LMFDB - Embedded newform 5292.2.l.h.361.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5292,2,Mod(361,5292)]

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, 4]))

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

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

chi := DirichletCharacter("5292.361");

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.l (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:	$\Z[a_1, \ldots, a_{29}]$
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			361.1
Root			$0.965975 + 1.43767i$ of defining polynomial
Character	$\chi$	$=$	5292.361
Dual form			5292.2.l.h.3313.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.90027 q^{5} +O(q^{10})$	$q-3.90027 q^{5} +4.26757 q^{11} +(2.71221 - 4.69768i) q^{13} +(2.49012 - 4.31301i) q^{17} +(0.222091 + 0.384673i) q^{19} -8.21213 q^{23} +10.2121 q^{25} +(-1.33850 - 2.31835i) q^{29} +(-0.425996 - 0.737847i) q^{31} +(4.90135 + 8.48939i) q^{37} +(-2.69402 + 4.66618i) q^{41} +(4.31078 + 7.46649i) q^{43} +(1.74623 - 3.02456i) q^{47} +(-2.83850 + 4.91642i) q^{53} -16.6447 q^{55} +(-0.947789 - 1.64162i) q^{59} +(5.62832 - 9.74853i) q^{61} +(-10.5783 + 18.3222i) q^{65} +(-1.29529 - 2.24350i) q^{67} -0.141862 q^{71} +(5.16595 - 8.94769i) q^{73} +(0.204714 - 0.354576i) q^{79} +(-2.08231 - 3.60666i) q^{83} +(-9.71213 + 16.8219i) q^{85} +(1.20625 + 2.08929i) q^{89} +(-0.866216 - 1.50033i) q^{95} +(3.80448 + 6.58955i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q+O(q^{10}) $ 12 * q	$ 12 q + 16 q^{11} + 24 q^{25} - 2 q^{29} + 6 q^{37} + 6 q^{43} - 20 q^{53} - 46 q^{65} - 12 q^{67} - 44 q^{71} + 6 q^{79} - 18 q^{85} - 28 q^{95}+O(q^{100}) $ 12 * q + 16 * q^11 + 24 * q^25 - 2 * q^29 + 6 * q^37 + 6 * q^43 - 20 * q^53 - 46 * q^65 - 12 * q^67 - 44 * q^71 + 6 * q^79 - 18 * q^85 - 28 * q^95

Display 100 coefficients 10 coefficients

Character values

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

$n$	$785$	$1081$	$2647$
$\chi(n)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{2}{3}\right)$	$1$

