LMFDB - Embedded newform 5292.2.l.d.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			$0.500000 - 2.05195i$ of defining polynomial
Character	$\chi$	$=$	5292.3313
Dual form			5292.2.l.d.361.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.05408 q^{5} +5.05408 q^{11} +(-0.500000 - 0.866025i) q^{13} +(-0.136673 - 0.236725i) q^{17} +(-2.69076 + 4.66053i) q^{19} +5.32743 q^{23} -0.780738 q^{25} +(-4.16372 + 7.21177i) q^{29} +(-5.08113 + 8.80077i) q^{31} +(-4.08113 + 7.06872i) q^{37} +(2.52704 + 4.37697i) q^{41} +(-2.30039 + 3.98439i) q^{43} +(-0.690757 - 1.19643i) q^{47} +(1.71780 + 2.97532i) q^{53} +10.3815 q^{55} +(-0.890369 + 1.54216i) q^{59} +(0.390369 + 0.676139i) q^{61} +(-1.02704 - 1.77889i) q^{65} +(4.19076 - 7.25860i) q^{67} +7.78074 q^{71} +(4.69076 + 8.12463i) q^{73} +(-6.47150 - 11.2090i) q^{79} +(2.86333 - 4.95943i) q^{83} +(-0.280738 - 0.486253i) q^{85} +(6.90856 - 11.9660i) q^{89} +(-5.52704 + 9.57312i) q^{95} +(-1.10963 + 1.92194i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	2.05408			0.918614			0.459307	−	0.888277i	$-0.348098\pi$
$5$	2.05408			0.918614			0.459307	+	0.888277i	$0.348098\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$		0			0
$10$		0			0
$11$	5.05408			1.52386			0.761932	−	0.647657i	$-0.224251\pi$
$11$	5.05408			1.52386			0.761932	+	0.647657i	$0.224251\pi$
$12$		0			0
$13$	−0.500000	−	0.866025i	−0.138675	−	0.240192i	0.788320	−	0.615265i	$-0.210951\pi$
$13$	−0.500000	−	0.866025i	−0.138675	−	0.240192i	−0.926995	+	0.375073i	$0.877618\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−0.136673	−	0.236725i	−0.0331481	−	0.0574142i	0.848975	−	0.528432i	$-0.177220\pi$
$17$	−0.136673	−	0.236725i	−0.0331481	−	0.0574142i	−0.882124	+	0.471018i	$0.843887\pi$
$18$		0			0
$19$	−2.69076	+	4.66053i	−0.617302	+	1.06920i	0.372674	+	0.927962i	$0.378441\pi$
$19$	−2.69076	+	4.66053i	−0.617302	+	1.06920i	−0.989976	+	0.141236i	$0.954892\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	5.32743			1.11085			0.555423	−	0.831568i	$-0.312556\pi$
$23$	5.32743			1.11085			0.555423	+	0.831568i	$0.312556\pi$
$24$		0			0
$25$	−0.780738			−0.156148
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−4.16372	+	7.21177i	−0.773183	+	1.33919i	0.162628	+	0.986687i	$0.448003\pi$
$29$	−4.16372	+	7.21177i	−0.773183	+	1.33919i	−0.935810	+	0.352504i	$0.885330\pi$
$30$		0			0
$31$	−5.08113	+	8.80077i	−0.912597	+	1.58066i	−0.102216	+	0.994762i	$0.532593\pi$
$31$	−5.08113	+	8.80077i	−0.912597	+	1.58066i	−0.810382	+	0.585903i	$0.800740\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−4.08113	+	7.06872i	−0.670933	+	1.16209i	0.306707	+	0.951804i	$0.400773\pi$
$37$	−4.08113	+	7.06872i	−0.670933	+	1.16209i	−0.977640	+	0.210286i	$0.932560\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	2.52704	+	4.37697i	0.394658	+	0.683567i	0.993057	−	0.117631i	$-0.0375299\pi$
$41$	2.52704	+	4.37697i	0.394658	+	0.683567i	−0.598400	+	0.801198i	$0.704197\pi$
$42$		0			0
$43$	−2.30039	+	3.98439i	−0.350806	+	0.607614i	−0.986391	−	0.164417i	$-0.947426\pi$
$43$	−2.30039	+	3.98439i	−0.350806	+	0.607614i	0.635585	+	0.772031i	$0.280759\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−0.690757	−	1.19643i	−0.100757	−	0.174517i	0.811240	−	0.584714i	$-0.198793\pi$
$47$	−0.690757	−	1.19643i	−0.100757	−	0.174517i	−0.911997	+	0.410197i	$0.865460\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$		0			0
$52$		0			0
$53$	1.71780	+	2.97532i	0.235958	+	0.408691i	0.959551	−	0.281536i	$-0.0908439\pi$
$53$	1.71780	+	2.97532i	0.235958	+	0.408691i	−0.723593	+	0.690227i	$0.757511\pi$
$54$		0			0
$55$	10.3815			1.39984
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−0.890369	+	1.54216i	−0.115916	+	0.200773i	−0.918146	−	0.396243i	$-0.870314\pi$
$59$	−0.890369	+	1.54216i	−0.115916	+	0.200773i	0.802229	+	0.597016i	$0.203647\pi$
$60$		0			0
$61$	0.390369	+	0.676139i	0.0499816	+	0.0865707i	0.889934	−	0.456090i	$-0.150750\pi$
$61$	0.390369	+	0.676139i	0.0499816	+	0.0865707i	−0.839952	+	0.542660i	$0.817417\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−1.02704	−	1.77889i	−0.127389	−	0.220644i
$66$		0			0
$67$	4.19076	−	7.25860i	0.511982	−	0.886780i	−0.487921	−	0.872888i	$-0.662245\pi$
$67$	4.19076	−	7.25860i	0.511982	−	0.886780i	0.999904	−	0.0138919i	$-0.00442207\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	7.78074			0.923404			0.461702	−	0.887035i	$-0.347239\pi$
$71$	7.78074			0.923404			0.461702	+	0.887035i	$0.347239\pi$
$72$		0			0
$73$	4.69076	+	8.12463i	0.549012	+	0.950916i	0.998343	+	0.0575506i	$0.0183291\pi$
$73$	4.69076	+	8.12463i	0.549012	+	0.950916i	−0.449331	+	0.893365i	$0.648338\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−6.47150	−	11.2090i	−0.728100	−	1.26111i	−0.957685	−	0.287818i	$-0.907070\pi$
$79$	−6.47150	−	11.2090i	−0.728100	−	1.26111i	0.229585	−	0.973289i	$-0.426263\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	2.86333	−	4.95943i	0.314291	−	0.544368i	−0.664996	−	0.746847i	$-0.731567\pi$
$83$	2.86333	−	4.95943i	0.314291	−	0.544368i	0.979287	+	0.202479i	$0.0648999\pi$
$84$		0			0
$85$	−0.280738	−	0.486253i	−0.0304503	−	0.0527415i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	6.90856	−	11.9660i	0.732306	−	1.26839i	−0.223590	−	0.974683i	$-0.571778\pi$
$89$	6.90856	−	11.9660i	0.732306	−	1.26839i	0.955895	−	0.293707i	$-0.0948890\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−5.52704	+	9.57312i	−0.567063	+	0.982181i
$96$		0			0
$97$	−1.10963	+	1.92194i	−0.112666	+	0.195143i	−0.916844	−	0.399245i	$-0.869272\pi$
