LMFDB - Embedded newform 5292.2.j.f.1765.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5292,2,Mod(1765,5292)]

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, 0]))

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

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

chi := DirichletCharacter("5292.1765");

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.j (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			1765.6
Root			$-0.965975 - 1.43767i$ of defining polynomial
Character	$\chi$	$=$	5292.1765
Dual form			5292.2.j.f.3529.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.95014 - 3.37774i) q^{5} +O(q^{10})$	$q+(1.95014 - 3.37774i) q^{5} +(-2.13378 - 3.69582i) q^{11} +(2.71221 - 4.69768i) q^{13} -4.98023 q^{17} -0.444182 q^{19} +(4.10607 - 7.11191i) q^{23} +(-5.10607 - 8.84396i) q^{25} +(-1.33850 - 2.31835i) q^{29} +(-0.425996 + 0.737847i) q^{31} -9.80270 q^{37} +(-2.69402 + 4.66618i) q^{41} +(4.31078 + 7.46649i) q^{43} +(1.74623 + 3.02456i) q^{47} +5.67700 q^{53} -16.6447 q^{55} +(-0.947789 + 1.64162i) q^{59} +(5.62832 + 9.74853i) q^{61} +(-10.5783 - 18.3222i) q^{65} +(-1.29529 + 2.24350i) q^{67} -0.141862 q^{71} -10.3319 q^{73} +(0.204714 + 0.354576i) q^{79} +(-2.08231 - 3.60666i) q^{83} +(-9.71213 + 16.8219i) q^{85} -2.41251 q^{89} +(-0.866216 + 1.50033i) q^{95} +(3.80448 + 6.58955i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q+O(q^{10}) $ 12 * q	$ 12 q - 8 q^{11} - 12 q^{25} - 2 q^{29} - 12 q^{37} + 6 q^{43} + 40 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 - 8 * q^11 - 12 * q^25 - 2 * q^29 - 12 * q^37 + 6 * q^43 + 40 * 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 $\left(\mathbb{Z}/5292\mathbb{Z}\right)^\times$.

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

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	1.95014	−	3.37774i	0.872128	−	1.51057i	0.0123362	−	0.999924i	$-0.496073\pi$
$5$	1.95014	−	3.37774i	0.872128	−	1.51057i	0.859791	−	0.510645i	$-0.170593\pi$
$6$		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	−0.984678	−	0.174384i	$-0.944207\pi$
$11$	−2.13378	−	3.69582i	−0.643360	−	1.11433i	0.341318	−	0.939948i	$-0.389127\pi$
$12$		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
$17$	−4.98023			−1.20788			−0.603942	−	0.797028i	$-0.706404\pi$
$17$	−4.98023			−1.20788			−0.603942	+	0.797028i	$0.706404\pi$
$18$		0			0
$19$	−0.444182			−0.101902			−0.0509512	−	0.998701i	$-0.516225\pi$
$19$	−0.444182			−0.101902			−0.0509512	+	0.998701i	$0.516225\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	4.10607	−	7.11191i	0.856174	−	1.48294i	−0.0193779	−	0.999812i	$-0.506169\pi$
$23$	4.10607	−	7.11191i	0.856174	−	1.48294i	0.875552	−	0.483124i	$-0.160498\pi$
$24$		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
$31$	−0.425996	+	0.737847i	−0.0765112	+	0.132521i	−0.901742	−	0.432274i	$-0.857711\pi$
$31$	−0.425996	+	0.737847i	−0.0765112	+	0.132521i	0.825231	+	0.564795i	$0.191045\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−9.80270			−1.61155			−0.805777	−	0.592219i	$-0.798252\pi$
$37$	−9.80270			−1.61155			−0.805777	+	0.592219i	$0.798252\pi$
$38$		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	0.964818	−	0.262919i	$-0.0846853\pi$
$47$	1.74623	+	3.02456i	0.254714	+	0.441178i	−0.710104	+	0.704097i	$0.751352\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$		0			0
