LMFDB - Embedded newform 5292.2.l.g.3313.3

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 = [6,0,0,0,6] 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:	$6$
Relative dimension:	$3$ over $\Q(\zeta_{3})$
Coefficient field:	6.0.309123.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 3x^{5} + 10x^{4} - 15x^{3} + 19x^{2} - 12x + 3 $ x^6 - 3x^5 + 10x^4 - 15x^3 + 19x^2 - 12*x + 3
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 3^{3} $
Twist minimal:	no (minimal twist has level 252)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			3313.3
Root			$0.500000 + 1.41036i$ of defining polynomial
Character	$\chi$	$=$	5292.3313
Dual form			5292.2.l.g.361.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+3.94282 q^{5} -0.942820 q^{11} +(0.500000 + 0.866025i) q^{13} +(2.80150 + 4.85235i) q^{17} +(-0.641315 + 1.11079i) q^{19} +4.66019 q^{23} +10.5458 q^{25} +(-3.83009 + 6.63392i) q^{29} +(-3.91423 + 6.77965i) q^{31} +(4.91423 - 8.51170i) q^{37} +(0.471410 + 0.816506i) q^{41} +(-4.63160 + 8.02217i) q^{43} +(-2.64132 - 4.57489i) q^{47} +(-4.61273 - 7.98947i) q^{53} -3.71737 q^{55} +(-4.77292 + 8.26693i) q^{59} +(5.27292 + 9.13296i) q^{61} +(1.97141 + 3.41458i) q^{65} +(0.858685 - 1.48729i) q^{67} -3.54583 q^{71} +(-1.35868 - 2.35331i) q^{73} +(8.18715 + 14.1806i) q^{79} +(-0.198495 + 0.343803i) q^{83} +(11.0458 + 19.1319i) q^{85} +(2.75404 - 4.77014i) q^{89} +(-2.52859 + 4.37965i) q^{95} +(6.77292 - 11.7310i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 6 q^{5} + 12 q^{11} + 3 q^{13} - 3 q^{19} + 12 q^{23} + 12 q^{25} - 15 q^{29} + 3 q^{31} + 3 q^{37} - 6 q^{41} - 3 q^{43} - 15 q^{47} - 18 q^{53} - 24 q^{55} - 3 q^{59} + 6 q^{61} + 3 q^{65} + 6 q^{67}+ \cdots + 15 q^{97}+O(q^{100}) $ 6 * q + 6 * q^5 + 12 * q^11 + 3 * q^13 - 3 * q^19 + 12 * q^23 + 12 * q^25 - 15 * q^29 + 3 * q^31 + 3 * q^37 - 6 * q^41 - 3 * q^43 - 15 * q^47 - 18 * q^53 - 24 * q^55 - 3 * q^59 + 6 * q^61 + 3 * q^65 + 6 * q^67 + 30 * q^71 - 9 * q^73 - 3 * q^79 - 18 * q^83 + 15 * q^85 + 6 * q^89 - 24 * q^95 + 15 * 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
$5$	3.94282			1.76328			0.881641	−	0.471920i	$-0.156439\pi$
$5$	3.94282			1.76328			0.881641	+	0.471920i	$0.156439\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−0.942820			−0.284271			−0.142135	−	0.989847i	$-0.545397\pi$
$11$	−0.942820			−0.284271			−0.142135	+	0.989847i	$0.545397\pi$
$12$		0			0
$13$	0.500000	+	0.866025i	0.138675	+	0.240192i	0.926995	−	0.375073i	$-0.122382\pi$
$13$	0.500000	+	0.866025i	0.138675	+	0.240192i	−0.788320	+	0.615265i	$0.789049\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	2.80150	+	4.85235i	0.679465	+	1.17687i	0.975142	+	0.221580i	$0.0711213\pi$
$17$	2.80150	+	4.85235i	0.679465	+	1.17687i	−0.295678	+	0.955288i	$0.595545\pi$
$18$		0			0
$19$	−0.641315	+	1.11079i	−0.147128	+	0.254833i	−0.930165	−	0.367142i	$-0.880336\pi$
$19$	−0.641315	+	1.11079i	−0.147128	+	0.254833i	0.783037	+	0.621975i	$0.213670\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	4.66019			0.971717			0.485858	−	0.874038i	$-0.338507\pi$
$23$	4.66019			0.971717			0.485858	+	0.874038i	$0.338507\pi$
$24$		0			0
$25$	10.5458			2.10917
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−3.83009	+	6.63392i	−0.711231	+	1.23189i	0.253165	+	0.967423i	$0.418529\pi$
$29$	−3.83009	+	6.63392i	−0.711231	+	1.23189i	−0.964395	+	0.264465i	$0.914805\pi$
$30$		0			0
$31$	−3.91423	+	6.77965i	−0.703016	+	1.21766i	0.264386	+	0.964417i	$0.414831\pi$
$31$	−3.91423	+	6.77965i	−0.703016	+	1.21766i	−0.967403	+	0.253243i	$0.918503\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	4.91423	−	8.51170i	0.807894	−	1.39931i	−0.106425	−	0.994321i	$-0.533940\pi$
$37$	4.91423	−	8.51170i	0.807894	−	1.39931i	0.914320	−	0.404994i	$-0.132726\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	0.471410	+	0.816506i	0.0736219	+	0.127517i	0.900486	−	0.434885i	$-0.143211\pi$
$41$	0.471410	+	0.816506i	0.0736219	+	0.127517i	−0.826864	+	0.562402i	$0.809878\pi$
$42$		0			0
$43$	−4.63160	+	8.02217i	−0.706312	+	1.22337i	0.259903	+	0.965635i	$0.416309\pi$
$43$	−4.63160	+	8.02217i	−0.706312	+	1.22337i	−0.966216	+	0.257734i	$0.917024\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−2.64132	−	4.57489i	−0.385275	−	0.667317i	0.606532	−	0.795059i	$-0.292560\pi$
$47$	−2.64132	−	4.57489i	−0.385275	−	0.667317i	−0.991807	+	0.127743i	$0.959227\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−4.61273	−	7.98947i	−0.633607	−	1.09744i	−0.986809	−	0.161892i	$-0.948241\pi$
$53$	−4.61273	−	7.98947i	−0.633607	−	1.09744i	0.353202	−	0.935547i	$-0.385093\pi$
$54$		0			0
$55$	−3.71737			−0.501250
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−4.77292	+	8.26693i	−0.621381	+	1.07626i	0.367848	+	0.929886i	$0.380095\pi$
$59$	−4.77292	+	8.26693i	−0.621381	+	1.07626i	−0.989229	+	0.146377i	$0.953239\pi$
$60$		0			0
$61$	5.27292	+	9.13296i	0.675128	+	1.16936i	0.976432	+	0.215827i	$0.0692448\pi$
$61$	5.27292	+	9.13296i	0.675128	+	1.16936i	−0.301304	+	0.953528i	$0.597422\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	1.97141	+	3.41458i	0.244523	+	0.423527i
$66$		0			0
$67$	0.858685	−	1.48729i	0.104905	−	0.181701i	−0.808794	−	0.588092i	$-0.799879\pi$
$67$	0.858685	−	1.48729i	0.104905	−	0.181701i	0.913699	+	0.406391i	$0.133213\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−3.54583			−0.420813			−0.210406	−	0.977614i	$-0.567479\pi$
$71$	−3.54583			−0.420813			−0.210406	+	0.977614i	$0.567479\pi$
$72$		0			0
$73$	−1.35868	−	2.35331i	−0.159022	−	0.275434i	0.775494	−	0.631355i	$-0.217501\pi$
