LMFDB - Embedded newform 5292.2.l.b.3313.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [5292,2,Mod(361,5292)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage: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")

magma://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:traces = [2,0,0,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	no
Analytic conductor:	$42.2568327497$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 252)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			3313.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	5292.3313
Dual form			5292.2.l.b.361.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.00000 q^{5} -4.00000 q^{11} +(1.50000 + 2.59808i) q^{13} +(-3.50000 - 6.06218i) q^{17} +(2.50000 - 4.33013i) q^{19} -4.00000 q^{23} -1.00000 q^{25} +(-0.500000 + 0.866025i) q^{29} +(-1.50000 + 2.59808i) q^{31} +(-5.50000 + 9.52628i) q^{37} +(4.50000 + 7.79423i) q^{41} +(-2.50000 + 4.33013i) q^{43} +(-1.50000 - 2.59808i) q^{47} +(1.50000 + 2.59808i) q^{53} -8.00000 q^{55} +(3.50000 - 6.06218i) q^{59} +(1.50000 + 2.59808i) q^{61} +(3.00000 + 5.19615i) q^{65} +(-6.50000 + 11.2583i) q^{67} +8.00000 q^{71} +(3.50000 + 6.06218i) q^{73} +(4.50000 + 7.79423i) q^{79} +(-0.500000 + 0.866025i) q^{83} +(-7.00000 - 12.1244i) q^{85} +(-7.50000 + 12.9904i) q^{89} +(5.00000 - 8.66025i) q^{95} +(-8.50000 + 14.7224i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{5} - 8 q^{11} + 3 q^{13} - 7 q^{17} + 5 q^{19} - 8 q^{23} - 2 q^{25} - q^{29} - 3 q^{31} - 11 q^{37} + 9 q^{41} - 5 q^{43} - 3 q^{47} + 3 q^{53} - 16 q^{55} + 7 q^{59} + 3 q^{61} + 6 q^{65}+ \cdots - 17 q^{97}+O(q^{100}) $ 2 * q + 4 * q^5 - 8 * q^11 + 3 * q^13 - 7 * q^17 + 5 * q^19 - 8 * q^23 - 2 * q^25 - q^29 - 3 * q^31 - 11 * q^37 + 9 * q^41 - 5 * q^43 - 3 * q^47 + 3 * q^53 - 16 * q^55 + 7 * q^59 + 3 * q^61 + 6 * q^65 - 13 * q^67 + 16 * q^71 + 7 * q^73 + 9 * q^79 - q^83 - 14 * q^85 - 15 * q^89 + 10 * q^95 - 17 * q^97

Character values

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

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

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	2.00000			0.894427			0.447214	−	0.894427i	$-0.352416\pi$
$5$	2.00000			0.894427			0.447214	+	0.894427i	$0.352416\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−4.00000			−1.20605			−0.603023	−	0.797724i	$-0.706037\pi$
$11$	−4.00000			−1.20605			−0.603023	+	0.797724i	$0.706037\pi$
$12$		0			0
$13$	1.50000	+	2.59808i	0.416025	+	0.720577i	0.995535	−	0.0943882i	$-0.0300895\pi$
$13$	1.50000	+	2.59808i	0.416025	+	0.720577i	−0.579510	+	0.814965i	$0.696756\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−3.50000	−	6.06218i	−0.848875	−	1.47029i	−0.882213	−	0.470850i	$-0.843947\pi$
$17$	−3.50000	−	6.06218i	−0.848875	−	1.47029i	0.0333386	−	0.999444i	$-0.489386\pi$
$18$		0			0
$19$	2.50000	−	4.33013i	0.573539	−	0.993399i	−0.422659	−	0.906289i	$-0.638903\pi$
$19$	2.50000	−	4.33013i	0.573539	−	0.993399i	0.996199	−	0.0871106i	$-0.0277634\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−4.00000			−0.834058			−0.417029	−	0.908893i	$-0.636929\pi$
$23$	−4.00000			−0.834058			−0.417029	+	0.908893i	$0.636929\pi$
$24$		0			0
$25$	−1.00000			−0.200000
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−0.500000	+	0.866025i	−0.0928477	+	0.160817i	−0.908708	−	0.417432i	$-0.862930\pi$
$29$	−0.500000	+	0.866025i	−0.0928477	+	0.160817i	0.815861	+	0.578249i	$0.196264\pi$
$30$		0			0
$31$	−1.50000	+	2.59808i	−0.269408	+	0.466628i	−0.968709	−	0.248199i	$-0.920161\pi$
$31$	−1.50000	+	2.59808i	−0.269408	+	0.466628i	0.699301	+	0.714827i	$0.253495\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−5.50000	+	9.52628i	−0.904194	+	1.56611i	−0.0821995	+	0.996616i	$0.526194\pi$
$37$	−5.50000	+	9.52628i	−0.904194	+	1.56611i	−0.821995	+	0.569495i	$0.807139\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	4.50000	+	7.79423i	0.702782	+	1.21725i	0.967486	+	0.252924i	$0.0813924\pi$
$41$	4.50000	+	7.79423i	0.702782	+	1.21725i	−0.264704	+	0.964330i	$0.585274\pi$
$42$		0			0
$43$	−2.50000	+	4.33013i	−0.381246	+	0.660338i	−0.991241	−	0.132068i	$-0.957838\pi$
$43$	−2.50000	+	4.33013i	−0.381246	+	0.660338i	0.609994	+	0.792406i	$0.291172\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−1.50000	−	2.59808i	−0.218797	−	0.378968i	0.735643	−	0.677369i	$-0.236880\pi$
$47$	−1.50000	−	2.59808i	−0.218797	−	0.378968i	−0.954441	+	0.298401i	$0.903547\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$		0			0
$52$		0			0
$53$	1.50000	+	2.59808i	0.206041	+	0.356873i	0.950464	−	0.310835i	$-0.100609\pi$
$53$	1.50000	+	2.59808i	0.206041	+	0.356873i	−0.744423	+	0.667708i	$0.767275\pi$
$54$		0			0
$55$	−8.00000			−1.07872
$56$		0			0
$57$		0			0
$58$		0			0
$59$	3.50000	−	6.06218i	0.455661	−	0.789228i	−0.543065	−	0.839691i	$-0.682736\pi$
$59$	3.50000	−	6.06218i	0.455661	−	0.789228i	0.998726	+	0.0504625i	$0.0160695\pi$
$60$		0			0
$61$	1.50000	+	2.59808i	0.192055	+	0.332650i	0.945931	−	0.324367i	$-0.105151\pi$
$61$	1.50000	+	2.59808i	0.192055	+	0.332650i	−0.753876	+	0.657017i	$0.771818\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	3.00000	+	5.19615i	0.372104	+	0.644503i
$66$		0			0
$67$	−6.50000	+	11.2583i	−0.794101	+	1.37542i	0.129307	+	0.991605i	$0.458725\pi$
$67$	−6.50000	+	11.2583i	−0.794101	+	1.37542i	−0.923408	+	0.383819i	$0.874609\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	8.00000			0.949425			0.474713	−	0.880141i	$-0.342552\pi$
$71$	8.00000			0.949425			0.474713	+	0.880141i	$0.342552\pi$
$72$		0			0
$73$	3.50000	+	6.06218i	0.409644	+	0.709524i	0.994850	−	0.101361i	$-0.0323196\pi$