$52$		0			0
$53$	5.67700			0.779795			0.389898	−	0.920858i	$-0.372510\pi$
$53$	5.67700			0.779795			0.389898	+	0.920858i	$0.372510\pi$
$54$		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	−0.921103	−	0.389319i	$-0.872710\pi$
$59$	−0.947789	+	1.64162i	−0.123392	+	0.213721i	0.797711	+	0.603039i	$0.206044\pi$
$60$		0			0
$61$	5.62832	+	9.74853i	0.720632	+	1.24817i	0.960747	+	0.277427i	$0.0894817\pi$
$61$	5.62832	+	9.74853i	0.720632	+	1.24817i	−0.240114	+	0.970745i	$0.577185\pi$
$62$		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	−0.934236	−	0.356656i	$-0.883917\pi$
$67$	−1.29529	+	2.24350i	−0.158244	+	0.274087i	0.775991	+	0.630744i	$0.217250\pi$
$68$		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
$73$	−10.3319			−1.20926			−0.604629	−	0.796507i	$-0.706678\pi$
$73$	−10.3319			−1.20926			−0.604629	+	0.796507i	$0.706678\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	0.204714	+	0.354576i	0.0230322	+	0.0398929i	0.877312	−	0.479921i	$-0.159335\pi$
$79$	0.204714	+	0.354576i	0.0230322	+	0.0398929i	−0.854280	+	0.519814i	$0.826001\pi$
$80$		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
$89$	−2.41251			−0.255725			−0.127863	−	0.991792i	$-0.540812\pi$
$89$	−2.41251			−0.255725			−0.127863	+	0.991792i	$0.540812\pi$
$90$		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
$101$	2.56185	+	4.43726i	0.254914	+	0.441524i	0.964872	−	0.262720i	$-0.0846196\pi$
$101$	2.56185	+	4.43726i	0.254914	+	0.441524i	−0.709958	+	0.704244i	$0.751286\pi$
$102$		0			0
$103$	6.48031	−	11.2242i	0.638524	−	1.10596i	−0.347233	−	0.937779i	$-0.612879\pi$
$103$	6.48031	−	11.2242i	0.638524	−	1.10596i	0.985757	−	0.168177i	$-0.0537880\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	11.5351			1.11514			0.557572	−	0.830129i	$-0.311733\pi$
$107$	11.5351			1.11514			0.557572	+	0.830129i	$0.311733\pi$
$108$		0			0
$109$	−0.409429			−0.0392162			−0.0196081	−	0.999808i	$-0.506242\pi$
$109$	−0.409429			−0.0392162			−0.0196081	+	0.999808i	$0.506242\pi$
$110$		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	$-0.776641\pi$
$131$	2.02775	−	3.51216i	0.177165	−	0.306859i	0.940908	+	0.338661i	$0.109974\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	4.57835	+	7.92993i	0.391155	+	0.677500i	0.992602	−	0.121413i	$-0.0387424\pi$
$137$	4.57835	+	7.92993i	0.391155	+	0.677500i	−0.601448	+	0.798912i	$0.705409\pi$
$138$		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
$143$	−23.1490			−1.93582
$144$		0			0
$145$	−10.4410			−0.867079
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−8.13378	+	14.0881i	−0.666346	+	1.15414i	0.312573	+	0.949894i	$0.398809\pi$
$149$	−8.13378	+	14.0881i	−0.666346	+	1.15414i	−0.978919	+	0.204251i	$0.934524\pi$
$150$		0			0
$151$	3.60607	+	6.24589i	0.293457	+	0.508283i	0.974625	−	0.223844i	$-0.0718608\pi$
$151$	3.60607	+	6.24589i	0.293457	+	0.508283i	−0.681167	+	0.732128i	$0.738527\pi$
$152$		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	−0.888738	−	0.458416i	$-0.848417\pi$
$157$	−0.593529	+	1.02802i	−0.0473688	+	0.0820451i	0.841369	+	0.540461i	$0.181750\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	21.0148			1.64601			0.823004	−	0.568035i	$-0.192296\pi$
$163$	21.0148			1.64601			0.823004	+	0.568035i	$0.192296\pi$
$164$		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	0.957918	−	0.287042i	$-0.0926718\pi$