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
$5$	−3.90027			−1.74426			−0.872128	−	0.489279i	$-0.837260\pi$
$5$	−3.90027			−1.74426			−0.872128	+	0.489279i	$0.837260\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$		0			0
$10$		0			0
$11$	4.26757			1.28672			0.643360	−	0.765564i	$-0.277540\pi$
$11$	4.26757			1.28672			0.643360	+	0.765564i	$0.277540\pi$
$12$		0			0
$13$	2.71221	−	4.69768i	0.752231	−	1.30290i	−0.194508	−	0.980901i	$-0.562311\pi$
$13$	2.71221	−	4.69768i	0.752231	−	1.30290i	0.946739	−	0.322001i	$-0.104355\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	2.49012	−	4.31301i	0.603942	−	1.04606i	−0.388276	−	0.921543i	$-0.626929\pi$
$17$	2.49012	−	4.31301i	0.603942	−	1.04606i	0.992218	−	0.124515i	$-0.0397374\pi$
$18$		0			0
$19$	0.222091	+	0.384673i	0.0509512	+	0.0882501i	0.890376	−	0.455226i	$-0.150441\pi$
$19$	0.222091	+	0.384673i	0.0509512	+	0.0882501i	−0.839425	+	0.543476i	$0.817108\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−8.21213			−1.71235			−0.856174	−	0.516688i	$-0.827165\pi$
$23$	−8.21213			−1.71235			−0.856174	+	0.516688i	$0.827165\pi$
$24$		0			0
$25$	10.2121			2.04243
$26$		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
$31$	−0.425996	−	0.737847i	−0.0765112	−	0.132521i	0.825231	−	0.564795i	$-0.191045\pi$
$31$	−0.425996	−	0.737847i	−0.0765112	−	0.132521i	−0.901742	+	0.432274i	$0.857711\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	4.90135	+	8.48939i	0.805777	+	1.39565i	0.915765	+	0.401714i	$0.131585\pi$
$37$	4.90135	+	8.48939i	0.805777	+	1.39565i	−0.109988	+	0.993933i	$0.535081\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−2.69402	+	4.66618i	−0.420735	+	0.728735i	−0.996012	−	0.0892242i	$-0.971561\pi$
$41$	−2.69402	+	4.66618i	−0.420735	+	0.728735i	0.575276	+	0.817959i	$0.304895\pi$
$42$		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
$47$	1.74623	−	3.02456i	0.254714	−	0.441178i	−0.710104	−	0.704097i	$-0.751352\pi$
$47$	1.74623	−	3.02456i	0.254714	−	0.441178i	0.964818	+	0.262919i	$0.0846853\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−2.83850	+	4.91642i	−0.389898	+	0.675323i	−0.992435	−	0.122768i	$-0.960823\pi$
$53$	−2.83850	+	4.91642i	−0.389898	+	0.675323i	0.602538	+	0.798090i	$0.294156\pi$
$54$		0			0
$55$	−16.6447			−2.24437
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−0.947789	−	1.64162i	−0.123392	−	0.213721i	0.797711	−	0.603039i	$-0.206044\pi$
$59$	−0.947789	−	1.64162i	−0.123392	−	0.213721i	−0.921103	+	0.389319i	$0.872710\pi$
$60$		0			0
$61$	5.62832	−	9.74853i	0.720632	−	1.24817i	−0.240114	−	0.970745i	$-0.577185\pi$
$61$	5.62832	−	9.74853i	0.720632	−	1.24817i	0.960747	−	0.277427i	$-0.0894817\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−10.5783	+	18.3222i	−1.31208	+	2.27259i
$66$		0			0
$67$	−1.29529	−	2.24350i	−0.158244	−	0.274087i	0.775991	−	0.630744i	$-0.217250\pi$
$67$	−1.29529	−	2.24350i	−0.158244	−	0.274087i	−0.934236	+	0.356656i	$0.883917\pi$
$68$		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	−0.387481	−	0.921878i	$-0.626655\pi$
$73$	5.16595	−	8.94769i	0.604629	−	1.04725i	0.992110	−	0.125370i	$-0.0400118\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	0.204714	−	0.354576i	0.0230322	−	0.0398929i	−0.854280	−	0.519814i	$-0.826001\pi$
$79$	0.204714	−	0.354576i	0.0230322	−	0.0398929i	0.877312	+	0.479921i	$0.159335\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−2.08231	−	3.60666i	−0.228563	−	0.395882i	0.728820	−	0.684706i	$-0.240069\pi$
$83$	−2.08231	−	3.60666i	−0.228563	−	0.395882i	−0.957382	+	0.288823i	$0.906736\pi$
$84$		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	$-0.125855\pi$
$89$	1.20625	+	2.08929i	0.127863	+	0.221464i	−0.794986	+	0.606628i	$0.792522\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−0.866216	−	1.50033i	−0.0888719	−	0.153931i
$96$		0			0
$97$	3.80448	+	6.58955i	0.386286	+	0.669067i	0.991947	−	0.126656i	$-0.0404244\pi$
$97$	3.80448	+	6.58955i	0.386286	+	0.669067i	−0.605661	+	0.795723i	$0.707091\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−5.12370			−0.509828			−0.254914	−	0.966964i	$-0.582047\pi$
$101$	−5.12370			−0.509828			−0.254914	+	0.966964i	$0.582047\pi$
$102$		0			0
$103$	−12.9606			−1.27705			−0.638524	−	0.769602i	$-0.720455\pi$
$103$	−12.9606			−1.27705			−0.638524	+	0.769602i	$0.720455\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−5.76757	−	9.98972i	−0.557572	−	0.965743i	−0.997698	−	0.0678070i	$-0.978400\pi$
$107$	−5.76757	−	9.98972i	−0.557572	−	0.965743i	0.440127	−	0.897936i	$-0.354934\pi$
$108$		0			0
$109$	0.204714	−	0.354576i	0.0196081	−	0.0339622i	−0.856055	−	0.516885i	$-0.827091\pi$
$109$	0.204714	−	0.354576i	0.0196081	−	0.0339622i	0.875663	+	0.482923i	$0.160425\pi$
$110$		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
$115$	32.0296			2.98677
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	7.21213			0.655648
$122$		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
$131$	−4.05549			−0.354330			−0.177165	−	0.984181i	$-0.556693\pi$
$131$	−4.05549			−0.354330			−0.177165	+	0.984181i	$0.556693\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−9.15669			−0.782309			−0.391155	−	0.920325i	$-0.627924\pi$
