.1		12	1.1	even	1	trivial
5292.2.i.h.1549.6		12	7.6	odd	2	inner
5292.2.i.h.2125.1		12	63.25	even	3	inner
5292.2.i.h.2125.6		12	63.52	odd	6	inner
5292.2.j.f.1765.1		12	7.2	even	3
5292.2.j.f.1765.6		12	7.5	odd	6
5292.2.j.f.3529.1		12	63.16	even	3
5292.2.j.f.3529.6		12	63.61	odd	6
5292.2.l.h.361.1		12	7.3	odd	6
5292.2.l.h.361.6		12	7.4	even	3
5292.2.l.h.3313.1		12	63.34	odd	6
5292.2.l.h.3313.6		12	9.7	even	3

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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 5292 = 2^{2} \cdot 3^{3} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5292.i (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.95014 - 3.37774i) q^{5} +O(q^{10})\)	\(q+(-1.95014 - 3.37774i) q^{5} +(-2.13378 + 3.69582i) q^{11} +(-2.71221 + 4.69768i) q^{13} +(-2.49012 - 4.31301i) q^{17} +(-0.222091 + 0.384673i) q^{19} +(4.10607 + 7.11191i) q^{23} +(-5.10607 + 8.84396i) q^{25} +(-1.33850 - 2.31835i) q^{29} -0.851993 q^{31} +(4.90135 - 8.48939i) q^{37} +(2.69402 - 4.66618i) q^{41} +(4.31078 + 7.46649i) q^{43} +3.49246 q^{47} +(-2.83850 - 4.91642i) q^{53} +16.6447 q^{55} -1.89558 q^{59} +11.2566 q^{61} +21.1567 q^{65} +2.59057 q^{67} -0.141862 q^{71} +(-5.16595 - 8.94769i) q^{73} -0.409429 q^{79} +(2.08231 + 3.60666i) q^{83} +(-9.71213 + 16.8219i) q^{85} +(-1.20625 + 2.08929i) q^{89} +1.73243 q^{95} +(-3.80448 - 6.58955i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q+O(q^{10}) \) 12 * q	\( 12 q - 8 q^{11} - 12 q^{25} - 2 q^{29} + 6 q^{37} + 6 q^{43} - 20 q^{53} + 92 q^{65} + 24 q^{67} - 44 q^{71} - 12 q^{79} - 18 q^{85} + 56 q^{95}+O(q^{100}) \) 12 * q - 8 * q^11 - 12 * q^25 - 2 * q^29 + 6 * q^37 + 6 * q^43 - 20 * q^53 + 92 * q^65 + 24 * q^67 - 44 * q^71 - 12 * q^79 - 18 * q^85 + 56 * q^95

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