$173$	3.03009	+	5.24828i	0.230374	+	0.399019i	−0.727544	+	0.686061i	$0.759338\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−8.83369			−0.660261			−0.330131	−	0.943935i	$-0.607093\pi$
$179$	−8.83369			−0.660261			−0.330131	+	0.943935i	$0.607093\pi$
$180$		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	0.956119	+	0.292978i	$0.0946462\pi$
$191$	10.1135	+	17.5171i	0.731786	+	1.26749i	−0.224333	+	0.974512i	$0.572020\pi$
$192$		0			0
$193$	6.19664	−	10.7329i	0.446044	−	0.772570i	−0.552081	−	0.833791i	$-0.686166\pi$
$193$	6.19664	−	10.7329i	0.446044	−	0.772570i	0.998124	+	0.0612205i	$0.0194993\pi$
$194$		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
$199$	10.5138			0.745301			0.372650	−	0.927972i	$-0.378449\pi$
$199$	10.5138			0.745301			0.372650	+	0.927972i	$0.378449\pi$
$200$		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
$215$	33.6264			2.29330
$216$		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
$227$	−9.01267	−	15.6104i	−0.598192	−	1.03610i	−0.993088	−	0.117373i	$-0.962553\pi$
$227$	−9.01267	−	15.6104i	−0.598192	−	1.03610i	0.394896	−	0.918726i	$-0.370781\pi$
$228$		0			0
$229$	2.71221	−	4.69768i	0.179228	−	0.310431i	−0.762389	−	0.647120i	$-0.775973\pi$
$229$	2.71221	−	4.69768i	0.179228	−	0.310431i	0.941616	+	0.336688i	$0.109307\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	2.84331			0.186271			0.0931356	−	0.995653i	$-0.470311\pi$
$233$	2.84331			0.186271			0.0931356	+	0.995653i	$0.470311\pi$
$234$		0			0
$235$	13.6216			0.888573
$236$		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	−0.992841	−	0.119445i	$-0.961888\pi$
$241$	−9.31237	−	16.1295i	−0.599863	−	1.03899i	0.392978	−	0.919548i	$-0.371445\pi$
$242$		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	$-0.572350\pi$
$251$	−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
$257$	−0.672148	+	1.16419i	−0.0419274	+	0.0726204i	−0.886228	−	0.463250i	$-0.846683\pi$
$257$	−0.672148	+	1.16419i	−0.0419274	+	0.0726204i	0.844300	+	0.535871i	$0.180016\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−12.1493	−	21.0432i	−0.749157	−	1.29758i	−0.948227	−	0.317592i	$-0.897126\pi$
$263$	−12.1493	−	21.0432i	−0.749157	−	1.29758i	0.199071	−	0.979985i	$-0.436208\pi$
$264$		0			0
$265$	11.0709	−	19.1754i	0.680081	−	1.17794i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−5.39979			−0.329231			−0.164615	−	0.986358i	$-0.552638\pi$
$269$	−5.39979			−0.329231			−0.164615	+	0.986358i	$0.552638\pi$
$270$		0			0
$271$	13.5718			0.824427			0.412213	−	0.911087i	$-0.364756\pi$
$271$	13.5718			0.824427			0.412213	+	0.911087i	$0.364756\pi$
$272$		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	0.860272	+	0.509835i	$0.170294\pi$
$277$	14.5074	+	25.1276i	0.871666	+	1.50977i	0.0113940	+	0.999935i	$0.496373\pi$
$278$		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	0.239789	+	0.970825i	$0.422922\pi$
$283$	−12.1268	+	21.0043i	−0.720864	+	1.24857i	−0.960654	+	0.277749i	$0.910412\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	7.80270			0.458982
$290$		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
$305$	43.9040			2.51393
$306$		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	−0.842500	−	0.538697i	$-0.818917\pi$
$311$	0.798442	−	1.38294i	0.0452755	−	0.0784195i	0.887775	+	0.460278i	$0.152250\pi$