$73$	3.50000	+	6.06218i	0.409644	+	0.709524i	−0.585206	+	0.810885i	$0.698986\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	4.50000	+	7.79423i	0.506290	+	0.876919i	0.999974	+	0.00727784i	$0.00231663\pi$
$79$	4.50000	+	7.79423i	0.506290	+	0.876919i	−0.493684	+	0.869641i	$0.664350\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−0.500000	+	0.866025i	−0.0548821	+	0.0950586i	−0.892161	−	0.451717i	$-0.850812\pi$
$83$	−0.500000	+	0.866025i	−0.0548821	+	0.0950586i	0.837279	+	0.546776i	$0.184145\pi$
$84$		0			0
$85$	−7.00000	−	12.1244i	−0.759257	−	1.31507i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−7.50000	+	12.9904i	−0.794998	+	1.37698i	0.127842	+	0.991795i	$0.459195\pi$
$89$	−7.50000	+	12.9904i	−0.794998	+	1.37698i	−0.922840	+	0.385183i	$0.874138\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$	5.00000	−	8.66025i	0.512989	−	0.888523i
$96$		0			0
$97$	−8.50000	+	14.7224i	−0.863044	+	1.49484i	0.00593185	+	0.999982i	$0.498112\pi$
$97$	−8.50000	+	14.7224i	−0.863044	+	1.49484i	−0.868976	+	0.494854i	$0.835222\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	2.00000			0.199007			0.0995037	−	0.995037i	$-0.468274\pi$
$101$	2.00000			0.199007			0.0995037	+	0.995037i	$0.468274\pi$
$102$		0			0
$103$	−8.00000			−0.788263			−0.394132	−	0.919054i	$-0.628955\pi$
$103$	−8.00000			−0.788263			−0.394132	+	0.919054i	$0.628955\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−1.50000	+	2.59808i	−0.145010	+	0.251166i	−0.929377	−	0.369132i	$-0.879655\pi$
$107$	−1.50000	+	2.59808i	−0.145010	+	0.251166i	0.784366	+	0.620298i	$0.212988\pi$
$108$		0			0
$109$	−3.50000	−	6.06218i	−0.335239	−	0.580651i	0.648292	−	0.761392i	$-0.275484\pi$
$109$	−3.50000	−	6.06218i	−0.335239	−	0.580651i	−0.983531	+	0.180741i	$0.942150\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−0.500000	−	0.866025i	−0.0470360	−	0.0814688i	0.841549	−	0.540181i	$-0.181644\pi$
$113$	−0.500000	−	0.866025i	−0.0470360	−	0.0814688i	−0.888585	+	0.458712i	$0.848311\pi$
$114$		0			0
$115$	−8.00000			−0.746004
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	5.00000			0.454545
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−12.0000			−1.07331
$126$		0			0
$127$		0			0			−	1.00000i	$-0.5\pi$
$127$		0			0				1.00000i	$0.5\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−4.00000			−0.349482			−0.174741	−	0.984614i	$-0.555909\pi$
$131$	−4.00000			−0.349482			−0.174741	+	0.984614i	$0.555909\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−14.0000			−1.19610			−0.598050	−	0.801459i	$-0.704058\pi$
$137$	−14.0000			−1.19610			−0.598050	+	0.801459i	$0.704058\pi$
$138$		0			0
$139$	−2.50000	−	4.33013i	−0.212047	−	0.367277i	0.740308	−	0.672268i	$-0.234680\pi$
$139$	−2.50000	−	4.33013i	−0.212047	−	0.367277i	−0.952355	+	0.304991i	$0.901346\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−6.00000	−	10.3923i	−0.501745	−	0.869048i
$144$		0			0
$145$	−1.00000	+	1.73205i	−0.0830455	+	0.143839i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	6.00000			0.491539			0.245770	−	0.969328i	$-0.420959\pi$
$149$	6.00000			0.491539			0.245770	+	0.969328i	$0.420959\pi$
$150$		0			0
$151$	−8.00000			−0.651031			−0.325515	−	0.945537i	$-0.605538\pi$
$151$	−8.00000			−0.651031			−0.325515	+	0.945537i	$0.605538\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−3.00000	+	5.19615i	−0.240966	+	0.417365i
$156$		0			0
$157$	−6.50000	+	11.2583i	−0.518756	+	0.898513i	0.481006	+	0.876717i	$0.340272\pi$
$157$	−6.50000	+	11.2583i	−0.518756	+	0.898513i	−0.999762	+	0.0217953i	$0.993062\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	9.50000	−	16.4545i	0.744097	−	1.28881i	−0.206518	−	0.978443i	$-0.566213\pi$
$163$	9.50000	−	16.4545i	0.744097	−	1.28881i	0.950615	−	0.310372i	$-0.100454\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−11.5000	−	19.9186i	−0.889897	−	1.54135i	−0.839996	−	0.542592i	$-0.817443\pi$
$167$	−11.5000	−	19.9186i	−0.889897	−	1.54135i	−0.0499004	−	0.998754i	$-0.515890\pi$
$168$		0			0
$169$	2.00000	−	3.46410i	0.153846	−	0.266469i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	0.500000	+	0.866025i	0.0380143	+	0.0658427i	0.884407	−	0.466717i	$-0.154563\pi$
$173$	0.500000	+	0.866025i	0.0380143	+	0.0658427i	−0.846392	+	0.532560i	$0.821230\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−10.5000	−	18.1865i	−0.784807	−	1.35933i	−0.929114	−	0.369792i	$-0.879429\pi$
$179$	−10.5000	−	18.1865i	−0.784807	−	1.35933i	0.144308	−	0.989533i	$-0.453905\pi$
$180$		0			0
$181$	6.00000			0.445976			0.222988	−	0.974821i	$-0.428419\pi$
$181$	6.00000			0.445976			0.222988	+	0.974821i	$0.428419\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−11.0000	+	19.0526i	−0.808736	+	1.40077i
$186$		0			0
$187$	14.0000	+	24.2487i	1.02378	+	1.77324i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	7.50000	+	12.9904i	0.542681	+	0.939951i	0.998749	+	0.0500060i	$0.0159241\pi$
$191$	7.50000	+	12.9904i	0.542681	+	0.939951i	−0.456068	+	0.889945i	$0.650743\pi$
$192$		0			0
$193$	0.500000	−	0.866025i	0.0359908	−	0.0623379i	−0.847469	−	0.530845i	$-0.821875\pi$
$193$	0.500000	−	0.866025i	0.0359908	−	0.0623379i	0.883460	+	0.468507i	$0.155208\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	26.0000			1.85242			0.926212	−	0.377004i	$-0.123046\pi$
$197$	26.0000			1.85242			0.926212	+	0.377004i	$0.123046\pi$
$198$		0			0
$199$	−6.50000	−	11.2583i	−0.460773	−	0.798082i	0.538227	−	0.842800i	$-0.319094\pi$
$199$	−6.50000	−	11.2583i	−0.460773	−	0.798082i	−0.999000	+	0.0447181i	$0.985761\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$	9.00000	+	15.5885i	0.628587	+	1.08875i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−10.0000	+	17.3205i	−0.691714	+	1.19808i
$210$		0			0