$73$	−1.35868	−	2.35331i	−0.159022	−	0.275434i	−0.934516	+	0.355920i	$0.884167\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	8.18715	+	14.1806i	0.921126	+	1.59544i	0.797676	+	0.603086i	$0.206062\pi$
$79$	8.18715	+	14.1806i	0.921126	+	1.59544i	0.123449	+	0.992351i	$0.460604\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−0.198495	+	0.343803i	−0.0217877	+	0.0377373i	−0.876714	−	0.481013i	$-0.840269\pi$
$83$	−0.198495	+	0.343803i	−0.0217877	+	0.0377373i	0.854926	+	0.518750i	$0.173602\pi$
$84$		0			0
$85$	11.0458	+	19.1319i	1.19809	+	2.07515i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	2.75404	−	4.77014i	0.291928	−	0.505634i	−0.682338	−	0.731037i	$-0.739037\pi$
$89$	2.75404	−	4.77014i	0.291928	−	0.505634i	0.974266	+	0.225403i	$0.0723700\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−2.52859	+	4.37965i	−0.259428	+	0.449342i
$96$		0			0
$97$	6.77292	−	11.7310i	0.687685	−	1.19111i	−0.284899	−	0.958557i	$-0.591960\pi$
$97$	6.77292	−	11.7310i	0.687685	−	1.19111i	0.972585	−	0.232549i	$-0.0747064\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	2.60301			0.259009			0.129505	−	0.991579i	$-0.458661\pi$
$101$	2.60301			0.259009			0.129505	+	0.991579i	$0.458661\pi$
$102$		0			0
$103$	15.9806			1.57461			0.787306	−	0.616562i	$-0.211475\pi$
$103$	15.9806			1.57461			0.787306	+	0.616562i	$0.211475\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−5.44282	+	9.42724i	−0.526177	+	0.911366i	0.473358	+	0.880870i	$0.343042\pi$
$107$	−5.44282	+	9.42724i	−0.526177	+	0.911366i	−0.999535	+	0.0304954i	$0.990291\pi$
$108$		0			0
$109$	1.64132	+	2.84284i	0.157209	+	0.272295i	0.933861	−	0.357635i	$-0.116417\pi$
$109$	1.64132	+	2.84284i	0.157209	+	0.272295i	−0.776652	+	0.629930i	$0.783084\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−3.22545	−	5.58664i	−0.303425	−	0.525547i	0.673485	−	0.739201i	$-0.264797\pi$
$113$	−3.22545	−	5.58664i	−0.303425	−	0.525547i	−0.976909	+	0.213654i	$0.931463\pi$
$114$		0			0
$115$	18.3743			1.71341
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−10.1111			−0.919190
$122$		0			0
$123$		0			0
$124$		0			0
$125$	21.8662			1.95577
$126$		0			0
$127$	−0.828460			−0.0735140			−0.0367570	−	0.999324i	$-0.511703\pi$
$127$	−0.828460			−0.0735140			−0.0367570	+	0.999324i	$0.511703\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−5.43147			−0.474550			−0.237275	−	0.971443i	$-0.576254\pi$
$131$	−5.43147			−0.474550			−0.237275	+	0.971443i	$0.576254\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	21.1488			1.80687			0.903434	−	0.428728i	$-0.141038\pi$
$137$	21.1488			1.80687			0.903434	+	0.428728i	$0.141038\pi$
$138$		0			0
$139$	0.923945	+	1.60032i	0.0783680	+	0.135737i	0.902546	−	0.430594i	$-0.141696\pi$
$139$	0.923945	+	1.60032i	0.0783680	+	0.135737i	−0.824178	+	0.566331i	$0.808362\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−0.471410	−	0.816506i	−0.0394213	−	0.0682797i
$144$		0			0
$145$	−15.1014	+	26.1563i	−1.25410	+	2.17217i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−12.1488			−0.995272			−0.497636	−	0.867386i	$-0.665798\pi$
$149$	−12.1488			−0.995272			−0.497636	+	0.867386i	$0.665798\pi$
$150$		0			0
$151$	−8.98057			−0.730828			−0.365414	−	0.930845i	$-0.619073\pi$
$151$	−8.98057			−0.730828			−0.365414	+	0.930845i	$0.619073\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−15.4331	+	26.7309i	−1.23962	+	2.14708i
$156$		0			0
$157$	5.90451	−	10.2269i	0.471232	−	0.816197i	−0.528227	−	0.849103i	$-0.677143\pi$
$157$	5.90451	−	10.2269i	0.471232	−	0.816197i	0.999458	+	0.0329062i	$0.0104763\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	6.14132	−	10.6371i	0.481025	−	0.833160i	−0.518738	−	0.854933i	$-0.673598\pi$
$163$	6.14132	−	10.6371i	0.481025	−	0.833160i	0.999763	+	0.0217737i	$0.00693132\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−11.4428	−	19.8195i	−0.885472	−	1.53368i	−0.845172	−	0.534495i	$-0.820502\pi$
$167$	−11.4428	−	19.8195i	−0.885472	−	1.53368i	−0.0403003	−	0.999188i	$-0.512831\pi$
$168$		0			0
$169$	6.00000	−	10.3923i	0.461538	−	0.799408i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	9.63160	+	16.6824i	0.732277	+	1.26834i	0.955908	+	0.293667i	$0.0948757\pi$
$173$	9.63160	+	16.6824i	0.732277	+	1.26834i	−0.223631	+	0.974674i	$0.571791\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−2.41586	−	4.18440i	−0.180570	−	0.312757i	0.761505	−	0.648159i	$-0.224461\pi$
$179$	−2.41586	−	4.18440i	−0.180570	−	0.312757i	−0.942075	+	0.335403i	$0.891128\pi$
$180$		0			0
$181$	3.26320			0.242552			0.121276	−	0.992619i	$-0.461301\pi$
$181$	3.26320			0.242552			0.121276	+	0.992619i	$0.461301\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	19.3759	−	33.5601i	1.42455	−	2.46739i
$186$		0			0
$187$	−2.64132	−	4.57489i	−0.193152	−	0.334549i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	7.34897	+	12.7288i	0.531753	+	0.921023i	0.999313	+	0.0370616i	$0.0117998\pi$
$191$	7.34897	+	12.7288i	0.531753	+	0.921023i	−0.467560	+	0.883961i	$0.654867\pi$
$192$		0			0
$193$	−5.05555	+	8.75646i	−0.363906	+	0.630304i	−0.988600	−	0.150566i	$-0.951890\pi$
$193$	−5.05555	+	8.75646i	−0.363906	+	0.630304i	0.624694	+	0.780870i	$0.285224\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−16.6375			−1.18537			−0.592686	−	0.805434i	$-0.701933\pi$
$197$	−16.6375			−1.18537			−0.592686	+	0.805434i	$0.701933\pi$
$198$		0			0
$199$	−6.55555	−	11.3545i	−0.464710	−	0.804902i	0.534478	−	0.845182i	$-0.320508\pi$
$199$	−6.55555	−	11.3545i	−0.464710	−	0.804902i	−0.999188	+	0.0402805i	$0.987175\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$	1.85868	+	3.21934i	0.129816	+	0.224848i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	0.604645	−	1.04728i	0.0418242	−	0.0724416i
$210$		0			0