$312$		0			0
$313$	2.43556	+	4.21851i	0.137666	+	0.238444i	0.926613	−	0.376017i	$-0.122707\pi$
$313$	2.43556	+	4.21851i	0.137666	+	0.238444i	−0.788947	+	0.614461i	$0.789373\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−4.18922	−	7.25594i	−0.235290	−	0.407534i	0.724067	−	0.689730i	$-0.242271\pi$
$317$	−4.18922	−	7.25594i	−0.235290	−	0.407534i	−0.959357	+	0.282195i	$0.908937\pi$
$318$		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
$325$	−55.3948			−3.07275
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−12.9168	−	22.3726i	−0.709974	−	1.22971i	−0.964866	−	0.262742i	$-0.915373\pi$
$331$	−12.9168	−	22.3726i	−0.709974	−	1.22971i	0.254892	−	0.966969i	$-0.417960\pi$
$332$		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
$341$	3.63594			0.196897
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−11.2047	+	19.4071i	−0.601501	+	1.04183i	0.391093	+	0.920351i	$0.372097\pi$
$347$	−11.2047	+	19.4071i	−0.601501	+	1.04183i	−0.992594	+	0.121479i	$0.961236\pi$
$348$		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
$353$	−5.70106	−	9.87453i	−0.303437	−	0.525568i	0.673475	−	0.739210i	$-0.264801\pi$
$353$	−5.70106	−	9.87453i	−0.303437	−	0.525568i	−0.976912	+	0.213642i	$0.931468\pi$
$354$		0			0
$355$	−0.276650	+	0.479171i	−0.0146830	+	0.0254318i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	18.7634			0.990296			0.495148	−	0.868809i	$-0.335114\pi$
$359$	18.7634			0.990296			0.495148	+	0.868809i	$0.335114\pi$
$360$		0			0
$361$	−18.8027			−0.989616
$362$		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	$-0.180525\pi$
$367$	−0.833806	−	1.44420i	−0.0435243	−	0.0753864i	−0.886967	+	0.461833i	$0.847192\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	12.5229	−	21.6903i	0.648412	−	1.12308i	−0.335090	−	0.942186i	$-0.608767\pi$
$373$	12.5229	−	21.6903i	0.648412	−	1.12308i	0.983502	−	0.180896i	$-0.0578998\pi$
$374$		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	−0.421864	−	0.906659i	$-0.638624\pi$
$383$	11.2385	−	19.4656i	0.574258	−	0.994644i	0.996122	−	0.0879849i	$-0.0280427\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−5.20471	−	9.01483i	−0.263889	−	0.457070i	0.703383	−	0.710811i	$-0.251672\pi$
$389$	−5.20471	−	9.01483i	−0.263889	−	0.457070i	−0.967272	+	0.253741i	$0.918339\pi$
$390$		0			0
$391$	−20.4492	+	35.4190i	−1.03416	+	1.79121i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	1.59688			0.0803480
$396$		0			0
$397$	2.62872			0.131932			0.0659659	−	0.997822i	$-0.478987\pi$
$397$	2.62872			0.131932			0.0659659	+	0.997822i	$0.478987\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−5.59865	+	9.69714i	−0.279583	+	0.484252i	−0.971281	−	0.237935i	$-0.923530\pi$
$401$	−5.59865	+	9.69714i	−0.279583	+	0.484252i	0.691698	+	0.722187i	$0.256863\pi$
$402$		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
$409$	15.8654	−	27.4797i	0.784495	−	1.35879i	−0.144805	−	0.989460i	$-0.546256\pi$
$409$	15.8654	−	27.4797i	0.784495	−	1.35879i	0.929300	−	0.369325i	$-0.120411\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$	−16.2431			−0.797343
$416$		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
$431$	9.32300			0.449073			0.224537	−	0.974466i	$-0.427913\pi$
$431$	9.32300			0.449073			0.224537	+	0.974466i	$0.427913\pi$
$432$		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	0.920340	−	0.391119i	$-0.127912\pi$