$211$	6.50000	+	11.2583i	0.447478	+	0.775055i	0.998221	−	0.0596196i	$-0.0189888\pi$
$211$	6.50000	+	11.2583i	0.447478	+	0.775055i	−0.550743	+	0.834675i	$0.685655\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−5.00000	+	8.66025i	−0.340997	+	0.590624i
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$	10.5000	−	18.1865i	0.706306	−	1.22336i
$222$		0			0
$223$	−3.50000	+	6.06218i	−0.234377	+	0.405953i	−0.959092	−	0.283096i	$-0.908638\pi$
$223$	−3.50000	+	6.06218i	−0.234377	+	0.405953i	0.724714	+	0.689050i	$0.241972\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−12.0000			−0.796468			−0.398234	−	0.917284i	$-0.630377\pi$
$227$	−12.0000			−0.796468			−0.398234	+	0.917284i	$0.630377\pi$
$228$		0			0
$229$	14.0000			0.925146			0.462573	−	0.886581i	$-0.346926\pi$
$229$	14.0000			0.925146			0.462573	+	0.886581i	$0.346926\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−14.5000	+	25.1147i	−0.949927	+	1.64532i	−0.204354	+	0.978897i	$0.565509\pi$
$233$	−14.5000	+	25.1147i	−0.949927	+	1.64532i	−0.745573	+	0.666424i	$0.767824\pi$
$234$		0			0
$235$	−3.00000	−	5.19615i	−0.195698	−	0.338960i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−10.5000	−	18.1865i	−0.679189	−	1.17639i	−0.975226	−	0.221213i	$-0.928999\pi$
$239$	−10.5000	−	18.1865i	−0.679189	−	1.17639i	0.296037	−	0.955176i	$-0.404335\pi$
$240$		0			0
$241$	10.0000			0.644157			0.322078	−	0.946713i	$-0.395619\pi$
$241$	10.0000			0.644157			0.322078	+	0.946713i	$0.395619\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	15.0000			0.954427
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0			−	1.00000i	$-0.5\pi$
$251$		0			0				1.00000i	$0.5\pi$
$252$		0			0
$253$	16.0000			1.00591
$254$		0			0
$255$		0			0
$256$		0			0
$257$	18.0000			1.12281			0.561405	−	0.827541i	$-0.310261\pi$
$257$	18.0000			1.12281			0.561405	+	0.827541i	$0.310261\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0			−	1.00000i	$-0.5\pi$
$263$		0			0				1.00000i	$0.5\pi$
$264$		0			0
$265$	3.00000	+	5.19615i	0.184289	+	0.319197i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	0.500000	+	0.866025i	0.0304855	+	0.0528025i	0.880866	−	0.473366i	$-0.156961\pi$
$269$	0.500000	+	0.866025i	0.0304855	+	0.0528025i	−0.850380	+	0.526169i	$0.823628\pi$
$270$		0			0
$271$	−1.50000	+	2.59808i	−0.0911185	+	0.157822i	−0.907982	−	0.419009i	$-0.862378\pi$
$271$	−1.50000	+	2.59808i	−0.0911185	+	0.157822i	0.816864	+	0.576831i	$0.195711\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	4.00000			0.241209
$276$		0			0
$277$	2.00000			0.120168			0.0600842	−	0.998193i	$-0.480863\pi$
$277$	2.00000			0.120168			0.0600842	+	0.998193i	$0.480863\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−8.50000	+	14.7224i	−0.507067	+	0.878267i	0.492899	+	0.870087i	$0.335937\pi$
$281$	−8.50000	+	14.7224i	−0.507067	+	0.878267i	−0.999967	+	0.00818015i	$0.997396\pi$
$282$		0			0
$283$	0.500000	−	0.866025i	0.0297219	−	0.0514799i	−0.850782	−	0.525519i	$-0.823871\pi$
$283$	0.500000	−	0.866025i	0.0297219	−	0.0514799i	0.880504	+	0.474039i	$0.157204\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−16.0000	+	27.7128i	−0.941176	+	1.63017i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−13.5000	−	23.3827i	−0.788678	−	1.36603i	−0.926777	−	0.375613i	$-0.877432\pi$
$293$	−13.5000	−	23.3827i	−0.788678	−	1.36603i	0.138098	−	0.990419i	$-0.455901\pi$
$294$		0			0
$295$	7.00000	−	12.1244i	0.407556	−	0.705907i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−6.00000	−	10.3923i	−0.346989	−	0.601003i
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	3.00000	+	5.19615i	0.171780	+	0.297531i
$306$		0			0
$307$	8.00000			0.456584			0.228292	−	0.973593i	$-0.426686\pi$
$307$	8.00000			0.456584			0.228292	+	0.973593i	$0.426686\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	15.5000	−	26.8468i	0.878924	−	1.52234i	0.0264017	−	0.999651i	$-0.491595\pi$
$311$	15.5000	−	26.8468i	0.878924	−	1.52234i	0.852523	−	0.522690i	$-0.175072\pi$
$312$		0			0
$313$	−8.50000	−	14.7224i	−0.480448	−	0.832161i	0.519300	−	0.854592i	$-0.326193\pi$
$313$	−8.50000	−	14.7224i	−0.480448	−	0.832161i	−0.999748	+	0.0224310i	$0.992859\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−4.50000	−	7.79423i	−0.252745	−	0.437767i	0.711535	−	0.702650i	$-0.248000\pi$
$317$	−4.50000	−	7.79423i	−0.252745	−	0.437767i	−0.964281	+	0.264883i	$0.914667\pi$
$318$		0			0
$319$	2.00000	−	3.46410i	0.111979	−	0.193952i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−35.0000			−1.94745
$324$		0			0
$325$	−1.50000	−	2.59808i	−0.0832050	−	0.144115i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	12.5000	+	21.6506i	0.687062	+	1.19003i	0.972784	+	0.231714i	$0.0744333\pi$
$331$	12.5000	+	21.6506i	0.687062	+	1.19003i	−0.285722	+	0.958313i	$0.592233\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−13.0000	+	22.5167i	−0.710266	+	1.23022i
$336$		0			0
$337$	−1.50000	−	2.59808i	−0.0817102	−	0.141526i	0.822274	−	0.569091i	$-0.192705\pi$
$337$	−1.50000	−	2.59808i	−0.0817102	−	0.141526i	−0.903985	+	0.427565i	$0.859372\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	6.00000	−	10.3923i	0.324918	−	0.562775i
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−7.50000	+	12.9904i	−0.402621	+	0.697360i	−0.994041	−	0.109003i	$-0.965234\pi$
$347$	−7.50000	+	12.9904i	−0.402621	+	0.697360i	0.591420	+	0.806363i	$0.298567\pi$
$348$		0			0
$349$	−2.50000	+	4.33013i	−0.133822	+	0.231786i	−0.925147	−	0.379610i	$-0.876058\pi$
$349$	−2.50000	+	4.33013i	−0.133822	+	0.231786i	0.791325	+	0.611396i	$0.209392\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	6.00000			0.319348			0.159674	−	0.987170i	$-0.448956\pi$