$211$	−1.06526	−	1.84509i	−0.0733355	−	0.127021i	0.827026	−	0.562164i	$-0.190031\pi$
$211$	−1.06526	−	1.84509i	−0.0733355	−	0.127021i	−0.900361	+	0.435143i	$0.856698\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−18.2616	+	31.6300i	−1.24543	+	2.15715i
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−2.80150	+	4.85235i	−0.188450	+	0.326404i
$222$		0			0
$223$	7.04583	−	12.2037i	0.471824	−	0.817223i	−0.527657	−	0.849458i	$-0.676929\pi$
$223$	7.04583	−	12.2037i	0.471824	−	0.817223i	0.999480	+	0.0322352i	$0.0102626\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−4.79071			−0.317971			−0.158985	−	0.987281i	$-0.550822\pi$
$227$	−4.79071			−0.317971			−0.158985	+	0.987281i	$0.550822\pi$
$228$		0			0
$229$	18.0917			1.19553			0.597765	−	0.801671i	$-0.296055\pi$
$229$	18.0917			1.19553			0.597765	+	0.801671i	$0.296055\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−1.57442	+	2.72698i	−0.103144	+	0.178650i	−0.912978	−	0.408008i	$-0.866223\pi$
$233$	−1.57442	+	2.72698i	−0.103144	+	0.178650i	0.809835	+	0.586658i	$0.199557\pi$
$234$		0			0
$235$	−10.4142	−	18.0380i	−0.679349	−	1.17667i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−6.16019	−	10.6698i	−0.398470	−	0.690170i	0.595068	−	0.803676i	$-0.297125\pi$
$239$	−6.16019	−	10.6698i	−0.398470	−	0.690170i	−0.993537	+	0.113506i	$0.963792\pi$
$240$		0			0
$241$	14.5458			0.936979			0.468490	−	0.883469i	$-0.344798\pi$
$241$	14.5458			0.936979			0.468490	+	0.883469i	$0.344798\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−1.28263			−0.0816118
$248$		0			0
$249$		0			0
$250$		0			0
$251$	2.28263			0.144078			0.0720392	−	0.997402i	$-0.477049\pi$
$251$	2.28263			0.144078			0.0720392	+	0.997402i	$0.477049\pi$
$252$		0			0
$253$	−4.39372			−0.276231
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−0.793980			−0.0495271			−0.0247636	−	0.999693i	$-0.507883\pi$
$257$	−0.793980			−0.0495271			−0.0247636	+	0.999693i	$0.507883\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−10.4887			−0.646758			−0.323379	−	0.946270i	$-0.604819\pi$
$263$	−10.4887			−0.646758			−0.323379	+	0.946270i	$0.604819\pi$
$264$		0			0
$265$	−18.1871	−	31.5011i	−1.11723	−	1.93509i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	5.16991	+	8.95454i	0.315215	+	0.545968i	0.979483	−	0.201526i	$-0.0645901\pi$
$269$	5.16991	+	8.95454i	0.315215	+	0.545968i	−0.664268	+	0.747494i	$0.731257\pi$
$270$		0			0
$271$	15.4142	−	26.6982i	0.936348	−	1.62180i	0.164135	−	0.986438i	$-0.447517\pi$
$271$	15.4142	−	26.6982i	0.936348	−	1.62180i	0.772213	−	0.635364i	$-0.219150\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−9.94282			−0.599575
$276$		0			0
$277$	14.2438			0.855825			0.427913	−	0.903820i	$-0.359249\pi$
$277$	14.2438			0.855825			0.427913	+	0.903820i	$0.359249\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−4.46169	+	7.72788i	−0.266162	+	0.461007i	−0.967868	−	0.251461i	$-0.919089\pi$
$281$	−4.46169	+	7.72788i	−0.266162	+	0.461007i	0.701705	+	0.712468i	$0.252422\pi$
$282$		0			0
$283$	11.6316	−	20.1465i	0.691427	−	1.19759i	−0.279944	−	0.960016i	$-0.590316\pi$
$283$	11.6316	−	20.1465i	0.691427	−	1.19759i	0.971370	−	0.237570i	$-0.0763509\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−7.19686	+	12.4653i	−0.423345	+	0.733255i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	11.9412	+	20.6827i	0.697611	+	1.20830i	0.969292	+	0.245911i	$0.0790871\pi$
$293$	11.9412	+	20.6827i	0.697611	+	1.20830i	−0.271681	+	0.962387i	$0.587580\pi$
$294$		0			0
$295$	−18.8187	+	32.5950i	−1.09567	+	1.89776i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	2.33009	+	4.03584i	0.134753	+	0.233399i
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	20.7902	+	36.0096i	1.19044	+	2.06190i
$306$		0			0
$307$	−7.37429			−0.420873			−0.210436	−	0.977608i	$-0.567489\pi$
$307$	−7.37429			−0.420873			−0.210436	+	0.977608i	$0.567489\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	4.58577	−	7.94279i	0.260035	−	0.450394i	−0.706216	−	0.707997i	$-0.749599\pi$
$311$	4.58577	−	7.94279i	0.260035	−	0.450394i	0.966251	+	0.257603i	$0.0829325\pi$
$312$		0			0
$313$	−8.05555	−	13.9526i	−0.455326	−	0.788648i	0.543381	−	0.839486i	$-0.317144\pi$
$313$	−8.05555	−	13.9526i	−0.455326	−	0.788648i	−0.998707	+	0.0508381i	$0.983811\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−2.13160	−	3.69204i	−0.119723	−	0.207366i	0.799935	−	0.600086i	$-0.204867\pi$
$317$	−2.13160	−	3.69204i	−0.119723	−	0.207366i	−0.919658	+	0.392721i	$0.871534\pi$
$318$		0			0
$319$	3.61109	−	6.25459i	0.202182	−	0.350190i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−7.18659			−0.399873
$324$		0			0
$325$	5.27292	+	9.13296i	0.292489	+	0.506605i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	6.14132	+	10.6371i	0.337557	+	0.584666i	0.983973	−	0.178319i	$-0.0570660\pi$
$331$	6.14132	+	10.6371i	0.337557	+	0.584666i	−0.646415	+	0.762986i	$0.723733\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	3.38564	−	5.86410i	0.184977	−	0.320390i
$336$		0			0
$337$	13.4903	+	23.3659i	0.734863	+	1.27282i	0.954784	+	0.297302i	$0.0960867\pi$
$337$	13.4903	+	23.3659i	0.734863	+	1.27282i	−0.219921	+	0.975518i	$0.570580\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	3.69041	−	6.39199i	0.199847	−	0.346145i
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−10.7524	+	18.6237i	−0.577219	+	0.999773i	0.418577	+	0.908181i	$0.362529\pi$
$347$	−10.7524	+	18.6237i	−0.577219	+	0.999773i	−0.995797	+	0.0915921i	$0.970804\pi$
$348$		0			0
$349$	2.54583	−	4.40951i	0.136275	−	0.236035i	−0.789809	−	0.613353i	$-0.789820\pi$
$349$	2.54583	−	4.40951i	0.136275	−	0.236035i	0.926084	+	0.377318i	$0.123154\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−9.56853			−0.509281			−0.254641	−	0.967036i	$-0.581957\pi$