$97$	−1.10963	+	1.92194i	−0.112666	+	0.195143i	0.804178	+	0.594388i	$0.202606\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	2.72665			0.271312			0.135656	−	0.990756i	$-0.456686\pi$
$101$	2.72665			0.271312			0.135656	+	0.990756i	$0.456686\pi$
$102$		0			0
$103$	−17.9823			−1.77185			−0.885924	−	0.463831i	$-0.846475\pi$
$103$	−17.9823			−1.77185			−0.885924	+	0.463831i	$0.846475\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	0.554084	−	0.959702i	0.0535653	−	0.0927779i	−0.837999	−	0.545671i	$-0.816275\pi$
$107$	0.554084	−	0.959702i	0.0535653	−	0.0927779i	0.891565	+	0.452893i	$0.149608\pi$
$108$		0			0
$109$	−1.69076	−	2.92848i	−0.161945	−	0.280497i	0.773621	−	0.633649i	$-0.218443\pi$
$109$	−1.69076	−	2.92848i	−0.161945	−	0.280497i	−0.935566	+	0.353151i	$0.885110\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	9.43560	+	16.3429i	0.887626	+	1.53741i	0.842673	+	0.538425i	$0.180980\pi$
$113$	9.43560	+	16.3429i	0.887626	+	1.53741i	0.0449531	+	0.998989i	$0.485686\pi$
$114$		0			0
$115$	10.9430			1.02044
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	14.5438			1.32216
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−11.8741			−1.06205
$126$		0			0
$127$	17.1623			1.52290			0.761452	−	0.648221i	$-0.224487\pi$
$127$	17.1623			1.52290			0.761452	+	0.648221i	$0.224487\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−17.8889			−1.56296			−0.781481	−	0.623930i	$-0.785535\pi$
$131$	−17.8889			−1.56296			−0.781481	+	0.623930i	$0.785535\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	4.49261			0.383829			0.191915	−	0.981412i	$-0.438530\pi$
$137$	4.49261			0.383829			0.191915	+	0.981412i	$0.438530\pi$
$138$		0			0
$139$	9.07227	+	15.7136i	0.769500	+	1.33281i	0.937834	+	0.347083i	$0.112828\pi$
$139$	9.07227	+	15.7136i	0.769500	+	1.33281i	−0.168334	+	0.985730i	$0.553839\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−2.52704	−	4.37697i	−0.211322	−	0.366020i
$144$		0			0
$145$	−8.55262	+	14.8136i	−0.710257	+	1.23020i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	4.50739			0.369260			0.184630	−	0.982808i	$-0.440891\pi$
$149$	4.50739			0.369260			0.184630	+	0.982808i	$0.440891\pi$
$150$		0			0
$151$	−10.9823			−0.893726			−0.446863	−	0.894602i	$-0.647459\pi$
$151$	−10.9823			−0.893726			−0.446863	+	0.894602i	$0.647459\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−10.4371	+	18.0775i	−0.838325	+	1.45202i
$156$		0			0
$157$	2.08998	−	3.61995i	0.166799	−	0.288904i	−0.770494	−	0.637447i	$-0.779990\pi$
$157$	2.08998	−	3.61995i	0.166799	−	0.288904i	0.937293	+	0.348544i	$0.113324\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	2.80924	−	4.86575i	0.220037	−	0.381115i	−0.734782	−	0.678303i	$-0.762716\pi$
$163$	2.80924	−	4.86575i	0.220037	−	0.381115i	0.954819	+	0.297188i	$0.0960489\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	5.44592	+	9.43260i	0.421418	+	0.729917i	0.996078	−	0.0884750i	$-0.0281993\pi$
$167$	5.44592	+	9.43260i	0.421418	+	0.729917i	−0.574661	+	0.818392i	$0.694866\pi$
$168$		0			0
$169$	6.00000	−	10.3923i	0.461538	−	0.799408i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−7.30039	−	12.6446i	−0.555038	−	0.961354i	−0.997901	−	0.0647648i	$-0.979370\pi$
$173$	−7.30039	−	12.6446i	−0.555038	−	0.961354i	0.442862	−	0.896590i	$-0.353963\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−11.7448	−	20.3427i	−0.877851	−	1.52048i	−0.853695	−	0.520774i	$-0.825644\pi$
$179$	−11.7448	−	20.3427i	−0.877851	−	1.52048i	−0.0241559	−	0.999708i	$-0.507690\pi$
$180$		0			0
$181$	1.39922			0.104003			0.0520017	−	0.998647i	$-0.483440\pi$
$181$	1.39922			0.104003			0.0520017	+	0.998647i	$0.483440\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−8.38298	+	14.5197i	−0.616329	+	1.06751i
$186$		0			0
$187$	−0.690757	−	1.19643i	−0.0505132	−	0.0874914i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	11.6819	+	20.2336i	0.845273	+	1.46406i	0.885384	+	0.464860i	$0.153895\pi$
$191$	11.6819	+	20.2336i	0.845273	+	1.46406i	−0.0401112	+	0.999195i	$0.512771\pi$
$192$		0			0
$193$	7.27188	−	12.5953i	0.523442	−	0.906628i	−0.476186	−	0.879345i	$-0.657981\pi$
$193$	7.27188	−	12.5953i	0.523442	−	0.906628i	0.999628	−	0.0272830i	$-0.00868552\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	17.3422			1.23558			0.617791	−	0.786343i	$-0.288028\pi$
$197$	17.3422			1.23558			0.617791	+	0.786343i	$0.288028\pi$
$198$		0			0
$199$	−5.77188	−	9.99720i	−0.409158	−	0.708682i	0.585638	−	0.810573i	$-0.300844\pi$
$199$	−5.77188	−	9.99720i	−0.409158	−	0.708682i	−0.994796	+	0.101891i	$0.967511\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$	5.19076	+	8.99066i	0.362538	+	0.627935i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−13.5993	+	23.5547i	−0.940684	+	1.62931i
$210$		0			0
$211$	12.2630	+	21.2402i	0.844222	+	1.46223i	0.886295	+	0.463120i	$0.153270\pi$
$211$	12.2630	+	21.2402i	0.844222	+	1.46223i	−0.0420736	+	0.999115i	$0.513396\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−4.72519	+	8.18427i	−0.322255	+	0.558163i
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−0.136673	+	0.236725i	−0.00919363	+	0.0159238i
$222$		0			0
$223$	4.28074	−	7.41446i	0.286659	−	0.496509i	−0.686351	−	0.727271i	$-0.740789\pi$
$223$	4.28074	−	7.41446i	0.286659	−	0.496509i	0.973010	+	0.230762i	$0.0741219\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−21.1986			−1.40700			−0.703501	−	0.710694i	$-0.748381\pi$
$227$	−21.1986			−1.40700			−0.703501	+	0.710694i	$0.748381\pi$
$228$		0			0