$353$	6.00000			0.319348			0.159674	+	0.987170i	$0.448956\pi$
$354$		0			0
$355$	16.0000			0.849192
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−15.5000	+	26.8468i	−0.818059	+	1.41692i	0.0890519	+	0.996027i	$0.471616\pi$
$359$	−15.5000	+	26.8468i	−0.818059	+	1.41692i	−0.907111	+	0.420892i	$0.861717\pi$
$360$		0			0
$361$	−3.00000	−	5.19615i	−0.157895	−	0.273482i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	7.00000	+	12.1244i	0.366397	+	0.634618i
$366$		0			0
$367$	−32.0000			−1.67039			−0.835193	−	0.549957i	$-0.814644\pi$
$367$	−32.0000			−1.67039			−0.835193	+	0.549957i	$0.814644\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	22.0000			1.13912			0.569558	−	0.821951i	$-0.307114\pi$
$373$	22.0000			1.13912			0.569558	+	0.821951i	$0.307114\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−3.00000			−0.154508
$378$		0			0
$379$	−20.0000			−1.02733			−0.513665	−	0.857991i	$-0.671713\pi$
$379$	−20.0000			−1.02733			−0.513665	+	0.857991i	$0.671713\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	16.0000			0.817562			0.408781	−	0.912633i	$-0.365954\pi$
$383$	16.0000			0.817562			0.408781	+	0.912633i	$0.365954\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−26.0000			−1.31825			−0.659126	−	0.752032i	$-0.729074\pi$
$389$	−26.0000			−1.31825			−0.659126	+	0.752032i	$0.729074\pi$
$390$		0			0
$391$	14.0000	+	24.2487i	0.708010	+	1.22631i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	9.00000	+	15.5885i	0.452839	+	0.784340i
$396$		0			0
$397$	−10.5000	+	18.1865i	−0.526980	+	0.912756i	0.472526	+	0.881317i	$0.343342\pi$
$397$	−10.5000	+	18.1865i	−0.526980	+	0.912756i	−0.999506	+	0.0314391i	$0.989991\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−30.0000			−1.49813			−0.749064	−	0.662497i	$-0.769497\pi$
$401$	−30.0000			−1.49813			−0.749064	+	0.662497i	$0.769497\pi$
$402$		0			0
$403$	−9.00000			−0.448322
$404$		0			0
$405$		0			0
$406$		0			0
$407$	22.0000	−	38.1051i	1.09050	−	1.88880i
$408$		0			0
$409$	11.5000	−	19.9186i	0.568638	−	0.984911i	−0.428063	−	0.903749i	$-0.640804\pi$
$409$	11.5000	−	19.9186i	0.568638	−	0.984911i	0.996701	−	0.0811615i	$-0.0258630\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$	−1.00000	+	1.73205i	−0.0490881	+	0.0850230i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−7.50000	−	12.9904i	−0.366399	−	0.634622i	0.622601	−	0.782540i	$-0.286076\pi$
$419$	−7.50000	−	12.9904i	−0.366399	−	0.634622i	−0.989000	+	0.147918i	$0.952743\pi$
$420$		0			0
$421$	4.50000	−	7.79423i	0.219317	−	0.379867i	−0.735283	−	0.677761i	$-0.762951\pi$
$421$	4.50000	−	7.79423i	0.219317	−	0.379867i	0.954599	+	0.297893i	$0.0962839\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	3.50000	+	6.06218i	0.169775	+	0.294059i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$	19.5000	+	33.7750i	0.939282	+	1.62688i	0.766814	+	0.641869i	$0.221841\pi$
$431$	19.5000	+	33.7750i	0.939282	+	1.62688i	0.172468	+	0.985015i	$0.444826\pi$
$432$		0			0
$433$	−2.00000			−0.0961139			−0.0480569	−	0.998845i	$-0.515303\pi$
$433$	−2.00000			−0.0961139			−0.0480569	+	0.998845i	$0.515303\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−10.0000	+	17.3205i	−0.478365	+	0.828552i
$438$		0			0
$439$	−10.5000	−	18.1865i	−0.501138	−	0.867996i	−0.999999	−	0.00131415i	$-0.999582\pi$
$439$	−10.5000	−	18.1865i	−0.501138	−	0.867996i	0.498861	−	0.866682i	$-0.333752\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−14.5000	−	25.1147i	−0.688916	−	1.19324i	−0.972189	−	0.234198i	$-0.924754\pi$
$443$	−14.5000	−	25.1147i	−0.688916	−	1.19324i	0.283273	−	0.959039i	$-0.408580\pi$
$444$		0			0
$445$	−15.0000	+	25.9808i	−0.711068	+	1.23161i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−2.00000			−0.0943858			−0.0471929	−	0.998886i	$-0.515028\pi$
$449$	−2.00000			−0.0943858			−0.0471929	+	0.998886i	$0.515028\pi$
$450$		0			0
$451$	−18.0000	−	31.1769i	−0.847587	−	1.46806i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−15.5000	−	26.8468i	−0.725059	−	1.25584i	−0.958950	−	0.283577i	$-0.908479\pi$
$457$	−15.5000	−	26.8468i	−0.725059	−	1.25584i	0.233890	−	0.972263i	$-0.424854\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−11.5000	+	19.9186i	−0.535608	+	0.927701i	0.463525	+	0.886084i	$0.346584\pi$
$461$	−11.5000	+	19.9186i	−0.535608	+	0.927701i	−0.999134	+	0.0416172i	$0.986749\pi$
$462$		0			0
$463$	2.50000	+	4.33013i	0.116185	+	0.201238i	0.918253	−	0.395995i	$-0.129600\pi$
$463$	2.50000	+	4.33013i	0.116185	+	0.201238i	−0.802068	+	0.597233i	$0.796267\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−8.50000	+	14.7224i	−0.393333	+	0.681273i	−0.992887	−	0.119062i	$-0.962011\pi$
$467$	−8.50000	+	14.7224i	−0.393333	+	0.681273i	0.599554	+	0.800334i	$0.295345\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	10.0000	−	17.3205i	0.459800	−	0.796398i
$474$		0			0
$475$	−2.50000	+	4.33013i	−0.114708	+	0.198680i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−8.00000			−0.365529			−0.182765	−	0.983157i	$-0.558505\pi$
$479$	−8.00000			−0.365529			−0.182765	+	0.983157i	$0.558505\pi$
$480$		0			0
$481$	−33.0000			−1.50467
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−17.0000	+	29.4449i	−0.771930	+	1.33702i
$486$		0			0
$487$	0.500000	+	0.866025i	0.0226572	+	0.0392434i	0.877132	−	0.480250i	$-0.159454\pi$
$487$	0.500000	+	0.866025i	0.0226572	+	0.0392434i	−0.854475	+	0.519493i	$0.826121\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−4.50000	−	7.79423i	−0.203082	−	0.351749i	0.746438	−	0.665455i	$-0.231763\pi$
$491$	−4.50000	−	7.79423i	−0.203082	−	0.351749i	−0.949520	+	0.313707i	$0.898429\pi$
$492$		0			0
$493$	7.00000			0.315264
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	4.00000			0.179065			0.0895323	−	0.995984i	$-0.471463\pi$