$353$	−9.56853			−0.509281			−0.254641	+	0.967036i	$0.581957\pi$
$354$		0			0
$355$	−13.9806			−0.742012
$356$		0			0
$357$		0			0
$358$		0			0
$359$	5.48113	−	9.49359i	0.289283	−	0.501052i	−0.684356	−	0.729148i	$-0.739917\pi$
$359$	5.48113	−	9.49359i	0.289283	−	0.501052i	0.973639	+	0.228096i	$0.0732499\pi$
$360$		0			0
$361$	8.67743	+	15.0297i	0.456707	+	0.791039i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−5.35705	−	9.27868i	−0.280401	−	0.485668i
$366$		0			0
$367$	4.69794			0.245230			0.122615	−	0.992454i	$-0.460872\pi$
$367$	4.69794			0.245230			0.122615	+	0.992454i	$0.460872\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	9.26320			0.479630			0.239815	−	0.970819i	$-0.422913\pi$
$373$	9.26320			0.479630			0.239815	+	0.970819i	$0.422913\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−7.66019			−0.394520
$378$		0			0
$379$	−13.5458			−0.695803			−0.347901	−	0.937531i	$-0.613106\pi$
$379$	−13.5458			−0.695803			−0.347901	+	0.937531i	$0.613106\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−12.2794			−0.627446			−0.313723	−	0.949515i	$-0.601576\pi$
$383$	−12.2794			−0.627446			−0.313723	+	0.949515i	$0.601576\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	7.51135			0.380841			0.190420	−	0.981703i	$-0.439015\pi$
$389$	7.51135			0.380841			0.190420	+	0.981703i	$0.439015\pi$
$390$		0			0
$391$	13.0555	+	22.6129i	0.660247	+	1.14358i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	32.2804	+	55.9114i	1.62421	+	2.81321i
$396$		0			0
$397$	11.4601	−	19.8494i	0.575164	−	0.996213i	−0.420860	−	0.907126i	$-0.638272\pi$
$397$	11.4601	−	19.8494i	0.575164	−	0.996213i	0.996024	−	0.0890875i	$-0.0283951\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	21.1261			1.05499			0.527495	−	0.849558i	$-0.323132\pi$
$401$	21.1261			1.05499			0.527495	+	0.849558i	$0.323132\pi$
$402$		0			0
$403$	−7.82846			−0.389963
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−4.63323	+	8.02500i	−0.229661	+	0.397784i
$408$		0			0
$409$	−1.71737	+	2.97457i	−0.0849185	+	0.147083i	−0.905356	−	0.424652i	$-0.860396\pi$
$409$	−1.71737	+	2.97457i	−0.0849185	+	0.147083i	0.820438	+	0.571736i	$0.193730\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$	−0.782630	+	1.35556i	−0.0384178	+	0.0665416i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−10.2157	−	17.6942i	−0.499071	−	0.864417i	0.500928	−	0.865489i	$-0.332992\pi$
$419$	−10.2157	−	17.6942i	−0.499071	−	0.864417i	−0.999999	+	0.00107202i	$0.999659\pi$
$420$		0			0
$421$	−7.90451	+	13.6910i	−0.385242	+	0.667260i	−0.991803	−	0.127778i	$-0.959216\pi$
$421$	−7.90451	+	13.6910i	−0.385242	+	0.667260i	0.606560	+	0.795037i	$0.292549\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	29.5442	+	51.1720i	1.43310	+	2.48221i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−13.3759	−	23.1678i	−0.644296	−	1.11595i	−0.984464	−	0.175588i	$-0.943817\pi$
$431$	−13.3759	−	23.1678i	−0.644296	−	1.11595i	0.340168	−	0.940365i	$-0.389516\pi$
$432$		0			0
$433$	−14.4347			−0.693689			−0.346845	−	0.937923i	$-0.612747\pi$
$433$	−14.4347			−0.693689			−0.346845	+	0.937923i	$0.612747\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−2.98865	+	5.17649i	−0.142967	+	0.247625i
$438$		0			0
$439$	19.3187	+	33.4610i	0.922033	+	1.59701i	0.796264	+	0.604949i	$0.206807\pi$
$439$	19.3187	+	33.4610i	0.922033	+	1.59701i	0.125769	+	0.992060i	$0.459860\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	5.28263	+	9.14978i	0.250985	+	0.434719i	0.963797	−	0.266636i	$-0.0859121\pi$
$443$	5.28263	+	9.14978i	0.250985	+	0.434719i	−0.712812	+	0.701355i	$0.752579\pi$
$444$		0			0
$445$	10.8587	−	18.8078i	0.514751	−	0.891575i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	9.37429			0.442400			0.221200	−	0.975228i	$-0.429003\pi$
$449$	9.37429			0.442400			0.221200	+	0.975228i	$0.429003\pi$
$450$		0			0
$451$	−0.444455	−	0.769818i	−0.0209286	−	0.0362493i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−10.3587	−	17.9418i	−0.484559	−	0.839281i	0.515284	−	0.857020i	$-0.327687\pi$
$457$	−10.3587	−	17.9418i	−0.484559	−	0.839281i	−0.999843	+	0.0177391i	$0.994353\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	17.6300	−	30.5360i	0.821109	−	1.42220i	−0.0837475	−	0.996487i	$-0.526689\pi$
$461$	17.6300	−	30.5360i	0.821109	−	1.42220i	0.904857	−	0.425716i	$-0.139978\pi$
$462$		0			0
$463$	−3.55555	−	6.15838i	−0.165240	−	0.286204i	0.771500	−	0.636229i	$-0.219507\pi$
$463$	−3.55555	−	6.15838i	−0.165240	−	0.286204i	−0.936741	+	0.350024i	$0.886173\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	2.51724	−	4.35999i	0.116484	−	0.201756i	−0.801888	−	0.597474i	$-0.796171\pi$
$467$	2.51724	−	4.35999i	0.116484	−	0.201756i	0.918372	+	0.395718i	$0.129504\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	4.36677	−	7.56346i	0.200784	−	0.347768i
$474$		0			0
$475$	−6.76320	+	11.7142i	−0.310317	+	0.537485i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−6.26647			−0.286322			−0.143161	−	0.989699i	$-0.545727\pi$
$479$	−6.26647			−0.286322			−0.143161	+	0.989699i	$0.545727\pi$
$480$		0			0
$481$	9.82846			0.448139
$482$		0			0
$483$		0			0
$484$		0			0
$485$	26.7044	−	46.2534i	1.21258	−	2.10026i
$486$		0			0
$487$	−7.03611	−	12.1869i	−0.318837	−	0.552242i	0.661409	−	0.750026i	$-0.269959\pi$
$487$	−7.03611	−	12.1869i	−0.318837	−	0.552242i	−0.980246	+	0.197784i	$0.936626\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	1.97141	+	3.41458i	0.0889685	+	0.154098i	0.907075	−	0.420968i	$-0.138310\pi$
$491$	1.97141	+	3.41458i	0.0889685	+	0.154098i	−0.818107	+	0.575066i	$0.804976\pi$
$492$		0			0