$137$	−9.15669			−0.782309			−0.391155	+	0.920325i	$0.627924\pi$
$138$		0			0
$139$	5.23869	−	9.07369i	0.444340	−	0.769620i	−0.553666	−	0.832739i	$-0.686771\pi$
$139$	5.23869	−	9.07369i	0.444340	−	0.769620i	0.998006	+	0.0631191i	$0.0201048\pi$
$140$		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
$149$	16.2676			1.33269			0.666346	−	0.745643i	$-0.267858\pi$
$149$	16.2676			1.33269			0.666346	+	0.745643i	$0.267858\pi$
$150$		0			0
$151$	−7.21213			−0.586915			−0.293457	−	0.955972i	$-0.594806\pi$
$151$	−7.21213			−0.586915			−0.293457	+	0.955972i	$0.594806\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	1.66150	+	2.87781i	0.133455	+	0.231151i
$156$		0			0
$157$	−0.593529	−	1.02802i	−0.0473688	−	0.0820451i	0.841369	−	0.540461i	$-0.181750\pi$
$157$	−0.593529	−	1.02802i	−0.0473688	−	0.0820451i	−0.888738	+	0.458416i	$0.848417\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−10.5074	−	18.1994i	−0.823004	−	1.42549i	−0.903435	−	0.428725i	$-0.858963\pi$
$163$	−10.5074	−	18.1994i	−0.823004	−	1.42549i	0.0804306	−	0.996760i	$-0.474370\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−10.0322	+	17.3763i	−0.776315	+	1.34462i	0.157738	+	0.987481i	$0.449580\pi$
$167$	−10.0322	+	17.3763i	−0.776315	+	1.34462i	−0.934053	+	0.357136i	$0.883753\pi$
$168$		0			0
$169$	−8.21213	−	14.2238i	−0.631702	−	1.09414i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	3.03009	−	5.24828i	0.230374	−	0.399019i	−0.727544	−	0.686061i	$-0.759338\pi$
$173$	3.03009	−	5.24828i	0.230374	−	0.399019i	0.957918	+	0.287042i	$0.0926718\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	4.41685	−	7.65020i	0.330131	−	0.571803i	−0.652407	−	0.757869i	$-0.726241\pi$
$179$	4.41685	−	7.65020i	0.330131	−	0.571803i	0.982537	+	0.186066i	$0.0595739\pi$
$180$		0			0
$181$	−24.1809			−1.79735			−0.898676	−	0.438614i	$-0.855470\pi$
$181$	−24.1809			−1.79735			−0.898676	+	0.438614i	$0.855470\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−19.1166	−	33.1109i	−1.40548	−	2.43436i
$186$		0			0
$187$	10.6267	−	18.4060i	0.777104	−	1.34598i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	10.1135	−	17.5171i	0.731786	−	1.26749i	−0.224333	−	0.974512i	$-0.572020\pi$
$191$	10.1135	−	17.5171i	0.731786	−	1.26749i	0.956119	−	0.292978i	$-0.0946462\pi$
$192$		0			0
$193$	6.19664	+	10.7329i	0.446044	+	0.772570i	0.998124	−	0.0612205i	$-0.0194993\pi$
$193$	6.19664	+	10.7329i	0.446044	+	0.772570i	−0.552081	+	0.833791i	$0.686166\pi$
$194$		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	$-0.954884\pi$
$199$	−5.25688	+	9.10518i	−0.372650	+	0.645449i	0.617322	+	0.786711i	$0.288218\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$	10.5074	−	18.1994i	0.733870	−	1.27110i
$206$		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
$221$	−13.5074	−	23.3955i	−0.908607	−	1.57375i
$222$		0			0
$223$	−12.6727	−	21.9497i	−0.848625	−	1.46986i	−0.882436	−	0.470433i	$-0.844098\pi$
$223$	−12.6727	−	21.9497i	−0.848625	−	1.46986i	0.0338111	−	0.999428i	$-0.489236\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	18.0253			1.19638			0.598192	−	0.801353i	$-0.295886\pi$
$227$	18.0253			1.19638			0.598192	+	0.801353i	$0.295886\pi$
$228$		0			0
$229$	−5.42441			−0.358455			−0.179228	−	0.983808i	$-0.557360\pi$
$229$	−5.42441			−0.358455			−0.179228	+	0.983808i	$0.557360\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−1.42165	−	2.46237i	−0.0931356	−	0.161316i	0.815693	−	0.578484i	$-0.196356\pi$
$233$	−1.42165	−	2.46237i	−0.0931356	−	0.161316i	−0.908829	+	0.417169i	$0.863022\pi$
$234$		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
$241$	18.6247			1.19973			0.599863	−	0.800103i	$-0.295222\pi$
$241$	18.6247			1.19973			0.599863	+	0.800103i	$0.295222\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	2.40943			0.153308
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−7.14015			−0.450682			−0.225341	−	0.974280i	$-0.572350\pi$
$251$	−7.14015			−0.450682			−0.225341	+	0.974280i	$0.572350\pi$
$252$		0			0
$253$	−35.0458			−2.20331
$254$		0			0
$255$		0			0
$256$		0			0
$257$	1.34430			0.0838549			0.0419274	−	0.999121i	$-0.486650\pi$
$257$	1.34430			0.0838549			0.0419274	+	0.999121i	$0.486650\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	24.2986			1.49831			0.749157	−	0.662393i	$-0.230459\pi$
$263$	24.2986			1.49831			0.749157	+	0.662393i	$0.230459\pi$
$264$		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	$-0.780695\pi$
$269$	2.69989	−	4.67636i	0.164615	−	0.285122i	0.936519	+	0.350618i	$0.114028\pi$
$270$		0			0
$271$	−6.78589	−	11.7535i	−0.412213	−	0.713974i	0.582918	−	0.812531i	$-0.301911\pi$
$271$	−6.78589	−	11.7535i	−0.412213	−	0.713974i	−0.995131	+	0.0985565i	$0.968577\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	43.5810			2.62803
$276$		0			0
$277$	−29.0148			−1.74333			−0.871666	−	0.490100i	$-0.836960\pi$
$277$	−29.0148			−1.74333			−0.871666	+	0.490100i	$0.836960\pi$
$278$		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
$283$	−12.1268	−	21.0043i	−0.720864	−	1.24857i	−0.960654	−	0.277749i	$-0.910412\pi$
$283$	−12.1268	−	21.0043i	−0.720864	−	1.24857i	0.239789	−	0.970825i	$-0.422922\pi$