$499$	4.00000			0.179065			0.0895323	+	0.995984i	$0.471463\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	24.0000			1.07011			0.535054	−	0.844818i	$-0.320291\pi$
$503$	24.0000			1.07011			0.535054	+	0.844818i	$0.320291\pi$
$504$		0			0
$505$	4.00000			0.177998
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−18.0000			−0.797836			−0.398918	−	0.916987i	$-0.630614\pi$
$509$	−18.0000			−0.797836			−0.398918	+	0.916987i	$0.630614\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−16.0000			−0.705044
$516$		0			0
$517$	6.00000	+	10.3923i	0.263880	+	0.457053i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−1.50000	−	2.59808i	−0.0657162	−	0.113824i	0.831295	−	0.555831i	$-0.187600\pi$
$521$	−1.50000	−	2.59808i	−0.0657162	−	0.113824i	−0.897011	+	0.442007i	$0.854267\pi$
$522$		0			0
$523$	14.5000	−	25.1147i	0.634041	−	1.09819i	−0.352677	−	0.935745i	$-0.614728\pi$
$523$	14.5000	−	25.1147i	0.634041	−	1.09819i	0.986718	−	0.162446i	$-0.0519382\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	21.0000			0.914774
$528$		0			0
$529$	−7.00000			−0.304348
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−13.5000	+	23.3827i	−0.584750	+	1.01282i
$534$		0			0
$535$	−3.00000	+	5.19615i	−0.129701	+	0.224649i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	16.5000	−	28.5788i	0.709390	−	1.22870i	−0.255693	−	0.966758i	$-0.582304\pi$
$541$	16.5000	−	28.5788i	0.709390	−	1.22870i	0.965084	−	0.261942i	$-0.0843630\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−7.00000	−	12.1244i	−0.299847	−	0.519350i
$546$		0			0
$547$	−16.5000	+	28.5788i	−0.705489	+	1.22194i	0.261026	+	0.965332i	$0.415939\pi$
$547$	−16.5000	+	28.5788i	−0.705489	+	1.22194i	−0.966515	+	0.256611i	$0.917394\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	2.50000	+	4.33013i	0.106504	+	0.184470i
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	5.50000	+	9.52628i	0.233042	+	0.403641i	0.958702	−	0.284413i	$-0.0917985\pi$
$557$	5.50000	+	9.52628i	0.233042	+	0.403641i	−0.725660	+	0.688054i	$0.758465\pi$
$558$		0			0
$559$	−15.0000			−0.634432
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−6.50000	+	11.2583i	−0.273942	+	0.474482i	−0.969868	−	0.243632i	$-0.921661\pi$
$563$	−6.50000	+	11.2583i	−0.273942	+	0.474482i	0.695925	+	0.718114i	$0.254994\pi$
$564$		0			0
$565$	−1.00000	−	1.73205i	−0.0420703	−	0.0728679i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	13.5000	+	23.3827i	0.565949	+	0.980253i	0.996961	+	0.0779066i	$0.0248236\pi$
$569$	13.5000	+	23.3827i	0.565949	+	0.980253i	−0.431011	+	0.902347i	$0.641843\pi$
$570$		0			0
$571$	1.50000	−	2.59808i	0.0627730	−	0.108726i	−0.832931	−	0.553377i	$-0.813339\pi$
$571$	1.50000	−	2.59808i	0.0627730	−	0.108726i	0.895704	+	0.444651i	$0.146672\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	4.00000			0.166812
$576$		0			0
$577$	3.50000	+	6.06218i	0.145707	+	0.252372i	0.929636	−	0.368478i	$-0.120121\pi$
$577$	3.50000	+	6.06218i	0.145707	+	0.252372i	−0.783930	+	0.620850i	$0.786788\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	−6.00000	−	10.3923i	−0.248495	−	0.430405i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−18.5000	+	32.0429i	−0.763577	+	1.32255i	0.177419	+	0.984135i	$0.443225\pi$
$587$	−18.5000	+	32.0429i	−0.763577	+	1.32255i	−0.940996	+	0.338418i	$0.890108\pi$
$588$		0			0
$589$	7.50000	+	12.9904i	0.309032	+	0.535259i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−13.5000	+	23.3827i	−0.554379	+	0.960212i	0.443573	+	0.896238i	$0.353711\pi$
$593$	−13.5000	+	23.3827i	−0.554379	+	0.960212i	−0.997952	+	0.0639736i	$0.979623\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−1.50000	+	2.59808i	−0.0612883	+	0.106155i	−0.895042	−	0.445983i	$-0.852854\pi$
$599$	−1.50000	+	2.59808i	−0.0612883	+	0.106155i	0.833753	+	0.552137i	$0.186188\pi$
$600$		0			0
$601$	17.5000	−	30.3109i	0.713840	−	1.23641i	−0.249565	−	0.968358i	$-0.580288\pi$
$601$	17.5000	−	30.3109i	0.713840	−	1.23641i	0.963405	−	0.268049i	$-0.0863789\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	10.0000			0.406558
$606$		0			0
$607$	48.0000			1.94826			0.974130	−	0.225989i	$-0.0725612\pi$
$607$	48.0000			1.94826			0.974130	+	0.225989i	$0.0725612\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	4.50000	−	7.79423i	0.182051	−	0.315321i
$612$		0			0
$613$	4.50000	+	7.79423i	0.181753	+	0.314806i	0.942478	−	0.334269i	$-0.108489\pi$
$613$	4.50000	+	7.79423i	0.181753	+	0.314806i	−0.760724	+	0.649075i	$0.775156\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	21.5000	+	37.2391i	0.865557	+	1.49919i	0.866493	+	0.499190i	$0.166369\pi$
$617$	21.5000	+	37.2391i	0.865557	+	1.49919i	−0.000935233		1.00000i	$0.500298\pi$
$618$		0			0
$619$	20.0000			0.803868			0.401934	−	0.915669i	$-0.368338\pi$
$619$	20.0000			0.803868			0.401934	+	0.915669i	$0.368338\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−19.0000			−0.760000
$626$		0			0
$627$		0			0
$628$		0			0
$629$	77.0000			3.07019
$630$		0			0
$631$	−16.0000			−0.636950			−0.318475	−	0.947931i	$-0.603171\pi$
$631$	−16.0000			−0.636950			−0.318475	+	0.947931i	$0.603171\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	2.00000			0.0789953			0.0394976	−	0.999220i	$-0.487424\pi$
$641$	2.00000			0.0789953			0.0394976	+	0.999220i	$0.487424\pi$
$642$		0			0
$643$	−6.50000	−	11.2583i	−0.256335	−	0.443985i	0.708922	−	0.705287i	$-0.249182\pi$
$643$	−6.50000	−	11.2583i	−0.256335	−	0.443985i	−0.965257	+	0.261301i	$0.915848\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−1.50000	−	2.59808i	−0.0589711	−	0.102141i	0.835033	−	0.550200i	$-0.185449\pi$