$493$	−42.9201			−1.93302
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	19.6569			0.879965			0.439982	−	0.898006i	$-0.354985\pi$
$499$	19.6569			0.879965			0.439982	+	0.898006i	$0.354985\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	10.2632			0.457613			0.228807	−	0.973472i	$-0.426518\pi$
$503$	10.2632			0.457613			0.228807	+	0.973472i	$0.426518\pi$
$504$		0			0
$505$	10.2632			0.456706
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−1.33981			−0.0593860			−0.0296930	−	0.999559i	$-0.509453\pi$
$509$	−1.33981			−0.0593860			−0.0296930	+	0.999559i	$0.509453\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$	63.0085			2.77649
$516$		0			0
$517$	2.49028	+	4.31330i	0.109523	+	0.189699i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	7.89699	+	13.6780i	0.345973	+	0.599244i	0.985530	−	0.169500i	$-0.0542154\pi$
$521$	7.89699	+	13.6780i	0.345973	+	0.599244i	−0.639557	+	0.768744i	$0.720882\pi$
$522$		0			0
$523$	5.33818	−	9.24599i	0.233422	−	0.404299i	−0.725391	−	0.688337i	$-0.758341\pi$
$523$	5.33818	−	9.24599i	0.233422	−	0.404299i	0.958813	+	0.284038i	$0.0916743\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−43.8629			−1.91070
$528$		0			0
$529$	−1.28263			−0.0557665
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−0.471410	+	0.816506i	−0.0204190	+	0.0353668i
$534$		0			0
$535$	−21.4601	+	37.1699i	−0.927799	+	1.60700i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−8.32846	+	14.4253i	−0.358068	+	0.620193i	−0.987638	−	0.156751i	$-0.949898\pi$
$541$	−8.32846	+	14.4253i	−0.358068	+	0.620193i	0.629570	+	0.776944i	$0.283231\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	6.47141	+	11.2088i	0.277205	+	0.480133i
$546$		0			0
$547$	−7.81875	+	13.5425i	−0.334305	+	0.579034i	−0.983351	−	0.181715i	$-0.941835\pi$
$547$	−7.81875	+	13.5425i	−0.334305	+	0.579034i	0.649046	+	0.760749i	$0.275168\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−4.91260	−	8.50886i	−0.209284	−	0.362490i
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−8.39536	−	14.5412i	−0.355723	−	0.616130i	0.631519	−	0.775361i	$-0.282432\pi$
$557$	−8.39536	−	14.5412i	−0.355723	−	0.616130i	−0.987241	+	0.159231i	$0.949099\pi$
$558$		0			0
$559$	−9.26320			−0.391792
$560$		0			0
$561$		0			0
$562$		0			0
$563$	7.58577	−	13.1389i	0.319702	−	0.553740i	−0.660724	−	0.750629i	$-0.729751\pi$
$563$	7.58577	−	13.1389i	0.319702	−	0.553740i	0.980426	+	0.196889i	$0.0630838\pi$
$564$		0			0
$565$	−12.7174	−	22.0271i	−0.535024	−	0.926688i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	17.7632	+	30.7668i	0.744672	+	1.28981i	0.950348	+	0.311190i	$0.100727\pi$
$569$	17.7632	+	30.7668i	0.744672	+	1.28981i	−0.205676	+	0.978620i	$0.565939\pi$
$570$		0			0
$571$	0.772915	−	1.33873i	0.0323455	−	0.0560240i	−0.849400	−	0.527750i	$-0.823036\pi$
$571$	0.772915	−	1.33873i	0.0323455	−	0.0560240i	0.881745	+	0.471726i	$0.156369\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	49.1456			2.04951
$576$		0			0
$577$	0.838175	+	1.45176i	0.0348937	+	0.0604377i	0.882945	−	0.469477i	$-0.155557\pi$
$577$	0.838175	+	1.45176i	0.0348937	+	0.0604377i	−0.848051	+	0.529914i	$0.822224\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	4.34897	+	7.53264i	0.180116	+	0.311970i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−19.7346	+	34.1813i	−0.814535	+	1.41082i	0.0951271	+	0.995465i	$0.469674\pi$
$587$	−19.7346	+	34.1813i	−0.814535	+	1.41082i	−0.909662	+	0.415350i	$0.863659\pi$
$588$		0			0
$589$	−5.02051	−	8.69578i	−0.206866	−	0.358303i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	11.4159	−	19.7729i	0.468793	−	0.811974i	−0.530570	−	0.847641i	$-0.678022\pi$
$593$	11.4159	−	19.7729i	0.468793	−	0.811974i	0.999364	+	0.0356669i	$0.0113556\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	22.6375	−	39.2093i	0.924943	−	1.60205i	0.133290	−	0.991077i	$-0.457446\pi$
$599$	22.6375	−	39.2093i	0.924943	−	1.60205i	0.791653	−	0.610971i	$-0.209221\pi$
$600$		0			0
$601$	−13.5253	+	23.4265i	−0.551709	+	0.955589i	0.446442	+	0.894813i	$0.352691\pi$
$601$	−13.5253	+	23.4265i	−0.551709	+	0.955589i	−0.998151	+	0.0607761i	$0.980642\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−39.8662			−1.62079
$606$		0			0
$607$	−18.6569			−0.757261			−0.378631	−	0.925548i	$-0.623605\pi$
$607$	−18.6569			−0.757261			−0.378631	+	0.925548i	$0.623605\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	2.64132	−	4.57489i	0.106856	−	0.185080i
$612$		0			0
$613$	14.4806	+	25.0811i	0.584865	+	1.01302i	0.994892	+	0.100942i	$0.0321857\pi$
$613$	14.4806	+	25.0811i	0.584865	+	1.01302i	−0.410028	+	0.912073i	$0.634481\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	18.8457	+	32.6417i	0.758699	+	1.31411i	0.943514	+	0.331332i	$0.107498\pi$
$617$	18.8457	+	32.6417i	0.758699	+	1.31411i	−0.184815	+	0.982773i	$0.559169\pi$
$618$		0			0
$619$	0.0194303			0.000780971			0.000390485	−	1.00000i	$-0.499876\pi$
$619$	0.0194303			0.000780971			0.000390485		1.00000i	$0.499876\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	33.4854			1.33942
$626$		0			0
$627$		0			0
$628$		0			0
$629$	55.0690			2.19574
$630$		0			0
$631$	28.4854			1.13399			0.566993	−	0.823723i	$-0.308107\pi$
$631$	28.4854			1.13399			0.566993	+	0.823723i	$0.308107\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−3.26647			−0.129626
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−44.5393			−1.75920			−0.879598	−	0.475718i	$-0.842188\pi$
$641$	−44.5393			−1.75920			−0.879598	+	0.475718i	$0.842188\pi$
$642$		0			0
$643$	−4.50972	−	7.81106i	−0.177846	−	0.308038i	0.763297	−	0.646048i	$-0.223579\pi$