$284$		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	−0.558050	−	0.829808i	$-0.688450\pi$
$293$	7.52491	−	13.0335i	0.439610	−	0.761426i	0.997659	+	0.0683813i	$0.0217835\pi$
$294$		0			0
$295$	3.69664	+	6.40276i	0.215226	+	0.372783i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−22.2730	+	38.5780i	−1.28808	+	2.23102i
$300$		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	$-0.536588\pi$
$307$	−4.01912			−0.229383			−0.114692	+	0.993401i	$0.536588\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	0.798442	+	1.38294i	0.0452755	+	0.0784195i	0.887775	−	0.460278i	$-0.152250\pi$
$311$	0.798442	+	1.38294i	0.0452755	+	0.0784195i	−0.842500	+	0.538697i	$0.818917\pi$
$312$		0			0
$313$	2.43556	−	4.21851i	0.137666	−	0.238444i	−0.788947	−	0.614461i	$-0.789373\pi$
$313$	2.43556	−	4.21851i	0.137666	−	0.238444i	0.926613	+	0.376017i	$0.122707\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−4.18922	+	7.25594i	−0.235290	+	0.407534i	−0.959357	−	0.282195i	$-0.908937\pi$
$317$	−4.18922	+	7.25594i	−0.235290	+	0.407534i	0.724067	+	0.689730i	$0.242271\pi$
$318$		0			0
$319$	−5.71213	−	9.89370i	−0.319818	−	0.553941i
$320$		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
$331$	−12.9168	+	22.3726i	−0.709974	+	1.22971i	0.254892	+	0.966969i	$0.417960\pi$
$331$	−12.9168	+	22.3726i	−0.709974	+	1.22971i	−0.964866	+	0.262742i	$0.915373\pi$
$332$		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
$347$	−11.2047	−	19.4071i	−0.601501	−	1.04183i	−0.992594	−	0.121479i	$-0.961236\pi$
$347$	−11.2047	−	19.4071i	−0.601501	−	1.04183i	0.391093	−	0.920351i	$-0.372097\pi$
$348$		0			0
$349$	−5.38804	−	9.33236i	−0.288415	−	0.499550i	0.685016	−	0.728528i	$-0.259795\pi$
$349$	−5.38804	−	9.33236i	−0.288415	−	0.499550i	−0.973432	+	0.228978i	$0.926462\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	11.4021			0.606874			0.303437	−	0.952852i	$-0.401866\pi$
$353$	11.4021			0.606874			0.303437	+	0.952852i	$0.401866\pi$
$354$		0			0
$355$	0.553299			0.0293661
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−9.38171	−	16.2496i	−0.495148	−	0.857621i	0.504837	−	0.863215i	$-0.331553\pi$
$359$	−9.38171	−	16.2496i	−0.495148	−	0.857621i	−0.999984	+	0.00559386i	$0.998219\pi$
$360$		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
$367$	1.66761			0.0870487			0.0435243	−	0.999052i	$-0.486141\pi$
$367$	1.66761			0.0870487			0.0435243	+	0.999052i	$0.486141\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−25.0458			−1.29682			−0.648412	−	0.761290i	$-0.724566\pi$
$373$	−25.0458			−1.29682			−0.648412	+	0.761290i	$0.724566\pi$
$374$		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
$383$	−22.4769			−1.14852			−0.574258	−	0.818674i	$-0.694709\pi$
$383$	−22.4769			−1.14852			−0.574258	+	0.818674i	$0.694709\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	10.4094			0.527779			0.263889	−	0.964553i	$-0.414995\pi$
$389$	10.4094			0.527779			0.263889	+	0.964553i	$0.414995\pi$
$390$		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	$-0.187680\pi$
$397$	−1.31436	−	2.27654i	−0.0659659	−	0.114256i	−0.897122	+	0.441783i	$0.854346\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	11.1973			0.559166			0.279583	−	0.960121i	$-0.409804\pi$
$401$	11.1973			0.559166			0.279583	+	0.960121i	$0.409804\pi$
$402$		0			0
$403$	−4.62156			−0.230216
$404$		0			0
$405$		0			0
$406$		0			0
$407$	20.9168	+	36.2290i	1.03681	+	1.79581i
$408$		0			0
$409$	15.8654	+	27.4797i	0.784495	+	1.35879i	0.929300	+	0.369325i	$0.120411\pi$
$409$	15.8654	+	27.4797i	0.784495	+	1.35879i	−0.144805	+	0.989460i	$0.546256\pi$
$410$		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	$0.547209\pi$
$419$	−19.0449	+	32.9867i	−0.930403	+	1.61151i	−0.782633	+	0.622483i	$0.786124\pi$
$420$		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
$425$	25.4294	−	44.0450i	1.23351	−	2.13650i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−4.66150	+	8.07396i	−0.224537	+	0.388909i	−0.956180	−	0.292778i	$-0.905420\pi$
$431$	−4.66150	+	8.07396i	−0.224537	+	0.388909i	0.731644	+	0.681687i	$0.238754\pi$
$432$		0			0
$433$	−15.3849			−0.739350			−0.369675	−	0.929161i	$-0.620531\pi$
$433$	−15.3849			−0.739350			−0.369675	+	0.929161i	$0.620531\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−1.82384	−	3.15899i	−0.0872462	−	0.151115i
$438$		0			0
$439$	2.54467	−	4.40750i	0.121451	−	0.210359i	−0.798889	−	0.601478i	$-0.794579\pi$
$439$	2.54467	−	4.40750i	0.121451	−	0.210359i	0.920340	+	0.391119i	$0.127912\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−17.3933	+	30.1260i	−0.826379	+	1.43133i	0.0744812	+	0.997222i	$0.476270\pi$
$443$	−17.3933	+	30.1260i	−0.826379	+	1.43133i	−0.900861	+	0.434109i	$0.857063\pi$
$444$		0			0
$445$	−4.70471	−	8.14880i	−0.223025	−	0.386290i
$446$		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
$457$	−8.41685	+	14.5784i	−0.393723	+	0.681949i	−0.992937	−	0.118640i	$-0.962147\pi$
$457$	−8.41685	+	14.5784i	−0.393723	+	0.681949i	0.599214	+	0.800589i	$0.295480\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−6.13780	−	10.6310i	−0.285866	−	0.495134i	0.686953	−	0.726702i	$-0.258948\pi$