$229$	4.56148			0.301431			0.150715	−	0.988577i	$-0.451842\pi$
$229$	4.56148			0.301431			0.150715	+	0.988577i	$0.451842\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	6.75370	−	11.6977i	0.442449	−	0.766345i	−0.555421	−	0.831569i	$-0.687443\pi$
$233$	6.75370	−	11.6977i	0.442449	−	0.766345i	0.997871	+	0.0652244i	$0.0207763\pi$
$234$		0			0
$235$	−1.41887	−	2.45756i	−0.0925571	−	0.160314i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−6.82743	−	11.8255i	−0.441630	−	0.764925i	0.556181	−	0.831061i	$-0.312266\pi$
$239$	−6.82743	−	11.8255i	−0.441630	−	0.764925i	−0.997811	+	0.0661361i	$0.978933\pi$
$240$		0			0
$241$	−3.21926			−0.207371			−0.103685	−	0.994610i	$-0.533064\pi$
$241$	−3.21926			−0.207371			−0.103685	+	0.994610i	$0.533064\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	5.38151			0.342418
$248$		0			0
$249$		0			0
$250$		0			0
$251$	4.38151			0.276559			0.138279	−	0.990393i	$-0.455843\pi$
$251$	4.38151			0.276559			0.138279	+	0.990393i	$0.455843\pi$
$252$		0			0
$253$	26.9253			1.69278
$254$		0			0
$255$		0			0
$256$		0			0
$257$	11.4533			0.714438			0.357219	−	0.934021i	$-0.383725\pi$
$257$	11.4533			0.714438			0.357219	+	0.934021i	$0.383725\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	6.83482			0.421453			0.210727	−	0.977545i	$-0.432417\pi$
$263$	6.83482			0.421453			0.210727	+	0.977545i	$0.432417\pi$
$264$		0			0
$265$	3.52850	+	6.11155i	0.216754	+	0.375429i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−4.83628	−	8.37669i	−0.294873	−	0.510736i	0.680082	−	0.733136i	$-0.261944\pi$
$269$	−4.83628	−	8.37669i	−0.294873	−	0.510736i	−0.974955	+	0.222400i	$0.928611\pi$
$270$		0			0
$271$	−6.41887	+	11.1178i	−0.389919	+	0.675359i	−0.992438	−	0.122745i	$-0.960830\pi$
$271$	−6.41887	+	11.1178i	−0.389919	+	0.675359i	0.602519	+	0.798104i	$0.294164\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−3.94592			−0.237948
$276$		0			0
$277$	11.5831			0.695959			0.347980	−	0.937502i	$-0.386868\pi$
$277$	11.5831			0.695959			0.347980	+	0.937502i	$0.386868\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−2.46410	+	4.26795i	−0.146996	+	0.254605i	−0.930116	−	0.367266i	$-0.880294\pi$
$281$	−2.46410	+	4.26795i	−0.146996	+	0.254605i	0.783120	+	0.621871i	$0.213627\pi$
$282$		0			0
$283$	−9.30039	+	16.1087i	−0.552851	+	0.957565i	0.445217	+	0.895423i	$0.353127\pi$
$283$	−9.30039	+	16.1087i	−0.552851	+	0.957565i	−0.998067	+	0.0621426i	$0.980207\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	8.46264	−	14.6577i	0.497802	−	0.862219i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	12.3801	+	21.4429i	0.723250	+	1.25271i	0.959690	+	0.281060i	$0.0906861\pi$
$293$	12.3801	+	21.4429i	0.723250	+	1.25271i	−0.236440	+	0.971646i	$0.575981\pi$
$294$		0			0
$295$	−1.82889	+	3.16774i	−0.106482	+	0.184433i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−2.66372	−	4.61369i	−0.154047	−	0.266817i
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	0.801851	+	1.38885i	0.0459138	+	0.0795251i
$306$		0			0
$307$	−21.9430			−1.25235			−0.626176	−	0.779681i	$-0.715381\pi$
$307$	−21.9430			−1.25235			−0.626176	+	0.779681i	$0.715381\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−13.5811	+	23.5232i	−0.770115	+	1.33388i	0.167384	+	0.985892i	$0.446468\pi$
$311$	−13.5811	+	23.5232i	−0.770115	+	1.33388i	−0.937499	+	0.347987i	$0.886865\pi$
$312$		0			0
$313$	−4.27188	−	7.39912i	−0.241461	−	0.418223i	0.719670	−	0.694317i	$-0.244293\pi$
$313$	−4.27188	−	7.39912i	−0.241461	−	0.418223i	−0.961131	+	0.276094i	$0.910960\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	0.199612	+	0.345738i	0.0112113	+	0.0194186i	0.871577	−	0.490259i	$-0.163098\pi$
$317$	0.199612	+	0.345738i	0.0112113	+	0.0194186i	−0.860365	+	0.509678i	$0.829765\pi$
$318$		0			0
$319$	−21.0438	+	36.4489i	−1.17822	+	2.04075i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	1.47102			0.0818495
$324$		0			0
$325$	0.390369	+	0.676139i	0.0216538	+	0.0375054i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	2.80924	+	4.86575i	0.154410	+	0.267446i	0.932844	−	0.360281i	$-0.117319\pi$
$331$	2.80924	+	4.86575i	0.154410	+	0.267446i	−0.778434	+	0.627726i	$0.783986\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	8.60817	−	14.9098i	0.470314	−	0.814609i
$336$		0			0
$337$	14.4911	+	25.0994i	0.789383	+	1.36725i	0.926345	+	0.376675i	$0.122933\pi$
$337$	14.4911	+	25.0994i	0.789383	+	1.36725i	−0.136962	+	0.990576i	$0.543734\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−25.6804	+	44.4798i	−1.39067	+	2.40872i
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	17.2345	−	29.8511i	0.925198	−	1.60249i	0.133955	−	0.990987i	$-0.457232\pi$
$347$	17.2345	−	29.8511i	0.925198	−	1.60249i	0.791243	−	0.611502i	$-0.209434\pi$
$348$		0			0
$349$	8.78074	−	15.2087i	0.470022	−	0.814102i	−0.529390	−	0.848378i	$-0.677579\pi$
$349$	8.78074	−	15.2087i	0.470022	−	0.814102i	0.999412	+	0.0342762i	$0.0109126\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	32.8889			1.75050			0.875250	−	0.483671i	$-0.160697\pi$
$353$	32.8889			1.75050			0.875250	+	0.483671i	$0.160697\pi$
$354$		0			0
$355$	15.9823			0.848252
$356$		0			0
$357$		0			0
$358$		0			0
$359$	1.48181	−	2.56657i	0.0782071	−	0.135459i	−0.824269	−	0.566198i	$-0.808414\pi$
$359$	1.48181	−	2.56657i	0.0782071	−	0.135459i	0.902476	+	0.430739i	$0.141747\pi$
$360$		0			0
$361$	−4.98035	−	8.62622i	−0.262124	−	0.454012i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	9.63521	+	16.6887i	0.504330	+	0.873525i
$366$		0			0