$643$	−4.50972	−	7.81106i	−0.177846	−	0.308038i	−0.941142	+	0.338010i	$0.890246\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	21.4126	+	37.0877i	0.841816	+	1.45807i	0.888358	+	0.459152i	$0.151847\pi$
$647$	21.4126	+	37.0877i	0.841816	+	1.45807i	−0.0465419	+	0.998916i	$0.514820\pi$
$648$		0			0
$649$	4.50000	−	7.79423i	0.176640	−	0.305950i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−5.60301			−0.219263			−0.109631	−	0.993972i	$-0.534967\pi$
$653$	−5.60301			−0.219263			−0.109631	+	0.993972i	$0.534967\pi$
$654$		0			0
$655$	−21.4153			−0.836765
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−3.26376	+	5.65299i	−0.127138	+	0.220209i	−0.922567	−	0.385838i	$-0.873912\pi$
$659$	−3.26376	+	5.65299i	−0.127138	+	0.220209i	0.795429	+	0.606047i	$0.207246\pi$
$660$		0			0
$661$	5.50972	−	9.54311i	0.214303	−	0.371184i	−0.738754	−	0.673976i	$-0.764585\pi$
$661$	5.50972	−	9.54311i	0.214303	−	0.371184i	0.953057	+	0.302792i	$0.0979186\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−17.8490	+	30.9153i	−0.691115	+	1.19705i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−4.97141	−	8.61073i	−0.191919	−	0.332414i
$672$		0			0
$673$	1.67743	−	2.90539i	0.0646602	−	0.111995i	−0.831883	−	0.554951i	$-0.812737\pi$
$673$	1.67743	−	2.90539i	0.0646602	−	0.111995i	0.896543	+	0.442956i	$0.146070\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−20.3187	−	35.1931i	−0.780913	−	1.35258i	−0.931411	−	0.363970i	$-0.881421\pi$
$677$	−20.3187	−	35.1931i	−0.780913	−	1.35258i	0.150498	−	0.988610i	$-0.451912\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−7.03775	−	12.1897i	−0.269292	−	0.466427i	0.699387	−	0.714743i	$-0.253456\pi$
$683$	−7.03775	−	12.1897i	−0.269292	−	0.466427i	−0.968679	+	0.248316i	$0.920123\pi$
$684$		0			0
$685$	83.3861			3.18602
$686$		0			0
$687$		0			0
$688$		0			0
$689$	4.61273	−	7.98947i	0.175731	−	0.304375i
$690$		0			0
$691$	21.9601	+	38.0359i	0.835400	+	1.44696i	0.893704	+	0.448656i	$0.148097\pi$
$691$	21.9601	+	38.0359i	0.835400	+	1.44696i	−0.0583042	+	0.998299i	$0.518569\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	3.64295	+	6.30977i	0.138185	+	0.239343i
$696$		0			0
$697$	−2.64132	+	4.57489i	−0.100047	+	0.173286i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−7.69578			−0.290666			−0.145333	−	0.989383i	$-0.546425\pi$
$701$	−7.69578			−0.290666			−0.145333	+	0.989383i	$0.546425\pi$
$702$		0			0
$703$	6.30314	+	10.9174i	0.237727	+	0.411756i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−0.111090	−	0.192414i	−0.00417209	−	0.00722626i	0.863932	−	0.503609i	$-0.167995\pi$
$709$	−0.111090	−	0.192414i	−0.00417209	−	0.00722626i	−0.868104	+	0.496383i	$0.834661\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−18.2411	+	31.5944i	−0.683133	+	1.18322i
$714$		0			0
$715$	−1.85868	−	3.21934i	−0.0695109	−	0.120396i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−1.32201	+	2.28980i	−0.0493028	+	0.0853950i	−0.889624	−	0.456694i	$-0.849033\pi$
$719$	−1.32201	+	2.28980i	−0.0493028	+	0.0853950i	0.840321	+	0.542089i	$0.182367\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−40.3915	+	69.9602i	−1.50010	+	2.59826i
$726$		0			0
$727$	−12.7427	+	22.0710i	−0.472600	+	0.818568i	−0.999508	−	0.0313547i	$-0.990018\pi$
$727$	−12.7427	+	22.0710i	−0.472600	+	0.818568i	0.526908	+	0.849922i	$0.323351\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−51.9018			−1.91966
$732$		0			0
$733$	1.86948			0.0690508			0.0345254	−	0.999404i	$-0.489008\pi$
$733$	1.86948			0.0690508			0.0345254	+	0.999404i	$0.489008\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−0.809585	+	1.40224i	−0.0298215	+	0.0516523i
$738$		0			0
$739$	−25.5819	−	44.3092i	−0.941047	−	1.62994i	−0.763479	−	0.645833i	$-0.776510\pi$
$739$	−25.5819	−	44.3092i	−0.941047	−	1.62994i	−0.177568	−	0.984108i	$-0.556823\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−19.3668	−	33.5442i	−0.710498	−	1.23062i	−0.964671	−	0.263459i	$-0.915137\pi$
$743$	−19.3668	−	33.5442i	−0.710498	−	1.23062i	0.254173	−	0.967159i	$-0.418197\pi$
$744$		0			0
$745$	−47.9007			−1.75495
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	6.19097			0.225912			0.112956	−	0.993600i	$-0.463968\pi$
$751$	6.19097			0.225912			0.112956	+	0.993600i	$0.463968\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−35.4088			−1.28866
$756$		0			0
$757$	−13.7174			−0.498566			−0.249283	−	0.968431i	$-0.580195\pi$
$757$	−13.7174			−0.498566			−0.249283	+	0.968431i	$0.580195\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−39.3204			−1.42536			−0.712681	−	0.701488i	$-0.752519\pi$
$761$	−39.3204			−1.42536			−0.712681	+	0.701488i	$0.752519\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−9.54583			−0.344680
$768$		0			0
$769$	−11.3646	−	19.6840i	−0.409817	−	0.709824i	0.585052	−	0.810996i	$-0.301074\pi$
$769$	−11.3646	−	19.6840i	−0.409817	−	0.709824i	−0.994869	+	0.101172i	$0.967741\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−14.3662	−	24.8830i	−0.516717	−	0.894979i	−0.999812	−	0.0194115i	$-0.993821\pi$
$773$	−14.3662	−	24.8830i	−0.516717	−	0.894979i	0.483095	−	0.875568i	$-0.339513\pi$
$774$		0			0
$775$	−41.2788	+	71.4970i	−1.48278	+	2.56825i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−1.20929			−0.0433273
$780$		0			0
$781$	3.34308			0.119625
$782$		0			0
$783$		0			0
$784$		0			0
$785$	23.2804	−	40.3229i	0.830915	−	1.43919i
$786$		0			0
$787$	27.0059	−	46.7756i	0.962656	−	1.66737i	0.246871	−	0.969048i	$-0.420598\pi$
$787$	27.0059	−	46.7756i	0.962656	−	1.66737i	0.715785	−	0.698321i	$-0.246069\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−5.27292	+	9.13296i	−0.187247	+	0.324321i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−0.103565	−	0.179381i	−0.00366848	−	0.00635399i	0.864185	−	0.503174i	$-0.167834\pi$