$461$	−6.13780	−	10.6310i	−0.285866	−	0.495134i	−0.972819	+	0.231568i	$0.925614\pi$
$462$		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	0.931318	+	0.364206i	$0.118660\pi$
$467$	16.8791	+	29.2354i	0.781071	+	1.35285i	−0.150248	+	0.988648i	$0.548007\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	18.3965	+	31.8637i	0.845874	+	1.46510i
$474$		0			0
$475$	2.26802	+	3.92833i	0.104064	+	0.180244i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−23.9186			−1.09287			−0.546434	−	0.837502i	$-0.684015\pi$
$479$	−23.9186			−1.09287			−0.546434	+	0.837502i	$0.684015\pi$
$480$		0			0
$481$	53.1739			2.42452
$482$		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	−0.723931	−	0.689873i	$-0.757666\pi$
$487$	5.19664	−	9.00084i	0.235482	−	0.407867i	0.959413	+	0.282006i	$0.0909998\pi$
$488$		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
$493$	−13.3321			−0.600446
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−19.2431			−0.861440			−0.430720	−	0.902486i	$-0.641740\pi$
$499$	−19.2431			−0.861440			−0.430720	+	0.902486i	$0.641740\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−22.3334			−0.995798			−0.497899	−	0.867235i	$-0.665895\pi$
$503$	−22.3334			−0.995798			−0.497899	+	0.867235i	$0.665895\pi$
$504$		0			0
$505$	19.9838			0.889269
$506$		0			0
$507$		0			0
$508$		0			0
$509$	40.2262			1.78299			0.891497	−	0.453027i	$-0.149656\pi$
$509$	40.2262			1.78299			0.891497	+	0.453027i	$0.149656\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$	50.5500			2.22750
$516$		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	−0.113108	−	0.993583i	$-0.536081\pi$
$521$	18.3497	−	31.7826i	0.803914	−	1.39242i	0.917022	−	0.398837i	$-0.130586\pi$
$522$		0			0
$523$	−14.1727	−	24.5479i	−0.619731	−	1.07341i	−0.989535	−	0.144296i	$-0.953908\pi$
$523$	−14.1727	−	24.5479i	−0.619731	−	1.07341i	0.369804	−	0.929110i	$-0.379425\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−4.24312			−0.184833
$528$		0			0
$529$	44.4391			1.93213
$530$		0			0
$531$		0			0
$532$		0			0
$533$	14.6135	+	25.3113i	0.632980	+	1.09635i
$534$		0			0
$535$	22.4951	+	38.9626i	0.972547	+	1.68450i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	12.3027	+	21.3089i	0.528934	+	0.916141i	0.999431	+	0.0337394i	$0.0107416\pi$
$541$	12.3027	+	21.3089i	0.528934	+	0.916141i	−0.470496	+	0.882402i	$0.655925\pi$
$542$		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
$551$	0.594537	−	1.02977i	0.0253281	−	0.0438696i
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	9.40877	−	16.2965i	0.398662	−	0.690503i	−0.594899	−	0.803801i	$-0.702808\pi$
$557$	9.40877	−	16.2965i	0.398662	−	0.690503i	0.993561	+	0.113297i	$0.0361412\pi$
$558$		0			0
$559$	46.7669			1.97803
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−6.43907	−	11.1528i	−0.271375	−	0.470035i	0.697839	−	0.716254i	$-0.254145\pi$
$563$	−6.43907	−	11.1528i	−0.271375	−	0.470035i	−0.969214	+	0.246220i	$0.920812\pi$
$564$		0			0
$565$	−27.0857	+	46.9138i	−1.13950	+	1.97368i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−13.5000	+	23.3827i	−0.565949	+	0.980253i	0.431011	+	0.902347i	$0.358157\pi$
$569$	−13.5000	+	23.3827i	−0.565949	+	0.980253i	−0.996961	+	0.0779066i	$0.975176\pi$
$570$		0			0
$571$	5.42492	+	9.39624i	0.227026	+	0.393221i	0.956925	−	0.290334i	$-0.0937664\pi$
$571$	5.42492	+	9.39624i	0.227026	+	0.393221i	−0.729899	+	0.683555i	$0.760433\pi$
$572$		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	$-0.687730\pi$
$577$	10.6085	−	18.3745i	0.441640	−	0.764942i	0.997811	+	0.0661250i	$0.0210636\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	−12.1135	+	20.9812i	−0.501689	+	0.868951i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−7.92684	−	13.7297i	−0.327176	−	0.566685i	0.654775	−	0.755824i	$-0.272764\pi$
$587$	−7.92684	−	13.7297i	−0.327176	−	0.566685i	−0.981950	+	0.189139i	$0.939430\pi$
$588$		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	−0.999992	−	0.00390123i	$-0.998758\pi$
$593$	−12.2580	−	21.2314i	−0.503375	−	0.871871i	0.496618	−	0.867969i	$-0.334575\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	2.21213	+	3.83152i	0.0903852	+	0.156552i	0.907673	−	0.419678i	$-0.137857\pi$
$599$	2.21213	+	3.83152i	0.0903852	+	0.156552i	−0.817288	+	0.576229i	$0.804524\pi$
$600$		0			0
$601$	−4.47075	−	7.74357i	−0.182366	−	0.315867i	0.760320	−	0.649549i	$-0.225042\pi$
$601$	−4.47075	−	7.74357i	−0.182366	−	0.315867i	−0.942686	+	0.333682i	$0.891709\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−28.1293			−1.14362
$606$		0			0
$607$	−37.8254			−1.53529			−0.767643	−	0.640878i	$-0.778571\pi$
$607$	−37.8254			−1.53529			−0.767643	+	0.640878i	$0.778571\pi$
$608$		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	−0.613638	−	0.789588i	$-0.710294\pi$
$613$	9.33369	−	16.1664i	0.376984	−	0.652956i	0.990622	+	0.136632i	$0.0436278\pi$
$614$		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
$619$	3.81578			0.153369			0.0766846	−	0.997055i	$-0.475567\pi$
$619$	3.81578			0.153369			0.0766846	+	0.997055i	$0.475567\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	28.2270			1.12908
$626$		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
$635$	−45.3273			−1.79876