$367$	−13.3638			−0.697585			−0.348792	−	0.937200i	$-0.613408\pi$
$367$	−13.3638			−0.697585			−0.348792	+	0.937200i	$0.613408\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	4.60078			0.238219			0.119110	−	0.992881i	$-0.461996\pi$
$373$	4.60078			0.238219			0.119110	+	0.992881i	$0.461996\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	8.32743			0.428884
$378$		0			0
$379$	−2.21926			−0.113996			−0.0569979	−	0.998374i	$-0.518153\pi$
$379$	−2.21926			−0.113996			−0.0569979	+	0.998374i	$0.518153\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−31.0335			−1.58574			−0.792868	−	0.609394i	$-0.791413\pi$
$383$	−31.0335			−1.58574			−0.792868	+	0.609394i	$0.791413\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	24.8348			1.25918			0.629588	−	0.776929i	$-0.283224\pi$
$389$	24.8348			1.25918			0.629588	+	0.776929i	$0.283224\pi$
$390$		0			0
$391$	−0.728116	−	1.26113i	−0.0368224	−	0.0637783i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−13.2930	−	23.0241i	−0.668843	−	1.15847i
$396$		0			0
$397$	8.86186	−	15.3492i	0.444764	−	0.770354i	−0.553272	−	0.833001i	$-0.686621\pi$
$397$	8.86186	−	15.3492i	0.444764	−	0.770354i	0.998036	+	0.0626467i	$0.0199541\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−30.1770			−1.50697			−0.753485	−	0.657466i	$-0.771629\pi$
$401$	−30.1770			−1.50697			−0.753485	+	0.657466i	$0.771629\pi$
$402$		0			0
$403$	10.1623			0.506218
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−20.6264	+	35.7259i	−1.02241	+	1.77087i
$408$		0			0
$409$	8.38151	−	14.5172i	0.414439	−	0.717830i	−0.580930	−	0.813953i	$-0.697311\pi$
$409$	8.38151	−	14.5172i	0.414439	−	0.717830i	0.995369	+	0.0961236i	$0.0306444\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$	5.88151	−	10.1871i	0.288712	−	0.500064i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−1.44445	−	2.50187i	−0.0705662	−	0.122224i	0.828583	−	0.559866i	$-0.189147\pi$
$419$	−1.44445	−	2.50187i	−0.0705662	−	0.122224i	−0.899150	+	0.437641i	$0.855814\pi$
$420$		0			0
$421$	0.0899807	−	0.155851i	0.00438539	−	0.00759572i	−0.863824	−	0.503793i	$-0.831937\pi$
$421$	0.0899807	−	0.155851i	0.00438539	−	0.00759572i	0.868210	+	0.496197i	$0.165271\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	0.106706	+	0.184820i	0.00517600	+	0.00896509i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−2.38298	−	4.12744i	−0.114784	−	0.198812i	0.802909	−	0.596101i	$-0.203284\pi$
$431$	−2.38298	−	4.12744i	−0.114784	−	0.198812i	−0.917693	+	0.397289i	$0.869951\pi$
$432$		0			0
$433$	27.7630			1.33421			0.667103	−	0.744965i	$-0.267534\pi$
$433$	27.7630			1.33421			0.667103	+	0.744965i	$0.267534\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−14.3348	+	24.8286i	−0.685728	+	1.18771i
$438$		0			0
$439$	−2.32889	−	4.03376i	−0.111152	−	0.192521i	0.805083	−	0.593162i	$-0.202121\pi$
$439$	−2.32889	−	4.03376i	−0.111152	−	0.192521i	−0.916235	+	0.400641i	$0.868787\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−1.38151	−	2.39285i	−0.0656377	−	0.113688i	0.831339	−	0.555766i	$-0.187575\pi$
$443$	−1.38151	−	2.39285i	−0.0656377	−	0.113688i	−0.896977	+	0.442078i	$0.854242\pi$
$444$		0			0
$445$	14.1908	−	24.5791i	0.672706	−	1.16516i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−19.9430			−0.941168			−0.470584	−	0.882355i	$-0.655957\pi$
$449$	−19.9430			−0.941168			−0.470584	+	0.882355i	$0.655957\pi$
$450$		0			0
$451$	12.7719	+	22.1216i	0.601405	+	1.04166i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−13.6908	−	23.7131i	−0.640427	−	1.10925i	−0.985338	−	0.170616i	$-0.945424\pi$
$457$	−13.6908	−	23.7131i	−0.640427	−	1.10925i	0.344911	−	0.938635i	$-0.387909\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	3.02558	−	5.24046i	0.140915	−	0.244072i	−0.786926	−	0.617047i	$-0.788329\pi$
$461$	3.02558	−	5.24046i	0.140915	−	0.244072i	0.927842	+	0.372975i	$0.121662\pi$
$462$		0			0
$463$	8.77188	+	15.1933i	0.407664	+	0.706095i	0.994628	−	0.103519i	$-0.0330101\pi$
$463$	8.77188	+	15.1933i	0.407664	+	0.706095i	−0.586964	+	0.809613i	$0.699677\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	11.8078	−	20.4517i	0.546399	−	0.946391i	−0.452119	−	0.891958i	$-0.649332\pi$
$467$	11.8078	−	20.4517i	0.546399	−	0.946391i	0.998517	−	0.0544328i	$-0.0173351\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−11.6264	+	20.1374i	−0.534580	+	0.925920i
$474$		0			0
$475$	2.10078	−	3.63865i	0.0963902	−	0.166953i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	38.2527			1.74781			0.873906	−	0.486096i	$-0.161579\pi$
$479$	38.2527			1.74781			0.873906	+	0.486096i	$0.161579\pi$
$480$		0			0
$481$	8.16225			0.372167
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−2.27928	+	3.94782i	−0.103497	+	0.179261i
$486$		0			0
$487$	3.28959	+	5.69774i	0.149066	+	0.258189i	0.930882	−	0.365319i	$-0.119040\pi$
$487$	3.28959	+	5.69774i	0.149066	+	0.258189i	−0.781817	+	0.623508i	$0.785707\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−1.02704	−	1.77889i	−0.0463498	−	0.0802801i	0.841920	−	0.539603i	$-0.181426\pi$
$491$	−1.02704	−	1.77889i	−0.0463498	−	0.0802801i	−0.888270	+	0.459323i	$0.848092\pi$
$492$		0			0
$493$	2.27627			0.102518
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−16.3245			−0.730785			−0.365393	−	0.930854i	$-0.619065\pi$
$499$	−16.3245			−0.730785			−0.365393	+	0.930854i	$0.619065\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−5.60078			−0.249726			−0.124863	−	0.992174i	$-0.539849\pi$
$503$	−5.60078			−0.249726			−0.124863	+	0.992174i	$0.539849\pi$