$797$	−0.103565	−	0.179381i	−0.00366848	−	0.00635399i	−0.867854	+	0.496820i	$0.834501\pi$
$798$		0			0
$799$	14.7993	−	25.6332i	0.523562	−	0.906836i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	1.28100	+	2.21875i	0.0452053	+	0.0782980i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−18.0270	−	31.2236i	−0.633794	−	1.09776i	−0.986769	−	0.162131i	$-0.948163\pi$
$809$	−18.0270	−	31.2236i	−0.633794	−	1.09776i	0.352975	−	0.935633i	$-0.385170\pi$
$810$		0			0
$811$	−2.01943			−0.0709118			−0.0354559	−	0.999371i	$-0.511288\pi$
$811$	−2.01943			−0.0709118			−0.0354559	+	0.999371i	$0.511288\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	24.2141	−	41.9401i	0.848183	−	1.46910i
$816$		0			0
$817$	−5.94063	−	10.2895i	−0.207836	−	0.359983i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−15.1871	−	26.3049i	−0.530035	−	0.918048i	−0.999386	−	0.0350359i	$-0.988845\pi$
$821$	−15.1871	−	26.3049i	−0.530035	−	0.918048i	0.469351	−	0.883012i	$-0.344488\pi$
$822$		0			0
$823$	6.56526	−	11.3714i	0.228851	−	0.396381i	−0.728617	−	0.684921i	$-0.759837\pi$
$823$	6.56526	−	11.3714i	0.228851	−	0.396381i	0.957468	+	0.288540i	$0.0931699\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	11.3236			0.393762			0.196881	−	0.980427i	$-0.436919\pi$
$827$	11.3236			0.393762			0.196881	+	0.980427i	$0.436919\pi$
$828$		0			0
$829$	5.53022	+	9.57863i	0.192073	+	0.332680i	0.945937	−	0.324351i	$-0.105146\pi$
$829$	5.53022	+	9.57863i	0.192073	+	0.332680i	−0.753864	+	0.657030i	$0.771812\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$	−45.1170	−	78.1449i	−1.56134	−	2.70432i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	5.27128	−	9.13013i	0.181985	−	0.315207i	−0.760572	−	0.649254i	$-0.775081\pi$
$839$	5.27128	−	9.13013i	0.181985	−	0.315207i	0.942556	+	0.334047i	$0.108414\pi$
$840$		0			0
$841$	−14.8393	−	25.7023i	−0.511698	−	0.886288i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	23.6569	−	40.9750i	0.813823	−	1.40958i
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$	22.9012	−	39.6661i	0.785045	−	1.35974i
$852$		0			0
$853$	13.6413	−	23.6275i	0.467070	−	0.808989i	−0.532223	−	0.846604i	$-0.678643\pi$
$853$	13.6413	−	23.6275i	0.467070	−	0.808989i	0.999292	+	0.0376160i	$0.0119764\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	2.43147			0.0830574			0.0415287	−	0.999137i	$-0.486777\pi$
$857$	2.43147			0.0830574			0.0415287	+	0.999137i	$0.486777\pi$
$858$		0			0
$859$	−31.7486			−1.08325			−0.541624	−	0.840621i	$-0.682190\pi$
$859$	−31.7486			−1.08325			−0.541624	+	0.840621i	$0.682190\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−13.9126	+	24.0973i	−0.473590	+	0.820282i	−0.999543	−	0.0302315i	$-0.990376\pi$
$863$	−13.9126	+	24.0973i	−0.473590	+	0.820282i	0.525953	+	0.850514i	$0.323709\pi$
$864$		0			0
$865$	37.9757	+	65.7758i	1.29121	+	2.23644i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−7.71900	−	13.3697i	−0.261849	−	0.453536i
$870$		0			0
$871$	1.71737			0.0581909
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−4.58685			−0.154887			−0.0774434	−	0.996997i	$-0.524676\pi$
$877$	−4.58685			−0.154887			−0.0774434	+	0.996997i	$0.524676\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−57.4465			−1.93542			−0.967711	−	0.252062i	$-0.918891\pi$
$881$	−57.4465			−1.93542			−0.967711	+	0.252062i	$0.918891\pi$
$882$		0			0
$883$	36.3937			1.22475			0.612373	−	0.790569i	$-0.290215\pi$
$883$	36.3937			1.22475			0.612373	+	0.790569i	$0.290215\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	43.4476			1.45883			0.729414	−	0.684072i	$-0.239793\pi$
$887$	43.4476			1.45883			0.729414	+	0.684072i	$0.239793\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$	6.77566			0.226739
$894$		0			0
$895$	−9.52532	−	16.4983i	−0.318396	−	0.551479i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−29.9837	−	51.9334i	−1.00001	−	1.73207i
$900$		0			0
$901$	25.8451	−	44.7651i	0.861027	−	1.49134i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	12.8662			0.427687
$906$		0			0
$907$	22.3138			0.740919			0.370459	−	0.928849i	$-0.379200\pi$
$907$	22.3138			0.740919			0.370459	+	0.928849i	$0.379200\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−7.67799	+	13.2987i	−0.254383	+	0.440604i	−0.964728	−	0.263250i	$-0.915206\pi$
$911$	−7.67799	+	13.2987i	−0.254383	+	0.440604i	0.710345	+	0.703854i	$0.248539\pi$
$912$		0			0
$913$	0.187145	−	0.324145i	0.00619360	−	0.0107276i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	18.4142	−	31.8944i	0.607429	−	1.05210i	−0.384233	−	0.923236i	$-0.625534\pi$
$919$	18.4142	−	31.8944i	0.607429	−	1.05210i	0.991662	−	0.128862i	$-0.0411326\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−1.77292	−	3.07078i	−0.0583562	−	0.101076i
$924$		0			0
$925$	51.8246	−	89.7629i	1.70398	−	2.95139i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−20.8646	−	36.1385i	−0.684545	−	1.18567i	−0.973580	−	0.228347i	$-0.926668\pi$
$929$	−20.8646	−	36.1385i	−0.684545	−	1.18567i	0.289035	−	0.957318i	$-0.406666\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−10.4142	−	18.0380i	−0.340582	−	0.589905i
$936$		0			0
$937$	−25.3743			−0.828942			−0.414471	−	0.910063i	$-0.636033\pi$
$937$	−25.3743			−0.828942			−0.414471	+	0.910063i	$0.636033\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	23.5712	−	40.8264i	0.768398	−	1.33090i	−0.170034	−	0.985438i	$-0.554388\pi$
$941$	23.5712	−	40.8264i	0.768398	−	1.33090i	0.938431	−	0.345465i	$-0.112279\pi$
$942$		0			0
$943$	2.19686	+	3.80507i	0.0715396	+	0.123910i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−17.2535	−	29.8839i	−0.560663	−	0.971097i	−0.997439	−	0.0715263i	$-0.977213\pi$