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	9.50415			0.375391			0.187696	−	0.982227i	$-0.439898\pi$
$641$	9.50415			0.375391			0.187696	+	0.982227i	$0.439898\pi$
$642$		0			0
$643$	−15.0611	+	26.0866i	−0.593952	+	1.02876i	0.399741	+	0.916628i	$0.369100\pi$
$643$	−15.0611	+	26.0866i	−0.593952	+	1.02876i	−0.993694	+	0.112128i	$0.964233\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−10.6267	+	18.4060i	−0.417780	+	0.723616i	−0.995716	−	0.0924659i	$-0.970525\pi$
$647$	−10.6267	+	18.4060i	−0.417780	+	0.723616i	0.577936	+	0.816082i	$0.303858\pi$
$648$		0			0
$649$	−4.04475	−	7.00572i	−0.158770	−	0.274999i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	0.450032			0.0176111			0.00880556	−	0.999961i	$-0.497197\pi$
$653$	0.450032			0.0176111			0.00880556	+	0.999961i	$0.497197\pi$
$654$		0			0
$655$	15.8175			0.618042
$656$		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
$661$	−17.0048	−	29.4532i	−0.661411	−	1.14560i	−0.980245	−	0.197786i	$-0.936625\pi$
$661$	−17.0048	−	29.4532i	−0.661411	−	1.14560i	0.318835	−	0.947810i	$-0.396709\pi$
$662$		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
$677$	−8.85745	+	15.3416i	−0.340420	+	0.589624i	−0.984511	−	0.175325i	$-0.943902\pi$
$677$	−8.85745	+	15.3416i	−0.340420	+	0.589624i	0.644091	+	0.764949i	$0.277236\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−16.2676	+	28.1763i	−0.622461	+	1.07813i	0.366565	+	0.930393i	$0.380534\pi$
$683$	−16.2676	+	28.1763i	−0.622461	+	1.07813i	−0.989026	+	0.147742i	$0.952800\pi$
$684$		0			0
$685$	35.7136			1.36455
$686$		0			0
$687$		0			0
$688$		0			0
$689$	15.3972	+	26.6687i	0.586586	+	1.01600i
$690$		0			0
$691$	−15.6910	+	27.1776i	−0.596914	+	1.03389i	0.396359	+	0.918095i	$0.370273\pi$
$691$	−15.6910	+	27.1776i	−0.596914	+	1.03389i	−0.993274	+	0.115791i	$0.963060\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−20.4323	+	35.3899i	−0.775043	+	1.34241i
$696$		0			0
$697$	13.4168	+	23.2387i	0.508199	+	0.880227i
$698$		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
$709$	−3.59057	+	6.21905i	−0.134847	+	0.233561i	−0.925539	−	0.378653i	$-0.876388\pi$
$709$	−3.59057	+	6.21905i	−0.134847	+	0.233561i	0.790692	+	0.612214i	$0.209721\pi$
$710$		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	$-0.0724222\pi$
$719$	7.82360	+	13.5509i	0.291771	+	0.505362i	−0.682458	+	0.730925i	$0.739089\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−13.6689	−	23.6753i	−0.507651	−	0.879277i
$726$		0			0
$727$	−22.5678	−	39.0886i	−0.836995	−	1.44972i	−0.892396	−	0.451253i	$-0.850977\pi$
$727$	−22.5678	−	39.0886i	−0.836995	−	1.44972i	0.0554015	−	0.998464i	$-0.482356\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	42.9374			1.58810
$732$		0			0
$733$	41.2334			1.52299			0.761495	−	0.648171i	$-0.224466\pi$
$733$	41.2334			1.52299			0.761495	+	0.648171i	$0.224466\pi$
$734$		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	0.379593	+	0.925153i	$0.376064\pi$
$739$	−16.6209	+	28.7882i	−0.611410	+	1.05899i	−0.991003	+	0.133839i	$0.957269\pi$
$740$		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
$745$	−63.4480			−2.32455
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	40.0768			1.46242			0.731212	−	0.682151i	$-0.238955\pi$
$751$	40.0768			1.46242			0.731212	+	0.682151i	$0.238955\pi$
$752$		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
$761$	−9.12135			−0.330649			−0.165324	−	0.986239i	$-0.552867\pi$
$761$	−9.12135			−0.330649			−0.165324	+	0.986239i	$0.552867\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−10.2824			−0.371276
$768$		0			0
$769$	−14.3654	+	24.8815i	−0.518028	+	0.897250i	0.481753	+	0.876307i	$0.340000\pi$
$769$	−14.3654	+	24.8815i	−0.518028	+	0.897250i	−0.999781	+	0.0209433i	$0.993333\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−19.7283	+	34.1705i	−0.709578	+	1.22903i	0.255435	+	0.966826i	$0.417781\pi$
$773$	−19.7283	+	34.1705i	−0.709578	+	1.22903i	−0.965014	+	0.262200i	$0.915552\pi$
$774$		0			0
$775$	−4.35033	−	7.53499i	−0.156268	−	0.270665i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−2.39327			−0.0857479
$780$		0			0
$781$	−0.605404			−0.0216631
$782$		0			0
$783$		0			0
$784$		0			0
$785$	2.31493	+	4.00957i	0.0826232	+	0.143108i
$786$		0			0
$787$	−5.51534	−	9.55286i	−0.196601	−	0.340523i	0.750823	−	0.660503i	$-0.229657\pi$
$787$	−5.51534	−	9.55286i	−0.196601	−	0.340523i	−0.947424	+	0.319981i	$0.896324\pi$
$788$		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	−0.844796	−	0.535089i	$-0.820278\pi$
$797$	1.15757	−	2.00497i	0.0410032	−	0.0710196i	0.885799	+	0.464070i	$0.153611\pi$
$798$		0			0
$799$	−8.69664	−	15.0630i	−0.307665	−	0.532891i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	22.0460	−	38.1849i	0.777988	−	1.34751i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−3.70471	+	6.41675i	−0.130251	+	0.225601i	−0.923773	−	0.382940i	$-0.874912\pi$
$809$	−3.70471	+	6.41675i	−0.130251	+	0.225601i	0.793522	+	0.608541i	$0.208245\pi$
$810$		0			0
$811$	12.8879			0.452555			0.226277	−	0.974063i	$-0.427344\pi$
$811$	12.8879			0.452555			0.226277	+	0.974063i	$0.427344\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	40.9818	+	70.9826i	1.43553	+	2.48641i