$647$	−1.50000	−	2.59808i	−0.0589711	−	0.102141i	−0.894004	+	0.448059i	$0.852115\pi$
$648$		0			0
$649$	−14.0000	+	24.2487i	−0.549548	+	0.951845i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	18.0000			0.704394			0.352197	−	0.935926i	$-0.385435\pi$
$653$	18.0000			0.704394			0.352197	+	0.935926i	$0.385435\pi$
$654$		0			0
$655$	−8.00000			−0.312586
$656$		0			0
$657$		0			0
$658$		0			0
$659$	12.5000	−	21.6506i	0.486931	−	0.843389i	−0.512956	−	0.858415i	$-0.671450\pi$
$659$	12.5000	−	21.6506i	0.486931	−	0.843389i	0.999887	+	0.0150258i	$0.00478303\pi$
$660$		0			0
$661$	−8.50000	+	14.7224i	−0.330612	+	0.572636i	−0.982632	−	0.185565i	$-0.940588\pi$
$661$	−8.50000	+	14.7224i	−0.330612	+	0.572636i	0.652020	+	0.758202i	$0.273922\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	2.00000	−	3.46410i	0.0774403	−	0.134131i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−6.00000	−	10.3923i	−0.231627	−	0.401190i
$672$		0			0
$673$	12.5000	−	21.6506i	0.481840	−	0.834571i	−0.517943	−	0.855415i	$-0.673302\pi$
$673$	12.5000	−	21.6506i	0.481840	−	0.834571i	0.999783	+	0.0208444i	$0.00663546\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−13.5000	−	23.3827i	−0.518847	−	0.898670i	−0.999760	−	0.0219013i	$-0.993028\pi$
$677$	−13.5000	−	23.3827i	−0.518847	−	0.898670i	0.480913	−	0.876768i	$-0.340305\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−10.5000	−	18.1865i	−0.401771	−	0.695888i	0.592168	−	0.805814i	$-0.298272\pi$
$683$	−10.5000	−	18.1865i	−0.401771	−	0.695888i	−0.993940	+	0.109926i	$0.964939\pi$
$684$		0			0
$685$	−28.0000			−1.06983
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−4.50000	+	7.79423i	−0.171436	+	0.296936i
$690$		0			0
$691$	5.50000	+	9.52628i	0.209230	+	0.362397i	0.951472	−	0.307735i	$-0.0995710\pi$
$691$	5.50000	+	9.52628i	0.209230	+	0.362397i	−0.742242	+	0.670132i	$0.766238\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−5.00000	−	8.66025i	−0.189661	−	0.328502i
$696$		0			0
$697$	31.5000	−	54.5596i	1.19315	−	2.06659i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	22.0000			0.830929			0.415464	−	0.909610i	$-0.363619\pi$
$701$	22.0000			0.830929			0.415464	+	0.909610i	$0.363619\pi$
$702$		0			0
$703$	27.5000	+	47.6314i	1.03718	+	1.79645i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	12.5000	+	21.6506i	0.469447	+	0.813107i	0.999390	−	0.0349269i	$-0.0111198\pi$
$709$	12.5000	+	21.6506i	0.469447	+	0.813107i	−0.529943	+	0.848034i	$0.677787\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	6.00000	−	10.3923i	0.224702	−	0.389195i
$714$		0			0
$715$	−12.0000	−	20.7846i	−0.448775	−	0.777300i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	7.50000	−	12.9904i	0.279703	−	0.484459i	−0.691608	−	0.722273i	$-0.743097\pi$
$719$	7.50000	−	12.9904i	0.279703	−	0.484459i	0.971311	+	0.237814i	$0.0764307\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$	0.500000	−	0.866025i	0.0185695	−	0.0321634i
$726$		0			0
$727$	−23.5000	+	40.7032i	−0.871567	+	1.50960i	−0.0111912	+	0.999937i	$0.503562\pi$
$727$	−23.5000	+	40.7032i	−0.871567	+	1.50960i	−0.860376	+	0.509661i	$0.829771\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	35.0000			1.29452
$732$		0			0
$733$	−30.0000			−1.10808			−0.554038	−	0.832492i	$-0.686914\pi$
$733$	−30.0000			−1.10808			−0.554038	+	0.832492i	$0.686914\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	26.0000	−	45.0333i	0.957722	−	1.65882i
$738$		0			0
$739$	10.5000	+	18.1865i	0.386249	+	0.669002i	0.991942	−	0.126696i	$-0.0404373\pi$
$739$	10.5000	+	18.1865i	0.386249	+	0.669002i	−0.605693	+	0.795699i	$0.707104\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−4.50000	−	7.79423i	−0.165089	−	0.285943i	0.771598	−	0.636111i	$-0.219458\pi$
$743$	−4.50000	−	7.79423i	−0.165089	−	0.285943i	−0.936687	+	0.350168i	$0.886124\pi$
$744$		0			0
$745$	12.0000			0.439646
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−24.0000			−0.875772			−0.437886	−	0.899030i	$-0.644273\pi$
$751$	−24.0000			−0.875772			−0.437886	+	0.899030i	$0.644273\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−16.0000			−0.582300
$756$		0			0
$757$	42.0000			1.52652			0.763258	−	0.646094i	$-0.223599\pi$
$757$	42.0000			1.52652			0.763258	+	0.646094i	$0.223599\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−2.00000			−0.0724999			−0.0362500	−	0.999343i	$-0.511541\pi$
$761$	−2.00000			−0.0724999			−0.0362500	+	0.999343i	$0.511541\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$	21.0000			0.758266
$768$		0			0
$769$	−2.50000	−	4.33013i	−0.0901523	−	0.156148i	0.817423	−	0.576038i	$-0.195402\pi$
$769$	−2.50000	−	4.33013i	−0.0901523	−	0.156148i	−0.907575	+	0.419890i	$0.862069\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−11.5000	−	19.9186i	−0.413626	−	0.716422i	0.581657	−	0.813434i	$-0.302405\pi$
$773$	−11.5000	−	19.9186i	−0.413626	−	0.716422i	−0.995283	+	0.0970125i	$0.969071\pi$
$774$		0			0
$775$	1.50000	−	2.59808i	0.0538816	−	0.0933257i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	45.0000			1.61229
$780$		0			0
$781$	−32.0000			−1.14505
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−13.0000	+	22.5167i	−0.463990	+	0.803654i
$786$		0			0
$787$	10.5000	−	18.1865i	0.374285	−	0.648280i	−0.615935	−	0.787797i	$-0.711222\pi$
$787$	10.5000	−	18.1865i	0.374285	−	0.648280i	0.990220	+	0.139517i	$0.0445550\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−4.50000	+	7.79423i	−0.159800	+	0.276781i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−7.50000	−	12.9904i	−0.265664	−	0.460143i	0.702074	−	0.712104i	$-0.252258\pi$
$797$	−7.50000	−	12.9904i	−0.265664	−	0.460143i	−0.967737	+	0.251961i	$0.918924\pi$
$798$		0			0