$504$		0			0
$505$	5.60078			0.249231
$506$		0			0
$507$		0			0
$508$		0			0
$509$	0.672570			0.0298111			0.0149056	−	0.999889i	$-0.495255\pi$
$509$	0.672570			0.0298111			0.0149056	+	0.999889i	$0.495255\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−36.9371			−1.62764
$516$		0			0
$517$	−3.49115	−	6.04684i	−0.153540	−	0.265940i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−13.2267	−	22.9092i	−0.579470	−	1.00367i	−0.995540	−	0.0943392i	$-0.969926\pi$
$521$	−13.2267	−	22.9092i	−0.579470	−	1.00367i	0.416070	−	0.909333i	$-0.363407\pi$
$522$		0			0
$523$	13.6534	−	23.6484i	0.597021	−	1.03407i	−0.396237	−	0.918148i	$-0.629684\pi$
$523$	13.6534	−	23.6484i	0.597021	−	1.03407i	0.993258	−	0.115923i	$-0.0369826\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	2.77781			0.121003
$528$		0			0
$529$	5.38151			0.233979
$530$		0			0
$531$		0			0
$532$		0			0
$533$	2.52704	−	4.37697i	0.109458	−	0.189587i
$534$		0			0
$535$	1.13814	−	1.97131i	0.0492059	−	0.0852271i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	9.66225	−	16.7355i	0.415413	−	0.719516i	−0.580059	−	0.814574i	$-0.696971\pi$
$541$	9.66225	−	16.7355i	0.415413	−	0.719516i	0.995472	+	0.0950586i	$0.0303038\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−3.47296	−	6.01534i	−0.148765	−	0.257669i
$546$		0			0
$547$	9.17111	−	15.8848i	0.392128	−	0.679186i	−0.600602	−	0.799548i	$-0.705072\pi$
$547$	9.17111	−	15.8848i	0.392128	−	0.679186i	0.992730	+	0.120362i	$0.0384056\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−22.4071	−	38.8102i	−0.954574	−	1.65337i
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	4.59931	+	7.96625i	0.194879	+	0.337541i	0.946861	−	0.321643i	$-0.104235\pi$
$557$	4.59931	+	7.96625i	0.194879	+	0.337541i	−0.751982	+	0.659184i	$0.770902\pi$
$558$		0			0
$559$	4.60078			0.194592
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−16.5811	+	28.7194i	−0.698811	+	1.21038i	0.270068	+	0.962841i	$0.412954\pi$
$563$	−16.5811	+	28.7194i	−0.698811	+	1.21038i	−0.968879	+	0.247535i	$0.920379\pi$
$564$		0			0
$565$	19.3815	+	33.5698i	0.815386	+	1.41229i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	13.1008	+	22.6912i	0.549213	+	0.951265i	0.998329	+	0.0577914i	$0.0184058\pi$
$569$	13.1008	+	22.6912i	0.549213	+	0.951265i	−0.449116	+	0.893474i	$0.648261\pi$
$570$		0			0
$571$	−4.89037	+	8.47037i	−0.204656	+	0.354474i	−0.950023	−	0.312180i	$-0.898941\pi$
$571$	−4.89037	+	8.47037i	−0.204656	+	0.354474i	0.745367	+	0.666654i	$0.232274\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−4.15933			−0.173456
$576$		0			0
$577$	18.1534	+	31.4426i	0.755736	+	1.30897i	0.945008	+	0.327048i	$0.106054\pi$
$577$	18.1534	+	31.4426i	0.755736	+	1.30897i	−0.189272	+	0.981925i	$0.560613\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	8.68190	+	15.0375i	0.359568	+	0.622789i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	12.0737	−	20.9123i	0.498336	−	0.863144i	−0.501662	−	0.865064i	$-0.667278\pi$
$587$	12.0737	−	20.9123i	0.498336	−	0.863144i	0.999998	+	0.00191995i	$0.000611139\pi$
$588$		0			0
$589$	−27.3442	−	47.3615i	−1.12670	−	1.95150i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−20.7448	+	35.9311i	−0.851889	+	1.47551i	0.0276133	+	0.999619i	$0.491209\pi$
$593$	−20.7448	+	35.9311i	−0.851889	+	1.47551i	−0.879502	+	0.475896i	$0.842124\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−11.3422	+	19.6453i	−0.463430	+	0.802685i	−0.999129	−	0.0417243i	$-0.986715\pi$
$599$	−11.3422	+	19.6453i	−0.463430	+	0.802685i	0.535699	+	0.844409i	$0.320048\pi$
$600$		0			0
$601$	−20.1249	+	34.8573i	−0.820912	+	1.42186i	0.0840927	+	0.996458i	$0.473201\pi$
$601$	−20.1249	+	34.8573i	−0.820912	+	1.42186i	−0.905004	+	0.425403i	$0.860133\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	29.8741			1.21456
$606$		0			0
$607$	−17.3245			−0.703180			−0.351590	−	0.936154i	$-0.614359\pi$
$607$	−17.3245			−0.703180			−0.351590	+	0.936154i	$0.614359\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−0.690757	+	1.19643i	−0.0279450	+	0.0484022i
$612$		0			0
$613$	16.4823	+	28.5482i	0.665713	+	1.15305i	0.979091	+	0.203421i	$0.0652060\pi$
$613$	16.4823	+	28.5482i	0.665713	+	1.15305i	−0.313378	+	0.949629i	$0.601461\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−13.4700	−	23.3308i	−0.542283	−	0.939262i	−0.998772	−	0.0495330i	$-0.984227\pi$
$617$	−13.4700	−	23.3308i	−0.542283	−	0.939262i	0.456489	−	0.889729i	$-0.349107\pi$
$618$		0			0
$619$	1.98229			0.0796750			0.0398375	−	0.999206i	$-0.487316\pi$
$619$	1.98229			0.0796750			0.0398375	+	0.999206i	$0.487316\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−20.4868			−0.819470
$626$		0			0
$627$		0			0
$628$		0			0
$629$	2.23112			0.0889606
$630$		0			0
$631$	−25.4868			−1.01461			−0.507306	−	0.861766i	$-0.669359\pi$
$631$	−25.4868			−1.01461			−0.507306	+	0.861766i	$0.669359\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	35.2527			1.39896
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	40.0846			1.58325			0.791623	−	0.611009i	$-0.209236\pi$
$641$	40.0846			1.58325			0.791623	+	0.611009i	$0.209236\pi$
$642$		0			0
$643$	3.50885	+	6.07751i	0.138376	+	0.239674i	0.926882	−	0.375353i	$-0.122479\pi$
$643$	3.50885	+	6.07751i	0.138376	+	0.239674i	−0.788506	+	0.615027i	$0.789145\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	5.90709	+	10.2314i	0.232232	+	0.402237i	0.958465	−	0.285212i	$-0.0920639\pi$
$647$	5.90709	+	10.2314i	0.232232	+	0.402237i	−0.726233	+	0.687449i	$0.758731\pi$