$816$		0			0
$817$	−1.91477	+	3.31648i	−0.0669894	+	0.116029i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−12.9371	+	22.4078i	−0.451510	+	0.782037i	−0.998480	−	0.0551144i	$-0.982448\pi$
$821$	−12.9371	+	22.4078i	−0.451510	+	0.782037i	0.546970	+	0.837152i	$0.315781\pi$
$822$		0			0
$823$	−22.0310	−	38.1588i	−0.767952	−	1.33013i	−0.938672	−	0.344812i	$-0.887943\pi$
$823$	−22.0310	−	38.1588i	−0.767952	−	1.33013i	0.170720	−	0.985320i	$-0.445391\pi$
$824$		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	$-0.880799\pi$
$829$	−4.27759	+	7.40900i	−0.148567	+	0.257325i	0.782131	+	0.623114i	$0.214133\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$	39.1283	−	67.7722i	1.35409	−	2.34535i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	18.0971	+	31.3451i	0.624781	+	1.08215i	0.988583	+	0.150676i	$0.0481450\pi$
$839$	18.0971	+	31.3451i	0.624781	+	1.08215i	−0.363803	+	0.931476i	$0.618522\pi$
$840$		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
$851$	−40.2505	−	69.7160i	−1.37977	−	2.38983i
$852$		0			0
$853$	3.97889	+	6.89164i	0.136235	+	0.235965i	0.926068	−	0.377356i	$-0.123167\pi$
$853$	3.97889	+	6.89164i	0.136235	+	0.235965i	−0.789834	+	0.613321i	$0.789833\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−45.5798			−1.55698			−0.778489	−	0.627658i	$-0.784014\pi$
$857$	−45.5798			−1.55698			−0.778489	+	0.627658i	$0.784014\pi$
$858$		0			0
$859$	−20.9035			−0.713219			−0.356609	−	0.934254i	$-0.616067\pi$
$859$	−20.9035			−0.713219			−0.356609	+	0.934254i	$0.616067\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−3.88586	−	6.73050i	−0.132276	−	0.229109i	0.792278	−	0.610161i	$-0.208895\pi$
$863$	−3.88586	−	6.73050i	−0.132276	−	0.229109i	−0.924554	+	0.381052i	$0.875562\pi$
$864$		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
$871$	−14.0523			−0.476145
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−39.3771			−1.32967			−0.664835	−	0.746990i	$-0.731498\pi$
$877$	−39.3771			−1.32967			−0.664835	+	0.746990i	$0.731498\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	26.7967			0.902805			0.451402	−	0.892320i	$-0.350924\pi$
$881$	26.7967			0.902805			0.451402	+	0.892320i	$0.350924\pi$
$882$		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
$887$	37.2506			1.25075			0.625377	−	0.780323i	$-0.284945\pi$
$887$	37.2506			1.25075			0.625377	+	0.780323i	$0.284945\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$	1.55129			0.0519120
$894$		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
$905$	94.3121			3.13504
$906$		0			0
$907$	−9.78655			−0.324957			−0.162478	−	0.986712i	$-0.551949\pi$
$907$	−9.78655			−0.324957			−0.162478	+	0.986712i	$0.551949\pi$
$908$		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
$913$	−8.88638	−	15.3917i	−0.294096	−	0.509390i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	8.61348	+	14.9190i	0.284133	+	0.492132i	0.972398	−	0.233326i	$-0.0749611\pi$
$919$	8.61348	+	14.9190i	0.284133	+	0.492132i	−0.688266	+	0.725459i	$0.741628\pi$
$920$		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
$929$	−13.4299	+	23.2612i	−0.440620	+	0.763176i	−0.997736	−	0.0672589i	$-0.978575\pi$
$929$	−13.4299	+	23.2612i	−0.440620	+	0.763176i	0.557116	+	0.830435i	$0.311908\pi$
$930$		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	$-0.181212\pi$
$937$	51.5653			1.68456			0.842282	+	0.539037i	$0.181212\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−20.9002	−	36.2002i	−0.681328	−	1.18009i	−0.974576	−	0.224058i	$-0.928070\pi$
$941$	−20.9002	−	36.2002i	−0.681328	−	1.18009i	0.293248	−	0.956036i	$-0.405264\pi$
$942$		0			0
$943$	22.1237	−	38.3193i	0.720445	−	1.24785i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	4.75207	−	8.23083i	0.154422	−	0.267466i	−0.778427	−	0.627736i	$-0.783982\pi$
$947$	4.75207	−	8.23083i	0.154422	−	0.267466i	0.932848	+	0.360269i	$0.117315\pi$
$948$		0			0
$949$	−28.0222	−	48.5360i	−0.909641	−	1.57554i
$950$		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
$961$	15.1371	−	26.2181i	0.488292	−	0.845747i
$962$		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	−0.988860	−	0.148850i	$-0.952443\pi$
$971$	−19.4238	−	33.6429i	−0.623338	−	1.07965i	0.365522	−	0.930803i	$-0.380891\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−21.0661	−	36.4876i	−0.673965	−	1.16734i	−0.976770	−	0.214289i	$-0.931257\pi$
$977$	−21.0661	−	36.4876i	−0.673965	−	1.16734i	0.302805	−	0.953052i	$-0.402077\pi$
$978$		0			0
$979$	5.14776	+	8.91619i	0.164523	+	0.284963i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	34.0569			1.08625			0.543123	−	0.839653i	$-0.317242\pi$
$983$	34.0569			1.08625			0.543123	+	0.839653i	$0.317242\pi$
$984$		0			0
$985$	56.1377			1.78870
$986$		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	0.327748	+	0.944765i	$0.393710\pi$
$991$	−20.5980	+	35.6768i	−0.654317	+	1.13331i	−0.982065	+	0.188544i	$0.939623\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	20.5033	−	35.5127i	0.649997	−	1.12583i
$996$		0			0
$997$	−31.4399			−0.995711			−0.497856	−	0.867260i	$-0.665879\pi$
$997$	−31.4399			−0.995711			−0.497856	+	0.867260i	$0.665879\pi$
$998$		0			0
$999$		0			0