$799$	−10.5000	+	18.1865i	−0.371463	+	0.643393i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−14.0000	−	24.2487i	−0.494049	−	0.855718i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−0.500000	−	0.866025i	−0.0175791	−	0.0304478i	0.857102	−	0.515147i	$-0.172263\pi$
$809$	−0.500000	−	0.866025i	−0.0175791	−	0.0304478i	−0.874681	+	0.484699i	$0.838929\pi$
$810$		0			0
$811$	20.0000			0.702295			0.351147	−	0.936320i	$-0.385792\pi$
$811$	20.0000			0.702295			0.351147	+	0.936320i	$0.385792\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	19.0000	−	32.9090i	0.665541	−	1.15275i
$816$		0			0
$817$	12.5000	+	21.6506i	0.437320	+	0.757460i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−26.5000	−	45.8993i	−0.924856	−	1.60190i	−0.791792	−	0.610791i	$-0.790852\pi$
$821$	−26.5000	−	45.8993i	−0.924856	−	1.60190i	−0.133064	−	0.991107i	$-0.542482\pi$
$822$		0			0
$823$	11.5000	−	19.9186i	0.400865	−	0.694318i	−0.592966	−	0.805228i	$-0.702043\pi$
$823$	11.5000	−	19.9186i	0.400865	−	0.694318i	0.993831	+	0.110910i	$0.0353764\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−12.0000			−0.417281			−0.208640	−	0.977992i	$-0.566904\pi$
$827$	−12.0000			−0.417281			−0.208640	+	0.977992i	$0.566904\pi$
$828$		0			0
$829$	9.50000	+	16.4545i	0.329949	+	0.571488i	0.982501	−	0.186256i	$-0.0596352\pi$
$829$	9.50000	+	16.4545i	0.329949	+	0.571488i	−0.652553	+	0.757743i	$0.726302\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$	−23.0000	−	39.8372i	−0.795948	−	1.37862i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	11.5000	−	19.9186i	0.397024	−	0.687666i	−0.596333	−	0.802737i	$-0.703376\pi$
$839$	11.5000	−	19.9186i	0.397024	−	0.687666i	0.993357	+	0.115071i	$0.0367096\pi$
$840$		0			0
$841$	14.0000	+	24.2487i	0.482759	+	0.836162i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	4.00000	−	6.92820i	0.137604	−	0.238337i
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$	22.0000	−	38.1051i	0.754150	−	1.30623i
$852$		0			0
$853$	17.5000	−	30.3109i	0.599189	−	1.03783i	−0.393753	−	0.919216i	$-0.628823\pi$
$853$	17.5000	−	30.3109i	0.599189	−	1.03783i	0.992941	−	0.118609i	$-0.0378434\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−18.0000			−0.614868			−0.307434	−	0.951569i	$-0.599470\pi$
$857$	−18.0000			−0.614868			−0.307434	+	0.951569i	$0.599470\pi$
$858$		0			0
$859$	−44.0000			−1.50126			−0.750630	−	0.660722i	$-0.770250\pi$
$859$	−44.0000			−1.50126			−0.750630	+	0.660722i	$0.770250\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	20.5000	−	35.5070i	0.697828	−	1.20867i	−0.271390	−	0.962470i	$-0.587483\pi$
$863$	20.5000	−	35.5070i	0.697828	−	1.20867i	0.969218	−	0.246204i	$-0.0791834\pi$
$864$		0			0
$865$	1.00000	+	1.73205i	0.0340010	+	0.0588915i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−18.0000	−	31.1769i	−0.610608	−	1.05760i
$870$		0			0
$871$	−39.0000			−1.32146
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−38.0000			−1.28317			−0.641584	−	0.767052i	$-0.721723\pi$
$877$	−38.0000			−1.28317			−0.641584	+	0.767052i	$0.721723\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	30.0000			1.01073			0.505363	−	0.862907i	$-0.331359\pi$
$881$	30.0000			1.01073			0.505363	+	0.862907i	$0.331359\pi$
$882$		0			0
$883$	−28.0000			−0.942275			−0.471138	−	0.882060i	$-0.656156\pi$
$883$	−28.0000			−0.942275			−0.471138	+	0.882060i	$0.656156\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	12.0000			0.402921			0.201460	−	0.979497i	$-0.435431\pi$
$887$	12.0000			0.402921			0.201460	+	0.979497i	$0.435431\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−15.0000			−0.501956
$894$		0			0
$895$	−21.0000	−	36.3731i	−0.701953	−	1.21582i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−1.50000	−	2.59808i	−0.0500278	−	0.0866507i
$900$		0			0
$901$	10.5000	−	18.1865i	0.349806	−	0.605881i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	12.0000			0.398893
$906$		0			0
$907$	36.0000			1.19536			0.597680	−	0.801735i	$-0.296089\pi$
$907$	36.0000			1.19536			0.597680	+	0.801735i	$0.296089\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−19.5000	+	33.7750i	−0.646064	+	1.11902i	0.337991	+	0.941149i	$0.390253\pi$
$911$	−19.5000	+	33.7750i	−0.646064	+	1.11902i	−0.984055	+	0.177866i	$0.943081\pi$
$912$		0			0
$913$	2.00000	−	3.46410i	0.0661903	−	0.114645i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	17.5000	−	30.3109i	0.577272	−	0.999864i	−0.418519	−	0.908208i	$-0.637451\pi$
$919$	17.5000	−	30.3109i	0.577272	−	0.999864i	0.995791	−	0.0916559i	$-0.0292160\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	12.0000	+	20.7846i	0.394985	+	0.684134i
$924$		0			0
$925$	5.50000	−	9.52628i	0.180839	−	0.313222i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−1.50000	−	2.59808i	−0.0492134	−	0.0852401i	0.840369	−	0.542014i	$-0.182338\pi$
$929$	−1.50000	−	2.59808i	−0.0492134	−	0.0852401i	−0.889583	+	0.456774i	$0.849005\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$	28.0000	+	48.4974i	0.915698	+	1.58604i
$936$		0			0
$937$	34.0000			1.11073			0.555366	−	0.831606i	$-0.312578\pi$
$937$	34.0000			1.11073			0.555366	+	0.831606i	$0.312578\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−29.5000	+	51.0955i	−0.961673	+	1.66567i	−0.243372	+	0.969933i	$0.578253\pi$
$941$	−29.5000	+	51.0955i	−0.961673	+	1.66567i	−0.718301	+	0.695733i	$0.755080\pi$
$942$		0			0
$943$	−18.0000	−	31.1769i	−0.586161	−	1.01526i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−12.5000	−	21.6506i	−0.406195	−	0.703551i	0.588264	−	0.808669i	$-0.299811\pi$
$947$	−12.5000	−	21.6506i	−0.406195	−	0.703551i	−0.994460	+	0.105118i	$0.966478\pi$
$948$		0			0
$949$	−10.5000	+	18.1865i	−0.340844	+	0.590360i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	26.0000			0.842223			0.421111	−	0.907009i	$-0.361640\pi$