$947$	−17.2535	−	29.8839i	−0.560663	−	0.971097i	0.436776	−	0.899570i	$-0.356120\pi$
$948$		0			0
$949$	1.35868	−	2.35331i	0.0441048	−	0.0763917i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	44.1833			1.43124			0.715619	−	0.698491i	$-0.246145\pi$
$953$	44.1833			1.43124			0.715619	+	0.698491i	$0.246145\pi$
$954$		0			0
$955$	28.9757	+	50.1873i	0.937631	+	1.62402i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−15.1424	−	26.2274i	−0.488464	−	0.846045i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−19.9331	+	34.5252i	−0.641669	+	1.11140i
$966$		0			0
$967$	−21.2330	−	36.7766i	−0.682806	−	1.18266i	−0.974121	−	0.226028i	$-0.927426\pi$
$967$	−21.2330	−	36.7766i	−0.682806	−	1.18266i	0.291314	−	0.956627i	$-0.405907\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	3.02696	−	5.24284i	0.0971396	−	0.168251i	−0.813360	−	0.581761i	$-0.802364\pi$
$971$	3.02696	−	5.24284i	0.0971396	−	0.168251i	0.910500	+	0.413510i	$0.135697\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−22.0264	+	38.1508i	−0.704687	+	1.22055i	0.262117	+	0.965036i	$0.415579\pi$
$977$	−22.0264	+	38.1508i	−0.704687	+	1.22055i	−0.966804	+	0.255518i	$0.917754\pi$
$978$		0			0
$979$	−2.59656	+	4.49738i	−0.0829866	+	0.143737i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	4.47033			0.142581			0.0712907	−	0.997456i	$-0.477288\pi$
$983$	4.47033			0.142581			0.0712907	+	0.997456i	$0.477288\pi$
$984$		0			0
$985$	−65.5986			−2.09015
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−21.5841	+	37.3848i	−0.686336	+	1.18877i
$990$		0			0
$991$	15.7524	+	27.2840i	0.500392	+	0.866704i	1.00000		0.000452297i	$0.000143971\pi$
$991$	15.7524	+	27.2840i	0.500392	+	0.866704i	−0.499608	+	0.866251i	$0.666523\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−25.8473	−	44.7689i	−0.819416	−	1.41927i
$996$		0			0
$997$	26.0000			0.823428			0.411714	−	0.911313i	$-0.364930\pi$
$997$	26.0000			0.823428			0.411714	+	0.911313i	$0.364930\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5292.2.l.g.3313.3		6
3.2	odd	2		1764.2.l.d.961.2		6
7.2	even	3		756.2.j.a.505.1		6
7.3	odd	6		5292.2.i.g.2125.3		6
7.4	even	3		5292.2.i.d.2125.1		6
7.5	odd	6		5292.2.j.e.3529.3		6
7.6	odd	2		5292.2.l.d.3313.1		6
9.4	even	3		5292.2.i.d.1549.1		6
9.5	odd	6		1764.2.i.f.373.3		6
21.2	odd	6		252.2.j.b.169.1	yes	6
21.5	even	6		1764.2.j.d.1177.3		6
21.11	odd	6		1764.2.i.f.1537.3		6
21.17	even	6		1764.2.i.e.1537.1		6
21.20	even	2		1764.2.l.g.961.2		6
28.23	odd	6		3024.2.r.i.2017.1		6
63.2	odd	6		2268.2.a.g.1.1		3
63.4	even	3	inner	5292.2.l.g.361.3		6
63.5	even	6		1764.2.j.d.589.3		6
63.13	odd	6		5292.2.i.g.1549.3		6
63.16	even	3		2268.2.a.j.1.3		3
63.23	odd	6		252.2.j.b.85.1	✓	6
63.31	odd	6		5292.2.l.d.361.1		6
63.32	odd	6		1764.2.l.d.949.2		6
63.40	odd	6		5292.2.j.e.1765.3		6
63.41	even	6		1764.2.i.e.373.1		6
63.58	even	3		756.2.j.a.253.1		6
63.59	even	6		1764.2.l.g.949.2		6
84.23	even	6		1008.2.r.g.673.3		6
252.23	even	6		1008.2.r.g.337.3		6
252.79	odd	6		9072.2.a.bz.1.3		3
252.191	even	6		9072.2.a.bt.1.1		3
252.247	odd	6		3024.2.r.i.1009.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
252.2.j.b.85.1	✓	6	63.23	odd	6
252.2.j.b.169.1	yes	6	21.2	odd	6
756.2.j.a.253.1		6	63.58	even	3
756.2.j.a.505.1		6	7.2	even	3
1008.2.r.g.337.3		6	252.23	even	6
1008.2.r.g.673.3		6	84.23	even	6
1764.2.i.e.373.1		6	63.41	even	6
1764.2.i.e.1537.1		6	21.17	even	6
1764.2.i.f.373.3		6	9.5	odd	6
1764.2.i.f.1537.3		6	21.11	odd	6
1764.2.j.d.589.3		6	63.5	even	6
1764.2.j.d.1177.3		6	21.5	even	6
1764.2.l.d.949.2		6	63.32	odd	6
1764.2.l.d.961.2		6	3.2	odd	2
1764.2.l.g.949.2		6	63.59	even	6
1764.2.l.g.961.2		6	21.20	even	2
2268.2.a.g.1.1		3	63.2	odd	6
2268.2.a.j.1.3		3	63.16	even	3
3024.2.r.i.1009.1		6	252.247	odd	6
3024.2.r.i.2017.1		6	28.23	odd	6
5292.2.i.d.1549.1		6	9.4	even	3
5292.2.i.d.2125.1		6	7.4	even	3
5292.2.i.g.1549.3		6	63.13	odd	6
5292.2.i.g.2125.3		6	7.3	odd	6
5292.2.j.e.1765.3		6	63.40	odd	6
5292.2.j.e.3529.3		6	7.5	odd	6
5292.2.l.d.361.1		6	63.31	odd	6
5292.2.l.d.3313.1		6	7.6	odd	2
5292.2.l.g.361.3		6	63.4	even	3	inner
5292.2.l.g.3313.3		6	1.1	even	1	trivial
9072.2.a.bt.1.1		3	252.191	even	6
9072.2.a.bz.1.3		3	252.79	odd	6

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+3.94282 q^{5} -0.942820 q^{11} +(0.500000 + 0.866025i) q^{13} +(2.80150 + 4.85235i) q^{17} +(-0.641315 + 1.11079i) q^{19} +4.66019 q^{23} +10.5458 q^{25} +(-3.83009 + 6.63392i) q^{29} +(-3.91423 + 6.77965i) q^{31} +(4.91423 - 8.51170i) q^{37} +(0.471410 + 0.816506i) q^{41} +(-4.63160 + 8.02217i) q^{43} +(-2.64132 - 4.57489i) q^{47} +(-4.61273 - 7.98947i) q^{53} -3.71737 q^{55} +(-4.77292 + 8.26693i) q^{59} +(5.27292 + 9.13296i) q^{61} +(1.97141 + 3.41458i) q^{65} +(0.858685 - 1.48729i) q^{67} -3.54583 q^{71} +(-1.35868 - 2.35331i) q^{73} +(8.18715 + 14.1806i) q^{79} +(-0.198495 + 0.343803i) q^{83} +(11.0458 + 19.1319i) q^{85} +(2.75404 - 4.77014i) q^{89} +(-2.52859 + 4.37965i) q^{95} +(6.77292 - 11.7310i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 6 q^{5} + 12 q^{11} + 3 q^{13} - 3 q^{19} + 12 q^{23} + 12 q^{25} - 15 q^{29} + 3 q^{31} + 3 q^{37} - 6 q^{41} - 3 q^{43} - 15 q^{47} - 18 q^{53} - 24 q^{55} - 3 q^{59} + 6 q^{61} + 3 q^{65} + 6 q^{67}+ \cdots + 15 q^{97}+O(q^{100}) \) 6 * q + 6 * q^5 + 12 * q^11 + 3 * q^13 - 3 * q^19 + 12 * q^23 + 12 * q^25 - 15 * q^29 + 3 * q^31 + 3 * q^37 - 6 * q^41 - 3 * q^43 - 15 * q^47 - 18 * q^53 - 24 * q^55 - 3 * q^59 + 6 * q^61 + 3 * q^65 + 6 * q^67 + 30 * q^71 - 9 * q^73 - 3 * q^79 - 18 * q^83 + 15 * q^85 + 6 * q^89 - 24 * q^95 + 15 * q^97

Introduction

L-functions

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