Twists

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

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

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-3.90027 q^{5} +O(q^{10})\)	\(q-3.90027 q^{5} +4.26757 q^{11} +(2.71221 - 4.69768i) q^{13} +(2.49012 - 4.31301i) q^{17} +(0.222091 + 0.384673i) q^{19} -8.21213 q^{23} +10.2121 q^{25} +(-1.33850 - 2.31835i) q^{29} +(-0.425996 - 0.737847i) q^{31} +(4.90135 + 8.48939i) q^{37} +(-2.69402 + 4.66618i) q^{41} +(4.31078 + 7.46649i) q^{43} +(1.74623 - 3.02456i) q^{47} +(-2.83850 + 4.91642i) q^{53} -16.6447 q^{55} +(-0.947789 - 1.64162i) q^{59} +(5.62832 - 9.74853i) q^{61} +(-10.5783 + 18.3222i) q^{65} +(-1.29529 - 2.24350i) q^{67} -0.141862 q^{71} +(5.16595 - 8.94769i) q^{73} +(0.204714 - 0.354576i) q^{79} +(-2.08231 - 3.60666i) q^{83} +(-9.71213 + 16.8219i) q^{85} +(1.20625 + 2.08929i) q^{89} +(-0.866216 - 1.50033i) q^{95} +(3.80448 + 6.58955i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q+O(q^{10}) \) 12 * q	\( 12 q + 16 q^{11} + 24 q^{25} - 2 q^{29} + 6 q^{37} + 6 q^{43} - 20 q^{53} - 46 q^{65} - 12 q^{67} - 44 q^{71} + 6 q^{79} - 18 q^{85} - 28 q^{95}+O(q^{100}) \) 12 * q + 16 * q^11 + 24 * q^25 - 2 * q^29 + 6 * q^37 + 6 * q^43 - 20 * q^53 - 46 * q^65 - 12 * q^67 - 44 * q^71 + 6 * q^79 - 18 * q^85 - 28 * q^95

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