$953$	26.0000			0.842223			0.421111	+	0.907009i	$0.361640\pi$
$954$		0			0
$955$	15.0000	+	25.9808i	0.485389	+	0.840718i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	11.0000	+	19.0526i	0.354839	+	0.614599i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	1.00000	−	1.73205i	0.0321911	−	0.0557567i
$966$		0			0
$967$	−23.5000	−	40.7032i	−0.755709	−	1.30893i	−0.945021	−	0.327009i	$-0.893959\pi$
$967$	−23.5000	−	40.7032i	−0.755709	−	1.30893i	0.189312	−	0.981917i	$-0.439374\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	7.50000	−	12.9904i	0.240686	−	0.416881i	−0.720224	−	0.693742i	$-0.755961\pi$
$971$	7.50000	−	12.9904i	0.240686	−	0.416881i	0.960910	+	0.276861i	$0.0892941\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	1.50000	−	2.59808i	0.0479893	−	0.0831198i	−0.841033	−	0.540984i	$-0.818052\pi$
$977$	1.50000	−	2.59808i	0.0479893	−	0.0831198i	0.889022	+	0.457864i	$0.151385\pi$
$978$		0			0
$979$	30.0000	−	51.9615i	0.958804	−	1.66070i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−36.0000			−1.14822			−0.574111	−	0.818778i	$-0.694652\pi$
$983$	−36.0000			−1.14822			−0.574111	+	0.818778i	$0.694652\pi$
$984$		0			0
$985$	52.0000			1.65686
$986$		0			0
$987$		0			0
$988$		0			0
$989$	10.0000	−	17.3205i	0.317982	−	0.550760i
$990$		0			0
$991$	18.5000	+	32.0429i	0.587672	+	1.01788i	0.994537	+	0.104389i	$0.0332887\pi$
$991$	18.5000	+	32.0429i	0.587672	+	1.01788i	−0.406865	+	0.913488i	$0.633378\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−13.0000	−	22.5167i	−0.412128	−	0.713826i
$996$		0			0
$997$	22.0000			0.696747			0.348373	−	0.937356i	$-0.386734\pi$
$997$	22.0000			0.696747			0.348373	+	0.937356i	$0.386734\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5292.2.l.b.3313.1		2
3.2	odd	2		1764.2.l.b.961.1		2
7.2	even	3		5292.2.j.b.3529.1		2
7.3	odd	6		756.2.i.a.613.1		2
7.4	even	3		5292.2.i.b.2125.1		2
7.5	odd	6		5292.2.j.c.3529.1		2
7.6	odd	2		756.2.l.a.289.1		2
9.4	even	3		5292.2.i.b.1549.1		2
9.5	odd	6		1764.2.i.b.373.1		2
21.2	odd	6		1764.2.j.a.1177.1		2
21.5	even	6		1764.2.j.c.1177.1		2
21.11	odd	6		1764.2.i.b.1537.1		2
21.17	even	6		252.2.i.a.25.1	✓	2
21.20	even	2		252.2.l.a.205.1	yes	2
28.3	even	6		3024.2.q.e.2881.1		2
28.27	even	2		3024.2.t.b.289.1		2
63.4	even	3	inner	5292.2.l.b.361.1		2
63.5	even	6		1764.2.j.c.589.1		2
63.13	odd	6		756.2.i.a.37.1		2
63.20	even	6		2268.2.k.a.1297.1		2
63.23	odd	6		1764.2.j.a.589.1		2
63.31	odd	6		756.2.l.a.361.1		2
63.32	odd	6		1764.2.l.b.949.1		2
63.34	odd	6		2268.2.k.b.1297.1		2
63.38	even	6		2268.2.k.a.1621.1		2
63.40	odd	6		5292.2.j.c.1765.1		2
63.41	even	6		252.2.i.a.121.1	yes	2
63.52	odd	6		2268.2.k.b.1621.1		2
63.58	even	3		5292.2.j.b.1765.1		2
63.59	even	6		252.2.l.a.193.1	yes	2
84.59	odd	6		1008.2.q.f.529.1		2
84.83	odd	2		1008.2.t.b.961.1		2
252.31	even	6		3024.2.t.b.1873.1		2
252.59	odd	6		1008.2.t.b.193.1		2
252.139	even	6		3024.2.q.e.2305.1		2
252.167	odd	6		1008.2.q.f.625.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
252.2.i.a.25.1	✓	2	21.17	even	6
252.2.i.a.121.1	yes	2	63.41	even	6
252.2.l.a.193.1	yes	2	63.59	even	6
252.2.l.a.205.1	yes	2	21.20	even	2
756.2.i.a.37.1		2	63.13	odd	6
756.2.i.a.613.1		2	7.3	odd	6
756.2.l.a.289.1		2	7.6	odd	2
756.2.l.a.361.1		2	63.31	odd	6
1008.2.q.f.529.1		2	84.59	odd	6
1008.2.q.f.625.1		2	252.167	odd	6
1008.2.t.b.193.1		2	252.59	odd	6
1008.2.t.b.961.1		2	84.83	odd	2
1764.2.i.b.373.1		2	9.5	odd	6
1764.2.i.b.1537.1		2	21.11	odd	6
1764.2.j.a.589.1		2	63.23	odd	6
1764.2.j.a.1177.1		2	21.2	odd	6
1764.2.j.c.589.1		2	63.5	even	6
1764.2.j.c.1177.1		2	21.5	even	6
1764.2.l.b.949.1		2	63.32	odd	6
1764.2.l.b.961.1		2	3.2	odd	2
2268.2.k.a.1297.1		2	63.20	even	6
2268.2.k.a.1621.1		2	63.38	even	6
2268.2.k.b.1297.1		2	63.34	odd	6
2268.2.k.b.1621.1		2	63.52	odd	6
3024.2.q.e.2305.1		2	252.139	even	6
3024.2.q.e.2881.1		2	28.3	even	6
3024.2.t.b.289.1		2	28.27	even	2
3024.2.t.b.1873.1		2	252.31	even	6
5292.2.i.b.1549.1		2	9.4	even	3
5292.2.i.b.2125.1		2	7.4	even	3
5292.2.j.b.1765.1		2	63.58	even	3
5292.2.j.b.3529.1		2	7.2	even	3
5292.2.j.c.1765.1		2	63.40	odd	6
5292.2.j.c.3529.1		2	7.5	odd	6
5292.2.l.b.361.1		2	63.4	even	3	inner
5292.2.l.b.3313.1		2	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+2.00000 q^{5} -4.00000 q^{11} +(1.50000 + 2.59808i) q^{13} +(-3.50000 - 6.06218i) q^{17} +(2.50000 - 4.33013i) q^{19} -4.00000 q^{23} -1.00000 q^{25} +(-0.500000 + 0.866025i) q^{29} +(-1.50000 + 2.59808i) q^{31} +(-5.50000 + 9.52628i) q^{37} +(4.50000 + 7.79423i) q^{41} +(-2.50000 + 4.33013i) q^{43} +(-1.50000 - 2.59808i) q^{47} +(1.50000 + 2.59808i) q^{53} -8.00000 q^{55} +(3.50000 - 6.06218i) q^{59} +(1.50000 + 2.59808i) q^{61} +(3.00000 + 5.19615i) q^{65} +(-6.50000 + 11.2583i) q^{67} +8.00000 q^{71} +(3.50000 + 6.06218i) q^{73} +(4.50000 + 7.79423i) q^{79} +(-0.500000 + 0.866025i) q^{83} +(-7.00000 - 12.1244i) q^{85} +(-7.50000 + 12.9904i) q^{89} +(5.00000 - 8.66025i) q^{95} +(-8.50000 + 14.7224i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{5} - 8 q^{11} + 3 q^{13} - 7 q^{17} + 5 q^{19} - 8 q^{23} - 2 q^{25} - q^{29} - 3 q^{31} - 11 q^{37} + 9 q^{41} - 5 q^{43} - 3 q^{47} + 3 q^{53} - 16 q^{55} + 7 q^{59} + 3 q^{61} + 6 q^{65}+ \cdots - 17 q^{97}+O(q^{100}) \) 2 * q + 4 * q^5 - 8 * q^11 + 3 * q^13 - 7 * q^17 + 5 * q^19 - 8 * q^23 - 2 * q^25 - q^29 - 3 * q^31 - 11 * q^37 + 9 * q^41 - 5 * q^43 - 3 * q^47 + 3 * q^53 - 16 * q^55 + 7 * q^59 + 3 * q^61 + 6 * q^65 - 13 * q^67 + 16 * q^71 + 7 * q^73 + 9 * q^79 - q^83 - 14 * q^85 - 15 * q^89 + 10 * q^95 - 17 * q^97

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