$439$	2.54467	+	4.40750i	0.121451	+	0.210359i	−0.798889	+	0.601478i	$0.794579\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−17.3933	−	30.1260i	−0.826379	−	1.43133i	−0.900861	−	0.434109i	$-0.857063\pi$
$443$	−17.3933	−	30.1260i	−0.826379	−	1.43133i	0.0744812	−	0.997222i	$-0.476270\pi$
$444$		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
$451$	22.9938			1.08274
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−8.41685	−	14.5784i	−0.393723	−	0.681949i	0.599214	−	0.800589i	$-0.295480\pi$
$457$	−8.41685	−	14.5784i	−0.393723	−	0.681949i	−0.992937	+	0.118640i	$0.962147\pi$
$458$		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
$467$	−33.7582			−1.56214			−0.781071	−	0.624443i	$-0.785326\pi$
$467$	−33.7582			−1.56214			−0.781071	+	0.624443i	$0.785326\pi$
$468$		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
$479$	11.9593	+	20.7141i	0.546434	+	0.946451i	0.998515	+	0.0544741i	$0.0173482\pi$
$479$	11.9593	+	20.7141i	0.546434	+	0.946451i	−0.452082	+	0.891977i	$0.649318\pi$
$480$		0			0
$481$	−26.5870	+	46.0500i	−1.21226	+	2.09970i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	29.6770			1.34756
$486$		0			0
$487$	−10.3933			−0.470964			−0.235482	−	0.971879i	$-0.575667\pi$
$487$	−10.3933			−0.470964			−0.235482	+	0.971879i	$0.575667\pi$
$488$		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	−0.566215	−	0.824257i	$-0.691593\pi$
$499$	9.62156	−	16.6650i	0.430720	−	0.746029i	0.996935	+	0.0782282i	$0.0249263\pi$
$500$		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
$509$	−20.1131	+	34.8369i	−0.891497	+	1.54412i	−0.0534152	+	0.998572i	$0.517011\pi$
$509$	−20.1131	+	34.8369i	−0.891497	+	1.54412i	−0.838081	+	0.545545i	$0.816323\pi$
$510$		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
$521$	−36.6993			−1.60783			−0.803914	−	0.594746i	$-0.797253\pi$
$521$	−36.6993			−1.60783			−0.803914	+	0.594746i	$0.797253\pi$
$522$		0			0
$523$	28.3455			1.23946			0.619731	−	0.784814i	$-0.287242\pi$
$523$	28.3455			1.23946			0.619731	+	0.784814i	$0.287242\pi$
$524$		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
$535$	22.4951	−	38.9626i	0.972547	−	1.68450i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−24.6054			−1.05787			−0.528934	−	0.848663i	$-0.677408\pi$
$541$	−24.6054			−1.05787			−0.528934	+	0.848663i	$0.677408\pi$
$542$		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
$557$	−18.8175			−0.797325			−0.398662	−	0.917098i	$-0.630525\pi$
$557$	−18.8175			−0.797325			−0.398662	+	0.917098i	$0.630525\pi$
$558$		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	−0.969214	−	0.246220i	$-0.920812\pi$
$563$	−6.43907	+	11.1528i	−0.271375	+	0.470035i	0.697839	+	0.716254i	$0.254145\pi$
$564$		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	−0.996961	−	0.0779066i	$-0.975176\pi$
$569$	−13.5000	−	23.3827i	−0.565949	−	0.980253i	0.431011	−	0.902347i	$-0.358157\pi$
$570$		0			0
$571$	5.42492	−	9.39624i	0.227026	−	0.393221i	−0.729899	−	0.683555i	$-0.760433\pi$
$571$	5.42492	−	9.39624i	0.227026	−	0.393221i	0.956925	+	0.290334i	$0.0937664\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−83.8634			−3.49734
$576$		0			0
$577$	−21.2171			−0.883279			−0.441640	−	0.897192i	$-0.645603\pi$
$577$	−21.2171			−0.883279			−0.441640	+	0.897192i	$0.645603\pi$