$648$		0			0
$649$	−4.50000	+	7.79423i	−0.176640	+	0.305950i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−0.273346			−0.0106969			−0.00534843	−	0.999986i	$-0.501702\pi$
$653$	−0.273346			−0.0106969			−0.00534843	+	0.999986i	$0.501702\pi$
$654$		0			0
$655$	−36.7453			−1.43576
$656$		0			0
$657$		0			0
$658$		0			0
$659$	7.39970	−	12.8167i	0.288251	−	0.499266i	−0.685141	−	0.728410i	$-0.740259\pi$
$659$	7.39970	−	12.8167i	0.288251	−	0.499266i	0.973392	+	0.229144i	$0.0735928\pi$
$660$		0			0
$661$	−4.50885	+	7.80956i	−0.175374	+	0.303757i	−0.940291	−	0.340372i	$-0.889447\pi$
$661$	−4.50885	+	7.80956i	−0.175374	+	0.303757i	0.764917	+	0.644129i	$0.222780\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−22.1819	+	38.4202i	−0.858887	+	1.48764i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	1.97296	+	3.41726i	0.0761652	+	0.131922i
$672$		0			0
$673$	−11.9803	+	20.7506i	−0.461809	+	0.799876i	−0.999051	−	0.0435519i	$-0.986133\pi$
$673$	−11.9803	+	20.7506i	−0.461809	+	0.799876i	0.537243	+	0.843428i	$0.319466\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	3.32889	+	5.76581i	0.127940	+	0.221598i	0.922878	−	0.385092i	$-0.125830\pi$
$677$	3.32889	+	5.76581i	0.127940	+	0.221598i	−0.794938	+	0.606690i	$0.792497\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−15.0364	−	26.0438i	−0.575351	−	0.996537i	−0.996003	−	0.0893152i	$-0.971532\pi$
$683$	−15.0364	−	26.0438i	−0.575351	−	0.996537i	0.420652	−	0.907222i	$-0.361801\pi$
$684$		0			0
$685$	9.22820			0.352591
$686$		0			0
$687$		0			0
$688$		0			0
$689$	1.71780	−	2.97532i	0.0654429	−	0.113351i
$690$		0			0
$691$	−1.63814	−	2.83733i	−0.0623176	−	0.107937i	0.833183	−	0.552997i	$-0.186516\pi$
$691$	−1.63814	−	2.83733i	−0.0623176	−	0.107937i	−0.895501	+	0.445060i	$0.853183\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	18.6352	+	32.2771i	0.706874	+	1.22434i
$696$		0			0
$697$	0.690757	−	1.19643i	0.0261643	−	0.0453179i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	32.2891			1.21954			0.609771	−	0.792578i	$-0.291261\pi$
$701$	32.2891			1.21954			0.609771	+	0.792578i	$0.291261\pi$
$702$		0			0
$703$	−21.9626	−	38.0404i	−0.828337	−	1.43472i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	24.5438	+	42.5111i	0.921761	+	1.59654i	0.796690	+	0.604388i	$0.206582\pi$
$709$	24.5438	+	42.5111i	0.921761	+	1.59654i	0.125071	+	0.992148i	$0.460084\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−27.0693	+	46.8855i	−1.01376	+	1.75588i
$714$		0			0
$715$	−5.19076	−	8.99066i	−0.194123	−	0.336231i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	20.9808	−	36.3399i	0.782453	−	1.35525i	−0.148056	−	0.988979i	$-0.547302\pi$
$719$	20.9808	−	36.3399i	0.782453	−	1.35525i	0.930509	−	0.366269i	$-0.119365\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$	3.25077	−	5.63050i	0.120731	−	0.209112i
$726$		0			0
$727$	−14.2434	+	24.6703i	−0.528258	+	0.914969i	0.471200	+	0.882027i	$0.343821\pi$
$727$	−14.2434	+	24.6703i	−0.528258	+	0.914969i	−0.999457	+	0.0329425i	$0.989512\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	1.25760			0.0465142
$732$		0			0
$733$	−28.5261			−1.05363			−0.526817	−	0.849979i	$-0.676615\pi$
$733$	−28.5261			−1.05363			−0.526817	+	0.849979i	$0.676615\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	21.1804	−	36.6856i	0.780192	−	1.35133i
$738$		0			0
$739$	−3.92967	−	6.80639i	−0.144555	−	0.250377i	0.784652	−	0.619937i	$-0.212842\pi$
$739$	−3.92967	−	6.80639i	−0.144555	−	0.250377i	−0.929207	+	0.369560i	$0.879508\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−3.37364	−	5.84332i	−0.123767	−	0.214371i	0.797483	−	0.603341i	$-0.206164\pi$
$743$	−3.37364	−	5.84332i	−0.123767	−	0.214371i	−0.921250	+	0.388970i	$0.872831\pi$
$744$		0			0
$745$	9.25856			0.339207
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	22.1800			0.809358			0.404679	−	0.914459i	$-0.367383\pi$
$751$	22.1800			0.809358			0.404679	+	0.914459i	$0.367383\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−22.5586			−0.820990
$756$		0			0
$757$	−20.3815			−0.740779			−0.370389	−	0.928877i	$-0.620776\pi$
$757$	−20.3815			−0.740779			−0.370389	+	0.928877i	$0.620776\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	40.6549			1.47374			0.736869	−	0.676036i	$-0.236304\pi$
$761$	40.6549			1.47374			0.736869	+	0.676036i	$0.236304\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$	1.78074			0.0642987
$768$		0			0
$769$	−16.9518	−	29.3615i	−0.611299	−	1.05880i	−0.991022	−	0.133701i	$-0.957314\pi$
$769$	−16.9518	−	29.3615i	−0.611299	−	1.05880i	0.379723	−	0.925100i	$-0.376019\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	4.37412	+	7.57620i	0.157326	+	0.272497i	0.933904	−	0.357525i	$-0.116379\pi$
$773$	4.37412	+	7.57620i	0.157326	+	0.272497i	−0.776577	+	0.630022i	$0.783046\pi$
$774$		0			0
$775$	3.96703	−	6.87110i	0.142500	−	0.246817i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−27.1986			−0.974492
$780$		0			0
$781$	39.3245			1.40714
$782$		0			0
$783$		0			0
$784$		0			0
$785$	4.29300	−	7.43569i	0.153224	−	0.265391i
$786$		0			0
$787$	4.64260	−	8.04122i	0.165491	−	0.286639i	−0.771339	−	0.636425i	$-0.780412\pi$
$787$	4.64260	−	8.04122i	0.165491	−	0.286639i	0.936830	+	0.349786i	$0.113746\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	0.390369	−	0.676139i	0.0138624	−	0.0240104i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−11.2271	−	19.4460i	−0.397685	−	0.688811i	0.595754	−	0.803167i	$-0.296853\pi$
$797$	−11.2271	−	19.4460i	−0.397685	−	0.688811i	−0.993440	+	0.114355i	$0.963520\pi$