$578$		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
$593$	24.5160			1.00675			0.503375	−	0.864068i	$-0.332092\pi$
$593$	24.5160			1.00675			0.503375	+	0.864068i	$0.332092\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	2.21213	−	3.83152i	0.0903852	−	0.156552i	−0.817288	−	0.576229i	$-0.804524\pi$
$599$	2.21213	−	3.83152i	0.0903852	−	0.156552i	0.907673	+	0.419678i	$0.137857\pi$
$600$		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
$605$	14.0646	+	24.3607i	0.571809	+	0.990402i
$606$		0			0
$607$	18.9127	−	32.7578i	0.767643	−	1.32960i	−0.171195	−	0.985237i	$-0.554763\pi$
$607$	18.9127	−	32.7578i	0.767643	−	1.32960i	0.938838	−	0.344359i	$-0.111904\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	18.9446			0.766415
$612$		0			0
$613$	−18.6674			−0.753968			−0.376984	−	0.926220i	$-0.623039\pi$
$613$	−18.6674			−0.753968			−0.376984	+	0.926220i	$0.623039\pi$
$614$		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	$-0.191100\pi$
$619$	−1.90789	−	3.30456i	−0.0766846	−	0.132822i	−0.901818	+	0.432117i	$0.857767\pi$
$620$		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	0.756786	−	0.653663i	$-0.226769\pi$
$641$	−4.75207	−	8.23083i	−0.187696	−	0.325098i	−0.944482	+	0.328564i	$0.893435\pi$
$642$		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
$647$	21.2535			0.835560			0.417780	−	0.908548i	$-0.362808\pi$
$647$	21.2535			0.835560			0.417780	+	0.908548i	$0.362808\pi$
$648$		0			0
$649$	8.08951			0.317541
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−0.225016	+	0.389739i	−0.00880556	+	0.0152517i	−0.870395	−	0.492355i	$-0.836136\pi$
$653$	−0.225016	+	0.389739i	−0.00880556	+	0.0152517i	0.861589	+	0.507606i	$0.169470\pi$
$654$		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
$661$	−17.0048	+	29.4532i	−0.661411	+	1.14560i	0.318835	+	0.947810i	$0.396709\pi$
$661$	−17.0048	+	29.4532i	−0.661411	+	1.14560i	−0.980245	+	0.197786i	$0.936625\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−21.9838			−0.851218
$668$		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	0.644091	−	0.764949i	$-0.277236\pi$
$677$	−8.85745	−	15.3416i	−0.340420	−	0.589624i	−0.984511	+	0.175325i	$0.943902\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	32.5351			1.24492			0.622461	−	0.782651i	$-0.286133\pi$
$683$	32.5351			1.24492			0.622461	+	0.782651i	$0.286133\pi$
$684$		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	−0.993274	−	0.115791i	$-0.963060\pi$
$691$	−15.6910	−	27.1776i	−0.596914	−	1.03389i	0.396359	−	0.918095i	$-0.370273\pi$
$692$		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
$703$	4.35419			0.164221
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−3.59057	−	6.21905i	−0.134847	−	0.233561i	0.790692	−	0.612214i	$-0.209721\pi$
$709$	−3.59057	−	6.21905i	−0.134847	−	0.233561i	−0.925539	+	0.378653i	$0.876388\pi$
$710$		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
$719$	−15.6472			−0.583542			−0.291771	−	0.956488i	$-0.594245\pi$
$719$	−15.6472			−0.583542			−0.291771	+	0.956488i	$0.594245\pi$
$720$		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
$731$	−21.4687	−	37.1848i	−0.794048	−	1.37533i
$732$		0			0
$733$	−20.6167	+	35.7092i	−0.761495	+	1.31895i	0.180585	+	0.983559i	$0.442201\pi$
$733$	−20.6167	+	35.7092i	−0.761495	+	1.31895i	−0.942080	+	0.335388i	$0.891133\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	11.0554			0.407232
$738$		0			0
$739$	33.2418			1.22282			0.611410	−	0.791314i	$-0.290603\pi$
$739$	33.2418			1.22282			0.611410	+	0.791314i	$0.290603\pi$
$740$		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	0.225154	+	0.974323i	$0.427712\pi$
$751$	−20.0384	+	34.7075i	−0.731212	+	1.26650i	−0.956366	+	0.292173i	$0.905622\pi$
$752$		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	$-0.780466\pi$
$761$	4.56067	−	7.89932i	0.165324	−	0.286350i	0.936770	+	0.349945i	$0.113800\pi$
$762$		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	$0.340000\pi$
$769$	−14.3654	+	24.8815i	−0.518028	+	0.897250i	−0.999781	+	0.0209433i	$0.993333\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	39.4566			1.41916			0.709578	−	0.704626i	$-0.248885\pi$
$773$	39.4566			1.41916			0.709578	+	0.704626i	$0.248885\pi$
$774$		0			0
$775$	8.70066			0.312537
$776$		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
$787$	−5.51534	+	9.55286i	−0.196601	+	0.340523i	−0.947424	−	0.319981i	$-0.896324\pi$
$787$	−5.51534	+	9.55286i	−0.196601	+	0.340523i	0.750823	+	0.660503i	$0.229657\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	61.0607			2.16833
$794$		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
$809$	7.40943			0.260502			0.130251	−	0.991481i	$-0.458422\pi$
$809$	7.40943			0.260502			0.130251	+	0.991481i	$0.458422\pi$
$810$		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	0.546970	−	0.837152i	$-0.315781\pi$
$821$	−12.9371	−	22.4078i	−0.451510	−	0.782037i	−0.998480	+	0.0551144i	$0.982448\pi$
$822$		0			0
$823$	−22.0310	+	38.1588i	−0.767952	+	1.33013i	0.170720	+	0.985320i	$0.445391\pi$
$823$	−22.0310	+	38.1588i	−0.767952	+	1.33013i	−0.938672	+	0.344812i	$0.887943\pi$
$824$		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
$829$	8.55517			0.297133			0.148567	−	0.988902i	$-0.452534\pi$
$829$	8.55517			0.297133			0.148567	+	0.988902i	$0.452534\pi$
$830$		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
$845$	−64.0591			−2.20370
$846$		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
$857$	22.7899	+	39.4733i	0.778489	+	1.34838i	0.932813	+	0.360362i	$0.117347\pi$
$857$	22.7899	+	39.4733i	0.778489	+	1.34838i	−0.154324	+	0.988020i	$0.549320\pi$
$858$		0			0
$859$	10.4518	−	18.1030i	0.356609	−	0.617666i	−0.630783	−	0.775960i	$-0.717266\pi$
$859$	10.4518	−	18.1030i	0.356609	−	0.617666i	0.987392	+	0.158294i	$0.0505994\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	7.77171			0.264552			0.132276	−	0.991213i	$-0.457771\pi$
$863$	7.77171			0.264552			0.132276	+	0.991213i	$0.457771\pi$
$864$		0			0
$865$	23.6364			0.803661
$866$		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	−0.314495	−	0.949259i	$-0.601835\pi$
$877$	19.6886	−	34.1016i	0.664835	−	1.15153i	0.979330	−	0.202269i	$-0.0648317\pi$
$878$		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
$887$	−18.6253	+	32.2600i	−0.625377	+	1.08318i	0.363091	+	0.931754i	$0.381721\pi$
$887$	−18.6253	+	32.2600i	−0.625377	+	1.08318i	−0.988468	+	0.151431i	$0.951612\pi$
$888$		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
$899$	2.28078			0.0760683
$900$		0			0
$901$	−28.2728			−0.941902
$902$		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	0.935757	−	0.352646i	$-0.114718\pi$
$907$	4.89327	+	8.47540i	0.162478	+	0.281421i	−0.773279	+	0.634067i	$0.781385\pi$