$798$		0			0
$799$	−0.188816	+	0.327039i	−0.00667983	+	0.0115698i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	23.7075	+	41.0626i	0.836619	+	1.44907i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−2.70107	−	4.67840i	−0.0949647	−	0.164484i	0.814629	−	0.579982i	$-0.196940\pi$
$809$	−2.70107	−	4.67840i	−0.0949647	−	0.164484i	−0.909594	+	0.415498i	$0.863607\pi$
$810$		0			0
$811$	0.0177088			0.000621841			0.000310920	−	1.00000i	$-0.499901\pi$
$811$	0.0177088			0.000621841			0.000310920		1.00000i	$0.499901\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	5.77042	−	9.99466i	0.202129	−	0.350098i
$816$		0			0
$817$	−12.3796	−	21.4420i	−0.433106	−	0.750162i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−0.528505	−	0.915397i	−0.0184449	−	0.0319476i	0.856656	−	0.515889i	$-0.172538\pi$
$821$	−0.528505	−	0.915397i	−0.0184449	−	0.0319476i	−0.875101	+	0.483941i	$0.839205\pi$
$822$		0			0
$823$	−6.76303	+	11.7139i	−0.235744	+	0.408321i	−0.959489	−	0.281747i	$-0.909086\pi$
$823$	−6.76303	+	11.7139i	−0.235744	+	0.408321i	0.723744	+	0.690068i	$0.242419\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	49.3068			1.71457			0.857283	−	0.514846i	$-0.172151\pi$
$827$	49.3068			1.71457			0.857283	+	0.514846i	$0.172151\pi$
$828$		0			0
$829$	−26.8530	−	46.5108i	−0.932644	−	1.61539i	−0.778783	−	0.627294i	$-0.784163\pi$
$829$	−26.8530	−	46.5108i	−0.932644	−	1.61539i	−0.153861	−	0.988093i	$-0.549171\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$	11.1864	+	19.3754i	0.387120	+	0.670512i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	18.7163	−	32.4176i	0.646160	−	1.11918i	−0.337873	−	0.941192i	$-0.609707\pi$
$839$	18.7163	−	32.4176i	0.646160	−	1.11918i	0.984032	−	0.177990i	$-0.0569593\pi$
$840$		0			0
$841$	−20.1730	−	34.9407i	−0.695622	−	1.20485i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	12.3245	−	21.3467i	0.423976	−	0.734348i
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−21.7419	+	37.6581i	−0.745303	+	1.29090i
$852$		0			0
$853$	−10.3092	+	17.8561i	−0.352982	+	0.611382i	−0.986770	−	0.162124i	$-0.948166\pi$
$853$	−10.3092	+	17.8561i	−0.352982	+	0.611382i	0.633789	+	0.773506i	$0.281499\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	20.8889			0.713551			0.356776	−	0.934190i	$-0.383876\pi$
$857$	20.8889			0.713551			0.356776	+	0.934190i	$0.383876\pi$
$858$		0			0
$859$	−26.8860			−0.917338			−0.458669	−	0.888607i	$-0.651674\pi$
$859$	−26.8860			−0.917338			−0.458669	+	0.888607i	$0.651674\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	13.4071	−	23.2218i	0.456383	−	0.790478i	−0.542384	−	0.840131i	$-0.682478\pi$
$863$	13.4071	−	23.2218i	0.456383	−	0.790478i	0.998767	+	0.0496527i	$0.0158114\pi$
$864$		0			0
$865$	−14.9956	−	25.9732i	−0.509866	−	0.883114i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−32.7075	−	56.6510i	−1.10953	−	1.92175i
$870$		0			0
$871$	−8.38151			−0.283997
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−37.9076			−1.28005			−0.640024	−	0.768355i	$-0.721076\pi$
$877$	−37.9076			−1.28005			−0.640024	+	0.768355i	$0.721076\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	7.47782			0.251934			0.125967	−	0.992034i	$-0.459797\pi$
$881$	7.47782			0.251934			0.125967	+	0.992034i	$0.459797\pi$
$882$		0			0
$883$	5.07472			0.170778			0.0853889	−	0.996348i	$-0.472787\pi$
$883$	5.07472			0.170778			0.0853889	+	0.996348i	$0.472787\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	18.5231			0.621946			0.310973	−	0.950419i	$-0.399345\pi$
$887$	18.5231			0.621946			0.310973	+	0.950419i	$0.399345\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$	7.43464			0.248791
$894$		0			0
$895$	−24.1249	−	41.7855i	−0.806406	−	1.39674i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−42.3127	−	73.2878i	−1.41121	−	2.44428i
$900$		0			0
$901$	0.469554	−	0.813291i	0.0156431	−	0.0270947i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	2.87412			0.0955391
$906$		0			0
$907$	−49.6490			−1.64857			−0.824284	−	0.566176i	$-0.808422\pi$
$907$	−49.6490			−1.64857			−0.824284	+	0.566176i	$0.808422\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	11.9808	−	20.7514i	0.396943	−	0.687525i	−0.596404	−	0.802684i	$-0.703405\pi$
$911$	11.9808	−	20.7514i	0.396943	−	0.687525i	0.993347	+	0.115159i	$0.0367379\pi$
$912$		0			0
$913$	14.4715	−	25.0654i	0.478937	−	0.829543i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	9.41887	−	16.3140i	0.310700	−	0.538148i	−0.667814	−	0.744328i	$-0.732770\pi$
$919$	9.41887	−	16.3140i	0.310700	−	0.538148i	0.978514	+	0.206180i	$0.0661032\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−3.89037	−	6.73832i	−0.128053	−	0.221794i
$924$		0			0
$925$	3.18629	−	5.51882i	0.104765	−	0.181458i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−7.45185	−	12.9070i	−0.244487	−	0.423464i	0.717500	−	0.696558i	$-0.245286\pi$
$929$	−7.45185	−	12.9070i	−0.244487	−	0.423464i	−0.961987	+	0.273094i	$0.911953\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−1.41887	−	2.45756i	−0.0464021	−	0.0803708i
$936$		0			0
$937$	−3.94299			−0.128812			−0.0644059	−	0.997924i	$-0.520515\pi$
$937$	−3.94299			−0.128812			−0.0644059	+	0.997924i	$0.520515\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	21.4056	−	37.0756i	0.697804	−	1.20863i	−0.271423	−	0.962460i	$-0.587494\pi$
$941$	21.4056	−	37.0756i	0.697804	−	1.20863i	0.969226	−	0.246171i	$-0.0791725\pi$
$942$		0			0