$908$		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
$919$	−17.2270			−0.568265			−0.284133	−	0.958785i	$-0.591706\pi$
$919$	−17.2270			−0.568265			−0.284133	+	0.958785i	$0.591706\pi$
$920$		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	0.557116	−	0.830435i	$-0.311908\pi$
$929$	−13.4299	−	23.2612i	−0.440620	−	0.763176i	−0.997736	+	0.0672589i	$0.978575\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$	82.8943			2.71093
$936$		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	0.293248	+	0.956036i	$0.405264\pi$
$941$	−20.9002	+	36.2002i	−0.681328	+	1.18009i	−0.974576	+	0.224058i	$0.928070\pi$
$942$		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	0.932848	−	0.360269i	$-0.117315\pi$
$947$	4.75207	+	8.23083i	0.154422	+	0.267466i	−0.778427	+	0.627736i	$0.783982\pi$
$948$		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
$955$	78.8907			2.55284
$956$		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
$971$	38.8475			1.24668			0.623338	−	0.781953i	$-0.285776\pi$
$971$	38.8475			1.24668			0.623338	+	0.781953i	$0.285776\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−21.0661	+	36.4876i	−0.673965	+	1.16734i	0.302805	+	0.953052i	$0.402077\pi$
$977$	−21.0661	+	36.4876i	−0.673965	+	1.16734i	−0.976770	+	0.214289i	$0.931257\pi$
$978$		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	−0.998722	−	0.0505312i	$-0.983909\pi$
$983$	−17.0284	−	29.4941i	−0.543123	−	0.940716i	0.455600	−	0.890185i	$-0.349425\pi$
$984$		0			0
$985$	−28.0688	+	48.6167i	−0.894348	+	1.54906i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	70.8014			2.25135
$990$		0			0
$991$	41.1960			1.30863			0.654317	−	0.756221i	$-0.272956\pi$
$991$	41.1960			1.30863			0.654317	+	0.756221i	$0.272956\pi$
$992$		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	$-0.000787640\pi$
$997$	15.7199	+	27.2277i	0.497856	+	0.862311i	−0.502141	+	0.864786i	$0.667454\pi$
$998$		0			0
$999$		0			0

Twists

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

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

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+(1.95014 - 3.37774i) q^{5} +O(q^{10})\)	\(q+(1.95014 - 3.37774i) q^{5} +(-2.13378 - 3.69582i) q^{11} +(2.71221 - 4.69768i) q^{13} -4.98023 q^{17} -0.444182 q^{19} +(4.10607 - 7.11191i) q^{23} +(-5.10607 - 8.84396i) q^{25} +(-1.33850 - 2.31835i) q^{29} +(-0.425996 + 0.737847i) q^{31} -9.80270 q^{37} +(-2.69402 + 4.66618i) q^{41} +(4.31078 + 7.46649i) q^{43} +(1.74623 + 3.02456i) q^{47} +5.67700 q^{53} -16.6447 q^{55} +(-0.947789 + 1.64162i) q^{59} +(5.62832 + 9.74853i) q^{61} +(-10.5783 - 18.3222i) q^{65} +(-1.29529 + 2.24350i) q^{67} -0.141862 q^{71} -10.3319 q^{73} +(0.204714 + 0.354576i) q^{79} +(-2.08231 - 3.60666i) q^{83} +(-9.71213 + 16.8219i) q^{85} -2.41251 q^{89} +(-0.866216 + 1.50033i) q^{95} +(3.80448 + 6.58955i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q+O(q^{10}) \) 12 * q	\( 12 q - 8 q^{11} - 12 q^{25} - 2 q^{29} - 12 q^{37} + 6 q^{43} + 40 q^{53} - 46 q^{65} - 12 q^{67} - 44 q^{71} + 6 q^{79} - 18 q^{85} - 28 q^{95}+O(q^{100}) \) 12 * q - 8 * q^11 - 12 * q^25 - 2 * q^29 - 12 * q^37 + 6 * q^43 + 40 * q^53 - 46 * q^65 - 12 * q^67 - 44 * q^71 + 6 * q^79 - 18 * q^85 - 28 * q^95

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