$943$	13.4626	+	23.3180i	0.438404	+	0.759338i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−13.5919	−	23.5419i	−0.441678	−	0.765009i	0.556136	−	0.831091i	$-0.312283\pi$
$947$	−13.5919	−	23.5419i	−0.441678	−	0.765009i	−0.997814	+	0.0660823i	$0.978950\pi$
$948$		0			0
$949$	4.69076	−	8.12463i	0.152268	−	0.263737i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−1.12295			−0.0363760			−0.0181880	−	0.999835i	$-0.505790\pi$
$953$	−1.12295			−0.0363760			−0.0181880	+	0.999835i	$0.505790\pi$
$954$		0			0
$955$	23.9956	+	41.5616i	0.776480	+	1.34490i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−36.1357	−	62.5889i	−1.16567	−	2.01900i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	14.9371	−	25.8717i	0.480841	−	0.832841i
$966$		0			0
$967$	4.75223	+	8.23111i	0.152822	+	0.264695i	0.932264	−	0.361780i	$-0.117831\pi$
$967$	4.75223	+	8.23111i	0.152822	+	0.264695i	−0.779442	+	0.626474i	$0.784497\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	12.2989	−	21.3024i	0.394691	−	0.683625i	−0.598370	−	0.801220i	$-0.704185\pi$
$971$	12.2989	−	21.3024i	0.394691	−	0.683625i	0.993062	+	0.117594i	$0.0375182\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−12.7016	+	21.9997i	−0.406359	+	0.703834i	−0.994479	−	0.104940i	$-0.966535\pi$
$977$	−12.7016	+	21.9997i	−0.406359	+	0.703834i	0.588120	+	0.808774i	$0.299868\pi$
$978$		0			0
$979$	34.9164	−	60.4770i	1.11593	−	1.93285i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	22.8535			0.728913			0.364457	−	0.931220i	$-0.381255\pi$
$983$	22.8535			0.728913			0.364457	+	0.931220i	$0.381255\pi$
$984$		0			0
$985$	35.6224			1.13502
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−12.2552	+	21.2266i	−0.389691	+	0.674965i
$990$		0			0
$991$	−12.2345	−	21.1908i	−0.388642	−	0.673149i	0.603625	−	0.797269i	$-0.293723\pi$
$991$	−12.2345	−	21.1908i	−0.388642	−	0.673149i	−0.992267	+	0.124120i	$0.960389\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−11.8559	−	20.5351i	−0.375858	−	0.651006i
$996$		0			0
$997$	−26.0000			−0.823428			−0.411714	−	0.911313i	$-0.635070\pi$
$997$	−26.0000			−0.823428			−0.411714	+	0.911313i	$0.635070\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5292.2.l.d.3313.3		6
3.2	odd	2		1764.2.l.g.961.3		6
7.2	even	3		5292.2.j.e.3529.1		6
7.3	odd	6		5292.2.i.d.2125.3		6
7.4	even	3		5292.2.i.g.2125.1		6
7.5	odd	6		756.2.j.a.505.3		6
7.6	odd	2		5292.2.l.g.3313.1		6
9.4	even	3		5292.2.i.g.1549.1		6
9.5	odd	6		1764.2.i.e.373.2		6
21.2	odd	6		1764.2.j.d.1177.2		6
21.5	even	6		252.2.j.b.169.2	yes	6
21.11	odd	6		1764.2.i.e.1537.2		6
21.17	even	6		1764.2.i.f.1537.2		6
21.20	even	2		1764.2.l.d.961.1		6
28.19	even	6		3024.2.r.i.2017.3		6
63.4	even	3	inner	5292.2.l.d.361.3		6
63.5	even	6		252.2.j.b.85.2	✓	6
63.13	odd	6		5292.2.i.d.1549.3		6
63.23	odd	6		1764.2.j.d.589.2		6
63.31	odd	6		5292.2.l.g.361.1		6
63.32	odd	6		1764.2.l.g.949.3		6
63.40	odd	6		756.2.j.a.253.3		6
63.41	even	6		1764.2.i.f.373.2		6
63.47	even	6		2268.2.a.g.1.3		3
63.58	even	3		5292.2.j.e.1765.1		6
63.59	even	6		1764.2.l.d.949.1		6
63.61	odd	6		2268.2.a.j.1.1		3
84.47	odd	6		1008.2.r.g.673.2		6
252.47	odd	6		9072.2.a.bt.1.3		3
252.103	even	6		3024.2.r.i.1009.3		6
252.131	odd	6		1008.2.r.g.337.2		6
252.187	even	6		9072.2.a.bz.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
252.2.j.b.85.2	✓	6	63.5	even	6
252.2.j.b.169.2	yes	6	21.5	even	6
756.2.j.a.253.3		6	63.40	odd	6
756.2.j.a.505.3		6	7.5	odd	6
1008.2.r.g.337.2		6	252.131	odd	6
1008.2.r.g.673.2		6	84.47	odd	6
1764.2.i.e.373.2		6	9.5	odd	6
1764.2.i.e.1537.2		6	21.11	odd	6
1764.2.i.f.373.2		6	63.41	even	6
1764.2.i.f.1537.2		6	21.17	even	6
1764.2.j.d.589.2		6	63.23	odd	6
1764.2.j.d.1177.2		6	21.2	odd	6
1764.2.l.d.949.1		6	63.59	even	6
1764.2.l.d.961.1		6	21.20	even	2
1764.2.l.g.949.3		6	63.32	odd	6
1764.2.l.g.961.3		6	3.2	odd	2
2268.2.a.g.1.3		3	63.47	even	6
2268.2.a.j.1.1		3	63.61	odd	6
3024.2.r.i.1009.3		6	252.103	even	6
3024.2.r.i.2017.3		6	28.19	even	6
5292.2.i.d.1549.3		6	63.13	odd	6
5292.2.i.d.2125.3		6	7.3	odd	6
5292.2.i.g.1549.1		6	9.4	even	3
5292.2.i.g.2125.1		6	7.4	even	3
5292.2.j.e.1765.1		6	63.58	even	3
5292.2.j.e.3529.1		6	7.2	even	3
5292.2.l.d.361.3		6	63.4	even	3	inner
5292.2.l.d.3313.3		6	1.1	even	1	trivial
5292.2.l.g.361.1		6	63.31	odd	6
5292.2.l.g.3313.1		6	7.6	odd	2
9072.2.a.bt.1.3		3	252.47	odd	6
9072.2.a.bz.1.1		3	252.187	even	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)

\(f(q)\)	\(=\)	\(q+2.05408 q^{5} +5.05408 q^{11} +(-0.500000 - 0.866025i) q^{13} +(-0.136673 - 0.236725i) q^{17} +(-2.69076 + 4.66053i) q^{19} +5.32743 q^{23} -0.780738 q^{25} +(-4.16372 + 7.21177i) q^{29} +(-5.08113 + 8.80077i) q^{31} +(-4.08113 + 7.06872i) q^{37} +(2.52704 + 4.37697i) q^{41} +(-2.30039 + 3.98439i) q^{43} +(-0.690757 - 1.19643i) q^{47} +(1.71780 + 2.97532i) q^{53} +10.3815 q^{55} +(-0.890369 + 1.54216i) q^{59} +(0.390369 + 0.676139i) q^{61} +(-1.02704 - 1.77889i) q^{65} +(4.19076 - 7.25860i) q^{67} +7.78074 q^{71} +(4.69076 + 8.12463i) q^{73} +(-6.47150 - 11.2090i) q^{79} +(2.86333 - 4.95943i) q^{83} +(-0.280738 - 0.486253i) q^{85} +(6.90856 - 11.9660i) q^{89} +(-5.52704 + 9.57312i) q^{95} +(-1.10963 + 1.92194i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{5} + 12 q^{11} - 3 q^{13} + 3 q^{19} + 12 q^{23} + 12 q^{25} - 15 q^{29} - 3 q^{31} + 3 q^{37} + 6 q^{41} - 3 q^{43} + 15 q^{47} - 18 q^{53} + 24 q^{55} + 3 q^{59} - 6 q^{61} + 3 q^{65} + 6 q^{67}+ \cdots - 15 q^{97}+O(q^{100}) \) 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

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more